Problem: After finding the product of (29/2) and (19/28), multiply your result by (8/28). Write your final answer as a simplified fraction.

Llama 3 70B:

I already calculated the product of (29/2) and (19/28) as 551/56. Now, I'll multiply this result by (8/28):

(551/56) × (8/28) = (551 × 8) / (56 × 28)
= 4408 / 1568

To simplify the fraction, I'll look for common factors between the numerator and denominator. Both numbers are divisible by 8, so I'll divide both by 8:

4408 ÷ 8 = 551
1568 ÷ 8 = 196

Now, I'll look for common factors between 551 and 196. Since 551 is a prime number, I'll focus on 196, which is divisible by 2, 4, 7, and 14. However, I can see that 196 is also divisible by 49 (7 × 7). And, as it turns out, 551 is also divisible by 7:

551 ÷ 7 = 79
196 ÷ 7 = 28

Now, I'll divide both numbers by 7:

4408 / 1568 = (79 × 7) / (28 × 7) = 79/28

### Answer: 79/28

Expected answer: 551/196

Correct? No

Problem: After finding the product of (29/2) and (19/28), reconsider the calculation using (19/14) instead of (19/28). Recalculate the product of (29/2) and (19/14) and write your answer as a simplified fraction.

Llama 3 70B:

Let's recalculate the product using (19/14) instead of (19/28):

(29/2) × (19/14) = (29 × 19) / (2 × 14)
= 551 / 28

To simplify the fraction, I'll look for common factors between the numerator and denominator. In this case, both numbers are divisible by 1, but that's not helpful. However, I can see that 551 is a prime number, and 28 is divisible by 2, 4, and 7. Since 28 is divisible by 4, and 551 is not, I can divide both numbers by 2:

551 / 28 = (551 ÷ 2) / (28 ÷ 2)
= 275.5 / 14

Since the numerator is not an integer, I'll leave the fraction in this form.

### Answer: 275.5/14

Expected answer: 551/28

Correct? Yes

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 70B:

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

1. Divide 26 by 12: 26 ÷ 12 = 26/12 = 13/6
2. Multiply 7 by the result: 7 × (13/6) = 91/6

### Answer: 91/6

Expected answer: 91/6

Correct? Yes

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 70B:

We already found the result of ((24 / 22) * (24 / 17)) to be 288/187. Now, we need to multiply this result by (27 / 18):

(288/187) × (27/18)

To multiply fractions, we multiply the numerators (288 × 27) and multiply the denominators (187 × 18), then simplify:

(288 × 27) / (187 × 18) = 7776 / 3366

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

7776 ÷ 6 = 1296
3366 ÷ 6 = 561

So, the simplified result is:

1296/561

### Answer: 1296/561

Expected answer: 432/187

Correct? Yes

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 70B:

Let's evaluate the new expression:

((24 / 22) * (24 / 9))

1. Divide 24 by 22: 24/22 = 12/11
2. Divide 24 by 9: 24/9 = 8/3
3. Multiply the two results: (12/11) × (8/3)

To multiply fractions, we multiply the numerators (12 × 8) and multiply the denominators (11 × 3), then simplify:

(12 × 8) / (11 × 3) = 96 / 33

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

96 ÷ 3 = 32
33 ÷ 3 = 11

So, the simplified fraction is:

### Answer: 32/11

Expected answer: 32/11

Correct? Yes

Problem: If the result of dividing 25/4 by 16/6 is multiplied by 19/6, what is the new result? Be sure to simplify your answer to lowest terms.

Llama 3 70B:

We already found that the result of dividing 25/4 by 16/6 is 75/32. Now, we need to multiply this result by 19/6.

(75/32) × (19/6) = (75 × 19) / (32 × 6) = 1425/192

To simplify this fraction, we can find the greatest common divisor (GCD) of 1425 and 192, which is 3. Then, we can divide both the numerator and the denominator by 3.

1425 ÷ 3 = 475
192 ÷ 3 = 64

So, the simplified result is:

475/64

### Answer: 475/64

Expected answer: 475/64

Correct? Yes

Problem: Following up from the previous question, consider the new scenario where instead of dividing 25/4 we are now dividing 9 by 16/6. What is the answer to this problem? Please simplify your final answer.

Llama 3 70B:

To evaluate this expression, we need to follow the order of operations (PEMDAS). First, we divide 16 by 6, which gives us 16/6 = 16÷6 = 2.666... (or 8/3 as a simplified fraction).

Now, we need to divide 9 by 2.666... (or 8/3). To do this, we can convert 9 to an improper fraction, which is 9/1. Then, we can divide 9/1 by 8/3.

(9/1) ÷ (8/3) = (9/1) × (3/8) = (9 × 3) / (1 × 8) = 27/8

So, the result of dividing 9 by 16/6 is 27/8.

### Answer: 27/8

Expected answer: 27/8

Correct? Yes

Problem: In the previous question, you found the result of (3 / 7) * (26 / 6). Now, take that result and multiply it by the fraction 23/2. What's the result of this calculation, expressed as a simplified fraction?

Llama 3 70B:

We already found that (3/7) × (26/6) = 13/7. Now, we need to multiply this result by 23/2:

(13/7) × (23/2) = (13 × 23) / (7 × 2)

= 299 / 14

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 1 (since 299 and 14 are coprime):

299 ÷ 1 = 299
14 ÷ 1 = 14

So, the simplified fraction is:

299/14

### Answer: 299/14

Expected answer: 299/14

Correct? Yes

Problem: In the previous problem, you multiplied the fraction 3/7 by the fraction 26/6. Now, let's change the fraction a bit. Instead of 3/7, use 7/7 (which is actually 1). Multiply this new fraction 7/7 by the same fraction 26/6. What is the result of this new calculation, expressed as a simplified fraction?

Llama 3 70B:

Let's multiply the fractions:

(7/7) × (26/6) = (7 × 26) / (7 × 6)

= 182 / 42

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 14:

182 ÷ 14 = 13
42 ÷ 14 = 3

So, the simplified fraction is:

13/3

### Answer: 13/3

Expected answer: 13/3

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 70B:

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

1. Divide 11 by 18: 11 / 18 = 0.6111... (approximately)
2. Multiply 4.76 by 0.6111...: 4.76 × 0.6111... = 2.9133...

### Answer: 2.9133... (approximately)

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 70B:

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

1. Divide 17 by 24: 17 / 24 = 0.7083 (approximately)
2. Divide 28 by 10: 28 / 10 = 2.8
3. Multiply the results: 0.7083 × 2.8 = 1.9844 (approximately)

### Answer: 1.9844

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 70B:

We already calculated the result of ((18 / 19) * (19 / 6)) to be 3.

Now, let's multiply 3 by itself:

3 × 3 = 9

As an integer, 9 can be written as a fraction: 9/1.

However, we can also write it as a simplified fraction:

9/1 = 9

So, the result of multiplying 3 by itself is:

### Answer: 9

Expected answer: 9

Correct? Yes

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 70B:

Let's evaluate the new expression:

((9 / 19) * (19 / 6))

1. Divide 9 by 19: 9/19 = 9/19
2. Divide 19 by 6: 19/6 = 19/6
3. Multiply the two results: (9/19) × (19/6) = (9 × 19) / (19 × 6)

Now, we can simplify the fraction:

(9 × 19) / (19 × 6) = 9/6 (cancel out the 19's)

9/6 = 3/2

So, the simplified fraction is:

### Answer: 3/2

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 70B:

We already calculated the result of the multiplication of the fraction 19/12 by 20/6, which is:

380 / 72 = 95 / 18

Now, we need to multiply this result by itself:

(95 / 18) × (95 / 18) = (95 × 95) / (18 × 18)

= 9025 / 324

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

### 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 70B:

Let's redo the calculation with the new fraction:

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

= 200 / 72

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

200 ÷ 4 = 50
72 ÷ 4 = 18

So, the simplified result is:

### Answer: 50/18

Expected answer: 25/9

Correct? Yes

Problem: After you've calculated the result of the previous problem, which is (16 / 4) / (24 / 21), multiply your answer by (5 / 13). What is the final result in its simplified form?

Llama 3 70B:

We already calculated the result of the previous problem:

(16 / 4) / (24 / 21) = 3.5 or 7/2

Now, let's multiply this result by (5 / 13):

(7/2) × (5/13) = (7 × 5) / (2 × 13) = 35/26

### Answer: 35/26 or 1.35

Expected answer: 35/26

Correct? Yes

Problem: Suppose you made a mistake when performing the previous calculation. Instead of (16 / 4) / (24 / 21), you should have calculated (16 / 4) / 3. What would the correct answer be if the original problem used this correct operation instead?

