Banker’s Gain known → Banker’s Discount (3 years @ 5%): The banker's gain on a sum due 3 years hence at 5% per annum is Rs. 90. What is the banker's discount?

Introduction / Context:
With simple interest, BG relates to face value as BG = F * (r * t)^2 / (1 + r * t). Banker's Discount is BD = F * r * t. Given BG and r, t, we can recover F, then BD.

Given Data / Assumptions:

• r = 5% per annum; t = 3 years ⇒ x = r * t = 0.15.
• BG = Rs. 90.

Concept / Approach:
Compute F from BG = F * x^2 / (1 + x); then BD = F * x.

Step-by-Step Solution:
BG = F * (0.15)^2 / 1.15 = F * 0.0225 / 1.15 ≈ F * 0.0195652.F ≈ BG / 0.0195652 ≈ 90 / 0.0195652 ≈ Rs. 4600.BD = F * x = 4600 * 0.15 = Rs. 690.

Verification / Alternative check:
TD = BD − BG = 690 − 90 = 600. Check TD formula: F * x / (1 + x) = 4600 * 0.15 / 1.15 = 690 / 1.15 = 600, consistent.

Why Other Options Are Wrong:
720 and 810 assume different x or F; 150 is the interest on Rs. 1000 at 5% for 3 years, unrelated; 600 is TD, not BD.

Common Pitfalls:
Using compound interest; rounding F before computing BD; mistaking TD for BD.

Final Answer:
Rs. 690