Mrs. McConaughy Geometry 1

LESSON 8.1: RATIOS AND PROPORTIONS

OBJECTIVE:OBJECTIVE: To write ratios and solve proportions

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Definition:: Ratio

A ratio is ________________________________________________. If a and b are two numbers, the

ratio of a to b can be written in the following ways:

a/b a:b a is to b

an expression thatcompares two quantities by division

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FINDING RATIOS:Determine the following ratios:

female to male studentsin this classroom ____

male to female studentsin this classroom ____

male to total studentsin this classroom ____

total students to femalestudents in this classroom ____

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FINDING RATIOS:Determine the following ratios:

Ratio of ♥ to ♣:__

♥♣♥♣♥♣♥♣♥♣♥♣♥♣♣

♣♣♣♥♥♣♣♣♥♥♣♣♣♥♥

♣♥♣♣♣♥♣♣♣♥♣♣♥♥♥

♥♥♥♣♣♥♥♥♥♥♥♣♣♥♥♥Ratio of shaded to unshaded area:__ Ratio of shaded area to area of the whole figure:__

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ExampleA scale model of a car is 4 inches long.

The actual car is 15 ft. long.

a.) What is the ratio of the length of the model to the length of the car? ________________

b.) What is the ratio of the length of the car to the length of the model? ______________

Write both measurements in the same units.

NOTE: ORDER is important

when writing ratios.

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Example A photo that is 8 inches

wide and 5 1/3 inches high is enlarged to poster size which is 2 ft. wide by 1 1/3 ft. high.

What is the ratio of the height of the photo to the height of the poster?

.

NOTE: ORDER is important

when writing ratios.

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ProportionProportion When two ratios are equal, you have

a ________.

.3 6= is a proportion4 8

proportion

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Proportion

A proportion is__________________________________

_______________________.

a statement of equality between two ratios (equal ratios).

a c = and a: b = c:d are examples of proportions b d

a : b = c : d extremes

means

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PROPERTIES OF PROPORTIONS1. Cross Product Property The product of the extremes equals the product of the means.

If a = c, then________. b d2. Reciprocal Property If two ratios are equal, then their reciprocals are also equal.

If a = c, then b d __________.

ad = bc

b = d

a c

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To solve the proportion, you find the value of the variable.

Solve the proportions. a. 4 = 5 b. 3 = 2 x 7 y +2 y

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Example Solving for a Variable

26 xSolve f or x in = 50 75.

Before you cross multiply, determine if you can reduce fractions. Example: 26 = __ 50

13

25

13 x= 25 75

How can you determine the answer without cross multiplying?

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Additional Properties of Proportions

Two more properties of proportions, which are especially useful in geometry, are:

c

a cI f is equivalent tob d

a a + b = _____ and = _____b

b

d

c + d

d

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NOTE: You can avoid memorizing many of the Properties of Proportions by remembering to write your ratio (comparison of two quantities by division) in words before setting up and solving your proportion by cross multiplying.

ORDER is important

when setting up proportions.

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Complete:

.a 12 bI f = , then =4 b 12

Write two expressions that m nare equivalent to = 4 11.

The Cross-Product Property states that________________.

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Solving for a Variable

x + 1 xSolve f or x in = 3 2

3x = 2 (x + 1)

3x = 2x + 2

x = 2

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USING PROPORTIONS If you earn $380 working for two

weeks, how much will you earn in 15 weeks?

KEY: Words SymbolsAlways remember to set up your proportion in words first!

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Example: (Always ask: What two quantities am I comparing?)

money = weeks

weeks _______ = _______money

380 x

2 15

2 15

380 x

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2x = (380)(15) 2x = 5700 x = 2850

You will earn $2,850 in 15 weeks.

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.

6 4 1 = = is an example of an24 16 4 extended proportion

Extended Proportions

An extended proportion is _______________________.

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The floor plan of a house is drawn to the scale of ¼ inch = 1 foot. The master bedroom measures 3 inches by 3 ¾ inches on the blueprints.

What are the actual measurements of the room?

USING PROPORTIONS

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f eetinches =

f eet _______ = _______inches

The actual size of the room is_______________ .12 feet by 15 feet

NOTE: Rather than writing an extended proportion, you may find it easier to write two separate proportions.

Pick your poison: Choose how you want to set up your two ratios in words first!

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Final Checks for Understanding

1. A ratio is_______________________.2. A proportion is ___________________.3. What is the most efficient way to solve 8 = x ? Explain. 24 94. Solve for x: 4 = x x 16.

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Summary: Properties of Proportions

a c= is equivalent tob d

b(1) ad = _____ (2) = _____aa a + b (3) = ___ (4) = ____c b

ac

d

c

b

d

c + d

d

Mrs. McConaughy Geometry 24

Homework Assignment:

Ratios and Proportions WS