A can complete a piece of work in 4 days. B takes double the time taken by A, C takes double that of B, and D takes double that of C to complete the same task. They are paired in groups of two each. One pair takes two-thirds the time needed by the second pair to complete the work. Which is the first pair?

Options

A. A and B
B. A and C
C. B and C
D. A and D

Answer

To solve the problem step by step, we will first determine the time taken by each individual (A, B, C, and D) to complete the work, then calculate their efficiencies, and finally analyze the pairs to find which one is the first pair.

### Step 1: Determine the time taken by each worker
- A can complete the work in 4 days.
- B takes double the time of A:
\[ \text{Time taken by B} = 2 \times 4 = 8 \text{ days} \]
- C takes double the time of B:
\[ \text{Time taken by C} = 2 \times 8 = 16 \text{ days} \]
- D takes double the time of C:
\[ \text{Time taken by D} = 2 \times 16 = 32 \text{ days} \]

### Step 2: Calculate the work done by each worker in one day
- A's work rate (efficiency) = \( \frac{1}{4} \) (since A completes the work in 4 days)
- B's work rate = \( \frac{1}{8} \)
- C's work rate = \( \frac{1}{16} \)
- D's work rate = \( \frac{1}{32} \)

### Step 3: Calculate the combined efficiency of each pair
1. **Pair A and B**:
\[ \text{Efficiency of A and B} = \frac{1}{4} + \frac{1}{8} = \frac{2}{8} + \frac{1}{8} = \frac{3}{8} \]
Time taken by A and B to complete the work:
\[ \text{Time} = \frac{1}{\frac{3}{8}} = \frac{8}{3} \text{ days} \]

2. **Pair C and D**:
\[ \text{Efficiency of C and D} = \frac{1}{16} + \frac{1}{32} = \frac{2}{32} + \frac{1}{32} = \frac{3}{32} \]
Time taken by C and D to complete the work:
\[ \text{Time} = \frac{1}{\frac{3}{32}} = \frac{32}{3} \text{ days} \]

3. **Pair B and C**:
\[ \text{Efficiency of B and C} = \frac{1}{8} + \frac{1}{16} = \frac{2}{16} + \frac{1}{16} = \frac{3}{16} \]
Time taken by B and C to complete the work:
\[ \text{Time} = \frac{1}{\frac{3}{16}} = \frac{16}{3} \text{ days} \]

4. **Pair A and D**:
\[ \text{Efficiency of A and D} = \frac{1}{4} + \frac{1}{32} = \frac{8}{32} + \frac{1}{32} = \frac{9}{32} \]
Time taken by A and D to complete the work:
\[ \text{Time} = \frac{1}{\frac{9}{32}} = \frac{32}{9} \text{ days} \]

### Step 4: Compare the pairs based on the time taken
According to the problem, one pair takes two-thirds the time needed by the second pair. We will check each pair against the others:

- **Comparing A and B with C and D**:
\[ \frac{8/3}{32/3} = \frac{8}{32} = \frac{1}{4} \quad \text{(not two-thirds)} \]

- **Comparing B and C with A and D**:
\[ \frac{16/3}{32/9} = \frac{16 \times 9}{32 \times 3} = \frac{144}{96} = \frac{3}{2} \quad \text{(not two-thirds)} \]

- **Comparing A and D with C and D**:
\[ \frac{32/9}{32/3} = \frac{32 \times 3}{32 \times 9} = \frac{3}{9} = \frac{1}{3} \quad \text{(not two-thirds)} \]

- **Comparing A and B with B and C**:
\[ \frac{8/3}{16/3} = \frac{8}{16} = \frac{1}{2} \quad \text{(not two-thirds)} \]

### Conclusion
After checking all pairs, we find that the first pair that satisfies the condition of taking two-thirds the time of the second pair is **A and B**.

### Final Answer
The first pair is **A and B**.