Section 3-1: Fraction Terminology
Learning Outcome 1
Identify the following as proper fractions, improper fractions, or mixed numbers:
3 8 5 2
, , ,4 .
7 3 5 3
3
8 5
2
, proper fraction; , , improper fractions; 4 , mixed number.
7
5 5
3
Write the following in decimal notation:
7
1
1
,
, 3
. 0.07; 0.1; 3.001.
100 10
1,000
Section 3-2: Multiples, Divisibility, and Factor Pairs
Learning Outcome 1
Find the first five multiples of 3: 3 x 1 = 3, 3 x 2 = 6, 3 x 3 = 9, 3 x 4 = 12, 3 x 5 = 15. The first five
multiples of 3 are 3, 6, 9, 12, and 15.
Learning Outcome 2
Is 732 divisible by 4? Yes, the last two digits form a number, 32, which is divisible by 4. That is,
32 ÷ 4 = 8.
Learning Outcome 3
Find all the factors of 48: Write factor pairs. 1 x 48, 2 x 24, 3 x 16, 4 x 12; 5 is not a factor; 6 x 8; 7
is not a factor; 8 x 6 is a repeat. Factors are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.
Section 3-3: Prime and Composite Numbers
Learning Outcome 1
Is 22 prime or composite? Factor pairs of 22 are 1 x 22 and 2 x 11; therefore 22 is composite.
Learning Outcome 2
Find the prime factorization of 45: 45 = 3 x 15 = 3 x 3 x 5 , so the prime factorization is 3 x 3 x 5
or 32 x 5.
Section 3-4: Least Common Multiple and Greatest Common Factor
Learning Outcome 1
Find the LCM for 10, 15, and 20: 10 = 2 x 5, 15 = 3 x 5, 20 = 2 x 2 x 5 or 2 2 x 5
LCM = 22 x 3 x 5 or 60
Learning Outcome 2
Find the GCF of 10, 25, and 35: 10 = 2 x 5, 25 = 5 2 , 35 = 5 x 7
GCF = 5
Section 3-5: Equivalent Fractions and Decimals
Learning Outcome 1
5
.
6
n
Multiplying by
is the same as multiplying by 1. So,
n
Write three fractions equivalent to
5 2 10
× =
6 2 12
5 3 15
× =
6 3 18
the original value is unchanged.
5 4 20
× =
6 4 24
Learning Outcome 2
12
Reduce:
32
12 12 4 3
=
÷ =
32 32 4 8
4
is the same as dividing by 1.
4
So, the original value is unchanged.
Dividing by
Learning Outcome 3
Write 1.27 as a fraction
27
1.27 = 1
100
Learning Outcome 4
4
to a decimal.
Convert
5
.8
5 4.0 That is, 0.8.
Divide the numerator by the denominator.
Section 3-6: Improper Fractions and Mixed Numbers
Learning Outcome 1
Convert the following to whole or mixed numbers:
Learning Outcome 2
Convert 6 2 to an improper fraction:
5
6
9 8 9
8
1
, .
= 3; = 1
3 7 3
7
7
2 (5 × 6) + 2 32
=
=
5
5
5
Section 3-7: Finding Common Denominators and Comparing Fractions
Learning Outcome 1
Find the lowest common denominator (LCD) for
4
5
and
5
9
5 = 1 x 5 9 = 3 x 3 or 32
LCD = 5 x 32= 5 x 9 = 45
The smallest number that can be divided evenly by both 5 and 9 is 45.
Learning Outcome 2
Which fraction is larger, 3 or 5 ?
8
16
3 6 5
5
=
=
8 16 16 16
Since
6
5
5 3
is larger than
is larger than
.
,
16
16
16 8
Section 3-8: Adding Fractions and Mixed Numbers
Learning Outcome 1
Add: 2 + 4 + 1 = 7
9 9 9 9
Add:
1 2 3 4 7
1
+ = + = =1
2 3 6 6 6
6
Learning Outcome 2
Add: 3 1 + 2 1 + 4 3
2
3
4
1
6
3 =3
2
12
1
4
2 =2
3
12
3
9
4 =4
4
12
19 19
7
7
7
9
= 1 , so 9 + 1 = 10
12 12
12
12
12
Add the numerators. Keep the comon
demoninator.
