The avanced practical arithmetic

216 Pages·2011·31.68 MB·English

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THE ADVANCED
PRACTICAL ARITHMETIC
• BY
FLETCHER DURELL, PH.D.
MATHEMATICAL MASTER IN TH-J I.AWRENCEVILLE SCHOOL
AND
EDWARD R. BOBBINS, A.B.
MATHEMATICAL MASTER IN THE WILLIAM PBNN CHARTER SCHOOL
NEW YORK
CHARLES E. MERRILL CO.
44-60 EAST TWENTY-THIRD STREET
Durell & Robbins'
Mathematical Series
P R E F A CE
Durell & Robbins—Elementary Practical
THE object in preparing this arithmetic has been the same
Arithmetic. 201 pages, 12mo, cloth, 35 cents
which the authors had in view when writing their school
Durell & Robbins—Advanced Practical
algebra, viz.: " to show more plainly, if possible, than has
Arithmetic. 363 pages, 12mo, cloth, 65 cents
been done heretofore, the practical or common sense reason
Durell & Robbins—A Grammar School Al- for each step or process." It is believed that this treatment
gebra. 287 pages, 12mo, half leather, 80 cents
is not only " adapted to the practical American spirit, but
Durell & Robbins—A School Algebra also gives the study of the subject a larger educational value."
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It is also believed that the scholarly possibilities and value
Durell & Robbins—A School Algebra Com- of the treatment, instead of being diminished, are increased
plete. 483 pages, 12mo, half leather, $1.12
thereby. For instance, it is hoped that the matter presented
Durell—Plane Geometry in the chapters on the Applications pf Percentage and In-
341 pages, 12mo, half leather 75 cents terest, and on Arithmetical History, will have a new value.
Durell—Solid Geometry
The main principle which has governed the authors in
213 pages, 12mo, half leather 75 cents
writing the book has also been the controlling factor in the
-Plane and Solid Geometry treatment of many matters of detail which are still in debate
iges, 12mo, half leather $1.25
among teachers of arithmetic and writers on the subject.
Trigonometry and Tables
Almost all are agreed that the study of arithmetic should begin
fe, 8vo, cloth $1.25
with the study of concrete objects; and that the use of geometric dia-
grams is a great help in presenting certain parts of the subject, as
;t 1901 by Charles E. Merrill Co. fractions. But the authors believe that it is a mistake to have the
concrete objects, or even pictures of them, constantly before the
[7]
pupil. Nor should diagrams be printed as part of the text. The
pure arithmetical processes should be made easy and natural as
ACERVO GENERAI: Boon as possible, on account of their brevity and simplicity. Con-
129557
crete objects or diagrams, if kept before the eye constantly, tend
to clog and hamper the mind; hence they should be recalled only
so often and at such places as may be necessary in order to make
the subject vivid and real.
In the same way algebraic symbols ami methods have been intro-
duced only when they give a clear and pronounced advantage, and TO THE TEACHER
thus arouse in the pupil the desire to know more of them.
It is believed that rules for processes are useful in many ways
when they are arrived at after the proper preliminary work and
1. The teacher should make sure at different times that the
when they are used with discretion.
pupil carries in mind the concrete object for which a symbol stalls.
In order to cultivate habits of analysis and exact statement, and
Now show the pupil, now have him show the concrete object.
yet to prevent these from becoming mere mechanical rote proc-
Show him diagrams illustrating the properties of fractions. Have
esses, different forms of analysis have been used adapted to dif-
him make these diagrams. But do this only occasionally, and
ferent kinds of work. These are indicated by the use of different
always for some good reason.
words as "Operation," " Explanation," "Solution," etc.
2. In oral work and explanations insist on careful and accurate state-
In like manner, oral exercises are sometimes put before written
ments. For instance, in oral work do not allow a pupil to give the
exercises, sometimes after them.
answer merely without a formal statement of the analysis or steps
The subject-matter is not spirally arranged, but is adapted
by which the result was obtained.
to spiral study. The subject is presented as an organic whole,
3. In written problems in which the analysis is difficult (as in Exer-
yet one which can be learned by successive steps (see p. 6).
cise 23) elicit the analysis from the pupil by oral questions, and
A large number of examples adapted to the theory of the afterward have the pupil write out the analysis and solution.
book has been made and carefully graded. Especial atten- 4. Insist on the use of cancellation wherever possible. Train
tion is also called to the chapter on the Metric System. the pupil to combine all the operations required in the solution
of a problem in a comprehensive plan or scheme; then to factor
The authors will be glad to receive any corrections or sug-
and cancel wherever possible; never to multiply till it is necessary
gestions from teachers using this book.
to do so.
