Problem: After finding the sum of (74 / 63) and (91 / 78), compute what will be the sum if you add the result to itself.

GPT-4o:

To find the sum of \(\frac{295}{126}\) and itself, we need to add \(\frac{295}{126}\) to \(\frac{295}{126}\):

\[
\frac{295}{126} + \frac{295}{126} = \frac{295 + 295}{126} = \frac{590}{126}
\]

Next, we simplify \(\frac{590}{126}\). We find the greatest common divisor (GCD) of 590 and 126. Using the Euclidean algorithm:

1. \(590 \div 126 = 4\) with a remainder of \(86\).
2. \(126 \div 86 = 1\) with a remainder of \(40\).
3. \(86 \div 40 = 2\) with a remainder of \(6\).
4. \(40 \div 6 = 6\) with a remainder of \(4\).
5. \(6 \div 4 = 1\) with a remainder of \(2\).
6. \(4 \div 2 = 2\) with a remainder of \(0\).

The GCD is 2. Therefore, we simplify \(\frac{590}{126}\) by dividing both the numerator and the denominator by 2:

\[
\frac{590 \div 2}{126 \div 2} = \frac{295}{63}
\]

Thus, the sum of \(\frac{295}{126}\) added to itself is \(\frac{295}{63}\).

### Answer: \(\frac{295}{63}\)

Expected answer: 295/63

Correct? Yes

Problem: In the previous problem you calculated the sum of (74 / 63) and (91 / 78). Now, assume instead of 74 in the first fraction, there were 35. What is the new sum when you add (35 / 63) and (91 / 78)?

GPT-4o:

To determine the value of the sum when adding \(\frac{35}{63}\) and \(\frac{91}{78}\), we first need to find a common denominator for the fractions.

1. The denominators are 63 and 78. We need to find the least common multiple (LCM) of these two numbers.
2. The prime factorization of 63 is \(63 = 3^2 \times 7\).
3. The prime factorization of 78 is \(78 = 2 \times 3 \times 13\).
4. The LCM is found by taking the highest power of each prime that appears in the factorizations:
   - The highest power of 2 is \(2^1\).
   - The highest power of 3 is \(3^2\).
   - The highest power of 7 is \(7^1\).
   - The highest power of 13 is \(13^1\).

Thus, the LCM of 63 and 78 is \(2^1 \times 3^2 \times 7^1 \times 13^1 = 2 \times 9 \times 7 \times 13 = 1638\).

Next, we convert each fraction to have the common denominator of 1638:

\[
\frac{35}{63} = \frac{35 \times 26}{63 \times 26} = \frac{910}{1638}
\]

\[
\frac{91}{78} = \frac{91 \times 21}{78 \times 21} = \frac{1911}{1638}
\]

Now, we add the two fractions:

\[
\frac{910}{1638} + \frac{1911}{1638} = \frac{910 + 1911}{1638} = \frac{2821}{1638}
\]

To simplify \(\frac{2821}{1638}\), we find the greatest common divisor (GCD) of 2821 and 1638. Using the Euclidean algorithm:

1. \(2821 \div 1638 = 1\) with a remainder of \(1183\).
2. \(1638 \div 1183 = 1\) with a remainder of \(455\).
3. \(1183 \div 455 = 2\) with a remainder of \(273\).
4. \(455 \div 273 = 1\) with a remainder of \(182\).
5. \(273 \div 182 = 1\) with a remainder of \(91\).
6. \(182 \div 91 = 2\) with a remainder of \(0\).

The GCD is 91. Therefore, we simplify \(\frac{2821}{1638}\) by dividing both the numerator and the denominator by 91:

\[
\frac{2821 \div 91}{1638 \div 91} = \frac{31}{18}
\]

Thus, the value of the sum is \(\frac{31}{18}\).

### Answer: \(\frac{31}{18}\)

Expected answer: 31/18

Correct? Yes

Problem: If the sum from the previous problem is then added to 73/89, what is the final result?

GPT-4o:

To add \(\frac{471}{25}\) and \(\frac{73}{89}\), we need a common denominator. The least common multiple (LCM) of 25 and 89 is 25 * 89 since they are relatively prime.

First, convert each fraction to have the common denominator of 2225 (25 * 89):

\[ \frac{471}{25} = \frac{471 \times 89}{25 \times 89} = \frac{41919}{2225} \]

\[ \frac{73}{89} = \frac{73 \times 25}{89 \times 25} = \frac{1825}{2225} \]

Now, add the two fractions:

\[ \frac{41919}{2225} + \frac{1825}{2225} = \frac{41919 + 1825}{2225} = \frac{43744}{2225} \]

### Answer: \(\frac{43744}{2225}\)

Expected answer: 43744/2225

Correct? Yes

Problem: If instead of adding 94/5 you had added 94/89 to 1/25 in the previous problem, what would have been the result?

GPT-4o:

To add the fractions \( \frac{94}{89} \) and \( \frac{1}{25} \), we need a common denominator. The least common multiple of 89 and 25 is 2225 (since 89 and 25 are coprime).

First, convert each fraction to have the common denominator of 2225:

\[ \frac{94}{89} = \frac{94 \times 25}{89 \times 25} = \frac{2350}{2225} \]

\[ \frac{1}{25} = \frac{1 \times 89}{25 \times 89} = \frac{89}{2225} \]

Now, we can add the two fractions:

\[ \frac{2350}{2225} + \frac{89}{2225} = \frac{2350 + 89}{2225} = \frac{2439}{2225} \]

### Answer: \(\frac{2439}{2225}\)

Expected answer: 2439/2225

Correct? Yes

Problem: Suppose we had made a mistake in the original problem and it was actually 48/14 we were to subtract from 49/8. What would the result have been in that case? Please answer as a simplified fraction.

GPT-4o:

To solve the problem of subtracting \(\frac{48}{14}\) from \(\frac{49}{8}\), we first need to find a common denominator for the fractions.

1. Simplify \(\frac{48}{14}\):
   \[
   \frac{48}{14} = \frac{24}{7}
   \]

2. The fractions we need to subtract are now \(\frac{49}{8}\) and \(\frac{24}{7}\).

3. Find the least common denominator (LCD) for 8 and 7, which is 56.

4. Convert each fraction to have the common denominator of 56:
   \[
   \frac{49}{8} = \frac{49 \times 7}{8 \times 7} = \frac{343}{56}
   \]
   \[
   \frac{24}{7} = \frac{24 \times 8}{7 \times 8} = \frac{192}{56}
   \]

5. Subtract the fractions:
   \[
   \frac{343}{56} - \frac{192}{56} = \frac{343 - 192}{56} = \frac{151}{56}
   \]

The result is \(\frac{151}{56}\), which is already in its simplest form.

### Answer: \(\frac{151}{56}\)

Expected answer: 151/56

Correct? Yes

Problem: After you've computed the sum of 22/49 and 72/35, add this result to 32/21. What is the final sum, presented as a simplified fraction?

GPT-4o:

We already computed the sum of \( \frac{22}{49} \) and \( \frac{72}{35} \) as \( \frac{614}{245} \).

Now, we need to add \( \frac{614}{245} \) to \( \frac{32}{21} \).

First, we find a common denominator for \( \frac{614}{245} \) and \( \frac{32}{21} \). The denominators are 245 and 21.

The least common multiple (LCM) of 245 and 21 can be found by determining their prime factorizations:
- 245 = \( 5 \times 7^2 \)
- 21 = \( 3 \times 7 \)

The LCM is the highest power of each prime that appears in these factorizations:
- LCM = \( 5 \times 7^2 \times 3 = 735 \)

Next, we convert each fraction to have the common denominator of 735:
- For \( \frac{614}{245} \):
  \[
  \frac{614}{245} = \frac{614 \times 3}{245 \times 3} = \frac{1842}{735}
  \]

- For \( \frac{32}{21} \):
  \[
  \frac{32}{21} = \frac{32 \times 35}{21 \times 35} = \frac{1120}{735}
  \]

Now, we add the two fractions:
\[
\frac{1842}{735} + \frac{1120}{735} = \frac{1842 + 1120}{735} = \frac{2962}{735}
\]

Finally, we simplify the fraction \( \frac{2962}{735} \) if possible. We check for the greatest common divisor (GCD) of 2962 and 735. The prime factorizations are:
- 2962 = \( 2 \times 1481 \)
- 735 = \( 3 \times 5 \times 7^2 \)

Since there are no common factors other than 1, the fraction \( \frac{2962}{735} \) is already in its simplest form.

### Answer: \(\frac{2962}{735}\)

Expected answer: 2962/735

Correct? Yes

Problem: Imagine you originally had 22/34 of a basket of ripe apples instead of 22/49, and you also have 72/35 of a basket of oranges. How much do you have in total, given these new amounts? Provide your answer in simplified form.

GPT-4o:

To find the sum of the fractions \( \frac{22}{34} \) and \( \frac{72}{35} \), we first need to find a common denominator. The denominators are 34 and 35.

The least common multiple (LCM) of 34 and 35 can be found by determining their prime factorizations:
- 34 = \( 2 \times 17 \)
- 35 = \( 5 \times 7 \)

The LCM is the product of the highest powers of all prime factors:
- LCM = \( 2 \times 17 \times 5 \times 7 = 1190 \)

Next, we convert each fraction to have the common denominator of 1190:
- For \( \frac{22}{34} \):
  \[
  \frac{22}{34} = \frac{22 \times 35}{34 \times 35} = \frac{770}{1190}
  \]

- For \( \frac{72}{35} \):
  \[
  \frac{72}{35} = \frac{72 \times 34}{35 \times 34} = \frac{2448}{1190}
  \]

Now, we add the two fractions:
\[
\frac{770}{1190} + \frac{2448}{1190} = \frac{770 + 2448}{1190} = \frac{3218}{1190}
\]

Finally, we simplify the fraction \( \frac{3218}{1190} \) if possible. We check for the greatest common divisor (GCD) of 3218 and 1190. Using the Euclidean algorithm:
- 3218 ÷ 1190 = 2 remainder 838
- 1190 ÷ 838 = 1 remainder 352
- 838 ÷ 352 = 2 remainder 134
- 352 ÷ 134 = 2 remainder 84
- 134 ÷ 84 = 1 remainder 50
- 84 ÷ 50 = 1 remainder 34
- 50 ÷ 34 = 1 remainder 16
- 34 ÷ 16 = 2 remainder 2
- 16 ÷ 2 = 8 remainder 0

The GCD is 2. Therefore, we divide both the numerator and the denominator by 2:
\[
\frac{3218 \div 2}{1190 \div 2} = \frac{1609}{595}
\]

### Answer: \(\frac{1609}{595}\)

Expected answer: 1609/595

Correct? Yes

Problem: After getting the sum of 49/79 and 18/21 from the previous problem, let's denote this sum as n. Calculate the result of adding 71/97 to n.

GPT-4o:

We previously found that the sum of \(\frac{49}{79}\) and \(\frac{18}{21}\) is \(\frac{817}{553}\). Let's denote this sum as \(n\). Now, we need to add \(\frac{71}{97}\) to \(n\).

First, we need to find a common denominator for \(\frac{817}{553}\) and \(\frac{71}{97}\). The least common multiple (LCM) of 553 and 97 will be our common denominator.

Since 553 and 97 are both prime numbers, we can find the LCM by multiplying them together:
\[
553 \times 97 = 53641
\]

Next, we convert each fraction to have the common denominator of 53641.

For \(\frac{817}{553}\):
\[
\frac{817}{553} = \frac{817 \times 97}{553 \times 97} = \frac{79249}{53641}
\]

For \(\frac{71}{97}\):
\[
\frac{71}{97} = \frac{71 \times 553}{97 \times 553} = \frac{39263}{53641}
\]

Now, we add the two fractions:
\[
\frac{79249}{53641} + \frac{39263}{53641} = \frac{79249 + 39263}{53641} = \frac{118512}{53641}
\]

Next, we simplify \(\frac{118512}{53641}\). We find the greatest common divisor (GCD) of 118512 and 53641. Using the Euclidean algorithm:

1. \(118512 \div 53641 \approx 2\) remainder \(111230\)
2. \(53641 \div 111230 \approx 0\) remainder \(53641\)

Since the remainder is 0, the GCD is 1. Therefore, \(\frac{118512}{53641}\) is already in its simplest form.

### Answer: \(\frac{118512}{53641}\)

Expected answer: 118512/53641

Correct? Yes

Problem: In the previous problem, we found the sum of 49/79 and 18/21. Now, let's alter the scenario a bit. Instead of having 49/79 part, let's say we have 49/3 part. Keeping the 18/21 part the same, what would be the new total? Calculate the sum of 49/3 and 18/21.

GPT-4o:

To calculate the sum of the fractions \(\frac{49}{3}\) and \(\frac{18}{21}\), we first need to find a common denominator. The least common multiple (LCM) of 3 and 21 will be our common denominator.

Since 21 is a multiple of 3, the LCM of 3 and 21 is 21.

Now, we convert each fraction to have the common denominator of 21.

For \(\frac{49}{3}\):
\[
\frac{49}{3} = \frac{49 \times 7}{3 \times 7} = \frac{343}{21}
\]

For \(\frac{18}{21}\):
\[
\frac{18}{21} = \frac{18}{21}
\]

Now, we add the two fractions:
\[
\frac{343}{21} + \frac{18}{21} = \frac{343 + 18}{21} = \frac{361}{21}
\]

Next, we check if \(\frac{361}{21}\) can be simplified. We find the greatest common divisor (GCD) of 361 and 21. Using the Euclidean algorithm:

1. \(361 \div 21 \approx 17\) remainder \(4\)
2. \(21 \div 4 \approx 5\) remainder \(1\)
3. \(4 \div 1 \approx 4\) remainder \(0\)

The GCD is 1. Therefore, \(\frac{361}{21}\) is already in its simplest form.

### Answer: \(\frac{361}{21}\)

Expected answer: 361/21

Correct? Yes

Problem: What would be the addition of the results if we add 95/46 and 65/31 together twice?

GPT-4o:

To find the sum of \(\frac{95}{46}\) and \(\frac{65}{31}\) together twice, we first need to determine the sum of these fractions once, and then double the result.

We already found that:
\[
\frac{95}{46} + \frac{65}{31} = \frac{5935}{1426}
\]

Now, we need to add this result to itself:
\[
\frac{5935}{1426} + \frac{5935}{1426} = \frac{5935 + 5935}{1426} = \frac{11870}{1426}
\]

To simplify \(\frac{11870}{1426}\), we find the greatest common divisor (GCD) of 11870 and 1426. The GCD is 2.

So, we divide both the numerator and the denominator by 2:
\[
\frac{11870 \div 2}{1426 \div 2} = \frac{5935}{713}
\]

### Answer: \(\frac{5935}{713}\)

Expected answer: 5935/713

Correct? Yes

Problem: In the previous question, we added 95/46 and 65/31. But what if we used 31/46 instead of 95/46? What's the result when we add 31/46 and 65/31?

GPT-4o:

To find the sum of the fractions \(\frac{31}{46}\) and \(\frac{65}{31}\), we first need to find a common denominator. The denominators are 46 and 31. The least common multiple (LCM) of 46 and 31 is 1426.

Next, we convert each fraction to have the common denominator of 1426:

1. For \(\frac{31}{46}\):
   \[
   \frac{31}{46} = \frac{31 \times 31}{46 \times 31} = \frac{961}{1426}
   \]

2. For \(\frac{65}{31}\):
   \[
   \frac{65}{31} = \frac{65 \times 46}{31 \times 46} = \frac{2990}{1426}
   \]

Now, we add the two fractions:
\[
\frac{961}{1426} + \frac{2990}{1426} = \frac{961 + 2990}{1426} = \frac{3951}{1426}
\]

To simplify \(\frac{3951}{1426}\), we find the greatest common divisor (GCD) of 3951 and 1426. The GCD is 1, so the fraction is already in its simplest form.

### Answer: \(\frac{3951}{1426}\)

Expected answer: 3951/1426

Correct? Yes

Problem: If you subtract 9 over 47 from the result obtained from summing 25 over 63 and 74 over 6, what new value do you get?

GPT-4o:

We already found that the sum of \(\frac{25}{63}\) and \(\frac{74}{6}\) is \(\frac{802}{63}\).

Now, we need to subtract \(\frac{9}{47}\) from \(\frac{802}{63}\). To do this, we need a common denominator. The least common multiple (LCM) of 63 and 47 is 2961.

First, convert each fraction to have the common denominator of 2961.

1. Convert \(\frac{802}{63}\) to a fraction with a denominator of 2961:
\[
\frac{802}{63} = \frac{802 \times 47}{63 \times 47} = \frac{37694}{2961}
\]

2. Convert \(\frac{9}{47}\) to a fraction with a denominator of 2961:
\[
\frac{9}{47} = \frac{9 \times 63}{47 \times 63} = \frac{567}{2961}
\]

Now, subtract the two fractions:
\[
\frac{37694}{2961} - \frac{567}{2961} = \frac{37694 - 567}{2961} = \frac{37127}{2961}
\]

Thus, the new value after subtracting \(\frac{9}{47}\) from \(\frac{802}{63}\) is \(\frac{37127}{2961}\).

### Answer: \(\frac{37127}{2961}\)

Expected answer: 37127/2961

Correct? Yes

Problem: What would be the new sum if we change the previous problem of adding 25 over 63 and 74 over 6 to now adding 25 over 63 and 6 over 6 instead?

GPT-4o:

To find the sum of the fractions \(\frac{25}{63}\) and \(\frac{6}{6}\), we first note that \(\frac{6}{6}\) is equal to 1.

Next, we convert 1 to a fraction with a denominator of 63:
\[
1 = \frac{63}{63}
\]

Now, we add the two fractions:
\[
\frac{25}{63} + \frac{63}{63} = \frac{25 + 63}{63} = \frac{88}{63}
\]

Thus, the sum of \(\frac{25}{63}\) and \(\frac{6}{6}\) is \(\frac{88}{63}\).

### Answer: \(\frac{88}{63}\)

Expected answer: 88/63

Correct? Yes

Problem: After finding the sum of 79/91 and 100/47, what is the result if you add this sum to itself?

GPT-4o:

We previously found that the sum of \(\frac{79}{91}\) and \(\frac{100}{47}\) is \(\frac{12813}{4277}\).

To add this sum to itself, we perform the following calculation:
\[ \frac{12813}{4277} + \frac{12813}{4277} \]

Since the denominators are the same, we can simply add the numerators:
\[ \frac{12813 + 12813}{4277} = \frac{25626}{4277} \]

Thus, the result of adding the sum to itself is \(\frac{25626}{4277}\).

### Answer: \(\frac{25626}{4277}\)

Expected answer: 25626/4277

Correct? Yes

Problem: What is the result when you subtract 57/52 from the sum of 86/62 and 99/41?

GPT-4o:

First, let's recall the sum of \(\frac{86}{62}\) and \(\frac{99}{41}\) that we calculated previously:

\[
\frac{86}{62} + \frac{99}{41} = \frac{4832}{1271}
\]

Next, we need to subtract \(\frac{57}{52}\) from this sum. To do this, we need a common denominator for \(\frac{4832}{1271}\) and \(\frac{57}{52}\). The least common multiple (LCM) of 1271 and 52 is 66172.

First, convert \(\frac{4832}{1271}\) to have the common denominator 66172:
\[
\frac{4832}{1271} = \frac{4832 \times 52}{1271 \times 52} = \frac{251264}{66172}
\]

Next, convert \(\frac{57}{52}\) to have the common denominator 66172:
\[
\frac{57}{52} = \frac{57 \times 1271}{52 \times 1271} = \frac{72447}{66172}
\]

Now, we can subtract the two fractions:
\[
\frac{251264}{66172} - \frac{72447}{66172} = \frac{251264 - 72447}{66172} = \frac{178817}{66172}
\]

Finally, we simplify the fraction \(\frac{178817}{66172}\). Both the numerator and the denominator can be divided by their greatest common divisor (GCD), which is 1 (since 178817 and 66172 are coprime):

\[
\frac{178817}{66172}
\]

Thus, the result when you subtract \(\frac{57}{52}\) from the sum of \(\frac{86}{62}\) and \(\frac{99}{41}\) is \(\frac{178817}{66172}\).

### Answer: \(\frac{178817}{66172}\)

Expected answer: 178817/66092

Correct? No

Problem: What would be the sum if the fraction was changed from 86/62 to 88/62? In other words, what is 88/62 + 99/41?

GPT-4o:

To find the sum of the fractions \(\frac{88}{62}\) and \(\frac{99}{41}\), we first need to find a common denominator. The least common multiple (LCM) of 62 and 41 is 2542.

Next, we convert each fraction to have this common denominator:

1. Convert \(\frac{88}{62}\):
\[
\frac{88}{62} = \frac{88 \times 41}{62 \times 41} = \frac{3608}{2542}
\]

2. Convert \(\frac{99}{41}\):
\[
\frac{99}{41} = \frac{99 \times 62}{41 \times 62} = \frac{6138}{2542}
\]

Now, we can add the two fractions:
\[
\frac{3608}{2542} + \frac{6138}{2542} = \frac{3608 + 6138}{2542} = \frac{9746}{2542}
\]

Finally, we simplify the fraction \(\frac{9746}{2542}\). Both the numerator and the denominator can be divided by their greatest common divisor (GCD), which is 2:
\[
\frac{9746 \div 2}{2542 \div 2} = \frac{4873}{1271}
\]

Thus, the sum of \(\frac{88}{62}\) and \(\frac{99}{41}\) is \(\frac{4873}{1271}\).

### Answer: \(\frac{4873}{1271}\)

Expected answer: 4873/1271

Correct? Yes

Problem: After you calculated the sum of 87/88 and 24/13, find the result when you add this sum to 36/9.

GPT-4o:

First, we need to simplify \(\frac{36}{9}\):

\[
\frac{36}{9} = 4
\]

Next, we add this to the previously calculated sum \(\frac{3243}{1144}\):

\[
4 = \frac{4 \times 1144}{1144} = \frac{4576}{1144}
\]

Now, we add the two fractions:

\[
\frac{3243}{1144} + \frac{4576}{1144} = \frac{3243 + 4576}{1144} = \frac{7819}{1144}
\]

Thus, the result when you add \(\frac{3243}{1144}\) to 4 (or \(\frac{4576}{1144}\)) is \(\frac{7819}{1144}\).

### Answer: \(\frac{7819}{1144}\)

Expected answer: 7819/1144

Correct? Yes

Problem: After you calculated the sum of 87/88 and 24/13 and added 36/9 to this sum, reconsider the problem. This time, let's say we started with 87/83 instead of 87/88. What would be the sum of 87/83 and 24/13?

GPT-4o:

To calculate the sum of the fractions \(\frac{87}{83}\) and \(\frac{24}{13}\), we need to find a common denominator. The least common multiple (LCM) of 83 and 13 is 1079.

First, we convert each fraction to have the common denominator of 1079:

\[
\frac{87}{83} = \frac{87 \times 13}{83 \times 13} = \frac{1131}{1079}
\]

\[
\frac{24}{13} = \frac{24 \times 83}{13 \times 83} = \frac{1992}{1079}
\]

Now, we add the two fractions:

\[
\frac{1131}{1079} + \frac{1992}{1079} = \frac{1131 + 1992}{1079} = \frac{3123}{1079}
\]

Thus, the sum of \(\frac{87}{83}\) and \(\frac{24}{13}\) is \(\frac{3123}{1079}\).

### Answer: \(\frac{3123}{1079}\)

Expected answer: 3123/1079

Correct? Yes

Problem: After you calculated the sum ((68 / 33) + (70 / 25)), now subtract (90 / 92) from the previous result. What is the new value?

GPT-4o:

To solve \(\left(\frac{802}{165}\right) - \left(\frac{90}{92}\right)\), we need to find a common denominator for the fractions.

The denominators are 165 and 92. The least common multiple (LCM) of 165 and 92 is 15180.

Next, we convert each fraction to have the common denominator of 15180:

\[
\frac{802}{165} = \frac{802 \times 92}{165 \times 92} = \frac{73784}{15180}
\]

\[
\frac{90}{92} = \frac{90 \times 165}{92 \times 165} = \frac{14850}{15180}
\]

Now, we subtract the second fraction from the first:

\[
\frac{73784}{15180} - \frac{14850}{15180} = \frac{73784 - 14850}{15180} = \frac{58934}{15180}
\]

To simplify \(\frac{58934}{15180}\), we find the greatest common divisor (GCD) of 58934 and 15180. The GCD is 2.

\[
\frac{58934 \div 2}{15180 \div 2} = \frac{29467}{7590}
\]

Thus, the simplest form of \(\left(\frac{802}{165}\right) - \left(\frac{90}{92}\right)\) is \(\frac{29467}{7590}\).

### Answer: \(\frac{29467}{7590}\)

Expected answer: 29467/7590

Correct? Yes

Problem: Previously, you had calculated ((68 / 33) + (70 / 25)). Now consider if the 70 in the second term was replaced by 39. What value would you get when you calculate ((68 / 33) + (39 / 25))?

GPT-4o:

To solve \(\left(\frac{68}{33}\right) + \left(\frac{39}{25}\right)\), we need to find a common denominator for the fractions.

The denominators are 33 and 25. The least common multiple (LCM) of 33 and 25 is 825.

Next, we convert each fraction to have the common denominator of 825:

\[
\frac{68}{33} = \frac{68 \times 25}{33 \times 25} = \frac{1700}{825}
\]

\[
\frac{39}{25} = \frac{39 \times 33}{25 \times 33} = \frac{1287}{825}
\]

Now, we add the two fractions:

\[
\frac{1700}{825} + \frac{1287}{825} = \frac{1700 + 1287}{825} = \frac{2987}{825}
\]

To simplify \(\frac{2987}{825}\), we find the greatest common divisor (GCD) of 2987 and 825. The GCD is 1, meaning the fraction is already in its simplest form.

Thus, the simplest form of \(\left(\frac{68}{33}\right) + \left(\frac{39}{25}\right)\) is \(\frac{2987}{825}\).

### Answer: \(\frac{2987}{825}\)

Expected answer: 2987/825

Correct? Yes

Problem: After finding the summation of 35/97 and 11/9, what would be the result if you add 11/25 to that sum? Express the result as a simplified fraction.

GPT-4o:

First, we start with the sum of \( \frac{35}{97} \) and \( \frac{11}{9} \), which we previously found to be \( \frac{1382}{873} \).

Next, we need to add \( \frac{11}{25} \) to \( \frac{1382}{873} \). To do this, we need a common denominator. The least common multiple (LCM) of 873 and 25 is 21825.

