Mr. Ghosh sold a certain number of $₹ 20$ shares paying $8 %$ dividend at $₹ 18$ and invested the proceeds in $₹ 10$ shares, paying $12 %$ dividend at $50 %$ premium. If the change in his annual income is $₹ 120$ , then find the number of shares sold by Mr. Ghosh.

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Solution

Step 1: Given data:
Nominal value of the shares sold, $N_{1} = ₹ 20$
Dividend% at which the shares were sold, $D_{1} = 8 %$
Market value of the shares sold, $M_{1} = ₹ 18$
Nominal value of the shares bought, $N_{2} = ₹ 10$
Dividend% of the shares bought, $D_{2} = 12 %$
Premium% of the shares bought, $P % = 50 %$
Change in Mr. Gosh's annual income $= ₹ 120$
Step 2: Calculate the sale proceeds of Mr. Gosh:
Let us assume that the number of shares sold was $n_{1}$ .
$S a l e p r o c e e d s = N u m b e r o f s h a r e s \times M a r k e t v a l u e$
$= n_{1} \times M_{1} = n_{1} \times 18 = ₹ 18 n_{1}$
Step 3: Calculate the number of shares bought at $50 %$ premium:
As the shares were bought at a premium value, the market value of these shares will be greater than the nominal value because
$M a r k e t v a l u e = N o m i n a l v a l u e + P r e m i u m % o f N o m i n a l v a l u e$
$M_{2} = N_{2} + P % o f N_{2} = 10 + \frac{50}{100} \times 10 = 10 + 5 = ₹ 15$
Now,
$N u m b e r o f s h a r e s = \frac{I n v e s t m e n t}{M a r k e t v a l u e}$
$n_{2} = \frac{S a l e p r o c e e d s}{M_{2}} = \frac{18 n_{1}}{15} shares$
Step 4: Calculate the annual income from the shares sold:
$A n n u a l i n c o m e = N u m b e r o f s h a r e s \times N o m i n a l v a l u e \times \frac{R a t e o f d i v i d e n d}{100}$
$A_{1} = n_{1} \times N_{1} \times \frac{D_{1}}{100} = n_{1} \times 20 \times \frac{8}{100} = ₹ \frac{8 n_{1}}{5}$
Step 5: Calculate the annual income from the shares bought:
$A n n u a l i n c o m e = N u m b e r o f s h a r e s \times N o m i n a l v a l u e \times \frac{R a t e o f d i v i d e n d}{100}$
$A_{2} = n_{2} \times N_{2} \times \frac{D_{2}}{100} = \frac{18 n_{1}}{15} \times 10 \times \frac{12}{100} = ₹ \frac{36 n_{1}}{25}$
Step 6: Calculate the difference in the annual incomes of Mr. Gosh:
$D i f f e r e n c e i n a n n u a l i n c o m e s = A_{1} - A_{2}$
$= \frac{8 n_{1}}{5} - \frac{36 n_{1}}{25} = \frac{8 n_{1} \times 5 - 36 n_{1}}{25} = \frac{40 n_{1} - 36 n_{1}}{25} = ₹ \frac{4 n_{1}}{25}$
According to the question, the change in Mr. Gosh's income is $₹ 120$
$\Rightarrow 120 = \frac{4 n_{1}}{25}$
$\Rightarrow n_{1} = \frac{120 \times 25}{4} \Rightarrow n_{1} = 750 shares$
Final Answer: Hence, the number of shares sold by Mr. Ghosh is $750$ .

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Mr. Ghosh sold a certain number of ₹20 shares paying 8% dividend at ₹18 and invested the proceeds in ₹10 shares, paying 12% dividend at 50% premium. If the change in his annual income is ₹120, then find the number of shares sold by Mr. Ghosh.

Mr. Ghosh sold a certain number of $₹ 20$ shares paying $8 %$ dividend at $₹ 18$ and invested the proceeds in $₹ 10$ shares, paying $12 %$ dividend at $50 %$ premium. If the change in his annual income is $₹ 120$ , then find the number of shares sold by Mr. Ghosh.