The word * percent* means * per hundred*. You divide something into
100 equal parts and then consider so many parts of it.

The key word here, and the item that causes the greatest confusion in
percentage calculations, is ** "something" **. The something of which
you calculate a certain number of percent is rarely identified as such
explicitly, and it often changes even within the same problem.

Consider this ** example**. Suppose you earn $10.- an hour and
you receive a raise of 20%, followed by another raise of 20%. The
first raise applies to your hourly wage of $10.-, and the
second to your new hourly wage. 20% of $10 is $2.-
and so your new hourly wage is $12.-. Your second raise is
20% of $12, which is $2.40, and so your hourly wage
after your second raise is $14.40. Note the phrase you receive
"a raise of 20%". It's pretty clear that it's 20% of your current
salary, and not your CEO's salary, or the gross national product, but
that fact is not stated explicitly. Notice also that even though both
raises are 20%, they translate into different dollar amounts because
they are applied to different hourly wages.

Of course we all know that raises of 20% are all too rare...

Here is another ** example** along the same lines: If you make a
certain income, and you receive a raise of 100%, followed by a pay cut
of 100%, then your new income is not what you started with, but rather
it is zero!

It is possible to have more than 100 percent of something. For example, your boss probably makes more than 100 percent of your income.

To make progress let's introduce some language. The number of which
we compute the percent (first $10, and then $12 in the above
example) is called the **base number**.
The number of percent (20 in the above example) is the **rate**, and the rate applied to the base number
is the **part**. Depending on the
application the part can have many different names. For example, when
the problem is about money, it could be a raise, a discount, a fee, a
commission, .etc.

Denoting the part by , the rate by , and the base number by , the basic equation governing all percent calculations is

Percentage problems differ by what is known and what is to be determined, and, to reiterate, their difficulty usually stems from describing the base number only implicitly. Another source of difficulty is that usually the part is not of interest in itself, but it needs to be added or subtracted to the base number to get the result of interest. In the above example, you are probably more interested in your hourly income after the raise than in the raise itself. The following examples illustrate situations in which one of the three key ingredients is unknown:

**Unknown part.**You buy a computer at a 25% discount. The base (list) price is $2,000. What is your total price? The base number is 2,000, the rate is 25, and so the part is**Unknown part.**You invest money at an annual interest rate of percent. Interest is paid monthly. Thus every month the bank adds*octillion*dollars) or thereabouts.**Unknown base number.**You buy a car for $18,750 and the dealer informs you that you purchased it at a 25% discount. What is the list price (the base number) of the car? Since we received a 25% discount we purchased the car for 75% of its list price. Denoting the list price by we obtain**Unknown Rate.**The population of your town is 17,000. A year later it is 17,678. At what (annual) rate is the population growing? The rate satisfies:

Percentage problems often involve more than one rate, base, and part. I recommend you work through every detail of the following example. It will go a long way to teach you about percent, and it will enable you to calculate your standing and your prospects in this class at any time.

Suppose your WeBWorK homeworks account for 40% of your grade, your midterm exams for 30%, and your final exam also for 30%. (Your actual numbers may be different, of course.) You obtain 90% on your homeworks, 70% on your midterms, and you need to get 80% overall to receive the grade of B+. The maximum point count on the final is 200. How many points do you need to receive (at least) on the final exam so as to obtain a B+ overall?

There are at least four sets of , , and in this
problem, your overall score, and one each for homeworks, midterm
exams, and the final exam. (In reality these can be broken down
farther since there are several homeworks and several midterm exams.)
It helps to make things clearer by thinking of the percent you get
overall (your primary concern) as something else, like ** points**
or
** cookies**. Let's use ** cookies** since* points* is
used for the score on individual assignments. You want to obtain 80
cookies. Your homeworks give you up to 40 cookies, but you get only
90% of those, for a total of 36 cookies. Similarly, the midterms give
you 70% of 30 possible cookies, or 21 cookies. So without the final
you have 36+21=57 cookies for sure. You need 23 more cookies to get
80 cookies total. The final exam can give you up to 30 cookies, and
23 is 77% of 30. So you need to get at least 77% of the possible
final exam score. The maximum possible score is 200 and 77% of 200 is
154. So you need to get at least 154 points on the final to get a B+
in this class. But there is no point in aiming low. So you decide to
prepare carefully for the final. Your preparation pays off and you get
200 points, or 100%, or 30 cookies, on the final. Your total in the
class is 87 cookies, and you receive the grade of A-.
Congratulations!

Here is a problem for you to ponder. E-mail me your answer and your reasoning.

You are commencing a career. Raises in your job are described as
percentages of your current salary. Suppose you are going to get a
certain set of raises. (If you like and if it helps fix your attention
on something like 15%, 10%, 0%, 12%.) Your objective is to maximize
your **final salary** (let's say because it determines your
retirement income, and you are going to be a retiree for many
decades). In order to obtain this objective, do you want the large
raises initially when they apply to small salaries, or do your want
them later, when they apply to large salaries? Why? What changes if
your objective is to maximize your **total earnings** rather than
your final salary? Summarize the result of your thinking in ten words
or less.