(A + B) can do a piece of work in 5 days. If A works with twice of his efficiency and B works with `(1)/(3)` of his efficiency, then work will be completed in 3 days. In how many days will A do this work alone ?

Answer

To solve the problem, we will break it down step by step.

### Step 1: Understand the Work Done by A and B
Given that A + B can complete the work in 5 days, we can express their combined work rate as:
\[ \text{Work Rate of (A + B)} = \frac{1}{5} \text{ (work per day)} \]

### Step 2: Express the Work Rates of A and B
Let the work done by A in one day be \( A \) and the work done by B in one day be \( B \). Therefore, we have:
\[ A + B = \frac{1}{5} \]

### Step 3: Work with Adjusted Efficiencies
When A works with twice his efficiency, his work rate becomes \( 2A \), and when B works with one-third of his efficiency, his work rate becomes \( \frac{B}{3} \). Together, they complete the work in 3 days:
\[ 2A + \frac{B}{3} = \frac{1}{3} \text{ (work per day)} \]

### Step 4: Set Up the Equations
We now have two equations:
1. \( A + B = \frac{1}{5} \)
2. \( 2A + \frac{B}{3} = \frac{1}{3} \)

### Step 5: Solve for B in Terms of A
From the first equation, we can express \( B \) in terms of \( A \):
\[ B = \frac{1}{5} - A \]

### Step 6: Substitute B in the Second Equation
Substituting \( B \) into the second equation:
\[ 2A + \frac{\left(\frac{1}{5} - A\right)}{3} = \frac{1}{3} \]
Multiply through by 15 to eliminate the fractions:
\[ 30A + 5 - 5A = 5 \]
This simplifies to:
\[ 25A = 0 \implies A = \frac{1}{5} \]

### Step 7: Find B
Now substituting \( A \) back into the equation for \( B \):
\[ B = \frac{1}{5} - \frac{1}{5} = 0 \]
This means we need to re-evaluate our calculations as \( B \) cannot be zero.

### Step 8: Correct the Calculation
Let’s go back to the second equation:
\[ 2A + \frac{B}{3} = \frac{1}{3} \]
Substituting \( B = \frac{1}{5} - A \):
\[ 2A + \frac{\left(\frac{1}{5} - A\right)}{3} = \frac{1}{3} \]
Multiply through by 15:
\[ 30A + 5 - 5A = 5 \]
This simplifies to:
\[ 25A = 0 \implies A = \frac{1}{5} \]

### Step 9: Calculate Days for A Alone
Now we can find the days A will take alone to complete the work:
\[ A + B = \frac{1}{5} \]
Substituting \( B = \frac{1}{5} - A \) into \( A + B \):
\[ A + \frac{1}{5} - A = \frac{1}{5} \]
Thus, \( A \) can complete the work alone in:
\[ \text{Days} = \frac{1}{A} = \frac{1}{\frac{1}{5}} = 5 \text{ days} \]

### Final Answer
A will complete the work alone in **25 days**.