- Demonstrate an understanding of how to translate phrases into algebraic expressions and equations
- Demonstrate an understanding of the six-step method used for solving applications of linear equations
- Learn how to solve motion word problems

## How to Solve Motion Word Problems

### Six-step method for Solving Word Problems with Linear Equations in One Variable

- Read the problem and determine what you are asked to find
- Assign a variable to represent the unknown
- If more than one unknown exists, we express the other unknowns in terms of this variable

- Write out an equation that describes the given situation
- Solve the equation
- State the answer using a nice clear sentence
- Check the result
- We need to make sure the answer is reasonable. In other words, if asked how many students were on a bus, the answer shouldn't be (-4) as we can't have a negative amount of students on a bus.

### Solving Motion Word Problems » d = r x t

At this point, most of us have seen motion word problems. These problems use the distance formula (d = r x t) in order to gain a solution. The distance formula:d » distance traveled

r » rate of speed

t » time traveled

This formula is one of the most intuitive. If we take a road trip and travel at an average rate of speed of 60 miles per hour for 12 hours, we can calculate our distance traveled as:

d = r • t

d = 60 • 12

d = 720

So in this example, we would have traveled a total distance of 720 miles. Let's look at an example.

Example 1: Solve each word problem.

Mishel can get to work in 15 minutes when she takes the bus. When she rides her bike to work, it takes 45 minutes. Her average speed when riding her bike is 10 miles per hour slower than the speed of the bus. What is the average speed of the bus?

Step 1) After reading the problem, it is clear that we need to find the average speed of the bus in miles per hour

Step 2) In this case, we have quite a few things that are unknown. We are given the time it takes to get to work by bus (15 minutes) and by bike (45 minutes). We are not given a rate for either scenario. We are just told that whatever the rate is for the bus, the bike is 10 miles per hour slower. If we let a variable like x be equal to the speed of the bus, then (x - 10) would be the speed of the bike.

let x = speed of the bus in miles per hour

then: x - 10 = speed of the bike in miles per hour

It is also important to note that for this type of problem we assume the distance to work is always the same. This may not be very realistic as a bike path in the real world may not be exactly the same as the path taken for a car. Here we will just say the distance to work is always the same.

Step 3) To make an equation, let's look at our information in a table. Remember our distance formula is: d = r x t. This means we can multiply rate of speed by time traveled to obtain a distance.

d | r | t | |
---|---|---|---|

Bike | 0.75(x - 10) | x - 10 | 0.75 |

Bus | 0.25x | x | 0.25 |

0.75(x - 10) = 0.25x

Step 4) Solve the equation:

0.75x - 7.5 = 0.25x

Multiply by 100 to clear the decimals:

75x - 750 = 25x

75x - 25x = 750

50x = 750

x = 15

Step 5) Since x represented the speed of the bus in miles per hour, this tells us the bus travels at an average speed of 15 miles per hour and the bike will travel at (15 - 10 = 5) 5 miles per hour. Let's state our answer as:

The bus travels at an average speed of 15 miles per hour, while Mishel's bike travels at an average speed of 5 miles per hour.

Step 6) To check our answer, let's see if our answer matches up with the information given.

The distance traveled to work should be the same in each case:

Bike:

0.75 hours • 5 miles per hour

Bus:

0.25 hours • 15 miles per hour

0.75 • 5 = 0.25 • 15

3.75 = 3.75 ✔

Example 2: Solve each word problem.

Beth made a trip to Jamestown and back. The trip there took three hours and the trip back took five hours. What was Beth's average speed on the trip there if she averaged 45 mph on the return trip?

1) The main objective here is to determine Beth's average speed on the trip to Jamestown.

2) We can assign a variable like x to represent her average speed on the trip to Jamestown.

let x = the average speed of Beth in miles per hour on her trip to Jamestown

3) We write our equation. Let's first think about the information using a table. This can sometimes help to organize our thoughts:

Trip | Rate (mph) | Time (hours) | Distance (miles) |
---|---|---|---|

There | x | 3 | 3x |

Back | 45 | 5 | 225 |

This sets up our equation as:

3x = 225 4) Let's solve our equation:

3x = 225

3/3 x = 225/3

x = 75

5) Since x = 75, this tells us Beth has traveled at an average speed of 75 mph on her trip there. Let's create a nice clean sentence:

Beth drove at an average speed of 75 mph on her trip to Jamestown.

6) Check

Let's check the distance to Jamestown on each leg of the trip.

Trip There: distance (225) = rate (75) x time (3)

225 = 75 x 3

225 = 225

Trip Back: distance (225) = rate (45) x time (5)

225 = 45 x 5

225 = 225 ✔

Our solution is correct. Beth drove at an average speed of 75 mph on her trip to Jamestown.

#### Skills Check:

Example #1

Solve each word problem.

Bill left home and drove toward the ferry office. Micaela left one hour later driving 10 miles per hour faster in an effort to catch up to him. After four hours Micaela finally caught up. What was Bill's average speed?

Please choose the best answer.

Example #2

Solve each word problem.

A freight train made a trip to St. Paul and back. The trip there took 15 hours and the trip back took 6 hours. It averaged 45 miles per hour faster on the return trip than on the outbound trip. Find the freight train's average speed on the outbound trip.

Please choose the best answer.

Example #3

Solve each word problem.

Alberto left home three hours before Ashley. They drove in opposite directions. Ashley drove at 30 miles per hour for three hours. After this time they were 480 miles apart. Find Alberto's speed.

Please choose the best answer.

Congrats, Your Score is 100%

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