To solve the problem step by step, we will first determine the work done by each pair of workers and then find the total work done by B, C, and D together.
### Step 1: Determine the work done by each pair of workers
1. **B and C can complete the work in 12 days.**
- Work done by B and C in one day = \( \frac{1}{12} \) of the work.
2. **C and D can complete the work in 18 days.**
- Work done by C and D in one day = \( \frac{1}{18} \) of the work.
3. **D and B can complete the work in 24 days.**
- Work done by D and B in one day = \( \frac{1}{24} \) of the work.
### Step 2: Set up equations for the work done
Let:
- Work done by B in one day = \( b \)
- Work done by C in one day = \( c \)
- Work done by D in one day = \( d \)
From the above information, we can write the following equations:
1. \( b + c = \frac{1}{12} \) (Equation 1)
2. \( c + d = \frac{1}{18} \) (Equation 2)
3. \( d + b = \frac{1}{24} \) (Equation 3)
### Step 3: Solve the equations
We can add all three equations to eliminate the variables:
\[
(b + c) + (c + d) + (d + b) = \frac{1}{12} + \frac{1}{18} + \frac{1}{24}
\]
This simplifies to:
\[
2b + 2c + 2d = \frac{1}{12} + \frac{1}{18} + \frac{1}{24}
\]
Now, we need to find the right side's common denominator. The LCM of 12, 18, and 24 is 72. Thus:
\[
\frac{1}{12} = \frac{6}{72}, \quad \frac{1}{18} = \frac{4}{72}, \quad \frac{1}{24} = \frac{3}{72}
\]
Adding these fractions gives:
\[
\frac{6 + 4 + 3}{72} = \frac{13}{72}
\]
So we have:
\[
2(b + c + d) = \frac{13}{72}
\]
Dividing both sides by 2:
\[
b + c + d = \frac{13}{144}
\]
### Step 4: Calculate the total work done by B, C, and D together
The total work done by B, C, and D in one day is \( \frac{13}{144} \).
### Step 5: Find the time taken to complete the work
To find the number of days B, C, and D together can complete the work, we take the reciprocal of their combined work rate:
\[
\text{Days} = \frac{1}{\frac{13}{144}} = \frac{144}{13}
\]
### Final Answer
Thus, B, C, and D together can complete the work in \( \frac{144}{13} \) days, which is approximately 11.08 days.
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