suppose that a cyclist began a 341 mile ride across a state

suppose that a cyclist began a 341 mile ride across a state at the western edge of the state at the same time that a car traveling toward it leaves the Eastern end of the state if the bicycle and a car met after 5.5 hours and the car travels 37.8 miles per hour faster than the bicycle find the average rate of each

Solution

Let the average speed of the bicycle be x mph. Then the average speed of the car is x + 37.8 mph. In 5.5 hours, the cyclist travels 5.5x miles and the car travels 5.5* (x + 37.8) miles. Together, the cyclist and the car have travelled a distance of 341 miles. Therefore, we have 5.5x + 5.5* (x + 37.8) = 341 or, 5.5 ( x + x + 37.8) = 341 or, 2x + 37.8 = 341/5.5 = 62 so that 2x = 62 - 37.8 = 24.2 and hence x = 24.2/2 = 12.1 mph. Then x + 37.8 = 12.1 + 37.8 = 49.9 mph. Thus, the average speed of the cyclist and the car are 12.1 mph and 49.9 mph respectively. We can verify the result by computing the distance travelled by the cyclist in 5.5 hours which is 5.5*12.1 = 66.55 miles and  the distance travelled by the car in 5.5 hours which is 5.5*49.9 = 274.45 miles. Since 66.55 + 274.45 = 341, our answer is correct.