Download Chapter 4 FRACTION NOTATION: ADDITION

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Chapter 4 FRACTION NOTATION: ADDITION, SUBTRACTION,
AND MIXED NUMERALS
4.1 Least Common Multiples
Learning Objectives
A Find the least common multiple, or LCM, of two or more numbers.
Key Terms
Use the terms listed below to complete each statement in Exercises 1–2. One of the terms
will not be used.
larger
smaller
least common multiple
1.
The ___________________ of two natural numbers is the smallest number that is a
multiple of both.
2.
If one natural number is a multiple of another, then the ___________________ number
is the LCM of the two numbers.
GUIDED EXAMPLES AND PRACTICE
Objective A
Find the least common multiple, or LCM, of two or more numbers.
Review these examples for Objective A:
1. Find the LCM of 15 and 18 using a list of
multiples.
Practice these exercises:
1. Find the LCM of 12 and 16
using a list of multiples.
1. 18 is the larger number, but it is not a multiple
of 15.
2. Check multiples of 18:
2 ⋅18 = 36 Not a multiple of 15
3 ⋅18 = 54 Not a multiple of 15
4 ⋅18 = 72 Not a multiple of 15
5 ⋅18 = 90 A multiple of both 15 and 18
The LCM is 90.
2.
Find the LCM of 9 and 21 using prime
factorizations.
2.
Find the LCM of 8 and 20
using prime factorizations.
We write the prime factorization of each number
in exponential notation.
9 = 3 ⋅ 3 = 32
32 is the greatest power of 3
21 = 3 ⋅ 7 = 31 ⋅ 71 71 is the greatest power of 7
LCM = = 32 ⋅ 71 = 63
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67
ADDITIONAL EXERCISES
Objective A Find the least common multiple, or LCM, of two or more numbers.
For extra help, see Examples 1–12 on pages 220–224 of your text and the Section 4.1
lecture video.
Find the LCM of each set of numbers or expressions.
1.
10, 35
2.
18, 45
3.
9, 12, 30
4.
8, 12, 25
5.
x 2 y , y 3 z , xz 4
68
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Section:
Chapter 4 FRACTION NOTATION: ADDITION, SUBTRACTION,
AND MIXED NUMERALS
4.2 Addition, Order, and Applications
Learning Objectives
A Add using fraction notation when denominators are the same.
B Add using fraction notation when denominators are different.
C Use < or > to form a true statement with fraction notation.
D Solve problems involving addition with fraction notation.
Key Terms
Use the terms listed below to complete each statement in Exercises 1–2.
keep
add
To add fractions when the denominators are the same
1.
_________________ the numerators,
2.
_________________ the denominators, and if possible, simplify.
GUIDED EXAMPLES AND PRACTICE
Objective A Add using fraction notation when denominators are the same.
Review this example for Objective A:
1. Add and simplify: −3 + 7 .
8
8
Adding numerators;
−3 + 7 = −3 + 7 = 4 the denominators
8
8
8
8 remain the same
Simplify by removing a
= 4⋅1
factor of 1: 4 = 1
4 2
4
1
=
2
Practice this exercise:
1.
Add and simplify: 1 + 7 .
12 12
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69
Objective B
Add using fraction notation when denominators are different.
Review this example for Objective B:
2. Add and simplify: 2 + 1 .
9 6
Practice this exercise:
2.
Add and simplify: 3 + −3 .
4 10
Since 9 = 3 ⋅ 3 and 6 = 2 ⋅ 3 , the LCM of 9 and 6
is 2 ⋅ 3 ⋅ 3, or 18. Thus the LCD is 18.
We multiply by 1,
2+1 = 2⋅2+1⋅3
using 2 and 3 .
9 6 9 2 6 3
2
3
4
3
= +
18 18
= 7
No simplification is necessary.
18
Objective C Use < or > to form a true statement with fraction notation.
Review this example for Objective C:
3. Use < or > for
to form a true sentence:
3
5.
5
8
Practice this exercise:
3. Use < or > for
to form a
5.
true sentence: 2
3
9
3 ⋅ 8 = 24 ;
5 8 40
5 ⋅ 5 = 25
8 5 40
Since 24 < 25, it follows that 3 < 5 .
