Topics
5. Fractions
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AustraliaNSW
Stage 3

5.04 Order fractions

Lesson
Practice
Lesson

Are you ready?

If we can find the  lowest common denominator  , it will help us be able to see which fraction is smaller or larger. Let's try this problem to review.

Examples

Example 1

Which fraction has a denominator that is a multiple of the denominator in \dfrac{2}{6}.

A
\dfrac{3}{5}
B
\dfrac{6}{9}
C
\dfrac{10}{12}
Worked Solution
Create a strategy

List the multiples of 6.

Apply the idea

The first 3 multiples of 6 are 6,\,12,\,18.

Among the choices, the fraction \dfrac{10}{12} has a denominator 12 that is in the list of multiples. So, the correct answer is Option C.

Idea summary

Look for common multiples between the denominators of two different fractions as this will help us to find common denominators.

Order fractions with benchmarks

Let's look at how to benchmark fractions to 0,\dfrac{1}{2} or 1.

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Examples

Example 2

Consider the following fractions: \dfrac{3}{5},\,\dfrac{7}{8},\, \dfrac{1}{6}.

a

Which of these fractions is closest to 0?

A
\dfrac{3}{5}
B
\dfrac{7}{8}
C
\dfrac{1}{6}
Worked Solution
Create a strategy

Choose the fraction with the smallest numerator and largest denominator.

Apply the idea

Among the choices, the fraction \dfrac{1}{6} has the smallest numerator and the largest denominator. The correct answer is Option C.

b

Which of these fractions is closest to 1?

A
\dfrac{7}{8}
B
\dfrac{3}{5}
C
\dfrac{1}{6}
Worked Solution
Create a strategy

Choose the fraction with both the numerator that is close in value to the denominator.

Apply the idea

The fraction \dfrac{7}{8} has a numerator that is only 1 less than the denominator. So, the correct answer is Option A.

Idea summary

To order fractions we can compare them to the benchmarks 0, \, \dfrac{1}{2}, \, 1.

Order fractions with related denominators

This video shows how to order fractions with related denominators.

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Examples

Example 3

Arrange the following in descending order: \dfrac{1}{2},\,\dfrac{1}{4},\,\dfrac{1}{3}.

Worked Solution
Create a strategy

To compare fractions with the same numerator, compare their denominators. The fraction with the bigger denominator is smaller.

Apply the idea

Among the fractions, the fraction with the biggest denominator is \dfrac{1}{4}, so this is the smallest fraction. The fraction with smallest denominator is \dfrac{1}{2}, so this is the biggest fraction.

Arranging the fractions in descending order, we have:\dfrac{1}{2},\,\dfrac{1}{3},\, \dfrac{1}{4}

Idea summary

In comparing fractions with the same numerators, the higher the denominator is, the smaller the value of the fraction.

Compare mixed numbers and improper fractions

This video looks at comparing mixed numbers and improper fractions.

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Examples

Example 4

We wish to arrange the following in descending order: 4\dfrac{1}{2},\,\dfrac{31}{4},\, \dfrac{15}{2}.

a

Rewrite all 3 fractions as improper fractions with the lowest common denominator of 4.

Worked Solution
Create a strategy

Convert each fraction to an improper fraction with the same denominator.

Apply the idea

\dfrac{31}{4} is already an improper fraction with a denominator of 4, so we can leave it.

First we must 4 \dfrac{1}{2} to an improper fraction:

\displaystyle 4\dfrac{1}{2}\displaystyle =\displaystyle \dfrac{4\times 2+1}{2}Multiply 4 by 2 and add 1
\displaystyle =\displaystyle \dfrac{9}{2}Find the numerator

Now we need to multiply the numerator and denominator by 2 to get a denominator of 4.

\displaystyle \dfrac{9}{2}\displaystyle =\displaystyle \dfrac{9\times 2}{2\times 2}Multiply both parts by 2
\displaystyle =\displaystyle \dfrac{18}{4}

For \dfrac{15}{2} we also need to multiply the numerator and denominator by 2 to get a denominator of 4.

\displaystyle \dfrac{15}{2}\displaystyle =\displaystyle \dfrac{15 \times 2}{2 \times 2}Multiply both parts by 2
\displaystyle =\displaystyle \dfrac{30}{4}

4\dfrac{1}{2} =\dfrac{18}{4},\quad \dfrac{31}{4},\quad \dfrac{15}{2} =\dfrac{30}{4}

b

Which of the following lists the fractions in descending order?

A
\dfrac{15}{2},4\dfrac{1}{2},\dfrac{31}{4}
B
4\dfrac{1}{2},\dfrac{15}{2},\dfrac{31}{4}
C
4\dfrac{1}{2},\dfrac{31}{4}, \dfrac{15}{2}
D
\dfrac{31}{4},\dfrac{15}{2},4\dfrac{1}{2}
Worked Solution
Create a strategy

Compare the results found from part (a).

Apply the idea

From part (a), by comparing the numerators we can see that: \dfrac{18}{4} \lt \dfrac{30}{4} \lt \dfrac{31}{4}.

Arranging the fractions in descending order, we have:\dfrac{31}{4},\,\dfrac{30}{4},\,\dfrac{18}{4} So the original fractions in descending order are:\dfrac{31}{4},\,\dfrac{15}{2},\,4\dfrac{1}{2}

The correct answer is Option D.

Idea summary

In order to compare fractions, having the same denominator helps enormously, so you may need to find equivalent fractions if they are not the same.

Converting mixed numbers to improper fractions might be needed as well, so remember to think of which steps can help you achieve the same denominators.

Outcomes

MA3-7NA

compares, orders and calculates with fractions, decimals and percentages

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