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

Gemini-1.5 Flash:

We already found that the sum of (74/63) and (91/78) is 3835/1638.

To add this result to itself, we simply multiply it by 2:

(3835/1638) * 2 = 7670 / 1638

### Answer: 7670/1638 

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)?

Gemini-1.5 Flash:

We already found the least common multiple (LCM) of 63 and 78 to be 1638.  Let's adjust the fractions:

* (35 / 63) = (35 * 26) / (63 * 26) = 910 / 1638
* (91 / 78) = (91 * 21) / (78 * 21) = 1911 / 1638

Now we add the fractions:

* (910 / 1638) + (1911 / 1638) = (910 + 1911) / 1638 = 2821 / 1638

### Answer: 2821/1638 

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?

Gemini-1.5 Flash:

We need to find a common denominator for 471/25 and 73/89.  The least common multiple of 25 and 89 is 2225.

* Convert 471/25 to an equivalent fraction with a denominator of 2225: (471/25) * (89/89) = 41969/2225
* Convert 73/89 to an equivalent fraction with a denominator of 2225: (73/89) * (25/25) = 1825/2225

Now we can add:

41969/2225 + 1825/2225 = 43794/2225

### Answer: 43794/2225 

Expected answer: 43744/2225

Correct? No

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?

Gemini-1.5 Flash:

The least common multiple of 89 and 25 is 2225. 

*  Convert 94/89 to an equivalent fraction with a denominator of 2225: (94/89) * (25/25) = 2350/2225
* Convert 1/25 to an equivalent fraction with a denominator of 2225: (1/25) * (89/89) = 89/2225

Now we can add:

2350/2225 + 89/2225 = 2439/2225

### Answer: 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.

Gemini-1.5 Flash:

Following the same approach as before, we find a common denominator for 14 and 8, which is 56.

* Convert 48/14 to an equivalent fraction with a denominator of 56: (48/14) * (4/4) = 192/56
* Convert 49/8 to an equivalent fraction with a denominator of 56: (49/8) * (7/7) = 343/56

Now, subtract the fractions: (343/56) - (192/56) = (343 - 192)/56 = 151/56

### Answer: 151/56 

Expected answer: 151/56

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.

Gemini-1.5 Flash:

We already know that the sum of 49/79 and 18/21 is 2451/1659.  Let's call this sum 'n':

n = 2451/1659

Now we need to add 71/97 to 'n':

n + 71/97 = 2451/1659 + 71/97

To add these fractions, we need a common denominator.  The least common multiple of 1659 and 97 is 160873.

Let's rewrite our fractions with the new denominator:

* (2451/1659) * (97/97) = 237847/160873
* (71/97) * (1659/1659) = 117749/160873

Now we can add them:

237847/160873 + 117749/160873 = 355596/160873

### Answer: 355596/160873 

Expected answer: 118512/53641

Correct? No

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.

Gemini-1.5 Flash:

To find the sum of 49/3 and 18/21, we need to find a common denominator.  The least common multiple (LCM) of 3 and 21 is 21.

Let's rewrite the fractions with the new denominator:

* (49/3) * (7/7) = 343/21
* (18/21) stays the same.

Now we can add the fractions:

343/21 + 18/21 = 361/21

### Answer: 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?

Gemini-1.5 Flash:

We already know that 95/46 + 65/31 = 5935/1426.

Adding this sum twice is the same as multiplying it by 2:

(5935/1426) * 2 = 11870/1426

This fraction can be simplified by dividing both numerator and denominator by their greatest common factor, which is 2:

11870/1426 = (11870/2) / (1426/2) = 5935/713

### Answer: 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?

Gemini-1.5 Flash:

Following the same process as before:

1. **Find the LCM:** The LCM of 46 and 31 is still 1426.

2. **Convert fractions:**
   * 31/46 * (31/31) = 961/1426
   * 65/31 * (46/46) = 2990/1426

3. **Add the fractions:**
   * 961/1426 + 2990/1426 = 3951/1426

This fraction can be simplified further, but it's not as obvious as the previous example.  We can use a calculator to find the simplified form:

* 3951/1426 ≈ 2.77

### Answer: 3951/1426 or approximately 2.77 

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?

Gemini-1.5 Flash:

We already know the sum of 25/63 and 74/6 is 802/63.  

Now we need to subtract 9/47 from that sum. To do this, we need a common denominator:

* The least common multiple of 63 and 47 is 2961.
* Convert 802/63: (802/63) * (47/47) = 37684/2961
* Convert 9/47: (9/47) * (63/63) = 567/2961

Now we can subtract:

37684/2961 - 567/2961 = 37117/2961

### Answer: 37117/2961 

Expected answer: 37127/2961

Correct? No

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?

Gemini-1.5 Flash:

We're now adding 25/63 and 6/6.  Since 6/6 is equivalent to 1, we can simply add 1 to 25/63:

25/63 + 1 = 25/63 + 63/63 = 88/63

### Answer: 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?

Gemini-1.5 Flash:

We already found that the sum of 79/91 and 100/47 is 12813/4277.  Adding this sum to itself is the same as multiplying it by 2:

(12813/4277) * 2 = 25626/4277

### Answer: 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?

Gemini-1.5 Flash:

We already know from the previous calculation that the sum of 86/62 and 99/41 is 4832/1271.

Now we need to subtract 57/52 from that. To do this, we need a common denominator:

* The least common multiple of 1271 and 52 is 66172.
* Multiply the numerator and denominator of 4832/1271 by 52: (4832/1271) * (52/52) = 251264/66172
* Multiply the numerator and denominator of 57/52 by 1271: (57/52) * (1271/1271) = 72597/66172

Now we can subtract: 251264/66172 - 72597/66172 = 178667/66172

### Answer: 178667/66172 

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?

Gemini-1.5 Flash:

Following the same steps as before, we find the least common multiple of 62 and 41, which is 2542.

*  To get 2542 in the denominator of the first fraction, we multiply both the numerator and denominator by 41:  (88/62) * (41/41) = 3608/2542
* To get 2542 in the denominator of the second fraction, we multiply both the numerator and denominator by 62:  (99/41) * (62/62) = 6138/2542

Now we can add the fractions: 3608/2542 + 6138/2542 = 9746/2542

We can simplify this fraction by dividing both the numerator and denominator by their greatest common factor, which is 2: 9746/2542 = 4873/1271

### Answer: 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.

Gemini-1.5 Flash:

We already know that the sum of 87/88 and 24/13 is 3243/1144. Now we need to add 36/9 to this.

First, simplify 36/9 to 4. 

Now we need to find a common denominator for 3243/1144 and 4:

* 3243/1144 stays the same
* 4 can be rewritten as (4 * 1144)/1144 = 4576/1144

Now we can add the fractions:

3243/1144 + 4576/1144 = (3243 + 4576)/1144 = 7819/1144

This fraction can be simplified to a mixed number: 7819/1144 = 6 847/1144

### Answer: 6 847/1144 

Expected answer: 7819/1144

Correct? No

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?

Gemini-1.5 Flash:

We need to find a common denominator for 83 and 13. The least common multiple is 1079.

*  Rewrite 87/83 as (87 * 13)/(83 * 13) = 1131/1079
*  Rewrite 24/13 as (24 * 83)/(13 * 83) = 1992/1079

Now, add the fractions:

1131/1079 + 1992/1079 = (1131 + 1992)/1079 = 3123/1079 

### Answer: 3123/1079 

Expected answer: 3123/1079

Correct? Yes

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

Gemini-1.5 Flash:

Our previous answer was 15/14.  To add 62/46 to that, we first need to find a common denominator. The least common multiple of 14 and 46 is 322.

* **Convert 15/14 to have a denominator of 322:** Multiply both the numerator and denominator by 23: (15/14) * (23/23) = 345/322
* **Convert 62/46 to have a denominator of 322:** Multiply both the numerator and denominator by 7: (62/46) * (7/7) = 434/322

Now we can add:

345/322 + 434/322 = (345 + 434) / 322 = 779/322

### Answer: 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?

Gemini-1.5 Flash:

Let's follow the same steps as before, but with 61/5 instead of 11/5.

