Define variables

Let the total distance be $D$. The bus travels $d_1 = \frac{2}{5}D$ at $v_1 = 50$ km/h and $d_2 = \frac{3}{5}D$ at $v_2 = 60$ km/h.

$d_1 = 0.4D, d_2 = 0.6D$

Calculate time for each segment

Time $t = \frac{\text{Distance}}{\text{Speed}}$. Thus, $t_1 = \frac{0.4D}{50} = \frac{2D}{250} = \frac{D}{125}$ and $t_2 = \frac{0.6D}{60} = \frac{6D}{600} = \frac{D}{100}$.

$t_1 = \frac{D}{125}, t_2 = \frac{D}{100}$

Calculate average speed

The average speed is total distance divided by total time: $V_{avg} = \frac{D}{t_1 + t_2} = \frac{D}{\frac{D}{125} + \frac{D}{100}}$.

$V_{avg} = \frac{1}{\frac{1}{125} + \frac{1}{100}} = \frac{1}{\frac{4+5}{500}} = \frac{500}{9}$

Final conversion

Converting $\frac{500}{9}$ into a mixed fraction gives $55 \frac{5}{9}$ km/h.

$\frac{500}{9} = 55 \frac{5}{9} \text{ km/h}$