A and B together can complete a work in 8 days and B and C together in 12 days. All of the three together can complete the work in 6 days. In how much time will A and C together complete the work?

Options

A. 8 days
B. 10 days
C. 12 days
D. 20 days

Answer

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

### Step 1: Determine the work done by A and B together.
A and B together can complete the work in 8 days. Therefore, the amount of work they can do in one day is:
\[ \text{Work rate of A + B} = \frac{1}{8} \text{ (work per day)} \]
If we consider the total work to be 24 units, then:
\[ \text{Work done by A + B in 1 day} = \frac{24}{8} = 3 \text{ units} \]

### Step 2: Determine the work done by B and C together.
B and C together can complete the work in 12 days. Therefore, their work rate is:
\[ \text{Work rate of B + C} = \frac{1}{12} \text{ (work per day)} \]
Thus, the work done by B and C in one day is:
\[ \text{Work done by B + C in 1 day} = \frac{24}{12} = 2 \text{ units} \]

### Step 3: Determine the work done by A, B, and C together.
A, B, and C together can complete the work in 6 days. Therefore, their work rate is:
\[ \text{Work rate of A + B + C} = \frac{1}{6} \text{ (work per day)} \]
Thus, the work done by A, B, and C in one day is:
\[ \text{Work done by A + B + C in 1 day} = \frac{24}{6} = 4 \text{ units} \]

### Step 4: Set up equations for the work rates.
From the above calculations, we have:
- A + B = 3 units/day
- B + C = 2 units/day
- A + B + C = 4 units/day

### Step 5: Find the work done by A and C together.
To find the work done by A and C together, we can use the following relationship:
\[ \text{(A + C)} = \text{(A + B + C)} - \text{(B)} \]
We can express B in terms of the other equations.

From A + B + C = 4 units and B + C = 2 units, we can find B:
\[ B = (A + B + C) - (A + C) = 4 - (A + C) \]
Now, substituting the value of B from B + C:
\[ B + C = 2 \implies (4 - (A + C)) + C = 2 \]
This simplifies to:
\[ 4 - A = 2 \implies A = 2 \text{ units} \]

### Step 6: Substitute to find C.
Now substituting A back into A + B = 3:
\[ 2 + B = 3 \implies B = 1 \text{ unit} \]
Now substituting B into B + C = 2:
\[ 1 + C = 2 \implies C = 1 \text{ unit} \]

### Step 7: Calculate A + C.
Now we can find A + C:
\[ A + C = 2 + 1 = 3 \text{ units/day} \]

### Step 8: Calculate the time taken by A and C to complete the work.
To find the time taken by A and C together to complete the work:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Work Rate of A + C}} = \frac{24}{3} = 8 \text{ days} \]

### Final Answer:
A and C together can complete the work in **8 days**.

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