InternLM-Math Base 7B 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: Complete the following comparison between two fractions. (1 / 8) _ (4 / 1).
InternLM-Math Base 7B:
Reasoning: 1/8 = 0.125, 4/1=4. 0.125<4.
Expected answer: <
Correct? Yes
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.
InternLM-Math Base 7B:
Reasoning: 6/8=0.75, 4/1=4. 0.75<4.
Expected answer: <
Correct? Yes
Problem: Determine which symbol appropriately fills in the blank (either '<', '=', or '>') to accurately compare these two fractions: (7 / 2) _ (10 / 6)
InternLM-Math Base 7B:
Expected answer: >
Correct? No
Problem: Complete the following comparison by filling in the blank (<, >, or =): (6 / 8) _ (8 / 7)
InternLM-Math Base 7B:
8/7 = 1 1/7 6/8 = 3/4 3/4 1 1/7 ### Answer: 3/4 < 1 1/7
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)
InternLM-Math Base 7B:
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: Compare the values of the two fractions below by filling in the blank with <, >, or =. (9 / 3) _ (7 / 1)
InternLM-Math Base 7B:
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
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.
InternLM-Math Base 7B:
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: Fill in the blank with one of the following signs: <, >, or = (5 / 3) _ (10 / 5)
InternLM-Math Base 7B:
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)
InternLM-Math Base 7B:
Expected answer: >
Correct? No
Problem: Compare the values of the two fractions by filling in the blank with <, >, or =. Here are the fractions: (3 / 2) _ (9 / 10)
InternLM-Math Base 7B:
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
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.
InternLM-Math Base 7B:
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: Compare the values of these fractions by filling in the blank with <, >, or =. (10 / 9) _ (5 / 6)
InternLM-Math Base 7B:
Reasoning: 10/9 = 1.1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
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)
InternLM-Math Base 7B:
Reasoning: 5/5=1. 10/9= 1.111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
Expected answer: >
Correct? Yes
Problem: Compare the values of the two fractions and fill in the blank with <, >, or =. (10 / 9) _ (5 / 10)
InternLM-Math Base 7B:
Expected answer: >
Correct? No
Problem: Compare the two fractions by completing the following statement with <, >, or =. (9 / 8) _ (6 / 4)
InternLM-Math Base 7B:
Answer: 9/8 = 6/4
Expected answer: <
Correct? No
Problem: Compare the two fractions below by filling in the gap with the appropriate symbol (<,>,=). (10 / 3) _ (10 / 2)
InternLM-Math Base 7B:
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?
InternLM-Math Base 7B:
Expected answer: >
Correct? No
Problem: Compare the two fractions by filling in the blank with <, >, or =. (2 / 1) _ (6 / 5)
InternLM-Math Base 7B:
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
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)
InternLM-Math Base 7B:
Reasoning: 2/2 = 1, 6/5 = 1.2. 1<1.2. 2/2<6/5.
Expected answer: <
Correct? Yes
Problem: Compare the fractions 5 / 4 and 10 / 3 by filling in the blank with one of the following symbols: <, >, or =.
InternLM-Math Base 7B:
Reasoning: 3/2 = 1 1/2. 5/4 = 1 1/4. 1 1/4 < 1 1/2.
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 =.
InternLM-Math Base 7B:
5/4 = 1 1/4. 8/3 = 2 2/3. 1 1/4 < 2 2/3.
Expected answer: <
Correct? Yes
Problem: Compare the values of these two fractions by filling in the blank with <, > or =: (7 / 5) _ (9 / 9)
InternLM-Math Base 7B:
Reasoning: 9/9 = 1. 7/5 = <<7/5=1.4>>1.4. 1.4 < 1. 7/5 < 9/9.
Expected answer: >
Correct? No
Problem: Compare the following two fractions by filling in the blank with <, >, or =: (3 / 7) _ (5 / 8)
InternLM-Math Base 7B:
Reasoning: 3/7 = <<3/7=0.428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428
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.
