Profits and Discounts
Every question with a fully worked solution.
- Profits and Discounts - Video - Profits and Discounts Watch
Theory
Profit, loss, mark-ups, discounts, commission and GST are all percentages of money. This Year 9 maths page revises the Year 8 work in financial contexts: finding a sale price, working out percentage profit or loss on the cost price, marking up a price, finding the original price after a discount, adding or extracting \(10\%\) GST, and calculating commission.
The cost price is what the seller paid. The selling price is what the buyer pays. If the selling price is higher there is a profit; if it is lower there is a loss.
Unless the question says otherwise, percentage profit or loss is worked out on the cost price: the profit (or loss) goes on top and the cost price underneath.
A discount is a percentage of the original price. The sale price is what is left: a \(35\%\) discount leaves \(65\%\). To find the original price from a sale price, divide by that multiplier.
GST (goods and services tax) in Australia is \(10\%\) of the price before tax. Adding GST multiplies the price by \(1.1\). In a price that already includes GST, the GST is one part in eleven.
A mark-up is the amount a shop adds to its cost price to set the selling price. A commission is pay worked out as a percentage of the value of sales, often at different rates for different parts of the sale.
Money answers are given to the nearest cent, with two decimal places: \(\$8.40\), not \(\$8.4\).
Profit, loss and their percentages:
Discounts and GST as multipliers:
Getting the base right is most of the work:
| Question asks for | Percentage is taken of |
|---|---|
| percentage profit or loss | the cost price |
| percentage discount | the original (marked) price |
| GST to add | the price before GST |
| GST already included | the total, divided by \(11\) |
How to solve a money percentage problem
- Name the prices: cost, selling, original, sale, before GST, after GST.
- Choose the base: cost price for profit and loss, original price for a discount, pre-GST price for GST.
- Calculate with a multiplier, or find the amount and then add or subtract it.
- Work backwards by dividing by the multiplier when the final price is given.
- Write money to two decimal places and check by going the other way.
(i) the discount
(ii) the sale price.
(i) The discount is \(35\%\) of the marked price:
| \(0.35 \times 180\) | \(=\) | \(\mathbf{\$63.00}\) |
(ii) What is left is \(100\% - 35\% = 65\%\):
| \(0.65 \times 180\) | \(=\) | \(\mathbf{\$117.00}\) |
Check: \(\$180 - \$63 = \$117\).
(ii) A phone bought for \(\$75\) is sold for \(\$63\). Find the percentage loss.
(iii) A cafe buys mugs for \(\$12\) each and marks them up by \(65\%\). Find the selling price.
(i) Profit \(= 81 - 60 = \$21\). Divide by the cost price, then multiply by \(100\%\):
| \(\dfrac{21}{60} \times 100\%\) | \(=\) | \(\mathbf{35\%}\) |
Dividing by \(\$81\) would answer a different question: profit as a percentage of the selling price.
(ii) Loss \(= 75 - 63 = \$12\). Again divide by the cost price:
| \(\dfrac{12}{75} \times 100\%\) | \(=\) | \(\mathbf{16\%}\) |
(iii) A mark-up is an increase on the cost price, so the multiplier is \(1 + 0.65 = 1.65\):
| \(12 \times 1.65\) | \(=\) | \(\mathbf{\$19.80}\) |
The cafe makes \(\$7.80\) profit on each mug.
The sale price is what is left after the discount: \(100\% - 40\% = 60\% = 0.6\). So \(\$126\) is the original multiplied by \(0.6\). Divide to undo it:
| \(\text{original}\) | \(=\) | \(126 \div 0.6\) |
| \(\) | \(=\) | \(\mathbf{\$210.00}\) |
Check: \(40\%\) of \(\$210\) is \(\$84\), and \(\$210 - \$84 = \$126\). Taking \(40\%\) of \(\$126\) and adding it on would be wrong, because the discount was taken of the original price.
(ii) A repair bill is \(\$451\), including GST. How much of it is GST?
(iii) A salesperson earns \(3\%\) commission on the first \(\$10\,000\) of her sales and \(2\%\) on the rest. One month she sells \(\$16\,500\). Find her commission.
(i) Adding \(10\%\) means multiplying by \(1.1\):
| \(420 \times 1.1\) | \(=\) | \(\mathbf{\$462.00}\) |
The GST is \(\$42\) of that.
(ii) A price that includes GST is \(110\%\) of the price before it. The GST is \(10\) parts of those \(110\), so divide by \(11\):
| \(451 \div 11\) | \(=\) | \(\mathbf{\$41.00}\) |
Check: the price before GST is \(451 - 41 = \$410\), and \(10\%\) of \(\$410\) is \(\$41\).
(iii) Split the sales into the two parts, work out each part at its own rate, then add:
| \(3\% \text{ of } 10\,000\) | \(=\) | \(\$300\) |
| \(2\% \text{ of } 6500\) | \(=\) | \(\$130\) |
| \(\text{commission}\) | \(=\) | \(300 + 130 = \mathbf{\$430.00}\) |
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 \(100\%\). Bought for \(\$60\) and sold for \(\$81\) is a \(\$21\) profit, which is \(35\%\).
How do you work out a sale price after a discount?
Multiply the original price by what is left. For a \(35\%\) discount, \(65\%\) is left, so multiply by \(0.65\). A \(\$180\) jacket becomes \(\$117\).
How do you find the GST in a price that includes GST?
Divide the total by \(11\). A bill of \(\$451\) including GST has \(\$41\) of GST, because the total is \(110\%\) and the GST is \(10\) of those \(110\) parts.
How do you find the original price before a discount?
Divide the sale price by the multiplier for what is left. After a \(40\%\) discount, \(60\%\) is left, so a \(\$126\) sale price came from \(126 \div 0.6 = \$210\).
Is 20 per cent off then 10 per cent off the same as 30 per cent off?
No. The second discount is taken off the already reduced price: \(0.8 \times 0.9 = 0.72\), so the total saving is \(28\%\), not \(30\%\).
What is a mark-up?
A mark-up is the amount added to the cost price to get the selling price, usually given as a percentage of the cost price. Multiply the cost price by \(1\) plus the mark-up as a decimal.