Percentages and Financial Applications

Key Ideas

Key Terms

percentage increase/decrease: A change in a quantity expressed as a percentage of the original value.
multiplier: A single number applied to the original to produce the new value — e.g. a 15% increase uses multiplier 1.15; a 25% decrease uses 0.75.
reverse percentage: Finding the original value before a percentage change by dividing the new value by the multiplier.
GST: Goods and Services Tax — a 10% tax added to prices in Australia; price with GST = original × 1.10.
discount: A reduction in price calculated as a percentage decrease of the original price.
commission: A payment calculated as a percentage of a total sales value, typically earned by salespeople.
profit: The amount earned above the cost price; profit % = (profit ÷ cost price) × 100.
loss: When the selling price is less than the cost price; loss % = (loss ÷ cost price) × 100.

Situation	Formula	Multiplier
% increase of r%	New = Original × (1 + r/100)	e.g. 15% → 1.15
% decrease of r%	New = Original × (1 − r/100)	e.g. 25% → 0.75
Reverse percentage	Original = New ÷ multiplier
Add 10% GST	Price with GST = Price × 1.10	1.10
Remove 10% GST	Pre-GST price = Price incl. GST ÷ 1.10	÷ 1.10
Profit/Loss %	(Profit or Loss ÷ Cost Price) × 100
Commission	Commission = Rate × Sales value

Hot Tip The multiplier method is the fastest approach. A 15% discount → multiply by 0.85. A 20% increase → multiply by 1.20. To reverse: divide by the multiplier. Don’t add/subtract 15% of the new price — always use the original as the base.

Worked Example — Percentage Discount

Question: A jacket has an original price of $80. It is reduced by 15%. What is the sale price?

Step 1 — Find the multiplier.
Decrease of 15% → multiplier = 1 − 0.15 = 0.85

Step 2 — Apply the multiplier.
Sale price = $80 × 0.85 = $68

Worked Example — Reverse Percentage

Question: After a 20% price increase, a fridge costs $960. What was the original price?

Step 1 — The multiplier for a 20% increase is 1.20.

Step 2 — Divide the new price by the multiplier.
Original = $960 ÷ 1.20 = $800

Finding a Percentage of a Quantity

To find a percentage of a quantity, convert the percentage to a decimal (divide by 100) and multiply. For example, 35% of $240 = 0.35 × $240 = $84. Alternatively, find 1% first (divide by 100), then multiply by the required percentage.

Real-world uses: calculating discounts in shops, finding commission, working out tax amounts.

Percentage Increase and Decrease

Percentage increase: multiply by (1 + rate). A 20% increase on $150: $150 × 1.20 = $180. The multiplier 1.20 adds the original plus 20%.

Percentage decrease: multiply by (1 − rate). A 15% discount on $200: $200 × 0.85 = $170. The multiplier 0.85 keeps 85% of the original.

Using a single multiplier is faster than calculating the percentage separately and then adding or subtracting, and it is less prone to error.

Finding the Original Amount (Reverse Percentage)

If you know the final amount after a percentage change and want to find the original, divide by the multiplier instead of multiplying.

Example: A jacket costs $136 after a 15% discount. What was the original price? The multiplier was 0.85, so original = $136 / 0.85 = $160.

A common mistake is to take 15% off the discounted price — but that is the wrong price to apply it to. Always divide by the correct multiplier.

GST (Goods and Services Tax)

In Australia, GST is a 10% tax added to the price of most goods and services. The multiplier for adding GST is 1.10.

Price including GST = original price × 1.10. Example: A tool costs $80 before GST. Price with GST = $80 × 1.10 = $88.

Removing GST (finding the pre-GST price): divide by 1.10. Example: A receipt shows $110. Pre-GST price = $110 ÷ 1.10 = $100. Note: the GST component is not 10% of $110; it is 10/110 = 1/11 of the total, which equals $10.

Key tip: A very common exam error is to remove GST by multiplying by 0.9 instead of dividing by 1.1. These give different answers! Multiplying $110 by 0.9 gives $99, which is wrong. Dividing $110 by 1.1 gives $100, which is correct. Always divide by the multiplier to reverse a percentage change — never multiply by the complement.

Mastery Practice

Calculate the new amount after each percentage change. Fluency

	Original Amount	Change
(a)	$250	increase by 12%
(b)	$340	decrease by 25%
(c)	480 kg	increase by 7.5%
(d)	$1200	decrease by 30%
(e)	$620	decrease by 15%
(f)	3500	increase by 4%
(g)	$760	increase by 2.5%
(h)	840 g	decrease by 12.5%

Find the original value before the percentage change. Fluency

	After the Change	What Happened
(a)	$180	20% increase
(b)	$45 000	10% decrease
(c)	$660 (incl. GST)	10% GST added
(d)	$117	35% discount
(e)	32 400 people	8% population increase
(f)	$18 700	15% loss on sale
(g)	$544	36% increase
(h)	$374	15% discount

Complete the GST and financial applications table. Fluency

	Situation	Answer
(a)	Plumber charges $350 + GST. Find total bill.
(b)	Laptop priced at $1320 incl. GST. Find the GST component.
(c)	Vase bought for $40, sold for $58. Find profit and profit %.
(d)	Car bought $25 000, sold $19 000. Find loss %.
(e)	Agent earns 3.5% commission on $28 000 of sales.
(f)	Agent earns 2% commission; received $1450. Find total sales.
(g)	Shirt wholesale $35, marked up 40%. Find selling price.
(h)	Textbook $55 incl. GST. Student gets 10% off pre-GST, then pays GST. Find total paid.

True or False? Write T or F, then briefly explain your reasoning. Fluency

	Statement	T / F
(a)	A 25% increase then a 25% decrease returns to the original value.
(b)	The GST component of a $110 item is $11.
(c)	A 50% discount and then another 50% discount gives a total discount of 100%.
(d)	Profit percentage is always calculated on the selling price.

Buying a TV. Understanding

Shopping scenario. An electronics store advertises a television for $880 including GST. During a sale, a 10% discount is applied to the GST-inclusive price. A member then receives an additional 5% off the already-discounted price.
1. What is the pre-GST price of the television?
2. What is the price after the 10% sale discount?
3. What does a member pay after both discounts?
4. What is the total percentage saving from the original $880?
Salary negotiations. Understanding

Pay rises. Maya earns $58 000 per year. She receives a 4% pay rise, and 12 months later receives a further 3% rise on her new salary.
1. What is Maya’s salary after the first rise?
2. What is her salary after the second rise?
3. What single percentage increase from her original salary achieves the same result? (Round to 2 decimal places.)
Real estate commission. Understanding

Agent’s earnings. A real estate agent earns a 2.5% commission on all property sales. One month the agent sells three properties: $420 000, $675 000, and $310 000.
1. Calculate the commission earned on each property.
2. Find the total commission for the month.
3. The agent’s monthly expenses are $4 200. Did the agent make a profit or a loss this month? By how much?
Comparing deals. Understanding

Two stores, same item. A $500 camera body is advertised at two stores. Store A offers “20% off, then a further 10% off the sale price.” Store B offers “28% off the original price.”
1. Find the final price at Store A.
2. Find the final price at Store B.
3. Which store is cheaper and by how much?
4. Explain why two successive percentage discounts of 20% and 10% do not equal a single 30% discount.
Property investment. Problem Solving

Long-term growth. A house was valued at $480 000 in 2020 and $552 000 in 2023.
1. Find the percentage increase in value over the 3-year period.
2. Assuming equal annual growth, find the average annual percentage increase. (Hint: if annual rate is r, then 480 000 × (1 + r)³ = 552 000. Solve for r.)
3. If the property continues to grow at this average annual rate, predict its value after a further 4 years. Round to the nearest dollar.
Business profit analysis. Problem Solving

Buying and selling in bulk. A business buys 200 units at $18 each and sells 180 of them at $28 each. The remaining 20 units are sold at a 40% discount on the original $28 selling price.
1. What is the total cost of all 200 units?
2. What is the revenue from the 180 units sold at full price?
3. What is the discounted price per unit for the remaining 20, and what is the revenue from those units?
4. Calculate the total revenue and the overall profit.
5. Express the overall profit as a percentage of the total cost (round to 1 decimal place).

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