Transcript 2.6 Ratio, Proportion, and Percent

2.6 Ratio, Proportion, and
Percent
Write ratios.
A ratio is a comparison of two quantities using a quotient.
Ratio
The ratio of the number a to the number b (b ≠ 0) is written
atob,
a : b,
or
a
.
b
The last way of writing a ratio is most common in algebra.
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EXAMPLE 1
Writing Word Phrases as Ratios
Write a ratio for each word phrase.
3 days to 2 weeks
Solution:
2weeks  7days  14days
3days
3days

14days
weeks
3

14
12 hr to 4 days
4days  24hours  96hours
12hours
1
hours


96hours
8
4days
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EXAMPLE 2
Finding Price per Unit
A supermarket charges the following prices for pancake syrup. Which size is
the best buy? What is the unit cost for that size?
Solution:
$3.89
 $0.108
36
$2.79
 $0.116
24
$1.89
 $0.158
12
The 36 oz. size is the best buy. The
unit price is $0.108 per oz.
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Solve proportions.
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
3
4
and
15
are equal.
20
In the proportion
a c
  b, d  0  ,
b d
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
a c as “a is to b as c is to d.”

b d
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Solve proportions. (cont’d)
Beginning with this proportion and multiplying each side by the common
denominator, bd, gives
a c

b d
a c
 bd      bd 
b d
ad  bc.
We can also find the products ad and bc by multiplying diagonally.
bc
a c

b d
ad
For this reason, ad and bc are called cross products.
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Solve proportions. (cont’d)
Cross Products
If
a c
the cross products ad and bc are equal—that is, the product
 then
,
b d
of the extremes equals the product of the means.
Also, ifad
If
 bc,then
a c
  where b, d  0  .
b d
a b then ad = cb, or ad = bc. This means that the two proportions
 ,
c d
are equivalent, and the proportion
a
ccan also be written as

b
d
a
b
  c, d  0  .
c
d
Sometimes one form is more convenient to work with than the other.
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EXAMPLE 3
Deciding Whether Proportions Are True
Decide whether the proportion is true or false.
21 62

15 45
13 91

17 119
15  62  930
Solution: False
21 45  945
17  91  1547
Solution: True
13119  1547
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EXAMPLE 4
Solve the proportion
Finding an Unknown in a Proportion
x 35
 .
6 42
Solution:
x 42  6  35
42 x 210

42
42
x 5
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.
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EXAMPLE 5
Solve
Solving an Equation by Using Cross Products
x6 2
 .
2
5
Solution:
 x  6  5  2  2
5 x  30  30  4  30
5 x 26

5
5
26
x
5
 26 
The solution set is    .
 5
When you set cross products equal to each other, you are really multiplying each
ratio in the proportion by a common denominator.
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Objective 3
Solve applied problems by using
proportions.
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EXAMPLE 6
Applying Proportions
Twelve gallons of diesel fuel costs $37.68. How much would 16.5 gal of the
same fuel cost?
Solution:
Let x = the price of 16.5 gal of fuel.
$37.68
x

12 gal 16.5 gal
12 x 621.72

12
12
x  51.81
16.5 gal of diesel fuel costs $51.81.
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Objective 4
Find percents and percentages.
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Write ratios.
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,
or
1
1% 
100
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EXAMPLE 7
Converting Between Decimals and Percents
Convert.
310% to a decimal
Solution:
3.1
8% to a decimal
.08
0.685 to a percent
68.5%
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EXAMPLE 8
Solving Percent Equations
Solve each problem.
What is 6% of 80?
Solution:
x  .06  80
x  4.8
16% of what number is 12?
.16 x  1200
1200
x
16
x  75
What percent of 75 is 90?
9000  x  75
9000
x
75
x  1.2 or 120%
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EXAMPLE 9
Solving Applied Percent Problems
Mark scored 34 points on a test, which was 85% of the possible points. How
many possible points were on the test?
Solution:
Let x = the number of possible points on the test.
34 85

x 100
3400 85 x

85
85
x  40
There were 40 possible points on the test.
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