🔗 Problem

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

Gemma 2B:

     Reasoning: 1/8 = 0.125, 4/1=4. 0.125<4.

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. 0.75<4.

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:

Expected answer: >

Correct? No

🔗 Problem

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

Gemma 2B:

     8/7 = 1 1/7
    6/8 = 3/4
    3/4 1 1/7
    ### Answer: 3/4 < 1 1/7

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:

      6/7 = 0 6/7
    6/8 = 3/4
    0 6/7 3/4
    ### Answer: 0 6/7 < 3/4

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 = <<9/3=3>>3, and 7/1 = <<7/1=7>>7.
    3 < 7, so 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 = <<9/3=3>>3, and 1/1 = <<1/1=1>>1.
    3 < 1, so 9/3 < 1/1.

Expected answer: >

Correct? No

🔗 Problem

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

Gemma 2B:

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:

Expected answer: >

Correct? No

🔗 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=1 1/2, 9/10=0 9/10. 1 1/2>0 9/10.>>1 1/2>0 9/10

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=0 3/7, 9/10=0 9/10. 0 3/7>0 9/10.>>0 3/7>0 9/10

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 = 1.

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: 5/5=1. 10/9= 1.

Expected answer: >

Correct? Yes

🔗 Problem

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

Gemma 2B:

Expected answer: >

Correct? No

🔗 Problem

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

Gemma 2B:

     Answer: 9/8 = 6/4

Expected answer: <

Correct? No

🔗 Problem

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

Gemma 2B:

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:

Expected answer: >

Correct? No

🔗 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, 6/5 = <<6/5=1.2>>1.2. 2>1.2. 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 = 1, 6/5 = 1.2. 1<1.2. 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: 3/2 = 1 1/2. 5/4 = 1 1/4. 1 1/4 < 1 1/2.

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:

     5/4 = 1 1/4. 8/3 = 2 2/3. 1 1/4 < 2 2/3.

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: 9/9 = 1. 7/5 = <<7/5=1.4>>1.4. 1.4 < 1. 7/5 < 9/9.

Expected answer: >

Correct? No

🔗 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.

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.

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:

     Answer:

Expected answer: >

Correct? No

🔗 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: 1 = 1

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 = <<6/5=1.2>>1.2. 1 < 1.2.
    ### Answer: 1 < 1.2

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=1.1666...>>1.1666... and 9/6 = <<9/6=1.5>>1.5. 1.1666... < 1.5. 7/6 < 9/6.

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=0.7777...>>0.7777... and 9/6 = <<9/6=1.5>>1.5. 0.7777... < 1.5. 7/9 < 9/6.

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:

     Answer: <<
    2/7=0.2857142857142857
    2/9=0.2222222222222222
    0.2857142857142857>0.2222222222222222
    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:

      Answer: <<
    2/7=0.2857142857142857
    2/4=0.5
    0.2857142857142857<0.5
    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:

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:

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:

    |user|

Expected answer: <

Correct? No

🔗 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 = <<8/9=0.

Expected answer: >

Correct? No

🔗 Problem

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

Gemma 2B:

     Reasoning: 9/10 = <<9/10=0.9>>0.9, 3/10 = <<3/10=0.3>>0.3, 0.9 > 0.3.

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.

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:

    |user|

Expected answer: <

Correct? No

🔗 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. 1.2>0.7, so 6/5>7/10.
    ### Answer: 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:

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 is <<1/9=0.

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 is <<1/2=0.5>>0.5.

Expected answer: >

Correct? No

🔗 Problem

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

Gemma 2B:

     Reasoning: 6/1=6, 7/1=7. 7/1=<<7/1=7>>7, so 7/1>6/1.

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:

     9/1=9, 6/1=6. 9/1=<<9/1=9>>9, so 9/1>6/1.

