← LearnYear 8 Maths · Number & MeasurementFree placement check
Year 8AC9M8N05

Percentages: Discount, Profit and GST

Use percentages in financial contexts to solve problems involving percentage increase and decrease, profit and loss, discounts and mark-ups.

Turn the percentage change into one number, then do it in one step.

Builds onYr 7 · Percentages of quantitiesYr 6 · Decimals

Why this works

Most you calculate the change and then add or subtract. There is a faster and more reliable way.

A 20% increase means you end up with the original 100% plus another 20% — that is 120% of the original.
100%+20%=120%=1.2100\% + 20\% = 120\% = 1.2
So multiplying by 1.21.2 does the whole job in one step.
$50×1.2=$60\$50 \times 1.2 = \$60
A 20% decrease leaves you with 100%−20%=80%100\% - 20\% = 80\%.
100%−20%=80%=0.8100\% - 20\% = 80\% = 0.8
Again, one multiplication.
$50×0.8=$40\$50 \times 0.8 = \$40
GST in Australia is 10%, so adding GST means multiplying by 1.11.1 — and removing GST means dividing by 1.11.1, not subtracting 10%.
price×1.1=price inc GST\text{price} \times 1.1 = \text{price inc GST}

That last point catches almost everyone. If a price already includes GST, taking 10% off does not get you back — because the 10% was calculated on the smaller pre-GST figure. You must divide by 1.1.

Worked example

e.g. A $80 jacket is discounted 25%. Find the sale price.\text{A } \$80 \text{ jacket is discounted } 25\%. \text{ Find the sale price.}
  1. Decide whether this is an increase or a decrease. A discount is a decrease.decrease of 25%\text{decrease of } 25\%
  2. Work out the multiplier: start from 100% and subtract.100%−25%=75%=0.75100\% - 25\% = 75\% = 0.75
  3. Multiply the price by it.$80×0.75=$60\$80 \times 0.75 = \$60
  4. Sanity-check: the sale price must be lower than $80\$80, and a quarter off $80\$80 is $20\$20 off. ✓$60\$60

Common mistakes

WrongRemoving GST by taking 10% off\text{Removing GST by taking } 10\% \text{ off}
RightDivide the GST-inclusive price by 1.1\text{Divide the GST-inclusive price by } 1.1
A $110\$110 inclusive price came from $100×1.1\$100 \times 1.1. Taking 10% off $110\$110 gives $99\$99, which is wrong. Dividing by 1.1 gives $100\$100, which is right.
WrongUp 10% then down 10% returns the original\text{Up } 10\% \text{ then down } 10\% \text{ returns the original}
Right1.1×0.9=0.99 — a 1% net loss1.1 \times 0.9 = 0.99 \ \text{— a } 1\% \text{ net loss}
The second percentage is calculated on a different amount, so the two changes do not cancel. $100→$110→$99\$100 \rightarrow \$110 \rightarrow \$99.
WrongFinding 25% and calling it the answer\text{Finding } 25\% \text{ and calling it the answer}
RightSubtract it, or multiply by 0.75\text{Subtract it, or multiply by } 0.75
Read the question. 'Find the discount' and 'find the sale price' want different numbers.
WrongPercentage profit calculated on the selling price\text{Percentage profit calculated on the selling price}
Rightpercentage profit=profitcost price×100\text{percentage profit} = \frac{\text{profit}}{\text{cost price}} \times 100
Percentage profit is always measured against what you paid, not what you sold it for.

Practice

FluencyGet quick and accurate at the method.
1
Increase $40 by 15%\text{Increase } \$40 \text{ by } 15\%
Answer$46\$46
2
Decrease $200 by 30%\text{Decrease } \$200 \text{ by } 30\%
Answer$140\$140
3
Add GST to $250\text{Add GST to } \$250
Answer$275\$275
4
Remove GST from $330\text{Remove GST from } \$330
Answer$300\$300
5
A $60 item, 20% off\text{A } \$60 \text{ item, } 20\% \text{ off}
Answer$48\$48
6
Increase $85 by 40%\text{Increase } \$85 \text{ by } 40\%
Answer$119\$119
7
Decrease $150 by 12%\text{Decrease } \$150 \text{ by } 12\%
Answer$132\$132
8
Bought $80, sold $100. Percentage profit?\text{Bought } \$80, \text{ sold } \$100. \text{ Percentage profit?}
Answer25%25\%
9
Bought $50, sold $40. Percentage loss?\text{Bought } \$50, \text{ sold } \$40. \text{ Percentage loss?}
Answer20%20\%
10
A $45 item marked up 60%\text{A } \$45 \text{ item marked up } 60\%
Answer$72\$72
ReasoningExplain why. Say it in your own words.
1
A shop raises a price by 10%, then drops it by 10%. Is the price back where it started?
AnswerNo — it is 1% lower. The multipliers are 1.1×0.9=0.991.1 \times 0.9 = 0.99. The 10% rise was calculated on the original price, but the 10% fall was calculated on the larger price, so more came off than went on.
2
Why can you not remove GST by subtracting 10% from the inclusive price?
AnswerThe 10% was worked out on the pre-GST price, which is smaller than the inclusive price. Taking 10% off the larger figure removes too much. The inclusive price is 1.1×1.1 \times the original, so you divide by 1.1 to reverse it.
3
Which is the better deal on a $200\$200 item: 25% off, or $40\$40 off then a further 10% off?
AnswerThe second is better, by $6\$6. A straight 25% off gives 200×0.75=$150200 \times 0.75 = \$150. Taking $40\$40 off first gives $160\$160, and a further 10% off that is 160×0.9=$144160 \times 0.9 = \$144. The lesson is that successive changes are applied to progressively different amounts, so you cannot judge these by intuition — compute both.
4
Why is percentage profit measured against the cost price rather than the selling price?
AnswerPercentage profit answers 'how much did my investment grow?', and the investment is what you paid. Measuring against the selling price would make identical trades look different depending on the mark-up, and would never reach 100%.
AppliedThe same maths, inside a real question.
1
A laptop costs $1200\$1200 before GST. What does a customer pay?
Answer$1320\$1320. 1200×1.1=13201200 \times 1.1 = 1320.
2
A receipt shows $96.80\$96.80 including GST. What was the price before GST?
Answer$88\$88. Divide by 1.1: 96.80÷1.1=8896.80 \div 1.1 = 88.
3
A shop buys shirts for $25\$25 and sells them for $40\$40. What is the percentage profit?
Answer60%. Profit is $15\$15, and 1525×100=60%\tfrac{15}{25} \times 100 = 60\%.
4
A $1500\$1500 bike is discounted 20%, then a further 10% off at the counter. What is the final price?
Answer$1080\$1080. 1500×0.8=12001500 \times 0.8 = 1200, then 1200×0.9=10801200 \times 0.9 = 1080. Note this is not 30% off, which would be $1050\$1050.
5
Mia's rent rises from $450\$450 to $486\$486 per week. What is the percentage increase?
Answer8%. The rise is $36\$36, and 36450×100=8%\tfrac{36}{450} \times 100 = 8\%.
Next step
Practise Percentages: Discount, Profit and GST with instant marking
A free 10-minute placement check finds which Year 8 topics to work on first, then Summit builds a weekly plan around them.
See where my child is →