Free Math Guide

Comparing Fractions Explained

Learn how to choose the simplest fraction-comparison strategy and determine whether one fraction is less than, greater than, or equal to another.

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Comparing fractions means determining which fraction has the greater value or whether two fractions are equal. The easiest method depends on the numbers in the problem. Before using cross multiplication, always check whether a simpler comparison strategy is available.

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The Basic Idea

What Does It Mean to Compare Fractions?

Comparing fractions means deciding which fraction represents more of a whole. The comparison depends on both the numerator and denominator.

2 5 < 4 5

Both fractions use fifths. Since four fifths contains more fifth-size pieces than two fifths, four fifths is greater.

Compare the Fraction Values

Do not compare the numerator or denominator by itself unless the fractions have a special matching structure.

Comparison Language

Less Than, Greater Than, and Equal To

Fraction comparisons use the same three symbols used to compare whole numbers.

Less Than

1/4 < 3/4

The symbol opens toward the greater value. One-fourth is less than three-fourths.

Greater Than

5/6 > 2/6

Five-sixths represents more of the whole than two-sixths.

Equal To

1/2 = 3/6

Equivalent fractions may use different numbers but represent the same value.

Read the Entire Statement

3/8 < 5/8

Read this as “three-eighths is less than five-eighths.”

Work Smarter

How to Choose a Fraction-Comparison Strategy

Before performing calculations, inspect the fractions and look for a simple relationship.

1

Check the Denominators

When the denominators match, compare the numerators.

2

Check the Numerators

When the numerators match, the fraction with the smaller denominator is greater.

3

Look for a Benchmark

Decide whether each fraction is below, equal to, or above one-half or one whole.

4

Check for Equivalence

One fraction may be a scaled version of the other.

5

Use a Common Denominator

Rewrite the fractions using equal-size parts and then compare their numerators.

6

Cross Multiply

When no simpler relationship is available, compare the cross products.

Use the Simplest Valid Method

Cross multiplication works for most fraction comparisons, but it may hide easier relationships that build stronger fraction understanding.

Strategy 1

Comparing Fractions With the Same Denominator

When two fractions have the same denominator, their pieces are the same size. Compare how many pieces each fraction has.

Same-Denominator Rule

When denominators are equal, the fraction with the greater numerator is greater.

Example 1

3/8 ___ 7/8
  1. The denominators are both 8.
  2. Compare the numerators 3 and 7.
  3. Since 3 is less than 7, 3/8 is less than 7/8.
3/8 < 7/8

Example 2

11/15 ___ 4/15
  1. The denominators are both 15.
  2. Compare the numerators 11 and 4.
  3. Since 11 is greater than 4, 11/15 is greater.
11/15 > 4/15

Strategy 2

Comparing Fractions With the Same Numerator

When two fractions have equal numerators, they contain the same number of pieces. The denominator tells you the size of each piece.

3 4

Three larger pieces

>
3 8

Three smaller pieces

Same-Numerator Rule

When numerators are equal, the fraction with the smaller denominator is greater because its pieces are larger.

Example: Compare 5/6 and 5/9

5/6 ___ 5/9
  1. The numerators are both 5.
  2. Sixths are larger pieces than ninths.
  3. Therefore, five-sixths is greater than five-ninths.
5/6 > 5/9

Strategy 3

Comparing Fractions Using Benchmarks

A benchmark fraction is a familiar value used as a reference. Common benchmarks include zero, one-half, and one whole.

Comparing With One-Half

Double the numerator. If the result is less than the denominator, the fraction is below one-half. If it is greater, the fraction is above one-half.

Example 1

3/10 ___ 7/12
  1. For 3/10, double 3: 3 × 2 = 6.
  2. Since 6 is less than 10, 3/10 is below one-half.
  3. For 7/12, double 7: 7 × 2 = 14.
  4. Since 14 is greater than 12, 7/12 is above one-half.
3/10 < 7/12

Example 2

4/8 ___ 5/9
  1. For 4/8, double 4: 4 × 2 = 8.
  2. Therefore, 4/8 equals one-half.
  3. For 5/9, double 5: 5 × 2 = 10.
  4. Since 10 is greater than 9, 5/9 is above one-half.
4/8 < 5/9

Strategy 4

Comparing Fractions Using Common Denominators

When fractions are rewritten using the same denominator, they are expressed using equal-size pieces. You can then compare the numerators.

1

Find the LCD

Find the least common multiple of the two denominators.

2

Rewrite the Fractions

Multiply each numerator and denominator by the required scale factor.

3

Compare Numerators

Once the denominators match, compare the new numerators.

4

Write the Symbol

Return to the original fractions and insert the correct comparison symbol.

