Short Answer 
The solution involves defining the distances of two cars, where the first car travels for ( t ) hours at 50 mph, and the second for ( t + 2 ) hours at 40 mph. By equating the distances and solving, we find that the first car traveled for 8 hours, and this is verified as both cars traveled the same distance of 400 miles.
Step 1: Define Variables and Set Up Equations
First, let’s define the variables for the problem. We will let d1 represent the distance traveled by the first car and d2 represent the distance traveled by the second car. The first car travels for t hours, while the second car travels for t + 2 hours at a speed of 40 mph. We can set up the equations for both distances:
- d1 = 50t
- d2 = 40(t + 2)
Step 2: Equate the Distances and Solve
Next, we equate the distances of both cars since they traveled the same distance. This gives us the equation: 50t = 40(t + 2). By expanding the right side, we get 50t = 40t + 80. We can rearrange this equation to isolate t:
- 50t – 40t = 80
- 10t = 80
- t = 8
Step 3: Conclusion and Verification
Finally, we find that the first car traveled for 8 hours, while the second car traveled for 10 hours. To verify, we can calculate the distances:
- d1 = 50 * 8 = 400 miles
- d2 = 40 * 10 = 400 miles
Both distances are equal, confirming that the solution is correct. The first car indeed traveled for 8 hours.