- •Study Skills Workshop
- •1.1 An Introduction to the Whole Numbers
- •1.2 Adding Whole Numbers
- •1.3 Subtracting Whole Numbers
- •1.4 Multiplying Whole Numbers
- •1.5 Dividing Whole Numbers
- •1.6 Problem Solving
- •1.7 Prime Factors and Exponents
- •1.8 The Least Common Multiple and the Greatest Common Factor
- •1.9 Order of Operations
- •THINK IT THROUGH Education Pays
- •2.1 An Introduction to the Integers
- •THINK IT THROUGH Credit Card Debt
- •2.2 Adding Integers
- •THINK IT THROUGH Cash Flow
- •2.3 Subtracting Integers
- •2.4 Multiplying Integers
- •2.5 Dividing Integers
- •2.6 Order of Operations and Estimation
- •Cumulative Review
- •3.1 An Introduction to Fractions
- •3.2 Multiplying Fractions
- •3.3 Dividing Fractions
- •3.4 Adding and Subtracting Fractions
- •THINK IT THROUGH Budgets
- •3.5 Multiplying and Dividing Mixed Numbers
- •3.6 Adding and Subtracting Mixed Numbers
- •THINK IT THROUGH
- •3.7 Order of Operations and Complex Fractions
- •Cumulative Review
- •4.1 An Introduction to Decimals
- •4.2 Adding and Subtracting Decimals
- •4.3 Multiplying Decimals
- •THINK IT THROUGH Overtime
- •4.4 Dividing Decimals
- •THINK IT THROUGH GPA
- •4.5 Fractions and Decimals
- •4.6 Square Roots
- •Cumulative Review
- •5.1 Ratios
- •5.2 Proportions
- •5.3 American Units of Measurement
- •5.4 Metric Units of Measurement
- •5.5 Converting between American and Metric Units
- •Cumulative Review
- •6.2 Solving Percent Problems Using Percent Equations and Proportions
- •6.3 Applications of Percent
- •6.4 Estimation with Percent
- •6.5 Interest
- •Cumulative Review
- •7.1 Reading Graphs and Tables
- •THINK IT THROUGH The Value of an Education
- •Cumulative Review
- •8.1 The Language of Algebra
- •8.2 Simplifying Algebraic Expressions
- •8.3 Solving Equations Using Properties of Equality
- •8.4 More about Solving Equations
- •8.5 Using Equations to Solve Application Problems
- •8.6 Multiplication Rules for Exponents
- •Cumulative Review
- •9.1 Basic Geometric Figures; Angles
- •9.2 Parallel and Perpendicular Lines
- •9.3 Triangles
- •9.4 The Pythagorean Theorem
- •9.5 Congruent Triangles and Similar Triangles
- •9.6 Quadrilaterals and Other Polygons
- •9.7 Perimeters and Areas of Polygons
- •THINK IT THROUGH Dorm Rooms
- •9.8 Circles
- •9.9 Volume
- •Cumulative Review
3.4 Adding and Subtracting Fractions |
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Solve We must find the sum of three fractions with different denominators. To find the LCD, we prime factor the denominators and use each prime factor the greatest number of times it appears in any one factorization:
6 2 ~3
4 2 2 ¶LCD 2 2 3 5 60 15 3 ~5
The LCD for |
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2 appears twice in the factorization of 4.
3 appears once in the factorization of 6 and 15.
5 appears once in the factorization of 15.
Build each fraction so that its denominator is 60.
1060 1560 2860
10 15 28
60
5360
Multiply the numerators. Multiply the denominators. The denominators are now the same.
Add the numerators and write the sum |
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over the common denominator 60. |
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This fraction is in simplest form. |
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State The fraction of the student body that watches 0 to 2 hours of TV daily is 5360 .
Check We can check by estimation. The result, 5360 , is approximately 5060 , which simplifies to 56 . The red, yellow, and blue shaded areas appear to shade
about 56 of the pie chart. The result seems reasonable.
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THINK IT THROUGH Budgets
“Putting together a budget is crucial if you don’t want to spend your way into serious problems.You’re also developing a habit that can serve you well throughout your life.”
Liz Pulliam Weston, MSN Money
The circle graph below shows a suggested budget for new college graduates as recommended by Springboard, a nonprofit consumer credit counseling service. What fraction of net take-home pay should be spent on housing?
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252 Chapter 3 Fractions and Mixed Numbers
S E C T I O N 3.4 STUDY SET
VOCABULARY
Fill in the blanks.
