What annual instalment will discharge a debt of ₹26,160 due in 4 years at 6% simple interest p.a.?
- (a)₹5,800
- (b)₹5,500
- (c)₹4,500
- (d)₹6,000
Answer
Why
Correct — D. Let the annual instalment be x, paid at the end of each of the 4 years, with the debt of ₹26,160 standing due at the end of year 4.
Each instalment then earns simple interest at 6% for the years still left:
1st instalment, 3 years to run: x(1 + 3 × 6⁄100) → 1.18x
2nd instalment, 2 years: 1.12x
3rd instalment, 1 year: 1.06x
4th instalment, paid on the due date: 1.00x
Add the multipliers:
1.18 + 1.12 + 1.06 + 1 = 4.36
4.36x = 26,160
x = 26,160 ⁄ 4.36 = ₹6,000 → option (d).
Check: 4 × ₹6,000 = ₹24,000 of instalments plus ₹2,160 of interest = ₹26,160.
Why the others are wrong
- (a)₹5,800 — Every rupee of instalment discharges ₹4.36 of debt, so ₹5,800 clears 5,800 × 4.36 = ₹25,288 — ₹872 short of the ₹26,160 due.
- (b)₹5,500 — ₹5,500 × 4.36 = ₹23,980, leaving ₹2,180 of the debt unpaid at the end of year 4. The instalment is too small by nearly a tenth.
- (c)₹4,500 — ₹4,500 is below even the interest-free floor. With no interest at all the instalment would be 26,160 ⁄ 4 = ₹6,540, and interest can only pull that down to ₹6,000, never to ₹4,500.
Concept
An annual instalment on a simple-interest debt is not the debt divided by the number of years. Each instalment is paid before the due date, so it keeps earning interest until then and does more work than its face value.
With n instalments of x at r% per annum, the first earns interest for n − 1 years, the second for n − 2, and the last for none.
Total discharged = nx + (r⁄100)x × [1 + 2 + … + (n − 1)], so the multiplier on x is n + (r⁄100) × n(n − 1)⁄2.
For n = 4 and r = 6 that is 4 + 0.06 × 6 = 4.36.
Read ₹26,160 as the amount due at the end of year 4, which is how the paper words "a debt due in 4 years". Discounting it back to a present value is a different formula and gives a different number; the key uses the accumulation reading.
Key facts
- The multiplier for n annual instalments at r% simple interest is n + (r⁄100) × n(n − 1)⁄2.
- For 4 years at 6% that multiplier is 4.36, so the instalment is the debt divided by 4.36.
- ₹6,000 × 4 = ₹24,000 of principal plus ₹2,160 of accumulated interest is exactly ₹26,160.
Study next
Common traps
- Dividing ₹26,160 by 4 to get ₹6,540 and picking the nearest option.
- Giving the final instalment a year of interest too, which turns 4.36 into 4.60 and the answer into ₹5,687.
- Writing the rate as 6 rather than 6⁄100 inside the multiplier, which turns 4.36 into 40 and the instalment into ₹654 — far below every option.
The numbers are chosen so the division comes out whole: 26,160 ⁄ 4.36 is exactly ₹6,000.
Use that as a check. If your division leaves a messy decimal, the multiplier is wrong — you have almost certainly credited the last instalment with interest it never earned.
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