Fractions and Mixed Numbers

Examples on fractions and mixed numbers along with more questions and their solutions .
The fractions and like fractions are defined using examples. The naming and saying of a list of fractions is presented as an example.
The definition ofproper and improper fractions , mixed numbers , the relationship between improper fractions and mixed numbers are also included.
The evaluation of fractions of quantities as well as equivalent fractions are explained through examples. Reducing fractions to lower terms and comparing fractions are also discussed by examples.


A fraction is used to express part of a whole. It may be written in the form \( \displaystyle \frac{a}{b} \) where \( a \) is called the numerator and \( b \) is called the denominator.


What Are Fractions?

Example 1
You have a wire and you cut it into 4 equal parts as shown in the diagram below. Each small part is \( \displaystyle \frac{1}{4} \): a whole represented by "1" divided by "4".

whole wire
You now take 3 parts (in red) of the 4 parts as shown in the diagram below

a fraction of a whole 3/4
This (red part) is represented mathematically as a fraction and written \(\displaystyle \frac{3}{4} \) (or \( 3/4 \) ). The denominator of this fraction is 4 and the numerator is 3.

Example 2
You have large square piece of paper that you cut into 8 equal parts as shown below. Each small part is \( \displaystyle \frac{1}{8} \).

whole square
You now paint (in red) 5 of the 8 parts red as shown in the diagram below.

a fraction of a whole 5/8
The painted part of the whole paper is represented diagrammatically by 5 small parts and mathematically as a fraction written as \( \displaystyle \frac{5}{8} \) (or \( 5/8 \) ). The denominator of this fraction is 8 and the numerator is 5.
Notes:
1) Fractions with the same denominator are called like fractions .
2) A fraction cannot have denominator equal to zero as it is undefined.



Naming and "Saying" Fractions

\( \displaystyle \frac{1}{2} \) : one half , 1 over 2

\( \displaystyle \frac{1}{3} \) : one third , 1 over 3

\( \displaystyle \frac{1}{4} \) : one-fourth (or a quarter) , 1 over 4

\( \displaystyle \frac{1}{5} \) : one-fifth , 1 over 5

\( \displaystyle \frac{1}{6} \) : one-sixth , 1 over 6

\( \displaystyle \frac{1}{7} \) : one-seventh , 1 over 7

\( \displaystyle \frac{1}{8} \) : one-eighth , 1 over 8

\( \displaystyle \frac{1}{9} \) : one-ninth , 1 over 9

\( \displaystyle \frac{1}{10} \) : one-tenth , 1 over 10

\( \displaystyle \frac{2}{3} \) : two-thirds , 2 over 3

\( \displaystyle \frac{3}{5} \) : three-fifths, 3 over 5

\( \displaystyle \frac{5}{6} \) : five-sixths , 5 over 6

\( \displaystyle \frac{3}{2} \) : three-halves , 3 over 2

\( \displaystyle \frac{8}{11} \) : eight-elevenths , 8 over 11

\( \displaystyle \frac{3}{8} \) : three-eighths , 3 over 8



Proper and Improper Fractions

A fraction with the numerator less than the denominator is called a proper fraction; otherwise it is called an improper fraction.
Example 3
The following are proper fractions: \( \displaystyle \frac{1}{2} \) , \( \displaystyle \frac{3}{4} \) , \( \displaystyle \frac{12}{23} \)

The following are improper fractions: \( \displaystyle \frac{2}{2} \) , \( \displaystyle \frac{6}{5} \) , \( \displaystyle \frac{25}{24} \)



