Percentages
Percentage increase and decrease, and real-life problems: profit and loss, discount, growth and depreciation, simple and compound interest, and salaries tax. Most of them are done by multiplying by one factor, and a money question appears in most DSE papers.
What you will learn
- 5.1 Percentage change: percentage increase and decrease.
- 5.2 Real-life problems:
You need first: Number and algebra basics
Key terms
| English | 中文 |
|---|---|
| percentage change | 百分變化 |
| cost price, selling price, marked price | 成本，售價，標價 |
| profit, loss, profit per cent | 盈利，虧蝕，盈利百分率 |
| discount | 折扣 |
| simple interest, compound interest | 單利息，複利息 |
| principal, amount | 本金，本利和 |
| interest rate per annum | 年利率 |
| depreciation | 折舊 |
| successive changes | 連續變化 |
| salaries tax | 薪俸稅 |
Standard steps
The one idea: multiply by a factor
| Change | Multiply by |
|---|---|
| increase by r% | (1 + r%) |
| decrease by r% | (1 − r%) |
| two changes in a row | multiply both factors |
Why: "increase by 20%" means "keep 100% and add 20%", which is 120% = 1.2.
Profit and loss
- Profit = selling price − cost price.
- Profit per cent = profit ÷ cost price × 100%.
- Discount per cent = discount ÷ marked price × 100%.
- Selling price = marked price × (1 − discount%).
Interest
- Simple interest: I = P × r × n.
- Compound interest: A = P(1 + r/k)^(nk), where k is the number of times per year interest is added (k = 4 for quarterly, 12 for monthly).
- Interest = A − P.
Growth and depreciation
New value = original × (1 ± r%)ⁿ.
Find the original value
Divide, don't subtract. If the price after a 20% increase is $600, the original is 600 ÷ 1.2 = $500.
Worked examples
marked price = 500 × 1.4 = $700
selling price = 700 × 0.8 = $560
profit = 560 − 500 = $60
profit % = 60 ÷ 500 × 100% = 12%A = 20 000 × (1 + 3%/4)^(2×4)
= 20 000 × 1.0075⁸
≈ $21 231.98
interest ≈ 21 231.98 − 20 000 ≈ $1 2321.2 × 0.75 = 0.9, so the price falls by 10%, not 5%.
Common mistakes
- Using the selling price instead of the cost price in profit per cent.
- Adding successive percentages (+20% then −25% is not −5%).
- Undoing an increase by subtracting the same percentage (600 − 20% of 600 is not the original).
- Using the annual rate with quarterly periods: the rate must be divided by 4 and the number of periods multiplied by 4.
- Rounding in the middle of a compound interest calculation.
How the DSE tests it
- A(1): a 4–5 mark question with two or three linked parts (cost → marked → selling price → profit or loss).
- Paper 2: compound interest, depreciation, successive changes, and "find the original" questions.
- Senior links: geometric sequences use the same factor idea (F5 sequences), and salaries tax returns in F6 further applications.
Questions you might not ask in class
Why isn't +20% then −20% back to the start?
The second 20% is taken from a bigger number. 100 → 120 → 96.
Profit per cent: over cost or over selling price?
Over cost, unless the question says otherwise. Think: "how much did my money grow?"
What does 'compounded quarterly' actually mean?
The bank adds interest 4 times a year. Each time, it adds ¼ of the yearly rate, and the next interest is worked out on the new total.