We convert each fraction to have the common denominator of 21825:

\[
\frac{1382}{873} = \frac{1382 \times 25}{873 \times 25} = \frac{34550}{21825}
\]

\[
\frac{11}{25} = \frac{11 \times 873}{25 \times 873} = \frac{9603}{21825}
\]

Now, we add the two fractions:

\[
\frac{34550}{21825} + \frac{9603}{21825} = \frac{34550 + 9603}{21825} = \frac{44153}{21825}
\]

Next, we simplify \( \frac{44153}{21825} \) by finding the greatest common divisor (GCD) of 44153 and 21825. Using the Euclidean algorithm:

1. \( 44153 \div 21825 \) gives a quotient of 2 and a remainder of 4978.
2. \( 21825 \div 4978 \) gives a quotient of 4 and a remainder of 3913.
3. \( 4978 \div 3913 \) gives a quotient of 1 and a remainder of 1065.
4. \( 3913 \div 1065 \) gives a quotient of 3 and a remainder of 718.
5. \( 1065 \div 718 \) gives a quotient of 1 and a remainder of 347.
6. \( 718 \div 347 \) gives a quotient of 2 and a remainder of 24.
7. \( 347 \div 24 \) gives a quotient of 14 and a remainder of 11.
8. \( 24 \div 11 \) gives a quotient of 2 and a remainder of 2.
9. \( 11 \div 2 \) gives a quotient of 5 and a remainder of 1.
10. \( 2 \div 1 \) gives a quotient of 2 and a remainder of 0.

The GCD is 1, meaning \( \frac{44153}{21825} \) is already in its simplest form.

### Answer: \(\frac{44153}{21825}\)

Expected answer: 44153/21825

Correct? Yes

Problem: Suppose instead of adding 35/97 and 11/9, we change the problem slightly to adding 35/97 and 11/15. What would be the result of the sum? Express it in simplest form.

GPT-4o:

To find the summation of \( \frac{35}{97} \) and \( \frac{11}{15} \), we need to find a common denominator. The least common multiple (LCM) of 97 and 15 is 1455 (since 97 is a prime number and does not share any factors with 15).

First, we convert each fraction to have the common denominator of 1455:

\[
\frac{35}{97} = \frac{35 \times 15}{97 \times 15} = \frac{525}{1455}
\]

\[
\frac{11}{15} = \frac{11 \times 97}{15 \times 97} = \frac{1067}{1455}
\]

Now, we add the two fractions:

\[
\frac{525}{1455} + \frac{1067}{1455} = \frac{525 + 1067}{1455} = \frac{1592}{1455}
\]

Next, we simplify \( \frac{1592}{1455} \) by finding the greatest common divisor (GCD) of 1592 and 1455. Using the Euclidean algorithm:

1. \( 1592 \div 1455 \) gives a quotient of 1 and a remainder of 137.
2. \( 1455 \div 137 \) gives a quotient of 10 and a remainder of 85.
3. \( 137 \div 85 \) gives a quotient of 1 and a remainder of 52.
4. \( 85 \div 52 \) gives a quotient of 1 and a remainder of 33.
5. \( 52 \div 33 \) gives a quotient of 1 and a remainder of 19.
6. \( 33 \div 19 \) gives a quotient of 1 and a remainder of 14.
7. \( 19 \div 14 \) gives a quotient of 1 and a remainder of 5.
8. \( 14 \div 5 \) gives a quotient of 2 and a remainder of 4.
9. \( 5 \div 4 \) gives a quotient of 1 and a remainder of 1.
10. \( 4 \div 1 \) gives a quotient of 4 and a remainder of 0.

The GCD is 1, meaning \( \frac{1592}{1455} \) is already in its simplest form.

### Answer: \(\frac{1592}{1455}\)

Expected answer: 1592/1455

Correct? Yes

Problem: What is the result when you add 62/46 to your previous answer from subtracting 79/70 from 11/5?

GPT-4o:

To solve the problem of adding \(\frac{62}{46}\) to \(\frac{15}{14}\), we first need to find a common denominator for the fractions.

The denominators are 46 and 14. The least common multiple (LCM) of 46 and 14 is 322.

Next, we convert both fractions to have a denominator of 322.

First, convert \(\frac{15}{14}\) to a fraction with a denominator of 322:
\[
\frac{15}{14} = \frac{15 \times 23}{14 \times 23} = \frac{345}{322}
\]

Next, convert \(\frac{62}{46}\) to a fraction with a denominator of 322:
\[
\frac{62}{46} = \frac{62 \times 7}{46 \times 7} = \frac{434}{322}
\]

Now we can add the two fractions:
\[
\frac{345}{322} + \frac{434}{322} = \frac{345 + 434}{322} = \frac{779}{322}
\]

### Answer: \(\frac{779}{322}\)

Expected answer: 779/322

Correct? Yes

Problem: Previously, we solved for the difference when you subtracted 79/70 from 11/5. Now, consider what happens if you originally started with 61/5 instead of 11/5. What would be the result when you subtract 79/70 from this new value?

GPT-4o:

To solve the problem of subtracting \(\frac{79}{70}\) from \(\frac{61}{5}\), we first need to find a common denominator for the fractions.

The denominators are 70 and 5. The least common multiple (LCM) of 70 and 5 is 70.

Next, we convert \(\frac{61}{5}\) to a fraction with a denominator of 70:
\[
\frac{61}{5} = \frac{61 \times 14}{5 \times 14} = \frac{854}{70}
\]

Now we can subtract \(\frac{79}{70}\) from \(\frac{854}{70}\):
\[
\frac{854}{70} - \frac{79}{70} = \frac{854 - 79}{70} = \frac{775}{70}
\]

We can simplify \(\frac{775}{70}\) by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 5:
\[
\frac{775 \div 5}{70 \div 5} = \frac{155}{14}
\]

### Answer: \(\frac{155}{14}\)

Expected answer: 155/14

Correct? Yes

Problem: Following the original problem of finding the sum of 68/16 and 52/78, now add 37/38 to that result. What is the new sum?

GPT-4o:

To find the new sum, we start with the result from the previous problem, which is \(\frac{59}{12}\), and add \(\frac{37}{38}\) to it.

First, we need to find a common denominator for \(\frac{59}{12}\) and \(\frac{37}{38}\). The least common multiple (LCM) of 12 and 38 is 228.

Convert each fraction to have a denominator of 228:
\[
\frac{59}{12} = \frac{59 \times 19}{12 \times 19} = \frac{1121}{228}
\]
\[
\frac{37}{38} = \frac{37 \times 6}{38 \times 6} = \frac{222}{228}
\]

Now, add the fractions:
\[
\frac{1121}{228} + \frac{222}{228} = \frac{1121 + 222}{228} = \frac{1343}{228}
\]

### Answer: \(\frac{1343}{228}\)

Expected answer: 1343/228

Correct? Yes

Problem: In the original problem, we added 68/16 and 52/78. Now, consider what happens if we start with 8/16 instead of 68/16. What is the sum of 8/16 and 52/78?

GPT-4o:

To find the sum of \(\frac{8}{16}\) and \(\frac{52}{78}\), we first simplify each fraction.

1. Simplify \(\frac{8}{16}\):
   \[
   \frac{8}{16} = \frac{8 \div 8}{16 \div 8} = \frac{1}{2}
   \]

2. Simplify \(\frac{52}{78}\):
   \[
   \frac{52}{78} = \frac{52 \div 26}{78 \div 26} = \frac{2}{3}
   \]

Next, we need to find a common denominator to add the fractions \(\frac{1}{2}\) and \(\frac{2}{3}\). The least common multiple (LCM) of 2 and 3 is 6.

Convert each fraction to have a denominator of 6:
\[
\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}
\]
\[
\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}
\]

Now, add the fractions:
\[
\frac{3}{6} + \frac{4}{6} = \frac{3 + 4}{6} = \frac{7}{6}
\]

### Answer: \(\frac{7}{6}\)

Expected answer: 7/6

Correct? Yes

Problem: Now that you calculated the sum of 21/75 and 44/47, can you find out what would be the total of two such sums added together?

GPT-4o:

To find the total of two such sums, we need to add \(\frac{1429}{1175}\) to itself.

First, let's add the fractions:
\[ \frac{1429}{1175} + \frac{1429}{1175} = \frac{1429 + 1429}{1175} = \frac{2858}{1175} \]

Next, we check if \(\frac{2858}{1175}\) can be simplified. We need to find the greatest common divisor (GCD) of 2858 and 1175.

Using the Euclidean algorithm:
\[ 2858 \div 1175 \approx 2 \quad \text{(remainder 508)} \]
\[ 1175 \div 508 \approx 2 \quad \text{(remainder 159)} \]
\[ 508 \div 159 \approx 3 \quad \text{(remainder 31)} \]
\[ 159 \div 31 = 5 \quad \text{(remainder 4)} \]
\[ 31 \div 4 = 7 \quad \text{(remainder 3)} \]
\[ 4 \div 3 = 1 \quad \text{(remainder 1)} \]
\[ 3 \div 1 = 3 \quad \text{(remainder 0)} \]

The GCD is 1, meaning \(\frac{2858}{1175}\) is already in its simplest form.

### Answer: \(\frac{2858}{1175}\)

Expected answer: 2858/1175

Correct? Yes

Problem: Previously, you computed the sum of 21/75 and 44/47. Now, imagine instead that the fraction was 21/63 rather than 21/75. What would the sum of 21/63 and 44/47 be?

GPT-4o:

To find the sum of the fractions \( \frac{21}{63} \) and \( \frac{44}{47} \), we first simplify \( \frac{21}{63} \).

The fraction \( \frac{21}{63} \) can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 21:
\[ \frac{21}{63} = \frac{21 \div 21}{63 \div 21} = \frac{1}{3} \]

Now, we need to find the sum of \( \frac{1}{3} \) and \( \frac{44}{47} \). To do this, we need a common denominator. The denominators are 3 and 47, which are relatively prime. Therefore, the common denominator is their product: \( 3 \times 47 = 141 \).

Convert each fraction to have this common denominator:
\[ \frac{1}{3} = \frac{1 \times 47}{3 \times 47} = \frac{47}{141} \]
\[ \frac{44}{47} = \frac{44 \times 3}{47 \times 3} = \frac{132}{141} \]

Now, add the two fractions:
\[ \frac{47}{141} + \frac{132}{141} = \frac{47 + 132}{141} = \frac{179}{141} \]

The fraction \( \frac{179}{141} \) is already in its simplest form because 179 and 141 have no common factors other than 1.

### Answer: \(\frac{179}{141}\)

Expected answer: 179/141

Correct? Yes

Problem: After finding the result of (77/5 + 4/11), add that result to 94/60. What is the final answer?

GPT-4o:

First, we have the result from the previous calculation:
\[ \frac{77}{5} + \frac{4}{11} = \frac{867}{55} \]

Now, we need to add this result to \(\frac{94}{60}\).

To add these fractions, we need a common denominator. The denominators are 55 and 60. We find the least common multiple (LCM) of 55 and 60.

Prime factorization:
\[ 55 = 5 \times 11 \]
\[ 60 = 2^2 \times 3 \times 5 \]

The LCM is:
\[ \text{LCM}(55, 60) = 2^2 \times 3 \times 5 \times 11 = 660 \]

Next, we convert each fraction to have this common denominator of 660.

For \(\frac{867}{55}\):
\[ \frac{867}{55} = \frac{867 \times 12}{55 \times 12} = \frac{10404}{660} \]

For \(\frac{94}{60}\):
\[ \frac{94}{60} = \frac{94 \times 11}{60 \times 11} = \frac{1034}{660} \]

Now, we can add the two fractions:
\[ \frac{10404}{660} + \frac{1034}{660} = \frac{10404 + 1034}{660} = \frac{11438}{660} \]

To simplify \(\frac{11438}{660}\), we find the greatest common divisor (GCD) of 11438 and 660. Using the Euclidean algorithm:

1. \( 11438 \div 660 \approx 17 \) remainder \( 158 \)
2. \( 660 \div 158 \approx 4 \) remainder \( 28 \)
3. \( 158 \div 28 \approx 5 \) remainder \( 18 \)
4. \( 28 \div 18 \approx 1 \) remainder \( 10 \)
5. \( 18 \div 10 \approx 1 \) remainder \( 8 \)
6. \( 10 \div 8 \approx 1 \) remainder \( 2 \)
7. \( 8 \div 2 = 4 \) remainder \( 0 \)

The GCD is 2.

Now, we simplify the fraction:
\[ \frac{11438 \div 2}{660 \div 2} = \frac{5719}{330} \]

Thus, the final answer is:
### Answer: \(\frac{5719}{330}\)

Expected answer: 5719/330

Correct? Yes

Problem: Instead of the original problem (77/5 + 4/11), consider a change such that now we are adding 77/5 and 4/79 instead. In the context of the previous word problem, if we were originally adding parts of a recipe, we could consider the change as modifying one part of the recipe. What happens if we modify the recipe in this way, by replacing the quantity we originally added with 4/79?

GPT-4o:

To solve the new expression \(\frac{77}{5} + \frac{4}{79}\), we need to find a common denominator.

First, let's find the least common multiple (LCM) of 5 and 79. Since 5 and 79 are both prime numbers, their LCM is simply their product:
\[ \text{LCM}(5, 79) = 5 \times 79 = 395 \]

Next, we convert each fraction to have this common denominator of 395.

For \(\frac{77}{5}\):
\[ \frac{77}{5} = \frac{77 \times 79}{5 \times 79} = \frac{6083}{395} \]

For \(\frac{4}{79}\):
\[ \frac{4}{79} = \frac{4 \times 5}{79 \times 5} = \frac{20}{395} \]

Now, we can add the two fractions:
\[ \frac{6083}{395} + \frac{20}{395} = \frac{6083 + 20}{395} = \frac{6103}{395} \]

To simplify \(\frac{6103}{395}\), we check if the numerator and the denominator have any common factors. Since 6103 and 395 do not have any common factors other than 1, the fraction is already in its simplest form.

Thus, the result of \(\frac{77}{5} + \frac{4}{79}\) is:
### Answer: \(\frac{6103}{395}\)

Expected answer: 6103/395

Correct? Yes

Problem: After finding the sum of 60/48 and 37/47, what would you get if you then added 35/35 to your result?

GPT-4o:

First, let's recall the sum of \(\frac{60}{48}\) and \(\frac{37}{47}\) which we found to be \(\frac{383}{188}\).

Next, we need to add \(\frac{35}{35}\) to this result. Note that \(\frac{35}{35}\) simplifies to 1.

To add \(\frac{383}{188}\) and 1, we need to express 1 with the same denominator as \(\frac{383}{188}\):
\[
1 = \frac{188}{188}
\]

Now, we can add the two fractions:
\[
\frac{383}{188} + \frac{188}{188} = \frac{383 + 188}{188} = \frac{571}{188}
\]

### Answer: \(\frac{571}{188}\)

Expected answer: 571/188

Correct? Yes

Problem: After reconsidering her grocery list, Jane decided to buy 60/45 kilograms of apples instead of 60/48. If she still buys 37/47 kilograms of oranges, what would be the total weight of the apples and oranges she buys?

GPT-4o:

To find the total weight of the apples and oranges Jane buys, we need to add the fractions \(\frac{60}{45}\) and \(\frac{37}{47}\).

1. Simplify \(\frac{60}{45}\):
   \[
   \frac{60}{45} = \frac{60 \div 15}{45 \div 15} = \frac{4}{3}
   \]

2. Now, we need to add \(\frac{4}{3}\) and \(\frac{37}{47}\). To do this, we find a common denominator. The least common multiple (LCM) of 3 and 47 is \(3 \times 47 = 141\).

3. Convert each fraction to have the common denominator of 141:
   \[
   \frac{4}{3} = \frac{4 \times 47}{3 \times 47} = \frac{188}{141}
   \]
   \[
   \frac{37}{47} = \frac{37 \times 3}{47 \times 3} = \frac{111}{141}
   \]

4. Add the fractions:
   \[
   \frac{188}{141} + \frac{111}{141} = \frac{188 + 111}{141} = \frac{299}{141}
   \]

### Answer: \(\frac{299}{141}\)

Expected answer: 299/141

Correct? Yes

Problem: In addition to the calculation you did in the last question, let's add another operation. Solve ((14 / 24) + ((14 / 72) + (25 / 6))).

GPT-4o:

To solve \(\left(\frac{14}{24} + \left(\frac{14}{72} + \frac{25}{6}\right)\right)\), we will follow these steps:

1. Simplify \(\frac{14}{24}\):
   \[
   \frac{14}{24} = \frac{14 \div 2}{24 \div 2} = \frac{7}{12}
   \]

2. From the previous calculation, we know:
   \[
   \frac{14}{72} + \frac{25}{6} = \frac{157}{36}
   \]

3. Convert \(\frac{7}{12}\) to a fraction with a denominator of 36:
   \[
   \frac{7}{12} = \frac{7 \times 3}{12 \times 3} = \frac{21}{36}
   \]

4. Now add the fractions:
   \[
   \frac{21}{36} + \frac{157}{36} = \frac{21 + 157}{36} = \frac{178}{36}
   \]

5. Simplify \(\frac{178}{36}\):
   \[
   \frac{178}{36} = \frac{178 \div 2}{36 \div 2} = \frac{89}{18}
   \]

### Answer: \(\frac{89}{18}\)

Expected answer: 89/18

Correct? Yes

Problem: In the previous problem, we calculated the sum ((14 / 72) + (25 / 6)). But what if we made a mistake and it wasn't 25/6 of something added, but it was actually 90/6? Given this new information, what would be the new value of ((14 / 72) + (90 / 6))?

GPT-4o:

To solve \(\left(\frac{14}{72} + \frac{90}{6}\right)\), we first need to find a common denominator for the fractions.

1. Simplify \(\frac{14}{72}\):
   \[
   \frac{14}{72} = \frac{14 \div 2}{72 \div 2} = \frac{7}{36}
   \]

2. The common denominator for \(\frac{7}{36}\) and \(\frac{90}{6}\) is 36.

3. Convert \(\frac{90}{6}\) to a fraction with a denominator of 36:
   \[
   \frac{90}{6} = \frac{90 \times 6}{6 \times 6} = \frac{540}{36}
   \]

4. Now add the fractions:
   \[
   \frac{7}{36} + \frac{540}{36} = \frac{7 + 540}{36} = \frac{547}{36}
   \]

### Answer: \(\frac{547}{36}\)

Expected answer: 547/36

Correct? Yes

Problem: After finding the sum of 84/97 and 25/2, subtract 100/71 from your result. What is the new result?

GPT-4o:

First, we need to find the sum of \(\frac{84}{97}\) and \(\frac{25}{2}\), which we previously calculated as \(\frac{2593}{194}\).

Next, we need to subtract \(\frac{100}{71}\) from \(\frac{2593}{194}\). To do this, we need a common denominator. The least common multiple (LCM) of 194 and 71 is \(194 \times 71 = 13774\).

Convert each fraction to have the common denominator of 13774:

\[
\frac{2593}{194} = \frac{2593 \times 71}{194 \times 71} = \frac{184103}{13774}
\]

\[
\frac{100}{71} = \frac{100 \times 194}{71 \times 194} = \frac{19400}{13774}
\]

Now, we can subtract the two fractions:

\[
\frac{184103}{13774} - \frac{19400}{13774} = \frac{184103 - 19400}{13774} = \frac{164703}{13774}
\]

### Answer: \(\frac{164703}{13774}\)

Expected answer: 164703/13774

Correct? Yes

Problem: After previously finding the sum of 84/97 and 25/2, now consider a new scenario. What if, instead of having 84 parts out of 97, we had 84 parts out of 96? What then would be the sum of these fractions: 84/96 and 25/2?

GPT-4o:

To find the sum of the fractions \(\frac{84}{96}\) and \(\frac{25}{2}\), we first need to find a common denominator. The least common multiple (LCM) of 96 and 2 is 96.

First, we convert each fraction to have the common denominator of 96:

\[
\frac{84}{96} \text{ is already in terms of 96.}
\]

\[
\frac{25}{2} = \frac{25 \times 48}{2 \times 48} = \frac{1200}{96}
\]

Now, we can add the two fractions:

\[
\frac{84}{96} + \frac{1200}{96} = \frac{84 + 1200}{96} = \frac{1284}{96}
\]

Next, we simplify the fraction \(\frac{1284}{96}\). We find the greatest common divisor (GCD) of 1284 and 96. The GCD is 12.

\[
\frac{1284 \div 12}{96 \div 12} = \frac{107}{8}
\]

### Answer: \(\frac{107}{8}\)

Expected answer: 107/8

Correct? Yes

Problem: If you double the sum of 95/57 and 28/80, what will you get?

GPT-4o:

We previously found that the sum of \(\frac{95}{57}\) and \(\frac{28}{80}\) is \(\frac{121}{60}\).

To find double this sum, we simply multiply \(\frac{121}{60}\) by 2:

\[
2 \times \frac{121}{60} = \frac{2 \times 121}{60} = \frac{242}{60}
\]

Next, we simplify \(\frac{242}{60}\):

1. Find the greatest common divisor (GCD) of 242 and 60. Using the Euclidean algorithm:
   - \(242 \div 60 \approx 4\) remainder 2
   - \(60 \div 2 = 30\) remainder 0

   The GCD is 2.

2. Simplify by dividing both the numerator and the denominator by 2:
   \[
   \frac{242 \div 2}{60 \div 2} = \frac{121}{30}
   \]

So, doubling the sum of \(\frac{95}{57}\) and \(\frac{28}{80}\) gives us \(\frac{121}{30}\).

### Answer: \(\frac{121}{30}\)

Expected answer: 121/30

Correct? Yes

Problem: In the previous problem you considered the sum of 95/57 and 28/80, now consider what would happen if you changed the 57 in the denominator of the first fraction to a 21. What would be the new sum of 95/21 and 28/80?

GPT-4o:

To find the sum of the fractions \(\frac{95}{21}\) and \(\frac{28}{80}\), we first need to find a common denominator. The least common multiple (LCM) of 21 and 80 will be our common denominator.

1. **Find the LCM of 21 and 80:**
   - The prime factorization of 21 is \(3 \times 7\).
   - The prime factorization of 80 is \(2^4 \times 5\).

   The LCM is found by taking the highest power of each prime that appears in the factorizations:
   - LCM = \(2^4 \times 3 \times 5 \times 7 = 16 \times 3 \times 5 \times 7\).

   Let's calculate this step-by-step:
   - \(16 \times 3 = 48\)
   - \(48 \times 5 = 240\)
   - \(240 \times 7 = 1680\)

   So, the LCM of 21 and 80 is 1680.

2. **Convert each fraction to have the common denominator 1680:**
   - For \(\frac{95}{21}\):
     \[
     \frac{95}{21} = \frac{95 \times 80}{21 \times 80} = \frac{7600}{1680}
     \]

   - For \(\frac{28}{80}\):
     \[
     \frac{28}{80} = \frac{28 \times 21}{80 \times 21} = \frac{588}{1680}
     \]

3. **Add the fractions:**
   \[
   \frac{7600}{1680} + \frac{588}{1680} = \frac{7600 + 588}{1680} = \frac{8188}{1680}
   \]

4. **Simplify the fraction \(\frac{8188}{1680}\):**
   - Find the greatest common divisor (GCD) of 8188 and 1680. Using the Euclidean algorithm:
     - \(8188 \div 1680 \approx 4\) remainder 468
     - \(1680 \div 468 \approx 3\) remainder 276
     - \(468 \div 276 \approx 1\) remainder 192
     - \(276 \div 192 \approx 1\) remainder 84
     - \(192 \div 84 \approx 2\) remainder 24
     - \(84 \div 24 \approx 3\) remainder 12
     - \(24 \div 12 = 2\) remainder 0

     The GCD is 12.

   - Simplify by dividing both the numerator and the denominator by 12:
     \[
     \frac{8188 \div 12}{1680 \div 12} = \frac{682}{140}
     \]

   - Further simplify \(\frac{682}{140}\) by dividing both the numerator and the denominator by their GCD, which is 2:
     \[
     \frac{682 \div 2}{140 \div 2} = \frac{341}{70}
     \]

So, the sum of \(\frac{95}{21}\) and \(\frac{28}{80}\) is \(\frac{341}{70}\).

### Answer: \(\frac{341}{70}\)

Expected answer: 2047/420

Correct? No

Problem: If you subtract 38/65 from the sum of 14/57 and 92/42, what would be the result?

GPT-4o:

First, we need to find the sum of \(\frac{14}{57}\) and \(\frac{92}{42}\). We already calculated this in the previous problem as \(\frac{324}{133}\).

Next, we need to subtract \(\frac{38}{65}\) from \(\frac{324}{133}\). To do this, we need a common denominator for the fractions \(\frac{324}{133}\) and \(\frac{38}{65}\).

The least common multiple (LCM) of 133 and 65 will serve as the common denominator. We find the prime factorizations of 133 and 65:
- 133 = 7 × 19
- 65 = 5 × 13

The LCM is the product of the highest powers of all prime factors present in either number:
- LCM = 5 × 7 × 13 × 19 = 8655

Next, we convert each fraction to have the common denominator of 8655:
\[
\frac{324}{133} = \frac{324 \times 65}{133 \times 65} = \frac{21060}{8655}
\]
\[
\frac{38}{65} = \frac{38 \times 133}{65 \times 133} = \frac{5054}{8655}
\]

Now, we subtract the two fractions:
\[
\frac{21060}{8655} - \frac{5054}{8655} = \frac{21060 - 5054}{8655} = \frac{16006}{8655}
\]

To simplify \(\frac{16006}{8655}\), we find the greatest common divisor (GCD) of 16006 and 8655. Using the Euclidean algorithm:
\[
16006 \div 8655 \approx 1.848 \quad \text{(quotient 1, remainder 7351)}
\]
\[
16006 - 1 \times 8655 = 7351
\]
\[
8655 \div 7351 \approx 1.177 \quad \text{(quotient 1, remainder 1304)}
\]
\[
8655 - 1 \times 7351 = 1304
\]
\[
7351 \div 1304 \approx 5.637 \quad \text{(quotient 5, remainder 671)}
\]
\[
7351 - 5 \times 1304 = 671
\]
\[
1304 \div 671 \approx 1.943 \quad \text{(quotient 1, remainder 633)}
\]
\[
1304 - 1 \times 671 = 633
\]
\[
671 \div 633 \approx 1.06 \quad \text{(quotient 1, remainder 38)}
\]
\[
671 - 1 \times 633 = 38
\]
\[
633 \div 38 = 16.658 \quad \text{(quotient 16, remainder 25)}
\]
\[
633 - 16 \times 38 = 25
\]
\[
38 \div 25 = 1.52 \quad \text{(quotient 1, remainder 13)}
\]
\[
38 - 1 \times 25 = 13
\]
\[
25 \div 13 = 1.923 \quad \text{(quotient 1, remainder 12)}
\]
\[
25 - 1 \times 13 = 12
\]
\[
13 \div 12 = 1.083 \quad \text{(quotient 1, remainder 1)}
\]
\[
13 - 1 \times 12 = 1
\]
\[
12 \div 1 = 12 \quad \text{(quotient 12, remainder 0)}
\]

The GCD is 1. Since the GCD is 1, \(\frac{16006}{8655}\) is already in its simplest form.