Expected answer: >

Correct? Yes

🔗 Problem

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

Gemma 2B:

     Answer: <<
    5/4=1.25
    9/8=1.125
    1.25>1.125
    5/4>9/8
    >>

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Please consider the following change to the previous problem and fill in the blank with <, >, or = to compare the fractions: If the value of the second fraction's denominator was 1 instead of 8, contrast the following fractions: (5 / 4) _ (9 / 1)

Gemma 2B:

      Answer: <<
    5/4=1.25
    9/1=9
    9>1.25
    5/4<9/1
    >>

Expected answer: <

Correct? Yes

🔗 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:

Expected answer: >

Correct? No

🔗 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 = <<2/1=2>>2, 4/9 = <<4/9=0.4444...>>0.4444...

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=1) 9/9=1>> 9/9 = 1. <<(2/2=1) 2/2=1>> 2/2 = 1. 9/9 = 2/2.
    ### 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) 9/3=3>> 9/3 = 3. 3/3 = 1. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 = 3/3. 9/3 =

Expected answer: >

Correct? No

🔗 Problem

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

Gemma 2B:

     6/1 = <<6/1=6>>6
    4/4 = <<4/4=1>>1
    6 _ 1

Expected answer: >

Correct? No

🔗 Problem

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

Gemma 2B:

     Reasoning: 8/8 = 1. 3/9 = 1/3. 1/3 < 1. 3/9 < 8/8.

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:

     3/9 = 1/3. 8/3 = 8/3. 8/3 is a larger number than 1/3. 3/9 < 8/3.

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=2/5>>2/5. 1/2 = <<1/2=1/2>>1/2. 2/5 < 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=4/5>>4/5. 4/5 > 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:

    |user|

Expected answer: <

Correct? No

🔗 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.

Expected answer: >

Correct? No

🔗 Problem

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

Gemma 2B:

     Reasoning: <<2/10=0.2>>0.2 = 2/10.
    <<6/4=1.5>>1.5 = 3/2.
    0.2 < 3/2.
    ### 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.
    6/4 = 3/2.
    3/5 = 0.6.
    3/2 = 1.5.
    0.6 < 1.5.
    ### 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:

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:

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=0.2 5/5=1>> 0.2 < 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=0.2 5/8=0.625 0.2<0.625>> 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:

     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:

      Answer: >

Expected answer: <

Correct? No

🔗 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: <<1/6=0.16666666666666666>>0.16666666666666666, and <<2/7=0.2857142857142857>>0.2857142857142857. 0.16666666666666666 < 0.2857142857142857. (2 / 7) is greater than (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: <<1/6=0.16666666666666666>>0.16666666666666666, and <<2/10=0.2>>0.2. 0.16666666666666666 < 0.2. (2 / 10) is greater than (1 / 6).

Expected answer: >

Correct? Yes

🔗 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=9/9=1>>1 is greater than or equal to <<9/10=0.9>>0.9.

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: The value of 9/9 is <<9/9=1>>1. The value of 9/5 is <<9/5=1.8>>1.8.

Expected answer: <

Correct? Yes

🔗 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
    5/2 = <<5/2=2.5>>2.5
    2 < 2.5
    ### Answer: 8/4 < 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
    5/2 = <<5/2=2.5>>2.5
    2.5 = 2.5
    ### Answer: 10/4 = 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: 1/10 = <<1/10=0.1>>0.1. 5/2 = <<5/2=2.5>>2.5. 2.5 > 0.1. 2.5 _ 0.1.
    ### Answer: 2.5 > 0.1

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: 3/2 = <<3/2=1.5>>1.5. 1.5 < 0.5. 3/2 < 1/2.
    ### Answer: 3/2 < 1/2

Expected answer: >

Correct? No

🔗 Problem

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

Gemma 2B:

     Reasoning: 9/2 = 4.5. 2/4 = 0.5. 0.5 < 4.5.

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: 9/10 = 0.9. 2/4 = 0.5. 0.9 > 0.5.

