Ahmad had $1000$ shares of a company with a face value of $₹ 50$ and paying $10 %$ dividend. He sold some of these shares at a discount of $10 %$ and invested the proceeds in $₹ 20$ shares at a premium of $50 %$ and paying $10 %$ dividend. If the changes in his income is $₹ 200$ , then find the number of shares sold by Ahmad.

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Solution

Step 1: Given Data
Number of shares sold, $N_{1} = 1000$
Face value of shares sold, $F_{1} = ₹ 50$
Dividend of shares sold, $D_{1} = 10 %$
Discount% on some shares sold, $D % = 10 %$
Face value of some shares bought, $F_{2} = ₹ 20$
Dividend of some shares bought, $D_{2} = 10 %$
Premium% on some shares bought, $P % = 50 %$
Change in Ahmad's annual income $= ₹ 200$
Step 2: Calculate the sale proceeds of Ahmad on shares sold
Let us assume that the number of shares sold be $x$ . Then,
$S a l e p r o c e e d s = N u m b e r o f s h a r e s \times M a r k e t v a l u e o f x s h a r e s$
$= x \times (F_{1} - D % \times F_{1}) [∵ M a r k e t v a l u e = F a c e v a l u e - D i s c o u n t % o f F a c e v a l u e] = x \times (50 - \frac{10}{100} \times 50) = x \times (50 - 5) = ₹ 45 x$
Step 3: Calculate the number of shares bought by Ahmad
Ahmad bought some shares from the sale proceeds earned from selling some of the shares he sold. He bought $₹ 20$ shares at a premium of $50 %$ .
∴ $M a r k e t v a l u e = F a c e v a l u e + P r e m i u m % o f F a c e v a l u e$
$= F_{2} + P % o f F_{2} = 20 + \frac{50}{100} \times 20 = 20 + 10 = ₹ 30$
$N u m b e r o f s h a r e s = \frac{I n v e s t m e n t}{M a r k e t v a l u e}$
$N_{2} = \frac{S a l e p r o c e e d s}{M_{2}} = \frac{45 x}{30} = \frac{3 x}{2} shares$
Step 4: Calculate the annual income from the shares sold
$A n n u a l i n c o m e = N u m b e r o f s h a r e s \times N o m i n a l v a l u e \times \frac{R a t e o f d i v i d e n d}{100}$
$A_{1} = x \times F_{1} \times \frac{D_{1}}{100} = x \times 50 \times \frac{10}{100} = ₹ 5 x$
Step 5: Calculate the annual income from the shares bought
$A n n u a l i n c o m e = N u m b e r o f s h a r e s \times N o m i n a l v a l u e \times \frac{R a t e o f d i v i d e n d}{100}$
$A_{2} = N_{2} \times F_{2} \times \frac{D_{2}}{100} = \frac{3 x}{2} \times 20 \times \frac{10}{100} = ₹ 3 x$
Step 6: Calculate the difference in the annual incomes of Ahmad
As it is mentioned that there is a change of $₹ 200$ in the annual income of Ahmad.
∴ $D i f f e r e n c e i n a n n u a l i n c o m e s = A_{1} - A_{2}$
$200 = 5 x - 3 x 200 = 2 x x = \frac{200}{2} x = 100 shares$
Final Answer: The number of shares sold by Ahmad is $100$ .

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