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

Gemini-1.5 Pro:

Reasoning: We found that (29/2)*(19/28) = 551/56. So, (551/56)*(8/28) = (551*8)/(56*28) = 4408/1568 = 551/196.
### Answer: 551/196 

Expected answer: 551/196

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: (29/2)*(19/14) = (29*19)/(2*14) = 551/28.
### Answer: 551/28 

Expected answer: 551/28

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: 
* First, we divide 26 by 12: 26/12 simplifies to 13/6. 
* Then, we multiply 7 by 13/6: 7 * (13/6) = 91/6.

### Answer: 91/6 

Expected answer: 91/6

Correct? Yes

Problem: Following on from the last question: If Susan made 23/4 batches of her recipe, how would the amount of water she used change? Calculate this using fraction multiplication: ((23 / 4) * ((30 / 30) * (29 / 27))).

Gemini-1.5 Pro:

Reasoning: We can calculate the amount of water Susan uses by multiplying the amount she uses for one batch by the number of batches. 
From the previous question, we know that Susan uses 29/27 parts of water for one batch. 
Therefore, for 23/4 batches, she would use (23/4) * (29/27) = 667/108 parts of water.
### Answer: 667/108 

Expected answer: 667/108

Correct? Yes

Problem: Following on from the last question: What happens if Susan decides to use 30 parts of flour out of an available 12, while still keeping the water proportion the same? Calculate this using fraction multiplication: ((30 / 12) * (29 / 27)).

Gemini-1.5 Pro:

Reasoning: This calculation doesn't directly relate to the water proportion. This calculates what proportion of the flour Susan will use, given that she has 12 parts available and wants to use 30.
(30/12) * (29/27) = (5/2) * (29/27) = 145/54
### Answer: 145/54 

Expected answer: 145/54

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: In the previous problem, we found that (24/22)*(24/17) = 288/187.  Multiplying this result by 27/18, we obtain:
(288/187)*(27/18) = (16/187)*(27/1) = (16*27)/(187*1) = 432/187.
### Answer: 432/187

Expected answer: 432/187

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: We have the following:
 (24/22)*(24/9) = (12/11)*(24/9) = (12/11)*(8/3) = (4/11)*(8/1) = 32/11.
 
 ### Answer: 32/11

Expected answer: 32/11

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: We know from the previous problem that (25/4)/(16/6) = 75/32.  To multiply this result by 19/6, we perform the following calculation: (75/32)*(19/6) = (75*19)/(32*6). Before multiplying, we can simplify by canceling common factors: 75 shares a common factor of 3 with 6, and 32 shares a common factor of 2 with 6.  Simplifying gives us (25*19)/(16*2) = 475/32.  
### Answer: 475/32 

Expected answer: 475/64

Correct? No

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

Gemini-1.5 Pro:

Reasoning: Dividing by a fraction is the same as multiplying by the reciprocal of that fraction. The reciprocal of 16/6 is 6/16. Therefore, 9 divided by 16/6 is the same as 9 multiplied by 6/16. 

9 / (16/6) = 9 * (6/16) = (9*6)/16 = 54/16 = 27/8

### Answer: 27/8

Expected answer: 27/8

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning:  The previous result was 13/7. Multiplying that by 23/2 gives us:
(13/7) * (23/2) = (13 * 23) / (7 * 2) = 299/14
This fraction cannot be simplified further. 
### Answer: 299/14 

Expected answer: 299/14

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: We multiply the numerators and the denominators: (7 * 26) / (7 * 6) = 182/42
Simplifying this fraction by dividing both numerator and denominator by their greatest common factor, 14, gives us 13/3.
### Answer: 13/3 

Expected answer: 13/3

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: The answer from the previous problem is 95/18. Multiplying this by itself, we get (95/18) * (95/18) = 9025/324. This fraction cannot be simplified further as the numerator and denominator do not have any common divisors other than 1.
### Answer: 9025/324 

Expected answer: 9025/324

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: Multiplying the numerators and the denominators of the fractions 10/12 and 20/6, we get (10 * 20) / (12 * 6) = 200/72. 
Simplifying the fraction by dividing the numerator and denominator by their greatest common divisor, which is 4, we get 50/18.
Simplifying further by again dividing the numerator and denominator by 2, we obtain 25/9. 
### Answer: 25/9 
 

Expected answer: 25/9

Correct? Yes

Problem: After calculating the total fractional representation of the cargo hauled in one day, suppose the ship needs to set aside 5/24 of this total fraction for an emergency supply run later in the week. What fraction of the overall cargo does this emergency supply run represent? To find this, calculate ((5 / 24) * your previous answer). Write your answer as a simplified fraction.

Gemini-1.5 Pro:

Reasoning: We need to calculate (5/24) * (22/3).  Multiplying the numerators, we get 5*22 = 110. Multiplying the denominators, we get 24*3 = 72. So the resulting fraction is 110/72. This can be simplified by dividing both numerator and denominator by 2, resulting in 55/36.
### Answer: 55/36 

Expected answer: 55/36

Correct? Yes

Problem: After recalculating the cargo hauled in one day with the new total of 29/21 representing the consumer goods, how does this change the overall cargo hauled in one day? If the cargo in the morning now represents 29/21 of its capacity, and the cargo in the afternoon still represents 22/4 of its capacity for industrial supplies, what is the new total fractional representation of these two types of cargo hauled in one day? Calculate ((29 / 21) * (22 / 4)). Write your answer as a simplified fraction.

Gemini-1.5 Pro:

Reasoning: We need to simplify the expression ((29 / 21) * (22 / 4)).  First, we can simplify the second fraction: 22/4 simplifies to 11/2 (because both 22 and 4 are divisible by 2).
Now our expression is (29/21) * (11/2). Multiplying the numerators gives us 29*11 = 319, and multiplying the denominators gives us 21*2 = 42.  So the resulting fraction is 319/42. 
### Answer: 319/42 

Expected answer: 319/42

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning:
* We previously calculated (16 / 4) / (24 / 21) = 7/2
* Now we multiply that result by (5/13):  (7/2) * (5/13)
* Multiplying the numerators and the denominators, we get: (7 * 5) / (2 * 13) = 35/26

### Answer: 35/26 

Expected answer: 35/26

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning:
* First, we simplify the division in the parentheses: (16 / 4) = 4.
* Our expression now becomes: 4 / 3.

