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Fractions are numbers created by dividing one integer, the
numerator, by another integer, the denominator.
Numerator / Denominator
For example, 3/11 is a fraction. You might ask, why not
just carry out the division and express it as a real number?
It is true that 3 / 11 = 0.27272727, but sometimes we might
want to leave it as a fraction. For one thing, expressing
it as 3/11 is more compact than giving the full repeating
decimal expansion; for another, if we leave it as 3/11,
we might find it easier to work with. There are several
operations we can do with fractions.
The most fundamental trick to working with fractions is
to multiply the numerator and the denominator by the same
quantity. The rationale behind this is that if you multiply
the top and bottom of a fraction by the same number, you
are actually only multiplying the fraction by 1. The fraction
might look different, but has the exact same value. Multiplying
any number by 1 leaves the number the same. Let's take a
look:
After our manipulation, the fraction looks different. But
its value is still the same. (To convince yourself, do the
division.)
This trick is helpful in many ways. An important one is
in reducing a fraction to its simplest form. A fraction
is fully reduced when there is no number that evenly divides
both the numerator and denominator. This is equivalent to
saying that the numerator and denominator have no common
factors.
For example, 44/112 is not in fully reduced form because:
But now that it is reduced to 11/28, it cannot be reduced
any further.
Often we need to add two fractions together. We will once
again use the trick of multiplying the top and bottom by
the same thing. This time the goal is to make sure that
two (or more) fractions we are adding together have the
same denominator. When fractions have the same denominator,
then they can be combined.
Example: How do you add 1/3 + 1/5?
You could just turn them into real numbers and get 0.3333...
+ 0.2 = 0.5333... .
That works fine. You could also manipulate 1/3 and 1/5
so that they have a common denominator. In this case, making
both fractions so that they are over 15 is a good choice.
Here's how it works:
A similar principle applies when you cross-multiply to
simplify an equation. (Equations are discussed in more detail
in the algebra section.) Suppose you have an equation of
the form:
Perhaps you want to get rid of the fractions so that this
equation looks cleaner and is easier to understand. The
best way is to multiply each side by a number that has the
denominators for factors. Once again you'll want a number
that has both of the denominators as factors. For example,
if each side of the above equation is multiplied by 21,
then the denominators on each side will cancel. Here's how
it works:
No more fractions on either side. And since we did the
same thing to each side, we actually haven't changed a thing.
Percentages
Percentages, we are happy to tell you, are nothing more
complicated than fractions with a denominator of 100. (Per
cent literally means per one hundred. )
So 70% of X is just the number you would get if you divided
X into 100 parts and then gathered together 70 of those
parts.
The phrase Y % of X is translated into numbers by taking:
Use the percentage formula to answer the following:
What is 30% of 60? (see answer below)
This is trickier: what % of 120 is 10? (see answer below)
Note that it makes sense to talk about percentages greater
than 100%. If a jury awards an accident victim 300% of the
damages he suffered, then they have awarded him three times
what the accident actually cost him.
The basic rule is this: taking less than 100% of something
makes it smaller; taking more than 100% of something makes
it bigger.
ANSWERS
What is 30% of 60?
What % of 120 is 10?
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