How To Subtract Fractions

Fractions are just a simple representation of the percentage of one item with respect to another.  Any fraction will include two numbers, one at the top and bottom, which are respectively, the numerator and denominator.  So if we look at the fraction 5/16, it can be easily recognized that the fraction represents 5 out of 16 or 31.3%.  It should be no surprise from this that when performing the subtraction operation for fractions that they must each have a common denominator.  Also, from this it should be noted that when the fractions are subtracted the denominator does not change but only the numerators are subtracted.

It is a very simple operation to perform the subtraction of two or more fractions when they have common denominators.  As an example, let’s look at 5/16 + 7/16.  We see that 5 + 7 = 12, so our new fraction is 12/16 = ¾ and this is our reduced final fraction.

For the situation where your fractions have different denominators, there is more effort you have to put into it.  These uncommon denominators can be made equivalent by simplification, reduction or both.  This is illustrated in the following example, 5/16 + 22/32 = 5/16 + 11/16 = 16/16 = 1.  In this subtraction operation, we divided two into both the denominator and numerator of 22/32 to then reduce it into11/16.  It should be noted that as you perform an operation with the numerator of a fraction, such as with this example, that it must also be performed with the denominator. 

The denominators in the fractions can also be equated by multiplying the denominator of the second fraction by the denominator and numerator of the first fraction and vice versa for the second fraction with the denominator of the first.  As you work through this you’ll see that you end up having new denominators, which are the same, in the fractions that are the same product of each denominator of the original fractions.

Here is an example illustrating this process.  Say we have the following:

11/16 – 9/20 = ?

(11 x 20)/(16 x 20) – (9 x 16)/(20 x 16) = 220/320 – 144/320 = 76/320 =19/80. 

Note that for reducing the final fraction that 76 and 320 were each divided by four to reduce them to 19 and 80, respectively.  By inspection you can see that each new fraction was obtained by multiplying both its denominator and numerator by the denominator of the other fraction.  This is effectively multiplying each fraction by 1 but only using a bigger denominator and numerator.  The method used here can also be carried out for more than two fractions, even at one time.  If you have three fractions to be used in a subtraction operation with uncommon denominators, you will multiply the denominator and numerator of each fraction by each denominator of the additional fractions.  Just as before, this process will give you a new denominator, being the same for each fraction, that is a product of three terms and results in a new numerator for each fraction also a product of three terms.  From this, you have three fractions with common denominators for which the subtraction operation can be performed.