Ch.4 Fractions and Mixed Numbers
4.1 An Introduction to Fractions
4.2 Multiplying Fractions
4.3 Dividing Fractions
4.4 Adding and Subtracting Fractions
4.5 Multiplying and Dividing Mixed Numbers
4.6 Adding and Subtracting Mixed Numbers
4.7 Order of Operations and Complex Fractions
4.8 Solving Equations that Involving Fractions
1
4.1 An Introduction to Fractions
Assume the knowledge of positive fractions
1. Identify the numerator and denominator of a fraction
2. Simplify special fraction forms
3. Define equivalent fractions
4. Build equivalent fractions
5. Simplify fractions
6. Build and simplify algebraic fractions
2
4.1 An Introduction to Fractions
1. Identify the numerator and denominator of a fraction
–
–
Numerator and denominator
Proper fraction and improper fraction
2. Simplify special fraction forms
e.g.3 Simplify:
a.
10
1
= 10
b.
9
9
=1
c.
0
17
=0
d.
21
0
undefined
3
4.1 An Introduction to Fractions
3. Define equivalent fractions
Equivalent Fractions
Two fractions are equivalent if they represent the same
number.
4. Build equivalent fractions
Multiplying Fractions
To multiply two fractions, multiply the numerators and
multiply the denominators
4
4.1 An Introduction to Fractions
4. Build equivalent fractions
e.g.4 Write
of 48.
5
6
as an equivalent fraction with a denominator
Ans.
40
48
e.g.5 Write 7 as an equivalent fraction with a denominator
of 4.
27
Ans.
4
5
4.1 An Introduction to Fractions
5. Simplify fractions
Simplest Form of a Fraction
A fraction is in simplest form, or lowest form, when the
numerator and denominator have no common factors other
than 1.
e.g.6 Are the following fractions in simplest form?
a.
14
18
Ans. no
b.
4
9
Ans. yes
6
4.1 An Introduction to Fractions
5. Simplify fractions
e.g.7 Simplify each fraction:
a.
9
15
Ans.
e.g.8 Simplify each fraction:
(using prime factorization)
b.
8
45
9
15
75
315
5
Ans.
21
a.
b.
2
18
Ans. simplest form
7
4.1 An Introduction to Fractions
6. Build and simplify algebraic fractions
Fractions that contains variable(s) in numerator and/or
denominators are called algebraic fractions.
3
8
e.g.10 Write as an equivalent fraction with a
12 x
denominator of 32x.
Ans.
32 x
8
4.1 An Introduction to Fractions
6. Build and simplify algebraic fractions
e.g.11 Simplify each fractions
a.
7
21m
Ans.
1
3m
b.
4r 4
11r 3
c.
30c 3d
12c5d
Ans.
5
2c 2
9
4.2 Multiplying Fractions
Assume the knowledge of multiplying positive fractions
1. Multiplying fractions
2. Simplify answers when multiplying fractions
3. Multiplying algebraic fractions
4. Evaluate exponential expressions that have fractional
bases
5. Solve application problem by multiplying fractions
6. Find the area of a triangle
10
4.2 Multiplying Fractions
1. Multiplying fractions
e.g.2 Multiply:
5⎛1⎞
− ⎜ ⎟
8⎝9⎠
Ans. −
5
72
Comments: the sign rules for multiplying integers also
holds for multiplying fractions!
1
e.g.3 Multiply:
⋅9
4
(integer with fraction)
Ans.
9
4
11
4.2 Multiplying Fractions
2. Simplify answers when multiplying fractions
e.g.5 Multiply:
2 ⎛ 15 ⎞⎛ 11 ⎞
⎜ − ⎟⎜ − ⎟
5 ⎝ 22 ⎠⎝ 26 ⎠
Ans.
3
26
Comments: don’t multiply the numerator and denominator
and then simplify.
12
4.2 Multiplying Fractions
3. Multiply algebraic fractions
e.g.6 Multiply and simplify:
a.
9 4b
⋅
16b 5
Ans.
9
20
b.
−
3c 28cd
⋅
35d 9c5
Ans. −
4
15c 2
e.g.7 Multiply and simplify: (integer with fraction)
a.
