Fraction multiplication
Two plain fractions are the purest form of fraction multiplication — no rewriting, no common denominators, just multiply across: 2/3 × 4/5 = (2 × 4)/(3 × 5) = 8/15. This page explains why the rule works, shows how to cancel before multiplying when it saves effort, and collects the mistakes students make most often.
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Interactive tool
Type the problem directly or adjust the factors below. The tool rewrites mixed numbers and whole numbers, simplifies when possible, and shows every multiplication step.
Result
2/5
Multiply fractions by multiplying across. Multiply the numerators, multiply the denominators, and simplify at the end.
Treat each fraction as equal-size pieces. Count how many pieces stay selected after multiplying, then simplify if possible.
Rewritten
2/3 x 3/5
Simplify first
2/1 x 1/5
Before final simplify
2/5
Mixed-number form
Not needed
Formula flow
Step by step
Multiply across
Multiply across: Fraction multiplication does not need common denominators. The operation tracks part of a part, so multiplying across keeps the size relationship intact.
Worked example
1/2 × 1/3 = (1 × 1)/(2 × 3) = 1/6
This is the purest case: two proper fractions, nothing that cancels, nothing to rewrite. Multiply the numerators (1 × 1 = 1) and the denominators (2 × 3 = 6). One half of one third really is one sixth — take a whole cut into thirds, then keep half of that third.
Worked example
2/3 × 3/4 = 6/12 = 1/2
Multiply across as always: 2 × 3 = 6 over 3 × 4 = 12. The product 6/12 shares the factor 6, so it simplifies to 1/2. You could also cancel first — the 3 in the second numerator cancels the 3 in the first denominator — and reach 2/1 × 1/4 = 1/2 with smaller numbers along the way.
Worked example
4/5 × 5/8 = 1/1 × 1/2 = 1/2
The 5 sitting in a denominator cancels the 5 in the other numerator, and then 4 and 8 share the factor 4. Cancel both pairs before multiplying and the problem collapses to 1 × 1/2. Multiply without cancelling and you get 20/40 — the same answer after a longer trip.
Worked example
3/7 × 2/9 = 1/7 × 2/3 = 2/21
The numerators are small, but the denominators 7 and 9 look intimidating. They are fine as they are — just check for cancellations first. The 3 in the first numerator shares a factor with the 9 in the second denominator, so cancel to 1 and 3, then multiply across to 2/21.
The word “of” turns a multiplication problem into a picture. 1/2 × 1/3 means one half of one third. Start with a whole cut into thirds, then keep only half of that third: the whole has now been cut into 2 × 3 = 6 equal pieces, and you are holding exactly one of them — 1/6.
That picture is also the proof of the rule. Each factor cuts the whole again, so the denominators multiply (2 × 3 = 6), and the kept pieces multiply, so the numerators multiply. This is why no common denominator is ever needed: different denominators are not an obstacle here — they are the two cuts that the multiplication performs.
Every fraction-by-fraction problem follows the same two steps. For 3/4 × 2/5:
3/4 × 2/5
= (3 × 2)/(4 × 5)
= 6/20
= 3/10
Step one is the multiplication itself: numerators together (3 × 2 = 6), denominators together (4 × 5 = 20). Step two is simplification: 6 and 20 share the factor 2, so the finished answer is 3/10. That is the entire method — whether the fractions are proper, improper, or a mix of both.
Simplifying at the end always works, but products can get large before they shrink. Cancelling first does the same reduction while the numbers are still small. Look for a numerator that shares a factor with either denominator — the pair can sit in different fractions. In 4/5 × 5/8, the 5s cancel across the fractions, and 4 and 8 reduce to 1 and 2.
Cancelling is optional, not a separate rule. If nothing shares a factor — as in 1/2 × 1/3 — just multiply across. The answer comes out the same either way; cancelling only decides how big the numbers get in between.
Students learn early that “multiplication makes numbers bigger,” and fraction multiplication breaks that expectation. Half of 3/4 is 3/8 — smaller than 3/4. Two thirds of 3/4 is 1/2 — smaller again. Nothing has gone wrong: multiplying by a proper fraction takes a part of something, and a part is always less than the whole.
Keeping this in mind turns a common panic moment into a built-in answer check. If you multiply two proper fractions and get an answer bigger than both of them, the multiplication — not the rule — went wrong somewhere.
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Common questions
