The difference between the simple interest and the true discount on a certain sum for 2 years at 15% per annum (simple interest) is ₹ 45. Determine the sum due (face value).

Introduction / Context:
This problem compares simple interest on the face value with the corresponding true discount at the same rate and time. Their difference allows us to compute the face value directly.

Given Data / Assumptions:

• Time t = 2 years
• Rate r = 15% per annum
• Difference (SI − TD) = ₹ 45

Concept / Approach:
With x = r*t, SI on the face value F is F * x and TD is F * x / (1 + x). Hence: SI − TD = F * x − F * x/(1 + x) = F * x^2/(1 + x). We solve for F.

Step-by-Step Solution:
x = r*t = 0.15 * 2 = 0.30. SI − TD = F * x^2/(1 + x) = F * 0.09/1.30. Given 45 = F * (0.09/1.30) → F = 45 * 1.30 / 0.09. F = 58.5 / 0.09 = 650.

Verification / Alternative check:
Compute SI = F * x = 650 * 0.30 = 195. TD = F * x/(1 + x) = 650 * 0.30/1.30 = 150. SI − TD = 195 − 150 = 45, exactly as given.

Why Other Options Are Wrong:
675, 700, 625, 600 do not reproduce the exact 45 difference when plugged into SI and TD formulas at 15% for 2 years.

Common Pitfalls:
Forgetting to square x in the numerator; using present worth instead of face value; or mixing up SI − TD with TD − SI.

Final Answer:
₹ 650