5 8
Objective D
Solve problems involving addition with fraction notation.
Review this example for Objective D:
4. Morton bought 1 lb of Vermont cheddar cheese
2
2
and lb of feta cheese. How many pounds of
3
cheese did he buy?
Practice this exercise:
4. Renza walked 3 mi to campus
4
3
and then mi to her part time
5
job. How far did she walk?
1. Familiarize. Let c = the number of pounds of
cheese Morton bought.
2. Translate.
Amount
Amount
Total
of
plus of
is
cheddar
feta
amount
↓
1
2
70
↓
+
↓
2
3
↓
↓
=
c
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3. Solve. We carry out the addition.
The LCM of the denominators is 6.
Multiplying by 1 to
1⋅3+ 2⋅2 =c
obtain the LCD
2 3 3 2
3+4 =c
6 6
7 =c
6
4. Check. As a partial check, note that the result
is larger than either of the individual amounts, so
the answer seems reasonable. We can also repeat
the calculation. The answer checks.
5. State. Morton bought 7 lb of cheese.
6
ADDITIONAL EXERCISES
Objective A Add using fraction notation when denominators are the same.
For extra help, see Examples 1–5 on pages 228–229 of your text and the Section 4.2 lecture
video.
Add, and if possible, simplify.
1.
3+ 9
8 8
2.
7 + −5
6
6
3.
1 x+ 5 x
7
7
4.
−4 + 9
a a
Objective B Add using fraction notation when denominators are different.
For extra help, see Examples 6–10 on pages 230–231 of your text and the Section 4.2
lecture video.
Add, and if possible, simplify.
5.
3+5
4 8
6.
4+0
9 7
7.
−5 x+ 3 x
6
10
8.
2+ 5 +1
3 12 6
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Objective C Use < or > to form a true statement with fraction notation.
For extra help, see Examples 11–12 on pages 231–232 of your text and the Section 4.2
lecture video.
Use < or > for
to form a true sentence.
9.
5
4
11. −5
9
3
2
10. 4
15
−7
10
12. − 7
6
3
10
1
12
Objective D Solve problems involving addition with fraction notation.
For extra help, see Example 13 on page 232 of your text and the Section 4.2 lecture video.
Solve.
13. Sally bought 1 lb of cheddar cheese
2
3
and lb of Swiss cheese. How
8
many pounds of cheese did she buy
altogether?
14. On Tuesday, Clark walked 7 mi,
8
5
9
mi. How
ran mi, and swam
6
10
many miles did he cover altogether?
15. Rodney bought 1 lb of Kona coffee and
4
9 lb of French Roast coffee. How
10
much coffee did he buy altogether?
72
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Section:
Chapter 4 FRACTION NOTATION: ADDITION, SUBTRACTION,
AND MIXED NUMERALS
4.3 Subtraction, Equation, and Applications
Learning Objectives
A Subtract using fraction notation.
B Solve equations of the type x + a = b and a + x = b, where a and b may be fractions.
C Solve applied problems involving subtraction with fraction notation.
GUIDED EXAMPLES AND PRACTICE
Objective A Subtract using fraction notation.
Review this example for Objective A:
1. Subtract and simplify, if possible: 2 − 1 .
3 4
Practice this exercise:
1.
Subtract and simplify: 3 − 4
8 5
The LCM of 3 and 4 is 12, so the LCD is 12.
We multiply by 1,
2−1 = 2⋅4−1⋅3
using 4 and 3 .
3 4 3 4 4 3
4
3
8
3
= −
12 12
= 5
No simplification is necessary.
12
Objective B
fractions.
Solve equations of the type x + a = b and a + x = b, where a and b may be
Review this example for Objective B:
2. Solve: x + 1 = 4 .
3 5
Practice this exercise:
2. Solve: x + 5 = 7 .
6 8
x+1 = 4
3 5
Using the addition principle:
adding − 13 to, or subtracting
1
from, both sides
3
Multiplying by 1 to
x +0 = 4 ⋅ 3− 1⋅ 5
obtain
the LCD, 15
5 3 3 5
x = 12 − 5 = 7
15 15 15
The solution is 7 .
