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Gemma 2B on 4.NF.A.2

Common Core Standard 4.NF.A.2

This standard covers the following:

Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

🔗 Problem

Problem: Complete the following comparison between two fractions. (1 / 8) _ (4 / 1).

Gemma 2B:

 Reasoning: 1/8 = 0.125, 4/1 = 4.0. 0.125 < 4.0.
(1 / 8) _ (4 / 1).
### Answer: <

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Using the same approach we used before, now think about the comparison between (6 / 8) and (4 / 1). Complete the comparison using the same symbols (<, >, =). (6 / 8) _ (4 / 1). This is similar to the previous problem but now you are comparing a different fraction with the same whole number.

Gemma 2B:

  Reasoning: 6/8 = 0.75, 4/1 = 4.0. 0.75 < 4.0.
(6 / 8) _ (4 / 1).
### Answer: <

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Determine which symbol appropriately fills in the blank (either '<', '=', or '>') to accurately compare these two fractions: (7 / 2) _ (10 / 6)

Gemma 2B:

 Reasoning: 7/2 = <<7/2=3.5>>3.5 and 10/6 = <<10/6=1.6666666666666667>>1.6666666666666667. 3.5 _ 1.6666666666666667. 3.5 > 1.6666666666666667.
### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Let's modify our previous problem slightly and continue the exercise. Fill in the blank with one of the following symbols: <, >, or =, to properly compare the two fractions. (6 / 2) _ (10 / 6)

Gemma 2B:

  Reasoning: 6/2 = <<6/2=3>>3 and 10/6 = <<10/6=1.6666666666666667>>1.6666666666666667. 3 _ 1.6666666666666667. 3 > 1.6666666666666667.
### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Complete the following comparison by filling in the blank (<, >, or =): (6 / 8) _ (8 / 7)

Gemma 2B:

 Reasoning: (6 / 8) = <<6/8=0.75>>0.75 and (8 / 7) = <<8/7=1.14>>1.14. Since 0.75 < 1.14, (6 / 8) < (8 / 7).
(6 / 8) < (8 / 7)

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Complete the following comparison by filling in the blank (<, >, or =). Consider what will happen if we are comparing the fraction 6/8 to the fraction 6/7 instead: (6 / 8) _ (6 / 7)

Gemma 2B:

  Reasoning: (6 / 8) = <<6/8=0.75>>0.75 and (6 / 7) = <<6/7=0.86>>0.86. Since 0.75 < 0.86, (6 / 8) < (6 / 7).
(6 / 8) < (6 / 7)

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Compare the values of the two fractions below by filling in the blank with <, >, or =. (9 / 3) _ (7 / 1)

Gemma 2B:

 Reasoning: 9/3 = 3 and 7/1 = 7. 3 < 7.
(9 / 3) < (7 / 1)

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Consider the original problem where we compared the fractions (9 / 3) and (7 / 1). Now, instead of comparing it with (7 / 1), let's compare it with the fraction (1 / 1). Fill in the blank in the new problem: (9 / 3) _ (1 / 1) Use the symbols <, >, or = to make the correct comparison.

Gemma 2B:

  Reasoning: 9/3 = 3 and 1/1 = 1. 3 > 1.
(9 / 3) > (1 / 1)

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Fill in the blank with one of the following signs: <, >, or = (5 / 3) _ (10 / 5)

Gemma 2B:

 Reasoning: 5/3 = <<5/3=1.6666666666666667>>1.6666666666666667 and 10/5 = <<10/5=2>>2. 1.6666666666666667 < 2.
(5 / 3) < (10 / 5)

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Consider the following situation: Previously, you compared the values of the fractions (5 / 3) and (10 / 5). Now, imagine if the second fraction was changed to (7 / 5) instead. Based on this new situation, fill in the blank with one of the following signs: <, >, or = (5 / 3) _ (7 / 5)

Gemma 2B:

 Reasoning: 5/3 = <<5/3=1.6666666666666667>>1.6666666666666667 and 7/5 = <<7/5=1.4>>1.4. 1.6666666666666667 > 1.4.
(5 / 3) > (7 / 5)

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Compare the values of the two fractions by filling in the blank with <, >, or =. Here are the fractions: (3 / 2) _ (9 / 10)

Gemma 2B:

 Reasoning: 3/2 = <<3/2=1.5>>1.5 and 9/10 = <<9/10=0.9>>0.9. Since 1.5 > 0.9, 3/2 > 9/10.
### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Let's consider a slight change to our previous fractions problem. This time, fill in the blank to compare these two fractions: (3 / 7) _ (9 / 10) Is the comparison the same or different than the comparison between 3 / 2 and 9 / 10? Think about the impact of decreasing the numerator of our first fraction on the comparison.

Gemma 2B:

  Reasoning: 3/7 = <<3/7=0.42857142857142855>>0.42857142857142855 and 9/10 = <<9/10=0.9>>0.9. Since 0.42857142857142855 < 0.9, 3/7 < 9/10. The comparison is different than the comparison between 3 / 2 and 9 / 10. The impact of decreasing the numerator of our first fraction is that the value of the fraction decreases, and the comparison changes.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Compare the values of these fractions by filling in the blank with <, >, or =. (10 / 9) _ (5 / 6)

Gemma 2B:

 Reasoning: 10/9 = <<10/9=1.1111111111111111>>1.1111111111111111 and 5/6 = <<5/6=0.8333333333333334>>0.8333333333333334. Since 1.1111111111111111 > 0.8333333333333334, 10/9 > 5/6.
(10 / 9) > (5 / 6)

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Now, consider the following fractions and fill in the blank with <, >, or =, to compare their values. (10 / 9) _ (5 / 5)

Gemma 2B:

  Reasoning: 10/9 = <<10/9=1.1111111111111111>>1.1111111111111111 and 5/5 = <<5/5=1>>1. Since 1.1111111111111111 = 1, 10/9 = 5/5.
(10 / 9) = (5 / 5)

Expected answer: >

Correct? No

🔗 Problem

Problem: Compare the values of the two fractions and fill in the blank with <, >, or =. (10 / 9) _ (5 / 10)

Gemma 2B:

 Reasoning: 10/9 = <<10/9=1.1111111111111111>>1.1111111111111111 and 5/10 = <<5/10=0.5>>0.5.
10/9 _ 5/10
### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Using the same logic as the previous problem, fill in the blank with <, >, or =. However, now consider the case where the second fraction is changed to 5/3. So the question is: (10 / 9) _ (5 / 3)

Gemma 2B:

  Reasoning: 10/9 = <<10/9=1.1111111111111111>>1.1111111111111111 and 5/3 = <<5/3=1.6666666666666665>>1.6666666666666665.
10/9 _ 5/3
### Answer: <

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Compare the two fractions by completing the following statement with <, >, or =. (9 / 8) _ (6 / 4)

Gemma 2B:

 Reasoning: 9/8 = <<9/8=1.125>>1.125 and 6/4 = <<6/4=1.5>>1.5. Since 1.125 < 1.5, 9/8 < 6/4.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Let's consider a modification to the previous problem. Now, compare the fractions in the following statement: (9 / 9) _ (6 / 4) What happens to the relational comparison when the numerator of first fraction changes from 9 to 9 (effectively making it 1), while the second fraction remains same? Fill in the blank with <, >, or =.

Gemma 2B:

  Reasoning: 9/9 = <<9/9=1>>1 and 6/4 = <<6/4=1.5>>1.5. Since 1 < 1.5, 9/9 < 6/4.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Compare the two fractions below by filling in the gap with the appropriate symbol (<,>,=). (10 / 3) _ (10 / 2)

Gemma 2B:

 Reasoning: 10/3 = <<10/3=3.3333333333333335>>3.3333333333333335 and 10/2 = <<10/2=5>>5.
3.3333333333333335 _ 5
### Answer: <

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Let's revisit the previous comparison of fractions, but this time consider the change: (10 / 3) _ (10 / 10) Complete the comparison by filling in the blank with the appropriate symbol (<, >, or =). What happens to the comparison when the denominator of the second fraction is increased to 10?

Gemma 2B:

  Reasoning: 10/3 = <<10/3=3.3333333333333335>>3.3333333333333335 and 10/10 = <<10/10=1>>1.
3.3333333333333335 _ 1
### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Compare the two fractions by filling in the blank with <, >, or =. (2 / 1) _ (6 / 5)

Gemma 2B:

 Reasoning: 2/1 = <<2/1=2>>2 and 6/5 = <<6/5=1.2>>1.2. Since 2 > 1.2, 2/1 > 6/5.
(2 / 1) > (6 / 5)

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Compare the two fractions by filling in the blank with <, >, or =. In the last example, you were comparing the fractions 2 / 1 and 6 / 5. Now consider how the result might change when you compare these fractions: (2 / 2) _ (6 / 5)

Gemma 2B:

  Reasoning: 2/2 = <<2/2=1>>1 and 6/5 = <<6/5=1.2>>1.2. Since 1 < 1.2, 2/2 < 6/5.
(2 / 2) < (6 / 5)

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Compare the fractions 5 / 4 and 10 / 3 by filling in the blank with one of the following symbols: <, >, or =.