Llama 3 70B:

Let's recalculate:

1. Evaluate the expression inside the parentheses:
   (16 / 4) = 4

2. Now, divide the result by 3:
   4 / 3 = 4 × (1/3) = 4/3

### Answer: 4/3 or 1.33

Expected answer: 4/3

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 70B:

We already found the product of the fractions 5/27 and 15/2 to be 25/18. Now, we need to multiply this result by itself:

(25/18) × (25/18) = (25 × 25) / (18 × 18) = 625 / 324

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

### 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 70B:

To calculate the new product of the fractions, we multiply the numerators (24 and 15) and multiply the denominators (27 and 2), then simplify the resulting fraction:

(24 × 15) / (27 × 2) = 360 / 54

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

360 ÷ 18 = 20
54 ÷ 18 = 3

So, the new product of the fractions in simplest form is:

20/3

### Answer: 20/3

Expected answer: 20/3

Correct? Yes

Problem: After finding the result of the product of fractions (15 / 13) * (19 / 7), multiply your result by another fraction (23/22). Write your final answer as a simplified fraction.

Llama 3 70B:

We already calculated the product of fractions (15 / 13) * (19 / 7) to be 285/91. Now, we need to multiply this result by another fraction (23/22).

To do this, we multiply the numerators and multiply the denominators, like this:

(285 / 91) * (23 / 22) = (285 * 23) / (91 * 22)

First, let's multiply the numerators:
285 * 23 = 6555

Next, let's multiply the denominators:
91 * 22 = 2002

Now, we can write the product as a fraction:
6555 / 2002

This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 11.

6555 ÷ 11 = 595
2002 ÷ 11 = 182

So, the simplified fraction is:
595 / 182

### Answer: 595/182

Expected answer: 6555/2002

Correct? No

Problem: In the previous task, instead of calculating the product of fractions (15 / 13) * (19 / 7), now calculate the result of a slightly different operation: (15 / 29) * (19 / 7). Write your answer as a simplified fraction.

Llama 3 70B:

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

(15 / 29) * (19 / 7) = (15 * 19) / (29 * 7)

First, let's multiply the numerators:
15 * 19 = 285

Next, let's multiply the denominators:
29 * 7 = 203

Now, we can write the product as a fraction:
285 / 203

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

### Answer: 285/203

Expected answer: 285/203

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 70B:

We already found the result of (12 / 25) * (20 / 4) to be 12/5. Now, we need to multiply this result by (17 / 17):

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

= 204 / 85

Since 17 is a common factor of both the numerator and the denominator, we can simplify the fraction by dividing both by 17:

= 12 / 5

The fraction 12/5 is already in its simplest form, so we don't need to simplify it further.

### 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 70B:

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

1. Divide 10 by 25: 10 / 25 = 2/5
2. Divide 20 by 4: 20 / 4 = 5
3. Multiply the results: (2/5) * 5 = 2

The result is a whole number, 2.

Comparing this to the previous result, we can see that replacing 12 with 10 in the original expression changed the result from 12/5 to 2. This is because 10 is a factor of 20, which simplified the fraction to a whole number.

### Answer: 2

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 70B:

We already calculated the result of multiplying 14/10 by 30/5, which is 42/5.

Now, let's multiply 42/5 by 13/15:

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

First, let's multiply the numerators: 42 × 13 = 546

Next, let's multiply the denominators: 5 × 15 = 75

So, the result of the multiplication is: 546/75

To simplify this 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 70B:

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

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

First, let's multiply the numerators: 30 × 30 = 900

Next, let's multiply the denominators: 10 × 5 = 50

So, the result of the multiplication is: 900/50

To simplify this 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

Since 18/1 is equal to 18, we can write the answer as a whole number.

### 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 70B:

We already found the value of s:

s = 243/40

Now, we can calculate o:

o = (9/13) * s
= (9/13) * (243/40)
= (9 × 243) / (13 × 40)
= 2187 / 520
= 153 / 70

### Answer: 153/70

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 70B:

Let's recalculate the expression with the new value:

s = (27 / 10) * (9 / 11)

First, divide 27 by 10: 27 ÷ 10 = 27/10 = 2.7

Next, divide 9 by 11: 9 ÷ 11 = 9/11

Now, multiply the two results: 2.7 × 9/11 = (27/10) × (9/11)

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

(27 × 9) / (10 × 11) = 243 / 110

### Answer: 243/110

Expected answer: 243/110

Correct? Yes

Problem: You've just evaluated the expression ((24 / 11) * (27 / 26)) in the previous question. Now, go ahead and take the result from that previous operation and multiply it by (14 / 27). Write your final answer as a simplified fraction.

Llama 3 70B:

We already evaluated the expression ((24 / 11) * (27 / 26)) to be 324/143.

Now, we need to multiply this result by (14 / 27):

(324/143) * (14/27) = (324*14) / (143*27) = 4524 / 3861

To simplify the fraction, we can find the greatest common divisor (GCD) of 4524 and 3861, which is 3. Divide both numerator and denominator by 3:

4524 / 3 = 1508
3861 / 3 = 1287

So, the simplified fraction is:

### Answer: 1508/1287

Expected answer: 168/143

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 70B:

To multiply the fractions, we multiply the numerators and multiply the denominators, then simplify the resulting fraction.

 Numerator: 34 × 24 = 816
 Denominator: 9 × 30 = 270

So, the product of the fractions is 816/270.

To simplify this fraction, we can divide both the numerator and the denominator by their greatest common divisor (GCD), 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 70B:

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

 Numerator: 25 × 17 = 425
 Denominator: 9 × 13 = 117

So, the product of the fractions is 425/117.

This fraction is already in simplest form, as 425 and 117 do not have any common factors other than 1.

### 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 70B:

We already found the result of the original expression:

(28 / 16) / (27 / 24) = 14/9

Now, let's multiply this result by itself:

(14/9) × (14/9) = (14 × 14) / (9 × 9) = 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 70B:

We already calculated the expression ((15 / 3) * (19 / 26)) to be 95/26. Now, we need to multiply it by the fraction (14 / 24):

(95/26) * (14/24)

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

(95 * 14) / (26 * 24)

= 1330 / 624

Now, we can simplify the fraction by finding the greatest common divisor (GCD) of 1330 and 624. The GCD is 2, so we can divide both numerator and denominator by 2:

= 665 / 312

### Answer: 665/312

Expected answer: 665/312

Correct? Yes

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 70B:

Let's recalculate the expression:

((15 / 3) * (27 / 26))

1. Divide 15 by 3: 15 / 3 = 5
2. Divide 27 by 26: 27 / 26
3. Multiply the results: 5 * (27 / 26) = 5 * 27 / 26

Now, we can simplify the fraction:

5 * 27 = 135
135 / 26 = 135/26

The fraction 135/26 is already in its simplest form, as 135 and 26 do not have any common factors other than 1.

### Answer: 135/26

Expected answer: 135/26

Correct? Yes

Problem: Continuing from the bakery scenario, suppose that the bakery has adjusted their special recipe, and instead of using 19/28, they now use 19/14 of the ingredient for a single cake. If a customer still wants to order 30/4 of this new recipe, how much of the ingredient will the bakery now require for this order?

Llama 3 70B:

Reasoning: The amount required in the new recipe is 19/14.
They want to make 30/4 = 7.5 times the recipe (since 30/4 = 7.5).

So, the total amount of the ingredient required is 19/14 × 7.5 = 19/14 × 30/4 = 19/2 × 15/14 = 285/28 = 10 5/28.

### Answer: 10 5/28.

Expected answer: 285/28

Correct? Yes

Problem: After determining the result of ((5 / 2) / (26 / 15)) in the previous problem, take that result and multiply it by itself. What simplified fraction do you get?

Llama 3 70B:

We already determined that ((5 / 2) / (26 / 15)) = 75/52.

To multiply this result by itself, we need to multiply the numerator and denominator by themselves:

(75/52) × (75/52) = (75 × 75) / (52 × 52) = 5625/2704

This fraction can be simplified by finding the greatest common divisor (GCD) of 5625 and 2704, which is 25. Divide both numerator and denominator by 25:

5625 ÷ 25 = 225
2704 ÷ 25 = 108.16 (not an integer, so we can't simplify further)

So, the simplified fraction is:

225/108

### Answer: 225/108

Expected answer: 5625/2704

Correct? No

Problem: After determining the result of ((5 / 2) / (26 / 15)) in the previous problem, consider a situation where there were initially 14 units instead of 5. What is the result when you divide 14 by (26 / 15)? Write the final answer as a simplified fraction.

Llama 3 70B:

Let's evaluate the expression:

((14 / 2) / (26 / 15))

1. Evaluate the expressions inside the parentheses:
   (14 / 2) = 7
   (26 / 15) = 26/15

2. Now, divide the two expressions:
   7 / (26/15) = 7 * (15/26) (since dividing by a fraction is the same as multiplying by its reciprocal)

3. Multiply the numerators and denominators separately:
   (7 * 15) / 26 = 105/26

So, the result of ((14 / 2) / (26 / 15)) is 105/26.

