Gopal has some `100rs` shares of company A, paying `10%` dividend. He sells a certain number of these shares at a discount of `20%` and invests the proceeds in `100rs` shares of `60rs` of company B paying `20%` dividend. If his income from the shares sold increases by `18000rs` find the number of shares sold by Gopal

Answer

To solve the problem step-by-step, we will define the variables and calculate the required values based on the information provided.

### Step 1: Define the Variables
Let \( x \) be the number of shares sold by Gopal.

### Step 2: Calculate the Income from Shares of Company A
Gopal holds shares of Company A, which are valued at \( 100 \) Rs each and pay a \( 10\% \) dividend. The income from these shares can be calculated as:
\[ \text{Income from Company A} = \text{Number of shares} \times \text{Dividend per share} = x \times 10\% \times 100 = 10x \text{ Rs} \]

### Step 3: Calculate the Selling Price of Shares
Gopal sells his shares at a \( 20\% \) discount. Therefore, the selling price of each share is:
\[ \text{Selling Price} = 100 - 20\% \times 100 = 100 - 20 = 80 \text{ Rs} \]
Thus, the total proceeds from selling \( x \) shares is:
\[ \text{Total Proceeds} = 80x \text{ Rs} \]

### Step 4: Invest Proceeds in Shares of Company B
Gopal invests the proceeds in shares of Company B, which are valued at \( 60 \) Rs each and pay a \( 20\% \) dividend. The number of shares he can buy from the proceeds is:
\[ \text{Number of shares bought} = \frac{\text{Total Proceeds}}{\text{Price per share}} = \frac{80x}{60} = \frac{4}{3}x \]

### Step 5: Calculate the Income from Shares of Company B
The income from the shares of Company B can be calculated as:
\[ \text{Income from Company B} = \text{Number of shares} \times \text{Dividend per share} = \frac{4}{3}x \times 20\% \times 100 = \frac{4}{3}x \times 20 = \frac{80}{3}x \text{ Rs} \]

### Step 6: Set Up the Equation for the Increase in Income
According to the problem, the increase in income from the shares sold is \( 18000 \) Rs. Therefore, we can set up the equation:
\[ \text{Income from Company B} - \text{Income from Company A} = 18000 \]
Substituting the incomes we calculated:
\[ \frac{80}{3}x - 10x = 18000 \]

### Step 7: Solve the Equation
To solve for \( x \), first, we need a common denominator:
\[ \frac{80}{3}x - \frac{30}{3}x = 18000 \]
This simplifies to:
\[ \frac{50}{3}x = 18000 \]
Now, multiply both sides by \( 3 \):
\[ 50x = 54000 \]
Now, divide by \( 50 \):
\[ x = \frac{54000}{50} = 1080 \]

### Conclusion
The number of shares sold by Gopal is \( 1080 \).

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