### Percentages (Economics)

#### Definition

Percent means “out of 100” and is denoted by the symbol $\%$. Percentages can be thought of as another way of writing fractions and are often easier to compare than fractions.

Percentages are closely related to decimals. To change a decimal to a percentage, move the decimal point two places to the right (i.e. multiply the decimal number by $100$) and add a $\%$ sign. For example $0.12$ as a percentage is $12\%$.

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

##### Fraction to Percentage

Method 1: The simplest method for converting a fraction into a percentage involves evaluating the fraction, multiplying the result by $100$ and adding a $\%$ sign.

Method 2: Another method involves remembering that percent means “out of $100$”, so try to convert the fraction into $\large{\frac{?}{100}}$ form by multiplying or dividing the numerator and denominator by the same number until the denominator is $100$. The numerator of this fraction is then the percentage we are after. Note that this method will only works if the fraction's denominator is a factor of $100$.

##### Percentage to Fraction

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.

#### Worked Example

###### Example 1

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

###### Solution

Method 1 Dividing $3$ by $5$ gives $0.6$ and $0.6 \times 100=60$. Thus $\frac{3}{5}=60\%$.

Method 2 $100=5\times20$ so we can write the given fraction in $\frac{?}{100}$ form by multiplying the numerator and denominator by $20$: $\dfrac{3\times20}{5\times20}=\frac{60}{100}$ so $\frac{3}{5}=60\%$.

###### 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 a Percentage of a Number

To find $X\%$ of $Y$, the calculation is: $\frac{X}{100}\times Y$ For instance, suppose we know that $10\%$ of firms in a small town made a profit last year, and we know that there are $120$ firms. We can then use the above formula with $X=10$ and $Y=120$ to work out exactly how many firms made a profit: $\frac{10}{100}\times 120=12$ 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

Find $12\%$ of $£25.40$.

###### Solution

Method 1

Using the formula we have:

$\frac{12}{100} \times 25.40 = 0.12 \times 25.40=3.048$

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 we can see that the two methods give the same answer.

#### 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 $X$:

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

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

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

###### Solution

$£699$ is the original cost ($100\%$) plus the VAT of $17.5\%$, so is a total of $117.5\%$ of the original price. $£699$ is the $Z$ from the above formula and $X=117.5\%$.

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 is $594.89$, 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. So $Z=127.50$ and $X=85$. 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 was $150$.

#### 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.

For instance, using the example from above suppose that instead we know that $12$ out of $120$ firms made a profit last year, but want to know what percentage of firms made a loss. We have $a=12$, $b$ and $\frac{a}{b}=0.1$ Multiplying this by $100$ gives: $100\times \frac{a}{b}=10$ So $10\%$ percent of the firms made a loss.

##### Worked Example
###### Example

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: $\textbf{Percentage change }=\frac{v_1-v_0}{v_0}\times100$

##### 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 $1\%$.

#### Elasticities

An important use of percentage changes in economics is to calculate elasticities.

##### Worked Example
###### Worked Example

Suppose that the manager of a company which produces sheep food wants to know whether increasing the price of its produce will result in an increase or a decrease in profit. To determine this, he must first assess how sensitive the demand for the sheep food is to changes in its price. That is, he must calculate the price elasticity of demand (PED) for the sheep food.

The formula for the PED is given by: $\textbf{PED }= \frac{\text{percentage change in quantity demanded}~}{\text{percentage change in price}~}$

To calculate the PED for its produce, the company decides to trail the effect of an increase in the price of the food by raising the price from $£16$ per bag to $£20$ per page for one month. In this time the number of bags of food purchased dropped from $55$ in the previous month to $44$ in the month with the higher price. What is the PED for the bags of sheep food?

###### Solution

Using the above formula for percentage changes, the percentage change in quantity demanded is: $100\times\frac{44-55}{55}=100\times\frac{-11}{55}=-20\%$ As this is negative we have a percentage decrease in the quantity demanded.

Using the formula again, the percentage change in price is: $100\times\frac{20-16}{16}=100\times\frac{4}{16}=25\%$ (a percentage increase).

Now, using the formula for the PED we have: $\text{PED }=\frac{-20}{25}=\frac{-4}{5}=-0.8$

So the PED for the company's sheep food is $0.8$ (since we ignore the sign). Since this value is between $0$ and $1$ demand is inelastic (not sensitive to changes in price) since the change in quantity demanded was smaller than the change in price.

#### 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.

#### More Support

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