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.

GPT-4o:

First, we already found the product of \(\frac{29}{2}\) and \(\frac{19}{28}\) to be \(\frac{551}{56}\).

Next, we need to multiply this result by \(\frac{8}{28}\):

\[
\frac{551}{56} \times \frac{8}{28} = \frac{551 \times 8}{56 \times 28}
\]

First, calculate the new numerator:

\[
551 \times 8 = 4408
\]

Next, calculate the new denominator:

\[
56 \times 28 = 1568
\]

So, the product is:

\[
\frac{4408}{1568}
\]

To simplify this fraction, we need to find the greatest common divisor (GCD) of 4408 and 1568. We can use the Euclidean algorithm to find the GCD.

1. Divide 4408 by 1568 and find the remainder:
\[
4408 \div 1568 \approx 2.81 \quad \text{(integer part is 2)}
\]
\[
4408 - 2 \times 1568 = 4408 - 3136 = 1272
\]

2. Now, divide 1568 by 1272 and find the remainder:
\[
1568 \div 1272 \approx 1.23 \quad \text{(integer part is 1)}
\]
\[
1568 - 1 \times 1272 = 1568 - 1272 = 296
\]

3. Next, divide 1272 by 296 and find the remainder:
\[
1272 \div 296 \approx 4.29 \quad \text{(integer part is 4)}
\]
\[
1272 - 4 \times 296 = 1272 - 1184 = 88
\]

4. Now, divide 296 by 88 and find the remainder:
\[
296 \div 88 \approx 3.36 \quad \text{(integer part is 3)}
\]
\[
296 - 3 \times 88 = 296 - 264 = 32
\]

5. Next, divide 88 by 32 and find the remainder:
\[
88 \div 32 \approx 2.75 \quad \text{(integer part is 2)}
\]
\[
88 - 2 \times 32 = 88 - 64 = 24
\]

6. Now, divide 32 by 24 and find the remainder:
\[
32 \div 24 \approx 1.33 \quad \text{(integer part is 1)}
\]
\[
32 - 1 \times 24 = 32 - 24 = 8
\]

7. Finally, divide 24 by 8 and find the remainder:
\[
24 \div 8 = 3 \quad \text{(integer part is 3)}
\]
\[
24 - 3 \times 8 = 24 - 24 = 0
\]

Since the remainder is now 0, the GCD of 4408 and 1568 is 8.

Now, we can simplify the fraction \(\frac{4408}{1568}\) by dividing both the numerator and the denominator by their GCD, which is 8:

\[
\frac{4408 \div 8}{1568 \div 8} = \frac{551}{196}
\]

### Answer: \(\frac{551}{196}\)

Expected answer: 551/196

Correct? Yes

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.

GPT-4o:

To find the product of \(\frac{29}{2}\) and \(\frac{19}{14}\), we multiply the numerators together and the denominators together:

\[
\frac{29}{2} \times \frac{19}{14} = \frac{29 \times 19}{2 \times 14}
\]

First, calculate the numerator:

\[
29 \times 19 = 551
\]

Next, calculate the denominator:

\[
2 \times 14 = 28
\]

So, the product is:

\[
\frac{551}{28}
\]

To check if this fraction can be simplified, we need to determine if there are any common factors between 551 and 28. 

The prime factorization of 551 is:
\[
551 = 19 \times 29
\]

The prime factorization of 28 is:
\[
28 = 2^2 \times 7
\]

Since there are no common factors between 551 and 28, the fraction \(\frac{551}{28}\) is already in its simplest form.

### Answer: \(\frac{551}{28}\)

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.

GPT-4o:

First, we need to divide 26 by 12.

\[
\frac{26}{12}
\]

Next, we simplify the fraction by finding the greatest common divisor (GCD) of 26 and 12. The GCD of 26 and 12 is 2. We divide both the numerator and the denominator by 2:

\[
\frac{26 \div 2}{12 \div 2} = \frac{13}{6}
\]

Now, we need to multiply 7 by \(\frac{13}{6}\):

\[
7 \times \frac{13}{6} = \frac{7 \times 13}{6} = \frac{91}{6}
\]

The fraction \(\frac{91}{6}\) is already in its simplest form.

### Answer: \(\frac{91}{6}\)

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.

GPT-4o:

We start with the simplified fraction from the previous problem, which is \(\frac{288}{187}\). We need to multiply this by \(\frac{27}{18}\):

\[
\frac{288}{187} \times \frac{27}{18}
\]

First, we multiply the numerators and the denominators:

\[
\frac{288 \times 27}{187 \times 18}
\]

Next, we calculate the products:

\[
288 \times 27 = 7776
\]
\[
187 \times 18 = 3366
\]

So, the expression becomes:

\[
\frac{7776}{3366}
\]

To simplify this fraction, we need to find the greatest common divisor (GCD) of 7776 and 3366. We can use the Euclidean algorithm to find the GCD.

1. Divide 7776 by 3366 and find the remainder:
\[
7776 \div 3366 \approx 2 \quad \text{(quotient)} \quad \text{remainder} = 7776 - 6732 = 1044
\]

2. Now, divide 3366 by 1044 and find the remainder:
\[
3366 \div 1044 \approx 3 \quad \text{(quotient)} \quad \text{remainder} = 3366 - 3132 = 234
\]

3. Next, divide 1044 by 234 and find the remainder:
\[
1044 \div 234 \approx 4 \quad \text{(quotient)} \quad \text{remainder} = 1044 - 936 = 108
\]

4. Now, divide 234 by 108 and find the remainder:
\[
234 \div 108 \approx 2 \quad \text{(quotient)} \quad \text{remainder} = 234 - 216 = 18
\]

5. Finally, divide 108 by 18 and find the remainder:
\[
108 \div 18 \approx 6 \quad \text{(quotient)} \quad \text{remainder} = 108 - 108 = 0
\]

Since the remainder is now 0, the GCD of 7776 and 3366 is 18.

We can now simplify the fraction \(\frac{7776}{3366}\) by dividing both the numerator and the denominator by their GCD, which is 18:

\[
\frac{7776 \div 18}{3366 \div 18} = \frac{432}{187}
\]

Thus, the simplified fraction is:

### Answer: \(\frac{432}{187}\)

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.

GPT-4o:

To solve the expression \(\left(\frac{24}{22}\right) \times \left(\frac{24}{9}\right)\), we first multiply the numerators and the denominators separately:

\[
\frac{24}{22} \times \frac{24}{9} = \frac{24 \times 24}{22 \times 9}
\]

Next, we calculate the products:

\[
24 \times 24 = 576
\]
\[
22 \times 9 = 198
\]

So, the expression becomes:

\[
\frac{576}{198}
\]

To simplify this fraction, we need to find the greatest common divisor (GCD) of 576 and 198. We can use the Euclidean algorithm to find the GCD.

1. Divide 576 by 198 and find the remainder:
\[
576 \div 198 \approx 2 \quad \text{(quotient)} \quad \text{remainder} = 576 - 2 \times 198 = 576 - 396 = 180
\]

2. Now, divide 198 by 180 and find the remainder:
\[
198 \div 180 \approx 1 \quad \text{(quotient)} \quad \text{remainder} = 198 - 180 = 18
\]

3. Next, divide 180 by 18 and find the remainder:
\[
180 \div 18 \approx 10 \quad \text{(quotient)} \quad \text{remainder} = 180 - 180 = 0
\]

Since the remainder is now 0, the GCD of 576 and 198 is 18.

We can now simplify the fraction \(\frac{576}{198}\) by dividing both the numerator and the denominator by their GCD, which is 18:

\[
\frac{576 \div 18}{198 \div 18} = \frac{32}{11}
\]

Thus, the simplified fraction is:

### Answer: \(\frac{32}{11}\)

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.

GPT-4o:

We already know that the result of dividing \( \frac{25}{4} \) by \( \frac{16}{6} \) is \( \frac{75}{32} \).

Now, we need to multiply \( \frac{75}{32} \) by \( \frac{19}{6} \):
\[ \frac{75}{32} \times \frac{19}{6} = \frac{75 \times 19}{32 \times 6} = \frac{1425}{192} \]

Next, we simplify \( \frac{1425}{192} \) by finding the greatest common divisor (GCD) of 1425 and 192. The GCD of 1425 and 192 is 3.

Divide both the numerator and the denominator by their GCD:
\[ \frac{1425 \div 3}{192 \div 3} = \frac{475}{64} \]

Thus, the simplified fraction is:
### Answer: \(\frac{475}{64}\)

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.

GPT-4o:

To solve the problem of dividing 9 by \( \frac{16}{6} \), we need to multiply 9 by the reciprocal of \( \frac{16}{6} \).

First, find the reciprocal of \( \frac{16}{6} \):
\[ \text{Reciprocal of } \frac{16}{6} = \frac{6}{16} \]

Now, multiply 9 by \( \frac{6}{16} \):
\[ 9 \times \frac{6}{16} = \frac{9 \times 6}{16} = \frac{54}{16} \]

Next, simplify \( \frac{54}{16} \) by finding the greatest common divisor (GCD) of 54 and 16. The GCD of 54 and 16 is 2.

Divide both the numerator and the denominator by their GCD:
\[ \frac{54 \div 2}{16 \div 2} = \frac{27}{8} \]

Thus, the simplified fraction is:
### Answer: \(\frac{27}{8}\)

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?

GPT-4o:

We previously found that \(\frac{3}{7} \times \frac{26}{6} = \frac{13}{7}\).

Now, we need to multiply this result by \(\frac{23}{2}\):

\[
\frac{13}{7} \times \frac{23}{2}
\]

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

1. Multiply the numerators:
   \[
   13 \times 23 = 299
   \]

2. Multiply the denominators:
   \[
   7 \times 2 = 14
   \]

So, the product of the fractions is:
\[
\frac{299}{14}
\]

Since 299 and 14 have no common factors other than 1, the fraction \(\frac{299}{14}\) is already in its simplest form.

### Answer: \(\frac{299}{14}\)

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?

GPT-4o:

To determine the result of multiplying the fractions \(\frac{7}{7}\) and \(\frac{26}{6}\), we follow these steps:

1. Multiply the numerators together:
   \[
   7 \times 26 = 182
   \]

2. Multiply the denominators together:
   \[
   7 \times 6 = 42
   \]

So, the product of the fractions is:
\[
\frac{182}{42}
\]

Next, we need to simplify this fraction. To do this, we find the greatest common divisor (GCD) of 182 and 42.

The prime factorization of 182 is:
\[
182 = 2 \times 7 \times 13
\]

The prime factorization of 42 is:
\[
42 = 2 \times 3 \times 7
\]

The common factors are 2 and 7. Therefore, the GCD of 182 and 42 is:
\[
2 \times 7 = 14
\]

We divide both the numerator and the denominator by their GCD (14):
\[
\frac{182 \div 14}{42 \div 14} = \frac{13}{3}
\]

Thus, the simplified fraction is:
\[
\frac{13}{3}
\]

### Answer: \(\frac{13}{3}\)

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.

GPT-4o:

We previously found that \((\frac{17}{10}) \times (\frac{28}{10}) = 4.76\).

Now, we need to multiply this result by \(\frac{11}{18}\):

\[
4.76 \times \frac{11}{18}
\]

First, convert 4.76 to a fraction:

\[
4.76 = \frac{476}{100}
\]

Now, multiply \(\frac{476}{100}\) by \(\frac{11}{18}\):

\[
\frac{476}{100} \times \frac{11}{18} = \frac{476 \times 11}{100 \times 18}
\]

Calculate the product of the numerators:

\[
476 \times 11 = 5236
\]

Calculate the product of the denominators:

\[
100 \times 18 = 1800
\]

So, the expression becomes:

\[
\frac{5236}{1800}
\]

To convert this fraction to decimal form, we divide the numerator by the denominator:

\[
\frac{5236}{1800} \approx 2.908888\ldots
\]

Rounding to two decimal places, we get:

\[
2.91
\]

### Answer: 2.91

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.

GPT-4o:

To solve the expression \((\frac{17}{24}) \times (\frac{28}{10})\), we first multiply the numerators and then the denominators:

\[
\frac{17 \times 28}{24 \times 10}
\]

Next, we calculate the product of the numerators:

\[
17 \times 28 = 476
\]

And the product of the denominators:

\[
24 \times 10 = 240
\]

So, the expression becomes:

\[
\frac{476}{240}
\]

To simplify this fraction, we can divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 476 and 240 is 4.

\[
\frac{476 \div 4}{240 \div 4} = \frac{119}{60}
\]

To convert this fraction to decimal form, we divide the numerator by the denominator:

\[
\frac{119}{60} \approx 1.9833\overline{3}
\]

### Answer: 1.9833

Expected answer: 119/60

Correct? No

Problem: The value of the expression ((29 / 14) * (23 / 6)) was calculated in the previous problem. Now, calculate the result of multiplying the previously found value by (21 / 21).

GPT-4o:

To calculate the result of multiplying the previously found value \(\frac{667}{84}\) by \(\frac{21}{21}\), we follow these steps:

1. Write the expression:
   \[
   \left(\frac{667}{84}\right) \times \left(\frac{21}{21}\right)
   \]

2. Multiply the numerators together:
   \[
   667 \times 21 = 14007
   \]

3. Multiply the denominators together:
   \[
   84 \times 21 = 1764
   \]

4. Combine the results to form the fraction:
   \[
   \frac{14007}{1764}
   \]

5. Simplify the fraction if possible. We need to check if 14007 and 1764 have any common factors.

   - The prime factorization of 14007 is \(14007 = 3^2 \times 7 \times 223\).
   - The prime factorization of 1764 is \(1764 = 2^2 \times 3^2 \times 7^2\).

   The greatest common divisor (GCD) of 14007 and 1764 is \(3^2 \times 7 = 63\).

6. Divide both the numerator and the denominator by their GCD (63):
   \[
   \frac{14007 \div 63}{1764 \div 63} = \frac{222}{28}
   \]

7. Simplify further if possible:
   - The prime factorization of 222 is \(222 = 2 \times 3 \times 37\).
   - The prime factorization of 28 is \(28 = 2^2 \times 7\).

   The GCD of 222 and 28 is 2.

8. Divide both the numerator and the denominator by their GCD (2):
   \[
   \frac{222 \div 2}{28 \div 2} = \frac{111}{14}
   \]

### Answer: \(\frac{111}{14}\)

Expected answer: 667/84

Correct? No

Problem: Recall the original problem, which dealt with the expression ((29 / 14) * (23 / 6)). Now, reconsider this problem, but replace the 6 in the denominator of the second fraction with a 2. That is, find the value of the expression ((29 / 14) * (23 / 2)). What's the outcome?

GPT-4o:

To determine the value of the new expression \(\left(\frac{29}{14}\right) \times \left(\frac{23}{2}\right)\), we follow these steps:

1. Multiply the numerators together:
   \[
   29 \times 23 = 667
   \]

2. Multiply the denominators together:
   \[
   14 \times 2 = 28
   \]

3. Combine the results to form the fraction:
   \[
   \frac{667}{28}
   \]

4. Simplify the fraction if possible. We need to check if 667 and 28 have any common factors other than 1.

   - The prime factorization of 667 is \(667 = 23 \times 29\).
   - The prime factorization of 28 is \(28 = 2^2 \times 7\).

Since there are no common factors between 667 and 28, the fraction \(\frac{667}{28}\) is already in its simplest form.

### Answer: \(\frac{667}{28}\)

Expected answer: 667/28

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?

GPT-4o:

We previously found that the product of \( \frac{19}{12} \) and \( \frac{20}{6} \) is \( \frac{95}{18} \).

Now, we need to multiply \( \frac{95}{18} \) by itself:
\[ \left( \frac{95}{18} \right) \times \left( \frac{95}{18} \right) \]

First, multiply the numerators:
\[ 95 \times 95 = 9025 \]

Next, multiply the denominators:
\[ 18 \times 18 = 324 \]

So, the product is:
\[ \frac{9025}{324} \]

Now, we need to simplify this fraction. To do this, we find the greatest common divisor (GCD) of 9025 and 324.

First, we find the prime factorizations:
- 9025: \( 9025 = 5^2 \times 19^2 \)
- 324: \( 324 = 2^2 \times 3^4 \)

Since there are no common prime factors between 9025 and 324, the GCD is 1.

Therefore, the fraction \( \frac{9025}{324} \) is already in its simplest form.

### Answer: \(\frac{9025}{324}\)

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?

GPT-4o:

To find the result of the multiplication of the fractions \( \frac{10}{12} \) and \( \frac{20}{6} \), we follow these steps:

1. Multiply the numerators together:
\[ 10 \times 20 = 200 \]

2. Multiply the denominators together:
\[ 12 \times 6 = 72 \]

So, the product of the fractions is:
\[ \frac{200}{72} \]

Next, we simplify the fraction \( \frac{200}{72} \). To do this, we find the greatest common divisor (GCD) of 200 and 72.

First, we find the prime factorizations:
- 200: \( 200 = 2^3 \times 5^2 \)
- 72: \( 72 = 2^3 \times 3^2 \)

The common factors are \( 2^3 \), so the GCD is \( 2^3 = 8 \).

Now, we divide both the numerator and the denominator by their GCD:
\[ \frac{200 \div 8}{72 \div 8} = \frac{25}{9} \]

Thus, the simplified fraction is:
\[ \frac{25}{9} \]

### Answer: \(\frac{25}{9}\)

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?

GPT-4o:

We previously determined that the result of \((16 / 4) / (24 / 21)\) is \(7/2\).

Now, we need to multiply this result by \(5/13\):
\[ \left(\frac{7}{2}\right) \times \left(\frac{5}{13}\right) \]

To multiply two fractions, we multiply the numerators together and the denominators together:
\[ \frac{7 \times 5}{2 \times 13} = \frac{35}{26} \]

The fraction \(\frac{35}{26}\) is already in its simplest form because 35 and 26 have no common factors other than 1.

### Answer: 35/26

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?

GPT-4o:

To determine the result of the expression \((16 / 4) / 3\), we will follow these steps:

1. Simplify the numerator.
2. Divide the simplified numerator by 3.

First, simplify the numerator:
\[ 16 / 4 = 4 \]

Next, divide the simplified numerator by 3:
\[ 4 / 3 \]

Thus, the result of the expression \((16 / 4) / 3\) is:
### Answer: 4/3

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?

GPT-4o:

We previously found that the product of \( \frac{5}{27} \) and \( \frac{15}{2} \) is \( \frac{25}{18} \).

Now, we need to multiply \( \frac{25}{18} \) by itself:

\[
\left( \frac{25}{18} \right) \times \left( \frac{25}{18} \right) = \frac{25 \times 25}{18 \times 18}
\]

First, calculate the numerators and denominators:

\[
25 \times 25 = 625
\]
\[
18 \times 18 = 324
\]

So, the product is:

\[
\frac{625}{324}
\]

Next, we check if this fraction can be simplified. We need to find the greatest common divisor (GCD) of 625 and 324. The prime factorizations are:

\[
625 = 5 \times 5 \times 5 \times 5
\]
\[
324 = 2 \times 2 \times 3 \times 3 \times 3 \times 3
\]

There are no common factors between 625 and 324, so the fraction is already in its simplest form.

### Answer: \(\frac{625}{324}\)

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.

