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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

}

V

V

SILVER, BURDETT AND COMPANY

NEW YORK BOSTON CHICAGO

c

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sducatios iiBa#

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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 â€” more particularly those in the early part

of the book â€” 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.

Digitized by VjOOQIC

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.

Digitized by VjOOQIC

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, â€” that side of the subject being left

to the succeeding book. The first thoughts are whaJt and

hxniOj â€” these so presented that the processes shall be easily

Digitized by VjOOQIC

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^ â€” 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 . .â€¢ 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 â€¢ 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 . â€¢ â€¢ . 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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Digitized by VjOOQIC

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 â€” ^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.

I ^ i I

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86 6, 40 6, 38 2, 10 4, 579

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

Digitized by VjOOQIC

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

Digitized by VjOOQIC

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.

Digitized by VjOOQIC

ROMAN NOTATION. 6

â€¢ 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,

to make the world's books discoverable online.

It has survived long enough for the copyright to expire and the book to enter the public domain. A public domain book is one that was never subject

to copyright or whose legal copyright term has expired. Whether a book is in the public domain may vary country to country. Public domain books

are our gateways to the past, representing a wealth of history, culture and knowledge that's often difficult to discover.

Marks, notations and other marginalia present in the original volume will appear in this file - a reminder of this book's long journey from the

publisher to a library and finally to you.

Usage guidelines

Google is proud to partner with libraries to digitize public domain materials and make them widely accessible. Public domain books belong to the

public and we are merely their custodians. Nevertheless, this work is expensive, so in order to keep providing this resource, we have taken steps to

prevent abuse by commercial parties, including placing technical restrictions on automated querying.

We also ask that you:

+ Make non-commercial use of the files We designed Google Book Search for use by individuals, and we request that you use these files for

personal, non-commercial purposes.

+ Refrain from automated querying Do not send automated queries of any sort to Google's system: If you are conducting research on machine

translation, optical character recognition or other areas where access to a large amount of text is helpful, please contact us. We encourage the

use of public domain materials for these purposes and may be able to help.

+ Maintain attribution The Google "watermark" you see on each file is essential for informing people about this project and helping them find

additional materials through Google Book Search. Please do not remove it.

+ Keep it legal Whatever your use, remember that you are responsible for ensuring that what you are doing is legal. Do not assume that just

because we believe a book is in the public domain for users in the United States, that the work is also in the public domain for users in other

countries. Whether a book is still in copyright varies from country to country, and we can't offer guidance on whether any specific use of

any specific book is allowed. Please do not assume that a book's appearance in Google Book Search means it can be used in any manner

anywhere in the world. Copyright infringement liability can be quite severe.

About Google Book Search

Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers

discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web

at http : //books . google . com/|

Digitized by VjOOQiC

'^w>.

^\-

fX^-^

X

U-

'jy\

^-'.i.

v;

â– H-vv

Digitized by VjOOQIC

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

^-'^

-y^'^ THE. NEW '^ ^i^

ADVANCED ARITHMETIC

JOHN W. COOK

raxswxtiT or illihoib btatx kobxai. CKiraMirr

MISS N. CROPSEY

ASSISTANT SUPSBINTSKDENT OF CITT SCHOOLS, INDIANAPOLIS, INDIANA

}

V

V

SILVER, BURDETT AND COMPANY

NEW YORK BOSTON CHICAGO

c

Digitized by VjOOQICx

u\.

f J ^ ' <^-'

>tr^O^

â– 4^i^ "Wt-W^^^

GoPTmiOHT, 1892. 1896, 1903, 1904, 1906,

Br SIL^^Â», BURDETT A^'D COMPANY.

sducatios iiBa#

Digitized by VjOOQIC

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 â€” more particularly those in the early part

of the book â€” 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.

Digitized by VjOOQIC

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, â€” that side of the subject being left

to the succeeding book. The first thoughts are whaJt and

hxniOj â€” 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^ â€” 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 . .â€¢ 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 â€¢ 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 . â€¢ â€¢ . 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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- ^

'

' \-'

â€¢^ '

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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 â€” ^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.

I ^ i I

tgS tiSao '2sÂ§ 'Sis I

M'Co M'^p ^Ss ^uas M

^3*3.2 73-9Â© '09.2 "O^S '^ t^ m

5a. g Sfls ^ a B Sao ggS

Â£S^ 30^ J2Â®3 3 ^ fi Â»SÂ®5

Whh whoq khs Whh Who

86 6, 40 6, 38 2, 10 4, 579

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

â€¢ 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,

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