### Answer: \(\frac{16006}{8655}\)

Expected answer: 16006/8645

Correct? No

Problem: In the previous problem, you found the result of adding 14/57 and 92/42. Now consider a different scenario where instead of adding 92/42 you add 32/42 to 14/57. What would be the new result?

GPT-4o:

To calculate the sum of the fractions \(\frac{14}{57}\) and \(\frac{32}{42}\), we first need to find a common denominator. The least common multiple (LCM) of 57 and 42 will serve as the common denominator.

First, we find the prime factorizations of 57 and 42:
- 57 = 3 × 19
- 42 = 2 × 3 × 7

The LCM is the product of the highest powers of all prime factors present in either number:
- LCM = 2 × 3 × 7 × 19 = 798

Next, we convert each fraction to have the common denominator of 798:
\[
\frac{14}{57} = \frac{14 \times 14}{57 \times 14} = \frac{196}{798}
\]
\[
\frac{32}{42} = \frac{32 \times 19}{42 \times 19} = \frac{608}{798}
\]

Now, we add the two fractions:
\[
\frac{196}{798} + \frac{608}{798} = \frac{196 + 608}{798} = \frac{804}{798}
\]

To simplify \(\frac{804}{798}\), we find the greatest common divisor (GCD) of 804 and 798. Using the Euclidean algorithm:
\[
804 \div 798 \approx 1.0075 \quad \text{(quotient 1, remainder 6)}
\]
\[
804 - 1 \times 798 = 6
\]
\[
798 \div 6 = 133 \quad \text{(quotient 133, remainder 0)}
\]

The GCD is 6. We simplify \(\frac{804}{798}\) by dividing both the numerator and the denominator by 6:
\[
\frac{804 \div 6}{798 \div 6} = \frac{134}{133}
\]

Thus, the sum of \(\frac{14}{57}\) and \(\frac{32}{42}\) is \(\frac{134}{133}\).

### Answer: \(\frac{134}{133}\)

Expected answer: 134/133

Correct? Yes

Problem: If we add the previously calculated result to 39/64, what fraction do we get?

GPT-4o:

To add \(\frac{13}{5}\) and \(\frac{39}{64}\), we need a common denominator. The least common multiple (LCM) of 5 and 64 is 320.

First, convert \(\frac{13}{5}\) to a fraction with a denominator of 320:
\[
\frac{13}{5} = \frac{13 \times 64}{5 \times 64} = \frac{832}{320}
\]

Next, convert \(\frac{39}{64}\) to a fraction with a denominator of 320:
\[
\frac{39}{64} = \frac{39 \times 5}{64 \times 5} = \frac{195}{320}
\]

Now, add the two fractions:
\[
\frac{832}{320} + \frac{195}{320} = \frac{832 + 195}{320} = \frac{1027}{320}
\]

So, the resulting fraction is \(\frac{1027}{320}\).

### Answer: \(\frac{1027}{320}\)

Expected answer: 1027/320

Correct? Yes

Problem: Let's change the previous problem a bit. Instead of adding 6/5, what if we added 6/79 to 77/55? What fraction would we get then?

GPT-4o:

To solve the expression \(\left(\frac{77}{55} + \frac{6}{79}\right)\), we need to find a common denominator and then add the fractions.

First, identify the common denominator for 55 and 79. Since 55 and 79 are relatively prime (they have no common factors other than 1), the common denominator is their product:
\[
55 \times 79 = 4345
\]

Next, convert each fraction to have this common denominator.

For \(\frac{77}{55}\):
\[
\frac{77}{55} = \frac{77 \times 79}{55 \times 79} = \frac{6083}{4345}
\]

For \(\frac{6}{79}\):
\[
\frac{6}{79} = \frac{6 \times 55}{79 \times 55} = \frac{330}{4345}
\]

Now, add the two fractions:
\[
\frac{6083}{4345} + \frac{330}{4345} = \frac{6083 + 330}{4345} = \frac{6413}{4345}
\]

So, the result of the expression is \(\frac{6413}{4345}\).

### Answer: \(\frac{6413}{4345}\)

Expected answer: 583/395

Correct? Yes

Problem: Previously, you found the sum of 23/20 and 82/97. Now, take that result and subtract 7/44 from it. What's the answer?

GPT-4o:

To subtract \( \frac{7}{44} \) from \( \frac{3871}{1940} \), we first need to find a common denominator for the fractions. The least common multiple (LCM) of 1940 and 44 is 21340.

First, we convert each fraction to have the common denominator of 21340:

\[
\frac{3871}{1940} = \frac{3871 \times 11}{1940 \times 11} = \frac{42581}{21340}
\]

\[
\frac{7}{44} = \frac{7 \times 485}{44 \times 485} = \frac{3395}{21340}
\]

Now, we can subtract the two fractions:

\[
\frac{42581}{21340} - \frac{3395}{21340} = \frac{42581 - 3395}{21340} = \frac{39186}{21340}
\]

Next, we simplify the fraction \( \frac{39186}{21340} \). We find the greatest common divisor (GCD) of 39186 and 21340, which is 2.

\[
\frac{39186 \div 2}{21340 \div 2} = \frac{19593}{10670}
\]

Thus, the result of subtracting \( \frac{7}{44} \) from \( \frac{3871}{1940} \) is \( \frac{19593}{10670} \).

### Answer: \(\frac{19593}{10670}\)

Expected answer: 19593/10670

Correct? Yes

Problem: Previously, you found the sum of 23/20 and 82/97. Now imagine instead of 23/20, you have 23/43. What would be the new result if you add 23/43 and 82/97?

GPT-4o:

To find the sum of the fractions \( \frac{23}{43} \) and \( \frac{82}{97} \), we need to find a common denominator. The least common multiple (LCM) of 43 and 97 is 4171, since 43 and 97 are relatively prime.

First, we convert each fraction to have the common denominator of 4171:

\[
\frac{23}{43} = \frac{23 \times 97}{43 \times 97} = \frac{2231}{4171}
\]

\[
\frac{82}{97} = \frac{82 \times 43}{97 \times 43} = \frac{3526}{4171}
\]

Now, we can add the two fractions:

\[
\frac{2231}{4171} + \frac{3526}{4171} = \frac{2231 + 3526}{4171} = \frac{5757}{4171}
\]

Thus, the sum of \( \frac{23}{43} \) and \( \frac{82}{97} \) is \( \frac{5757}{4171} \).

### Answer: \(\frac{5757}{4171}\)

Expected answer: 5757/4171

Correct? Yes

Problem: After finding the sum of 54/88 and 28/10, add this to 21/13 and find the resulting value.

GPT-4o:

We previously found that the sum of \(\frac{54}{88}\) and \(\frac{28}{10}\) is \(\frac{751}{220}\).

Now, we need to add \(\frac{751}{220}\) to \(\frac{21}{13}\).

First, we need to find a common denominator for the fractions \(\frac{751}{220}\) and \(\frac{21}{13}\).

The least common multiple (LCM) of 220 and 13 is 2860.

Convert each fraction to have the common denominator of 2860:
- \(\frac{751}{220} = \frac{751 \times 13}{220 \times 13} = \frac{9763}{2860}\).
- \(\frac{21}{13} = \frac{21 \times 220}{13 \times 220} = \frac{4620}{2860}\).

Now, add the fractions:
\[
\frac{9763}{2860} + \frac{4620}{2860} = \frac{9763 + 4620}{2860} = \frac{14383}{2860}.
\]

The fraction \(\frac{14383}{2860}\) is already in its simplest form because 14383 is a prime number and does not share any common factors with 2860.

### Answer: \(\frac{14383}{2860}\).

Expected answer: 14383/2860

Correct? Yes

Problem: After finding the result of the operation: 7 / 69 + 71 / 47, what is the outcome when you add 61 / 7 to that result?

GPT-4o:

First, we have the result of the previous operation:

\[
\frac{7}{69} + \frac{71}{47} = \frac{5228}{3243}
\]

Now, we need to add \(\frac{61}{7}\) to this result. To do this, we need a common denominator for \(\frac{5228}{3243}\) and \(\frac{61}{7}\).

The denominators are 3243 and 7. The least common multiple (LCM) of 3243 and 7 is their product, since 3243 and 7 are relatively prime (they have no common factors other than 1).

So, the common denominator is \( 3243 \times 7 = 22701 \).

Next, we convert each fraction to have this common denominator:

\[
\frac{5228}{3243} = \frac{5228 \times 7}{3243 \times 7} = \frac{36696}{22701}
\]

\[
\frac{61}{7} = \frac{61 \times 3243}{7 \times 3243} = \frac{197823}{22701}
\]

Now, we can add the two fractions:

\[
\frac{36696}{22701} + \frac{197823}{22701} = \frac{36696 + 197823}{22701} = \frac{234519}{22701}
\]

To simplify \(\frac{234519}{22701}\), we check for any common factors. The greatest common divisor (GCD) of 234519 and 22701 is 3.

Dividing both the numerator and the denominator by 3:

\[
\frac{234519 \div 3}{22701 \div 3} = \frac{78173}{7567}
\]

Thus, the simplified result of the operation is:

\[
\frac{78173}{7567}
\]

### Answer: \(\frac{78173}{7567}\)

Expected answer: 234419/22701

Correct? No

Problem: Following the previous question where you found the result of 7 / 69 + 71 / 47, now let's change the second fraction to 58 / 47 instead of 71 / 47. What is the result of the new operation: 7 / 69 + 58 / 47?

GPT-4o:

To solve the expression \( \frac{7}{69} + \frac{58}{47} \), we need to find a common denominator for the fractions.

The denominators are 69 and 47. The least common multiple (LCM) of 69 and 47 is their product, since 69 and 47 are relatively prime (they have no common factors other than 1).

So, the common denominator is \( 69 \times 47 = 3243 \).

Next, we convert each fraction to have this common denominator:

\[
\frac{7}{69} = \frac{7 \times 47}{69 \times 47} = \frac{329}{3243}
\]

\[
\frac{58}{47} = \frac{58 \times 69}{47 \times 69} = \frac{4002}{3243}
\]

Now, we can add the two fractions:

\[
\frac{329}{3243} + \frac{4002}{3243} = \frac{329 + 4002}{3243} = \frac{4331}{3243}
\]

To simplify \( \frac{4331}{3243} \), we check for any common factors. The greatest common divisor (GCD) of 4331 and 3243 is 1, so the fraction is already in its simplest form.

Thus, the result of the operation is:

\[
\frac{4331}{3243}
\]

### Answer: \(\frac{4331}{3243}\)

Expected answer: 4331/3243

Correct? Yes

Problem: What is the sum of 87/26 and the result of the previous problem ((15/32) + (65/15))?

GPT-4o:

To find the sum of \( \frac{87}{26} \) and \( \frac{461}{96} \), we first need to convert both fractions to have a common denominator. The least common denominator (LCD) of 26 and 96 can be found by determining the least common multiple (LCM) of these two numbers.

First, we find the prime factorizations:
- \( 26 = 2 \times 13 \)
- \( 96 = 2^5 \times 3 \)

The LCM is the product of the highest powers of all prime factors:
\[
\text{LCM}(26, 96) = 2^5 \times 3 \times 13 = 32 \times 3 \times 13 = 1248
\]

Next, we convert each fraction to have this common denominator:

1. Convert \( \frac{87}{26} \) to a fraction with a denominator of 1248:
   \[
   \frac{87}{26} = \frac{87 \times 48}{26 \times 48} = \frac{4176}{1248}
   \]

2. Convert \( \frac{461}{96} \) to a fraction with a denominator of 1248:
   \[
   \frac{461}{96} = \frac{461 \times 13}{96 \times 13} = \frac{5993}{1248}
   \]

Now, we can add the two fractions:
\[
\frac{4176}{1248} + \frac{5993}{1248} = \frac{4176 + 5993}{1248} = \frac{10169}{1248}
\]

Thus, the sum of \( \frac{87}{26} \) and \( \frac{461}{96} \) is \( \frac{10169}{1248} \).

### Answer: \(\frac{10169}{1248}\)

Expected answer: 10169/1248

Correct? Yes

Problem: After finding the sum of 4/31 and 76/11, from that result subtract the fraction 63/35. What is the final result?

GPT-4o:

First, we already found the sum of \(\frac{4}{31}\) and \(\frac{76}{11}\) to be \(\frac{2400}{341}\).

Next, we need to subtract \(\frac{63}{35}\) from \(\frac{2400}{341}\). To do this, we need a common denominator. The least common multiple (LCM) of 341 and 35 is 11935 (since 341 and 35 are relatively prime).

First, we convert each fraction to have the common denominator of 11935:

\[
\frac{2400}{341} = \frac{2400 \times 35}{341 \times 35} = \frac{84000}{11935}
\]

\[
\frac{63}{35} = \frac{63 \times 341}{35 \times 341} = \frac{21483}{11935}
\]

Now, we can subtract the two fractions:

\[
\frac{84000}{11935} - \frac{21483}{11935} = \frac{84000 - 21483}{11935} = \frac{62517}{11935}
\]

Thus, the final result after subtracting \(\frac{63}{35}\) from the sum of \(\frac{4}{31}\) and \(\frac{76}{11}\) is \(\frac{62517}{11935}\).

### Answer: \(\frac{62517}{11935}\)

Expected answer: 8931/1705

Correct? Yes

Problem: In the previous problem, instead of adding 4/31 to 76/11, what would the result be if we started with 42/31 and added it to 76/11 instead?

GPT-4o:

To find the sum of the fractions \(\frac{42}{31}\) and \(\frac{76}{11}\), we again need to find a common denominator. As before, the least common multiple (LCM) of 31 and 11 is 341.

First, we convert each fraction to have the common denominator of 341:

\[
\frac{42}{31} = \frac{42 \times 11}{31 \times 11} = \frac{462}{341}
\]

\[
\frac{76}{11} = \frac{76 \times 31}{11 \times 31} = \frac{2356}{341}
\]

Now, we can add the two fractions:

\[
\frac{462}{341} + \frac{2356}{341} = \frac{462 + 2356}{341} = \frac{2818}{341}
\]

Thus, the sum of \(\frac{42}{31}\) and \(\frac{76}{11}\) is \(\frac{2818}{341}\).

### Answer: \(\frac{2818}{341}\)

Expected answer: 2818/341

Correct? Yes

Problem: Now, in addition to the previous question, what is the sum when you add the fraction 46/83 to the result of adding 82/15 and 100/12?

GPT-4o:

We already found that the sum of \( \frac{82}{15} \) and \( \frac{100}{12} \) is \( \frac{69}{5} \).

Now, we need to add \( \frac{46}{83} \) to \( \frac{69}{5} \).

To add these fractions, we need a common denominator. The least common multiple (LCM) of 5 and 83 is 415.

First, we convert each fraction to have the denominator of 415:

\[ \frac{69}{5} \]
To convert \( \frac{69}{5} \) to a fraction with a denominator of 415, we multiply both the numerator and the denominator by 83:
\[ \frac{69 \times 83}{5 \times 83} = \frac{5727}{415} \]

\[ \frac{46}{83} \]
To convert \( \frac{46}{83} \) to a fraction with a denominator of 415, we multiply both the numerator and the denominator by 5:
\[ \frac{46 \times 5}{83 \times 5} = \frac{230}{415} \]

Now, we add the two fractions:
\[ \frac{5727}{415} + \frac{230}{415} = \frac{5727 + 230}{415} = \frac{5957}{415} \]

Thus, the sum of \( \frac{69}{5} \) and \( \frac{46}{83} \) is \( \frac{5957}{415} \).

### Answer: \( \frac{5957}{415} \)

Expected answer: 5957/415

Correct? Yes

Problem: Now suppose we made an adjustment. Instead of adding 82/15 to 100/12, we are going to add 61/15 to 100/12. What would be the new sum?

GPT-4o:

To find the sum of \( \frac{61}{15} \) and \( \frac{100}{12} \), we again need to find a common denominator. The least common multiple (LCM) of 15 and 12 is 60.

First, we convert each fraction to have the denominator of 60:

\[ \frac{61}{15} \]
To convert \( \frac{61}{15} \) to a fraction with a denominator of 60, we multiply both the numerator and the denominator by 4:
\[ \frac{61 \times 4}{15 \times 4} = \frac{244}{60} \]

\[ \frac{100}{12} \]
To convert \( \frac{100}{12} \) to a fraction with a denominator of 60, we multiply both the numerator and the denominator by 5:
\[ \frac{100 \times 5}{12 \times 5} = \frac{500}{60} \]

Now, we add the two fractions:
\[ \frac{244}{60} + \frac{500}{60} = \frac{244 + 500}{60} = \frac{744}{60} \]

Next, we simplify \( \frac{744}{60} \). We find the greatest common divisor (GCD) of 744 and 60, which is 12.

\[ \frac{744 \div 12}{60 \div 12} = \frac{62}{5} \]

Thus, the sum of \( \frac{61}{15} \) and \( \frac{100}{12} \) is \( \frac{62}{5} \).

### Answer: \( \frac{62}{5} \)

Expected answer: 62/5

Correct? Yes

Problem: After calculating the result of the expression ((92 / 29) + (41 / 33)), what do you get when you add the result to (55/26)? Write your answer as a simplified fraction or a whole number.

GPT-4o:

First, let's recall the result of the expression \(\left(\frac{92}{29}\right) + \left(\frac{41}{33}\right)\), which we found to be \(4 + \frac{397}{957}\).

Now, we need to add this result to \(\frac{55}{26}\).

First, convert \(4 + \frac{397}{957}\) to a single fraction:
\[
4 + \frac{397}{957} = \frac{4 \times 957}{957} + \frac{397}{957} = \frac{3828}{957} + \frac{397}{957} = \frac{3828 + 397}{957} = \frac{4225}{957}
\]

Next, we add \(\frac{4225}{957}\) to \(\frac{55}{26}\). To do this, we need a common denominator. The least common multiple (LCM) of 957 and 26 is 24882.

Convert each fraction to have the common denominator of 24882:
\[
\frac{4225}{957} = \frac{4225 \times 26}{957 \times 26} = \frac{109850}{24882}
\]
\[
\frac{55}{26} = \frac{55 \times 957}{26 \times 957} = \frac{52635}{24882}
\]

Now, add the fractions:
\[
\frac{109850}{24882} + \frac{52635}{24882} = \frac{109850 + 52635}{24882} = \frac{162485}{24882}
\]

Next, we simplify \(\frac{162485}{24882}\). We find the greatest common divisor (GCD) of 162485 and 24882. Using the Euclidean algorithm, we find that the GCD is 1, meaning the fraction is already in its simplest form.

Thus, the final answer is:
\[
\frac{162485}{24882}
\]

### Answer: \(\frac{162485}{24882}\)

Expected answer: 162485/24882

Correct? Yes

Problem: After finding the result from the previous problem: 70/25 + 83/28, what is the result when you subtract 47/13 from your answer?

GPT-4o:

To solve the problem, we need to subtract \( \frac{47}{13} \) from the result of \( \frac{70}{25} + \frac{83}{28} \), which we previously found to be \( \frac{807}{140} \).

First, we need to express \( \frac{47}{13} \) with a common denominator of 140.

1. Convert \( \frac{47}{13} \) to a fraction with a denominator of 140:
   \[
   \frac{47}{13} = \frac{47 \times 140}{13 \times 140} = \frac{47 \times 140}{1820} = \frac{6580}{1820}
   \]

2. Simplify \( \frac{6580}{1820} \):
   \[
   \frac{6580}{1820} = \frac{47 \times 140}{13 \times 140} = \frac{6580}{1820} = \frac{47}{13}
   \]

Next, we need to find a common denominator for \( \frac{807}{140} \) and \( \frac{6580}{1820} \). The least common multiple (LCM) of 140 and 1820 is 1820.

3. Convert \( \frac{807}{140} \) to a fraction with a denominator of 1820:
   \[
   \frac{807}{140} = \frac{807 \times 13}{140 \times 13} = \frac{10491}{1820}
   \]

Now, subtract the two fractions:
\[
\frac{10491}{1820} - \frac{6580}{1820} = \frac{10491 - 6580}{1820} = \frac{3911}{1820}
\]

Finally, simplify \( \frac{3911}{1820} \) if possible. Since 3911 and 1820 have no common factors other than 1, the fraction is already in its simplest form.

### Answer: \(\frac{3911}{1820}\)

Expected answer: 3911/1820

Correct? Yes

Problem: Let's revise the problem from before. If instead of having 70/25 oranges, you had 70/25 apples, and instead of having 83/28 more apples, you had 83/8 more oranges. What is the total of apples and oranges you have now?

GPT-4o:

To solve this problem, we need to find the total number of apples and oranges separately and then add them together.

1. Calculate the number of apples:
   \[
   \frac{70}{25} \text{ apples}
   \]
   Simplify \( \frac{70}{25} \):
   \[
   \frac{70}{25} = \frac{70 \div 5}{25 \div 5} = \frac{14}{5} \text{ apples}
   \]

2. Calculate the number of oranges:
   \[
   \frac{83}{8} \text{ oranges}
   \]

Next, we need to add the number of apples and oranges together. Since they are different units (apples and oranges), we simply add the fractions as they are:

Total apples:
\[
\frac{14}{5} \text{ apples}
\]

Total oranges:
\[
\frac{83}{8} \text{ oranges}
\]

Thus, the total number of apples and oranges is:
\[
\frac{14}{5} \text{ apples} + \frac{83}{8} \text{ oranges}
\]

### Answer: \(\frac{14}{5} \text{ apples} + \frac{83}{8} \text{ oranges}\)

Expected answer: 527/40

Correct? No

Problem: If you add 83/35 to the result of adding the fractions 50/73 and 20/23, what is the total value?

GPT-4o:

First, let's recall the result of adding the fractions \(\frac{50}{73}\) and \(\frac{20}{23}\), which we previously found to be \(\frac{2610}{1679}\).

Now, we need to add \(\frac{83}{35}\) to \(\frac{2610}{1679}\). To do this, we need a common denominator. The least common multiple (LCM) of 1679 and 35 is 58765.

1. Convert \(\frac{2610}{1679}\) to have the common denominator 58765:
\[
\frac{2610}{1679} = \frac{2610 \times 35}{1679 \times 35} = \frac{91350}{58765}
\]

2. Convert \(\frac{83}{35}\) to have the common denominator 58765:
\[
\frac{83}{35} = \frac{83 \times 1679}{35 \times 1679} = \frac{139357}{58765}
\]

Now, we add the two fractions:
\[
\frac{91350}{58765} + \frac{139357}{58765} = \frac{91350 + 139357}{58765} = \frac{230707}{58765}
\]

Since 230707 and 58765 have no common factors other than 1, the fraction \(\frac{230707}{58765}\) is already in its simplest form.

### Answer: \(\frac{230707}{58765}\)

Expected answer: 230707/58765

Correct? Yes

Problem: In the previous scenario, instead of having 50/73 of an hour left to complete your project, you have 50/57 of an hour left. Now, if you still have the additional 20/23 of an hour, how much total time do you have now?

GPT-4o:

To add the fractions \(\frac{50}{57}\) and \(\frac{20}{23}\), we need to find a common denominator. The least common multiple (LCM) of 57 and 23 is 1311, since 57 and 23 are both prime numbers.

First, we convert each fraction to have the common denominator of 1311:

1. Convert \(\frac{50}{57}\):
\[
\frac{50}{57} = \frac{50 \times 23}{57 \times 23} = \frac{1150}{1311}
\]

2. Convert \(\frac{20}{23}\):
\[
\frac{20}{23} = \frac{20 \times 57}{23 \times 57} = \frac{1140}{1311}
\]

Now, we add the two fractions:
\[
\frac{1150}{1311} + \frac{1140}{1311} = \frac{1150 + 1140}{1311} = \frac{2290}{1311}
\]

Since 2290 and 1311 have no common factors other than 1, the fraction \(\frac{2290}{1311}\) is already in its simplest form.

### Answer: \(\frac{2290}{1311}\)

Expected answer: 2290/1311

Correct? Yes

Problem: After finding the value of (49 / 57) + (89 / 99), what would be the sum if you add the resultant value to itself?

GPT-4o:

We previously found that \(\left(\frac{49}{57} + \frac{89}{99}\right) = \frac{3308}{1881}\).

To find the sum of this value added to itself, we perform the following calculation:
\[
\frac{3308}{1881} + \frac{3308}{1881} = \frac{3308 + 3308}{1881} = \frac{6616}{1881}
\]

Thus, the sum of \(\frac{3308}{1881}\) added to itself is \(\frac{6616}{1881}\).

### Answer: \(\frac{6616}{1881}\)

Expected answer: 6616/1881

Correct? Yes

Problem: Earlier you found out the sum of (49 / 57) + (89 / 99). Now suppose the numerator of the first fraction was increased from 49 to 59. What is the sum of (59 / 57) + (89 / 99)?

GPT-4o:

To find the value of \(\left(\frac{59}{57} + \frac{89}{99}\right)\), we will follow a similar process as before by finding a common denominator and then adding the fractions.

We already know that the least common multiple (LCM) of 57 and 99 is 1881.

Next, we convert each fraction to have the common denominator of 1881:
- \(\frac{59}{57} = \frac{59 \times 33}{57 \times 33} = \frac{1947}{1881}\)
- \(\frac{89}{99} = \frac{89 \times 19}{99 \times 19} = \frac{1691}{1881}\)

Now, we can add the fractions:
\[
\frac{1947}{1881} + \frac{1691}{1881} = \frac{1947 + 1691}{1881} = \frac{3638}{1881}
\]

Thus, the value of \(\left(\frac{59}{57} + \frac{89}{99}\right)\) is \(\frac{3638}{1881}\).