GPT-4o:

To find the product of the fractions \( \frac{24}{27} \) and \( \frac{15}{2} \), we multiply the numerators together and the denominators together:

\[
\frac{24}{27} \times \frac{15}{2} = \frac{24 \times 15}{27 \times 2}
\]

First, calculate the numerators and denominators:

\[
24 \times 15 = 360
\]
\[
27 \times 2 = 54
\]

So, the product is:

\[
\frac{360}{54}
\]

Next, we simplify the fraction by finding the greatest common divisor (GCD) of 360 and 54. The prime factorizations are:

\[
360 = 2^3 \times 3^2 \times 5
\]
\[
54 = 2 \times 3^3
\]

The common factors are \(2\) and \(3^2\). The GCD is \(2 \times 3^2 = 18\). Divide both the numerator and the denominator by 18:

\[
\frac{360 \div 18}{54 \div 18} = \frac{20}{3}
\]

Thus, the simplified form of the product is:

### Answer: \(\frac{20}{3}\)

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.

GPT-4o:

First, we start with the product of the fractions \(\frac{15}{13} \times \frac{19}{7}\), which we previously calculated as \(\frac{285}{91}\).

Next, we need to multiply this result by another fraction, \(\frac{23}{22}\):

\[
\frac{285}{91} \times \frac{23}{22} = \frac{285 \times 23}{91 \times 22}
\]

First, we calculate the product of the numerators:

\[
285 \times 23 = 6555
\]

Next, we calculate the product of the denominators:

\[
91 \times 22 = 2002
\]

So, the product of the fractions is:

\[
\frac{6555}{2002}
\]

To simplify this fraction, we need to find the greatest common divisor (GCD) of 6555 and 2002. We can use the Euclidean algorithm to find the GCD.

1. Divide 6555 by 2002 and find the remainder:
\[
6555 \div 2002 = 3 \quad \text{remainder} \quad 6555 - 3 \times 2002 = 1549
\]

2. Now, divide 2002 by 1549 and find the remainder:
\[
2002 \div 1549 = 1 \quad \text{remainder} \quad 2002 - 1 \times 1549 = 453
\]

3. Next, divide 1549 by 453 and find the remainder:
\[
1549 \div 453 = 3 \quad \text{remainder} \quad 1549 - 3 \times 453 = 190
\]

4. Now, divide 453 by 190 and find the remainder:
\[
453 \div 190 = 2 \quad \text{remainder} \quad 453 - 2 \times 190 = 73
\]

5. Next, divide 190 by 73 and find the remainder:
\[
190 \div 73 = 2 \quad \text{remainder} \quad 190 - 2 \times 73 = 44
\]

6. Now, divide 73 by 44 and find the remainder:
\[
73 \div 44 = 1 \quad \text{remainder} \quad 73 - 1 \times 44 = 29
\]

7. Next, divide 44 by 29 and find the remainder:
\[
44 \div 29 = 1 \quad \text{remainder} \quad 44 - 1 \times 29 = 15
\]

8. Now, divide 29 by 15 and find the remainder:
\[
29 \div 15 = 1 \quad \text{remainder} \quad 29 - 1 \times 15 = 14
\]

9. Next, divide 15 by 14 and find the remainder:
\[
15 \div 14 = 1 \quad \text{remainder} \quad 15 - 1 \times 14 = 1
\]

10. Finally, divide 14 by 1:
\[
14 \div 1 = 14 \quad \text{remainder} \quad 14 - 14 \times 1 = 0
\]

Since the remainder is now 0, the GCD is the last non-zero remainder, which is 1. Therefore, the fraction \(\frac{6555}{2002}\) is already in its simplest form.

### Answer: \(\frac{6555}{2002}\)

Expected answer: 6555/2002

Correct? Yes

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.

GPT-4o:

To calculate the product of the fractions \(\frac{15}{29} \times \frac{19}{7}\), we multiply the numerators together and the denominators together:

\[
\frac{15}{29} \times \frac{19}{7} = \frac{15 \times 19}{29 \times 7}
\]

First, we calculate the product of the numerators:

\[
15 \times 19 = 285
\]

Next, we calculate the product of the denominators:

\[
29 \times 7 = 203
\]

So, the product of the fractions is:

\[
\frac{285}{203}
\]

To simplify this fraction, we need to find the greatest common divisor (GCD) of 285 and 203. We can use the Euclidean algorithm to find the GCD.

1. Divide 285 by 203 and find the remainder:
\[
285 \div 203 = 1 \quad \text{remainder} \quad 285 - 1 \times 203 = 82
\]

2. Now, divide 203 by 82 and find the remainder:
\[
203 \div 82 = 2 \quad \text{remainder} \quad 203 - 2 \times 82 = 39
\]

3. Next, divide 82 by 39 and find the remainder:
\[
82 \div 39 = 2 \quad \text{remainder} \quad 82 - 2 \times 39 = 4
\]

4. Now, divide 39 by 4 and find the remainder:
\[
39 \div 4 = 9 \quad \text{remainder} \quad 39 - 9 \times 4 = 3
\]

5. Finally, divide 4 by 3 and find the remainder:
\[
4 \div 3 = 1 \quad \text{remainder} \quad 4 - 1 \times 3 = 1
\]

Since the remainder is now 1, the GCD is 1. Therefore, the fraction \(\frac{285}{203}\) is already in its simplest form.

### Answer: \(\frac{285}{203}\)

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.

GPT-4o:

We already found that \(\frac{12}{5}\) is the result of \((12 / 25) * (20 / 4)\).

Now, we need to multiply this result by \(\frac{17}{17}\):

1. Write the multiplication:
   \[
   \frac{12}{5} \times \frac{17}{17}
   \]

2. Multiply the numerators and the denominators:
   \[
   \frac{12 \times 17}{5 \times 17} = \frac{204}{85}
   \]

3. Simplify the fraction \(\frac{204}{85}\):
   - Find the greatest common divisor (GCD) of 204 and 85. The GCD is 1.
   - Since the GCD is 1, the fraction is already in its simplest form.

### Answer: \(\frac{204}{85}\)

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.

GPT-4o:

First, we need to perform the division and multiplication operations step by step for the new problem.

1. Calculate \( \frac{10}{25} \):
   \[
   \frac{10}{25} = \frac{2}{5} \quad \text{(simplified by dividing both numerator and denominator by 5)}
   \]

2. Calculate \( \frac{20}{4} \):
   \[
   \frac{20}{4} = 5
   \]

3. Multiply the two fractions:
   \[
   \frac{2}{5} \times 5
   \]

4. Convert 5 to a fraction:
   \[
   5 = \frac{5}{1}
   \]

5. Perform the multiplication:
   \[
   \frac{2}{5} \times \frac{5}{1} = \frac{2 \times 5}{5 \times 1} = \frac{10}{5}
   \]

6. Simplify the fraction \( \frac{10}{5} \):
   - Divide both the numerator and the denominator by their GCD, which is 5:
   \[
   \frac{10 \div 5}{5 \div 5} = \frac{2}{1} = 2
   \]

### 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.

GPT-4o:

We already calculated the multiplication of \(\frac{14}{10}\) by \(\frac{30}{5}\) and found it to be \(\frac{42}{5}\).

Now, we need to multiply this result by \(\frac{13}{15}\):

\[
\frac{42}{5} \times \frac{13}{15} = \frac{42 \times 13}{5 \times 15}
\]

Calculate the products:

\[
42 \times 13 = 546
\]
\[
5 \times 15 = 75
\]

So, we have:

\[
\frac{546}{75}
\]

Next, we simplify the fraction by finding the greatest common divisor (GCD) of 546 and 75. The GCD of 546 and 75 is 3.

Now, divide both the numerator and the denominator by their GCD:

\[
\frac{546 \div 3}{75 \div 3} = \frac{182}{25}
\]

Thus, the simplified fraction is:

\[
\frac{182}{25}
\]

### Answer: \(\frac{182}{25}\)

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?

GPT-4o:

To solve the problem, we need to multiply the two fractions and then simplify the result.

First, let's multiply the numerators and the denominators:

\[
\frac{30}{10} \times \frac{30}{5} = \frac{30 \times 30}{10 \times 5}
\]

Calculate the products:

\[
30 \times 30 = 900
\]
\[
10 \times 5 = 50
\]

So, we have:

\[
\frac{900}{50}
\]

Next, we simplify the fraction by finding the greatest common divisor (GCD) of 900 and 50. The GCD of 900 and 50 is 50.

Now, divide both the numerator and the denominator by their GCD:

\[
\frac{900 \div 50}{50 \div 50} = \frac{18}{1}
\]

Thus, the simplified fraction is:

\[
\frac{18}{1} = 18
\]

### Answer: 18

Expected answer: 18

Correct? Yes

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

GPT-4o:

First, we need to use the value of \( s \) from the previous problem, which is \( \frac{243}{40} \).

Now, we need to calculate \( o \) where \( o = \left( \frac{9}{13} \right) * \left( \frac{243}{40} \right) \).

1. Multiply the numerators together:
\[ 9 \times 243 = 2187 \]

2. Multiply the denominators together:
\[ 13 \times 40 = 520 \]

So, the expression becomes:
\[ \frac{2187}{520} \]

Next, we check if the fraction can be simplified. The greatest common divisor (GCD) of 2187 and 520 is 1, meaning the fraction is already in its simplest form.

### Answer: \(\frac{2187}{520}\)

Expected answer: 2187/520

Correct? Yes

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.

GPT-4o:

To evaluate the expression \((27 / 10) * (9 / 11)\), we follow these steps:

1. Multiply the numerators together:
\[ 27 \times 9 = 243 \]

2. Multiply the denominators together:
\[ 10 \times 11 = 110 \]

So, the expression becomes:
\[ \frac{243}{110} \]

Next, we check if the fraction can be simplified. The greatest common divisor (GCD) of 243 and 110 is 1, meaning the fraction is already in its simplest form.

### Answer: \(\frac{243}{110}\)

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.

GPT-4o:

To solve this problem, we start with the result from the previous operation, which is \(\frac{324}{143}\), and multiply it by \(\frac{14}{27}\).

The expression to evaluate is:
\[
\left(\frac{324}{143}\right) \times \left(\frac{14}{27}\right)
\]

1. Multiply the numerators together:
   \[
   324 \times 14 = 4536
   \]

2. Multiply the denominators together:
   \[
   143 \times 27 = 3861
   \]

So, the expression becomes:
\[
\frac{4536}{3861}
\]

Next, we need to simplify this fraction. We do this by finding the greatest common divisor (GCD) of 4536 and 3861.

First, we find the prime factorizations of both numbers:
- 4536: \(4536 = 2^3 \times 3 \times 3 \times 3 \times 7 \times 3 = 2^3 \times 3^5 \times 7\)
- 3861: \(3861 = 3 \times 1287 = 3 \times 3 \times 429 = 3^2 \times 429 = 3^2 \times 3 \times 143 = 3^3 \times 143\)

The common factor is \(3^3\).

Now, divide both the numerator and the denominator by their GCD (which is 27):
\[
\frac{4536 \div 27}{3861 \div 27} = \frac{168}{143}
\]

Since 168 and 143 have no common factors other than 1, the fraction \(\frac{168}{143}\) is in its simplest form.

### Answer: \(\frac{168}{143}\)

Expected answer: 168/143

Correct? Yes

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

GPT-4o:

To solve the multiplication of the fractions \(\frac{34}{9}\) and \(\frac{24}{30}\), we follow these steps:

1. Multiply the numerators together:
   \[
   34 \times 24 = 816
   \]

2. Multiply the denominators together:
   \[
   9 \times 30 = 270
   \]

So, the product of the fractions is:
\[
\frac{816}{270}
\]

Next, we need to simplify this fraction. We do this by finding the greatest common divisor (GCD) of 816 and 270.

First, we use the Euclidean algorithm to find the GCD:
- Divide 816 by 270, which gives a quotient of 3 and a remainder of 6:
  \[
  816 = 270 \times 3 + 6
  \]
- Next, divide 270 by 6, which gives a quotient of 45 and a remainder of 0:
  \[
  270 = 6 \times 45 + 0
  \]

Since the remainder is now 0, the GCD is the last non-zero remainder, which is 6.

Now, we divide both the numerator and the denominator of \(\frac{816}{270}\) by their GCD, 6:
\[
\frac{816 \div 6}{270 \div 6} = \frac{136}{45}
\]

Thus, the simplified form of the product is:
\[
\frac{136}{45}
\]

### Answer: \(\frac{136}{45}\)

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?

GPT-4o:

To solve the multiplication of the fractions \(\frac{25}{9}\) and \(\frac{17}{13}\), we follow these steps:

1. Multiply the numerators together:
   \[
   25 \times 17 = 425
   \]

2. Multiply the denominators together:
   \[
   9 \times 13 = 117
   \]

So, the product of the fractions is:
\[
\frac{425}{117}
\]

Next, we need to simplify this fraction. We do this by finding the greatest common divisor (GCD) of 425 and 117.

First, we use the Euclidean algorithm to find the GCD:
- Divide 425 by 117, which gives a quotient of 3 and a remainder of 74:
  \[
  425 = 117 \times 3 + 74
  \]
- Next, divide 117 by 74, which gives a quotient of 1 and a remainder of 43:
  \[
  117 = 74 \times 1 + 43
  \]
- Then, divide 74 by 43, which gives a quotient of 1 and a remainder of 31:
  \[
  74 = 43 \times 1 + 31
  \]
- Next, divide 43 by 31, which gives a quotient of 1 and a remainder of 12:
  \[
  43 = 31 \times 1 + 12
  \]
- Then, divide 31 by 12, which gives a quotient of 2 and a remainder of 7:
  \[
  31 = 12 \times 2 + 7
  \]
- Finally, divide 12 by 7, which gives a quotient of 1 and a remainder of 5:
  \[
  12 = 7 \times 1 + 5
  \]
- Continue dividing 7 by 5, which gives a quotient of 1 and a remainder of 2:
  \[
  7 = 5 \times 1 + 2
  \]
- Finally, divide 5 by 2, which gives a quotient of 2 and a remainder of 1:
  \[
  5 = 2 \times 2 + 1
  \]
- Lastly, divide 2 by 1, which gives a quotient of 2 and a remainder of 0:
  \[
  2 = 1 \times 2 + 0
  \]

Since the remainder is now 0, the GCD is the last non-zero remainder, which is 1.

Since the GCD is 1, the fraction \(\frac{425}{117}\) is already in its simplest form.

### Answer: \(\frac{425}{117}\)

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.

GPT-4o:

We previously found that the result of the expression \(\frac{28}{16} \div \frac{27}{24}\) is \(\frac{14}{9}\).

Now, we need to multiply this result by itself:

\[
\left(\frac{14}{9}\right) \times \left(\frac{14}{9}\right) = \frac{14 \times 14}{9 \times 9} = \frac{196}{81}
\]

The fraction \(\frac{196}{81}\) is already in its simplest form because 196 and 81 have no common factors other than 1.

Thus, the outcome of multiplying the result by itself is:

### Answer: \(\frac{196}{81}\)

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.

GPT-4o:

We start with the result from the previous calculation:
\[
\frac{95}{26}
\]

Now, we need to multiply this by the fraction \(\frac{14}{24}\):
\[
\frac{95}{26} \times \frac{14}{24}
\]

First, multiply the numerators and the denominators:
\[
\frac{95 \times 14}{26 \times 24} = \frac{1330}{624}
\]

Next, we simplify the fraction \(\frac{1330}{624}\). To do this, we find the greatest common divisor (GCD) of 1330 and 624.

1. Prime factorization of 1330:
   \[
   1330 = 2 \times 5 \times 7 \times 19
   \]

2. Prime factorization of 624:
   \[
   624 = 2^4 \times 3 \times 13
   \]

The common factor is 2. Therefore, the GCD is 2.

Now, divide both the numerator and the denominator by their GCD:
\[
\frac{1330 \div 2}{624 \div 2} = \frac{665}{312}
\]

We check if \(\frac{665}{312}\) can be simplified further. The GCD of 665 and 312 is 1, so the fraction is already in its simplest form.

### Answer: \(\frac{665}{312}\)

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.

GPT-4o:

First, we simplify the expression step by step.

1. Simplify the division inside the parentheses:
   \[
   \frac{15}{3} = 5
   \]

2. Now, multiply the results:
   \[
   5 \times \frac{27}{26} = \frac{5 \times 27}{26} = \frac{135}{26}
   \]

3. Simplify the fraction \(\frac{135}{26}\). We need to find the greatest common divisor (GCD) of 135 and 26. The GCD of 135 and 26 is 1, so the fraction is already in its simplest form.

### Answer: \(\frac{135}{26}\)

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?

GPT-4o:

To determine how much of the ingredient the bakery will require with the new recipe, we need to multiply the amount required for one recipe by the number of recipes they want to make.

The new recipe requires \( \frac{19}{14} \) of the ingredient, and they want to make \( \frac{30}{4} \) of this recipe.

First, simplify \( \frac{30}{4} \):
\[ \frac{30}{4} = \frac{15}{2} \]

Now, multiply \( \frac{19}{14} \) by \( \frac{15}{2} \):
\[ \frac{19}{14} \times \frac{15}{2} = \frac{19 \times 15}{14 \times 2} = \frac{285}{28} \]

To simplify \( \frac{285}{28} \), we check for any common factors. The greatest common divisor (GCD) of 285 and 28 is 1, so the fraction is already in its simplest form.

Thus, the bakery will require \( \frac{285}{28} \) of the ingredient.

### Answer: \(\frac{285}{28}\)

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?

GPT-4o:

We previously determined that \(\left(\frac{5}{2}\right) \div \left(\frac{26}{15}\right) = \frac{75}{52}\).

Now, we need to multiply \(\frac{75}{52}\) by itself:
\[
\left(\frac{75}{52}\right) \times \left(\frac{75}{52}\right) = \frac{75 \times 75}{52 \times 52} = \frac{5625}{2704}
\]

Next, we need to check if \(\frac{5625}{2704}\) can be simplified. We do this by finding the greatest common divisor (GCD) of 5625 and 2704.

The prime factorization of 5625 is:
\[
5625 = 3^2 \times 5^4
\]

The prime factorization of 2704 is:
\[
2704 = 2^4 \times 13^2
\]

Since there are no common prime factors between 5625 and 2704, the fraction \(\frac{5625}{2704}\) is already in its simplest form.

### Answer: \(\frac{5625}{2704}\)

Expected answer: 5625/2704

Correct? Yes

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.

GPT-4o:

To determine the result of \(14 \div \left(\frac{26}{15}\right)\), we can use the property of division of fractions, which states that dividing by a fraction is equivalent to multiplying by its reciprocal.

First, find the reciprocal of \(\frac{26}{15}\):
\[
\text{Reciprocal of } \frac{26}{15} = \frac{15}{26}
\]

Next, multiply 14 by \(\frac{15}{26}\):
\[
14 \times \frac{15}{26} = \frac{14 \times 15}{26} = \frac{210}{26}
\]

Now, simplify the fraction \(\frac{210}{26}\). We find the greatest common divisor (GCD) of 210 and 26, which is 2.

Divide both the numerator and the denominator by their GCD:
\[
\frac{210 \div 2}{26 \div 2} = \frac{105}{13}
\]

The fraction \(\frac{105}{13}\) is in its simplest form.

### Answer: \(\frac{105}{13}\)

Expected answer: 105/13

Correct? Yes

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.

GPT-4o:

First, we start with the result from the previous problem:
\[
\frac{81}{28}
\]

Next, we need to calculate the square of this fraction:
\[
\left( \frac{81}{28} \right)^2
\]

To square a fraction, we square both the numerator and the denominator:
\[
\left( \frac{81}{28} \right)^2 = \frac{81^2}{28^2}
\]

Calculate the squares:
\[
81^2 = 6561
\]
\[
28^2 = 784
\]

So, the squared fraction is:
\[
\frac{6561}{784}
\]

Now, we check if this fraction can be simplified. The greatest common divisor (GCD) of 6561 and 784 is 1, so the fraction is already in its simplest form.

### Answer: \(\frac{6561}{784}\)

Expected answer: 6561/784

Correct? Yes

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.

