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Et)e StannarD Snieis at ^at))ematiciS ^

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ADVANCED ARITHMETIC



JOHN W. COOK
raxswxtiT or illihoib btatx kobxai. CKiraMirr

MISS N. CROPSEY

ASSISTANT SUPSBINTSKDENT OF CITT SCHOOLS, INDIANAPOLIS, INDIANA

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NEW YORK BOSTON CHICAGO



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PREFACE TO THE REVISED EDITION.

A NUMBER of changes have been thought desirable in
the plan of "The New Advanced Arithmetic"; all of
them, however, are intended to recognize the latest and
best phases of class-room- work in arithmetic. Some of
these changes — more particularly those in the early part
of the book — have been necessary the better to relate
the Advanced book of the Series to the Elementary book,
which has at the same time undergone revision.

Much emphasis is given to Factoring, and its applica-
tion to Cancellation, the Highest Common Factor, the
Lowest Common Multiple, and Evolution. The equation
is introduced in a simple form much earlier tlian in the
previous edition. This is done in order Jx) simplify the
treatment of other important topics which follow it.
Percentage and its applications, as well as Mensuration,
are treated more fully than before.

A large number of new problems have been added,
which will, it is thought, be found interesting and
profitable.

Cube Root, Compound Proportion, Equation of Pay-
ments, and True Discount are among the subjects which
have been omitted.

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iv PREFACE TO THE REVISED EDITION

It is hoped that the revised book is fairly representa-
tive of what is best and most progressive in present-day
methods, and that teachers generally will find it well
adapted to their needs.

The authors and publishers desire to express their sin-
cere appreciation of the assistance and cooperation given
in the preparation of this edition by Dr. Robert J. Aley,
Professor of Mathematics in Indiana University, and Oscar
L. Kelso, Professor of Mathematics in the Indiana State
Normal School.



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PREFACE TO FIRST EDITION.

It has seemed to the authors of the Normal Goubsb m
Number that there is room for another series of Arithmetics,
notwithstanding the fact that there are many admirable books
on the subject already in the field.

The Elementary Arithmetic is the result of the expe-
rience of a supervisor of primary schools in a leading Ameri-
can city. Finding it quite impossible to secure satisfactory
results by the use of such elementary arithmetics as were
available, she began the experiment of supplying supple-
mentary material. An effort was made to prepare problems
that should be in the highest degree practical, that should
develop the subject systematically, and that should appeal
constantly to the child's ability to think. Believing that
abundant practice is a prime necessity to the acquisition of
skill, the number of problems was made unusually large.
The accumulations of several years have been carefully re-
examined, re-arranged, and supplemented, and are now pre-
sented to the public for its candid consideration. Not the
least valuable feature of this book is the careful gradation
of the examples, securing thereby a natural and logical devel-
opment of number work. No space is occupied with the pres-
entation of theory, — that side of the subject being left
to the succeeding book. The first thoughts are whaJt and
hxniOj — these so presented that the processes shall be easily



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VI PREFACE TO FIRST EDITION.

comprehended and mastered. Subsequently, the why may be
intelligently considered and readily understood.

The Advanced Arithmetic is the outgrowth of a somewhat
similar experience in the class-room of a teachers' training-
school. For many years an opportunity was afforded to study
the effects upon large numbers of pupils of the current methods
of instruction in arithmetic. The result of such observation
was the conviction that the ratibnal side of the subject had
been seriously neglected. An effort was made to supplement
the ordinary text-book by a study of principles and by explana-
tions of processes. The accumulations of fifteen years have
been edited with all of the discrimination of which the authors
are capable. Great care has been exercised in the presenta-
tion of principles and in the formulation of processes, to the
end that the learner shall have every facility for the use of
his reasoning powers, and at no point be relieved from the
proper exercise of his mental activity and acumen. It is
hoped that the book may contribute somewhat to the move-
ment, now so happily going on, that looks toward the dis-
establishment of the method of pure authority, and the
establishment of a method that makes its appeal to intelli-
gence and reason.

The authors desire to express their appreciation of the
excellent suggestions offered by many friends; but especial
thanks are due Professor David Felmley, of the Illinois
State Normal School, for his discriminating criticisms and
valuable assistance.

THE AUTHORS.



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INTRODUCTION.

Arithmetic amply justifies its place in the cumculum
because of its utilitarian and culture values.

No subject surpasses it in usefulness. It is so closely
connected with our lives that the blotting out of all
aritKmetical notions would stop at once all commercial
and industrial activity. To be a member of the social
world one must know something of arithmetic.

