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# Percentage change

Percentages are used to tell how a quantity will increase or decrease.

• Percentage change from an old number $\Tblue{O}$ to a new one $\Tgreen{N}$ is $\Tred{p} = \frac{\Tgreen{N}- \Tblue{O}}{\Tblue{O}} = \frac{\Tgreen{N}}{\Tblue{O}} - 1$

The price of a shirt went from $\Tblue{\11}$ to $\Tgreen{\12}$. It increased by $\frac{\Tgreen{12}- \Tblue{11}}{\Tblue{11}} = \frac{1}{\Tblue{11}} =\Tred{0.09} = \Tred{9\%}.$

• To know a value after the change $p$, multiply by $1+p$ $\Tgreen{N} = \Tblue{O} \times (1+\Tred{p})$

A population of $\Tblue{35}$ million increased by $\Tred{2\%}$. It is now $\Tblue{35}\cdot (1+\Tred{2\%}) = \Tblue{35} \times 1\Tred{.02} = \Tgreen{35.7}$

• To find the value before the change, divide by $1+p$. $\Tblue{O} =\Tgreen{N} \div (1+\Tred{p})$

After a $\Tred{20\%}$ discount, the price of a toy is $\Tgreen{\ 8}$. Before, it was $\frac{\Tgreen{8}}{1\Tred{-20\%}}=\frac{\Tgreen{8}}{1\Tred{-0.2}} = \frac{\Tgreen{8}}{0.8} = \Tblue{\ 10}$

Discounts in sales are often described using percentages. The new price is sometimes not given, so you need to know how to work it out.