16.02 Financial applications of percentages

Percentage increase and decrease

There are many financial applications for percentages, including the value of something either increasing or decreasing.

A common example of a percentage decrease is in retail, when an item is advertised as being on sale. If an item is advertised as $20%$20% off, calculations can be made to find the new price of the item when given the original value. Or vice versa, the original price when given only the sale price.

When applying a percentage decrease $%$%

decrease in value  = $\frac{%}{100}\times$%100​× original value

new value = original value - decrease in value

                      = original value $\times\left(100-decrease\right)%$×(100−decrease)%

An item can also increase in value. This is sometimes referred to as a mark-up, where the value of an item is increased by a certain percentage. A restaurant, for example, may introduce a $15%$15% surcharge on Sundays, meaning that the total bill is increased by $15%$15%.

When applying a percentage increase $%$%

increase in value = $\frac{%}{100}\times$%100​× original value

new value = original value + increase in value

                  = original value $\times\left(100+increase\right)%$×(100+increase)%

 

Practice questions

QUESTION 1

QUESTION 2

Percentage change

Percentage can be useful when comparing a change in values. For example the cost of bread at a store may have increased due to drought from $\$2.00$$2.00 to $\$2.30$$2.30. To compare this increase to another store it is useful to use percentage rather than dollars. The bread has increased by $30$30 cents. Comparing this increase to the starting value:

$\frac{0.30}{2.00}=0.15$0.302.00​=0.15

By converting this to a percentage we can say the bread has increased by $15%$15%

To find the percentage change

Percentage change $=\frac{new\ value-\ original\ value}{original\ value}\times100$=new value− original valueoriginal value​×100 %

Practice question

QUESTION 3

Calculating original value

The original value of an item can be calculated when given a new amount and a percentage change (increase or decrease).

For example, the original value of an item when given a new amount of $\$63$$63 and a percentage increase or decrease of $3$3 %.

Finding original value given new amount (percentage increase)

original value = new value $\times\frac{100}{100-%\ change}$×100100−% change​

Finding original value given new amount (percentage decrease)

original value = new value $\times\frac{100}{100+%\ change}$×100100+% change​

Practice question

QUESTION 4

Goods and services tax (GST)

Goods and services tax (GST) is a  tax of $10%$10% on most goods, services and other items sold or consumed in Australia. Businesses charge the customer an additional $10%$10% of the original price as a GST amount. For example, if the original price of an item was $\$20$$20, the GST on this item would be $\$2$$2 since $10%$10% of $\$20$$20 is $\$2$$2. The total price then charged would be $\$22$$22 .

For tax reasons, businesses need to keep track of how much GST they pay and receive, so it is important to be able to calculate prices before and after GST, as well as the amount and rate of GST.

Calculations including GST

cost including GST $=\cos t\ including\ GST\times1.1$=cost including GST×1.1

amount of GST  $=\frac{\cos t\ including\ GST}{11}$=cost including GST11​

Calculations excluding GST

cost excluding GST $=\frac{\cos t\ including\ GST}{1.1}$=cost including GST1.1​

amount of GST  $=\frac{\cos t\ excluding\ GST}{10}$=cost excluding GST10​

 

Practice question

Question 5

Shares, dividends and yields

A popular method of investing is through buying shares.

• the expectation that over time the value of the company (and thus the value of each share) will increase, meaning a small piece of the company is now worth more. 
• Dividends are paid to shareholders when a company makes a profit.

Price-to-earnings ratio (P/E ratio)

As a shareholder we are interested in calculating the amount of profit and how this profit compared to the amount invested in the company.

Calculating the price-to-earnings ratio allows investors to compare different companies or different dividends. The lower the price-to-earnings ratio the better the result for an investor as it shows that they have invested less for each dollar of profit.

Price-to-earnings ratio

P/E ratio$=\frac{current\ share\ price}{dividend\ per\ share}$=current share pricedividend per share​

Practice question

QUESTION 6

Dividends

Companies share their profits with their shareholders by paying out dividends either as

• a dollar value per share or
• a percentage of current price of the share which is known as a dividend yield.

Dividends and yields

Dividend per share $=\frac{company\ profit}{total\ number\ of\ shares}$=company profittotal number of shares​

Dividend yield $=\frac{dividend\ per\ share}{current\ share\ price}\times\frac{100}{1}$=dividend per sharecurrent share price​×1001​ %

Practice questions

Question 7

Question 8