• Simplify fractions

Simplify Fractions

A fraction is considered simplified if there are no common factors, other than [latex]1[/latex], in the numerator and denominator. If a fraction does have common factors in the numerator and denominator, we can reduce the fraction to its simplified form by removing the common factors.

simplified fraction

A fraction is considered simplified, or reduced, if there are no common factors in the numerator and denominator.

For example,

• [latex]\Large\frac{2}{3}[/latex] is simplified because there are no common factors of [latex]2[/latex] and [latex]3[/latex].
• [latex]\Large\frac{10}{15}[/latex] is not simplified because [latex]5[/latex] is a common factor of [latex]10[/latex] and [latex]15[/latex].

The process of simplifying a fraction is often called reducing the fraction. We can use the Equivalent Fractions Property in reverse to simplify fractions.

equivalent fractions property

If [latex]a,b,c[/latex] are numbers where [latex]b\ne 0,c\ne 0[/latex], then [latex]{\Large\frac{a}{b}}={\Large\frac{a\cdot c}{b\cdot c}}\text{ and }{\Large\frac{a\cdot c}{b\cdot c}}={\Large\frac{a}{b}}[/latex].

How to: Simplify/Reduce a Fraction

1. Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers.
2. Simplify, using the equivalent fractions property, by removing common factors.
3. Multiply any remaining factors.

Note: To simplify a negative fraction, we use the same process as above. Remember to keep the negative sign.

After simplifying a fraction, it is always important to check the result to make sure that the numerator and denominator do not have any more factors in common. Remember, the definition of a simplified fraction: a fraction is considered simplified if there are no common factors in the numerator and denominator.

Let’s simplify the fraction we saw earlier.

Simplify: [latex]\Large\frac{10}{15}[/latex]

Show Solution

To simplify the fraction, we look for any common factors in the numerator and the denominator.

Notice that [latex]5[/latex] is a factor of both [latex]10[/latex] and [latex]15[/latex].	[latex]\Large\frac{10}{15}[/latex]
Factor the numerator and denominator.	[latex]\Large\frac{2\cdot5}{3\cdot5}[/latex]
Remove the common factors.	[latex]\Large\frac{2\cdot\color{red}{5}}{3\cdot\color{red}{5}}[/latex]
Simplify.	[latex]\Large\frac{2}{3}[/latex]

To simplify a negative fraction, we use the same process as in the previous example. Remember to keep the negative sign.

Simplify: [latex]\Large-\frac{18}{24}[/latex]

Show Answer

We notice that 18 and 24 both have factors of 6.	[latex]\Large-\frac{18}{24}[/latex]
Rewrite the numerator and denominator showing the common factor.	[latex]\Large\frac{3\cdot6}{4\cdot6}[/latex]
Remove common factors.	[latex]\Large-\frac{3\cdot\color{red}{6}}{4\cdot\color{red}{6}}[/latex]
Simplify.	[latex]\Large-\frac{3}{4}[/latex]