What you'll learn
This topic covers working with fractions — adding, subtracting, multiplying and dividing them — and converting between fractions, decimals and percentages. In this guide you will learn how to find common denominators, how to multiply and divide fractions, how to handle mixed numbers, and how to switch between the three forms. These number skills appear throughout maths.
Key terms and definitions
Numerator — the top number of a fraction.
Denominator — the bottom number of a fraction.
Equivalent fractions — fractions with the same value (e.g. ½ = 2/4).
Mixed number — a whole number and a fraction together (e.g. 2¾).
Improper fraction — a fraction where the numerator is larger than the denominator.
Core concepts
Adding and subtracting fractions
To add or subtract, the fractions need the same denominator. Find a common denominator (often the lowest common multiple), convert each fraction, then add or subtract the numerators only. For example, ½ + ⅓ = 3/6 + 2/6 = 5/6.
Multiplying fractions
To multiply, multiply the numerators together and the denominators together, then simplify. For example, ⅔ × ¾ = 6/12 = ½. You can cancel common factors before multiplying to keep numbers small.
Dividing fractions
To divide, multiply by the reciprocal (flip the second fraction): ⅔ ÷ ¾ = ⅔ × 4/3 = 8/9. "Keep, change, flip" is a useful reminder.
Mixed numbers
Convert mixed numbers to improper fractions before multiplying or dividing — for example, 2¾ = 11/4. Convert back to a mixed number at the end if required.
Fractions, decimals and percentages
The three forms are interchangeable: divide numerator by denominator for a decimal (¾ = 0.75), multiply a decimal by 100 for a percentage (0.75 = 75%), and write a percentage as a fraction over 100 and simplify. Knowing common equivalents speeds up calculations.
Worked examples
Example 1: Adding
Work out ¼ + ⅔.
Common denominator 12: 3/12 + 8/12 = 11/12.
Example 2: Dividing
Work out ⅗ ÷ ²⁄₃.
Keep, change, flip: ⅗ × 3/2 = 9/10.
Example 3: To a percentage
Write ⅖ as a percentage.
⅖ = 0.4 = 40%.
Common mistakes and how to avoid them
Adding denominators. Only the numerators are added, over a common denominator.
Forgetting to flip when dividing. Multiply by the reciprocal.
Not simplifying. Cancel down to lowest terms.
Multiplying mixed numbers directly. Convert to improper fractions first.
Confusing the conversions. Divide for decimals; ×100 for percentages.
Exam technique for Fractions
Find a common denominator before adding or subtracting.
Multiply tops and bottoms to multiply; flip and multiply to divide.
Convert mixed numbers to improper fractions first.
Simplify your final answer.
Switch forms by dividing (decimal) and multiplying by 100 (percentage).
Quick revision summary
To add or subtract fractions, find a common denominator, convert each fraction, then combine the numerators only. To multiply, multiply the numerators and denominators (cancelling first if you can); to divide, multiply by the reciprocal ("keep, change, flip"). Convert mixed numbers to improper fractions before multiplying or dividing, and back again at the end. The three forms are interchangeable: divide numerator by denominator for a decimal (¾ = 0.75), multiply by 100 for a percentage (0.75 = 75%), and write a percentage over 100 and simplify for a fraction. Always simplify the final answer. The common errors are adding denominators, forgetting to flip when dividing, not simplifying, and multiplying mixed numbers directly. Use common denominators to add, multiply tops and bottoms, flip to divide, convert mixed numbers, and switch easily between fractions, decimals and percentages.