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Comparing fractions

Comparing Fractions by Cross Multiplication

To cross multiply fractions, multiply along each diagonal and compare the two products — the bigger product marks the bigger fraction. It is the fastest way to compare fractions with unlike denominators when no benchmark like one-half stands out, and this page shows exactly why it works and when to reach for it.

Cross multiply tool

Cross multiply and let the products decide

Each diagonal multiplies one fraction's numerator by the other's denominator. Both products share the same hidden denominator, so the bigger product really is the bigger fraction.

Numerators 1–20, denominators 2–20. Improper fractions are welcome; mixed numbers should be converted first.

Fraction A

Fraction B

5634A: 5 × 4 = 20B: 3 × 6 = 18

Result

5/6 > 3/4

5 × 4 = 20 beats 3 × 6 = 18, so 5/6 is greater.

Steps

  1. 1.Multiply along each diagonal: 5 × 4 = 20 and 3 × 6 = 18.
  2. 2.Compare the cross products: 20 is greater than 18.
  3. 3.Both products are numerators over the same hidden denominator (6 × 4 = 24), so the bigger product belongs to the bigger fraction: 5/6 is greater.

Best method

Cross products

Compare the cross products

Multiply 5/6 by 4/4 and 3/4 by 6/6 — both multipliers equal 1, so the amounts never change. Both fractions now share the denominator 24, and the cross products are their numerators. Same-size parts: the bigger numerator wins.

Explain to a child

Both fractions secretly get the same denominator, so the two products are the parts they would count. More parts of the same size means the bigger product belongs to the bigger fraction.

How it works

Three steps and the cross product formula

For fractions a/b and c/d, the cross product formula compares a × d against c × b. That is all the butterfly picture is doing: one diagonal computes each product.

  1. 1. Multiply along each diagonal

    Multiply the top of the left fraction by the bottom of the right fraction, then the top of the right fraction by the bottom of the left. These are the two cross products.

  2. 2. Compare the two cross products

    The cross products are plain numbers, so compare them directly. One will be greater, or they will be equal.

  3. 3. Let the bigger product crown its fraction

    Whichever fraction sits at the diagonal with the bigger cross product is the greater fraction. Equal products mean the fractions are equal.

Why it works

A common denominator in disguise

Cross multiplication is the common-denominator method with the writing skipped — and the skipped step is what makes it fair. To compare a/b and c/d, multiply the first fraction by d/d and the second by b/b. Each multiplier equals 1, so neither amount changes; only the names do. Both fractions now carry the same denominator b × d: they become ad/bd and cb/bd.

Same denominator means same-size parts, so comparing the fractions comes down to comparing the numerators: ad against cb. If ad is the bigger numerator, then a/b is the bigger fraction — and the same logic favors c/d when cb is bigger. Those two numerators, a × d and c × b, are exactly the two cross products the butterfly computes.

That is also why the method never fails on proper fractions, improper fractions, or close calls: the rewrite always makes the parts the same size, so the comparison is always fair. Read more on the common denominators method in the comparing fractions topic guide.

The hidden rewrite

Multiply 5/6 by 4/4 and 3/4 by 6/6. Both multipliers equal 1, so the amounts never change — but now both fractions are twenty-fourths: 20/24 and 18/24. Same-size parts, so the numerators decide: 20 against 18.

Now look at where those numerators came from: 20 = 5 × 4 and 18 = 3 × 6. They are exactly the two cross products — which is why the butterfly never needs the rewrite written down.

When to use it

When do you cross multiply fractions?

Reach for it when

  • The denominators differ and no benchmark like one-half helps — 5/6 vs 3/4.
  • The pair is too close to call by picture — 4/7 vs 5/9.
  • You want a fast check after reasoning it out another way.
  • Improper fractions are in play — 3/2 vs 5/6 works the same way.

Skip it when

  • The denominators already match — compare numerators directly.
  • The numerators already match — the smaller denominator wins.
  • You are adding or subtracting — cross multiplication never applies there.
  • The numbers are mixed — compare wholes first, or convert before cross multiplying.

Worked examples

The formula in action

Classic unlike denominators: 5/6 vs 3/4

How to read it: Multiply 5 × 4 = 20 and 3 × 6 = 18. The first product is bigger, so 5/6 is the greater fraction — no rewriting, no common denominators on paper.

Conclusion: 5/6 > 3/4

A close call the eye cannot judge: 4/7 vs 5/9

How to read it: Both fractions sit near one-half, and pictures barely tell them apart. Cross multiplication settles it exactly: 4 × 9 = 36 beats 5 × 7 = 35 by a single step.

Conclusion: 4/7 > 5/9

Equal products: 2/4 vs 1/2

How to read it: 2 × 2 = 4 and 1 × 4 = 4. Equal cross products mean the fractions are equivalent — the same check works in reverse to prove two fractions are equal.

Conclusion: 2/4 = 1/2

Two different “cross” ideas

Cross multiplication is not cross cancelling

They share a word and an X-shaped picture, but they solve different problems. Cross multiplication compares two fractions by multiplying each numerator into the other denominator. Cross cancelling simplifies before multiplying fractions, dividing one numerator and the other denominator by a shared factor.

If your goal is to multiply fractions faster, that is cross cancelling — see it step by step on the multiplying fractions page.

Cross multiplication — comparing

5/6 vs 3/4 → 5 × 4 = 20, 3 × 6 = 18 → 5/6 > 3/4. No multiplication of the fractions themselves ever happens.

Cross cancelling — multiplying

5/6 × 3/10 → the 5 and the 10 share a factor → 1/6 × 3/2 = 3/12 = 1/4. Here the fractions really are multiplied.

Common mistakes

Watch for these slips

Adding along the diagonal instead of multiplying

Cross multiplication means multiply. Adding 5 + 4 across a diagonal produces a number that means nothing about the fractions' sizes.

Cross multiplying three fractions at once

The method compares exactly two fractions. For three or more, compare them in pairs, or group them with a benchmark first.

Cross multiplying mixed numbers directly

A mixed number such as 2 1/4 must be rewritten as the improper fraction 9/4 first. Cross multiplying the "2" and the "1/4" separately mixes up the whole parts.

Common questions

Cross Multiplication FAQ

Use it when two fractions have different denominators and no easy benchmark stands out. Skip it when the denominators match or the numerators match — faster methods already exist. It never applies to adding or subtracting fractions.

Related pages

Keep going with fractions