GPT-4o:

First, we simplify the expression step by step.

1. Simplify the first fraction:
\[
\frac{9}{3} = 3
\]

2. Now, multiply the simplified result by the second fraction:
\[
3 \times \frac{23}{28}
\]

3. To multiply, we multiply the numerator by the numerator and the denominator by the denominator:
\[
3 \times \frac{23}{28} = \frac{3 \times 23}{28} = \frac{69}{28}
\]

4. Check if the fraction \(\frac{69}{28}\) can be simplified further. The greatest common divisor (GCD) of 69 and 28 is 1, so the fraction is already in its simplest form.

### Answer: \(\frac{69}{28}\)

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.

GPT-4o:

First, we already determined that:

\[
\left(\frac{15}{13}\right) \times \left(\frac{24}{24}\right) = \frac{15}{13}
\]

Now, we need to multiply this result by \(\frac{22}{7}\):

\[
\frac{15}{13} \times \frac{22}{7}
\]

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

\[
\frac{15 \times 22}{13 \times 7} = \frac{330}{91}
\]

Next, we check if the fraction \(\frac{330}{91}\) can be simplified. We find the greatest common divisor (GCD) of 330 and 91. The prime factorization of 330 is \(2 \times 3 \times 5 \times 11\), and the prime factorization of 91 is \(7 \times 13\). Since there are no common factors, the fraction is already in its simplest form.

### Answer: \(\frac{330}{91}\)

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.

GPT-4o:

First, we simplify the expression inside the parentheses:

\[
\left(\frac{15}{2}\right) \times \left(\frac{24}{24}\right)
\]

Next, we recognize that \(\frac{24}{24} = 1\). Therefore, the expression simplifies to:

\[
\left(\frac{15}{2}\right) \times 1 = \frac{15}{2}
\]

Thus, the simplified fraction is:

### Answer: \(\frac{15}{2}\)

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?

GPT-4o:

To solve the multiplication of the fractions \(\frac{15}{14} \times \frac{15}{15}\), we follow these steps:

1. Multiply the numerators together:
\[ 15 \times 15 = 225 \]

2. Multiply the denominators together:
\[ 14 \times 15 = 210 \]

So, the product of the fractions is:
\[ \frac{225}{210} \]

Next, we simplify the fraction by finding the greatest common divisor (GCD) of 225 and 210.

The prime factorization of 225 is:
\[ 225 = 3^2 \times 5^2 \]

The prime factorization of 210 is:
\[ 210 = 2 \times 3 \times 5 \times 7 \]

The common factors are 3 and 5. Therefore, the GCD is:
\[ 3 \times 5 = 15 \]

Now, we divide both the numerator and the denominator by their GCD:
\[ \frac{225 \div 15}{210 \div 15} = \frac{15}{14} \]

Thus, the simplified fraction is:
\[ \frac{15}{14} \]

### Answer: \(\frac{15}{14}\)

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?

GPT-4o:

To solve the multiplication of the fractions \(\frac{9}{7} \times \frac{25}{21}\), we follow these steps:

1. Multiply the numerators together:
\[ 9 \times 25 = 225 \]

2. Multiply the denominators together:
\[ 7 \times 21 = 147 \]

So, the product of the fractions is:
\[ \frac{225}{147} \]

Next, we simplify the fraction by finding the greatest common divisor (GCD) of 225 and 147.

The prime factorization of 225 is:
\[ 225 = 3^2 \times 5^2 \]

The prime factorization of 147 is:
\[ 147 = 3 \times 7^2 \]

The common factor is 3. Therefore, the GCD is:
\[ 3 \]

Now, we divide both the numerator and the denominator by their GCD:
\[ \frac{225 \div 3}{147 \div 3} = \frac{75}{49} \]

Thus, the simplified fraction is:
\[ \frac{75}{49} \]

### Answer: \(\frac{75}{49}\)

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))).

GPT-4o:

We need to calculate \(\left(\frac{29}{21}\right) \times \left(\frac{81}{80}\right)\).

First, multiply the numerators and the denominators:
\[
\frac{29 \times 81}{21 \times 80}
\]

Calculate the products:
\[
29 \times 81 = 2349
\]
\[
21 \times 80 = 1680
\]

So, we have:
\[
\frac{2349}{1680}
\]

Next, we need to simplify this fraction. To do this, we find the greatest common divisor (GCD) of 2349 and 1680. Using the Euclidean algorithm, we find that the GCD is 3.

Divide both the numerator and the denominator by their GCD:
\[
\frac{2349 \div 3}{1680 \div 3} = \frac{783}{560}
\]

Thus, the simplified fraction is:
\[
\frac{783}{560}
\]

### Answer: \(\frac{783}{560}\)

Expected answer: 783/560

Correct? Yes

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.

GPT-4o:

To solve the problem \(\frac{9}{\frac{20}{18}}\), we can use the property of division of fractions, which states that dividing by a fraction is equivalent to multiplying by its reciprocal.

First, find the reciprocal of \(\frac{20}{18}\):
\[
\text{Reciprocal of } \frac{20}{18} = \frac{18}{20}
\]

Now, multiply 9 by \(\frac{18}{20}\):
\[
9 \times \frac{18}{20}
\]

Next, express 9 as a fraction:
\[
9 = \frac{9}{1}
\]

Now, multiply the fractions:
\[
\frac{9}{1} \times \frac{18}{20} = \frac{9 \times 18}{1 \times 20} = \frac{162}{20}
\]

To simplify \(\frac{162}{20}\), find the greatest common divisor (GCD) of 162 and 20. The GCD of 162 and 20 is 2.

Divide both the numerator and the denominator by their GCD:
\[
\frac{162 \div 2}{20 \div 2} = \frac{81}{10}
\]

Thus, the simplified fraction is:
\[
\frac{81}{10}
\]

### Answer: \(\frac{81}{10}\)

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?

GPT-4o:

To find the new result, we need to multiply the previous answer \(\frac{45}{13}\) by \(\frac{14}{4}\).

First, write the multiplication of the two fractions:
\[ \frac{45}{13} \times \frac{14}{4} \]

Next, multiply the numerators together and the denominators together:
\[ \frac{45 \times 14}{13 \times 4} \]

Calculate the products:
\[ 45 \times 14 = 630 \]
\[ 13 \times 4 = 52 \]

So, the result is:
\[ \frac{630}{52} \]

Now, simplify the fraction by finding the greatest common divisor (GCD) of 630 and 52. The GCD of 630 and 52 is 2.

Divide both the numerator and the denominator by their GCD:
\[ \frac{630 \div 2}{52 \div 2} = \frac{315}{26} \]

Thus, the new result is:
\[ \frac{315}{26} \]

### Answer: \(\frac{315}{26}\)

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?

GPT-4o:

We already determined that the result from the previous problem is 5.

Now, we need to multiply this result by the fraction 8/20:

\[ 5 \times \frac{8}{20} \]

First, simplify the fraction 8/20:

\[ \frac{8}{20} = \frac{8 \div 4}{20 \div 4} = \frac{2}{5} \]

Now, multiply:

\[ 5 \times \frac{2}{5} = \frac{5 \times 2}{5} = \frac{10}{5} = 2 \]

### 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?

GPT-4o:

First, we need to divide 30 by 4:

\[ \frac{30}{4} = 7.5 \]

Next, we multiply the result by the fraction 18/6:

\[ 7.5 \times \frac{18}{6} \]

Simplify the fraction 18/6:

\[ \frac{18}{6} = 3 \]

Now, multiply:

\[ 7.5 \times 3 = 22.5 \]

Comparing this to our original calculation, which was 5, we see that 22.5 is significantly larger.

### 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.

GPT-4o:

First, we need to use the result from the previous calculation, which was \(\frac{10}{9}\).

Now, we need to evaluate the new expression step by step:

1. Calculate the division inside the first set of parentheses:
   \[
   27 / 4 = \frac{27}{4}
   \]

2. Multiply this result by the previous result \(\frac{10}{9}\):
   \[
   \left(\frac{27}{4}\right) * \left(\frac{10}{9}\right) = \frac{27 * 10}{4 * 9} = \frac{270}{36}
   \]

3. Simplify the fraction \(\frac{270}{36}\):
   \[
   \frac{270}{36} = \frac{270 \div 18}{36 \div 18} = \frac{15}{2}
   \]

Thus, the simplified fraction is \(\frac{15}{2}\).

### Answer: \(\frac{15}{2}\)

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.

GPT-4o:

First, we need to evaluate the expression step by step.

1. Calculate the division inside the first set of parentheses:
   \[
   20 / 2 = 10
   \]

2. Calculate the division inside the second set of parentheses:
   \[
   15 / 18 = \frac{15}{18}
   \]

3. Now, multiply the results from steps 1 and 2:
   \[
   10 * \frac{15}{18} = \frac{10 * 15}{18} = \frac{150}{18}
   \]

4. Simplify the fraction \(\frac{150}{18}\):
   \[
   \frac{150}{18} = \frac{150 \div 6}{18 \div 6} = \frac{25}{3}
   \]

Thus, the simplified fraction is \(\frac{25}{3}\).

### Answer: \(\frac{25}{3}\)

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.

GPT-4o:

We start with the result from the previous calculation:
\[ \frac{319}{54} \]

Now, we need to multiply this by the fraction \(\frac{11}{5}\):
\[ \left(\frac{319}{54}\right) \times \left(\frac{11}{5}\right) \]

First, multiply the numerators:
\[ 319 \times 11 = 3509 \]

Next, multiply the denominators:
\[ 54 \times 5 = 270 \]

So, the product of the fractions is:
\[ \frac{3509}{270} \]

To simplify this fraction, we need to find the greatest common divisor (GCD) of 3509 and 270. After checking, we find that the GCD is 1, meaning the fraction is already in its simplest form.

### Answer: \(\frac{3509}{270}\)

Expected answer: 3509/270

Correct? Yes

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.

GPT-4o:

To solve the problem \(\left(\frac{21}{9}\right) \times \left(\frac{29}{6}\right)\), we first multiply the numerators together and the denominators together.

First, multiply the numerators:
\[ 21 \times 29 = 609 \]

Next, multiply the denominators:
\[ 9 \times 6 = 54 \]

So, the product of the fractions is:
\[ \frac{609}{54} \]

To simplify this fraction, we need to find the greatest common divisor (GCD) of 609 and 54. We can use the Euclidean algorithm to find the GCD.

1. Divide 609 by 54:
\[ 609 \div 54 = 11 \text{ remainder } 15 \]
So, \( 609 = 54 \times 11 + 15 \).

2. Now, find the GCD of 54 and 15:
\[ 54 \div 15 = 3 \text{ remainder } 9 \]
So, \( 54 = 15 \times 3 + 9 \).

3. Next, find the GCD of 15 and 9:
\[ 15 \div 9 = 1 \text{ remainder } 6 \]
So, \( 15 = 9 \times 1 + 6 \).

4. Now, find the GCD of 9 and 6:
\[ 9 \div 6 = 1 \text{ remainder } 3 \]
So, \( 9 = 6 \times 1 + 3 \).

5. Finally, find the GCD of 6 and 3:
\[ 6 \div 3 = 2 \text{ remainder } 0 \]
So, \( 6 = 3 \times 2 + 0 \).

Since the remainder is now 0, the GCD is the last non-zero remainder, which is 3.

Now, we divide both the numerator and the denominator of \(\frac{609}{54}\) by their GCD, which is 3:
\[ \frac{609 \div 3}{54 \div 3} = \frac{203}{18} \]

### Answer: \(\frac{203}{18}\)

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.

GPT-4o:

To solve the expression \(\frac{480}{133} \times \frac{22}{7}\), we follow these steps:

1. Write the expression as a single fraction:
\[
\frac{480}{133} \times \frac{22}{7} = \frac{480 \times 22}{133 \times 7}
\]

2. Calculate the numerator and the denominator:
\[
480 \times 22 = 10560
\]
\[
133 \times 7 = 931
\]

3. Combine these results into a single fraction:
\[
\frac{10560}{931}
\]

4. Check if the fraction can be simplified. The greatest common divisor (GCD) of 10560 and 931 is 1, so the fraction is already in its simplest form.

### Answer: \(\frac{10560}{931}\)

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.

GPT-4o:

To solve the expression \((21 / 7) * (16 / 19)\), we follow these steps:

1. First, simplify the fraction \(\frac{21}{7}\):
\[
\frac{21}{7} = 3
\]

2. Now, multiply the simplified fraction by \(\frac{16}{19}\):
\[
3 \times \frac{16}{19} = \frac{3 \times 16}{19} = \frac{48}{19}
\]

3. Check if the fraction can be simplified. The greatest common divisor (GCD) of 48 and 19 is 1, so the fraction is already in its simplest form.

### Answer: \(\frac{48}{19}\)

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.

GPT-4o:

We previously found that the product of \(\frac{25}{22}\) and \(\frac{20}{3}\) is \(\frac{250}{33}\).

Now, we need to multiply this result by \(\frac{28}{29}\):

\[
\frac{250}{33} \times \frac{28}{29} = \frac{250 \times 28}{33 \times 29}
\]

First, calculate the numerator:

\[
250 \times 28 = 7000
\]

Next, calculate the denominator:

\[
33 \times 29 = 957
\]

So, the product is:

\[
\frac{7000}{957}
\]

To simplify this fraction, we need to find the greatest common divisor (GCD) of 7000 and 957. The prime factorization of 7000 is \(2^3 \times 5^3 \times 7\), and the prime factorization of 957 is \(3 \times 11 \times 29\). There are no common factors between 7000 and 957 other than 1.

Since the GCD is 1, the fraction \(\frac{7000}{957}\) is already in its simplest form.

### Answer: \(\frac{7000}{957}\)

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.

GPT-4o:

To find the product of the fractions \(\frac{25}{15}\) and \(\frac{20}{3}\), we multiply the numerators together and the denominators together:

\[
\frac{25}{15} \times \frac{20}{3} = \frac{25 \times 20}{15 \times 3}
\]

First, calculate the numerator:

\[
25 \times 20 = 500
\]

Next, calculate the denominator:

\[
15 \times 3 = 45
\]

So, the product is:

\[
\frac{500}{45}
\]

To simplify this fraction, we need to find the greatest common divisor (GCD) of 500 and 45. The prime factorization of 500 is \(2^2 \times 5^3\), and the prime factorization of 45 is \(3^2 \times 5\). The common factor is 5.

Divide both the numerator and the denominator by their GCD, which is 5:

\[
\frac{500 \div 5}{45 \div 5} = \frac{100}{9}
\]

Thus, the simplified fraction is:

\[
\frac{100}{9}
\]

### Answer: \(\frac{100}{9}\)

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.

GPT-4o:

First, we start with the result from the previous calculation:
\[
\left(\frac{29}{2}\right) \div \left(\frac{23}{19}\right) = \frac{551}{46}
\]

Next, we need to multiply this result by \(\frac{29}{18}\):
\[
\frac{551}{46} \times \frac{29}{18}
\]

To multiply these fractions, we multiply the numerators together and the denominators together:
\[
\frac{551 \times 29}{46 \times 18}
\]

Calculate the products:
\[
551 \times 29 = 15979
\]
\[
46 \times 18 = 828
\]

So, the fraction becomes:
\[
\frac{15979}{828}
\]

Now, we need to simplify this fraction. We check for the greatest common divisor (GCD) of 15979 and 828. Using the Euclidean algorithm:

1. \(15979 \div 828 \approx 19\) remainder \(307\)
2. \(828 \div 307 \approx 2\) remainder \(214\)
3. \(307 \div 214 \approx 1\) remainder \(93\)
4. \(214 \div 93 \approx 2\) remainder \(28\)
5. \(93 \div 28 \approx 3\) remainder \(9\)
6. \(28 \div 9 \approx 3\) remainder \(1\)
7. \(9 \div 1 = 9\) remainder \(0\)

The GCD is 1, meaning the fraction \(\frac{15979}{828}\) is already in its simplest form.

### Answer: \(\frac{15979}{828}\)

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?

GPT-4o:

First, we start with the result from the previous question, which is \(\frac{45}{13}\).

Now, we need to multiply this by \(\frac{8}{14}\):

\[
\frac{45}{13} \times \frac{8}{14}
\]

We multiply the numerators together and the denominators together:

\[
\frac{45 \times 8}{13 \times 14} = \frac{360}{182}
\]

Next, we simplify the fraction \(\frac{360}{182}\). We find the greatest common divisor (GCD) of 360 and 182. The prime factorization of 360 is \(2^3 \times 3^2 \times 5\), and the prime factorization of 182 is \(2 \times 7 \times 13\). The common factor is 2.

Divide both the numerator and the denominator by their GCD, which is 2:

\[
\frac{360 \div 2}{182 \div 2} = \frac{180}{91}
\]

Since 180 and 91 have no common factors other than 1, \(\frac{180}{91}\) is already in its simplest form.

### Answer: \(\frac{180}{91}\)

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.

GPT-4o:

To solve the expression \((\frac{5}{3}) \times (\frac{27}{9})\), we multiply the numerators together and the denominators together:

\[
\frac{5}{3} \times \frac{27}{9} = \frac{5 \times 27}{3 \times 9}
\]

First, calculate the product of the numerators:

\[
5 \times 27 = 135
\]

Next, calculate the product of the denominators:

\[
3 \times 9 = 27
\]

So, the expression becomes:

\[
\frac{135}{27}
\]

Now, we need to simplify this fraction. We find the greatest common divisor (GCD) of 135 and 27. The prime factorization of 135 is \(3^3 \times 5\), and the prime factorization of 27 is \(3^3\). The common factor is \(3^3 = 27\).

Divide both the numerator and the denominator by their GCD, which is 27:

\[
\frac{135 \div 27}{27 \div 27} = \frac{5}{1}
\]

Since \(\frac{5}{1}\) simplifies to 5, the final answer is:

### Answer: 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?

GPT-4o:

First, we simplify \( \frac{27}{27} \):
\[ \frac{27}{27} = 1 \]

Next, we use the answer from the previous problem, which is 1, and multiply it by the simplified result:
\[ 1 \times 1 = 1 \]

### 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?

GPT-4o:

First, we simplify each part of the expression separately.

1. Simplify \( \frac{29}{6} \):
\[ \frac{29}{6} \]

2. Simplify \( \frac{1}{1} \):
\[ \frac{1}{1} = 1 \]

Now, multiply the simplified results:
\[ \frac{29}{6} \times 1 = \frac{29}{6} \]

### Answer: \(\frac{29}{6}\)

Expected answer: 29/6

Correct? Yes

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?

GPT-4o:

First, we already found the product of the fractions \( \frac{11}{19} \) and \( \frac{22}{7} \):

\[
\frac{11}{19} \times \frac{22}{7} = \frac{242}{133}
\]

Next, we need to multiply this result by \( \frac{25}{4} \):

\[
\frac{242}{133} \times \frac{25}{4} = \frac{242 \times 25}{133 \times 4}
\]

First, calculate the numerator:

\[
242 \times 25 = 6050
\]

Next, calculate the denominator:

\[
133 \times 4 = 532
\]

So, the product of the fractions is:

\[
\frac{6050}{532}
\]

To simplify this fraction, we need to find the greatest common divisor (GCD) of 6050 and 532. We can use the Euclidean algorithm to find the GCD:

1. Divide 6050 by 532, which gives a quotient of 11 and a remainder of 258:
   \[
   6050 = 532 \times 11 + 258
   \]

2. Divide 532 by 258, which gives a quotient of 2 and a remainder of 16:
   \[
   532 = 258 \times 2 + 16
   \]

3. Divide 258 by 16, which gives a quotient of 16 and a remainder of 2:
   \[
   258 = 16 \times 16 + 2
   \]

4. Finally, divide 16 by 2, which gives a quotient of 8 and a remainder of 0:
   \[
   16 = 2 \times 8 + 0
   \]

Since the remainder is now 0, the GCD is the last non-zero remainder, which is 2. Therefore, we can simplify the fraction \( \frac{6050}{532} \) by dividing both the numerator and the denominator by 2:

\[
\frac{6050 \div 2}{532 \div 2} = \frac{3025}{266}
\]

### Answer: \(\frac{3025}{266}\)

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.

