Problem: Let's reconsider the previous question. Instead of multiplying 7 by the result of dividing 27 by 12, calculate the result when you multiply 7 by the division of 26 by 12. Write your answer as a simplified fraction.

Llama 3 8B:

Reasoning: First, we need to divide 26 by 12.
26 ÷ 12 = 2.17
Now, we multiply 7 by the result:
7 × 2.17 = 15.19
To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 1.
So, the simplified fraction is:
15.19 = 15.19/1
### Answer: 15 19/19

Expected answer: 91/6

Correct? No

Problem: After finding the result of ((24 / 22) * (24 / 17)) in the previous problem, multiply the answer you found by (27 / 18). Write the result as a simplified fraction.

Llama 3 8B:

Reasoning: We already found the result of ((24 / 22) * (24 / 17)) to be 288/187.
Now, we multiply the result by (27 / 18):
(288/187) * (27/18) = (288*27)/(187*18) = 7776/3342
We can simplify the fraction by dividing both numerator and denominator by their greatest common divisor, which is 6:
7776/3342 = (7776/6)/(3342/6) = 1296/558
### Answer: 1296/558

Expected answer: 432/187

Correct? No

Problem: Since the previous problem had you evaluating ((24 / 22) * (24 / 17)), consider this change: What if instead of multiplying (24 / 22) by (24 / 17), you multiplied it by (24 / 9)? Calculate the result of ((24 / 22) * (24 / 9)) and write your answer as a simplified fraction.

Llama 3 8B:

Reasoning: First, we calculate the innermost operations:
(24 / 22) = 24/22 = 12/11
(24 / 9) = 24/9 = 8/3
Now, we multiply the two results:
(12/11) * (8/3) = (12*8)/(11*3) = 96/33
### Answer: 96/33

Expected answer: 32/11

Correct? Yes

Problem: Following the previous question, you found the result of (17 / 10) * (28 / 10). Now, multiply that result by (11 / 18). What will the new result be? Write your answer in decimal form.

Llama 3 8B:

Reasoning: We already found the result of (17 / 10) * (28 / 10) to be 4.76. Now, we need to multiply this result by (11 / 18):

1. Divide 11 by 18: 11 / 18 = 0.61
2. Multiply the result by 0.61: 4.76 * 0.61 = 2.9136

### Answer: 2.9136

Expected answer: 1309/450

Correct? No

Problem: In the previous problem, you were asked to calculate the result of (17 / 10) * (28 / 10). But consider this variation: calculate the result of the expression (17 / 24) * (28 / 10) instead. What would be the new result? Write your answer in decimal form.

Llama 3 8B:

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

1. Divide 17 by 24: 17 / 24 = 0.708333 (approximately)
2. Divide 28 by 10: 28 / 10 = 2.8
3. Multiply the results: 0.708333 * 2.8 = 1.984936 (approximately)

### Answer: 1.98

Expected answer: 119/60

Correct? No

Problem: After calculating the solution for ((18 / 19) * (19 / 6)), take the result and multiply it by itself. What do you get? Write your answer as a simplified fraction.

Llama 3 8B:

We previously calculated that:

((18 / 19) * (19 / 6)) = 57/19

Now, let's multiply this result by itself:

(57/19) × (57/19) = (57 × 57) / (19 × 19)

= 3249 / 361

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 9:

(3249 / 361) = (363 / 41)

So, the result of multiplying the original expression by itself is:

### Answer: 363/41

Expected answer: 9

Correct? No

Problem: After you've calculated the solution for ((18 / 19) * (19 / 6)), now consider what would happen if the fraction was instead ((9 / 19) * (19 / 6)). What would be the result? Write your answer as a simplified fraction.

Llama 3 8B:

Let's calculate the result for ((9 / 19) * (19 / 6)):

((9 / 19) * (19 / 6)) = ((9 * 19) / (19 * 6)) = (171 / 114)

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

(171 / 114) = (57 / 38)

So, the simplified result is:

### Answer: 57/38

Expected answer: 3/2

Correct? Yes

Problem: Taking the answer from the previous problem (19/12 * 20/6), multiply this result with itself. What is the resulting fraction when simplified?

Llama 3 8B:

We are given the result of the previous problem:

190/36

To multiply this result by itself, we square it:

(190/36) × (190/36) = (190 × 190) / (36 × 36)

= 36100 / 1296

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

36100 ÷ 4 = 9025
1296 ÷ 4 = 324

So, the simplified fraction is:

9025 / 324

### Answer: 9025/324

Expected answer: 9025/324

Correct? Yes

Problem: In the original problem, you multiplied the fractions 19/12 and 20/6. Now suppose you replace 19/12 with 10/12 and redo the calculation. What is the resulting fraction when it is simplified?

Llama 3 8B:

Let's redo the calculation with the new fraction 10/12:

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

= 200 / 72

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

200 ÷ 4 = 50
72 ÷ 4 = 18

So, the simplified fraction is:

50 / 18

### Answer: 50/18

Expected answer: 25/9

Correct? Yes

Problem: After finding the product of the fractions 5/27 and 15/2, take that result and multiply it by itself. What is the result when you write it in simplest form?

Llama 3 8B:

Reasoning: We previously found that the product of the fractions 5/27 and 15/2 is 25/18.
To square this result, we multiply it by itself:
(25/18) × (25/18) = 25^2 / 18^2 = 625 / 324.
To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 1.
625/1 = 625 and 324/1 = 324.
So, the simplified result is 625/324.
### Answer: 625/324

Expected answer: 625/324

Correct? Yes

Problem: After previously calculating the product of the fractions 5/27 and 15/2, what if the original number of the numerator in the first fraction was 24 instead of 5? Calculate the new product of the fractions 24/27 and 15/2 and write your answer in simplest form.

Llama 3 8B:

Reasoning: To calculate the new product of the fractions, we multiply the numerators and multiply the denominators.
The product of the numerators is 24*15 = 360.
The product of the denominators is 27*2 = 54.
So, the product of the fractions is 360/54.
To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 18.
360/18 = 20 and 54/18 = 3.
So, the simplified product is 20/3.
### Answer: 20/3

Expected answer: 20/3

Correct? Yes

Problem: After finding the result of (12 / 25) * (20 / 4), now multiply this result by (17 / 17). Report your final answer as a simplified fraction.

Llama 3 8B:

Reasoning: We already calculated the result of (12 / 25) * (20 / 4) as 12/5. Now, we multiply this result by (17 / 17):

(12/5) * (17/17) = (12*17)/(5*17) = 204/85

To simplify the result, we can divide both the numerator and the denominator by their greatest common divisor, which is 17:

204/85 = (204/17)/(85/17) = 12/5

### Answer: 12/5

Expected answer: 12/5

Correct? Yes

Problem: After resolving the previous problem, let us consider a slightly different situation. Now, calculate the result of (10 / 25) * (20 / 4) instead of (12 / 25) * (20 / 4). What difference does this make to the result? Write your new answer as a simplified fraction.

Llama 3 8B:

Reasoning: First, we calculate the division operations inside the parentheses:

(10 / 25) = 0.4
(20 / 4) = 5

Then, we multiply the two results:

0.4 * 5 = 2

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

2 = 2/1 = (2/2)/(1/1) = 1/0.5 = 2/1

### Answer: 2/1

Expected answer: 2

Correct? Yes

Problem: After calculating the multiplication of 14/10 by 30/5, multiply that result by the fraction 13/15. Write your answer as a simplified fraction.

Llama 3 8B:

We previously calculated the result of multiplying 14/10 by 30/5 to be 42/5.

Now, we multiply 42/5 by 13/15:

(42/5) × (13/15) = (42 × 13) / (5 × 15) = 546 / 75

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

546 ÷ 3 = 182
75 ÷ 3 = 25

So, the simplified fraction is:

182/25

### Answer: 182/25

Expected answer: 182/25

Correct? Yes

Problem: In the previous problem, suppose that instead of 14/10 we had 30/10. Now, the problem becomes: calculate the multiplication of 30/10 by 30/5. What would be the simplified fraction as an answer?

Llama 3 8B:

To multiply the fractions, we multiply the numerators (30 and 30) and multiply the denominators (10 and 5), then simplify the resulting fraction.

