Directions: There are 30 questions in this section. Each question is followed by five alternatives (a), (b), (c),(d) and (e). You are required to choose the best alternative from these five alternatives.81. P, Q and E start a joint venture, where in they make an annual profit. P invested one-third of thecapital for one-fourth of the time, Q invested one-fourth of the capital for one-half of the time and Rinvested 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 hasput and also to the square of the number of months for which each has put their sum in the venture. Ifin 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

To solve this problem, let's break it down step by step:

1. We are given that P, Q, and E start a joint venture and make an annual profit.
2. P invests one-third of the capital for one-fourth of the time.
3. Q invests one-fourth of the capital for one-half of the time.
4. R invests the remainder of the capital for the entire year.
5. P is a working partner and receives a salary of `10,000 per month.
6. 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.
7. It is given that in a year, P earns `60,000 more than Q.
8. We need to find out how much P earns.

Let's calculate the investments and time periods for each partner:

Let the total capital be C.

P's investment = (1/3) * C P's time period = (1/4) * 12 months = 3 months

Q's investment = (1/4) * C Q's time period = (1/2) * 12 months = 6 months

R's investment = C - (P's investment + Q's investment) R's time period = 12 months

Now, let's calculate the profit shares for each partner:

P's profit share = (P's investment) * (P's time period)^2 Q's profit share = (Q's investment) * (Q's time period)^2 R's profit share = (R's investment) * (R's time period)^2

Given that P earns `60,000 more than Q, we can set up the equation:

P's profit share - Q's profit share = `60,000

Substituting the values, we get:

[(P's investment) * (P's time period)^2] - [(Q's investment) * (Q's time period)^2] = `60,000

Simplifying the equation, we get:

[(1/3) * C * (3 months)^2] - [(1/4) * C * (6 months)^2] = `60,000

Solving this equation will give us the value of C, which is the total capital.

Once we have the value of C, we can calculate P's earnings:

P's earnings = P's profit share + P's salary

Substituting the values, we get:

P's earnings = [(P's investment) * (P's time period)^2] + `10,000 * 12 months

Finally, we can compare P's earnings with the given options to find the correct answer.

I hope this helps!