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Places A and B are 80 km apart from each other on a highway. A car starts from A and another from B at the same time. If they move in the same direction they meet in 8 hours and if they move towards each other they meet in 1 hour and 20 minutes. Find the speeds of the cars.

Answer
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Hint: Here, we will be using the general formula of distance-speed-time which is ${\text{Speed}} = \dfrac{{{\text{Distance}}}}{{{\text{Time}}}}{\text{ }}$ and also the concept of relative speed between the two cars.

Complete step-by-step answer:

Let us suppose the speed of a car starting from A is $x$ km per hour and the speed of car starting from B is $y$ km per hour.
Given, distance between the places A and B$ = 80$ km
As we know that ${\text{Speed}} = \dfrac{{{\text{Distance}}}}{{{\text{Time}}}}{\text{ }} \to {\text{(1)}}$
First case- When both the cars are moving in the same direction.
Given, time taken$ = 8$ hours
In this case, the relative speed between the two cars starting from A and B will be subtracted.
Relative speed$ = \left( {x - y} \right)$ km per hour
Using formula given by equation (1), we get
$\left( {x - y} \right) = \dfrac{{{\text{80}}}}{{\text{8}}} \Rightarrow x - y = 10{\text{ }} \to {\text{(2)}}$
Second case- When both the cars are moving towards each other i.e., in opposite directions.
Given, time taken$ = 1{\text{ hr 20 min}} = 1 + \dfrac{{20}}{{60}} = 1 + \dfrac{1}{3} = \dfrac{4}{3}$ hours
In this case, the relative speed between the two cars starting from A and B will be added.
Relative speed$ = \left( {x + y} \right)$ km per hour
Using formula given by equation (1), we get
$\left( {x + y} \right) = \dfrac{{{\text{80}}}}{{\dfrac{4}{3}}} = \dfrac{{80 \times 3}}{4} = 60 \Rightarrow x + y = 60{\text{ }} \to {\text{(3)}}$
Adding equations (2) and (1), we get
$ \Rightarrow x - y + x + y = 10 + 60 \Rightarrow 2x = 70 \Rightarrow x = 35$
Put the above value of $x$ in equation (2), we get
$ \Rightarrow 35 - y = 10 \Rightarrow y = 35 - 10 = 25$
Therefore, the speed of car starting from place A is 35 km per hour and the speed of the car starting from place B is 25 km per hour.

Note: In these types of problems, we have calculated relative speed because both the cars are moving and through relative speed we can actually obtain relation between their speeds. When they are moving in the same direction, the relative speed is the difference between their individual speeds and when they are moving in opposite directions, the relative speed is the sum of their individual speeds.