Fractions

Notes

Key idea: a fraction is a number of equal parts out of a whole. The top number (the numerator) says how many parts you have. The bottom number (the denominator) says how many equal parts the whole is split into.

Equivalent fractions and simplifying

Equivalent fractions have the same value. Multiply or divide the top and bottom by the same number.

$$\frac{3}{4} = \frac{6}{8} = \frac{9}{12}$$

To simplify a fraction, divide the top and bottom by their highest common factor (HCF).

Example: simplify $\frac{18}{24}$. The HCF of 18 and 24 is 6, so divide both by 6:

$$\frac{18}{24} = \frac{3}{4}$$

Mixed numbers and improper fractions

An improper fraction has a numerator bigger than its denominator, such as $\frac{17}{5}$. A mixed number has a whole number part and a fraction part, such as $3\frac{2}{5}$.

• Mixed to improper: multiply the whole number by the denominator, then add the numerator. $3\frac{2}{5} = \frac{3 \times 5 + 2}{5} = \frac{17}{5}$
• Improper to mixed: divide the numerator by the denominator. $\frac{23}{4}$: 23 ÷ 4 = 5 remainder 3, so $5\frac{3}{4}$.

Adding and subtracting

You need a common denominator. Find a common multiple of the denominators (the lowest one keeps the numbers small), change each fraction, then add or subtract the numerators only.

Example: $\frac{2}{3} + \frac{3}{4}$

The lowest common multiple of 3 and 4 is 12.

$$\frac{2}{3} + \frac{3}{4} = \frac{8}{12} + \frac{9}{12} = \frac{17}{12} = 1\frac{5}{12}$$

Example: $\frac{5}{6} - \frac{1}{4} = \frac{10}{12} - \frac{3}{12} = \frac{7}{12}$

For mixed numbers, either convert to improper fractions or deal with the whole numbers and fractions separately. Converting is usually safer.

Multiplying

Multiply the numerators, multiply the denominators, then simplify. You can cancel before you multiply to keep the numbers small.

$$\frac{2}{5} \times \frac{3}{4} = \frac{6}{20} = \frac{3}{10}$$

For mixed numbers, convert to improper fractions first:

$$2\frac{1}{3} \times 1\frac{1}{2} = \frac{7}{3} \times \frac{3}{2} = \frac{21}{6} = \frac{7}{2} = 3\frac{1}{2}$$

Dividing

Keep, Change, Flip: keep the first fraction, change $\div$ to $\times$, flip the second fraction.

$$\frac{3}{4} \div \frac{2}{3} = \frac{3}{4} \times \frac{3}{2} = \frac{9}{8} = 1\frac{1}{8}$$

To divide by a whole number, treat it as a fraction over 1. So $\frac{2}{3} \div 4 = \frac{2}{3} \times \frac{1}{4}$.

Fraction of an amount

Divide by the denominator, then multiply by the numerator.

Example: find $\frac{3}{5}$ of £40. 40 ÷ 5 = 8, then 8 × 3 = £24.

Common mistakes

• Adding the denominators: $\frac{1}{2} + \frac{1}{3}$ is not $\frac{2}{5}$. It is $\frac{5}{6}$.
• Needing a common denominator for multiplying or dividing. You don't.
• Forgetting to simplify the final answer, and forgetting to turn an improper fraction back into a mixed number when the question uses mixed numbers.