Chapter 20 - Handling Percentages

A percent age is merely a two-place decimal without the decimal point shown.

Except that it seems to be the cause of so much general lip-biting, we would dismiss percentages with the above definition. 82% is exactly the same as .82. 6% is no more and no less than .06 (two places, remember). 4½% is .04½, or .045.

A decimal-form fraction with two digits to the right of the point is in hundredths—a "1" followed by as many 0's as there are digits to the right of the point. The term per cent comes from the same root as century (a hundred years) and cent (one-hundredth of a dollar): the Latin word for a hundred. Per cent is our contraction of the original per centum— per hundred.

So if you say you will pay interest on a loan at the rate of 7% a year, for instance, you are saying that for each 100 parts of the loan you will pay 7 parts a year in interest. If the loan is for $300, you will pay $21 a year; there are 3 100's, and you will pay 7 for each of them. You get precisely the same result if you multiply 300 by .07.

Since we often handle percentages in different ways, let us explore some of the basic relationships and processes involved.

Finding a Percentage of a Number

Finding a percentage of a number is what we just did, and it is the simplest of all percentage calculations. Just multiply the number by the decimal equivalent of the percentage, and you have the answer.

Try one yourself: find 36% of 298. Here, in no-carry multiplication, is the way you work it out. 36% is, by definition, the same as 36/100, or .36:

How do we place the decimal? Remember the decimal rule. The answer has the same number of digits (to the left of the point) as do the two numbers multiplied (to the left of the point). 298 has three places, .36 has none, so the answer has three digits to the left of the point including the first digit of the first partial product, even if it is a 0. The answer is 107.28.

Do one more:

Cover the solution with your pad until you have finished this to your satisfaction.

8% is the equivalent of .08, and our solution looks like this:

Note that there seems to be a spare 0 in the answer. This is to aid the placing of the point in the answer, since the multiplier (.08) has in effect minus one places before the point. If we include the 0 in .08 in writing our answer, the correct handling of the point is automatic. We place it two spaces to the right because there are two places to the left of the points in the numbers multiplied.

Finding What Per Cent A Number Is

Often you need to find what per cent one number is of another. You might have, for instance, the two numbers 15 and 75, and be required to express one of them as a percentage of the other.

The important thing is to make very sure which number is which. Do you want to know what per cent 15 is of 75, or what per cent 75 is of 15? It makes a big difference.

Recall at this point that a per cent is only a special way of writing a decimal, and that a decimal is a special form of fraction. So in either of the above cases, you are really being asked to show a fraction in percentage form.

If you want to know what per cent 15 is of 75, you need to convert into decimal (and therefore percentage) form the fraction 15/75. If you are required to state what per cent 75 is of 15, you again must convert into decimal and percentage form the fraction 75/15.

Another way of keeping your relationships absolutely straight, in case this conversion does not lock itself memorably in your mind, is that one of the numbers always follows the word of. You always ask "what per cent is this number of that?" The number following the "of" is always the base—the base of which you are figuring a percentage—and the base is always the bottom of the fraction.

You know perfectly well how to convert any fraction to decimal form. You divide the top by the bottom. To convert this decimal fraction to a per cent, move the decimal point two places to the right.

What per cent is 15 of 75?

The fraction to which we want a percentage answer is 1%5- Using the other key, the number following "of" is 75, and the base is the bottom—again, 15/75. Now convert:

 
Move the point two places to the right, and we have the answer 20%. 15 is 20% of 75.

Turn the relationship around. What per cent is 75 of 15? Here the fraction expressing the relationship is 75/15. Or, again, the number following "of" is 15 and therefore the base and the bottom. Divide:

In order to convert this in turn to a percentage, move the point two places to the right—adding 0's as necessary. So 75 is 500% of 15.

500% means that for each 100 parts of the other number, you have 500 parts of this one. Wiping out the 100's, you see that 500% is the same as five times as much.

Try one on your own now. Cover the explanation below with your pad and work out both sides of this relationship:

20 is what per cent of 50?

50 is what per cent of 20?