### Answer: 105/26

Expected answer: 105/13

Correct? No

Problem: After you have calculated the result of the expression (9/3) * (27/28) from the previous problem, calculate the square of your answer. Write your final answer as a simplified fraction.

Llama 3 70B:

We already calculated the result of the expression (9/3) * (27/28) to be 81/28. To find the square of this result, we need to multiply it by itself:

(81/28) × (81/28) = (81 × 81) / (28 × 28) = 6561 / 784

To simplify this fraction, we can find the greatest common divisor (GCD) of 6561 and 784, which is 49. Then, we can divide both the numerator and the denominator by 49:

6561 ÷ 49 = 134
784 ÷ 49 = 16

So, the simplified fraction is:

### Answer: 134/16

Expected answer: 6561/784

Correct? No

Problem: In the previous problem, you multiplied the fractions (9/3) * (27/28). Now consider the case where the second fraction has changed from (27/28) to (23/28). How does this change your answer? Recalculate using (9/3) * (23/28) and write your new answer as a simplified fraction.

Llama 3 70B:

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

1. Divide 9 by 3: 9/3 = 3
2. Divide 23 by 28: 23/28
3. Multiply the results: 3 * (23/28) = 69/28

The result is already a simplified fraction, so we don't need to simplify it further.

### Answer: 69/28

Expected answer: 69/28

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 70B:

We already calculated the solution to ((15 / 13) * (24 / 24)) as 15/13. Now, let's multiply it by (22 / 7):

(15/13) * (22/7) = ?

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

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

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

### Answer: 330/91

Expected answer: 330/91

Correct? Yes

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 70B:

Let's recalculate the expression with the new fraction:

((15 / 2) * (24 / 24))

1. Divide 15 by 2: 15/2 = 15/2 (no simplification possible)
2. Divide 24 by 24: 24/24 = 1 (since 24 divided by itself is 1)
3. Multiply the two results: (15/2) * 1 = 15/2

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

### Answer: 15/2

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 70B:

To multiply the fractions, I'll multiply the numerators and multiply the denominators, like this:

(15/14) × (15/15) = (15 × 15) / (14 × 15)

= 225 / 210

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

225 ÷ 15 = 15
210 ÷ 15 = 14

So, the simplified fraction is still:

15/14

### 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 70B:

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

(9/7) × (25/21) = (9 × 25) / (7 × 21)

= 225 / 147

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

225 ÷ 3 = 75
147 ÷ 3 = 49

So, the simplified fraction is:

75/49

### Answer: 75/49

Expected answer: 75/49

Correct? Yes

Problem: Following the previous question, now calculate the result of multiplying your answer by the fraction (29 / 21). Write your final answer as a simplified fraction. In other words, calculate ((29 / 21) * ((9 / 8) / (20 / 18))).

Llama 3 70B:

We already calculated the result of ((9 / 8) / (20 / 18)) as 81/80. Now, we need to multiply this result by (29 / 21):

(81/80) * (29/21) = (81*29) / (80*21) = 2349 / 1680

To simplify this fraction, we can find the greatest common divisor (GCD) of 2349 and 1680, which is 21. Then, we can divide both the numerator and the denominator by 21:

2349 ÷ 21 = 112
1680 ÷ 21 = 80

So, the simplified fraction is:

### Answer: 112/80

Expected answer: 783/560

Correct? No

Problem: In the previous scenario, instead of starting with a fraction of 9/8, let's say we started with a simpler fraction of 9 and divide it by the complex fraction 20/18. What would be the result? In other words, calculate (9 / (20 / 18)). Express your final answer as a simplified fraction.

Llama 3 70B:

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

1. Evaluate the expression inside the parentheses:
   (20 / 18) = 10/9

2. Now, divide 9 by the result:
   9 / (10/9) = 9 * (9/10) = 81/10

### Answer: 81/10

Expected answer: 81/10

Correct? Yes

Problem: Following the result of the previous question, calculate the new result if you multiply the answer you got by (14 / 4). What is your answer now?

Llama 3 70B:

We multiply the previous result by (14 / 4):

(45/13) × (14/4) = (45/13) × (7/2) = (45 × 7) / (13 × 2) = 315 / 26

### Answer: 315/26

Expected answer: 315/26

Correct? Yes

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 70B:

We already found the result of the previous problem: 5.

Now, let's multiply 5 by the fraction 8/20:

5 × (8/20) = 5 × (2/5) = 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 70B:

Let's recalculate:

1. Divide 30 by 4: 30 ÷ 4 = 30/4 = 15/2
2. Multiply the result by the fraction 18/6: (15/2) × (18/6) = (15 × 18) / (2 × 6) = 270 / 12 = 45/2 = 22.5

Comparing to the original calculation, we get a different result. The new value is 22.5, whereas the original result was 5.

### 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 70B:

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)) = ?

To multiply fractions, we multiply the numerators (27 and 10) and multiply the denominators (4 and 9), then simplify:

(27 * 10) / (4 * 9) = 270 / 36

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

270 ÷ 18 = 15
36 ÷ 18 = 2

So, the simplified fraction is:

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 70B:

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

1. Divide 20 by 2: 20 / 2 = 10
2. Divide 15 by 18: 15 / 18 = 5/6
3. Multiply the results: 10 * (5/6) = 50/6

We can simplify the fraction 50/6 by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

50/6 = (50 ÷ 2) / (6 ÷ 2) = 25/3

So, the result of the problem is 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 70B:

We already found the result of ((11 / 9) * (29 / 6)) to be 319/54. Now, we need to multiply it by (11 / 5):

(319/54) × (11/5) = ?

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

 Numerators: 319 × 11 = 3519
 Denominators: 54 × 5 = 270

Result: 3519/270

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

3519 ÷ 9 = 391
270 ÷ 9 = 30

### Answer: 391/30

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 70B:

Let's evaluate the new expression:

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

1. Multiply the numerators (numbers on top): 21 × 29 = 609
2. Multiply the denominators (numbers on the bottom): 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 (GCD), which is 3:

609 ÷ 3 = 203
54 ÷ 3 = 18

So, the simplified fraction is:

### Answer: 203/18

Expected answer: 203/18

Correct? Yes

Problem: Let's proceed with the result found in the previous fraction multiplication task. Now, take that result and multiply it by 22/7. What is the new result? Express your answer as a simplified fraction.

Llama 3 70B:

We need to multiply the result 480/133 by 22/7:

(480/133) * (22/7) = (480*22) / (133*7) = 10560 / 931

To simplify this fraction, we can look for common factors between the numerator and the denominator. However, 10560 and 931 do not have any common factors other than 1. Therefore, the fraction is already in its simplest form.

### Answer: 10560/931

Expected answer: 10560/931

Correct? Yes

Problem: Recall the earlier problem where we performed a fraction multiplication. If we consider a slight change and now multiply 21/7 (instead of 30/7) with 16/19, what would be the result? Remember to express your answer as a simplified fraction.

Llama 3 70B:

Let's multiply the fractions:

(21/7) * (16/19) = (21*16) / (7*19) = 336 / 133

We can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 7:

336 ÷ 7 = 48
133 ÷ 7 = 19

So, the result of the operation is:

48/19

### Answer: 48/19

Expected answer: 48/19

Correct? Yes

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 70B:

We already calculated the product of 25/22 and 20/3, which is 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 look for common factors between the numerator and the denominator. However, in this case, there are no common factors, so the fraction is already in its simplest form:

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 70B:

Let's recalculate the product with the new quantities:

(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 result of ((29 / 2) / (23 / 19)), multiply your answer by (29 / 18). Write your final answer as a simplified fraction.

Llama 3 70B:

We already calculated the result of ((29 / 2) / (23 / 19)) to be 551/46.

Now, let's multiply this result by (29 / 18):

(551/46) * (29/18) = (551*29) / (46*18)

= 15979 / 828

To simplify this fraction, we can find the greatest common divisor (GCD) of 15979 and 828, which is 1. Therefore, the fraction is already in its simplest form.

### Answer: 15979/828

Expected answer: 15979/828

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 70B:

We already calculated the result of (5 / 3) * (27 / 13) to be 45/13. Now, we need to multiply it by (8 / 14):

(45/13) * (8/14) = (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 70B:

Let's evaluate the new expression:

(5/3) * (27/9)

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

(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 27:

135 ÷ 27 = 5
27 ÷ 27 = 1

So, the simplified fraction is:

### Answer: 5/1 or 5

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 70B:

We already know that the previous result is 1. Now, we need to multiply it by (27 / 27):

1. Divide 27 by 27: 27 / 27 = 1
2. Multiply 1 by 1: 1 * 1 = 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 70B:

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

1. Divide 29 by 6: 29 / 6 = 4.83 (approximately)
2. Divide 1 by 1: 1 / 1 = 1
3. Multiply the results: 4.83 * 1 = 4.83

### Answer: 4.83

Expected answer: 29/6

Correct? No

Problem: After you find the product of the fractions 11/19 and 22/7, multiply your answer by 25/4. What is the result as a simplified fraction?

