🔗 Problem

Problem: Determine the product when you multiply the fraction 8/9 by the fraction 19/3. Please provide your answer as a simplified fraction.

Gemma 2B:

    To multiply fractions, we multiply the numerators (numbers on top) and multiply the denominators (numbers on the bottom), then simplify the resulting fraction.

     Numerator: 8 × 19 = 152
     Denominator: 9 × 3 = 27

    So, the product is 152/27.

    ### Answer: 152/27

Expected answer: 152/27

Correct? Yes

🔗 Follow-up

Problem: After determining the product when multiplying the fraction 8/9 by the fraction 19/3, take your result and then multiply it by 2. What does this new calculation result in when written as a simplified fraction?

Gemma 2B:

    We already found the product of 8/9 and 19/3 to be 152/27.

    Now, we multiply 152/27 by 2:

     Numerator: 152 × 2 = 304
     Denominator: 27 × 1 = 27 (since multiplying by 2 is the same as multiplying by 2/1)

    So, the new product is 304/27.

    ### Answer: 304/27

Expected answer: 304/27

Correct? Yes

🔗 Follow-up

Problem: Imagine you made an error in the original multiplication calculation where you incorrectly used 8/9 instead of the correct fraction 8/15. Can you correct this mistake and find the new product if you multiply 8/15 by 19/3? Please provide your answer as a simplified fraction.

Gemma 2B:

    Let's correct the mistake.

    To multiply fractions, we multiply the numerators (numbers on top) and multiply the denominators (numbers on the bottom), then simplify the resulting fraction.

     Numerator: 8 × 19 = 152
     Denominator: 15 × 3 = 45

    So, the product is 152/45.

    We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 1. Therefore, the simplified product is:

    152/45

    ### Answer: 152/45

Expected answer: 152/45

Correct? Yes

🔗 Problem

Problem: Julian is cutting sections of rope for a project. Each section is 13/2 feet long. He is only able to use 5/7 of each section. How many feet are in the usable part of each section of rope? Calculate this by multiplying 13/2 by 5/7.

Gemma 2B:

    To find the usable part of each section of rope, we need to multiply 13/2 by 5/7.

    First, we can multiply the numerators (the numbers on top): 13 × 5 = 65

    Next, we can multiply the denominators (the numbers on the bottom): 2 × 7 = 14

    Now, we can write the result as a fraction: 65/14

    We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 1. So, the simplified fraction is still 65/14.

    ### Answer: 65/14

Expected answer: 65/14

Correct? Yes

🔗 Follow-up

Problem: Suppose Julian is considering replacing the rope sections with new ones. Each new section is 17/2 feet long and he is still able to use only 5/7 of each section. How many feet are in the usable part of each new section of rope? Calculate this by multiplying 17/2 by 5/7.

Gemma 2B:

    To find the usable part of each new section of rope, we need to multiply 17/2 by 5/7.

    First, we can multiply the numerators (the numbers on top): 17 × 5 = 85

    Next, we can multiply the denominators (the numbers on the bottom): 2 × 7 = 14

    Now, we can write the result as a fraction: 85/14

    We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 1. So, the simplified fraction is still 85/14.

    ### Answer: 85/14

Expected answer: 85/14

Correct? Yes

🔗 Problem

Problem: What is the result when you multiply the whole number 12 by 1?

Gemma 2B:

    Reasoning: When you multiply a whole number by 1, the result is the same whole number. In this case, multiplying 12 by 1 does not change its value.

    ### Answer: 12

Expected answer: 12

Correct? Yes

🔗 Follow-up

Problem: If you take the product of 12 and 1, and then multiply that result by the fraction 7/16, what would be the final result? Write your answer as a simplified fraction or a whole number, if possible.

Gemma 2B:

    Reasoning: First, we multiply 12 by 1, which gives us 12. Then, we multiply 12 by the fraction 7/16.

    12 × (7/16) = (12 × 7) / 16 = 84 / 16

    We can simplify the fraction 84/16 by dividing both the numerator and the denominator by their greatest common divisor, which is 4.

    84 ÷ 4 = 21
    16 ÷ 4 = 4

    So, the simplified fraction is 21/4.

    ### Answer: 21/4

Expected answer: 21/4

Correct? Yes

🔗 Follow-up

Problem: Previously, you found the result of multiplying 12 by 1. Now, let's change the scenario and replace the 12 with 11. What would be the result when you multiply the whole number 11 by 1?

Gemma 2B:

    Reasoning: When you multiply a whole number by 1, the result is the same whole number. In this case, multiplying 11 by 1 does not change its value.

    ### Answer: 11

Expected answer: 11

Correct? Yes

🔗 Problem

Problem: Find the answer of the given math problem: 14 / 15 * 12 / 4, performing the multiplication operation on fractions and whole numbers. Provide your answer as a simplified fraction or, if possible, as a whole number.

Gemma 2B:

    To evaluate this expression, we need to follow the order of operations (PEMDAS):

    1. Divide 14 by 15: 14/15 = 14/15 (as a fraction)
    2. Divide 12 by 4: 12/4 = 3 (as a whole number)
    3. Multiply the two results: (14/15) × 3 = (14 × 3) / 15 = 42/15

    To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 3:

    42 ÷ 3 = 14
    15 ÷ 3 = 5

    So, the simplified fraction is: 14/5

    ### Answer: 14/5

Expected answer: 14/5

Correct? Yes

🔗 Follow-up

Problem: After finding the solution to 14 / 15 * 12 / 4 in the previous problem, now, use that answer and multiply it by 12 / 14. Provide your answer as a simplified fraction or, if possible, as a whole number.

Gemma 2B:

    We already found that 14 / 15 * 12 / 4 = 14/5. Now, we need to multiply this result by 12/14:

    (14/5) × (12/14) = ?

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

     Numerator: 14 × 12 = 168
     Denominator: 5 × 14 = 70

    So, the result is: 168/70

    To simplify this fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 14:

    168 ÷ 14 = 12
    70 ÷ 14 = 5

    So, the simplified fraction is: 12/5

    ### Answer: 12/5

Expected answer: 12/5

Correct? Yes

🔗 Follow-up

Problem: After finding the solution to the previous problem where you multiplied 14 / 15 * 12 / 4, let's change the scenario slightly. Suppose instead of having 15 parts of a whole, you only had 5 parts to begin with, making the original fraction 14 / 5 before multiplying by 12 / 4. How does this affect the final outcome? Calculate 14 / 5 * 12 / 4 and provide your answer as a simplified fraction or a whole number, if possible.

Gemma 2B:

    Let's recalculate the expression with the new fraction:

    1. Multiply 14/5 by 12/4:

    (14/5) × (12/4) = ?

    2. Multiply the numerators (numbers on top): 14 × 12 = 168
    3. Multiply the denominators (numbers on the bottom): 5 × 4 = 20
    4. Write the result as a fraction: 168/20

    To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 4:

    168 ÷ 4 = 42
    20 ÷ 4 = 5

    So, the simplified fraction is: 42/5

    ### Answer: 42/5

Expected answer: 42/5

Correct? Yes

🔗 Problem

Problem: Compute the value for the following expression: 15 * (5 / 17). Write your answer as a decimal rounded to the nearest hundredth.

Gemma 2B:

    To evaluate this expression, we need to follow the order of operations (PEMDAS):

    1. Divide 5 by 17: 5 / 17 = 0.29 (rounded to the nearest hundredth)
    2. Multiply 15 by the result: 15 * 0.29 = 4.35

    ### Answer: 4.35

Expected answer: 75/17

Correct? No

🔗 Problem

Problem: Find the result of the following calculation: 16 * (12 / 19)

Gemma 2B:

    To evaluate this expression, we need to follow the order of operations (PEMDAS):

    1. Divide 12 by 19: 12 / 19 = 0.63
    2. Multiply 16 by the result: 16 * 0.63 = 10.08

    ### Answer: 10.08

Expected answer: 192/19

Correct? No

🔗 Problem

Problem: If a farmer has 10/17 of a barrel of apples and he decides to multiply that amount by 18/5 barrels, how many barrels of apples will he have in total? Calculate your answer keeping it as a simplified fraction.

Gemma 2B:

    Reasoning: To find the total number of barrels of apples, we need to multiply the initial amount by the multiplier.

    Initial amount: 10/17 barrels
    Multiplier: 18/5 barrels

    Total amount = Initial amount × Multiplier
    = (10/17) × (18/5)
    = (10 × 18) / (17 × 5)
    = 180 / 85
    = 36 / 17 (simplified fraction)

    ### Answer: 36/17

Expected answer: 36/17

Correct? Yes

🔗 Follow-up

Problem: Given the farmer's new total of apples from the previous question, he decides to increase his total number of barrels by a factor of 7. Calculate the new total number of barrels of apples the farmer will have. Represent your answer as a simplified fraction where appropriate.

Gemma 2B:

    Reasoning: To find the new total number of barrels of apples, we need to multiply the current total by the factor of increase.

    Current total: 36/17 barrels
    Factor of increase: 7

    New total = Current total × Factor of increase
    = (36/17) × 7
    = (36 × 7) / 17
    = 252 / 17

    ### Answer: 252/17

Expected answer: 252/17

Correct? Yes

🔗 Follow-up

Problem: Suppose the farmer initially started with 10/17 of a barrel of apples but this time he decided to increase that amount by multiplying it with 19/5 barrels instead of 18/5. Calculate how many barrels of apples will he have now. Represent your answer as a simplified fraction.

Gemma 2B:

    Reasoning: To find the total number of barrels of apples, we need to multiply the initial amount by the multiplier.

    Initial amount: 10/17 barrels
    Multiplier: 19/5 barrels

    Total amount = Initial amount × Multiplier
    = (10/17) × (19/5)
    = (10 × 19) / (17 × 5)
    = 190 / 85
    = 38 / 17 (simplified fraction)

    ### Answer: 38/17

Expected answer: 38/17

Correct? Yes

🔗 Problem

Problem: Simplify the expression: (20 / 4) * (1 / 5). What is the result?

Gemma 2B:

    To simplify the expression, we need to follow the order of operations (PEMDAS):

    1. Evaluate the expressions inside the parentheses:
       (20 / 4) = 5
       (1 / 5) = 1/5

    2. Multiply the results:
       5 * (1/5) = 1

    ### Answer: 1

Expected answer: 1

Correct? Yes

🔗 Follow-up

Problem: After you have found the result of (20 / 4) * (1 / 5), multiply it by (16 / 3). What is the final result?

Gemma 2B:

    We already found the result of (20 / 4) * (1 / 5) to be 1.

    Now, we need to multiply 1 by (16 / 3):

    1 * (16 / 3) = 1 * (16/3) = 16/3

    ### Answer: 16/3

Expected answer: 16/3

Correct? Yes

🔗 Problem

Problem: Julia has 11 boxes and she decides to fill 8/11 of each box with apples. If each box can hold up to 1 bushel of apples, how many bushels of apples did she use in total?

Gemma 2B:

    Reasoning: Julia has 11 boxes and she decides to fill 8/11 of each box with apples.
    So, the total amount of apples she used is 11 * (8/11) = 8 bushels.
    ### Answer: 8

Expected answer: 8

Correct? Yes

🔗 Follow-up

Problem: In the continuation of the previous situation, Julia decided to sell 7/14, or half, of all the bushels of apples she had. How many bushels of apples did Julia sell?

Gemma 2B:

    Reasoning: Julia had 8 bushels of apples and she decided to sell half of them.
    Half of 8 is 8 * (1/2) = 8 * (7/14) = 4 bushels.
    ### Answer: 4

Expected answer: 4

Correct? Yes

🔗 Follow-up

Problem: In continuation to the previous scenario, suppose Julia initially had 6 boxes instead of 11. How many bushels of apples would she have if she still fills 8/11 of each box with apples?

Gemma 2B:

    Reasoning: Julia has 6 boxes and she decides to fill 8/11 of each box with apples.
    So, the total amount of apples she used is 6 * (8/11) = 48/11 = 4 4/11 bushels.
    Since we can't have a fraction of a bushel, we can say she used 4 full bushels and had 4/11 of a bushel left over.
    ### Answer: 4

Expected answer: 48/11

Correct? No

🔗 Problem

Problem: In a bakery, a baker uses 9/4 cups of sugar for a cake and 16/15 cups for a batch of cookies. If he needs the equal amount of sugar for the cake and the cookies, how much sugar in cups will he use in total? Calculate it by multiplying the two fractions.

Gemma 2B:

    To find the total amount of sugar used, we need to multiply the two fractions:

    (9/4) × (16/15) = ?

    To multiply fractions, we multiply the numerators (numbers on top) and multiply the denominators (numbers on the bottom), like this:

    (9 × 16) / (4 × 15) = 144 / 60

    We can simplify the fraction 144/60 by dividing both numbers by their greatest common divisor, which is 12:

    144 ÷ 12 = 12
    60 ÷ 12 = 5

    So, the simplified fraction is:

    12/5

    To convert this to a mixed number, we can divide the numerator by the denominator:

    12 ÷ 5 = 2 with a remainder of 2

    So, the mixed number is:

    2 2/5

    To convert this to a decimal, we can divide the numerator by the denominator:

    2 2/5 = 2.4

    ### Answer: 2.4 cups

Expected answer: 12/5

Correct? No

🔗 Problem

Problem: If Peter was making home-made pizza and he needed 19/9 cups of flour and 14/18 cups of sugar for his pizza dough, how many cups of ingredients in total did he need? Multiply the amounts of flour and sugar together to find the total.

Gemma 2B:

    To find the total amount of ingredients, we need to multiply the amounts of flour and sugar together.

    First, let's simplify the fractions:

    19/9 = 2 1/9 cups of flour
    14/18 = 7/9 cups of sugar (since 14 ÷ 2 = 7 and 18 ÷ 2 = 9)

    Now, let's multiply the amounts together:

    (2 1/9) × (7/9) = ?

    To multiply mixed numbers, we need to convert them to improper fractions:

    2 1/9 = (2 × 9 + 1) / 9 = 19/9
    7/9 = 7/9 (already an improper fraction)

    Now, multiply the numerators (the numbers on top) and multiply the denominators (the numbers on the bottom):

    (19 × 7) / (9 × 9) = 133 / 81

    So, Peter needs 133/81 cups of ingredients in total.

    ### Answer: 133/81

Expected answer: 133/81

Correct? Yes

🔗 Follow-up

Problem: In a new experiment, Peter decides to change his recipe slightly. Instead of using 19/9 cups of flour, he now uses 14/9 cups of flour but keeps the amount of sugar the same (14/18 cups). How would this affect the overall amount of ingredients used for his pizza dough? Multiply the new amount of flour and the original amount of sugar together to find the new total.

Gemma 2B:

    Let's multiply the new amount of flour and the original amount of sugar together:

    14/9 cups of flour × 14/18 cups of sugar = ?

    To multiply fractions, we multiply the numerators (the numbers on top) and multiply the denominators (the numbers on the bottom):

    (14 × 14) / (9 × 18) = 196 / 162

    We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

    196 ÷ 2 = 98
    162 ÷ 2 = 81

    So, the new total amount of ingredients is:

    98 / 81 cups

    ### Answer: 98/81

Expected answer: 98/81

Correct? Yes

🔗 Problem

Problem: What is the result when you multiply the fractions 14/7 and 16/4?

Gemma 2B:

    To multiply fractions, we multiply the numerators (the numbers on top) and multiply the denominators (the numbers on the bottom), then simplify the result.

    In this case, we have:

    (14/7) × (16/4) = (14 × 16) / (7 × 4)

    = 224 / 28

    We can simplify the fraction 224/28 by dividing both the numerator and the denominator by their greatest common divisor, which is 28.

    224 ÷ 28 = 8
    28 ÷ 28 = 1

    So, the result is:

    8/1 = 8

    ### Answer: 8

Expected answer: 8

Correct? Yes

🔗 Follow-up

Problem: Continuing from our previous problem, instead of multiplying 14/7 and 16/4, let's consider we have 14/7 and 13/4. What is the result when you multiply these two fractions?

Gemma 2B:

    To multiply fractions, we multiply the numerators (the numbers on top) and multiply the denominators (the numbers on the bottom), then simplify the result.