(30/10) × (30/5) = (30 × 30) / (10 × 5) = 900 / 50

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 50.

900 ÷ 50 = 18
50 ÷ 50 = 1

So, the simplified fraction is:

18/1

Which can be further simplified to:

18

### Answer: 18

Expected answer: 18

Correct? Yes

Problem: After solving for s in the previous problem, now calculate a new value o, where o = (9 / 13) * s. Write your answer as a simplified fraction.

Llama 3 8B:

We know that s = 1215/200. Now, we need to calculate o:

o = (9/13) * (1215/200)

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

o = (9*1215) / (13*200)

o = 10875 / 2600

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 25:

10875 / 2600 = 435 / 104

### Answer: 435/104

Expected answer: 2187/520

Correct? No

Problem: After having calculated the previous problem where you multiplied (27 / 10) * (9 / 4), consider what would happen if instead of multiplying by (9 / 4), you multiplied by (9 / 11). Calculate the new value of s = ((27 / 10) * (9 / 11)) and write your answer as a simplified fraction.

Llama 3 8B:

Let's evaluate the new expression:

1. Divide 27 by 10: 27 / 10 = 2.7
2. Divide 9 by 11: 9 / 11 = 0.81
3. Multiply the two results: 2.7 * 0.81 = 2.187

To simplify the result, we can write it as a fraction:

2.187 = 2187/1000

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

2187/1000 = 727/333

### Answer: 727/333

Expected answer: 243/110

Correct? No

Problem: Having solved the previous problem, now multiply your answer by the fraction 24/30. What is the result in simplest form?

Llama 3 8B:

We multiply the simplified fraction 34/9 by the fraction 24/30:

(34/9) × (24/30) = (34 × 24) / (9 × 30) = 816 / 270

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 6.

816 ÷ 6 = 136
270 ÷ 6 = 45

So, the simplified fraction is:

136/45

### Answer: 136/45

Expected answer: 136/45

Correct? Yes

Problem: If instead of multiplying the fraction 26/9 by the fraction 17/13, you multiplied 25/9 by 17/13, what would the result be in simplest form?

Llama 3 8B:

To multiply the fractions, we multiply the numerators (25 and 17) and multiply the denominators (9 and 13), then simplify the resulting fraction.

(25/9) × (17/13) = (25 × 17) / (9 × 13) = 425 / 117

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 13.

425 ÷ 13 = 32.69 (not a whole number)
117 ÷ 13 = 9

Since the numerator is not a whole number multiple of the denominator, we cannot simplify the fraction further. The result is already in simplest form.

### Answer: 425/117

Expected answer: 425/117

Correct? Yes

Problem: After solving for the result of the original expression (28 / 16) / (27 / 24), now multiply this result by itself. What is the outcome? Write your answer as a simplified fraction.

Llama 3 8B:

We previously found that the result of the original expression is 14/9. To multiply this result by itself, we need to square it:

(14/9) × (14/9) = (14 × 14) / (9 × 9) = 196/81

To simplify the fraction 196/81, we can divide both the numerator and the denominator by their greatest common divisor, which is 1:

196 ÷ 1 = 196
81 ÷ 1 = 81

So, the simplified fraction is:

196/81

### Answer: 196/81

Expected answer: 196/81

Correct? Yes

Problem: After you calculated ((15 / 3) * (19 / 26)), take that answer and multiply it by the fraction (14 / 24). Write your answer as a simplified fraction.

Llama 3 8B:

Reasoning: We already calculated the expression ((15 / 3) * (19 / 26)) and got 95/26.

Now, we multiply 95/26 by (14 / 24):

(95/26) * (14/24) = (95 * 14) / (26 * 24)

= 1330 / 624

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

1330 / 624 = 665 / 312

Since the numerator and denominator are not coprime, we can further simplify the fraction by dividing both by their greatest common divisor, which is 13:

665 / 312 = 65 / 31

### Answer: 65/31

Expected answer: 665/312

Correct? No

Problem: Let's consider a situation where instead of multiplying (15 / 3) by (19 / 26), you are now multiplying it by (27 / 26). What would be the result then? Write your answer as a simplified fraction.

Llama 3 8B:

Reasoning: First, we evaluate the expressions inside the parentheses:
(15 / 3) = 5
(27 / 26) = 27/26

Then, we multiply the two expressions:
5 * (27/26) = (5 * 27) / 26 = 135 / 26

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 1:
135 / 26 = 135/26

Since the numerator and denominator are already coprime, the fraction is already in its simplest form.

### Answer: 135/26

Expected answer: 135/26

Correct? Yes

Problem: After calculating the solution to ((15 / 13) * (24 / 24)), multiply your answer with (22 / 7) and write that answer as a simplified fraction.

Llama 3 8B:

We already calculated the solution to ((15 / 13) * (24 / 24)):

(15/13) * 1 = 15/13

Now, we need to multiply this result by (22 / 7):

(15/13) * (22/7) = (15*22)/(13*7) = 330/91

To simplify this fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 11:

330 ÷ 11 = 30
91 ÷ 11 = 8

So, the simplified solution is:

### Answer: 30/8

Expected answer: 330/91

Correct? No

Problem: After calculating the solution to ((15 / 2) * (24 / 24)), consider what would happen if you originally had 15 halves instead of 15 thirteenths. Multiply this new fraction with 1 (represented by (24 / 24)) and write your answer as a simplified fraction.

Llama 3 8B:

Let's calculate the solution:

1. Evaluate the expressions inside the parentheses:
	* (15 / 2) = 7.5 (since 15 divided by 2 is 7.5)
	* (24 / 24) = 1
2. Multiply the two expressions:
	* 7.5 * 1 = 7.5

Since we can't simplify 7.5 further, the answer remains:

### Answer: 7.5

Expected answer: 15/2

Correct? Yes

Problem: Using the answer you obtained from the previous problem, multiply it by 15/15. What simplified fraction do you get?

Llama 3 8B:

To multiply the fraction 15/14 by 15/15, we multiply the numerators and multiply the denominators:

(15/14) * (15/15) = (15 * 15) / (14 * 15)
= 225 / 210

Since the numerator and denominator are the same as before, the result is still:

225 / 210

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

225 / 210 = (225 ÷ 15) / (210 ÷ 15)
= 15 / 14

So, the result is still the same:

### Answer: 15/14

Expected answer: 15/14

Correct? Yes

Problem: The previous problem involved calculating (9 / 7) * (25 / 30). Now calculate (9 / 7) * (25 / 21) which is similar but with a slight change in the second fraction. What simplified fraction do you get in this scenario?

Llama 3 8B:

To multiply fractions, we multiply the numerators (9 and 25) and multiply the denominators (7 and 21), then simplify the result.

(9/7) * (25/21) = (9 * 25) / (7 * 21)
= 225 / 147

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 21.

225 / 147 = (225 ÷ 21) / (147 ÷ 21)
= 15 / 7

### Answer: 15/7

Expected answer: 75/49

Correct? No

Problem: If you take the result from the previous problem (30 divided by 18, multiplied by 18/6) and multiply this by 8/20, what is the resulting value?

Llama 3 8B:

Reasoning: We already found the result of dividing 30 by 18 and multiplying by 18/6 to be 5.
Now, we multiply 5 by the fraction 8/20, which gives us 5 × (8/20) = 5 × 2/5 = 10/5 = 2.
### Answer: 2

Expected answer: 2

Correct? Yes

Problem: If we adjust the original problem and now divide 30 by 4 and then multiply by the fraction 18/6, how does the result compares to our original calculation? What is the new value?

Llama 3 8B:

Reasoning: First, we divide 30 by 4, which gives us 30/4 = 7.5.
Then, we multiply 7.5 by the fraction 18/6, which gives us 7.5 × (18/6) = 7.5 × 3 = 22.5.
Comparing this to the original calculation, we see that the result is different because the initial division by 18 changed the value of the number being multiplied by the fraction.
### Answer: 22.5

Expected answer: 45/2

Correct? Yes

Problem: After calculating the result of ((20 / 2) * (2 / 18)), now find the answer for ((27 / 4) * that result). Write your final answer as a simplified fraction.

