Adding and Subtracting Fractions
It is not difficult to visualize what it means to add two
fractions, as in
As explained before, each of the two fractions represents
dividing a whole into a certain number of equalsized pieces, and
including a certain number of those pieces. So, we can represent
the two fractions above pictorially as
Thus, 3 / 4 represents 3 pieces, each of which is 1 / 4 of the
whole, and 2 / 3 represents 2 pieces, each 1 / 3 of the whole.
Then, to form the sum
we need to come up with a fraction representing how much of a
whole we get when we combine all five shaded pieces in the
diagrams above. The difficulty is that we can’t just count
up the number of shaded pieces to be 5 and the total number of
pieces to be 7, and so declare the answer to be 5 / 7 (from ), since the pieces are
not all the same size to begin with. In fact, you can see that
this answer would clearly be wrong, because it represents less
than a complete whole, and yet it is obvious from the pictures
that the combined shaded areas are much more than a complete
whole.
There are several errors in the incorrect procedure just
described. The most obvious is that the five smaller shaded bits
in the diagram are not all the same size, and so simply counting
a total of 5 pieces is not a correct reflection of the combined
size of the two fractions. However, there is a way around this
problem.
First, divide each of the quarters in the diagram for 3 / 4
into three smaller, equalsized pieces:
In fact, this diagram is now illustrating both 3 / 4 as well
as 9 / 12, because now the whole can also be viewed as being
partitioned into 12 equal pieces, of which 9 are shaded.
Secondly, divide each of the thirds in the diagram for 2 / 3
into four smaller equalsized pieces:
Again, this results in the whole being divided into twelve
equal pieces of which eight are shaded.
But now, both diagrams have shaded pieces which are the same
size! Each is 1 / 12 of the whole. When the total number of these
equalsized shaded pieces is tallied, we get
9 + 8 = 17
shaded pieces, each of size 1 / 12 of the whole. Thus, we
apparently get that
You can verify that this is a plausible result by using your
calculator to turn the fractions into decimal numbers and
checking the addition.
What this really shows is that we can add two fractions
together only if they have the same denominator (and the same
rule will apply to subtraction):
because for the sum of the numerators (or the difference of
the numerators) to be meaningful, they must refer to pieces of
the whole all of which are the same size (that is the same thing
as saying that the denominators of the fractions are the same).
