Three partners P, Q and R invested their amounts in ratio of $2 : 5 : 7$. At the end of 6 months, ‘P’ added some more amount such that his investment becomes equal to half of the sum of ‘Q’ and ‘R’ initial investments. If at the end of the year, Q’s share in profit is Rs 425, then find the total profit.

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Accepted Answer

**Solution:**

1. Let the initial investments of P, Q, and R be $2x$, $5x$, and $7x$ respectively.

2. After 6 months, P adds an extra amount $y$ such that:
   $$2x + y = \frac{1}{2} (5x + 7x) = 6x$$
   So,
   $$y = 6x - 2x = 4x$$

3. Calculate the effective investments considering time (in months):
   - P invests $2x$ for 6 months and $(2x + 4x) = 6x$ for the next 6 months.
   - Q invests $5x$ for 12 months.
   - R invests $7x$ for 12 months.

4. Calculate the investment-time products:
   - P: $2x 	imes 6 + 6x 	imes 6 = 12x + 36x = 48x$
   - Q: $5x 	imes 12 = 60x$
   - R: $7x 	imes 12 = 84x$

5. The ratio of profits is:
   $$P : Q : R = 48x : 60x : 84x = 4 : 5 : 7$$

6. Given Q's share of profit is Rs 425, corresponding to 5 parts.
   - 1 part = $\frac{425}{5} = 85$
   - Total profit = $(4 + 5 + 7) 	imes 85 = 16 	imes 85 = 1360$

**Answer:** Rs 1360

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