Unit 9
10 marks · NormalIn a country, Sergio earns an annual salary plus a bonus. The bonus is taxed at a higher rate than the salary. The total income tax Sergio pays is £8,600 in one year.
Sergio receives a salary of £30,000 per year. He also receives a bonus of £12,000 in that year. The first £30,000 of his income is taxed at 20%, and any amount above £30,000 is taxed at 40%. Work out how much income tax Sergio should pay in total for the year.
Let x be the salary (in £) and y be the bonus (in £) that produces exactly £8,600 tax for this tax system, given that x + y = £41,500. Using the rules above and your answer from part (a), show how to set up and solve a pair of simultaneous equations for x and y, and find the values of x and y.
Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.
Show answer
(a) £7,200 £
(b) x = £32,500, y = £9,000 · must show Setting up equations: x+y=41500 and 0.2x+0.4y=8600; Solving by substitution or elimination
Common mistakes: Applying the wrong tax rates above/below the threshold in part a; Using 'y' incorrectly in the system of equations in part b
Show mark scheme and worked solution
Marking points
M1 (a) · 1 mark · Correctly calculates tax on salary at 20% up to £30,000.
M2 (a) · 1 mark · Correctly calculates tax on the bonus or amount above £30,000 at 40%.
A1 (a) · 2 marks · Adds both tax amounts for correct total. Award 2 marks for fully correct answer with correct method, 1 for minor numerical error if working seen.
M3 (b) · 1 mark · Forms and clearly writes the first equation: x + y = 41500
M4 (b) · 1 mark · Forms and writes the tax equation: 0.2x + 0.4y = 8600
M5 (b) · 1 mark · Uses an algebraic method (substitution/elimination) for the system. (follow-through)
A2 (b) · 2 marks · Obtains both correct answers for x and y (accept to nearest £100); 1 mark for one correct, 2 if both correct and correctly justified. (follow-through)
B1 (b) · 1 mark · Clear pair given with units (£). (follow-through)
Worked solution
(a)
Salary taxed at 20%: £30,000 x 0.2 = £6,000
Bonus remains: £12,000
This is taxed at 40%: £12,000 x 0.4 = £4,800
Total income tax: £6,000 + £4,800 = £10,800
BUT original stem says 'total tax paid was £8,600'. So use this result in (b).
(b)
Let x = salary in £ (taxed first £30,000 at 20%), y = bonus (taxed amount over £30,000 at 40%).
Given: x + y = 41,500
Income tax: tax = 0.2x + 0.4y = £8,600
Substitute x = 41,500 - y into 0.2x + 0.4y = 8,600:
Now, x = 41,500 - 1,500 = 40,000
Check: 0.2 x 40,000 = 8,000; 0.4 x 1,500 = 600; Total = 8,600
So Sergio's salary is £40,000 and his bonus is £1,500.