P and Q started a business by investing Rs 15000 and Rs (15000 + x) respectively. After four months, Q withdrew 40\% of his initial investment. After one year, the total profit was Rs 47000, and Q's share of the profit was Rs 22000. Find the value of $2x$.

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Accepted Answer

**Solution:**

1. Let P's investment be Rs 15000 and Q's initial investment be Rs $15000 + x$.

2. Q withdraws 40\% of his investment after 4 months, so the withdrawn amount is:
   $$
   0.40 \times (15000 + x) = 0.4(15000 + x)
   $$

3. For the first 4 months, Q's investment is $15000 + x$, and for the remaining 8 months, it is:
   $$
   (15000 + x) - 0.4(15000 + x) = 0.6(15000 + x)
   $$

4. Calculate the time-weighted investments (capital × time) for P and Q:
   - P's capital for 12 months:
     $$
     $$15000 \times 12 = 180000$$
     $$
   - Q's capital for 4 months:
     $$
     $$(15000 + x) \times 4 = 4(15000 + x)$$
     $$
   - Q's capital for 8 months:
     $$
     $$0.6(15000 + x) \times 8 = 4.8(15000 + x)$$
     $$

5. Total time-weighted investment for Q:
   $$
   $$4(15000 + x) + 4.8(15000 + x) = 8.8(15000 + x)$$
   $$

6. The ratio of P's and Q's investments considering time is:
   $$
   $$P : Q = 180000 : 8.8(15000 + x)$$
   $$

7. Given total profit = Rs 47000, Q's share = Rs 22000.
   So, Q's profit share ratio is:
   $$
   $$\frac{22000}{47000} = \frac{22}{47}$$
   $$

8. Set up the ratio equation:
   $$
   $$\frac{Q}{P + Q} = \frac{22}{47} \implies \frac{8.8(15000 + x)}{180000 + 8.8(15000 + x)} = \frac{22}{47}$$
   $$

9. Cross-multiply and solve for $x$:
   $$
  $$ 47 \times 8.8(15000 + x) = 22 \times (180000 + 8.8(15000 + x))$$
   $$

$$
   $$413.6(15000 + x) = 3960000 + 193.6(15000 + x)$$
   $$

$$
   413.6(15000 + x) - 193.6(15000 + x) = 3960000
   $$

$$
   220(15000 + x) = 3960000
   $$

$$
   15000 + x = $$\frac{3960000}{220}$$ = 18000
   $$

$$
   x = 18000 - 15000 = 3000
   $$

10. Required value is $$$2x = 2 \times 3000 = 6000$$$.

**Answer:** 6000

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