For the first comparison, the number following the "of," and therefore our base, is 50. The fraction is 20/50. Dividing by the bottom, we get

We move the point two places to the right, and find that 20 is 40% of 50.

Reversing the question, we have a base of 20—the number following the "of." The fraction is 50/20. The division is

Again we move the point two places to the right. 50 is 250% of 20.

In these examples, we have not bothered to reduce each fraction to its simplest form before dividing because showing the division with the original numbers in the question seems to make the process clearer. In practice, of course, you would consider these numbers 2 and 5 rather than 20 and 50.

Finding An Unknown Base

One of the most baffling operations in percentage seems to be finding an unknown base. If you have a clear grasp of the relationships, however, it becomes quite easy.
An example of this situation might be the question, "90 is 45% of what?"

We know the number that is a percentage of another. We know the percentage. But we do not know the base.

Let us approach the method through logical conversion of the methods we already understand. Once you know why, you are not likely to forget how.

We have three numbers: 90, 45%, and "what." The number (unknown) following "of" is "what," so "what" is the base.

The fraction, therefore, is
 
We know the answer to the fraction, but we do not know the fraction itself. In order to convert a fraction to a decimal, and therefore a percentage, we divide the top by the bottom. So we will set up the problem, along with the answer we know:

Now, if someone asked you, without confusing matters by including words such as percentage and decimals, the question, "What divided into 90 gives the answer .45?" you would answer without a second thought, "Divide .45 into 90 and find out."

Divider multiplied by answer must give number divided. Number divided, divided by the answer, must give the divider.

So we simply divide the number we have by the percentage, and we find the base:

Note how the decimal point was moved over, following the rule in the chapter on decimals.

90 is 45% of 200.

The reason we developed this method step by step is to emphasize the logical reasoning behind the general rule:

To find an unknown base, convert the percentage to a decimal and divide it into the known number.

Reinforce this rule at once by trying another example. 68 is 20% of what?

Convert the percentage into a decimal and divide it into the known number:

68 is 20% of 340.

Try one by yourself. Cover up the solution with your pad. 87 is 30% of what?

To find the unknown base, convert the percentage into a decimal and divide it into the known number:

87 is 30% of 290.

Percentage of Change

Business arithmetic often involves a percentage of change or difference. Rather than asking what per cent 18 is of 360, the business world is more apt to ask, "How much more is 500 than 475?" or, "How much less is 390 than 415?"

Suppose that sales in territory #8 were $350,000 last year, and are $375,000 this year. What is the percentage of increase?

The first step is to find the raw amount of the difference in plain numbers. It is $25,000, found by subtracting the total last year from the total this year.

Now our problem is, "$25,000 is what per cent of $350,000?"

This is familiar. You did similar problems a few pages ago. The dollar signs and 0's do not change the principle. In fact, you can simplify matters by dropping both the dollar signs and the same number of 0's: 25 is what per cent of 350?

Work out the division to convert this fraction to decimal form in shorthand division:
 
Remember your base, the number following "of." The fraction is

The answer is not precise, but we can round it off to 7%. Territory # 8 is 7% ahead of last year. The general rule, then, is this:

Find the difference, and divide it by the base.

Sometimes the base is the smaller of the two numbers; sometimes it is the larger. After all, sales in territory #8 might have gone down this year. Then the base would be the larger of the two figures.

Do this one on your pad:

Sales last year $320,000
Sales this year $307,200

What is the percentage of decrease?

When we find a percentage of decrease, our base is the GES
larger number. The difference in sales, by subtraction, is $12,800. Dividing by the base—dropping thousands and dollar signs—we have.

This territory is, unhappily, 4% behind last year in sales.

Note especially that sometimes you figure the percentage of difference on the smaller of two numbers, and sometimes on the larger. The difference, as a percentage, will be larger when based on the smaller number—and smaller when based on the larger number.

The saving grace, perhaps, is that an increase in sales from $100,000 to $150,000 will show up as a 50% increase, while a decline from $150,000 to $100,000 is only 33%!

Now that we have covered decimals and percentage, we are equipped to cover the more common business expressions such as discount and interest and some of the other yardsticks most frequently used in the commercial world.

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