A mark up indicates that the price of the product has increased from its original price and the percentage of the mark up indicates by how much. For example: if the price of a jumper has been marked up by 20\%, then its price has been increased by 20\%.

Similarly, the percentage of a discount indicates how much the price of a product has decreased from its original price.

Since mark ups and discounts are simply percentage increases and decreases , we can calculate them using the same methods.

Examples

Example 1

A pair of boots priced at \$80 is discounted by 35\%. What is the discounted price of the pair of boots?

Worked Solution

Create a strategy

To find the discounted price multiply the original price by (100-35)\%=65\%.

Apply the idea

\displaystyle \text{Discount price}	\displaystyle =	\displaystyle 65\% \times 80	Multiply by 65\%
	\displaystyle =	\displaystyle 0.65 \times 80	Convert the percentage to a decimal
	\displaystyle =	\displaystyle \$52	Evaluate

Idea summary

A mark up is a percentage increase in the price of a product.

A discount is a percentage decrease in the price of a product.

Multiple mark ups and discounts

Suppose a video game was marked up by 20\%, later discounted by 25\% and finally discounted again by 40\%. If the original price of the video game was \$50, what is its current price?

The video game starts at the price of \$50.
After a mark up of 20\%, its price is \$50\times 120\% which is equal to \$60.
After a discount of 25\%, its price is \$60\times 75\% which is equal to \$45.
After a final discount 40\%, its price is \$45\times 60\% which is equal to \$27.

So the current price of the game is \$27.

However, this is a lot of steps. Notice that the only operation we applied to the price was multiplication. Since multiplication does not need to be applied one at a time, we can reach the same answer by calculating:\text{Current price} = \$50\times 120\%\times 75\%\times 60\%

The advantages of this approach are that we can cancel out factors if we convert the percentages into fractions, or simply use a calculator to evaluate the expression in one step. Since the order in which we multiply doesn't matter either, we can also choose which percentage changes we want to apply first to make calculations easier.

Mark ups and discounts of the same percentage do not cancel each other out.

For example: if an item is marked up by 10\% and then discounted by 10\%, its final price will be:\text{Final price}=\text{Original price}\times 110\% \times 90\%

Evaluating the left hand side, we get:\text{Final price}=\text{Original price} \times 99\%

As we can see, the final price and the original price are not the same.

Examples

Example 2

A bladesmith began her career selling her swords for \$1500. After a few months she marked up the price by 16\%.

A few weeks after that a video of her making the swords went viral, and she marked up the price by 11\%.

In a few days she is due to appear on a popular podcast, and in anticipation of the increased demand she marks up the price by 13\%.

What is the price of the bladesmith's swords after all the mark ups?

Worked Solution

Create a strategy

Multiply the original price by the three percentage changes for each mark up.

Apply the idea

To mark up the price by 16\% then 11\% then 13\% we need to multiply by 116\%, \, 111\% and 113\%.

\displaystyle \text {Final price}	\displaystyle =	\displaystyle 1500\times 116\%\times 111\%\times 113\%	Multiply by the percentages
	\displaystyle =	\displaystyle 1500\times 1.16\times 1.11\times 1.13	Convert the percentages to decimals
	\displaystyle =	\displaystyle \$ 2182.48	Evaluate

Idea summary

We can find the final price after multiple markups and discounts by multiplying the original price by all of the equivalent percentage changes.

For example: if an item is marked up by 10\% and then discounted by 10\%, its final price will be:\text{Final price}=\text{Original price}\times 110\% \times 90\%

GST

Almost everything we buy has a compulsory \$10 mark-up applied to it, called goods and services tax (GST). The extra money we pay goes to the government, and they can spend it on essential things like schools and roads. Some items we buy are excluded from GST. The most common examples are fresh fruit and vegetables, as well as other basic foods such as bread, flour, oil, and dairy products.

We can calculate the total price, inclusive of GST, in the same way we calculate percentage increases.

GST is a 10\% price increase.

Examples

Example 3

Quentin is buying items he needs for school. The prices marked for the items are all pre-GST.

He buys: a laptop for \$1360, exercise books for \$18, a pack of pens for \$7, and a calculator for \$38.

Find the total price Quentin pays including GST.

Worked Solution

Create a strategy

Add the prices and increase the total by 10\%.

Apply the idea

\displaystyle \text{Total price}	\displaystyle =	\displaystyle 1360+18+7+38	Add the prices
	\displaystyle =	\displaystyle 1423	Evaluate
\displaystyle \text{Total including GST }	\displaystyle =	\displaystyle 110\% \times 1423	Multiply by 110\%
	\displaystyle =	\displaystyle \$1565.30	Evaluate

Example 4

A book has a price including GST of \$55. Find the pre-GST price of the book.

Worked Solution

Create a strategy

To calculate a pre-GST price divide the final price to 110\%.

Apply the idea

\displaystyle \text{Original price }	\displaystyle =	\displaystyle \text{Final price} \div 110\%	Divide by 110\%
	\displaystyle =	\displaystyle 55\div 110\%	Substitute the value
	\displaystyle =	\displaystyle 55\div 1.1	Convert the percentage to a decimal
	\displaystyle =	\displaystyle \$50	Evaluate

Idea summary

GST is a 10\% price increase. \text{Price including GST} =\text{Original price} \times 110\%

2.04 Consumer percentages

Ideas

Mark ups and discounts

Examples

Example 1

Multiple mark ups and discounts

Examples

Example 2

GST

Examples

Example 3

Example 4

Outcomes

MA4-5NA

MA4-6NA

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