Amit, Bharat, and Chirag started a business with their investments in the ratio $1:2:4$. After 6 months, Amit invested half the amount more than before; Bharat invested twice the amount more than before, while Chirag withdrew $\frac{1}{4}$ of his investment. What will be the ratio of their profits at the end of the year?

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**Solution:**

1. Let the initial investments of Amit, Bharat, and Chirag be $x$, $2x$, and $4x$ respectively.

2. For the first 6 months, their investments remain the same:
   - Amit: $x 	imes 6 = 6x$
   - Bharat: $2x 	imes 6 = 12x$
   - Chirag: $4x 	imes 6 = 24x$

3. After 6 months:
   - Amit invests half the amount more than before, so new investment = $x + \frac{1}{2}x = \frac{3x}{2}$
   - Bharat invests twice the amount more than before, so new investment = $2x + 2 	imes 2x = 2x + 4x = 6x$
   - Chirag withdraws $\frac{1}{4}$ of his investment, so new investment = $4x - \frac{1}{4} 	imes 4x = 4x - x = 3x$

4. For the next 6 months, their investments are:
   - Amit: $\frac{3x}{2} 	imes 6 = 9x$
   - Bharat: $6x 	imes 6 = 36x$
   - Chirag: $3x 	imes 6 = 18x$

5. Total investment over the year:
   - Amit: $6x + 9x = 15x$
   - Bharat: $12x + 36x = 48x$
   - Chirag: $24x + 18x = 42x$

6. The ratio of their profits will be the same as the ratio of their total investments:
   $$15x : 48x : 42x = 15 : 48 : 42$$

7. Simplify the ratio by dividing each term by 3:
   $$5 : 16 : 14$$

**Answer:** $5:16:14$

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