15
x+ 1−1 = 4−1
3 3 5 3
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Objective C
Solve applied problems involving subtraction with fraction notation.
Review this example for Objective C:
3. Bert spent 7 hr doing his chemistry and English
4
assignments. He spent 5 hr on the chemistry
6
assignment. How long did he spend on the
English assignment?
Practice this exercise:
3. Mary has walked 3 mi and
4
will stop walking when she has
walked 9 mi. How much
8
farther does she have to walk?
1. Familiarize. Let t = the number hours Bert
spent on his English assignment.
2. Translate. This is a “how much more”
situation.
Chemistry
English Total
plus
is
time
time
time
N
↓
5
6
↓
↓
↓
+
t
=
↓
7
4
3. Solve. We subtract 5 on both sides of the
6
equation.
5 +t = 7
6
4
5 +t− 5 = 7 − 5
6
6 4 6
Multiplying by 1 to
t +0 = 7 ⋅3− 5⋅ 2
obtain
the LCD, 12
4 3 6 2
t = 21 − 10 = 11
12 12 12
4. Check. We return to the original problem and
add:
5 + 11 = 5 ⋅ 2 + 11
6 12 6 2 12
= 10 + 11
12 12
= 21 = 3 ⋅ 7
12 3 4
= 7.
4
The answer checks.
5. State. Bert spent 11 hr on his English
12
assignment.
74
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ADDITIONAL EXERCISES
Objective A Subtract using fraction notation.
For extra help, see Examples 1–8 on pages 237–238 of your text and the Section 4.3 lecture
video.
Subtract, and if possible, simplify.
1.
7−2
9 9
2.
15 − 7
16 16
3.
3− 5
8 6
4.
1−1
8 3
5.
3 x− 7 x
4
12
Objective B Solve equations of the type x + a = b and a + x = b, where a and b may be
fractions.
For extra help, see Examples 9–10 on page 239 of your text and the Section 4.3 lecture
video.
Solve.
6.
t+1 = 5
3 6
7.
7+x=1
4
5
8.
x− 5 = 1
4 4
9.
n+ 3 =−3
10
8
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Objective C Solve applied problems involving subtraction with fraction notation.
For extra help, see Example 11 on page 240 of your text and the Section 4.3 lecture video.
Solve.
10. A 2 -cup mixture of cinnamon and sugar 11. Chris has 1 gal of peppermint ice cream
3
2
contains 1 cup of cinnamon. How much
and serves 5 gal to guests at a dinner
4
16
sugar is in the mixture?
party. How many gallons remain?
12. Charlie is walking 9 mi to the library
10
3
and has mi left to walk. How far has
8
he walked already?
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Section:
Chapter 4 FRACTION NOTATION: ADDITION, SUBTRACTION,
AND MIXED NUMERALS
4.4 Solving Equations: Using the Principles Together
Learning Objectives
A Solve equations that involve fractions and require use of both the addition principle and
the multiplication principle.
B Solve equations by using the multiplication principle to clear fractions.
GUIDED EXAMPLES AND PRACTICE
Objective A Solve equations that involve fractions and require use of both the
addition principle and the multiplication principle.
Review this example for Objective A:
1. Solve: 3 x − 1 = 3 .
4
7 7
Practice this exercise:
1. Solve: 5 x − 1 = 7 .
8
3 3
3 x−1 = 3
4
7 7
3 x− 1 + 1 = 3 + 1
Adding 1 to both sides
4
7 7 7 7
7
Note that the reciprocal of
3x=4
3
is 43 .
4
7
4
4 ⋅ 3 x = 4 ⋅ 4 Multiplying both sides by 4
3 4
3 7
3
16
x=
21
The solution is 16 .
21
Objective B
Solve equations by using the multiplication principle to clear fractions.
Review this example for Objective B:
2. Solve by using the multiplication principle to
clear fractions: 1 x − 1 = 1 .
3
6 2
The LCD is 6, so we begin by multiplying both
sides of the equation by 6.
