Converting Improper Fractions to Mixed Numbers

Improper fractions are fractions that are greater than a whole, where the whole number and the fraction are expressed in one, top-heavy, fraction.  Improper fractions always have numerators that are bigger than their denominators, that's how you know they contain at least one whole. 

How do improper fractions work?

Any time fractions have numerators and denominators that are equal, they equal 1 whole:

$\dfrac{2}{2}=1$

$\dfrac{5}{5}=1$

$\dfrac{14}{14}=1$

$\dfrac{x}{x}=1$

So, if the numerator of a fraction is greater than the denominator, then the fraction (which is an improper fraction) is more than one. To make our answers clearer, we often want to change that improper fraction into a mixed number.  So, how do you figure out how much more than 1 an improper fraction is?

Remember, every fraction is a division problem, that reads $\mathbf{\text{numerator}\div\text{denominator}}$.

So, you do the division problem

• The whole  number answer is the number of whole numbers in the mixed number.
• The remainder is the numerator of the mixed number's fraction.
• The denominator of the old fraction is the denominator of the mixed number's fraction. 

Example:

Convert $\dfrac{11}{4}$ to a mixed number.

$$\eqalign{\dfrac{11}{4}=& 11 \div 4 \quad&&\text{Turn the fraction into a division problem}\\11 \div 4 =& 2 \text{ remainder } 3&&\text{Do the division problem} \\=&2\dfrac{3}{4}&&\text{The answer is the whole number, the remainder the numerator}}$$

The most common question that students ask is, why does the denominator stay the same?

The denominator stays the same because the item represented by the fraction is still cut into the same fractional pieces.  If you have $\dfrac{11}{4}$ of a pizza, you have pizzas that are cut into fourths.  You just have more than 1 whole pizza that is cut into fourths.  You can break the problem down this way:

$$\eqalign{\dfrac{11}{4}&= \dfrac{4}{4} + \dfrac{4}{4} + \dfrac{3}{4}\\&= 1 + 1 + \dfrac {3}{4}\\&=2\dfrac{3}{4}}$$

When you convert an improper fraction into a mixed number, you are just taking out the wholes (any groups in the numerator that are the same size as the denominator) and then leaving what is left over as a fraction.