🔗 Problem

Problem: John has been learning to add and subtract fractions at school. He was given a problem during his homework that puzzled him. He had to take the sum of 48/56 and 95/23 and then subtract 34/26 from the result. What was the value that John found for this operation? Answer in simplest form.

Gemma 2B:

    Here's how to solve the problem step-by-step:

    **1. Find the Least Common Multiple (LCM) of the denominators:**

    * The denominators are 56, 23, and 26.
    * The LCM of these numbers is 8008.

    **2. Convert each fraction to an equivalent fraction with the LCM as the denominator:**

    * 48/56 = (48 * 143) / (56 * 143) = 6864/8008
    * 95/23 = (95 * 348) / (23 * 348) = 33060/8008
    * 34/26 = (34 * 308) / (26 * 308) = 10472/8008

    **3. Add the first two fractions:**

    * 6864/8008 + 33060/8008 = 39924/8008

    **4. Subtract the third fraction:**

    * 39924/8008 - 10472/8008 = 29452/8008

    **5. Simplify the result:**

    * Both the numerator and denominator can be divided by 8.
    * 29452/8008 = (29452/8) / (8008/8) = 3681/1001

    **Therefore, the value John found for the operation is 3681/1001.**
    ### Answer: 3681/1001

Expected answer: 7702/2093

Correct? No

🔗 Problem

Problem: First, add the fractions 88/9 and 28/65. Then subtract the fraction 18/37 from the result. What is the result expressed as a simple fraction?

Gemma 2B:

    Here's how to solve the problem step-by-step:

    1. **Find a common denominator for 88/9 and 28/65:**
       - The least common multiple of 9 and 65 is 585.
       - Convert 88/9 to an equivalent fraction with a denominator of 585: (88/9) * (65/65) = 5720/585
       - Convert 28/65 to an equivalent fraction with a denominator of 585: (28/65) * (9/9) = 252/585

    2. **Add the fractions:**
       - (5720/585) + (252/585) = 5972/585

    3. **Find a common denominator for 5972/585 and 18/37:**
       - The least common multiple of 585 and 37 is 21645.
       - Convert 5972/585 to an equivalent fraction with a denominator of 21645: (5972/585) * (37/37) = 221044/21645
       - Convert 18/37 to an equivalent fraction with a denominator of 21645: (18/37) * (585/585) = 10530/21645

    4. **Subtract the fractions:**
       - (221044/21645) - (10530/21645) = 210514/21645

    5. **Simplify the fraction:**
       - Both the numerator and denominator are divisible by 2, so we can simplify to 105257/10823.

    ### Answer: 105257/10823

Expected answer: 210434/21645

Correct? No

🔗 Problem

Problem: What is the result of (63 / 41) + (44 / 50) when simplified to the lowest terms?

Gemma 2B:

    Here's how to simplify the expression:

    1. **Calculate the fractions:**
       * 63 / 41 = 1.5366 (approximately)
       * 44 / 50 = 0.88

    2. **Add the results:**
       * 1.5366 + 0.88 = 2.4166 (approximately)

    3. **Convert to a fraction:**
       * We can approximate 2.4166 as 24166/10000

    4. **Simplify the fraction:**
       * Both the numerator and denominator are divisible by 2: 24166/10000 = 12083/5000

    **Therefore, the simplified result of (63 / 41) + (44 / 50) is 12083/5000.**

    ### Answer: 12083/5000

Expected answer: 2477/1025

Correct? No

🔗 Problem

Problem: John had 71/41 of a cake left and Emily added 41/77 of another cake to it. How much cake is there in total? Give your answer as a simplified fraction.

Gemma 2B:

    Reasoning: To add fractions, they need to have the same denominator.  The least common multiple of 41 and 77 is 3167. We can rewrite the fractions as:

    * 71/41 = (71 * 77) / (41 * 77) = 5487 / 3167
    * 41/77 = (41 * 41) / (77 * 41) = 1681 / 3167

    Now we can add the fractions:

    * 5487/3167 + 1681/3167 = (5487 + 1681)/3167 = 7168/3167

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

    * 7168/3167 = (7168 / 1) / (3167 / 1) = 7168/3167

    ### Answer: 7168/3167

Expected answer: 7148/3157

Correct? No

🔗 Problem

Problem: Calculate the result of 10/65 + 56/35. Answer as a simple fraction.

Gemma 2B:

    To add fractions, they need to have the same denominator.  The least common multiple of 65 and 35 is 910.

    * **10/65:**  Multiply numerator and denominator by 14: (10 * 14) / (65 * 14) = 140/910
    * **56/35:** Multiply numerator and denominator by 26: (56 * 26) / (35 * 26) = 1456/910

    Now we can add the fractions:
    140/910 + 1456/910 = (140 + 1456)/910 = 1596/910

    We can simplify this fraction by dividing both numerator and denominator by their greatest common factor, which is 2:
    (1596 / 2) / (910 / 2) = 798/455

    ### Answer: 798/455

Expected answer: 114/65

Correct? Yes

🔗 Follow-up

Problem: Following the result of the previous calculation (10/65 + 56/35), subtract your answer from 34. What's the resultant value? Answer as a simple fraction.

Gemma 2B:

    We need to subtract 798/455 from 34. To do this, we need a common denominator.

    1.  **Represent 34 as a fraction with the same denominator as 798/455:**
        34 = (34 * 455)/455 = 15570/455

    2. **Subtract the fractions:**
        (15570/455) - (798/455) = (15570 - 798) / 455 = 14772/455

    ### Answer: 14772/455

Expected answer: 2096/65

Correct? No

🔗 Follow-up

Problem: Given your previous calculation with the 10/65 + 56/35, imagine instead that you started with 25 instead of 10 in the first fraction. What would be the outcome of 25/65 + 56/35? Answer as a simplified fraction.

Gemma 2B:

    Following the same process as before:

    * **25/65:** Multiply numerator and denominator by 14: (25 * 14) / (65 * 14) = 350/910
    * **56/35:** Multiply numerator and denominator by 26: (56 * 26) / (35 * 26) = 1456/910

    Now we can add the fractions:
    350/910 + 1456/910 = (350 + 1456)/910 = 1806/910

    We can simplify this fraction by dividing both numerator and denominator by their greatest common factor, which is 2:
    (1806 / 2) / (910 / 2) = 903/455

    ### Answer: 903/455

Expected answer: 129/65

Correct? Yes

🔗 Problem

Problem: What is the result of adding 75/81 and 97/47? Leave your answer as an unsimplified fraction.

Gemma 2B:

    To add fractions, they need to have the same denominator.  We find the least common multiple (LCM) of 81 and 47, which is 3807.

    *  We multiply 75/81 by 47/47: (75/81) * (47/47) = 3525/3807
    *  We multiply 97/47 by 81/81: (97/47) * (81/81) = 7857/3807

    Now we can add the fractions: 3525/3807 + 7857/3807 = (3525 + 7857)/3807

    ### Answer: 11382/3807

Expected answer: 3794/1269

Correct? Yes

🔗 Follow-up

Problem: After calculating the result of adding 75/81 and 97/47, what do you get when you add that result to 52/71? Provide your answer as an unsimplified fraction.

Gemma 2B:

    We already know that 75/81 + 97/47 = 11382/3807.

    To add this to 52/71, we need a common denominator. The least common multiple of 3807 and 71 is 269857.

    * Multiply 11382/3807 by 71/71: (11382/3807) * (71/71) = 808242/269857
    * Multiply 52/71 by 3807/3807: (52/71) * (3807/3807) = 197964/269857

    Now we can add the fractions: 808242/269857 + 197964/269857 = (808242 + 197964)/269857

    ### Answer: 1006206/269857

Expected answer: 335362/90099

Correct? No

🔗 Problem

Problem: What is 88/1 - 38/53? Give your answer as a simplified fraction.

Gemma 2B:

    Reasoning:
    1.  First, we need to find a common denominator for the fractions. The least common multiple of 1 and 53 is 53.
    2.  We can rewrite 88/1 as (88 * 53) / (1 * 53) = 4664/53.
    3.  Now we have: (4664/53) - (38/53)
    4.  Since they have the same denominator, we can simply subtract the numerators: (4664 - 38) / 53 = 4626/53

    ### Answer: 4626/53

Expected answer: 4626/53

Correct? Yes

🔗 Follow-up

Problem: After calculating the result of 88/1 - 38/53, add it to 25/79. What is the final value, and answer as a simplified fraction?

Gemma 2B:

    Reasoning:

    1. We already know that 88/1 - 38/53 = 4626/53.
    2. Now we need to add 25/79 to this result: (4626/53) + (25/79)
    3. To add fractions, they need a common denominator. The least common multiple of 53 and 79 is 4187.
    4.  We rewrite each fraction with the common denominator:
       * (4626/53) * (79/79) = 366294/4187
       * (25/79) * (53/53) = 1325/4187
    5. Now we can add the numerators: (366294 + 1325) / 4187 = 367619/4187

    ### Answer: 367619/4187

Expected answer: 366779/4187

Correct? No

🔗 Follow-up

Problem: Previous problem: What is 88/1 - 38/53? Now consider if the second fraction (38/53) was changed to 38/93 instead. So, what is 88/1 - 38/93? Answer as a simplified fraction.

Gemma 2B:

    Reasoning:
    1.  The least common multiple of 1 and 93 is 93.
    2.  Rewrite 88/1 as (88 * 93) / (1 * 93) = 8184/93
    3.  Now we have: (8184/93) - (38/93)
    4.  Since the denominators are the same, we can subtract the numerators: (8184 - 38) / 93 = 8146/93

    ### Answer: 8146/93

Expected answer: 8146/93

Correct? Yes

🔗 Problem

Problem: Evaluate the expression ((80 / 94) + (78 / 59)). What is the answer in simplified form?

Gemma 2B:

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

    1. **Parentheses/Brackets:**  We have two sets of parentheses, so we need to evaluate the fractions inside them first.
       * 80 / 94 = 0.851 (approximately)
       * 78 / 59 = 1.322 (approximately)

    2. **Addition:** Now we add the results of the two fractions.
       * 0.851 + 1.322 = 2.173 (approximately)

    Therefore, the simplified form of the expression is approximately **2.173**.

    ### Answer: 2.173 (approximately)

Expected answer: 6026/2773

Correct? No

🔗 Problem

Problem: What is the sum of 89/56 and 48/50? Answer as a simplified fraction.

Gemma 2B:

    Reasoning: To add fractions, they need to have the same denominator.  The least common multiple of 56 and 50 is 1400.
    * 89/56 = (89*25)/(56*25) = 2225/1400
    * 48/50 = (48*28)/(50*28) = 1344/1400

    Now we can add: 2225/1400 + 1344/1400 = (2225+1344)/1400 = 3569/1400

    We can simplify this fraction by dividing both numerator and denominator by their greatest common factor, which is 11.

    3569/1400 = (3569/11) / (1400/11) = 324.45/127.27

    Since 324.45/127.27 is not a simplified fraction, we will leave the answer as it is.

    ### Answer: 3569/1400

Expected answer: 3569/1400

Correct? Yes

🔗 Follow-up

Problem: After finding the sum of 89/56 and 48/50, add the result to 2/65. What is the final simplified fraction?

Gemma 2B:

    We already know the sum of 89/56 and 48/50 is 3569/1400. Now we need to add 2/65 to this result.

    To add these fractions, we need a common denominator. The least common multiple of 1400 and 65 is 18200.

    * 3569/1400 = (3569*13)/(1400*13) = 46397/18200
    * 2/65 = (2*280)/(65*280) = 560/18200

    Now we can add: 46397/18200 + 560/18200 = (46397 + 560)/18200 = 46957/18200

    We can simplify this fraction by dividing both numerator and denominator by their greatest common factor, which is 11.

    46957/18200 = (46957/11) / (18200/11) = 4269/1655

    ### Answer: 4269/1655

Expected answer: 46957/18200

Correct? No

🔗 Follow-up

Problem: Suppose in the original problem, instead of having 89/12, we only have 89/12, and the number of parts out of 50 remain the same. So, what is the sum of 89/12 and 48/50 this time? Answer as a simplified fraction.

Gemma 2B:

    Reasoning: To add fractions, they need to have the same denominator. The least common multiple of 12 and 50 is 300.
    * 89/12 = (89*25)/(12*25) = 2225/300
    * 48/50 = (48*6)/(50*6) = 288/300

    Now we can add: 2225/300 + 288/300 = (2225+288)/300 = 2513/300

    This fraction can be simplified by dividing both numerator and denominator by their greatest common factor, which is 1.

    ### Answer: 2513/300

Expected answer: 2513/300

Correct? Yes

🔗 Problem

Problem: What is the result when you subtract 38/87 from the sum of 96/11 and 12/5? Please leave your answer as a fraction.

Gemma 2B:

    First, we need to find the sum of 96/11 and 12/5. To do this, we need a common denominator:

    * The least common multiple of 11 and 5 is 55.
    * 96/11 * (5/5) = 480/55
    * 12/5 * (11/11) = 132/55

    Now we add the two fractions:

    * 480/55 + 132/55 = 612/55

    Finally, we subtract 38/87 from the sum:

    * To subtract fractions, we need a common denominator. The least common multiple of 55 and 87 is 4785.
    * 612/55 * (87/87) = 53244/4785
    * 38/87 * (55/55) = 2090/4785

    Now we can subtract:

    * 53244/4785 - 2090/4785 = 51154/4785

    ### Answer: 51154/4785

Expected answer: 51154/4785

Correct? Yes

🔗 Follow-up

Problem: After you find the result subtracting 38/87 from the sum of 96/11 and 12/5, add that result back to the initial sum of 96/11 and 12/5. What is the final result? Please leave your answer as a fraction.

Gemma 2B:

    We already know the sum of 96/11 and 12/5 is 612/55.  We also know the result of subtracting 38/87 from that sum is 51154/4785.

    Now, we need to add those two fractions back together:

    * 612/55 * (87/87) = 53244/4785
    * 53244/4785 + 51154/4785 = 104398/4785

    ### Answer: 104398/4785

Expected answer: 104398/4785

Correct? Yes

🔗 Follow-up

Problem: After considering the fruit you have, suppose you actually started off with a fraction of the fruits, having 96/58 oranges, and still having 12/5 apples. If you subtracted 38/87 of your total fruits for charity, what fraction of fruits would you have left? Please leave your answer as a fraction.

Gemma 2B:

    Here's how to solve this problem step-by-step:

    1. **Find the total initial fraction of fruits:**
       * Add the fractions of oranges and apples: 96/58 + 12/5 = (96*5 + 12*58) / (58*5) = 912/290

    2. **Calculate the fraction of fruits donated:**
       * Multiply the total initial fraction by 38/87: (912/290) * (38/87) = 3456/25230

    3. **Subtract the donated fraction from the initial fraction:**
       * 912/290 - 3456/25230 = (912*87 - 3456*290) / (290*87) = -73920/25230

    4. **Simplify the result (if possible):**
       * The fraction can be simplified by dividing both numerator and denominator by their greatest common factor, which is 18480.
       * (-73920 / 18480) / (25230 / 18480) = -4/13.5

    **Important Note:** You can't have a negative fraction of fruits. This result suggests that the amount donated was more than the total initial amount of fruits.  There might be an error in the problem setup.

    **Assuming the problem meant to subtract 38/87 of the *oranges* for charity, we can solve as follows:**

    1. **Calculate the fraction of oranges donated:**
       * (96/58) * (38/87) = 1536/4761

    2. **Subtract the donated oranges from the initial oranges:**
       * 96/58 - 1536/4761 = (96*87 - 1536*58) / (58*87) = -63360/5046

    3. **Simplify the fraction:**
       * (-63360 / 18480) / (5046 / 18480) = -3.5/2.75

    4. **Add the remaining apples to find the total fraction left:**
       * -3.5/2.75 + 12/5 = (-3.5*5 + 12*2.75) / (2.75*5) = 16.5/13.75

    5. **Simplify the fraction:**
       * (16.5/13.75) = (33/27.5) = (66/55)

    ### Answer: 66/55

Expected answer: 1574/435

Correct? No

🔗 Problem

Problem: If you have a number that's 82 divided by 9 and from this you subtract 82 divided by 63, what's the resulting value? Answer as a simplified fraction.

Gemma 2B:

    Here's how to solve this problem:

    1. **Calculate the first division:** 82 divided by 9 is 82/9.

    2. **Calculate the second division:** 82 divided by 63 is 82/63.

    3. **Subtract the second division from the first:** (82/9) - (82/63)

    4. **Find a common denominator:** The least common denominator of 9 and 63 is 63.  To get a denominator of 63 for the first fraction, multiply both numerator and denominator by 7: (82/9) * (7/7) = 574/63.

    5. **Subtract the fractions:** (574/63) - (82/63) = (574-82)/63 = 492/63.

    6. **Simplify the fraction:** Both 492 and 63 are divisible by 9.  (492/9) / (63/9) = 54/7

    ### Answer: 54/7

Expected answer: 164/21

Correct? No

🔗 Problem

Problem: Calculate the value of 23/3 minus the sum of 80/91 and 26/24. Answer as a simple fraction.

Gemma 2B:

    First, we need to find a common denominator for 80/91 and 26/24. The least common multiple of 91 and 24 is 2184.

    * 80/91 * (24/24) = 1920/2184
    * 26/24 * (91/91) = 2366/2184

    Now we can add these fractions: 1920/2184 + 2366/2184 = 4286/2184

    Next, we simplify this fraction by dividing both numerator and denominator by their greatest common factor, which is 2: 4286/2184 = 2143/1092

    Now we can subtract this simplified fraction from 23/3: 23/3 - 2143/1092

    To subtract these fractions, we need a common denominator:

    * 23/3 * (364/364) = 8382/1092
    * 2143/1092 stays the same

    Now we can subtract: 8382/1092 - 2143/1092 = 6239/1092

    Finally, we simplify this fraction: 6239/1092 = 2079/364

    ### Answer: 2079/364

Expected answer: 6229/1092

Correct? No

🔗 Problem

Problem: What is the sum of 37/18 and 24/60 when simplified to the lowest terms?

Gemma 2B:

    Reasoning:
    1. Find a common denominator for 18 and 60. The least common multiple of 18 and 60 is 180.
    2. Convert 37/18 to an equivalent fraction with a denominator of 180: (37/18) * (10/10) = 370/180.
    3. Convert 24/60 to an equivalent fraction with a denominator of 180: (24/60) * (3/3) = 72/180.
    4. Add the fractions: 370/180 + 72/180 = 442/180.
    5. Simplify the fraction to its lowest terms by dividing both numerator and denominator by their greatest common factor, which is 2: (442/180) / (2/2) = 221/90.

    ### Answer: 221/90

Expected answer: 221/90

Correct? Yes

🔗 Follow-up

Problem: If the initial sum of 37/18 and 24/60 is added to the fraction 92/28, what would the new total be? Answer as a simplified fraction.

Gemma 2B:

    Reasoning:
    1. We already know the sum of 37/18 and 24/60 is 221/90.
    2. Find a common denominator for 90 and 28. The least common multiple of 90 and 28 is 1260.
    3. Convert 221/90 to an equivalent fraction with a denominator of 1260: (221/90) * (14/14) = 3094/1260.
    4. Convert 92/28 to an equivalent fraction with a denominator of 1260: (92/28) * (45/45) = 4140/1260.
    5. Add the fractions: 3094/1260 + 4140/1260 = 7234/1260.
    6. Simplify the fraction to its lowest terms by dividing both numerator and denominator by their greatest common factor, which is 2: (7234/1260) / (2/2) = 3617/630.

