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.

LLemma 34B:

 Reasoning: 6/8 = <<6/8=0.75>>0.75 and 4/1 = <<4/1=4>>4.
6/8 < 4/1.
### Answer: <

Expected answer: <

Correct? Yes

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)

LLemma 34B:

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

Expected answer: >

Correct? Yes

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)

LLemma 34B:

 Reasoning: 6/8 = <<6/8=0.75>>0.75 and 6/7 = <<6/7=0.8571428571428571>>0.8571428571428571.
0.75 < 0.8571428571428571
### Answer: <

Expected answer: <

Correct? Yes

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.

LLemma 34B:

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

Expected answer: >

Correct? Yes

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)

LLemma 34B:

 Reasoning: (5 / 3) = <<5/3=1.6666666666666667>>1.6666666666666667
(7 / 5) = <<7/5=1.4>>1.4
1.6666666666666667 > 1.4
### Answer: >

Expected answer: >

Correct? Yes

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.

LLemma 34B:

 Reasoning: 3/7 = <<3/7=0.42857142857142855>>0.42857142857142855 and 9/10 = <<9/10=0.9>>0.9.
0.42857142857142855 < 0.9
### Answer: <

Expected answer: <

Correct? Yes

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

LLemma 34B:

  Reasoning: 10/9 = <<10/9=1.1111111111111112>>1.1111111111111112 and 5/5 = <<5/5=1.0>>1.0.
1.1111111111111112 > 1.0
### Answer: >

Expected answer: >

Correct? Yes

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)

LLemma 34B:

 Reasoning: 10/9 = <<10/9=1.1111111111111112>>1.1111111111111112 and 5/3 = <<5/3=1.6666666666666667>>1.6666666666666667.
1.1111111111111112 < 1.6666666666666667
### Answer: <

Expected answer: <

Correct? Yes

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 =.

LLemma 34B:

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

Expected answer: <

Correct? Yes

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?

LLemma 34B:

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

Expected answer: >

Correct? Yes

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 =.

LLemma 34B:

 Reasoning: 5/4 = <<5/4=1.25>>1.25 and 8/3 = <<8/3=2.666666666666667>>2.666666666666667.
5/4 < 8/3
### Answer: <

Expected answer: <

Correct? Yes

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)

LLemma 34B:

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

Expected answer: >

Correct? Yes

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.

LLemma 34B:

 Reasoning: 3/7 = <<3/7=0.42857142857142855>>0.42857142857142855
6/8 = <<6/8=0.75>>0.75
3/7 < 6/8
### Answer: <

Expected answer: <

Correct? Yes

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 =.

LLemma 34B:

  Reasoning: 
(3 / 1) _ (5 / 5)

3/1 = 3
5/5 = 1

3 > 1
### Answer: >

Expected answer: >

Correct? Yes

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.

LLemma 34B:

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

Expected answer: <

Correct? Yes

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?

LLemma 34B:

 Reasoning: 2/7 = <<2/7=0.2857142857142857>>0.2857142857142857 and 2/4 = <<2/4=0.5>>0.5.

2/7 < 2/4
### Answer: <

Expected answer: <

Correct? Yes

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?

LLemma 34B:

 Reasoning:

(9 / 10) = <<9/10=0.9>>0.9

(7 / 1) = <<7/1=7>>7

(9 / 10) < (7 / 1)

### Answer: <

Expected answer: <

Correct? Yes

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)

LLemma 34B:

 Reasoning: 4/8 = <<4/8=1/2>>1/2 and 2/6 = <<2/6=1/3>>1/3.
1/2 > 1/3
### Answer: >

Expected answer: >

Correct? Yes

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)

LLemma 34B:

  Reasoning: 9/4 is greater than 3/10.
### Answer: >

Expected answer: >

Correct? Yes

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)

LLemma 34B:

  Reasoning: (3 / 7) = <<3/7=0.42857142857142855>>0.42857142857142855
(5 / 5) = <<5/5=1.0>>1.0
(3 / 7) < (5 / 5)
### Answer: <

Expected answer: <

Correct? Yes

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)

LLemma 34B:

  Reasoning: 1/5 = <<1/5=0.2>>0.2 and 7/10 = <<7/10=0.7>>0.7.
0.2 < 0.7
### Answer: <

Expected answer: <

Correct? Yes

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)

LLemma 34B:

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

Expected answer: >

Correct? Yes

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)

LLemma 34B:

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

Expected answer: >

Correct? Yes

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.

