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2
Summary
DEFINITION /PROCEDURE
EXAMPLE
REFERENCE
Prime Numbers and Divisibility
Section 2.1
Prime Number Any whole number that has exactly two
factors, 1 and itself.
7, 13, 29, and 73 are prime
numbers.
p. 130
Composite Number Any whole number greater than 1 that is
not prime.
8, 15, 42, and 65 are composite
numbers.
p. 131
Zero and 1 0 and 1 are not classified as prime or composite
numbers.
p. 131
Divisibility Tests
By 2
A whole number is divisible by 2 if its last digit is 0, 2, 4, 6,
or 8.
932 is divisible by 2; 1347 is
not.
p. 131
By 3
A whole number is divisible by 3 if the sum of its digits is
divisible by 3.
546 is divisible by 3; 2357 is
not.
p. 132
865 is divisible by 5; 23,456 is
not.
p. 132
By 5
A whole number is divisible by 5 if its last digit is 0 or 5.
Factoring Whole Numbers
Prime Factorization
To find the prime factorization of a number, divide the
number by a series of primes until the final quotient is a
prime number. The prime factors include each prime divisor
and the final quotient.
Section 2.2
2B 630
3B 315
3B 105
5B35
7
So 630 2 3 3 5 7.
© 2001 McGraw-Hill Companies
Greatest Common Factor (GCF) The GCF is the largest
number that is a factor of each of a group of numbers.
To Find the GCF
Step 1 Write the prime factorization for each of the numbers
in the group.
p. 139
p. 140
To find the GCF of 24, 30,
and 36:
Step 2
Locate the prime factors that are common to all the
numbers.
24 2 2 2 3
Step 3
The greatest common factor (GCF) will be the
product of all of the common prime factors. If there
are no common prime factors, the GCF is 1.
36 2 2 3 3
30 2 3 5
The GCF is 2 3 6
p. 141
Continued
209
210
CHAPTER 2
MULTIPLYING AND DIVIDING FRACTIONS
DEFINITION /PROCEDURE
EXAMPLE
REFERENCE
Fraction Basics
Section 2.3
Fraction Fractions name a number of equal parts of a unit or
a
whole. A fraction is written in the form , in which a and b are
b
whole numbers and b cannot be zero.
p. 147
Denominator The number of equal parts into which the whole
is divided.
Numerator
5
8
Numerator The number of parts of the whole that are used.
Denominator
p. 147
Proper Fraction A fraction whose numerator is less than its
denominator. It names a number less than 1.
2
11
are proper fractions.
and
3
15
p. 149
Improper Fraction A fraction whose numerator is greater than
or equal to its denominator. It names a number greater than or
equal to 1.
7 21
8
, , and are improper
5 20
8
fractions.
p. 149
Mixed Number The sum of a whole number and a proper
fraction.
7
1
2 and 5 are mixed numbers.
3
8
1
1
Note that 2 means 2 .
3
3
p. 150
To Change an Improper Fraction into a Mixed Number
1. Divide the numerator by the denominator. The quotient is
the whole-number portion of the mixed number.
22
to a mixed
To change
5
number:
2. If there is a remainder, write the remainder over the original
To Change a Mixed Number to an Improper Fraction
1. Multiply the denominator of the fraction by the
whole-number portion of the mixed number.
2. Add the numerator of the fraction to that product.
3. Write that sum over the original denominator to form the
improper fraction.
4
5B22
Quotient
20
2
Remainder
22
2
4
5
5
Denominator
p. 151
Whole number
Numerator
3
(4 5) 3
23
5 4
4
4
Denominator
p. 152
© 2001 McGraw-Hill Companies
denominator. This gives the fractional portion of the mixed
number.
SUMMARY
DEFINITION /PROCEDURE
EXAMPLE
REFERENCE
Simplifying Fractions
Section 2.4
Equivalent Fractions Two fractions that are equivalent (have
equal value) are different names for the same number.
p. 159
Cross Products
a
b
c
d
a d and b c are called the
cross products.