    In this case, we have:

    (14/7) × (13/4) = (14 × 13) / (7 × 4)

    = 182 / 28

    We can simplify the fraction 182/28 by dividing both the numerator and the denominator by their greatest common divisor, which is 14.

    182 ÷ 14 = 13
    28 ÷ 14 = 2

    So, the result is:

    13/2

    ### Answer: 13/2

Expected answer: 13/2

Correct? Yes

🔗 Problem

Problem: Calculate the result of (17 / 18) multiplied by (13 / 8). Write the answer as a simplified fraction.

Gemma 2B:

    To calculate the result, we need to multiply the two fractions:

    (17 / 18) × (13 / 8) = (17 × 13) / (18 × 8)

    = 221 / 144

    We can simplify this fraction by finding the greatest common divisor (GCD) of 221 and 144, which is 1. Therefore, the fraction is already in its simplest form.

    ### Answer: 221/144

Expected answer: 221/144

Correct? Yes

🔗 Follow-up

Problem: After calculating the result of (17 / 18) multiplied by (13 / 8), multiply the answer you got by itself. Write the final result as a simplified fraction.

Gemma 2B:

    We already calculated the result of (17 / 18) multiplied by (13 / 8) to be 221/144. Now, we need to multiply this result by itself:

    (221/144) × (221/144) = (221 × 221) / (144 × 144)

    = 48841 / 20736

    To simplify this fraction, we can find the greatest common divisor (GCD) of 48841 and 20736, which is 1. Therefore, the fraction is already in its simplest form.

    ### Answer: 48841/20736

Expected answer: 48841/20736

Correct? Yes

🔗 Problem

Problem: Evaluate the following expression: ((15 / 15) * 18) and write your answer.

Gemma 2B:

    To evaluate this expression, we need to follow the order of operations (PEMDAS):

    1. Evaluate the expression inside the parentheses: 15 / 15 = 1
    2. Multiply 1 by 18: 1 * 18 = 18

    ### Answer: 18

Expected answer: 18

Correct? Yes

🔗 Follow-up

Problem: For this next question, let's slightly adjust our variables. Instead of doing (15 / 15) * 18 as in the previous problem, let's try (15 / 16) * 18. What's the result of this multiplication?

Gemma 2B:

    To evaluate this expression, we need to follow the order of operations (PEMDAS):

    1. Evaluate the expression inside the parentheses: 15 / 16 = 0.9375 (approximately)
    2. Multiply 0.9375 by 18: 0.9375 * 18 = 16.875

    ### Answer: 16.875

Expected answer: 135/8

Correct? Yes

🔗 Problem

Problem: A group of 18 students is planning a pizza party. If each student will eat 17/20 of a pizza, how many pizzas should they order?

Gemma 2B:

    Reasoning: To find the total amount of pizza needed, we multiply the number of students by the amount each student will eat: 18 × (17/20) = 18 × 0.85 = 15.3 pizzas.

    Since we can't order a fraction of a pizza, we round up to the nearest whole number to ensure everyone gets enough pizza.

    ### Answer: 16

Expected answer: 153/10

Correct? No

🔗 Problem

Problem: Sophie picked 9 pounds of apples. She decided to make apple pies for a community gathering, and each pie requires about 1/14 of the total apples she picked. Sophie made 7 pies. How many pounds of apples did she use in total? Calculate it using the multiplication of fraction and a whole number, specifically ((9 / 14) * 7). Write your answer as a simplified fraction or a whole number.

Gemma 2B:

    Reasoning: Sophie used 1/14 of the total apples she picked for each pie, and she made 7 pies. To find the total amount of apples used, we can multiply the fraction of apples used per pie by the number of pies: ((9 / 14) * 7).

    First, multiply the numerator (9) by 7: 9 * 7 = 63.

    Next, multiply the denominator (14) by 1 (since we're not changing the denominator): 14 * 1 = 14.

    Now, rewrite the expression as a fraction: 63/14.

    To simplify the fraction, divide both the numerator and the denominator by their greatest common divisor, which is 7: 63 ÷ 7 = 9 and 14 ÷ 7 = 2.

    So, the simplified fraction is 9/2.

    To convert the fraction to a mixed number, divide the numerator by the denominator: 9 ÷ 2 = 4 with a remainder of 1. Therefore, the mixed number is 4 1/2.

    ### Answer: 4 1/2 pounds

Expected answer: 9/2

Correct? Yes

🔗 Follow-up

Problem: Sophie initially thought she had 9 pounds of apples, but upon rechecking, she found she actually had 18 pounds. Sophie still decided to make pies for the community gathering, with each pie requiring 1/14 of the total amount of apples. If Sophie still made 7 pies just like before, calculate how many pounds of apples she used this time. Solve the problem using the multiplication of the fraction and the whole number ((18 / 14) * 7). Write your answer as a simplified fraction or a whole number.

Gemma 2B:

    Reasoning: Sophie used 1/14 of the total apples she picked for each pie, and she made 7 pies. To find the total amount of apples used, we can multiply the fraction of apples used per pie by the number of pies: ((18 / 14) * 7).

    First, multiply the numerator (18) by 7: 18 * 7 = 126.

    Next, multiply the denominator (14) by 1 (since we're not changing the denominator): 14 * 1 = 14.

    Now, rewrite the expression as a fraction: 126/14.

    To simplify the fraction, divide both the numerator and the denominator by their greatest common divisor, which is 14: 126 ÷ 14 = 9 and 14 ÷ 14 = 1.

    So, the simplified fraction is 9/1, which is equal to 9.

    ### Answer: 9 pounds

Expected answer: 9

Correct? Yes

🔗 Problem

Problem: John bought 2 boxes of cookies with 6/8 of a dozen cookies in each box. How many dozens of cookies did John buy in total?

Gemma 2B:

    Reasoning: Since there are 12 cookies in a dozen, 6/8 of a dozen is equivalent to (6/8) × 12 = 9 cookies.
    John bought 2 boxes with 9 cookies in each box, so he bought a total of 2 × 9 = 18 cookies.
    Since there are 12 cookies in a dozen, John bought 18/12 = 1.5 dozens of cookies.
    ### Answer: 1.5

Expected answer: 3/2

Correct? Yes

🔗 Follow-up

Problem: John initially bought 2 boxes of cookies with 6/8 of a dozen cookies in each box. However, instead of 2 boxes, suppose that John decided to buy 7 identical boxes. How many dozens of cookies would he have in total?

Gemma 2B:

    Reasoning: Since there are 12 cookies in a dozen, 6/8 of a dozen is equivalent to (6/8) × 12 = 9 cookies.
    John bought 7 boxes with 9 cookies in each box, so he bought a total of 7 × 9 = 63 cookies.
    Since there are 12 cookies in a dozen, John bought 63/12 = 5.25 dozens of cookies.
    ### Answer: 5.25

Expected answer: 21/4

Correct? No

🔗 Problem

Problem: Tom has a box that could potentially hold 6 gallons of paint. However, due to the design of the box, it can only be filled up to 16/17 of its capacity. How many gallons of paint can Tom actually store in the box?

Gemma 2B:

    Reasoning: The box can hold 6 gallons of paint, but it can only be filled up to 16/17 of its capacity. To find the actual amount of paint Tom can store, we need to multiply the capacity of the box by the fraction of its capacity that it can be filled.

    6 gallons x (16/17) = 6 x 16/17 = 96/17 = 5.647058823529412 (approximately)

    Since we can't have a fraction of a gallon, we'll round down to the nearest whole number.

    ### Answer: 5 gallons

Expected answer: 96/17

Correct? No

🔗 Problem

Problem: If Mary has 14 marbles and she gives away 4/10 of them, how many marbles does Mary give away?

Gemma 2B:

    Reasoning: To find the number of marbles Mary gives away, we need to find 4/10 of 14. To do this, we can multiply 14 by 4/10.

    14 × (4/10) = 14 × 0.4 = 5.6

    Since we can't give away a fraction of a marble, we'll round down to the nearest whole number. Mary gives away 5 marbles.

    ### Answer: 5

Expected answer: 28/5

Correct? No

🔗 Problem

Problem: Calculate the product of 19/6 and 4/4. Write the resulting answer as a simplified fraction.

Gemma 2B:

    To calculate the product, we multiply the numerators (19 and 4) and multiply the denominators (6 and 4), then simplify the resulting fraction:

    (19/6) × (4/4) = (19 × 4) / (6 × 4) = 76 / 24

    We can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 4:

    76 ÷ 4 = 19
    24 ÷ 4 = 6

    So, the simplified fraction is:

    19/6

    ### Answer: 19/6

Expected answer: 19/6

Correct? Yes

🔗 Follow-up

Problem: Use the answer you obtained from the previous word problem as a fraction, multiply it by itself, and write the resulting answer as a simplified fraction.

Gemma 2B:

    We multiply the fraction 19/6 by itself:

    (19/6) × (19/6) = (19 × 19) / (6 × 6) = 361 / 36

    We can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 1 (since they are coprime):

    361 ÷ 1 = 361
    36 ÷ 1 = 36

    So, the simplified fraction is:

    361/36

    ### Answer: 361/36

Expected answer: 361/36

Correct? Yes

🔗 Follow-up

Problem: In the previous word problem, use 19/6 and 9/4 instead of 19/6 and 4/4. Multiply these two fractions and write the resulting answer as a simplified fraction.

Gemma 2B:

    To calculate the product, we multiply the numerators (19 and 9) and multiply the denominators (6 and 4), then simplify the resulting fraction:

    (19/6) × (9/4) = (19 × 9) / (6 × 4) = 171 / 24

    We can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3:

    171 ÷ 3 = 57
    24 ÷ 3 = 8

    So, the simplified fraction is:

    57/8

    ### Answer: 57/8

Expected answer: 57/8

Correct? Yes

🔗 Problem

Problem: What is the result when you divide 16 by 8 and then multiply the result by 10?

Gemma 2B:

    Reasoning: When we divide 16 by 8, we get 16 ÷ 8 = 2.
    Then, we multiply the result by 10, so 2 × 10 = 20.
    ### Answer: 20

Expected answer: 20

Correct? Yes

🔗 Follow-up

Problem: Using the result you obtained from the previous question, what is the outcome when you multiply it by 1/17?

Gemma 2B:

    Reasoning: We multiply 20 by 1/17, which is equivalent to dividing 20 by 17.
    So, 20 × (1/17) = 20 ÷ 17 = 20/17.
    ### Answer: 20/17

Expected answer: 20/17

Correct? Yes

🔗 Follow-up

Problem: What would the result have been in the original problem if you had divided 16 by 10 instead of 8 before multiplying by 10?

Gemma 2B:

    Reasoning: When we divide 16 by 10, we get 16 ÷ 10 = 1.6.
    Then, we multiply the result by 10, so 1.6 × 10 = 16.
    ### Answer: 16

Expected answer: 16

Correct? Yes

🔗 Problem

Problem: What is the product when 18 is multiplied by 1/11? Please provide your answer as a decimal or a simplified fraction.

Gemma 2B:

    To find the product, I'll multiply 18 by 1/11:

    18 × 1/11 = 18 × 0.090909... (converting 1/11 to a decimal)
    = 1.636363...

    To simplify the answer, I can convert it back to a fraction:

    1.636363... = 18/11

    ### Answer: 18/11 or 1.636363...

Expected answer: 18/11

Correct? Yes

🔗 Follow-up

Problem: Following up on the previous question, after you have found the product of 18 and 1/11, now multiply that result by 7. What is the final output? Please provide your answer as a decimal or a simplified fraction.

Gemma 2B:

    I'll multiply the result by 7:

    (18/11) × 7 = (18 × 7) / 11
    = 126 / 11
    = 11.454545... (converting to a decimal)

    To simplify the answer, I can keep it as a decimal or rewrite it as a mixed number:

    11.454545... = 11 5/11

    ### Answer: 126/11 or 11.454545... or 11 5/11

Expected answer: 126/11

Correct? Yes

🔗 Follow-up

Problem: In continuation of the previous problem, suppose instead of 1/11, we multiply 18 by 4/11. What would be the new product? Provide your answer as a decimal or a simplified fraction.

Gemma 2B:

    To find the product, I'll multiply 18 by 4/11:

    18 × 4/11 = 18 × 4 × 1/11
    = 18 × 4 × 0.090909... (converting 1/11 to a decimal)
    = 18 × 0.363636...
    = 6.545454...

    To simplify the answer, I can convert it back to a fraction:

    6.545454... = 72/11

    ### Answer: 72/11 or 6.545454...

Expected answer: 72/11

Correct? Yes

🔗 Problem

Problem: Mary has a garden with apple trees. Each tree produces 3 bushels of apples, and she has 6 trees in total. However, due to pests, only half of her trees produce apples this year. How many bushels of apples does Mary harvest from her garden this year?

Gemma 2B:

    Reasoning: Since only half of Mary's trees produce apples, she has 6/2 = 3 trees producing apples this year.
    Each tree produces 3 bushels of apples, so Mary harvests 3 x 3 = 9 bushels of apples this year.
    ### Answer: 9

Expected answer: 9

Correct? Yes

🔗 Problem

Problem: Calculate the result of the following problem: (13 / 10) * (14 / 11). Please write your answer as a simplified fraction.

Gemma 2B:

    To evaluate this expression, we need to follow the order of operations (PEMDAS):

    1. Divide 13 by 10: 13 / 10 = 13/10
    2. Divide 14 by 11: 14 / 11 = 14/11
    3. Multiply the two results: (13/10) * (14/11) = (13*14) / (10*11) = 182 / 110

    To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 2:

    182 ÷ 2 = 91
    110 ÷ 2 = 55

    So, the simplified fraction is:

    ### Answer: 91/55

Expected answer: 91/55

Correct? Yes

🔗 Follow-up

Problem: Taking your answer from the previous problem, (13 / 10) * (14 / 11), calculate the square of that answer. Write your answer as a simplified fraction.

Gemma 2B:

    To calculate the square of the answer, we need to multiply the answer by itself:

    (91/55) × (91/55) = (91 × 91) / (55 × 55) = 8281 / 3025

    To simplify the fraction, we can look for common factors between the numerator and the denominator. In this case, both numbers are divisible by 7:

    8281 ÷ 7 = 1183
    3025 ÷ 7 = 433

    So, the simplified fraction is:

    ### Answer: 1183/433

Expected answer: 8281/3025

Correct? No

🔗 Follow-up

Problem: Given the previous problem, now imagine that instead of multiplying by (14/11), we are multiplying by (14/4). What would be the result of (13 / 10) * (14 / 4)? Please give your answer as a simplified fraction.

Gemma 2B:

    Let's evaluate the expression:

    1. Divide 13 by 10: 13 / 10 = 13/10
    2. Divide 14 by 4: 14 / 4 = 14/4 = 7/2 (since 14 is divisible by 4)
    3. Multiply the two results: (13/10) * (7/2) = (13*7) / (10*2) = 91 / 20

    The result is already in its simplest form, so we don't need to simplify further.

    ### Answer: 91/20

Expected answer: 91/20

Correct? Yes

🔗 Problem

Problem: James bought 20 very thin fabrics. Each piece of fabric has 2/19 meters of thread. How many meters of thread are there in total in 20 pieces of fabric? Calculate it by multiplying 2/19 by 20.

Gemma 2B:

    Reasoning: To find the total meters of thread, we need to multiply the number of pieces of fabric (20) by the amount of thread in each piece (2/19 meters).

    To multiply a whole number by a fraction, we can multiply the whole number by the numerator (2) and then divide by the denominator (19).

    So, we multiply 20 by 2: 20 × 2 = 40

    Then, we divide 40 by 19: 40 ÷ 19 = 2 2/19

    ### Answer: 2 2/19 or 2.11 (approximately)

Expected answer: 40/19

Correct? Yes

🔗 Problem

Problem: What is the result of 1 times 16?

Gemma 2B:

    Reasoning: The product of 1 and 16 is simply the value of 16 itself, since multiplying any number by 1 does not change its value.

    ### Answer: 16

Expected answer: 16

Correct? Yes

🔗 Follow-up

Problem: Taking the result you found from multiplying 1 times 16, what do you get if you then multiply that result by 2/11?

Gemma 2B:

    Reasoning: We multiply 16 by 2/11, which is equivalent to multiplying 16 by 2 and then dividing by 11.

