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Places A and B are 100km apart on a highway. One car starts at 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 two cars?

Answer
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Hint: We solve this problem by assuming the meeting points in two different conditions as C and D then the rough figure of given information will be
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We assume the speeds of two cars as some variables then we use the formula of time as
Time=Distancespeed
We use the two conditions given in the question to find the speeds of two cars.

Complete step by step answer:
We are given that the distance between two points A and B as 100km.
Let us assume that the speed of car that started at A as Va
Let us assume that the speed of car that started at B as Vb
Now, let us consider the first case that is they travel in same direction.
(i) Travel in same direction.
From the figure we can see that the two cars meet at point C which is x distance from B.
So, we can say that first car covered a total distance of 100+x and second car covered a distance of x to meet at point C.
Let us assume that the time taken by first car to reach point C as t1
We know that the formula for time as
Time=Distancespeed
By using this formula we get
t1=100+xVa
Similarly, by assuming the time taken by second car to reach point C as t2 we get
t2=xVb
We are given that the two cars meet in 5 hours when moved in same direction which means that both cars took 5 hours to meet at point C

So, we can write from the above condition that is
t1=t2=5100+xVa=xVb=5
Now, by separating the terms of unknown variables to constant we get
100+xVa=5100+x=5Vax=5Va100
Now, by taking the other terms we get
xVb=5x=5Vb
By substituting the value of x in above equation we get
5Va100=5VbVaVb=20........equation(i)
Now let us go for the second condition that is the cars move in opposite direction
(ii) Travel in opposite direction.
From the figure we can see that the two cars meet at point D which is y distance from A.
So, we can say that first car covered a total distance of y and second car covered a distance of 100y to meet at point D.
Let us assume that the time taken by first car to reach point D as t3
We know that the formula for time as
Time=Distancespeed
By using this formula we get
t3=yVa

Similarly, by assuming the time taken by second car to reach point C as t4 we get
t4=100yVb
We are given that the two cars meet in 1 hour when moved in same direction which means that both cars took 1 hour to meet at point D
So, we can write from the above condition that is
t3=t4=1yVa=100yVb=1
Now, by separating the terms of unknown variables to constant we get
y=Va
Now, by taking the other terms we get
100yVb=1
By substituting the value of y in above equation we get
100Va=VbVa+Vb=100........equation(ii)
Now, by adding the equation (i) and equation (ii) we get
(VaVb)+(Va+Vb)=20+100Va=1202=60kmph
By substituting the above value in equation (i) we get
60Vb=20Vb=40kmph

Therefore the speeds of two cars are 60kmph and 40kmph.

Note: We have a shortcut for two conditions that is relative velocity.
Let us assume that the speed of car that started at A as Va
Let us assume that the speed of car that started at B as Vb
If t1 is the time taken by two cars to meet when they move in same direction then by using the relative velocity we get
t1=dVaVb
Here, d is the distance between two cars that is 100km.
By using this formula we get
5=100VaVbVaVb=20.......equation(i)
Similarly, if t2 is the time taken by two cars to meet when they travel in opposite direction hen the relative velocity is given as
t2=dVa+Vb
Here, d is the distance between two cars that is 100km.
By using the above formula we get
1=100Va+VbVa+Vb=100......equation(ii)
Now, by adding the equation (i) and equation (ii) we get
(VaVb)+(Va+Vb)=20+100Va=1202=60kmph
By substituting the above value in equation (i) we get
60Vb=20Vb=40kmph
Therefore the speeds of two cars are 60kmph and 40kmph.