LLemma 34B:

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

Expected answer: >

Correct? Yes

Problem: Now, let's consider a change in the fractions. Use the correct symbol (<, >, or =) to complete the following comparison: (2 / 1) _ (8 / 9)

LLemma 34B:

 Reasoning: 2/1 = <<2/1=2>>2 and 8/9 = <<8/9=0.8888888888888888>>0.8888888888888888.
2 > 0.8888888888888888.
### Answer: >

Expected answer: >

Correct? Yes

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 =.

LLemma 34B:

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

Expected answer: >

Correct? Yes

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?

LLemma 34B:

 Reasoning: 6/1 = <<6/1=6>>6 and 4/3 = <<4/3=1.3333333333333333>>1.3333333333333333.
6 > 1.3333333333333333.
### Answer: >

Expected answer: >

Correct? Yes

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.

LLemma 34B:

  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: 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 =.

LLemma 34B:

 Reasoning: 8/10 = <<8/10=4/5>>4/5 and 1/2 = <<1/2=1/2>>1/2.
4/5 > 1/2
### Answer: >

Expected answer: >

Correct? Yes

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)?

LLemma 34B:

 Reasoning: (6 / 4) = <<6/4=1.5>>1.5 and (6 / 3) = <<6/3=2>>2.
1.5 < 2
### Answer: <

Expected answer: <

Correct? Yes

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)

LLemma 34B:

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

Expected answer: >

Correct? Yes

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?

LLemma 34B:

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

Expected answer: <

Correct? Yes

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)

LLemma 34B:

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

Expected answer: >

Correct? No

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)

LLemma 34B:

  Reasoning: 2/9 = <<2/9=0.2222222222222222>>0.2222222222222222 and 5/8 = <<5/8=0.625>>0.625.

2/9 < 5/8
### Answer: 2/9 < 5/8

Expected answer: <

Correct? Yes

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 =.

LLemma 34B:

 Reasoning: 1/9 = <<1/9=0.1111111111111111>>0.1111111111111111 and 7/7 = <<7/7=1.0>>1.0.
0.1111111111111111 < 1.0
### Answer: <

Expected answer: <

Correct? Yes

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 =.

LLemma 34B:

 Reasoning: (2 / 10) = <<2/10=0.2>>0.2
(1 / 6) = <<1/6=0.16666666666666666>>0.16666666666666666
(2 / 10) is greater than (1 / 6).
### Answer: (2 / 10) > (1 / 6)

Expected answer: >

Correct? Yes

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.

LLemma 34B:

 Reasoning: 9/9 = 1 and 9/5 = 1.8.
1 > 1.8.
### Answer: >

Expected answer: <

Correct? No

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.

LLemma 34B:

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

Expected answer: =

Correct? No

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.

LLemma 34B:

 Reasoning: 5/2 = <<5/2=2.5>>2.5 and 1/2 = <<1/2=0.5>>0.5.
2.5 > 0.5
### Answer: >

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.

(3 / 2) _ (1 / 2)

Expected answer: >

Correct? Yes

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)

LLemma 34B:

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

Expected answer: <

Correct? No

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)

LLemma 34B:

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

Expected answer: <

Correct? Yes

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?

LLemma 34B:

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

Expected answer: >

Correct? Yes

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)

LLemma 34B:

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

Expected answer: >

Correct? Yes

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)

LLemma 34B:

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

Expected answer: >

Correct? Yes

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?

LLemma 34B:

 Reasoning: 6/4 = <<6/4=1.5>>1.5 and 2/10 = <<2/10=0.2>>0.2.
1.5 > 0.2.
### Answer: >

Expected answer: >

Correct? Yes

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?

