# Percentages

## Converting a fraction to a percentage

To convert a fraction to a percentage:

$\boxed{\text{multiply by 100%}}$ which is the same as $\boxed{\text{multiply by }\frac{100\%}1.}$

### Examples

1. What is $\frac12$ as a percentage?

$\frac12$ as a percentage is $\frac12\times\frac{100\%}1 = \frac{1\times100\%}{2\times1} = \frac{100\%}{2} = 50\%$.

2. What is $\frac34$ as a percentage?

$\frac34$ as a percentage is $\frac34\times\frac{100\%}1 = \frac{3\times100\%}{4\times1} = \frac{300\%}{4} = 75\%$.

3. Ali scored $9$ out of $1$5 for a test. What is this as a percentage?

It's $\frac9{15}\times\frac{100\%}1 = \frac{9\times100\%}{15\times1} = \frac{900\%}{15} = 60\%$.

The quick way to do this on a calculator is to enter $9$ divided by $15$ times $100$.

## Converting a percentage to a fraction

To convert a percentage to a fraction:

$\boxed{\text{put it over 100}}$ which is the same as $\boxed{\text{divide by 100.}}$

### Examples

1. What is $60\%$ as a fraction?

It is $\frac{60}{100} = \frac{3}{5}$. As a decimal this is $0.6$.

2. What is $50\%$ as a fraction?

It is $\frac{50}{100} = \frac{1}{2}$. As a decimal this is $0.5$.

## Finding a percentage of a quantity

To find a percentage of a quantity, write the percentage as a fraction and multiply by the quantity:

$\boxed{\frac{\text{percentage}}{100}\times \text{quantity}.}$

In this context "of" means "multiply".

### Examples

1. Find $60\% {\color{red}{\text{ of }}} 15$.

This is $\frac{60}{100}{\color{red}{\times}}15 = \frac{3}{5}\times15 = \frac{3\times15}{5} = \frac{45}5 = 9$. Note that the "${\color{red}{\text{of}}}$" in the question was replaced by "${\color{red}{\times}}$" in the calculation.

2. A machine is bought for $\$5,660$. GST of$10\%$has to be added to this. How much is the GST? What is the total cost? The GST is$\frac{10}{100}$of \$5,660. This is $\frac{10}{100}\times5660 = \frac{1}{10}\times5660 = \frac{5660}{10} = 566$. It is $\$566$. The total cost is$\$5,660 + \$566 =  \$6,226$.

3. Jules scored $40\%$ in a test out of $30$. What mark did Jules get?

The mark is $40\%$ of $30$, that is, $\frac{40}{100}\times 30 = 12$.

Jules scored $12$ out of $30$.

### Miscellaneous examples

1. Jessie paid $\$6,006$for equipment (including GST of$10\%$). Jessie will get a refund for the GST paid. How much will this refund be? (Be careful, the answer is not$\$600.60$.)

Write $n$ for the cost before GST is added. When the GST, which is $10\%$ of $n$, is added we get $\$6,006$. The GST paid is$\frac{10}{100}\times n = \frac{n}{10}$. So the total Jessie paid is$n+\frac{n}{10} = 6006$. 2. We must solve for$n$:$\frac{10n}{10}+\frac{n}{10} = 6006$, so$\frac{10n+n}{10} = 6006$which is$\frac{11n}{10} = 6006$. Therefore$n = \frac{6006\times 10}{11} = \frac{60060}{11} = 5460$. As the GST paid is$\frac{n}{10}$, the GST is$\frac{5460}{10} = \$546$.

3. A project is estimated to be $15\%$ complete and thus far has taken staff $300$  hours. How many staff hours will the whole project take?

Two different methods will be given.

Method 1. $15\%$ of the project takes $300$ hours, therefore

$\phantom{Method 1.}1\%$ of the project takes $\frac{300}{15} = 20$ hours (divide both numbers by $15$), so

$\phantom{Method 1.}100\%$ of the project takes $20 \times100 = 2000$ hours (multiply both by $100$).

Method 2. Write $n$ for the total number of hours needed by staff for the whole project. That $15\%$ of the total number of hours staff need is $300$ tells us that $\frac{15}{100}\times n = 300$.

We must solve for $n$: $\frac{15n}{100} = 300$, so $15n = 100\times 300$ and therefore $n = \frac{100\times 300}{15}= 2000$ hours.

4. ACME Products pays $\$62.50$for each widget they buy from the importer. They sell each widget for$\$80.00$. Finding the percentage profit as a proportion of cost price.

The profit for each widget is $\$80.00 - \$62.50 = \$17.50$. The profit as a proportion of cost price is$\frac{17.50}{62.50}$so the percentage profit as a proportion of cost price is$\frac{17.50}{62.50}\times100 = 28\%\$.