Zero-loss investment rate against 10% banker’s discount: A bill is discounted at 10% per annum (banker’s discount) for its full 1-year term. At what simple-interest rate should the proceeds be invested to avoid any loss at maturity?

Introduction / Context:
For term t under banker’s discount rate r, proceeds are P = S(1 − r t). To have no loss, invest P at rate i for t so that P(1 + i t) = S → (1 − r t)(1 + i t) = 1.

Given Data / Assumptions:
r = 10% p.a.; t = 1 year; simple interest on proceeds.

Concept / Approach:
i = r / (1 − r t). With t = 1 → i = r / (1 − r).

Step-by-Step Solution:

Verification / Alternative check:
Proceeds = 0.9 S. Invest at 11 1/9 % for 1 year: multiplier 1 + 1/9 = 10/9. Product = 0.9 * (10/9) = 1.00 → S ✔️.

Why Other Options Are Wrong:
10% or 11% do not return exactly S; 9 1/9 % is too small.

Common Pitfalls:
Setting i = r (incorrect); mixing simple and compound interest.

Final Answer:
11 1/9 % per annum