The formula for distance problems is: distance = rate × time or d = r × t
Things to watch out for: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.
The following diagrams give the steps to solve DistanceRateTime Problems. Scroll down the page for examples and solutions. We will show you how to solve distance problems by the following examples:
Traveling At Different Rates
Traveling In Different Directions
Given Total Time
Wind and Current 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:
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)
Use distributive property
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.
How to find the 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.
How to solve Wind Word Problems?Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with stepbystep explanations.
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