chapter 2 section 6 copyright © 2008 pearson education, inc. publishing as pearson addison-wesley
TRANSCRIPT
Chapter Chapter 22Section Section 66
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Ratios and Proportions
11
33
22
2.62.62.62.6Write ratios.Solve proportions.Solve applied problems by using proportions.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 11
Slide 2.6 - 3
Write ratios.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Write ratios.
A ratio is a comparison of two quantities using a quotient.
The last way of writing a ratio is most common in algebra.
Percents are ratios in which the second number is always 100. For example, 50% represents the ratio 50 to 100, 27% represents the ratio 27 to 100, and so on.
Slide 2.6 - 4
The ratio of the number a to the number b (b ≠ 0) is written
,a to b ,:a b or .a
b
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Write a ratio for each word phrase.
3 days to 2 weeks
12 hr to 4 days
EXAMPLE 1Writing Word Phrases as Ratios
Solution:
Solution:
Slide 2.6 - 5
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
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EXAMPLE 2
Solution:
Finding Price per Unit
$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
Slide 2.6 - 6
The local supermarket charges the following prices for a popular brand of pancake syrup. Which size is the best buy? What is the unit cost for that size?
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Objective 22
Solve proportions.
Slide 2.6 - 7
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Solve proportions.
Slide 2.6 - 8
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
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Beginning with this proportion and multiplying each side by the common denominator, bd, gives
Solve proportions. (cont’d)
We can also find the products ad and bc by multiplying diagonally.
For this reason, ad and bc are called cross products.Slide 2.6 - 9
d bda
bb
c
d
ad
bca c
b d
a c
b d .ad bc
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Solve proportions. (cont’d)
Slide 2.6 - 10
If then the cross products ad and bc are equal.
Also, if then
,a c
b d
,ad bc , 0 .a c
b db d
From this rule, if then ad = bc; that is, the product of
the extremes equals the product of the means.
a c
b 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 b
c d
a c
b d
, .0a b
c dc d
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13 119 1547
17 91 1547
21 45 945
15 62 930
EXAMPLE 3
Solution: False
Deciding whether Proportions Are True
21 62
15 45
Slide 2.6 - 11
13 91
17 119
Solution: True
Decide whether the proportion is true or false.
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EXAMPLE 4
Solution:
Finding an Unknown in a Proportion
35
6 42
x
42 6 35x
Slide 2.6 - 12
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.
Solve the proportion .
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EXAMPLE 5
The solution set is
Solving an Equation by Using Cross Products
Slide 2.6 - 13
6 2.
2 5
a
6 5 2 2a 3030 4 05 3a 5 2
5 5
6a
26
5a
Solution:
26.
5
When you set cross products equal to each other, you are really multiplying each ratio in the proportion by a common denominator.
Solve
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Objective 33
Solve applied problems with proportions.
Slide 2.6 - 14
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EXAMPLE 6
Solution: Let x = the price of 16.5 gal of fuel.
Applying Proportions
Slide 2.6 - 15
$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 gal of diesel fuel costs $37.68. How much would 16.5 gal of the same fuel cost?