mixed numbers and improper fractions - diablo valley...

Free Pre-Algebra Lesson 11 page 1

© 2010 Cheryl Wilcox

Lesson 11

Mixed Numbers and Improper Fractions

If a line is longer than 2 inches, but not as long as 3 inches, the measurement can be given as 2 whole inches plus another

fraction of an inch. When we measure a line to be 2 1

2 inches, the number

2 1

2 is called a mixed number – it has a

whole number part (2) and a fraction part (1/2). The fraction part of a mixed number is greater than 0 and less than 1.

Mixed Number Length

Mixed Number Area

Mixed Number Volume

Mixed Number Groups

Mixed Number Money

Mixed Number Time

1:00 to 2:00 2:00 to 3:00 3:00 to 3:30



Improper Alternative Any mixed number or whole number can also be written as an improper fraction (as opposed to the proper fractions between 0 and 1). These are formed by counting the total number of parts of the size indicated by the denominator. Improper fractions have the numerator greater than or equal to the denominator.

Improper Fraction Length There are 5 lengths of 1/2 inch, or 5/2 inches.

Improper Fraction Area

5 half fields is 5/2 fields, the same as 2 and 1/2 fields.

Improper Fraction Volume

There are 5 of the half loaves, equal to 2 and 1/2 loaves.

Improper Fraction Groups

Each band has two half bands, so there are 5 half bands in all.

Improper Fraction Money

Two fifty-cent pieces make one dollar.

Improper Fraction Time

1:00 to 1:30 1:30 to 2:00 2:00 to 2:30 2:30 to 3:00 3:00 to 3:30



Fractions Equivalent to Whole Numbers When the numerator of a fraction is equal to its denominator, the fraction is equal to one. Since the denominator stands for the number of parts in one whole, and the numerator stands for the number of parts you have, if they are equal, you have all the parts in one whole.

2/2 parts are blue. 3/3 parts are blue. 4/4 parts are blue. 1 whole rectangle is blue. 1 whole rectangle is blue. 1 whole rectangle is blue. Writing Whole Numbers as Improper Fractions Any whole number can be written as an improper fraction with any denominator. The denominator tells the number of parts in 1 whole, so if you multiply the denominator by the number of wholes, you get the total number of parts.

Example: How many 1/4ths in 3 wholes?

The parts are 4ths, so each of the 3 wholes has 4 parts. There are 4 • 3 = 12 fourths in all.

Example: How many 1/10ths in 4 wholes?

The parts are 10ths, so each of the 4 wholes has 10 parts. There are 10 • 4 = 40 tenths in all.

How To Change a Mixed Number to an Improper Fraction

Example: Change the mixed number 2 3/8 to improper fraction form.

Step 1: The denominator of the fraction part is the same as the denominator of the improper fraction.

Step 2: Change the whole number part to an improper fraction by multiplying it by the denominator.

Step 3: Add the numerators of the two fractions to get the new improper fraction.

2 3

8

?

8

2

8 • 2

8

16

8

2 3

8

16

8

3

8

19

8

Here the denominator is 8. The parts are 8ths, so each whole has 8 parts. 8 • 2 = 16 parts total in the 2 wholes.

16 eighths + 3 eighths = 19 eighths. (2 wholes)

3

4 • 3

4

12

4

4

10 • 4

10

40

10



Example: Change the mixed number 11 2/7 to improper fraction form.

Step 1: The denominator of the fraction part is the same as the denominator of the improper fraction.

Step 2: Change the whole number part to an improper fraction by multiplying it by the denominator.

Step 3: Add the numerators of the two fractions to get the new improper fraction.

112

7

?

7

11

7 • 11

7

77

7

112

7

77

7

2

7

79

7

Here the denominator is 7. The parts are 7ths, so each whole has 7 parts. 7 • 11 = 77 parts total in the 11 wholes.

77 sevenths + 2 sevenths = 79 sevenths.

How To Change an Improper Fraction to a Mixed (or Whole) Number

Example: Change the improper fraction 9/4 to a mixed or whole number.

Picturing What’s Happening

The numerator is 9, and the denominator is 4. You could picture this problem as changing 9 quarters into dollars (whole number) and quarters (fraction part).

Each set of 4 quarters makes one whole dollar. There are two sets of 4 quarters, so we have two dollar bills.

9

4

4

4

4

4

1

4

1 1

1

42

1

4

Since we are finding how many 4s in 9, the arithmetic operation is division. 9 4 2r1 There are two 4s in 9, leaving a remainder of 1. The two 4s become the whole number part, and the 1 remaining is the numerator of the fraction part.

9

42

1

4

To review division with remainders, see any of these resources: video; text.

http://www.youtube.com/watch?v=ITloMQvj-lU&feature=related

http://www.mathsisfun.com/long_division2.html




Instead of using a picture this time, we work directly with the fraction notation. 21 7 3 , and there is no remainder. Since there are three 7s in 21, there are three wholes in 21/7. There are no 7ths left over, so

21

73 .


47 5 9r2

47

59

2

5

More Fraction Notation

Notation Explanation Example Picture

N

N1

Since there are N parts in one whole, and we are considering all N of them, we are considering one whole.

3

31

3/3 parts are blue

N

1N

There is only 1 part in each whole, and we are considering N parts, so we are considering N wholes.

3

13

3 wholes with 1 part each are blue.

0

K0

There are K parts in one whole, but we are considering none (0) of them, so we are considering 0 wholes. The mnemonic 0/K helps you differentiate between this and the next example.

0

30

0 of the 3 parts are blue.

It is impossible to divide a whole into zero parts. The expression is meaningless. The mnemonic N/0 helps you remember that this expression is mathematically undefined.

3

0 is undefined.

ERROR

Long Division with Remainder on the Calculator

1 Divide the numbers as usual, and

see the decimal result.

