Zero-loss investment rate against banker’s discount: A bill is discounted at 5% per annum (banker’s discount) for its full term of 1 year. At what annual simple-interest rate should the proceeds be invested for the same term so that there is no loss at maturity?

Introduction / Context:
When a bill is discounted at rate r for term t, the holder receives the proceeds P = S(1 − r t). To avoid loss, investing P for time t at rate i should grow exactly back to S. This determines a relationship between i and r.

Given Data / Assumptions:

• Banker’s discount rate r = 5% p.a.
• Term t = 1 year.
• Simple interest for the re-investment.

Concept / Approach:
Set future value of proceeds equal to face value: (1 − r t)(1 + i t) = 1. Solve i = r / (1 − r t). For t = 1, i = r / (1 − r).

Step-by-Step Solution:

Verification / Alternative check:
Check: Proceeds = 0.95 S. Investing at 100/19 % for 1 year gives multiplier 1 + 1/19 = 20/19. Product: 0.95 * (20/19) = 0.95 * 1.052631… = 1.0000 = S ✔️.

Why Other Options Are Wrong:
5% or 10% do not satisfy the exact no-loss condition; 55/19 % ≈ 2.894% is far too low.

Common Pitfalls:
Using i = r instead of i = r / (1 − r t); applying compound interest instead of simple interest on the proceeds.

Final Answer:
100/19 % per annum (≈ 5.263%)