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

Llama 3 70B:

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

Now, we multiply 152/27 by 2:

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

So, the new product is 304/27.

### Answer: 304/27

Expected answer: 304/27

Correct? Yes

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

Llama 3 70B:

Let's correct the mistake.

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

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

So, the product is 152/45.

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

152/45

### Answer: 152/45

Expected answer: 152/45

Correct? Yes

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

Llama 3 70B:

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

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

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

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

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

### Answer: 85/14

Expected answer: 85/14

Correct? Yes

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

Llama 3 70B:

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

12 × (7/16) = (12 × 7) / 16 = 84 / 16

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

84 ÷ 4 = 21
16 ÷ 4 = 4

So, the simplified fraction is 21/4.

### Answer: 21/4

Expected answer: 21/4

Correct? Yes

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

Llama 3 70B:

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

### Answer: 11

Expected answer: 11

Correct? Yes

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

Llama 3 70B:

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

(14/5) × (12/14) = ?

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

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

So, the result is: 168/70

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

168 ÷ 14 = 12
70 ÷ 14 = 5

So, the simplified fraction is: 12/5

### Answer: 12/5

Expected answer: 12/5

Correct? Yes

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

Llama 3 70B:

Let's recalculate the expression with the new fraction:

1. Multiply 14/5 by 12/4:

(14/5) × (12/4) = ?

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

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

168 ÷ 4 = 42
20 ÷ 4 = 5

So, the simplified fraction is: 42/5

### Answer: 42/5

Expected answer: 42/5

Correct? Yes

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

Llama 3 70B:

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

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

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

### Answer: 252/17

Expected answer: 252/17

Correct? Yes

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

Llama 3 70B:

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

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

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

### Answer: 38/17

Expected answer: 38/17

Correct? Yes

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

Llama 3 70B:

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

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

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

### Answer: 16/3

Expected answer: 16/3

Correct? Yes

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

Llama 3 70B:

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

Expected answer: 4

Correct? Yes

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

Llama 3 70B:

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

Expected answer: 48/11

Correct? No

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

Llama 3 70B:

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

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

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

(14 × 14) / (9 × 18) = 196 / 162

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

196 ÷ 2 = 98
162 ÷ 2 = 81

So, the new total amount of ingredients is:

98 / 81 cups

### Answer: 98/81

Expected answer: 98/81

Correct? Yes

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

Llama 3 70B:

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

In this case, we have:

(14/7) × (13/4) = (14 × 13) / (7 × 4)

= 182 / 28

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

182 ÷ 14 = 13
28 ÷ 14 = 2

So, the result is:

13/2

### Answer: 13/2

Expected answer: 13/2

Correct? Yes

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

Llama 3 70B:

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

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

= 48841 / 20736

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

### Answer: 48841/20736

Expected answer: 48841/20736

Correct? Yes

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

Llama 3 70B:

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

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

### Answer: 16.875

Expected answer: 135/8

Correct? Yes

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

Llama 3 70B:

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

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

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

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

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

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

### Answer: 9 pounds

Expected answer: 9

Correct? Yes

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

Llama 3 70B:

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

Expected answer: 21/4

Correct? No

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

Llama 3 70B:

We multiply the fraction 19/6 by itself:

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

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

361 ÷ 1 = 361
36 ÷ 1 = 36

So, the simplified fraction is:

361/36

### Answer: 361/36

Expected answer: 361/36

Correct? Yes

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

Llama 3 70B:

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

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

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

171 ÷ 3 = 57
24 ÷ 3 = 8

So, the simplified fraction is:

57/8

### Answer: 57/8

Expected answer: 57/8

Correct? Yes

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

Llama 3 70B:

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

Expected answer: 20/17

Correct? Yes

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

Llama 3 70B:

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

Expected answer: 16

Correct? Yes

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

Llama 3 70B:

I'll multiply the result by 7:

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

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

11.454545... = 11 5/11

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

Expected answer: 126/11

Correct? Yes

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

Llama 3 70B:

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

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

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

6.545454... = 72/11

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

Expected answer: 72/11

Correct? Yes

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

Llama 3 70B:

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

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

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

8281 ÷ 7 = 1183
3025 ÷ 7 = 433

So, the simplified fraction is:

### Answer: 1183/433

Expected answer: 8281/3025

Correct? No

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

Llama 3 70B:

Let's evaluate the expression:

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

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

### Answer: 91/20

Expected answer: 91/20

Correct? Yes

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

Llama 3 70B:

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

16 × 2 = 32
32 ÷ 11 = 32/11

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

Expected answer: 32/11

Correct? Yes

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

Llama 3 70B:

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

### Answer: 11

Expected answer: 11

Correct? Yes

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

Llama 3 70B:

