Divide ₹ 50,760 into two parts such that if one part is invested in 8%, ₹ 100 shares at 8% discount and the other in 9%, ₹ 100 shares at 8% premium, the annual incomes from both the investments are equal.

Given,

Total Investment = ₹ 50,760

Let the first part invested in 8%, ₹ 100 shares at 8% discount be ₹ x.

Second part = ₹ 50,760 − ₹ x

For the first investment :

Face Value = ₹ 100

Discount Rate = 8%

Discount = 8% of 100 = $\dfrac{8}{100} \times 100$ = ₹ 8

Market Value = Face Value - Discount = ₹ 92

Dividend Rate = 8%

By formula,

Number of shares = $\dfrac{\text{Investment}}{\text{Market value of each share}} = \dfrac{x}{92}$

By formula,

$\text{Income from first part} =\text{ No. of shares × Rate of div. × N.V. of 1 share}\\[1em] = \dfrac{x}{92} \times \dfrac{8}{100} \times {100}\\[1em] = \dfrac{8x}{92}.$

For the second investment :

Face Value = ₹ 100

Premium Rate = 8%

Premium = 8% of 100 = $\dfrac{8}{100} \times 100$ = ₹ 8

Market Value = Face Value + Premium = ₹ 108

Dividend Rate = 9%

Number of shares = $\dfrac{\text{Investment}}{\text{Market value of each share}} = \dfrac{50760 - x}{108}$

By formula,

$\text{Income from second part} = \text{No. of shares × Rate of div. × N.V. of 1 share}\\[1em] = \dfrac{50760 - x}{108} \times \dfrac{9}{100} \times {100} \\[1em] = \dfrac{50760 - x}{108} \times 9$

Given,

Income from the both the investments are equal.

$\therefore \dfrac{8x}{92} = \dfrac{50760 - x}{108} \times 9 \\[1em] \Rightarrow 8 \times 108x = 9 \times 92(50760 - x) \\[1em] \Rightarrow 864x = 828(50760 - x) \\[1em] \Rightarrow 864x = 42029280 - 828x \\[1em] \Rightarrow 864x + 828x = 42029280 \\[1em] \Rightarrow 1692x = 42029280 \\[1em] \Rightarrow x = \dfrac{42029280}{1692} = ₹24,840.$

First part = x = ₹ 24,840

Second part = ₹ (50,760 − x) = ₹ 25,920

Hence, first part = ₹ 24,840 and second part = ₹ 25,920.