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 been completed in 3 days. To complete the job alone. A would require

Options

A. `5(1)/(5)` days
B. `6(1)/(4)` days
C. `7(1)/(2)` days
D. `8(3)/(4)` days

Answer

To solve the problem step by step, we will define the efficiencies of workers A and B, set up equations based on the given information, and solve for the time A would take to complete the job alone.

### Step-by-Step Solution:

1. **Define Variables:**
Let the efficiency of worker A be \( x \) (work done per day) and the efficiency of worker B be \( y \) (work done per day).

2. **Total Work Done:**
Since A and B together complete the job in 5 days, the total work \( W \) can be expressed as:
\[ W = (x + y) \cdot 5 \]

3. **Modified Efficiencies:**
If A worked twice as efficiently, his new efficiency would be \( 2x \). If B worked at one-third of his efficiency, his new efficiency would be \( \frac{y}{3} \).

4. **New Work Completion Time:**
With these modified efficiencies, they would complete the work in 3 days:
\[ W = \left(2x + \frac{y}{3}\right) \cdot 3 \]

5. **Set Up Equations:**
From the two expressions for \( W \), we can set them equal to each other:
\[ (x + y) \cdot 5 = \left(2x + \frac{y}{3}\right) \cdot 3 \]

6. **Simplifying the Equation:**
Expanding both sides gives:
\[ 5x + 5y = 6x + \frac{y}{1} \]
Rearranging terms leads to:
\[ 5y - \frac{y}{1} = 6x - 5x \]
This simplifies to:
\[ 4y = x \]

7. **Substituting Back:**
Substitute \( x = 4y \) back into the equation for \( W \):
\[ W = (4y + y) \cdot 5 = 5y \cdot 5 = 25y \]

8. **Finding Time for A Alone:**
To find how long A would take to complete the work alone:
\[ \text{Time for A} = \frac{W}{x} = \frac{25y}{4y} = \frac{25}{4} = 6.25 \text{ days} \]

### Final Answer:
A would require \( 6 \frac{1}{4} \) days to complete the job alone.

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