The correct answer is option (D):
1,50,000
Let's break down this profit-sharing problem step by step.
First, define the variables. Let C be the total capital. The investment times are given in terms of a year, which is 12 months.
P invested (1/3)C for 1/4 of the year, which is 12/4 = 3 months.
Q invested (1/4)C for 1/2 of the year, which is 12/2 = 6 months.
R invested the remainder of the capital, which is 1 - 1/3 - 1/4 = 5/12 of the capital, for the entire year, which is 12 months.
The profit after P's salary is proportional to the investment amount and the square of the time. Therefore, let's represent the profit shares before salary. Let 'k' be the constant of proportionality.
P's profit share (before salary) = k * (C/3) * (3^2) = 3kC
Q's profit share = k * (C/4) * (6^2) = 9kC
R's profit share = k * (5C/12) * (12^2) = 60kC
The total profit share after paying P's salary is = 3kC + 9kC + 60kC = 72kC
Now, let's consider P's salary. P earns a salary of 10,000 per month, and since the time period is a year, P earns a total salary of 10,000 * 12 = 120,000
Let's assume the profit is X. P earns salary plus share.
P's total earnings = 120,000 + (3kC/72kC) * X = 120,000 + (1/24)* X
Q's total earnings = (9kC/72kC) * X = (1/8)* X
P earns 60,000 more than Q. Therefore:
P's total earnings - Q's total earnings = 60,000
[120,000 + (1/24)X] - (1/8)X = 60,000
120,000 - (2/24)X = 60,000
120,000 - (1/12)X = 60,000
(1/12)X = 60,000
X = 720,000
P's total earnings = 120,000 + (1/24) * 720,000
P's total earnings = 120,000 + 30,000
P's total earnings = 150,000
Therefore, P earns 150,000.