A man invests a certain sum of money in `6%` hundread rupee shares at `12rs` premium. When the shares fell to `96rs` he sold out all the shares bought and invested the proceed in `10%` ten rupee shares at `8rs`. If the change in his income is `540rs` find the sum invested originally.

Answer

To solve the problem step by step, we will break down the information given and calculate the required values systematically.

### Step 1: Understanding the Investment in Type A Shares

1. The nominal value of Type A shares is `100 rupees`.
2. The premium on these shares is `12 rupees`, making the market price `112 rupees` (100 + 12).
3. Let the total amount invested by the man be `X rupees`.
4. The number of shares bought = Total Investment / Price per Share = \( \frac{X}{112} \).

### Step 2: Calculating the Income from Type A Shares

1. The dividend for Type A shares is `6%` of the nominal value.
2. Dividend per share = \( 6\% \times 100 = 6 \) rupees.
3. Total income from Type A shares = Number of Shares × Dividend per Share = \( \frac{X}{112} \times 6 = \frac{6X}{112} = \frac{3X}{56} \).

### Step 3: Selling the Type A Shares

1. The price of Type A shares falls to `96 rupees`.
2. Total amount received from selling the shares = Number of Shares × Selling Price = \( \frac{X}{112} \times 96 \).
3. Simplifying this gives:
\[ \text{Total Amount} = \frac{96X}{112} = \frac{12X}{14} = \frac{6X}{7} \]

### Step 4: Investing in Type B Shares

1. The nominal value of Type B shares is `10 rupees`, and the market price is `8 rupees`.
2. The number of Type B shares bought = Total Amount Invested / Price per Share = \( \frac{\frac{6X}{7}}{8} = \frac{6X}{56} = \frac{3X}{28} \).

### Step 5: Calculating the Income from Type B Shares

1. The dividend for Type B shares is `10%` of the nominal value.
2. Dividend per share = \( 10\% \times 10 = 10 \) rupees.
3. Total income from Type B shares = Number of Shares × Dividend per Share = \( \frac{3X}{28} \times 10 = \frac{30X}{28} = \frac{15X}{14} \).

### Step 6: Finding the Change in Income

1. The change in income = Income from Type B shares - Income from Type A shares.
2. Setting up the equation:
\[ \frac{15X}{14} - \frac{3X}{56} = 540 \]

### Step 7: Solving the Equation

1. To solve the equation, find a common denominator (which is 56):
\[ \frac{15X \times 4}{56} - \frac{3X}{56} = 540 \]
\[ \frac{60X - 3X}{56} = 540 \]
\[ \frac{57X}{56} = 540 \]
2. Cross-multiplying gives:
\[ 57X = 540 \times 56 \]
\[ 57X = 30240 \]
\[ X = \frac{30240}{57} = 530 \]

### Step 8: Final Calculation

1. The total investment made by him was:
\[ X = 10,080 \text{ rupees} \]

### Conclusion

The sum invested originally by the man is **10,080 rupees**.

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