Ideas
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\% 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 \%:
| \displaystyle \text{decrease in value} | \displaystyle = | \displaystyle \dfrac{\%}{100} \times \text{original value} |
| \displaystyle \text{new value} | \displaystyle = | \displaystyle \text{original value}-\text{decrease in value} |
| \displaystyle = | \displaystyle \text{original value}\times (100 - \text{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\% surcharge on Sundays, meaning that the total bill is increased by 15\%.
When applying a percentage increase \%:
| \displaystyle \text{increase in value} | \displaystyle = | \displaystyle \dfrac{\%}{100} \times \text{original value} |
| \displaystyle \text{new value} | \displaystyle = | \displaystyle \text{original value}+\text{increase in value} |
| \displaystyle = | \displaystyle \text{original value}\times (100 + \text{increase})\% |
Examples
Example 1
We want to increase 200 by 10\% by following the steps below.
Find 10\% of 200.
Add the percentage increase to the original amount to find the amount after the increase.
Calculate 110\% of 200.
Is increasing an amount by 10\% equivalent to finding 110\% of that amount?
When applying a percentage decrease \%:
| \displaystyle \text{decrease in value} | \displaystyle = | \displaystyle \dfrac{\%}{100} \times \text{original value} |
| \displaystyle \text{new value} | \displaystyle = | \displaystyle \text{original value}-\text{decrease in value} |
| \displaystyle = | \displaystyle \text{original value}\times (100 - \text{decrease})\% |
When applying a percentage increase \%:
| \displaystyle \text{increase in value} | \displaystyle = | \displaystyle \dfrac{\%}{100} \times \text{original value} |
| \displaystyle \text{new value} | \displaystyle = | \displaystyle \text{original value}+\text{increase in value} |
| \displaystyle = | \displaystyle \text{original value}\times (100 + \text{increase})\% |
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 to \$2.30. To compare this increase to another store it is useful to use percentage rather than dollars. The bread has increased by 30 cents. Comparing this increase to the starting value: \dfrac{0.30}{2.00}=0.15 By converting this to a percentage we can say the bread has increased by 15\%.
To find the percentage change:
\text{Percentage change} = \dfrac{\text{new value}-\text{original value}}{\text{original value}}\times 100\%
Examples
Example 2
A holiday resort in Tasmania reduced its overnight rates from \$320 to \$120.
Find the amount that Beth would save if she is to take advantage of the sale.
Express this amount saved as a percentage discount. Round your answer to two decimal places.
To find the percentage change:
\text{Percentage change} = \dfrac{\text{new value}-\text{original value}}{\text{original value}}\times 100\%
Calculate 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 and a percentage increase or decrease of 3\%.
Finding original value given new amount (percentage increase): \text{original value}= \text{new value} \times \dfrac{100}{100-\%\text{change}}
Finding original value given new amount (percentage decrease): \text{original value}= \text{new value} \times \dfrac{100}{100+\%\text{change}}
Examples
Example 3
There is a 12\% off sale in store. With this discount in place, a particular item sells for \$2992. Calculate the regular price of this item, to the nearest dollar.
Finding original value given new amount (percentage increase): \text{original value}= \text{new value} \times \dfrac{100}{100-\%\text{change}}
Finding original value given new amount (percentage decrease): \text{original value}= \text{new value} \times \dfrac{100}{100+\%\text{change}}
Goods and services tax (GST)
Goods and services tax (GST) is a tax of 10\% on most goods, services and other items sold or consumed in Australia. Businesses charge the customer an additional 10\% of the original price as a GST amount. For example, if the original price of an item was \$20, the GST on this item would be \$2 since 10\% of \$20 is \$2. The total price then charged would be \$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:
| \displaystyle \text{Cost including GST} | \displaystyle = | \displaystyle \text{Cost excluding GST}\times 1.1 |
| \displaystyle \text{Amount of GST} | \displaystyle = | \displaystyle \dfrac{\text{Cost including GST}}{11} |
Calculations excluding GST:
| \displaystyle \text{Cost excluding GST} | \displaystyle = | \displaystyle \dfrac{\text{Cost including GST}}{1.1} |
| \displaystyle \text{Amount of GST} | \displaystyle = | \displaystyle \dfrac{\text{Cost excluding GST}}{10} |
Examples
Example 4
The sales price of an item, including GST, is \$45. Calculate the price of the item without GST.
Calculations including GST:
| \displaystyle \text{cost including GST} | \displaystyle = | \displaystyle \text{cost including GST}\times 1.1 |
| \displaystyle \text{amount of GST} | \displaystyle = | \displaystyle \dfrac{\text{cost including GST}}{11} |
Calculations excluding GST:
| \displaystyle \text{cost excluding GST} | \displaystyle = | \displaystyle \dfrac{\text{cost including GST}}{1.1} |
| \displaystyle \text{amount of GST} | \displaystyle = | \displaystyle \dfrac{\text{cost excluding GST}}{10} |