If the cross products for two fractions are equal, the two
fractions are equivalent.
The Fundamental Principle of Fractions
a
For the fraction , and any nonzero number c,
b
ac
a
b
bc
4
2
because
3
6
2634
p. 159
8
84
2
12
12 4
3
In words: We can divide the numerator and denominator of a
fraction by the same nonzero number. The result will be an
equivalent fraction.
2
8
and are equivalent
12
3
fractions.
p. 160
Simplest Form A fraction is in simplest form, or in lowest
terms, if the numerator and denominator have no common
factors other than 1. This means that the fraction has the
smallest possible numerator and denominator.
2
is in simplest form.
3
12
is not in simplest form.
18
The numerator and denominator
have the common factor 6.
p. 160
To Write a Fraction in Simplest Form
Divide the numerator and denominator by any common factor
greater than 1 to reduce a fraction to an equivalent fraction in
lower terms.
211
10
10 5
2
15
15 5
3
Multiplying Fractions
p. 160
Section 2.5 and 2.6
To Multiply Two Fractions
1. Multiply numerator by numerator. This gives the numerator
of the product.
5
3
53
15
8
7
87
56
2. Multiply denominator by denominator. This gives the
denominator of the product.
© 2001 McGraw-Hill Companies
3. Simplify the resulting fraction if possible.
In multiplying fractions it is usually easiest to divide by any
common factors in the numerator and denominator before
multiplying.
1
1
3
2
5
3
53
1
9
10
9 10
6
pp. 169, 172
Continued
212
MULTIPLYING AND DIVIDING FRACTIONS
CHAPTER 2
DEFINITION /PROCEDURE
EXAMPLE
REFERENCE
Dividing Fractions
Section 2.7
To Divide Two Fractions
Invert the divisor and multiply.
4
3
5
15
3
7
5
7
4
28
Multiplying or Dividing Mixed Numbers
Convert any mixed or whole numbers to improper fractions.
Then multiply or divide the fractions as before.
p. 189
4
1
20
16
2
6 3 3
5
3
5
1
64
1
21
3
3
Computer-Related Applications: Time
Section 2.8
Unit
1
Microsecond (ms)
s/ms
1,000,000
Nanosecond (ns)
Picosecond (ps)
1
s/ns
1,000,000,000
1
s/ps
1,000,000,000,000
1000 ms/s
2s 2000 ms
1,000,000 ms/s
3s 3,000,000 ms
1,000,000,000 ns/s
4ms 4000 ns
1,000,000,000,000 ps/s
5ms 5,000,000 ps
pp. 202, 203, 205
© 2001 McGraw-Hill Companies
Conversion Units
1
Millisecond (ms)
s/ms
1000
p. 180, 190
Summary Exercises
You should now be reviewing the material in Chapter 2. The following exercises will help in that process. Work all the
exercises carefully. References are provided to the chapter and section for each exercise. If you have difficulty with any
exercises, go back and review the related material.
[2.1]
In exercises 1 and 2, list all the factors of the given numbers.
1. 52
2. 41
In exercise 3, use the group of numbers 2, 5, 7, 11, 14, 17, 21, 23, 27, 39, and 43.
3. List the prime numbers; then list the composite numbers.
In exercises 4 and 5, use the divisibility tests to determine which, if any, of the numbers 2, 3, and 5 are factors of the
following numbers.
4. 2350
[2.2]
5. 33,451
In exercises 6 to 9, find the prime factorization for the given numbers.
6. 48
7. 420
8. 2640
9. 2250
In exercises 10 to 15, find the greatest common factor (GCF).
10. 15 and 20
11. 30 and 31
12. 24 and 40
13. 39 and 65
14. 49, 84, and 119
15. 77, 121, and 253
[2.3]
© 2001 McGraw-Hill Companies
16.
Identify the numerator and denominator of each fraction.
5
9
17.
17
23
Give the fractions that name the shaded portions of the following diagrams. Identify the numerator and the denominator.
18.
