Fractions

Equivalent fractions are worth the same, such as ½, 2/4 and 5/10. In a fraction, the top number is the numerator and the bottom number is the denominator.

1. To make an equivalent fraction, multiply or divide the top and bottom by the same number.
2. To write a fraction in its simplest form, divide the top and bottom by their HCF.

Example: write 36/48 in its simplest form.

1. The HCF of 36 and 48 is 12.
2. 36 ÷ 12 = 3 and 48 ÷ 12 = 4
3. ¾

You can also divide in smaller steps: 36/48 = 18/24 = 9/12 = ¾. The fraction is in its simplest form when the top and bottom have no common factor bigger than 1.

A mixed number has a whole-number part and a fraction, such as 2 ¾. An improper fraction has a top bigger than its bottom, such as 11/4.

To change a mixed number to an improper fraction:

1. Multiply the whole number by the denominator.
2. Add the numerator.
3. Put the result over the same denominator.

To change an improper fraction to a mixed number, divide the top by the bottom; the remainder goes over the denominator.

Example: write 2 ¾ as an improper fraction, and 17/5 as a mixed number.

1. 2 × 4 = 8
2. 8 + 3 = 11, so 2 ¾ = 11/4
3. 17 ÷ 5 = 3 remainder 2
4. 17/5 = 3 2/5

You can only add or subtract fractions with the same denominator.

1. Change mixed numbers to improper fractions.
2. Find a common denominator (the LCM of the denominators).
3. Change each fraction to an equivalent one with that denominator.
4. Add or subtract the numerators. Keep the denominator.
5. Simplify, and change back to a mixed number if needed.

Example: work out 2/3 + ¾.

1. Common denominator: 12
2. 2/3 = 8/12 and ¾ = 9/12
3. 8/12 + 9/12 = 17/12
4. 1 5/12

Harder example: work out 3 ½ − 1 2/3.

1. 3 ½ = 7/2 and 1 2/3 = 5/3
2. Common denominator: 6, so 21/6 − 10/6
3. 21/6 − 10/6 = 11/6
4. 1 5/6

1. Change mixed numbers to improper fractions.
2. Multiply the numerators together and the denominators together.
3. Simplify.

You can cancel before multiplying: divide a top number and a bottom number by a common factor. It keeps the numbers small.

Example: work out 4/9 × 3/8.

1. 4 × 3 = 12 and 9 × 8 = 72, so 12/72
2. The HCF of 12 and 72 is 12.
3. 1/6

Example: work out 1 ⅓ × 2 ¼.

1. 1 ⅓ = 4/3 and 2 ¼ = 9/4
2. 4/3 × 9/4 = 36/12
3. 3

A whole number can be written over 1: 5 × 2/3 = 5/1 × 2/3 = 10/3 = 3 ⅓.

To divide by a fraction, multiply by its reciprocal (turn it upside down).

1. Change mixed numbers to improper fractions.
2. Keep the first fraction.
3. Flip the second fraction.
4. Change ÷ to ×, then multiply.
5. Simplify.

Example: work out 3/5 ÷ 9/10.

1. 3/5 × 10/9
2. 3 × 10 = 30 and 5 × 9 = 45, so 30/45
3. 2/3

Harder example: work out 2 2/3 ÷ 1 3/5.

1. 2 2/3 = 8/3 and 1 3/5 = 8/5
2. 8/3 × 5/8 = 40/24
3. 40/24 = 5/3
4. 1 2/3

To find a fraction of an amount:

1. Divide the amount by the denominator. This gives one part.
2. Multiply by the numerator.

To write one number as a fraction of another, put the part over the whole and simplify.

Example: find 3/8 of 96.

1. 96 ÷ 8 = 12
2. 12 × 3 = 36

Example: a class has 30 students and 12 walk to school. What fraction walk?

1. Part over whole: 12/30
2. The HCF of 12 and 30 is 6.
3. 2/5

Harder example: Sam spends ¼ of £60 on a game and 2/5 of what is left on a book. How much is left?

1. ¼ of £60 = £15, so £60 − £15 = £45 left
2. 2/5 of £45 = 45 ÷ 5 × 2 = £18
3. £45 − £18 = £27