Arranging Fractions in Order of Size

Topic Content:

Fractions with Same Denominators: 

The four shapes below are of the same size, and they are divided in different ways. 

 You can see that:

\(\frac{4}{5}\) is greater than \(\frac{3}{5}\)

\(\frac{3}{5}\) is greater than \(\frac{2}{5}\)

When a set of fractions has the same denominator, the fraction with the largest numerator is the greatest fraction, while the fraction with the smallest numerator is the least.

⇒ the order of size is:

\( \frac{1}{5}, \: \frac{2}{5}, \:\frac{3}{5}, \:\frac{4}{5}\) in ascending order. 

\( \frac{4}{5}, \: \frac{3}{5}, \:\frac{2}{5}, \:\frac{1}{5}\) in descending order. 

Fractions with Different Denominators:

Three shapes of the same size are shown below. It is obvious from these shapes that: 

\(\frac{1}{2}\) is larger than \(\frac{1}{4}\)

\(\frac{1}{4}\) is larger than \(\frac{2}{9}\)

\(\frac{1}{2}, \: \frac{1}{4}, \: \frac{2}{9}\) in descending order 

Worked Example 7.3.1:

Which is greater \(\frac{4}{5}\) or \(\frac{2}{3} \scriptsize\: ?\)

Solution:

Find the LCM of the denominators, i.e. LCM of 3 and 5 is 15. 

Write each fraction with the same denominator by using equivalent fractions.

\( \frac {2}{3}= \frac {2}{3} \: \times \: \frac {5}{5}= \frac {10}{15} \)

\( \frac {4}{5} = \frac {4}{5} \: \times \: \frac {3}{3}= \frac {12}{15} \)

This means \(\normalsize \frac{2}{3} = \frac{10}{15} \scriptsize \: and \: \normalsize \frac{4}{5} = \frac{12}{15} \)

Comparing the size of the numerators of the equivalent fraction, \( \frac{4}{5}\) is greater than \( \frac{2}{3}\).

Worked Example 7.3.2:

Arrange the following factors in ascending order:  \( \frac {2}{3}, \frac {3}{8}, \frac {1}{4}, \: \frac {5}{6} \)

Solution: 

Find the LCM of the denominators 3, 8, 4 and 6. The LCM is 24.

Therefore, the equivalent fractions are: 

\( \frac {2}{3} = \frac {2}{3} \: \times \: \frac {8}{8}= \frac {16}{24} \)

\( \frac {3}{8} = \frac {3}{8} \: \times \: \frac {3}{3}= \frac {9}{24} \)

\( \frac {1}{4} = \frac {1}{4} \: \times \: \frac {6}{6}= \frac {6}{24} \)

\( \frac {5}{6} = \frac {5}{6} \: \times \: \frac {4}{4}= \frac {20}{24} \)

By comparing the sizes of the numerators of the equivalent fractions in ascending order, we have   

\( \frac {6}{24}, \frac {9}{24}, \frac {16}{24}, \frac {20}{24} \)

Here, the given fractions in ascending order give:   

\( \frac {1}{4}, \frac {3}{8}, \frac {2}{3}, \frac {5}{6} \)