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Simplifying Fractions
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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

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.

Image with the number 126 at the top. Under 126 are two lines pointing down and outward, one to the right and one to the left. The left line points to the number 2. The right line points to the number 63. 

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 \(3\times3\).

Image with the number 126 at the top. Under 126 are two lines pointing down and outward, one to the right and one to the left. The left line points to the number 2. The right line points to the number 63. Under 63 are two lines pointing down and outward, one to the right and one to the left. The left line points to the number 9. The right line points to the number 7. Under 9 are two lines pointing down and outward, one to the right and one to the left. The left line points to the number 3. The right line points to the number 3.

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.

Image with the number 210 at the top. Under 210 are two lines pointing down and outward, one to the right and one to the left. The left line points to the number 2. The right line points to the number 105. Under 105 are two lines pointing down and outward, one to the right and one to the left. The left line points to the number 5. The right line points to the number 21. Under 21 are two lines pointing down and outward, one to the right and one to the left. The left line points to the number 3. The right line points to the number 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{navy}\small\text{Simplify the fraction}\\\\&\frac{2\times3\times3\times7}{2\times3\times5\times7} &\color{navy}\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{navy}\small\text{Cancel common prime factors}\\\\&1\cdot1\cdot 1\cdot \frac{3}{5}&\color{navy}\small\text{Any number divided by itself is 1 \(\)}\\\\&\frac{3}{5}&\color{navy}\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 \(2\times3\times11\).

Image with two factorizations. The first factorization starts with the number 12 at the top. Under 12 are two lines pointing down and outward, one to the right and one to the left. The left line points to the number 2. The right line points to the number 6. Under 6 are two lines pointing down and outward, one to the right and one to the left. The left line points to the number 2. The right line points to the number 3. Under this factorization it says 2 times 2 times 3. The second factorization starts with the number 66 at the top. Under 66 are two lines pointing down and outward, one to the right and one to the left. The left line points to the number 6. The right line points to the number 11. Under 6 are two lines pointing down and outward, one to the right and one to the left. The left line points to the number 2. The right line points to the number 3. Under this factorization it says 2 times 3 times 11.

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{navy}\small\text{Simplify the fraction}\\\\&\frac{2\times2\times3}{2\times3\times11} &\color{navy}\small\text{Use prime factorization}\\\\&\frac{\cancel 2}{\cancel 2}\cdot \frac{\cancel 3}{\cancel 3}\cdot \frac{2}{11}&\color{navy}\small\text{Cancel common prime factors}\\\\&1\cdot1\cdot \frac{2}{11}&\color{navy}\small\text{Any number divided by itself is \(1\)}\\\\&\frac{2}{11}&\color{navy}\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{navy}\small\text{Simplify the fraction}\\\\&\frac{2 \cdot \cancel{12}} {4\cdot \cancel {12}} &\color{navy}\small\text{Identify a common factor}\\\\&\frac{2}{4} &\color{navy}\small\text{Cancel the common factor}\\\\& \frac{\cancel{2}\cdot1}{\cancel{2}\cdot2} &\color{navy}\small\text{Identify a common factor}\\\\&\frac{1}{2} &\color{navy}\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:
  1. \(\dfrac{4}{6}\) (
    Solution
    x
    Solution: \(\displaystyle \frac{4}{6}=\frac{2\times2}{3\times2}=\frac{2}{3}\)
    Details:
    This fraction bar represents 4 out of 6 or \(\dfrac{4}{6}\). Four are shaded orange out of a total of six.
    This image shows a fraction: four over six. Four is highlighted in orange.A long horizontal rectangle divided into 6 sections. The left four sections are shaded orange. The right two sections are white.

    To simplify the fraction, divide the numerator and denominator by 2.
    This is the image of a fraction. The top or numerator contains 4 divided by 2. The denominator or bottom contains 6 divided by 2.

