Banker’s gain on a long-dated bill: The face value of a bill due 5 years hence is ₹13800. At 5% per annum simple interest, compute the banker’s gain on discounting the bill.

Introduction / Context:
Banker’s Gain (BG) is the difference between the Banker’s Discount (BD) and True Discount (TD). It measures the extra advantage to the banker arising because BD is calculated on the face value whereas the true loss to the holder is TD on present worth.

Given Data / Assumptions:

• A = ₹13800
• t = 5 years
• r = 5% = 0.05 per annum

Concept / Approach:
Useful identities:

Step-by-Step Solution:

Verification / Alternative check:
Compute BD = 13800 * 0.25 = ₹3450; PW = A/(1 + 0.25) = 13800/1.25 = ₹11040; TD = A − PW = ₹2760; BG = 3450 − 2760 = ₹690 (consistent).

Why Other Options Are Wrong:
Other values do not match BG = A*(r t)^2/(1 + r t) using r t = 0.25.

Common Pitfalls:
Confusing BD with BG, or computing TD incorrectly by omitting the (1 + r t) term in the denominator.

Final Answer:
₹ 690