Practice Maths

Percentages and Applications

Key Terms

To convert a fraction to a percentage: divide numerator by denominator then multiply by 100.
To convert a decimal to a percentage: multiply by 100 (e.g. 0.73 = 73%).
Percentage of a quantity
change % to a decimal, then multiply (e.g. 15% of $240 = 0.15 × 240 = $36).
Percentage increase
new value = original × (1 + rate). E.g. 8% increase → multiply by 1.08.
Percentage decrease
new value = original × (1 − rate). E.g. 12% decrease → multiply by 0.88.
GST (Goods and Services Tax)
in Australia is 10%. GST-inclusive price = original × 1.1. To find the pre-GST price: divide by 1.1.
Discount
amount off = original × discount rate. Sale price = original × (1 − discount rate).
Markup
selling price = cost price × (1 + markup rate).
Profit
profit = selling price − cost price.   Loss: loss = cost price − selling price.
Key Formulas
• % change = (change ÷ original) × 100
• New value after increase = original × (1 + r)   where r = rate as a decimal
• New value after decrease = original × (1 − r)
• GST-inclusive price = original × 1.1
• Pre-GST price = GST-inclusive price ÷ 1.1
• Profit/Loss % = (profit or loss ÷ cost price) × 100
Operation Multiplier Example
10% increase (GST)× 1.10$80 × 1.10 = $88
20% discount× 0.80$150 × 0.80 = $120
25% markup× 1.25$60 × 1.25 = $75
15% of quantity× 0.15$340 × 0.15 = $51
Find % change(change ÷ original)×100$20 rise on $80 = 25%

Visual: $200 item with 20% discount → $160 sale price + 10% GST → $176

Original: $200 $200 After 20% discount: $160 (saved $40) $160 -$40 +GST → $176
Hot Tip When applying a percentage change, always use a multiplier rather than calculating the amount and then adding or subtracting. It is faster and avoids errors. For a 15% discount: multiply by 0.85 in one step. For 10% GST: multiply by 1.1 in one step. To undo GST, divide by 1.1 (not by 1.1 − 0.1 = 1.0).

Worked Example 1 — Percentage increase and GST

Question: A jacket has a pre-GST price of $145. Find (a) the GST amount and (b) the GST-inclusive price.

(a) GST = 10% of $145 = 0.10 × $145 = $14.50

(b) GST-inclusive price = $145 × 1.1 = $159.50

Check: $145 + $14.50 = $159.50 ✓

Worked Example 2 — Percentage change and profit

Question: A shopkeeper buys a watch for $80 and sells it for $116. Find (a) the profit and (b) the profit as a percentage of cost price.

(a) Profit = $116 − $80 = $36

(b) Profit % = (36 ÷ 80) × 100 = 45%

The shopkeeper made a 45% profit on cost price.

The Meaning of "Per Cent"

The word "percent" literally means per hundred — it is a fraction with a denominator of 100. So 37% simply means 37 out of every 100 parts, which as a decimal is 0.37 and as a fraction is 37/100. Understanding this connection is the foundation for every percentage calculation you will ever do.

Because percentages are just another way of writing fractions or decimals, every percentage problem can be solved by converting to a decimal first. The phrase "percentage of" always means multiply. So "30% of $200" means 0.30 × 200 = $60. There is no separate rule to memorise — just convert and multiply.

This proportional thinking also explains why percentage increase and decrease use multipliers. A 15% increase means the result is 100% + 15% = 115% of the original, and 115% as a decimal is 1.15. So a single multiplication by 1.15 gives you the new value in one step — faster and more reliable than calculating 15% separately and adding.

The Percentage Change Formula — and Why We Divide by the OLD Value

The formula for percentage change is:

% change = (new − old) ÷ old × 100

The key question students often ask is: why do we divide by the old value, not the new one? The reason is that percentage change is always measured relative to where you started. If a price rises from $80 to $100, the change is $20. We want to know what fraction of the starting price ($80) that change represents, so we divide by 80. The result, 25%, means the price increased by 25% relative to its starting value.

If you accidentally divided by the new value (100), you would get 20%, which is a different ratio — it tells you what fraction of the final price the change is, not how much the price changed from the start. This distinction matters in finance, retail, and data analysis.

GST and Tax Calculations in Australia

Australia's Goods and Services Tax (GST) is a flat 10% added to most goods and services. Because businesses are required to collect GST on behalf of the government, understanding these calculations is a practical life skill as well as an exam requirement.

Adding GST: Multiply the pre-GST price by 1.1 (i.e. 100% + 10%). A $180 item becomes $180 × 1.1 = $198 including GST.

Removing GST (finding the pre-GST price): Divide the GST-inclusive price by 1.1. So if a receipt shows $198 total, the pre-GST price is $198 ÷ 1.1 = $180, and the GST component is $198 − $180 = $18.

This is where a very common error occurs — some students calculate 10% of the GST-inclusive price and subtract it, thinking that gives the pre-GST price. For example: $198 × 0.10 = $19.80, and $198 − $19.80 = $178.20. That is wrong. The correct method is always to divide by 1.1. Businesses, accountants, and the ATO all use divide-by-1.1 because it is mathematically exact.

