Subtracting Fractions
Expressed in symbols, the rule for subtracting one fraction
from another is as follows:
![](./articles_imgs/1288/pic1.gif)
Lets break this down to see everything that is expressed
in this rule. The numerator of the sum is a · d - b · c . This
is almost exactly the same as the pattern of cross-multiplying.
The only difference is that because the fractions are subtracted,
a minus sign now joins the a · d to the b · c .
![](./articles_imgs/1288/pic2.gif)
To get the denominator of the sum, you just multiply the two
denominators ( b and d ) together.
Example
Work out each of the following differences of fractions.
![](./articles_imgs/1288/pic3.gif)
Solution
(a) You can work out this difference in the
two fractions using the given rule for finding the difference of
two fractions:
![](./articles_imgs/1288/pic4.gif)
This fraction subtraction can also be accomplished in a
slightly more efficient way by noticing that the two denominators
that are involved (i.e. 7 and 14) are
related because 14 is a multiple of 7
(i.e. 7 × 2 = 14).
![](./articles_imgs/1288/pic5.gif)
(b) You can work out this difference of two
fractions using the standard rule for subtracting two fractions:
![](./articles_imgs/1288/pic6.gif)
You can then simplify further by FOILing and collecting like
terms. Note that when - 1 · ( x + 3) is expanded the negative
sign multiplies both the x (to create - x ) and the
+3 (to create the - 3).
![](./articles_imgs/1288/pic7.gif)
You could also FOIL out the ( x + 3)
that appears in the denominator, but it is a matter of opinion as
to whether or not that actually makes the fraction any simpler.
There is a more efficient way of subtracting these two fractions.
This more efficient way is possible because the two denominators
(i.e. ( x + 3) and ( x + 3)
) are related because ( x + 3)
is a multiple of ( x + 3). The more efficient thing to do is to
multiply both the numerator and denominator of by ( x + 3). Doing this:
![](./articles_imgs/1288/pic11.gif)
Although the two answers look different, they are actually the
same because:
![](./articles_imgs/1288/pic12.gif)
(c) Although the subtraction:
![](./articles_imgs/1288/pic13.gif)
does not initially appear to be a subtraction that can be
carried out using the usual rule for fraction subtraction, it is
possible if you re-write the 2 · x as a fraction by putting it
over a denominator of one. Doing this:
![](./articles_imgs/1288/pic14.gif)
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