### Answer: \(\frac{3638}{1881}\)

Expected answer: 3638/1881

Correct? Yes

Problem: After finding the result of 94 / 35 - 23 / 49, add the result to the fraction 27 / 72. What is the final simplified fraction?

GPT-4o:

First, we start with the result from the previous calculation:
\[ \frac{543}{245} \]

Next, we need to add this to the fraction \( \frac{27}{72} \). To do this, we first simplify \( \frac{27}{72} \):
\[ \frac{27}{72} = \frac{27 \div 9}{72 \div 9} = \frac{3}{8} \]

Now, we need a common denominator for \( \frac{543}{245} \) and \( \frac{3}{8} \). The least common multiple (LCM) of 245 and 8 can be found by their prime factorizations:
- 245 = 5 × 7 × 7
- 8 = 2 × 2 × 2

The LCM is \( 5 \times 7 \times 7 \times 2 \times 2 \times 2 = 1225 \).

Next, we convert each fraction to have the common denominator of 1225:
\[ \frac{543}{245} = \frac{543 \times 5}{245 \times 5} = \frac{2715}{1225} \]
\[ \frac{3}{8} = \frac{3 \times 153}{8 \times 153} = \frac{459}{1225} \]

Now, we add the fractions:
\[ \frac{2715}{1225} + \frac{459}{1225} = \frac{2715 + 459}{1225} = \frac{3174}{1225} \]

Finally, we simplify the fraction \( \frac{3174}{1225} \). We check for the greatest common divisor (GCD) of 3174 and 1225. The prime factorization of 3174 is \( 2 \times 3 \times 3 \times 3 \times 59 \), and the prime factorization of 1225 is \( 5 \times 5 \times 7 \times 7 \). Since they have no common factors other than 1, the fraction is already in its simplest form.

### Answer: \(\frac{3174}{1225}\)

Expected answer: 5079/1960

Correct? No

Problem: Instead of 94 / 35, assume there were only 54 / 35 of something. You still subtract the same amount of 23 / 49. What is the new result as a simplified fraction?

GPT-4o:

To solve the expression \( \frac{54}{35} - \frac{23}{49} \), we first need to find a common denominator for the fractions.

The denominators are 35 and 49. The least common multiple (LCM) of 35 and 49 can be found by their prime factorizations:
- 35 = 5 × 7
- 49 = 7 × 7

The LCM is \( 5 \times 7 \times 7 = 245 \).

Next, we convert each fraction to have the common denominator of 245:
\[ \frac{54}{35} = \frac{54 \times 7}{35 \times 7} = \frac{378}{245} \]
\[ \frac{23}{49} = \frac{23 \times 5}{49 \times 5} = \frac{115}{245} \]

Now, we subtract the fractions:
\[ \frac{378}{245} - \frac{115}{245} = \frac{378 - 115}{245} = \frac{263}{245} \]

Finally, we simplify the fraction \( \frac{263}{245} \). We check for the greatest common divisor (GCD) of 263 and 245. The prime factorization of 263 is \( 263 \) (since it is a prime number), and the prime factorization of 245 is \( 5 \times 7 \times 7 \). Since they have no common factors other than 1, the fraction is already in its simplest form.

### Answer: \(\frac{263}{245}\)

Expected answer: 263/245

Correct? Yes

Problem: If the sum of 58/30 and 49/37 is represented by the variable 't', what would be the value of 't' added to 't'?

GPT-4o:

Given that the sum of \(\frac{58}{30}\) and \(\frac{49}{37}\) is represented by the variable \( t \), we previously calculated:

\[
t = \frac{1808}{555}
\]

To find the value of \( t + t \), we simply double the fraction:

\[
t + t = 2t = 2 \times \frac{1808}{555} = \frac{2 \times 1808}{555} = \frac{3616}{555}
\]

Thus, the value of \( t + t \) is \(\frac{3616}{555}\).

### Answer: \(\frac{3616}{555}\)

Expected answer: 3616/555

Correct? Yes

Problem: After adding 45/53 and 71/50, add this sum to an additional 8/100. What is the final result?

GPT-4o:

First, we already have the sum of \(\frac{45}{53}\) and \(\frac{71}{50}\) as \(\frac{6013}{2650}\).

Next, we need to add \(\frac{8}{100}\) to this sum. To do this, we need a common denominator. The least common multiple (LCM) of 2650 and 100 is 265000.

Convert each fraction to have the common denominator of 265000:

\[
\frac{6013}{2650} = \frac{6013 \times 100}{2650 \times 100} = \frac{601300}{265000}
\]

\[
\frac{8}{100} = \frac{8 \times 2650}{100 \times 2650} = \frac{21200}{265000}
\]

Now, we add the two fractions:

\[
\frac{601300}{265000} + \frac{21200}{265000} = \frac{601300 + 21200}{265000} = \frac{622500}{265000}
\]

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 622500 and 265000 is 500.

\[
\frac{622500 \div 500}{265000 \div 500} = \frac{1245}{530}
\]

Thus, the final result after adding \(\frac{45}{53}\), \(\frac{71}{50}\), and \(\frac{8}{100}\) is \(\frac{1245}{530}\).

### Answer: \(\frac{1245}{530}\)

Expected answer: 249/106

Correct? Yes

Problem: Instead of adding 45/53 and 71/50, let's now consider if we add 45/53 and 71/72. What result do we get?

GPT-4o:

To add the fractions \(\frac{45}{53}\) and \(\frac{71}{72}\), we need to find a common denominator. The least common multiple (LCM) of 53 and 72 is 3816.

First, we convert each fraction to have the common denominator of 3816:

\[
\frac{45}{53} = \frac{45 \times 72}{53 \times 72} = \frac{3240}{3816}
\]

\[
\frac{71}{72} = \frac{71 \times 53}{72 \times 53} = \frac{3763}{3816}
\]

Now, we add the two fractions:

\[
\frac{3240}{3816} + \frac{3763}{3816} = \frac{3240 + 3763}{3816} = \frac{7003}{3816}
\]

Thus, the sum of \(\frac{45}{53}\) and \(\frac{71}{72}\) is \(\frac{7003}{3816}\).

### Answer: \(\frac{7003}{3816}\)

Expected answer: 7003/3816

Correct? Yes

Problem: After calculating the sum of 36/72 and 49/1, subtract 48/10 from the result. What is the final answer?

GPT-4o:

First, we have already calculated the sum of \( \frac{36}{72} \) and \( \frac{49}{1} \) as \( \frac{99}{2} \).

Next, we need to subtract \( \frac{48}{10} \) from \( \frac{99}{2} \). To do this, we need a common denominator. The least common multiple (LCM) of 2 and 10 is 10. We convert both fractions to have a denominator of 10:

\[
\frac{99}{2} = \frac{99 \times 5}{2 \times 5} = \frac{495}{10}
\]

Now, we subtract \( \frac{48}{10} \) from \( \frac{495}{10} \):

\[
\frac{495}{10} - \frac{48}{10} = \frac{495 - 48}{10} = \frac{447}{10}
\]

Thus, the final answer is \( \frac{447}{10} \).

### Answer: \(\frac{447}{10}\)

Expected answer: 447/10

Correct? Yes

Problem: Earlier, you calculated the sum of 36/72 and 49/1. But what if instead of adding 49/1, you only added 14/1? What would be the new sum?

GPT-4o:

To calculate the sum of \( \frac{36}{72} \) and \( \frac{14}{1} \), we first simplify the fraction \( \frac{36}{72} \).

The greatest common divisor (GCD) of 36 and 72 is 36. Therefore, we can simplify \( \frac{36}{72} \) as follows:
\[
\frac{36}{72} = \frac{36 \div 36}{72 \div 36} = \frac{1}{2}
\]

Next, we add \( \frac{1}{2} \) and \( \frac{14}{1} \). To do this, we need a common denominator. The common denominator for 2 and 1 is 2. We convert \( \frac{14}{1} \) to a fraction with a denominator of 2:
\[
\frac{14}{1} = \frac{14 \times 2}{1 \times 2} = \frac{28}{2}
\]

Now, we add the two fractions:
\[
\frac{1}{2} + \frac{28}{2} = \frac{1 + 28}{2} = \frac{29}{2}
\]

Thus, the sum of \( \frac{36}{72} \) and \( \frac{14}{1} \) is \( \frac{29}{2} \).

### Answer: \(\frac{29}{2}\)

Expected answer: 29/2

Correct? Yes

Problem: After finding the sum of 61/20 and 29/15, add that result to 58/31. What is the final answer?

GPT-4o:

We already found that the sum of \( \frac{61}{20} \) and \( \frac{29}{15} \) is \( \frac{299}{60} \).

Now, we need to add \( \frac{299}{60} \) to \( \frac{58}{31} \).

To do this, we need a common denominator. The least common multiple (LCM) of 60 and 31 is 1860.

Next, we convert each fraction to have the common denominator of 1860:

1. For \( \frac{299}{60} \):
   \[
   \frac{299}{60} = \frac{299 \times 31}{60 \times 31} = \frac{9269}{1860}
   \]

2. For \( \frac{58}{31} \):
   \[
   \frac{58}{31} = \frac{58 \times 60}{31 \times 60} = \frac{3480}{1860}
   \]

Now, we can add the two fractions:
\[
\frac{9269}{1860} + \frac{3480}{1860} = \frac{9269 + 3480}{1860} = \frac{12749}{1860}
\]

The fraction \( \frac{12749}{1860} \) can be simplified if there are common factors. However, 12749 and 1860 do not have common factors other than 1, so the fraction is already in its simplest form.

### Answer: \(\frac{12749}{1860}\)

Expected answer: 12749/1860

Correct? Yes

Problem: Previously you found the sum of 61/20 and 29/15. Now let's change the 15 in the denominator of the second fraction to 67, so that we are adding 61/20 and 29/67. What will the new sum be?

GPT-4o:

To find the sum of the fractions \( \frac{61}{20} \) and \( \frac{29}{67} \), we need to find a common denominator. The least common multiple (LCM) of 20 and 67 is 1340, since 67 is a prime number and does not share any factors with 20.

Next, we convert each fraction to have the common denominator of 1340:

1. For \( \frac{61}{20} \):
   \[
   \frac{61}{20} = \frac{61 \times 67}{20 \times 67} = \frac{61 \times 67}{1340} = \frac{4087}{1340}
   \]

2. For \( \frac{29}{67} \):
   \[
   \frac{29}{67} = \frac{29 \times 20}{67 \times 20} = \frac{29 \times 20}{1340} = \frac{580}{1340}
   \]

Now, we can add the two fractions:
\[
\frac{4087}{1340} + \frac{580}{1340} = \frac{4087 + 580}{1340} = \frac{4667}{1340}
\]

The fraction \( \frac{4667}{1340} \) is already in its simplest form because 4667 and 1340 have no common factors other than 1.

### Answer: \(\frac{4667}{1340}\)

Expected answer: 4667/1340

Correct? Yes

Problem: In continuation of the previous question, what would you get if you subtract 7/89 from the result of ((33 / 90) + (77 / 79))? Please give your answer as an unsimplified fraction.

GPT-4o:

We already have the result of \(\left(\frac{33}{90} + \frac{77}{79}\right)\) as \(\frac{9537}{7110}\).

Now, we need to subtract \(\frac{7}{89}\) from this result. To do this, we need a common denominator. The common denominator will be the product of the two denominators, \(7110\) and \(89\).

First, we rewrite the fractions with the common denominator:
\[
\frac{9537}{7110} - \frac{7}{89} = \frac{9537 \cdot 89}{7110 \cdot 89} - \frac{7 \cdot 7110}{89 \cdot 7110}
\]

This gives us:
\[
\frac{9537 \cdot 89 - 7 \cdot 7110}{7110 \cdot 89}
\]

Now, we calculate the numerators:
\[
9537 \cdot 89 = 849393
\]
\[
7 \cdot 7110 = 49770
\]

Subtracting these results:
\[
849393 - 49770 = 799623
\]

Thus, the expression becomes:
\[
\frac{799623}{7110 \cdot 89}
\]

Calculating the denominator:
\[
7110 \cdot 89 = 632790
\]

So, the final unsimplified fraction is:
\[
\frac{799623}{632790}
\]

### Answer: \(\frac{799623}{632790}\)

Expected answer: 266341/210930

Correct? No

Problem: Taking the previous problem into consideration, what if the fraction you were initially adding to 33/90 was 98/79 instead of 77/79? Solve the expression ((33 / 90) + (98 / 79)) and give your answer as an unsimplified fraction.

GPT-4o:

To solve the expression \(\left(\frac{33}{90} + \frac{98}{79}\right)\), we will again use the product of the denominators as the common denominator.

First, we rewrite the fractions with a common denominator:
\[
\frac{33}{90} + \frac{98}{79} = \frac{33 \cdot 79}{90 \cdot 79} + \frac{98 \cdot 90}{79 \cdot 90}
\]

This gives us:
\[
\frac{33 \cdot 79 + 98 \cdot 90}{90 \cdot 79}
\]

Now, we calculate the numerators:
\[
33 \cdot 79 = 2607
\]
\[
98 \cdot 90 = 8820
\]

Adding these results:
\[
2607 + 8820 = 11427
\]

Thus, the expression becomes:
\[
\frac{11427}{90 \cdot 79}
\]

Calculating the denominator:
\[
90 \cdot 79 = 7110
\]

So, the final unsimplified fraction is:
\[
\frac{11427}{7110}
\]

### Answer: \(\frac{11427}{7110}\)

Expected answer: 3809/2370

Correct? Yes

Problem: After finding the sum of 30/79 and 29/6, subtract 44/75 from it. What is the result?

GPT-4o:

First, we have the sum of \(\frac{30}{79}\) and \(\frac{29}{6}\) which is \(\frac{2471}{474}\).

Next, we need to subtract \(\frac{44}{75}\) from \(\frac{2471}{474}\). To do this, we need a common denominator. The least common multiple (LCM) of 474 and 75 is 11850.

Convert each fraction to have the common denominator of 11850:

\[
\frac{2471}{474} = \frac{2471 \times 25}{474 \times 25} = \frac{61775}{11850}
\]

\[
\frac{44}{75} = \frac{44 \times 158}{75 \times 158} = \frac{6952}{11850}
\]

Now, we subtract the two fractions:

\[
\frac{61775}{11850} - \frac{6952}{11850} = \frac{61775 - 6952}{11850} = \frac{54823}{11850}
\]

Thus, the result of subtracting \(\frac{44}{75}\) from the sum of \(\frac{30}{79}\) and \(\frac{29}{6}\) is \(\frac{54823}{11850}\).

### Answer: \(\frac{54823}{11850}\)

Expected answer: 54823/11850

Correct? Yes

Problem: After recalculating, you realized you had 30/97 instead of 30/79. Now, add that to 29/6. What is the new result?

GPT-4o:

To find the sum of the fractions \(\frac{30}{97}\) and \(\frac{29}{6}\), we need to find a common denominator. The least common multiple (LCM) of 97 and 6 is 582.

First, we convert each fraction to have the common denominator of 582:

\[
\frac{30}{97} = \frac{30 \times 6}{97 \times 6} = \frac{180}{582}
\]

\[
\frac{29}{6} = \frac{29 \times 97}{6 \times 97} = \frac{2813}{582}
\]

Now, we add the two fractions:

\[
\frac{180}{582} + \frac{2813}{582} = \frac{180 + 2813}{582} = \frac{2993}{582}
\]

Thus, the sum of \(\frac{30}{97}\) and \(\frac{29}{6}\) is \(\frac{2993}{582}\).

### Answer: \(\frac{2993}{582}\)

Expected answer: 2993/582

Correct? Yes

Problem: What is the sum of 63/51 and the result of subtracting 11/66 from 59/2?

GPT-4o:

To find the sum of \( \frac{63}{51} \) and the result of subtracting \( \frac{11}{66} \) from \( \frac{59}{2} \), we need to follow these steps:

1. First, simplify \( \frac{63}{51} \):
   \[
   \frac{63}{51} = \frac{63 \div 3}{51 \div 3} = \frac{21}{17}
   \]

2. Next, we already found the result of subtracting \( \frac{11}{66} \) from \( \frac{59}{2} \) in the previous problem, which is \( \frac{968}{33} \).

3. Now, we need to find the sum of \( \frac{21}{17} \) and \( \frac{968}{33} \). To do this, we need a common denominator. The least common multiple of 17 and 33 is 561.

4. Convert \( \frac{21}{17} \) to a fraction with a denominator of 561:
   \[
   \frac{21}{17} = \frac{21 \times 33}{17 \times 33} = \frac{693}{561}
   \]

5. Convert \( \frac{968}{33} \) to a fraction with a denominator of 561:
   \[
   \frac{968}{33} = \frac{968 \times 17}{33 \times 17} = \frac{16456}{561}
   \]

6. Add the two fractions:
   \[
   \frac{693}{561} + \frac{16456}{561} = \frac{693 + 16456}{561} = \frac{17149}{561}
   \]

Thus, the sum of \( \frac{63}{51} \) and the result of subtracting \( \frac{11}{66} \) from \( \frac{59}{2} \) is \( \frac{17149}{561} \).

### Answer: \( \frac{17149}{561} \)

Expected answer: 1559/51

Correct? Yes

Problem: What would the result be if we subtracted 11/31 instead of 11/66 from 59/2 in the previous problem?

GPT-4o:

To solve the problem of subtracting \( \frac{11}{31} \) from \( \frac{59}{2} \), we first need to find a common denominator. The least common multiple of 31 and 2 is 62.

1. Convert \( \frac{59}{2} \) to a fraction with a denominator of 62:
   \[
   \frac{59}{2} = \frac{59 \times 31}{2 \times 31} = \frac{1829}{62}
   \]

2. Convert \( \frac{11}{31} \) to a fraction with a denominator of 62:
   \[
   \frac{11}{31} = \frac{11 \times 2}{31 \times 2} = \frac{22}{62}
   \]

3. Now, we have the fractions \( \frac{1829}{62} \) and \( \frac{22}{62} \).

4. Subtract \( \frac{22}{62} \) from \( \frac{1829}{62} \):
   \[
   \frac{1829}{62} - \frac{22}{62} = \frac{1829 - 22}{62} = \frac{1807}{62}
   \]

5. Simplify \( \frac{1807}{62} \) by finding the greatest common divisor (GCD) of 1807 and 62. The GCD is 1, so the fraction is already in its simplest form:
   \[
   \frac{1807}{62}
   \]

Thus, the simplified form of the fraction is \( \frac{1807}{62} \).

### Answer: \( \frac{1807}{62} \)

Expected answer: 1807/62

Correct? Yes

Problem: We previously calculated the sum of the fractions 58/23 and 25/26. Now, suppose we had started with 45/23 instead of 58/23. How does the sum of 45/23 and 25/26 compare to the original calculation?

GPT-4o:

To compare the sum of \(\frac{45}{23}\) and \(\frac{25}{26}\) with the original sum of \(\frac{58}{23}\) and \(\frac{25}{26}\), we need to calculate the new sum and then compare it to the original sum.

1. Find the least common multiple (LCM) of the denominators 23 and 26:
   \[
   \text{LCM}(23, 26) = 23 \times 26 = 598
   \]

2. Convert each fraction to have the common denominator 598:
   \[
   \frac{45}{23} = \frac{45 \times 26}{23 \times 26} = \frac{1170}{598}
   \]
   \[
   \frac{25}{26} = \frac{25 \times 23}{26 \times 23} = \frac{575}{598}
   \]

3. Add the fractions:
   \[
   \frac{1170}{598} + \frac{575}{598} = \frac{1170 + 575}{598} = \frac{1745}{598}
   \]

Now, let's compare this to the original sum:
   \[
   \frac{58}{23} + \frac{25}{26} = \frac{1508}{598} + \frac{575}{598} = \frac{2083}{598}
   \]

We can see that:
   \[
   \frac{1745}{598} < \frac{2083}{598}
   \]

Thus, the sum of \(\frac{45}{23}\) and \(\frac{25}{26}\) is less than the sum of \(\frac{58}{23}\) and \(\frac{25}{26}\).

### Answer: The sum of \(\frac{45}{23}\) and \(\frac{25}{26}\) is less than the sum of \(\frac{58}{23}\) and \(\frac{25}{26}\).

Expected answer: 1745/598

Correct? No

Problem: After calculating the sum of the fractions 70/13 and 17/52, subtract 35/87 from the result. What is your new answer?

GPT-4o:

First, we need to find the sum of the fractions \( \frac{70}{13} \) and \( \frac{17}{52} \), which we previously calculated as \( \frac{297}{52} \).

Next, we need to subtract \( \frac{35}{87} \) from \( \frac{297}{52} \). To do this, we need a common denominator for the fractions \( \frac{297}{52} \) and \( \frac{35}{87} \).

The least common multiple (LCM) of 52 and 87 is 4524.

Convert \( \frac{297}{52} \) to a fraction with a denominator of 4524:
\[ \frac{297}{52} = \frac{297 \times 87}{52 \times 87} = \frac{25839}{4524} \]

Convert \( \frac{35}{87} \) to a fraction with a denominator of 4524:
\[ \frac{35}{87} = \frac{35 \times 52}{87 \times 52} = \frac{1820}{4524} \]

Now, subtract the two fractions:
\[ \frac{25839}{4524} - \frac{1820}{4524} = \frac{25839 - 1820}{4524} = \frac{24019}{4524} \]

The fraction \( \frac{24019}{4524} \) is already in its simplest form because 24019 and 4524 have no common factors other than 1.

### Answer: \(\frac{24019}{4524}\)

Expected answer: 24019/4524

Correct? Yes

Problem: After reconsidering the quantities, suppose that originally there were 70/93 oranges instead of 70/13. Recalculate the sum of the fractions with this quantity of oranges and 17/52 apples. What is your new answer?

GPT-4o:

To calculate the sum of the fractions \( \frac{70}{93} \) and \( \frac{17}{52} \), we first need to find a common denominator. The least common multiple (LCM) of 93 and 52 is 4824.

Next, we convert each fraction to have a denominator of 4824.

For \( \frac{70}{93} \):
\[ \frac{70}{93} = \frac{70 \times 52}{93 \times 52} = \frac{3640}{4824} \]

For \( \frac{17}{52} \):
\[ \frac{17}{52} = \frac{17 \times 93}{52 \times 93} = \frac{1581}{4824} \]

Now, we can add the two fractions:
\[ \frac{3640}{4824} + \frac{1581}{4824} = \frac{3640 + 1581}{4824} = \frac{5221}{4824} \]

The fraction \( \frac{5221}{4824} \) is already in its simplest form because 5221 and 4824 have no common factors other than 1.

### Answer: \(\frac{5221}{4824}\)

Expected answer: 5221/4836

Correct? No

Problem: Next, what is the result if you add the sum of 84/34 to the result of the previous problem (the sum of 78/44 and 17/91)?

GPT-4o:

To add \(\frac{84}{34}\) to the previous result \(\frac{3923}{2002}\), we first need to simplify \(\frac{84}{34}\) and then find a common denominator.

1. Simplify \(\frac{84}{34}\):
   - The GCD of 84 and 34 is 2.
   - Simplify by dividing both the numerator and the denominator by 2:
     \[
     \frac{84 \div 2}{34 \div 2} = \frac{42}{17}
     \]

2. Find the least common multiple (LCM) of the denominators 17 and 2002:
   - The prime factorization of 17 is \(17\).
   - The prime factorization of 2002 is \(2 \times 7 \times 11 \times 13\).
   - Since 17 is a prime number and does not share any factors with 2002, the LCM is:
     \[
     LCM = 17 \times 2002 = 34034
     \]

3. Convert each fraction to have the common denominator 34034:
   - For \(\frac{42}{17}\):
     \[
     \frac{42}{17} = \frac{42 \times 2002}{17 \times 2002} = \frac{84084}{34034}
     \]
   - For \(\frac{3923}{2002}\):
     \[
     \frac{3923}{2002} = \frac{3923 \times 17}{2002 \times 17} = \frac{66791}{34034}
     \]

4. Add the fractions:
   \[
   \frac{84084}{34034} + \frac{66791}{34034} = \frac{84084 + 66791}{34034} = \frac{150875}{34034}
   \]

5. Simplify the resulting fraction if possible:
   - The GCD of 150875 and 34034 is 1, so the fraction is already in its simplest form.

### Answer: \(\frac{150875}{34034}\)

Expected answer: 150775/34034

Correct? No

Problem: Suppose that instead of the previous problem where we added 78/44 and 17/91, we actually had to add 78/44 and 17/67 instead. What would be the new sum?

GPT-4o:

To find the sum of the fractions \(\frac{78}{44}\) and \(\frac{17}{67}\), we first need to find a common denominator.

1. Simplify the fractions if possible:
   - \(\frac{78}{44}\) can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 2:
     \[
     \frac{78 \div 2}{44 \div 2} = \frac{39}{22}
     \]
   - \(\frac{17}{67}\) is already in its simplest form since 17 and 67 have no common factors other than 1.

2. Find the least common multiple (LCM) of the denominators 22 and 67:
   - The prime factorization of 22 is \(2 \times 11\).
   - The prime factorization of 67 is \(67\) (since 67 is a prime number).
   - The LCM is the product of the highest powers of all prime factors:
     \[
     LCM = 2 \times 11 \times 67 = 1474
     \]

3. Convert each fraction to have the common denominator 1474:
   - For \(\frac{39}{22}\):
     \[
     \frac{39}{22} = \frac{39 \times 67}{22 \times 67} = \frac{2613}{1474}
     \]
   - For \(\frac{17}{67}\):
     \[
     \frac{17}{67} = \frac{17 \times 22}{67 \times 22} = \frac{374}{1474}
     \]

4. Add the fractions:
   \[
   \frac{2613}{1474} + \frac{374}{1474} = \frac{2613 + 374}{1474} = \frac{2987}{1474}
   \]

5. Simplify the resulting fraction if possible:
   - The GCD of 2987 and 1474 is 1, so the fraction is already in its simplest form.

### Answer: \(\frac{2987}{1474}\)

Expected answer: 2987/1474

Correct? Yes

Problem: What is the sum of 43/4 and 71/4 plus 92/79?

GPT-4o:

To find the sum of \(\frac{43}{4}\), \(\frac{71}{4}\), and \(\frac{92}{79}\), we first add the fractions with the same denominator:

\[
\frac{43}{4} + \frac{71}{4} = \frac{43 + 71}{4} = \frac{114}{4}
\]

Next, we simplify \(\frac{114}{4}\):

\[
\frac{114}{4} = \frac{57}{2}
\]

Now, we need to add \(\frac{57}{2}\) and \(\frac{92}{79}\). To do this, we need a common denominator. The least common multiple (LCM) of 2 and 79 is 158.