1. **Convert 61/5 to have a denominator of 70:**  Multiply both the numerator and denominator by 14: (61/5) * (14/14) = 854/70

2. **Subtract:** 854/70 - 79/70 = (854 - 79) / 70 = 775/70

3. **Simplify:**  Divide both numerator and denominator by their greatest common factor, 5: 
775/70 = (775/5) / (70/5) = 155/14

### Answer: 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?

Gemini-1.5 Flash:

We already know the sum of 68/16 and 52/78 is 767/156. Now we need to add 37/38 to this result.

To do this, we again need a common denominator.  The LCM of 156 and 38 is 2964.

* 767/156 = (767 * 19) / (156 * 19) = 14573/2964
* 37/38 = (37 * 78) / (38 * 78) = 2886/2964

Now we can add the fractions:

14573/2964 + 2886/2964 = (14573 + 2886)/2964 = 17459/2964

### Answer: 17459/2964 

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?

Gemini-1.5 Flash:

Following the same process as before, we need to find a common denominator for 16 and 78, which is 312.

* 8/16 = (8 * 19.5) / (16 * 19.5) = 156/312
* 52/78 = (52 * 4) / (78 * 4) = 208/312

Now, we add the fractions:

156/312 + 208/312 = (156 + 208)/312 = 364/312

Finally, we simplify by dividing both numerator and denominator by their greatest common factor, 4:

364/312 = (364/4) / (312/4) = 91/78

### Answer: 91/78 

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?

Gemini-1.5 Flash:

We know that the sum of 21/75 and 44/47 is 1429/1175.  

To find the total of two such sums, we simply multiply this sum by 2:

(1429/1175) * 2 = 2858/1175

### Answer: 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?

Gemini-1.5 Flash:

Let's follow the same steps as before:

1. **Find the LCM of the denominators:**
   * The LCM of 63 and 47 is 2961.

2. **Convert each fraction to an equivalent fraction with the LCM as the denominator:**
   * 21/63 * (47/47) = 987/2961
   * 44/47 * (63/63) = 2772/2961

3. **Add the numerators:**
   * 987/2961 + 2772/2961 = 3759/2961

4. **Simplify the answer (if possible):**
    * The fraction 3759/2961 cannot be simplified further as they have no common factors other than 1.

### Answer: 3759/2961 

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?

Gemini-1.5 Flash:

We already know that (77/5 + 4/11) = 867/55. 

Now we need to add 94/60 to this result.

1. **Find the Least Common Multiple (LCM) of 55 and 60:** The LCM of 55 and 60 is 330.

2. **Convert the fractions:**
   * (867/55) * (6/6) = 5202/330
   * (94/60) * (5.5/5.5) = 517/330

3. **Add the fractions with the common denominator:**
   * 5202/330 + 517/330 = 5719/330

4. **Simplify the result (optional):** You can leave the answer as an improper fraction (5719/330) or convert it to a mixed number (17 109/330).

### Answer: 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?

Gemini-1.5 Flash:

Let's think of this in terms of a recipe.  Imagine you're baking a cake, and the original recipe called for 77/5 cups of flour and 4/11 cups of sugar.  

Now, instead of 4/11 cups of sugar, you're using 4/79 cups of sugar.  What's the impact?

Since 4/79 is much smaller than 4/11, you're using significantly less sugar in the new recipe. This means your cake will likely be less sweet. 

**The overall impact:** The cake will be less sweet, but it's hard to say exactly how much less sweet without knowing the specific sweetness of the other ingredients. It could be a subtle difference, or it could be quite noticeable. 

It's important to note that modifying a recipe like this can change the balance of flavors.  What was originally intended as a balanced cake may now have a more pronounced floury taste due to the reduced sweetness. 

**In the context of the math problem:** Replacing 4/11 with 4/79 will result in a smaller overall sum. The change is significant because it reduces the amount added by a considerable margin.  

Expected answer: 6103/395

Correct? No

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

Gemini-1.5 Flash:

Here's how to solve the problem step-by-step:

1. **Solve the inner parentheses first:**  ((14 / 72) + (25 / 6))
   * We already know from the previous calculation that this equals 157/36.

2. **Now we have:** (14 / 24) + (157 / 36)

3. **Find a common denominator for the fractions:**
   * The least common multiple of 24 and 36 is 72.
   * (14 / 24) becomes (14 * 3) / (24 * 3) = 42 / 72
   * (157 / 36) becomes (157 * 2) / (36 * 2) = 314 / 72

4. **Add the fractions:**
   (42 / 72) + (314 / 72) = (42 + 314) / 72 = 356 / 72

5. **Simplify (optional):**
   * Both numerator and denominator are divisible by 4:
   * (356 / 72) = (89 / 18)

### Answer: 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))?

Gemini-1.5 Flash:

Let's recalculate the sum with the new information:

1. **Find a common denominator:**
   * The least common multiple of 72 and 6 is 72.
   * (14 / 72) stays the same.
   * (90 / 6) becomes (90 * 12) / (6 * 12) = 1080 / 72

2. **Add the fractions:**
   (14 / 72) + (1080 / 72) = (14 + 1080) / 72 = 1094 / 72

3. **Simplify (optional):**
   * We can simplify by dividing both numerator and denominator by their greatest common factor (2):
   * (1094 / 72) = (547 / 36)

### Answer: 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?

Gemini-1.5 Flash:

We already know the sum of 84/97 and 25/2 is 2593/194.  Now we need to subtract 100/71 from that.

To do this, we need a common denominator for 194 and 71.  The least common multiple (LCM) of 194 and 71 is 13778.

* **For 2593/194:** Multiply numerator and denominator by 71: (2593 * 71) / (194 * 71) = 184253/13778
* **For 100/71:** Multiply numerator and denominator by 194: (100 * 194) / (71 * 194) = 19400/13778

Now we can subtract: 184253/13778 - 19400/13778 = (184253 - 19400)/13778 = 164853/13778

### Answer: 164853/13778 

Expected answer: 164703/13774

Correct? No

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?

Gemini-1.5 Flash:

Let's follow the same steps as before:

* **Simplify 84/96:** Both numerator and denominator are divisible by 12, so 84/96 simplifies to 7/8.
* **Find the LCM:** The LCM of 8 and 2 is 8.
* **Convert 7/8:** It already has the denominator 8.
* **Convert 25/2:** Multiply numerator and denominator by 4: (25 * 4) / (2 * 4) = 100/8.

Now we can add the fractions: 7/8 + 100/8 = (7 + 100)/8 = 107/8

### Answer: 107/8 

Expected answer: 107/8

Correct? Yes

Problem: After finding the difference between the fractions 66/20 and 4/78, what would the result be if you add 6/15 to your answer?

Gemini-1.5 Flash:

We found the difference between 66/20 and 4/78 to be 1267/390. Now we need to add 6/15 to this result.

1. **Find a common denominator for 1267/390 and 6/15:**  The LCM of 390 and 15 is 780.

2. **Convert the fractions:**
   * 1267/390 = (1267 * 2) / (390 * 2) = 2534/780
   * 6/15 = (6 * 52) / (15 * 52) = 312/780

3. **Add the fractions:**  2534/780 + 312/780 = 2846/780

4. **Simplify (if possible):**  Both the numerator and denominator are divisible by 2. This gives us 1423/390. 

### Answer: 1423/390 

Expected answer: 1423/390

Correct? Yes

Problem: Suppose that instead of having 66/20 in the original problem, we have 66/21. What would be the difference between the fractions 66/21 and 4/78?

Gemini-1.5 Flash:

Let's follow the same steps as before, but with 66/21 instead of 66/20:

1. **Find the LCM of the denominators:** The LCM of 21 and 78 is 546.
2. **Convert the fractions:**
    * 66/21 = (66 * 26) / (21 * 26) = 1716/546
    * 4/78 = (4 * 7) / (78 * 7) = 28/546
3. **Subtract the fractions:** 1716/546 - 28/546 = 1688/546
4. **Simplify:** Both the numerator and denominator are divisible by 2, giving us 844/273.

### Answer: 844/273 

Expected answer: 844/273

Correct? Yes

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

Gemini-1.5 Flash:

We need to add 13/5 and 39/64. To do this, they need to have the same denominator.

1. **Find the Least Common Multiple (LCM) of 5 and 64:**
   * The LCM of 5 and 64 is 320.

2. **Convert the fractions to have a denominator of 320:**
   * 13/5 * (64/64) = 832/320
   * 39/64 * (5/5) = 195/320

3. **Add the fractions:**
   * 832/320 + 195/320 = 1027/320

**Therefore, the result of adding 13/5 and 39/64 is 1027/320.**
### Answer: 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?