Gemma 2B:

 Reasoning: 5 / 4 = <<5/4=1.25>>1.25 and 10 / 3 = <<10/3=3.3333333333333335>>3.3333333333333335.
5 / 4 < 10 / 3
### Answer: 5 / 4 < 10 / 3

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Consider now the fractions 5 / 4 and 8 / 3. Compare these two fractions by filling in the blank with one of the following symbols: <, >, or =.

Gemma 2B:

  Reasoning: 5 / 4 = <<5/4=1.25>>1.25 and 8 / 3 = <<8/3=2.6666666666666665>>2.6666666666666665.
5 / 4 < 8 / 3
### Answer: 5 / 4 < 8 / 3

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Compare the values of these two fractions by filling in the blank with <, > or =: (7 / 5) _ (9 / 9)

Gemma 2B:

 Reasoning: 7/5 = <<7/5=1.4>>1.4 and 9/9 = <<9/9=1>>1.
1.4 _ 1
### Answer: 1.4 > 1

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Now consider this situation: Instead of dividing 7 by 5, you divided 7 by 10 to form a new fraction. How would this new fraction compare to the fraction 9 / 10? Fill in the blank using <, >, or =: (7 / 5) _ (9 / 10)

Gemma 2B:

  Reasoning: 7/5 = <<7/5=1.4>>1.4 and 9/10 = <<9/10=0.9>>0.9.
1.4 _ 0.9
### Answer: 1.4 > 0.9

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Compare the following two fractions by filling in the blank with <, >, or =: (3 / 7) _ (5 / 8)

Gemma 2B:

 Reasoning: 3/7 = <<3/7=0.42857142857142855>>0.42857142857142855 and 5/8 = <<5/8=0.625>>0.625. Since 0.42857142857142855 < 0.625, 3/7 < 5/8.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Following up on the previous comparison, let's consider a similar one: Compare the value of the fractions (3 / 7) and (6 / 8) by filling in the blank with <, >, or =: (3 / 7) _ (6 / 8) This comparison would tell you how the situation would change if we considered 6 parts out of 8 instead of 5 parts out of 8 as on the previous case.

Gemma 2B:

  Reasoning: 3/7 = <<3/7=0.42857142857142855>>0.42857142857142855 and 6/8 = <<6/8=0.75>>0.75. Since 0.42857142857142855 < 0.75, 3/7 < 6/8.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Question: Compare these two fractions: (3 / 1) _ (5 / 9) Hint: Fill the blank with one of the following: <, >, or =.

Gemma 2B:

 Reasoning: 3/1 = 3 and 5/9 = <<5/9=0.5555555555555556>>0.5555555555555556.
3 > 0.5555555555555556
### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Question: In the previous question where we compared two fractions, we had (3 / 1) _ (5 / 9). Now, consider a change in the second number we are comparing. Instead of (5 / 9), we now have (5 / 5). Compare these two fractions: (3 / 1) _ (5 / 5) Hint: Fill the blank with one of the following: <, >, or =.

Gemma 2B:

  Reasoning: 3/1 = 3 and 5/5 = <<5/5=1>>1.
3 > 1
### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Compare the values of the following fractions: (2 / 2) _ (6 / 6) Fill in the blank with <, >, or = .

Gemma 2B:

 Reasoning: 2/2 = 1 and 6/6 = 1.
1 = 1
### Answer: =

Expected answer: =

Correct? Yes

🔗 Follow-up

Problem: Consider the change to the previous problem: instead of comparing (2 / 2) with (6 / 6), we now have (2 / 2) and (6 / 5). Now, fill in the blank with <, >, or = to compare these two fractions: (2 / 2) _ (6 / 5)

Gemma 2B:

 Reasoning: 2/2 = 1 and 6/5 = 1.2.
1 < 1.2
### Answer: <

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Compare the following fractions by inserting "<", ">" or "=" between them. (7 / 6) _ (9 / 6)

Gemma 2B:

 Reasoning: 7/6 = <<7/6=1.1666666666666667>>1.1666666666666667 and 9/6 = <<9/6=1.5>>1.5. Since 1.1666666666666667 < 1.5, 7/6 < 9/6.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Now let's consider a slight change to the fractions. Please fill in the blank with <, >, or =: (7 / 9) _ (9 / 6) This is just like in our previous problem, but this time we're looking at 7/9 instead of 7/6.

Gemma 2B:

  Reasoning: 7/9 = <<7/9=0.7777777777777778>>0.7777777777777778 and 9/6 = <<9/6=1.5>>1.5. Since 0.7777777777777778 < 1.5, 7/9 < 9/6.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Fill in the blank in the following question with <, >, or = : (2 / 7) _ (2 / 9) to express whether the first fraction is greater than, less than, or equal to the second fraction.

Gemma 2B:

 Reasoning: 2/7 = <<2/7=0.2857142857142857>>0.2857142857142857 and 2/9 = <<2/9=0.2222222222222222>>0.2222222222222222. Since 0.2857142857142857 > 0.2222222222222222, 2/7 > 2/9.
(2 / 7) > (2 / 9)

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Now let's consider a slight alteration to our previous question. Using the same comparison methods, compare the following fractions: (2 / 7) _ (2 / 4) As you did previously, fill in the blank with <, >, or = signs to correctly compare the two fractions. Does the outcome change if we adjust the denominator in our second fraction from a 9 to a 4?

Gemma 2B:

  Reasoning: 2/7 = <<2/7=0.2857142857142857>>0.2857142857142857 and 2/4 = <<2/4=0.5>>0.5. Since 0.2857142857142857 < 0.5, 2/7 < 2/4.
(2 / 7) < (2 / 4)

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Determine the relation between the two following fractions by filling in the blank with <, >, or =: (9 / 4) _ (7 / 1)

Gemma 2B:

 Reasoning: 9/4 = <<9/4=2.25>>2.25 and 7/1 = <<7/1=7>>7.
9/4 _ 7/1
### Answer: <

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Consider now a different set of fractions: (9 / 10) _ (7 / 1) Fill in the blank with <, >, or = to correctly compare these two fractions. How do these fractions compare to the previous fractions we worked with? Has our answer changed?

Gemma 2B:

  Reasoning: 9/10 = <<9/10=0.9>>0.9 and 7/1 = <<7/1=7>>7.
9/10 _ 7/1
### Answer: <

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Fill in the blank with one of the symbols <, >, or = to complete the comparison between the two fractions. (4 / 8) _ (2 / 1)

Gemma 2B:

 Reasoning: 4/8 = 1/2 and 2/1 = 2. 1/2 < 2.
(4 / 8) < (2 / 1)

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Consider the previous problem where we compared (4 / 8) with (2 / 1). Now, imagine we change the second fraction to (2 / 6) instead of (2 / 1). Fill in the blank with one of the symbols <, >, or = to complete this new comparison: (4 / 8) _ (2 / 6)

Gemma 2B:

  Reasoning: 4/8 = 1/2 and 2/6 = 1/3. 1/2 > 1/3.
(4 / 8) > (2 / 6)

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Fill in the blank with either <, >, or = in the following expression: (9 / 9) _ (8 / 9), to properly compare the two fractions.

Gemma 2B:

 Reasoning: 9/9 = 1 and 8/9 < 1.
(9 / 9) > (8 / 9).
### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Continuing from the previous question, let's adjust the fractions slightly. Now consider: (9 / 2) _ (8 / 9) What symbol (<, >, =) would you use in the blank to accurately compare these two fractions?

Gemma 2B:

  Reasoning: 9/2 = 4.5 and 8/9 < 4.5.
(9 / 2) > (8 / 9).
### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Compare the following fractions by filling in the blank with <, >, or =. (9 / 10) _ (3 / 10)

Gemma 2B:

 Reasoning: 9/10 = <<9/10=9/10>>9/10, 3/10 = <<3/10=3/10>>3/10, 9/10 > 3/10.
(9 / 10) > (3 / 10)

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Consider the values of two different fractions instead of the ones in the previous problem. Fill in the blank with <, >, or = to compare the fractions: (9 / 4) _ (3 / 10)

Gemma 2B:

  Reasoning: 9/4 = <<9/4=2.25>>2.25, 3/10 = <<3/10=0.3>>0.3, 2.25 > 0.3.
(9 / 4) > (3 / 10)

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Determine if the first fraction is less than, equal to, or greater than the second. Fill in the blank with one of the following: <, >, or =. (3 / 7) _ (5 / 3)

Gemma 2B:

 Reasoning: 3/7 = <<3/7=0.42857142857142855>>0.42857142857142855 and 5/3 = <<5/3=1.6666666666666665>>1.6666666666666665. 0.42857142857142855 < 1.6666666666666665.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Consider the same fractions as in the previous problem, but with a change in the latter fraction. Determine if the first fraction is less than, equal to, or greater than the second. Fill in the blank with one of the following: <, >, or =. (3 / 7) _ (5 / 5)

Gemma 2B:

  Reasoning: 3/7 = <<3/7=0.42857142857142855>>0.42857142857142855 and 5/5 = <<5/5=1>>1. 0.42857142857142855 < 1.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Compare the values of the following fractions by filling in the blank with <, >, or =. (6 / 5) _ (7 / 10)

