Question

P, Q and E start a joint venture, wherein 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 Rs. 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 Rs. 60,000 more than Q, then how much does P earn?

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

Hint:

Work out each partner's capital share and months, form the weight capital times months squared, then use the salary plus share condition for P and Q.

Answer 1

The Correct Option is D

Step 1: Find each partner's capital fraction and time in months.
P puts in one third of the capital for one fourth of the year, so P's time is 3 months.
Q puts in one fourth of the capital for one half of the year, so Q's time is 6 months.
R puts in the remaining capital, which is \( 1 - \frac{1}{3} - \frac{1}{4} = \frac{5}{12} \) of the total, for the full 12 months.

Step 2: Build a profit weight for each partner.
The profit after P's salary is proportional to capital times the square of the months invested.
Weight of P \( = \frac{1}{3} \times 3^2 = 3 \) units, taking the total capital as C.
Weight of Q \( = \frac{1}{4} \times 6^2 = 9 \) units.
Weight of R \( = \frac{5}{12} \times 12^2 = 60 \) units.
Total weight \( = 3 + 9 + 60 = 72 \) units.

Step 3: Write P's and Q's share of the divisible profit.
Let the profit left after paying P's salary be \( P_0 \).
P's share \( = \frac{3}{72}P_0 = \frac{P_0}{24} \).
Q's share \( = \frac{9}{72}P_0 = \frac{P_0}{8} \).

Step 4: Use the earnings condition to find \( P_0 \).
P's yearly salary is \( 12 \times 10{,}000 = \) Rs. 1,20,000.
P's total earning \( = 1{,}20{,}000 + \frac{P_0}{24} \), and Q's total earning \( = \frac{P_0}{8} \).
P earns Rs. 60,000 more than Q, so \( 1{,}20{,}000 + \frac{P_0}{24} - \frac{P_0}{8} = 60{,}000 \).
This gives \( \frac{P_0}{24} - \frac{3P_0}{24} = -60{,}000 \), so \( -\frac{P_0}{12} = -60{,}000 \), hence \( P_0 = 7{,}20{,}000 \).

Step 5: Compute P's total earning.
P's share of the profit \( = \frac{7{,}20{,}000}{24} = 30{,}000 \).
P's total earning \( = 1{,}20{,}000 + 30{,}000 = 1{,}50{,}000 \).

Final Answer:
P earns Rs. 1,50,000 in the year. \[ \boxed{Rs.\ 1,50,000} \]