Question

Two brothers A and B invest Rs. 16,000 each in buying shares of two companies. A buys 3% hundred-rupee shares at 80 and B buys ten-rupee shares at par. If they both receive equal dividend at the end of the year, find the rate percent of the dividend received by B. \[\]

Answer 1

Hint: We find the amount stocks bought by A as ${{n}_{A}}=\dfrac{{{I}_{A}}}{M{{V}_{A}}}$where ${{I}_{A}}$ is the investment of A and $M{{V}_{A}}$ is the market value for A. We find total nominal values $TN{{V}_{A}}={{n}_{A}}\times N{{V}_{A}}=n\times 100$ and the dividend amount ${{D}_{A}}=\dfrac{D\%}{100}\times TN{{V}_{A}}$. We then find the number of stocks bought by B as ${{n}_{B}}=\dfrac{{{I}_{B}}}{M{{V}_{B}}}$ and total nominal value$TN{{V}_{B}}={{n}_{b}}\times N{{V}_{b}}={{n}_{b}}\times 100$. We find rate of dividend for B as $\%{{D}_{B}}=\dfrac{{{D}_{B}}}{TN{{V}_{B}}}\times 100$ using given ${{D}_{A}}={{D}_{B}}$. \[\]

Complete step-by-step answer:
We know that the amount of money that is used to run a company is called stock capital and the stock capital is divided into small units called stocks or share. The company issues a share certificate where the value of share is written called nominal value (NV) of face value. Then the shares are sold in the open market at stock exchanges at the market value (MV). If the market value and the nominal value are the same it is called at par. The profit annually distributed among shareholders is called dividend (D) in percentage of total nominal value of purchased shares. \[\]
Let us consider the investment of brother A. He buys hundred-rupee shares at $M{{V}_{A}}=80$ rupees from the market spending ${{I}_{A}}=16000$ rupees. So the number of stocks he has bought is ${{n}_{A}}=\dfrac{{{I}_{A}}}{M{{V}_{A}}}=\dfrac{16000}{80}=200$. The nominal value of each share is $N{{V}_{A}}=100$rupees then the nominal value of 200 shares is $TN{{V}_{A}}={{n}_{A}}\times N{{V}_{A}}=200\times 100=20000$ rupees. His profit percentage is 3% dividend. So his profit in amount 3% of total nominal value that is ${{D}_{A}}=\dfrac{3}{100}\times 20000=600$ rupees.\[\]
Let us consider brother B. He buys ten rupees shares at par which means the nominal value and market value of the share is same and is equal to 10 rupees $\left( M{{V}_{B}}=N{{V}_{B}}=10 \right)$. He also has spent ${{I}_{B}}=16000$ rupees to buy stocks. So the number of sticks B has purchased is ${{n}_{b}}=\dfrac{{{I}_{B}}}{M{{V}_{B}}}=\dfrac{16000}{10}=1600$.The total nominal value of all 1600 shares bought by brother B is $1600\times 10=16000=TN{{V}_{B}}$ rupees. We are given the question that he and A both earn the same amount of dividend which is ${{D}_{B}}={{D}_{A}}=600$rupees as obtained before. So the rate of dividend is
\[\dfrac{{{D}_{B}}}{TM{{V}_{B}}}\times 100=\dfrac{600}{16000}\times 100=3.75\%\]

Note: We note that the question assumes that there is no brokerage charge which is added when the stock is sold on the selling price or bought on the cost price as percentage. If the market value is above nominal value it is called at premium or above par. If the market value is below nominal value it is called at discount or below par. We must be careful that the nominal value never changes but the market value changes from, time to time.