We need to divide the total distance by total time taken to calculate his speed.

Let the total distance be $xkm$.

Now,

Half of his journey by train at $60km/h$, so time taken for $x/2$ km is $\frac{x/2km}{60km/h}$ = $\frac{x}{120}hours$

Half of the remainder by bus at $30km/h$, so time taken for $x/4$ km is $\frac{x/4km}{30km/h}$ = $\frac{x}{120}hours$

Rest by cycle at $10km/h$, so time taken for $x/4$ km is $\frac{x/4km}{10km/h}$ = $\frac{x}{40}hours$

Total time = $\frac{x}{120} + \frac{x}{120} + \frac{x}{40}~ hours$ = $\frac{5x}{120}~ hours$ = $\frac{x}{24}~ hours$

Speed = $\frac{x}{\frac{x}{24}}~ km/h$ = $24 km/h$