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Fractions of Whole 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 fractions of whole numbers. By the end of the module students should be able to understand the concept of fractions of whole numbers and recognize the parts of a fraction.

Section Overview

  • More Numbers on the Number Line
  • Fractions of Whole Numbers
  • The Parts of a Fraction
  • Reading and Writing Fractions

More Numbers on the Number Line

In Chapters (Reference), (Reference), and (Reference), we studied the whole numbers and methods of combining them. We noted that we could visually display the whole numbers by drawing a number line and placing closed circles at whole number locations.

A number line with dots on the whole numbers.

By observing this number line, we can see that the whole numbers do not account for every point on the line. What numbers, if any, can be associated with these points? In this section we will see that many of the points on the number line, including the points already associated with whole numbers, can be associated with numbers called fractions.

Fractions of Whole Numbers

The Nature of the Positive Fractions

We can extend our collection of numbers, which now contains only the whole numbers, by including fractions of whole numbers. We can determine the nature of these fractions using the number line.

If we place a pencil at some whole number and proceed to travel to the right to the next whole number, we see that our journey can be broken into different types of equal parts as shown in the following examples.

  1. 1 part.
     A number line. Two marks: one on the left, labeled, whole number, and one on the right, labeled next whole number. An arrow is drawn from the whole number to the next whole number.

  2. 2 equal parts.
     A number line. Two marks: one on the left, labeled, whole number, and one on the right, labeled next whole number. In between the two marks is the midway point. An arrow is drawn from the whole number to the midway point, and then from the midway point to the next whole number.

  3. 3 equal parts.
     A number line. Two marks: one on the left, labeled, whole number, and one on the right, labeled next whole number. There are three arrows, connecting the two whole numbers and two evenly-spaced hash marks in between the whole numbers.

  4. 4 equal parts.
     A number line. Two marks: one on the left, labeled, whole number, and one on the right, labeled next whole number. There are four arrows, connecting the two whole numbers and three evenly-spaced hash marks in between the whole numbers.

The Latin Word Fractio

Notice that the number of parts, 2, 3, and 4, that we are breaking the original quantity into is always a nonzero whole number. The idea of breaking up a whole quantity gives us the word fraction. The word fraction comes from the Latin word "fractio" which means a breaking, or fracture.

Suppose we break up the interval from some whole number to the next whole number into five equal parts.

A number line. Two marks: one on the left, labeled whole number, and one on the right, labeled next whole number. There are four hash marks in between the two whole numbers, creating five spaces of equal width, labeled Part 1 through Part 5.

After starting to move from one whole number to the next, we decide to stop after covering only two parts. We have covered 2 parts of 5 equal parts. This situation is described by writing 2525 size 12{ { {2} over {5} } } {}.

A number line. Two marks: one on the left, labeled whole number, and one on the right, labeled next whole number. There are four evenly-spaced hash marks in between the two marks. There is an arrow from the whole number to the first hash mark, and an arrow from the first hash mark to the second hash mark.

Positive Fraction

A number such as 25 25 size 12{ { {2} over {5} } } {} is called a positive fraction, or more simply, a fraction.

The Parts of a Fraction

A fraction has three parts.

  1. The fraction bar               
    .

    Fraction Bar

    The fraction bar serves as a grouping symbol. It separates a quantity into individual groups. These groups have names, as noted in 2 and 3 below.

  2. The nonzero number below the fraction bar.

    Denominator

    This number is called the denominator of the fraction, and it indicates the number of parts the whole quantity has been divided into. Notice that the denominator must be a nonzero whole number since the least number of parts any quantity can have is one.

  3. The number above the fraction bar.

    Numerator

    This number is called the numerator of the fraction, and it indicates how many of the specified parts are being considered. Notice that the numerator can be any whole number (including zero) since any number of the specified parts can be considered.

    whole number nonzero whole number ↔ numerator denominator whole number nonzero whole number ↔ numerator denominator

Sample Set A

The diagrams in the following problems are illustrations of fractions.

Example 1

Diagrams 1
(a) A whole circle(b) The whole circle divided into 3 equal parts(c) 1 of the 3 equal parts
A whole circle.A whole circle divided into 3 equal parts.A whole circle divided into 3 equal parts. One of the parts is shaded.

