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EATON & FRaZEE'S
COMMERCIAL ARITHMETIC
FOB
|h-. ■ ., ■
SCHOOLS, ACADEMIES, COLLEGES,
BAHKERS, MERCKIIITS imD MECHimiCS,
KMBBACINO . , J ^ V
AN EXTbNSIYE COURSE OF THEORY AND PRACTICE.
>
COMPILED B\
T. A. BRYCE, M. A., LL. D.
BEYISED AND COBREOTED BY
%
. ^K.iv-
A. H. EATON,
PBINCIPAL OP THE COMMERCIAL COLLEGK,
, / ST. JOHN, N. B.» ^
"■ ' -. *"■ ; AND .:•■ f ■•'■'" ^
■.;'i^/.
J. C. P. FRAZEE,
PRINCIPAL OF THE COMMERCIAL COLLEGE,
HALIFAX, N. S.
.."tff..-,K-- V
iir
.fi -f ArT? •
BT. JOHX. N. B.. AKD HALIFAX. K. 8..
PUBLISHED BY EATON & FRAZEK.
1874.
REVISOES' PREFACE.
«♦>
This Work, originally compiled by T. A. Brycf, M. A.,
LL. D., has undergone extensive alterations in the hands
of the proprietors.
In order to make it more useful as a Text Book in the
Schools and Acadevnies of these Provinces, where it is already
extensively used, it has been entirely re-written as far as the
Exercises in Simple Proportion. The Explanatioi-S have
been shortened, and put in such a shape as to be more easily
understood by the unassisted student. The Exercises are
also nearly all new, and are of such a nature as, it is hoped,
will more speedily conduce to the advancement of those into
whose hands the work may be placed.
The articles on Equation of Payments and Averaging
Accounts have also been re-written, and these subjects are
handled in a manner that cannot fail to ,be appreciated by
Teachers, Accountants, and Mercantile men.
The remaining partF of the work have merely been cor-
rected where it was found necessary.
4,? ' •; ;i ,,f- ^ :■
X^i
n
\\
PREFACE.
THOTTOn elementary works on Arithmetic are in abundance, yel
it seems desirable that there should bo added to this an extensive
treatise on the commercial rales, and commercial laws and usages.
It is not enough that the school-boy should be provided with o
course suited to his age. There must bo supplied to him something
higher as ho advances in age and progress, and ncars the period
when ho is to enter on real business life.
The Author's aim has, therefore, been to combine these two
objects, and to produce a work adequate to carry the learner from
the very elements up to the highest rules required by those preparing
for business. As the work proceeded, it was found necessary t"*
exf'nd the original programme considerably, and, therefore, also
the limits of the book, so as to make it useful to all classes in the
community.
In carrying out this plan, much care has been taken to unfold
tho theory of Arithmetic as a science in as concise a manner aa
seemed consistent with clearness, and at the same time to show its
applications as an Art. Every effort has been made to render tho
business part 30 copious and practical as to afford the young student
ample information and discipline in all the principles and usages of
commercial intercourse. For the same reason some articles on
Commercial Law have been introduced, as it was a prominent part
of the Author's aim to produce a work which should be found useful,
not only in the class-room, and the learner's study, but also on the
merchant's table, and the accountant's desk. The Author begs to
tender his best thanks to J. Smith Homans, Esq., New York City,
Editor and Proprietor of the "Banker's Magazine and Statistical
Register," for the able manner in which he supplied this part of the
work.
Throughout the work particular care has been taken not to
enunciate any rule without explaining the reason of the operation,
for, without a knowledge of the principle, the operator is a mere
•calculating machine that can work but z certain round, and is almost
«are to be at fault when any novel case arises. The ezplanations
PREFACE.
are, of course, more or less tho result of reading, but, nevertheless,
they aro mainly derived from personal study and experience in
teaching. The great mass of the exercises arc likewise entirely new.
though the Author hus not scrupled to make selections from some el
the most approved works on tho subject ; but in doing so, he haa
confined himself almost entirely to such questions as are to he fount?
in nearly all popular books, and which, therefore, are to bo looked
upon as the common property of science.
Algebraic forms have been avoided as much as possible, as being
unsuited to a largo proportion of those for whom tho hookas in-
tended, and to many altogether unintelligible, and besides, those
who understand Algebraic modes will have all tho less difficulty in
understanding tho Arithmetical ones. Even in the more purely
mathematical parts the subject has been popularized as much as
possible.
In arranging the subjects it was necessary to follow a certain
logical order, but tho intelligent teacher aad learner will often find
it necessary to depart from that order. (See suggestions to teachers.)
Every one will admit that rules and definitions should be ex-
pressed in the smallest possible number of \words, consistent
with perspicuity and accuracy. Great pains have been taken to
carry out this principle in every case. Indeed, it might be desirable,
if practicable, not to enunciate any rules, but simply to illustrate
each case by a few examples, and leave the learner to take the prin-
ciple into his mind, as his rule, without the encumbrance of words.
Copious exercises are appended to each rule, and especially to-
the most important, such as Fractions, Analysis, Fercentage, with
its applications, &c. Besides these, there have been introduced
extensive collections of mixed exercises throughout the body of th&
woirk, besides a lai^e number at the end. The utility of such
miscellaneous questions will be readily admitted by all, but the
reason why ^ey are of so much importance seems strangely over-
looked or misunderstood even by writers on the subject. They are
spoken of as review exercises, but their great value depends on some*
thing still more important. An illustration will best serve here.
A class is working questions on a certain rule, and each member
of the class has jvst heard the rule enunciated and explained, and
therefore readily applies it. So far one important object is attained,
jAm., freedom of operation. But something more is necessary. The
fuefage.
't.
'learner nmst be taught to disoem what rule is to be applied for the
solution of each question proposed. The pupil, under careful teach'
ing, may bo able to understand fully every rule, ard never con-
found any one wLu any other, and yet be doubtful what rule is to
bo applied to an individual case. Tho iniDcollaneous problems,
therefore, are intended not so much as exercises on tho ojyerationt
of tho d-lferent rules as on tho mode of applying these rules ; or, in
other words, to practice the pupil^ in perceiving of what rule any
proposed question is a particular case. Great importance should be
attached to this by the practical educator, not only as regards readi^
ness in real business, but also as a mental exercise to the young
student. ■i''. » -. v^- ; t ■.■i/,.: .
Tho Author is far from supposing, much less asserting, that tho
work is complete, especially us the whole has been prepared in less
than the short space of six months. It is presented, however, to tho
public in the confident expectation that it will meet, in a great
degree at least, the necessities of the times. With this view, there
are given OAtensivo collections of examples and exercises, involving
money in dollars and cents, with, however, a number in pounds,
shillings and pence, sufficient for tho purpose of illustration. This
seems necessary, as many must have mercantile transactions with
Britain and British America.
The Rule for finding the Greatest Common Measure, though
not new, is given in a new, and it is hoped, a concise and convenient
form of operation.
The llule for finding tho Cube B«ot is a modification of that
given by Dr. Hinds, and will be found ready and short.
In treating of Common Fractions, Multiplication and Division
have been placed before Addition and Subtraction, for two reasons.
First, — In Common Fractions, Multiplication and Division present
much less difficulty than Addition and Subtraction ; and, secondly,
as in Whole Numbers Addition is the Bule that regulates all others ;
50 in Fractions, which originate from Division, we see, in like
manner, ':hat all other operations result from Division, and, in con-
nection with it, Multiplication.
Several subjects, commonly treated of in works on Arithmetic,
have 1 3i;n omitted in order to leave space for more important matter
bearing on commercial subjects. Duodecimals, for example, have
been omitted, as that mode of calculation i» now virtually superseded
?i.
PREFACE.
by that of Dooimals. Barter, too, has been passed by, as qnestion?
of that olass can easily be solved by the Rule of Proportion, i»I:ich
has been fully explained.
The sabjoot of Analysis has been gone into at considerable
length, and it is hoped that the new manner in which tho oxplana*
tions and solutions are presented, and the extensive collection of
exercises appended, will contribute to make this a valuable part of
the treatise.
The view given of Decimal Fractions seems the only true one,
and calculated to give the student clear notions regarding tho nature
of the notation, as a simple extension of the common Arabic system,
and also appropriate to show the convenience and utility of Decimals.
The distinction beeween Dedrnvah and Decimal Fractions has been
ignored as being " A distinction without a difference." D-cimah is
merely a short way of writing Decimal Fractions ; thus, .7 is merely
a convenient mode of writing /g. These differ in form only, but
otiherwise are as perfectly identical as f and g.
The contracted methods of Multiplication and Division will be
found, after soiiio practice, extremely useful and expeditious in
Decimals expressed by long lines of figures.
The averaging of Accounts and Equations of Payments, Gash
Balance and Partnership Settlements, have been introduced as
essential parts of a oommeroiai eduo&tion, and, it is hoped, will form
a most important and usefUl study for those preparing for business^,
and probably a safe guide to many in business who have not sys-
tematically studied the subject
■'X.
';''*.'»^
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SUGGESTIONS TO TEACHERS.
'..i
.')•.:(•
Ths author would first refer to the remark made in the Prsfaoe
that he does not expect that the Teacher will follow the logioal
order adopted '.q the ]ix)ok, and oven advises that ho should not do
tui in many cases. He knows by ozperionco that the same order
does not suit all students any more than the samo medical treat-
ment suits all patients. The ooure requires to bo varied according
to age, ability and acquirements. The greatest difficulties generally
present themselves at the earliest stages. What more serious diffi-
culty, for example, has a child to encounter than the learning of the
alphabet ? Though this is perhaps the extreme case, yet others will
be found to be in proportion. For b^inners, therefore, wo recom-
mend the following course.
Let the elementary rules be carefully explained and illustrated by
simple examples, and the pupil shown how to work easy exercises ;
this done, let the whole be reviewed, rud exercises of a more difficult
kind pro^K)8ed. The decimal coinage should then be taken up. In ex-
plaining this part of the subject the teacher ought to notice carefully
that the operations in this case di£fer in no way from ^hose already
j(one through in reference to whole numbers, except in the preserv^'ng
of the mark that separates the cents from the dollars, usually called
the deoinal point. The next step ought to be the whole subject of
denominate numbers, and in illustration and application, the rule of
practice. After a thorough review of all the ground now gone over,
Simple Proportion may be entered upon, using such questions as do
not involve Fractions. Then, after a course of Fractions has been
gone through, Proportion should be reviewed, and questions which
involve Fractions proposed. After this it will generally be found de-
arable to study Percentage, with its applications.
The order in which the rest of the course shall be taken is com-
paratively unimportant, as the student has now realized a capital on
which he can draw upon for any purpose. The author would, in the
strongest manner possible, impress on thv minds of teachers the great
ut"'.ty of frequent reviews, and especiallv of constant ezeroise in the
addition of money oolomns
T
viii.
RUOOE8TION8 TO TEACnERS.
To make the oxoroiscs under each rule of progressive diffioulty, as
far 08 poflsiblo, has boon an object kopt oonstantlj in view, as also to give
caeh oxoreisc the semblance of a real question, for all persons, ospceially
the young, take greater interest in exercises that assume the form of
reality than in such as are merely abstruot ; and, besides, this is a
preparatory exercise to the application of the rules afterwards. At
every stage the greatest care should be taken that the leamor
thoroughly understands the meaning of each rulq, and the conditions
of each question and the terms in which it is expressed, before ho
attempts to solve it.
The Teacher should not always be talking or working on tho
black-board ; ho should require the pupils to speak a good deal in
answer to questions, and also work much on their slates, and each in
his turn on tho board for illustration to the rest. '.^t :^i\ ><>
Finally, it is suggested to every Teacher to keep constantly be-
fore his mind both of tho two chief works he has to accoiuplish.
Fir it, tho dovolopomont of tho mental powers of his pupil ; and,
secondly, imparting to him such knowledge ad he will require to
use when he enters upon life, either as a professional man, or a mer-
chant or clerk. Some seem to consider these two objects incompati-
ble, as if taking up time in mental training left insufficient time for
tho imparting of actual knowledge. This is a palpable errror, for the
more the mental powers are cultivated, the more readily and rapidly
will any species of knowledge bo apprehended, and the more surely,
too, will it be retained when it has been mastered. Mental culture
is at once the foundation and the means ; the other is the super-
structure raised on that foundation and by that means ; or it may
be compared to a great capital judiciously embarked in trade, and
often turned, and therefore yielding good profits. It frequently
happens, however, from the peculiar circumstances of individuals and
families, and even communities, that young men require to be hurried
into business, so as to be able to support themselves ; but even in
such cases the desired object will be much more readily and securely
attained by such a course than by what is usually and not inappro-
priately called " Cramming." Every effort has been made to give to
this book the character here recoiunended, especially in the explana*
torypartfl. ^vw^ a
•"Ml-
i '- >i ,
i'lAi. *%•* **iv^J''
SUGGESTIONS TO COMMERCIAL STUDENTS.
ii ,
Tbk foregoing suggestions are addressed dircetly to the Teacher,
but a careful eonsidoration of thoui by the Student will, it is hoped,
be found highly profitable. A few additional hints are Hubjoined
for the benetit of those seeking a liberal and praetical oommeroial
education.
As in all branches, so in Arithmetic, it is of the utmost conse-
quence to digest the rules of the art thoroughly, and storo them in
the memory, to be reproduced when required, and applied with
acouroey. But this is not enough ; something more is needed by
the Student. To be an eminent accountant he roust acquire rapidity
of operation. Accuracy, it is true, should be attained first, especially
as it is the direct means of arriving at readiness and rapidity. Acou<-
racy may be called the foundation, readiness and rapidity the two
wings of the superstructure. Either of these acquirements is indeed
valuable in itself, but it is the combination of them that constitutes
real efieotive skill, and makes the possessor relied upon, and looked
up to in mercantile circles. Some one may ask, " How are these to
be acquired? " The answer is as simple as it is undeniably true;
only by extentive practice, not in the counting-house or warehouse,
indeed, though these will improve and mature them, but in the
school and college, so that you may take them with you to the busi-
ness office when you go to your first day's duty. Oo prepared is a
maxim that all intelligent business men will affirm. Be so prepared
thaC you will not keep your customers waiting restlessly in your
office or warehouse while you are puzzling through the account you are
to render to him, but strive rather to surprise him by having your
bill ready so soon.
Another important help to the attaining of this rapidity, as no*
tieed in the note at foot of page 18, is not to use the tongue in oalcu-
lating but the eye and the mind.
Nor should the course of self-disoipline end here. ?^o be an ex-
pert aocountant even, is but or' part, though an important one of a
qualification for business. Study Oommeroial geography— oommer^
oial and international relations— politioal eoonomy — iadSs, &o., &o.
mm
Z.
SUGGESTIONS TO OOMMEBCIAL STUDETS.
Study even politics, not for their own sake but on account of the
manner in which they affect trade and commerce.
Do not, except in the case, of some aerious di£Boulty, indulge in
the indolent habit of asking your teacher or fellow student to work
the question for you ; work it out yourself ^rely upon your self, and
aim at the freedom and correctness which 'U give you confidence in
yourself, or rather in your powers and acqairements. Another can*
tion will not be out of place. Many students follow the practice of
keeping the text book beside them to see what the answo^ is; this
has th'^ same effect as a leading question in an examination, being a
ffsxde to tho mode by seeing the result. Study and use the mode to
come at tho result ; gain that knowled^i,^ of principles and correotnese
of operation that will inspire the confidence that your answer is cor-
rect without knowing what answer the text bouk or the teacher may
assign to it.
Th'iro are t-;70 things of such constant ooonrrenoe and requiring
such extreme accuracy that they must be specially mentioned, — they
are tho addition of money columns &nd the making of Bills of Par*
eels. Too muc^ care and practice can scarcely be bestowed on these.
'/, ;,; ik.
TABLE OF CONTENTS. '''''
•••
h*
bi.
Arithmetic, ... ^•
Nunieraticn, ... -
Notation,
Addition,
Subtraction,
Multiplication, ...
Division, ... .. >*•¥'
Canada Currency,
Properties of Numbers,
Greatest Conimon Divisor,
Least Comnaon Multiple,
Fractions,
Beductiou of Fractions,
Multiplication of Fractions,
Division of Fjactions,
Least Common Denominator,
Addition of Fractions,
Subtraction of Fractions,
Decimal Fractions,
Addition and Subtraction of Docimnls,
Multiplication of Decimals,
Division of Decimals,
Denominate Numbers,
Tables of Weights and Measures.
Canada and U.S. Money,
Avoirdupois,
Produce,
Long or Linear, „,
Square Measure, " ..;
Land Measure, . «..
Cubic or Solid, - ....
Measure of Capacity,
Liquid and Dry,
X^UXci ••• St* •••
Angular and Circular,
Miscellaneous, ... ...
Metric, ... ..;
Lo^ Measure,
Surfaces, ... '" :.;<#!. :'
Weights,
Capacity,
Beduction of Denominate Nv js, ...
Addition "
Subtraction, " "
Multiplication "
Division "
The Ctatal,
Longitude and Time,
Batio and Proportion,
Simple Proportion,
Oompoond rroportion,
Page.
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lOL
^■H^pmnmi
zu.
INDEX.
Analysis and SynthesiB,
Fntcttce,
Accounts and Invoices,
Percentage,
Interest,
Simple •••
Commercial Paper,
Partial Payments,
C!ompound Interest,
Discount and Present Worth,
Banks and Banking,
Bank Discount,
Commission, ... .
Brokerage, ... ;- ... '
Insurance, ... ...
Life ; '. ...
Profit and Loss, " ..♦
Storage, ... ...
General Average,
Taxes and Customs Duties,
Stocks and Bonds,
Partnership, • ..
Bankruptcy, .t
Equation of Payments,
Averaging Accounts,
Cash,Balance,
Account of Solos,
Alligation, ... ..
Medial
Alternate
Money-rlts Natui.^ and Value,
Paper Currency,
Exchange,
American
Sterling
Arbitration
Involution,
Evolution,
Square Boot, .. ..
Cube Root, . . . .
Progression, . . ^ .
by a common difference,
by Batio, . .
Annuities,
Partnership Settlements,
Conamercial Questions,
Mensuration,
of Solids,
Piling of Balls and Shells,
Mensuration of Timber,
Miscellaneous Exercises,
Foreign Qold Coins,
Silver Coins,
■■''■
Fxes.
1
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• • t
• ft
Ill
i
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• ••
• ••
119
ll
•t*
• ••
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122
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•w ■ l'
ARITHMETIC. '^
•."V::r:.;.^'
Article. 1. — Quantity is anything which can be increased
or diminished. Thus, numbers, lines, space, time, motion,
and weipfht are quantities.
Art. 2. — Mathematics is the science of quantity.
Art. 3. — The fundamental branches of Mathematics are
Arithmetic, Algebra and Geometry.
Art. 4. — Arithmetic is the science of niuubers, and the art
of computing by them.
Art. 5. — A Number is a unit or a collection of units.
Art. 6. — A Unit is a single thing, or one.
Art. 7. — A Problem is a question proposed for solution.
• Art. 8. — A Theorem is a truth to be proved.
f Art. 9. — A Demonstration is a process of reasoning by
which a proposition is shown to be true.
Art. 10. — An Axiom is a self-evident truth: that is, a
proposition so evident that it cannot be made plainer by any
demonstration ; as, — the whole is greater than a part.
AiiT.^ 11. — Arithmetic is founded on Notation, and its oper-
ations are carried on by means of Addition, Subtraction, Mul-
tiplication and Division. These are the fundamental rules
of Arithmetic.
Art. 1 2. — ^Notation is the art of representing numbers, by
figures, letters or other characters.
Art. 13. — There are two systems of Notation in coTimon
use, — the Arabic and the Koman.
Art. 14. — The Arabic system (made known through the
Arabs) represents numbers by ten characters or symbols, called
figures, viz : — 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. The first nine are
called significant figures, because they indicate some value.
The cipher, 0, when alone, indicates no value.
Art. 15. — Every number is represented by one of these
figures, or a combination of them.
