Fractions:

Converting Between Improper Fractions and Mixed Numbers

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:

Proper Fraction = A fraction whose numerator is smaller than the denominator. Example: $\frac{3}{4}$
Improper Fraction = A fraction whose numerator is larger 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 whole and $\frac{1}{3}$ 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: $2\frac{1}{3}=\left ( \frac{2}{1}\times\frac{3}{3} \right )+\frac{1}{3}=\frac{2\times3}{3}+\frac{1}{3}=\frac{6}{3}+\frac{1}{3}=\frac{7}{3}$

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: $\frac{11}{5}=11\div5=2 {\text{ with 1 divided by 5 left over = 2}}\frac{1}{5}$

Additional Resources

Khan Academy: Writing mixed numbers as improper fractions (06:42 mins)
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

Convert the following mixed number to an improper fraction:

$1\frac{3}{4}$

$5\frac{1}{8}$

$3\frac{2}{5}$

$\frac{11}{4}$

$\frac{13}{6}$

$\frac{32}{3}$

Solutions

$\frac{7}{4}$ (Written Solution)

Step 1: Rewrite the whole number as a fraction with the same denominator as the fraction.

$This image contains two pie circles. The first circle is not divided and is fully shaded. Under the first circle is number 1. The second circle is divided into 4 pieces and three of them are shaded. Under the second circle, there's the fraction 3 over 4.$

Right now we have $1\frac{3}{4}$ ,which is the same as $1+\frac{3}{4}$. 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 $\frac{3}{4}$ has a denominator of 4. $\frac{4}{4}$ is equal to 1 because 4 divided by 4 equals 1.

We now have:

$\frac{4}{4}+\frac{3}{4}$

$This image contains two pie circles. The first circle is divided in 4 pieces and all of the pieces are shaded. Under the first circle is fraction 4 over 4. The second circle is divided into 4 pieces and three of them are shaded. Under the second circle, there's the fraction 3 over 4. Between the two fractions there is the addition sign.$

Step 2: Add the fractions

$\frac{4}{4}+\frac{3}{4}=\frac{7}{4}$

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 $\frac{1}{4}$.

$\frac{41}{8}$ (Solution Video | Transcript)

$\frac{17}{5}$

$2\frac{3}{4}$ (Solution Video | Transcript)

$2\frac{1}{6}$ (Written Solution)

To convert $\frac{13}{6}$ 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, 2 times with a remainder of 1. So $\frac{13}{6}$ is equivalent to 2 wholes with a remainder of $\frac{1}{6}$ of a whole, or $2\frac{1}{6}$.

We can also represent this visually:

$This image contains three pie circles. Each circle is divided into six parts. The first two circles have all six pieces shaded in. The third circle has only one of the six pieces shaded in. Above the first two circles is the number one. Above the third circle is the fraction one over six. If all of the numbers are added together, they add up to two and one over six. Under the first two circles are the fractions six over six. Under the third circle is the fraction one over six. If all three of the fractions are added together, then they add up to thirteen over six. This is to show the two ways fractions can be added together.$

$10\frac{2}{3}$