Section 1.3 and intro to 1.4
(Proportions)
1. Use unit rates and dimensional analysis to solve
real-life problems.
2. Begin to solve proportions.
A ratio is the ________________________________.
For example: Your school’s basketball team has won 7 games
and lost 3 games. What is the ratio of wins to
losses?
Because we are comparing wins to losses the first
number in our ratio should be the number of wins and
the second number is the number of losses.
The ratio is ___________
games won ______
7
= 7 games = __
games lost
3 games
3
In a ratio, if the numerator and denominator are measured in
different units then the ratio is called a rate.
A unit rate is a rate per one given unit, like 60 miles per 1 hour.
Example: You can travel 120 miles on 60 gallons of gas.
What is your fuel efficiency in miles per gallon?
________
miles = ________
Rate = 120
60 gallons 1 gallon
Your fuel efficiency is _______ miles per
gallon.
Notes: Convert Rates
Customary Units of Measure
Smaller
Larger
12 inches
1 foot
16 ounces
1 pound
8 pints
1 gallon
3 feet
1 yard
5,280 feet
1 mile
Notes: Convert Rates
Metric Units of Measure
Smaller
Larger
100 centimeters
1 meter
1,000 grams
1 kilogram
1,000 milliliters
1 liter
10 milliliters
1 centimeter
1,000 milligrams
1 gram
Notes: Convert Rates
Each of the relationships in the tables
can be written as a _____________.
Like a unit rate, a unit ____ is one in
which the denominator is 1 unit.
Below are three examples of unit ratios:
12 inches
1 foot
16 ounces 100 centimeters
1 pound
1 meter
Notes: Convert Rates
The ______ and __________ of each of the unit
ratios shown are equal. So, the value of each
ratio is ______.
You can convert one rate to an equivalent rate
by multiplying by a unit ratio or its reciprocal.
When you convert rates, you include the
_______ in your computation.
The process of including units of measure as
factors when you compute is called
____________.
Writing the units when comparing each unit of a rate is called
dimensional analysis.
You can multiply and divide units just like you would multiply and
divide numbers. When solving problems involving rates, you can
use unit analysis to determine the correct units for the answer.
Example: How many minutes are in 5 hours?
5 hours • 60
________
minutes = 300 minutes
1 hour
To solve this problem we need a unit rate that relates minutes to
hours. Because there are 60 minutes in an hour, the unit rate we
choose is 60 minutes per hour.
Notes: Convert Rates
Example: A remote control car travels at a rate of
10 feet per second. How many inches per second
is this?
Steps:
10 ft = 10 ft 12 in
Use 1 foot=12 inches
1s
1s
1 ft
= 10 ft
1s
12 in
1 ft
Divide out common units
Notes: Convert Rates
= 10
1s
12 in
1
= 120 in
1s
Simplify
Simplify
So, 10 feet per second equals 120 inches
per second.
Dimensional Analysis Examples
1. A gull can fly at a speed of 22 miles per hour.
About how many feet per hour can a gull fly? (Use
the chart)
Essential Question
Explain why the ratio 3 feet has a value of 1.
1 yard
______________________________________
______________________________________
______________________________________
______________________________________
______________________________________
Dimensional Analysis Examples
2. An AMTRAK train travels 125 miles per hour.
Convert the speed to miles per minute. Round
to the nearest tenth. (Use the chart)
An equation in which two ratios are equal is called a proportion.
A proportion can be written using colon notation like this
a:b::c:d
or as the more recognizable (and useable) equivalence of two fractions.
___
a = ___
c
b
d
When Ratios are written in this order, a and d are the extremes, or
outside values, of the proportion, and b and c are the means, or
middle values, of the proportion.
___
a = ___
c
b
d
a:b::c:d
Extremes
Means
To solve problems which require the use of a proportion we can use one
of two properties.
The reciprocal property of proportions.
If two ratios are equal, then their reciprocals are equal.
The cross product property of proportions.
The product of the extremes equals the product of the means
Example:
5 35

3 x
Write the original proportion.
3 x

5 35
Use the reciprocal property.
3 x
Multiply both sides by 35 to isolate
35    35
the variable, then simplify.
5 35
21  x
Example:
2 6

x 9
Write the original proportion.
__________
_ Use the cross product property.
_________
______  x
Divide both sides by 6 to isolate the
variable, then simplify.