Dividing fractions

The rule: multiply by the reciprocal

Dividing by a fraction is the same as multiplying by its reciprocal (you turn the fraction round):

a/b : c/d = a/b × d/c

What is the reciprocal?

The inverse of a fraction a/b is b/a (where a and b are not zero). For example, the inverse of 3/4 is 4/3, and the inverse of 5 (i.e. 5/1) is 1/5.

Example

2/3 : 4/5 = 2/3 × 5/4 = 10/12 = 5/6 (after simplifying by 2)

Dividing by an integer

Dividing by an integer n is the same as multiplying by 1/n. For example:

3/4 : 2 = 3/4 × 1/2 = 3/8

Common pitfall

The most common mistake is to invert the first fraction instead of the second. The rule is simple: you always invert the fraction that follows the ‘:’ sign (the divisor), never the first fraction (the dividend).

Summary table

Operation	Method
Multiplication	multiply the numerators together, multiply the denominators together
Division	multiply by the reciprocal of the divisor

With practice, these two operations become as natural as addition.