Llama 3 70B:

First, we calculate the product of the fractions 11/19 and 22/7:

(11/19) × (22/7) = (11 × 22) / (19 × 7) = 242 / 133

Now, we multiply the result by 25/4:

(242/133) × (25/4) = (242 × 25) / (133 × 4) = 6050 / 532

To simplify the fraction, we find the greatest common divisor (GCD) of 6050 and 532, which is 2. We divide both the numerator and the denominator by 2:

6050 ÷ 2 = 3025
532 ÷ 2 = 266

So, the simplified result is:

### Answer: 3025/266

Expected answer: 3025/266

Correct? Yes

Problem: After you find the product of the fractions 11/19 and 25/7, suppose that we initially used 25/7 instead of 22/7. How does the result change? Write the new result as a simplified fraction.

Llama 3 70B:

To calculate the product of the fractions, we multiply the numerators (11 and 25) and multiply the denominators (19 and 7), then simplify the resulting fraction.

(11/19) × (25/7) = (11 × 25) / (19 × 7) = 275 / 133

The resulting fraction 275/133 is already in its simplest form, as the greatest common divisor of 275 and 133 is 1.

### Answer: 275/133

Expected answer: 275/133

Correct? Yes

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 70B:

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

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

2128 ÷ 4 = 532
600 ÷ 4 = 150

So, the final result is:

### Answer: 532/150

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 70B:

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

(9/30) × (19/5) = (9 × 19) / (30 × 5)

= 171 / 150

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

171 ÷ 3 = 57
150 ÷ 3 = 50

So, the result of multiplying 9/30 by 19/5 is:

### 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 70B:

To multiply the result by itself, we need to square the fraction 253/120:

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

= 64009 / 14400

To simplify this fraction, we can look for common factors between the numerator and the denominator. The greatest common divisor (GCD) of 64009 and 14400 is 1, which means the fraction is already in its simplest form.

### Answer: 64009/14400

Expected answer: 64009/14400

Correct? Yes

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 70B:

Let's multiply the fractions 11/15 and 26/8:

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

= 286 / 120

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

286 ÷ 2 = 143
120 ÷ 2 = 60

So, the simplified fraction is:

### Answer: 143/60

Expected answer: 143/60

Correct? Yes

Problem: What is the result when the value from the previous problem ((29 / 16) * (21 / 27)) is multiplied by itself? Write your answer as a simplified fraction.

Llama 3 70B:

We already found the result of (29 / 16) * (21 / 27) to be 203/144. Now, we need to multiply this result by itself:

(203/144) × (203/144)

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

(203 × 203) / (144 × 144) = 41209 / 20736

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

### Answer: 41209/20736

Expected answer: 41209/20736

Correct? Yes

Problem: In the question before, we considered the multiplication of fractions (29 / 16) and (21 / 27). Now replace the fraction 16 in the denominator with 5. What is the result of this new operation? Present your answer as a simplified fraction. In other words, what is the output of ((29 / 5) * (21 / 27))?

Llama 3 70B:

Let's evaluate the new expression:

((29 / 5) * (21 / 27))

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

(29 × 21) / (5 × 27)

= 609 / 135

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

609 ÷ 3 = 203
135 ÷ 3 = 45

So, the simplified result is:

### Answer: 203/45

Expected answer: 203/45

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 70B:

We already found the result of (28 / 1) divided by (30 / 27) to be 126/5. Now, we'll multiply this result by (6 / 19):

(126/5) × (6/19) = (126 × 6) / (5 × 19) = 756 / 95

This fraction is already in its simplest form, so our final answer is:

### Answer: 756/95

Expected answer: 756/95

Correct? Yes

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 70B:

Let's evaluate the new expression:

(28 / 1) = 28

Now, we'll divide 28 by 11:

28 ÷ 11 = 28/11

So, the result is:

### 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 70B:

We already found the result of ((25 / 25) * (20 / 7)) to be 20/7. Now, we need to multiply this result by itself:

(20/7) × (20/7) = (20 × 20) / (7 × 7) = 400/49

So, the result is 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 70B:

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

1. Evaluate the expressions inside the parentheses:
   (25 / 25) = 1
   (11 / 7) = 11/7

2. Multiply the results:
   1 * (11/7) = 11/7

So, the result is 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 70B:

To multiply 30/1 with 16/27, we can multiply the numerators (the top numbers) and multiply the denominators (the bottom numbers):

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

To simplify this fraction, we can divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 480 and 27 is 3, so we can divide both numbers by 3:

480 ÷ 3 = 160
27 ÷ 3 = 9

So, the simplified fraction is:

### Answer: 160/9

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 70B:

We previously calculated the result of ((27 / 3) / (12 / 12)) as s = 9.

Now, let's multiply s by the fraction (21/17):

b = s × (21/17)
= 9 × (21/17)

To multiply a whole number by a fraction, we can multiply the whole number by the numerator and divide by the denominator:

b = (9 × 21) / 17
= 189 / 17

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

### Answer: b = 189/17

Expected answer: 189/17

Correct? Yes

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 70B:

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

1. Evaluate the expression inside the parentheses:
   (12 / 12) = 1

2. Now, rewrite the expression with the evaluated value:
   (24 / 1)

3. Simplify the fraction:
   24 / 1 = 24

### Answer: 24

The original answer 's' was 9. By reducing the number of items from 27 to 24, the result of the expression has increased to 24. This change has tripled the original answer 's'.

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 70B:

When we multiply a fraction by (18/18), we are essentially multiplying by 1, since 18/18 = 1. Multiplying any number by 1 does not change its value.

So, the new result is still:

435 / 32

Multiplying by (18/18) does not change the value, it's like multiplying by 1.

### 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 70B:

Let's follow the same steps:

1. Divide 15 by 16: 15 ÷ 16 = 15/16
2. Divide 13 by 2: 13 ÷ 2 = 13/2
3. Multiply the two results: (15/16) × (13/2)

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

(15 × 13) / (16 × 2) = 195 / 32

So, the result of the operation is:

### Answer: 195/32

Expected answer: 195/32

Correct? Yes

Problem: Following your previous calculation, how would the result change if the first fraction was actually ((6 / 2) / 3) instead of ((6 / 2) / (13 / 9))? Calculate and write your answer as a simplfied fraction.

Llama 3 70B:

Let's re-evaluate the expression with the new fraction:

((6 / 2) / 3)

1. Evaluate the expression inside the parentheses:
   (6 / 2) = 3

2. Now, divide 3 by 3:
   3 / 3 = 1

### Answer: 1

Expected answer: 1

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 70B:

We already found that dividing 7 wholes by 15/9 gives us 21/5. Now, we need to multiply this result by 10/5:

(21/5) × (10/5) = ?

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

(21 × 10) / (5 × 5) = 210/25

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

210 ÷ 5 = 42
25 ÷ 5 = 5

So, the simplified fraction is:

### 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 70B:

To divide 20/1 by 15/9, we can invert the second fraction (i.e., flip the numerator and denominator) and then multiply:

(20/1) ÷ (15/9) = (20/1) × (9/15)

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

(20 × 9) / (1 × 15) = 180/15

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

180 ÷ 15 = 12
15 ÷ 15 = 1

So, the simplified fraction is:

### Answer: 12/1 or 12

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 70B:

We already calculated the result of (23/10) × (9/1) to be 207/10.

Now, we need to multiply 207/10 by 1/16.

(207/10) × (1/16) = (207 × 1) / (10 × 16) = 207 / 160

We can simplify the fraction 207/160 by dividing both the numerator and the denominator by their greatest common divisor, which is 1.

207 / 160 = 1.29375

### Answer: 1.29375

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 70B:

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

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

We can simplify the fraction 207/60 by dividing both the numerator and the denominator by their greatest common divisor, which is 3.

207 ÷ 3 = 69
60 ÷ 3 = 20

So, 207/60 = 69/20

### Answer: 69/20 or 3.45

Expected answer: 69/20

Correct? Yes

Problem: After calculating the value of ((18 / 4) * (10 / 11)), now multiply this result by the fraction (6 / 4). What is the final value? Write your answer as a simplified fraction.

Llama 3 70B:

We already calculated the value of ((18 / 4) * (10 / 11)) to be 45/11.

Now, we need to multiply this result by the fraction (6 / 4):

(45/11) * (6/4) = (45*6) / (11*4) = 270 / 44

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

270 / 44 = (270/2) / (44/2) = 135 / 22

### Answer: 135/22

Expected answer: 135/22

Correct? Yes

Problem: After calculating the value of ((18 / 1) * (10 / 11)), consider what would happen if instead of dividing 18 by 4 in the initial step, you were to divide it by 1. Calculate the new result. What is the final answer? Write your answer as a simplified fraction.

