Distance problems are word problems that involve the distance an object will travel at a certain average rate for a given period of time.
The formula for distance problems is: distance = rate × time or d = r × t
Things to watch out for:
Make sure that you change the units when necessary. For example, if the rate is given in miles per hour and the time is given in minutes then change the units appropriately.
It would be helpful to use a table to organize the information for distance problems. A table helps you to think about one number at a time instead being confused by the question.
We will show you how distance problems are solved by the following examples:
Traveling At Different Rates
Traveling In Different Directions
Given Total Time
Wind and Current Problems
Related Topics: More Algebra Word Problems
Example:
A bus traveling at an average rate of 50 kilometers per hour made the trip to town in 6 hours. If it had traveled at 45 kilometers per hour, how many more minutes would it have taken to make the trip?
Solution:
Step 1: Set up a rtd table.
r 
t 
d 

Case 1 

Case 2 
Step 2: Fill in the table with information given in the question.
A bus traveling at an average rate of 50 kilometers per hour made the trip to town in 6 hours. If it had traveled at 45 kilometers per hour, how many more minutes would it have taken to make the trip?
Let t = time to make the trip in Case 2.
r 
t 
d 

Case 1 
50 
6 

Case 2 
45 
t 
Step 3: Fill in the values for d using the formula d = rt
r 
t 
d 

Case 1 
50 
6 
50 × 6 = 300 
Case 2 
45 
t 
45t 
Step 4: Since the distances traveled in both cases are the same, we get the equation:
45t = 300
Isolate variable t
Step 5: Beware  the question asked for “how many more minutes would it have taken to make the trip”, so we need to deduct the original 6 hours taken.
Answer: The time taken would have been 40 minutes longer.
Example:
A bus and a car leave the same place and traveled in opposite directions. If the bus is traveling at 50 mph and the car is traveling at 55 mph, in how many hours will they be 210 miles apart?
Solution:
Step 1: Set up a rtd table.
r 
t 
d 

bus 

car 
Step 2: Fill in the table with information given in the question.
If the bus is traveling at 50 mph and the car is traveling at 55 mph, in how many hours will they be 210 miles apart?
Let t = time when they are 210 miles apart.
r 
t 
d 

bus 
50 
t 

car 
55 
t 
Step 3: Fill in the values for d using the formula d = rt
r 
t 
d 

bus 
50 
t 
50t 
car 
55 
t 
55t 
Step 4: Since the total distance is 210, we get the equation:
50t + 55t = 210
105t = 210
Isolate variable t
Answer: They will be 210 miles apart in 2 hours.
Example:
This motion problem (or distance rate time problem or uniform rate problem) involves one object traveling in one direction and the other in the opposite direction, solving for "how long" (or the amount of time) two moving objects traveling until they are certain distance apart.
"Two planes leave the same point at 8 AM. Plane 1 heads East at 600 mph and Plane 2 heads West at 450 mph. How long will they be 1400 miles apart? At what time will they be 1400 miles apart? How far has each plane traveled?"
Example:
John took a drive to town at an average rate of 40 mph. In the evening, he drove back at 30 mph. If he spent a total of 7 hours traveling, what is the distance traveled by John?
Solution:
Step 1: Set up a rtd table.
r 
t 
d 

Case 1 

Case 2 
Step 2: Fill in the table with information given in the question.
John took a drive to town at an average rate of 40 mph. In the evening, he drove back at 30 mph. If he spent a total of 7 hours traveling, what is the distance traveled by John?
Let t = time to travel to town.
7 – t = time to return from town.
r 
t 
d 

Case 1 
40 
t 

Case 2 
30 
7 – t 
Step 3: Fill in the values for d using the formula d = rt
r 
t 
d 

Case 1 
40 
t 
40t 
Case 2 
30 
7 – t 
30(7 – t) 
Step 4: Since the distances traveled in both cases are the same, we get the equation:
40t = 30(7 – t)
40t = 210 – 30t
Isolate variable t
40t + 30t = 210
70t = 210
Step 5: The distance traveled by John to town is
40t = 120
The distance traveled by John to go back is also 120
So, the total distance traveled by John is 240
Answer: The distance traveled by John is 240 miles.
Find total distance given total time and two rates
There is another group of distancetime problems that involves the speed of the water current or the speed of wind affecting the speed of the vehicle. The following video shows an example of such a problem.
Into the headwind, the plane flew 2000 miles in 5 hours. With a tailwind, the return trip took 4 hours. Find the speed of the plane in still air and the speed of the wind.
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