1 x− 1 = 1
3
6 2
1
1
6 x − = 6 ⋅ 1 Multiplying both sides by 6
3
6
2
(
Practice this exercise:
2. Solve by using the
multiplication principle to clear
fractions: 3 x − 1 = 7 .
8
4 8
)
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6⋅ 1 x −6⋅ 1 = 6⋅ 1
3
6
2
6 ⋅1 2 ⋅ 3⋅1
2 ⋅ 3 ⋅1
=
x−
2
3
6
2 x −1 = 3
2 x −1 + 1 = 3 + 1
2x = 4
2x = 4
2
2
x=2
The solution is 2.
Using the distributive law
Factoring and simplifying
Fractions are cleared.
Adding 1 to both sides
Dividing both sides by 2
Simplifying
ADDITIONAL EXERCISES
Objective A Solve equations that involve fractions and require use of both the
addition principle and the multiplication principle.
For extra help, see Examples 1–3 on pages 246–248 of your text and the Section 4.4 lecture
video.
Solve.
1.
5x − 9 = 6
2.
7− 2 a = 3
5
3.
15 = y + 9
5
4.
15 x − 3 = 7
8
4 12
Objective B Solve equations by using the multiplication principle to clear fractions.
For extra help, see Examples 4–5 on page 249 of your text and the Section 4.4 lecture video.
Solve.
5.
−4 = 5 x + 1
12
6.
1 x+ 3 = 7
4
8 8
7.
−9 =− 7 − 7 s
8
20 4
8.
7−2t= 4
2 5
3
78
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Section:
Chapter 4 FRACTION NOTATION: ADDITION, SUBTRACTION,
AND MIXED NUMERALS
4.5 Mixed Numerals
Learning Objectives
A Convert between mixed numerals and fraction notation.
B Divide, writing the quotient as a mixed numeral.
Key Terms
Use the terms listed below to complete the statement in Exercise 1.
fraction
1.
mixed numeral
To convert from a _________________ to a _________________ , divide.
GUIDED EXAMPLES AND PRACTICE
Objective A Convert between mixed numerals and fraction notation.
Review these examples for Objective A:
1. Convert 5 3 to fraction notation.
8
5 3 = 43
5 ⋅ 8 = 40, 40 + 3 = 43
8
8
2.
Convert 13 to a mixed numeral.
3
4
13 = 4 1
3 13
3
3
12
1
Objective B
Practice these exercises:
1. Convert 3 4 to fraction
5
notation.
2.
Convert 11 to a mixed
6
numeral.
Divide, writing the quotient as a mixed numeral.
Review this example for Objective B:
3. Divide. Write a mixed numeral for the quotient.
5 2367
We first divide as usual.
473
The answer is 473 R 2, or 473 2 .
5 2367
5
20
36
35
17
15
2
Practice this exercise:
3. Divide. Write a mixed numeral
for the quotient.
7 4115
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ADDITIONAL EXERCISES
Objective A Convert between mixed numerals and fraction notation.
For extra help, see Examples 1–12 on pages 252–255 of your text and the Section 4.5
lecture video.
Convert to fraction notation.
1.
43
5
2.
−3 5
9
4.
−47
6
Convert to a mixed numeral.
3.
37
15
Objective B Divide, writing the quotient as a mixed numeral.
For extra help, see Examples 13–14 page 255 of your text and the Section 4.5 lecture video.
Divide. Write a mixed numeral for the answer.
5.
6.
4 4229
313 ÷ (−16)
Find the average of the numbers. Write a mixed numeral for the answer.
7.
80
32, 39, 46, 50
8.
15, 16, 25, 31, 47
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Section:
Chapter 4 FRACTION NOTATION: ADDITION, SUBTRACTION,
AND MIXED NUMERALS
4.6 Addition and Subtraction of Mixed Numerals; Applications
Learning Objectives
A Add using mixed numerals.
B Subtract using mixed numerals.
C Solve applied problems involving addition and subtraction with mixed numerals.
D Add and subtract using negative mixed numerals.
