Introduction

We saw how to compare fraction models using common denominators in a previous lesson. In this lesson, we'll learn how to carry out these processes without models.

For example, suppose we have the following two fractions:

\dfrac{1}{\color{blue}3} \qquad \dfrac{4}{\color{red}6}

Which of them is larger?

At the moment, it's difficult to compare them as they don't have a common denominator. Therefore, we must create a common denominator.

Notice that our denominators are {\color{blue}3} and {\color{red}{6}}, and {\color{red}{6}} is a multiple of {\color{blue}3}. In fact,

{\color{red}{6}} = {\color{purple}{2}} \times {\color{blue}3}.

Therefore, we can create a common denominator by multiplying the numerator and denominator of the first fraction by {\color{purple}{2}}, as follows:

\begin{align*} \dfrac{1}{\color{blue}3} &=\dfrac{1\times {\color{purple}{2}}}{{\color{blue}3}\times {\color{purple}{2}}} = \dfrac{2}{\color{red}6} \end{align*}

So, we now have the following two fractions:

\dfrac{2}{\color{red}6} \qquad \dfrac{4}{\color{red}6}

From here, it's easy to see that the second fraction is larger. So, we have

\dfrac{2}{\color{red}6} < \dfrac{4}{\color{red}6}\,.

Therefore, we conclude that

\dfrac{1}{\color{blue}3} < \dfrac{4}{\color{red}6}.

Let's see another example.

FLAG
Example: Comparing Fractions When One Denominator Is a Multiple of the Other

\dfrac{10}{12} \,\square\, \dfrac{3}{4}

Which symbols could replace the empty box above to make the statement true?

  1. <
  2. =
  3. >
EXPLANATION

We cannot compare the fractions directly as they don't have a common denominator.

Notice that our denominators are 12 and 4, and 12 is a multiple of 4. In fact,

12 = {\color{purple}{3}}\times 4

Therefore, we can make a common denominator of 12 by multiplying the numerator and denominator of the second fraction by {\color{purple}{3}}{:}

\dfrac{3}{4} = \dfrac{3 \times {\color{purple}{3}}}{4 \times {\color{purple}{3}}} = \dfrac{9}{12}

So, we now have the following:

\dfrac{10}{12} \,\square\, \dfrac{9}{12}

Since 10 > 9, the correct statement must be:

\dfrac{10}{12} > \dfrac{9}{12}

So, the correct answer is "III only."

FLAG
Practice: Comparing Fractions When One Denominator Is a Multiple of the Other

\[ \dfrac{2}{7} \,\square\, \dfrac{3}{14} \]

Which symbols could replace the empty box above to make the statement true?

  1. $>$
  2. $< $
  3. $=$
a
 III only
b
 I and III only
c
 II and III only
d
 I only
e
 II only
Practice: Comparing Fractions When One Denominator Is a Multiple of the Other

\[ \dfrac{3}{5} \,\square\, \dfrac{8}{10} \]

Which symbols could replace the empty box above to make the statement true?

  1. $>$
  2. $=$
  3. $< $
a
 II only
b
 II and III only
c
 I and II only
d
 I only
e
 III only
Cases Where the Least Common Denominator Is a Product of the Denominators

Consider the following fractions:

\dfrac{1}{\color{blue}2} \qquad \dfrac{1}{\color{red}3}

Let's compare these fractions using common denominators.

Notice that {\color{red}3} is not a multiple of {\color{blue}2}. However, we can create a common denominator by taking the product of the denominators.

In this case, our common denominator will be

{\color{blue}2} \times {\color{red}3} = 6.

  • To put \dfrac12 over a denominator of 6, we multiply the numerator and denominator by {\color{red}3}{:} \begin{align*} \dfrac{1}{2} = \dfrac{1\times {\color{red}3}}{2 \times {\color{red}3}} = \dfrac36 \end{align*}

  • To put \dfrac13 over a denominator of 6, we multiply the numerator and denominator by {\color{blue}2}{:} \begin{align*} \dfrac{1}{3} = \dfrac{1\times {\color{blue}2}}{3 \times {\color{blue}2}} = \dfrac26 \end{align*}

So, we now have the following two fractions:

\dfrac36 \qquad\dfrac26

From here, it's easy to see that the first fraction is larger. So, we have

\dfrac36 > \dfrac26

Therefore, we conclude that

\dfrac12 > \dfrac13.

Let's see another example.

FLAG
Example: Comparing Fractions When the Least Common Denominator Equals the Product of Denominators

\dfrac{1}{3} \,\square\, \dfrac{2}{4}

Which symbols could replace the empty box above to make the statement true?

  1. <
  2. =
  3. >
EXPLANATION

We cannot compare the fractions directly as they don't have a common denominator.

Notice that 4 is not a multiple of 3. However, we can create a common denominator by taking the product of the denominators.

In this case, we can make a common denominator of 3 \times 4 = 12.

