Simple Interest – Weighted-rate portfolio on one principal: Two-thirds of a sum is lent at 3% per annum, one-sixth at 6% per annum, and the remaining one-sixth at 12% per annum, all at simple interest. If the annual income is ₹ 25, what is the total principal?

Introduction / Context:
This is a classic weighted-average rate problem under simple interest: different fractions of the same principal earn different rates, producing a combined annual interest figure.

Given Data / Assumptions:

• S is the total principal
• Fractions: 2/3 at 3%, 1/6 at 6%, 1/6 at 12%
• Annual simple interest (1 year) = ₹ 25

Concept / Approach:
Total annual SI = S * ( (2/3)*0.03 + (1/6)*0.06 + (1/6)*0.12 ).

Step-by-Step Solution:
(2/3)*0.03 = 0.02(1/6)*0.06 = 0.01(1/6)*0.12 = 0.02Weighted rate = 0.02 + 0.01 + 0.02 = 0.05 per yearSo 0.05 * S = 25 → S = 25 / 0.05 = 500

Verification / Alternative check:
Compute each part’s interest on ₹ 500 for 1 year: ₹ (2/3*500)*3% = 333.33*0.03 ≈ 10; ₹ (1/6*500)*6% ≈ 0.06*83.33 = 5; ₹ (1/6*500)*12% ≈ 10; total ≈ 25.

Why Other Options Are Wrong:
₹ 450, ₹ 600, ₹ 650 give annual interest 22.5, 30, and 32.5 respectively at the same weighted rate.

Common Pitfalls:
Adding the nominal rates without weighting by fractions; mixing monthly and annual reasoning.

Final Answer:
₹ 500