LLemma 34B:

  Reasoning: 1/5 is less than 2/1.
### Answer: <

Expected answer: <

Correct? Yes

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)

LLemma 34B:

 Reasoning: 7/1 = 7 and 4/7 = <<4/7=0.5714285714285714>>0.5714285714285714.
7/1 > 4/7.
### Answer: >

Expected answer: >

Correct? Yes

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)?

LLemma 34B:

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

Expected answer: >

Correct? Yes

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?

LLemma 34B:

  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: 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)

LLemma 34B:

 Reasoning: 8/9 = <<8/9=0.8888888888888888>>0.8888888888888888 and 7/3 = <<7/3=2.3333333333333335>>2.3333333333333335.
8/9 < 7/3
### Answer: <

Expected answer: <

Correct? Yes

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?

LLemma 34B:

 Reasoning: 5/1 = 5 and 7/2 = 3.5.
5/1 > 7/2.
### Answer: >

Expected answer: >

Correct? Yes

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)

LLemma 34B:

 Reasoning: 5/4 = <<5/4=1.25>>1.25 and 3/3 = <<3/3=1>>1.
1.25 > 1
### Answer: >

Expected answer: >

Correct? Yes

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.

LLemma 34B:

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

Expected answer: <

Correct? Yes

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?

LLemma 34B:

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

Expected answer: <

Correct? Yes

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.

LLemma 34B:

 Reasoning: 5/4 = <<5/4=1.25>>1.25 and 2/10 = <<2/10=0.2>>0.2.
1.25 > 0.2
### Answer: >

Expected answer: >

Correct? Yes

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)

LLemma 34B:

 Reasoning: 6/5 = 3/2.5 and 6/10 = 3/5.
6/5 > 6/10
### Answer: >

Expected answer: >

Correct? Yes

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 =.

LLemma 34B:

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

Expected answer: >

Correct? Yes

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

LLemma 34B:

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

Expected answer: <

Correct? Yes

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)

LLemma 34B:

  Reasoning: 2/1 = <<2/1=2>>2 and 9/4 = <<9/4=2.25>>2.25.
2/1 > 9/4
### Answer: >

Expected answer: <

Correct? No

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?

LLemma 34B:

 Reasoning: 3/2 = <<3/2=1.5>>1.5 and 9/6 = <<9/6=1.5>>1.5.
1.5 = 1.5
### Answer: =

Expected answer: =

Correct? Yes

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)

LLemma 34B:

 Reasoning: 8/2 is less than 1/1.
### Answer: <

Expected answer: >

Correct? No

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?

LLemma 34B:

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

Expected answer: >

Correct? Yes

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 "=" ?

LLemma 34B:

 Reasoning: 3/5 = <<3/5=0.6>>0.6 and 9/6 = <<9/6=1.5>>1.5.
0.6 < 1.5
### Answer: <

Expected answer: <

Correct? Yes

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.

LLemma 34B:

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

Expected answer: >

Correct? Yes

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)

LLemma 34B:

 Reasoning: 10/8 = <<10/8=5/4>>5/4 and 10/3 = <<10/3=10/3>>10/3.
5/4 < 10/3
### Answer: <

Expected answer: <

Correct? Yes

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 =.

LLemma 34B:

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

Expected answer: <

Correct? Yes

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)

LLemma 34B:

  Reasoning: 7/7 = <<7/7=1.0>>1.0 and 4/6 = <<4/6=0.6666666666666666>>0.6666666666666666.
1.0 > 0.6666666666666666
### Answer: >

Expected answer: >

Correct? Yes

Problem: Consider the two fractions from before: (5 / 5) and (4 / 6). Now, change the latter fraction's numerator to 2 instead of 4. Fill in the blank with the correct symbol (<, >, =) to compare these two fractions: (5 / 5) _ (2 / 6)

LLemma 34B:

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

Expected answer: >

Correct? Yes

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?

