Dividing a fraction by a fraction
Dividing by a fraction looks hard, but there is a simple trick behind it: instead of dividing, you multiply by the reciprocal (the flipped fraction).
The main rule
To divide by a fraction, multiply by its reciprocal.
In symbols:
a/b ÷ c/d = a/b · d/c
The reciprocal of c/d is d/c — just swap the numerator and the denominator.
Step by step
Work out 2/3 ÷ 4/5.
- Flip the second fraction: 4/5 → 5/4.
- Change ÷ to ·: 2/3 · 5/4.
- Multiply: numerators together (2 · 5 = 10), denominators together (3 · 4 = 12). You get 10/12.
- Simplify: 10/12 = 5/6.
Result: 2/3 ÷ 4/5 = 5/6.
Why it works
Dividing anything by a half gives you double. (How many halves are in 6? Twelve.) That is exactly what flipping does: dividing by 1/2 becomes multiplying by 2.
Checking your answer
- Multiply back: result · second fraction = first fraction. (5/6) · (4/5) = 20/30 = 2/3. ✓
- Size of the result: dividing by a number smaller than 1 makes the result larger; dividing by a number larger than 1 makes it smaller.
Common traps
- Flipping the wrong fraction. Always flip the second one, not the first.
- Forgetting to change ÷ to ·. After flipping, the sign must change too.
- Not simplifying. Write the result in lowest terms.