No-loss investment after discounting: A banker discounts a 4-month bill at a simple discount rate of 3% per annum. To ensure nothing is lost overall, at what simple annual rate should the banker invest the proceeds for those 4 months so that the maturity value equals the bill’s face value?

Introduction / Context:
When a bill is discounted at a simple discount rate, the banker receives the proceeds (face less discount). If those proceeds are invested until maturity, the investment rate must be high enough to recover exactly the face value so that there is “no loss.” This links simple discounting and simple interest growth over the same period.

Given Data / Assumptions:

• Discount rate d = 3% per annum (simple).
• Time t = 4 months = 1/3 year.
• Face value F (symbolic); proceeds P = F * (1 − d * t).

Concept / Approach:
Require P * (1 + i * t) = F, where i is the simple annual investment rate. Solve for i using the proceeds from discounting and the condition that the invested proceeds grow back to F at maturity.

Step-by-Step Solution:

Verification / Alternative check:
Using fractions: 1/0.99 = 100/99; so i/3 = 1/99 ⇒ i = 3/99 = 1/33 ≈ 3.0303%, i.e., 3 1/3%.

Why Other Options Are Wrong:
3% under-recovers the face; 4% over-recovers; 3 1/36% and 2 2/3% do not satisfy 0.99(1 + i/3) = 1.

Common Pitfalls:
Treating the discount as simple interest on proceeds or compounding over 4 months; the setup requires simple-rate equivalence over the same period.

Final Answer:
3 1/3%