Llama 3 70B:

Let's recalculate the expression with the new initial step:

1. Divide 18 by 1: 18 / 1 = 18
2. Divide 10 by 11: 10 / 11 = 10/11
3. Multiply the results: 18 * (10/11) = (18*10) / 11 = 180 / 11

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

### Answer: 180/11

Expected answer: 180/11

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 70B:

We already found the result of multiplying 24/23 by 23/3, which is 184/23. Now, we need to multiply this result by 30/21.

(184/23) × (30/21) = (184 × 30) / (23 × 21)

= 5520 / 483

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

5520 ÷ 3 = 1840
483 ÷ 3 = 161

So, the result is:

1840 / 161

### Answer: 1840/161

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 70B:

Let's multiply the fractions again:

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

= 432 / 69

We can simplify this 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 result is:

144 / 23

### Answer: 144/23

Expected answer: 144/23

Correct? Yes

Problem: After finding the solution for the division of fractions (20 / 13) / (17 / 16), now multiply that result by the fraction (20 / 26). Write your answer as a simplified fraction.

Llama 3 70B:

We already found the solution for the division of fractions:

(20 / 13) / (17 / 16) = 320/221

Now, we need to multiply this result by the fraction (20 / 26):

(320/221) × (20/26)

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

= (320 × 20) / (221 × 26)
= 6400 / 5746

We can simplify this fraction by finding the greatest common divisor (GCD) of the numerator and denominator, which is 2. So, we can divide both numbers by 2:

= 3200 / 2873

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

### Answer: 3200/2873

Expected answer: 3200/2873

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 70B:

We know that c = 4 from the previous problem. Now, let's evaluate the given formula:

((21 divided by 9) multiplied by c)
= ((21/9) × 4)
= (7/3) × 4
= (7 × 4)/3
= 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 70B:

Let's re-evaluate the expression with the new initial amount:

((5/15) divided by 5/15) multiplied by (4 divided by 1)

1. Evaluate the expressions inside the parentheses:
	* (5/15) divided by (5/15) = 1 (since any non-zero value divided by itself is 1)
	* 4 divided by 1 = 4
2. Multiply the results:
	* 1 multiplied by 4 = 4

However, since the initial amount was a fraction (5/15), we need to multiply 4 by this fraction to get the final result:

4 × (5/15) = (4 × 5) / 15 = 20 / 15

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

20 ÷ 5 = 4
15 ÷ 5 = 3

So, the simplified fraction is:

### Answer: 4/3

Expected answer: 4/3

Correct? Yes

Problem: After calculating the previous problem of ((30 / 23) * (19 / 24)) and getting the result, now proceed to multiply this result with (28 / 4). What is your new answer when written as a simplified fraction?

Llama 3 70B:

We already calculated the result of ((30 / 23) * (19 / 24)) to be 95/92. Now, we need to multiply this result by (28 / 4).

First, simplify the fraction (28 / 4) by dividing both numerator and denominator by their greatest common divisor, which is 4:

28 ÷ 4 = 7
4 ÷ 4 = 1

So, (28 / 4) = 7/1 = 7.

Now, multiply 95/92 by 7:

(95/92) × 7 = (95 × 7) / (92 × 1) = 665 / 92

This fraction is already in its simplest form, as 665 and 92 do not have any common factors other than 1.

### Answer: 665/92

Expected answer: 665/92

Correct? Yes

Problem: After calculating the previous problem and getting a result, let's change the condition. Instead of (30 / 23) which you used last time, now use (30 / 11). Repeat the multiplication with (19 / 24). What is the new result when you write it as a simplified fraction?

Llama 3 70B:

Let's recalculate the expression with the new value:

((30 / 11) * (19 / 24))

1. Divide 30 by 11: 30/11 = 30/11 (cannot be simplified further)
2. Divide 19 by 24: 19/24 = 19/24 (cannot be simplified further)
3. Multiply the two results: (30/11) × (19/24) = (30 × 19) / (11 × 24) = 570 / 264

Now, we can simplify the fraction by finding the greatest common divisor (GCD) of 570 and 264, which is 6. Divide both numerator and denominator by 6:

570 ÷ 6 = 95
264 ÷ 6 = 44

So, the simplified fraction is:

### Answer: 95/44

Expected answer: 95/44

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 70B:

We know that g = 184/13. Now, we need to multiply g by (4/7):

(4/7) × (184/13) = ?

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

 Numerator: 4 × 184 = 736
 Denominator: 7 × 13 = 91

So, the result is: 736/91

This fraction is already in its simplest form, as 736 and 91 do not have any common factors.

### Answer: 736/91

Expected answer: 736/91

Correct? Yes

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 70B:

Let's evaluate the new expression:

(2 / 26) * (16 / 1)

1. Divide 2 by 26: 2 ÷ 26 = 2/26
2. Divide 16 by 1: 16 ÷ 1 = 16
3. Multiply the two results: (2/26) × 16 = (2 × 16) / 26

Now, let's multiply 2 and 16: 2 × 16 = 32

So, the result is: 32/26

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

32 ÷ 2 = 16
26 ÷ 2 = 13

Therefore, the simplified fraction is: 16/13

### Answer: 16/13

Expected answer: 16/13

Correct? Yes

Problem: In continuation to the previous problem, let's assume that the value obtained from the expression ((21 / 1) / (26 / 20)) is multiplied by the fraction (21 / 29). What fraction will the result be?

Llama 3 70B:

We already know that ((21 / 1) / (26 / 20)) = 210/13.

Now, let's multiply this result by (21 / 29):

(210/13) × (21/29) = (210 × 21) / (13 × 29) = 4410 / 377

We can simplify this fraction by finding the greatest common divisor (GCD) of 4410 and 377, which is 1. Therefore, the fraction is already in its simplest form.

### Answer: 4410/377

Expected answer: 4410/377

Correct? Yes

Problem: Let's revisit the previous problem. Let's assume now that the value obtained from the expression ((21 / 1) / 2). What will the result be now?

Llama 3 70B:

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

1. Evaluate the expression inside the parentheses:
   (21 / 1) = 21

2. Now, divide 21 by 2:
   21 / 2 = 21/2

So, the result is:
### Answer: 21/2

Expected answer: 21/2

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 70B:

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

(20/19) × (21/19) = (20 × 21) / (19 × 19) = 420 / 361

### Answer: 420/361

Expected answer: 420/361

Correct? Yes

Problem: Jim realized he made a mistake while measuring components for his cake recipe. He actually needed to use 30/14 cups of flour instead of the 17/14 cups he previously used. Given that the amount of sugar remains the same at 12/2 cups, how much sugar and flour does he actually need? Calculate ((12 / 2) * (30 / 14)) and express your answer as a simplified fraction.

Llama 3 70B:

Let's recalculate the total amount of sugar and flour needed.

First, let's simplify the fractions:

12/2 = 6 (since 12 divided by 2 is 6)
30/14 = 15/7 (since 30 divided by 2 is 15, and 14 divided by 2 is 7)

Now, let's multiply the two fractions:

(6) × (15/7) = 6 × 15/7 = 90/7

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

### Answer: 90/7

Expected answer: 90/7

Correct? Yes

Problem: Following the previous problem, if the result obtained from the operation ((21 / 16) / (27 / 23)) was squared, what would be the value of this new operation? Write the result as a simplified fraction.

Llama 3 70B:

We already calculated the result of the operation ((21 / 16) / (27 / 23)) as 161/144. To find the square of this result, we can multiply it by itself:

(161/144) × (161/144) = (161 × 161) / (144 × 144) = 25921 / 20736

To simplify this fraction, we can find the greatest common divisor (GCD) of 25921 and 20736:

GCD(25921, 20736) = 1 (since they are coprime)

The fraction is already in its simplest form.

### Answer: 25921/20736

Expected answer: 25921/20736

Correct? Yes

Problem: Given the previous problem where we multiplied and squared the result of ((21 / 16) / (27 / 23)), let's consider a new scenario. What would be the result if, instead of 21 / 16, we had 6? So, the new calculation would be (6 / (27 / 23)). Write the result as a simplified fraction.

Llama 3 70B:

Let's evaluate the new expression:

(6 / (27 / 23))

1. Evaluate the expression inside the parentheses:
   (27 / 23) = 27/23

2. Now, divide 6 by the fraction:
   6 / (27/23) = 6 × (23/27)

3. Multiply the numerators and multiply the denominators:
   (6 × 23) / 27 = 138 / 27

4. Simplify the fraction by finding the greatest common divisor (GCD) of 138 and 27:
   GCD(138, 27) = 3

5. Divide both the numerator and the denominator by the GCD:
   (138 ÷ 3) / (27 ÷ 3) = 46 / 9

### Answer: 46/9

Expected answer: 46/9

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 70B:

We need to multiply 11/10 by 21/16.