### Answer: 4/3 

Expected answer: 4/3

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning:
We found that (5/27)*(15/2) = 25/18.
Now, we need to square that result: (25/18)*(25/18) = (25*25)/(18*18).
There are no common factors to simplify, so we calculate the result directly: (25*25)/(18*18) = 625/324.

### Answer: 625/324

Expected answer: 625/324

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: 
We calculate the new product: (24/27) * (15/2) = (24*15) / (27*2).
We can simplify this by canceling common factors: (24*15)/(27*2) = (8*3*3*5)/(3*9*2) = (8*5)/(3*2) = (4*2*5)/(3*2) = (4*5)/3.
Calculating the final result: (4*5)/3 = 20/3.

### Answer: 20/3 

Expected answer: 20/3

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: We first calculate the product (15 / 13) * (19 / 7) = 285/91 as we did previously. 
Then, we multiply this result by (23/22): (285/91) * (23/22) = (285 * 23) / (91 * 22) = 6555/2002.
This fraction can be simplified by dividing numerator and denominator by 13: 6555/2002 = (6555/13) / (2002/13) = 504/154.
### Answer: 504/154 

Expected answer: 6555/2002

Correct? No

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

Gemini-1.5 Pro:

Reasoning:  We multiply the numerators and the denominators of the fractions:
(15/29) * (19/7) = (15 * 19) / (29 * 7) = 285/203.
### Answer: 285/203

Expected answer: 285/203

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning:
From the previous calculation, we know (12 / 25) * (20 / 4) simplifies to 12/5.
Now we have (12/5) * (17/17).
Any number divided by itself is 1, so (17/17) = 1.
Therefore, our calculation is (12/5) * 1, which is simply 12/5.

### Answer: 12/5

Expected answer: 12/5

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning:
Let's break down the calculation step-by-step:

1. Simplify within the multiplication: (10/25) can be simplified to 2/5, and (20/4) simplifies to 5/1.
2. This gives us: (2/5) * (5/1)
3. Multiply the numerators and the denominators: (2 * 5) / (5 * 1) = 10/5
4. Simplify the fraction: 10/5 = 2

Comparing this to the previous result of 12/5, we see that changing the '12' to a '10' in the original problem results in a simpler answer and a whole number. 

### Answer: 2 

Expected answer: 2

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: We know from the previous problem that (14/10)*(30/5) = 42/5. So now we need to calculate (42/5)*(13/15).
(42/5)*(13/15) = (42*13)/(5*15). 
We can simplify this by dividing both the numerator and denominator by 3: (14*13)/(5*5) = 182/25.
### Answer: 182/25

Expected answer: 182/25

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: (30/10)*(30/5) = (30*30)/(10*5).
We can simplify this by dividing both the numerator and denominator by 10: (3*30)/(1*5) = 90/5.
Furthermore, we can simplify this by dividing both the numerator and denominator by 5: (90/5)/(5/5) = 18/1 = 18.
### Answer: 18 

Expected answer: 18

Correct? Yes

Problem: After completing the cookie orders for the event, the baker considers a change in his recipe that will impact how much sugar he uses. Now, instead of 26 cups of sugar per batch, he only plans to use 16 cups of sugar in the recipe. If he still only has enough sugar on hand for 29/14 of a full batch, how many batches can he now prepare with the amount of sugar he has on hand? Calculate this as (16 / (29 / 14)). Write your answer as a simplified fraction.

Gemini-1.5 Pro:

Reasoning: Dividing by a fraction is the same as multiplying by the inverse of that fraction. Therefore (16 / (29 / 14)) is equivalent to (16/1)*(14/29).
(16/1)*(14/29) = 224/29
224/29 simplified is 7 21/29
### Answer: 7 21/29 

Expected answer: 224/29

Correct? Yes

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

Gemini-1.5 Pro:

You did not provide a previous problem where a value for "s" was solved. Please provide the previous problem or the value of "s" so I can calculate "o". 

Expected answer: 2187/520

Correct? No

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

Gemini-1.5 Pro:

Reasoning:
We multiply the numerators: 27 * 9 = 243
We multiply the denominators: 10 * 11 = 110
This results in the fraction 243/110.

### Answer: 243/110

Expected answer: 243/110

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: We need to multiply 34/9 by 24/30.  Before multiplying, we can simplify by canceling common factors: (34/9)*(24/30) = (34*8)/(3*15) = 272/45.
### Answer: 272/45 

Expected answer: 136/45

Correct? No

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

Gemini-1.5 Pro:

Reasoning:  To multiply fractions, multiply the numerators and the denominators. (25/9)*(17/13) = (25*17)/(9*13). This results in the fraction 425/117. 
### Answer: 425/117 

Expected answer: 425/117

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: We found that (28/16)/(27/24) simplified to 14/9. 

Now we need to multiply this result by itself: (14/9) * (14/9)

Multiply the numerators and the denominators: (14 * 14) / (9 * 9) = 196/81

### Answer: 196/81

Expected answer: 196/81

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning:
We already know from the previous calculation that ((15 / 3) * (19 / 26)) = 95/26
Now we need to multiply that result by 14/24:  (95/26)*(14/24) 
This can be written as (95*14)/(26*24)
Before calculating, we can simplify by dividing both 14 and 24 by their common factor of 2, yielding: (95*7)/(26*12)
Now, multiplying the numerators and denominators gives us: 665/312
### Answer: 665/312 

Expected answer: 665/312

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: 
We already know (15 / 3) = 5.
Multiplying: 5 * (27 / 26) = (5 * 27) / 26 = 135/26 
### Answer: 135/26 

Expected answer: 135/26

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: We need to find (30/4) of (19/14), which means multiplying: (30/4)*(19/14) = (15*19)/(2*14) = 285/28.
### Answer: 285/28 

Expected answer: 285/28

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: The result of ((5 / 2) / (26 / 15)) was 75/52.  Multiplying this fraction by itself gives us:

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

### Answer: 5625/2704

Expected answer: 5625/2704

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: We need to calculate 14 / (26 / 15).  Again, dividing by a fraction is the same as multiplying by its reciprocal:

14 / (26 / 15) = 14 * (15 / 26) = (14 * 15) / 26 = 210 / 26

This simplifies to 105/13.