Gemini-1.5 Flash:

Here's how to solve the problem:

1. **Simplify the first fraction:**
   * 77/55 simplifies to 7/5 (divide both numerator and denominator by 11)

2. **Find a common denominator:**
   * The least common multiple of 5 and 79 is 395.
   * To get a denominator of 395 for 7/5, multiply both numerator and denominator by 79: (7 * 79) / (5 * 79) = 553/395
   * To get a denominator of 395 for 6/79, multiply both numerator and denominator by 5: (6 * 5) / (79 * 5) = 30/395

3. **Add the fractions:**
   * (553/395) + (30/395) = 583/395

**Therefore, the result of ((77 / 55) + (6 / 79)) is 583/395.**
### Answer: 583/395 

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?

Gemini-1.5 Flash:

We found that the sum of 23/20 and 82/97 is 3871/1940.  Now we need to subtract 7/44 from that.

Again, we need a common denominator. The LCM of 1940 and 44 is 21340.

* 3871/1940 = (3871 * 11) / (1940 * 11) = 42581/21340
* 7/44 = (7 * 485) / (44 * 485) = 3395/21340

Now we subtract:

42581/21340 - 3395/21340 = (42581 - 3395) / 21340 = 39186/21340

### Answer: 39186/21340 

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?

Gemini-1.5 Flash:

We follow the same process as before, finding the LCM of 43 and 97, which is 4161.

* 23/43 = (23 * 97) / (43 * 97) = 2231/4161
* 82/97 = (82 * 43) / (97 * 43) = 3526/4161

Adding the fractions:

2231/4161 + 3526/4161 = (2231 + 3526) / 4161 = 5757/4161

### Answer: 5757/4161 

Expected answer: 5757/4171

Correct? No

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

Gemini-1.5 Flash:

We already know that the sum of 54/88 and 28/10 is 751/220. Now we need to add 21/13 to this.

To add these fractions, we need a common denominator. The least common multiple of 220 and 13 is 2860.

* Convert 751/220 to a fraction with a denominator of 2860:
  (751/220) * (13/13) = 9763/2860

* Convert 21/13 to a fraction with a denominator of 2860:
  (21/13) * (220/220) = 4620/2860

Now, we can add the fractions: 
9763/2860 + 4620/2860 = 14383/2860

### Answer: 14383/2860 

Expected answer: 14383/2860

Correct? Yes

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

Gemini-1.5 Flash:

We know that (15/32) + (65/15) = 461/96. Now we need to add this to 87/26.

1. **Find a common denominator:** The least common multiple of 96 and 26 is 2496.

2. **Convert the fractions:**
   * (461/96) * (26/26) = 11966/2496
   * (87/26) * (96/96) = 8352/2496

3. **Add the fractions:** 11966/2496 + 8352/2496 = 20318/2496

4. **Simplify (if possible):** Both 20318 and 2496 are divisible by 2, so we can simplify:
   * 20318/2496 = (20318/2) / (2496/2) = 10159/1248 

### Answer: 10159/1248 

Expected answer: 10169/1248

Correct? No

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

Gemini-1.5 Flash:

We already know that the sum of 4/31 and 76/11 is 2400/341.  Let's subtract 63/35 from this result.

First, we need to find a common denominator for 341 and 35. The least common multiple of 341 and 35 is 11935.

* Convert 2400/341 to an equivalent fraction with a denominator of 11935: (2400/341) * (35/35) = 84000/11935
* Convert 63/35 to an equivalent fraction with a denominator of 11935: (63/35) * (341/341) = 21503/11935

Now we can subtract: 84000/11935 - 21503/11935 = 62497/11935

### Answer: 62497/11935 

Expected answer: 8931/1705

Correct? No

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?

Gemini-1.5 Flash:

Following the same steps as before, we need to find equivalent fractions with a common denominator of 341:

*  (42/31) * (11/11) = 462/341
*  (76/11) * (31/31) = 2356/341

Now we add the fractions: 462/341 + 2356/341 = 2818/341

### Answer: 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?

Gemini-1.5 Flash:

We already know that 82/15 + 100/12 = 138/10.

Now we need to add 46/83 to that:

* **Find a common denominator:** The least common multiple of 10 and 83 is 830.
* **Convert the fractions:**
    * (138/10) * (83/83) = 11454/830
    * (46/83) * (10/10) = 460/830
* **Add the fractions:** 11454/830 + 460/830 = 11914/830

### Answer: 11914/830 

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?

Gemini-1.5 Flash:

We follow the same steps as before:

* **Find a common denominator:** The least common multiple of 15 and 12 is 60.
* **Convert the fractions:**
    *  (61/15) * (4/4) = 244/60
    * (100/12) * (5/5) = 500/60
* **Add the fractions:** 244/60 + 500/60 = 744/60
* **Simplify:** 744/60 can be simplified to 124/10

### Answer: 124/10 

Expected answer: 62/5

Correct? Yes

Problem: If you subtract 78/92 from the sum of 57/29 and 14/98 (from the previous problem), what do you get?

Gemini-1.5 Flash:

We know the sum of 57/29 and 14/98 is 428/203 (from the previous problem). Now we need to subtract 78/92.

1. **Simplify:** 78/92 can be simplified by dividing both numerator and denominator by their greatest common factor, which is 2. This gives us 39/46.

2. **Find the Least Common Multiple (LCM):** The LCM of 203 and 46 is 4669.

3. **Convert Fractions:**
   * (428/203) * (23/23) = 9864/4669
   * (39/46) * (101/101) = 3939/4669

4. **Subtract Fractions:**
   * 9864/4669 - 3939/4669 = 5925/4669

### Answer: 5925/4669 

Expected answer: 11771/9338

Correct? No

Problem: If instead of the sum of 57/29 and 14/98, you calculated the sum of 57/29 and 14/26, what would be the new answer?

Gemini-1.5 Flash:

Let's follow the same steps as before, but this time with the fractions 57/29 and 14/26.

1. **Simplify:** The fraction 14/26 can be simplified by dividing both numerator and denominator by their greatest common factor, which is 2. This gives us 7/13.

2. **Find the Least Common Multiple (LCM):** The LCM of 29 and 13 is 377.

3. **Convert Fractions:**
   * (57/29) * (13/13) = 741/377
   * (7/13) * (29/29) = 203/377

4. **Add Fractions:**
   * 741/377 + 203/377 = 944/377

### Answer: 944/377 

Expected answer: 944/377

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?

Gemini-1.5 Flash:

We know from the previous calculation that 70/25 + 83/28 = 807/140. 

Now we need to subtract 47/13 from that result.  

**1. Find a common denominator:**

* The least common denominator for 140 and 13 is 1820.

**2. Convert the fractions to equivalent fractions with the common denominator:**

* (807/140) * (13/13) = 10491/1820
* (47/13) * (140/140) = 6580/1820

**3. Subtract the fractions:**

* 10491/1820 - 6580/1820 = 3911/1820

**4. Simplify the result (if possible):**

* 3911/1820 cannot be simplified further.

**Therefore:**

(70/25 + 83/28) - 47/13 = 3911/1820

### Answer: 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?

Gemini-1.5 Flash:

Here's the breakdown:

1. **Apples:** You have 70/25 apples. This simplifies to 14/5 apples.

2. **Oranges:** You have 83/8 oranges. This fraction is already in its simplest form.

3. **Total:** To find the total, we need to add the apples and oranges. However, we can't directly add fractions with different denominators.

4. **Common Denominator:**  The least common denominator for 5 and 8 is 40.

5. **Convert Fractions:**
   * (14/5) * (8/8) = 112/40 apples
   * (83/8) * (5/5) = 415/40 oranges

6. **Add:**
   * 112/40 + 415/40 = 527/40

**Therefore, you have a total of 527/40 apples and oranges.**

### Answer: 527/40 

Expected answer: 527/40

Correct? Yes

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

Gemini-1.5 Flash:

We already know that 50/73 + 20/23 = 2610/1679. 

Now, we need to add 83/35 to this result. To do that, we need a common denominator:

1. **Find the LCM of 1679 and 35:** The LCM is 58765.

