Skill: comparing-fractions

Comparing Fractions

Decide which fraction is bigger using benchmarks, common denominators, and the golden rule: with the same numerator, the smaller denominator wins.

1 How to compare any two fractions

Comparing fractions is easy once both fractions speak the same language: same denominators, same numerators, or the same benchmark.

Strategy 1: same denominator

With the same size pieces, more pieces wins: 58 > 38 because 5 > 3.

Strategy 2: same numerator

With the same number of pieces, bigger pieces win: 34 > 38 because fourths are bigger than eighths. Same numerator → the smaller denominator is the bigger fraction.

Strategy 3: common denominator

Nothing matches? Rename both with the least common denominator. 23 vs 35: the LCD of 3 and 5 is 15, so 1015 vs 915 → 2/3 wins.

Strategy 4: benchmarks 0, 1/2, 1

Is it close to 0, 1/2, or 1? 78 is near 1; 29 is near 0 — no common denominator needed.

2 See it: same numerator, different pieces

Both bars shade 3 pieces — but the pieces are different sizes.

Same numerator → compare the piece size: smaller denominator means bigger pieces.

3 Worked examples

Pick the strategy that fits: same bottom, same top, LCD, or benchmark.

Example 1 Same denominator: 5/8 vs 3/8

Thinking: same denominator 8 → same-size pieces.
Compare numerators: 5 > 3.
More same-size pieces wins.

5/8 > 3/8

Example 2 Same numerator: 3/4 vs 3/8

Thinking: same numerator 3 → same number of pieces.
Compare piece size: fourths are bigger than eighths.
Same numerator → smaller denominator wins.

3/4 > 3/8

Example 3 Common denominator: 2/3 vs 3/5

Thinking: neither tops nor bottoms match. LCD of 3 and 5 is 15.
2/3 = 10/15; 3/5 = 9/15.
10 > 9 with the same denominator → 2/3 wins.

2/3 > 3/5

Example 4 Benchmark: closest to 1 among 2/9, 7/8, 4/7

Thinking: 2/9 is near 0 (tiny). 4/7 is near 1/2.
7/8 is one small eighth away from the whole — nearest to 1.

7/8 is closest to 1

4 Common mistakes

Two traps catch almost everyone.

Mistake 1: bigger denominator = bigger fraction

Wrong

18 > 14 — “8 is bigger than 4.”

Right

14 > 18 — a bigger denominator cuts smaller pieces.

Fix: with the same numerator, the SMALLER denominator is the bigger fraction.

Mistake 2: cross-multiplying without comparing correctly

Wrong

For 2/3 vs 3/5, writing 2 × 5 = 10 and 3 × 3 = 9 but comparing 10 < 9 by accident.

Right

Cross products 10 and 9 are the new numerators over 15: 10 > 9, so 2/3 > 3/5.

Fix: the cross product beside the FIRST fraction’s numerator belongs to the FIRST fraction — label them.

5 Key vocabulary

Common denominator: A shared bottom number both fractions can be renamed with.
Least common denominator (LCD): The smallest such shared bottom number.
Benchmark: A friendly reference fraction like 0, 1/2, or 1.
Cross-multiply: Comparing a/b and c/d via a×d and b×c.