Relating simple interest to true discount: If the simple interest on Rs. 50 at 4 1/2% per annum equals the true discount on Rs. 59 for the same time at the same rate, for how many years is the latter sum due?

Introduction / Context: True discount (TD) on a sum due at a future time is different from simple interest (SI) on a present sum, though both use the same rate and time. The relationship TD = S * r * t / (1 + r * t) is crucial, where S is the future due amount. This problem equates SI on a smaller principal to TD on a larger amount to find the time.

Given Data / Assumptions:

• SI on Rs. 50 at r = 4.5% for time t equals TD on Rs. 59 for same r and t.
• r = 4.5% per annum = 0.045; t in years.

Concept / Approach: SI on Rs. 50 = 50 * r * t. TD on Rs. 59 = 59 * r * t / (1 + r * t). Equate them and solve for r * t, then for t since r is known.

Step-by-Step Solution:

Verification / Alternative check: Substitute back: r * t = 0.18 ⇒ TD factor = 0.18/1.18 = 0.152542…; 59 * 0.152542… ≈ 9; SI on 50 = 50 * 0.045 * 4 = 9; they match.

Why Other Options Are Wrong: 2, 3, 5, and 6 years produce different r * t values that break the equality.

Common Pitfalls: Using TD = S * r * t (ignoring the denominator), or forgetting that TD is computed on the future amount.

Final Answer: 4 years