Multiplying fractions
Multiplying fractions is the easiest of the four operations: multiply the numerators, multiply the denominators, simplify. No common denominator needed.
Working
- Multiply the numerators together and the denominators together.
(2 × 3) / (3 × 4) = 6/12 - Simplify: divide top and bottom by their greatest common factor, 6.
6/12 = 1/2
How to multiply fractions, step by step
- Turn mixed numbers and whole numbers into fractions. 1 1/2 becomes 3/2; 4 becomes 4/1.
- Cancel if you can (optional). Divide any numerator and any denominator by a common factor. In 2/3 × 3/4 the 3s cancel and 2 and 4 share a 2, leaving 1/1 × 1/2.
- Multiply straight across. Top × top, bottom × bottom: 2/3 × 3/4 = 6/12.
- Simplify. 6/12 = 1/2 — the same answer cancelling gave directly.
Multiplying fractions is finding an area
"2/3 × 3/4" means "2/3 of 3/4". Shade 2/3 of a square across and 3/4 of it down. The part where both overlap is the answer — and counting cells shows why you multiply tops and bottoms.
Overlap = 2 × 3 = 6 cells. Whole square = 3 × 4 = 12 cells. So 2/3 × 3/4 = 6/12.
Multiplying a fraction by a whole number
Put the whole number over 1: 5 × 2/3 = 5/1 × 2/3 = 10/3 = 3 1/3. In words, five lots of two-thirds.
Multiplying mixed numbers
Always convert first. 1 1/2 × 2 2/3 = 3/2 × 8/3 = 24/6 = 4. Multiplying the whole parts and fraction parts separately (1 × 2 and 1/2 × 2/3) gives the wrong answer, because it skips the cross terms.
Why the answer is smaller
Multiplying by a fraction less than 1 takes a part of something, so 1/2 × 3/4 is less than 3/4. "Of" in a word problem usually means multiply: half of three-quarters of a pizza is 3/8 of the pizza.
Common mistakes
- Finding a common denominator first — it is not wrong, just unnecessary work.
- Multiplying mixed numbers part by part instead of converting.
- Cross-multiplying (that is for comparing fractions or solving proportions, not for multiplying).
Questions people ask
Do you need a common denominator to multiply fractions?
No. Multiply the numerators together and the denominators together, then simplify.
What is 1/2 × 1/2?
1/4. Half of a half is a quarter.
What is 2/3 of 12?
2/3 × 12/1 = 24/3 = 8.
What does cross-cancelling mean?
Dividing a numerator of one fraction and the denominator of the other by the same factor before multiplying. It gives the same answer with smaller numbers.