Question

Points A and B are $90$km apart from each other on a highway. A car starts from A and another from B at the same time. If they go in the same direction they meet in $9$ hours and if they go in opposite directions they meet in $\frac{9}{7}$ hours. Find their speeds.

Answer 1

Let X and Y be two cars starting from points A and B respectively. Let the speed of car X be x km/hr and that of car Y be y km/hr.

Case I When two cars move in the same directions:

Suppose two cars meet at point Q. Then,
Distance travelled by car $X=$AQ,
Distance travelled by car $Y=$BQ
It is given that two cars meet in $9$ hours.

$\therefore$ Distance travelled by car X in $9$ hours $=9x$ km.
$\Rightarrow AQ=9x$
Distance travelled by car y in $9$ hours $=9y$ km.
$\Rightarrow BQ=9y$

Clearly, $AQ-BQ=AB$
$\Rightarrow 9x-9y=90$ $[\because AB=90km]$
$\Rightarrow x-y=10$ (i)

Case II When two cars move in opposite directions:

Suppose two cars meet at point P. Then,
Distance travelled by car X $=$ AP,
Distance travelled by car Y $=$ BP
In this case, two cars meet in $\frac{9}{7}$ hours.

$\therefore$ Distance travelled by car X in $\frac{9}{7}$ hours $=\frac{9}{7}x$km
$\Rightarrow BP=\frac{9}{7}y$

Clearly, $AP+BP=AB$

$\Rightarrow \frac{9}{7}x+\frac{9}{7}y=90$

$\Rightarrow \frac{9}{7}(x+y)=90$

$\Rightarrow x+y=70$ .(ii)

Adding equations (i) and (ii), we get
$2x+0=80\Rightarrow x=40$

Putting in (ii), we get,

$y=70-40=30$

$\therefore x=40$ and $y=30$.

Hence, the speed of car X is $40$ km/hr and speed of car Y is $30$ km/hr.