Speed Mathematics Simplified (62 page)

Each of these is a little different, but all operate on exactly the same system.

      
1. The point in the answer will be between the 7 and 8 of the number divided, because we move the point one place to the right.

      
2. The point in the answer will be directly above the point in the number divided, since there is no point in the divider.

      
3. The point in the answer will be after the final 6 in the number divided. Moved two places.

      
4. The point in the answer will be between the 3 and 9 in the number divided. Moved one place.

Other than placing your decimal point properly in the answer, there is no more to dividing with decimal numbers than there is to any division. Once you have determined the right place for the point, simply ignore all the points in the original problem. Your answer will be correct.

Only one other aspect needs special mention. We demonstrated it before, but it should be spelled out too. If you have to move the point in your answer way beyond the end of the number divided, simply do it. Fill in with 0's as needed. For instance:

Decimal Remainders

Depending on the particular problem and the particular field in which your answer will be used, you may work out a division problem that has a remainder in either fractional or decimal form.

The making of a decimal remainder is very simple. It makes no difference how many 0's you add after the last digit to the right of the point, any more than it makes any difference how many 0's you add to the left of a whole number.

There is one special meaning to 0's following the last digit to the right of a decimal point, however, and you should be aware of it. By common agreement, the 0 you place to the right means that the number is accurate to this point.

The number 4.6 might be a rounded-off number anywhere from 4.56 to 4.64. But the number 4.60 means that any rounding off was done beyond the 0. The convention in mathematics goes further, incidentally, and often places a plus or minus sign at the end of a number that has been rounded off, to indicate that it is not a precise quantity.

To make a decimal remainder, then, you simply keep mentally bringing down 0's as long as you have to in order to get an exact answer or the accuracy you need. With your mastery of shorthand division, you do not even have to note the 0's in the number divided; just bring down imaginary ones:

If the final division here had not come out even, you would keep bringing down imaginary 0's until you had no remainder, or had as complete an answer as you needed. If you divide 3 into 10, you will never get a complete answer. But at some point you will have as complete an answer as you need.

Converting from Fractions

A fraction, as we have said, is only a special way of writing a division problem. It expresses a specific quantity, but one that (except by decimals) we have no other way of showing with the numbers available than as a division of two known numbers. 3/8 is the same as 3 ÷ 8 or 8
. The fraction has a different purpose from the division, however; it says, in effect, “this is a quantity,” rather than “here is a problem,” because for many purposes 3/8 is more convenient than other expressions of that quantity.

Often, however, you want to convert a fraction to a decimal form. The method is simplicity itself. Simply carry out the implied division, and use a decimal remainder.

To convert 3/8 to a decimal, for instance, you do this:

The decimal equivalent of 3/8 is 0.375. In this case, it is an exact equivalent, and it should sound familiar: 375 is one of the basic aliquots.

Now you convert 6/7 to a decimal. Get out your pad and cover the answer. Express 6/7 as a decimal accurate to the nearest 10,000th.

Here is how the conversion looks in shorthand division:

The nearest 10,000th means four places after the point. We worked it out to five places so we could round off, and the last 4 indicates that the rounded-off form is .8571.

Sometimes you find it necessary to convert decimals back to fractions for particular purposes. In some problems, fractions are easier to handle. This, in fact, is part of the basis of the aliquot short cut.

For decimals other than aliquots, the process for converting to a fraction is to write it in fractional form and then see if it can be reduced. The decimal .1 can be written 1/10 and .45 can be written as 45/100.

You reduce this resulting fraction exactly as you reduce any other fraction: divide both top and bottom by any number that will divide both exactly, if there is any. Try reducing the example above, .45.

45 is exactly divisible by 5 or by 9. 100, however, is divisible by 5 but not by 9. Dividing both top and bottom by 5, we reduce 45/100 to 9/20. No further reduction is possible.

Convert the following decimals to fractions:

The last one, admittedly, is a dilly. But it can be reduced
quite substantially. Cover the reductions with your pad until you are satisfied.

Your answers should read 1/4, 13/16, 5/8, and 31/32.

The next chapter will take up decimals in another and quite special form, percentage. Before going on to that chapter, reflect for a moment or two on the entire decimal method of expressing fractions—and its firm foundation on the point made several times before in this book that each digit decreases in importance by a factor of 10 as it moves each place to the right. This is true right across the decimal point—which is the end of the whole number.

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 3100'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: