A man is twice as fast as a woman. Together the man and the woman do the piece of work in 8 days. In how many days each will do the work if engaged alone?

Let the man alone do the work in $\mathrm{x}$days. 

Then, the woman alone do the work in $2\mathrm{x}$ days.

Their one day's work$= \frac{1}{8}$ th part of whole work

$\text{ i.e. }\frac{1}{\mathrm{x}}+\frac{1}{2\mathrm{x}}=\frac{1}{8}$

$\Rightarrow \mathrm{x}=12$ days

$\therefore$ man takes $12$days and woman $2\mathrm{x}= 24$ days.