5. Train the pupil also to make a rough estimate or forecast of the
FLETCHER DURELL,
EDWARD R. ROBBINS. answer before beginning the exact numerical work. This not only
LAWRENCEVTLLE, N. J., tends to eliminate large errors, but is also a valuable habit, since, in
i
PHILADELPHIA, PA., practical life, fully one-half the applications of arithmetic are
May 1, 1901.
made in this way.
6. Impress upon the minds of pupils in various ways the local
value of digits and the limitations in the accuracy of all arithmetical
work based on measurements (see Arts. 19, 77, 78).
7. Study to vary methods to suit the needs of efferent pupils,
both in presenting topics and in meeting difficulties. It is to be
remembered that pupils as they come from different homes prob-
CONTENTS.
ably have more varied capacities with respect to the subject of
arithmetic than to any other.
8. Do not be satisfied till by long practice and working innu- CHAPTER I. CHAPTER IX.
merable examples, if necessary, the pupil has become a rapid and PAGE PAGK
COMMON FRACTIONS 103
NUMBER. NUMERATION. NOTA-
accurate computer. The power of handling figures with facility Transformations of Fractions . 108
TION 9
and accuracy is of the first importance both in practical life and Number 9 Operations with Fractions . . 114
I. Addition of Fractions . . 114
in its influence on the further educational development of the Numeration 12 II. Subtraction of Fractions . 117
Notation I4
pupil. TTT. Multiplication of Frac-
Roman Notation 20
tions 120
9. REVIEW CONSTANTLY.
CHAPTER II. IV. Division of Fractions . 124
V. Complex Fractions . . . 126
23
SCHEDULE RECOMMENDED IN USING DURELL AND ROBBINS* ADDITION VI. G. C. D. and L. C. M. of
ARITHMETIC. CHAPTER III. Fractions 129
CHAPTER X.
TEAK. FIRST HALF-YEAR. SECOND HALF-YEAR. SUBTRACTION 33
1 Oral work vrilhoiU text-book. El(emmuecnhtaorfy itPreraacdt.t opAurpithil. . byp pte. ac1h-e3r2; CHAPTER IV. DECOIMpeArLa tiFoRnsA CwTiIthO NDS ecimals . . 134382
supplemented by other oral work). MULTIPLICATION 42 Relation of Decimal Fractions
2 Elementary Pract. Arith., pp. 1-32. Elementary Pract. Arith., pp. 33-77 to Common Fractions . . . 148
Review and second course. (ommoitrtee ddi).f ficult parta of some lessons CHAPTER V. Applications of the Decimal
3 Elementary Pract. Arith., pp. 33-77. Elementary Pract. Arith., pp. 78-121 DIVISION 56 System 150
Review and second course. (more difficult parts of some lessons CHAPTER XI.
omitted). CHAPTER VI.
4 EleRmeevnietawreyd Praondd. Acorimthp..l eptepd. 5i5n- 12d1e-. Ele(mleeandtianrgy pPrriancct.i pAlersi tha.,n dp pe.a s1i2er2 -1e9x4- ABBREVIATED PROCESSES . .. 72 COMPOUI. NDM eNaUsuMreBsE oRfS Weight . . 116508
tails. ercises). Abbreviated Multiplication . 72 IT. Measures of Length . . 165
5 Elementary Pract. Arith., pp 91-194. Advanced Pract. Arith. to p. 158 Abbreviated Division . .. 76 in. Measures of Surface . 167
Reviewed and completed in details. (leading principles and easier ex- Combinations of Operations . 79 IV. Measures of Volume and
ercises).
Capacity 1 169
6 Ad(vfaunllceerd coPurrascet).. Arith. to p. 158 Ad(vlaenacdeidng Pprarcint.c iAplreisth .a, ndp pe. as1i5er8 -2e5x5- CHAPTER VII. V. Measures of Value . 173
ercises). FACTORS AND ANALYSIS . . .. 82 VI. Measures of Time . .. 175
7 Advanced Pract. Arith., pp. 158-255 Advanced Pract. Arith., pp. 256 toend Factors 82 VII. Circular and Angular
(fuller course). (leading principles and easier ex- Analysis 89 Measure 1~8
ercises).
VIII. Miscellaneous Units 179
8 Advanced Pract. Arith. Review. Elementary Algebra. CHAPTER VIII. Operations with Compound
Rapid review to p. 256. Geometrical Drawing.
Fill in details, pp. 256 to end. Q. C. D. AND LC.M 94 Numbers 181
Greatest Common Divisor . . 94 Application to Longitude and
Least Common Multiple . , • Time 183
7
PAGE % PAGE
Common Frétions and De- Proportion 286
nominate Numbers . . .. 194 Compound Proportion . . .. 289
Decimal Fractions and De- Proportional Parts 291
nominate Numbers . . .. 195 Partnership 292 ARITHMETIC.