2. **Rewrite each fraction with the LCM as the denominator:**
   * (2610/1679) * (35/35) = 91350/58765
   * (83/35) * (1679/1679) = 139497/58765

3. **Add the numerators:** 91350/58765 + 139497/58765 = 230847/58765

4. **Simplify (if possible):**  The fraction 230847/58765 doesn't simplify further. 

### Answer: 230847/58765 

Expected answer: 230707/58765

Correct? No

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?

Gemini-1.5 Flash:

We need to add the two fractions 50/57 and 20/23 to find the total time. 

1. **Find the Least Common Multiple (LCM) of the denominators:** The LCM of 57 and 23 is 1311.

2. **Rewrite each fraction with the LCM as the denominator:**
   * (50/57) * (23/23) = 1150/1311
   * (20/23) * (57/57) = 1140/1311

3. **Add the numerators:** 1150/1311 + 1140/1311 = 2290/1311

### Answer: 2290/1311 

Expected answer: 2290/1311

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?

Gemini-1.5 Flash:

We already know from the previous calculation that 94/35 - 23/49 = 543/245.

1. **Simplify 27/72:**
   * Both 27 and 72 are divisible by 9.
   * 27/72 = (27/9) / (72/9) = 3/8

2. **Find a common denominator for 543/245 and 3/8:**
   * The least common multiple of 245 and 8 is 1960.
   * To get 1960 in the denominator of 543/245, multiply both numerator and denominator by 8: (543 * 8) / (245 * 8) = 4344/1960
   * To get 1960 in the denominator of 3/8, multiply both numerator and denominator by 245: (3 * 245) / (8 * 245) = 735/1960

3. **Add the fractions:**
   * (4344/1960) + (735/1960) = (4344 + 735) / 1960 = 5079/1960

4. **Simplify the result:**
   * Both 5079 and 1960 are divisible by 39.
   * 5079/1960 = (5079/39) / (1960/39) = 130/50

### Answer: 130/50 

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?

Gemini-1.5 Flash:

Let's follow the same steps as before:

1. **Simplify the fractions:**
    * 54/35 is already in its simplest form.
    * 23/49 is also in its simplest form.

2. **Find a common denominator:**
    * The least common multiple of 35 and 49 is 245.
    * To get 245 in the denominator of 54/35, multiply both numerator and denominator by 7: (54 * 7) / (35 * 7) = 378/245
    * To get 245 in the denominator of 23/49, multiply both numerator and denominator by 5: (23 * 5) / (49 * 5) = 115/245

3. **Subtract the fractions:**
    * (378/245) - (115/245) = (378 - 115) / 245 = 263/245

### Answer: 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'?

Gemini-1.5 Flash:

We know that 't' represents the sum of 58/30 and 49/37, which we calculated earlier to be 1808/555.

Therefore, 't' + 't' is the same as (1808/555) + (1808/555).

Adding fractions with the same denominator is straightforward: (1808 + 1808)/555 = 3616/555

### Answer: 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?

Gemini-1.5 Flash:

We already know the sum of 45/53 and 71/50 is 6013/2650.  

Let's simplify 8/100: 8/100 = 2/25

Now, we need to find the LCM of 2650 and 25. The LCM is 2650.

Let's convert the fractions:

* 6013/2650 * (1/1) = 6013/2650
* 2/25 * (106/106) = 212/2650

Now we can add:

6013/2650 + 212/2650 = 6225/2650

### Answer: 6225/2650 

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?

Gemini-1.5 Flash:

To add 45/53 and 71/72, we need a common denominator. The least common multiple of 53 and 72 is 3816.

* 45/53 * (72/72) = 3240/3816
* 71/72 * (53/53) = 3763/3816

Now we add the fractions:

3240/3816 + 3763/3816 = 7003/3816

### Answer: 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?

Gemini-1.5 Flash:

Reasoning: We already know that the sum of 36/72 and 49/1 is 99/2.  Now we need to subtract 48/10 from this sum.

First, let's simplify 48/10 by dividing both numerator and denominator by their greatest common factor, which is 2: 48/10 = 24/5.

To subtract fractions, they need to have the same denominator. The least common denominator for 2 and 5 is 10.

We can rewrite 99/2 as 99*5/2*5 = 495/10.

Now we can subtract: 495/10 - 24/5 = 495/10 - 48/10 = (495-48)/10 = 447/10.

### Answer: 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?

Gemini-1.5 Flash:

Reasoning: We can use the same approach as before, finding a common denominator and then adding the fractions.

The least common denominator for 72 and 1 is still 72.  

We rewrite 14/1 as 14*72/1*72 = 1008/72.

Now we can add the fractions: 36/72 + 1008/72 = (36+1008)/72 = 1044/72.

Finally, we can simplify the fraction by dividing both numerator and denominator by their greatest common factor, which is 36: 1044/72 = 29/2.

### Answer: 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?

Gemini-1.5 Flash:

We already know the sum of 61/20 and 29/15 is 299/60.  Now we need to add 58/31 to that. 

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

* Convert 299/60 to an equivalent fraction with a denominator of 1860: (299/60) * (31/31) = 9269/1860
* Convert 58/31 to an equivalent fraction with a denominator of 1860: (58/31) * (60/60) = 3480/1860

Now we can add the fractions: 9269/1860 + 3480/1860 = 12749/1860

This fraction can be simplified, but it's a bit tedious. You could leave it as an improper fraction, or use a calculator to get the decimal approximation.

### Answer: 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?

Gemini-1.5 Flash:

To add fractions, they need a common denominator.  The least common multiple of 20 and 67 is 1340.

* Convert 61/20 to an equivalent fraction with a denominator of 1340: (61/20) * (67/67) = 4087/1340
* Convert 29/67 to an equivalent fraction with a denominator of 1340: (29/67) * (20/20) = 580/1340

Now we can add the fractions: 4087/1340 + 580/1340 = 4667/1340

This fraction cannot be simplified further. 

### Answer: 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.

Gemini-1.5 Flash:

Reasoning: We already know that ((33 / 90) + (77 / 79)) is equal to 9537/7110.  Now we need to subtract 7/89 from that result. 

To subtract fractions, they must have a common denominator.  The least common denominator of 7110 and 89 is 7110 * 89 = 632790.

* (9537/7110) * (89/89) = 848333/632790
* (7/89) * (7110/7110) = 50770/632790

Now, we can subtract the fractions:
* (848333/632790) - (50770/632790) = (848333 - 50770)/632790

### Answer: 797563/632790

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.

Gemini-1.5 Flash:

Reasoning:  We follow the same process as before.  The least common denominator of 90 and 79 is 90 * 79 = 7110.

* (33/90) * (79/79) = 2607/7110
* (98/79) * (90/90) = 8820/7110

Now, we can add the fractions:
* (2607/7110) + (8820/7110) = (2607 + 8820)/7110

### Answer: 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?

Gemini-1.5 Flash:

We already found the sum of 30/79 and 29/6 to be 2471/474.  Now we need to subtract 44/75.

To subtract fractions, they need a common denominator. The LCM of 474 and 75 is 11850.

*  To get 11850 in the denominator of 2471/474, multiply both numerator and denominator by 25: (2471/474) * (25/25) = 61775/11850
*  To get 11850 in the denominator of 44/75, multiply both numerator and denominator by 158: (44/75) * (158/158) = 6952/11850

Now we subtract: 61775/11850 - 6952/11850 = (61775 - 6952) / 11850

Simplifying the numerator: 54823 / 11850

### Answer: 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?

Gemini-1.5 Flash:

We need to find the least common multiple (LCM) of 97 and 6, which is 582.

* To get 582 in the denominator of 30/97, we multiply both numerator and denominator by 6: (30/97) * (6/6) = 180/582
* To get 582 in the denominator of 29/6, we multiply both numerator and denominator by 97: (29/6) * (97/97) = 2813/582

Now we can add the fractions: 180/582 + 2813/582 = (180 + 2813) / 582

Simplifying the numerator: 2993/582

### Answer: 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?

Gemini-1.5 Flash:

We already know from the previous answer that subtracting 11/66 from 59/2 results in 968/33.

Now, we need to add 63/51 to 968/33.  To do this, we need a common denominator:

* The least common multiple of 51 and 33 is 561.

