# Price Change and New Price in Two Steps

Percentages are used very often in daily life. A shop might have a sale—for example, they’re offering a $30$ % discount)—or the price of a bus ticket might be increased by $6$ %.

Let’s begin by practicing how to find the new price without thinking about percentages:

### Discount and Price Increase

New price = Old price - discount

New price = Old price + price increase

I will deal with discounts given as a percentage first.

**A jacket used to cost $\text{\$}\text{}250\text{}$, but it is now being sold at a $\text{}20\text{}\phantom{\rule{0.17em}{0ex}}\text{\%}$ discount. What is the current price of the jacket? **

You can figure out the answer in two steps:

The discount is

$$\begin{array}{llll}\hfill 20\phantom{\rule{0.17em}{0ex}}\text{\%}\times \text{\$}250& =20\times \frac{1}{100}\times \text{\$}250\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\text{\$}50\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$ That helps give you the new price:

$$\begin{array}{llll}\hfill \text{Oldprice}-\text{discount}& =\text{\$}250-\text{\$}50\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\text{\$}2005\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$ At other times, the price might be increasing by a given percentage. You find this new price in a similar way, you just have to add instead of subtracting.

**A monthly bus ticket used to cost $\text{\$}\text{}32\text{}$, but then the price increased by $\text{}15\text{}\phantom{\rule{0.17em}{0ex}}\text{\%}$. What is the new price? **

You can do this in two steps the following way:

The increase in price is

$$\begin{array}{llll}\hfill 15\phantom{\rule{0.17em}{0ex}}\text{\%}\times \text{\$}32& =\frac{15}{100}\times \text{\$}32\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\text{\$}\frac{15\times 32}{100}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\text{\$}4.80\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$ The new price is: $\text{\$}32+\text{\$}4.80=\text{\$}36.80$