### Answer: 105/13 

Expected answer: 105/13

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: From the previous problem, we know that ((15 / 13) * (24 / 24)) simplifies to 15/13.  We then multiply this result by 22/7: (15/13)*(22/7) = (15*22)/(13*7) = 330/91
### Answer: 330/91 

Expected answer: 330/91

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning:  We begin with the fraction 15/2 and multiply by 24/24.  Any number multiplied by 1 is itself. Therefore, multiplying 15/2 by 1 results in 15/2.
### Answer: 15/2

Expected answer: 15/2

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning:
We need to multiply the fraction 15/14 (the answer from the previous problem) by 15/15. 

This simplifies easily because 15/15 equals 1, and any number multiplied by 1 remains the same.

Therefore: (15/14) * (15/15) = (15/14) * 1 = 15/14

### Answer: 15/14 

Expected answer: 15/14

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning:
Let's simplify each fraction before multiplying:
(9/7) remains as it is.
(25/21) can be simplified as 5*5 / 3*7

Now our multiplication is: (9/7) * (5*5 / 3*7) 
We can simplify by cancelling the common factor of 7: (9 / 7) * (5 * 5 / 3 * 7) = (9/1) * (5*5 / 3*7) = (9/1) * (25/21)
Now, multiply the numerators and the denominators: (9 * 25) / (1 * 21) = 225 / 21

Finally, simplify the resulting fraction by dividing both numerator and denominator by their greatest common divisor, which is 3:  (225/3) / (21/3) = 75/7

### Answer: 75/7 

Expected answer: 75/49

Correct? No

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

Gemini-1.5 Pro:

Reasoning: From the previous question, we know that ((9 / 8) / (20 / 18)) simplifies to 81/80.  Therefore, we need to calculate:

(29 / 21) * (81 / 80)

Multiplying the numerators and denominators:
(29 * 81) / (21 * 80) = 2349 / 1680

This fraction can be simplified by dividing both numerator and denominator by their greatest common divisor, which is 3:
(2349 / 3) / (1680 / 3) = 783 / 560

### Answer: 783/560 

Expected answer: 783/560

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning:  We can think of 9 as 9/1. So the problem becomes: (9/1) / (20/18)

Dividing by a fraction is the same as multiplying by its inverse: (9/1) * (18/20)

Multiplying the numerators and denominators: (9 * 18) / (1 * 20) = 162 / 20

Simplifying the fraction by dividing by the greatest common divisor (2): (162 / 2) / (20 / 2) = 81/10

### Answer: 81/10 

Expected answer: 81/10

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: 
The previous answer was 45/13. We need to multiply this fraction by 14/4:

(45/13) * (14/4) 

We can simplify this multiplication by canceling out common factors:

(45/13) * (7/2)  = 315 / 26

### Answer: 315/26

Expected answer: 315/26

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: In the previous problem, we found that (30 / 18) * (18 / 6) = 5.  Multiplying this result by 8/20 gives us 5 * (8/20) = 40/20 = 2.
### Answer: 2 

Expected answer: 2

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: Let's calculate the new value: 30 divided by 4 is 30/4 = 7.5.  Multiplying 7.5 by 18/6 (which simplifies to 3) gives us 7.5 * 3 = 22.5. 
The original calculation resulted in 5, and our new calculation is 22.5.  Therefore, the result of the new calculation is larger than the original result. 
### Answer: The new result is larger than the original result. The new value is 22.5. 

Expected answer: 45/2

Correct? Yes

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

Gemini-1.5 Pro:

From the previous problem, we know that (20/2)*(2/18) simplifies to 10/9.
Therefore, we need to calculate (27/4)*(10/9).
Before multiplying, we can simplify: 27 and 9 share a common factor of 9, and 10 and 4 share a common factor of 2. Simplifying gives us (3/2)*(5/1).
Multiplying the numerators and denominators gives us 15/2. 
### Answer: 15/2 

Expected answer: 15/2

Correct? Yes

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

Gemini-1.5 Pro:

First, we simplify inside the parentheses: 20/2 = 10. 
Our problem becomes: (10) * (15/18).
We can simplify 15/18 by dividing both numerator and denominator by 3, resulting in 5/6.
The problem is now: (10) * (5/6) = 50/6
We can simplify 50/6 by dividing both numerator and denominator by 2, resulting in 25/3.
### Answer: 25/3 

Expected answer: 25/3

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: We already know from the previous problem that ((11 / 9) * (29 / 6)) = 319/54.
Now, we multiply that result by (11/5): (319/54) * (11/5) = (319 * 11) / (54 * 5) = 3509 / 270.
The fraction 3509/270 can't be simplified further.
### Answer: 3509/270 

Expected answer: 3509/270

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning:  Let's calculate the new result: (21 / 9) * (29 / 6).
First, we can simplify within the multiplication: (7/3) * (29/6).
Then, we multiply the numerators and the denominators: (7 * 29) / (3 * 6) = 203 / 18.
The fraction 203/18 cannot be simplified further.
### Answer: 203/18

Expected answer: 203/18

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning:
We start with the result from the previous calculation, 480/133. 
We need to multiply this fraction by 22/7.
To multiply fractions, we multiply the numerators and the denominators.
Finally, we simplify the resulting fraction, if possible, by finding the greatest common divisor of the numerator and denominator. 

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

The greatest common divisor of 10560 and 931 is 1. Therefore, the fraction cannot be simplified further.

### Answer: 10560/931

Expected answer: 10560/931

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: 
We follow a similar process as before. First, simplify the fraction 21/7, as it divides cleanly. Then, multiply the simplified fraction with 16/19 by multiplying the numerators and the denominators. Finally, check if the resulting fraction can be simplified.

Calculations:
* 21/7 = 3
* 3 * (16/19) = (3 * 16) / 19
* = 48/19

The GCD of 48 and 19 is 1, so the fraction cannot be simplified further.