Llama 3 8B:

Reasoning: We already calculated the result of ((20 / 2) * (2 / 18)) to be 10/9.

Now, we need to multiply this result by ((27 / 4)):

((27 / 4) * (10/9)) = (27/4) * (10/9)

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

(27/4) * (10/9) = (27*10)/(4*9)

= 270/36

Now, we can simplify the fraction:

270/36 = 15/2

### Answer: 15/2

Expected answer: 15/2

Correct? Yes

Problem: After calculating the result of ((20 / 2) * (2 / 18)), reconsider your calculation, but this time using (15 / 18) instead of (2 / 18). Thus, find the answer for ((20 / 2) * (15 / 18)). Write your final answer as a simplified fraction.

Llama 3 8B:

Reasoning: First, we need to follow the order of operations (PEMDAS) and evaluate the expressions inside the parentheses.

((20 / 2) = 10)

So, the expression becomes: (10 * (15 / 18))

Now, we multiply 10 by 15/18:

10 * 15/18 = 5 * 15/9

We can simplify the fraction 15/9 by dividing both numerator and denominator by their greatest common divisor, which is 3:

5 * 5/3 = 25/3

### Answer: 25/3

Expected answer: 25/3

Correct? Yes

Problem: After finding the result of ((11 / 9) * (29 / 6)), multiply your answer by the fraction (11 / 5). Write your final answer as a simplified fraction.

Llama 3 8B:

First, let's multiply the result of ((11 / 9) * (29 / 6)) by the fraction (11 / 5):

(319/54) × (11/5) = (319 × 11) / (54 × 5) = 3491 / 270

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

3491/270

### Answer: 3491/270

Expected answer: 3509/270

Correct? No

Problem: After finding the result of ((21 / 9) * (29 / 6)), consider what would have happened if the initial calculation involved (21 / 9) instead of (11 / 9). Calculate the new result and write your answer as a simplified fraction.

Llama 3 8B:

Let's calculate the new result:

((21 / 9) * (29 / 6)) = ?

1. Multiply the numerators (21 and 29): 21 × 29 = 609
2. Multiply the denominators (9 and 6): 9 × 6 = 54
3. Write the result as a fraction: 609/54

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

207/18

### Answer: 207/18

Expected answer: 203/18

Correct? No

Problem: Following the previous calculation, take the product result of 25/22 and 20/3. Now multiply that result with 28/29 and write your final answer as a simplified fraction.

Llama 3 8B:

Reasoning: We previously calculated the product of 25/22 and 20/3 as 250/33. Now, we multiply this result by 28/29:

(250/33) × (28/29) = (250 × 28) / (33 × 29)

= 7000 / 957

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 1:

7000 ÷ 1 = 7000
957 ÷ 1 = 957

So, the simplified fraction is:

7000/957

### Answer: 7000/957

Expected answer: 7000/957

Correct? Yes

Problem: After reconsidering the calculation from the previous step, if we actually had 25/15 instead of 25/22. Recalculate the product with these new quantities. What is the product of 25/15 and 20/3? Write your answer as a simplified fraction.

Llama 3 8B:

Reasoning: To calculate the product, we multiply the numerators and multiply the denominators:

(25/15) × (20/3) = (25 × 20) / (15 × 3)

= 500 / 45

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 5:

500 ÷ 5 = 100
45 ÷ 5 = 9

So, the simplified fraction is:

100/9

### Answer: 100/9

Expected answer: 100/9

Correct? Yes

Problem: After calculating the solution for the previous question ((5 / 3) * (27 / 13)), multiply the result you obtained by (8 / 14) and simplify the fraction that you get. What is your answer?

Llama 3 8B:

We previously calculated the result of (5 / 3) * (27 / 13) to be 45/13.

Now, we need to multiply this result by (8 / 14):

(45/13) * (8/14) = ?

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

(45 * 8) / (13 * 14) = 360 / 182

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

360 ÷ 2 = 180
182 ÷ 2 = 91

So, the simplified fraction is:

### Answer: 180/91

Expected answer: 180/91

Correct? Yes

Problem: After considering the previous problem, there is a slight change. Instead of multiplying (5/3) by (27/13), let's make it a little easy and multiply (5/3) by (27/9). Calculate the new solution and write your answer as a simplified fraction.

Llama 3 8B:

Let's evaluate the new expression:

(5/3) * (27/9) = ?

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

(5 * 27) / (3 * 9) = 135/27

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 9:

135 ÷ 9 = 15
27 ÷ 9 = 3

So, the simplified fraction is:

### Answer: 15/3

Expected answer: 5

Correct? Yes

Problem: Now, use your answer from the last problem and multiply it by (27 / 27) to solve this follow-up problem. What is the result?

Llama 3 8B:

Reasoning: We already found that (29 / 29) * (1 / 1) = 1.
Now, we need to multiply 1 by (27 / 27).
When we divide 27 by 27, we get 1, because any number divided by itself is 1. So, (27 / 27) = 1.
Now, we multiply 1 by 1, which gives us 1.
### Answer: 1

Expected answer: 1

Correct? Yes

Problem: Now, let's change the problem a little bit. Instead of (29 / 29) * (1 / 1), let's use (29 / 6) * (1 / 1). What will the result be?

Llama 3 8B:

Reasoning: When we divide 29 by 6, we get 4.83 (approximately), because 29 is not exactly divisible by 6.
When we divide 1 by 1, we get 1, because any number divided by 1 is itself. So, (1 / 1) = 1.
Now, we multiply 4.83 by 1, which gives us 4.83.
### Answer: 4.83

Expected answer: 29/6

Correct? No

Problem: If you further multiply the result you got from the last problem ((21/30) * (19/5)) by 16/12, what would be the final result? Write your answer as a simplified fraction.

Llama 3 8B:

We already found that (21/30) * (19/5) = 133/50. Now, we need to multiply this result by 16/12.

(133/50) * (16/12) = (133 × 16) / (50 × 12)

= 2128 / 600

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 8.

2128 ÷ 8 = 266
600 ÷ 8 = 75

So, the simplified fraction is:

266/75

### Answer: 266/75

Expected answer: 266/75

Correct? Yes

Problem: If in the previous problem, instead of multiplying 21/30 by 19/5, you had to multiply 9/30 by 19/5, what would be the new result? Write your answer as a simplified fraction.

Llama 3 8B:

To multiply 9/30 by 19/5, we need to multiply the numerators (9 and 19) and multiply the denominators (30 and 5), then simplify the resulting fraction.

9/30 × 19/5 = (9 × 19) / (30 × 5) = 171 / 150

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

171 ÷ 3 = 57
150 ÷ 3 = 50

So, the simplified fraction is:

57/50

### Answer: 57/50

Expected answer: 57/50

Correct? Yes

Problem: Based on your previous calculation, what is the result when you multiply the answer you found for 11/15 * 23/8 by itself? Write your answer as a simplified fraction.

Llama 3 8B:

Let's multiply the result (253/120) by itself:

(253/120) × (253/120) = (253 × 253) / (120 × 120)

= 64389 / 14400

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

64389 ÷ 3 = 21463
14400 ÷ 3 = 4800

So, the simplified fraction is:

21463 / 4800

### Answer: 21463/4800

Expected answer: 64009/14400

Correct? No

Problem: In the previous problem, you multiplied the fractions 11/15 and 23/8. But what if instead, you multiply the fractions 11/15 and 26/8? What would be the result of this change? Write your answer as a simplified fraction.

Llama 3 8B:

To multiply the fractions, we multiply the numerators (11 and 26) and multiply the denominators (15 and 8), then simplify the result:

(11/15) × (26/8) = (11 × 26) / (15 × 8) = 286 / 120

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

286 / 120 = 143 / 60

Since the numerator and denominator are not coprime, we can further simplify the fraction by dividing both by their greatest common divisor, which is 1:

143 / 60 = 143 / 60

Since the numerator and denominator are already coprime, the fraction is already in its simplest form.

### Answer: 143/60

Expected answer: 143/60

Correct? Yes

Problem: After finding the result of (28 / 1) divided by (30 / 27), what is the new result when this is multiplied by (6 / 19)? Write your final answer as a simplified fraction.

