G. A. (George Albert) Wentworth.

A grammar shcool arithmetic online

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A



GRAMMAR SCHOOL



ARITHMETIC.



BT



G. A. ;p3NTW0ETH, A.M.,

PBOFB880R OT MATHIBHATICS IN PHILLIPS EXSTBB ACADEMY.



BOSTON:

PUBLISHED BY QINN & COMPANY.

1886.



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Entered, Mjoording to Act of Congress, in the year 1835, by

\'\\: GEORGE A. WENTWORTH,
• ■ •

In^tbe Office of the Librarian of Congress, at Washington.



J. S. CnsHiNo & Co., Printbrs, Boston.



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PEEFAOE.



rriHIS Arithmetic is designed to give pupils of the grammar-school
age an intelligent knowledge of the subject and a moderate
power of independent thought.

Whether Arithmetic is studied for mental discipline or for practical
mastery over the every-day problems of common life, mechanical pro-
cesses and routine methods are of no lvalue. Pupils can ba trained
to logical habits of mind and stimulated to a high degree of intel-
lectual energy by solving problems adapted to their capacities.
They become practical arithmeticians, not by learning special
busiaess forms, but by founding their knowledge on reasoning
which they fully comprehend, and by being so thoroughly exer-
cised ia logical analysis that they are independent of arbitrary
rules.

The book contains a great number of well-graded and progres-
sive problems, made up for youths from ten to fourteen years of
age. Definitions and explanations are made as brief and simple
as possible. It is not intended that definitions shall be committed
to memory, but that they shall be simply discussed by teacher
and pupils. Every teacher, of course, will be at liberty to give
better definitions, and to make a better presentation of methods,
than those given in the book. In short, the chief object in view
will be gained if pupils are trained to solve the problems by neat
and intelligent methods, and are kept free from set rules and
formulas.

A great many number-problems are given in the first pages of
the book, so that the necessary facility and accuracy in computing



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iv PREFACE.



under the four fundamental rales may be acquired; as want of
accuracy and rapidity in mere calculations distracts the attention
which should be given to the investigation and correct statement
of clothed exercises. The pupil should be required to do only so
many of these number-problems as are found to be necessary to
give him facility and accuracy in the four fundamental operations.

The chapter on the Metric System is put near the end of the
book because many grammar-school pupils have no time for it,
while those who have time can as well learn the system at this stage
of their progress as earlier.

The chapter on Mensuration is suited to the ability of beginners.
The intention is not to give a system of Geometry, but to render
familiar the notions of geomftry that are indispensable for practical
purposes. The whole subject has been illustrated and enforced by
many practical examples.

The chapter on Miscellaneous Problems and Examination Papers
is intended as a review of the subject-matter of Arithmetic and as
a test of the learner's knowledge.

The author is under obligations to many teachers who have
given valuable suggestions and assistance in the preparation of
this work.

G. A. WENTWORTH.
Phillips Exetee Academy, Sept., 1885.



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CONTENTS.



PAGE

CHAPTER I. Notation and Numeeation .... 1

II. Addition 12

III. Subtraction 31

IV. Multiplication 44

V. Division 61

VI. Decimals 80

VII. Multiples and Measures .... 105

VIII. Common Fractions 119

IX. Compound Quantities 163

X. Percentage 211

XI. Interest and Discount 231

XII. Proportion 260

XIII. Powers and Roots . . . . . . 272

XIV. Metric System 284

XV. Mensuration. 294

XVI. Miscellaneous Problems .... 332
Examination Papers . . ., . .361



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VOCABULARY.



Abstract number. This phrase is employed to designate nuinbers
used without reference to any particular unit, as 8, 10, 21. But
all numbers are in themselves abstract whether the kind of thing
numbered is or is not mentioned.

Addition. The process of combining two or more numbers so as to
form a single number.

Aliquot part. A number which is contained an integral number of
times in a given number. Thus, 5, 6 J, 12}, 16 J, are aliquot parts
of 100.

Amount. The sum of two or more numbers. In Interest, the sum
of principal and interest.

Analysis. The separation of a question into parts, to be examined
each by itself.

Antecedent. The first of the two terms named in a ratio.

Area of a sorface. The ratio of the surface to another surface
assumed as the unit of measure ; usually the square of the linear
unit.

Arithmetic. The science that treats of numbers and the methods
of using them.

Assets. All the property belonging to an estate, individual, or cor-
poration.

Average. The mean of several unequal numbers, so that, if substi-
tuted for each, the aggregate would be the same.

Bank. An establishment for the custody, loaning, and exchange of
money ; and often for the issue of money.

