A bill is due 3 years hence at 5% p.a. The banker’s gain (BG) is ₹90. Find the banker’s discount (BD).

Introduction / Context:
We are given BG and the period and rate. From BG we first recover TD using x = r * t. Then BD follows from BD = (1 + x) * TD or directly via BD = F * x once F is known. The compact route is via TD and the (1 + x) multiplier.

Given Data / Assumptions:

• r = 5% p.a., t = 3 years ⇒ x = 0.15.
• BG = ₹90.

Concept / Approach:
Key identities: BG = x * TD and BD = (1 + x) * TD. Hence TD = BG / x and then BD = (1 + x) * TD.

Step-by-Step Solution:

Verification / Alternative check:
With BD = ₹690 and TD = ₹600, BG = BD − TD = ₹90, consistent.

Why Other Options Are Wrong:
₹550, ₹600, and ₹650 do not maintain BD − TD = ₹90 with x = 0.15.

Common Pitfalls:
Using BD = x * TD (incorrect); the correct relation is BD = (1 + x) * TD and BG = x * TD.

Final Answer:
Rs. 690