Mohan had some shares with face- value Rs 100 paying 8% dividend . He sold them for Rs 130 each and invested the proceed in Rs 50 shares of some other company at Rs 75 and paying `12%` divident. If by doing so his annual income is increased by Rs 360, find the number of shares Mohan had.

Answer

To solve the problem step by step, we will break down the information given and perform the necessary calculations.

### Step 1: Define the variables
Let the number of shares Mohan had be \( a \).

### Step 2: Calculate the annual income from the original shares
The face value of each share is Rs 100, and the dividend rate is 8%.
The annual income from one share can be calculated as:
\[ \text{Dividend per share} = \frac{8}{100} \times 100 = 8 \text{ Rs} \]
Thus, the total annual income from \( a \) shares is:
\[ \text{Total income from original shares} = 8a \text{ Rs} \]

### Step 3: Calculate the proceeds from selling the shares
Mohan sold each share for Rs 130. Therefore, the total proceeds from selling \( a \) shares is:
\[ \text{Total proceeds} = 130a \text{ Rs} \]

### Step 4: Calculate the number of new shares purchased
Mohan invested the proceeds in Rs 50 shares of another company, which are priced at Rs 75 each. The number of new shares he can buy is:
\[ \text{Number of new shares} = \frac{\text{Total proceeds}}{\text{Price per new share}} = \frac{130a}{75} = \frac{26a}{15} \]

### Step 5: Calculate the annual income from the new shares
The new shares pay a dividend of 12%. The dividend per new share can be calculated as:
\[ \text{Dividend per new share} = \frac{12}{100} \times 50 = 6 \text{ Rs} \]
Thus, the total annual income from the new shares is:
\[ \text{Total income from new shares} = 6 \times \left(\frac{26a}{15}\right) = \frac{156a}{15} = \frac{52a}{5} \text{ Rs} \]

### Step 6: Set up the equation for the increase in income
According to the problem, the increase in annual income is Rs 360. Therefore, we can set up the equation:
\[ \text{New income} - \text{Old income} = 360 \]
Substituting the expressions we calculated:
\[ \frac{52a}{5} - 8a = 360 \]

### Step 7: Solve the equation
To solve for \( a \), first, we need to express \( 8a \) with a common denominator:
\[ \frac{52a}{5} - \frac{40a}{5} = 360 \]
This simplifies to:
\[ \frac{12a}{5} = 360 \]
Now, multiply both sides by 5:
\[ 12a = 1800 \]
Now, divide by 12:
\[ a = \frac{1800}{12} = 150 \]

### Conclusion
The number of shares Mohan had is \( \boxed{150} \).