By selling at ₹92, some 2.5% ₹100 shares and investing the proceeds in 5% ₹100 shares at ₹115, a person increased his annual income by ₹90. Find:

(i) the number of shares sold.

(ii) the number of shares purchased.

(iii) the new income.

(iv) the rate percent which he earns on his investment.

(i)
Let the number of shares sold by the man be x

Selling price of one share = ₹92
∴ Sales proceeds = ₹92x

Number of ₹100 5% shares purchased by the man

$= \dfrac{92x}{115} \\[0.5em] = \dfrac{4x}{5}$

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

$= x \times \dfrac{2.5}{100} \times 100 \\[0.5em] = \dfrac{25x}{10} \\[0.5em] = \bold{₹\dfrac{5x}{2}}$

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

$= \dfrac{4x}{5} \times \dfrac{5}{100} \times 100 \\[0.5em] = \bold{₹4x}$

As per the given,

$4x - \dfrac{5x}{2} = 90 \\[0.5em] \Rightarrow \dfrac{8x - 5x}{2} = 90 \\[0.5em] \Rightarrow \dfrac{3x}{2} = 90 \\[0.5em] \Rightarrow x = \dfrac{90 \times 2}{3} \\[0.5em] \Rightarrow x = 60 \\[0.5em]$

∴ Number of shares sold = 60

(ii)

$\text{No. of shares purchased} = \dfrac{4x}{5} \\[0.5em] = \dfrac{4 \times 60}{5} \\[0.5em] = 48$

∴ Number of shares purchased = 48

(iii)
Annual Income from new shares = ₹4x [From part (i) above]
= ₹(4 x 60)
= ₹240

(iv)
Total Investment = ₹(48 x 115) = ₹5520

$\% \text{Return} = \Big(\dfrac{\text{Annual Inc.}}{\text{Investment}} \times 100\Big)\% \\[0.5em] = \Big(\dfrac{240}{5520} \times 100\Big) \% \\[0.5em] = \Big(\dfrac{2400}{552} \Big) \% \\[0.5em] = \Big(\dfrac{100}{23} \Big) \% \\[0.5em] = \bold{4\dfrac{8}{23}\%}$