How can the GCF be used to write a fraction?

May 6, 2016

Divide both the numerator and denominator by their GCF to get a fraction in lowest terms.

Explanation:

The GCF (greatest common factor) can be used to express a fraction in simplest terms:

• Find the GCF of the numerator and denominator.

• Divide both the numerator and denominator by the GCF.

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For example, to express $\frac{70}{42}$ in lowest terms, first find the GCF of $70$ and $42$.

My favourite method to find the GCF of two numbers goes as follows:

• Divide the larger number by the smaller to get a quotient and remainder.

• If the remainder is zero, then the smaller number is the GCF.

• Otherwise, repeat with the smaller number and the remainder.

So in my example:

$\frac{70}{42} = 1$ with remainder $28$

$\frac{42}{28} = 1$ with remainder $14$

$\frac{28}{14} = 2$ with remainder $0$

So the GCF of $70$ and $42$ is $14$.

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Having found the GCF, we can now write:

$\frac{70}{42} = \frac{\left(\frac{70}{14}\right)}{\left(\frac{42}{14}\right)} = \frac{5}{3}$

Alternatively, we can express the division implicitly:

$\frac{70}{42} = \frac{5 \times \textcolor{red}{\cancel{\textcolor{b l a c k}{14}}}}{3 \times \textcolor{red}{\cancel{\textcolor{b l a c k}{14}}}} = \frac{5}{3}$