Mr. Gupta has a choice to invest in ten-rupee shares of two firms at `13rs` or at `16rs`. If the first firm pays `5%` dividend and the second firm pays `6%` dividend per annum, find :
(i) which firm is paying better
(ii) if Mr. Gupta invests equally in both the firms and the difference between the returns from them is `30rs`, find how much, in all does he invest ?

Answer

To solve the problem step by step, we will analyze the investments in both firms and calculate the income percentages to determine which firm is paying better. We will also find out how much Mr. Gupta invests in total if the difference in returns is Rs. 30.

### Step 1: Identify the nominal and market values of shares
- **Nominal value of each share**: Rs. 10
- **Market value of shares**:
- Firm 1: Rs. 13
- Firm 2: Rs. 16

### Step 2: Calculate the dividend for each firm
- **Dividend for Firm 1**:
- Dividend percentage = 5%
- Dividend = 5% of Rs. 10 = \( \frac{5}{100} \times 10 = Rs. 0.50 \)

- **Dividend for Firm 2**:
- Dividend percentage = 6%
- Dividend = 6% of Rs. 10 = \( \frac{6}{100} \times 10 = Rs. 0.60 \)

### Step 3: Calculate the income percentage for each firm
- **Income percentage for Firm 1**:
\[ \text{Income Percentage} = \left( \frac{\text{Income}}{\text{Investment}} \right) \times 100 \]
- Income = Rs. 0.50
- Investment = Rs. 13
\[ \text{Income Percentage for Firm 1} = \left( \frac{0.50}{13} \right) \times 100 \approx 3.846\% \]

- **Income percentage for Firm 2**:
- Income = Rs. 0.60
- Investment = Rs. 16
\[ \text{Income Percentage for Firm 2} = \left( \frac{0.60}{16} \right) \times 100 \approx 3.75\% \]

### Step 4: Compare the income percentages
- Firm 1: 3.846%
- Firm 2: 3.75%

**Conclusion for Part (i)**: Firm 1 is paying better than Firm 2.

### Step 5: Calculate the total investment if Mr. Gupta invests equally in both firms
Let the amount invested in each firm be \( x \).

- **Returns from Firm 1**:
\[ \text{Number of shares} = \frac{x}{13} \]
\[ \text{Return from Firm 1} = \text{Number of shares} \times \text{Dividend} = \frac{x}{13} \times 0.50 = \frac{0.50x}{13} \]

- **Returns from Firm 2**:
\[ \text{Number of shares} = \frac{x}{16} \]
\[ \text{Return from Firm 2} = \text{Number of shares} \times \text{Dividend} = \frac{x}{16} \times 0.60 = \frac{0.60x}{16} \]

### Step 6: Set up the equation for the difference in returns
According to the problem, the difference in returns is Rs. 30:
\[ \left| \frac{0.50x}{13} - \frac{0.60x}{16} \right| = 30 \]

### Step 7: Solve the equation
1. Calculate the common denominator (which is 208):
\[ \frac{0.50x \cdot 16}{208} - \frac{0.60x \cdot 13}{208} = 30 \]
\[ \frac{8x - 7.8x}{208} = 30 \]
\[ \frac{0.2x}{208} = 30 \]
\[ 0.2x = 30 \times 208 \]
\[ 0.2x = 6240 \]
\[ x = \frac{6240}{0.2} = 31200 \]

### Step 8: Calculate total investment
Since Mr. Gupta invests equally in both firms:
\[ \text{Total Investment} = x + x = 2x = 2 \times 31200 = 62400 \]

**Conclusion for Part (ii)**: Mr. Gupta invests a total of Rs. 62,400.