3.3 Percentages & Financial Maths

Percentage questions appear in nearly every ICAS paper and they grow with you: percentage and ratio problems feature in Papers E-F, while Papers I-J add simple and compound interest. The unifying idea is simple — a percentage is just a fraction out of 100 — but marks are lost on the base of the percentage, so this section drills both the conversions and the traps.

Per Cent Means Per Hundred

45% means 45 out of 100, so 45% = 45/100 = 0.45. The three conversions to have at your fingertips:

• Percentage to decimal: divide by 100 (shift the point two places left). 7% = 0.07 — a classic ICAS trap is writing 0.7.
• Decimal to percentage: multiply by 100. 0.075 = 7.5%.
• Fraction to percentage: scale to a denominator of 100 or go via the decimal. 3/5 = 60/100 = 60%.

Percentage of a Quantity

Two methods, both worth knowing. The decimal method: 35% of 240 = 0.35 × 240 = 84. The mental 10%-chunk method: 10% of 240 = 24, so 30% = 3 × 24 = 72 and 5% = 24 ÷ 2 = 12, totalling 72 + 12 = 84. Without a personal calculator, the chunk method is faster and less error-prone. For awkward percentages go via 1%: 4% of 350 means 1% = 3.5, so 4% = 14.

Percentage Increase and Decrease

Multiply by (1 + rate) to increase and by (1 − rate) to decrease. A $60 game rises 15%: 60 × 1.15 = $69. An $80 jacket falls 25%: 80 × 0.75 = $60. Key trap: a 10% rise followed by a 10% fall does not return to the starting value — 100 → 110 → 99, because the second percentage acts on a different base.

Discounts and GST

A discount is a percentage decrease: 30% off $45 gives 45 × 0.70 = $31.50. Read the question carefully — if it asks for the sale price, subtracting is essential; if it asks how much is saved, the answer is the 30% itself ($13.50).

Australia's GST is 10%. Adding GST: multiply by 1.10. Removing GST from an inclusive price: divide by 1.10, never subtract 10%. A $110 item includes $10 of GST, because 110 ÷ 1.10 = 100; subtracting 10% of $110 would wrongly give $11.

Profit and Loss

Profit = selling price − cost price, and percentage profit is measured on the cost: buy at $40, sell at $50, profit $10, so percentage profit = 10/40 = 25%. Dividing by the selling price instead (giving 20%) is the classic wrong answer ICAS includes among the options. Percentage loss works identically on the cost price.

One Quantity as a Percentage of Another

Use part ÷ whole × 100%. Scoring 34 out of 40 gives 34/40 = 0.85 = 85%. Watch the units: to find what percentage 45 centimetres is of 2 metres, convert first — 45/200 = 22.5%.

Reverse Percentages: Finding the Original

ICAS also asks you to work backwards from the changed value. After a 20% increase, a membership costs $72. What was the original price? The new price is 120% of the original, so divide: 72 ÷ 1.20 = $60. The mental route works too: 120% = $72, so 10% = $6 and 100% = $60. The trap is taking 20% off $72 (giving $57.60) — the 20% was measured on the original, not on $72, so you must divide rather than subtract a percentage.

Simple Interest (Papers I-J)

Simple interest pays the same amount every year: I = P × r × t, where P is the principal, r the annual rate as a decimal and t the time in years. $2,000 at 5% p.a. for 3 years earns 2000 × 0.05 × 3 = $300, for a total of $2,300.

Compound Interest (Papers I-J)

With compound interest, each year's interest is added to the balance before the next year is calculated: A = P(1 + r)^t. Working year by year for $2,000 at 5% p.a.:

• After year 1: 5% of 2000 = 100, so the balance is $2,100
• After year 2: 5% of 2100 = 105, so the balance is $2,205
• After year 3: 5% of 2205 = 110.25, so the balance is $2,315.25

Compare the two over three years:

Compounding earns $15.25 more because interest itself earns interest. Without a calculator, work year by year using 10% halved for 5%; and remember compound interest always beats simple interest after the first year, which lets you eliminate options instantly.

Common Traps

• Writing 5% as 0.5 instead of 0.05.
• Finding 30% of the price when the question asks for the discounted price (subtract!).
• Measuring percentage change or profit against the wrong base.
• Treating compound interest as simple interest, or adding the rate instead of compounding it.