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 \(\displaystyle \begin{aligned}1-\frac{16}{100}&=0.84\\\text{original price}&=\$159.60\div0.84\\&=\$190\end{aligned}\)
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 9 Mathematics, chapter 1: Number and financial mathematics
In this chapter
Evaluate an integer expressionFactors, multiples, HCF, LCM and primesRound to decimal places or significant figuresEstimate an expression using rounded inputsConvert between fractions, mixed numbers and decimalsClassify real numbers and compare exact valuesAdd, subtract, multiply and divide fractions and mixed numbersWrite a ratio from countsWrite ratios in simplest formComplete equivalent ratiosUse a constant rateDivide a quantity in a given ratioDecide which option is the better valueConvert between fractions, decimals and percentagesFind a percentage of a quantityExpress one quantity as a percentage of another1F Percentages and money: Find the original amount from a percentageIncrease or decrease a quantity by a percentageFind percentage changePercentage error against a reference value1G Percentage change and percentage error: Find the original amount from a percentageProfit, mark-up and discountIncome and taxSimple interestCompound interest (optional extension)