Two workers A and B are engaged to do a piece of work. Working alone, A takes 8 hours more to complete the work than if both worked together. On the other hand, working alone, B would need `4(1)/(2)` hours more to complete the work than if both worked together. How much time would they take to complete the job working together?

Options

A. 4Hours
B. 5Hours
C. 6 Hours
D. 7 Hours

Explanation

To solve the problem, we need to find the time \( T \) that workers A and B would take to complete the job when working together. Let's break down the solution step by step.

### Step 1: Define Variables
Let \( T \) be the time taken by A and B together to complete the work.

### Step 2: Express Individual Times
According to the problem:
- Worker A takes \( T + 8 \) hours to complete the work alone.
- Worker B takes \( T + 4.5 \) hours to complete the work alone (since \( 4.5 \) hours is \( 4 \frac{1}{2} \) hours).

### Step 3: Write Work Rates
The work done by A in one hour is:
\[ \text{Rate of A} = \frac{1}{T + 8} \]
The work done by B in one hour is:
\[ \text{Rate of B} = \frac{1}{T + 4.5} \]
The combined work rate of A and B working together is:
\[ \text{Rate of A + B} = \frac{1}{T} \]

### Step 4: Set Up the Equation
Since the work done by A and B together in one hour should equal their combined work rate, we can set up the equation:
\[ \frac{1}{T + 8} + \frac{1}{T + 4.5} = \frac{1}{T} \]

### Step 5: Clear the Denominators
To eliminate the fractions, multiply through by \( T(T + 8)(T + 4.5) \):
\[ T(T + 4.5) + T(T + 8) = (T + 8)(T + 4.5) \]

### Step 6: Expand the Equation
Expanding both sides:
\[ T^2 + 4.5T + T^2 + 8T = T^2 + 12.5T + 36 \]
Combine like terms:
\[ 2T^2 + 12.5T = T^2 + 12.5T + 36 \]

### Step 7: Rearrange the Equation
Rearranging gives:
\[ 2T^2 - T^2 + 12.5T - 12.5T - 36 = 0 \]
This simplifies to:
\[ T^2 - 36 = 0 \]

### Step 8: Solve for T
Factoring gives:
\[ (T - 6)(T + 6) = 0 \]
Thus, \( T = 6 \) or \( T = -6 \). Since time cannot be negative, we take:
\[ T = 6 \]

### Conclusion
The time taken by A and B working together to complete the job is \( \boxed{6} \) hours.