Question

P. Q and E start a joint venture, where in they make an annual profit. P invested one-third of the capital for one-fourth of the time, Q invested one-fourth of the capital for one-half of the time and R invested the remainder of the capital for the entire year. P is a working partner and gets a salary of 10,000 per month. The profit after paying P's salary is directly proportional to the sum each one has put and also to the square of the number of months for which each has put their sum in the venture. If in a year P earns 60,000 more than Q, then how much does P earn?

• A. 1,00,000
• B. 1,20,000
• C. 1,35,000
• D. 1,50,000
• E. 1,80,000

Answer 1

The Correct Option is D

To solve the problem of determining how much P earns, we need to analyze the profit distribution and P's additional earnings.

Given:

• P invested one-third of the capital for one-fourth of the time.
• Q invested one-fourth of the capital for one-half of the time.
• R invested the remainder of the capital for the entire year.

P receives a salary of 10,000 per month, totaling 120,000 for the year as a working partner.

The profit distribution is directly proportional to:

• The sum each invested (capital).
• The square of the number of months for which each invested.

Let's represent the total capital as \( C \).

Investment terms:

• P invested \(\frac{C}{3}\) for \( \frac{1}{4} \) of the year (3 months).
• Q invested \(\frac{C}{4}\) for \( \frac{1}{2} \) of the year (6 months).
• R invested the remaining capital \(\frac{5C}{12}\) for the entire 12 months.

The distribution formula involves both capital and time squared:

• For P: \(\frac{C}{3} \times (3)^2 = 3C\)
• For Q: \(\frac{C}{4} \times (6)^2 = 9C\)
• For R: \(\frac{5C}{12} \times (12)^2 = 60C\)

The combined ratio of profit distribution based on these factors is:

\(\text{P : Q : R} = 3C : 9C : 60C \Rightarrow 1 : 3 : 20\)

Let's denote the total profit after paying P's salary as \( X \).

The proportionate shares in the profit:

• P's share: \(\frac{1}{24} X\)
• Q's share: \(\frac{3}{24} X\)
• R's share: \(\frac{20}{24} X\)

According to the problem, P earns 60,000 more than Q:

\(\frac{1}{24}X - \frac{3}{24}X = 60,000\)

Solving the above equation gives:

\[ \frac{-2}{24}X = 60,000 \Rightarrow X = 720,000 \]

P's total earnings = Salary + Profit share

So, P's total earnings = 120,000 + \(\frac{1}{24}X\)

\[ P's \, total \, earnings = 120,000 + \frac{1}{24} \times 720,000 = 120,000 + 30,000 = 150,000 \]

Hence, P earns a total of 1,50,000 in the year. Therefore, the correct answer is 1,50,000.