Gagan invested `80%` of his savings in `10%` `100rs` shares at `20%` premium and the rest of his savings in `20%` `50rs` shares at `20%` discount. If his incomes from these shares is `5600rs` calculate
(i) his investment in shares on the whole
(ii) the number of shares of first kind that he bought.
(iii) percentage return on the shares bought on the whole

Answer

To solve the problem step by step, we will break down the calculations for Gagan's investments and returns from the shares.

### Step 1: Calculate the Market Value of Each Share Type
1. **For the 10% Rs. 100 shares at 20% premium:**
- Market Value (MV) = Nominal Value + Premium
- MV = 100 + (20% of 100) = 100 + 20 = Rs. 120

2. **For the 20% Rs. 50 shares at 20% discount:**
- Market Value (MV) = Nominal Value - Discount
- MV = 50 - (20% of 50) = 50 - 10 = Rs. 40

### Step 2: Define the Total Savings
Let the total savings be denoted as \( TS \).

### Step 3: Calculate Investments in Each Share Type
1. **Investment in 10% Rs. 100 shares:**
- Investment = 80% of total savings = \( 0.8 \times TS \)

2. **Investment in 20% Rs. 50 shares:**
- Investment = 20% of total savings = \( 0.2 \times TS \)

### Step 4: Calculate the Number of Shares Purchased
1. **Number of 10% Rs. 100 shares:**
- Number of shares = Investment / Market Value
- Number of shares = \( \frac{0.8 \times TS}{120} \)

2. **Number of 20% Rs. 50 shares:**
- Number of shares = Investment / Market Value
- Number of shares = \( \frac{0.2 \times TS}{40} \)

### Step 5: Calculate Income from Each Share Type
1. **Income from 10% Rs. 100 shares:**
- Income = Number of shares × Dividend per share
- Income = \( \left(\frac{0.8 \times TS}{120}\right) \times 10 \)

2. **Income from 20% Rs. 50 shares:**
- Income = Number of shares × Dividend per share
- Income = \( \left(\frac{0.2 \times TS}{40}\right) \times 10 \)

### Step 6: Set Up the Total Income Equation
According to the problem, the total income from both types of shares is Rs. 5600.
- Total Income = Income from 10% shares + Income from 20% shares
- \( \left(\frac{0.8 \times TS}{120}\right) \times 10 + \left(\frac{0.2 \times TS}{40}\right) \times 10 = 5600 \)

### Step 7: Simplify the Equation
- \( \frac{8 \times TS}{120} + \frac{2 \times TS}{40} = 5600 \)
- Convert to a common denominator (120):
- \( \frac{8 \times TS}{120} + \frac{6 \times TS}{120} = 5600 \)
- \( \frac{14 \times TS}{120} = 5600 \)

### Step 8: Solve for Total Savings (TS)
- \( 14 \times TS = 5600 \times 120 \)
- \( TS = \frac{5600 \times 120}{14} \)
- \( TS = 48000 \)

### Step 9: Calculate Investment in Shares on the Whole
- Total Investment = \( TS = 48000 \)

### Step 10: Calculate the Number of Shares of First Kind
- Number of 10% Rs. 100 shares = \( \frac{0.8 \times 48000}{120} \)
- Number of shares = \( \frac{38400}{120} = 320 \)

### Step 11: Calculate Percentage Return on the Shares Bought
- Total Income = Rs. 5600
- Percentage Return = \( \frac{Total Income}{Total Investment} \times 100 \)
- Percentage Return = \( \frac{5600}{48000} \times 100 = \frac{35}{3} \approx 11.67\% \)

### Final Answers:
(i) Total Investment in shares = Rs. 48000
(ii) Number of shares of first kind (10% Rs. 100 shares) = 320
(iii) Percentage return on the shares bought = \( \approx 11.67\% \)