AR,t. 16. — ^The Boman system of Notation represents num-
bers by letters. The letter I represents one; V, five; X^
ten ; L, fifty ; G, one hundred ; D, five hundred, and M, one
thousand.
Art. 17. — The other numbers are represented according^
to the following principles : — 1. Every time a letter is re-
peated its value is repeated, thus,: — ^11, denotes two; XX»
14
ARITHMETIC.
twenty, &c. 2. When a letter of less value is placed before
one of greater value, the less is taken from the greater.
Thus, IV denotes four, while VI denotes six ; IX denotes ninO;
while XI denotes eleven, &c. A bar, — , placed over a letter
multiplies its value by one thousand; thus, v denotes five
thousand.
'';•::>
NUMERATION.
Art. 18. — Numeration is the art of reading numbers ex-
pressed by figures, or letters.
Art. 19. — There are two methods of Numeration, the
French and the English.
Art. 20. — The French method is almost universally used.
It separates the figures into groups of three figures each,
<!alled periods, with a distinct name to each period.
ii
o^
III
r>th Foriod.
Trillions.
00
FRKirCH NUMEBATION TABLE.
I ^
pq I
•73 .
' — r^
4th Period.
BUlioiu.
/^-.'-isV^
3rd Period.
Millions.
2nd Period.
Thousands.
-3
WEiP
Ist Period.
Units.
The periods above Trillions are Quadrillions, Quintillions,
Sextillions, Septillions, Octillions, Nonillions, Decillions,
Undecillions, Duodecillions, Tredecillions, ^c.
Art. 21. — In the English method, which is seldom used,
the figures are separated into periods of six places each. The
first period is regarded as units and thousands of units ; the
sucond, as millions and thousands of millions ; the third, as
billions and thousands of billions, and so on.
Art. 22. — Rule fof. Numeration. — Begin at the rightj
and point off in periods of three jlgv/rea each; then, begin
at the left and read vn aucceaaion eachperiod ivith ita name.
NOTATION.
15
EXERCISES IN NUMEBATIOir.
Example.— 36827 1 927. Read thus :—
^'2
dp *» S
I
•^ -S 1 1
• V 3 6 8, 2
125. 58763.
372. 86552.
864. 155731.
1076. 196472.
1884. 251103.
2750. 564989.
5890. 2285432.
9759. 2711511.
10864. 5318754.
17651. 9871832.
42414. 11867438.
'f'S
?0 H
1,9
2 7
25643287.
/ 87418389.
234656431.
761118445.
• 4519876314.
37965432819.
98740811087.
880195038604.
9108630106543.
86419038765789.
386480967318640.
..>■' •v.-
- NOTATION.
Art. 23. — Rule for Notation. — Write first the figures
'of tJie highest period, then of the other periods in their
proper succession, filling vacant places with ciphers.
Note.— Eveiy period (except sometimes the highest) must have three
figures, and if any period is omitted in the given number, its place must
be supplied with three ciphers.
BXEBCI8E8 IN NOTATION. '
Write in figures the following ^umbers i—
1. Forty-six thousand, seven hundred and one. i
2. Six thousand, six hundred and sixty.
3. Eight hundred and eighty-eight thousand, eight hun-
dred and eighty-nine.
4. Eight hundred and eighty-eight thousand, eight hun-
dred and nine.
5. Eight hundred thousand and nine.
6. Ten millions, ten thousand and ten.
7. Ten millions and ten.
8. Ninety millions, nine thousand and ninety.
16
ARITHMETIC.
9. Ninety millions, nine hundred and nine.
10. Seven hundred and seventy billions, tive thousand and
seven.
11. Eleven millions and eleven. ~
12. Eleven billions, eleven millions, one himdred and
eleven. ,
13. Two trillions, thirty millions and thirty.
14. Nine quadrillions, twenty trillions, five hundred bil-
lions, two hundred millions, three thousand and thirty-three.
MATHEMATICAL SIGNS.
Art. 24. — For the sake of brevity, characters called Signs
v^e used in Mathematics. Those most commm in Arith-
metic are +j — , X, -T-, =.
Art. 25. — The + , called plus,* is the sign of Addition^
and denotes that the numbers between which it is placed are
to be added together. Thus 4+5 equals 9 ; 3 + 1+4 equals 8.
Art. 26. — The sign — , called minus* or less, is the sigii
of Subtraction, and denotes that the number which follow^
it is to be subtracted from that which precedes it. Thus
7 — 3 equals 4.
Art. 27. — The sign x , read multiplied by, is the sign of
Multiplication, and denotes that the niunbers between which
it is placed are to be multiplied together. Thus 5x5 equals 25.
Art. 28. — The sign -^, read- divided by, is the sign of
Division, and denotes that the number which precedes it is
to be divided by that which follows it. Thus 30-^5 equals 6.
Division is also frequently represented thus ^ meaning
that 30 is to be divided by 5.
Art. 29. — ^The sign =, is the sign of equality, and de-
notes that the quantities between which it is placed are
equal. Thus 3+5=8; 3x5=15.
ADDITION.
Art. 30. — ^Addition is 'the process of combining two or
more niunbers into one, which shall be equal to the whole.
The result is called the sum. Thus 13 is the sum of 5 and 8.
* Plus and fniii«« are two Latin words signifying more and loss.
ADDITION.
17
Art. 31. — Numbers to be added must be of the same kind,
or such as may be brought under the same denomination,
and the figures of the same order of units. Thus 9 horses
and 4 horses can be added because tliey are of the same kind ;
but 9 horses and $4 cannot be added, because they are not of
the same kind. Again, 3 cows and 4 sheep cannot be added
to make 7 cows or 7 'sheep, but they may be brought under
the same denomination and added to make seven animals.
What is the sum of 82, 543 and 639 ?
Solution. — Having arranged the numbers as in 82
the margin, placing units under units, tens under 543
tens, &c., because the figures to be added must be 639
of the same order of units, say 9 and 3 are 12, and
2 are 14 imits ; 1 ten and 4 units, write the 4 units 1264
beneath, and add the 1 ten with the column of tens. ^
Then one and 3 are 4, and 4 are 8, and 8 are 16 t?ns, that
is 6 tens to be written beneatli, and 1 hundred to be added
to the column of hundreds. Lastly, 1 and 6 are 7 and 5 are
12 hundred, that is 2 hundreds to bo written under the column
of hundreds, and 1 thousand to occupy the thousand's place.
BULE FOR ADDINO SISIPLE NUMBERS.
Write the numbers to he added so tliat figures of the same
order may stand in column^— units under units, tens
under tens, hundreds under hundreds, &c. Begin at the
right, and add each coluimn separately, placing the units
of e^^h sum under the column added, and adding the tens
with the next column. At the last column, set drn'm the comf
plete sum.
TsooF. — Begin at the top and add the figures downwards.
If the result is the same the work is probably correct.
EXERCISES. -J
Find the sum
of the following quantities:-
— .
^^}
. (2)
(3)
(4)
895763
99876
49176
987654321
63879
89765324
283527
123456789
54387
42356798
659845
908760504
789
56798423
7984
890705063
137568
23567989
31659
759086391
278652
79842356
968438
670998767
85945
65324897
2896392
4340661835
721096
357655787
2
1
•
\
•
-
i
18
ARITHMEIIO.
•
1 *
(6)
(')
738
1
78563
659
471
897
47986
12345
•
658
5798
67890
918273
856
19843
98765
651928
789
56479
43219
374859
978
28795
87654
263748
654
Zr. 897
32169
597485
999
: 1984
78912
986879
888
68195
65439
98765
777
3879
98765
9876
666
* 698
43288
987
555
k
5879
77877
456879
897
17985
98989
345678
978
% 336981
805312
4705357
12460
(9)
(10)
(11)
(12)
189
1298
976
98
47
764
85
89
96
5837
73
76
88
6495
338
67 •
59
789
793
281
74
638
49
592
82
546
75
678
97
98
218
58
68
475
1 365
67
75
394
> 113
98
49
89
279
149
76
157
^•;-v ^^
67
.54
638
^ 76
54
78
594
' 84
72
69
789
1 1379
1 ^1^9
298
2744
87
1044
114
19715
SUDTBACTION.
19
SUBTRACTION.
t
Art. 32. — SUBTBACTION is the process of finding the dif-
ference lietwoen two numbers.
The hirj^er number is culled the minuend; the less, the
subtrahend; the result, the difference^ or remainder.
Subtraction is the converse of Addition, and since none
but numbers of the same kind can be added, it follows that
unless the minuend and subtrahend be of the b ne kind
Subtraction cannot be performed.
Subtract 3742 from 8396.
SoLUTioN.-^Haviug arranged the numbers as in 8396
the margin, Ixjgin at the right and say 2 from 6 3742
leaves 4 ; 4 from 9 leaves 5 ; 7 from 3 you can not,
bon'ow 1 from 8, which, beiAg in the fourth place, 4654
is 1 thousand, and, considered as hundreds, is 10,
add the 10 to 3 = 13, then 7 from 13 leaves 6; 3 from 7 and
4, or, add 1 to 3 is 4, 4 from 8 leaves 4.
BULB FOR STTBTRAOTIOir.
Write the leas number under the greater, placing units
t{.nder units, tens under tens, &c. ; begin at tlie right, sub~
troAit each figure from, tlie one above it,placing the remainder
beneath. If any figure in tlie subtrahend exceeds the one
above it, add ten, to the upper oTie, subtract the lower from,
tlte sum, and eWier diminish tJie next figure in the mi/nur'
end by one, or increase the next in tlie svijbtrahend by one
aa you proceed.
^ ' EXEBCISES.
(1) (2) (3)
Minuend, 847639021 1010305061 59638743
Subtrahend, 476584359 670685093 18796854
Difference, 371054662 339619968 40841889
4. From 7813257 take 3745679= 4067578.
5.
6.
7.
8.
9.
10.
((
((
111111111 «
8420163 «
910486 "
350000147 "
7130426 «
33333333 "
98657293=12453818.
78590= 8341573.
91364= 819122.
250197319=99802828.
9999= 7120427.
23333334= 9999999.
MIXED EXERCISES IN THE APPLICATION OP ADDITION
AND SUBTRACTION.
1. I have in my desk $43 ; my neighbor, A, owes me $8 ;
so
ARITHMETIC.
B,owes rae )?;147; C, $409; D, $649; K, $961; F, $91;:
liow many doUurs will I have if they all pay me ?
An8. $2308.
2. Last night I had $2308 ; to-day I received nothing, but
paid away $973 ; how many dollars have I l6ft ? Ann. $1 335.
3. A farmer gathered from his orchard 1C83 bushels of
apples, and sold 558 bushels ; how many bushels had ho left ?
Ans. 525.
4. A farmer fatted and took to market 7 hogs ; one weighed
163 pounds, another 270 pounds, another 328 pounds, another
197 pounds, another 449 pounds, another 95 pounds, and
another 256 pounds; how many pounds of pork did lie have-
to sell? ■ Ans. 1758.
5. A wholesale merchant received 8 notes from his custo-
mers in one day; the Ist was for $725, the 2nd for $197,
the 3rd for $75, the 4th for $19, thd 5th for $473, the 6th
for $690, the 7th for $84, and the 8th for $69 ; what was
the value of the whole ? Ans. $2332.
6. In 1871, the population of Nova Scotia was 387800,
and that of New Brunswick was 285594, how many more
people were there in Nova Scotia than in New Brunswick ?
Ans. 102206.
7. If a man were worth $3112 on new year's day, and
gained during the year $849, how much yroiUd he be worth
the next new year's day ? Ans. $3961.
8. If a man were worth $4000 on new year's day, and lost
$1943 during the year, how much would he be worth the
next new year's day ? Ana. 2057.
9. A man intending to move from the country to the city
sold his farm for $1743, his horses for $395, his cows for $98,
his sheep for $137, his farming utensils tor $249, his hay for
$217, his grain for $75, and his poultry for $29, how many
dollars worth did he sell altogether ? Ans. $2943.
10. In the last question, how much more did the man get
for his hay than for his grain ? Ans. $142.
11. In 1871 the population of the pounties of New Bruns-
wick was as follows : St. John 52120, Charlotte 25888, King's
24593, Queen's 13847, Sunbury 6824, York 27140, Carleton
19938, Victoria 11641, Eestigouche 5575, Gloucester 18810,
Northumberland 20116, Kent 19101, Westmorland 29335,
Albert 10672, what was the population of the whole Pro-
vince ? Ans. 285594.
MULTIPLICATION.
f
31
MULTIPLICATION.
Art. 33. — Is the process of findin.'( the result of repeating
a given number as often as there are units in another given
number.
Tlie number tr» be repeated is the Multiplicand.
The number which shows how often the multiplicand is to
be repeated is the Multiplier.
The result of the operation is called the Product.
The multiplicand and the multiplier are called Factors of
the Product.
AuT. 34. — The product of two numbers is the same, which-
ever factor is taken as the multiplier.
Art. 35. — The product is of the same name as the mul-
tiplicand.
Art. 36. — The multiplier must always be considered as an
abstract number. It is absurd to talk of multiplying dollars
by dollars. It would be as rational to propose to find the
product of 5 apples by 4 potatoes.
MULTIPLICATION TABLE.
Twice
1 are 2
2— 4
3— 6
4— 8
6—10
6—12
7-14
8 — 16
9 — 18
10 — 20
11 — 22
-12 — 24
3 times |
li
ftreS
2
- 6
8
- 9*
4
-12
6
"15
6
-18
7-
-21
8
-24
9
-27
10
-30
11
-33
12
-36
4 times
1 are 4
2— 8
3 — 12
4 — 16
6 — 20
6 — 24
7 — 28
8 — 32
9 — 86
10 — 40
11 —44
12 — 48
6 times
1 are 6
2 — 10
3 — 15
4 — 20
6 — 26
6 — 80
7 — 35
8 — 40
9 — 46
10—60
11 — 66
12 — 60
6 times
1 are 6
2—12
3 — 18
4 — 24
6 — 30
6 — 36
7 — 42
8 — 48
9 — 64
10 — 60
11—66
12-72
7 times
1 are 7
2 — 14
3 — 21
4 — 28
6 — 35
6 — 42
7—49
8 — 66
9 — 63
10 — 70
11-77
12 — 84
8 times
1 are8
2—16
3 — 24
4 — 32
5 — 40
6 — 48
7 — 66
8 — 64
9 — 72
10 — 80
11-88
12 — 96
9 times
10 times
11 times
12 times
1 are 9
larelO
lareli
1 are 12
2—18
2 — 20
2 — 22
2 — 24
8 — 27
3-30
8 — 33
3 — 86
4 — 36
4 — 40
4-44
4-48
6 — 46
6 — 60
6 — 66
6 — 60
6 — 64
6 — 60
6 — 66
6-72
7 — 63
7 — 70
7 — 77
7-84
8-72
8 — 80
8 — 88
8 — 96
9 — 81
9 — 90
9 — 99
9-108
10 — 90
10-100
10-110
10-120
11-99
11 -110
11 -121
11 -182
12-108
12-120
12-132
12-144
S2
ARITHMETIC. '
Multiply 9246 by 7.
Solution. — For convenience set the Multiplier 924<
under the Multiplicand as in the marp;in. Multiply 7
each figure of the multiplicand by the multipher, •■ ■ - ■
thus, 7 times 6 units are 42 units, write 2, and re- 64722
serve the 4 to be added to the product of the tens ;
then 7 times 4 tens are 28 tens, and 4 tens are 32 tens, write
2 in the tons place, and reserve the 3 to be added to the pro-
duct of tlie hundreds; then 7 times 2 hundred are 14 hundred,
add 3 from last product makes 17 hundreds, write 7 in the
hundreds place, and reserve the 1 to be added to the product
of the thousands; lastly, 7 times thousands are 63 thousands,
and 1 are 64 thousands, which you will write in full.
Multiply 34.C186 by 268.
Solution. — Set the multiplier under the 345186
multiplicand as before, units under imits, t^ns 268
under tens,&c. Begin at the right, and multi-
ply each figure of the multiplicand by 8, sett- 2761488
ing down the product as in the preceding 2071116
example ; then by 6, which by netting the pro- 690372
duct one place to the left, gives the product
by 60, or 6 tens ; then by 2, which by setting 92509848
the product two places to the left, gives the
Eroduct by 200. The three lines are then, 1st, the product
y 8, 2nd, the product by 60, 3rd, the product by 200, and
their sum, the product by 268 which was required.
RULE FOR MULTIPLICATION.
Set the multiplier under the multiplicandf units under •
units, tens under tens, &c., then,
1. When the multiplier consists of only one figure, mul-
tiply each figure of the multiplicand by the multiplier,
writing under each tJie right hand figure of its prockbct
increased by the rermainviig figure or figures of the product
immediately p)xcedlng it, observing to xvrite the last pro-
dtLct so increased in full.
2. When the multiplier consists of more than one figure,
multiply tlie multiplicand by each figure of the multiplier
in succession, beginning each partial pi^od^ct under the
figure which produces it. The stim of these partial pro-
ducts will be the total product.
Proof. — Multiply the multiplier by the multiplicand. If
the result is the same the work may be deemed correct.
Multiplicand,
Multiplier,
Product,
MULTIPUCATION.
EXERCISES.
(1) (2)
7896 581067
5 8
(3)
938746
4
38
(4)
193764
7
5. 391876 X 9
6. 987456 X 6
7. 496783x52
8. 719864x43
9. 375967x64
15. Find the
16. Find the
17. Find the
■ 18. Find the
39480 4655736 3754984 1356488
= 3526884 10. 27859 x 29= 807911
= 5924736 11. 679854 x 83= 56427882
=25832716 12. 7596d4 x 187=142060908
=30954152 13. 5372x1634= 8777848
=24061888 14. 7986x3795= 30306870
squaru or second power of 389. Ans. 151321
cube or third power of 538. Ans. 155720872
fourth power of 144. Ans. 429981696
cube 0/991 970299
Art. 37. — Operations in multiplication may sometimes be
shortened, as in the following case : —
When the multiplier is a composite number, that is a num-
ber which is the product of two or more whole numbers, each
greater than. one,
BuLE. — Separate tJie multiplier into two or jnore factoTB.
Multiply first by one of these fadorsj then this prod/act by
another, and so on till each factor has been used as a mul-
tiplier. The Uzst product is the complete product required.
I EXERCISES.
1. Multiply 7325 by 24.
24 is a composite number, and is the product of the factors
4 and 6, therefore if 7326 be multiplied by 4, and that pro-
duct by 6, the required product will be obtained. Thus:
Multiplicand, - - - - 7325
1st factor of the multiplier, 4
■■!(>.
2nd
«
(i
29300
6
i .i
Product by 24, 175800
2. Multiply 1728 by 64.
3. Multiply 5673 by 42*
4. Multiply 7916 by 72.
5. Mtdtiply 19743 by 88.
6. Multiply 9173 by 144. kmi
-•g.|li;,.
Ans. 110592
Ans. 238266
Ans. 569952
Ans. 1737384
Ans. 1320912
i!
u
ABITHMEriC.
I:
Ans. 15936
Ans. 674478
Ana. 4462577
Ans. 686208
7. Multiple 498 by 32.
8. Multiply 10706 by 63.
9.. Multiply 91073 by 49.
10. Multiply 7148 by 96.
Art. 38. — When the multiplier is 1 with ciphers annexed,
as 10, 100, 1000, &c..
Rule. — Annex to the multipliccmd as many cvphera as
there are in tJie mvMiplier ; the result will be the required
product,
EXERCISES.
1. Multiply 7394 by 10.
2. « , 5786 100.
a. *< 7120 1000.
4. Multiply 648 by 10000
5. « 7863 100000
6. *• 9104 1000000
Art. 39. — When ciphers ate at the right of one or both
factors.
Rule. — Multiply as usual, but omitting th^e ciphers r«-
f erred to ; then annex to the product as many cyphers aa
are to tlib right of both fa>ctors.
Find the product of 4300 by 1600.
SOLUTION.—Multiply 43 by 16, and to the 4300
product 688 annex four ciphers, viz: the num- 1600
ber to the right of both factors.
258
43
EXERCISES.
^s^ 1. 738000 X 7300. 4.
2. 5900 X 2700. 5.