InternLM-Math Base 7B:
Reasoning: 3/7 = <<3/7=0.4285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714
Expected answer: <
Correct? Yes
Problem: Question: Compare these two fractions: (3 / 1) _ (5 / 9) Hint: Fill the blank with one of the following: <, >, or =.
InternLM-Math Base 7B:
Answer:
Expected answer: >
Correct? No
Problem: Compare the values of the following fractions: (2 / 2) _ (6 / 6) Fill in the blank with <, >, or = .
InternLM-Math Base 7B:
Reasoning: 2/2 = 1, and 6/6 = 1. 1 = 1. ### Answer: 1 = 1
Expected answer: =
Correct? Yes
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)
InternLM-Math Base 7B:
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: Compare the following fractions by inserting "<", ">" or "=" between them. (7 / 6) _ (9 / 6)
InternLM-Math Base 7B:
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
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.
InternLM-Math Base 7B:
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: 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.
InternLM-Math Base 7B:
Answer: << 2/7=0.2857142857142857 2/9=0.2222222222222222 0.2857142857142857>0.2222222222222222 2/7>2/9 >>
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?
InternLM-Math Base 7B:
Answer: << 2/7=0.2857142857142857 2/4=0.5 0.2857142857142857<0.5 2/7<2/4 >>
Expected answer: <
Correct? Yes
Problem: Determine the relation between the two following fractions by filling in the blank with <, >, or =: (9 / 4) _ (7 / 1)
InternLM-Math Base 7B:
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?
InternLM-Math Base 7B:
Expected answer: <
Correct? Yes
Problem: Fill in the blank with one of the symbols <, >, or = to complete the comparison between the two fractions. (4 / 8) _ (2 / 1)
InternLM-Math Base 7B:
|user|
Expected answer: <
Correct? No
Problem: Fill in the blank with either <, >, or = in the following expression: (9 / 9) _ (8 / 9), to properly compare the two fractions.
InternLM-Math Base 7B:
Reasoning: 9/9 = 1 and 8/9 = <<8/9=0.8888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888
Expected answer: >
Correct? No
Problem: Compare the following fractions by filling in the blank with <, >, or =. (9 / 10) _ (3 / 10)
InternLM-Math Base 7B:
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
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)
InternLM-Math Base 7B:
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: 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)
InternLM-Math Base 7B:
|user|
Expected answer: <
Correct? No
Problem: Compare the values of the following fractions by filling in the blank with <, >, or =. (6 / 5) _ (7 / 10)
InternLM-Math Base 7B:
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
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)
InternLM-Math Base 7B:
Expected answer: <
Correct? Yes
Problem: Fill in the blank in the following equation with one of these symbols: <, >, or =. (1 / 9) _ (1 / 4)
InternLM-Math Base 7B:
Reasoning: 1/9 is <<1/9=0.11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
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)
InternLM-Math Base 7B:
Reasoning: 1/2 is <<1/2=0.5>>0.5.
Expected answer: >
Correct? No
Problem: Compare the following fractions and fill in the blank with <, >, or =: (7 / 1) _ (6 / 1)
InternLM-Math Base 7B:
Reasoning: 6/1=6, 7/1=7. 7/1=<<7/1=7>>7, so 7/1>6/1.
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)
InternLM-Math Base 7B:
9/1=9, 6/1=6. 9/1=<<9/1=9>>9, so 9/1>6/1.
Expected answer: >
Correct? Yes
Problem: Please fill in the blank with <, >, or = to compare the following fractions. (5 / 4) _ (9 / 8)
InternLM-Math Base 7B:
Answer: << 5/4=1.25 9/8=1.125 1.25>1.125 5/4>9/8 >>
Expected answer: >
Correct? Yes
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)
InternLM-Math Base 7B:
Answer: << 5/4=1.25 9/1=9 9>1.25 5/4<9/1 >>
Expected answer: <
Correct? Yes
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)
InternLM-Math Base 7B:
Expected answer: >
Correct? No
Problem: Compare the following fractions by choosing the correct symbol (<, >, or =) to fill in the blank: (2 / 1) _ (4 / 9)
InternLM-Math Base 7B:
Reasoning: 2/1 = <<2/1=2>>2, 4/9 = <<4/9=0.4444...>>0.4444...
