### Percentages

#### Definition

Percent means 'out of 100' and is denoted by the symbol %. Percentages are closely related to fractions and decimals.

To change a fraction to a percentage, divide the numerator by the denominator and multiply by 100 - move the decimal point two places to the left. To change a decimal to a percentage, move the decimal point two places to the right.

Some common percentage amounts and their fraction and decimal equivalients.

Percentage

Fraction

Reduced fraction

Decimal

$75\%$

$\tfrac{75}{100}$

$\tfrac{3}{4}$

$0.75$

$50\%$

$\tfrac{50}{100}$

$\tfrac{1}{2}$

$0.5$

$25\%$

$\tfrac{25}{100}$

$\tfrac{1}{4}$

$0.25$

$10\%$

$\tfrac{10}{100}$

$\tfrac{1}{10}$

$0.1$

$5\%$

$\tfrac{5}{100}$

$\tfrac{1}{20}$

$0.05$

#### Converting Between Fractions and Percentages

To convert a fraction into a percentage, evaluate the fraction and multiply the result by $100$.

To convert a percentage into a fraction, write the percentage as a fraction with $100$ as the denominator and write this fraction in its simplest form.

###### Example 1

Write $\frac{3}{5}$ as a percentage.

###### Solution

Dividing $3$ by $5$ gives $0.6$ and $0.6 \times 100=60$. Thus $\frac{3}{5}=60\%$.

###### Example 2

Convert $75\%$ into a fraction.

###### Solution

Writing $75$ as a fraction with $100$ as the denominator gives $\frac{75}{100}$. The highest common factor of $75$ and $100$ is $25$. Dividing top and bottom by $25$ gives $\frac{3}{4}$ so we have $75\%=\frac{3}{4}$.

#### Finding Percentage Amounts

To find $X\%$ of $Y$, the calculation is: $\frac{X}{100}\times Y$ This is the best method to use if the numbers are difficult - it can just be typed into a calculator.

It is useful to note that:

• $50\%$ of a number can be found by dividing that number by $2$;
• $25\%$ by dividing by $4$;
• $10\%$ by dividing by $10$;
• $5\%$ by dividing by $20$;
• $1\%$ by dividing by $100$.

These can then be used to find more difficult percentages, for example to find $35\%$, find $25\%$ and $10\%$ and add them together, or $2\%$ can be found by finding $1\%$ and doubling it. This is the best method if you can't use a calculator, and often quicker than using a calculator, if it is a simple question.

##### Worked Example
###### Example 1

Find $12\%$ of $£25.40$.

###### Solution

Method 1

Using the formula on a calculator we have:

$\frac{12}{100} \times 25.40 = 3.048$

This number must be written $2$ decimal places since it is money. Rounding $3.048$ to $2$d.p. gives $3.05$. So $12\%$ of $£25.40$ is $£3.05$.

Method 2 Using the table above, we have:

$10\% \text{ of }25.40 = 2.54$, and

$1\% \text{ of }25.40= 0.254$

Adding $10\%$ of $25.40$ to two lots of $1\%$ of $25.40$ gives $12\%$ of $25.40$:

$2.54 + 0.254 + 0.254 = 3.048.$

So $12\%$ of $£25.40$ is $£3.05$.

#### Finding the Original Amount

If we know that $Z$ is $X\%$ of a number and we want to know what that number is, we multiply $Z$ by $100$ and divide by $X$:

$\frac{Z\times100}{X}$

###### Example 1

The cost of a computer is $£699$ including VAT ($17.5\%$). Calculate the cost before VAT.

###### Solution

$£699$ is the cost of the original $100\%$ plus the VAT of $17.5\%$ which is a total of $117.5\%$ of the original price.

Start by multiplying $699$ by $100$ to get $69,900$. Then divide $69,900$ by $117.5$. To $2$ decimal places (since we are working with money) this gives $594.89$, so the cost before VAT is $£594.89$.

###### Example 2

The cost of a coat which has been reduced by $15\%$ in the sale is $£127.50$. What was the original cost of the coat?

###### Solution

The (reduced) price $£127.50$ is $100\%-15\%=85\%$ of the original amount. To obtain the original amount, we must therefore multiply $127.50$ by $100$ and then divide the result by $85$: \begin{align} &127.50\times100=12,750 &\frac{12,750}{85}=150 \end{align}

So the original cost of the coat is $150$.

#### Expressing a Number as a Percentage of Another Number

Suppose $a$ and $b$ are numbers. To express $a$ as a percentage of $b$ we divide $a$ by $b$ to produce a fraction and use the above rule to convert this fraction into a percentage.

##### Example
###### Example 1

Write $4$ as a percentage of $25$.

###### Solution

Dividing $4$ by $25$ gives $\frac{4}{25}$. To express this fraction as a percentage we must first evaluate the fraction and then multiply the result by $100$: \begin{align} &\frac{4}{25}=0.16\\ \text{and }&0.16\times100=16 \end{align}

so $4$ is equal to $16\%$ of $25$.

#### Expressing a Change as a Percentage

A percentage change is a way of expressing a change in a value or quantity. In particular, the percent change expresses the change from the “old” to the “new” value as a percentage of the old value.

Let $v_0$ denote the old value and $v_1$ denote the new value. Then the percentage change from $v_0$ to $v_1$ is given by: $\textbf{Percentage change }=\frac{v_0-v_1}{v_0}\times100$

##### Worked Examples
###### Example 1

Four years ago, a house was bought for $£180,000$. It is now valued at $£350,000$. Calculate the percentage increase in the value of the house to the nearest $1\%$.

###### Solution

Using the formula we have: \begin{align} \text{Percentage increase} &= \frac{350,000 - 180,000}{180,\!000} \times 100\\ &=\frac{170,000}{180,000} \times 100\\ &= 94\% \text{ to the nearest }1\% \end{align}

###### Example 2

A car cost $£12,000$. After three years, it is now worth $£8,000$. Calculate the percentage decrease to the nearest $1\%$.

###### Solution

Using the formula we have: \begin{align} \text{Percentage decrease} &= \frac{12,000-8,000}{12,000} \times 100\\ &=\frac{4,000}{12,000} \times 100\\\ &= 33\% \text{ to the nearest} 1\% \end{align}

#### Using a Calculator

Using the percent button on a calculator can have a different effect depending on where it is used. If you are unsure, do not use it, and just use the formula above.

$x$ ÷ $y$ % finds $x$ as a percentage of $y$.
$x$÷ $y$ × $z$ % finds $z\%$ of $\frac{x}{y}$.
$x$ × $y$ % finds $y\%$ of $x$.
$x$ % × $y$ finds $x \times y$. (The percent button has no effect).
##### Examples

a) 48 ÷ 400 % = gives $12$. So $48$ is $12\%$ of $400$.

b) 1 ÷ 2 × 300 % = gives $1.5$. So $300\%$ of $\frac{1}{2}$ is $1.5$.

c) 400 × 50 % = gives $200$. So $400\%$ of $50$ is $200$.

d) 50 % × 400 = gives $20,000$.But $50\times400=20,000$ so here pressing the % button has no effect.

#### Workbook

This workbook produced by HELM is a good revision aid, containing key points for revision and many worked examples.