Divide `50760rs` into two parts such that if one part is invested in `8%` `100rs` shares at `8%` discount and the other in `9%` `100rs` shares at `8%` premium the annual incomes from both the investments are equal

Answer

To solve the problem of dividing ₹50,760 into two parts such that the annual incomes from both investments are equal, we can follow these steps:

### Step 1: Define the Variables
Let:
- \( x \) = amount invested in 8% shares at 8% discount
- \( 50,760 - x \) = amount invested in 9% shares at 8% premium

### Step 2: Calculate the Effective Price of the Shares
For the first investment:
- The face value of the shares = ₹100
- Discount = 8% of ₹100 = ₹8
- Effective price of shares = ₹100 - ₹8 = ₹92

For the second investment:
- The face value of the shares = ₹100
- Premium = 8% of ₹100 = ₹8
- Effective price of shares = ₹100 + ₹8 = ₹108

### Step 3: Calculate the Annual Income from Each Investment
The annual income from the first investment (8% shares):
- Income = \( \frac{8}{100} \times \frac{x}{92} \times 100 \)
- Simplifying this gives:
\[ \text{Income}_1 = \frac{8x}{92} \]

The annual income from the second investment (9% shares):
- Income = \( \frac{9}{100} \times \frac{50,760 - x}{108} \times 100 \)
- Simplifying this gives:
\[ \text{Income}_2 = \frac{9(50,760 - x)}{108} \]

### Step 4: Set the Incomes Equal
According to the problem, the incomes from both investments are equal:
\[ \frac{8x}{92} = \frac{9(50,760 - x)}{108} \]

### Step 5: Cross Multiply to Solve for \( x \)
Cross multiplying gives:
\[ 8x \cdot 108 = 9(50,760 - x) \cdot 92 \]
Expanding both sides:
\[ 864x = 9 \cdot 50,760 \cdot 92 - 9 \cdot 92x \]
Combining like terms:
\[ 864x + 828x = 9 \cdot 50,760 \cdot 92 \]
\[ 1692x = 9 \cdot 50,760 \cdot 92 \]

### Step 6: Calculate the Value of \( x \)
Calculating the right side:
\[ 9 \cdot 50,760 \cdot 92 = 42,000,000 \]
Now, dividing both sides by 1692:
\[ x = \frac{42,000,000}{1692} \approx 24,840 \]

### Step 7: Calculate the Second Investment
The second investment is:
\[ 50,760 - x = 50,760 - 24,840 = 25,920 \]

### Final Answer
Thus, the amounts invested are:
- First investment (8% shares at 8% discount): ₹24,840
- Second investment (9% shares at 8% premium): ₹25,920