Mathematics • Year 10 • Unit 1 • Lesson 7
Depreciation
Build fluency with the two methods from Lesson 7: straight-line depreciation (constant dollar loss each year) and reducing balance depreciation (constant percentage loss each year, formula Book Value = P(1 − r)ⁿ).
1. I do, fully worked example
Read every line. The lesson uses both methods, we'll do straight-line here, then reducing balance in Section 2.
Problem. A delivery van is purchased for $45,000 with an expected scrap value of $5,000 after 8 years (straight-line depreciation). Find the annual depreciation and the book value after 3 years.
Step 1, Identify what we have.
Original value = $45,000 Scrap value = $5,000 Useful life = 8 years
Reason: straight-line needs three inputs, the starting cost, the end-of-life value, and the years between.
Step 2, Total loss over the asset's life.
Total depreciation = 45,000 − 5,000 = 40,000
Reason: the van drops from $45,000 to $5,000 over its full useful life.
Step 3, Annual depreciation: spread the loss evenly.
Annual depreciation = 40,000 ÷ 8 = 5,000 per year
Reason: that's exactly what "straight-line" means, the same dollar amount every year.
Step 4, Book value after 3 years.
Book value = 45,000 − (5,000 × 3) = 45,000 − 15,000 = 30,000
Reason: subtract three years' worth of annual depreciation from the original value.
Answer: Annual depreciation = $5,000; book value after 3 years = $30,000.
2. We do, fill in the missing steps
Same structure as Section 1, but using reducing balance depreciation: Book Value = P(1 − r)ⁿ. Fill in each blank. 4 marks
Problem. A laptop costs $2,400 and depreciates at 25% p.a. using reducing balance. Find the book value after 2 years and the total depreciation.
Step 1, Identify P, r and n.
P = $ __________ r = ______ (as a decimal) n = ______ years
Step 2, Substitute into Book Value = P(1 − r)ⁿ:
Book Value = 2,400 × (1 − ______)____ = 2,400 × ( ______ )____
Step 3, Evaluate the bracket and power:
(0.75)² = ______________
Step 4, Compute the book value, then the total depreciation:
Book Value = $ ______________
Total depreciation = 2,400 − ______________ = $ ______________
3. You do, independent practice
Show your working. The first four are foundation (straight-line). The middle two are standard (reducing balance). The last two are extension (mixed/compare).
Foundation, straight-line
3.1 A machine costs $25,000 and has a scrap value of $5,000 after 5 years. Find the annual straight-line depreciation. 1 mark
3.2 Office furniture costs $12,000 and depreciates by $1,500 per year (straight-line). Find the book value after 4 years. 1 mark
3.3 A photocopier is bought for $8,000 with a scrap value of zero after 10 years (straight-line). Find the book value after 6 years. 2 marks
3.4 A printer is purchased for $1,800 and depreciates by $300 per year (straight-line). After how many whole years does the book value first drop below $500? 2 marks
Standard, reducing balance
3.5 A laptop costs $2,000 and depreciates at 20% p.a. reducing balance. Find the book value after 3 years. 2 marks
3.6 A 4WD costs $55,000 and depreciates at 18% p.a. reducing balance. Find the book value after 5 years and the total depreciation over the 5 years. 3 marks
Extension, push your thinking
3.7 A car bought for $30,000 depreciates over 5 years. Compare: (a) straight-line with scrap value $10,000; (b) reducing balance at 20% p.a. Which method gives the lower book value after 3 years, and by how much? 3 marks
3.8 A tablet's book value drops from $1,200 to $614.40 in 2 years using reducing balance. Find the annual depreciation rate r. 2 marks
Section 2, We do (laptop, reducing balance)
Step 1: P = $2,400; r = 0.25; n = 2.
Step 2: Book Value = 2,400 × (1 − 0.25)2 = 2,400 × (0.75)2.
Step 3: (0.75)² = 0.5625.
Step 4: Book Value = 2,400 × 0.5625 = $1,350. Total depreciation = 2,400 − 1,350 = $1,050.
3.1, Machine $25,000 → $5,000 over 5 years
Annual depreciation = (25,000 − 5,000) ÷ 5 = 20,000 ÷ 5 = $4,000 per year.
3.2, Furniture $12,000, $1,500/year for 4 years
Book Value = 12,000 − (1,500 × 4) = 12,000 − 6,000 = $6,000.
3.3, Photocopier $8,000 → $0 over 10 years, after 6 years
Annual depreciation = 8,000 ÷ 10 = $800. After 6 years: 8,000 − (800 × 6) = 8,000 − 4,800 = $3,200.
3.4, Printer $1,800, $300/year, below $500
Book Value = 1,800 − 300n. Need 1,800 − 300n < 500, so 300n > 1,300, n > 4.33. Smallest whole year: n = 5 years. (At n = 5, book value = $300, which is below $500.)
3.5, Laptop $2,000 at 20% reducing balance, 3 years
Book Value = 2,000 × (1 − 0.20)³ = 2,000 × (0.80)³ = 2,000 × 0.512 = $1,024.
3.6-4WD $55,000 at 18% reducing balance, 5 years
Book Value = 55,000 × (0.82)⁵ = 55,000 × 0.370738 = $20,390.62.
Total depreciation = 55,000 − 20,390.62 = $34,609.38.
3.7, Compare methods (car $30,000, 3 years)
(a) Straight-line: annual = (30,000 − 10,000) ÷ 5 = $4,000. After 3 years: 30,000 − 12,000 = $18,000.
(b) Reducing balance 20%: 30,000 × (0.80)³ = 30,000 × 0.512 = $15,360.
Reducing balance gives the lower book value, by 18,000 − 15,360 = $2,640. This matches the lesson's claim that reducing balance is more realistic for cars (faster early loss).
3.8, Tablet $1,200 → $614.40 in 2 years
Set up: 1,200 × (1 − r)² = 614.40.
(1 − r)² = 614.40 ÷ 1,200 = 0.512.
1 − r = √0.512 ≈ 0.71554.
r = 1 − 0.71554 ≈ 0.2845 ≈ 28.4% p.a.
Check: 1,200 × (0.71554)² = 1,200 × 0.512 = $614.40 ✓
A 28.4% annual depreciation rate is realistic for a fast-depreciating tablet.