A takes twice the time taken by B and thrice the time taken by C to do a particular piece of work. Working together, they can complete the work in 2 days. Find the number of days taken by A, B and C respectively to complete the work alone

Options

A. 12,6 and 4
B. 18,9 and 6
C. 24,12 and 8
D. 6,3 and 2

Answer

To solve the problem step by step, we will denote the time taken by B to complete the work alone as \( x \) days.

### Step 1: Define the time taken by A and C
According to the problem:
- A takes twice the time taken by B, so the time taken by A is \( 2x \) days.
- A takes thrice the time taken by C, so the time taken by C can be expressed as \( \frac{2x}{3} \) days.

### Step 2: Calculate the work done by A, B, and C in one day
Now we can calculate the work done by each person in one day:
- Work done by A in one day = \( \frac{1}{2x} \)
- Work done by B in one day = \( \frac{1}{x} \)
- Work done by C in one day = \( \frac{3}{2x} \)

### Step 3: Combine the work done by A, B, and C
When A, B, and C work together, their combined work done in one day is:
\[ \text{Total work in one day} = \frac{1}{2x} + \frac{1}{x} + \frac{3}{2x} \]

### Step 4: Simplify the combined work
To combine these fractions, we need a common denominator, which is \( 2x \):
\[ \text{Total work in one day} = \frac{1}{2x} + \frac{2}{2x} + \frac{3}{2x} = \frac{1 + 2 + 3}{2x} = \frac{6}{2x} = \frac{3}{x} \]

### Step 5: Set up the equation based on the total work done in 2 days
According to the problem, working together they can complete the work in 2 days. Therefore:
\[ \text{Total work} = 1 \quad \text{(the whole work)} \]
So, the equation becomes:
\[ \frac{3}{x} \times 2 = 1 \]
This simplifies to:
\[ \frac{6}{x} = 1 \]

### Step 6: Solve for x
Now, we solve for \( x \):
\[ 6 = x \quad \Rightarrow \quad x = 6 \]

### Step 7: Find the time taken by A and C
Now that we have \( x \):
- Time taken by B = \( x = 6 \) days
- Time taken by A = \( 2x = 2 \times 6 = 12 \) days
- Time taken by C = \( \frac{2x}{3} = \frac{2 \times 6}{3} = 4 \) days

### Final Answer
Thus, the number of days taken by A, B, and C respectively to complete the work alone are:
- A: 12 days
- B: 6 days
- C: 4 days