OpenStax-CNX module: m34912
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Introduction to Fractions and
Multiplication and Division of
Fractions: Proper Fractions,
Improper Fractions, and Mixed
Numbers
∗
Wade Ellis
Denny Burzynski
This work is produced by OpenStax-CNX and licensed under the
Creative Commons Attribution License 3.0
†
Abstract
This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr.
module discusses proper fractions, improper fractions, and mixed numbers.
This
By the end of the module
students should be able to distinguish between proper fractions, improper fractions, and mixed numbers,
convert an improper fraction to a mixed number and convert a mixed number to an improper fraction.
1 Section Overview
•
•
•
•
•
•
Positive Proper Fractions
Positive Improper Fractions
Positive Mixed Numbers
Relating Positive Improper Fractions and Positive Mixed Numbers
Converting an Improper Fraction to a Mixed Number
Converting a Mixed Number to an Improper Fraction
Now that we know what positive fractions are, we consider three types of positive fractions: proper fractions,
improper fractions, and mixed numbers.
2 Positive Proper Fractions
Positive Proper Fraction
Fractions in which the whole number in the numerator is strictly less than the whole number in the denominator are called positive proper fractions. On the number line, proper fractions are located in the
interval from 0 to 1. Positive proper fractions are always less than one.
∗ Version
1.2: Aug 18, 2010 8:37 pm -0500
† http://creativecommons.org/licenses/by/3.0/
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The closed circle at 0 indicates that 0 is included, while the open circle at 1 indicates that 1 is not
included.
Some examples of positive proper fractions are
106
1 3 20
2 , 5 , 27 , and 255
Note that 1 < 2, 3 < 5, 20 < 27, and 106 < 225.
3 Positive Improper Fractions
Positive Improper Fractions
Fractions in which the whole number in the numerator is greater than or equal to the whole number in the
denominator are called positive improper fractions. On the number line, improper fractions lie to the
right of (and including) 1. Positive improper fractions are always greater than or equal to 1.
Some examples of positive improper fractions are
105
3 8 4
2 , 5 , 4 , and 16
Note that 3 ≥ 2, 8 ≥ 5, 4 ≥ 4, and 105 ≥ 16.
4 Positive Mixed Numbers
Positive Mixed Numbers
A number of the form
nonzero whole number + proper fraction
is called a positive mixed number. For example, 2 35 is a mixed number. On the number line, mixed
numbers are located in the interval to the right of (and including) 1. Mixed numbers are always greater than
or equal to 1.
5 Relating Positive Improper Fractions and Positive Mixed Numbers
A relationship between improper fractions and mixed numbers is suggested by two facts. The rst is that
improper fractions and mixed numbers are located in the same interval on the number line. The second fact,
that mixed numbers are the sum of a natural number and a fraction, can be seen by making the following
observations.
Divide a whole quantity into 3 equal parts.
Now, consider the following examples by observing the respective shaded areas.
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3
2
In the
shaded region, there are 2 one thirds, or 3 .
2
1
3
=
2
3
There
3 one thirds, or 33 , or 1.
are
3
1
3 3 = 3 or 1
Thus,
3
3
=1
Improper fraction = whole number.
There
4 one thirds, or 34 , or 1 and 13 .
are
4
1
4 3 = 3 or 1 and 13
The terms 1 and 13 can be represented as 1 +
Thus,
4
1
3 = 13.
Improper fraction = mixed number.
There
5 one thirds, or 35 , or 1 and 23 .
are
1
5
5 3 = 3 or 1 and 23
The terms 1 and 23 can be represented as 1 +
Thus,
5
2
3 = 13.
Improper fraction = mixed number.
1
3
or 1 31
2
3
or 1 32 .
6
There
are 6 one thirds, or 3 , or 2.
6
1
3
Thus,
6
3
=
6
3
=2
=2
Improper fraction = whole number.
The following important fact is illustrated in the preceding examples.
Mixed Number = Natural Number + Proper Fraction
Mixed numbers are the sum of a natural number and a proper fraction.
Mixed number = (natural number)
+ (proper fraction)
For example 1 13 can be expressed as 1 + 13 The fraction 5 87 can be expressed as 5 + 78 .
It is important to note that a number such as 5 + 87 does not indicate multiplication. To indicate
multiplication, we would need to use a multiplication symbol (such as ·)
note:
5 78 means 5 +
7
8
and not5 · 78 , which means 5 times
7
8
or 5 multiplied by 87 .