    16 × 2 = 32
    32 ÷ 11 = 32/11

    ### Answer: 32/11 or 2 10/11

Expected answer: 32/11

Correct? Yes

🔗 Follow-up

Problem: If instead of 16, you were to multiply 1 by 11. What would the result be?

Gemma 2B:

    Reasoning: The product of 1 and 11 is simply the value of 11 itself, since multiplying any number by 1 does not change its value.

    ### Answer: 11

Expected answer: 11

Correct? Yes

🔗 Problem

Problem: Calculate the result of the following math problem: (18 / 12) * 6. Write your answer as a simplified fraction or a whole number.

Gemma 2B:

    To evaluate this expression, we need to follow the order of operations (PEMDAS):

    1. Divide 18 by 12: 18 ÷ 12 = 3/2
    2. Multiply the result by 6: (3/2) × 6 = (3 × 6) / 2 = 18/2
    3. Simplify the fraction: 18/2 = 9

    ### Answer: 9

Expected answer: 9

Correct? Yes

🔗 Follow-up

Problem: After calculating the result of the first problem as e = (18 / 12) * 6, perform an additional calculation. Multiply the result e by 9/15. Write the result as a simplified fraction or a whole number.

Gemma 2B:

    We already calculated the value of e: e = (18 / 12) * 6 = 9

    Now, we need to multiply e by 9/15:

    e × (9/15) = 9 × (9/15) = 81/15

    To simplify this fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 3:

    81 ÷ 3 = 27
    15 ÷ 3 = 5

    So, the simplified result is:

    ### Answer: 27/5

Expected answer: 27/5

Correct? Yes

🔗 Follow-up

Problem: After completing the first calculation, let's modify it a bit. Originally we had (18 / 12) * 6. Now let's change the dividend to 10, keeping the divisor and the multiplier the same. So we have (10 / 12) * 6. Calculate the result and write your answer as a simplified fraction or a whole number.

Gemma 2B:

    Let's evaluate the modified expression:

    1. Divide 10 by 12: 10 ÷ 12 = 5/6
    2. Multiply the result by 6: (5/6) × 6 = (5 × 6) / 6 = 30/6
    3. Simplify the fraction: 30/6 = 5

    ### Answer: 5

Expected answer: 5

Correct? Yes

🔗 Problem

Problem: Here's a math problem about multiplying fractions and whole numbers: Johnny has 8 friends. He promised to give each of his friends 13/8 pounds of candy. How many pounds of candy will Johnny need in total?

Gemma 2B:

    Reasoning: To find the total amount of candy Johnny needs, we need to multiply the number of friends he has by the amount of candy he promised to give each friend.

    Since Johnny has 8 friends, and he promised to give each friend 13/8 pounds of candy, we can multiply 8 by 13/8:

    8 × (13/8) = 8 × 13 / 8

    To multiply a whole number by a fraction, we can multiply the whole number by the numerator (13) and then divide by the denominator (8):

    8 × 13 = 104

    Now, divide by the denominator:

    104 ÷ 8 = 13

    So, Johnny will need 13 pounds of candy in total.

    ### Answer: 13

Expected answer: 13

Correct? Yes

🔗 Problem

Problem: Mathew was cutting up some square blocks for a craft project. Each block was 16/18th of a foot long. Mathew needed 16/7th of a foot long blocks for his project. If he uses all the blocks, how long in feet would the blocks be in total? Solve this by using multiplication of fractions.

Gemma 2B:

    Let's break this problem down step by step.

    We know that each block is 16/18th of a foot long, and Mathew needs 16/7th of a foot long blocks. To find the total length of the blocks, we need to multiply the number of blocks by the length of each block.

    First, let's convert the mixed fraction 16/7 to an improper fraction: 16/7 = 16/1 ÷ 1/7 = 16/1 × 7/1 = 112/7.

    Now, we can multiply the number of blocks by the length of each block:

    (Number of blocks) × (Length of each block) = Total length
    = (112/7) × (16/18)
    = (112 × 16) / (7 × 18)
    = 1792 / 126
    = 14 4/7 (simplified)

    So, the total length of the blocks would be 14 4/7 feet.

    ### Answer: 14 4/7 feet.

Expected answer: 128/63

Correct? No

🔗 Problem

Problem: What is the result when you multiply the fraction 8/8 by 15?

Gemma 2B:

    Reasoning: The fraction 8/8 is equal to 1, since 8 divided by 8 is 1.
    Multiplying 1 by 15 gives us 15.
    ### Answer: 15

Expected answer: 15

Correct? Yes

🔗 Follow-up

Problem: After you've calculated the result of 8/8 multiplied by 15, what do you get if you further multiple that result by 19/18?

Gemma 2B:

    Reasoning: We already know that 8/8 multiplied by 15 is 15.
    To multiply 15 by 19/18, we can rewrite the fraction as a decimal: 19/18 = 1.0556 (approximately).
    Now, multiply 15 by 1.0556: 15 × 1.0556 = 15.8333 (approximately).
    ### Answer: 15.83 (approximately)

Expected answer: 95/6

Correct? No

🔗 Follow-up

Problem: Suppose instead of having a fraction of 8/8 in the previous question, there was a fraction of 8/7. What would be the result when that fraction is multiplied by 15?

Gemma 2B:

    Reasoning: To multiply a fraction by a whole number, we multiply the numerator (the top number) by the whole number, and keep the denominator (the bottom number) the same.
    So, multiplying 8/7 by 15 gives us (8 × 15)/7 = 120/7.
    ### Answer: 120/7

Expected answer: 120/7

Correct? Yes

🔗 Problem

Problem: A recipe for apple pie requires 10 apples. However, you only have two thirds of the amount needed. How many apples do you have if you multiply 10 apples by 4/6?

Gemma 2B:

    Reasoning: To find two thirds of 10 apples, we can multiply 10 by the fraction 2/3. Since 4/6 is equivalent to 2/3, we can multiply 10 by 4/6.

    10 × (4/6) = 10 × 2/3 = 20/3 = 6.67 (approximately)

    Since we can't have a fraction of an apple, we'll round down to the nearest whole number.

    ### Answer: 6

Expected answer: 20/3

Correct? No

🔗 Problem

Problem: What is the product when you multiply the fraction 18/12 by the whole number 11?

Gemma 2B:

    To multiply a fraction by a whole number, we can multiply the numerator (18) by the whole number (11), and keep the denominator (12) the same.

    So, we get: (18 × 11) / 12 = 198 / 12

    We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 6.

    198 ÷ 6 = 33
    12 ÷ 6 = 2

    So, the simplified product is: 33/2

    ### Answer: 33/2

Expected answer: 33/2

Correct? Yes

🔗 Follow-up

Problem: You found that the product of 18/12 and 11 was 'x'. Now, multiply the result 'x' by the fraction 14/13. What is the new result? Write your answer as a simplified fraction.

Gemma 2B:

    We already found that x = 33/2. Now, we need to multiply x by 14/13.

    To multiply two fractions, we multiply the numerators and multiply the denominators:

    (33/2) × (14/13) = (33 × 14) / (2 × 13) = 462 / 26

    We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2.

    462 ÷ 2 = 231
    26 ÷ 2 = 13

    So, the simplified result is: 231/13

    ### Answer: 231/13

Expected answer: 231/13

Correct? Yes

🔗 Follow-up

Problem: In the previous scenario, we calculated the product of the fraction 18/12 and the whole number 11. Now, consider what would happen if instead of multiplying by 11, we multiplied by 10. What would be the product of 18/12 and 10? Please write your answer as a simplified fraction or whole number as appropriate.

Gemma 2B:

    To multiply the fraction 18/12 by 10, we can multiply the numerator (18) by 10, and keep the denominator (12) the same.

    So, we get: (18 × 10) / 12 = 180 / 12

    We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 6.

    180 ÷ 6 = 30
    12 ÷ 6 = 2

    So, the simplified product is: 30/2

    We can further simplify this by dividing both the numerator and the denominator by 2.

    30 ÷ 2 = 15
    2 ÷ 2 = 1

    So, the final simplified product is: 15

    ### Answer: 15

Expected answer: 15

Correct? Yes

🔗 Problem

Problem: In a pizza shop, the mushroom pizza, which is very popular, usually sells out in just 9/20 of an hour. The pizza chef can make 16 pizzas in just one hour. How many mushroom pizzas can the chef make before they sell out?

Gemma 2B:

    Reasoning: The pizza chef can make 16 pizzas in 1 hour. Since the mushroom pizzas sell out in 9/20 of an hour, we need to find the number of pizzas the chef can make in 9/20 of an hour.

    To do this, we can multiply the number of pizzas the chef can make in 1 hour by the fraction of the hour: 16 × (9/20) = 16 × 0.45 = 7.2.

    Since we can't make a fraction of a pizza, we round down to the nearest whole number, as we can't make a part of a pizza.

    ### Answer: 7

Expected answer: 36/5

Correct? No

🔗 Problem

Problem: A baker wants to bake enough cakes for a large party. Each cake requires 5/12 of a kilogram of flour. If the baker needs to bake 16 cakes, how many kilograms of flour will he need? Use the formula s = (16 * (5 / 12)) to calculate the answer.

Gemma 2B:

    Let's break down the problem step by step:

    We know that each cake requires 5/12 of a kilogram of flour, and the baker needs to bake 16 cakes. To find the total amount of flour needed, we can multiply the number of cakes by the amount of flour required per cake.

    The formula given is s = (16 * (5 / 12)), where s represents the total amount of flour needed.

    To evaluate this expression, we need to follow the order of operations (PEMDAS):

    1. Divide 5 by 12: 5 / 12 = 0.4167 (approximately)
    2. Multiply 16 by the result: 16 * 0.4167 = 6.6667 (approximately)

    So, the baker will need approximately 6.67 kilograms of flour.

    ### Answer: 6.67 kilograms

Expected answer: 20/3

Correct? No

🔗 Problem

Problem: Determine the result of the following calculation: multiply 13 by 3/12. Remember to write your answer as a fraction or a decimal number.

Gemma 2B:

    To evaluate this expression, we need to follow the order of operations (PEMDAS). First, we multiply 13 by the fraction 3/12.

    13 × (3/12) = 13 × 3 × (1/12) = 39 × (1/12) = 39/12

    We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3.

    39 ÷ 3 = 13
    12 ÷ 3 = 4

    So, the result of the calculation is:

    39/12 = 13/4 = 3.25

    ### Answer: 3.25 or 13/4.

Expected answer: 13/4

Correct? Yes

🔗 Follow-up

Problem: After calculating the result from the previous problem, take your answer and square it (multiply it by itself). Write your final answer as a fraction, decimal, or a whole number if possible.

Gemma 2B:

    Let's take the result from the previous problem, which is 3.25 or 13/4, and square it.

    (3.25)² = 3.25 × 3.25 = 10.5625

    To express this as a fraction, we can convert the decimal to a fraction:

    10.5625 = 10656/1024 = 4221/409 = 1681/169

    So, the result of squaring 3.25 or 13/4 is:

    ### Answer: 10.5625 or 1681/169.

Expected answer: 169/16

Correct? Yes

🔗 Follow-up

Problem: After previously calculating 13 multiplied by 3/12, now consider if instead we actually needed to multiply 13 by 3/15 instead of 3/12. What would be the result of this new calculation? Remember to write your answer as a fraction, decimal, or a whole number if possible.

Gemma 2B:

    To evaluate this new expression, we multiply 13 by the fraction 3/15.

    13 × (3/15) = 13 × 3 × (1/15) = 39 × (1/15) = 39/15

    We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3.

    39 ÷ 3 = 13
    15 ÷ 3 = 5

    So, the result of the calculation is:

    39/15 = 13/5 = 2.6

    ### Answer: 2.6 or 13/5.

Expected answer: 13/5

Correct? Yes

🔗 Problem

Problem: Find the product when 8 is multiplied by 1. Show your work and provide your answer.

Gemma 2B:

    To find the product, I will multiply 8 by 1:

    8 × 1 = ?

    To multiply, I will add 8 together as many times as the multiplier (1) indicates:

    8

    Since I am adding 8 together only 1 time, the result is simply 8.

    ### Answer: 8

Expected answer: 8

Correct? Yes

🔗 Follow-up

Problem: After finding the product when 8 is multiplied by 1, take that result and multiply it by 2. What is the final result? Show your work process.

Gemma 2B:

    I already found the product when 8 is multiplied by 1, which is 8. Now, I will multiply 8 by 2:

    8 × 2 = ?

    To multiply, I will add 8 together as many times as the multiplier (2) indicates:

    8 + 8 = ?

    Adding 8 and 8 together gives me:

    16

    So, the result of multiplying 8 by 2 is 16.

    ### Answer: 16

Expected answer: 16

Correct? Yes

🔗 Follow-up

Problem: In the same situation as the previous question, consider that originally there were 16 units instead of 8. Multiply the 16 units by 1. What will the new result be? Show your calculations and the final figure.

Gemma 2B:

    To find the product, I will multiply 16 by 1:

    16 × 1 = ?

    To multiply, I will add 16 together as many times as the multiplier (1) indicates:

    16

    Since I am adding 16 together only 1 time, the result is simply 16.

    ### Answer: 16

Expected answer: 16

Correct? Yes

🔗 Problem

Problem: What is the result when you multiply 9 by the fraction 9/17?

Gemma 2B:

    To multiply 9 by the fraction 9/17, we can multiply 9 by the numerator (9) and divide by the denominator (17).

    9 × 9 = 81
    81 ÷ 17 = 4.76

    So, the result of multiplying 9 by the fraction 9/17 is 4.76.

    ### Answer: 4.76

Expected answer: 81/17

Correct? No

🔗 Problem

Problem: Tom is planning to share his 13 boxes with his friends. He has decided each of them will get 12/13 of a box. How many boxes, in total, will Tom's friends receive?

Gemma 2B:

    Reasoning: Since Tom has 13 boxes and each friend will get 12/13 of a box, the total number of boxes Tom's friends will receive is 13 x (12/13) = 12.
    ### Answer: 12

Expected answer: 12

Correct? Yes

🔗 Follow-up

Problem: Tom has decided to donate some of the boxes to a local school after sharing with his friends. He is now giving away 9/15 of the total boxes his friends received. How many boxes are being donated to the local school?

Gemma 2B:

    Reasoning: Tom's friends received 12 boxes in total. Tom is donating 9/15 of these boxes to the local school. To find the number of boxes being donated, multiply the total number of boxes Tom's friends received by the fraction being donated: 12 x (9/15) = 12 x (3/5) = 12/5 x 3 = 2.4 x 3 = 7.2. Since you can't donate a fraction of a box, we'll round down to the nearest whole number, as you can't donate a fraction of a box. So, Tom is donating 7 boxes to the local school.
    ### Answer: 7

Expected answer: 36/5

Correct? No

🔗 Follow-up

Problem: Tom originally wanted to divide 13 boxes among his friends with each of them getting 12/13 of a box. However, he changed his mind and decided to give each friend 17/13 of a box instead. How many boxes in total will his friends receive now?

Gemma 2B:

    Reasoning: Since Tom has 13 boxes and each friend will get 17/13 of a box, the total number of boxes Tom's friends will receive is 13 x (17/13) = 17.
    ### Answer: 17

Expected answer: 17

Correct? Yes

🔗 Problem

Problem: What is the product of multiplying 16 and 1?

Gemma 2B:

    Reasoning: The product of multiplying 16 and 1 is simply 16 × 1, which equals 16.
    ### Answer: 16

Expected answer: 16

Correct? Yes

🔗 Follow-up

Problem: After finding the product of 16 and 1, what is the result when this product is multiplied by 3/8? Write your answer as a simplified fraction.

Gemma 2B:

    Reasoning: We already found the product of 16 and 1, which is 16. Now, we need to multiply 16 by 3/8.

    To do this, we can multiply 16 by the numerator (3) and divide by the denominator (8):

    16 × 3 = 48
    48 ÷ 8 = 6

    So, the result is 6.

    ### Answer: 6

Expected answer: 6

Correct? Yes

🔗 Follow-up

Problem: After finding the product of 16 and 1 and multiplying it by 3/8, consider a similar action with different numbers. What is the product of 15 and 1?