Gemma 2B:

 Reasoning: 6/5 = <<6/5=1.2>>1.2 and 7/10 = <<7/10=0.7>>0.7. Since 1.2 > 0.7, 6/5 > 7/10.
(6 / 5) > (7 / 10)

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Consider a change in the fractions previously compared. Now, compare the values of the following fractions by filling in the blank <, >, or =. (1 / 5) _ (7 / 10)

Gemma 2B:

  Reasoning: 1/5 = <<1/5=0.2>>0.2 and 7/10 = <<7/10=0.7>>0.7. Since 0.2 < 0.7, 1/5 < 7/10.
(1 / 5) < (7 / 10)

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Fill in the blank in the following equation with one of these symbols: <, >, or =. (1 / 9) _ (1 / 4)

Gemma 2B:

 Reasoning: 1/9 = <<1/9=0.11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Considering the previous comparison between 1/9 and 1/4, let's now change the first fraction to 1/2 instead. This means modifying the comparison as per the following instruction. Fill in the blank in the following equation with one of these symbols: <, >, or =. (1 / 2) _ (1 / 4)

Gemma 2B:

  Reasoning: 1/2 = <<1/2=0.5>>0.5 and 1/4 = <<1/4=0.25>>0.25. 0.5 > 0.25.
1/2 > 1/4.
### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Compare the following fractions and fill in the blank with <, >, or =: (7 / 1) _ (6 / 1)

Gemma 2B:

 Reasoning: 7/1 = <<7/1=7>>7 and 6/1 = <<6/1=6>>6. 7 > 6.
### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Continue from the previous problem and now consider the fractions (9 / 1) and (6 / 1). Fill in the blank with <, >, or =: (9 / 1) _ (6 / 1)

Gemma 2B:

  Reasoning: 9/1 = <<9/1=9>>9 and 6/1 = <<6/1=6>>6. 9 > 6.
### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Please fill in the blank with <, >, or = to compare the following fractions. (5 / 4) _ (9 / 8)

Gemma 2B:

 Reasoning: 5/4 = <<5/4=5/4>>5/4, 9/8 = <<9/8=9/8>>9/8, 5/4 _ 9/8 = <<5/4<9/8>>5/4 < 9/8.
### Answer: <

Expected answer: >

Correct? No

🔗 Problem

Problem: Determine the correct symbol to place between the two fractions in the following expression. Fill in the blank with <, >, or =. (8 / 9) _ (3 / 4)

Gemma 2B:

 Reasoning: 8/9 = <<8/9=0.8888888888888888>>0.8888888888888888 and 3/4 = <<3/4=0.75>>0.75. 0.8888888888888888 > 0.75, so 8/9 > 3/4.
### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Consider the following comparison: (8 / 2) _ (3 / 4) Compared to the previous scenario where the fractions were (8 / 9) and (3 / 4), we've now changed the denominator of the first fraction from 9 to 2. Fill in the blank with <, >, or = to show how this change affects the comparison of these two fractions.

Gemma 2B:

  Reasoning: 8/2 = <<8/2=4>>4 and 3/4 = <<3/4=0.75>>0.75. 4 > 0.75, so 8/2 > 3/4.
### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Compare the following fractions by choosing the correct symbol (<, >, or =) to fill in the blank: (2 / 1) _ (4 / 9)

Gemma 2B:

 Reasoning: 2/1 = 2, 4/9 = 4/3. 2 < 4/3.
### Answer: <

Expected answer: >

Correct? No

🔗 Problem

Problem: Compare the following fractions by inserting either <, >, or = in the space provided: (9 / 9) _ (2 / 2)

Gemma 2B:

 Reasoning: 9/9 = 2/2 = 1
### Answer: =

Expected answer: =

Correct? Yes

🔗 Follow-up

Problem: Now consider the following fractions: (9 / 3) _ (2 / 2) Perform the same comparison operation as in the previous problem. Recall in the previous problem we compared the fractions (9 / 9) and (2 / 2). Consider what happens if we change the denominator of the first fraction from 9 to 3. Would this make the first fraction greater than, less than, or equal to the second fraction (2 / 2)? Fill in the blank with <, >, or =.

Gemma 2B:

  Reasoning: 9/3 = 3, 2/2 = 1
### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Determine the relationship between the following two fractions by filling the blank with <, >, or =: (6 / 1) _ (4 / 4)

Gemma 2B:

 Reasoning: 6/1 = <<6/1=6>>6 and 4/4 = <<4/4=1>>1. 6 > 1, so 6/1 > 4/4.
### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Now consider this scenario: Determine the relationship between the following two fractions by filling the blank with <, >, or =: (6 / 1) _ (4 / 3) How does this result compare to the previous one?

Gemma 2B:

  Reasoning: 6/1 = <<6/1=6>>6 and 4/3 = <<4/3=4/3>>4/3. 6 > 4/3, so 6/1 > 4/3.
### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Compare the values of the following fractions by filling in the blank with <, >, or =. (3 / 9) _ (8 / 8)

Gemma 2B:

 Reasoning: 3/9 = <<3/9=1/3>>1/3 and 8/8 = <<8/8=1>>1. 1/3 < 1.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Compare the values of the following fractions by filling in the blank with <, >, or =. (3 / 9) _ (8 / 3) Reflect on the previous question and this question and see how changing the denominator in the second fraction affects the relation between the two fractions.

Gemma 2B:

  Reasoning: 3/9 = <<3/9=1/3>>1/3 and 8/3 = <<8/3=2 2/3>>2 2/3. 1/3 < 2 2/3.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Compare the values of the following fractions by filling in the blank with <, >, or =. (4 / 10) _ (1 / 2)

Gemma 2B:

 Reasoning: 4/10 = <<4/10=0.4>>0.4 and 1/2 = <<1/2=0.5>>0.5. Since 0.4 < 0.5, 4/10 < 1/2.
(4 / 10) < (1 / 2)

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Consider this change to the previous problem: (8 / 10) _ (1 / 2) Now, suppose instead of 4/10, we now have 8/10. How would this change the comparison? Fill in the blank with <, >, or =.

Gemma 2B:

  Reasoning: 8/10 = <<8/10=0.8>>0.8 and 1/2 = <<1/2=0.5>>0.5. Since 0.8 > 0.5, 8/10 > 1/2.
(8 / 10) > (1 / 2)

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Compare the values of the two fractions below by filling in the blank space with one of the following symbols: <, >, or =. (6 / 6) _ (6 / 3)

Gemma 2B:

 Reasoning: 6/6 = 1 and 6/3 = 2. 1 _ 2.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Now consider a slight variation: (6 / 4) _ (6 / 3) Reassigning the values while maintaining the idea of comparison, frame the above expression into a question. Does the change in denominator affect the outcome as compared to the previous question, (6 / 6) _ (6 / 3)?

Gemma 2B:

  Reasoning: 6/4 = 3/2 and 6/3 = 2. 3/2 _ 2.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Determine which sign (<, >, or =) correctly fills the blank when comparing these two fractions: (2 / 3) _ (2 / 10)

Gemma 2B:

 Reasoning: 2/3 = <<2/3=0.6666666666666666>>0.6666666666666666 and 2/10 = <<2/10=0.2>>0.2. 0.6666666666666666 > 0.2.
### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Following from the previous question, what would happen if we changed the first fraction from 2 / 3 to 10 / 3? Fill in the blank in the question with <, >, or = to compare the two fractions: (10 / 3) _ (2 / 10)

Gemma 2B:

  Reasoning: 10/3 = <<10/3=3.3333333333333335>>3.3333333333333335 and 2/10 = <<2/10=0.2>>0.2. 3.3333333333333335 > 0.2.
### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Compare the following fractions by filling in the blank with either <, =, or >: (2 / 10) _ (6 / 4)

Gemma 2B:

 Reasoning: 2/10 = 1/5 and 6/4 = 3/2. 1/5 < 3/2. Therefore, 2/10 < 6/4.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Now consider a slight change to the previous problem. Use the symbol <, =, or > to compare these two fractions: (6 / 10) _ (6 / 4) What happens if we increase the numerator of the first fraction from 2 to 6?

