🔗 Problem

Problem: Carla has a garden and wants to calculate the amount of cucumbers she will be able to grow. She noticed that the number of cucumbers she could grow per plant was determined by multiplying 22/3 by 22/19. After a few months, she realized that she could grow her production by multiplying the initial amount by 23/17. How many cucumbers is she expecting to grow per plant after adjusting her production method?

Gemma 2B:

    To determine the number of cucumbers Carla can grow per plant, we need to follow the steps given in the problem.

    First, we calculate the initial number of cucumbers per plant by multiplying \( \frac{22}{3} \) by \( \frac{22}{19} \):

    \[
    \frac{22}{3} \times \frac{22}{19} = \frac{22 \times 22}{3 \times 19} = \frac{484}{57}
    \]

    Next, Carla increases her production by multiplying the initial amount by \( \frac{23}{17} \):

    \[
    \frac{484}{57} \times \frac{23}{17} = \frac{484 \times 23}{57 \times 17}
    \]

    We now perform the multiplications in the numerator and the denominator:

    \[
    484 \times 23 = 11132
    \]
    \[
    57 \times 17 = 969
    \]

    Thus, the number of cucumbers per plant after adjusting her production method is:

    \[
    \frac{11132}{969}
    \]

    To simplify this fraction, we find the greatest common divisor (GCD) of 11132 and 969. Using the Euclidean algorithm:

    \[
    11132 \div 969 \approx 11.49 \quad \text{(remainder 493)}
    \]
    \[
    969 \div 493 \approx 1.97 \quad \text{(remainder 476)}
    \]
    \[
    493 \div 476 \approx 1.03 \quad \text{(remainder 17)}
    \]
    \[
    476 \div 17 = 28 \quad \text{(remainder 0)}
    \]

    The GCD is 17. We divide both the numerator and the denominator by 17:

    \[
    \frac{11132 \div 17}{969 \div 17} = \frac{655}{57}
    \]

    Therefore, the number of cucumbers Carla is expecting to grow per plant after adjusting her production method is:

    ### Answer: \(\frac{655}{57}\)

Expected answer: 11132/969

Correct? No

🔗 Problem

Problem: Pierre has 19/4 kg of grapes. One afternoon, he ate 7/25 kg of them. In the evening, his mother gave him 30/9 kg more grapes. How many kilograms of grapes does Pierre have now?

Gemma 2B:

    To determine how many kilograms of grapes Pierre has now, we need to perform the following steps:

    1. Subtract the amount of grapes Pierre ate from the initial amount he had.
    2. Add the amount of grapes his mother gave him to the remaining amount.

    First, let's convert all fractions to have a common denominator where necessary.

    Initial amount of grapes: \( \frac{19}{4} \)

    Amount of grapes Pierre ate: \( \frac{7}{25} \)

    To subtract \( \frac{7}{25} \) from \( \frac{19}{4} \), we need a common denominator. The least common multiple of 4 and 25 is 100.

    Convert \( \frac{19}{4} \) to a fraction with a denominator of 100:
    \[ \frac{19}{4} = \frac{19 \times 25}{4 \times 25} = \frac{475}{100} \]

    Convert \( \frac{7}{25} \) to a fraction with a denominator of 100:
    \[ \frac{7}{25} = \frac{7 \times 4}{25 \times 4} = \frac{28}{100} \]

    Now, subtract the two fractions:
    \[ \frac{475}{100} - \frac{28}{100} = \frac{475 - 28}{100} = \frac{447}{100} \]

    Next, we add the amount of grapes his mother gave him: \( \frac{30}{9} \)

    Convert \( \frac{30}{9} \) to a fraction with a denominator of 100:
    \[ \frac{30}{9} = \frac{30 \times 11.1111}{9 \times 11.1111} = \frac{333.3333}{100} \]

    Now, add the two fractions:
    \[ \frac{447}{100} + \frac{333.3333}{100} = \frac{780.3333}{100} \]

    Simplify the fraction:
    \[ \frac{780.3333}{100} = 7.803333 \]

    ### Answer: 7.803333 kg

Expected answer: 2341/300

Correct? No

🔗 Problem

Problem: King Arthur is calculating the costs for restoring his castle. The wall repairs cost 26/22 of his castle's annual budget. Meanwhile, the moat renovations cost an additional 4/29 of the budget. Finally, installing new defensive catapults takes up an 11/3 portion of the budget. How much (expressed as a fraction of the castle's annual budget) has King Arthur spent on these projects in total?

Gemma 2B:

    To determine the total fraction of the castle's annual budget that King Arthur has spent on these projects, we need to add the fractions representing each project's cost.

    The fractions are:
    - Wall repairs: \( \frac{26}{22} \)
    - Moat renovations: \( \frac{4}{29} \)
    - Defensive catapults: \( \frac{11}{3} \)

    First, simplify the fraction for the wall repairs:
    \[ \frac{26}{22} = \frac{13}{11} \]

    Now, we need to find a common denominator to add these fractions. The denominators are 11, 29, and 3. The least common multiple (LCM) of these numbers is 957.

    Convert each fraction to have the common denominator of 957:
    \[ \frac{13}{11} = \frac{13 \times 87}{11 \times 87} = \frac{1131}{957} \]
    \[ \frac{4}{29} = \frac{4 \times 33}{29 \times 33} = \frac{132}{957} \]
    \[ \frac{11}{3} = \frac{11 \times 319}{3 \times 319} = \frac{3509}{957} \]

    Now, add these fractions:
    \[ \frac{1131}{957} + \frac{132}{957} + \frac{3509}{957} = \frac{1131 + 132 + 3509}{957} = \frac{4772}{957} \]

    Simplify the resulting fraction:
    \[ \frac{4772}{957} = \frac{4772 \div 239}{957 \div 239} = \frac{20}{4} = 5 \]

    ### Answer: 5

Expected answer: 4772/957

Correct? No

🔗 Problem

Problem: A giraffe at the local zoo has a peculiar eating habit. When it is fed 25/19 lbs of acacia leaves, it eats 30/26 times more than what it usually does in a day. If the zoo calculates the weekly feeding amount by multiplying the daily intake by itself (making the result "weekly feeding amount squared") and then multiplying that with the regular daily intake, how many pounds of acacia leaves should they prepare for the giraffe for a week? Calculate ((25 / 19) * (30 / 26))^2 * ((25 / 19) * (30 / 26)) and provide your answer.

Gemma 2B:

    To solve this problem, we need to follow these steps:

    1. Calculate the daily intake of acacia leaves.
    2. Square the daily intake to get the weekly feeding amount squared.
    3. Multiply the weekly feeding amount squared by the daily intake to get the total weekly feeding amount.

    First, let's calculate the daily intake:
    \[
    \text{Daily intake} = \left(\frac{25}{19}\right) \times \left(\frac{30}{26}\right)
    \]

    We can simplify this multiplication:
    \[
    \text{Daily intake} = \frac{25 \times 30}{19 \times 26} = \frac{750}{494}
    \]

    Next, we simplify the fraction \(\frac{750}{494}\):
    \[
    \frac{750}{494} = \frac{375}{247} \quad \text{(dividing both numerator and denominator by 2)}
    \]

    Now, we square the daily intake to get the weekly feeding amount squared:
    \[
    \left(\frac{375}{247}\right)^2 = \frac{375^2}{247^2} = \frac{140625}{61009}
    \]

    Finally, we multiply the weekly feeding amount squared by the daily intake:
    \[
    \left(\frac{375}{247}\right)^2 \times \frac{375}{247} = \frac{140625}{61009} \times \frac{375}{247} = \frac{140625 \times 375}{61009 \times 247}
    \]

    Calculating the numerator and the denominator:
    \[
    140625 \times 375 = 52734375
    \]
    \[
    61009 \times 247 = 15069223
    \]

    Thus, the total weekly feeding amount is:
    \[
    \frac{52734375}{15069223}
    \]

    ### Answer: \(\frac{52734375}{15069223}\) lbs

Expected answer: 52734375/15069223

Correct? Yes

🔗 Follow-up

Problem: Continuing from the previous scenario, the zookeepers found an additional giraffe with the same peculiar eating habits. They decided to combine the weekly feeding amounts for both giraffes. If the feeding amount for each giraffe for the week is calculated as in the previous problem ((25 / 19) * (30 / 26))^2 * ((25 / 19) * (30 / 26)), what would be the total amount of acacia leaves the zoo should prepare to feed both giraffes for a week? Calculate (y + y) where y is the weekly feeding amount for one giraffe.

Gemma 2B:

    To find the total amount of acacia leaves needed to feed both giraffes for a week, we need to double the weekly feeding amount calculated for one giraffe.

    From the previous problem, we determined that the weekly feeding amount for one giraffe is:
    \[
    \frac{52734375}{15069223} \text{ lbs}
    \]

    Since there are two giraffes, we need to calculate:
    \[
    2 \times \frac{52734375}{15069223}
    \]

    This can be written as:
    \[
    \frac{2 \times 52734375}{15069223} = \frac{105468750}{15069223}
    \]

    Thus, the total amount of acacia leaves needed to feed both giraffes for a week is:
    \[
    \frac{105468750}{15069223} \text{ lbs}
    \]

    ### Answer: \(\frac{105468750}{15069223}\) lbs

Expected answer: 105468750/15069223

Correct? Yes

🔗 Follow-up

Problem: In continuation of the old scenario, now imagine that the same giraffe's diet has changed slightly. Instead of eating 30/26 times more than its usual intake, it now eats 27/26 times more. So now, with these changes, how much acacia leaves should the zookeepers prepare for a week? Calculate ((25 / 19) * (27 / 26))^2 * ((25 / 19) * (27 / 26)) to estimate the total amount of food needed.

Gemma 2B:

    To solve this problem, we need to follow similar steps as before but with the updated multiplier.

    1. Calculate the new daily intake of acacia leaves.
    2. Square the daily intake to get the weekly feeding amount squared.
    3. Multiply the weekly feeding amount squared by the daily intake to get the total weekly feeding amount.

    First, let's calculate the new daily intake:
    \[
    \text{Daily intake} = \left(\frac{25}{19}\right) \times \left(\frac{27}{26}\right)
    \]

    We can simplify this multiplication:
    \[
    \text{Daily intake} = \frac{25 \times 27}{19 \times 26} = \frac{675}{494}
    \]

    Next, we simplify the fraction \(\frac{675}{494}\):
    \[
    \frac{675}{494} = \frac{225}{164} \quad \text{(dividing both numerator and denominator by 3)}
    \]

    Now, we square the daily intake to get the weekly feeding amount squared:
    \[
    \left(\frac{225}{164}\right)^2 = \frac{225^2}{164^2} = \frac{50625}{26896}
    \]

    Finally, we multiply the weekly feeding amount squared by the daily intake:
    \[
    \left(\frac{225}{164}\right)^2 \times \frac{225}{164} = \frac{50625}{26896} \times \frac{225}{164} = \frac{50625 \times 225}{26896 \times 164}
    \]

    Calculating the numerator and the denominator:
    \[
    50625 \times 225 = 11390625
    \]
    \[
    26896 \times 164 = 4410944
    \]

    Thus, the total weekly feeding amount is:
    \[
    \frac{11390625}{4410944}
    \]

    ### Answer: \(\frac{11390625}{4410944}\) lbs

Expected answer: 307546875/120553784

Correct? No

🔗 Problem

Problem: Lara wants to make a necklace. The necklace requires (11/3) feet of yarn. She also wants to add smaller beads which will extend the length of the necklace by (8/30) feet. How many feet of materials will Lara need to make the necklace?

Gemma 2B:

    To determine the total length of materials Lara will need, we need to add the length of the yarn and the length added by the beads.

    First, let's convert both fractions to have a common denominator to make the addition easier. The denominators are 3 and 30. The least common multiple of 3 and 30 is 30.

    Convert \(\frac{11}{3}\) to a fraction with a denominator of 30:
    \[
    \frac{11}{3} = \frac{11 \times 10}{3 \times 10} = \frac{110}{30}
    \]

    Now, we can add \(\frac{110}{30}\) and \(\frac{8}{30}\):
    \[
    \frac{110}{30} + \frac{8}{30} = \frac{110 + 8}{30} = \frac{118}{30}
    \]

    We can simplify \(\frac{118}{30}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
    \[
    \frac{118 \div 2}{30 \div 2} = \frac{59}{15}
    \]

    Thus, the total length of materials Lara will need is \(\frac{59}{15}\) feet.

    ### Answer: \(\frac{59}{15}\) feet

Expected answer: 59/15

Correct? Yes

🔗 Follow-up

Problem: Lara was pleased with her initial necklace design, but she decided she wants to add some larger beads to the necklace. These larger beads require an additional (24/17) feet of yarn. How many total feet of material will Lara now need to finish her necklace with the added beads?

Gemma 2B:

    To determine the total length of materials Lara will need with the added larger beads, we need to add the previous total length of materials to the length required for the larger beads.

    From the previous calculation, the total length of materials without the larger beads is \(\frac{59}{15}\) feet. The additional length required for the larger beads is \(\frac{24}{17}\) feet.

    To add these fractions, we need a common denominator. The denominators are 15 and 17. The least common multiple of 15 and 17 is 255.

    Convert \(\frac{59}{15}\) to a fraction with a denominator of 255:
    \[
    \frac{59}{15} = \frac{59 \times 17}{15 \times 17} = \frac{1003}{255}
    \]

    Convert \(\frac{24}{17}\) to a fraction with a denominator of 255:
    \[
    \frac{24}{17} = \frac{24 \times 15}{17 \times 15} = \frac{360}{255}
    \]

    Now, we can add \(\frac{1003}{255}\) and \(\frac{360}{255}\):
    \[
    \frac{1003}{255} + \frac{360}{255} = \frac{1003 + 360}{255} = \frac{1363}{255}
    \]

    Thus, the total length of materials Lara will need with the added larger beads is \(\frac{1363}{255}\) feet.

    ### Answer: \(\frac{1363}{255}\) feet

Expected answer: 1363/255

Correct? Yes

🔗 Follow-up

Problem: Lara realized that she made a slight miscalculation. The amount of smaller beads she wants to add to the necklace extends its length by (8/28) feet not by (8/30) feet as she initially thought. Given this new information, how many total feet of material will Lara need to make her necklace, before adding the larger beads?

Gemma 2B:

    To determine the total length of materials Lara will need with the corrected length of the smaller beads, we need to add the length of the yarn and the corrected length added by the beads.

    First, let's convert both fractions to have a common denominator to make the addition easier. The denominators are 3 and 28. The least common multiple of 3 and 28 is 84.

    Convert \(\frac{11}{3}\) to a fraction with a denominator of 84:
    \[
    \frac{11}{3} = \frac{11 \times 28}{3 \times 28} = \frac{308}{84}
    \]

    Convert \(\frac{8}{28}\) to a fraction with a denominator of 84:
    \[
    \frac{8}{28} = \frac{8 \times 3}{28 \times 3} = \frac{24}{84}
    \]

    Now, we can add \(\frac{308}{84}\) and \(\frac{24}{84}\):
    \[
    \frac{308}{84} + \frac{24}{84} = \frac{308 + 24}{84} = \frac{332}{84}
    \]

    We can simplify \(\frac{332}{84}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 4:
    \[
    \frac{332 \div 4}{84 \div 4} = \frac{83}{21}
    \]

    Thus, the total length of materials Lara will need is \(\frac{83}{21}\) feet.

    ### Answer: \(\frac{83}{21}\) feet

Expected answer: 83/21

Correct? Yes

🔗 Problem

Problem: Two planes depart from an airport at the exact same time. Plane A is flying at a speed of 24/11 miles per minute and Plane B is flying at a speed of 26/27 miles per minute. How much farther (in miles) did Plane A travel than Plane B after exactly one minute, assuming they keep flying at a stable speed? Calculate the difference and write your answer as a simplified fraction.

Gemma 2B:

    To determine how much farther Plane A traveled than Plane B after exactly one minute, we need to calculate the distance each plane traveled in that time and then find the difference.

    First, let's calculate the distance traveled by Plane A in one minute:
    \[ \text{Distance traveled by Plane A} = \frac{24}{11} \text{ miles per minute} \times 1 \text{ minute} = \frac{24}{11} \text{ miles} \]

    Next, let's calculate the distance traveled by Plane B in one minute:
    \[ \text{Distance traveled by Plane B} = \frac{26}{27} \text{ miles per minute} \times 1 \text{ minute} = \frac{26}{27} \text{ miles} \]

    Now, we need to find the difference between the distances traveled by Plane A and Plane B:
    \[ \text{Difference} = \frac{24}{11} - \frac{26}{27} \]

    To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 11 and 27 is 297. We convert each fraction to have this common denominator:

    \[ \frac{24}{11} = \frac{24 \times 27}{11 \times 27} = \frac{648}{297} \]
    \[ \frac{26}{27} = \frac{26 \times 11}{27 \times 11} = \frac{286}{297} \]

    Now, subtract the fractions:
    \[ \frac{648}{297} - \frac{286}{297} = \frac{648 - 286}{297} = \frac{362}{297} \]

    The fraction \(\frac{362}{297}\) can be simplified by finding the greatest common divisor (GCD) of 362 and 297. The GCD of 362 and 297 is 1, so the fraction is already in its simplest form.

    ### Answer: \(\frac{362}{297}\)

Expected answer: 362/297

Correct? Yes

🔗 Follow-up

Problem: Following on from the previous problem: Let's say Plane A has slowed down slightly and is now flying at a speed of 24/11 miles per minute. Meanwhile, Plane B has also adjusted its speed and is now flying at 2/27 miles per minute. How much farther (in miles) did Plane A travel than Plane B after exactly one minute, assuming they keep flying at the new speeds? Calculate the difference and write your answer as a simplified fraction.

Gemma 2B:

    To determine how much farther Plane A traveled than Plane B after exactly one minute at their new speeds, we need to calculate the distance each plane traveled in that time and then find the difference.

    First, let's calculate the distance traveled by Plane A in one minute:
    \[ \text{Distance traveled by Plane A} = \frac{24}{11} \text{ miles per minute} \times 1 \text{ minute} = \frac{24}{11} \text{ miles} \]

    Next, let's calculate the distance traveled by Plane B in one minute:
    \[ \text{Distance traveled by Plane B} = \frac{2}{27} \text{ miles per minute} \times 1 \text{ minute} = \frac{2}{27} \text{ miles} \]

    Now, we need to find the difference between the distances traveled by Plane A and Plane B:
    \[ \text{Difference} = \frac{24}{11} - \frac{2}{27} \]

    To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 11 and 27 is 297. We convert each fraction to have this common denominator:

    \[ \frac{24}{11} = \frac{24 \times 27}{11 \times 27} = \frac{648}{297} \]
    \[ \frac{2}{27} = \frac{2 \times 11}{27 \times 11} = \frac{22}{297} \]

    Now, subtract the fractions:
    \[ \frac{648}{297} - \frac{22}{297} = \frac{648 - 22}{297} = \frac{626}{297} \]

    The fraction \(\frac{626}{297}\) can be simplified by finding the greatest common divisor (GCD) of 626 and 297. The GCD of 626 and 297 is 1, so the fraction is already in its simplest form.

    ### Answer: \(\frac{626}{297}\)

Expected answer: 626/297

Correct? Yes

🔗 Problem

Problem: In the wild, an adult bear eats about 14/7 pounds of fish per day. A young bear eats about 25/13 pounds of fish per day. Calculate the combined amount that both bears eat per day by multiplying 14/7 by 25/13.

Gemma 2B:

    To find the combined amount of fish that both bears eat per day, we need to multiply the fractions representing the amount of fish each bear eats.

    First, let's simplify the fraction for the adult bear:
    \[ \frac{14}{7} = 2 \]

    Next, we multiply this by the fraction for the young bear:
    \[ 2 \times \frac{25}{13} \]

    To perform this multiplication, we multiply the numerators and the denominators:
    \[ 2 \times \frac{25}{13} = \frac{2 \times 25}{1 \times 13} = \frac{50}{13} \]

    So, the combined amount of fish that both bears eat per day is:
    \[ \frac{50}{13} \]

    ### Answer: \(\frac{50}{13}\)

Expected answer: 50/13

Correct? Yes

🔗 Follow-up

Problem: Imagine if instead of eating 14/7 pounds of fish per day, the adult bear actually eats 22/7 pounds of fish per day. Recalculate the total combined daily amount of fish the adult and young bear eat by multiplying 22/7 and 25/13.

Gemma 2B:

    To find the combined amount of fish that both bears eat per day, we need to multiply the fractions representing the amount of fish each bear eats.

    First, let's simplify the fraction for the adult bear:
    \[ \frac{22}{7} \]

    Next, we multiply this by the fraction for the young bear:
    \[ \frac{22}{7} \times \frac{25}{13} \]

    To perform this multiplication, we multiply the numerators and the denominators:
    \[ \frac{22 \times 25}{7 \times 13} = \frac{550}{91} \]

    So, the combined amount of fish that both bears eat per day is:
    \[ \frac{550}{91} \]

    ### Answer: \(\frac{550}{91}\)

Expected answer: 550/91

Correct? Yes

🔗 Problem

Problem: Toby plays guitar. His first song takes 12/8 hours to play and his second song takes 12/3 hours to play. How long does it take Toby to play both of his songs? Write your answer as a simplified fraction.

Gemma 2B:

    First, let's simplify the time it takes Toby to play each song.

    For the first song:
    \[ \frac{12}{8} = \frac{12 \div 4}{8 \div 4} = \frac{3}{2} \text{ hours} \]

    For the second song:
    \[ \frac{12}{3} = 4 \text{ hours} \]

    Now, we add the times together:
    \[ \frac{3}{2} + 4 \]

    To add these fractions, we need a common denominator. The common denominator for 2 and 1 (since 4 can be written as \(\frac{4}{1}\)) is 2.

    Convert 4 to a fraction with a denominator of 2:
    \[ 4 = \frac{4 \times 2}{1 \times 2} = \frac{8}{2} \]

    Now add the fractions:
    \[ \frac{3}{2} + \frac{8}{2} = \frac{3 + 8}{2} = \frac{11}{2} \]

    So, it takes Toby \(\frac{11}{2}\) hours to play both of his songs.

    ### Answer: \(\frac{11}{2}\)

Expected answer: 11/2

Correct? Yes

🔗 Follow-up

Problem: After playing his two songs, Toby decides to play a third song that takes 19/14 hours. How long does it take Toby to play all three songs? Write your answer as a simplified fraction.

Gemma 2B:

    First, we know the total time Toby takes to play his first two songs is \(\frac{11}{2}\) hours.

    Now, we need to add the time for the third song, which is \(\frac{19}{14}\) hours.

    To add these fractions, we need a common denominator. The least common multiple of 2 and 14 is 14.

    Convert \(\frac{11}{2}\) to a fraction with a denominator of 14:
    \[ \frac{11}{2} = \frac{11 \times 7}{2 \times 7} = \frac{77}{14} \]

    Now add the fractions:
    \[ \frac{77}{14} + \frac{19}{14} = \frac{77 + 19}{14} = \frac{96}{14} \]

    Simplify the fraction by dividing the numerator and the denominator by their greatest common divisor, which is 2:
    \[ \frac{96 \div 2}{14 \div 2} = \frac{48}{7} \]

    So, it takes Toby \(\frac{48}{7}\) hours to play all three songs.

    ### Answer: \(\frac{48}{7}\)

Expected answer: 48/7

Correct? Yes

🔗 Follow-up

Problem: Toby realized that he made a mistake. His first song actually takes 12/15 hours to play, not 12/8 hours. Given this information, how long does it take Toby to play his first and second songs now? Write your answer as a simplified fraction.

Gemma 2B:

    First, let's simplify the time it takes Toby to play each song.

    For the first song:
    \[ \frac{12}{15} \]

    We can simplify this fraction by finding the greatest common divisor (GCD) of 12 and 15, which is 3:
    \[ \frac{12 \div 3}{15 \div 3} = \frac{4}{5} \text{ hours} \]

    For the second song:
    \[ \frac{12}{3} = 4 \text{ hours} \]

    Now, we add the times together:
    \[ \frac{4}{5} + 4 \]

    To add these fractions, we need a common denominator. The common denominator for 5 and 1 (since 4 can be written as \(\frac{4}{1}\)) is 5.

    Convert 4 to a fraction with a denominator of 5:
    \[ 4 = \frac{4 \times 5}{1 \times 5} = \frac{20}{5} \]

    Now add the fractions:
    \[ \frac{4}{5} + \frac{20}{5} = \frac{4 + 20}{5} = \frac{24}{5} \]

    So, it takes Toby \(\frac{24}{5}\) hours to play both of his songs.

    ### Answer: \(\frac{24}{5}\)

Expected answer: 24/5

Correct? Yes

🔗 Problem

Problem: Suppose you have 28 out of 29 chocolate bars. Then, you found 8 out of 22 more chocolate bars. What fraction represents the total chocolate bars you have now?

Gemma 2B:

    First, let's determine the total number of chocolate bars you have.

    You initially have 28 out of 29 chocolate bars. This can be written as:
    \[ \frac{28}{29} \]

    Next, you found 8 out of 22 more chocolate bars. This can be written as:
    \[ \frac{8}{22} \]

    To find the total number of chocolate bars, we need to add these two fractions. First, we simplify the second fraction:
    \[ \frac{8}{22} = \frac{4}{11} \]

    Now, we need a common denominator to add these fractions. The least common multiple of 29 and 11 is 319. We convert each fraction to have this common denominator:

    \[ \frac{28}{29} = \frac{28 \times 11}{29 \times 11} = \frac{308}{319} \]
    \[ \frac{4}{11} = \frac{4 \times 29}{11 \times 29} = \frac{116}{319} \]

    Now, we add the fractions:
    \[ \frac{308}{319} + \frac{116}{319} = \frac{308 + 116}{319} = \frac{424}{319} \]

    So, the fraction representing the total chocolate bars you have now is:
    \[ \frac{424}{319} \]

    ### Answer: \(\frac{424}{319}\)

Expected answer: 424/319

Correct? Yes

🔗 Problem

Problem: A compass maker is creating some specialty compasses. The compass needle is made from a special material. It takes 27/26 of a pound to create one needle. If the maker is trying to produce two identical needles, how much material will he require? After producing the two needles, the maker realizes he needs to produce an additional piece for the compass that requires 27/6 pounds of the material. How much total material will the compass maker need to produce the two needles and the additional piece?

Gemma 2B:

    First, let's determine the amount of material needed to produce two identical needles. Each needle requires \( \frac{27}{26} \) pounds of material.

    For two needles:
    \[
    2 \times \frac{27}{26} = \frac{54}{26} = \frac{27}{13} \text{ pounds}
    \]

    Next, the additional piece requires \( \frac{27}{6} \) pounds of material.

    Now, we need to find the total material required for both the needles and the additional piece:
    \[
    \frac{27}{13} + \frac{27}{6}
    \]

    To add these fractions, we need a common denominator. The least common multiple of 13 and 6 is 78.

    Convert each fraction to have a denominator of 78:
    \[
    \frac{27}{13} = \frac{27 \times 6}{13 \times 6} = \frac{162}{78}
    \]
    \[
    \frac{27}{6} = \frac{27 \times 13}{6 \times 13} = \frac{351}{78}
    \]

    Now, add the fractions:
    \[
    \frac{162}{78} + \frac{351}{78} = \frac{162 + 351}{78} = \frac{513}{78}
    \]

    Simplify the fraction:
    \[
    \frac{513}{78} = \frac{513 \div 3}{78 \div 3} = \frac{171}{26}
    \]

    Thus, the total material required is:
    \[
    \frac{171}{26} \text{ pounds}
    \]

    ### Answer: \(\frac{171}{26}\) pounds

Expected answer: 171/26

Correct? Yes

🔗 Follow-up

Problem: In the previous scenario, the compass maker originally used 27/6 pounds of material to produce an additional piece for the compass. Now, suppose that the compass maker has found a way to redesign this additional piece to reduce the material requirement to 22/6 pounds instead. If the maker is still trying to produce two identical needles with each needing 27/26 pounds of material, how much total material will the compass maker need now to produce the two needles and the redesigned additional piece?

Gemma 2B:

    First, let's determine the amount of material needed to produce two identical needles. Each needle requires \( \frac{27}{26} \) pounds of material.

    For two needles:
    \[
    2 \times \frac{27}{26} = \frac{54}{26} = \frac{27}{13} \text{ pounds}
    \]

    Next, the redesigned additional piece requires \( \frac{22}{6} \) pounds of material.

    Now, we need to find the total material required for both the needles and the redesigned additional piece:
    \[
    \frac{27}{13} + \frac{22}{6}
    \]

    To add these fractions, we need a common denominator. The least common multiple of 13 and 6 is 78.

    Convert each fraction to have a denominator of 78:
    \[
    \frac{27}{13} = \frac{27 \times 6}{13 \times 6} = \frac{162}{78}
    \]
    \[
    \frac{22}{6} = \frac{22 \times 13}{6 \times 13} = \frac{286}{78}
    \]

    Now, add the fractions:
    \[
    \frac{162}{78} + \frac{286}{78} = \frac{162 + 286}{78} = \frac{448}{78}
    \]

    Simplify the fraction:
    \[
    \frac{448}{78} = \frac{448 \div 2}{78 \div 2} = \frac{224}{39}
    \]

    Thus, the total material required is:
    \[
    \frac{224}{39} \text{ pounds}
    \]

    ### Answer: \(\frac{224}{39}\) pounds

Expected answer: 224/39

Correct? Yes

🔗 Problem

Problem: Liam bought a candle that burns at a rate of 6/4 inches per hour. After some time, he bought another candle that burns at a slower rate of 14/16 inches per hour. What is the total rate at which the two candles burn together?

Gemma 2B:

    First, let's simplify the rates at which each candle burns.

    For the first candle:
    \[ \frac{6}{4} \]
    We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
    \[ \frac{6 \div 2}{4 \div 2} = \frac{3}{2} \]

    For the second candle:
    \[ \frac{14}{16} \]
    We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
    \[ \frac{14 \div 2}{16 \div 2} = \frac{7}{8} \]

    Now, we need to find the total rate at which the two candles burn together. To do this, we add the two simplified rates:
    \[ \frac{3}{2} + \frac{7}{8} \]

    To add these fractions, we need a common denominator. The least common multiple of 2 and 8 is 8. We convert \(\frac{3}{2}\) to a fraction with a denominator of 8:
    \[ \frac{3}{2} = \frac{3 \times 4}{2 \times 4} = \frac{12}{8} \]

    Now we can add the two fractions:
    \[ \frac{12}{8} + \frac{7}{8} = \frac{12 + 7}{8} = \frac{19}{8} \]

    So, the total rate at which the two candles burn together is:
    \[ \frac{19}{8} \] inches per hour.

    ### Answer: \(\frac{19}{8}\) inches per hour.

Expected answer: 19/8

Correct? Yes

🔗 Follow-up

Problem: Liam reevaluated and decided to change the second candle he bought, which now burns at a rate of 26/16 inches per hour, instead of the initial slow rate. Considering the rate of the first candle, what is the total rate at which the two candles now burn?

Gemma 2B:

    First, let's simplify the rates at which each candle burns.

    For the first candle:
    \[ \frac{6}{4} \]
    We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
    \[ \frac{6 \div 2}{4 \div 2} = \frac{3}{2} \]

    For the second candle:
    \[ \frac{26}{16} \]
    We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
    \[ \frac{26 \div 2}{16 \div 2} = \frac{13}{8} \]

    Now, we need to find the total rate at which the two candles burn together. To do this, we add the two simplified rates:
    \[ \frac{3}{2} + \frac{13}{8} \]

    To add these fractions, we need a common denominator. The least common multiple of 2 and 8 is 8. We convert \(\frac{3}{2}\) to a fraction with a denominator of 8:
    \[ \frac{3}{2} = \frac{3 \times 4}{2 \times 4} = \frac{12}{8} \]

    Now we can add the two fractions:
    \[ \frac{12}{8} + \frac{13}{8} = \frac{12 + 13}{8} = \frac{25}{8} \]

    So, the total rate at which the two candles burn together is:
    \[ \frac{25}{8} \] inches per hour.

    ### Answer: \(\frac{25}{8}\) inches per hour.

Expected answer: 25/8

Correct? Yes

🔗 Problem

Problem: Sophia has 17/20 of a dollar. Her friend gives her another 1/4 of a dollar. After some time, her brother gives her 1 dollar. How much money does Sophia have now in dollars?

Gemma 2B:

    To determine how much money Sophia has now, we need to add the amounts she received from her friend and her brother to the amount she initially had.

    1. Sophia initially has \(\frac{17}{20}\) of a dollar.
    2. Her friend gives her \(\frac{1}{4}\) of a dollar.

    First, we need to add these two fractions. To do this, we need a common denominator. The least common multiple of 20 and 4 is 20.

    Convert \(\frac{1}{4}\) to a fraction with a denominator of 20:
    \[
    \frac{1}{4} = \frac{1 \times 5}{4 \times 5} = \frac{5}{20}
    \]

    Now add \(\frac{17}{20}\) and \(\frac{5}{20}\):
    \[
    \frac{17}{20} + \frac{5}{20} = \frac{17 + 5}{20} = \frac{22}{20}
    \]

    Simplify \(\frac{22}{20}\):
    \[
    \frac{22}{20} = \frac{11}{10} = 1.1
    \]

    3. Her brother gives her 1 dollar.

    Now add this to the previous total:
    \[
    1.1 + 1 = 2.1
    \]

    ### Answer: 2.1

Expected answer: 21/10

Correct? Yes

🔗 Follow-up

Problem: After collecting the money from her friend and brother, Sophia decided to invest a fraction of it, exactly 10/14, into a new toy. How much money did she spend on the new toy?

Gemma 2B:

    To determine how much money Sophia spent on the new toy, we need to calculate \(\frac{10}{14}\) of the total amount of money she has, which is 2.1 dollars.

    First, simplify the fraction \(\frac{10}{14}\):
    \[
    \frac{10}{14} = \frac{5}{7}
    \]

    Now, calculate \(\frac{5}{7}\) of 2.1 dollars:
    \[
    \frac{5}{7} \times 2.1
    \]

    Convert 2.1 to a fraction for easier multiplication:
    \[
    2.1 = \frac{21}{10}
    \]

    Now multiply the fractions:
    \[
    \frac{5}{7} \times \frac{21}{10} = \frac{5 \times 21}{7 \times 10} = \frac{105}{70}
    \]

    Simplify \(\frac{105}{70}\):
    \[
    \frac{105}{70} = \frac{3}{2} = 1.5
    \]

    So, Sophia spent 1.5 dollars on the new toy.

    ### Answer: 1.5

Expected answer: 3/2

Correct? No

🔗 Follow-up

Problem: Suppose instead of having 17/20 of a dollar originally, Sophia actually had 17/19 of a dollar. Then her friend gave her another 1/4 of a dollar, and her brother gave her 1 dollar. How much money does she have now in total?

Gemma 2B:

    To determine how much money Sophia has now, we need to add the amounts she received from her friend and her brother to the amount she initially had.

    1. Sophia initially has \(\frac{17}{19}\) of a dollar.
    2. Her friend gives her \(\frac{1}{4}\) of a dollar.

    First, we need to add these two fractions. To do this, we need a common denominator. The least common multiple of 19 and 4 is 76.

    Convert \(\frac{17}{19}\) and \(\frac{1}{4}\) to fractions with a denominator of 76:
    \[
    \frac{17}{19} = \frac{17 \times 4}{19 \times 4} = \frac{68}{76}
    \]
    \[
    \frac{1}{4} = \frac{1 \times 19}{4 \times 19} = \frac{19}{76}
    \]

    Now add \(\frac{68}{76}\) and \(\frac{19}{76}\):
    \[
    \frac{68}{76} + \frac{19}{76} = \frac{68 + 19}{76} = \frac{87}{76}
    \]

    Simplify \(\frac{87}{76}\):
    \[
    \frac{87}{76} = 1 + \frac{11}{76} = 1.144736842105263
    \]

    3. Her brother gives her 1 dollar.

    Now add this to the previous total:
    \[
    1.144736842105263 + 1 = 2.144736842105263
    \]

    ### Answer: 2.144736842105263

Expected answer: 163/76

Correct? No

🔗 Problem

Problem: Imagine you're a music producer who is working on a new song. Your resources are divided into different sections. - First, you dedicate 16/4 hours to composing the music and another 16/29 hours to writing the lyrics. This is your initial time investment for the song (let's call it p). - Then, you spend 28/13 hours for mixing and mastering the song in addition to the initial time investment (let's call this new total time b). - Suddenly, a drummer you hired for a session uses 18/4 hours less because he finished his parts quickly. What is the final time you spent on producing the song, if your total time is expressed as (b - (18 / 4))?

Gemma 2B:

    Let's break down the problem step by step.

    1. **Initial Time Investment (p):**
       - Time spent composing the music: \( \frac{16}{4} \) hours
       - Time spent writing the lyrics: \( \frac{16}{29} \) hours

       Calculate the total initial time investment \( p \):
       \[
       p = \frac{16}{4} + \frac{16}{29}
       \]

       Simplify \( \frac{16}{4} \):
       \[
       \frac{16}{4} = 4
       \]

       So,
       \[
       p = 4 + \frac{16}{29}
       \]

    2. **Total Time Including Mixing and Mastering (b):**
       - Additional time for mixing and mastering: \( \frac{28}{13} \) hours

       Calculate the new total time \( b \):
       \[
       b = p + \frac{28}{13}
       \]

       Substitute \( p \) into the equation:
       \[
       b = 4 + \frac{16}{29} + \frac{28}{13}
       \]

    3. **Final Time After Drummer's Adjustment:**
       - Drummer's time reduction: \( \frac{18}{4} \) hours

       Calculate the final time:
       \[
       \text{Final Time} = b - \frac{18}{4}
       \]

       Substitute \( b \) into the equation:
       \[
       \text{Final Time} = \left( 4 + \frac{16}{29} + \frac{28}{13} \right) - \frac{18}{4}
       \]

    4. **Simplify the Expression:**
       - Convert all fractions to a common denominator if necessary.

       First, find a common denominator for \( \frac{16}{29} \) and \( \frac{28}{13} \):
       \[
       \text{Common denominator} = 29 \times 13 = 377
       \]

       Convert \( \frac{16}{29} \) and \( \frac{28}{13} \) to have the common denominator:
       \[
       \frac{16}{29} = \frac{16 \times 13}{29 \times 13} = \frac{208}{377}
       \]
       \[
       \frac{28}{13} = \frac{28 \times 29}{13 \times 29} = \frac{812}{377}
       \]

       Combine the fractions:
       \[
       \frac{208}{377} + \frac{812}{377} = \frac{208 + 812}{377} = \frac{1020}{377}
       \]

       So,
       \[
       b = 4 + \frac{1020}{377}
       \]

       Now, convert \( 4 \) to a fraction with the same denominator:
       \[
       4 = \frac{4 \times 377}{377} = \frac{1508}{377}
       \]

       Combine:
       \[
       b = \frac{1508}{377} + \frac{1020}{377} = \frac{2528}{377}
       \]

       Now, subtract the drummer's time:
       \[
       \frac{18}{4} = 4.5
       \]

       Convert \( 4.5 \) to a fraction with the same denominator:
       \[
       4.5 = \frac{4.5 \times 377}{377} = \frac{1696.5}{377}
       \]

       Subtract:
       \[
       \text{Final Time} = \frac{2528}{377} - \frac{1696.5}{377} = \frac{831.5}{377}
       \]

       Simplify the fraction if possible:
       \[
       \frac{831.5}{377} = 2.205
       \]

    ### Answer: 2.205 hours

Expected answer: 1663/754

Correct? No

🔗 Problem

Problem: Monica has 21/15 dozens of cookies at her bakery shop. Later in the day, she bakes additional 29/17 dozens of cookies to add into her stock. At the end of the day, she decides to increase her total stock of cookies by a ratio of 7/5 for an upcoming cookie festival. How many dozens of cookies will Monica have for the cookie festival?

Gemma 2B:

    First, we need to find the total number of dozens of cookies Monica has before increasing her stock. We start by adding the two fractions:

    \[
    \frac{21}{15} + \frac{29}{17}
    \]

    To add these fractions, we need a common denominator. The least common multiple (LCM) of 15 and 17 is 255. We convert each fraction to have this common denominator:

    \[
    \frac{21}{15} = \frac{21 \times 17}{15 \times 17} = \frac{357}{255}
    \]

    \[
    \frac{29}{17} = \frac{29 \times 15}{17 \times 15} = \frac{435}{255}
    \]

    Now, we add the two fractions:

    \[
    \frac{357}{255} + \frac{435}{255} = \frac{357 + 435}{255} = \frac{792}{255}
    \]

    Next, we simplify the fraction \(\frac{792}{255}\). We find the greatest common divisor (GCD) of 792 and 255, which is 3:

    \[
    \frac{792 \div 3}{255 \div 3} = \frac{264}{85}
    \]

    So, Monica has \(\frac{264}{85}\) dozens of cookies before increasing her stock. Now, she decides to increase her total stock by a ratio of \(\frac{7}{5}\):

    \[
    \frac{264}{85} \times \frac{7}{5} = \frac{264 \times 7}{85 \times 5} = \frac{1848}{425}
    \]

    We simplify the fraction \(\frac{1848}{425}\). The GCD of 1848 and 425 is 1, so the fraction is already in its simplest form.

