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VCE Foundation Mathematics (Year 12) · Number and Arithmetic · 10m read

Percentages and Rates

Ever wondered why that "50% off" sale price doesn't actually save you half the money once GST is added back in? Percentages and rates are the maths behind price tags, discounts, and how quantities change — skills you'll use every time you shop or check your payslip. Mastering percentage increases and decreases, plus understanding GST, unlocks the practical maths VCE examiners test constantly.

Learning objective: Calculate percentage increases and decreases, add and remove GST, and interpret rates in everyday contexts.

By the end of this lesson you can:

  • I can multiply by 1.1 to add GST and divide by 1.1 to remove it.
  • I can find percentage change using change ÷ original × 100.
  • I apply discounts and GST in the correct order.

Percentages are everyday shopping maths. Discounts, GST, tips and pay rises all use percentages. Master them and you'll never get ripped off at the checkout.

What are percentages and rates

A percentage is a fraction with a denominator of 100 — "how much out of every hundred". A rate compares two different quantities like km/h or $/h.

The Australian GST is 10%. To add GST to a price, multiply by 1.10. To remove GST from a GST-inclusive price, divide by 1.10 (which is the same as dividing the price by 11 to find the GST portion). Get these two moves automatic and every financial word problem gets faster.

Percentage and rate moves

  • % of an amount: Amount × percentage ÷ 100.
  • % increase: Original × (1 + rate). +25% → ×1.25.
  • % decrease: Original × (1 − rate). −30% → ×0.70.
  • Add GST: Price × 1.10.
  • Remove GST: Price ÷ 1.10.

Percentage and GST formulas

  • Percent of an amount: A \times \dfrac{p}{100}
  • Percentage change: \dfrac{\text{new} - \text{old}}{\text{old}} \times 100\%
  • Add GST: \text{ex-GST} \times 1.10 = \text{GST inclusive}
  • GST inside an inclusive price: \text{inclusive} \div 11

Quick reference

| Calculation | Example |
|---|---|
| 10% of $45 | $4.50 |
| 25% off $80 | $60 |
| Add 10% GST to $50 | $55 |
| Rate | $18/h × 7.5 h = $135 |
| Unit price | $3.60 ÷ 1.2 kg = $3/kg |

Worked example (Medium)

A phone originally costs $850. It is on sale for 20% off. After the discount, 10% GST is added. What is the final price?

  1. Discount amount. Find 20% of the original price. $850 × 0.20 = $170
  2. Discounted price. Subtract the discount. $850 − $170 = $680
  3. Add GST. Multiply by 1.10. $680 × 1.10 = $748

Answer: The final price is $748.

Practice questions

  1. A worker earns $22.40 per hour. What are their earnings for a 7.5 hour shift?

    1. $168
    2. $157.80
    3. $179.20
    4. $164.50

    (2 marks)

  2. A TV is reduced from $960 to $720. What is the percentage discount?

    1. 25%
    2. 33%
    3. 20%
    4. 30%

    (2 marks)

  3. A restaurant bill is $180 before GST. The group adds a 10% tip on the GST-inclusive price. What is the total paid?

    1. $217.80
    2. $214.20
    3. $198.00
    4. $207.90

    (3 marks)

  4. Maya buys a laptop priced at $1,200 (before GST). The retailer gives a $150 trade-in off the pre-GST price, then GST is added on the discounted amount. How much does Maya pay?

    (3 marks)

  5. A $220 GST-inclusive invoice is issued. What is the GST component, and what is the GST-exclusive (ex-GST) price?

    1. GST $20, ex-GST $200
    2. GST $22, ex-GST $198
    3. GST $20, ex-GST $220
    4. GST $44, ex-GST $176

    (2 marks)

  6. A $45/h award rate increases by 4%. What is the new hourly rate?

    1. $46.80
    2. $49.00
    3. $45.40
    4. $48.60

    (2 marks)

Worked example (Hard)

A jacket is marked at $240 (GST inclusive). It is reduced by 15% in a sale, and a $20 voucher is then applied. How much does the customer pay, and what percentage have they saved compared with the original $240?

  1. Discount amount. Find 15% of the marked price. $240 × 0.15 = $36
  2. Sale price. Subtract discount. $240 − $36 = $204
  3. After voucher. Subtract the voucher. $204 − $20 = $184
  4. Total saved. Original − final paid. $240 − $184 = $56
  5. Saving %. Saving ÷ original × 100. $56 ÷ $240 × 100 ≈ 23.3%

Answer: Customer pays $184 and saves about 23.3% on the original price.

Common mistakes

  • Multiplying by 0.10 to add GST instead of by 1.10 Multiplying by 0.10 gives only the GST itself (10%), not the GST-inclusive total (110%). Fix: To ADD GST: × 1.10. To FIND the GST inside an inclusive price: ÷ 11.
  • Stacking discount and GST in the wrong order Adding GST first then discounting gives a different answer to discounting then adding GST. Fix: Read the question carefully — usually discount applies first, then GST.
  • 20% off followed by 20% on returns to the original It does not. $100 → $80 → $96. You lose 4% overall. Fix: Successive percentages multiply: 0.80 × 1.20 = 0.96, a net 4% decrease.

Going deeper — percentage change vs percentage points

If unemployment goes from 4% to 6%, that's a rise of 2 *percentage points*, but a 50% *percentage change* (because (6 − 4)/4 = 0.5).

VCAA (and the news) frequently mix these up. Use the formula: percentage change = (new − old) ÷ old × 100. Use percentage points when subtracting two rates that are themselves percentages.

Best-buys also lean on this: a "30% extra free" pack is cheaper per gram than a "20% off" pack of the same listed price, even though 30 looks bigger because the comparison bases differ.

Summary

  • Percentages convert to decimals: 25% → 0.25, 4% → 0.04.
  • Add GST: × 1.10. Remove GST from an inclusive price: ÷ 1.10 (or "÷ 11" gives the GST itself).
  • Apply discounts BEFORE GST unless the question states otherwise.
  • Percentage change = (new − old) ÷ old × 100.

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