Working with Fractions
Misunderstanding the rules for working with fractions is by
far the most common reason for mistakes in Calculus problems.
That is more than enough to justify repeating these rules.
• The best thing about fractions is being able to reduce
them
Notice that all bets are off when the denominator is zero,
since we’re fundamentally unable to deal with that
situation. Here’s an example: “how many piles of
quarters can you make with $10?” has an answer but “how
many piles of nothing can you make with $10?” has no
comprehensible answer. Thus, we avoid the whole problem by
defining a fraction as
Remember, division by zero never makes sense, even in limits!
• To multiply fractions, we simply multiply “tops
and bottoms” or numerators and denominators
Notice that and since we wrote the first two fractions, and this
means
by the Zero Factor Law, so the resulting fraction also makes
sense.
• To divide fractions, we flip the one we’re
dividing by and then multiply those
In this case, we have to assume and then again by
the Zero Factor Law. However c = 0 would mean we were secretly
trying to divide by , which we already know is not allowed.
• Unfortunately, addition of fractions is harder,
basically because addition and division don’t go together
very well. Up to reducing fractions, the rule is still pretty
simple though:
We should expect to reduce this answer in most cases.
The right way to think of this rule is that we know how to add
fractions with the same denominator
and we can always “unreduce” two fractions to get
their product as a common denominator
.
• The rule for subtracting fractions is basically the
same
although we should expect to reduce this answer in most cases.
• It is often important to find out which of two
fractions is larger. However, you can always “unreduce”
to get either the numerators or denominators to be the same
(whichever is easier in your specific case). If the numerators
are the same, the fraction with the larger denominator is smaller
, because you are dividing the same numerator into more pieces.
If the denominators are the same, the fraction with the larger
numerator is larger , since you are dividing a bigger numerator
into the same number of pieces.
• If there is a plus or minus sign in the denominator of
a fraction, there is no rule telling us how to break the fraction
into two reasonable pieces.
Let’s look at some examples:
Some examples of comparing two fractions:
Now some examples to show that plus/minus signs in the
denominator are not something we can just break up into two
pieces.