On the culture side, arithmetic is fitted by its nature
to do certain things. The three most important things
that it can do well are :

1. To train in scientific reasoning.

Arithmetic is a science ; that is, it is a body of organ-
ized truths. Its conclusions are the results of logical
reasonings. Every step is taken because of some pre-
ceding step or fundamental assumption. The value of
the training that comes from this sort of study can hardly
be overestimated.

2. To train in concentration.

The nature of arithmetical work is such that the whole
mind must be given to the problem under consideration.
If, in adding a column of figures, the student allows his
mind to wander, he finds that he must pay the penalty
for so doing. He must add the column again. No other
subject in the common school course equals arithmetic in
its power to train the attention to concentrated effort.

3. To train in accuracy,

â–¼il

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vm INTRODUCTION.

Arithmetic is an exact science. The results of arith-
metic are fixed and infallible. The multiplication table
will never cease to be true. It is of great value to a
growing mind to work at a subject in which the results
are exact.

The purpose of the teacher in teaching arithmetic should
be to put the pupil in possession of arithmetic as a tool
for use in everyday life, and, in addition, to give him all
the training in scientific reasoning, concentration, and
accuracy that the subject is capable of furnishing.

The arithmetical operations employed in ordinary busi-
ness affairs are simple, but they must be performed with
absolute accuracy and with rapidity. They are based, pri-
marily, upon the memory. Neither accuracy nor rapidity
is possible without a thorough mastery of the primary work.
This mastery is acquired ' through constant repetition of
the old in connection with the acquisition of the new.

The tables of the fundamental operations must be
learned so thoroughly that* their application can be made
without much conscious thought. To this end much of
the work in the elementary arithmetic should be abstract
and with rather large numbers.

The mechanical side of arithmetic consists of the four
fundamental operations of addition; subtraction, multipli-
cation, and division. These need to be thoroughly mas-
tered so that in the higher phases of work upon problems
the whole strength of the mind may be given to the
reasoning.

In view of the nature of arithmetic and the purpose in
teaching it, the following things should be observed by
the teacher:

1. Arithmetic is a unity.

AH the parts of arithmetic are so related that they



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INTRODUCTION. ix

make a unity. This unity is a part of the larger unity,
mathematics. At no stage of the development of the
subject should the student learn anything which it will
be necessary to unlearn at a later stage. If the student
learns a definition of multiplication, it should be true not
only throughout arithmetic, but also throughout the whole
of mathematics.

2. There are certain truths wh^ch are organic.

The organic truths of arithmetic are the threads of
unity which run through it and about which it is organ-
ized. The %cale relation, which is the fundamental thing
in all number systems, is organic. It binds together
simple numbers, compound numbers, and fractions. That
"multiplying or dividing both dividend and divisor by
the same number does not affect the quotient" is an
organic truth which binds together division, fractions,
and ratio.

3. Arithmetic is twofold^ — pure and applied.

Pure arithmetic has to do with the properties of num-
ber. It includes the fundamental operations, factors,
multiples, divisors, powers, roots, and ratios, with all the
principles and relations pertaining to them. Pure arith-
metic could exist in perfection without any application.
Applied arithmetic is pure arithmetic answering the ques-
tions that come to man in his material experiences. The
culture value of arithmetic is found in both its phases.
The bread and butter value is found only in the applied
side. The teacher must know that no applied arithmetic
is possible except as it rests upon the sure foundation of
the pure arithmetic.

4. There is no definite boundary/ between arithmetic and
algebra.

Algebra and arithmetic overlap. No one has yet been



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X INTRODUCTION.

wise enough to mark the exact place where the one ends
and the other begins. The simple form of the equation
belongs as much to arithmetic as it does to algebra.
There is no reason why the equation in arithmetic should
be treated in a stilted, complicated manner. Its principles
are as simple and easy as those of multiplication or divi-
sion. It is to the student of arithmetic what improved
machinery is to the farmer.

The observation of the four facts above enumerated
will have an important effect upon the method of teach-
ing the subject. Among the many results, the following
may be enumerated :

1. There will be teaching instead of hearing of lesions,

2. The teacher will see the end from the beginning,
and will direct his instruction to that end.

3. The law of apperception will be followed, and each
new subject will be related to the preceding ones.

4. In applied arithmetic the pupil will be led into the
particular field of experience in which the problems occur,
and made familiar with the activities in that field before
attempting to apply the pure arithmetic.

5. The value of mental arithmetic will be appreciated,
and the pupils given ample opportunity to grow through
mental problems.

6. The value of direct, masterly methods of solution
will be understood, and the pupils encouraged to use
every possible short cut.

7. The pupils, when solving problems, will be urged
to think first and to figure as little as possible.



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CONTENTS.