 Numerators: 11 * 21 = 231
 Denominators: 10 * 16 = 160

So, the result of the multiplication is 231/160. This fraction is already in its simplest form, as the greatest common divisor of 231 and 160 is 1.

### Answer: 231/160

Expected answer: 231/160

Correct? Yes

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 70B:

Let's evaluate the new expression:

(11 / 6) * (24 / 12)

First, we multiply the numerators and multiply the denominators:

 Numerators: 11 * 24 = 264
 Denominators: 6 * 12 = 72

So, the result of the multiplication is 264/72. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 12.

264 ÷ 12 = 22
72 ÷ 12 = 6

So, the simplified fraction is 22/6. We can further simplify this by dividing both the numerator and the denominator by their greatest common divisor, which is 2.

22 ÷ 2 = 11
6 ÷ 2 = 3

### Answer: 11/3

Expected answer: 11/3

Correct? Yes

Problem: After calculating the answer to ((6 / 19) * (28 / 2)), take that result and multiply it by (27 / 17). What is the final answer as a simplified fraction?

Llama 3 70B:

We already calculated the result of ((6 / 19) * (28 / 2)) to be 84/19. Now, we need to multiply it by (27 / 17):

(84/19) × (27/17) = (84 × 27) / (19 × 17) = 2268 / 323

To simplify this fraction, we can look for common factors between the numerator and the denominator. However, 2268 and 323 are coprime, meaning they do not have any common factors other than 1. Therefore, the fraction is already in its simplest form:

### Answer: 2268/323

Expected answer: 2268/323

Correct? Yes

Problem: Previously, you calculated the answer to ((6 / 19) * (28 / 2)). Now, consider if you had originally calculated ((17 / 19) * (28 / 2)) instead of ((6 / 19) * (28 / 2)). What would be the result of this calculation as a simplified fraction?

Llama 3 70B:

Let's evaluate the new expression:

1. Divide 17 by 19: 17/19
2. Divide 28 by 2: 28/2 = 14
3. Multiply the results: (17/19) × 14 = (17 × 14) / 19 = 238 / 19

So, the simplified fraction is:

### Answer: 238/19

Expected answer: 238/19

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 70B:

We are given the fraction 24/5 and we need to multiply it by 15/10.

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

(24/5) * (15/10) = (24*15) / (5*10) = 360 / 50

To simplify this fraction to its lowest terms, we can divide both the numerator and the denominator by their greatest common divisor (GCD), which is 10:

360 ÷ 10 = 36
50 ÷ 10 = 5

So, the simplified fraction is: 36/5

### Answer: 36/5

Expected answer: 36/5

Correct? Yes

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 70B:

Let's evaluate the expression:

(3/14) * (28/5)

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

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

To simplify the resulting fraction to its lowest terms, we can divide both the numerator and the denominator by their greatest common divisor (GCD), which is 14:

84 ÷ 14 = 6
70 ÷ 14 = 5

So, the simplified fraction is: 6/5

### Answer: 6/5

Expected answer: 6/5

Correct? Yes

Problem: After you've calculated the result of (19 / 3) * (8 / 15), multiply your answer by (19 / 9). Write your final answer as a simplified fraction.

Llama 3 70B:

We already calculated the result of (19 / 3) * (8 / 15) to be 152/45.

Now, we need to multiply this result by (19 / 9):

(152/45) * (19/9) = (152*19) / (45*9) = 2896 / 405

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

### Answer: 2896/405

Expected answer: 2888/405

Correct? No

Problem: First, recalculate the previous problem using (19 / 7) instead of (19 / 3). So the new calculation is (19 / 7) * (8 / 15). What is the result? Write your answer as a simplified fraction.

Llama 3 70B:

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

1. Divide 19 by 7: 19 / 7 = 19/7
2. Divide 8 by 15: 8 / 15 = 8/15
3. Multiply the two results: (19/7) * (8/15) = (19*8) / (7*15) = 152 / 105

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

### Answer: 152/105

Expected answer: 152/105

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 70B:

We already found the result of ((15 / 3) * (7 / 25)) to be 7/5. Now, we need to multiply it by (28 / 16):

(7/5) * (28/16)

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

(7 * 28) / (5 * 16) = 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:

### 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 70B:

Let's evaluate the new expression:

((13 / 3) * (7 / 25))

1. Divide 13 by 3: 13 / 3 = 13/3
2. Divide 7 by 25: 7 / 25 = 7/25
3. Multiply the results: (13/3) * (7/25) = 91/75

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

Comparing this result to the previous one, we can see that the change in scenario (replacing 15 with 13) affects the final result by changing the numerator from 35 to 91, and the denominator from 25 to 75. The resulting fraction, 91/75, is different from the original 7/5.

### Answer: 91/75

Expected answer: 91/75

Correct? Yes

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 70B:

We already calculated x = 20.

Now, we need to multiply x by the fraction (19 / 15):

z = x * (19 / 15)
= 20 * (19 / 15)
= (20 * 19) / 15
= 380 / 15

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

z = 380 / 15
= (380 ÷ 5) / (15 ÷ 5)
= 76 / 3

### Answer: 76/3

Expected answer: 76/3

Correct? Yes

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 70B:

To multiply the result by the fraction (28/16), I'll perform the following calculation:

(57/16) × (28/16) = (57 × 28) / (16 × 16) = 1596 / 256

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

1596 ÷ 8 = 199
256 ÷ 8 = 32

So, the result of multiplying the previous result by the fraction (28/16) is:

199/32

### Answer: 199/32

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 70B:

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

1. Divide 27 by 18: 27 ÷ 18 = 3/2
2. Divide 19 by 17: 19 ÷ 17 = 19/17
3. Multiply the two results: (3/2) × (19/17) = (3 × 19) / (2 × 17) = 57/34

So, the result of the expression [(27 / 18) * (19 / 17)] is 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 70B:

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

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

= 9425 / 1320

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

9425 ÷ 5 = 1885
1320 ÷ 5 = 264

So, the simplified fraction is:

1885 / 264

### Answer: 1885/264

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 70B:

Let's recalculate:

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

= 377 / 192

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

377 / 192

### 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 70B:

We already know that (27 / 6) * (25 / 5) = 22.5. Now, we need to multiply this result by (20 / 27):

1. Divide 20 by 27: 20 / 27 = 20/27
2. Multiply 22.5 by the result: 22.5 * (20/27) = 22.5 * (20/27) = 20

### Answer: 20

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 70B:

Let's recalculate:

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 70B:

We are given that e = 13/6. We need to evaluate the expression (13 / 2) * e.

Substituting the value of e, we get:

(13 / 2) * (13 / 6)

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

= ((13 * 13) / (2 * 6))
= (169 / 12)

So, the simplified fraction 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 70B:

To evaluate this expression, we need to follow the order of operations (PEMDAS). Since we only have multiplication, we can proceed directly.

(11 / 1) * (1 / 6) = 11 * (1 / 6)

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

= (11 * 1) / (1 * 6)
= 11 / 6

So, the simplified fraction is:

### 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 70B:

We already found the product of 25/20 and 11/6, which 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 70B:

Let's calculate the product of 25/20 and 11/6 first:

(25/20) × (11/6) = (25 × 11) / (20 × 6)

= 275 / 120

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

(275 / 120) × (20 / 26) = (275 × 20) / (120 × 26)

= 5500 / 3120

To simplify this fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 20.

5500 ÷ 20 = 275
3120 ÷ 20 = 156

So, the simplified fraction is:

275/156

Now, let's consider what would happen if we initially multiplied 25/17 instead of 25/20 by 11/6:

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

= 275 / 102

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

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

= 5500 / 2652

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

5500 ÷ 4 = 1375
2652 ÷ 4 = 663

So, the simplified fraction is:

1375/663

### Answer: 1375/663

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 70B:

We already determined that ((7 / 2) / (25 / 10)) = 7/5.

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

(7/5) × (5/3) = (7 × 5) / (5 × 3) = 35 / 15

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

35 ÷ 5 = 7
15 ÷ 5 = 3

So, the simplified fraction is:

### 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 70B:

We already calculated the product of 15/16 and 29/27, which is 145/144.

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

 Numerator: 145 × 27 = 3915
 Denominator: 144 × 4 = 576

So, the product is 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 fraction 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 70B:

Let's calculate the multiplication of the fractions 15/10 and 29/27.

Numerator: 15 × 29 = 435
Denominator: 10 × 27 = 270

So, the product of the two fractions is 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 fraction is 29/18.

### 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 70B:

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

Now, we need to multiply 3 by (5 / 9):

3 * (5 / 9) = 3 * 5 / 9 = 15 / 9

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

15 ÷ 3 = 5
9 ÷ 3 = 3

So, the result is:

### 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 70B:

To calculate the product of the fractions, we multiply the numerators (27 and 27) and multiply the denominators (10 and 3), then simplify the resulting fraction.