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:

    |user| The first fraction, 6/3, is equal to 2. The second fraction, 10/2, is also equal to 2. Since the two fractions are equal, the sign that should be in the blank is =.

Expected answer: <

Correct? No

🔗 Problem

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

Gemma 2B:

     Reasoning: 8/6 = <<8/6=1.3333...>>1.3333... and 6/3 = <<6/3=2>>2.
    (6/3) < (8/6)

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 = 0.75 and 6/5 = 1.2. 0.75 < 1.2.

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 = 0.75 and 6/9 = 0.66. 0.75 > 0.66.

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: 3/1 = <<3/1=3>>3, 1/1 = <<1/1=1>>1. 3>1.

Expected answer: <

Correct? No

🔗 Problem

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

Gemma 2B:

     Reasoning: <<7/6=1.1666666666666667, 4/5=0.8, 7/6>4/5.>>

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=2.3333333333333335, 4/5=0.8, 7/3>4/5.>>

Expected answer: >

Correct? Yes

🔗 Problem

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

Gemma 2B:

     Reasoning: 2/2 = 1. 6/4 = 1.5. 1.5 > 1. 6/4 > 2/2.

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: 1.5 > 0.2. 6/4 > 2/10.

Expected answer: >

Correct? Yes

🔗 Problem

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

Gemma 2B:

    |user|

Expected answer: >

Correct? No

🔗 Problem

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

Gemma 2B:

     Reasoning: <<1/5=0.2, 1/1=1.0>>0.2<1.0
    ### Answer: <

Expected answer: <

Correct? Yes

🔗 Follow-up

Problem: Now, let's consider a slight change in the problem. Fill in the blank (1 / 5) _ (2 / 1) with <, >, or = to compare the two fractions. This time, suppose you're comparing one fifth of your initially available sweets against two whole sweets someone offered you. Which is greater or are they equal?

Gemma 2B:

      Reasoning: <<1/5=0.2, 2/1=2.0>>0.2<2.0
    ### Answer: <

Expected answer: <

Correct? Yes

🔗 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 = <<7/1=7>>7, and 4/2 = <<4/2=2>>2.
    7/1 = 4/2, so the correct symbol is =.

Expected answer: >

Correct? No

🔗 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=0.571428571428571428571428571428571428571428571428571428571428571428>>0.571428571428571428571428571428571428571428571428571428571428571428
    9/4 = 2.25
    0.571428571428571428571428571428571428571428571428571428571428571428 < 2.25
    ### 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=0.571428571428571428571428571428571428571428571428571428571428571428>>0.571428571428571428571428571428571428571428571428571428571428571428
    2/4 = 0.5
    0.571428571428571428571428571428571428571428571428571428571428571428 < 0.5
    ### Answer: <

Expected answer: >

Correct? No

🔗 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) < (4 / 7) because 5/9 is less than 4/7.>>

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) = (4 / 10) because 5/9 and 4/10 are equivalent fractions.>>

Expected answer: >

Correct? No

🔗 Problem

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

Gemma 2B:

     Answer: <<

Expected answer: >

Correct? No

🔗 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>>5 and 2/2 = <<2/2=1>>1.
    5 is __ 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>>5 and 7/2 = <<7/2=3.5>>3.5.
    5 is __ 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=0.625>>0.625, 3/3 = <<3/3=1>>1
    0.625 < 1
    5/8 < 3/3

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=1.25>>1.25, 3/3 = <<3/3=1>>1
    1.25 > 1
    5/4 > 3/3

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:

Expected answer: >

Correct? No

🔗 Problem

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

Gemma 2B:

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

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

Expected answer: <

Correct? Yes

🔗 Problem

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

Gemma 2B:

     5/4 = <<5/4=1.25>>1.25
    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:

      5/4 = <<5/4=1.25>>1.25
    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=0.6, 6/10=0.6. 0.6=0.6. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/10. 6/10=6/

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=1.2, 6/10=0.6. 1.2>0.6. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6/10. 6/5>6

Expected answer: >

Correct? Yes

🔗 Problem

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

Gemma 2B:

     Reasoning: 2/9 = <<2/9=0.22>>0.22
    6/8 = <<6/8=0.75>>0.75
    0.22 < 0.75
    6/8 < 2/9
    ### Answer: 6/8 < 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:

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:

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:

     9/4 = <<9/4=2.25>>2.25, and 2/7 = <<2/7=0.28>>0.28. 9/4 is greater than 2/7.

