A and B can do a piece of work in 12 days, B and C in 15 days and C and A in 20 days. In how many days can they do it, all working together?

Answer

To solve the problem step by step, we will first express the work done by each pair of workers in terms of their individual work rates, and then find the total work rate when all three work together.

### Step 1: Define the work rates
Let:
- A's work rate = \( a \) (work done by A in one day)
- B's work rate = \( b \) (work done by B in one day)
- C's work rate = \( c \) (work done by C in one day)

From the problem statement:
1. A and B together can complete the work in 12 days:
\[ a + b = \frac{1}{12} \]
2. B and C together can complete the work in 15 days:
\[ b + c = \frac{1}{15} \]
3. C and A together can complete the work in 20 days:
\[ c + a = \frac{1}{20} \]

### Step 2: Set up the equations
We now have three equations:
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 for individual work rates
We will solve these equations to find the values of \( a \), \( b \), and \( c \).

**From Equation 1 and Equation 2:**
Subtract Equation 1 from Equation 2:
\[ (b + c) - (a + b) = \frac{1}{15} - \frac{1}{12} \]
This simplifies to:
\[ c - a = \frac{1}{15} - \frac{1}{12} \]
Finding a common denominator (60):
\[ c - a = \frac{4 - 5}{60} = -\frac{1}{60} \implies a - c = \frac{1}{60} \quad \text{(Equation 4)} \]

**From Equation 3 and Equation 4:**
Now, we can add Equation 4 to Equation 3:
\[ (c + a) + (a - c) = \frac{1}{20} + \frac{1}{60} \]
This simplifies to:
\[ 2a = \frac{3 + 1}{60} = \frac{4}{60} = \frac{1}{15} \implies a = \frac{1}{30} \]

### Step 4: Find values of b and c
**Substituting \( a \) back into Equation 1:**
\[ \frac{1}{30} + b = \frac{1}{12} \]
Finding \( b \):
\[ b = \frac{1}{12} - \frac{1}{30} \]
Finding a common denominator (60):
\[ b = \frac{5 - 2}{60} = \frac{3}{60} = \frac{1}{20} \]

**Substituting \( b \) back into Equation 2:**
\[ \frac{1}{20} + c = \frac{1}{15} \]
Finding \( c \):
\[ c = \frac{1}{15} - \frac{1}{20} \]
Finding a common denominator (60):
\[ c = \frac{4 - 3}{60} = \frac{1}{60} \]

### Step 5: Total work rate when A, B, and C work together
Now we have:
- \( a = \frac{1}{30} \)
- \( b = \frac{1}{20} \)
- \( c = \frac{1}{60} \)

The total work rate when A, B, and C work together is:
\[ a + b + c = \frac{1}{30} + \frac{1}{20} + \frac{1}{60} \]
Finding a common denominator (60):
\[ = \frac{2}{60} + \frac{3}{60} + \frac{1}{60} = \frac{6}{60} = \frac{1}{10} \]

### Step 6: Calculate the total time taken
If they can complete \( \frac{1}{10} \) of the work in one day, the total time taken to complete the work together is:
\[ \text{Total time} = \frac{1}{\frac{1}{10}} = 10 \text{ days} \]

### Final Answer:
A, B, and C can complete the work together in **10 days**.