Llama 3 8B:

Reasoning: We already found the result of (28 / 1) divided by (30 / 27) to be 25.2. Now, we need to multiply this result by (6 / 19):

25.2 × (6 / 19) = 25.2 × (6/19) = 151.2/19

To simplify the result, we can divide both the numerator and the denominator by their greatest common divisor, which is 1:

151.2/19 = 8

So, the result is 8.

### Answer: 8

Expected answer: 756/95

Correct? No

Problem: Suppose instead of the original calculation, we now have to perform (28 / 1) divided by 11, how would the result change? Write your updated answer as a simplified fraction.

Llama 3 8B:

Reasoning: We can simplify the original calculation:

(28 / 1) = 28

Now, we can divide 28 by 11:

28 ÷ 11 = 28/11

To simplify the result, we can divide both the numerator and the denominator by their greatest common divisor, which is 1:

28/11 = 28/11

So, the result is still an improper fraction.

### Answer: 28/11

Expected answer: 28/11

Correct? Yes

Problem: After finding the result of ((25 / 25) * (20 / 7)), what is the result when you multiply this product by itself? Write your answer as a simplified fraction.

Llama 3 8B:

Reasoning: We already found the result of ((25 / 25) * (20 / 7)) to be 20/7.

Now, we multiply this result by itself:
(20/7) * (20/7) = (20*20) / (7*7)

Simplifying the numerator and denominator:
(400) / (49) = 400/49

### Answer: 400/49

Expected answer: 400/49

Correct? Yes

Problem: After reevaluating the previous problem, if you found the result of ((25 / 25) * (11 / 7)) instead of ((25 / 25) * (20 / 7)), what would be the new result? Write your answer as a simplified fraction.

Llama 3 8B:

Reasoning: First, we simplify the numerator and denominator of the first fraction:
(25 / 25) = 1

Then, we simplify the numerator and denominator of the second fraction:
(11 / 7) = 11/7

Now, we multiply the two fractions:
1 * (11/7) = 11/7

### Answer: 11/7

Expected answer: 11/7

Correct? Yes

Problem: Suppose you initially started with 30 (instead of 5), which is equivalent to the fraction 30/1. You are then required to multiply this with the fraction 16/27. How does this change the result from the previous problem? Write your answer as a simplified fraction.

Llama 3 8B:

If we multiply 30/1 with 16/27, we get:

30/1 × 16/27 = (30 × 16) / (1 × 27) = 480/27

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

480 ÷ 3 = 160
27 ÷ 3 = 9

So, the simplified fraction is:

160/9

This is different from the previous result of 80/27, which was obtained by multiplying 5/1 with 16/27.

Expected answer: 160/9

Correct? Yes

Problem: You previously calculated the result of ((27 / 3) / (12 / 12)) and labeled that answer as 's'. Now, compute the result of multiplying your answer 's' by the fraction (21/17) and report the result as a simplified fraction. This new result will be labeled as 'b'.

Llama 3 8B:

Reasoning: We previously calculated the result of ((27 / 3) / (12 / 12)) as 9. Now, we multiply 9 by the fraction (21/17):

9 × (21/17) = (9 × 21) / 17 = 189 / 17

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

189 ÷ 3 = 63
17 ÷ 3 = 5.67 (not an integer, so we cannot simplify further)

So, the result of multiplying 9 by the fraction (21/17) is:
### Answer: 63/17

Expected answer: 189/17

Correct? No

Problem: Earlier, you calculated the result of ((27 / 3) / (12 / 12)) and labeled that answer as 's'. Consider the same scenario, but this time, instead of having 27 items, you only have 24. Calculate the result of (24 / (12 / 12)) and report the answer as a simplified fraction. What effect does this change have on the answer 's'?

Llama 3 8B:

Reasoning: First, we evaluate the expression inside the parentheses:
(12 / 12) = 1
So, the expression becomes:
24 / 1 = 24

Since 24 is an integer, it cannot be simplified further, so the final answer is:
### Answer: 24

Comparing this result to the original answer 's' (9), we can see that the change from 27 to 24 has a significant effect on the result. The original answer 's' was 9, but with the change to 24, the result becomes 24. This is because the value of the numerator (24) has decreased, which has a direct impact on the result of the division.

Expected answer: 24

Correct? Yes

Problem: If the result from the previous problem is multiplied by (18/18), what is the new result? Write your answer as a simplified fraction.

Llama 3 8B:

Reasoning: We multiply the result from the previous problem by (18/18):
(435/32) * (18/18) = (435*18)/(32*18)

Now, we simplify the fraction:
(435*18)/(32*18) = 435/32

Notice that the denominator (32*18) cancels out with the numerator (18), leaving us with the original result:
435/32

### Answer: 435/32

Expected answer: 435/32

Correct? Yes

Problem: In the previous problem, instead of multiplying (15/16) by (29/2), what would be the result if we multiply it by (13/2) instead? Write your answer as a simplified fraction.

Llama 3 8B:

Reasoning: We multiply (15/16) by (13/2):
(15/16) * (13/2) = (15*13)/(16*2)

Now, we simplify the fraction:
(15*13)/(16*2) = 195/32

### Answer: 195/32

Expected answer: 195/32

Correct? Yes

Problem: Following up on the last question, once you have found the result of dividing seven wholes by 15/9, multiply your answer by the fraction 10/5. What do you get as a simplified fraction?

Llama 3 8B:

We previously found that dividing 7/1 by 15/9 gives us 21/5.

To multiply 21/5 by 10/5, we can multiply the numerators and multiply the denominators:

(21 × 10) / (5 × 5) = 210 / 25

To simplify the fraction 210/25, we can divide both the numerator and the denominator by their greatest common divisor, which is 5.

210 ÷ 5 = 42
25 ÷ 5 = 5

So, the simplified fraction is:

42/5

### Answer: 42/5

Expected answer: 42/5

Correct? Yes

Problem: Previously, you evaluated the result of dividing seven wholes, or 7/1, by 15/9. Now, consider what would happen if instead of seven whole, there were twenty wholes, or 20/1. What would be the result if you divide twenty wholes by 15/9? Write your answer as a simplified fraction.

Llama 3 8B:

To divide 20/1 by 15/9, we can multiply 20/1 by the reciprocal of 15/9, which is 9/15.

20/1 × 9/15 = (20 × 9) / (1 × 15) = 180/15

To simplify the fraction 180/15, we can divide both the numerator and the denominator by their greatest common divisor, which is 15.

180 ÷ 15 = 12
15 ÷ 15 = 1

So, the simplified fraction is:

12/1

### Answer: 12/1

Expected answer: 12

Correct? Yes

Problem: You have already calculated the result of (23/10) multiplied by (9/1). Now, multiply this result by (1/16). What is your answer?

Llama 3 8B:

Reasoning: We already calculated the result of (23/10) multiplied by (9/1) to be 20.7. Now, we need to multiply this result by (1/16).

20.7 × (1/16) = 20.7/16

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

20.7/16 = 1.296875

### Answer: 1.296875

Expected answer: 207/160

Correct? No

Problem: Previously, you computed the result of (23/10) multiplied by (9/1). Now, consider what would have happened if instead of (9/1), you had (9/6). Multiply (23/10) by this new fraction. What is your answer?

Llama 3 8B:

Reasoning: To multiply two fractions, we multiply the numerators (23 and 9) and multiply the denominators (10 and 6), then simplify the result.

(23/10) × (9/6) = (23 × 9) / (10 × 6) = 207/60

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

207/60 = 69/20

### Answer: 69/20

Expected answer: 69/20

Correct? Yes

Problem: Continuing from the previous problem, if the result is then multiplied by the fraction 30/21, what is the final numerical outcome?

Llama 3 8B:

Reasoning: We already simplified the result to 24/3. Now, we multiply this fraction by the new fraction 30/21.

(24/3) × (30/21) = (24 × 30) / (3 × 21) = 720 / 63

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

720 ÷ 3 = 240
63 ÷ 3 = 21

So, the simplified fraction is 240/21.