Bank discount. An allowance received by a bank for the loan of
money, paid at the time of lending as interest on the sum lent.

Bonds. Written contracts under seal to pay specified sums of money
at specified times, issued by national governments, states, cities,
and other corporations.

Cancellation. The striking out of a common factor from the divi-
dend and divisor.



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viii yOCABULARY.

Commiasion. Compensation for the transaction of business, reck-
oned at some per cent of the money employed in the transaction.

Common denominator. A denominator common to two or more
fractions.

Common factor. A factor common to two or more numbers.

Common multiple. A multiple common to two or more numbers.

Complex fraction. A fraction that has a fraction in one or both of
its terms.

CompoEdte number. The product of two or more integral factors,
each factor being greater than unity.

Compound denominations. Several denominations used to express
parts of one quantity.

Compound interest. When the interest due is left unpaid, and con-
sidered as an increase made to the principal, the whole interest,
accruing in any time, is called compound interest.

Compound fraction. A fraction of another fraction.

Concrete number. A phrase used to denote numbers applied to
specified things ; as 6 horses, 8 desks.

Consequent. The second of the two terms named in a ratio.

Consignee. The person or firm to whom goods are sent.

Consignor. The person or firm who sends goods to another.

Corporation. An association of individuals authorized by law to
transact business as a single person.

Couplet. The two terms of a ratio taken together.

Coupon. A certificate of interest attached to a bond, to be cut off
when due and presented for payment.

Creditor. A person or firm to whom money is due.

Cube root. One of the three equal factors of a number.

Customs. Duties or taxes imposed by law on merchandise imported,
and sometimes on merchandise exported.

Debtor. A person who owes money to another.

Decimal fractions. Fractions of which only the numerators are
written, and the denominators are ten or some power of ten.

Decimal point. A dot placed after the units' figure to mark its place.

Decimal system. The common system of numbers founded on their
relations to ten, ten tens, etc.

Denominator. The number which shows into how many equal parts
a unit is divided.

Difference. The number which, added to a given number, makes a
sum equal to another given number.



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VOCABULARY. ix

Discount. Allowance made for the payment of money before it be-
comes due. Also, the amount wh^ch the market value is below
the face or nominal value.

Dividend. In division, the given number which is equal to the
product of a given factor (called divisor) and required factor
(called quotient). In business, the share of profits which belongs
to each owner of stock, on his proportion of the capital.

Division, The operation by which, when a product and one of its
factors are given, the other factor is found.

Divisor. The number by which a given dividend is to be divided.

Draft. A written order directing one person to pay a specified sum
of money to another.

Drawee of a draft. The person to whose order the sum of money
named in a draft is to be paid.

Drawer of a draft. The person who signs the draft.

Duty. A sum of money required by government to be paid on the
importation, exportation, or consumption of goods.

Equation. A statement that two expressions of number are equal.

Equation of payments. The finding of an average time at which
several payments may be justly made.

Exchange. A system of paying debts, due to persons living at a
distance, by transmitting drafts instead of money.

Exponent. A small figure placed at the right of a number to show
how many times the number is taken as a factor.

Extremes. The first and last terms of a proportion.

Evolution. The process of finding the root of a number.

Factors. The factors of a number are a set of numbers whose prod-
uct is the given number ; they are assumed to be integral, except
in the extraction of roots. In commerce, agents employed by
merchants to transact business.

Fig^es. Symbols used to represent numbers in the common system
of notation. Also diagrams used to represent geometrical forms.

Firm. The name under which a company transact business.

Fractions. One or more of the equal parts into which the unit is
divided.

Grace. An allowance of three days, after the date a note becomes
due, within which to pay the note.

Gram. The unit of weight in the metric system.

Greatest common measure. The greatest number which is a com-
mon factor of two or more given numbers.



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X VOCABULARY.

Improper fraction. A fraction whose numerator equals or exceeds
the denominator.

Index. A figure written at the left and above the radical sign to
show what root of the number under the radical sign is required.
A fraction written at the right of a number, of which the nume-
rator shows the required power of that number, and the denomi-
nator the required root of that power.

Instalment. A payment in part.

Insurance. A guarantee of a specified sum of money in the event
of loss of property by fire, storm at sea, or other disaster ; or of
loss of life.

Integral number. A number which denotes whole things.

Interest. The sum paid for the use of money.

Involution. The process of finding a power of a number.

Latitude of a point. The angle made by the vertical line at that
point with the plane of the equator.

Least common multiple. The least number which is a common
multiple of several given numbers.

Liability. A debt, or obligation to pay.

Line. Length without breadth or thickness. The path of a moving
point.