CHAPTER XII. CHAPTER XVIII.
PRACTICAL MEASUREMENTS . . 200
INVOLUTION AND EVOLUTION . 295
Applications to Areas . . .. 201
Involution 295
Applications to Volumes . . 208 CHAPTER I.
Evolution 296
CHAPTER XIII. Square Root 297
Cube Root 302 N U M B E R. N U M E R A T I O N. N O T A T I O N.
PERCENTAGE 217
Other Methods 305 1. Units.—For many purposes the most convenient way
CHAPTER XIV.
CHAPTER XIX of dealing with quantity (as, for instance, with the length
APPLICATIONS OP PERCENTAGE . 231
Profit and Loss 231 MENSURATION 309 of a given line) is to take a certain definite part of the given
I. Mensuration of Lines . . 309
Trade Discounts 233 quantity as a unit, and determine the number of times the
II. Mensuration of Plane
Commission and Brokerage . 235
Taxes 238 Areas 312 unit must be used in order to make up the given quantity
Customs or Duties 240 III. Mensuration of Surfaces of (or line).
Insurance 242 Solid Figures 316 Thus, in determining the length of a given linear object,
Stocks and Bonds 244 IV. Mensuration of Solids . . 320
V. Lines, Areas and Volumes as a rope, we do not depend merely on general impressions
CHAPTER XV. of Similar Figures . . .323 of its magnitude (formed by the eye or by moving the hand
INTEREST 251 CHAPTER XX. over it), but by taking a unit, as one inch or one foot, and
I. When Time is Exact Num-
ber of Years or Months 252 METRIC SYSTEM 326 determining the number of times the unit must be used in
Tables 327 order to make up the line.
II. Six Per Cent. Method . . 253
III. Exact Interest 256 Notation. Numeration. Re- A boy dealing with a quantity of marbles in his possession
duction 331
IV. Interest Table 257 does not do so merely by means of the aggregate impression
Operations with Metric Num-
Problems in Interest . . .. 258
bers 333 which they make in his pocket, but by taking a single marble
CHAPTER XVI. Metric Equivalents 337 as a unit, and counting the number of marbles which he has.
APPI.ICATIONS OF INTEREST . . 264 CHAPTER XXI. This method of regarding quantity as made up of units
Promissory Notes 264
ARITHMETICAL HISTORY . . . 345 gives greater ease and' precision in all the ordinary uses made
Partial Payments 269
Bank Discount 273 History of Numeration and of an aggregate of material.
Compound Interest 277 Notation 345 A unit is a certain quantity taken as a standard of refer-
Annual Interest 278 History of Arithmetical Opera-
Exchange 279 tions 348 ence when dealing with quantity of the same kind.
Equation of Payments . . .. 283 History of Fractions 351 2. Kinds of Units.—Units are of different kinds.
History of Compound Quanti- Natural units are those which occur in the world about
CHAPTER XVN. ties 353
us, as one apple, one man, one year, one day.
RATIO AND PROPORTION . . .. 285 History of Other Topics and
Ratio 285 J Processes 355 Artificial units do not occur naturally, but are devised
9
PAGE % PAGE
Common Frétions and De- Proportion 286
nominate Numbers . . .. 194 Compound Proportion . . .. 289
Decimal Fractions and De- Proportional Parts 291
nominate Numbers . . .. 195 Partnership 292 ARITHMETIC.
CHAPTER XII. CHAPTER XVIII.
PRACTICAL MEASUREMENTS . . 200
INVOLUTION AND EVOLUTION . 295
Applications to Areas . . .. 201
Involution 295
Applications to Volumes . . 208 CHAPTER I.
Evolution 296
CHAPTER XIII. Square Root 297
Cube Root 302 N U M B E R. N U M E R A T I O N. N O T A T I O N.
PERCENTAGE 217
Other Methods 305 1. Units.—For many purposes the most convenient way
CHAPTER XIV.
CHAPTER XIX of dealing with quantity (as, for instance, with the length
APPLICATIONS OP PERCENTAGE . 231
Profit and Loss 231 MENSURATION 309 of a given line) is to take a certain definite part of the given
I. Mensuration of Lines . . 309
Trade Discounts 233 quantity as a unit, and determine the number of times the
II. Mensuration of Plane
Commission and Brokerage . 235
Taxes 238 Areas 312 unit must be used in order to make up the given quantity
Customs or Duties 240 III. Mensuration of Surfaces of (or line).
Insurance 242 Solid Figures 316 Thus, in determining the length of a given linear object,
Stocks and Bonds 244 IV. Mensuration of Solids . . 320
V. Lines, Areas and Volumes as a rope, we do not depend merely on general impressions
CHAPTER XV. of Similar Figures . . .323 of its magnitude (formed by the eye or by moving the hand
INTEREST 251 CHAPTER XX. over it), but by taking a unit, as one inch or one foot, and
I. When Time is Exact Num-
ber of Years or Months 252 METRIC SYSTEM 326 determining the number of times the unit must be used in
Tables 327 order to make up the line.