### Answer: 240/21

Expected answer: 80/7

Correct? Yes

Problem: Continuing from the previous problem, if we replace the original fraction of 23/3 with 18/3 and keep the 24/23 fraction the same, what now happens to the numerical result when we perform the multiplication?

Llama 3 8B:

Reasoning: We are keeping the same fraction 24/23 and replacing the other fraction 23/3 with 18/3. When we multiply these fractions, we multiply the numerators (24 and 18) and multiply the denominators (23 and 3), then simplify the resulting fraction.

24/23 × 18/3 = (24 × 18) / (23 × 3) = 432 / 69

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

432 ÷ 3 = 144
69 ÷ 3 = 23

So, the simplified fraction is 144/23.

### Answer: 144/23

Expected answer: 144/23

Correct? Yes

Problem: Given that the result from the previous problem is c, calculate the result of the following formula: ((21 divided by 9) multiplied by c). Write your answer as a simplified fraction.

Llama 3 8B:

Reasoning: First, we need to calculate the division operation:
(21 divided by 9) = 21/9 = 7/3

Then, we multiply the result by c:
(7/3) × c = (7/3) × 4 = 28/3

### Answer: 28/3

Expected answer: 28/3

Correct? Yes

Problem: Assuming that in the previous problem, the initial amount of oranges was 5 but was changed to 5/15, how would this affect the outcome if it is multiplied by the same fraction (4/1)? Write your answer as a simplified fraction.

Llama 3 8B:

Reasoning: The initial amount of oranges was changed from 5 to 5/15, which is equivalent to multiplying the original amount by 1/3 (since 5/15 = 5 × 1/3).

So, the new amount of oranges is 1 × (5/15) = 1/3.

Then, we multiply the new amount by the same fraction (4/1):
(1/3) × (4/1) = 4/3

### Answer: 4/3

Expected answer: 4/3

Correct? Yes

Problem: Given that you previously calculated (23 / 26) * (16 / 1) as 'g', now find the value of (4 / 7) * g. Express your answer as a simplified fraction.

Llama 3 8B:

Reasoning: We previously calculated g as 184/13. Now, we need to multiply g by (4 / 7):
(4 / 7) * (184/13) = (4 * 184) / (7 * 13)
= 736 / 91
We can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 13:
736/91 = 56/7
### Answer: 56/7

Expected answer: 736/91

Correct? No

Problem: In the previous problem, you calculated (23 / 26) * (16 / 1) as 'g'. Now assume that instead of 23/26, you had 2/26. How would that change the value of 'g'? Compute g as (2 / 26) * (16 / 1) and express your answer as a simplified fraction.

Llama 3 8B:

Reasoning: First, we evaluate the expressions inside the parentheses:
(2 / 26) = 2/26
(16 / 1) = 16
Then, we multiply the two expressions:
(2/26) * (16) = 32/26
We can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
32/26 = 16/13
### Answer: 16/13

Expected answer: 16/13

Correct? Yes

Problem: Considering the previous problem, let's make a small change to it. Instead of multiplying 24 over 19, let's multiply 20 over 19 by 21 over 19 but without simplifying. Write your answer as a fraction.

Llama 3 8B:

To calculate the product, we need to multiply the numerators and multiply the denominators:

(20/19) × (21/19) = (20 × 21) / (19 × 19)

= 420 / 361

### Answer: 420/361

Expected answer: 420/361

Correct? Yes

Problem: Continue the previous problem with this next step: Multiply your answer by (21 / 16) and write the result as a simplified fraction.

Llama 3 8B:

We already found that (11 / 20) * (24 / 12) = 11/10. Now, we need to multiply this result by (21 / 16):

(11/10) * (21/16) = (11*21) / (10*16)

This gives us:

231 / 160

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 11:

(231 ÷ 11) / (160 ÷ 11) = 21 / 14

So, the simplified answer is:

### Answer: 21/14

Expected answer: 231/160

Correct? No

Problem: In the previous problem, you calculated (11 / 20) * (24 / 12). Now, consider what would happen if you were to calculate (11 / 6) * (24 / 12) instead. What would be the result, expressed as a simplified fraction?

Llama 3 8B:

Let's calculate the result:

(11 / 6) * (24 / 12) = (11 * 24) / (6 * 12)

This gives us:

(264) / (72)

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

(264 ÷ 12) / (72 ÷ 12) = 22 / 6

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

(22 ÷ 2) / (6 ÷ 2) = 11 / 3

So, the simplified answer is:

### Answer: 11/3

Expected answer: 11/3

Correct? Yes

Problem: You were given a fraction after calculating the result of (12 / 14) * (28 / 5) in a previous problem. Now multiply your result by 15/10. What new fraction do you get? Simplify your answer to the lowest terms.

Llama 3 8B:

Reasoning: We multiply the fraction 6/1.25 by 15/10:

(6/1.25) * (15/10) = (6*15)/(1.25*10) = 90/12.5

To simplify the resulting fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 5:

90 ÷ 5 = 18
12.5 ÷ 5 = 2.5

So, the simplified result is:

18/2.5

### Answer: 18/2.5

Expected answer: 36/5

Correct? No

Problem: In the previous problem, you calculated the result of (12 / 14) * (28 / 5) and then multiplied your result by 15/10. Now consider the situation where instead of 12/14, you have 3/14. What is the result when you multiply 3/14 by 28/5? Simplify your answer to the lowest terms.

Llama 3 8B:

Reasoning: We can calculate the result of the multiplication:

(3/14) * (28/5) = (3*28)/(14*5) = 84/70

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

84 ÷ 14 = 6
70 ÷ 14 = 5

So, the simplified result is:

6/5

### Answer: 6/5

Expected answer: 6/5

Correct? Yes

Problem: After finding the result of ((15 / 3) * (7 / 25)), take that result and multiply it by (28 / 16). Write your answer as a simplified fraction.

Llama 3 8B:

Reasoning: We already found the result of ((15 / 3) * (7 / 25)) to be 7/5.

Now, we multiply this result by (28 / 16):
(7/5) * (28/16) = (7*28) / (5*16)

First, we multiply the numerators:
7*28 = 196

Then, we multiply the denominators:
5*16 = 80

So, the result is:
196/80

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 4:
196 ÷ 4 = 49
80 ÷ 4 = 20

So, the simplified fraction is:
49/20

### Answer: 49/20

Expected answer: 49/20

Correct? Yes

Problem: Given the result of the previous problem, let's consider a new scenario. Suppose instead of (15 / 3), we have (13 / 3). Now, multiply (13 / 3) * (7 / 25) and write your answer as a simplified fraction. How does this change in scenario affect the final result?

Llama 3 8B:

Reasoning: First, we evaluate the expressions inside the parentheses:
(13 / 3) = 13/3

Then, we multiply the two results:
(13/3) * (7/25) = 91/75

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 13:
91 ÷ 13 = 7
75 ÷ 13 = 5

So, the simplified fraction is:
7/5

Comparing this result to the previous one, we can see that the change in scenario (from 15/3 to 13/3) does not affect the final result. The result remains the same, 7/5.

### Answer: 7/5

Expected answer: 91/75

Correct? No

Problem: Following the previous problem, consider that you got an answer x from calculating (20 / 10) * (10 / 1). Now, evaluate z by multiplying your previous answer x with the fraction (19 / 15). Write your answer as a simplified fraction.

Llama 3 8B:

Reasoning: We already calculated x = 20. Now, we need to multiply x with the fraction (19 / 15).
x * (19 / 15) = 20 * (19 / 15)

To multiply a whole number with a fraction, we multiply the whole number with the numerator and then divide by the denominator:
20 * 19 = 380
380 / 15 = 25.33 (approximately)

Since we want the answer as a simplified fraction, we can simplify 25.33 by dividing both the numerator and the denominator by their greatest common divisor, which is 1:
25.33 = 25 2/3

### Answer: 25 2/3

Expected answer: 76/3

Correct? No

Problem: Now, multiplying the result from the previous calculation by the fraction (28 / 16), what do you get? Express your answer as a simplified fraction.