Let's rewrite both fractions:

* 63/51 = (63 * 11) / (51 * 11) = 693/561
* 968/33 = (968 * 17) / (33 * 17) = 16456/561

Now we can add them:

* (693/561) + (16456/561) = (693 + 16456)/561 = 17149/561

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

* 17149/561 = (17149/3) / (561/3) = 5716 1/3 / 187

### Answer: 5716 1/3 / 187 

Expected answer: 1559/51

Correct? No

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

Gemini-1.5 Flash:

We'll follow the same steps as before, but this time we need to find the least common multiple of 31 and 2, which is 62.

* 59/2 can be rewritten as (59 * 31) / (2 * 31) = 1829/62
* 11/31 can be rewritten as (11 * 2) / (31 * 2) = 22/62

Now we can subtract:

* (1829/62) - (22/62) = (1829 - 22) / 62 = 1807/62

### Answer: 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?

Gemini-1.5 Flash:

Let's analyze the change:

* We replaced 58/23 with 45/23, which means we decreased the first fraction by 58/23 - 45/23 = 13/23.

Since we are adding the fractions, this decrease in the first fraction will directly decrease the overall sum.  Therefore, the new sum (45/23 + 25/26) will be **smaller** than the original sum (58/23 + 25/26).

### Answer: The sum of 45/23 and 25/26 is smaller than the sum of 58/23 and 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?

Gemini-1.5 Flash:

We already know the sum of 70/13 and 17/52 is 297/52.  Now we need to subtract 35/87 from this.

To subtract fractions, they need a common denominator.  The least common multiple of 52 and 87 is 4524.

* 297/52 is equivalent to (297 * 87) / (52 * 87) = 25809 / 4524
* 35/87 is equivalent to (35 * 52) / (87 * 52) = 1820 / 4524

Now we can subtract:

* 25809/4524 - 1820/4524 = (25809 - 1820) / 4524 = 23989/4524

### Answer: 23989/4524 

Expected answer: 24019/4524

Correct? No

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?

Gemini-1.5 Flash:

We need to find a common denominator for 93 and 52. The least common multiple of 93 and 52 is 4836.

*  Rewrite 70/93 as (70 * 52) / (93 * 52) = 3640/4836.
*  Rewrite 17/52 as (17 * 93) / (52 * 93) = 1581/4836.

Now we can add the fractions:

*  3640/4836 + 1581/4836 = (3640 + 1581) / 4836 = 5221/4836

### Answer: 5221/4836 

Expected answer: 5221/4836

Correct? Yes

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

Gemini-1.5 Flash:

First, we need to find a common denominator for all three fractions. The least common multiple of 4 and 79 is 316. 

* To convert 43/4, we multiply the numerator and denominator by 79: (43/4) * (79/79) = 3407/316
* To convert 71/4, we multiply the numerator and denominator by 79: (71/4) * (79/79) = 5629/316
* To convert 92/79, we multiply the numerator and denominator by 4: (92/79) * (4/4) = 368/316

Now we can add the fractions:

3407/316 + 5629/316 + 368/316 = (3407 + 5629 + 368)/316 = 9404/316

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

9404/316 = (9404/4)/(316/4) = 2351/79

### Answer: 2351/79 

Expected answer: 4687/158

Correct? No

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

Gemini-1.5 Flash:

To add fractions, they need to have the same denominator.  The least common multiple of 4 and 89 is 356.

* **Convert 43/4:** Multiply the numerator and denominator by 89: (43/4) * (89/89) = 3827/356
* **Convert 71/89:** Multiply the numerator and denominator by 4: (71/89) * (4/4) = 284/356

Now we can add the fractions:

3827/356 + 284/356 = (3827 + 284) / 356 = 4111/356

### Answer: 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?

Gemini-1.5 Flash:

We need to find a common denominator for 1998 and 29. The least common multiple is 57942.

* Convert 4381/1998: (4381/1998) * (29/29) = 127069/57942
* Convert 23/29: (23/29) * (1998/1998) = 45954/57942

Now we can subtract: 127069/57942 - 45954/57942 = 81115/57942

### Answer: 81115/57942 

Expected answer: 81095/57942

Correct? No

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?

Gemini-1.5 Flash:

We start with 28/74 of the basket filled with oranges. 

We are adding 53/27 of the basket filled with apples. 

To find the total fraction of the basket filled, we need to add these two fractions:

28/74 + 53/27

To add these fractions, we need a common denominator:

* The least common multiple (LCM) of 74 and 27 is 1998.
* Convert 28/74: (28/74) * (27/27) = 756/1998
* Convert 53/27: (53/27) * (74/74) = 3922/1998

Now we can add:

756/1998 + 3922/1998 = 4678/1998

### Answer: 4678/1998 

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?

Gemini-1.5 Flash:

Reasoning:
We previously found that (76/76) + (96/16) = 7.
Adding 7 to itself gives us 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?

Gemini-1.5 Flash:

Reasoning:
* (76/76) still simplifies to 1.
* Now we need to calculate 96/33. This does not simplify to a whole number, but we can approximate it as 2.91 (rounded to two decimal places).
* Therefore, (76/76) + (96/33) is approximately 1 + 2.91 = 3.91.

### Answer: 3.91 

Expected answer: 43/11

Correct? No

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?

Gemini-1.5 Flash:

Let's recalculate with the new fraction:

1. **Simplify the fractions:**
   * 94/14 simplifies to 47/7.
   * 50/19 is already in its simplest form.

2. **Find a common denominator:**
   * The least common denominator for 7 and 19 is 133.
   * Convert 47/7 to 901/133 (multiply numerator and denominator by 19).
   * Convert 50/19 to 350/133 (multiply numerator and denominator by 7).

3. **Subtract the fractions:**
   * (901/133) - (350/133) = 551/133

### Answer: 551/133 

Expected answer: 543/133

Correct? No

Problem: After calculating the value of 39/48 + 19/84, what would be the result if you added the obtained result to itself?

Gemini-1.5 Flash:

We previously found that 39/48 + 19/84 = 1 13/336.  

Adding this result to itself is the same as multiplying it by 2:

(1 13/336) * 2 = (349/336) * 2 = 698/336

Simplifying this fraction, we get:

698/336 = 2 32/336 = 2 1/168

### Answer: 2 1/168 

Expected answer: 349/168

Correct? No

Problem: If instead of adding 39/48 and 19/84, you were to add 39/48 and 19/13, what would be the result?

Gemini-1.5 Flash:

Let's follow the same steps as before:

1. **Find the Least Common Multiple (LCM) of the denominators:**
   - The LCM of 48 and 13 is 624.

2. **Convert each fraction to have the LCM as the denominator:**
   - 39/48 = (39 * 13) / (48 * 13) = 507/624
   - 19/13 = (19 * 48) / (13 * 48) = 912/624

3. **Add the numerators, keeping the common denominator:**
   - 507/624 + 912/624 = (507 + 912) / 624 = 1419/624

4. **Simplify if possible:**
   - The fraction 1419/624 can be simplified to 2 167/624.

### Answer: 2 167/624 

Expected answer: 473/208

Correct? No

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?

Gemini-1.5 Flash:

Reasoning:  We already know that the sum of 38/8 and 4/16 is 5. Now we need to add 32/64 to this.

First, we can simplify 32/64 by dividing both numerator and denominator by 32: 32/64 = 1/2

Then, we add 1/2 to the previous sum of 5: 5 + 1/2 = 11/2

Therefore, the new total is 11/2.

### Answer: 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?

Gemini-1.5 Flash:

Reasoning: We need to find a common denominator for 12/8 and 4/16. The least common denominator is 16. 

To convert 12/8 to an equivalent fraction with denominator 16, we multiply both numerator and denominator by 2:
 12/8 * 2/2 = 24/16

Now we have:
 24/16 + 4/16

Adding the numerators, we get:
 (24 + 4)/16 = 28/16

Simplifying the fraction, we get:
 28/16 = 7/4 

### Answer: 7/4 

Expected answer: 7/4

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?

Gemini-1.5 Flash:

We know from the previous calculation that subtracting 19/44 from 33/5 results in 1357/220.

Now, we need to add 32/76 to this fraction.

1. **Simplify 32/76:** Both the numerator and denominator have a common factor of 8. Dividing both by 8, we get 4/19.

2. **Find the LCM of the denominators:** The LCM of 220 and 19 is 4180.

3. **Convert the fractions:**
   * (1357/220) * (19/19) = 25883/4180
   * (4/19) * (220/220) = 880/4180

4. **Add the fractions:**
   * 25883/4180 + 880/4180 = 26763/4180

### Answer: 26763/4180 

Expected answer: 27543/4180

Correct? No

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?

