Comparing Fractions Worksheet - Free Printable

Problem Description:

The task involves comparing pairs of fractions and determining whether one fraction is greater than, less than, or equal to the other. The provided image shows a worksheet with 24 problems, each comparing two fractions. The answers are already filled in using the symbols \( > \), \( < \), or \( = \).

Solution Explanation:

To compare two fractions, we can use one of the following methods:
1. Find a Common Denominator: Rewrite both fractions with the same denominator and then compare the numerators.
2. Cross-Multiply: Multiply the numerator of the first fraction by the denominator of the second fraction and vice versa. Compare the products.
3. Convert to Decimals: Convert both fractions to decimal form and compare the decimal values.

Below, I will explain the reasoning for a few sample problems to illustrate the process.

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#### Sample Problem 1:
Compare \( \frac{1}{6} \) and \( \frac{1}{10} \).

- Method: Cross-multiply.
- \( 1 \times 10 = 10 \)
- \( 1 \times 6 = 6 \)
- Since \( 10 > 6 \), \( \frac{1}{6} > \frac{1}{10} \).

Answer: \( > \)

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#### Sample Problem 2:
Compare \( \frac{2}{3} \) and \( \frac{1}{2} \).

- Method: Cross-multiply.
- \( 2 \times 2 = 4 \)
- \( 1 \times 3 = 3 \)
- Since \( 4 > 3 \), \( \frac{2}{3} > \frac{1}{2} \).

Answer: \( > \)

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#### Sample Problem 3:
Compare \( \frac{7}{12} \) and \( \frac{3}{4} \).

- Method: Find a common denominator.
- The least common denominator (LCD) of 12 and 4 is 12.
- Rewrite \( \frac{3}{4} \) with a denominator of 12: \( \frac{3}{4} = \frac{9}{12} \).
- Compare \( \frac{7}{12} \) and \( \frac{9}{12} \).
- Since \( 7 < 9 \), \( \frac{7}{12} < \frac{3}{4} \).

Answer: \( < \)

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#### Sample Problem 4:
Compare \( \frac{3}{5} \) and \( \frac{11}{15} \).

- Method: Cross-multiply.
- \( 3 \times 15 = 45 \)
- \( 11 \times 5 = 55 \)
- Since \( 45 < 55 \), \( \frac{3}{5} < \frac{11}{15} \).

Answer: \( < \)

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#### Sample Problem 5:
Compare \( \frac{1}{6} \) and \( \frac{2}{12} \).

- Method: Simplify the second fraction.
- \( \frac{2}{12} = \frac{1}{6} \) (dividing numerator and denominator by 2).
- Since both fractions are equal, \( \frac{1}{6} = \frac{2}{12} \).

Answer: \( = \)

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#### Sample Problem 6:
Compare \( \frac{3}{7} \) and \( \frac{1}{2} \).

- Method: Cross-multiply.
- \( 3 \times 2 = 6 \)
- \( 1 \times 7 = 7 \)
- Since \( 6 < 7 \), \( \frac{3}{7} < \frac{1}{2} \).

Answer: \( < \)

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#### Sample Problem 7:
Compare \( \frac{6}{8} \) and \( \frac{3}{4} \).

- Method: Simplify the first fraction.
- \( \frac{6}{8} = \frac{3}{4} \) (dividing numerator and denominator by 2).
- Since both fractions are equal, \( \frac{6}{8} = \frac{3}{4} \).

Answer: \( = \)

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#### Sample Problem 8:
Compare \( \frac{5}{7} \) and \( \frac{5}{8} \).

- Method: Compare the denominators.
- Both fractions have the same numerator (5).
- A larger denominator means a smaller fraction.
- Since \( 7 < 8 \), \( \frac{5}{7} > \frac{5}{8} \).

Answer: \( > \)

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#### Sample Problem 9:
Compare \( \frac{1}{5} \) and \( \frac{3}{10} \).

- Method: Cross-multiply.
- \( 1 \times 10 = 10 \)
- \( 3 \times 5 = 15 \)
- Since \( 10 < 15 \), \( \frac{1}{5} < \frac{3}{10} \).

Answer: \( < \)

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#### Sample Problem 10:
Compare \( \frac{7}{12} \) and \( \frac{3}{4} \).

- Method: Find a common denominator.
- The least common denominator (LCD) of 12 and 4 is 12.
- Rewrite \( \frac{3}{4} \) with a denominator of 12: \( \frac{3}{4} = \frac{9}{12} \).
- Compare \( \frac{7}{12} \) and \( \frac{9}{12} \).
- Since \( 7 < 9 \), \( \frac{7}{12} < \frac{3}{4} \).

Answer: \( < \)

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Final Answer:

The worksheet provides the correct answers for all 24 problems. The solutions are consistent with the methods explained above.

\[
\boxed{\text{The answers are correct as provided in the worksheet.}}
\]