A man has some shares of ₹100 par value paying 6% dividend. He sells half of these at a discount of 10% and invests the proceeds in 7% ₹50 shares at a premium of ₹10. This transaction decreases his income from dividends by ₹120.

Calculate:

(i) the number of shares before the transaction.

(ii) the number of shares he sold.

(iii) his initial annual income from shares.

Let the number of 6% ₹100 shares held by the man be x

Number of shares sold by the man = x/2

As the 6% ₹100 shares were at par,
∴ Nominal Value = Market Value = ₹100

As the shares were sold at a discount of 10%,
∴ Selling price of one share = ₹100 - 10% of ₹100 = ₹100 - ₹10 = ₹90

$\therefore \text{Sales proceeds} = 90 \times \dfrac{x}{2} \\[0.5em] = ₹45x$

Market Value of 7% ₹50 shares at a premium of ₹10 = ₹50 + ₹10 = ₹60

Number of 7% ₹50 shares purchased by the man

$= \dfrac{45x}{60} \\[0.5em] = \dfrac{3x}{4}$

Annual Income from previous shares = No. of shares x Rate of Dividend x Nominal Value per share

$= x \times \dfrac{6}{100} \times 100 \\[0.5em] = \bold{₹6x}$

Annual Income from new shares = No. of shares x Rate of Dividend x Nominal Value per share

$= \dfrac{3x}{4} \times \dfrac{7}{100} \times 50 \\[0.5em] = \bold{₹\dfrac{21x}{8}}$

New Annual Income = Annual income from (x/2) 6% ₹100 shares + Annual income from (3x/4) 7% ₹50 shares

$= \dfrac{6x}{2} + \dfrac{21x}{8} \\[0.5em] = 3x + \dfrac{21x}{8} \\[0.5em] = \dfrac{24x + 21x}{8} \\[0.5em] = \dfrac{45x}{8}$

As per the given,

$6x - \dfrac{45x}{8} = 120 \\[0.5em] \Rightarrow \dfrac{48x - 45x}{8} = 120 \\[0.5em] \Rightarrow \dfrac{3x}{8} = 120 \\[0.5em] \Rightarrow x = \dfrac{120 \times 8}{3} \\[0.5em] \Rightarrow x = 320 \\[0.5em]$

(i) Number of shares before the transaction = x = 320

(ii) Number of shares sold = x / 2 = 160

(iii) Initial Income = 6x = 6 x 320 = 1920