MathCAMPS

🔗 Problem

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

DeepSeek 67B:

To compare the fractions (1/8) and (4/1), we need to find a common denominator. The common denominator for 8 and 1 is 8.

So, we rewrite the fractions with the common denominator:

(1/8) = (1 * 1)/(8 * 1) = (1/8)
(4/1) = (4 * 8)/(1 * 8) = (32/8)

Now, we can compare the fractions:

(1/8) _ (32/8)

Since the denominators are the same, we can compare the numerators:

1 _ 32

Since 1 is less than 32, we can conclude that:

(1/8) < (4/1)

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

To compare these two fractions, we first need to find a common denominator. The least common multiple of 2 and 6 is 6. So, we can rewrite the fractions with a common denominator of 6:

(7 / 2) = (7 * 3) / (2 * 3) = (21 / 6)
(10 / 6) = (10 / 6)

Now, we can compare the fractions:

(21 / 6) _ (10 / 6)

Since 21 is greater than 10, the fraction (21 / 6) is greater than (10 / 6). Therefore, the symbol that appropriately fills in the blank is '>':

(7 / 2) > (10 / 6)

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

To compare the fractions (6/8) and (8/7), we need to find a common denominator. The least common multiple of 8 and 7 is 56. So, we can rewrite the fractions as:

(6/8) = (6 * 7) / (8 * 7) = 42 / 56
(8/7) = (8 * 8) / (7 * 8) = 64 / 56

Now, we can compare the fractions:

(42 / 56) _ (64 / 56)

Since 42 < 64, we can conclude that:

(6 / 8) < (8 / 7)

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the two fractions, we can simplify them and then compare their values.

The first fraction, 9/3, can be simplified by dividing both the numerator and denominator by their greatest common divisor, which is 3.
9 � 3 = 3
3 � 3 = 1
So, 9/3 simplifies to 3/1.

The second fraction, 7/1, cannot be simplified any further.

Now, we can compare the two simplified fractions:

3/1 _ 7/1

Since the denominators are the same, we can compare the numerators.

3 _ 7

Since 3 is less than 7, we can conclude that 3/1 is less than 7/1.

Therefore, the answer is <.

### Answer:

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

To determine which sign to use, we need to compare the two fractions.

(5 / 3) is equivalent to 1.67 (rounded to two decimal places).
(10 / 5) is equivalent to 2.

Since 1.67 is less than 2, we can conclude that (5 / 3) is less than (10 / 5).

Therefore, the correct sign to use is "<".

### Answer:

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the fractions, we need to find a common denominator. The least common multiple of 2 and 10 is 10. So, we can rewrite the fractions with a common denominator of 10:

(3 / 2) = (3 * 5) / (2 * 5) = 15 / 10
(9 / 10) = 9 / 10

Now, we can compare the fractions:

15 / 10 _ 9 / 10

Since 15 is greater than 9, the fraction 15 / 10 is greater than 9 / 10.

### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

Reasoning:
To compare the fractions, we need to find a common denominator. The least common multiple of 7 and 10 is 70. So, we can rewrite the fractions with a common denominator of 70:

(3 / 7) = (3 * 10) / (7 * 10) = 30 / 70
(9 / 10) = 9 / 10

Now, we can compare the fractions:

30 / 70 _ 9 / 10

Since 30 is less than 9, the fraction 30 / 70 is less than 9 / 10.

### Answer: <

The comparison is different than the comparison between 3 / 2 and 9 / 10. In the previous problem, the fraction with the larger numerator (15 / 10) was greater than the fraction with the smaller numerator (9 / 10). In this problem, the fraction with the smaller numerator (30 / 70) is less than the fraction with the larger numerator (9 / 10). The impact of decreasing the numerator of our first fraction (from 15 to 30) has changed the comparison.

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

To compare the fractions (10/9) and (5/6), we can find a common denominator. The least common multiple of 9 and 6 is 18. So, we can rewrite the fractions as:

(10/9) = (10 * 2) / (9 * 2) = 20/18
(5/6) = (5 * 3) / (6 * 3) = 15/18

Now, we can compare the fractions:

20/18 _ 15/18

Since 20 is greater than 15, we can conclude that:

(10/9) > (5/6)

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the two fractions, we need to find a common denominator. The least common multiple of 9 and 10 is 90. So, we can rewrite the fractions with the common denominator of 90:

(10 / 9) = (10 * 10) / (9 * 10) = 100 / 90
(5 / 10) = (5 * 9) / (10 * 9) = 45 / 90

Now, we can compare the two fractions:

100 / 90 > 45 / 90

So, (10 / 9) > (5 / 10)

### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

Reasoning:
To compare the two fractions, we need to find a common denominator. The least common multiple of 9 and 3 is 9. So, we can rewrite the fractions with the common denominator of 9:

(10 / 9) = (10 * 1) / (9 * 1) = 10 / 9
(5 / 3) = (5 * 3) / (3 * 3) = 15 / 9

Now, we can compare the two fractions:

10 / 9 < 15 / 9

So, (10 / 9) < (5 / 3)

### Answer:

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

To compare the two fractions, we need to find a common denominator. The least common multiple of 8 and 4 is 8. So, we can rewrite the fractions as:

(9 / 8) _ (6 / 4) = (9 / 8) _ (3 / 2)

Now, we can compare the numerators:

9 _ 3

Since 9 is greater than 3, we can conclude that:

(9 / 8) > (6 / 4)

Expected answer: <

Correct? No

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the two fractions, we can convert them to a common denominator. The least common multiple of 3 and 2 is 6. So, we can rewrite the fractions as:

(10 / 3) = (10 * 2) / (3 * 2) = 20 / 6
(10 / 2) = (10 * 3) / (2 * 3) = 30 / 6

Now, we can compare the fractions:

20 / 6 _ 30 / 6

Since the denominators are the same, we can compare the numerators:

20 _ 30

Since 20 is less than 30, the fraction (10 / 3) is less than the fraction (10 / 2).

### Answer:

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

To compare the two fractions, we can multiply the first fraction by the reciprocal of the second fraction.

(2 / 1) * (5 / 6) = 10 / 6 = 5 / 3

Since 5/3 is greater than 1, we can conclude that (2 / 1) > (6 / 5).

### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

To compare the two fractions, we can multiply the first fraction by the reciprocal of the second fraction.

(2 / 2) * (5 / 6) = 5 / 6

Since 5/6 is less than 1, we can conclude that (2 / 2) < (6 / 5).

### Answer:

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

5 / 4 = 1.25
10 / 3 = 3.33
Since 1.25 is less than 3.33, the correct symbol is <.
### Answer:

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the fractions, we need to find a common denominator. In this case, the common denominator is 45.

(7 / 5) = (7 * 9) / (5 * 9) = 63 / 45
(9 / 9) = (9 * 5) / (9 * 5) = 45 / 45

Now we can compare the fractions:
(63 / 45) _ (45 / 45)

Since 63 is greater than 45, the fraction (7 / 5) is greater than (9 / 9).

### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

Reasoning:
To compare the fractions, we need to find a common denominator. In this case, the common denominator is 50.

(7 / 5) = (7 * 10) / (5 * 10) = 70 / 50
(9 / 10) = (9 * 5) / (10 * 5) = 45 / 50

Now we can compare the fractions:
(70 / 50) _ (45 / 50)

Since 70 is greater than 45, the fraction (7 / 5) is greater than (9 / 10).

### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the fractions (3/7) and (5/8), we can find a common denominator and then compare the numerators.

The least common multiple of 7 and 8 is 56. So, we can rewrite the fractions with the common denominator of 56:

(3/7) = (3 * 8) / (7 * 8) = 24 / 56
(5/8) = (5 * 7) / (8 * 7) = 35 / 56

Now, we can compare the numerators:

24 < 35

So, (3/7) < (5/8).

### Answer:

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the two fractions, we need to find a common denominator. The least common multiple of 1 and 9 is 9. So, we can rewrite the fractions with the common denominator:

(3 / 1) = (3 * 9) / (1 * 9) = 27 / 9
(5 / 9) = (5 * 1) / (9 * 1) = 5 / 9

Now, we can compare the fractions:

(27 / 9) _ (5 / 9)

Since 27 is greater than 5, the fraction (27 / 9) is greater than (5 / 9).

### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

Reasoning:
To compare the two fractions, we need to find a common denominator. The least common multiple of 1 and 5 is 5. So, we can rewrite the fractions with the common denominator:

(3 / 1) = (3 * 5) / (1 * 5) = 15 / 5
(5 / 5) = (5 * 1) / (5 * 1) = 5 / 5

Now, we can compare the fractions:

(15 / 5) _ (5 / 5)

Since 15 is greater than 5, the fraction (15 / 5) is greater than (5 / 5).

### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
Both fractions have the same denominator, which is 2.
The numerator of the first fraction is 2, and the numerator of the second fraction is 6.
Since 2 is less than 6, the first fraction is smaller than the second fraction.
Therefore, (2 / 2) < (6 / 6).
The answer is:

Expected answer: =

Correct? No

🔗 Problem

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

DeepSeek 67B:

(7 / 6) < (9 / 6)

Expected answer: <

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

(7 / 9) < (9 / 6)

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

(2 / 7) < (2 / 9)

Expected answer: >

Correct? No

🔗 Problem

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

DeepSeek 67B:

To determine the relation between the two fractions, we need to compare their values.

First, we can simplify the fractions:

(9 / 4) = 2.25
(7 / 1) = 7

Now, we can compare the values:

2.25 _ 7

Since 2.25 is less than 7, we can fill in the blank with "<":

(9 / 4) < (7 / 1)

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the two fractions, we need to find a common denominator. The least common multiple of 8 and 1 is 8. So, we can rewrite the fractions with the common denominator:

(4 / 8) = (1 / 2)
(2 / 1) = (16 / 8)

Now, we can compare the fractions:

(1 / 2) _ (16 / 8)

Since 16 is greater than 1, we can conclude that:

(1 / 2) < (16 / 8)

So, the correct symbol to fill in the blank is "<".
### Answer:

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the two fractions, we can simplify them.
(9 / 9) simplifies to 1, and (8 / 9) remains the same.
Now, we can compare the two simplified fractions:
1 _ (8 / 9)
Since 1 is greater than (8 / 9), we can fill in the blank with ">".
### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

Reasoning:
To compare the two fractions, we can simplify them.
(9 / 2) simplifies to 4.5, and (8 / 9) remains the same.
Now, we can compare the two simplified fractions:
4.5 _ (8 / 9)
Since 4.5 is greater than (8 / 9), we can fill in the blank with ">".
### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the fractions, we can simplify them to have the same denominator. In this case, the denominators are already the same (10). So, we can compare the numerators directly.

9 (numerator of the first fraction) > 3 (numerator of the second fraction)

Therefore, (9 / 10) > (3 / 10)

### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

Reasoning:
To compare the fractions, we need to have the same denominator. We can do this by finding the least common multiple (LCM) of the denominators, which in this case is 20. Now, we can rewrite the fractions with the common denominator:

(9 / 4) = (9 * 5) / (4 * 5) = 45 / 20
(3 / 10) = (3 * 2) / (10 * 2) = 6 / 20

Now, we can compare the numerators:

45 (numerator of the first fraction) > 6 (numerator of the second fraction)

Therefore, (9 / 4) > (3 / 10)

### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

To determine if the first fraction is less than, equal to, or greater than the second, we can compare their values.

First, let's find a common denominator for both fractions. The least common multiple of 7 and 3 is 21.

Now, let's rewrite both fractions with the common denominator of 21:

(3 / 7) = (3 * 3) / (7 * 3) = 9 / 21
(5 / 3) = (5 * 7) / (3 * 7) = 35 / 21

Now, we can compare the two fractions:

9 / 21 _ 35 / 21

Since 9 is less than 35, the first fraction is less than the second fraction.

Therefore, the answer is:

(3 / 7) < (5 / 3)

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the fractions, we need to find a common denominator. The least common multiple of 5 and 10 is 10. So, we can rewrite the fractions with a common denominator of 10:

(6 / 5) = (6 * 2) / (5 * 2) = 12 / 10
(7 / 10) = 7 / 10

Now, we can compare the fractions:

12 / 10 _ 7 / 10

Since 12 is greater than 7, we can conclude that:

(6 / 5) > (7 / 10)

### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

Reasoning:
To compare the fractions, we need to find a common denominator. The least common multiple of 5 and 10 is 10. So, we can rewrite the fractions with a common denominator of 10:

(1 / 5) = (1 * 2) / (5 * 2) = 2 / 10
(7 / 10) = 7 / 10

Now, we can compare the fractions:

2 / 10 _ 7 / 10

Since 2 is less than 7, we can conclude that:

(1 / 5) < (7 / 10)

### Answer:

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

To determine which symbol to use, we need to compare the two fractions.

(1 / 9) is equivalent to 0.1111...
(1 / 4) is equivalent to 0.25

Since 0.1111... is less than 0.25, we can conclude that (1 / 9) is less than (1 / 4).

Therefore, the correct symbol to use is "<".

### Answer:

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

The fractions (7 / 1) and (6 / 1) can be compared by comparing their numerators. Since 7 is greater than 6, the fraction (7 / 1) is greater than the fraction (6 / 1).

Therefore, (7 / 1) > (6 / 1).

Expected answer: >

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

The fractions (9 / 1) and (6 / 1) can be compared by comparing their numerators. Since 9 is greater than 6, the fraction (9 / 1) is greater than the fraction (6 / 1).

Therefore, (9 / 1) > (6 / 1).

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

(5 / 4) > (9 / 8)

Reasoning:
To compare the fractions, we can find a common denominator. In this case, the least common multiple of 4 and 8 is 8.

