A, B, and C started a business with their investments in the ratio of $2:3:4$ respectively. After one year, A doubled his investment while C withdrew half of his investment. What is the ratio of the profit earned by B and C at the end of two years?

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Accepted Answer

**Solution:**

1. Initial investments of A, B, and C are in the ratio $2:3:4$.

2. For the first year, investments remain the same:
   - A's investment = $2$
   - B's investment = $3$
   - C's investment = $4$

3. For the second year:
   - A doubles his investment, so A's investment = $2 		imes 2 = 4$
   - B's investment remains $3$
   - C withdraws half of his investment, so C's investment = $4 		imes rac{1}{2} = 2$

4. Calculate the total investment for each person over two years (investment × time):
   - A's total investment = $2 		imes 1 + 4 		imes 1 = 2 + 4 = 6$
   - B's total investment = $3 		imes 2 = 6$
   - C's total investment = $4 		imes 1 + 2 		imes 1 = 4 + 2 = 6$

5. The profit sharing ratio is proportional to the total investments:
   - A : B : C = $6 : 6 : 6 = 1 : 1 : 1$

6. The question asks for the ratio of profit earned by B and C, which is:
   $$1 : 1$$

**Answer:** $1:1$

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