Gemini-1.5 Flash:

We already know that 33/5 is equivalent to 1452/220.  Let's find the equivalent fraction for 17/44:

* (17/44) * (5/5) = 85/220

Now, subtract the fractions:

* 1452/220 - 85/220 = 1367/220

### Answer: 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?

Gemini-1.5 Flash:

We already know the sum of 49/75 and 88/58 is 4721/2175.  Now we need to add 3/82 to this sum.

To do this, we need to find a common denominator for all three fractions. The LCM of 2175, 82, and the previous denominator of 4350 (used to add the first two fractions) is 178350.

Let's convert each fraction:

* (4721/2175) * (82/82) = 387002/178350
* (3/82) * (2175/2175) = 6525/178350

Now we can add all three fractions:

387002/178350 + 6525/178350 + 6600/4350 = 393527/178350

This fraction can be simplified by dividing both numerator and denominator by their greatest common factor, which is 1:

393527/178350 = 393527/178350

Therefore, the sum of 49/75, 88/58, and 3/82 is 393527/178350.

### Answer: 393527/178350

Expected answer: 393647/178350

Correct? No

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?

Gemini-1.5 Flash:

Following the same process as before, we need to find a common denominator for 12/75 and 88/58.  The LCM of 75 and 58 is still 4350.

Let's convert the fractions:

* (12/75) * (58/58) = 696/4350
* (88/58) * (75/75) = 6600/4350

Now we can add:

696/4350 + 6600/4350 = 7296/4350

Simplifying by dividing both numerator and denominator by their greatest common factor, 4:

7296/4350 = (7296/4) / (4350/4) = 1824/1087.5

Therefore, the sum of 12/75 and 88/58 is 1824/1087.5

### Answer: 1824/1087.5

Expected answer: 1216/725

Correct? Yes

Problem: First, calculate (36 / 37) + (8 / 4). Then, add the result to itself. What is the final result?

Gemini-1.5 Flash:

We already know that (36 / 37) + (8 / 4) = 110/37. 

Adding this result to itself: 

(110/37) + (110/37) = 220/37

### Answer: 220/37 

Expected answer: 220/37

Correct? Yes

Problem: Initially, you calculated (36 / 37) + (8 / 4) and added the result to itself. Now, suppose the initial calculation was actually (36 / 37) + (8 / 56) instead. What is the new result?

Gemini-1.5 Flash:

First, simplify the second fraction: 8/56 = 1/7.

Now we have: (36 / 37) + (1 / 7)

To add these, we need a common denominator. The common denominator is 37 * 7 = 259.

(36 * 7 / 259) + (1 * 37 / 259) = (252 / 259) + (37 / 259)

Now we can add the numerators: (252 + 37) / 259 = 289 / 259

### Answer: 289/259 

Expected answer: 289/259

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?

Gemini-1.5 Flash:

We already calculated that (57 / 22) - (72 / 48) = 12/11.

Now, adding this result to itself:

12/11 + 12/11 = (12 + 12) / 11 = 24/11

### Answer: 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?

Gemini-1.5 Flash:

Let's recalculate the expression with the corrected second fraction:

(57/22) - (72/100)

We can simplify 72/100 by dividing both numerator and denominator by their greatest common factor, 4:

(72/100) = (18/25)

Now we need to find the least common denominator of 22 and 25. The least common denominator is 550.  We convert each fraction to have a denominator of 550:

* (57/22) * (25/25) = 1425/550
* (18/25) * (22/22) = 396/550

Now we can subtract:

(1425/550) - (396/550) = (1425-396)/550 = 1029/550

We can simplify this fraction by dividing both numerator and denominator by their greatest common factor, 9:

(1029/9) / (550/9) = 114/61

### Answer: 114/61 

Expected answer: 1029/550

Correct? No

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?

Gemini-1.5 Flash:

Reasoning: We already know the sum of 48 divided by 16 and 12 divided by 29 is 99/29.
Adding this variable to itself is the same as multiplying it by 2: (99/29) * 2 = <<(99/29)*2=198/29>>198/29.
### Answer: 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?

Gemini-1.5 Flash:

Reasoning: 12 divided by 32 is 12/32.
The sum of 3 and 12/32 is 3+12/32 = <<3+12/32=(96+12)/32=108/32>>108/32.
### Answer: 108/32 

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?

Gemini-1.5 Flash:

We already know the sum of 37/20 and 16/75 is 619/300.  Now we need to add 61/80 to this.

First, we find a common denominator for 300 and 80.  The least common multiple is 1200.

* Convert 619/300 to an equivalent fraction with a denominator of 1200: (619/300) * (4/4) = 2476/1200

* Convert 61/80 to an equivalent fraction with a denominator of 1200: (61/80) * (15/15) = 915/1200

Now we can add all three fractions:

* 2476/1200 + 915/1200 = 3391/1200

### Answer: 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.

Gemini-1.5 Flash:

First, simplify the initial amount of cake: 60/20 = 3

Now, find a common denominator for 3 and 16/75.  The least common multiple of 1 and 75 is 75.

*  3 can be written as 3/1, so  (3/1) * (75/75) = 225/75

Now add the fractions:

* 225/75 + 16/75 = 241/75

### Answer: 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?

Gemini-1.5 Flash:

We already know that 83/40 + 6/8 = 113/40.  

Now we need to subtract 24/64.  To subtract fractions, they must have a common denominator. The least common multiple of 40 and 64 is 320.

* Convert 113/40 to a fraction with a denominator of 320: (113/40) * (8/8) = 904/320
* Convert 24/64 to a fraction with a denominator of 320: (24/64) * (5/5) = 120/320

Now we can subtract:

904/320 - 120/320 = (904 - 120) / 320 = 784/320

Simplifying by dividing both numerator and denominator by 8: 784/320 = 98/40

Simplifying further by dividing both numerator and denominator by 2: 98/40 = 49/20

### Answer: 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?

Gemini-1.5 Flash:

To add these fractions, we need a common denominator. The least common multiple of 18 and 8 is 72.

* Convert 83/18: (83/18) * (4/4) = 332/72
* Convert 6/8: (6/8) * (9/9) = 54/72

Now we can add the fractions:

332/72 + 54/72 = (332 + 54)/72 = 386/72

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

386/72 = (386/2) / (72/2) = 193/36

### Answer: 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?

Gemini-1.5 Flash:

We already know that the sum of 37/69 and 67/61 is 6880/4209.  

Now, to add 100/7 to this sum, we need to find a common denominator for all three fractions. The least common multiple of 4209, 7, and 1 is 4209. 

* We already have 6880/4209.
* We multiply the numerator and denominator of 100/7 by 601 to get (100 * 601) / (7 * 601) = 60100/4209.

Now we can add all three fractions: 6880/4209 + 60100/4209 = (6880 + 60100) / 4209 = 66980/4209.

### Answer: 66980/4209 

Expected answer: 469060/29463

Correct? No

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?

Gemini-1.5 Flash:

We found the sum of 51/78 and 44/83 to be 7665/6474.  

Now we need to add 71/92 to this. Again, we need a common denominator:

* The LCM of 6474 and 92 is 595728.
* 7665/6474 = (7665 * 92) / (6474 * 92) = 705380 / 595728
* 71/92 = (71 * 6474) / (92 * 6474) = 459694 / 595728

Now we can add:

* 705380 / 595728 + 459694 / 595728 = (705380 + 459694) / 595728 
* = 1165074 / 595728

This fraction can be simplified by dividing both numerator and denominator by their greatest common factor, which is 2:

* 1165074 / 595728 = 582537 / 297864

### Answer: 582537 / 297864 

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?

Gemini-1.5 Flash:

Let's recalculate the sum with 51/74 instead of 51/78.

First, we need to find the least common multiple (LCM) of 74 and 83:

* 74 = 2 x 37
* 83 is a prime number

The LCM of 74 and 83 is 74 x 83 = 6142.

Now, we'll convert each fraction:

* 51/74 = (51 x 83) / (74 x 83) = 4233 / 6142
* 44/83 = (44 x 74) / (83 x 74) = 3256 / 6142

Finally, we can add the fractions:

* 4233/6142 + 3256/6142 = (4233 + 3256) / 6142 = 7489 / 6142

### Answer: 7489/6142 

Expected answer: 7489/6142

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.

