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Textbook by: Denny Burzynski, Wade Ellis. E-mail the authors

# Proper Fractions, Improper Fractions, and Mixed Numbers

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses proper fractions, improper fractions, and mixed numbers. 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.

## 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.

## 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.

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

{}1212 size 12{ { {1} over {2} } } {}, 3535 size 12{ { {3} over {5} } } {}, 20272027 size 12{ { {"20"} over {"27"} } } {}, and 106255106255 size 12{ { {"106"} over {"255"} } } {}

Note that 1 < 21 < 2 size 12{"1 "<" 2"} {}, 3 < 53 < 5 size 12{"3 "<" 5"} {}, 20 < 2720 < 27 size 12{"20 "<" 27"} {}, and 106 < 225106 < 225 size 12{"106 "<" 225"} {}.

## 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

3232 size 12{ { {3} over {2} } } {}, 8585 size 12{ { {8} over {5} } } {}, 4444 size 12{ { {4} over {4} } } {}, and 1051610516 size 12{ { {"105"} over {"16"} } } {}

Note that 3232 size 12{3 >= 2} {}, 8585 size 12{8 >= 5} {}, 4444 size 12{4 >= 4} {}, and 1051610516 size 12{"105" >= "16"} {}.

## Positive Mixed Numbers

### Positive Mixed Numbers

A number of the form

nonzero whole number + proper fraction nonzero whole number + proper fraction size 12{"nonzero whole number "+" proper fraction"} {}

is called a positive mixed number. For example, 235235 size 12{2 { {3} over {5} } } {} 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.

## Relating Positive Improper Fractions and Positive Mixed Numbers

A relationship between improper fractions and mixed numbers is suggested by two facts. The first 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.

In the shaded region, there are 2 one thirds, or 2323 size 12{ { {2} over {3} } } {}.

2 1 3 = 2 3 2 1 3 = 2 3 size 12{2 left ( { {1} over {3} } right )= { {2} over {3} } } {}

There are 3 one thirds, or 3333 size 12{ { {3} over {3} } } {}, or 1.

3 1 3 = 3 3 or 1 3 1 3 = 3 3 or 1 size 12{3 left ( { {1} over {3} } right )= { {3} over {3} }  ital "or"1} {}

Thus,

3 3 = 1 3 3 = 1 size 12{ { {3} over {3} } =1} {}

Improper fraction = whole number.

There are 4 one thirds, or 4343 size 12{ { {4} over {3} } } {}, or 1 and 1313 size 12{ { {1} over {3} } } {}.

413=43413=43 size 12{4 left ( { {1} over {3} } right )= { {4} over {3} } } {}or 1and131and13 size 12{1 ital "and" { {1} over {3} } } {}

The terms 1 and 1313 size 12{ { {1} over {3} } } {} can be represented as 1+131+13 size 12{1+ { {1} over {3} } } {} or 113113 size 12{1 { {1} over {3} } } {}

Thus,

43=11343=113 size 12{ { {4} over {3} } =1 { {2} over {3} } } {}.

Improper fraction = mixed number.

There are 5 one thirds, or 5353 size 12{ { {5} over {3} } } {}, or 1 and 2323 size 12{ { {2} over {3} } } {}.

5 1 3 = 5 3 or 1 and 2 3 5 1 3 = 5 3 or 1 and 2 3 size 12{5 left ( { {1} over {3} } right )= { {5} over {3} }  ital "or"1 ital "and" { {2} over {3} } } {}

The terms 1 and 2323 size 12{ { {2} over {3} } } {} can be represented as 1+231+23 size 12{1+ { {2} over {3} } } {} or 123123 size 12{1 { {2} over {3} } } {}.

Thus,

53=12353=123 size 12{ { {5} over {3} } =1 { {2} over {3} } } {}.

Improper fraction = mixed number.

There are 6 one thirds, or 6363 size 12{ { {6} over {3} } } {}, or 2.

6 1 3 = 6 3 = 2 6 1 3 = 6 3 = 2 size 12{6 left ( { {1} over {3} } right )= { {6} over {3} } =2} {}

Thus,

6 3 = 2 6 3 = 2 size 12{ { {6} over {3} } =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 113113 size 12{1 { {1} over {3} } } {} can be expressed as 1+131+13 size 12{1+ { {1} over {3} } } {} The fraction 578578 size 12{5 { {7} over {8} } } {} can be expressed as 5+785+78 size 12{5+ { {7} over {8} } } {}.

It is important to note that a number such as 5+785+78 size 12{5+ { {7} over {8} } } {} does not indicate multiplication. To indicate multiplication, we would need to use a multiplication symbol (such as ⋅)

### Note:

578578 size 12{5 { {7} over {8} } } {} means 5+785+78 size 12{5+ { {7} over {8} } } {} and not 578578 size 12{5 cdot { {7} over {8} } } {}, which means 5 times 7878 size 12{ { {7} over {8} } } {} or 5 multiplied by 7878 size 12{ { {7} over {8} } } {}.