Example: Compare 3/4 and 5/8

3/4 ___ 5/8
  1. The least common denominator of 4 and 8 is 8.
  2. Rewrite 3/4 as 6/8 by multiplying by 2/2.
  3. The second fraction is already 5/8.
  4. Since 6 is greater than 5, 6/8 is greater than 5/8.
3/4 = 6/8
6/8 > 5/8
3/4 > 5/8

Strategy 5

Recognizing Equivalent Fractions

Equivalent fractions use different numerators or denominators but represent the same amount.

2/3 = 4/6 = 6/9

You may recognize equivalence through a scale factor, simplification, or equal cross products.

Use a Scale Factor

3/5 ___ 12/20
  1. Three was multiplied by 4 to produce 12.
  2. Five was also multiplied by 4 to produce 20.
  3. Both parts changed by the same scale factor.
3/5 = 12/20

Simplify Both Fractions

8/12 ___ 14/21
  1. Simplify 8/12 by dividing by 4: 2/3.
  2. Simplify 14/21 by dividing by 7: 2/3.
  3. Both fractions simplify to the same value.
8/12 = 14/21

Strategy 6

Comparing Fractions With Cross Multiplication

Cross multiplication compares two fractions without first finding a common denominator.

Compare 4/7 and 5/9

First Cross Product

Multiply the first numerator by the second denominator.

4 × 9 = 36

Second Cross Product

Multiply the second numerator by the first denominator.

5 × 7 = 35
Since 36 > 35, 4/7 > 5/9

Cross-Product Direction

The product created from the first numerator corresponds to the first fraction. The product created from the second numerator corresponds to the second fraction.

Example With Equal Cross Products

3/8 ___ 9/24
  1. Multiply 3 × 24 = 72.
  2. Multiply 9 × 8 = 72.
  3. The cross products are equal.
  4. Therefore, the fractions are equivalent.
3/8 = 9/24

More Difficult Comparisons

Comparing Improper Fractions and Mixed Numbers

Improper fractions have numerators greater than or equal to their denominators. Mixed numbers contain a whole-number part and a fractional part.

Compare Whole-Number Parts

2 1/4 ___ 1 7/8
  1. The first mixed number contains 2 wholes.
  2. The second mixed number contains only 1 whole.
  3. No fractional calculation is necessary.
2 1/4 > 1 7/8

Equal Whole-Number Parts

3 2/5 ___ 3 3/8
  1. Both mixed numbers contain 3 wholes.
  2. Compare 2/5 and 3/8.
  3. Cross products: 2 × 8 = 16 and 3 × 5 = 15.
  4. Therefore, 2/5 is greater than 3/8.
3 2/5 > 3 3/8

Comparing Improper Fractions to One Whole

9/8 ___ 7/6

Both fractions are greater than one, so comparing each fraction only to one whole does not settle the comparison. Use cross multiplication.

9 × 6 = 54
7 × 8 = 56
9/8 < 7/6

What to Watch For

Common Fraction-Comparison Mistakes

Comparing Numerators Only

4/9 > 3/4 because 4 > 3

The denominator changes the size of each piece. Numerators can be compared directly only when the denominators are equal.

Assuming a Larger Denominator Means a Larger Fraction

1/8 > 1/4 because 8 > 4

With equal numerators, a larger denominator creates smaller pieces. Therefore, one-eighth is less than one-fourth.

Reversing Cross Products

Each cross product must remain associated with the numerator used to create it.

For a/b and c/d:
a × d corresponds to a/b
c × b corresponds to c/d

Creating Incorrect Equivalent Fractions

2/3 = 4/5

Multiplying only the numerator does not preserve the fraction's value. Multiply the numerator and denominator by the same number.

Using the Denominator Comparison Backward

5/12 > 5/8 because 12 > 8

With equal numerators, the smaller denominator produces the greater fraction. Therefore, 5/8 is greater.

Converting to Decimals Too Early

Decimal conversion can work, but repeating decimals may make the comparison less clear. Look for a simpler fraction relationship first.

Check Your Work

Fraction-Comparison Checklist

Use these questions before submitting your answer.

Same Denominators?
Compare the numerators directly.
Same Numerators?
The fraction with the smaller denominator is greater.
Useful Benchmark?
Compare each fraction with one-half or one whole.
Equivalent Fractions?
Check scale factors, simplified forms, or cross products.
Common Denominator?
Rewrite both fractions using the least common denominator.
Cross Products Correct?
Verify both multiplication calculations and preserve their direction.
Symbol Facing Correctly?
The open side of the comparison symbol faces the greater value.

Ready to Practice?

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Practice choosing the best comparison strategy, rewriting fractions, using benchmarks, calculating cross products, and selecting the correct comparison symbol with guided feedback.

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