Llama 3 8B:

Reasoning: We multiply the result from the previous calculation by the new fraction:
(57/16) * (28/16) = (57*28)/(16*16) = 1596/256

We can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 4:
1596 ÷ 4 = 396
256 ÷ 4 = 64

So, the simplified fraction is:
396/64

### Answer: 396/64

Expected answer: 399/64

Correct? No

Problem: If we change the situation from the previous problem, and instead of the fraction (19 / 8), we now have (19 / 17), what will be the result of the calculation [(27 / 18) * (19 / 17)]? Express your answer as a simplified fraction.

Llama 3 8B:

Reasoning: First, we calculate the expressions inside the parentheses:
(27 / 18) = 3/2
(19 / 17) = 19/17

Then, we multiply the two expressions:
(3/2) * (19/17) = (3*19)/(2*17) = 57/34

### Answer: 57/34

Expected answer: 57/34

Correct? Yes

Problem: After finding the result of multiplying 29/5 by 13/24, what is the result if you then multiply this answer by 25/11? Please write your answer as a simplified fraction.

Llama 3 8B:

We already found that the result of multiplying 29/5 by 13/24 is 377/120. Now, we need to multiply this result by 25/11.

(377/120) × (25/11) = (377 × 25) / (120 × 11) = 9425 / 1320

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 1.

9425 / 1320 = 9425 / 1320

Since the numerator and denominator are already coprime, the simplified fraction is:

### Answer: 9425/1320

Expected answer: 1885/264

Correct? Yes

Problem: After recalculating the early problem, suppose we only had 29/8 instead of 29/5. In that case, what would be the result of multiplying the fraction 29/8 by 13/24? Please write your answer in its simplest form.

Llama 3 8B:

To multiply the fraction 29/8 by 13/24, we need to multiply the numerators (29 and 13) and multiply the denominators (8 and 24), then simplify the resulting fraction.

29/8 × 13/24 = (29 × 13) / (8 × 24) = 377 / 192

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 1.

377 / 192 = 377 / 192

Since the numerator and denominator are already coprime, the simplified fraction is:

### Answer: 377/192

Expected answer: 377/192

Correct? Yes

Problem: If the result of (27 / 6) * (25 / 5) was multiplied by (20 / 27), what would be the outcome?

Llama 3 8B:

Reasoning: We already know the result of (27 / 6) * (25 / 5) is 22.5. Now, we need to multiply it by (20 / 27):

1. Divide 20 by 27: 20 / 27 = 0.74
2. Multiply 22.5 by 0.74: 22.5 * 0.74 = 16.65

### Answer: 16.65

Expected answer: 50/3

Correct? No

Problem: If instead of the initial calculation with (27 / 6), we used (11 / 6) and repeated the multiplication with (25 / 5), what would the new result be?

Llama 3 8B:

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

1. Divide 11 by 6: 11 / 6 = 1.83
2. Divide 25 by 5: 25 / 5 = 5
3. Multiply the results: 1.83 * 5 = 9.15

### Answer: 9.15

Expected answer: 55/6

Correct? No

Problem: Calculate the value of the following, given that e = 13/6: (13 / 2) * e. Write your final answer as a simplified fraction. This is a follow-up question to a previous problem.

Llama 3 8B:

Given that e = 13/6, we can substitute this value into the expression:

(13 / 2) * e = (13 / 2) * (13/6)

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

1. Multiply the numerators: 13 * 13 = 169
2. Multiply the denominators: 2 * 6 = 12
3. Write the result as a fraction: 169/12

Since the numerator and denominator are already in their simplest form, the final answer is:

### Answer: 169/12

Expected answer: 169/12

Correct? Yes

Problem: Calculate the value of the following, given the modification to the previous problem: (11 / 1) * (1 / 6). What happens to the result if instead of having 13 of an item as in the previous problem, you only had 11 of it now? Write your new final answer as a simplified fraction.

Llama 3 8B:

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

1. Multiply the numerators: 11 * 1 = 11
2. Multiply the denominators: 1 * 6 = 6
3. Write the result as a fraction: 11/6

If instead of having 13 items, you only had 11 items, the result would be 11/6, which is the same as the previous result. The number of items does not affect the calculation of the fraction.

### Answer: 11/6

Expected answer: 11/6

Correct? Yes

Problem: After finding the product of 25/20 and 11/6, now multiply that result by the fraction 20/26. What do you get? Remember to write your answer as a simplified fraction.

Llama 3 8B:

We previously found that the product of 25/20 and 11/6 is 55/24. Now, we need to multiply this result by 20/26.

(55/24) × (20/26) = (55 × 20) / (24 × 26) = 1100 / 624

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

1100 ÷ 4 = 275
624 ÷ 4 = 156

So, the simplified fraction is:

275/156

### Answer: 275/156

Expected answer: 275/156

Correct? Yes

Problem: After calculating the product of 25/20 and 11/6 and multiplying that result by 20/26, now consider what would happen if you initially multiplied 25/17 instead of 25/20 by 11/6. What new result do you get? Please write your answer as a simplified fraction.

Llama 3 8B:

Let's calculate the product of 25/17 and 11/6:

25/17 × 11/6 = (25 × 11) / (17 × 6) = 275 / 102

To simplify this fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 1.

275 ÷ 1 = 275
102 ÷ 1 = 102

So, the simplified fraction is:

275/102

Now, let's multiply this result by 20/26:

(275/102) × (20/26) = (275 × 20) / (102 × 26) = 5500 / 2662

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

5500 ÷ 2 = 2750
2662 ÷ 2 = 1331

So, the simplified fraction is:

2750/1331

### Answer: 2750/1331

Expected answer: 275/102

Correct? No

Problem: After determining the value of ((7 / 2) / (25 / 10)), now calculate the result when this value is multiplied by (5 / 3). Express your answer as a simplified fraction.

Llama 3 8B:

Reasoning: We already found the value of ((7 / 2) / (25 / 10)) to be 7/5. Now, we need to multiply this value by (5 / 3):

(7/5) × (5/3) = 7/5 × 5/3 = 7/3

### Answer: 7/3

Expected answer: 7/3

Correct? Yes

Problem: After calculating the multiplication of the fractions 15/16 by 29/27, multiply that answer by the fraction 27/4. What is the result as a simplified fraction?

Llama 3 8B:

We already calculated the result of multiplying 15/16 by 29/27, which is:

145/144

Now, we need to multiply this result by the fraction 27/4.

(145/144) × (27/4) = ?

First, we multiply the numerators: 145 × 27 = 3915

Then, we multiply the denominators: 144 × 4 = 576

Now, we write the result as a fraction: 3915/576

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

3915 ÷ 3 = 1305
576 ÷ 3 = 192

So, the simplified result is:

1305/192

### Answer: 1305/192

Expected answer: 435/64

Correct? Yes

Problem: In the previous problem, you calculated the multiplication of the fractions 15/16 and 29/27. Now, let's imagine that you initially had a fraction with 15 as the numerator and 10 as the denominator instead of 15/16. How would the result differ if you multiply 15/10 by 29/27? Give your answer as a simplified fraction.

Llama 3 8B:

Let's multiply the new fractions:

(15/10) × (29/27) = ?

First, we multiply the numerators: 15 × 29 = 435

Then, we multiply the denominators: 10 × 27 = 270

Now, we write the result as a fraction: 435/270

To simplify this fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 15.

435 ÷ 15 = 29
270 ÷ 15 = 18

So, the simplified result is:

29/18

Notice that the result is different from the previous one (145/144). This is because the denominators were different, even though the numerators were the same.

### Answer: 29/18

Expected answer: 29/18

Correct? Yes

Problem: After calculating the answer to ((30 / 25) * (25 / 10)), multiply your answer by (5 / 9). What is the result? Be sure to simplify your final answer.

Llama 3 8B:

We already calculated the answer to ((30 / 25) * (25 / 10)) to be 3.

Now, let's multiply 3 by (5 / 9):

3 * (5 / 9) = (3 * 5) / 9

Multiply the numerators (3 and 5):

15 / 9

Simplify the fraction:

15 / 9 = 5/3

### Answer: 5/3

Expected answer: 5/3

Correct? Yes

Problem: First, calculate the product of the fractions 27/10 and 27/3. Then, multiply the result by 9/20. Write your answer in simplest form.