Expected answer: >
Correct? No
Problem: Compare the following fractions by inserting either <, >, or = in the space provided: (9 / 9) _ (2 / 2)
InternLM-Math Base 7B:
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
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 =.
InternLM-Math Base 7B:
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: Determine the relationship between the following two fractions by filling the blank with <, >, or =: (6 / 1) _ (4 / 4)
InternLM-Math Base 7B:
6/1 = <<6/1=6>>6 4/4 = <<4/4=1>>1 6 _ 1
Expected answer: >
Correct? No
Problem: Compare the values of the following fractions by filling in the blank with <, >, or =. (3 / 9) _ (8 / 8)
InternLM-Math Base 7B:
Reasoning: 8/8 = 1. 3/9 = 1/3. 1/3 < 1. 3/9 < 8/8.
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.
InternLM-Math Base 7B:
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: Compare the values of the following fractions by filling in the blank with <, >, or =. (4 / 10) _ (1 / 2)
InternLM-Math Base 7B:
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
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 =.
InternLM-Math Base 7B:
Reasoning: 8/10 = <<8/10=4/5>>4/5. 4/5 > 1/2.
Expected answer: >
Correct? Yes
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)
InternLM-Math Base 7B:
|user|
Expected answer: <
Correct? No
Problem: Determine which sign (<, >, or =) correctly fills the blank when comparing these two fractions: (2 / 3) _ (2 / 10)
InternLM-Math Base 7B:
Reasoning: 2/3 = <<2/3=0.66666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666
Expected answer: >
Correct? No
Problem: Compare the following fractions by filling in the blank with either <, =, or >: (2 / 10) _ (6 / 4)
InternLM-Math Base 7B:
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
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?
InternLM-Math Base 7B:
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: Compare the values of the following fractions and indicate either <, >, or = to complete the equation. (5 / 10) _ (6 / 9)
InternLM-Math Base 7B:
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)
InternLM-Math Base 7B:
Expected answer: >
Correct? No
Problem: Compare the fractions by filling in the blank with <, >, or =. (2 / 9) _ (5 / 5)
InternLM-Math Base 7B:
Reasoning: <<2/9=0.2 5/5=1>> 0.2 < 1 ### Answer: <
Expected answer: <
Correct? Yes
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)
InternLM-Math Base 7B:
Reasoning: <<2/9=0.2 5/8=0.625 0.2<0.625>> 2/9 < 5/8 ### Answer: <
Expected answer: <
Correct? Yes
Problem: Fill in the blank with <, >, or = to correctly compare the two fractions: (1 / 9) _ (7 / 9)
InternLM-Math Base 7B:
Answer: <
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 =.
InternLM-Math Base 7B:
Answer: >
Expected answer: <
Correct? No
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 =.
InternLM-Math Base 7B:
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
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 =.
InternLM-Math Base 7B:
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: In the following expression, properly insert either <, >, or = in order to correctly compare the two fractions: (9 / 9) _ (9 / 10)
InternLM-Math Base 7B:
Reasoning: <<9/9=9/9=1>>1 is greater than or equal to <<9/10=0.9>>0.9.
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.
InternLM-Math Base 7B:
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: 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)
InternLM-Math Base 7B:
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
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.
InternLM-Math Base 7B:
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: Compare the values of the following fractions by deciding which symbol (<, >, or =) to fill in the blank with: (5 / 2) _ (1 / 10)
InternLM-Math Base 7B:
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
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.
InternLM-Math Base 7B:
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: Fill in the blank with <, >, or = to compare the two fractions in the following question: (2 / 4) _ (9 / 2)
InternLM-Math Base 7B:
Reasoning: 9/2 = 4.5. 2/4 = 0.5. 0.5 < 4.5.
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)
InternLM-Math Base 7B:
Reasoning: 9/10 = 0.9. 2/4 = 0.5. 0.9 > 0.5.
