Question

P, Q and R were partners in a firm sharing profits and losses in the ratio of $3:4:1$. On $31^{st}$ March, 2022, R retired. R surrendered $\frac{1}{3}$ of his share in favour of P and the remaining share in favour of Q. Calculate the new profit sharing ratio of P and Q.

Hint:

When a partner retires, his share is distributed among the remaining partners in the ratio in which they acquire it. Add the gained share to their old share to find the new ratio.

Answer 1

Step 1: Determine the old profit sharing ratio.
The old profit sharing ratio of the partners was: \[ P : Q : R = 3 : 4 : 1 \] Total parts $=3+4+1=8$ Thus, \[ P=\frac{3}{8}, \quad Q=\frac{4}{8}, \quad R=\frac{1}{8} \]
Step 2: Determine the share of R.
R’s share in the firm is: \[ \frac{1}{8} \] R gives $\frac{1}{3}$ of his share to P. \[ \text{Share received by P}=\frac{1}{3}\times\frac{1}{8}=\frac{1}{24} \] The remaining share goes to Q. \[ \text{Share received by Q}=\frac{2}{3}\times\frac{1}{8}=\frac{2}{24}=\frac{1}{12} \]
Step 3: Calculate the new shares of P and Q.
New share of P: \[ \frac{3}{8}+\frac{1}{24} \] Convert into common denominator: \[ \frac{3}{8}=\frac{9}{24} \] \[ P=\frac{9}{24}+\frac{1}{24}=\frac{10}{24} \] New share of Q: \[ \frac{4}{8}+\frac{1}{12} \] Convert into common denominator: \[ \frac{4}{8}=\frac{1}{2}=\frac{12}{24}, \quad \frac{1}{12}=\frac{2}{24} \] \[ Q=\frac{12}{24}+\frac{2}{24}=\frac{14}{24} \]
Step 4: Determine the new profit sharing ratio.
\[ P:Q=\frac{10}{24}:\frac{14}{24} \] \[ P:Q=10:14 \] Divide by $2$: \[ P:Q=5:7 \]
Step 5: Conclusion.
Thus, after the retirement of R, the new profit sharing ratio of P and Q becomes $5:7$.
Final Answer: $5:7$