A and B can do a piece of work in 40 days, B and C in 30 days, and C and A in 24 days.
How long will it take them to do the work, working together ?

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the given information
We know:
- A and B can complete the work in 40 days.
- B and C can complete the work in 30 days.
- C and A can complete the work in 24 days.

### Step 2: Write the work done by each pair in one day
The work done by A and B in one day is:
\[ \text{Work done by A and B} = \frac{1}{40} \]

The work done by B and C in one day is:
\[ \text{Work done by B and C} = \frac{1}{30} \]

The work done by C and A in one day is:
\[ \text{Work done by C and A} = \frac{1}{24} \]

### Step 3: Set up the equation for total work done by A, B, and C
If we add the work done by all pairs, we get:
\[ (A + B) + (B + C) + (C + A) = \frac{1}{40} + \frac{1}{30} + \frac{1}{24} \]

This simplifies to:
\[ 2A + 2B + 2C = \frac{1}{40} + \frac{1}{30} + \frac{1}{24} \]

### Step 4: Calculate the right-hand side
To add the fractions, we need a common denominator. The LCM of 40, 30, and 24 is 120.

Now, convert each fraction:
\[ \frac{1}{40} = \frac{3}{120}, \quad \frac{1}{30} = \frac{4}{120}, \quad \frac{1}{24} = \frac{5}{120} \]

Adding these gives:
\[ \frac{3}{120} + \frac{4}{120} + \frac{5}{120} = \frac{12}{120} = \frac{1}{10} \]

### Step 5: Solve for A + B + C
Now, we have:
\[ 2A + 2B + 2C = \frac{1}{10} \]

Dividing both sides by 2 gives:
\[ A + B + C = \frac{1}{20} \]

### Step 6: Interpret the result
The equation \(A + B + C = \frac{1}{20}\) means that together, A, B, and C can complete \(\frac{1}{20}\) of the work in one day.

### Step 7: Calculate the total time taken to complete the work
If they can complete \(\frac{1}{20}\) of the work in one day, the total time taken to complete the work is:
\[ \text{Time} = \frac{1}{\frac{1}{20}} = 20 \text{ days} \]

### Final Answer
Thus, A, B, and C together will take **20 days** to complete the work.

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