Profit, mark-up and discount
Choose the base before calculating a percentage. Profit and loss compare selling price with cost. A discount is taken from the price to which the offer applies. Other expenses count only when the question includes them
\[\begin{gathered}\text{signed percentage profit or loss}\\=\frac{\text{selling price}-\text{cost price}}{\text{cost price}}\times100\%\end{gathered}\]
- Interpret the sign Subtract cost from selling price. A negative signed change means a loss of that magnitude \(\displaystyle \begin{aligned}\text{change}&=\$63-\$72\\&=-\$9\end{aligned}\)
- Use cost for percentage profit or loss The amount gained or lost is compared with what the seller paid \(\displaystyle \frac{-9}{72}\times100\%=-12.5\%\)
- Find the marked price A mark-up adds a percentage of cost. Add the mark-up amount or use the corresponding increase multiplier \(\displaystyle \begin{aligned}\text{marked price}&=\$132\times1.35\\&=\$178.20\end{aligned}\)
- Apply each discount to its stated base A further percentage discount normally uses the already reduced price. Apply the changes in sequence rather than adding the rates \(\displaystyle \begin{aligned}\text{sale price}&=\$178.20\times0.80\\&=\$142.56\end{aligned}\)
- Reverse a discount Identify the fraction of the original price that remains, then divide by it \(\displaystyle \begin{aligned}1-\frac{16}{100}&=0.84\\\text{original price}&=\$159.60\div0.84\\&=\$190\end{aligned}\)
Worked example
A stall pays $132 for a set of display stands. It marks up the cost by 35%, then discounts the marked price by 20%. Calculate its profit on the sale, ignoring other expenses
\[\text{profit}=\text{sale price}-\text{cost price}\]
- \(\displaystyle 1+\frac{35}{100}=1.35\)The marked price includes the full cost and the mark-up
- \(\displaystyle \$132\times1.35=\$178.20\)Apply the increase to cost to find the marked price
- \(\displaystyle 1-\frac{20}{100}=0.80\)Find the fraction of the marked price that remains
- \(\displaystyle \$178.20\times0.80=\$142.56\)Apply the discount to the marked price
- \(\displaystyle \$142.56-\$132=\$10.56\)Subtract cost from the sale price. The positive result is the profit on this item
Common mistakes
Using selling price as the profit base
✕\(\displaystyle \frac{70-56}{70}\times100\%=20\%\)✓\(\displaystyle \frac{70-56}{56}\times100\%=25\%\)
The first calculation measures profit relative to selling price. Percentage profit on cost requires cost as the denominatorTaking a sale discount from cost
✕\(\displaystyle \$144\times1.25-\$144\times0.10=\$165.60\)✓\(\displaystyle \begin{aligned}\$144\times1.25\times0.90&=\$180\times0.90\\&=\$162\end{aligned}\)
The discount is a percentage of the marked price, which includes the mark-upAdding another percentage to reverse a discount
✕\(\displaystyle \$159.60\times1.16=\$185.136\)✓\(\displaystyle \$159.60\div0.84=\$190\)
Adding the same percentage to the smaller sale price does not restore the original price. Divide by the fraction that remainsAdding successive discount rates
✕\(\displaystyle \$240\times\left(1-\frac{20+15}{100}\right)=\$156\)✓\(\displaystyle \begin{aligned}\$240\times0.80\times0.85&=\$192\times0.85\\&=\$163.20\end{aligned}\)
The further discount uses the reduced price. Adding the rates applies both discounts to the original priceDifficulty levels
introFind a profit, loss or percentage discount amount
standardFind percentage profit or loss, a mark-up amount or a marked-up selling price
challengeFollow successive price changes and find profit or loss, or recover a price before a discount
examChoose the correct comparison of final prices under two discount offers
Questions about this topic
Which price is the base for percentage profit?+
Use the cost price. The calculation compares the amount gained with what the seller paid
Does a sale below the marked price always make a loss?+
No. Compare the sale price with cost. It can remain above cost after the discount
Can equal mark-up and discount rates cancel?+
Equal positive rates on different bases give different dollar amounts. A positive discount on the marked-up price removes more than the equal-rate mark-up added
How do I reverse a discount?+
Find the fraction of the original price that remains and divide the sale price by that fraction
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 2: Consumer arithmetic
In this chapter
Convert between fractions, decimals and percentagesFind a percentage of a quantityExpress one quantity as a percentage of anotherProfit, mark-up and discountIncrease or decrease a quantity by a percentageFind the original amount from a percentageAdd GST to a priceFind the GST and the pre-GST price from an inclusive priceIncome from wages, overtime, shifts and commissionIncome tax payableBudgeting with percentages and fixed valuesDecide which option is the better valueSimple interest, unknowns and repaymentsCompound interest and non-annual periodsPurchase cost from deposit and repaymentsBank interest on a minimum monthly balanceCompare simple and compound interest