ch 4 sec 5: slide #1 columbus state community college chapter 4 section 5 problem solving: mixed...

Ch 4 Sec 5: Slide #1

Columbus State Community College

Chapter 4 Section 5

Problem Solving: Mixed Numbers and Estimating



1. Identify mixed numbers and graph them on a number line.

2. Rewrite mixed numbers as improper fractions, or the reverse.

3. Estimate the answer and multiply or divide mixed numbers.

4. Estimate the answer and add or subtract mixed numbers.

5. Solve application problems containing mixed numbers.


Illustrating a Mixed Number with a Diagram

EXAMPLE 1 Illustrating a Mixed Number with a Diagram

1 whole

1 whole

of a whole14

4 shaded parts

4 shaded parts

1 shaded part

214

whole parts shaded is equivalent to shaded parts.94


85

– 35

– 1

0 1 2-2 -1

Illustrating a Mixed Number with a Number Line

EXAMPLE 1 Illustrating Mixed Numbers with a Number Line

12345678

85

– 35

– 1is equivalent to


Mixed Numbers

NOTE

12

4 represents .12

4 +

– represents – – , which can also be written as

– – .

12

4 4 + 12

4

12


Mixed Numbers

NOTE

In algebra we usually work with the improper fraction form of mixed numbers, especially for negative mixed numbers. However, positive mixed numbers are frequently used in daily life, so it’s important to know how to work with them.

For example, we usually say inches rather than inches.12

4 92


34

5

Writing a Mixed Number as an Improper Fraction


Step 1 Multiply the denominator of the fraction times the whole number and add the numerator of the fraction to the product.

Step 2 Write the result of Step 1 as the numerator and keep the original denominator.

4 • 5 = 20 20 + 3 = 23 234

34

5 =



Write as an improper fraction.

EXAMPLE 2 Writing a Mixed Number as an Improper Fraction

29

8

Step 1 8 • 9 = 72

Step 2 =

29

8 Then 72 + 2 = 74

29

8 ( 8 • 9 ) + 2749

Same denominator


Writing an Improper Fraction as a Mixed Number

Writing an Improper Fraction as a Mixed Number

To write an improper fraction as a mixed number, divide the numerator by the denominator. The quotient is the whole number part (of the mixed number), the remainder is the numerator of the fraction part, and the denominator remains the same.

Divide 38 by 5.385

5 38

35

3

735

7 Remainder Original

denominator


Writing Improper Fractions as Mixed Numbers

(a) Write as a mixed number.

EXAMPLE 3 Writing Improper Fractions as Mixed Numbers

283

Divide 28 by 3.

3 28

27

1

913


denominator


Writing Improper Fractions as Mixed Numbers

(b) Write as a mixed number.

EXAMPLE 3 Writing Improper Fractions as Mixed Numbers

429

Divide 42 by 9.

9 42

36

6

469


denominator

23

4=

Write in lowest terms.69


Estimating Mixed Numbers

Estimating Mixed Numbers

To estimate answers, first round each mixed number to the nearest whole number. If the numerator is half of the denominator or more, round up the whole number part. If the numerator is less than half the denominator, leave the whole number as it is.


Rounding Mixed Numbers to the Nearest Whole Number

(a) Round

EXAMPLE 4 Rounding Mixed Numbers to the Nearest Whole Number

47

9

rounds up to 1047

9

Half of 7 is 12

3

4 is more than 12

3


Rounding Mixed Numbers to the Nearest Whole Number

(b) Round

EXAMPLE 4 Rounding Mixed Numbers to the Nearest Whole Number

38

6

rounds to 638

6

Half of 8 is 4

3 is less than 4


Multiplying and Dividing Mixed Numbers

Multiplying and Dividing Mixed Numbers

Step 1 Rewrite each mixed number as an improper fraction.

Step 2 Multiply or divide the improper fractions.

Step 3 Write the answer in lowest terms and change it to a mixed number or whole number where possible. This step gives you an answer that is in simplest form.


Estimating the Answer and Multiplying Mixed Numbers

EXAMPLE 5 Estimating the Answer and Multiplying Mixed Numbers

(a) •34

3 45

2

34

3 45

2rounds to 4 rounds to 3.and

Estimate the answer by rounding the mixed numbers.

4 • 3 = 12 Estimated answer


3

1

7

2



To find the exact answer, first rewrite each mixed number

as an improper fraction.