Mixed Numbers

The sum of a whole number and a proper fraction is called a mixed number.
Example 4
Let us assume that each square in the diagram below represents one unit. There are 3 whole squares (or units) painted red plus a fraction of \( \displaystyle \frac{3}{4} \) also painted red. Using mixed numbers, the total area painted red is:
\( 3 \) units + \( \displaystyle \frac{3}{4} \) of one unit or simply written as a mixed number as follows: \[ \quad 3 \displaystyle \frac{3}{4} \]

meaning of mixed number using a diagram



Relationship Between and Improper Fractions and Mixed Numbers

Another way of representing the area painted red in the diagram above is to split each of the 3 units into 4 equal parts as shown in the diagram below. We now count the total number of the (small) equal parts and express the area in read using improper fractions as
\[ \displaystyle \frac{15}{4} \]
improper fraction
In the above example the relationship between the mixed number and improper fraction is

\( \displaystyle \frac{15}{4} = 3 \displaystyle \frac{3}{4} \)

1 - How Convert a mixed number into an improper fraction?
Example 5
Convert \( 3 \displaystyle \frac{3}{4} \) into an improper fraction.
Solution to Example 5

Write the mixed number as a sum of a whole number and a fraction
\( 3 \displaystyle \frac{3}{4} = 3 + \displaystyle \frac{3}{4}\)
then convert the whole number into a fraction using the same denominator 4 as the fraction
\( 3 = \displaystyle \frac{12}{4} \)
We now add the two fractions
\( 3 \displaystyle \frac{3}{4} = 3 + \displaystyle \frac{3}{4} = \displaystyle \frac{12}{4} + \displaystyle \frac{3}{4} = \displaystyle \frac{15}{4}\)



2 - How Convert an improper fraction into a Mixed Number?
Example 6
Convert the improper fraction \( \displaystyle \frac{15}{4} \) into a mixed number.
Solution to Example 6

To convert \( \displaystyle \frac{15}{4} \) into a mixed number, first divide \(15\) by \(4\) using the long division.
The quotient of the division is the whole number, the remainder of the division is the numerator of the fraction and the denominator of the fraction is the divisor 4.
Divide 15 by 4, you get a quotient 3 and a remainder equal to 3. Hence
\( \displaystyle \frac{15}{4} = quotient + \displaystyle \frac{remainder}{4} = 3 + \displaystyle \frac{3}{4} = 3\displaystyle \frac{3}{4}\)



Evaluate Fractions of Quantities

Example 7
What is \( \displaystyle \frac{2}{5} \) of \( 500 \) ?
Solution to Example 7

\( \displaystyle \frac{2}{5} \) of \( 500 \) is written as \[ \displaystyle \frac{2}{5} \times 500 \] and evaluated as follows \[ \displaystyle \frac{2}{5} \times 500 = \frac{2 \times 500}{5} = \frac{1000}{5} = 1000 \div 5 = 200 \]

Example 8
What is \( \displaystyle \frac{3}{7} \) of \( 700\)?
Solution to Example 8

\( \displaystyle \frac{3}{7} \) of \( 700 \) is written as \[ \displaystyle \frac{3}{7} \times 700 \] and evaluated as follows \[ \displaystyle \frac{3}{7} \times 700 = \frac{3 \times 700}{7} = 2100 \div 7 = 300 \]



Equivalent Fractions

If we multiply or divide the numerator and denominator of a fraction by the same number not equal to zero, we obtain an equivalent fraction.
Example 8
The fractions \( \displaystyle \frac{2}{4} \) , \( \displaystyle \frac{6}{12} \) and \( \displaystyle \frac{1}{2} \) are equivalent because:

Divide numerator and denominator of \( \displaystyle \frac{2}{4} \) by 2 gives: \( \displaystyle \frac{2}{4} = \displaystyle \frac{2\div2}{4\div2} = \displaystyle \frac{1}{2} \)

Multiply numerator and denominator of \( \displaystyle \frac{1}{2} \) by 6 gives: \( \displaystyle \frac{1}{2} = \displaystyle \frac{1\times6}{2\times6} = \displaystyle \frac{6}{12} \)



Reduce Fractions to Lowest Terms

A fraction is reduced to lower terms if its numerator and denominator have no common factor other than 1.
Example 9

The fraction \( \displaystyle \frac{3}{14} \) is in reduced form because the numerator \( 3 \) and the denominator \( 14 \) have no common factors other than 1

The fraction\( \displaystyle \frac{11}{17} \) is in reduced form because the numerator \( 11 \) and the denominator \( 17 \) have no common factors other than 1.

