Use the three steps to solve the problem.
A "Local" train leaves a station and runs at an average rate of 35 mph. An hour and a half later, an "Express" train leaves the station and travels at an average rate of 56 mph on a parallel track. How many hours after the Express train starts will it overtake the Local? Write the time as a decimal.
{
hrs.}
Step 1: Define the variables:
Let x be the number of hours that the Express train runs.
Step 2: Set up an equation:
Since the Local train started an hour and a half earlier, it has already been running for x + 1.5 hours when the Express train starts. Distance = Rate × Time, so:
35(x + 1.5) = 56x
Step 3: Solve the equation:
35x + 52.5 = 56x
52.5 = 56x - 35x
52.5 = 21x
x = 52.5/21
x ≈ 2.5
Therefore, the Express train will overtake the Local train approximately 2.5 hours after it starts.
Step 1: Determine the time it takes for the Local train to cover the initial distance.
The Local train travels at an average rate of 35 mph. It takes an hour and a half (1.5 hours) for the Express train to start after the Local train. Therefore, the Local train has a head start of 1.5 hours.
The distance covered by the Local train during this time is given by the formula distance = rate × time.
Distance covered by the Local train = 35 mph × 1.5 hours = 52.5 miles.
Step 2: Determine the relative speed between the Express and Local trains.
The Express train travels at an average rate of 56 mph, which is faster than the Local train by 56 mph - 35 mph = 21 mph.
Step 3: Use the relative speed to calculate the time it takes for the Express train to overtake the Local train.
To calculate the time it takes for the Express train to overtake the Local train, we can use the formula time = distance / rate. In this case, the distance is equal to the head start distance of the Local train, which is 52.5 miles, and the rate is the relative speed between the two trains, which is 21 mph.
Time taken by the Express train to overtake the Local train = 52.5 miles / 21 mph = 2.5 hours.
Therefore, the Express train will overtake the Local train 2.5 hours after it starts.
To solve this problem using the three steps, we need to follow these steps:
Step 1: Identify the known quantities and variables:
- Average rate of the Local train = 35 mph
- Average rate of the Express train = 56 mph
- Time elapsed before the Express train starts running = 1.5 hours
- Time it takes for the Express train to overtake the Local train = unknown (let's call it "t" hours)
Step 2: Set up the equation based on the information given:
We need to determine the time it takes for the Express train to overtake the Local train. Both trains have the same starting point and run in parallel tracks, so when the Express train overtakes the Local train, they will have covered the same distance.
The distance the Local train travels in "t" hours is given by:
Distance_Local = Rate_Local * Time_Local
Distance_Local = 35 * t
The distance the Express train travels in "t" hours is given by:
Distance_Express = Rate_Express * Time_Express
Since the Express train starts 1.5 hours later than the Local train, its time is "t - 1.5" hours.
Distance_Express = 56 * (t - 1.5)
The distances covered by both trains when the Express overtakes the Local are the same:
Distance_Local = Distance_Express
35 * t = 56 * (t - 1.5)
Step 3: Solve the equation:
Now we can solve the equation to find the value of "t" in hours:
35t = 56t - 84
84 = 21t
t = 84 / 21
t = 4
Therefore, it will take the Express train 4 hours after it starts to overtake the Local train. The answer is written as a decimal, so it will be 4.0 hours.