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Introduction to Fractions and Multiplication and Division of Fractions: 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.

A number line. 0 is marked with a black dot, and 1 is marked with a hollow dot. The distance between the two is labeled, all proper fractions are located in this interval.

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.

A number line. 0 is labeled, and 1 is marked with a hollow dot. An arrow is drawn to the right, labeled Positive improper fractions.

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.

A number line. 0 is labeled, and 1 is marked with a hollow dot. An arrow is drawn to the right, labeled Positive mixed numbers.

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.

A rectangle divided into three equal parts with vertical bars. Each part contains the fraction, one-third.

Now, consider the following examples by observing the respective shaded areas.

A rectangle divided into three equal parts with vertical bars. Each part contains the fraction, one-third. The two leftmost parts are shaded.

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} } } {}

A rectangle divided into three equal parts with vertical bars. Each part contains the fraction, one-third. All three parts are shaded.

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.

A rectangle divided into three equal parts with vertical bars. Each part contains the fraction, one-third. All three parts are shaded. A rectangle divided into three equal parts with vertical bars. Each part contains the fraction, one-third. One part is shaded.

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.

A rectangle divided into three equal parts with vertical bars. Each part contains the fraction, one-third. All three parts are shaded. A rectangle divided into three equal parts with vertical bars. Each part contains the fraction, one-third. The two leftmost parts are shaded.

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.

A rectangle divided into three equal parts with vertical bars. Each part contains the fraction, one-third. All three parts are shaded. A rectangle divided into three equal parts with vertical bars. Each part contains the fraction, one-third. All three parts are shaded.

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} } } {}.

Two rectangles, each divided into three equal parts with vertical bars. Each part contains the fraction, one-third. Under the rectangle on the left is a bracket grouping all three parts together to make one. Under the rectangle on the right is a bracket under only one of the three parts, making one third. The two bracketed segments are added together.

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.

Long division. 5 divided by 3 is one, with a remainder of 2. 1 is the whole number part, 2 is the numerator of the fractional part, and 3 is the denominator of the fractional part.

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

A number line with marks for 0, 1, and 2. In between 1 and 2 is a dot for five thirds, or one and two thirds.

Example 2

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

Long division. 46 divided by 9 is 5, with a remainder of 1. 5 is the whole number part, 1 is the numerator of the fractional part, and 9 is the denominator of the fractional part.

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

A number line with marks for 0, 5, and 6. In between 5 and 6 is a dot showing the location of forty-six ninths, or five and one ninth.

Example 3

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

Long division. 83 divided by 11 is 7, with a remainder of 6. 7 is the whole number part, 6 is the numerator of the fractional part, and 11 is the denominator of the fractional part.

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

A number line with marks for 0, 7, and 8. In between 7 and 8 is a dot showing the location of eighty-three elevenths, or seven and six elevenths.

Example 4

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

Long division. 104 divided by 4 is 26, with a remainder of 0. 26 is the whole number part, 0 is the numerator of the fractional part, and 4 is the denominator of the fractional part.

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"} {}.

A number line with marks for 0, 25, 26, and 27. 26 is marked with a dot, showing the location of one hundred four fourths.

Practice Set A

Convert each improper fraction to its corresponding mixed number.

Exercise 1

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

Solution

412412 size 12{4 { {1} over {2} } } {}

Exercise 2

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

Solution

323323 size 12{3 { {2} over {3} } } {}

Exercise 3

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

Solution

13111311 size 12{1 { {3} over {"11"} } } {}

Exercise 4

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

Solution

25132513 size 12{2 { {5} over {"13"} } } {}

Exercise 5

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

Solution

19341934 size 12{"19" { {3} over {4} } } {}

Exercise 6

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

Solution

62

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.

Two rectangles, each divided into three equal parts with vertical bars. Each part contains the fraction, one-third. Under the rectangle on the left is a bracket grouping all three parts together to make one. Under the rectangle on the right is a bracket under two of the three parts, making two thirds. The two bracketed segments are added together.

one and two thirds is equivalent to one plus two thirds. One can be expanded to three thirds, making the original number equivalent to the sum of five one-thirds, or five thirds.

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} {}

one and two thirds, with an arrow drawn from the denominator to the one.

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} } } {}.

A number line showing the location of five and seven eigths, or 47 eights.

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} } } {}

Solution

334334 size 12{ { {"33"} over {4} } } {}

Exercise 8

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

Solution

285285 size 12{ { {"28"} over {5} } } {}

Exercise 9

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

Solution

19151915 size 12{ { {"19"} over {"15"} } } {}

Exercise 10

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

Solution

867867 size 12{ { {"86"} 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} } } {}

Solution

improper fraction

Exercise 12

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

Exercise 13

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

Solution

proper fraction

Exercise 14

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

Exercise 15

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

Solution

mixed number

Exercise 16

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

Exercise 17

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

Solution

improper fraction

Exercise 18

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

Exercise 19

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

Solution

mixed number

Exercise 20

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

Exercise 21

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

Solution

mixed number

Exercise 22

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

Exercise 23

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

Solution

proper fraction

Exercise 24

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

Exercise 25

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

Solution

mixed number

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} } } {}

Solution

423423 size 12{4 { {2} over {3} } } {}

Exercise 28

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

Exercise 29

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

Solution

834834 size 12{8 { {3} over {4} } } {}

Exercise 30

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

Exercise 31

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

Solution

99 size 12{9} {}

Exercise 32

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

Exercise 33

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

Solution

1391213912 size 12{"13" { {9} over {"12"} } } {} or 13341334 size 12{"13" { {3} over {"4"} } } {}

Exercise 34

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

Exercise 35

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

Solution

5555955559 size 12{"555" { {5} over {9} } } {}

Exercise 36

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

Exercise 37

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

Solution

36123612 size 12{"36" { {1} over {2} } } {}

Exercise 38

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

Exercise 39

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

Solution

7294172941 size 12{7 { {"29"} 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} } } {}

Solution

338338 size 12{ { {"33"} over {8} } } {}

Exercise 42

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

Exercise 43

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

Solution

619619 size 12{ { {"61"} over {9} } } {}

Exercise 44

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

Exercise 45

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

Solution

1151111511 size 12{ { {"115"} over {"11"} } } {}

Exercise 46

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

Exercise 47

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

Solution

263263 size 12{ { {"26"} over {3} } } {}

Exercise 48

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

Exercise 49

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

Solution

10751075 size 12{ { {"107"} over {5} } } {}

Exercise 50

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

Exercise 51

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

Solution

2092120921 size 12{ { {"209"} over {"21"} } } {}

Exercise 52

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

Exercise 53

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

Solution

90011009001100 size 12{ { {"9001"} over {"100"} } } {}

Exercise 54

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

Exercise 55

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

Solution

15981598 size 12{ { {"159"} 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.

Solution

… because it may be written as 5 0n 50n , where nn is any positive whole 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"} } } {}

Solution

1,652611,65261 size 12{ { {1,"652"} over {"61"} } } {}

Exercise 60

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

Exercise 61

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

Solution

2,436232,43623 size 12{ { {2,"436"} over {"23"} } } {}

Exercise 62

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

Exercise 63

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

Solution

20,4192520,41925 size 12{ { {"20","419"} 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"} } } {}

Solution

48,803,6208,11748,803,6208,117 size 12{ { {"48","803","620"} 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"} {}.

Solution

1,009,020

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.

Solution

252

Exercise 70

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

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