How to calculate a percentage

Per cent means "out of 100". So $35\%$ means $\dfrac{35}{100}$, which is $0.35$ as a decimal.

To change a percentage to a decimal, divide by 100 (move the decimal point two places left). To change a decimal to a percentage, multiply by 100.

In maths, "of" means multiply. On a calculator you can type $15 \div 100 \times 80$, or $0.15 \times 80$.

Make sure both amounts are in the same units first. To write 30 cents as a percentage of $2, use 30 cents out of 200 cents: $\dfrac{30}{200} \times 100 = 15\%$.

There are two ways. Either find the percentage and add it on (or take it off), or multiply by a multiplier in one step.

• Increase by $X\%$: multiply by $1 + \dfrac{X}{100}$. Increasing by $25\%$ means multiplying by $1.25$.
• Decrease by $X\%$: multiply by $1 - \dfrac{X}{100}$. Taking $30\%$ off means multiplying by $0.7$.

Percentage difference. In most school questions, "percentage difference" means the same thing as percentage change: compare to the original value. In science, it sometimes means the difference divided by the average of the two values. Check which one your question wants.

Percentage error (Year 8) works the same way: $\dfrac{\text{measured} - \text{actual}}{\text{actual}} \times 100$, usually written without the sign.

If you know the amount after a percentage change, divide by the multiplier to undo it. Don't take the percentage of the new amount.

GST. In Australia GST is $10\%$, so a price including GST is $110\%$ of the price without it. To take the GST out, divide by $1.1$: $110 including GST is $100 before GST, and the GST is $10 (which is the total divided by 11).

Start from easy percentages and build the one you need.

• $50\%$: halve it.
• $25\%$: halve it, then halve again.
• $10\%$: divide by 10.
• $5\%$: half of $10\%$.
• $1\%$: divide by 100.

Example: $35\%$ of $240$. $10\%$ is $24$, so $30\%$ is $72$. $5\%$ is $12$. So $35\%$ is $72 + 12 = 84$.

The swap trick: $X\%$ of $Y$ is the same as $Y\%$ of $X$. So $8\%$ of $25$ is $25\%$ of $8$, which is $2$.

• Dividing by the new value in percentage change. Always divide by the original (starting) value.
• Undoing a percentage by reversing it. Going up $10\%$ then down $10\%$ doesn't get you back: $100 \to 110 \to 99$. To undo a change, divide by the multiplier.
• Forgetting to add or subtract. $25\%$ of $120$ is $30$, but "increase $120$ by $25\%$" is $150$.
• Mixing units. Convert to the same units before writing one amount as a percentage of another.