Gemma 2B:

  Reasoning: 6/10 = 3/5 and 6/4 = 3/2. 3/5 < 3/2. Therefore, 6/10 < 6/4.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Compare the values of the following fractions and indicate either <, >, or = to complete the equation. (5 / 10) _ (6 / 9)

Gemma 2B:

 Reasoning: 5/10 = 1/2 and 6/9 = 2/3. 1/2 < 2/3.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Consider the change and answer this follow-up question: How does the comparison change when the value of the second fraction is slightly reduced? Fill in the blank with any of <, >, or =. (5 / 10) _ (4 / 9)

Gemma 2B:

  Reasoning: 5/10 = 1/2 and 4/9 = 4/9. 1/2 < 4/9.
### Answer: <

Expected answer: >

Correct? No

🔗 Problem

Problem: Compare the fractions by filling in the blank with <, >, or =. (2 / 9) _ (5 / 5)

Gemma 2B:

 Reasoning: 2/9 = <<2/9=2/9>>2/9, 5/5 = <<5/5=1>>1, 2/9 < 1.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Consider the follow-up question: Now let's change our fractional values. Instead of comparing 2/9 to a whole, as in the previous problem, let's compare it to 5/8. Fill in the blank with <, >, or =. (2 / 9) _ (5 / 8)

Gemma 2B:

  Reasoning: 2/9 = <<2/9=2/9>>2/9, 5/8 = <<5/8=5/8>>5/8, 2/9 < 5/8.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Fill in the blank with <, >, or = to correctly compare the two fractions: (1 / 9) _ (7 / 9)

Gemma 2B:

 Reasoning: 1/9 = <<1/9=1/9>>1/9, 7/9 = <<7/9=7/9>>7/9, 1/9 < 7/9.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: For the follow-up question, consider now comparing the fractions again, but this time in this specific situation: (1 / 9) _ (7 / 7) Keep in mind the comparison you made in the previous question when 1/9 was compared with 7/9. How does the comparison change when the second fraction is in this new form, 7/7? Fill the blank with the correct symbol, <, > or =.

Gemma 2B:

  Reasoning: 1/9 = <<1/9=1/9>>1/9, 7/7 = <<7/7=1>>1, 1/9 < 7/7.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Analyze and compare the two fractions by determining whether (2 / 7) is either less than, greater than, or equal to (1 / 6). Use the symbols <, >, or =.

Gemma 2B:

 Reasoning: 2/7 = <<2/7=0.2857142857142857>>0.2857142857142857 and 1/6 = <<1/6=0.16666666666666666>>0.16666666666666666.
2/7 is greater than 1/6.
### Answer: 2/7 > 1/6

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Consider the following change to the previous problem: Use the fractions (2 / 10) and (1 / 6) instead of (2 / 7) and (1 / 6). Compare these two fractions by determining whether (2 / 10) is either less than, greater than, or equal to (1 / 6), filling in the blank with the symbols <, >, or =.

Gemma 2B:

  Reasoning: 2/10 = <<2/10=0.2>>0.2 and 1/6 = <<1/6=0.16666666666666666>>0.16666666666666666.
2/10 is less than 1/6.
### Answer: 2/10 < 1/6

Expected answer: >

Correct? No

🔗 Problem

Problem: In the following expression, properly insert either <, >, or = in order to correctly compare the two fractions: (9 / 9) _ (9 / 10)

Gemma 2B:

 Reasoning: 9/9 = 1 and 9/10 < 1.
(9 / 9) _ (9 / 10)
### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Consider the following changes in the fractions: (9 / 9) _ (9 / 5) Based on the adjustments made, fill in the blank with <, >, or = to appropriately compare the values of the two fractions.

Gemma 2B:

  Reasoning: 9/9 = 1 and 9/5 > 1.
(9 / 9) _ (9 / 5)
### Answer: >

Expected answer: <

Correct? No

🔗 Problem

Problem: Compare the following two fractions by determining if the first fraction is greater than, equal to, or less than the second fraction: (8 / 4) _ (5 / 2)

Gemma 2B:

 Reasoning: 8/4 = <<8/4=2>>2 and 5/2 = <<5/2=2.5>>2.5. Since 2 is less than 2.5, 8/4 is less than 5/2.
### Answer: 8/4 is less than 5/2

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Now consider a slight change to the values. Instead of the first fraction being (8 / 4), let's change it to (10 / 4). So, we have this comparison: (10 / 4) _ (5 / 2) Compare these two fractions as you did before by determining if the first fraction is greater than, equal to, or less than the second fraction.

Gemma 2B:

  Reasoning: 10/4 = <<10/4=2.5>>2.5 and 5/2 = <<5/2=2.5>>2.5. Since 2.5 is equal to 2.5, 10/4 is equal to 5/2.
### Answer: 10/4 is equal to 5/2

Expected answer: =

Correct? Yes

🔗 Problem

Problem: Compare the values of the following fractions by deciding which symbol (<, >, or =) to fill in the blank with: (5 / 2) _ (1 / 10)

Gemma 2B:

 Reasoning: 5/2 = <<5/2=2.5>>2.5 and 1/10 = <<1/10=0.1>>0.1. 2.5 > 0.1.
(5 / 2) > (1 / 10)

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Continuing from our previous comparison, now consider a new set of fractions. Compare the values of the following fractions by filling in the blank with <, >, or =. (5 / 2) _ (1 / 2) Now, think about what would happen if the first fraction was changed from (5 / 2) to (3 / 2). How would this change affect the comparison? Write down a similar comparison question replacing 5 with 3.

Gemma 2B:

  Reasoning: 5/2 = <<5/2=2.5>>2.5 and 1/2 = <<1/2=0.5>>0.5. 2.5 > 0.5.
(5 / 2) > (1 / 2)

If the first fraction was changed from (5 / 2) to (3 / 2), the comparison would be: (3 / 2) _ (1 / 2)

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Fill in the blank with <, >, or = to compare the two fractions in the following question: (2 / 4) _ (9 / 2)

Gemma 2B:

 Reasoning: 2/4 = 1/2 and 9/2 = 4.5. 1/2 < 4.5, so 2/4 < 9/2.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Now suppose we change the fractions a bit in the previous problem. Fill in the blank with <, >, or = to compare the ratios in the following question: (2 / 4) _ (9 / 10)

Gemma 2B:

  Reasoning: 2/4 = 1/2 and 9/10 = 0.9. 1/2 > 0.9, so 2/4 > 9/10.
### Answer: >

Expected answer: <

Correct? No

🔗 Problem

Problem: Fill in the blank in the following equation with either <, >, or = to show the comparison between the two fractions: (6 / 3) _ (10 / 2)

Gemma 2B:

 Reasoning: 6/3 = 2 and 10/2 = 5. 2 _ 5. 2 < 5.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Now consider the fractions (6 / 8) and (10 / 2). Fill in the blank in the following equation with either <, >, or = to show the comparison between the two fractions: (6 / 8) _ (10 / 2)

Gemma 2B:

  Reasoning: 6/8 = 3/4 and 10/2 = 5. 3/4 _ 5. 3/4 < 5.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Compare the given fractions by filling in the blank with the correct symbol (<, >, or =). (6 / 3) _ (8 / 6)

Gemma 2B:

 Reasoning: 6/3 = <<6/3=2>>2 and 8/6 = <<8/6=4/3>>4/3. 2 _ <<2<_>4/3>> 4/3. 
### Answer: <

Expected answer: >

Correct? No

🔗 Problem

Problem: Compare the values of the following fractions by filling in the blank with <, >, or =. (3 / 4) _ (6 / 5)

Gemma 2B:

 Reasoning: 3/4 = <<3/4=0.75>>0.75 and 6/5 = <<6/5=1.2>>1.2. Since 0.75 < 1.2, 3/4 < 6/5.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Compare the values of the following fractions by filling in the blank with <, >, or =. (3 / 4) _ (6 / 9) Consider what happens if the second fraction changes, and 6 is divided by 9 instead of 5. How does this change affect the comparison between the two fractions?

Gemma 2B:

  Reasoning: 3/4 = <<3/4=0.75>>0.75 and 6/9 = <<6/9=0.6666666666666666>>0.67. Since 0.75 > 0.67, 3/4 > 6/9.
### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Compare the two fractions below by filling in the blank with <, >, or =: (1 / 1) _ (3 / 1)

Gemma 2B:

 Reasoning: 1/1 = <<1/1=1>>1 and 3/1 = <<3/1=3>>3. Since 1 < 3, 1/1 < 3/1.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: After comparing (1 / 1) and (3 / 1), let's consider a new scenario. How would the comparison change if we replaced (3 / 1) with (3 / 7)? Fill in the blank with <, >, or = to compare the two fractions: (1 / 1) _ (3 / 7)

Gemma 2B:

  Reasoning: 1/1 = <<1/1=1>>1 and 3/7 = <<3/7=0.42857142857142855>>0.42857142857142855. Since 1 > 0.42857142857142855, 1/1 > 3/7.
### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Compare the following fractions by filling in the blank with <, >, or =. (7 / 6) _ (4 / 5)

Gemma 2B:

 Reasoning: 7/6 = <<7/6=1.1666666666666667>>1.1666666666666667 and 4/5 = <<4/5=0.8>>0.8. Since 1.1666666666666667 > 0.8, 7/6 > 4/5.
### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Consider the situation from the previous problem. Now, think about what would happen if the first fraction is 7 / 3 instead of 7 / 6. Fill in the blank to compare the two fractions: (7 / 3) _ (4 / 5)

Gemma 2B:

  Reasoning: 7/3 = <<7/3=2.3333333333333335>>2.3333333333333335 and 4/5 = <<4/5=0.8>>0.8. Since 2.3333333333333335 > 0.8, 7/3 > 4/5.
### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Compare the following two fractions by filling the blank with <, >, or =. (6 / 4) _ (2 / 2)

Gemma 2B:

 Reasoning: 6/4 = <<6/4=3/2>>3/2 and 2/2 = <<2/2=1>>1.
3/2 _ 1
### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Now consider a change in the problem. Compare the following two fractions by filling the blank with <, >, or =. (6 / 4) _ (2 / 10) In this scenario, suppose we replace the whole number in the denominator of the second fraction from 2 to 10. How does this change the comparison?