### Answer: 48/19 

Expected answer: 48/19

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: From the previous calculation, the product of 25/22 and 20/3 is 250/33. Multiplying this result by 28/29 gives us (250/33)*(28/29) = (250*28)/(33*29) = 7000/957.
### Answer: 7000/957

Expected answer: 7000/957

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: (25/15)*(20/3) = (25*20)/(15*3) = 500/45 = 100/9
### Answer: 100/9 

Expected answer: 100/9

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: We previously determined that (29/2)/(23/19) = 551/46.  Multiplying this result by (29/18) gives us:
(551/46) * (29/18) = (551*29)/(46*18) = 15979/828
This fraction cannot be simplified any further.
### Answer: 15979/828 

Expected answer: 15979/828

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: The previous answer is 45/13, so we need to calculate (45/13) * (8/14). Before multiplying, we can simplify by canceling common factors. Notice that 45 and 15 share a common factor of 5, and 8 and 14 share a common factor of 2: (̶4̶5̶9 / 13) * (̶8̶4 / ̶1̶4̶7) = (9/13)*(4/7).  Now we can multiply the numerators and denominators: (9 * 4) / (13 * 7) = 36/91.
### Answer: 36/91

Expected answer: 180/91

Correct? No

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

Gemini-1.5 Pro:

Reasoning:  We can simplify this expression by canceling common factors in the numerator and denominator before multiplying. Notice that 27 in the numerator and 3 in the denominator share a common factor of 3, and 9 in the denominator and 9 in the numerator share a common factor of 9: (̶5̶5/̶3̶1) * (̶2̶7̶3/̶9̶1) = (5/1)*(3/1). Now we can multiply the numerators and the denominators: (5 * 3) / (1 * 1) = 15/1.
### Answer: 15 

Expected answer: 5

Correct? No

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

Gemini-1.5 Pro:

Reasoning: From the previous problem, we know that (29/29) * (1/1) = 1.  Anything divided by itself is 1, so (27/27) also equals 1. Multiplying our previous result by 1 gives us 1 * 1 = 1. 
### Answer: 1 

Expected answer: 1

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning:  Anything divided by 1 is itself. So, 1/1 = 1.  Our problem simplifies to (29/6) * 1. Anything multiplied by 1 is itself.  Therefore, our final answer is 29/6. We could also calculate this out to 4.83.  
### Answer: 29/6 or 4.83

Expected answer: 29/6

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: We already know (11/19)*(22/7) = 242/133. Now we need to calculate (242/133)*(25/4).
This equals (242*25)/(133*4) = 6050/532. 
Both the numerator and denominator share a common factor of 2, so the simplified fraction is 3025/266.
### Answer: 3025/266

Expected answer: 3025/266

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: Following the same logic as before, (11/19) * (25/7) = (11*25) / (19*7).
This simplifies to 275/133.
### Answer: 275/133 

Expected answer: 275/133

Correct? Yes

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

Gemini-1.5 Pro:

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

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

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

(133*16)/(50*12) = (133*4)/(25*3) = 532/75

### Answer: 532/75 

Expected answer: 266/75

Correct? No

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

Gemini-1.5 Pro:

Reasoning: To multiply fractions, we multiply the numerators and the denominators.
(9/30)*(19/5) = (9*19)/(30*5)
We can simplify this by dividing the numerator and denominator by their greatest common factor, which is 3. 
(9*19)/(30*5) = (3*19)/(10*5) = 57/50
### Answer: 57/50 

Expected answer: 57/50

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: We need to calculate (253/120) * (253/120).
Multiplying the numerators and denominators, we get (253 * 253) / (120 * 120) = 64009 / 14400.
This fraction cannot be simplified further. 
### Answer: 64009/14400 

Expected answer: 64009/14400

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: Multiplying the numerators and the denominators, we get (11 * 26) / (15 * 8) = 286 / 120.
This fraction can be simplified by dividing the numerator and denominator by their greatest common divisor, which is 2: 286/2 / 120/2 = 143/60. 
### Answer: 143/60

Expected answer: 143/60

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: The result of a fraction multiplied by itself is the same as squaring the fraction.  Therefore, we need to calculate (203/144)^2. This is equivalent to (203 * 203) / (144 * 144). Since 203 is a prime number and shares no common factors with 144, the result is 41209/20736. 
### Answer: 41209/20736

Expected answer: 41209/20736

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning:  The new operation is (29/5) * (21/27).  Again, we multiply the numerators and the denominators: (29 * 21) / (5 * 27). We can simplify this by canceling the common factor of 3 between 21 and 27: (29 * 7) / (5 * 9) = 203/45.
### Answer: 203/45 

Expected answer: 203/45

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: 
From the previous problem, we know (25/25)*(20/7) = 20/7
Multiplying this result by itself, we get (20/7)*(20/7) = 400/49
### Answer: 400/49

Expected answer: 400/49

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning:
First, we simplify (25/25) which equals 1.
Then, we are left with 1 * (11/7) = 11/7
### Answer: 11/7

Expected answer: 11/7

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: Instead of 5/1, our initial fraction is now 30/1. We multiply this by 16/27: (30/1) * (16/27) = (30 * 16) / (1 * 27).
This simplifies to 480/27. Both 480 and 27 are divisible by 3, so we can simplify further: (480/3) / (27/3) = 160/9.
### Answer: 160/9

Expected answer: 160/9

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: Multiplying a fraction by a fraction equivalent to 1 (like 18/18) doesn't change its value. This is because any number multiplied by 1 remains the same. Therefore, the result of (435/32) * (18/18) is still (435/32).
### Answer: 435/32 

Expected answer: 435/32

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: We need to calculate (15/16) * (13/2). To multiply fractions, we multiply the numerators and the denominators: (15 * 13) / (16 * 2). This simplifies to 195/32. This fraction cannot be simplified further.
### Answer: 195/32 

Expected answer: 195/32

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning:
Let's break down the calculation step-by-step. 

First, solve the division within the parentheses: 6 / 2 = 3. 
Our expression now becomes: 3 / 3.
Finally, 3 divided by 3 equals 1.

### Answer: 1 

Expected answer: 1

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: We previously found that (7/1) / (15/9) = 21/5. Now we multiply that result by 10/5.
(21/5) * (10/5) = 210/25 = 42/5
### Answer: 42/5 

Expected answer: 42/5

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: Dividing by a fraction is the same as multiplying by its inverse. Therefore, dividing 20/1 by 15/9 is the same as multiplying 20/1 by 9/15.

