**Introduction**

In this lesson, you will learn how to simplify fractions.

To simplify a fraction, you will first do the prime factorization of the numerator and denominator. Any factors that are on both the top and the bottom will “cancel out,” which means they divide to equal 1. You can cross out the numbers that cancel out and get rid of them.

These videos illustrate the lesson material below. Watching the videos is optional.

- Simplifying Fractions (07:01 mins) | Transcript
- Examples of Simplifying Fractions (04:42 mins) | Transcript

**Simplifying Fractions**

**Example 1**

\(\frac{126}{210}\)

Break down both of these numbers into their prime factorization. Start with the numerator, or the top number. Please use a calculator as needed when performing the prime factorization of large numbers.

Since 126 is an even number, you know that it has at least one 2 in it. When you divide 126 by 2, you get 63. 63 is not a prime number, so you need to break it down further.

Figure 1

Any number that is a multiple of nine has digits that add up to equal nine. \(6+3=9\), indicating that 9 is a factor of the number 63. One of the multiplication facts that you memorized is \(9\times7=63\), so you can further factor this number into 7 and 9.

7 is a prime number, but 9 is not; it can be further broken down into \(3\times3\).

Figure 2

The prime factorization of 126 is \(2\times3\times3\times7\).

Next complete the prime factorization of the denominator**.** 210 is an even number, so you know there is a 2 in it. \(210\div2=105\), and since there is a 5 at the end of 105, you can factor it by 5. \(105\div5=21\), and 21 can be further factored into 3 and 7.

Figure 3

The prime factorization of \(210\) is \(2\times3\times5\times7\).

Now you can simplify \(\frac{126}{210}\) by using the prime factors for the numerator and denominator.

\begin{align*}&\frac{126}{210} &\color{red}\small\text{Simplify the fraction}\\\\&\frac{2\times3\times3\times7}{2\times3\times5\times7} &\color{red}\small\text{Use prime factorization}\\\\&\frac{\cancel 2}{\cancel 2}\cdot \frac{\cancel 3}{\cancel 3}\cdot \frac{\cancel 7}{\cancel 7}\cdot \frac{3}{5}&\color{red}\small\text{Cancel common prime factors}\\\\&1\cdot1\cdot 1\cdot \frac{3}{5}&\color{red}\small\text{Any number divided by itself is 1 \(\)}\\\\&\frac{3}{5}&\color{red}\small\text{Simplify the fraction}\\\\\end{align*}

Any factors that are the same on the top and the bottom will “cancel out,” which means they divide to equal 1. This is one of the division rules that any number divided by itself is always 1. This allows you to put your fraction in its simplest form. What remains is \(\frac{3}{5}\).

To check your work, you can compute \(\frac{126}{210}\) on a calculator, which gives you 0.6. When you compute \(\frac{3}{5}\), it also gives you 0.6. This confirms that you have simplified the fraction correctly.

**Additional Examples of Simplifying Fractions**

**Example 2**

\(\frac{12}{66}\)

Breaking 12 down into its prime factorization gives you \(2\times2\times3\). Breaking 66 down into its prime factorization gives you \(2\times3\times11\).

Figure 4

Use the concept of canceling common factors to cancel out one 2 and one 3 from both the top and the bottom.

\begin{align*}&\frac{12}{66} &\color{red}\small\text{Simplify the fraction}\\\\&\frac{2\times2\times3}{2\times3\times11} &\color{red}\small\text{Use prime factorization}\\\\&\frac{\cancel 2}{\cancel 2}\cdot \frac{\cancel 3}{\cancel 3}\cdot \frac{2}{11}&\color{red}\small\text{Cancel common prime factors}\\\\&1\cdot1\cdot \frac{2}{11}&\color{red}\small\text{Any number divided by itself is \(1\)}\\\\&\frac{2}{11}&\color{red}\small\text{Simplify the fraction}\\\\\end{align*}

What remains is \(\frac{2}{11}\). This means that the simplest form of \(\frac{12}{66}\) is \(\frac{2}{11}\).

**Simplifying Fractions using Factors that are Not Prime**

Sometimes you don’t have to break down a fraction to its prime factorization to get to its simplest form. You may recognize a common factor that isn’t a prime number, and you can use that factor instead.

**Example 3**

\(\frac{24}{48}\)

In this example, you may recognize that 12 is a common factor for both 24 and 48.

\begin{align*}&\frac{24}{48} &\color{red}\small\text{Simplify the fraction}\\\\&\frac{2 \cdot \cancel{12}} {4\cdot \cancel {12}} &\color{red}\small\text{Identify a common factor}\\\\&\frac{2}{4} &\color{red}\small\text{Cancel the common factor}\\\\& \frac{\cancel{2}\cdot1}{\cancel{2}\cdot2} &\color{red}\small\text{Identify a common factor}\\\\&\frac{1}{2} &\color{red}\small\text{Cancel the common factor}\\\\\end{align*}

The most simplified form of \(\frac{24}{48}\) is \(\frac{1}{2}\).

**Simplifying Fractions with No Common Factor**

What about fractions that don’t have any common factors like \(\frac{20}{21}\)?

Factoring out 20 gives you \(2\times2\times5\), and factoring out 21 gives you \(3\times7\).

\begin{align*} \frac{20}{21} = \frac{2\cdot2\cdot 5}{3\cdot7} = \frac{20}{21} \end{align*}

There are no common factors between the numerator and the denominator. This means that \(\frac{20}{21}\) is as simplified as it can be. Even though neither one of the numbers are prime, they are considered prime to one another because they do not have any common factors.

**Things to Remember**

- Please use a calculator as needed to perform prime factorization.
- Any number that is a multiple of nine has digits that add up to equal nine.
- When everything cancels out in the numerator and denominator, the simplified fraction is 1, not 0.
- Use the greatest common factor to simplify fractions.

### Practice Problems

**Simplify the following fractions to the lowest terms:**

- \(\dfrac{4}{6}\) (Solution
- \(\dfrac{10}{25}\) (Solution
- \(\dfrac{4}{7}\) (Solution
- \(\dfrac{30}{48}\) (Video Solution
- \(\dfrac{42}{70}\) (Video Solution
- \(\dfrac{12}{84}\) (Solution

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