Expected answer: <
Correct? No
Problem: Fill in the blank in the following equation with either <, >, or = to show the comparison between the two fractions: (6 / 3) _ (10 / 2)
InternLM-Math Base 7B:
|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: Compare the given fractions by filling in the blank with the correct symbol (<, >, or =). (6 / 3) _ (8 / 6)
InternLM-Math Base 7B:
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: Compare the values of the following fractions by filling in the blank with <, >, or =. (3 / 4) _ (6 / 5)
InternLM-Math Base 7B:
Reasoning: 3/4 = 0.75 and 6/5 = 1.2. 0.75 < 1.2.
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?
InternLM-Math Base 7B:
Reasoning: 3/4 = 0.75 and 6/9 = 0.66. 0.75 > 0.66.
Expected answer: >
Correct? Yes
Problem: Compare the two fractions below by filling in the blank with <, >, or =: (1 / 1) _ (3 / 1)
InternLM-Math Base 7B:
Reasoning: 3/1 = <<3/1=3>>3, 1/1 = <<1/1=1>>1. 3>1.
Expected answer: <
Correct? No
Problem: Compare the following fractions by filling in the blank with <, >, or =. (7 / 6) _ (4 / 5)
InternLM-Math Base 7B:
Reasoning: <<7/6=1.1666666666666667, 4/5=0.8, 7/6>4/5.>>
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)
InternLM-Math Base 7B:
Reasoning: <<7/3=2.3333333333333335, 4/5=0.8, 7/3>4/5.>>
Expected answer: >
Correct? Yes
Problem: Compare the following two fractions by filling the blank with <, >, or =. (6 / 4) _ (2 / 2)
InternLM-Math Base 7B:
Reasoning: 2/2 = 1. 6/4 = 1.5. 1.5 > 1. 6/4 > 2/2.
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?
InternLM-Math Base 7B:
Reasoning: 1.5 > 0.2. 6/4 > 2/10.
Expected answer: >
Correct? Yes
Problem: Compare the following fractions by filling in the blank with <, >, or =. (5 / 5) _ (9 / 10)
InternLM-Math Base 7B:
|user|
Expected answer: >
Correct? No
Problem: Fill in the blank (1 / 5) _ (1 / 1) with <, >, or = to compare the two fractions.
InternLM-Math Base 7B:
Reasoning: <<1/5=0.2, 1/1=1.0>>0.2<1.0 ### 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?
InternLM-Math Base 7B:
Reasoning: <<1/5=0.2, 2/1=2.0>>0.2<2.0 ### Answer: <
Expected answer: <
Correct? Yes
Problem: Fill the blank with either <, >, or = to compare the two fractions in the following equation: (7 / 1) _ (4 / 2)
InternLM-Math Base 7B:
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: Compare the following fractions by using <, >, or = to fill in the blank: (4 / 7) _ (9 / 4)
InternLM-Math Base 7B:
Reasoning: 4/7 = <<4/7=0.571428571428571428571428571428571428571428571428571428571428571428>>0.571428571428571428571428571428571428571428571428571428571428571428 9/4 = 2.25 0.571428571428571428571428571428571428571428571428571428571428571428 < 2.25 ### 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)?
InternLM-Math Base 7B:
Reasoning: 4/7 = <<4/7=0.571428571428571428571428571428571428571428571428571428571428571428>>0.571428571428571428571428571428571428571428571428571428571428571428 2/4 = 0.5 0.571428571428571428571428571428571428571428571428571428571428571428 < 0.5 ### Answer: <
Expected answer: >
Correct? No
Problem: Fill in the blank in the question (5 / 9) _ (4 / 7) with <, >, or = to compare the two fractions.
InternLM-Math Base 7B:
Reasoning: <<(5 / 9) < (4 / 7) because 5/9 is less than 4/7.>>
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?