    Thus, Monica will have \(\frac{1848}{425}\) dozens of cookies for the cookie festival.

    ### Answer: \(\frac{1848}{425}\) dozens

Expected answer: 1848/425

Correct? Yes

🔗 Follow-up

Problem: In the previous scenario, Monica initially had 21/15 dozens of cookies in her bakery shop. Suppose now she initially has the same amount but later bakes 29/26 dozens of cookies instead of 29/17 dozens. At the end of the day, she still decides to increase the total stock of cookies by a ratio of 7/5 for a different event. How many dozens of cookies will Monica have now for this event?

Gemma 2B:

    First, we need to find the total number of dozens of cookies Monica has before increasing her stock. We start by adding the two fractions:

    \[
    \frac{21}{15} + \frac{29}{26}
    \]

    To add these fractions, we need a common denominator. The least common multiple (LCM) of 15 and 26 is 390. We convert each fraction to have this common denominator:

    \[
    \frac{21}{15} = \frac{21 \times 26}{15 \times 26} = \frac{546}{390}
    \]

    \[
    \frac{29}{26} = \frac{29 \times 15}{26 \times 15} = \frac{435}{390}
    \]

    Now, we add the two fractions:

    \[
    \frac{546}{390} + \frac{435}{390} = \frac{546 + 435}{390} = \frac{981}{390}
    \]

    Next, we simplify the fraction \(\frac{981}{390}\). We find the greatest common divisor (GCD) of 981 and 390, which is 3:

    \[
    \frac{981 \div 3}{390 \div 3} = \frac{327}{130}
    \]

    So, Monica has \(\frac{327}{130}\) dozens of cookies before increasing her stock. Now, she decides to increase her total stock by a ratio of \(\frac{7}{5}\):

    \[
    \frac{327}{130} \times \frac{7}{5} = \frac{327 \times 7}{130 \times 5} = \frac{2289}{650}
    \]

    We simplify the fraction \(\frac{2289}{650}\). The GCD of 2289 and 650 is 1, so the fraction is already in its simplest form.

    Thus, Monica will have \(\frac{2289}{650}\) dozens of cookies for the event.

    ### Answer: \(\frac{2289}{650}\) dozens

Expected answer: 2289/650

Correct? Yes

🔗 Problem

Problem: In a bakery, a certain cake recipe requires 26/24 kg of sugar per cake. If a chef starts with 28 kg of sugar, how many kilograms of sugar will they have left after making one cake and then adding 10/6 kg of sugar to their supply?

Gemma 2B:

    First, let's determine how much sugar is used to make one cake. The recipe requires \( \frac{26}{24} \) kg of sugar per cake. We can simplify this fraction:

    \[
    \frac{26}{24} = \frac{13}{12} \text{ kg}
    \]

    Next, the chef starts with 28 kg of sugar. After making one cake, the amount of sugar left is:

    \[
    28 - \frac{13}{12}
    \]

    To subtract these, we need a common denominator. The common denominator for 28 and \( \frac{13}{12} \) is 12. We convert 28 to a fraction with a denominator of 12:

    \[
    28 = \frac{28 \times 12}{12} = \frac{336}{12}
    \]

    Now, we subtract \( \frac{13}{12} \) from \( \frac{336}{12} \):

    \[
    \frac{336}{12} - \frac{13}{12} = \frac{336 - 13}{12} = \frac{323}{12}
    \]

    Next, the chef adds \( \frac{10}{6} \) kg of sugar to their supply. We simplify \( \frac{10}{6} \):

    \[
    \frac{10}{6} = \frac{5}{3}
    \]

    We need a common denominator to add \( \frac{323}{12} \) and \( \frac{5}{3} \). The common denominator for 12 and 3 is 12. We convert \( \frac{5}{3} \) to a fraction with a denominator of 12:

    \[
    \frac{5}{3} = \frac{5 \times 4}{3 \times 4} = \frac{20}{12}
    \]

    Now, we add \( \frac{323}{12} \) and \( \frac{20}{12} \):

    \[
    \frac{323}{12} + \frac{20}{12} = \frac{323 + 20}{12} = \frac{343}{12}
    \]

    Finally, we simplify \( \frac{343}{12} \):

    \[
    \frac{343}{12} \approx 28.5833 \text{ kg}
    \]

    ### Answer: \( \frac{343}{12} \) kg or approximately 28.5833 kg

Expected answer: 343/12

Correct? Yes

🔗 Follow-up

Problem: Following the previous scenario in the bakery, after the chef adds 10/6 kg of sugar to their supply, they then use 13/23 kg of sugar to make a glaze. How many kilograms of sugar does the chef have left after all of these operations?

Gemma 2B:

    We previously determined that after making one cake and adding \( \frac{10}{6} \) kg of sugar, the chef has \( \frac{343}{12} \) kg of sugar.

    Next, the chef uses \( \frac{13}{23} \) kg of sugar to make a glaze. To find out how much sugar is left, we need to subtract \( \frac{13}{23} \) from \( \frac{343}{12} \).

    First, we need a common denominator for the fractions \( \frac{343}{12} \) and \( \frac{13}{23} \). The least common multiple (LCM) of 12 and 23 is 276.

    Convert \( \frac{343}{12} \) to a fraction with a denominator of 276:

    \[
    \frac{343}{12} = \frac{343 \times 23}{12 \times 23} = \frac{7889}{276}
    \]

    Convert \( \frac{13}{23} \) to a fraction with a denominator of 276:

    \[
    \frac{13}{23} = \frac{13 \times 12}{23 \times 12} = \frac{156}{276}
    \]

    Now, subtract \( \frac{156}{276} \) from \( \frac{7889}{276} \):

    \[
    \frac{7889}{276} - \frac{156}{276} = \frac{7889 - 156}{276} = \frac{7733}{276}
    \]

    Finally, simplify \( \frac{7733}{276} \) if possible. Since 7733 and 276 have no common factors other than 1, the fraction is already in its simplest form.

    ### Answer: \( \frac{7733}{276} \) kg

Expected answer: 7733/276

Correct? Yes

🔗 Follow-up

Problem: Continuing the bakery scenario, let's adjust the sugar quantity in the cake recipe. Now, suppose the cake recipe only requires 24/24 kg of sugar per cake, instead of 26/24 kg as in the previous scenario. If the chef starts with 28 kg of sugar, how many kilograms of sugar will they have left after making one cake and then adding 10/6 kg of sugar to their supply?

Gemma 2B:

    First, let's determine how much sugar is used to make one cake with the new recipe. The recipe now requires \( \frac{24}{24} \) kg of sugar per cake, which simplifies to:

    \[
    \frac{24}{24} = 1 \text{ kg}
    \]

    Next, the chef starts with 28 kg of sugar. After making one cake, the amount of sugar left is:

    \[
    28 - 1 = 27 \text{ kg}
    \]

    Next, the chef adds \( \frac{10}{6} \) kg of sugar to their supply. We simplify \( \frac{10}{6} \):

    \[
    \frac{10}{6} = \frac{5}{3}
    \]

    To add \( 27 \) kg and \( \frac{5}{3} \) kg, we need a common denominator. The common denominator for 27 and \( \frac{5}{3} \) is 3. We convert 27 to a fraction with a denominator of 3:

    \[
    27 = \frac{27 \times 3}{3} = \frac{81}{3}
    \]

    Now, we add \( \frac{81}{3} \) and \( \frac{5}{3} \):

    \[
    \frac{81}{3} + \frac{5}{3} = \frac{81 + 5}{3} = \frac{86}{3}
    \]

    Finally, we simplify \( \frac{86}{3} \):

    \[
    \frac{86}{3} \approx 28.6667 \text{ kg}
    \]

    ### Answer: \( \frac{86}{3} \) kg or approximately 28.6667 kg

Expected answer: 86/3

Correct? Yes

🔗 Problem

Problem: On a plane trip, Emily watched a total of 16 / 4 hours of movies. During this time, she spent 6 / 3 hours watching a comedy film. How many hours did she spend watching movies other than the comedy film?

Gemma 2B:

    First, let's determine the total time Emily spent watching movies. The total time is given as \( \frac{16}{4} \) hours.

    \[
    \frac{16}{4} = 4 \text{ hours}
    \]

    Next, we need to find out how much time she spent watching the comedy film. This time is given as \( \frac{6}{3} \) hours.

    \[
    \frac{6}{3} = 2 \text{ hours}
    \]

    To find the time she spent watching movies other than the comedy film, we subtract the time spent on the comedy film from the total time spent watching movies.

    \[
    4 \text{ hours} - 2 \text{ hours} = 2 \text{ hours}
    \]

    ### Answer: 2 hours

Expected answer: 2

Correct? Yes

🔗 Follow-up

Problem: On the same plane trip, Emily watched a thriller film for 7 / 3 hours. With her total watching time staying at 16 / 4 hours, how many hours did she spend watching movies other than the thriller film?

Gemma 2B:

    First, let's determine the total time Emily spent watching movies. The total time is given as \( \frac{16}{4} \) hours.

    \[
    \frac{16}{4} = 4 \text{ hours}
    \]

    Next, we need to find out how much time she spent watching the thriller film. This time is given as \( \frac{7}{3} \) hours.

    \[
    \frac{7}{3} \approx 2.33 \text{ hours}
    \]

    To find the time she spent watching movies other than the thriller film, we subtract the time spent on the thriller film from the total time spent watching movies.

    \[
    4 \text{ hours} - \frac{7}{3} \text{ hours} = 4 - 2.33 \approx 1.67 \text{ hours}
    \]

    ### Answer: 1.67 hours

Expected answer: 5/3

Correct? No

🔗 Problem

Problem: Princess Lily was dining at a finely set table and in front of her, two dishes were placed, one with slices of apple pie and the other with slices of blueberry pie. She can eat pie slices endlessly! She ate 10 out of the 12 slices of the apple pie and 10 out of the 29 slices of the blueberry pie. How much of the pies (combined) did she eat? Represent your answer as a fraction.

Gemma 2B:

    To determine how much of the pies Princess Lily ate, we need to add the fractions of each pie she consumed.

    First, let's calculate the fraction of the apple pie she ate:
    \[ \text{Fraction of apple pie eaten} = \frac{10}{12} \]

    Next, let's calculate the fraction of the blueberry pie she ate:
    \[ \text{Fraction of blueberry pie eaten} = \frac{10}{29} \]

    To find the total amount of pie she ate, we need to add these two fractions:
    \[ \frac{10}{12} + \frac{10}{29} \]

    To add these fractions, we need a common denominator. The least common multiple (LCM) of 12 and 29 is 348. We convert each fraction to have this common denominator:

    \[ \frac{10}{12} = \frac{10 \times 29}{12 \times 29} = \frac{290}{348} \]
    \[ \frac{10}{29} = \frac{10 \times 12}{29 \times 12} = \frac{120}{348} \]

    Now, we add the fractions:
    \[ \frac{290}{348} + \frac{120}{348} = \frac{290 + 120}{348} = \frac{410}{348} \]

    This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 410 and 348 is 2:

    \[ \frac{410 \div 2}{348 \div 2} = \frac{205}{174} \]

    Thus, the fraction representing the total amount of pie Princess Lily ate is:
    \[ \frac{205}{174} \]

    ### Answer: \(\frac{205}{174}\)

Expected answer: 205/174

Correct? Yes

🔗 Follow-up

Problem: As Princess Lily was about to leave the dining room, she noticed another plate - this one filled with 29 slices of peach pie. She decided to help herself and ate 10 of the slices. Before her peach pie adventure, she had already eaten an entire 10-slice chocolate cake. Now, how much of the total food (chocolate cake and peach pie) did Princess Lily eat? Please represent your answer as a fraction.

Gemma 2B:

    To determine how much of the total food Princess Lily ate, we need to consider both the chocolate cake and the peach pie.

    First, let's calculate the fraction of the peach pie she ate:
    \[ \text{Fraction of peach pie eaten} = \frac{10}{29} \]

    Next, let's consider the chocolate cake. Since she ate the entire 10-slice chocolate cake, the fraction of the chocolate cake she ate is:
    \[ \text{Fraction of chocolate cake eaten} = \frac{10}{10} = 1 \]

    Now, we need to add these two fractions to find the total amount of food she ate:
    \[ 1 + \frac{10}{29} \]

    To add these fractions, we need a common denominator. The common denominator for 1 and \(\frac{10}{29}\) is 29. We convert the whole number 1 to a fraction with this common denominator:
    \[ 1 = \frac{29}{29} \]

    Now, we add the fractions:
    \[ \frac{29}{29} + \frac{10}{29} = \frac{29 + 10}{29} = \frac{39}{29} \]

    Thus, the fraction representing the total amount of food Princess Lily ate is:
    \[ \frac{39}{29} \]

    ### Answer: \(\frac{39}{29}\)

Expected answer: 300/29

Correct? No

🔗 Problem

Problem: Marta has a box of colored pencils. She uses 20 of the colored pencils every 11 days on her art projects. However, she only uses about 23 out of every 26 colored pencils for shading techniques. How many colored pencils does Marta use for shading in her art projects over an 11-day period? Evaluate by performing the operation (20 / 11) * (23 / 26) and write your answer as a simplified fraction.

Gemma 2B:

    To determine how many colored pencils Marta uses for shading in her art projects over an 11-day period, we need to evaluate the expression \((20 / 11) * (23 / 26)\).

    First, let's break down the expression:

    1. Calculate the rate of colored pencils used per day:
       \[
       \frac{20}{11}
       \]

    2. Calculate the fraction of colored pencils used for shading:
       \[
       \frac{23}{26}
       \]

    3. Multiply these two fractions together:
       \[
       \left(\frac{20}{11}\right) \times \left(\frac{23}{26}\right)
       \]

    To multiply the fractions, multiply the numerators together and the denominators together:
    \[
    \frac{20 \times 23}{11 \times 26} = \frac{460}{286}
    \]

    Next, we simplify the fraction \(\frac{460}{286}\). We find the greatest common divisor (GCD) of 460 and 286. The GCD of 460 and 286 is 2.

    Divide both the numerator and the denominator by their GCD:
    \[
    \frac{460 \div 2}{286 \div 2} = \frac{230}{143}
    \]

    Thus, the simplified fraction is:
    \[
    \frac{230}{143}
    \]

    ### Answer: \(\frac{230}{143}\)

Expected answer: 230/143

Correct? Yes

🔗 Follow-up

Problem: In addition to the colored pencils Marta uses for her art projects, she also has a pack of special metallic colored pencils. She uses 8 of these special pencils every 6 days. Adding these two amounts together, how many colored pencils in total does Marta use over an 11-day period? To find this, you need to add the fraction of colored pencils she uses from the previous problem to the amount of metallic pencils she uses. Evaluate this by performing the operation (8 / 6) + ((20 / 11) * (23 / 26)) and write your answer as a simplified fraction.

Gemma 2B:

    To find the total number of colored pencils Marta uses over an 11-day period, we need to add the fraction of colored pencils she uses for shading to the amount of metallic pencils she uses.

    From the previous problem, we have:
    \[
    \left(\frac{20}{11}\right) \times \left(\frac{23}{26}\right) = \frac{230}{143}
    \]

    Now, we need to calculate the fraction of metallic pencils she uses:
    \[
    \frac{8}{6}
    \]

    First, simplify \(\frac{8}{6}\):
    \[
    \frac{8}{6} = \frac{4}{3}
    \]

    Next, we add the two fractions:
    \[
    \frac{4}{3} + \frac{230}{143}
    \]

    To add these fractions, we need a common denominator. The least common multiple (LCM) of 3 and 143 is 429.

    Convert each fraction to have the common denominator of 429:
    \[
    \frac{4}{3} = \frac{4 \times 143}{3 \times 143} = \frac{572}{429}
    \]
    \[
    \frac{230}{143} = \frac{230 \times 3}{143 \times 3} = \frac{690}{429}
    \]

    Now, add the fractions:
    \[
    \frac{572}{429} + \frac{690}{429} = \frac{572 + 690}{429} = \frac{1262}{429}
    \]

    Finally, simplify the fraction \(\frac{1262}{429}\). The greatest common divisor (GCD) of 1262 and 429 is 1, so the fraction is already in its simplest form.

    ### Answer: \(\frac{1262}{429}\)

Expected answer: 1262/429

Correct? Yes

🔗 Follow-up

Problem: Marta realized that she actually only uses 23 out of 24 colored pencils for shading techniques, not 23 out of 26 as previously mentioned. Now, how many colored pencils does Marta use for shading in her art projects over an 11-day period? To find this, you need to adjust the previous calculation. Evaluate this by performing the operation (20 / 11) * (23 / 24) and write your answer as a simplified fraction.

Gemma 2B:

    To determine how many colored pencils Marta uses for shading in her art projects over an 11-day period with the updated information, we need to evaluate the expression \((20 / 11) * (23 / 24)\).

    First, let's break down the expression:

    1. Calculate the rate of colored pencils used per day:
       \[
       \frac{20}{11}
       \]

    2. Calculate the fraction of colored pencils used for shading:
       \[
       \frac{23}{24}
       \]

    3. Multiply these two fractions together:
       \[
       \left(\frac{20}{11}\right) \times \left(\frac{23}{24}\right)
       \]

    To multiply the fractions, multiply the numerators together and the denominators together:
    \[
    \frac{20 \times 23}{11 \times 24} = \frac{460}{264}
    \]

    Next, we simplify the fraction \(\frac{460}{264}\). We find the greatest common divisor (GCD) of 460 and 264. The GCD of 460 and 264 is 4.

    Divide both the numerator and the denominator by their GCD:
    \[
    \frac{460 \div 4}{264 \div 4} = \frac{115}{66}
    \]

    Thus, the simplified fraction is:
    \[
    \frac{115}{66}
    \]

    ### Answer: \(\frac{115}{66}\)

Expected answer: 115/66

Correct? Yes

🔗 Problem

Problem: A police investigator is analyzing two evidence specimens. The first specimen has a unique property that decreases its value by 26 out of every 6 units. The second specimen decreases its value by 25 out of every 17 units. The investigator needs to find the ratio of these changes, referred to as indicator i. Additionally, he needs to figure out a set of changes in another pair of evidence specimens. The first one of this pair increases its value by 18 out of every 8 units and the second one increases its value by 17 out of every 23 units. The investigator has to find the difference between these two rates and refer to it as indicator h. He then combines these two indicators (h and i) by multiplying them to get a final evidence factor, referred to as indicator f. This will help determine the likelihood of a particular suspect’s involvement. Calculate the value of f.

Gemma 2B:

    To solve this problem, we need to calculate the indicators \( i \) and \( h \) and then multiply them to find the final evidence factor \( f \).

    First, let's calculate indicator \( i \):

    1. The first specimen decreases its value by 26 out of every 6 units. This can be expressed as a ratio:
       \[
       \frac{26}{6}
       \]

    2. The second specimen decreases its value by 25 out of every 17 units. This can be expressed as a ratio:
       \[
       \frac{25}{17}
       \]

    3. The ratio of these changes (indicator \( i \)) is:
       \[
       i = \frac{\frac{26}{6}}{\frac{25}{17}} = \frac{26}{6} \times \frac{17}{25} = \frac{26 \times 17}{6 \times 25} = \frac{442}{150}
       \]

    4. Simplify the fraction:
       \[
       \frac{442}{150} = \frac{221}{75}
       \]

    Next, let's calculate indicator \( h \):

    1. The first specimen increases its value by 18 out of every 8 units. This can be expressed as a ratio:
       \[
       \frac{18}{8}
       \]

    2. The second specimen increases its value by 17 out of every 23 units. This can be expressed as a ratio:
       \[
       \frac{17}{23}
       \]

    3. The difference between these two rates (indicator \( h \)) is:
       \[
       h = \frac{18}{8} - \frac{17}{23}
       \]

    4. To subtract these fractions, we need a common denominator. The least common multiple of 8 and 23 is 184:
       \[
       \frac{18}{8} = \frac{18 \times 23}{8 \times 23} = \frac{414}{184}
       \]
       \[
       \frac{17}{23} = \frac{17 \times 8}{23 \times 8} = \frac{136}{184}
       \]

    5. Subtract the fractions:
       \[
       h = \frac{414}{184} - \frac{136}{184} = \frac{278}{184}
       \]

    6. Simplify the fraction:
       \[
       \frac{278}{184} = \frac{139}{92}
       \]

    Finally, we combine these two indicators by multiplying them to get the final evidence factor \( f \):

    1. Multiply \( i \) and \( h \):
       \[
       f = \frac{221}{75} \times \frac{139}{92} = \frac{221 \times 139}{75 \times 92} = \frac{30719}{6900}
       \]

    2. Simplify the fraction:
       \[
       \frac{30719}{6900}
       \]

    Since 30719 and 6900 have no common factors other than 1, the fraction is already in its simplest form.

    ### Answer: \(\frac{30719}{6900}\)

Expected answer: 30719/6900

Correct? Yes

🔗 Problem

Problem: In a wildlife park, there are lions and giraffes. During one particular year, the gestation period of a lioness is represented as 20/22 of a year, while a giraffe's gestation period is 15 years. Find the total gestation period of both animals together. Later in the same year, a lioness is pregnant again but this time her gestation period reduces by 15/20 of a year compared to the earlier total gestation period of both animals. How long will it take for the lioness to give birth this second time?

Gemma 2B:

    First, let's find the total gestation period of both animals together.

    1. The gestation period of a lioness is given as \( \frac{20}{22} \) of a year.
    2. The gestation period of a giraffe is given as 15 years.

    To find the total gestation period, we add these two periods together:
    \[ \text{Total gestation period} = \frac{20}{22} + 15 \]

    Simplify \( \frac{20}{22} \):
    \[ \frac{20}{22} = \frac{10}{11} \]

    So, the total gestation period is:
    \[ \frac{10}{11} + 15 \]

    Next, we need to find the reduced gestation period for the lioness in the second instance. The reduction is given as \( \frac{15}{20} \) of a year.

    Simplify \( \frac{15}{20} \):
    \[ \frac{15}{20} = \frac{3}{4} \]

    Now, subtract this reduction from the total gestation period of both animals:
    \[ \text{Reduced gestation period} = \left( \frac{10}{11} + 15 \right) - \frac{3}{4} \]

    First, convert \( 15 \) to a fraction with a common denominator with \( \frac{10}{11} \):
    \[ 15 = \frac{165}{11} \]

    So, the total gestation period in terms of a common denominator is:
    \[ \frac{10}{11} + \frac{165}{11} = \frac{175}{11} \]

    Now, convert \( \frac{3}{4} \) to a fraction with a common denominator with \( \frac{175}{11} \):
    \[ \frac{3}{4} = \frac{3 \times 11}{4 \times 11} = \frac{33}{44} \]

    Convert \( \frac{175}{11} \) to a fraction with a denominator of 44:
    \[ \frac{175}{11} = \frac{175 \times 4}{11 \times 4} = \frac{700}{44} \]

    Subtract \( \frac{33}{44} \) from \( \frac{700}{44} \):
    \[ \frac{700}{44} - \frac{33}{44} = \frac{700 - 33}{44} = \frac{667}{44} \]

    Simplify \( \frac{667}{44} \):
    \[ \frac{667}{44} = 15.159 \]

    So, the reduced gestation period for the lioness is approximately 15.159 years.

    ### Answer: 15.159 years

Expected answer: 667/44

Correct? No

🔗 Problem

Problem: A rectangle's length is 30/7 meters and its width is 19/6 meters. The area of the rectangle is found by multiplying the length and the width. If a similar rectangle has an area which is 18/17 times bigger than the original rectangle, what is the area of the similar rectangle?

Gemma 2B:

    First, we need to find the area of the original rectangle. The area \( A \) of a rectangle is given by the formula:

    \[ A = \text{length} \times \text{width} \]

    Given the length is \( \frac{30}{7} \) meters and the width is \( \frac{19}{6} \) meters, we can calculate the area as follows:

    \[ A = \frac{30}{7} \times \frac{19}{6} \]

    To multiply these fractions, we multiply the numerators together and the denominators together:

    \[ A = \frac{30 \times 19}{7 \times 6} \]

    \[ A = \frac{570}{42} \]

    We can simplify this fraction by finding the greatest common divisor (GCD) of 570 and 42. The GCD of 570 and 42 is 6. So, we divide both the numerator and the denominator by 6:

    \[ A = \frac{570 \div 6}{42 \div 6} \]

    \[ A = \frac{95}{7} \]

    Now, we need to find the area of the similar rectangle, which is \( \frac{18}{17} \) times bigger than the original rectangle. We multiply the area of the original rectangle by \( \frac{18}{17} \):

    \[ \text{Area of similar rectangle} = \frac{95}{7} \times \frac{18}{17} \]

    Again, we multiply the numerators together and the denominators together:

    \[ \text{Area of similar rectangle} = \frac{95 \times 18}{7 \times 17} \]

    \[ \text{Area of similar rectangle} = \frac{1710}{119} \]

    We can check if this fraction can be simplified. The GCD of 1710 and 119 is 1, so the fraction is already in its simplest form.

    ### Answer: \(\frac{1710}{119}\)

Expected answer: 1710/119

Correct? Yes

🔗 Follow-up

Problem: A rectangle's length is 30/7 meters and its width is 19/6 meters. The area of the rectangle is found by multiplying the length and the width. A similar rectangle has an area which is 18/17 times larger than the original rectangle. If we subtract 19/2 square meters from the area of the similar rectangle, what is the remaining area?

Gemma 2B:

    First, we need to find the area of the original rectangle. The area \( A \) of a rectangle is given by the formula:

    \[ A = \text{length} \times \text{width} \]

    Given the length is \( \frac{30}{7} \) meters and the width is \( \frac{19}{6} \) meters, we can calculate the area as follows:

    \[ A = \frac{30}{7} \times \frac{19}{6} \]

    To multiply these fractions, we multiply the numerators together and the denominators together:

    \[ A = \frac{30 \times 19}{7 \times 6} \]

    \[ A = \frac{570}{42} \]

    We can simplify this fraction by finding the greatest common divisor (GCD) of 570 and 42. The GCD of 570 and 42 is 6. So, we divide both the numerator and the denominator by 6:

    \[ A = \frac{570 \div 6}{42 \div 6} \]

    \[ A = \frac{95}{7} \]

    Now, we need to find the area of the similar rectangle, which is \( \frac{18}{17} \) times larger than the original rectangle. We multiply the area of the original rectangle by \( \frac{18}{17} \):

    \[ \text{Area of similar rectangle} = \frac{95}{7} \times \frac{18}{17} \]

    Again, we multiply the numerators together and the denominators together:

    \[ \text{Area of similar rectangle} = \frac{95 \times 18}{7 \times 17} \]

    \[ \text{Area of similar rectangle} = \frac{1710}{119} \]

    We can simplify this fraction by finding the greatest common divisor (GCD) of 1710 and 119. The GCD of 1710 and 119 is 1, so the fraction is already in its simplest form.

    Next, we need to subtract \( \frac{19}{2} \) square meters from the area of the similar rectangle. To do this, we need a common denominator for the fractions \( \frac{1710}{119} \) and \( \frac{19}{2} \). The least common multiple (LCM) of 119 and 2 is 238.

    Convert both fractions to have a denominator of 238:

    \[ \frac{1710}{119} = \frac{1710 \times 2}{119 \times 2} = \frac{3420}{238} \]

    \[ \frac{19}{2} = \frac{19 \times 119}{2 \times 119} = \frac{2261}{238} \]

    Now, subtract the fractions:

    \[ \frac{3420}{238} - \frac{2261}{238} = \frac{3420 - 2261}{238} = \frac{1159}{238} \]

    ### Answer: \(\frac{1159}{238}\)

Expected answer: 1159/238

Correct? Yes

🔗 Follow-up

Problem: A rectangle's length is 30/7 meters and its width is 19/6 meters. The area of the rectangle is found by multiplying the length and the width. Previously, we considered a similar rectangle that was 18/17 times larger than the original one. Now, let's consider a different scenario. If a similar rectangle has an area which is 15/17 times larger than the original rectangle, what would its area be?

Gemma 2B:

    First, we need to find the area of the original rectangle. The area \( A \) of a rectangle is given by the formula:

    \[ A = \text{length} \times \text{width} \]

    Given the length is \( \frac{30}{7} \) meters and the width is \( \frac{19}{6} \) meters, we can calculate the area as follows:

    \[ A = \frac{30}{7} \times \frac{19}{6} \]

    To multiply these fractions, we multiply the numerators together and the denominators together:

    \[ A = \frac{30 \times 19}{7 \times 6} \]

    \[ A = \frac{570}{42} \]

    We can simplify this fraction by finding the greatest common divisor (GCD) of 570 and 42. The GCD of 570 and 42 is 6. So, we divide both the numerator and the denominator by 6:

    \[ A = \frac{570 \div 6}{42 \div 6} \]

    \[ A = \frac{95}{7} \]

    Now, we need to find the area of the similar rectangle, which is \( \frac{15}{17} \) times larger than the original rectangle. We multiply the area of the original rectangle by \( \frac{15}{17} \):

    \[ \text{Area of similar rectangle} = \frac{95}{7} \times \frac{15}{17} \]

    Again, we multiply the numerators together and the denominators together:

    \[ \text{Area of similar rectangle} = \frac{95 \times 15}{7 \times 17} \]

    \[ \text{Area of similar rectangle} = \frac{1425}{119} \]

    We can check if this fraction can be simplified. The GCD of 1425 and 119 is 1, so the fraction is already in its simplest form.

    ### Answer: \(\frac{1425}{119}\)

Expected answer: 1425/119

Correct? Yes

🔗 Problem

Problem: A soap factory produces 23/8 tonnes of soap each day. One day, they decided to increase production by an extra 4/20 of a tonne. However, later that same day, they reduce the production by 19/24 of a tonne due to a sudden power cut. How many tonnes of soap does the factory produce that day after these adjustments?

Gemma 2B:

    To determine the total production of soap after the adjustments, we need to follow these steps:

    1. **Initial Production**: The factory produces \(\frac{23}{8}\) tonnes of soap each day.

    2. **Increase in Production**: The factory increases production by \(\frac{4}{20}\) tonnes. Simplify \(\frac{4}{20}\):
       \[
       \frac{4}{20} = \frac{1}{5}
       \]

    3. **Total Production after Increase**: Add the increased amount to the initial production:
       \[
       \frac{23}{8} + \frac{1}{5}
       \]

       To add these fractions, find a common denominator. The least common multiple of 8 and 5 is 40. Convert each fraction:
       \[
       \frac{23}{8} = \frac{23 \times 5}{8 \times 5} = \frac{115}{40}
       \]
       \[
       \frac{1}{5} = \frac{1 \times 8}{5 \times 8} = \frac{8}{40}
       \]

       Now add the fractions:
       \[
       \frac{115}{40} + \frac{8}{40} = \frac{123}{40}
       \]

    4. **Reduction in Production**: The factory reduces production by \(\frac{19}{24}\) tonnes. Convert \(\frac{123}{40}\) and \(\frac{19}{24}\) to a common denominator. The least common multiple of 40 and 24 is 120. Convert each fraction:
       \[
       \frac{123}{40} = \frac{123 \times 3}{40 \times 3} = \frac{369}{120}
       \]
       \[
       \frac{19}{24} = \frac{19 \times 5}{24 \times 5} = \frac{95}{120}
       \]

       Now subtract the fractions:
       \[
       \frac{369}{120} - \frac{95}{120} = \frac{274}{120}
       \]

    5. **Simplify the Result**: Simplify \(\frac{274}{120}\) by finding the greatest common divisor (GCD) of 274 and 120, which is 2:
       \[
       \frac{274 \div 2}{120 \div 2} = \frac{137}{60}
       \]

    Thus, the factory produces \(\frac{137}{60}\) tonnes of soap that day after the adjustments.

    ### Answer: \(\frac{137}{60}\) tonnes

Expected answer: 137/60

Correct? Yes

🔗 Follow-up

Problem: In the same soap factory, consider a new scenario where instead of reducing production by 19/24 of a tonne due to a power cut, they only reduce production by 8/24 of a tonne. Following the same adjustment of increasing production by an extra 4/20 of a tonne earlier in the day, how many tonnes of soap does the factory produce that day?

Gemma 2B:

    To determine the total production of soap after the new adjustments, we need to follow these steps:

    1. **Initial Production**: The factory produces \(\frac{23}{8}\) tonnes of soap each day.

    2. **Increase in Production**: The factory increases production by \(\frac{4}{20}\) tonnes. Simplify \(\frac{4}{20}\):
       \[
       \frac{4}{20} = \frac{1}{5}
       \]

    3. **Total Production after Increase**: Add the increased amount to the initial production:
       \[
       \frac{23}{8} + \frac{1}{5}
       \]

       To add these fractions, find a common denominator. The least common multiple of 8 and 5 is 40. Convert each fraction:
       \[
       \frac{23}{8} = \frac{23 \times 5}{8 \times 5} = \frac{115}{40}
       \]
       \[
       \frac{1}{5} = \frac{1 \times 8}{5 \times 8} = \frac{8}{40}
       \]

       Now add the fractions:
       \[
       \frac{115}{40} + \frac{8}{40} = \frac{123}{40}
       \]

    4. **Reduction in Production**: The factory reduces production by \(\frac{8}{24}\) tonnes. Simplify \(\frac{8}{24}\):
       \[
       \frac{8}{24} = \frac{1}{3}
       \]

       Convert \(\frac{123}{40}\) and \(\frac{1}{3}\) to a common denominator. The least common multiple of 40 and 3 is 120. Convert each fraction:
       \[
       \frac{123}{40} = \frac{123 \times 3}{40 \times 3} = \frac{369}{120}
       \]
       \[
       \frac{1}{3} = \frac{1 \times 40}{3 \times 40} = \frac{40}{120}
       \]

       Now subtract the fractions:
       \[
       \frac{369}{120} - \frac{40}{120} = \frac{329}{120}
       \]

    5. **Simplify the Result**: Simplify \(\frac{329}{120}\) if possible. The greatest common divisor (GCD) of 329 and 120 is 1, so the fraction is already in its simplest form.

    Thus, the factory produces \(\frac{329}{120}\) tonnes of soap that day after the adjustments.

    ### Answer: \(\frac{329}{120}\) tonnes

Expected answer: 329/120

Correct? Yes

🔗 Problem

Problem: You have a paint mixing machine that creates specific shades of color. The machine starts by mixing 24/8 gallons of red paint. It then pours away 12/8 gallons of the mix. Let 'd' represent the amount of red paint left in the machine. Finally, the machine adds another 1/28 gallon of blue pigment to lighten the color, subtract that from 'd' to find 'x' in gallons, which is the total amount of paint. Find the value of 'x'.

Gemma 2B:

    First, let's determine the initial amount of red paint mixed by the machine:
    \[ \frac{24}{8} = 3 \text{ gallons} \]

    Next, the machine pours away:
    \[ \frac{12}{8} = 1.5 \text{ gallons} \]

    So, the amount of red paint left in the machine, \( d \), is:
    \[ d = 3 - 1.5 = 1.5 \text{ gallons} \]

    The machine then adds \( \frac{1}{28} \) gallon of blue pigment. The total amount of paint, \( x \), is:
    \[ x = d + \frac{1}{28} \]
    \[ x = 1.5 + \frac{1}{28} \]

    To add these, we need a common denominator. The common denominator for 1.5 (which is \( \frac{3}{2} \)) and \( \frac{1}{28} \) is 28:
    \[ 1.5 = \frac{3}{2} = \frac{3 \times 14}{2 \times 14} = \frac{42}{28} \]

    Now, add the fractions:
    \[ x = \frac{42}{28} + \frac{1}{28} = \frac{42 + 1}{28} = \frac{43}{28} \]

    ### Answer: \(\frac{43}{28}\) gallons

Expected answer: 41/28

Correct? No

🔗 Problem

Problem: A banana bread recipe requires you to multiply the fractions (27 / 14) and (17 / 13) together to determine how many ounces of mashed bananas you'll need. Compute the product of these fractions and write your answer as a simplified fraction to find out how many ounces are required.

Gemma 2B:

    To find the product of the fractions \(\frac{27}{14}\) and \(\frac{17}{13}\), we multiply the numerators together and the denominators together:

    \[
    \frac{27}{14} \times \frac{17}{13} = \frac{27 \times 17}{14 \times 13}
    \]

    First, calculate the numerator:

    \[
    27 \times 17 = 459
    \]

    Next, calculate the denominator:

    \[
    14 \times 13 = 182
    \]

    So, the product of the fractions is:

    \[
    \frac{459}{182}
    \]

    To simplify this fraction, we need to find the greatest common divisor (GCD) of 459 and 182. We can use the Euclidean algorithm to find the GCD.

    1. Divide 459 by 182 and find the remainder:
    \[
    459 \div 182 \approx 2.52 \quad \text{(integer part is 2)}
    \]
    \[
    459 - 2 \times 182 = 459 - 364 = 95
    \]

    2. Now, divide 182 by 95 and find the remainder:
    \[
    182 \div 95 \approx 1.91 \quad \text{(integer part is 1)}
    \]
    \[
    182 - 1 \times 95 = 182 - 95 = 87
    \]

    3. Next, divide 95 by 87 and find the remainder:
    \[
    95 \div 87 \approx 1.09 \quad \text{(integer part is 1)}
    \]
    \[
    95 - 1 \times 87 = 95 - 87 = 8
    \]

    4. Finally, divide 87 by 8 and find the remainder:
    \[
    87 \div 8 \approx 10.875 \quad \text{(integer part is 10)}
    \]
    \[
    87 - 10 \times 8 = 87 - 80 = 7
    \]

    5. Divide 8 by 7 and find the remainder:
    \[
    8 \div 7 \approx 1.14 \quad \text{(integer part is 1)}
    \]
    \[
    8 - 1 \times 7 = 8 - 7 = 1
    \]

    Since the remainder is now 1, the GCD of 459 and 182 is 1. Therefore, the fraction \(\frac{459}{182}\) is already in its simplest form.

    ### Answer: \(\frac{459}{182}\)

Expected answer: 459/182

Correct? Yes

🔗 Follow-up

Problem: After finding out how many ounces of mashed bananas you'll need for the banana bread, you realize that you need to scale up the recipe by the factor represented by the fraction (8 / 4). Compute the product of the original fraction and the scaling factor (8 / 4) to determine the new amount of mashed bananas that will be needed. Write your answer as a simplified fraction.

Gemma 2B:

    To determine the new amount of mashed bananas needed, we need to multiply the original fraction \(\frac{459}{182}\) by the scaling factor \(\frac{8}{4}\).

    First, simplify the scaling factor \(\frac{8}{4}\):

    \[
    \frac{8}{4} = 2
    \]

    Now, multiply the original fraction by 2:

    \[
    \frac{459}{182} \times 2 = \frac{459 \times 2}{182}
    \]

    Calculate the numerator:

    \[
    459 \times 2 = 918
    \]

    So, the new fraction is:

    \[
    \frac{918}{182}
    \]

    Next, we simplify this fraction. We need to find the greatest common divisor (GCD) of 918 and 182. We can use the Euclidean algorithm to find the GCD.

    1. Divide 918 by 182 and find the remainder:
    \[
    918 \div 182 \approx 5.04 \quad \text{(integer part is 5)}
    \]
    \[
    918 - 5 \times 182 = 918 - 910 = 8
    \]

    2. Now, divide 182 by 8 and find the remainder:
    \[
    182 \div 8 \approx 22.75 \quad \text{(integer part is 22)}
    \]
    \[
    182 - 22 \times 8 = 182 - 176 = 6
    \]

    3. Next, divide 8 by 6 and find the remainder:
    \[
    8 \div 6 \approx 1.33 \quad \text{(integer part is 1)}
    \]
    \[
    8 - 1 \times 6 = 8 - 6 = 2
    \]

    4. Finally, divide 6 by 2 and find the remainder:
    \[
    6 \div 2 = 3 \quad \text{(integer part is 3)}
    \]
    \[
    6 - 3 \times 2 = 6 - 6 = 0
    \]

    Since the remainder is now 0, the GCD of 918 and 182 is 2. Therefore, we can simplify the fraction \(\frac{918}{182}\) by dividing both the numerator and the denominator by 2:

    \[
    \frac{918 \div 2}{182 \div 2} = \frac{459}{91}
    \]

    ### Answer: \(\frac{459}{91}\)

Expected answer: 459/91

Correct? Yes

🔗 Follow-up

Problem: After reevaluating the recipe, you discover that the amount of mashed bananas required is actually determined by multiplying the fractions (27 / 14) and (17 / 15), not (27 / 14) and (17 / 13) as you initially thought. Recompute the product of these fractions to find out the correct amount of mashed bananas needed for the banana bread. Write your answer as a simplified fraction.

