Speed Mathematics Simplified

SPEED
MATHEMATICS
SIMPLIFIED

Edward Stoddard

DOVER PUBLICATIONS, INC.

New York

Bibliographical Note

This Dover edition, first published in 1994, is an unabridged and unaltered republication of the second printing (1965) of the work first published by The Dial Press, New York, in 1962.

Library of Congress Cataloging-in-Publication Data

Stoddard, Edward.

Speed mathematics simplified / by Edward Stoddard. —Dover ed.
p.
cm.

Originally published: New York : Dial, 1962.

Includes bibliographical references.

ISBN 0-486-27887-5

1. Ready-reckoners. I. Title.

QA111.S85 1994

513′.9—dc20

93-46724
CIP

Manufactured in the United States by Courier Corporation
27887507
www.doverpublications.com

CONTENTS

INTRODUCTION

  
1
   
NUMBER SENSE

  
2
   
COMPLEMENT ADDITION

  
3
   
BUILDING SPEED IN ADDITION

  
4
   
COMPLEMENT SUBTRACTION

  
5
   
BUILDING SPEED IN SUBTRACTION

  
6
   
NO-CARRY MULTIPLICATION

  
7
   
BUILDING SPEED IN MULTIPLICATION

  
8
   
SHORT-HAND DIVISION

  
9
   
BUILDING SPEED IN DIVISION

10
   
ACCURACY: THE QUICK CHECK

11
   
ACCURACY: THE BACK-UP CHECK

12
   
HOW TO USE SHORT CUTS

13
   
BREAKDOWN

14
   
ALIQUOTS

15
   
FACTORS

16
   
PROPORTIONATE CHANGE

17
   
CHOOSING AND COMBINING SHORT CUTS

18
   
MASTERING FRACTIONS

19
   
SPEED AND EASE IN DECIMALS

20
   
HANDLING PERCENTAGES

21
   
BUSINESS ARITHMETIC

BIBLIOGRAPHY

INTRODUCTION

W
HETHER 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.

With the techniques in this book, you will find yourself doing many problems in your head that formerly required pencil and paper. More complex problems that still need pencil and paper will get done in a fraction of the former time, and in many cases you will simply jot down two or three numbers rather than copy down the whole problem.

When a quick estimate or accurate guess is needed, you will be the one who can glance at a column of figures or a complicated multiplication and give a rapid approximation accurate to any number of places needed.

If all this sounds too good to be true, let me hasten to point out that there are some things this book
cannot
do:

This book cannot make you a “number genius” who multiplies a six-digit number by a twelve-digit number in his head and gives the complete answer in ten seconds flat. There are such people, but they are born—not trained. There are mighty few of them, at that.

This book cannot hand you mastery of streamlined math on a silver platter. It can show you the techniques, explain each of them as clearly and simply as possible, and encourage you to do the pleasantest possible kind of practice. But only you can decide to spend the necessary time the explanations and the practice will inevitably take.

You have already taken the first major step in mastering speed math. You bought or borrowed this book because you want to become better at figures. Wanting to learn is basic. If your interest ever flags, if the practice ever seems irksome, it might be well to remind yourself why you picked up the book in the first place. Keeping the goal in mind is the best way to keep your feet firmly on the path.

There are at least half a dozen books in print on “speed” or “short-cut” mathematics.

Why, then, this one?

There are a number of good reasons. First, almost all books on the subject rely primarily on a number of standard short cuts. The use of these devices, which include such simple conversions as aliquot parts and factoring, can often save a great deal of time. As far as I have been able to find out, however, no book has yet attempted to relate them to each other and show the ways to pick out the most useful in each case. Here you will find the most valuable of the classic short cuts explained quite simply and arrranged for sensible, rapid selection and use.

Beyond this, the book introduces an entirely new system of basic figuring that works in all cases. This approach builds on the arithmetic you already know. It takes your present training in numbers and streamlines it, cutting down the number of steps you take in solving each problem. By combining this approach with the best of the classic short cuts, you will compound your speed and ease.

This new system is a development of a little-known oriental technique growing directly out of abacus theory. The abacus is a startlingly efficient machine, for all the jokes made about it, mainly because it forced on the orientals who perfected the modern version a simplified approach to numbers.

The chapter on addition will go more fully into the contributions of the abacus to this system.

One more point about this book. Simply reading through it will accomplish little. Practice is required to master any activity, whether it be streamlined mathematics or water skiing. I have already mentioned the importance of this, but very few of us have the patience to work out small-print examples or the self-control to avoid peeking at the answers printed right beside them.