Gemini-1.5 Flash:

We already know that 97/56 - 31/52 = 827/728.

Now let's simplify 100/25:

100/25 = 4

To add 827/728 and 4, we need a common denominator:

4 = 4 * (728/728) = 2912/728

Now we can add the fractions:

827/728 + 2912/728 = (827 + 2912) / 728 = 3739/728

### Answer: 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.

Gemini-1.5 Flash:

We already know that 97/56 is equivalent to 1261/728. 

Let's simplify 26/52: 26/52 = 1/2

We need to find a common denominator for 1/2 and 1261/728.  The least common multiple of 2 and 728 is 728. 

* (1/2) * (364/364) = 364/728

Now we can subtract:

1261/728 - 364/728 = (1261 - 364) / 728 = 897/728

### Answer: 897/728 

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?

Gemini-1.5 Flash:

We know from the previous calculation that (78 / 74) + (72 / 51) = 4653 / 1887

To find the sum when we add this result to itself, we simply double the fraction:

(4653 / 1887) + (4653 / 1887) = (4653 + 4653) / 1887 = 9306 / 1887

### Answer: 9306 / 1887 

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)?

Gemini-1.5 Flash:

Let's follow the same steps as before, but this time with (56/51) instead of (72/51).

1. **Find the LCM:**
   * The LCM of 74 and 51 remains the same: 3774.

2. **Convert to equivalent fractions:**
   * (78 / 74) = (78 * 51 / 74 * 51) = 3978 / 3774
   * (56 / 51) = (56 * 74 / 51 * 74) = 4144 / 3774

3. **Add the numerators:**
   * (3978 / 3774) + (4144 / 3774) = (3978 + 4144) / 3774 = 8122 / 3774

4. **Simplify the fraction:**
   * The fraction 8122 / 3774 can be simplified by dividing both numerator and denominator by their greatest common factor, which is 2.
   * (8122 / 2) / (3774 / 2) = 4061 / 1887

### Answer: 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.

Gemini-1.5 Flash:

We already know that (33 / 23) - (12 / 75) = 733 / 575. 

Now, let's add 24 / 81:

1. **Simplify 24 / 81:**
   * Both 24 and 81 are divisible by 3, so we can simplify: 24/81 = 8/27

2. **Find a common denominator for 575 and 27:**
   * The least common denominator is 575 * 27 = 15525

3. **Rewrite the fractions with the common denominator:**
   * (733 / 575) * (27 / 27) = 19761 / 15525
   * (8 / 27) * (575 / 575) = 4600 / 15525

4. **Add the fractions:**
   * (19761 / 15525) + (4600 / 15525) = (19761 + 4600) / 15525

5. **Simplify the result:**
   * (19761 + 4600) / 15525 = 24361 / 15525

### Answer: 24361/15525 

Expected answer: 24391/15525

Correct? No

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.

Gemini-1.5 Flash:

We already know the sum of 57/66 and 39/64 is 3111/2112.  Now we need to add 10/29.

To add fractions, they need a common denominator.  The least common multiple of 2112 and 29 is 61248.

* We multiply the numerator and denominator of 3111/2112 by 29 to get (3111*29)/(2112*29) = 90229/61248
* We multiply the numerator and denominator of 10/29 by 2112 to get (10*2112)/(29*2112) = 21120/61248

Now we can add: 90229/61248 + 21120/61248 = (90229 + 21120)/61248 = 111349/61248

### Answer: 111349/61248 

Expected answer: 37113/20416

Correct? No

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?

Gemini-1.5 Flash:

You are absolutely right!  Let's recalculate the sum using the correct fraction of 49/64.

We need to find the least common multiple (LCM) of 66 and 64, which is 2112. 

*  To get a denominator of 2112 for 57/66, we multiply the numerator and denominator by 32: (57 * 32) / (66 * 32) = 1824/2112

*  To get a denominator of 2112 for 49/64, we multiply the numerator and denominator by 33: (49 * 33) / (64 * 33) = 1617/2112

Now we can add the fractions: 1824/2112 + 1617/2112 = (1824 + 1617)/2112 = 3441/2112 

### Answer: 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?

Gemini-1.5 Flash:

We already know that 80/13 + 2/48 = 1933/312.  

To find the sum when we add this result to itself, we simply multiply it by 2:

(1933/312) * 2 = 3866/312

### Answer: 3866/312 

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?

Gemini-1.5 Flash:

Let's break down the problem:

* **Simplify 2/48:**  2/48 simplifies to 1/24.
* **Find a common denominator:** The least common multiple of 5 and 24 is 120.
* **Convert 80/5:** (80/5) * (24/24) = 1920/120
* **Convert 1/24:** (1/24) * (5/5) = 5/120
* **Add the fractions:** 1920/120 + 5/120 = 1925/120

**Simplify the result:** 1925/120 can be simplified to 77/48.

### Answer: 77/48 

Expected answer: 385/24

Correct? No

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?

Gemini-1.5 Flash:

Let's pick up where we left off. We found that ((55 / 30) - (1 / 72)) = 131/72.

Now, we need to add 73/92 to this result.  To do that, we'll follow these steps:

1. **Find a common denominator for 131/72 and 73/92:** The least common denominator is 3456.

2. **Convert the fractions to have the common denominator:**
   - (131/72) * (48/48) = 6288/3456
   - (73/92) * (37/37) = 2701/3456

3. **Add the fractions:** (6288/3456) + (2701/3456) = 8989/3456

4. **Simplify the result (optional):** You can leave the answer as 8989/3456, or convert it to a mixed number (2 1081/3456) or a decimal (approximately 2.60).

### Answer: 8989/3456 

Expected answer: 4327/1656

Correct? No

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))?

Gemini-1.5 Flash:

Let's analyze how changing the numerator from 55 to 80 affects the result:

1. **Original calculation:** (55/30) - (1/72) = 131/72 

2. **New calculation:** (80/30) - (1/72)

3. **Simplifying the first fraction:** 80/30 simplifies to 8/3.

4. **Finding a common denominator:** The least common denominator for 8/3 and 1/72 is 72.

   -  8/3 * (24/24) = 192/72

5. **Subtracting the fractions:** (192/72) - (1/72) = 191/72

**Comparing the results:**

- The original result was 131/72.
- The new result is 191/72.

**Conclusion:**  Increasing the numerator of the first fraction from 55 to 80  **increases** the overall result of the operation. The difference between the new and original results is 60/72, which simplifies to 5/6.

### Answer: The result will increase by 5/6. 

Expected answer: 191/72

Correct? No

Problem: After finding the sum of the fractions 69/21 and 71/86, add the result to 86/62. What is the result?

Gemini-1.5 Flash:

We found that the sum of $\frac{69}{21}$ and $\frac{71}{86}$ is $\frac{7425}{1806}$.  Now we need to add $\frac{86}{62}$ to this.  First, we need to find a common denominator.

The prime factorization of 1806 is $2\times3\times7\times43$, and the prime factorization of 62 is $2\times31$.  The least common multiple is $2\times3\times7\times31\times43=54942$.  

We can rewrite our fractions with this denominator:

$\frac{7425}{1806}=\frac{7425\times31}{1806\times31}=\frac{230075}{54942}$

$\frac{86}{62}=\frac{86\times882}{62\times882}=\frac{75972}{54942}$

Now we add the fractions:

$\frac{230075}{54942}+\frac{75972}{54942}=\frac{230075+75972}{54942}=\frac{306047}{54942}$

### Answer: 306047/54942 

Expected answer: 102611/18662

Correct? No

Problem: Suppose instead that the fractions in the previous problem were 69/60 and 71/86. What is the sum of these fractions?

Gemini-1.5 Flash:

To add fractions, they must have the same denominator.  We find the least common multiple (LCM) of 60 and 86.

The prime factorization of 60 is $2^2\times3\times5$, and the prime factorization of 86 is $2\times43$.  Their LCM is $2^2\times3\times5\times43=2580$.

We can rewrite our fractions with the denominator 2580:

$\frac{69}{60}=\frac{69\times43}{60\times43}=\frac{2967}{2580}$

$\frac{71}{86}=\frac{71\times30}{86\times30}=\frac{2130}{2580}$

Now we add the fractions:

$\frac{2967}{2580}+\frac{2130}{2580}=\frac{2967+2130}{2580}=\frac{5097}{2580}$

### Answer: 5097/2580 

Expected answer: 1699/860

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.