Convert \(\frac{57}{2}\) to a fraction with a denominator of 158:

\[
\frac{57}{2} = \frac{57 \times 79}{2 \times 79} = \frac{4503}{158}
\]

Convert \(\frac{92}{79}\) to a fraction with a denominator of 158:

\[
\frac{92}{79} = \frac{92 \times 2}{79 \times 2} = \frac{184}{158}
\]

Now, add the two fractions:

\[
\frac{4503}{158} + \frac{184}{158} = \frac{4503 + 184}{158} = \frac{4687}{158}
\]

Thus, the sum of \(\frac{43}{4}\), \(\frac{71}{4}\), and \(\frac{92}{79}\) is \(\frac{4687}{158}\).

### Answer: \(\frac{4687}{158}\)

Expected answer: 4687/158

Correct? Yes

Problem: What if instead of adding 71/4, we added 71/89 to 43/4? What would be the new total?

GPT-4o:

To add the fractions \(\frac{43}{4}\) and \(\frac{71}{89}\), we need to find a common denominator. The least common multiple (LCM) of 4 and 89 is 356, since 89 is a prime number and does not share any factors with 4.

First, we convert each fraction to have the common denominator of 356:

\[
\frac{43}{4} = \frac{43 \times 89}{4 \times 89} = \frac{3827}{356}
\]

\[
\frac{71}{89} = \frac{71 \times 4}{89 \times 4} = \frac{284}{356}
\]

Now, we can add the two fractions:

\[
\frac{3827}{356} + \frac{284}{356} = \frac{3827 + 284}{356} = \frac{4111}{356}
\]

Thus, the sum of \(\frac{43}{4}\) and \(\frac{71}{89}\) is \(\frac{4111}{356}\).

### Answer: \(\frac{4111}{356}\)

Expected answer: 4111/356

Correct? Yes

Problem: If the result from the previous calculation is subtracted by 23/29, what is the final result?

GPT-4o:

To subtract \( \frac{23}{29} \) from \( \frac{4381}{1998} \), we need to find a common denominator. The least common multiple (LCM) of 1998 and 29 will be our common denominator.

First, let's find the prime factorizations of 1998 and 29:
- 1998 = 2 × 3^3 × 37
- 29 is a prime number

The LCM of 1998 and 29 is the product of the highest powers of all prime factors involved:
- LCM = 2^1 × 3^3 × 37^1 × 29^1 = 2 × 27 × 37 × 29

Calculating the LCM:
\[ 2 \times 27 = 54 \]
\[ 54 \times 37 = 1998 \]
\[ 1998 \times 29 = 57942 \]

So, the common denominator is 57942.

Next, we convert each fraction to have this common denominator:
\[ \frac{4381}{1998} = \frac{4381 \times 29}{1998 \times 29} = \frac{127049}{57942} \]
\[ \frac{23}{29} = \frac{23 \times 1998}{29 \times 1998} = \frac{45954}{57942} \]

Now, we subtract the two fractions:
\[ \frac{127049}{57942} - \frac{45954}{57942} = \frac{127049 - 45954}{57942} = \frac{81095}{57942} \]

The fraction \( \frac{81095}{57942} \) can be simplified by finding the greatest common divisor (GCD) of 81095 and 57942. Using the Euclidean algorithm:

1. \( 81095 \div 57942 \approx 1 \) remainder \( 23153 \)
2. \( 57942 \div 23153 \approx 2 \) remainder \( 11636 \)
3. \( 23153 \div 11636 \approx 1 \) remainder \( 11017 \)
4. \( 11636 \div 11017 \approx 1 \) remainder \( 619 \)
5. \( 11017 \div 619 \approx 17 \) remainder \( 508 \)
6. \( 619 \div 508 \approx 1 \) remainder \( 111 \)
7. \( 508 \div 111 \approx 4 \) remainder \( 64 \)
8. \( 111 \div 64 \approx 1 \) remainder \( 47 \)
9. \( 64 \div 47 \approx 1 \) remainder \( 17 \)
10. \( 47 \div 17 \approx 2 \) remainder \( 13 \)
11. \( 17 \div 13 \approx 1 \) remainder \( 4 \)
12. \( 13 \div 4 \approx 3 \) remainder \( 1 \)
13. \( 4 \div 1 = 4 \) remainder \( 0 \)

The GCD is 1, so the fraction \( \frac{81095}{57942} \) is already in its simplest form.

### Answer: \(\frac{81095}{57942}\)

Expected answer: 81095/57942

Correct? Yes

Problem: If initially, instead of having 17/74 of the fruit basket filled with oranges, we had 28/74 of it filled with oranges, and we still added the same 53/27 fraction of the basket filled with apples, then what would the total fraction of the fruit basket be when filled with these fruits?

GPT-4o:

To find the total fraction of the fruit basket filled with oranges and apples, we need to add the fractions \( \frac{28}{74} \) and \( \frac{53}{27} \).

First, we need to find a common denominator. As before, the least common multiple (LCM) of 74 and 27 is 1998.

Next, we convert each fraction to have this common denominator:
\[ \frac{28}{74} = \frac{28 \times 27}{74 \times 27} = \frac{756}{1998} \]
\[ \frac{53}{27} = \frac{53 \times 74}{27 \times 74} = \frac{3922}{1998} \]

Now, we add the two fractions:
\[ \frac{756}{1998} + \frac{3922}{1998} = \frac{756 + 3922}{1998} = \frac{4678}{1998} \]

To simplify \( \frac{4678}{1998} \), we need to find the greatest common divisor (GCD) of 4678 and 1998. Using the Euclidean algorithm:

1. \( 4678 \div 1998 \) gives a quotient of 2 and a remainder of 682.
2. \( 1998 \div 682 \) gives a quotient of 2 and a remainder of 634.
3. \( 682 \div 634 \) gives a quotient of 1 and a remainder of 48.
4. \( 634 \div 48 \) gives a quotient of 13 and a remainder of 10.
5. \( 48 \div 10 \) gives a quotient of 4 and a remainder of 8.
6. \( 10 \div 8 \) gives a quotient of 1 and a remainder of 2.
7. \( 8 \div 2 \) gives a quotient of 4 and a remainder of 0.

The GCD is 2.

Now, we simplify \( \frac{4678}{1998} \) by dividing both the numerator and the denominator by their GCD:
\[ \frac{4678 \div 2}{1998 \div 2} = \frac{2339}{999} \]

### Answer: \(\frac{2339}{999}\)

Expected answer: 2339/999

Correct? Yes

Problem: After finding the value of (76/76) + (96/16), what is the result when you add the solution to itself?

GPT-4o:

We have already determined that the value of \((76/76) + (96/16)\) is 7.

Now, we need to add this solution to itself:
\[
7 + 7 = 14
\]

### Answer: 14

Expected answer: 14

Correct? Yes

Problem: You originally calculated (76/76) + (96/16). Now reconsider your thought, what would be the new total if the second fraction was 96/33 instead of 96/16?

GPT-4o:

First, let's simplify each term in the new expression \((76/76) + (96/33)\).

1. Simplify \(76/76\):
\[
76/76 = 1
\]

2. Simplify \(96/33\):
\[
96/33 = \frac{96}{33}
\]

Now, add the simplified values together:
\[
1 + \frac{96}{33}
\]

To add these, we need a common denominator. The common denominator is 33. So, we convert 1 to a fraction with a denominator of 33:
\[
1 = \frac{33}{33}
\]

Now, add the fractions:
\[
\frac{33}{33} + \frac{96}{33} = \frac{33 + 96}{33} = \frac{129}{33}
\]

Simplify \(\frac{129}{33}\) if possible. Both 129 and 33 are divisible by 3:
\[
\frac{129 \div 3}{33 \div 3} = \frac{43}{11}
\]

### Answer: \(\frac{43}{11}\)

Expected answer: 43/11

Correct? Yes

Problem: After calculating the result of ((94 / 14) - (50 / 90)), reconsider the calculation by replacing (50 / 90) with (50 / 19) instead. What is the result of ((94 / 14) - (50 / 19))? What is the final result as a simplified fraction?

GPT-4o:

First, we need to simplify each fraction separately.

1. Simplify \( \frac{94}{14} \):
   \[
   \frac{94}{14} = \frac{47}{7} \quad \text{(dividing both the numerator and the denominator by 2)}
   \]

2. Simplify \( \frac{50}{19} \):
   \[
   \frac{50}{19} \quad \text{(this fraction is already in its simplest form)}
   \]

Next, we need to subtract these two fractions:
\[
\frac{47}{7} - \frac{50}{19}
\]

To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 7 and 19 is 133.

Convert each fraction to have the common denominator of 133:
\[
\frac{47}{7} = \frac{47 \times 19}{7 \times 19} = \frac{893}{133}
\]
\[
\frac{50}{19} = \frac{50 \times 7}{19 \times 7} = \frac{350}{133}
\]

Now, subtract the fractions:
\[
\frac{893}{133} - \frac{350}{133} = \frac{893 - 350}{133} = \frac{543}{133}
\]

The fraction \(\frac{543}{133}\) is already in its simplest form because 543 and 133 have no common factors other than 1.

### Answer: \(\frac{543}{133}\)

Expected answer: 543/133

Correct? Yes

Problem: Now, after you found the result of subtracting 29/79 from 33/7, what would you get if you subtract 61/76 from that earlier result?

GPT-4o:

To solve the problem of subtracting \(\frac{61}{76}\) from \(\frac{2404}{553}\), we need to follow these steps:

1. **Find a common denominator**: The denominators are 553 and 76. The least common multiple (LCM) of 553 and 76 can be found by their prime factorizations:
   - 553 = 7 × 79
   - 76 = 2² × 19

   The LCM is \(7 \times 79 \times 2² \times 19 = 105064\).

2. **Convert each fraction to have the common denominator**:
   - For \(\frac{2404}{553}\):
     \[
     \frac{2404}{553} = \frac{2404 \times 76}{553 \times 76} = \frac{182704}{105064}
     \]
   - For \(\frac{61}{76}\):
     \[
     \frac{61}{76} = \frac{61 \times 553}{76 \times 553} = \frac{33733}{105064}
     \]

3. **Subtract the fractions**:
   \[
   \frac{182704}{105064} - \frac{33733}{105064} = \frac{182704 - 33733}{105064} = \frac{148971}{105064}
   \]

4. **Simplify the fraction if possible**: In this case, \(\frac{148971}{105064}\) is already in its simplest form because 148971 and 105064 have no common factors other than 1.

### Answer: \(\frac{148971}{105064}\)

Expected answer: 148971/42028

Correct? No

Problem: Suppose in the previous problem, instead of subtracting 29/79 from 33/7, we subtract 11/79 from 33/7. What would be the new result?

GPT-4o:

To solve the problem of subtracting \(\frac{11}{79}\) from \(\frac{33}{7}\), we follow similar steps as before:

1. **Find a common denominator**: The denominators are 79 and 7. The least common multiple (LCM) of 79 and 7 is \(79 \times 7 = 553\).

2. **Convert each fraction to have the common denominator**:
   - For \(\frac{33}{7}\):
     \[
     \frac{33}{7} = \frac{33 \times 79}{7 \times 79} = \frac{2607}{553}
     \]
   - For \(\frac{11}{79}\):
     \[
     \frac{11}{79} = \frac{11 \times 7}{79 \times 7} = \frac{77}{553}
     \]

3. **Subtract the fractions**:
   \[
   \frac{2607}{553} - \frac{77}{553} = \frac{2607 - 77}{553} = \frac{2530}{553}
   \]

4. **Simplify the fraction if possible**: In this case, \(\frac{2530}{553}\) is already in its simplest form because 2530 and 553 have no common factors other than 1.

### Answer: \(\frac{2530}{553}\)

Expected answer: 2530/553

Correct? Yes

Problem: What is the new total if you add 5/11 to the previous result of 60/79 and 53/38?

GPT-4o:

To find the new total, we need to add \( \frac{5}{11} \) to the previous result of \( \frac{6467}{3002} \).

1. **Find a common denominator for \( \frac{6467}{3002} \) and \( \frac{5}{11} \):**
   - The denominators are 3002 and 11.
   - The least common multiple (LCM) of 3002 and 11 is \( 3002 \times 11 \) since 11 is a prime number and does not divide 3002.

2. **Calculate the LCM:**
   \[
   3002 \times 11 = 33022
   \]

3. **Convert each fraction to have the common denominator 33022:**
   - For \( \frac{6467}{3002} \):
     \[
     \frac{6467}{3002} = \frac{6467 \times 11}{3002 \times 11} = \frac{71137}{33022}
     \]
   - For \( \frac{5}{11} \):
     \[
     \frac{5}{11} = \frac{5 \times 3002}{11 \times 3002} = \frac{15010}{33022}
     \]

4. **Add the fractions:**
   \[
   \frac{71137}{33022} + \frac{15010}{33022} = \frac{71137 + 15010}{33022} = \frac{86147}{33022}
   \]

Thus, the new total when you add \( \frac{5}{11} \) to the previous result of \( \frac{6467}{3002} \) is \( \frac{86147}{33022} \).

### Answer: \(\frac{86147}{33022}\)

Expected answer: 86147/33022

Correct? Yes

Problem: What would be the new sum if instead of 60/79 in the previous problem, we had 79/79? So, what is 79/79 + 53/38?

GPT-4o:

To find the sum of \( \frac{79}{79} \) and \( \frac{53}{38} \), we first simplify \( \frac{79}{79} \):

\[
\frac{79}{79} = 1
\]

Now, we need to add 1 to \( \frac{53}{38} \). To do this, we convert 1 to a fraction with the same denominator as \( \frac{53}{38} \):

\[
1 = \frac{38}{38}
\]

Now, we can add the two fractions:

\[
\frac{38}{38} + \frac{53}{38} = \frac{38 + 53}{38} = \frac{91}{38}
\]

Thus, the sum of \( \frac{79}{79} \) and \( \frac{53}{38} \) is \( \frac{91}{38} \).

### Answer: \(\frac{91}{38}\)

Expected answer: 91/38

Correct? Yes

Problem: Given the sum you found in the last problem, add that number to 86/82. What is the result?

GPT-4o:

First, let's simplify \( \frac{86}{82} \):

1. Simplify \( \frac{86}{82} \):
   - The greatest common divisor (GCD) of 86 and 82 is 2.
   - Divide both the numerator and the denominator by 2:
     \[
     \frac{86 \div 2}{82 \div 2} = \frac{43}{41}
     \]

Now, we need to add \( 20.186046511627907 \) to \( \frac{43}{41} \).

First, convert \( \frac{43}{41} \) to a decimal:
\[
\frac{43}{41} \approx 1.048780487804878
\]

Now, add this decimal to \( 20.186046511627907 \):
\[
20.186046511627907 + 1.048780487804878 \approx 21.234827
\]

### Answer: 21.234827

Expected answer: 37437/1763

Correct? No

Problem: What if we change the previous problem so that instead of sharing 75/86 of a pie, we share 75/86 of it, and instead of adding 60/3 slices, we still add 60/3 slices to it. What is the total number of slices now?

GPT-4o:

To find the total number of slices when we share \( \frac{75}{86} \) of a pie and add \( \frac{60}{3} \) slices to it, we first simplify each fraction.

1. Simplify \( \frac{60}{3} \):
   - Divide the numerator by the denominator:
     \[
     \frac{60}{3} = 20
     \]

Now, we need to add \( \frac{75}{86} \) and 20. To do this, we write 20 as a fraction with a denominator of 1:
\[
\frac{75}{86} + 20 = \frac{75}{86} + \frac{20 \cdot 86}{1 \cdot 86} = \frac{75}{86} + \frac{1720}{86}
\]

Now, add the numerators:
\[
\frac{75 + 1720}{86} = \frac{1795}{86}
\]

Finally, simplify \( \frac{1795}{86} \):
\[
1795 \div 86 \approx 20.872093023255813
\]

### Answer: 20.872093023255813

Expected answer: 1795/86

Correct? No

Problem: After finding the result of 95 / 16 - 43 / 60, add the result to 80 / 79. What is the final answer?

GPT-4o:

We previously found that \( \frac{95}{16} - \frac{43}{60} = \frac{1253}{240} \).

Now, we need to add \( \frac{1253}{240} \) to \( \frac{80}{79} \).

1. **Find the least common multiple (LCM) of 240 and 79:**
   - Since 79 is a prime number, the LCM of 240 and 79 is \( 240 \times 79 = 18960 \).

2. **Convert each fraction to have the common denominator of 18960:**
   - For \( \frac{1253}{240} \):
     \[
     \frac{1253}{240} = \frac{1253 \times 79}{240 \times 79} = \frac{99087}{18960}
     \]
   - For \( \frac{80}{79} \):
     \[
     \frac{80}{79} = \frac{80 \times 240}{79 \times 240} = \frac{19200}{18960}
     \]

3. **Add the fractions:**
   \[
   \frac{99087}{18960} + \frac{19200}{18960} = \frac{99087 + 19200}{18960} = \frac{118287}{18960}
   \]

4. **Simplify the fraction if possible:**
   - The greatest common divisor (GCD) of 118287 and 18960 is 3.
   - Divide both the numerator and the denominator by 3:
     \[
     \frac{118287 \div 3}{18960 \div 3} = \frac{39429}{6320}
     \]

### Answer: \(\frac{39429}{6320}\)

Expected answer: 118187/18960

Correct? No

Problem: After finding the result of 95 / 16 - 43 / 60 in the previous problem, let's change it a bit. What if instead of 95/16, you had 91/16 and you subtracted 43/60 from it? What would the new result be?

GPT-4o:

To solve the expression \( \frac{91}{16} - \frac{43}{60} \), we need to find a common denominator for the fractions.

1. **Find the least common multiple (LCM) of 16 and 60:**
   - The prime factorization of 16 is \( 2^4 \).
   - The prime factorization of 60 is \( 2^2 \times 3 \times 5 \).
   - The LCM is found by taking the highest power of each prime factor:
     \[
     \text{LCM} = 2^4 \times 3 \times 5 = 16 \times 3 \times 5 = 240
     \]

2. **Convert each fraction to have the common denominator of 240:**
   - For \( \frac{91}{16} \):
     \[
     \frac{91}{16} = \frac{91 \times 15}{16 \times 15} = \frac{1365}{240}
     \]
   - For \( \frac{43}{60} \):
     \[
     \frac{43}{60} = \frac{43 \times 4}{60 \times 4} = \frac{172}{240}
     \]

3. **Subtract the fractions:**
   \[
   \frac{1365}{240} - \frac{172}{240} = \frac{1365 - 172}{240} = \frac{1193}{240}
   \]

4. **Simplify the fraction if possible:**
   - The greatest common divisor (GCD) of 1193 and 240 is 1, so the fraction is already in its simplest form.

### Answer: \(\frac{1193}{240}\)

Expected answer: 1193/240

Correct? Yes

Problem: Recalling the fraction calculation from the previous problem (38/8 and 4/16), what is the new total if you were to add 32/64 to this sum?

GPT-4o:

First, let's recall the sum of the fractions \( \frac{38}{8} \) and \( \frac{4}{16} \) from the previous problem, which we simplified to 5.

Now, we need to add \( \frac{32}{64} \) to this sum.

1. Simplify \( \frac{32}{64} \):
   - The greatest common divisor of 32 and 64 is 32.
   - Simplify by dividing both the numerator and the denominator by 32:
     \[
     \frac{32 \div 32}{64 \div 32} = \frac{1}{2}
     \]

2. Add \( \frac{1}{2} \) to the previous sum of 5:
   - Convert 5 to a fraction with a common denominator of 2:
     \[
     5 = \frac{10}{2}
     \]
   - Now add \( \frac{10}{2} \) and \( \frac{1}{2} \):
     \[
     \frac{10}{2} + \frac{1}{2} = \frac{10 + 1}{2} = \frac{11}{2}
     \]

### Answer: \(\frac{11}{2}\)

Expected answer: 11/2

Correct? Yes

Problem: In the previous problem, we considered the sum of the fractions 38/8 and 4/16 and then added 32/64. Now, instead of 38/8, let's consider 12/8. What would be the new sum if you added 12/8 and 4/16 together?

GPT-4o:

To find the sum of the fractions \( \frac{12}{8} \) and \( \frac{4}{16} \), we first need to simplify the fractions and then find a common denominator.

1. Simplify the fractions if possible:
   - \( \frac{12}{8} \) can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 4:
     \[
     \frac{12 \div 4}{8 \div 4} = \frac{3}{2}
     \]
   - \( \frac{4}{16} \) can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 4:
     \[
     \frac{4 \div 4}{16 \div 4} = \frac{1}{4}
     \]

2. Find a common denominator for the fractions \( \frac{3}{2} \) and \( \frac{1}{4} \):
   - The least common denominator (LCD) of 2 and 4 is 4.

3. Convert \( \frac{3}{2} \) to a fraction with a denominator of 4:
   - Multiply both the numerator and the denominator by 2:
     \[
     \frac{3}{2} = \frac{3 \times 2}{2 \times 2} = \frac{6}{4}
     \]

4. Now, add the fractions \( \frac{6}{4} \) and \( \frac{1}{4} \):
   - Since they have the same denominator, we can directly add the numerators:
     \[
     \frac{6}{4} + \frac{1}{4} = \frac{6 + 1}{4} = \frac{7}{4}
     \]

### Answer: \(\frac{7}{4}\)

Expected answer: 7/4

Correct? Yes

Problem: After calculating 67 / 15 - 27 / 12 from the previous problem, what would be the result if you subtract 7 / 35 from your answer?

GPT-4o:

First, we need to subtract \(\frac{7}{35}\) from the previous result \(\frac{133}{60}\).

1. Simplify \(\frac{7}{35}\):
   \[
   \frac{7}{35} = \frac{1}{5}
   \]

2. Convert \(\frac{1}{5}\) to have the same denominator as \(\frac{133}{60}\):
   \[
   \frac{1}{5} = \frac{1 \times 12}{5 \times 12} = \frac{12}{60}
   \]

3. Subtract the fractions:
   \[
   \frac{133}{60} - \frac{12}{60} = \frac{133 - 12}{60} = \frac{121}{60}
   \]

4. Simplify the fraction if possible. In this case, \(\frac{121}{60}\) is already in its simplest form.

### Answer: \(\frac{121}{60}\)

Expected answer: 121/60

Correct? Yes

Problem: In the previous question, we considered the difference between 67 / 15 and 27 / 12. Now, if there were only 27 / 50 instead of 27 / 12, what would the difference between the 67 / 15 and this new value be?

GPT-4o:

To solve the expression \( \frac{67}{15} - \frac{27}{50} \), we need to find a common denominator for the fractions.

1. The denominators are 15 and 50. The least common multiple (LCM) of 15 and 50 is 150.

2. Convert each fraction to have the common denominator of 150:
   \[
   \frac{67}{15} = \frac{67 \times 10}{15 \times 10} = \frac{670}{150}
   \]
   \[
   \frac{27}{50} = \frac{27 \times 3}{50 \times 3} = \frac{81}{150}
   \]

3. Subtract the fractions:
   \[
   \frac{670}{150} - \frac{81}{150} = \frac{670 - 81}{150} = \frac{589}{150}
   \]

4. Simplify the fraction if possible. In this case, \(\frac{589}{150}\) is already in its simplest form.

### Answer: \(\frac{589}{150}\)

Expected answer: 589/150

Correct? Yes

Problem: If you add 84/30 to the sum of 38/59 and 80/94, what is the resultant value?

GPT-4o:

To find the resultant value when adding \(\frac{84}{30}\) to the sum of \(\frac{38}{59}\) and \(\frac{80}{94}\), we first need to calculate the sum of \(\frac{38}{59}\) and \(\frac{80}{94}\) as we did previously.

From the previous calculation, we found:
\[
\frac{38}{59} + \frac{80}{94} = \frac{4146}{2773}
\]

Next, we need to add \(\frac{84}{30}\) to this result. First, let's simplify \(\frac{84}{30}\):
\[
\frac{84}{30} = \frac{84 \div 6}{30 \div 6} = \frac{14}{5}
\]

Now, we need a common denominator to add \(\frac{4146}{2773}\) and \(\frac{14}{5}\). The least common multiple (LCM) of 2773 and 5 is \(2773 \times 5 = 13865\).

Convert each fraction to have this common denominator:
\[
\frac{4146}{2773} = \frac{4146 \times 5}{2773 \times 5} = \frac{20730}{13865}
\]
\[
\frac{14}{5} = \frac{14 \times 2773}{5 \times 2773} = \frac{38822}{13865}
\]

Now, add the two fractions:
\[
\frac{20730}{13865} + \frac{38822}{13865} = \frac{20730 + 38822}{13865} = \frac{59552}{13865}
\]

Next, we simplify the fraction \(\frac{59552}{13865}\). We find the greatest common divisor (GCD) of 59552 and 13865. Using the Euclidean algorithm:
\[
59552 \div 13865 \approx 4.29 \quad \text{(integer part is 4)}
\]
\[
59552 - 4 \times 13865 = 4062
\]
\[
13865 \div 4062 \approx 3.41 \quad \text{(integer part is 3)}
\]
\[
13865 - 3 \times 4062 = 1679
\]
\[
4062 \div 1679 \approx 2.42 \quad \text{(integer part is 2)}
\]
\[
4062 - 2 \times 1679 = 704
\]
\[
1679 \div 704 \approx 2.38 \quad \text{(integer part is 2)}
\]
\[
1679 - 2 \times 704 = 271
\]
\[
704 \div 271 \approx 2.60 \quad \text{(integer part is 2)}
\]
\[
704 - 2 \times 271 = 162
\]
\[
271 \div 162 \approx 1.67 \quad \text{(integer part is 1)}
\]
\[
271 - 1 \times 162 = 109
\]
\[
162 \div 109 \approx 1.49 \quad \text{(integer part is 1)}
\]
\[
162 - 1 \times 109 = 53
\]
\[
109 \div 53 \approx 2.06 \quad \text{(integer part is 2)}
\]
\[
109 - 2 \times 53 = 3
\]
\[
53 \div 3 \approx 17.67 \quad \text{(integer part is 17)}
\]
\[
53 - 17 \times 3 = 2
\]
\[
3 \div 2 \approx 1.5 \quad \text{(integer part is 1)}
\]
\[
3 - 1 \times 2 = 1
\]
\[
2 \div 1 = 2
\]
\[
2 - 2 \times 1 = 0
\]

The GCD is 1. Therefore, \(\frac{59552}{13865}\) is already in its simplest form.

