Question 4 (c)
Rate of depreciation from cost, scrap value and life
6 marks
- Time to write
- About 12 minutes
Question
Question 4 (c) · 6 marksA machine costing ₹6400 will depreciate to a scrap value of ₹2302 in 8 years. Find the rate of depreciation.
Reworded from the university paper: names and figures differ, the task is the same. The original paper belongs to the University of Delhi. See DU's question bank.
Before you write
Read off the question and its marks
Read from the question
- FindFormula first, then the working, then the answer with its unit (₹, %, units).
What the paper leaves open
Part of "Solve any three"; 6 marks.
Write it in this order
- Read the data; note exactly what is asked 1 min
- 1 Depreciation relation 2 marks 3 min
- 2 Logarithmic solution 3 marks 5 min
- 3 Rate with check 1 mark 2 min
- Check: arithmetic, units (₹, %), final answer underlined 1 min
Short on time? Step 2 carries 3 of the 6 marks: guard its minutes.
Try it first. About 12 minutes for 6 marks. Write it out, then open the solution.
Solution
Hover or tap a marked figure, like 72,000, to see how it is made and where else it appears.
On the reducing balance, depreciation each year is a fixed rate of the value at the beginning of that year. The compound amount after n periods is Aₙ = P(1 + i)ⁿ. A value that falls by rate r each year is, by our reasoning, that relation with i = −r, so it leaves a value C(1 − r)ⁿ after n years.
6,400(1 − r)⁸ = 2,302 ⟹ (1 − r)⁸ = 2,302/6,400 = 0.3596875.
Taking logarithms: 8 log(1 − r) = log 0.3596875 = −0.444075 ⟹ log(1 − r) = −0.055509 ⟹ 1 − r = 0.880016.
r = 0.119984, i.e. the rate of depreciation is 12% a year (to two decimals, 12.00%).
Check: 6,400 × (0.88)⁸ = 6,400 × 0.359635 = ₹2,301.66, i.e. ₹2,302 to the nearest rupee.
| End of year | 0 | 2 | 4 | 8 |
|---|---|---|---|---|
| Value at 12% reducing balance (₹) | 6,400.00 | 4,956.16 | 3,838.05 | 2,301.66 |
Where the 6 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | Depreciation relation | 2 |
| 2 | Logarithmic solution | 3 |
| 3 | Rate with check | 1 |
| Total | 6 |
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