From PW and true discount, find banker’s discount and banker’s gain: The present worth (PW) of a bill due sometime hence is Rs 1600, and its true discount (TD) is Rs 160. Find the banker’s discount (BD) and the extra gain (banker’s gain, BG).

Introduction / Context:
Relations among present worth (PW), true discount (TD), banker’s discount (BD), and banker’s gain (BG) under simple interest are standard results used in banker’s discount problems.

Given Data / Assumptions:

• PW = Rs 1600.
• TD = Rs 160.
• Face value S = PW + TD = 1760.

Concept / Approach:
Let x = r * t (rate × time). Then TD = S * x / (1 + x) and BD = S * x. Also BG = BD − TD = S * x^2 / (1 + x). We can find x from PW and TD without knowing r or t individually.

Step-by-Step Solution:

Verification / Alternative check:
TD = S * x / (1 + x) = 1760 * 0.1 / 1.1 = 160 ✔️; BG = S * x^2 / (1 + x) = 1760 * 0.01 / 1.1 = 16 ✔️.

Why Other Options Are Wrong:
Rs 176, Rs 18 gives a BG inconsistent with the identity BG = BD − TD; Rs 186, Rs 16 implies x > 0.1 and contradicts PW.

Common Pitfalls:
Using BD = TD or mixing up PW with face value; forgetting that x = r * t and cancels in ratios.

Final Answer:
Banker’s Discount = Rs 176; Banker’s Gain = Rs 16