A and B can do a piece of work in 40 days. B and C can do a piece of work in 30 days. C and A can do a piece of work in 24 days. How long would they all take to do the same work?

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 how to approach the problem step by step:

### Step 1: Determine the work rates of A and B
A and B can complete the work in 40 days. Therefore, their combined work rate is:
\[ \text{Work rate of A + B} = \frac{1}{40} \text{ (work per day)} \]

### Step 2: Determine the work rates of B and C
B and C can complete the work in 30 days. Therefore, their combined work rate is:
\[ \text{Work rate of B + C} = \frac{1}{30} \text{ (work per day)} \]

### Step 3: Determine the work rates of C and A
C and A can complete the work in 24 days. Therefore, their combined work rate is:
\[ \text{Work rate of C + A} = \frac{1}{24} \text{ (work per day)} \]

### Step 4: Set up the equations
Let the work rates of A, B, and C be represented as \( a, b, \) and \( c \) respectively. We can write the following equations based on the combined work rates:
1. \( a + b = \frac{1}{40} \)
2. \( b + c = \frac{1}{30} \)
3. \( c + a = \frac{1}{24} \)

### Step 5: Solve the equations
To find the individual work rates, we can add all three equations:
\[ (a + b) + (b + c) + (c + a) = \frac{1}{40} + \frac{1}{30} + \frac{1}{24} \]
This simplifies to:
\[ 2a + 2b + 2c = \frac{1}{40} + \frac{1}{30} + \frac{1}{24} \]
Now, we need to find a common denominator for the right side. The least common multiple of 40, 30, and 24 is 120. Converting 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:
\[ \frac{3}{120} + \frac{4}{120} + \frac{5}{120} = \frac{12}{120} = \frac{1}{10} \]
Thus, we have:
\[ 2(a + b + c) = \frac{1}{10} \implies a + b + c = \frac{1}{20} \]

### Step 6: Calculate the time taken by A, B, and C together
The combined work rate of A, B, and C is \( \frac{1}{20} \) work per day. Therefore, the time taken to complete the work when all three work together is:
\[ \text{Time} = \frac{1}{\text{Work rate}} = \frac{1}{\frac{1}{20}} = 20 \text{ days} \]

### Final Answer
Thus, A, B, and C together can complete the work in **20 days**.