The banker's gain on a certain sum of money, due 2 years hence at 10% per annum simple interest, is Rs 24. What is the present worth (that is, the present value) of this sum?

Introduction:
In banker's discount problems, three related quantities are very important: the banker's discount (BD), the true discount (TD), and the banker's gain (BG). This question gives us the banker's gain on a bill and asks for the present worth, or present value, of the sum using the given rate and time period.

Given Data / Assumptions:
Banker's gain (BG) = Rs 24. Time period t = 2 years. Rate of simple interest r = 10% per annum. We must find: Present worth (PW) of the sum.

Concept / Approach:
For a sum with face value P due after time t at rate r: True discount: TD = P * r * t / (100 + r * t). Banker's discount: BD = P * r * t / 100. Banker's gain: BG = BD − TD. There is also a direct relationship: BG = TD * r * t / 100. We will first find TD from the banker's gain, then find the face value P, and finally subtract TD from P to get the present worth.

Step-by-Step Solution:
Step 1: Compute r * t = 10 * 2 = 20. Step 2: Use BG = TD * r * t / 100. 24 = TD * 20 / 100. TD = 24 * 100 / 20 = 24 * 5 = Rs 120. Step 3: Use TD formula to find the face value P. TD = P * r * t / (100 + r * t). 120 = P * 20 / (100 + 20) = P * 20 / 120. 120 = P / 6 ⇒ P = 120 * 6 = Rs 720. Step 4: Present worth PW = P − TD = 720 − 120 = Rs 600.

Verification / Alternative check:
We can verify quickly: BD = P * r * t / 100 = 720 * 20 / 100 = Rs 144. TD = Rs 120 (as found). BG = BD − TD = 144 − 120 = Rs 24, which matches the given banker's gain.

Why Other Options Are Wrong:
Rs 400: This would give inconsistent discount and gain values when formulas are applied. Rs 800: This would lead to a much larger banker's discount and a different banker's gain. Rs 500: Does not satisfy the exact banker's gain relationship BG = Rs 24. Rs 720: This is actually the face value, not the present worth.

Common Pitfalls:
Many learners confuse true discount with banker's discount or forget the relation BG = TD * r * t / 100. Another common mistake is to use the simple interest formula directly on the present worth instead of solving step by step for TD, P, and finally PW.

Final Answer:
The present worth of the sum is Rs 600.