Percentages

Without a calculator, build from easy percentages: 10% is one tenth, 1% is one hundredth, 5% is half of 10%. So 35% of 80 is $3\times 8+4=28$.

With a calculator, convert to a decimal: 35% of 80 is $0.35\times 80=28$.

To write one quantity as a percentage of another, divide and multiply by 100: 18 out of 24 is $\frac{18}{24}\times 100=75\%$.

Increase by $r\%$: multiply by $1+\frac{r}{100}$. A 15% increase is $\times 1.15$.

Decrease by $r\%$: multiply by $1-\frac{r}{100}$. A 15% decrease is $\times 0.85$.

Successive changes multiply: a 10% increase followed by a 10% decrease is $\times 1.1\times 0.9=\times 0.99$, an overall 1% decrease.

Percentage change $=\frac{\text{change}}{\text{original}}\times 100$.

Always divide by the $\text{original}$ value, not the new one.

Percentage profit and loss use the same formula, with the cost price as the original.

If you know the value $\text{after}$ a percentage change, divide by the multiplier to find the original.

After a 20% discount, the sale price is 80% of the original. If that is £68, the original is $68÷0.8$.

Do not find 20% of the sale price and add it on — that uses the wrong 100%.