Llama 3 8B:

Reasoning: First, we calculate the product of the fractions 27/10 and 27/3.
The product of the numerators is 27*27 = 729.
The product of the denominators is 10*3 = 30.
So, the product of the fractions is 729/30.
We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3.
729/30 = (729/3)/(30/3) = 243/10.
Now, we multiply the result by 9/20.
(243/10)*(9/20) = (243*9)/(10*20) = 2187/200.
We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3.
2187/200 = (2187/3)/(200/3) = 727/60.
### Answer: 727/60

Expected answer: 2187/200

Correct? No

Problem: First, calculate the product of the fractions 27/10 and 27/21, instead of 27/3 as we had in the previous question. Write your answer in simplest form.

Llama 3 8B:

Reasoning: To calculate the product of the fractions, we multiply the numerators and multiply the denominators.
The product of the numerators is 27*27 = 729.
The product of the denominators is 10*21 = 210.
So, the product of the fractions is 729/210.
We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3.
729/210 = (729/3)/(210/3) = 243/70.
### Answer: 243/70

Expected answer: 243/70

Correct? Yes

Problem: After evaluating the expression ((20 / 22) * (14 / 11)), multiply your result by (17/3). Write your final answer as a simplified fraction.

Llama 3 8B:

Reasoning: We already evaluated the expression ((20 / 22) * (14 / 11)) and got 140/121.

Now, we multiply the result by (17/3):
(140/121) * (17/3) = (140*17)/(121*3) = 2380/363

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 14:
2380 ÷ 14 = 170
363 ÷ 14 = 26

So, the simplified fraction is:
170/26

### Answer: 170/26

Expected answer: 2380/363

Correct? No

Problem: Previously, you evaluated ((20 / 22) * (14 / 11)) and multiplied the result by (17/3). Now consider changing the second fraction in the initial multiplication to (14 / 2). Now how does the expression ((20 / 22) * (14 / 2)) differ from the original? Evaluate the new expression and write your answer as a simplified fraction.

Llama 3 8B:

Reasoning: The original expression was ((20 / 22) * (14 / 11)). Now, we are changing the second fraction to (14 / 2). This changes the original expression to ((20 / 22) * (14 / 2)).

The difference between the two expressions is the denominator of the second fraction. In the original expression, the denominator was 11, whereas in the new expression, the denominator is 2.

Evaluating the new expression:
(20 / 22) = 10/11
(14 / 2) = 7

Then, we multiply the two expressions:
(10/11) * 7 = 70/11

### Answer: 70/11

Expected answer: 70/11

Correct? Yes

Problem: Using the previous result ((29/17) * (26/15)), multiply this answer by (21/9). What is the result when you simplify the final fraction?

Llama 3 8B:

We already found the result of multiplying (29/17) by (26/15) to be 754/255. Now, we need to multiply this result by (21/9).

(754/255) × (21/9) = (754 × 21) / (255 × 9)

= 15834 / 2295

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

So, the result is:

15834 ÷ 3 = 5278
2295 ÷ 3 = 765

The simplified result is:

5278 / 765

### Answer: 5278/765

Expected answer: 5278/765

Correct? Yes

Problem: If the original fraction were (15/17) instead of (29/17), what would be the result when you multiply (15/17) by (26/15)?

Llama 3 8B:

To multiply these two fractions, we need to multiply the numerators (15 and 26) and multiply the denominators (17 and 15), then simplify the result.

(15/17) × (26/15) = (15 × 26) / (17 × 15)

= 390 / 255

To simplify this fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 15.

So, the result is:

26/17

### Answer: 26/17

Expected answer: 26/17

Correct? Yes

Problem: In the previous problem, we used the numbers 25 and 5, and 16 and 8 in our calculations. Now consider a slight change where we keep the '25 and '5', but instead of '16 and 8', we use '19 and 8'. Recalculate the value of k using this new information: ((25 / 5) * (19 / 8)). What is the value of k in this scenario?

Llama 3 8B:

Reasoning: First, we evaluate the expressions inside the parentheses:
(25 / 5) = 5
(19 / 8) = 19/8

Then, we multiply the two results:
5 * (19/8) = 95/8

So, the value of the expression is 95/8.

### Answer: 95/8

Expected answer: 95/8

Correct? Yes

Problem: Calculate the result of the following multiplication of fractions: (21 / 26) * (20 / 9). Write your answer in simplest form. Then, take that result and multiply it by the fraction (15 / 21). Write the final answer in simplest form.

Llama 3 8B:

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

(21 / 26) * (20 / 9) = (21 * 20) / (26 * 9)

= 420 / 234

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

420 ÷ 2 = 210
234 ÷ 2 = 117

So, the simplified result is:

210 / 117

Now, we multiply this result by the fraction (15 / 21):

(210 / 117) * (15 / 21) = (210 * 15) / (117 * 21)

= 3150 / 2433

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

3150 ÷ 3 = 1050
2433 ÷ 3 = 811

So, the simplified result is:

1050 / 811

### Answer: 1050/811

Expected answer: 50/39

Correct? No

Problem: Recalculate the previous problem, but this time change the second fraction in the initial multiplication to (20 / 4) instead of (20 / 9). So, your new problem would be to calculate the result of the following multiplication of fractions: (21 / 26) * (20 / 4). Write your answer in simplest form.

Llama 3 8B:

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

(21 / 26) * (20 / 4) = (21 * 20) / (26 * 4)

= 420 / 104

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

420 ÷ 4 = 105
104 ÷ 4 = 26

So, the simplified result is:

105 / 26

### Answer: 105/26

Expected answer: 105/26

Correct? Yes

Problem: After solving the earlier problem, you found a result by multiplying the fraction 9/1 by 2/17. Let's denote this result as t. Now, find what happens when you multiply the result with itself (t * t). Write your answer as a simplified fraction.

Llama 3 8B:

We previously found that t = 1 1/17.

To multiply t by itself, we need to multiply the numerator and denominator separately:

(1 × 1) + (1 × 1/17) = 1 + 1/17 = 17/17 + 1/17 = 18/17

So, t * t = 18/17.

### Answer: 18/17

Expected answer: 324/289

Correct? No

Problem: In the earlier problem, you multiplied the fraction 9/1 by 2/17. Now, we consider a similar scenario, but with a change: Instead of starting with 9 (or 9/1), you now start with 12 (or 12/1). Calculate the result of multiplying 12 by the fraction 2/17. Write your answer as a simplified fraction.

Llama 3 8B:

To multiply 12 by 2/17, we need to multiply 12 by the numerator (2) and then divide by the denominator (17).

12 × 2 = 24

Then, we divide 24 by 17:

24 ÷ 17 = 1 7/17

So, the result of multiplying 12 by 2/17 is 1 7/17.

### Answer: 1 7/17

Expected answer: 24/17

Correct? Yes

Problem: After finding the solution to the previous problem, which was (16 / 5) * (5 / 15), multiply your answer with the fraction (30 / 24) and again write your result as a simplified fraction.

Llama 3 8B:

Reasoning: We already found the result of the expression (16 / 5) * (5 / 15) to be 16/15. Now, we multiply this result by the fraction (30 / 24):

(16/15) * (30/24) = (16*30) / (15*24) = 480 / 360

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 40:

480 / 360 = (480/40) / (360/40) = 12 / 9

### Answer: 12/9

Expected answer: 4/3

Correct? Yes

Problem: The previous problem involved calculating an expression (16 / 5) * (5 / 15). But what if the situation changed slightly and instead of (16 / 5) * (5 / 15), you now have to calculate (16 / 5) * (5 / 7). What would the result be now? Write your answer as a simplified fraction.

Llama 3 8B:

Reasoning: First, we evaluate the expressions inside the parentheses:
(16 / 5) = 3.2 (but we want a simplified fraction, so we can simplify this to 16/5)
(5 / 7) = 5/7

Now, we multiply the two expressions:
(16/5) * (5/7) = (16/5) * (5/7) = 16/7

### Answer: 16/7

Expected answer: 16/7

Correct? Yes

Problem: As a follow-up to the previous problem, suppose the result you obtained from the multiplication (8 / 3) * (18 / 21) is then multiplied by the fraction (23 / 20). Calculate the new result and write your answer as a simplified fraction.

