MULTIPLYING AND DIVIDING FRACTIONS MIXED NUMBERS · 81 Activity 2.4 — Multiplying and Dividing Fractions and Mixed Numbers Model 3 Special Case: Product of More than Two Fractions

77

2.4 MULTIPLYING AND DIVIDING FRACTIONS AND MIXED NUMBERS

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Bringit on!

• the terminology and notation used when multiplying and dividing fractions

• the process of reducing before multiplying

• how to deal with mixed numbers in multiplication and division

• the division process—how and why division is turned into multiplication

• in what form to present a fi nal answer

• how to validate the answer to a multiplication or division problem involving fractions or mixed numbers

How wide must the shelf be?

______________________ inches

Greg is building a simple shelf to hold his DVD collection. He has 64 DVDs in his collection and he knows that each DVD case is 5/8 of an inch wide.

He’s not sure how wide the shelf needs to be, in order for it to hold his complete collection.

• Multiplying any given combinations of fractions and mixed numbers correctly– presentation of the fi nal answer in lowest terms– validation of the answer

• Dividing any given combinations of fractions and mixed numbers correctly– presentation of the fi nal answer in lowest terms– validation of the answer

40

78

Chapter 2 — Fractions

Steps in the Methodology Example 1 Example 2

Step 1

Set up the problem.

Set up the problem horizontally for ease of calculation.

78

445

×

Step 2

Convert mixed numbers.

Convert the mixed numbers to improper fractions and rewrite the problem.

Whole number factor(s)(see Model 1)

Special Case:

78

245

×

Step 3

Prime factor and cancel.

Simplify before multiplying.Determine the prime factorizations of both numerators and denominators, then cancel all common factors.

Quick reduction(see Model 2)

Shortcut:

Product of more than two fractions (see Model 3)

Special Case:

7

2 2 2

2 2 2 351 1 1

1 1 1

• •×

• • •

Step 4

Multiply across.

Multiply the remaining numerators and use the product as the new numerator. Multiply the remaining denominators, and use the product as the new denominator.

7 35

215

•=

Step 5

Convert to a mixed number (if necessary).

If the product is an improper fraction, convert it to a mixed number.

215

415

=

Simply multiplying the numerators and denominators of two fractions to fi nd their product will often result in a fraction that must be reduced to lowest terms. The Methodology for Multiplication uses canceling before fi nding the product so as not to end up with large numbers to reduce for the fi nal answer. It also addresses how to effi ciently multiply factors that are mixed numbers.

Example 1: Multiply by .

Example 2: Multiply: Try It!

78

4 45

3 34

115

×

79

Activity 2.4 — Multiplying and Dividing Fractions and Mixed Numbers


Step 6

Verify that the fraction is reduced.

Verify that the fraction is fully reduced.

Note: If you canceled all common factors in Step 3, it will be fully reduced. If not, reduce fully now.

15

is fully reduced.

Step 7

Present the answer.

Present your answer.4

15

Step 8

Validate your answer.

Validate the fi nal answer by division, using the original fractions and/or mixed numbers.

415

445

215

245

215

524

3 7

5

5

2 2 2 378

1

1

1

1

÷

= ÷

= ×

=•

×• • •

=

Model 1

Multiply:

Special Case: Whole Number Factor(s)

5 2 23

×

Step 1 5 223

×

Step 251×

83

Step 351

83

× no common factors

Steps 4 & 55 8

3403

1313

×= =

Step 613

is fully reduced

Step 7 Answer : 1313

Step 8 Validate:

1313

223

403

83

40

3

3

851

5

5

1

1

1

÷

= ÷

= ×

= =

In a fraction problem, if a factor is a whole number, write it in its improper form and proceed from there.

80


Model 2 Shortcut: Quick Reduction

Multiply

Step 1 Step 1

Step 2 Step 2

Step 3 Step 3

Steps 4 & 5

Step 6

Step 7

Step 8 Validate: Step 8 Validate:

Shortcut version (optional)17

92 5

8 by

Cancel the factors (not necessarily

prime factors) you recognize as common to both numerator and denominator.

179

258

× 179

258

×

169

218

×169

218

×

1 1 1

1

1

1 1 1

2 2 2 2

3 3

3 7

2 2 2

• • •

•×

•

• •

2

3

7

1

16

9

21

8×

8 is a factor of both 8 and 16.

