Equivalent fractions and simplification

What is an equivalent fraction?

Two fractions are equivalent if they represent the same quantity, even if they are written differently. For example, 1/2 and 2/4 represent exactly the same part of a whole.

How to find an equivalent fraction

Multiply (or divide) both the numerator AND the denominator by the same non-zero number.

Example: 1/2 = (1 × 3)/(2 × 3) = 3/6

Similarly: 6/8 = (6 : 2)/(8 : 2) = 3/4

Simplifying a fraction

Simplifying a fraction means rewriting it using smaller numbers by dividing both the numerator and the denominator by a common divisor, until there are no further divisors left (irreducible fraction).

Example: simplify 12/18

• 12 and 18 are both divisible by 6.
• 12/18 = (12 ÷ 6)/(18 ÷ 6) = 2/3

2/3 is the irreducible form of 12/18, as 2 and 3 have no common divisor other than 1.

Step-by-step example

Simplify 20/30:

1. Find a common divisor: 10 works (20 = 10 × 2, 30 = 10 × 3).
2. 20/30 = 2/3.
3. 2 and 3 have no further common divisors: the fraction has been simplified as far as possible.

Common pitfall to avoid

A common mistake is to simplify by subtracting the same number from both the numerator and the denominator (for example, thinking that 12/18 = 6/12 by removing 6 from both). This is incorrect: you must divide, never subtract, to obtain an equivalent fraction.