P, Q, and R entered into a partnership. After ten months, P and Q left the business. Find the total profit at the end of the year.

Statement I: The combined profit share of P and R is 55\% more than that of Q.

Statement II: The annual profit of R is Rs. 450 less than that of P, and the annual profit of Q is 80\% of R.

Statement III: The total amount invested by R and Q are in the ratio 5:4 respectively.

Which of the following statements or combinations of statements are sufficient to find the total profit?

Question

P, Q, and R entered into a partnership. After ten months, P and Q left the business. Find the total profit at the end of the year.

Statement I: The combined profit share of P and R is 55\% more than that of Q.

Statement II: The annual profit of R is Rs. 450 less than that of P, and the annual profit of Q is 80\% of R.

Statement III: The total amount invested by R and Q are in the ratio 5:4 respectively.

Which of the following statements or combinations of statements are sufficient to find the total profit?

Accepted Answer

**Solution:**

1. We need to find the total profit at the end of the year.

2. Statement I says the combined profit share of P and R is 55\% more than that of Q. This gives a relation between their profit shares but not the absolute values.

3. Statement II provides the difference between the annual profits of P and R (Rs. 450) and states that Q's profit is 80\% of R's profit. This gives relative profit amounts but not the total profit.

4. Statement III gives the ratio of investments of R and Q as 5:4, which helps relate their capital contributions.

5. Combining Statement I and II:
   - From Statement I, $ P + R = 1.55Q $
   - From Statement II, $ R = P - 450 $ and $ Q = 0.8R $
   These relations allow us to solve for P, Q, and R's profits in absolute terms, hence total profit.

6. Combining Statement II and III:
   - Statement II relates profits.
   - Statement III relates investments.
   Since profit shares depend on investment and time, knowing investment ratios and profit relations allows calculation of total profit.

7. Statement I and III alone are insufficient because without the exact profit difference or relation from Statement II, we cannot find absolute profit values.

**Answer:** Either statement I & II together or statement II & III together

Answer

Answer

Answer