Chapter 3 FRACTION NOTATION: MULTIPLICATION AND DIVISION

Name: Date: Instructor: Section:

Copyright © 2012 Pearson Education, Inc. Publishing as Addison-Wesley 45

Chapter 3 FRACTION NOTATION: MULTIPLICATION AND DIVISION 3.1 Multiples and Divisibility

Learning Objectives A Find some multiples of a number and determine whether a number is divisible by

another number. B Test to see if a number is divisible by 2, 3, 5, 6, 9, or 10. Key Terms Use the vocabulary terms listed below to complete each statement in Exercises 1–2.

divisible by another number multiple of a number

1. A number b is said to be ___________________ if b is a multiple of a.

2. A ___________________ is a product of that number and an integer. GUIDED EXAMPLES AND PRACTICE Objective A Find some multiples of a number and determine whether a number is divisible by another number.

Review these examples for Objective A: 1. Multiply by 1, 2, 3, and so on to find ten multiples

of 8.

1 8 8 6 8 482 8 16 7 8 563 8 24 8 8 644 8 32 9 8 725 8 40 10 8 80

⋅ = ⋅ =⋅ = ⋅ =⋅ = ⋅ =⋅ = ⋅ =⋅ = ⋅ =

Practice these exercises: 1. Multiply by 1, 2, 3, and so on

to find ten multiples of 13.

2. Determine whether 86 is divisible by 2 and by 4.

43 212 86 4 86

8 8 6 6 6 4 0 2

Since the remainder is 0 when 86 is divided by 2, 86 is divisible by 2. When 86 is divided by 4, the remainder is not 0, so 86 is not divisible by 4.

2. Determine whether 188 is divisible by 8.

46 Copyright © 2012 Pearson Education, Inc. Publishing as Addison-Wesley

Objective B Test to see if a number is divisible by 2, 3, 5, 6, 9, or 10.

Review this example for Objective B: 3. Determine whether 56,340 is divisible by 2, 3, 5,

6, 9, or 10.

The ones digit, 0, is even so 56,340 is divisible by 2. 5 + 6 + 3 + 4 + 0 = 18 and 18 is divisible by 3, so 56,340 is divisible by 3. The ones digit is 0, so 56,340 is divisible by 5. The ones digit is even and the sum of the digits, 18, is divisible by 3, so 56,340 is divisible by 6. The sum of the digits, 18, is divisible by 9, so 56,340 is divisible by 9. The ones digit is 0, so 56,340 is divisible by 10.

Practice this exercise: 3. Determine whether 18,225 is

divisible by 2, 3, 5, 6, 9, or 10.

ADDITIONAL EXERCISES Objective A Find some multiples of a number and determine whether a number is divisible by another number. For extra help, see Examples 1–3 on pages 152–153 of your text and the Section 3.1 lecture video. Multiply by 1, 2, 3, and so on, to find ten multiples of each number.

1. 5 2. 9



Objective B Test to see if a number is divisible by 2, 3, 5, 6, 9, or 10. For extra help, see Examples 4–21 on pages 153–155 of your text and the Section 3.1 lecture video. Test each number for divisibility by 2, 3, 5, 6, 9, and 10.

5. 732 6. 1845

Consider the following numbers. Use the tests for divisibility. 57 171 95 487 48 7317 5001

7. Which of the above are divisible by 3? 8. Which of the above are divisible by 5?



Chapter 3 FRACTION NOTATION: MULTIPLICATION AND DIVISION 3.2 Factorizations

Learning Objectives A Find the factors of a number. B Given a number from 1 to 100, tell whether it is prime, composite or neither. C Find the prime factorization of a composite number. Key Terms Use the vocabulary terms listed below to complete each statement in Exercises 1–2.

factor factorization

1. The _________________ of a expresses a as a product of two or more numbers.

2. A number c is a _________________ of a if a is divisible by c. GUIDED EXAMPLES AND PRACTICE Objective A Find the factors of a number.

Review this example for Objective A: 1. List all the factors of 36.

36 1 36 36 4 936 2 18 36 6 636 3 12

= ⋅ = ⋅= ⋅ = ⋅= ⋅

Factors: 1, 2, 3, 4, 6, 9, 12, 18, 36

Practice this exercise: 1. List all the factors of 20.

