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"Distance" Word Problems (page 1 of 2)

Just a note: Make sure that the units for time and distance agree with the units for the rate. For instance, if they give you a rate of feet per second, your time must be in seconds and your distance must be in feet. Sometimes they try to trick you by using the wrong units, and you have to catch this and convert to the correct units.

• A 555-mile, 5-hour plane trip was flown at two speeds. For the first part of the trip, the average speed was 105 mph. Then the tailwind picked up, and the remainder of the trip was flown at an average speed of 115 mph. For how long did the plane fly at each speed?

Let's set up a grid:   Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

	d	r	t
first part	d	105	t
second part	555 – d	115	5 – t
total	555	---	5

Using "d = rt", the first row gives us d = 105t and the second row gives us:

555 – d = 115(5 – t)

Since the two distances add up to 555, add the two distances:

555 = 105t + 115(5 – t)

Then we get:

555 = 105t + 575 – 115t
555 = 575 – 10t
–20 = –10t
2 = t

According to our grid, "t" stands for the time spent on the first part of the trip, so the answer is "The plane flew for two hours at 105 mph and three hours at 115 mph."

You can add distances and you can add times, but you cannot add rates. Think about it: If you drive 20 mph on one street, and 40 mph on another street, does that mean you averaged 60 mph?

As you can see, the actual math involved is often quite simple. It's the set-up that's the hard part. So what follows are some more examples, but only with the set-up; no solutions.

• An executive drove from home at an average speed of 30 mph to an airport where a helicopter was waiting. The executive boarded the helicopter and flew to the corporate offices at an average speed of 60 mph. The entire distance was 150 miles; the entire trip took three hours. Find the distance from the airport to the corporate offices.
     

	d	r	t
driving	d	30	t
flying	150 – d	60	3 – t
total	150	---	3

The first row gives us d = 30t. The second row gives us:

150 – d = 60(3 – t)

Adding, we get:

150 = 30t + 60(3 – t)

Solve for t; interpret the value; state the final answer.

• A car and a bus set out a 2 p.m. from the same point, headed in the same direction. The average speed of the car is 30 mph slower than twice the speed of the bus. In two hours, the car is 20 miles ahead of the bus. Find the rate of the car.
     

	d	r	t
car	d + 20	2r – 30	2
bus	d	r	2
total	---	---	---

(As it turns out, we don't need the "total" row this time.) The first row gives us:

d + 20 = 2(2r – 30)

This is not terribly helpful. The second row gives us:

d = 2r

We can use this to simplify the first equation. Substitute "2r" for "d", and solve for "r". Then interpret the value, and state the final answer.

• A passenger train leave the train depot 2 hours after a freight train leaves the same depot. The freight train is traveling 20 mph slower than the passenger train. Find the rate of each train, if the passenger train overtakes the freight train in three hours.
     

	d	r	t
passenger train	d	r	3
freight train	d	r – 20	3 + 2 = 5
total	---	---	---

(As it turns out, we don't need the "total" row this time.) Why is the distance just "d" for both trains? Partly, because the problem doesn't say how far the trains actually went. But mostly because they went the same distance, as far as we're concerned, because we're only counting from the depot to wherever they met. After that, we don't care. And how did we get those times? We know that the passenger train drove for three hours to catch up to the freight train; that's how we got the "3". But note that the freight train had a two-hour head start. That means that the freight train was going for five hours.

	d	r	t
passenger train	d = 3r	r	3
freight train	d = 5(r – 20)	r – 20	3 + 2 = 5
total	---	---	---

Now that we have this information, we can try to find our equation. Using the fact that d = rt, the first row gives us d = 3r (note the revised table above). The second row gives us:

d = 5(r – 20)

Since the distances are equal, set the equations equal:

3r = 5(r – 20)

Solve for r; interpret the value, and state the final answer.

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