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Algebra: Distance Problems




 
What are distance word problems or distance rate time problems?
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 to solve distance problems 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

How to Solve Distance Problems: Traveling At Different Rates

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
t=300/45

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.

2/3 hrs 40 min

Answer: The time taken would have been 40 minutes longer.




How to Solve Distance Problems: Two objects traveling in the same direction

Example:
This motion problem (or distance rate time problem or uniform rate problem) involves traveling in the same direction, solving for "how long" one moving object traveling until it meets up with the second moving object. It uses d = rt (distance equals rate times time).

Car 1 starts from point A and heads for point B at 60 mph. Fifteen minutes later, car 2 leaves the same point A and heads for point B at 75 mph. How long before car 2 overtakes car 1? How to Solve Distance Problems: Two objects traveling in the opposite directions

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
210/105

Answer: They will be 210 miles apart in 2 hours.



 
Objects traveling at opposite directions, calculate how long it takes for them to be a given distance apart
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.
Example:
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? How to Solve Distance Problems: Given the Total Time

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.

7t = 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
210/70

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?
Example:
Roy took 5 hours to complete a journey. For the first 2 hours, he traveled at an average speed of 65 km/h. For the rest of the journey, he traveled at an average speed of 78 km/h. What was the total distance of the journey?

How to solve Wind and Current Word Problems?

There is another group of distance-time 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?
Example: 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. How to solve Current Word Problems?
Example: Traveling downstream, Elmo can go 6 km in 45 minutes. On the return trip, it takes him 1.5 hours. What is the boat's speed in still water and what is the rate of the current?


 
How to find the speed of the current of a stream?
Example: The speed of a boat in still water is 10 mph. It travels 24 miles upstream and 24 miles downstream in 5 hours. What is the speed of the current?

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 step-by-step explanations.


You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.


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