Unit 9
9 marks · Normal(a) Rafik earns $135,000 per annum in the country of Kalculus (tax bands: 20% on the first $50,000, 40% on the rest).
Calculate Rafik's annual income tax bill.
The equation of a straight line is given by . Rafik's income after tax, as a function of his pre-tax salary , is linear above $50,000.
Write down an equation for the income after tax for all , using the figures from part (a).
On a set of axes, sketch the graph of against for using your answer from (b). Clearly indicate the point at which the tax band changes.
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) $38,000 dollars · must show Working for tax in both bands
(b) y = 0.6x + 20000 · must show Equation correct for x > 50000
(c) A graph with slope 0.8 for 0 ≤ x ≤ 50000, slope 0.6 for x > 50000, join at x = 50000 · must show two linear regions, kink at x=50,000
Common mistakes: Forgetting to split tax calculation into two bands; Writing a one-piece equation for income after tax
Show mark scheme and worked solution
Marking points
M1 (a) · 1 mark · Method to split salary into bands and apply correct tax rates
A1 (a) · 2 marks · Accurate tax for both bands, correct final answer (follow-through)
M2 (b) · 1 mark · Identifying that after-tax income for x > 50000 is linear with gradient 0.6
A2 (b) · 2 marks · Correct equation derived, using tax rates and band point correctly (follow-through)
M3 (c) · 1 mark · Sketch with correct gradients for both regions and a clear kink at x = 50000
A3 (c) · 2 marks · Axes labelled, points (0,0), (50000,40000), (150000,110000) located, shape matches formulas (follow-through)
Worked solution
(a) Tax on first $50,000 = $50,000 × 0.20 = $10,000
Amount over $50,000: $135,000 − $50,000 = $85,000
Tax on excess = $85,000 × 0.40 = $34,000
Total tax = $10,000 + $34,000 = $44,000
Annual after-tax income = $135,000 − $44,000 = $91,000
(b) Let x be salary. Up to $50,000: tax is 20%, after-tax = 0.8x. Above $50,000: y = 0.8×50,000 + 0.6(x−50,000) = 40,000 + 0.6x − 30,000 = 0.6x + 10,000
But y must be continuous at x = 50,000, so constant term is 40,000 (answer: y = 0.6x + 10,000 for x > 50,000).
(c) Graph: Line from (0,0) to (50,000,40,000), then bend: from (50,000,40,000), line gradient 0.6, up to (135,000,91,000). Kink at x = 50,000. Axes labelled.