Start from the left side of each drawing and shade in the fraction shown.

This shape is the whole thing:

Shade [latex]\dfrac{1}{2}[/latex]:

Shade [latex]\dfrac{2}{4}[/latex]:

Shade [latex]\dfrac{3}{6}[/latex]:

Shade [latex]\dfrac{4}{8}[/latex]:

Shade [latex]\dfrac{5}{10}[/latex]:

Did you notice that the amount you shaded was the same in each drawing?

The fractions that you were asked to shade are equivalent fractions. Equivalent fractions are fractions that are equal.

Now shade the fractions asked for in these drawings the same way.

This shape is the whole thing:

Shade [latex]\dfrac{1}{3}[/latex]:

Shade [latex]\dfrac{2}{6}[/latex]

Shade [latex]\dfrac{3}{9}[/latex]:

Shade [latex]\dfrac{4}{12}[/latex]:

Shade [latex]\dfrac{5}{15}[/latex]:

The above examples are all equivalent fractions.

[latex]\dfrac{1}{3}[/latex] = [latex]\dfrac{2}{6}[/latex] = [latex]\dfrac{3}{9}[/latex] = [latex]\dfrac{4}{12}[/latex] = [latex]\dfrac{5}{15}[/latex]

Working with common fractions often requires using an equivalent fraction in place of the given fraction. You will learn about several processes to help you to find equivalent fractions.

Factors

Factors are the numbers which are multiplied together to make a product. An understanding of factors is needed to express fractions in the lowest terms.

Some numbers only have two factors: 1 and the number itself. These numbers are called prime numbers.

Look at Table 3.5.1 for some prime numbers.

Table 3.5.1: Factors of Prime Numbers

Prime Numbers	Factors
1	1, 1
2	1, 2
3	1, 3
5	1, 5
7	1, 7
11	1, 11
13	1, 13
17	1, 17
19	1, 19
23	1, 23
29	1, 29

Add other prime numbers to the chart as you find them.

Finding Common Factors

A common factor is a number used to reduce the numerator and denominator.

Expressing Fractions in Lower Terms

Express means to say it or write it.

Lower terms mean to express equivalent fractions with smaller (lower) denominators.

The words simplify and reduce are another way to say “express fractions in lower (or lowest) terms.”

To express a fraction in the lowest terms, do this:

Step 1: Find the greatest common factor (GCF) of the numerator and denominator.

• [latex]\dfrac{4}{12}[/latex]   The factors of 4 are 1, 2, 4
• The factors of 12 are 1, 2, 3, 4, 6, 12

The GCF is 4.

Step 2: Divide the numerator and the denominator by the greatest common factor.

• [latex]\dfrac{4}{12}[/latex] [latex]\dfrac{\div 4}{\div 4}[/latex]   = [latex]\dfrac{1}{3}[/latex]
• [latex]\dfrac{4}{12}[/latex] = [latex]\dfrac{1}{3}[/latex]

There are several reasons lower terms are used:

• The math is usually easier with lower numbers.
• Is it easier to think of [latex]\tfrac{1}{2}[/latex] an apple or [latex]\tfrac{15}{30}[/latex] of an apple? ([latex]\tfrac{1}{2}[/latex] = [latex]\tfrac{15}{30}[/latex])
• Do you want to think about [latex]\tfrac{155}{620}[/latex] of your pay cheque or [latex]\tfrac{1}{4}[/latex] of your pay cheque? ([latex]\tfrac{1}{4}[/latex] = [latex]\tfrac{155}{620}[/latex])
• Always express fractions in lowest terms!

Dividing the numerator and denominator by the GCF will give an equivalent fraction in lower terms.

Expressing Fractions in Higher Terms

Higher Terms are needed when you add and subtract fractions with different denominators.

You have learned that dividing the numerator and denominator of a fraction by a common factor gives an equivalent fraction in lower terms. You know that dividing and multiplying are opposite operations, so this next rule will match the one you just learned for reducing:

Multiplying the numerator and denominator of a fraction by the same number (a common factor) will give an equivalent fraction in higher terms.

Are the Fractions Equivalent?

If the denominators are the same, you can easily judge if the fractions are equivalent by comparing the numerators.

Compare [latex]\dfrac{4}{5}[/latex] and [latex]\dfrac{3}{5}[/latex]:                    [latex]\dfrac{4}{5}[/latex] ≠ [latex]\dfrac{3}{5}[/latex] (≠ means ‘not equal’)

Compare [latex]\dfrac{12}{20}[/latex] and [latex]\dfrac{12}{20}[/latex]:                     [latex]\dfrac{12}{20} = \dfrac{12}{20}[/latex]

If the denominators are different, you might be able to rewrite one or more of the fractions so they have the same denominator.