Gemma 2B:

    Reasoning: The product of multiplying 15 and 1 is simply 15 × 1, which equals 15.
    ### Answer: 15

Expected answer: 15

Correct? Yes

🔗 Problem

Problem: Multiply 10 by the fraction 11/10. What is the result?

Gemma 2B:

    To multiply 10 by the fraction 11/10, we can follow the order of operations (PEMDAS) and multiply 10 by the numerator (11) and then divide by the denominator (10).

    10 × 11 = 110

    Now, divide 110 by 10:

    110 ÷ 10 = 11

    ### Answer: 11

Expected answer: 11

Correct? Yes

🔗 Follow-up

Problem: After finding the result of 10 * 11/10, now multiply this result by 2/19. What is the final result when you write it as a simplified fraction?

Gemma 2B:

    We already found that 10 * 11/10 = 11. Now, let's multiply 11 by 2/19:

    11 × 2/19 = 22/19

    This fraction is already in its simplest form, as the greatest common divisor of 22 and 19 is 1.

    ### Answer: 22/19

Expected answer: 22/19

Correct? Yes

🔗 Follow-up

Problem: Now instead of 10 * 11/10, let's change the fraction to 11/17. Multiply 10 by 11/17. What is the new result?

Gemma 2B:

    To multiply 10 by the fraction 11/17, we can follow the order of operations (PEMDAS) and multiply 10 by the numerator (11) and then divide by the denominator (17).

    10 × 11 = 110

    Now, divide 110 by 17:

    110 ÷ 17 = 110/17 = 6 8/17

    ### Answer: 6 8/17

Expected answer: 110/17

Correct? Yes

🔗 Problem

Problem: Calculate the product of 10 and the fraction 8/14. Write your answer as a simplified fraction or a decimal.

Gemma 2B:

    To calculate the product, I'll multiply 10 by the fraction 8/14:

    10 × (8/14) = 10 × 8/14 = 80/14

    To simplify the fraction, I'll divide both the numerator and the denominator by their greatest common divisor, which is 2:

    80 ÷ 2 = 40
    14 ÷ 2 = 7

    So, the simplified fraction is:

    40/7

    As a decimal, this is equivalent to:

    40 ÷ 7 = 5.71 (rounded to two decimal places)

    ### Answer: 40/7 or 5.71

Expected answer: 40/7

Correct? Yes

🔗 Follow-up

Problem: After calculating the product of 10 and the fraction 8/14, multiply the result by 9/6. Write your answer as a simplified fraction or a decimal.

Gemma 2B:

    I already calculated the product of 10 and the fraction 8/14:

    10 × (8/14) = 80/14

    Now, I'll multiply the result by 9/6:

    (80/14) × (9/6) = (80 × 9) / (14 × 6) = 720 / 84

    To simplify the fraction, I'll divide both the numerator and the denominator by their greatest common divisor, which is 12:

    720 ÷ 12 = 60
    84 ÷ 12 = 7

    So, the simplified fraction is:

    60/7

    As a decimal, this is equivalent to:

    60 ÷ 7 = 8.57 (rounded to two decimal places)

    ### Answer: 60/7 or 8.57

Expected answer: 60/7

Correct? Yes

🔗 Follow-up

Problem: After calculating the product of 10 and the fraction 8/14 in the previous problem, now consider if the number was not 10 but 18 instead. Calculate the product of 18 and the fraction 8/14. Write your answer as a simplified fraction or a decimal.

Gemma 2B:

    To calculate the product, I'll multiply 18 by the fraction 8/14:

    18 × (8/14) = 18 × 8/14 = 144/14

    To simplify the fraction, I'll divide both the numerator and the denominator by their greatest common divisor, which is 2:

    144 ÷ 2 = 72
    14 ÷ 2 = 7

    So, the simplified fraction is:

    72/7

    As a decimal, this is equivalent to:

    72 ÷ 7 = 10.29 (rounded to two decimal places)

    ### Answer: 72/7 or 10.29

Expected answer: 72/7

Correct? Yes

🔗 Problem

Problem: John has a pile of 20 assignment papers to grade. However, he has only managed to grade 8/18 of the pile. How many papers has John graded?

Gemma 2B:

    Reasoning: To find the number of papers John has graded, we need to multiply the total number of papers by the fraction of papers he has graded.

    Let's convert the fraction 8/18 to a decimal by dividing the numerator by the denominator: 8 ÷ 18 = 0.44 (rounded to two decimal places).

    Now, multiply the total number of papers by the decimal: 20 × 0.44 = 8.8.

    Since we can't grade a fraction of a paper, we'll round down to the nearest whole number. John has graded 8 papers.

    ### Answer: 8

Expected answer: 80/9

Correct? No

🔗 Problem

Problem: Calculate the result of multiplying 16 by the fraction 13/12.

Gemma 2B:

    To multiply 16 by the fraction 13/12, we can follow the order of operations (PEMDAS) and multiply 16 by the numerator (13) and then divide by the denominator (12).

    16 × 13 = 208

    Now, divide 208 by 12:

    208 ÷ 12 = 17.33

    ### Answer: 17.33

Expected answer: 52/3

Correct? No

🔗 Problem

Problem: Maddison is dividing a piece of wood to build a bird house. The piece of wood is initially 16 units long. If she uses 2/19 of the piece of wood for each part of the bird house, how long is each part?

Gemma 2B:

    Reasoning: To find the length of each part, we need to multiply the total length of the wood by the fraction used for each part.

    Length of each part = Total length of wood × Fraction used for each part
    = 16 × (2/19)
    = 16 × 2 ÷ 19
    = 32 ÷ 19
    = 1.684 (approximately)

    Since the length cannot be a fraction, we can round it to 1.7 units.

    ### Answer: 1.7 units

Expected answer: 32/19

Correct? No

🔗 Problem

Problem: What is the result of multiplying the fraction twelve divided by twelve with the whole number sixteen?

Gemma 2B:

    Reasoning: The fraction twelve divided by twelve is equal to 1, since 12/12 = 1.
    Multiplying 1 by 16 gives us 1 × 16 = 16.
    ### Answer: 16

Expected answer: 16

Correct? Yes

🔗 Follow-up

Problem: After finding the result of multiplying twelve divided by twelve with the whole number sixteen, what is the new result when that value is further multiplied by four divided by fourteen?

Gemma 2B:

    Reasoning: We already found that multiplying twelve divided by twelve with the whole number sixteen gives us 16.
    The fraction four divided by fourteen is equal to 4/14, which can be simplified to 2/7.
    Multiplying 16 by 2/7 gives us 16 × (2/7) = 32/7 = 4 4/7.
    ### Answer: 4 4/7

Expected answer: 32/7

Correct? Yes

🔗 Follow-up

Problem: After multiplying twelve divided by twelve with sixteen in the previous problem, let's consider a scenario where instead of sixteen, we have eight. What is the result of twelve divided by twelve multiplied by eight?

Gemma 2B:

    Reasoning: The fraction twelve divided by twelve is equal to 1, since 12/12 = 1.
    Multiplying 1 by 8 gives us 1 × 8 = 8.
    ### Answer: 8

Expected answer: 8

Correct? Yes

🔗 Problem

Problem: Albert plans to make miniature models for his project and needs certain amount of material. He knows that he needs 8/7th of a pound of clay per model and he wants to create 17/3rd models. Calculate amount of clay in pounds he will need. What is it when the fractions are multiplied?

Gemma 2B:

    Reasoning: To find the total amount of clay Albert needs, we need to multiply the amount of clay per model by the number of models he wants to create.

    Amount of clay per model = 8/7 pounds
    Number of models = 17/3

    Total amount of clay = (8/7) × (17/3)

    To multiply fractions, we multiply the numerators (numbers on top) and multiply the denominators (numbers at the bottom), then simplify the fraction:

    Total amount of clay = (8 × 17) / (7 × 3)
    = 136 / 21
    = 64/3 + 4/21 (simplified)

    Since we can't have a fraction of a pound, we can convert the mixed number to an improper fraction:

    Total amount of clay = 64/3 + 4/21
    = (64 × 7 + 4 × 3) / (3 × 7)
    = 452 / 21

    ### Answer: 452/21 pounds

Expected answer: 136/21

Correct? No

🔗 Problem

Problem: Mrs. Parker makes her famous apple pie using 14/12 hours for preparing and baking. If she decides to make 2 pies, how many total hours will she spend? Please calculate and express your answer as a mixed number.

Gemma 2B:

    Reasoning: To make one pie, Mrs. Parker spends 14/12 hours. To make 2 pies, she will spend 2 × (14/12) hours.

    To multiply a fraction by an integer, we multiply the numerator by the integer: 2 × (14/12) = (2 × 14)/12 = 28/12.

    To express 28/12 as a mixed number, we divide the numerator by the denominator: 28 ÷ 12 = 2 with a remainder of 4. So, 28/12 = 2 4/12.

    We can simplify the fraction part by dividing both the numerator and the denominator by their greatest common divisor, which is 4: 4/12 = 1/3.

    So, Mrs. Parker will spend 2 1/3 hours making 2 pies.

    ### Answer: 2 1/3

Expected answer: 7/3

Correct? Yes

🔗 Follow-up

Problem: Mrs. Parker continues with her baking spree and decides to make 13 of her apple pies for a local charity event. Each apple pie still takes 14/12 hours to prepare and bake. How long, in hours, will she spend on making these pies? Please calculate and express your answer as a mixed number.

Gemma 2B:

    Reasoning: To make one pie, Mrs. Parker spends 14/12 hours. To make 13 pies, she will spend 13 × (14/12) hours.

    To multiply a fraction by an integer, we multiply the numerator by the integer: 13 × (14/12) = (13 × 14)/12 = 182/12.

    To express 182/12 as a mixed number, we divide the numerator by the denominator: 182 ÷ 12 = 15 with a remainder of 2. So, 182/12 = 15 2/12.

    We can simplify the fraction part by dividing both the numerator and the denominator by their greatest common divisor, which is 2: 2/12 = 1/6.

    So, Mrs. Parker will spend 15 1/6 hours making 13 pies.

    ### Answer: 15 1/6

Expected answer: 91/6

Correct? Yes

🔗 Problem

Problem: Evaluate the following expression and write your answer as a simplified fraction: (18/11) * (17/8)

Gemma 2B:

    To evaluate this expression, we need to multiply the two fractions.

    First, we multiply the numerators (the numbers on top):
    18 × 17 = 306

    Next, we multiply the denominators (the numbers on the bottom):
    11 × 8 = 88

    Now, we write the result as a fraction:
    306/88

    We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
    306 ÷ 2 = 153
    88 ÷ 2 = 44

    So, the simplified fraction is:
    153/44

    ### Answer: 153/44

Expected answer: 153/44

Correct? Yes

🔗 Follow-up

Problem: After solving the expression (18/11) * (17/8) in the earlier problem, multiply the result by the fraction (12/19). What is your answer as a simplified fraction?

Gemma 2B:

    We already evaluated the expression (18/11) * (17/8) and got the result 153/44.

    Now, we need to multiply this result by the fraction (12/19).

    To do this, we multiply the numerators:
    153 × 12 = 1836

    And we multiply the denominators:
    44 × 19 = 836

    Now, we write the result as a fraction:
    1836/836

    We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 4:
    1836 ÷ 4 = 459
    836 ÷ 4 = 209

    So, the simplified fraction is:
    459/209

    ### Answer: 459/209

Expected answer: 459/209

Correct? Yes

🔗 Follow-up

Problem: After previously solving (18/11) * (17/8), let's consider what would happen if we instead have a different fraction: (10/11). So, recalculate the expression, but this time use (10/11) * (17/8). What is your answer as a simplified fraction?

Gemma 2B:

    To evaluate this expression, we need to multiply the two fractions.

    First, we multiply the numerators (the numbers on top):
    10 × 17 = 170

    Next, we multiply the denominators (the numbers on the bottom):
    11 × 8 = 88

    Now, we write the result as a fraction:
    170/88

    We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
    170 ÷ 2 = 85
    88 ÷ 2 = 44

    So, the simplified fraction is:
    85/44

    ### Answer: 85/44

Expected answer: 85/44

Correct? Yes

🔗 Problem

Problem: You are eating healthy and have decided to only eat half of an apple. Each apple costs $2. How much would you essentially be paying for half an apple?

Gemma 2B:

    Reasoning: If each apple costs $2, and you're only eating half of it, you're essentially paying for half of the cost of the apple. To find half of $2, you can divide $2 by 2, which gives you $1.
    ### Answer: $1.

Expected answer: 1

Correct? Yes

🔗 Follow-up

Problem: You've decided to extend your health regiment and you want to include bananas into your diet. If each banana costs $2 and you only consume 7/10 of each banana, what would be the cost for the portion of banana you consume?

Gemma 2B:

    Reasoning: If each banana costs $2 and you only consume 7/10 of each banana, you need to find 7/10 of $2. To do this, you can multiply $2 by 7/10, which gives you $1.40.
    ### Answer: $1.40.

Expected answer: 7/5

Correct? No

🔗 Problem

Problem: What is the product when you multiply 1 by 8?

Gemma 2B:

    Reasoning: When we multiply 1 by 8, we are essentially adding 8 together 1 time. Therefore, the product is 1 × 8 = 8.

    ### Answer: 8

Expected answer: 8

Correct? Yes

🔗 Follow-up

Problem: After finding the product of 1 and 8, what is the value when you multiply this product with 9/8?

Gemma 2B:

    Reasoning: We already found the product of 1 and 8, which is 8. Now, we need to multiply 8 by 9/8.

    To do this, we can multiply 8 by 9 and then divide by 8:

    8 × 9 = 72

    72 ÷ 8 = 9

    So, the value when you multiply the product by 9/8 is 9.

    ### Answer: 9

Expected answer: 9

Correct? Yes

🔗 Follow-up

Problem: If instead of multiplying 1 by 8, we multiply 1 by 6, what would the new product be?

Gemma 2B:

    Reasoning: When we multiply 1 by 6, we are essentially adding 6 together 1 time. Therefore, the product is 1 × 6 = 6.

    ### Answer: 6

Expected answer: 6

Correct? Yes

🔗 Problem

Problem: If a recipe calls for 5 cups of flour, but you are only using a scoop that is 14/16 the size of a cup, how many scoops do you need to have 5 whole cups of flour? Calculate it using the formula: p = 5 * (14 / 16)

Gemma 2B:

    Let's break it down step by step!

    We are given the formula: p = 5 * (14 / 16)

    First, let's simplify the fraction inside the parentheses: 14 / 16 = 7 / 8

    Now, multiply 5 by the simplified fraction: p = 5 * (7 / 8)

    To multiply a whole number by a fraction, we can multiply the whole number by the numerator (7) and then divide by the denominator (8): p = (5 * 7) / 8

    p = 35 / 8

    To find the number of scoops, we need to divide 35 by 8: p = 35 ÷ 8

    p = 4.375

    Since we can't have a fraction of a scoop, we need to round up to the nearest whole number to ensure we have enough flour. Therefore, we need:

    p = 5 scoops

    ### Answer: 5

Expected answer: 35/8

Correct? No

🔗 Problem

Problem: William has a recipe that requires 9/10 of a cup of sugar. However, he wants to triple the recipe. How many cups of sugar will William need? Follow the operation ((9 / 10) * 3) to answer the question.

Gemma 2B:

    To triple the recipe, we need to multiply the original amount of sugar by 3. The original amount is 9/10 of a cup, so we'll multiply that by 3.

    First, we'll multiply the numerator (9) by 3: 9 × 3 = 27

    Next, we'll keep the denominator (10) the same: 27/10

    To convert this to a mixed number, we can divide the numerator by the denominator: 27 ÷ 10 = 2 with a remainder of 7. So, we can write the result as 2 7/10 cups.

    ### Answer: 2 7/10 cups

Expected answer: 27/10

Correct? Yes

🔗 Follow-up

Problem: William decided to try a new recipe that only needed 9/17 of a cup of sugar for one batch. Just like the previous time, he wants to triple the recipe. How much sugar will he need this time? Use the operation in ((9 / 17) * 3) to answer the question.

Gemma 2B:

    To triple the recipe, we need to multiply the original amount of sugar by 3. The original amount is 9/17 of a cup, so we'll multiply that by 3.