Fraction
Numerator
Denominator
213
214
CHAPTER 2
MULTIPLYING AND DIVIDING FRACTIONS
19.
Fraction:
Numerator:
Denominator
20. From the following group of numbers:
2 5 3 45 7 4 9 7 12 2
, ,2 , , ,3 , , , ,5
3 4 7 8 7 5 1 10 5 9
List the proper fractions.
List the improper fractions
List the improper fractions
Convert to mixed or whole numbers.
21.
41
6
22.
32
8
23.
23
3
24.
47
4
Convert to improper fractions.
25. 7
5
8
[2.4]
29.
26. 4
3
10
27. 5
2
7
8
13
28. 12
Determine whether each of the following pairs of fractions are equivalent.
5 7
,
8 12
30.
8 32
,
15 60
33.
140
180
36.
32
?
40
5
Write each fraction in simplest form.
24
36
32.
45
75
34.
16
21
Find the missing numerators.
35.
15
?
25
5
[2.5]
37.
Multiply.
7
5
15
21
38.
10
9
27
20
39. 4 3
8
40. 3
2
5
5
8
© 2001 McGraw-Hill Companies
31.
SUMMARY EXERCISES
41. 5
1
4
1
3
5
42. 1
5
8
12
43. 3
215
1
7
6
2
5
8
7
[2.1]–[2.6] Solve the following applications.
3
4
44. Distance. The scale on a map is 1 inch (in.) 80 miles (mi). If two cities are 2 in. apart on the map, what is the
actual distance between the cities?
1
4
1
3
45. Cost of linoleum. A kitchen measures 5 by 4 yards (yd). If you purchase linoleum costing $9 per square yard
(yd2), what will it cost to cover the floor?
1
2
2
3
46. Cost of carpet. Your living room measures 6 by 4 yards (yd). If you purchase carpeting at $18 per square yard
(yd2), what will it cost to carpet the room?
47. Earnings. Maria earns $72 per day. If she works
5
of a day, how much will she earn?
8
2
5
48. Miles traveled. David drove at an average speed of 65 mi/h for 2 h. How many miles did he travel?
2
5
49. Distance. The scale on a map is 1 in. 120 mi. What actual distance, in miles, does 3 in. on the map
represent?
50. Student numbers. At a college,
2
1
of the students take a science course. Of the students taking science, take
5
4
biology. What fraction of the students take biology?
51. Student workers. A student survey found that
jobs,
3
of the students have jobs while going to school. Of those who have
4
5
work more than 20 h per week. What fraction of those surveyed work more than 20 h per week?
6
2
3
1
2
52. Area. A living room has dimensions 5 by 4 yd. How much carpeting must be purchased to cover the
© 2001 McGraw-Hill Companies
room?
[2.7]
53.
Divide.
5
5
12
8
3
8
56. 3 2
1
4
54.
7
14
15
25
57. 3
3
8
7
55.
9
2
2
20
5
58. 6
1
3
7
14
216
CHAPTER 2
MULTIPLYING AND DIVIDING FRACTIONS
Solve the following applications.
3
4
59. Length of wire. A piece of wire 3 ft long is to be cut into five pieces of the same length. How long will each
piece be?
3
4
60. Quantity. A blouse pattern requires 1 yd of fabric. How many blouses can be made from a piece of silk that is 28 yd
long?
1
4
61. Speed. If you drive 126 mi in 2 h, what is your average speed?
1
4
62. Average speed. If you drive 117 mi in 2 h, what is your average speed?
63. Number of lots. An 18-acre piece of land is to be subdivided into home lots that are each
be formed?
3
acre. How many lots can
8
[2.8]* Complete each statement.
64. 20 s ____________ ms
65. 15 ms ____________ ns
66. 50 ns ____________ ps
67. 25 ms ____________ ps
Solve the following applications.
68. How far will light travel in 7 s? Express your answer in miles.
*Optional section
© 2001 McGraw-Hill Companies
69. How far will sound travel in 7 s? Express your answer in feet.