Question 4 (c)
Equation of value: finding the time of the final payment
6 marks
- Time to write
- About 12 minutes
Question
Question 4 (c) · 6 marksA loan of ₹70,000 due 9 years from now is instead to be repaid by three payments: ₹20,000 now, ₹11,000 in four years and a final payment of ₹11,000 at the end of n years. With the rate of interest at 8% compounded annually, find the value of n.
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 "Attempt any three"; 6 marks, from the marker printed after part (d).
Write it in this order
- Read the data; note exactly what is asked 1 min
- 1 Equation of value 2 marks 3 min
- 2 Present values of the known amounts 2 marks 3 min
- 3 Solving for n 2 marks 3 min
- Check: arithmetic, units (₹, %), final answer underlined 2 min
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.
Equation of value at the present time. The three payments must have the same present value, at 8% compounded annually, as the ₹70,000 due in 9 years. A sum due in n years has present value A/(1 + i)ⁿ.
20,000 + 11,000(1.08)⁻⁴ + 11,000(1.08)⁻ⁿ = 70,000(1.08)⁻⁹
| Item | Working | Present value (₹) |
|---|---|---|
| Payment now | 20,000.00 | |
| Payment in 4 years | 11,000/1.360489 | 8,085.33 |
| Final payment (balancing figure) | 6,932.10 | |
| Debt due in 9 years | 70,000/1.999005 | 35,017.43 |
11,000(1.08)⁻ⁿ = 6,932.10 ⟹ (1.08)ⁿ = 11,000/6,932.10 = 1.586821
n = log 1.586821 / log 1.08 = 0.200528/0.033424 = 6.00 years.
Answer: the final payment of ₹11,000 is to be made at the end of 6 years.
Check: 20,000 + 8,085.33 + 11,000/1.586874 (= 6,931.87) = ₹35,017.20, which equals the present value of the debt, ₹35,017.43, to within the rounding of n to 6 years (n = 5.9996).
Where the 6 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | Equation of value | 2 |
| 2 | Present values of the known amounts | 2 |
| 3 | Solving for n | 2 |
| Total | 6 |
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