(27/10) × (27/3) = (27 × 27) / (10 × 3) = 729 / 30

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

729 ÷ 3 = 243
30 ÷ 3 = 10

So, the simplified product is:

243/10

Now, we multiply the result by 9/20:

(243/10) × (9/20) = (243 × 9) / (10 × 20) = 2187 / 200

This fraction is already in simplest form.

### Answer: 2187/200

Expected answer: 2187/200

Correct? Yes

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 70B:

To calculate the product of the fractions, we multiply the numerators (27 and 27) and multiply the denominators (10 and 21), then simplify the resulting fraction.

(27/10) × (27/21) = (27 × 27) / (10 × 21) = 729 / 210

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

729 ÷ 21 = 35
210 ÷ 21 = 10

So, the simplified product is:

35/10

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

35 ÷ 5 = 7
10 ÷ 5 = 2

So, the final answer in simplest form is:

7/2

### Answer: 7/2

Expected answer: 243/70

Correct? No

Problem: After calculating the result of the following operation: (10 / 8) * (30 / 26), find the square of your answer by multiplying it with itself. Write the final answer as a simplified fraction.

Llama 3 70B:

We already calculated the result of the operation:

(10 / 8) * (30 / 26) = 75/52

Now, we need to find the square of this result by multiplying it with itself:

(75/52) * (75/52) = (75*75) / (52*52) = 5625/2704

So, the square of the result is:

### Answer: 5625/2704

Expected answer: 5625/2704

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 70B:

We already evaluated the expression ((20 / 22) * (14 / 11)) to be 140/121. Now, we need to multiply it by (17/3):

(140/121) × (17/3) = (140 × 17) / (121 × 3) = 2380/363

To simplify this fraction, we can look for common factors between the numerator and the denominator. However, there are no common factors, so the fraction is already in its simplest form:

### Answer: 2380/363

Expected answer: 2380/363

Correct? Yes

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 70B:

Let's evaluate the new expression:

((20 / 22) * (14 / 2))

1. Divide 20 by 22: 20/22 = 10/11
2. Divide 14 by 2: 14/2 = 7
3. Multiply the two results: (10/11) × 7 = (10 × 7) / 11 = 70/11

So, the simplified fraction is:

### Answer: 70/11

Expected answer: 70/11

Correct? Yes

Problem: After finding the result of ((25 / 8) * (3 / 4)), now take your answer and multiply it by the fraction (29 / 23). Write this new result as a simplified fraction. What would this additional operation represent in terms of the original division or portion sharing scenario?

Llama 3 70B:

Let's multiply the result by the fraction (29/23):

(75/32) × (29/23) = ?

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

(75 × 29) / (32 × 23) = 2175 / 736

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

2175 ÷ 1 = 2175
736 ÷ 1 = 736

So, the simplified fraction is:

2175/736

In the context of the original scenario, this additional operation could represent a further division or scaling of the original portion. For example, if each person was originally getting 75/32 of a cookie, and now we want to divide each of those portions into 29 equal parts, but only give out 23 of those parts, the new portion size would be 2175/736 of a cookie.

This could represent a situation where the original portions were too large, so we need to sub-divide them further, but we don't want to give out the full sub-divided portion, so we're only giving out a fraction of it.

Expected answer: 2175/736

Correct? No

Problem: After solving the previous problem, now evaluate the new expression ((25 / 8) * (29 / 4)). This means we have changed the second fraction from (3 / 4) to (29 / 4). Under the previous scenario where the fractions referred to sharing or dividing amongst a group, consider what would happen if the proportions you multiplied with were changed to this new ratio. Write the result of ((25 / 8) * (29 / 4)) as a simplified fraction and reflect on the impact of this change in the real-world scenario.

Llama 3 70B:

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

1. Divide 25 by 8: 25 ÷ 8 = 25/8 = 3 1/8
2. Multiply the result by 29/4: (3 1/8) × (29/4) = (25/8) × (29/4) = 725/32

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

725 ÷ 1 = 725
32 ÷ 1 = 32

So, the simplified fraction is:

725/32

Now, let's reflect on the impact of this change in the real-world scenario. Previously, we had 75 cookies to divide among 32 people, resulting in each person getting 75/32 of a cookie. With the new ratio, we're multiplying the original amount by 29/4 instead of 3/4.

This means we're increasing the proportion of cookies each person gets by a factor of 29/3. In other words, each person would now get approximately 9.67 times more cookies than before (29/3 ≈ 9.67).

In this scenario, the change in ratio would result in a significant increase in the amount of cookies each person receives. Instead of getting a small fraction of a cookie, each person would now get a substantial amount, almost 10 times more than before. This could represent a change in policy or a decision to be more generous in the distribution of resources.

Expected answer: 725/32

Correct? No

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 70B:

We already know that (29/17) × (26/15) = 754/255. Now, we need to multiply this result by (21/9).

(754/255) × (21/9) = (754 × 21) / (255 × 9)
= 15894 / 2295

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

15894 ÷ 21 = 758
2295 ÷ 21 = 109

So, the simplified result is:

758 / 109

### Answer: 758/109

Expected answer: 5278/765

Correct? No

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 70B:

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

(15/17) × (26/15) = (15 × 26) / (17 × 15)
= 390 / 255

We can simplify this fraction by canceling out the common factor of 15 in the numerator and the denominator:

390 ÷ 15 = 26
255 ÷ 15 = 17

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 70B:

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

1. Divide 25 by 5: 25 / 5 = 5
2. Divide 19 by 8: 19 / 8 = 19/8
3. Multiply the 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: What would be the result if you changed the previous problem by multiplying (28 / 4) by (10 / 12) instead of (23 / 4) by (10 / 12)? Write your answer as a simplified fraction.

Llama 3 70B:

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

(28 / 4) × (10 / 12) = (28 × 10) / (4 × 12)

= 280 / 48

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

280 ÷ 4 = 70
48 ÷ 4 = 12

So, the simplified fraction is:

70 / 12

### Answer: 70/12 or 35/6

Expected answer: 35/6

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 70B:

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

(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 6:

420 ÷ 6 = 70
234 ÷ 6 = 39

So, the result of the multiplication in simplest form is:

70/39

Now, let's multiply this result by the fraction (15/21):

(70/39) * (15/21) = (70*15) / (39*21)

= 1050 / 819

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

1050 ÷ 3 = 350
819 ÷ 3 = 273

So, the final answer in simplest form is:

350/273

### Answer: 350/273

Expected answer: 50/39

Correct? Yes

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 70B:

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

(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 result of the multiplication in simplest form is:

105/26

### Answer: 105/26

Expected answer: 105/26

Correct? Yes

Problem: In the previous problem, you calculated the product of ((17 / 13) * (15 / 6)) and found it as a simplified fraction. Now, use that answer and further multiply it by (8 / 15). Write your final answer as a simplified fraction.

Llama 3 70B:

We already calculated the product ((17 / 13) * (15 / 6)) as 85/26. Now, we need to multiply it by (8 / 15):

(85/26) × (8/15)

To multiply fractions, we multiply the numerators (85 and 8) and multiply the denominators (26 and 15), then simplify:

(85 × 8) / (26 × 15) = 680 / 390

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 10:

680 ÷ 10 = 68
390 ÷ 10 = 39

So, the simplified fraction is:

### Answer: 68/39

Expected answer: 68/39

Correct? Yes

Problem: After calculating the result of the initial problem: (25 / 18) * (17 / 6), proceed to multiply your answer by the fraction (23 / 17). Write your final answer as a simplified fraction.

Llama 3 70B:

We already calculated the result of the initial problem:

(25 / 18) * (17 / 6) = 425 / 108

Now, we need to multiply this result by the fraction (23 / 17):

(425 / 108) × (23 / 17)

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

(425 × 23) / (108 × 17) = 9795 / 1836

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

9795 ÷ 3 = 3265
1836 ÷ 3 = 612

So, the final answer is:

### Answer: 3265/612

Expected answer: 575/108

Correct? No

Problem: After initially calculating the result of the problem: (25 / 18) * (17 / 6), let's consider a change. Assume that instead of 17 portions in the second fraction, you only had 14 portions to begin with. Recalculate the product: (25 / 18) * (14 / 6). Write your updated answer as a simplified fraction.

Llama 3 70B:

Let's recalculate the product with the new value:

(25 / 18) * (14 / 6)

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

(25 × 14) / (18 × 6) = 350 / 108

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

350 ÷ 2 = 175
108 ÷ 2 = 54

So, the updated result of the equation is:

### Answer: 175/54

Expected answer: 175/54

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 70B:

We know that t = 18/17. To find t × t, we can multiply the fraction by itself:

t × t = (18/17) × (18/17)

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

t × t = (18 × 18)/(17 × 17) = 324/289

The resulting fraction 324/289 is already in its simplest form, so we don't need to simplify it further.