Gemma 2B:

    To find the product of the fractions \(\frac{27}{14}\) and \(\frac{17}{15}\), we multiply the numerators together and the denominators together:

    \[
    \frac{27}{14} \times \frac{17}{15} = \frac{27 \times 17}{14 \times 15}
    \]

    First, calculate the numerator:

    \[
    27 \times 17 = 459
    \]

    Next, calculate the denominator:

    \[
    14 \times 15 = 210
    \]

    So, the product of the fractions is:

    \[
    \frac{459}{210}
    \]

    To simplify this fraction, we need to find the greatest common divisor (GCD) of 459 and 210. We can use the Euclidean algorithm to find the GCD.

    1. Divide 459 by 210 and find the remainder:
    \[
    459 \div 210 \approx 2.19 \quad \text{(integer part is 2)}
    \]
    \[
    459 - 2 \times 210 = 459 - 420 = 39
    \]

    2. Now, divide 210 by 39 and find the remainder:
    \[
    210 \div 39 \approx 5.38 \quad \text{(integer part is 5)}
    \]
    \[
    210 - 5 \times 39 = 210 - 195 = 15
    \]

    3. Next, divide 39 by 15 and find the remainder:
    \[
    39 \div 15 \approx 2.6 \quad \text{(integer part is 2)}
    \]
    \[
    39 - 2 \times 15 = 39 - 30 = 9
    \]

    4. Finally, divide 15 by 9 and find the remainder:
    \[
    15 \div 9 \approx 1.67 \quad \text{(integer part is 1)}
    \]
    \[
    15 - 1 \times 9 = 15 - 9 = 6
    \]

    5. Divide 9 by 6 and find the remainder:
    \[
    9 \div 6 \approx 1.5 \quad \text{(integer part is 1)}
    \]
    \[
    9 - 1 \times 6 = 9 - 6 = 3
    \]

    6. Divide 6 by 3 and find the remainder:
    \[
    6 \div 3 = 2 \quad \text{(integer part is 2)}
    \]
    \[
    6 - 2 \times 3 = 6 - 6 = 0
    \]

    Since the remainder is now 0, the GCD of 459 and 210 is 3. Therefore, we can simplify the fraction \(\frac{459}{210}\) by dividing both the numerator and the denominator by their GCD, which is 3:

    \[
    \frac{459 \div 3}{210 \div 3} = \frac{153}{70}
    \]

    ### Answer: \(\frac{153}{70}\)

Expected answer: 153/70

Correct? Yes

🔗 Problem

Problem: Sonia uses 21/12 oz of toothpaste to brush her teeth every day. Her brother Anthony uses less toothpaste, just 6/27 oz daily. Add both quantities of toothpaste to get the total daily amount used by Sonia and Anthony. Now, suppose their mom includes an amount of toothpaste for herself to the total - she uses 3/18 oz of toothpaste daily. Add this amount to the total used by Sonia and Anthony to get the final total daily toothpaste usage for the whole family. Solve each fraction separately and give the answer as a simplified fraction.

Gemma 2B:

    First, let's add the quantities of toothpaste used by Sonia and Anthony.

    Sonia uses \( \frac{21}{12} \) oz of toothpaste daily.
    Anthony uses \( \frac{6}{27} \) oz of toothpaste daily.

    To add these fractions, we need a common denominator. The least common multiple (LCM) of 12 and 27 is 108.

    Convert \( \frac{21}{12} \) to a fraction with a denominator of 108:
    \[ \frac{21}{12} = \frac{21 \times 9}{12 \times 9} = \frac{189}{108} \]

    Convert \( \frac{6}{27} \) to a fraction with a denominator of 108:
    \[ \frac{6}{27} = \frac{6 \times 4}{27 \times 4} = \frac{24}{108} \]

    Now, add the two fractions:
    \[ \frac{189}{108} + \frac{24}{108} = \frac{189 + 24}{108} = \frac{213}{108} \]

    Next, simplify \( \frac{213}{108} \):
    The greatest common divisor (GCD) of 213 and 108 is 3.
    \[ \frac{213 \div 3}{108 \div 3} = \frac{71}{36} \]

    Now, let's add the amount of toothpaste used by their mom, which is \( \frac{3}{18} \).

    Convert \( \frac{3}{18} \) to a fraction with a denominator of 36:
    \[ \frac{3}{18} = \frac{3 \times 2}{18 \times 2} = \frac{6}{36} \]

    Now, add this to the previous total:
    \[ \frac{71}{36} + \frac{6}{36} = \frac{71 + 6}{36} = \frac{77}{36} \]

    Finally, simplify \( \frac{77}{36} \):
    The GCD of 77 and 36 is 1, so the fraction is already in its simplest form.

    ### Answer: \(\frac{77}{36}\)

Expected answer: 77/36

Correct? Yes

🔗 Follow-up

Problem: Sonia uses 21/12 oz of toothpaste to brush her teeth every day, and her brother Anthony uses 6/27 oz daily. Their mom includes her daily usage of 3/18 oz of toothpaste to the total. Now, their visiting grandma also starts using toothpaste daily. She uses a different toothpaste amount, specifically, she uses 24/28 oz daily. Add grandma's toothpaste usage to the already calculated total daily toothpaste usage for Sonia, Anthony, and their mom to get the final total daily toothpaste usage for the whole family including grandma. Give your answer as a simplified fraction.

Gemma 2B:

    We previously calculated the total daily toothpaste usage for Sonia, Anthony, and their mom as \( \frac{77}{36} \).

    Now, we need to add grandma's daily usage of \( \frac{24}{28} \) oz.

    First, simplify \( \frac{24}{28} \):
    The greatest common divisor (GCD) of 24 and 28 is 4.
    \[ \frac{24 \div 4}{28 \div 4} = \frac{6}{7} \]

    Next, we need a common denominator to add \( \frac{77}{36} \) and \( \frac{6}{7} \). The least common multiple (LCM) of 36 and 7 is 252.

    Convert \( \frac{77}{36} \) to a fraction with a denominator of 252:
    \[ \frac{77}{36} = \frac{77 \times 7}{36 \times 7} = \frac{539}{252} \]

    Convert \( \frac{6}{7} \) to a fraction with a denominator of 252:
    \[ \frac{6}{7} = \frac{6 \times 36}{7 \times 36} = \frac{216}{252} \]

    Now, add the two fractions:
    \[ \frac{539}{252} + \frac{216}{252} = \frac{539 + 216}{252} = \frac{755}{252} \]

    Finally, simplify \( \frac{755}{252} \):
    The GCD of 755 and 252 is 1, so the fraction is already in its simplest form.

    ### Answer: \(\frac{755}{252}\)

Expected answer: 755/252

Correct? Yes

🔗 Follow-up

Problem: In our ongoing toothpaste saga, consider the situation where Sonia is still using 21/12 oz of toothpaste to brush her teeth daily but Anthony decides to use more toothpaste and now uses 26/27 oz daily instead. Their mom's daily toothpaste usage remains at 3/18 oz. Calculate the new total daily toothpaste usage for Sonia, Anthony, and their mom. Give your answer as a simplified fraction.

Gemma 2B:

    First, let's add the quantities of toothpaste used by Sonia and Anthony.

    Sonia uses \( \frac{21}{12} \) oz of toothpaste daily.
    Anthony now uses \( \frac{26}{27} \) oz of toothpaste daily.

    To add these fractions, we need a common denominator. The least common multiple (LCM) of 12 and 27 is 108.

    Convert \( \frac{21}{12} \) to a fraction with a denominator of 108:
    \[ \frac{21}{12} = \frac{21 \times 9}{12 \times 9} = \frac{189}{108} \]

    Convert \( \frac{26}{27} \) to a fraction with a denominator of 108:
    \[ \frac{26}{27} = \frac{26 \times 4}{27 \times 4} = \frac{104}{108} \]

    Now, add the two fractions:
    \[ \frac{189}{108} + \frac{104}{108} = \frac{189 + 104}{108} = \frac{293}{108} \]

    Next, let's add the amount of toothpaste used by their mom, which is \( \frac{3}{18} \).

    Convert \( \frac{3}{18} \) to a fraction with a denominator of 108:
    \[ \frac{3}{18} = \frac{3 \times 6}{18 \times 6} = \frac{18}{108} \]

    Now, add this to the previous total:
    \[ \frac{293}{108} + \frac{18}{108} = \frac{293 + 18}{108} = \frac{311}{108} \]

    Finally, simplify \( \frac{311}{108} \):
    The GCD of 311 and 108 is 1, so the fraction is already in its simplest form.

    ### Answer: \(\frac{311}{108}\)

Expected answer: 311/108

Correct? Yes

🔗 Problem

Problem: Let's suppose for a special recipe sandwich, you need to divide a wheel of cheese into 14 equal parts and a pickle jar into 18 equal parts. To make one special sandwich, you need 21 parts of the wheel of cheese and 7 parts from the pickle jar. You've just made these sandwiches and you are down to ((7 / 18) + (21 / 14)) of your original ingredients. Later, you decided to make a simpler version of the sandwich which required half of the wheel of cheese and did not use any pickles. What fraction of the original ingredients do you have left after making the simpler sandwich? Calculate the answer by subtracting (2 / 4) from the fraction you had after making the first sandwich.

Gemma 2B:

    To solve this problem, we need to follow these steps:

    1. Calculate the fraction of the original ingredients left after making the special sandwiches.
    2. Subtract the fraction used for the simpler sandwich from the remaining fraction.

    First, let's calculate the fraction of the original ingredients left after making the special sandwiches:

    Given:
    - The fraction of the wheel of cheese left is \(\frac{21}{14}\).
    - The fraction of the pickle jar left is \(\frac{7}{18}\).

    We need to add these fractions together:
    \[
    \frac{21}{14} + \frac{7}{18}
    \]

    First, simplify \(\frac{21}{14}\):
    \[
    \frac{21}{14} = \frac{21 \div 7}{14 \div 7} = \frac{3}{2}
    \]

    Now, add \(\frac{3}{2}\) and \(\frac{7}{18}\). To do this, we need a common denominator. The least common multiple of 2 and 18 is 18.

    Convert \(\frac{3}{2}\) to a fraction with a denominator of 18:
    \[
    \frac{3}{2} = \frac{3 \times 9}{2 \times 9} = \frac{27}{18}
    \]

    Now add the fractions:
    \[
    \frac{27}{18} + \frac{7}{18} = \frac{27 + 7}{18} = \frac{34}{18}
    \]

    Simplify \(\frac{34}{18}\):
    \[
    \frac{34}{18} = \frac{34 \div 2}{18 \div 2} = \frac{17}{9}
    \]

    Next, we need to subtract the fraction used for the simpler sandwich. The simpler sandwich uses half of the wheel of cheese, which is \(\frac{1}{2}\) or \(\frac{2}{4}\).

    Convert \(\frac{2}{4}\) to a fraction with a denominator of 9:
    \[
    \frac{2}{4} = \frac{2 \times 9}{4 \times 9} = \frac{18}{36} = \frac{1}{2}
    \]

    Now, subtract \(\frac{1}{2}\) from \(\frac{17}{9}\). Convert \(\frac{1}{2}\) to a fraction with a denominator of 9:
    \[
    \frac{1}{2} = \frac{1 \times 9}{2 \times 9} = \frac{9}{18}
    \]

    Subtract:
    \[
    \frac{17}{9} - \frac{9}{18}
    \]

    Convert \(\frac{9}{18}\) to a fraction with a denominator of 9:
    \[
    \frac{9}{18} = \frac{9 \div 9}{18 \div 9} = \frac{1}{2} = \frac{9}{18}
    \]

    Now, subtract:
    \[
    \frac{17}{9} - \frac{9}{18} = \frac{34}{18} - \frac{9}{18} = \frac{34 - 9}{18} = \frac{25}{18}
    \]

    ### Answer: \(\frac{25}{18}\)

Expected answer: 25/18

Correct? Yes

🔗 Follow-up

Problem: In the same scenario, let's say you found a jar of pickles in the back of your fridge, so now you have 19 parts of the pickle jar instead of the 7 you thought you had. So, for the initial special recipe sandwiches, you now have ((19 / 18) + (21 / 14)) of your original ingredients. Just like before, later, you decided to make a simpler version of the sandwich that required half of the wheel of cheese and didn't use any pickles. What fraction of the original ingredients do you now have left after making the simpler sandwich? Calculate the answer by subtracting (2 / 4) from the fraction you had after making the first special recipe sandwiches with the extra pickles.

Gemma 2B:

    To solve this problem, we need to follow these steps:

    1. Calculate the fraction of the original ingredients left after making the special sandwiches with the extra pickles.
    2. Subtract the fraction used for the simpler sandwich from the remaining fraction.

    First, let's calculate the fraction of the original ingredients left after making the special sandwiches with the extra pickles:

    Given:
    - The fraction of the wheel of cheese left is \(\frac{21}{14}\).
    - The fraction of the pickle jar left is \(\frac{19}{18}\).

    We need to add these fractions together:
    \[
    \frac{21}{14} + \frac{19}{18}
    \]

    First, simplify \(\frac{21}{14}\):
    \[
    \frac{21}{14} = \frac{21 \div 7}{14 \div 7} = \frac{3}{2}
    \]

    Now, add \(\frac{3}{2}\) and \(\frac{19}{18}\). To do this, we need a common denominator. The least common multiple of 2 and 18 is 18.

    Convert \(\frac{3}{2}\) to a fraction with a denominator of 18:
    \[
    \frac{3}{2} = \frac{3 \times 9}{2 \times 9} = \frac{27}{18}
    \]

    Now add the fractions:
    \[
    \frac{27}{18} + \frac{19}{18} = \frac{27 + 19}{18} = \frac{46}{18}
    \]

    Simplify \(\frac{46}{18}\):
    \[
    \frac{46}{18} = \frac{46 \div 2}{18 \div 2} = \frac{23}{9}
    \]

    Next, we need to subtract the fraction used for the simpler sandwich. The simpler sandwich uses half of the wheel of cheese, which is \(\frac{1}{2}\) or \(\frac{2}{4}\).

    Convert \(\frac{2}{4}\) to a fraction with a denominator of 9:
    \[
    \frac{2}{4} = \frac{2 \times 9}{4 \times 9} = \frac{18}{36} = \frac{1}{2}
    \]

    Now, subtract \(\frac{1}{2}\) from \(\frac{23}{9}\). Convert \(\frac{1}{2}\) to a fraction with a denominator of 9:
    \[
    \frac{1}{2} = \frac{1 \times 9}{2 \times 9} = \frac{9}{18}
    \]

    Subtract:
    \[
    \frac{23}{9} - \frac{9}{18}
    \]

    Convert \(\frac{9}{18}\) to a fraction with a denominator of 9:
    \[
    \frac{9}{18} = \frac{9 \div 9}{18 \div 9} = \frac{1}{2} = \frac{9}{18}
    \]

    Now, subtract:
    \[
    \frac{23}{9} - \frac{9}{18} = \frac{46}{18} - \frac{9}{18} = \frac{46 - 9}{18} = \frac{37}{18}
    \]

    ### Answer: \(\frac{37}{18}\)

Expected answer: 37/18

Correct? Yes

🔗 Problem

Problem: A window washer needs to clean two windows in a large building. The first window is 15/6 meters tall while the other one is 7/16 meters tall. How tall are the windows in total?

Gemma 2B:

    To find the total height of the two windows, we need to add their heights together. The heights are given as fractions: \( \frac{15}{6} \) meters and \( \frac{7}{16} \) meters.

    First, let's simplify the fraction \( \frac{15}{6} \):
    \[ \frac{15}{6} = \frac{15 \div 3}{6 \div 3} = \frac{5}{2} \]

    Now, we need to add \( \frac{5}{2} \) and \( \frac{7}{16} \). To do this, we need a common denominator. The least common multiple of 2 and 16 is 16.

    Convert \( \frac{5}{2} \) to a fraction with a denominator of 16:
    \[ \frac{5}{2} = \frac{5 \times 8}{2 \times 8} = \frac{40}{16} \]

    Now, add \( \frac{40}{16} \) and \( \frac{7}{16} \):
    \[ \frac{40}{16} + \frac{7}{16} = \frac{40 + 7}{16} = \frac{47}{16} \]

    So, the total height of the two windows is \( \frac{47}{16} \) meters.

    ### Answer: \( \frac{47}{16} \) meters

Expected answer: 47/16

Correct? Yes

🔗 Follow-up

Problem: The window washer noticed that he made an error in measuring the first window. It is actually 15/5 meters tall, not 15/6 meters tall as he initially recorded. How tall are the windows in total now?

Gemma 2B:

    To find the total height of the two windows with the corrected measurement, we need to add their heights together. The corrected heights are \( \frac{15}{5} \) meters and \( \frac{7}{16} \) meters.

    First, let's simplify the fraction \( \frac{15}{5} \):
    \[ \frac{15}{5} = 3 \]

    Now, we need to add 3 meters and \( \frac{7}{16} \) meters. To do this, we need to convert 3 to a fraction with a denominator of 16:
    \[ 3 = \frac{3 \times 16}{1 \times 16} = \frac{48}{16} \]

    Now, add \( \frac{48}{16} \) and \( \frac{7}{16} \):
    \[ \frac{48}{16} + \frac{7}{16} = \frac{48 + 7}{16} = \frac{55}{16} \]

    So, the total height of the two windows is \( \frac{55}{16} \) meters.

    ### Answer: \( \frac{55}{16} \) meters

Expected answer: 55/16

Correct? Yes

🔗 Problem

Problem: Jan is mixing some fruit juices for a party. She mixes 16/6 liters of apple juice with 20/12 liters of orange juice. To add some tartness, she also adds 2/3 liters of cranberry juice. How many liters of juice does she have in total?

Gemma 2B:

    To find the total amount of juice Jan has, we need to add the quantities of each type of juice together.

    First, let's simplify the fractions where possible:

    1. Simplify \( \frac{16}{6} \):
       \[
       \frac{16}{6} = \frac{8}{3}
       \]

    2. Simplify \( \frac{20}{12} \):
       \[
       \frac{20}{12} = \frac{5}{3}
       \]

    Now, we add the simplified fractions together along with the cranberry juice:

    \[
    \frac{8}{3} + \frac{5}{3} + \frac{2}{3}
    \]

    Since the denominators are the same, we can add the numerators directly:

    \[
    \frac{8 + 5 + 2}{3} = \frac{15}{3} = 5
    \]

    So, Jan has a total of 5 liters of juice.

    ### Answer: 5 liters

Expected answer: 5

Correct? Yes

🔗 Problem

Problem: In a puzzle game, each level gives you 4/3 points. If you have played 13/10 of these levels, how many points have you earned? Standardize your response as a simplified fraction.

Gemma 2B:

    To determine the total points earned, we need to multiply the points per level by the number of levels played.

    Given:
    - Points per level = \( \frac{4}{3} \)
    - Levels played = \( \frac{13}{10} \)

    We multiply these fractions:

    \[
    \frac{4}{3} \times \frac{13}{10} = \frac{4 \times 13}{3 \times 10} = \frac{52}{30}
    \]

    Next, we simplify the fraction \( \frac{52}{30} \). We find the greatest common divisor (GCD) of 52 and 30, which is 2.

    \[
    \frac{52 \div 2}{30 \div 2} = \frac{26}{15}
    \]

    Thus, the simplified fraction is \( \frac{26}{15} \).

    ### Answer: \(\frac{26}{15}\)

Expected answer: 26/15

Correct? Yes

🔗 Follow-up

Problem: In the same puzzle game, let's say now each level gives you the same amount of points, 4/3, but you've only played the game completely (13/13 of the levels). How many points have you earned now? Write your response as a simplified fraction.

Gemma 2B:

    To determine the total points earned, we need to multiply the points per level by the number of levels played.

    Given:
    - Points per level = \( \frac{4}{3} \)
    - Levels played = \( \frac{13}{13} \)

    Since \( \frac{13}{13} \) is equivalent to 1 (as it represents a whole), we can simplify the calculation:

    \[
    \frac{4}{3} \times 1 = \frac{4}{3}
    \]

    Thus, the total points earned is \( \frac{4}{3} \).

    ### Answer: \(\frac{4}{3}\)

Expected answer: 4/3

Correct? Yes

🔗 Problem

Problem: An astronaut on a mission has a routine where he spends 18/8 hours a day cycling on the spaceship's exercise bike and 26/2 hours a day conducting scientific experiments. The total amount of time spent on these activities in a day makes up what he calls his "work". One day he decides to add another activity to his routine around 30/8 hours of reviewing data from the mission so far. He adds this time to his previous "work" routine. How many hours does the astronaut spend on his whole daily routine now?

Gemma 2B:

    First, let's calculate the time the astronaut spends on each activity:

    1. Time spent cycling:
    \[ \frac{18}{8} \text{ hours} = 2.25 \text{ hours} \]

    2. Time spent conducting scientific experiments:
    \[ \frac{26}{2} \text{ hours} = 13 \text{ hours} \]

    3. Time spent reviewing data:
    \[ \frac{30}{8} \text{ hours} = 3.75 \text{ hours} \]

    Next, we sum these times to find the total time spent on the whole daily routine:

    \[ 2.25 \text{ hours} + 13 \text{ hours} + 3.75 \text{ hours} \]

    Adding these together:

    \[ 2.25 + 13 + 3.75 = 19 \text{ hours} \]

    ### Answer: 19 hours

Expected answer: 19

Correct? Yes

🔗 Follow-up

Problem: The astronaut then decides to reduce the time he spends on scientific experiments from 26/2 hours a day to 11/2 hours a day, but keeps his exercise and data review routines the same. After making these adjustments, how many hours does the astronaut spend on his routine now?

Gemma 2B:

    First, let's recalculate the time spent on each activity with the new adjustment:

    1. Time spent cycling:
    \[ \frac{18}{8} \text{ hours} = 2.25 \text{ hours} \]

    2. Time spent conducting scientific experiments (adjusted):
    \[ \frac{11}{2} \text{ hours} = 5.5 \text{ hours} \]

    3. Time spent reviewing data:
    \[ \frac{30}{8} \text{ hours} = 3.75 \text{ hours} \]

    Next, we sum these times to find the total time spent on the whole daily routine after the adjustment:

    \[ 2.25 \text{ hours} + 5.5 \text{ hours} + 3.75 \text{ hours} \]

    Adding these together:

    \[ 2.25 + 5.5 + 3.75 = 11.5 \text{ hours} \]

    ### Answer: 11.5 hours

Expected answer: 23/2

Correct? No

🔗 Problem

Problem: Jeremy went to a sandwich shop that slices cheese by weight measured in ounces. He ordered slices of cheese that weighed 30/9 ounces in total. However, upon weighing, he found that there was 17/22 ounces less cheese than he requested. Calculate how much cheese he actually received. Use the equation 'd = n - (17 / 22)' to find the answer.

Gemma 2B:

    To determine how much cheese Jeremy actually received, we start with the total weight of cheese he ordered, which is \( \frac{30}{9} \) ounces. We need to subtract the amount of cheese that was missing, which is \( \frac{17}{22} \) ounces.

    First, simplify \( \frac{30}{9} \):
    \[ \frac{30}{9} = \frac{10}{3} \]

    Next, we need to subtract \( \frac{17}{22} \) from \( \frac{10}{3} \). To do this, we need a common denominator. The least common multiple of 3 and 22 is 66.

    Convert \( \frac{10}{3} \) to a fraction with a denominator of 66:
    \[ \frac{10}{3} = \frac{10 \times 22}{3 \times 22} = \frac{220}{66} \]

    Convert \( \frac{17}{22} \) to a fraction with a denominator of 66:
    \[ \frac{17}{22} = \frac{17 \times 3}{22 \times 3} = \frac{51}{66} \]

    Now, subtract the two fractions:
    \[ \frac{220}{66} - \frac{51}{66} = \frac{220 - 51}{66} = \frac{169}{66} \]

    Thus, Jeremy actually received \( \frac{169}{66} \) ounces of cheese.

    ### Answer: \(\frac{169}{66}\) ounces

Expected answer: 169/66

Correct? Yes

🔗 Follow-up

Problem: Following the previous problem, Jeremy decided to make sandwiches for his friends. Each of his friends wanted a sandwich that had (30 / 18) ounces of the cheese Jeremy received from the shop. Calculate the total amount of cheese in ounces that Jeremy will have to use if he uses 'd' ounces of cheese from what he received for each sandwich. Use the formula 'k = (30 / 18) * d' to find your answer.

Gemma 2B:

    To determine the total amount of cheese Jeremy will use for each sandwich, we start with the formula \( k = \frac{30}{18} \times d \), where \( d \) is the amount of cheese Jeremy received, which we previously calculated as \( \frac{169}{66} \) ounces.

    First, simplify \( \frac{30}{18} \):
    \[ \frac{30}{18} = \frac{5}{3} \]

    Now, substitute \( d = \frac{169}{66} \) into the formula:
    \[ k = \frac{5}{3} \times \frac{169}{66} \]

    To multiply these fractions, multiply the numerators and the denominators:
    \[ k = \frac{5 \times 169}{3 \times 66} = \frac{845}{198} \]

    Thus, the total amount of cheese Jeremy will use for each sandwich is \( \frac{845}{198} \) ounces.

    ### Answer: \(\frac{845}{198}\) ounces

Expected answer: 845/198

Correct? Yes

🔗 Follow-up

Problem: Based on the previous problem, suppose now that when Jeremy weighed his cheese, he found that there was actually 20/22 ounces less cheese than he requested, instead of 17/22 ounces. With this change, calculate how much cheese he actually received now. Use the equation 'd = n - (20 / 22)' to find the answer.

Gemma 2B:

    To determine how much cheese Jeremy actually received with the new discrepancy, we start with the total weight of cheese he ordered, which is \( \frac{30}{9} \) ounces. We need to subtract the new amount of cheese that was missing, which is \( \frac{20}{22} \) ounces.

    First, simplify \( \frac{30}{9} \):
    \[ \frac{30}{9} = \frac{10}{3} \]

    Next, we need to subtract \( \frac{20}{22} \) from \( \frac{10}{3} \). To do this, we need a common denominator. The least common multiple of 3 and 22 is 66.

    Convert \( \frac{10}{3} \) to a fraction with a denominator of 66:
    \[ \frac{10}{3} = \frac{10 \times 22}{3 \times 22} = \frac{220}{66} \]

    Convert \( \frac{20}{22} \) to a fraction with a denominator of 66:
    \[ \frac{20}{22} = \frac{20 \times 3}{22 \times 3} = \frac{60}{66} \]

    Now, subtract the two fractions:
    \[ \frac{220}{66} - \frac{60}{66} = \frac{220 - 60}{66} = \frac{160}{66} \]

    Simplify the fraction \( \frac{160}{66} \):
    \[ \frac{160}{66} = \frac{80}{33} \]

    Thus, Jeremy actually received \( \frac{80}{33} \) ounces of cheese.

    ### Answer: \(\frac{80}{33}\) ounces

Expected answer: 80/33

Correct? Yes

🔗 Problem

Problem: Sophia is a researcher who studies sunglasses. She was examining a batch of sunglasses and trying to calculate the average light blocking efficiency. In the first test, she found that 13 out of every 4 sunglasses blocked all UV rays completely, while in the second test, 20 out of every 18 sunglasses also blocked all UV rays. Sophia decided to add these numbers to her total. Next, she added the additional 21 out of 17 sunglasses from another batch that also blocked all UV rays. Later, she realized she made a mistake and needed to subtract the 16 out of 30 sunglasses that actually did not block all UV rays. What is the final corrected number of sunglasses that blocked all UV rays, according to Sophia's tests and calculations? Please give your answer as a simplified fraction.

Gemma 2B:

    To solve this problem, we need to add and subtract the fractions representing the sunglasses that blocked all UV rays and those that did not.

    First, let's convert all the given fractions to a common denominator to make the calculations easier.

    1. The first test result is \( \frac{13}{4} \).
    2. The second test result is \( \frac{20}{18} \), which simplifies to \( \frac{10}{9} \).
    3. The additional batch result is \( \frac{21}{17} \).
    4. The incorrect batch result is \( \frac{16}{30} \), which simplifies to \( \frac{8}{15} \).

    Now, we need to find a common denominator for these fractions. The least common multiple (LCM) of 4, 9, 17, and 15 is 3060.

    Let's convert each fraction to have the denominator 3060:

    1. \( \frac{13}{4} = \frac{13 \times 765}{4 \times 765} = \frac{9945}{3060} \)
    2. \( \frac{10}{9} = \frac{10 \times 340}{9 \times 340} = \frac{3400}{3060} \)
    3. \( \frac{21}{17} = \frac{21 \times 180}{17 \times 180} = \frac{3780}{3060} \)
    4. \( \frac{8}{15} = \frac{8 \times 204}{15 \times 204} = \frac{1632}{3060} \)

    Now, we add the fractions that represent sunglasses blocking all UV rays and subtract the fraction that represents sunglasses not blocking all UV rays:

    \[
    \frac{9945}{3060} + \frac{3400}{3060} + \frac{3780}{3060} - \frac{1632}{3060}
    \]

    Combine the numerators:

    \[
    9945 + 3400 + 3780 - 1632 = 15493
    \]

    So, the combined fraction is:

    \[
    \frac{15493}{3060}
    \]

    This fraction is already in its simplest form because 15493 and 3060 have no common factors other than 1.

    ### Answer: \(\frac{15493}{3060}\)

Expected answer: 15493/3060

Correct? Yes

🔗 Follow-up

Problem: After making her final corrections, Sophia wanted to estimate the impact of the sunglasses' UV blocking efficiency. She decided to conduct an experiment, where she subjected 13 out of 6 pairs of sunglasses from the final corrected number to a special UV light test, and then multiplied the results by the rest. What fraction represents this result? Please give your answer as a simplified fraction.

Gemma 2B:

    To solve this problem, we need to multiply the fraction representing the final corrected number of sunglasses that blocked all UV rays by the fraction representing the pairs of sunglasses subjected to the special UV light test.

    The final corrected number of sunglasses that blocked all UV rays is represented by the fraction \(\frac{15493}{3060}\).

    The fraction of pairs of sunglasses subjected to the special UV light test is \(\frac{13}{6}\).

    We need to multiply these two fractions:

    \[
    \frac{15493}{3060} \times \frac{13}{6}
    \]

    First, multiply the numerators and the denominators:

    \[
    \frac{15493 \times 13}{3060 \times 6} = \frac{201409}{18360}
    \]

    Next, we need to simplify this fraction. To do this, we find the greatest common divisor (GCD) of 201409 and 18360. Using the Euclidean algorithm:

    1. \(201409 \div 18360 \approx 10\) with a remainder of \(201409 - 183600 = 17809\)
    2. \(18360 \div 17809 \approx 1\) with a remainder of \(18360 - 17809 = 551\)
    3. \(17809 \div 551 \approx 32\) with a remainder of \(17809 - 17632 = 177\)
    4. \(551 \div 177 \approx 3\) with a remainder of \(551 - 531 = 20\)
    5. \(177 \div 20 \approx 8\) with a remainder of \(177 - 160 = 17\)
    6. \(20 \div 17 \approx 1\) with a remainder of \(20 - 17 = 3\)
    7. \(17 \div 3 \approx 5\) with a remainder of \(17 - 15 = 2\)
    8. \(3 \div 2 \approx 1\) with a remainder of \(3 - 2 = 1\)
    9. \(2 \div 1 \approx 2\) with a remainder of \(2 - 2 = 0\)

    The GCD is 1, so the fraction \(\frac{201409}{18360}\) is already in its simplest form.

    ### Answer: \(\frac{201409}{18360}\)

Expected answer: 201409/18360

Correct? Yes

🔗 Follow-up

Problem: Sophia revisited her calculations and identified an error. She had originally noted that 13 out of every 4 sunglasses blocked all UV rays. However, upon double-checking, she discovered that actually 13 out of every 28 sunglasses blocked all UV rays. She wants to correct this while keeping the rest of her operations the same. After correcting this mistake, and continuing to add the 20 out of 18 sunglasses from the second test, adding an additional 21 out of 17 sunglasses from another batch, and finally subtracting the 16 out of 30 sunglasses that did not meet the standard, what is the final corrected number of sunglasses that blocked all UV rays, according to Sophia's tests and calculations? Please give your answer as a simplified fraction.

Gemma 2B:

    To correct the calculations, we need to update the fraction for the first test and then proceed with the same operations as before.

    1. The corrected first test result is \( \frac{13}{28} \).
    2. The second test result remains \( \frac{20}{18} \), which simplifies to \( \frac{10}{9} \).
    3. The additional batch result remains \( \frac{21}{17} \).
    4. The incorrect batch result remains \( \frac{16}{30} \), which simplifies to \( \frac{8}{15} \).

    We need to find a common denominator for these fractions. The least common multiple (LCM) of 28, 9, 17, and 15 is 21420.

    Let's convert each fraction to have the denominator 21420:

    1. \( \frac{13}{28} = \frac{13 \times 765}{28 \times 765} = \frac{9945}{21420} \)
    2. \( \frac{10}{9} = \frac{10 \times 2380}{9 \times 2380} = \frac{23800}{21420} \)
    3. \( \frac{21}{17} = \frac{21 \times 1260}{17 \times 1260} = \frac{26460}{21420} \)
    4. \( \frac{8}{15} = \frac{8 \times 1428}{15 \times 1428} = \frac{11424}{21420} \)

    Now, we add the fractions that represent sunglasses blocking all UV rays and subtract the fraction that represents sunglasses not blocking all UV rays:

    \[
    \frac{9945}{21420} + \frac{23800}{21420} + \frac{26460}{21420} - \frac{11424}{21420}
    \]

    Combine the numerators:

    \[
    9945 + 23800 + 26460 - 11424 = 48781
    \]

    So, the combined fraction is:

    \[
    \frac{48781}{21420}
    \]

    This fraction is already in its simplest form because 48781 and 21420 have no common factors other than 1.

    ### Answer: \(\frac{48781}{21420}\)

Expected answer: 48781/21420

Correct? Yes

🔗 Problem

Problem: Shawn is making sandwiches for a picnic. The recipe for the sandwiches requires different amounts of mayonnaise and sandwich spread. To make a sandwich, Shawn uses 23/28 cups of mayonnaise and 6/9 cups of sandwich spread. How many cups of these two ingredients does Shawn use in total to make a sandwich? While preparing the sandwiches, Shawn realizes that he has to add 18/12 cups of cheese spread to his recipe. Calculate the total amount of ingredients Shawn now needs to make one sandwich which includes mayonnaise, sandwich spread, and cheese spread. Finally, if Shawn has to make an equal number of sandwiches using 4/4 cups of all the ingredients combined, how many cups of ingredients will he use in total to make those sandwiches?

Gemma 2B:

    First, let's find the total amount of mayonnaise and sandwich spread Shawn uses for one sandwich.

    1. **Mayonnaise:**
       \[
       \frac{23}{28} \text{ cups}
       \]

    2. **Sandwich spread:**
       \[
       \frac{6}{9} = \frac{2}{3} \text{ cups}
       \]

    To add these fractions, we need a common denominator. The least common multiple of 28 and 3 is 84.

    Convert \(\frac{23}{28}\) to a fraction with a denominator of 84:
    \[
    \frac{23}{28} = \frac{23 \times 3}{28 \times 3} = \frac{69}{84}
    \]

    Convert \(\frac{2}{3}\) to a fraction with a denominator of 84:
    \[
    \frac{2}{3} = \frac{2 \times 28}{3 \times 28} = \frac{56}{84}
    \]

    Now, add the fractions:
    \[
    \frac{69}{84} + \frac{56}{84} = \frac{69 + 56}{84} = \frac{125}{84}
    \]

    So, the total amount of mayonnaise and sandwich spread is:
    \[
    \frac{125}{84} \text{ cups}
    \]

    Next, we add the cheese spread:
    \[
    \frac{18}{12} = \frac{3 \times 6}{2 \times 6} = \frac{3}{2} = \frac{126}{84}
    \]

    Now, add this to the previous total:
    \[
    \frac{125}{84} + \frac{126}{84} = \frac{125 + 126}{84} = \frac{251}{84}
    \]

    So, the total amount of ingredients for one sandwich is:
    \[
    \frac{251}{84} \text{ cups}
    \]

    Finally, if Shawn uses 4/4 cups of all ingredients combined to make an equal number of sandwiches, we need to determine how many sandwiches he can make with \(\frac{4}{4} = 1\) cup of ingredients.

    To find the number of sandwiches:
    \[
    \frac{1}{\frac{251}{84}} = \frac{84}{251}
    \]

    Thus, the total amount of ingredients used to make these sandwiches is:
    \[
    \frac{84}{251} \times 1 = \frac{84}{251} \text{ cups}
    \]

    ### Answer: \(\frac{84}{251}\) cups

Expected answer: 251/84

Correct? No

🔗 Problem

Problem: Cassandra collected keys for a hobby. At first she had a collection equivalent to 7/6 of a key set. After she lost 1/21 of a key set, how many key sets does she have now?

Gemma 2B:

    To determine how many key sets Cassandra has now, we need to follow these steps:

    1. Start with the initial number of key sets Cassandra had, which is \( \frac{7}{6} \) of a key set.
    2. Subtract the number of key sets she lost, which is \( \frac{1}{21} \) of a key set.

    First, we need a common denominator to subtract these fractions. The least common multiple of 6 and 21 is 42.

    Convert \( \frac{7}{6} \) to a fraction with a denominator of 42:
    \[ \frac{7}{6} = \frac{7 \times 7}{6 \times 7} = \frac{49}{42} \]

    Convert \( \frac{1}{21} \) to a fraction with a denominator of 42:
    \[ \frac{1}{21} = \frac{1 \times 2}{21 \times 2} = \frac{2}{42} \]

    Now, subtract \( \frac{2}{42} \) from \( \frac{49}{42} \):
    \[ \frac{49}{42} - \frac{2}{42} = \frac{49 - 2}{42} = \frac{47}{42} \]

    So, Cassandra now has \( \frac{47}{42} \) of a key set.

    ### Answer: \(\frac{47}{42}\)

Expected answer: 47/42

Correct? Yes

🔗 Follow-up

Problem: Let's consider a slight variation. Instead, Cassandra started her key-hobby with a collection that equaled 23/6 of a key set. After losing 1/21 of a key set, just as before, how many key sets remains in her collection now?

Gemma 2B:

    To determine how many key sets Cassandra has now, we need to follow these steps:

    1. Start with the initial number of key sets Cassandra had, which is \( \frac{23}{6} \) of a key set.
    2. Subtract the number of key sets she lost, which is \( \frac{1}{21} \) of a key set.

    First, we need a common denominator to subtract these fractions. The least common multiple of 6 and 21 is 42.

    Convert \( \frac{23}{6} \) to a fraction with a denominator of 42:
    \[ \frac{23}{6} = \frac{23 \times 7}{6 \times 7} = \frac{161}{42} \]

    Convert \( \frac{1}{21} \) to a fraction with a denominator of 42:
    \[ \frac{1}{21} = \frac{1 \times 2}{21 \times 2} = \frac{2}{42} \]

    Now, subtract \( \frac{2}{42} \) from \( \frac{161}{42} \):
    \[ \frac{161}{42} - \frac{2}{42} = \frac{161 - 2}{42} = \frac{159}{42} \]

    So, Cassandra now has \( \frac{159}{42} \) of a key set. This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3:
    \[ \frac{159}{42} = \frac{159 \div 3}{42 \div 3} = \frac{53}{14} \]

    ### Answer: \(\frac{53}{14}\)

Expected answer: 53/14

Correct? Yes

🔗 Problem

Problem: A songwriter is composing a new piece. In the first section, he uses a ratio of 29 notes every 23 measures. In the second section, he uses a ratio of 19 notes every 17 measures. Calculate the difference between the sum of the ratios of the two sections ((29 / 23) + (19 / 17)) and the ratio of a rest, which is equivalent to one whole measure (26/26), used in the third section of his composition.

Gemma 2B:

    To solve this problem, we need to calculate the sum of the ratios of the two sections and then find the difference between this sum and the ratio of a rest used in the third section.

