A man invests equal amounts of money in two companies A and B. Company A pays a dividend of 15% and its Rs100 shares are available at 20% discount. The shares of company B has a nominal value of Rs25 and are available at 20% premium. If at the end of one year, the man gets equal dividends from both the companies, find the rate of dividend paid by company B.

Answer

To solve the problem step by step, we will analyze the investments in both companies A and B, calculate the dividends, and then find the rate of dividend paid by company B.

### Step 1: Define the investment in company A
Let the amount invested in each company be \( S \).

### Step 2: Calculate the market value of shares in company A
The nominal value of shares in company A is Rs. 100, and they are available at a 20% discount. Therefore, the market value of shares in company A is:
\[ \text{Market Value of A} = 100 - (20\% \text{ of } 100) = 100 - 20 = 80 \]

### Step 3: Calculate the number of shares purchased in company A
The number of shares purchased in company A can be calculated as:
\[ \text{Number of shares in A} = \frac{S}{\text{Market Value of A}} = \frac{S}{80} \]

### Step 4: Calculate the dividend income from company A
The dividend paid by company A is 15%. Therefore, the dividend income from company A is:
\[ \text{Income from A} = \text{Number of shares in A} \times \text{Dividend} \times \frac{1}{100} = \frac{S}{80} \times 15 \times \frac{1}{100} \times 100 \]
This simplifies to:
\[ \text{Income from A} = \frac{15S}{80} = \frac{3S}{16} \]

### Step 5: Define the investment in company B
The nominal value of shares in company B is Rs. 25, and they are available at a 20% premium. Therefore, the market value of shares in company B is:
\[ \text{Market Value of B} = 25 + (20\% \text{ of } 25) = 25 + 5 = 30 \]

### Step 6: Calculate the number of shares purchased in company B
The number of shares purchased in company B can be calculated as:
\[ \text{Number of shares in B} = \frac{S}{\text{Market Value of B}} = \frac{S}{30} \]

### Step 7: Calculate the dividend income from company B
Let the rate of dividend paid by company B be \( D \)%. The dividend income from company B is:
\[ \text{Income from B} = \text{Number of shares in B} \times \text{Dividend} \times \frac{1}{100} = \frac{S}{30} \times D \times \frac{1}{100} \times 25 \]
This simplifies to:
\[ \text{Income from B} = \frac{SD}{120} \]

### Step 8: Set the income from both companies equal
According to the problem, the income from both companies is equal:
\[ \frac{3S}{16} = \frac{SD}{120} \]

### Step 9: Solve for \( D \)
We can cancel \( S \) from both sides (assuming \( S \neq 0 \)):
\[ \frac{3}{16} = \frac{D}{120} \]
Cross-multiplying gives:
\[ 3 \times 120 = 16D \]
\[ 360 = 16D \]
\[ D = \frac{360}{16} = 22.5 \]

### Conclusion
The rate of dividend paid by company B is \( 22.5\% \).