At 3:25 p.m., a passenger train and a freight train both left Kalamazoo, Michigan. Right now, the passenger train is 61.5 miles west of Kalamazoo. The freight train is now 49.5 mil... At 3:25 p.m., a passenger train and a freight train both left Kalamazoo, Michigan. Right now, the passenger train is 61.5 miles west of Kalamazoo. The freight train is now 49.5 miles east of Kalamazoo. The passenger train averages 16 miles per hour faster than the freight train. What time is it now?
Understand the Problem
The question describes a scenario involving two trains leaving Kalamazoo at the same time and asks for the current time based on their positions relative to Kalamazoo. We need to analyze the distance covered by each train and their speed to determine how long they have been traveling and consequently the current time.
Answer
The current time is 4:10 PM.
Answer for screen readers
The current time is 4:10 PM.
Steps to Solve
- Determine the total distance between the trains and Kalamazoo
The passenger train is 61.5 miles west of Kalamazoo, while the freight train is 49.5 miles east. The total distance from the passenger train to Kalamazoo plus the distance from the freight train to Kalamazoo is:
$$ \text{Total Distance} = 61.5 + 49.5 = 111 \text{ miles} $$
- Define the speeds of the trains
Let the speed of the freight train be $x$ miles per hour. Therefore, the speed of the passenger train is:
$$ \text{Speed of Passenger Train} = x + 16 \text{ miles per hour} $$
- Set up the time equation for both trains
Since both trains left Kalamazoo at the same time, the time traveled by each train can be equated:
- The time for the freight train to travel the distance of 49.5 miles is:
$$ \text{Time}_{\text{freight}} = \frac{49.5}{x} $$
- The time for the passenger train to travel the distance of 61.5 miles is:
$$ \text{Time}_{\text{passenger}} = \frac{61.5}{x + 16} $$
Set the times equal:
$$ \frac{49.5}{x} = \frac{61.5}{x + 16} $$
- Cross-multiply and solve for x
Cross-multiplying gives:
$$ 49.5(x + 16) = 61.5x $$
Expanding this equation:
$$ 49.5x + 792 = 61.5x $$
Rearranging terms to isolate x:
$$ 792 = 61.5x - 49.5x $$
$$ 792 = 12x $$
- Calculate the speed of the freight train
Now, divide by 12:
$$ x = \frac{792}{12} = 66 \text{ miles per hour}$$
This means the freight train travels at 66 miles per hour. The passenger train travels at:
$$ 66 + 16 = 82 \text{ miles per hour}$$
- Calculate the time traveled by either train
Using the speed of the freight train to find the time traveled:
$$ \text{Time} = \frac{49.5}{66} \approx 0.75 \text{ hours} $$
Converting 0.75 hours into minutes:
$$ 0.75 \times 60 = 45 \text{ minutes} $$
- Determine the current time
Adding 45 minutes to the initial time of 3:25 PM:
3:25 PM + 45 minutes = 4:10 PM
The current time is 4:10 PM.
More Information
This problem involves setting up a system of equations based on distances and speeds to find the unknown time.
Tips
- Misunderstanding the direction of distances (east vs. west).
- Forgetting to convert hours into minutes when needed.
- Incorrectly setting up the equation by not properly equating the time traveled for both trains.
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