# How Can You Calculate Percentages?

Here you will learn about the calculation of percentages. It is often useful to know how to convert numbers to percentages and back. For example, you have probably been shopping while there has been a sale. How much will something cost if you get a $40$ % discount? How do you calculate that? That is exactly what we are going to look at here.

Rule

When you want to find the percentage of a number:

Example 1

Calculate $\text{}40\text{}\phantom{\rule{0.17em}{0ex}}\text{%}$ of 300.

 $300\cdot 40\phantom{\rule{0.17em}{0ex}}\text{%}=\frac{3\cdot \text{100}\cdot 40}{\text{100}}=3\cdot 40=120$

This means $40\phantom{\rule{0.17em}{0ex}}\text{%}$ of $300$ is $120$.

Example 2

Find $\text{}75\text{}\phantom{\rule{0.17em}{0ex}}\text{%}$ of 80.

$\begin{array}{llll}\hfill 80\cdot 75\phantom{\rule{0.17em}{0ex}}\text{%}& =\frac{80\cdot 75}{100}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\frac{\text{80}\cdot 75}{\text{100}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\frac{8\cdot 75}{10}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =8\cdot 7.5\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =60\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$ This means $75\phantom{\rule{0.17em}{0ex}}\text{%}$ of $80$ is $60$.

What should you do if you want to add a percentage? I will show you how with two methods: The long way and the short way. The short method uses a growth factor, and you will find that it becomes your best friend. Let us try an example.

Example 3

You want to add $\text{}15\text{}\phantom{\rule{0.17em}{0ex}}\text{%}$ to 400.

### The long way

Begin the same way you did in the examples above. First find what $15$ % of $400$ is:
$\begin{array}{llll}\hfill 400\cdot 15\phantom{\rule{0.17em}{0ex}}\text{%}& =\frac{400\cdot 15}{100}=\frac{4\cdot \text{100}\cdot 15}{\text{100}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =4\cdot 15=60\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

 $400\cdot 15\phantom{\rule{0.17em}{0ex}}\text{%}=\frac{400\cdot 15}{100}=\frac{4\cdot \text{100}\cdot 15}{\text{100}}=4\cdot 15=60$

Since you want to add $15$ % to $400$, you’ll have to add $60$. You’ll find that
 $400+60=460.$

### The easy way: growth factor

You do both calculations in one move:
 $400\cdot \phantom{\rule{-0.17em}{0ex}}\left(1+\frac{15}{100}\right)=400\cdot 1.15=460$

Example 4

You are out shopping during a sale. You find a jacket with a discount that says $\text{}30\text{}\phantom{\rule{0.17em}{0ex}}\text{%}$ off. The regular price is $70. How much do you have to pay? We’re going to use the growth factor method: $\begin{array}{llll}\hfill 70\cdot \phantom{\rule{-0.17em}{0ex}}\left(1-\frac{30}{100}\right)& =70\cdot \phantom{\rule{-0.17em}{0ex}}\left(1-0.30\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =70\cdot 0.7=49\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$  $70\cdot \phantom{\rule{-0.17em}{0ex}}\left(1-\frac{30}{100}\right)=70\cdot \phantom{\rule{-0.17em}{0ex}}\left(1-0.30\right)=70\cdot 0.7=49$ This means that a $30$ % discount means you have to pay$49.