3, 1070 X 79000. 6.
6880000
270 X 860.
11 1000 X 610.
790000x43000.
APPLIC
1. If one yard of cloth cost
25 yards cost ?
2. If 1 pound of cheese cost
I give for 9 pounds.
3. If 7 boxes contain 144
all?
4. If a laborer earn $7 a
he earn in 35 weeks ?
5. How many bricks would
if he took 1625 at a load ?
ATIONS.
75 cents, how many cents will
Ans 1875 cents.
18 cents, how many cents must
Ans. 162 cents.
pens each, how many in them
Ans. 1008.
week, how many dollars would
Ans. $24^.
a teamster remove at 23 loads,
Ans. 37376.
DIVISION.
25
* 6. If a wagon wheel make 586 revolutions in a mile, how
many revolutions would it make in a journey of 67 miles ?
Ans. 39262.
V. An ordinary clock strikes 156 strokes in a day, how
many strokes does it strike in a year of 365 days ?
Ans. 56940.
8. A bushel of potatoes weighs 60 pounds, what is the
weight of 350 bushels ? Ans. 21000 pounds.
9. At $15 per acre, what would be the price of a field
measuring 29 acres ? Ans. $435»-
10. If an acre of land yield 47 bushels of wheat, how many
bushels will 109 acres yield ? Ans. 5123.
DIVISION. '
Akt. 40. — Division' is the process of finding how often one
number is contained in another ; or it is the process of find-
ing one of two factors of a given niunber when the other is
known.
The number to be divided is the Dividend. ' "
The number by which to divide is the Divisor.
The result, which shows how often the divisor is contained
in the dividend, is the Quotient.
The number sometimes left, and which shows the excess
-of the dividend over the Divisor repeated as often as there
are units in the quotient, is called the Bemainder.
Note. — ^Tbe remainder is always of the same kind as the dividend, and
must always be less than the divisor.
How often is 3 contained in 672 ?
, Solution. — For convenience Divisor 3)672 Dividend,
write the divisor on the left of ^24 Quotient,
the dividend as in the margin ;
then say 3 in 6 (hundred) 2 (hundred) times ; write two in
the hundreds' place ; then 3 in 7 (tens) 2 (tens) times and 1
• '(ten) over, write 2 in the tens' place, and add 10 to the next
figure, 2=12 ; then 3 in 12, 4 times, write 4 in the units'
place. The quotient is 224.
In practice we. do not name the orders of units, but pro-
ceed thus, 3 iu 6, twice, write 2 ; 3 into 7, twice, and 1 over,
write 2 and prefix 1 to the next figure in the dividend, making
12 ; 3 in 12, 4 times, write 4.
26
ARrinMKnc.
Again, divide 533965 by 7. '—^
Solution. — Arrange the numbers as be- 7)533965 ^;
fore ; then 7 in 5, times, but 5 remains 76280-5
undivided ; join it with the next figure ; 7
in 53, 7 times and 4 over, write 7, and mentally prefix the 4
to the next figure 3 ; 7 in 43, 6 times and 1 over, write 6 and
prefix 1 to 9, the next figure ; 7 in 19, twice, and 5 over,
write 2, and prefix 5 to 6, the next figure; 7 in 56, 8 times
even, write 8 ; 7 in 5, times, and 5 over, write 0, and indi-
cate the division of 5 thus, | and annex it to the other figures.
Wlien division is performed in this way, which is the case
when the divisor is not greater than 12, the process is called
Short Division.
^ y RULE FOR SHORT DIVISION.
1. Tfriie the divisor on the left of the dividend, with a
curved line between them; divide successively each figure of
the dividend by the divisor, and set the quotient beneath
tlie figure divided.
2. Wien^vei' there is a remainder, prefix it, mentally, to
the next figure in the dividend, and divide as before,
3. If any partial dividend does not contai/n the divisor,
'filace a dphsr beneath, and form a new dividend by iwc/Zoj-
^ng the old one to the succeeding figure, and divide asoefore,
4. If there is a remainder after dividing the last figure,
indicate its division by placing the divisor under it, and
annex it to tJie quotient.
?J1*.»
EXERCISES.
1.
Divide
7936427 by 3.
Ans.
26454751
2.
((
87965328 « 4.
Ans.
21991332
3.
((
7963821 « 5.
Ans.
1592764^
4.
it
6875324 « 6.
Ans.
1145887f
5.
((
3987654 " 7.
Ans.
569664|
e.
((
19876532 « 8.
Ans.
2484566|-
7.
C(
2976532 « 9.
Ans.
330725J.
8.
((
4967854 « 10.
Ans.
496785 jL
D.
((
46879352 "11.
Ans.
4261759:1
10.
((
18765314 « 12.
Ans.
1563776^
Methods of Proof. — 1. Multiply the quotient by the
DIVISION^'^
4
divisor, or the divisor by the quotient, adding the remainder,
if any, to the product. Tlie sum should be the dividend.
2. — Subtract the remainder, if any, from the dividend;,
the result divided by the quotient should be tlie divisor.
EXAMPLE.
9) 77 .
i'\ ?■■ .,^■'
8—5.
1st Proof.
9x8=72 and 72 + 5=77.
2nd t»roof.
77—5=72, and 72-^8=9.
Divisor. Dividend. Quotient.
15)3465(231 -
30 ,H •.;,,.•.,.. .
46 . ;
45 ;.-:
15
15
00 remainder
Art. 41. — ^When the divisor exceeds 12, the process em-
ployed is generally such as is shown below, and is called Long.
Division.
Divide 3465 by 15.
Solution. — Write the divisor on the
left of the dividend as in the margin.
Then as 15 is not contained in 3 (thous-
ands), the quotient has no thousands.
Next see how often 15 is contained in
34 (hundreds) and set the number 2
(hundreds) in the quotient on the right
of the dividend; multiply the divisor
■ by this figure (2) and set the product
under 34, and subtract, the remainder
is 4 (hundreds). Then bring down the
next figure of the dividend which is 6 (tens), and take 46
(tens) for the next partial dividend. 15 is contained in 46
(tens), 3 (tens) times; multiply the divisor by 3, set the pro-
duct under 46 and subtract, — the remainder is 1 (ten), to
which annex the next figure of the dividend (5), v hicL makes
the last partial dividend 15. 15, the divisor, is contained in
this once; set one in the quotient, multiply the divisor by it,
and subtract the product as in other cases. The remainder
is nothing. 15 is therefore contained in 3465, 231 times
which is the quotient.
RULE FOR LONG DIVISION.
1. — Draiv a curved line on the left, and one on the Hght
of tJie dividend, and set the divisor on the left.
2. — See how often the divisor is contained, in the feivest
of the left hand figures of the dividend that will contain it,
a7id set the number in the quotient on the right of the divir^
dend.
■■i
as
ARITUMETIC.
3. — Multiply the divisor by this quotient figure, and set
the product under the figures ueed hi tlie dividend, cmd
subtract it from tliem.
4,— Annex to the remainder tJie next figure of the divi-
dend, and divide as before, continuing the process until
ea-ch figure of tfie dividend lias been used.
5.—^ at any time, after a figure of the dividend has been
brought down, the number thus formed is too small to con-
tain the divisor, a cipher must be placed in the quotient,
and anotJier figure brougid down, afte/r which divide as
before.
6. — If tJiere is a final remainder, write the divisor under
it as in Short Division.
Proof. — The same as in Short Division.
Note 1. — In finding any quotient figure, tria! niay be made of a num-
ber which, when the Divisor is multiplied by it, will give a product
greater than the partial dividend used. If so, the quotient figure is too
large. Try the next smaller figure, and so on till the product obtained is
not greater than the partial dividend.
Note 2. — ^When the poduct of the divisor by any quotient figure sub-
tracted from the partial dividend gives a remrinder greater than the
divisor, the quotient figure is too small. Try the next greater figure, and
80 on till a remainder less than the divisor is obtained.
EXERCISES.
1. Divide
2. ««
3. «
4.
5.
6.
7.
8.
9.
10.
«
C(
■it
cc
(i
(C
i(
11.
12.
13.
14.
35190 by
81306 «
66683 «
273284 «
666774 «
2317565 «
7910683 ««
9621304 «
10045643 «
57300847 «
41807164-
89709786-
37418007-
45676827-
15. 170064915561-v- 759=
15.
18.
21.
47.
93.
275.
74.
396.
571.
1078.
^3727:
^4509:
^5763:
^7609:
759:
Ans. 2346
Ans. 4517
Ana. 3175-8-
91
Ans. 581411
41
Ans. 7169»1
• 8
Ans. 84271*0
Ans. 106901-1-
74
Ans. 24296_»«-
898
Ans. 17593*1
671
Ans. 53154A»»-
1078
112l71*»i.
3787
19895^U1
4609
6492*111
6788
6003
2240644473M
769
if4
i
DIVISION.
29
COXTBACTIONS IN DIVISION.
Case 1. — When the Divisor is u composite number.
Divide 5775 by 25.
25 is the product of two factors, 5 x 5.
Rule. — Divide the dividend by one of the 5)5775
factors, and that quotient by another, and 5)1155
80 on till each factor ho a been used as a di- 231
visor. The last quotient luill be the one required.
To find the true remainder, multiply each remainder by
all the preceding divisors except the one which produced it,
and to the sum of the products add the remainder from the
first division, if there be one. Thus, divide 247 by 15.
3)247
2x3=6, and 6 + 1=7, true remainder.
1. Divide
2. «
3.
4.
5.
6.
7.
((
((
((
((
(t
EXERCISES.
90144 by 24.
69090 « 35.
2507639 « 42.
3827864 « 63.
57834722 "81.
55937133 " 56.
497021 " 72.
5) 82-1.
16. 2.
Ana. 3756
Ans. 1974
Ans. 59705M
43
Ans. 6075911
63
Ans. 7140081*
81
Ans. 998877»i
58
Ans. 6903-1-
73
From! the above it appears that if the divisor and the div-
idend both be divided by the same nimiber, the quotient will
not be changed.
Case 2. — When the r'livisor is 1 with a cipher or ciphers
annexed, as 10, 100, 1000, &c.,
Rule. — C^U of as many figures from the right of the
dividend as there are ciphers in the divisor; the figures
out off will be the remavnder, and tlie others %vill be t1\a
quotient.
EXERCISES.
1. Divide 35243 by 100.
110 0)352143
352-11
100
Or simply from inspeotion, write all the figures except the
mmmmm
30
ABITHMEnC.
t
I
I
two last for the quotient, and place the two right hand figures
above the divisor and annex thereto.
2. Divide 7639 by 10.
3. « 154736 " 100.
4. " 791084 « 1000.
5. « 9018765 " 10000.
6. « 5587391 « lOOOOO.
Case 3. — When there are ciphers on the right of the di-
visor, and the remaining part greater than 1.
KULE. — Gut off the cipJiera from the right of the divi80i\
and as wZbriy figures from, the right of the dividend^ and
divide the rem^avning jUgurea of the dividend by the remain-
ing figures of the divisor. To the remainder annex tJie
Jigurea cut off from the dividend, and you will have th6
true remxiinder.
EXEBCISES.
1. Divide 54300 by 130.
52 .
23
13
100
91
90
130
2. D
ivide
47689 by 170.
3.
((
916«>345 « 29000.
4.
cc
6008074 « 7500.
^
«
7961745 «« 60.
6.
(C
15728 « 20. '
7.
i(
9186437 « 70.
8.
((
820186 " 40.
9.
((
7310745 "1290000.
lOL
u
750900 « 500.
•fk-
Ans. 280-»-»-
170
Ans. 316 MM
89000
Ans. 801 -»11-
1600
Ans. 132695M
•
Ans. 786-1-
80
Ans. 13123411
70
Ans. 20504M
40
Ans. 5-MMM.
1S90000
Ans. 1501i
CANADA CUBRKNCY.
81
APPLICATI0N3.
1. What is the half of 9786? Ana. 4893.
2. What is the one third of 768594 ? Ana. 256198.
3. Find one eighth of 673915 ? Ans. 84239|.
4. Find the one fortieth of 976183 ? Ans. 24404j§
5. If 5 barrels of apples cost $20, what is the price of
one barrel ? Ans. $4.
6. If a clock strike 1092 strokes in a week, how many
strokes does it strike each day ? Ans. 1 56.
7. If 19 yds. of cloth cost 1805 cents, wliat is the price per
yard ? Ans. 95 cents.
8. If an orchard of 27 trees produce 5103 apples, how
many on each tree, on an average ? Ans. 189.
9. If 56 men earn $30072 in a year, what is the salary of
«ach man on an average ? Ans. $537.
i» 10. If there he 54432 pens in 378 boxes of equal size, how
many in each box ? Ans. 144.
11. How many acres in a field which produces 4277 but^h-
els of oats, at the rate of 29 bushels to the acre.
Ans. 14711
90
12. If each family in a city consume 72 eggs in a year,
and it require 1229688 eggs to supply the town, how many
families in the town ? Ans. 17079.
13. If 379 bushels of com cost 13265 cents, how much is
that per bushel ? Ans. 35 cents.
CANADA CURRENCY.
Art. 42. — The currency of Canada is of two denomina-
tions, viz : Dollars and Cents.
The dollar is the unit of value, of which the cent, from
centum — a hundred, is the one hundredth part.
When dollars and cents are to be expressed, the cents are
written to the right of the dollars, and separated from thepi
only by a period (.) called the decimal point. The sign of
dollars ($ ) is prefixed to the whole. Thus, two dollars, forty-
five cents are written, $2.45.
We may estimate the value of the figures in any expres-
sion of dollars and cents thus : —
Take $436.75 ; the 6, that is the first figure on the left of
the point, represents units of dollars, the next, tens of dollars^
•
'>
»M
(i
82
ARITHMEnC.
and 80 OD, as explained under the head of Numeration.
The 7, that is, the first figure on the right of the point, re-
presents tenths of a dollar, and as a dollar is 100 cents,
one tenth of a dollar is 10 cents, and 7 tenths are 70 cents,
therefore the 7 represents 70 cents. The next figure repre-
sents hundredths of a dollar, and as the hundredth part of a
dollar is a cent, the 5 represents 5 cents. Or thus : —
The right hand figure represents imits of cents, or 5 cents ;
the second figure tens of cents, and as 7 tens are 70, the 7
represents 70 cents ; the third represents hundreds of cents,
and as a hundred cents are a dollar, the 6 represents 6 dol-
lars; the fourth, tens of dollars, and so on as explained
under the head of Numeration.
Thui^ it will be seen that the figures throughout bear the
same relation to one another as in simple numbers, and may
therefore be added, subtracted, multiplied and divided by the-
rules already given.
ADDITION. •
Place the quantities to be added so that the decimal points
shall be directly under one another, add as in simple num-
bers, and cut off two figures from the right of the sum for
cents.
1. Add $125.75, $98.50, $25.15, $76.05, $9 l.lH,$43.87i,
$84.20, $67.62^, $39.80, and $17.37^. Ans. $669.44.
2. What is the sum of $13.19, $14.16, $85.92, $64.15,
$37.25, $91.20, $18.75, $29.10, $47.85, $55.55, and $72.63.
Ans. $529.75.
3. Find the sum of $85.50, $49.63, $92.18, $37.09, $8.92,
$76.45, $25.75, $64.16, $18.60, $59.11, $148.17, and $265.90.
Ans. $931.46.
4. Add together $116.20,$291.45, $89.75, $365.84, $91.50^,
$76.15, $485.00, $157.92, $263.75, $188.25, $39.48, and
$136.13. Ans. 2301.42.
5. Add together $175.18, $1.74, $2864.91, $3.24, $876.45,,
$79.79, $0.85, $278.01, $5371.56, $17.20, and $740.95.
Ans. 10409.88.
SUBTRACTION.
Place the quantities so that the decimal point in the sub-
trahend shall be directly under that of the minuend, subtract
as in simple nun^bers, and cut o£f from the di£ference the
two right hand figures for cents.
CANADA CURRENCY.
1. Subtract $278056.80 from $567810.83.
Ans. 288862.94
2. From $83756.17 take $76480.71. Ans. 7266!46'.
8 What is the di£ference between $17423.37^ and
$9645.634 ? Ans. $7777.74.
4. Find the difference between $100623.40 and $9781.37.
Ans. $00842.03.
MULTIPLICATION.
Multiply as in simple numbers, and cut off two figures
from the right of the product for cents.
1. Multiply $365.75 by 7.
2. Multiply $1873.47 by 60.
3. Multiply $865.63 by 03.
4. Multiply $24786.38 by 145.
Ans. $2560.25.
Ans. $120260.43.
Ans. $80503.50.
Ans. $3504025.10.
DIVISION.
Divide as in simple numbers, and point off two figures
from the right of the quotient for cents.
1. Divide $60500.68 by 7.
2. Divide $28642.14 by 20.
. 3. Divide $37133.34 by 87.
" 4. Divide $1043243.55 by 083.
Ans. $8644.24.
Ans. $087.66;
Ans. $426.82.
Ans. 1076.85.
MIXED EXERCISES IN THE APPLICATION OF MULTIPLICATION
AND DIVISION.
1. What is the cost of 17 acres of land at $52.50 per
acre? Ans. $802.50.
. 2. A hammer factory turns out 37440 hammers in a yfor
of 52 weeks, how many is that per week on an average ?
Ans. 720.
3. How many yards of calico at 8 cents a yard can I buy
for $2.80? Ans. 35.
4. How many yards of ribbon at 25 cents per yard can
be purchased for $3 ? Ans. 12.
5. I sold 15 kegs of butter,,each containing 25 pounds,
for $60, how much was that a pound? Ans. 16 cents.
6. Bought 21 barrels of apples at $1.05 a barrel, what did
they cost ? Ans. $22.05.
7. If 11 tons of hay cost $214.50, what will 1 ton costf
What will 27 tons cost ? Ans. I ton, $10.50.
27 tons, $526.50;
S4
ARITHMETIC.
8. 1125 bbls. fish were sold for $5906.25, how much per
barrel ? Ans. $5,*4S,
0. 269 persons pay a tax of $1312.72, what is the average
tax on each ? Ans. $4.88.
10. Huppose a manufacturing company employs 250 men,
and pays them on an average $1.75 per day, what is the cost
to the company for 1 day ? for 1 week ? for 1 year ?
Ans. For a day, $437.50.
For a week, $2625.00.
For a year, $136500.00.
1 1 . If the liouses in a town are worth on an average $950
each, and their total value is $1168500, how many houses in
the town ? Ans. 1230.
12. If the total value of 1230 houses be $1039350, what is
the value of each house on an average ? Ans. $845.
13. What would be the value of 1230 houses, if the aver-
age value were $845 each ? Ans. $1039350.
14. What is the value of 437 sheep at $4.75 each ?
Ans. $2075.75.
15. If a man travel 3 miles an hour every day for 40 days
of 12 hours each, how many miles will he travel ? Ans. 1440.
16. If a railway train runs 264 miles in 12 hours, what the
average rate per hour ? Ans. 22 miles.
17. At 45 cents per bushel, what must be paid for 1195
bushels potatoes ? Ans. $537.75.
18. A cargo of 4700 bushels oats sold for $1504, how much
is that per liushel? Ans. 32 c^nts.
19. What is the weight of a cargo of 5000 bushels of wheat
weighing 60 pounds per bushel ? Ans. 300000 pounds.
20. 180 chaldrons of coal were sold for $1035, what was
the price for 1 chaldron? Ans. $5.75.
PROPERTIES OF NUMBERS.
Abt. 43. — 1. An Integer i» a whole number, as 1,2, 3, &c.
2. An Even number is one that can be evenly divided by
2,08 2,4, 6, 8,10, &c.
3. An Odd number is one that cannot be evenly divided
by 2, as 1, 3, 5, 7, 9, 11, &c.
4. A Prime number is one that cannot be exactly divided
by any whole number except itself and 1, as 1, 2, 3, 5, 7»
11, 13, 17, 19, &c.
rUOPEHTIES OF MUMBKIU.
85
5. A Composite number is one that can ))e exactly divided
by some wliole number boHideH itHctlf and 1, as 4, 6, 8, 9, 12,
14, 1.5,&c.
fi. Every composite num})er in tbe product of two or more
prime numbers.