(a) •34

3 45

2

and34

3 = 154

45

2 = 145

Step 1

34

3 •45

2 =145

•154

Step 2 = 212

12

10=




Estimate

(a) •34

3 45

2

12

10

Exact

12

The exact answer is reasonable.




(b) •16

1 47

5

16

1 47



1 • 6 = 6 Estimated answer


1

1

13

2




as an improper fraction.

and16

1 = 76

47

5 = 397

Step 1

16

1 •47

5 =397

•76

Step 2 = 132

12

6=

(b) •16

1 47

5




Estimate

12

6

Exact

6


(b) •16

1 47

5


Estimating the Answer and Dividing Mixed Numbers

EXAMPLE 6 Estimating the Answer and Dividing Mixed Numbers

(a) ÷35

7 25

2

35

7 25



8 ÷ 2 = 4 Estimated answer


19

6

1

1



To find the exact answer, first rewrite each

mixed number as an improper fraction.

and35

7 = 385

25

2 = 125

Step 1

35

7 ÷25

2 =125

÷385

Step 2 = 196

16

3=

(a) ÷35

7 25

2

=512

•385




Estimate

16

3

Exact

4


(a) ÷35

7 25

2




(b) ÷13

2 5

Write 2 ÷ 5 using

a fraction bar.

13

2 ÷ 5

First, round the numbers and estimate the answer.

2 ÷ 5 25




Now find the exact answer.

and13

2 = 73

5 = 51

Step 1

13

2 ÷ 5 =51

÷73

Step 2 = 715

=15

•73

(b) ÷13

2 5


715

25



Estimate Exact

The estimate and the exact answer are close to one-half.

Therefore, the exact answer is reasonable.

(b) ÷13

2 5


Estimating the Answer and Adding Mixed Numbers

EXAMPLE 7 Estimating the Answer and Adding Mixed Numbers

(a) +23

3 16

5

23

3 16



4 + 5 = 9 Estimated answer





as an equivalent improper fraction.

(a) +23

3 16

5

23

3 +16

5 =316

+113

= 536

56

8==316

+226

Rewrite each improper fraction with the LCD of 6.Add the numerators. Keep the common denominator.Write your answer in simplest form.




Estimate

56

8

Exact

9


(a) +23

3 16

5


Estimating the Answer and Subtracting Mixed Numbers

EXAMPLE 7 Estimating the Answer and Subtracting Mixed Numbers

9(b) – 47

2

9 47



9 – 3 = 6 Estimated answer


Write your answer in simplest form.Rewrite each improper fraction with the LCD of 7.To find the exact answer, first rewrite each mixed number

as an equivalent improper fraction.

Subtract the numerators. Keep the common denominator.



9 –47

2 =187–

91

= 457

37

6==187–

637

9(b) – 47

2




Estimate

37

6

Exact

6


9(b) – 47

2


Solving Application Problems with Mixed Numbers

EXAMPLE 8 Solving Application Problems: Mixed Numbers

(a) Mike started his trip with gallons of gas in his car. After

his trip, he had gallons remaining. How many gallons of

gas did Mike use on his trip?

35

20

12

5

To help understand the mathematical operation needed to solve this problem, read it again using rounded numbers.

Mike started his trip with 21 gallons of gas in his car. After his

trip he had 6 gallons remaining. How many gallons of gas did

Mike use on his trip? 21 – 6 = 15 gallons Estimate




(a) Mike started his trip with gallons of gas in his car. After

his trip, he had gallons remaining. How many gallons of

gas did Mike use on his trip?

35

20

12

5

Mike used gallons of gas on his trip. This result is close

to the estimate of 15 gallons.

110

15



(b) Mary’s recipe for chocolate chip cookies calls for cups

of flour per batch. If she has cups of flour available,

how many batches of cookies can Mary make?

34

15

14

2

To help understand the mathematical operation needed to solve this problem, read it again using rounded numbers.

Mary’s recipe for chocolate chip cookies calls for 2 cups of flour per

batch. If she has 16 cups of flour available, how many batches of

cookies can Mary make? 16 ÷ 2 = 8 batches Estimate





Mary can make 7 batches of cookies. This result is close to the

estimate of 8 batches of cookies.

(b) Mary’s recipe for chocolate chip cookies calls for cups

of flour per batch. If she has cups of flour available,

how many batches of cookies can Mary make?

34

15

14

2



Chapter 4 Section 5 – Completed

Written by John T. Wallace

ch 4 sec 5: slide #1 columbus state community college chapter 4 section 5 problem solving: mixed...

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