Mathematics Advanced • Year 12 • Module 7 • Lesson 4
Depreciation
Practise HSC-style writing on depreciation, including a structured extended response.
1. Short-answer questions
1.1 A car worth $35,000 depreciates at 18% p.a. flat rate. Find (a) the annual depreciation D, (b) the book value after 4 years, and (c) the total depreciation over the 4 years. 3 marks Band 3
1.2 Equipment was purchased for $65,000 and has a book value of $18,000 after 5 years under reducing balance. Find the annual depreciation rate r, to two decimal places. 3 marks Band 3-4
1.3 An $80,000 asset is depreciated at 20% per year. (a) Find the book value after 3 years using both flat rate and reducing balance. (b) Recommend, with a 1-line reason, which method best models a technology asset like a fleet of laptops. 4 marks Band 4
2. Extended response
2.1 A small business buys a delivery van for $45,000 at the start of the financial year. The accountant must choose between two depreciation methods, both at 15% per year, for the next four years.
Method A, Flat rate: S = V₀ − rV₀ · n.
Method B, Reducing balance: S = V₀(1 − r)ⁿ.
(a) Find the book value at the end of years 1, 2, 3 and 4 under each method.
(b) Compute the dollar depreciation expense for year 1 alone under each method, and explain why they are equal.
(c) Compute the dollar depreciation expense for year 4 alone under each method, and explain why method B's year-4 expense is smaller.
(d) In 3-4 sentences, assess whether the schedules support preferring method B for tax minimisation. Distinguish the annual timing from the four-year totals, and state what additional information a tax conclusion would require. 8 marks Band 5-6
Explicit marking criteria
Part (a), 2 marks
• 1 mark flat-rate values (38,250 / 31,500 / 24,750 / 18,000).
• 1 mark RB values (38,250 / 32,512.50 / 27,635.63 / 23,490.28). Round to nearest cent.
Part (b), 1 mark
• Year-1 expense = $6,750 in both methods, because RB's first year is r × V₀, the same as flat rate's annual amount.
Part (c), 2 marks
• 1 mark year-4 expense values: flat = $6,750; RB = 15% of the unrounded year-4 opening balance, $27,635.625, which is $4,145.34 to the nearest cent.
• 1 mark explains RB shrinks each year because depreciation is r × current book value, not r × V₀.
Part (d), 3 marks
• 1 mark observes flat rate's total depreciation ($27,000) exceeds RB's ($21,509.72).
• 1 mark compares the annual schedules accurately: year 1 is equal, but flat rate is larger in years 2–4, so RB does not front-load deductions on these figures.
• 1 mark states that tax treatment and the business's circumstances are needed before drawing a tax conclusion; a general timing/value-of-money point may be discussed conditionally.
Your response:
How did this worksheet feel?
What I'll revisit before next class:
1.1, $35,000 at 18% flat for 4 yr (3 marks)
Sample response. (a) D = 0.18 × 35,000 = $6,300 per year. (b) S4 = 35,000 − 6,300 × 4 = $9,800. (c) Total depreciation = 35,000 − 9,800 = $25,200.
Marking notes. 0.5 each for D, S, total; 1.5 marks for explicit working showing the substitution at each step.
1.2, $65,000 → $18,000 in 5 yr RB (3 marks)
Sample response. 18,000 = 65,000(1 − r)⁵. (1 − r)⁵ = 18,000/65,000 = 0.27692. 1 − r = 0.276921/5 = 0.76851. r = 1 − 0.76851 = 0.23149 ≈ 23.15% p.a.
Marking notes. 1 mark, correct equation set-up. 1 mark, correct 5th-root step. 1 mark, final answer to 2 dp as a percentage. Common error: students forget the "1 − r" structure and solve r⁵ = ratio.
1.3, $80,000 at 20% for 3 yr (4 marks)
(a) Sample. Flat: S = 80,000 − 16,000 × 3 = $32,000. RB: S = 80,000(0.80)³ = 80,000 × 0.512 = $40,960.
(b) Sample. Reducing balance better models a fleet of laptops because technology loses the biggest dollar amount in year 1 (rapid obsolescence), exactly the steep early drop that exponential decay produces.
Marking notes. 1 mark each for the two book values, 1 mark for the recommendation, 1 mark for a justification linked to year-1 obsolescence or "steep early drop".
2.1, Extended response (8 marks): sample Band-6 response with annotations
Sample Band-6 response.
(a) Book values at years 1-4.
| Year | Flat rate | Reducing balance |
|---|---|---|
| 1 | $38,250.00 | $38,250.00 |
| 2 | $31,500.00 | $32,512.50 |
| 3 | $24,750.00 | $27,635.63 |
| 4 | $18,000.00 | $23,490.28 |
[2 marks, full table to nearest cent.]
(b) Year-1 depreciation expense. Both methods deduct $6,750 in year 1. Reason: RB year-1 expense = r × V₀ = 0.15 × 45,000 = $6,750, which equals the flat-rate annual amount. The two methods only diverge from year 2 onwards. [1 mark]
(c) Year-4 depreciation expense. Flat: still $6,750 (constant). RB: 0.15 × $27,635.625 = $4,145.34375, which rounds to $4,145.34. (This uses the unrounded opening balance; subtracting the two displayed, cent-rounded balances gives a one-cent rounding difference.) Reason: RB calculates depreciation as 15% of the current book value, which has shrunk every year, so the dollar amount of depreciation shrinks too. [2 marks, value + reason.]
(d) Assessing the tax claim. Flat rate gives $6,750 of depreciation in each of years 1–4, while reducing balance gives $6,750, $5,737.50, $4,876.88 and $4,145.34. The first-year amounts are equal and flat rate is larger in each later year; its four-year total is $27,000 compared with $21,509.72 for reducing balance. These schedules do not support saying that method B front-loads deductions or is preferred for minimising tax. A tax conclusion would require the applicable tax rules, the business's circumstances and the tax value of deductions in each year; those are not specified by these mathematical models. [3 marks, total and annual comparison + accurate timing conclusion + identifies limits of the model.]
Total: 8/8.
Band descriptors for marker.
Band 3: Table partially correct, year-1 equivalence not noticed, year-4 calculations missing or wrong. ≈ 3-4 marks.
Band 4: Table correct, year-1 expense explained, year-4 expense given but reason vague ("RB decreases"). ≈ 5-6 marks.
Band 5: Table correct, year-1 and year-4 expenses with structural reasoning; tax discussion compares the totals but gives an incomplete account of annual timing or the model's limits. ≈ 6-7 marks.
Band 6: Full table, year-1 equivalence explicitly justified, year-4 contrast with "depreciation on current book value", and a tax discussion that accurately compares annual and total deductions while identifying the information the mathematical model omits. 8/8.