Percentages, profit and loss

Percentage change is a multiplier

Increasing an amount by $r\%$ leaves $(100 + r)\%$ of it, so multiply by $1 + \frac{r}{100}$. Decreasing by $r\%$ multiplies by $1 - \frac{r}{100}$. A $20\%$ increase is $\times 1.2$; a $15\%$ decrease is $\times 0.85$.

Try Find the original: the new value is $(100 \pm r)\%$ of the original, so you divide by the multiplier. Then try Two reductions: they multiply, and don’t simply add.

Profit and loss

Profit or loss is always a percentage of the cost price (what the seller paid):

So a $20\%$ profit means selling price $= 1.2 \times$ cost price, and a $15\%$ loss means selling price $= 0.85 \times$ cost price.

Marked price and discount

A discount is a reduction on the marked price (the price on the label). A shop that marks goods up and then gives a discount makes profit = selling price − cost price, as a percentage of the cost price.

Two changes in a row

Each change applies to the amount after the previous one, so multiply the multipliers.

Your turn