### Answer: \(\frac{59552}{13865}\)

Expected answer: 59552/13865

Correct? Yes

Problem: If instead of 38/59, the fraction was 38/39, what would be the new sum when combined with 80/94?

GPT-4o:

To calculate the sum of the fractions \(\frac{38}{39}\) and \(\frac{80}{94}\), we first need to find a common denominator. The least common multiple (LCM) of 39 and 94 will serve as the common denominator.

First, let's factorize the denominators:
- 39 can be factored into \(3 \times 13\).
- 94 can be factored into \(2 \times 47\).

Since 39 and 94 do not share any common factors, the LCM of 39 and 94 is \(39 \times 94\).

Now, let's convert each fraction to have this common denominator:
\[
\frac{38}{39} = \frac{38 \times 94}{39 \times 94} = \frac{3572}{3666}
\]
\[
\frac{80}{94} = \frac{80 \times 39}{94 \times 39} = \frac{3120}{3666}
\]

Now, we can add the two fractions:
\[
\frac{3572}{3666} + \frac{3120}{3666} = \frac{3572 + 3120}{3666} = \frac{6692}{3666}
\]

Next, we simplify the fraction \(\frac{6692}{3666}\). We find the greatest common divisor (GCD) of 6692 and 3666. Using the Euclidean algorithm:
\[
6692 \div 3666 \approx 1.825 \quad \text{(integer part is 1)}
\]
\[
6692 - 3666 = 3026
\]
\[
3666 \div 3026 \approx 1.211 \quad \text{(integer part is 1)}
\]
\[
3666 - 3026 = 640
\]
\[
3026 \div 640 \approx 4.728 \quad \text{(integer part is 4)}
\]
\[
3026 - 4 \times 640 = 466
\]
\[
640 \div 466 \approx 1.373 \quad \text{(integer part is 1)}
\]
\[
640 - 466 = 174
\]
\[
466 \div 174 \approx 2.678 \quad \text{(integer part is 2)}
\]
\[
466 - 2 \times 174 = 118
\]
\[
174 \div 118 \approx 1.475 \quad \text{(integer part is 1)}
\]
\[
174 - 118 = 56
\]
\[
118 \div 56 \approx 2.107 \quad \text{(integer part is 2)}
\]
\[
118 - 2 \times 56 = 6
\]
\[
56 \div 6 \approx 9.333 \quad \text{(integer part is 9)}
\]
\[
56 - 9 \times 6 = 2
\]
\[
6 \div 2 = 3
\]
\[
6 - 3 \times 2 = 0
\]

The GCD is 2. Therefore, we simplify \(\frac{6692}{3666}\) by dividing both the numerator and the denominator by 2:
\[
\frac{6692 \div 2}{3666 \div 2} = \frac{3346}{1833}
\]

### Answer: \(\frac{3346}{1833}\)

Expected answer: 3346/1833

Correct? Yes

Problem: After finding the result of subtracting 19/44 from 33/5, what will be the resulting fraction when you add 32/76 to the previous outcome?

GPT-4o:

First, we need to add \( \frac{32}{76} \) to the previous result \( \frac{1357}{220} \). To do this, we need a common denominator for the fractions.

1. Simplify \( \frac{32}{76} \):
\[
\frac{32}{76} = \frac{8}{19}
\]

2. Find the least common multiple (LCM) of 220 and 19. Since 19 is a prime number and does not divide 220, the LCM is \( 220 \times 19 = 4180 \).

3. Convert both fractions to have the common denominator of 4180:

\[
\frac{1357}{220} = \frac{1357 \times 19}{220 \times 19} = \frac{25783}{4180}
\]

\[
\frac{8}{19} = \frac{8 \times 220}{19 \times 220} = \frac{1760}{4180}
\]

4. Add the two fractions:
\[
\frac{25783}{4180} + \frac{1760}{4180} = \frac{25783 + 1760}{4180} = \frac{27543}{4180}
\]

Thus, the resulting fraction when you add \( \frac{32}{76} \) to \( \frac{1357}{220} \) is \( \frac{27543}{4180} \).

### Answer: \(\frac{27543}{4180}\)

Expected answer: 27543/4180

Correct? Yes

Problem: Previously, we calculated the result of subtracting 19/44 from 33/5. But what if instead of subtracting 19/44, we subtracted 17/44 from 33/5? What would be the new result?

GPT-4o:

To subtract \( \frac{17}{44} \) from \( \frac{33}{5} \), we again need to find a common denominator. The least common multiple (LCM) of 44 and 5 is 220.

Next, we convert each fraction to have the common denominator of 220:

1. Convert \( \frac{33}{5} \):
\[
\frac{33}{5} = \frac{33 \times 44}{5 \times 44} = \frac{1452}{220}
\]

2. Convert \( \frac{17}{44} \):
\[
\frac{17}{44} = \frac{17 \times 5}{44 \times 5} = \frac{85}{220}
\]

Now, we subtract the two fractions:
\[
\frac{1452}{220} - \frac{85}{220} = \frac{1452 - 85}{220} = \frac{1367}{220}
\]

Thus, the result of subtracting \( \frac{17}{44} \) from \( \frac{33}{5} \) is \( \frac{1367}{220} \).

### Answer: \(\frac{1367}{220}\)

Expected answer: 1367/220

Correct? Yes

Problem: Following the previous problem, suppose we now also add 3/82 to the sum of 49/75 and 88/58, what would the new sum be?

GPT-4o:

To find the new sum of \(\frac{49}{75}\), \(\frac{88}{58}\), and \(\frac{3}{82}\), we will follow these steps:

1. **Find the common denominator for all three fractions:**
   - We already have the least common multiple (LCM) of 75 and 58, which is 4350.
   - Now, we need to find the LCM of 4350 and 82.
   - The prime factorization of 82 is \(2 \times 41\).

   The LCM of 4350 and 82 is:
   \[
   \text{LCM}(4350, 82) = 2 \times 3 \times 5^2 \times 29 \times 41 = 178350
   \]

2. **Convert each fraction to have the common denominator 178350:**
   - For \(\frac{49}{75}\):
     \[
     \frac{49}{75} = \frac{49 \times 2378}{75 \times 2378} = \frac{116522}{178350}
     \]

   - For \(\frac{88}{58}\):
     \[
     \frac{88}{58} = \frac{88 \times 3075}{58 \times 3075} = \frac{270600}{178350}
     \]

   - For \(\frac{3}{82}\):
     \[
     \frac{3}{82} = \frac{3 \times 2175}{82 \times 2175} = \frac{6525}{178350}
     \]

3. **Add the fractions:**
   \[
   \frac{116522}{178350} + \frac{270600}{178350} + \frac{6525}{178350} = \frac{116522 + 270600 + 6525}{178350} = \frac{393647}{178350}
   \]

4. **Simplify the resulting fraction:**
   - To simplify \(\frac{393647}{178350}\), we find the greatest common divisor (GCD) of 393647 and 178350.
   - Using the Euclidean algorithm:
     \[
     \text{GCD}(393647, 178350) = 1
     \]

   Since the GCD is 1, the fraction \(\frac{393647}{178350}\) is already in its simplest form.

Thus, the new sum of \(\frac{49}{75}\), \(\frac{88}{58}\), and \(\frac{3}{82}\) is \(\frac{393647}{178350}\).

### Answer: \(\frac{393647}{178350}\)

Expected answer: 393647/178350

Correct? Yes

Problem: Now consider a different scenario where we started with 12/75 instead of 49/75, and we still add 88/58. What would we get as the sum in this case?

GPT-4o:

To find the sum of the fractions \(\frac{12}{75}\) and \(\frac{88}{58}\), we first need to find a common denominator. The least common multiple (LCM) of 75 and 58 will serve as the common denominator.

1. **Find the LCM of 75 and 58:**
   - The prime factorization of 75 is \(3 \times 5^2\).
   - The prime factorization of 58 is \(2 \times 29\).

   The LCM is the product of the highest powers of all prime factors involved:
   \[
   \text{LCM}(75, 58) = 2 \times 3 \times 5^2 \times 29 = 2 \times 3 \times 25 \times 29 = 4350
   \]

2. **Convert each fraction to have the common denominator 4350:**
   - For \(\frac{12}{75}\):
     \[
     \frac{12}{75} = \frac{12 \times 58}{75 \times 58} = \frac{696}{4350}
     \]

   - For \(\frac{88}{58}\):
     \[
     \frac{88}{58} = \frac{88 \times 75}{58 \times 75} = \frac{6600}{4350}
     \]

3. **Add the fractions:**
   \[
   \frac{696}{4350} + \frac{6600}{4350} = \frac{696 + 6600}{4350} = \frac{7296}{4350}
   \]

4. **Simplify the resulting fraction:**
   - To simplify \(\frac{7296}{4350}\), we find the greatest common divisor (GCD) of 7296 and 4350.
   - Using the Euclidean algorithm:
     \[
     \text{GCD}(7296, 4350) = 6
     \]

   - Divide both the numerator and the denominator by their GCD:
     \[
     \frac{7296 \div 6}{4350 \div 6} = \frac{1216}{725}
     \]

Thus, the sum of \(\frac{12}{75}\) and \(\frac{88}{58}\) is \(\frac{1216}{725}\).

### Answer: \(\frac{1216}{725}\)

Expected answer: 1216/725

Correct? Yes

Problem: Following up on our previous question, what is the result when you add (72 / 81) to the difference between (50 / 2) and (50 / 32)?

GPT-4o:

To solve the expression \((72 / 81) + ((50 / 2) - (50 / 32))\), we need to perform the operations step by step.

1. First, calculate \(50 / 2\):
   \[
   50 / 2 = 25
   \]

2. Next, calculate \(50 / 32\):
   \[
   50 / 32 = 1.5625
   \]

3. Find the difference between \(50 / 2\) and \(50 / 32\):
   \[
   25 - 1.5625 = 23.4375
   \]

4. Now, calculate \(72 / 81\):
   \[
   72 / 81 = 0.8888888888888888 \approx 0.8889
   \]

5. Finally, add \(72 / 81\) to the difference calculated in step 3:
   \[
   0.8889 + 23.4375 = 24.3264
   \]

### Answer: 24.3264

Expected answer: 3503/144

Correct? No

Problem: Following the previous problem, consider this change, what would the result be if instead of dividing by 32, you divide 50 by 91? What would be the new result for ((50 / 2) - (50 / 91))?

GPT-4o:

To solve the expression \(((50 / 2) - (50 / 91))\), we need to perform the operations step by step.

1. First, calculate \(50 / 2\):
   \[
   50 / 2 = 25
   \]

2. Next, calculate \(50 / 91\):
   \[
   50 / 91 \approx 0.5494505494505494
   \]

3. Finally, subtract the result of \(50 / 91\) from the result of \(50 / 2\):
   \[
   25 - 0.5494505494505494 \approx 24.45054945054945
   \]

### Answer: 24.45054945054945

Expected answer: 2225/91

Correct? No

Problem: What is the result when you add the sum of ((77 / 30) + (18 / 51)) to itself?

GPT-4o:

To find the result of adding the sum of \(\left(\frac{77}{30} + \frac{18}{51}\right)\) to itself, we first need to determine the sum \(\left(\frac{77}{30} + \frac{18}{51}\right)\).

From the previous calculation, we found:
\[
\left(\frac{77}{30} + \frac{18}{51}\right) = \frac{1489}{510}.
\]

Now, we need to add this result to itself:
\[
\frac{1489}{510} + \frac{1489}{510} = \frac{1489 + 1489}{510} = \frac{2978}{510}.
\]

Next, we simplify \(\frac{2978}{510}\):
- The greatest common divisor (GCD) of 2978 and 510 is 2.
- So, \(\frac{2978}{510} = \frac{2978 \div 2}{510 \div 2} = \frac{1489}{255}\).

### Answer: \(\frac{1489}{255}\).

Expected answer: 1489/255

Correct? Yes

Problem: What is the result if instead of adding (18 / 51), we add (18 / 74) to (77 / 30)?

GPT-4o:

To solve the expression \(\left(\frac{77}{30} + \frac{18}{74}\right)\), we need to find a common denominator for the fractions.

First, let's simplify \(\frac{18}{74}\):
- The greatest common divisor (GCD) of 18 and 74 is 2.
- So, \(\frac{18}{74} = \frac{18 \div 2}{74 \div 2} = \frac{9}{37}\).

Now, we have the expression \(\left(\frac{77}{30} + \frac{9}{37}\right)\).

Next, we find the least common multiple (LCM) of the denominators 30 and 37:
- Since 37 is a prime number and does not divide 30, the LCM of 30 and 37 is \(30 \times 37 = 1110\).

Now, we convert each fraction to have the common denominator of 1110:
- \(\frac{77}{30} = \frac{77 \times 37}{30 \times 37} = \frac{2849}{1110}\).
- \(\frac{9}{37} = \frac{9 \times 30}{37 \times 30} = \frac{270}{1110}\).

Now, we can add the fractions:
\[
\frac{2849}{1110} + \frac{270}{1110} = \frac{2849 + 270}{1110} = \frac{3119}{1110}.
\]

The fraction \(\frac{3119}{1110}\) is already in its simplest form because 3119 and 1110 have no common factors other than 1.

### Answer: \(\frac{3119}{1110}\).

Expected answer: 3119/1110

Correct? Yes

Problem: Calculate the result of (57 / 22) - (72 / 48) expressing your answer as a fraction in the simplest form. Then, add this result to itself. What is the final simplified fraction?

GPT-4o:

First, we already calculated \(\frac{57}{22} - \frac{72}{48}\) in the previous problem and found it to be \(\frac{12}{11}\).

Now, we need to add this result to itself:
\[
\frac{12}{11} + \frac{12}{11}
\]

To add these fractions, we simply add the numerators since the denominators are the same:
\[
\frac{12 + 12}{11} = \frac{24}{11}
\]

The fraction \(\frac{24}{11}\) is already in its simplest form.

### Answer: \(\frac{24}{11}\)

Expected answer: 24/11

Correct? Yes

Problem: Originally, you calculated the result of (57 / 22) - (72 / 48) and expressed your answer as a fraction in the simplest form. Now, consider the scenario where you miscalculated and the second fraction was actually (72 / 100) instead of (72 / 48). Recalculate the expression (57 / 22) - (72 / 100). What is the final simplified fraction in this case?

GPT-4o:

To solve \(\frac{57}{22} - \frac{72}{100}\), we first simplify each fraction individually.

1. The fractions \(\frac{57}{22}\) and \(\frac{72}{100}\) are already in their simplest forms.

2. To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 22 and 100 is 1100.

3. Convert each fraction to have a denominator of 1100:
   \[
   \frac{57}{22} = \frac{57 \times 50}{22 \times 50} = \frac{2850}{1100}
   \]
   \[
   \frac{72}{100} = \frac{72 \times 11}{100 \times 11} = \frac{792}{1100}
   \]

4. Now, we can subtract the fractions:
   \[
   \frac{2850}{1100} - \frac{792}{1100} = \frac{2850 - 792}{1100} = \frac{2058}{1100}
   \]

5. Simplify \(\frac{2058}{1100}\):
   - Find the greatest common divisor (GCD) of 2058 and 1100. The GCD is 2.
   \[
   \frac{2058}{1100} = \frac{2058 \div 2}{1100 \div 2} = \frac{1029}{550}
   \]

### Answer: \(\frac{1029}{550}\)

Expected answer: 1029/550

Correct? Yes

Problem: If the sum of 48 divided by 16 and 12 divided by 29 is represented as a variable, what would be the value if we add this variable to itself?

GPT-4o:

Let's denote the sum of \( \frac{48}{16} \) and \( \frac{12}{29} \) as a variable \( x \).

From the previous calculation, we found:
\[ x = \frac{99}{29} \]

Now, we need to find the value of \( x + x \):
\[ x + x = 2x \]

Substitute \( x \) with \( \frac{99}{29} \):
\[ 2x = 2 \times \frac{99}{29} = \frac{2 \times 99}{29} = \frac{198}{29} \]

### Answer: \(\frac{198}{29}\)

Expected answer: 198/29

Correct? Yes

Problem: Earlier, we found the sum of 48 divided by 16 and 12 divided by 29. Now, consider if the second fraction was 12 divided by 32 instead of 12 divided by 29. What would be the new sum?

GPT-4o:

To find the new sum of \( \frac{48}{16} \) and \( \frac{12}{32} \), we first need to perform the individual divisions.

1. Calculate \( \frac{48}{16} \):
\[ \frac{48}{16} = 3 \]

2. Calculate \( \frac{12}{32} \):
\[ \frac{12}{32} = \frac{3}{8} \]

Now, add the two results together:
\[ 3 + \frac{3}{8} \]

To add these, we need a common denominator. The integer 3 can be written as \( \frac{24}{8} \):
\[ 3 = \frac{24}{8} \]

Now, add the fractions:
\[ \frac{24}{8} + \frac{3}{8} = \frac{24 + 3}{8} = \frac{27}{8} \]

### Answer: \(\frac{27}{8}\)

Expected answer: 27/8

Correct? Yes

Problem: In addition to the previously computed sum of 37/20 and 16/75, now add the fraction 61/80 to this sum. What is the total when all three fractions are added together?

GPT-4o:

To add the fractions \(\frac{619}{300}\) and \(\frac{61}{80}\), we need to find a common denominator. The least common multiple (LCM) of 300 and 80 is 1200.

First, we convert each fraction to have the denominator of 1200:

\[
\frac{619}{300} = \frac{619 \times 4}{300 \times 4} = \frac{2476}{1200}
\]

\[
\frac{61}{80} = \frac{61 \times 15}{80 \times 15} = \frac{915}{1200}
\]

Now, we add the two fractions:

\[
\frac{2476}{1200} + \frac{915}{1200} = \frac{2476 + 915}{1200} = \frac{3391}{1200}
\]

Next, we check if \(\frac{3391}{1200}\) can be simplified. The greatest common divisor (GCD) of 3391 and 1200 is 1, so the fraction is already in its simplest form.

### Answer: \(\frac{3391}{1200}\)

Expected answer: 3391/1200

Correct? Yes

Problem: If there originally were 60/20 portions of a cake and you added 16/75 portions of another cake to it, how much cake would you have in total? Please write your answer as a simplified fraction.

GPT-4o:

First, let's simplify the fraction \(\frac{60}{20}\):

\[
\frac{60}{20} = 3
\]

Now, we need to add \(\frac{16}{75}\) to 3. To do this, we convert 3 to a fraction with a denominator of 75:

\[
3 = \frac{3 \times 75}{1 \times 75} = \frac{225}{75}
\]

Now, we add the two fractions:

\[
\frac{225}{75} + \frac{16}{75} = \frac{225 + 16}{75} = \frac{241}{75}
\]

Next, we check if \(\frac{241}{75}\) can be simplified. The greatest common divisor (GCD) of 241 and 75 is 1, so the fraction is already in its simplest form.

### Answer: \(\frac{241}{75}\)

Expected answer: 241/75

Correct? Yes

Problem: After finding the result of 83/40 + 6/8, subtract 24/64 from it. What do you get?

GPT-4o:

First, let's recall the result of \( \frac{83}{40} + \frac{6}{8} \):

\[ \frac{83}{40} + \frac{6}{8} = \frac{113}{40} \]

Next, we need to subtract \( \frac{24}{64} \) from \( \frac{113}{40} \).

1. Simplify \( \frac{24}{64} \):
\[ \frac{24}{64} = \frac{3}{8} \]

2. Find a common denominator for \( \frac{113}{40} \) and \( \frac{3}{8} \). The least common multiple (LCM) of 40 and 8 is 40.

3. Convert \( \frac{3}{8} \) to a fraction with a denominator of 40:
\[ \frac{3}{8} = \frac{3 \times 5}{8 \times 5} = \frac{15}{40} \]

4. Now, subtract the fractions:
\[ \frac{113}{40} - \frac{15}{40} = \frac{113 - 15}{40} = \frac{98}{40} \]

5. Simplify \( \frac{98}{40} \):
\[ \frac{98}{40} = \frac{49}{20} \]

### Answer: \(\frac{49}{20}\)

Expected answer: 49/20

Correct? Yes

Problem: After adjusting our original calculation, we now have 83/18 instead of 83/40. So now, what is the result of 83/18 + 6/8?

GPT-4o:

To calculate the summation of \( \frac{83}{18} + \frac{6}{8} \), we first need to find a common denominator.

1. Simplify \( \frac{6}{8} \):
\[ \frac{6}{8} = \frac{3}{4} \]

2. Find the least common multiple (LCM) of the denominators 18 and 4. The LCM of 18 and 4 is 36.

3. Convert \( \frac{83}{18} \) to a fraction with a denominator of 36:
\[ \frac{83}{18} = \frac{83 \times 2}{18 \times 2} = \frac{166}{36} \]

4. Convert \( \frac{3}{4} \) to a fraction with a denominator of 36:
\[ \frac{3}{4} = \frac{3 \times 9}{4 \times 9} = \frac{27}{36} \]

5. Now, add the fractions:
\[ \frac{166}{36} + \frac{27}{36} = \frac{166 + 27}{36} = \frac{193}{36} \]

So, the summation is:
\[ \frac{193}{36} \]

### Answer: \(\frac{193}{36}\)

Expected answer: 193/36

Correct? Yes

Problem: After finding the result of the addition of the fractions 37/69 and 67/61, what would you get if you further added 100/7 to the result?

GPT-4o:

To add \(\frac{6880}{4209}\) and \(\frac{100}{7}\), we need to find a common denominator. The least common multiple (LCM) of 4209 and 7 is their product, since 7 is a prime number and does not share any common factors with 4209.

First, let's find the LCM:
\[ \text{LCM}(4209, 7) = 4209 \times 7 = 29463 \]

Next, we convert each fraction to have this common denominator:
\[ \frac{6880}{4209} = \frac{6880 \times 7}{4209 \times 7} = \frac{48160}{29463} \]
\[ \frac{100}{7} = \frac{100 \times 4209}{7 \times 4209} = \frac{420900}{29463} \]

Now, we can add the fractions:
\[ \frac{48160}{29463} + \frac{420900}{29463} = \frac{48160 + 420900}{29463} = \frac{469060}{29463} \]

The fraction \(\frac{469060}{29463}\) can be simplified if there are common factors between the numerator and the denominator. Let's check for common factors:
- The greatest common divisor (GCD) of 469060 and 29463 is 1 (since they have no common prime factors).

Thus, the fraction \(\frac{469060}{29463}\) is already in its simplest form.

### Answer: \(\frac{469060}{29463}\)

Expected answer: 469060/29463

Correct? Yes

Problem: After finding the sum of 89/84 and 75/34 from the previous problem, what would be the result if you added that fraction to itself?

GPT-4o:

To find the result of adding the fraction \(\frac{4663}{1428}\) to itself, we perform the following calculation:

\[
\frac{4663}{1428} + \frac{4663}{1428} = \frac{4663 + 4663}{1428} = \frac{9326}{1428}
\]

Next, we simplify the fraction \(\frac{9326}{1428}\). We need to find the greatest common divisor (GCD) of 9326 and 1428 to simplify the fraction.

Using the Euclidean algorithm:
1. \(9326 \div 1428 \approx 6.53\) (integer part is 6)
2. \(9326 - 6 \times 1428 = 9326 - 8568 = 758\)
3. \(1428 \div 758 \approx 1.88\) (integer part is 1)
4. \(1428 - 1 \times 758 = 1428 - 758 = 670\)
5. \(758 \div 670 \approx 1.13\) (integer part is 1)
6. \(758 - 1 \times 670 = 758 - 670 = 88\)
7. \(670 \div 88 \approx 7.61\) (integer part is 7)
8. \(670 - 7 \times 88 = 670 - 616 = 54\)
9. \(88 \div 54 \approx 1.63\) (integer part is 1)
10. \(88 - 1 \times 54 = 88 - 54 = 34\)
11. \(54 \div 34 \approx 1.59\) (integer part is 1)
12. \(54 - 1 \times 34 = 54 - 34 = 20\)
13. \(34 \div 20 \approx 1.7\) (integer part is 1)
14. \(34 - 1 \times 20 = 34 - 20 = 14\)
15. \(20 \div 14 \approx 1.43\) (integer part is 1)
16. \(20 - 1 \times 14 = 20 - 14 = 6\)
17. \(14 \div 6 \approx 2.33\) (integer part is 2)
18. \(14 - 2 \times 6 = 14 - 12 = 2\)
19. \(6 \div 2 = 3\) (integer part is 3)
20. \(6 - 3 \times 2 = 6 - 6 = 0\)

The GCD of 9326 and 1428 is 2. We divide both the numerator and the denominator by 2:

\[
\frac{9326}{1428} = \frac{9326 \div 2}{1428 \div 2} = \frac{4663}{714}
\]

### Answer: \(\frac{4663}{714}\)

Expected answer: 4663/714

Correct? Yes

Problem: Assume the original problem was about adding fractions from two recipes. The first recipe demanded 89/84 gallons of milk and the second demanded 75/34 gallons. Now consider a slightly altered scenario where the first recipe demands 94/84 gallons of milk instead of 89/84 gallons. What is the new total amount of milk needed for both recipes in this altered scenario?

GPT-4o:

To find the new total amount of milk needed, we need to add the fractions \(\frac{94}{84}\) and \(\frac{75}{34}\). As before, we need a common denominator. The least common multiple (LCM) of 84 and 34 is 1428.

First, we convert each fraction to have the common denominator of 1428:
- For \(\frac{94}{84}\):
  \[
  \frac{94}{84} = \frac{94 \times 17}{84 \times 17} = \frac{1598}{1428}
  \]

- For \(\frac{75}{34}\):
  \[
  \frac{75}{34} = \frac{75 \times 42}{34 \times 42} = \frac{3150}{1428}
  \]

Now, we add the two fractions:
\[
\frac{1598}{1428} + \frac{3150}{1428} = \frac{1598 + 3150}{1428} = \frac{4748}{1428}
\]

Next, we simplify the fraction \(\frac{4748}{1428}\). We find the greatest common divisor (GCD) of 4748 and 1428. Using the Euclidean algorithm:
- 4748 ÷ 1428 = 3 remainder 464
- 1428 ÷ 464 = 3 remainder 36
- 464 ÷ 36 = 12 remainder 28
- 36 ÷ 28 = 1 remainder 8
- 28 ÷ 8 = 3 remainder 4
- 8 ÷ 4 = 2 remainder 0

The GCD is 4. Therefore, we simplify \(\frac{4748}{1428}\) by dividing both the numerator and the denominator by 4:
\[
\frac{4748 ÷ 4}{1428 ÷ 4} = \frac{1187}{357}
\]

### Answer: \(\frac{1187}{357}\)

Expected answer: 1187/357

Correct? Yes

Problem: If you already found the sum of 51/78 and 44/83, what would be the result if you added that sum to 71/92?

