## Simplifying Fractions

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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 \(1,\) 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,

- \(\frac{2}{3}\) is simplified because there are no common factors of \(2\) and \(3.\)
- \(\frac{10}{15}\) is not simplified because \(5\) is a common factor of \(10\) and \(15.\)

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 \(a,b,c\) are numbers where \(b\ne 0,c\ne 0,\) then

Notice that \(c\) is a common factor in the **numerator** and **denominator**. Anytime we have a common factor in the numerator and denominator, it can be removed.

- 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: \(\frac{10}{15}.\)

### Solution

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

Notice that 5 is a factor of both 10 and 15. | \(\frac{10}{15}\) |

Factor the numerator and denominator. | |

Remove the common factors. | |

Simplify. | \(\frac{2}{3}\) |

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

## Example

Simplify: \(-\frac{18}{24}.\)

### Solution

We notice that 18 and 24 both have factors of 6. | \(-\frac{18}{24}\) |

Rewrite the numerator and denominator showing the common factor. | |

Remove common factors. | |

Simplify. | \(-\frac{3}{4}\) |

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: \(-\frac{56}{32}.\)

### Solution

\(-\frac{56}{32}\) | |

Rewrite the numerator and denominator, showing the common factors, 8. | |

Remove common factors. | |

Simplify. | \(-\frac{7}{4}\) |

### Simplifying 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

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: \(\frac{210}{385}.\)

### Solution

Use factor trees to factor the numerator and denominator. | \(\frac{210}{385}\) |

Rewrite the numerator and denominator as the product of the primes. | \(\frac{210}{385}=\frac{2\cdot 3\cdot 5\cdot 7}{5\cdot 7\cdot 11}\) |

Remove the common factors. | |

Simplify. | \(\frac{2\cdot 3}{11}\) |

Multiply any remaining factors. | \(\frac{6}{11}\) |

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: \(\frac{5xy}{15x}.\)

### Solution

\(\frac{5xy}{15x}\) | |

Rewrite numerator and denominator showing common factors. | \(\frac{5·x·y}{3·5·x}\) |

Remove common factors. | \(\require{cancel}\frac{\cancel{5}·\cancel{x}·y}{3·\cancel{5}·\cancel{x}}\) |

Simplify. | \(\frac{y}{3}\) |