7. Two numbers ure prime to each other when 1 is the
only number wliich will exactly divide both, as 5 and 9.
8. A number which will exactly divide two or more num-
bers is called a common factor of titem. Thus, 3 is a com-
mon factor of 6, 9, 12 and 15. ■ 7 is a common factor of 14
and 35,
9. A Prime Factor of a number is a prime number which
will exactly divide it. Thus, the prime factors of 21 are 3
and 7. The prime factors of 24 are 2, 2, 2, and 3.
To resolve a composite number into its prime factors,
Rule. — Divide me given numher hy any pHme number
r/reater than 1 that tvill exactly divide it, repeat the process
^vith the quotient^ and so on till a prime number ia ob-
tained ; the divisors and remaining numbers are tlw prime
factors required,
EXERCISES.
1. What are the prime factors of 90 ? ■■ '
2 )90
3>45 .„ ;,
3)15
5
2. What are the prime factors of 35 ?
3. What are the prime factors of 75 ?
4. Besolve 651 into its prime factors.
5. Besolve 1764 into its prime factors.
Ans. 3, 3, 4, 7, and 7.
6. What are the 'prime factors of 198?
Ans. 3, 3, 2, and 11.
7. What are the prime factors of 171 ? Ans. 3, 3, and 19.
8. What are the prime factors of 210 ?
X Ans. 2, 3, 5, and 7.
9. What are the prime factors of 2310 ?, i
Ans. 2, 3, 5, 7, and 11.
10. What are the prime factors of 362880 ?
Ans. 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 5, and 7.
11. Find the prime factors of 180642.
Ans. 2, 3, 7, 11, 17, and 23.
Ans. 2, 3, 3, and 5.
Ans. 5 and 7.
Ans. 3, 5, and 5.
Ans. .3, 7, and 31.
Jo ARITHMBTIC.
12. What uro iho prime factora of 51051 ?
Ans. 3, 7, 11, 13, and 17^
ORBATKST COMMON DIVI80R.
A (Common Divisor of two or more niunlieni in a number
which will divido eiieh of them without a remuider. Thus,
3 in a common divisor of 12', IM, 24 and 30.
The Oreateflt Common Divisor of two or more numbers is
the ffreatest numlier that >vill divide each of them without
a remainder. Thu^ 4 is the greatest common divisor of H,
12, and IH.
To find the greatest common divisor of two or more
numl)ers,
Rule 1. — Resolve the fjiven numhera into tJteir prime
factoi'8 ; the product of the factora commaii to all %mll be
the greatest cormnon divlaor.
EXERCISES.
1. Find tlie greatest common divisor of 18, 27 and 36.
The prime factors, common to
18=2x3x3.
27=3x3x3.
36=2x2x3x3.
all, are 3 and 3. 3 x 3=9, the
greatest common divisor.
;
2. Find the greatest common divisor of 16, 40 and 72.
Ans. 8.
3. Find the greatest common div' or of 36, 54, 90 and 72.
Ans. 18.
4. Find the grerttest common divisor of 126, 210, 84 and
168. Ans. 42.
5. Find the greatest common divisor of 175 and 245.
Ans. 35.
Rule 2. — If there be onhj two numhera, divide the greater
by tJie leasy that divlaor by the remainder, and ao on till there
ia no remainder. The laat diviaor will be tJie greateat com'
mon diviaor.
When there are more than ttvo nwmbera,find the greateat
common diviaor of two, f:hen pf tJiat and another, and
so on to t/ie laat. TJie last greoiteat com/mon diviaor wiU
be the greateat common diviaor of tht. whole.
riujFKKTiKri or numbkus.
EXERCISRS.
1. Find the greatest common divisor of 21 and 98.
21)98(4
H4 ' > ; - , ^ ;. .
IT
•jf,'
14)21(1
14
7)14 ' ■ : .^. '■=: ,
2
The last divisor, 7, is the greatest common divisor.
2. Find the greatest common divisor of 52 and 299.
Ans. 13.
3. Find the greatest common divisor of 108 and 9342.
Ans. 54*.
4. What is the greatest common divisor of 1 638 and 2106 ?
Ans. 234.
5. What is the greatest common divisor of 31185 and
50457? Ans. 33.
6. Required the greatest common divisor of 24, 108, and
464. Ans. 4.
7. Required the greatest common divisor of 576, 1344, and
2592. Ans. 96.
8. Required the greatest common divisor of 576, 768, 480
and 1360. Ans. 16.
9. What is the largest price per acre, in whole dollars, at
wliich each of three farms, containing respectively 128, 224,
and 376 acres, can be bought or sold ? Ans. $8.
10. A gentleman's garden is 162 feet long, and 138 feet
wide. He wishes to set posts for fencing at the greatest dis-
tance apart that will make equal spaces on all sides. Required
the number of feet from centre to centre of posts. Ans. 6 ft.
LEAST COMMON MULTIPLE.
A Multiple of a number is a number which can be divided
by it witho!'t a remainder. Thus, 15 is a multiple of 3 ; so
is 12 ; so is 21.
A Common Multiple of two or more numbers is a number
which can be divided by each of them without a remainder.
Thus, 15 is a common multiple of 3 and 5 ; 24 is a common
anultiple of 2, 4, and 6.
The Least Common Multiple of two or more numbers ia
t
^
ARITHMETIC.
|i
I
the least numl)er that can be divided by each of them with-
out a remainder. Thus, 45 is the least common multiple of
A and 9 ; 12 is the least common multiple of 2, 4, and 6.
To find the least commjn multiple of several numbers, — >
BuLE. — Arrange the nttmbera in a horizontal line, omit-
ting such of t1ie,n as are factm'8 of any of the others;
divide by any prime number that will divide two or more
of them witJtout a remainder, vn'iting the quotients and
undivided numbers in a line below ; contimue tlie 'process
till a line is obtained the numbers in which are altprvme
to each other. Multiply together the divisors and the numr
bers in the last UnCy and the product will be tfie least com-
mon multiple required.
Note 1. — This process gives the product of all the prime factors in the
given numbers.
Note 2.— If the given numbers are prime to each other, their product
is their least common multiple.
'■ EXERCISES.
1. What is the least common multiple of 6, 8, 12, and 15 ?
2)8, 12, 15. Omit 6 because it is a factor of 12. The
divisors are all prime numbers, and the num-
bers in the last line are prime to each other.
Then 2x2x3x2x5=120, which is the
least common multiple of 6, 8, 12, and 15.
2. Find the least common multiple of 12, 16, 18, 30, and'
48. Ans. 720.
3. Find the least common multiple of 3, 4, 5, 6, and 7.
Ans. 420.
4. What is the least common multiple of 2, 4, 7, 12, 16,
21, and 56 ? Ans. 336.
5. What is the least common multiple of 2, 9, 11, and
33 ? --»i.*^ ........ . • - Ajjg^ i9g^
6. Find the least coihmon multiple of 2, 3, 4, 5, 6, 7, 8,
and 9c Ans. 2520.
7. Find the least common multiple of 8, 12, 16, 24,
and 33? Ans. 528.
8. What is the least number into which 2, 4, 8, 16, 32, 64,
and 128 will divide without a remainder? Ans. 128.
9. F' d the least common multiple of 3, 9,27, 81, 243,
and 729. Ans. 729.
10. Wliat is the least common multiple of 2, S, 5, 7, 11 ?-'
Ans. 2310. .
2 )4, 6, 15.
3 )2, 3, 15.
2, 1, 5.
FRACTIONS.
FRACTIONS.
,. V
■Ui/
Art. 4C — A Fraction is a part of a unit or whole thing,'
supposed to be divided into equal parts.
Fraction is derived from the Latin, /roc^tts, broken.
Fractions are divided into two classes, Common and
Becimal.
A Common Fraction is expressed by two numbers, one
above the other with a horizontal line between them, as ^,
read two-thirds.
The number below the line is called the Denominator.
The number above the line is called the Numerator.
The namvj of a fraction and the value of its parts depend
on the number of parts into which the imit is divided. When
the unit is divided into 2, 3, 4, 5 or 6 equal parts, the frac-
tion is named halves, thirds, fourths, fifths or sixths.
The Denominator gives a name to the fraction, and shows
the number of equal parts into which the unit is divided.
The Numerator shows how many of these parts are ex-
pressed by the fraction.
Thus, in the expression | of a mile, a mile is supposed to
be divided into four equal parts, called fourths, three of which
are expressed by the fraction, |.
A fraction is either Proper or Improper.
A Proper Fraction has its numerator less than its denomi-
nator, 8 s |.
An Improper Fraction has its numerator equal to, or
greater than its denominator, as ^, i .
A Mixed Number is a whole number with a fraction an-
nexed, as 4^
I Again, a Fraction is either Simple, Compound or Complex.
^^j A Simple Fraction has but one numerator and one denom-
inator, and may be proper or improper. Thus f and ^ are
both simple fractions.
A Compound Fraction is a fraction of a fraction, that is,
it is two or more fractions connected by the word of, as f of
I- of h
A Com.plex Fraction has a fraction in its numerator or
denominator, or in both. Thus, i,?, f, ^,£,?i
are all complex fractions. 4| 1 3 3^ 9^
40
ABITHMETIC.
All common fractions represent division, the numerator
being the dividend, and the denominator the divisor. The
value of the fraction is the quotient arising from perform-
ing the operation of division. Thus the fraction f is 4.
When the fraction is proper the division cannot be performed
but is merely indicated, and the quotient can only be ex-
pressed in the fractional form.
REDUCTION OP FRACTIONS.
Reduction of Fractions consists in changing their forms
without altering their values.
A fraction is in its lowest terms when its numerator and
denominator are prime to each other, as f , ^, f , but not f .
Since the nimierator and denominator of a fraction are a
dividend and divisor, they may be divided by the same num-
ber without changing the quotient, or value of the fraction.
Therefore,
To reduce a fraction to its lowest terms, ' '
Rule. — Divide both terms by any common factor, and
continue the process till tJiey are prim^ to each other. Or,
divide both terms by their greatest common divisor.
EXERCISES. ^ .i,..,^::H ^■^■-'
1. Reduce ff to its lowest terms. '-^^
2|ff=U and 41^=4 Ans. Or, 8l^=f
Divide both terms by 2, which is a common factor, then
the resulting terms by any common factor of them, say 4,
which makes a fraction, ^, the terms of which are prime to
each other. | is the fraction ff , in its lowest terms.
2. Reduce -J-| to its lowest terms.
3. Reduce -j^ to its lowest terms. ^ ^ .. /'
4. Reduce -^^^ to its lowest terms. ■'' '■
6. Reduce |4f to its lowest terms.
■ 6. Reduce -jV^s" ^^ ^^^ lowest teims. v ..
7. Reduce ^j^y to its lowest terms. & ,^;
8. Reduce -^/f^ to its lowest terms. v
9. Reduce ^^\ to its lowest terms. ,.. .\^
10. Reduce ^o~f4 ^^ ^^ lowest terms.
11. Reduce ^^^ to its lowest terms.
12. Rfeduce flM to its lowest terms.
13. Reduce ^|y ^ to its lowest terms.
14. Reduce xVWA^ *^ ^^ lowest terms.
15. Reduce |4^dF ^ ^^^ lowest terms.
16. Reduce f !^j| to its lowest terms.
Ans. ^.
Ans. ^,
Ans. -^.
Ans. ^i
Ans. I-.
Ans. 1^.
REDUCTION OF FRACTIONS.
41
17. Reduce tV^Ws ^ ^^^ lowest terms.
18. Reduce ^VoWo ^^ ^^^ lowest terms.
19. Reduce H^ ^ ^^^ lowest terms.
20. Reduce j||^| to its lowest terms.
)
To reduce an improper fraction to a whole or mixed
number,
Rule. — Divide the numerator by the denominator; the
quotient will be the whole or mixed number.
Note.— If there be a fraction in the answer, reduce it to its lowest
terms.
EXERCISES.
1. Reduce f^ to a whole or mixed number.
The denominator shows that the unit is divided into 16
equal parts ; hence 16 sixteenths make 1, and there are as
many units in || as 16 is contained times in 87, and
16)87(5^ *. ^ i 'i'
. ... <■>,;.■. • • 80 ,..■-... ■ ■ •' ■■
/;■
;..vL. .
2. Reduce *^i to a whole or mixed number. Ans. 49.
3. Reduce "^ to a whole or mixed number. Ans. 71.
4. Reduce ^ to a whole or mixed munber. Ans. 4^.
5. Reduce i-J* to a whole or mixed number. Ans. 19^.
6. Reduce ff to a whole or mixed number. Ans. 7-^.
7. Reduce i-f^* to a whole or mixed number. Ans. 83|-.
8. I?educe \^ to a whole or mixed nuAiber. Ans, 54.
9 }i*iduce ^^ to a whole or mixed number. Ans. 29f.
W. liriiice ^^VVu' *® * whole or mixed number.
Ans. 5U^.
11. Reauce V ^^^ dollar to dollars. Ans. $3|.
12. In y of a dollar how many dollars? r|^i«. Ans. 9^.
13. How many bushels in if of a bushel? Ans. 17|.'
14. In '^^ of a gallon how many gallons ? Ans. 38f .
To reduce a mixed number to an improper fraction,
Bulb. — Multiply the whole nurftber by the denominator
of the fraction, to the product add the numerator, and write
the denominator under the sum.
A whole number may be expressed fractionally by writing
1 under it for a denominator.
A whole number may be reduced to a fraction having any
proposed denominator, by multiplying it by the proposed
42
ARITHMETIC.
denominator for a numerator, and writing the denominator
under it.
EXERCISES.
1. Beduce 4^ to an improper fraction.
4^ The denominator of the fraction is to become the de-
} nominator of the answer, therefore, the answer is to be
halves, and as two halves make a whole, there will be
twice as many halves as whole ones, that is twice 4, and the
1 half expressed ])y the fraction makes 9 halves or f .
2. Reduce 4| to an improper fraction.
3. Reduce 27^ to un improper fraction.
i.f 4. Reduce 66^^ to an improper fraction.
5. Reduce 18f to an improper fraction.
6. How many sevenths in 9| ?
V 7. In $7^ liow many eighths of a dollar?
8. In 17§ gallons how many thirds of a gallon ?
9. Change 27 to a fraction .
10. Express 9 as a fraction having 7 for its denominator
Ans. ^*,
11. Reduce 19 to twelfths. Ans. ^^.
12. Reduce 28 to a fraction having 19 for its denominator.
Ans. «^,
Ans. y.
Ans. Y'
Ans. «-|«.
Ans. i|''
Ans. ^.
Ans. y..
Ans. y*
Ans. V-^
MULTIPLICATION OP FRACTIONS.
* Multiplication o^ Fractions is the process of multiplica-
tion when one or both of the factors are fractional.
To multiply a fraction by a whole number,
Rule. — Multiply tlie numerator of the fraction by the
whole number, and set the product over the denominator.
Or, when it can be done without a remainder.
Divide the denominator by tJie whole number, and set the
quotient under the nuTneratoi'.
NoTK. — Resulting fractions are, in all cases, if improper, to be reduced
to whole or mixed numbers, if proper, to their lowest terms.
EXERCISES.
.irj 1. Multiply ^ by 3.
2. Multiply I by 8.
3. Multiply -^ by 9.
^.4. Multiply -iV by 8.
Ans. f
x3=V=li.
Ans. ^=6f..
Ans. 3^.
Ans. 3|>
MULTIPLICATION OF FRACTIONS.
To multiply a whole numheir by a fraction,
Rule. — Multiply the whole number by the numerator of
the fraction, and set tJie product over tlw denominator.
EXERCISES.
1. Multiply 7 by g.
2. Multiply 12 by |.
3. Multiply 18 by ».
4. Multiply 5 by ^j,, ' '
5. Multiply 4783 by ^g.
:<S"
Ans. 7xi=V=2f.
Ans. V=4-
Ans. 12f.
Ans. 1^.
- Ans. 19921^.
To multiply one fraction by another, or several fractions
together.
Rule — Multiply all the numerator's together for a iiew nu-
merator, and all the denominators for a new denominator.
NoTR. — When some of the factors {ive mixed numbers, they must be
reduced to simple fractions.
EXERCISES. \ i J n
1. Multiply f by|.
Ist. Multiply § by 5=^5; but as 5 is
I x^=^ Ans. 7 times the multiplier, |, the product, *g*,^
is 7 times the required product ; in other
words, the required product must be \ as much as ^. Now,
if ^ be divided into 7 equal parts, each of the parts will be
•X, or \ as much as ^ ; therefore, ^g is ^ as much as y which
IS what is required. Therefore, 2nd, multiply the denomi-
nator, 8, by the denominator, 7, for the denominator of the
product.
2. Multiply I by I . • Ans. M.
3. Multiply I by f . , , Ans. |4.
4. Multiply I, |- and 5- together. ' " Ans. •^^.
5. Multiply ^, I, ^ and f- together. Ans. |.
Observe the following methods of solving the last question*
2nd Method.
2 3
Ist. Divide the numerator, 4, and
the denominator, 8, by 4, which will
divide both exactly ; next cancel the
numerator 5 and the denominator 5 ;
next cancel the two 7*8 ; lastly divide
the numerator 3 and the denominator
9 by 3. Then multiply the quotients
instead of the original number.
1st Method.
i/Wir=i
by reducing the frac-
tion ^syv by the Neat-
est common divisor of
its terms, viz : 420.
44 ARiTUHITIC.
The second method is called
CANCELLING,
and depends upon the principle already stated, — that if the
numerator and denominator of a fraction, and therefore of
the factors which make up a fraction be divided by the same
number, the value of the fraction will not be changed.
BuLE. — Divide any numerator and any denominator of
the fractions to he multiplied by any number that will di-
vide both without a remainder, aiul continue the 'process
until no numerator and denominator can be exactly ddvided
by any number greater than 1. Or, cancel equal factors
in numerators and denominators, MuUvply the resultvng
figures, a/ad tlve product vjill be obtained in its lowest terms.
Note.— When the quotient is 1, it may be omitted.
6. Multiply I, i, f and f- together.
7. Multiply ^, 3|, 2^ and | together.
8. Multiply 5|, 4| and -}-f together.
9. Multiply 3f , ^, ^^ and 6{ together.
Ans. ^.
Ans. 4|.
Ans. 24.
Ans. 3^.
A Compound'S^Fraction is '^'isentially an expression of
multiplication of fractions. Thus f of | is equivalent to
I X f . The latter expression indicates that | is to be repeated
as often as there are imits in f . Now, f does not contain a
'Unit, but only two-thirds of a unit ; therefore, | is to be taken
two-thirds of once, which will be equal to f of itself, or f of
}. Therefore,
To reduce a compound fraction to a simple fraction.
Rule. — Multiply all the numerators for a new numera-
tor, and all the denominators for a neio denominator, can-
celling as before, whenever practicahle.
10. Beduce f of 44 ^<^ ^ simple fraction. Ans. ■^.
11. Beduce -^ of 4^ of -^ to a simple fraction. Ans. |.
12. What is the value of 4 of ^ of ^f of ^ ? Ans. ^.
13. What is the value of | of 4 of | of || ? Ans. -^,
14. What is the value of | of 5^ of 4 of -j^ of 4 ? Ans. 1 J.
15. What is the cost of f of | of a pound of tea at -g- of
a dollar per pound ? Ans. -^^ of a dollar.
MULTIPUCATION OF FRACTIONH.
lb
To multiply a whole niimljer by a mixed number.
BuLE. — multiply by tlie fractional part and the whole
number separately^ and add the products.
NoT£.-nt will be found more convenient to multiply by the Iractioiii
flrat.
EXERCISES.
1. Multiply 320 by 8^.
2)320
160
2560
^aj-
*■ :*■ -
■ KP-!'--
2720 Ans.
2. Multiply 3629 by 5^.
3. Multiply 198 by 6|.
4. Multiply 7960 by 3|.
Ans. 24688.
Ans. 1216f.
Ans. 29850..