CHAPTER L

PAGB

NUMEBATION AND NOTATION . 1-6

Arabic Notation .... 3

Roman Notation .... 5
Different Readings for a

Number 6

CHAPTER IL

The Fundamental Opeba-

TION8 7-36

Addition and Subtraction . 7

Multiplication and Division . 11

The Law of Signs .... 16

Compound Numbers ... 19
Compound Addition and

Subtraction 22

Compound Multiplication

and Division 25

Extension of Addition and

Subtraction 27

The Area of a Rectangle . 28
The Volume of a Rectangu-
lar Solid 30

Miscellaneous Problems . 33

CHAPTER III.

Factobs, Divisors, and Mul-
tiples ...... 37-53

Tests of Divisibility ... 39

Factoring 40

Cancellation 41

The Highest Common Factor 43



The Lowest Common Mul-
tiple 45

Miscellaneous Problems . 47

CHAPTER IV.

Longitude and Time . . 54-61
Standard Time 59

CHAPTER V.

Fractions 62-97

Reduction of Fractions . . 65

Addition of Fractions . . 69

Subtraction of Fractions . . 74

Multiplication of Fractions . 76

Division of Fractions ... 82

Complex Fractions ... 86
To Find the Part which One

Number is of Another . . 87
To Find a Number when a

Specified Part of it is given 89

Miscellaneous Problems . 89

CHAPTER VL

Decimal Fractions . . 98-128
Reading Decimal Fractions . 99
Writing Decimal Fractions . 100
Reduction of Decimal Frac-
tions 102

Addition of Decimal Frac-
tions 105

Subtraction of Decimal Frac-
tions . .• 106



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Xll



CONTENTS.



PiiGS

Multiplication of Decimal

Fractions 106

Division of Decimal Frac-
tions 109

The Metric System ... 112

Reduction of Metric Numbers 115

Specific Gravity . . . .117

Miscellaneous Problems . 119

CHAPTER Vn. *

The Equation . . . 120-146
The Use of a; in Problems . 129
Equations containing Frac-
tions 134

Verifying an Equation . . 135

Miscellaneous Problems . 138

CHAPTER Vin.

Pbrcektaoe .... 147-162
Miscellaneous Problems . 160

CHAPTER IX.

Applications op Percent- '

age 163-185

Profit and Loss 163

Commission 167

Commercial or Trade Dis-
count 170

Taxes 174

Insurance 180

Property Insurance . . . 181
Life Insurance 182

Miscellaneous Problems . 183

CHAPTER X.

Applications of Percent-
age {continued) . 186-226

Interest 186

Notes ........ 195



FAOS

Partial Payments .... 196
The Method of finding In-
terest by Days .... 201
General Problems in Simple

Interest 203

Banks and Banking . . . 205

Exchange 211

Stocks 214

Bonds 215

Miscellaneous Problems . 221

CHAPTER XI.

Ratio and Proportion 226-241
The Relation of Numbers . 226

Ratio 228

Proportion 233

Miscellaneous Problems • 238

CHAPTER XII.

Involution and Evolution

242-253

Powers 242

Roots 243

The Relation of the Squares

of Consecutive Numbers . %ib
The Relation of the Squares

of Any Two Numbers . . 247
The Number of Figures in a

Product 250

The Number of Figures in

the Square Root of a

Number 251

The Extraction of Square

Root 252

CHAPTER Xin.

Mensuration .... 254-302

Carpeting 257

Papering ....... 259



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CONTENTS.



xui



PAOB

Plastering 261

The Parallelogram .... 262
The Triangle ..... 264
The Right-angled Triangle . 266

The Trapezoid 269

United States Surveys . . 271

The Circle 272

The Area of a Circle ... 275

Solids 277

Wood Measure 277

Lnmber Measure . • • . 279



PAOB

Masonry and Brickwork . . 281
The Inverse Problem in Men-
suration 284

Prisms and Pyramids . . . 287
Cylinders, Cones, and
Spheres 290

MiSCBLLANEOnS PROBLEMS . 294

General Reviews . . 802-313
Appendix 815-^27



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THE NEW ADVANCED ARITHMETIC.

CHAPTER I.
NUMERATION AND NOTATION.

1. Numeration is the naming or reading of numbers.

2. The group system is universally used in naming
numbers. The numbers of a small group are given indi-
vidual names, and then a systematic process of repetition
follows. In naming numbers we use the group ten. The
numbers from one to ten have distinct individual names.
The names eleven and twelve seem to be individual, but in
their early forms they were the combination of one and
ten^ and two and ten respectively. The word thirteen is a
combination of three and fen, the idea of three being in
thir^ and of ten in teen. The same relation is seen in the
remaining teenB, The word twenty is made from the com-
bination two tena^ the idea of two being in twen^ and ten in
ty. The same idea is in tliirty, forty, fifty, sixty, seventy,
eighty, and ninety.

