Fractions

Simplify fractions, then add, subtract, multiply and divide them, including mixed numbers and fractions of amounts.

Also known as: adding fractions, subtracting fractions, multiplying fractions, dividing fractions, mixed numbers, equivalent fractions.

GCSE Maths · Number IGCSE Maths · Number

Key ideas

Equivalent fractions: multiply or divide the top and bottom by the same number, e.g. $\frac{2}{3} = \frac{8}{12}$ .
To add or subtract, first write both fractions with a common denominator, then add or subtract the numerators only.
To multiply, multiply the numerators and multiply the denominators: $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$ .
To divide, multiply by the reciprocal of the second fraction: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$ .
Turn mixed numbers into improper fractions before multiplying or dividing.

Equivalent fractions and simplifying

A fraction doesn't change if you multiply or divide the numerator (top) and denominator (bottom) by the same number: $\frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8}.$ To simplify a fraction, divide the top and bottom by their highest common factor. For $\frac{18}{24}$ the highest common factor is $6$ , so $\frac{18}{24} = \frac{3}{4}$ . A fraction is in its simplest form when the top and bottom have no common factor other than $1$ .

Adding and subtracting fractions

You can only add or subtract fractions when the denominators are the same. If they're different, rewrite both fractions over a common denominator, ideally the lowest common multiple of the two denominators.

For $\frac{2}{3} + \frac{3}{4}$ , the lowest common multiple of $3$ and $4$ is $12$ : $\frac{2}{3} + \frac{3}{4} = \frac{8}{12} + \frac{9}{12} = \frac{17}{12} = 1\tfrac{5}{12}.$ Only the numerators are added. The denominator tells you the size of the pieces, and that stays the same.

Multiplying fractions

Multiply the numerators together and the denominators together, then simplify: $\frac{2}{5} \times \frac{15}{8} = \frac{30}{40} = \frac{3}{4}.$ It's often quicker to cancel common factors before multiplying: the $5$ and $15$ share a factor of $5$ , and the $2$ and $8$ share a factor of $2$ , so $\frac{2}{5} \times \frac{15}{8} = \frac{1}{1} \times \frac{3}{4} = \frac{3}{4}$ .

"Of" means multiply: $\frac{3}{5}$ of $40$ is $\frac{3}{5} \times 40 = 24$ .

Dividing fractions

Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). Keep the first fraction, change $\div$ to $\times$ , and flip the second fraction: $\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8} = 1\tfrac{7}{8}.$ This works because dividing asks "how many $\frac{2}{5}$ s fit into $\frac{3}{4}$ ?", and multiplying by $\frac{5}{2}$ answers exactly that.

Mixed numbers

A mixed number such as $2\tfrac{1}{3}$ has a whole-number part and a fraction part. To convert it to an improper fraction, multiply the whole number by the denominator and add the numerator: $2\tfrac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{7}{3}.$ Always convert mixed numbers to improper fractions before multiplying or dividing. For adding and subtracting you can work with the parts, but converting first is the most reliable method.

Worked examples

Example 1 · Adding with different denominators

Work out $\frac{5}{6} + \frac{3}{8}$ . Give your answer as a mixed number.

Step 1.The lowest common multiple of $6$ and $8$ is $24$ .
Step 2. $\frac{5}{6} = \frac{20}{24}$ and $\frac{3}{8} = \frac{9}{24}$ .
Step 3. $\frac{20}{24} + \frac{9}{24} = \frac{29}{24}$ .
Step 4. $\frac{29}{24} = 1\tfrac{5}{24}$ .

Answer: $1\tfrac{5}{24}$

Example 2 · Subtracting mixed numbers

Work out $3\tfrac{1}{2} - 1\tfrac{2}{3}$ .

Step 1.Convert to improper fractions: $3\tfrac{1}{2} = \frac{7}{2}$ and $1\tfrac{2}{3} = \frac{5}{3}$ .
Step 2.Common denominator $6$ : $\frac{7}{2} = \frac{21}{6}$ and $\frac{5}{3} = \frac{10}{6}$ .
Step 3. $\frac{21}{6} - \frac{10}{6} = \frac{11}{6}$ .
Step 4. $\frac{11}{6} = 1\tfrac{5}{6}$ .

Answer: $1\tfrac{5}{6}$

Example 3 · Dividing a mixed number by a fraction

Work out $2\tfrac{1}{4} \div \frac{3}{8}$ .

Step 1.Convert: $2\tfrac{1}{4} = \frac{9}{4}$ .
Step 2.Multiply by the reciprocal: $\frac{9}{4} \times \frac{8}{3}$ .
Step 3.Cancel: $\frac{9}{3} = 3$ and $\frac{8}{4} = 2$ , so the result is $3 \times 2 = 6$ .

Answer: $6$

Common mistakes

Adding the denominators: $\frac{1}{2} + \frac{1}{3}$ is $\frac{5}{6}$ , not $\frac{2}{5}$ .
Flipping the first fraction instead of the second when dividing.
Multiplying mixed numbers part by part: $2\tfrac{1}{2} \times 2\tfrac{1}{2}$ is $6\tfrac{1}{4}$ , not $4\tfrac{1}{4}$ . Convert to improper fractions first.
Forgetting to simplify the final answer.
Changing only the denominator when finding an equivalent fraction (the numerator must change by the same factor).

Try it yourself

Original practice questions. Your answer is checked instantly.

Question 1FractionsEasier

Which fraction is equivalent to $\frac{3}{5}$ ?

Question 2FractionsEasier

Write $\frac{18}{24}$ in its simplest form.