 Three persons A, B and C started a business. They invest in the ratio of 
(1/3) ∶ (1/2) ∶ (1/4). After 6 months A's investment becomes (3/2)th of his 
initial amount and B investment becomes (1/6)th of his initial amount. At 
the end of the year, the difference in the profit share of A and B is Rs. 
2400. Find C's share

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.

---
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!

Three persons A, B, and C started a business. They invest in the ratio of (1/3) ∶ (1/2) ∶ (1/4). After 6 months A's investment becomes (3/2)th of his initial amount and B's investment becomes (1/6)th of his initial amount. At the end of the year, the difference in the profit share of A and B is Rs. 2400. Find C's share.

Learn how to calculate the profit share of C in a business where A and B modify their investments after 6 months. The solution involves ratio simplification, profit-sharing formulas, and proportional distribution. This detailed step-by-step guide is ideal for Grades 10-12.

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