Difference between simple interest and true discount: For a certain sum due after 6 months at 4% per annum simple interest, the difference between simple interest and true discount is Rs. 15. Find the sum due.

Introduction / Context:
For a future amount A due after time t at rate r, banker’s discount (BD) = A * r * t and true discount (TD) = A * (r * t) / (1 + r * t). The difference BD − TD equals A * (r * t)^2 / (1 + r * t). This identity lets us back out A when the difference is known.

Given Data / Assumptions:

• Difference (BD − TD) = 15.
• t = 6 months = 0.5 year.
• r = 4% p.a. => r * t = 0.04 * 0.5 = 0.02.

Concept / Approach:
Use BD − TD = A * (r * t)^2 / (1 + r * t). Solve for A. Plug in r * t = 0.02.

Step-by-Step Solution:
BD − TD = A * 0.02^2 / (1 + 0.02) = A * 0.0004 / 1.02.So 15 = A * 0.00039215686… ⇒ A ≈ 15 / 0.00039215686 ≈ 38265.3.Nearest option provided = Rs. 38250.

Verification / Alternative check:
At A = 38250: r * t = 0.02, BD = 38250 * 0.02 = 765; TD = 38250 * 0.02 / 1.02 = 750; BD − TD = 15, exactly matching.

Why Other Options Are Wrong:

• 32850 and 28250 give smaller differences than 15.
• 38350 leads to a difference slightly over 15.

Common Pitfalls:

• Confusing SI on the principal with BD on the amount; in this identity, use the future amount A.
• Ignoring the denominator 1 + r * t in TD.

Final Answer:
Rs. 38250