7960
3*
23880
5970
•
or,
2)7960
3f
2)3980
1990
29850 Ans.
29850 Ans.
5. If a ton of hay cost $17.60, what is the price of 3 J
tons ? Ans. $66.
Note. — ObsArre that where the whole number and the numerator of the
fraction are the same, you must only multiply once.
6. "What is the price of 14| barrels ")f apples at $5.50 per
barrel? Ans. $81.81 J.
7. Find the value of 324^ acres of Jand at $35.85 per
acre. Ans. $11635.31f.
8. What will 4| bushels of wheat cost at $1.75 per
bushel ? Ans.-$7.97f ►
9. Multiply 7598 by 2f ; by 3i ; by 4^; by 7f
46
▲BITHMEnC.
DIVISION OP PEACTIONS.
Division of Fractions is the process of division when the
divisor or dividend or both are fractional.
To divide a fraction by a whole number, •
Rule. — Divide the numerator by the whole number, if
it can be done ivithout a remainder, and set the quotient
over the denominator. Or, multiply the denominator" by
the whole iiumher, ajid set tJie numerator over the product.
1. Divide ^ by 2.
EXERCISES.
|-^2=f Ans. or, | x 2=^*^=1 Ans.
To divide any quantity by 2 gives the half of that quan-
tity ; and as the half of ^ is |, the correctness of the first
method is evident.
The second method may be explained thus :— ^ indicates
that the unit is divided into five equal parts of* which 4 are
expressed. If the denominator 5 be multiplied by 2, the
fraction will be expressed in tenths, and shows that the unit
is divided into twice as many parts as before, and the parts
are therefore only half the value of fifths ; if then the same
number of parts be taken the fraction (-j^) will represent
half of 4, which was required, and the principle holds good
in all cases. '
2. Divide -j^ by 5.
3. Divide ^^f by 4.
4. Divide f by 7.
5. Divide l^ by 9.
6. Divide f^ by 12.
Ans. ^.
Ans. 3^.
Ans. •^.
Ans. ^.
Ans. ^.
To divide a whole number by a fraction,
SuLE. — Divide the whole number by the numerator of
the fraction, if it can be done without a remainder, and
m/ultiply the quotient by the denominator. Or, Tnultiply
the whole nunAtir by the denominator, and divide the pro-
duct by the num^ator.
* EXERCISES.
I.Divide 8 by f
8^2=4, and 4 x 3=12 Ans.
Or, 8x3^24=12 Ans.
2 2
DIVISION OF FRACTIONS.
4?
.»''*>«'•'>,
Ist. Divide 8 by 2=4; but 2 is three times the given
cliviHor, and if 2 is contained 4 times in 8, one-third of 2 is
contained 3 times us often. Therefore, 2nd, multiply 4 by
3=12 Ans.
The second method is merely reversing the order of the
operations for the sake of convenience.
2. Divide 9 by ?.
3. Divide 18 by ^V- •
4. Divide 27 by f .
5. Divide 43 by i.
6. Divide 78 by ||.
To divide one fraction by another.
Rule. — Invert the divisor and proceed as in multipU'
cation.
Note 1.— If the divisor or dividend or both be mixed numbers they
must be reduced to improper fractions.
Note 2,— The rule for this case will solve any case in division of frac-
tions ; but t!\e operations are sometimes more tedious than by the niles laid
down for other coses.
EXERCISES.
1. Divide § by |. ♦ .
7 J m ■■,:■■-. n'r V t
Ans. 21.
Ans. 39.
Ans. 33|.
Ans. 49|.
Ans. 84.
This operation may be explained by reference to the pre-
ceding rules, thus : 1st., divide | by 3, that is (case 1) mul-
tiply the denominator by 3 ; now we have divided by a num-
l)er 4 times the given divisor, hence the quotient -^ is only
^ of what is required, therefore, next, nidtiply by 4 which
give^ ||.=-J=l-j^, ans. By cancelling i-s above the work is
shortened.
■^.'
2. Divide I by 3^.
3. Divide || by |f.
4. Divide || by ||.
5. Divide iH- ^J M-
6. Divide 4^ by |.
7. Divide 9^ by ||-.
8. Divide ^ by l}.
9. Divide ^V by 4}.
10. Divide 5^ by 4f
11. Divide 4^ by 5|.
When it is required to divide the product
tions by the product of several others.
Ans. fj
Ans. Ir^
Ans. i^
. t, Ans. Jf
Ans. 5^
Ans. 13^
Ans. 1^
Ans. -^
Ans. l4
Ans. -^
of several frac-
48
IBrrHMETIC.
Invert all the faetor$ of the divieor and nvultiply alt
together.
12. Divide the product of |, | and \ by the product of ^,
I and •^. Ans. 1{.
Jg
13. Divide the product of |, 4^ and f by the product of 2,
1^, 3^ and I. Ans. 11.
14. Divide ft of 2f by f of 4||. Ans. f .
15. Divide f of | by J of ||. Ans. 1^^^-
A Complex Fraction is an expression of division of Frac-
tions, — the denominator being the divisor, and the numera-
tor the dividend. Hence,
To reduce a complex fraction to a simple fraction,
Divide the numerator by the denominator.
16. Find the value of £
1
4^$-4-"**-
17. What is the value of 4?
18. Reduce | to a simple fraction.
19. Find the value of 7.
20. Seduce -I to a simple fraction.
H
21. Find the value of 7^^.
22. Beduce 9| to a simple fraction.
12f
23. Divide ^ of ^ of 5^ by Sf of | of 18.
2i 6i
To divide a mixed number by a vrhole number, when the-
dividend is greater than the divisor.
Rule. — Divide the integral part of the dividend by the
dnmaor, Tfie remainder with the fraction^ or the fraction
alonej if there be no remainder, will form the nwmerator of
a complex fraction of which the divisor is the denominator.
Beduee this complex fraction to a simple one, and arme»-
it to the qvatient.
Ans. 1^.
Ans. 2|
Ans. ^,
Ans. 6\
Ans. f '
Ans. 18
Ans. 14
Ans. T^.
DIVISION OF FRACTIONS.
49
KXKKCI8KS.
1. Divide 5876§ by 3.
Bemaindcr, 2K_h
3" »•"
2. Divide 7918| })y .1.
3. Divide 491 8 J bv ».
4. Divide 68355| by 7.
5. Divide 19864? by 27.
6. Divide 9131} by 51.
To dividu a whole or mixed number by a mixed number,
when the dividend is greater than the divisor,
BuLE. — Multiply both the dlvinor mid the diindend by
the dencyminator of tJm fractioa in the divisor, and prO"
oeed by the last rule.
EXERCISES.
1. Divide 372 by 4^.
3).'587({§
" 1958| Ans.
AnM. 15831
AuH. 546f|
Ans. 9705^^
An8. 735|
Ans. 17}-^!
1^.
372
2
,•
9
744
82if Ana. .
4'' ' '■
2. Divide 5973| bv 8J.
Ans. 6941^.
3. Divide 386g by' 5f
Ans. 68,»j^,
4. Divide 5987 by ^.
Ans. 1651f|.
5. Divide 9176 by 5^.
Ans. 1668^
6. Divide 763^ by 2^.
Ans. 286f
NoTK.— The probloms in the laHt two cases, as well an all others in divi-
■ion of fractions, may be solved by the f^eneral rule under CuKe 3. The
methods here mven are deductions from that rule ; and when the dividend
is a large number they are very convenient, and those generally adopted.
LEAST COMMON DENOMINATOR.
It has already been shown that to divide both terms of a
fraction by the same nmnber does not change the value of the
fraction. Hence, also.
To multiply both terms of a fraction by the same uimiber
does not change the value of the fraction.
Two or more fractions have a common denominator when
their denominators are alike. Thus f , ^ and | have a com-
mon denominator, 7.
Any two or more fractions may be reduced to equivalent
fractions, having a common denominator.
A common denominator of two or more fractions must be
a common multiple of their denominators, in order that the
6
50
AIlITUMKriO.
flquivalcnt fmctioug ))uvin{{ the common ilenominator shall
be simple tViictiunB.
Thus u coiinnun denominator for the fractions } and |
must be u common multiple of 3 and 4, as 12, 24, 36, &o.,
and ^ and '} muy be reduced to equivalent fractious having
12, 24, 3() oi' any other common multiple of 3 and 4 for their
common denominator.
The leitHt, common denominator of two or more fractions
\H the Least Common Multiple of their denominators.
To re(hice two or more fractions to equivalent fractions
having u common denominator.
lieduce f and j to equivalent fractions having a common
denominator.
2x4=8
3x4=12
3x3= n
4x3=12
like r
vason
Tak(? 1 2 for a common denominator. Then,
liintje the numerator and denominator of a frac-
tion may lie multiplied by tlie fiame number
without alterin^if itn value, multiply both terms
of ^ by 4, l)ecau8e it makes the denominator
12; and multiply botli tiTiriH of ^ by 3, for a
and we obtain -,^y and ^\j as equivalent fractions
havinj^ a common denominator.
RuLK. — Multiply both tennn of e,ach fraction by the pro-
duct of all the deiiomlnatorti except its own,
2. Reduc(i § , 3 and 5 to equivalent fractions having a com-
mon denominator.
5x8x3=120
3x6x3= 54
2x6x«= 96
8 = 111-
•i — 00
a— 14 4-
M-
6x8x3=144
3. Reduce ■§-, 4^ and g to equivalent fractions having a
common denominator. Ans. -f^, ||^, -^j.
To reduce two or more fractions to <M|uivalent fractions
having the least common denominator.
' Rule. — Ftnd the least comm&n 'multiple of the denomi-
nators which ivill be tlie least common denominator.
For the numerators, dh'Ule tlie least common denominct-
natoi' by the (hiwmlnator of each fraction, and multiply
the quotmit by tlie corxesponding numerator.
JXotr. — ^If the fractions are not all niinplc tlipy must bo reduced to
aiiuple fractions and to tlieir lowest terms.
ADDITION OF FllA(TlOXS.
51
KXRRCI8KS.
1. Reduce! -^, f mul } to their o<tuivulontM with thoir least
4>oniTnon denominator.
The leaHt common (Umominutor la 30. Then,
8)3
«■'
3)r«o
5)30
thatiH, ^^jj^
The proctwH in eiiuul to tho foHowinjjf : — <
1 X 1 ;>= U 2 X 10=20 3 X = 18
2x15=30' 3x10=30' 5xfi=3d'
and the multiplier for each fraction la fomul by dividin{( the
leatit common denominator hy the denominator of the fraction.
2. Reduce ^, J, 3, to their least common denominator.
3. Reduce ^, J, -j^y, to their least common den iminator.
Alls. -jV fj, tV-
4. Reduce ^, J, ^'»J^, ^J-, to their lejist common denomina-
tor. AnU'lhihU^U-
5. Reduce ^j -fV> *^°^ ^o t^* their least common denomi-
nator. Ans. II, f|, V4*.
6. Reduce J, 2^ and ^'j- to their least common denomi-
nator. Au8.^f, W»li-
7. Reduce } J, 1 -^'g-, 7^ and ^''^ to their least common de-
nominator. Ana. «}, ]§, 8,"5/>, f|.
8. Reduce §, ^ of 3^- and § of ^ to their 'least common
denominator. Ans. ;f{|, i/^^, 1^.
9. Reduce f of «, f of §, ^ of | of ? of 2g to their least
common denominator. Ana. f^g, f^J' HJ*
10. Reduce -} of ^, -^ of 4^ and ^ to their least common
denominator. 9 Ans. j%-, |^«, x^.
ADDITION OP FRACTIONS.
Addition of F'ractions is the process of finding the sum of
several fractions.
To add fractions,
Rule. — If the fractions to he aililed have the 8anie rfe-
iiominator, adtl t/telr numerators, and write tl^ sum over
fliecoriUium ilenomlnator. If the f radio ns have not the
naine il^.noniinator, reduce tJiein to a coiti/niou ilenmnvlncir-
tor; add tJie new numerators, and set the sum over tlie
common denominator.
52
ARITHMCTIC.
>H
EXERCISES.
1. Add together S, 5, ^ and |.
6 3 ,7
rr» 1 f > If
2. Add together , , .
3. Add togetlier H, f »-, 4« and
4. Add together -j and -J. ,,,
and r^p
it*
X
4
-4- X ,,
4— «
- !» ly + lJ— IT— *Tf
5. Find the Kr.m of i and ^. * , ./'• <
H. Find the sum of ,^. and -|^.
7. Find the sum of ^ and }-^.
8. Find tlie sum of \^ J- and }-»] .
9. Find the sum of |, j} and -j^.
10. What is the sum of ^, f ,
-J^ and I ?
Ans. Y=:2..
Ans. 2^^
Ans. l|..
Ans. 1 J..
Ans. -j-j.
Ans. Iff-.
Ans. 2u\f
Ans. l|-»r.
Ans. 2|.
When there are mixed numbers it is as well to add the
fractions and whole numbers separately, and add their sums.
11. Find the sum of 1§ and 2^.
X2 =
3 y 3 _ 9
t
IT
10 I 9
15
1+2=3
Ans. 5-§.
Ans. 10-^^.
Ans. 11|-J.
Ans. 40^.
Ans. 21^'^.
4j^ Ans.
12. What is the sum. of 2^ and 3^ ?
13. Wliat is the sura of 2^^, 3f and 41 ?
14. Add together U, 2^, 3^ and 4j^.
15. Add together 16|, 12^, 8f and 2^.
16. Find the sum of f, 4^, -/^r, 9|^- and o||.
17. Add together ^ of f and | of -f of -/v. Ans. -^.
1 8. What is the value of ^ of 6f + -^ of f of 7^ ? Ans. 4:J^.
1 9. Find the value of f of 96^ + § of -]^ of 5 J- ? Ans. 59 J|.
20. Add together f|-, 7^?'^, «- of 4 J- and o^V ^^g jg »7
8k .' T'*^*
The following will be foimd \;seful : —
To adf^ two fractions both of which have 1 for a numerator.
Rule. — Add the denominators for a numerator, and
multipli/ them for a denominator. , -
EXERCISES.
1.
Add ^ and ^.
7. Add 1 and y.
2.
Add ^ and ^.
. 5. 8. Add ^t|-and ^V.
3.
Add ^ end ^.
' ' 9. Add ^and^.
4.
Add ^ and 4.
10. Add 1^ and i.
5.
Add ^ and f .
■ \. Add J^ and y*g-.
6.
Add ^ and ^V-
-2. Add 1 and ^\r.
SUmUACTION OF FllACTIONS.
53
I SUBTRACTION OF FRACTIONS. v .,
.'.i Subtraction of Fractions is the process of finding the dif-
ference between tw<j fractions.
1<TJLE. — // the fractiona have the sartie (lenominator, 8vh-
tract the smaller mtmerator frmti the larger and set tJis
remainilcr over the common detiominator. If they have
.not the same dencmDuitor^ reduce them to a common de-
noi7iinat«r, take the difference between the new numerators,
and set it over tJie common denominator.
EXERCISES.
ir sums.
1. From -j'j- take
n
■ff
iV
-A-
•^j- Ans.
2. From -J take ^.
3. From -jV take -jOf.
4. From |4 take ^.
5. From ^ take |.
3^1 — 21
2^5 — t a
- A 5 — ,j—
and 1^-
■y}=}i Ans.
6. From ^ take §.
7. Find the difference between ^ and -^tr.
Ans. ^.
Ans. -j^j.
Ans. -^.
Ans. -g^.
Atih ^^
Ans. ^\.
Ans. -^.
Ans. ||.
■ 8. Take .% from }; »-.
9. Find the difference between § and -^^
10. From -j-^- subtract ^^.
Wften a mixed niomber occurs it may be reduced to an
hnprojper fraidion and the subtraction performed accord-
ing to the rule. • '^
Or,the fractions and ivhole numbers may be subtracted
■ separately ; but it must be ^^bserved to add 1 to the fraction
ill the minuend, if it be less than that in the subtrahend,
'and carry 1 to the units Jigure of the stibtrahend.
11. From 4f take 24,
45
12. From 5| take 3f .
13. From IS\-^ take 12^^.
14. Take 437 y\ from 9(i3n
15. From 3^ .subtract If.
16. From 8^ tiike 3§.
17. From 7| take 4|.
Ans. 2i.
Ans. 2-^-.
Ans. 6t^.
'24'
Ans. 5263
12*
> Ans. 1|.
Ans. 4f.
Ans. 2-^-.
54
.'* ARriHMETIC.
18. From IG subtract 3^. ^ ' '' ' ' '^ Ans. I25.
Here it is easier to subtract the fraction and whole num-
bers separately. Thus, '■'
16 Subtract ^ from 1 (§") and | remain, carry
M
3j^ 1 to 3=4 ; 4 from 16=12.
' "'' \2i Ans.
19. From 391§ take 14Vf.
39 1 § Add 1 to f = 1 1 = i[ ; then.
\"i*:
r>:s^ri'X\i<
Ans. 243^.
147$
-i^x
4-^3
n
243 1-^ Ans. f x l| x-^V and f^— 1^=1^- Carry 1
to the whole nuiubors.
20. From 320} take 249f . Ans. 70^»
21. A has $725| and B has $690f how much more has A
than B ? Ans. $34|.
22. A man pwned |-§ of a ship, and sold ^ of his share ;
how mucli had he left ? Ai "
23. What is the difference between g of 1|- and 4^^
Ans. -f^g.
Ans. 7^^«
24. After selliiig 4- of f +i of | of a farpa what part of
it remained ? Ans. ^-g.
When the numerator of both fractions is 1,
The difference of the denoininatcyra will he the numerator
and their product the denominator of tJie difference.
EXERCISES. •
1. From ^ take 4.
2. From | take -?y.
3. From ^ take i. ' ■ - ' '
4. From ^ take -J.
5. From ^ take J . ' '
6. From ^ take -jU-. ^ ' '
DECIMAL FRACTIONS.
*
A Decimal Fraction is one that has 10, or some power of
10 for its denominator, as -^^, -j-|^, yj-^ir, &c.
The word decimal is derived from the Latin, decern, ten.
Observe the relation. betwv.en the decimal fractions, -^j
Tinr» 1 (1 ' ^^*-
The first is six tenths, or -j*^ of units ; the second, six
hundredths, or -j'^^ of -^^^ ; the third six thonsandths, or -^ of
Hence, it is seen that these fractions bear the same relation
1 )I X'lM AI. FU ACTIONS.
m.
to one another as exists between the same (lijj;it3 in a whole
number, and also tliat the first of the series })ear8 the same
relation to 6 units. It' therefore the numerators of these
fractions be arranj^ed side by side, thus, 666, tliey form an
extension of the Arabic system, and may be used alone, or
annexed to whohj numbers without their denominators.
Decimals are generally so written, and »re known by being
preceded by a period (.) called the decimal point.
TABLE OF DECIMAL OKDEBS.
s . ■ - ■
i : « J ■'■■'.■- "^ :- • .
■ ' it il
1st place .5 read 5 tenths.
2nd " .04 « 4 hundredths.
3rd " .006 " 6 thousandths.
4th " .0007 " 7 ten-thousandths.
5th " .00003 " 3 hundred thousandths.
6th « .000002 " 2millionths.
7th « .0000009.... " 9 ten-millionths.
8th " .00000001.. " 1 hundredth millionths.
9th " .000000008 " 8 billionths.
Sum— .546732918 " five hundred and forty-six
million, seven hundred and thirty-two thousand, nine hun-
dred and eighteen billionths.
And it will be found that if each of these decimals be ex-
pressed in the common fractional form, viz.: -j*^, -j-ffs, n/W»
&c., and added together by the rule for adding common frao-
tions the sum will be JlA«:7^_s_w^_.
By examing the above table and what has l)een said it will
be seen that the value of a decimal figure depends on the
place it occupies, diminishing in a one tenth ratio for every
place it is removed farther from the decimal point.
Hence to place a ciplier on the right of a decimal does
not alter its value. r)ecause tlie cipher is nothing in itself,
and, so placed, does not change the place of the figures.
But a cipher placed on the left, between the decimal and the
point, removes the figures one place to the right, and thus
divides the value of the decimal by 10.
56
AIirrUMETIC.