(5 / 4) = (5 * 2) / (4 * 2) = 10 / 8
(9 / 8) = (9 * 1) / (8 * 1) = 9 / 8

Now we can compare the fractions:
10 / 8 > 9 / 8

So, (5 / 4) > (9 / 8)

Expected answer: >

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

(5 / 4) < (9 / 1)

Reasoning:
To compare the fractions, we can find a common denominator. In this case, the least common multiple of 4 and 1 is 4.

(5 / 4) = (5 * 1) / (4 * 1) = 5 / 4
(9 / 1) = (9 * 4) / (1 * 4) = 36 / 4

Now we can compare the fractions:
5 / 4 < 36 / 4

So, (5 / 4) < (9 / 1)

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

To determine the correct symbol to place between the two fractions, we need to compare their values.

First, we can find a common denominator for the two fractions. The least common multiple of 9 and 4 is 36.

Next, we can rewrite the fractions with the common denominator:

(8 / 9) = (8 * 4) / (9 * 4) = 32 / 36
(3 / 4) = (3 * 9) / (4 * 9) = 27 / 36

Now, we can compare the two fractions:

32 / 36 > 27 / 36

Therefore, the correct symbol to place between the two fractions is >.

### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

To determine the correct symbol to place between the two fractions, we need to compare their values.

First, we can find a common denominator for the two fractions. The least common multiple of 2 and 4 is 4.

Next, we can rewrite the fractions with the common denominator:

(8 / 2) = (8 * 2) / (2 * 2) = 16 / 4
(3 / 4) = (3 * 1) / (4 * 1) = 3 / 4

Now, we can compare the two fractions:

16 / 4 > 3 / 4

Therefore, the correct symbol to place between the two fractions is >.

### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the fractions, we can multiply the first fraction by 9/9 and the second fraction by 2/2 to get a common denominator of 9.

(2/1) * (9/9) = (18/9)
(4/9) * (2/2) = (8/9)

Now we can compare the fractions:

(18/9) > (8/9)

So, (2/1) > (4/9)

### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

Reasoning:
To compare the fractions, we can multiply the first fraction by 9/9 and the second fraction by 2/2 to get a common denominator of 9.

(2/1) * (9/9) = (18/9)
(8/9) * (2/2) = (16/9)

Now we can compare the fractions:

(18/9) > (16/9)

So, (2/1) > (8/9)

### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

(9 / 9) = (2 / 2)

Reasoning:
Both fractions have the same denominator, which is 9. To compare the fractions, we need to compare the numerators. The numerator of the first fraction is 9, and the numerator of the second fraction is 2. Since 9 is greater than 2, the first fraction is greater than the second fraction.

### Answer: >

Expected answer: =

Correct? No

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the two fractions, we need to have a common denominator. In this case, the common denominator is 4.

(6 / 1) = (6 * 4) / (1 * 4) = 24 / 4
(4 / 4) = 4 / 4

Now we can compare the two fractions:

(24 / 4) _ (4 / 4)

Since 24 is greater than 4, the relationship between the two fractions is:

(24 / 4) > (4 / 4)

### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

Reasoning:
To compare the two fractions, we need to have a common denominator. In this case, the common denominator is 3.

(6 / 1) = (6 * 3) / (1 * 3) = 18 / 3
(4 / 3) = 4 / 3

Now we can compare the two fractions:

(18 / 3) _ (4 / 3)

Since 18 is greater than 4, the relationship between the two fractions is:

(18 / 3) > (4 / 3)

### Answer: >

Comparing the results of the two scenarios, we can see that in both cases, the relationship between the fractions is >. This means that the fraction (6 / 1) is greater than both (4 / 4) and (4 / 3).

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the fractions, we need to find a common denominator. In this case, the common denominator is 9.

(3 / 9) = (3 * 1) / (9 * 1) = 3 / 9
(8 / 8) = (8 * 1) / (8 * 1) = 8 / 9

Now we can compare the fractions:

3 / 9 < 8 / 9

So, (3 / 9) < (8 / 8)

### Answer:

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

To compare the fractions, we need to find a common denominator. The least common multiple of 10 and 2 is 10. So, we can rewrite the fractions as:

(4 / 10) = (4 / 10) * (1 / 1) = (4 / 10) * (5 / 5) = (20 / 50)
(1 / 2) = (1 / 2) * (5 / 5) = (5 / 10)

Now, we can compare the fractions:

(20 / 50) _ (5 / 10)

Since 20 is greater than 5, we can conclude that:

(20 / 50) > (5 / 10)

So, the answer is:

(4 / 10) > (1 / 2)

Expected answer: <

Correct? No

🔗 Problem

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

DeepSeek 67B:

To compare the two fractions, we need to find a common denominator. The least common multiple of 6 and 3 is 6. So, we can rewrite the fractions as:

(6 / 6) = (6 / 6)
(6 / 3) = (6 / 3) * (2 / 2) = (12 / 6)

Now, we can compare the fractions:

(6 / 6) _ (12 / 6)

Since the denominators are the same, we can compare the numerators:

6 _ 12

Since 6 is less than 12, we can conclude that:

(6 / 6) < (6 / 3)

So, the answer is:

(6 / 6) < (6 / 3)

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the fractions (2/3) and (2/10), we need to find a common denominator. The least common multiple of 3 and 10 is 30. So, we can rewrite the fractions with the common denominator of 30:

(2/3) = (2 * 10) / (3 * 10) = (20/30)
(2/10) = (2 * 3) / (10 * 3) = (6/30)

Now, we can compare the fractions:

(20/30) _ (6/30)

Since 20 is greater than 6, the correct sign is >.

### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

Reasoning:
To compare the fractions (10/3) and (2/10), we need to find a common denominator. The least common multiple of 3 and 10 is 30. So, we can rewrite the fractions with the common denominator of 30:

(10/3) = (10 * 10) / (3 * 10) = (100/30)
(2/10) = (2 * 3) / (10 * 3) = (6/30)

Now, we can compare the fractions:

(100/30) _ (6/30)

Since 100 is greater than 6, the correct sign is >.

### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

To compare the fractions (2/10) and (6/4), we need to find a common denominator. The least common multiple of 10 and 4 is 20. So, we can rewrite the fractions as:

(2/10) = (2 * 2) / (10 * 2) = 4/20
(6/4) = (6 * 5) / (4 * 5) = 30/20

Now, we can compare the fractions:

(4/20) _ (30/20)

Since the denominators are the same, we can compare the numerators:

4 _ 30

Since 4 is less than 30, we can conclude that:

(2/10) < (6/4)

So, the answer is:

(2 / 10) < (6 / 4)

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the fractions, we need to find a common denominator. The least common multiple of 10 and 9 is 90. So, we can rewrite the fractions with the common denominator of 90:

(5 / 10) = (5 * 9) / (10 * 9) = 45 / 90
(6 / 9) = (6 * 10) / (9 * 10) = 60 / 90

Now, we can compare the fractions:

45 / 90 _ 60 / 90

Since 45 is less than 60, we can conclude that:

45 / 90 < 60 / 90

So, the equation is:

(5 / 10) < (6 / 9)

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

To compare the fractions (2/9) and (5/5), we need to find a common denominator. The least common multiple of 9 and 5 is 45. So, we can rewrite the fractions as:

(2/9) = (2 * 5) / (9 * 5) = 10 / 45
(5/5) = (5 * 9) / (5 * 9) = 45 / 45

Now, we can compare the fractions:

(10 / 45) _ (45 / 45)

Since 10 is less than 45, we can conclude that:

(2 / 9) < (5 / 5)

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

The answer is: >

Expected answer: <

Correct? No

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the two fractions, we can find a common denominator and then compare the numerators. The least common multiple of 7 and 6 is 42, so we can rewrite the fractions with a denominator of 42:

(2 / 7) = (2 * 6) / (7 * 6) = 12 / 42
(1 / 6) = (1 * 7) / (6 * 7) = 7 / 42

Now we can compare the numerators:

12 / 42 > 7 / 42

So, (2 / 7) is greater than (1 / 6).

### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

Reasoning:
To compare the two fractions, we can find a common denominator and then compare the numerators. The least common multiple of 10 and 6 is 30, so we can rewrite the fractions with a denominator of 30:

(2 / 10) = (2 * 3) / (10 * 3) = 6 / 30
(1 / 6) = (1 * 5) / (6 * 5) = 5 / 30

Now we can compare the numerators:

6 / 30 > 5 / 30

So, (2 / 10) is greater than (1 / 6).

### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the two fractions, we need to determine if they are equal, if the first fraction is greater than the second, or if the second fraction is greater than the first.

The first fraction is 9/9, which simplifies to 1.
The second fraction is 9/10.

Since 1 is greater than 9/10, we can conclude that the first fraction is greater than the second.

Therefore, the correct comparison is:

(9 / 9) > (9 / 10)

Expected answer: >

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

Reasoning:
To compare the two fractions, we need to determine if they are equal, if the first fraction is greater than the second, or if the second fraction is greater than the first.

The first fraction is 9/9, which simplifies to 1.
The second fraction is 9/5.

Since 1 is less than 9/5, we can conclude that the first fraction is less than the second.

Therefore, the correct comparison is:

(9 / 9) < (9 / 5)

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the two fractions, we can simplify them first.
The first fraction, 8/4, can be simplified by dividing both the numerator and denominator by 4. This gives us 2/1, which is equal to 2.
The second fraction, 5/2, cannot be simplified any further.
Now, we can compare the two fractions. Since 2 is less than 5, the first fraction is less than the second fraction.
### Answer: Less than

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

To compare the fractions, we need to find a common denominator. The least common multiple of 2 and 10 is 10. So, we can rewrite the fractions with a common denominator of 10:

(5 / 2) = (5 * 5) / (2 * 5) = 25 / 10
(1 / 10) = 1 / 10

Now, we can compare the fractions:

25 / 10 > 1 / 10

So, the correct symbol to fill in the blank is >.

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

To compare the two fractions, we need to find a common denominator. The least common multiple of 4 and 2 is 4. So, we can rewrite the fractions as:

(2 / 4) = (1 / 2)
(9 / 2) = (18 / 4)

Now, we can compare the fractions:

(1 / 2) _ (18 / 4)

Since 1 is less than 18, we can conclude that:

(1 / 2) < (18 / 4)

So, the answer is <.

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

To compare the two fractions, we need to find a common denominator. The least common multiple of 3 and 2 is 6. So, we can rewrite the fractions as:

(6 / 3) = (6 * 2) / (3 * 2) = 12 / 6
(10 / 2) = (10 * 3) / (2 * 3) = 30 / 6

Now, we can compare the fractions:

12 / 6 _ 30 / 6

Since 12 is less than 30, we can say that:

(6 / 3) < (10 / 2)

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the fractions, we need to find a common denominator. The least common multiple of 3 and 6 is 6. So, we can rewrite the fractions with the common denominator:

(6 / 3) = (6 * 2) / (3 * 2) = 12 / 6
(8 / 6) = (8 * 1) / (6 * 1) = 8 / 6

Now, we can compare the fractions:

12 / 6 _ 8 / 6

Since 12 is greater than 8, the fraction 12 / 6 is greater than the fraction 8 / 6.

### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

Problem: Consider the previous question where you compared the fractions (6 / 3) and (8 / 6). Now let's change the second fraction from (8 / 6) to (8 / 4). How would this affect the comparison? Fill in the blank with the correct symbol (<, >, or =). (6 / 3) _ (8 / 4)

DeepSeek 67B:

Reasoning:
To compare the fractions, we need to find a common denominator. The least common multiple of 3 and 4 is 12. So, we can rewrite the fractions with the common denominator:

(6 / 3) = (6 * 4) / (3 * 4) = 24 / 12
(8 / 4) = (8 * 3) / (4 * 3) = 24 / 12

Now, we can compare the fractions:

24 / 12 _ 24 / 12

Since both fractions are equal, the comparison is equal.

### Answer: =

Expected answer: =

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the fractions, we need to find a common denominator. The least common multiple of 4 and 5 is 20. So, we can rewrite the fractions with the common denominator of 20:

(3 / 4) = (3 * 5) / (4 * 5) = 15 / 20
(6 / 5) = (6 * 4) / (5 * 4) = 24 / 20

Now, we can compare the fractions:

15 / 20 _ 24 / 20

Since 15 is less than 24, we can conclude that:

(3 / 4) < (6 / 5)

### Answer:

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

The two fractions are 1/1 and 3/1.
To compare them, we can look at the numerators.
The numerator of the first fraction is 1, and the numerator of the second fraction is 3.
Since 1 is less than 3, we can conclude that 1/1 is less than 3/1.
Therefore, the answer is <.
### Answer:

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

To compare the fractions (7/6) and (4/5), we can find a common denominator and then compare the numerators.

The least common multiple of 6 and 5 is 30, so we can rewrite the fractions with a denominator of 30:

(7/6) = (7 \* 5) / (6 \* 5) = 35/30
(4/5) = (4 \* 6) / (5 \* 6) = 24/30

Now we can compare the numerators:

35/30 > 24/30

So, (7/6) > (4/5).

