The rule
To multiply fractions and find their product, multiply the numerators together and the denominators together:
a/b × c/d = (a × c)/(b × d)
Example: 2/3 × 4/5 = 8/15
No common denominator needed! Multiplying is easier than adding.
What it means
"2/3 × 4/5" means "2/3 of 4/5". Picture a rectangle:
- Shade 4/5 of it with vertical strips.
- Shade 2/3 of it with horizontal strips.
- The double-shaded part is 8 out of 15 equal pieces: 8/15.
That’s why the denominators multiply (3 × 5 = 15 pieces) and the numerators multiply (2 × 4 = 8 double-shaded).
Fractions times whole numbers
Write the whole number over 1.
- 3/4 × 12 = 3/4 × 12/1 = 36/4 = 9
Or think: 1/4 of 12 is 3, so 3/4 is 9.
Simplify first: cross-cancelling
You can divide a numerator and any denominator by a common factor before multiplying. This is called cross-cancelling and keeps numbers small.
Example: 6/7 × 14/9
- 6 and 9 share 3 → 2 and 3
- 14 and 7 share 7 → 2 and 1
- Now: 2/1 × 2/3 = 4/3 = 1 1/3
Without cancelling: 84/63, which also simplifies to 4/3, but with more work.
Mixed numbers
Change mixed numbers to improper fractions first.
Example: 2 1/2 × 1 1/3
- 5/2 × 4/3 = 20/6 = 10/3 = 3 1/3
Never multiply the whole parts and fraction parts separately: 2 × 1 + 1/2 × 1/3 gives the wrong answer.
Does multiplying make things smaller?
When you multiply by a fraction less than 1, the answer is smaller than the number you started with. Half of 10 is 5. That surprises people who think "multiply always makes bigger".
| Multiply by | Effect |
|---|---|
| more than 1 | gets bigger |
| exactly 1 | stays the same |
| between 0 and 1 | gets smaller |
Worked examples
Example 1: 3/5 × 2/7 = 6/35
Example 2: 5/8 × 4/15 → cancel 4 & 8 (→ 1, 2) and 5 & 15 (→ 1, 3) → 1/2 × 1/3 = 1/6
Example 3: A recipe uses 3/4 cup of sugar. You make half the recipe. How much sugar?
1/2 × 3/4 = 3/8 cup.
Common mistakes
- Finding a common denominator first. Not needed for multiplication.
- Multiplying mixed numbers part by part.
- Cancelling two numerators with each other (you can only cancel a numerator with a denominator).
Try it yourself
- 1/2 × 3/5
- 4/9 × 3/8
- 2/3 × 18
- 1 3/4 × 2 2/7
- 3/5 of a 40 km trip
Answers: 1) 3/10 2) 1/6 3) 12 4) 4 5) 24 km
Quick recap
- Multiply numerators, multiply denominators.
- Cross-cancel first to keep numbers small.
- Convert mixed numbers to improper fractions; "of" means multiply.