Divide ₹ 35,400 into two parts such that if one part is invested in 9%, ₹ 100 shares at 4% discount, and the other in 12%, ₹ 50 shares at 8% premium, the annual incomes from both investments are equal.

Given,

Total Investment = ₹ 35,400

Let the investments be ₹ x and ₹ 35,400 - x.

For the first investment,

Face Value = ₹ 100

Discount Rate = 4%

Discount = 4% of 100 = $\dfrac{4}{100} \times 100 = ₹ 4$

Market Value = Face Value - Discount = ₹ 96

Dividend Rate = 9%

By formula,

Number of shares = $\dfrac{\text{Investment}}{\text{Market value of each share}} = \dfrac{x}{96}$

$\text{Income from first part} = \text{No. of shares × Rate of div. × N.V. of 1 share}\\[1em] = \dfrac{x}{96} \times \dfrac{9}{100} \times {100}\\[1em]= \dfrac{9x}{96} \\[1em] = \dfrac{3x}{32}.$

For the second investment,

Face Value = ₹ 50

Premium Rate = 8%

Premium = 8% of 50 = $\dfrac{8}{100} \times 50$ = ₹ 4

Market Value = Face Value + Premium = ₹ 54

Dividend Rate = 12%

By formula,

Number of shares = $\dfrac{\text{Investment}}{\text{Market value of each share}} = \dfrac{35400 - x}{54}$

$\text{Income from second part} =\text{ No. of shares × Rate of div. × N.V. of 1 share}\\[1em] =\dfrac{35400 - x}{54} \times \dfrac{12}{100} \times 50\\[1em] = \dfrac{35400 - x}{54} \times 6 \\[1em] = \dfrac{35400 - x}{9}.$

Given,

Income from the both the investments are equal.

$\therefore \dfrac{3x}{32} = \dfrac{35400 - x}{9} \\[1em] \Rightarrow 9 \times 3x = 32(35400 - x) \\[1em] \Rightarrow 27x = 1132800 - 32x \\[1em] \Rightarrow 27x + 32x = 1132800 \\[1em] \Rightarrow 59x = 1132800 \\[1em] \Rightarrow x = \dfrac{1132800}{59} \\[1em] \Rightarrow x = ₹ 19,200.$

First part = x = ₹ 19,200

Second part = ₹ (35,400 - x) = ₹ 35,400 - ₹ 19,200 = ₹ 16,200

Hence, first part = ₹ 19,200 and second part = ₹ 16,200.