(20/1) / (15/9) = (20/1) * (9/15) = 180/15 = 12/1 = 12

### Answer: 12

Expected answer: 12

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: We previously calculated (23/10)*(9/1) = 207/10. Now we must calculate (207/10)*(1/16).
(207/10)*(1/16) =  207/160.
### Answer: 207/160 

Expected answer: 207/160

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: (23/10) * (9/6) is equivalent to (23 * 9) / (10 * 6).
(23*9)/(10*6) = 207/60. This fraction can be simplified by dividing both numerator and denominator by their greatest common factor, 3.
This simplifies to 69/20.
### Answer: 69/20 

Expected answer: 69/20

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning:
From the previous calculation, we know (18/4)*(10/11) simplifies to 45/11.
Now we multiply this result by 6/4:  (45/11)*(6/4) 
Before multiplying, we can simplify by dividing 6 and 4 by their common factor of 2: (45/11)*(3/2)
Now, multiplying numerators and denominators: (45*3)/(11*2) = 135/22

### Answer: 135/22 

Expected answer: 135/22

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning:
Following the order of operations, we first simplify the division: 18/1 = 18.  The expression becomes (18) * (10/11).
Multiplying the numerators and denominators, we have: (18*10)/(1*11) = 180/11
This fraction cannot be simplified further.

### Answer: 180/11 

Expected answer: 180/11

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: We need to multiply the previous result, 8, by the fraction 30/21. This can be written as (8/1) * (30/21).
Multiplying the numerators and denominators gives us (8 * 30) / (1 * 21) = 240/21.
This fraction can be simplified by dividing both numerator and denominator by their greatest common factor, 3, resulting in 80/7.

### Answer: 80/7

Expected answer: 80/7

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: Our new problem is (24/23)*(18/3). We can simplify 18/3 to 6/1.  Multiplying the numerators and the denominators gives us (24*6)/(23*1), or 144/23. 

### Answer: 144/23

Expected answer: 144/23

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: We found that (20/13)/(17/16) is 320/221. Now we need to multiply that result by 20/26.
(320/221)*(20/26) = (320*20)/(221*26) = 6400/5746
This simplifies to 3200/2873
### Answer: 3200/2873

Expected answer: 3200/2873

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: 
1. We know from the previous problem that c = 4. 
2. Substitute c with 4 in the formula: ((21 divided by 9) multiplied by 4).
3. Calculate the division within the parentheses: 21 divided by 9, which simplifies to the fraction 7/3.
4. Now we have: (7/3) multiplied by 4.
5. Multiply the fraction by 4: (7/3) * 4 = 28/3.

### Answer: 28/3 

Expected answer: 28/3

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning:
1. Our problem is now: (5/15) * (4/1). 
2. We multiply the numerators: 5 * 4 = 20
3. We multiply the denominators: 15 * 1 = 15
4. This gives us the fraction: 20/15
5. Simplifying by dividing numerator and denominator by their greatest common factor, 5, gives us: 4/3

### Answer: 4/3 

Expected answer: 4/3

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning:
* We already know from the previous calculation that ((30 / 23) * (19 / 24)) simplifies to 95/92. 
* Now we multiply this result by (28/4):  (95/92) * (28/4)
* Before multiplying, we can simplify by dividing 28 and 4 by their greatest common divisor, which is 4: (95/92) * (28/4) = (95/92) * (7/1)
* Now we multiply the numerators and denominators: (95 * 7) / (92 * 1) = 665/92
### Answer: 665/92 

Expected answer: 665/92

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning:
* We need to calculate (30/11) * (19/24).
* Performing the multiplications: (30 * 19) / (11 * 24) = 570 / 264
* Simplifying the fraction by dividing both numerator and denominator by their greatest common divisor, which is 6: (570/6) / (264/6) = 95/44
### Answer: 95/44 

Expected answer: 95/44

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: We know g = (23 / 26) * (16 / 1) = 184/13.  Substituting for 'g', we get (4/7) * (184/13).  Multiplying the numerators and denominators, we have (4 * 184) / (7 * 13) = 736/91.
### Answer: 736/91 

Expected answer: 736/91

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning:  We can simplify 2/26 to 1/13. Then, we can simplify (1/13) * (16/1) by dividing both 16 and 13 by their greatest common factor, 1 to get (1/13) * 16. Finally, we multiply the numerators and denominators to get 16/13.
### Answer: 16/13 

Expected answer: 16/13

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: We know from the previous problem that ((21 / 1) / (26 / 20)) simplifies to 210/13.  Multiplying this by (21/29) gives us:

(210/13) * (21/29) = (210 * 21) / (13 * 29)

= 4410 / 377

This fraction cannot be simplified further.

### Answer: 4410/377 

Expected answer: 4410/377

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: First, we simplify the expression inside the parentheses: 
 (21 / 1) = 21.
 
 Then, we perform the division: 
 21 / 2 = 10.5
 
 ### Answer: 10.5 

Expected answer: 21/2

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: Multiplying 20/19 by 21/19 is the same as (20*21)/(19*19).
### Answer: 420/361

Expected answer: 420/361

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: We already know from the previous problem that ((21 / 16) / (27 / 23)) is equal to 161/144.
 
 To square this fraction, we need to multiply it by itself:
 
 (161/144)^2 = (161/144) * (161/144) 
 
 This results in:
 
 (161 * 161) / (144 * 144) = 25921 / 20736
 
 ### Answer: 25921/20736 
 
 

Expected answer: 25921/20736

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: First, we need to solve the division inside the parenthesis:
 
 6 / (27/23) 
 
 Dividing by a fraction is the same as multiplying by its reciprocal:
 
 6 * (23/27) = (6*23)/27
 
 Now, we can simplify by canceling common factors:
 
 (6*23)/27 = (2*3*23)/(3*9) = (2*23)/9
 
 Finally, we get:
 
 (2*23)/9 = 46/9 
 
 ### Answer: 46/9 

Expected answer: 46/9

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: We know from the previous problem that ((6 / 19) * (28 / 2)) = 84/19. 
Now, multiply that result by (27/17): (84/19)*(27/17) = 2268/323
Finally, simplify this fraction: 2268/323 = 6 310/323
### Answer: 6 310/323 

Expected answer: 2268/323

Correct? No

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

Gemini-1.5 Pro:

Reasoning: First, simplify inside the parenthesis: (28/2) = 14.
Then, multiply the fraction (17/19) by 14: (17/19)*(14/1) = 238/19
Finally, simplify the fraction: 238/19 = 12 10/19
### Answer: 12 10/19 

Expected answer: 238/19

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: The previous result was 24/5. Now we need to multiply this by 15/10: (24/5) * (15/10).

Before multiplying, we can simplify: 15/10 can be simplified to 3/2.

Now the operation becomes: (24/5) * (3/2) = (24*3) / (5*2) = 72/10.

