Let's solve this step by step.

Step 1: Investment Ratio of A, B, and C

The initial investment ratio of A, B, and C is given as:

$A : B : C = \frac{1}{3} : \frac{1}{2} : \frac{1}{4}$

To simplify this ratio, let's find a common denominator for 3, 2, and 4. The least common multiple (LCM) is 12. So we multiply each term by 12:

$A : B : C = \left( \frac{1}{3} \times 12 \right) : \left( \frac{1}{2} \times 12 \right) : \left( \frac{1}{4} \times 12 \right)$

$A : B : C = 4 : 6 : 3$

Thus, the initial investment ratio is $4 : 6 : 3$.

Step 2: Changes in Investment after 6 Months

• After 6 months, A's investment becomes $\frac{3}{2}$ times his initial investment.
• B's investment becomes $\frac{1}{6}$ of his initial investment.
• C's investment remains unchanged.

So, we need to calculate the new effective investments for A and B for the second half of the year.

A's Investment:

For the first 6 months, A's investment is $4$. For the last 6 months, A's investment becomes $\frac{3}{2}$ times 4, which is:

$\text{A's new investment} = 4 \times \frac{3}{2} = 6$

The total contribution of A over the year is:

$\text{A's total investment} = 4 \times 6 \, \text{months} + 6 \times 6 \, \text{months} = 24 + 36 = 60$

B's Investment:

For the first 6 months, B's investment is $6$. For the last 6 months, B's investment becomes $\frac{1}{6}$ of 6, which is:

$\text{B's new investment} = 6 \times \frac{1}{6} = 1$

The total contribution of B over the year is:

$\text{B's total investment} = 6 \times 6 \, \text{months} + 1 \times 6 \, \text{months} = 36 + 6 = 42$

C's Investment:

C's investment remains unchanged throughout the year. So, for 12 months, C's total contribution is:

$\text{C's total investment} = 3 \times 12 = 36$

Step 3: Profit Sharing

The total investment of A, B, and C combined is:

$\text{Total investment} = 60 + 42 + 36 = 138$

The profit share ratio will be proportional to the total investments of A, B, and C, i.e., the ratio of profits is:

$\text{A : B : C} = 60 : 42 : 36$

Simplifying this ratio by dividing by 6:

$\text{A : B : C} = 10 : 7 : 6$

Step 4: Given Information about the Difference in Profit

We are given that the difference in profit share between A and B is Rs. 2400.

$\text{A's profit share} - \text{B's profit share} = 2400$

Let the total profit be $P$. According to the profit-sharing ratio:

$\frac{10}{23}P - \frac{7}{23}P = 2400$

Simplifying:

$\frac{3}{23}P = 2400$

$P = \frac{2400 \times 23}{3} = 18400$

Step 5: C's Profit Share

C's profit share is $\frac{6}{23}$ of the total profit. Therefore, C's share is:

$\text{C's share} = \frac{6}{23} \times 18400 = 4800$

Final Answer:

C's profit share is Rs. 4800.

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Would you like a more detailed explanation of any step? Here are some related questions for you to explore:

1. How do changes in investments affect profit-sharing?
2. What is the general formula for calculating profit share based on investments?
3. How can we simplify ratios of fractions in different contexts?
4. How would the solution change if C also changed his investment after 6 months?
5. What if the total profit were unknown and had to be calculated from multiple data points?

Tip: Always reduce ratios to their simplest form to make further calculations easier!