The four operations with fractions, converting between fractions, decimals and percentages, and finding percentages and percentage change of an amount.

What this dot point is asking

AQA wants you to calculate confidently with fractions using all four operations, convert freely between fractions, decimals and percentages, and find percentages and percentage changes of an amount. These number skills run through almost every other topic, from probability to compound interest, and the non-calculator paper tests fraction arithmetic directly.

The four operations with fractions

For addition, $\dfrac{1}{4} + \dfrac{2}{5} = \dfrac{5}{20} + \dfrac{8}{20} = \dfrac{13}{20}$. For multiplication, $\dfrac{3}{4} \times \dfrac{2}{9} = \dfrac{6}{36} = \dfrac{1}{6}$ after simplifying. For division, $\dfrac{2}{3} \div \dfrac{4}{5} = \dfrac{2}{3} \times \dfrac{5}{4} = \dfrac{10}{12} = \dfrac{5}{6}$. With mixed numbers, convert to improper fractions first: $1\tfrac{1}{2} \times 2\tfrac{2}{3} = \tfrac{3}{2} \times \tfrac{8}{3} = \tfrac{24}{6} = 4$.

Converting between the three forms

A fraction becomes a decimal by dividing: $\tfrac{3}{8} = 3 \div 8 = 0.375$. A decimal becomes a percentage by multiplying by $100$: $0.375 \to 37.5\%$. To go from a percentage to a fraction, write it over $100$ and simplify: $40\% = \tfrac{40}{100} = \tfrac{2}{5}$. Recurring decimals also convert to fractions at Higher tier: $0.\overline{3} = \tfrac{1}{3}$, and $0.\overline{45} = \tfrac{45}{99} = \tfrac{5}{11}$.

Finding percentages and percentage change

The fastest method uses a decimal multiplier. To find $18\%$ of $250$, compute $250 \times 0.18 = 45$. For a percentage change, multiply by one plus or minus the rate.

To find a percentage change from two values, use $\dfrac{\text{change}}{\text{original}} \times 100$. If a price rises from $\pounds 40$ to $\pounds 50$, the change is $\pounds 10$, so the percentage increase is $\dfrac{10}{40} \times 100 = 25\%$.

Ordering mixed forms

A common question gives a mixture of fractions, decimals and percentages and asks you to order them. The reliable method is to convert everything to the same form, usually decimals, then compare. For $\tfrac{3}{5}$, $0.58$ and $62\%$: convert to $0.6$, $0.58$ and $0.62$, so the order from smallest is $0.58$, then $\tfrac{3}{5}$, then $62\%$. Working in one consistent form removes the guesswork and is faster than trying to compare across forms by eye.

Fractions of amounts and the reverse

Finding a fraction of an amount means dividing by the denominator and multiplying by the numerator: $\tfrac{3}{4}$ of $\pounds 60$ is $60 \div 4 \times 3 = \pounds 45$. The reverse problem, where you are told a fraction equals a value, undoes this: if $\tfrac{2}{5}$ of a number is $18$, then one fifth is $9$, so the whole number is $45$. This mirrors the reverse-percentage idea and the two are often examined together, since a percentage is just a fraction over $100$.