Simplifying Fractions
Learning Outcomes
- Simplify fractions by finding common factors between the numerator and denominator
- Simplify fractions containing variables
Simplify Fractions
In working with equivalent fractions, you saw that there are many ways to write fractions that have the same value, or represent the same part of the whole. 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 [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 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. In the previous section, we used the Equivalent Fractions Property to find equivalent fractions. 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.
Simplify a fraction.
- Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers.
- Simplify, using the equivalent fractions property, by removing common factors.
- 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
[ohm_question height="400"]146014[/ohm_question]
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]
Answer:
Solution:
| 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] |
Try it
[ohm_question height="400"]146015[/ohm_question]
Watch the following video to see another example of how to simplify a fraction.
https://youtu.be/_2Wk7jXf3Ok
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.
When we simplify an improper fraction, there is no need to change it to a mixed number.
Example
Simplify: [latex]\Large-\frac{56}{32}[/latex]
Answer:
Solution:
|
[latex]\Large-\frac{56}{32}[/latex] |
| Rewrite the numerator and denominator, showing the common factors, 8. |
[latex]\Large\frac{7\cdot8}{4\cdot8}[/latex] |
| Remove common factors. |
[latex]\Large\frac{7\cdot\color{red}{8}}{4\cdot\color{red}{8}}[/latex] |
| Simplify. |
[latex]\Large-\frac{7}{4}[/latex] |
Try it
[ohm_question height="400"]146017[/ohm_question]
Sometimes it may not be easy to find common factors of the numerator and denominator. A good idea, then, is to factor the numerator and the denominator into prime numbers. (You may want to use the factor tree method to identify the prime factors.) Then divide out the common factors using the Equivalent Fractions Property.
Example
Simplify: [latex]\Large\frac{210}{385}[/latex]
Answer:
Solution:
| Use factor trees to factor the numerator and denominator. |
[latex]
\Large\frac{210}{385}[/latex]
 |
| Rewrite the numerator and denominator as the product of the primes. |
[latex]{\Large\frac{210}{385}}={\Large\frac{2\cdot 3\cdot 5\cdot 7}{5\cdot 7\cdot 11}}[/latex] |
| Remove the common factors. |
[latex]\Large\frac{2\cdot 3\cdot\color{blue}{5}\cdot\color{red}{7}}{\color{blue}{5}\cdot\color{red}{7}\cdot 11}[/latex] |
| Simplify. |
[latex]\Large\frac{2\cdot 3}{11}[/latex] |
| Multiply any remaining factors. |
[latex]\Large\frac{6}{11}[/latex] |
Try it
[ohm_question height="400"]146018[/ohm_question]
We can also simplify fractions containing variables. If a variable is a common factor in the numerator and denominator, we remove it just as we do with an integer factor.
Example
Simplify: [latex]\Large\frac{5xy}{15x}[/latex]
Answer:
Solution:
|
[latex]\Large\frac{5xy}{15x}[/latex] |
| Rewrite numerator and denominator showing common factors. |
[latex]\Large\frac{5\cdot x\cdot y}{3\cdot 5\cdot x}[/latex] |
| Remove common factors. |
[latex]\Large\frac{\overline{)5}\cdot \overline{)x}\cdot y}{3\cdot \overline{)5}\cdot \overline{)x}}[/latex] |
| Simplify. |
[latex]\Large\frac{y}{3}[/latex] |
try it
[ohm_question height="400"]146019[/ohm_question]
Here's a video with another example of how to simplify a fraction that contains variables.
https://youtu.be/tLgfPeecGe0
Licenses & Attributions
CC licensed content, Original
- Question ID: 146014, 146015, 146017, 146018, 146019. Authored by: Alyson Day. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.
CC licensed content, Shared previously
- Ex 1: Simplify Fractions. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
- Ex 3: Simplify Fractions Containing Variables. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
CC licensed content, Specific attribution