    First, let's calculate the sum of the ratios of the two sections:

    1. The ratio of notes to measures in the first section is \( \frac{29}{23} \).
    2. The ratio of notes to measures in the second section is \( \frac{19}{17} \).

    We add these two ratios together:

    \[
    \frac{29}{23} + \frac{19}{17}
    \]

    To add these fractions, we need a common denominator. The least common multiple (LCM) of 23 and 17 is 391 (since 23 and 17 are both prime numbers, their LCM is their product).

    Convert each fraction to have the common denominator of 391:

    \[
    \frac{29}{23} = \frac{29 \times 17}{23 \times 17} = \frac{493}{391}
    \]

    \[
    \frac{19}{17} = \frac{19 \times 23}{17 \times 23} = \frac{437}{391}
    \]

    Now, add the fractions:

    \[
    \frac{493}{391} + \frac{437}{391} = \frac{493 + 437}{391} = \frac{930}{391}
    \]

    Next, we need to find the difference between this sum and the ratio of a rest, which is \( \frac{26}{26} \) (equivalent to 1):

    \[
    \frac{930}{391} - \frac{26}{26}
    \]

    Since \( \frac{26}{26} = 1 \), we can rewrite the expression as:

    \[
    \frac{930}{391} - 1
    \]

    Convert 1 to a fraction with the same denominator:

    \[
    1 = \frac{391}{391}
    \]

    Now, subtract the fractions:

    \[
    \frac{930}{391} - \frac{391}{391} = \frac{930 - 391}{391} = \frac{539}{391}
    \]

    Thus, the difference between the sum of the ratios of the two sections and the ratio of a rest used in the third section is:

    ### Answer: \(\frac{539}{391}\)

Expected answer: 539/391

Correct? Yes

🔗 Follow-up

Problem: In the same song composition, the songwriter decided to revise the second section. Instead of having a ratio of 19 notes every 17 measures, he decided to keep it consistent with the first section and have a ratio of 19 notes every 23 measures just like in the first section. Calculate the difference between the sum of the ratios of the two sections ((29 / 23) + (19 / 23)) and the ratio of a rest, which is equivalent to one whole measure (26/26), used in the third section of his composition after this change.

Gemma 2B:

    To solve this problem, we need to calculate the sum of the revised ratios of the two sections and then find the difference between this sum and the ratio of a rest used in the third section.

    First, let's calculate the sum of the revised ratios of the two sections:

    1. The ratio of notes to measures in the first section is \( \frac{29}{23} \).
    2. The revised ratio of notes to measures in the second section is \( \frac{19}{23} \).

    We add these two ratios together:

    \[
    \frac{29}{23} + \frac{19}{23}
    \]

    Since the denominators are the same, we can directly add the numerators:

    \[
    \frac{29 + 19}{23} = \frac{48}{23}
    \]

    Next, we need to find the difference between this sum and the ratio of a rest, which is \( \frac{26}{26} \) (equivalent to 1):

    \[
    \frac{48}{23} - \frac{26}{26}
    \]

    Since \( \frac{26}{26} = 1 \), we can rewrite the expression as:

    \[
    \frac{48}{23} - 1
    \]

    Convert 1 to a fraction with the same denominator:

    \[
    1 = \frac{23}{23}
    \]

    Now, subtract the fractions:

    \[
    \frac{48}{23} - \frac{23}{23} = \frac{48 - 23}{23} = \frac{25}{23}
    \]

    Thus, the difference between the sum of the revised ratios of the two sections and the ratio of a rest used in the third section is:

    ### Answer: \(\frac{25}{23}\)

Expected answer: 25/23

Correct? Yes

🔗 Problem

Problem: You're trying to solve a puzzle that requires you to multiply two fractions together. The fractions are 22/25 and 22/8. Calculate the product to get the answer you need to advance in the puzzle.

Gemma 2B:

    To find the product of the fractions \( \frac{22}{25} \) and \( \frac{22}{8} \), we multiply the numerators together and the denominators together.

    First, multiply the numerators:
    \[ 22 \times 22 = 484 \]

    Next, multiply the denominators:
    \[ 25 \times 8 = 200 \]

    So, the product of the fractions is:
    \[ \frac{484}{200} \]

    To simplify this fraction, we find the greatest common divisor (GCD) of 484 and 200. The GCD of 484 and 200 is 4.

    Now, divide both the numerator and the denominator by their GCD:
    \[ \frac{484 \div 4}{200 \div 4} = \frac{121}{50} \]

    Thus, the simplified product of the fractions is:
    \[ \frac{121}{50} \]

    ### Answer: \(\frac{121}{50}\)

Expected answer: 121/50

Correct? Yes

🔗 Follow-up

Problem: You solved the first part of the puzzle with the result from multiplying the fractions 22/25 and 22/8. Now, you've come across a key with the number 23/13 etched on it. To open the next door, you need to add the result from the step you just completed to the number on the key. What is the sum of these two values?

Gemma 2B:

    To find the sum of \(\frac{121}{50}\) and \(\frac{23}{13}\), we need to add these two fractions. First, we need a common denominator.

    The denominators are 50 and 13. The least common multiple (LCM) of 50 and 13 is 650.

    Next, we convert each fraction to have the common denominator of 650.

    For \(\frac{121}{50}\):
    \[ \frac{121}{50} = \frac{121 \times 13}{50 \times 13} = \frac{1573}{650} \]

    For \(\frac{23}{13}\):
    \[ \frac{23}{13} = \frac{23 \times 50}{13 \times 50} = \frac{1150}{650} \]

    Now, we can add the two fractions:
    \[ \frac{1573}{650} + \frac{1150}{650} = \frac{1573 + 1150}{650} = \frac{2723}{650} \]

    Thus, the sum of \(\frac{121}{50}\) and \(\frac{23}{13}\) is:
    \[ \frac{2723}{650} \]

    ### Answer: \(\frac{2723}{650}\)

Expected answer: 2723/650

Correct? Yes

🔗 Follow-up

Problem: Previously, you multiplied fractions 22/25 and 22/8 in the first part of the puzzle. Suppose instead of the 22/25 that you initially saw, the actual fraction in the text was 13/25. Now, you wonder how this change in information might affect your product. Recalculate the result using the correct fractions, which are now 13/25 and 22/8. What is the new product of these fractions?

Gemma 2B:

    To find the product of the fractions \( \frac{13}{25} \) and \( \frac{22}{8} \), we multiply the numerators together and the denominators together.

    First, multiply the numerators:
    \[ 13 \times 22 = 286 \]

    Next, multiply the denominators:
    \[ 25 \times 8 = 200 \]

    So, the product of the fractions is:
    \[ \frac{286}{200} \]

    To simplify this fraction, we find the greatest common divisor (GCD) of 286 and 200. The GCD of 286 and 200 is 2.

    Now, divide both the numerator and the denominator by their GCD:
    \[ \frac{286 \div 2}{200 \div 2} = \frac{143}{100} \]

    Thus, the simplified product of the fractions is:
    \[ \frac{143}{100} \]

    ### Answer: \(\frac{143}{100}\)

Expected answer: 143/100

Correct? Yes

🔗 Problem

Problem: James is making a decorative pillow and would like to add a matching border. He needs to calculate the length of border material he will need. The pillow is 23/9 feet long. He had an additional 22/25 feet of length due to the fluffing of the pillow. The total length is then multiplied by 16/13 to account for overlap and corners. Calculate the length of border material James will need for his pillow.

Gemma 2B:

    To determine the length of border material James will need, we need to follow these steps:

    1. **Add the lengths of the pillow and the additional fluffing:**
       \[
       \text{Total length} = \frac{23}{9} + \frac{22}{25}
       \]

    2. **Find a common denominator to add the fractions:**
       The least common multiple of 9 and 25 is 225. Convert each fraction to have this common denominator:
       \[
       \frac{23}{9} = \frac{23 \times 25}{9 \times 25} = \frac{575}{225}
       \]
       \[
       \frac{22}{25} = \frac{22 \times 9}{25 \times 9} = \frac{198}{225}
       \]

    3. **Add the fractions:**
       \[
       \frac{575}{225} + \frac{198}{225} = \frac{575 + 198}{225} = \frac{773}{225}
       \]

    4. **Multiply the total length by the factor to account for overlap and corners:**
       \[
       \text{Length of border material} = \frac{773}{225} \times \frac{16}{13}
       \]

    5. **Multiply the fractions:**
       \[
       \frac{773 \times 16}{225 \times 13} = \frac{12368}{2925}
       \]

    6. **Simplify the fraction if possible:**
       The greatest common divisor (GCD) of 12368 and 2925 is 1, so the fraction is already in its simplest form.

    Thus, the length of border material James will need is:
    \[
    \frac{12368}{2925} \text{ feet}
    \]

    ### Answer: \(\frac{12368}{2925}\) feet

Expected answer: 12368/2925

Correct? Yes

🔗 Follow-up

Problem: James realizes he made a mistake when measuring the length of his pillow. The pillow is actually 12/9 feet long, not 23/9 feet as previously calculated. Using this corrected length, and still considering the added 22/25 feet for fluffing plus the factor of 16/13 for overlap and corners, how much border material will he need for his pillow now?

Gemma 2B:

    To determine the correct length of border material James will need with the corrected pillow length, we follow these steps:

    1. **Add the corrected lengths of the pillow and the additional fluffing:**
       \[
       \text{Total length} = \frac{12}{9} + \frac{22}{25}
       \]

    2. **Simplify the fraction \(\frac{12}{9}\):**
       \[
       \frac{12}{9} = \frac{4}{3}
       \]

    3. **Find a common denominator to add the fractions:**
       The least common multiple of 3 and 25 is 75. Convert each fraction to have this common denominator:
       \[
       \frac{4}{3} = \frac{4 \times 25}{3 \times 25} = \frac{100}{75}
       \]
       \[
       \frac{22}{25} = \frac{22 \times 3}{25 \times 3} = \frac{66}{75}
       \]

    4. **Add the fractions:**
       \[
       \frac{100}{75} + \frac{66}{75} = \frac{100 + 66}{75} = \frac{166}{75}
       \]

    5. **Multiply the total length by the factor to account for overlap and corners:**
       \[
       \text{Length of border material} = \frac{166}{75} \times \frac{16}{13}
       \]

    6. **Multiply the fractions:**
       \[
       \frac{166 \times 16}{75 \times 13} = \frac{2656}{975}
       \]

    7. **Simplify the fraction if possible:**
       The greatest common divisor (GCD) of 2656 and 975 is 1, so the fraction is already in its simplest form.

    Thus, the length of border material James will need is:
    \[
    \frac{2656}{975} \text{ feet}
    \]

    ### Answer: \(\frac{2656}{975}\) feet

Expected answer: 2656/975

Correct? Yes

🔗 Problem

Problem: A car travels 21/4 miles every hour. One day, the car was needed to travel an extra 16/13 miles, but due to a detour, it only traveled an additional 1/21 miles. How many more miles was the car supposed to travel in total that day, given that it ran continuously at the same speed? Calculate by multiplying the additional distance needed by the average miles the car travels in an hour.

Gemma 2B:

    First, let's determine the additional distance the car was supposed to travel. The car was supposed to travel an extra \( \frac{16}{13} \) miles, but it only traveled an additional \( \frac{1}{21} \) miles.

    To find the difference, we need to subtract \( \frac{1}{21} \) from \( \frac{16}{13} \):

    \[
    \frac{16}{13} - \frac{1}{21}
    \]

    To subtract these fractions, we need a common denominator. The least common multiple of 13 and 21 is 273. We convert each fraction:

    \[
    \frac{16}{13} = \frac{16 \times 21}{13 \times 21} = \frac{336}{273}
    \]

    \[
    \frac{1}{21} = \frac{1 \times 13}{21 \times 13} = \frac{13}{273}
    \]

    Now, subtract the fractions:

    \[
    \frac{336}{273} - \frac{13}{273} = \frac{336 - 13}{273} = \frac{323}{273}
    \]

    Next, we need to calculate how many more miles the car was supposed to travel in total that day. The car travels \( \frac{21}{4} \) miles every hour. We multiply this speed by the additional distance:

    \[
    \frac{21}{4} \times \frac{323}{273}
    \]

    First, simplify the multiplication:

    \[
    \frac{21 \times 323}{4 \times 273}
    \]

    Calculate the numerator and the denominator:

    \[
    21 \times 323 = 6783
    \]

    \[
    4 \times 273 = 1092
    \]

    So, the fraction becomes:

    \[
    \frac{6783}{1092}
    \]

    Simplify the fraction by finding the greatest common divisor (GCD) of 6783 and 1092. The GCD is 21:

    \[
    \frac{6783 \div 21}{1092 \div 21} = \frac{323}{52}
    \]

    Thus, the car was supposed to travel an additional \( \frac{323}{52} \) miles.

    ### Answer: \(\frac{323}{52}\) miles.

Expected answer: 323/52

Correct? Yes

🔗 Follow-up

Problem: Continuing from the original scenario, instead of an extra 16/13 miles, suppose the car was actually supposed to travel an additional 16/6 miles, but still it only traveled an additional 1/21 miles due to the detour. How many more miles was the car supposed to travel in total that day, given that it ran continuously at the same speed? Calculate this by multiplying the new additional distance needed by the average miles the car travels in an hour.

Gemma 2B:

    First, let's determine the new additional distance the car was supposed to travel. The car was supposed to travel an extra \( \frac{16}{6} \) miles, but it only traveled an additional \( \frac{1}{21} \) miles.

    To find the difference, we need to subtract \( \frac{1}{21} \) from \( \frac{16}{6} \):

    \[
    \frac{16}{6} - \frac{1}{21}
    \]

    First, simplify \( \frac{16}{6} \):

    \[
    \frac{16}{6} = \frac{8}{3}
    \]

    Now, we need a common denominator to subtract these fractions. The least common multiple of 3 and 21 is 21. We convert each fraction:

    \[
    \frac{8}{3} = \frac{8 \times 7}{3 \times 7} = \frac{56}{21}
    \]

    \[
    \frac{1}{21} = \frac{1}{21}
    \]

    Now, subtract the fractions:

    \[
    \frac{56}{21} - \frac{1}{21} = \frac{56 - 1}{21} = \frac{55}{21}
    \]

    Next, we need to calculate how many more miles the car was supposed to travel in total that day. The car travels \( \frac{21}{4} \) miles every hour. We multiply this speed by the additional distance:

    \[
    \frac{21}{4} \times \frac{55}{21}
    \]

    First, simplify the multiplication:

    \[
    \frac{21 \times 55}{4 \times 21}
    \]

    The 21s cancel out:

    \[
    \frac{55}{4}
    \]

    Thus, the car was supposed to travel an additional \( \frac{55}{4} \) miles.

    ### Answer: \(\frac{55}{4}\) miles.

Expected answer: 55/4

Correct? Yes

🔗 Problem

Problem: In a book publishing company, the manager has to look after various departments. The editing team takes 20/5 hours to edit a manuscript of a book, while the typesetting team needs 18/17 hours to adjust the layout of a page. Some break delay happens that results in a collective loss of 6/15 hours. After these delays and losses, the manager, to compensate, extended the working hours by the same amount of time as the initial time span of the process. On a particular day, they continue to work. What's the total extended time in hours, if you multiply the time required for the entire process and the extended time after the reduction of the lost hours with the initial total time required?

Gemma 2B:

    To solve this problem, we need to follow these steps:

    1. Calculate the time taken by the editing team to edit a manuscript.
    2. Calculate the time taken by the typesetting team to adjust the layout of a page.
    3. Sum these times to get the initial total time required for the process.
    4. Subtract the collective loss due to breaks.
    5. Determine the extended time, which is equal to the initial total time required.
    6. Multiply the initial total time required by the extended time after the reduction of the lost hours.

    Let's go through each step in detail:

    1. **Time taken by the editing team:**
       \[
       \text{Time taken by the editing team} = \frac{20}{5} \text{ hours} = 4 \text{ hours}
       \]

    2. **Time taken by the typesetting team:**
       \[
       \text{Time taken by the typesetting team} = \frac{18}{17} \text{ hours}
       \]

    3. **Initial total time required:**
       \[
       \text{Initial total time required} = 4 \text{ hours} + \frac{18}{17} \text{ hours}
       \]
       To add these times, we need a common denominator:
       \[
       4 = \frac{68}{17}
       \]
       So,
       \[
       \text{Initial total time required} = \frac{68}{17} + \frac{18}{17} = \frac{86}{17} \text{ hours}
       \]

    4. **Subtract the collective loss due to breaks:**
       \[
       \text{Collective loss} = \frac{6}{15} \text{ hours} = \frac{2}{5} \text{ hours}
       \]
       Subtracting this from the initial total time:
       \[
       \text{Adjusted time} = \frac{86}{17} - \frac{2}{5}
       \]
       To subtract these, we need a common denominator:
       \[
       \frac{86}{17} = \frac{86 \times 5}{17 \times 5} = \frac{430}{85}
       \]
       \[
       \frac{2}{5} = \frac{2 \times 17}{5 \times 17} = \frac{34}{85}
       \]
       So,
       \[
       \text{Adjusted time} = \frac{430}{85} - \frac{34}{85} = \frac{396}{85} \text{ hours}
       \]

    5. **Extended time (equal to the initial total time required):**
       \[
       \text{Extended time} = \frac{86}{17} \text{ hours}
       \]

    6. **Multiply the initial total time required by the extended time after the reduction of the lost hours:**
       \[
       \text{Total extended time} = \left(\frac{86}{17}\right) \times \left(\frac{396}{85}\right)
       \]
       Simplify the multiplication:
       \[
       \frac{86}{17} = 5.058823529
       \]
       \[
       \frac{396}{85} = 4.658823529
       \]
       \[
       5.058823529 \times 4.658823529 = 23.56
       \]

    ### Answer: 23.56

Expected answer: 34056/1445

Correct? No

🔗 Problem

Problem: Jenny plants 13/11 rows of potatoes in her garden. Each row yields approximately 21/3 bushels. If Jenny sells her yield at the farmer's market where each bushel earns her 21/15 dollars, how many dollars will she make in total by selling all her potatoes?

Gemma 2B:

    To determine how much Jenny will make in total by selling all her potatoes, we need to follow these steps:

    1. Calculate the total number of bushels Jenny yields.
    2. Calculate the total earnings from selling the bushels.

    First, let's find the total number of bushels Jenny yields:

    Jenny plants \( \frac{13}{11} \) rows, and each row yields \( \frac{21}{3} \) bushels.

    \[
    \text{Total bushels} = \frac{13}{11} \times \frac{21}{3}
    \]

    Simplify \( \frac{21}{3} \):

    \[
    \frac{21}{3} = 7
    \]

    Now, multiply:

    \[
    \frac{13}{11} \times 7 = \frac{13 \times 7}{11} = \frac{91}{11} = 8.2727 \text{ (approximately)}
    \]

    Next, calculate the total earnings from selling the bushels:

    Each bushel earns \( \frac{21}{15} \) dollars.

    \[
    \text{Total earnings} = \frac{91}{11} \times \frac{21}{15}
    \]

    Simplify \( \frac{21}{15} \):

    \[
    \frac{21}{15} = \frac{7}{5}
    \]

    Now, multiply:

    \[
    \frac{91}{11} \times \frac{7}{5} = \frac{91 \times 7}{11 \times 5} = \frac{637}{55}
    \]

    Simplify \( \frac{637}{55} \):

    \[
    \frac{637}{55} \approx 11.5818 \text{ (approximately)}
    \]

    Thus, Jenny will make approximately 11.5818 dollars by selling all her potatoes.

    ### Answer: 11.5818 dollars

Expected answer: 637/55

Correct? No

🔗 Problem

Problem: A jeweler uses 14/28 of a kilogram of gold to craft a certain type of rings. Additionally, he uses 13/26 of a kilogram of silver to design another type of rings. What is the sum of the amount of gold and silver used by the jeweler in kilograms?

Gemma 2B:

    First, let's simplify the fractions to make the calculations easier.

    For the gold:
    \[ \frac{14}{28} = \frac{14 \div 14}{28 \div 14} = \frac{1}{2} \]

    For the silver:
    \[ \frac{13}{26} = \frac{13 \div 13}{26 \div 13} = \frac{1}{2} \]

    Now, we add the simplified fractions:
    \[ \frac{1}{2} + \frac{1}{2} = 1 \]

    So, the sum of the amount of gold and silver used by the jeweler is 1 kilogram.

    ### Answer: 1 kilogram

Expected answer: 1

Correct? Yes

🔗 Follow-up

Problem: The jeweler decided to use a more expensive gold making process for his rings, which requires a slightly larger amount of gold, specifically 21/28 of a kilogram instead of the original 14/28. Taking this into account, now how much total weight of gold and silver is he using to craft his rings?

Gemma 2B:

    First, let's simplify the new fraction for the gold:

    \[ \frac{21}{28} = \frac{21 \div 7}{28 \div 7} = \frac{3}{4} \]

    We already know the amount of silver used is:

    \[ \frac{13}{26} = \frac{1}{2} \]

    Now, we need to add these two fractions. To do this, we need a common denominator. The least common denominator of 4 and 2 is 4.

    Convert \(\frac{1}{2}\) to a fraction with a denominator of 4:

    \[ \frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4} \]

    Now, add the two fractions:

    \[ \frac{3}{4} + \frac{2}{4} = \frac{3 + 2}{4} = \frac{5}{4} \]

    So, the total weight of gold and silver used by the jeweler is \(\frac{5}{4}\) kilograms, which is equivalent to 1.25 kilograms.

    ### Answer: 1.25 kilograms

Expected answer: 5/4

Correct? No

🔗 Problem

Problem: In a beehive, there are sections for honey storage and bee rearing. Each section is classified into many sub-sections. One day, the beekeeper discovered that each honey storage sub-section had (25 / 9) gallons of honey. He also found the same amount of honey in each bee rearing sub-section. If he combined the honey from both a honey storage sub-section and a bee rearing sub-section, the beekeeper would have a total quantity represented by 'p'. However, one honey storage sub-section had an additional (2 / 26) gallons of honey. So, if the beekeeper was calculating the total amount from this sub-section plus a bee rearing sub-section, the total amount of honey would be defined as 's'. What quantity of honey ('s') would the beekeeper have in total from this scenario?

Gemma 2B:

    To solve this problem, we need to determine the total amount of honey in the combined sub-sections under the given conditions.

    First, let's denote the amount of honey in each honey storage sub-section and each bee rearing sub-section as \( h \).

    Given:
    \[ h = \frac{25}{9} \text{ gallons} \]

    When the beekeeper combines the honey from one honey storage sub-section and one bee rearing sub-section, the total amount of honey \( p \) is:
    \[ p = h + h = \frac{25}{9} + \frac{25}{9} = \frac{50}{9} \text{ gallons} \]

    Now, consider the scenario where one honey storage sub-section has an additional \( \frac{2}{26} \) gallons of honey. The total amount of honey in this honey storage sub-section is:
    \[ h + \frac{2}{26} = \frac{25}{9} + \frac{2}{26} \]

    To add these fractions, we need a common denominator. The least common multiple of 9 and 26 is 234. We convert each fraction to have this common denominator:
    \[ \frac{25}{9} = \frac{25 \times 26}{9 \times 26} = \frac{650}{234} \]
    \[ \frac{2}{26} = \frac{2 \times 9}{26 \times 9} = \frac{18}{234} \]

    Now, add these fractions:
    \[ \frac{650}{234} + \frac{18}{234} = \frac{650 + 18}{234} = \frac{668}{234} \]

    Simplify the fraction:
    \[ \frac{668}{234} = \frac{334}{117} \]

    Thus, the total amount of honey \( s \) in this scenario, combining the honey from this honey storage sub-section and a bee rearing sub-section, is:
    \[ s = \frac{334}{117} + \frac{25}{9} \]

    Convert \( \frac{25}{9} \) to have the same denominator:
    \[ \frac{25}{9} = \frac{25 \times 13}{9 \times 13} = \frac{325}{117} \]

    Add the fractions:
    \[ s = \frac{334}{117} + \frac{325}{117} = \frac{334 + 325}{117} = \frac{659}{117} \]

    Simplify the fraction:
    \[ \frac{659}{117} = \frac{659 \div 1}{117 \div 1} = \frac{659}{117} \]

    Thus, the quantity of honey \( s \) the beekeeper would have in total from this scenario is:
    \[ s = \frac{659}{117} \]

    ### Answer: \(\frac{659}{117}\) gallons

Expected answer: 659/117

Correct? Yes

🔗 Follow-up

Problem: After reflecting on the honey production in the beehive, the beekeeper discovered new data. Instead of each sub-section having (25 / 9) gallons of honey, they each actually had (25 / 4) gallons of honey. This new quantity is represented by 'r'. When he combines the honey from a honey storage sub-section and a bee rearing sub-section, the total amount of honey is now represented by 'p'. However, the particular honey storage sub-section that had an additional (2 / 26) gallons of honey has changed. This new quantity is represented by 's'. Using the updated quantities, how much honey ('s') does the beekeeper now have in total from this scenario?

Gemma 2B:

    To solve this problem with the updated quantities, we need to determine the total amount of honey in the combined sub-sections under the new conditions.

    First, let's denote the amount of honey in each honey storage sub-section and each bee rearing sub-section as \( r \).

    Given:
    \[ r = \frac{25}{4} \text{ gallons} \]

    When the beekeeper combines the honey from one honey storage sub-section and one bee rearing sub-section, the total amount of honey \( p \) is:
    \[ p = r + r = \frac{25}{4} + \frac{25}{4} = \frac{50}{4} = 12.5 \text{ gallons} \]

    Now, consider the scenario where one honey storage sub-section has an additional \( \frac{2}{26} \) gallons of honey. The total amount of honey in this honey storage sub-section is:
    \[ r + \frac{2}{26} = \frac{25}{4} + \frac{2}{26} \]

    To add these fractions, we need a common denominator. The least common multiple of 4 and 26 is 52. We convert each fraction to have this common denominator:
    \[ \frac{25}{4} = \frac{25 \times 13}{4 \times 13} = \frac{325}{52} \]
    \[ \frac{2}{26} = \frac{2 \times 2}{26 \times 2} = \frac{4}{52} \]

    Now, add these fractions:
    \[ \frac{325}{52} + \frac{4}{52} = \frac{325 + 4}{52} = \frac{329}{52} \]

    Thus, the total amount of honey \( s \) in this scenario, combining the honey from this honey storage sub-section and a bee rearing sub-section, is:
    \[ s = \frac{329}{52} + \frac{25}{4} \]

    Convert \( \frac{25}{4} \) to have the same denominator:
    \[ \frac{25}{4} = \frac{25 \times 13}{4 \times 13} = \frac{325}{52} \]

    Add the fractions:
    \[ s = \frac{329}{52} + \frac{325}{52} = \frac{329 + 325}{52} = \frac{654}{52} \]

    Simplify the fraction:
    \[ \frac{654}{52} = \frac{654 \div 2}{52 \div 2} = \frac{327}{26} \]

    Thus, the quantity of honey \( s \) the beekeeper would have in total from this scenario is:
    \[ s = \frac{327}{26} \]

    ### Answer: \(\frac{327}{26}\) gallons

Expected answer: 327/26

Correct? Yes

🔗 Problem

Problem: In prehistoric times, a herbivorous dinosaur eats 16/8 bunches of leaves per hour during the day. But at night, it only eats 6/17 bunches of leaves every hour because of reduced visibility. Calculate the total bunches of leaves the dinosaur eats in an hour if it is eating day and night.

Gemma 2B:

    First, let's simplify the fraction for the bunches of leaves the dinosaur eats during the day:
    \[ \frac{16}{8} = 2 \]

    Next, we know the dinosaur eats \( \frac{6}{17} \) bunches of leaves every hour at night.

    To find the total bunches of leaves the dinosaur eats in an hour if it is eating both day and night, we add the two amounts together:
    \[ 2 + \frac{6}{17} \]

    To add these, we need a common denominator. The common denominator for 2 (which can be written as \( \frac{34}{17} \)) and \( \frac{6}{17} \) is 17:
    \[ 2 = \frac{34}{17} \]

    Now, add the fractions:
    \[ \frac{34}{17} + \frac{6}{17} = \frac{34 + 6}{17} = \frac{40}{17} \]

    ### Answer: \(\frac{40}{17}\)

Expected answer: 40/17

Correct? Yes

🔗 Follow-up

Problem: If this dinosaur finds a field where, due to unique plant growth conditions, it can eat an additional 27/13 bunches of leaves per hour, regardless of whether it's day or night, how many bunches of leaves would the dinosaur eat in total in an hour? Include the quantities it was previously eating during the day and night.

Gemma 2B:

    To determine the total bunches of leaves the dinosaur eats in an hour, we need to add the additional bunches of leaves it can eat per hour to the amount it was already eating.

    From the previous calculation, the dinosaur eats:
    \[ \frac{40}{17} \text{ bunches of leaves per hour} \]

    Now, it can eat an additional:
    \[ \frac{27}{13} \text{ bunches of leaves per hour} \]

    To add these fractions, we need a common denominator. The least common multiple of 17 and 13 is 221.

    Convert each fraction to have the common denominator of 221:
    \[ \frac{40}{17} = \frac{40 \times 13}{17 \times 13} = \frac{520}{221} \]
    \[ \frac{27}{13} = \frac{27 \times 17}{13 \times 17} = \frac{459}{221} \]

    Now, add the fractions:
    \[ \frac{520}{221} + \frac{459}{221} = \frac{520 + 459}{221} = \frac{979}{221} \]

    ### Answer: \(\frac{979}{221}\)

Expected answer: 979/221

Correct? Yes

🔗 Problem

Problem: A bear ate 28/6 pounds of fish one day and 21/18 pounds the next day. On the third day, the bear ate 24/22 pounds of fish. How many total pounds of fish did the bear eat in these three days, written as a fraction?

Gemma 2B:

    To find the total pounds of fish the bear ate over the three days, we need to add the fractions:

    1. First day: \( \frac{28}{6} \)
    2. Second day: \( \frac{21}{18} \)
    3. Third day: \( \frac{24}{22} \)

    Let's simplify each fraction first:

    1. Simplify \( \frac{28}{6} \):
       \[
       \frac{28}{6} = \frac{14}{3}
       \]

    2. Simplify \( \frac{21}{18} \):
       \[
       \frac{21}{18} = \frac{7}{6}
       \]

    3. Simplify \( \frac{24}{22} \):
       \[
       \frac{24}{22} = \frac{12}{11}
       \]

    Now, we need to add these fractions together. To do this, we need a common denominator. The least common multiple (LCM) of 3, 6, and 11 is 66.

    Convert each fraction to have a denominator of 66:

    1. \( \frac{14}{3} \):
       \[
       \frac{14}{3} = \frac{14 \times 22}{3 \times 22} = \frac{308}{66}
       \]

    2. \( \frac{7}{6} \):
       \[
       \frac{7}{6} = \frac{7 \times 11}{6 \times 11} = \frac{77}{66}
       \]

    3. \( \frac{12}{11} \):
       \[
       \frac{12}{11} = \frac{12 \times 6}{11 \times 6} = \frac{72}{66}
       \]

    Now, add the fractions:
    \[
    \frac{308}{66} + \frac{77}{66} + \frac{72}{66} = \frac{308 + 77 + 72}{66} = \frac{457}{66}
    \]

    Thus, the total pounds of fish the bear ate over the three days is:
    \[
    \frac{457}{66}
    \]

    ### Answer: \(\frac{457}{66}\)

Expected answer: 457/66

Correct? Yes

🔗 Follow-up

Problem: In the previously mentioned problem, suppose on the second day, the bear ate only 5/18 pounds of fish rather than 21/18 pounds. With this change, how many total pounds of fish did the bear eat in those three days? Write your answer as a simplified fraction.

Gemma 2B:

    To find the total pounds of fish the bear ate over the three days with the updated amount for the second day, we need to add the fractions:

    1. First day: \( \frac{28}{6} \)
    2. Second day: \( \frac{5}{18} \)
    3. Third day: \( \frac{24}{22} \)

    Let's simplify each fraction first:

    1. Simplify \( \frac{28}{6} \):
       \[
       \frac{28}{6} = \frac{14}{3}
       \]

    2. \( \frac{5}{18} \) is already in its simplest form.

    3. Simplify \( \frac{24}{22} \):
       \[
       \frac{24}{22} = \frac{12}{11}
       \]

    Now, we need to add these fractions together. To do this, we need a common denominator. The least common multiple (LCM) of 3, 18, and 11 is 198.

    Convert each fraction to have a denominator of 198:

    1. \( \frac{14}{3} \):
       \[
       \frac{14}{3} = \frac{14 \times 66}{3 \times 66} = \frac{924}{198}
       \]

    2. \( \frac{5}{18} \):
       \[
       \frac{5}{18} = \frac{5 \times 11}{18 \times 11} = \frac{55}{198}
       \]

    3. \( \frac{12}{11} \):
       \[
       \frac{12}{11} = \frac{12 \times 18}{11 \times 18} = \frac{216}{198}
       \]

    Now, add the fractions:
    \[
    \frac{924}{198} + \frac{55}{198} + \frac{216}{198} = \frac{924 + 55 + 216}{198} = \frac{1195}{198}
    \]

    Thus, the total pounds of fish the bear ate over the three days is:
    \[
    \frac{1195}{198}
    \]

    ### Answer: \(\frac{1195}{198}\)

Expected answer: 1195/198

Correct? Yes

🔗 Problem

Problem: Heather was preparing a special three-course dinner for her family. For the first course, she had a bread recipe that required 11/17 of a cup of sugar. In addition, she baked a custard for dessert that required another 6/2 cups of sugar. As her family loves sweet dishes, she decided to increase the total quantity of sugar by a factor of 29/21. How many cups of sugar did Heather use in total?

Gemma 2B:

    First, let's determine the total amount of sugar Heather used in her recipes before increasing the quantity.

    For the bread recipe, she used:
    \[ \frac{11}{17} \text{ cups of sugar} \]

    For the custard, she used:
    \[ \frac{6}{2} = 3 \text{ cups of sugar} \]

    Adding these amounts together:
    \[ \frac{11}{17} + 3 \]

    To add these fractions, we need a common denominator. The common denominator for 17 and 1 (since 3 can be written as \(\frac{3}{1}\)) is 17.

    Convert 3 to a fraction with a denominator of 17:
    \[ 3 = \frac{3 \times 17}{1 \times 17} = \frac{51}{17} \]

    Now add the fractions:
    \[ \frac{11}{17} + \frac{51}{17} = \frac{11 + 51}{17} = \frac{62}{17} \]

    Next, Heather decided to increase the total quantity of sugar by a factor of \(\frac{29}{21}\). So, we multiply the total amount of sugar by this factor:
    \[ \frac{62}{17} \times \frac{29}{21} \]

    Multiply the numerators and the denominators:
    \[ \frac{62 \times 29}{17 \times 21} = \frac{1798}{357} \]

    To simplify \(\frac{1798}{357}\), we find the greatest common divisor (GCD) of 1798 and 357. The GCD is 1, so the fraction is already in its simplest form.

    Thus, the total amount of sugar Heather used is:
    \[ \frac{1798}{357} \approx 5.04 \text{ cups} \]

    ### Answer: \(\frac{1798}{357}\) cups

Expected answer: 1798/357

Correct? Yes

🔗 Follow-up

Problem: Suppose while preparing the dinner, Heather discovered that instead of 11/17 cups of sugar, the bread recipe only required 11/25 cups of sugar. With this adjustment, and keeping the increased factor of 29/21 the same, how many cups of sugar did Heather use in total for her special three-course dinner?

Gemma 2B:

    First, let's determine the total amount of sugar Heather used in her recipes with the adjusted amount for the bread.

    For the bread recipe, she used:
    \[ \frac{11}{25} \text{ cups of sugar} \]

    For the custard, she used:
    \[ \frac{6}{2} = 3 \text{ cups of sugar} \]

    Adding these amounts together:
    \[ \frac{11}{25} + 3 \]

    To add these fractions, we need a common denominator. The common denominator for 25 and 1 (since 3 can be written as \(\frac{3}{1}\)) is 25.

    Convert 3 to a fraction with a denominator of 25:
    \[ 3 = \frac{3 \times 25}{1 \times 25} = \frac{75}{25} \]

    Now add the fractions:
    \[ \frac{11}{25} + \frac{75}{25} = \frac{11 + 75}{25} = \frac{86}{25} \]

    Next, Heather decided to increase the total quantity of sugar by a factor of \(\frac{29}{21}\). So, we multiply the total amount of sugar by this factor:
    \[ \frac{86}{25} \times \frac{29}{21} \]

    Multiply the numerators and the denominators:
    \[ \frac{86 \times 29}{25 \times 21} = \frac{2494}{525} \]

    To simplify \(\frac{2494}{525}\), we find the greatest common divisor (GCD) of 2494 and 525. The GCD is 1, so the fraction is already in its simplest form.

    Thus, the total amount of sugar Heather used is:
    \[ \frac{2494}{525} \approx 4.75 \text{ cups} \]

    ### Answer: \(\frac{2494}{525}\) cups

Expected answer: 2494/525

Correct? Yes

🔗 Problem

Problem: Ms. Parker, a math teacher, was grading papers and noticed that a student did the calculation ((7 / 8) + (21 / 18)), but she didn't write down the final answer. What is the result of this calculation as a simplified fraction?

Gemma 2B:

    To solve the expression \(\left(\frac{7}{8} + \frac{21}{18}\right)\), we need to find a common denominator for the fractions.

    1. The denominators are 8 and 18. The least common multiple (LCM) of 8 and 18 is 72.

    2. Convert each fraction to have the common denominator of 72:
       - For \(\frac{7}{8}\):
         \[
         \frac{7}{8} = \frac{7 \times 9}{8 \times 9} = \frac{63}{72}
         \]
       - For \(\frac{21}{18}\):
         \[
         \frac{21}{18} = \frac{21 \times 4}{18 \times 4} = \frac{84}{72}
         \]

    3. Now add the fractions:
       \[
       \frac{63}{72} + \frac{84}{72} = \frac{63 + 84}{72} = \frac{147}{72}
       \]

    4. Simplify the fraction \(\frac{147}{72}\):
       - Find the greatest common divisor (GCD) of 147 and 72. The GCD is 3.
       - Divide both the numerator and the denominator by their GCD:
         \[
         \frac{147 \div 3}{72 \div 3} = \frac{49}{24}
         \]

    ### Answer: \(\frac{49}{24}\)

Expected answer: 49/24

Correct? Yes

🔗 Follow-up

Problem: In the previous exercise, the student was initially solving the problem ((7 / 8) + (21 / 18)). However, upon double-checking their work, they realised that the second part of their calculation had been wrong, it wasn't (21 / 18), but instead (25 / 18). What would be the answer to ((7 / 8) + (25 / 18)) as a simplified fraction?

Gemma 2B:

    To solve the expression \(\left(\frac{7}{8} + \frac{25}{18}\right)\), we need to find a common denominator for the fractions.

    1. The denominators are 8 and 18. The least common multiple (LCM) of 8 and 18 is 72.

    2. Convert each fraction to have the common denominator of 72:
       - For \(\frac{7}{8}\):
         \[
         \frac{7}{8} = \frac{7 \times 9}{8 \times 9} = \frac{63}{72}
         \]
       - For \(\frac{25}{18}\):
         \[
         \frac{25}{18} = \frac{25 \times 4}{18 \times 4} = \frac{100}{72}
         \]

    3. Now add the fractions:
       \[
       \frac{63}{72} + \frac{100}{72} = \frac{63 + 100}{72} = \frac{163}{72}
       \]

    4. Simplify the fraction \(\frac{163}{72}\):
       - The greatest common divisor (GCD) of 163 and 72 is 1, so the fraction is already in its simplest form.

    ### Answer: \(\frac{163}{72}\)

Expected answer: 163/72

Correct? Yes

🔗 Problem

Problem: King Arthur is planning a grand feast at his castle. He estimates that each guest will consume 15/7 gallons of water and 15/24 gallons of wine. On the other hand, the royal baker needs 26/20 gallons of milk for each of the cakes he's baking. Calculate the total amount of liquid (v) that will be consumed if the baker's requirement is multiplied by the guests' consumption (i), and then add this to the guests' consumption (v) to find the total amount of liquid (x) needed for the feast.

Gemma 2B:

    To solve this problem, we need to follow these steps:

    1. Calculate the total amount of liquid each guest will consume.
    2. Calculate the total amount of liquid the baker needs for each cake.
    3. Multiply the baker's requirement by the guests' consumption.
    4. Add the result to the guests' consumption to find the total amount of liquid needed.

    Let's start with the guests' consumption:

    1. Each guest consumes \( \frac{15}{7} \) gallons of water and \( \frac{15}{24} \) gallons of wine.
       - Total liquid per guest:
       \[
       \frac{15}{7} + \frac{15}{24}
       \]

       To add these fractions, we need a common denominator. The least common multiple of 7 and 24 is 168.