Finally, we simplify 72/10 by dividing both numerator and denominator by their greatest common divisor, which is 2: 72/2 = 36 and 10/2 = 5.

### Answer: 36/5

Expected answer: 36/5

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: We can simplify the fractions before multiplying. The fraction 3/14 cannot be simplified further. The fraction 28/5 also cannot be simplified. However, when multiplying 3/14 by 28/5, notice that 28 (the numerator of the second fraction) and 14 (the denominator of the first fraction) share a common factor of 14. We can therefore simplify: 28/14 = 2 and 14/14 = 1. 

The multiplication then becomes: (3/1) * (2/5) = (3*2)/(1*5) = 6/5

### Answer: 6/5 

Expected answer: 6/5

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning:
From the previous calculation, we know that ((15 / 3) * (7 / 25)) simplifies to 7/5.
Now we need to calculate (7/5) * (28/16).
Before multiplying, we can simplify by canceling common factors: 7/5 * 28/16 = 7/5 * (7*4)/(8*2) = 7/5 * 7/4. 
Now we multiply the numerators and the denominators: (7*7)/(5*4) = 49/20.

### Answer: 49/20

Expected answer: 49/20

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning:
Let's calculate the new expression: (13 / 3) * (7 / 25). 
Unlike the previous problem, (13 / 3) does not simplify to a whole number. So, we multiply the numerators and the denominators directly: (13 * 7) / (3 * 25) = 91 / 75.

This fraction cannot be simplified further as there are no common divisors for 91 and 75 other than 1. 

Comparing to the previous result of 7/5, we see that changing (15/3) to (13/3) leads to a different final result and doesn't allow for simplification into a whole number in the intermediary steps. This highlights how even small changes in the numbers can significantly impact the outcome of calculations.

### Answer: 91/75

Expected answer: 91/75

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: We know from the previous problem that x = 20.  So, we need to calculate 20 * (19/15).
This can be simplified by first calculating 20 * 19 = 380, and then dividing by 15: 380/15.
Both 380 and 15 are divisible by 5, so we can simplify the fraction: (380/5)/(15/5) = 76/3. 
### Answer: 76/3 

Expected answer: 76/3

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: We need to calculate (57/16) * (28/16). Before multiplying, we can simplify:
- Both 28 and 16 are divisible by 4: 28/4 = 7 and 16/4 = 4 

The calculation becomes: (57/4) * (7/4)
Multiplying numerators and denominators: (57 * 7) / (4 * 4) = 399/16

### Answer: 399/16 

Expected answer: 399/64

Correct? No

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

Gemini-1.5 Pro:

Reasoning: As before, we simplify 27/18 to 3/2. The fraction 19/17 cannot be simplified further. 
Our expression becomes: (3/2) * (19/17)
Multiplying the numerators and denominators, we get: (3 * 19) / (2 * 17) = 57/34 

### Answer: 57/34 

Expected answer: 57/34

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: We know (29/5)*(13/24) = 377/120. So (377/120)*(25/11) = (377*25)/(120*11). Cancelling a factor of 5 and a factor of 11 gives us (7*11*5*5)/(12*11*5) = (7*5)/(12) = 35/12
### Answer: 35/12 

Expected answer: 1885/264

Correct? No

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

Gemini-1.5 Pro:

Reasoning: (29/8)*(13/24) = (29*13)/(8*24) = 377/192
### Answer: 377/192 

Expected answer: 377/192

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: We substitute e = 13/6 into the expression: (13/2)*(13/6) 
When multiplying fractions, we multiply the numerators and the denominators: (13 * 13) / (2 * 6) = 169/12
The fraction 169/12 cannot be simplified any further. 
### Answer: 169/12 

Expected answer: 169/12

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: Following the same principle as before, we multiply the numerators and the denominators: (11 * 1) / (1 * 6) = 11/6
The fraction 11/6 cannot be simplified any further. 
We see that the result is smaller than before, as we are now multiplying 1/6 by 11 instead of 13.
### Answer: 11/6 

Expected answer: 11/6

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: From the previous problem, we know that (25/20)*(11/6) = 55/24.  Multiplying that result by 20/26 gives us (55/24)*(20/26) = (55*20)/(24*26) = 1100/624.  This simplifies to 550/312 = 275/156.
### Answer: 275/156 

Expected answer: 275/156

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: (25/17)*(11/6) = (25*11)/(17*6) = 275/102
### Answer: 275/102 

Expected answer: 275/102

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: We know from the previous problem that (15/16)*(29/27) = 145/144.  Multiplying this by 27/4 gives us (145/144)*(27/4) = (145*27)/(144*4). Before calculating this, notice that 144 is divisible by 4: 144/4 = 36.  Therefore, we can simplify our calculation to 145*27/(36*4) = (145*27)/144 = 3915/144.  Finally, we see that both 3915 and 144 are divisible by 3: 3915/3 = 1305 and 144/3 = 48. So, the simplified fraction is 1305/48.
### Answer: 1305/48 

Expected answer: 435/64

Correct? No

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

Gemini-1.5 Pro:

Reasoning:  Multiplying 15/10 by 29/27, we get (15 * 29) / (10 * 27). This simplifies to 435/270. We can simplify this fraction by dividing both numerator and denominator by their greatest common divisor, which is 45. This gives us (435/45) / (270/45) = 29/18.
### Answer: 29/18

Expected answer: 29/18

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: We previously found that (30/25)*(25/10) = 3.
Now we need to multiply that answer by 5/9: 3 * (5/9).
We can rewrite 3 as a fraction: 3 = 3/1.
Now we can multiply: (3/1)*(5/9) = (3*5)/(1*9) = 15/9.
Finally, we can simplify the fraction 15/9 by dividing the numerator and denominator by their greatest common factor, 3: 15/9 = (15/3)/(9/3) = 5/3.
### Answer: 5/3 

Expected answer: 5/3

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: First, we calculate (27/10)*(27/3). (27/10)*(27/3) = (27*27)/(10*3). 27*27 = 729, and 10*3 = 30, so (27/10)*(27/3)=729/30. 729 and 30 share a common factor of 3, so we can simplify the fraction: 729/30 = (729/3)/(30/3) = 243/10.

