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 (W_{2}) (unit: grams) 
10

20

40

60

80

100

Distance (D_{2}) (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 (W_{2}) (unit: grams) 
10 
50 
200 
1 
4 
8 
Distance (D_{2}) (unit: cm) 
40 
8 
2 
400 
100 
50 
Part B
 4.2 hours
 15 mph
 1.4 hours
 $100,750
 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
© Copyright
Area 10 Mathematics and Technology Professional Development Center
Permission is granted to duplicate these materials for classroom use.
Last updated on 1/30/1999
Comments: egalindo@indiana.edu
http://www.indiana.edu/~atmat/units/ratio/ratio_t7.htm
