 # Price Change and New Price in Two Steps

Percentage is used very often in daily life. For example, when a shop is having a sale (say $30$ % discount), or when the price of a bus ticket is increased by $6$ %.

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

Rule

### DiscountandPriceIncrease

New price = Old price - discount

New price = Old price + price increase

I will deal with discounts given as a percentage first.

Example 1

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

You can do this in two steps the following way:

The discount is $\begin{array}{llll}\hfill 20\phantom{\rule{0.17em}{0ex}}\text{%}×\text{}250& =20×\frac{1}{100}×\text{}250\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\text{}50.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

That gives you the new price

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.

Example 2

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 $\begin{array}{llll}\hfill 15\phantom{\rule{0.17em}{0ex}}\text{%}×\text{}32& =\frac{15}{100}×\text{}32\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\text{}\frac{15×32}{100}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\text{}4.8.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

The new price is $\text{}32+\text{}4.8=\text{}36.8$.