Comparing and Ordering Fractions

Comparing fractions means figuring out which is larger, which is smaller, or whether they are equal. This is straightforward when fractions share the same denominator, but requires an extra step when they do not. This page covers three reliable methods so you can choose whichever feels most natural.

Method 1: Common Denominators

When two fractions have the same denominator, the one with the larger numerator is larger — just like comparing 5 slices to 3 slices when both pizzas are cut the same way.

$\frac{5}{8} > \frac{3}{8}$

When the denominators are different, convert both fractions to equivalent fractions with a common denominator, then compare the numerators.

Example 1: Compare $\frac{3}{4}$ and $\frac{5}{6}$

Step 1: Find the LCD of 4 and 6. The LCD is 12.

Step 2: Convert both fractions:

$\frac{3}{4} = \frac{9}{12} \qquad \frac{5}{6} = \frac{10}{12}$

Step 3: Compare numerators: $9 < 10$

$\frac{3}{4} < \frac{5}{6}$

Example 2: Compare $\frac{2}{3}$ and $\frac{4}{6}$

LCD of 3 and 6 is 6:

$\frac{2}{3} = \frac{4}{6} \qquad \frac{4}{6} = \frac{4}{6}$

The numerators are equal, so $\frac{2}{3} = \frac{4}{6}$. These are equivalent fractions.

Method 2: Cross-Multiplication

Cross-multiplication is a quick shortcut. To compare $\frac{a}{b}$ and $\frac{c}{d}$:

1. Compute $a \times d$ (left cross-product)
2. Compute $b \times c$ (right cross-product)
3. The fraction on the side with the larger product is the larger fraction

Example 3: Compare $\frac{4}{7}$ and $\frac{3}{5}$

$4 \times 5 = 20 \qquad 7 \times 3 = 21$

Left product (20) < Right product (21), so:

$\frac{4}{7} < \frac{3}{5}$

Example 4: Compare $\frac{5}{9}$ and $\frac{7}{12}$

$5 \times 12 = 60 \qquad 9 \times 7 = 63$

$60 < 63$, so $\frac{5}{9} < \frac{7}{12}$

Method 3: Benchmark Fractions

Benchmark fractions ($0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1$) give you a quick way to estimate without any computation. Decide whether each fraction is less than, equal to, or greater than the benchmark.

Example 5: Compare $\frac{3}{8}$ and $\frac{5}{7}$

• $\frac{3}{8}$: Is $3$ less than half of $8$? Half of 8 is 4, and $3 < 4$, so $\frac{3}{8} < \frac{1}{2}$
• $\frac{5}{7}$: Is $5$ more than half of $7$? Half of 7 is 3.5, and $5 > 3.5$, so $\frac{5}{7} > \frac{1}{2}$

Since $\frac{3}{8}$ is below $\frac{1}{2}$ and $\frac{5}{7}$ is above $\frac{1}{2}$:

$\frac{3}{8} < \frac{5}{7}$

Comparing Fractions with the Same Numerator

When two fractions have the same numerator, the one with the smaller denominator is larger. Fewer pieces means each piece is bigger.

$\frac{3}{5} > \frac{3}{8}$

Think of it this way: $\frac{3}{5}$ means 3 out of 5 equal parts (larger pieces), while $\frac{3}{8}$ means 3 out of 8 equal parts (smaller pieces).

Ordering Multiple Fractions

To sort a list of fractions from least to greatest (or greatest to least):

1. Find the LCD of all the denominators
2. Convert every fraction to the common denominator
3. Order by numerator

Example 6: Order $\frac{2}{3}$, $\frac{1}{4}$, $\frac{5}{6}$, $\frac{1}{2}$ from least to greatest

Step 1: LCD of 3, 4, 6, 2 is 12.

Step 2: Convert all fractions:

$\frac{2}{3} = \frac{8}{12} \qquad \frac{1}{4} = \frac{3}{12} \qquad \frac{5}{6} = \frac{10}{12} \qquad \frac{1}{2} = \frac{6}{12}$

Step 3: Order the numerators: 3, 6, 8, 10

Answer: $\frac{1}{4} < \frac{1}{2} < \frac{2}{3} < \frac{5}{6}$

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Same denominator: compare numerators directly
• Different denominators: use common denominators, cross-multiplication, or benchmarks
• Same numerator: the smaller denominator makes the larger fraction
• To order multiple fractions: convert all to a common denominator and sort by numerator
• Benchmark fractions ($\frac{1}{2}$, etc.) give quick estimates without computation

Return to Arithmetic for more foundational math topics.