A person invested one-fifth of the capital at 5% per annum, one-sixth of the capital at 6% per annum and the rest at 10% per annum simple interest. If the annual interest received on his investment is Rs. 150, then find the capital. (in Rs. )

Options

A. 1000
B. 1500
C. 2000
D. 1800

Explanation

To solve the problem step by step, let's break it down:

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

### Step 2: Calculate the Investments
1. The amount invested at 5% per annum:
\[ \text{Investment}_1 = \frac{1}{5}x \]

2. The amount invested at 6% per annum:
\[ \text{Investment}_2 = \frac{1}{6}x \]

3. The remaining amount invested at 10% per annum:
\[ \text{Investment}_3 = x - \left(\frac{1}{5}x + \frac{1}{6}x\right) \]

### Step 3: Simplify the Remaining Investment
To simplify \( Investment_3 \):
\[ \text{Investment}_3 = x - \left(\frac{6}{30}x + \frac{5}{30}x\right) = x - \frac{11}{30}x = \frac{19}{30}x \]

### Step 4: Calculate the Interests
1. Interest from the first investment:
\[ \text{Interest}_1 = \frac{1}{5}x \times \frac{5}{100} = \frac{x}{100} \]

2. Interest from the second investment:
\[ \text{Interest}_2 = \frac{1}{6}x \times \frac{6}{100} = \frac{x}{100} \]

3. Interest from the third investment:
\[ \text{Interest}_3 = \frac{19}{30}x \times \frac{10}{100} = \frac{19x}{300} \]

### Step 5: Total Interest
The total interest received from all investments is given as Rs. 150:
\[ \text{Total Interest} = \text{Interest}_1 + \text{Interest}_2 + \text{Interest}_3 \]
\[ \frac{x}{100} + \frac{x}{100} + \frac{19x}{300} = 150 \]

### Step 6: Combine the Interests
To combine the terms, find a common denominator (which is 300):
\[ \frac{3x}{300} + \frac{3x}{300} + \frac{19x}{300} = 150 \]
\[ \frac{25x}{300} = 150 \]

### Step 7: Solve for \( x \)
Multiply both sides by 300:
\[ 25x = 150 \times 300 \]
\[ 25x = 45000 \]
Now, divide by 25:
\[ x = \frac{45000}{25} = 1800 \]

### Conclusion
The capital is Rs. 1800.