Converting Mixed Numbers to Improper Fractions

Mixed numbers are numbers that include whole numbers and fractions, such as:

$1\dfrac{1}{2}$

$3\dfrac{4}{5}$

$25\dfrac{3}{4}$

When we think about numbers that involve whole numbers and fractions, we tend to think in mixed numbers.  How many pizzas are left? Two and a half.  How many hours long is the movie?  1 and a quarter.

Mixed numbers are easy to understand.  But, in math, we often prefer to work with improper fractions, fractions where the whole number and the fraction are expressed in one, top-heavy, fraction.  If you are multiplying or dividing fractions, you definitely want to convert any mixed numbers into improper fractions before you begin!

When you have a mixed number and you want to convert it to an improper fraction, you pull out the whole numbers, turn them into fractions, then add the fraction, and you have an improper fraction.  Because you can create whole number with any denonimator you want (just make the numerator equal to the denominator for 1), it's easiest to work in the denominator that the mixed number already uses.

Let's use $3\dfrac{4}{5}$.

$$\eqalign{3\dfrac{4}{5}&= 1 + 1 + 1+ \dfrac{4}{5}\quad &&\text{Break out the whole number into wholes}\\&=\dfrac{5}{5}+\dfrac{5}{5}+\dfrac{5}{5}+\dfrac{4}{5}&&\text{Turn the wholes into fractions}\\&=\dfrac{19}{5}&&\text{Add your numerators to create an improper fraction}}$$

What we did above is exactly right, but there's a quicker way to do it.  Rather than adding all of those individual wholes, why not multiply them?  The basic algorithm for turning mixed numbers into improper fractions is to multiply the whole number times the denominator, then add the numerator from the fraction, and put the new numerator on top of the denominator.

Example:

$$\eqalign{3\dfrac{4}{5}&= \dfrac{3 \times 5}{5}+ \dfrac{4}{5}\quad &&\text{Multiply the whole by the denominator}\\&=\dfrac{15}{5}+\dfrac{4}{5}&&\text{Add the improper fraction to the other fraction}\\&=\dfrac{19}{5}&&\text{There's your improper fraction}}$$

Basically, to turn a mixed number into an improper fraction, find the improper fraction that would capture the whole number (whole number times denominator, over denominator) then add in fraction.  Voila: an improper fraction that captures your whole number and your fraction.