# How do I find a fraction that is smaller than any other fraction?Please explain the steps for me. Thanks

cburr | Certified Educator

It sounds like you need to find which one of several fractions is the smallest, since there is no such thing as THE smallest fraction.

To compare different fractions, you CAN'T just look to see which one has the largest number in the denominator, because that one may NOT be the smallest. It is true that 1/big number is smaller than 1/small number. However, if the numerators are larger numbers, you can't go by the size of the denominator.

The way to compare fractions is to convert them all to fractions with the same denominator. To do this, you have to find a number that is a multiple of all the denominators you are working with.

Let's take an example. Which is smallest?

1/2 4/9 3/8

Look at the tables for 2, 9, 8 and find the smallest number that is common to all of them.

Any even number is a multiple of 2, so we can focus on 9 and 8.

9: 9 18 27 36 45 54 63 72

8: 8 16 24 32 40 48 56 64 72

In this case, 9 x 8 is the smallest that works, and it is an even number so it works for 2 as well.

To convert 1/2 into 72nds, we have to divide 72 by 2 = 36. Now just multiply top and bottom by that:

numerator: 1 x 36 = 36

denominator 2 x 36 = 72

equivalent fraction = 36/72

To convert 4/9 into 72, multiply top and bottom by 8.

num: 4 x 8 = 32

denom: 8 x 9 = 72

equiv fraction = 32/72

To convert 3/8, multiply by 9.

num: 3 x 9 = 27

denom: 8 x 9 = 72

equiv fraction = 27/72

So, we have three new fractions: 36/72, 32/72, 27/72. So, 27/72 -- which is equiv to 3/8 -- is the smallest.

Sometimes it is even harder to find the common denominator, and you have to break each denominator down into its prime factors. Here's an example of how to do that with 24.

It's an even number, so start by dividing by 2.

24 / 2 = 12 / 2 = 6 / 2 = 3

So, the prime factors of 24 are 2 x 2 x 2 x 3 or 2^3 x 3.

You can find the prime factors of all the denominators. Then, to find the lowest common multiple, find every prime factor that shows up in any, and use it to the highest power in any.

So, let's say your 3 denominators are 18, 120 and 28.  The prime factors of your three denominators are:

18:    2 x 3^2

120:  2^3 x 3 x 5

28:    2^2 x 7

The lowest common multiple would be:

2^3 x 3^2 x 5 x 7 = 2520

To find out what to multiply the fractions by to get the appropriate equivalent fractions, take that number's prime factors out and see what's left.

For 18, take out 2 and 3^2, leaving 2^2 x 5 x 7 = 140.  So, for the fraction in 18ths, you would multiply top and bottom by 140.