    To simplify the fraction bar, group 2 rectangles into one bar. This is the same as to divide by 2.
    A long horizontal rectangle divided into 6 sections. The left four sections are shaded orange. The right two sections are white.A long horizontal rectangle divided into 3 sections. The left two sections are shaded orange. The right section is white.

    The resulting fraction is \(\dfrac{2}{3}\); and 2 out of 3 bars are shaded orange.
    An equation of 4 divided by 2 over 6 divided by 2 equal 2 over 3.A long horizontal rectangle divided into 3 sections. The left two sections are shaded orange. The right section is white.
    )
  2. \(\dfrac{10}{25}\) (
    Solution
    x
    Solution: \(\displaystyle \frac{10}{25}=\frac{2\times5}{5\times5}=\frac{2}{5}\)
    Details: You can simplify a fraction if the numerator and the denominator share at least one common factor. You must first find the factorizations of these two numbers.

    Prime factorization of 10:
    10 is divisible by 2 since it is an even number.
    \(2 \times 5 = 10\)

    Image with the number 10 at the top. Under 10 are two lines pointing down and outward, one to the right and one to the left. The left line points to the number 2, which is orange. The right line points to the number 5, which is also orange. There is a multiplication sign between the numbers 2 and 5 indicating that 2 times 5 is 10.

    Both the number 2 and the number 5 are prime numbers so the prime factorization of 10 is \(2 \times 5\).

    Prime factorization of 25:
    25 ends in a 5 and any number that ends in a 5 or a 0 is divisible by 5.
    \(5 \times 5 = 25\)

    Image with the number 25 at the top. Under 25 are two lines pointing down and outward, one to the right and one to the left. The left line points to the number 5, which is orange. The right line points to the number 5, which is also orange. There is a multiplication sign between the numbers 5 and 5 indicating that 5 times 5 is 25.

    The number 5 is a prime number so the prime factorization of 25 is simply \(5 \times 5\).

    Now that you have the prime factorization of both the numerator and the denominator, you can rewrite the fraction like this: \(\displaystyle \frac{10}{25}=\frac{2\times5}{5\times5}\)

    Since the numerator and the denominator both share one 5, and since \(\dfrac{5}{5}=1\) you can rewrite the fraction as: \(\displaystyle \frac{2\times5}{5\times5}=\frac{2}{5}\times1\)

    Anything multiplied by 1 is just itself, so \(\displaystyle \frac{2}{5}\times1=\frac{2}{5}\).

    There are no other common factors between 2 and 5 so this fraction is as simplified as it can be.

    The final solution: \(\dfrac{2}{5}\)
    )
  3. \(\dfrac{4}{7}\) (
    Solution
    x
    Solution: \(\dfrac{4}{7}\) is already simplified to lowest terms.
    Details:
    The 4 and 7 do not share a common factor other than 1, therefore this answer is already written in its simplest or lowest terms.
    )
  4. \(\dfrac{30}{48}\) (
    Video Solution
    x
    Solution: \(\displaystyle \frac{30}{48}=\frac{2\times3\times5}{2\times2\times2\times2\times3}=\frac{5}{8}\)
    Details:

    | Transcript)
  5. \(\dfrac{42}{70}\) (
    Video Solution
    x
    Solution:\(\displaystyle \frac{42}{70}=\frac{2\times3\times7}{2\times5\times7}=\frac{3}{5}\)
    Details:

     | Transcript)
  6. \(\dfrac{12}{84}\) (
    Solution
    x
    Solution: \(\displaystyle \frac{12}{84}=\frac{2\times2\times3}{2\times2\times3\times7}=\frac{1}{7}\)
    Details:
    If the numerator and denominator of a fraction have any common factors, the fraction can be simplified.

    First, find the prime factorization of the numerator:
    12 is divisible by 2 because it is an even number and all even numbers are divisible by 2:
    \(2 \times 6 = 12\)

    Image with 12 at top and two lines pointing down to the left and right. The left line points to a 2, which is orange, and the right arrow points to a 6. There is a multiplication sign between the 2 and 6, indicating that 2 times 6 is 12.

