Arithmetic Made Simple

180 Pages·1988·16.232 MB·English

by  Abraham P. Sperling,  Samuel D. Levison, Robert R. Belge

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Revised Edition
!
A. P Sperling and Samuel D. Levison
Revised by Robert R. Beige
|
Over 1/2 million copies in use, the r^iost effective step-bv-step quide
available to learn the basics of arithmetic

ARITHMETIC
MADE
SIMPLE
by
A. P. SPERLING, Ph.D.
and
SAMUEL D. LEVINSON, M.S.
Revised by Robert R. Beige,
Department ofElectrical and Computer Engineering,
Syracuse University
BOOKS
A MADE SIMPLE BOOK
DOUBLEDAY
NEW YORK LONDON TORONTO SYDNEY AUCKLAND
AMadeSimpleBook
PUBLISHED BY DOUBLEDAY
adivisionofBantamDoubleday Dell PublishingGroup, Inc.
1540Broadway, NewYork, NewYork 10036
MadeSimpleand Doubleday are trademarksofDoubleday,
adivisionofBantamDoubledayDellPublishingGroup, Inc.
Library ofCongress Cataloging-in-Publication Data
Sperling, A. P. (Abraham Paul), 1912-
Arithmetic made simple.
ISBN 0-385-23938-6
Includes index.
—
1. Arithmetic 1961- . I. Levinson, Samuel D.
II. Beige, Robert R. III. Title.
QA107.S63 1988 513 87-24716
Copyright ® 1960, 1988 by Doubleday,
a division ofBantam Doubleday DellPublishing Group. Inc.
ALL RIGHTS RESERVED
PRINTED IN THE UNITED STATES OF AMERICA
DECEMBER 1988
8 9
1
CONTENTS
1 HOW MATHEMATICIANSSOLVEPROBLEMS 5
2 LEARNINGTO USEOUR NUMBERSYSTEM 9
3 ADDITION ANDSUBTRACTION OFWHOLE
NUMBERS 15
4 MULTIPLICATION ANDDIVISION OF
WHOLE NUMBERS 24
5 ALLABOUTFRACTIONS 39
6 LEARNTOUSE DECIMALSWITH EASE 55
7 PERCENTAGE 69
8 HOW PERCENTSAREUSED IN DAILY
BUSINESS 78
9 HOW MONEY IS USEDTOEARN MONEY 88
10 MEASUREMENTOFDISTANCE, WEIGHT,
ANDTIME 97
1 MEASURESOFUNES, ANGLES, AND
PERIMETERSOFPLANEFIGURES 109
12

CHAPTERONE
HOW MATHEMATICIANS SOLVE PROBLEMS
DON'T BE AFRAID these words, you have the intelligence todo
Most of us subconsciously think that we all the mathematics in this book.
should know everything. We also think that Let's assume we have three containers,
one measuring 5 cups, one measuring 3
a facility with mathematics is a sign of
superior intelligence and ignorance of cups, and one very large container that is
mathematics isequivalenttostupidity. We, unmarked. Our problem is to mark the
ofcourse, know that this is ridiculous, yet large container with lines indicating mea-
these subconscious misconceptions can surements from 1 to 10 cups so it will look
and often do have a debilitating effect on like this:
our ability to learn. Along with these mis-
conceptions often comes fear that the world
will find us out, fear that with failure I will
have to admit my stupidity even to myself.
Let's begin by saying, "I know not!" We
are engaged in this endeavor to learn. The
wise man knows that he knows not. If we
were not ignorant, we would know every-
thing and there would be nothing to learn.
Sotolearn we must be ignorant and there is
noshame in that. There is no shame in fail-
We
ure, either. learn from failure, from our
failures we will discover our errors, and Figure 1.
this IS learning. Shame only comes from
not trying. But we only have two measurements, the
3-cup and 5-cup, sowe must figureout how
to use these 3- and 5-cup measures to mark
H-I-N-T
all the lines on the large container.
Number 1 We could fill the 3-cup measure, pour it
in the large container, and draw a mark for
Whenstudyingmathematics, it'sagoodideato 3 cups. Now we can fill the 3-cup measure
useapencilandpapertowritedownideasand
syooluurtihoenas.d bDoenc'atusteryittoiskeeaespyatolgotetofcotnhfiunsgesd.in amgaarikn,it.poNuorwitwienthoatvheemlaarrgkescfoonrt3aicnuepr,s aanndd
6 cups.
One further misconception: that there Mathematicians like to express solu-
are those who have a facility for mathemat- tions like this in an abstract way, known as
ics and those whodo not. This is simply not "equations." For the problem just de-
scribed, they would write an equation like
true. It is true that afacility in mathematics
is a function of innate intellectual ability. this:
But if you have the intelligence to read 3 + 3 = 6
ArithmeticMadeSimple
This is read as "3 added to 3 equals 6" or it! (Check your answer under D.) Remem-
"3 plus 3 equals 6." This can also be ber, our problem was to mark off the large
expressed as: containerin 1-cup intervals. Do you see the
2X3 = general principle here? Once we have a
6
way to make a mark for 1 cup, wecan make
This is read as "2 times 3 equals 6." all the other marks too. How? Just add 1
Now it's your turn. Can you figure out cup to 1 cup and we have two, add 1 cup to
how to make marks for 5 cups and then 10 2 and we have 3, and so on.
cups using only the 3-cup and 5-cup mea- All right! Let's change the problem a lit-
sure? Can you write the equation for how tle. We again want to mark offa large con-
you found the 10-cup mark? (Check your tainer in 1-cup intervals as before, but now
answer at the end ofthis chapter under A.) insteadofthe 5-cupand3-cupmeasure, we
Nowfind away to make a mark on the large have a 2-cup and a 6-cup container.
container for 8 cups. Write an equation for
this also. (Check your answer under B at
the end ofthe chapter.)
To make a mark for 2 cups involves a
slightly different approach. First, we would
fill the 5-cup container, then pourthewater
into the 3-cup container until it's full. That
which is left in the 5-cup container is 2
cups! We would then pour this 2-cup
amount into the large container and mark
it. The equation for this is: Figure3.
5-3 =
2
What would be the "equation" for a 4-cup
This is read as "5 minus 3 equals 2." mark? How about a 10-cup mark? That's
Using this 2-cup mark and one of our correct, 2 + 2 + 6 = 10. Notice our
original 3-cup or 5-cup measures, how equation here has three numbers to the left
would weget 7 cups marked offon the large of the "=" sign. We could have many
container? Write the equation. (Check it numbers strung together by "-|-" signs,
under C.) Our large container now looks indicating that we should simply add them
like this: all together. For example, we could get the
8-cup mark by 2 + 2 + 2 + 2 = 8.
Now, anotherway toget the 10-cup mark
would be to fill the 6-cup measure, pour it
8 into the large container, then fill the 6-cup
container again but pour enough out to fill
the 2-cup container. We would then have 4
cups left in the 6-cup measure, which we
would add to the large container, giving us
10 cups in the large container. Can you
figure out the equation that would describe
Figure2. this? (Lx)ok under E at the end of the
chapter.)
Now do you think you can figure out how Remember, now, we are trying to find a
to get a mark for 1 cup? Don't be afraid, try way to get all the marks from 1 to 10 on the
HowMalhemaiiciansSolveProblems
large container. This means we must get a in the8. Now the8-cup measure hasonly 2
1-cup mark. We know that ifwe can get a cups in it.
1-cup mark we can get all the marks. BUT
now we are workingwith a2-cup and 6-cup 8 - (2 X 3) = 2
measure instead of the 3-cup and 5-cup
measure. Put the 2 into the large container and mark
Let's fill the 6-cup measure and pour off it; now with the 2- and 3-cup measure we
into the 2-cup measure. Write the equation can get the 1-cup. Suppose oneofthe origi-
for this. What if we now empty the 2-cup nal two measures was 6 and the other 3?
measure and again pour some of the Try this on yourown. (The answer is at the
remaining liquid from the 6-cup measure end under F.) Now try to guess a general
into the 2-cup measure until it's full. The principle about what the relationship be-
equation for this is: tween the numbers must be tobeable toget
a 1-cup mark.
6 - (2 X 2) = 2 These are the kinds ofquestions mathe-
maticians ask themselves; this is what we
Theparentheses mean we multiply before call "logical thinking." It's just the kind of
wweesfiulbltarancdte. mDpotyyofurosmeeontheatconnotamiantetrertohtohwe otfhintkhiinngkipnegopYleOUdo heavevrey bdaeye.nItd'ositnhge. kiWned
other, we cannot make the sum or differ- think ofa problem, then guess and wonder
ence between them come out to 1 cup? what it all means. What can we conclude
Something different is happening here. from it? How does it apply to the simple
Whenwehad the3-cupandthe 5-cupmea- case? How does it apply to the more com-
sure, we were able to find the 1-cup mark plicated case? How does it apply to the
without toomuchdifficulty. Alsorecall that general case? We guess, we wonder, and
the key to getting all the m£u:ks was to get we guess some more; this is how we make
the 1-cup mark. With both known contain- discoveries.
ers being of odd numbers (3 and 5), we The only difference between you and the
could find away to get the 1-cup mark. But so-called mathematician is how many basic
now we have measures of even numbers arithmetic and mathematical fundamentals
(2 and 6), and no matter how many differ- they have learned. So all you need is to
ent ways we try, we still can't get the begin learning the fundamentals and this
1-cup mark. The numbers 1, 3, 5, 7, 9, book will help you dojust that.
11, and so on are called odd numbers; the Here are the answers to the practice
numbers 2, 4, 6, 8, 10, and so on are questions in this chapter.
called even numbers. Even numbers
always differ by at least 2. So we cannot A. 5 + 5 = 10
combine them by addition orsubtraction to B. 5 + 3 = 8
get an odd number and the number 1 is an
odd number. C. 2 + 5 = 7
What if the two original measures are
—
both odd but d—ifferent f—rom 3 and 5? What D. Fill the 3-cup container pour it into the
if on—e is odd—say, 3 and the other is large—container until water is—just up to the 2-cup
even say, 8 could we then get a 1-cup mark empty large container pour the remainder
ofthe 3-cup container into the large container and
mthaernk?poWuirthfrtohmet3hean8di8ntwoethceou3l,ddfiullmtphet8h,e mark it. 3 - 2 = 1
3, and then fill the 3 again from what's left E. 6 + 6 - 2 = 10
8 ArithmeticMadeSimple
F. Theproblemwithnumberslike3and6,even
H-l-N-T
though one isodd and theother iseven, is thatone
isamultipleoftheother. Nomatterhowweaddand Number 2
subtractintegralmultiplesofthesetwonumbers, we
willalwaysgetaneven numberandweneedanodd When studying a subject like mathematics, it
number, 1. is wise not to look too far ahead in the book.
You need to leam thesesubjectsstepbystep,
In the later chapters, practice exercises
and material that you are not ready forcan be
will follow each chapter. The answers to frighteningand discouraging. You will under-
those exercises can be found beginning on
stand the latermaterial as you get to it.
page 154.