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

Introduction:
This question involves the banker's gain (BG) on a bill at a given rate and time, and asks us to find the present worth (PW) of the sum. We must use the relationships between banker's gain, true discount, face value, and present worth in simple-interest discounting.

Given Data / Assumptions:
Banker's gain BG = Rs 200. Rate r = 5% per annum. Time t = 4 years. We must find: Present worth PW.

Concept / Approach:
Let P be the face value of the bill. Then: TD = true discount. BD = banker's discount. BG = BD − TD. Key formulas: BG = TD * r * t / 100. TD = P * r * t / (100 + r * t). PW = P − TD. We will first find TD from BG, then find P using the TD formula, and finally subtract TD from P to obtain the present worth PW.

Step-by-Step Solution:
Step 1: Compute r * t. r * t = 5 * 4 = 20. Step 2: Find TD from BG. BG = TD * r * t / 100 ⇒ 200 = TD * 20 / 100. TD = 200 * 100 / 20 = 200 * 5 = Rs 1000. Step 3: Use TD to find P. TD = P * r * t / (100 + r * t). 1000 = P * 20 / (100 + 20) = P * 20 / 120. 1000 = P / 6 ⇒ P = 1000 * 6 = Rs 6000. Step 4: Present worth PW = P − TD = 6000 − 1000 = Rs 5000.

Verification / Alternative check:
We can compute BD from P: BD = P * r * t / 100 = 6000 * 20 / 100 = Rs 1200. Then BG should be: BG = BD − TD = 1200 − 1000 = Rs 200, which matches the given banker's gain, confirming that PW = Rs 5000 is correct.

Why Other Options Are Wrong:
Rs 4500, Rs 4000: Both are lower than the correct present worth and would imply different values of TD and BG. Rs 6000: This is the face value, not the present worth. Rs 5200: Does not satisfy the exact relationships among TD, BD, and BG at 5% for 4 years.

Common Pitfalls:
Often, learners confuse face value with present worth, or they attempt to find BD directly from BG without first finding TD. Remember, BG is tied most directly to TD through BG = TD * r * t / 100. Once TD is known, the rest of the values follow systematically.

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