Simple Interest – Mixed investments at 7%, 8%, and 10% yield ₹ 561 yearly: A man invests 1/3 of his capital at 7%, 1/4 at 8%, and the remainder at 10%. His total annual income is ₹ 561. Find the total capital.

Introduction / Context:
This is a weighted-average simple-interest problem: parts of the capital earn different rates, and the total annual income is known. By summing the contributions, we can solve for the total capital.

Given Data / Assumptions:

• Fractions: 1/3 at 7%, 1/4 at 8%, remainder at 10%
• Total annual income = ₹ 561
• Let total capital be C

Concept / Approach:
Compute the effective rate on the whole capital as the weighted sum: (1/3)*7% + (1/4)*8% + (remainder)*10%. The remainder fraction is 1 − 1/3 − 1/4 = 5/12. Multiply the effective rate by C to match the given income.

Step-by-Step Solution:
Effective rate = (1/3)*7 + (1/4)*8 + (5/12)*10= 7/3 + 2 + 50/12 = 2.333... + 2 + 4.166... = 8.5%Thus 0.085 * C = 561 ⇒ C = 561 / 0.085 = 6,600

Verification / Alternative check:
Income from parts: (C/3)*7% + (C/4)*8% + (5C/12)*10% = C*(0.085) = 561. Substituting C = 6,600 gives 561, confirming the result.

Why Other Options Are Wrong:
Other totals produce annual incomes not equal to ₹ 561 at the computed effective rate of 8.5%.

Common Pitfalls:
Forgetting to convert percentages to decimals, miscomputing the remainder fraction, or adding rates directly without weighting can cause mistakes.

Final Answer:
₹ 6,600