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the fractions, we need to find a common denominator. In this case, the common denominator is 4.
(6 / 4) = 3 / 2
(2 / 2) = 1
So, (3 / 2) _ (1)
Since 3/2 is greater than 1, we can fill the blank with >.
### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

Reasoning:
To compare the fractions, we need to find a common denominator. In this case, the common denominator is 20.
(6 / 4) = 15 / 10
(2 / 10) = 2 / 10
So, (15 / 10) _ (2 / 10)
Since 15/10 is greater than 2/10, we can fill the blank with >.
### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the fractions, we need to find a common denominator. The least common multiple of 5 and 10 is 10. So, we can rewrite the fractions with the common denominator:

(5 / 5) = (5 * 2) / (5 * 2) = 10 / 10
(9 / 10) = (9 * 1) / (10 * 1) = 9 / 10

Now, we can compare the fractions:

10 / 10 > 9 / 10

So, (5 / 5) > (9 / 10)

### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

Reasoning:
To compare the fractions, we need to find a common denominator. The least common multiple of 5 and 10 is 10. So, we can rewrite the fractions with the common denominator:

(4 / 5) = (4 * 2) / (5 * 2) = 8 / 10
(9 / 10) = (9 * 1) / (10 * 1) = 9 / 10

Now, we can compare the fractions:

8 / 10 < 9 / 10

So, (4 / 5) < (9 / 10)

### Answer:

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

(1 / 5) < (1 / 1)

Expected answer: <

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

(1 / 5) < (2 / 1)

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

The answer is: >

Expected answer: >

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

The answer is: >

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

To compare the fractions (4/7) and (9/4), we need to find a common denominator. The least common multiple of 7 and 4 is 28. So, we can rewrite the fractions as:

(4/7) = (4 * 4) / (7 * 4) = 16 / 28
(9/4) = (9 * 7) / (4 * 7) = 63 / 28

Now, we can compare the fractions:

16 / 28 _ 63 / 28

Since 16 is less than 63, we can conclude that:

(4/7) < (9/4)

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

(5 / 9) < (4 / 7)

Expected answer: <

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

When we change the denominator of the second fraction from 7 to 10, the fraction becomes (4/10). To compare the two fractions, we can find a common denominator. The least common multiple of 9 and 10 is 90. So, we can rewrite the fractions as:

(5/9) = (5 * 10) / (9 * 10) = 50/90
(4/10) = (4 * 9) / (10 * 9) = 36/90

Now, we can compare the fractions:

50/90 > 36/90

So, (5/9) > (4/10)

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the fractions, we need to find a common denominator. In this case, the common denominator is 3.

Now, let's compare the numerators:
8 (numerator of the first fraction) > 7 (numerator of the second fraction)

Since the numerator of the first fraction is greater than the numerator of the second fraction, the first fraction is greater.

Therefore, (8 / 3) > (7 / 3)

### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

Reasoning:
To compare the fractions, we need to find a common denominator. In this case, the common denominator is 9.

Now, let's compare the numerators:
8 (numerator of the first fraction) > 7 (numerator of the second fraction)

Since the numerator of the first fraction is greater than the numerator of the second fraction, the first fraction is greater.

However, we also need to consider the denominators. In this case, the denominator of the first fraction (9) is greater than the denominator of the second fraction (3).

When the denominator is greater, the fraction is smaller. So, we need to compare the fractions with the same denominator.

Let's rewrite the fractions with the common denominator of 9:
(8 / 9) = (8 / 9)
(7 / 3) = (21 / 9)

Now, let's compare the numerators:
8 (numerator of the first fraction) < 21 (numerator of the second fraction)

Since the numerator of the first fraction is less than the numerator of the second fraction, the first fraction is less.

Therefore, (8 / 9) < (7 / 3)

### Answer:

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the two fractions, we need to find the value of each fraction and then compare them.

The first fraction is 5/1, which simplifies to 5.
The second fraction is 2/2, which simplifies to 1.

Now, we can compare the two values:
5 > 1

So, the correct sign to use is >.

### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

Reasoning:
To compare the two fractions, we need to find the value of each fraction and then compare them.

The first fraction is 5/1, which simplifies to 5.
The second fraction is 7/2, which simplifies to 3.5.

Now, we can compare the two values:
5 > 3.5

So, the correct sign to use is >.

### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the fractions, we need to find a common denominator. In this case, the common denominator is 8.

(5 / 8) = (5 / 8)
(3 / 3) = (3 * 2) / (3 * 2) = 6 / 8

Now we can compare the fractions:
(5 / 8) _ (6 / 8)

Since 5 is less than 6, we can conclude that:
(5 / 8) < (6 / 8)

### Answer:

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the fractions (5/8) and (6/10), we need to find a common denominator. The least common multiple of 8 and 10 is 40. So, we can rewrite the fractions with the common denominator:

(5/8) = (5 * 5) / (8 * 5) = 25/40
(6/10) = (6 * 4) / (10 * 4) = 24/40

Now, we can compare the fractions:

25/40 _ 24/40

Since 25 is greater than 24, the fraction (5/8) is greater than (6/10).

### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

Reasoning:
To compare the fractions (5/8) and (6/5), we need to find a common denominator. The least common multiple of 8 and 5 is 40. So, we can rewrite the fractions with the common denominator:

(5/8) = (5 * 5) / (8 * 5) = 25/40
(6/5) = (6 * 8) / (5 * 8) = 48/40

Now, we can compare the fractions:

25/40 _ 48/40

Since 25 is less than 48, the fraction (5/8) is less than (6/5).

### Answer:

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the two fractions, we need to find a common denominator. In this case, the common denominator is 10.

(5 / 10) = (5 * 1) / (10 * 1) = 5 / 10
(6 / 6) = (6 * 1) / (6 * 1) = 6 / 10

Now we can compare the fractions:

5 / 10 _ 6 / 10

Since 5 is less than 6, the fraction 5 / 10 is less than the fraction 6 / 10.

### Answer:

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the fractions, we need to find a common denominator. In this case, the common denominator is 4.

(5 / 4) = (5 * 1) / (4 * 1) = 5 / 4
(2 / 4) = (2 * 1) / (4 * 1) = 2 / 4

Now we can compare the fractions:
5 / 4 > 2 / 4

So, (5 / 4) > (2 / 4)

### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

Reasoning:
To compare the fractions, we need to find a common denominator. In this case, the common denominator is 40.

(5 / 4) = (5 * 10) / (4 * 10) = 50 / 40
(2 / 10) = (2 * 4) / (10 * 4) = 8 / 40

Now we can compare the fractions:
50 / 40 > 8 / 40

So, (5 / 4) > (2 / 10)

### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

The fractions (6 / 10) and (6 / 10) are equal, so the correct symbol to fill in the blank is =.
### Answer: =

Expected answer: =

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

The fraction (6 / 5) is greater than (6 / 10) because 5 is a smaller denominator than 10, so the correct symbol to fill in the blank is >.
### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

To compare these two fractions, we need to find a common denominator. The least common multiple of 8 and 9 is 72. So, we can rewrite the fractions as:

(6 / 8) = (6 * 9) / (8 * 9) = 54 / 72
(2 / 9) = (2 * 8) / (9 * 8) = 16 / 72

Now, we can compare the fractions:

(54 / 72) _ (16 / 72)

Since 54 is greater than 16, we can conclude that:

(54 / 72) > (16 / 72)

So, the answer is >.

