Question

Places $A$ and $B$ are $100$ km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds, they meet in $5$ hours. If they travel towards each other, they meet in $1$ hour. What are the speeds of the two cars?

Answer 1

Hint: Here in this question we know the distance the two cars are apart from each other and we know the time taken when they travel in two different directions so we can calculate their speeds.
As we know that, $speed = \dfrac{{dis\tan ce}}{{time}}$

Step by step solution:
Here in this question we will start solving the question after enlisting the given information.
So, it is given in this that
Distance between the point $A$ and point $B = 100$km
Now, let us assume that the two cars are named as $Ca{r_A}$and $Ca{r_B}$.
That means $Ca{r_A}$ starts from point $A$ and $Ca{r_B}$ starts from point $B$.
And also here we have to assume the speeds of these cars.
So, let Speed of $Ca{r_A} = x\dfrac{{km}}{h}$ and Speed of $Ca{r_B} = y\dfrac{{km}}{h}$
So, now according to question it is given that
If the cars travel in the same direction at different speeds, they meet in $5$hours
And if they travel towards each other, they meet in $1$ hour.
Now firstly we will consider the case when both the cars are moving in same direction
So, in this case the Speed of $Ca{r_A} = x\dfrac{{km}}{h}$ and Speed of $Ca{r_B} = y\dfrac{{km}}{h}$
So, we are given that in this case they will meet after travelling for $5$hours
It means that the relative distance travelled by them will be $100$km in $5$hours.
So, as we know that $speed = \dfrac{{dis\tan ce}}{{time}}$
$\Rightarrow speed \times time = dis\tan ce$
So, by putting values we will get
$5x - 5y = 100$
$\Rightarrow x - y = 20$ ……….(i)
Now, we will consider the second case when both the cars travel in the opposite direction that is moving towards each other.
In this case the Speed of $Ca{r_A} = x\dfrac{{km}}{h}$ and Speed of $Ca{r_B} = - y\dfrac{{km}}{h}$
Speed of $Ca{r_B}$is negative because we have assumed that it is moving in the opposite direction.
So, here the equation formed will be
$x - \left( { - y} \right) = 100$
$\Rightarrow x + y = 100$ ………….(ii)
So, now we have two equations {(i)&(ii)} and two variables {$x\& y$}
$\therefore$After adding both the equations we will get
$x - y = 20$ …….(i)
$x + y = 100$ ………(ii)
$\Rightarrow 2x = 120$
$\Rightarrow x = 60\dfrac{{km}}{h}$
Now, putting the value of $x$in equation (i)
$x - y = 20 \\\ \Rightarrow 60 - y = 20 \\\ \Rightarrow - y = 20 - 60 \\\ \Rightarrow - y = - 40 \\\ \Rightarrow y = 40\dfrac{{km}}{h} \\\$
$\therefore$ The speeds of two cars are $60\dfrac{{km}}{h}$ and $40\dfrac{{km}}{h}$.

Note: These types of questions can also be done by making a proper diagram by using the given information that will give you an exact idea of what is happening in the question.
In this type of questions students generally make mistakes while considering the speed of cars in the second case when both the cars are travelling towards each other. This leads to the formation of wrong equations and that will lead to the wrong answer.