RATIO AND PROPORTION

Inverse Proportion
(Teacher Copy)

### Activity 12: An Example of Inverse Proportion

• Students use a fulcrum to find inverse proportions.

• To set up and solve inverse proportions.

### Introduction:

Think back to earlier lessons in this unit and answer the following questions:

• What happens to the circumference of a circle when its diameter increases?  The circumference increases.
• What happens to the distance traveled by your bicycle when the number of wheel revolutions increases?  Distance increases.  What happens when the number of pedal revolutions decreases?  Distance  decreases.
Think about your everyday experiences.  In each of the cases below, what happens to the quantity on the right when the quantity on the left increases?  And when it decreases?

 unit cost total cost time traveled (constant speed) distance traveled distance in feet distance in meters size of a gas tank cost to fill up

Because these are all direct proportions, both either increase together or decrease together.  More specifically, they increase or decrease at the same rate.

There are other quantities that are also related in a different way.  Consider the following variables:

 speed of a car time spent traveling number of workers time needed to finish a task length of a hammer handle effort needed to pull a nail

How are they related?  What happens to the quantities on the right when those on the left increase?  When one quantity increases, the other decreases.

### Part A

 Weight (W2) (unit: grams) 10 20 40 60 80 100 Distance (D2) (unit: mm) 40 20 10 6 2/3 5 4

This table of weight and distance is not of direct proportions because as weight increases, distance decreases (distance would increase if these were directly proportional).

### Part B

In the first table of Part B, W2 X D2 = 400 in each case.
Students should say that as weight increases, distance decreases.
In the second table of Part B, W2 X D2 = 600 in each case.

### Closing Discussion

How can you tell when a relationship between two numbers is a direct proportion, an inverse proportion, or not a proportion at all?  If dividing values in corresponding columns of a table always gives the same number, there is a direct proportion.  If  multiplying values in the corresponding columns of a table always gives the same number, there is a direct  proportion.  Otherwise, there is no proportion.

Think of a situation where two quantities both increase but are not in direct proportion.  Explain.  An example of such a proportion is your height over time.  Both the amount of time and your height increase, but you do not grow at the same rate over your lifetime.

Think of a situation where one quantity goes up while another goes down but the two quantities are not an inverse proportion.  Explain.  An example of this type of proportion is your ability to run over a long distance.  As the amount of distance increases, you run more and more slowly, but your running speed decreases at a faster rate than the rate of increase in distance.

### Activity  13:  Solving Inverse Proportions

• Students set up and solve word problems involving inverse proportions.

To set up and solve inverse proportions.

### Introduction:

At this point, you should have realized that in an inverse proportion, the two terms of each ratio (the numerator and denominator if the ratio is written as a fraction) always multiply to give the same value.  Suppose you think it will take you 6 hours to paint your room by yourself without stopping.  We can write this as:

The two terms of the fraction (1 and 6) multiply to give you 6.  With a friend, it should only take 3 hours.  When two people work, you can write

Again, the product of the two terms is 6.  This relationship always holds in an inverse proportion, so you can always solve an inverse proportion problem by looking for the number that is the product of the two terms.

Let's try one more example.  Suppose Tom and Vickie are working together on a term paper.  Tom, who can type 30 words per minute, can type the paper in 2.6 hours.  If Vickie, who can type 22 words per minute, decides to help Tom by working on the second part of the paper, how long will it take the two of them working together to type the paper?  We can write a fraction representing Tom working alone.

Multiply the two terms of the fraction together.  What did you get? When Tom and Vickie work together, their combined typing speed is 52 words per minute.  Knowing that the typing speed multiplied by the number of hours must equal 78, find the time needed for Tom and Vickie to type the paper together.

### Part A

 Weight (W2) (unit: grams) 10 50 200 1 4 8 Distance (D2) (unit: cm) 40 8 2 400 100 50

### Part B

1. 4.2 hours
2. 15 mph
3. 1.4 hours
4. \$100,750
a. 12.1 hours
b. »7 hours (6.98 hours)
5. 500 cubic feet of water

### Closing Discussion

Look back to question 2 of Part B.  Compare the different ways that each group solved the problem.  Did all groups get the same answer?  Was one way faster?  Which of the procedures used is easiest to remember?  Answers will vary

Area 10 Mathematics and Technology Professional Development Center
Permission is granted to duplicate these materials for classroom use.

Last updated on 1/30/1999