    6 is not prime so you must factor 6 as well.

    Because 6 is even, it can be divided by 2:
    \(2 \times 3 = 6\)

    The numbers 2 and 3 are both prime, so this is as far as you can factor this number.

    Image with 12 at top and two lines pointing down to the left and right. The left line points to a 2, which is orange, and the right line points to a 6. There is a multiplication sign between the 2 and 6, indicating that 2 times 6 is 12. Under the 6 are two more lines pointing down to the left and right. The left line points to an orange 2 and the right line points to an orange 3. There is a multiplication sign between the 2 and 3, indicating that 2 times 3 is 6. 

    The prime factorization of 12 is \(2 \times 2 \times 3\).

    Next, find the prime factorization of the denominator:
    84 is even so you can start by dividing it by 2:
    \(2 \times 42 = 84\)

    Image with 84 at top and two lines pointing down to the left and right. The left line points to an orange 2, and the right line points to a 42. There is a multiplication sign between the 2 and 42, indicating that 2 times 42 is 84.

    The number 2 is prime but 42 is not so you still need to find the factors of 42.

    42 is even so you can divide it by 2:
    \(2 \times 21 = 42\)

    Image with 84 at top and two lines pointing down to the left and right. The left line points to an orange 2, and the right line points to a 42. There is a multiplication sign between the 2 and 42, indicating that 2 times 42 is 84. Under the 42 are two more lines pointing down to the left and right. The left line points to an orange 2 and the right line points to a 21. There is a multiplication sign between the 2 and 21, indicating that 2 times 21 is 42. 

    The number 2 is prime, but 21 is not prime, so you need to factor 21.

    21 is divisible by 3:
    \(3 \times 7 = 21\)

    Image with 84 at top and two lines pointing down to the left and right. The left line points to an orange 2, and the right line points to a 42. There is a multiplication sign between the 2 and 42, indicating that 2 times 42 is 84. Under the 42 are two more lines pointing down to the left and right. The left line points to an orange 2 and the right points to a 21. There is a multiplication sign between the 2 and 21, indicating that 2 times 21 is 42. Under the 21 are two more lines pointing down to the left and right. The left line points to an orange 3 and the right points to an orange 7. There is a multiplication sign between the 3 and 7, indicating that 3 times 7 is 21.

    Both 3 and 7 are prime, so this is as far as you can factor.

    The prime factorization of 21 is \(2 \times 2 \times 3 \times 7\).

    Use the prime factorizations to determine if there are any common factors in the numerator and the denominator.

    \(\displaystyle \frac{12}{84}=\frac{2\times2\times3}{2\times2\times3\times7}\)

    Here you see that both the numerator and the denominator have two 2s and a 3 in common. You can rewrite the fraction like this:

    \(\displaystyle \frac{12}{84}=\frac{2\times2\times3}{2\times2\times3\times7}=\frac{2}{2}\times\frac{2}{2}\times\frac{3}{3}\times\frac{1}{7}\)

    Note: You might think that there is only 0 left in the numerator after separating out the \(2 \times 2 \times 3\), but there is still a 1 because if there was a 0 in the numerator, the numerator would equal 0. There is always an invisible 1 being multiplied to everything. \(2 \times 2 \times 3 \times {\color{green}1} = 12\).

    Anything divided by itself is equal to 1.

    \(\displaystyle \frac{2}{2}\times\frac{2}{2}\times\frac{3}{3}\times\frac{1}{7}=1\times1\times1\times\frac{1}{7}\)

    Anything multiplied by 1 is just itself, so you are left with \(\dfrac{1}{7}\).

    \(\dfrac{12}{84}\) simplifies to \(\dfrac{1}{7}\).
    )
Introduction to Fractions
Addition and Subtraction of Fractions with Common Denominators