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 rates of A, B, and C
1. **Work done by A and B together in 10 days:**
- Work rate of A and B together = 1/10 (since they complete the work in 10 days).
2. **Work done by B and C together in 6 days:**
- Work rate of B and C together = 1/6 (since they complete the work in 6 days).
3. **Work done by A and C together in 12 days:**
- Work rate of A and C together = 1/12 (since they complete the work in 12 days).
### Step 2: Set up equations based on the work rates
Let:
- Rate of A = a (work done by A in one day)
- Rate of B = b (work done by B in one day)
- Rate of C = c (work done by C in one day)
From the information:
1. \( a + b = \frac{1}{10} \) (Equation 1)
2. \( b + c = \frac{1}{6} \) (Equation 2)
3. \( a + c = \frac{1}{12} \) (Equation 3)
### Step 3: Solve the equations
To find the individual work rates, we can manipulate these equations.
**Step 3.1: Add all three equations**
\[
(a + b) + (b + c) + (a + c) = \frac{1}{10} + \frac{1}{6} + \frac{1}{12}
\]
This simplifies to:
\[
2a + 2b + 2c = \frac{1}{10} + \frac{1}{6} + \frac{1}{12}
\]
**Step 3.2: Find a common denominator for the right side**
The least common multiple of 10, 6, and 12 is 60. Therefore:
\[
\frac{1}{10} = \frac{6}{60}, \quad \frac{1}{6} = \frac{10}{60}, \quad \frac{1}{12} = \frac{5}{60}
\]
Adding these gives:
\[
\frac{6 + 10 + 5}{60} = \frac{21}{60} = \frac{7}{20}
\]
So, we have:
\[
2a + 2b + 2c = \frac{7}{20}
\]
Dividing by 2:
\[
a + b + c = \frac{7}{40} \quad (Equation 4)
\]
### Step 4: Calculate the combined work rate of A, B, and C
The combined work rate of A, B, and C is:
\[
a + b + c = \frac{7}{40}
\]
This means A, B, and C together can complete \(\frac{7}{40}\) of the work in one day.
### Step 5: Calculate the time taken by A, B, and C to complete the work
To find the number of days taken to complete the work, we take the reciprocal of the combined work rate:
\[
\text{Days} = \frac{1}{a + b + c} = \frac{1}{\frac{7}{40}} = \frac{40}{7} \text{ days}
\]
### Final Answer
A, B, and C together can complete the work in \(\frac{40}{7}\) days, which is approximately 5.71 days.
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