1.Because the denominators of 38 and 78 are the same number, we say that they have a
denominator.
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CONCEPTS
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5. To add (or subtract) fractions that have the same |
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6.To add (or subtract) fractions that have different denominators, we express each fraction as an
equivalent fraction that has the for its
denominator. Then we use the rule for adding (subtracting) fractions that have the denominator.
7.When adding (or subtracting) two fractions with different denominators, if the smaller denominator is a factor of the larger denominator, the
denominator is the LCD.
8.Write the subtraction as addition of the opposite:
18 a 58b
9.Consider 34 . By what form of 1 should we multiply the numerator and denominator to express it as an
equivalent fraction with a denominator of 36?
10.The denominators of two fractions are given. Find the least common denominator.
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6 and 36 |
11. Consider the following prime factorizations:
24 2 2 2 3
90 2 3 3 5
For any one factorization, what is the greatest number of times
a.a 5 appears?
b.a 3 appears?
c.a 2 appears?
12.The denominators of two fractions have their primefactored forms shown below. Fill in the blanks to find the LCD for the fractions.
20 2 |
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5 fLCD |
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13.The denominators of three fractions have their primefactored forms shown below. Fill in the blanks to find the LCD for the fractions.
20 2 2 5
30 2 3 5 ¶LCD
90 2 3 3 5
14.Place a or symbol in the blank to make a true statement.
3231
a.35 35
b.1317 1117
NOTATION
Fill in the blanks to complete each solution.
15.25 17 25 17 55
35 5
35
35
16.78 23 78 33 23
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GUIDED PRACTICE
Perform each operation and simplify, if possible. See Example 1.
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Perform the operations and simplify, if possible. See Example 3.
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3.4 |
Adding and Subtracting Fractions |
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Chapter 3 |
Fractions and Mixed Numbers |
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83. |
9 |
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1 |
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84. |
5 |
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3 |
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20 |
30 |
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6 |
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10 |
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85. |
27 |
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5 |
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86. |
49 |
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15 |
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50 |
16 |
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50 |
16 |
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87. |
13 |
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1 |
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88. |
71 |
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1 |
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20 |
5 |
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100 |
10 |
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89. |
37 |
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17 |
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90. |
54 |
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52 |
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103 |
103 |
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53 |
53 |
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91. |
3 |
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5 |
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92. |
2 |
7 |
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8 |
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93. |
4 |
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1 |
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94. |
8 |
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7 |
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27 |
6 |
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95. |
7 |
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19 |
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96. |
73 |
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31 |
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30 |
75 |
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75 |
30 |
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97. |
Find the difference of |
11 |
and |
2 |
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60 |
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45 |
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97
98.Find the sum of 48 and 40 .
52
99.Subtract 12 from 15 .
100. What is the sum of |
11 |
and |
7 |
increased by |
5 |
? |
24 |
36 |
48 |
APPLICATIONS
101.BOTANY To determine the effects of smog on tree development, a scientist cut down a pine tree and measured the width of the growth rings for the last two years.
a.What was the growth over this two-year period?
b.What is the difference in the widths of the two rings?
5 |
1 |
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–– in. |
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–– in. |
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32 |
16 |
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102.GARAGE DOOR OPENERS What is the difference in strength between a 13 -hp and a 12 -hp garage door opener?
103.MAGAZINE COVERS The page design for the magazine cover shown below includes a blank strip at the top, called a header, and a blank strip at the bottom of the page, called a footer. How much page length is lost because of the header and footer?
3 in. header
–
FRAUD & SAT EVALUATION | jon cheater 8
THE TRUTH BEHIND COLLEGE TESTING | issac icue WHAT REALLY HAPPENS IN DORMS | laura life lesson
colleg e life
Page TODAY
length The
TRUTH about college
A Real
Student
talking with with kids all over America and in depth intreviews with Colby students and teachers
all the news that’s fit to print and quite a bit that isn’t PLUS
articles and lots of pictures gossip and trash and misinformation
5 in. footer
––
16
104.DELIVERY TRUCKS A truck can safely carry a one-ton load. Should it be used to deliver one-half ton of sand, one-third ton of gravel, and one-fifth ton of cement in one trip to a job site?
105.DINNERS A family bought two large pizzas for dinner. Some pieces of each pizza were not eaten, as shown.
a.What fraction of the first pizza was not eaten?
b.What fraction of the second pizza was not eaten?
c.What fraction of a pizza was left?
d.Could the family have been fed with just one pizza?