Liter. The unit of capacity in the metric system equal in volume
to a cube each edge of which is one-tenth of a meter.

Long division. The method of dividing in which the processes are
written in full.

Longitude of a point. The angle between two planes supposed to
pass through the centre of the earth and to contain, the one the
meridian of that point, and the other the standard meridian.

Loss. The excess of the cost price above the selling price.

Maturity of a note. The date at which a note legally becomes due.

Mean proportional. A number which is both the second and third
terms of a proportion.

Means. The terms of a proportion between the extremes.

Meter. The unit of length in the metric system.

Minuend. The given number in subtraction which is equal to the
sum of another given number called the subtrahend, and a
required number called the difference or remainder.

Mixed niunber. A number that expresses both entire things and
parts of things taken together.



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VOCABULARY. xi

MnlUple of a nninber. The product obtained by taking the given

number an integral number of times.
Multiplicand. The number to be multiplied by another.
Multiplication. The operation of finding a number bearing the

same ratio to the multiplicand which the multiplier bears to unity.
Multiplier. The number by which the multiplicand is multiplied.
Net proceeds. The amount that remains of the money received for

property after paying all expenses incurred in disposing of it.
Notation. A system of expressing numbers by symbols.
Note. A written agreement to pay a specified sum of money at a

specified time.
Number. The answer to the question, How many?
Numeration. A system of naming numbers.
Obligation. A debt, or liability to pay.
Order of units. A name used to designate the number of things

in a group, as tens, hundreds, thousands, etc.
Partial payment. Fart payment on a note.
Partnership. An association of two or more persons to carry on

business.
Par value. Face or nominal value.
Pendulum. A body suspended by a straight line from a fixed point,

and moving freely about that point as a centre.
Percentage. A part of any given number reckoned at some rate

per cent.
Period. A group of three figures.
Policy. A written contract of insurance.
Poll tax. A tax levied by the head or poll.
Power. The product of two or more equal factors.
Premium. The sum paid for insurance computed at some rate per

cent of the amount insured. Also the excess of market value

above par value.
Present worth. The present value of a debt due at some future day.
Prime number. A number which has no integral factors except

itself and one.
Principal. The sum of money drawing interest.
Problem. A question to be solved.
Product. The result obtained by multiplying th^ multiplicand by

the mul tiplier.
Profii The excess ^f selling price above cost.



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xii VOCABULAEY.

Proof The evidence by which the accuracy of any resuU is estab-
lished.

Proper fraction. A fraction, the numerator of which is less than
the denominator.

Proportion. A statement that two ratios are equal.

Quantity. The answer to the question, How much ?

Quotient. The number sought in division.

Rate per cent. Rate by the hundred.

Ratio. The relative magnitude of two numbers or of two quantities.

Reciprocal of a number. One divided by that number.

Reduction. The process of changing the unit in which a quantity
is expressed without changing the valiLe of the quantity.

Remainder. The number which, added to the subtrahend, gives a
sum equal to the minuend.

Root of a number. One of the equal factors of the number.

Rule. The statement of a prescribed method.

Security. Property used to guarantee the payment of any debt.

Share. One of a certain number of equal parts into which the capi-
tal of a company is divided.

Short division. The method of dividing in which the operations of
multiplying and subtracting are performed mentally.

Solid. A magnitude which has length, breadth, and thickness.

Solution. The process by which the answer to a question is obtained.

Specific gravity of a substance. The ratio of the weight of a
given volume of it to that of an equal volume of water.

Square root. One of two equal factors.

Stock. Capital invested in business.

Subtraction. The process of finding a number which added to one
of two given numbers will produce the other.

Sum. The number which results from combining two or more num-
bers together.

Surd. An indicated root the value of which cannot be exactly ex-
pressed in figures.

Surface. That which has only length and breadth.

Thermometer. An instrument for measuring heat.

Unit. A single thing. Also, an arbitrary length, adopted as a stan-
dard of measure, in terms of which all measurements are expressed.

Verify. To establish, by experiment, the truth of any statement.

Volume of a solid. The ratio of a solid to an assumed unit of
measure ; usually a cube of the linear unit.



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A
GRAMMAR SCHOOL ARITHMETIC.



CHAPTER I.
PRELIMINARY DEFINITIONS.

1. A COLLECTION of several similar objects (as a collection
of apples) or the repetition of the same event (as successive
peals of thunder) gives the idea of Kumber.

2. The idea of number presents itself also when we wish
to express the values of quantities in terms of some well-
known value.

3. A Unit is a fixed value with which we compare all
quantities of the same kind. Each kind of quantity has
its own unit. Thus :

The unit of length is the yard.
The unit of surface is the square yard.
The unit of capacity is the quart.
The unit of weight is the pound.
The unit of money is the dollar.