    ### Answer: 3617/630

Expected answer: 3617/630

Correct? Yes

🔗 Follow-up

Problem: If instead of the original sum of 37/18 and 24/60, we now have the sum of 37/18 and 24/72, what would the new total be? Please express your answer as a simplified fraction.

Gemma 2B:

    Reasoning:
    1. Find a common denominator for 18 and 72. The least common multiple of 18 and 72 is 72.
    2. Convert 37/18 to an equivalent fraction with a denominator of 72: (37/18) * (4/4) = 148/72.
    3. Keep 24/72 as it already has the desired denominator.
    4. Add the fractions: 148/72 + 24/72 = 172/72.
    5. Simplify the fraction to its lowest terms by dividing both numerator and denominator by their greatest common factor, which is 4: (172/72) / (4/4) = 43/18.

    ### Answer: 43/18

Expected answer: 43/18

Correct? Yes

🔗 Problem

Problem: Determine the sum of 33/41 and 95/62. Write your answer as a simplified fraction.

Gemma 2B:

    To add fractions, they need to have a common denominator.  The least common multiple of 41 and 62 is 2542.

    * Convert 33/41 to have a denominator of 2542: (33/41) * (62/62) = 2046/2542
    * Convert 95/62 to have a denominator of 2542: (95/62) * (41/41) = 3895/2542

    Now we can add the fractions:

    2046/2542 + 3895/2542 = 5941/2542

    ### Answer: 5941/2542

Expected answer: 5941/2542

Correct? Yes

🔗 Follow-up

Problem: After finding the sum of 33/41 and 95/62, add that sum to 95/62 once again. What is the final simplified fraction?

Gemma 2B:

    We already know the sum of 33/41 and 95/62 is 5941/2542.

    Now we need to add 95/62 to that sum.  To do this, we need to convert 95/62 to have a denominator of 2542:

    (95/62) * (41/41) = 3895/2542

    Finally, add the two fractions:

    5941/2542 + 3895/2542 = 9836/2542

    This fraction can be simplified by dividing both numerator and denominator by their greatest common factor, which is 2:

    9836/2542 = (9836/2) / (2542/2) = 4918/1271

    ### Answer: 4918/1271

Expected answer: 4918/1271

Correct? Yes

🔗 Follow-up

Problem: After reconsidering the previous problem, suppose instead of 95/62, we now have 83/62. What would the sum of 33/41 and 83/62 be? Write your answer as a simplified fraction.

Gemma 2B:

    We follow the same process as before.  The least common multiple of 41 and 62 is 2542.

    * Convert 33/41 to have a denominator of 2542: (33/41) * (62/62) = 2046/2542
    * Convert 83/62 to have a denominator of 2542: (83/62) * (41/41) = 3403/2542

    Now we can add the fractions:

    2046/2542 + 3403/2542 = 5449/2542

    ### Answer: 5449/2542

Expected answer: 5449/2542

Correct? Yes

🔗 Problem

Problem: Solve ((30 / 11) + (72 / 38)) and then add the result to (81 / 34). Answer as a simplified fraction.

Gemma 2B:

    Here's how to solve the problem:

    **1. Simplify the fractions inside the parentheses:**

    * (30 / 11)  is already in its simplest form.
    * (72 / 38) can be simplified by dividing both numerator and denominator by their greatest common factor, 2: (72/38) = (36/19)

    **2. Add the simplified fractions inside the parentheses:**

    * To add fractions, they need a common denominator. The least common denominator of 11 and 19 is 209.
    * (30/11) + (36/19) = (30 * 19 / 11 * 19) + (36 * 11 / 19 * 11) = (570 / 209) + (396 / 209) = (966 / 209)

    **3. Add the result to (81/34):**

    * Find a common denominator for 209 and 34. The least common denominator is 7086.
    * (966 / 209) + (81 / 34) = (966 * 34 / 209 * 34) + (81 * 209 / 34 * 209) = (32844 / 7086) + (16929 / 7086) = (49773 / 7086)

    **4. Simplify the final fraction:**

    * The greatest common factor of 49773 and 7086 is 1. Therefore, the fraction is already in its simplest form.

    ### Answer: 49773/7086

Expected answer: 49773/7106

Correct? No

🔗 Problem

Problem: What is the result of adding (64 / 34) + (3 / 24) and then adding the result to (41 / 96)? Answer as a simplified fraction.

Gemma 2B:

    Here's how to solve the problem step-by-step:

    1. **Simplify the fractions:**
       - (64 / 34) can be simplified to (32 / 17) by dividing both numerator and denominator by 2.
       - (3 / 24) can be simplified to (1 / 8) by dividing both numerator and denominator by 3.

    2. **Find a common denominator for (32 / 17) and (1 / 8):**
       - The least common multiple of 17 and 8 is 136.
       - Convert (32 / 17) to (256 / 136) by multiplying both numerator and denominator by 8.
       - Convert (1 / 8) to (17 / 136) by multiplying both numerator and denominator by 17.

    3. **Add the fractions:**
       - (256 / 136) + (17 / 136) = (273 / 136)

    4. **Find a common denominator for (273 / 136) and (41 / 96):**
       - The least common multiple of 136 and 96 is 1088.
       - Convert (273 / 136) to (849 / 1088) by multiplying both numerator and denominator by 8.
       - Convert (41 / 96) to (461 / 1088) by multiplying both numerator and denominator by 11.

    5. **Add the fractions:**
       - (849 / 1088) + (461 / 1088) = (1310 / 1088)

    6. **Simplify the fraction:**
       - Both 1310 and 1088 are divisible by 2, so we get (655 / 544).

    ### Answer: 655/544

Expected answer: 3973/1632

Correct? No

🔗 Problem

Problem: Daniel has to solve a math problem for his homework that includes fractions. Initially, he has been asked to add 83/73 and 53/77. The result of this step he saved as 'w'. Then, he needs to add 80/76 to the result 'w'. How would he calculate the final answer in simplified fraction form?

Gemma 2B:

    Here's how Daniel can calculate the final answer in simplified fraction form:

    **Step 1: Add the first two fractions**

    * Find a common denominator for 73 and 77. The least common multiple is 5621.
    * Convert the fractions:
        * 83/73 * (77/77) = 6391/5621
        * 53/77 * (73/73) = 3869/5621
    * Add the fractions: 6391/5621 + 3869/5621 = 10260/5621
    * This is the value of 'w'.

    **Step 2: Add 80/76 to 'w'**

    * Find a common denominator for 5621 and 76. The least common multiple is 426876.
    * Convert the fractions:
        * 10260/5621 * (76/76) = 779760/426876
        * 80/76 * (5621/5621) = 449680/426876
    * Add the fractions: 779760/426876 + 449680/426876 = 1229440/426876

    **Step 3: Simplify the result**

    * Find the greatest common factor (GCD) of 1229440 and 426876.  The GCD is 15368.
    * Divide both numerator and denominator by 15368:
        * (1229440 / 15368) / (426876 / 15368) = 80/28

    **Step 4: Further simplification**

    * The fraction 80/28 can be further simplified by dividing both numerator and denominator by 4:
        * (80 / 4) / (28 / 4) = 20/7

    ### Answer: 20/7

Expected answer: 307360/106799

Correct? No

🔗 Problem

Problem: Add the following fractions and simplify your answer: 27/20 and 14/66.

Gemma 2B:

    To add fractions, they need to have the same denominator.  We find the least common multiple (LCM) of 20 and 66:

    * Factors of 20: 1, 2, 4, 5, 10, 20
    * Factors of 66: 1, 2, 3, 6, 11, 22, 33, 66

    The LCM of 20 and 66 is 660.

    Now, we rewrite each fraction with a denominator of 660:

    * (27/20) * (33/33) = 891/660
    * (14/66) * (10/10) = 140/660

    Finally, we add the fractions:

    * 891/660 + 140/660 = (891 + 140)/660 = 1031/660

    This fraction can be simplified by finding the greatest common factor (GCF) of 1031 and 660.  The GCF is 1, so the fraction is already in its simplest form.

    ### Answer: 1031/660

Expected answer: 1031/660

Correct? Yes

🔗 Follow-up

Problem: You previously added the fractions 27/20 and 14/66. Now, add the result to the fraction 58/18 and simplify your answer.

Gemma 2B:

    We previously found that 27/20 + 14/66 = 1031/660.

    Now we need to add 58/18 to this result.  Again, we need a common denominator:

    * The LCM of 660 and 18 is 1980.
    * (1031/660) * (3/3) = 3093/1980
    * (58/18) * (110/110) = 6380/1980

    Now we can add:

    * 3093/1980 + 6380/1980 = (3093 + 6380)/1980 = 9473/1980

    This fraction can be simplified by finding the GCF of 9473 and 1980.  The GCF is 33.  Dividing both numerator and denominator by 33:

    * (9473/33) / (1980/33) = 287/60

    ### Answer: 287/60

Expected answer: 9473/1980

Correct? No

🔗 Follow-up

Problem: Previously, you added the fractions 27/20 and 14/66. Let's change the problem slightly. Now, instead of 14/66, consider a fraction of 14/41. Add 27/20 and 14/41 together and simplify your answer.

Gemma 2B:

    To add the fractions 27/20 and 14/41, we need to find a common denominator. The least common multiple (LCM) of 20 and 41 is 820.

    Now, we rewrite each fraction with a denominator of 820:

    * (27/20) * (41/41) = 1107/820
    * (14/41) * (20/20) = 280/820

    Finally, we add the fractions:

    * 1107/820 + 280/820 = (1107 + 280)/820 = 1387/820

    This fraction cannot be simplified further as 1387 and 820 share no common factors other than 1.

    ### Answer: 1387/820

Expected answer: 1387/820

Correct? Yes

🔗 Problem

Problem: What is the result of the expression ((32 / 14) - (22 / 47))? Write your answer as a simplified fraction.

Gemma 2B:

    Reasoning:
    First, we need to simplify the fractions:
    32/14 = 16/7
    22/47 cannot be simplified further

    Now, we need to find a common denominator for the two fractions. The least common multiple of 7 and 47 is 329.

    (16/7) * (47/47) = 752/329
    (22/47) * (7/7) = 154/329

    Finally, we subtract the fractions:
    (752/329) - (154/329) = (752-154)/329 = 598/329

    ### Answer: 598/329

Expected answer: 598/329

Correct? Yes

🔗 Follow-up

Problem: If you take the result of ((32 / 14) - (22 / 47)) you calculated earlier, what would be the value you get when you add this result to itself? Write your answer as a simplified fraction.

Gemma 2B:

    Reasoning:
    We already know that the result of ((32 / 14) - (22 / 47)) is 598/329.

    Adding this fraction to itself is the same as multiplying it by 2:

    (598/329) * 2 = (598 * 2) / 329 = 1196 / 329

    ### Answer: 1196/329

Expected answer: 1196/329

Correct? Yes

🔗 Follow-up

Problem: Suppose instead of having 32/14 part of the original mixture, you actually had 71/14 part, and the same fraction (22/47) was subtracted from it. In that case, what would be the new result of the calculation: ((71 / 14) - (22 / 47))? Write your answer as a simplified fraction.

Gemma 2B:

    Reasoning:
    First, we need to simplify the fractions:
    71/14 cannot be simplified further
    22/47 cannot be simplified further

    Now, we need to find a common denominator for the two fractions. The least common multiple of 14 and 47 is 658.

    (71/14) * (47/47) = 3337/658
    (22/47) * (14/14) = 308/658

    Finally, we subtract the fractions:
    (3337/658) - (308/658) = (3337-308)/658 = 3029/658

    ### Answer: 3029/658

Expected answer: 3029/658

Correct? Yes

🔗 Problem

Problem: What is the sum of 74/61 and 41/53 expressed in fractions?

Gemma 2B:

    To add fractions, they must have a common denominator.  We find the least common multiple (LCM) of 61 and 53.  Since 61 and 53 are both prime numbers, their LCM is their product: 61 * 53 = 3233.

    Now we rewrite each fraction with a denominator of 3233:

    * (74/61) * (53/53) = 3922/3233
    * (41/53) * (61/61) = 2501/3233

    Finally, we add the numerators:

    3922/3233 + 2501/3233 = (3922 + 2501)/3233 = 6423/3233

    ### Answer: 6423/3233

Expected answer: 6423/3233

Correct? Yes

🔗 Follow-up

Problem: Assuming you already calculated the sum of 74/61 and 41/53, what would be the total if you added 83/4 to the result? Answer as a simplified fraction.

Gemma 2B:

    We already found that the sum of 74/61 and 41/53 is 6423/3233.

    To add 83/4, we need a common denominator. The LCM of 3233 and 4 is 3233 * 4 = 12932.

    * (6423/3233) * (4/4) = 25692/12932
    * (83/4) * (3233/3233) = 268279/12932

    Now we add the numerators:

    25692/12932 + 268279/12932 = (25692 + 268279)/12932 = 293971/12932

    This fraction can be simplified by dividing both numerator and denominator by their greatest common factor, which is 1:

    293971/12932 = **293971/12932**

    ### Answer: 293971/12932

Expected answer: 294031/12932

Correct? No

🔗 Follow-up

Problem: Previously you calculated the sum of 74/61 and 41/53. Now, let's consider if the second fraction was actually 41/88 instead of 41/53. What would be the new total of 74/61 and 41/88? Answer as a simplified fraction.

Gemma 2B:

    We need to find the least common multiple (LCM) of 61 and 88.  The prime factorizations are:

    * 61 = 61 (already prime)
    * 88 = 2 * 2 * 2 * 11 = 2³ * 11

    The LCM is the product of the highest powers of all prime factors: 2³ * 11 * 61 = 5368

    Now we rewrite each fraction with a denominator of 5368:

    * (74/61) * (88/88) = 6512/5368
    * (41/88) * (61/61) = 2501/5368

    Finally, we add the numerators:

    6512/5368 + 2501/5368 = (6512 + 2501)/5368 = 9013/5368

    ### Answer: 9013/5368

Expected answer: 9013/5368

Correct? Yes

🔗 Problem

Problem: First, find out the result of 94 / 2 - 74 / 57. Then add the result to 56 / 97. What fraction do you get?

Gemma 2B:

    Let's break down the problem step-by-step:

    1. **Calculate 94 / 2:** 94 divided by 2 equals 47.

    2. **Calculate 74 / 57:** This fraction cannot be simplified further.

    3. **Subtract the results:** 47 - (74/57) = (2679 - 74) / 57 = 2605 / 57

    4. **Add 56/97:** (2605/57) + (56/97) = (252465 + 3092) / 5529 = 255557 / 5529

    ### Answer: 255557/5529

Expected answer: 255877/5529

Correct? No

🔗 Problem

Problem: What is the sum of 14/89 and the sum of 24/34 and 44/87? Give your answer as a simplified fraction.

Gemma 2B:

    First, we need to find the sum of 24/34 and 44/87. To do this, we need a common denominator. The least common multiple of 34 and 87 is 2958.

    * 24/34 * 87/87 = 2088/2958
    * 44/87 * 34/34 = 1496/2958

    Now we can add the two fractions: 2088/2958 + 1496/2958 = 3584/2958

    Next, we need to add 14/89 to this sum. Again, we need a common denominator. The least common multiple of 2958 and 89 is 263462.

    * 3584/2958 * 89/89 = 318976/263462
    * 14/89 * 2958/2958 = 41412/263462

    Now we can add the two fractions: 318976/263462 + 41412/263462 = 360388/263462

    Finally, we can simplify this fraction by dividing both numerator and denominator by their greatest common factor, which is 2:

    360388/263462 = 180194/131731

    ### Answer: 180194/131731

Expected answer: 180194/131631

Correct? No

🔗 Problem

Problem: What would you get if you add 73/30 to 8/60?

Gemma 2B:

    First, we need to find a common denominator for the two fractions. The least common multiple of 30 and 60 is 60.

    * We can rewrite 73/30 as (73 * 2) / (30 * 2) = 146/60

    Now we can add the fractions:

    * 146/60 + 8/60 = (146 + 8) / 60 = 154/60

    Finally, we can simplify the fraction by dividing both numerator and denominator by their greatest common factor, which is 2:

    * 154/60 = (154 / 2) / (60 / 2) = 77/30

    ### Answer: 77/30

Expected answer: 77/30

Correct? Yes

🔗 Follow-up

Problem: If you take the result of adding 73/30 to 8/60 and then add it to itself again, what fraction do you get?

Gemma 2B:

    We already know that adding 73/30 and 8/60 gives us 77/30.

    To add 77/30 to itself, we simply multiply it by 2:

    * (77/30) * 2 = 154/30

    Finally, we can simplify this fraction by dividing both numerator and denominator by their greatest common factor, which is 2:

    * 154/30 = (154 / 2) / (30 / 2) = 77/15

    ### Answer: 77/15

Expected answer: 77/15

Correct? Yes

🔗 Follow-up

Problem: Suppose you originally had a fraction of 73/30. Now, instead of adding 8/60 to it, you decide to add 49/60. What fraction will you get in this case?

Gemma 2B:

    We need to find a common denominator for 73/30 and 49/60. The least common multiple of 30 and 60 is 60.

    * We can rewrite 73/30 as (73 * 2) / (30 * 2) = 146/60

    Now we can add the fractions:

    * 146/60 + 49/60 = (146 + 49) / 60 = 195/60

    Finally, we can simplify the fraction by dividing both numerator and denominator by their greatest common factor, which is 3:

    * 195/60 = (195 / 3) / (60 / 3) = 65/20

    We can simplify further by dividing both numerator and denominator by 5:

    * 65/20 = (65 / 5) / (20 / 5) = 13/4

    ### Answer: 13/4

Expected answer: 13/4

Correct? Yes

🔗 Problem

Problem: Suppose you have the following three fractions: 82/49, 45/60, and 99/59. If you add the first fraction (82/49) with the second one (45/60), call this result 'd'. Then, add 'd' to the third fraction (99/59). What would be the resulting sum? Please provide your answer as a simplified fraction.

Gemma 2B:

    Here's how to solve the problem step-by-step:

    1. **Simplify the second fraction:** 45/60 simplifies to 3/4 by dividing both numerator and denominator by 15.

    2. **Find a common denominator for the first two fractions:** The least common denominator for 49 and 4 is 196.
       * 82/49 becomes (82 * 4) / (49 * 4) = 328/196
       * 3/4 becomes (3 * 49) / (4 * 49) = 147/196

    3. **Add the first two fractions:** 328/196 + 147/196 = 475/196

    4. **Find a common denominator for 'd' and the third fraction:** The least common denominator for 196 and 59 is 11524.
       * 475/196 becomes (475 * 59) / (196 * 59) = 28075/11524
       * 99/59 becomes (99 * 196) / (59 * 196) = 19404/11524

    5. **Add 'd' and the third fraction:** 28075/11524 + 19404/11524 = 47479/11524

    6. **Simplify the resulting fraction:** The greatest common factor of 47479 and 11524 is 1. Therefore, the simplified fraction is **47479/11524**.

    ### Answer: 47479/11524

Expected answer: 47429/11564

Correct? No

🔗 Problem

Problem: Solve the following two steps: first, subtract 76/30 from 91/16 and second, add the answer to 18/44. Round the final answer to the nearest hundredth.

Gemma 2B:

    First, we need to find a common denominator for 76/30 and 91/16. The least common multiple of 30 and 16 is 240.

     76/30 = (76 * 8)/(30 * 8) = 608/240
     91/16 = (91 * 15)/(16 * 15) = 1365/240

     Now, we can subtract:

     1365/240 - 608/240 = (1365 - 608)/240 = 757/240

     Next, we need to add this result to 18/44. We can simplify 18/44 to 9/22. The least common multiple of 240 and 22 is 2640.

