Question:hard

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?

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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.
Updated On: Jul 21, 2026
  • Rs. 1,00,000
  • Rs. 1,20,000
  • Rs. 1,35,000
  • Rs. 1,50,000
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The Correct Option is D

Solution and Explanation

Step 1: List capital share and duration for each partner.
P: one third of capital for 3 months. Q: one fourth of capital for 6 months. R: the rest, $\frac{5}{12}$ of capital, for 12 months.

Step 2: Form the ratio of capital times months squared, then simplify.
P's number $= \frac{1}{3}\times 9 = 3$. Q's number $= \frac{1}{4}\times 36 = 9$. R's number $= \frac{5}{12}\times 144 = 60$.
Ratio P : Q : R $= 3 : 9 : 60$, which simplifies to $1 : 3 : 20$ after dividing by 3.

Step 3: Split the post salary profit into 24 equal parts.
Total parts $= 1+3+20 = 24$. So P's share is $\frac{1}{24}$ of the divisible profit and Q's share is $\frac{3}{24} = \frac{1}{8}$ of it, matching the weight ratio.

Step 4: Set up the earnings equation.
P's yearly salary is $12 \times 10000 = Rs. 120000$. Let the divisible profit be $x$.
P's total pay is $120000 + \frac{x}{24}$ and Q's total pay is $\frac{3x}{24}$.
Since P earns Rs. 60000 more than Q: $120000 + \frac{x}{24} - \frac{3x}{24} = 60000$.

Step 5: Solve for x and then for P's earning.
$120000 - \frac{2x}{24} = 60000$, so $\frac{x}{12} = 60000$, giving $x = 720000$.
P's share of profit $= \frac{720000}{24} = 30000$. P's total earning $= 120000+30000 = 150000$.

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