In general, we simplify fractions by dividing the numerator and denominator by the same number.
It is usually easiest to start by dividing using smaller numbers such as 2, 3, 5 or 10.
You'll notice the easy numbers to divide by are all prime.
Before we try to compare fractions, we should try to reduce them down to their simplest form.
Simplifying Fractions by Dividing Using Prime Numbers 
Let's take a look at simplifying the fraction . 



Since both 36 and 48 are even, we know we can divide them by 2. If we divide 36 by 2, we will get a result of 18. If we divide 48 by 2, we will get a result of 24. 



Since 18 and 24 are both even, we can divide them by 2. 18 divided by 2 gives us a result of 9. 24 divided by 2 gives us a result of 12. 



9 and 12 are not both even, so we can't divide them both by 2. We should move on to the next prime number and check if we can divide by it. As it turns out, we can divide both 9 and 12 by 3. 9 divided by 3 gives us 3. 12 divided by 3 gives us 4. 



We were able to simplify our fraction down to something much smaller by dividing the numerator and denominator by our smallest prime numbers. is the simplest form of the fraction . You'll find that we can use this technique to simplify many fractions. 

Simplifying Fractions by Dividing Using Composite Numbers 
Another way to think about how we simplify fractions is using composite numbers. 



Since both 36 and 48 are even, we know we can divide them by 2. We can rewrite each number as you see on the right. 



At this point, since we have 2 on top and 2 on bottom, we can cancel them out from each other and be left with . 
We can rewrite 18 as 2 times 9. And 24 as 2 times 12. We can cancel out the 2 from the numerator and the denominator and we will be left with . 



We can rewrite 9 as 3 times 3. And 12 as 3 times 4. 

Comparing Fractions 
So what if we wanted to compare the fractions and ? Which is greater? Before we can solve this problem, we need to reduce our fractions down to simplest form. Luckily, we already saw that will turn into , and is already in its simplest form. So, we have to compare and and decide which is larger. 



Before we can decide which is largest, we need to convert their denominators to be the same. In this case, the lowest common denominator will be 4 since that is the first multiple 2 and 4 have in common. To convert to have a denominator of 4, we will need to multiply it by 2. However, we always have to multiply both the numerator and denominator by the same thing. So, we will multiply our fraction by . 

At this point, we can clearly see which fraction is largest. is greater than . 

To recap, we simplify fractions by dividing the the numerator and denominator by factors that they both have in common.
Before we can compare two fractions, we should simplify them down to their simplest form first, and then convert them to have the same denominators.