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
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Hint: Assume two variables for the speeds of the two cars. Then, use the given information to form two equations involving the variables using the relation \[d = v \times t\]. Solve the two equations involving two unknowns by substitution and find the speeds of the car.
Complete step by step answer: Let the speed of the car starting from A be x km/h and the speed of the car starting from B be y km/h.
It is given that the length of the highway between A and B is 100 km.
The distance d covered in time t, when traveling at the speed of v is given as follows:
\[d = v \times t..............(1)\]
If the two cars travel in the same direction, they meet in 5 hours. Let us take the speed of car A is greater than B. Then, the distance covered by car at A in 5 hours is the sum of the distance of the highway and the distance covered by car B. Hence, using equation (1), we have:
\[5x = 100 + 5y\]
Simplifying, by dividing both sides by 5, we have:
\[x = 20 + y..............(2)\]
If the two cars travel in the opposite direction, they meet in 1 hour. The total distance covered by both the cars together is equal to the length of the highway. Hence, using equation (1), we have:
\[x + y = 100.............(3)\]
Hence, we have two equations involving two unknowns.
Substituting equation (2) in equation (3), we get:
\[20 + y + y = 100\]
\[20 + 2y = 100\]
Solving for y, we have:
\[2y = 100 - 20\]
\[2y = 80\]
\[y = \dfrac{{80}}{2}\]
\[y = 40km/hr............(4)\]
Using equation (4) in equation (2), we get:
\[x = 20 + 40\]
\[x = 60km/hr\]
Hence, the speeds of the car are 60 km/hr and 40 km/hr.
Note: When writing the equation for the cars traveling in the same direction, consider the length of the highway also to be covered, otherwise, your answer will be wrong.