GPT-4o:

To find the product of the fractions \( \frac{11}{19} \) and \( \frac{25}{7} \), we multiply the numerators together and the denominators together:

\[
\frac{11}{19} \times \frac{25}{7} = \frac{11 \times 25}{19 \times 7}
\]

First, calculate the numerator:

\[
11 \times 25 = 275
\]

Next, calculate the denominator:

\[
19 \times 7 = 133
\]

So, the product of the fractions is:

\[
\frac{275}{133}
\]

To simplify this fraction, we need to find the greatest common divisor (GCD) of 275 and 133. We can use the Euclidean algorithm to find the GCD:

1. Divide 275 by 133, which gives a quotient of 2 and a remainder of 9:
   \[
   275 = 133 \times 2 + 9
   \]

2. Divide 133 by 9, which gives a quotient of 14 and a remainder of 7:
   \[
   133 = 9 \times 14 + 7
   \]

3. Divide 9 by 7, which gives a quotient of 1 and a remainder of 2:
   \[
   9 = 7 \times 1 + 2
   \]

4. Divide 7 by 2, which gives a quotient of 3 and a remainder of 1:
   \[
   7 = 2 \times 3 + 1
   \]

5. Finally, divide 2 by 1, which gives a quotient of 2 and a remainder of 0:
   \[
   2 = 1 \times 2 + 0
   \]

Since the remainder is now 0, the GCD is the last non-zero remainder, which is 1. Therefore, the fraction \( \frac{275}{133} \) is already in its simplest form.

### Answer: \(\frac{275}{133}\)

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.

GPT-4o:

To find the result of multiplying \(\frac{133}{50}\) by \(\frac{16}{12}\), we follow these steps:

1. Multiply the numerators together:
\[ 133 \times 16 = 2128 \]

2. Multiply the denominators together:
\[ 50 \times 12 = 600 \]

So, the product of the fractions is:
\[ \frac{2128}{600} \]

Next, we need to simplify this fraction. We do this by finding the greatest common divisor (GCD) of 2128 and 600.

First, we find the prime factorizations:
- 2128: \( 2128 = 2^4 \times 133 = 2^4 \times 7 \times 19 \)
- 600: \( 600 = 2^3 \times 3 \times 5^2 \)

The common factor is \( 2^3 = 8 \).

Now, divide both the numerator and the denominator by their GCD (8):
\[ \frac{2128 \div 8}{600 \div 8} = \frac{266}{75} \]

Thus, the simplified fraction is:
\[ \frac{266}{75} \]

### Answer: \(\frac{266}{75}\)

Expected answer: 266/75

Correct? Yes

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

GPT-4o:

To find the result of multiplying the fractions \( \frac{9}{30} \) and \( \frac{19}{5} \), we follow these steps:

1. Multiply the numerators together:
\[ 9 \times 19 = 171 \]

2. Multiply the denominators together:
\[ 30 \times 5 = 150 \]

So, the product of the fractions is:
\[ \frac{171}{150} \]

Next, we need to simplify this fraction. We do this by finding the greatest common divisor (GCD) of 171 and 150.

First, we find the prime factorizations:
- 171: \( 171 = 3 \times 57 = 3 \times 3 \times 19 \)
- 150: \( 150 = 2 \times 75 = 2 \times 3 \times 25 = 2 \times 3 \times 5 \times 5 \)

The common factor is 3.

Now, divide both the numerator and the denominator by their GCD (3):
\[ \frac{171 \div 3}{150 \div 3} = \frac{57}{50} \]

Thus, the simplified fraction is:
\[ \frac{57}{50} \]

### Answer: \(\frac{57}{50}\)

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.

GPT-4o:

To find the result of multiplying \( \frac{253}{120} \) by itself, we follow these steps:

1. Multiply the numerators together:
\[ 253 \times 253 = 253^2 = 64009 \]

2. Multiply the denominators together:
\[ 120 \times 120 = 120^2 = 14400 \]

So, the product of the fractions is:
\[ \frac{64009}{14400} \]

Next, we check if the fraction can be simplified. We need to find the greatest common divisor (GCD) of 64009 and 14400.

- The prime factors of 64009 are 64009 itself (since 64009 is a prime number).
- The prime factors of 14400 are \( 2^6 \times 3^2 \times 5^2 \).

Since 64009 and 14400 have no common prime factors, their GCD is 1. Therefore, the fraction \( \frac{64009}{14400} \) is already in its simplest form.

### Answer: \(\frac{64009}{14400}\)

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.

GPT-4o:

To find the result of multiplying the fractions \( \frac{11}{15} \) and \( \frac{26}{8} \), we follow these steps:

1. Multiply the numerators together:
\[ 11 \times 26 = 286 \]

2. Multiply the denominators together:
\[ 15 \times 8 = 120 \]

So, the product of the fractions is:
\[ \frac{286}{120} \]

Next, we need to simplify the fraction. We do this by finding the greatest common divisor (GCD) of 286 and 120.

- The prime factors of 286 are 2 and 143 (since \( 286 = 2 \times 143 \)).
- The prime factors of 120 are 2, 3, and 5 (since \( 120 = 2^3 \times 3 \times 5 \)).

The common factor between 286 and 120 is 2. We divide both the numerator and the denominator by their GCD, which is 2:

\[ \frac{286 \div 2}{120 \div 2} = \frac{143}{60} \]

Since 143 and 60 have no common factors other than 1, the fraction \( \frac{143}{60} \) is in its simplest form.

### Answer: \(\frac{143}{60}\)

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.

GPT-4o:

We previously found that \(\frac{29}{16} \times \frac{21}{27} = \frac{203}{144}\).

Now, we need to multiply \(\frac{203}{144}\) by itself:

\[
\left(\frac{203}{144}\right)^2 = \frac{203 \times 203}{144 \times 144}
\]

First, calculate the products:

\[
203 \times 203 = 41209
\]
\[
144 \times 144 = 20736
\]

So, we have:

\[
\frac{41209}{20736}
\]

Next, we need to check if this fraction can be simplified. We do this by finding the greatest common divisor (GCD) of 41209 and 20736. 

Using the Euclidean algorithm:
1. \(41209 \div 20736 = 1\) with a remainder of \(20473\)
2. \(20736 \div 20473 = 1\) with a remainder of \(263\)
3. \(20473 \div 263 = 77\) with a remainder of \(0\)

Since the GCD is 1, the fraction \(\frac{41209}{20736}\) is already in its simplest form.

### Answer: \(\frac{41209}{20736}\)

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))?

GPT-4o:

To find the result of \(\frac{29}{5} \times \frac{21}{27}\), we first multiply the numerators together and then the denominators together:

\[
\frac{29}{5} \times \frac{21}{27} = \frac{29 \times 21}{5 \times 27}
\]

Next, we calculate the products:

\[
29 \times 21 = 609
\]
\[
5 \times 27 = 135
\]

So, we have:

\[
\frac{609}{135}
\]

Now, we need to simplify this fraction. We do this by finding the greatest common divisor (GCD) of 609 and 135. 

First, we find the prime factorizations:
- 609: \(609 = 3 \times 7 \times 29\)
- 135: \(135 = 3 \times 3 \times 3 \times 5 = 3^3 \times 5\)

The common factor is 3. We divide both the numerator and the denominator by 3:

\[
\frac{609 \div 3}{135 \div 3} = \frac{203}{45}
\]

Since 203 and 45 have no common factors other than 1, the fraction is already in its simplest form.

### Answer: \(\frac{203}{45}\)

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.

GPT-4o:

We already found that \(\frac{28}{1} \div \frac{30}{27} = \frac{126}{5}\).

Now, we need to multiply this result by \(\frac{6}{19}\):

1. Start with the expression:
   \[
   \frac{126}{5} \times \frac{6}{19}
   \]

2. Multiply the numerators together and the denominators together:
   \[
   \frac{126 \times 6}{5 \times 19} = \frac{756}{95}
   \]

3. Simplify the fraction \(\frac{756}{95}\). We need to check if there is a common factor between 756 and 95. The prime factorization of 756 is \(2^2 \times 3^3 \times 7\), and the prime factorization of 95 is \(5 \times 19\). Since there are no common factors, the fraction is already in its simplest form.

Thus, the simplified fraction is:
### Answer: \(\frac{756}{95}\)

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.

GPT-4o:

To solve the problem \(\frac{28}{1} \div 11\), we need to understand that dividing by a whole number is equivalent to multiplying by its reciprocal.

1. Start with the given expression:
   \[
   \frac{28}{1} \div 11
   \]

2. Replace the division with multiplication by the reciprocal of 11:
   \[
   \frac{28}{1} \times \frac{1}{11}
   \]

3. Multiply the numerators together and the denominators together:
   \[
   \frac{28 \times 1}{1 \times 11} = \frac{28}{11}
   \]

Thus, the simplified fraction is:
### Answer: \(\frac{28}{11}\)

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.

GPT-4o:

We already found that:
\[
\left( \frac{25}{25} \times \frac{20}{7} \right) = \frac{20}{7}
\]

Now, we need to multiply this result by itself:
\[
\left( \frac{20}{7} \right) \times \left( \frac{20}{7} \right)
\]

To multiply two fractions, we multiply the numerators together and the denominators together:
\[
\frac{20 \times 20}{7 \times 7} = \frac{400}{49}
\]

Since \(\frac{400}{49}\) is already in its simplest form, the final answer is:
### Answer: \(\frac{400}{49}\)

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.

GPT-4o:

Let's simplify the new expression step by step.

1. Simplify the first fraction:
\[
\frac{25}{25} = 1
\]

2. Now, multiply this result by the second fraction:
\[
1 \times \frac{11}{7} = \frac{11}{7}
\]

Since \(\frac{11}{7}\) is already in its simplest form, the final answer is:
### Answer: \(\frac{11}{7}\)

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.

GPT-4o:

To multiply the fraction \( \frac{30}{1} \) by \( \frac{16}{27} \), we follow these steps:

1. Multiply the numerators: \( 30 \times 16 = 480 \).
2. Multiply the denominators: \( 1 \times 27 = 27 \).

So, the product is \( \frac{480}{27} \).

Next, we simplify the fraction \( \frac{480}{27} \). We need to find the greatest common divisor (GCD) of 480 and 27.

- The prime factorization of 480 is \( 2^5 \times 3 \times 5 \).
- The prime factorization of 27 is \( 3^3 \).

The common factor is 3.

Now, divide both the numerator and the denominator by their GCD, which is 3:

\[ \frac{480 \div 3}{27 \div 3} = \frac{160}{9} \]

Thus, the simplified fraction is \( \frac{160}{9} \).

### Answer: \(\frac{160}{9}\)

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'.

GPT-4o:

First, recall that the value of \( s \) from the previous calculation is 9.

Now, we need to compute the result of multiplying \( s \) by the fraction \( \frac{21}{17} \):
\[ s \times \frac{21}{17} = 9 \times \frac{21}{17} \]

To perform this multiplication, we multiply the numerator of the fraction by 9:
\[ 9 \times 21 = 189 \]

So, the product is:
\[ \frac{189}{17} \]

Since 189 and 17 have no common factors other than 1, the fraction is already in its simplest form.

Therefore, the result \( b \) is:
### Answer: \frac{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'?

GPT-4o:

Let's calculate the new expression step by step.

1. Simplify the denominator: \( 12 / 12 \)
\[ 12 / 12 = 1 \]

2. Now, we have the expression:
\[ 24 / 1 \]

Since dividing by 1 does not change the value, the expression simplifies to:
\[ 24 \]

Therefore, the result of the expression \( 24 / (12 / 12) \) is:
### Answer: 24

Comparing this to the previous answer 's' which was 9, we see that the new result is larger. Specifically, the new result is 24, which is 24/9 = 8/3 times the previous result '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.

GPT-4o:

To find the new result, we multiply the previous result \(\frac{435}{32}\) by \(\frac{18}{18}\):

\[
\frac{435}{32} \times \frac{18}{18}
\]

Since \(\frac{18}{18} = 1\), multiplying by this fraction does not change the value of the original fraction. However, let's perform the multiplication explicitly:

\[
\frac{435 \times 18}{32 \times 18}
\]

The \(18\) in the numerator and the \(18\) in the denominator cancel each other out:

\[
\frac{435 \times 18}{32 \times 18} = \frac{435}{32}
\]

Thus, the new result remains the same as the original fraction.

### Answer: \(\frac{435}{32}\)

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.

GPT-4o:

To solve the operation \(\left(\frac{15}{16}\right) \times \left(\frac{13}{2}\right)\), we multiply the numerators together and the denominators together:

\[
\frac{15 \times 13}{16 \times 2}
\]

First, calculate the numerator:

\[
15 \times 13 = 195
\]

Next, calculate the denominator:

\[
16 \times 2 = 32
\]

So, the fraction is:

\[
\frac{195}{32}
\]

Since 195 and 32 have no common factors other than 1, the fraction is already in its simplest form.

### Answer: \(\frac{195}{32}\)

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.

GPT-4o:

To solve the equation \(\left(\frac{6}{2}\right) \div 3\), we will follow these steps:

1. Simplify the expression inside the parentheses:
   \[
   \frac{6}{2} = 3
   \]

2. Perform the division:
   \[
   3 \div 3 = 1
   \]

The result is already in its simplest form.

### Answer: \(\frac{1}{1}\)

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?

GPT-4o:

We previously found that dividing seven wholes by \( \frac{15}{9} \) results in \( \frac{21}{5} \).

Now, we need to multiply \( \frac{21}{5} \) by \( \frac{10}{5} \):

\[ \frac{21}{5} \times \frac{10}{5} = \frac{21 \times 10}{5 \times 5} = \frac{210}{25} \]

Next, we simplify \( \frac{210}{25} \) by finding the greatest common divisor (GCD) of 210 and 25. The GCD of 210 and 25 is 5.

Divide both the numerator and the denominator by their GCD:
\[ \frac{210 \div 5}{25 \div 5} = \frac{42}{5} \]

Thus, the simplified fraction is:
### Answer: \(\frac{42}{5}\)

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.

GPT-4o:

To solve the problem of dividing \( \frac{20}{1} \) by \( \frac{15}{9} \), we need to multiply \( \frac{20}{1} \) by the reciprocal of \( \frac{15}{9} \).

First, find the reciprocal of \( \frac{15}{9} \):
\[ \text{Reciprocal of } \frac{15}{9} = \frac{9}{15} \]

Next, multiply \( \frac{20}{1} \) by \( \frac{9}{15} \):
\[ \frac{20}{1} \times \frac{9}{15} = \frac{20 \times 9}{1 \times 15} = \frac{180}{15} \]

Now, simplify \( \frac{180}{15} \) by finding the greatest common divisor (GCD) of 180 and 15. The GCD of 180 and 15 is 15.

Divide both the numerator and the denominator by their GCD:
\[ \frac{180 \div 15}{15 \div 15} = \frac{12}{1} \]

Thus, the simplified fraction is:
### Answer: \(\frac{12}{1}\) 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?

GPT-4o:

To find the result of multiplying \(\frac{207}{10}\) by \(\frac{1}{16}\), we follow these steps:

1. Multiply the numerators: \(207 \times 1 = 207\).
2. Multiply the denominators: \(10 \times 16 = 160\).

So, the product is \(\frac{207}{160}\).

### Answer: \(\frac{207}{160}\)

Expected answer: 207/160

Correct? Yes

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?

GPT-4o:

To find the result of multiplying \(\frac{23}{10}\) by \(\frac{9}{6}\), we follow these steps:

1. Multiply the numerators: \(23 \times 9 = 207\).
2. Multiply the denominators: \(10 \times 6 = 60\).

So, the product is \(\frac{207}{60}\).

Next, we simplify the fraction \(\frac{207}{60}\) by finding the greatest common divisor (GCD) of 207 and 60. The GCD of 207 and 60 is 3.

Now, divide both the numerator and the denominator by their GCD:

\[
\frac{207 \div 3}{60 \div 3} = \frac{69}{20}
\]

Thus, the simplified form of \(\frac{207}{60}\) is \(\frac{69}{20}\).

### Answer: \(\frac{69}{20}\)

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.

GPT-4o:

We already calculated \(\left(\frac{18}{4}\right) \times \left(\frac{10}{11}\right) = \frac{45}{11}\).

Now, we need to multiply this result by \(\frac{6}{4}\):

\[
\frac{45}{11} \times \frac{6}{4} = \frac{45 \times 6}{11 \times 4} = \frac{270}{44}
\]

Next, we simplify \(\frac{270}{44}\):

1. Find the greatest common divisor (GCD) of 270 and 44. The GCD is 2.
2. Divide both the numerator and the denominator by their GCD:

\[
\frac{270 \div 2}{44 \div 2} = \frac{135}{22}
\]

Thus, the simplified fraction is \(\frac{135}{22}\).

### Answer: \(\frac{135}{22}\)

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.

GPT-4o:

First, we need to perform the division and multiplication operations within the expression \(\left(\frac{18}{1}\right) \times \left(\frac{10}{11}\right)\).

1. Simplify \(\frac{18}{1}\):
   \[
   \frac{18}{1} = 18
   \]

2. Now, multiply the simplified fraction by \(\frac{10}{11}\):
   \[
   18 \times \frac{10}{11} = \frac{18 \times 10}{11} = \frac{180}{11}
   \]

Thus, the simplified fraction is \(\frac{180}{11}\).

### Answer: \(\frac{180}{11}\)

Expected answer: 180/11

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.

GPT-4o:

We already found that \(\frac{20}{13} \div \frac{17}{16} = \frac{320}{221}\).

Now, we need to multiply this result by \(\frac{20}{26}\):

\[
\frac{320}{221} \times \frac{20}{26}
\]

First, multiply the numerators together and the denominators together:

\[
\frac{320 \times 20}{221 \times 26} = \frac{6400}{5746}
\]

Next, we simplify the fraction \(\frac{6400}{5746}\). We need to find the greatest common divisor (GCD) of 6400 and 5746. 

Using the Euclidean algorithm:
1. \(6400 \div 5746 = 1\) with a remainder of \(654\).
2. \(5746 \div 654 = 8\) with a remainder of \(494\).
3. \(654 \div 494 = 1\) with a remainder of \(160\).
4. \(494 \div 160 = 3\) with a remainder of \(14\).
5. \(160 \div 14 = 11\) with a remainder of \(6\).
6. \(14 \div 6 = 2\) with a remainder of \(2\).
7. \(6 \div 2 = 3\) with a remainder of \(0\).

The GCD is 2.

Now, we divide both the numerator and the denominator by their GCD:

\[
\frac{6400 \div 2}{5746 \div 2} = \frac{3200}{2873}
\]

Thus, the simplified fraction is:

### Answer: \(\frac{3200}{2873}\)

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.

GPT-4o:

First, we need to determine the value of \( c \) from the previous problem, which is 4.

Next, we calculate \( \frac{21}{9} \):
\[ \frac{21}{9} = \frac{7}{3} \]
(simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3).

Now, we multiply \( \frac{7}{3} \) by \( c \):
\[ \frac{7}{3} \times 4 = \frac{7 \times 4}{3} = \frac{28}{3} \]

### Answer: \(\frac{28}{3}\)

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.