II. Six Per Cent. Method . . 253
III. Exact Interest 256 Notation. Numeration. Re- A boy dealing with a quantity of marbles in his possession
duction 331
IV. Interest Table 257 does not do so merely by means of the aggregate impression
Operations with Metric Num-
Problems in Interest . . .. 258
bers 333 which they make in his pocket, but by taking a single marble
CHAPTER XVI. Metric Equivalents 337 as a unit, and counting the number of marbles which he has.
APPLICATIONS OF INTEREST . . 264 CHAPTER XXI. This method of regarding quantity as made up of units
Promissory Notes 264
ARITHMETICAL HISTORY . . . 345 gives greater ease and precision in all the ordinary uses made
Partial Payments 269
Bank Discount 273 History of Numeration and of an aggregate of material.
Compound Interest 277 Notation 345 A unit is a certain quantity taken as a standard of refer-
Annual Interest 278 History of Arithmetical Opera-
Exchange 279 tions 348 ence when dealing with quantity of the same kind.
Equation of Payments . . .. 283 History of Fractions 351 2. Kinds of Units.—Units are of different kinds.
History of Compound Quanti- Natural units are those which occur in the world about
CHAPTER XVII. ties 353
us, as one apple, one man, one year, one day.
RATIO AND PROPORTION . . .. 285 History of Other Topics and
Ratio 285 j Processes 355 Artificial units do not occur naturally, but are devised
9
by man so as to extend the advantages arising from the use
the number idea thus formed, to other persons, and thus
of units as widely as possible, as one foot, one-third of an
give them a definite conception of the quantity dealt with,
apple, etc.
without labor on their part. Hence, words are useful by
A primary unit is a single unit of a given kind, as one
which to designate different aggregates of units, or numbers.
dollar.
Number words are useful also to the person using them, in
A derived unit is an aggregate of single units, as five dol-
calling up the precise ideas connected with each aggregate
lars (a " V ») ; or a part of a unit, regarded as a new unit, as
of units.
one-third of a dollar.
The words used for the different aggregates of units (begin-
A unit of one kind may become, in certain relations, a unit of another
ning with a single unit) are—
kind. Thus, an artificial unit maj- become, in some senses, a naUiral unit,
as one dollar. Also, a derived unit may come to be regarded as a primary one, two, three, four, five, six, seven, eight, nine, ten.
unit, as one week, one quarter (of a dollar). For larger aggregates of units a system of grouping units
and naming the groups formed is used, which is explained
EXERCISE 1.
later.
1. What unit of length is used in measuring the length of a room ? The
length of a pencil? Of a quantity of cloth? 6. Counting is the process ot affixing to any group of units
2. What unit of length is used in measuring the distance between two the number word belonging to that group, beginning with
cities? The diameter of the earth? unity, and affixing its number word to each group, till the
3. With what unit of capacity is milk measured? Grain? Strawberries? last unit of the entire group dealt with is reached.
Potatoes?
7. Number Symbols.—Further economies and additional
4. Which unit of area is used in stating the size of a farm ? Of a county ?
power in dealing with numbers are obtained by using a dis-
5. Which of the following units are natural and which are artificial:
year, second, week, foot, quart, yard, peck, mile, month, degree? tinct symbol for each number apart from its number word.
Thus, for the number words
3. Number is a unit, or collection of like units.
one, two, three, four, five, six, seven, eight, nine,
When quantity is regarded as made up of like units, it
we use 1, 2, 3, 4, 5, 6, 7, 8, 9.
becomes a number. Thus, when an aggregate of apples is
These number symbols, or figures, have advantages as
regarded as made up of distinct apples, it becomes a number
of apples. compared with number words, in that they are easier to
write, and to recognize when written. They have many other
Thus, also, when a line is regarded as made up of inches,
it becomes a number of inches. derived advantages, when used in combinations, both in de-
noting and in operating with, numbers larger than nine.
4. Arithmetic is the science which treats of number. It
The number symbols 1, 2, 3, 4, 5, 6, 7, 8, 9 are called the
investigates the most advantageous ways of expressing quan-
nine digits. The absence of number is denoted by a sym-
tities as numbers, and of using numbers when formed.
bol, 0, called zero, naught, or cipher.
5. Number Words.—When we have determined a quan-
Zero is sometimes regarded as a number.
tity as made up of units, and ascertained the number of
8. Large Numbers.—In order to denote large numbers
the units in a given quantity, it is often useful to transfer
by words and symbols, it is necessary to devise a plan of so