Gemini-1.5 Flash:

We already know the sum of 24/60 and 42/45 is 4/3. Now we need to add 66/12 to it.

1. **Find a common denominator:** The least common denominator for 3 and 12 is 12.
   * To get 12 in the denominator of the first fraction, multiply both the numerator and denominator by 4: (4/3) * (4/4) = 16/12

2. **Add the fractions:** Now that they have the same denominator, we can add the numerators: 16/12 + 66/12 = 82/12

3. **Simplify the fraction:** Both the numerator and denominator are divisible by 2: (82/12) ÷ (2/2) = 41/6

### Answer: 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.

Gemini-1.5 Flash:

Here's how we can solve this:

1. **Find a common denominator:** The least common denominator for 60 and 67 is 4020.
   * To get 4020 in the denominator of the first fraction, multiply both the numerator and denominator by 67: (24/60) * (67/67) = 1608/4020
   * To get 4020 in the denominator of the second fraction, multiply both the numerator and denominator by 60: (42/67) * (60/60) = 2520/4020

2. **Add the fractions:**  Now that they have the same denominator, we can add the numerators: 1608/4020 + 2520/4020 = 4128/4020

3. **Simplify the fraction:** Both the numerator and denominator are divisible by 4: (4128/4020) ÷ (4/4) = 1032/1005

### Answer: 1032/1005 

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?

Gemini-1.5 Flash:

We already know the sum of 64/76 and 35/33 is 1193/627.  Now we need to add 47/38 to this.

To add fractions, they need the same denominator. The least common multiple (LCM) of 627 and 38 is 23826.

* We convert 1193/627 by multiplying the numerator and denominator by 38: 
   (1193/627) * (38/38) = 45334/23826

* We convert 47/38 by multiplying the numerator and denominator by 627: 
   (47/38) * (627/627) = 29469/23826

Now we can add the fractions: 
45334/23826 + 29469/23826 = 74803/23826

### Answer: 74803/23826 

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?

Gemini-1.5 Flash:

Let's follow the same steps as before:

1. **Find the LCM:** The LCM of 76 and 33 is 2508.
2. **Convert fractions:**
    * (43/76) * (33/33) = 1419/2508
    * (35/33) * (76/76) = 2660/2508
3. **Add the fractions:** 1419/2508 + 2660/2508 = 4079/2508

Therefore, the new sum of 43/76 of an orange and 35/33 of an apple is 4079/2508.

### Answer: 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?

Gemini-1.5 Flash:

We already know the sum of 47/16 and 96/10 is 1003/80.  Now we need to subtract 72/22.

First, let's find a common denominator for 80 and 22. The least common multiple is 880.

* Convert 1003/80 to an equivalent fraction with a denominator of 880: (1003/80) * (11/11) = 11033/880
* Convert 72/22 to an equivalent fraction with a denominator of 880: (72/22) * (40/40) = 2880/880

Now we can subtract:

11033/880 - 2880/880 = 8153/880

This fraction can be simplified to a mixed number:

8153/880 = 9 233/880 

### Answer: 9 233/880 

Expected answer: 8153/880

Correct? Yes

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?

Gemini-1.5 Flash:

Following the same process as before, we need to find a common denominator for 16 and 10. The least common multiple is 80.

* Convert 47/16 to an equivalent fraction with a denominator of 80: (47/16) * (5/5) = 235/80
* Convert 64/10 to an equivalent fraction with a denominator of 80: (64/10) * (8/8) = 512/80

Now we can add the fractions:

235/80 + 512/80 = 747/80

To express this as a mixed number:

747/80 = 9 27/80

### Answer: 9 27/80 

Expected answer: 747/80

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?

Gemini-1.5 Flash:

We already know the sum of 26/90 and 50/26 is 1294/585.  Now we need to subtract 92/89 from that.

To subtract fractions, they need a common denominator. The LCM of 585 and 89 is 52065.

* (1294/585) * (89/89) = 115066/52065
* (92/89) * (585/585) = 53820/52065

Now we can subtract:

* 115066/52065 - 53820/52065 = 61246/52065

This fraction can be simplified by dividing both numerator and denominator by their GCF, which is 2:

* 61246/52065 = (2 * 30623) / (2 * 26032.5) = 30623/26032.5 

### Answer: 30623/26032.5 

Expected answer: 61346/52065

Correct? No

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.

Gemini-1.5 Flash:

Let's first find the sum of 26/90 and 50/38. 

* The LCM of 90 and 38 is 1710.
* (26/90) * (19/19) = 494/1710
* (50/38) * (45/45) = 2250/1710
* 494/1710 + 2250/1710 = 2744/1710

Simplifying, we get: 2744/1710 = (2 * 1372) / (2 * 855) = 1372/855

Now, let's compare this result to the previous sum (1294/585):

* 1294/585 is greater than 1372/855 because the numerator and denominator are both smaller.

Therefore, adding 26/90 and 50/38 results in a **smaller** sum than adding 26/90 and 50/26.

### Answer: The result would be smaller. 

Expected answer: 1372/855

Correct? No

Problem: What would be the result if you added together the sum of 89/44 and 69/5 to itself?

Gemini-1.5 Flash:

We already know that the sum of 89/44 and 69/5 is 3481/220. 

Adding this sum to itself is the same as multiplying it by 2:

(3481/220) * 2 = 6962/220

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

(6962/2) / (220/2) = 3481/110

### Answer: 3481/110 

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?

Gemini-1.5 Flash:

To add 89/44 and 69/38, we need a common denominator.  The LCM of 44 and 38 is 836.

* **Convert 89/44:** Multiply both numerator and denominator by 19: (89 * 19) / (44 * 19) = 1691/836
* **Convert 69/38:** Multiply both numerator and denominator by 22: (69 * 22) / (38 * 22) = 1518/836

Now we can add the fractions:

1691/836 + 1518/836 = (1691 + 1518) / 836 = 3209/836

### Answer: 3209/836 

Expected answer: 3209/836

Correct? Yes

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.

Gemini-1.5 Flash:

We already know that 54/1 + 62/30 = 841/15.

Now we need to subtract 53/41 from this result.

* **Find a common denominator:**
    * The least common multiple of 15 and 41 is 615.
    * (841/15) * (41/41) = 34521/615
    * (53/41) * (15/15) = 795/615

* **Subtract the fractions:**
    * 34521/615 - 795/615 = 33726/615

### Answer: 33726/615 

Expected answer: 33686/615

Correct? No

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.

Gemini-1.5 Flash:

Here's how to solve the problem:

* **Simplify the fraction:** 62/30 simplifies to 31/15 (divide both numerator and denominator by 2).

* **Add the whole number and the fraction:**
   * 93 + 31/15

* **Find a common denominator:**
   * 93 can be written as (93 * 15)/15 = 1395/15
   * Now we have 1395/15 + 31/15 = 1426/15

### Answer: 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?

Gemini-1.5 Flash:

We already know that 85/29 + 47/7 = 1958/203.  

Now we need to add 81/42 to that result.  First, we need a common denominator:

* **Find the least common multiple of 203 and 42:** The least common multiple is 2682.
* **Convert 1958/203:** Multiply numerator and denominator by 13: (1958 * 13) / (203 * 13) = 25454/2682
* **Convert 81/42:** Multiply numerator and denominator by 64: (81 * 64) / (42 * 64) = 5184/2682

Now we can add:

25454/2682 + 5184/2682 = (25454 + 5184) / 2682 = 30638/2682

### Answer: 30638/2682 

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?

Gemini-1.5 Flash:

To find the total weight, we need to add the weight of the oranges and the weight of the apples. 

* **Oranges:** 85/64 pounds
* **Apples:** 47/7 pounds

To add these fractions, we need a common denominator. The least common multiple of 64 and 7 is 448.

* **Convert 85/64:** Multiply the numerator and denominator by 7: (85 * 7) / (64 * 7) = 595/448
* **Convert 47/7:** Multiply the numerator and denominator by 64: (47 * 64) / (7 * 64) = 3008/448

Now we can add the fractions:

595/448 + 3008/448 = (595 + 3008) / 448 = 3603/448

### Answer: 3603/448 pounds 

Expected answer: 3603/448

Correct? Yes