GPT-4o:

First, we need to calculate the new initial amount of oranges, which is \( \frac{5}{15} \).

Next, we multiply this fraction by \( \frac{4}{1} \):

\[ \frac{5}{15} \times \frac{4}{1} \]

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

\[ \frac{5 \times 4}{15 \times 1} = \frac{20}{15} \]

Now, we simplify the fraction \( \frac{20}{15} \). The greatest common divisor (GCD) of 20 and 15 is 5. We divide both the numerator and the denominator by 5:

\[ \frac{20 \div 5}{15 \div 5} = \frac{4}{3} \]

### Answer: \(\frac{4}{3}\)

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?

GPT-4o:

To solve the new expression \(\left(\frac{95}{92}\right) \times \left(\frac{28}{4}\right)\), we follow these steps:

1. Simplify the fraction \(\frac{28}{4}\):
   \[
   \frac{28}{4} = 7
   \]

2. Multiply the simplified fraction by \(\frac{95}{92}\):
   \[
   \frac{95}{92} \times 7 = \frac{95 \times 7}{92} = \frac{665}{92}
   \]

3. Simplify the resulting fraction \(\frac{665}{92}\). We need to check if there is a common divisor for 665 and 92. 

   - The prime factorization of 665 is \(5 \times 7 \times 19\).
   - The prime factorization of 92 is \(2^2 \times 23\).

   Since there are no common prime factors, the fraction \(\frac{665}{92}\) is already in its simplest form.

### Answer: \(\frac{665}{92}\)

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?

GPT-4o:

To solve the new expression \(\left(\frac{30}{11}\right) \times \left(\frac{19}{24}\right)\), we follow these steps:

1. Multiply the numerators together:
   \[
   30 \times 19 = 570
   \]

2. Multiply the denominators together:
   \[
   11 \times 24 = 264
   \]

3. Combine the results to form a single fraction:
   \[
   \frac{570}{264}
   \]

4. Simplify the fraction by finding the greatest common divisor (GCD) of 570 and 264. We can use the Euclidean algorithm to find the GCD:

   - \(570 \div 264 = 2\) with a remainder of \(42\)
   - \(264 \div 42 = 6\) with no remainder

   Therefore, the GCD of 570 and 264 is 42.

5. Divide both the numerator and the denominator by their GCD:
   \[
   \frac{570 \div 42}{264 \div 42} = \frac{13.5714}{6.2857} = \frac{95}{44}
   \]

### Answer: \(\frac{95}{44}\)

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.

GPT-4o:

We previously calculated \( g = \frac{184}{13} \).

Now, we need to find the value of \( \left(\frac{4}{7}\right) \times g \).

Substitute \( g \) with \(\frac{184}{13}\):

\[
\left(\frac{4}{7}\right) \times \left(\frac{184}{13}\right)
\]

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

\[
\frac{4 \times 184}{7 \times 13}
\]

First, calculate the numerator:

\[
4 \times 184 = 736
\]

Next, calculate the denominator:

\[
7 \times 13 = 91
\]

So, the expression becomes:

\[
\frac{736}{91}
\]

To simplify this fraction, we need to check if there is a common divisor between 736 and 91. We can use the Euclidean algorithm to find the GCD:

1. Divide 736 by 91 to get the quotient and remainder:
   \[
   736 \div 91 = 8 \quad \text{remainder} \, 8
   \]

2. Now, divide 91 by the remainder 8:
   \[
   91 \div 8 = 11 \quad \text{remainder} \, 3
   \]

3. Next, divide 8 by the remainder 3:
   \[
   8 \div 3 = 2 \quad \text{remainder} \, 2
   \]

4. Finally, divide 3 by the remainder 2:
   \[
   3 \div 2 = 1 \quad \text{remainder} \, 1
   \]

Since the remainder is now 1, the GCD is 1.

Since the GCD is 1, the fraction \(\frac{736}{91}\) is already in its simplest form.

### Answer: \(\frac{736}{91}\)

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.

GPT-4o:

To evaluate the expression \(\frac{2}{26} \times \frac{16}{1}\), we multiply the numerators together and the denominators together:

\[
\frac{2 \times 16}{26 \times 1}
\]

First, calculate the numerator:

\[
2 \times 16 = 32
\]

Next, calculate the denominator:

\[
26 \times 1 = 26
\]

So, the expression becomes:

\[
\frac{32}{26}
\]

To simplify this fraction, we need to find the greatest common divisor (GCD) of 32 and 26. We can use the Euclidean algorithm to find the GCD:

1. Divide 32 by 26 to get the quotient and remainder:
   \[
   32 \div 26 = 1 \quad \text{remainder} \, 6
   \]

2. Now, divide 26 by the remainder 6:
   \[
   26 \div 6 = 4 \quad \text{remainder} \, 2
   \]

3. Next, divide 6 by the remainder 2:
   \[
   6 \div 2 = 3 \quad \text{remainder} \, 0
   \]

Since the remainder is now 0, the GCD is the last non-zero remainder, which is 2.

Now, divide both the numerator and the denominator by their GCD (2):

\[
\frac{32 \div 2}{26 \div 2} = \frac{16}{13}
\]

Thus, the simplified fraction is:

\[
\frac{16}{13}
\]

### Answer: \(\frac{16}{13}\)

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?

GPT-4o:

We previously found that \(\left(\frac{21}{1}\right) \div \left(\frac{26}{20}\right) = \frac{210}{13}\).

Now, we need to multiply this result by the fraction \(\frac{21}{29}\):
\[
\frac{210}{13} \times \frac{21}{29}
\]

To multiply these fractions, we multiply the numerators together and the denominators together:
\[
\frac{210 \times 21}{13 \times 29} = \frac{4410}{377}
\]

Next, we check if the fraction \(\frac{4410}{377}\) can be simplified. We need to find the greatest common divisor (GCD) of 4410 and 377. 

Using the Euclidean algorithm:
1. \(4410 \div 377 \approx 11.7\) (integer part is 11)
2. \(4410 - 377 \times 11 = 4410 - 4147 = 263\)
3. \(377 \div 263 \approx 1.43\) (integer part is 1)
4. \(377 - 263 \times 1 = 377 - 263 = 114\)
5. \(263 \div 114 \approx 2.3\) (integer part is 2)
6. \(263 - 114 \times 2 = 263 - 228 = 35\)
7. \(114 \div 35 \approx 3.26\) (integer part is 3)
8. \(114 - 35 \times 3 = 114 - 105 = 9\)
9. \(35 \div 9 \approx 3.89\) (integer part is 3)
10. \(35 - 9 \times 3 = 35 - 27 = 8\)
11. \(9 \div 8 \approx 1.125\) (integer part is 1)
12. \(9 - 8 \times 1 = 9 - 8 = 1\)
13. \(8 \div 1 = 8\) (integer part is 8)
14. \(8 - 1 \times 8 = 8 - 8 = 0\)

The GCD is 1, meaning \(\frac{4410}{377}\) is already in its simplest form.

### Answer: \(\frac{4410}{377}\)

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?

GPT-4o:

To solve the expression \(\left(\frac{21}{1}\right) \div 2\), we can rewrite the division by 2 as multiplication by the reciprocal of 2, which is \(\frac{1}{2}\).

So, we have:
\[
\left(\frac{21}{1}\right) \div 2 = \left(\frac{21}{1}\right) \times \frac{1}{2}
\]

Now, we perform the multiplication:
\[
\frac{21}{1} \times \frac{1}{2} = \frac{21 \times 1}{1 \times 2} = \frac{21}{2}
\]

Thus, the result of the expression is:
\[
\frac{21}{2}
\]

### Answer: \(\frac{21}{2}\)

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.

GPT-4o:

To multiply two fractions, you multiply the numerators together and the denominators together.

Given fractions: \(\frac{20}{19}\) and \(\frac{21}{19}\)

First, multiply the numerators:
\[ 20 \times 21 = 420 \]

Next, multiply the denominators:
\[ 19 \times 19 = 361 \]

So, the product of the two fractions is:
\[ \frac{420}{361} \]

### Answer: \(\frac{420}{361}\)

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.

GPT-4o:

First, let's simplify the amount of sugar Jim needs for the cake:

The amount of sugar required is:
\[ \frac{12}{2} = 6 \text{ cups} \]

The corrected amount of flour required is:
\[ \frac{30}{14} \text{ cups} \]

Now, we need to multiply these two quantities together:
\[ 6 \times \frac{30}{14} \]

To perform this multiplication, we can write 6 as a fraction:
\[ 6 = \frac{6}{1} \]

Now, multiply the fractions:
\[ \frac{6}{1} \times \frac{30}{14} = \frac{6 \times 30}{1 \times 14} = \frac{180}{14} \]

Next, we simplify the fraction \(\frac{180}{14}\). We find the greatest common divisor (GCD) of 180 and 14, which is 2.

Divide both the numerator and the denominator by their GCD:
\[ \frac{180 \div 2}{14 \div 2} = \frac{90}{7} \]

Thus, the total amount of sugar and flour Jim needs is:
\[ \frac{90}{7} \]

### Answer: \(\frac{90}{7}\)

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.

GPT-4o:

To find the value of the new operation, we need to square the result obtained from the previous problem, which was \(\frac{161}{144}\).

First, we square both the numerator and the denominator:

\[
\left(\frac{161}{144}\right)^2 = \frac{161^2}{144^2}
\]

Next, we calculate the squares of the numerator and the denominator:

\[
161^2 = 25921
\]
\[
144^2 = 20736
\]

So, the expression becomes:

\[
\frac{25921}{20736}
\]

To simplify this fraction, we need to check if there is a common divisor for 25921 and 20736. We can use the Euclidean algorithm to find the GCD of 25921 and 20736:

1. \(25921 \div 20736 = 1\) with a remainder of \(5185\)
2. \(20736 \div 5185 = 4\) with a remainder of \(1\)
3. \(5185 \div 1 = 5185\) with a remainder of \(0\)

The GCD is \(1\), which means the fraction \(\frac{25921}{20736}\) is already in its simplest form.

### Answer: \(\frac{25921}{20736}\)

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.

GPT-4o:

To solve the expression \(\frac{6}{\frac{27}{23}}\), we need to remember that dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore, we can rewrite the expression as:

\[
\frac{6}{\frac{27}{23}} = 6 \times \frac{23}{27}
\]

Next, we multiply the numerator by the reciprocal of the denominator:

\[
6 \times \frac{23}{27} = \frac{6 \times 23}{27}
\]

Calculating the product:

\[
6 \times 23 = 138
\]

So, the expression becomes:

\[
\frac{138}{27}
\]

To simplify this fraction, we need to find the greatest common divisor (GCD) of 138 and 27. We can use the Euclidean algorithm to find the GCD:

1. \(138 \div 27 = 5\) with a remainder of \(3\)
2. \(27 \div 3 = 9\) with a remainder of \(0\)

The GCD is \(3\).

Now, we divide both the numerator and the denominator by their GCD:

\[
\frac{138 \div 3}{27 \div 3} = \frac{46}{9}
\]

Thus, the simplified fraction is:

\[
\frac{46}{9}
\]

### Answer: \(\frac{46}{9}\)

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.

GPT-4o:

To continue, we need to multiply the previous result \(\frac{11}{10}\) by \(\frac{21}{16}\):

\[
\frac{11}{10} \times \frac{21}{16} = \frac{11 \times 21}{10 \times 16} = \frac{231}{160}
\]

Next, we check if \(\frac{231}{160}\) can be simplified. We need to find the greatest common divisor (GCD) of 231 and 160. 

The prime factorization of 231 is:
\[
231 = 3 \times 7 \times 11
\]

The prime factorization of 160 is:
\[
160 = 2^5 \times 5
\]

Since there are no common factors between 231 and 160, the fraction \(\frac{231}{160}\) is already in its simplest form.

### Answer: \(\frac{231}{160}\)

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?

GPT-4o:

To solve the problem \(\left(\frac{11}{6}\right) \times \left(\frac{24}{12}\right)\), we first simplify the fractions if possible.

The fraction \(\frac{24}{12}\) can be simplified:
\[
\frac{24}{12} = 2
\]

Now, we multiply the simplified fraction by \(\frac{11}{6}\):
\[
\left(\frac{11}{6}\right) \times 2 = \frac{11 \times 2}{6} = \frac{22}{6}
\]

Next, we simplify \(\frac{22}{6}\) by finding the greatest common divisor (GCD) of 22 and 6, which is 2:
\[
\frac{22 \div 2}{6 \div 2} = \frac{11}{3}
\]

Thus, the simplified fraction is:
### Answer: \(\frac{11}{3}\)

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?

GPT-4o:

We previously calculated that \(\left(\frac{6}{19}\right) \times \left(\frac{28}{2}\right) = \frac{84}{19}\).

Now, we need to multiply this result by \(\frac{27}{17}\):

\[
\frac{84}{19} \times \frac{27}{17}
\]

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

\[
\frac{84 \times 27}{19 \times 17}
\]

First, calculate the numerator:

\[
84 \times 27 = 2268
\]

Next, calculate the denominator:

\[
19 \times 17 = 323
\]

So, the fraction is:

\[
\frac{2268}{323}
\]

We need to check if this fraction can be simplified. We do this by finding the greatest common divisor (GCD) of 2268 and 323. 

Using the Euclidean algorithm:
1. \(2268 \div 323 \approx 7\) with a remainder of \(2268 - 7 \times 323 = 2268 - 2261 = 7\).
2. \(323 \div 7 = 46\) with a remainder of \(323 - 46 \times 7 = 323 - 322 = 1\).
3. \(7 \div 1 = 7\) with a remainder of \(0\).

Since the GCD is 1, the fraction \(\frac{2268}{323}\) is already in its simplest form.

### Answer: \(\frac{2268}{323}\)

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?

GPT-4o:

To solve the expression \(\left(\frac{17}{19}\right) \times \left(\frac{28}{2}\right)\), we first simplify the fraction \(\frac{28}{2}\):

\[
\frac{28}{2} = 14
\]

Now, the expression becomes:

\[
\left(\frac{17}{19}\right) \times 14
\]

We can write 14 as a fraction:

\[
14 = \frac{14}{1}
\]

So the expression is now:

\[
\left(\frac{17}{19}\right) \times \left(\frac{14}{1}\right)
\]

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

\[
\frac{17 \times 14}{19 \times 1} = \frac{238}{19}
\]

Next, we simplify the fraction \(\frac{238}{19}\). We perform the division:

\[
238 \div 19 = 12.5263
\]

Since 238 is exactly divisible by 19, we get:

\[
\frac{238}{19} = 12
\]

### Answer: 12

Expected answer: 238/19

Correct? No

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.

GPT-4o:

We start with the fraction obtained from the previous problem, which is \(\frac{24}{5}\). We need to multiply this fraction by \(\frac{15}{10}\):

\[
\frac{24}{5} \times \frac{15}{10}
\]

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

\[
\frac{24 \times 15}{5 \times 10}
\]

Next, we calculate the products in the numerator and the denominator:

\[
24 \times 15 = 360
\]
\[
5 \times 10 = 50
\]

So, the fraction becomes:

\[
\frac{360}{50}
\]

Now, we need to simplify this fraction to its lowest terms. We do this by finding the greatest common divisor (GCD) of 360 and 50. 

First, we find the prime factorizations of both numbers:
- The prime factorization of 360 is \( 2^3 \times 3^2 \times 5 \).
- The prime factorization of 50 is \( 2 \times 5^2 \).

The common factors are \( 2 \) and \( 5 \). Therefore, the GCD is:

\[
2 \times 5 = 10
\]

We divide both the numerator and the denominator by their GCD (10):

\[
\frac{360 \div 10}{50 \div 10} = \frac{36}{5}
\]

Thus, the simplified fraction is:

\[
\frac{36}{5}
\]

### Answer: \(\frac{36}{5}\)

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.

GPT-4o:

First, we start by performing the multiplication of the two fractions:

\[
\left( \frac{3}{14} \right) \times \left( \frac{28}{5} \right)
\]

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

\[
\frac{3 \times 28}{14 \times 5}
\]

Next, we calculate the products in the numerator and the denominator:

\[
3 \times 28 = 84
\]
\[
14 \times 5 = 70
\]

So, the fraction becomes:

\[
\frac{84}{70}
\]

Now, we need to simplify this fraction to its lowest terms. We do this by finding the greatest common divisor (GCD) of 84 and 70. 

First, we find the prime factorizations of both numbers:
- The prime factorization of 84 is \( 2^2 \times 3 \times 7 \).
- The prime factorization of 70 is \( 2 \times 5 \times 7 \).

The common factors are \( 2 \) and \( 7 \). Therefore, the GCD is:

\[
2 \times 7 = 14
\]

We divide both the numerator and the denominator by their GCD (14):

\[
\frac{84 \div 14}{70 \div 14} = \frac{6}{5}
\]

Thus, the simplified fraction is:

\[
\frac{6}{5}
\]

### Answer: \(\frac{6}{5}\)

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.

GPT-4o:

First, we start with the result of \((19 / 3) * (8 / 15)\):

\[
\frac{152}{45}
\]

Next, we need to multiply this result by \(\frac{19}{9}\):

\[
\left( \frac{152}{45} \right) \times \left( \frac{19}{9} \right)
\]

We multiply the numerators together:

\[
152 \times 19 = 2888
\]

And we multiply the denominators together:

\[
45 \times 9 = 405
\]

So, the expression becomes:

\[
\frac{2888}{405}
\]

Now, we simplify the fraction by finding the greatest common divisor (GCD) of 2888 and 405. 

First, we list the factors:
- Factors of 2888: 1, 2, 4, 7, 14, 28, 103, 206, 412, 721, 1442, 2888
- Factors of 405: 1, 3, 5, 9, 15, 27, 45, 81, 135, 405

The common factor is 1, so the fraction \(\frac{2888}{405}\) is already in its simplest form.

### Answer: \(\frac{2888}{405}\)

Expected answer: 2888/405

Correct? Yes

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.

GPT-4o:

To solve the expression \((19 / 7) * (8 / 15)\), we follow these steps:

1. Multiply the numerators together:
\[ 19 \times 8 = 152 \]

2. Multiply the denominators together:
\[ 7 \times 15 = 105 \]

So, the expression becomes:
\[ \frac{152}{105} \]

Next, we simplify the fraction by finding the greatest common divisor (GCD) of 152 and 105. 

First, we list the factors:
- Factors of 152: 1, 2, 4, 8, 19, 38, 76, 152
- Factors of 105: 1, 3, 5, 7, 15, 21, 35, 105

The common factors are 1, and since 1 is the greatest common factor, the fraction \(\frac{152}{105}\) is already in its simplest form.

### Answer: \(\frac{152}{105}\)

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.

GPT-4o:

We already found that \(\left(\frac{15}{3} \times \frac{7}{25}\right) = \frac{7}{5}\).

Now, we need to multiply this result by \(\frac{28}{16}\):

\[
\frac{7}{5} \times \frac{28}{16}
\]

First, multiply the numerators and the denominators:

\[
\frac{7 \times 28}{5 \times 16} = \frac{196}{80}
\]

Next, simplify the fraction \(\frac{196}{80}\). We need to find the greatest common divisor (GCD) of 196 and 80. The GCD of 196 and 80 is 4.

Now, divide both the numerator and the denominator by their GCD:

\[
\frac{196 \div 4}{80 \div 4} = \frac{49}{20}
\]

Thus, the simplified fraction is \(\frac{49}{20}\).

### Answer: \(\frac{49}{20}\)

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?

GPT-4o:

Let's evaluate the new expression step by step.

