The discount on Rs. 3050 equals the simple interest on Rs. 3000 for the same time at 5% per annum. Find the time.

Introduction / Context:
Here “discount” refers to true discount unless otherwise stated, and it is equated to the simple interest on another sum for the same time and rate. We solve for time.

Given Data / Assumptions:

• TD on 3050 equals SI on 3000
• Rate r = 5% per annum
• Time t = ? (years)

Concept / Approach:
TD = F * (r * t)/(1 + r * t). SI(3000) = 3000 * r * t. Set them equal and solve for t.

Step-by-Step Solution:
3050 * 0.05 t /(1 + 0.05 t) = 3000 * 0.05 t. Cancel 0.05 t (t ≠ 0): 3050/(1 + 0.05 t) = 3000. 1 + 0.05 t = 3050/3000 = 1.016666... 0.05 t = 0.016666... → t = 0.3333... years = 4 months.

Verification / Alternative check:
Let t = 1/3 year. SI on 3000 = 3000 * 0.05 * (1/3) = 50. TD = 3050 * 0.05 * (1/3) / (1 + 0.05/3) = 50 exactly.

Why Other Options Are Wrong:
6 or 3 months do not make the equality hold; “None” is incorrect since 4 months satisfies the condition.

Common Pitfalls:
Treating “discount” as banker’s discount; forgetting to cancel t carefully; unit conversion errors.

Final Answer:
4 months