       - Convert \( \frac{15}{7} \) to a fraction with a denominator of 168:
       \[
       \frac{15}{7} = \frac{15 \times 24}{7 \times 24} = \frac{360}{168}
       \]

       - Convert \( \frac{15}{24} \) to a fraction with a denominator of 168:
       \[
       \frac{15}{24} = \frac{15 \times 7}{24 \times 7} = \frac{105}{168}
       \]

       - Add the fractions:
       \[
       \frac{360}{168} + \frac{105}{168} = \frac{360 + 105}{168} = \frac{465}{168}
       \]

    2. The baker needs \( \frac{26}{20} \) gallons of milk for each cake.
       - Simplify \( \frac{26}{20} \):
       \[
       \frac{26}{20} = \frac{13}{10}
       \]

    3. Multiply the baker's requirement by the guests' consumption:
       - Multiply \( \frac{13}{10} \) by \( \frac{465}{168} \):
       \[
       \frac{13}{10} \times \frac{465}{168} = \frac{13 \times 465}{10 \times 168} = \frac{6045}{1680}
       \]

    4. Add the result to the guests' consumption:
       - Convert \( \frac{465}{168} \) to a fraction with a denominator of 1680:
       \[
       \frac{465}{168} = \frac{465 \times 10}{168 \times 10} = \frac{4650}{1680}
       \]

       - Add the fractions:
       \[
       \frac{4650}{1680} + \frac{6045}{1680} = \frac{4650 + 6045}{1680} = \frac{10695}{1680}
       \]

       - Simplify \( \frac{10695}{1680} \):
       \[
       \frac{10695}{1680} = \frac{10695 \div 15}{1680 \div 15} = \frac{713}{112}
       \]

    ### Answer: \(\frac{713}{112}\)

Expected answer: 713/112

Correct? Yes

🔗 Follow-up

Problem: In preparation for a subsequent feast at King Arthur's castle, the royal baker has tweaked his recipes and now needs 19/20 gallons of milk for each cake he's baking, as opposed to the previous requirement of 26/20 gallons. Calculate the new total amount of liquid (v) that will be consumed if the baker's revised requirement is multiplied by the guests' consumption (i), and then add this to the guests' original consumption (v). How much total liquid (x) will now be needed for the feast?

Gemma 2B:

    To solve this problem, we need to follow these steps:

    1. Calculate the total amount of liquid each guest will consume.
    2. Calculate the total amount of liquid the baker needs for each cake with the revised requirement.
    3. Multiply the baker's revised requirement by the guests' consumption.
    4. Add the result to the guests' consumption to find the total amount of liquid needed.

    Let's start with the guests' consumption:

    1. Each guest consumes \( \frac{15}{7} \) gallons of water and \( \frac{15}{24} \) gallons of wine.
       - Total liquid per guest:
       \[
       \frac{15}{7} + \frac{15}{24}
       \]

       To add these fractions, we need a common denominator. The least common multiple of 7 and 24 is 168.

       - Convert \( \frac{15}{7} \) to a fraction with a denominator of 168:
       \[
       \frac{15}{7} = \frac{15 \times 24}{7 \times 24} = \frac{360}{168}
       \]

       - Convert \( \frac{15}{24} \) to a fraction with a denominator of 168:
       \[
       \frac{15}{24} = \frac{15 \times 7}{24 \times 7} = \frac{105}{168}
       \]

       - Add the fractions:
       \[
       \frac{360}{168} + \frac{105}{168} = \frac{360 + 105}{168} = \frac{465}{168}
       \]

    2. The baker now needs \( \frac{19}{20} \) gallons of milk for each cake.
       - This fraction is already in its simplest form.

    3. Multiply the baker's revised requirement by the guests' consumption:
       - Multiply \( \frac{19}{20} \) by \( \frac{465}{168} \):
       \[
       \frac{19}{20} \times \frac{465}{168} = \frac{19 \times 465}{20 \times 168} = \frac{8835}{3360}
       \]

    4. Add the result to the guests' consumption:
       - Convert \( \frac{465}{168} \) to a fraction with a denominator of 3360:
       \[
       \frac{465}{168} = \frac{465 \times 20}{168 \times 20} = \frac{9300}{3360}
       \]

       - Add the fractions:
       \[
       \frac{9300}{3360} + \frac{8835}{3360} = \frac{9300 + 8835}{3360} = \frac{18135}{3360}
       \]

       - Simplify \( \frac{18135}{3360} \):
       \[
       \frac{18135}{3360} = \frac{18135 \div 15}{3360 \div 15} = \frac{1209}{224}
       \]

    ### Answer: \(\frac{1209}{224}\)

Expected answer: 1209/224

Correct? Yes

🔗 Problem

Problem: Teddy is a bear who loves to play with numbers. One day, Teddy was playing with his favorite number, which just so happened to be 30/17. While playing, he saw another number, 18/15, and decided to add it to his favorite number. With this new number in mind, he then saw another, 24/13, and decided to multiply it by his new number. What is Teddy's final number?

Gemma 2B:

    To solve this problem, we need to follow the steps of addition and multiplication with fractions.

    First, let's add the two fractions \( \frac{30}{17} \) and \( \frac{18}{15} \).

    1. Find a common denominator for the fractions \( \frac{30}{17} \) and \( \frac{18}{15} \). The least common multiple (LCM) of 17 and 15 is 255.

    2. Convert each fraction to have the common denominator:
       \[
       \frac{30}{17} = \frac{30 \times 15}{17 \times 15} = \frac{450}{255}
       \]
       \[
       \frac{18}{15} = \frac{18 \times 17}{15 \times 17} = \frac{306}{255}
       \]

    3. Add the fractions:
       \[
       \frac{450}{255} + \frac{306}{255} = \frac{450 + 306}{255} = \frac{756}{255}
       \]

    4. Simplify the fraction \( \frac{756}{255} \):
       \[
       \frac{756 \div 3}{255 \div 3} = \frac{252}{85}
       \]

    Next, we need to multiply this result by \( \frac{24}{13} \).

    5. Multiply the fractions:
       \[
       \frac{252}{85} \times \frac{24}{13} = \frac{252 \times 24}{85 \times 13} = \frac{6048}{1105}
       \]

    Thus, Teddy's final number is \( \frac{6048}{1105} \).

    ### Answer: \(\frac{6048}{1105}\)

Expected answer: 6048/1105

Correct? Yes

🔗 Follow-up

Problem: In the previous calculation, Teddy the bear started with the number 30/17. But what if Teddy had started with the number 20/17 instead? Then he added 18/15 to this number, and multiplied the result by 24/13. What is Teddy's final number now?

Gemma 2B:

    To solve this problem, we need to follow the steps of addition and multiplication with fractions, starting with \( \frac{20}{17} \) instead of \( \frac{30}{17} \).

    First, let's add the two fractions \( \frac{20}{17} \) and \( \frac{18}{15} \).

    1. Find a common denominator for the fractions \( \frac{20}{17} \) and \( \frac{18}{15} \). The least common multiple (LCM) of 17 and 15 is 255.

    2. Convert each fraction to have the common denominator:
       \[
       \frac{20}{17} = \frac{20 \times 15}{17 \times 15} = \frac{300}{255}
       \]
       \[
       \frac{18}{15} = \frac{18 \times 17}{15 \times 17} = \frac{306}{255}
       \]

    3. Add the fractions:
       \[
       \frac{300}{255} + \frac{306}{255} = \frac{300 + 306}{255} = \frac{606}{255}
       \]

    4. Simplify the fraction \( \frac{606}{255} \):
       \[
       \frac{606 \div 3}{255 \div 3} = \frac{202}{85}
       \]

    Next, we need to multiply this result by \( \frac{24}{13} \).

    5. Multiply the fractions:
       \[
       \frac{202}{85} \times \frac{24}{13} = \frac{202 \times 24}{85 \times 13} = \frac{4848}{1105}
       \]

    Thus, Teddy's final number is \( \frac{4848}{1105} \).

    ### Answer: \(\frac{4848}{1105}\)

Expected answer: 4848/1105

Correct? Yes

🔗 Problem

Problem: Johnny is building a scooter from scratch. He needs to drill holes into the scooter deck to attach the wheels. The size of each hole should be approximately (20/6) cm in diameter. Unfortunately, his drill bit only has a diameter of (11/21) cm. If he uses his current drill bit, he needs to drill a hole, move it over and drill again to achieve the diameter he needs. If he drills each hole exactly once with his drill bit, calculate what fraction of the desired diameter would each hole have? Don't simplify the fraction, keep as (20/6) * (11 / 21).

Gemma 2B:

    To determine what fraction of the desired diameter each hole would have if Johnny uses his current drill bit, we need to compare the diameter of the drill bit to the desired diameter.

    The desired diameter is given as:
    \[ \frac{20}{6} \text{ cm} \]

    The diameter of the drill bit is:
    \[ \frac{11}{21} \text{ cm} \]

    To find the fraction of the desired diameter that each hole would have, we divide the diameter of the drill bit by the desired diameter:
    \[ \frac{\frac{11}{21}}{\frac{20}{6}} \]

    To divide by a fraction, we multiply by its reciprocal:
    \[ \frac{11}{21} \times \frac{6}{20} \]

    Now, we multiply the numerators and the denominators:
    \[ \frac{11 \times 6}{21 \times 20} = \frac{66}{420} \]

    Thus, each hole drilled with the current drill bit would be:
    \[ \frac{66}{420} \]

    ### Answer: \(\frac{66}{420}\)

Expected answer: 110/63

Correct? No

🔗 Problem

Problem: Aaron has a small model car collection. His favorite car model takes exactly 17/10 hours to assemble. He started assembling a new car model and realized that it was simpler, taking only 12/23 hours to complete. How much less time, in hours, will the second model take than the first one? Write your answer as a simplified fraction or decimal.

Gemma 2B:

    To determine how much less time the second model takes compared to the first one, we need to subtract the time taken for the second model from the time taken for the first model.

    First, let's convert the fractions to a common denominator to make the subtraction easier. The denominators are 10 and 23. The least common multiple (LCM) of 10 and 23 is 230.

    Convert \( \frac{17}{10} \) to a fraction with a denominator of 230:
    \[ \frac{17}{10} = \frac{17 \times 23}{10 \times 23} = \frac{391}{230} \]

    Convert \( \frac{12}{23} \) to a fraction with a denominator of 230:
    \[ \frac{12}{23} = \frac{12 \times 10}{23 \times 10} = \frac{120}{230} \]

    Now, subtract the two fractions:
    \[ \frac{391}{230} - \frac{120}{230} = \frac{391 - 120}{230} = \frac{271}{230} \]

    The fraction \( \frac{271}{230} \) is already in its simplest form because 271 is a prime number and does not share any common factors with 230.

    Thus, the second model takes \( \frac{271}{230} \) hours less than the first model.

    ### Answer: \( \frac{271}{230} \)

Expected answer: 271/230

Correct? Yes

🔗 Follow-up

Problem: Aaron realized he had made a mistake in his timekeeping. Assembling his favorite car model actually took 18/10 hours, not 17/10 hours as he initially thought. How much less time, in hours, will it now take to assemble the second model as compared to the new time for the first model? Write your answer as a simplified fraction or decimal.

Gemma 2B:

    To determine how much less time the second model takes compared to the corrected time for the first model, we need to subtract the time taken for the second model from the corrected time for the first model.

    First, let's convert the fractions to a common denominator to make the subtraction easier. The denominators are 10 and 23. The least common multiple (LCM) of 10 and 23 is 230.

    Convert \( \frac{18}{10} \) to a fraction with a denominator of 230:
    \[ \frac{18}{10} = \frac{18 \times 23}{10 \times 23} = \frac{414}{230} \]

    Convert \( \frac{12}{23} \) to a fraction with a denominator of 230:
    \[ \frac{12}{23} = \frac{12 \times 10}{23 \times 10} = \frac{120}{230} \]

    Now, subtract the two fractions:
    \[ \frac{414}{230} - \frac{120}{230} = \frac{414 - 120}{230} = \frac{294}{230} \]

    Simplify the fraction \( \frac{294}{230} \):
    The greatest common divisor (GCD) of 294 and 230 is 2.
    \[ \frac{294 \div 2}{230 \div 2} = \frac{147}{115} \]

    Thus, the second model takes \( \frac{147}{115} \) hours less than the corrected time for the first model.

    ### Answer: \( \frac{147}{115} \)

Expected answer: 147/115

Correct? Yes

🔗 Problem

Problem: Carlos is very handy, and he decided to start a business making and selling chairs. He usually makes each chair using 20/15 units of wood. For a special order, he decided to design a larger, sturdier chair that uses (24/5) times (7/6) units more wood than usual. In total, how many units of wood would Carlos need to make one of these special chairs? Please give your answer in simplest form.

Gemma 2B:

    First, let's determine how much additional wood Carlos needs for the special chair. The additional wood required is given by the product of \(\frac{24}{5}\) and \(\frac{7}{6}\).

    \[
    \frac{24}{5} \times \frac{7}{6} = \frac{24 \times 7}{5 \times 6} = \frac{168}{30}
    \]

    Next, we simplify \(\frac{168}{30}\) by finding the greatest common divisor (GCD) of 168 and 30. The GCD of 168 and 30 is 6.

    \[
    \frac{168 \div 6}{30 \div 6} = \frac{28}{5}
    \]

    So, the additional wood required is \(\frac{28}{5}\) units.

    Now, we need to add this to the usual amount of wood Carlos uses for a chair, which is \(\frac{20}{15}\). We simplify \(\frac{20}{15}\) first:

    \[
    \frac{20}{15} = \frac{4}{3}
    \]

    Now, we add \(\frac{4}{3}\) and \(\frac{28}{5}\). To do this, we need a common denominator. The least common multiple (LCM) of 3 and 5 is 15.

    \[
    \frac{4}{3} = \frac{4 \times 5}{3 \times 5} = \frac{20}{15}
    \]

    \[
    \frac{28}{5} = \frac{28 \times 3}{5 \times 3} = \frac{84}{15}
    \]

    Now, we add the two fractions:

    \[
    \frac{20}{15} + \frac{84}{15} = \frac{20 + 84}{15} = \frac{104}{15}
    \]

    Thus, Carlos needs \(\frac{104}{15}\) units of wood to make one of these special chairs.

    ### Answer: \(\frac{104}{15}\)

Expected answer: 104/15

Correct? Yes

🔗 Follow-up

Problem: Carlos realized he could be using a more efficient design for his chairs. Instead of using 20/15 units of wood as he was previously, he can now use only 20/8 units for the basic chair design. The larger design with the special feature still requires (24/5) times (7/6) units more wood than the basic chair. With this more efficient design for the basic chair, how many units of wood would Carlos now need in total to make one of these larger special design chairs? Please express your answer as a simplified fraction.

Gemma 2B:

    First, let's determine the amount of wood required for the basic chair with the new design. The new basic chair uses \(\frac{20}{8}\) units of wood. We simplify this fraction:

    \[
    \frac{20}{8} = \frac{5}{2}
    \]

    Next, we calculate the additional wood required for the special feature, which is \(\frac{24}{5} \times \frac{7}{6}\):

    \[
    \frac{24}{5} \times \frac{7}{6} = \frac{24 \times 7}{5 \times 6} = \frac{168}{30}
    \]

    We simplify \(\frac{168}{30}\) by finding the greatest common divisor (GCD) of 168 and 30, which is 6:

    \[
    \frac{168 \div 6}{30 \div 6} = \frac{28}{5}
    \]

    Now, we add the amount of wood required for the basic chair (\(\frac{5}{2}\)) to the additional wood required for the special feature (\(\frac{28}{5}\)). To do this, we need a common denominator. The least common multiple (LCM) of 2 and 5 is 10.

    \[
    \frac{5}{2} = \frac{5 \times 5}{2 \times 5} = \frac{25}{10}
    \]

    \[
    \frac{28}{5} = \frac{28 \times 2}{5 \times 2} = \frac{56}{10}
    \]

    Now, we add the two fractions:

    \[
    \frac{25}{10} + \frac{56}{10} = \frac{25 + 56}{10} = \frac{81}{10}
    \]

    Thus, Carlos now needs \(\frac{81}{10}\) units of wood to make one of these larger special design chairs.

    ### Answer: \(\frac{81}{10}\)

Expected answer: 81/10

Correct? Yes

🔗 Problem

Problem: Officer Davis was monitoring traffic for speeders. He was measuring the speed of cars passing by him using a radar gun. The gun showed one car was going 13/4 miles per minute. Suddenly, the car slows down and reduces his speed by 14/15 miles per minute. Officer Davis noted down the new speed of the car. Suddenly, the car increased its speed again by 30/15 miles per minute. What was the final speed of the car according to Officer Davis's radar gun?

Gemma 2B:

    To determine the final speed of the car, we need to follow these steps:

    1. **Initial Speed**: The car was initially going at \( \frac{13}{4} \) miles per minute.

    2. **Speed Reduction**: The car then reduced its speed by \( \frac{14}{15} \) miles per minute. We need to subtract this from the initial speed:
       \[
       \frac{13}{4} - \frac{14}{15}
       \]

       To subtract these fractions, we need a common denominator. The least common multiple of 4 and 15 is 60. We convert both fractions to have this common denominator:
       \[
       \frac{13}{4} = \frac{13 \times 15}{4 \times 15} = \frac{195}{60}
       \]
       \[
       \frac{14}{15} = \frac{14 \times 4}{15 \times 4} = \frac{56}{60}
       \]

       Now, subtract the fractions:
       \[
       \frac{195}{60} - \frac{56}{60} = \frac{195 - 56}{60} = \frac{139}{60}
       \]

    3. **Speed Increase**: The car then increased its speed by \( \frac{30}{15} \) miles per minute. Simplify \( \frac{30}{15} \):
       \[
       \frac{30}{15} = 2
       \]

       We need to add this to the current speed:
       \[
       \frac{139}{60} + 2
       \]

       Convert 2 to a fraction with a denominator of 60:
       \[
       2 = \frac{120}{60}
       \]

       Now, add the fractions:
       \[
       \frac{139}{60} + \frac{120}{60} = \frac{139 + 120}{60} = \frac{259}{60}
       \]

    Thus, the final speed of the car according to Officer Davis's radar gun is \( \frac{259}{60} \) miles per minute.

    ### Answer: \(\frac{259}{60}\) miles per minute

Expected answer: 259/60

Correct? Yes

🔗 Follow-up

Problem: The car that Officer Davis was monitoring suddenly picked up a passenger who urged the driver to speed up a bit. The car's speed increased again by an additional 14/29 miles per minute. What was the final speed of the car now, according to Officer Davis's radar gun?

Gemma 2B:

    To determine the final speed of the car after the additional increase, we need to add \( \frac{14}{29} \) miles per minute to the current speed of \( \frac{259}{60} \) miles per minute.

    1. **Current Speed**: The current speed of the car is \( \frac{259}{60} \) miles per minute.

    2. **Additional Increase**: The car's speed increased by \( \frac{14}{29} \) miles per minute.

    To add these fractions, we need a common denominator. The least common multiple of 60 and 29 is 1740. We convert both fractions to have this common denominator:

    \[
    \frac{259}{60} = \frac{259 \times 29}{60 \times 29} = \frac{7511}{1740}
    \]

    \[
    \frac{14}{29} = \frac{14 \times 60}{29 \times 60} = \frac{840}{1740}
    \]

    Now, add the fractions:

    \[
    \frac{7511}{1740} + \frac{840}{1740} = \frac{7511 + 840}{1740} = \frac{8351}{1740}
    \]

    Thus, the final speed of the car according to Officer Davis's radar gun is \( \frac{8351}{1740} \) miles per minute.

    ### Answer: \(\frac{8351}{1740}\) miles per minute

Expected answer: 8351/1740

Correct? Yes

🔗 Follow-up

Problem: After reviewing his radar readings, Officer Davis realized he made an error in his measurements. Instead of the car increasing its speed by 30/15 miles per minute after it slowed down, it had actually only increased its speed by 19/15 miles per minute. What would the corrected final speed of the car be, according to Officer Davis's radar gun?

Gemma 2B:

    To determine the corrected final speed of the car, we need to follow these steps:

    1. **Initial Speed**: The car was initially going at \( \frac{13}{4} \) miles per minute.

    2. **Speed Reduction**: The car then reduced its speed by \( \frac{14}{15} \) miles per minute. We need to subtract this from the initial speed:
       \[
       \frac{13}{4} - \frac{14}{15}
       \]

       To subtract these fractions, we need a common denominator. The least common multiple of 4 and 15 is 60. We convert both fractions to have this common denominator:
       \[
       \frac{13}{4} = \frac{13 \times 15}{4 \times 15} = \frac{195}{60}
       \]
       \[
       \frac{14}{15} = \frac{14 \times 4}{15 \times 4} = \frac{56}{60}
       \]

       Now, subtract the fractions:
       \[
       \frac{195}{60} - \frac{56}{60} = \frac{195 - 56}{60} = \frac{139}{60}
       \]

    3. **Corrected Speed Increase**: The car then increased its speed by \( \frac{19}{15} \) miles per minute. Simplify \( \frac{19}{15} \):
       \[
       \frac{19}{15} = \frac{19 \times 4}{15 \times 4} = \frac{76}{60}
       \]

       We need to add this to the current speed:
       \[
       \frac{139}{60} + \frac{76}{60}
       \]

       Now, add the fractions:
       \[
       \frac{139}{60} + \frac{76}{60} = \frac{139 + 76}{60} = \frac{215}{60}
       \]

    Thus, the corrected final speed of the car according to Officer Davis's radar gun is \( \frac{215}{60} \) miles per minute.

    ### Answer: \(\frac{215}{60}\) miles per minute

Expected answer: 43/12

Correct? Yes

🔗 Problem

Problem: Jake is trying to figure out how many grapes he can get from a certain vineyard. - The vineyard can usually produce 19/2 baskets of grapes every season. However, due to weather conditions, only 19/29 of the usual amount were produced this season. How many baskets of grapes were produced this season? - But Jake isn't the only one interested in these grapes. He has to share the grapes with others. He gets to keep only 7/11 of the total amount of grapes. How many baskets of grapes does he get? - Out of the grapes that he got, Jake wants to make some grape juice. But he only uses 9/10 of his grapes to make the juice. How many baskets of grapes does he use for the juice?

Gemma 2B:

    Let's break down the problem step by step.

    1. **Calculating the number of baskets produced this season:**
       - The usual production is \( \frac{19}{2} \) baskets.
       - This season, only \( \frac{19}{29} \) of the usual amount was produced.
       - To find the number of baskets produced this season, we multiply the usual production by the fraction produced this season:
         \[
         \text{Baskets produced this season} = \left( \frac{19}{2} \right) \times \left( \frac{19}{29} \right)
         \]
       - Simplifying the multiplication:
         \[
         \left( \frac{19}{2} \right) \times \left( \frac{19}{29} \right) = \frac{19 \times 19}{2 \times 29} = \frac{361}{58}
         \]
       - Simplifying the fraction:
         \[
         \frac{361}{58} = \frac{361 \div 29}{58 \div 29} = \frac{12.4483}{2} = 6.2241
         \]
       - Therefore, the vineyard produced approximately 6.2241 baskets of grapes this season.

    2. **Calculating the number of baskets Jake gets to keep:**
       - Jake gets to keep \( \frac{7}{11} \) of the total amount of grapes.
       - To find the number of baskets Jake gets, we multiply the total baskets produced by \( \frac{7}{11} \):
         \[
         \text{Baskets Jake gets} = 6.2241 \times \frac{7}{11}
         \]
       - Simplifying the multiplication:
         \[
         6.2241 \times \frac{7}{11} = 6.2241 \times 0.6364 = 3.961
         \]
       - Therefore, Jake gets approximately 3.961 baskets of grapes.

    3. **Calculating the number of baskets Jake uses for grape juice:**
       - Jake uses \( \frac{9}{10} \) of his grapes to make juice.
       - To find the number of baskets Jake uses for juice, we multiply the number of baskets he gets by \( \frac{9}{10} \):
         \[
         \text{Baskets used for juice} = 3.961 \times \frac{9}{10}
         \]
       - Simplifying the multiplication:
         \[
         3.961 \times \frac{9}{10} = 3.961 \times 0.9 = 3.5649
         \]
       - Therefore, Jake uses approximately 3.5649 baskets of grapes to make juice.

    ### Answer: 3.5649

Expected answer: 22743/6380

Correct? No

🔗 Problem

Problem: Joel has a grape farm. One day, he picked 9/2 bushels of red grapes and 10/22 bushels of green grapes from his farm. Calculate the total number of bushels of grapes Joel picked that day.

Gemma 2B:

    To find the total number of bushels of grapes Joel picked, we need to add the bushels of red grapes and green grapes together.

    First, let's convert the fractions to a common denominator if necessary. However, in this case, we can directly add the fractions since they are already in a simple form.

    The red grapes picked are:
    \[ \frac{9}{2} \]

    The green grapes picked are:
    \[ \frac{10}{22} \]

    We can simplify \(\frac{10}{22}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
    \[ \frac{10 \div 2}{22 \div 2} = \frac{5}{11} \]

    Now, we need to add \(\frac{9}{2}\) and \(\frac{5}{11}\). To do this, we need a common denominator. The least common multiple of 2 and 11 is 22.

    Convert \(\frac{9}{2}\) to a fraction with a denominator of 22:
    \[ \frac{9}{2} = \frac{9 \times 11}{2 \times 11} = \frac{99}{22} \]

    Now we can add the two fractions:
    \[ \frac{99}{22} + \frac{5}{11} = \frac{99}{22} + \frac{5 \times 2}{11 \times 2} = \frac{99}{22} + \frac{10}{22} = \frac{99 + 10}{22} = \frac{109}{22} \]

    So, the total number of bushels of grapes Joel picked that day is:
    \[ \frac{109}{22} \]

    ### Answer: \(\frac{109}{22}\)

Expected answer: 109/22

Correct? Yes

🔗 Follow-up

Problem: Joel then sold a unique juice blend at a farmer's market. He used a ratio of 25/17 bushels of juice per bushel of grapes. Calculate how many bushels of juice Joel made for the market using the total number of bushels of grapes he picked.

Gemma 2B:

    To determine how many bushels of juice Joel made, we need to multiply the total number of bushels of grapes he picked by the ratio of bushels of juice per bushel of grapes.

    From the previous problem, we know that Joel picked \(\frac{109}{22}\) bushels of grapes.

    The ratio of bushels of juice per bushel of grapes is \(\frac{25}{17}\).

    To find the total bushels of juice, we multiply these two fractions:
    \[
    \frac{109}{22} \times \frac{25}{17}
    \]

    We multiply the numerators together and the denominators together:
    \[
    \frac{109 \times 25}{22 \times 17} = \frac{2725}{374}
    \]

    So, the total number of bushels of juice Joel made is:
    \[
    \frac{2725}{374}
    \]

    ### Answer: \(\frac{2725}{374}\)

Expected answer: 2725/374

Correct? Yes

🔗 Follow-up

Problem: Suppose Joel made an error and actually only picked 9/3 bushels of red grapes instead of 9/2 bushels. He still picked 10/22 bushels of green grapes. Calculate the corrected total number of bushels of grapes Joel picked.

Gemma 2B:

    To find the corrected total number of bushels of grapes Joel picked, we need to add the bushels of red grapes and green grapes together with the corrected amount for the red grapes.

    First, let's convert the fractions to a common denominator if necessary. However, in this case, we can directly add the fractions since they are already in a simple form.

    The corrected red grapes picked are:
    \[ \frac{9}{3} \]

    The green grapes picked are:
    \[ \frac{10}{22} \]

    We can simplify \(\frac{10}{22}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
    \[ \frac{10 \div 2}{22 \div 2} = \frac{5}{11} \]

    Now, we need to add \(\frac{9}{3}\) and \(\frac{5}{11}\). To do this, we need a common denominator. The least common multiple of 3 and 11 is 33.

    Convert \(\frac{9}{3}\) to a fraction with a denominator of 33:
    \[ \frac{9}{3} = \frac{9 \times 11}{3 \times 11} = \frac{99}{33} \]

    Convert \(\frac{5}{11}\) to a fraction with a denominator of 33:
    \[ \frac{5}{11} = \frac{5 \times 3}{11 \times 3} = \frac{15}{33} \]

    Now we can add the two fractions:
    \[ \frac{99}{33} + \frac{15}{33} = \frac{99 + 15}{33} = \frac{114}{33} \]

    So, the corrected total number of bushels of grapes Joel picked that day is:
    \[ \frac{114}{33} \]

    ### Answer: \(\frac{114}{33}\)

Expected answer: 38/11

Correct? Yes

🔗 Problem

Problem: A snowman is constructed by stacking two different sized snowballs on top of each other. The larger snowball at the bottom has a diameter of 17/8 feet, and the smaller snowball on top has a diameter of 1/20 feet. What is the total height of the snowman in feet when these two snowballs are stacked on top of each other?

Gemma 2B:

    To find the total height of the snowman, we need to add the diameters of the two snowballs.

    First, let's convert the diameters to a common fraction format for easier addition:

    1. The diameter of the larger snowball is \( \frac{17}{8} \) feet.
    2. The diameter of the smaller snowball is \( \frac{1}{20} \) feet.

    Next, we need to find a common denominator to add these fractions. The least common multiple (LCM) of 8 and 20 is 40.

    Convert \( \frac{17}{8} \) to a fraction with a denominator of 40:
    \[ \frac{17}{8} = \frac{17 \times 5}{8 \times 5} = \frac{85}{40} \]

    Convert \( \frac{1}{20} \) to a fraction with a denominator of 40:
    \[ \frac{1}{20} = \frac{1 \times 2}{20 \times 2} = \frac{2}{40} \]

    Now, add the two fractions:
    \[ \frac{85}{40} + \frac{2}{40} = \frac{85 + 2}{40} = \frac{87}{40} \]

    Thus, the total height of the snowman is \( \frac{87}{40} \) feet.

    To convert this to a mixed number:
    \[ \frac{87}{40} = 2 \frac{7}{40} \]

    ### Answer: \( 2 \frac{7}{40} \) feet

Expected answer: 87/40

Correct? Yes

🔗 Follow-up

Problem: After constructing the snowman with two snowballs, we decide to add a unusually tall hat with a height of 14/19 feet to the snowman. What is the total height of the snowman now, with the two snowballs and the hat?

Gemma 2B:

    To find the total height of the snowman with the hat, we need to add the height of the hat to the previously calculated height of the snowman.

    The height of the snowman without the hat is \( \frac{87}{40} \) feet.

    The height of the hat is \( \frac{14}{19} \) feet.

    To add these two fractions, we need a common denominator. The least common multiple (LCM) of 40 and 19 is 760.

    Convert \( \frac{87}{40} \) to a fraction with a denominator of 760:
    \[ \frac{87}{40} = \frac{87 \times 19}{40 \times 19} = \frac{1653}{760} \]

    Convert \( \frac{14}{19} \) to a fraction with a denominator of 760:
    \[ \frac{14}{19} = \frac{14 \times 40}{19 \times 40} = \frac{560}{760} \]

    Now, add the two fractions:
    \[ \frac{1653}{760} + \frac{560}{760} = \frac{1653 + 560}{760} = \frac{2213}{760} \]

    Thus, the total height of the snowman with the hat is \( \frac{2213}{760} \) feet.

    To convert this to a mixed number:
    \[ \frac{2213}{760} = 2 \frac{693}{760} \]

    ### Answer: \( 2 \frac{693}{760} \) feet

Expected answer: 2213/760

Correct? Yes

🔗 Follow-up

Problem: In our previous snowman, the larger snowball at the bottom had a diameter of 17/8 feet. Now imagine that the larger snowball had a larger diameter of 17/6 feet instead. The smaller snowball on top still has a diameter of 1/20 feet. What is the total height of the snowman now, with these two adjusted snowballs stacked on top of each other?

Gemma 2B:

    To find the total height of the snowman with the new diameters, we need to add the diameters of the two snowballs.

    1. The diameter of the larger snowball is now \( \frac{17}{6} \) feet.
    2. The diameter of the smaller snowball remains \( \frac{1}{20} \) feet.

    First, we need to find a common denominator to add these fractions. The least common multiple (LCM) of 6 and 20 is 60.

    Convert \( \frac{17}{6} \) to a fraction with a denominator of 60:
    \[ \frac{17}{6} = \frac{17 \times 10}{6 \times 10} = \frac{170}{60} \]

    Convert \( \frac{1}{20} \) to a fraction with a denominator of 60:
    \[ \frac{1}{20} = \frac{1 \times 3}{20 \times 3} = \frac{3}{60} \]

    Now, add the two fractions:
    \[ \frac{170}{60} + \frac{3}{60} = \frac{170 + 3}{60} = \frac{173}{60} \]

    Thus, the total height of the snowman is \( \frac{173}{60} \) feet.

    To convert this to a mixed number:
    \[ \frac{173}{60} = 2 \frac{53}{60} \]

    ### Answer: \( 2 \frac{53}{60} \) feet

Expected answer: 173/60

Correct? Yes

🔗 Problem

Problem: In a sandbox, a child splits up 24/7 parts of the sand in a fair way. Later on, another child takes away 2/16 part of the sand. Calculate the remaining part of sand in the sandbox.

Gemma 2B:

    First, let's simplify the fractions involved.

    The initial amount of sand is \( \frac{24}{7} \).

    The amount of sand taken away is \( \frac{2}{16} \). Simplifying \( \frac{2}{16} \):
    \[ \frac{2}{16} = \frac{1}{8} \]

    Now, we need to subtract \( \frac{1}{8} \) from \( \frac{24}{7} \).

    To perform this subtraction, we need a common denominator. The least common multiple of 7 and 8 is 56.

    Convert \( \frac{24}{7} \) to a fraction with a denominator of 56:
    \[ \frac{24}{7} = \frac{24 \times 8}{7 \times 8} = \frac{192}{56} \]

    Convert \( \frac{1}{8} \) to a fraction with a denominator of 56:
    \[ \frac{1}{8} = \frac{1 \times 7}{8 \times 7} = \frac{7}{56} \]

    Now, subtract the two fractions:
    \[ \frac{192}{56} - \frac{7}{56} = \frac{192 - 7}{56} = \frac{185}{56} \]

    Thus, the remaining part of the sand in the sandbox is \( \frac{185}{56} \).

    ### Answer: \(\frac{185}{56}\)

Expected answer: 185/56

Correct? Yes

🔗 Follow-up

Problem: Suppose now the child originally splits up 24/16 parts of the sand in the sandbox. Later on, the same amount of 2/16 part of the sand is taken away again. How much sand remains in the sandbox this time?

Gemma 2B:

    First, let's simplify the fractions involved.

    The initial amount of sand is \( \frac{24}{16} \).

    Simplify \( \frac{24}{16} \):
    \[ \frac{24}{16} = \frac{3}{2} \]

    The amount of sand taken away is \( \frac{2}{16} \). Simplify \( \frac{2}{16} \):
    \[ \frac{2}{16} = \frac{1}{8} \]

    Now, we need to subtract \( \frac{1}{8} \) from \( \frac{3}{2} \).

    To perform this subtraction, we need a common denominator. The least common multiple of 2 and 8 is 8.

    Convert \( \frac{3}{2} \) to a fraction with a denominator of 8:
    \[ \frac{3}{2} = \frac{3 \times 4}{2 \times 4} = \frac{12}{8} \]

    Now, subtract the two fractions:
    \[ \frac{12}{8} - \frac{1}{8} = \frac{12 - 1}{8} = \frac{11}{8} \]

    Thus, the remaining part of the sand in the sandbox is \( \frac{11}{8} \).

    ### Answer: \(\frac{11}{8}\)

Expected answer: 11/8

Correct? Yes

🔗 Problem

Problem: In the jungle, a lion ate 28/26 of a zebra in one day and 23/9 of a wildebeest the next day. How much less of the zebra did the lion eat compared with the wildebeest? Calculate your answer as a simplified fraction.

Gemma 2B:

    To determine how much less of the zebra the lion ate compared to the wildebeest, we need to subtract the fraction of the zebra eaten from the fraction of the wildebeest eaten.

    First, let's write down the fractions:
    - Zebra: \( \frac{28}{26} \)
    - Wildebeest: \( \frac{23}{9} \)

    We need to subtract \( \frac{28}{26} \) from \( \frac{23}{9} \). To do this, we need a common denominator. The least common multiple (LCM) of 26 and 9 is 234.

    Convert each fraction to have the common denominator of 234:
    - For the zebra:
      \[
      \frac{28}{26} = \frac{28 \times 9}{26 \times 9} = \frac{252}{234}
      \]
    - For the wildebeest:
      \[
      \frac{23}{9} = \frac{23 \times 26}{9 \times 26} = \frac{598}{234}
      \]

    Now, subtract the fractions:
    \[
    \frac{598}{234} - \frac{252}{234} = \frac{598 - 252}{234} = \frac{346}{234}
    \]

    Simplify the fraction \( \frac{346}{234} \):
    - The greatest common divisor (GCD) of 346 and 234 is 2.
    - Divide both the numerator and the denominator by 2:
      \[
      \frac{346 \div 2}{234 \div 2} = \frac{173}{117}
      \]

    Thus, the simplified fraction is \( \frac{173}{117} \).

    ### Answer: \(\frac{173}{117}\)

Expected answer: 173/117

Correct? Yes

🔗 Follow-up

Problem: In the jungle, a lion ate 28/26 of a zebra in one day and 23/9 of a wildebeest the next day. The lion realized it was still hungry, so it ate 21/4 as much as the difference between the wildebeest and the zebra the next day. How much did the lion eat the next day? Please express your answer as a simplified fraction.

Gemma 2B:

    To determine how much the lion ate the next day, we first need to find the difference between the amount of wildebeest and zebra it ate, and then calculate \( \frac{21}{4} \) of that difference.

    First, let's write down the fractions:
    - Zebra: \( \frac{28}{26} \)
    - Wildebeest: \( \frac{23}{9} \)

    We need to subtract \( \frac{28}{26} \) from \( \frac{23}{9} \). To do this, we need a common denominator. The least common multiple (LCM) of 26 and 9 is 234.

    Convert each fraction to have the common denominator of 234:
    - For the zebra:
      \[
      \frac{28}{26} = \frac{28 \times 9}{26 \times 9} = \frac{252}{234}
      \]
    - For the wildebeest:
      \[
      \frac{23}{9} = \frac{23 \times 26}{9 \times 26} = \frac{598}{234}
      \]

    Now, subtract the fractions:
    \[
    \frac{598}{234} - \frac{252}{234} = \frac{598 - 252}{234} = \frac{346}{234}
    \]

    Simplify the fraction \( \frac{346}{234} \):
    - The greatest common divisor (GCD) of 346 and 234 is 2.
    - Divide both the numerator and the denominator by 2:
      \[
      \frac{346 \div 2}{234 \div 2} = \frac{173}{117}
      \]

    Thus, the simplified fraction is \( \frac{173}{117} \).

    Next, we need to calculate \( \frac{21}{4} \) of this difference:
    \[
    \frac{21}{4} \times \frac{173}{117}
    \]

    Multiply the numerators and the denominators:
    \[
    \frac{21 \times 173}{4 \times 117} = \frac{3633}{468}
    \]

    Simplify the fraction \( \frac{3633}{468} \):
    - The greatest common divisor (GCD) of 3633 and 468 is 3.
    - Divide both the numerator and the denominator by 3:
      \[
      \frac{3633 \div 3}{468 \div 3} = \frac{1211}{156}
      \]

    Thus, the simplified fraction is \( \frac{1211}{156} \).

    ### Answer: \(\frac{1211}{156}\)

Expected answer: 1211/156

Correct? Yes

🔗 Problem

Problem: A chocolate factory makes a batch of chocolate by combining 14/4 kilograms of cocoa and 21/11 liters of milk for every kilogram. How many kilograms of chocolate will the factory produce by combining these ingredients? Calculate your answer to the following expression, ((14 / 4) * (21 / 11)).

Gemma 2B:

    To find out how many kilograms of chocolate the factory will produce, we need to calculate the expression \(\left(\frac{14}{4}\right) \times \left(\frac{21}{11}\right)\).

    First, simplify the fractions if possible:
    \[
    \frac{14}{4} = \frac{7}{2}
    \]

    Now, multiply the simplified fractions:
    \[
    \left(\frac{7}{2}\right) \times \left(\frac{21}{11}\right)
    \]

    Multiply the numerators together and the denominators together:
    \[
    \frac{7 \times 21}{2 \times 11} = \frac{147}{22}
    \]

    Next, simplify the fraction \(\frac{147}{22}\) if possible. Since 147 and 22 have no common factors other than 1, the fraction is already in its simplest form.

