# Percentages

## In a nutshell

Percentages are used very often in the real world as they are very easy to interpret and understand. It is important to know how to find one quantity as a percentage of another as well as how to find percentages of a quantity.

### Interpreting percentages

The percent symbol ($\%$) means "out of $100$". So, $35\%$ literally means "$35$ out of $100$". Another way of writing this is $\frac{35}{100}$, or $0.35$.

## Finding percentages of a number

A common problem is to find percentages of a number. To approach these questions, use the fraction interpretation of the percentage symbol. It is also helpful to read the word "of" as "times".

##### Example 1

*What is $35\%$ of $160$?*

*Write the percentage symbol as $\frac{}{100}$ and the "of" as the multiplication symbol:*

$35\%$ *of $160$ $= \frac{35}{100}\times160$*

*Put this into a calculator:*

$\frac{35}{100}\times160=56$

$35\%$* of *$160$* is *$\underline{56}$.

*However, if you don't have a calculator you can follow this method: *

$35\%$ *of* $160= \dfrac{35}{100}\times160\\ \ \\= \dfrac{35}{100}\times\dfrac{160}{1}\\ \ \\=\dfrac{5600}{100}=56$

## Expressing one number as a percentage of another

You may be asked to compare one number as a percentage of another number. To do this, first write what you want to find as a fraction. Then, convert it to a decimal, and then convert it to a percentage.

##### Example 2

*A quiz has $60$ questions. If I answered $54$ questions correctly, what percentage of questions did I get correct?*

*First, find the relevant fraction:*

*I answered $54$ questions correct out of $60$ questions in total. *

*So, the relevant fraction is $\frac{54}{60}$.*

*To convert this to a percentage, first convert it to a decimal:*

$\frac{54}{60}=54\div60=0.9$

*Now, convert the decimal to a percentage:*

*$0.9=(0.9\times100)\%=90\%$*

*I answered *$\underline{90\%}$* of questions correctly.*

*Note: **In this example, you also could have simplified the fraction to $\frac{9}{10}$*. *Using your common conversions, you should know that $\frac{9}{10}=90\%$.*