Example 10
Reduce the fraction \( \displaystyle \frac{12}{18} \) to lowest terms.

Solution to Example 10

Step 1: Find the greatest common factor (GCF) of the numerator \( 12 \) and denominator \( 18 \)
GCF of \( (12 \text{ and } 18) = 6 \)
Step 2: Divide numerator and denominator by the GCF found above
\( \displaystyle \frac{12}{18} = \displaystyle \frac{12\div6}{18\div8} = \displaystyle \frac{2}{3} \)



Comparing Fractions

Comparing like fractions (fractions with the same denominator) is straightforward; we just compare the numerators.
Example 11
\( \displaystyle \frac{3}{8} > \displaystyle \frac{2}{8} \)
To compare fractions with different denominators, we first rewrite them with the same denominator and then compare them.
Example 12

Compare the fractions \( \displaystyle \frac{6}{5} \) and \( \displaystyle \frac{3}{2} \)
Step 1: Find the lowest common multiple (LCM) of the denominators \( 5 \) and \( 2 \)
LCM of \( (5 \text { and } 2) = 10 \)
Rewrite each fraction with the common denominator equal to LCM
\( \displaystyle \frac{6}{5} = \displaystyle \frac{6\times 2}{5\times2} = \displaystyle \frac{12}{10} \)

\( \displaystyle \frac{3}{2} = \displaystyle \frac{3\times5}{2\times5} = \displaystyle \frac{15}{10} \)
Hence comparing numerators we deduce that
\( \displaystyle \frac{6}{5} \lt \displaystyle \frac{3}{2} \)



Questions

  1. Represent each of the colored (in red) parts as fraction, a mixed number or both.

    1. A whole circle is one unit.
      fraction diagram exercise 1-a

    2. A whole square is one unit.
      fraction diagram exercise 1-b

    3. A whole square is one unit.
      fraction diagram exercise 1-c

    4. A whole square is one unit.
      fraction diagram exercise 1-d


  2. Which of the following is a proper fraction, improper or a mixed number?
    a) \( \displaystyle \frac{1}{6} \) ,     b) \( 4\displaystyle \frac{3}{5} \) ,     c) \( \displaystyle \frac{7}{7} \) ,     d) \( \displaystyle \frac{12}{11} \) ,     e) \( \displaystyle \frac{0}{3} \) ,     f) \( 1\displaystyle \frac{9}{13} \)

  3. Convert the given improper fractions into mixed numbers.
    a) \( \displaystyle \frac{10}{6} \) ,     b) \( \displaystyle \frac{123}{11} \) ,     c) \( \displaystyle \frac{4}{3} \)

  4. Convert the given mixed numbers into improper fractions.
    a) \( 2\displaystyle \frac{1}{6} \) ,     b) \( 2\displaystyle \frac{2}{5} \) ,     c) \( 1\displaystyle \frac{4}{9} \)

  5. Evaluate the following:
    1. \( \displaystyle \frac{1}{10} \) of 2500 people

    2. \( \displaystyle \frac{2}{4} \) of 1.02 grams

    3. \( \displaystyle \frac{3}{5} \) of 1000 students

    4. \( \displaystyle \frac{1}{9} \) of 270 cars


  6. Reduce the following fractions to lowest terms.
    a) \( \displaystyle \frac{16}{24} \) ,     b) \( \displaystyle \frac{14}{30} \) ,     c) \( \displaystyle \frac{28}{52} \)

  7. Order the following fractions and mixed numbers from smallest to largest.
    a) \( 5\displaystyle \frac{1}{6} \) ,     b) \( \displaystyle \frac{21}{5} \) ,     c) \( 3\displaystyle \frac{4}{9} \)

