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 $$\frac{9}{10}=90\%$$.