Free Pre-Algebra
Lesson 11  page 1
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 21 inches, the number 2 21 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
© 2010 Cheryl Wilcox
2:00 to 3:00
3:00 to 3:30
Free Pre-Algebra
Lesson 11  page 2
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 Money
Two fifty-cent pieces make one dollar.
© 2010 Cheryl Wilcox
Improper Fraction Groups
Each band has two half bands,
so there are 5 half bands in all.
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
Free Pre-Algebra
Lesson 11  page 3
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.
1 whole rectangle is blue.
3/3 parts are blue.
1 whole rectangle is blue.
4/4 parts are 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.
3
4•3
4
12
4
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.
10 • 4
10
4
40
10
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.
2 38
?
8
Here the denominator is 8.
© 2010 Cheryl Wilcox
Step 2: Change the whole number part
to an improper fraction by multiplying it
by the denominator.
2
8•2
8
16
8
The parts are 8ths, so each whole has
8 parts. 8 • 2 = 16 parts total in the 2
wholes.
Step 3: Add the numerators of the two
fractions to get the new improper
fraction.
2
16
8
3
8
3
8
19
8
16 eighths + 3 eighths = 19 eighths.
(2 wholes)
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Lesson 11  page 4
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.
?
7
1127
Step 2: Change the whole number part
to an improper fraction by multiplying it
by the denominator.
11
Here the denominator is 7.
7 • 11
7
Step 3: Add the numerators of the two
fractions to get the new improper
fraction.
77
7
77
7
1127
The parts are 7ths, so each whole has
7 parts. 7 • 11 = 77 parts total in the 11
wholes.
2
7
79
7
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).
9
4
4
4
4
4
Each set of 4 quarters makes one whole dollar. There are
two sets of 4 quarters, so we have two dollar bills.
1
4
1 1
1
4
2
1
4
Since we are finding how many 4s in 9, the arithmetic operation is division. 9 4 2r 1 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
4
2
1
4
To review division with remainders, see any of these resources: video; text.
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 11  page 5
Example: Change the improper fraction 21/7 to a mixed or whole number.
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
3.
7
Example: Change the improper fraction 47/5 to a mixed or whole number.
47 5
47
5
9r 2
9
2
5
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.
More Fraction Notation
Notation
N
N
1
Explanation
Since there are N parts in one whole, and we
are considering all N of them, we are
considering one whole.
Example
3
3
Picture
1
3/3 parts are blue
N
1
N
There is only 1 part in each whole, and we are
considering N parts, so we are considering N
wholes.
3
1
3
3 wholes with 1 part each are blue.
0
K
0
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.
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.

© 2010 Cheryl Wilcox
0
3
0
0 of the 3 parts are blue.
3
is undefined.
0
ERROR
Free Pre-Algebra
Lesson 11  page 6
Lesson 11: Mixed Numbers and Improper Fractions
Worksheet
Name __________________________________________
1. Divide the rectangle into 5ths.
2. Divide each rectangle into 3rds.
How many 1/5ths in 1 whole?
How many 1/3rds in 2 wholes?
Write 1 as an improper fraction with denominator 5.
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.
5. Write an improper fraction for the number of blue
rectangles. Each rectangle is 1 whole.
To write an improper fraction, divide each of the whole
rectangles into ___ parts.
Write a mixed number for this amount.
The improper fraction name for this amount is ______.
5. Change each mixed or whole number to an improper
fraction.
6. Change each improper fraction to a mixed or whole
number.
1
7
a.
9
8
b. 6
3
5
b.
23
4
c. 2
5
16
c.
90
9
a. 4
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.
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 11  page 7
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
to an improper fraction.
12
Change 6 feet 1 inches to inches.
How were those two problems related?
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 11  page 8
Lesson 11: Mixed Numbers and Improper Fractions
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.
2. Draw a line of the given length.
a. 1 and 1/4 inches
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
b. 2 and 7/16 inches
c. Use the formulas to figure out your share of the apartment
cost if Rent = $850, Utilities = $104, and Number of
Roommates = 3.
c. 2 and 9/16 inches
3. a. Find all the factor pairs and the GCF for the given
numbers.
112
126
4. a. Find the prime factorizations of the given numbers.
87
116
b. Use the prime factorizations to write the fractions in lowest
terms.
b. Use the GCF to write the given fractions in lowest terms.