3 is a factor of both 9 and 21.

2 73

143

423

•= =

23

is fully reduced

423

258

143

218

143

821

2 73

2 2 2

3 7169

179

1

1

÷

= ÷

= ×

=•

×• •

•

= =

423

258

143

218

143

8

21169

179

2

3

÷

= ÷

= ×

= =

Answer : 423

7 is a factor of 14 and 21.

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Model 3 Special Case: Product of More than Two Fractions

Step 1

Step 2

Step 3

Steps 4 & 5

Step 6

Step 7

Step 8 Validate with two divisions:

Find the product of , , and .310

49

2 17

310

49

217

× ×

= × ×3

1049

157

= × ×1

2 3

33

10

4

9

157

The common factor of 3 and 9 is 3.

The common factor of 15 and 10 is 5.

Rewrite and cancel again1

2

4

3

371

2

1

1

× ×

2 is a factor of 4 and 2. 3 and 3 cancel.

=27

, proper fraction

27

is fully reduced

Answer : 27

27

217

49

27

157

49

2

7

7

15

9

4

310

1

1

1

5

3

2

÷ ÷ = ÷ ÷

= × × =

= × ×1

1

3

10

4

9

1571

1

2 3

32

The numerator and denominator in which you recognize a common factor do not have to be in adjacent fractions.

Continue canceling common factors.

OR

Dividing Fractions and Mixed Numbers versusMultiplying Fractions and Mixed Numbers: Critical Diff erences

It is important to note that the following methodology, Dividing Fractions and Mixed Numbers, is very similar to the methodology for Multiplying Fractions and Mixed Numbers. There are only two differences between the methodologies; the most critical is that there is an additional step in the Division Methodology wherein the divisor is inverted and the operation is changed from division to multiplication. The second difference is that the validation process for division uses multiplication.

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Step 1

Set up the problem.

Set up the problem horizontally with the dividend fi rst. 6

38

112

÷

Step 2

Convert mixed numbers.

Convert mixed numbers to improper fractions and rewrite the problem.

Whole number divisor ordividend (see Model 1)

Special Case:

518

32

÷

Step 3

Invert the divisor and multiply.

Invert the divisor (the second fraction) and change the operation to multiplication.

Division is defi ned as the inverse operation of multiplication. This means that dividing a number by a second number is the same as multiplying the fi rst number by the inverse of the second number. For example:

15 ÷ 3 can be written as 15

3

and 15

3 is the same as 15 ×

1

3

518

×23

Step 4

Cancel.

Cancel the common factors by prime factoring fi rst or by using the quick reduction shortcut.

1

1

1

1

17

4

1

1

3 17

2 2 2

2

3

51

8

2

3

•

• •×

×

or

Step 5

Multiply across.

Multiply the remaining numerators and denominators.

172 2

174•

=

Step 6

Convert to a mixed number.

Convert to a mixed number, if necessary. 174

414

=

Example 1: Divide

Example 2: Divide: Try It!

The methodology below converts a given division problem into a multiplication problem to solve. While Example 1 is worked out, step by step, you are welcome to complete Example 2 as a running problem.

6 38

112

by .

8 34

178

÷

83



Step 7

Verify the fraction is reduced.

Verify that the fraction is fully reduced. 14

is fully reduced

Step 8

Present the answer.

Present your answer.4

14

Step 9

Validate your answer.

Validate your fi nal answer by multiplication, using the original fractions and/or mixed numbers.

no common factors to cancel

414×

= ×

= =

112

174

32

518

638

Model 1 Special Case: Whole Number Divisor or Dividend

Divide 10 by 425

.

Step 1

Step 2

Step 3

Steps 4, 5 & 6

Step 7

Step 8

Step 9 Validate:

1025

÷4

525

÷41

In a fraction problem, if the divisor or dividend is a whole number, write it as “the whole number” and proceed from there.

1525

×14

13525

135

235

×1

41 = =

35

is fully reduced

Answer : 235

235

4

135

41

525

1035

×

= × = =

84


Model 2 Model 3

Divide by 8 34

78

.