Key Terms
Use the terms listed below to complete the statement in Exercise 1.
whole numbers
1.
fractions
To add mixed numerals, first add the _________________ ; then add the
_________________ , and if possible, simplify the fraction part.
GUIDED EXAMPLES AND PRACTICE
Objective A Add using mixed numerals.
Review this example for Objective A:
1. Add: 3 5 + 4 1 .
8
2
Practice this exercise:
1. Add: 5 2 + 1 3 .
3
4
35
35
=
8
8
4
1
4
+4 ⋅ = +4
8
2 4
7 9 = 7+ 9
8
8
= 7 +11
8
1
=8
8
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Objective B
Subtract using mixed numerals.
Review this example for Objective B:
2. Subtract: 6 1 − 4 1 .
3
2
3
2
1
2
6 ⋅ =
6 2 To subtract 6 from 6 ,
3 2
6 we borrow 1, or 6 from 6:
6
1
3
− 4 ⋅ = − 4 3 6 2 = 5 +1+ 2 = 5 + 6 + 2
6
6
6
6
2 3
6
8
=56
We can write this as
62 =
58
6
6
3
− 4 = − 43
6
6
15
6
Practice this exercise:
2. Subtract: 9 3 − 3 3 .
8
4
Objective C Solve applied problems involving addition and subtraction with mixed
numerals.
Review this example for Objective C:
3. Melanie bought 1 1 lb of apples and 2 3 lb of
2
4
pears. What was the total weight of the fruit?
1. Familiarize. Let w = the total weight of the
fruit, in pounds.
2. Translate.
Weight
Weight
Total
plus
is
of apples
of pears weight
Practice this exercise:
3. Sam is 73 1 in. tall, and Ray is
4
70 1 in. tall. How much taller
2
is Sam?
↓
↓
↓
↓
↓
1
3
1
2
+
= w
2
4
3. Solve. The translation tells us what to do. We
add. The LCD is 4.
11 ⋅ 2 =
12
2 2
4
3
3
+2
= +2
4
4
35 = 41
4
4
4. Check. We repeat the calculation. The answer
checks.
5. State. The total weight of the fruit was 4 1 lb.
4
82
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Objective D Add and subtract using negative mixed numerals.
Review this example for Objective D:
4. Subtract: 1 1 − 3 1 .
5
2
Practice this exercise:
4. Subtract: −2 1 − 3 3 .
3
4
We rewrite the subtraction as addition:
1 1 + −3 1 . The difference in absolute values is
5
2
31 ⋅5 = 3 5
2 5
10
1
2
−1 ⋅ = −1 2
5 2
10
2 3
10
Since −3 1 has the larger absolute value, we
2
make the answer negative.
Thus, 1 1 − 3 1 = −2 3 .
5
2
10
( )
ADDITIONAL EXERCISES
Objective A Add using mixed numerals.
For extra help, see Examples 1–2 on page 260 of your text and the Section 4.6 lecture video.
Add. Write a mixed numeral for each answer.
1.
85
7
+ 23
7
2.
12 3
8
+ 37
12
3.
55
6
+ 11 5
8
4.
Combine like terms: 3 4 x + 2 3 x .
5
10
Objective B Subtract using mixed numerals.
For extra help, see Examples 3–7 on pages 261–262 of your text and the Section 4.6 lecture
video.
Subtract. Write a mixed numeral for each answer.
5.
72
5
4
−3
5
6.
45 5
12
− 13 7
8
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83
7.
22 1
6
− 18 5
9
8.
Combine like terms: 32 7 x − 18 5 x .
12
6
Objective C Solve applied problems involving addition and subtraction with mixed
numerals.
For extra help, see Examples 8–9 on pages 262–263 of your text and the Section 4.6 lecture
video.
Solve.
9.
A plumber uses pipes of lengths 5 5 ft
8
3
and 8 ft in the installation of a spa.
4
How much pipe was used?
10. A 7 2 -ft pole is set 1 7 ft into the
3
8
ground. How many feet of the pole are
above ground?
11. Loni worked 14 1 hr during a two-day
3
period. If she worked 8 1 hr the first
4
day, how long did she work the second
day?
Objective D Add and subtract using negative mixed numerals.