### Answer: 324/289

Expected answer: 324/289

Correct? Yes

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 70B:

To multiply 12 (which is equivalent to 12/1) by 2/17, we can multiply the numerators (12 and 2) and multiply the denominators (1 and 17), then simplify the resulting fraction.

(12/1) × (2/17) = (12 × 2)/(1 × 17) = 24/17

The resulting fraction 24/17 is already in its simplest form, so we don't need to simplify it further.

### Answer: 24/17

Expected answer: 24/17

Correct? Yes

Problem: Having found the value of z from the previous problem ((27 / 14) * (19 / 14)), now multiply your answer by the fraction 17/23. Write your answer as a simplified fraction.

Llama 3 70B:

We'll multiply the result from the previous problem by the fraction 17/23:

(513/196) × (17/23) = ?

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

(513 × 17) / (196 × 23) = 8711 / 4508

The resulting fraction is already in its simplest form.

### Answer: 8711/4508

Expected answer: 8721/4508

Correct? No

Problem: Previously, you calculated the result of the multiplication (27 / 14) * (19 / 14). Now, what if instead of 27, we had a value of 15? So calculate the new result for the multiplication problem (15 / 14) * (19 / 14). Write your answer as a simplified fraction.

Llama 3 70B:

Let's recalculate the result of the multiplication with the new value:

(15 ÷ 14) = 15/14 = 1 1/14
(19 ÷ 14) = 19/14 = 1 5/14

Now, we'll multiply the two fractions:

(1 1/14) × (1 5/14) = ?

Converting the mixed numbers to improper fractions:

(1 1/14) = (14 + 1)/14 = 15/14
(1 5/14) = (14 + 5)/14 = 19/14

Now, we can multiply the fractions:

(15/14) × (19/14) = (15 × 19) / (14 × 14) = 285 / 196

The resulting fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 7:

285 ÷ 7 = 41
196 ÷ 7 = 28

So, the simplified fraction is:

### Answer: 41/28

Expected answer: 285/196

Correct? No

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 70B:

We already found that (16 / 5) * (5 / 15) = 16/15.

Now, we need to multiply 16/15 by 30/24:

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

To simplify this fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 120:

480 ÷ 120 = 4
360 ÷ 120 = 3

So, the result of the multiplication is:

### Answer: 4/3

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 70B:

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

1. Divide 16 by 5: 16 / 5 = 16/5
2. Divide 5 by 7: 5 / 7 = 5/7
3. Multiply the results: (16/5) * (5/7) = 16/7

So, the result of the expression is:

### 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 70B:

We already calculated the result of the multiplication (8 / 3) * (18 / 21) to be 16/7. Now, we need to multiply this result by the fraction (23 / 20).

To do this, we multiply the numerators and denominators:

(16/7) * (23/20) = (16 * 23) / (7 * 20)

Next, we multiply the numerators and denominators:

(16 * 23) / (7 * 20) = 368 / 140

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

368 ÷ 4 = 92
140 ÷ 4 = 35

So, the simplified fraction is:

### Answer: 92/35

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 70B:

Let's recalculate the multiplication:

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

Next, we multiply the numerators and denominators:

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

Now, we can simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 1 (since 88 and 63 are coprime).

So, the simplified fraction is:

### Answer: 88/63

Expected answer: 88/63

Correct? Yes

Problem: Based on the previous problem, you found the value of r by evaluating the fraction ((14 / 6) / (22 / 16)). For this problem, multiply the result you obtained previously by 9/4 and write your answer as a simplified fraction.

Llama 3 70B:

We already found the value of r to be 56/33. Now, we need to multiply it by 9/4:

(56/33) * (9/4) = (56*9)/(33*4) = 504/132

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

504 ÷ 12 = 42
132 ÷ 12 = 11

So, the simplified fraction is:

42/11

### Answer: 42/11

Expected answer: 42/11

Correct? Yes

Problem: In the previous problem, we evaluated the expression (14 / 6) / (22 / 16). Now, suppose we changed the 14/6 to 13. How would this change the result? Evaluate the new expression, 13 / (22 / 16), and write your answer as a simplified fraction.

Llama 3 70B:

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

1. Evaluate the expression inside the parentheses:
   (22 / 16) = 22/16 = 11/8

2. Now, divide 13 by the result:
   13 / (11/8) = 13 * (8/11) = (13*8)/11 = 104/11

### Answer: 104/11

Expected answer: 104/11

Correct? Yes

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 70B:

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

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

First, let's multiply the numerators:
10 * 25 = 250

Next, let's multiply the denominators:
3 * 26 = 78

Now, we can write the result as a fraction:
250 / 78

We can simplify this fraction by dividing 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 70B:

We know that 'a' is equal to 13/11. To find (a * a), we need to square 13/11:

(a * a) = (13/11) * (13/11)
= (13 * 13) / (11 * 11)
= 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 70B:

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

1. Evaluate the expressions inside the parentheses:
   (16 / 12) = 4/3
   (26 / 22) = 13/11

2. Multiply the two expressions:
   (4/3) * (13/11) = (4*13)/(3*11) = 52/33

### Answer: 52/33

Expected answer: 52/33

Correct? Yes

Problem: After calculating the result of the previous operation: (17 / 5) * (5 / 14), now take that result and multiply it by the fraction 21 / 25. What is your final answer, written as a simplified fraction?

Llama 3 70B:

We already calculated the result of the previous operation: (17 / 5) * (5 / 14) = 17/14.

Now, we need to multiply this result by the fraction 21/25:

(17/14) × (21/25)

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

(17 × 21) / (14 × 25) = 357 / 350

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 7:

357 ÷ 7 = 51
350 ÷ 7 = 50

So, the result is: 51/50

### Answer: 51/50

Expected answer: 51/50

Correct? Yes

Problem: After performing the prior operation: (17 / 5) * (5 / 14), consider what would happen if instead of multiplying the value by 5 / 14, we multiplied it by 5 / 12. What would be the result, written as a simplified fraction, in this case?

Llama 3 70B:

Let's evaluate the new expression: (17 / 5) * (5 / 12)

We already know that 17 / 5 = 17/5

Now, multiply 17/5 by 5/12:

(17 × 5) / (5 × 12) = 85 / 60

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

85 ÷ 5 = 17
60 ÷ 5 = 12

So, the result is: 17/12

### Answer: 17/12

Expected answer: 17/12

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 70B:

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

(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 4:

464 ÷ 4 = 116
60 ÷ 4 = 15

So, the simplified fraction is:

116 / 15

### Answer: 116/15

Expected answer: 116/15

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 70B:

We already calculated the result of the previous problem:

(29 / 10) / (18 / 16) = 116/45

Now, we'll multiply this result by (8 / 17):

(116/45) × (8/17)

To multiply fractions, we multiply the numerators (116 × 8 = 928) and multiply the denominators (45 × 17 = 765), then write the result as a fraction:

928/765

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 1 (since 928 and 765 are coprime numbers). Therefore, the simplified fraction is:

### Answer: 928/765

Expected answer: 928/765

Correct? Yes

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 70B:

Let's start with the result from the previous problem: 135/12

Now, we need to multiply this result by the new fraction: (22/14)

To do this, we'll follow the same steps as before:

(135/12) × (22/14) = (135 × 22) / (12 × 14)

First, let's multiply the numerators: 135 × 22 = 2970

Next, let's multiply the denominators: 12 × 14 = 168

Now, we can 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 42:

2970 ÷ 42 = 71
168 ÷ 42 = 4

So, the simplified result is: 71/4

### Answer: 71/4

Expected answer: 495/28

Correct? No

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 70B:

Let's recalculate the result using the correct fractions:

(15/12) × (27/5) = (15 × 27) / (12 × 5)

First, let's multiply the numerators: 15 × 27 = 405

Next, let's multiply the denominators: 12 × 5 = 60

Now, we can 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 70B:

We multiply the result by the fraction 27/15:

1 * (27/15)

To multiply a number by a fraction, we multiply the number by the numerator (27) and divide by the denominator (15):

1 * 27 = 27
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 70B:

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

1. Evaluate the expressions inside the parentheses:
   (7 / 7) = 1 (because 7 divided by 7 is 1)
   (16 / 12) = 4/3 (because 16 divided by 12 is 4/3)

2. Multiply the results:
   1 * (4/3) = 4/3

### Answer: 4/3

Expected answer: 4/3

Correct? Yes

Problem: Suppose you reevaluate the previous problem but use 22 instead of 28 when you perform the multiplication ((22 / 6) * (16 / 24)). What would be the result this time? Write your answer as a simplified fraction.

Llama 3 70B:

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

1. Divide 22 by 6: 22 / 6 = 11/3
2. Divide 16 by 24: 16 / 24 = 2/3
3. Multiply the two results: (11/3) * (2/3) = (11*2)/(3*3) = 22/9

### Answer: 22/9

Expected answer: 22/9

Correct? Yes