  8. Group these fractions into equivalent fractions.
    a) \( \displaystyle \frac{2}{8} \) ,     b) \( \displaystyle \frac{3}{5} \) ,     c) \( \displaystyle \frac{1}{4} \) ,     d) \( \displaystyle \frac{21}{35} \) ,     e) \( \displaystyle \frac{4}{16} \) ,     f) \( \displaystyle \frac{9}{15} \)


Solutions to the Exercises

    1. As an improper fraction: \( \displaystyle \frac{9}{4} \) or a mixed number: \( 2 \displaystyle \frac{1}{4} \)

    2. As a proper fraction: \( \displaystyle \frac{7}{8} \)

    3. As an improper fraction: \( \displaystyle \frac{8}{8} \) or a whole number: \( 1 \)

    4. As an improper fraction: \( \displaystyle \frac{21}{8} \) or a mixed number: \( 2 \displaystyle \frac{5}{8} \)



  1. Proper fractions: a) and e)
    Improper fractions: c) and d)
    Mixed numbers: b) and f)


  2. a) \( \displaystyle \frac{10}{6} = 1 \frac{4}{6} = 1 \frac{2}{3}\)

    b) \( \displaystyle \frac{123}{11} = 11 \frac{2}{11} \)

    c) \( \displaystyle \frac{4}{3} = 1 \frac{1}{3} \)


  3. a) \( 2\displaystyle \frac{1}{6} = \frac{13}{6} \)

    b) \( 2\displaystyle \frac{2}{5} = \frac{12}{5} \)

    c) \( 1\displaystyle \frac{4}{9} = \frac{13}{9} \)

    1. \( \displaystyle \frac{1}{10} \) of 2500 people = \( \displaystyle \frac{1}{10} \times 2500 = \frac{1 \times 2500}{10} = 250 \) people

    2. \( \displaystyle \frac{2}{4} \) of 1.02 grams = \( \displaystyle \frac{2}{4} \times 1.02 = \frac{2 \times 1.01}{4} = 0.505 \) grams

    3. \( \displaystyle \frac{3}{5} \) of 1000 students = \( \displaystyle \frac{3}{5} \times 1000 = \frac{3 \times 1000}{5} = 600 \) students

    4. \( \displaystyle \frac{1}{9} \) of 270 cars = \( \displaystyle \frac{1}{9} \times 270 = \frac{1 \times 270}{9} = 30 \) cars



  4. a) \( \displaystyle \frac{16}{24} = \frac{2}{3} \)

    b) \( \displaystyle \frac{14}{30} = \frac{7}{15} \)

    c) \( \displaystyle \frac{28}{52} = \frac{7}{13} \)


  5. Convert mixed number into improper fractions then convert all fractions to the same denominator using the LCM of the denominators: 6, 5 and 9.

    a) \( 5\displaystyle \frac{1}{6} = \frac{31}{6} = \frac{31 \times 15}{6 \times 15} = \frac{465}{90} \)

    b) \( \displaystyle \frac{21}{5} = \frac{21 \times 18}{5 \times 18 } = \frac{378}{90} \)

    c) \( 3\displaystyle \frac{4}{9} = \frac{31}{9} = \frac{31 \times 10}{9 \times 10} = \frac{310}{90} \)

    Order from smallest to largest: \( 3\displaystyle \frac{4}{9} , \quad \frac{21}{5} , \quad 5 \frac{1}{6} \)



  6. Group 1: \( \quad \displaystyle \frac{2}{8} , \quad \frac{1}{4} , \quad \frac{4}{16} \) are equivalent

    Group 2: \( \quad \displaystyle \frac{3}{5} , \quad \frac{21}{35} , \quad \frac{9}{15} \) are equivalent



More References and Links

Fraction (mathematics)
Adding Fractions
Fraction Multiplication
Fraction Division
Reduce Fractions
Complex Fractions with Variables
Greatest Common Factor
Lowest Common Multiple
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