1. Calculate the division inside the first set of parentheses:
   \[
   \frac{13}{3}
   \]

2. Calculate the division inside the second set of parentheses:
   \[
   \frac{7}{25}
   \]

3. Now, multiply the results from steps 1 and 2:
   \[
   \frac{13}{3} \times \frac{7}{25} = \frac{13 \times 7}{3 \times 25} = \frac{91}{75}
   \]

4. Simplify the fraction \(\frac{91}{75}\):
   - The greatest common divisor (GCD) of 91 and 75 is 1, so the fraction is already in its simplest form.

Thus, the simplified fraction is \(\frac{91}{75}\).

### Answer: \(\frac{91}{75}\)

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.

GPT-4o:

First, recall the previous answer \( x = 20 \).

Now, we need to multiply \( x \) by the fraction \( \frac{19}{15} \):

\[
z = 20 \times \frac{19}{15}
\]

To perform this multiplication, we can write 20 as a fraction:

\[
20 = \frac{20}{1}
\]

Now, multiply the fractions:

\[
\frac{20}{1} \times \frac{19}{15} = \frac{20 \times 19}{1 \times 15} = \frac{380}{15}
\]

Next, we simplify the fraction \( \frac{380}{15} \). We find the greatest common divisor (GCD) of 380 and 15, which is 5.

Divide both the numerator and the denominator by their GCD:

\[
\frac{380 \div 5}{15 \div 5} = \frac{76}{3}
\]

Thus, the simplified fraction is:

\[
\frac{76}{3}
\]

### Answer: \(\frac{76}{3}\)

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.

GPT-4o:

To solve the expression \(\frac{57}{16} \times \frac{28}{16}\), we first multiply the numerators and the denominators:

\[
\frac{57 \times 28}{16 \times 16} = \frac{1596}{256}
\]

Next, we simplify the fraction \(\frac{1596}{256}\). To do this, we need to find the greatest common divisor (GCD) of 1596 and 256.

1. Prime factorization of 1596:
   - 1596 is even, so divide by 2: \(1596 \div 2 = 798\)
   - 798 is even, so divide by 2: \(798 \div 2 = 399\)
   - 399 is divisible by 3: \(399 \div 3 = 133\)
   - 133 is divisible by 7: \(133 \div 7 = 19\)
   - 19 is a prime number.

   Therefore, the prime factorization of 1596 is \(2^2 \times 3 \times 7 \times 19\).

2. Prime factorization of 256:
   - 256 is a power of 2: \(256 = 2^8\).

The common factors are \(2^2\). Therefore, the GCD of 1596 and 256 is \(2^2 = 4\).

Now, divide both the numerator and the denominator by their GCD (4):

\[
\frac{1596 \div 4}{256 \div 4} = \frac{399}{64}
\]

The fraction \(\frac{399}{64}\) is already in its simplest form.

### Answer: \(\frac{399}{64}\)

Expected answer: 399/64

Correct? Yes

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.

GPT-4o:

To solve the expression \(\left(\frac{27}{18}\right) \times \left(\frac{19}{17}\right)\), we first simplify each fraction individually if possible.

1. Simplify \(\frac{27}{18}\):
   - The greatest common divisor (GCD) of 27 and 18 is 9.
   - \(\frac{27}{18} = \frac{27 \div 9}{18 \div 9} = \frac{3}{2}\).

2. The fraction \(\frac{19}{17}\) is already in its simplest form.

Now, multiply the simplified fractions:
\[
\left(\frac{3}{2}\right) \times \left(\frac{19}{17}\right) = \frac{3 \times 19}{2 \times 17} = \frac{57}{34}
\]

The fraction \(\frac{57}{34}\) is already in its simplest form.

### Answer: \(\frac{57}{34}\)

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.

GPT-4o:

We already found that the result of multiplying \( \frac{29}{5} \) by \( \frac{13}{24} \) is \( \frac{377}{120} \).

Now, we need to multiply \( \frac{377}{120} \) by \( \frac{25}{11} \).

1. Multiply the numerators together:
\[ 377 \times 25 = 9425 \]

2. Multiply the denominators together:
\[ 120 \times 11 = 1320 \]

So, the product of the fractions is:
\[ \frac{9425}{1320} \]

Next, we simplify the fraction \( \frac{9425}{1320} \). We need to find the greatest common divisor (GCD) of 9425 and 1320.

- The prime factors of 9425 are 5, 5, 13, and 29.
- The prime factors of 1320 are 2, 2, 2, 3, 5, and 11.

The common factor is 5.

Divide both the numerator and the denominator by 5:
\[ \frac{9425 \div 5}{1320 \div 5} = \frac{1885}{264} \]

We check if \( \frac{1885}{264} \) can be simplified further. The prime factors of 1885 are 5, 13, and 29, and the prime factors of 264 are 2, 2, 2, 3, and 11. There are no common factors other than 1.

Thus, \( \frac{1885}{264} \) is in its simplest form.

### Answer: \(\frac{1885}{264}\)

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.

GPT-4o:

To find the result of multiplying the fractions \( \frac{29}{8} \) and \( \frac{13}{24} \), we follow these steps:

1. Multiply the numerators together:
\[ 29 \times 13 = 377 \]

2. Multiply the denominators together:
\[ 8 \times 24 = 192 \]

So, the product of the fractions is:
\[ \frac{377}{192} \]

Next, we check if the fraction can be simplified. We need to find the greatest common divisor (GCD) of 377 and 192.

- The prime factors of 377 are 1 and 377 (since 377 is a prime number).
- The prime factors of 192 are 2, 2, 2, 2, 2, and 3.

Since 377 and 192 have no common factors other than 1, the fraction \( \frac{377}{192} \) is already in its simplest form.

### Answer: \(\frac{377}{192}\)

Expected answer: 377/192

Correct? Yes

Problem: After finding the value of the expression ((29 / 17) / (27 / 22)), now calculate the result of that value when it is multiplied by (27 / 2). Write your answer as a simplified fraction.

GPT-4o:

We previously found that the value of the expression \(\left(\frac{29}{17}\right) \div \left(\frac{27}{22}\right)\) is \(\frac{638}{459}\).

Now, we need to multiply this result by \(\frac{27}{2}\):

\[
\frac{638}{459} \times \frac{27}{2}
\]

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

\[
\frac{638 \times 27}{459 \times 2}
\]

Calculating the products:

\[
638 \times 27 = 17226
\]
\[
459 \times 2 = 918
\]

So, the expression simplifies to:

\[
\frac{17226}{918}
\]

To simplify this fraction, we need to find the greatest common divisor (GCD) of 17226 and 918. Using the Euclidean algorithm:

1. \(17226 \div 918\) gives a quotient of 18 and a remainder of \(17226 - 918 \times 18 = 0\).

Since the remainder is 0, the GCD is 918.

Now, we divide both the numerator and the denominator by their GCD:

\[
\frac{17226 \div 918}{918 \div 918} = \frac{18}{1}
\]

Thus, the simplified fraction is:

\[
18
\]

### Answer: 18

Expected answer: 319/17

Correct? No

Problem: After finding the value of the expression ((29 / 17) / (27 / 22)), consider how the result would change if instead of 29/17, we had 4/1. Calculate the value of this new expression, 4 / (27 / 22). Write your answer as a simplified fraction.

GPT-4o:

To solve the new expression \(\frac{4}{\left(\frac{27}{22}\right)}\), we can use the property of division of fractions, which states that dividing by a fraction is equivalent to multiplying by its reciprocal.

First, we find the reciprocal of \(\frac{27}{22}\), which is \(\frac{22}{27}\).

Now, we rewrite the expression as:
\[
4 \times \left(\frac{22}{27}\right)
\]

Next, we multiply the numerator by the fraction:
\[
4 \times \frac{22}{27} = \frac{4 \times 22}{27} = \frac{88}{27}
\]

The fraction \(\frac{88}{27}\) is already in its simplest form because 88 and 27 have no common factors other than 1.

### Answer: \(\frac{88}{27}\)

Expected answer: 88/27

Correct? Yes

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

GPT-4o:

First, we already know the result of \((\frac{27}{6}) \times (\frac{25}{5})\) is 22.5.

Next, we need to calculate \(\frac{20}{27}\).

Now, we multiply 22.5 by \(\frac{20}{27}\):

\[ 22.5 \times \frac{20}{27} \]

To simplify this, we can convert 22.5 to a fraction:
\[ 22.5 = \frac{45}{2} \]

So the expression becomes:
\[ \frac{45}{2} \times \frac{20}{27} \]

Multiply the numerators and the denominators:
\[ \frac{45 \times 20}{2 \times 27} = \frac{900}{54} \]

Now, simplify \(\frac{900}{54}\):
\[ \frac{900}{54} = \frac{50}{3} \]

Converting \(\frac{50}{3}\) to a decimal:
\[ \frac{50}{3} \approx 16.67 \]

### Answer: 16.67

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?

GPT-4o:

First, we need to perform the division operations inside the parentheses.

1. Calculate \( \frac{11}{6} \):
\[ \frac{11}{6} \approx 1.8333 \]

2. Calculate \( \frac{25}{5} \):
\[ \frac{25}{5} = 5 \]

Next, we multiply the results of these two divisions:
\[ 1.8333 \times 5 \approx 9.1665 \]

### Answer: 9.1665

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.

GPT-4o:

Given that \( e = \frac{13}{6} \), we need to calculate the value of \(\left(\frac{13}{2}\right) \times e\).

First, substitute \( e \) with \(\frac{13}{6}\):

\[
\left(\frac{13}{2}\right) \times \left(\frac{13}{6}\right)
\]

Next, multiply the numerators and the denominators:

\[
\frac{13 \times 13}{2 \times 6} = \frac{169}{12}
\]

The fraction \(\frac{169}{12}\) is already in its simplest form because 169 and 12 have no common factors other than 1.

### Answer: \(\frac{169}{12}\)

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.

GPT-4o:

To solve the expression \((11 / 1) * (1 / 6)\), we follow these steps:

1. Simplify the first fraction: \(\frac{11}{1}\) is already in its simplest form.
2. Multiply the numerators: \(11 \times 1 = 11\).
3. Multiply the denominators: \(1 \times 6 = 6\).

So, the product of the fractions is:

\[
\frac{11 \times 1}{1 \times 6} = \frac{11}{6}
\]

The fraction \(\frac{11}{6}\) is already in its simplest form because 11 and 6 have no common factors other than 1.

### Answer: \(\frac{11}{6}\)

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.

GPT-4o:

We already found that the product of \( \frac{25}{20} \) and \( \frac{11}{6} \) is \( \frac{55}{24} \).

Now, we need to multiply \( \frac{55}{24} \) by \( \frac{20}{26} \):

\[
\frac{55}{24} \times \frac{20}{26} = \frac{55 \times 20}{24 \times 26}
\]

First, calculate the numerator:

\[
55 \times 20 = 1100
\]

Next, calculate the denominator:

\[
24 \times 26 = 624
\]

So, the product is:

\[
\frac{1100}{624}
\]

Now, we need to simplify this fraction. We find the greatest common divisor (GCD) of 1100 and 624. The prime factorization of 1100 is \( 2^2 \times 5^2 \times 11 \), and the prime factorization of 624 is \( 2^4 \times 3 \times 13 \). The common factor is \( 2^2 = 4 \).

Divide both the numerator and the denominator by their GCD, which is 4:

\[
\frac{1100 \div 4}{624 \div 4} = \frac{275}{156}
\]

Since 275 and 156 have no common factors other than 1, the fraction is already in its simplest form.

### Answer: \(\frac{275}{156}\)

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.

GPT-4o:

First, let's calculate the product of \( \frac{25}{17} \) and \( \frac{11}{6} \).

\[
\frac{25}{17} \times \frac{11}{6} = \frac{25 \times 11}{17 \times 6}
\]

Calculate the numerator:

\[
25 \times 11 = 275
\]

Calculate the denominator:

\[
17 \times 6 = 102
\]

So, the product is:

\[
\frac{275}{102}
\]

Next, we need to simplify this fraction. We find the greatest common divisor (GCD) of 275 and 102. The prime factorization of 275 is \( 5^2 \times 11 \), and the prime factorization of 102 is \( 2 \times 3 \times 17 \). There are no common factors between 275 and 102 other than 1.

Since 275 and 102 have no common factors other than 1, the fraction is already in its simplest form.

### Answer: \(\frac{275}{102}\)

Expected answer: 275/102

Correct? Yes

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.

GPT-4o:

We have already determined that \(\frac{7}{2} \div \frac{25}{10} = \frac{7}{5}\).

Now, we need to multiply this result by \(\frac{5}{3}\):
\[
\frac{7}{5} \times \frac{5}{3}
\]

Multiply the numerators together and the denominators together:
\[
\frac{7 \times 5}{5 \times 3} = \frac{35}{15}
\]

Next, simplify the fraction \(\frac{35}{15}\) by finding the greatest common divisor (GCD) of 35 and 15, which is 5:
\[
\frac{35 \div 5}{15 \div 5} = \frac{7}{3}
\]

Thus, the simplified fraction is:
### Answer: \(\frac{7}{3}\)

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?

GPT-4o:

First, we start with the result from the previous multiplication:
\[ \frac{145}{144} \]

Now, we need to multiply this fraction by \( \frac{27}{4} \).

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

First, let's multiply the numerators:
\[ 145 \times 27 = 3915 \]

Next, let's multiply the denominators:
\[ 144 \times 4 = 576 \]

So, the product of the fractions is:
\[ \frac{3915}{576} \]

Now, we need to simplify this fraction. To do this, we find the greatest common divisor (GCD) of 3915 and 576.

The prime factorization of 3915 is:
\[ 3915 = 3 \times 5 \times 261 = 3 \times 5 \times 3 \times 87 = 3^2 \times 5 \times 29 \]

The prime factorization of 576 is:
\[ 576 = 2^6 \times 3^2 \]

The common factor between 3915 and 576 is \( 3^2 = 9 \).

Now, we divide both the numerator and the denominator by their GCD, which is 9:
\[ \frac{3915 \div 9}{576 \div 9} = \frac{435}{64} \]

Thus, the simplified fraction is:
\[ \frac{435}{64} \]

### Answer: \(\frac{435}{64}\)

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.

GPT-4o:

To multiply the fractions \(\frac{15}{10}\) and \(\frac{29}{27}\), we follow the same process of multiplying the numerators and the denominators.

First, let's multiply the numerators:
\[ 15 \times 29 = 435 \]

Next, let's multiply the denominators:
\[ 10 \times 27 = 270 \]

So, the product of the fractions is:
\[ \frac{435}{270} \]

Now, we need to simplify this fraction. To do this, we find the greatest common divisor (GCD) of 435 and 270.

The prime factorization of 435 is:
\[ 435 = 3 \times 5 \times 29 \]

The prime factorization of 270 is:
\[ 270 = 2 \times 3^3 \times 5 \]

The common factors between 435 and 270 are 3 and 5.

The GCD of 435 and 270 is:
\[ 3 \times 5 = 15 \]

Now, we divide both the numerator and the denominator by their GCD, which is 15:
\[ \frac{435 \div 15}{270 \div 15} = \frac{29}{18} \]

Thus, the simplified fraction is:
\[ \frac{29}{18} \]

### Answer: \(\frac{29}{18}\)

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.

GPT-4o:

We previously found that \(\left(\frac{30}{25}\right) \times \left(\frac{25}{10}\right) = 3\).

Now, we need to multiply this result by \(\frac{5}{9}\):
\[
3 \times \frac{5}{9} = \frac{3 \times 5}{9} = \frac{15}{9}
\]

Next, we simplify \(\frac{15}{9}\):
\[
\frac{15}{9} = \frac{15 \div 3}{9 \div 3} = \frac{5}{3}
\]

### Answer: \(\frac{5}{3}\)

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.

GPT-4o:

First, we calculate the product of the fractions \(\frac{27}{10}\) and \(\frac{27}{3}\):

\[
\frac{27}{10} \times \frac{27}{3} = \frac{27 \times 27}{10 \times 3} = \frac{729}{30}
\]

Next, we multiply this result by \(\frac{9}{20}\):

\[
\frac{729}{30} \times \frac{9}{20} = \frac{729 \times 9}{30 \times 20}
\]

First, calculate the numerator:

\[
729 \times 9 = 6561
\]

Next, calculate the denominator:

\[
30 \times 20 = 600
\]

So, the product of the fractions is:

\[
\frac{6561}{600}
\]

To simplify this fraction, we need to find the greatest common divisor (GCD) of 6561 and 600. The prime factorization of 6561 is \(3^8\), and the prime factorization of 600 is \(2^3 \times 3 \times 5^2\). The only common factor is 3.

Divide both the numerator and the denominator by 3:

\[
\frac{6561 \div 3}{600 \div 3} = \frac{2187}{200}
\]

Since 2187 and 200 have no common factors other than 1, \(\frac{2187}{200}\) is in its simplest form.

### Answer: \(\frac{2187}{200}\)

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.

GPT-4o:

To find the product of the fractions \(\frac{27}{10}\) and \(\frac{27}{21}\), we multiply the numerators together and the denominators together:

\[
\frac{27}{10} \times \frac{27}{21} = \frac{27 \times 27}{10 \times 21}
\]

First, calculate the numerator:

\[
27 \times 27 = 729
\]

Next, calculate the denominator:

\[
10 \times 21 = 210
\]

So, the product of the fractions is:

\[
\frac{729}{210}
\]

To simplify this fraction, we need to find the greatest common divisor (GCD) of 729 and 210. The prime factorization of 729 is \(3^6\), and the prime factorization of 210 is \(2 \times 3 \times 5 \times 7\). The only common factor is 3.

Divide both the numerator and the denominator by 3:

\[
\frac{729 \div 3}{210 \div 3} = \frac{243}{70}
\]

Since 243 and 70 have no common factors other than 1, \(\frac{243}{70}\) is in its simplest form.

### Answer: \(\frac{243}{70}\)

Expected answer: 243/70

Correct? Yes

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.

GPT-4o:

First, we start with the result of the previous operation:

\[
\frac{75}{52}
\]

Next, we need to find the square of this fraction by multiplying it with itself:

\[
\left( \frac{75}{52} \right)^2 = \frac{75 \times 75}{52 \times 52} = \frac{5625}{2704}
\]

Now, we need to check if this fraction can be simplified. We find the greatest common divisor (GCD) of 5625 and 2704. 

The prime factorization of 5625 is:
\[
5625 = 3^2 \times 5^4
\]

The prime factorization of 2704 is:
\[
2704 = 2^4 \times 13^2
\]

Since there are no common prime factors between 5625 and 2704, the fraction \(\frac{5625}{2704}\) is already in its simplest form.

### Answer: \(\frac{5625}{2704}\)

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.

GPT-4o:

We previously evaluated the expression \(\left(\frac{20}{22}\right) \times \left(\frac{14}{11}\right)\) and found it to be \(\frac{140}{121}\).

Now, we need to multiply this result by \(\frac{17}{3}\):
\[
\frac{140}{121} \times \frac{17}{3} = \frac{140 \times 17}{121 \times 3}
\]

First, calculate the numerator:
\[
140 \times 17 = 2380
\]

Next, calculate the denominator:
\[
121 \times 3 = 363
\]

So, the product is:
\[
\frac{2380}{363}
\]

Now, we need to simplify this fraction. To do this, we find the greatest common divisor (GCD) of 2380 and 363. 

Using the Euclidean algorithm:
1. \(2380 \div 363 \approx 6\) with a remainder of \(2380 - 6 \times 363 = 2380 - 2178 = 202\).
2. \(363 \div 202 \approx 1\) with a remainder of \(363 - 202 = 161\).
3. \(202 \div 161 \approx 1\) with a remainder of \(202 - 161 = 41\).
4. \(161 \div 41 \approx 3\) with a remainder of \(161 - 3 \times 41 = 161 - 123 = 38\).
5. \(41 \div 38 \approx 1\) with a remainder of \(41 - 38 = 3\).
6. \(38 \div 3 \approx 12\) with a remainder of \(38 - 12 \times 3 = 38 - 36 = 2\).
7. \(3 \div 2 \approx 1\) with a remainder of \(3 - 2 = 1\).
8. \(2 \div 1 \approx 2\) with a remainder of \(2 - 2 \times 1 = 2 - 2 = 0\).

