Dividing Fractions
Keep, change, flip — and the reason it is not an arbitrary trick.
The explanation
To divide by a fraction, flip the second one and multiply. 3/4 ÷ 2/5 becomes 3/4 × 5/2 = 15/8.
Why flipping works: dividing asks "how many of these fit inside that?" 1 ÷ 1/4 asks how many quarters fit in one whole. Four of them. And indeed 1 × 4/1 = 4.
Only the second fraction flips. The first one stays exactly as it is.
Division is defined as multiplication by the multiplicative inverse: x ÷ y := x · y⁻¹ for y ≠ 0. For a fraction, (c/d)⁻¹ = d/c because (c/d)·(d/c) = cd/dc = 1. So:
(a/b) ÷ (c/d) = (a/b)·(d/c) = ad/bc
You can also see it from the complex fraction (a/b)/(c/d) by multiplying numerator and denominator by bd, which clears both inner denominators and leaves ad/bc directly.
Note the standing restriction c ≠ 0. In algebra, dividing by a rational expression carries the same condition on every factor of the divisor, which is where domain restrictions on rational functions come from.
Worked example
Compute 5/8 ÷ 15/16.
- Flip the divisor: 5/8 × 16/15.
- Cancel 5 into 15 and 8 into 16: 1/1 × 2/3.
- Multiply: 2/3.
Answer: 2/3
Common mistakes
- Flipping the first fraction instead of the second.
- Flipping and then still dividing.