Comparing Fractions — Definition, Formula & Examples

Comparing fractions is the process of determining which of two fractions is greater, less, or whether they are equal. You do this by rewriting the fractions so they share a common denominator, then comparing the numerators.

Given two fractions $\frac{a}{b}$ and $\frac{c}{d}$ with positive denominators, $\frac{a}{b} < \frac{c}{d}$ if and only if $a \times d < c \times b$ . Equivalently, both fractions can be expressed with a common denominator and their numerators compared directly.

How It Works

There are two main methods. The first is to find a common denominator, rewrite each fraction with that denominator, and compare the numerators — the fraction with the larger numerator is the larger fraction. The second method is cross multiplication: multiply each numerator by the other fraction's denominator and compare the two products. If the denominators are already the same, just compare the numerators directly. If the numerators are the same, the fraction with the smaller denominator is actually the larger fraction, since each piece is bigger.

Worked Example

Problem: Which is greater:

\frac{3}{4}

\frac{5}{6}

Find a common denominator: The least common denominator of 4 and 6 is 12.

\text{LCD} = 12

Rewrite each fraction: Multiply the numerator and denominator of each fraction to get a denominator of 12.

\frac{3}{4} = \frac{9}{12}, \quad \frac{5}{6} = \frac{10}{12}

Compare the numerators: Since 9 < 10, the first fraction is smaller.

9 < 10 \implies \frac{3}{4} < \frac{5}{6}

Answer:

\frac{5}{6}

is greater than

\frac{3}{4}

Why It Matters

Comparing fractions comes up whenever you need to decide which portion is larger — splitting food fairly, reading measurements, or choosing the better deal at a store. It also builds the foundation for ordering rational numbers and working with inequalities in later math courses.

Common Mistakes

Mistake: Comparing numerators without making the denominators the same (e.g., saying

\frac{3}{4} > \frac{5}{6}

because 3 and 4 are closer together).

Correction: You can only compare numerators directly when the denominators are equal. Always find a common denominator first, or use cross multiplication.