    Thus, the factory will produce \(\frac{147}{22}\) kilograms of chocolate.

    ### Answer: \(\frac{147}{22}\) kilograms

Expected answer: 147/22

Correct? Yes

🔗 Follow-up

Problem: After the chocolate factory combines the ingredients to make the chocolate, they find out that some of it has spoiled. If the spoiled chocolate weighed 12/4 kilograms, calculate the new amount of the chocolate they have left. Use the expression (o - (12 / 4)) to determine this.

Gemma 2B:

    First, we need to determine the amount of spoiled chocolate. The spoiled chocolate weighs \(\frac{12}{4}\) kilograms.

    Simplify the fraction:
    \[
    \frac{12}{4} = 3 \text{ kilograms}
    \]

    Next, we need to subtract this amount from the total amount of chocolate produced, which we previously calculated as \(\frac{147}{22}\) kilograms.

    The expression to calculate the new amount of chocolate is:
    \[
    \frac{147}{22} - 3
    \]

    To perform this subtraction, we need to express 3 as a fraction with the same denominator as \(\frac{147}{22}\):
    \[
    3 = \frac{3 \times 22}{1 \times 22} = \frac{66}{22}
    \]

    Now, subtract the fractions:
    \[
    \frac{147}{22} - \frac{66}{22} = \frac{147 - 66}{22} = \frac{81}{22}
    \]

    Thus, the new amount of chocolate they have left is \(\frac{81}{22}\) kilograms.

    ### Answer: \(\frac{81}{22}\) kilograms

Expected answer: 81/22

Correct? Yes

🔗 Follow-up

Problem: Let's say the chocolate factory decided to reduce their cocoa to milk ratio in the chocolate in order to experiment with the taste. If they decrease the amount of milk used in each batch of chocolate from 21/11 liters to 6/11 liters while keeping the quantity of cocoa constant at 14/4 kilograms, calculate the amount of chocolate they will produce now. Use the expression ((14 / 4) * (6 / 11)) to find the new amount of chocolate.

Gemma 2B:

    To find the new amount of chocolate the factory will produce, we need to calculate the expression \(\left(\frac{14}{4}\right) \times \left(\frac{6}{11}\right)\).

    First, simplify the fraction \(\frac{14}{4}\):
    \[
    \frac{14}{4} = \frac{7}{2}
    \]

    Now, multiply the simplified fraction by \(\frac{6}{11}\):
    \[
    \left(\frac{7}{2}\right) \times \left(\frac{6}{11}\right)
    \]

    Multiply the numerators together and the denominators together:
    \[
    \frac{7 \times 6}{2 \times 11} = \frac{42}{22}
    \]

    Next, simplify the fraction \(\frac{42}{22}\). Both 42 and 22 can be divided by their greatest common divisor, which is 2:
    \[
    \frac{42 \div 2}{22 \div 2} = \frac{21}{11}
    \]

    Thus, the factory will produce \(\frac{21}{11}\) kilograms of chocolate with the new ratio.

    ### Answer: \(\frac{21}{11}\) kilograms

Expected answer: 21/11

Correct? Yes

🔗 Problem

Problem: In the tropical rainforest, there are two different species of banana trees. One species produces 12 bananas every 29 days, while another species produces 30 bananas every 10 days. If a day is chosen at random, what is the expected number of bananas that the two species together will have produced on that day? Use the formula ((12 / 29) + (30 / 10)) to calculate the answer.

Gemma 2B:

    To find the expected number of bananas produced by the two species of banana trees on any given day, we use the given formula:

    \[
    \text{Expected number of bananas} = \left(\frac{12}{29}\right) + \left(\frac{30}{10}\right)
    \]

    First, we calculate the contribution from the first species:

    \[
    \frac{12}{29} \approx 0.4138
    \]

    Next, we calculate the contribution from the second species:

    \[
    \frac{30}{10} = 3
    \]

    Adding these two contributions together gives us the total expected number of bananas produced per day:

    \[
    0.4138 + 3 = 3.4138
    \]

    ### Answer: 3.4138

Expected answer: 99/29

Correct? No

🔗 Problem

Problem: A giraffe needs to eat a total of 11/11 ton of acacia leaves plus 26/14 ton of grasses a day. Calculate the total ton of food the giraffe needs to eat in a day.

Gemma 2B:

    First, let's simplify the fractions given:

    1. The fraction \( \frac{11}{11} \) simplifies to 1 because any number divided by itself is 1.
    2. The fraction \( \frac{26}{14} \) can be simplified by finding the greatest common divisor (GCD) of 26 and 14, which is 2.

    So, we divide both the numerator and the denominator by 2:
    \[ \frac{26 \div 2}{14 \div 2} = \frac{13}{7} \]

    Now, we need to add the simplified fractions:
    \[ 1 + \frac{13}{7} \]

    To add these fractions, we convert 1 to a fraction with the same denominator as \( \frac{13}{7} \):
    \[ 1 = \frac{7}{7} \]

    Now, we can add the fractions:
    \[ \frac{7}{7} + \frac{13}{7} = \frac{7 + 13}{7} = \frac{20}{7} \]

    Thus, the total ton of food the giraffe needs to eat in a day is:
    \[ \frac{20}{7} \]

    ### Answer: \(\frac{20}{7}\) tons

Expected answer: 20/7

Correct? Yes

🔗 Follow-up

Problem: In addition to the acacia leaves and grasses, the giraffe also drinks 10/30 ton of water per day. What is the total weight of food and water that the giraffe consumes in a day?

Gemma 2B:

    First, let's simplify the fraction for the water consumption:

    \[ \frac{10}{30} \]

    We can simplify this fraction by finding the greatest common divisor (GCD) of 10 and 30, which is 10. So, we divide both the numerator and the denominator by 10:

    \[ \frac{10 \div 10}{30 \div 10} = \frac{1}{3} \]

    Now, we need to add this to the total weight of food the giraffe consumes, which we previously calculated as \( \frac{20}{7} \) tons.

    To add \( \frac{20}{7} \) and \( \frac{1}{3} \), we need a common denominator. The least common multiple (LCM) of 7 and 3 is 21.

    Convert each fraction to have a denominator of 21:

    \[ \frac{20}{7} = \frac{20 \times 3}{7 \times 3} = \frac{60}{21} \]

    \[ \frac{1}{3} = \frac{1 \times 7}{3 \times 7} = \frac{7}{21} \]

    Now, add the fractions:

    \[ \frac{60}{21} + \frac{7}{21} = \frac{60 + 7}{21} = \frac{67}{21} \]

    Thus, the total weight of food and water that the giraffe consumes in a day is:

    ### Answer: \(\frac{67}{21}\) tons

Expected answer: 67/21

Correct? Yes

🔗 Follow-up

Problem: In the previous scenario, suppose the giraffe ate 24/14 ton of grasses, instead of 26/14 ton. How much total food does the giraffe eat per day now?

Gemma 2B:

    First, let's simplify the fractions given:

    1. The fraction \( \frac{11}{11} \) simplifies to 1 because any number divided by itself is 1.
    2. The fraction \( \frac{24}{14} \) can be simplified by finding the greatest common divisor (GCD) of 24 and 14, which is 2.

    So, we divide both the numerator and the denominator by 2:
    \[ \frac{24 \div 2}{14 \div 2} = \frac{12}{7} \]

    Now, we need to add the simplified fractions:
    \[ 1 + \frac{12}{7} \]

    To add these fractions, we convert 1 to a fraction with the same denominator as \( \frac{12}{7} \):
    \[ 1 = \frac{7}{7} \]

    Now, we can add the fractions:
    \[ \frac{7}{7} + \frac{12}{7} = \frac{7 + 12}{7} = \frac{19}{7} \]

    Thus, the total ton of food the giraffe needs to eat in a day is:
    \[ \frac{19}{7} \]

    ### Answer: \(\frac{19}{7}\) tons

Expected answer: 19/7

Correct? Yes

🔗 Problem

Problem: A bear spends 18 hours of the day sleeping. Given that each day has 24 hours, this bear spends 18/24th of each day asleep. For 27 days of the month, the bear's activity is tracked. Assume that the bear has a regular sleeping pattern. Calculate the fraction of time the bear has spent sleeping over the 27 days assuming each month has 13 weeks. Your answer should be the result of (18 / 24) * (27 / 13). Write your answer as a decimal.

Gemma 2B:

    First, let's break down the problem step by step.

    1. The bear spends 18 hours out of 24 hours each day sleeping. This can be expressed as a fraction:
       \[
       \frac{18}{24}
       \]

    2. Simplify the fraction:
       \[
       \frac{18}{24} = \frac{3}{4}
       \]

    3. The bear's activity is tracked for 27 days. We need to find the fraction of time the bear has spent sleeping over these 27 days. We are given that each month has 13 weeks, but this information is not necessary for our calculation.

    4. Calculate the fraction of time the bear has spent sleeping over 27 days:
       \[
       \left(\frac{3}{4}\right) \times 27
       \]

    5. Convert the fraction to a decimal:
       \[
       \frac{3}{4} = 0.75
       \]

    6. Multiply by 27:
       \[
       0.75 \times 27 = 20.25
       \]

    Therefore, the bear has spent 20.25 days sleeping over the 27 days.

    ### Answer: 20.25

Expected answer: 81/52

Correct? No

🔗 Problem

Problem: In a beautiful forest, the park rangers recorded that 9 out of every 14 trees were pine trees. They also noticed an exceptional occurrence where 19 out of every 2 trees were birches. If you count all the pine trees and birch trees together, what fraction of the trees in the forest are either pine or birch? Use the formula f = ((9 / 14) + (19 / 2)) to solve this.

Gemma 2B:

    To solve the problem, we need to calculate the fraction of the trees that are either pine or birch using the given formula \( f = \left(\frac{9}{14}\right) + \left(\frac{19}{2}\right) \).

    First, let's convert the fractions to have a common denominator. The least common multiple of 14 and 2 is 14.

    1. Convert \(\frac{19}{2}\) to a fraction with a denominator of 14:
       \[
       \frac{19}{2} = \frac{19 \times 7}{2 \times 7} = \frac{133}{14}
       \]

    2. Now, add the two fractions:
       \[
       \frac{9}{14} + \frac{133}{14} = \frac{9 + 133}{14} = \frac{142}{14}
       \]

    3. Simplify the fraction:
       \[
       \frac{142}{14} = \frac{142 \div 2}{14 \div 2} = \frac{71}{7}
       \]

    Thus, the fraction of the trees in the forest that are either pine or birch is \(\frac{71}{7}\).

    ### Answer: \(\frac{71}{7}\)

Expected answer: 71/7

Correct? Yes

🔗 Follow-up

Problem: In the same forest, the park rangers conducted another seasonal survey. This time, they found that the number of pine trees made up 9 out of every 14 trees. In an unexpected turn of events, the number of birch trees significantly increased to 30 out of every 2 trees. Given these new numbers, what fraction of the trees in the forest are either pine or birch now? Use the updated formula f = ((9 / 14) + (30 / 2)) to solve this.

Gemma 2B:

    To solve the problem, we need to calculate the fraction of the trees that are either pine or birch using the updated formula \( f = \left(\frac{9}{14}\right) + \left(\frac{30}{2}\right) \).

    First, let's convert the fractions to have a common denominator. The least common multiple of 14 and 2 is 14.

    1. Convert \(\frac{30}{2}\) to a fraction with a denominator of 14:
       \[
       \frac{30}{2} = \frac{30 \times 7}{2 \times 7} = \frac{210}{14}
       \]

    2. Now, add the two fractions:
       \[
       \frac{9}{14} + \frac{210}{14} = \frac{9 + 210}{14} = \frac{219}{14}
       \]

    Thus, the fraction of the trees in the forest that are either pine or birch is \(\frac{219}{14}\).

    ### Answer: \(\frac{219}{14}\)

Expected answer: 219/14

Correct? Yes

🔗 Problem

Problem: On planet Zog, aliens use an interesting system to track their energy levels. At the beginning of the day, Zogorian calculates his energy level to be (14 / 2) / (23 / 17) zogs. Later in the day, after harnessing energy from their primary star, his energy level increases by (16 / 24) zogs. Calculate the Zogorian's energy level after the increase.

Gemma 2B:

    First, we need to calculate the initial energy level of the Zogorian.

    The initial energy level is given by:
    \[
    \frac{\left(\frac{14}{2}\right)}{\left(\frac{23}{17}\right)}
    \]

    First, simplify the numerator and the denominator separately:
    \[
    \frac{14}{2} = 7
    \]
    \[
    \frac{23}{17} \text{ remains as it is.}
    \]

    Now, the expression becomes:
    \[
    \frac{7}{\frac{23}{17}}
    \]

    To divide by a fraction, multiply by its reciprocal:
    \[
    \frac{7}{\frac{23}{17}} = 7 \times \frac{17}{23} = \frac{7 \times 17}{23} = \frac{119}{23}
    \]

    Next, we need to add the increase in energy level, which is:
    \[
    \frac{16}{24}
    \]

    Simplify \(\frac{16}{24}\):
    \[
    \frac{16}{24} = \frac{2 \times 8}{3 \times 8} = \frac{2}{3}
    \]

    Now, add this to the initial energy level:
    \[
    \frac{119}{23} + \frac{2}{3}
    \]

    To add these fractions, find a common denominator. The least common multiple of 23 and 3 is 69:
    \[
    \frac{119}{23} = \frac{119 \times 3}{23 \times 3} = \frac{357}{69}
    \]
    \[
    \frac{2}{3} = \frac{2 \times 23}{3 \times 23} = \frac{46}{69}
    \]

    Now, add the fractions:
    \[
    \frac{357}{69} + \frac{46}{69} = \frac{357 + 46}{69} = \frac{403}{69}
    \]

    Thus, the Zogorian's energy level after the increase is:
    \[
    \frac{403}{69}
    \]

    ### Answer: \(\frac{403}{69}\)

Expected answer: 403/69

Correct? Yes

🔗 Follow-up

Problem: After the Zogorian's energy level increased, he used a special device that multiplies the current energy level by a factor of (18 / 11) to store the additional energy for later use. Calculate the Zogorian's new energy level in zogs after using the device.

Gemma 2B:

    To find the Zogorian's new energy level after using the device, we need to multiply the current energy level by the given factor.

    The current energy level is:
    \[
    \frac{403}{69}
    \]

    The factor by which the energy level is multiplied is:
    \[
    \frac{18}{11}
    \]

    Now, multiply these two fractions:
    \[
    \frac{403}{69} \times \frac{18}{11}
    \]

    Multiply the numerators and the denominators:
    \[
    \frac{403 \times 18}{69 \times 11}
    \]

    Calculate the products:
    \[
    403 \times 18 = 7254
    \]
    \[
    69 \times 11 = 759
    \]

    So, the new energy level is:
    \[
    \frac{7254}{759}
    \]

    To simplify this fraction, we need to find the greatest common divisor (GCD) of 7254 and 759. Using the Euclidean algorithm:

    1. \(7254 \div 759 \approx 9.55\), so the integer part is 9.
    2. \(7254 - 759 \times 9 = 7254 - 6831 = 423\)
    3. \(759 \div 423 \approx 1.79\), so the integer part is 1.
    4. \(759 - 423 = 336\)
    5. \(423 \div 336 \approx 1.26\), so the integer part is 1.
    6. \(423 - 336 = 87\)
    7. \(336 \div 87 \approx 3.86\), so the integer part is 3.
    8. \(336 - 87 \times 3 = 336 - 261 = 75\)
    9. \(87 \div 75 \approx 1.16\), so the integer part is 1.
    10. \(87 - 75 = 12\)
    11. \(75 \div 12 \approx 6.25\), so the integer part is 6.
    12. \(75 - 12 \times 6 = 75 - 72 = 3\)
    13. \(12 \div 3 = 4\), so the integer part is 4.
    14. \(12 - 3 \times 4 = 0\)

    The GCD is 3. Now, divide both the numerator and the denominator by 3:
    \[
    \frac{7254 \div 3}{759 \div 3} = \frac{2418}{253}
    \]

    Thus, the Zogorian's new energy level is:
    \[
    \frac{2418}{253}
    \]

    ### Answer: \(\frac{2418}{253}\)

Expected answer: 2418/253

Correct? Yes

🔗 Follow-up

Problem: Let's continue tracking this Zogorian's energy levels. Suppose instead, after harnessing energy from their primary star, his energy level had increased by (23 / 24) zogs instead of (16 / 24) zogs. Calculate the Zogorian's energy level after this larger increase.

Gemma 2B:

    First, we need to recall the initial energy level of the Zogorian, which we previously calculated as:
    \[
    \frac{119}{23}
    \]

    Now, we need to add the new increase in energy level, which is:
    \[
    \frac{23}{24}
    \]

    To add these fractions, we need a common denominator. The least common multiple of 23 and 24 is 552:
    \[
    \frac{119}{23} = \frac{119 \times 24}{23 \times 24} = \frac{2856}{552}
    \]
    \[
    \frac{23}{24} = \frac{23 \times 23}{24 \times 23} = \frac{529}{552}
    \]

    Now, add the fractions:
    \[
    \frac{2856}{552} + \frac{529}{552} = \frac{2856 + 529}{552} = \frac{3385}{552}
    \]

    Thus, the Zogorian's energy level after the larger increase is:
    \[
    \frac{3385}{552}
    \]

    ### Answer: \(\frac{3385}{552}\)

Expected answer: 3385/552

Correct? Yes

🔗 Problem

Problem: Teddy, the bear, loves to play hide and seek with his friends. One day, he decided to play a game where he hides some number of his stuffed animal friends. Teddy hides 18/11 of his friends in the forest and 8/21 of his friends at the beach. First calculate how many more friends Teddy hides in the forest than at the beach. Let's call this number 'f'. Then square 'f' to find 'n', the total number of hiding spots Teddy needs. How many hiding spots does Teddy need?

Gemma 2B:

    To find how many more friends Teddy hides in the forest than at the beach, we need to subtract the fraction of friends hidden at the beach from the fraction hidden in the forest.

    First, let's find a common denominator for the fractions \(\frac{18}{11}\) and \(\frac{8}{21}\). The least common multiple of 11 and 21 is 231.

    Convert \(\frac{18}{11}\) to a fraction with a denominator of 231:
    \[
    \frac{18}{11} = \frac{18 \times 21}{11 \times 21} = \frac{378}{231}
    \]

    Convert \(\frac{8}{21}\) to a fraction with a denominator of 231:
    \[
    \frac{8}{21} = \frac{8 \times 11}{21 \times 11} = \frac{88}{231}
    \]

    Now, subtract the two fractions:
    \[
    \frac{378}{231} - \frac{88}{231} = \frac{378 - 88}{231} = \frac{290}{231}
    \]

    So, the number of friends Teddy hides more in the forest than at the beach is \(\frac{290}{231}\).

    Next, we need to square this number to find the total number of hiding spots, \(n\):
    \[
    \left(\frac{290}{231}\right)^2 = \frac{290^2}{231^2} = \frac{84100}{53361}
    \]

    Thus, the total number of hiding spots Teddy needs is \(\frac{84100}{53361}\).

    ### Answer: \(\frac{84100}{53361}\)

Expected answer: 84100/53361

Correct? Yes

🔗 Follow-up

Problem: After hiding all his friends, Teddy, the bear, thinks about a new challenge. He decides to tack on an extra task that each of his friends must complete. The task involves going through an obstacle course, and only 18/28 of his friends decide to take on the new challenge. Teddy decides that the new challenge will require as many action points as there were hiding spots, but multiplied by the fraction of his friends who decided to take on the challenge. This is represented by 'o' = (18/28) * 'n'. So, calculate 'o', how many total action points are needed for Teddy's new challenge?

Gemma 2B:

    To calculate the total action points needed for Teddy's new challenge, we need to multiply the fraction of friends who decided to take on the challenge by the number of hiding spots, \(n\).

    First, let's simplify the fraction \(\frac{18}{28}\):
    \[
    \frac{18}{28} = \frac{9}{14}
    \]

    We previously calculated the number of hiding spots, \(n\), as \(\frac{84100}{53361}\).

    Now, we need to multiply \(\frac{9}{14}\) by \(\frac{84100}{53361}\):
    \[
    o = \frac{9}{14} \times \frac{84100}{53361}
    \]

    To perform the multiplication of these fractions:
    \[
    o = \frac{9 \times 84100}{14 \times 53361} = \frac{756900}{747054}
    \]

    We can simplify this fraction by finding the greatest common divisor (GCD) of 756900 and 747054. Using the Euclidean algorithm, we find that the GCD is 6.

    So, we simplify the fraction:
    \[
    o = \frac{756900 \div 6}{747054 \div 6} = \frac{126150}{124509}
    \]

    Thus, the total action points needed for Teddy's new challenge is \(\frac{126150}{124509}\).

    ### Answer: \(\frac{126150}{124509}\)

Expected answer: 42050/41503

Correct? Yes

🔗 Follow-up

Problem: As Teddy's game continues, he changes the number of friends he initially hides. This time, he hides 21/11 of his friends in the forest instead of 18/11, while still hiding 8/21 of his friends at the beach. Now first calculate how many more friends Teddy hides in the forest than at the beach, using this new number. Let's still call this number 'f'. Then square 'f' to find 'n', the new total number of hiding spots Teddy needs. What is the new total number of hiding spots Teddy needs?

Gemma 2B:

    To find how many more friends Teddy hides in the forest than at the beach with the new numbers, we need to subtract the fraction of friends hidden at the beach from the fraction hidden in the forest.

    First, let's find a common denominator for the fractions \(\frac{21}{11}\) and \(\frac{8}{21}\). The least common multiple of 11 and 21 is 231.

    Convert \(\frac{21}{11}\) to a fraction with a denominator of 231:
    \[
    \frac{21}{11} = \frac{21 \times 21}{11 \times 21} = \frac{441}{231}
    \]

    Convert \(\frac{8}{21}\) to a fraction with a denominator of 231:
    \[
    \frac{8}{21} = \frac{8 \times 11}{21 \times 11} = \frac{88}{231}
    \]

    Now, subtract the two fractions:
    \[
    \frac{441}{231} - \frac{88}{231} = \frac{441 - 88}{231} = \frac{353}{231}
    \]

    So, the number of friends Teddy hides more in the forest than at the beach is \(\frac{353}{231}\).

    Next, we need to square this number to find the total number of hiding spots, \(n\):
    \[
    \left(\frac{353}{231}\right)^2 = \frac{353^2}{231^2} = \frac{124609}{53361}
    \]

    Thus, the new total number of hiding spots Teddy needs is \(\frac{124609}{53361}\).

    ### Answer: \(\frac{124609}{53361}\)

Expected answer: 124609/53361

Correct? Yes

🔗 Problem

Problem: A robot is performing a maintenance task. It starts with 25/10 units of energy. However, it quickly uses 27/23 units of energy to complete a series of complex computations. Calculate how much energy it has left.

Gemma 2B:

    To determine how much energy the robot has left, we need to subtract the energy used from the initial energy.

    First, let's convert the fractions to a common denominator to make the subtraction easier. The initial energy is \( \frac{25}{10} \) and the energy used is \( \frac{27}{23} \).

    1. Simplify \( \frac{25}{10} \):
    \[ \frac{25}{10} = 2.5 \]

    2. Convert \( 2.5 \) to a fraction with the same denominator as \( \frac{27}{23} \):
    \[ 2.5 = \frac{2.5 \times 23}{1 \times 23} = \frac{57.5}{23} \]

    3. Now, subtract \( \frac{27}{23} \) from \( \frac{57.5}{23} \):
    \[ \frac{57.5}{23} - \frac{27}{23} = \frac{57.5 - 27}{23} = \frac{30.5}{23} \]

    So, the robot has \( \frac{30.5}{23} \) units of energy left.

    ### Answer: \(\frac{30.5}{23}\) units of energy.

Expected answer: 61/46

Correct? No

🔗 Problem

Problem: During rainy weather, a small snail moves at a snail-pace covering 17/21 of a meter in an hour. Another slightly faster snail covers 20/24 of a meter in the same time during such weather. One rainy day, the first snail started an hour earlier than the second. At the time when the second snail began, a third snail that covers 18/17 of a meter per hour also started moving. What is their total distance in meters 1 hour after all three have started moving? Compute the result using the operation ((17 / 21) + (20 / 24)) for the first two snails' distance, and multiply this result by (18 / 17) for the total distance , including the third snail.

Gemma 2B:

    To solve this problem, we need to calculate the distance covered by each snail and then sum these distances.

    First, let's simplify the distances covered by the first two snails in one hour.

    1. The first snail covers \( \frac{17}{21} \) meters in one hour.
    2. The second snail covers \( \frac{20}{24} \) meters in one hour. Simplifying \( \frac{20}{24} \):
       \[
       \frac{20}{24} = \frac{5}{6}
       \]

    Now, we add the distances covered by the first two snails:
    \[
    \frac{17}{21} + \frac{5}{6}
    \]

    To add these fractions, we need a common denominator. The least common multiple of 21 and 6 is 42.

    Convert each fraction to have a denominator of 42:
    \[
    \frac{17}{21} = \frac{17 \times 2}{21 \times 2} = \frac{34}{42}
    \]
    \[
    \frac{5}{6} = \frac{5 \times 7}{6 \times 7} = \frac{35}{42}
    \]

    Now add the fractions:
    \[
    \frac{34}{42} + \frac{35}{42} = \frac{34 + 35}{42} = \frac{69}{42}
    \]

    Next, we need to consider the third snail, which covers \( \frac{18}{17} \) meters in one hour.

    The total distance covered by all three snails in one hour is:
    \[
    \left( \frac{69}{42} \right) + \left( \frac{18}{17} \right)
    \]

    To find the total distance, we need to add these two fractions. First, we need a common denominator for \( \frac{69}{42} \) and \( \frac{18}{17} \). The least common multiple of 42 and 17 is 714.

    Convert each fraction to have a denominator of 714:
    \[
    \frac{69}{42} = \frac{69 \times 17}{42 \times 17} = \frac{1173}{714}
    \]
    \[
    \frac{18}{17} = \frac{18 \times 42}{17 \times 42} = \frac{756}{714}
    \]

    Now add the fractions:
    \[
    \frac{1173}{714} + \frac{756}{714} = \frac{1173 + 756}{714} = \frac{1929}{714}
    \]

    Simplify the fraction:
    \[
    \frac{1929}{714} = \frac{1929 \div 3}{714 \div 3} = \frac{643}{238}
    \]

    ### Answer: \(\frac{643}{238}\)

Expected answer: 207/119

Correct? No

🔗 Problem

Problem: Johnny had 24/14 of a yard of fabric that he used to make stuffed animals. After creating a few, he found he only had 9/29 of a yard left. He then got a new type of fabric and used 15/2 times the difference of the original and remaining fabric to make more stuffed animals. How many yards of the new fabric did Johnny use?

Gemma 2B:

    First, let's determine the difference between the original amount of fabric and the remaining fabric.

    Original fabric: \(\frac{24}{14}\) yards
    Remaining fabric: \(\frac{9}{29}\) yards

    To find the difference, we need a common denominator. The least common multiple of 14 and 29 is 406.

    Convert \(\frac{24}{14}\) to a fraction with a denominator of 406:
    \[
    \frac{24}{14} = \frac{24 \times 29}{14 \times 29} = \frac{696}{406}
    \]

    Convert \(\frac{9}{29}\) to a fraction with a denominator of 406:
    \[
    \frac{9}{29} = \frac{9 \times 14}{29 \times 14} = \frac{126}{406}
    \]

    Now, subtract the remaining fabric from the original fabric:
    \[
    \frac{696}{406} - \frac{126}{406} = \frac{696 - 126}{406} = \frac{570}{406}
    \]

    Simplify \(\frac{570}{406}\):
    \[
    \frac{570}{406} = \frac{285}{203}
    \]

    Next, Johnny used \(\frac{15}{2}\) times this difference to make more stuffed animals. Multiply \(\frac{15}{2}\) by \(\frac{285}{203}\):
    \[
    \frac{15}{2} \times \frac{285}{203} = \frac{15 \times 285}{2 \times 203} = \frac{4275}{406}
    \]

    Simplify \(\frac{4275}{406}\):
    \[
    \frac{4275}{406} = \frac{4275 \div 5}{406 \div 5} = \frac{855}{81.2} = \frac{855 \div 5}{81.2 \div 5} = \frac{171}{16.24}
    \]

    Thus, Johnny used \(\frac{171}{16.24}\) yards of the new fabric.

    ### Answer: \(\frac{171}{16.24}\) yards

Expected answer: 4275/406

Correct? Yes

🔗 Follow-up

Problem: Now let's consider that Johnny had originally miscalculated the amount of fabric he had left after creating the first few stuffed animals. Instead of having 9/29 of a yard left, he actually had 8/29 of a yard left. Keeping the calculation for the new fabric the same, i.e., using 15/2 times the difference of the original and the remaining fabric to make more stuffed animals, how many yards of the new fabric did Johnny use this time?

Gemma 2B:

    First, let's determine the difference between the original amount of fabric and the corrected remaining fabric.

    Original fabric: \(\frac{24}{14}\) yards
    Corrected remaining fabric: \(\frac{8}{29}\) yards

    To find the difference, we need a common denominator. The least common multiple of 14 and 29 is 406.

    Convert \(\frac{24}{14}\) to a fraction with a denominator of 406:
    \[
    \frac{24}{14} = \frac{24 \times 29}{14 \times 29} = \frac{696}{406}
    \]

    Convert \(\frac{8}{29}\) to a fraction with a denominator of 406:
    \[
    \frac{8}{29} = \frac{8 \times 14}{29 \times 14} = \frac{112}{406}
    \]

    Now, subtract the corrected remaining fabric from the original fabric:
    \[
    \frac{696}{406} - \frac{112}{406} = \frac{696 - 112}{406} = \frac{584}{406}
    \]

    Simplify \(\frac{584}{406}\):
    \[
    \frac{584}{406} = \frac{292}{203}
    \]

    Next, Johnny used \(\frac{15}{2}\) times this difference to make more stuffed animals. Multiply \(\frac{15}{2}\) by \(\frac{292}{203}\):
    \[
    \frac{15}{2} \times \frac{292}{203} = \frac{15 \times 292}{2 \times 203} = \frac{4380}{406}
    \]

    Simplify \(\frac{4380}{406}\):
    \[
    \frac{4380}{406} = \frac{2190}{203}
    \]

    Thus, Johnny used \(\frac{2190}{203}\) yards of the new fabric.

    ### Answer: \(\frac{2190}{203}\) yards

Expected answer: 2190/203

Correct? Yes

🔗 Problem

Problem: In a park, there are some kids lining up for the swings. Part of the waiting line consists of 2/10 of the total number of kids in the park, while another part is made up of 21/4 kids. What is the total number of kids waiting for the swing? Add together 2/10 and 21/4 to find your answer.

Gemma 2B:

    To find the total number of kids waiting for the swings, we need to add the fractions \( \frac{2}{10} \) and \( \frac{21}{4} \).

    First, let's simplify \( \frac{2}{10} \):
    \[ \frac{2}{10} = \frac{1}{5} \]

    Next, we need a common denominator to add \( \frac{1}{5} \) and \( \frac{21}{4} \). The least common multiple of 5 and 4 is 20.

    Convert \( \frac{1}{5} \) to a fraction with a denominator of 20:
    \[ \frac{1}{5} = \frac{1 \times 4}{5 \times 4} = \frac{4}{20} \]

    Convert \( \frac{21}{4} \) to a fraction with a denominator of 20:
    \[ \frac{21}{4} = \frac{21 \times 5}{4 \times 5} = \frac{105}{20} \]

    Now, add the two fractions:
    \[ \frac{4}{20} + \frac{105}{20} = \frac{4 + 105}{20} = \frac{109}{20} \]

    So, the total number of kids waiting for the swings is \( \frac{109}{20} \).

    ### Answer: \(\frac{109}{20}\)

Expected answer: 109/20

Correct? Yes

🔗 Follow-up

Problem: Now consider that the number of children forming a part of the waiting line is not 2/10 but 2/27 of the total number of children in the park. The other part of the line is still made up of 21/4 children. What is the new total number of children waiting for the swing? To find out, add together 2/27 and 21/4.

Gemma 2B:

    To find the new total number of children waiting for the swings, we need to add the fractions \( \frac{2}{27} \) and \( \frac{21}{4} \).

    First, we need a common denominator to add \( \frac{2}{27} \) and \( \frac{21}{4} \). The least common multiple of 27 and 4 is 108.

    Convert \( \frac{2}{27} \) to a fraction with a denominator of 108:
    \[ \frac{2}{27} = \frac{2 \times 4}{27 \times 4} = \frac{8}{108} \]

    Convert \( \frac{21}{4} \) to a fraction with a denominator of 108:
    \[ \frac{21}{4} = \frac{21 \times 27}{4 \times 27} = \frac{567}{108} \]

    Now, add the two fractions:
    \[ \frac{8}{108} + \frac{567}{108} = \frac{8 + 567}{108} = \frac{575}{108} \]

    So, the new total number of children waiting for the swings is \( \frac{575}{108} \).

    ### Answer: \(\frac{575}{108}\)

Expected answer: 575/108

Correct? Yes

🔗 Problem

Problem: An alien spacecraft has 18 fuel tanks out of a total of 30 fuel tanks filled. Each fuel tank can power the spacecraft for 22/6 hours. How many hours of power does the spacecraft currently have? Calculate (18 / 30) * (22 / 6) to find out.

Gemma 2B:

    To determine how many hours of power the spacecraft currently has, we need to calculate \(\frac{18}{30} \times \frac{22}{6}\).

    First, simplify \(\frac{18}{30}\):
    \[
    \frac{18}{30} = \frac{3 \times 6}{5 \times 6} = \frac{3}{5}
    \]

    Next, multiply \(\frac{3}{5}\) by \(\frac{22}{6}\):
    \[
    \frac{3}{5} \times \frac{22}{6} = \frac{3 \times 22}{5 \times 6} = \frac{66}{30}
    \]

    Now, simplify \(\frac{66}{30}\):
    \[
    \frac{66}{30} = \frac{33 \times 2}{15 \times 2} = \frac{33}{15} = \frac{11 \times 3}{5 \times 3} = \frac{11}{5}
    \]

    Convert \(\frac{11}{5}\) to a decimal:
    \[
    \frac{11}{5} = 2.2
    \]

    Thus, the spacecraft currently has 2.2 hours of power.

    ### Answer: 2.2

Expected answer: 11/5

Correct? Yes

🔗 Follow-up

Problem: In the previous problem, we found out how many hours of power the alien spacecraft has when there were 18 fuel tanks filled out of 30. Now suppose a mechanism cleaned the fuel tanks in the spacecraft, enabling 18 out of 5 tanks to be filled instead. If each fuel tank can still power the spacecraft for 22/6 hours, how many hours of power does the spacecraft currently have? Just as before, calculate ((18 / 5) * (22 / 6)) to find out.

Gemma 2B:

    To determine how many hours of power the spacecraft currently has with 18 out of 5 tanks filled, we need to calculate \(\frac{18}{5} \times \frac{22}{6}\).

    First, simplify \(\frac{22}{6}\):
    \[
    \frac{22}{6} = \frac{11 \times 2}{3 \times 2} = \frac{11}{3}
    \]

    Next, multiply \(\frac{18}{5}\) by \(\frac{11}{3}\):
    \[
    \frac{18}{5} \times \frac{11}{3} = \frac{18 \times 11}{5 \times 3} = \frac{198}{15}
    \]

    Now, simplify \(\frac{198}{15}\):
    \[
    \frac{198}{15} = \frac{66 \times 3}{5 \times 3} = \frac{66}{5}
    \]

    Convert \(\frac{66}{5}\) to a decimal:
    \[
    \frac{66}{5} = 13.2
    \]

    Thus, the spacecraft currently has 13.2 hours of power.

    ### Answer: 13.2

Expected answer: 66/5

Correct? No

🔗 Problem

Problem: A cat is fed twice a day. In the morning, it eats 16/12 of a can of cat food. in the evening, it eats 6/10 of a can. How much of a can in total does the cat eat in one day?

Gemma 2B:

    To find out how much of a can the cat eats in one day, we need to add the amount it eats in the morning to the amount it eats in the evening.

    First, let's simplify the fractions if possible:

    Morning: \( \frac{16}{12} \)
    Evening: \( \frac{6}{10} \)

    Simplify \( \frac{16}{12} \):
    \[ \frac{16}{12} = \frac{4 \times 4}{4 \times 3} = \frac{4}{3} \]

    Simplify \( \frac{6}{10} \):
    \[ \frac{6}{10} = \frac{3 \times 2}{5 \times 2} = \frac{3}{5} \]

    Now, we need to add \( \frac{4}{3} \) and \( \frac{3}{5} \). To do this, we need a common denominator. The least common multiple of 3 and 5 is 15.

    Convert \( \frac{4}{3} \) to a fraction with a denominator of 15:
    \[ \frac{4}{3} = \frac{4 \times 5}{3 \times 5} = \frac{20}{15} \]

    Convert \( \frac{3}{5} \) to a fraction with a denominator of 15:
    \[ \frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15} \]

    Now add the two fractions:
    \[ \frac{20}{15} + \frac{9}{15} = \frac{20 + 9}{15} = \frac{29}{15} \]

    So, the cat eats \( \frac{29}{15} \) of a can of cat food in one day.

    ### Answer: \(\frac{29}{15}\)

Expected answer: 29/15

Correct? Yes

🔗 Follow-up

Problem: Continuing from the original problem, suppose the morning serving size changes and the cat now eats 16/21 of a can of cat food in the morning rather than the original 16/12 of a can. Maintaining the same evening portion of 6/10 of a can, how much does the cat eat in total in one day with the new morning serving size?

Gemma 2B:

    To determine the total amount of cat food the cat eats in one day with the new morning serving size, we need to add the new morning portion to the evening portion.

    The new morning portion is \( \frac{16}{21} \) of a can.
    The evening portion remains \( \frac{6}{10} \) of a can.

    First, let's simplify \( \frac{6}{10} \):
    \[ \frac{6}{10} = \frac{3}{5} \]

    Next, we need to add \( \frac{16}{21} \) and \( \frac{3}{5} \). To do this, we need a common denominator. The least common multiple of 21 and 5 is 105.

    Convert \( \frac{16}{21} \) to a fraction with a denominator of 105:
    \[ \frac{16}{21} = \frac{16 \times 5}{21 \times 5} = \frac{80}{105} \]

    Convert \( \frac{3}{5} \) to a fraction with a denominator of 105:
    \[ \frac{3}{5} = \frac{3 \times 21}{5 \times 21} = \frac{63}{105} \]

    Now add the two fractions:
    \[ \frac{80}{105} + \frac{63}{105} = \frac{80 + 63}{105} = \frac{143}{105} \]

    So, the cat eats \( \frac{143}{105} \) of a can of cat food in one day with the new morning serving size.

    ### Answer: \(\frac{143}{105}\)

Expected answer: 143/105

Correct? Yes

🔗 Problem

Problem: A starfish laid 12 eggs in 3 nests and 11 eggs in 4 other nests. How much is the difference between the average number of eggs per nest for these two groups of nests?

Gemma 2B:

    First, let's calculate the average number of eggs per nest for the first group of nests.

    The first group has 3 nests with 12 eggs in each nest:
    \[ \text{Total eggs in the first group} = 3 \times 12 = 36 \]
    \[ \text{Average number of eggs per nest in the first group} = \frac{36}{3} = 12 \]

    Next, let's calculate the average number of eggs per nest for the second group of nests.

    The second group has 4 nests with 11 eggs in each nest:
    \[ \text{Total eggs in the second group} = 4 \times 11 = 44 \]
    \[ \text{Average number of eggs per nest in the second group} = \frac{44}{4} = 11 \]

    Now, we find the difference between the average number of eggs per nest for these two groups:
    \[ \text{Difference} = 12 - 11 = 1 \]

    ### Answer: 1

Expected answer: 5/4

Correct? No

🔗 Problem

Problem: A monkey finds a pile of bananas and ends up eating (22 / 7) times (10 / 25) of the pile. Later, he gives (3 / 28) of the pile to his friend. How much of the pile does the monkey have left?

Gemma 2B:

    To determine how much of the pile the monkey has left, we need to calculate the fractions of the pile that the monkey ate and gave away, and then subtract these from the whole pile.

    First, let's calculate the fraction of the pile the monkey ate:
    \[
    \frac{22}{7} \times \frac{10}{25}
    \]

    We can simplify this multiplication by simplifying the fractions first:
    \[
    \frac{22}{7} \times \frac{10}{25} = \frac{22 \times 10}{7 \times 25} = \frac{220}{175}
    \]

    Next, we simplify \(\frac{220}{175}\):
    \[
    \frac{220}{175} = \frac{220 \div 5}{175 \div 5} = \frac{44}{35}
    \]

    So, the monkey ate \(\frac{44}{35}\) of the pile.

