A,B and C enter into a partnership in the ratio `(7)/(2):(4)/(3):(6)/(5)`. After 4 months A increase his share `50%`. If the total profit at the end of one year be `₹21,600`, then B's share in the profit is :

Options

A. `₹2100`
B. `₹ 2400`
C. `₹ 3600`
D. `₹4000`

Explanation

To solve the problem step by step, we will follow the outlined process to find B's share in the profit.

### Step 1: Determine the initial ratio of investments
The partnership ratio given is:
\[ A : B : C = \frac{7}{2} : \frac{4}{3} : \frac{6}{5} \]

### Step 2: Find a common denominator
To simplify the ratios, we need to find a common denominator for the fractions. The denominators are 2, 3, and 5. The least common multiple (LCM) of these numbers is 30.

### Step 3: Convert the ratios to a common base
Now we will convert each fraction to have the common denominator of 30:
- For A:
\[ \frac{7}{2} = \frac{7 \times 15}{2 \times 15} = \frac{105}{30} \]
- For B:
\[ \frac{4}{3} = \frac{4 \times 10}{3 \times 10} = \frac{40}{30} \]
- For C:
\[ \frac{6}{5} = \frac{6 \times 6}{5 \times 6} = \frac{36}{30} \]

Thus, the ratio of investments becomes:
\[ A : B : C = 105 : 40 : 36 \]

### Step 4: Calculate the effective investment considering time
A invests for 12 months, but after 4 months, A increases his share by 50%. Therefore, we need to calculate the effective investment for A:
- For the first 4 months, A's investment is \( 105 \).
- For the next 8 months, A's investment becomes \( 105 \times 1.5 = 157.5 \).

Now, we calculate the total investment for each partner:
- A's total investment:
\[ 105 \times 4 + 157.5 \times 8 = 420 + 1260 = 1680 \]
- B's investment for 12 months:
\[ 40 \times 12 = 480 \]
- C's investment for 12 months:
\[ 36 \times 12 = 432 \]

### Step 5: Calculate the total investment
Now we sum up the total investments:
\[ \text{Total Investment} = 1680 + 480 + 432 = 2592 \]

### Step 6: Calculate the profit share of B
The total profit is given as ₹21,600. To find B's share, we first find the ratio of B's investment to the total investment:
- B's share in the total investment:
\[ \text{B's Share} = \frac{480}{2592} \]

Now, we calculate B's share in the profit:
\[ \text{B's Profit} = \text{Total Profit} \times \text{B's Share} = 21600 \times \frac{480}{2592} \]

Calculating this gives:
\[ \text{B's Profit} = 21600 \times \frac{480}{2592} = 4800 \]

### Final Answer
Thus, B's share in the profit is:
\[ \text{B's Share} = ₹4000 \]