Ratios and Proportions

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Keystone Geometry

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A ratio is a comparison of two numbers such as a : b. Ratio:

Ratio

?What is the ratio of AB to CB

When writing a ratio, always express it in simplest form.

** Ratios must be compared using the same units.

A ratio can be expressed: 1. As a fraction

2. As a ratio 3 : 7

3. Using the word “to” 3 to 7

3

7

Example: What is the ratio of side AB to side CB in the triangle?

10

6

AB

CB

5:3.ratio of AB to CB

3

A

BC

D3.6

6

8

4.8

10Now try to reduce the fraction.

10 5

6 3

Example: What is the ratio of side DB to side CD in the triangle?

DB

CD

3.6

4.8

3.6

4.8

36

48

3

4 ratio of DB to CD 3 : 4.

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The baseball player’s batting average is 0.307 which means he is getting approximately one hit every three times at bat.

A baseball player goes to bat 348 times and gets 107 hits. What is the players batting average?

Solution:Set up a ratio that compares the number of hits to the number of times he goes to bat.

Convert this fraction to a decimal rounded to three decimal places.

Example ……….

Ratio: 107

348

Decimal: 1070.307

348

Proportion• Definition: A proportion is an equation stating that two

ratios are equal.

• For example,

a c

b d

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First Term

Second Term

Third Term

Fourth Term

Terms of a Proportion

: :a b c dFirst Term

Second TermThird Term

Fourth Term

Means and Extremes• The first and last terms of a proportion are called

extremes.• The middle terms are called the means.

618

3.

2

918ex

** The product of the means is always equal to the product of the extremes.

Properties of Proportions is equal to:

Cross-multiplication Switching the means

ReciprocalsAdd one to both

sides

Example: If , then…

5y = _____2x 5 2 7

2 2

** Special Note: The easiest way to decide if two proportions are equal is to apply the mean-extremes property (cross multiplication).

However, all of the other properties work as well, provided your initial proportion is true.

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Solve the proportion using cross multiplication.

Example 1:

4x = 364 • x = 12 • 3

4x = 36 4 4

x = 9

Proportions- examples….

Some to try…1.

2.

3.

4.

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Find the value of x.

Example 2: Use a proportion to solve for the missing piece of a triangle.

84 yards

2 ftx

356 yards

First! Multiply by 3 to change yards into feet. The units must

be the same.

2

1068 252

x

252 2136x

8.5x feet

Examples: Find the measure of each angle.• Two complementary angles have measures in the ratio 2 : 3.

• Two supplementary angles have measures in the ratio 3 : 7.

• The measures of the angles of a triangle are in a ratio of 2 : 2 : 5.

• The perimeter of a triangle is 48cm and the lengths of the sides are in a ratio of 3 : 4 : 5. Find the length of each side.

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36 and 54

54 and 126

40, 40, and 100

12cm, 16cm, and 20cm