    Next, let's calculate the fraction of the pile the monkey gave to his friend:
    \[
    \frac{3}{28}
    \]

    Now, we need to find the total fraction of the pile that the monkey has given away:
    \[
    \frac{44}{35} + \frac{3}{28}
    \]

    To add these fractions, we need a common denominator. The least common multiple of 35 and 28 is 140. We convert each fraction to have this common denominator:
    \[
    \frac{44}{35} = \frac{44 \times 4}{35 \times 4} = \frac{176}{140}
    \]
    \[
    \frac{3}{28} = \frac{3 \times 5}{28 \times 5} = \frac{15}{140}
    \]

    Now, we add the fractions:
    \[
    \frac{176}{140} + \frac{15}{140} = \frac{176 + 15}{140} = \frac{191}{140}
    \]

    This is the total fraction of the pile that the monkey has eaten and given away. To find out how much of the pile the monkey has left, we subtract this from the whole pile (which is 1):
    \[
    1 - \frac{191}{140}
    \]

    We convert 1 to a fraction with the same denominator:
    \[
    1 = \frac{140}{140}
    \]

    Now, we perform the subtraction:
    \[
    \frac{140}{140} - \frac{191}{140} = \frac{140 - 191}{140} = \frac{-51}{140}
    \]

    Since a negative fraction doesn't make sense in this context, it indicates that the monkey has eaten and given away more than the whole pile, which suggests an error in the problem setup or interpretation. However, based on the calculations, the monkey has effectively consumed and given away more than the entire pile.

    ### Answer: \(\frac{-51}{140}\)

Expected answer: 23/20

Correct? No

🔗 Problem

Problem: A mouse runs at a speed of 24/20 feet per second. If the mouse runs for 23/2 seconds, how many feet does the mouse run?

Gemma 2B:

    To determine how many feet the mouse runs, we need to multiply the speed of the mouse by the time it runs.

    The speed of the mouse is given as \( \frac{24}{20} \) feet per second. Simplifying this fraction, we get:
    \[ \frac{24}{20} = \frac{6}{5} \text{ feet per second} \]

    The time the mouse runs is given as \( \frac{23}{2} \) seconds.

    Now, we multiply the speed by the time:
    \[ \text{Distance} = \left( \frac{6}{5} \text{ feet per second} \right) \times \left( \frac{23}{2} \text{ seconds} \right) \]

    To multiply these fractions, we multiply the numerators together and the denominators together:
    \[ \text{Distance} = \frac{6 \times 23}{5 \times 2} = \frac{138}{10} \]

    Simplifying the fraction \( \frac{138}{10} \):
    \[ \frac{138}{10} = 13.8 \text{ feet} \]

    ### Answer: 13.8 feet

Expected answer: 69/5

Correct? No

🔗 Problem

Problem: At a party, the drinks are served in glasses that contain 29/9 ounces of beverage each. If 26/25 of them were consumed, how many ounces of beverage were consumed? Now imagine that the party goers then drink an additional 2/19 ounces of beverages. How many total ounces of beverages were consumed at the party in total?

Gemma 2B:

    First, we need to determine how many ounces of beverage were consumed initially. Each glass contains \( \frac{29}{9} \) ounces, and \( \frac{26}{25} \) of these glasses were consumed.

    To find the total ounces consumed initially, we multiply:
    \[
    \frac{29}{9} \times \frac{26}{25} = \frac{29 \times 26}{9 \times 25} = \frac{754}{225}
    \]

    Next, we need to add the additional \( \frac{2}{19} \) ounces consumed later:
    \[
    \frac{754}{225} + \frac{2}{19}
    \]

    To add these fractions, we need a common denominator. The least common multiple of 225 and 19 is 4275. We convert each fraction to have this common denominator:
    \[
    \frac{754}{225} = \frac{754 \times 19}{225 \times 19} = \frac{14326}{4275}
    \]
    \[
    \frac{2}{19} = \frac{2 \times 225}{19 \times 225} = \frac{450}{4275}
    \]

    Now, we add the fractions:
    \[
    \frac{14326}{4275} + \frac{450}{4275} = \frac{14326 + 450}{4275} = \frac{14776}{4275}
    \]

    Thus, the total ounces of beverages consumed at the party is:
    \[
    \frac{14776}{4275}
    \]

    ### Answer: \(\frac{14776}{4275}\) ounces

Expected answer: 14776/4275

Correct? Yes

🔗 Follow-up

Problem: Revisiting the previous party scenario, suppose the glasses contain 29/9 ounces of beverage each and 26/25 of them were consumed just like before. However, this time around, the party goers decide to drink an additional 4/19 ounces of beverages instead of 2/19 ounces. How many total ounces of beverages were consumed at the party now?

Gemma 2B:

    First, we need to determine how many ounces of beverage were consumed initially. Each glass contains \( \frac{29}{9} \) ounces, and \( \frac{26}{25} \) of these glasses were consumed.

    To find the total ounces consumed initially, we multiply:
    \[
    \frac{29}{9} \times \frac{26}{25} = \frac{29 \times 26}{9 \times 25} = \frac{754}{225}
    \]

    Next, we need to add the additional \( \frac{4}{19} \) ounces consumed later:
    \[
    \frac{754}{225} + \frac{4}{19}
    \]

    To add these fractions, we need a common denominator. The least common multiple of 225 and 19 is 4275. We convert each fraction to have this common denominator:
    \[
    \frac{754}{225} = \frac{754 \times 19}{225 \times 19} = \frac{14326}{4275}
    \]
    \[
    \frac{4}{19} = \frac{4 \times 225}{19 \times 225} = \frac{900}{4275}
    \]

    Now, we add the fractions:
    \[
    \frac{14326}{4275} + \frac{900}{4275} = \frac{14326 + 900}{4275} = \frac{15226}{4275}
    \]

    Thus, the total ounces of beverages consumed at the party is:
    \[
    \frac{15226}{4275}
    \]

    ### Answer: \(\frac{15226}{4275}\) ounces

Expected answer: 15226/4275

Correct? Yes

🔗 Problem

Problem: You decide to start a toothbrush collection and you add 7/15 of a new set of toothbrushes to your current collection. Suddenly, a friend gifts you 9 more sets. After grouping everything, you realized you can only fit 26/27 of them in your display case. How many sets of toothbrushes can you fit in your display case?

Gemma 2B:

    Let's break down the problem step by step.

    1. **Initial Addition to Collection:**
       You add \( \frac{7}{15} \) of a new set of toothbrushes to your current collection.

    2. **Gift from Friend:**
       Your friend gifts you 9 more sets of toothbrushes.

    3. **Total Sets Before Display Case Limitation:**
       The total number of sets you have is:
       \[
       \frac{7}{15} + 9
       \]
       To add these, we need a common denominator. The fraction \( \frac{7}{15} \) can be converted to a fraction with a denominator of 15:
       \[
       9 = \frac{135}{15}
       \]
       So, adding these together:
       \[
       \frac{7}{15} + \frac{135}{15} = \frac{142}{15}
       \]

    4. **Fitting in the Display Case:**
       You can fit \( \frac{26}{27} \) of the total sets in your display case. Therefore, the number of sets you can fit is:
       \[
       \frac{26}{27} \times \frac{142}{15}
       \]
       Multiply the numerators and the denominators:
       \[
       \frac{26 \times 142}{27 \times 15} = \frac{3692}{405}
       \]

    5. **Simplifying the Fraction:**
       To simplify \( \frac{3692}{405} \), we find the greatest common divisor (GCD) of 3692 and 405. The GCD is 1, so the fraction is already in its simplest form.

    6. **Converting to a Mixed Number:**
       Divide 3692 by 405:
       \[
       3692 \div 405 \approx 9.11
       \]
       Since we are dealing with sets of toothbrushes, we consider the integer part of the division:
       \[
       9 \text{ sets}
       \]

    ### Answer: 9

Expected answer: 3692/405

Correct? No

🔗 Problem

Problem: Joan has a backpack which weighs 27/20 kilograms when it's empty. She adds a book which weighs 1/6 kilograms. Calculate the total weight of the backpack in kilograms now. Please write the answer as a simplified fraction.

Gemma 2B:

    To find the total weight of the backpack after adding the book, we need to add the weight of the empty backpack to the weight of the book.

    The weight of the empty backpack is \( \frac{27}{20} \) kilograms.
    The weight of the book is \( \frac{1}{6} \) kilograms.

    First, we need a common denominator to add these fractions. The least common multiple of 20 and 6 is 60.

    Convert \( \frac{27}{20} \) to a fraction with a denominator of 60:
    \[ \frac{27}{20} = \frac{27 \times 3}{20 \times 3} = \frac{81}{60} \]

    Convert \( \frac{1}{6} \) to a fraction with a denominator of 60:
    \[ \frac{1}{6} = \frac{1 \times 10}{6 \times 10} = \frac{10}{60} \]

    Now, add the two fractions:
    \[ \frac{81}{60} + \frac{10}{60} = \frac{81 + 10}{60} = \frac{91}{60} \]

    The total weight of the backpack is \( \frac{91}{60} \) kilograms.

    ### Answer: \(\frac{91}{60}\)

Expected answer: 91/60

Correct? Yes

🔗 Follow-up

Problem: Let's revise our original situation. This time, Joan's backpack weighs 27/20 kilograms when it's empty. She then adds a heavier book which weighs 21/6 kilograms. What's the total weight of her backpack now? Please write your answer as a simplified fraction.

Gemma 2B:

    To find the total weight of the backpack after adding the heavier book, we need to add the weight of the empty backpack to the weight of the book.

    The weight of the empty backpack is \( \frac{27}{20} \) kilograms.
    The weight of the book is \( \frac{21}{6} \) kilograms.

    First, we need a common denominator to add these fractions. The least common multiple of 20 and 6 is 60.

    Convert \( \frac{27}{20} \) to a fraction with a denominator of 60:
    \[ \frac{27}{20} = \frac{27 \times 3}{20 \times 3} = \frac{81}{60} \]

    Convert \( \frac{21}{6} \) to a fraction with a denominator of 60:
    \[ \frac{21}{6} = \frac{21 \times 10}{6 \times 10} = \frac{210}{60} \]

    Now, add the two fractions:
    \[ \frac{81}{60} + \frac{210}{60} = \frac{81 + 210}{60} = \frac{291}{60} \]

    Next, we simplify the fraction \( \frac{291}{60} \). The greatest common divisor (GCD) of 291 and 60 is 3.

    Divide both the numerator and the denominator by their GCD:
    \[ \frac{291 \div 3}{60 \div 3} = \frac{97}{20} \]

    The total weight of the backpack is \( \frac{97}{20} \) kilograms.

    ### Answer: \(\frac{97}{20}\)

Expected answer: 97/20

Correct? Yes

🔗 Problem

Problem: A soda company is changing its bottle design. The new design will change the volume of the bottles so that they become (7 / 7) * (25 / 22) times larger. What will be the multiplier of the new bottle compared to the old one?

Gemma 2B:

    To determine the multiplier of the new bottle compared to the old one, we need to calculate the product of the given fractions.

    First, simplify the fraction \(\frac{7}{7}\):
    \[
    \frac{7}{7} = 1
    \]

    Next, multiply this result by the second fraction \(\frac{25}{22}\):
    \[
    1 \times \frac{25}{22} = \frac{25}{22}
    \]

    Thus, the multiplier of the new bottle compared to the old one is \(\frac{25}{22}\).

    ### Answer: \(\frac{25}{22}\)

Expected answer: 25/22

Correct? Yes

🔗 Follow-up

Problem: The soda company decided to further increase the size of the new bottle design. Now the bottles will be enlarged by a multiplier of (28 / 13) times the previous size change. What will be the new multiplier of the updated bottle compared to the original one?

Gemma 2B:

    To find the new multiplier of the updated bottle compared to the original one, we need to multiply the previous multiplier \(\frac{25}{22}\) by the new multiplier \(\frac{28}{13}\).

    First, we perform the multiplication of the two fractions:
    \[
    \frac{25}{22} \times \frac{28}{13} = \frac{25 \times 28}{22 \times 13}
    \]

    Next, we calculate the numerator and the denominator:
    \[
    25 \times 28 = 700
    \]
    \[
    22 \times 13 = 286
    \]

    So, the new multiplier is:
    \[
    \frac{700}{286}
    \]

    To simplify this fraction, we find the greatest common divisor (GCD) of 700 and 286. The GCD of 700 and 286 is 14.

    Now, we divide both the numerator and the denominator by their GCD:
    \[
    \frac{700 \div 14}{286 \div 14} = \frac{50}{20.42857142857143}
    \]

    Thus, the simplified fraction is:
    \[
    \frac{50}{20.42857142857143}
    \]

    ### Answer: \(\frac{50}{20.42857142857143}\)

Expected answer: 350/143

Correct? Yes

🔗 Follow-up

Problem: The soda company revises their decision and decides to slightly reduce the increase in the size of the new bottle design. Now, the bottles will be (7 / 7) * (25 / 12) times larger than the original design. What will be the new multiplier for the revised bottle compared to the origianl one?

Gemma 2B:

    To determine the new multiplier for the revised bottle compared to the original one, we need to calculate the product of the given fractions.

    First, simplify the fraction \(\frac{7}{7}\):
    \[
    \frac{7}{7} = 1
    \]

    Next, multiply this result by the second fraction \(\frac{25}{12}\):
    \[
    1 \times \frac{25}{12} = \frac{25}{12}
    \]

    Thus, the new multiplier for the revised bottle compared to the original one is \(\frac{25}{12}\).

    ### Answer: \(\frac{25}{12}\)

Expected answer: 25/12

Correct? Yes

🔗 Problem

Problem: Turtle Tilly is collecting food for the winter. She has gathered 25/9 kg of sea plants, and she then finds 8/13 kg more. In a close-by rock, she spots another food stash weighing 24/27 kg. After gobbling up these finds, Tilly stumbles upon yet another food source weighing 21/26 kg. If Tilly combines all her found food, how many kilograms does she now have in total? Answer as a simplified fraction.

Gemma 2B:

    To find the total amount of food Turtle Tilly has gathered, we need to add all the fractions together. The fractions are:

    \[
    \frac{25}{9}, \frac{8}{13}, \frac{24}{27}, \frac{21}{26}
    \]

    First, we need to find a common denominator for these fractions. The denominators are 9, 13, 27, and 26. The least common multiple (LCM) of these numbers is 6318.

    Next, we convert each fraction to have the common denominator of 6318:

    1. \(\frac{25}{9}\):
    \[
    \frac{25}{9} = \frac{25 \times 702}{9 \times 702} = \frac{17550}{6318}
    \]

    2. \(\frac{8}{13}\):
    \[
    \frac{8}{13} = \frac{8 \times 486}{13 \times 486} = \frac{3888}{6318}
    \]

    3. \(\frac{24}{27}\):
    \[
    \frac{24}{27} = \frac{24 \times 234}{27 \times 234} = \frac{5616}{6318}
    \]

    4. \(\frac{21}{26}\):
    \[
    \frac{21}{26} = \frac{21 \times 243}{26 \times 243} = \frac{5103}{6318}
    \]

    Now, we add these fractions together:

    \[
    \frac{17550}{6318} + \frac{3888}{6318} + \frac{5616}{6318} + \frac{5103}{6318} = \frac{17550 + 3888 + 5616 + 5103}{6318} = \frac{32157}{6318}
    \]

    To simplify \(\frac{32157}{6318}\), we find the greatest common divisor (GCD) of 32157 and 6318. The GCD is 3.

    \[
    \frac{32157 \div 3}{6318 \div 3} = \frac{10719}{2106}
    \]

    We simplify further by finding the GCD of 10719 and 2106, which is 3.

    \[
    \frac{10719 \div 3}{2106 \div 3} = \frac{3573}{702}
    \]

    We simplify further by finding the GCD of 3573 and 702, which is 3.

    \[
    \frac{3573 \div 3}{702 \div 3} = \frac{1191}{234}
    \]

    We simplify further by finding the GCD of 1191 and 234, which is 3.

    \[
    \frac{1191 \div 3}{234 \div 3} = \frac{397}{78}
    \]

    Thus, the simplified fraction is:

    ### Answer: \(\frac{397}{78}\)

Expected answer: 397/78

Correct? Yes

🔗 Follow-up

Problem: In the previous problem, Turtle Tilly initially found a food source weighing 21/26 kg. However, if Tilly instead stumbled upon a food source weighing 29/26 kg, how many kilograms of food would she have in total? Answer as a simplified fraction.

Gemma 2B:

    To find the total amount of food Turtle Tilly has gathered with the new food source, we need to add the fractions together. The fractions are now:

    \[
    \frac{25}{9}, \frac{8}{13}, \frac{24}{27}, \frac{29}{26}
    \]

    First, we need to find a common denominator for these fractions. The denominators are 9, 13, 27, and 26. The least common multiple (LCM) of these numbers is 6318.

    Next, we convert each fraction to have the common denominator of 6318:

    1. \(\frac{25}{9}\):
    \[
    \frac{25}{9} = \frac{25 \times 702}{9 \times 702} = \frac{17550}{6318}
    \]

    2. \(\frac{8}{13}\):
    \[
    \frac{8}{13} = \frac{8 \times 486}{13 \times 486} = \frac{3888}{6318}
    \]

    3. \(\frac{24}{27}\):
    \[
    \frac{24}{27} = \frac{24 \times 234}{27 \times 234} = \frac{5616}{6318}
    \]

    4. \(\frac{29}{26}\):
    \[
    \frac{29}{26} = \frac{29 \times 243}{26 \times 243} = \frac{7047}{6318}
    \]

    Now, we add these fractions together:

    \[
    \frac{17550}{6318} + \frac{3888}{6318} + \frac{5616}{6318} + \frac{7047}{6318} = \frac{17550 + 3888 + 5616 + 7047}{6318} = \frac{34101}{6318}
    \]

    To simplify \(\frac{34101}{6318}\), we find the greatest common divisor (GCD) of 34101 and 6318. The GCD is 3.

    \[
    \frac{34101 \div 3}{6318 \div 3} = \frac{11367}{2106}
    \]

    We simplify further by finding the GCD of 11367 and 2106, which is 3.

    \[
    \frac{11367 \div 3}{2106 \div 3} = \frac{3789}{702}
    \]

    We simplify further by finding the GCD of 3789 and 702, which is 3.

    \[
    \frac{3789 \div 3}{702 \div 3} = \frac{1263}{234}
    \]

    We simplify further by finding the GCD of 1263 and 234, which is 3.

    \[
    \frac{1263 \div 3}{234 \div 3} = \frac{421}{78}
    \]

    Thus, the simplified fraction is:

    ### Answer: \(\frac{421}{78}\)

Expected answer: 421/78

Correct? Yes

🔗 Problem

Problem: Captain Alf undertakes multi-step calculations to ensure the safe navigation of his boat. First, he calculates the difference in his current location and the distance of a nearby island. The distance to the island is 17/2 nautical miles, and his boat is currently 4/2 nautical miles away from the island. Next, Captain Alf adjusts for the tide, which he has estimated to push his boat off course by 14/9 nautical miles. He adds this to his previous calculations. Finally, when considering his fuel efficiency rated at 22/15 nautical miles per gallon, Captain Alf multiplies his adjusted distance by this rate to figure out his fuel requirements. What does Captain Alf’s final calculation come to? Use these steps and corresponding fractions to find your answer.

Gemma 2B:

    Let's break down the problem step by step.

    1. **Calculate the difference in distance to the island:**
       - Distance to the island: \( \frac{17}{2} \) nautical miles
       - Current distance from the island: \( \frac{4}{2} \) nautical miles

       The difference in distance is:
       \[
       \frac{17}{2} - \frac{4}{2} = \frac{13}{2} \text{ nautical miles}
       \]

    2. **Adjust for the tide:**
       - Tide adjustment: \( \frac{14}{9} \) nautical miles

       Adding this to the previous distance:
       \[
       \frac{13}{2} + \frac{14}{9}
       \]

       To add these fractions, we need a common denominator. The least common multiple of 2 and 9 is 18.

       Convert each fraction:
       \[
       \frac{13}{2} = \frac{13 \times 9}{2 \times 9} = \frac{117}{18}
       \]
       \[
       \frac{14}{9} = \frac{14 \times 2}{9 \times 2} = \frac{28}{18}
       \]

       Now add the fractions:
       \[
       \frac{117}{18} + \frac{28}{18} = \frac{145}{18} \text{ nautical miles}
       \]

    3. **Calculate fuel requirements:**
       - Fuel efficiency: \( \frac{22}{15} \) nautical miles per gallon

       Multiply the adjusted distance by the fuel efficiency rate:
       \[
       \frac{145}{18} \times \frac{15}{22}
       \]

       Simplify the multiplication:
       \[
       \frac{145 \times 15}{18 \times 22} = \frac{2175}{396}
       \]

       Simplify the fraction by finding the greatest common divisor (GCD) of 2175 and 396. The GCD is 3.

       Divide both the numerator and the denominator by 3:
       \[
       \frac{2175 \div 3}{396 \div 3} = \frac{725}{132}
       \]

       Therefore, Captain Alf’s final calculation for the fuel requirement is:
       \[
       \frac{725}{132} \text{ gallons}
       \]

    ### Answer: \(\frac{725}{132}\) gallons

Expected answer: 319/27

Correct? No

🔗 Problem

Problem: A forest ranger patrolled 14/3 miles of trails in the morning. In the afternoon, he patrolled 2/18 mile less than what he walked in the morning. How many miles did the forest ranger patrol in the afternoon?

Gemma 2B:

    To determine how many miles the forest ranger patrolled in the afternoon, we start by calculating the distance he patrolled in the morning, which is given as \( \frac{14}{3} \) miles.

    Next, we need to find the distance he patrolled in the afternoon, which is \( \frac{2}{18} \) mile less than the morning distance. First, simplify \( \frac{2}{18} \):

    \[
    \frac{2}{18} = \frac{1}{9}
    \]

    Now, subtract \( \frac{1}{9} \) from \( \frac{14}{3} \). To do this, we need a common denominator. The least common multiple of 3 and 9 is 9. Convert \( \frac{14}{3} \) to a fraction with a denominator of 9:

    \[
    \frac{14}{3} = \frac{14 \times 3}{3 \times 3} = \frac{42}{9}
    \]

    Now, subtract \( \frac{1}{9} \) from \( \frac{42}{9} \):

    \[
    \frac{42}{9} - \frac{1}{9} = \frac{42 - 1}{9} = \frac{41}{9}
    \]

    Thus, the forest ranger patrolled \( \frac{41}{9} \) miles in the afternoon.

    ### Answer: \(\frac{41}{9}\) miles

Expected answer: 41/9

Correct? Yes

🔗 Follow-up

Problem: Instead of the 14/3 miles, suppose the forest ranger actually patrolled 29/3 miles of trails in the morning. Considering he covered 2/18 mile less in the afternoon, how many miles did he patrol in the afternoon now?

Gemma 2B:

    To determine how many miles the forest ranger patrolled in the afternoon, we start by calculating the distance he patrolled in the morning, which is given as \( \frac{29}{3} \) miles.

    Next, we need to find the distance he patrolled in the afternoon, which is \( \frac{2}{18} \) mile less than the morning distance. First, simplify \( \frac{2}{18} \):

    \[
    \frac{2}{18} = \frac{1}{9}
    \]

    Now, subtract \( \frac{1}{9} \) from \( \frac{29}{3} \). To do this, we need a common denominator. The least common multiple of 3 and 9 is 9. Convert \( \frac{29}{3} \) to a fraction with a denominator of 9:

    \[
    \frac{29}{3} = \frac{29 \times 3}{3 \times 3} = \frac{87}{9}
    \]

    Now, subtract \( \frac{1}{9} \) from \( \frac{87}{9} \):

    \[
    \frac{87}{9} - \frac{1}{9} = \frac{87 - 1}{9} = \frac{86}{9}
    \]

    Thus, the forest ranger patrolled \( \frac{86}{9} \) miles in the afternoon.

    ### Answer: \(\frac{86}{9}\) miles

Expected answer: 86/9

Correct? Yes

🔗 Problem

Problem: Arthur practices piano for 20/21 of an hour every day. On weekends, he practices an additional 11/29 of an hour per day. How many hours in total does Arthur spend practicing the piano each day on the weekend?

Gemma 2B:

    To find the total time Arthur spends practicing the piano each day on the weekend, we need to add the time he practices daily to the additional time he practices on weekends.

    Arthur practices daily for \( \frac{20}{21} \) of an hour. On weekends, he practices an additional \( \frac{11}{29} \) of an hour.

    First, we need to find a common denominator to add these fractions. The least common multiple (LCM) of 21 and 29 is 609.

    Convert \( \frac{20}{21} \) to a fraction with a denominator of 609:
    \[ \frac{20}{21} = \frac{20 \times 29}{21 \times 29} = \frac{580}{609} \]

    Convert \( \frac{11}{29} \) to a fraction with a denominator of 609:
    \[ \frac{11}{29} = \frac{11 \times 21}{29 \times 21} = \frac{231}{609} \]

    Now, add the two fractions:
    \[ \frac{580}{609} + \frac{231}{609} = \frac{580 + 231}{609} = \frac{811}{609} \]

    Thus, Arthur spends \( \frac{811}{609} \) hours practicing the piano each day on the weekend.

    ### Answer: \( \frac{811}{609} \) hours

Expected answer: 811/609

Correct? Yes

🔗 Follow-up

Problem: In addition to his daily piano practice, Arthur spends an extra 4/20 of an hour on music theory each weekend day. How many hours in total does Arthur spend on piano practice and music theory each weekend day?

Gemma 2B:

    To find the total time Arthur spends on piano practice and music theory each weekend day, we need to add the time he spends on piano practice to the time he spends on music theory.

    From the previous calculation, we know Arthur spends \( \frac{811}{609} \) hours on piano practice each weekend day.

    Arthur also spends \( \frac{4}{20} \) hours on music theory each weekend day. Simplify \( \frac{4}{20} \):
    \[ \frac{4}{20} = \frac{1}{5} \]

    Convert \( \frac{1}{5} \) to a fraction with a denominator of 609:
    \[ \frac{1}{5} = \frac{1 \times 609}{5 \times 609} = \frac{609}{3045} \]

    Now, we need to convert \( \frac{811}{609} \) to a fraction with a denominator of 3045:
    \[ \frac{811}{609} = \frac{811 \times 5}{609 \times 5} = \frac{4055}{3045} \]

    Now, add the two fractions:
    \[ \frac{4055}{3045} + \frac{609}{3045} = \frac{4055 + 609}{3045} = \frac{4664}{3045} \]

    Thus, Arthur spends \( \frac{4664}{3045} \) hours on piano practice and music theory each weekend day.

    ### Answer: \( \frac{4664}{3045} \) hours

Expected answer: 4664/3045

Correct? Yes

🔗 Follow-up

Problem: Suppose that instead of practicing the piano for an additional 11/29 of an hour on the weekends, Arthur changes his routine and now practices an extra 11/24 of an hour. How many hours in total does Arthur spend practicing the piano each day on the weekend now?

Gemma 2B:

    To find the total time Arthur spends practicing the piano each day on the weekend with the new routine, we need to add the time he practices daily to the additional time he practices on weekends.

    Arthur practices daily for \( \frac{20}{21} \) of an hour. On weekends, he now practices an additional \( \frac{11}{24} \) of an hour.

    First, we need to find a common denominator to add these fractions. The least common multiple (LCM) of 21 and 24 is 168.

    Convert \( \frac{20}{21} \) to a fraction with a denominator of 168:
    \[ \frac{20}{21} = \frac{20 \times 8}{21 \times 8} = \frac{160}{168} \]

    Convert \( \frac{11}{24} \) to a fraction with a denominator of 168:
    \[ \frac{11}{24} = \frac{11 \times 7}{24 \times 7} = \frac{77}{168} \]

    Now, add the two fractions:
    \[ \frac{160}{168} + \frac{77}{168} = \frac{160 + 77}{168} = \frac{237}{168} \]

    Thus, Arthur spends \( \frac{237}{168} \) hours practicing the piano each day on the weekend. This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3:
    \[ \frac{237 \div 3}{168 \div 3} = \frac{79}{56} \]

    ### Answer: \( \frac{79}{56} \) hours

Expected answer: 79/56

Correct? Yes

🔗 Problem

Problem: Lucas has a shirt store. Initially, he had 18 shirts which he divided equally among his 3 siblings. Later, he bought 27 more shirts but wanted to sell them at a higher price, so he decided to keep only 1/19 of those shirts for each sibling. How many more or fewer shirts does each sibling now have as compared to before? Calculate the difference using the expression ((18 / 3) - (27 / 19)) and find out the result.

Gemma 2B:

    First, let's determine how many shirts each sibling initially received. Lucas had 18 shirts and divided them equally among his 3 siblings:

    \[
    \frac{18}{3} = 6
    \]

    So, each sibling initially received 6 shirts.

    Next, Lucas bought 27 more shirts and decided to keep only \(\frac{1}{19}\) of those shirts for each sibling. We need to calculate how many shirts each sibling received from the new batch:

    \[
    \frac{27}{19}
    \]

    Now, we need to find the difference in the number of shirts each sibling has now compared to before. This is given by the expression:

    \[
    \left( \frac{18}{3} \right) - \left( \frac{27}{19} \right)
    \]

    Substituting the values we calculated:

    \[
    6 - \frac{27}{19}
    \]

    To perform this subtraction, we need a common denominator. The common denominator for 6 and \(\frac{27}{19}\) is 19. We convert 6 to a fraction with a denominator of 19:

    \[
    6 = \frac{6 \times 19}{19} = \frac{114}{19}
    \]

    Now, we subtract the fractions:

    \[
    \frac{114}{19} - \frac{27}{19} = \frac{114 - 27}{19} = \frac{87}{19}
    \]

    Finally, we simplify \(\frac{87}{19}\):

    \[
    \frac{87}{19} = 4.578947368421053
    \]

    So, each sibling now has approximately 4.58 fewer shirts compared to before.

    ### Answer: 4.58 fewer shirts

Expected answer: 87/19

Correct? No

🔗 Problem

Problem: A painter mixed 7 gallons of white paint with 2/5 gallon of blue paint to make a light blue paint. After using 18/4 gallons of the light blue paint for a project, how many gallons does the painter have left?

Gemma 2B:

    First, let's determine the total amount of light blue paint the painter initially made by mixing the white and blue paint.

    The painter mixed:
    - 7 gallons of white paint
    - 2/5 gallon of blue paint

    Total initial amount of light blue paint:
    \[ 7 + \frac{2}{5} \]

    To add these, we need a common denominator. The common denominator for 1 and 5 is 5. So, we convert 7 to a fraction with a denominator of 5:
    \[ 7 = \frac{35}{5} \]

    Now, add the fractions:
    \[ \frac{35}{5} + \frac{2}{5} = \frac{37}{5} \]

    So, the total initial amount of light blue paint is:
    \[ \frac{37}{5} \text{ gallons} \]

    Next, the painter used \( \frac{18}{4} \) gallons of the light blue paint. Simplify \( \frac{18}{4} \):
    \[ \frac{18}{4} = \frac{9}{2} \]

    Convert \( \frac{9}{2} \) to a fraction with a denominator of 5 to subtract from \( \frac{37}{5} \):
    \[ \frac{9}{2} = \frac{9 \times 5}{2 \times 5} = \frac{45}{10} \]

    Convert \( \frac{37}{5} \) to a fraction with a denominator of 10:
    \[ \frac{37}{5} = \frac{37 \times 2}{5 \times 2} = \frac{74}{10} \]

    Now, subtract the fractions:
    \[ \frac{74}{10} - \frac{45}{10} = \frac{74 - 45}{10} = \frac{29}{10} \]

    So, the painter has \( \frac{29}{10} \) gallons of light blue paint left. Converting this to a decimal:
    \[ \frac{29}{10} = 2.9 \]

    ### Answer: 2.9 gallons

Expected answer: 29/10

Correct? No

🔗 Problem

Problem: Jayson went to the roller rink and spent 24/7 hours skating on the first day. On the second day, he decided to skate for an additional 30/20 hours. How many hours in total did Jayson spend skate skating over the two days?

Gemma 2B:

    To find the total number of hours Jayson spent skating over the two days, we need to add the hours he skated on each day.

    First, let's convert the fractions to a common denominator to make the addition easier.

    For the first day:
    \[ \frac{24}{7} \]

    For the second day:
    \[ \frac{30}{20} \]

    We can simplify \(\frac{30}{20}\):
    \[ \frac{30}{20} = \frac{3}{2} \]

    Now, we need to add \(\frac{24}{7}\) and \(\frac{3}{2}\). To do this, we need a common denominator. The least common multiple of 7 and 2 is 14.

    Convert \(\frac{24}{7}\) to a fraction with a denominator of 14:
    \[ \frac{24}{7} = \frac{24 \times 2}{7 \times 2} = \frac{48}{14} \]

    Convert \(\frac{3}{2}\) to a fraction with a denominator of 14:
    \[ \frac{3}{2} = \frac{3 \times 7}{2 \times 7} = \frac{21}{14} \]

    Now, add the two fractions:
    \[ \frac{48}{14} + \frac{21}{14} = \frac{48 + 21}{14} = \frac{69}{14} \]

    So, the total number of hours Jayson spent skating over the two days is:
    \[ \frac{69}{14} \]

    ### Answer: \(\frac{69}{14}\) hours

Expected answer: 69/14

Correct? Yes

🔗 Follow-up

Problem: Initially we said Jayson had spent 24/7 hours skating at the roller rink on the first day. However, his friend Jill remembered that they had actually only spent 24/22 hours skating that day. Given this new information, how many hours did Jayson spend roller skating during the first two days including the 30/20 hours on the second day?

Gemma 2B:

    To find the total number of hours Jayson spent skating over the two days with the corrected information, we need to add the hours he skated on each day.

    First, let's convert the fractions to a common denominator to make the addition easier.

    For the first day:
    \[ \frac{24}{22} \]

    For the second day:
    \[ \frac{30}{20} \]

    We can simplify \(\frac{30}{20}\):
    \[ \frac{30}{20} = \frac{3}{2} \]

    Now, we need to add \(\frac{24}{22}\) and \(\frac{3}{2}\). To do this, we need a common denominator. The least common multiple of 22 and 2 is 22.

    Convert \(\frac{3}{2}\) to a fraction with a denominator of 22:
    \[ \frac{3}{2} = \frac{3 \times 11}{2 \times 11} = \frac{33}{22} \]

    Now, add the two fractions:
    \[ \frac{24}{22} + \frac{33}{22} = \frac{24 + 33}{22} = \frac{57}{22} \]

    So, the total number of hours Jayson spent skating over the two days is:
    \[ \frac{57}{22} \]

    ### Answer: \(\frac{57}{22}\) hours

Expected answer: 57/22

Correct? Yes

🔗 Problem

Problem: Dolphins can leap approximately 18/3 feet out of the water. If a dolphin only manages to jump 3/4 of his usual height one day, how high did the dolphin jump?

Gemma 2B:

    First, we need to determine the usual height a dolphin can leap out of the water. This is given by the expression \( \frac{18}{3} \).

    \[
    \frac{18}{3} = 6 \text{ feet}
    \]

    Next, we need to find out how high the dolphin jumped when it managed only \( \frac{3}{4} \) of its usual height. We calculate this by multiplying the usual height by \( \frac{3}{4} \).

    \[
    6 \times \frac{3}{4} = 6 \times 0.75 = 4.5 \text{ feet}
    \]

    ### Answer: 4.5 feet

Expected answer: 9/2

Correct? No

🔗 Problem

Problem: Callie owns a pen business. Last month, she sold 20 pens for every 15 she produced. At the same time, she sold an additional 16 pens for every 22 she put out for sale. Meanwhile, she found that for every 22 pens she advertised, 27 were interested customers. For each 23 pens she produced this week, she was able to sell 8. What is the result when she multiplies the ratio of last month's sales to production (20/15 + 16/22) and this week's figure for the ratio of interested customers to production (22/27 * 23/8)? Leave your answer as an unsimplified fraction.

Gemma 2B:

    To solve this problem, we need to calculate two ratios and then multiply them together.

    First, let's find the ratio of last month's sales to production. This ratio is given by the sum of two fractions:
    \[ \frac{20}{15} + \frac{16}{22} \]

    We can simplify each fraction individually:
    \[ \frac{20}{15} = \frac{4}{3} \]
    \[ \frac{16}{22} = \frac{8}{11} \]

    Now, we need to add these two fractions. To do this, we need a common denominator. The least common multiple of 3 and 11 is 33:
    \[ \frac{4}{3} = \frac{4 \times 11}{3 \times 11} = \frac{44}{33} \]
    \[ \frac{8}{11} = \frac{8 \times 3}{11 \times 3} = \frac{24}{33} \]

    Adding these fractions together:
    \[ \frac{44}{33} + \frac{24}{33} = \frac{44 + 24}{33} = \frac{68}{33} \]

    Next, let's find the ratio of interested customers to production for this week:
    \[ \frac{22}{27} \times \frac{23}{8} \]

    We multiply these fractions directly:
    \[ \frac{22 \times 23}{27 \times 8} = \frac{506}{216} \]

    Now, we multiply the two results together:
    \[ \frac{68}{33} \times \frac{506}{216} = \frac{68 \times 506}{33 \times 216} = \frac{34408}{7128} \]

    Thus, the result when she multiplies the ratio of last month's sales to production and this week's figure for the ratio of interested customers to production is:
    ### Answer: \(\frac{34408}{7128}\)

Expected answer: 391/81

Correct? Yes

🔗 Follow-up

Problem: Continuing from the previous problem, Callie decides to adjust her production strategy for next month. She plans on increasing her production by a factor of 6/3 (which is actually doubling her production). What would be the combined impact on her business of last month's sales ratio (20/15 + 16/22), this week's interest to production ratio (22/27 * 23/8), and her planned production increase (6/3), considering they multiply together? Leave your answer as an unsimplified fraction.

Gemma 2B:

    To determine the combined impact on Callie's business, we need to multiply the three ratios together: last month's sales ratio, this week's interest to production ratio, and her planned production increase.

    From the previous problem, we already have:
    1. Last month's sales ratio: \(\frac{68}{33}\)
    2. This week's interest to production ratio: \(\frac{506}{216}\)

    Now, we need to consider her planned production increase, which is a factor of \(\frac{6}{3}\). Simplifying this, we get:
    \[ \frac{6}{3} = 2 \]

    Next, we multiply all three ratios together:
    \[ \frac{68}{33} \times \frac{506}{216} \times 2 \]

    First, let's multiply the first two ratios:
    \[ \frac{68}{33} \times \frac{506}{216} = \frac{68 \times 506}{33 \times 216} = \frac{34408}{7128} \]

    Now, we multiply this result by 2:
    \[ \frac{34408}{7128} \times 2 = \frac{34408 \times 2}{7128} = \frac{68816}{7128} \]

    Thus, the combined impact on her business, considering all factors, is:
    ### Answer: \(\frac{68816}{7128}\)

Expected answer: 782/81

Correct? Yes

🔗 Follow-up

Problem: In the following month, Callie evaluates her pen business again. This time, for every 15 pens she produced, she sold 14. Just like the previous month, she managed to sell an additional 16 pens for every 22 she displayed for sale. The ratio of interested customers to the number of pens she produced this week remained the same as the previous week, with 22 for every 27 that were interested. Out of all the pens produced, she was able to sell 23 for every 8 she made. What would now be the result when she takes this month's sales to production ratio (14/15 + 16/22) and multiplies it by this week's ratio of interested customers to production (22/27 * 23/8)? Leave your answer as an unsimplified fraction.

Gemma 2B:

    To solve this problem, we need to calculate two ratios and then multiply them together.

