To solve the problem, we need to find the time taken by A, B, and C to complete the work together. We will start by determining the work done by each pair of workers and then find the combined work rate of A, B, and C.
### Step 1: Determine the work rates of each pair
1. A and B can complete the work in 12 days. Therefore, their combined work rate is:
\[
\text{Work rate of A and B} = \frac{1}{12} \text{ (work per day)}
\]
2. B and C can complete the work in 15 days. Therefore, their combined work rate is:
\[
\text{Work rate of B and C} = \frac{1}{15} \text{ (work per day)}
\]
3. C and A can complete the work in 20 days. Therefore, their combined work rate is:
\[
\text{Work rate of C and A} = \frac{1}{20} \text{ (work per day)}
\]
### Step 2: Set up equations for individual work rates
Let the work rates of A, B, and C be \(a\), \(b\), and \(c\) respectively. We can write the following equations based on the work rates calculated above:
1. \(a + b = \frac{1}{12}\) (Equation 1)
2. \(b + c = \frac{1}{15}\) (Equation 2)
3. \(c + a = \frac{1}{20}\) (Equation 3)
### Step 3: Solve the equations
We will add all three equations:
\[
(a + b) + (b + c) + (c + a) = \frac{1}{12} + \frac{1}{15} + \frac{1}{20}
\]
This simplifies to:
\[
2a + 2b + 2c = \frac{1}{12} + \frac{1}{15} + \frac{1}{20}
\]
Now, we need to find a common denominator for the right side. The least common multiple of 12, 15, and 20 is 60. We convert each fraction:
\[
\frac{1}{12} = \frac{5}{60}, \quad \frac{1}{15} = \frac{4}{60}, \quad \frac{1}{20} = \frac{3}{60}
\]
Adding these gives:
\[
\frac{5}{60} + \frac{4}{60} + \frac{3}{60} = \frac{12}{60} = \frac{1}{5}
\]
Thus, we have:
\[
2a + 2b + 2c = \frac{1}{5}
\]
Dividing by 2:
\[
a + b + c = \frac{1}{10} \text{ (Equation 4)}
\]
### Step 4: Find the time taken by A, B, and C together
The combined work rate of A, B, and C is \(\frac{1}{10}\) of the work per day. Therefore, the time taken by A, B, and C to complete the work together is:
\[
\text{Time} = \frac{1}{\text{Work rate}} = \frac{1}{\frac{1}{10}} = 10 \text{ days}
\]
### Final Answer
The time taken by A, B, and C to do the same work is **10 days**.
