Question

Arun, Kamal and Vinay invest Rs. 8000, Rs. 4000 and Rs. 8000 respectively in a business. Arun left after six months. If after eight months, there was a gain of Rs. 4005. What will be the share of Kamal?
a. Rs. 890
b. Rs. 730
c. Rs. 800
d. Rs. 990

Answer 1

In this question, we have to calculate the share of each person in business in two parts, first for six months, when Arun, Kamal and Vinay all of them are in business. And, the second, when for the next two months, when Arun left. So, for the first six months, according to their investments, we calculate their share and then we assume that the profit for the first six months is Rs. x per month and then calculate the share of Kamal in profit for the first six months. Then, after Arun left, investment will decrease, so profit will decrease, and the share of Kamal and Vinay will change, then we will calculate the profit of this two months according to the investment and then the share of Kamal and Vinay in this profit. Now, the total profit is given to us in the question. So, we will calculate profit in the first six months and Kamal’s share in that six months and the profit in the next two months and Kamal’s share in two months. Then add both the shares of Kamal in profit, we will get Kamal’s share. For calculating share of a person we will use the formula, $\text{share =}\dfrac{\text{investment by that person}}{\text{total investment}}\text{ }\\!\\!\times\\!\\!\text{ 100}$.

Complete step-by-step answer :
Now, according to the question, the data given to us are,
Amount Arun invested in business = Rs. 8000
Amount Kamal invested in business = Rs. 4000
Amount Vinay invested in business = Rs. 8000
So, the total investment in business = Rs. 8000 + Rs. 4000 + Rs. 8000 = Rs. 20000.
Now, to find the share of a person we will use the formula, $\text{share =}\dfrac{\text{investment by that person}}{\text{total investment}}\text{ }\\!\\!\times\\!\\!\text{ 100}$.

So, share of Arun in business = $\dfrac{8000}{20000}\times 100=40\%$
Share of Kamal in business = $\dfrac{4000}{20000}\times 100=20\%$
Share of Vinay in business = $\dfrac{8000}{20000}\times 100=40\%$
Now, let us assume that the profit in business of Rs. 20000 investment in one month is Rs. x per month.
So, after six months, the total profit will be,
$\begin{aligned} & Rs.6\times x \\\ & =Rs.6x \\\ \end{aligned}$
But after six months, Arun left the business, so the investment in the business will decrease and also the profit will decrease.
So, after Arun left, the rest investment in the business = Rs. 20000 – Rs. 8000 = Rs. 12000
So, now, the share of Kamal will be = $\dfrac{4000}{12000}\times 100=33.33\%$
And the share of Vinay will be = $\dfrac{8000}{12000}\times 100=66.67\%$
And now, we know that with Rs. 20000 investment, the profit in business = Rs. x per month.
So, with Rs. 1 investment, the profit will be = $Rs.\dfrac{x}{20000}$ per month.
So, with Rs. 12000 investment, the profit will be = $Rs.\dfrac{x}{20000}\times 12000$ per month, which is equal to $Rs.\dfrac{3x}{5}$ per month.
So, for the next two months, after Arun left, the total profit will be = $Rs.2\times \dfrac{3x}{5}=Rs.\dfrac{6x}{5}$.
So, total profit after eight months = profit of first six months + profit of two months after Arun left.
So, total profit after eight months is equal to,
$\begin{aligned} & Rs.6x+Rs.\dfrac{6x}{5} \\\ & =Rs.\dfrac{30x+6x}{5} \\\ & =Rs.\dfrac{36x}{5} \\\ \end{aligned}$
And, total profit after eight months is given in the question and its equal to Rs. 4005.
So, we can write,
$4005=\dfrac{36x}{5}$
On multiplying both the sides of the above equation by $\dfrac{5}{36}$, we get,
$\begin{aligned} & \dfrac{4005\times 5}{36}=x \\\ & x=\dfrac{6675}{12} \\\ & x=Rs.556.25 \\\ \end{aligned}$
So, for the first six months, when Arun was in business, the profit was Rs. 556.25 per month and the total profit in that six months is equal to,
$\begin{aligned} & 6x=6\times 556.25 \\\ & =Rs.3337.5 \\\ \end{aligned}$
In that six months, Kamal’s share was in business = 20%
So, in that profit of six months, Kamal’s share will be equal to,
$\begin{aligned} & 3337.5\times 20\% \\\ & =3337.5\times \dfrac{20}{100} \\\ & =Rs.667.5 \\\ \end{aligned}$
So, in the first six months, Kamal’s share in profit = Rs. 667.5.
Now, after six months, when Arun left, profit was,
$\begin{aligned} & Rs.\dfrac{3x}{5} \\\ & =\dfrac{3\times 556.25}{5} \\\ & =Rs.333.75\text{ per month} \\\ \end{aligned}$
So, after Arun left, for two months, the total profit will be,
$\begin{aligned} & 2\times Rs.333.75 \\\ & =Rs.667.5 \\\ \end{aligned}$
So, in this two months, Kamal’s share in profit will be,
$\begin{aligned} & =667.5\times 33.33\% \\\ & =667.5\times \dfrac{33.33}{100} \\\ & =\dfrac{667.5}{3} \\\ & =Rs.222.5 \\\ \end{aligned}$
So, the total share of Kamal in profit after eight months = share of Kamal in profit after six months when Arun was in business + share of Kamal in profit after Arun left.
= Rs. 667.5 + Rs. 222.5
= Rs. 890.00
Therefore, the total share of Kamal in profit after eight months will be Rs. 890.
Hence, option (a) is the correct answer.

Note : In this question, we have to calculate the profit according to the investment in the business. Here many students may make mistakes as they consider the same profit per month for eight months, which is totally incorrect. Here, we have to keep in mind that, after Arun left, the share of Kamal and Vinay will change, it will not remain the same, so profit share will be calculated according to the new share.