Mathematics Standard • Year 11 • Module 3 • Lesson 13
Depreciation
Apply depreciation methods to real business and consumer scenarios — match the model to the wording, then calculate and conclude.
Problem 1 — Delivery van comparison (both methods)
A courier company buys a delivery van for $54,000. The accountant considers two depreciation methods to estimate its book value at the end of year 5.
Method A: Straight-line, $7,800 per year.
Method B: Declining balance, 18% per annum.
Set up: What are we solving for?
(i) Calculate the van's value after 5 years under Method A. 1 mark
(ii) Calculate the van's value after 5 years under Method B. Show (1 − r)5 as a separate line. 2 marks
(iii) State which method gives the higher salvage value and by how much. Write a conclusion sentence. 2 marks
Stuck? Revisit lesson § Worked Example 3 — Comparing Both Methods.Problem 2 — Farm tractor and the "minimum value" question
A farm tractor is purchased for $120,000 and depreciates by $9,600 per year using the straight-line method. The owner wants to sell when the book value first drops below $60,000.
Set up: What are we solving for?
(i) Write the inequality S < $60,000 in terms of n. 1 mark
(ii) Solve for n and state the smallest whole number of complete years that satisfies the inequality. 2 marks
(iii) Verify your answer by calculating S at that year and at the year before. 2 marks
Stuck? Revisit lesson § Worked Example 1 (b) — round UP because we need the first year the value is fully below the threshold.Problem 3 — Dental chair (declining balance + total depreciation)
A dental practice buys a new chair for $35,000. It depreciates at 15% per annum declining balance.
Set up: What are we solving for?
(i) Calculate the chair's value after 6 years. 2 marks
(ii) Calculate the total depreciation over the 6 years. 1 mark
(iii) A junior accountant computes "average annual depreciation = $35,000 × 0.15 × 6 = $31,500". Explain in one sentence why this is wrong, then state the correct average annual depreciation. 2 marks
Stuck? Under declining balance, 15% is taken from the CURRENT value each year — not from $35,000 every time. Average annual = total ÷ 6.Problem 4 — Finding the depreciation rate from a trade-in
Maya bought a car for $32,000. After 5 years, the dealer offers her $14,500 as a trade-in. The dealer states this is consistent with declining balance depreciation at a constant annual rate.
Set up: What are we solving for?
(i) Set up the equation $14,500 = $32,000 × (1 − r)5. Divide both sides by $32,000 to isolate the bracket. 1 mark
(ii) Take the 5th root (raise to the power 1/5 = 0.2) to find (1 − r). 2 marks
(iii) Find r as a percentage to 2 decimal places. Then verify by computing $32,000 × (1 − r)5. 2 marks
Stuck? Revisit lesson § Worked Example 4 — keep full calculator precision until the final step.Problem 5 — Two pieces of equipment, different methods
A small printing business buys two items in the same year:
Item 1: A guillotine for $9,500 — straight-line depreciation of $1,100 per year.
Item 2: A digital press for $42,000 — declining balance depreciation at 20% per annum.
Set up: What are we solving for?
(i) Calculate the book value of each item after 4 years. 2 marks
(ii) Calculate the total combined book value of both items after 4 years. 1 mark
(iii) Calculate the total combined depreciation over the 4 years across both items, and state which item lost the larger dollar amount and which lost the larger percentage of its original value. 3 marks
Stuck? Find each item separately first, then combine. "Larger dollar amount" and "larger percentage" can have different answers.How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — Delivery van $54,000 over 5 years
Set up. Calculate salvage value under each method, then compare.
(i) Method A: S = 54,000 − (7,800 × 5) = 54,000 − 39,000 = $15,000.00.
(ii) Method B: (1 − 0.18) = 0.82; (0.82)5 = 0.370711…; S = 54,000 × 0.370711 = $20,018.39.
(iii) Method B gives a higher salvage value by $20,018.39 − $15,000.00 = $5,018.39. Method B (declining balance) gives the higher salvage value by $5,018.39 after 5 years.
Problem 2 — Farm tractor: when does S first fall below $60,000?
Set up. Set up the inequality, solve for n, round up to the next complete year.
(i) 120,000 − 9,600n < 60,000.
(ii) 9,600n > 60,000 ⇒ n > 6.25 ⇒ smallest whole n is n = 7 complete years.
(iii) Check year 6: S = 120,000 − 57,600 = $62,400 (still ≥ $60,000). Check year 7: S = 120,000 − 67,200 = $52,800 (first time < $60,000). ✓
Problem 3 — Dental chair $35,000 at 15% DB over 6 years
Set up. Find S after 6 years, then total depreciation, then critique the bad calculation.
(i) (1 − 0.15) = 0.85; (0.85)6 = 0.377150…; S = 35,000 × 0.377150 = $13,200.24.
(ii) Total depreciation = 35,000 − 13,200.24 = $21,799.76.
(iii) The junior accountant treats 15% as a fixed dollar amount of $35,000 every year and multiplies by 6, which is straight-line logic applied to a declining balance rate. Under DB the dollar amount shrinks each year, so the true total is less than $31,500. Correct average annual = $21,799.76 ÷ 6 = $3,633.29 per year.
Problem 4 — Maya's car: find r from V₀ = $32,000, S = $14,500, n = 5
Set up. Rearrange S = V₀(1 − r)n to isolate r.
(i) (1 − r)5 = 14,500 ÷ 32,000 = 0.453125.
(ii) 1 − r = (0.453125)1/5 = (0.453125)0.2 ≈ 0.852737.
(iii) r = 1 − 0.852737 ≈ 0.147263 = 14.73% per annum (to 2 d.p.). Verify: 32,000 × (0.852737)5 = 32,000 × 0.453125 ≈ $14,500. ✓
Problem 5 — Guillotine + digital press after 4 years
Set up. Calculate each item's value separately, then combine for the total book value and total depreciation.
(i) Guillotine (SL): S = 9,500 − (1,100 × 4) = 9,500 − 4,400 = $5,100.00.
Digital press (DB 20%): (0.80)4 = 0.4096; S = 42,000 × 0.4096 = $17,203.20.
(ii) Combined book value = $5,100.00 + $17,203.20 = $22,303.20.
(iii) Guillotine total dep. = 9,500 − 5,100 = $4,400 (46.3% of original). Press total dep. = 42,000 − 17,203.20 = $24,796.80 (59.0% of original). Combined total depreciation = $29,196.80. The digital press lost the larger dollar amount ($24,796.80 vs $4,400) AND the larger percentage (59.0% vs 46.3%) of its original value.