1
⋅ 5m
5
Ans. m
b.
⎛ 1 ⎞
12m ⎜
⎟
6
m
⎠
⎝
Ans. 2
13
4.2 Multiplying Fractions
4. Evaluate exponential expressions that have fractional
bases
e.g.8 Evaluate each expression:
a.
⎛4⎞
⎜ ⎟
⎝ 5⎠
3
b.
e.g.9 Find the power:
⎛ 3⎞
⎜− ⎟
⎝ 7⎠
2
⎛ 2x ⎞
⎜− ⎟
⎝ 3 ⎠
9
Ans.
49
c.
⎛ 3⎞
−⎜ ⎟
⎝7⎠
2
Ans. −
9
49
2
4 x2
Ans.
9
14
4.2 Multiplying Fractions
5. Solve application problem by multiplying fractions
3
4
e.g.10 COLLEGE LIFE It takes of active members of
Alpha Gamma Sigma to vote to accept a pledge into
their fraternity. How many votes are needed if the
fraternity has 84 active members?
Ans. 63 votes
15
4.2 Multiplying Fractions
6. Find the area of a triangle
e.g.11 Find the area of the triangle shown below.
6 ft
11 ft
Ans. 33 square feet
16
4.3 Dividing Fractions
1. Find the reciprocal of a fraction
2. Dividing fractions
3. Dividing algebraic fractions
4. Solve application problems by dividing fraction
17
4.3 Dividing Fractions
1. Find the reciprocal of a fraction
Reciprocals
Two numbers are reciprocals if their product is 1.
e.g.1 For each number, find its reciprocal, and show that
their product is 1.
a.
11
12
b.
−
4
5
Ans. −
5
4
c. 2
Ans.
1
2
18
4.3 Dividing Fractions
2. Divide fractions
Dividing Fractions (just like before)
To divide two fractions, multiply the 1st fraction by the
reciprocal of the 2nd fraction. Simplify if possible.
e.g.3 Divide and simplify:
3 9
÷
7 28
e.g.4 Divide and simplify:
90 ÷
30
17
Ans.
4
3
Ans. 51
19
4.3 Dividing Fractions
2. Divide fractions
e.g.5 Divide and simplify:
2 ⎛ 25 ⎞
÷ ⎜−
⎟
9 ⎝ 18 ⎠
e.g.6 Divide and simplify:
−
64
÷ ( − 8)
35
Ans. −
4
25
Ans.
8
35
20
4.3 Dividing Fractions
3. Dividing algebraic fractions
We can find reciprocal of an algebraic fraction in exactly
the same way as we find reciprocal of an fraction. (flip)
e.g.7 Divide:
9 4
÷
5 x
e.g.8 Divide and simplify:
Ans.
9x
20
6 a 3 15 a
÷
11b 22 b 2
4a 2 b
Ans.
5
21
4.3 Dividing Fractions
4. Solve application problems by dividing fraction
1
16
e.g.9 GOLD COINS How many -ounce coins can be
7
cast from a 8 -ounce bar of gold?
Ans. 14 coins
22
4.4 Adding and Subtracting Fractions
1. Add and subtract fractions that have the same
denominator
2. Add and subtract fractions that have different
denominators
3. Find LCD to add and subtract fractions
4. Add and subtract algebraic fractions
5. Identify the greater of two fractions
6. Solving application problems by adding and subtracting
fractions
23
4.4 Adding and Subtracting Fractions
1. Add and subtract fractions that have the same
denominator
e.g.2 Subtract:
−
9 ⎛ 2⎞
− ⎜− ⎟
5 ⎝ 5⎠
Ans. −
7
5
e.g.3 Perform the operations and simplify:
(from left to right)
Ans.
23 11 7
−
−
30 30 30
1
6
24
4.4 Adding and Subtracting Fractions
2. Add and subtract fractions that have different
denominators (Just like in Math12)
Least Common Denominator
The Least Common Denominator (LCD) for a set of
fractions is the smallest number each denominator will
divide exactly (divide with no remainder).
