Fractions
Fractions turn up on every paper, often hidden inside a ratio, probability or algebra question. Four skills cover almost all of it: simplify, add or subtract over a common denominator, multiply straight across, and divide by flipping the second fraction.
Part 1 of 3: Learn it
In short
- To add or subtract, rewrite both fractions over a common denominator first.
- To multiply, multiply the tops and multiply the bottoms. Turn mixed numbers into improper fractions first.
- To divide, keep the first fraction, change ÷ to ×, and flip the second fraction.
Where this is in your specification
Spec points: DfE N2, N8 and N11
| Board | Topic: Fractions, decimals and percentages |
|---|---|
| DfE content | N8, N10, N11, N12, R9 |
| AQA 8300 | Number; Ratio, proportion and rates of change |
| Edexcel 1MA1 | Number; Ratio, proportion and rates of change |
| OCR J560 | 2 Fractions, decimals and percentages |
| Eduqas C300 | HN8, HN10 to HN12, HR10 |
| Cambridge IGCSE 0580 | E1.4, E1.13 |
| National 5 C847 75 | Numerical skills: fractions, percentages, appreciation and depreciation |
Equivalent fractions and simplest form
Multiplying or dividing the numerator and the denominator by the same number keeps the value the same: 2/5 = 4/10 = 12/30.
A fraction is in its simplest form when the only number that divides both the top and the bottom is 1. Divide both by their highest common factor: 36/48 ÷ 12 gives 3/4.
Mixed numbers and improper fractions
- Mixed to improper: multiply the whole number by the denominator and add the numerator. 3 2/7 = (21 + 2)/7 = 23/7.
- Improper to mixed: divide, and the remainder sits over the same denominator. 29/6 = 4 remainder 5, so 4 5/6.
Adding and subtracting
Find a common denominator (the lowest common multiple of the denominators is neatest), convert each fraction, then add or subtract the numerators only. The denominator stays the same.
With mixed numbers, the safest method is to change them to improper fractions first. Simplify the answer at the end, and change it back to a mixed number if the question started with them.
Multiplying and dividing
| Operation | Method | Example |
|---|---|---|
| Multiply | tops times tops, bottoms times bottoms | 2/3 × 9/10 = 18/30 = 3/5 |
| Divide | flip the second fraction, then multiply | 4/9 ÷ 2/3 = 4/9 × 3/2 = 12/18 = 2/3 |
| Fraction of an amount | divide by the bottom, multiply by the top | 3/8 of 56 = 56 ÷ 8 × 3 = 21 |
You can cancel before you multiply: in 2/3 × 9/10, the 3 and 9 share a factor of 3 and the 2 and 10 share a factor of 2, which leaves 1/1 × 3/5.
Work out 5/6 − 3/8.
Show the answer
Common denominator 24: 20/24 − 9/24 = 11/24.
Part 2 of 3: See it worked
Worked examples
Example 1
Work out 2 3/5 + 1 3/4. Give your answer as a mixed number.
- Change to improper fractions: 2 3/5 = 13/5 and 1 3/4 = 7/4
- Common denominator 20: 13/5 = 52/20 and 7/4 = 35/20
- 52/20 + 35/20 = 87/20
- 87 ÷ 20 = 4 remainder 7
Answer: 4 7/20.
Example 2
Work out 3 1/3 ÷ 1 1/4.
- 3 1/3 = 10/3 and 1 1/4 = 5/4
- Flip the second fraction and multiply: 10/3 × 4/5
- = 40/15 = 8/3
Answer: 8/3, which is 2 2/3.
Common mistakes
- Adding the denominators: 1/3 + 1/4 is not 2/7. Convert to twelfths first: 4/12 + 3/12 = 7/12.
- Flipping the first fraction instead of the second when dividing.
- Multiplying mixed numbers by splitting them into whole numbers and fractions. Change them to improper fractions first.
- Leaving an answer unsimplified when the question asks for its simplest form.
Work out 1 1/2 × 2 2/3.
Show the answer
3/2 × 8/3 = 24/6 = 4.
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
Work out 5/6 − 3/8.
Common denominator 24: 20/24 − 9/24 = 11/24.
Work out 1 1/2 × 2 2/3.
3/2 × 8/3 = 24/6 = 4.
Find 2/7 of 63.
63 ÷ 7 = 9, then 9 × 2 = 18.
Jobs that use this
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
Full lessons and marked practice for this course are coming soon to Brainlag Learn. See courses