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Simplifying Fractions

In this section we learn how to simplify fractions, that is how to write a fraction in its simplest form.
So, for example, by the end of this section we'll know how to show that: \[\frac{21}{28} = \frac{3}{4}\] Simplifying fractions is important as when we talk, or refer to, fractions of quantities keeping the denominator as small as possible makes our undedrstanding/visualization easier.

Indeed it is far easier to picture what \(\frac{3}{4}\) (three quarters) of a pizza is than it is to picture \(\frac{21}{28}\) (twenty-one twenty-eighths) of a pizza.
We start by learning a three-step method for simplifying fractions, we then watch a tutorial and finally we'll work our way several questions (with answers).

Tutorial

In the following tutorial we review the method for simplifying fractions, using the highest common factor, HCF. Make sure to watch it now, before working through the exercise below.

The method, for simplifying fractions, we've just seen is summarized in the three steps shown below.

Method

Given a fraction \(\frac{a}{b}\), for example \(\frac{8}{12}\), we can write it in its simplest form in three steps:

  • Step 1: Find the highest common factor, HCF of the numerator and the denominator.

    The highest common factor of \(8\) and \(12\) is \(4\): \[HCF(8,12)=4\]

  • Step 2: Write both the numerator and the denominator in factored form.

    We do this by placing the \(HCF\) as a factor on both the numerator and the denominator: \[\frac{8}{12} = \frac{4\times 2}{4\times 3}\]

  • Step 3: Cancel-out the HCF and write the simplified fraction.

    \[\begin{aligned} \frac{8}{12} &= \frac{4\times 2}{4\times 3} \\ & = \frac{2}{3} \end{aligned} \]

Exercise

Simplify each of the following fractions as much as possible:

  1. \(\frac{8}{12}\)

  2. \(\frac{5}{35}\)

  3. \(\frac{3}{12}\)

  4. \(\frac{12}{20}\)

  5. \(\frac{9}{21}\)

  6. \(\frac{48}{120}\)

  7. \(\frac{30}{42}\)

  8. \(\frac{55}{66}\)

Answers Without Working

  1. \(\frac{8}{12} = \frac{2}{3}\)

  2. \(\frac{5}{35} = \frac{1}{7}\)

  3. \(\frac{3}{12} = \frac{1}{4}\)

  4. \(\frac{12}{20} = \frac{3}{5}\)

  5. \(\frac{9}{21} = \frac{3}{7}\)

  6. \(\frac{48}{120} = \frac{2}{5}\)

  7. \(\frac{30}{42} = \frac{5}{7}\)

  8. \(\frac{55}{66} = \frac{5}{6}\)

Solutions With Working

  1. We simplify \(\frac{8}{12}\) as follows:

    • Step 1: find the highest common factor of \(8\) and \(12\): \[HCF(8,12) = 4\]

    • Step 2: write the fraction in factored form: \[\frac{8}{12} = \frac{4\times 2}{4 \times 3}\]

    • Step 3: cancel-out the highest common factor and simplify the fraction: \[\begin{aligned} \frac{8}{12} &= \frac{4\times 2}{4 \times 3}\\ & = \frac{2}{3} \end{aligned}\]

  2. We simplify \(\frac{5}{35}\) as follows:

    • Step 1: find the highest common factor of \(5\) and \(35\): \[HCF(5,35) = 5\]

    • Step 2: write the fraction in factored form: \[\frac{5}{35} = \frac{5\times 1}{5 \times 7}\]

    • Step 3: cancel-out the highest common factor and simplify the fraction: \[\begin{aligned} \frac{5}{35} &= \frac{5\times 1}{5 \times 7}\\ & = \frac{1}{5} \end{aligned}\]

  3. We simplify \(\frac{3}{12}\) as follows:

    • Step 1: find the highest common factor of \(3\) and \(12\): \[HCF(3,12) = 3\]

    • Step 2: write the fraction in factored form: \[\frac{3}{12} = \frac{3\times 1}{3\times 4}\]

    • Step 3: cancel-out the highest common factor and simplify the fraction: \[\begin{aligned} \frac{3}{12} =& \frac{3\times 1}{3\times 4} \\ & = \frac{1}{4} \end{aligned}\]

  4. We simplify \(\frac{12}{20}\) as follows:
    • Step 1: find the highest common factor of \(12\) and \(20\): \[HCF(12,20=4)\]

    • Step 2: write the fraction in factored form: \[\frac{12}{20} = \frac{4\times 3}{4\times 5}\]

    • Step 3: cancel-out the highest common factor and simplify the fraction: \[\begin{aligned} \frac{12}{20} &= \frac{4\times 3}{4\times 5}\\ & = \frac{3}{5} \end{aligned}\]

  5. We simplify \(\frac{9}{21}\) as follows:
    • Step 1: find the highest common factor of \(9\) and \(21\): \[HCF(9,21)=3\]

    • Step 2: write the fraction in factored form: \[\frac{9}{21} = \frac{3\times 3}{3\times 7}\]

    • Step 3: cancel-out the highest common factor and simplify the fraction: \[\begin{aligned} \frac{9}{21} &= \frac{3\times 3}{3\times 7}\\ & = \frac{3}{7} \end{aligned}\]

  6. We simplify \(\frac{48}{120}\) as follows:
    • Step 1: find the highest common factor of \(48\) and \(120\): \[HCF(48,120)=24\]

    • Step 2: write the fraction in factored form: \[\frac{48}{120} = \frac{24\times 2}{24\times 5}\]

    • Step 3: cancel-out the highest common factor and simplify the fraction: \[\begin{aligned} \frac{48}{120} &= \frac{24\times 2}{24\times 5}\\ & = \frac{2}{5} \end{aligned}\]

  7. We simplify \(\frac{30}{42}\) as follows:
    • Step 1: find the highest common factor of \(30\) and \(42\): \[HCF(30,42) = 6\]

    • Step 2: write the fraction in factored form: \[\frac{30}{42} = \frac{6\times 5}{6\times 7}\]

    • Step 3: cancel-out the highest common factor and simplify the fraction: \[\begin{aligned} \frac{30}{42} &= \frac{6\times 5}{6\times 7}\\ & = \frac{5}{7} \end{aligned}\]

  8. We simplify \(\frac{55}{66}\) as follows:
    • Step 1: find the highest common factor of \(55\) and \(66\): \[HCF(55,66)=11\]

    • Step 2: write the fraction in factored form: \[\frac{55}{66} = \frac{11\times 5}{11\times 6}\]

    • Step 3: cancel-out the highest common factor and simplify the fraction: \[\begin{aligned} \frac{55}{66} &= \frac{11\times 5}{11\times 6}\\ & = \frac{5}{6} \end{aligned} \]