No-loss investment rate after banker’s discount: A bill is discounted at 5% per annum by banker’s discount (BD). At what annual simple-interest rate should the proceeds be invested so that the amount at maturity equals the face value (i.e., no loss)? Assume 1 year to maturity for clarity.

Introduction / Context:
Under banker’s discount, the bank deducts BD = S * d * t / 100 from the face value S upfront, where d is the discount rate and t is time (years). The borrower receives proceeds P = S − BD. If these proceeds are invested at a rate i for t years to exactly reach S (so there is no loss), we can solve for i in terms of d and t.

Given Data / Assumptions:

• d = 5% per annum.
• Assume t = 1 year (standard framing for this classic result).
• Proceeds P = S(1 − d t/100).
• No loss condition: P * (1 + i t) = S.

Concept / Approach:
From P(1 + i t) = S, rearrange i = (S/P − 1)/t. With t = 1 and P = S(1 − d/100), we get i = [1/(1 − d/100)] − 1. Substitute d = 5 to find i in percent.

Step-by-Step Solution:

Verification / Alternative check:
Check numerically: If S = 100, proceeds P = 95. Investing at 5.263…% for one year gives 95 * 1.05263… = 100.

Why Other Options Are Wrong:

• 5%: Falls short (95 * 1.05 = 99.75).
• 4 19/20 %, 10%, 4%: Do not produce S from P over one year.

Common Pitfalls:
Confusing banker’s discount with true discount; using i = d instead of solving from P(1 + i t) = S.

Final Answer:
5 5/9 %