Difference between SI and TD: The difference between simple interest and true discount on a certain sum due after 6 months at 6% per annum is Rs. 27. Find the face value (sum due).

Introduction / Context:
For a sum S due after time t at rate r, simple interest (SI) on S and true discount (TD) on S have a known difference: SI − TD = S * (r * t)^2 / (1 + r * t). This relation enables solving for S directly when SI − TD, r, and t are known.

Given Data / Assumptions:

• Time t = 6 months = 0.5 years.
• Rate r = 6% per annum = 0.06.
• Difference SI − TD = Rs. 27.

Concept / Approach:
Use SI − TD = S * (r t)^2 / (1 + r t). Compute r t, then rearrange to S = (SI − TD) * (1 + r t) / (r t)^2.

Step-by-Step Solution:

Verification / Alternative check:
Plug S ≈ 30,900 back: SI − TD ≈ 30,900 * 0.000873786 ≈ 27. It matches the given difference.

Why Other Options Are Wrong:
Rs. 39,000, Rs. 30,600, and Rs. 20,900 deviate from the precise computation; Rs. 25,000 is also incorrect.

Common Pitfalls:
Forgetting to convert months to years; squaring r t incorrectly; using SI + TD instead of the required difference formula.

Final Answer:
Rs. 30900