The fraction one-third. This would be read, 1 of 3 equal parts.

The fraction 1313 size 12{ { {1} over {3} } } {} is read as "one third."

Example 2

Figure 2
(a) A whole rectangle(b) The whole rectangle divided into 5 equal parts(c) 3 of the 5 equal parts
A rectangle.A rectangle divided into five equal parts by vertical bars.A rectangle divided into five equal parts by vertical bars. The three leftmost bars are shaded.

The fraction three-fifths. This would be read, 3 of 5 equal parts.

The fraction 3535 "is read as "three fifths."

Example 3

Figure 3
(a) The number line between 0 and 1
A number line with two marks, 0 and 1.
(b) The number line between 0 and 1 divided into 7 equal parts
A number line with two marks, 0, and 1, and six hash marks in between.
(c) 6 of the 7 equal parts
A number line with two marks, 0, and 1, and six hash marks in between. There is an arrow drawn from the zero to the first hash mark, and then from that hash mark to the next, until the sixth hash mark, where the arrows stop.

The fraction six-sevenths. This would be read, 6 of the 7 equal parts.

The fraction 6767 is read as "six sevenths."

Example 4

Figure 4
(a) A whole circle(b) The whole circle divided into 4 equal parts(c) 4 of the 4 equal parts
A whole circle.A whole circle divided into four equal parts.A whole circle divided into four equal parts. All four parts are shaded.

The fraction four-fourths. This would be read, 4 of the 4 equal parts.

When the numerator and denominator are equal, the fraction represents the entire quantity, and its value is 1.

nonzero whole number same nonzero whole number = 1 nonzero whole number same nonzero whole number =1

Practice Set A

Specify the numerator and denominator of the following fractions.

Exercise 1

4747 size 12{ { {4} over {7} } } {}

Solution

4, 7

Exercise 2

5858 size 12{ { {5} over {8} } } {}

Solution

5, 8

Exercise 3

10151015 size 12{ { {"10"} over {"15"} } } {}

Solution

10, 15

Exercise 4

1919 size 12{ { {1} over {9} } } {}

Solution

1, 9

Exercise 5

0202 size 12{ { {0} over {2} } } {}

Solution

0, 2

Reading and Writing Fractions

In order to properly translate fractions from word form to number form, or from number form to word form, it is necessary to understand the use of the hyphen.

Use of the Hyphen

One of the main uses of the hyphen is to tell the reader that two words not ordinarily joined are to be taken in combination as a unit. Hyphens are always used for numbers between and including 21 and 99 (except those ending in zero).

Sample Set B

Write each fraction using whole numbers.

Example 5

Fifty three-hundredths.
The hyphen joins the words three and hundredths and tells us to consider them as a single unit. Therefore,
fifty three-hundredths translates as 5030050300 size 12{ { {"50"} over {"300"} } } {}

Example 6

Fifty-three hundredths.
The hyphen joins the numbers fifty and three and tells us to con­sider them as a single unit. Therefore,
fifty-three hundredths translates as 5310053100 size 12{ { {"53"} over {"100"} } } {}

Example 7

Four hundred seven-thousandths.
The hyphen joins the words seven and thousandths and tells us to consider them as a single unit. Therefore,
four hundred seven-thousandths translates as 4007,0004007,000 size 12{ { {"400"} over {"7000"} } } {}

Example 8

Four hundred seven thousandths.
The absence of hyphens indicates that the words seven and thousandths are to be considered individually.
four hundred seven thousandths translates as 40710004071000 size 12{ { {"407"} over {"1000"} } } {}

Write each fraction using words.

Example 9

21852185 size 12{ { {"21"} over {"85"} } } {} translates as twenty-one eighty-fifths.

Example 10

2003,0002003,000 size 12{ { {"200"} over {"3000"} } } {} translates as two hundred three-thousandths.
A hyphen is needed between the words three and thousandths to tell the reader that these words are to be considered as a single unit.

Example 11

2031,0002031,000 size 12{ { {"203"} over {"1000"} } } {} translates as two hundred three thousandths.

Practice Set B

Write the following fractions using whole numbers.