For extra help, see Examples 10–12 on pages 264–265 of your text and the Section 4.6
lecture video.
Subtract. Write a mixed numeral for each answer.
12. 4 3 − 8 1
4
4
13. 7 5 − 10 1
6
3
14. −5 7 − 1 9
10
10
15. −6 1 − −4 5
7
7
84
( )
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Section:
Chapter 4 FRACTION NOTATION: ADDITION, SUBTRACTION,
AND MIXED NUMERALS
4.7 Multiplication and Division of Mixed Numerals; Applications
Learning Objectives
A Multiply using mixed numerals.
B Divide using mixed numerals.
C Evaluate expressions using mixed numerals.
D Solve applied problems involving multiplication and division with mixed numerals.
Key Terms
Use the terms listed below to complete the statement in Exercise 1.
fraction notation
1.
mixed numerals
To multiply using _________________ , first convert to _________________ .
GUIDED EXAMPLES AND PRACTICE
Objective A
Multiply using mixed numerals.
Review this example for Objective A:
1. Multiply: 1 3 ⋅ 5 2 .
8 3
Practice this exercise:
1. Multiply: 6 2 ⋅ 2 3 .
5 4
3 ⋅ 5 ⋅17
1 7 ⋅ 5 2 = 15 ⋅ 17 = 15 ⋅17 =
8 3 8 3
8⋅3
2⋅2⋅2⋅ 3
85
5
= = 10
8
8
Objective B
Divide using mixed numerals.
Review this example for Objective B:
2. Divide: 4 2 ÷ 2 2 .
3
5
Practice this exercise:
2. Divide: 9 1 ÷ 3 1 .
8
4
4 2 ÷ 2 2 = 14 ÷ 12 = 14 ⋅ 5
3
5
3
5
3 12
2 ⋅7 ⋅5
= 14 ⋅ 5 =
3 ⋅12 3 ⋅ 2 ⋅ 2 ⋅ 3
35
=
= 1 17
18
18
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85
Objective C Evaluate expressions using mixed numerals.
Review this examples for Objective C:
3. Evaluate a − bc for a = 1 1 , b = 4 1 and c = 2 .
3
2
a − bc = 1 1 − 4 1 ⋅ 2
3
2
1
9
= 1 − ⋅ 2 Multiply first; then add.
3 2 1
1
=1 − 9⋅2
3 2 ⋅1
1
=1 −9
3
= −7 2
3
Objective D
numerals.
Practice this exercise:
3. Evaluate wz
for w = −4 2 and z = 5 1 .
3
4
Solve applied problems involving multiplication and division with mixed
Review this example for Objective D:
4. A cookie recipe calls for 1 3 cups of sugar. How
4
much sugar is needed to double the recipe?
1. Familiarize. Let s = the number of cups of
sugar in two recipes.
2. Translate. The situation translates to the
multiplication sentence
c = 2 ⋅1 3 .
4
3. Solve. We carry out the multiplication.
c = 2 ⋅1 3 = 2 ⋅ 7
4
4
2 ⋅7
⋅
2
7
=
=
4
2 ⋅2
7
1
= =3
2
2
Practice this exercise:
4.
A car travels 276 mi on 11 5
10
gal of gas. How many miles per
gallon did it get?
4. Check. We can do a partial check by
estimating: 2 ⋅1 3 ≈ 2 ⋅ 2 = 4 ≈ 3 1 . We can also
4
2
repeat the calculation. The answer checks.
5. State. To double the recipe, 3 1 cups of sugar
2
are needed?
86
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Date:
Section:
ADDITIONAL EXERCISES
Objective A Multiply using mixed numerals.
For extra help, see Examples 1–4 on page 271 of your text and the Section 4.7 lecture video.
Multiply. Write a mixed numeral for each answer whenever possible.
1.
6⋅3 1
4
2.
4 3⋅2 2
5 3
3.
−8 2 ⋅ 3 2
9 5
4.
−4 7 −5 1
8
3
( )
Objective B Divide using mixed numerals.
For extra help, see Examples 5–7 on page 272 of your text and the Section 4.7 lecture video.
Divide. Write a mixed numeral for each answer whenever possible.