Next, we multiply that result by 9/20: (243/10)*(9/20) = (243*9)/(10*20). 243*9=2187, and 10*20=200, so (243/10)*(9/20)=2187/200. This fraction cannot be simplified further.
### Answer: 2187/200

Expected answer: 2187/200

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: (27/10)*(27/21) = (27*27)/(10*21). 27*27 = 729, and 10*21 = 210, so (27/10)*(27/21) = 729/210. 729 and 210 share a common factor of 3, so we can simplify the fraction: 729/210 = (729/3)/(210/3) = 243/70.
### Answer: 243/70 

Expected answer: 243/70

Correct? Yes

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

Gemini-1.5 Pro:

First, we need to calculate (10 / 8) * (30 / 26) which simplifies to (5/4)*(15/13) = 75/52.

Now, we square the result: (75/52) * (75/52)

Multiplying the numerators and denominators: (75*75) / (52*52) = 5625/2704

### Answer: 5625/2704 

Expected answer: 5625/2704

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: We know from the previous problem that ((20 / 22) * (14 / 11)) simplifies to 140/121.
So, our problem becomes (140/121) * (17/3) = 2380/363. 
The fraction 2380/363 cannot be simplified any further. 
### Answer: 2380/363

Expected answer: 2380/363

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: Changing the second fraction from (14/11) to (14/2) means we are now dividing by 2 instead of 11. Dividing by a smaller number (2 instead of 11) will result in a larger quotient.

Let's simplify and evaluate the new expression:

1. Simplify (20/22) to (10/11)
2. Simplify (14/2) to (7/1)
3. Multiply the simplified fractions: (10/11)*(7/1) = 70/11

### Answer: 70/11 

Expected answer: 70/11

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: We previously found (25/8)*(3/4) = 75/32. Now we multiply that result by 29/23: (75/32) * (29/23) = (75*29)/(32*23) = 2175/736. 
If the initial fraction, 75/32, represented dividing something like cookies amongst a group, multiplying by 29/23 could signify that the initial share, represented by 75/32, is further divided or multiplied by a factor of 29/23. This could mean, for instance, that each person originally slated to receive a portion of the 75/32 share now receives 29/23 of that original portion.
### Answer: 2175/736 

Expected answer: 2175/736

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: Multiplying the numerators, we get 25 * 29 = 725. Multiplying the denominators, we have 8 * 4 = 32. So the resulting fraction is 725/32.

In the context of our scenario of dividing items among people, increasing the second fraction from 3/4 to 29/4 significantly impacts the outcome.  Instead of having 75 units to divide, we now have 725 units. This could mean, for example, that instead of sharing 75 cookies amongst 32 people, we're now dealing with 725 cookies amongst the same number of people. This dramatically changes the portion each person receives, highlighting how even small adjustments to proportions can have substantial effects in real-world distributions. 

### Answer: 725/32 

Expected answer: 725/32

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: We have (754/255)*(21/9) = (754*21)/(255*9) = 15834/2295.  Both numerator and denominator share 51 as a common factor: 15834/51 = 310.47 and 2295/51 = 45. So, the simplified fraction is 310.47/45, which can be represented as the mixed number 6 40.47/45.
### Answer: 6 40.47/45 

Expected answer: 5278/765

Correct? No

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

Gemini-1.5 Pro:

Reasoning: (15/17)*(26/15) = (15*26)/(17*15).  Notice that the 15 in the numerator and the 15 in the denominator cancel out, leaving us with 26/17.
### Answer: 26/17 

Expected answer: 26/17

Correct? Yes

Problem: What would be the result if you changed the previous problem by multiplying (28 / 4) by (10 / 12) instead of (23 / 4) by (10 / 12)? Write your answer as a simplified fraction.

Gemini-1.5 Pro:

Reasoning: 
* We can simplify (28/4) to 7 and (10/12) to 5/6
* Multiplying these we get 7 * (5/6) = 35/6

### Answer: 35/6 

Expected answer: 35/6

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: First, we multiply (21/26)*(20/9).  To multiply fractions, we multiply the numerators and the denominators: (21*20)/(26*9). 
We simplify by canceling common factors. 21 and 9 share a common factor of 3, and 20 and 26 share a common factor of 2: (7*3*10*2)/(13*2*3*3). 
This simplifies to (7*10)/(13*3) = 70/39.

Next, we multiply this result by (15/21): (70/39)*(15/21).
Again, we multiply the numerators and denominators: (70*15)/(39*21).
70 and 21 share a common factor of 7, and 15 and 39 share a common factor of 3: (10*7*5*3)/(13*3*7*3).
Canceling the common factors, we get (10*5)/(13*3) = 50/39.
### Answer: 50/39 

Expected answer: 50/39

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: First, we simplify the fraction 20/4 to 5/1.
Then, to multiply fractions, we multiply the numerators and the denominators: (21/26)*(5/1) = (21*5)/(26*1).
This simplifies to 105/26. 
### Answer: 105/26

Expected answer: 105/26

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: We will multiply our previous answer, 85/26, by 8/15.
(85/26)*(8/15) 
We can simplify by canceling common factors: 85 and 15 share a common factor of 5, and 8 and 26 share a common factor of 2.
(85/26)*(8/15) = (17/13)*(4/3)
Now we can multiply the numerators and the denominators: 
(17*4)/(13*3) = 68/39
### Answer: 68/39 

Expected answer: 68/39

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: The initial calculation resulted in 425/108. We now need to multiply this by 23/17: (425/108) * (23/17).
Multiplying the numerators and denominators gives us (425 * 23) / (108 * 17) = 9775 / 1836.
This fraction cannot be simplified further. 
### Answer: 9775/1836 

Expected answer: 575/108

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: We multiply the numerators and the denominators: (25 * 14) / (18 * 6).
This simplifies to 350 / 108.
Both numbers are divisible by 2, so the simplified fraction is 175/54.
### Answer: 175/54

Expected answer: 175/54

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning:
We know from the previous problem that t = 18/17.  Therefore, t * t = (18/17) * (18/17).

Multiplying fractions involves multiplying the numerators and the denominators: (18 * 18) / (17 * 17) = 324/289

### Answer: 324/289 

Expected answer: 324/289

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning:
Similar to the previous problem, we multiply the numerator of the fraction (2) by the whole number (12).

This gives us: 12 * (2/17) = (12 * 2) / 17 = 24/17

### Answer: 24/17 

Expected answer: 24/17

Correct? Yes

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

Gemini-1.5 Pro:

From the previous problem, we know z = 513/196.