The GCD is 1, so the fraction \(\frac{2380}{363}\) is already in its simplest form.

### Answer: \(\frac{2380}{363}\)

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.

GPT-4o:

Let's first evaluate the new expression \(\left(\frac{20}{22}\right) \times \left(\frac{14}{2}\right)\).

1. Simplify \(\frac{20}{22}\):
   - The greatest common divisor (GCD) of 20 and 22 is 2.
   - \(\frac{20}{22} = \frac{20 \div 2}{22 \div 2} = \frac{10}{11}\).

2. Simplify \(\frac{14}{2}\):
   - The greatest common divisor (GCD) of 14 and 2 is 2.
   - \(\frac{14}{2} = \frac{14 \div 2}{2 \div 2} = \frac{7}{1} = 7\).

Now, multiply the simplified fractions:
\[
\left(\frac{10}{11}\right) \times 7 = \frac{10 \times 7}{11 \times 1} = \frac{70}{11}
\]

The fraction \(\frac{70}{11}\) is already in its simplest form because 70 and 11 have no common factors other than 1.

### Answer: \(\frac{70}{11}\)

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?

GPT-4o:

To find the result of \(\left(\frac{75}{32} \times \frac{29}{23}\right)\), we follow these steps:

1. Multiply the numerators together:
   \[
   75 \times 29 = 2175
   \]

2. Multiply the denominators together:
   \[
   32 \times 23 = 736
   \]

3. Combine the results to form a single fraction:
   \[
   \frac{2175}{736}
   \]

4. Simplify the fraction if possible. We need to check if 2175 and 736 have any common factors. 

First, let's find the greatest common divisor (GCD) of 2175 and 736. Using the Euclidean algorithm:

\[
2175 \div 736 \approx 2.955 \quad \text{(integer part is 2)}
\]
\[
2175 - 2 \times 736 = 2175 - 1472 = 703
\]

Next, apply the Euclidean algorithm to 736 and 703:

\[
736 \div 703 \approx 1.047 \quad \text{(integer part is 1)}
\]
\[
736 - 1 \times 703 = 736 - 703 = 33
\]

Next, apply the Euclidean algorithm to 703 and 33:

\[
703 \div 33 \approx 21.303 \quad \text{(integer part is 21)}
\]
\[
703 - 21 \times 33 = 703 - 693 = 10
\]

Next, apply the Euclidean algorithm to 33 and 10:

\[
33 \div 10 \approx 3.3 \quad \text{(integer part is 3)}
\]
\[
33 - 3 \times 10 = 33 - 30 = 3
\]

Finally, apply the Euclidean algorithm to 10 and 3:

\[
10 \div 3 \approx 3.333 \quad \text{(integer part is 3)}
\]
\[
10 - 3 \times 3 = 10 - 9 = 1
\]

Since the GCD is 1, the fraction \(\frac{2175}{736}\) is already in its simplest form.

Thus, the simplified fraction is \(\frac{2175}{736}\).

In terms of the original division or portion sharing scenario, this additional operation represents further distributing or scaling the portions. If \(\frac{75}{32}\) was the portion each group received initially, multiplying by \(\frac{29}{23}\) adjusts the portion size by a factor of \(\frac{29}{23}\), either increasing or decreasing the amount each group gets based on the value of the fraction.

### Answer: \(\frac{2175}{736}\)

Expected answer: 2175/736

Correct? Yes

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.

GPT-4o:

To evaluate the expression \(\left(\frac{25}{8} \times \frac{29}{4}\right)\), we follow these steps:

1. Multiply the numerators together:
   \[
   25 \times 29 = 725
   \]

2. Multiply the denominators together:
   \[
   8 \times 4 = 32
   \]

3. Combine the results to form a single fraction:
   \[
   \frac{725}{32}
   \]

4. Simplify the fraction if possible. In this case, 725 and 32 have no common factors other than 1, so the fraction is already in its simplest form.

Thus, the simplified fraction is \(\frac{725}{32}\).

Reflecting on the real-world scenario, if the fractions represent sharing or dividing amongst a group, changing the second fraction from \(\frac{3}{4}\) to \(\frac{29}{4}\) significantly increases the amount being multiplied. This means that if you were distributing 25 units of something among 8 groups and then further dividing by 4 groups, the new ratio \(\frac{29}{4}\) would result in a much larger portion per group compared to the previous ratio \(\frac{3}{4}\). Essentially, each group would receive a larger share of the total units.

### Answer: \(\frac{725}{32}\)

Expected answer: 725/32

Correct? Yes

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

GPT-4o:

To find the result of multiplying \(\frac{754}{255}\) by \(\frac{21}{9}\), we multiply the numerators together and the denominators together:

\[
\frac{754}{255} \times \frac{21}{9} = \frac{754 \times 21}{255 \times 9}
\]

First, calculate the numerator:

\[
754 \times 21 = 15834
\]

Next, calculate the denominator:

\[
255 \times 9 = 2295
\]

So, the product is:

\[
\frac{15834}{2295}
\]

To simplify this fraction, we need to find the greatest common divisor (GCD) of 15834 and 2295. We can use the Euclidean algorithm to find the GCD.

1. Divide 15834 by 2295 and find the remainder:
\[
15834 \div 2295 \approx 6 \quad \text{(quotient)} \quad \text{remainder} = 15834 - 6 \times 2295 = 1954
\]

2. Divide 2295 by 1954 and find the remainder:
\[
2295 \div 1954 \approx 1 \quad \text{(quotient)} \quad \text{remainder} = 2295 - 1 \times 1954 = 341
\]

3. Divide 1954 by 341 and find the remainder:
\[
1954 \div 341 \approx 5 \quad \text{(quotient)} \quad \text{remainder} = 1954 - 5 \times 341 = 249
\]

4. Divide 341 by 249 and find the remainder:
\[
341 \div 249 \approx 1 \quad \text{(quotient)} \quad \text{remainder} = 341 - 1 \times 249 = 92
\]

5. Divide 249 by 92 and find the remainder:
\[
249 \div 92 \approx 2 \quad \text{(quotient)} \quad \text{remainder} = 249 - 2 \times 92 = 65
\]

6. Divide 92 by 65 and find the remainder:
\[
92 \div 65 \approx 1 \quad \text{(quotient)} \quad \text{remainder} = 92 - 1 \times 65 = 27
\]

7. Divide 65 by 27 and find the remainder:
\[
65 \div 27 \approx 2 \quad \text{(quotient)} \quad \text{remainder} = 65 - 2 \times 27 = 11
\]

8. Divide 27 by 11 and find the remainder:
\[
27 \div 11 \approx 2 \quad \text{(quotient)} \quad \text{remainder} = 27 - 2 \times 11 = 5
\]

9. Divide 11 by 5 and find the remainder:
\[
11 \div 5 \approx 2 \quad \text{(quotient)} \quad \text{remainder} = 11 - 2 \times 5 = 1
\]

10. Divide 5 by 1 and find the remainder:
\[
5 \div 1 \approx 5 \quad \text{(quotient)} \quad \text{remainder} = 5 - 5 \times 1 = 0
\]

Since the remainder is now 0, the GCD is the last non-zero remainder, which is 1. Therefore, the fraction \(\frac{15834}{2295}\) is already in its simplest form.

### Answer: \(\frac{15834}{2295}\)

Expected answer: 5278/765

Correct? Yes

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

GPT-4o:

To find the result of multiplying \(\frac{15}{17}\) by \(\frac{26}{15}\), we multiply the numerators together and the denominators together:

\[
\frac{15}{17} \times \frac{26}{15} = \frac{15 \times 26}{17 \times 15}
\]

First, calculate the numerator:

\[
15 \times 26 = 390
\]

Next, calculate the denominator:

\[
17 \times 15 = 255
\]

So, the product is:

\[
\frac{390}{255}
\]

To simplify this fraction, we need to find the greatest common divisor (GCD) of 390 and 255. We can use the Euclidean algorithm to find the GCD.

1. Divide 390 by 255 and find the remainder:
\[
390 \div 255 \approx 1 \quad \text{(quotient)} \quad \text{remainder} = 390 - 1 \times 255 = 135
\]

2. Divide 255 by 135 and find the remainder:
\[
255 \div 135 \approx 1 \quad \text{(quotient)} \quad \text{remainder} = 255 - 1 \times 135 = 120
\]

3. Divide 135 by 120 and find the remainder:
\[
135 \div 120 \approx 1 \quad \text{(quotient)} \quad \text{remainder} = 135 - 1 \times 120 = 15
\]

4. Divide 120 by 15 and find the remainder:
\[
120 \div 15 \approx 8 \quad \text{(quotient)} \quad \text{remainder} = 120 - 8 \times 15 = 0
\]

Since the remainder is now 0, the GCD is the last non-zero remainder, which is 15. Therefore, we can simplify the fraction \(\frac{390}{255}\) by dividing both the numerator and the denominator by 15:

\[
\frac{390 \div 15}{255 \div 15} = \frac{26}{17}
\]

### Answer: \(\frac{26}{17}\)

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?

GPT-4o:

First, we need to evaluate the expression inside the parentheses.

1. Calculate \( \frac{25}{5} \):
\[
\frac{25}{5} = 5
\]

2. Calculate \( \frac{19}{8} \):
\[
\frac{19}{8} = 2.375
\]

Next, we multiply the results of these two calculations:
\[
5 \times 2.375 = 11.875
\]

Thus, the value of \( k \) in this scenario is:
### Answer: 11.875

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.

GPT-4o:

To find the result of multiplying \(\frac{28}{4}\) by \(\frac{10}{12}\), we follow these steps:

1. Simplify \(\frac{28}{4}\):
   \[
   \frac{28}{4} = 7
   \]

2. Multiply the simplified fraction by \(\frac{10}{12}\):
   \[
   7 \times \frac{10}{12} = \frac{7 \times 10}{12} = \frac{70}{12}
   \]

Next, we need to simplify \(\frac{70}{12}\). We do this by finding the greatest common divisor (GCD) of 70 and 12.

- The prime factorization of 70 is \(2 \times 5 \times 7\).
- The prime factorization of 12 is \(2^2 \times 3\).

The common factor is 2.

Now, divide both the numerator and the denominator by their GCD, which is 2:
\[
\frac{70 \div 2}{12 \div 2} = \frac{35}{6}
\]

Since 35 and 6 have no common factors other than 1, \(\frac{35}{6}\) is already in its simplest form.

### Answer: \(\frac{35}{6}\)

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.

GPT-4o:

First, we calculate the multiplication of the fractions \(\frac{21}{26} \times \frac{20}{9}\).

1. Multiply the numerators together:
\[ 21 \times 20 = 420 \]

2. Multiply the denominators together:
\[ 26 \times 9 = 234 \]

So, the product of the fractions is:
\[ \frac{420}{234} \]

Next, we simplify this fraction. We find the greatest common divisor (GCD) of 420 and 234.

The prime factorization of 420 is \( 2^2 \times 3 \times 5 \times 7 \).
The prime factorization of 234 is \( 2 \times 3^2 \times 13 \).

The common prime factors are 2 and 3. Therefore, the GCD is:
\[ 2 \times 3 = 6 \]

Now, we divide both the numerator and the denominator by their GCD:
\[ \frac{420 \div 6}{234 \div 6} = \frac{70}{39} \]

Thus, the simplified form of the fraction is:
\[ \frac{70}{39} \]

Next, we take this result and multiply it by the fraction \(\frac{15}{21}\).

1. Multiply the numerators together:
\[ 70 \times 15 = 1050 \]

2. Multiply the denominators together:
\[ 39 \times 21 = 819 \]

So, the product of the fractions is:
\[ \frac{1050}{819} \]

Next, we simplify this fraction. We find the greatest common divisor (GCD) of 1050 and 819.

The prime factorization of 1050 is \( 2 \times 3 \times 5^2 \times 7 \).
The prime factorization of 819 is \( 3 \times 3 \times 7 \times 13 \).

The common prime factors are 3 and 7. Therefore, the GCD is:
\[ 3 \times 7 = 21 \]

Now, we divide both the numerator and the denominator by their GCD:
\[ \frac{1050 \div 21}{819 \div 21} = \frac{50}{39} \]

Thus, the final simplified form of the fraction is:
\[ \frac{50}{39} \]

### Answer: \(\frac{50}{39}\)

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.

GPT-4o:

To solve the multiplication of the fractions \(\frac{21}{26} \times \frac{20}{4}\), we follow these steps:

1. Simplify the second fraction \(\frac{20}{4}\):
\[ \frac{20}{4} = 5 \]

So, the problem now becomes:
\[ \frac{21}{26} \times 5 \]

2. Multiply the numerator of the first fraction by 5:
\[ 21 \times 5 = 105 \]

3. The denominator remains the same:
\[ 26 \]

Thus, the product of the fractions is:
\[ \frac{105}{26} \]

Next, we need to simplify this fraction. To do this, we find the greatest common divisor (GCD) of 105 and 26.

First, we find the prime factorizations of both numbers:
- The prime factorization of 105 is \( 3 \times 5 \times 7 \)
- The prime factorization of 26 is \( 2 \times 13 \)

Since there are no common prime factors, the GCD is 1.

Therefore, the fraction \(\frac{105}{26}\) is already in its simplest form.

### Answer: \(\frac{105}{26}\)

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.

GPT-4o:

To solve the new expression \(\left(\frac{85}{26}\right) \times \left(\frac{8}{15}\right)\), we first multiply the numerators together and then the denominators together:

\[
\frac{85 \times 8}{26 \times 15}
\]

Calculating the numerator:

\[
85 \times 8 = 680
\]

Calculating the denominator:

\[
26 \times 15 = 390
\]

So, the expression simplifies to:

\[
\frac{680}{390}
\]

Next, we need to simplify this fraction by finding the greatest common divisor (GCD) of 680 and 390. We can use the Euclidean algorithm to find the GCD.

1. \(680 \div 390 = 1\) with a remainder of \(680 - 1 \times 390 = 290\)
2. \(390 \div 290 = 1\) with a remainder of \(390 - 1 \times 290 = 100\)
3. \(290 \div 100 = 2\) with a remainder of \(290 - 2 \times 100 = 90\)
4. \(100 \div 90 = 1\) with a remainder of \(100 - 1 \times 90 = 10\)
5. \(90 \div 10 = 9\) with a remainder of \(90 - 9 \times 10 = 0\)

The GCD is the last non-zero remainder, which is 10.

Now, we divide both the numerator and the denominator by their GCD:

\[
\frac{680 \div 10}{390 \div 10} = \frac{68}{39}
\]

Thus, the simplified fraction is:

### Answer: \(\frac{68}{39}\)

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.

GPT-4o:

First, we start with the result of the initial problem:

\[
\left(\frac{25}{18}\right) \times \left(\frac{17}{6}\right) = \frac{425}{108}
\]

Next, we need to multiply this result by the fraction \(\frac{23}{17}\):

\[
\frac{425}{108} \times \frac{23}{17}
\]

We multiply the numerators together and the denominators together:

\[
\frac{425 \times 23}{108 \times 17}
\]

First, calculate the numerator:

\[
425 \times 23 = 9775
\]

Next, calculate the denominator:

\[
108 \times 17 = 1836
\]

So, the fraction is:

\[
\frac{9775}{1836}
\]

To simplify this fraction, we need to find the greatest common divisor (GCD) of 9775 and 1836. We can use the Euclidean algorithm to find the GCD.

1. Divide 9775 by 1836 to get the remainder:
\[
9775 \div 1836 \approx 5 \quad \text{(remainder 1595)}
\]

2. Now, divide 1836 by 1595 to get the remainder:
\[
1836 \div 1595 \approx 1 \quad \text{(remainder 241)}
\]

3. Next, divide 1595 by 241 to get the remainder:
\[
1595 \div 241 \approx 6 \quad \text{(remainder 149)}
\]

4. Then, divide 241 by 149 to get the remainder:
\[
241 \div 149 \approx 1 \quad \text{(remainder 92)}
\]

5. Next, divide 149 by 92 to get the remainder:
\[
149 \div 92 \approx 1 \quad \text{(remainder 57)}
\]

6. Then, divide 92 by 57 to get the remainder:
\[
92 \div 57 \approx 1 \quad \text{(remainder 35)}
\]

7. Next, divide 57 by 35 to get the remainder:
\[
57 \div 35 \approx 1 \quad \text{(remainder 22)}
\]

8. Then, divide 35 by 22 to get the remainder:
\[
35 \div 22 \approx 1 \quad \text{(remainder 13)}
\]

9. Next, divide 22 by 13 to get the remainder:
\[
22 \div 13 \approx 1 \quad \text{(remainder 9)}
\]

10. Then, divide 13 by 9 to get the remainder:
\[
13 \div 9 \approx 1 \quad \text{(remainder 4)}
\]

11. Next, divide 9 by 4 to get the remainder:
\[
9 \div 4 \approx 2 \quad \text{(remainder 1)}
\]

12. Finally, divide 4 by 1 to get the remainder:
\[
4 \div 1 = 4 \quad \text{(remainder 0)}
\]

Since the remainder is now 0, the GCD is the last non-zero remainder, which is 1. Therefore, the fraction \(\frac{9775}{1836}\) is already in its simplest form.

### Answer: \(\frac{9775}{1836}\)

Expected answer: 575/108

Correct? Yes

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.

GPT-4o:

To solve the updated equation \(\left(\frac{25}{18}\right) \times \left(\frac{14}{6}\right)\), we multiply the numerators together and the denominators together:

\[
\frac{25 \times 14}{18 \times 6}
\]

First, calculate the numerator:

\[
25 \times 14 = 350
\]

Next, calculate the denominator:

\[
18 \times 6 = 108
\]

So, the fraction is:

\[
\frac{350}{108}
\]

To simplify this fraction, we need to find the greatest common divisor (GCD) of 350 and 108. We can use the Euclidean algorithm to find the GCD.

1. Divide 350 by 108 to get the remainder:
\[
350 \div 108 \approx 3 \quad \text{(remainder 26)}
\]

2. Now, divide 108 by 26 to get the remainder:
\[
108 \div 26 \approx 4 \quad \text{(remainder 4)}
\]

3. Next, divide 26 by 4 to get the remainder:
\[
26 \div 4 \approx 6 \quad \text{(remainder 2)}
\]

4. Then, divide 4 by 2 to get the remainder:
\[
4 \div 2 = 2 \quad \text{(remainder 0)}
\]

Since the remainder is now 0, the GCD is the last non-zero remainder, which is 2. Therefore, we divide both the numerator and the denominator by 2:

\[
\frac{350 \div 2}{108 \div 2} = \frac{175}{54}
\]

Thus, the simplified fraction is:

### Answer: \(\frac{175}{54}\)

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.

GPT-4o:

We previously found that \( t = \frac{18}{17} \).

Now, we need to find \( t \times t \), which is \( \left( \frac{18}{17} \right) \times \left( \frac{18}{17} \right) \).

Using the rule for multiplying fractions:

\[
\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}
\]

Here, \( a = 18 \), \( b = 17 \), \( c = 18 \), and \( d = 17 \). Substituting these values into the formula, we get:

\[
\frac{18}{17} \times \frac{18}{17} = \frac{18 \times 18}{17 \times 17} = \frac{324}{289}
\]

The fraction \( \frac{324}{289} \) is already in its simplest form because 324 and 289 have no common factors other than 1.

