A,B and C enter into a partnership . A invest X+8000, B invest `2X+ 2000` and C invest `3X+4000` for one year if B share is 4000 from total profit of 16000 then find the difference between investment of A and C .

Options

A. A)4000
B. B)5000
C. C)6000
D. D)2000

Answer

To solve the problem step by step, we will follow the information given in the question and derive the necessary values.

### Step 1: Define the investments
Let the investment of A, B, and C be defined as follows:
- A's investment = \( X + 8000 \)
- B's investment = \( 2X + 2000 \)
- C's investment = \( 3X + 4000 \)

### Step 2: Total profit and B's share
The total profit is given as 16000, and B's share of the profit is given as 4000.

### Step 3: Calculate the ratio of investments
Since the profit is distributed in the same ratio as the investments, we can set up the ratio of their investments:
- Total investments = \( (X + 8000) + (2X + 2000) + (3X + 4000) \)
- Simplifying this gives:
\[ = X + 8000 + 2X + 2000 + 3X + 4000 = 6X + 14000 \]

### Step 4: Set up the equation for B's share
B's share of the profit can be expressed as:
\[ \frac{B's \, investment}{Total \, investment} = \frac{4000}{16000} = \frac{1}{4} \]
Thus,
\[ \frac{2X + 2000}{6X + 14000} = \frac{1}{4} \]

### Step 5: Cross-multiply to solve for X
Cross-multiplying gives:
\[ 4(2X + 2000) = 1(6X + 14000) \]
Expanding both sides:
\[ 8X + 8000 = 6X + 14000 \]

### Step 6: Rearranging the equation
Rearranging the equation to isolate X:
\[ 8X - 6X = 14000 - 8000 \]
\[ 2X = 6000 \]
\[ X = 3000 \]

### Step 7: Calculate A's and C's investments
Now that we have the value of X, we can find the actual investments:
- A's investment:
\[ A's \, investment = X + 8000 = 3000 + 8000 = 11000 \]
- C's investment:
\[ C's \, investment = 3X + 4000 = 3(3000) + 4000 = 9000 + 4000 = 13000 \]

### Step 8: Find the difference between A's and C's investments
Now, we can find the difference between A's and C's investments:
\[ Difference = C's \, investment - A's \, investment = 13000 - 11000 = 2000 \]

### Final Answer
The difference between the investments of A and C is **2000**.

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