Now we need to calculate (513/196) * (17/23).

* Numerator: 513 * 17 = 8721
* Denominator: 196 * 23 = 4508

Therefore, the result of the multiplication is 8721/4508. This fraction cannot be simplified further.

### Answer: 8721/4508

Expected answer: 8721/4508

Correct? Yes

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

Gemini-1.5 Pro:

We will multiply the numerators and the denominators separately. 

* Numerator: 15 * 19 = 285
* Denominator: 14 * 14 = 196

Therefore, the result of the multiplication is 285/196.

### Answer: 285/196 

Expected answer: 285/196

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: The previous answer, 16/15, multiplied by 30/24 can be simplified: (16/15) * (30/24) = (16/~~15~~) * (~~15~~*2 / 8*3) = (~~16~~/1) * (2/~~8~~*3) = (~~2~~/1) * (1/1*3) = 2/3.
### Answer: 2/3

Expected answer: 4/3

Correct? No

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

Gemini-1.5 Pro:

Reasoning: We can simplify the expression before multiplying: (16 / 5) * (5 / 7) = (16/~~5~~) * (~~5~~/7) = 16/7
### Answer: 16/7 

Expected answer: 16/7

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: The problem asks us to calculate (16/7) * (23/20).
Multiplying the numerators and denominators, we get: (16 * 23) / (7 * 20)
This simplifies to: 368 / 140
Both numerator and denominator are divisible by 4, simplifying the fraction to: 92 / 35
### Answer: 92/35

Expected answer: 92/35

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: With the modified fraction, the multiplication becomes: (8 * 11) / (3 * 21)
This results in: 88 / 63
There are no common factors between 88 and 63, so this fraction cannot be further simplified. 
### Answer: 88/63 

Expected answer: 88/63

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: We found that r = 56/33.  Multiplying this result by 9/4 gives us:

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

= 504/132

= 14/3

### Answer: 14/3

Expected answer: 42/11

Correct? No

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

Gemini-1.5 Pro:

Reasoning: We can think of this problem as  13 / (22/16), which is the same as 13 * (16/22). Multiplying 13 by the fraction (16/22) will result in a different answer than multiplying (14/6) by (16/22), as we are starting with a different initial value.

Let's calculate the new expression:

13 / (22/16) = 13 * (16/22)

= (13*16)/22

= 208/22

= 104/11

### Answer: 104/11 

Expected answer: 104/11

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: Following the same principle as before, we multiply the numerators and the denominators: (10 * 25) / (3 * 26)
This gives us: 250 / 78
Simplifying by dividing both numerator and denominator by their greatest common divisor, 2, we get: 125 / 39 
### Answer: 125/39 

Expected answer: 125/39

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: We know from the previous problem that (16/16)*(26/22) simplifies to 13/11. Therefore, we need to calculate (13/11)*(13/11). When multiplying fractions, we multiply the numerators and the denominators: (13 * 13) / (11 * 11) = 169/121.
### Answer: 169/121 

Expected answer: 169/121

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: First, we can simplify each fraction before multiplying.  (16/12) simplifies to 4/3 by dividing the numerator and denominator by their greatest common factor, 4.  (26/22) simplifies to 13/11 by dividing by their greatest common factor, 2. 

Now, the problem becomes (4/3) * (13/11).  Multiplying the numerators and denominators, we get 52/33. 

### Answer: 52/33

Expected answer: 52/33

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: From the previous calculation, we know (17/5)*(5/14) = 17/14. Now we need to multiply this result by 21/25: (17/14) * (21/25) = (17 * 21) / (14 * 25). Before calculating, we can simplify by recognizing the common factors: 17 * 21 = 17 * (3 * 7) and 14 * 25 = (2 * 7) * 25.  Therefore, (17 * 21) / (14 * 25) = (17 * 3 * 7) / (2 * 7 * 25) = (17 * 3) / (2 * 25) = 51 / 50.
### Answer: 51/50 

Expected answer: 51/50

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: We start with the fraction 17/5 (the result of the first operation). Multiplying by 5/12 gives us: (17/5) * (5/12) = (17*5)/(5*12). We can simplify by canceling the common factor of 5: (17*5)/(5*12) = 17/12
### Answer: 17/12

Expected answer: 17/12

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: (16/6)*(29/10) = (16*29)/(6*10). We can simplify this by dividing the numerator and denominator by 2: (16*29)/(6*10) = (8*29)/(3*5) = 232/15.
### Answer: 232/15 

Expected answer: 116/15

Correct? No

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

Gemini-1.5 Pro:

Reasoning: The car will consume (16/10)*(13/7) = 208/70 gallons of petrol for the whole route.
We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2. That gives us (208/2)/(70/2) = 104/35. 
### Answer: 104/35 

Expected answer: 104/35

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: Jeremy's previous result was 45/4.  He now needs to multiply this by 22/14:  (45/4) * (22/14). 
This becomes (45 * 22) / (4 * 14) = 990/56.
We can simplify this by recognizing both the numerator and denominator are divisible by 2. This gives us 495/28.
Both 495 and 28 are divisible by 7, giving us a final simplified fraction of 70/4.

### Answer: 70/4 

Expected answer: 495/28

Correct? No

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

Gemini-1.5 Pro:

Reasoning: We need to calculate (15/12) * (27/5). Multiplying the numerators and the denominators, we get (15 * 27) / (12 * 5) = 405/60.
Simplifying by dividing both numerator and denominator by 15, we get 27/4. 

### Answer: 27/4

Expected answer: 27/4

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: Our previous answer was 1. Multiplying any number by 1 results in the same number: 1 * (27/15) = 27/15. We can simplify this fraction by dividing both numerator and denominator by their greatest common factor, which is 3: (27/3) / (15/3) = 9/5.
### Answer: 9/5 

Expected answer: 9/5

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: (7/7) = 1. Then, (16/12) can be simplified by dividing both numerator and denominator by 4, resulting in (4/3). So the problem becomes 1 * (4/3).
### Answer: 4/3 

Expected answer: 4/3

Correct? Yes

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

Gemini-1.5 Pro:

Reasoning: First, we can simplify the fractions within the parentheses: 22/6 simplifies to 11/3 and 16/24 simplifies to 2/3. So the expression becomes (11/3)*(2/3). Multiplying the numerators and the denominators gives us 22/9. 
### Answer: 22/9 

Expected answer: 22/9

Correct? Yes