Let's break down the problem step by step to find the total profit:

Step 1: Understanding the investments

• The ratio of investments of P, Q, and R at the beginning is given as $2 : 5 : 7$.
• Let the initial investments of P, Q, and R be $2x$, $5x$, and $7x$, respectively.

Step 2: P increases his investment after 6 months

• After 6 months, P increases his investment such that his total investment becomes equal to half of the sum of Q's and R's initial investments.
• The sum of Q's and R's initial investments = $5x + 7x = 12x$.
• Half of this sum is $\frac{12x}{2} = 6x$.
• So, after 6 months, P's investment becomes $6x$.

Step 3: Effective investments for 12 months

Since P increased his investment after 6 months, we need to account for this change when calculating the effective investments for the entire year. Here’s how we do it:

• P's investment for the first 6 months is $2x$.

• For the next 6 months, P’s investment is $6x$.

• The effective investment for P is: $\text{Effective investment of P} = (2x \times 6) + (6x \times 6) = 12x + 36x = 48x$

• Q's investment is constant throughout the year, so the effective investment for Q is: $\text{Effective investment of Q} = 5x \times 12 = 60x$

• Similarly, R’s investment is constant throughout the year, so the effective investment for R is: $\text{Effective investment of R} = 7x \times 12 = 84x$

Step 4: Total effective investment

The total effective investment is: $\text{Total effective investment} = 48x + 60x + 84x = 192x$

Step 5: Profit distribution

It is given that Q’s share of the profit is Rs 425. Since profit is distributed in the ratio of their effective investments, we first find Q's share in terms of the total profit: $\text{Q's share in the total profit} = \frac{60x}{192x} \times \text{Total profit}$ $\frac{60}{192} \times \text{Total profit} = 425$ $\frac{5}{16} \times \text{Total profit} = 425$ $\text{Total profit} = 425 \times \frac{16}{5} = 425 \times 3.2 = 1360$

Step 6: Conclusion

The total profit is Rs 1360.

Answer:

(b) Rs 1360

Would you like more details or have any questions?

Relative Questions:

1. How would the total profit change if R also altered his investment midway?
2. What if the ratio of investments had been different, say 3:4:5?
3. How would the calculation differ if P's investment increase occurred after 3 months instead of 6?
4. What if Q’s share in the profit was given as Rs 500? How would the total profit change?
5. How does the time period over which investments are held affect the total profit calculation?

Tip:

Always consider the period for which investments are held when calculating profit shares, especially if investments change during the period.