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

To compare the fractions, we need to find a common denominator. The least common multiple of 10 and 2 is 10. So, we can rewrite the fractions as:

(2 / 10) = (2 / 10) * (1 / 1) = (2 / 10)
(6 / 2) = (6 / 2) * (5 / 5) = (30 / 10)

Now, we can compare the fractions:

(2 / 10) _ (30 / 10)

Since 2 is less than 30, we can conclude that:

(2 / 10) < (30 / 10)

So, the answer is <.

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

To determine which fraction is greater, we can compare the numerators and denominators.
The numerator of the first fraction is 2, and the numerator of the second fraction is 9.
The denominator of the first fraction is 7, and the denominator of the second fraction is 4.
Since 9 is greater than 2 and 4 is less than 7, the second fraction is greater.
Therefore, (2 / 7) < (9 / 4).
### Answer:

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

To compare the fractions, we need to find a common denominator. The least common multiple of 1 and 10 is 10. So, we can rewrite the fractions as:

(1 / 1) = (10 / 10)
(2 / 10)

Now, we can compare the fractions:

(10 / 10) _ (2 / 10)

Since 10 is greater than 2, we can conclude that:

(1 / 1) > (2 / 10)

So, the answer is >.

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the fractions, we need to find a common denominator. The least common multiple of 2 and 9 is 18. So, we can rewrite the fractions with the common denominator of 18:

(8 / 2) = (8 * 9) / (2 * 9) = 72 / 18
(10 / 9) = (10 * 2) / (9 * 2) = 20 / 18

Now, we can compare the fractions:

72 / 18 _ 20 / 18

Since 72 > 20, we can conclude that:

(8 / 2) > (10 / 9)

### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

Reasoning:

We can rewrite the fractions with the common denominator of 18:

(8 / 2) = (8 * 9) / (2 * 9) = 72 / 18
(9 / 9) = (9 * 2) / (9 * 2) = 18 / 18

Now, we can compare the fractions:

72 / 18 _ 18 / 18

Since 72 > 18, we can conclude that:

(8 / 2) > (9 / 9)

### Answer: >

When the denominator of the second fraction is 9 instead of 10, the value of the second fraction becomes 1 (since 9/9 = 1). This means that the second fraction is now equal to 1, and the comparison is between the first fraction (8/2) and 1. Since 8/2 is greater than 1, the result is >.

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

To determine the relationship between the fractions, we can simplify the fractions and compare their values.

(3 / 2) = 1.5
(9 / 8) = 1.125

Since 1.5 is greater than 1.125, we can fill in the blank with ">".

### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

To determine the relationship between the fractions, we can simplify the fractions and compare their values.

(3 / 2) = 1.5
(9 / 6) = 1.5

Since 1.5 is equal to 1.5, we can fill in the blank with "=".

### Answer: =

If the denominator of the second fraction was 6 instead of 8, the fraction would still be equal to 1.5. Therefore, the relationship between the fractions would still be "=".

Expected answer: =

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

To compare the two quantities, we can simplify the fractions and then compare them.

(4 / 6) = 2 / 3
(6 / 10) = 3 / 5

Now, we can compare the two fractions:

2 / 3 _ 3 / 5

To compare the fractions, we can find a common denominator. The least common multiple of 3 and 5 is 15. So, we can rewrite the fractions with a common denominator of 15:

(2 / 3) = (10 / 15)
(3 / 5) = (9 / 15)

Now, we can compare the fractions:

10 / 15 _ 9 / 15

Since 10 is greater than 9, we can conclude that:

(4 / 6) > (6 / 10)

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

(8 / 10) < (1 / 1)

Reasoning:
To compare the fractions, we need to find a common denominator. The least common multiple of 10 and 1 is 10. So, we can rewrite the fractions with the common denominator:

(8 / 10) = (8 / 10)
(1 / 1) = (10 / 10)

Now, we can compare the fractions:

(8 / 10) < (10 / 10)

Since 8 is less than 10, the fraction (8 / 10) is less than the fraction (1 / 1).

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

The fraction (5 / 1) is greater than the fraction (5 / 8).

Reasoning:
When comparing fractions, we can compare them by finding a common denominator. In this case, the common denominator is 8.

(5 / 1) = (5 * 8) / (1 * 8) = 40 / 8
(5 / 8) = 5 / 8

Now we can compare the fractions:

40 / 8 > 5 / 8

So, (5 / 1) > (5 / 8)

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

To determine the correct symbol, we need to compare the two fractions (6/5) and (9/6).

First, let's simplify the fractions:

6/5 = 1.2
9/6 = 1.5

Now, let's compare the two values:

1.2 < 1.5

So, the correct symbol to use in the statement is "<".

### Answer:

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the fractions (7/5) and (1/10), we need to find a common denominator. The least common multiple of 5 and 10 is 10. So, we can rewrite the fractions with a common denominator of 10:

(7/5) = (7 * 2) / (5 * 2) = 14 / 10
(1/10) = 1 / 10

Now, we can compare the fractions:

(14 / 10) > (1 / 10)

So, (7/5) > (1/10).

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the two fractions, we need to find a common denominator. The least common multiple of 8 and 1 is 8. So, we can rewrite the fractions with the common denominator of 8:

(10 / 8) = (10 * 1) / (8 * 1) = 10 / 8
(10 / 1) = (10 * 8) / (1 * 8) = 80 / 8

Now, we can compare the two fractions:

10 / 8 _ 80 / 8

Since 10 is less than 80, the fraction 10 / 8 is less than the fraction 80 / 8.

### Answer:

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

To compare the fractions (3/2) and (4/6), we need to find a common denominator. The least common multiple of 2 and 6 is 6. So, we can rewrite the fractions as:

(3/2) = (3 * 3) / (2 * 3) = 9/6

Now, we can compare the fractions:

(9/6) _ (4/6)

Since 9 is greater than 4, we can conclude that:

(9/6) > (4/6)

So, the answer is:

(3 / 2) > (4 / 6)

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the fractions (9/7) and (4/6), we need to find a common denominator. The least common multiple of 7 and 6 is 42. So, we can rewrite the fractions with the common denominator of 42 as follows:

(9/7) = (9 * 6) / (7 * 6) = 54 / 42
(4/6) = (4 * 7) / (6 * 7) = 28 / 42

Now, we can compare the fractions:

(54 / 42) _ (28 / 42)

Since 54 > 28, we can conclude that:

(9 / 7) > (4 / 6)

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the two fractions, we need to find a common denominator. The least common multiple of 7 and 8 is 56. So, we rewrite the fractions with the common denominator:

(8 / 7) = (8 * 8) / (7 * 8) = 64 / 56
(8 / 8) = (8 * 7) / (8 * 7) = 56 / 56

Now, we can compare the fractions:

64 / 56 _ 56 / 56

Since 64 is greater than 56, the fraction 64 / 56 is greater than the fraction 56 / 56.

### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

Reasoning:

We have the following fractions:

(7 / 7) = 1
(8 / 8) = 1

Now, we can compare the fractions:

1 _ 1

Since both fractions are equal to 1, they are equal.

### Answer: =

Expected answer: =

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare these two fractions, we need to find a common denominator. The least common multiple of 5 and 6 is 30. So, we can rewrite the fractions with the common denominator of 30:

(5 / 5) = (5 * 6) / (5 * 6) = 30 / 30
(4 / 6) = (4 * 5) / (6 * 5) = 20 / 30

Now, we can compare the fractions:

(30 / 30) _ (20 / 30)

Since 30 is greater than 20, the fraction (30 / 30) is greater than (20 / 30).

### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

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)

DeepSeek 67B:

Reasoning:
To compare these two fractions, we need to find a common denominator. The least common multiple of 5 and 6 is 30. So, we can rewrite the fractions with the common denominator of 30:

(5 / 5) = (5 * 6) / (5 * 6) = 30 / 30
(2 / 6) = (2 * 5) / (6 * 5) = 10 / 30

Now, we can compare the fractions:

(30 / 30) _ (10 / 30)

Since 30 is greater than 10, the fraction (30 / 30) is greater than (10 / 30).

### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
The fractions (10 / 10) and (4 / 10) can be compared by comparing their numerators.
The numerator of the first fraction is 10, and the numerator of the second fraction is 4.
Since 10 is greater than 4, the first fraction is greater than the second fraction.
Therefore, (10 / 10) > (4 / 10).
### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

Reasoning:
The fractions (10 / 10) and (4 / 9) can be compared by comparing their numerators.
The numerator of the first fraction is 10, and the numerator of the second fraction is 4.
Since 10 is greater than 4, the first fraction is greater than the second fraction.
However, the denominators are different, so we need to compare the fractions with a common denominator.
The least common multiple of 10 and 9 is 90, so we can rewrite the fractions with a common denominator of 90:
(10 / 10) = (10 * 9) / (10 * 10) = 90 / 100
(4 / 9) = (4 * 10) / (9 * 10) = 40 / 90
Now we can compare the numerators:
90 > 40
Therefore, (10 / 10) > (4 / 9).
### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

To compare the fractions, we need to find a common denominator. The least common multiple of 1 and 9 is 9. So, we can rewrite the fractions as:

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

(4 / 9) = (4 * 1) / (9 * 1) = 4 / 9

Now, we can compare the fractions:

(72 / 9) _ (4 / 9)

Since 72 is greater than 4, we can conclude that:

(72 / 9) > (4 / 9)

So, the answer is >.

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

To compare the two fractions, we need to find a common denominator. The least common multiple of 3 and 8 is 24. So, we can rewrite the fractions as:

(9 / 3) = (9 * 8) / (3 * 8) = 72 / 24
(4 / 8) = (4 * 3) / (8 * 3) = 12 / 24

Now, we can compare the fractions:

72 / 24 > 12 / 24

So, the answer is >.

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the fractions, we need to find a common denominator. The least common multiple of 7 and 6 is 42. So, we rewrite the fractions with the common denominator:

(10 / 7) = (10 * 6) / (7 * 6) = 60 / 42
(9 / 6) = (9 * 7) / (6 * 7) = 63 / 42

Now, we can compare the fractions:

60 / 42 < 63 / 42

So, (10 / 7) _ (9 / 6) = <.

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the fractions, we need to find a common denominator. The least common multiple of 8 and 1 is 8. So, we can rewrite the fractions with the common denominator:

(10 / 8) = (10 * 1) / (8 * 1) = 10 / 8
(3 / 1) = (3 * 8) / (1 * 8) = 24 / 8

Now, we can compare the fractions:

10 / 8 < 24 / 8

So, the statement is:

(10 / 8) < (3 / 1)

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

To determine the relationship between the two fractions, we need to compare their values.

First, we can find a common denominator for the fractions. The least common multiple of 2 and 5 is 10.

Now, we can rewrite the fractions with the common denominator:

(3 / 2) = (3 * 5) / (2 * 5) = 15 / 10
(3 / 5) = (3 * 2) / (5 * 2) = 6 / 10

Now, we can compare the two fractions:

15 / 10 > 6 / 10

So, the relationship between the two fractions is:

(3 / 2) > (3 / 5)

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the two fractions, we need to find a common denominator. In this case, the common denominator is 10.

(4 / 10) = (4 / 10)
(2 / 2) = (10 / 10)

Now we can compare the fractions:
(4 / 10) _ (10 / 10)

Since 4 is less than 10, we can conclude that:
(4 / 10) < (10 / 10)

### Answer:

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the two fractions, we need to find a common denominator. The least common multiple of 7 and 10 is 70. So, we can rewrite the fractions with the common denominator of 70:

(6 / 7) = (6 * 10) / (7 * 10) = 60 / 70
(2 / 10) = (2 * 7) / (10 * 7) = 14 / 70

Now, we can compare the two fractions:

60 / 70 _ 14 / 70

Since 60 is greater than 14, the fraction 60 / 70 is greater than the fraction 14 / 70.

### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

Reasoning:
To compare the two fractions, we need to find a common denominator. The least common multiple of 7 and 10 is 70. So, we can rewrite the fractions with the common denominator of 70:

(5 / 7) = (5 * 10) / (7 * 10) = 50 / 70
(2 / 10) = (2 * 7) / (10 * 7) = 14 / 70

Now, we can compare the two fractions:

50 / 70 _ 14 / 70

Since 50 is greater than 14, the fraction 50 / 70 is greater than the fraction 14 / 70.

### Answer: >

However, to compare the original fraction (6 / 7) to the new fraction (5 / 7), we can rewrite the fractions with the common denominator of 7:

(6 / 7) = 6 / 7
(5 / 7) = 5 / 7

Now, we can compare the two fractions:

6 / 7 _ 5 / 7

Since 6 is greater than 5, the fraction 6 / 7 is greater than the fraction 5 / 7.

### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the fractions, we can find a common denominator and then compare the numerators. The least common multiple of 7 and 3 is 21. So, we can rewrite the fractions with the common denominator of 21:

(9 / 7) = (9 * 3) / (7 * 3) = 27 / 21
(10 / 3) = (10 * 7) / (3 * 7) = 70 / 21

Now, we can compare the numerators:

27 _ 70

Since 27 is less than 70, we can conclude that:

(9 / 7) < (10 / 3)

### Answer:

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the two fractions, we need to determine which fraction is greater.
The first fraction is 8/8, which simplifies to 1.
The second fraction is 2/8, which simplifies to 1/4.
Since 1 is greater than 1/4, we can conclude that the first fraction is greater than the second fraction.
Therefore, (8 / 8) > (2 / 8).
The answer is: >

Expected answer: >

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

Reasoning:
To compare the two fractions, we need to determine which fraction is greater.
The first fraction is 8/10, which simplifies to 4/5.
The second fraction is 2/8, which simplifies to 1/4.
Since 4/5 is greater than 1/4, we can conclude that the first fraction is greater than the second fraction.
Therefore, (8 / 10) > (2 / 8).
The answer is: >

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

To compare the two fractions, we need to find a common denominator. The least common multiple of 3 and 8 is 24. So, we can rewrite the fractions as:

(4 / 3) = (4 * 8) / (3 * 8) = 32 / 24
(1 / 8) = (1 * 3) / (8 * 3) = 3 / 24

Now, we can compare the two fractions:

32 / 24 > 3 / 24

So, (4 / 3) > (1 / 8)

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the fractions, we need to find a common denominator. The least common multiple of 6 and 10 is 30. So, we can rewrite the fractions with the common denominator of 30:

(4 / 6) = (4 * 5) / (6 * 5) = 20 / 30
(6 / 10) = (6 * 3) / (10 * 3) = 18 / 30

Now, we can compare the fractions:

(20 / 30) _ (18 / 30)

Since 20 is greater than 18, the fraction (20 / 30) is greater than (18 / 30).

### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

Reasoning:
To compare the ratios, we can rewrite the fractions with the same denominator. The least common multiple of 6 and 1 is 6. So, we can rewrite the fractions with the common denominator of 6:

(4 / 6) = (4 * 1) / (6 * 1) = 4 / 6
(6 / 1) = (6 * 6) / (1 * 6) = 36 / 6

Now, we can compare the fractions:

(4 / 6) _ (36 / 6)

Since 4 is less than 36, the ratio of hawks to total birds is less than the ratio of ravens to total birds.

### Answer:

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the two fractions, we need to find a common denominator. The least common multiple of 6 and 8 is 24. So, we can rewrite the fractions as:

(9 / 6) = (9 * 4) / (6 * 4) = 36 / 24
(6 / 8) = (6 * 3) / (8 * 3) = 18 / 24

Now, we can compare the fractions:

36 / 24 > 18 / 24

So, (9 / 6) > (6 / 8)

### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

Reasoning:
To compare the two fractions, we need to find a common denominator. The least common multiple of 6 and 8 is 24. So, we can rewrite the fractions as:

(9 / 6) = (9 * 4) / (6 * 4) = 36 / 24
(4 / 8) = (4 * 3) / (8 * 3) = 12 / 24

Now, we can compare the fractions:

36 / 24 > 12 / 24

So, (9 / 6) > (4 / 8)

### Answer: >

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

To compare the fractions (2/6) and (8/7), we need to find a common denominator. The least common multiple of 6 and 7 is 42, so we can rewrite the fractions as follows:

(2/6) = (2 \* 7) / (6 \* 7) = 14 / 42
(8/7) = (8 \* 6) / (7 \* 6) = 48 / 42

Now we can compare the fractions:

(14 / 42) _ (48 / 42)

Since 14 is less than 48, we can conclude that:

(2 / 6) < (8 / 7)

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

To compare the fractions (9/6) and (4/10), we need to find a common denominator. The least common multiple of 6 and 10 is 30. So, we can rewrite the fractions as:

(9/6) = (9 * 5) / (6 * 5) = 45/30
(4/10) = (4 * 3) / (10 * 3) = 12/30

Now, we can compare the fractions:

45/30 _ 12/30

Since 45 is greater than 12, we can conclude that:

(9/6) > (4/10)

Expected answer: >

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the fractions, we need to find a common denominator. The least common multiple of 8 and 1 is 8. So, we can rewrite the fractions with the common denominator of 8:

(6 / 8) = (6 * 1 / 8 * 1) = (6 / 8)
(5 / 1) = (5 * 8 / 1 * 8) = (40 / 8)

Now, we can compare the fractions:

(6 / 8) _ (40 / 8)

Since 6 < 40, we can conclude that:

(6 / 8) < (40 / 8)

So, the correct answer is:

(6 / 8) < (5 / 1)

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the two fractions, we need to find a common denominator. The least common multiple of 4 and 10 is 20. So, we can rewrite the fractions with the common denominator of 20:

(8 / 4) = (8 * 5) / (4 * 5) = 40 / 20
(1 / 10) = (1 * 2) / (10 * 2) = 2 / 20

Now, we can compare the fractions:

40 / 20 > 2 / 20

So, (8 / 4) > (1 / 10)

### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

Reasoning:
We can rewrite the fractions with the common denominator of 20:

(2 / 4) = (2 * 5) / (4 * 5) = 10 / 20
(1 / 10) = (1 * 2) / (10 * 2) = 2 / 20

Now, we can compare the fractions:

10 / 20 = 2 / 20

So, (2 / 4) = (1 / 10)

### Answer: =

Expected answer: >

Correct? No

🔗 Problem

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

DeepSeek 67B:

To compare the values of the expressions, we need to simplify each expression and then compare the results.

(6 / 3) = 2
(8 / 2) = 4

Therefore, (6 / 3) < (8 / 2)

### Answer:

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

To compare the fractions (4/6) and (6/3), we need to find a common denominator. The least common multiple of 6 and 3 is 6. So, we can rewrite the fractions as:

(4/6) = (4 * 1) / (6 * 1) = 4/6
(6/3) = (6 * 2) / (3 * 2) = 12/6

Now, we can compare the fractions:

4/6 _ 12/6

Since the denominators are the same, we can compare the numerators:

4 _ 12

Since 4 is less than 12, we can conclude that:

4/6 < 12/6

So, the answer is:

(4 / 6) < (6 / 3)

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

6 / 7 < 3 / 1

Expected answer: <

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

6 / 7 < 9 / 1

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

Reasoning:
To compare the fractions, we can simplify them to their lowest terms.
6/10 can be simplified to 3/5 by dividing both the numerator and denominator by 2.
9/10 cannot be simplified further.
Now we can compare the fractions:
(3/5) _ (9/10)
Since 3/5 is less than 9/10, we can fill in the blank with "<".
### Answer:

Expected answer: <

Correct? Yes

🔗 Problem

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

DeepSeek 67B:

To compare the two fractions, we can find a common denominator and then compare the numerators.

The common denominator for 8 and 9 is 72.

So, we can rewrite the fractions as:

(6 / 8) = (6 * 9) / (8 * 9) = 54 / 72
(4 / 9) = (4 * 8) / (9 * 8) = 32 / 72

Now, we can compare the numerators:

54 > 32

Therefore, (6 / 8) > (4 / 9)

### Answer: >

Expected answer: >

Correct? Yes

🔗 Follow-up

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

DeepSeek 67B:

To compare the two fractions, we can find a common denominator and then compare the numerators.

The common denominator for 8 and 4 is 8.

So, we can rewrite the fractions as:

(6 / 8) = 6 / 8
(4 / 4) = 4 / 8

Now, we can compare the numerators:

6 > 4

Therefore, (6 / 8) > (4 / 4)

### Answer: >

Expected answer: <

Correct? No