Crosses, then anchors

Judge size without slow division: cross-multiply pairs and simplify singles first. For 3/5 versus 5/8, the crosses are 3 x 8 = 24 and 5 x 5 = 25, so 5/8 is larger by 1/40. For 12/18, divide by 6 to get 2/3 before any comparison. For positive denominators, cross-products settle any pair exactly. First check equal denominators or an obvious benchmark: these can settle a pair with less arithmetic. Normalize denominator signs before cross-multiplying.

Why it works

For positive denominators, comparing a/b with c/d is the same as comparing a x d with c x b, and dividing numerator and denominator by their gcd preserves the value.

a/b < c/d exactly when a x d < c x b (b, d > 0); (a/g)/(b/g) = a/b

Spot it: Two positive fractions need ordering and the denominators are small (2 to 12).

Technique reference and sources

Picture 3/5 versus 5/8 as two rectangles scaled to the shared denominator 40: 24/40 against 25/40. The cross products 24 and 25 are those scaled numerators, so 25 wins by one fortieth. When a fraction is close to a benchmark such as 1/2, read the gap instead: 5/9 exceeds 1/2 by 1/18, so it is above half without any partner fraction.

When this method fits

Two positive fractions need ordering and the denominators are small (2 to 12).
A fraction needs simplifying before comparing, adding, or percenting.
A benchmark such as 1/2 or 1 tells you the side without a partner fraction.
You can hold two cross products (each under about 100) without losing the order of subtraction.

Simplify each fraction first. Divide by the gcd: 12/18 becomes 2/3 (gcd 6). 15/25 becomes 3/5 (gcd 5). Smaller numbers make every later step cheaper.
Cross-multiply the pair. For 3/5 versus 5/8: 3 x 8 = 24 and 5 x 5 = 25. The larger cross sits on the larger fraction side.
Read the verdict with the gap. 25 exceeds 24, so 5/8 exceeds 3/5. The gap is 25 - 24 = 1 over 5 x 8 = 40, so the difference is 1/40.
Check against a benchmark. 3/5 = 0.6 exceeds 1/2 and 5/8 = 0.625 exceeds it further. If the crosses disagree with the benchmarks, recompute the crosses.

Simplify by the gcd: Divide numerator and denominator by their greatest common divisor before comparing.

12/18 = 2/3 (gcd 6)
15/25 = 3/5 (gcd 5)
14/21 = 2/3 (gcd 7)

Cross products for pairs: Compare a x d with c x b; the larger cross marks the larger fraction.

3/5 vs 5/8: 24 vs 25, so 5/8 wins
4/7 vs 5/9: 36 vs 35, so 4/7 wins
2/3 vs 3/4: 8 vs 9, so 3/4 wins

Benchmarks for singles: Compare with 1/2 by doubling the numerator against the denominator; compare with 1 by comparing numerator with denominator. Converting both fractions to decimals is the alternative; it loses for isolated pairs with small denominators because two short products beat two divisions, while decimals win when placing five or more fractions on one scale.

5/9 vs 1/2: 10 vs 9, so above half
7/12 vs 1/2: 14 vs 12, so above half
5/6 vs 7/8: 40 vs 42, so 7/8 wins

Common traps

Comparing numerators only: 5/9 beats 4/7 because 5 exceeds 4. Denominators differ, so crosses decide: 4 x 9 = 36 beats 5 x 7 = 35, so 4/7 is larger. Numerator-only comparison works only with a shared denominator.

Reading the cross order backwards and crowning 3/5 over 5/8. Each cross sits opposite its fraction: 3 x 8 = 24 belongs to 3/5 and 5 x 5 = 25 belongs to 5/8. Larger cross 25 means 5/8 is larger.

Comparing 12/18 with 2/3 by crossing 36 against 36 and calling it luck. Simplify first: 12/18 = 2/3 exactly (gcd 6), so equality is structural, not luck. Crossing unsimplified numbers wastes working memory.

Another example: Which is larger: 4/7 or 5/9?

Left cross: 4 x 9 = 36
Right cross: 5 x 7 = 35
36 exceeds 35, so 4/7 exceeds 5/9 by 1/63

Another way

Benchmarks before cross-products

For fractions with positive denominators, try a shared benchmark such as 1/2 or 1 first. If they lie on opposite sides, the comparison is settled. Equal denominators need only numerator comparison. If neither helps, use cross-products. Normalize negative denominators before using the rule.

1.4/9 < 1/2 < 5/8, so 4/9 < 5/8.
2.3/5 and 5/8 both exceed 1/2; 3 × 8 = 24 < 25 = 5 × 5 settles the pair.

Wins when: Fractions straddle an obvious benchmark or share a denominator, avoiding multiplication.

Loses when: Both are close on the same side of the benchmark; cross-products give an exact verdict.

Source notes

These explanations and questions are written for QuantArc. Links show the underlying mathematics and learning design.

Gerard W. Kelly, Short-Cut Math (Dover 1984), Chapter VI
Chapter VI on fractions, mixed numbers, and percentages (contents page seen). Lesson examples are original.

IES: Organizing Instruction and Study to Improve Student Learning
Supports worked examples, retrieval, explanations and spaced study. Does not establish QuantArc's exact lesson length, mastery threshold or review intervals.