112
126
© 2010 Cheryl Wilcox
126
112
87
116
116
87
Free Pre-Algebra
5. Write five fractions equivalent to the given fraction.
7
10
Lesson 11  page 9
6. Write two fraction names for the blue part of each
rectangle.
a.
9
16
b.
7. Write the fraction in lowest terms. Use either the GCF or
prime factorization method.
8. Write the whole numbers as improper fractions using the
given denominator.
45
60
3
48
64
5
51
68
13
9. Write the mixed numbers as improper fractions.
10. Write the improper fractions as mixed numbers.
7
1
3
47
5
1
15
32
39
16
9
15
17
421
100
20
37
100
© 2010 Cheryl Wilcox
93
3
4
7
1
Free Pre-Algebra
Lesson 11  page 10
Lesson 11: Mixed Numbers and Improper Fractions
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.
2. Draw a line of the given length.
a. 1 and 1/4 inches
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
b. 2 and 7/16 inches
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
c. 2 and 9/16 inches
S = A / N = 954 / 3 = 318
Your share would be $318.
3. a. Find all the factor pairs and the GCF for the given
numbers.
4. a. Find the prime factorizations of the given numbers.
87 3 • 29
116 2 • 2 • 29
b. Use the prime factorizations to write the fractions in lowest
terms.
b. Use the GCF to write the given fractions in lowest terms.
112
126
112 14
126 14
© 2010 Cheryl Wilcox
8
9
126
112
126 14
112 14
9
8
87
116
3 • 29
2 • 2 • 29
3
2•2
3
4
116
87
2 • 2 • 29
3 • 29
2•2
3
4
3
Free Pre-Algebra
Lesson 11  page 11
5. Write five fractions equivalent to the given fraction.
7
10
14
20
21
30
28
40
35
50
42
60
9
16
18
32
27
48
32
64
45
80
54
96
6. Write two fraction names for the blue part of each
rectangle.
a.
b.
1
2
3
6
2
3
4
6
7. Write the fraction in lowest terms. Use either the GCF or
prime factorization method.
8. Write the whole numbers as improper fractions using the
given denominator.
45
60
2•2• 3 • 5
3
12
4
3• 4
12
48
64
48 16
64 16
3
4
5
35
7
5•7
35
51
68
51 17
68 17
3
4
13
3• 3 • 5
3
4
9. Write the mixed numbers as improper fractions.
7
1
3
1
15
32
32
32
15
17
153
17
9
20
21 1
3 3
37
100
22
3
15
32
15
17
2000
100
© 2010 Cheryl Wilcox
47
32
168
17
37
100
13 • 1 13
10. Write the improper fractions as mixed numbers.
47
5
47 5
39
16
39 16
421
100
2037
100
13
1
93
3
2r 7
421 100
94 3
47
5
9r 2
39
16
4r 21
31r 0
94
3
9
2
5
2
7
16
421
100
31
4
21
100
Free Pre-Algebra
Lesson 11  page 12
Lesson 11: Mixed Numbers and Improper Fractions
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
2. Draw a line of the given length.
a. 1 and 3/8 inches
b. The travel expenses are the product of the price per mile
times the number of miles traveled.
Travel Expenses = Price per Mile • Miles
b. 2 and 1/16 inches
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.
c. 2 and 7/8 inches
3. a. Find all the factor pairs and the GCF for the given
numbers.
91
169
4. a. Find the prime factorizations of the given numbers.
84
210
b. Use the prime factorizations to write the fractions in lowest
terms.
b. Use the GCF to write the given fractions in lowest terms.
91
169
© 2010 Cheryl Wilcox
169
91
210
84
210
84
Free Pre-Algebra
5. Write five fractions equivalent to the given fraction.
8
15
Lesson 11 ! page 13
6. Write two fraction names for the blue part of each
rectangle.
a.
4
17
b.
7. Write the fraction in lowest terms. Use either the GCF or
prime factorization method.
8. Write the whole numbers as improper fractions using the
given denominator.
21
35
3=
52
65
5=
66
121
20 =
9. Write the mixed numbers as improper fractions.
10. Write the improper fractions as mixed numbers.
4
2
3
36
7
5
7
32
39
18
11
12
19
973
100
75
89
100
125
25
© 2010 Cheryl Wilcox
7
6
1