Mr. Ram Gopal invested `8000rs` in `7%` `100rs` shares at `80rs`. After a year, he sold these shares at `75rs`each and invested the proceeds (including his dividend ) in `18%`, `25rs` shares at `41rs` .Find
(i) his dividend for the first year
(ii) his annual income in the second year
(iii) the percentage increase in his return on his original investment.

Answer

To solve the problem step by step, we will break down each part of the question systematically.

### Given Data:
- Initial Investment = ₹8000
- Dividend Rate for first year = 7%
- Face Value of shares = ₹100
- Market Price of shares = ₹80
- Selling Price after one year = ₹75
- Dividend Rate for second year = 18%
- Face Value of second shares = ₹25
- Market Price of second shares = ₹41

### Step 1: Calculate the number of shares purchased in the first year.
The number of shares can be calculated using the formula:
\[ \text{Number of Shares} = \frac{\text{Investment}}{\text{Market Price}} \]
Substituting the values:
\[ \text{Number of Shares} = \frac{8000}{80} = 100 \]

### Step 2: Calculate the dividend for the first year.
The dividend per share can be calculated as:
\[ \text{Dividend per Share} = \frac{\text{Dividend Rate}}{100} \times \text{Face Value} \]
Substituting the values:
\[ \text{Dividend per Share} = \frac{7}{100} \times 100 = 7 \]
Now, the total dividend for the first year is:
\[ \text{Total Dividend} = \text{Number of Shares} \times \text{Dividend per Share} = 100 \times 7 = 700 \]

### Step 3: Calculate the total proceeds from selling the shares after one year.
The total proceeds from selling the shares can be calculated as:
\[ \text{Sale Proceeds} = \text{Selling Price} \times \text{Number of Shares} \]
Substituting the values:
\[ \text{Sale Proceeds} = 75 \times 100 = 7500 \]

### Step 4: Calculate the total investment for the second year.
The total investment for the second year includes the sale proceeds and the dividend earned from the first year:
\[ \text{Total Investment for Second Year} = \text{Sale Proceeds} + \text{Total Dividend} = 7500 + 700 = 8200 \]

### Step 5: Calculate the number of shares purchased in the second year.
Using the same formula as before:
\[ \text{Number of Shares in Second Year} = \frac{\text{Total Investment}}{\text{Market Price of Second Shares}} = \frac{8200}{41} = 200 \]

### Step 6: Calculate the dividend per share for the second year.
The dividend per share for the second year can be calculated as:
\[ \text{Dividend per Share} = \frac{18}{100} \times 25 = 4.5 \]

### Step 7: Calculate the total annual income in the second year.
The total annual income in the second year is:
\[ \text{Total Income} = \text{Number of Shares} \times \text{Dividend per Share} = 200 \times 4.5 = 900 \]

### Step 8: Calculate the percentage increase in return on the original investment.
The return on the original investment in the first year was ₹700 (dividend). The return in the second year is ₹900. The percentage increase can be calculated as:
\[ \text{Percentage Increase} = \left(\frac{\text{New Return} - \text{Old Return}}{\text{Old Return}}\right) \times 100 = \left(\frac{900 - 700}{700}\right) \times 100 = \frac{200}{700} \times 100 \approx 28.57\% \]

### Final Answers:
(i) His dividend for the first year = ₹700
(ii) His annual income in the second year = ₹900
(iii) The percentage increase in his return on his original investment = 28.57%