1 12 5

110

××

=

78÷8

34

78÷

354

78

×4

35

1

2

7

8×

1

5

4

35

110

is fully reduced

Answer : 1

10

110

834

110

354

72 4

78

2

7

×

= ×

=×

=

Step 1

Step 2

Step 3

Step 4

Step 5

Step 6 proper

Step 7

Step 8

Step 9 Validate:

Divide: 38

114

÷

Steps 1 & 2

Step 3

Steps 4 & 5

Step 6

Step 7

Step 8

Step 9 Validate:

38

114

÷ no mixed numbers to convert

38

×141

3

8

141

3 74

2144

7

× =×

=

14

is fully reduced

Answer : 514

=514

514

114

214

1

143

4 238

3

2

×

= ×

=×

=

Divide: 3 4 57

÷

711

457

7

11

33

7

31

31

1

3

1

×

= × = =

Steps 1 & 2 3 457

31

337

÷ = ÷

Step 3 =31

×733

Steps 4 & 5 = × =1

11

31

7

33

711

Steps 6 & 7 proper fraction, fully reduced

Step 7 Answer : 7

11

Step 8 Validate:

85


Make Your Own Model

Problem: _________________________________________________________________________

Either individually or as a team exercise, create a model demonstrating how to solve the most diffi cult problem you can think of.

Answers will vary.

86


1. What is the fi rst critical step when multiplying or dividing mixed numbers?

2. How are whole numbers converted to fractions for multiplying and dividing?

3. How do you convert a division of fractions into a multiplication of fractions?

4. What can you do to simplify a multiplication of fractions problem before computing the fi nal answer?

5. What is the result when all factors in the numerators cancel out?

6. What is the result when all factors in the denominators cancel out?

7. When you multiply a proper fraction by a second number, will the product be greater or less than the second number? Explain.

8. What aspect of the model you created is the most diffi cult to explain to someone else? Explain why.

It will always be less than the second number. A fractional part of any number is always smaller than the original number.

The numbers involved must all be made into improper fractions or proper fractions.

Whole numbers are converted to fractions by making the denominator a one. The whole number is the numerator and the denominator is a “1.”

Replace the divisor with its reciprocal and set up as a multiplication: i.e. invert the divisor and change the operation to multiplication.

“Canceling” (dividing out) can be done with ANY common factor, not just prime factors.

The result is a fraction with 1 in the numerator. Example: 14

The denominator will be one and the result then will be a whole number. Example: 6

16=

Answers will vary.

87


Problem Worked Solution Validation

1)

1235

730

×

2)

23

18

45

35

× × ×

3)

38

27

÷

4)

558

3÷

Solve each problem and validate your answer.

88


Problem Worked Solution Validation

5)

634

849

×

6)

312

514

÷

7)

613

÷

8) Bruno’s share of the profi ts from a land sale is to be 2/7 of $280,000. Calculate his share.

89


Perform the indicated operations and validate your answers.

1.

2.

3.

4.

3 29

35

2 78

114

37

25

1415

1011

2 16

4 78

×

×

× × ×

÷

5..

6.

7.

8. 1

3 19

112

2 112

12 112

4 13

2 12

57

÷

÷

× ×

÷

In the second column, identify the error(s) you fi nd in each of the following worked solutions. If the answer appears to be correct, validate it in the second column and label it “Correct.” If the worked solution is incorrect, solve the problem correctly in the third column and validate your answer in the last column.

Worked SolutionWhat is Wrong Here?

Identify Errors or Validate Correct Process Validation

1)

1216

623

×You must change toimproper fractions,reduce, thenmultiply to get theanswer.

)

12 16

x 6 23

736

x 203

= 7309

9 730 8 1

72 10

= 3

10

19

9 1

Answer

81 19

6 23

7309

203

73 09

32 0

= 736

= 12 16

= x 3

1

÷

÷

11415

31932

855

49

22

27

113

78

1712

90


Worked SolutionWhat is Wrong Here?

Identify Errors or Validate Correct Process Validation

2) 535

318

÷You must change division to multiplication .

You do this by multiplying by the reciprocal of the second number. Then reduce.

3)

Find the product of

, and 35

415

58

, .

Reduced incorrectly.

Cannot use 5 twice in the denominators.

4)

117

514

×CORRECT

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MULTIPLYING AND DIVIDING FRACTIONS MIXED NUMBERS · 81 Activity 2.4 — Multiplying and Dividing Fractions and Mixed Numbers Model 3 Special Case: Product of More than Two Fractions