# Simplifying Fractions

To simplify a fraction means to write the fraction in **
simplest form**. A

fraction is **"simplified"** when the numerator and denominator have

no common factors other than1.

Consider the following fractions. Can they be simplified?

To see if they can, we use** divisibility rules** and
**prime factorization** to

**factor **the numerator and denominator. Then we cancel common factors.

Divisibility Rule for 2

Rule for 2 Divisibility rules are based on patterns of multiplication. When we list the multiples of 2,we get even numbers, numbers that end in 2,4,6,8 or 0: Multiples of 2: 2, 4, 6, 8, 10, ... This means all |
Example 1: Simplify
The
numerator and denominator are |

Divisibility Rule for 3

Rule for 3 If a number's digits add to a multiple of 3,the number can be divided by 3. For example, 84 is divisible by 3 because its digits add to a multiple of 3:
97 is not divisible by 3 because its |
Example 2: Simplify
The
numerator and denominator can To simplify the fraction, divide each |

Divisibility Rules for 5 and 10

Rule for 5 If a number ends in 0 or 5,divide the number by 5. Multiples of 5: 5, 10, 15, 20, … Rule for 10 If the number ends in 0, divide by 10. Multiples of 10: 10, 20, 30, … |
Example 3: Simplify
Divide each number by 5 and cancel the common factor: Example 4: Simplify Canceling the 0's,divides each number by 10: |

Applying Divisibility Rules

Example5:Can the fraction be simplified?

Since the numerator,51,can be divided by 3 and the denominator,85,

can be divided by 5,we can apply each rule separately. Notice the

the numbers share a common factor of 17:

divide 51 by 3 | |

divide 85 by 5 |

Practice Exercise 1

Simplify each fraction.

Check answers

Solution − Practice Exercise 1

Simplify each fraction.

←When the number in the numerator cancels, we must write a"1." | |

← Cancel the 0's first. |

Prime Factorization

A **prime number** is divisible by itself and1.Prime numbers include,

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, …

A number that is **not **prime is called** composite** because the number

can be written as a **product** of its **prime factors**. For example, the

number6 is composite because we can factor 6 and write it as a

product of its prime factors,2 and 3:

6 = 2ยท 3