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Introduction - Whether you are an executive concerned with inventories and markups and profit ratios or a carpenter who works with board feet and squares of shingles—whether you do your figuring in gallons and pennies or tons and dollars— this book will show you new ways to do that figuring with dispatch and authority.

01. Number Sense - Number sense is our name for a "feel" for figures—an ability to sense relationships and to visualize completely and clearly that numbers only symbolize real situations. They have no life of their own, except as a game .

Almost all of us disliked arithmetic in school. Most of us still find it a chore.

There are two main reasons for this. One is that we were usually taught the hardest, slowest way to do problems because it was the easiest way to teach. The other is that numbers often seem utterly cold, impersonal, and foreign.

02. Complement Addition - It has been estimated by experts that, for the average business, the total time spent in arithmetical computations breaks down to 70% addition, 5% subtraction, 20% multiplication, and 5% division.

These exact proportions may or may not hold in your particular business or profession. But chances are that they are not far wrong if you include all the number work you do.

03. Speed Addition - In the last chapter you have had a taste of one of the newest and most exciting developments in the whole field of speed mathematics. Its sheer beauty and rapidity will grow on you as you begin to make it a habit.

Part of making it a habit is plain old-fashioned practice. There is simply no way of learning high-speed arithmetic without a pretty fair dose of practice. You cannot begin to master the systems without using them enough times to feel at ease with them.

04. Complement Subtraction - Subtraction is merely the other side of the coin of addition.

For most of us, however, it causes far more trouble. There are probably two reasons for this. While many of us learned our "addition tables" by heart in school, few of us really mastered the conversion of these into "subtraction tables" with anything approaching the same thoroughness. More important, however, the traditional process of "borrowing" is a tricky concept. Many of us find ourselves forgetting to borrow, or borrowing twice, because it is basically unnatural.

05. Speed Subtraction - Certain 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.

06. No-Carry Multiplication - Multiplying, according to the same estimates we mentioned before, averages about 20% of the figuring done in normal business.

But while it is used less than addition, multiplication is dreaded by more people, and done poorly or inaccurately by more people, than either addition or subtraction.

07. Speed Multiplication - You recall that we stated three basic secrets for speed in multiplication:

First, work from left to right (possible only with this system).

Second, "see" the result of each multiplication of two digits, rather than the problem.

Third, use the no-carry method.

The second point is the one that obviously requires the most practice. The foundation of all your speed is the easy, natural, painless use of the no-carry system—but the way to make it easy and painless is to make as automatic and unthinking as possible the process of "seeing" 8 x 7 as "50's" and "ends in 6."

08. Short-Hand Division - So far, we have covered three out of the four basic arithmetical computations.

In adding, we learned to use complements for the tougher combinations—those that would add over ten if we ever added over ten—and to record tens in such a way that the answer forms naturally in our mind, just as, on the modern abacus, the answer forms naturally on the board.

09. Speed Division - The last exercise in the last chapter was a stunner. It was, just from the quantity of digits to be handled, the most tedious situation you are likely to face in arithmetic. The numbers go on and on, and if you need a full answer there is simply no way to avoid dealing with every single digit.

10. Quick Check - Producing a quick answer is not always the end of a problem in arithmetic. A wrong answer can be worse than none at all.

When you balance your checkbook, you care whether every stub was done perfectly because otherwise you face an unpleasant hour or so finding out why your balance does not agree with the bank statement. When you make out your income-tax return, you check and double-check every operation to make sure you are not paying too much, or else getting yourself into trouble with wrong arithmetic.

11. Back-Up Check - The digit-sum, or "casting out nines," method is the quickest and easiest way to check any problem. Once you become fully accustomed to it, you will find yourself checking a problem about as quickly as you could read it over.

It is not, however, completely foolproof. The last chapter explained the types of errors to which it is quite blind. As someone once pointed out, the digit-sum method will tell you that a problem is wrong, but it will not tell you for sure that it is right.

12. Use Short Cuts - In this chapter we shift gears entirely, and learn how to build on our simplified arithmetic a different, but very useful, system of conversion.

This second section does not ignore the first. Indeed, the swift and confident working of these new principles depends very much on a smooth and automatic handling of your basic number combinations—the practical, fast, fully integrated system of speed arithmetic based on the phenomenal simplicity of the modern Japanese abacus.

13. Breakdown - It is hardly likely that you would ever multiply a number by ten by putting down the number with ten under it, and multiplying out digit by digit like this:

4 6 3 0

Instead, you know that in order to multiply any number by ten you simply add a zero. If the number has a decimal point in it, you move the decimal point to the right instead of adding the zero.

14. ALIQUOTS - This fascinating and useful technique of conversion suffers under a traditional and foreign-sounding name. "Aliquot" means, simply, an exact fraction. The word is derived from a Latin word meaning some, or several. It is usually used as an adjective (aliquot parts, meaning exact parts), but since it is also a noun we will save words.

The key word in the definition is exact. 8 is an aliquot of 16, because it is contained within 16 exactly twice and leaves no remainder.

15. FACTORS - Most of us remember, from our school days, the word "factors." Chances are you have not encountered the word or the process since. Instead of considering them merely an exercise for students, however, we will show how they can short-cut many problems in multiplication and division.

16. Proportionate Change - The fourth generally useful type of short cut has no traditional name. Because the phrase most accurately describes what we do, we will call it proportionate change.

The technique is simply that: proportionate change. You change one number of a problem into a simpler form in any way you wish (double it, triple it, cut it to one-third, or whatever) but change the other number in proportion so the essential relationship remains the same.

17. Short Cuts - You have learned and practiced the four most generally A useful short cuts. There are others, but they are quite specialized. The most complete assortment can be found in the books listed in the bibliography. With the four short cuts you have learned, however, you can convert a great deal of your multiplying and dividing into simpler forms.

18. Mastering Fractions - Of all the specialized branches of mathematics, fractions seem to be greeted with more general panic than all the others put together.

It does not have to be so. Fractions are really not much more complicated than multiplying or dividing. Perhaps the reason for their general unpopularity is that they are taught, to an even greater extent than is true for the other processes, almost entirely by rote. The rote itself simply has to have a few more steps and rules than do whole numbers.

19. Decimals - Fractions are one way of expressing quantities of less than 1, or more than 1 but not reaching an exact digit. Decimals are another way of doing the same thing.

Of the two, decimals are usually by far the easier and more convenient way to express fractional quantities. If our measuring systems were based on our ten-base counting system (as is the Continental system of meters, grams, litres, and so on) we would perhaps face fractions only a very few times in our lives.

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.

21. Business Arithmetic - This chapter will cover once over lightly the more common JL business expression involving arithmetic.

The first of these is discount, or mark-down. Retail stores figure the discount they get from the manufacturer or wholesaler with the retail price as the base (book stores, hardware stores, most specialized stores) or—just the opposite in some fields— with the net, discounted price as the base (department stores, chain stores, etc.). When the net price is the base, the store figures mark on or mark up, rather than mark down.

Bibliography - AMUSEMENTS IN MATHEMATICS
H. E. Dudeney; Dover Publications, N.Y.

ARITHMETICAL EXCURSIONS
Henry Bowers & Joan E. Bowers; Dover Publications,N.Y.

HIGH-SPEED MATH
Lester Meyers; D. Van Nostrand Company, Inc., N.Y.

HOW TO CALCULATE QUICKLY
Henry Sticker; Dover Publications, N.Y.

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