Equivalent Fractions and Simplifying

Equivalent fractions are fractions that look different but are worth exactly the same. Cut a pizza into 2 equal pieces and take 1, or cut the same pizza into 8 pieces and take 4: you get the same amount of pizza. So 1/2 = 4/8.

A fraction wall shows this well. Each row is one whole, split into halves, thirds, quarters and so on. Any fractions that line up exactly, such as 2/3, 4/6 and 8/12, are equivalent.

To make an equivalent fraction, multiply (or divide) the numerator and the denominator by the same number.

For 2/3, multiplying both by 4 gives 8/12. In picture terms, you have cut every third into 4 smaller pieces. There are now 4 times as many pieces, each a quarter of the size, so the amount you have has not changed. In number terms, you have multiplied by 4/4, which equals 1, and multiplying by 1 never changes a number.

Adding the same number to the top and bottom does not work. Adding 1 to both parts of 1/2 gives 2/3, which is bigger than 1/2.

A common question is: 3/5 = ?/20. Look at the denominators first. What turns 5 into 20? Multiplying by 4. Do the same to the numerator: 3 × 4 = 12. So 3/5 = 12/20.

It also works with division: 15/25 = 3/?. The numerator went from 15 to 3, which is dividing by 5, so 25 ÷ 5 = 5, and 15/25 = 3/5.

Simplifying (also called cancelling or reducing) means finding the equivalent fraction with the smallest possible whole numbers. A fraction in its simplest form or lowest terms has no common factor bigger than 1 in its numerator and denominator.

The fastest way is to divide both numbers by their highest common factor (HCF), the biggest number that divides exactly into both.

Take 18/24:

1. Factors of 18: 1, 2, 3, 6, 9, 18.
2. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
3. The biggest number in both lists is 6, so the HCF is 6.
4. 18 ÷ 6 = 3 and 24 ÷ 6 = 4, so 18/24 = 3/4.

For bigger numbers, prime factors help. 36 = 2 × 2 × 3 × 3 and 48 = 2 × 2 × 2 × 2 × 3. The primes they share are 2 × 2 × 3 = 12, so the HCF is 12 and 36/48 = 3/4.

You do not have to find the HCF straight away. Divide by any common factor you can spot, then keep going. 18/24 → divide by 2 → 9/12 → divide by 3 → 3/4. You reach the same answer, it just takes more steps. Stop when the only number that divides into both is 1. Small primes such as 2, 3, 5 and 7 are good ones to try.

Equivalent fractions also let you compare fractions that have different denominators. Which is bigger, 5/8 or 2/3? Rename both with the lowest common multiple of 8 and 3, which is 24: 5/8 = 15/24 and 2/3 = 16/24. Now the parts are the same size, so 2/3 is bigger, but only by 1/24. The same idea is the first step of adding and subtracting fractions, the next part of this series.