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:
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?
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.