LLemma 34B:

  Reasoning: 10/10 = 1 and 4/9 = 0.4444444444444444.
1 > 0.4444444444444444.
### Answer: >

Expected answer: >

Correct? Yes

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)

LLemma 34B:

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

Expected answer: >

Correct? Yes

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?

LLemma 34B:

 Reasoning: 9/3 = <<9/3=3>>3 and 4/2 = <<4/2=2>>2.
3 > 2.
### Answer: >

Expected answer: >

Correct? Yes

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?

LLemma 34B:

 Reasoning: (10 / 7) _ (9 / 3) = (10 / 7) _ (3) = (10 / 7) _ (3) = 4.2857142857142855.
(10 / 7) _ (9 / 3) = (10 / 7) _ (3) = (10 / 7) _ (3) = 4.2857142857142855.
### Answer: >

Expected answer: <

Correct? No

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.

LLemma 34B:

 Reasoning: 10/7 = <<10/7=10/7>>10/7 and 3/1 = <<3/1=3>>3.
10/7 < 3.
### Answer: <

Expected answer: <

Correct? Yes

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 =.

LLemma 34B:

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

Expected answer: >

Correct? Yes

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 =.

LLemma 34B:

 Reasoning: 4/10 = <<4/10=2/5>>2/5 and 9/2 = <<9/2=4.5>>4.5.
2/5 < 4.5.
### Answer: <

Expected answer: <

Correct? Yes

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)

LLemma 34B:

  Reasoning: 5/7 = <<5/7=0.7142857142857143>>0.7142857142857143 and 2/10 = <<2/10=0.2>>0.2.
5/7 > 2/10
### Answer: >

Expected answer: >

Correct? Yes

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?

LLemma 34B:

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

Expected answer: <

Correct? Yes

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.

LLemma 34B:

 Reasoning: 8/10 = 4/5 and 2/8 = 1/4.
4/5 > 1/4.
### Answer: >

Expected answer: >

Correct? Yes

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?

LLemma 34B:

  Reasoning:

(4 / 9) _ (1 / 8)

(4 / 9) = (4 * 8) / (9 * 8) = <<4*8=32>>32 / (9 * 8) = <<9*8=72>>32 / 72 = <<32/72=4/9>>4 / 9

(1 / 8) = (1 * 9) / (8 * 9) = <<1*9=9>>9 / (8 * 9) = <<8*9=72>>9 / 72 = <<9/72=1/8>>1 / 8

(4 / 9) < (1 / 8)

### Answer: <

Expected answer: >

Correct? No

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.

LLemma 34B:

  Reasoning:

(9 / 6) = (9 / 6) * (2 / 2) = (18 / 12)

(4 / 8) = (4 / 8) * (1 / 1) = (4 / 8)

(18 / 12) > (4 / 8)

### Answer: >

Expected answer: >

Correct? Yes

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?

LLemma 34B:

 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
### Answer: <

Expected answer: <

Correct? Yes

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.

LLemma 34B:

 Reasoning: 9/6 = <<9/6=1.5>>1.5 and 4/6 = <<4/6=0.6666666666666666>>0.6666666666666666.
1.5 > 0.6666666666666666.
### Answer: >

Expected answer: >

Correct? Yes

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?

LLemma 34B:

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

Expected answer: <

Correct? Yes

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 =.

LLemma 34B:

 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: 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)

LLemma 34B:

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

Expected answer: >

Correct? Yes

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?

LLemma 34B:

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

Expected answer: <

Correct? Yes

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.

LLemma 34B:

 Reasoning: 6/7 = <<6/7=0.8571428571428571>>0.8571428571428571 and 9/1 = <<9/1=9>>9.
6/7 < 9/1
### Answer: 6/7 < 9/1

Expected answer: <

Correct? Yes

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)?

LLemma 34B:

 Reasoning: 1/10 = <<1/10=0.1>>0.1 and 9/10 = <<9/10=0.9>>0.9.
0.1 < 0.9.
### Answer: <

Expected answer: <

Correct? Yes