In these lessons, we will learn how to solve algebra word problems that involve motion.

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Rate, Time and Distance Word Problems

Rate, Time and Distance

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**What are Motion or Distance Word Problems?**

Motion problems are based on the formula

d = rt

where d = distance, r = rate and t = time.

**How to solve Motion or Distance Word Problems?**

Step 1: Draw a diagram to represent the relationship between the distances involved in the problem.

Step 2: Set up a chart based on the formula: rate × time = distance.

Step 3: Use the chart to set up one or more equations.

Step 4: Solve the equations.

We will look at three types of Motion Word Problems:

- Two objects going in opposite directions.
- Both objects going in the same direction, but one goes further.
- One object going and returning at different rates.

**Solve Motion Word Problems: Two objects going in opposite directions**

**Example:**

John and Philip who live 14 miles apart start at noon to walk toward each other at rates of 3 mph and 4 mph
respectively. After how many hours will they meet?

**Solution:**

Let x = time walked.

r | t | d | |

John | 3 | x | 3x |

Philip | 4 | x | 4x |

3x + 4x = 14

7x = 14

x = 2

They will meet in 2 hours.

**How to solve motion word problems with objects traveling in opposite directions?**

**Example:**

Two cars leave from the same place at the same time and travel in opposite direction. One car travels at 55 mph
and the other at 75 mph. After how many hours will there be 520 miles 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?

**Solve Motion Word Problems: Two objects going in the same direction**

**Example:**

Aaron left L.A. to drive at 55 mph towards Las Vegas. Mike left L.A. an hour after Aaron (also towards Las Vegas),
driving at 70 mph. How long will it take Mike to overtake Aaron?

**How to solve motion word problems with objects traveling in the same direction?**

**Example:**

John left his house at 3.00 pm to drive 60 mph to drive towards Michigan. Phoebe left the same house at 5.00 pm,
driving 80 mph in the same direction as John. How long will it take Phoebe to overtake John?

**Solve Motion Word Problems: One object going and returning at different rates**

**Example:**

In still water, Peter’s boat goes 4 times as fast as the current in the river. He takes a 15-mile trip up the
river and returns in 4 hours. Find the rate of the current.

**Solution:**

Let x = rate of the current.

r | t | d | |

down river | 4x + x | 15 / 5x | 15 |

up river | 4x - x | 15 / 3x | 15 |

The rate of the current is 2 mph.

**Example:**

Gordon rode his bike at 15 mph to get his car. He then drove back at 45 mph. If the entire trip took him 8
hours, how far away was his car?

**Motion Word Problems**

This is how to set up motion problems for Algebra.

Three Types of Problems

- Both going the same direction but one going further
- Two going in opposite directions
- Going in one direction and then returning at a different rate.

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