Objective B Given a number from 1 to 100, tell whether it is prime, composite or neither.

Review this example for Objective B: 2. Classify each of the numbers 1, 19, and 24 as

prime, composite, or neither.

1 does not have two different factors. It is neither prime nor composite.

19 has only the factors 1 and 19. It is prime.

24 has more than two different factors. It is composite.

Practice this exercise: 2. Classify 57 as prime,

composite, or neither.


Objective C Find the prime factorization of a composite number.

Review this example for Objective C: 3. Find the prime factorization of 60.

We can use a string of divisions or a factor tree.

53 15

2 30

2 60

60 4 15

2 2 3 5

60 2 2 3 5= ⋅ ⋅ ⋅

Practice this exercise: 3. Find the prime factorization of

63.

ADDITIONAL EXERCISES Objective A Find the factors of a number. For extra help, see Example 1 on page 159 of your text and the Section 3.2 lecture video. List all the factors of each number.

1. 32 2. 60

3. 28 4. 50

Objective B Given a number from 1 to 100, tell whether it is prime, composite or neither. For extra help, see Example 2 on page 160 of your text and the Section 3.2 lecture video. Classify each number as prime, composite, or neither.

5. 15 6. 38

7. 37 8. 1

Objective C Find the prime factorization of a composite number. For extra help, see Examples 3–7 on pages 161–162 of your text and the Section 3.2 lecture video. Find the prime factorization of each number.

9. 48 10. 54

11. 270 12. 66



Chapter 3 FRACTION NOTATION: MULTIPLICATION AND DIVISION 3.3 Fractions and Fraction Notation

Learning Objectives A Identify the numerator and the denominator of a fraction and write fraction notation for

part of an object or part of a set of objects as a ratio. B Simplify fraction notation like n/n to 1, 0/n to 0, and n/1 to n. Key Terms Use the vocabulary terms listed below to complete each statement in Exercises 1–5.

denominator numerator ratio zero undefined

1. A _________________ is a quotient of two quantities.

2. The top number in a fraction is the _________________ .

3. We say that 0x

is _________________ .

4. The bottom number in a fraction is the _________________ .

5. We say that 0x is _________________ .

GUIDED EXAMPLES AND PRACTICE Objective A Identify the numerator and the denominator of a fraction and write fraction notation for part of an object or part of a set of objects as a ratio.

Review these examples for Objective A:

1. Identify the numerator and denominator: 7 .12

The top number, 7, is the numerator; the bottom number, 12, is the denominator.

Practice these exercises: 1. Identify the numerator and

denominator: 5 .6−

2. What part is shaded?

The object is divided into 6 parts of the same size

and 4 of them are shaded. Thus, 46

of the object is

shaded.

2. What part is shaded?


3. In a class of 31 students, 17 are men. What is the ratio of men to the total number of students?

There are 31 people in the class and 17 are men,

so the desired ratio is 17 .31

3. In a bouquet of 15 tulips, 4 are white. Find the ratio of white tulips to the total number of tulips.

Objective B Simplify fraction notation like n/n to 1, 0/n to 0, and n/1 to n.

Review this example for Objective B:

4. Simplify: 6 0 3, , and .6 10 1

6 0 31, 0, 36 10 1= = =

Practice this exercise:

4. Simplify: 5 12 0, , and .1 12 2

ADDITIONAL EXERCISES Objective A Identify the numerator and the denominator of a fraction and write fraction notation for part of an object or part of a set of objects as a ratio. For extra help, see Examples 1–8 on pages 165–168 of your text and the Section 3.3 lecture video.

1. Identify the numerator and denominator: 2 .9ab

What part of the object or set of objects is shaded?

2.

3.

Consider a group of 5 women and 6 men.

4. What is the ratio of women to the total number of people in the group?

5. What is the ratio of men to women?

Objective B Simplify fraction notation like n/n to 1, 0/n to 0, and n/1 to n. For extra help, see Examples 9–18 on page 169 of your text and the Section 3.3 lecture video. Simplify. Assume that all variables are nonzero.

6. 019

7. 91

x−

8. 70

9. 41a



Chapter 3 FRACTION NOTATION: MULTIPLICATION AND DIVISION 3.4 Multiplication and Applications

Learning Objectives A Multiply an integer and a fraction. B Multiply using fraction notation. C Solve problems involving multiplication of fractions. GUIDED EXAMPLES AND PRACTICE Objective A Multiply an integer and a fraction.

