Fractions

When you add fractions, you need to make sure that the denominators are the same.  

Why? The denominators of a fraction tell you how many pieces the whole is cut into.  If the denominators are different, then the sizes of the pieces are different, so you can't just add them together.

Fractions show the number of pieces a whole is cut into (the denominator) and the number of those pieces that you have (numerator).  So, if you have $\dfrac{1}{4}$ of a pizza, you know that the pizza is cut into 4 equal pieces and you have one of them. 

Because fractions are numbers, you can add, subtract, multiply, and divide them. 

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:

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

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

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

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

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.

There are infinite ways to write a fraction or ratio.

Look at the fractions below.  They are all equivalent.  They all represent one half. 

$\require{cancel}\displaystyle{\frac{1}{2}=\frac{2}{4}=\frac{3}{6}=\frac{4}{8}=\frac{5}{10}}$

Do you see the patterns in these numbers? In each fraction, the numerator (the number on the top) is half of the denominator (the number on the bottom). Each of the fractions above represents half. 

There are many ways to compare fractions.  Some methods are quick and easy.  But a lot of methods require a lot of math.  The best way to compare a group of fractions is to figure out the best method for any given set of fractions (and, in a large set, you might employ different methods to compare different specific fractions!)

Fractions are ways to show a portion.

Fractions are divided into two parts: the numerator and the denominator.  The numerator goes on top and tells you the part or portion you are interested in.  The denominator goes on the bottom, and it tells you the total number of possible parts or pieces.

$\displaystyle{\frac{\text{Numerator}}{\text{Denominator}}=\frac{\text{The part you are interested in}}{\text{The total number of parts}}}$

Example: