A,B and C started a business with their investments in the ratio `1:2:4`, after 6 monts.A invested the half amount more as before and B invested twice the amount more as before, while C withdraws `(1)/( 4)` th of their investments. Find the ratio of their profit at the end of the year.

Options

A. `5:10:15`
B. `15:20:25`
C. `5:16:20`
D. `5:16:14`

Answer

To solve the problem step by step, we will calculate the investments and profits of A, B, and C over the year.

### Step 1: Initial Investments
Let the initial investments of A, B, and C be represented as:
- A's investment = K
- B's investment = 2K
- C's investment = 4K

### Step 2: Calculate the Profit Contribution for the First 6 Months
For the first 6 months, the profits will be calculated based on their initial investments.

- A's contribution for 6 months = K * 6 = 6K
- B's contribution for 6 months = 2K * 6 = 12K
- C's contribution for 6 months = 4K * 6 = 24K

### Step 3: Changes in Investments After 6 Months
After 6 months:
- A invests half of K more, which is K/2. So, A's new investment = K + K/2 = (3/2)K.
- B invests twice the amount he initially invested, which is 2 * 2K = 4K. So, B's new investment = 2K + 4K = 6K.
- C withdraws 1/4 of his investment, which is (1/4) * 4K = K. So, C's new investment = 4K - K = 3K.

### Step 4: Calculate the Profit Contribution for the Last 6 Months
Now we calculate the contributions for the next 6 months based on their new investments.

- A's contribution for the next 6 months = (3/2)K * 6 = 9K
- B's contribution for the next 6 months = 6K * 6 = 36K
- C's contribution for the next 6 months = 3K * 6 = 18K

### Step 5: Total Profit Contributions
Now we add the contributions from both periods to find the total profit contributions for each partner.

- Total profit contribution of A = 6K + 9K = 15K
- Total profit contribution of B = 12K + 36K = 48K
- Total profit contribution of C = 24K + 18K = 42K

### Step 6: Ratio of Profits
Now, we find the ratio of their profits:

- A's profit = 15K
- B's profit = 48K
- C's profit = 42K

The ratio of their profits is:
\[ A : B : C = 15K : 48K : 42K \]

### Step 7: Simplifying the Ratio
To simplify the ratio, we can divide each term by K:
\[ 15 : 48 : 42 \]

Now, we can simplify this further:
- Divide each term by 3:
\[ 5 : 16 : 14 \]

### Final Answer
The ratio of their profits at the end of the year is:
\[ \text{Ratio of profits} = 5 : 16 : 14 \]

---