A boy goes to a school from his home at a speed of  $x$  km/hr and comes back at a speed of y km/hr, then the average speed is given by

Let the distance between the school and the home 

be $s$ km. Then, total time taken in going to and coming from the

 school is given by 

   $t=\frac{s}{x}+\frac{s}{y}\Rightarrow t=\left(\frac{x+y}{xy}\right) s\Rightarrow \frac{s}{t}=\frac{xy}{x+y}$ 

  Average speed $=\frac{2s}{t}=\frac{2xy}{x+y}=$H. M. of x and y