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Converting Between Improper Fractions and Mixed Numbers
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Introduction

In this lesson, you will learn how to convert between improper fractions and mixed numbers.


This video illustrates the lesson material below. Watching the video is optional.


Different Fractions

An important part of learning about fractions is becoming comfortable understanding what they mean. Being able to convert between improper fractions and mixed numbers is a great way to understand fractions and recognize how large or small a fraction is. Here are some math terms that will help you to understand this lesson better:

  • Proper Fraction: A fraction whose numerator is less than the denominator.
    • Example: \(\frac{3}{4}\)
  • Improper Fraction: A fraction whose numerator is greater than the denominator.
    • Example: \(\frac{4}{3}\)
  • Mixed Number: An integer combined with a proper fraction showing how many wholes and how many parts are in the number.
    • Example: \(2\frac{1}{3}\) means 2 wholes and \(\frac{1}{3}\) pieces, pronounced "two and one-third."

Converting Fractions and Mixed Numbers

Being able to convert between improper fractions and mixed numbers is a great way to be able to understand fractions and recognize how large or small a fraction is.

Example 1
Suppose you went to a party with lots of pizza. At the end, there was some pizza leftover. Below is an image of how much pizza was left over.

Circles showing pizzas with missing pieces

Figure 1

If you combine all of the leftover slices, how many pizzas will be left over? These are the fractions you have:
\begin{align*} \frac{4}{8}+\frac{3}{8}+\frac{5}{8}+\frac{2}{8}+\frac{1}{8}+\frac{6}{8}+\frac{8}{8} \end{align*}

Circles showing pizzas with missing pieces and fractions next to each circle.

Figure 2

Because all of these fractions have the same denominator (because all of the pizzas had the same amount of slices at the start) you can add the numerators to see how many slices you have leftover.

\begin{align*} Numerators: 4+3+5+2+1+6+8=29 \end{align*}

Now that you know the total of leftover slices is 29, you also know that the fraction of leftover pizza is \(\frac{29}{8}\). This is an example of an improper fraction, meaning the numerator is greater than the denominator.

To get a better understanding of what \(\frac{29}{8}\) really means, you can demonstrate it visually. If you take another look at each of the pizzas, you can combine the leftover slices to create whole pizzas.

You can combine the \(\frac{4}{8}\), the \(\frac{3}{8}\), and the \(\frac{1}{8}\) to get \(\frac{8}{8}\), making one whole pizza.

Partial pizzas being added to make a whole pizza.

Figure 3

Similarly, you can combine the \(\frac{2}{8}\) with the \(\frac{6}{8}\) to get a whole pizza.

Partial pizzas being added to make a whole pizza.

Figure 4

Now you can see that there are three full pizzas and \(\frac{5}{8}\) leftover. So \(\frac{29}{8}\) is the same as \(3\frac{5}{8}\). This is an example of a mixed number because there is an integer, 3, and a proper fraction, \(\frac{5}{8}\), where the numerator is smaller than the denominator.

Three full pizzas and one partial pizza.

Figure 5

Breaking \(3\frac{5}{8}\) down into fractions would make it look like this:

\begin{align*} \frac{8}{8}+\frac{8}{8}+\frac{8}{8}+\frac{5}{8} = \frac{29}{8} \end{align*}

Converting a Mixed Number to an Improper Fraction

When converting from a mixed number to an improper fraction:

  • Multiply the integer by the denominator
  • Add the numerator to get the new numerator.
  • Keep the denominator the same.

Example 2
Find the improper fraction of \(2\frac{1}{3} \).

\begin{align*} 2\times3 &= 6 &\color{red}\small\text{Multiply the integer by the denominator} \\\\ 6+ 1 &= 7&\color{red}\small\text{Add the numerator} \\\\ & \frac{7}{3} &\color{red}\small\text{New numerator over the denominator} \end{align*}

\(\frac{7}{3}\) is the improper fraction of \(2\frac{1}{3} \).

Example 3
Find the improper fraction of \(3\frac{5}{8} \).

\begin{align*} 3\times8 &= 24 &\color{red}\small\text{Multiply the integer by the denominator} \\\\ 24+ 5 &= 29&\color{red}\small\text{Add the numerator} \\\\ & \frac{29}{8} &\color{red}\small\text{New numerator over the denominator} \end{align*}

Another way to look at it: \(\frac{3\times8+5}{8} = \frac{29}{8} \).

Example 4
Find the improper fraction of \(2\frac{1}{4}\).

\begin{align*} 2\frac{1}{4} = \frac{4\times2+1}{4} =\frac{9}{4} \end{align*}

Because the denominator is 4, you know you are talking about something that has four parts. In this case, that means:

\begin{align*} \frac{4}{4}+&\frac{4}{4}+\frac{1}{4} = \frac{8}{4}+\frac{1}{4} = \frac{9}{4} \end{align*}

Converting an Improper Fraction to a Mixed Number

When converting from an improper fraction to a mixed number:

  • Divide the numerator by the denominator.
  • Keep the remainder as the numerator of the new fraction part.
  • The denominator stays the same.

