### Numbers: Ratios

### Percentages

In everyday life we often see percentages, for example when shopping. Almost all products include VAT (value added tax). This is a percentage that is added to the price. In addition, we often see discounted products, where the discount is given using a percentage. Discount is a percentage that is being deducted from the price.

Percent literally means 'per #100#'. When #35# out of #100# parking spots are occupied, we can say that #35# percent of the parking spots are occupied. We write this as #35\%#.

When #350# out of #500# parking spots are occupied, this is the same ratio as #70# out of #100# parking spots being occupied. Therefore, we can say that #70\%# of the parking spots are occupied.

A percentage always indicates for which part of #100# a specific characteristic holds.

**Examples**

#1\%# is #1# out of #100#

#40\%# is #40# out of #100#

#9\%# is #9# out of #100#

#36\%# is #36# out of #100#

#99\%# is #99# out of #100#

We will now look at three important rules for working with percentages.

We can convert a percentage to a decimal number by dividing by #100#.

Therefore, #5\%=\dfrac{5}{100}=5:100=0.05#.

**Examples**

#28\%=0.28#

#77\%=0.77#

When we convert percentages to decimals, we can calculate how much a particular #\blue{\text{percentage}}# is of a #\green{\text{number}}#.

For example, #\blue{28}\%# of #\green{2600}# equals: #0.28 \times \green{2600}=728#.

In general we can state the following:

*To calculate a certain #\blue{\textit{percentage}}# of a #\green{\textit{number}}#, we calculate:*

*#\frac{\blue{\textit{percentage}}}{100} \times \green{\textit{number}}#*

**Example**

How much is #\blue{59}\%# of #\green{600}#?

\[\begin{array}{rcl}\\ \frac{\blue{59}}{100} \times \green{600}&=&0.59 \times \green{600} \\ &=& 354\end{array}\]

We also want to be able to calculate what percentage a specific #\blue{\text{part}}# is of a specific #\green{\text{whole}}#. Suppose we have a group of #\green{79}# people. #\blue{19}# people have blue eyes, and we want to know what percentage of the group has blue eyes.

The entire group of #\green{79}# people equals #100\%#.

Therefore, #1# person is #\frac{1}{\green{79}}\times 100\% \approx 1.27\%#.

To calculate what percentage #\blue{19}# people is, we multiply #\frac{1}{\green{79}}\times 100\%# by #\blue{19}#. Therefore, #\blue{19}# of #\green{79}# equals: #\frac{\blue{19}}{\green{79}} \times 100 \%\approx 24.05\%#

In general we can state:

*To calculate what percentage a specific #\blue{\textit{part}}# is of a specific #\green{\textit{whole}}#, we calculate:*

*#\frac{\blue{\textit{part}}}{\green{\textit{whole}}} \times 100\%#*

**Examples**

\[\begin{array}{rcl}\blue{18} &\text{ of} & \green{29} \\&\text{ equals }& \\\dfrac{\blue{18}}{\green{29}} \times 100\% &\approx& 62.1\% \\ \\ \\ \\ \blue{52} &\text{ of }&\green{128}\\&\text{ equals }& \\\dfrac{\blue{52}}{\green{128}} \times 100\% &\approx &40.6\%\end{array}\]

We convert a percentage to a decimal number by dividing by #100#. When dividing by #100#, the decimal point moves two places to the left.

Therefore, #10\%=\frac{10}{100}=10:100=0.10#.

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