Simple interest on ₹ 2000 at 5% per annum equals the true discount on ₹ 2500 for the same time and at the same rate. Find the time (in years).

Introduction / Context:
Here the equality of a simple interest on one principal and the true discount on another principal (same time and rate) allows solving for time without finding the rate separately.

Given Data / Assumptions:

• SI on ₹ 2000 at 5% per annum for time t equals TD on ₹ 2500 at the same rate and time.

Concept / Approach:
SI = P * r * t. TD = F * (r * t)/(1 + r * t). Set these equal and solve for t (note r cancels).

Step-by-Step Solution:
2000 * 0.05 * t = 2500 * (0.05 * t) / (1 + 0.05 * t). 100 t = 125 t/(1 + 0.05 t). Cross-multiply: 100 t (1 + 0.05 t) = 125 t. 100 + 5 t = 125 → 5 t = 25 → t = 5 years.

Verification / Alternative check:
Let t = 5. SI on 2000 = 2000 * 0.05 * 5 = 500. TD on 2500 with r t = 0.25 is 2500 * 0.25/1.25 = 500. Match confirmed.

Why Other Options Are Wrong:
Any time other than 5 years fails the equality because the TD has the denominator (1 + r t), which uniquely balances at t = 5 under these principals.

Common Pitfalls:
Cancelling t too early (t ≠ 0); ignoring the (1 + r t) factor; or equating SI on face value directly to SI on present worth instead of TD.

Final Answer:
5 years