Further applications
Longer real-life and mathematical problems that combine different topics, such as taxation, instalments and data from investigations. There is no new content; the unit is about choosing a strategy and linking ideas.
What you will learn
- Solve multi-step problems that need two or more units together.
- Read a real-life context and decide which maths applies.
- Simple financial problems: salaries tax (with the rates given), instalment payments, interest.
- Analyse and interpret data.
You need first: Percentages, Arithmetic and geometric sequences, Measures of dispersion
Key terms
| English | 中文 |
|---|---|
| salaries tax | 薪俸稅 |
| net chargeable income | 應課稅入息實額 |
| allowance, deduction | 免稅額，扣除 |
| tax rate (progressive rates) | 稅率（累進稅率） |
| instalment | 分期付款 |
| down payment | 首期 |
| model, assumption | 模型，假設 |
Standard steps
Any long problem: the four steps
- Understand: underline the numbers and the question. Draw a picture or table.
- Choose: which unit does this look like? List two or three possible methods.
- Do: carry it out step by step; write each step clearly.
- Check: is the answer sensible (units, size, sign)? Does it answer the question that was asked?
Salaries tax (rates are always given)
- Net chargeable income = income − deductions − allowances.
- Apply the given rates band by band (progressive rates).
- Compare with the standard rate if the question says to; the tax is the lower one.
Instalments
Total paid = down payment + (number of instalments × amount). Extra cost = total paid − cash price. If interest is involved, use the compound interest or GS sum formula.
Worked examples
The question gives: first $50 000 at 2%, next $50 000 at 6%, next $50 000 at 10%, the rest at 14%. Net chargeable income is $180 000.
50 000 × 2% = 1 000
50 000 × 6% = 3 000
50 000 × 10% = 5 000
30 000 × 14% = 4 200
tax = $13 200Total = 24 × 560 = $13 440. Paying by instalments costs $1 440 more.
Common mistakes
- Applying the top tax rate to the whole income instead of band by band.
- Forgetting allowances before applying the rates.
- Answering a different question from the one asked.
- Giving an answer without units.
How the DSE tests it
- Long questions in A(2) and Section B often combine units in a real context.
- Financial contexts appear in Paper 1 and Paper 2; tax rates are given in the question.
Questions you might not ask in class
I don't know where to start on long questions.
Start with what you can find from the given information, even if it's not what's asked. Each fact you find is often a step towards the answer, and partial working still earns method marks.
Do I need to memorise the tax rates?
No. They are given in the question.