Two workers A and B working together completed a job in 5
days. If A worked twice as efficiently as he actually did and
B worked 1/3 as efficiently as he actually did, the work would
have completed in 3 days. Fine the time for A to complete
the job alone.

Options

A. `6 1/4` days
B. `5 3/4` days
C. 5 days
D. 3 days

Explanation

To solve the problem step by step, let's denote the time taken by worker A to complete the job alone as \( A \) days and the time taken by worker B to complete the job alone as \( B \) days.

### Step 1: Set up the equations based on the given information.
From the problem, we know that A and B together complete the job in 5 days. Therefore, their combined work rate can be expressed as:
\[ \frac{1}{A} + \frac{1}{B} = \frac{1}{5} \]

### Step 2: Analyze the second condition.
If A worked twice as efficiently, he would complete the job in \( \frac{A}{2} \) days. If B worked one-third as efficiently, he would take \( 3B \) days to complete the job. Thus, their combined work rate under these new conditions can be expressed as:
\[ \frac{2}{A} + \frac{1}{3B} = \frac{1}{3} \]

### Step 3: Rewrite the equations.
Now we have two equations:
1. \(\frac{1}{A} + \frac{1}{B} = \frac{1}{5}\) (Equation 1)
2. \(\frac{2}{A} + \frac{1}{3B} = \frac{1}{3}\) (Equation 2)

### Step 4: Solve Equation 1 for \(\frac{1}{B}\).
From Equation 1, we can express \(\frac{1}{B}\) in terms of \(\frac{1}{A}\):
\[ \frac{1}{B} = \frac{1}{5} - \frac{1}{A} \]

### Step 5: Substitute \(\frac{1}{B}\) into Equation 2.
Substituting \(\frac{1}{B}\) into Equation 2 gives:
\[ \frac{2}{A} + \frac{1}{3\left(\frac{1}{5} - \frac{1}{A}\right)} = \frac{1}{3} \]

### Step 6: Simplify the equation.
To simplify the second term:
\[ \frac{1}{3\left(\frac{1}{5} - \frac{1}{A}\right)} = \frac{1}{3} \cdot \frac{5A}{A - 5} \]
Thus, our equation becomes:
\[ \frac{2}{A} + \frac{5A}{3(A - 5)} = \frac{1}{3} \]

### Step 7: Clear the fractions by multiplying through by \( 3A(A - 5) \).
Multiplying through by \( 3A(A - 5) \) gives:
\[ 6(A - 5) + 5A^2 = A(A - 5) \]

### Step 8: Rearranging the equation.
Expanding and rearranging gives:
\[ 6A - 30 + 5A^2 = A^2 - 5A \]
\[ 4A^2 + 11A - 30 = 0 \]

### Step 9: Solve the quadratic equation.
Using the quadratic formula \( A = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ A = \frac{-11 \pm \sqrt{11^2 - 4 \cdot 4 \cdot (-30)}}{2 \cdot 4} \]
\[ A = \frac{-11 \pm \sqrt{121 + 480}}{8} \]
\[ A = \frac{-11 \pm \sqrt{601}}{8} \]

### Step 10: Calculate the positive root.
Since time cannot be negative, we take the positive root:
\[ A = \frac{-11 + \sqrt{601}}{8} \]

### Final Answer:
Thus, the time taken by worker A to complete the job alone is approximately \( 25/4 \) days or 6.25 days.