The true discount on a bill of Rs. 2160 is Rs. 360 for a certain time and rate of interest. For the same conditions, what is the corresponding banker's discount on this bill?

Introduction / Context:
This problem uses the standard relationship between true discount, banker's discount and present worth of a bill. You are given the true discount on a known face value and asked to determine the banker's discount for the same rate and time. Understanding how these quantities are related is important in questions about commercial mathematics, banking and bill discounting.

Given Data / Assumptions:

• Face value (sum due) S = Rs. 2160.
• True discount TD = Rs. 360.
• Rate of interest and time are the same for both TD and BD (but not numerically specified).
• We assume simple interest and standard discount formulas.

Concept / Approach:
Key concepts:

• Present worth (P) = S − TD.
• Banker's discount (BD) is calculated on the face value S.
• A useful relation connecting BD, TD and P is: BD = (S * TD) / P.
• Another derived relation is banker's gain = BD − TD = TD^2 / P.

We will use P = S − TD to compute the present worth and then apply BD = (S * TD) / P.

Step-by-Step Solution:
Step 1: Compute present worth: P = S − TD = 2160 − 360 = 1800.Step 2: Use the relation BD = (S * TD) / P.Step 3: Substitute: BD = (2160 * 360) / 1800.Step 4: Simplify: 2160 / 1800 = 1.2, so BD = 1.2 * 360 = 432.Step 5: Therefore the banker's discount is Rs. 432.

Verification / Alternative check:
If we compute the banker's gain, we have BG = BD − TD = 432 − 360 = 72. Using the theoretical relation BG = TD^2 / P, we check: TD^2 / P = 360^2 / 1800 = 129600 / 1800 = 72. Since this matches, our computed BD is consistent with all standard formulas, confirming that Rs. 432 is correct.

Why Other Options Are Wrong:
Rs. 422, Rs. 412 and Rs. 442 do not satisfy the relation BD = (S * TD) / P when S = 2160 and P = 1800. If any of these values were taken as BD, the derived banker's gain would not equal TD^2 / P, contradicting the theoretical definitions. Rs. 452 is even farther away from the true value and clearly inconsistent with the algebraic relationships.

Common Pitfalls:

• Using simple interest directly on the present worth instead of on the face value.
• Forgetting that present worth is S − TD, not S + TD.
• Not knowing the key formula BD = (S * TD) / P and trying to guess based only on intuitive reasoning.
• Confusing banker's discount with true discount and treating them as equal.

Final Answer:
The banker's discount on the bill is Rs. 432.