Find time when BD(1600 at 15%) equals TD(1680 at 15%) for same term: The banker’s discount on Rs 1600 at 15% p.a. equals the true discount on Rs 1680 for the same time and rate. Find the time.

Introduction / Context:
Use BD = S1 * r * t and TD = S2 * r * t / (1 + r * t). Given BD(1600) = TD(1680), solve for t.

Given Data / Assumptions:
S1 = 1600; S2 = 1680; r = 15% p.a.; t = ?

Concept / Approach:
1600 * r * t = 1680 * r * t / (1 + r t) → 1600(1 + r t) = 1680 → 1 + r t = 1680/1600 = 1.05 → r t = 0.05.

Step-by-Step Solution:

Verification / Alternative check:
Check equality numerically at t = 1/3 year; both sides match.

Why Other Options Are Wrong:
3, 5, 6 months do not satisfy 1 + r t = 1.05.

Common Pitfalls:
Canceling t improperly; forgetting both sides share r * t.

Final Answer:
4 months