### Answer: \(\frac{324}{289}\)

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.

GPT-4o:

To calculate the result of multiplying \( \frac{12}{1} \) by \( \frac{2}{17} \), we use the rule for multiplying fractions:

\[
\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}
\]

Here, \( a = 12 \), \( b = 1 \), \( c = 2 \), and \( d = 17 \). Substituting these values into the formula, we get:

\[
\frac{12}{1} \times \frac{2}{17} = \frac{12 \times 2}{1 \times 17} = \frac{24}{17}
\]

The fraction \( \frac{24}{17} \) is already in its simplest form because 24 and 17 have no common factors other than 1.

### Answer: \(\frac{24}{17}\)

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.

GPT-4o:

We previously found that:
\[
z = \frac{513}{196}
\]

Now, we need to multiply this by \(\frac{17}{23}\):
\[
\frac{513}{196} \times \frac{17}{23}
\]

First, multiply the numerators:
\[
513 \times 17 = 8721
\]

Next, multiply the denominators:
\[
196 \times 23 = 4508
\]

So, the product of the fractions is:
\[
\frac{8721}{4508}
\]

Next, we need to simplify this fraction by finding the greatest common divisor (GCD) of 8721 and 4508.

- The prime factorization of 8721 is \(3 \times 2907\), and further factorizing 2907 gives \(3 \times 969\), and further factorizing 969 gives \(3 \times 323\), and further factorizing 323 gives \(17 \times 19\). So, \(8721 = 3^3 \times 17 \times 19\).
- The prime factorization of 4508 is \(2^2 \times 1127\). 

Since there are no common factors between 8721 and 4508 other than 1, the fraction \(\frac{8721}{4508}\) is already in its simplest form.

### Answer: \(\frac{8721}{4508}\)

Expected answer: 8721/4508

Correct? Yes

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.

GPT-4o:

To solve the multiplication \(\left(\frac{15}{14}\right) \times \left(\frac{19}{14}\right)\), we follow these steps:

1. Multiply the numerators together:
   \[
   15 \times 19 = 285
   \]

2. Multiply the denominators together:
   \[
   14 \times 14 = 196
   \]

So, the product of the fractions is:
\[
\frac{285}{196}
\]

Next, we need to check if this fraction can be simplified. We do this by finding the greatest common divisor (GCD) of 285 and 196.

- The prime factorization of 285 is \(3 \times 5 \times 19\).
- The prime factorization of 196 is \(2^2 \times 7^2\).

Since there are no common factors between 285 and 196 other than 1, the fraction \(\frac{285}{196}\) is already in its simplest form.

### Answer: \(\frac{285}{196}\)

Expected answer: 285/196

Correct? Yes

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

GPT-4o:

We start with the result from the previous problem, which is \(\frac{16}{15}\). We need to multiply this by \(\frac{30}{24}\):

\[
\frac{16}{15} \times \frac{30}{24}
\]

First, we multiply the numerators together and the denominators together:

\[
\frac{16 \times 30}{15 \times 24}
\]

Next, we simplify the fraction. We can start by simplifying the numerator and the denominator separately:

\[
16 \times 30 = 480
\]
\[
15 \times 24 = 360
\]

So, we have:

\[
\frac{480}{360}
\]

To simplify this fraction, we find the greatest common divisor (GCD) of 480 and 360. The GCD of 480 and 360 is 120. We divide both the numerator and the denominator by 120:

\[
\frac{480 \div 120}{360 \div 120} = \frac{4}{3}
\]

Thus, the simplified fraction is:

### Answer: \(\frac{4}{3}\)

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.

GPT-4o:

Let's start by calculating the new expression step by step:

\[
(16 / 5) * (5 / 7)
\]

We can multiply the numerators together and the denominators together:

\[
\frac{16 \times 5}{5 \times 7}
\]

Next, we simplify the fraction by canceling out the common factor of 5 in the numerator and the denominator:

\[
\frac{16 \times 5}{5 \times 7} = \frac{16}{7}
\]

Thus, the simplified fraction is:

### Answer: \(\frac{16}{7}\)

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.

GPT-4o:

To solve the multiplication of the fractions \(\frac{16}{7} \times \frac{23}{20}\), we follow these steps:

1. Multiply the numerators together:
   \[
   16 \times 23 = 368
   \]

2. Multiply the denominators together:
   \[
   7 \times 20 = 140
   \]

So, the product of the fractions is:
\[
\frac{368}{140}
\]

Next, we need to simplify this fraction. We do this by finding the greatest common divisor (GCD) of 368 and 140.

First, we find the prime factorizations:
- 368: \(2^4 \times 23\)
- 140: \(2^2 \times 5 \times 7\)

The common factor is \(2^2 = 4\), so the GCD is 4.

Now, we divide both the numerator and the denominator by their GCD:
\[
\frac{368 \div 4}{140 \div 4} = \frac{92}{35}
\]

Thus, the simplified fraction is:
\[
\frac{92}{35}
\]

### Answer: \(\frac{92}{35}\)

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.

GPT-4o:

To solve the multiplication of the fractions \(\frac{8}{3} \times \frac{11}{21}\), we follow these steps:

1. Multiply the numerators together:
   \[
   8 \times 11 = 88
   \]

2. Multiply the denominators together:
   \[
   3 \times 21 = 63
   \]

So, the product of the fractions is:
\[
\frac{88}{63}
\]

Next, we need to simplify this fraction. We do this by finding the greatest common divisor (GCD) of 88 and 63.

First, we find the prime factorizations:
- 88: \(2^3 \times 11\)
- 63: \(3^2 \times 7\)

There are no common factors between 88 and 63 other than 1, so the GCD is 1.

Since the GCD is 1, the fraction \(\frac{88}{63}\) is already in its simplest form.

### Answer: \(\frac{88}{63}\)

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.

GPT-4o:

To solve this problem, we need to multiply the result from the previous problem, \(\frac{56}{33}\), by \(\frac{9}{4}\).

1. Write the multiplication of the fractions:
   \[
   \frac{56}{33} \times \frac{9}{4}
   \]

2. Multiply the numerators and the denominators:
   \[
   \frac{56 \times 9}{33 \times 4}
   \]

3. Calculate the products:
   \[
   56 \times 9 = 504
   \]
   \[
   33 \times 4 = 132
   \]
   So, the fraction is:
   \[
   \frac{504}{132}
   \]

4. Simplify the fraction by finding the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 504 and 132 is 12.

5. Divide both the numerator and the denominator by the GCD:
   \[
   \frac{504 \div 12}{132 \div 12} = \frac{42}{11}
   \]

6. Check if the fraction \(\frac{42}{11}\) can be simplified further. Since 42 and 11 have no common factors other than 1, the fraction is already in its simplest form.

### Answer: \(\frac{42}{11}\)

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.

GPT-4o:

To evaluate the new expression \( \frac{13}{\frac{22}{16}} \), we can follow these steps:

1. Rewrite the division of fractions as multiplication by the reciprocal:
   \[
   \frac{13}{\frac{22}{16}} = 13 \times \frac{16}{22}
   \]

2. Multiply the numerators and the denominators:
   \[
   13 \times \frac{16}{22} = \frac{13 \times 16}{22}
   \]

3. Calculate the product in the numerator:
   \[
   13 \times 16 = 208
   \]
   So, the fraction is:
   \[
   \frac{208}{22}
   \]

4. Simplify the fraction by finding the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 208 and 22 is 2.

5. Divide both the numerator and the denominator by the GCD:
   \[
   \frac{208 \div 2}{22 \div 2} = \frac{104}{11}
   \]

6. Check if the fraction \(\frac{104}{11}\) can be simplified further. Since 104 and 11 have no common factors other than 1, the fraction is already in its simplest form.

### Answer: \(\frac{104}{11}\)

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?

GPT-4o:

To solve the multiplication of the fractions \(\frac{10}{3} \times \frac{25}{26}\), we multiply the numerators together and the denominators together.

First, multiply the numerators:
\[ 10 \times 25 = 250 \]

Next, multiply the denominators:
\[ 3 \times 26 = 78 \]

So, the product of the fractions is:
\[ \frac{250}{78} \]

To simplify this fraction, we need to find the greatest common divisor (GCD) of 250 and 78. We can use the Euclidean algorithm to find the GCD.

1. Divide 250 by 78 and find the remainder:
\[ 250 \div 78 = 3 \text{ remainder } 16 \]
2. Now, divide 78 by the remainder 16:
\[ 78 \div 16 = 4 \text{ remainder } 14 \]
3. Next, divide 16 by the remainder 14:
\[ 16 \div 14 = 1 \text{ remainder } 2 \]
4. Finally, divide 14 by the remainder 2:
\[ 14 \div 2 = 7 \text{ remainder } 0 \]

Since the remainder is now 0, the GCD is the last non-zero remainder, which is 2.

Now, divide both the numerator and the denominator of \(\frac{250}{78}\) by their GCD, which is 2:
\[ \frac{250 \div 2}{78 \div 2} = \frac{125}{39} \]

### Answer: \(\frac{125}{39}\)

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.

GPT-4o:

The result of the previous problem was \(\frac{13}{11}\). We need to square this fraction.

To square a fraction, we square both the numerator and the denominator:
\[
\left(\frac{13}{11}\right)^2 = \frac{13^2}{11^2}
\]

Calculate the squares:
\[
13^2 = 169
\]
\[
11^2 = 121
\]

So, we have:
\[
\left(\frac{13}{11}\right)^2 = \frac{169}{121}
\]

### Answer: \(\frac{169}{121}\)

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.

GPT-4o:

To solve the problem \((\frac{16}{12}) * (\frac{26}{22})\), we first simplify each fraction individually.

1. Simplify \(\frac{16}{12}\):
We find the greatest common divisor (GCD) of 16 and 12, which is 4. We then divide both the numerator and the denominator by their GCD:
\[
\frac{16}{12} = \frac{16 \div 4}{12 \div 4} = \frac{4}{3}
\]

2. Simplify \(\frac{26}{22}\):
We find the greatest common divisor (GCD) of 26 and 22, which is 2. We then divide both the numerator and the denominator by their GCD:
\[
\frac{26}{22} = \frac{26 \div 2}{22 \div 2} = \frac{13}{11}
\]

Now, we multiply the simplified fractions:
\[
\frac{4}{3} * \frac{13}{11} = \frac{4 * 13}{3 * 11} = \frac{52}{33}
\]

The fraction \(\frac{52}{33}\) is already in its simplest form since 52 and 33 have no common factors other than 1.

### Answer: \(\frac{52}{33}\)

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?

GPT-4o:

We start with the result from the previous operation, which is \(\frac{17}{14}\). We need to multiply this by \(\frac{21}{25}\):

\[
\frac{17}{14} \times \frac{21}{25}
\]

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

\[
\frac{17 \times 21}{14 \times 25} = \frac{357}{350}
\]

Next, we simplify the fraction \(\frac{357}{350}\). We find the greatest common divisor (GCD) of 357 and 350, which is 7. We divide both the numerator and the denominator by their GCD:

\[
\frac{357 \div 7}{350 \div 7} = \frac{51}{50}
\]

Thus, the simplest form of the fraction is:

### Answer: \(\frac{51}{50}\)

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?

GPT-4o:

First, we start with the initial fractions:

\[
\frac{17}{5} \quad \text{and} \quad \frac{5}{12}
\]

Next, we multiply these two fractions:

\[
\frac{17}{5} \times \frac{5}{12}
\]

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

\[
\frac{17 \times 5}{5 \times 12} = \frac{85}{60}
\]

Now, we simplify the fraction \(\frac{85}{60}\). We find the greatest common divisor (GCD) of 85 and 60, which is 5. We divide both the numerator and the denominator by their GCD:

\[
\frac{85 \div 5}{60 \div 5} = \frac{17}{12}
\]

Thus, the simplest form of the fraction is:

### Answer: \(\frac{17}{12}\)

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.

GPT-4o:

To find the fraction resulting from the multiplication of \(\frac{16}{6}\) and \(\frac{29}{10}\), we follow these steps:

1. Multiply the numerators:
\[
16 \times 29 = 464
\]

2. Multiply the denominators:
\[
6 \times 10 = 60
\]

So, the product of the fractions is:
\[
\frac{464}{60}
\]

Next, we simplify this fraction by finding the greatest common divisor (GCD) of 464 and 60. We can use the Euclidean algorithm to find the GCD.

First, divide 464 by 60 and find the remainder:
\[
464 \div 60 \approx 7 \quad \text{(quotient)} \quad \text{remainder} = 464 - 7 \times 60 = 464 - 420 = 44
\]

Next, divide 60 by 44:
\[
60 \div 44 \approx 1 \quad \text{(quotient)} \quad \text{remainder} = 60 - 1 \times 44 = 60 - 44 = 16
\]

Next, divide 44 by 16:
\[
44 \div 16 \approx 2 \quad \text{(quotient)} \quad \text{remainder} = 44 - 2 \times 16 = 44 - 32 = 12
\]

Next, divide 16 by 12:
\[
16 \div 12 \approx 1 \quad \text{(quotient)} \quad \text{remainder} = 16 - 1 \times 12 = 16 - 12 = 4
\]

Finally, divide 12 by 4:
\[
12 \div 4 \approx 3 \quad \text{(quotient)} \quad \text{remainder} = 12 - 3 \times 4 = 12 - 12 = 0
\]

Since the remainder is now 0, the GCD of 464 and 60 is 4.

Now, we simplify \(\frac{464}{60}\) by dividing both the numerator and the denominator by their GCD, which is 4:
\[
\frac{464 \div 4}{60 \div 4} = \frac{116}{15}
\]

Thus, the fraction in simplest form is:
\[
\frac{116}{15}
\]

### Answer: \(\frac{116}{15}\)

Expected answer: 116/15

Correct? Yes

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

GPT-4o:

To determine the amount of petrol consumed, we need to multiply the petrol consumption rate per mile by the new distance driven.

The petrol consumption rate is given as \(\frac{16}{10}\) gallons per mile, and the new distance is \(\frac{13}{7}\) of a mile.

First, we multiply these two fractions:

\[
\frac{16}{10} \times \frac{13}{7}
\]

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

\[
\frac{16 \times 13}{10 \times 7} = \frac{208}{70}
\]

Next, we simplify the fraction \(\frac{208}{70}\). We find the greatest common divisor (GCD) of 208 and 70. The GCD of 208 and 70 is 2.

We divide both the numerator and the denominator by their GCD:

\[
\frac{208 \div 2}{70 \div 2} = \frac{104}{35}
\]

We check if \(\frac{104}{35}\) can be simplified further. The GCD of 104 and 35 is 1, so the fraction is already in its simplest form.

### Answer: \(\frac{104}{35}\)

Expected answer: 104/35

Correct? Yes

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

GPT-4o:

We start with the result from the previous problem, which is \(\frac{116}{45}\). We need to multiply this by \(\frac{8}{17}\):

\[
\frac{116}{45} \times \frac{8}{17}
\]

We multiply the numerators together and the denominators together:

\[
\frac{116 \times 8}{45 \times 17} = \frac{928}{765}
\]

Next, we simplify the fraction \(\frac{928}{765}\). We do this by finding the greatest common divisor (GCD) of 928 and 765.

First, we find the prime factorizations:
- 928: \(2^4 \times 58\)
- 765: \(3 \times 5 \times 17\)

Since there are no common prime factors between 928 and 765, the GCD is 1.

Thus, the fraction \(\frac{928}{765}\) is already in its simplest form.

### Answer: \(\frac{928}{765}\)

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.

GPT-4o:

To solve the problem of multiplying \(\frac{45}{4}\) by \(\frac{22}{14}\), we follow these steps:

1. Multiply the numerators together:
\[ 45 \times 22 = 990 \]

2. Multiply the denominators together:
\[ 4 \times 14 = 56 \]

So, the product of the fractions is:
\[ \frac{990}{56} \]

Next, we need to simplify this fraction. We do this by finding the greatest common divisor (GCD) of 990 and 56.

First, we find the prime factorizations:
- 990: \( 990 = 2 \times 3^2 \times 5 \times 11 \)
- 56: \( 56 = 2^3 \times 7 \)

The common prime factor is \(2\), and the lowest power of \(2\) in both factorizations is \(2\).

Thus, the GCD is:
\[ 2 \]

Now, we divide both the numerator and the denominator by their GCD:
\[ \frac{990 \div 2}{56 \div 2} = \frac{495}{28} \]

Next, we check if \(\frac{495}{28}\) can be simplified further. We need to find the GCD of 495 and 28.

Prime factorizations:
- 495: \( 495 = 3^2 \times 5 \times 11 \)
- 28: \( 28 = 2^2 \times 7 \)

There are no common prime factors between 495 and 28, so the fraction \(\frac{495}{28}\) is already in its simplest form.

### Answer: \(\frac{495}{28}\)

Expected answer: 495/28

Correct? Yes

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

GPT-4o:

To solve the problem of multiplying \(\frac{15}{12}\) by \(\frac{27}{5}\), we follow these steps:

1. Multiply the numerators together:
\[ 15 \times 27 = 405 \]

2. Multiply the denominators together:
\[ 12 \times 5 = 60 \]

So, the product of the fractions is:
\[ \frac{405}{60} \]

Next, we need to simplify this fraction. We do this by finding the greatest common divisor (GCD) of 405 and 60.

First, we find the prime factorizations:
- 405: \( 405 = 5 \times 3^4 \)
- 60: \( 60 = 2^2 \times 3 \times 5 \)

The common prime factors are \(3\) and \(5\), and the lowest power of each common factor is:
- \(3^1\)
- \(5^1\)

Thus, the GCD is:
\[ 3 \times 5 = 15 \]

Now, we divide both the numerator and the denominator by their GCD:
\[ \frac{405 \div 15}{60 \div 15} = \frac{27}{4} \]

So, the simplified fraction is:
\[ \frac{27}{4} \]

### Answer: \(\frac{27}{4}\)

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?

GPT-4o:

First, let's multiply the previous answer by the given fraction:

\[
1 \times \frac{27}{15} = \frac{27}{15}
\]

Next, we simplify the fraction \(\frac{27}{15}\). To do this, we find the greatest common divisor (GCD) of 27 and 15. The GCD of 27 and 15 is 3.

Now, we divide both the numerator and the denominator by their GCD:

\[
\frac{27 \div 3}{15 \div 3} = \frac{9}{5}
\]

### Answer: \(\frac{9}{5}\)

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)?

GPT-4o:

First, let's simplify each fraction individually:

\[
\frac{7}{7} = 1
\]

Next, simplify the second fraction:

\[
\frac{16}{12}
\]

To simplify \(\frac{16}{12}\), we find the greatest common divisor (GCD) of 16 and 12, which is 4. We then divide both the numerator and the denominator by 4:

\[
\frac{16 \div 4}{12 \div 4} = \frac{4}{3}
\]

Now, we multiply the simplified results:

\[
1 \times \frac{4}{3} = \frac{4}{3}
\]

### Answer: \(\frac{4}{3}\)

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.

GPT-4o:

First, we simplify each fraction individually:

\[
\frac{22}{6} = \frac{22 \div 2}{6 \div 2} = \frac{11}{3}
\]

\[
\frac{16}{24} = \frac{16 \div 8}{24 \div 8} = \frac{2}{3}
\]

Next, we multiply the simplified fractions:

\[
\left( \frac{11}{3} \right) \times \left( \frac{2}{3} \right) = \frac{11 \times 2}{3 \times 3} = \frac{22}{9}
\]

The fraction \(\frac{22}{9}\) is already in its simplest form.

### Answer: \(\frac{22}{9}\)

Expected answer: 22/9

Correct? Yes