Comparing Fractions – Which Is Bigger?
Comparing fractions means deciding which of two fractions is larger, smaller, or whether they are equal. With like fractions it is immediate; with unlike fractions you need one of the four methods below.
Cross-multiplication, the fastest of the four methods below, relies on a simple piece of algebra: for positive fractions a/b and c/d, a/b is greater than c/d exactly when a × d is greater than b × c. This shortcut avoids finding a common denominator altogether and was already in use by medieval Islamic mathematicians working with ratios centuries before it appeared in European textbooks. Comparing fractions correctly matters outside school too — deciding whether a 3/8-full tank or a 2/5-full tank has more fuel, or whether a “buy 2/3 off” sale beats a “half price” sale, both come down to comparing fractions.
Case 1 – Same Denominator (Like Fractions)
Compare the numerators directly. The larger numerator gives the larger fraction.
Case 2 – Same Numerator
The fraction with the smaller denominator is larger (larger pieces).
Method 1 – Common Denominator
Convert both fractions to like fractions using the LCM, then compare numerators.
LCM(4,6) = 12. 3/4 = 9/12. 5/6 = 10/12. 10 > 9, so 5/6 > 3/4.
Method 2 – Cross-Multiplication
Multiply diagonally. Compare the two products.
2 × 5 = 10. 3 × 3 = 9. 10 > 9 → 2/3 is on the left → 2/3 > 3/5.
Method 3 – Convert to Decimals
5/8 = 0.625. 3/5 = 0.6. 0.625 > 0.6 → 5/8 > 3/5.
Method 4 – Benchmark Fractions
Compare each fraction to 1/2 (or 0 or 1) as a reference point.
Symbols Reminder
| Symbol | Meaning | Example |
|---|---|---|
| > | Greater than | 3/4 > 1/2 |
| < | Less than | 1/3 < 1/2 |
| = | Equal to | 2/4 = 1/2 |
Key Takeaways
- Like fractions: compare numerators. Same numerator: smaller denominator wins.
- Common denominator: convert then compare.
- Cross-multiplication is fast and reliable for any two fractions.
- Benchmark fractions (using 0, 1/2, 1) give quick estimates.