Two workers A and B are engaged to do a piece of work. A working alone would take 8 hours more to complete the work that when work together. If B worked alone, would take `4(1)/(2)` hours more than when work together. The time required to finish the work together is

Options

A. 1. 5 hours
B. 2. 8 hours
C. 3. 4 hours
D. 4. 6 hours

Answer

To solve the problem step by step, we will define the variables and use the information given in the question.

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

### Step 2: Express Individual Times
According to the problem:
- A takes \( x + 8 \) hours to complete the work alone.
- B takes \( x + 4.5 \) hours to complete the work alone.

### Step 3: Calculate Work Rates
The work done in one hour by each worker can be expressed as:
- Work rate of A: \( \frac{1}{x + 8} \)
- Work rate of B: \( \frac{1}{x + 4.5} \)

When A and B work together, their combined work rate is:
\[ \frac{1}{x + 8} + \frac{1}{x + 4.5} = \frac{1}{x} \]

### Step 4: Set Up the Equation
Now, we can set up the equation based on the combined work rate:
\[ \frac{1}{x + 8} + \frac{1}{x + 4.5} = \frac{1}{x} \]

### Step 5: Clear the Fractions
To eliminate the fractions, we can multiply through by \( x(x + 8)(x + 4.5) \):
\[ x(x + 4.5) + x(x + 8) = (x + 8)(x + 4.5) \]

### Step 6: Expand the Equation
Expanding both sides:
- Left side:
\[ x^2 + 4.5x + x^2 + 8x = 2x^2 + 12.5x \]
- Right side:
\[ x^2 + 4.5x + 8x + 36 = x^2 + 12.5x + 36 \]

### Step 7: Combine Like Terms
Now we have:
\[ 2x^2 + 12.5x = x^2 + 12.5x + 36 \]
Subtract \( x^2 + 12.5x \) from both sides:
\[ 2x^2 - x^2 + 12.5x - 12.5x = 36 \]
This simplifies to:
\[ x^2 = 36 \]

### Step 8: Solve for \( x \)
Taking the square root of both sides:
\[ x = 6 \]

### Conclusion
The time required to finish the work together is **6 hours**.