     757/240 = (757 * 11)/(240 * 11) = 8327/2640
     9/22 = (9 * 120)/(22 * 120) = 1080/2640

     Now, we can add:

     8327/2640 + 1080/2640 = (8327 + 1080)/2640 = 9407/2640

     Finally, we round 9407/2640 to the nearest hundredth:

     9407/2640 ≈ 3.56

     ### Answer: 3.56

Expected answer: 9407/2640

Correct? No

🔗 Problem

Problem: Calculate the result of ((8 / 91) + (49 / 5)) and subtract (32 / 76) from the result. Provide your answer as a simplified fraction.

Gemma 2B:

    Here's how to solve the problem step-by-step:

    1. **Calculate (8 / 91) + (49 / 5):**
       - Find a common denominator for the fractions: 91 * 5 = 455
       - Convert the fractions: (8 * 5) / 455 + (49 * 91) / 455
       - Add the numerators: (40 + 4459) / 455
       - Simplify: 4509 / 455

    2. **Calculate (32 / 76):**
       - Simplify the fraction by dividing both numerator and denominator by their greatest common factor (4): 8/19

    3. **Subtract (8/19) from (4509/455):**
       - Find a common denominator: 455 * 19 = 8645
       - Convert the fractions: (4509 * 19) / 8645 - (8 * 455) / 8645
       - Subtract the numerators: (85671 - 3640) / 8645
       - Simplify: 82031 / 8645

    ### Answer: 82031/8645

Expected answer: 81841/8645

Correct? No

🔗 Problem

Problem: Calculate the value of ((75 / 45) + (5 / 49)) first. Then add this value with 3 / 58. Provide the answer as a simple fraction.

Gemma 2B:

    Here's how we can solve this step-by-step:

    1. **Simplify the first fraction:** 75 / 45 can be simplified by dividing both numerator and denominator by 15, resulting in 5/3.

    2. **Simplify the second fraction:** 5/49 is already in its simplest form.

    3. **Add the first two fractions:**  (5/3) + (5/49) requires a common denominator. The least common multiple of 3 and 49 is 147.
        * (5/3) * (49/49) = 245/147
        * (5/49) * (3/3) = 15/147
        * Adding these: (245/147) + (15/147) = 260/147

    4. **Add the third fraction:** (260/147) + (3/58) requires another common denominator. The least common multiple of 147 and 58 is 8526.
        * (260/147) * (58/58) = 15080/8526
        * (3/58) * (147/147) = 441/8526
        * Adding these: (15080/8526) + (441/8526) = 15521/8526

    5. **Simplify the final fraction:**  Both 15521 and 8526 are divisible by 3, giving us 5173/2842.

    ### Answer: 5173/2842

Expected answer: 15521/8526

Correct? No

🔗 Problem

Problem: Calculate the answer to ((43 / 15) - (33 / 95)) and simplify it to its lowest terms.

Gemma 2B:

    First, we need to find a common denominator for the two fractions.  The least common multiple of 15 and 95 is 285.

    *  (43/15) * (19/19) = 817/285
    *  (33/95) * (3/3) = 99/285

    Now we can subtract the fractions:

    * (817/285) - (99/285) = (817 - 99) / 285 = 718 / 285

    Finally, we can simplify this fraction by dividing both numerator and denominator by their greatest common factor, which is 19:

    * (718 / 19) / (285 / 19) = 38/15

    ### Answer: 38/15

Expected answer: 718/285

Correct? No

🔗 Problem

Problem: What is the sum of ((78 / 45) + (89 / 36)) added to itself? Answer as a simple fraction.

Gemma 2B:

    Here's how to solve the problem:

    1. **Find a common denominator for the fractions:**  The least common multiple of 45 and 36 is 180.
       * (78/45) * (4/4) = 312/180
       * (89/36) * (5/5) = 445/180

    2. **Add the fractions:** 312/180 + 445/180 = 757/180

    3. **Double the sum:** (757/180) * 2 = 1514/180

    4. **Simplify the fraction:** Both the numerator and denominator are divisible by 2.
       * 1514/180 = 757/90

    ### Answer: 757/90

Expected answer: 757/90

Correct? Yes

🔗 Follow-up

Problem: After finding the sum of ((78 / 45) + (89 / 36)) added to itself, what is the resultant value when that sum is added to the original ((78 / 45) + (89 / 36))? Answer as a simple fraction.

Gemma 2B:

    We already know that the sum of ((78/45) + (89/36)) added to itself is 757/90.

    Now we need to add the original ((78/45) + (89/36)) to that result.  Let's do it step-by-step:

    1. **Convert the original fractions to a common denominator (180):**
       * (78/45) * (4/4) = 312/180
       * (89/36) * (5/5) = 445/180

    2. **Add the original fractions:**
       * 312/180 + 445/180 = 757/180

    3. **Add the result to the sum we found earlier (757/90):**
       * 757/180 + 757/90 = 757/180 + 1514/180 = 2271/180

    4. **Simplify the fraction:** Both the numerator and denominator are divisible by 3.
       * 2271/180 = 757/60

    ### Answer: 757/60

Expected answer: 757/60

Correct? Yes

🔗 Follow-up

Problem: Previously, you calculated the sum of ((78 / 45) + (89 / 36)) added to itself. But what if the first fraction was instead (78 / 93)? In that case, what would be the new result of ((78 / 93) + (89 / 36)) added to itself? Please give your answer as a simple fraction.

Gemma 2B:

    Here's how to solve the problem with the new fraction (78/93):

    1. **Find a common denominator:** The least common multiple of 93 and 36 is 3348.
        * (78/93) * (36/36) = 2808/3348
        * (89/36) * (93/93) = 8277/3348

    2. **Add the fractions:** 2808/3348 + 8277/3348 = 11085/3348

    3. **Double the sum:** (11085/3348) * 2 = 22170/3348

    4. **Simplify the fraction:** Both the numerator and denominator are divisible by 2.
        * 22170/3348 = 11085/1674

    ### Answer: 11085/1674

Expected answer: 3695/558

Correct? Yes

🔗 Problem

Problem: Let's consider that 93 is divided by 34 to get a fraction value, from which is subtracted another fraction 77/74. What would be the result as a simplified fraction?

Gemma 2B:

    Here's how to solve this problem step-by-step:

    1. **Divide 93 by 34:**  93 / 34 = 2.735 (approximately)

    2. **Convert the decimal to a fraction:**  2.735 can be written as 2735/1000.  We can simplify this fraction by dividing both numerator and denominator by their greatest common factor, which is 5: (2735/5) / (1000/5) = 547/200.

    3. **Subtract the second fraction:** (547/200) - (77/74)

    4. **Find a common denominator:** The least common denominator for 200 and 74 is 3700.  We get:
       * (547/200) * (18.5/18.5) = 10106.5 / 3700
       * (77/74) * (50/50) = 3850 / 3700

    5. **Subtract the fractions:** (10106.5 / 3700) - (3850 / 3700) = 6256.5 / 3700

    6. **Simplify the fraction:** Both the numerator and denominator are divisible by 0.5.
      * (6256.5 / 0.5) / (3700 / 0.5) = 12513 / 7400

    7. **Further simplification:** The numerator and denominator have a common factor of 13.
       * (12513 / 13) / (7400 / 13) = **963/569**

    ### Answer: 963/569

Expected answer: 1066/629

Correct? No

🔗 Problem

Problem: Find the sum of 86/35 and 68/85. Express your answer as a simplified fraction.

Gemma 2B:

    To add fractions, they need to have a common denominator.  The least common multiple of 35 and 85 is 595.

    *  We multiply the numerator and denominator of 86/35 by 17 to get (86 * 17)/(35 * 17) = 1462/595
    * We multiply the numerator and denominator of 68/85 by 7 to get (68 * 7)/(85 * 7) = 476/595

    Now we can add the fractions: 1462/595 + 476/595 = (1462 + 476)/595 = 1938/595

    Finally, we simplify the fraction by dividing both numerator and denominator by their greatest common factor, which is 19: 1938/595 = (1938/19)/(595/19) = 102/31

    ### Answer: 102/31

Expected answer: 114/35

Correct? No

🔗 Problem

Problem: What is the sum of 99/53 and 67/94 in simplest form?

Gemma 2B:

    To add fractions, they need to have the same denominator.  We find the least common multiple (LCM) of 53 and 94.  Since 53 is prime, the LCM is simply 53 * 94 = 4982.

    * We multiply 99/53 by 94/94: (99/53)*(94/94) = 9306/4982
    * We multiply 67/94 by 53/53: (67/94)*(53/53) = 3551/4982

    Now we can add the fractions: 9306/4982 + 3551/4982 = 12857/4982

    This fraction can be simplified by dividing both numerator and denominator by their greatest common factor, which is 1.  So the answer is:

    ### Answer: 12857/4982

Expected answer: 12857/4982

Correct? Yes

🔗 Follow-up

Problem: What if you were to add the fraction 50/56 to the sum you previously found (of 99/53 and 67/94)? What is the resulting value in its simplest form?

Gemma 2B:

    We've already determined that 99/53 + 67/94 = 12857/4982.

    Now let's add 50/56 to this sum. First, we need a common denominator for all three fractions. The LCM of 4982, 53, 94, and 56 is 70148.

    *  12857/4982 * (14/14) = 180000/70148
    *  50/56 * (1252/1252) = 62600/70148

    Now we can add all three fractions:

    180000/70148 + 62600/70148 + 3551/4982 * (14/14) = 180000/70148 + 62600/70148 + 49714/70148 = 292314/70148

    This fraction can be simplified by dividing both numerator and denominator by their greatest common factor, which is 2.

    ### Answer: 146157/35074

Expected answer: 242273/69748

Correct? No

🔗 Problem

Problem: What is the simplified form of the fraction that results from 74/27 minus 56/33?

Gemma 2B:

    To subtract fractions, they must have a common denominator.  The least common multiple of 27 and 33 is 297.

    *  We multiply 74/27 by 11/11 to get (74 * 11) / (27 * 11) = 814/297.
    *  We multiply 56/33 by 9/9 to get (56 * 9) / (33 * 9) = 504/297.

    Now we can subtract: 814/297 - 504/297 = (814 - 504) / 297 = 310/297.

    This fraction can be simplified by dividing both the numerator and denominator by their greatest common factor, which is 2:

    310/297 = (310/2) / (297/2) = 155/148.5

    ### Answer: 155/148.5

Expected answer: 310/297

Correct? Yes

🔗 Problem

Problem: Find the result when you subtract 91/65 from the sum of 96/40 and 11/99. Leave your answer as a simplified fraction.

Gemma 2B:

    Here's how to solve the problem step-by-step:

    1. **Find the sum of 96/40 and 11/99:**
       * Simplify 96/40:  96/40 = 24/10 = 12/5
       * Find a common denominator for 12/5 and 11/99. The least common multiple of 5 and 99 is 495.
       * Convert 12/5: (12/5) * (99/99) = 1188/495
       * Convert 11/99: (11/99) * (5/5) = 55/495
       * Add the fractions: 1188/495 + 55/495 = 1243/495

    2. **Subtract 91/65 from the sum:**
       * Simplify 91/65: 91/65 = 13/5
       * Find a common denominator for 1243/495 and 13/5. The least common multiple of 495 and 5 is 495.
       * Convert 13/5: (13/5) * (99/99) = 1287/495
       * Subtract the fractions: 1243/495 - 1287/495 = -44/495

    3. **Simplify the result:**
       * The fraction -44/495 can be simplified by dividing both numerator and denominator by their greatest common factor, which is 11:
       * -44/495 = (-4*11)/(45*11) = -4/45

    ### Answer: -4/45

Expected answer: 10/9

Correct? No

🔗 Problem

Problem: Calculate the sum of the following expressions: 1) Divide 10 by 5 and add that result to 35 divided by 27. This will give you a value h. 2) Then, add 47 divided by 99 to h. This will give you a value p. What is the value of p?

Gemma 2B:

    Let's break down the calculation step by step:

    1. **Calculate h:**
       - Divide 10 by 5: 10 / 5 = 2
       - Divide 35 by 27: 35 / 27 ≈ 1.296
       - Add the results: 2 + 1.296 = 3.296
       - Therefore, h = 3.296

    2. **Calculate p:**
       - Divide 47 by 99: 47 / 99 ≈ 0.475
       - Add this result to h: 3.296 + 0.475 = 3.771

    ### Answer: 3.771

Expected answer: 1120/297

Correct? No

🔗 Problem

Problem: Consider the fraction 90/81. Add it to 2/66 and simplify your answer to the simplest form. What is the result?

Gemma 2B:

    Let's break down the problem step-by-step:

    1. **Simplify the fractions:**
        - 90/81 can be simplified by dividing both numerator and denominator by 9, giving us 10/9.
        - 2/66 can be simplified by dividing both numerator and denominator by 2, giving us 1/33.

    2. **Find a common denominator:**
        - The least common multiple of 9 and 33 is 99.
        - To get a denominator of 99 for 10/9, we multiply both numerator and denominator by 11: (10*11)/(9*11) = 110/99
        - To get a denominator of 99 for 1/33, we multiply both numerator and denominator by 3: (1*3)/(33*3) = 3/99

    3. **Add the fractions:**
        - 110/99 + 3/99 = (110+3)/99 = 113/99

    Therefore, the simplified answer is 113/99.

    ### Answer: 113/99

Expected answer: 113/99

Correct? Yes

🔗 Follow-up

Problem: Imagine again the fraction 90/81. We already added it to 2/66. Now, pretend to add that sum to another fraction, 77/62. If you have to simplify your answer to the simplest form, what would be the result?

Gemma 2B:

    We already know that 90/81 + 2/66 simplifies to 113/99.  Now let's add 77/62 to this:

    1. **Find a common denominator:**
        - The least common multiple of 99 and 62 is 6138.
        - To get a denominator of 6138 for 113/99, we multiply both numerator and denominator by 62: (113*62)/(99*62) = 7006/6138
        - To get a denominator of 6138 for 77/62, we multiply both numerator and denominator by 99: (77*99)/(62*99) = 7623/6138

    2. **Add the fractions:**
        - 7006/6138 + 7623/6138 = (7006 + 7623)/6138 = 14629/6138

    3. **Simplify the fraction:**
        - Both the numerator and denominator are divisible by 3: (14629/3)/(6138/3) = 4876 1/3 / 2046

    ### Answer: 4876 1/3 / 2046

Expected answer: 14629/6138

Correct? No

🔗 Follow-up

Problem: Previously, you calculated the sum of the fraction 90/81 and 2/66. In a new scenario, what if you started with 90/39 instead of 90/81 then added it with 2/66? What would be the result in the simplest form?

Gemma 2B:

    Let's follow the same steps as before:

    1. **Simplify the fractions:**
        - 90/39 can be simplified by dividing both numerator and denominator by 3, giving us 30/13.
        - 2/66 can be simplified by dividing both numerator and denominator by 2, giving us 1/33.

    2. **Find a common denominator:**
        - The least common multiple of 13 and 33 is 429.
        - To get a denominator of 429 for 30/13, we multiply both numerator and denominator by 33: (30*33)/(13*33) = 990/429.
        - To get a denominator of 429 for 1/33, we multiply both numerator and denominator by 13: (1*13)/(33*13) = 13/429.

    3. **Add the fractions:**
        - 990/429 + 13/429 = (990+13)/429 = 1003/429.

    Therefore, the simplified answer is 1003/429.

    ### Answer: 1003/429

Expected answer: 1003/429

Correct? Yes

🔗 Problem

Problem: If Oliver has a piece of ribbon that is 56/41 yards long and he gets another piece of the same length, how long would both pieces of ribbon be together?

Gemma 2B:

    Reasoning: Oliver would have two pieces of ribbon, each 56/41 yards long. To find the total length, we need to add the length of both pieces. So, we add 56/41 yards + 56/41 yards. Since the denominators are the same, we can directly add the numerators.
    Total length = (56 + 56)/41 = 112/41 yards.
    ### Answer: 112/41 yards.

Expected answer: 112/41

Correct? Yes

🔗 Follow-up

Problem: If Oliver receives two more pieces of ribbon, with each being 56/41 yards long, how long would all the pieces of ribbon be when added together?

Gemma 2B:

    Reasoning: Oliver already has two pieces of ribbon, each 56/41 yards long. He receives two more pieces of the same length. So, he has a total of 2+2 = 4 pieces.

    The total length of all the pieces would be 4 * (56/41) yards.

    Total length = 224/41 yards.

    ### Answer: 224/41 yards.

Expected answer: 224/41

Correct? Yes

🔗 Follow-up

Problem: If Oliver initially had a piece of ribbon that was 63/41 yards long instead of 56/41 yards, and he still receives a second piece of the same length, how long would both pieces of ribbon be when put together?

Gemma 2B:

    Reasoning: Oliver now has two pieces of ribbon, each 63/41 yards long. To find the total length, we add the length of both pieces: 63/41 yards + 63/41 yards.  Since the denominators are the same, we can directly add the numerators.
    Total length = (63 + 63)/41 = 126/41 yards.

    ### Answer: 126/41 yards.

Expected answer: 126/41

Correct? Yes

🔗 Problem

Problem: What is the result of 92/75 + 66/92? Please give your answer as a simplified fraction.

Gemma 2B:

    To add fractions, they need to have the same denominator.  Here's how to find the solution:

    1. **Find the Least Common Multiple (LCM) of the denominators:**
       * The LCM of 75 and 92 is 6900.

    2. **Convert each fraction to have the denominator of 6900:**
       * 92/75 = (92 * 92) / (75 * 92) = 8464 / 6900
       * 66/92 = (66 * 75) / (92 * 75) = 4950 / 6900

    3. **Add the fractions:**
       * 8464/6900 + 4950/6900 = (8464 + 4950) / 6900 = 13414 / 6900

    4. **Simplify the fraction:**
       * Both the numerator and denominator are divisible by 2: 13414 / 6900 = 6707 / 3450

    ### Answer: 6707/3450

Expected answer: 6707/3450

Correct? Yes

🔗 Follow-up

Problem: Following the previous problem, what is the result of 97/17 minus the sum of 92/75 + 66/92? Give your answer as a simplified fraction.

Gemma 2B:

    We already know from the previous problem that 92/75 + 66/92 = 6707/3450.

    Now, let's subtract that sum from 97/17:

    1. **Convert 97/17 to have the same denominator as 6707/3450:**
        * 97/17 = (97 * 203) / (17 * 203) = 19701 / 3450

    2. **Subtract the fractions:**
        * 19701/3450 - 6707/3450 = (19701 - 6707) / 3450 = 12994 / 3450

    3. **Simplify the fraction:**
        * Both the numerator and denominator are divisible by 2: 12994 / 3450 = 6497 / 1725

    ### Answer: 6497/1725

Expected answer: 220631/58650

Correct? No

🔗 Follow-up

Problem: Following the previous problem, consider instead if there were 83/75 of something instead of 92/75. What would be the new total when you add 66/92 to it? Please provide your answer as a simplified fraction.

Gemma 2B:

    Let's follow the same steps as before:

    1. **Find the Least Common Multiple (LCM) of the denominators:**
       * The LCM of 75 and 92 is 6900.

