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 4¹/₂ hours more to complete the job than they both working together. What time would they take to do the work together ?

Let A and B together take x hours to complete the work.
Then, A alone takes (x + 8) hours
And
B alone takes $$\left( {{\text{x}} + \frac{9}{2}} \right)$$   hours to complete the work
Then,
$$\eqalign{ & \Rightarrow \frac{1}{{\left( {x + 8} \right)}} + \frac{1}{{\left( {x + \frac{9}{2}} \right)}} = \frac{1}{x} \cr & \Rightarrow \frac{1}{{\left( {x + 8} \right)}} + \frac{2}{{\left( {2x + 9} \right)}} = \frac{1}{x} \cr & \Rightarrow x\left( {4x + 25} \right) = \left( {x + 8} \right)\left( {2x + 9} \right) \cr & \Rightarrow 2{x^2} = 72 \cr & \Rightarrow {x^2} = 36 \cr & \Rightarrow x = 6 \cr} $$