Thus, mixed numbers may be represented by improper fractions, and improper fractions may be represented by mixed numbers.

## Converting Improper Fractions to Mixed Numbers

To understand how we might convert an improper fraction to a mixed number, let's consider the fraction, 4343 size 12{ { {4} over {3} } } {}.

43 = 13 + 13 + 13 1 + 13 = 1 + 13 = 1 13 43 = 13 + 13 + 13 1 + 13 = 1 + 13 = 1 13

Thus, 43=11343=113 size 12{ { {4} over {3} } =1 { {1} over {3} } } {}.

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).

### Sample Set A

Convert each improper fraction to its corresponding mixed number.

#### Example 1

5353 size 12{ { {5} over {3} } } {} Divide 5 by 3.

The improper fraction 53=12353=123 size 12{ { {5} over {3} } =1 { {2} over {3} } } {}.

#### Example 2

469469 size 12{ { {"46"} over {9} } } {}. Divide 46 by 9.

The improper fraction 469=519469=519 size 12{ { {"46"} over {9} } =5 { {1} over {9} } } {}.

#### Example 3

83118311 size 12{ { {"83"} over {"11"} } } {}. Divide 83 by 11.

The improper fraction 8311=76118311=7611 size 12{ { {"83"} over {"11"} } =7 { {6} over {"11"} } } {}.

#### Example 4

10441044 size 12{ { {"104"} over {4} } } {}Divide 104 by 4.

104 4 = 26 0 4 = 26 104 4 = 26 0 4 = 26 size 12{ { {"104"} over {4} } ="26" { {0} over {4} } ="26"} {}

The improper fraction 1044=261044=26 size 12{ { {"104"} over {4} } ="26"} {}.

### Practice Set A

Convert each improper fraction to its corresponding mixed number.

#### Exercise 1

9292 size 12{ { {9} over {2} } } {}

#### Exercise 2

113113 size 12{ { {"11"} over {3} } } {}

#### Exercise 3

14111411 size 12{ { {"14"} over {"11"} } } {}

#### Exercise 4

31133113 size 12{ { {"31"} over {"13"} } } {}

#### Exercise 5

794794 size 12{ { {"79"} over {4} } } {}

#### Exercise 6

49684968 size 12{ { {"496"} over {8} } } {}

## 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 5353 size 12{ { {5} over {3} } } {} can be obtained from 123123 size 12{1 { {2} over {3} } } {} using multiplication in the following way.

Multiply: 31=331=3 size 12{3 cdot 1 - 3} {}

Add: 3+2=53+2=5 size 12{3+2=5} {}. Place the 5 over the 3: 5353 size 12{ { {5} over {3} } } {}

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.

### Sample Set B

Convert each mixed number to an improper fraction.

#### Example 5

5 7 8 5 7 8 size 12{5 { {7} over {8} } } {}

1. Multiply: 85=4085=40 size 12{8 cdot 5="40"} {}.
2. Add: 40 + 7 = 4740 + 7 = 47 size 12{"40 "+" 7 "=" 47"} {}.
3. Place 47 over 8: 478478 size 12{ { {"47"} over {8} } } {}.

Thus, 578=478578=478 size 12{5 { {7} over {8} } = { {"47"} over {8} } } {}.

#### Example 6

16 2 3 16 2 3 size 12{"16" { {2} over {3} } } {}

1. Multiply: 3 16 = 483 16 = 48 size 12{"3 " cdot " 16 "=" 48"} {}.
2. Add: 48 + 2 = 5048 + 2 = 50 size 12{"48 "+" 2 "=" 50"} {}.
3. Place 50 over 3: 503503 size 12{ { {"50"} over {3} } } {}

Thus, 1623=5031623=503 size 12{"16" { {2} over {3} } = { {"50"} over {3} } } {}

### Practice Set B

Convert each mixed number to its corresponding improper fraction.

#### Exercise 7

814814 size 12{8 { {1} over {4} } } {}

#### Exercise 8

535535 size 12{5 { {3} over {5} } } {}

#### Exercise 9

14151415 size 12{1 { {4} over {"15"} } } {}

#### Exercise 10

12271227 size 12{"12" { {2} over {7} } } {}

## Exercises

For the following 15 problems, identify each expression as a proper fraction, an improper fraction, or a mixed number.