Common Mistake — Removing GST by Multiplying by 0.1: To find the pre-GST price from an inclusive price, you must divide by 1.1, not subtract 10%. Subtracting 10% of the inclusive price gives the wrong answer because GST is 10% of the pre-GST price, not 10% of the final price. For example: pre-GST price of $120 + GST of $12 = $132 inclusive. Dividing $132 by 1.1 = $120. But $132 × 0.1 = $13.20, and $132 − $13.20 = $118.80 — wrong.

Markup and Discount in Retail

In retail, there are two common percentage adjustments applied to prices — and it is important to know what each is calculated on.

Markup is a percentage increase applied to the cost price (what the business paid). A shop that buys a jacket for $60 and applies a 40% markup sells it for $60 × 1.40 = $84. The markup percentage is always based on cost price.

Discount is a percentage decrease applied to the selling price (what the product is advertised at). A $84 jacket offered at 25% off sells for $84 × 0.75 = $63. The discount is always based on the marked (advertised) price.

The profit percentage is calculated relative to the cost price: profit % = (selling price − cost price) ÷ cost price × 100. This tells you how much the business made for every dollar it spent.

Reverse Percentage — Finding the Original Value

Sometimes you know the result of a percentage change and need to work backwards to the original value. This is called a reverse percentage or back-calculation problem.

The key insight is: after a percentage change, the new value equals the original multiplied by some factor. To reverse the change, divide by that factor.

Example 1: A price has been increased by 20% to give $144. The multiplier applied was 1.20, so the original price = $144 ÷ 1.20 = $120.

Example 2: After a 30% discount, a dress costs $77. The multiplier applied was 0.70, so the original price = $77 ÷ 0.70 = $110.

Example 3 (GST removal): A GST-inclusive price of $44 ÷ 1.10 = $40 pre-GST.

Exam Tip — Successive Percentage Changes Are Not Additive: A 10% increase followed by a 10% decrease does NOT return to the original. The multipliers are 1.10 × 0.90 = 0.99 — a net 1% decrease. Always multiply the chain of multipliers together; never add or subtract percentage changes directly.

Mastery Practice

See Answers ➔
  1. Fluency

    Convert each of the following to a percentage.

    1. (a) 3/5
    2. (b) 7/8
    3. (c) 0.46
    4. (d) 1.025
    5. (e) 11/4
  2. Fluency

    Calculate the following amounts.

    1. (a) 20% of $350
    2. (b) 8% of $1 200
    3. (c) 12.5% of $640
    4. (d) 2.5% of $8 400
  3. Fluency

    Apply a single multiplier to find the new price.

    1. (a) Increase $240 by 15%
    2. (b) Decrease $580 by 25%
    3. (c) Add 10% GST to $95
    4. (d) Reduce $1 600 by 7.5%
  4. Fluency

    Calculate the percentage change in each situation.

    1. (a) Price rises from $50 to $65
    2. (b) Wage falls from $800 to $680
    3. (c) Population grows from 12 000 to 13 500
    4. (d) Score drops from 90 to 72
  5. Understanding

    GST calculations.

    Reminder: GST-inclusive price = pre-GST × 1.1. To find pre-GST price from GST-inclusive price, divide by 1.1.
    1. (a) A pair of shoes has a pre-GST price of $120. Find the GST-inclusive price.
    2. (b) A receipt shows a GST-inclusive total of $231. What was the pre-GST price? How much GST was paid?
    3. (c) A laptop is advertised for $1 298 including GST. A business can claim back the GST. How much does the business pay (excluding GST)?
  6. Understanding

    Discounts and markups.

    Note: Markup is applied to cost price. Discount is applied to the selling/marked price.
    1. (a) A surfboard costs a shop $340 wholesale. They apply a 40% markup. What is the selling price?
    2. (b) A television marked at $890 is discounted by 18%. What is the sale price?
    3. (c) After a 30% discount, a dress costs $84. What was the original marked price?
  7. Understanding

    Profit and loss.

    1. (a) A dealer buys a car for $12 500 and sells it for $15 000. Find the profit and the profit percentage on cost price.
    2. (b) A market stall buys mangoes for $2.40 per kg and sells them for $1.80 per kg after they overripen. Find the loss per kg and the loss percentage.
    3. (c) A house was bought for $420 000 and sold for $378 000. Find the percentage loss.
  8. Understanding

    Successive percentage changes.

    Key idea: Apply multipliers one after the other. A 10% increase then a 10% decrease does NOT return to the original value.
    1. (a) A salary of $56 000 increases by 3%, then increases by 2% the following year. What is the final salary?
    2. (b) A phone was discounted 20%, then a further 15% off the discounted price. What single percentage discount is this equivalent to?
    3. (c) A share price rises 25% then falls 20%. What is the overall percentage change?
  9. Problem Solving

    Back-calculating the original price.

    Challenge. After a percentage change, you are given the final value. Find the original.
    1. (a) After a 35% increase, a price is $675. Find the original price.
    2. (b) After adding 10% GST, a price is $484. Find the pre-GST price.
    3. (c) A store increases a price by 20% to account for markup, then adds 10% GST. The final price is $158.40. What was the original cost price (before markup and GST)?
  10. Problem Solving

    Real-world percentage problem.

    Challenge. A retailer buys a product for $x, applies a 60% markup, then offers a 25% discount in a sale. The sale price (before GST) is $108.
    1. (a) Write an equation for the sale price in terms of x.
    2. (b) Solve to find the cost price x.
    3. (c) What is the final price the customer pays including 10% GST?
    4. (d) Did the retailer make a profit or loss on cost price after the discount? By what percentage?