Change to equivalent fractions that have common
denominators.
Add fractional parts and whole-number parts. Combine
the results.
Section 3-9: Subtracting Fractions and Mixed Numbers
Learning Outcome 1
Subtract: 3 - 2
4 5
3 15
=
4 20
2 8
=
5 20
7
20
Change to equivalent fractions that have common
denominators.
Subtract numerators.
Learning Outcome 2
Subtract: 7 2 - 4 7
5
10
2
4
14
=7
= 6
10
5
10
7
7
7
4
=4
= 4
10
10
10
7
2
10
7
Borrow 1 from 7 and add it as
10
4
to
.
10
10
Section 3-10: Multiplying Fractions and Mixed Numbers
Learning Outcome 1
Multiply: 2 3 1
× ×
3 4 2
1 1
2/ 3/ 1 1
× × =
3/ 4/ 2 4
1 2
Reduce numerators and denominators before multiplying.
Multiply numerators and multiply denominators.
Learning Outcome 2
Multiply:
1
1
×2 ×3
5
2
1
1 7 3/ 7
2
× × = =1
5 3/ 1 5
5
1
Write whole or mixed numbers as improper fractions.
Reduce numerator and denominator before multiplying.
Learning Outcome 3
4
Simplify:  3 
5
Write as repeated factors.
3 3 3 3 81
• • • =
5 5 5 5 625
Multiply.
Section 3-11: Dividing Fractions and Mixed Numbers
Learning Outcome 1
Find the reciprocal of 7, 7 , 1.1 .
8 4
The reciprocal of 7 or
7
1
7
8
is ; the reciprocal of is ;
1
7
8
7
Write whole or mixed numbers as
improper fractions.
Interchange the numerator and
denominator.
the reciprocal of 1 1 or 5 is 4 .
4
4
5
Learning Outcome 2
5 2
Divide: ÷
8 3
5 2 5 3 15
÷ = × =
8 3 8 2 16
Multiply
5
2
3
by the reciprocal of , that is, by . .
8
3
2
Learning Outcome 3
Divide: 3 3 ÷ 2 1
4
2
3
1 15 5
÷2 =
÷ =
4
2 4 2
1
15 2/ 15
5
1
× =
=1 =1
4/ 5 10
10
2
2
3
Change mixed numbers to improper fractions as a first step.
Multiply the dividend (first number) by the reciprocal of the
divisor (second number).
Learning Outcome 4
Simplify the following complex fraction.
1 7
3 = 3 = 7 ÷ 3 = 7 × 2/ = 14 = 1 5
1 3 3 2 3 3 9
9
1
2 2
2
Change mixed numbers to improper fractions and write
complex fraction as division.
Change division to equivalent multiplication and multiply.
Section 3-12: Signed Fractions and Decimals
Learning Outcome 1
3
Write three equivalent signed fractions for - 8
-
3
+3 , which can be written as +3 or -3 or -3 .
=-8
+8
8
+8
-8
Learning Outcome 2
Add: 2 + -4 .
9 9
2 -4 -2
2
+
=
=9 9
9
9
Learning Outcome 3
Add: −3.25 + 2.65
−3.25 + 2.65 = −0.6
Use rules for adding fractions and for adding signed numbers.
Use rules for adding decimals and for adding signed numbers.
Add: 7.45 + (−3.15)
7.45 – 3.15 = 4.3
Learning Outcome 4
3  10  3  10 
Simplify:
 −  + −  − .
5  21  7  2 
1
2
 10  3  10 
−
 + + +  =
 21  7  2 
1
7
2 3
− + +5 =
7 7
1
+5 =
7
1
5
7
3
5
Change any two of the three signs
of the fraction.
Multiply and divide.
Add.