Review this example for Objective A:

1. Multiply: 24 .5

− ×

2 4 2 8 84 , or 5 5 5 5

− × −− × = = −


1. Multiply: 36 .7

×

Objective B Multiply using fraction notation.


2. Multiply: 3 5 .4 2⋅

3 5 3 5 154 2 4 2 8

⋅⋅ = =⋅


2. Multiply: ( )5 7 .8 6

− ×

Objective C Solve problems involving multiplication of fractions.

Review this example for Objective C:

3. A rectangular rug is 76

m long and 56

m wide.

What is the area?

1. Familiarize. We make a drawing.

5 m 6

7 m6

Recall that the area of a rectangle is length times width. Let A = the area of the rug.


3. A cookie recipe calls for 12

cup of brown sugar. How much brown sugar is needed to make 12

of a recipe?


2. Translate. We translate to an equation.

Area is length times width

7 56 6

A

↓ ↓ ↓ ↓ ↓

= ×

3. Solve. We multiply.

7 5 7 5 356 6 6 6 36

A ×= × = =×

4. Check. We repeat the calculation. The answer checks.

5. State. The area of the rug is 235 m .36

ADDITIONAL EXERCISES Objective A Multiply an integer and a fraction. For extra help, see Examples 1–4 on page 175 of your text and the Section 3.4 lecture video. Multiply.

1. ( ) 123

− × 2. 5 56⋅

3. ( )5 29

− 4. 473−− ⋅

5. 58

x ⋅

Objective B Multiply using fraction notation. For extra help, see Examples 5–8 on pages 176–177 of your text and the Section 3.4 lecture video. Multiply.

6. ( )1 13 7

− × 7. 6 57 11⋅



8. ( )( )3 34 8

− − 9. 1 27b⋅

10. 13 517 9− ⋅

Objective C Solve problems involving multiplication of fractions. For extra help, see Examples 9–11 on pages 177–178 of your text and the Section 3.4 lecture video. Solve.

11. A recipe calls for 3 cups of flour. How

much flour is needed to make 12

of the

recipe?

12. Three-fourths of Sean’s first grade students attend an after-school program. Of these, five-eighths are girls. What fractional part of Sean’s students are girls who attend an after-school program?

13. It takes 13

yd of fabric to make a napkin.

How much fabric is needed to make 8 napkins?




Chapter 3 FRACTION NOTATION: MULTIPLICATION AND DIVISION 3.5 Simplifying

Learning Objectives A Multiply by 1 to find an equivalent expression using a different denominator. B Simplify fraction notation. C Test to determine whether two fractions are equivalent. Key Terms Use the vocabulary terms listed below to complete each statement in Exercises 1–2.

cross products equivalent fractions

1. To test two fractions for equality, we compare _________________ .

2. Two fractions that name the same number are _________________ . GUIDED EXAMPLES AND PRACTICE Objective A Multiply by 1 to find an equivalent expression using a different denominator.


1. Find a number equivalent to 23

with a

denominator of 12.

Since 12 3 4,÷ = we multiply by 44

:

2 2 4 2 4 83 3 4 3 4 12

⋅= ⋅ = =⋅


1. Find a number equivalent to 34

with a denominator of 20.

Objective B Simplify fraction notation.


2. Simplify: 16 .36

−

16 4 4 4 4 4 4136 9 4 9 4 9 9

4 Removing a factor equal to 1: 14

⋅− =− =− ⋅ =− ⋅ =−⋅

=


2. Simplify: 9 .24

−


Objective C Test to determine whether two fractions are equivalent.

Review this example for Objective C: 3. Use = or ≠ for to write a true sentence:

5 7 .7 10

We multiply these We multiply thesetwo numbers: two numbers: 5 10 50 7 7 49⋅ = ⋅ =

5 7 7 10

Since 5 750 49, .7 10

≠ ≠

Practice this exercise: 3. Use = or ≠ for to write

a true sentence: 4 12 .5 15

ADDITIONAL EXERCISES Objective A Multiply by 1 to find an equivalent expression using a different denominator. For extra help, see Examples 1–3 on pages 181–182 of your text and the Section 3.5 lecture video. Find an equivalent expression for each number, using the denominator indicated. Use multiplication by 1.