' To read decimals expressed by figures, ^m, n,r
BuLE. — Read tlie declmat as a. whole nuihber and give it
the name of the right hand figure.
48.7804.
, 83.0084.
A.J .
EXERCISES.
Read the following : —
.2. . ,.. .8004. , ,
.04. ., .4010. h ,
.138. .21042. 121.18006.
.4531. .000014. 345.000018.
.0098. .1743196. 909.000999.
.00006. .0008980. 1203.080764.
To write decimals in figures.
Rule. — Write the decimal figures as a whole number;
then place the point so that the right liand figure shall have
its expressed value, x>lacing ciphers to the left of the signif-
icant figures if necessary.
EXERCISES.
Write decimally the following quantities : — J
1. Five tenths.
2. Twenty-two hundredths. '^ .,, ■',.
3. Eight^'-seven thousandths. , - .v * ,
4. Fiity-six ten-thousandths. ' ' ^' ,-.• . .'-.
5. Tliree hundred and four ten-thousandths.
6. Five thousand three lumdred and forty-seven ten
thousandths. - j - - «
7. Eighty-eight millionth s.
8. Eight lumdred and eight, and eiglit thousand and
eijiht millionths.
9. Ten tliousand and fifty-seven hundred-thousandths.
10. One lumdred and twenty-one, and one hundred and
twenty-one thousand, one hundred and one millionths.
11. Seven thousand and seven ten-millionths.
12. Twelve thousand, and twelve thousand, one hundred
and one ten millionths.
1 3. Six liundred thousand six hundred and seven millionths.
14. Twenty-seven thousand nine hundred and five, and
forty thousand and four millionths.
15. Ninety-seven million, four hundred and fifty-three
thousand, one hundred and sixty-eiglit billionths.
A decimal is deduced from a common fraction by changing
DECIMAL FRACTIONS.
57
seven ten
the unit of the fraction to tenths, hundreths, &c., and per-
forming the division indicated. Thus ^ means ^ of 1, or ^
of 4 imits ; but 4 units equal 40 tenths, hence ^ of 4 units
=-J- of 40 tenths, or Y tenths =8 tenths, or .8. Again, | of
of a unit=''^^ tentli8=.8, and g- of a tenth over, that is *^
hundredths =.07 and -*• of a hundreth over, that is ^ thous-
andths =.005, and adding these three parts together they
make .875 as a decimal equivalent to |. Therefore,
To deduce a decimal from a common fraction,
RuLK. — Annex ciphers to the numerator and divide by
the deaaminator, placing the 'point in the quotient so as
to make as many decimal fir/urea as ciphers annexed to
the 7humerator.
■ o' t
EXERCISES.
1. Deduce an equivalent decimal from \.
4 )1.00
.25
2. Deduce an equivalent decimal from ^.
3. What decimal is \ equal to ?
4. Reduce ^ to a decimal.
5. Reduce % to a decimal.
6. Reduce ^ to a decimal.
7. Deduce a decimal from -j\. .
Ans.
.25.
8. Deduce a decimal from ^ ;^.
9. What decimal is \1^ e([ual to?
10. What decimal is \% equal to ?
11. Reduce \ to a decimal of tux places.
12. Reduce \ of ^^,T- to its equivalent decimal.
13. What decimal is equal to -^^^
• Ans. .6.
Ans. .75.
Ans. .5.
Ans. .375.
Ans. .625.
Ans. .4375.
Ans. .52.
Ans. .53125.
Ans. .47916 + .
Ans. .714285 + .
Ans. .3125.
US-
Ans. 3.6.
In deducing decimals from common fractions when any
quotient figure or figures are found to continually repeat, as
in exercises 9 and 10 above, the decimal is called an Infinite
or Circulating decimal.
The part of the decimal which repeats is called a Ke-
petend.
A repetend may be terminated at any point where it begins
or ends by taking it as the numerator of a common fraction,
and as many 9*3 as there are repeating figures for a denomi-
nator and annexing the fraction to the preceding decimal
if any.
Thus I is equal to .8333, &c., in which the figure three is
M
ARITHMCTIC.
a repetend. This decimal is correctly expressed thus, — .8^>
or .83;|, or .833^, &c., that is the ^ is ^ reduced to its lowest
terms.
Again, | is equal to .714285 repeated ad infinUumy
and is correctly expressed -J^^^H = \, or .71 4285 1-, or
.714285714285^, &c.
A repetend of one figure is distinguished by a point placed
above it, thus .83.
A repetend of more than one figure is denoted by a point
over both the first arid the last figure, thus .712485.
14. Reduce ^ to a decimal.
15. Reduce § to a decimal.
16. Reduce % to a decimal.
V 1
17. Reduce f^ to a decimal.
18. Reduce •j'j- to a decimal.
19. Reduce y^ to a decimal.
20. Reduce \^ to a decimal.
21. Reduce ^ to a decimal.
22. Reduce -j^j to a decimal.
Ans. .3, or .3^.
Ans. .6, or .6f .
Ans. .5, or.5-§.
/ Ans. .8, or .81.
Ans. .63, or .63-j^.
Ans. .583, or .58^.
Ans. .916, or .91|.
Ans. .428571, or .42857 If
Ans. .461538, or. 46 1538yV
To reduce a decimal to a common fraction.
Rule. — Write tJie decimal for a numerator, omitti/ng the
point and cipJtera on tJie left ; and for a denominator, 1
with 0,8 many cipJisrs annexed as there are figures in the
decimal, and reduce the fraction to its lowest terms.
EXERCISES. -^
1. Reduce .5 to a common fraction.
2. Reduce .25 to a common fraction.
3. Reduce .75 to a common fraction.
4. Reduce .875 to a common fraction.
5. Reduce .0625 to a common fraction.
6. Reduce .125 to a common fraction.
7. Reduce .3125 to a common fraction.
8. Reduce 2.125 to a common fraction.
9. Reduce 16.002 to a common fraction.
10. Reduce .0175 to a common fraction.
11. Reduce .390625 to a common fraction.
12. Reduce .003125 to a common fraction.
13. Reduce .15234375 to a common fraction.
Ans. ^.
Ans. \.
'■"- Ans. |.
Ans. |.
Ans. -^.
Ans. \.
Ans- j5({.
Ans. 2|.
Ans. IQ-sl-^,
Ans. ^J-^.
Ana. f 2.
Ans. 3^.
Ans. 5^^.
DKCIMAL FRACTION'S.
m.
Wien tfie decimal is a rfipetendf make the decimal with
the point omitted t/ie numerator, and as many 9'« aa tliere
are repeatiiuj Jirjurea for denominator, and reduce the
I fraction as before.
Ans. ^.
14. Reduce .3 to a common fraction.
15. Reduce .8 to a common fraction.
16. Reduce .8888 to a common fraction.
17. Reduce .72 to a common fraction.
18. Reduce .307694 to a common fraction.
19. Reduce .857142 to a common fraction.
'.'t''
Ans. 5.
Ans. -J.
Ans. -^3.
Ans. 1.-
When the decimal is composed of a finite paH and a re~
petend, convert the repetend into a cmnmon fraction, and
annex it to the finite part ; under this write the denomi-
nator of the decimal, and reduce the complex fraction to a
simple one. j; 'V
20. Reduce .83 to a common fraction.
.83=.8|=8i, that is 8t^=8^-^10=lg^ x ^\.:
21. Reduce .916 to a common fraction.
22. Find a common fraction equal to .583.
23. Find a common fraction equal to .7083.
24. Reduce .027 to a common fraction.
25. Reduce .78545 to a common fraction.
: J. Ans.
Ans. \^.
Ans. ^.
Ans. ^,
Ans. ^.
The followiuf^ rule deduced from tlie above will be found
[Convenient in solving (juestions like the last six.
Rule. — Srblract the finite part of the decimal from tJie
Iwhole, use the remainder for a numerator, and for a de-
\nominator as many 9'« as there arejiijures in the repetend
\ and as many ciphers annexed as thsre are fifjures in the
initepaH. „ ... '< . ■ -j ■■ '.
ADDITION AND SUBTRACTION OF DECIMALS.
As decimals are an extension of tlie common Arabic system,
they are added and subtracted in the same manner as whole
numbers ; and it should be remembered that fif/ures of the
same order must be placed under one another, that is tenths
under tenths, hundredths under hundredths, &c. In other
words,
«60
ARITHMETIC.
Arrcmge the quantities to be added or ftubtraded ao that
the decimal points eJiall stand i^i a vertical column, add
or subtract as in whole numbers, and place the decvmal
point in the sum or difference directly under those in the
numhers added or subtracted.
*t EXERCISES.
Add together .575, .0456, .73, and .1642i).
Observe the decimal points in a column, so
that tenths are under tentlis, Imndredths under
hundredths^ &c. The column of tenths, with
what is carried to it, amounts to 15 tenths, that
is, 1 unit and 5 tenths.
.575
.0456
.73
.16425
1.51485
2. Addtogether 21.611,6888.32, 3.4167. Ans. 6913.3477.
3. Add 6.6J, 636.1, 6516.14, 67.1234, 1233.
Ans. 8458.9734.
4. Add 14.034, .25. .0000625, and .0034.
Ans. 39.0374625.
5. Add 16.75, .375, 5, 3.4375 and .000875.
* Ans. 25.563375.
6. Add 173, 7000.0005, 1.7, 125.728 and .0005.
: , Ans. 7300.429.
7. Add .16, 39.5, .7283. Ans. 40.3949(i.
y 8. Add 700.83, 16.765, .72835, 81.9. Ans. 800.227238.
9. Add .142857, .0186, 920, .0139428571. Ans. 920.1754.
10. Reduce to decimals and find the sum of 2^, 4| and 5-^^^. |
( < ■.■'^■■:yr'->:.^i ■■ Ans. 12.775.'
>C 11. What is the sum of .76, .416, .46, .648, .23 ?
Ans. 2.52087.
1
\¥\i
Find the sum of .427, .416, 1.328, 3.029, and 5.476 J
Ans. 10.67803712.
13. Find the sum of 35 units, 35 tenths, 35 hundredths!
and 35 thousandths. Ans. 38.885. |
14. From 8.53 subtract 3.643.
Arrange the numl)er8 so that the points
shall be in the same column, and subtract as
in whole numbers. Tlie place of thousandths
being vacant in the minuend, we borrow one
8.53
3.643
4.887 Ans.!
DKCIMAL FIIACTIONS.
Gl
[from tlu; hundredtliH, which U 10 thouHundthH, subtract 3
[thousandths i.ud carry one as in whole numlxjrs.
15. From 20.03()| subtract 8.77^.
20.0365 20.03(>J
8.7733 or 8.773|
■>.>\' .(V
'1 ;
11.2«32
Ans.
11.2631 . \.
16. From 24.0042 take 13.7013. Ans. 10.3029.
17. From 170.0035 take 68.00181. Ans. 102.00169.
18. From .0142 take .005. Ans. .0092.
19* What is the diflferencc }»etween .05 and .0024?
Ans. .0476..
20. What is the diflference between 72.01 and 72.0001 ?
Ans. .0099.
21. From 19 take 8.9991. ;: "^ Ans. 10.0008, or 10.00(^5.
22. From .4 take. 04i^. ,' Ans. .356, or .35§.
23. From 2^ take ij. Ans. .95.
24. From 1.169? take .93^V Ans. .238857142, or .238^.
25. Wliat is the ditference between 24^ tenths and 3701
[thousandths? Ans. 1.251.
26. Subtract 1^ liundredtlis from 49§ tenths.
Ans. 4.9225,
MULTIPLICATION OF DECIMALS,
Multiply .375 liy 7.
Operation by common fractions.
Opprallon,
.375
7 "'■
' ,\r
mn v7 — 3636 — onss — 9 fi'j/i
To 0' ^ T — J 000 — 'to — — "'i''
Multiply 2.75 by .9 , operation.
Operation by common fractions. ^ 2.75
\ .J
•^Tfftr— ro (T ^ Iff— to'i) — ^To ——•*'«>
Hence, to multiply decimals.
Rule. — Multlplt/ as in ■whole nvmhers, and point off in
Ithe product as inany decimal places as there are in the
Imultiplicand and multiplier tor/ether. If there be not
\ enough figures in the product to give the required number
lof decimal pluces, supply the deficiency by prefixing ciphers.
62
ARITHMETIC.
1. Multiply 2.54 by .34
yji. Multiply 4.16 by .014.
,3. Multiply 4.5 ])y 4.
4. Multiply .01 by .15.
5. Multiply .08 by 80.
«. Multiply 18.46 by 1.007.
7. Multiply .00076 by .0015.
. H. Multiply 7.49 by 63.1.
9. Multiply .0021 by 21.
10. Find the continual product of .2, .2, .2, .2, .2, .2.
^ Ans. .000064.
11. Find the continual product of .101, .011, .11, 1.1, and
11. AnB. .001478 741
Ans. .020736.
Ans. .86.36.
Ans. .05824.
Ana. 18.
Ana. .0015.
Ann. 6.4.
AnB. 18.58922.
Ana. .00000114.
Ans. 472.619.
Ans. .0441.
12. Multiply . 144 by .144.
13. Multiply 14.583 by 2.75
«:J
14.583
2.75
72916
1020833
2916666
40.10416
33 &c.
66 &c.
33 &c.
G6&C.
In this exercise the last figure in the
midtiplicand in a repetend, and muat be
treated as such. In multiplying by 5
we must carry 1 from the product of
3 understood on the right, and the 6 in
the product is a repetend. In a similar
manner we carry two when we begin to
multiply by 7, and the 3 in the product
is also a repetend for which reason we till up the place on the
right of the product usually left blank. Also in multiplying
by 2, as the 3 in the multiplicand is a repetend so is the 6 in
the product, and we must fill up the two places on the right
with 6'8. Then in adding the partial products, we must allow
for other columns on the right, made up of the repeating
figures, and so carry 1 at the beginning.
Note. — The above method answers very well when the
multiplicand alone contains a repetend of only one figure ;
but when the repetend consists of more than one figure, or
when there is a repetend in both multiplicand and multiplier,
the process becomes complicated, and it is usual to proceed
by the following
RuLP]. — Reduce the decimals to common fradiona and
perform tm multiplication required; then reduce tJie frac-
tion, if any. In the product to a decimal,
14. Multiply 7.416 by 8.5. Ans. 63.041*6.
15. MiUtiply .078 by 7. Ans. .552.
> "
Ml LTI PLICATION OF DECIMAI-S.
68
1(». Multiply ;).<53H ]>y .27.54. » Ana. 1.5">295.
17.Mnltiply.7aby2.fi. ' Ana. 1.95.
y,lH. Multiply r>.7.3fi by .41(5. Ana. 2.39015.
19. Multiply 9.4*>7142Hhy ..)384f>i. Ann. 5.0923()7fi.
To Multiply by 10 or any power of 10, as 100, 1000,
10000, &e. , . : ; ,, ; - .,; , .,
RuLK. — Af<n<fi the iledmal point aa many jAacea to the
r'ujht as there, are ciphers in the innltiplirr.
KXKIICISES.
1. Multiply 4.5 by 10.
2. Multiply .007 by 100.
3. Multiply 170.5 by UK).
4. Multiply .0(125 by 1000.
5. Multiply 4.8(1 by 1000.
(>. One poiuiil Hterlinj; is worth ^4.H(i§ ; what is the value
ofLlOO? Aus. !t^48G.6G§.
7. What will 1000 barrels of Flour cost at $6.75 per barrel ?
Ana. ^()750.
8. What is the cost of 100 acres of land at $17.37^ per
acre? Ans. $1737.50.
9. What is a million pounds Bterlin^ worth at $4.86f
each? . , - Ans. $486(>G()6.fi6§.
10. Multiply 6^ by 100000. Ana. 671428.'571428.
To multiply by 15.
Move the point one place to the rir/ht, taJce the half and
add it. ^
To multiply by 25.
Move the point two places to the right and divide by 4.
To multiply liy 250.
Move the point threp: places to the ri(fht, and divide 4.
To multiply by 75.
Move the point two places to the right and suhstract a
a fourth part.
To multiply by 7^. . ,,
More the point one pkice to the right and suhtract a
Jourth part.
lo multiply by 12^.
Move the point two places to the right and divide by 8.
To Multiply by 2^-
■Move the point one pla^e to tJie right and divide 4. •
64
ABmiNi>7no.
'ii
KXKRCI8B8,
1. Multiply
2. Multiply
3. Multiply
4. Multiply
5. Multiply
G. Multiply
7. Multiply
8. Multiply
9. Multiply
10. Multiply
2r).7fi4 by 25.
.0890 by 1.5.
.798.5 by 2.10.
240.8 by 7^.
3.987 by 75.
19.50 by 2^.
100.5 by 1.50.
.0(X)32 by 2.50.
73.5 by 750.
99 by 2^.
Ana. 644. K
Adb. 1.344.
An8. 199.625.
AuB. 1806.
AnH. 299.025
Ans. 48.7.5.
An8. 24075.
AnH. .08.
An8.5512.5.
Ana. 247.5.
• DIVISION OF DECIMALS.
Division is the converse and proof of multiplieution, — the
product becoming the dividend, the midtiplier or multipli-
cand, the divisor, and the multiplicand or multiplier, the
quotient.
Hence, since the product contains as many decimal places
as the two factors to<j;ether, it follows that the dividend con-
tains iis many decimal places as the r' isor and quotient
together, or
The quotient miiM contain aa many decimal places as
the dividetul has more than the divisor.
From this, again, it follows that the dividend must con-
tain, at least, as many decimal iigures as the divisor.
Therefore,
KuLK. — Wlien the dividend does not contain as many de-
cimal fif/iircs as the divlsm' annex ciphers to make up the
number. Tlusn divide as in luJiole nuimhers, and the quo-
tient will he a whole number. If there he no quotient sofar^
or if there be a remainder, and it be desired to carry the
division fartJter, annex as many more cipfiera as necessary ,
continue thedivision, and the additional figures obtained
in the quotient will be decimal.
Note. — When there is not enough figures in the quotient
to give the required number of decimal places, the deficiency
must be supplied by prefixing ciphers.
UIVIHIUN UF DHCIMALH.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
KXKUCISKS.
Divide 1728 by .12.
.12 )1728. 00
144W Ann.
i'.»
Divide 28 by .4.
Divide 21 by .5. ^.
Divide 86.07o by 27.5.
Divide 24.73704 ])y 3.44.
Divide .21318 by .19. .
Divide 9.9 by .0225.
Divide 81.2d9r» by 1.28.
Divide 3.15 b> 875.
Divide 8735.724 by .9.
Divide 724.573 by .7.
Divide 573.183 by .6.
Divide 6927.8516 by 78.5 to seven places of deciinalt).
Ans. 88.2528874-}-.
Divide 9.6 by .55. Ans. 17.57.
Divide 12173.9583 ])y .3,1416.
3.1416)12173.958'33(3875 Ans.
Ans. 70.
Ans. 42.
Ans. .3.13.
Ans. 7.191.
Ans. 1.122.
Ant*. 440.
Ans. 63.445.
Ans. .0084.
Ans. 9706.36.
Ana. 10,35.104 + .
Ans. 955.305.
942490 »«•
«?-.,.,': I I
V.l ;'
2748958 3 3
2513.333 3 3
2356250
219916««
1570833
1570838
',i>
■«» f
The above method of dividing, when the divisor contains
a repetend, is somewhat tedious and requires great care.
The more usual method is to reduce the repetend to a coip-
mon fraction, and then divide by the mixed numb«r^
Thus — ''.,
3.141§)12173.9583(3875 Ans. •■ «;4f«|!ufM Ji
3 3 . .•J.V^tWJMliH 1^
9.425 36521.875. n- >
66
ARITHMETIC,
16. Divide .8 by 2.6.
17. Divide 6020.06 by 4.86.
18. Divide 1.77975 by 25425.
AnB. .3.
Ans. 1237.
Ad8. .00007.
To divide by 10 or any power of 10, as 100, 1000, 10000,
1000000, &c.
Rule. — Move the decimal point as many places to the
left aa there are ciphers in the divisor.
Ans. 342.55.
EXERCISES.