    2. **Convert each fraction to have the denominator of 6900:**
       * 83/75 = (83 * 92) / (75 * 92) = 7636 / 6900
       * 66/92 = (66 * 75) / (92 * 75) = 4950 / 6900

    3. **Add the fractions:**
       * 7636/6900 + 4950/6900 = (7636 + 4950) / 6900 = 12586 / 6900

    4. **Simplify the fraction:**
       * Both the numerator and denominator are divisible by 2: 12586 / 6900 = 6293 / 3450

    ### Answer: 6293/3450

Expected answer: 6293/3450

Correct? Yes

🔗 Problem

Problem: What is the sum of 32/65 and 73/22? Leave your answer as an unsimplified fraction.

Gemma 2B:

    Reasoning: To add fractions, they must have the same denominator. The least common multiple of 65 and 22 is 65 * 22 = 1430.

    * We need to multiply 32/65 by 22/22 to get (32 * 22)/(65 * 22) = 704/1430.
    * We need to multiply 73/22 by 65/65 to get (73 * 65)/(22 * 65) = 4745/1430.

    Now we can add the fractions: 704/1430 + 4745/1430 = (704 + 4745)/1430

    ### Answer: 5449/1430

Expected answer: 5449/1430

Correct? Yes

🔗 Follow-up

Problem: Following up on the previous question, what is the sum of your answer from that problem and 24/71? Leave your answer as an unsimplified fraction.

Gemma 2B:

    Reasoning: We need to find a common denominator for 5449/1430 and 24/71.  The least common multiple of 1430 and 71 is 1430 * 71 = 101630.

    * Multiply 5449/1430 by 71/71: (5449 * 71)/(1430 * 71) = 386719/101630
    * Multiply 24/71 by 1430/1430: (24 * 1430)/(71 * 1430) = 34320/101630

    Now we can add the fractions: 386719/101630 + 34320/101630 = (386719 + 34320)/101630

    ### Answer: 421039/101630

Expected answer: 421199/101530

Correct? No

🔗 Follow-up

Problem: In the earlier problem, we considered the sum of 32/65 and 73/22. But what if the fraction was 53/65 instead of 32/65? What would the new sum be? Please leave your answer as an unsimplified fraction.

Gemma 2B:

    Reasoning:  We already know the least common multiple of 65 and 22 is 1430.

    * We need to multiply 53/65 by 22/22 to get (53 * 22)/(65 * 22) = 1166/1430.
    * We need to multiply 73/22 by 65/65 to get (73 * 65)/(22 * 65) = 4745/1430.

    Now we can add the fractions: 1166/1430 + 4745/1430 = (1166 + 4745)/1430

    ### Answer: 5911/1430

Expected answer: 5911/1430

Correct? Yes

🔗 Problem

Problem: What is the difference between 51/34 and 12/86? Express your answer in its simplest form.

Gemma 2B:

    Reasoning: To find the difference, we need to subtract the smaller fraction from the larger fraction. However, to do that, they need to have the same denominator. The least common multiple of 34 and 86 is 172.

    To get a denominator of 172 for the first fraction, we multiply both the numerator and denominator by 5:
    (51/34) * (5/5) = 255/172

    To get a denominator of 172 for the second fraction, we multiply both the numerator and denominator by 2:
    (12/86) * (2/2) = 24/172

    Now we can subtract the fractions:
    255/172 - 24/172 = 231/172

    Finally, we can simplify the fraction by dividing both the numerator and denominator by their greatest common factor, which is 3:
    231/172 = (231/3) / (172/3) = 77/57

    ### Answer: 77/57

Expected answer: 117/86

Correct? No

🔗 Problem

Problem: First take the whole number 51 and add it to the fraction 93/8. Take that sum and add it to the fraction 58/44. What is the final result as a simplified fraction?

Gemma 2B:

    Here's how to solve this step-by-step:

    1. **Convert the whole number to a fraction:** 51 can be written as 51/1.

    2. **Find a common denominator for 51/1 and 93/8:** The least common denominator is 8.
       * 51/1 * 8/8 = 408/8

    3. **Add the fractions:** 408/8 + 93/8 = 501/8

    4. **Find a common denominator for 501/8 and 58/44:** The least common denominator is 88.
       * 501/8 * 11/11 = 5511/88
       * 58/44 * 2/2 = 116/88

    5. **Add the fractions:** 5511/88 + 116/88 = 5627/88

    **Therefore, the final result as a simplified fraction is 5627/88.**
    ### Answer: 5627/88

Expected answer: 5627/88

Correct? Yes

🔗 Follow-up

Problem: After completing the initial calculation, you realize there was an extra component to your math problem. You must now also add the fraction 38/53 to your previous result. What is the new total, expressed as a simplified fraction?

Gemma 2B:

    We need to add 38/53 to our previous result of 5627/88.

    1. **Find a common denominator for 5627/88 and 38/53:** The least common denominator is 4624.
       * 5627/88 * 53/53 = 298071/4624
       * 38/53 * 88/88 = 3344/4624

    2. **Add the fractions:** 298071/4624 + 3344/4624 = 301415/4624

    **Therefore, the new total, expressed as a simplified fraction, is 301415/4624.**

    ### Answer: 301415/4624

Expected answer: 301575/4664

Correct? No

🔗 Problem

Problem: Calculate the result of the following fraction subtraction: 82/5 - 98/25. Write the answer in the simplest form.

Gemma 2B:

    To subtract fractions, they must have a common denominator. The least common multiple of 5 and 25 is 25.

    We can rewrite 82/5 as (82 * 5) / (5 * 5) = 410/25.

    Now we can perform the subtraction:

    410/25 - 98/25 = (410 - 98) / 25 = 312/25

    The fraction 312/25 can be simplified as a mixed number:

    312/25 = 12 12/25

    ### Answer: 12 12/25

Expected answer: 312/25

Correct? Yes

🔗 Follow-up

Problem: Given the result you obtained previously from calculating 82/5 - 98/25, what would you get if you added this result to itself?

Gemma 2B:

    We previously found that 82/5 - 98/25 = 12 12/25.

    Adding this result to itself means:

    12 12/25 + 12 12/25 = (12 + 12) + (12/25 + 12/25) = 24 + 24/25

    This can be written as a mixed number:

    24 + 24/25 = 24 24/25

    ### Answer: 24 24/25

Expected answer: 624/25

Correct? Yes

🔗 Follow-up

Problem: If you originally had 82/5 oranges and instead of taking away 98/25 of them, you took away 98/90 oranges, how many oranges would you have left? Write the answer in the simplest form.

Gemma 2B:

    First, we need to make sure the fractions have a common denominator to subtract them. The least common multiple of 5 and 90 is 90.

    * 82/5 becomes (82 * 18) / (5 * 18) = 1476/90
    * 98/90 stays the same.

    Now we subtract:

    1476/90 - 98/90 = (1476 - 98)/90 = 1378/90

    This fraction can be simplified by dividing both numerator and denominator by their greatest common factor, which is 2:

    1378/90 = (1378/2) / (90/2) = 689/45

    This is a proper fraction, so it's already in the simplest form.

    ### Answer: 689/45

Expected answer: 689/45

Correct? Yes

🔗 Problem

Problem: Calculate (64 / 50) + (70 / 22) and provide your answer as an unsimplified fraction.

Gemma 2B:

    Here's how to calculate the expression:

    1. **Find a common denominator:** The least common denominator for 50 and 22 is 550.
    2. **Convert the fractions:**
       * (64/50) * (11/11) = 704/550
       * (70/22) * (25/25) = 1750/550
    3. **Add the fractions:** 704/550 + 1750/550 = 2454/550

    ### Answer: 2454/550

Expected answer: 1227/275

Correct? Yes

🔗 Follow-up

Problem: In the previous question, we considered a situation where we added the fractions (64 / 50) and (70 / 22). Now imagine we made a slight error in our calculations, and the initial fraction was actually (64 / 49), not (64 / 50). How would that affect our total sum? Calculate (64 / 49) + (70 / 22) and provide your answer as an unsimplified fraction.

Gemma 2B:

    Let's recalculate with the corrected fraction:

    1. **Find a common denominator:** The least common denominator for 49 and 22 is 1078.
    2. **Convert the fractions:**
       * (64/49) * (22/22) = 1408/1078
       * (70/22) * (49/49) = 3430/1078
    3. **Add the fractions:** 1408/1078 + 3430/1078 = 4838/1078

    ### Answer: 4838/1078

Expected answer: 2419/539

Correct? Yes

🔗 Problem

Problem: John calculated s by adding 24 divided by 6 to 40 divided by 53. He then added s to itself to get b. Can you find the value of b?

Gemma 2B:

    Reasoning:
    1. First, we need to calculate 24 divided by 6: 24 / 6 = 4.
    2. Then, we calculate 40 divided by 53: 40 / 53 ≈ 0.75 (rounded to two decimal places).
    3. Now we add these two results to find s: 4 + 0.75 = 4.75.
    4. Finally, we add s to itself to find b: 4.75 + 4.75 = 9.5.

    ### Answer: 9.5

Expected answer: 504/53

Correct? No

🔗 Problem

Problem: Calculate ((80 / 90) + (52 / 11)) and express your answer as a mixed number.

Gemma 2B:

    Here's how to calculate the expression and express the answer as a mixed number:

    1. **Simplify the fractions:**
       * 80/90 simplifies to 8/9
       * 52/11 remains as is (it's already in simplest form)

    2. **Find a common denominator:**
       * The least common multiple of 9 and 11 is 99.
       * (8/9) * (11/11) = 88/99
       * (52/11) * (9/9) = 468/99

    3. **Add the fractions:**
       * 88/99 + 468/99 = 556/99

    4. **Convert to a mixed number:**
       * 556 divided by 99 is 5 with a remainder of 51.
       * Therefore, 556/99 is equal to 5 51/99.

    ### Answer: 5 51/99

Expected answer: 556/99

Correct? No

🔗 Problem

Problem: Calculate the sum of 12/18 and 91/74. Express your answer as a simplified fraction.

Gemma 2B:

    To add fractions, they must have a common denominator.  The least common multiple of 18 and 74 is 666.

    * We multiply the numerator and denominator of $\frac{12}{18}$ by $\frac{37}{37}$ to get $\frac{444}{666}$.
    * We multiply the numerator and denominator of $\frac{91}{74}$ by $\frac{9}{9}$ to get $\frac{819}{666}$.

    Now we can add the fractions:

    $$\frac{444}{666} + \frac{819}{666} = \frac{444+819}{666} = \frac{1263}{666}$$

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

    $$\frac{1263}{666} = \frac{1263 \div 3}{666 \div 3} = \boxed{\frac{421}{222}}$$

Expected answer: 421/222

Correct? Yes

🔗 Problem

Problem: Find the sum of the fractions 7/2 and 51/25. Write your answer as a simple fraction.

Gemma 2B:

    To add fractions, they need to have the same denominator. The least common multiple of 2 and 25 is 50.

    * Convert 7/2 to an equivalent fraction with a denominator of 50: (7/2) * (25/25) = 175/50
    * Convert 51/25 to an equivalent fraction with a denominator of 50: (51/25) * (2/2) = 102/50

    Now, we can add the fractions:

    175/50 + 102/50 = 277/50

    ### Answer: 277/50

Expected answer: 277/50

Correct? Yes

🔗 Follow-up

Problem: Following up on the previous question, add the fraction 86/39 to the sum of 7/2 and 51/25. Express your answer as a simple fraction.

Gemma 2B:

    We already know that the sum of 7/2 and 51/25 is 277/50.  Now we need to add 86/39 to that.

    To add these fractions, we need a common denominator.  The least common multiple of 50 and 39 is 1950.

    * Convert 277/50 to an equivalent fraction with a denominator of 1950: (277/50) * (39/39) = 10803/1950
    * Convert 86/39 to an equivalent fraction with a denominator of 1950: (86/39) * (50/50) = 4300/1950

    Now we can add:

    10803/1950 + 4300/1950 = 15103/1950

    ### Answer: 15103/1950

Expected answer: 15103/1950

Correct? Yes

🔗 Follow-up

Problem: In the previous problem, you were asked to find the sum of 7/2 and 51/25. Now suppose that instead of 7/2, the fraction was 7/22. How would this change the result? Calculate 7/22 + 51/25 as a simplified fraction.

Gemma 2B:

    To add the fractions 7/22 and 51/25, we need a common denominator.  The least common multiple of 22 and 25 is 550.

    * Convert 7/22 to an equivalent fraction with a denominator of 550: (7/22) * (25/25) = 175/550
    * Convert 51/25 to an equivalent fraction with a denominator of 550: (51/25) * (22/22) = 1122/550

    Now, we can add the fractions:

    175/550 + 1122/550 = 1297/550

    ### Answer: 1297/550

Expected answer: 1297/550

Correct? Yes

🔗 Problem

Problem: Calculate the following expression: ((68 / 69) + (93 / 7)) is a monomial. Add this to (3 / 46). Provide the resulting value as a simple fraction.

Gemma 2B:

    Let's break down the calculation step-by-step:

    1. **Calculate the fractions inside the parentheses:**
       - (68 / 69) ≈ 0.9855
       - (93 / 7) ≈ 13.2857

    2. **Add the two results:**
       - 0.9855 + 13.2857 ≈ 14.2712

    3. **Add the result to (3 / 46):**
       - 14.2712 + (3 / 46) ≈ 14.3372

    4. **Convert the decimal result to a fraction:**
       - Since the decimal is recurring, it's impossible to express it as a simple fraction. Therefore, we leave the result as it is.

    ### Answer: 14.3372

Expected answer: 13849/966

Correct? No

🔗 Problem

Problem: What is the result of adding 31/98 to 60/4? Answer as a simplified fraction.

Gemma 2B:

    Reasoning: To add fractions, they need to have the same denominator.  We can find a common denominator by finding the least common multiple (LCM) of 98 and 4.

    The LCM of 98 and 4 is 98. To get a denominator of 98 for 60/4, we multiply both numerator and denominator by 24.5:

    (60/4) * (24.5/24.5) = 1470/98

    Now we can add the fractions:

    31/98 + 1470/98 = (31 + 1470)/98 = 1501/98

    This fraction cannot be simplified further.

    ### Answer: 1501/98

Expected answer: 1501/98

Correct? Yes

🔗 Follow-up

Problem: Following the previous problem, what's the result when you subtract 41/52 from the sum of 31/98 and 60/4? Answer as a simplified fraction.

Gemma 2B:

    Reasoning: We already know the sum of 31/98 and 60/4 is 1501/98. Now we need to subtract 41/52 from that.

    To subtract fractions, they must have a common denominator. The LCM of 98 and 52 is 2548.

    *  Multiplying 1501/98 by 26/26 (to get a denominator of 2548): (1501/98) * (26/26) = 39026/2548
    *  Multiplying 41/52 by 49/49 (to get a denominator of 2548): (41/52) * (49/49) = 2009/2548

    Now we can subtract:

    39026/2548 - 2009/2548 = (39026 - 2009)/2548 = 37017/2548

    This fraction can be simplified by dividing both numerator and denominator by their greatest common factor, which is 13:

    37017/2548 = (37017/13) / (2548/13) = 2847/196

    ### Answer: 2847/196

Expected answer: 37017/2548

Correct? No

🔗 Follow-up

Problem: In the previous situation, instead of adding 31/98 to 60/4, what would happen if you added 31/98 to 83/4? What would the new sum be, in simplified fraction form?

Gemma 2B:

    Reasoning:  Similar to the previous problem, we need a common denominator. The LCM of 98 and 4 is 98.

    We multiply both numerator and denominator of 83/4 by 24.5 to get a denominator of 98:

    (83/4) * (24.5/24.5) = 2033.5/98

    Now we can add the fractions:

    31/98 + 2033.5/98 = (31 + 2033.5)/98 = 2064.5/98

    To simplify, we can divide both numerator and denominator by their greatest common factor, which is 0.5:

    (2064.5 / 0.5) / (98 / 0.5) = 4129/196

    ### Answer: 4129/196

Expected answer: 4129/196

Correct? Yes

🔗 Problem

Problem: Calculate ((3 / 2) + (24 / 36)) and state your answer as a simplified fraction.

Gemma 2B:

    First, simplify the fractions:
    * 24/36 simplifies to 2/3 (dividing both numerator and denominator by 12).

    Now we have:
    (3/2) + (2/3)

    To add fractions, they need a common denominator.  The least common denominator of 2 and 3 is 6.

    * (3/2) * (3/3) = 9/6
    * (2/3) * (2/2) = 4/6

    Now we can add:
    9/6 + 4/6 = 13/6

    ### Answer: 13/6

Expected answer: 13/6

Correct? Yes

🔗 Follow-up

Problem: After completing the calculation for ((3 / 2) + (24 / 36)), add the result to 7/2 and give your answer as a simplified fraction.

Gemma 2B:

    We already know that (3/2) + (24/36) = 13/6.

    Now we need to add 7/2:

    13/6 + 7/2

    To add these fractions, we need a common denominator. The least common denominator of 6 and 2 is 6.

    * (7/2) * (3/3) = 21/6

    Now we can add:

    13/6 + 21/6 = 34/6

    Finally, we simplify the fraction by dividing both numerator and denominator by their greatest common factor, which is 2:

    34/6 = 17/3

    ### Answer: 17/3

Expected answer: 17/3

Correct? Yes

🔗 Follow-up

Problem: Suppose instead of having 3/2, we had (32/36) as part of the calculation along with what you previously calculated. Can you determine the new total if you calculate ((3 / 2) + (32 / 36))? Answer as a simplified fraction.

Gemma 2B:

    First, simplify 32/36. Both the numerator and denominator can be divided by 4, giving us 8/9.

    Now we have:
    (8/9) + (2/3)

    To add fractions, we need a common denominator. The least common denominator of 9 and 3 is 9.

    * (2/3) * (3/3) = 6/9

    Now we can add:
    8/9 + 6/9 = 14/9

    ### Answer: 14/9

Expected answer: 43/18

Correct? No

🔗 Problem

Problem: Firstly, calculate the sum of 92/54 and 90/13, then from the result take away 46/41. What's the result as a simplified fraction?

Gemma 2B:

    Here's how to solve this problem step-by-step:

    1. **Find a common denominator for 92/54 and 90/13:**
       * The least common multiple of 54 and 13 is 702.
       * Convert 92/54: (92/54) * (13/13) = 1196/702
       * Convert 90/13: (90/13) * (54/54) = 4860/702

    2. **Add the fractions:**
       * (1196/702) + (4860/702) = 6056/702

    3. **Simplify the fraction:**
       * Both 6056 and 702 are divisible by 4.
       * (6056/4) / (702/4) = 1514/175.5

    4. **Subtract 46/41 from the result:**
       * To subtract fractions, we need a common denominator. The least common multiple of 175.5 and 41 is 7200.5.
       * Convert 1514/175.5: (1514/175.5) * (41/41) = 62074/7200.5
       * Convert 46/41: (46/41) * (175.5/175.5) = 8071/7200.5
       * Subtract: (62074/7200.5) - (8071/7200.5) = 54003/7200.5

    5. **Simplify the final fraction:**
       * Both 54003 and 7200.5 are divisible by 1.5.
       * (54003/1.5) / (7200.5/1.5) = 36002/4800.33333...

    **Therefore, the final result as a simplified fraction is approximately 36002/4800.33333...**

Expected answer: 108002/14391

Correct? No

🔗 Problem

Problem: What is twice the value of 29/23? Simplify your answer to the simplest form.

Gemma 2B:

    Twice the value of 29/23 means multiplying it by 2:

    (29/23) * 2 = 58/23

    This fraction is already in its simplest form because 58 and 23 share no common factors other than 1.

    ### Answer: 58/23

Expected answer: 58/23

Correct? Yes

🔗 Follow-up

Problem: After calculating the value of 2*(29/23), subtract this result from 32/5. What is the resulting fraction in simplest form?