1. 2 ?3 12= 2. 7 ?

15 45− =−

3. 11 ?21 147− = 4. 10 ?

3 21a=

Objective B Simplify fraction notation. For extra help, see Examples 4–11 on pages 182–183 of your text and the Section 3.5 lecture video. Simplify.

5. 618− 6. 56

7−

7. 250325

8. 64aba

Objective C Test to determine whether two fractions are equivalent. For extra help, see Examples 12–14 on page 185 of your text and the Section 3.5 lecture video.

Use = or ≠ for to write a true sentence.

9. 4 32 9 56

10. 3 12 2 9

11. 5 25 8 40− − 12. 10 12

15 18−−



Chapter 3 FRACTION NOTATION: MULTIPLICATION AND DIVISION 3.6 Multiplying, Simplifying, and More with Area

Learning Objectives A Multiply and simplify using fraction notation. B Solve applied problems involving multiplication. Key Terms Use the terms listed below to complete each statement in Exercises 1–2.

find the area of a triangle simplify a fraction

1. We _________________ by removing a factor equal to 1.

2. We _________________ by finding half the length of the based times the height. GUIDED EXAMPLES AND PRACTICE Objective A Multiply and simplify using fraction notation.


1. Multiply and simplify: 3 2 .4 9⋅

3 2 3 2 3 2 14 9 4 9 2 2 3 3

Removing a factor3 2 1 11 3 2equal to 1: 13 2 2 3 2 3

3 21 1

2 3 6

⋅ ⋅ ⋅⋅ = =⋅ ⋅ ⋅ ⋅

⋅= ⋅ = ⋅ ⋅ =⋅ ⋅ ⋅ ⋅= =

⋅


1. Multiply and simplify: 5 4 .6 15

−⋅

Objective B Solve applied problems involving multiplication.

Review this example for Objective B: 2. On a map, 1 in. represents 150 miles. How much

does 35

in. represent?

1. Familiarize. Let m = the number of miles

represented by 35

in.

Practice this exercise: 2. Kim earns $56 for working a

full day. How much does she

earn for working 34

of a day?


2. Translate. The problem translates to the

following equation: 3 150.5

m = ⋅

3. Solve. We carry out the multiplication.

3 3 1501505 53 5 30 5 3 30

5 1 5 190

m ⋅= ⋅ =

⋅ ⋅ ⋅= = ⋅⋅

=

4. Check. We can repeat the calculation. We can also do a partial check by thinking about the

reasonableness of the answer. Since 35

in. is less

than 1 in., the distance should be less than 150. And 90 is less than 150, so the answer does seem reasonable.

5. State. 35

in. on the map represents 90 mi.

ADDITIONAL EXERCISES Objective A Multiply and simplify using fraction notation. For extra help, see Examples 1–4 on pages 190–191 of your text and the Section 3.6 lecture video. Factor. Don’t forget to simplify, if possible.

1. 8 37 10⋅ 2. 12 9

5 16⋅

−

3. yxy x⋅ 4. ( )48 9

7 16−

Objective B Solve applied problems involving multiplication. For extra help, see Examples 5–8 on pages 191–193 of your text and the Section 3.6 lecture video. Solve.

5. On a map, 1 in. represents 180 miles.

How much does 23

in. represent?

6. Roberto can earn $600 per week and

saves 112

of this amount each week.

How much does he save each week?



Find the area.

7.

8.

9.




Chapter 3 FRACTION NOTATION: MULTIPLICATION AND DIVISION 3.7 Reciprocals and Division

Learning Objectives A Find the reciprocal of a number. B Divide and simplify using fraction notation.

Key Terms Use the terms listed below to complete each statement in Exercises 1–2.

one zero

1. _________________ has no reciprocal.

2. If the product of two numbers is __________ , the numbers are reciprocals of each other. GUIDED EXAMPLES AND PRACTICE Objective A Find the reciprocal of a number.


1. Find the reciprocals of 5 1, 4, and .9 6

The reciprocal of 5 9 is .9 5

Note that 5 9 1.9 5⋅ =

The reciprocal of 14 is .4

Note that 1 44 1.4 4⋅ = =

The reciprocal of 1 is 6.6

Note that 1 66 1.6 6⋅ = =

Practice this exercise: 1. Find the reciprocal of 13.

