A and B can complete a work in 20 days. B and C can complete the same work in 30 days. C and A can complete the same work in 40 days. In how many days they working together to complete the work?

Answer

To solve the problem step by step, we will first determine the individual work rates of A, B, and C based on the information given.

### Step 1: Determine the work rates of A, B, and C

1. **Work done by A and B together**:
- A and B can complete the work in 20 days.
- Therefore, their combined work rate is \( \frac{1}{20} \) of the work per day.

2. **Work done by B and C together**:
- B and C can complete the work in 30 days.
- Therefore, their combined work rate is \( \frac{1}{30} \) of the work per day.

3. **Work done by C and A together**:
- C and A can complete the work in 40 days.
- Therefore, their combined work rate is \( \frac{1}{40} \) of the work per day.

### Step 2: Set up equations based on the work rates

Let:
- The work rate of A = \( a \)
- The work rate of B = \( b \)
- The work rate of C = \( c \)

From the above information, we can set up the following equations:
1. \( a + b = \frac{1}{20} \) (Equation 1)
2. \( b + c = \frac{1}{30} \) (Equation 2)
3. \( c + a = \frac{1}{40} \) (Equation 3)

### Step 3: Solve the equations

To find the individual work rates, we can add all three equations together:

\[ (a + b) + (b + c) + (c + a) = \frac{1}{20} + \frac{1}{30} + \frac{1}{40} \]

This simplifies to:

\[ 2a + 2b + 2c = \frac{1}{20} + \frac{1}{30} + \frac{1}{40} \]

### Step 4: Find a common denominator

The least common multiple of 20, 30, and 40 is 120. We can convert each fraction:

\[ \frac{1}{20} = \frac{6}{120}, \quad \frac{1}{30} = \frac{4}{120}, \quad \frac{1}{40} = \frac{3}{120} \]

Adding these fractions gives:

\[ \frac{6}{120} + \frac{4}{120} + \frac{3}{120} = \frac{13}{120} \]

### Step 5: Substitute back into the equation

Now we have:

\[ 2(a + b + c) = \frac{13}{120} \]

Dividing by 2:

\[ a + b + c = \frac{13}{240} \]

### Step 6: Find the total work rate when A, B, and C work together

The combined work rate of A, B, and C is \( \frac{13}{240} \) of the work per day.

### Step 7: Calculate the time taken to complete the work together

To find the number of days they will take to complete the work together, we take the reciprocal of their combined work rate:

\[ \text{Time} = \frac{1}{\frac{13}{240}} = \frac{240}{13} \approx 18.46 \text{ days} \]

### Final Answer

A, B, and C together can complete the work in approximately **18.46 days**.

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