GPT-4o:

We previously found that the sum of \(\frac{51}{78}\) and \(\frac{44}{83}\) is \(\frac{2555}{2158}\). Now, we need to add this result to \(\frac{71}{92}\).

First, we need to find a common denominator for \(\frac{2555}{2158}\) and \(\frac{71}{92}\). The least common multiple (LCM) of 2158 and 92 will be our common denominator.

To find the LCM, we first find the prime factorizations of 2158 and 92:

- 2158: \( 2158 = 2 \times 1079 \) (1079 is a prime number)
- 92: \( 92 = 2^2 \times 23 \)

The LCM is the product of the highest powers of all prime factors present in the factorizations:

\[ \text{LCM}(2158, 92) = 2^2 \times 23 \times 1079 = 4 \times 23 \times 1079 = 99148 \]

Next, we convert each fraction to have this common denominator:

\[ \frac{2555}{2158} = \frac{2555 \times 46}{2158 \times 46} = \frac{117530}{99148} \]
\[ \frac{71}{92} = \frac{71 \times 1079}{92 \times 1079} = \frac{76509}{99148} \]

Now, we can add the fractions:

\[ \frac{117530}{99148} + \frac{76509}{99148} = \frac{117530 + 76509}{99148} = \frac{194039}{99148} \]

Finally, we simplify the fraction \(\frac{194039}{99148}\). We find the greatest common divisor (GCD) of 194039 and 99148. Using the Euclidean algorithm:

1. \( 194039 \div 99148 \approx 1 \) remainder \( 94991 \)
2. \( 99148 \div 94991 \approx 1 \) remainder \( 4157 \)
3. \( 94991 \div 4157 \approx 22 \) remainder \( 4157 \)
4. \( 4157 \div 4157 = 1 \) remainder \( 0 \)

The GCD is 1, meaning the fraction \(\frac{194039}{99148}\) is already in its simplest form.

### Answer: \(\frac{194039}{99148}\)

Expected answer: 194139/99268

Correct? No

Problem: You previously calculated the sum of 51/78 and 44/83. What if you made a mistake and the first fraction was actually 51/74 instead of 51/78? What would be the new sum of 51/74 and 44/83?

GPT-4o:

To add the fractions \(\frac{51}{74}\) and \(\frac{44}{83}\), we need to find a common denominator. The least common multiple (LCM) of 74 and 83 will be our common denominator.

First, let's check if 74 and 83 have any common factors other than 1. Since 83 is a prime number, it does not share any factors with 74 other than 1. Therefore, the LCM of 74 and 83 is simply their product:

\[ 74 \times 83 = 6142 \]

Next, we convert each fraction to have this common denominator:

\[ \frac{51}{74} = \frac{51 \times 83}{74 \times 83} = \frac{4233}{6142} \]
\[ \frac{44}{83} = \frac{44 \times 74}{83 \times 74} = \frac{3256}{6142} \]

Now, we can add the fractions:

\[ \frac{4233}{6142} + \frac{3256}{6142} = \frac{4233 + 3256}{6142} = \frac{7489}{6142} \]

Finally, we simplify the fraction \(\frac{7489}{6142}\). We find the greatest common divisor (GCD) of 7489 and 6142. Using the Euclidean algorithm:

1. \( 7489 \div 6142 \approx 1 \) remainder \( 1347 \)
2. \( 6142 \div 1347 \approx 4 \) remainder \( 754 \)
3. \( 1347 \div 754 \approx 1 \) remainder \( 593 \)
4. \( 754 \div 593 \approx 1 \) remainder \( 161 \)
5. \( 593 \div 161 \approx 3 \) remainder \( 110 \)
6. \( 161 \div 110 \approx 1 \) remainder \( 51 \)
7. \( 110 \div 51 \approx 2 \) remainder \( 8 \)
8. \( 51 \div 8 \approx 6 \) remainder \( 3 \)
9. \( 8 \div 3 \approx 2 \) remainder \( 2 \)
10. \( 3 \div 2 \approx 1 \) remainder \( 1 \)
11. \( 2 \div 1 \approx 2 \) remainder \( 0 \)

The GCD is 1. Therefore, the fraction \(\frac{7489}{6142}\) is already in its simplest form.

### Answer: \(\frac{7489}{6142}\)

Expected answer: 7489/6142

Correct? Yes

Problem: If you instead added the fractions 99/85 and 96/55, what would the new sum be?

GPT-4o:

To calculate the sum of the fractions \( \frac{99}{85} \) and \( \frac{96}{55} \), we again need to find a common denominator. As before, the least common multiple (LCM) of 85 and 55 is 935.

First, we convert each fraction to have the common denominator of 935:
- For \( \frac{99}{85} \):
  \[
  \frac{99}{85} = \frac{99 \times 11}{85 \times 11} = \frac{1089}{935}
  \]

- For \( \frac{96}{55} \):
  \[
  \frac{96}{55} = \frac{96 \times 17}{55 \times 17} = \frac{1632}{935}
  \]

Now, we add the two fractions:
\[
\frac{1089}{935} + \frac{1632}{935} = \frac{1089 + 1632}{935} = \frac{2721}{935}
\]

To simplify \( \frac{2721}{935} \), we check for any common factors. The greatest common divisor (GCD) of 2721 and 935 is 1, so the fraction is already in its simplest form.

### Answer: \(\frac{2721}{935}\)

Expected answer: 2721/935

Correct? Yes

Problem: After finding the value of 97 / 56 - 31 / 52, what is the result of adding it to 100 / 25? Write your final answer as a simplified fraction.

GPT-4o:

First, we already found that:
\[ \frac{97}{56} - \frac{31}{52} = \frac{827}{728} \]

Next, we need to add this result to \( \frac{100}{25} \).

First, simplify \( \frac{100}{25} \):
\[ \frac{100}{25} = 4 \]

Now, we need to add \( 4 \) to \( \frac{827}{728} \). To do this, we convert \( 4 \) to a fraction with the same denominator as \( \frac{827}{728} \):
\[ 4 = \frac{4 \times 728}{728} = \frac{2912}{728} \]

Now, add the two fractions:
\[ \frac{827}{728} + \frac{2912}{728} = \frac{827 + 2912}{728} = \frac{3739}{728} \]

Finally, we check if the fraction \(\frac{3739}{728}\) can be simplified. Since 3739 and 728 have no common factors other than 1, the fraction is already in its simplest form.

### Answer: \(\frac{3739}{728}\)

Expected answer: 3739/728

Correct? Yes

Problem: If instead of 31/52 in the previous problem you had 26/52 to subtract from 97/56, what would the new result be? Please express your answer as a simplified fraction.

GPT-4o:

To find the value of \( \frac{97}{56} - \frac{26}{52} \), we first need to find a common denominator for the fractions.

The denominators are 56 and 52. The least common multiple (LCM) of 56 and 52 can be found by determining their prime factorizations:
- 56 = \( 2^3 \times 7 \)
- 52 = \( 2^2 \times 13 \)

The LCM will be the highest power of each prime that appears in the factorizations:
- LCM = \( 2^3 \times 7 \times 13 = 8 \times 7 \times 13 = 728 \)

Next, we convert each fraction to have the common denominator of 728:
\[ \frac{97}{56} = \frac{97 \times 13}{56 \times 13} = \frac{1261}{728} \]
\[ \frac{26}{52} = \frac{26 \times 14}{52 \times 14} = \frac{364}{728} \]

Now, we subtract the fractions:
\[ \frac{1261}{728} - \frac{364}{728} = \frac{1261 - 364}{728} = \frac{897}{728} \]

Finally, we check if the fraction \(\frac{897}{728}\) can be simplified. We find the greatest common divisor (GCD) of 897 and 728. Using the Euclidean algorithm:
- 897 ÷ 728 = 1 remainder 169
- 728 ÷ 169 = 4 remainder 52
- 169 ÷ 52 = 3 remainder 13
- 52 ÷ 13 = 4 remainder 0

The GCD is 13. Therefore, we simplify \(\frac{897}{728}\) by dividing both the numerator and the denominator by 13:
\[ \frac{897 \div 13}{728 \div 13} = \frac{69}{56} \]

### Answer: \(\frac{69}{56}\)

Expected answer: 69/56

Correct? Yes

Problem: After you have calculated the result of the previous operation (78 / 74) + (72 / 51), find the sum if you add the result of the operation to itself. What is the final answer?

GPT-4o:

We previously calculated the result of the operation \(\frac{78}{74} + \frac{72}{51}\) to be \(\frac{1551}{629}\).

Now, we need to add this result to itself:
\[
\frac{1551}{629} + \frac{1551}{629}
\]

Since the denominators are the same, we can simply add the numerators:
\[
\frac{1551 + 1551}{629} = \frac{3102}{629}
\]

Thus, the final answer is \(\frac{3102}{629}\).

### Answer: \(\frac{3102}{629}\)

Expected answer: 3102/629

Correct? Yes

Problem: Suppose in our previous operation, instead of adding (72 / 51), we were to add (56 / 51) to (78 / 74). What would be the new result of the operation (78 / 74) + (56 / 51)?

GPT-4o:

To solve the fraction operation \(\frac{78}{74} + \frac{56}{51}\), we first simplify each fraction if possible.

1. Simplify \(\frac{78}{74}\):
   - The greatest common divisor (GCD) of 78 and 74 is 2.
   - \(\frac{78}{74} = \frac{78 \div 2}{74 \div 2} = \frac{39}{37}\).

2. Simplify \(\frac{56}{51}\):
   - The greatest common divisor (GCD) of 56 and 51 is 1 (since they have no common factors other than 1).
   - \(\frac{56}{51}\) is already in its simplest form.

Next, we need to add the two fractions \(\frac{39}{37} + \frac{56}{51}\). To do this, we find a common denominator.

The least common multiple (LCM) of 37 and 51 is 1887 (since 37 and 51 are both prime numbers, their LCM is their product).

Convert each fraction to have the common denominator 1887:
- \(\frac{39}{37} = \frac{39 \times 51}{37 \times 51} = \frac{1989}{1887}\)
- \(\frac{56}{51} = \frac{56 \times 37}{51 \times 37} = \frac{2072}{1887}\)

Now, add the fractions:
\[
\frac{1989}{1887} + \frac{2072}{1887} = \frac{1989 + 2072}{1887} = \frac{4061}{1887}
\]

Thus, the result of the fraction operation is \(\frac{4061}{1887}\).

### Answer: \(\frac{4061}{1887}\)

Expected answer: 4061/1887

Correct? Yes

Problem: After calculating the result of (33 / 23) - (12 / 75), now find the sum when you add this result to 24 / 81. Express your answer as a fraction.

GPT-4o:

To find the sum of \(\frac{2199}{1725}\) and \(\frac{24}{81}\), we first need to express both fractions with a common denominator.

The denominators are 1725 and 81. We need to find the least common multiple (LCM) of these two numbers. 

First, we find the prime factorizations:
- 1725 = 3 × 5^2 × 23
- 81 = 3^4

The LCM is found by taking the highest power of each prime that appears in the factorizations:
- LCM = 3^4 × 5^2 × 23 = 207,675

Next, we convert each fraction to have the common denominator of 207,675:

\[
\frac{2199}{1725} = \frac{2199 \times 120}{1725 \times 120} = \frac{263,880}{207,675}
\]

\[
\frac{24}{81} = \frac{24 \times 2565}{81 \times 2565} = \frac{61,560}{207,675}
\]

Now, we add the two fractions:

\[
\frac{263,880}{207,675} + \frac{61,560}{207,675} = \frac{263,880 + 61,560}{207,675} = \frac{325,440}{207,675}
\]

To simplify \(\frac{325,440}{207,675}\), we find the greatest common divisor (GCD) of 325,440 and 207,675. Using the Euclidean algorithm, we find that the GCD is 15.

Dividing both the numerator and the denominator by 15:

\[
\frac{325,440 \div 15}{207,675 \div 15} = \frac{21,696}{13,845}
\]

Thus, the sum of \(\frac{2199}{1725}\) and \(\frac{24}{81}\) is:

### Answer: \(\frac{21,696}{13,845}\)

Expected answer: 24391/15525

Correct? No

Problem: Suppose we add 1/88 to the sum of 19/30 and 75/77. What is the resultant fraction? Simplify your answer.

GPT-4o:

First, we need to find the sum of \( \frac{19}{30} \) and \( \frac{75}{77} \) as we did previously:

\[
\frac{19}{30} = \frac{19 \times 77}{30 \times 77} = \frac{1463}{2310}
\]

\[
\frac{75}{77} = \frac{75 \times 30}{77 \times 30} = \frac{2250}{2310}
\]

Adding these two fractions:

\[
\frac{1463}{2310} + \frac{2250}{2310} = \frac{1463 + 2250}{2310} = \frac{3713}{2310}
\]

Next, we add \( \frac{1}{88} \) to \( \frac{3713}{2310} \). To do this, we need a common denominator for 2310 and 88. The least common multiple (LCM) of 2310 and 88 is 101640.

Convert each fraction to have the common denominator of 101640:

\[
\frac{3713}{2310} = \frac{3713 \times 44}{2310 \times 44} = \frac{163372}{101640}
\]

\[
\frac{1}{88} = \frac{1 \times 1155}{88 \times 1155} = \frac{1155}{101640}
\]

Now, add the two fractions:

\[
\frac{163372}{101640} + \frac{1155}{101640} = \frac{163372 + 1155}{101640} = \frac{164527}{101640}
\]

Next, we simplify the fraction \( \frac{164527}{101640} \). We need to check if there is a common factor between 164527 and 101640. 

The prime factorization of 101640 is:
\[
101640 = 2^3 \times 3 \times 5 \times 13 \times 29
\]

The prime factorization of 164527 is:
\[
164527 = 7 \times 23503
\]

Since 164527 and 101640 have no common prime factors, the fraction \( \frac{164527}{101640} \) is already in its simplest form.

### Answer: \(\frac{164527}{101640}\)

Expected answer: 14957/9240

Correct? Yes

Problem: Previously, we added 19/30 and 75/77. Now suppose that instead of adding 75/77 we add 75/68. What will be the new sum of these fractions? Please give your answer as a simplified fraction.

GPT-4o:

To find the sum of the fractions \( \frac{19}{30} \) and \( \frac{75}{68} \), we need to find a common denominator. The least common multiple (LCM) of 30 and 68 is 1020.

First, we convert each fraction to have the common denominator of 1020:

\[
\frac{19}{30} = \frac{19 \times 34}{30 \times 34} = \frac{646}{1020}
\]

\[
\frac{75}{68} = \frac{75 \times 15}{68 \times 15} = \frac{1125}{1020}
\]

Now, we add the two fractions:

\[
\frac{646}{1020} + \frac{1125}{1020} = \frac{646 + 1125}{1020} = \frac{1771}{1020}
\]

Next, we simplify the fraction \( \frac{1771}{1020} \). We need to check if there is a common factor between 1771 and 1020.

The prime factorization of 1020 is:
\[
1020 = 2^2 \times 3 \times 5 \times 17
\]

The prime factorization of 1771 is:
\[
1771 = 7 \times 251
\]

Since 1771 and 1020 have no common prime factors, the fraction \( \frac{1771}{1020} \) is already in its simplest form.

### Answer: \(\frac{1771}{1020}\)

Expected answer: 1771/1020

Correct? Yes

Problem: After finding the sum of 57/66 and 39/64, what is its sum with 10/29? Please express your answer as a fraction.

GPT-4o:

To find the sum of \(\frac{3111}{2112}\) and \(\frac{10}{29}\), we need to find a common denominator. The least common multiple (LCM) of 2112 and 29 will be our common denominator.

First, let's find the prime factorizations of 2112 and 29:
- 2112 = 2^6 × 3 × 11
- 29 is a prime number.

The LCM is the product of the highest powers of all primes that appear in the factorizations:
- LCM = 2^6 × 3 × 11 × 29 = 2112 × 29 = 61248

Now, we convert each fraction to have the common denominator of 61248:
\[
\frac{3111}{2112} = \frac{3111 \times 29}{2112 \times 29} = \frac{90219}{61248}
\]
\[
\frac{10}{29} = \frac{10 \times 2112}{29 \times 2112} = \frac{21120}{61248}
\]

Next, we add the two fractions:
\[
\frac{90219}{61248} + \frac{21120}{61248} = \frac{90219 + 21120}{61248} = \frac{111339}{61248}
\]

Thus, the sum of \(\frac{3111}{2112}\) and \(\frac{10}{29}\) is \(\frac{111339}{61248}\).

### Answer: \(\frac{111339}{61248}\)

Expected answer: 37113/20416

Correct? Yes

Problem: When we earlier calculated the sum of two fractions, we used 57/66 and 39/64. However, there was a mistake in our calculation and the second fraction should have been 49/64 instead. Could you please recalculate and find the correct sum of the fractions 57/66 and 49/64?

GPT-4o:

To find the sum of the fractions \(\frac{57}{66}\) and \(\frac{49}{64}\), we need to find a common denominator. The least common multiple (LCM) of 66 and 64 will be our common denominator.

First, let's find the prime factorizations of 66 and 64:
- 66 = 2 × 3 × 11
- 64 = 2^6

The LCM is the highest power of each prime that appears in the factorizations:
- LCM = 2^6 × 3 × 11 = 64 × 3 × 11 = 2112

Now, we convert each fraction to have the common denominator of 2112:
\[
\frac{57}{66} = \frac{57 \times 32}{66 \times 32} = \frac{1824}{2112}
\]
\[
\frac{49}{64} = \frac{49 \times 33}{64 \times 33} = \frac{1617}{2112}
\]

Next, we add the two fractions:
\[
\frac{1824}{2112} + \frac{1617}{2112} = \frac{1824 + 1617}{2112} = \frac{3441}{2112}
\]

Thus, the sum of \(\frac{57}{66}\) and \(\frac{49}{64}\) is \(\frac{3441}{2112}\).

### Answer: \(\frac{3441}{2112}\)

Expected answer: 1147/704

Correct? Yes

Problem: What is the sum if you add the result of 80/13 + 2/48 to itself?

GPT-4o:

To find the sum when adding the result of \( \frac{80}{13} + \frac{2}{48} \) to itself, we first use the previously calculated result:

\[
\frac{80}{13} + \frac{2}{48} = \frac{1933}{312}
\]

Now, we add this result to itself:

\[
\frac{1933}{312} + \frac{1933}{312} = \frac{1933 + 1933}{312} = \frac{3866}{312}
\]

Next, we simplify the fraction \( \frac{3866}{312} \). We find the greatest common divisor (GCD) of 3866 and 312. Using the Euclidean algorithm:

1. \( 3866 \div 312 \approx 12 \) remainder \( 122 \)
2. \( 312 \div 122 \approx 2 \) remainder \( 68 \)
3. \( 122 \div 68 \approx 1 \) remainder \( 54 \)
4. \( 68 \div 54 \approx 1 \) remainder \( 14 \)
5. \( 54 \div 14 \approx 3 \) remainder \( 12 \)
6. \( 14 \div 12 \approx 1 \) remainder \( 2 \)
7. \( 12 \div 2 \approx 6 \) remainder \( 0 \)

The GCD is 2. Therefore, we simplify \( \frac{3866}{312} \) by dividing both the numerator and the denominator by 2:

\[
\frac{3866 \div 2}{312 \div 2} = \frac{1933}{156}
\]

### Answer: \(\frac{1933}{156}\)

Expected answer: 1933/156

Correct? Yes

Problem: What would be the result if you were to add 80/5 and 2/48 instead of 80/13 and 2/48?

GPT-4o:

To find the sum of \( \frac{80}{5} \) and \( \frac{2}{48} \), we first simplify the fractions if possible and then find a common denominator.

1. Simplify \( \frac{80}{5} \):
   \[
   \frac{80}{5} = 16
   \]

2. Simplify \( \frac{2}{48} \):
   \[
   \frac{2}{48} = \frac{1}{24}
   \]

3. Convert 16 to a fraction with the same denominator as \( \frac{1}{24} \):
   \[
   16 = \frac{16 \times 24}{1 \times 24} = \frac{384}{24}
   \]

4. Add the fractions:
   \[
   \frac{384}{24} + \frac{1}{24} = \frac{384 + 1}{24} = \frac{385}{24}
   \]

### Answer: \(\frac{385}{24}\)

Expected answer: 385/24

Correct? Yes

Problem: In the previous problem, you calculated the result of the operation ((55 / 30) - (1 / 72)). What would be the result if you added the fraction (73 / 92) to your answer?

GPT-4o:

To find the result of adding \(\frac{73}{92}\) to \(\frac{131}{72}\), we need to perform the following steps:

1. Find a common denominator for the fractions \(\frac{131}{72}\) and \(\frac{73}{92}\). The least common multiple (LCM) of 72 and 92 is 3312.

2. Convert \(\frac{131}{72}\) to a fraction with a denominator of 3312:
   \[
   \frac{131}{72} = \frac{131 \times 46}{72 \times 46} = \frac{6026}{3312}
   \]

3. Convert \(\frac{73}{92}\) to a fraction with a denominator of 3312:
   \[
   \frac{73}{92} = \frac{73 \times 36}{92 \times 36} = \frac{2628}{3312}
   \]

4. Add the two fractions:
   \[
   \frac{6026}{3312} + \frac{2628}{3312} = \frac{6026 + 2628}{3312} = \frac{8654}{3312}
   \]

5. Simplify the fraction \(\frac{8654}{3312}\) by finding the greatest common divisor (GCD) of 8654 and 3312. The GCD is 2.

6. Divide both the numerator and the denominator by their GCD:
   \[
   \frac{8654 \div 2}{3312 \div 2} = \frac{4327}{1656}
   \]

Thus, the result of adding \(\frac{73}{92}\) to \(\frac{131}{72}\) is \(\frac{4327}{1656}\).

### Answer: \(\frac{4327}{1656}\)

Expected answer: 4327/1656

Correct? Yes

Problem: In the previous problem, we started with the operation ((55 / 30) - (1 / 72)). Suppose now that we started with 80 instead of 55 in the first fraction. How will the result change when you calculate the new operation ((80 / 30) - (1 / 72))?

GPT-4o:

To solve the new expression \(\left(\frac{80}{30}\right) - \left(\frac{1}{72}\right)\), we need to follow similar steps as before:

1. Simplify \(\frac{80}{30}\):
   \[
   \frac{80}{30} = \frac{80 \div 10}{30 \div 10} = \frac{8}{3}
   \]

2. Convert both fractions to have a common denominator. The least common multiple (LCM) of 3 and 72 is 72.

3. Convert \(\frac{8}{3}\) to a fraction with a denominator of 72:
   \[
   \frac{8}{3} = \frac{8 \times 24}{3 \times 24} = \frac{192}{72}
   \]

4. Now, subtract \(\frac{1}{72}\) from \(\frac{192}{72}\):
   \[
   \frac{192}{72} - \frac{1}{72} = \frac{192 - 1}{72} = \frac{191}{72}
   \]

Thus, the result of the operation \(\left(\frac{80}{30}\right) - \left(\frac{1}{72}\right)\) is \(\frac{191}{72}\).

### Answer: \(\frac{191}{72}\)

Expected answer: 191/72

Correct? Yes

Problem: If we take the sum of 24/60 and 42/45 from the previous problem, and add it to the fraction 66/12, what is the new total? Answer as a simplified fraction.

GPT-4o:

First, we need to add the fraction \( \frac{66}{12} \) to the sum \( \frac{4}{3} \) from the previous problem.

1. Simplify \( \frac{66}{12} \):
   - The greatest common divisor (GCD) of 66 and 12 is 6.
   - Divide both the numerator and the denominator by 6:
     \[
     \frac{66 \div 6}{12 \div 6} = \frac{11}{2}
     \]

Next, we need a common denominator to add the fractions \( \frac{4}{3} \) and \( \frac{11}{2} \). The least common multiple (LCM) of 3 and 2 is 6.

2. Convert \( \frac{4}{3} \) to a fraction with a denominator of 6:
   - Multiply both the numerator and the denominator by 2:
     \[
     \frac{4 \times 2}{3 \times 2} = \frac{8}{6}
     \]

3. Convert \( \frac{11}{2} \) to a fraction with a denominator of 6:
   - Multiply both the numerator and the denominator by 3:
     \[
     \frac{11 \times 3}{2 \times 3} = \frac{33}{6}
     \]

Now, add the fractions \( \frac{8}{6} \) and \( \frac{33}{6} \):
\[
\frac{8}{6} + \frac{33}{6} = \frac{8 + 33}{6} = \frac{41}{6}
\]

### Answer: \(\frac{41}{6}\)

Expected answer: 41/6

Correct? Yes

Problem: In the previous problem, suppose the fraction we added to 24/60 was actually 42/67, not 42/45. What would the new total be? Answer as a simplified fraction.

GPT-4o:

To find the sum of the fractions \( \frac{24}{60} \) and \( \frac{42}{67} \), we first simplify \( \frac{24}{60} \).

1. Simplify \( \frac{24}{60} \):
   - The greatest common divisor (GCD) of 24 and 60 is 12.
   - Divide both the numerator and the denominator by 12:
     \[
     \frac{24 \div 12}{60 \div 12} = \frac{2}{5}
     \]

Next, we need a common denominator to add the fractions \( \frac{2}{5} \) and \( \frac{42}{67} \). The least common multiple (LCM) of 5 and 67 is \( 5 \times 67 = 335 \), since 67 is a prime number.

2. Convert \( \frac{2}{5} \) to a fraction with a denominator of 335:
   - Multiply both the numerator and the denominator by 67:
     \[
     \frac{2 \times 67}{5 \times 67} = \frac{134}{335}
     \]

3. Convert \( \frac{42}{67} \) to a fraction with a denominator of 335:
   - Multiply both the numerator and the denominator by 5:
     \[
     \frac{42 \times 5}{67 \times 5} = \frac{210}{335}
     \]

Now, add the fractions \( \frac{134}{335} \) and \( \frac{210}{335} \):
\[
\frac{134}{335} + \frac{210}{335} = \frac{134 + 210}{335} = \frac{344}{335}
\]

Since \( \frac{344}{335} \) is already in its simplest form (the GCD of 344 and 335 is 1), this is the final answer.

