Comparing two fractions

Same denominator: direct comparison

When two fractions have the same denominator, you simply need to compare the numerators. The larger the numerator, the larger the fraction.

Example: 3/7 and 5/7 have the same denominator (7). As 5 > 3, we have 5/7 > 3/7.

Same numerator: be careful with the reverse

When two fractions have the same numerator, it is the smallest denominator that gives the largest fraction. This is because dividing a whole into fewer parts results in larger parts.

Example: 1/3 and 1/5. As 3 < 5, we have 1/3 > 1/5 (one part out of 3 is larger than one part out of 5).

General case: reducing to the same denominator

To compare fractions with different numerators and denominators, we bring them to the same denominator by cross-multiplying.

Example: compare 2/3 and 3/5.

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

As 10/15 > 9/15, we conclude that 2/3 > 3/5.

Summary table

Situation	Rule
Same denominator	Compare the numerators
Same numerator	The smaller denominator gives the larger fraction
General case	Convert to the same denominator, then compare

Common pitfall to avoid

Never compare numerators directly if the denominators are different: 3/4 is not smaller than 5/8 simply because 3 < 5. You must first reduce to the same denominator (3/4 = 6/8, and 6/8 > 5/8).