BMCC: Math Tutorials: Fractions:
Subtraction of Fractions
Quick links to topics on this page:
Three Quick Definitions
Subtracting Fractions with the Same Denominator
Subtracting Fractions with Different Denominators
Subtracting Two Mixed Numbers Without "Borrowing"
Subtracting Two Mixed Numbers Using "Borrowing"
Subtracting a Mixed Number from a Whole Number Using "Borrowing"
Before we start discussing subtraction of fractions, you need to know three terms relating to subtraction. Consider the general expression a - b = c . In a subtraction expression, the number being subtracted from (a in this example) is called the minuend. The number being subtracted (b in this example) is called the subtrahend. The "answer," or result (c in this example), is called the difference.
Here's how it looks using the words themselves:
These terms will be used throughout this tutorial.
Example: Consider 13 - 5 = 8 .
13 is the minuend
5 is the subtrahend
8 is the difference
The approach used to subtract one fraction from another with the same denominator is the same as for adding two fractions with the same denominator. The denominator of the difference (the result of the subtraction) will be the same as that of the two original fractions. The numerator of the difference is obtained by subtracting the numerators of the original fractions. Shown in general terms,
Examples:
Note that, as in the second example, even if both initial fractions were in simplest form, the final difference may still need to be reduced to simplest form.
As with whole numbers, the final difference after subtracting two fractions may be negative. For instance, consider subtracting 
from 
:
where 
is more commonly written as 
.
To subtract fractions with different denominators, they must first be rewritten as equivalent fractions with a common denominator. The common denominator will be the least common multiple of the two original denominators. The fractions are then subtracted as described just above.
Example: Subtract 
.
The denominators are 5 and 6, which you can show to have a least common multiple of 30. The two original fractions can be rewritten as equivalent fractions with a common denominator of 30 like this:
Now we just need to subtract 
:
Sometimes, the fraction obtained may still need to be reduced to simplest form, although in this case 
is already in simplest form.
And a second example: Subtract 
.
The denominators are 12 and 8, which have a least common multiple of 24. Writing the original fractions as equivalent fractions with a denominator of 24 gives:
Substituting the equivalent fractions into the original equation, we now need to subtract 
as follows:
Since 
is already in simplest form, no further manipulation is necessary.
Subtracting two mixed numbers involves three main steps, just like adding two mixed numbers, and some of the steps may involve additional processes as shown in the examples below. The general procedure for subtracting two mixed numbers is:
Example: Subtract 
.
Step 1: Subtract the two fractional portions. The fractional portions of the two mixed numbers are 
and 
. Since these have different denominators, each must be converted to an equivalent fraction so they have common denominators. The common denominator we want to use is the least common multiple. For the two original denominators of 8 and 5, the least common multiple is 40. Rewriting the two fractions as equivalent fractions with 40 for a denominator gives
Substituting the equivalent fractions for the original fractional portions, we can now subtract the fractional portions:
The fraction 
is already in simplest form, and does not need to be reduced further.
Step 2: Subtract the whole number portions of the mixed numbers.
Subtracting 6 - 4 gives a difference of 2 .
Step 3: Combine the fractional portion difference from step 1, which is , with the whole number difference from step 2, which is 2, into a mixed number. This gives an answer of 
.
The method outlined just above for subtracting two mixed numbers is fairly straightforward, as long as the fractional portion of the subtrahend has a smaller value than the fractional portion of the minuend. In the previous example we were subtracting (value = 0.4) from (value = 0.875), so the subtraction could be done without borrowing.
But what if the fractional portion of the subtrahend is larger than the fractional portion of the minuend? This can occur even if the subtrahend itself has a smaller value than the minuend, as shown here.
Consider the subtraction 
. The minuend mixed number is larger than the subtrahend mixed number, so the difference after subtracting will be a positive number. However, the minuend fractional portion (
) has a smaller value than the subtrahend fractional portion (
). To perform this subtraction you still go through the same three steps outlined above, but there is a modification in the first step.
Step 1: Subtract the two fractional portions. We need to subtract 
. If this was the entire problem, that is, if we were simply subtracting one fraction from another, we would proceed as we did earlier on this page, and the resulting difference would be a negative fraction. Because these fractions are only portions of mixed numbers, though, we need to proceed differently. We will make the minuend fractional portion larger than the subtrahend fractional portion by borrowing from the minuend whole number portion.
First, though, the fractions must have a common denominator. The original denominators of 4 and 3 have a least common multiple of 12. Rewriting the original fractional portions with a common denominator of 12 gives:
So our subtraction expression is 
. Now we want to make the minuend larger than the subtrahend by borrowing some of the value of the whole number portion of the original minuend 
. This is done in a way that does not change the overall value of the original minuend.
The original minuend can be rewritten using the common denominator of 12 like this:
To see how we can borrow some of the whole number portion of 
and transfer it to the fractional portion, let's rewrite the mixed number in different ways:
In the sequence above, we are "borrowing" 1 from the value of the whole number portion of the mixed number, and adding it to the fractional portion to convert the fractional portion from a proper fraction into an improper fraction. Note that the overall value of the mixed number remains unchanged in each of the expressions in the above sequence.
Now our original subtraction problem looks like 
. Remember that we're still in step 1! The task is still to subtract the fractional portions of the mixed numbers: 
. Now that the minuend fractional portion is larger than the subtrahend fractional portion, the subtraction can proceed normally:
Since 
is already in simplest form, no further reduction is necessary.
Whew! We've finally finished step 1.
Step 2: Subtract the whole number portions of the two mixed numbers. This is easy, except you need to remember that the whole number portion of the minuend in the original subtraction problem was reduced by 1 during the borrowing procedure, so it's value is now 6 instead of the original 7. Step 2 thus consists of subtracting 6 - 2 , leaving a whole number portion difference of 4 .
Step 3: Combine the whole number difference from step 2 (4) and the fractional difference from step 1 () . This gives us 
for the final result, which does not need to be reduced further.
In order to subtract a mixed number from a whole number, it's easiest to first convert the whole number to a mixed number by borrowing a portion of the whole number value and converting the borrowed portion to a fractional component. In this case there will be no need to convert the denominators to a common denominator. Since you can select any denominator you want when you first express the whole number as a mixed number, you just make the denominator the same as the denominator in the fractional component of the original subtrahend mixed number.
Example: Subtract 
.
We first need to convert the whole number 6 to a mixed number. Since the denominator of the fractional portion of the original subtrahend is 12, we also want the denominator of the fractional portion of the minuend to be 12 . We can convert the minuend to a mixed number by borrowing part of the whole number value like this:
With the original whole number 6 now rewritten as the mixed number 
, the original subtraction expression reads as
From this point on, the procedure is the same as subtracting two mixed numbers, as outlined farther up the page.
Step 1: Subtract the fractional portions of the two mixed numbers. This gives us
Step 2: Subtract the whole number portions of the two mixed numbers. Remember that the whole number portion for the minuend is now 5, rather than the original 6, because we borrowed 1 from the original whole number value to create the fractional portion 
. The subtraction of the whole number portions now gives us 5 - 2 = 3 .
Step 3: Combine the whole number difference and the fractional portion difference into a mixed number, and reduce the final sum to simplest form if necessary. From step 1 we have 
, and from step 2 we have 3 . Combining these gives us the final difference of 
. Since is already in simplest form, no further reduction is necessary.
Next: Multiplication of Fractions
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