The banker's gain on a certain sum due 4 years hence at 5% per annum is Rs 200. What is the present worth of this sum (in rupees)?

Introduction / Context:
This problem focuses on computing the present worth of a sum when banker's gain is known for a fixed rate and time. Banker's gain is the excess of banker's discount over true discount. For fixed rate and time, banker's gain is a fixed fraction of the face value of the bill. Using this relationship we can determine face value first and then true discount and present worth.

Given Data / Assumptions:
- Time until the sum is due t = 4 years
- Rate of simple interest r = 5 percent per annum
- Banker's gain BG = Rs 200
- Let face value of the bill be F
- Present worth PW must be determined

Concept / Approach:
For a bill of face value F at rate r for time t, banker's discount is BD = F × r × t / 100 and true discount is TD = F × r × t / (100 + r × t). Banker's gain is BG = BD − TD. This expression can be rewritten as a constant fraction of F that depends only on r t. Once we find F using BG, the true discount and present worth follow from standard true discount relations: PW = F − TD, and TD = F × r t / (100 + r t).

Step-by-Step Solution:
Step 1: Compute r × t = 5 × 4 = 20.Step 2: Banker's discount BD = F × 20 / 100 = F / 5.Step 3: True discount TD = F × 20 / (100 + 20) = F × 20 / 120 = F / 6.Step 4: Banker's gain BG = BD − TD = F / 5 − F / 6 = F / 30.Step 5: Given BG = 200, so F / 30 = 200 and therefore F = 200 × 30 = Rs 6000.Step 6: True discount TD = F / 6 = 6000 ÷ 6 = Rs 1000.Step 7: Present worth PW = face value − true discount = 6000 − 1000 = Rs 5000.

Verification / Alternative check:
We can verify by recomputing BD and BG from the found values. Banker's discount BD = F / 5 = 6000 ÷ 5 = Rs 1200. True discount TD is Rs 1000. Therefore banker's gain BG = BD − TD = 1200 − 1000 = Rs 200, which matches the given value. Since both BD and TD satisfy the interest relationships at 5 percent for 4 years, the present worth of Rs 5000 is fully consistent.

Why Other Options Are Wrong:
A present worth of Rs 4500 or Rs 4000 would give a much smaller true discount and would no longer produce a banker's gain of Rs 200 at the given rate and time. A value of Rs 6000 is the face value itself and would wrongly imply zero true discount. Rs 4800 is also inconsistent when checked through the formulas. Only Rs 5000 leads to internal consistency between BD, TD and BG.

Common Pitfalls:
Students sometimes confuse present worth with face value or try to compute BD directly from BG without first deriving the correct relationship. Another mistake is to compute true discount on the wrong base. Always express BD, TD and BG in terms of F using r t first, then use the given banker's gain to find F and from that determine present worth accurately.

Final Answer:
The present worth of the sum is Rs. 5000.