Exercise 6

one tenth

Solution

110110 size 12{ { {1} over {"10"} } } {}

Exercise 7

eleven fourteenths

Solution

11141114 size 12{ { {"11"} over {"14"} } } {}

Exercise 8

sixteen thirty-fifths

Solution

16351635 size 12{ { {"16"} over {"35"} } } {}

Exercise 9

eight hundred seven-thousandths

Solution

8007,0008007,000 size 12{ { {"800"} over {7,"000"} } } {}

Write the following using words.

Exercise 10

3838 size 12{ { {3} over {8} } } {}

Solution

three eighths

Exercise 11

110110 size 12{ { {1} over {"10"} } } {}

Solution

one tenth

Exercise 12

32503250 size 12{ { {3} over {"250"} } } {}

Solution

three two hundred fiftieths

Exercise 13

1143,1901143,190 size 12{ { {"114"} over {"3190"} } } {}

Solution

one hundred fourteen three thousand one hundred ninetieths

Name the fraction that describes each shaded portion.

Exercise 14

A whole circle divided into eight equal parts, with three parts shaded.

Solution

3 8 3 8 size 12{ { {3} over {8} } } {}

Exercise 15

A whole circle divided into sixteen equal parts, with one part shaded.

Solution

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

In the following 2 problems, state the numerator and denominator, and write each fraction in words.

Exercise 16

The number 5959 size 12{ { {5} over {9} } } {} is used in converting from Fahrenheit to Celsius.

Solution

5, 9, five ninths

Exercise 17

A dime is 110110 size 12{ { {1} over {"10"} } } {} of a dollar.

Solution

1, 10, one tenth

Exercises

For the following 10 problems, specify the numerator and denominator in each fraction.

Exercise 18

3434 size 12{ { {3} over {4} } } {}

Solution

numerator, 3; denominator, 4

Exercise 19

910910 size 12{ { {9} over {"10"} } } {}

Exercise 20

1515 size 12{ { {1} over {5} } } {}

Solution

numerator, 1; denominator, 5

Exercise 21

5656 size 12{ { {5} over {6} } } {}

Exercise 22

7777 size 12{ { {7} over {7} } } {}

Solution

numerator, 7; denominator, 7

Exercise 23

4646 size 12{ { {4} over {6} } } {}

Exercise 24

012012 size 12{ { {0} over {"12"} } } {}

Solution

numerator, 0; denominator, 12

Exercise 25

25252525 size 12{ { {"25"} over {"25"} } } {}

Exercise 26

181181 size 12{ { {"18"} over {1} } } {}

Solution

numerator, 18; denominator, 1

Exercise 27

016016 size 12{ { {0} over {"16"} } } {}

For the following 10 problems, write the fractions using whole numbers.

Exercise 28

four fifths

Solution

4545 size 12{ { {4} over {5} } } {}

Exercise 29

two ninths

Exercise 30

fifteen twentieths

Solution

15201520 size 12{ { {"15"} over {"20"} } } {}

Exercise 31

forty-seven eighty-thirds

Exercise 32

ninety-one one hundred sevenths

Solution

9110791107 size 12{ { {"91"} over {"107"} } } {}

Exercise 33

twenty-two four hundred elevenths

Exercise 34

six hundred five eight hundred thirty-fourths

Solution

605834605834 size 12{ { {"658"} over {"134"} } } {}

Exercise 35

three thousand three forty-four ten-thousandths

Exercise 36

ninety-two one-millionths

Solution

921,000,000921,000,000 size 12{ { {"92"} over {1,"000","000"} } } {}

Exercise 37

one three-billionths

For the following 10 problems, write the fractions using words.

Exercise 38

5959 size 12{ { {5} over {9} } } {}

Solution

five ninths

Exercise 39

610610 size 12{ { {6} over {"10"} } } {}

Exercise 40

815815 size 12{ { {8} over {"15"} } } {}

Solution

eight fifteenths

Exercise 41

10131013 size 12{ { {"10"} over {"13"} } } {}

Exercise 42

7510075100 size 12{ { {"75"} over {"100"} } } {}

Solution

seventy-five one hundredths

Exercise 43

8613586135 size 12{ { {"86"} over {"135"} } } {}

Exercise 44

9161,0149161,014 size 12{ { {"916"} over {"1014"} } } {}

Solution

nine hundred sixteen one thousand fourteenths

Exercise 45

50110,00150110,001 size 12{ { {"501"} over {"10001"} } } {}

Exercise 46

1831,6081831,608 size 12{ { {"18"} over {"31608"} } } {}

Solution

eighteen thirty-one thousand six hundred eighths

Exercise 47

1500,0001500,000 size 12{ { {1} over {"500000"} } } {}

For the following 4 problems, name the fraction corresponding to the shaded portion.

