Percentages and Money
Shops, traders and buyers all use percentages to set prices and measure results. This NSW Year 9 Advanced maths subtopic revises Stage 4 money percentages (NESA outcome MA4‑FRC‑C‑01): discounts, profit and loss, and GST.
You will find percentage discounts, profit and loss, work back from a sale price to the original or cost price, and find the GST in a price that includes it. You will apply a mark-up and then a discount, compare two discounts in a row with one larger discount, calculate commission in two bands, and find profit as a percentage of both cost and selling price. One problem follows a bike sold twice. It is part of Year 9 Advanced Working with Numbers.
Every question with a fully worked solution.
- Percentages and Money - Video - Percentage Discounts Watch
Theory
Shops use percentages for mark-ups, discounts and GST, and traders measure profit and loss as a percentage of the cost price. This Year 9 Advanced maths page revises the skills from earlier years and extends them to chains of changes, working backwards to an original price, removing GST, commission and overall profit or loss.
The cost price is what a seller pays for an item, including extra costs. The selling price is what the buyer pays. Profit is selling price minus cost price; if this is negative it is a loss.
A mark-up is the amount added to the cost price to set the marked price. A discount is taken off the marked price. Both are usually given as percentages and work as multipliers.
GST (Goods and Services Tax) is \(10\%\) of the price before GST. Prices in Australian shops usually include it, so a price with GST is \(110\%\) of the price before GST, and the GST is one-eleventh of that price.
Commission is a percentage of the value of sales, paid to the person who made the sale. Some rates change above a set amount.
Profit and loss are measured from the cost price:
A negative profit is a loss. Discount percentage uses the original (marked) price: \(\dfrac{\text{discount}}{\text{marked price}} \times 100\%\).
GST at \(10\%\):
How to solve money problems with percentages
- Find the cost price, including any extra costs such as repairs or parts.
- Write each mark-up, discount or GST as a multiplier and apply them in order.
- Work backwards by dividing when the final price is given: divide by \(0.8\) to undo \(20\%\) off, by \(1.1\) to remove GST.
- Compare with the right base: profit and loss with the cost price, discount with the marked price, unless told otherwise.
- Round money to the nearest cent and check by working forwards.
(i) the profit
(ii) the profit as a percentage of the cost price
(iii) the profit as a percentage of the selling price, to one decimal place.
(i) The cost price includes the parts.
| \(\text{cost price}\) | \(=\) | \(380 + 45 = \$425\) |
| \(\text{profit}\) | \(=\) | \(510 - 425 = \mathbf{\$85}\) |
| \(\text{(ii) } \dfrac{85}{425} \times 100\%\) | \(=\) | \(\mathbf{20\%}\) |
| \(\text{(iii) } \dfrac{85}{510} \times 100\%\) | \(=\) | \(16.66\ldots\% \approx \mathbf{16.7\%}\) |
The same \(\$85\) is a smaller percentage of the larger selling price.
(i) A repair costs \(\$64\) before GST. Find the price including GST.
(ii) A heater costs \(\$264\) including GST. Find the GST and the price before GST.
(i) Adding \(10\%\) means multiplying by \(1.1\).
| \(64 \times 1.1\) | \(=\) | \(\mathbf{\$70.40}\) |
(ii) \(\$264\) is \(110\%\) of the price before GST, so the GST is one-eleventh of it.
| \(\text{GST}\) | \(=\) | \(264 \div 11 = \mathbf{\$24}\) |
| \(\text{price before GST}\) | \(=\) | \(264 - 24 = \mathbf{\$240}\) |
Check: \(240 \times 1.1 = 264\). Taking \(10\%\) of \(\$264\) gives \(\$26.40\), which is wrong: that is \(10\%\) of the total, not of the price before GST.
(i) It marks the lamp up by \(40\%\), then sells it at \(15\%\) off. Find the selling price and the percentage profit.
(ii) What marked price would give a \(30\%\) profit after a \(20\%\) discount?
(i) Apply the two multipliers in order.
| \(\text{marked price}\) | \(=\) | \(56 \times 1.4 = \$78.40\) |
| \(\text{selling price}\) | \(=\) | \(78.40 \times 0.85 = \mathbf{\$66.64}\) |
| \(\text{profit}\ \%\) | \(=\) | \(\dfrac{66.64 - 56}{56} \times 100\% = \mathbf{19\%}\) |
Quick check: \(1.4 \times 0.85 = 1.19\), a \(19\%\) profit.
(ii) Work backwards. Find the selling price needed, then undo the discount.
| \(\text{selling price}\) | \(=\) | \(56 \times 1.3 = \$72.80\) |
| \(0.8 \times \text{marked price}\) | \(=\) | \(72.80\) |
| \(\text{marked price}\) | \(=\) | \(72.80 \div 0.8 = \mathbf{\$91}\) |
Each percentage is based on that phone's cost price, so find each cost by dividing.
| \(1.1 \times C_1\) | \(=\) | \(660 \Rightarrow C_1 = \$600\) |
| \(0.88 \times C_2\) | \(=\) | \(660 \Rightarrow C_2 = \$750\) |
| \(\text{total cost}\) | \(=\) | \(\$1350\) |
| \(\text{total sales}\) | \(=\) | \(2 \times 660 = \$1320\) |
\(1320 - 1350 = -30\), so she makes an overall loss of \(\$30\). The loss is a percentage of the dearer phone, so it outweighs the profit.
Common pitfalls
Frequently asked questions
How do you calculate percentage profit?
Subtract the cost price from the selling price to get the profit, divide by the cost price, then multiply by \(100\%\). Buying for \(\$425\) and selling for \(\$510\) gives \(\dfrac{85}{425} = 20\%\).
How do you work out GST from a total price?
In Australia GST is \(10\%\), so a GST-inclusive price is \(110\%\) of the price before GST. Divide the total by \(11\) to get the GST, and divide by \(1.1\) to get the price before GST.
How do you find the original price after a discount?
Find the percentage that is left after the discount, write it as a decimal and divide the sale price by it. After \(20\%\) off, divide by \(0.8\).
What is the difference between mark-up and profit?
A mark-up is what a shop adds to the cost price to set its marked price. Profit is what it actually makes when the item sells, which can be less than the mark-up if the item is discounted.
Is 20% off and then 10% off the same as 30% off?
No. The second discount is taken from the reduced price. \(0.8 \times 0.9 = 0.72\), so the total is \(28\%\) off, not \(30\%\).