Which fraction is bigger?
Comparing fractions is trickier than comparing whole numbers, because the bigger numbers don’t always mean the bigger fraction. There are several smart ways to compare.
Method 1: Same denominators
When fractions have like denominators, the parts are the same size. Just compare the numerators.
- 5/8 > 3/8
Method 2: Same numerators
When numerators are the same, you have the same number of parts. The one with the smaller denominator has bigger parts, so it’s bigger.
- 3/5 > 3/8
Method 3: Benchmark fractions
Compare each fraction to a benchmark fraction: 0, 1/2 or 1.
| Fraction | Compared to 1/2 |
|---|---|
| 3/8 | less (half of 8 is 4) |
| 5/9 | more (half of 9 is 4.5) |
So 3/8 < 5/9.
You can also use "distance from 1": 7/8 is 1/8 away from 1, and 5/6 is 1/6 away. Since 1/8 is smaller, 7/8 is closer to 1, so 7/8 > 5/6.
Method 4: Common denominators
Rewrite both fractions with a common denominator, then compare numerators.
Example: 2/3 vs 3/4
- Common denominator: 12.
- 2/3 = 8/12; 3/4 = 9/12.
- 9/12 > 8/12, so 3/4 > 2/3.
Method 5: Cross-multiply
Multiply each numerator by the other fraction’s denominator.
Example: 4/7 vs 5/9
- 4 × 9 = 36 (goes with 4/7)
- 5 × 7 = 35 (goes with 5/9)
- 36 > 35, so 4/7 > 5/9.
This is really a shortcut for Method 4.
Ordering several fractions
To order fractions, convert them all to a common denominator (or to decimals), then line them up.
Example: Order 1/2, 3/8, 3/4 from least to greatest.
- Use eighths: 4/8, 3/8, 6/8.
- Order: 3/8, 1/2, 3/4.
Worked examples
Example 1: Which is greater: 2/5 or 2/7?
Same numerators; fifths are bigger than sevenths. 2/5.
Example 2: Compare 5/6 and 7/12.
5/6 = 10/12. 10/12 > 7/12, so 5/6 > 7/12.
Example 3: Order 2/3, 1/6, 5/12 from greatest to least.
Twelfths: 8/12, 2/12, 5/12 → 2/3, 5/12, 1/6.
Common mistakes
- Comparing only the denominators, thinking 1/8 > 1/4 because 8 > 4.
- Comparing only the numerators when denominators differ.
- Forgetting to change the numerator when finding a common denominator.
Try it yourself
- 3/10 ◯ 7/10
- 4/9 ◯ 4/5
- 5/8 ◯ 3/5
- Order from least to greatest: 1/3, 1/4, 1/2.
- Order: 3/4, 5/8, 7/8.
Answers: 1) < 2) < 3) > (25/40 vs 24/40) 4) 1/4, 1/3, 1/2 5) 5/8, 3/4, 7/8
Quick recap
- Like denominators: compare numerators.
- Same numerators: smaller denominator is bigger.
- Use benchmark fractions, a common denominator or cross-multiply for others.