Fractions and Mixed Numbers

Examples on fractions and mixed numbers along with more questions and their solutions are presented.
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 of proper and improper fractions, mixed numbers, and the relationship between improper fractions and mixed numbers are also included.
Equivalent fractions, the evaluation of fractions of quantities, and the reduction of fractions are also included.

What Are Fractions?

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

Example 1
You have a wire and you cut it into 4 equal parts as shown in the diagram below. Each small part has a length of \[ \dfrac{1}{4} \] of the whole length 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 \[ \dfrac{3}{4} \; \text{or} \; 3/4 \]

Example 2
You have a large square piece of paper that you cut into 8 equal parts as shown below. Each small part has an area \[ \dfrac{1}{8} \] of the area of the whole square.

whole square

You now paint (in red) 5 of the 8 parts 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 \[ \dfrac{5}{8} \; \text{or} \; 5/8 \] 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 a 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 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 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 to convert a mixed number into an improper fraction?
Example 5
Convert \( 3 \displaystyle \frac{3}{4} \) into an improper fraction.

Solution

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 to convert an improper fraction into a mixed number?
Example 6
Convert the improper fraction \( \displaystyle \frac{15}{4} \) into a mixed number.

Solution

Solution to Example 6
To convert \( \displaystyle \frac{15}{4} \) into a mixed number, first divide \(15\) by \(4\) using long division.
The quotient of the division is the whole number, the remainder is the numerator of the fraction, and the denominator is the divisor 4.
Divide 15 by 4, you get a quotient of 3 and a remainder of 3. Hence:
\( \displaystyle \frac{15}{4} = \text{quotient} + \displaystyle \frac{\text{remainder}}{4} = 3 + \displaystyle \frac{3}{4} = 3\displaystyle \frac{3}{4} \)

Comparing Fractions

Comparing like fractions (fractions with the same denominator) is straightforward; we just compare the numerators.
Example 7
\( \displaystyle \frac{3}{8} > \displaystyle \frac{2}{8} \)

To compare fractions with different denominators, we first rewrite them with the same denominator and then compare them.

Example 8
Compare the fractions \( \displaystyle \frac{6}{5} \) and \( \displaystyle \frac{3}{2} \)

Solution

Solution
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:
\( \displaystyle \frac{6}{5} = \displaystyle \frac{6 \times 2}{5 \times 2} = \displaystyle \frac{12}{10} \)

\( \displaystyle \frac{3}{2} = \displaystyle \frac{3 \times 5}{2 \times 5} = \displaystyle \frac{15}{10} \)
Hence comparing numerators, we deduce that:
\( \displaystyle \frac{6}{5} < \displaystyle \frac{3}{2} \)

Questions and Step-by-Step Solutions

  1. Question 1: Represent each of the colored (in red) parts as a 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. Solution
    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: \( 2 \displaystyle \frac{5}{8} \)
  2. Question 2: Which of the following is a proper fraction, improper fraction, 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. Solution

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

  3. Question 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. Solution

    a) \( \displaystyle \frac{10}{6} = 1 \frac{4}{6} = 1 \frac{2}{3} \)
    b) \( \displaystyle \frac{123}{11} = 11 \frac{2}{11} \)
    c) \( \displaystyle \frac{4}{3} = 1 \frac{1}{3} \)

  4. Question 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. Solution

    a) \( 2\displaystyle \frac{1}{6} = \frac{13}{6} \)
    b) \( 2\displaystyle \frac{2}{5} = \frac{12}{5} \)
    c) \( 1\displaystyle \frac{4}{9} = \frac{13}{9} \)

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

    Convert mixed numbers 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: \( 3\displaystyle \frac{4}{9} , \quad \frac{21}{5} , \quad 5 \frac{1}{6} \)