    First, we'll multiply the numerator (9) by 3: 9 × 3 = 27

    Next, we'll keep the denominator (17) the same: 27/17

    This is an improper fraction, but we can simplify it by dividing both the numerator and the denominator by their greatest common divisor, which is 1. So, the result remains 27/17.

    ### Answer: 27/17 cups

Expected answer: 27/17

Correct? Yes

🔗 Problem

Problem: Compute the following operation: multiply 9 by 5/14. Write your answer as a simplified fraction.

Gemma 2B:

    To compute the operation, I'll follow the order of operations (PEMDAS):

    1. Multiply 9 by the fraction 5/14:

    9 × (5/14) = 9 × 5 / 14

    2. Multiply 9 by 5:

    9 × 5 = 45

    3. Divide the result by 14:

    45 ÷ 14 = 45/14

    To simplify the fraction, I'll look for the greatest common divisor (GCD) of 45 and 14. The GCD is 1, so the fraction is already in its simplest form:

    45/14

    ### Answer: 45/14

Expected answer: 45/14

Correct? Yes

🔗 Follow-up

Problem: In the previous problem, you multiplied 9 by 5/14. Now, reconsider the problem with 18 instead of 9. Calculate the new result when you multiply 18 by 5/14. Write your answer as a simplified fraction.

Gemma 2B:

    To compute the operation, I'll follow the order of operations (PEMDAS):

    1. Multiply 18 by the fraction 5/14:

    18 × (5/14) = 18 × 5 / 14

    2. Multiply 18 by 5:

    18 × 5 = 90

    3. Divide the result by 14:

    90 ÷ 14 = 90/14

    To simplify the fraction, I'll look for the greatest common divisor (GCD) of 90 and 14. The GCD is 2, so I can divide both numerator and denominator by 2:

    90 ÷ 2 = 45
    14 ÷ 2 = 7

    The simplified fraction is:

    45/7

    ### Answer: 45/7

Expected answer: 45/7

Correct? Yes

🔗 Problem

Problem: Michael wants to adjust a recipe that originally serves 16 servings to only serve 12 servings. He found a new vineyard that produces a type of grape juice that is 15/6 more concentrated than the one he originally uses. How much of the new grape juice should he use in comparison to the old recipe, if he calculates it as (12/16) * (15/6)?

Gemma 2B:

    Reasoning: To adjust the recipe to serve 12 servings, Michael needs to multiply the original amount by 12/16, which is the ratio of the new number of servings to the original number of servings.

    Since the new grape juice is 15/6 more concentrated, Michael needs to use 15/6 times less of the new grape juice to achieve the same effect as the original recipe.

    To find the amount of new grape juice Michael should use, we multiply the adjustment ratio by the concentration ratio:

    (12/16) * (15/6) = (12*15) / (16*6) = 180 / 96 = 15/8

    So, Michael should use 15/8 as much of the new grape juice as the original recipe called for.

    ### Answer: 15/8

Expected answer: 15/8

Correct? Yes

🔗 Follow-up

Problem: Now, let's say that Michael tasted the juice made with the adjusted recipe and found it too concentrated for his liking. He decides to dilute the juice by adding water, making the juice only 7/13 as concentrated as before. In comparison to the old undiluted recipe, how much of the new grape juice should he use now, if he calculates it as (7/13) * the previous fraction ((12/16) * (15/6))?

Gemma 2B:

    Reasoning: Michael wants to find the new amount of grape juice to use, taking into account the dilution. He can multiply the previous fraction by the dilution factor.

    The previous fraction was (12/16) * (15/6) = 15/8.

    The dilution factor is 7/13, which means the juice is now 7/13 as concentrated as before.

    To find the new amount of grape juice, Michael multiplies the previous fraction by the dilution factor:

    (7/13) * (15/8) = (7*15) / (13*8) = 105 / 104 = 105/104

    So, Michael should use 105/104 as much of the new grape juice as the original recipe called for.

    ### Answer: 105/104

Expected answer: 105/104

Correct? Yes

🔗 Follow-up

Problem: Let's tweak the scenario again. This time, Michael found an even more concentrated grape juice. This new variant is 15/2 times as strong as the juice he originally used. Now, how much of this new juice should he use for 12 servings instead of 16, in comparison to the old recipe? So, he calculates it as (12/16) * (15/2).

Gemma 2B:

    Reasoning: To adjust the recipe to serve 12 servings, Michael needs to multiply the original amount by 12/16, which is the ratio of the new number of servings to the original number of servings.

    Since the new grape juice is 15/2 times more concentrated, Michael needs to use 15/2 times less of the new grape juice to achieve the same effect as the original recipe.

    To find the amount of new grape juice Michael should use, we multiply the adjustment ratio by the concentration ratio:

    (12/16) * (2/15) = (12*2) / (16*15) = 24 / 240 = 1/10

    So, Michael should use 1/10 as much of the new grape juice as the original recipe called for.

    ### Answer: 1/10

Expected answer: 45/8

Correct? No

🔗 Problem

Problem: If a baker uses 9 cups of flour for a cake recipe and each cup of flour weighs 19/19 pounds, how many pounds of flour does the baker use for the cake?

Gemma 2B:

    Reasoning: Since each cup of flour weighs 19/19 pounds, which is equal to 1 pound, the baker uses 9 cups of flour, each weighing 1 pound. Therefore, the total weight of flour used is 9 pounds.

    ### Answer: 9

Expected answer: 9

Correct? Yes

🔗 Follow-up

Problem: If the baker decides to make a smaller cake that only uses 6/18 of the original amount of flour, how many pounds of flour would the baker use for the smaller cake?

Gemma 2B:

    Reasoning: The original amount of flour used was 9 pounds. To find the amount of flour used for the smaller cake, we need to find 6/18 of 9 pounds.

    First, we can simplify the fraction 6/18 by dividing both numerator and denominator by their greatest common divisor, which is 6. This gives us 1/3.

    Now, we can find 1/3 of 9 pounds by multiplying 9 by 1/3, which is equal to 3 pounds.

    ### Answer: 3

Expected answer: 3

Correct? Yes

🔗 Follow-up

Problem: If the baker decides to use just 1 cup of flour instead of 9 for the cake recipe, and each cup of flour still weighs 19/19 pounds, how many pounds of flour does the baker use now?

Gemma 2B:

    Reasoning: Since each cup of flour weighs 19/19 pounds, which is equal to 1 pound, the baker uses 1 cup of flour, which weighs 1 pound.

    ### Answer: 1

Expected answer: 1

Correct? Yes

🔗 Problem

Problem: A baker needs 5 times the amount of sugar required in one recipe. If one recipe needs 9/4 cups of sugar, how much sugar does the baker need in cups? Do the multiplication and state your answer as a fraction in simplest form.

Gemma 2B:

    Reasoning: The baker needs 5 times the amount of sugar required in one recipe, which is 9/4 cups. To find the total amount of sugar needed, we multiply 5 by 9/4.

    5 × (9/4) = (5 × 9) / 4 = 45 / 4

    To write the answer in simplest form, we can simplify the fraction:

    45 / 4 = 9 1/4

    ### Answer: 9 1/4 cups

Expected answer: 45/4

Correct? No

🔗 Problem

Problem: Julia is baking a cake that requires 12/8 cups of flour. However, she only needs to make 9/13 of the recipe. How much flour does she need for the portion of the recipe she is making?

Gemma 2B:

    Reasoning: Julia needs 12/8 cups of flour for the full recipe. To find out how much flour she needs for 9/13 of the recipe, we can multiply the amount of flour needed for the full recipe by 9/13.

    (12/8) × (9/13) = (12 × 9) / (8 × 13) = 108 / 104 = 27/26

    So Julia needs 27/26 cups of flour for 9/13 of the recipe.

    ### Answer: 27/26 cups

Expected answer: 27/26

Correct? Yes

🔗 Follow-up

Problem: Julia’s sister also wants to bake a cake, but she's adapting a different recipe that requires 12 cups of flour. However, she only intends to make 9/13 of the recipe. How much flour does Julia's sister need for the portion of the recipe she is making?

Gemma 2B:

    Reasoning: Julia's sister needs 12 cups of flour for the full recipe. To find out how much flour she needs for 9/13 of the recipe, we can multiply the amount of flour needed for the full recipe by 9/13.

    12 × (9/13) = 108/13

    So Julia's sister needs 108/13 cups of flour for 9/13 of the recipe.

    ### Answer: 108/13 cups

Expected answer: 108/13

Correct? Yes

🔗 Problem

Problem: Calculate the following expression: 9 * (20/11). Write the answer as a mixed number.

Gemma 2B:

    To evaluate this expression, we need to follow the order of operations (PEMDAS):

    1. Divide 20 by 11: 20/11 = 1 9/11
    2. Multiply 9 by the result: 9 × (1 9/11) = 9 × 1 + 9 × 9/11 = 9 + 81/11 = 9 + 7 4/11 = 16 4/11

    ### Answer: 16 4/11

Expected answer: 180/11

Correct? Yes

🔗 Follow-up

Problem: After calculating the previous expression, which was 9 * (20/11), take your result and then multiply it by 4/5. What is the final result? Write your answer in the simplified form.

Gemma 2B:

    We already calculated the expression 9 * (20/11) to be 16 4/11. Now, we need to multiply this result by 4/5:

    (16 4/11) × (4/5) = ?

    First, let's convert the mixed number to an improper fraction:
    16 4/11 = (16 × 11 + 4) / 11 = 180/11

    Now, multiply the fractions:
    (180/11) × (4/5) = (180 × 4) / (11 × 5) = 720 / 55

    Simplify the fraction:
    720 ÷ 20 = 36, and 55 ÷ 5 = 11, so the simplified fraction is:
    36/11

    ### Answer: 36/11

Expected answer: 144/11

Correct? No

🔗 Follow-up

Problem: In the previous problem, you calculated 9 * (20/11). But what if instead of multiplying by 20/11, you multiply by 7/11? Recalculate the expression now being 9 * (7/11). What is the new result? Write your answer in the simplified form.

Gemma 2B:

    To evaluate the expression 9 * (7/11), I'll follow the same steps:

    1. Multiply 9 by the numerator (7): 9 × 7 = 63
    2. Divide the result by the denominator (11): 63 ÷ 11 = 5 8/11

    ### Answer: 5 8/11

Expected answer: 63/11

Correct? Yes

🔗 Problem

Problem: Solve the following problem: Divide 20 by 5 and then multiply the result by the fraction 6/10. Write down your answer.

Gemma 2B:

    Reasoning: First, let's divide 20 by 5: 20 ÷ 5 = 4.

    Next, we need to multiply 4 by the fraction 6/10. To do this, we can convert the fraction to a decimal by dividing the numerator by the denominator: 6 ÷ 10 = 0.6.

    Now, we can multiply 4 by 0.6: 4 × 0.6 = 2.4.

    ### Answer: 2.4

Expected answer: 12/5

Correct? Yes

🔗 Follow-up

Problem: After solving the previous problem, what if instead of dividing 20 by 5, you divided 16 by 5 and kept the rest of the problem the same, where the result is then multiplied by 6/10? Calculate this new problem and provide your answer.

Gemma 2B:

    Reasoning: First, let's divide 16 by 5: 16 ÷ 5 = 3.2.

    Next, we need to multiply 3.2 by the fraction 6/10. To do this, we can convert the fraction to a decimal by dividing the numerator by the denominator: 6 ÷ 10 = 0.6.

    Now, we can multiply 3.2 by 0.6: 3.2 × 0.6 = 1.92.

    ### Answer: 1.92

Expected answer: 48/25

Correct? No

🔗 Problem

Problem: In a school where there are 17 classrooms and each classroom has a bin for paper recycling, 5/14 of the total classrooms forgot to recycle their papers during a specific day. What fraction of the total classrooms didn't recycle their papers?

Gemma 2B:

    Reasoning: Since 5/14 of the total classrooms forgot to recycle, we can find the number of classrooms that forgot to recycle by multiplying the total number of classrooms by the fraction: 17 × (5/14) = 60/14 = 30/7.

    So, 30/7 classrooms forgot to recycle. To find the fraction of the total classrooms that didn't recycle, we can divide the number of classrooms that forgot to recycle by the total number of classrooms: (30/7) ÷ 17 = 30/(7 × 17) = 30/119.

    ### Answer: 30/119

Expected answer: 85/14

Correct? No

🔗 Problem

Problem: Calculate the answer to the following problem: what is (19 / 11) * (13 / 17)? Write your answer as a simplified fraction.

Gemma 2B:

    To evaluate this expression, we need to follow the order of operations (PEMDAS):

    1. Divide 19 by 11: 19 ÷ 11 = 19/11
    2. Divide 13 by 17: 13 ÷ 17 = 13/17
    3. Multiply the two results: (19/11) × (13/17)

    To multiply fractions, we multiply the numerators (19 and 13) and multiply the denominators (11 and 17), then simplify:

    (19 × 13) / (11 × 17) = 247 / 187

    The resulting fraction is already in its simplest form, so we don't need to simplify further.

    ### Answer: 247/187

Expected answer: 247/187

Correct? Yes

🔗 Follow-up

Problem: Following up to the last problem, take the result of (19 / 11) * (13 / 17) and multiply it by itself. Write your answer as a simplified fraction.

Gemma 2B:

    We already calculated the result of (19 / 11) * (13 / 17) to be 247/187. Now, we need to multiply this result by itself:

    (247/187) × (247/187)

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

    (247 × 247) / (187 × 187) = 61009 / 34969

    The resulting fraction is already in its simplest form, so we don't need to simplify further.

    ### Answer: 61009/34969

Expected answer: 61009/34969

Correct? Yes

🔗 Follow-up

Problem: Given the previous problem setup, re-calculate the operation if we now consider as initial terms 19/8 and 13/17 instead. What is the result of multiplying (19 / 8) * (13 / 17)? Write your answer as a simplified fraction.

Gemma 2B:

    To evaluate this expression, we need to follow the order of operations (PEMDAS):

    1. Multiply the two fractions: (19/8) × (13/17)

    To multiply fractions, we multiply the numerators (19 and 13) and multiply the denominators (8 and 17), then simplify:

    (19 × 13) / (8 × 17) = 247 / 136

    The resulting fraction is already in its simplest form, so we don't need to simplify further.

    ### Answer: 247/136

Expected answer: 247/136

Correct? Yes

🔗 Problem

Problem: Samantha has a rope that is 9 feet long. She uses 3/11 of the rope to secure a tent. How long is the piece of rope that Samantha uses?

Gemma 2B:

    Reasoning: To find the length of the rope Samantha uses, we need to multiply the total length of the rope by the fraction of the rope she uses.

    Length of rope Samantha uses = Total length of rope × Fraction of rope used
    = 9 feet × (3/11)
    = 9 × (3/11)
    = 27/11
    = 2 5/11 feet

    ### Answer: 2 5/11 feet

Expected answer: 27/11

Correct? Yes

🔗 Follow-up

Problem: Continuing from the previous problem, let's say Samantha had initially a longer, 18 feet rope instead of a 9 feet one. If she still cut out 3/11 of the length from this rope, how long is the new piece of Samantha's rope?

Gemma 2B:

    Reasoning: To find the length of the rope Samantha uses, we need to multiply the total length of the rope by the fraction of the rope she uses.

    Length of rope Samantha uses = Total length of rope × Fraction of rope used
    = 18 feet × (3/11)
    = 18 × (3/11)
    = 54/11
    = 4 10/11 feet

    ### Answer: 4 10/11 feet

Expected answer: 54/11

Correct? Yes

🔗 Problem

Problem: What is the result of the multiplication when 1 is multiplied by 7?

Gemma 2B:

    Reasoning: When 1 is multiplied by 7, the result is the product of 1 and 7, which is 1 × 7 = 7.

    ### Answer: 7

Expected answer: 7

Correct? Yes

🔗 Follow-up

Problem: What is the result when you multiply the previous answer (result of 1 multiplied by 7) by the fraction 5/5?

Gemma 2B:

    Reasoning: The previous answer was 7. Multiplying 7 by the fraction 5/5 is equivalent to multiplying 7 by 1, since 5/5 is equal to 1. Therefore, the result is 7 × 1 = 7.

