A and B can do a piece of work in 8 days, B and C can do the same work in 12 days and A and C complete it in 8 days. In how many days A, B and C can complete the whole work, working together?

Options

A. 4
B. 6
C. 12
D. 9

Answer

To solve the problem step by step, we will first determine the efficiencies of A, B, and C based on the information provided.

### Step 1: Determine the efficiencies of pairs A & B, B & C, and A & C
- A and B can complete the work in 8 days. Therefore, their combined efficiency is:
\[ \text{Efficiency of A and B} = \frac{1 \text{ work}}{8 \text{ days}} = \frac{1}{8} \]

- B and C can complete the work in 12 days. Therefore, their combined efficiency is:
\[ \text{Efficiency of B and C} = \frac{1 \text{ work}}{12 \text{ days}} = \frac{1}{12} \]

- A and C can complete the work in 8 days. Therefore, their combined efficiency is:
\[ \text{Efficiency of A and C} = \frac{1 \text{ work}}{8 \text{ days}} = \frac{1}{8} \]

### Step 2: Set up equations based on efficiencies
Let the efficiencies of A, B, and C be represented as \( a \), \( b \), and \( c \) respectively. We can set up the following equations based on the efficiencies calculated:

1. \( a + b = \frac{1}{8} \) (from A and B)
2. \( b + c = \frac{1}{12} \) (from B and C)
3. \( a + c = \frac{1}{8} \) (from A and C)

### Step 3: Solve the equations
We can solve these equations to find the individual efficiencies of A, B, and C.

From equation (1):
\[ a + b = \frac{1}{8} \quad \text{(1)} \]

From equation (3):
\[ a + c = \frac{1}{8} \quad \text{(3)} \]

Subtract equation (1) from equation (3):
\[ (a + c) - (a + b) = \frac{1}{8} - \frac{1}{8} \]
\[ c - b = 0 \implies c = b \quad \text{(4)} \]

Now substitute \( c = b \) into equation (2):
\[ b + b = \frac{1}{12} \]
\[ 2b = \frac{1}{12} \implies b = \frac{1}{24} \]

Now substitute \( b \) back into equation (4) to find \( c \):
\[ c = \frac{1}{24} \]

Now substitute \( b \) into equation (1) to find \( a \):
\[ a + \frac{1}{24} = \frac{1}{8} \]
Convert \( \frac{1}{8} \) to a fraction with a denominator of 24:
\[ \frac{1}{8} = \frac{3}{24} \]
Thus,
\[ a + \frac{1}{24} = \frac{3}{24} \implies a = \frac{3}{24} - \frac{1}{24} = \frac{2}{24} = \frac{1}{12} \]

### Step 4: Calculate the total efficiency of A, B, and C
Now we have:
- \( a = \frac{1}{12} \)
- \( b = \frac{1}{24} \)
- \( c = \frac{1}{24} \)

The total efficiency of A, B, and C working together is:
\[ \text{Total Efficiency} = a + b + c = \frac{1}{12} + \frac{1}{24} + \frac{1}{24} \]

To add these fractions, we need a common denominator, which is 24:
\[ \frac{1}{12} = \frac{2}{24} \]
Thus,
\[ \text{Total Efficiency} = \frac{2}{24} + \frac{1}{24} + \frac{1}{24} = \frac{4}{24} = \frac{1}{6} \]

### Step 5: Calculate the time taken by A, B, and C to complete the work
The time taken to complete the work is the reciprocal of the total efficiency:
\[ \text{Time} = \frac{1}{\text{Total Efficiency}} = \frac{1}{\frac{1}{6}} = 6 \text{ days} \]

### Final Answer
A, B, and C can complete the whole work together in **6 days**.

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