### Exercise 11

3232 size 12{ { {3} over {2} } } {}

### Exercise 12

4949 size 12{ { {4} over {9} } } {}

### Exercise 13

5757 size 12{ { {5} over {7} } } {}

### Exercise 14

1818 size 12{ { {1} over {8} } } {}

### Exercise 15

614614 size 12{6 { {1} over {4} } } {}

### Exercise 16

118118 size 12{ { {"11"} over {8} } } {}

### Exercise 17

1,001121,00112 size 12{ { {1,"001"} over {"12"} } } {}

### Exercise 18

1914519145 size 12{"191" { {4} over {5} } } {}

### Exercise 19

19131913 size 12{1 { {9} over {"13"} } } {}

### Exercise 20

31673167 size 12{"31" { {6} over {7} } } {}

### Exercise 21

31403140 size 12{3 { {1} over {"40"} } } {}

### Exercise 22

55125512 size 12{ { {"55"} over {"12"} } } {}

### Exercise 23

0909 size 12{ { {0} over {9} } } {}

### Exercise 24

8989 size 12{ { {8} over {9} } } {}

### Exercise 25

101111101111 size 12{"101" { {1} over {"11"} } } {}

For the following 15 problems, convert each of the improper fractions to its corresponding mixed number.

### Exercise 26

116116 size 12{ { {"11"} over {6} } } {}

### Exercise 27

143143 size 12{ { {"14"} over {3} } } {}

### Exercise 28

254254 size 12{ { {"25"} over {4} } } {}

### Exercise 29

354354 size 12{ { {"35"} over {4} } } {}

### Exercise 30

718718 size 12{ { {"71"} over {8} } } {}

### Exercise 31

637637 size 12{ { {"63"} over {7} } } {}

### Exercise 32

1211112111 size 12{ { {"121"} over {"11"} } } {}

### Exercise 33

1651216512 size 12{ { {"165"} over {"12"} } } {}

### Exercise 34

3461534615 size 12{ { {"346"} over {"15"} } } {}

### Exercise 35

5,00095,0009 size 12{ { {5,"000"} over {9} } } {}

### Exercise 36

235235 size 12{ { {"23"} over {5} } } {}

### Exercise 37

732732 size 12{ { {"73"} over {2} } } {}

### Exercise 38

192192 size 12{ { {"19"} over {2} } } {}

### Exercise 39

3164131641 size 12{ { {"316"} over {"41"} } } {}

### Exercise 40

80038003 size 12{ { {"800"} over {3} } } {}

For the following 15 problems, convert each of the mixed num­bers to its corresponding improper fraction.

### Exercise 41

418418 size 12{4 { {1} over {8} } } {}

### Exercise 42

15121512 size 12{1 { {5} over {"12"} } } {}

### Exercise 43

679679 size 12{6 { {7} over {9} } } {}

### Exercise 44

15141514 size 12{"15" { {1} over {4} } } {}

### Exercise 45

1051110511 size 12{"10" { {5} over {"11"} } } {}

### Exercise 46

1531015310 size 12{"15" { {3} over {"10"} } } {}

### Exercise 47

823823 size 12{8 { {2} over {3} } } {}

### Exercise 48

434434 size 12{4 { {3} over {4} } } {}

### Exercise 49

21252125 size 12{"21" { {2} over {5} } } {}

### Exercise 50

1791017910 size 12{"17" { {9} over {"10"} } } {}

### Exercise 51

9202192021 size 12{9 { {"20"} over {"21"} } } {}

### Exercise 52

51165116 size 12{5 { {1} over {"16"} } } {}

### Exercise 53

901100901100 size 12{"90" { {1} over {"100"} } } {}

### Exercise 54

300431,000300431,000 size 12{"300" { {"43"} over {1,"000"} } } {}

### Exercise 55

19781978 size 12{"19" { {7} over {8} } } {}

### Exercise 56

Why does 047047 size 12{0 { {4} over {7} } } {} not qualify as a mixed number?

#### Hint:

See the definition of a mixed number.

### Exercise 57

Why does 5 qualify as a mixed number?

#### Note:

See the definition 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

351112351112 size 12{"35" { {"11"} over {"12"} } } {}

### Exercise 59

2756127561 size 12{"27" { {5} over {"61"} } } {}

### Exercise 60

834041834041 size 12{"83" { {"40"} over {"41"} } } {}

### Exercise 61

10521231052123 size 12{"105" { {"21"} over {"23"} } } {}

### Exercise 62

7260560672605606 size 12{"72" { {"605"} over {"606"} } } {}

### Exercise 63

81619258161925 size 12{"816" { {"19"} over {"25"} } } {}

### Exercise 64

70842517084251 size 12{"708" { {"42"} over {"51"} } } {}

### Exercise 65

6,0124,2168,1176,0124,2168,117 size 12{6,"012" { {4,"216"} over {8,"117"} } } {}

### Exercises For Review

#### Exercise 66

((Reference)) Round 2,614,000 to the nearest thousand.

#### Exercise 67

((Reference)) Find the product. 1,004 1,0051,004 1,005 size 12{"1,004 " cdot "1,005"} {}.

#### Exercise 68

((Reference)) Determine if 41,826 is divisible by 2 and 3.

#### Exercise 69

((Reference)) Find the least common multiple of 28 and 36.

#### Exercise 70

((Reference)) Specify the numerator and denominator of the fraction 12191219 size 12{ { {"12"} over {"19"} } } {}.

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