A train left point $A$ at noon sharp. Two hours later another train started from point $A$ in the same direction. It overtook the first train at $8$ p.m. Find the average speeds of the trains if the sum of their average speeds is $70$ kmph.
Let the average speed of the first train be $= s_1$ kmph
and the average speed of the second train be $= s_2$ kmph
Given: $\;$ $s_1 + s_2 = 70$ kmph $\;\;\; \cdots \; (1)$
Time from noon sharp to $8$ p.m. $= 8$ hours
Let the first train cover a distance $d$ km in time $t_1 = 8$ hr
Then $\;$ $d = s_1 \times t_1 = 8 \times s_1$ $\;\;\; \cdots \; (2)$
Since the second train starts $2$ hours later and overtakes the first train at $8$ p.m.,
$\implies$ the second train covers the distance $d$ km in time $t_2 = 6$ hours
$\therefore \;$ Distance covered by the second train $= d = s_2 \times t_2 = 6 \times s_2$ $\;\;\; \cdots \; (3)$
$\therefore \;$ We have from equations $(2)$ and $(3)$
$8 \times s_1 = 6 \times s_2$
i.e. $\;$ $s_2 = \dfrac{4 \; s_1}{3}$ $\;\;\; \cdots \; (4)$
Substituting the value of $s_2$ in equation $(1)$ gives
$s_1 + \dfrac{4 \; s_1}{3} = 70$
i.e. $\;$ $\dfrac{7 \; s_1}{3} = 70$ $\implies$ $s_1 = 30$
Substituing the value of $s_1$ in equation $(1)$ gives
$s_2 = 70 - 30 = 40$
$\therefore \;$ Speed of the first train $= s_1 = 30$ kmph
and speed of the second train $= s_2 = 40$ kmph