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 be able 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:

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

The following video will show how this conversion can be done:

Video Source (10:18 mins) | Transcript

When converting from a Mixed Number to an Improper Fraction:

• Multiply the integer by the denominator and add the numerator to get the new numerator.
• Keep the denominator the same.
• Example: \(\displaystyle2\frac{1}{3}=\left ( \frac{2}{1}\times\frac{3}{3} \right )+\frac{1}{3}=\frac{2\times3}{3}+\frac{1}{3}=\frac{6}{3}+\frac{1}{3}=\frac{7}{3}\)

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

Additional Resources

• Khan Academy: Writing mixed numbers as improper fractions (06:42 mins | Transcript)
• Khan Academy: Writing improper fractions as mixed numbers (04:01 mins | Transcript)

There are additional resource videos to the left of the screen on each of these links above.

Practice Problems

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.
   
   
   Right now we have \(\displaystyle 1\frac{3}{4}\), which is the same as \(\displaystyle 1+\frac{3}{4}\). To write this as an improper fraction, we will change 1 to a fraction with a denominator of 4. We know that we 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.
   
   We now have the following:
   \(\displaystyle \frac{4}{4}+\frac{3}{4}\)
   
   
   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 we count how many parts are shaded, we have a total of 7 sections of size \(\dfrac{1}{4}\).
   )
2. \(\displaystyle 5\frac{1}{8}\) (
   Video Solution

   x

   Solution: \(\dfrac{41}{8}\)
   Details:
   

   
   (Video Source | Transcript)
   )
3. \(\displaystyle 3\frac{2}{5}\) (
   Solution

   x

   Solution:
   \(\dfrac{17}{5}\)
   )
4. \(\dfrac{11}{4}\) (
   Video Solution

   x

   Solution: \(\displaystyle 2\frac{3}{4}\)
   Details:
   

   
   (Video Source | Transcript)
   )
5. \(\dfrac{13}{6}\) (
   Solution

   x

   Solution: \(\displaystyle 2\frac{1}{6}\)
   Details: To convert \(\dfrac{13}{6}\) into a mixed number we start by dividing 13 by 6:
   
   We know that 6 times 2 equals 12:
   
   Subtract 12 which gives us a remainder of 1:
   
   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 \(\displaystyle 2\frac{1}{6}\).
   We can also represent this visually:
   
   )
6. \(\dfrac{32}{3}\) (
   Solution

   x

   Solution:
   \(\displaystyle 10\frac{2}{3}\)
   )

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