1. Divide 3425.5 by 10.
2. Divide 57.75 by 100.
3. Divide 1444.755 by 1000.
4 Divide 8.J39 by 100.
5. Divide .75 by 10000.
6. Divide 5863.72 by 100000.
7. If it cost $7000 to furnish a meal for an army of
100000 men, what is the cost of each man's meal ?
Ans. 7 cents.
h
S ■'
DENOMINATE NUMBERS.
An abstract number is simply a number without reference
to any object, as, 7, 16, 39, &c.
A Concrete number is a number in connection with some
object oro})ject8 named, as, 1 h'jrse, 7 men, 39 ships, &c.
Denominate numbers are coi.crete numbers applied to the
denominations of weights and measures.
TABLES OF WEIGHTS AND MEASURES.
CANADA C
100 cents (cts.) are 1 dollar
BRITISH OR STERLING
CURRENCY.
4 farthings aro 1 perny (d.)
12 pence " 1 shilling (s.)
20 shillings " 1 pound (.i\)
5 shillings *' 1 crown.
21 shillings " 1 guinea.
URRENCY.
(S).
UNITED STATES CURRENCY.
10 mills
li) cents
10 dimes
^0 dollars
are
a
1 cent (ct.)
1 dime (d.)
1 dollar ($.)
1 eagle (ea.)
Although the abovo is in theory
the U. »S. table of currency, in prac-
tice it is thtt 8Amc as that of Canada.
■-k
■"I
DENOMINATK NUMMUW.
67
AVOIRDUPOIS WEIGHT.
The Imperial pound avoirdupois, containing 7000 grains,
is the standard for weighing heavy articles, as meat, groceries,
vegetables, grain, &c.
16 drams (drs.) are 1 ounce (oz).
■ 16 ounces " 1 pound (flj).
100 pounds " 1 hundred weight (cwt.) or cental.
20 hundredweight " 1 ton (t).
In Great Britain 28 lbs. are 1 quarter, and 4 quarters, or
112 lbs. 1 cwt. This manner of reckoning weight of heavy
articles has been used in this country, and still is, in some
exceptional cases, but it is gradually gro Mng less prevalent.
In the Dominion of Canada and the United St;ites a quarter
means 25 l!)s. ; but it is not much used. 112 lbs. is the
weight of a. quintal of fish ; and a ton of coal is 2240 lbs., or
20 cwt. of 1 1 2 1)8. each . ' The "Weights and Measures Act" of
1873, which fixes the ton at 2000 flis. will probably have the
effect of doing away with the " long ton," in weighing coal
as well as other articles. In Great Britain, 14 lbs. make 1
stone.
TROY WEIGHT.
The Troy ounce (480 grains) is the standard for weighing
gold, silver, platina and precious stones.
24 grains (gi's.) are
20 pennyweights "
12 ounces
a
1 pennyweight (dwt.)
1 ounce (oz.")
1 pound (lb.)
APOTHECAllIES WEIGHT.
Apothecaries mix their medicines by this weig'it, using
the Troy ounce, but they buy and sell by Avoirdupois.
20 grains (grs.) are 1 scruple ( 3).
3 scruples " 1 drum ( 3 ).
8 drams " 1 ounce ( ? )•
* 12 ounces " 1 pound (tti.)
■•■;*-'^ ^i^i... . , ■ . - .
PUODUCE WEIGHT TABLE.
By the "Weights and Measures Act 1873*' the weights of
produce are fixed as follows : —
68
ARITHMETIC.
To the
bU!<llvl.
Wheat. 60 lbs.
((
<(
((
II
Indian Corn 56
Rye 56
Peas 60
Bailey 48
Oats 34
Beans 60
Clover Seed 60
Timothy Seed 48
Buckwheat 48
Flaxseed 50
Hemp Seed 44
After the first of January, 1874, whenever any of the
above articles are bought and sold by weight, they are to be
specified by the cental (100 fi)s.) and parts of a cental.
Bushel in the above table means the Winchester bushel.
To the
bui.hcl.
Blue Grass Seed 14 lbs.
Castor Beans 40
Potatoes 60
Turnips 60
Carrots 60
Parsnips 60
"Beets 60
Onions 60
Salt 56
Dried Apples 22
Dried Peaches 33
Malt 36
(C
LINEAR OR LONG MEASURE.
\
ii
If ■';
i
■ i
The Imperial yard is the standard measure of length, from
which all other measures of length, whether lineal, superfi-
cial or solid, are derived.
12 inches (in.)
3 feet
5^ yards
40 rods
8 furlongs
3 miles
are
1 foot (ft.)
1 yard (yd.)
1 rod, pole, or perch (rd.)
1 furlong (fur.)
1 mile (m.)
1 league (league.)
11
i
i
I
1
'■1-
SQUARE MEASURE.
144 inches
9 feet
30^ yards
40 rods
4 roods
are
1 foot.
1 yard.
1 rod.
1 rood.
1 acre.
LAND MEASURE.
LENGTH.
'i-^\ inches
25 links « 1 rod.
4 rods or 100 links " 1 chain.
80 chains " 1 mile.
are 1 link. 10000 sq. links are I sq. chain.
AREA.
10 sq. chains " 1 acre.
DKXOMIXATE NUMBERS.
69
CUBIC OR
1728 inches
27 feet
128 feet
SOLID MEASURE,
are 1 foot.
" 1 yard.
1 cord of wood.
((
Dry goods are measured by the yard and fractions of a
yard, the fractions used being ^, {, ^ and ^q.
MEASURE OF CAPACITY.
By the " Weights and Measures Act, 1873," the Imperial
gallon, containing 10 pounds weight of distilled water weighed
in air at a temperature of 62 degrees Fahrenh-'it, and the
barometer standing at .30 inches, is made the standard meas-
ure for liquids ; and the Imperial bushel, containing eight
Imperial or standard gallons, is made the standard measure
of capacity for commodities sold by dry measure, instead of
the Wine gallon and the Winchester bushel which are the
standards in the United States, and have been in common
use in this country. But it is provided that for the period
of seven years the Wine gallon and the Winchester bi!°hel
may be used by special understanding between the parties to
any contract or agreement ; and during the said period the
ratio which such measures shall bear to the standard measures
shall be 8,8 follows : — 12 wine gallons=10 standard gallons,
and Ijfjj'f^ Winchester bushels =1 standard bushel.
The Wine gallon contains 231 cubic inches and the Win-
chester bushel 2150^^ cubic inches. The Imperial gallon
contains 277.274 cubic inche , and the Imperial bushel
2218.192 cubic inches. .
LIQUID MEASURE. 1
4 gills are 1 pint (pt.)
2 pints " 1 (piart (<irt.)
4 quarts " 1 gallon (gal.)
63 gallons " 1 hogshead(hhd. )
2 hhds. " 1 pipe (pi.)
2 pipes " 1 tun (tun.)
DRY MEASURE.
2 pints are 1 quart (qrt.)
t quarts
- gallons
4 jecks
are
1 gallon (gal.)
1 peck (nk.)
1 bushel (bu.)
In England a quarter of grain is 8
Imperial buahels, or 600 Iba.
MEASURE OF TIME.
60 seconds (s.) are 1 minute (m.)
60 minutes
24 hours - "
365 days , ,
366 days
lA
a
(t
1 hour (h.)
1 day (d.)
1 common year (y.)
1 leap year.
Also, 7 days are 1 week ; 52 weeks and 1 day are 1 year;
12 calendar anonths are 1 year: 100 years arc 1 century.
70
January
February
March
April
May
June
AlUTHMETIC.
THE CALENDAR MONTHS OF THE TEAR.
has
a
a
«
31 days.
28
31
30
31
30
July
August
September
October
November
December
has
((
((
((
31
days
31
30
31
30
31
led leap year, in which February
Every fourth year is cai
has 29 days.
The above division of time makes tlie average year consist
of 365 d., 6h., while the true solar year, which is the time
the earth is performing a complete revolution round the sun,
is 365d., 5h., 48m., 49.7s. Thus, the average year is llm.,
10.3s. too long. A farther correction of about 3 days in
every 400 years is therefore necessary. Hence only every
fourth centennial year is a leap year.
Rule for Finding the Leap Year. — Divide the two
right hand figures of the number denoting the year by 4 ;
if there be no remainder it is leap year.
Exception. — No centennial year, that is, no year whose
number ends in two ciphers, is leap year, except its number
can be divided by 400 without a remainder.
marine, angular, or circular measure.
60 seconds ( ' ') are 1 minute or mile ( ')
60 minutes " 1 degree C)
360 degrees " 1 circle.
Also, \\°"\5' makes 1 point of the Compass, and 32 points,
1 Circle. Among seamen a fathom is 6 feet, and a knot is a
division of the log line, about 47 feet in length, used in ex-
pressing the rate of a vessel's speed. When a ship sails at
the rate of 6 miles an hour, her speed is said to be 6 knots.
The Marine or nautical mile is ^^j- of a degree of latitude^
or about 2025 yards.
■-><}[■
miscellaneous ^jeasures.
make
12 articles
20 «
144 «
24 sheets of paper "
20 quires
196 fcs. Flour
1 dozen.
((
t(
1
1
1
1
1
score,
gross,
quire,
ream,
barrel.
200 " Beefor Pork make 1 barrel.
■ f
31 days. 'l
31 " 1
30 << 1
31 " 1
30 " m
31 <« m
February H
iar consist h|
3 the time ^
d the sun, 1
r is llm., M
3 days in M
m\y every ^
fe the two ^W
ear by 4 ; ^
ear whose H
ts number H
UKNOMIXATH NUMLKUS.
OF HOOKS.
A sheet folded in 2 leaves is called a folio.
A sheet folded in 4
A slieet folded in H
A sheet folded in 12
A sheet folded in 18
71
a
, a'
quarto, or 4 to.
a
n
an octavo or 8 vo.
a
^i
a duodecimo or 12 mo.
4b
a
an 18 mo.
METRIC WEIGHTS AND MEASURES.
The following are the principal tables of tho Metric sys-
tem. The use of these weijahts and measures has been
made legal by Act of Parliament; but as it will likely
be many years before they come into general use in thi»
country, the following brief synopsis of them is all that is
deemed necessary in this work.
10 millimetres
10 centimetres
10 decimetres
10 metres
10 decametres
1 hectometres
10. kilometres
LONG MEASURE.
1 centimetre =
1 decimetre =
1 MKTIIE =
1 decametre =
1 hectometre =
1 kilometre =
.010939 yards.
.109394 "
1.093944 «
10.939444 "
109.394444 "
1093.944444 «
1 miriametre = 10939.444444 "
The metre is the unit for measuring common distances,
and the kilometre, the unit for long distances.
SURFACE MEASURE.
100 centiares =
1 ARE = 119.6714 sq.
yards.
10 ares =
1 decare = 1196.7144 "
(C
^ 10 decares =
1 hectare = 11967.1444 «
((
The hectare is a
little less than 2^ acres.
Sr- :; ■:: --.^: : ■ I
WEIGHTS.
10 milligrams =
1 centigram = .0000220 fcs. aroir.
10 centigrams =
1 decigram = .0002204
(( u
10 decigrams =
1 GRAM = .002204
U ((
10 grams =
1 decagram = .022046
(( ((
10 decagrams =
1 hectogram = .220462
(( «
10 hectograms =
1 kilogram = 2.204621
U it
10 kilograms =
1 myriagram = 22.046212
(k (i
10 myriagrams =
1 quintal = 220.46212
u ((
10 quintals =
1 millier =2204.62125
or tonneau
(( u
S'l'i i
f%
ARITHMETIC.
MEASURE OF CAPACITY.
10 centilitres = 1 decilitre
10 decilitres = 1 litre
10 litres = 1 decalitre
10 decalitres = 1 hectolitre
10 liectolitrcs = 1 kilolitre
The Litre is used in measuring liquida.
is used in measuring grains, &c.
= .022 Imp. gal.
— .2202 " "
= 2.2024 « «
= 22.0244 « «
= 220.2443 « «
The Hectolitre
REDUCTION OP DENOMINATE NUMBERS.
Reduction is the process of changing the denomination of
a quantity without altering its value.
Reduction may be considered as of two kinds — REDUCTION
Descknding and Reduction Ascending.
Reduction Descending consists in reducing a quantity to
a lower denomination than that in which it is expressed.
Thus, reducing dollars to cents, pounds to shillings, tons to
ounces, bushels to quarts, &c., is Reduction Descending.
Reduction Ascending consists in reducing a quantity to a
higher denomination than that in which it is expressed.
Thus, reducing cents to dollars, pence to shillings, ounces
to pounds, quarts to gallons, 4&c., is Reduction Ascending.
Rule for Reduction Descending. — Multiply by that
nuTnber which expresses Jioiv many of tJie lower name make
one of the higher. .:
Reduce £26 to shillings. £'2Q
20
520 shillings.
We multiply by 20 because there are 20 shillings in £1,
that is, 20 of the lower name make one of the higher. Or,
because, since there are 20 shillings in £1, there are 26 timea
20, or 20 times 26 shillings in £'26.
Reduce 18 days, lOh. 23m. 40 sec. to seconds.
^.i fr\
..■*"
d.
18
24
ii.
10
m.
23
40
: ^^ ^ w;
442
60
26543
60
1592620
Multiply by 24, because
24 hours niaku 1 da^. Add
in 10 hours. Multiply by
yO, because 60ni. make 1
hour. Add in 23 minutes.
Multiply by 60, because
60 sec. make 1 minute.
Add in 40u8ecoDd«,
DKNOMINATE NUMBERS.
73
1.
2.
(( (( .' 1
3.
u (( •■
4.
(( ((
5.
Hectolitre
6.
7.
8.
ERS.
9.
nination of
10.
11.
Reduction
12.
juantity to
expressed.
gs, tons to
nding.
13.
14.
15.
mtity to a
expressed.
igs, ounces
icending.
\y by that
icume make
16.
17.
18.
19.
20.
i
21.
"
22.
lillings.
ings in £1,
gher. Or,
re 26 times
23.
24.
25.
26.
, ■ ■- :~ ■, :-- -;■- -. iv
21.
, .;i j« -^ U^
28.
. ^u 41 ^'t
29.
«-4t--?4 -^ m
30.
1
31.
ii..,U-M-^-Sfi 1
32.
fti^iim-il^: ■
33.
- ' EXERCISES.
Reduce £25 12u. to pence.
How many pence in £'325 1 9s. 7d. ?
Reduce i-l9 to farthings.
Keduce £27 17s. l^d. to farthings.
Reduce $273 to cents.
How many cents in $478.25 ?
Reduce $16 to mills.
Reduce 17ea. 7dol. 3 dimes to cents.
Reduce 37 tons to cwts.
How many pounds in 3 tons 17 cwt. ?
Ans. 6144 d.
Ans. 78235 d.
Ans. 18240 far.
Ans. 26781 far.
Ans. 27300 cts.
Ans. 47825 cts.
Ans. 16000 mills.
Ans. 17730 cts.
Ans. 740 cwt.
Ans. 7700 fl)s.
Reduce 7 cwt. 59 lbs. 7 oz. 12 drs. to drams.
Ans. 1 94428 drs.
How many ounces in 18 tons? Ans. 576000 oz.
Reduce 25Ibs. Troy to grains. Ans. 144000 grs.
Reduce 6lbs. 8oz. 15dwt. to pennyweights.
Ans. 1615 dwt.
How many grains in 3oz. i6dwt. 18grs. ? Ans. 1842grs.
Reduce 251bs to grains by Apothecaries' weight.
Ans. 144000 grs.
Reduce 5fljs. 6oz. 4dr8. Iscr. 8grs. to grs.
Ans. 31948 grs.
How many scruples in 7oz ? Ans. 1 68 scr.
How nrany lbs. in 25 bushels of wheat. Ans. 1500 B)s.
What IS the weight of 245 bushels of oats ?
Ans. 8330Ibs.
Reduce 17 bushels of potatoes to lbs.
How many rods in 7 miles ?
How many yards in 40 rods ?
How many feet in 47 miles ?
Reduce 15 miles 5 fur. 35 rods, 3 yds. 1 ft. 7in. to inches.
Ans. 997057 in.
How many square yards in 3 acres ? Ans. 14520 yds.
How many rods in 2 acres, 3 roods, 20 rods ?
Ans. 460 rods.
Reduce 5a. Ir. 37rds. 20yds. 6ft. 112in. to square inches.
Ans. 34408804 sq. in.
How many chains in 20 miles ? , , Ans. 1600 chs.
How many inches in 4 chains ? ' ' Ans. 3168 in.
How many square chains in 27 acres ? Ans. 270 sq. chs.
Reduce 12 cubic yards to cubic inches.
Ans. 559872 cu. in.
How many pints in 75 gallons ? . Ans. 600 pts.
Ans. 1020Ibs.
Ans. 2240 rods.
Ans. 220 yds.
Ans. 248160 ft.
74
AmTUMETIC.
I!
34. How many ^illsin 9 hlids. ? Ans. 18144 gills.
3o. How inauy <(uarts in 17 busliels? Ans. 544 qrts.
36. Keduce 12bu. Spks. 5qrta. to pints. Ans. 82G pts.
37. How many days in 1873 years. Am. 084113^ days.
38. Reduce 5y. 24Ud. 12h. 42m. 36s. to seconds.
Ans. 178569756 sec.
39. How many seconds in 47<^, 50 ', 25 " ? Ans. 1 72225 ' '.
40. How many days in the first six months of tjje year ?
Ans. 181.
41. How many days in the last six months of the year?
. , .. . . Ans. 184.
Rule fou Reduction AHCJENDiNn. — Divide by tJiat num-
ber which expresses how many of the loiver name make one
of the higher.
2,0)52,08.
£26 Ans.
Reduce 520 shillings to pounds.
We divide by 20 because there are 20
shillings in a pound, that is, 20 of the
lower name make one of the higher. Or,
since there are 20 shillings in £1^ there are in any number
of shillings ^ as many pounds, which is found by dividing
by 20.
Reduce 1592620 seconds to days. ' ,
Ist step, from seconds to 6,0)159262,0 s. ' • ' '
2nd « " minutes to 6,0)2654,3 m. 40s.
hours
3rd
((
((
to
24)442 h. 23m. 40s.
24 (1 8d. lOh. 23m. 408. Ans.
202
192
lOh. ' '
The remainder after each division is of the same name
as the dividend.
^- ' EXERCISES. ^■
1. Reduce 6144 pence to pounds.
2. Reduce 78235 pence to pounds.
3. How many pounds in 18240 farthings? t ,> .
. 4. Reduce 26781 farthings to pounds.
> 5. Reduce 27300 cents to dollars. ,,;.(,;' *
6. How many dollars in 47825 cents ? i- i ^ >f ;^
•J: 7. Reduce 16000 mills to dollars.
.» 8. Reduce 17730 cents to eagles, &c. ,' > "^ c
DICNOMINATK NUMBKRH.
75
9. IT<i'.7 m.niy iouA in 740 cwt. ? ■ : . ■ < • »
10. K(;,liii'e 7700 Ihs. to tons.
11. In ID-HiiH (Inims Avoirdupois liow many cvvt. &c. ?
12. Til fjTil'HK) ouiicH^a liow many toiiH ?
13. How muiy podiuls Troy in 14:4000 grains?
14. llvdwvo l(il,)(lwt. toll)V.
15. In 1842 {grains liow many ounces ?
16. In 14400()grain8l>ow many pounds Apothecaries' weijjht?
17. In 31948 grains how many poundd, etc., Apotliecarie**
weight ?
18. How many onnccw in 108 scruplos?
19. In 1500 lbs. wheat how many bushels ?
20. How many bushels in 8330 il)s. of oats?
21. In 1020 lbs. of potatoes how many bushels?
22. Reduce 2240 rods to miles.
23. Express 220 yards in rods.
24. How many miles are e<juivalent to 248160 feet ?
25. Reduce 997057 inches to miles, &c.
26. In 14520 sq. yards how many acres ?
27. How many acres, &c.,'are equivalent to 460 rods ?
28. Reduce 34 108804 b ].. inches to acres, roods, &c.
29. In 1600 chains how many miles, &c. ?
30. Reduce 3168 inches to chains.
31. How many acres in 270 scj. chains ?
32. How many cubic yards equal 559872 cubic inches ? /
33. How many gallons in 600 pints ?
34. Reduce 18144 gills to hogsheads.
35. Reduce 544 quarts to bus-hels.
36. Reduce 826 pints to bushels.
37. In 684113;^ days how nrany years, reckoning the year
at365i?
