Find the annual rate when TD and SI are given for the same sum: The true discount on a certain sum due 4 years hence is ₹ 75. The simple interest on the same sum for the same time and at the same rate is ₹ 225. Find the annual rate of interest.

Introduction / Context:
For a future amount A due after time t at annual rate r, banker’s discount (BD) = A * r * t and true discount (TD) = A * (r * t) / (1 + r * t). If BD and TD are both given for the same A, r, t, their ratio reveals r * t directly.

Given Data / Assumptions:

• TD = 75.
• SI on the same sum, same time, same rate = 225. (This SI equals BD on the future amount.)
• t = 4 years.

Concept / Approach:
Ratio: BD / TD = (A * r * t) / [A * r * t / (1 + r * t)] = 1 + r * t. Hence 225 / 75 = 3 = 1 + r * t. Solve for r * t, then r.

Step-by-Step Solution:
225 / 75 = 3 ⇒ 1 + r * t = 3 ⇒ r * t = 2.Given t = 4 ⇒ r = (r * t) / t = 2 / 4 = 0.5 = 50% p.a.

Verification / Alternative check:
At r = 50%, t = 4: BD = A * 2; TD = A * 2 / 3; BD / TD = 3, agreeing with the given 225/75.

Why Other Options Are Wrong:

• 25%, 31%, 45% do not satisfy BD / TD = 3 at t = 4 years.

Common Pitfalls:

• Confusing SI on present worth (which equals TD) with SI on the future amount (which equals BD); here, the given SI corresponds to BD.

Final Answer:
50%