    ### Answer: 7

Expected answer: 7

Correct? Yes

🔗 Follow-up

Problem: What would be the result of the multiplication if the original number was not 1 multiplied by 7, but rather 1 multiplied by 19?

Gemma 2B:

    Reasoning: When 1 is multiplied by 19, the result is the product of 1 and 19, which is 1 × 19 = 19.

    ### Answer: 19

Expected answer: 19

Correct? Yes

🔗 Problem

Problem: Find the product of the following calculation: 1 times 9.

Gemma 2B:

    Reasoning: To find the product, we multiply the two numbers together.

    1 × 9 = 9

    ### Answer: 9

Expected answer: 9

Correct? Yes

🔗 Follow-up

Problem: Take your answer from the previous problem which was the product of 1 and 9. Now multiply that result by the fraction 13/18. What is your new output?

Gemma 2B:

    Reasoning: We take the result from the previous problem, which is 9, and multiply it by the fraction 13/18.

    9 × (13/18) = 9 × 0.7222... (converting the fraction to a decimal)
    = 6.5

    ### Answer: 6.5

Expected answer: 13/2

Correct? Yes

🔗 Follow-up

Problem: In the previous question, instead of multiplying 1 by 9, suppose you need to multiply 1 by 4. What would be the outcome in this case?

Gemma 2B:

    Reasoning: To find the product, we multiply the two numbers together.

    1 × 4 = 4

    ### Answer: 4

Expected answer: 4

Correct? Yes

🔗 Problem

Problem: Calculate the product of the fraction (18 / 18) and the fraction (11 / 5). Write your answer as a simplified fraction.

Gemma 2B:

    To calculate the product of the fractions, we multiply the numerators (the numbers on top) and multiply the denominators (the numbers on the bottom), like this:

    (18/18) × (11/5) = (18 × 11) / (18 × 5)

    First, let's simplify the fraction 18/18, which is equal to 1.

    So, we have:
    1 × (11/5) = 11/5

    The numerator 11 is not divisible by the denominator 5, so the fraction 11/5 is already in its simplest form.

    ### Answer: 11/5

Expected answer: 11/5

Correct? Yes

🔗 Follow-up

Problem: Continuing from the previous problem where you calculated the product of (18 / 18) and (11 / 5), what would be the result if you multiplied this produced value by itself? Report your answer as a simplified fraction.

Gemma 2B:

    We already calculated the product of (18/18) and (11/5) to be 11/5. Now, we need to multiply this value by itself:

    (11/5) × (11/5) = (11 × 11) / (5 × 5)
    = 121/25

    The numerator 121 is not divisible by the denominator 25, so the fraction 121/25 is already in its simplest form.

    ### Answer: 121/25

Expected answer: 121/25

Correct? Yes

🔗 Follow-up

Problem: Let's modify the previous problem slightly. If instead of (18 / 18), you used (9 / 18), what would be the product of this fraction and (11 / 5)? Report your answer as a simplified fraction.

Gemma 2B:

    To calculate the product of the fractions, we multiply the numerators (the numbers on top) and multiply the denominators (the numbers on the bottom), like this:

    (9/18) × (11/5) = (9 × 11) / (18 × 5)

    First, let's simplify the fraction 9/18, which is equal to 1/2.

    So, we have:
    (1/2) × (11/5) = (1 × 11) / (2 × 5) = 11/10

    The numerator 11 is not divisible by the denominator 10, so the fraction 11/10 is already in its simplest form.

    ### Answer: 11/10

Expected answer: 11/10

Correct? Yes

🔗 Problem

Problem: Calculate the answer for the following equation: (5 / 16) * 13. Please express your answer as a decimal.

Gemma 2B:

    To evaluate this expression, we need to follow the order of operations (PEMDAS):

    1. Divide 5 by 16: 5 ÷ 16 = 0.3125
    2. Multiply the result by 13: 0.3125 × 13 = 4.0625

    ### Answer: 4.0625

Expected answer: 65/16

Correct? Yes

🔗 Problem

Problem: Find the result of the following calculation: Multiply the fraction 4/5 by the whole number 15. Write your answer in simplest form.

Gemma 2B:

    To multiply a fraction by a whole number, we can multiply the numerator (the top number) by the whole number, and keep the denominator (the bottom number) the same.

    So, we multiply 4 by 15:

    4 × 15 = 60

    The denominator remains 5, so our result is:

    60/5

    We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 5.

    60 ÷ 5 = 12
    5 ÷ 5 = 1

    So, the result of the calculation is:

    12/1 = 12

    ### Answer: 12

Expected answer: 12

Correct? Yes

🔗 Follow-up

Problem: After finding the result of 4/5 multiplied by 15, multiply this result by the fraction 3/15. Write your answer in simplest form.

Gemma 2B:

    We already found that 4/5 multiplied by 15 is 12. Now, we need to multiply 12 by the fraction 3/15.

    To do this, we can convert 12 to a fraction by dividing by 1:

    12 = 12/1

    Then, we can multiply the fractions:

    (12/1) × (3/15) = (12 × 3) / (1 × 15) = 36/15

    We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3.

    36 ÷ 3 = 12
    15 ÷ 3 = 5

    So, the result of the calculation is:

    12/5

    ### Answer: 12/5

Expected answer: 12/5

Correct? Yes

🔗 Follow-up

Problem: Suppose that you have adjusted your previous calculation. Instead of multiplying 4/5 by 15, you now need to multiply 4/5 by 6. What is the result of that calculation? Please write your answer in the simplest form.

Gemma 2B:

    To multiply a fraction by a whole number, we can multiply the numerator (the top number) by the whole number, and keep the denominator (the bottom number) the same.

    So, we multiply 4 by 6:

    4 × 6 = 24

    The denominator remains 5, so our result is:

    24/5

    This fraction is already in simplest form, as 24 and 5 do not have any common factors other than 1.

    ### Answer: 24/5

Expected answer: 24/5

Correct? Yes

🔗 Problem

Problem: What will be the result if you multiply the fraction 20/12 by the fraction 19/16?

Gemma 2B:

    To multiply fractions, we multiply the numerators (the numbers on top) and multiply the denominators (the numbers on the bottom), then simplify the result.

     Numerator: 20 × 19 = 380
     Denominator: 12 × 16 = 192

    So, the result of multiplying the fractions is 380/192.

    We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 4.

    380 ÷ 4 = 95
    192 ÷ 4 = 48

    So, the simplified result is 95/48.

    ### Answer: 95/48

Expected answer: 95/48

Correct? Yes

🔗 Follow-up

Problem: In continuation from the previous problem, what will be the result if you square the obtained result? In other words, multiply the result of the previous operation with itself.

Gemma 2B:

    To square the result, we multiply the fraction 95/48 by itself.

    (95/48) × (95/48) = ?

    To multiply fractions, we multiply the numerators and multiply the denominators.

    Numerator: 95 × 95 = 9025
    Denominator: 48 × 48 = 2304

    So, the result of squaring the fraction is 9025/2304.

    ### Answer: 9025/2304

Expected answer: 9025/2304

Correct? Yes

🔗 Follow-up

Problem: Instead of multiplying 20/12 by 19/16 as in the previous problem, what would be the result if you multiplied 20/12 by another fraction of similar value, namely 19/12?

Gemma 2B:

    To multiply fractions, we multiply the numerators (the numbers on top) and multiply the denominators (the numbers on the bottom), then simplify the result.

     Numerator: 20 × 19 = 380
     Denominator: 12 × 12 = 144

    So, the result of multiplying the fractions is 380/144.

    We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 4.

    380 ÷ 4 = 95
    144 ÷ 4 = 36

    So, the simplified result is 95/36.

    ### Answer: 95/36

Expected answer: 95/36

Correct? Yes

🔗 Problem

Problem: Calculate the result of the following multiplication operation: 17/20 * 17/13. Write your answer as a simplified fraction.

Gemma 2B:

    To multiply fractions, we multiply the numerators (the numbers on top) and multiply the denominators (the numbers on the bottom), like this:

    (17/20) × (17/13) = (17 × 17) / (20 × 13)

    = 289 / 260

    We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 1. Therefore, the simplified fraction is:

    289 / 260

    ### Answer: 289/260

Expected answer: 289/260

Correct? Yes

🔗 Follow-up

Problem: For the next part of your calculation, multiply the result of 17/20 * 17/13, denoted as 'f', by 12/11. What is the result of this operation? Please write your answer as a simplified fraction.

Gemma 2B:

    We are given that f = 289/260. To multiply f by 12/11, we multiply the numerators and multiply the denominators, like this:

    f × (12/11) = (289/260) × (12/11)

    = (289 × 12) / (260 × 11)

    = 3478 / 2860

    We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2. Therefore, the simplified fraction is:

    1739 / 1430

    ### Answer: 1739/1430

Expected answer: 867/715

Correct? No

🔗 Follow-up

Problem: Suppose instead of 17/20 in the previous problem, the fraction was actually 16/20. What would the result of the multiplication operation 16/20 * 17/13 be? Please write your answer as a simplified fraction.

Gemma 2B:

    To multiply fractions, we multiply the numerators (the numbers on top) and multiply the denominators (the numbers on the bottom), like this:

    (16/20) × (17/13) = (16 × 17) / (20 × 13)

    = 272 / 260

    We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 4. Therefore, the simplified fraction is:

    68 / 65

    ### Answer: 68/65

Expected answer: 68/65

Correct? Yes

🔗 Problem

Problem: Solve the following problem: 5 * (13 / 12) and give your answer in the form of a mixed number or fraction where necessary.

Gemma 2B:

    To evaluate this expression, we need to follow the order of operations (PEMDAS):

    1. Divide 13 by 12: 13 / 12 = 1 1/12
    2. Multiply 5 by the result: 5 * (1 1/12) = 5 * 1 + 5 * (1/12) = 5 + 5/12 = 5 5/12

    ### Answer: 5 5/12

Expected answer: 65/12

Correct? Yes

🔗 Follow-up

Problem: Continuing from our previous problem, now, multiply the answer you got by (14 / 14). What is your new result as a mixed number or fraction?

Gemma 2B:

    We'll multiply the previous result by (14 / 14):

    5 5/12 × (14 / 14) = 5 5/12 × 1 = 5 5/12

    Since (14 / 14) is equal to 1, multiplying by it doesn't change the value. The result remains the same:

    ### Answer: 5 5/12

Expected answer: 65/12

Correct? Yes

🔗 Follow-up

Problem: Continuing from the first problem, suppose we used 5 * (13 / 10) instead of 5 * (13 / 12). Calculate the new problem and give your answer in the form of a mixed number or fraction where necessary. What happens to the result when we use 13 / 10 instead of 13 / 12 in the problem?

Gemma 2B:

    To evaluate the new expression, we follow the same steps:

    1. Divide 13 by 10: 13 / 10 = 1.3 = 1 3/10
    2. Multiply 5 by the result: 5 * (1 3/10) = 5 * 1 + 5 * (3/10) = 5 + 3/2 = 5 + 1.5 = 6.5 = 6 1/2

    ### Answer: 6 1/2

    When we use 13 / 10 instead of 13 / 12, the result increases. This is because 13 / 10 is greater than 13 / 12, so multiplying 5 by this larger value yields a larger product. In this case, the result increases by 6 1/2 - 5 5/12 = 1 1/12.

Expected answer: 13/2

Correct? Yes

🔗 Problem

Problem: Jack had a full bar of chocolate. If he ate 3/4 of it and each piece of the bar represented 1/4, how many pieces did he eat?

Gemma 2B:

    Reasoning: Since each piece of the bar represents 1/4, there are 4 pieces in the full bar (1/4 + 1/4 + 1/4 + 1/4 = 1 whole bar).

    Jack ate 3/4 of the bar, which means he ate 3 out of the 4 pieces (3/4 = 3 pieces).

    ### Answer: 3

Expected answer: 3

Correct? Yes

🔗 Follow-up

Problem: Following the previous scenario, Jack found an even bigger bar of chocolate that had 17 pieces in total. If he ate the same amount (3/4) of this bar as he did with the previous one, how many pieces of the new chocolate bar did he eat?

Gemma 2B:

    Reasoning: Since Jack ate 3/4 of the new bar, and the new bar has 17 pieces, we can find the number of pieces he ate by multiplying 17 by 3/4.

    To do this, we can convert the fraction 3/4 to a decimal by dividing 3 by 4, which gives us 0.75. Then, we can multiply 17 by 0.75:

    17 × 0.75 = 12.75

    Since we can't eat a fraction of a piece of chocolate, we round down to the nearest whole number. Therefore, Jack ate 12 pieces of the new chocolate bar.

    ### Answer: 12

Expected answer: 51/4

Correct? No

🔗 Problem

Problem: Calculate the product of 17 and 9/18 without simplifying the fraction. What is the result?

Gemma 2B:

    To calculate the product, I'll multiply 17 by the numerator (9) and then divide by the denominator (18).

    17 × 9 = 153

    Now, divide by the denominator:

    153 ÷ 18 = 8.5

    ### Answer: 8.5

Expected answer: 17/2

Correct? Yes

🔗 Follow-up

Problem: Reconsider the previous problem, but this time replace 9/18 by 13/18. So, now calculate the product of 17 and 13/18 without simplifying the fraction. What is the result now?

Gemma 2B:

    To calculate the product, I'll multiply 17 by the numerator (13) and then divide by the denominator (18).

    17 × 13 = 221

    Now, divide by the denominator:

    221 ÷ 18 = 12.28 (rounded to two decimal places)

    ### Answer: 12.28

Expected answer: 221/18

Correct? No

🔗 Problem

Problem: Calculate the result of multiplying the fraction 15/15 by the whole number 19. Leave your answer as a simplified fraction or whole number, as appropriate.

Gemma 2B:

    Reasoning: When we multiply a fraction by a whole number, we can multiply the numerator (the top number) by the whole number, and keep the denominator (the bottom number) the same.

    So, multiplying 15/15 by 19, we get:

    (15 × 19) / 15 = 285 / 15

    We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 15.

    285 ÷ 15 = 19
    15 ÷ 15 = 1

    So, the simplified result is:

    19/1 = 19

    ### Answer: 19

Expected answer: 19

Correct? Yes

🔗 Follow-up

Problem: Following your previous calculation, now multiply the result by the fraction 6/10. Write down the result as a simplified fraction or a whole number, as appropriate.

Gemma 2B:

    Reasoning: We need to multiply 19 by the fraction 6/10.

    19 × (6/10) = (19 × 6) / 10 = 114 / 10

    We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2.

    114 ÷ 2 = 57
    10 ÷ 2 = 5

    So, the simplified result is:

    57/5

    ### Answer: 57/5 or 11 2/5

Expected answer: 57/5

Correct? Yes

🔗 Follow-up

Problem: Following your previous calculation, reconsider the problem assuming the whole number you previously used was 20 instead of 19. Calculate the result of multiplying the fraction 15/15 by this new whole number. Write down the result as a simplified fraction or a whole number, as appropriate.

Gemma 2B:

    Reasoning: When we multiply a fraction by a whole number, we can multiply the numerator (the top number) by the whole number, and keep the denominator (the bottom number) the same.

    So, multiplying 15/15 by 20, we get:

    (15 × 20) / 15 = 300 / 15

    We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 15.

    300 ÷ 15 = 20
    15 ÷ 15 = 1

    So, the simplified result is:

    20/1 = 20

    ### Answer: 20

Expected answer: 20

Correct? Yes

🔗 Problem

Problem: Mary has 16 apples. She shares 3/5 of them with her friends. How many apples does Mary share with her friends?

Gemma 2B:

    Reasoning: To find the number of apples Mary shares with her friends, we need to find 3/5 of 16.

    To do this, we can multiply 16 by 3/5:

    16 × (3/5) = 16 × 0.6 = 9.6

    Since we can't share a fraction of an apple, we'll round down to the nearest whole number. Mary shares 9 apples with her friends.

    ### Answer: 9

Expected answer: 48/5

Correct? No

🔗 Problem

Problem: What is the product when you divide 12 by 3 and then multiply the result by 3?

Gemma 2B:

    Reasoning: When we divide 12 by 3, we get 12 ÷ 3 = 4.
    Then, we multiply the result by 3, so we have 4 × 3 = 12.
    ### Answer: 12

Expected answer: 12

Correct? Yes

🔗 Follow-up

Problem: Reconsider the original problem, where 12 was divided by 3, and the result was multiplied by 3. Now, replace 12 with 2. What would be the product now?

