Present Worth & True Discount known → Banker’s Gain: The present worth of a sum due sometime hence is Rs. 1600 and the true discount is Rs. 160. What is the banker's gain?

Introduction / Context:
Given present worth (P) and true discount (TD), we can reconstruct face value F = P + TD and the combined interest factor (1 + r * t). From these, Banker's Discount (BD) and hence Banker's Gain (BG = BD − TD) follow directly.

Given Data / Assumptions:

• P = Rs. 1600.
• TD = Rs. 160.
• Simple interest; time and rate combined as x = r * t (unknown).

Concept / Approach:
We use P = F / (1 + x) and TD = F − P = F * x / (1 + x). Knowing P and TD gives F and 1 + x immediately, then BD = F * x and BG = BD − TD.

Step-by-Step Solution:
F = P + TD = 1600 + 160 = Rs. 1760.Since P = F / (1 + x) ⇒ 1600 = 1760 / (1 + x) ⇒ 1 + x = 1760 / 1600 = 11/10 ⇒ x = 0.1.BD = F * x = 1760 * 0.1 = Rs. 176.BG = BD − TD = 176 − 160 = Rs. 16.

Verification / Alternative check:
TD formula: F * x / (1 + x) = 1760 * 0.1 / 1.1 = 176 / 1.1 = 160, consistent.

Why Other Options Are Wrong:
10, 20, 24, 14 do not match BG after computing the correct BD and TD relation.

Common Pitfalls:
Mixing up TD with BD; forgetting that x = r * t is dimensionless; rounding prematurely.

Final Answer:
Rs. 16