Gemma 2B:

  Reasoning: 6/4 = <<6/4=3/2>>3/2 and 2/10 = <<2/10=1/5>>1/5.
3/2 _ 1/5
### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Compare the following fractions by filling in the blank with <, >, or =. (5 / 5) _ (9 / 10)

Gemma 2B:

 Reasoning: 5/5 = 1 and 9/10 < 1. Therefore, 5/5 > 9/10.
### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Consider this follow-up to our previous fraction comparison question: Now, compare these fractions by filling in the blank with <, >, or =. (4 / 5) _ (9 / 10) Think of it like this: if you had 4 parts of a cake that was sliced into 5 equal pieces versus someone who had 9 parts of a cake that was sliced into 10 equal pieces, which one would have more cake?

Gemma 2B:

  Reasoning: 4/5 < 9/10.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Fill in the blank (1 / 5) _ (1 / 1) with <, >, or = to compare the two fractions.

Gemma 2B:

 Reasoning: 1/5 = <<1/5=1/5>>1/5
### Answer: =

Expected answer: <

Correct? No

🔗 Problem

Problem: Fill the blank with either <, >, or = to compare the two fractions in the following equation: (7 / 1) _ (4 / 2)

Gemma 2B:

 Reasoning: 7/1 = 7 and 4/2 = 2. 7 > 2.
### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Consider if instead of dividing 4 by 2, we divided it by 7 in the previous question. Fill in the blank in the following equation with <, >, or = to compare the new fractions: (7 / 1) _ (4 / 7)

Gemma 2B:

  Reasoning: 7/1 = 7 and 4/7 = 4/7. 7 > 4/7.
### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Compare the following fractions by using <, >, or = to fill in the blank: (4 / 7) _ (9 / 4)

Gemma 2B:

 Reasoning: 4/7 = <<4/7=0.5714285714285714>>0.5714285714285714 and 9/4 = <<9/4=2.25>>2.25. Since 0.5714285714285714 < 2.25, 4/7 < 9/4.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Consider the change in fraction comparison by filling in the blank with <, >, or =. Now, observe the relationship between the fractions (4 / 7) and (2 / 4). How does the comparison change from the previous problem with the fraction (9 / 4)?

Gemma 2B:

  Reasoning: 4/7 = <<4/7=0.5714285714285714>>0.5714285714285714 and 2/4 = <<2/4=0.5>>0.5. Since 0.5714285714285714 > 0.5, 4/7 > 2/4.
### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Fill in the blank in the question (5 / 9) _ (4 / 7) with <, >, or = to compare the two fractions.

Gemma 2B:

 Reasoning: 5/9 = <<5/9=0.5555555555555556>>0.5555555555555556 and 4/7 = <<4/7=0.5714285714285714>>0.5714285714285714. 0.5555555555555556 < 0.5714285714285714.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Now, consider a slight change to the previous question: Compare the following fractions by filling in the blank with <, >, or =: (5 / 9) _ (4 / 10) What happens when we change the denominator of the second fraction from 7 to 10?

Gemma 2B:

  Reasoning: 5/9 = <<5/9=0.5555555555555556>>0.5555555555555556 and 4/10 = <<4/10=0.4>>0.4. 0.5555555555555556 > 0.4.
### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Compare the following fractions and fill in the blank with <, >, or =: (8 / 3) _ (7 / 3)

Gemma 2B:

 Reasoning: 8/3 = <<8/3=2 2/3>>2 2/3 and 7/3 = <<7/3=2 1/3>>2 1/3. 2 2/3 > 2 1/3, so 8/3 > 7/3.
(8 / 3) > (7 / 3)

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Continuing from the previous problem, now consider what happens when the fraction changes. Using the same symbols, <, >, or =, compare the following fractions: (8 / 9) _ (7 / 3)

Gemma 2B:

  Reasoning: 8/9 = <<8/9=0 8/9>>0 8/9 and 7/3 = <<7/3=2 1/3>>2 1/3. 0 8/9 < 2 1/3, so 8/9 < 7/3.
(8 / 9) < (7 / 3)

Expected answer: <

Correct? Yes

🔗 Problem

Problem: In the equation: (5 / 1) _ (2 / 2), replace the underscore with one of the following signs: <, >, or = to compare the two fractions.

Gemma 2B:

 Reasoning: (5 / 1) = (5 / 1) = <<5/1=5>>5
(2 / 2) = (1 / 1) = <<2/2=1>>1
5 > 1
### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Given that you found the relation between the fractions (5 / 1) and (2 / 2) earlier, now consider if we replace the (2 / 2) with a new fraction. In the updated equation: (5 / 1) _ (7 / 2), replace the underscore with one of the following signs: <, >, or = to compare the two fractions. How does this change the result, if at all?

Gemma 2B:

  Reasoning: (5 / 1) = (5 / 1) = <<5/1=5>>5
(7 / 2) = (3.5 / 1) = <<7/2=3.5>>3.5
5 > 3.5
### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Compare the following fractions and fill in the blank with either <, >, or =: (5 / 8) _ (3 / 3)

Gemma 2B:

 Reasoning: 5/8 = <<5/8=5/8>>5/8, 3/3 = <<3/3=1>>1, 5/8 _ 1 = <<5/8<1>><.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: For a follow-up question: In our previous question, we compared (5 / 8) to (3 / 3), but what if the first fraction was a little different? Consider this scenario where you now have 5 fourths of a pizza instead of 5 eighths. Keeping the second fraction the same at (3 / 3), fill in the blank with <, >, or = to appropriately compare the fractions: (5 / 4) _ (3 / 3)

Gemma 2B:

  Reasoning: 5/4 = <<5/4=5/4>>5/4, 3/3 = <<3/3=1>>1, 5/4 _ 1 = <<5/4>1><.
### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Determine which symbol : <, >, or =, is appropriate to fill the blank in the following comparison: (5 / 8) _ (6 / 10)

Gemma 2B:

 Reasoning: (5 / 8) = <<5/8=0.625>>0.625 and (6 / 10) = <<6/10=0.6>>0.6.
(5 / 8) > (6 / 10)
### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: In the same vein as the previous problem, consider a new comparison: (5 / 8) _ (6 / 5). Which symbol: <, >, or =, correctly complete the comparison? Keep in mind how your answer might change compared to the previous problem because of the different denominator in the second fraction.

Gemma 2B:

  Reasoning: (5 / 8) = <<5/8=0.625>>0.625 and (6 / 5) = <<6/5=1.2>>1.2.
(5 / 8) < (6 / 5)
### Answer: <

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Compare the two fractions by filling in the blank with <, >, or =: (5 / 10) _ (6 / 6)

Gemma 2B:

 Reasoning: 5/10 = 1/2 and 6/6 = 1. 1/2 < 1. 5/10 < 6/6. 
### Answer: <

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Now consider the change in fractions: (5 / 10) _ (6 / 1) Fill in the blank with <, >, or = to compare these two fractions. Suppose that out of 10 parts, you only have 5 parts, like the 5 apples from the previous problem. Compare this to a situation where you have 6 whole items, like the oranges. Which one has more parts or items?

Gemma 2B:

  Reasoning: 5/10 = 1/2 and 6/1 = 6. 1/2 < 6. 5/10 < 6/1. 
### Answer: <

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Compare these two fractions by filling in the blank with <, >, or =. (5 / 4) _ (2 / 4)

Gemma 2B:

 Reasoning: 5/4 = <<5/4=1.25>>1.25 and 2/4 = <<2/4=0.5>>0.5. 1.25 > 0.5.
(5 / 4) > (2 / 4)

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Considering the previous comparison between the fractions, what would be the result if we changed the fractions to: (5 / 4) _ (2 / 10) Fill in the blank with <, >, or = to make the comparison.

Gemma 2B:

  Reasoning: 5/4 = <<5/4=1.25>>1.25 and 2/10 = <<2/10=0.2>>0.2. 1.25 > 0.2.
(5 / 4) > (2 / 10)

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Compare the values of the following fractions by using <, >, or = to fill in the blank: (6 / 10) _ (6 / 10)

Gemma 2B:

 Reasoning: 6/10 = 6/10
### Answer: =

Expected answer: =

Correct? Yes

🔗 Follow-up

Problem: Following the last problem, consider a change such that instead of having 6 tenths of a pizza, you now have 6 fifths of a pizza. Compare this with the previous scenario by using <, >, or = to fill in the blank: (6 / 5) _ (6 / 10)

Gemma 2B:

  Reasoning: 6/5 > 6/10
### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Please compare these two fractions and fill the blank with <, >, or =. (6 / 8) _ (2 / 9)

Gemma 2B:

 Reasoning: 6/8 = 3/4 and 2/9 = 2/9. 3/4 > 2/9.
(6 / 8) > (2 / 9)

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Now, let's consider a slight change to the previous comparison. Instead of having a fraction of 6/8, let's change the numerator to 6 and denominator to 10. The new comparison now is: (6 / 10) _ (2 / 9) Please compare these two fractions and fill the blank with <, >, or =.