Gemma 2B:

    Reasoning: When we divide 2 by 3, we get 2 ÷ 3 = 2/3.
    Then, we multiply the result by 3, so we have (2/3) × 3 = 2.
    ### Answer: 2

Expected answer: 2

Correct? Yes

🔗 Problem

Problem: If each box of chocolates contains 12 pieces and each piece is split into 8 equal parts for tasting, how many whole pieces of chocolate are there if there are 6 boxes? Use the equation t = (6 * (12 / 8)) to calculate your answer.

Gemma 2B:

    Let's break down the problem step by step:

    We have 6 boxes of chocolates, and each box contains 12 pieces of chocolate. So, the total number of whole pieces of chocolate is 6 x 12 = 72.

    However, each piece is split into 8 equal parts for tasting. To find the total number of parts, we can divide the total number of whole pieces by 8: 72 ÷ 8 = 9.

    Now, let's use the given equation t = (6 * (12 / 8)) to calculate the answer:

    t = (6 * (12 / 8))
    t = (6 * 1.5)
    t = 9

    The equation indeed gives us the correct answer: there are 9 times as many parts as there are boxes.

    ### Answer: 72

Expected answer: 9

Correct? No

🔗 Problem

Problem: Emily uses 10/19th of her weekly allowance on Monday. On Tuesday, she spends an equivalent amount which is 14/7 times more compared to what she spent on Monday. How much of her weekly allowance did she spend on Tuesday? Calculate: (10 / 19) * (14 / 7)

Gemma 2B:

    Reasoning: To find out how much Emily spent on Tuesday, we need to multiply the fraction of her weekly allowance she spent on Monday by the factor of how much more she spent on Tuesday.

    (10/19) is the fraction of her weekly allowance she spent on Monday.
    (14/7) is the factor of how much more she spent on Tuesday.

    To multiply these fractions, we multiply the numerators (10 and 14) and multiply the denominators (19 and 7), then write the product as a fraction:

    (10*14) / (19*7) = 140 / 133

    So, Emily spent 140/133 of her weekly allowance on Tuesday.
    ### Answer: 140/133

Expected answer: 20/19

Correct? Yes

🔗 Follow-up

Problem: Following the previous problem, Emily notices she has been overspending and decides to decrease her spendings on Tuesday by 4 times less than she had spent on Tuesday. How much of her weekly allowance does she plan on spending for the upcoming Tuesday? Calculate by multiplying ((10 / 19) * (14 / 7)) by 4.

Gemma 2B:

    Reasoning: We already calculated the fraction of her weekly allowance Emily spent on Tuesday: (10/19) * (14/7) = 140/133.

    To decrease her spendings on Tuesday by 4 times less, we need to divide the fraction by 4:

    (140/133) ÷ 4 = (140/133) × (1/4) = (140 × 1) / (133 × 4) = 140 / 532

    So, Emily plans on spending 140/532 of her weekly allowance for the upcoming Tuesday.
    ### Answer: 140/532

Expected answer: 80/19

Correct? No

🔗 Follow-up

Problem: After her initial plan, Emily decides to modify her strategy for spending her actions on Tuesdays. She changes her spending to 10/12th of her weekly allowance, but still spends 14/7 times more than the new amount. What fraction of her weekly allowance does she now plan on spending? Calculate: (10 / 12) * (14 / 7)

Gemma 2B:

    Reasoning: To find out what fraction of her weekly allowance Emily now plans to spend, we need to multiply the new fraction of her weekly allowance she spends on Monday by the factor of how much more she spends on Tuesday.

    (10/12) is the new fraction of her weekly allowance she spends on Monday.
    (14/7) is the factor of how much more she spends on Tuesday.

    To multiply these fractions, we multiply the numerators (10 and 14) and multiply the denominators (12 and 7), then write the product as a fraction:

    (10*14) / (12*7) = 140 / 84

    We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 4:

    140 ÷ 4 = 35
    84 ÷ 4 = 21

    So, Emily now plans to spend 35/21 of her weekly allowance on Tuesday.
    ### Answer: 35/21

Expected answer: 5/3

Correct? Yes

🔗 Problem

Problem: What is the product of 2 and 2?

Gemma 2B:

    To find the product of 2 and 2, we need to multiply them together.

    2 × 2 = 4

    ### Answer: 4

Expected answer: 4

Correct? Yes

🔗 Follow-up

Problem: After finding the product of 2 and 2, what would you get if you multiply that result by 5/2?

Gemma 2B:

    We already found the product of 2 and 2, which is 4. Now, we need to multiply 4 by 5/2.

    To do this, we can multiply 4 by 5, which gives us 20, and then divide the result by 2.

    4 × 5 = 20
    20 ÷ 2 = 10

    So, multiplying 4 by 5/2 gives us 10.

    ### Answer: 10

Expected answer: 10

Correct? Yes

🔗 Problem

Problem: You have 11 boxes of cookies. Each box has 8/5 dozen cookies in it. How many dozens of cookies do you have in total?

Gemma 2B:

    Reasoning: First, we need to find the total number of cookies in each box. Since each box has 8/5 dozen cookies, we can convert this to a whole number by multiplying by 12 (since there are 12 cookies in a dozen): (8/5) × 12 = 192/5 = 38.4 cookies per box.

    Since we have 11 boxes, we can multiply the number of cookies per box by 11: 38.4 × 11 = 422.4 cookies.

    Finally, we can convert this back to dozens by dividing by 12: 422.4 ÷ 12 = 35.2 dozens.

    ### Answer: 35.2

Expected answer: 88/5

Correct? No

🔗 Problem

Problem: Compute the result of the following operation: multiply the fraction 6/18 by the whole number 3. Write your answer in its simplest form.

Gemma 2B:

    To multiply a fraction by a whole number, we can multiply the numerator (6) by the whole number (3), and keep the denominator (18) the same.

    So, we get: (6 × 3) / 18 = 18 / 18

    We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 18.

    18 ÷ 18 = 1
    18 ÷ 18 = 1

    So, the result of the operation is 1.

    ### Answer: 1

Expected answer: 1

Correct? Yes

🔗 Follow-up

Problem: After finding the result of the operation 6/18 * 3, calculate the square of that result. Write your answer in the simplest form.

Gemma 2B:

    We already found the result of the operation 6/18 * 3, which is 1.

    To find the square of 1, we multiply it by itself:

    1 × 1 = 1

    So, the square of the result is still 1.

    ### Answer: 1

Expected answer: 1

Correct? Yes

🔗 Follow-up

Problem: After finding the result of the previous operation 6/18 * 3, consider a new situation where you multiply the fraction 17/18 by the whole number 3 instead. What is the result of this new operation? Write your answer in its simplest form.

Gemma 2B:

    To multiply a fraction by a whole number, we can multiply the numerator (17) by the whole number (3), and keep the denominator (18) the same.

    So, we get: (17 × 3) / 18 = 51 / 18

    We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3.

    51 ÷ 3 = 17
    18 ÷ 3 = 6

    So, the result of the operation is 17/6.

    ### Answer: 17/6

Expected answer: 17/6

Correct? Yes

🔗 Problem

Problem: Lucy had 18 candies which she evenly distributed into 8 bags, she then gave 5 bags to her friends. How many candies does she give away in total? Use the formula (18 / 8) * 5 to find the answer.

Gemma 2B:

    Reasoning: Lucy had 18 candies and distributed them evenly into 8 bags, so each bag contains 18 / 8 = 2.25 candies. She then gave 5 bags away, so the total number of candies she gave away is (18 / 8) * 5 = 2.25 * 5 = 11.25 candies.

    ### Answer: 11.25

Expected answer: 45/4

Correct? Yes

🔗 Follow-up

Problem: If Lucy originally had 13 candies instead of 18, evenly distributed into 8 bags, and she then gave 5 bags to her friends, how many candies does she give away in total? Use the formula (13 / 8) * 5 to find the answer.

Gemma 2B:

    Reasoning: Lucy had 13 candies and distributed them evenly into 8 bags, so each bag contains 13 / 8 = 1.625 candies. She then gave 5 bags away, so the total number of candies she gave away is (13 / 8) * 5 = 1.625 * 5 = 8.125 candies.

    ### Answer: 8.125

Expected answer: 65/8

Correct? No

🔗 Problem

Problem: Calculate the following multiplication of a whole number and a fraction. What is the result of 5 * (14 / 10)?

Gemma 2B:

    To evaluate this expression, we need to follow the order of operations (PEMDAS). First, we need to divide 14 by 10, which gives us 1.4. Then, we multiply 5 by 1.4, which gives us 7.

    ### Answer: 7

Expected answer: 7

Correct? Yes

🔗 Follow-up

Problem: After finding the result of 5 * (14 / 10), continue with the next step of the problem by multiplying this result by the fraction 5/6. What is the final result?

Gemma 2B:

    Now that we have the result of 5 * (14 / 10) = 7, we need to multiply it by the fraction 5/6.

    To do this, we multiply 7 by the numerator 5, which gives us 35. Then, we divide the result by the denominator 6, which gives us 35/6 = 5.83.

    ### Answer: 5.83

Expected answer: 35/6

Correct? No

🔗 Follow-up

Problem: After initially calculating the result of 5 * (14 / 10), consider a different scenario where you have to calculate 5 * (14 / 13) instead. What would the result be in this case?

Gemma 2B:

    To evaluate this expression, we need to follow the order of operations (PEMDAS). First, we need to divide 14 by 13, which gives us 14/13. Then, we multiply 5 by 14/13, which gives us 5*(14/13) = 70/13 = 5 5/13.

    ### Answer: 5 5/13

Expected answer: 70/13

Correct? Yes

🔗 Problem

Problem: What is the result of multiplying 6 by the fraction (20/7)? Write your answer as a mixed number, if applicable.

Gemma 2B:

    To multiply 6 by the fraction (20/7), we can follow the order of operations (PEMDAS) and multiply 6 by the numerator (20) and then divide by the denominator (7).

    6 × (20/7) = (6 × 20) / 7
    = 120 / 7
    = 17 1/7

    ### Answer: 17 1/7

Expected answer: 120/7

Correct? Yes

🔗 Follow-up

Problem: After finding the result of multiplying 6 by the fraction (20/7), what would you get if you multiply this result by (10/10)? Write your answer as a mixed number, if applicable.

Gemma 2B:

    We already found the result of multiplying 6 by the fraction (20/7) to be 17 1/7. Now, we need to multiply this result by (10/10).

    (17 1/7) × (10/10) = (17 1/7) × 1
    = 17 1/7

    Since (10/10) is equal to 1, multiplying by it does not change the value. Therefore, the result remains the same.

    ### Answer: 17 1/7

Expected answer: 120/7

Correct? Yes

🔗 Follow-up

Problem: After finding the result of multiplying 5/8 by 3, what would be the result if instead of multiplying (20/7) by 6, you multiplied (20/10) by 6? Write your answer as a mixed number, if applicable.

Gemma 2B:

    First, let's find the result of multiplying 5/8 by 3:

    (5/8) × 3 = (5 × 3) / 8
    = 15 / 8
    = 1 7/8

    Now, let's multiply (20/10) by 6:

    (20/10) × 6 = (20 × 6) / 10
    = 120 / 10
    = 12

    ### Answer: 12

Expected answer: 12

Correct? Yes

🔗 Problem

Problem: Calculate: What is the product of 8/19 times 6? Give your answer as a simplified fraction or a whole number, if possible.

Gemma 2B:

    To calculate the product, I'll multiply the numerator (8) by 6, and the denominator (19) will remain the same.

    8 × 6 = 48

    So, the product is:

    48/19

    This fraction is already in its simplest form, as 48 and 19 do not have any common factors.

    ### Answer: 48/19

Expected answer: 48/19

Correct? Yes

🔗 Follow-up

Problem: For the following calculation, consider the same scenario as in the previous problem, but this time, instead of having 8/19, you now have 8/8 times 6. How would this change the result? Note: Carry out the multiplication based on this alteration and provide your answer.

Gemma 2B:

    Since we have 8/8, which is equal to 1, we can rewrite the expression as:

    1 × 6

    Multiplying 1 by 6 gives us:

    6

    So, the product is simply 6.

    ### Answer: 6

Expected answer: 6

Correct? Yes

🔗 Problem

Problem: Calculate the answer to the following problem. What is the result of multiplying 8 by 1/1?

Gemma 2B:

    Reasoning: When multiplying a number by a fraction, we multiply the number by the numerator (the top number) and divide by the denominator (the bottom number). In this case, we multiply 8 by 1 and divide by 1.

    8 × 1 = 8
    8 ÷ 1 = 8

    So, the result of multiplying 8 by 1/1 is 8.

    ### Answer: 8

Expected answer: 8

Correct? Yes

🔗 Follow-up

Problem: After finding the result of multiplying 8 by 1/1 in the previous problem, what is the result when we multiply that answer by 3/2?

Gemma 2B:

    Reasoning: We multiply the result of the previous problem (8) by the numerator (3) and divide by the denominator (2).

    8 × 3 = 24
    24 ÷ 2 = 12

    So, the result of multiplying 8 by 1/1 and then multiplying the answer by 3/2 is 12.

    ### Answer: 12

Expected answer: 12

Correct? Yes

🔗 Follow-up

Problem: After finding the result of multiplying 8 by 1/1 in the previous problem, what would be the result if you were originally working with 20 instead of 8? In other words, what would be the outcome of multiplying 20 by 1/1?

Gemma 2B:

    Reasoning: Since multiplying a number by 1/1 doesn't change the value of the number, we can apply the same logic as before.

    20 × 1 = 20
    20 ÷ 1 = 20

    So, the result of multiplying 20 by 1/1 is 20.

    ### Answer: 20

Expected answer: 20

Correct? Yes

🔗 Problem

Problem: Jack has a long sheet of paper that he wants to use for his art project. The sheet has a length of 5/15 meters. If Jack cuts 9 identical pieces from the sheet, what would be the total length of all the 9 pieces? Calculate the answer and write it as a fraction or a whole number.

Gemma 2B:

    Reasoning: Let's first find the length of each piece. Since Jack cuts 9 identical pieces from the sheet, each piece would have a length of 5/15 ÷ 9. To divide a fraction by a whole number, we can multiply the fraction by the reciprocal of the whole number: (5/15) × (1/9) = 5/(15×9) = 5/135.

    Now, we can find the total length of all 9 pieces by multiplying the length of each piece by 9: (5/135) × 9 = 5/15.

    ### Answer: 5/15 or 1/3

Expected answer: 3

Correct? No

🔗 Problem

Problem: Suppose the length of a book is 1 foot, and there are 13 books side by side on a shelf. How long is the shelf in feet? To find out, calculate the product of 1 and 13.

Gemma 2B:

    Reasoning: To find the length of the shelf, we need to multiply the length of one book by the number of books on the shelf. Since each book is 1 foot long, and there are 13 books, we can multiply 1 by 13 to get the total length of the shelf.

    1 x 13 = 13

    ### Answer: 13

Expected answer: 13

Correct? Yes

🔗 Follow-up

Problem: In the previous problem, the total length of the shelf was found to be 13 feet. Now, suppose that only 12/15 of that shelf is filled with books. What would be the length of the shelf that is filled with books? To find out, multiply the total length of the shelf by 12/15.

Gemma 2B:

    Reasoning: To find the length of the shelf that is filled with books, we need to multiply the total length of the shelf (13 feet) by the fraction of the shelf that is filled with books (12/15).

    13 x (12/15) = 13 x 0.8 = 10.4

    ### Answer: 10.4

Expected answer: 52/5

Correct? Yes

🔗 Problem

Problem: Calculate the product when you multiply the fraction 6/5 by the whole number 1. Write your answer as a simplified fraction or whole number.

Gemma 2B:

    To multiply a fraction by a whole number, we can multiply the numerator (the top number) by the whole number, and keep the denominator (the bottom number) the same.

    So, multiplying 6/5 by 1, we get:

    (6 × 1) / 5 = 6/5

    Since 1 is a multiplicative identity, multiplying by 1 doesn't change the value of the fraction. Therefore, the product is still 6/5, which is already in its simplest form.

