# Mathematics 1010 online

## Percent

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

If you understand this one equation, and you understand and appreciate that the base number is often specified only implicitly, and sometimes not very clearly, then you will able to solve all percent problems that you'll encounter in this class!

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 Your purchase price is • Unknown part. You invest money at an annual interest rate of percent. Interest is paid monthly. Thus every month the bank adds to which is the amount of money in your account at the beginning of the month. Equivalently, every month your money is multiplied with the factor If you invest a penny at 7% interest for 1,000 years, your descendants 1,000 years hence will own (more than 20 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

and hence

• 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:

Hence

### A Complicated Problem

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.