Adding and Subtracting Fractions

What you will learn

• Add and subtract fractions with the same denominator.
• Use a common denominator when the denominators differ.
• Simplify answers and handle answers bigger than one.

Before you start: you should be able to find equivalent fractions and the lowest common multiple. See Equivalent Fractions and Simplifying and HCF and LCM.

The idea

Two eighths plus three eighths is five eighths, just as two apples plus three apples is five apples. The parts are the same size, so you count them. The denominator names the size and stays the same.

Different denominators

Halves and thirds are different sizes, so you cannot count them together. Cut both into the same size first. The lowest common multiple of $2$ and $3$ is $6$, so use sixths: $\frac{1}{2} = \frac{3}{6}$ and $\frac{1}{3} = \frac{2}{6}$.

Subtracting

Subtracting works the same way. Make the denominators match, then subtract the tops.

The steps

1. Find a common denominator, usually the lowest common multiple.
2. Rewrite each fraction over it.
3. Add or subtract the numerators. Keep the denominator.
4. Simplify, and write as a mixed number if it is bigger than one.

Worked examples

Example 1: Add and simplify

Work out $\frac{2}{5} + \frac{1}{10}$.

1. The lowest common multiple of $5$ and $10$ is $10$.
2. $\frac{2}{5} = \frac{4}{10}$.
3. $\frac{4}{10} + \frac{1}{10} = \frac{5}{10}$.
4. Simplify: $\frac{5}{10} = \frac{1}{2}$.

Answer: $\frac{1}{2}$

Example 2: Subtract

Work out $\frac{5}{6} - \frac{1}{4}$.

1. The lowest common multiple of $6$ and $4$ is $12$.
2. $\frac{5}{6} = \frac{10}{12}$ and $\frac{1}{4} = \frac{3}{12}$.
3. $\frac{10}{12} - \frac{3}{12} = \frac{7}{12}$.

Answer: $\frac{7}{12}$

Example 3: Mixed numbers

Work out $1\frac{1}{2} + 2\frac{3}{4}$.

1. Add the wholes: $1 + 2 = 3$.
2. Add the fractions: $\frac{1}{2} + \frac{3}{4} = \frac{2}{4} + \frac{3}{4} = \frac{5}{4} = 1\frac{1}{4}$.
3. Combine: $3 + 1\frac{1}{4} = 4\frac{1}{4}$.

Answer: $4\frac{1}{4}$

Watch out: never add the denominators. $\frac{1}{2} + \frac{1}{2}$ is one whole, not $\frac{2}{4}$.

Why it matters

Recipes, carpentry, sharing a bill and fitting jobs into a day all use adding and subtracting fractions. The same common-denominator idea reappears in algebra, where you add expressions such as $\frac{1}{x} + \frac{1}{y}$.