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___________games lost

= 7 games______3 games

= 7__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?

Rate = 120 miles________60 gallons= ________

1 gallon

Your fuel efficiency is _______ miles per gallon.

Notes: Convert RatesCustomary 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 RatesEach 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 centimeters1 foot 1 pound 1 meter

Notes: Convert RatesThe ______ 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?

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.

5 hours • 60 minutes________1 hour

= 300 minutes

Notes: Convert RatesExample: 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

1 s 1 s 1 ft

= 10 ft 12 in Divide out common units

1 s 1 ft

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.

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 QuestionExplain 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___ ___=b

cd

a:b::c:d a___ ___=b

cd

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.

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:

x

35

3

5

355

3 x

Write the original proportion.

Use the reciprocal property.

35355

335

x Multiply both sides by 35 to isolate the variable, then simplify.

x21

Example:

9

62

x

___________

Write the original proportion.

Use the cross product property.

_________ Divide both sides by 6 to isolate the variable, then simplify.

x______