Simplify Fractions

Learning Outcomes

  • Simplify fractions by finding common factors between the numerator and denominator

Simplify Fractions

There are a red and a orange pepper in a sack and lemons in a second sack on a couch.

There are many ways to write fractions that have the same value, or represent the same part of the whole. For example, suppose a bag contains 8 peppers, of which 3 are yellow, 4 are green, and 1 is red. Without looking, we reach in and randomly select a pepper. To find the chance we select a green pepper we need the fraction of green peppers in the bag.  Since 4 out of the 3+4+1=8 peppers are green, the fraction of green peppers is [latex]\frac{4}{8}[/latex]. But 4 is half of 8, so we could also say the fraction of green peppers is [latex]\frac{1}{2}[/latex].

How do you know which one to use? Often, we’ll use the fraction that is in simplified form.

A fraction is considered simplified if there are no common factors, other than 1, 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 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 also use the Equivalent Fractions Property in reverse to simplify fractions. We rewrite the property to show both forms together.

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].

Notice that [latex]c[/latex] is a common factor in the numerator and denominator. Anytime we have a common factor in the numerator and denominator, it can be removed. Sometimes we say we cancel the common factor, [latex]c[/latex].

Simplify 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.

Example

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

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]

 

try it

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

Example

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

 

Try it

Watch the following video to see another example of how to simplify a fraction.

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.