Simplifying Fractions

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. 

But, we typically prefer to display fractions in "simplest" or "reduced" form, so that they are easy to read and work with.

How do you simplify a fraction?

To simplify fractions, divide the numerator and denominator by the same number until you cannot divide the numerator or denominator by the same number any longer.

Example:

Simplify $\displaystyle{\frac{24}{96}}$

The fastest way to simplify a fraction is to find the GCF (greatest common factor) of the numerator and denominator and then divide by that number. 

The greatest common factor of 24 and 96 is 24.

$\displaystyle{\frac{24}{96}=\frac{24 \div 24}{96 \div 24}=\mathbf{\frac{1}{4}}}$

However, sometimes we don't know the GCF of two numbers.  In that case, it's fine to divide each number by any common factor. If you don't hit the GCF on the first try, you'll just have to divide several times. 

$\displaystyle{\frac{24}{96}=\frac{24 \div 12}{96 \div 12}=\frac{2}{8}=\frac{2 \div 2}{8 \div 2}=\mathbf{\frac{1}{4}}}$

In a case when you can't come up with any large factors, start with the basics.  If you have even numbers, you can always divide by 2.  Once the numbers get smaller, you might find other factors that work. Try that!

$\displaystyle{\frac{24}{96}=\frac{24 \div 2}{96 \div 2}=\frac{12}{48}=\frac{12 \div 2}{48 \div 2}=\frac{6}{24}=\frac{6 \div 2}{24 \div 2}=\frac{3}{12}=\frac{3 \div 3}{23 \div 3}=\mathbf{\frac{1}{4}}}$

The basic rule of simplifying fractions is: you can divide the numerator and denominator by anything, as long as you divide the top and the bottom by the same number!

Why does this work? It's just math. Remember how, in fractions, if the numerator and the denominator are equal, the fraction is equal to 1.

$\displaystyle{\frac{9}{9}=1}$    and     $\displaystyle{\frac{343}{343}=1}$

That means that, if you can pull the same factors out of both the numerator and denominator of a fraction, they make 1, and when you multiply anything by 1, it stays itself.  If you pull "ones" out of a fraction without changing the value of the fraction, so you can cancel out factors as long as they are in both the numerator and denominator.  Watch how this works:

$\displaystyle{\frac{6}{9}=\frac{3 \times 2}{3 \times 3}=\frac{\cancel{3} \times 2}{\cancel{3} \times 3}=\mathbf{\frac{2}{3}}}$

This process works for any kind of fraction, even one with variables in it!

Example:

Simplify $\displaystyle{\frac{4xy^2}{6x^2y^2}}$

First, we'll break out the factors in these fractions.  Then we will cancel out the factors that are on top and bottom

$\displaystyle{\frac{4xy^2}{6x^2y^2}=\frac{2\cdot 2 \cdot x\cdot y \cdot y}{3\cdot 2 \cdot x\cdot x \cdot y \cdot y}=\frac{2\cdot \cancel{2} \cdot \cancel{x}\cdot \cancel{y} \cdot \cancel{y}}{3\cdot \cancel{2} \cdot \cancel{x} \cdot x \cdot \cancel{y} \cdot \cancel{y}}=\frac{2}{3x}}$

Any terms that are the same on the top and the bottom can be canceled out:

Example:

Simplify $\displaystyle{\frac{4(x+y)}{5(x+y)}}$

$\displaystyle{\frac{4(x+y}{5(x+y)}=\frac{4\cancel{(x+y)}}{5\cancel{(x+y)}}=\mathbf{\frac{4}{5}}}$

All fractions follow these rules.  Just use them carefully and you'll make your fractions much easier to work with!