Introduction
Connections to Fractions
Proportion Story Problems
Unit Pricing
MATH 113
Section 6.1: Ratio and Proportions
Prof. Jonathan Duncan
Walla Walla College
Winter Quarter, 2007
Conclusion
Introduction
Connections to Fractions
Outline
1
Introduction
2
Connections to Fractions
3
Proportion Story Problems
4
Unit Pricing
5
Conclusion
Proportion Story Problems
Unit Pricing
Conclusion
Introduction
Connections to Fractions
Proportion Story Problems
Unit Pricing
Conclusion
Comparing in Mathematics
One activity of interest in mathematics is comparing things.
Consider the following example.
Example
Two children decide to step off the distance from the drinking
fountain to their classroom door. Child A takes 9 steps to reach
the door and child B takes 6 steps to reach the door. Compare the
size of the children’s steps.
Additive Comparison
Child A took 3 more steps than did child B.
Multiplicative Comparison
Child A took 9 steps to child B’s 6 steps. The ratio of child A’s
steps to child B’s is 9:6 or 3:2. Alternatively, child A took 50%
more steps than did child B.
Introduction
Connections to Fractions
Proportion Story Problems
Unit Pricing
Conclusion
Terminology
Before we start studying ratios and proportions it is important to
define these terms and other related words.
Defining Terminology
Ratio
A ratio is a relationship between two amounts. It is written as
a : b or ba .
Proportion
A proportion is two ratios which are equal to each other. It is
written as ba = dc .
Rate
A rate is a ratio in which the two amounts being compared
represent different units such as miles/gallon or feet/second.
Introduction
Connections to Fractions
Proportion Story Problems
Unit Pricing
A New Kind of Number
Dealing with ratios and proportions can be an adjustment.
Numbers as Quantities
Up to this point, numbers have been quantities which we can
visualize such as:
60 feet
20 gallons
5 people
Numbers as Rates or Comparisons
Ratios express a relationship which is harder to visualize such as:
60 feet per second
30 miles per gallon
5 people per group
Conclusion
Introduction
Connections to Fractions
Proportion Story Problems
Unit Pricing
Real World Examples
Ratios and proportions, while new to us in this class, have many
real-world applications.
Real World Applications
Below are some real-world applications for ratios and proportions.
Comparing like quantities (3 steps to 2 steps)
Stating rates (miles per gallon)
Dividing things up (students per group)
Others?
Conclusion
Introduction
Connections to Fractions
Proportion Story Problems
Unit Pricing
Conclusion
Differences between Ratios and Fractions
Ratios and fractions, while related, are not exactly the same thing.
Ratios vs. Fractions
Ratios can be used to express “part:part” or “whole:whole”
relationships. On the other hand, fractions are used to represent
“part:whole” relationships. While at times either a fraction or a
ratio can be used, there are times when fractions can not be used.
Example
The ratio of the distance traveled by your car to the amount of gas
used is 30 miles to 1 gallon (30:1 or 30 mpg). Can this be
expressed as a fraction?
Introduction
Connections to Fractions
Proportion Story Problems
Unit Pricing
Conclusion
Conversion between Ratios and Fractions
The following examples can be stated as ratios or as fractions.
Draw a diagram to represent the ratio and convert it into a
fraction.
Example
The ratio of Democrats to Republicans in the US Senate is 51
to 49.
The ratio of a window’s height to its width is 6 feet to 9 feet.
The ratio of juice to water in condensed orange juice is 1 can
to 3 cans.
Which Ratios can be Fractions?
Note that ratios which express “whole:whole” relationships can not
be translated into fractions.
Introduction
Connections to Fractions
Proportion Story Problems
Unit Pricing
Conclusion
Painting
In the next several slides we will look at story problems involving
ratios and proportions.
Example
There are 2 painters. The first, John, can paint a wall in 6 minutes
and the second, Harry, can paint the wall in 3 minutes. How long
does it take them to paint the wall working together?
Express painting rates as ratios
In x minutes they paint the entire wall
Solve for x to find time
Introduction
Connections to Fractions
Proportion Story Problems
Unit Pricing
Conclusion
A Mountain Run
For the next problem, we will need to remember the relationship
between distance, rate, and time. That is, d = rt.
Example
A runner can make it up a certain mountain road at 3 km/hr (at a
constant rate of speed). Once she reaches the top of the road, she
turns around and runs down at 9 km/hr. (again at a constant
speed). What is the average speed of the runner?
Answer is not 6 km/hr
Distance traveled up-hill and down is the same, say d
Time spent differs, say t1 uphill and t2 downhill
Average speed is the rate r =
d
t
where t is the total time
Introduction
Connections to Fractions
Proportion Story Problems
Unit Pricing
Conclusion
Running a Bridge
This last problem is a challenging one!
Example
Daphne is running across a narrow bridge. When she gets 58 of the
way across the bridge, Daphne notices a car approaching the
bridge at 80 km/hr. She can sprint at a constant speed to the end
of the bridge and arrive there at the same time as the car, or she
can turn around and sprint (at the same constant speed) back to
the beginning of the bridge and arrive at that end of the bridge at
the same time as the car. How fast can Daphne sprint?
Introduction
Connections to Fractions
Proportion Story Problems
Unit Pricing
Conclusion
Running a Bridge - Solution
Bridge Solution
Yes, there is enough information to solve this!
The car is at the far end of the bridge when Daphne has
covered 38 ths of the bridge.
The car, at 80 km/hr, covers the length of the bridge while
Daphne covers 82 ths of the bridge.
So using proportions,
rate.
2
8
is to Daphne’s rate as 1 is to the car’s
Introduction
Connections to Fractions
Proportion Story Problems
Unit Pricing
What is Unit Pricing?
Many times it is useful to reduct ratios to their lowest terms. On
instance in which this is true is in unit pricing.
Example
An 8 ounce brick of cheese costs $2.30 and a 2 pound brick (32
ounces) costs $6.00. Which is the better deal?
If these prices were to be equal, then we would have the
6.00
proportions 2.50
8 = 32 , but is this the case?
Solution #1
To check, we will find a common denominator for the two ratios.
10.00
2.50 4
· =
8
4
32
We would expect to pay $10.00 for a 32 ounce block of cheese.
Conclusion
Introduction
Connections to Fractions
Proportion Story Problems
Unit Pricing
Conclusion
Approaches to Unit Pricing Problems
The solution to the example given on the previous slide is not the
only possible way to solve this problem.
Solution #2
Another option is to convert to decimal.
2.50
6.00
= 0.3125
= 0.1875
8
32
These are called “unit prices” as they represent the price per ounce.
Solution #3
Finally, we could look at the reciprocals.
8
32
= 3.2
= 5.33
2.50
6.00
These are ounces per dollar, so we look for higher numbers.
Introduction
Connections to Fractions
Proportion Story Problems
Unit Pricing
Important Concepts
Things to Remember from Section 6.1
1
Ratio and Proportion Terminology
2
Relationship between Ratios and Fractions
3
Solving Story Problems involving Ratios and Proportions
4
Unit Pricing
Conclusion