Income Tax and Salaries
Income tax questions in IGCSE Maths are percentage problems in disguise, and this page shows you exactly how to unpick them. You will learn how to calculate wages, salaries and overtime pay, then how a personal allowance and tax bands decide how much of a salary actually gets taxed. Work through the method and worked examples, then test yourself on the auto-marked practice questions below; every set is randomly generated, so you can keep practising until the steps are automatic.
Wages, Overtime and Salaries
A wage is pay worked out from an hourly rate: the more hours you work, the more you earn. A salary is a fixed amount for the year, usually stated per annum (p.a.) and paid in equal monthly instalments.
To work out a week's pay that includes overtime:
- Multiply the basic hours by the basic hourly rate.
- Find the overtime rate: time-and-a-half means \(1.5 \times\) the basic rate; double time means \(2 \times\) the basic rate.
- Multiply the overtime hours by the overtime rate.
- Add the basic pay and the overtime pay.
\(\text{annual salary} = 12 \times \text{monthly pay}\)
\(\text{annual pay} = 52 \times \text{weekly pay}\)
Read carefully which period the question gives you and which it asks for; use the fact above in whichever direction the question needs.
How a week's pay with overtime builds up
The overtime slice is short in hours but each of those hours pays 1.5 times as much.
How Income Tax Works: the Personal Allowance
Income tax is charged on money you earn, but almost every tax system lets you earn a certain amount tax-free first. That tax-free amount is the personal allowance. Only the income above the allowance, called the taxable income, gets taxed.
\(\text{taxable income} = \text{total income} - \text{personal allowance}\)
\(\text{income tax} = \text{taxable income} \times \dfrac{\text{rate}}{100}\)
So in a country with a personal allowance of 14,000 dalers and a tax rate of 22%, someone earning 14,000 dalers or less pays no income tax at all, and someone earning more pays 22% of only the part above 14,000.
Which part of a 20,000 daler income is actually taxed
The grey slice is never taxed, however much you earn. Only the blue slice is, which is why the tax is far less than 22% of the whole income.
Banded Income Tax
Most governments believe higher earners should pay a larger share, so taxable income is cut into bands, and each band has its own rate. Think of the salary as a bar: the first slice is tax-free, the next slice is taxed at the basic rate, and anything above that is taxed at a higher rate. Each slice is taxed separately; moving into a higher band never changes the tax on the earlier slices.
A 52,000 crown salary cut into tax bands (Norvette's system)
The slice method, step by step:
- Subtract the personal allowance to find the taxable income.
- Fill the first band: tax as much of the taxable income as fits in band 1, at the band 1 rate.
- Tax whatever is left over at the next rate (and keep going if there is a third band).
- Add the slices to get the total bill. Divide by 12 if the question asks for a monthly bill.
Core Ideas
The first slice of income, on which no tax is paid at all.
Total income minus the allowance. Tax rates only ever apply to this part.
A slice of taxable income with its own rate. Each slice is taxed separately, then the slices are added.
Time-and-a-half pays \(1.5 \times\) the basic hourly rate; double time pays \(2 \times\) it.
Worked Examples
💡 Example 1: A Week with Overtime
A warehouse assistant is paid £11.20 an hour for a basic 35-hour week. Overtime is paid at time-and-a-half. One week he works 41 hours. Work out his total pay for that week.
Overtime hours: \(41 - 35 = 6\)
\[ \begin{array}{rcl} \text{basic pay} &=& 35 \times 11.20 \\ &=& 392.00 \\ \text{overtime rate} &=& 1.5 \times 11.20 \\ &=& 16.80 \\ \text{overtime pay} &=& 6 \times 16.80 \\ &=& 100.80 \\ \text{total pay} &=& 392.00 + 100.80 \\ &=& 492.80 \end{array} \]His total pay is £492.80.
What's happening?
Only 6 of the 41 hours are overtime, so split the week before touching the money.
Time-and-a-half multiplies the rate, not the hours: each overtime hour is worth £16.80.
Finish by adding the two parts of the pay.
💡 Example 2: Tax with an Allowance
In Avaria, the personal allowance is 14,000 dalers and all income above it is taxed at 22%. Mira earns 39,200 dalers a year. Work out her monthly income tax bill.
Mira pays 462 dalers a month.
What's happening?
The first 14,000 dalers are free of tax, so subtract the allowance before using the rate.
22% as a multiplier is 0.22.
The question asks for a monthly bill, so divide the annual tax by 12 at the end.
💡 Example 3: Banded Tax
Norvette's tax system: the first 12,000 crowns are tax-free, the next 24,000 crowns are taxed at 18%, and everything above that is taxed at 35%. Kaia earns 52,000 crowns a year. Work out her annual income tax bill.
Taxable income: \(52\,000 - 12\,000 = 40\,000\)
\[ \begin{array}{rcl} \text{band 1 tax} &=& 24\,000 \times 0.18 \\ &=& 4\,320 \\ \text{band 2 slice} &=& 40\,000 - 24\,000 \\ &=& 16\,000 \\ \text{band 2 tax} &=& 16\,000 \times 0.35 \\ &=& 5\,600 \\ \text{total tax} &=& 4\,320 + 5\,600 \\ &=& 9\,920 \end{array} \]Kaia pays 9,920 crowns a year. The bar diagram above shows exactly this calculation.
What's happening?
Band 1 holds 24,000 crowns at most, and Kaia's taxable 40,000 more than fills it.
Only the 16,000 that spills over is taxed at the higher 35% rate.
The classic error is 40,000 at 35%. The bands protect the lower slices.
💡 Example 4: Working Backwards
In Belmara, the personal allowance is 15,000 florins and income above it is taxed at a single rate of 25%. Zia paid 4,850 florins of income tax last year. Work out Zia's salary.
Zia's salary is 34,400 florins.
What's happening?
The tax is 25% of the taxable income, so dividing the tax by 0.25 undoes the multiplication.
That recovers the taxable income, not the salary: add the allowance back on as the final step.
Check: \(34\,400 - 15\,000 = 19\,400\), and 25% of that is 4,850. It works.
🔑 Key Points
- Subtract the personal allowance first; tax rates only ever touch the taxable income.
- Banded tax works slice by slice: fill band 1, tax the spill-over at the next rate, then add.
- Time-and-a-half and double time multiply the hourly rate (by 1.5 or 2).
- Annual, monthly, weekly: 12 months or 52 weeks in a year, and always check which one the question wants.
- To work backwards from a tax bill, divide by the rate (as a decimal) to recover the taxable income, then add the allowance back.
⚠️ Common Pitfalls
- Taxing the whole salary at the top rate. Only the slice inside each band gets that band's rate.
- Forgetting to subtract the allowance before multiplying by the rate.
- Answering with the annual figure when the question asked for a monthly one, or the other way round.
- Treating time-and-a-half as extra hours instead of a higher rate per hour.
- Working backwards and forgetting to add the allowance back on: dividing by the rate only gives the taxable income.