Geometry

The hour hand will move at a rate of 

$360°  \space \text{in} \space 12 \space \text{hours} \\ 30° \space \text{in}  \space 1 \space \text{hour} \\ 30°/ 60  =  0.5° \space \text{in}  \text{ one minute}$

So  at   4:15, the hour  hand has  moved   $4 (30°) +  15(.5°)  =  [120+ 7.5]° = 127.5°$from 12 o'clock

The minute hand  will move  at a rate of $360°/ 60  = 6°$ per minute

So at 4:15 it will have moved  $[15* 6]°  = 90°$  from the top of the hour

So...the smaller angle formed by the hands$= 127.5 - 90 = 37.5^\circ$

May have made an error....

Thanks! :)