The image produced by a concave mirror is one quarter the size of the object. If the object is moved b = 5 cm closer to the mirror the image will only be half the size of the object. Find the focal length of the mirror in mm?

If $u_1$ and  $v_1$are the initial distance of the object and the image from the mirror , then
 $\begin{array}{l}\frac{v_1}{u_1}=-\frac{1}{4}\mathrm{....}\left(1\right)\\ \frac{1}{v_1}+\frac{1}{u_1}=\frac{1}{f}\mathrm{.....}\left(2\right)\end{array}$
When object moves towards the mirror a distance b then 
 $\frac{v_2}{(u_1-b)}=-\frac{1}{2}\mathrm{......}\left(3\right)$
And  $\frac{1}{v_2}+\frac{1}{(u_1-b)}=\frac{1}{f}\mathrm{......}\left(4\right)$
On substitution the values and solving above equations, we get  $f=2.5cm$