Lesson Travel and Distance problems

Travel and Distance problems

In this lesson simple typical word problems on Travel and Distance are presented to show the approach and the methodology of their solutions.

Problem 1. Two objects moving toward each other Two cars entered an Interstate highway at the same time and traveled toward each other       (see Figure 1). The initial distance between cars was 390 miles. First car was going 70 miles per hour, the second car was going 60 miles per hour. How long will it take for two cars to pass each other? What distance each car will travel before passing?

Figure 1. Two objects moving toward each other

Solution
We will solve the problem by reducing it to the simple linear equation.
Let us denote as the unknown time in hours for two cars to pass each other.
Since the rate of the first car is 70 miles per hour, the distance it travels for time is equal to miles.
Since the rate of the second car is 60 miles per hour, the distance it travels for time is equal to miles.
At the moment two cars pass each other, the sum of these distances is equal to the initial distance between two cars, i.e. 390 miles.
This gives us an equation
.
Simplify the left side of the equation using the distributive law:
.
So, our equation takes the form
.
Divide both sides of the equation by 130. You get
.
Thus, it will take 3 hours for two cars to pass each other.
Now, you can check the solution.
The distance that the first car will travel for 3 hours is equal to miles.
The distance that the second car will travel for 3 hours is equal to miles.
Since , the solution is correct.

Answer. It will take 3 hours for two cars to pass each other.
First car will travel 210 miles, and the second car will travel 180 miles before passing.

Note. For those who prefer more physical thinking, the following alternative solution may seem more suitable.
The value of miles per hour is the rate of decreasing the distance between two cars, so the time before passing is equal to hours. Based on this, the distance traveled by the first car for this time is equal to miles, and the distance traveled by the second car for this time is equal to miles.
Surely, the results are the same.

Problem 2. Two objects moving in the same direction Two cars entered an Interstate highway at the same time at different locations and traveled   in the same direction as shown in Figure 2. The initial distance between cars was 30 miles. First car was going 70 miles per hour, the second car was going 60 miles per hour. How long will it take for the first car to catch the second one? What distance will each car travel before the first car catches the second one?

Figure 2. Two objects moving in the same direction

Solution
I will show you first how to solve the problem by reducing it to the simple linear equation.
Let us denote as the unknown time in hours for the first car to catch the second one.
Since the rate of the first car is 70 miles per hour, the distance it travels for time is equal to miles.
Since the rate of the second car is 60 miles per hour, the distance it travels for time is equal to miles.
At the moment when the first car catches the second one, the difference of these distances is equal to the initial distance between two cars, i.e. 30 miles.
This gives us an equation
.
Simplify the left side of the equation using the distributive law:
.
So, our equation takes a form
.
Divide both sides of the equation by 10. You get
.
Thus, it takes 3 hours for the first car to catch the second one.
You can check the solution.
The distance first car will travel for 3 hours is equal to miles.
The distance the second car will travels for 3 hours is equal to miles.
Since , the solution is correct.

Answer. It will take 3 hours for the first car to catch the second one.
First car will travel 210 miles, and the second car will travel 180 miles before the first car catches the second one.

Note. Again, there is alternative physical solution.
The value of miles per hour is the rate of decreasing the distance between two cars in this case, so the time before passing is equal to hours. Based on this, the distance traveled by the first car for this time is equal to miles, and the distance traveled by the second car for this time is equal to miles.
The results are the same.

Problem 3. Two objects moving toward each other Two cars entered an Interstate highway at the same time and traveled toward each other       (see Figure 3). The initial distance between cars was 390 miles. The speed of the first car was in 10 miles per hour greater than that of the second car. It took 3 hours for two cars to pass each other. What was the speed of each car? What distance each car traveled before they pass each other?

Figure 3. Two objects moving toward each other

Solution
The situation is similar to that of the Problem 1, but the time to get passing is given and the speed is under the question in this case.
Again, we will solve the problem by reducing it to the simple linear equation.
Let us denote as the unknown speed (in miles per hour) of the first car.
Then the distance the first car traveled for 3 hours is equal to miles.
Since the rate of the first car is in 10 miles per hour greater than that of the second car, the rate of the second car is equal to miles per hour.
Then the distance the second car traveled for 3 hours is equal to miles.
At the moment two cars pass each other, the sum of these distances is equal to the initial distance between two cars, i.e. 390 miles.
This gives us an equation
.
Simplify the left side of the equation by opening brackets and collecting like terms:
.
So, our equation takes the form
.
Transfer the constant term from the left side to the right and collect like terms. You get
.
Divide both sides by 6. You get
.
Thus, the speed of the first car was equal to 70 miles per hour.
Hence, the speed of the second car was equal to miles per hour.
The distance first car traveled for 3 hours was equal to miles.
The distance the second car traveled for 3 hours was equal to miles.
Since , the solution is correct.

Answer. The speed of the first car was 70 miles per hour, the speed of the second car was 60 miles per hour.
First car traveled 210 miles, and the second car traveled 180 miles before they pass each other.