A and B do a work in 18 days, B and C do it in 15 days while C and A in 20 days. In how many days A, B and C all working together can finish the same work?

Answer

To solve the problem, we need to find the individual work rates of A, B, and C based on the information provided about their combined work rates. Let's break it down step by step.

### Step 1: Determine the work rates of A+B, B+C, and C+A
- A and B can complete the work in 18 days.
- B and C can complete the work in 15 days.
- C and A can complete the work in 20 days.

### Step 2: Calculate the work done per day by each pair
- Work done by A and B in one day = \( \frac{1}{18} \) (since they complete the work in 18 days)
- Work done by B and C in one day = \( \frac{1}{15} \)
- Work done by C and A in one day = \( \frac{1}{20} \)

### Step 3: Set up equations based on the work rates
Let the work done by A in one day be \( a \), by B be \( b \), and by C be \( c \).
From the information given, we can write the following equations:
1. \( a + b = \frac{1}{18} \) (Equation 1)
2. \( b + c = \frac{1}{15} \) (Equation 2)
3. \( c + a = \frac{1}{20} \) (Equation 3)

### Step 4: Solve the equations
We can add all three equations:
\[ (a + b) + (b + c) + (c + a) = \frac{1}{18} + \frac{1}{15} + \frac{1}{20} \]
This simplifies to:
\[ 2a + 2b + 2c = \frac{1}{18} + \frac{1}{15} + \frac{1}{20} \]
Now, we need to find a common denominator for the right side. The least common multiple of 18, 15, and 20 is 180. Converting each fraction:
- \( \frac{1}{18} = \frac{10}{180} \)
- \( \frac{1}{15} = \frac{12}{180} \)
- \( \frac{1}{20} = \frac{9}{180} \)

Adding these gives:
\[ \frac{10 + 12 + 9}{180} = \frac{31}{180} \]
Thus, we have:
\[ 2(a + b + c) = \frac{31}{180} \]
Dividing both sides by 2:
\[ a + b + c = \frac{31}{360} \]

### Step 5: Calculate the time taken by A, B, and C together
The combined work rate of A, B, and C is \( \frac{31}{360} \). To find the time taken to complete the work together, we take the reciprocal of their combined work rate:
\[ \text{Time} = \frac{1}{\frac{31}{360}} = \frac{360}{31} \]

### Final Answer
Thus, A, B, and C together can finish the work in \( \frac{360}{31} \) days, which is approximately 11.61 days.

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