Speed Mathematics Simplified (12 page)

The illustrations are admittedly extreme. A first-column-only estimate of the addition would give a rough total of 50, while actually the real total is 95. A first-column-only approximation of the subtraction would be 60, while the real answer is 51.

The reason why the first digit of the addition can be changed in this case by 4, and the first digit of the subtraction is changed only by 1, is that you might be adding any quantity of numbers and any
two
of them can add up to more than ten—carrying back as much as ten for each two numbers. In subtraction, you never deal with more than two numbers and the maximum amount that can be canceled is one ten.

In subtraction, the safe approach is to work out your subtraction to one more digit than you really need, and round off. In adding, carry your addition
at least
one more place than you really need and assume that the final digit is raised by one for each two numbers you have added, then round off; or else carry it two digits beyond the accuracy required and round off.

Try one estimate in addition at this point. Give a rounded-off three-digit approximation of the following problem (The section immediately following takes up rounding off, in case you are not acquainted with the technique.):

If you worked this out to four digits and assumed the last digit would be raised by 3 (since you added six numbers), your working figures would be 2874 plus 3, or 2877. This you would round off to 1,880,000,000. If you went to five digits, they would be 28770. You would still round off to 2,880,000,000.

A properly rounded-off three-digit estimate can never at worst be more than one per cent wrong, incidentally, and more usually is restricted to no more than one-half of one per cent. The maximum error would be in an estimate of 100 when the accurate answer is 101. An estimate of 999, if properly rounded off, cannot be wrong by more than one-tenth of one per cent. Numbers in between have a maximum possible error that increases as the first digit decreases, from 9 to 1, but it cannot go over one per cent. This, once again, is because each digit becomes just one-tenth as important as it moves one place to the right.

How to Round Off

If anyone doesn't know how to round off, he has missed one of the greatest time- and energy-savers in modern business. Traditional accountants kicked and dragged their heels until they had worked with it a bit, then became its most enthusiastic supporters.

Rounding off simply means expressing any quantity to the nearest standard unit. The standard unit may be whatever you say it is. In the three-digit estimates you just did, we in effect determined that the standard unit would be one in which there could not be an error greater than one per cent.

The standard unit in a U. S. personal income-tax report is one dollar. $3.99 is rounded off to $4.00. $3.01 becomes $3.00. To become a little subtler, $3.51 becomes $4.00 and $3.49 becomes $3.00. The usual rule is to give away an even half, and call $3.50 an even $4.00.

Any other standard unit that makes sense for a particular situation can be adopted. The operating and financial statements of many companies are rounded off to even thousands. $357,800 is expressed on the statement as 358—with a note at the top of the report, of course, that all figures are in thousands of dollars. Smaller companies may round off to tens or hundreds of dollars. Very large corporations may even round off to hundreds of thousands or, for certain purposes, to the nearest million!

At the other extreme, there is an almost forgotten currency value in this country of one mil—a tenth of a cent. It was used primarily in state sales taxes, before sales taxes went up to much higher rates. Naturally, people working with quantities of mils soon learned to round off their reports—to the nearest cent!

The most accurate way to estimate in adding or subtracting, as we have said, is to work out your figures to one place more than the accuracy needed, and round off. If the extra (not needed) digit is 5 or more, raise the preceding digit by one before reporting the estimate. If the extra digit is 4 or less, leave the final significant digit alone.

The theory is that roundings-off tend to cancel each other out in practice. You will add half or less to just about as many numbers as those from which you subtract less than half. To the surprise of many old-line accountants and bookkeepers, several test-runs of complicated reports and statements proved this to be completely true. The error is hardly ever likely to be larger than a single rounding-off.

Review quickly now the three secrets of speed in subtraction, before going on to practice that speed. The three major secrets—in addition to the over-all speed-math secret of “seeing only the answer”—are

      
1. Subtract from left to right.

      
2. Never subtract a larger digit from a smaller. Instead, add the complement of the larger digit to the smaller digit and—

      
3. Cancel tens in the answer by slashing, rather than “borrowing” in the larger number.

5

BUILDING SPEED IN SUBTRACTION

C
ERTAIN parts of this book may seem repetitious.

This is intentional. Repeating the basic points is the easiest and most painless form of review. Doing one essential exercise over several times—but not over and over in succession—is the most effective way to build the automatic response that is the foundation of high-speed mathematics.

Read the following line as if it were a sentence of words. But instead of words, read the complements of these digits:

Before going on to some necessary practice in complement subtraction, reinforce your understanding of the principle at work by describing in words completely different from any used in this book precisely what a complement is.

Now explain to yourself, as if you had never heard of the idea before, how you can subtract 7 from 12 by adding 3 plus 2—and doing something else in the answer. It might be a good idea to set up, on your pad, the three expressions 12 – 7, 12 – 10 + 3, and 12 + 3 (cancel).

Your speed and ease with numbers will depend not only on how easily and automatically you “sense” these new techniques, but also on how easily and automatically you see only the answer to any digit combination. We will now go through the basic vocabulary of subtraction. It will not take very long, because the combinations are really the same ones you have already practiced for addition. They are all pairs you recognize at sight, but in this case one-half of each pair and the addition-answer are given, and you must respond with the missing number. 3 + 5 is a pair you should be starting to read at sight as “8” instead of “3 + 5 is 8”; the same pair will show up here as 8 – 5 (see 3) and 8 – 3 (see 5).

Work for speed with these combinations. School yourself to think not about the digits you see, but only the answer. Where the bottom digit is smaller than the top digit, see only the answer. Where the bottom digit is larger than the top, work at seeing the result of adding the complement of the bottom digit to the top digit, and mentally slash an imaginary digit to the left in the answer. Therefore you “see” 6 – 7 as “3 + 6 (9)—slash.”

Just as in the practice tables in adding, every possible digit combination has been include in these section. If you learn to read the answers to these without effort, you know you will never handle a single combination that you did not have a chance to practice.

See how quickly and automatically you can subtract these:

A certain amount of your speed at handling these must be pure habit, of course. There is no way to avoid developing the “automatic response” that only practice can bring. But the number sense at which you worked in
Chapter II
will be a substantial help here. The better you can visualize the relationships of numbers, the more quickly you will develop astonishing mastery of basic mathematical figuring.

Remember, too, that while this practice series shows 94 pairs (the 45 pairs, the 45 pairs upside down, plus the four hardest pairs repeated just to make it come out even), you need only be concerned with those twenty easiest combinations, plus the five complement pairs. Looked at this way, it should certainly be a reasonable task to master fully and automatically. What you are really doing, in effect, is learning to recognize those twenty easiest pairs whether they show up in simple smaller-from-larger form or disguised in complement applications.

Use your complements faithfully, and you will never deal with any combinations adding to more than ten—or subtract a larger digit from a smaller.

This finishes up all the possibilities: