A and B can do a piece of work in 3 days. B and C can do the same workin 9days, while C and A can do it in 12 days. Find the time in which A, B and C can finish the work, working together.

Answer

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

### Step 1: Determine the work rates of A and B
A and B can complete the work in 3 days. Therefore, their combined work rate is:
\[ \text{Work rate of A + B} = \frac{1}{3} \text{ (work per day)} \]

### Step 2: Determine the work rates of B and C
B and C can complete the work in 9 days. Therefore, their combined work rate is:
\[ \text{Work rate of B + C} = \frac{1}{9} \text{ (work per day)} \]

### Step 3: Determine the work rates of C and A
C and A can complete the work in 12 days. Therefore, their combined work rate is:
\[ \text{Work rate of C + A} = \frac{1}{12} \text{ (work per day)} \]

### Step 4: Set up equations
Let the work rates of A, B, and C be \(a\), \(b\), and \(c\) respectively. We can set up the following equations based on the information we have:
1. \(a + b = \frac{1}{3}\) (Equation 1)
2. \(b + c = \frac{1}{9}\) (Equation 2)
3. \(c + a = \frac{1}{12}\) (Equation 3)

### Step 5: Solve the equations
Now, we can add all three equations together:
\[ (a + b) + (b + c) + (c + a) = \frac{1}{3} + \frac{1}{9} + \frac{1}{12} \]
This simplifies to:
\[ 2a + 2b + 2c = \frac{1}{3} + \frac{1}{9} + \frac{1}{12} \]

### Step 6: Find a common denominator
To add the fractions on the right side, we need a common denominator. The least common multiple of 3, 9, and 12 is 36. Thus, we convert each fraction:
\[ \frac{1}{3} = \frac{12}{36}, \quad \frac{1}{9} = \frac{4}{36}, \quad \frac{1}{12} = \frac{3}{36} \]
Adding these gives:
\[ \frac{12}{36} + \frac{4}{36} + \frac{3}{36} = \frac{19}{36} \]

### Step 7: Solve for \(a + b + c\)
Now we can simplify the equation:
\[ 2(a + b + c) = \frac{19}{36} \]
Dividing both sides by 2:
\[ a + b + c = \frac{19}{72} \]

### Step 8: Find the time taken by A, B, and C together
The combined work rate of A, B, and C is \(\frac{19}{72}\) work per day. Therefore, the time taken to complete the work when A, B, and C work together is the reciprocal of their combined work rate:
\[ \text{Time} = \frac{1}{\frac{19}{72}} = \frac{72}{19} \text{ days} \]

### Final Answer
Thus, A, B, and C can finish the work together in \(\frac{72}{19}\) days, which is approximately 3.79 days.

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