Except for slight changes in form made to produce
better sounding words, no new number name occurs after
ten, until we reach ten tens, when the new name hundred
appears. Repetition and combination then continue to
ten hundreds, when the new name thousand occurs.

In the number eight hundred seventy-four, we have a com-
bination of the words eight, hundred, seventy, and four,

1

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2 NUMERATION AND NOTATION.

After the word thousand is introduced, no new name is
needed until the number one thousand thousand occurs, to
which the name million is given. 'The next new number
name is billion^ given to the number a thousand million.

3. It will be noticed that all this number naming
centers about ten. The new names are for numbers ten
times as large as those already named. A hundred is ten
times a ten, a thousand is ten times a hundred, a million
is ten times a hundred thousand, and a billion is ten times
a hundred million.

4. For the reason that in our system everything centers
about ten, it is called the decimal system (^decern — ^ten).

5. Ones are grouped into tens^ tens into hundreds^ hun-
dreds into thousands^ and so on. Ones are units of the
first order, tens are units of the second order, hundreds are
units of the third order, and so on.

6. From the formation of our number names we see
that ten units of one order make one of the next higher.
Ten is, therefore, the scale (or radix) of our system.

7. Three orders form a period. The orders from left
to right of any period are hundreds, tens, and units of that
period.

8. Arrangement of orders and periods :

Trillions. Billions. Millions. Thousands. Units.



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Note, Learn the names of the periods in their order from left to righL



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ARABIC NOTATION. 8

9. Notation is the expression of numbers by means of
characters.

10. There are three methods of notation in use: the
word, the Arabic, and the Roman.

11. In the word method the number names are written
out in full, e.ff. May eleventh, eighteen hundred sixty-three.

ARABIC NOTATION.

12. The Arabic notation represents numbers by the use
of the ten characters 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. These
character are called figures or digiU^ and are in' almost
universal use.

The Arabic notation possesses three marked character-
istics :

(1) Its number of characters corresponds to the scale of the
system ; that is, there are ten characters, and the scale of the
system is ten. This makes possible the same sort of combina-
tion and repetition in the writing of numbers that occurs in the
naming of numbers.

(2) It uses the idea of place value ; that is, the same char-
acter in different places in a number represents different values.

In the number 6666 the first 6, beginning at the left, represents, because
of its place, six thousand ; the second 6, six hundred ; the third 6, sixty ;
and the fourth 6, six.

(3) It has one character, (zero), used to fill vacant places.

13. Moving a character one place to the left multiplies
its value by ten, while moving it one place to the right
divides its value by ten.

In the number 666 the middle 6 is ten times the right-hand
6 and one tenth of the left-hand 6.

14. To read a number, group the figures into periods,
beginning at the right, and separating the periods by



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4 NUMERATION AND NOTATION

commas. Beginning at the left, read the number in each
period as if it stood alone; then add the name of the
period.

81,367,937 is read " eighty-one million, three hundred sixty-
seven thousand, nine hundred thirty-seven."

Note i. The name of the last period, units, is generally omitted.
Note 2, In reading whole numbers the word and is unnecessary.

EXERCISE I.
Eead the following numbers :

1. 2345. 6. 250849. 11. 683471.

2. 4638. 7. 381307. 12. 829406.

3. 7912. a 408391. 13. 200619.

4. 3105. 9. 716004. 14. 100054.

5. 26853. 10. 500836. 15. 973070.

EXERCISE 2.

Before writing the following numbers, tell how each will
appear when written.

Thus, three thousand eight hundred seven is expressed by writing
the following : three, comma, eight, cipher, seven.

1. Forty thousand six.

2. Ninety-seven thousand five hundred twelve.

3. Three hundred sixty-nine thousand twenty-four.

4. Four million eight thousand two.

5. Fifty-six million nineteen thousand thirty-three.

6. Eighty -one million five hundred thirteen thousand two
hundred fifty-one.

7. Three hundred million ninety thousand four.

8. Five billion six million seven thousand eight.

9. Seventy-two billion six hundred thirty-five thousand two
hundred fifty-one.



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ROMAN NOTATION. 6

• 10. One hundred three billion two million seventeen thousand
one hundred four.

To the Teacher. If needed by the pupils, the teacher should supply
further exercises in reading and writing numbers.

ROKAN NOTATION.

15. The Roman notation expresses numbers by means
of the characters I, V, X, L, C, D, M, and .

This notation is used very little. It is being rapidly re-
placed in dates and chapter numbers by the Arabic notation,

1 = 1, V = 6, X = 10, L = 60, C = 100, D = 600,