A and B complete a piece of work in 24 days, B and C do the same work in 36 days, and A, B and C together finish it in 18 days. In how many days will :
A and C together, complete the work ?

Answer

To solve the problem step by step, we will first determine the efficiencies of A, B, and C based on the information given, and then find out how long A and C together will take to complete the work.

### Step 1: Determine the efficiencies of A, B, and C

1. **Efficiency of A and B together**:
- A and B can complete the work in 24 days.
- Efficiency of A and B = Total work / Time = 1 / 24.

2. **Efficiency of B and C together**:
- B and C can complete the work in 36 days.
- Efficiency of B and C = Total work / Time = 1 / 36.

3. **Efficiency of A, B, and C together**:
- A, B, and C can complete the work in 18 days.
- Efficiency of A, B, and C = Total work / Time = 1 / 18.

### Step 2: Calculate the efficiency of A

To find the efficiency of A, we can use the formula:
\[ \text{Efficiency of A} = \text{Efficiency of (A + B + C)} - \text{Efficiency of (B + C)} \]
Substituting the values:
\[ \text{Efficiency of A} = \frac{1}{18} - \frac{1}{36} \]
Finding a common denominator (LCM of 18 and 36 is 36):
\[ \text{Efficiency of A} = \frac{2}{36} - \frac{1}{36} = \frac{1}{36} \]

### Step 3: Calculate the efficiency of C

Similarly, to find the efficiency of C:
\[ \text{Efficiency of C} = \text{Efficiency of (A + B + C)} - \text{Efficiency of (A + B)} \]
We need to find the efficiency of A + B first:
\[ \text{Efficiency of (A + B)} = \frac{1}{24} \]
Now substituting:
\[ \text{Efficiency of C} = \frac{1}{18} - \frac{1}{24} \]
Finding a common denominator (LCM of 18 and 24 is 72):
\[ \text{Efficiency of C} = \frac{4}{72} - \frac{3}{72} = \frac{1}{72} \]

### Step 4: Calculate the efficiency of A and C together

Now we can find the combined efficiency of A and C:
\[ \text{Efficiency of (A + C)} = \text{Efficiency of A} + \text{Efficiency of C} \]
Substituting the values:
\[ \text{Efficiency of (A + C)} = \frac{1}{36} + \frac{1}{72} \]
Finding a common denominator (LCM of 36 and 72 is 72):
\[ \text{Efficiency of (A + C)} = \frac{2}{72} + \frac{1}{72} = \frac{3}{72} = \frac{1}{24} \]

### Step 5: Calculate the time taken by A and C together

To find the time taken by A and C to complete the work:
\[ \text{Time} = \frac{1}{\text{Efficiency of (A + C)}} \]
Substituting the efficiency:
\[ \text{Time} = \frac{1}{\frac{1}{24}} = 24 \text{ days} \]

### Final Answer
A and C together will complete the work in **24 days**.