5.
12 ÷ 7 1
2
6.
6 2 ÷5
3
7.
−4 1 ÷ 5 1
5
3
8.
10 1 ÷ (−7)
2
Objective C Evaluate expressions using mixed numerals.
For extra help, see Examples 8–9 on page 273 of your text and the Section 4.7 lecture video.
Evaluate.
9.
ab, for a = 4 3 and b = 20
5
10. x − yz, for x = 12 3 , y = 2, and z = 4 2
4
5
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87
11. a ⋅ b ÷ c,
for a = 8 7 , b = −3, and c = 2 2
8
3
12. M + N ⋅ P,
for M = 4 1 , N = 3 3 , and P = 5
6
8
Objective D Solve applied problems involving multiplication and division with mixed
numerals.
For extra help, see Examples 10–11 on pages 274–276 of your text and the Section 4.7
lecture video.
Solve.
13. The weight of water is 62 1 lb per cubic
2
foot. What is the weight of 3 3 cubic
4
feet of water?
15. A satellite orbits Earth every 1 1 hr.
2
How many orbits are made every 24 hr?
88
14. A carton of textbooks weighs 84 lb. If
each book weighs 3 1 lb, how many
2
books are in the carton?
16. Find the area of a rectangle with length
6 1 ft and width 3 3 ft.
3
4
Copyright © 2012 Pearson Education, Inc. Publishing as Addison-Wesley
Name:
Instructor:
Date:
Section:
Chapter 4 FRACTION NOTATION: ADDITION, SUBTRACTION,
AND MIXED NUMERALS
4.8 Order of Operations and Complex Fractions
Learning Objectives
A Simplify expressions containing fraction notation using the rules for order of operations.
B Simplify complex fractions.
GUIDED EXAMPLES AND PRACTICE
Objective A Simplify expressions containing fraction notation using the rules for
order of operations.
Review this example for Objective A:
1. Simplify: 5 ÷ 6 − 2 − 4 .
8 7
3 9
5 ÷ 5− 2−4
9 7
3 9
Writing with the LCD,
= 5 ÷ 5− 6−4
9,
in order to subtract
9 7 9 9
= 5 ÷ 5 − 2 Subtracting within the parentheses
9 7 9
Doing the division by multiplying
= 5⋅7−2
by
the reciprocal of 57
9 5 9
5 ⋅7 2
−
=
Multiplying and factoring
9⋅ 5 9
= 7−2
Simplifying
9 9
=5
Subtracting
9
(
(
Objective B
(
)
)
Practice this exercise:
2
1. Simplify: 3 ÷ 1 − 1 .
8 4
2
()
)
Simplify complex fractions.
Review this example for Objective B:
8
2. Simplify: 9 .
x
12
8
Rewriting using a
9 =8÷ x
x
division symbol
9 12
12
Practice this exercise:
5
2. Simplify: 36 .
7
9
Copyright © 2012 Pearson Education, Inc. Publishing as Addison-Wesley
89
Multiplying by the
= 8 ⋅ 12
reciprocal of the divisor
9 x
2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3 Multiplying numerators
=
and denominators; factoring
3 ⋅ 3⋅ x
Removing a factor equal
= 32
to 1 and simplifying
3x
ADDITIONAL EXERCISES
Objective A Simplify expressions containing fraction notation using the rules for
order of operations.
For extra help, see Examples 1–3 on pages 284–285 of your text and the Section 4.8 lecture
video.
Simplify.
( )
1.
1 − 2 −5
4 5 8
3.
− 11 4 − 3
15 5 10
(
)
2.
5÷ 2− 5 ⋅4
9 3 12 5
4.
( ) (
2
2
−1 −4 2 − 1
3
9 3
)
ADDITIONAL EXERCISES
Objective B Simplify complex fractions.
For extra help, see Examples 4–7 on pages 286–287 of your text and the Section 4.8 lecture
video.
Simplify.
5.
4
11
3
11
7.
3
8
−2 1
4
90
6.
−5
7
35
8.
1−1
4 5
1
2x
Copyright © 2012 Pearson Education, Inc. Publishing as Addison-Wesley