Expected answer: <

Correct? No

🔗 Problem

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

Gemma 2B:

     Reasoning: 1/1 = <<1/1=1>>1 and 2/10 = <<2/10=0.2>>0.2
    0.2 < 1
    ### Answer:

Expected answer: >

Correct? No

🔗 Problem

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

Gemma 2B:

     8/2 = <<8/2=4>>4, 10/9 = <<10/9=1.1...>>1.1...
    4 < 1.1...

Expected answer: >

Correct? No

🔗 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=1.5, 9/8=1.125, 3/2>9/8>>3/2>9/8

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=1.5, 9/6=1.5, 3/2=9/6>>3/2=9/6

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 = 2/3, 6/10 = 3/5, 2/3 < 3/5
    ### Answer: 2/3 < 3/5

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:

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:

Expected answer: >

Correct? No

🔗 Problem

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

Gemma 2B:

     Reasoning: 5/8 = <<5/8=0.625>>0.625. 5/1 = <<5/1=5>>5. 0.625<5. 5/8<5/1.

Expected answer: >

Correct? No

🔗 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=1.2, 9/6=1.5. 9/6>6/5.>>

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=0.6, 9/6=1.5. 9/6>3/5.>>

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 = 1.4, 1/10 = 0.1. 1.4 > 0.1. The first fraction is greater.
    ### Answer: 7/5 > 1/10

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 = 0.2, 1/10 = 0.1. 0.2 > 0.1. The first fraction is greater.
    ### Answer: 1/5 > 1/10

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
    10/1 = <<10/1=10>>10
    1.25 < 10
    ### 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
    10/3 = <<10/3=3.33>>3.33
    1.25 < 3.33
    ### 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 = 2/3 = <<2/3=0.6666...>>0.6666...
    1.5 < 0.6666...
    ### Answer: <

Expected answer: >

Correct? No

🔗 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=1.28>>1.28, 4/6 = <<4/6=0.66>>0.66. 1.28 > 0.66.
    ### Answer: 9/7>4/6

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, 4/6 = <<4/6=0.66>>0.66. 1>0.66.
    ### Answer: 7/7>4/6

Expected answer: >

Correct? Yes

🔗 Problem

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

Gemma 2B:

Expected answer: >

Correct? No

🔗 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.

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 and 4/10 = <<4/10=0.4>>0.4
    1 < 0.4
    10/10 < 4/10

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 and 4/9 = 4/9. 8/1 = 8 is greater than 4/9.

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 and 4/9 = 4/9. 5/1 = 5 is less than 4/9.

Expected answer: >

Correct? No

🔗 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 = <<9/3=3>>3 and 4/8 = <<4/8=1/2>>1/2. 3 is greater than 1/2. So, 9/3 > 4/8.

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 = <<9/3=3>>3 and 4/2 = <<4/2=2>>2. 3 is greater than 2. So, 9/3 > 4/2.

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=1 2/7. 9/6=1 3/6. 1 2/7>1 3/6.>> The answer is >.

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=1 3/7. 9/3=3. 1 3/7<3.>> The answer is <.

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 = <<10/8=1.25>>1.25, and 3/1 = <<3/1=3>>3. 3 is more than 1.25.
    ### Answer: 3

Expected answer: <

Correct? No

🔗 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=1.5>>1.5 is __ 3/5.