    First, let's find the ratio of this month's sales to production. This ratio is given by the sum of two fractions:
    \[ \frac{14}{15} + \frac{16}{22} \]

    We can simplify each fraction individually:
    \[ \frac{14}{15} \]
    \[ \frac{16}{22} = \frac{8}{11} \]

    Now, we need to add these two fractions. To do this, we need a common denominator. The least common multiple of 15 and 11 is 165:
    \[ \frac{14}{15} = \frac{14 \times 11}{15 \times 11} = \frac{154}{165} \]
    \[ \frac{8}{11} = \frac{8 \times 15}{11 \times 15} = \frac{120}{165} \]

    Adding these fractions together:
    \[ \frac{154}{165} + \frac{120}{165} = \frac{154 + 120}{165} = \frac{274}{165} \]

    Next, let's find the ratio of interested customers to production for this week:
    \[ \frac{22}{27} \times \frac{23}{8} \]

    We multiply these fractions directly:
    \[ \frac{22 \times 23}{27 \times 8} = \frac{506}{216} \]

    Now, we multiply the two results together:
    \[ \frac{274}{165} \times \frac{506}{216} = \frac{274 \times 506}{165 \times 216} = \frac{138644}{35640} \]

    Thus, the result when she multiplies this month's sales to production ratio and this week's ratio of interested customers to production is:
    ### Answer: \(\frac{138644}{35640}\)

Expected answer: 3151/810

Correct? Yes

🔗 Problem

Problem: Consider a band practice session in which a musician played a song for exactly 30 minutes. After dividing this time equally between 15 songs, how much time did they spend on each song? From this average time per song, the band reduced the time spent on a particular song by 1/17 of an hour. Can you calculate the final duration of that song?

Gemma 2B:

    First, we need to determine the time spent on each song during the practice session. The total time spent is 30 minutes, and this time is divided equally among 15 songs.

    To find the time spent on each song, we divide the total time by the number of songs:
    \[
    \text{Time per song} = \frac{30 \text{ minutes}}{15 \text{ songs}} = 2 \text{ minutes per song}
    \]

    Next, we need to convert the time per song into hours since the reduction is given in hours. There are 60 minutes in an hour, so:
    \[
    2 \text{ minutes} = \frac{2}{60} \text{ hours} = \frac{1}{30} \text{ hours}
    \]

    The band reduced the time spent on a particular song by \(\frac{1}{17}\) of an hour. We need to subtract this reduction from the original time per song:
    \[
    \text{Final duration} = \frac{1}{30} \text{ hours} - \frac{1}{17} \text{ hours}
    \]

    To perform this subtraction, we need a common denominator. The least common multiple of 30 and 17 is 510. We convert both fractions to have this common denominator:
    \[
    \frac{1}{30} = \frac{17}{510}
    \]
    \[
    \frac{1}{17} = \frac{30}{510}
    \]

    Now we subtract the fractions:
    \[
    \frac{17}{510} - \frac{30}{510} = \frac{17 - 30}{510} = \frac{-13}{510} = -\frac{13}{510}
    \]

    Since we are dealing with time, we should consider the absolute value:
    \[
    \left| -\frac{13}{510} \right| = \frac{13}{510}
    \]

    Simplifying \(\frac{13}{510}\):
    \[
    \frac{13}{510} = \frac{1}{39.23} \approx 0.0255 \text{ hours}
    \]

    Converting this back to minutes:
    \[
    0.0255 \text{ hours} \times 60 \text{ minutes/hour} \approx 1.53 \text{ minutes}
    \]

    ### Answer: 1.53 minutes

Expected answer: 33/17

Correct? No

🔗 Problem

Problem: Josh rented a scooter for a day. He rode the scooter for a total of 18 hours during the first day. The next day, he rode the scooter for an additional 1/12 of an hour. Later that afternoon, his friend Jack also rented a scooter and rode it for 7/17 of an hour. How many hours did they both ride scooters in total (expressed as a fraction)? Combine the hours they both rode and simplify the fraction to get the answer.

Gemma 2B:

    First, let's convert the hours Josh rode the scooter into a fraction. Josh rode the scooter for 18 hours and an additional \( \frac{1}{12} \) of an hour.

    To combine these, we need to express 18 as a fraction with a common denominator of 12:
    \[ 18 = \frac{18 \times 12}{12} = \frac{216}{12} \]

    Adding the additional \( \frac{1}{12} \) hour:
    \[ \frac{216}{12} + \frac{1}{12} = \frac{216 + 1}{12} = \frac{217}{12} \]

    Next, we consider the time Jack rode the scooter, which is \( \frac{7}{17} \) of an hour.

    To find the total time both Josh and Jack rode the scooters, we need to add \( \frac{217}{12} \) and \( \frac{7}{17} \). To do this, we need a common denominator. The least common multiple (LCM) of 12 and 17 is 204.

    Convert both fractions to have the denominator of 204:
    \[ \frac{217}{12} = \frac{217 \times 17}{12 \times 17} = \frac{3689}{204} \]
    \[ \frac{7}{17} = \frac{7 \times 12}{17 \times 12} = \frac{84}{204} \]

    Now, add the two fractions:
    \[ \frac{3689}{204} + \frac{84}{204} = \frac{3689 + 84}{204} = \frac{3773}{204} \]

    The fraction \( \frac{3773}{204} \) is already in its simplest form because 3773 and 204 have no common factors other than 1.

    ### Answer: \(\frac{3773}{204}\)

Expected answer: 3773/204

Correct? Yes

🔗 Follow-up

Problem: Continuing from the previous situation, suppose a second friend, Maria, decided to rent a scooter too. She rode the scooter for an additional 1/21 of an hour. Adding up the time Maria rode her scooter, how many total hours did all three friends ride scooters, expressed as a fraction? Make sure to simplify any fractions used in your final answer.

Gemma 2B:

    To find the total time all three friends rode scooters, we need to add the time Maria rode her scooter, \( \frac{1}{21} \) of an hour, to the total time Josh and Jack rode their scooters, which we previously calculated as \( \frac{3773}{204} \).

    First, we need a common denominator for \( \frac{3773}{204} \) and \( \frac{1}{21} \). The least common multiple (LCM) of 204 and 21 is 1428.

    Convert both fractions to have the denominator of 1428:
    \[ \frac{3773}{204} = \frac{3773 \times 7}{204 \times 7} = \frac{26411}{1428} \]
    \[ \frac{1}{21} = \frac{1 \times 68}{21 \times 68} = \frac{68}{1428} \]

    Now, add the two fractions:
    \[ \frac{26411}{1428} + \frac{68}{1428} = \frac{26411 + 68}{1428} = \frac{26479}{1428} \]

    The fraction \( \frac{26479}{1428} \) is already in its simplest form because 26479 and 1428 have no common factors other than 1.

    ### Answer: \(\frac{26479}{1428}\)

Expected answer: 26479/1428

Correct? Yes

🔗 Problem

Problem: A lamp store owner is doing inventory of all the lamps in his store. He found that he has 4/26 of lamps that are floor lamps, 15/11 that are desk lamps, he also discovered he has 23/28 of table lamps and finally 8/19 of lamps that are decorative. What fraction of his total inventory do these lamps constitute?

Gemma 2B:

    To find the total fraction of the lamps in the store, we need to add the fractions of each type of lamp together. The fractions given are:

    - Floor lamps: \(\frac{4}{26}\)
    - Desk lamps: \(\frac{15}{11}\)
    - Table lamps: \(\frac{23}{28}\)
    - Decorative lamps: \(\frac{8}{19}\)

    First, we simplify the fractions where possible:

    \[
    \frac{4}{26} = \frac{2}{13}
    \]

    Next, we add the fractions together. To do this, we need a common denominator. The least common multiple (LCM) of 13, 11, 28, and 19 is 7604. We convert each fraction to have this common denominator:

    \[
    \frac{2}{13} = \frac{2 \times 584}{13 \times 584} = \frac{1168}{7604}
    \]

    \[
    \frac{15}{11} = \frac{15 \times 691}{11 \times 691} = \frac{10365}{7604}
    \]

    \[
    \frac{23}{28} = \frac{23 \times 271}{28 \times 271} = \frac{6233}{7604}
    \]

    \[
    \frac{8}{19} = \frac{8 \times 400}{19 \times 400} = \frac{3200}{7604}
    \]

    Now, we add these fractions together:

    \[
    \frac{1168}{7604} + \frac{10365}{7604} + \frac{6233}{7604} + \frac{3200}{7604} = \frac{1168 + 10365 + 6233 + 3200}{7604} = \frac{20966}{7604}
    \]

    We simplify the resulting fraction:

    \[
    \frac{20966}{7604} = \frac{20966 \div 2}{7604 \div 2} = \frac{10483}{3802}
    \]

    Since 10483 and 3802 have no common factors other than 1, the fraction is already in its simplest form.

    ### Answer: \(\frac{10483}{3802}\)

Expected answer: 209967/76076

Correct? No

🔗 Problem

Problem: Oliver has a kite string that is 26/19 meters long and another kite string that is 7/11 meters long. He decides to add these two strings to create an even longer kite string. Later on, he decides to use only 5/8 of this combined string. How long is the kite string that he is using now? Do not simplify your fractions within your calculation.

Gemma 2B:

    First, we need to add the lengths of the two kite strings.

    The lengths of the kite strings are:
    \[ \frac{26}{19} \text{ meters} \]
    \[ \frac{7}{11} \text{ meters} \]

    To add these fractions, we need a common denominator. The least common multiple (LCM) of 19 and 11 is 209.

    Convert each fraction to have the common denominator of 209:
    \[ \frac{26}{19} = \frac{26 \times 11}{19 \times 11} = \frac{286}{209} \]
    \[ \frac{7}{11} = \frac{7 \times 19}{11 \times 19} = \frac{133}{209} \]

    Now, add the fractions:
    \[ \frac{286}{209} + \frac{133}{209} = \frac{286 + 133}{209} = \frac{419}{209} \]

    Next, Oliver decides to use only \(\frac{5}{8}\) of this combined string. We need to multiply \(\frac{419}{209}\) by \(\frac{5}{8}\):
    \[ \frac{419}{209} \times \frac{5}{8} = \frac{419 \times 5}{209 \times 8} = \frac{2095}{1672} \]

    Thus, the length of the kite string that Oliver is using now is:
    ### Answer: \(\frac{2095}{1672}\) meters.

Expected answer: 2095/1672

Correct? Yes

🔗 Follow-up

Problem: After using the combined kite string, Oliver realizes he needs a similar length of string for another kite. He decides to use the same length of string that he used earlier, effectively doubling his previous length. What is the total length of the kite string Oliver is now using? Note: Do not simplify your fractions within your calculation.

Gemma 2B:

    To find the total length of the kite string Oliver is now using, we need to double the length of the string he used earlier. The length of the string he used earlier was \(\frac{2095}{1672}\) meters.

    To double this length, we multiply by 2:
    \[ 2 \times \frac{2095}{1672} = \frac{2 \times 2095}{1672} = \frac{4190}{1672} \]

    Thus, the total length of the kite string Oliver is now using is:
    ### Answer: \(\frac{4190}{1672}\) meters.

Expected answer: 2095/836

Correct? Yes

🔗 Follow-up

Problem: Previously, Oliver used a string that was 5/8 of the combined total. However, now, he decides to use 26/8 of the same combined total length (26/19 meters and 7/11 meters) for a larger kite. What would be the length of the string that he is using now? Note: Do not simplify your fractions within your calculation.

Gemma 2B:

    First, we need to find the combined total length of the two kite strings.

    The lengths of the kite strings are:
    \[ \frac{26}{19} \text{ meters} \]
    \[ \frac{7}{11} \text{ meters} \]

    To add these fractions, we need a common denominator. The least common multiple (LCM) of 19 and 11 is 209.

    Convert each fraction to have the common denominator of 209:
    \[ \frac{26}{19} = \frac{26 \times 11}{19 \times 11} = \frac{286}{209} \]
    \[ \frac{7}{11} = \frac{7 \times 19}{11 \times 19} = \frac{133}{209} \]

    Now, add the fractions:
    \[ \frac{286}{209} + \frac{133}{209} = \frac{286 + 133}{209} = \frac{419}{209} \]

    Next, Oliver decides to use \(\frac{26}{8}\) of this combined string. We need to multiply \(\frac{419}{209}\) by \(\frac{26}{8}\):
    \[ \frac{419}{209} \times \frac{26}{8} = \frac{419 \times 26}{209 \times 8} = \frac{10894}{1672} \]

    Thus, the length of the kite string that Oliver is using now is:
    ### Answer: \(\frac{10894}{1672}\) meters.

Expected answer: 5447/836

Correct? Yes

🔗 Problem

Problem: In a snow fight, Mike threw 9/19 of his snowballs and his friend, John threw 6/8 of his snowballs. After a while, Mike found 4 more snowballs to throw. The question is, if 'u' represents the number of snowballs thrown by both Mike and John and 'j' is the total number of snowballs Mike had after finding more, how many snowballs did Mike have in total? Solve the following equations to find the variable 'j': u = 9/19 + 6/8 j = u + 4/1

Gemma 2B:

    To solve the problem, we need to find the total number of snowballs Mike had after finding more. We will solve the given equations step by step.

    First, let's solve for \( u \):

    \[ u = \frac{9}{19} + \frac{6}{8} \]

    To add these fractions, we need a common denominator. The least common multiple of 19 and 8 is 152.

    Convert each fraction to have a denominator of 152:

    \[ \frac{9}{19} = \frac{9 \times 8}{19 \times 8} = \frac{72}{152} \]
    \[ \frac{6}{8} = \frac{6 \times 19}{8 \times 19} = \frac{114}{152} \]

    Now add the fractions:

    \[ u = \frac{72}{152} + \frac{114}{152} = \frac{72 + 114}{152} = \frac{186}{152} \]

    Simplify the fraction:

    \[ \frac{186}{152} = \frac{93}{76} \]

    So, \( u = \frac{93}{76} \).

    Next, we need to find \( j \):

    \[ j = u + \frac{4}{1} \]

    Substitute \( u \) with \( \frac{93}{76} \):

    \[ j = \frac{93}{76} + \frac{4}{1} \]

    Convert \( \frac{4}{1} \) to have a denominator of 76:

    \[ \frac{4}{1} = \frac{4 \times 76}{1 \times 76} = \frac{304}{76} \]

    Now add the fractions:

    \[ j = \frac{93}{76} + \frac{304}{76} = \frac{93 + 304}{76} = \frac{397}{76} \]

    So, the total number of snowballs Mike had after finding more is:

    ### Answer: \(\frac{397}{76}\)

Expected answer: 397/76

Correct? Yes

🔗 Follow-up

Problem: Continuing from the initial problem, suppose that instead of finding 4 more snowballs, Mike now finds 5 more snowballs instead. Now, if 'u' represents the number of snowballs thrown by both Mike and John initially, and 'j' is the total number of snowballs Mike has after finding more, how many snowballs does Mike have in total now? Solve the following equations to find the updated value of 'j': u = 9/19 + 6/8 j = u + 5/1

Gemma 2B:

    To solve the updated problem, we will follow similar steps as before but with the new value of 5 more snowballs found by Mike.

    First, let's solve for \( u \):

    \[ u = \frac{9}{19} + \frac{6}{8} \]

    We already found the common denominator and converted the fractions in the previous solution:

    \[ \frac{9}{19} = \frac{72}{152} \]
    \[ \frac{6}{8} = \frac{114}{152} \]

    Adding these fractions:

    \[ u = \frac{72}{152} + \frac{114}{152} = \frac{186}{152} \]

    Simplify the fraction:

    \[ \frac{186}{152} = \frac{93}{76} \]

    So, \( u = \frac{93}{76} \).

    Next, we need to find \( j \):

    \[ j = u + \frac{5}{1} \]

    Substitute \( u \) with \( \frac{93}{76} \):

    \[ j = \frac{93}{76} + \frac{5}{1} \]

    Convert \( \frac{5}{1} \) to have a denominator of 76:

    \[ \frac{5}{1} = \frac{5 \times 76}{1 \times 76} = \frac{380}{76} \]

    Now add the fractions:

    \[ j = \frac{93}{76} + \frac{380}{76} = \frac{93 + 380}{76} = \frac{473}{76} \]

    So, the total number of snowballs Mike has after finding 5 more is:

    ### Answer: \(\frac{473}{76}\)

Expected answer: 473/76

Correct? Yes

🔗 Problem

Problem: In a chocolate factory, one worker packages 26 pieces of chocolate in 18 minutes, while another worker can package 15 pieces of chocolate every 15 minutes. Find out the total amount of chocolate they can package in one minute.

Gemma 2B:

    To determine the total amount of chocolate both workers can package in one minute, we need to find their individual rates and then sum them up.

    First, let's find the rate for the first worker:
    - The first worker packages 26 pieces of chocolate in 18 minutes.
    - The rate of the first worker is \( \frac{26 \text{ pieces}}{18 \text{ minutes}} \).

    To find the rate per minute, we divide 26 by 18:
    \[ \frac{26}{18} = \frac{13}{9} \text{ pieces per minute} \]

    Next, let's find the rate for the second worker:
    - The second worker packages 15 pieces of chocolate in 15 minutes.
    - The rate of the second worker is \( \frac{15 \text{ pieces}}{15 \text{ minutes}} \).

    To find the rate per minute, we divide 15 by 15:
    \[ \frac{15}{15} = 1 \text{ piece per minute} \]

    Now, we sum the rates of both workers to find the total amount of chocolate they can package in one minute:
    \[ \frac{13}{9} + 1 = \frac{13}{9} + \frac{9}{9} = \frac{22}{9} \text{ pieces per minute} \]

    ### Answer: \(\frac{22}{9}\) pieces per minute

Expected answer: 22/9

Correct? Yes

🔗 Follow-up

Problem: In our previous scenario at the chocolate factory, one worker was packaging 26 pieces of chocolate in 18 minutes, and another worker was packaging 15 pieces of chocolate every 15 minutes. Now let's consider a small change: the second worker's speed has slightly decreased, and they are now able to package only 14 pieces of chocolate every 15 minutes instead of 15. How does this change the total amount of chocolate the two workers can package together in one minute?

Gemma 2B:

    To determine the new total amount of chocolate both workers can package in one minute, we need to find their individual rates with the updated information and then sum them up.

    First, let's find the rate for the first worker (unchanged):
    - The first worker packages 26 pieces of chocolate in 18 minutes.
    - The rate of the first worker is \( \frac{26 \text{ pieces}}{18 \text{ minutes}} \).

    To find the rate per minute, we divide 26 by 18:
    \[ \frac{26}{18} = \frac{13}{9} \text{ pieces per minute} \]

    Next, let's find the new rate for the second worker:
    - The second worker now packages 14 pieces of chocolate in 15 minutes.
    - The rate of the second worker is \( \frac{14 \text{ pieces}}{15 \text{ minutes}} \).

    To find the rate per minute, we divide 14 by 15:
    \[ \frac{14}{15} \text{ pieces per minute} \]

    Now, we sum the rates of both workers to find the total amount of chocolate they can package in one minute:
    \[ \frac{13}{9} + \frac{14}{15} \]

    To add these fractions, we need a common denominator. The least common multiple of 9 and 15 is 45. We convert each fraction:

    \[ \frac{13}{9} = \frac{13 \times 5}{9 \times 5} = \frac{65}{45} \]
    \[ \frac{14}{15} = \frac{14 \times 3}{15 \times 3} = \frac{42}{45} \]

    Now, we add the fractions:
    \[ \frac{65}{45} + \frac{42}{45} = \frac{65 + 42}{45} = \frac{107}{45} \]

    ### Answer: \(\frac{107}{45}\) pieces per minute

Expected answer: 107/45

Correct? Yes

🔗 Problem

Problem: Yesterday, Joanna washed 24/20 batches of small towels and 25/15 batches of large towels. Find the total number of batches of towels she washed.

Gemma 2B:

    First, we need to simplify the fractions for the number of batches Joanna washed.

    For the small towels:
    \[ \frac{24}{20} = \frac{24 \div 4}{20 \div 4} = \frac{6}{5} \]

    For the large towels:
    \[ \frac{25}{15} = \frac{25 \div 5}{15 \div 5} = \frac{5}{3} \]

    Next, we add the simplified fractions to find the total number of batches:
    \[ \frac{6}{5} + \frac{5}{3} \]

    To add these fractions, we need a common denominator. The least common multiple of 5 and 3 is 15.

    Convert each fraction to have a denominator of 15:
    \[ \frac{6}{5} = \frac{6 \times 3}{5 \times 3} = \frac{18}{15} \]
    \[ \frac{5}{3} = \frac{5 \times 5}{3 \times 5} = \frac{25}{15} \]

    Now, add the fractions:
    \[ \frac{18}{15} + \frac{25}{15} = \frac{18 + 25}{15} = \frac{43}{15} \]

    ### Answer: \(\frac{43}{15}\)

Expected answer: 43/15

Correct? Yes

🔗 Follow-up

Problem: Joanna originally thought she washed 25/15 batches of large towels. However, she realized that she had actually washed 25/7 batches. Taking this into account, with the 24/20 batches of small towels she washed, how many total batches of towels did Joanna wash?

Gemma 2B:

    First, we need to simplify the fractions for the number of batches Joanna washed.

    For the small towels:
    \[ \frac{24}{20} = \frac{24 \div 4}{20 \div 4} = \frac{6}{5} \]

    For the large towels:
    \[ \frac{25}{7} \]

    Next, we add the simplified fractions to find the total number of batches:
    \[ \frac{6}{5} + \frac{25}{7} \]

    To add these fractions, we need a common denominator. The least common multiple of 5 and 7 is 35.

    Convert each fraction to have a denominator of 35:
    \[ \frac{6}{5} = \frac{6 \times 7}{5 \times 7} = \frac{42}{35} \]
    \[ \frac{25}{7} = \frac{25 \times 5}{7 \times 5} = \frac{125}{35} \]

    Now, add the fractions:
    \[ \frac{42}{35} + \frac{125}{35} = \frac{42 + 125}{35} = \frac{167}{35} \]

    ### Answer: \(\frac{167}{35}\)

Expected answer: 167/35

Correct? Yes

🔗 Problem

Problem: Debra is baking cookies and she realizes she has two different recipes for the same type of cookie. She's wondering if she combines the ingredients from both, if she will end up with even more cookies or not. The first recipe requires (27 / 3) cups flour and (11 / 7) cups sugar. She determines the ratio of flour to sugar by dividing the amount of flour by the amount of sugar. She then subtracts (21 / 26) from this ratio because that's how many cups of sugar her friend says makes the cookies too sweet. After that, she adds (22 / 9) cups of flour from the second recipe to the new ratio to see how many total cups of flour she will need. How many cups of flour will she have in the end?

Gemma 2B:

    Let's break down the problem step by step.

    1. Calculate the amount of flour and sugar in the first recipe:
       - Flour: \( \frac{27}{3} = 9 \) cups
       - Sugar: \( \frac{11}{7} \) cups

    2. Determine the ratio of flour to sugar:
       \[
       \text{Ratio} = \frac{\text{Flour}}{\text{Sugar}} = \frac{9}{\frac{11}{7}} = 9 \times \frac{7}{11} = \frac{63}{11}
       \]

    3. Subtract \( \frac{21}{26} \) from this ratio:
       \[
       \frac{63}{11} - \frac{21}{26}
       \]

       To subtract these fractions, we need a common denominator. The least common multiple of 11 and 26 is 286.

       Convert each fraction to have the common denominator:
       \[
       \frac{63}{11} = \frac{63 \times 26}{11 \times 26} = \frac{1638}{286}
       \]
       \[
       \frac{21}{26} = \frac{21 \times 11}{26 \times 11} = \frac{231}{286}
       \]

       Now subtract the fractions:
       \[
       \frac{1638}{286} - \frac{231}{286} = \frac{1407}{286}
       \]

    4. Simplify the fraction \( \frac{1407}{286} \):
       \[
       \frac{1407 \div 13}{286 \div 13} = \frac{108.23}{22} \approx 4.92
       \]

    5. Add \( \frac{22}{9} \) cups of flour from the second recipe to the new ratio:
       \[
       4.92 + \frac{22}{9}
       \]

       Convert \( \frac{22}{9} \) to a decimal:
       \[
       \frac{22}{9} \approx 2.44
       \]

       Add the two values:
       \[
       4.92 + 2.44 = 7.36
       \]

    ### Answer: 7.36

Expected answer: 18955/2574

Correct? No

🔗 Problem

Problem: There are 26/13 chickens in one farm and 9/16 chickens in another farm. Farmer John calculated the difference between the number of chickens in the two farms. After a week, the number of chickens multiplied by 19/21 due to new chickens born. How many chickens are there now in Farmer John's farm?

Gemma 2B:

    First, let's calculate the number of chickens in each farm.

    For the first farm:
    \[
    \frac{26}{13} = 2 \text{ chickens}
    \]

    For the second farm:
    \[
    \frac{9}{16} \text{ chickens}
    \]

    Next, we find the difference between the number of chickens in the two farms:
    \[
    2 - \frac{9}{16}
    \]

    To subtract these fractions, we need a common denominator. The common denominator for 2 (which is \(\frac{32}{16}\)) and \(\frac{9}{16}\) is 16:
    \[
    2 = \frac{32}{16}
    \]

    So, the difference is:
    \[
    \frac{32}{16} - \frac{9}{16} = \frac{23}{16} \text{ chickens}
    \]

    After a week, the number of chickens multiplied by \(\frac{19}{21}\):
    \[
    \frac{23}{16} \times \frac{19}{21} = \frac{23 \times 19}{16 \times 21} = \frac{437}{336}
    \]

    To simplify \(\frac{437}{336}\), we check for common factors. Since 437 and 336 have no common factors other than 1, the fraction is already in its simplest form.

    ### Answer: \(\frac{437}{336}\)

Expected answer: 437/336

Correct? Yes

🔗 Follow-up

Problem: Farmer John was excited to find that after one week, not only did the number of chickens multiplied by 19/21 due to new chickens being born, but also an additional chicken (representing 28/28) wandered into his farm. How many chickens does Farmer John have on his farm now?

Gemma 2B:

    We previously calculated that the number of chickens after multiplying by \(\frac{19}{21}\) was \(\frac{437}{336}\).

    Now, we need to add the additional chicken, which is represented as \(\frac{28}{28}\).

    First, let's convert \(\frac{28}{28}\) to a fraction with the same denominator as \(\frac{437}{336}\). We need to find a common denominator. The least common multiple of 336 and 28 is 336.

    \[
    \frac{28}{28} = \frac{28 \times 12}{28 \times 12} = \frac{336}{336}
    \]

    Now, we add the two fractions:
    \[
    \frac{437}{336} + \frac{336}{336} = \frac{437 + 336}{336} = \frac{773}{336}
    \]

    So, the total number of chickens Farmer John has on his farm now is \(\frac{773}{336}\).

    ### Answer: \(\frac{773}{336}\)

Expected answer: 773/336

Correct? Yes

🔗 Follow-up

Problem: After reconsidering his calculations, Farmer John realized that instead of multiplying the number of chickens by 19/21 , he should have multiplied by 19/4 due to a miscalculation in the number of new chickens born. How many chickens does Farmer John have on his farm now?

Gemma 2B:

    Let's start from the difference in the number of chickens between the two farms, which we previously calculated as:
    \[
    \frac{23}{16} \text{ chickens}
    \]

    Now, instead of multiplying by \(\frac{19}{21}\), we need to multiply by \(\frac{19}{4}\):
    \[
    \frac{23}{16} \times \frac{19}{4} = \frac{23 \times 19}{16 \times 4} = \frac{437}{64}
    \]

    To simplify \(\frac{437}{64}\), we check for common factors. Since 437 and 64 have no common factors other than 1, the fraction is already in its simplest form.

    ### Answer: \(\frac{437}{64}\)

Expected answer: 437/64

Correct? Yes

🔗 Problem

Problem: A helicopter traveled a distance of 24/14 miles for a training exercise. During a second exercise, the helicopter flew the same distance. If we let 'w' be the distance flown for each exercise and 't' be the total distance flown for both exercises represented as the product of 'w' times 'w', what is the value of 't'?

Gemma 2B:

    First, let's identify the distance flown for each exercise, which is given as \( w = \frac{24}{14} \) miles.

    To simplify \( w \):
    \[ w = \frac{24}{14} = \frac{12}{7} \]

    The total distance flown for both exercises is represented as the product of \( w \) times \( w \):
    \[ t = w \times w \]

    Substituting \( w \) with \( \frac{12}{7} \):
    \[ t = \left( \frac{12}{7} \right) \times \left( \frac{12}{7} \right) \]
    \[ t = \frac{12 \times 12}{7 \times 7} \]
    \[ t = \frac{144}{49} \]

    ### Answer: \(\frac{144}{49}\)

Expected answer: 144/49

Correct? Yes

🔗 Follow-up

Problem: Let's consider a modification in our previous problem. Instead of initially flying a distance of 24/14 miles for each exercise, the helicopter now travels a distance of 17/14 miles for each exercise. If we still let 'w' be the distance flown for each exercise and 't' be the total distance flown for both exercises, which is represented as the product of 'w' times 'w', what is the new value of 't'?

Gemma 2B:

    First, let's identify the new distance flown for each exercise, which is given as \( w = \frac{17}{14} \) miles.

    The total distance flown for both exercises is represented as the product of \( w \) times \( w \):
    \[ t = w \times w \]

    Substituting \( w \) with \( \frac{17}{14} \):
    \[ t = \left( \frac{17}{14} \right) \times \left( \frac{17}{14} \right) \]
    \[ t = \frac{17 \times 17}{14 \times 14} \]
    \[ t = \frac{289}{196} \]

    ### Answer: \(\frac{289}{196}\)

Expected answer: 289/196

Correct? Yes

🔗 Problem

Problem: A UFO has made two fuel stops. During the first refuel, the UFO was able to fill 20/16 of its fuel tank. At the second fuel stop, the UFO filled another 24/25 of its tank. Calculate the total fraction of the tank that the UFO was able to refill during these two stops.

Gemma 2B:

    To find the total fraction of the tank that the UFO refilled during the two stops, we need to add the fractions from each stop.

    First refuel: \(\frac{20}{16}\)
    Second refuel: \(\frac{24}{25}\)

    First, simplify \(\frac{20}{16}\):
    \[
    \frac{20}{16} = \frac{5}{4}
    \]

    Now, we need to add \(\frac{5}{4}\) and \(\frac{24}{25}\). To do this, we need a common denominator. The least common multiple of 4 and 25 is 100.

    Convert \(\frac{5}{4}\) to a fraction with a denominator of 100:
    \[
    \frac{5}{4} = \frac{5 \times 25}{4 \times 25} = \frac{125}{100}
    \]

    Convert \(\frac{24}{25}\) to a fraction with a denominator of 100:
    \[
    \frac{24}{25} = \frac{24 \times 4}{25 \times 4} = \frac{96}{100}
    \]

    Now, add the two fractions:
    \[
    \frac{125}{100} + \frac{96}{100} = \frac{125 + 96}{100} = \frac{221}{100}
    \]

    Thus, the total fraction of the tank that the UFO refilled during the two stops is:
    \[
    \frac{221}{100}
    \]

    ### Answer: \(\frac{221}{100}\)

Expected answer: 221/100

Correct? Yes

🔗 Follow-up

Problem: On the next mission, the UFO only managed to refill 30/16 of its fuel tank at the first stop, while at the second fuel stop, it still refilled 24/25 of its tank. What is the total fraction of the tank that the UFO was able to refill during these two stops on this mission?

Gemma 2B:

    To find the total fraction of the tank that the UFO refilled during the two stops, we need to add the fractions from each stop.

    First refuel: \(\frac{30}{16}\)
    Second refuel: \(\frac{24}{25}\)

    First, simplify \(\frac{30}{16}\):
    \[
    \frac{30}{16} = \frac{15}{8}
    \]

    Now, we need to add \(\frac{15}{8}\) and \(\frac{24}{25}\). To do this, we need a common denominator. The least common multiple of 8 and 25 is 200.

    Convert \(\frac{15}{8}\) to a fraction with a denominator of 200:
    \[
    \frac{15}{8} = \frac{15 \times 25}{8 \times 25} = \frac{375}{200}
    \]

    Convert \(\frac{24}{25}\) to a fraction with a denominator of 200:
    \[
    \frac{24}{25} = \frac{24 \times 8}{25 \times 8} = \frac{192}{200}
    \]

    Now, add the two fractions:
    \[
    \frac{375}{200} + \frac{192}{200} = \frac{375 + 192}{200} = \frac{567}{200}
    \]

    Thus, the total fraction of the tank that the UFO refilled during the two stops is:
    \[
    \frac{567}{200}
    \]

    ### Answer: \(\frac{567}{200}\)

Expected answer: 567/200

Correct? Yes

🔗 Problem

Problem: Professor Astro is working on his telescope. He believes the power of his telescope is directly proportional to the sum of two fractions. The first fraction is the light gathering power of the lens, which is 24 out of 27. The second fraction is the zooming potential, which is 30 out of 26. So, Professor Astro calculates the sum of these two fractions ((24 / 27) + (30 / 26)). What does he find?

Gemma 2B:

    To find the sum of the two fractions \(\frac{24}{27}\) and \(\frac{30}{26}\), we first need to find a common denominator. The denominators are 27 and 26, and their least common multiple (LCM) is 27 * 26 = 702.

    Next, we convert each fraction to have this common denominator:

    1. Convert \(\frac{24}{27}\):
    \[
    \frac{24}{27} = \frac{24 \times 26}{27 \times 26} = \frac{624}{702}
    \]

    2. Convert \(\frac{30}{26}\):
    \[
    \frac{30}{26} = \frac{30 \times 27}{26 \times 27} = \frac{810}{702}
    \]

    Now, we add the two fractions:
    \[
    \frac{624}{702} + \frac{810}{702} = \frac{624 + 810}{702} = \frac{1434}{702}
    \]

    To simplify \(\frac{1434}{702}\), we find the greatest common divisor (GCD) of 1434 and 702. Using the Euclidean algorithm:

    1. \(1434 \div 702 = 2\) with a remainder of \(1434 - 2 \times 702 = 30\)
    2. \(702 \div 30 = 23\) with a remainder of \(702 - 23 \times 30 = 12\)
    3. \(30 \div 12 = 2\) with a remainder of \(30 - 2 \times 12 = 6\)
    4. \(12 \div 6 = 2\) with a remainder of \(12 - 2 \times 6 = 0\)

    The GCD is 6. We divide both the numerator and the denominator by 6:
    \[
    \frac{1434 \div 6}{702 \div 6} = \frac{239}{117}
    \]

    Thus, the simplified sum of the fractions is:
    \[
    \frac{239}{117}
    \]

    ### Answer: \(\frac{239}{117}\)

Expected answer: 239/117

Correct? Yes

🔗 Follow-up

Problem: Upon recalculating his telescope's potential with more accurate data, Professor Astro realizes that the light gathering power of his lens is actually 12 out of 27, not 24 out of 27 as previously calculated. Professor Astro recalculates the sum of these two fractions ((12 / 27) + (30 / 26)). What is the resulting sum now?

Gemma 2B:

    To find the sum of the two fractions \(\frac{12}{27}\) and \(\frac{30}{26}\), we first need to find a common denominator. The denominators are 27 and 26, and their least common multiple (LCM) is 27 * 26 = 702.

    Next, we convert each fraction to have this common denominator:

    1. Convert \(\frac{12}{27}\):
    \[
    \frac{12}{27} = \frac{12 \times 26}{27 \times 26} = \frac{312}{702}
    \]

    2. Convert \(\frac{30}{26}\):
    \[
    \frac{30}{26} = \frac{30 \times 27}{26 \times 27} = \frac{810}{702}
    \]

    Now, we add the two fractions:
    \[
    \frac{312}{702} + \frac{810}{702} = \frac{312 + 810}{702} = \frac{1122}{702}
    \]

    To simplify \(\frac{1122}{702}\), we find the greatest common divisor (GCD) of 1122 and 702. Using the Euclidean algorithm:

    1. \(1122 \div 702 = 1\) with a remainder of \(1122 - 1 \times 702 = 420\)
    2. \(702 \div 420 = 1\) with a remainder of \(702 - 1 \times 420 = 282\)
    3. \(420 \div 282 = 1\) with a remainder of \(420 - 1 \times 282 = 138\)
    4. \(282 \div 138 = 2\) with a remainder of \(282 - 2 \times 138 = 6\)
    5. \(138 \div 6 = 23\) with a remainder of \(138 - 23 \times 6 = 0\)

    The GCD is 6. We divide both the numerator and the denominator by 6:
    \[
    \frac{1122 \div 6}{702 \div 6} = \frac{187}{117}
    \]

    Thus, the simplified sum of the fractions is:
    \[
    \frac{187}{117}
    \]

    ### Answer: \(\frac{187}{117}\)

Expected answer: 187/117

Correct? Yes

🔗 Problem

Problem: Kiki is a lively little monkey. She loves eating bananas. She ate 11/6 bananas in the morning and the same amount in the afternoon. How many bananas did she eat in total? Express your answer as an improper fraction.

Gemma 2B:

    To find the total number of bananas Kiki ate, we need to add the amount she ate in the morning to the amount she ate in the afternoon.

    Kiki ate \( \frac{11}{6} \) bananas in the morning and \( \frac{11}{6} \) bananas in the afternoon.

    To find the total, we add these two fractions:
    \[
    \frac{11}{6} + \frac{11}{6}
    \]

    Since the denominators are the same, we can simply add the numerators:
    \[
    \frac{11 + 11}{6} = \frac{22}{6}
    \]

    Thus, Kiki ate \( \frac{22}{6} \) bananas in total.

    ### Answer: \(\frac{22}{6}\)

Expected answer: 11/3

Correct? Yes

🔗 Follow-up

Problem: Kiki realized that she miscounted the amount of bananas she ate. Instead of eating 11/6 bananas in the morning, she actually ate 11 whole bananas in the morning and another 11 in the afternoon. Given this new information, how many bananas did Kiki eat in total?

Gemma 2B:

    Given the new information, Kiki ate 11 whole bananas in the morning and another 11 whole bananas in the afternoon.

    To find the total number of bananas she ate, we simply add the two amounts:
    \[
    11 + 11 = 22
    \]

    Thus, Kiki ate a total of 22 bananas.

    ### Answer: 22

Expected answer: 22

Correct? Yes

🔗 Problem

Problem: Jaden is a dairy farmer who milks his cows twice a day. In the morning, his cows give him 26/14 gallons of milk per cow, and in the evening, they give 28/20 gallons of milk per cow. If he multiplies these two fractions together, how much milk is he getting from one cow in one day?

Gemma 2B:

    To find out how much milk Jaden gets from one cow in one day, we need to multiply the two fractions representing the morning and evening milk yields.

    First, let's simplify the fractions if possible:
    - The morning yield is \( \frac{26}{14} \). We can simplify this by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
      \[
      \frac{26}{14} = \frac{26 \div 2}{14 \div 2} = \frac{13}{7}
      \]

    - The evening yield is \( \frac{28}{20} \). We can simplify this by dividing both the numerator and the denominator by their greatest common divisor, which is 4:
      \[
      \frac{28}{20} = \frac{28 \div 4}{20 \div 4} = \frac{7}{5}
      \]

    Now, we multiply the simplified fractions:
    \[
    \frac{13}{7} \times \frac{7}{5}
    \]

    When multiplying fractions, we multiply the numerators together and the denominators together:
    \[
    \frac{13 \times 7}{7 \times 5} = \frac{91}{35}
    \]

    Next, we simplify \( \frac{91}{35} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 7:
    \[
    \frac{91}{35} = \frac{91 \div 7}{35 \div 7} = \frac{13}{5}
    \]

    Thus, Jaden gets \( \frac{13}{5} \) gallons of milk from one cow in one day.

    ### Answer: \(\frac{13}{5}\) gallons

Expected answer: 13/5

Correct? Yes

🔗 Follow-up

Problem: Following the previous problem, let's consider that Jaden has made some changes on his farm that affected the milk production in the evening. Now, his cows give him 28/10 gallons of milk per cow in the evening instead of the previous 28/20 gallons. If we still take the morning milk production as 26/14 gallons per cow, how much milk is he getting from one cow in one day now?

Gemma 2B:

    To determine the new total amount of milk Jaden gets from one cow in one day, we need to multiply the morning and evening milk yields.

    First, let's simplify the fractions if possible:
    - The morning yield is \( \frac{26}{14} \). We can simplify this by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
      \[
      \frac{26}{14} = \frac{26 \div 2}{14 \div 2} = \frac{13}{7}
      \]

    - The new evening yield is \( \frac{28}{10} \). We can simplify this by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
      \[
      \frac{28}{10} = \frac{28 \div 2}{10 \div 2} = \frac{14}{5}
      \]

    Now, we multiply the simplified fractions:
    \[
    \frac{13}{7} \times \frac{14}{5}
    \]

    When multiplying fractions, we multiply the numerators together and the denominators together:
    \[
    \frac{13 \times 14}{7 \times 5} = \frac{182}{35}
    \]

    Next, we simplify \( \frac{182}{35} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 7:
    \[
    \frac{182}{35} = \frac{182 \div 7}{35 \div 7} = \frac{26}{5}
    \]

    Thus, Jaden gets \( \frac{26}{5} \) gallons of milk from one cow in one day.

    ### Answer: \(\frac{26}{5}\) gallons

Expected answer: 26/5

Correct? Yes