The difference between the Simple Interest and the True Discount on a certain sum of money for 6 months at a rate of 12.5 percent per annum is Rs 25. What is the sum (amount due at the end of 6 months)?

Introduction / Context:
This is a standard question from the topic of true discount and simple interest. It involves the relationship between Simple Interest (calculated on the face value of a bill) and True Discount (the difference between the amount due and its present value). The difference between these two is given and we are asked to find the sum due at the end of the period. Such questions often appear in banking and finance related aptitude sections.

Given Data / Assumptions:
- Time period = 6 months = 0.5 years.
- Rate of interest = 12.5 percent per annum (which is 12.5 per year in percentage terms).
- Difference between Simple Interest and True Discount on the same sum for this time and rate is Rs 25.
- The interest is computed on a yearly basis using simple interest assumptions.

Concept / Approach:
In true discount problems, there are three related quantities: the sum due at maturity (amount), the present value, and the true discount. In addition, many exam questions refer to Simple Interest on the face value, which is often called the bank discount. For a sum A due after time t at rate r percent per annum, the simple interest on A is A * r * t / 100. The true discount TD is A * r * t / (100 + r * t). The difference between simple interest on the face value and true discount equals A * r^2 * t^2 / (100 * (100 + r * t)). We use this relationship to form an equation in A and solve for the sum.

Step-by-Step Solution:
Step 1: Let A be the sum due after 6 months. Step 2: Rate r = 12.5 percent, time t = 0.5 year. Step 3: Compute r * t = 12.5 * 0.5 = 6.25. Step 4: The difference between Simple Interest on A and the True Discount on A is given by A * r^2 * t^2 / (100 * (100 + r * t)). Step 5: Compute r^2 * t^2. Since r = 12.5 and t = 0.5, r^2 * t^2 = 12.5^2 * 0.5^2 = 156.25 * 0.25 = 39.0625. Step 6: Denominator = 100 * (100 + 6.25) = 100 * 106.25 = 10625. Step 7: Therefore difference = A * 39.0625 / 10625. Step 8: This difference is given as Rs 25, so we set A * 39.0625 / 10625 = 25. Step 9: Solve for A: A = 25 * 10625 / 39.0625 = 6800.

Verification / Alternative check:
To verify, compute Simple Interest and True Discount separately for A = 6800. First, r * t = 6.25 percent. Simple Interest on A = 6800 * 6.25 / 100 = 425. True Discount on A = 6800 * 6.25 / (100 + 6.25) = 425 * 100 / 106.25 = 400. The difference between Simple Interest and True Discount = 425 − 400 = 25, which matches the given condition. This confirmation shows that the sum due is indeed Rs 6800.

Why Other Options Are Wrong:
Option Rs 6500, Rs 6000 and Rs 6200 each lead to different values of Simple Interest and True Discount whose difference is not equal to Rs 25 when checked using the same formulas. Therefore these amounts cannot satisfy the given relationship between Simple Interest and True Discount for the given time and rate.

Common Pitfalls:
Many learners confuse True Discount with Simple Interest on the present value instead of on the face value, or they forget the correct formula for True Discount. Another common mistake is to ignore that the difference is given between Simple Interest on the sum due and the True Discount, not between Simple Interest and present value. Using consistent formulas, correctly handling the decimal rate 12.5 percent, and carefully solving the resulting equation avoids these errors.

Final Answer:
The sum due (amount) is Rs 6800.