# Percentage

A percentage is one-hundredth of a number. It uses the symbol $$\%$$. A percentage is a fraction where the denominator is $$\Tblue{100}$$.

$$ \Tred{50}\% = \frac{\Tred{50}}{\Tblue{100}} = \frac{1}{2} = 0.5,\quad \Tred{40}\% = \frac{\Tred{40}}{\Tblue{100}} = \frac{2}{5} = 0.4.$$ To express a number as a percentage, write it in **decimal form** and multiply it by $$\Tblue{100}$$.

Percentages are often used to represent a **fraction of a quantity**.

Half a cake is the same as $$50\%$$ of the cake.

$$25\%$$ of $$247$$ is $$\Torange{25}\%\times 247 = 0\Torange{.25}\times 247 = 61.75 $$

$$26$$ is $$14.6\%$$ of $$178$$ because $$\displaystyle \frac{26}{178} = 0\Torange{.14}6 = \Torange{14.}6\%. $$

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 $$ \frac{\Tgreen{8}}{1\Tred{-20\%}}=\frac{\Tgreen{8}}{1\Tred{-0.2}} = \frac{\Tgreen{8}}{0.8} = \Tblue{\$ 10} $$

A change in percentage can be **repeated** several times.

The money I invest in a savings account increases by $$4\%$$ each year due to interest payment.

For a yearly percentage change of $$\Tred{p\%}$$, the new value $$\Tgreen{N}$$ after $$\Torange{n}$$ years from an old value $$\Tblue{O}$$ is $$ \Tgreen{N} = \Tblue{O} \times (1+\Tred{\frac{p}{100}})^\Torange{n}. $$

I invest $$\Tblue{1000}$$ in a bank account with an interest rate of $$\Tred{4\%}$$. After $$\Torange{3}$$ years, I will have a balance of $$\Tblue{1000} \times (1 + \Tred{4\%})^\Torange{3} = \Tblue{1000} \times 1\Tred{.04}^\Torange{3} = \Tblue{1000}\times 1.125 = \Tgreen{1125}.$$