### Answer: \(\frac{344}{335}\)

Expected answer: 344/335

Correct? Yes

Problem: After finding the sum of 64/76 and 35/33, what is the result when you add 47/38 to that sum?

GPT-4o:

To find the result of adding \(\frac{47}{38}\) to the sum \(\frac{1193}{627}\), we first need to find a common denominator for \(\frac{1193}{627}\) and \(\frac{47}{38}\).

The least common multiple (LCM) of 627 and 38 will be our common denominator.

First, we find the prime factorizations of 627 and 38:
- 627 = 3 * 11 * 19
- 38 = 2 * 19

The LCM is the product of the highest powers of all prime factors:
- LCM = 2 * 3 * 11 * 19 = 1254

Next, we convert each fraction to have the common denominator of 1254:
- \(\frac{1193}{627} = \frac{1193 \times 2}{627 \times 2} = \frac{2386}{1254}\)
- \(\frac{47}{38} = \frac{47 \times 33}{38 \times 33} = \frac{1551}{1254}\)

Now, we add the two fractions:
\[
\frac{2386}{1254} + \frac{1551}{1254} = \frac{2386 + 1551}{1254} = \frac{3937}{1254}
\]

To simplify \(\frac{3937}{1254}\), we find the greatest common divisor (GCD) of 3937 and 1254. Using the Euclidean algorithm:
- 3937 ÷ 1254 = 3 remainder 175
- 1254 ÷ 175 = 7 remainder 29
- 175 ÷ 29 = 6 remainder 1
- 29 ÷ 1 = 29 remainder 0

The GCD is 1. Therefore, \(\frac{3937}{1254}\) is already in its simplest form.

Thus, the result of adding \(\frac{47}{38}\) to the sum \(\frac{1193}{627}\) is \(\frac{3937}{1254}\).

### Answer: \(\frac{3937}{1254}\)

Expected answer: 3937/1254

Correct? Yes

Problem: After reconsidering the previous problem, suppose initially there were 43/76 of an orange instead of 64/76. What will be the new sum when you add it to the same 35/33 of an apple?

GPT-4o:

To find the sum of the fractions \(\frac{43}{76}\) and \(\frac{35}{33}\), we first need to find a common denominator. The least common multiple (LCM) of 76 and 33 will be our common denominator.

First, we find the prime factorizations of 76 and 33:
- 76 = 2^2 * 19
- 33 = 3 * 11

The LCM is the product of the highest powers of all prime factors:
- LCM = 2^2 * 3 * 11 * 19 = 4 * 3 * 11 * 19 = 2508

Next, we convert each fraction to have the common denominator of 2508:
- \(\frac{43}{76} = \frac{43 \times 33}{76 \times 33} = \frac{1419}{2508}\)
- \(\frac{35}{33} = \frac{35 \times 76}{33 \times 76} = \frac{2660}{2508}\)

Now, we add the two fractions:
\[
\frac{1419}{2508} + \frac{2660}{2508} = \frac{1419 + 2660}{2508} = \frac{4079}{2508}
\]

To simplify \(\frac{4079}{2508}\), we find the greatest common divisor (GCD) of 4079 and 2508. Using the Euclidean algorithm:
- 4079 ÷ 2508 = 1 remainder 1571
- 2508 ÷ 1571 = 1 remainder 937
- 1571 ÷ 937 = 1 remainder 634
- 937 ÷ 634 = 1 remainder 303
- 634 ÷ 303 = 2 remainder 28
- 303 ÷ 28 = 10 remainder 23
- 28 ÷ 23 = 1 remainder 5
- 23 ÷ 5 = 4 remainder 3
- 5 ÷ 3 = 1 remainder 2
- 3 ÷ 2 = 1 remainder 1
- 2 ÷ 1 = 2 remainder 0

The GCD is 1. Therefore, \(\frac{4079}{2508}\) is already in its simplest form.

### Answer: \(\frac{4079}{2508}\)

Expected answer: 4079/2508

Correct? Yes

Problem: Following up on the last problem, after you find the sum of 47/16 and 96/10, subtract 72/22 from that sum. What is the result?

GPT-4o:

First, let's recall the sum of \( \frac{47}{16} \) and \( \frac{96}{10} \) that we previously calculated:
\[ \frac{47}{16} + \frac{96}{10} = \frac{1003}{80} \]

Next, we need to subtract \( \frac{72}{22} \) from \( \frac{1003}{80} \). To do this, we first convert \( \frac{72}{22} \) to a fraction with a common denominator of 80.

1. Simplify \( \frac{72}{22} \):
\[ \frac{72}{22} = \frac{36}{11} \]

2. Convert \( \frac{36}{11} \) to a fraction with a denominator of 80:
\[ \frac{36}{11} = \frac{36 \times 80}{11 \times 80} = \frac{2880}{880} \]

Now, we need to convert \( \frac{1003}{80} \) to a fraction with a denominator of 880:
\[ \frac{1003}{80} = \frac{1003 \times 11}{80 \times 11} = \frac{11033}{880} \]

Now, we subtract \( \frac{2880}{880} \) from \( \frac{11033}{880} \):
\[ \frac{11033}{880} - \frac{2880}{880} = \frac{11033 - 2880}{880} = \frac{8153}{880} \]

To express \( \frac{8153}{880} \) as a mixed fraction, we divide 8153 by 880:
\[ 8153 \div 880 = 9 \text{ remainder } 353 \]

So, \( \frac{8153}{880} \) can be written as:
\[ 9 \frac{353}{880} \]

### Answer: 9 \(\frac{353}{880}\)

Expected answer: 8153/880

Correct? No

Problem: Following the last problem where you added 47/16 and 96/10, let's consider a slightly different scenario: What if you were to add 47/16 and 64/10 instead? What would be the sum?

GPT-4o:

To find the sum of \( \frac{47}{16} \) and \( \frac{64}{10} \), we first need to find a common denominator. The least common multiple (LCM) of 16 and 10 is 80.

Next, we convert each fraction to have the denominator of 80:

1. Convert \( \frac{47}{16} \) to a fraction with a denominator of 80:
\[ \frac{47}{16} = \frac{47 \times 5}{16 \times 5} = \frac{235}{80} \]

2. Convert \( \frac{64}{10} \) to a fraction with a denominator of 80:
\[ \frac{64}{10} = \frac{64 \times 8}{10 \times 8} = \frac{512}{80} \]

Now, we add the two fractions:
\[ \frac{235}{80} + \frac{512}{80} = \frac{235 + 512}{80} = \frac{747}{80} \]

To express \( \frac{747}{80} \) as a mixed fraction, we divide 747 by 80:
\[ 747 \div 80 = 9 \text{ remainder } 27 \]

So, \( \frac{747}{80} \) can be written as:
\[ 9 \frac{27}{80} \]

### Answer: 9 \(\frac{27}{80}\)

Expected answer: 747/80

Correct? Yes

Problem: After you have calculated the result of the operation (58 / 2) + (31 / 32), subtract 90 / 58 from your answer. What is the final result?

GPT-4o:

First, let's recall the result of the operation \((58 / 2) + (31 / 32)\):
\[
\frac{959}{32}
\]

Next, we need to subtract \(\frac{90}{58}\) from this result. First, simplify \(\frac{90}{58}\):
\[
\frac{90}{58} = \frac{45}{29}
\]

Now, we need to subtract \(\frac{45}{29}\) from \(\frac{959}{32}\). To do this, we need a common denominator. The least common multiple of 32 and 29 is \(32 \times 29 = 928\).

Convert both fractions to have this common denominator:
\[
\frac{959}{32} = \frac{959 \times 29}{32 \times 29} = \frac{27811}{928}
\]
\[
\frac{45}{29} = \frac{45 \times 32}{29 \times 32} = \frac{1440}{928}
\]

Now, subtract the two fractions:
\[
\frac{27811}{928} - \frac{1440}{928} = \frac{27811 - 1440}{928} = \frac{26371}{928}
\]

Thus, the final result is:
\[
\frac{26371}{928}
\]

### Answer: \(\frac{26371}{928}\)

Expected answer: 26371/928

Correct? Yes

Problem: In the problem of adding (58 / 2) + (31 / 32), what would the result be if we originally had (58 / 2) + (10 / 32) instead? Now, compute the value.

GPT-4o:

First, we need to perform the division operations separately.

1. Calculate \( \frac{58}{2} \):
\[
\frac{58}{2} = 29
\]

2. Calculate \( \frac{10}{32} \):
\[
\frac{10}{32} \text{ (this fraction can be simplified)}
\]
\[
\frac{10}{32} = \frac{5}{16}
\]

Next, we add the results of these two operations:
\[
29 + \frac{5}{16}
\]

To add these, we need a common denominator. The integer 29 can be written as a fraction with a denominator of 16:
\[
29 = \frac{29 \times 16}{16} = \frac{464}{16}
\]

Now, add the two fractions:
\[
\frac{464}{16} + \frac{5}{16} = \frac{464 + 5}{16} = \frac{469}{16}
\]

Thus, the result of the operation is:
\[
\frac{469}{16}
\]

### Answer: \(\frac{469}{16}\)

Expected answer: 469/16

Correct? Yes

Problem: After you have calculated the sum of 26/90 and 50/26, subtract 92/89 from the result of that sum. What is the final value as a fraction?

GPT-4o:

We previously calculated the sum of \(\frac{26}{90}\) and \(\frac{50}{26}\) as \(\frac{1294}{585}\).

Now, we need to subtract \(\frac{92}{89}\) from \(\frac{1294}{585}\).

To perform this subtraction, we need a common denominator. The least common multiple (LCM) of 585 and 89 is 52065.

1. Convert \(\frac{1294}{585}\) to have a denominator of 52065:
   \[
   \frac{1294}{585} = \frac{1294 \times 89}{585 \times 89} = \frac{115166}{52065}
   \]

2. Convert \(\frac{92}{89}\) to have a denominator of 52065:
   \[
   \frac{92}{89} = \frac{92 \times 585}{89 \times 585} = \frac{53820}{52065}
   \]

Now, subtract the two fractions:
\[
\frac{115166}{52065} - \frac{53820}{52065} = \frac{115166 - 53820}{52065} = \frac{61346}{52065}
\]

Finally, simplify \(\frac{61346}{52065}\) if possible:
- The GCD of 61346 and 52065 is 1, so the fraction is already in its simplest form.

### Answer: \(\frac{61346}{52065}\)

Expected answer: 61346/52065

Correct? Yes

Problem: After calculating the sum of 26/90 and 50/26, suppose instead you added 26/90 and 50/38. How would the result change? Please give the answer as a simplified fraction.

GPT-4o:

To find the sum of the fractions \(\frac{26}{90}\) and \(\frac{50}{38}\), we first simplify each fraction if possible.

1. Simplify \(\frac{26}{90}\):
   - The greatest common divisor (GCD) of 26 and 90 is 2.
   - \(\frac{26}{90} = \frac{26 \div 2}{90 \div 2} = \frac{13}{45}\).

2. Simplify \(\frac{50}{38}\):
   - The GCD of 50 and 38 is 2.
   - \(\frac{50}{38} = \frac{50 \div 2}{38 \div 2} = \frac{25}{19}\).

Next, we need to find a common denominator for \(\frac{13}{45}\) and \(\frac{25}{19}\). The least common multiple (LCM) of 45 and 19 is 855.

3. Convert \(\frac{13}{45}\) to have a denominator of 855:
   - \(\frac{13}{45} = \frac{13 \times 19}{45 \times 19} = \frac{247}{855}\).

4. Convert \(\frac{25}{19}\) to have a denominator of 855:
   - \(\frac{25}{19} = \frac{25 \times 45}{19 \times 45} = \frac{1125}{855}\).

Now, add the two fractions:
\[
\frac{247}{855} + \frac{1125}{855} = \frac{247 + 1125}{855} = \frac{1372}{855}
\]

Finally, simplify \(\frac{1372}{855}\) if possible:
- The GCD of 1372 and 855 is 1, so the fraction is already in its simplest form.

### Answer: \(\frac{1372}{855}\)

Expected answer: 1372/855

Correct? Yes

Problem: After calculating the sum of 80/98 and 92/82, what would the result be if we added this sum to itself?

GPT-4o:

We previously calculated the sum of \(\frac{80}{98}\) and \(\frac{92}{82}\) to be \(\frac{3894}{2009}\).

To find the result of adding this sum to itself, we perform the following calculation:
\[
\frac{3894}{2009} + \frac{3894}{2009} = \frac{3894 + 3894}{2009} = \frac{7788}{2009}
\]

Next, we check if \(\frac{7788}{2009}\) can be simplified. We find the greatest common divisor (GCD) of 7788 and 2009.

Using the Euclidean algorithm:
1. \(7788 \div 2009 = 3\) with a remainder of \(1761\).
2. \(2009 \div 1761 = 1\) with a remainder of \(248\).
3. \(1761 \div 248 = 7\) with a remainder of \(17\).
4. \(248 \div 17 = 14\) with a remainder of \(10\).
5. \(17 \div 10 = 1\) with a remainder of \(7\).
6. \(10 \div 7 = 1\) with a remainder of \(3\).
7. \(7 \div 3 = 2\) with a remainder of \(1\).
8. \(3 \div 1 = 3\) with a remainder of \(0\).

The GCD is 1, so \(\frac{7788}{2009}\) is already in its simplest form.

### Answer: \(\frac{7788}{2009}\)

Expected answer: 7788/2009

Correct? Yes

Problem: What would be the result if you added together the sum of 89/44 and 69/5 to itself?

GPT-4o:

To find the result of adding the sum of \( \frac{89}{44} \) and \( \frac{69}{5} \) to itself, we first need to determine the sum, which we previously calculated as \( \frac{3481}{220} \).

Now, we add \( \frac{3481}{220} \) to itself:

\[
\frac{3481}{220} + \frac{3481}{220} = \frac{3481 + 3481}{220} = \frac{6962}{220}
\]

Thus, the result of adding the sum of \( \frac{89}{44} \) and \( \frac{69}{5} \) to itself is \( \frac{6962}{220} \).

### Answer: \(\frac{6962}{220}\)

Expected answer: 3481/110

Correct? Yes

Problem: What would be the result if we instead added 89/44 and 69/38, rather than 89/44 and 69/5 as was done in the original problem?

GPT-4o:

To calculate the sum of \( \frac{89}{44} \) and \( \frac{69}{38} \), we need to find a common denominator. The least common multiple (LCM) of 44 and 38 is 836.

First, convert each fraction to have the common denominator of 836:

\[
\frac{89}{44} = \frac{89 \times 19}{44 \times 19} = \frac{1691}{836}
\]

\[
\frac{69}{38} = \frac{69 \times 22}{38 \times 22} = \frac{1518}{836}
\]

Now, add the two fractions:

\[
\frac{1691}{836} + \frac{1518}{836} = \frac{1691 + 1518}{836} = \frac{3209}{836}
\]

Thus, the sum of \( \frac{89}{44} \) and \( \frac{69}{38} \) is \( \frac{3209}{836} \).

### Answer: \(\frac{3209}{836}\)

Expected answer: 3209/836

Correct? Yes

Problem: Let's extend the previous problem. If you take the result of 79/32 - 52/55 and add it to 42/99, what is the new result?

GPT-4o:

To solve the problem, we need to add the result of \( \frac{79}{32} - \frac{52}{55} \) to \( \frac{42}{99} \).

From the previous problem, we have:
\[ \frac{79}{32} - \frac{52}{55} = \frac{2681}{1760} \]

Now, we need to add \( \frac{2681}{1760} \) to \( \frac{42}{99} \). To do this, we need a common denominator for the fractions.

The denominators are 1760 and 99. The least common multiple (LCM) of 1760 and 99 is 174240.

Next, we convert each fraction to have the common denominator of 174240:

1. Convert \( \frac{2681}{1760} \):
\[ \frac{2681}{1760} = \frac{2681 \times 99}{1760 \times 99} = \frac{265419}{174240} \]

2. Convert \( \frac{42}{99} \):
\[ \frac{42}{99} = \frac{42 \times 1760}{99 \times 1760} = \frac{73920}{174240} \]

Now, add the two fractions:
\[ \frac{265419}{174240} + \frac{73920}{174240} = \frac{265419 + 73920}{174240} = \frac{339339}{174240} \]

Thus, the result of adding \( \frac{2681}{1760} \) to \( \frac{42}{99} \) is \( \frac{339339}{174240} \).

### Answer: \(\frac{339339}{174240}\)

Expected answer: 10283/5280

Correct? Yes

Problem: Suppose instead of the original problem where we subtracted 52/55 from 79/32, we now subtract 7/55 from 79/32 instead. What is the new result?

GPT-4o:

To calculate the expression \( \frac{79}{32} - \frac{7}{55} \), we need to find a common denominator for the fractions.

The denominators are 32 and 55. The least common multiple (LCM) of 32 and 55 is 1760.

Next, we convert each fraction to have the common denominator of 1760:

1. Convert \( \frac{79}{32} \):
\[ \frac{79}{32} = \frac{79 \times 55}{32 \times 55} = \frac{4345}{1760} \]

2. Convert \( \frac{7}{55} \):
\[ \frac{7}{55} = \frac{7 \times 32}{55 \times 32} = \frac{224}{1760} \]

Now, subtract the two fractions:
\[ \frac{4345}{1760} - \frac{224}{1760} = \frac{4345 - 224}{1760} = \frac{4121}{1760} \]

Thus, the result of the expression \( \frac{79}{32} - \frac{7}{55} \) is \( \frac{4121}{1760} \).

### Answer: \(\frac{4121}{1760}\)

Expected answer: 4121/1760

Correct? Yes

Problem: Following the previous problem, add the result of 99/92 + 15/74 to 27/71. What is the new sum as a simplified fraction?

GPT-4o:

To add \(\frac{4353}{3404}\) and \(\frac{27}{71}\), we first need to find a common denominator. The least common multiple (LCM) of 3404 and 71 will serve as the common denominator.

First, we find the prime factorizations of 3404 and 71:
- 3404 = 2^2 * 23 * 37
- 71 is a prime number.

The LCM is the product of the highest powers of all prime factors:
- LCM = 2^2 * 23 * 37 * 71

Calculating the LCM:
- 2^2 = 4
- 4 * 23 = 92
- 92 * 37 = 3404
- 3404 * 71 = 241684

So, the common denominator is 241684.

Next, we convert each fraction to have this common denominator:
\[
\frac{4353}{3404} = \frac{4353 \times 71}{3404 \times 71} = \frac{309063}{241684}
\]
\[
\frac{27}{71} = \frac{27 \times 3404}{71 \times 3404} = \frac{91908}{241684}
\]

Now, we add the two fractions:
\[
\frac{309063}{241684} + \frac{91908}{241684} = \frac{309063 + 91908}{241684} = \frac{400971}{241684}
\]

To simplify \(\frac{400971}{241684}\), we need to find the greatest common divisor (GCD) of 400971 and 241684. Using the Euclidean algorithm:
1. 400971 ÷ 241684 = 1 remainder 159287
2. 241684 ÷ 159287 = 1 remainder 82397
3. 159287 ÷ 82397 = 1 remainder 76890
4. 82397 ÷ 76890 = 1 remainder 5507
5. 76890 ÷ 5507 = 13 remainder 0

The GCD is 5507, so we divide both the numerator and the denominator by 5507:
\[
\frac{400971 ÷ 5507}{241684 ÷ 5507} = \frac{72}{44}
\]

Simplifying \(\frac{72}{44}\) further by dividing both the numerator and the denominator by their GCD, which is 4:
\[
\frac{72 ÷ 4}{44 ÷ 4} = \frac{18}{11}
\]

### Answer: \(\frac{18}{11}\)

Expected answer: 400971/241684

Correct? No

Problem: Instead of the previous calculation, what would be the sum of 99/92 and 15/54 as a simplified fraction?

GPT-4o:

To find the sum of the fractions \(\frac{99}{92}\) and \(\frac{15}{54}\), we first need to find a common denominator. The least common multiple (LCM) of 92 and 54 will serve as the common denominator.

First, we find the prime factorizations of 92 and 54:
- 92 = 2^2 * 23
- 54 = 2 * 3^3

The LCM is the product of the highest powers of all prime factors:
- LCM = 2^2 * 3^3 * 23

Calculating the LCM:
- 2^2 = 4
- 3^3 = 27
- 4 * 27 = 108
- 108 * 23 = 2484

So, the common denominator is 2484.

Next, we convert each fraction to have this common denominator:
\[
\frac{99}{92} = \frac{99 \times 27}{92 \times 27} = \frac{2673}{2484}
\]
\[
\frac{15}{54} = \frac{15 \times 46}{54 \times 46} = \frac{690}{2484}
\]

Now, we add the two fractions:
\[
\frac{2673}{2484} + \frac{690}{2484} = \frac{2673 + 690}{2484} = \frac{3363}{2484}
\]

To simplify \(\frac{3363}{2484}\), we need to find the greatest common divisor (GCD) of 3363 and 2484. Using the Euclidean algorithm:
1. 3363 ÷ 2484 = 1 remainder 879
2. 2484 ÷ 879 = 2 remainder 726
3. 879 ÷ 726 = 1 remainder 153
4. 726 ÷ 153 = 4 remainder 108
5. 153 ÷ 108 = 1 remainder 45
6. 108 ÷ 45 = 2 remainder 18
7. 45 ÷ 18 = 2 remainder 9
8. 18 ÷ 9 = 2 remainder 0

The GCD is 9, so we divide both the numerator and the denominator by 9:
\[
\frac{3363 \div 9}{2484 \div 9} = \frac{373}{276}
\]

To further simplify \(\frac{373}{276}\), we check if there are any common factors. Since 373 is a prime number and does not divide 276, the fraction is already in its simplest form.

### Answer: \(\frac{373}{276}\)

Expected answer: 1121/828

Correct? No

Problem: If we subtract 53/41 from the result of 54/1 + 62/30, what is the new result? Express your answer in simplest terms.

GPT-4o:

First, let's recall the result of \( \frac{54}{1} + \frac{62}{30} \) which we previously calculated as \( \frac{841}{15} \).

Next, we need to subtract \( \frac{53}{41} \) from \( \frac{841}{15} \).

To perform this subtraction, we need a common denominator. The least common multiple (LCM) of 15 and 41 is 615 (since 15 and 41 are coprime).

We convert both fractions to have this common denominator:
\[
\frac{841}{15} = \frac{841 \times 41}{15 \times 41} = \frac{34481}{615}
\]
\[
\frac{53}{41} = \frac{53 \times 15}{41 \times 15} = \frac{795}{615}
\]

Now, we subtract the second fraction from the first:
\[
\frac{34481}{615} - \frac{795}{615} = \frac{34481 - 795}{615} = \frac{33686}{615}
\]

To simplify \( \frac{33686}{615} \), we need to find the greatest common divisor (GCD) of 33686 and 615. Using the Euclidean algorithm, we find that the GCD is 1, meaning the fraction is already in its simplest form.

### Answer: \(\frac{33686}{615}\)

Expected answer: 33686/615

Correct? Yes

Problem: If we had originally started with 93 instead of 54, and added 62/30 to that, what would be the new result? Express the answer in simplest terms.

GPT-4o:

First, let's calculate each term separately.

1. The first term is \( \frac{93}{1} \):
\[
\frac{93}{1} = 93
\]

2. The second term is \( \frac{62}{30} \). We can simplify this fraction by finding the greatest common divisor (GCD) of 62 and 30. The GCD of 62 and 30 is 2. So, we divide both the numerator and the denominator by 2:
\[
\frac{62 \div 2}{30 \div 2} = \frac{31}{15}
\]

Now, we add the two terms together:
\[
93 + \frac{31}{15}
\]

To add these, we need a common denominator. The common denominator for 93 (which can be written as \( \frac{93}{1} \)) and \( \frac{31}{15} \) is 15. We convert 93 to a fraction with a denominator of 15:
\[
93 = \frac{93 \times 15}{1 \times 15} = \frac{1395}{15}
\]

Now, we add the fractions:
\[
\frac{1395}{15} + \frac{31}{15} = \frac{1395 + 31}{15} = \frac{1426}{15}
\]

Thus, the result in simplest terms is:
\[
\frac{1426}{15}
\]

### Answer: \(\frac{1426}{15}\)

Expected answer: 1426/15

Correct? Yes

Problem: After finding the result of adding 85/29 and 47/7, what is the new total if you also add 81/42?

GPT-4o:

We already found that the sum of \( \frac{85}{29} \) and \( \frac{47}{7} \) is \( \frac{1958}{203} \).

Now, we need to add \( \frac{81}{42} \) to this result. To do this, we need a common denominator for \( \frac{1958}{203} \) and \( \frac{81}{42} \). The least common multiple (LCM) of 203 and 42 is 4263.

First, convert each fraction to have the common denominator of 4263:

\[
\frac{1958}{203} = \frac{1958 \times 21}{203 \times 21} = \frac{41118}{4263}
\]

\[
\frac{81}{42} = \frac{81 \times 101}{42 \times 101} = \frac{8181}{4263}
\]

Now, add the two fractions:

\[
\frac{41118}{4263} + \frac{8181}{4263} = \frac{41118 + 8181}{4263} = \frac{49299}{4263}
\]

Simplify the fraction if possible. In this case, 49299 and 4263 have a common factor of 3:

\[
\frac{49299 \div 3}{4263 \div 3} = \frac{16433}{1421}
\]

### Answer: \(\frac{16433}{1421}\)

Expected answer: 4699/406

Correct? No

Problem: After calculating the result from the previous problem, consider a new situation: You have now 85/64 of a pound of oranges and 47/7 pounds of apples. What is the new total weight of your fruits in pounds?

GPT-4o:

To find the total weight of the fruits, we need to add the fractions \( \frac{85}{64} \) and \( \frac{47}{7} \). First, we need to find a common denominator. The least common multiple (LCM) of 64 and 7 is 448.

Convert each fraction to have the common denominator of 448:

\[
\frac{85}{64} = \frac{85 \times 7}{64 \times 7} = \frac{595}{448}
\]

\[
\frac{47}{7} = \frac{47 \times 64}{7 \times 64} = \frac{3008}{448}
\]

Now, add the two fractions:

\[
\frac{595}{448} + \frac{3008}{448} = \frac{595 + 3008}{448} = \frac{3603}{448}
\]

Simplify the fraction if possible. In this case, 3603 and 448 have no common factors other than 1, so the fraction is already in its simplest form.

### Answer: \(\frac{3603}{448}\)

Expected answer: 3603/448

Correct? Yes