Thus, mixed numbers may be represented by improper fractions, and improper fractions may be represented
by mixed numbers.
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6 Converting Improper Fractions to Mixed Numbers
To understand how we might convert an improper fraction to a mixed number, let's consider the fraction, 34 .
4
3
=
1 1 1
+ + +
3 3}
|3 {z
=
1+
=
1 13
1
3
1
1
3
Thus, 34 = 1 13 .
We can illustrate a procedure for converting an improper fraction to a mixed number using this example.
However, the conversion is more easily accomplished by dividing the numerator by the denominator and
using the result to write the mixed number.
Converting an Improper Fraction to a Mixed Number
To convert an improper fraction to a mixed number, divide the numerator by the denominator.
1. The whole number part of the mixed number is the quotient.
2. The fractional part of the mixed number is the remainder written over the divisor (the denominator
of the improper fraction).
6.1 Sample Set A
Convert each improper fraction to its corresponding mixed number.
Example 1
5
3
Divide 5 by 3.
The improper fraction
5
3
= 1 23 .
Example 2
46
9
. Divide 46 by 9.
The improper fraction
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46
9
= 5 91 .
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Example 3
83
11
. Divide 83 by 11.
The improper fraction
83
11
6
= 7 11
.
Example 4
104
4
Divide 104 by 4.
104
0
4 = 26 4 = 26
The improper fraction
104
4
= 26.
6.2 Practice Set A
Convert each improper fraction to its corresponding mixed number.
Exercise 1
(Solution on p. 11.)
9
2
Exercise 2
(Solution on p. 11.)
11
3
Exercise 3
(Solution on p. 11.)
Exercise 4
(Solution on p. 11.)
Exercise 5
(Solution on p. 11.)
14
11
31
13
79
4
Exercise 6
496
8
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(Solution on p. 11.)
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7 Converting Mixed Numbers to Improper Fractions
To understand how to convert a mixed number to an improper fraction, we'll recall
mixed number = (natural number) + (proper fraction)
and consider the following diagram.
Recall that multiplication describes repeated addition.
Notice that 53 can be obtained from 1 23 using multiplication in the following way.
Multiply: 3 · 1 = 3
Add: 3 + 2 = 5. Place the 5 over the 3: 53
The procedure for converting a mixed number to an improper fraction is illustrated in this example.
Converting a Mixed Number to an Improper Fraction
To convert a mixed number to an improper fraction,
1. Multiply the denominator of the fractional part of the mixed number by the whole number part.
2. To this product, add the numerator of the fractional part.
3. Place this result over the denominator of the fractional part.
7.1 Sample Set B
Convert each mixed number to an improper fraction.
Example 5
5 78
1. Multiply: 8 · 5 = 40.
2. Add: 40 + 7 = 47.
3. Place 47 over 8: 47
8 .
Thus, 5 78 =
47
8
.
Example 6
16 23
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1. Multiply: 3 · 16 = 48.
2. Add: 48 + 2 = 50.
3. Place 50 over 3: 50
3
Thus, 16 23 =
50
3
7.2 Practice Set B
Convert each mixed number to its corresponding improper fraction.
Exercise 7
(Solution on p. 11.)
8 14
Exercise 8
(Solution on p. 11.)
5 53
Exercise 9
(Solution on p. 11.)
4
1 15
Exercise 10
12 72
(Solution on p. 11.)
8 Exercises
For the following 15 problems, identify each expression as a proper fraction, an improper fraction, or a mixed
number.
Exercise 11
(Solution on p. 11.)
3
2
Exercise 12
4
9
Exercise 13
(Solution on p. 11.)
5
7
Exercise 14
1
8
Exercise 15
(Solution on p. 11.)
6 41
Exercise 16
11
8
Exercise 17
1,001
(Solution on p. 11.)
12
Exercise 18
191 54
Exercise 19
(Solution on p. 11.)
9
1 13
Exercise 20
31 76
Exercise 21
1
3 40
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(Solution on p. 11.)
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Exercise 22
55
12
Exercise 23
(Solution on p. 11.)
0
9
Exercise 24
8
9
Exercise 25
(Solution on p. 11.)
1
101 11
For the following 15 problems, convert each of the improper fractions to its corresponding mixed number.
Exercise 26
11
6
Exercise 27
(Solution on p. 11.)
14
3
Exercise 28
25
4
Exercise 29
(Solution on p. 11.)
35
4
Exercise 30
71
8
Exercise 31
(Solution on p. 11.)