Gemma 2B:

  Reasoning: 6/10 = 3/5 and 2/9 = 2/9. 3/5 < 2/9.
(6 / 10) < (2 / 9)

Expected answer: >

Correct? No

🔗 Problem

Problem: Fill in the blank with <, =, or > after comparing the fractions below: (2 / 10) _ (6 / 2)

Gemma 2B:

 Reasoning: 2/10 = 1/5 and 6/2 = 3. 1/5 < 3.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Let's modify the previous problem slightly. Fill in the blank with <, =, or > after comparing the fractions: (2 / 10) _ (6 / 9)

Gemma 2B:

  Reasoning: 2/10 = 1/5 and 6/9 = 2/3. 1/5 < 2/3.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Determine which of the following fractions is greater by filling in the blank with <, >, or =. (2 / 7) _ (9 / 4)

Gemma 2B:

 Reasoning: (2 / 7) = <<2/7=0.2857142857142857>>0.2857142857142857 and (9 / 4) = <<9/4=2.25>>2.25. Since 0.2857142857142857 < 2.25, (2 / 7) < (9 / 4).
### Answer: <

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Now suppose we change one of the fractions in the previous problem. Fill in the blank to compare the two fractions: (2 / 1) _ (9 / 4)

Gemma 2B:

  Reasoning: (2 / 1) = <<2/1=2>>2 and (9 / 4) = <<9/4=2.25>>2.25. Since 2 < 2.25, (2 / 1) < (9 / 4).
### Answer: <

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Compare the following fractions by filling in the blank with <, >, or =. (1 / 1) _ (2 / 10)

Gemma 2B:

 Reasoning: 1/1 = 1, 2/10 = 0.2, 1 > 0.2, so 1/1 > 2/10.
### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Consider the previous comparison between the fractions (1 / 1) and (2 / 10). Now let's change things a bit. Fill in the blank with <, >, or = to compare the following fractions: (10 / 1) _ (2 / 10)

Gemma 2B:

  Reasoning: 10/1 = 10, 2/10 = 0.2, 10 > 0.2, so 10/1 > 2/10.
### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Compare the results of the following fractions by filling the blank with <, >, or =. (8 / 2) _ (10 / 9)

Gemma 2B:

 Reasoning: 8/2 = <<8/2=4>>4 and 10/9 = <<10/9=1.1111111111111111>>1.1111111111111111. 4 _ 1.1111111111111111 = <<4>1.1111111111111111>>>. 
### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Given the results you obtained from the previous question, let's alter the fractions slightly. Now, compare the following expressions: (8 / 2) _ (9 / 9) Fill in the blank with <, >, or = to indicate whether the first fraction is less than, greater than, or equal to the second fraction. What changes when the denominator of the second fraction is 9 instead of 10?

Gemma 2B:

  Reasoning: 8/2 = <<8/2=4>>4 and 9/9 = <<9/9=1>>1. 4 _ 1 = <<4>1>>>. 
### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Determine the relationship between the fractions by filling in the blank in the following equation with either <, >, or =: (3 / 2) _ (9 / 8)

Gemma 2B:

 Reasoning: (3 / 2) = <<3/2=1.5>>1.5 and (9 / 8) = <<9/8=1.125>>1.125. Since 1.5 > 1.125, (3 / 2) > (9 / 8).
### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Determine the relationship between the fractions by filling in the blank in the following equation with either <, >, or =: (3 / 2) _ (9 / 6) Consider what happens if the denominator of the second fraction was 6 instead of 8. How would that affect your answer?

Gemma 2B:

  Reasoning: (3 / 2) = <<3/2=1.5>>1.5 and (9 / 6) = <<9/6=1.5>>1.5. Since 1.5 = 1.5, (3 / 2) = (9 / 6).
### Answer: =

Expected answer: =

Correct? Yes

🔗 Problem

Problem: Compare the following quantities by filling in the blank with <, >, or =: (4 / 6) _ (6 / 10)

Gemma 2B:

 Reasoning: 4/6 = <<4/6=2/3>>2/3 and 6/10 = <<6/10=3/5>>3/5. 2/3 _ 3/5 = <<2/3<3/5>><.
### Answer: <

Expected answer: >

Correct? No

🔗 Problem

Problem: Fill in the blank with the correct symbol to compare the following fractions: (8 / 10) _ (1 / 1)

Gemma 2B:

 Reasoning: 8/10 = 4/5, 1/1 = 1/1. 4/5 < 1/1.
(8 / 10) < (1 / 1)

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Let's consider a slight change to our earlier comparison. What if instead of having 8 out of 10 parts, you now have 8 divided by 2 parts? Fill in the blank with the correct symbol to compare the following fractions: (8 / 2) _ (1 / 1)

Gemma 2B:

  Reasoning: 8/2 = 4, 1/1 = 1. 4 > 1.
(8 / 2) > (1 / 1)

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Using either <, >, or =, compare the following fractions by filling in the blank: (5 / 1) _ (5 / 8)

Gemma 2B:

 Reasoning: 5/1 = 5 and 5/8 < 5.
5/1 > 5/8

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Following up from the previous question, let's change the fractions and compare again. Using either <, >, or =, complete the new mathematical comparison below: (5 / 3) _ (5 / 8) Think of it this way, if the basket had initially 5/3 of the apples instead of 5 whole apples, and we still have 5/8 of the apples from the other basket. Would the first basket have more, less or the same number of apples than the second?

Gemma 2B:

  Reasoning: 5/3 > 5/8.
5/3 > 5/8

Expected answer: >

Correct? Yes

🔗 Problem

Problem: What is the correct symbol to use in the following statement: (6 / 5) _ (9 / 6) Is it "<", ">", or "=" ? Fill in the blank with the correct symbol.

Gemma 2B:

 Reasoning: (6 / 5) = <<6/5=1.2>>1.2, (9 / 6) = <<9/6=1.5>>1.5, 1.2 _ 1.5.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Considering the previous question, what if we change the first fraction to 3 / 5 instead of 6 / 5? So the question becomes: (3 / 5) _ (9 / 6) Again, fill in the blank with the correct symbol – is it "<", ">", or "=" ?

Gemma 2B:

  Reasoning: (3 / 5) = <<3/5=0.6>>0.6, (9 / 6) = <<9/6=1.5>>1.5, 0.6 _ 1.5.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Compare the following fractions by filling in the blank: (7 / 5) _ (1 / 10) You can use <, > or =.

Gemma 2B:

 Reasoning: 7/5 = <<7/5=1.4>>1.4 and 1/10 = <<1/10=0.1>>0.1. Since 1.4 > 0.1, 7/5 > 1/10.
### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Let's consider a variation of the previous problem. Now require the comparison of two different fractions: (1 / 5) _ (1 / 10) Use <, > or = to complete the comparison. This question asks you to think about what happens when we change the number of parts the whole is divided into.

Gemma 2B:

  Reasoning: 1/5 = <<1/5=0.2>>0.2 and 1/10 = <<1/10=0.1>>0.1. Since 0.2 > 0.1, 1/5 > 1/10.
### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Compare the two fractions and fill in the blank with <, >, or =. (10 / 8) _ (10 / 1)

Gemma 2B:

 Reasoning: 10/8 = <<10/8=1.25>>1.25 and 10/1 = <<10/1=10>>10.
10/8 _ 10/1
### Answer: <

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Reevaluate the comparison from the previous problem, but this time compare the fractions (10 / 8) and (10 / 3). Fill in the blank with <, >, or =. (10 / 8) _ (10 / 3)

Gemma 2B:

  Reasoning: 10/8 = <<10/8=1.25>>1.25 and 10/3 = <<10/3=3.3333333333333335>>3.3333333333333335.
10/8 _ 10/3
### Answer: <

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Compare the values of the following fractions by filling in the blank with <, >, or =: (3 / 2) _ (4 / 6)

Gemma 2B:

 Reasoning: 3/2 = <<3/2=1.5>>1.5 and 4/6 = <<4/6=0.6666666666666666>>0.6666666666666666. 1.5 _ 0.6666666666666666 = <<1.5>0.6666666666666666>>>. 
### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Consider the change from the original problem. Now instead of having 3 halves, imagine it was changed to 3 ninths. Complete the comparison: (3 / 9) _ (4 / 6) Fill the blank with <, >, or =.

