(A + B) can do a piece of work in 5 days. If A works with twice of his efficiency and B works with half of his efficiency, then work will be completed in 4 days. In how many days will A and B do this work alone ?

Answer

To solve the problem step by step, we will first define the efficiencies of A and B, then use the information given to find out how long each person would take to complete the work alone.

### Step 1: Define the efficiencies of A and B
Let the efficiency of A be \( A \) units of work per day and the efficiency of B be \( B \) units of work per day.

### Step 2: Set up the equation based on the first condition
According to the problem, A and B together can complete the work in 5 days. Therefore, the total work can be expressed as:
\[ \text{Total Work} = (A + B) \times 5 \]

### Step 3: Set up the equation based on the second condition
When A works with twice his efficiency and B works with half his efficiency, they complete the same work in 4 days. This can be expressed as:
\[ \text{Total Work} = (2A + \frac{B}{2}) \times 4 \]

### Step 4: Equate the two expressions for total work
Since both expressions represent the same total work, we can set them equal to each other:
\[ (A + B) \times 5 = (2A + \frac{B}{2}) \times 4 \]

### Step 5: Simplify the equation
Expanding both sides gives:
\[ 5A + 5B = 8A + 2B \]
Rearranging the equation leads to:
\[ 5B - 2B = 8A - 5A \]
This simplifies to:
\[ 3B = 3A \]
Thus, we find:
\[ A = B \]

### Step 6: Substitute A in terms of B
Let’s denote both A and B as \( K \) (since they are equal):
\[ A = K \quad \text{and} \quad B = K \]

### Step 7: Find the total work in terms of K
Substituting \( A \) and \( B \) back into the total work equation:
\[ \text{Total Work} = (K + K) \times 5 = 10K \]

### Step 8: Calculate the time taken by A and B alone
Now, substituting back into the equation for total work:
\[ 10K = (2K + \frac{K}{2}) \times 4 \]
Simplifying the right side:
\[ 10K = (2K + 0.5K) \times 4 = (2.5K) \times 4 = 10K \]
This confirms our calculations are consistent.

### Step 9: Calculate the time taken by A and B alone
Since \( A \) and \( B \) both work at \( K \) units per day, the time taken by A alone to complete the work is:
\[ \text{Time taken by A} = \frac{\text{Total Work}}{A} = \frac{10K}{K} = 10 \text{ days} \]
Similarly, the time taken by B alone is:
\[ \text{Time taken by B} = \frac{\text{Total Work}}{B} = \frac{10K}{K} = 10 \text{ days} \]

### Conclusion
Thus, A will take 10 days to complete the work alone, and B will also take 10 days to complete the work alone.

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