InternLM-Math Base 7B:
Reasoning: <<(5 / 9) = (4 / 10) because 5/9 and 4/10 are equivalent fractions.>>
Expected answer: >
Correct? No
Problem: Compare the following fractions and fill in the blank with <, >, or =: (8 / 3) _ (7 / 3)
InternLM-Math Base 7B:
Answer: <<
Expected answer: >
Correct? No
Problem: In the equation: (5 / 1) _ (2 / 2), replace the underscore with one of the following signs: <, >, or = to compare the two fractions.
InternLM-Math Base 7B:
Reasoning: 5/1 = <<5/1=5>>5 and 2/2 = <<2/2=1>>1. 5 is __ 1. ### 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?
InternLM-Math Base 7B:
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: Compare the following fractions and fill in the blank with either <, >, or =: (5 / 8) _ (3 / 3)
InternLM-Math Base 7B:
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
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)
InternLM-Math Base 7B:
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: Determine which symbol : <, >, or =, is appropriate to fill the blank in the following comparison: (5 / 8) _ (6 / 10)
InternLM-Math Base 7B:
Reasoning:
Expected answer: >
Correct? No
Problem: Compare the two fractions by filling in the blank with <, >, or =: (5 / 10) _ (6 / 6)
InternLM-Math Base 7B:
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
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?
InternLM-Math Base 7B:
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: Compare these two fractions by filling in the blank with <, >, or =. (5 / 4) _ (2 / 4)
InternLM-Math Base 7B:
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
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.
InternLM-Math Base 7B:
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: Compare the values of the following fractions by using <, >, or = to fill in the blank: (6 / 10) _ (6 / 10)
InternLM-Math Base 7B:
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
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)
InternLM-Math Base 7B:
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: Please compare these two fractions and fill the blank with <, >, or =. (6 / 8) _ (2 / 9)
InternLM-Math Base 7B:
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: Fill in the blank with <, =, or > after comparing the fractions below: (2 / 10) _ (6 / 2)
InternLM-Math Base 7B:
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)
InternLM-Math Base 7B:
Expected answer: <
Correct? Yes
Problem: Determine which of the following fractions is greater by filling in the blank with <, >, or =. (2 / 7) _ (9 / 4)
InternLM-Math Base 7B:
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: Compare the following fractions by filling in the blank with <, >, or =. (1 / 1) _ (2 / 10)
InternLM-Math Base 7B:
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: Compare the results of the following fractions by filling the blank with <, >, or =. (8 / 2) _ (10 / 9)
InternLM-Math Base 7B:
8/2 = <<8/2=4>>4, 10/9 = <<10/9=1.1...>>1.1... 4 < 1.1...
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 / 8)
InternLM-Math Base 7B:
Reasoning: <<3/2=1.5, 9/8=1.125, 3/2>9/8>>3/2>9/8
Expected answer: >
Correct? Yes
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?
InternLM-Math Base 7B:
Reasoning: <<3/2=1.5, 9/6=1.5, 3/2=9/6>>3/2=9/6
Expected answer: =
Correct? Yes
Problem: Compare the following quantities by filling in the blank with <, >, or =: (4 / 6) _ (6 / 10)
InternLM-Math Base 7B:
Reasoning: 4/6 = 2/3, 6/10 = 3/5, 2/3 < 3/5 ### Answer: 2/3 < 3/5
Expected answer: >
Correct? No
Problem: Fill in the blank with the correct symbol to compare the following fractions: (8 / 10) _ (1 / 1)
InternLM-Math Base 7B:
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)
InternLM-Math Base 7B:
Expected answer: >
Correct? No
Problem: Using either <, >, or =, compare the following fractions by filling in the blank: (5 / 1) _ (5 / 8)
InternLM-Math Base 7B:
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: 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.
InternLM-Math Base 7B:
Reasoning: <<6/5=1.2, 9/6=1.5. 9/6>6/5.>>
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 "=" ?
InternLM-Math Base 7B:
Reasoning: <<3/5=0.6, 9/6=1.5. 9/6>3/5.>>
Expected answer: <
Correct? Yes
Problem: Compare the following fractions by filling in the blank: (7 / 5) _ (1 / 10) You can use <, > or =.
