A and B started a business by investing Rs 8000 and Rs 12000 respectively. After 4 months, A withdrew two-fifths of his investment, while B increased his investment by Rs 2000. If at the end of the year, A's share of the profit is Rs 11000, what is the total profit (in Rs)?

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**Solution:**

1. Calculate the effective investment of A and B over the 12 months:

- A invested Rs 8000 for the first 4 months.
- After 4 months, A withdrew two-fifths of his investment, so the remaining investment is:
  $$8000 - \frac{2}{5} \times 8000 = 8000 - 3200 = 4800$$
- A's investment for the remaining 8 months is Rs 4800.

- B invested Rs 12000 for the first 4 months.
- After 4 months, B added Rs 2000, so new investment is Rs 14000.
- B's investment for the remaining 8 months is Rs 14000.

2. Calculate the weighted investments (investment × time) for A and B:

- A's total investment time value:
  $$8000 \times 4 + 4800 \times 8 = 32000 + 38400 = 70400$$

- B's total investment time value:
  $$12000 \times 4 + 14000 \times 8 = 48000 + 112000 = 160000$$

3. Find the ratio of A's and B's investment time values:

$$\text{Ratio} = 70400 : 160000 = 44 : 100$$

4. Given A's profit share is Rs 11000, let total profit be $ P $.

- A's share of profit = $ $$\frac{44}{44 + 100} \times P = \frac{44}{144} \times P = \frac{11}{36} P $$$

- So, $$$ \frac{11}{36} P = 11000 $$$

- Multiply both sides by 36:
  $$11P = 11000 \times 36 = 396000$$

- Divide both sides by 11:
  $$P = \frac{396000}{11} = 36000$$

**Answer:** Rs 36000

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