4 To measure a quantity is to find the number of times
the quantity contains its unit.

5. Number results from measuring a quantity. If the
unit is contained in a quantity several times without
remainder, the result is an Litegral Number. Thus, th«
integral number three will represent the length of a line,
if the line contains the yard, three times exactly.



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NOTATION AND NUMERATION.



6. If the quantity to be measured is less than the unit,
we divide the unit into equal parts and find how many
times one of these parts is contained in the given quantity.
Thus, to measure a line less than a yard, we can apply to
this line a third part of a yard, and if this third is con-
tained twice in the line exactly, the length of the line is
expressed by two-thirds. The expression two-thirds is called
a Fraction.

7. If a line contains the yard five times and one-fourth
of a yard three times, the length of the line is expressed
by five and three-fourths. The expression five and three-
fourths is called a Mixed Kurnber.

8. Arithmetio comprises all questions that can be pro-
posed upon numbers.



NOTATION AND NUMERATION.

9. The first numbers have special names, as follows :
one, two, three, four, five, six, seven, eight, nine, ten.

10. The first nine of these numbers are called Simple
TInitSi or units of the first order.

11. The group of ten units has received the name of a
ten, or a unit of the second order; and we count by tens as
by units ; thus :

one ten, two tens, three tens ... nine tens, ten tens.

12. The group of ten tens has received the name of a
Hundred, or a unit of the third order; and we count by
hundreds, as by tens and units ; thus :

one hundred, two hundreds ...ten hundreds.



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NOTATION AND NUMERATION.



3



13. A group of ten hundreds is called a ThonBaad, or a
unit of the fourth order.

14. From ten units of the fourth order is formed a ten
thxmsand, or a unit of the fifth order; and from ten units of
the fifth order is formed a hundred thousand^ or a unit of
the sixth order,

15. Units of the seventh order are called Millioxui ; of the
eighth order, ten millions; of the ninth order, hundred
millions. Finally, units of the tenth order are called
Billions; units of the thirteenth order, Trillions; and so on.

16. The table of units of different orders is as follows :



First order,
Second order,
Third order.
Fourth order.
Fifth order,
Sixth order,
Seventh order.
Eighth order.
Ninth order,
Tenth order,
Eleventh order,
Twelfth order,
Thirteenth order.



simple uniiSy

tens of units,

hundreds of units,

tkousaTids,

tens of thousands,

hundreds of thousands,

millions,

tens of millions,

hundreds of millions,

billions,

tens of biUions,

hundreds of billions,

trillions,



first class.



second class.



third class.



fourth class.



fifth class.



17. The group of the first three orders is called the first
class of units, and the group of the three following orders,
the second class, and so on.

18. The unit of the second class is equal to a thousand
units of the first class, and a unit of the third class is equal
to a thousand units of the second class, and so on.



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NOTATION AND NUMERATION.



19. To read a number we decompose it into units of the
different orders, and state how many groups there are of
each kind, commencing with the highest order. Thus, for
example, two millions, three thousands, five hundreds,
seven tens, and four units.

20. It is clear that the names of all numbers up to a
billion are formed by combining the names of the first nine
numbers with the words ten, hundred, thousand, million.

21. Usage sanctions the following irregularities :

I. Instead of saying two tens, three tens, four tens, five
tens, six tens, seven tens, eight tens, nine tens, we say
twenty, thirty, forty, fifty, sixty, seventy, eighty, ninety.

II. The names of the numbers between ten and twenty
are eleven, twelve, thirteen, fourteen, fifteen, sixteen,
seventeen, eighteen, nineteen.

22. The names of the numbers between twenty and a
hundred are :

twenty-one, twenty-two, twenty-three ... twenty-nine,
thirty-one, thirty-two, thirty-three ... thirty-nine,

ninety-one, ninety-two, ninety-three ... ninety-nine.

23. The names of the numbers between a hundred and
a thousand are :

hundred one, hundred two ... hundred ninety-nine,
two hundred one ... two hundred ninety-nine,

nine hundred one ... nine hundred ninety-nine.

24. The common system of notation employs ten figures
or digits :

1, 2, 3, 4, 6, 6, 7, 8, 9, 0.



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NOTATION AND NUMERATION.



The first nine of these figures represent the first nine num-
bers ; the last, which is called Zero, Naught, or Cipher, is
used to denote the absence of units of the order in which
it stands. It is possible to express all numbers by these


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Online LibraryG. A. (George Albert) WentworthA grammar shcool arithmetic → online text (page 1 of 21)