The present worth of a sum due sometime in the future is Rs. 576 and the banker's gain on this transaction is Rs. 16. What is the true discount on the sum?

Introduction / Context:
This question tests your understanding of the relationship between present worth, true discount and banker's gain. You are given the present worth of the sum and the banker's gain and are required to find the true discount. This scenario is typical in discounting problems involving bills, where the bank earns a gain because it discounts the bill on its face value instead of on present worth.

Given Data / Assumptions:

• Present worth P = Rs. 576.
• Banker's gain BG = Rs. 16.
• Let TD be the true discount.
• Standard relations for simple interest and discount apply.

Concept / Approach:
A very useful relationship in discounting theory is:

• Banker's gain: BG = TD^2 / P.

This connects banker's gain directly with true discount and present worth, without explicitly needing the rate of interest or time. Knowing BG and P, we can solve for TD easily. Once TD is found, all other related quantities like face value and banker's discount can also be inferred if needed.

Step-by-Step Solution:
Step 1: Use the relation BG = TD^2 / P.Step 2: Substitute BG = 16 and P = 576 into the formula: 16 = TD^2 / 576.Step 3: Rearranging gives TD^2 = 16 * 576.Step 4: Compute 16 * 576 = 9216.Step 5: Thus TD^2 = 9216, so TD = √9216 = 96.Step 6: Therefore, the true discount is Rs. 96.

Verification / Alternative check:
Once TD = 96 is known, the face value S of the bill can be found as S = P + TD = 576 + 96 = 672. If we wanted, we could now compute the banker's discount BD using BD = S * TD / P = 672 * 96 / 576 = 112. The banker's gain BG = BD − TD = 112 − 96 = 16, which matches the given value. This reverse check confirms that TD = 96 is consistent with both present worth and banker's gain.

Why Other Options Are Wrong:
Values such as Rs. 78, Rs. 85, Rs. 105 or Rs. 64 do not satisfy the equation TD^2 = BG * P when BG = 16 and P = 576. For example, if TD were 78, TD^2 would be 6084, and TD^2 / P would not equal 16. Only TD = 96 produces TD^2 / 576 = 9216 / 576 = 16, which matches the required banker's gain.

Common Pitfalls:

• Not knowing or forgetting the relation BG = TD^2 / P and instead trying to introduce unnecessary unknowns like rate and time.
• Computing S first and then working backwards, which is longer and more error-prone.
• Mistakes in squaring or taking the square root of large numbers.
• Confusing present worth P with the face value S.

Final Answer:
The true discount on the sum is Rs. 96.