Banker’s gain to banker’s discount relation: On a sum due in 3 years at 10% per annum (simple discounting convention), the banker’s gain (BG) is $180. Find the banker’s discount (BD).

Introduction / Context:
For a face value S, time t, and discount rate d% (per annum), banker’s discount (BD) is S * d * t / 100. True discount (TD) equals S − S/(1 + d t/100). Banker’s gain (BG) is BD − TD. Given BG, we can solve for S and then find BD straightforwardly.

Given Data / Assumptions:

• t = 3 years, d = 10% per annum.
• BG = $180.

Concept / Approach:
Compute TD in terms of S: TD = S − S/(1 + 0.10 * 3) = S − S/1.3 = S * (0.3/1.3). BD = 0.3S. Then BG = BD − TD = 0.3S − S*(0.3/1.3). Solve for S, then find BD = 0.3S.

Step-by-Step Solution:

Verification / Alternative check:
TD ≈ 2600*(0.3/1.3) ≈ $600. BG = 780 − 600 = $180, consistent.

Why Other Options Are Wrong:

• $680, $580, $480: Do not satisfy BG = BD − TD with d = 10% and t = 3 years.

Common Pitfalls:
Confusing TD formula with BD. TD is based on present-worth relation; BD is simple-interest on face value.

Final Answer:
$ 780