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, we need to analyze the contributions of P, Q, and R to the joint venture and calculate the total earnings of P. Let's break this down step-by-step:

1. Understanding Contributions:

   • 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.

2. Time and Monetary Contributions:

   • P's investment duration: \( \frac{1}{4} \times 12 = 3 \) months.
   • Q's investment duration: \( \frac{1}{2} \times 12 = 6 \) months.
   • R invests for the full 12 months.

3. Profit Distribution:

   The profit after paying P’s salary is directly proportional to both the sum invested and to the square of the months invested. Therefore:

   • Share of P in profit: \( \frac{1}{3} \times (3^2) = 3 \).
   • Share of Q in profit: \( \frac{1}{4} \times (6^2) = 9 \).
   • Share of R in profit: \( \frac{5}{12} \times (12^2) = 60 \).

4. Calculating Shares:

   • Total share of profit distribution = \( 3 + 9 + 60 = 72 \).
   • Proportionate share of P = \( \frac{3}{72} = \frac{1}{24} \).
   • Proportionate share of Q = \( \frac{9}{72} = \frac{1}{8} \).

5. Salary of P:

   • P gets a salary of ₹10,000 per month.
   • Annual salary of P = \( 10,000 \times 12 = ₹120,000 \).

6. Profit Earnings:

   • Let the profit after paying salary be \( x \).
   • P's profit = \( \frac{1}{24} x \) and Q's profit = \( \frac{1}{8} x \).
   • Given, P earns ₹60,000 more than Q:
   • \( \frac{1}{24}x + 120,000 = \frac{1}{8}x + 60,000 \).
   • Simplifying, we find: \( \frac{1}{24}x - \frac{1}{8}x = 60,000 - 120,000 \).
   • \(\frac{-1}{12}x = -60,000 \)
   • This simplifies to \( x = ₹720,000 \).

7. Total Earnings of P:

   • P's share of profit = \( \frac{1}{24} \times 720,000 = ₹30,000 \).
   • Total earnings of P = \( ₹120,000 + ₹30,000 = ₹150,000 \).

Therefore, P's total earnings are ₹150,000, which corresponds to option 1,50,000.