A man invested one-third of his capital at 7%, one-fourth at 8%, and the remainder at 10% simple interest, respectively. If his annual income is Rs. 561, the capital is worth of:
A) Rs. 5400
B) Rs. 6000
C) Rs. 6600
D) Rs. 7200

To solve this problem, we need to find the total capital based on the income generated from different investments at different interest rates.

Let the total capital be C.

Step 1: Divide the capital into three parts
One-third of the capital is invested at 7% simple interest:
Amount invested = (1/3)C
Interest = (1/3)C * 7% = (1/3)C * (7/100) = (7/300)C

One-fourth of the capital is invested at 8% simple interest:
Amount invested = (1/4)C
Interest = (1/4)C * 8% = (1/4)C * (8/100) = (8/400)C = (2/100)C = (1/50)C

The remaining amount is invested at 10% simple interest:
The remaining capital is the rest of the total capital, which is:
Remaining capital = C - (1/3)C - (1/4)C
Finding a common denominator:
C - (1/3)C - (1/4)C = (12/12)C - (4/12)C - (3/12)C = (5/12)C
Interest = (5/12)C * 10% = (5/12)C * (10/100) = (50/1200)C = (1/24)C

Step 2: Add the interests
The total annual income is the sum of the interests from all three investments, which is given as Rs. 561.

Thus,
(7/300)C + (1/50)C + (1/24)C = 561

Step 3: Solve the equation
To solve this equation, we need a common denominator. The least common denominator of 300, 50, and 24 is 600.

Rewriting the terms with a denominator of 600:

(7/300)C = (14/600)C
(1/50)C = (12/600)C
(1/24)C = (25/600)C

So the equation becomes:

(14/600)C + (12/600)C + (25/600)C = 561
(51/600)C = 561
C = 561 * (600/51)
C = 561 * 11.7647
C = 6600

Final Answer:
The total capital is Rs. 6600.

Thus, the correct answer is C) Rs. 6600.