Chapter 2 Section 6. Objectives 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Ratio, Proportion, and Percent Write ratios. Solve proportions

Chapter 2 Section 6

Objectives

1

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

Ratio, Proportion, and Percent

Write ratios.

Solve proportions.

Solve applied problems by using proportions.

Find percents and percentages.

2.6

2

3

4


Objective 1

Write ratios.

Slide 2.6-3


A ratio is a comparison of two quantities using a quotient.

The last way of writing a ratio is most common in algebra.

RatioThe ratio of the number a to the number b (b ≠ 0) is written

,a to b : ,a b or .a

b

Slide 2.6-4

Write ratios.


Write a ratio for each word phrase.

3 days to 2 weeks

12 hr to 4 days

Solution:

2 7 14weeks days days 3 days

weeks

3

14

days

days

3

14

4 24 96days hours hours 4

hours

days

12

96

hours

hours

1

8

Slide 2.6-5

EXAMPLE 1 Writing Word Phrases as Ratios


Solution:

$2.79$0.116

24

$3.89$0.108

36

The 36 oz. size is the best buy. The unit price is $0.108 per oz.

$1.89$0.158

12

The supermarket charges the following prices for pancake syrup. Which size is the best buy? What is the unit cost for that size?

Slide 2.6-6

EXAMPLE 2 Finding Price per Unit


Objective 2

Solve proportions.

Slide 2.6-7


A ratio is used to compare two numbers or amounts. A proportion says that two ratios are equal, so it is a special type of equation. For example,

3 15

4 20

is a proportion which says that the ratios and are equal.3

4

15

20

In the proportion

a, b, c, and d are the terms of the proportion. The terms a and d are

called the extremes, and the terms b and c are called the means. We

read the proportions as “a is to b as c is to d.”

, ,0a c

b db d

a c

b d

Slide 2.6-8

Solve proportions.


We can also find the products ad and bc by multiplying diagonally.

For this reason, ad and bc are called cross products.

ad

bca c

b d

Beginning with this proportion and multiplying each side by the common denominator, bd, gives

d bda

bb

c

d

a c

b d .ad bc

Slide 2.6-9

Solve proportions. (cont’d)


Cross Products

If then the cross products ad and bc are equal—that is,

the product of the extremes equals the product of the means.

Also, if then

,a c

b d

,ad bc where , 0 .a c

b db d

Slide 2.6-10

Solve proportions. (cont’d)

If then ad = cb, or ad = bc. This means that the two proportions

are equivalent, and the proportion

can also be written as

Sometimes one form is more convenient to work with than the other.

a c

b d

,a b

c d

, 0 .a b

c dc d


13 119 1547

17 91 1547

21 45 945

15 62 930 Solution: False21 62

15 45

13 91

17 119

Solution: True

Decide whether the proportion is true or false.

Slide 2.6-11

EXAMPLE 3 Deciding Whether Proportions Are True


Solution:

42 6 35x

4

42 42

2 210x

5x

The solution set is {5}.

The cross-product method cannot be used directly if there is more than one term on either side of the equals symbol.

35.

6 42

xSolve the proportion

Slide 2.6-12

EXAMPLE 4 Finding an Unknown in a Proportion


6 5 2 2a 3030 4 05 3a 5 2

5 5

6a

26

5a

Solution:

The solution set is 26

.5

When you set cross products equal to each other, you are really multiplying each ratio in the proportion by a common denominator.

6 2.

2 5

x Solve

Slide 2.6-13

EXAMPLE 5 Solving an Equation by Using Cross Products


Objective 3

Solve applied problems by using proportions.

Slide 2.6-14


Solution:

Let x = the price of 16.5 gal of fuel.

$37.68

12 16.5

x

gal gal

12 621.

12 12

72x

51.81x

16.5 gal of diesel fuel costs $51.81.

Twelve gallons of diesel fuel costs $37.68. How much would 16.5 gal of the same fuel cost?

Slide 2.6-15

EXAMPLE 6 Applying Proportions


Objective 4

Find percents and percentages.

Slide 2.6-16


A percent is a ratio where the second number is always 100.

Since the word percent means “per 100,” one percent means “one per one hundred.”

1% 0.01, or1

1%100

Slide 2.6-17

Write ratios.


Convert.

310% to a decimal

8% to a decimal

0.685 to a percent

Solution:

3.1

Slide 2.6-18

EXAMPLE 7 Converting Between Decimals and Percents

.08

68.5%


Solve each problem.

What is 6% of 80?

16% of what number is 12?

What percent of 75 is 90?

Solution:

90

75x

12

.16x

1.2 or 120%x

Slide 2.6-19

EXAMPLE 8 Solving Percent Equations

.06 80x

.16 12x

90 75x

75x

4.8x


Solution:

Let x = the number of possible points on the test.

85

34 0

5

0 85

8

x

34 8

00

5

1x

40x

There were 40 possible points on the test.

Mark scored 34 points on a test, which was 85% of the possible points. How many possible points were on the test?

Slide 2.6-20

EXAMPLE 9 Solving Applied Percent Problems

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Chapter 2 Section 6. Objectives 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Ratio, Proportion, and Percent Write ratios. Solve proportions