Algebra: Distance 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 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

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.



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?



Distance Problems: Traveling In Different 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.

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?"



Distance Problems: Given 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.

Find total distance given total time and two rates



Wind and Current 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.

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.



Algebra Word Problem: Distance, Rate, and Time. This is an example about finding the speed of the current of stream!







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