To solve the problem, we need to determine the individual efficiencies of A, B, C, and D based on the information provided. Let's break it down step by step.
### Step 1: Determine the total work done
We know that:
- A, B, and C can complete the work in 5 days.
- B, C, and D can complete the work in 10 days.
- C, D, and A can complete the work in 15 days.
- D, A, and B can complete the work in 30 days.
To make calculations easier, we can assume the total work is 60 units (the least common multiple of the days).
### Step 2: Calculate the efficiencies
1. **Efficiency of A, B, and C:**
\[
\text{Work done by A, B, C in 5 days} = 60 \text{ units}
\]
\[
\text{Daily work} = \frac{60}{5} = 12 \text{ units}
\]
So, \( A + B + C = 12 \).
2. **Efficiency of B, C, and D:**
\[
\text{Work done by B, C, D in 10 days} = 60 \text{ units}
\]
\[
\text{Daily work} = \frac{60}{10} = 6 \text{ units}
\]
So, \( B + C + D = 6 \).
3. **Efficiency of C, D, and A:**
\[
\text{Work done by C, D, A in 15 days} = 60 \text{ units}
\]
\[
\text{Daily work} = \frac{60}{15} = 4 \text{ units}
\]
So, \( C + D + A = 4 \).
4. **Efficiency of D, A, and B:**
\[
\text{Work done by D, A, B in 30 days} = 60 \text{ units}
\]
\[
\text{Daily work} = \frac{60}{30} = 2 \text{ units}
\]
So, \( D + A + B = 2 \).
### Step 3: Set up equations
Now we have the following equations:
1. \( A + B + C = 12 \) (1)
2. \( B + C + D = 6 \) (2)
3. \( C + D + A = 4 \) (3)
4. \( D + A + B = 2 \) (4)
### Step 4: Solve the equations
We can add all four equations together:
\[
(A + B + C) + (B + C + D) + (C + D + A) + (D + A + B) = 12 + 6 + 4 + 2
\]
This simplifies to:
\[
3A + 3B + 3C + 3D = 24
\]
Dividing the entire equation by 3 gives:
\[
A + B + C + D = 8
\]
### Step 5: Calculate the time taken by A + B + C + D
Since \( A + B + C + D = 8 \) units of work can be done in one day, we can find the number of days required to complete the total work of 60 units:
\[
\text{Time taken} = \frac{\text{Total Work}}{\text{Daily Work}} = \frac{60}{8} = 7.5 \text{ days}
\]
### Final Answer
The time taken by A + B + C + D to complete the work is **7.5 days**.