25
4.4 Adding and Subtracting Fractions
2. Add and subtract fractions that have different
denominators (Read the book. Just like in Math12)
1) Find LCD
2) Build all fractions so that all have LCD as denominator
3) Add or subtract
4) Simplify
e.g.6 Subtract:
e.g.7 Add:
−3+
1 17
−
3 6
11
12
Ans. −
Ans. −
25
12
5
2
26
4.4 Adding and Subtracting Fractions
3. Find LCD to add and subtract fractions
•
•
The Least Common Multiple (LCM)
The Least Common Denominator (LCD)
–
–
–
Listing method
√ Prime factorization method (apply the algebraic fractions)
*Group factor method (math12)
e.g.8 Add and simplify:
e.g.9 Subtract and simplify:
1 5
+
8 6
13
3
−
24 40
Ans.
23
24
Ans.
7
15
27
4.4 Adding and Subtracting Fractions
4. Add and subtract algebraic fractions
e.g.10 Add or subtract, simplify if possible:
a.
2x 4x
+
15 15
2x
Ans.
5
b.
14
7
−
9d 9d
Ans.
7
9d
e.g.11 Add or subtract, simplify if possible:
a.
x 1
+
4 5
5x + 4
Ans.
20
b.
6 7
−
n 3
Ans.
18 − 7 n
3n
28
4.4 Adding and Subtracting Fractions
5. Identify the greater of two fractions (did this in math12)
e.g.12 Which fractions is larger:
Ans.
3
7
or
1
2
?
1
2
29
4.4 Adding and Subtracting Fractions
6. Solving application problems by adding and subtracting
fractions
4 or more hrs
1
30
3 hr 1 hr
1
12
2 hr
7
15
1
4
No TV
1
6
e.g.13 The pie chart shows how much time students are
watching TV daily in a college. Find the fraction of
student body that watches from 1 to 3 hrs of TV daily.
Ans.
4
5
30
4.5 Multiplying and Dividing Mixed Numbers
1. Identify the whole-number and fractional parts of a
mixed number
2. Write mixed numbers as improper fractions
3. Write improper fractions as mixed numbers
4. Graph fractions and mixed numbers on a number line
5. Multiply and divide mixed numbers
6. Solving application problems by multiplying and
dividing mixed numbers
31
4.5 Multiplying and Dividing Mixed Numbers
1. Identify the whole-number and fractional parts of a
mixed number
We did it in math12.
e.g.1 In the illustration on the right,
each square represents one whole.
Write an improper fraction and a mixed
number to represent the shaded portion.
Ans.
23
3
, 5
4
4
32
4.5 Multiplying and Dividing Mixed Numbers
2. Write mixed numbers as improper fractions
e.g.2 write
3
4
9
as an improper fraction.
Ans.
39
9
3. Write improper fractions as mixed numbers
e.g.3 write each improper fraction as a mixed number or a
whole number.
a. 319 Ans. 3 94
c. 813 Ans. 27
d. − 94 Ans. − 2 14
33
4.5 Multiplying and Dividing Mixed Numbers
4. Graph fractions and mixed numbers on a number line
e.g.4 Graph
3 8
1 9
, − ,
2 ,
4 3
8 10
−
−3
−2
−1
on a number line.
9
10
3
4
0
1
2
2
1 8
2
=2
8 3
3
3
34
4.5 Multiplying and Dividing Mixed Numbers
5. Multiply and divide mixed numbers (review)
Strategy: convert to improper fractions, then multiply or
divide.
e.g.5 Multiply and simplify, if possible:
a.
1 1
2 ⋅1
3 4
Ans. 2
11
12
b.
1 ⎛ 3⎞
3 ⎜5 ⎟
2⎝ 7⎠
c.
5
− 4 ( 3)
6
Ans. − 14
1
2
35
4.5 Multiplying and Dividing Mixed Numbers
5. Multiply and divide mixed numbers (review)
e.g.6 Multiply and simplify, if possible:
a.
4⎞
1 ⎛
−8 ÷ ⎜− 4 ⎟
9⎠
3 ⎝
Ans. 1
7
8
b.