Expected answer: >

Correct? No

🔗 Problem

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

Gemma 2B:

     Reasoning: 2/2 = 1, so 2/2 < 4/10.

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: 9/2 = 4.5, so 4/10 < 9/2.

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=0.85714285714285714285714285714286, 2/10=0.2. 0.85714285714285714285714285714286>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=0.71428571428571428571428571428571, 2/10=0.2. 0.71428571428571428571428571428571>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 = 1 2/7
    10/3 = 3 1/3
    3 1/3 > 1 2/7
    ### Answer: 3 1/3

Expected answer: <

Correct? No

🔗 Problem

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

Gemma 2B:

     Reasoning: 8/8 = 1, 2/8 = <<2/8=0.25>>0.25. 8/8 = <<8/8=1>>1. 1 > 0.25.
    ### Answer: 8/8>2/8

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Consider the following fractions: (8 / 10) _ (2 / 8) How does the comparison of these two fractions change from the previous problem where you compared 8 / 8 and 2 / 8? Fill in the blank with <, >, or = to express your answer.

Gemma 2B:

      Reasoning: 8/10 = 0.8, 2/8 = 0.25. 0.8 > 0.25.
    ### Answer: 8/10>2/8

Expected answer: >

Correct? Yes

🔗 Problem

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

Gemma 2B:

     Reasoning: 1/8 = <<1/8=0.125>>0.125
    4/3 = <<4/3=1.3333...>>1.3333...
    0.125 < 1.3333
    (1/8) < (4/3)
    ### 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 = 2/3 and 6/10 = 3/5. 2/3 < 3/5.

Expected answer: >

Correct? No

🔗 Problem

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

Gemma 2B:

     Answer: 9/6 = <<9/6=1.5>>1.5, and 6/8 = <<6/8=0.75>>0.75. 1.5 > 0.75.

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:

      Answer: 9/6 = <<9/6=1.5>>1.5, and 4/8 = <<4/8=0.5>>0.5. 1.5 > 0.5.

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 = <<2/6=1/3>>1/3, and 8/7 = <<8/7=1 1/7>>1 1/7. 1/3 < 1 1/7, so 2/6 < 8/7.

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 = <<2/7=2/7>>2/7, and 8/7 = <<8/7=1 1/7>>1 1/7. 2/7 = 1 1/7, so 2/7 = 8/7.

Expected answer: <

Correct? No

🔗 Problem

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

Gemma 2B:

     Reasoning: 9/6 = <<9/6=1.5>>1.5 and 4/10 = <<4/10=0.4>>0.4.
    1.5>0.4, so 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 = 1.5 and 4/6 = <<4/6=0.

Expected answer: >

Correct? No

🔗 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 = <<6/8=3/4>>3/4 and 5/1 = <<5/1=5>>5. 3/4 < 5.

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 = <<6/9=2/3>>2/3 and 5/1 = <<5/1=5>>5. 2/3 < 5.

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
    ### 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
    ### 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: 8/2=4. 6/3=2. 2<4. 6/3<8/2.

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: 1/2=0.5. 6/3=2. 0.5<2. 6/3>1/2.

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 = <<4/6=2/3>>2/3, and 6/3 = 2. 2/3 < 2. The symbol < should be in the blank.

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 2/3. 1 2/3 > 2/3. The symbol > should be in the blank. The result is that the second fraction is greater than the first fraction.

Expected answer: <

Correct? No

🔗 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=0.8571...>>0.8571. 3/1 = <<3/1=3>>3. 0.8571... < 3. 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=0.8571...>>0.8571. 9/1 = <<9/1=9>>9. 0.8571... < 9. 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.

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.

Expected answer: <

Correct? Yes

🔗 Problem

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

Gemma 2B:

     Answer: <<
    6/8=3/4
    4/9=2/3
    3/4>2/3
    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:

      Answer: <<
    6/8=3/4
    4/4=1
    3/4>1
    6/8>4/4
    >>

Expected answer: <

Correct? Yes