  • To put \dfrac{1}{3} over a denominator of 12 , we multiply the numerator and denominator by 4 : \dfrac{1}{3} = \dfrac{1 \times 4}{3 \times 4} = \dfrac{4}{12}

  • To put \dfrac{2}{4} over a denominator of 12, we multiply the numerator and denominator by 3 : \dfrac{2}{4} = \dfrac{2 \times 3}{4 \times 3} = \dfrac{6}{12}

This gives the following:

\dfrac{4}{12} \,\square\, \dfrac{6}{12}

We can now compare the fractions \dfrac{4}{12} and \dfrac{6}{12} by looking at the numerators only.

Since 4 < 6, the correct statement must be

\dfrac{4}{12} < \dfrac{6}{12}.

So, the correct answer is "I only."

FLAG
Practice: Comparing Fractions When the Least Common Denominator Equals the Product of Denominators

\[ \dfrac{2}{3} \,\square\, \dfrac{3}{5} \]

Which symbols could replace the empty box above to make the statement true?

  1. $>$
  2. $< $
  3. $=$
a
 I and III only
b
 II and III only
c
 III only
d
 II only
e
 I only
Practice: Comparing Fractions When the Least Common Denominator Equals the Product of Denominators

\[ \dfrac{1}{4} \,\square\, \dfrac{2}{5} \]

Which symbols could replace the empty box above to make the statement true?

  1. $< $
  2. $=$
  3. $>$
a
 I and II only
b
 II and III only
c
 III only
d
 II only
e
 I only
Using Lowest Common Multiples

Whenever two denominators have a common factor greater than 1, the most efficient way to find a least common denominator is to look at the denominators' multiples.

For example, consider the following fractions:

\dfrac{2}{\color{blue}6} \qquad \dfrac{5}{\color{red}9}

Notice that {\color{blue}6} and {\color{red}9} have 3 as a common factor. This suggests that we can create a common denominator by looking at their least common multiples:

  • Multiples of 6{:}\qquad 6,12,{\color{purple}{18}},24,\ldots

  • Multiples of 9{:}\qquad 9,{\color{purple}{18}},27,36,\ldots

Therefore, our least common denominator will be {\color{purple}{18}}.

  • To put \dfrac26 over a denominator of 18, we multiply the numerator and denominator by 3{:} \begin{align*} \dfrac{2}{6} = \dfrac{2\times 3}{6 \times 3} = \dfrac{6}{18} \end{align*}

  • To put \dfrac59 over a denominator of 18, we multiply the numerator and denominator by 2{:} \begin{align*} \dfrac{5}{9} = \dfrac{5\times 2}{9 \times 2} = \dfrac{10}{18} \end{align*}

So, we now have the following two fractions:

\dfrac{6}{18} \qquad\dfrac{10}{18}

From here, it's easy to see that the second fraction is larger. So, we have

\dfrac{6}{18}< \dfrac{10}{18}.

Therefore, we conclude that

\dfrac26 < \dfrac59.

Let's see another example.

FLAG
Example: Comparing Fractions When the Denominators Have a Common Factor

\dfrac{5}{6} \,\square\, \dfrac{6}{8}

Which symbols could replace the empty box above to make the statement true?

  1. >
  2. =
  3. <
EXPLANATION

We cannot compare the fractions directly as they don't have a common denominator.

Notice that the denominators 6 and 8 have 2 as a common factor. This suggests that we can create a common denominator by looking at their least common multiples:

  • Multiples of 6{:}\qquad 6,12,18,{\color{purple}{24}},\ldots

  • Multiples of 8{:}\qquad 8,16,{\color{purple}{24}},\ldots

Therefore, our least common denominator will be {\color{purple}{24}}.

  • To put \dfrac{5}{6} over a denominator of 24, we multiply the numerator and denominator by 4 : \dfrac{5}{6} = \dfrac{5 \times 4}{6 \times 4} = \dfrac{20}{24}

  • To put \dfrac{6}{8} over a denominator of 24, we multiply the numerator and denominator by 3 : \dfrac{6}{8} = \dfrac{6 \times 3}{8 \times 3} = \dfrac{18}{24}

We now have the following:

\dfrac{20}{24} \,\square\, \dfrac{18}{24}

We can now compare the fractions \dfrac{20}{24} and \dfrac{18}{24} by looking at the numerators only.

Since 20 > 18, the correct statement must be

\dfrac{20}{24} > \dfrac{18}{24}.

So, the correct answer is "I only."

FLAG
Practice: Comparing Fractions When the Denominators Have a Common Factor

\[ \dfrac{3}{4} \,\square\, \dfrac{5}{6} \]

Which symbols could replace the empty box above to make the statement true?

  1. $>$
  2. $< $
  3. $=$
a
 I only
b
 I and III only
c
 III only
d
 II and III only
e
 II only
Practice: Comparing Fractions When the Denominators Have a Common Factor

\[ \dfrac{4}{6} \,\square\, \dfrac{5}{8} \]

Which symbols could replace the empty box above to make the statement true?

  1. $>$
  2. $< $
  3. $=$
a
 III only
b
 I only
c
 II and III only
d
 I and III only
e
 II only
Flag Content
Did you notice an error, or do you simply believe that something could be improved? Please explain below.
SUBMIT
CANCEL