2
11 5
÷
12 6
Ans. 3
1
2
36
4.5 Multiplying and Dividing Mixed Numbers
6. Solving application problems by multiplying and
dividing mixed numbers
e.g.7 KLEENEX When unfolded and laid flat, a standard
1
8
sheet of Kleenex brand tissue is 5 inches wide by
2
8 in long. Find the area of a sheet of Kleenex.
5
Ans. 6 8
22
in 2
25
37
4.5 Multiplying and Dividing Mixed Numbers
6. Solving application problems by multiplying and
dividing mixed numbers
e.g.8 HIGHWAYS It took workers 3 days to paint the
yellow centerline on a new 17 12 mile stretch of
highway. If they covered the same distance each day,
how much did they do in one day?
Ans. 5
5
mi
6
38
4.6 Adding and Subtracting Mixed Numbers
1. Add mixed number
2. Add mixed number in vertical form
3. Subtract mixed number
4. Solving application problems by adding and subtracting
mixed numbers
39
4.6 Adding and Subtracting Mixed Numbers
1. Add mixed number
Strategy 1: convert to improper fractions, then add or
subtract.
e.g.2
−3
2
1
+1
5
10
Ans. − 1
7
10
Strategy 2: when add two positive mixed number, add the
whole part and fraction part separately. Then
combine (good strategy for large numbers)
e.g.3 353 3 + 71 1 Ans. 424 34
7
9
63
40
4.6 Adding and Subtracting Mixed Numbers
2. Add mixed number in vertical form
Comments: for adding positive mixed numbers.
e.g.5 Add:
44
2
1
1
+ 91 + 27
15
5
3
Ans. 1 62
2
3
41
4.6 Adding and Subtracting Mixed Numbers
3. Subtract mixed number
You can always use the “converting to improper fraction”
strategy. However, for large numbers, the “treat whole
number part and fraction part separately” method has some
advantages.
Here we use the 2nd method (need minuend > subtrahend).
e.g.8 Subtract and simplify:
219
10
1
− 63
11
4
Ans. 1 5 5
15
44
42
4.6 Adding and Subtracting Mixed Numbers
3. Subtract mixed number
e.g.9 Subtract and simplify:
643 − 81
3
28
Ans. 5 61
25
28
43
4.6 Adding and Subtracting Mixed Numbers
4. Solving application problems by adding and subtracting
mixed numbers
e.g.10 BAKING A baker wants to make a cookie recipe
that calls for 1 18 cups of flour, a bread recipe that
3
3
calls for 4 cups of flour, and a cake recipe that calls
for 2 12 cups. How many cups of flour does she need
to make all recipes?
Ans. 7
3
cups
8
44
4.6 Adding and Subtracting Mixed Numbers
4. Solving application problems by adding and subtracting
mixed numbers
e.g.11 REFRESHMENT How much punch is left in a
20-gallon container if 9 12 gallons have been used?
Ans. 1 0
1
gallons
2
45
4.7 Order of Operations and Complex Fractions
1. Use the order of operations rule
2. Solving application problems by using the order of
operations rule
3. Evaluate formulas
4. Simplify complex fractions
46
4.7 Order of Operations and Complex Fractions
1. Use the order of operations rule
2(3) − 4
,
52 + 2
A comment. When evaluating fraction like
evaluate the expression above the fraction bar and the
expression below the fraction bar separately. Then perform
the division if possible.
e.g.1
1 5⎛ 1⎞
+ ⎜− ⎟
6 4 ⎝ 3⎠
2
Ans.
11
36
47
4.7 Order of Operations and Complex Fractions
1. Use the order of operations rule
e.g.2
7⎞
⎛ 29 1 ⎞ ⎛
− ⎟ ÷ ⎜−1 ⎟
⎜
8⎠
⎝ 32 8 ⎠ ⎝
e.g.3 Add
4
1
6
Ans. −
5
12
7
9
3
4
to the difference of and .
Ans. 4
7
36
48
4.7 Order of Operations and Complex Fractions
2. Solving application problems by using the order of
operations rule
e.g.4 MASONRY Find the height of a planter if 8 layers
1
2
(called courses) of 2 -inch-high bricks are held
together by 83 -inch-thick layers of mortar.
