Convert

1
Convert
Unit
____
Section 1.3 and intro to 1.4 (Proportions)
2
Objectives

1. Use unit rates and dimensional analysis to solve
   real-life problems.
2. Begin to solve proportions.

3
Ratios
A ratio is the ________________________________.
For example
Your schools 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 games
______
7
__


games lost
3 games
3
4
Rates
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?
120 miles
________
________
Rate

60 gallons
1 gallon
Your fuel efficiency is _______ miles per gallon.
5
Notes Convert Rates
Customary Units of Measure 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
6
Notes Convert Rates
Metric Units of Measure 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
7
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 16 ounces 100 centimeters
• 1 foot 1 pound 1 meter

8
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 ____________.

9
Dimensional Analysis
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.
10
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 foot12
  inches
• 1 s 1 s 1 ft
• 10 ft 12 in Divide out common
  units
• 1 s 1 ft

11
Notes Convert Rates

• 10 12 in Simplify
• 1 s 1
• 120 in Simplify
• 1 s
• So, 10 feet per second equals 120 inches per
  second.

12
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)

13
Essential Question

• Explain why the ratio 3 feet has a value of 1.
• 1 yard
• __________________________________________________
  __________________________________________________
  __________________________________________________
  ________________________________________

14
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)

15
Proportion
An equation in which two ratios are equal is
called a proportion.
A proportion can be written using colon notation
like this
abcd
or as the more recognizable (and useable)
equivalence of two fractions.
a
___ ___
c

b
d
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Proportion
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
abcd

b
d
Extremes
Means
17
Proportion
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
18
Proportion
Example
Write the original proportion.
Use the reciprocal property.
Multiply both sides by 35 to isolate the
variable, then simplify.
19
Proportion
Example
Write the original proportion.
Use the cross product property.
Divide both sides by 6 to isolate the variable,
then simplify.