Places A and B are 240 km apart. 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 12 hours. If they travel towards each other, they meet in 2 hours. Given that the car starting from A is faster, find the speed of the two cars.
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Explanation:
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.
When they travel in the same direction, their relative speed is \(x - y\) km/h (faster car minus slower car). The time taken to meet in this scenario is 12 hours.
Distance = Speed × Time
\(240 = (x - y) \times 12\)
When they travel towards each other, their relative speed is \(x + y\) km/h. The time taken to meet in this scenario is 2 hours.
Distance = Speed × Time
\(240 = (x + y) \times 2\)
We have a system of equations:
1) \(240 = 12x - 12y\)
2) \(240 = 2x + 2y\)
Let's solve this system of equations to find \(x\) and \(y\). First, simplify the equations:
1) \(20 = x - y\)
2) \(120 = x + y\)
Now, solve these equations simultaneously. Adding both equations will eliminate \(y\):
\(20 + 120 = x - y + x + y\)
\(140 = 2x\)
\(x = 70\) km/h
Substitute \(x = 70\) into \(120 = x + y\):
\(120 = 70 + y\)
\(y = 50\) km/h
Therefore, the speed of the car starting from A is 70 km/h, and the speed of the car starting from B is 50 km/h.