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

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

e × (9/15) = 9 × (9/15) = 81/15

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

81 ÷ 3 = 27
15 ÷ 3 = 5

So, the simplified result is:

### Answer: 27/5

Expected answer: 27/5

Correct? Yes

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

Llama 3 70B:

Let's evaluate the modified expression:

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

### Answer: 5

Expected answer: 5

Correct? Yes

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

Llama 3 70B:

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

Expected answer: 95/6

Correct? No

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

Llama 3 70B:

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

Expected answer: 120/7

Correct? Yes

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

Llama 3 70B:

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

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

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

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

462 ÷ 2 = 231
26 ÷ 2 = 13

So, the simplified result is: 231/13

### Answer: 231/13

Expected answer: 231/13

Correct? Yes

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

Llama 3 70B:

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

So, we get: (18 × 10) / 12 = 180 / 12

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

180 ÷ 6 = 30
12 ÷ 6 = 2

So, the simplified product is: 30/2

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

30 ÷ 2 = 15
2 ÷ 2 = 1

So, the final simplified product is: 15

### Answer: 15

Expected answer: 15

Correct? Yes

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

Llama 3 70B:

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

(3.25)² = 3.25 × 3.25 = 10.5625

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

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

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

### Answer: 10.5625 or 1681/169.

Expected answer: 169/16

Correct? Yes

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

Llama 3 70B:

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

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

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

39 ÷ 3 = 13
15 ÷ 3 = 5

So, the result of the calculation is:

39/15 = 13/5 = 2.6

### Answer: 2.6 or 13/5.

Expected answer: 13/5

Correct? Yes

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

Llama 3 70B:

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

8 × 2 = ?

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

8 + 8 = ?

Adding 8 and 8 together gives me:

16

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

### Answer: 16

Expected answer: 16

Correct? Yes

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

Llama 3 70B:

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

16 × 1 = ?

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

16

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

### Answer: 16

Expected answer: 16

Correct? Yes

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

Llama 3 70B:

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

Expected answer: 36/5

Correct? No

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

Llama 3 70B:

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

Expected answer: 17

Correct? Yes

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

Llama 3 70B:

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

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

16 × 3 = 48
48 ÷ 8 = 6

So, the result is 6.

### Answer: 6

Expected answer: 6

Correct? Yes

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

Llama 3 70B:

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

Expected answer: 15

Correct? Yes

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

Llama 3 70B:

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

11 × 2/19 = 22/19

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

### Answer: 22/19

Expected answer: 22/19

Correct? Yes

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

Llama 3 70B:

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

10 × 11 = 110

Now, divide 110 by 17:

110 ÷ 17 = 110/17 = 6 8/17

### Answer: 6 8/17

Expected answer: 110/17

Correct? Yes

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

Llama 3 70B:

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

10 × (8/14) = 80/14

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

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

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

720 ÷ 12 = 60
84 ÷ 12 = 7

So, the simplified fraction is:

60/7

As a decimal, this is equivalent to:

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

### Answer: 60/7 or 8.57

Expected answer: 60/7

Correct? Yes

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

Llama 3 70B:

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

18 × (8/14) = 18 × 8/14 = 144/14

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

144 ÷ 2 = 72
14 ÷ 2 = 7

So, the simplified fraction is:

72/7

As a decimal, this is equivalent to:

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

### Answer: 72/7 or 10.29

Expected answer: 72/7

Correct? Yes

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

Llama 3 70B:

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

Expected answer: 32/7

Correct? Yes

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

Llama 3 70B:

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

Expected answer: 8

Correct? Yes

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

Llama 3 70B:

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

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

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

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

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

### Answer: 15 1/6

Expected answer: 91/6

Correct? Yes

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

Llama 3 70B:

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

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

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

And we multiply the denominators:
44 × 19 = 836

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

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

So, the simplified fraction is:
459/209

### Answer: 459/209

Expected answer: 459/209

Correct? Yes

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

Llama 3 70B:

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

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

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

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

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

So, the simplified fraction is:
85/44

### Answer: 85/44

Expected answer: 85/44

Correct? Yes

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

Llama 3 70B:

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

Expected answer: 7/5

Correct? No

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

Llama 3 70B:

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

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

8 × 9 = 72

72 ÷ 8 = 9

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

### Answer: 9

Expected answer: 9

Correct? Yes

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

Llama 3 70B:

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

### Answer: 6

Expected answer: 6

Correct? Yes

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

Llama 3 70B:

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

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

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

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

### Answer: 27/17 cups

Expected answer: 27/17

Correct? Yes

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

Llama 3 70B:

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

1. Multiply 18 by the fraction 5/14:

18 × (5/14) = 18 × 5 / 14

2. Multiply 18 by 5:

18 × 5 = 90

3. Divide the result by 14:

90 ÷ 14 = 90/14

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

90 ÷ 2 = 45
14 ÷ 2 = 7

The simplified fraction is:

45/7

### Answer: 45/7

Expected answer: 45/7

Correct? Yes

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

Llama 3 70B:

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

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

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

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

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

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

### Answer: 105/104

Expected answer: 105/104

Correct? Yes

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

Llama 3 70B:

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

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

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

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

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

### Answer: 1/10

Expected answer: 45/8

Correct? No

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

Llama 3 70B:

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

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

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

### Answer: 3

Expected answer: 3

Correct? Yes

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

Llama 3 70B:

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

### Answer: 1

Expected answer: 1

Correct? Yes

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

Llama 3 70B:

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

12 × (9/13) = 108/13

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

### Answer: 108/13 cups

Expected answer: 108/13

Correct? Yes

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

Llama 3 70B:

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

(16 4/11) × (4/5) = ?

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

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

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

### Answer: 36/11

Expected answer: 144/11

Correct? No

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

Llama 3 70B:

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

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

### Answer: 5 8/11

Expected answer: 63/11

Correct? Yes

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

Llama 3 70B:

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

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

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

### Answer: 1.92

Expected answer: 48/25

Correct? No

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

Llama 3 70B:

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

(247/187) × (247/187)

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

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

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

### Answer: 61009/34969

Expected answer: 61009/34969

Correct? Yes

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

Llama 3 70B:

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

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

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

(19 × 13) / (8 × 17) = 247 / 136

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

### Answer: 247/136

Expected answer: 247/136

Correct? Yes

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

Llama 3 70B:

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

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

### Answer: 4 10/11 feet

Expected answer: 54/11

Correct? Yes

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

Llama 3 70B:

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

### Answer: 7

Expected answer: 7

Correct? Yes

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

Llama 3 70B:

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

### Answer: 19

Expected answer: 19

Correct? Yes

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

Llama 3 70B:

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

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

### Answer: 6.5

Expected answer: 13/2

Correct? Yes

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

Llama 3 70B:

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

1 × 4 = 4

### Answer: 4

Expected answer: 4

Correct? Yes

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

Llama 3 70B:

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

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

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

### Answer: 121/25

Expected answer: 121/25

Correct? Yes

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

Llama 3 70B:

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

(9/18) × (11/5) = (9 × 11) / (18 × 5)

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

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

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

### Answer: 11/10

Expected answer: 11/10

Correct? Yes

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

Llama 3 70B:

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

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

12 = 12/1

Then, we can multiply the fractions:

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

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

36 ÷ 3 = 12
15 ÷ 3 = 5

So, the result of the calculation is:

12/5

### Answer: 12/5

Expected answer: 12/5

Correct? Yes

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

Llama 3 70B:

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

So, we multiply 4 by 6:

4 × 6 = 24

The denominator remains 5, so our result is:

24/5

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

### Answer: 24/5

Expected answer: 24/5

Correct? Yes

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

Llama 3 70B:

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

(95/48) × (95/48) = ?

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

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

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

### Answer: 9025/2304

Expected answer: 9025/2304

Correct? Yes

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

Llama 3 70B:

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

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

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

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

380 ÷ 4 = 95
144 ÷ 4 = 36

So, the simplified result is 95/36.

### Answer: 95/36

Expected answer: 95/36

Correct? Yes

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

Llama 3 70B:

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

f × (12/11) = (289/260) × (12/11)

= (289 × 12) / (260 × 11)

= 3478 / 2860

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

1739 / 1430

### Answer: 1739/1430

Expected answer: 867/715

Correct? No

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

Llama 3 70B:

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

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

= 272 / 260

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

68 / 65

### Answer: 68/65

Expected answer: 68/65

Correct? Yes

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

Llama 3 70B:

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

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

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

### Answer: 5 5/12

Expected answer: 65/12

Correct? Yes

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

Llama 3 70B:

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

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

### Answer: 6 1/2

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

Expected answer: 13/2

Correct? Yes

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

Llama 3 70B:

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

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

17 × 0.75 = 12.75

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

### Answer: 12

Expected answer: 51/4

Correct? No

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

Llama 3 70B:

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

17 × 13 = 221

Now, divide by the denominator:

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

### Answer: 12.28

Expected answer: 221/18

Correct? No

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

Llama 3 70B:

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

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

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

114 ÷ 2 = 57
10 ÷ 2 = 5

So, the simplified result is:

57/5

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

Expected answer: 57/5

Correct? Yes

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

Llama 3 70B:

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

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

(15 × 20) / 15 = 300 / 15

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

300 ÷ 15 = 20
15 ÷ 15 = 1

So, the simplified result is:

20/1 = 20

### Answer: 20

Expected answer: 20

Correct? Yes

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

Llama 3 70B:

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

Expected answer: 2

Correct? Yes

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

Llama 3 70B:

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

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

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

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

Expected answer: 80/19

Correct? No

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

Llama 3 70B:

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

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

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

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

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

140 ÷ 4 = 35
84 ÷ 4 = 21

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

Expected answer: 5/3

Correct? Yes

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

Llama 3 70B:

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

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

4 × 5 = 20
20 ÷ 2 = 10

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

### Answer: 10

Expected answer: 10

Correct? Yes

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

Llama 3 70B:

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

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

1 × 1 = 1

So, the square of the result is still 1.

### Answer: 1

Expected answer: 1

Correct? Yes

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

Llama 3 70B:

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

So, we get: (17 × 3) / 18 = 51 / 18

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

51 ÷ 3 = 17
18 ÷ 3 = 6

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

### Answer: 17/6

Expected answer: 17/6

Correct? Yes

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

Llama 3 70B:

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

### Answer: 8.125

Expected answer: 65/8

Correct? No

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

Llama 3 70B:

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

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

### Answer: 5.83

Expected answer: 35/6

Correct? No

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

Llama 3 70B:

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

### Answer: 5 5/13

Expected answer: 70/13

Correct? Yes

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

Llama 3 70B:

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

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

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

### Answer: 17 1/7

Expected answer: 120/7

Correct? Yes

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

Llama 3 70B:

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

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

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

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

### Answer: 12

Expected answer: 12

Correct? Yes

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

Llama 3 70B:

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

1 × 6

Multiplying 1 by 6 gives us:

6

So, the product is simply 6.

### Answer: 6

Expected answer: 6

Correct? Yes

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

Llama 3 70B:

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

8 × 3 = 24
24 ÷ 2 = 12

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

### Answer: 12

Expected answer: 12

Correct? Yes

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

Llama 3 70B:

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

20 × 1 = 20
20 ÷ 1 = 20

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

### Answer: 20

Expected answer: 20

Correct? Yes

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

Llama 3 70B:

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

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

### Answer: 10.4

Expected answer: 52/5

Correct? Yes

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

Llama 3 70B:

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

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

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

### Answer: 36/25

Expected answer: 36/25

Correct? Yes

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

Llama 3 70B:

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

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

So, 1 × 6/6 = 6/6.

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

6 ÷ 6 = 1
6 ÷ 6 = 1

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

### Answer: 1

Expected answer: 1

Correct? Yes

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

Llama 3 70B:

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

Expected answer: 16/5

Correct? Yes

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

Llama 3 70B:

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

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

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

54 ÷ 6 = 9
12 ÷ 6 = 2

So, the simplified fraction is:

### Answer: 9/2

Expected answer: 9/2

Correct? Yes

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

Llama 3 70B:

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

Expected answer: 52/7

Correct? Yes

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

Llama 3 70B:

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

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

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

= 120 / 24

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

= 120 ÷ 12 / 24 ÷ 12
= 10 / 2
= 5

So, the value of the expression is 5.

### Answer: 5

Expected answer: 5

Correct? Yes

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

Llama 3 70B:

Let's break this down step by step!

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

2 × 6 = 12

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

(19/14) × (2 × 6)

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

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

Now, let's calculate the product:

19 × 12 = 228

So, we have:

(228) / 14

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

(228 ÷ 2) / (14 ÷ 2) = 114 / 7

### Answer: 114/7

Expected answer: 114/7

Correct? Yes

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

Llama 3 70B:

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

3 × 6 = 18

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

### Answer: 18

Expected answer: 18

Correct? Yes

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

Llama 3 70B:

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

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

So, the value of 'a' is:

### Answer: 4.9

Expected answer: 49/10

Correct? No

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

Llama 3 70B:

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

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

So, the final answer is:

### Answer: 2.3338 (approximately)

Expected answer: 7/3

Correct? No

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

Llama 3 70B:

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

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

(45 × 9) / (7 × 7) = 405 / 49

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

### Answer: 405/49

Expected answer: 405/49

Correct? Yes

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

Llama 3 70B:

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

(5 × 17) / 14 = 85 / 14

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

So, the result is:

85 / 14

### Answer: 85/14

Expected answer: 85/14

Correct? Yes

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

Llama 3 70B:

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

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

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

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

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

55 ÷ 5 = 11
40 ÷ 5 = 8

So, the simplified fraction is 11/8.

### Answer: 11/8.

Expected answer: 11/8

Correct? Yes

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

Llama 3 70B:

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

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

### Answer: 99/8.

Expected answer: 99/8

Correct? Yes