Gemma 2B:

  Reasoning: 3/9 = <<3/9=0.3333333333333333>>0.3333333333333333 and 4/6 = <<4/6=0.6666666666666666>>0.6666666666666666. 0.3333333333333333 _ 0.6666666666666666 = <<0.3333333333333333<0.6666666666666666>>>. 
### Answer: <

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Compare the following two fractions by filling in the blank with <, >, or =. (9 / 7) _ (4 / 6)

Gemma 2B:

 Reasoning: 9/7 = <<9/7=1.2857142857142857>>1.2857142857142857 and 4/6 = <<4/6=0.6666666666666666>>0.6666666666666666. Since 1.2857142857142857 > 0.6666666666666666, 9/7 > 4/6.
### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Consider the following scenario. What happens if we change the math problem (9 / 7) _ (4 / 6) to consider a different set of fractions? Fill in the blank with <, >, or = to compare these two new fractions. (7 / 7) _ (4 / 6)

Gemma 2B:

  Reasoning: 7/7 = <<7/7=1>>1 and 4/6 = <<4/6=0.6666666666666666>>0.6666666666666666. Since 1 > 0.6666666666666666, 7/7 > 4/6.
### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Fill in the blank in the following comparison with a <, >, or = sign. (8 / 7) _ (8 / 8)

Gemma 2B:

 Reasoning: 8/7 = <<8/7=1.1428571428571428>>1.1428571428571428 and 8/8 = <<8/8=1>>1. 
8/7 _ 8/8
### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Consider the following change in fractions: (7 / 7) _ (8 / 8) Replace the earlier fractions with these new numbers. How do they compare now? Use the symbols <, >, or = to indicate your answer.

Gemma 2B:

  Reasoning: 7/7 = <<7/7=1>>1 and 8/8 = <<8/8=1>>1. 
7/7 _ 8/8
### Answer: =

Expected answer: =

Correct? Yes

🔗 Problem

Problem: Fill in the blank with the correct symbol (<, >, =) to compare these two fractions: (5 / 5) _ (4 / 6)

Gemma 2B:

 Reasoning: 5/5 = 1 and 4/6 = 2/3. 1 < 2/3. 5/5 < 4/6.
### Answer: <

Expected answer: >

Correct? No

🔗 Problem

Problem: Compare the following fractions and use <, >, or = to fill the blank: (10 / 10) _ (4 / 10)

Gemma 2B:

 Reasoning: 10/10 = 1, 4/10 = 0.4, 1 > 0.4, so 10/10 > 4/10.
### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Consider the following fractions and use <, >, or = to fill the blank: (10 / 10) _ (4 / 9) This is very similar to the last problem where we compared (10 / 10) to (4 / 10). What changes when the denominator of the second fraction is now 9 instead of 10?

Gemma 2B:

  Reasoning: 10/10 = 1, 4/9 = 0.4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444

Expected answer: >

Correct? No

🔗 Problem

Problem: Compare the following fractions and fill in the blank with <, >, or =. (8 / 1) _ (4 / 9)

Gemma 2B:

 Reasoning: 8/1 = <<8/1=8>>8 and 4/9 = <<4/9=0.4444444444444444>>0.4444444444444444. 8 > 0.4444444444444444.
8/1 > 4/9
### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Consider the values of the following fractions and fill in the blank with <, >, or =. How would your answer change if instead of 8 in the previous problem, we had 5? (5 / 1) _ (4 / 9)

Gemma 2B:

  Reasoning: 5/1 = <<5/1=5>>5 and 4/9 = <<4/9=0.4444444444444444>>0.4444444444444444. 5 > 0.4444444444444444.
5/1 > 4/9
### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Compare the values of the two fractions below by choosing either <, >, or = to fill in the blank: (9 / 3) _ (4 / 8)

Gemma 2B:

 Reasoning: 9/3 = 3 and 4/8 = 1/2. 3 > 1/2, so (9/3) > (4/8).
### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Now reconsider the previous fractions, only this time they have slightly changed: Compare the values of the following fractions by completing the sentence with one of the following options: <, >, or = : (9 / 3) _ (4 / 2) Think of this like the previous situation, but instead of the second fraction having a numerator of 4 and a denominator of 8, it now has a numerator of 4 and denominator of 2. How does this change the comparison?

Gemma 2B:

  Reasoning: 9/3 = 3 and 4/2 = 2. 3 > 2, so (9/3) > (4/2).
### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Compare the fractions in the expression (10 / 7) _ (9 / 6). Fill in the blank with <, >, or =.

Gemma 2B:

 Reasoning: 10/7 = <<10/7=1.4285714285714286>>1.4285714285714286 and 9/6 = <<9/6=1.5>>1.5. Since 1.4285714285714286 < 1.5, 10/7 < 9/6.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Now, let's consider a change to the previous problem. Instead of comparing (10 / 7) and (9 / 6), compare the fractions (10 / 7) _ (9 / 3). Using the same method as before, fill in the blank with <, >, or =. Is this new ratio greater than, less than, or equal to the first one?

Gemma 2B:

  Reasoning: 10/7 = <<10/7=1.4285714285714286>>1.4285714285714286 and 9/3 = <<9/3=3>>3. Since 1.4285714285714286 < 3, 10/7 < 9/3.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Compare the following fractions, and complete the statement: (10 / 8) _ (3 / 1) Fill the blank space with <, >, or =.

Gemma 2B:

 Reasoning: 10/8 = 5/4, 3/1 = 3. 5/4 < 3. 
### Answer: <

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Consider the following modification to our previous comparison: (10 / 7) _ (3 / 1) How does it affect the comparison? Place <, >, or = in the blank to make the statement true.

Gemma 2B:

  Reasoning: 10/7 = 5/7/3, 3/1 = 3. 5/7 < 3. 
### Answer: <

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Determine the relationship between the two fractions by filling in the blank with <, >, or =. The question is as follows: (3 / 2) _ (3 / 5)

Gemma 2B:

 Reasoning: 3/2 = <<3/2=1.5>>1.5 and 3/5 = <<3/5=0.6>>0.6. 1.5 > 0.6, so 3/2 > 3/5.
### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Consider the following scenario: You had 4 cups of blueberries and 2 cups of strawberries (4/2) and now have replaced the blueberries with 3 cups of raspberries. Keeping the strawberries still 5 cups, tell whether the new ratio of raspberries to strawberries (3/5) is more than, less than, or equal to the original ratio of blueberries to strawberries. Fill in the blank in the question (4 / 2) _ (3 / 5) with <, >, or =.

Gemma 2B:

 Reasoning: 4/2 = <<4/2=2>>2 and 3/5 = <<3/5=0.6>>0.6. 2 > 0.6, so 4/2 > 3/5.
### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Compare the following two fractions by filling in the blank with <, >, or =. (4 / 10) _ (2 / 2)

Gemma 2B:

 Reasoning: 4/10 = <<4/10=0.4>>0.4 and 2/2 = <<2/2=1>>1. 0.4<_>1. Therefore, 4/10<_>2/2.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: After comparing (4 / 10) and (2 / 2), consider the scenario where instead of (2 / 2), the comparison fraction becomes (9 / 2). Now, compare these two fractions: (4 / 10) _ (9 / 2) Fill in the blank with either <, >, or =.

Gemma 2B:

  Reasoning: 4/10 = <<4/10=0.4>>0.4 and 9/2 = <<9/2=4.5>>4.5. 0.4<_>4.5. Therefore, 4/10<_>9/2.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Compare the two fractions and fill in the blank with <, >, or =: (6 / 7) _ (2 / 10)

Gemma 2B:

 Reasoning: 6/7 = <<6/7=0.8571428571428571>>0.8571428571428571 and 2/10 = <<2/10=0.2>>0.2. 0.8571428571428571 > 0.2.
(6 / 7) > (2 / 10)

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Following the previous question, now consider if the first fraction had been (5 / 7) instead. How would this change compare to the original (6 / 7)? Fill in the blank with <, >, or =: (5 / 7) _ (2 / 10)

Gemma 2B:

  Reasoning: 5/7 = <<5/7=0.7142857142857143>>0.7142857142857143 and 2/10 = <<2/10=0.2>>0.2. 0.7142857142857143 > 0.2.
(5 / 7) > (2 / 10)

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Compare the values of the following fractions by filling in the blank with <, >, or =. (9 / 7) _ (10 / 3)

Gemma 2B:

 Reasoning: 9/7 = <<9/7=1.2857142857142857>>1.2857142857142857 and 10/3 = <<10/3=3.3333333333333335>>3.3333333333333335. Since 1.2857142857142857 < 3.3333333333333335, 9/7 < 10/3.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Consider a continuation of the previous comparison of fractions. Now, compare the following fractions by filling in the blank with <, >, or =. (9 / 7) _ (10 / 7) In other words, what would happen if the denominator of the second fraction was 7 instead of 3, keeping the ratio of 9 to 7 in the first fraction the same? How would this change affect the comparison?

Gemma 2B:

  Reasoning: 9/7 = <<9/7=1.2857142857142857>>1.2857142857142857 and 10/7 = <<10/7=1.4285714285714286>>1.4285714285714286. Since 1.2857142857142857 < 1.4285714285714286, 9/7 < 10/7.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Compare the following two fractions by filling in the blank with <, >, or =. (8 / 8) _ (2 / 8)

Gemma 2B:

 Reasoning: 8/8 = 1 and 2/8 = 1/4. 1/4 is less than 1. 1<_4.
### Answer: <

Expected answer: >

Correct? No

🔗 Problem

Problem: Evaluate the following expression by comparing the two fractions using <, >, or =: (4 / 3) _ (1 / 8)

Gemma 2B:

 Reasoning: 4/3 = <<4/3=1.3333333333333333>>1.3333333333333333 and 1/8 = <<1/8=0.125>>0.125. Since 1.3333333333333333 > 0.125, 4/3 > 1/8.
### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Consider the following problem: (4 / 9) _ (1 / 8) In the previous scenario, we compared the differences between 4/3 and 1/8. Now, let's change the first fraction from 4/3 to 4/9. Fill in the blank by comparing the fractions using <, >, or =. Would the relationship between the fractions be the same, or it would be different?