63
7
Exercise 32
121
11
Exercise 33
(Solution on p. 11.)
165
12
Exercise 34
346
15
Exercise 35
5,000
9
(Solution on p. 11.)
Exercise 36
23
5
Exercise 37
(Solution on p. 11.)
73
2
Exercise 38
19
2
Exercise 39
(Solution on p. 11.)
316
41
Exercise 40
800
3
For the following 15 problems, convert each of the mixed numbers to its corresponding improper fraction.
Exercise 41
4 81
Exercise 42
5
1 12
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(Solution on p. 12.)
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Exercise 43
9
(Solution on p. 12.)
6 79
Exercise 44
15 41
Exercise 45
5
10 11
(Solution on p. 12.)
Exercise 46
3
15 10
Exercise 47
(Solution on p. 12.)
8 32
Exercise 48
4 43
Exercise 49
21 52
(Solution on p. 12.)
Exercise 50
9
17 10
Exercise 51
(Solution on p. 12.)
20
9 21
Exercise 52
1
5 16
Exercise 53
1
90 100
(Solution on p. 12.)
Exercise 54
300 1,43
000
Exercise 55
19 87
(Solution on p. 12.)
Exercise 56
Why does 0 47 not qualify as a mixed number?
Hint:
See the denition of a mixed number.
Exercise 57
Why does 5 qualify as a mixed number?
note:
(Solution on p. 12.)
See the denition of a mixed number.
Calculator Problems
For the following 8 problems, use a calculator to convert each mixed number to its corresponding improper
fraction.
Exercise 58
11
35 12
Exercise 59
5
27 61
(Solution on p. 12.)
Exercise 60
40
83 41
Exercise 61
21
105 23
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(Solution on p. 12.)
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Exercise 62
72 605
606
Exercise 63
(Solution on p. 12.)
19
816 25
Exercise 64
42
708 51
Exercise 65
(Solution on p. 12.)
4,216
6, 012 8,
117
8.1 Exercises For Review
Exercise 66
() Round 2,614,000 to the nearest thousand.
Exercise 67
(Solution on p. 12.)
() Find the product. 1,004 · 1,005.
Exercise 68
() Determine if 41,826 is divisible by 2 and 3.
Exercise 69
(Solution on p. 12.)
() Find the least common multiple of 28 and 36.
Exercise 70
() Specify the numerator and denominator of the fraction
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19
.
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Solutions to Exercises in this Module
Solution to Exercise (p. 5)
4 12
Solution to Exercise (p. 5)
3 23
Solution to Exercise (p. 5)
3
1 11
Solution to Exercise (p. 5)
5
2 13
Solution to Exercise (p. 5)
19 43
Solution to Exercise (p. 5)
62
Solution to Exercise (p. 7)
33
4
Solution to Exercise (p. 7)
28
5
Solution to Exercise (p. 7)
19
15
Solution to Exercise (p. 7)
86
7
Solution to Exercise (p. 7)
improper fraction
Solution to Exercise (p. 7)
proper fraction
Solution to Exercise (p. 7)
mixed number
Solution to Exercise (p. 7)
improper fraction
Solution to Exercise (p. 7)
mixed number
Solution to Exercise (p. 7)
mixed number
Solution to Exercise (p. 8)
proper fraction
Solution to Exercise (p. 8)
mixed number
Solution to Exercise (p. 8)
4 23
Solution to Exercise (p. 8)
8 34
Solution to Exercise (p. 8)
9
Solution to Exercise (p. 8)
9
13 12
or 13 34
Solution to Exercise (p. 8)
555 59
Solution to Exercise (p. 8)
36 12
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Solution to Exercise (p. 8)
7 29
41
Solution to Exercise (p. 8)
33
8
Solution to Exercise (p. 9)
61
9
Solution to Exercise (p. 9)
115
11
Solution to Exercise (p. 9)
26
3
Solution to Exercise (p. 9)
107
5
Solution to Exercise (p. 9)
209
21
Solution to Exercise (p. 9)
9001
100
Solution to Exercise (p. 9)
159
8
Solution to Exercise (p. 9)
. . . because it may be written as 5 n0 , where n is any positive whole number.
Solution to Exercise (p. 9)
1,652
61
Solution to Exercise (p. 9)
2,436
23
Solution to Exercise (p. 10)
20,419
25
Solution to Exercise (p. 10)
48,803,620
8,117
Solution to Exercise (p. 10)
1,009,020
Solution to Exercise (p. 10)
252
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