That is why you will find a different method of practice here. It bears some similarity to the new theories of “teaching machines” in that it requires you to produce the answer and, immediately after, tells you whether you were right or wrong. In addition, I have kept the practice as varied as possible, and tried to give it some pace as well. The method is designed to give you enough basic practice as you go to begin mastery of each step.

Please do not skip these sections. They are absolutely essential to learning how to
use
streamlined math. They carry you from knowing how it is done to knowing how to do it—quite a different thing, really.

This is how these sections work:

As you come to an example or series of speed-practice steps, you will be asked to cover the answer (if it is on the same page) with a bit of working paper you should always keep on hand. Use the paper for any pencil figuring involved. I would recommend that you tuck a dozen blank file cards into the book for this purpose, or a thin pad no larger than the book. It can serve the additional use of a bookmark, too. A good idea would be to stop for a moment and get hold of a pencil and pad or cards right now.

When you come to a demonstration or practice problem, read it. Be sure you understand the specific technique to be used. Then work it out, keeping your paper over the answer. If a pencil is needed, work it out on the paper. Then, and only then, look at the answer. If you made a mistake, stop to see why before going on.

Do this faithfully if you want to get all the good from this book.

As in learning any new skill, you may feet a bit awkward and slow at first. This is entirely natural. Repetition and time will cure the awkwardness. The only way to learn to ice-skate is to ice-skate. The only way to learn speed mathematics is to use (not merely read about) speed mathematics.

By the time you have finished this book, your speed and ease with figures should easily have doubled. From then on, as you make these techniques automatic and habitual, your skill will continue to improve. You can ensure this in two ways:

First, consciously use the new ways you have learned for every number problem you run across in business or personal life. At the beginning you will have to strain a bit to break the old habits, and the process will take a little longer because it is new. But soon you will find yourself using these techniques comfortably and quickly. As you continue using them, you will find yourself approaching any number in this new way without even thinking about it.

Second, do a bit of special practice now and then just for fun. Instead of doing a crossword puzzle on the train, run through a few random problems using your new techniques. Instead of reading an old magazine while waiting for an appointment, do some mental exercising with the phone number or street address of the office where you are waiting. Instead of killing half an hour with a TV program you don't especially want to look at anyway, go through one of the speed-developer chapters in this book again.

Do all of these things cheerfully and conscientiously, making a game of them, and with only a reasonable amount of time and patience you will find yourself becoming truly a whiz at figures.

SPEED
MATHEMATICS
SIMPLIFIED

1

NUMBER SENSE

N
UMBER 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.

W. W. Sawyer expresses it this way in his book
Mathematician's Delight:
“The fear of mathematics is a tradition handed down from days when the majority of teachers knew little about human nature, and nothing at all about the nature of mathematics itself. What they did teach was an imitation.”

By “imitation,” Mr. Sawyer means the parrot repetition of rules, the memorizing of addition tables or multiplication tables without understanding of the simple truths behind them.

Actually, of course, in real life we are never faced with an abstract number four. We always deal with four tomatoes, or four cats, or four dollars. It is only in order to learn how to deal conveniently with the tomatoes or the cats or the dollars that we
practice
with an abstract four.

In recent years, teachers of mathematics have begun to express concern about popular understanding of numbers. Some advances have been made, especially in the teaching of fractions by diagrams and by colored bars of different lengths to help students visualize the relationships.

About the problem-solving methods, however, very little has been done. Most teaching is of methods directly contrary to speed and ease with numbers.

When I coached my son in his multiplication tables a year ago, for instance, I was horrified at the way he had been instructed to recite them. I had made up some flash cards and was trying to train him to “see only the answer”—a basic technique in speed mathematics explained in the next few pages. He hesitated, obviously ill at ease. Finally he blurted out the trouble:

“They don't let me do it that way in school, Daddy,” he said. “I'm not allowed to look at 6 x 7 and just say ‘42.' I have to say ‘six times seven is forty-two.'”

It is to be hoped that this will change soon—no fewer than three separate professional groups of mathematics teachers are re-examining current teaching methods—but meanwhile, we who went through this method of learning have to start from where we are.

Relationships

Even though arithmetic is basically useful only to serve us in dealing with solid objects, be they stocks, cows, column inches, or kilowatts, the fact that the same basic number system applies to all these things makes it possible to isolate “number” from “thing.”

This is both the beauty and—to schoolboys, at least—the terror of arithmetic. In order fully to grasp its
entire
application, we study it as a thing apart.

For practice purposes, at least, we forget about the tomatoes and think of the abstract concept “4” as if it had a real existence of its own. It exists at all, of course, only in the method of thinking about the tools we call “numbers” that we have slowly and painstakingly built up through thousands of years.