Gemma 2B:

    First, we calculate 2 * (29/23) = 58/23.

    Then, we subtract this from 32/5:

    32/5 - 58/23 = (32 * 23)/(5 * 23) - (58 * 5)/(23 * 5) = 736/115 - 290/115

    Now, we can subtract the numerators since they have the same denominator:

    (736 - 290)/115 = 446/115

    This fraction can be simplified by dividing both numerator and denominator by their greatest common factor, which is 23:

    446/115 = (446/23) / (115/23) = 19/5

    ### Answer: 19/5

Expected answer: 446/115

Correct? No

🔗 Follow-up

Problem: If the initial number was 36/23 instead of 29/23 in our previous problem, what would now be the value of two times this new fraction? Simplify your answer to the simplest form.

Gemma 2B:

    Two times the value of 36/23 means multiplying it by 2:

    (36/23) * 2 = 72/23

    This fraction is already in its simplest form because 72 and 23 share no common factors other than 1.

    ### Answer: 72/23

Expected answer: 72/23

Correct? Yes

🔗 Problem

Problem: What is the sum of 29/20 and 17/72? Answer as a simplified fraction.

Gemma 2B:

    To add fractions, they must have a common denominator.  The least common multiple of 20 and 72 is 360.

    *  Convert 29/20 to an equivalent fraction with a denominator of 360:  (29/20) * (18/18) = 522/360
    *  Convert 17/72 to an equivalent fraction with a denominator of 360: (17/72) * (5/5) = 85/360

    Now we can add:  522/360 + 85/360 = 607/360

    This fraction can be simplified: 607/360 = 1 247/360

    ### Answer: 1 247/360

Expected answer: 607/360

Correct? Yes

🔗 Follow-up

Problem: After finding the sum of 29/20 and 17/72 in the previous question, now subtract 9/68 from that sum to obtain a new value. What is the simplified fraction of this new answer?

Gemma 2B:

    We know the sum of 29/20 and 17/72 is 607/360.  Now we need to subtract 9/68 from that.

    To do this, we need a common denominator.  The least common multiple of 360 and 68 is 6120.

    * Convert 607/360 to an equivalent fraction with a denominator of 6120: (607/360) * (17/17) = 10319/6120
    * Convert 9/68 to an equivalent fraction with a denominator of 6120: (9/68) * (90/90) = 810/6120

    Now subtract: 10319/6120 - 810/6120 = 9509/6120

    ### Answer: 9509/6120

Expected answer: 9509/6120

Correct? Yes

🔗 Follow-up

Problem: In the previous problem, you found the sum of 29/20 and 17/72. Now, imagine that instead of having 29/20 of a quantity, you only had 29/5. What would be the sum of 29/5 and 17/72? Keep your answer as a simplified fraction.

Gemma 2B:

    We need to find a common denominator for 5 and 72. The least common multiple of 5 and 72 is 360.

    * Convert 29/5 to an equivalent fraction with a denominator of 360: (29/5) * (72/72) = 2088/360
    * Convert 17/72 to an equivalent fraction with a denominator of 360: (17/72) * (5/5) = 85/360

    Now we can add: 2088/360 + 85/360 = 2173/360

    This fraction can be simplified: 2173/360 = 6 53/360

    ### Answer: 6 53/360

Expected answer: 2173/360

Correct? No

🔗 Problem

Problem: Calculate the value of this expression: ((36 / 27) + (80 / 10)) - (73 / 79). Simplify your answer as much as possible.

Gemma 2B:

    Here's how to solve the expression step-by-step:

    1. **Simplify the fractions inside the parentheses:**
       * (36 / 27) = 4/3
       * (80 / 10) = 8
       * (73 / 79)  (This fraction cannot be simplified further)

    2. **Calculate the expression within the first set of parentheses:**
       * (4/3) + 8 = 28/3

    3. **Subtract the simplified fraction:**
       * (28/3) - (73/79) = (2184/237) - (219/237) = 1965/237

    4. **Simplify the final fraction:**
       * 1965/237 = (3 * 655)/(3 * 79) = 655/79

    ### Answer: 655/79

Expected answer: 1993/237

Correct? No

🔗 Problem

Problem: What is the answer to (86 / 71) + (91 / 75)? Answer as a decimal to the nearest hundredth.

Gemma 2B:

    First, we'll calculate the individual divisions:

    * 86 / 71 ≈ 1.21
    * 91 / 75 ≈ 1.21

    Now, we add these results:

    * 1.21 + 1.21 = 2.42

    ### Answer: 2.42

Expected answer: 12911/5325

Correct? No

🔗 Problem

Problem: Calculate the following, and express the result as a simplified fraction: ((21 / 75) + (74 / 43)) + (10 / 62)

Gemma 2B:

    Let's break this down step by step:

    1. **Simplify each fraction:**
       * 21/75 can be simplified by dividing both numerator and denominator by 3: 7/25
       * 74/43 is already in its simplest form.
       * 10/62 can be simplified by dividing both numerator and denominator by 2: 5/31

    2. **Find a common denominator for the first two fractions:**
       * The least common multiple of 25 and 43 is 1075.
       * Convert 7/25 to have a denominator of 1075: (7/25) * (43/43) = 301/1075
       * Convert 74/43 to have a denominator of 1075: (74/43) * (25/25) = 1850/1075

    3. **Add the first two fractions:**
       * (301/1075) + (1850/1075) = 2151/1075

    4. **Find a common denominator for the result and the third fraction:**
       * The least common multiple of 1075 and 31 is 33325
       * Convert 2151/1075 to have a denominator of 33325: (2151/1075) * (31/31) = 66681/33325
       * Convert 5/31 to have a denominator of 33325: (5/31) * (1075/1075) = 5375/33325

    5. **Add the final two fractions:**
       * (66681/33325) + (5375/33325) = 72056/33325

    6. **Simplify the result:**
       * Both 72056 and 33325 are divisible by 9, giving us 8006/3703.

    ### Answer: 8006/3703

Expected answer: 72056/33325

Correct? No

🔗 Problem

Problem: Compute ((75 / 66) + (81 / 61)), and give your answer as a fraction reduced to its lowest term.

Gemma 2B:

    Here's how to compute the expression:

    1. **Find a common denominator for the fractions:**
       * The least common multiple of 66 and 61 is 4026.
       *  (75/66) * (61/61) = 4575/4026
       *  (81/61) * (66/66) = 5346/4026

    2. **Add the fractions:**
       * (4575/4026) + (5346/4026) = 9921/4026

    3. **Simplify the fraction:**
       * Both 9921 and 4026 are divisible by 3.
       * (9921/3) / (4026/3) = 3307/1342

    ### Answer: 3307/1342

Expected answer: 3307/1342

Correct? Yes

🔗 Follow-up

Problem: After you have computed ((75 / 66) + (81 / 61)), what is the result if you additionally add (47 / 19) to that sum? Give your response as a simplified fraction.

Gemma 2B:

    We already know that ((75 / 66) + (81 / 61)) = 3307/1342.

    Now, we need to add (47/19) to this fraction.

    1. **Find a common denominator:**
       * The least common multiple of 1342 and 19 is 25498.
       * (3307/1342) * (19/19) = 62833/25498
       * (47/19) * (1342/1342) = 63054/25498

    2. **Add the fractions:**
       * (62833/25498) + (63054/25498) = 125887/25498

    3. **Simplify the fraction:**
       * Both 125887 and 25498 are divisible by 3.
       * (125887/3) / (25498/3) = 41962/8499

    ### Answer: 41962/8499

Expected answer: 125907/25498

Correct? No

🔗 Problem

Problem: Consider two identical fractions, t, each equal to sum of 53/18 and 21/99. Find the total sum, l, of these fractions and present your answer as a simple fraction.

Gemma 2B:

    First, we need to find a common denominator for 53/18 and 21/99. The least common multiple of 18 and 99 is 594.

    * Convert 53/18 to an equivalent fraction with a denominator of 594: (53/18) * (33/33) = 1749/594
    * Convert 21/99 to an equivalent fraction with a denominator of 594: (21/99) * (6/6) = 126/594

    Now, add the two fractions: 1749/594 + 126/594 = 1875/594

    Since each identical fraction, t, is equal to this sum, we have:

    t = 1875/594

    Finally, the total sum, l, of these two fractions is:

    l = t + t = 2 * (1875/594) = 3750/594

    We can simplify this fraction by dividing both numerator and denominator by their greatest common factor, which is 18:

    l = (3750/18) / (594/18) = 208.333 / 33

    ### Answer: 208.333/33

Expected answer: 625/99

Correct? No

🔗 Problem

Problem: First, calculate the result of 58/49 + 31/34. Then, add that result to the fraction 74/38. What is the final result as a simple fraction?

Gemma 2B:

    First, we need to find a common denominator for 58/49 and 31/34. The least common multiple of 49 and 34 is 1666.

    * (58/49) * (34/34) = 1972/1666
    * (31/34) * (49/49) = 1519/1666

    Now we can add the two fractions:

    * 1972/1666 + 1519/1666 = 3491/1666

    Next, we need to add 74/38 to this result. Again, we need a common denominator. The least common multiple of 1666 and 38 is 31674.

    * (3491/1666) * (19/19) = 66329/31674
    * (74/38) * (834/834) = 61676/31674

    Now we can add the two fractions:

    * 66329/31674 + 61676/31674 = 128005/31674

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

    * 128005/31674 = 128005/31674

    ### Answer: 128005/31674

Expected answer: 127971/31654

Correct? No

🔗 Problem

Problem: Calculate the sum of 100 divided by 76 and 29 divided by 14. Please provide your answer as a simplified fraction or decimal.

Gemma 2B:

    First, we calculate each division separately:

    * 100 divided by 76 = 100/76 = 25/19
    * 29 divided by 14 = 29/14

    Now we need to add these fractions. To do that, they need to have the same denominator.  The least common multiple of 19 and 14 is 266.

    * (25/19) * (14/14) = 350/266
    * (29/14) * (19/19) = 551/266

    Finally, we add the fractions:

    * 350/266 + 551/266 = 901/266

    This fraction can't be simplified further.

    ### Answer: 901/266

Expected answer: 901/266

Correct? Yes

🔗 Follow-up

Problem: For a step further, consider the sum you got from the previous question (100 divided by 76 plus 29 divided by 14), and then subtract 98 divided by 84 from this sum. What is the new final value? Write your answer as a simplified fraction or a decimal.

Gemma 2B:

    We already know that 100/76 + 29/14 = 901/266.

    Now let's calculate 98/84:

    * 98/84 = 49/42

    To subtract these fractions, they need the same denominator. The least common multiple of 266 and 42 is 5604.

    * (901/266) * (21/21) = 18921/5604
    * (49/42) * (133/133) = 6517/5604

    Now we can subtract:

    * 18921/5604 - 6517/5604 = 12404/5604

    This fraction can be simplified by dividing both numerator and denominator by their greatest common factor, which is 4:

    * 12404/4 = 3101
    * 5604/4 = 1401

    ### Answer: 3101/1401

Expected answer: 886/399

Correct? No

🔗 Follow-up

Problem: Taking into account the result from the previous question, let's modify it a little bit. This time, instead of summing up (100 divided by 76), let's use (100 divided by 76) + (29 divided by 54). What would be the new result? Write your answer as a simplified fraction or decimal.

Gemma 2B:

    We know from the previous calculation that 100 divided by 76 is 25/19.

    Now we need to calculate 29 divided by 54, which is 29/54.

    To add these fractions, we need a common denominator. The least common multiple of 19 and 54 is 1026.

    * (25/19) * (54/54) = 1350/1026
    * (29/54) * (19/19) = 551/1026

    Adding the fractions:

    * 1350/1026 + 551/1026 = 1901/1026

    This fraction can't be simplified further.

    ### Answer: 1901/1026

Expected answer: 1901/1026

Correct? Yes

🔗 Problem

Problem: Given that Evan has read 51/18 of his novel, and Olivia has read (8/42) after reading the same amount of novel as Evan, how much of the novel has Olivia read in total? Answer as a simplified fraction.

Gemma 2B:

    Here's how to solve this problem:

    1. **Find a common denominator for the fractions representing Evan's reading:**
       * The least common multiple of 18 and 42 is 126.
       * Convert 51/18 to an equivalent fraction with a denominator of 126: (51/18) * (7/7) = 357/126

    2. **Determine the fraction representing the amount Olivia read after Evan:**
       * Olivia read the same amount as Evan, so she also read 357/126 of the novel.

    3. **Add the fractions to find Olivia's total reading:**
       * Olivia's total reading is (357/126) + (8/42)
       * To add these fractions, we need a common denominator (126): (357/126) + (24/126) = 381/126

    4. **Simplify the fraction:**
       * Both 381 and 126 are divisible by 9: (381/126) ÷ (9/9) = 42.33/14

    ### Answer: 42.33/14

Expected answer: 127/42

Correct? No

🔗 Problem

Problem: What is the sum of 48 divided by 13 and 31 divided by 22? Answer as a fraction or a decimal.

Gemma 2B:

    First, we'll perform the division operations:

    * 48 divided by 13 is approximately 3.6923 (rounded to four decimal places).
    * 31 divided by 22 is approximately 1.4091 (rounded to four decimal places).

    Now, we'll add these results:

    3.6923 + 1.4091 = 5.1014

    Therefore, the sum of 48 divided by 13 and 31 divided by 22 is approximately **5.1014**.

Expected answer: 1459/286

Correct? No

🔗 Problem

Problem: First, find the value of 24/86 + 76/77. Subtract your answer from 27/6. Write out your process and show your work.

Gemma 2B:

    First, we need to find a common denominator for 24/86 and 76/77. The least common multiple of 86 and 77 is 6622.

    *  24/86 = (24 * 77) / (86 * 77) = 1848/6622
    *  76/77 = (76 * 86) / (77 * 86) = 6536/6622

    Now, we can add the fractions:

    * 1848/6622 + 6536/6622 = (1848 + 6536) / 6622 = 8384/6622

    Next, we need to simplify the fraction 8384/6622. Both the numerator and denominator are divisible by 4.

    * 8384/6622 = (8384/4) / (6622/4) = 2096/1655.5

    Finally, we need to subtract this result from 27/6.

    * 27/6 - 2096/1655.5 = (27 * 1655.5 - 2096 * 6) / (6 * 1655.5) = (44749.5 - 12576) / 9933 = 32173.5 / 9933

    ### Answer: 32173.5 / 9933

Expected answer: 21415/6622

Correct? No

🔗 Problem

Problem: Find the sum of 73/39 and 49/14, and express your answer as a simplified fraction.

Gemma 2B:

    To add fractions, they need to have the same denominator. The least common multiple of 39 and 14 is 546.

    * We can rewrite 73/39 as (73*14)/(39*14) = 1022/546
    * We can rewrite 49/14 as (49*39)/(14*39) = 1911/546

    Now we can add the fractions: 1022/546 + 1911/546 = 2933/546

    This fraction can be simplified by dividing both numerator and denominator by their greatest common factor, which is 3: 2933/546 = (2933/3)/(546/3) = 977/182

    ### Answer: 977/182

Expected answer: 419/78

Correct? No

🔗 Problem

Problem: What is the sum of 75/65 and 7/39? Simplify your answer.

Gemma 2B:

    To add fractions, they need to have the same denominator. The least common multiple of 65 and 39 is 390.

    * We can rewrite 75/65 as (75*6)/(65*6) = 450/390.
    * We can rewrite 7/39 as (7*10)/(39*10) = 70/390.

    Now we can add the fractions: 450/390 + 70/390 = 520/390.

    Finally, we simplify the fraction by dividing both numerator and denominator by their greatest common factor, which is 10: 520/390 = (520/10)/(390/10) = 52/39.

    ### Answer: 52/39

Expected answer: 4/3

Correct? Yes

🔗 Follow-up

Problem: You previously calculated the sum of 75/65 and 7/39. Now, add the fraction 65/97 to your result. What is your new total?

Gemma 2B:

    We previously found that the sum of 75/65 and 7/39 is 52/39. Now we need to add 65/97 to this.

    To add fractions, they need a common denominator. The least common multiple of 39 and 97 is 3783.

    * We can rewrite 52/39 as (52 * 97)/(39 * 97) = 5044/3783.
    * We can rewrite 65/97 as (65 * 39)/(97 * 39) = 2535/3783.

    Now we can add the fractions: 5044/3783 + 2535/3783 = 7579/3783.

    This fraction cannot be simplified further.

    ### Answer: 7579/3783

Expected answer: 583/291

Correct? Yes

🔗 Follow-up

Problem: In a previous calculation, you found the sum of 75/65 and 7/39. Now, suppose we had originally used 75/65 and 7/47 instead. What would be the sum of these two fractions?

Gemma 2B:

    We need to find the least common multiple (LCM) of 65 and 47. Since 65 and 47 are both prime numbers, their LCM is simply their product: 65 * 47 = 3055.

    Now, we rewrite the fractions with the common denominator:

    * 75/65 = (75 * 47) / (65 * 47) = 3525/3055
    * 7/47 = (7 * 65) / (47 * 65) = 455/3055

    Finally, we add the fractions:

    3525/3055 + 455/3055 = 3980/3055

    We can simplify this fraction by dividing both numerator and denominator by their greatest common factor, which is 5:

    3980/3055 = (3980/5) / (3055/5) = 796/611

    ### Answer: 796/611

Expected answer: 796/611

Correct? Yes

🔗 Problem

Problem: First, calculate 79/1 - 30/92. Then, subtract 61/63 from the result. What value do you get?

Gemma 2B:

    Here's how we can solve this problem step-by-step:

    1. **Calculate 79/1 - 30/92:**
       * 79/1 = 79
       * 30/92 simplifies to approximately 0.33 (you can use a calculator for this)
       * 79 - 0.33 = 78.67

    2. **Subtract 61/63 from the result:**
       * 61/63 is approximately 0.97 (again, using a calculator)
       * 78.67 - 0.97 = 77.7

    ### Answer: 77.7

Expected answer: 225191/2898

Correct? No

🔗 Problem

Problem: Calculate (58 / 26) + (58 / 26), and express your answer in simplest form.

Gemma 2B:

    We can simplify this problem by recognizing that we are adding the same fraction to itself.  This is equivalent to multiplying the fraction by 2.

    Therefore: (58/26) + (58/26) = 2 * (58/26)

    Now, we can simplify the fraction 58/26 by dividing both the numerator and denominator by their greatest common factor, which is 2. This gives us:

    2 * (58/26) = 2 * (29/13)

    Finally, we multiply the whole number by the numerator:

    2 * (29/13) = 58/13

    ### Answer: 58/13

Expected answer: 58/13

Correct? Yes

🔗 Follow-up

Problem: After you have calculated twice the fraction (58 / 26), what would be the result if you then added (49 / 66) to that sum? Express your final answer as a simplified fraction.

Gemma 2B:

    We already know that twice the fraction (58/26) is 58/13.  Now we need to add 49/66 to that.

    To add fractions, they need to have a common denominator. The least common multiple of 13 and 66 is 858.

    Let's convert each fraction to have a denominator of 858:

    * (58/13) * (66/66) = 3828/858
    * (49/66) * (13/13) = 637/858

    Now we can add the fractions:

    3828/858 + 637/858 = 4465/858

    To simplify this fraction, we need to find the greatest common factor of 4465 and 858.  The greatest common factor is 9.

    Dividing both the numerator and denominator by 9, we get:

    4465/858 = (4465/9) / (858/9) = 496/95

    ### Answer: 496/95

Expected answer: 4465/858

Correct? No

🔗 Follow-up

Problem: If originally instead of 58 parts, there were 54 parts in every batch of 26, and two such fractions were added, what would be the sum? Express your answer in simplest form.

Gemma 2B:

    We are asked to calculate (54/26) + (54/26).  Similar to the previous problem, we can simplify this by multiplying the fraction by 2:

    2 * (54/26)

    Now, we can simplify the fraction 54/26 by dividing both the numerator and denominator by their greatest common factor, which is 2:

    2 * (54/26) = 2 * (27/13)

    Finally, we multiply the whole number by the numerator:

    2 * (27/13) = 54/13

    ### Answer: 54/13

Expected answer: 54/13

Correct? Yes

🔗 Problem

Problem: If Alex has 69/63 gallons of orange juice and he bought additional 77/5 gallons, giving him a total of 'k' gallons. Later, he received another 47/71 gallons of orange juice from Sally. How many gallons did Alex have in total?

Gemma 2B:

    First, we need to add the initial amount of orange juice Alex had to the amount he bought: 69/63 + 77/5.

    To add fractions, they need to have the same denominator. The least common multiple of 63 and 5 is 315.  So, we need to convert each fraction:

    * (69/63) * (5/5) = 345/315
    * (77/5) * (63/63) = 4851/315

    Now we can add: 345/315 + 4851/315 = 5196/315

    This is the total amount of orange juice Alex had after buying more.

    Next, we need to add the orange juice he received from Sally: 5196/315 + 47/71

    Again, we need to find a common denominator. The least common multiple of 315 and 71 is 22485.

    * (5196/315) * (71/71) = 368556/22485
    * (47/71) * (315/315) = 14805/22485

    Now we can add: 368556/22485 + 14805/22485 = 383361/22485

    This is the final amount of orange juice Alex has, which we can call 'k'.

    ### Answer: k = 383361/22485

Expected answer: 127907/7455

Correct? No

🔗 Problem

Problem: What is the result of ((100 / 61) + ((39 / 6) + (67 / 49)))? Answer as a simplified fraction.

Gemma 2B:

    Let's break this problem down step-by-step:

    1. **Calculate the fractions within the parentheses:**
       - 100 / 61 = 100/61 (this fraction is already simplified)
       - 39 / 6 = 13/2 (simplifying by dividing both numerator and denominator by 3)
       - 67 / 49 = 67/49 (this fraction is already simplified)

    2. **Add the fractions inside the second set of parentheses:**
       - (13/2) + (67/49) = (13 * 49) / (2 * 49) + (67 * 2) / (49 * 2) = 637/98 + 134/98 = 771/98

    3. **Add the results from steps 1 and 2:**
       - (100/61) + (771/98) = (100 * 98) / (61 * 98) + (771 * 61) / (98 * 61) = 9800/5978 + 47001/5978 = 56801/5978

    4. **Simplify the resulting fraction (if possible):**
       - The fraction 56801/5978 doesn't simplify further, as there are no common factors between the numerator and denominator.

    ### Answer: 56801/5978

Expected answer: 56831/5978

Correct? No

🔗 Problem

Problem: What is the sum of 85/1 and 83/69? Simplify your answer as much as possible.

Gemma 2B:

    Reasoning:
    * 85/1 is simply 85.
    * To add fractions, they need a common denominator. The least common denominator for 1 and 69 is 69.
    * So, we need to rewrite 85/1 as (85*69)/69 = 5865/69.
    * Now we can add: 5865/69 + 83/69 = (5865+83)/69 = 5948/69.
    * We can simplify this fraction by dividing both numerator and denominator by their greatest common factor, which is 3: (5948/3) / (69/3) = 1982/23

    ### Answer: 1982/23

Expected answer: 5948/69

Correct? No

🔗 Problem

Problem: What is the sum of 92/13 and 43/86 as a simplified fraction?

Gemma 2B:

    To add fractions, they need to have the same denominator.

    1. **Find the Least Common Multiple (LCM) of the denominators:**
       The LCM of 13 and 86 is 1118.

    2. **Convert each fraction to have the LCM as the denominator:**
       * (92/13) * (86/86) = 7912/1118
       * (43/86) * (13/13) = 559/1118

    3. **Add the numerators:**
       7912/1118 + 559/1118 = 8471/1118

    4. **Simplify the fraction (if possible):**
       The fraction 8471/1118 cannot be simplified further.

    ### Answer: 8471/1118

Expected answer: 197/26

Correct? Yes

🔗 Follow-up

Problem: You solved the sum of 92/13 and 43/86 in the previous problem. Now, additional 36/72 to that sum. What is the total as a simplified fraction?

Gemma 2B:

    We already know the sum of 92/13 and 43/86 is 8471/1118. Now, let's add 36/72 to that.

    1. **Simplify 36/72:**
       Both 36 and 72 are divisible by 36, so 36/72 simplifies to 1/2.

    2. **Find the LCM of 1118 and 2:**
       The LCM of 1118 and 2 is 1118.

    3. **Convert 1/2 to have a denominator of 1118:**
       (1/2) * (559/559) = 559/1118

    4. **Add the fractions:**
       8471/1118 + 559/1118 = 9030/1118

    5. **Simplify the fraction:**
       Both 9030 and 1118 are divisible by 2, so 9030/1118 simplifies to 4515/559.

    ### Answer: 4515/559

Expected answer: 105/13

Correct? Yes

🔗 Follow-up

Problem: In the previous problem, you found the sum of 92/13 and 43/86. Now suppose the original problem actually was about adding 61/13, not 92/13, to 43/86. What would be the new sum as a simplified fraction?

Gemma 2B:

    Let's follow the same steps as before, but with 61/13 instead of 92/13:

    1. **Find the Least Common Multiple (LCM) of the denominators:**
       The LCM of 13 and 86 is still 1118.

    2. **Convert each fraction to have the LCM as the denominator:**
       * (61/13) * (86/86) = 5246/1118
       * (43/86) * (13/13) = 559/1118

    3. **Add the numerators:**
       5246/1118 + 559/1118 = 5805/1118

    4. **Simplify the fraction (if possible):**
       The fraction 5805/1118 cannot be simplified further.

    ### Answer: 5805/1118

Expected answer: 135/26

Correct? Yes

🔗 Problem

Problem: What is the sum of 67/32 and 10/98, expressed as a simplified fraction?

Gemma 2B:

    To add fractions, they must have a common denominator.

    * Find the least common multiple (LCM) of 32 and 98. The LCM of 32 and 98 is 1568.
    * Rewrite each fraction with the denominator 1568:
        * (67/32) * (49/49) = 3283/1568
        * (10/98) * (16/16) = 160/1568
    * Add the numerators: 3283/1568 + 160/1568 = 3443/1568
    * Simplify the fraction (if possible): The greatest common factor of 3443 and 1568 is 1, so the fraction is already simplified.

    ### Answer: 3443/1568

Expected answer: 3443/1568

Correct? Yes

🔗 Follow-up

Problem: Now, to the sum of 67/32 and 10/98, add 14/59. What is the final result as a simplified fraction?

Gemma 2B:

    We already know that 67/32 + 10/98 = 3443/1568.  Now we need to add 14/59.

    1. **Find a common denominator:** The least common multiple of 1568 and 59 is 92672.

    2. **Rewrite each fraction with the denominator 92672:**
       * (3443/1568) * (59/59) = 203257/92672
       * (14/59) * (1568/1568) = 21952/92672

    3. **Add the numerators:** 203257/92672 + 21952/92672 = 225209/92672

    4. **Simplify:** The greatest common factor of 225209 and 92672 is 1, so the fraction is already simplified.

    ### Answer: 225209/92672

Expected answer: 225089/92512

Correct? No

🔗 Follow-up

Problem: Let's reconsider the previous problem. Instead of a fraction of 67/32, consider a fraction of 43/32. Keeping the second fraction of 10/98 the same, what would be the new sum expressed as a simplified fraction?

Gemma 2B:

    We follow the same process as before, but with the new fraction 43/32:

    1. **Find the LCM:** The LCM of 32 and 98 is still 1568.
    2. **Rewrite fractions with the LCM:**
        * (43/32) * (49/49) = 2107/1568
        * (10/98) * (16/16) = 160/1568
    3. **Add numerators:** 2107/1568 + 160/1568 = 2267/1568
    4. **Simplify:** The greatest common factor of 2267 and 1568 is 1, so the fraction is already simplified.

    ### Answer: 2267/1568

Expected answer: 2267/1568

Correct? Yes

🔗 Problem

Problem: What is the sum of 59/52 and 70/32? Please provide your answer as a simplified fraction.

Gemma 2B:

    To add fractions, they need to have a common denominator.  The least common multiple of 52 and 32 is 416.

    *  We multiply 59/52 by 8/8 (which equals 1, so we don't change the value):
       (59/52) * (8/8) = 472/416

    * We multiply 70/32 by 13/13:
       (70/32) * (13/13) = 910/416

    Now we can add the fractions:

    472/416 + 910/416 = 1382/416

    This fraction can be simplified by dividing both numerator and denominator by their greatest common factor, which is 2:

    1382/416 = (1382/2) / (416/2) = 691/208

    ### Answer: 691/208

Expected answer: 691/208

Correct? Yes

🔗 Follow-up

Problem: After finding the sum of 59/52 and 70/32, add this to 42/41. What is the final sum as a simplified fraction?

Gemma 2B:

    We already know the sum of 59/52 and 70/32 is 691/208.  Now we need to add 42/41.

    To add fractions, they must have a common denominator. The least common multiple of 208 and 41 is 208.

    Since 42/41 already has a denominator of 41, we can just multiply it by 5/5:

    (42/41) * (5/5) = 210/208

    Now we can add the two fractions:

    691/208 + 210/208 = 901/208

    This fraction cannot be simplified further, as the numerator and denominator have no common factors besides 1.

    ### Answer: 901/208

Expected answer: 37067/8528

Correct? No

🔗 Follow-up

Problem: After recalculating the original problem with 74/52 instead of 59/52 and still adding this to 70/32, what is the new sum as a simplified fraction?

Gemma 2B:

    Let's follow the same steps as before, but using 74/52 instead of 59/52.

    1. **Find the Least Common Multiple (LCM):** The LCM of 52 and 32 is still 416.

    2. **Convert the fractions to have the LCM as the denominator:**

       * (74/52) * (8/8) = 592/416
       * (70/32) * (13/13) = 910/416

    3. **Add the fractions:**

       592/416 + 910/416 = 1502/416

    4. **Simplify the fraction:**  The greatest common factor of 1502 and 416 is 2.

       1502/416 = (1502/2) / (416/2) = 751/208

    ### Answer: 751/208

Expected answer: 751/208

Correct? Yes

🔗 Problem

Problem: Let's first perform this operation: 48 divided by 5 subtracted from 11 divided by 65. Then, add the result to 28 divided by 2. What is the answer?

Gemma 2B:

    Here's how to solve the problem step-by-step:

    1. **48 divided by 5:** 48 / 5 = 9.6
    2. **11 divided by 65:** 11 / 65 ≈ 0.169
    3. **Subtract the result from step 1 from the result of step 2:** 0.169 - 9.6 = -9.431
    4. **28 divided by 2:** 28 / 2 = 14
    5. **Add the result from step 3 to the result of step 4:** -9.431 + 14 = 4.569

    ### Answer: 4.569

Expected answer: 1523/65

Correct? No

🔗 Problem

Problem: What is the result when you add 20/52 to 75/82? Simplify your answer to the smallest possible fraction.

Gemma 2B:

    To add fractions, they need to have the same denominator.  We find the least common multiple (LCM) of 52 and 82:

    * Prime factorize 52: 2 x 2 x 13
    * Prime factorize 82: 2 x 41

    The LCM is the product of the highest powers of all the prime factors: 2² x 13 x 41 = 2132

    Now we adjust the fractions:

    * (20/52) * (41/41) = 820/2132
    * (75/82) * (26/26) = 1950/2132

    Now we can add:

    * 820/2132 + 1950/2132 = 2770/2132

    Finally, simplify by dividing both numerator and denominator by their greatest common factor, which is 2:

    * 2770/2132 = (2 x 1385) / (2 x 1066) = 1385/1066

    ### Answer: 1385/1066

Expected answer: 1385/1066

Correct? Yes

🔗 Follow-up

Problem: What will the result be if you take the sum obtained from adding 20/52 to 75/82, and then add 41/75 to it? Simplify your answer to the smallest possible fraction.

Gemma 2B:

    We already know that 20/52 + 75/82 = 1385/1066. Now we need to add 41/75 to this sum.

    First, find the LCM of 1066 and 75:

    * Prime factorize 1066: 2 x 13 x 41
    * Prime factorize 75: 3 x 5²

    The LCM is 2 x 3 x 5² x 13 x 41 = 79350

    Now adjust the fractions:

    * (1385/1066) * (75/75) = 103875/79350
    * (41/75) * (1066/1066) = 43726/79350

    Finally, add the fractions:

    * 103875/79350 + 43726/79350 = 147601/79350

    Simplify by dividing both numerator and denominator by their greatest common factor, which is 1:

    * 147601/79350 = 147601/79350

    ### Answer: 147601/79350

Expected answer: 147581/79950

Correct? No

🔗 Follow-up

Problem: What would the result be if you initially had 20/9 instead of 20/52, and you added this to 75/82? Remember to simplify your answer to the smallest possible fraction.

Gemma 2B:

    Following the same steps as before, we'll find the least common multiple (LCM) of 9 and 82:

    * Prime factorize 9: 3 x 3
    * Prime factorize 82: 2 x 41

    The LCM is the product of all prime factors: 2 x 3² x 41 = 738

    Now we adjust the fractions:

    * (20/9) * (82/82) = 1640/738
    * (75/82) * (9/9) = 675/738

    Now we can add:

    * 1640/738 + 675/738 = 2315/738

    Finally, simplify by dividing both numerator and denominator by their greatest common factor, which is 3:

    * 2315/738 = (3 x 771.66...) / (3 x 246) = 771.66.../246

    Since 771.66... is not a whole number, we can't simplify further.

    ### Answer: 771.66.../246

Expected answer: 2315/738

Correct? No

🔗 Problem

Problem: What is the result of subtracting 92/84 from 31/5? Answer as a simplified fraction.

Gemma 2B:

    First, we need to find a common denominator for the two fractions. The least common multiple of 5 and 84 is 420.

    *  31/5  is equivalent to (31 * 84) / (5 * 84) = 2604/420
    *  92/84 is equivalent to (92 * 5) / (84 * 5) = 460/420

    Now we can subtract the fractions:

    2604/420 - 460/420 = (2604 - 460) / 420 = 2144/420

    Finally, we simplify the fraction by dividing both numerator and denominator by their greatest common factor, which is 4:

    2144/420 = (2144/4) / (420/4) = 536/105

    ### Answer: 536/105

Expected answer: 536/105

Correct? Yes

🔗 Follow-up

Problem: In the previous problem, we subtracted 67/84 from 31/5. The result was a certain simplified fraction, but what if instead of 92/84, we had only subtracted 67/84 from 31/5? What would the result be as a simplified fraction with this modification?

Gemma 2B:

    We've already established that 31/5 is equivalent to 2604/420. Now, let's find the equivalent of 67/84:

    * 67/84 is equivalent to (67 * 5) / (84 * 5) = 335/420

    Now we can subtract:

    2604/420 - 335/420 = (2604 - 335) / 420 = 2269/420

    Finally, simplify by dividing both numerator and denominator by their greatest common factor, which is 3:

    2269/420 = (2269/3) / (420/3) = 756 1/3 / 140

    Since the fraction is improper (the numerator is greater than the denominator), we can rewrite it as a mixed number:

    756 1/3 / 140 = 5 56 1/3 / 140

    ### Answer: 5 56 1/3 / 140

Expected answer: 2269/420

Correct? No

🔗 Problem

Problem: Calculate the following: Double the sum of 81/90 and 94/49.

Gemma 2B:

    Here's how to solve the problem step-by-step:

    1. **Find the sum of the fractions:**
       - 81/90 + 94/49 = (81 * 49) / (90 * 49) + (94 * 90) / (49 * 90)
       - = 3969 / 4410 + 8460 / 4410
       - = 12429 / 4410

    2. **Double the sum:**
       - (12429 / 4410) * 2 = 24858 / 4410

    3. **Simplify (optional):**
       - The fraction can be simplified by dividing both numerator and denominator by their greatest common factor, which is 2.
       - 24858 / 4410 = 12429 / 2205

    ### Answer: 12429 / 2205

Expected answer: 1381/245

Correct? Yes

🔗 Follow-up

Problem: After you have calculated the sum of 81/90 and 94/49 and doubled it, subtract 36/43 from this value. What is the result?

Gemma 2B:

    Let's break this down step by step:

    1. **Calculate the sum of 81/90 and 94/49:**
       - As we found in the previous calculation, the sum is 12429/4410.

    2. **Double the sum:**
       - (12429/4410) * 2 = 24858/4410

    3. **Subtract 36/43 from the doubled sum:**
       - (24858/4410) - (36/43) = (24858 * 43) / (4410 * 43) - (36 * 4410) / (43 * 4410)
       - = 1068914 / 189630 - 158760 / 189630
       - = 910154 / 189630

    4. **Simplify (optional):**
       - The fraction can be simplified by dividing both numerator and denominator by their greatest common factor, which is 2.
       - 910154 / 189630 = 455077 / 94815

    ### Answer: 455077 / 94815

Expected answer: 50563/10535

Correct? No

🔗 Follow-up

Problem: After revising the quantities in the previous problem, calculate the following with the new numbers: Double the sum of 98/90 and 94/49. What is the result?

Gemma 2B:

    Here's the solution:

    1. **Find the sum of the fractions:**
       - 98/90 + 94/49 = (98 * 49) / (90 * 49) + (94 * 90) / (49 * 90)
       - = 4802 / 4410 + 8460 / 4410
       - = 13262 / 4410

    2. **Double the sum:**
       - (13262 / 4410) * 2 = 26524 / 4410

    3. **Simplify (optional):**
       - The fraction can be simplified by dividing both numerator and denominator by their greatest common factor, which is 2.
       - 26524 / 4410 = 13262 / 2205

    ### Answer: 13262 / 2205

Expected answer: 13262/2205

Correct? Yes

🔗 Problem

Problem: Find the difference between 69/21 and 10/92. Write your answer as a simple fraction.

Gemma 2B:

    To find the difference, we need to subtract the second fraction from the first:

    69/21 - 10/92

    Before we can subtract fractions, they need to have a common denominator. The least common multiple of 21 and 92 is 1932.

    *  69/21 * 92/92 = 6348/1932
    *  10/92 * 21/21 = 210/1932

    Now we can subtract:

    6348/1932 - 210/1932 = 6138/1932

    Finally, we can simplify the fraction by dividing both numerator and denominator by their greatest common factor, which is 2:

    6138/1932 = 3069/966

    ### Answer: 3069/966

Expected answer: 1023/322

Correct? Yes

🔗 Follow-up

Problem: Given that the difference between 69/21 and 10/92 is noted as a certain value, what will be the new value if you add 24/46 to it? Write your answer as a simplified fraction.

Gemma 2B:

    We already know that the difference between 69/21 and 10/92 is 3069/966.  Now we need to add 24/46 to this value.

    First, let's simplify 24/46 by dividing both numerator and denominator by their greatest common factor, which is 2:

    24/46 = 12/23

    Now we need to find a common denominator for 3069/966 and 12/23. The least common multiple of 966 and 23 is 22218.

    * 3069/966 * 23/23 = 70557/22218
    * 12/23 * 966/966 = 11592/22218

    Now we can add the fractions:

    70557/22218 + 11592/22218 = 82149/22218

    Finally, we can simplify the fraction by dividing both numerator and denominator by their greatest common factor, which is 3:

    82149/22218 = 27383/7406

    ### Answer: 27383/7406

Expected answer: 1191/322

Correct? No

🔗 Follow-up

Problem: In the previous problem, you found the difference between certain fractions. Now, consider a slight change. If the first fraction was 63/21 instead of 69/21, what would the new difference be? Calculate 63/21 - 10/92, and write your answer as a simplified fraction.

Gemma 2B:

    Let's calculate the new difference:

    63/21 - 10/92

    First, we simplify 63/21:

    63/21 = 3

    Now we need to find a common denominator for 3 and 10/92.  The least common multiple of 1 and 92 is 92.

    *  3 = 3/1 * 92/92 = 276/92

    Now we can subtract:

    276/92 - 10/92 = 266/92

    Finally, we can simplify the fraction by dividing both numerator and denominator by their greatest common factor, which is 2:

    266/92 = 133/46

    ### Answer: 133/46

Expected answer: 133/46

Correct? Yes

🔗 Problem

Problem: First, find the result of (26 / 87) + (92 / 88). Then add the result to (26 / 57). What's the answer?

Gemma 2B:

    First, let's calculate (26 / 87) + (92 / 88):

    * 26 / 87 ≈ 0.299
    * 92 / 88 ≈ 1.045
    * 0.299 + 1.045 ≈ 1.344

    Now, let's add this result to (26 / 57):

    * 26 / 57 ≈ 0.456
    * 1.344 + 0.456 ≈ 1.8

    ### Answer: 1.8

Expected answer: 21825/12122

Correct? No

🔗 Problem

Problem: What is the result if you add 78/66, 28/6, and 90/34 together? Please provide your answer as a simplified fraction.

Gemma 2B:

    To add fractions, they must have a common denominator.  Let's find the least common multiple (LCM) of 66, 6, and 34:

    * 66 = 2 x 3 x 11
    * 6 = 2 x 3
    * 34 = 2 x 17

    The LCM is 2 x 3 x 11 x 17 = 1122

    Now, let's convert each fraction:

    * 78/66 = (78 x 17) / (66 x 17) = 1326 / 1122
    * 28/6 = (28 x 187) / (6 x 187) = 5236 / 1122
    * 90/34 = (90 x 33) / (34 x 33) = 2970 / 1122

    Now, we can add the fractions:

    (1326 + 5236 + 2970) / 1122 = 9532 / 1122

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

    9532 / 1122 = (9532 / 2) / (1122 / 2) = 4766 / 561

    ### Answer: 4766/561

Expected answer: 4766/561

Correct? Yes

🔗 Follow-up

Problem: For the result you obtained from adding 78/66, 28/6, and 90/34 together, what would be the new result if you now subtract 56/61 from it? Please provide your answer as a simplified fraction.

Gemma 2B:

    We found the sum of the three fractions to be 4766/561.  To subtract 56/61, we need a common denominator.  The LCM of 561 and 61 is 34261.

    Let's convert the fractions:

    * 4766/561 = (4766 * 61) / (561 * 61) = 290526 / 34261
    * 56/61 = (56 * 561) / (61 * 561) = 31536 / 34261

    Now we can subtract:

    (290526 - 31536) / 34261 = 259000 / 34261

    We can simplify this fraction by dividing both numerator and denominator by their greatest common factor, which is 1:

    259000 / 34261 = 259000 / 34261

    ### Answer: 259000/34261

Expected answer: 259310/34221

Correct? No

🔗 Follow-up

Problem: Previously you added 78/66, 28/6, and 90/34 together. Now consider what would the result be if you instead added 78/66, 53/6, and 90/34 together? Please provide your answer as a simplified fraction.

Gemma 2B:

    Following the same process as before, we'll find the LCM of 66, 6, and 34:

    * 66 = 2 x 3 x 11
    * 6 = 2 x 3
    * 34 = 2 x 17

    The LCM is 2 x 3 x 11 x 17 = 1122

    Now, let's convert each fraction:

    * 78/66 = (78 x 17) / (66 x 17) = 1326 / 1122
    * 53/6 = (53 x 187) / (6 x 187) = 9911 / 1122
    * 90/34 = (90 x 33) / (34 x 33) = 2970 / 1122

    Now, we can add the fractions:

    (1326 + 9911 + 2970) / 1122 = 14207 / 1122

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

    14207 / 1122 = (14207 / 3) / (1122 / 3) = 4735.67 / 374

    ### Answer: 4735.67/374

Expected answer: 14207/1122

Correct? No

🔗 Problem

Problem: Calculate: (12 / 7) - (9 / 23). Please simplify your answer to the lowest terms.

Gemma 2B:

    To subtract fractions, they need to have the same denominator.  We find the least common multiple (LCM) of 7 and 23, which is 161.

    * (12/7) * (23/23) = 276/161
    * (9/23) * (7/7) = 63/161

    Now we can subtract:

    * (276/161) - (63/161) = (276-63)/161 = 213/161

    ### Answer: 213/161

Expected answer: 213/161

Correct? Yes

🔗 Follow-up

Problem: After finding the result of (12 / 7) - (9 / 23), use your answer as t. Then, calculate (40 / 11) - t. What do you find? Please express your answer as a simplified fraction.

Gemma 2B:

    We already found that (12/7) - (9/23) = 213/161. Let's call this value *t*.

    Now we need to calculate (40/11) - *t*, which is:

    (40/11) - (213/161)

    To subtract these fractions, we need a common denominator. The least common multiple of 11 and 161 is 1771.

    * (40/11) * (161/161) = 6440/1771
    * (213/161) * (11/11) = 2343/1771

    Now we can subtract:

    * (6440/1771) - (2343/1771) = (6440 - 2343)/1771 = 4097/1771

    ### Answer: 4097/1771

Expected answer: 4097/1771

Correct? Yes

🔗 Follow-up

Problem: In the previous problem, you calculated (12 / 7) - (9 / 23) which accounted for certain quantities of two different fruits. Now suppose you actually had 12/7th of a crate of oranges but only 9/48th of a crate of apples. In that case, what would be the resulting difference in fractions between the two fruit quantities? Simplify your answer to the lowest terms.

Gemma 2B:

    To find the difference between the two quantities, we need to subtract the fraction representing the apples from the fraction representing the oranges.

    First, we simplify the fraction representing the apples: 9/48 simplifies to 3/16 (dividing both numerator and denominator by 3).

    Now we have: (12/7) - (3/16)

    To subtract fractions, they need a common denominator.  The least common multiple (LCM) of 7 and 16 is 112.

    * (12/7) * (16/16) = 192/112
    * (3/16) * (7/7) = 21/112

    Now we can subtract:

    * (192/112) - (21/112) = (192 - 21) / 112 = 171/112

    ### Answer: 171/112

Expected answer: 171/112

Correct? Yes

🔗 Problem

Problem: Calculate the sum of 95/18 and 51/61, and represent your answer as a simplified fraction.

Gemma 2B:

    To add fractions, they must have a common denominator. The least common multiple of 18 and 61 is 1098.

    * Convert 95/18: (95/18) * (61/61) = 5795/1098
    * Convert 51/61: (51/61) * (18/18) = 918/1098

    Now we can add the fractions: 5795/1098 + 918/1098 = 6713/1098

    This fraction can be simplified by finding the greatest common divisor (GCD) of 6713 and 1098, which is 3.

    Simplifying the fraction: (6713/3) / (1098/3) = 2237/366

    ### Answer: 2237/366

Expected answer: 6713/1098

Correct? No

🔗 Problem

Problem: Calculate the following subtraction of fractions: (93 / 46) - (51 / 84). Answer as a simplified fraction.

Gemma 2B:

    To subtract fractions, they need to have the same denominator. We'll find the least common multiple (LCM) of 46 and 84.

    * Prime factorization of 46: 2 x 23
    * Prime factorization of 84: 2 x 2 x 3 x 7
    * LCM of 46 and 84: 2 x 2 x 3 x 7 x 23 = 1932

    Now we'll adjust the fractions:

    * (93 / 46) * (42/42) = 3906/1932
    * (51 / 84) * (23/23) = 1173/1932

    Now we can subtract:

    * (3906 / 1932) - (1173 / 1932) = (3906 - 1173) / 1932 = 2733 / 1932

    Finally, we simplify the fraction by dividing both numerator and denominator by their greatest common factor, which is 3:

    * (2733 / 3) / (1932 / 3) = 911/644

    ### Answer: 911/644

Expected answer: 911/644

Correct? Yes

🔗 Follow-up

Problem: Following the previous question, now calculate the result of the subtraction (93 / 46) - (51 / 84) and then add the fraction 35 / 27 to the result. What is the final answer when simplified?

Gemma 2B:

    We already know that (93 / 46) - (51 / 84) = 911/644.

    Now, we need to add 35/27 to this result.  To do that, we need a common denominator.  The least common multiple of 644 and 27 is 17388.

    * (911/644) * (27/27) = 24597 / 17388
    * (35/27) * (644/644) = 22540 / 17388

    Now we can add the fractions:

    * (24597 / 17388) + (22540 / 17388) = (24597 + 22540) / 17388 = 47137 / 17388

    Finally, we need to simplify this fraction. The greatest common factor of 47137 and 17388 is 1. Therefore, the simplified answer is:

    ### Answer: 47137/17388

Expected answer: 47137/17388

Correct? Yes

🔗 Follow-up

Problem: Following the previous question, let's imagine that instead of subtracting (51/84), you were actually subtracting (78/84) from (93/46). What would be the result in that case? Simplify your answer to the simplest form.

Gemma 2B:

    We already know the LCM of 46 and 84 is 1932. So, let's adjust the fractions:

    * (93 / 46) * (42/42) = 3906/1932
    * (78 / 84) * (23/23) = 1794/1932

    Now we can subtract:

    * (3906 / 1932) - (1794 / 1932) = (3906 - 1794) / 1932 = 2112 / 1932

    Finally, we simplify the fraction by dividing both numerator and denominator by their greatest common factor, which is 4:

    * (2112 / 4) / (1932 / 4) = 528 / 483

    ### Answer: 528/483

Expected answer: 176/161

Correct? Yes

🔗 Problem

Problem: Calculate the sum of the fractions 12/70 and 84/9. Please provide your answer in simplified form.

Gemma 2B:

    To add fractions, they need to have the same denominator. We need to find the least common multiple (LCM) of 70 and 9.

    The LCM of 70 and 9 is 630.

    Now, let's convert each fraction to have a denominator of 630:

    * 12/70 = (12 * 9) / (70 * 9) = 108/630
    * 84/9 = (84 * 70) / (9 * 70) = 5880/630

    Now, we can add the fractions:

    108/630 + 5880/630 = (108 + 5880)/630 = 5988/630

    Finally, we simplify the fraction by dividing both numerator and denominator by their greatest common factor, which is 6:

    5988/630 = (5988/6)/(630/6) = **998/105**

    ### Answer: 998/105

Expected answer: 998/105

Correct? Yes

🔗 Follow-up

Problem: If you remember, you previously calculated the sum of the fractions 12/70 and 84/9. Now, add this new fraction to the result: 21/86. What is the new total? Again, please present your answer in simplified form.

Gemma 2B:

    You are right! We previously found the sum of 12/70 and 84/9 to be 998/105.

    Now, we need to add 21/86 to that sum. To do this, we need a common denominator. The least common multiple of 105 and 86 is 9030.

    Let's convert each fraction:

    * 998/105 = (998 * 86) / (105 * 86) = 85828/9030
    * 21/86 = (21 * 105) / (86 * 105) = 2205/9030

    Now we can add the fractions:

    85828/9030 + 2205/9030 = (85828 + 2205)/9030 = 88033/9030

    Finally, we simplify by dividing numerator and denominator by their greatest common factor, which is 3:

    88033/9030 = (88033/3) / (9030/3) = **29344/3010**

    ### Answer: 29344/3010

Expected answer: 88033/9030

Correct? No

🔗 Follow-up

Problem: Let's rethink a detail from your previous calculation. Instead of adding the fractions 12/70 and 84/9, suppose you need to consider a situation where you add 12/70 and 40/9 instead. What is the sum of these two fractions? Provide your answer as a simplified fraction.

Gemma 2B:

    Following the same steps as before, we need to find the LCM of 70 and 9, which is 630.  We then convert each fraction to have a denominator of 630:

    * 12/70 = (12 * 9) / (70 * 9) = 108/630
    * 40/9 = (40 * 70) / (9 * 70) = 2800/630

    Now we can add the fractions:

    108/630 + 2800/630 = (108 + 2800)/630 = 2908/630

    Finally, we simplify the fraction by dividing both numerator and denominator by their greatest common factor, which is 6:

    2908/630 = (2908/6)/(630/6) = **484/105**

    ### Answer: 484/105

Expected answer: 1454/315

Correct? No

🔗 Problem

Problem: If you subtract 40/92 from 32/3 and double the result, what would you get?

Gemma 2B:

    Here's how to solve the problem step-by-step:

    1. **Find a common denominator for 32/3 and 40/92:**
       - The least common multiple of 3 and 92 is 276.
       - Convert 32/3 to 276ths: (32/3) * (92/92) = 2944/276
       - Convert 40/92 to 276ths: (40/92) * (3/3) = 120/276

    2. **Subtract the fractions:**
       - (2944/276) - (120/276) = 2824/276

    3. **Double the result:**
       - (2824/276) * 2 = 5648/276

    4. **Simplify (if possible):**
       - Both numerator and denominator are divisible by 4.
       - (5648/276) / (4/4) = 1412/69

    5. **The final answer:**
       - We can simplify further by dividing both numerator and denominator by 3.
       - (1412/69) / (3/3) = **470.67** (rounded to two decimal places).

    ### Answer: 470.67

Expected answer: 1412/69

Correct? No

🔗 Problem

Problem: What is the sum of 79/24 and 79/24? Keep your answer as a simplified fraction.

Gemma 2B:

    When adding fractions with the same denominator, we simply add the numerators and keep the denominator the same.

    So, 79/24 + 79/24 = (79 + 79) / 24 = 158/24

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

    158/24 = (158/2)/(24/2) = 79/12

    ### Answer: 79/12

Expected answer: 79/12

Correct? Yes

🔗 Follow-up

Problem: In continuation to the previous problem, now find out what is the sum of two times the previously found sum. In other words, calculate (79/24 + 79/24) + (79/24 + 79/24). Please answer as a simplified fraction.

Gemma 2B:

    We already know that 79/24 + 79/24 = 79/12.

    Therefore, (79/24 + 79/24) + (79/24 + 79/24) is the same as 79/12 + 79/12.

    Adding these fractions:

    79/12 + 79/12 = (79 + 79) / 12 = 158/12

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

    158/12 = (158/2)/(12/2) = 79/6

    ### Answer: 79/6

Expected answer: 79/6

Correct? Yes

🔗 Follow-up

Problem: In the previous problem, you added together two fractions of 79/24. Now, consider if those were fractions of 79/20 instead. What would be the total if you added together two fractions of 79/20? Please express your answer as a simplified fraction.

Gemma 2B:

    Just like before, we add the numerators and keep the denominator the same.

    79/20 + 79/20 = (79 + 79) / 20 = 158/20

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

    158/20 = (158/2)/(20/2) = 79/10

    ### Answer: 79/10

Expected answer: 79/10

Correct? Yes

🔗 Problem

Problem: What is the sum of 91/76 and 12/22? Please simplify your fraction.

Gemma 2B:

    To add fractions, they need to have a common denominator.  Let's find the least common multiple (LCM) of 76 and 22:

    * Factors of 76: 1, 2, 4, 19, 38, 76
    * Factors of 22: 1, 2, 11, 22

    The LCM of 76 and 22 is 836.

    Now we'll convert each fraction:

    * 91/76 * (11/11) = 1001/836
    * 12/22 * (38/38) = 456/836

    Finally, add the fractions:

    1001/836 + 456/836 = 1457/836

    This fraction can be simplified by dividing both numerator and denominator by their greatest common factor, which is 11:

    1457/836 = (1457/11) / (836/11) = 132.45 / 76.00

    ### Answer: 132.45/76.00

Expected answer: 1457/836

Correct? No

🔗 Problem

Problem: First, add 88/93 to 6/47. Then, add the result to 92/4. What do you get?

Gemma 2B:

    Here's how to solve this problem step-by-step:

    **1. Find a common denominator for 88/93 and 6/47:**

    * The least common multiple (LCM) of 93 and 47 is 4371.
    * Convert 88/93: (88 * 47) / (93 * 47) = 4136 / 4371
    * Convert 6/47: (6 * 93) / (47 * 93) = 558 / 4371

    **2. Add the fractions:**

    * (4136 / 4371) + (558 / 4371) = 4694 / 4371

    **3. Find a common denominator for 4694/4371 and 92/4:**

    * The least common multiple (LCM) of 4371 and 4 is 17484.
    * Convert 4694/4371: (4694 * 4) / (4371 * 4) = 18776 / 17484
    * Convert 92/4: (92 * 4371) / (4 * 4371) = 402212 / 17484

    **4. Add the fractions:**

    * (18776 / 17484) + (402212 / 17484) = 420988 / 17484

    **5. Simplify the fraction (optional):**

    * The fraction can be simplified by dividing both numerator and denominator by their greatest common factor (GCF), which is 4.
    * 420988 / 4 = 105247
    * 17484 / 4 = 4371
    * Simplified fraction: 105247/4371

    ### Answer: 105247/4371

Expected answer: 105227/4371

Correct? No

🔗 Problem

Problem: Calculate the sum of 95/83 and the result of 52/21 + 75/67. Write your answer as a simplified fraction.

Gemma 2B:

    First, we need to find a common denominator for 52/21 and 75/67. The least common multiple of 21 and 67 is 1407.

    * 52/21 * (67/67) = 3484/1407
    * 75/67 * (21/21) = 1575/1407

    Now we can add the two fractions:

    3484/1407 + 1575/1407 = 5059/1407

    Next, we add 95/83 to this result.  We again need to find a common denominator, which is 116511.

    * 5059/1407 * (83/83) = 420247/116511
    * 95/83 * (1407/1407) = 133665/116511

    Finally, we add the two fractions:

    420247/116511 + 133665/116511 = 553912/116511

    This fraction can be simplified by dividing both the numerator and denominator by their greatest common factor, which is 8.

    553912/116511 = (553912/8) / (116511/8) = **69239/14564**

    ### Answer: 69239/14564

Expected answer: 553562/116781

Correct? No

🔗 Problem

Problem: What is the value of the expression ((73 / 27) - (23 / 48))? Answer as a simplified fraction.

Gemma 2B:

    To subtract fractions, they need to have a common denominator.

    1. **Find the Least Common Multiple (LCM) of the denominators:**
       - The LCM of 27 and 48 is 432.

    2. **Convert the fractions to equivalent fractions with the LCM as the denominator:**
       - (73/27) * (16/16) = 1168/432
       - (23/48) * (9/9) = 207/432

    3. **Subtract the fractions:**
       - (1168/432) - (207/432) = 961/432

    ### Answer: 961/432

Expected answer: 961/432

Correct? Yes

🔗 Follow-up

Problem: What is the value of the original expression ((73 / 27) - (23 / 48)) minus the fraction 61/59? Answer as a simplified fraction.

Gemma 2B:

    We already know that ((73/27) - (23/48)) simplifies to 961/432.

    Now we need to subtract 61/59 from that:

    1. **Find a common denominator:**
       - The LCM of 432 and 59 is 25488.