    ### Answer: 6/5

Expected answer: 6/5

Correct? Yes

🔗 Follow-up

Problem: Once you have calculated the product of 6/5 and 1, multiply the result by itself. Write your answer as a simplified fraction or whole number.

Gemma 2B:

    We already calculated the product of 6/5 and 1, which is 6/5. Now, we need to multiply 6/5 by itself:

    (6/5) × (6/5) = (6 × 6) / (5 × 5) = 36/25

    This fraction is already in its simplest form, so we don't need to simplify it further.

    ### Answer: 36/25

Expected answer: 36/25

Correct? Yes

🔗 Problem

Problem: Solve the following problem: Recognize that 5/5 is the same as 1, then multiply that result by 17/6. Write your answer as a simplified fraction.

Gemma 2B:

    Reasoning: Since 5/5 is equal to 1, we can rewrite the problem as 1 × 17/6.

    To multiply a whole number by a fraction, we can multiply the whole number by the numerator (the top number) and keep the denominator (the bottom number) the same.

    So, 1 × 17/6 = 17/6.

    Since 17 is a prime number and 6 is not a factor of 17, the fraction 17/6 is already in its simplest form.

    ### Answer: 17/6

Expected answer: 17/6

Correct? Yes

🔗 Follow-up

Problem: In the previous problem, we multiplied 5/5 with 17/6. This time, let's consider that instead of 17/6, we want to multiply 5/5 by 6/6. What would be the result of this multiplication? Write down your answer as a simplified fraction.

Gemma 2B:

    Reasoning: Since 5/5 is equal to 1, we can rewrite the problem as 1 × 6/6.

    To multiply a whole number by a fraction, we can multiply the whole number by the numerator (the top number) and keep the denominator (the bottom number) the same.

    So, 1 × 6/6 = 6/6.

    Since 6 is a common factor of both the numerator and the denominator, we can simplify the fraction by dividing both numbers by 6.

    6 ÷ 6 = 1
    6 ÷ 6 = 1

    So, 6/6 simplifies to 1/1, which is equal to 1.

    ### Answer: 1

Expected answer: 1

Correct? Yes

🔗 Problem

Problem: What is the result of multiplying 8 by the result of 8 divided by 6?

Gemma 2B:

    Reasoning: First, let's calculate the result of 8 divided by 6: 8 ÷ 6 = 8/6 = 4/3.
    Now, let's multiply 8 by the result: 8 × (4/3) = 8 × 4/3 = 32/3 = 10 2/3.
    ### Answer: 10 2/3.

Expected answer: 32/3

Correct? Yes

🔗 Follow-up

Problem: Following the result of the previous problem, what would the result be if that value got multiplied by 6 divided by 20?

Gemma 2B:

    Reasoning: We need to multiply 10 2/3 by 6 divided by 20.
    First, let's convert the mixed number to an improper fraction: 10 2/3 = 32/3.
    Now, let's calculate 6 divided by 20: 6 ÷ 20 = 6/20 = 3/10.
    Next, let's multiply 32/3 by 3/10: (32/3) × (3/10) = (32 × 3) / (3 × 10) = 96 / 30 = 32 / 10 = 3.2.
    ### Answer: 3.2.

Expected answer: 16/5

Correct? Yes

🔗 Problem

Problem: Susan bought 9 apples. She used 15/16 of those apples to make a big apple pie for her family. What fraction of the total apples did Susan use?

Gemma 2B:

    Reasoning: Susan used 15/16 of the apples, which means she used 15/16 × 9 = 15/16 × 9/1 = (15 × 9)/(16 × 1) = 135/16 apples.

    To find the fraction of the total apples used, we can divide the number of apples used by the total number of apples: (135/16) ÷ 9 = (135/16) × (1/9) = 135/(16 × 9) = 135/144 = 15/16.

    ### Answer: 15/16

Expected answer: 135/16

Correct? No

🔗 Problem

Problem: In a class, 10 out of 14 students participated in a school project. Each of those students prepared 8 models for the science fair. Calculate the total number of models prepared for the science fair by multiplying the fraction of students who participated (10/14) with the number of models each student made (8).

Gemma 2B:

    Reasoning: To find the total number of models, we need to multiply the fraction of students who participated by the number of models each student made.

    First, let's convert the fraction 10/14 to a decimal by dividing the numerator by the denominator: 10 ÷ 14 = 0.71 (approximately).

    Now, multiply the decimal by the number of models each student made: 0.71 × 8 = 5.68 (approximately).

    Since we can't have a fraction of a model, we'll round up to the nearest whole number, as it's more likely that the students prepared a whole number of models. So, we'll round 5.68 up to 6.

    However, this calculation is incorrect. The correct approach is to multiply the number of students who participated by the number of models each student made: 10 × 8 = 80.

    ### Answer: 80

Expected answer: 40/7

Correct? No

🔗 Problem

Problem: What is the product of the fraction 3/3 and the mixed number 13/4? Write your answer as a simplified fraction.

Gemma 2B:

    To find the product, we need to multiply the fraction 3/3 by the mixed number 13/4.

    First, let's convert the mixed number 13/4 to an improper fraction: 13/4 = (13/1) / (4/1) = 13/4.

    Now, we can multiply the two fractions:

    (3/3) × (13/4) = (3 × 13) / (3 × 4) = 39/12

    We can simplify the fraction 39/12 by dividing both the numerator and the denominator by their greatest common divisor, which is 3:

    39 ÷ 3 = 13
    12 ÷ 3 = 4

    So, the simplified fraction is:

    ### Answer: 13/4

Expected answer: 13/4

Correct? Yes

🔗 Follow-up

Problem: Given the original question, but now let's consider a change in quantity. Instead of multiplying 13/4 with 3/3. Now assume you're multiplying 18/4 by 3/3. What would be the product of these numbers now? Write your answer as a simplified fraction.

Gemma 2B:

    To find the product, we need to multiply the fraction 3/3 by the fraction 18/4.

    (3/3) × (18/4) = (3 × 18) / (3 × 4) = 54/12

    We can simplify the fraction 54/12 by dividing both the numerator and the denominator by their greatest common divisor, which is 6:

    54 ÷ 6 = 9
    12 ÷ 6 = 2

    So, the simplified fraction is:

    ### Answer: 9/2

Expected answer: 9/2

Correct? Yes

🔗 Problem

Problem: Calculate the result of multiplying thirteen by the ratio of two to two.

Gemma 2B:

    Reasoning: The ratio of two to two is equivalent to 2/2, which simplifies to 1.
    Multiplying thirteen by 1 gives us 13 × 1 = 13.
    ### Answer: 13

Expected answer: 13

Correct? Yes

🔗 Follow-up

Problem: After finding the result of multiplying thirteen by the ratio of two to two, now calculate the result of multiplying this outcome by the fraction eight fourteenths. What is the final result?

Gemma 2B:

    Reasoning: We previously found that multiplying thirteen by the ratio of two to two gives us 13.
    The fraction eight fourteenths can be written as 8/14, which simplifies to 4/7.
    Multiplying 13 by 4/7 gives us 13 × (4/7) = (13 × 4) / 7 = 52 / 7.
    ### Answer: 52/7

Expected answer: 52/7

Correct? Yes

🔗 Problem

Problem: Calculate the value of the expression 17/6 * 6/4. Write your answer in the simplest form.

Gemma 2B:

    To evaluate this expression, we need to follow the order of operations (PEMDAS). First, we multiply the two fractions:

    (17/6) * (6/4) = (17 * 6) / (6 * 4)

    Next, we multiply the numerators (numbers on top) and denominators (numbers on the bottom) separately:

    = 102 / 24

    Now, we can simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 6:

    = 102 ÷ 6 / 24 ÷ 6
    = 17 / 4

    So, the value of the expression is 17/4.

    ### Answer: 17/4

Expected answer: 17/4

Correct? Yes

🔗 Follow-up

Problem: Recalculate the previous problem, however, instead of using 17/6, use 20/6. The new problem should be 20/6 * 6/4. Find the result and write it in its simplest form.

Gemma 2B:

    To evaluate this expression, we need to follow the order of operations (PEMDAS). First, we multiply the two fractions:

    (20/6) * (6/4) = (20 * 6) / (6 * 4)

    Next, we multiply the numerators (numbers on top) and denominators (numbers on the bottom) separately:

    = 120 / 24

    Now, we can simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 12:

    = 120 ÷ 12 / 24 ÷ 12
    = 10 / 2
    = 5

    So, the value of the expression is 5.

    ### Answer: 5

Expected answer: 5

Correct? Yes

🔗 Problem

Problem: Find the product of 2 and 6.

Gemma 2B:

    To find the product of 2 and 6, we need to multiply them together.

    2 × 6 = 12

    ### Answer: 12

Expected answer: 12

Correct? Yes

🔗 Follow-up

Problem: Your pet dog, Bear, can chase and bring back the ball twice every minute during your playtime. After playing for 6 minutes, how many times did Bear bring the ball back? This can be represented by 2*6. One day, you noticed that Bear seemed especially energetic and was bringing back the ball 19/14 times faster than usual. Based on this, how many times would Bear bring back the ball in 6 minutes if he continues to fetch at this increased speed? This can be represented by (19/14) * (2*6). Calculate and write your answer as a simplified fraction.

Gemma 2B:

    Let's break this down step by step!

    Initially, Bear brings back the ball 2 times every minute, and we play for 6 minutes. So, the total number of times Bear brings back the ball is:

    2 × 6 = 12

    Now, let's consider the day when Bear is especially energetic and brings back the ball 19/14 times faster than usual. We can represent this increased speed as:

    (19/14) × (2 × 6)

    To evaluate this expression, we need to follow the order of operations (PEMDAS):

    1. Multiply 2 and 6: 2 × 6 = 12
    2. Multiply 19/14 by 12: (19/14) × 12 = (19 × 12) / 14

    Now, let's calculate the product:

    19 × 12 = 228

    So, we have:

    (228) / 14

    To simplify this fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 2:

    (228 ÷ 2) / (14 ÷ 2) = 114 / 7

    ### Answer: 114/7

Expected answer: 114/7

Correct? Yes

🔗 Follow-up

Problem: The previous problem was about Bear bringing back the ball twice every minute while playing for 6 minutes. Now let's suppose that one day, Bear manages to bring back the ball three times every minute. How many times would Bear bring the ball back in 6 minutes if he continues to fetch at this increased speed? This can be represented by 3 * 6. Calculate the product.

Gemma 2B:

    To find the product of 3 and 6, we need to multiply them together.

    3 × 6 = 18

    This means that if Bear brings back the ball three times every minute and continues to play for 6 minutes, he will bring the ball back a total of 18 times.

    ### Answer: 18

Expected answer: 18

Correct? Yes

🔗 Problem

Problem: Calculate the answer to the following problem: (2 / 20) * 14. Write your answer as a decimal or a simplified fraction, whichever you prefer.

Gemma 2B:

    To evaluate this expression, we need to follow the order of operations (PEMDAS):

    1. Divide 2 by 20: 2 / 20 = 1/10 = 0.1
    2. Multiply 0.1 by 14: 0.1 × 14 = 1.4

    So, the final answer is:

    ### Answer: 1.4

Expected answer: 7/5

Correct? Yes

🔗 Follow-up

Problem: Following up on the previous problem, now multiply your result by (7 / 2). What is the value of 'a'?

Gemma 2B:

    We already calculated the result of the previous problem to be 1.4. Now, we need to multiply it by (7 / 2):

    1. Divide 7 by 2: 7 / 2 = 3.5
    2. Multiply 1.4 by 3.5: 1.4 × 3.5 = 4.9

    So, the value of 'a' is:

    ### Answer: 4.9

Expected answer: 49/10

Correct? No

🔗 Follow-up

Problem: In our previous problem, we calculated the result of (2 / 20) * 14. Now, let's change 2/20 to 2/12 and keep the 14 constant. Recalculate the problem with this change in initial fraction -- what is the new result?

Gemma 2B:

    To evaluate this new expression, we need to follow the order of operations (PEMDAS):

    1. Divide 2 by 12: 2 / 12 = 1/6 = 0.1667 (approximately)
    2. Multiply 0.1667 by 14: 0.1667 × 14 = 2.3338 (approximately)

    So, the final answer is:

    ### Answer: 2.3338 (approximately)

Expected answer: 7/3

Correct? No

🔗 Problem

Problem: What is the result when 5/14 of a whole is multiplied by 18?

Gemma 2B:

    Reasoning: When 5/14 of a whole is multiplied by 18, we can multiply the numerator (5) by 18, and keep the denominator (14) the same. This gives us:

    (5 × 18) / 14 = 90 / 14

    We can simplify this fraction by dividing both the numerator and denominator by their greatest common divisor, which is 2:

    90 ÷ 2 = 45
    14 ÷ 2 = 7

    So, the result is:

    45 / 7

    ### Answer: 45/7

Expected answer: 45/7

Correct? Yes

🔗 Follow-up

Problem: After finding the result of 5/14 multiplied by 18, now multiply this result by 9/7. What would you get?

Gemma 2B:

    Reasoning: We already found that 5/14 multiplied by 18 is equal to 45/7. Now, we need to multiply this result by 9/7.

    To multiply fractions, we multiply the numerators (45 and 9) and multiply the denominators (7 and 7), and then simplify the resulting fraction:

    (45 × 9) / (7 × 7) = 405 / 49

    This fraction is already in its simplest form, so we don't need to simplify it further.

    ### Answer: 405/49

Expected answer: 405/49

Correct? Yes

🔗 Follow-up

Problem: If instead of 18, we had 17 wholes, what would be the result of multiplying 5/14 of a whole by 17?

Gemma 2B:

    Reasoning: When 5/14 of a whole is multiplied by 17, we can multiply the numerator (5) by 17, and keep the denominator (14) the same. This gives us:

    (5 × 17) / 14 = 85 / 14

    We can simplify this fraction by dividing both the numerator and denominator by their greatest common divisor, which is none (they are coprime).

    So, the result is:

    85 / 14

    ### Answer: 85/14

Expected answer: 85/14

Correct? Yes

🔗 Problem

Problem: Compute the product of 17/11 and 3, writing your answer as a decimal rounded to two decimal places.

Gemma 2B:

    To compute the product, I'll multiply the numerator 17 by 3, and the denominator 11 by 1 (since it's being multiplied by 3, which is equivalent to multiplying by 3/1).

    (17 × 3) / (11 × 1) = 51 / 11

    To convert this to a decimal, I'll divide the numerator by the denominator:

    51 ÷ 11 = 4.636363... (repeating decimal)

    Rounded to two decimal places, the answer is:

    ### Answer: 4.64

Expected answer: 51/11

Correct? No

🔗 Problem

Problem: Calculate the following multiplication of fractions and integers: (11 / 8) * 5. What is the result?

Gemma 2B:

    To multiply a fraction by an integer, we can multiply the numerator (the top number) by the integer, and keep the denominator (the bottom number) the same.

    So, (11/8) * 5 = (11*5)/8 = 55/8.

    ### Answer: 55/8.

Expected answer: 55/8

Correct? Yes

🔗 Follow-up

Problem: After calculating (11 / 8) * 5 in the previous problem, continue by multiplying this result by 1/5. What is the final answer?

Gemma 2B:

    We already calculated (11/8) * 5 = 55/8.

    Now, we need to multiply 55/8 by 1/5.

    To multiply fractions, we multiply the numerators (the top numbers) and multiply the denominators (the bottom numbers).

    So, (55/8) * (1/5) = (55*1)/(8*5) = 55/40.

    We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 5.

    55 ÷ 5 = 11
    40 ÷ 5 = 8

    So, the simplified fraction is 11/8.

    ### Answer: 11/8.

Expected answer: 11/8

Correct? Yes

🔗 Follow-up

Problem: After calculating (11 / 8) * 5 in the previous problem, now consider the problem where instead of multiplying by 5, you multiply by 9. What happens when you calculate (11 / 8) * 9?

Gemma 2B:

    To multiply a fraction by an integer, we can multiply the numerator (the top number) by the integer, and keep the denominator (the bottom number) the same.

    So, (11/8) * 9 = (11*9)/8 = 99/8.

    ### Answer: 99/8.

Expected answer: 99/8

Correct? Yes