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, we need to find the individual work rates of A, B, and C based on the information given about their combined work rates. Here’s the step-by-step solution:

### Step 1: Determine the work rates of A, B, and C
Let the total work be represented as 1 unit.

- A and B together can complete the work in 12 days.
- Work rate of A and B = 1/12 units of work per day.

- B and C together can complete the work in 15 days.
- Work rate of B and C = 1/15 units of work per day.

- C and A together can complete the work in 20 days.
- Work rate of C and A = 1/20 units of work per day.

### Step 2: Set up equations based on work rates
Let the work rates of A, B, and C be represented as:
- Work rate of A = a
- Work rate of B = b
- Work rate of C = c

From the above information, we can set up the following 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 the equations
To find the individual work rates, we can add all three equations together:
\[ (a + b) + (b + c) + (c + a) = \frac{1}{12} + \frac{1}{15} + \frac{1}{20} \]
This simplifies to:
\[ 2a + 2b + 2c = \frac{1}{12} + \frac{1}{15} + \frac{1}{20} \]

### Step 4: Calculate the right-hand side
To add the fractions, we need a common denominator. The least common multiple of 12, 15, and 20 is 60.
- Convert each fraction:
- \( \frac{1}{12} = \frac{5}{60} \)
- \( \frac{1}{15} = \frac{4}{60} \)
- \( \frac{1}{20} = \frac{3}{60} \)

Now, add them:
\[ \frac{5}{60} + \frac{4}{60} + \frac{3}{60} = \frac{12}{60} = \frac{1}{5} \]

### Step 5: Solve for a + b + c
Now we have:
\[ 2(a + b + c) = \frac{1}{5} \]
Dividing both sides by 2 gives:
\[ a + b + c = \frac{1}{10} \]

### Step 6: Calculate the total time for A, B, and C to work together
The combined work rate of A, B, and C is \( \frac{1}{10} \) units of work per day. Therefore, the time taken for all three to complete the work together is:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Combined Work Rate}} = \frac{1}{\frac{1}{10}} = 10 \text{ days} \]

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

---