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**Hint:**To find the number of shares purchased by Rajeev we use the formula:

Number of share bought \[=\dfrac{\text{Investment}}{\text{Market Value}}\]

And to find the yearly income, we get the formula as:

The yearly income generated as \[=\text{Number of Share}\times \text{Rate of Dividend}\times \text{Nominal Value}\]

And to find the final part of the gain when \[60%\] of the share is:

\[\left( \text{Number of Share}\times \text{Market Value}{{\text{e}}_{90}} \right)-\left( \text{Number of Share}\times \text{Market Value}{{\text{e}}_{65}} \right)\]

**Complete step by step solution:**

According to the question given, the investment given by Rajeev is Rs.\[9750\]. The dividend paid by the investment is given as \[15.5%\].

Now to find the number of shares by Rajeev, we use the formula where we divide the sum invested by the market value of a single share:

Number of share bought \[=\dfrac{\text{Investment}}{\text{Market Value}}\]

Number of share bought \[=\dfrac{Rs.9750}{Rs.65}\]

Number of share bought \[=150\]

Placing the values, we get the total number of shares as \[150\].

The total yearly income of the sum invested by Rajeev is found by the product of total number of shares, rate of dividend and nominal value of one share as:

The yearly income generated as \[=\text{Number of Share}\times \text{Rate of Dividend}\times\text{Nominal Value}\]

Placing the values in the above formula we get the yearly income as:

The yearly income generated as \[=\text{150 }\!\!\times\!\!\text{ }\dfrac{15.5}{100}\text{

}\!\!\times\!\!\text{ 50}\]

\[\Rightarrow \text{150 }\!\!\times\!\!\text{ }\dfrac{15.5}{100}\text{ }\!\!\times\!\!\text{ 50}\]

\[\Rightarrow \] Rs.\[1162.50\]

Now as we have got the income and the number of shares, we now find the gain on his \[60%\] of his shares when price rises to Rs.\[90\] is:

Now the total number of shares from his \[60%\] share is:

\[\Rightarrow \dfrac{60}{100}\times 150\]

\[\Rightarrow \dfrac{60}{100}\times 150=90\] shares

Therefore, the total price gain is \[=\left( \text{Number of Share}\times \text{Market Value}{{\text{e}}_{90}} \right)-\left( \text{Number of Share}\times \text{Market Value}{{\text{e}}_{65}}

\right)\].

\[\Rightarrow 90\times 90-90\times 65\]

\[\Rightarrow 2250\] Rupees

**Therefore, the profit gained when Rajeev sold \[60%\] of his share is Rs. \[2250\].**

**Note:**Dividend is the annual profit of a company, shareholders when shares are bought at a certain market value. The market value of a share depends upon the company’s performance.

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