Compare [latex]\dfrac{4}{5}[/latex] and [latex]\dfrac{6}{10}[/latex]:                     [latex]\dfrac{6}{10}\dfrac{\div 2}{\div 2}[/latex] = [latex]\dfrac{3}{5}[/latex]                    So: [latex]\dfrac{4}{5}\neq \tfrac{3}{5}[/latex]

Compare [latex]\dfrac{12}{16}[/latex] and [latex]\dfrac{5}{8}[/latex]:                    [latex]\dfrac{5}{8} \dfrac{\times 2}{\times 2}[/latex] = [latex]\dfrac{10}{16}[/latex]                    So: [latex]\dfrac{12}{16}\neq \tfrac{10}{16}[/latex]

or you could do this:            [latex]\dfrac{12}{16}[/latex][latex]\dfrac{\div 2}{\div 2}= \dfrac{6}{8}[/latex]                    So: [latex]\dfrac{6}{8}\neq\dfrac{5}{8}[/latex]

A quick method is to cross-multiply:

1. Multiply the numerator of one fraction by the denominator of the second fraction
2. Multiply the numerator of the other fraction by the denominator of the first fraction These are called the cross-products.
   If the cross products are the same, then the fraction is equivalent.

Rounding Common Fractions to Whole Numbers

When rounding to a whole number, if a fraction is less than do not change the whole number.

Examples:

• [latex]2\dfrac{4}{7}≈ 2[/latex]
• [latex]23\dfrac{1}{3} ≈ 23[/latex]
• [latex]\dfrac{1}{4} ≈ 0[/latex]
• [latex]5\dfrac{3}{8}≈ 5[/latex]

If the fraction is [latex]\tfrac{1}{2}[/latex] or more, consider the fraction as another one that must be added to the whole number.

Examples:

• [latex]2\dfrac{1}{2}[/latex] ≈ 3
• [latex]15\dfrac{4}{5}[/latex] ≈ 16
• [latex]6\dfrac{7}{8}[/latex] ≈ 7
• [latex]\dfrac{3}{4}[/latex] ≈ 1

If you are not sure if a fraction is more or less than [latex]\tfrac{1}{2}[/latex], you can compare it to [latex]\tfrac{1}{2}[/latex], by making equivalent fractions with a common denominator.

3.5:  Practice Questions

1. Define.
   1. 1. Equivalent                
      2. Prime number                
      3. Greatest common factor (GCF)                

    

2. Complete the chart.
   

   Table 3.5.2: Question 2 Complete the Chart

   Numbers	Factors	Common Factors	Greatest Common Factor
   12 18	—	—	—
   15 30	—	—	—
   7 28	—	—	—
   6 16	—	—	—
   18 27	—	—	—

    

3. Express in lowest terms.
   1. [latex]\dfrac{10}{15}[/latex] =                    .
   2. [latex]\dfrac{14}{16}[/latex] =                    .
   3. [latex]\dfrac{8}{12}[/latex] =                    .

    

4. State if each pair of fractions is equivalent (=) or not equivalent (≠).
   1. [latex]\dfrac{5}{9}[/latex]                  [latex]\dfrac{15}{27}[/latex]
   2. [latex]\dfrac{3}{7}[/latex]                   [latex]\dfrac{15}{35}[/latex]

    

5. Round to the nearest whole number.
   1. [latex]4\dfrac{5}{8}[/latex] ≈                      
   2. [latex]19\dfrac{4}{10}[/latex] ≈                      
   3. [latex]\dfrac{1}{2}[/latex] ≈                      
   4. [latex]6\dfrac{3}{4}[/latex] ≈                      
   5. [latex]\dfrac{1}{3}[/latex] ≈                      

3.5: Practice Answers

1. Define.
   1. Check your definitions in the glossary
2. Complete the chart.
   

   Table 3.5.3: Question 2 Answers

   Numbers	Factors	Common Factors	Greatest Common Factor
   12 18	… of 12 are 1, 2, 3, 4, 6, 12 … of 18 are 1, 2, 3, 6, 9, 18	2, 3, 6	6
   15 30	… of 15 are 1, 3, 5, 15 … of 30 are 1, 2, 3, 5, 6, 10, 15, 30	3, 5, 15	15
   7 28	… of 7 are 1, 7 … of 28 are 1, 2, 4, 7, 14, 28	7	7
   6 16	… of 6 are 1, 2, 3, 6 … of 16 are 1, 2, 4, 8, 16	2	2
   18 27	… of 18 are 1, 2, 3, 6, 9, 18 … of 27 are 1, 3, 9, 27	3, 9	9

    

3. Express in lowest terms.
   1. [latex]\dfrac{2}{5}[/latex]
   2. [latex]\dfrac{7}{8}[/latex]
   3. [latex]\dfrac{2}{3}[/latex]

    

4. State if each pair of fractions is equivalent (=) or not equivalent (≠).
   1. =
   2. =
   3. ≠
   4. ≠

    

5. Round to the nearest whole number.
   1. 5
   2. 19
   3. 1
   4. 7
   5. 0

Attribution

All figures in this chapter are from Topic A: Equivalent Fractions in Adult Literacy Fundamental Mathematics: Book 5 – 2nd Edition by Liz Girard, Wendy Tagami, and Leanne Caillier-Smith, via BCcampus.

This chapter has been adapted from Topic A: Equivalent Fractions in Adult Literacy Fundamental Mathematics: Book 5 – 2nd Edition (BCcampus) by Liz Girard, Wendy Tagami, and Leanne Caillier-Smith (2023), which is under a CC BY 4.0 license.