Gemma 2B:

  Reasoning: 4/9 = <<4/9=0.4444444444444444>>0.4444444444444444 and 1/8 = <<1/8=0.125>>0.125. Since 0.4444444444444444 < 0.125, 4/9 < 1/8.
### Answer: <

Expected answer: >

Correct? No

🔗 Problem

Problem: Compare the following fractions by completing the sentence with the appropriate symbol (<, >, or =): (4 / 6) _ (6 / 10)

Gemma 2B:

 Reasoning: 4/6 = <<4/6=2/3>>2/3 and 6/10 = <<6/10=3/5>>3/5. 2/3 _ <<2/3>3/5>> > 3/5.
### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Now consider this scenario: There are some large birds in a grove: 4 of every 6 are hawks, while the rest are falcons. In comparison, at another location 6 out of every 1 bird is a raven. Compare the ratios of hawks to total birds and of ravens to total birds. Fill in replacing _ with the correct sign (either <, >, or =). (4 / 6) _ (6 / 1)

Gemma 2B:

  Reasoning: 4/6 = <<4/6=2/3>>2/3 and 6/1 = <<6/1=6>>6. 2/3 _ <<2/3<6>> < 6.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Compare the two fractions by filling the blank using <, >, or =. Here's the question: (9 / 6) _ (6 / 8)

Gemma 2B:

 Reasoning: 9/6 = <<9/6=3/2>>3/2 and 6/8 = <<6/8=3/4>>3/4. 3/2 = <<3/2=1.5>>1.5 and 3/4 = <<3/4=0.75>>0.75. 1.5 > 0.75.
(9 / 6) > (6 / 8)

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Considering the same comparison concept as the previous problem, but this time let's change the fractions. Still filling the blank with <, >, or =, answer this question: (9/6) _ (4/8) Think about how this change, reducing the number in the numerator of the second fraction, affects the comparison between the two fractions.

Gemma 2B:

  Reasoning: 9/6 = <<9/6=3/2>>3/2 and 4/8 = <<4/8=1/2>>1/2. 3/2 = <<3/2=1.5>>1.5 and 1/2 = <<1/2=0.5>>0.5. 1.5 > 0.5.
(9 / 6) > (4 / 8)

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Compare the following fractions by filling in the blank with one of the following symbols: <, >, or =. (2 / 6) _ (8 / 7)

Gemma 2B:

 Reasoning: 2/6 = 1/3 and 8/7 = 1 1/7. 1/3 < 1 1/7, so 2/6 < 8/7.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Now think about a slightly different situation: Compare the two fractions below by filling the blank with <, >, or =. (2 / 7) _ (8 / 7) This time, instead of 2 out of 6 parts of a pizza, suppose you only got 2 out of 7 parts while the rest was still 8 out of 7. What would be the comparison between the two fractions then?

Gemma 2B:

  Reasoning: 2/7 < 8/7.
### Answer: <

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Compare the following fractions by filling in the blank with <, >, or =. (9 / 6) _ (4 / 10)

Gemma 2B:

 Reasoning: 9/6 = <<9/6=3/2>>3/2 and 4/10 = <<4/10=2/5>>2/5. 3/2 > 2/5, so 9/6 > 4/10.
(9 / 6) > (4 / 10)

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Now consider the new scenario. Instead of comparing the fractions (9 / 6) and (4 / 10), we want to compare (9 / 6) and (4 / 6). Fill in the blank with <, >, or = to compare these two fractions.

Gemma 2B:

  Reasoning: 9/6 = <<9/6=3/2>>3/2 and 4/6 = <<4/6=2/3>>2/3. 3/2 > 2/3, so 9/6 > 4/6.
(9 / 6) > (4 / 6)

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Fill in the blank in the following question with <, >, or = to correctly indicate which fraction is greater in value: (6 / 8) _ (5 / 1)

Gemma 2B:

 Reasoning: 6/8 = 3/4 and 5/1 = 5. 3/4 < 5.
(6 / 8) < (5 / 1)
### Answer: <

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Consider the modified question from before, but this time with different values in the fractions: (6 / 9) _ (5 / 1) Fill in the blank in the question with <, >, or = to correctly compare the two fractions. Just like we compared the earlier fractions, can you tell which of these two fractions is greater?

Gemma 2B:

  Reasoning: 6/9 = 2/3 and 5/1 = 5. 2/3 < 5.
(6 / 9) < (5 / 1)
### Answer: <

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Compare these two fractions by filling in the blank with <, >, or =. (8 / 4) _ (1 / 10)

Gemma 2B:

 Reasoning: 8/4 = <<8/4=2>>2 and 1/10 = <<1/10=0.1>>0.1.
2 _ 0.1
2 > 0.1
### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Now consider the comparison, (2 / 4) _ (1 / 10). How does this comparison change if we replace the fraction 8 / 4 from the previous problem with 2 / 4? Fill in the blank with <, >, or =.

Gemma 2B:

  Reasoning: 2/4 = <<2/4=0.5>>0.5 and 1/10 = <<1/10=0.1>>0.1.
0.5 _ 0.1
0.5 > 0.1
### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Compare the values of the following expressions by filling in the blank with <, >, or =. (6 / 3) _ (8 / 2)

Gemma 2B:

 Reasoning: (6 / 3) = <<6/3=2>>2 and (8 / 2) = <<8/2=4>>4. Since 2 < 4, (6 / 3) < (8 / 2).
### Answer: <

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: If we change the numbers a bit in our previous comparison and now consider (6 / 3) and (1 / 2), fill in the blank with <, >, or = to compare the two new values. (6 / 3) _ (1 / 2)

Gemma 2B:

  Reasoning: (6 / 3) = <<6/3=2>>2 and (1 / 2) = <<1/2=0.5>>0.5. Since 2 > 0.5, (6 / 3) > (1 / 2).
### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

Problem: Please compare the following two fractions and fill the blank with one of the symbols: <, >, or =: (4 / 6) _ (6 / 3)

Gemma 2B:

 Reasoning: 4/6 = 2/3 and 6/3 = 2. 2/3 is less than 2.
(4 / 6) < (6 / 3)
### Answer: <

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Great! Now, let's consider a minor change. Compare the following two fractions and fill in the blank with one of the symbols <, >, or =: (4 / 6) _ (5 / 3) How does the result change compared to the previous problem when we increase the numerator in the second fraction from 6 to 5?

Gemma 2B:

  Reasoning: 4/6 = 2/3 and 5/3 = 1.667. 2/3 is less than 1.667.
(4 / 6) < (5 / 3)
### Answer: <

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Fill in the blank with <, >, or = to compare the fractions 6 / 7 and 3 / 1 properly.

Gemma 2B:

 Reasoning: 6 / 7 = <<6/7=6*1/7*1=6/7>>6/7, 3 / 1 = <<3/1=3*7/1*7=21/7>>21/7, 6 / 7 < 3 / 1.
### Answer: 6 / 7 < 3 / 1

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Now let's change up the fractions a little. Let's consider 6 / 7 and 9 / 1 this time. Fill in the blank with <, >, or = to correctly compare these two fractions.

Gemma 2B:

  Reasoning: 6 / 7 = <<6/7=6*1/7*1=6/7>>6/7, 9 / 1 = <<9/1=9*7/1*7=63/7>>63/7, 6 / 7 < 9 / 1.
### Answer: 6 / 7 < 9 / 1

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Compare the values of the following fractions by filling in the blank with <, >, or =. (6 / 10) _ (9 / 10)

Gemma 2B:

 Reasoning: 6/10 = <<6/10=0.6>>0.6 and 9/10 = <<9/10=0.9>>0.9. 0.6 < 0.9, so 6/10 < 9/10.
(6 / 10) < (9 / 10)

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Now, consider if the fraction had been different. Compare the following fractions: (1 / 10) _ (9 / 10) How would your answer change from the previous question where you compared (6 / 10) and (9 / 10)?

Gemma 2B:

  Reasoning: 1/10 = <<1/10=0.1>>0.1 and 9/10 = <<9/10=0.9>>0.9. 0.1 < 0.9, so 1/10 < 9/10.
(1 / 10) < (9 / 10)

Expected answer: <

Correct? Yes

🔗 Problem

Problem: Compare the two fractions by filling in the blank with <, >, or =. (6 / 8) _ (4 / 9)

Gemma 2B:

 Reasoning: 6/8 = 3/4 and 4/9 = 8/36. 3/4 is greater than 8/36, so 6/8 is greater than 4/9.
(6 / 8) > (4 / 9)

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: As a follow-up to the previous question, now consider these two fractions. Fill in the blank with <, > or = to compare the values. (6 / 8) _ (4 / 4)

Gemma 2B:

  Reasoning: 6/8 = 3/4 and 4/4 = 1. 3/4 is less than 1, so 6/8 is less than 4/4.
(6 / 8) < (4 / 4)

Expected answer: <

Correct? Yes