InternLM-Math Base 7B:
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
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.
InternLM-Math Base 7B:
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: Compare the two fractions and fill in the blank with <, >, or =. (10 / 8) _ (10 / 1)
InternLM-Math Base 7B:
Reasoning: 10/8 = <<10/8=1.25>>1.25 10/1 = <<10/1=10>>10 1.25 < 10 ### 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)
InternLM-Math Base 7B:
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: Compare the values of the following fractions by filling in the blank with <, >, or =: (3 / 2) _ (4 / 6)
InternLM-Math Base 7B:
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: Compare the following two fractions by filling in the blank with <, >, or =. (9 / 7) _ (4 / 6)
InternLM-Math Base 7B:
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
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)
InternLM-Math Base 7B:
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: Fill in the blank in the following comparison with a <, >, or = sign. (8 / 7) _ (8 / 8)
InternLM-Math Base 7B:
Expected answer: >
Correct? No
Problem: Fill in the blank with the correct symbol (<, >, =) to compare these two fractions: (5 / 5) _ (4 / 6)
InternLM-Math Base 7B:
Reasoning: 5/5 = 1, and 4/6 = 2/3. 1 < 2/3.
Expected answer: >
Correct? No
Problem: Compare the following fractions and use <, >, or = to fill the blank: (10 / 10) _ (4 / 10)
InternLM-Math Base 7B:
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: Compare the following fractions and fill in the blank with <, >, or =. (8 / 1) _ (4 / 9)
InternLM-Math Base 7B:
Reasoning: 8/1 = 8 and 4/9 = 4/9. 8/1 = 8 is greater than 4/9.
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)
InternLM-Math Base 7B:
Reasoning: 5/1 = 5 and 4/9 = 4/9. 5/1 = 5 is less than 4/9.
Expected answer: >
Correct? No
Problem: Compare the values of the two fractions below by choosing either <, >, or = to fill in the blank: (9 / 3) _ (4 / 8)
InternLM-Math Base 7B:
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
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?
InternLM-Math Base 7B:
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: Compare the fractions in the expression (10 / 7) _ (9 / 6). Fill in the blank with <, >, or =.
InternLM-Math Base 7B:
Reasoning: <<10/7=1 2/7. 9/6=1 3/6. 1 2/7>1 3/6.>> The answer is >.
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?
InternLM-Math Base 7B:
Reasoning: <<10/7=1 3/7. 9/3=3. 1 3/7<3.>> The answer is <.
Expected answer: <
Correct? Yes
Problem: Compare the following fractions, and complete the statement: (10 / 8) _ (3 / 1) Fill the blank space with <, >, or =.
InternLM-Math Base 7B:
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: Determine the relationship between the two fractions by filling in the blank with <, >, or =. The question is as follows: (3 / 2) _ (3 / 5)
InternLM-Math Base 7B:
Reasoning: <<3/2=1.5>>1.5 is __ 3/5.
Expected answer: >
Correct? No
Problem: Compare the following two fractions by filling in the blank with <, >, or =. (4 / 10) _ (2 / 2)
InternLM-Math Base 7B:
Reasoning: 2/2 = 1, so 2/2 < 4/10.
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 =.
InternLM-Math Base 7B:
Reasoning: 9/2 = 4.5, so 4/10 < 9/2.
Expected answer: <
Correct? Yes
Problem: Compare the two fractions and fill in the blank with <, >, or =: (6 / 7) _ (2 / 10)
InternLM-Math Base 7B:
Reasoning: <<6/7=0.85714285714285714285714285714286, 2/10=0.2. 0.85714285714285714285714285714286>0.2. 6/7>2/10.>>
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)
InternLM-Math Base 7B:
Reasoning: <<5/7=0.71428571428571428571428571428571, 2/10=0.2. 0.71428571428571428571428571428571>0.2. 5/7>2/10.>>
Expected answer: >
Correct? Yes
Problem: Compare the values of the following fractions by filling in the blank with <, >, or =. (9 / 7) _ (10 / 3)
InternLM-Math Base 7B:
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: Compare the following two fractions by filling in the blank with <, >, or =. (8 / 8) _ (2 / 8)
InternLM-Math Base 7B:
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
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.
