speed = distance / time
Therefore,
time = distance/speed
Let's assign some symbols to these quantities...
time = t
speed = v (from 'velocity')
distance = s (don't ask... people just like using s for distance. it crops up a lot in maths too for path lengths and things like that)
The equation above described in symbols is:
$\displaystyle t = \frac{s}{v}$
Let's use the numbers 1, 2 and 3 to designate the three parts of a journey. These are called "indices".
Time for first part of journey:
$\displaystyle t_1 = \frac{s_1}{v_1}$
Time for second part of journey:
$\displaystyle t_2 = \frac{s_2}{v_2}$
Time for third part of journey:
$\displaystyle t_3 = \frac{s_3}{v_3}$
Total time for journey:
$\displaystyle t_{total} = t_1 + t_2 + t_3$
Substitute for individual times, we get:
$\displaystyle t_{total} = \frac{s_1}{v_1} + \frac{s_2}{v_2} + \frac{s_3}{v_3}$
The total distance is:
$\displaystyle s_{total} = s_1 + s_2 + s_3$
Therefore, the average speed is:
$\displaystyle \bar{v} = \frac{s_{total}}{t_{total}} = \frac{s_1 + s_2 + s_3}{\frac{s_1}{v_1} + \frac{s_2}{v_2} + \frac{s_3}{v_3}}$
