A man invested `(2)/(5)` of his capital at 8% p.a., `(3)/(8)` of his capital at 10% p.a. and the remaining at 12% p.a. If his annual income at simple interest is Rs 965, then his capital is _____

Options

A. Rs 8000
B. Rs 9000
C. Rs 10000
D. Rs 11000

Answer

To solve the problem step by step, we will break down the investments and their corresponding interests, and then set up an equation based on the total annual income.

### Step 1: Define the total capital
Let the total capital be \( X \).

### Step 2: Calculate the amounts invested at different rates
1. The amount invested at 8%:
\[ \text{Investment at 8%} = \frac{2}{5}X \]

2. The amount invested at 10%:
\[ \text{Investment at 10%} = \frac{3}{8}X \]

3. Calculate the remaining amount invested at 12%:
\[ \text{Remaining investment} = X - \left(\frac{2}{5}X + \frac{3}{8}X\right) \]

### Step 3: Find a common denominator to combine the fractions
To combine \(\frac{2}{5}X\) and \(\frac{3}{8}X\), we need a common denominator. The least common multiple of 5 and 8 is 40.

1. Convert \(\frac{2}{5}X\):
\[ \frac{2}{5}X = \frac{16}{40}X \]

2. Convert \(\frac{3}{8}X\):
\[ \frac{3}{8}X = \frac{15}{40}X \]

3. Now combine:
\[ \frac{16}{40}X + \frac{15}{40}X = \frac{31}{40}X \]

4. Thus, the remaining investment at 12% is:
\[ \text{Investment at 12%} = X - \frac{31}{40}X = \frac{9}{40}X \]

### Step 4: Calculate the simple interest for each investment
1. Simple interest from the investment at 8%:
\[ SI_1 = \frac{2}{5}X \cdot \frac{8}{100} \cdot 1 = \frac{16}{500}X = \frac{16X}{500} \]

2. Simple interest from the investment at 10%:
\[ SI_2 = \frac{3}{8}X \cdot \frac{10}{100} \cdot 1 = \frac{30}{800}X = \frac{3X}{80} \]

3. Simple interest from the investment at 12%:
\[ SI_3 = \frac{9}{40}X \cdot \frac{12}{100} \cdot 1 = \frac{108}{4000}X = \frac{27X}{1000} \]

### Step 5: Set up the equation for total annual income
The total annual income from all investments is given as Rs 965:
\[ SI_1 + SI_2 + SI_3 = 965 \]
Substituting the values:
\[ \frac{16X}{500} + \frac{3X}{80} + \frac{27X}{1000} = 965 \]

### Step 6: Find a common denominator for the left-hand side
The least common multiple of 500, 80, and 1000 is 2000.

1. Convert each term:
- \(\frac{16X}{500} = \frac{64X}{2000}\)
- \(\frac{3X}{80} = \frac{75X}{2000}\)
- \(\frac{27X}{1000} = \frac{54X}{2000}\)

2. Combine:
\[ \frac{64X + 75X + 54X}{2000} = \frac{193X}{2000} \]

### Step 7: Solve for \( X \)
Setting the equation:
\[ \frac{193X}{2000} = 965 \]
Cross-multiplying gives:
\[ 193X = 965 \times 2000 \]
Calculating the right-hand side:
\[ 965 \times 2000 = 1930000 \]
Thus:
\[ X = \frac{1930000}{193} = 10000 \]

### Final Answer
The total capital is Rs 10,000.

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