2 Multiply the whole number part of the

answer, 9, by the divisor 5.

3 Subtract the result of your

multiplication from the original number.

47 ÷ 5 is 9 remainder 2.



Lesson 11: Mixed Numbers and Improper Fractions

Worksheet Name __________________________________________

1. Divide the rectangle into 5ths.

How many 1/5ths in 1 whole?

Write 1 as an improper fraction with denominator 5.

2. Divide each rectangle into 3rds.

How many 1/3rds in 2 wholes?

Write 2 as an improper fraction with denominator 3.

4. Write a mixed number for the number of blue rectangles. Each rectangle is 1 whole.

To write an improper fraction, divide each of the whole rectangles into ___ parts.

The improper fraction name for this amount is ______.

5. Write an improper fraction for the number of blue rectangles. Each rectangle is 1 whole.

Write a mixed number for this amount.

5. Change each mixed or whole number to an improper fraction.

a. 4

1

7

b. 6

3

5

c. 2

5

16

6. Change each improper fraction to a mixed or whole number.

a.

9

8

b.

23

4

c.

90

9

7. Mark the number line in 4ths. Draw a line 5 1/4 units long on the number line. Write the length as an improper fraction.



8. Each loaf of bread has 20 slices. Write 93/20 loaves of bread as a mixed number, and explain what this means in terms of loaves and slices of bread.

9. Each hour has 60 minutes. Write 218/60 hours as a mixed number, and explain what this means in terms of hours and minutes.

10. There are 12 inches in a foot. Write 65/12 feet as a mixed number, and explain what this means in terms of feet and inches.

11. Change 6

1

12 to an improper fraction.

Change 6 feet 1 inches to inches.

How were those two problems related?




Homework 11A Name ________________________________________

1. a. Your apartment cost is the sum of your rent and your utilities costs. Write a formula using variables for your apartment cost.

Apartment Cost = Rent + Utilities

b. You decide to save money by finding some roommates to share costs. Write a formula using variables for your share of the apartment costs.

Your Share = Apartment Cost ÷ Number of Roomates

c. Use the formulas to figure out your share of the apartment cost if Rent = $850, Utilities = $104, and Number of Roommates = 3.

2. Draw a line of the given length.

a. 1 and 1/4 inches

b. 2 and 7/16 inches

c. 2 and 9/16 inches

3. a. Find all the factor pairs and the GCF for the given numbers.

112 126

b. Use the GCF to write the given fractions in lowest terms.

112

126

126

112

4. a. Find the prime factorizations of the given numbers.

87

116

b. Use the prime factorizations to write the fractions in lowest terms.

87

116

116

87



5. Write five fractions equivalent to the given fraction.

7

10

9

16

6. Write two fraction names for the blue part of each rectangle.

a. b.

7. Write the fraction in lowest terms. Use either the GCF or prime factorization method.

45

60

48

64

51

68

8. Write the whole numbers as improper fractions using the given denominator.

3

4

5

7

13

1

9. Write the mixed numbers as improper fractions.

7

1

3

1

15

32

9

15

17

20

37

100

10. Write the improper fractions as mixed numbers.

47

5

39

16

421

100

93

3




Homework 11A Answers

1. a. Your apartment cost is the sum of your rent and your utilities costs. Write a formula using variables for your apartment cost.

Apartment Cost = Rent + Utilities

A = R + U

b. You decide to save money by finding some roommates to share costs. Write a formula using variables for your share of the apartment costs.

Your Share = Apartment Cost ÷ Number of Roomates

S = A / N

c. Use the formulas to figure out your share of the apartment cost if Rent = $850, Utilities = $104, and Number of Roommates = 3.

A = R + U = 850 + 105 = 954

S = A / N = 954 / 3 = 318

Your share would be $318.


a. 1 and 1/4 inches


c. 2 and 9/16 inches



112

126

112 14

126 14

8

9

126

112

126 14

112 14

9

8


87 3 • 29

116 2 • 2 • 29


87

116

3 • 29

2 • 2 • 29

3

2 • 2

3

4

116

87

2 • 2 • 29

3 • 29

2 • 2

3

4

3




7

10

14

20

21

30

28

40

35

50

42

60

9

16

18

32

27

48

32

64

45

80

54

96


a. b.

1

2

3

6

2

3

4

6


45

60

3 • 3 • 5

2 • 2 • 3 • 5

3

4

48

64

48 16

64 16

3

4

51

68

51 17

68 17

3

4


3

12

43 • 4 12

5

35

75 • 7 35

13

13

113 • 1 13


7

1

3

21

3

1

3

22

3

1

15

32

32

32

15

32

47

32

9

15

17

153

17

15

17

168

17

20

37

100

2000

100

37

100

2037

100


47

547 5 9r2

47

59

2

5

39

1639 16 2r7

39

162

7

16

421

100421 100 4r21

421

1004

21

100

93

394 3 31r0

94

331




Homework 11B Name _________________________________________

1. a. The cost of showing your dog is the sum of the entry fee and your travel expenses.

Dog Show Cost = Entry Fee + Travel Expenses

b. The travel expenses are the product of the price per mile times the number of miles traveled.

Travel Expenses = Price per Mile • Miles

c. Use the formulas to figure out the cost of a show if the entry fee is $200, the price per mile is $2, and the number of miles is 180.


a. 1 and 3/8 inches


c. 2 and 7/8 inches


91 169


91

169

169

91


84

210


210

84

210

84

Free Pre-Algebra Lesson 11 ! page 13



8

15

4

17


a. b.


21

35

52

65

66

121


3 =7

5 =6

20 =1


42

3

57

32

1112

19

7589

100


36

7

39

18

973

100

125

25

mixed numbers and improper fractions - diablo valley...

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