Ans. 23 inches
49
4.7 Order of Operations and Complex Fractions
3. Evaluate formulas
a
Trapezoid:
h
b
e.g.5 The formula for the area of a trapezoid is
3
1
1
1
2
4
Find A if h = 8 ft, a = 4 ft, and b = 4 ft.
Ans. A = 6
1
A = h ( a + b ).
2
3 2
ft
8
50
4.7 Order of Operations and Complex Fractions
4. Simplify complex fractions
Complex Fraction
A complex fraction is a fraction whose numerator and
denominator, or both, contain one or more fractions or
mixed numbers.
e.g.
− 14 −
2 25
4
5
Numerator
Main fraction bar
Denominator
1
3
1
3
+
−
1
4
1
4
51
4.7 Order of Operations and Complex Fractions
4. Simplify complex fractions
Main fraction bar means division, e.g.
1
3
1
3
e.g.6 Simplify:
e.g.7 Simplify:
+
−
1
4
1
4
1
8
2
7
− 89 + 43
5
1
6 − 9
=
( 13 + 14 ) ÷ ( 13 − 14 )
Ans.
7
16
Ans. −
5
26
52
4.7 Order of Operations and Complex Fractions
4. Simplify complex fractions
e.g.8 Simplify:
8 − 14
1 87
Ans.
62
15
53
4.8 Solving Equations That Involve Fractions
1. Use the addition and subtraction properties of equality
to solve equations that involve fractions
2. Use reciprocals to solve equations
3. Clear equations of fractions
4. Use equations to solve application problems that involve
fractions
54
4.8 Solving Equations That Involve Fractions
1. Use the addition and subtraction properties of equality
to solve equations that involve fractions
This is review.
e.g.1 Solve:
13
1
x−
=
18 18
Ans. x =
7
9
e.g.2 Solve:
7
1
=m+
8
3
Ans. m =
13
24
55
4.8 Solving Equations That Involve Fractions
2. Use reciprocals to solve equations
e.g.3 Solve and check the result:
a.
3
y = 15
5
b.
5
− y=3
8
Ans. y = −
5
24
3
e.g.4 Solve y = 15 using a two-step process.
5
Ans.
y = 25
56
4.8 Solving Equations That Involve Fractions
2. Use reciprocals to solve equations
1
23
e.g.5 Solve a =
and check the result.
6
36
Ans. a =
23
e.g.6 Solve 36 d = −
and check the result.
3
Ans. d = −
23
6
23
108
Lastly, we use both properties of equality.
e.g.7 Solve
8
n − 7 = − 63 and check the result.
9
Ans.
n = –63
57
4.8 Solving Equations That Involve Fractions
3. Clear equations of fractions
7
1
e.g.8 Solve = m + by first clearing the equation of
8
3
fractions.
Ans. m =
13
24
x x
e.g.9 Solve − = − 4 and check the result.
5 7
Ans. x = –70
3
8
7
y
−
=
y and check the result.
e.g.10 Solve
2
5 10
Ans. y = 2
58
4.8 Solving Equations That Involve Fractions
3. Clear equations of fractions
2) To 5) are from section 3.5
Strategy for Solving Equations (for detail read the book)
1) clear the equation of fractions (new)
2) simplify each side of equation
3) isolate the variable term on one side
4) isolate the variable
5) Check the result
59
4.8 Solving Equations That Involve Fractions
4. Use equations to solve application problems that involve
fractions
e.g.11 AMERICAN LITERATURE A student has read the
first 768 pages of the paperback version of the novel
Gone With The Wind. If that is three-fourths of the
book, how many pages are there in the paperback
version?
Ans. 1024 pages
60
4.8 Solving Equations That Involve Fractions
4. Use equations to solve application problems that involve
fractions
e.g.12 TRAFFIC TICKET DISMISSAL A traffic school
class has three parts. In the first part, a film is shown
that takes one-sixth of the class time. In the second
part, the instructor lectures for 105 minutes. In the
final part, a test is given that takes one-fourth of the
class time. How many minutes long is the traffic
school class?
Ans. 180 minutes
61