Example 5
\( \frac{11}{5}=11\div5=2\;\) with 1 divided by 5 left over.

11 divided by 5 is 2 and 1/5

Figure 6

The mixed fraction is written as \(2\frac{1}{5}\).

Example 6
Change \(\frac{9}{4}\) into a mixed number.

To do this, take the numerator, 9, and divide it by the denominator, 4.

9 divided by 4 is 2 and 1/4

Figure 7

9 divided by 4 is 2 with a remainder of 1. The remainder 1 becomes the new numerator. The mixed fraction is written as \(2\frac{1}{4}\).

Example 7
Change \(\frac{29}{8}\) into a mixed number.

29 divided by 8 is 3 and 5/8

Figure 8

The mixed fraction is \(3\frac{5}{8}\).


Things to Remember


  • An improper fraction has a numerator that's larger than the denominator.
  • A mixed number is an integer combined with a proper fraction showing how many wholes and how many parts are in the number. To convert a mixed number to an improper fraction, take the integer and multiply it by the denominator, and then add the numerator. This becomes the new numerator, and the denominator stays the same.
  • To convert an improper fraction to a mixed number, take the numerator and divide it by the denominator.

Practice Problems

Convert the following mixed numbers into improper fractions:
  1. \(\displaystyle 1\frac{3}{4}\) (
    Solution
    x
    Solution: \(\dfrac{7}{4}\)
    Details:

    Step 1: Rewrite the whole number as a fraction with the same denominator as the fraction.
    This image contains two pie circles. The first circle is not divided and is fully shaded. Under the first circle is number 1. The second circle is divided into 4 pieces and three of them are shaded. Under the second circle, there is the fraction 3 over 4.
    Right now you have \(\displaystyle 1\frac{3}{4}\), which is the same as \(\displaystyle 1+\frac{3}{4}\). To write this as an improper fraction, change 1 to a fraction with a denominator of 4. You know that you need a denominator of 4 because \(\dfrac{3}{4}\) has a denominator of 4. \(\dfrac{4}{4}\) is equal to 1 because 4 divided by 4 equals 1.

    You now have the following:
    \(\displaystyle \frac{4}{4}+\frac{3}{4}\)
    This image contains two pie circles. The first circle is divided in 4 pieces and all of the pieces are shaded. Under the first circle is fraction 4 over 4. The second circle is divided into 4 pieces and three of them are shaded. Under the second circle, there's the fraction 3 over 4. Between the two fractions there is the addition sign.
    Step 2: Add the fractions.

    \(\displaystyle \frac{4}{4}+\frac{3}{4}=\frac{7}{4}\)

    The first circle has 4 shaded sections or 1 whole, and the second has 3 out of 4 shaded sections. When you count how many parts are shaded, you have a total of 7 sections of size \(\dfrac{1}{4}\).
    )
  2. \(\displaystyle 5\frac{1}{8}\) (
    Video Solution
    )
  3. \(\displaystyle 3\frac{2}{5}\) (
    Solution
    x
    Solution:
    \(\dfrac{17}{5}\)
    )
  4. \(\dfrac{11}{4}\) (
    Video Solution
    x
    )
  5. \(\dfrac{13}{6}\) (
    Solution
    x
    Solution: \(\displaystyle 2\frac{1}{6}\)
    Details:
    To convert \(\dfrac{13}{6}\) into a mixed number, start by dividing 13 by 6:
    Division bracket with dividend 13 inside the bracket and divisor 6 to the left of the bracket
    You know that 6 times 2 equals 12:
    Division bracket with dividend 13 inside the bracket and divisor 6 to the left of the bracket. A 2 is placed on top the bracket above the 3 in 13 and 12 is placed directly under the 13 under the bracket
    Subtract 12 which gives you a remainder of 1:
    Division bracket with dividend 13 inside the bracket and divisor 6 to the left of the bracket
    This means that 6 divides into 13 two times with a remainder of 1. So \(\dfrac{13}{6}\) is equivalent to 2 wholes with a remainder of \(\dfrac{1}{6}\) of a whole, or \(\displaystyle 2\frac{1}{6}\).
    You can also represent this visually:
    This image contains three pie circles. Each circle is divided into six parts. The first two circles have all six pieces shaded in. The third circle has only one of the six pieces shaded in. Above the first two circles is the number one. Above the third circle is the fraction one over six. If all of the numbers are added together, they add up to two and one over six. Under the first two circles are the fractions six over six. Under the third circle is the fraction one over six. If all three of the fractions are added together, then they add up to thirteen over six. This is to show the two ways fractions can be added together.
    )
  6. \(\dfrac{32}{3}\) (
    Solution
    x
    Solution:
    \(\displaystyle 10\frac{2}{3}\)
    )

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