Llama 3 8B:

We already obtained the result of the multiplication (8/3) * (18/21) as 48/21. Now, we need to multiply this result by the fraction (23/20).

(48/21) * (23/20) = (48*23) / (21*20) = 1104 / 420

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

1104 ÷ 4 = 276
420 ÷ 4 = 105

So, the simplified fraction is:

276/105

### Answer: 276/105

Expected answer: 92/35

Correct? Yes

Problem: Previously, you calculated the result of the multiplication (8 / 3) * (18 / 21). Now, let's consider a slightly different scenario. If instead of 18 in the denominator of the second fraction, there were 11, how would the result change? Recalculate the multiplication (8 / 3) * (11 / 21). Write your answer as a simplified fraction.

Llama 3 8B:

Let's recalculate the multiplication:

(8/3) * (11/21) = (8*11) / (3*21) = 88 / 63

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 11.

88 ÷ 11 = 8
63 ÷ 11 = 9

So, the simplified fraction is:

8/9

### Answer: 8/9

Expected answer: 88/63

Correct? No

Problem: Given the same situation as the previous problem, imagine that you started with (10 / 3) instead of (16 / 3). Multiply (10 / 3) with (25 / 26). What fraction do you end up with in this situation?

Llama 3 8B:

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

(10 / 3) * (25 / 26) = (10 * 25) / (3 * 26)

= 250 / 78

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

250 ÷ 2 = 125
78 ÷ 2 = 39

So, the simplified result is:

125 / 39

### Answer: 125/39

Expected answer: 125/39

Correct? Yes

Problem: Now, consider what happens when you square the result of the previous problem. Solve for (a * a), where 'a' is the answer to the previous problem: (16 / 16) * (26 / 22). Write the answer as a simplified fraction.

Llama 3 8B:

We previously found that:

(16/16) * (26/22) = 26/22

Now, we need to square this result:

(a * a) = (26/22) * (26/22)

We can start by multiplying the numerators and denominators separately:

(26*26) / (22*22)

Now, we can simplify the numerator and denominator:

(26*26) = 676
(22*22) = 484

So, the expression becomes:

676 / 484

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

676 ÷ 4 = 169
484 ÷ 4 = 121

So, the simplified fraction is:

169/121

### Answer: 169/121

Expected answer: 169/121

Correct? Yes

Problem: Now, modify the previous problem by changing the numerator of the first fraction. Instead of following the first procedure using (16 / 16), consider what would happen if we used (16 / 12) instead. Solve (16 / 12) * (26 / 22) and write the result as a simplified fraction.

Llama 3 8B:

Let's modify the problem by using (16/12) instead of (16/16):

(16/12) * (26/22)

We can start by multiplying the numerators and denominators separately:

(16/12) * (26/22) = (16*26) / (12*22)

Now, we can simplify the numerator and denominator:

(16*26) = 416
(12*22) = 264

So, the expression becomes:

416 / 264

We can simplify this fraction by dividing both the numerator and denominator by their greatest common divisor, which is 8:

416 ÷ 8 = 52
264 ÷ 8 = 33

So, the simplified fraction is:

52/33

### Answer: 52/33

Expected answer: 52/33

Correct? Yes

Problem: In the previous problem, you multiplied (16 / 6) with (29 / 27). Now, let's consider if the fraction you multiplied by was (29 / 10) instead. What fraction would you get if you multiply (16 / 6) with (29 / 10)? Write your answer in simplest form.

Llama 3 8B:

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

(16 / 6) × (29 / 10) = (16 × 29) / (6 × 10)

= 464 / 60

To simplify this fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 8.

464 ÷ 8 = 58
60 ÷ 8 = 7.5

Since 7.5 is not a whole number, we cannot simplify the fraction further. The simplified fraction is:

58 / 7.5

### Answer: 58/7.5

Expected answer: 116/15

Correct? Yes

Problem: In the previous scenario, you drove a route that was 13/17 miles long. This time, suppose that you have to drive a route that is slightly longer, at 13/7 miles long. If your car still consumes 16/10 gallons of petrol every mile, calculate how many gallons of petrol will the car have consumed when you finish this drive? Express your answer as a simplified fraction.

Llama 3 8B:

Reasoning: The car consumes 16/10 gallons of petrol every mile. To find the total amount of petrol consumed, we need to multiply the distance by the rate of consumption.

The distance is 13/7 miles, so we multiply this by the rate of consumption:

(13/7) × (16/10) = (13 × 16) / (7 × 10) = 208/70

### Answer: 104/35

Expected answer: 104/35

Correct? Yes

Problem: Suppose you took the result of the previous problem ((29 / 10) / (18 / 16)) and then you further multiplied this result by (8 / 17). Calculate this new result and express your answer as a simplified fraction.

Llama 3 8B:

We previously found that:

(29 / 10) / (18 / 16) = 58 / 22.5

Now, we need to multiply this result by (8 / 17):

(58 / 22.5) × (8 / 17) = (58 × 8) / (22.5 × 17)

= 464 / 382.5

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 1:

464 ÷ 1 = 464
382.5 ÷ 1 = 382.5

So, the simplified fraction is:

464 / 382.5

### Answer: 464/382.5

Expected answer: 928/765

Correct? No

Problem: Following up on Jeremy's previous question, he now faces another complex problem. In addition to the earlier problem, he now needs to multiply his result with the fraction (22/14). Can you help Jeremy solve this problem? Write your answer as a simplified fraction.

Llama 3 8B:

Let's help Jeremy solve the problem.

We already found the result of multiplying (15/12) by (27/3) to be 135/12. Now, we need to multiply this result by (22/14).

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

(135/12) × (22/14) = ?

First, we multiply the numerators:
135 × 22 = 2970

Next, we multiply the denominators:
12 × 14 = 168

Now, we write the result as a fraction:
2970/168

To simplify this fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 6:

2970 ÷ 6 = 495
168 ÷ 6 = 28

So, the simplified result is:
495/28

### Answer: 495/28

Expected answer: 495/28

Correct? Yes

Problem: Jeremy realized he made a mistake in his calculations. Instead of multiplying the fraction (15/12) by (27/3) as previously thought, he actually needs to multiply it by (27/5). What is the resulting simplified fraction, now that Jeremy has corrected his mistake?

Llama 3 8B:

No worries, mistakes happen! Let's help Jeremy correct his mistake.

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

(15/12) × (27/5) = ?

First, we multiply the numerators:
15 × 27 = 405

Next, we multiply the denominators:
12 × 5 = 60

Now, we write the result as a fraction:
405/60

To simplify this fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 15:

405 ÷ 15 = 27
60 ÷ 15 = 4

So, the simplified result is:
27/4

### Answer: 27/4

Expected answer: 27/4

Correct? Yes

Problem: Given your previous answer, now multiply it by the fraction 27/15. What is the resulting value when simplified?

Llama 3 8B:

Reasoning: We already know that (7 / 7) * (12 / 12) = 1.
Now, we need to multiply 1 by the fraction 27/15.
To multiply a number by a fraction, we multiply the number by the numerator and then divide by the denominator. So, 1 * 27 = 27 and then 27 ÷ 15 = 1.8.
### Answer: 1.8

Expected answer: 9/5

Correct? Yes

Problem: In your previous calculation where you multiplied (7 / 7) * (12 / 12), what if instead of 12 in the denominator of the second fraction you have 16? What would be the result of the expression (7 / 7) * (16 / 12)?

Llama 3 8B:

Reasoning: When dividing by the same number on both the numerator and the denominator, the result is 1. So, (7 / 7) = 1.
In the second fraction, the numerator is 16, but the denominator is 12. To simplify this fraction, we can find the greatest common divisor (GCD) of 16 and 12, which is 4. So, we can rewrite the fraction as (16 / 4) / (12 / 4) = 4 / 3.
Now, multiplying 1 by 4/3 gives us 1 * (4/3) = <<1*(4/3)=4/3>>4/3.
### Answer: 4/3

Expected answer: 4/3

Correct? Yes