Two workers A and B are engaged to do a work. A working alone takes 8 hours more to complete the job than if both worked together. If B worked alone, he would need $4\frac{1}{2}$ hours more to complete the job than they both working together. What time would they take to do the work together ?

### Concept & The $\sqrt{ab}$ Shortcut When the individual times of two workers are given as excessive time compared to their combined time, a direct relationship exists. If A takes $(x + a)$ time and B takes $(x + b)$ time, where $x$ is their combined time, then: $$ x = \sqrt{a \times b} $$ ### Step-by-Step Solution * **Define Variables:** Let the time taken by A and B working together be $x$ hours. Time taken by A alone = $(x + 8)$ hours. Time taken by B alone = $(x + 4.5)$ hours (since $4\frac{1}{2} = 4.5$). * **Apply the Formula:** Using the standard identity $x = \sqrt{a \times b}$: Here, excess time $a = 8$ and excess time $b = 4.5$. * **Calculate x:** $x = \sqrt{8 \times 4.5}$ $x = \sqrt{36}$ $x = 6$ hours. ### Exam Strategy & Shortcut This specific problem archetype appears frequently. Memorizing the $x = \sqrt{ab}$ theorem is the ultimate shortcut. It skips the process of setting up the fractional equation $\frac{1}{x+8} + \frac{1}{x+4.5} = \frac{1}{x}$ and solving the resulting quadratic equation. ### Common Pitfall Attempting to solve the quadratic equation manually ($\frac{1}{x+8} + \frac{1}{x+4.5} = \frac{1}{x}$) without the shortcut formula often leads to algebraic errors under time pressure. ### Final Answer Therefore, the correct answer is **6 hours**.