Two workers A and B are engaged to do a piece of work . A working alone would take 14 hours more to complete the work than when A and B work together B working alone would take ` 3 (1)/(2)` hours more than when A and B work together . The time required to finish the work together is

Options

A. 7 hours
B. 5 hours
C. 6 hours
D. 8 hours

Answer

To solve the problem, let's denote the time taken by A and B to complete the work together as \( x \) hours.

1. **Express the time taken by A and B:**
- If A works alone, he takes \( x + 14 \) hours.
- If B works alone, he takes \( x + 3.5 \) hours.

2. **Set up the work equations:**
- The work done by A in one hour is \( \frac{1}{x + 14} \).
- The work done by B in one hour is \( \frac{1}{x + 3.5} \).
- The work done by A and B together in one hour is \( \frac{1}{x} \).

3. **Combine the work equations:**
\[ \frac{1}{x + 14} + \frac{1}{x + 3.5} = \frac{1}{x} \]

4. **Clear the fractions by multiplying through by \( x(x + 14)(x + 3.5) \):**
\[ x(x + 3.5) + x(x + 14) = (x + 14)(x + 3.5) \]

5. **Expand and simplify:**
- Left side: \( x^2 + 3.5x + x^2 + 14x = 2x^2 + 17.5x \)
- Right side: \( x^2 + 3.5x + 14x + 49 = x^2 + 17.5x + 49 \)

So we have:
\[ 2x^2 + 17.5x = x^2 + 17.5x + 49 \]

6. **Rearranging the equation:**
\[ 2x^2 - x^2 + 17.5x - 17.5x - 49 = 0 \]
\[ x^2 - 49 = 0 \]

7. **Solve for \( x \):**
\[ x^2 = 49 \implies x = 7 \text{ (since time cannot be negative)} \]

Thus, the time required to finish the work together is **7 hours**.