Exercise 48

A whole circle divided into two equal parts, with one part shaded.

Solution

1 2 1 2 size 12{ { {1} over {2} } } {}

Exercise 49

A whole circle divided into six equal parts, with two parts shaded.

Exercise 50

A rectangle divided into seven equal parts, with four parts shaded.

Solution

4 7 4 7 size 12{ { {4} over {7} } } {}

Exercise 51

A rectangle divided into three equal parts, with zero parts shaded.

For the following 4 problems, shade the portion corresponding to the given fraction on the given figure.

Exercise 52

3535
A rectangle divided into five equal parts.

Solution

A rectangle divided into five equal parts, with three parts shaded.

Exercise 53

1818
A circle divided into eight equal parts.

Exercise 54

6666
A rectangle divided into six equal parts.

Solution

A rectangle divided into six equal parts, with all six parts shaded.

Exercise 55

0303
 A circle divided into three equal parts.

State the numerator and denominator and write in words each of the fractions appearing in the state­ments for the following 10 problems.

Exercise 56

A contractor is selling houses on 1414 size 12{ { {1} over {4} } } {} acre lots.

Solution

Numerator, 1; denominator, 4; one fourth

Exercise 57

The fraction 227227 size 12{ { {"22"} over {7} } } {} is sometimes used as an approximation to the number π π. (The symbol is read “pi.")

Exercise 58

The fraction 4343 size 12{ { {4} over {3} } } {} is used in finding the volume of a sphere.

Solution

Numerator, 4; denominator, 3; four thirds

Exercise 59

One inch is 112112 size 12{ { {1} over {"12"} } } {} of a foot.

Exercise 60

About 2727 size 12{ { {2} over {7} } } {} of the students in a college statistics class received a “B” in the course.

Solution

Numerator, 2; denominator, 7; two sevenths

Exercise 61

The probability of randomly selecting a club when drawing one card from a standard deck of 52 cards is 13521352 size 12{ { {"13"} over {"52"} } } {}.

Exercise 62

In a box that contains eight computer chips, five are known to be good and three are known to be defective. If three chips are selected at random, the probability that all three are defective is 156156 size 12{ { {1} over {"56"} } } {}.

Solution

Numerator, 1; denominator, 56; one fifty-sixth

Exercise 63

In a room of 25 people, the probability that at least two people have the same birthdate (date and month, not year) is 56910005691000 size 12{ { {"569"} over {"1000"} } } {}.

Exercise 64

The mean (average) of the numbers 21, 25, 43, and 36 is 12541254 size 12{ { {"125"} over {4} } } {}.

Solution

Numerator, 125; denominator, 4; one hundred twenty-five fourths

Exercise 65

If a rock falls from a height of 20 meters on Jupiter, the rock will be 32253225 size 12{ { {"32"} over {"25"} } } {} meters high after 6565 size 12{ { {6} over {5} } } {} seconds.

Exercises For Review

Exercise 66

((Reference)) Use the numbers 3 and 11 to illustrate the commutative property of addition.

Solution

3+11=11+3=143+11=11+3=14 size 12{3+"11"="11"+3="14"} {}

Exercise 67

((Reference)) Find the quotient. 676÷26676÷26 size 12{"676" div "26"} {}

Exercise 68

((Reference)) Write 7⋅7⋅7⋅7⋅77⋅7⋅7⋅7⋅7 size 12{7 cdot 7 cdot 7 cdot 7 cdot 7} {} using exponents.

Solution

7575 size 12{7 rSup { size 8{5} } } {}

Exercise 69

((Reference)) Find the value of 8⋅6+208+3⋅6+16228⋅6+208+3⋅6+1622 size 12{ { {8 cdot left (6+"20" right )} over {8} } + { {3 cdot left (6+"16" right )} over {"22"} } } {}.

Exercise 70

((Reference)) Find the least common multiple of 12, 16, and 18.

Solution

144

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