Simple Interest — Reverse present worth via SI discounting: A person closes an investment by withdrawing ₹ 10000 today, ₹ 6000 one year ago, and ₹ 5000 two years ago. Three years ago there was no withdrawal. Assuming 10% p.a. simple interest since opening, what was the approximate initial deposit 4 years ago?
Correct Answer: ₹ 15600
Introduction / Context:With simple interest, the present worth of withdrawals at different times (counted from opening) uses linear discounting: PV at opening = Withdrawal / (1 + r * t). Summing the present worths of all withdrawals reconstructs the initial deposit, assuming the scheme accrued SI uniformly from the opening on the original principal.
Given Data / Assumptions:
- Annual simple interest r = 10% = 0.10
- Opening was 4 years ago
- Withdrawals: ₹ 5000 at t = 2 years, ₹ 6000 at t = 3 years, ₹ 10000 at t = 4 years
- Goal: approximate initial deposit (no other cash flows)
Concept / Approach:Under SI, accumulation to a future date uses A = P * (1 + r * t). Conversely, discounting a known future amount W back to opening uses PV = W / (1 + r * t). The initial principal is the sum of the PVs of all withdrawals.
Step-by-Step Solution:
PV(₹ 5000 at t = 2) = 5000 / (1 + 0.10 * 2) = 5000 / 1.20 ≈ 4166.67PV(₹ 6000 at t = 3) = 6000 / (1 + 0.10 * 3) = 6000 / 1.30 ≈ 4615.38PV(₹ 10000 at t = 4) = 10000 / (1 + 0.10 * 4) = 10000 / 1.40 ≈ 7142.86Initial deposit ≈ 4166.67 + 4615.38 + 7142.86 ≈ ₹ 15924.91.Verification / Alternative check:
Rounding to the nearest option, ₹ 15600 is closest (diff ≈ ₹ 325), consistent with “approximately”.Why Other Options Are Wrong:
- ₹ 16500 and ₹ 17280 deviate farther from the computed present worth.
- ₹ 15000 underestimates the required initial deposit.
Common Pitfalls:
- Using compound discounting; SI requires linear discount factors 1 + r * t.
- Mistiming the withdrawals (ensure t = 2, 3, 4 from opening).
Final Answer:Approximately ₹ 15600.