InternLM-Math Base 7B:
Reasoning: 8/10 = 0.8, 2/8 = 0.25. 0.8 > 0.25. ### Answer: 8/10>2/8
Expected answer: >
Correct? Yes
Problem: Evaluate the following expression by comparing the two fractions using <, >, or =: (4 / 3) _ (1 / 8)
InternLM-Math Base 7B:
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: Compare the following fractions by completing the sentence with the appropriate symbol (<, >, or =): (4 / 6) _ (6 / 10)
InternLM-Math Base 7B:
Reasoning: 4/6 = 2/3 and 6/10 = 3/5. 2/3 < 3/5.
Expected answer: >
Correct? No
Problem: Compare the two fractions by filling the blank using <, >, or =. Here's the question: (9 / 6) _ (6 / 8)
InternLM-Math Base 7B:
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
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.
InternLM-Math Base 7B:
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: Compare the following fractions by filling in the blank with one of the following symbols: <, >, or =. (2 / 6) _ (8 / 7)
InternLM-Math Base 7B:
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
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?
InternLM-Math Base 7B:
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: Compare the following fractions by filling in the blank with <, >, or =. (9 / 6) _ (4 / 10)
InternLM-Math Base 7B:
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
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.
InternLM-Math Base 7B:
Reasoning: 9/6 = 1.5 and 4/6 = <<4/6=0.6666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666
Expected answer: >
Correct? No
Problem: Fill in the blank in the following question with <, >, or = to correctly indicate which fraction is greater in value: (6 / 8) _ (5 / 1)
InternLM-Math Base 7B:
Reasoning: 6/8 = <<6/8=3/4>>3/4 and 5/1 = <<5/1=5>>5. 3/4 < 5.
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?
InternLM-Math Base 7B:
Reasoning: 6/9 = <<6/9=2/3>>2/3 and 5/1 = <<5/1=5>>5. 2/3 < 5.
Expected answer: <
Correct? Yes
Problem: Compare these two fractions by filling in the blank with <, >, or =. (8 / 4) _ (1 / 10)
InternLM-Math Base 7B:
Reasoning: 8/4 = <<8/4=2>>2 and 1/10 = <<1/10=0.1>>0.1 2_0.1 ### 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 =.
InternLM-Math Base 7B:
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: Compare the values of the following expressions by filling in the blank with <, >, or =. (6 / 3) _ (8 / 2)
InternLM-Math Base 7B:
Reasoning: 8/2=4. 6/3=2. 2<4. 6/3<8/2.
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)
InternLM-Math Base 7B:
Reasoning: 1/2=0.5. 6/3=2. 0.5<2. 6/3>1/2.
Expected answer: >
Correct? Yes
Problem: Please compare the following two fractions and fill the blank with one of the symbols: <, >, or =: (4 / 6) _ (6 / 3)
InternLM-Math Base 7B:
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
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?
InternLM-Math Base 7B:
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: Fill in the blank with <, >, or = to compare the fractions 6 / 7 and 3 / 1 properly.
InternLM-Math Base 7B:
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
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.
InternLM-Math Base 7B:
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: Compare the values of the following fractions by filling in the blank with <, >, or =. (6 / 10) _ (9 / 10)
InternLM-Math Base 7B:
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
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)?
InternLM-Math Base 7B:
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: Compare the two fractions by filling in the blank with <, >, or =. (6 / 8) _ (4 / 9)
InternLM-Math Base 7B:
Answer: << 6/8=3/4 4/9=2/3 3/4>2/3 6/8>4/9 >>
Expected answer: >
Correct? Yes
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)
InternLM-Math Base 7B:
Answer: << 6/8=3/4 4/4=1 3/4>1 6/8>4/4 >>
Expected answer: <
Correct? Yes