Objective B Divide and simplify using fraction notation.


2. Divide and simplify: 5 25 .4 16÷

−

( )

5 25 5 16 Multiplying by the reciprocal of the divisor4 16 4 255 4 4 Factoring and identifying common factors4 5 5

Removing a factor equal5 4 4 5 4 to 1: 15 4 5 5 445

−÷ = ⋅−

⋅ − ⋅=⋅ ⋅

⋅ − ⋅= ⋅ =⋅ ⋅=−


2. Divide and simplify: 2 8 .3 9÷


ADDITIONAL EXERCISES Objective A Find the reciprocal of a number. For extra help, see Examples 1–5 on page 199 of your text and the Section 3.7 lecture video. Find the reciprocal.

1. 67

2. 19a

3. 1211−

4. 13y

−

5. 19x

Objective B Divide and simplify using fraction notation. For extra help, see Examples 6–9 on page 200 of your text and the Section 3.7 lecture video. Divide. Don’t forget to simplify when possible. Assume that all variables are nonzero.

6. 4 43 5÷ 7. 3 7

5 10÷

−

8. ( ) ( )7 2124 5

− ÷ − 9. 56 1455 15

÷

10.

56

169

−

−



Chapter 3 FRACTION NOTATION: MULTIPLICATION AND DIVISION 3.8 Solving Equations: The Multiplication Principle

Learning Objectives A Use the multiplication principle to solve equations. B Solve problems by using the multiplication principle. Key Terms Use the vocabulary terms listed below to complete each statement in Exercises 1–2.

coefficient multiplication principle

1. The _________________ states that for any numbers a, b and c, with 0, c a b≠ = is equivalent to .a c b c⋅ = ⋅

2. In the expression 2 ,3

x the constant factor 23

is the _________________ .

GUIDED EXAMPLES AND PRACTICE Objective A Use the multiplication principle to solve equations.


1. Solve: 954 .2

y=−

We multiply by the reciprocal of 9 2, or ,2 9

− − on

both sides of the equation to get y alone.

( )9542

2 2 9549 9 2

2 6 9

y

y

=−

− ⋅ =− −

⋅ ⋅−9 1

Removing a factor 912 equal to 1: 19

y

y

=⋅

− = =


1. Solve: 5 30.6

x =−


Objective B Solve problems by using the multiplication principle.


2. Jason uses 23

oz of dishwashing liquid each time

he washes the dishes. If Jason buys a 24 oz bottle of dishwashing liquid, how many times will he be able to wash dishes?

1. Familiarize. Repeated addition applies here. Let d = the number of times Jason can wash dishes. We visualize the situation.

2 2 2 oz oz ... oz3 3 3

2 oz portions3

d

2. Translate. We translate to an equation.

2 243

d ⋅ =

3. Solve. We divide by 23

on both sides of the

equation.

2 3 3243 2 2

2 12 32 1

2 12 32 136

d

d

d

d

⋅ ⋅ = ⋅

⋅ ⋅=⋅⋅= ⋅

=

4. Check. We repeat the calculation. The answer checks.

5. State. Jason can wash dishes 36 times.


2. How many 1 -cup4

salt shakers

can be filled from 4 cups of salt?

ADDITIONAL EXERCISES Objective A Use the multiplication principle to solve equations. For extra help, see Examples 1–3 on pages 204–205 of your text and the Section 3.8 lecture video. Use the multiplication principle to solve each equation. Don’t forget to check!

1. 5 408

x =− 2. 1473

a =



3. 3 95 10

x = 4. 16 14t− =−

Objective B Solve problems by using the multiplication principle. For extra help, see Examples 4–5 on pages 205–206 of your text and the Section 3.8 lecture video. Solve.

5. A piece of copper 56

yd long is cut into

4 equal pieces. How long is each piece?

6. A gas tank held 21 gal when it was 34

full. How much gas can it hold when it is full?

7. After driving 150 mi, Sheila notes

that she has completed 23

of her trip.

What is the length of the trip?

8. Patrick borrowed $4800 to pay 45

of his tuition. What is the total

amount of his tuition?


Chapter 3 FRACTION NOTATION: MULTIPLICATION AND DIVISION

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