    2. **Convert the fractions to equivalent fractions with the LCM as the denominator:**
       - (961/432) * (59/59) = 56759/25488
       - (61/59) * (432/432) = 26352/25488

    3. **Subtract the fractions:**
       - (56759/25488) - (26352/25488) = 30407/25488

    ### Answer: 30407/25488

Expected answer: 30347/25488

Correct? No

🔗 Follow-up

Problem: In the previous question, we calculated the value of the expression ((73 / 27) - (23 / 48)). Now, suppose we made a mistake and the original fraction was actually 65/27, not 73/27. What would the value of the new expression be? Answer as a simplified fraction.

Gemma 2B:

    Let's recalculate the expression using 65/27 instead of 73/27:

    1. **Substitute the new fraction:**
       - ((65/27) - (23/48))

    2. **Find the LCM of the denominators:**
       - The LCM of 27 and 48 is 432.

    3. **Convert the fractions to equivalent fractions with the LCM as the denominator:**
       - (65/27) * (16/16) = 1040/432
       - (23/48) * (9/9) = 207/432

    4. **Subtract the fractions:**
       - (1040/432) - (207/432) = 833/432

    ### Answer: 833/432

Expected answer: 833/432

Correct? Yes

🔗 Problem

Problem: Solve the expression (78 / 2) - (70 / 23). Answer as a simplified decimal or fraction.

Gemma 2B:

    Here's how to solve the expression:

    1. **Divide 78 by 2:**  78 / 2 = 39
    2. **Divide 70 by 23:** This results in a decimal, approximately 3.04.
    3. **Subtract:** 39 - 3.04 = 35.96

    ### Answer: 35.96

Expected answer: 827/23

Correct? No

🔗 Problem

Problem: What is the result when you add 34/1 to 53/4? Express your answer as a mixed number.

Gemma 2B:

    Reasoning: 34/1 is equal to 34.  To add fractions, they must have the same denominator.  We can rewrite 34 as 34/1 and then multiply both the numerator and denominator by 4 to get 136/4.  Now we can add the fractions:

    136/4 + 53/4 = (136+53)/4 = 189/4

    To express this as a mixed number, we divide 189 by 4:

    189 ÷ 4 = 47 with a remainder of 1

    So, 189/4 is equal to 47 1/4.

    ### Answer: 47 1/4

Expected answer: 189/4

Correct? Yes

🔗 Follow-up

Problem: Following the result of the previous problem, what is the sum of 24/67 and the result you got from adding 34/1 to 53/4? Express your answer as a mixed number.

Gemma 2B:

    Reasoning: We know from the previous problem that 34/1 + 53/4 = 47 1/4.  To add fractions, they must have the same denominator.  We can rewrite 47 1/4 as (47*4 + 1)/4 = 189/4.  Now we can add 24/67:

    189/4 + 24/67 = (189*67 + 24*4) / (4*67) = 12693/268

    To express this as a mixed number, we divide 12693 by 268:

    12693 ÷ 268 = 47 with a remainder of 109

    So, 12693/268 is equal to 47 109/268.

    ### Answer: 47 109/268

Expected answer: 12759/268

Correct? No

🔗 Follow-up

Problem: Let's reconsider the problem where you were adding fractions. Previously, it was asked to add 34/1 to 53/4. Now consider whether it may have been 34/97 instead of 34/1. What will be the result if you add 34/97 to 53/4? Write your answer as a mixed number.

Gemma 2B:

    Reasoning: To add fractions, they must have the same denominator. The least common multiple of 97 and 4 is 388.

    *  We multiply the numerator and denominator of 34/97 by 4 to get (34*4)/(97*4) = 136/388
    * We multiply the numerator and denominator of 53/4 by 97 to get (53*97)/(4*97) = 5141/388

    Now we can add the fractions:

    136/388 + 5141/388 = (136 + 5141)/388 = 5277/388

    To express this as a mixed number, we divide 5277 by 388:

    5277 ÷ 388 = 13 with a remainder of 203

    So, 5277/388 is equal to 13 203/388.

    ### Answer: 13 203/388

Expected answer: 5277/388

Correct? No

🔗 Problem

Problem: What is the sum of 3/36 and 97/96? Answer as a simplified fraction.

Gemma 2B:

    To add fractions, they must have a common denominator.  The least common multiple of 36 and 96 is 288.

    * **Convert 3/36:**  Multiply numerator and denominator by 8: (3 * 8) / (36 * 8) = 24/288

    * **Convert 97/96:** Multiply numerator and denominator by 3: (97 * 3) / (96 * 3) = 291/288

    Now we can add: 24/288 + 291/288 = 315/288

    Finally, simplify the fraction by dividing both numerator and denominator by their greatest common factor, which is 3: 315/288 = (315/3) / (288/3) = 105/96

    This fraction can be further simplified by dividing both numerator and denominator by 3:  105/96 = (105/3) / (96/3) = **35/32**

    ### Answer: 35/32

Expected answer: 35/32

Correct? Yes

🔗 Follow-up

Problem: After finding the sum of 3/36 and 97/96, add your answer to 55/69. What is your result reduced to its simplest form?

Gemma 2B:

    We previously found the sum of 3/36 and 97/96 to be 35/32. Now we need to add 55/69.

    To add these fractions, they need a common denominator. The least common multiple of 32 and 69 is 2208.

    * **Convert 35/32:** Multiply numerator and denominator by 69: (35 * 69) / (32 * 69) = 2415/2208

    * **Convert 55/69:** Multiply numerator and denominator by 32: (55 * 32) / (69 * 32) = 1760/2208

    Now we can add: 2415/2208 + 1760/2208 = 4175/2208

    This fraction cannot be simplified further as 4175 and 2208 share no common factors other than 1.

    ### Answer: 4175/2208

Expected answer: 4175/2208

Correct? Yes

🔗 Follow-up

Problem: After the initial calculation with 3/36 and 97/96, now consider you made an error and the actual fraction was 3/38 not 3/36. Recalculate the sum of 3/38 and 97/96. What is the new sum as a simplified fraction?

Gemma 2B:

    We need to find a common denominator for 38 and 96. The least common multiple is 1824.

    * **Convert 3/38:** Multiply numerator and denominator by 48: (3 * 48) / (38 * 48) = 144/1824

    * **Convert 97/96:** Multiply numerator and denominator by 19: (97 * 19) / (96 * 19) = 1843/1824

    Now we can add: 144/1824 + 1843/1824 = 1987/1824

    This fraction can be simplified by dividing both numerator and denominator by their greatest common factor, which is 3: 1987/1824 = (1987/3) / (1824/3) = **662/608**

    ### Answer: 662/608

Expected answer: 1987/1824

Correct? No

🔗 Problem

Problem: Calculate ((92 / 95) + (4 / 74)). Answer as a simplified fraction.

Gemma 2B:

    Here's how to calculate the expression and simplify the answer:

    1. **Calculate the first fraction:** 92 / 95. This fraction is already in its simplest form.

    2. **Calculate the second fraction:** 4 / 74. We can simplify this by dividing both numerator and denominator by their greatest common factor, which is 2: (4/2) / (74/2) = 2/37

    3. **Add the two fractions:**  (92/95) + (2/37). To add fractions, they must have a common denominator.  The least common multiple of 95 and 37 is 3515.

       *  (92/95) * (37/37) = 3404/3515
       *  (2/37) * (95/95) = 190/3515

    4. **Add the numerators:** (3404/3515) + (190/3515) = 3594/3515

    5. **Simplify the fraction:**  We can simplify this fraction by dividing both numerator and denominator by their greatest common factor, which is 2: (3594/2) / (3515/2) = 1797/1757.5

    ### Answer: 1797/1757.5

Expected answer: 3594/3515

Correct? Yes

🔗 Problem

Problem: If the value of a variable 'v' is 69/47, what is the value of 'd' if 'd' equals 88/72 + 'v'? Please write your answer as a fraction.

Gemma 2B:

    First, we need to simplify 88/72. Both the numerator and denominator are divisible by 8, giving us 11/9.

    Now, we can substitute the value of 'v' into the equation for 'd':

    d = 11/9 + 69/47

    To add these fractions, we need a common denominator. The least common multiple of 9 and 47 is 423.

    * Multiply 11/9 by 47/47: (11/9) * (47/47) = 517/423
    * Multiply 69/47 by 9/9: (69/47) * (9/9) = 621/423

    Now we can add the fractions:

    d = 517/423 + 621/423 = 1138/423

    ### Answer: 1138/423

Expected answer: 1138/423

Correct? Yes

🔗 Follow-up

Problem: Following from the last question, if we now have a new variable 'r' which is defined as 92/22 + 'd', what is the value of 'r'? Please write your answer as a fraction.

Gemma 2B:

    We know that 'd' = 1138/423.  Let's substitute this into the equation for 'r':

    r = 92/22 + 1138/423

    First, simplify 92/22. Both the numerator and denominator are divisible by 2, giving us 46/11.

    Now we need a common denominator to add the fractions. The least common multiple of 11 and 423 is 4653.

    * Multiply 46/11 by 423/423: (46/11) * (423/423) = 19458/4653
    * Multiply 1138/423 by 11/11: (1138/423) * (11/11) = 12518/4653

    Now we can add the fractions:

    r = 19458/4653 + 12518/4653 = 31976/4653

    ### Answer: 31976/4653

Expected answer: 31976/4653

Correct? Yes

🔗 Follow-up

Problem: In the problem from before, let's consider that variable 'v' no longer represents 69/47, but is now represented by 69/8. Given that change, what is the new value for 'd' if 'd' continues to be defined as the sum of 88/72 and 'v'? Write your answer as a fraction.

Gemma 2B:

    We'll start by simplifying 88/72. Both the numerator and denominator are divisible by 8, giving us 11/9.

    Now, we can substitute the new value of 'v' into the equation for 'd':

    d = 11/9 + 69/8

    To add these fractions, we need a common denominator. The least common multiple of 9 and 8 is 72.

    * Multiply 11/9 by 8/8: (11/9) * (8/8) = 88/72
    * Multiply 69/8 by 9/9: (69/8) * (9/9) = 621/72

    Now we can add the fractions:

    d = 88/72 + 621/72 = 709/72

    ### Answer: 709/72

Expected answer: 709/72

Correct? Yes

🔗 Problem

Problem: Let's say there is a wholesaler who has 41/26 barrels of olive oil in their warehouse. They sell a portion of the oil which is equivalent to 41/72 barrels. How much olive oil do they have now? Express your answer as a simple fraction.

Gemma 2B:

    Here's how to solve the problem:

    1. **Find the difference:** To determine how much oil is left, subtract the amount sold from the initial amount: (41/26) - (41/72)

    2. **Find a common denominator:** The least common multiple of 26 and 72 is 936.  Rewrite the fractions with this denominator:
       * (41/26) = (41 * 36) / (26 * 36) = 1476/936
       * (41/72) = (41 * 13) / (72 * 13) = 533/936

    3. **Subtract the fractions:** Now that they have the same denominator, we can subtract:
       (1476/936) - (533/936) = (1476 - 533) / 936 = 943/936

    **Therefore, the wholesaler has 943/936 barrels of olive oil left.**
    ### Answer: 943/936

Expected answer: 943/936

Correct? Yes

🔗 Follow-up

Problem: The wholesaler, after their sale, receives a new shipment of olive oil. This shipment contains 68/47 barrels of oil. After adding the new shipment to their existing stock, how much olive oil does the wholesaler have in total? Express your answer as a simple fraction.

Gemma 2B:

    Here's how to calculate the total olive oil after the shipment:

    1. **Add the new shipment to the existing stock:**
       (943/936) + (68/47)

    2. **Find a common denominator:** The least common multiple of 936 and 47 is 43992. Rewrite the fractions with this denominator:
       * (943/936) = (943 * 47) / (936 * 47) = 44321/43992
       * (68/47) = (68 * 936) / (47 * 936) = 63648/43992

    3. **Add the fractions:** Now that they have the same denominator, we can add:
       (44321/43992) + (63648/43992) = (44321 + 63648) / 43992 = 107969/43992

    **Therefore, the wholesaler has a total of 107969/43992 barrels of olive oil.**

    ### Answer: 107969/43992

Expected answer: 107969/43992

Correct? Yes

🔗 Follow-up

Problem: Let's change the start of our olive oil wholesaler's scenario. Now, they originally had 71/26 barrels of olive oil in their warehouse. They still sell a portion of the oil which is equivalent to 41/72 barrels. How much olive oil do they have now? Express your answer as a simple fraction.

Gemma 2B:

    Here's how to solve this problem:

    1. **Find the difference:**  To determine how much oil is left, subtract the amount sold from the initial amount: (71/26) - (41/72)

    2. **Find a common denominator:** The least common multiple of 26 and 72 is 936. Rewrite the fractions with this denominator:
       * (71/26) = (71 * 36) / (26 * 36) = 2556/936
       * (41/72) = (41 * 13) / (72 * 13) = 533/936

    3. **Subtract the fractions:** Now that they have the same denominator, we can subtract:
       (2556/936) - (533/936) = (2556 - 533) / 936 = 2023/936

    **Therefore, the wholesaler has 2023/936 barrels of olive oil left.**
    ### Answer: 2023/936

Expected answer: 2023/936

Correct? Yes

🔗 Problem

Problem: What is the sum of 85/75 and 88/56? Write your answer in decimal form.

Gemma 2B:

    To find the sum, we first need to find a common denominator for the fractions. The least common multiple of 75 and 56 is 2100.

    *  85/75 is equivalent to (85 * 28) / (75 * 28) = 2380/2100
    *  88/56 is equivalent to (88 * 37.5) / (56 * 37.5) = 3300/2100

    Now we can add the fractions:

    2380/2100 + 3300/2100 = 5680/2100

    Finally, we convert this fraction to decimal form:

    5680/2100 = 2.70476190476

    ### Answer: 2.70476190476

Expected answer: 284/105

Correct? No

🔗 Problem

Problem: What is the result of adding 12 over 33 to 49 over 2? Write your answer as a mixed fraction.

Gemma 2B:

    Reasoning: To add fractions, they must have a common denominator. The least common multiple of 33 and 2 is 66. We can rewrite the fractions with this denominator:
    * 12/33 = (12/33)*(2/2) = 24/66
    * 49/2 = (49/2)*(33/33) = 1617/66

    Now we can add the fractions: 24/66 + 1617/66 = (24+1617)/66 = 1641/66.

    We can convert this improper fraction to a mixed number by dividing 1641 by 66. 1641/66 = 24 with a remainder of 57.

    Therefore, the result is 24 57/66.

    ### Answer: 24 57/66

Expected answer: 547/22

Correct? Yes

🔗 Follow-up

Problem: What is the result when you add the fraction 17 over 94 to the solution obtained from the previous calculation (i.e., the sum of 12 over 33 and 49 over 2)? Write your answer as a mixed fraction.

Gemma 2B:

    Reasoning:  We already found that 12/33 + 49/2 = 24 57/66. To add 17/94 to this, we need a common denominator. The least common multiple of 66 and 94 is 3003.

    Let's rewrite the fractions:

    * 24 57/66 = 24 + (57/66) * (47/47) = 24 + 2679/3003
    * 17/94 = (17/94) * (32/32) = 544/3003

    Now we can add:

    24 + 2679/3003 + 544/3003 = 24 + (2679 + 544)/3003 = 24 + 3223/3003

    Since 3223 is greater than 3003, we can simplify this to a mixed number:

    24 + (3223/3003) = 24 + 1 220/3003 = 25 220/3003

    ### Answer: 25 220/3003

Expected answer: 12948/517

Correct? No

🔗 Follow-up

Problem: What if we originally started with 12 out of 84 parts instead of 33 parts, as in the first scenario? What would the sum of this fraction and 49 over 2 be? State your answer as a mixed fraction.

Gemma 2B:

    Reasoning:  Let's simplify the fraction 12/84 before we start. Both 12 and 84 are divisible by 12, so we can simplify the fraction to 1/7.

    Now, we need to find a common denominator for 1/7 and 49/2. The least common multiple of 7 and 2 is 14.

    * 1/7 = (1/7) * (2/2) = 2/14
    * 49/2 = (49/2) * (7/7) = 343/14

    Adding the fractions: 2/14 + 343/14 = (2 + 343)/14 = 345/14.

    We can convert this improper fraction to a mixed number by dividing 345 by 14. 345/14 = 24 with a remainder of 9.

    Therefore, the sum of 12/84 and 49/2 is 24 9/14.

    ### Answer: 24 9/14

Expected answer: 345/14

Correct? Yes

🔗 Problem

Problem: What is the result of adding two fractions, where each fraction is 78/65?

Gemma 2B:

    Here's how to add two fractions that are both 78/65:

    * **Understanding the Problem:** We're essentially multiplying the fraction 78/65 by 2.
    * **Adding Fractions:** Since the fractions have the same denominator (65), we can simply add the numerators: 78 + 78 = 156
    * **Simplifying:** The resulting fraction is 156/65. This can be simplified by dividing both numerator and denominator by their greatest common factor (13): 156/13 = 12, and 65/13 = 5.

    Therefore, the result is 12/5.

    ### Answer: 12/5

Expected answer: 12/5

Correct? Yes

🔗 Follow-up

Problem: After obtaining the result from the previous problem, where you added two fractions both equal to 78/65, what would you get if you add the result to 71/32?

Gemma 2B:

    We need to add the result from the previous problem (12/5) to 71/32.

    **1. Finding a Common Denominator:**

    * The least common multiple of 5 and 32 is 160.
    *  To get a denominator of 160 for 12/5, multiply both numerator and denominator by 32: (12 * 32) / (5 * 32) = 384/160
    * To get a denominator of 160 for 71/32, multiply both numerator and denominator by 5: (71 * 5) / (32 * 5) = 355/160

    **2. Adding the Fractions:**

    * Now that both fractions have the same denominator, we can add them: (384/160) + (355/160) = (384 + 355) / 160 = 739/160

    **3. Simplifying (Optional):**

    * The fraction 739/160 can be simplified to a mixed number: 4 99/160

    Therefore, the result of adding the sum of the two 78/65 fractions (12/5) to 71/32 is 739/160 or 4 99/160.

    ### Answer: 739/160 or 4 99/160

Expected answer: 739/160

Correct? Yes

🔗 Follow-up

Problem: Suppose that instead of having two fractions both equal to 78/65 like in the previous problem, they were both equal to 78/78. What would be the result of adding these two fractions together?

Gemma 2B:

    Here's how to solve this:

    * **Understanding the Problem:**  78/78 is equivalent to 1 (any number divided by itself equals 1).  We are essentially adding 1 + 1.
    * **Adding Whole Numbers:** 1 + 1 = 2

    Therefore, the result of adding two fractions, each equal to 78/78, is 2.

    ### Answer: 2

Expected answer: 2

Correct? Yes

🔗 Problem

Problem: Calculate the sum of 16/7 and 11/90. Answer as a simplified fraction.

Gemma 2B:

    To add fractions, they must have a common denominator.  The least common multiple of 7 and 90 is 630.

    * Convert 16/7 to an equivalent fraction with a denominator of 630: (16/7) * (90/90) = 1440/630

    * Convert 11/90 to an equivalent fraction with a denominator of 630: (11/90) * (7/7) = 77/630

    Now we can add the fractions:

    1440/630 + 77/630 = 1517/630

    This fraction can be simplified by dividing both numerator and denominator by their greatest common factor, which is 7:

    1517/630 = (1517/7) / (630/7) = 217/90

    ### Answer: 217/90

Expected answer: 1517/630

Correct? No