106.GASOLINE BARRELS Three identical-sized barrels are shown below. If their contents of the two of the barrels are poured into the empty third barrel, what fraction of the third barrel will be filled?
107.WEIGHTS AND MEASURES A consumer protection agency determines the accuracy of butcher shop scales by placing a known three- quarter-pound weight on the scale and then comparing that to the scale’s readout. According to the illustration, by how much is this scale off? Does it result in undercharging or overcharging customers on their meat purchases?
3
– pound 4
weight
1
–
2
3.4 Adding and Subtracting Fractions |
255 |
108.FIGURE DRAWING As an aid in drawing the human body, artists divide the body into three
parts. Each part is then expressed as a fraction of the total body height. For example, the torso is 154 of the body height. What fraction of body height
is the head?
Head
Torso:
4
––
15
Below the waist:
3
–
5
109. Suppose you work as a |
from Campus to Careers |
|
school guidance counselor |
School Guidance Counselor |
iStockphoto.com/Monkeybusinessimages |
at a community college |
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and your department has |
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conducted a survey of the |
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full-time students to learn |
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more about their study |
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habits. As part of a Power |
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Point presentation of the |
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survey results to the |
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school board, you show the following circle |
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graph. At that time, you are asked, “What |
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fraction of the full-time students study 2 hours |
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|
or more daily?” What would you answer? |
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More than 2 hr |
2 hr |
|
3 |
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–– |
2 |
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10 |
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– |
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5 |
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1 |
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–– |
1 |
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Less than 1 hr 10 |
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– |
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|
5 |
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1 hr |
0 |
1 pound |
256Chapter 3 Fractions and Mixed Numbers
110.HEALTH STATISTICS The circle graph below
shows the leading causes of death in the United States for 2006. For example, 1350 of all of the deaths that year were caused by heart disease. What
fraction of all the deaths were caused by heart disease, cancer, or stroke, combined?
|
Alzheimer’s |
|
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|
disease |
|
Diabetes |
|
3 |
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––– |
|
3 |
|
100 |
|
––– |
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100 |
|
Other |
|
Heart |
|
13 |
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disease |
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–– |
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50 |
|
13 |
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–– |
Respiratory |
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50 |
diseases |
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1 |
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–– |
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20 |
|
Cancer |
Flu |
Accidents |
|
6 |
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1 |
||
|
–– |
||
1 |
|
25 |
–– |
–– |
Stroke |
|
50 |
20 |
|
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3 |
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–– |
|
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50 |
|
|
Source: National Center for Health Statistics
111.MUSICAL NOTES The notes used in music have fractional values. Their names and the symbols used to represent them are shown in illustration (a). In common time, the values of the notes in each measure must add to 1. Is the measure in illustration
(b) complete?
Half |
Quarter |
Eighth |
Sixteenth |
note |
note |
note |
note |
(a)
(b)
112.TOOLS A mechanic likes to hang his wrenches above his tool bench in order of narrowest to widest. What is the proper order of the wrenches in the illustration?
113.TIRE TREAD A mechanic measured the tire tread depth on each of the tires on a car and recorded them on the form shown below. (The letters LF stand for left front, RR stands for right rear, and so on.)
a.Which tire has the most tread?
b.Which tire has the least tread?
|
|
Measure of tire tread depth |
1/4 in. |
LF |
RF 5/16 in. |
7/32 in. |
LR |
RR 21/64 in. |
114.HIKING The illustration below shows the length of each part of a three-part hike. Rank the lengths of the parts from longest to shortest.
|
4 |
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– |
C |
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5 mi |
|
3 |
B |
5 |
– mi |
||
– mi |
|
8 |
4 |
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D |
A |
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WRITING
115.Explain why we cannot add or subtract the fractions 29 and 25 as they are written.
116.To multiply fractions, must they have the same denominators? Explain why or why not. Give an example.
REVIEW
Perform each operation and simplify, if possible.
117. a. |
1 |
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1 |
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b. |
1 |
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c. |
1 |
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1 |
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d. |
1 |
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1 |
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4 |
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8 |
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4 |
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8 |
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118. a. |
5 |
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3 |
b. |
5 |
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3 |
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21 |
14 |
21 |
14 |
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c. |
5 |
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d. |
5 |
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3 |
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21 |
14 |
|
21 |
14 |
1 |
in. |
3 |
in. |
3 |
5 |
– |
– |
–– in. |
–– in. |
||
4 |
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8 |
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16 |
32 |