A and B invested Rs 2000 and Rs $ (2000 + x) $ respectively in a business. B withdrew his investment after 8 months. If at the end of the year, B's share of the profit is 20\% less than A's share, find the value of $ x $.

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**Solution:**

1. Let the investments of A and B be Rs 2000 and Rs $2000 + x$ respectively.

2. A invests for the entire 12 months, so the time period is 12 months.

3. B invests for 8 months only.

4. The profit share is proportional to the product of investment and time invested.

- A's share = $$2000 \times 12 = 24000$$
   - B's share = $($2000 + x) \times 8 = 8(2000 + x) = 16000 + 8x$$

5. According to the problem, B's profit is 20\% less than A's profit, so:

$$
   $\text{B's share} = 0.8 \times \text{A's share}$
   $$

Substitute the values:

$$
   $16000 + 8x = 0.8 \times 24000 = 19200$
   $$

6. Solve for $x$:

$$
   8x = 19200 - 16000 = 3200
   $$

$$
   $x = \frac{3200}{8} = 400$
   $$

**Answer:** Rs 400

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