38. Reduce 178569756 seconds to years, days, &c.
39. Reduce 172225 '' to degrees, &c.
The answers to the forep^oinp; 39 questions are to be found in the gues«
tions with corresponding nnnibers under Beduction Descending.
MISCELLANEOUS EXERCISES. /
1. How many pounds in 2 tons, 16cwt. 711bs. ?
Ans. 567lflj8. •
2. Reduce 14796fts. to tons, &c. Ans. 7 tons, 7cwt. 96%8.
3. Reduce 3057200 ounces to tons, &c.
Ans. 95 tons, lOcwt. 75ft8.
4. How many grains are there in 17Ib8. lloz. 18dwt
22gr8.? Ans. 103654gr8.
76
ARTTTIMCTTr.
5. Reduce 96 miles, 5fur. 30 ro.ls to rodw. An». 3 1 .'590 rods.
6. How many acrcH, &c., in 47i*68o97l Hq. iucheH ?
Ans. TOa. 1 rood, 3i5rd«. 19ydH. 2t't. I19in.
7. Reduce 527168 feet to miles, &c.
Ans. 99ra. 6tur. 29rd8. 3yd8. Oft. 6in.
8. How many nq. yards in 17a. 2 roods, 18 rods ?
Ans. 85244^ sq. yds.
9. Reduce ^ of a pound Avoirdupois to ounces.
The rules already given are good for fractions as well aa
whole numbers.
Ans. §x 10=9,2 = 10lfoz.
10. Reduce § of a dollar to cents. Ans. t>2i ct'nts.
11. Reduce ^\ of a ton to ounces. Ans. 1400yoz.
1 2. Whut is the value of § of a pound sterling ?
5
:(:^x^0=f=l2is.
^ 'sx
\
Ans. 12s. 6d.
13. Reduce j\ of an acre to sq. rods. Aus. 56 rods.
14. Reduce | of a shilling to the fraction of a pound.
6
=^\V-
Ans.
15. What is the value of -,7^ of a ton. Ans. llcwt. 66§B)s.
16. What is the value of -^jy of a yard. Ans. 2ft. 8|^in.
17. What is the value of ^ a pound Troy. Ans. 8oz.
18. Find the value of -j^g- of a shilling. Ans. ■'>j'jd.
19. Reduce -| of a dollar to its value in cents. Ans. 88^ts.
20. Reduce 401bs to the fraction of a cwt.
Ans. Divide 40 by 100, thus, V'b'V=fcwt.
21. Reduce 12 shillings to the fraction of a pound. Ans. -}.
22. Reduce 9d. to the fraction of u shilling. Ans. |.
23. Reduce 6oz. to the fraction of a pound, Avoirdupois.
Ans. §.
to the fraction of a pound.
Ans. ^j.
25. Reduce 35Ibs. to the fraction of a ton. Ans. ^^-j-.
26. Reduce 5 days to the fraction of a year. Ans. y'g-.
27. What part of a bushel of wheat is 251bs. ? Ans. ^y.
28. Reduce 4235 sq. yards to acres. Aus. |.
29. Reduce 1 28. 6d. to the fraction of a pound.
Begin with the lowest denomination, 6d., reduce it to shil-
24. Reduce lOd. sterlin;^
UKNUJllNATJi NLMUKIiS.
77
lin^H a« you would any other numl>or of pence, tliat in, diviflo
it by 12. Ni)W ^lic only way you can divide it l»y 12 in by
makiiij;; 'J tlu* nuiiuTator and twflvo the denominator of a
fniction, thun, /y, \vhi<'h, when rcducoil iw ^. l^s, (Id., there-
fore i« 12i^H. liediice this to pound by dividing,' by 20, that
is, make 12^ the? luinunator of u fr.ution and 20 the ^denom-
inator, thuH, '*' a t'omnU'x fraction ; reduce it, and it becomes
3, which i.s the answer. Hee tlni work ; —
i'^-=i;;i^ = f5 = {l Ans.
Or, reduce the whoh» (piantity to the lowest denomination,
and divide by th(^ number of tliat denomination which makes
one of tlie hijjher name to which it is to be reduced. Thus —
12s. (id.
12
L)0
240
= i, Ans.
8'
Ans. |.
Ans. ^,[..
Ans. jj.
Ans. }{.
Ans. -[|.
Ans. -,V\t-
Ans. -j.
Ane. f ^J.
Ans. ;jy^.
.30. Reduce 17s. (Id. to tlie fraction of a poimd.
M. Reduce .^s. (Id. to the fracti«m of a pound.
.32. Reduce 7s. (Id. to the fraction of a ])ound.
33. Reduce 4^d. to the fra<'tion of a sliillin^jf.
34. Reduce 9^d. to the fraction of a shilling.
3.5. Reduce 1 Is. 7-^d. to the fraction of a pound.
3(1. What part of a dollar is 40 cents ?
37. Reduce 56Ih3. Hoz. to the fr;«5tion of a cwt.
38. Reduce 65 lbs. to the fraction of a ton.
39. Reduce 19cwt. 281bs. 12oz. to the fraction of a ton.
4rtu t S 4 3
40. Reduce Goz. 13 dwt. 8grs. to lbs. Ans. §.
41. Reduce 3fur. 4 rods, 2yd8. 1ft. 4in. to the fraction of
a mile. ' ,;
42. Reduce o.'J days to the fraction of a year.
43. What is the value of .875 of a pound sterling ?
» Reduce as in the whole; number, observing to point off the
decimals properly. Thus, —
£.875 ~ ,
20 Ans. 17g. 6d.
Ans. -j\.
Ans. 1^-.
'-'•M
17.500 shillings.
12
6.(X)0 pence.
n
AUITHMETIO.
:: \
; I
44. Reduce £.02/) to its vahiP in MJiillingR Jin<l in'iico.
AllH. IL'H. ()fl.
45. Find ilie value of £.8r»87r). . Ahh. 17h. 4||d.
40. What is the value of £.58125 ? AnH. 1 1h. 7 Jd.
47. Find tlio value of £.3 ' Auh. Oh. 8d.
48. Find the value of £.0 Anw. 1:Jh. 4d.
4!). What iH the value of £.410? ' An8. 88. 4d.
60. JJeduce .79085 of a ton to its value.
Ans. 15cwt 93.71b,
51. What i8 the value of .1778125 of a cwt ?
Ans. I71t)8. 12oz. Hdrs.
52. What is the value of .89453125 of a lb. Avoirdupois ?
Ans. 14oz. 5flrs.
53. Find the value of .075 of a pound Troy.
Ans. 8oz. 2dwt.
54. Find the value of .97025 of an ounce Troy.
Ans. IJidwt. 12.0p;r8.
55^ Find the value of .79125 of an ounce Apolhecarioa'
■^freight. Ans. Cdrs. 08cr.'19.8grs.
50. What is the value of .170825 of a pound Apothecaries*
weight? Ans. 2oz. Odrs. 2scr. 18.512gr8.
67. What is the value of £.475 ? Ans. 93. Od.
58. What is the value of .7 of a cwt ?
Ans. 771b8. 12oz. 7|dr8.
69. Find the value of .5410 of a shilling sterling.
Ans. C^d.
■ 60. Find the value of .0845 of a cwt.
Ans. e8Ib8. 7oz. 3.2dr8.
61. Reduce 5s. lOJd. to the decimal of a pound.
Begin with the lowest denomination, and reduce it to the
next higher, thus 2 farthings, divided by 4=4)2.0
."II
f'f
' ' .5d., to this
prefix the number of pence, and it becomes 10.5d ; divide thii»
by 12 and reduce it to shillings, thus, 12)10.5d. ^
7875, to this
again prefix the shillings, and reduce the whole to pounds,
20)5.875
■~j»-
£.29375 which is the answer. The work is as follows-^
I>KN( )M I N ATK NUMBKIW.
70
IS follows — '
>r«
4)2.0 flirt liitigfl.
12)l()..'5(X)d.
2t)).'5.H7.'5H.
i,'. 2937/5 AnH.
f)2. Reiluco 10^(1. totluMhwimalof apound. A n». £.04375.
C3. Reduce l.'Js. })Jd. to ilio ducimul of :i pound.
Ans. £.790r>25.
04. Reduce 3 rooda, 1 1 rods to the dccirnul of an acre.
Ans. .H1H7.5.
fl5. Reduce 3cwt. 32lt)8. to tlie decimal of a ton. Anri. .166.
66. Reduce 37 rods to the decimal of a mile.
Ans. .115625.
67. Reduce 7oz. 4dwt;. to the decimal of a pound. Ans. .6.
68. Reduce 5 hours, 48min. 4i).7s«'(^ to th(? decimal of a
day. . Ans. .2422419 nearly.
ADDITION OF DENOMINATE NUMBERS.
Rule. — Write the quantitiea to he added 8o that numbers
of the same denomination may stand in cohimn. Begin
at the right hand^ or lowest denomination, add each denom-
ination separatehjf reducing each sum, to the next higher
denomination, the nunther of which cai'ry to the colujnn
to which it belongSf arid set tJte remainder, if any, under
the column added.
EXERCISES.
(1)
ydR. ft. in.
12.2. 9
16.1.11
27.3. 8
36.3. 6
(2)
£76.18. 4
17.11. 4i
99.19. 9
11.11.11
67.15.10i
79.19. 9
28.12. 1
63. 8. 4|
(3)
lb«. oz. (In.
13.14.10
15.11.10
11. 4. 9
8.12.13
15. 7. 8
10.13.11
8. 9. 6
4.15.15
t. cwt. qn. IbSk
26.17.3.21
18.11.0.19
25.15.1.16
13.17.2.20
39. 4.1.23
28.16.3.14
04. 2. 10 .445.17. 5^ 89.10. 2 153. 3.2.13
80
ARITHMETIC.
5. Add together 31bs. lloz. 16dwt. 21gr8. : 51b8. 8oz. 7dwt^
llgrs; 7fl5s. Goz. 18dwt. 23grH.; lifts. lOoz. 15dwt. ITgrs. ;
12ft8. 7oz. 9dwt., Sgrs. ; IGfts. lOoz. lldwt. 22gr8.; ISfts.
8oz. 19dwt. ISgrs. * Ans. 77fcs. 8oz. Odwt. Ogrs.
6. Find the sum of 5fr)8. lloz. 7drs. 28cr. 19grs. ; 4fts.
lOoz. 4drs. Isci'. 7grs. ; Sfts. lloz. 6dr8. 28cr, Idgrs. ; 1ft.
9oz. Sdrs. Iscr. 12gr8. ; 2fts. 4oz. Sdrs. lOgrs. ; Gfts. 7oz.
2dr8. 28cr. 9gr8. ; 2fts. 8oz. Idr. Iscr. 13grs.
Ans. 28ft8. 4oz. Odrs. Iscr. 4grs.
7. What is the sum of 176m. 7fur. 39 rods, .'>yd8. ; S.'jm.
4fur. 20 rods, 1yd. ; 79m. 6fur. 29 rods, 3yds. ; 42m. 3fur.
8 rods, 2yds.; 67m. Ifur. 11 rods, 2yds. ; 118m. 3fur. 10 rods,
3yds. ; 81m. 2fur. 31 rods, 1yd. ; 79m. 21 rods, 2yds. ; 18m.
3fur. 33 rods, 3yds ? Ans. 749m. 2fur. 6 rods.
8. Find the sum of 18yds. 2ft. llin. ; 14yd^. 2ft. 7in ;
8yd8. 1ft. lOin. ; 11yds. 7m.; 7yds. 2ft. 8in.; 16yds. 2ft. 9in.;
8yds. 1ft. 7in. Ans. 86yds. 2ft. llin.
9. Add together 29a. 3 roods, 39 rods; 57a. 2 roods, 18
rods; 118a. 26 rods; 7oa., 3 roods, 11 rods; 51a. 1 rood,
8 rods; 94a. 1 rood, 19 rods ; 63a. 2 roods, 21 rods ; 78a.
1 rood, 15 rods ; 19a. 3 roods, 33 rods.
Ans. 589a. roods, 30 rods.
10. Add together 39 rods, 30yds. 8ft. 143in.; 18 rods, llyds.
4ft. 68in. ; 24 rods, 4yd8. 7ft. 118in. ; 11 rods, 21yds. 2ft.
96in. : 15 rods, 27yd8. 124in. ; 27 rods, 6yds. 3ft. 87in.; 19
rods, 25yds, 2ft. 38in., — square measure.
Ans. 157 rods, 6yds. 3ft. 98in.
11. What is the area of 7 farms, measuring as follows : —
the 1st., 79 acres, 9 chains, ^999 links'; the 2nd,, li7a. 4ch.
36501.; the 3rd., 47a. 5ch., 941 1; the 4th., 56a. 2ch. 11821.;
the 5th., 27a., 7ch.. 28131.; the 6th., 36a. Ich. 7711. ; and
the 7th., 84a. 8ch. 11601. ? Ans. 449a. 8ch. 516 links.
12. Find the sum of 35 bushels, 3 pecks, 1 gallon, 3 quarts,
1 pint; 18b. 2pk8. Iqrt, Ipt. ; 7b. Ipk. Igal 1 pt; 26b.
Iqrt. ; 18b. Igal. Ipt. Ans. 106b. 3qtfl.
13. Add together 6 tuns, 1 pipe, Ihhd, 39gal. 3qrts. Ipt.
3 gills ; 4 tuns, 1 hhd. 47gals. 2qrts. 2 gills ; 5 tuns, Ipi.
Iqrt. 1 pt. 1 gill; Ihhd. 52 gals. Iqrt.
Ans. 17 tuns, 1 hhd. 14 gals. Ipt. 2 gills.
14. Find the sum of 4 tons, 7cwt. 86ft8. ; 2 tons, 9cwt^
DENOMINATE NUMBERS.
81
43lb8. ; 1 ton, 8cwt. OOlbg. ; 1 ton, 16cwt. 33B»s. ; 4 tons, 8cwt.
41fl)8. ; 2 tons, 17cwt. 89D)8. Ans. 17 tons, Scwt. 82ft)8,
15. What is the sum of 359^ 59', 59/'; 153", 40', 45''.
270«',0',0'/; 179°, 45', 30''; 81^30', 10'/; 89^59',59".
Aus. 1134^56', 23/'.
16. It is required to find the sum of the following^ pe-
riods : — 33 years, S64 days, 23 hours, 59 minutes, 59 seconds ;
28y. 113d. llh. 48m. 488. ; 17y. 97d. 12h.; ly 307d. 23h.
48m. 498.; 12y. 114d. Ans. 93y. 267d. 23h. 37m. 368.
To add denominate fractions,
Red/uce the fractions to tlieir value in lower d&nomvna^
UonSy and add the results.
17. Add f of a pound to f of a shilling.
.8. d.
|x*» = Vd.= 0.10
13.4 Ans.
18. Add 4^ of a ton to -^ of a cwt.
Ans. 12cwt. Ifl). 3oz. O^fdra.
19. Add together ^ of a mile, ^ of a furlong, and -^^ of a
rod. Ans. 4fur. 13 rods, 4yds. 2ft. 9|in.
20. Add together i^^j^ of a cv^t. f of a ton, and f of a pound.
Ans. 12cwt. 945)8. 6oz. lOfdra.
21. Add together £-^^ and .875 of a shilling.
Ans. 12s. l^d.
22. What is the sum of .79685 of a ton, and .1778125 of
a cwt. ? Ans. 16cwt. lllbs.- 7oz. 11.2drs.
23. Find the sum of .675 of a pound Troy, and .97625 of
an ounce Troy. Ans. 9oz. Idwt. 12.6gr3.
24. Add together £.790625, .5416 of a shilling, and .75d.
Ans. 16s. 5d,
SUBTRACTION OF DKNOMINATB NUMBPlllS.
Etjle. — Write the smaller quantity under the largely
setting numbers of the same denomination under each
other.
Begin at the right, and take the numbers in ths suifro-
hbiidfrom those immediately above them in tfie minu,Gnd.
ARITHMETIC.
When any number in tJte subtrahend exceeds that of the
same denomination in the minuend^ add to the number in
the minuend^ as rnaiiy of that denomination as make one of
the next hlf/her, subtract the number in the subtrahend from
the sum, and carry one to the next denomination as ^ you
proceed; or consider the next number in the minuend
diminished by \.
(1.)
£ s. d.
From 1573, 11, 4^
Take 976, 15, 10^
EXERCISES.
(2.)
tons. cwt. fb».
47, 17, 43.
29, 18, 97.
£ 596, 15, 6.
17, 18, 46.
(3.)
miles, fur. rods.
1407, 1, 16.
161, 1, 20.
1245, 7, S6.
4. A farmer possessed 1279 acres, 2 roods, 21 rods of land,
and l)y his will left 789 acres, 3 roods, 36 rods to the elder
of his two sons ; how much was left for the younger ?
Ans, 489 acres, 2 roods, 25 rods,
5. The latitude of London, (England) is 51^^, 30 ', 49 '/ N.,
and tliat of Gibraltar 3(5*^, (> ', 30^ ' N, ; how many degress is
Gibraltar south of London ? Ans. 15*^, 24 ', 19 ".
6. The earth performs a revolution round the sun in about
365 days, 5 hours, 48 minutes, 49 seconds, and the planet
Jupiter in about 4332 days, 14 hours, 26 minutes, 55 seconds;
how much longer does it take Jupiter to perform a revolu-
tion than the earth? Ans. 3967d., 8h., 38m., 6s.
7. What is the difference between 21 hours, 19 min., 24
eec, and 15 hours, 37 min., 45 sec?
Ans. 5 hours, 41 min., 39 sec.
8. IIow many months and days from August 29th, 1872,
to April 15th, 1873?
Ans. 7 mos., 17 davs.
9. How many ihonths and days from December 3rd, 1872,
to October 2nd, 1873? ' Ans. 9 mos., 29 days.
10. What is the difference in time between March 3rd, 5
hours, 36 min., 42 sec, and March 2nd, 21 hours, 52 min.,
47 sec? Ans. 7 hours, 43 m'n., 55 sec.
11. From 107", 40', 33'', take 69^ 50', 19"?
Ans. 37^50^ 14".
12. A man who owes you £19 lis. 5^d. gives you £20;
how much have you to give him back ? Ans. 8s. 6^d.
i ^
h
DENOMINATE NUMBEIIS.
83
To subtract denominate fractions,
Rule. — Reduce the fractions to their values in loiver de-
nominations, and then subtract.
13. What is the difference between j'j- of a mile and ^- of
a furlonj^ ? ' ' ,
niile. fur.
^"j-x 8=:]f = 6fur. 21 rods, 4yds. 1ft. Gin.
far rods.
-^X 4o = 3.oo=:Ofur. 28 rods, 3yds. Oft. 5|in.
5fur. 33 rods, 1yd. 1ft. O^in. Ana.
14. P^rom j'^Tj- of a ton take -^ of a cwt. Ans. 7cwt. 44^H)s.
15. AVhat is the difference between | of a 11). Troy, and
j^^ of <'m ounce Troy ? Ans. 8oz. 1 Gdwt. Gfj^rs.
IG. Find tlio ditterence between ^-of a bushel, and -^ of a
peck. Ans. lijrt. O^^T^pt.
17. What is tiie difference between -^^ of a pound and -J
of a shilling;? - Ans. 4s. 9^d.
18. Find the difference between i'H and £.4G25.
Ans. 3s. lOd.
19. Find the difference between jC.7G825 and .925 of a
shillin^^ Ans. 14s. 5.28d.
20. From .G90484375 of a ton, take .87790875 of a cwt.
Ans. 12cwt. 931bs. 2oz. 12drs.
21. Find th(} ditTereuce between .875 of a quart and
.90625 of a gallon. Ans. 2qrts. Ipt. 2 gills.
MULTIPLICATION OF DENOMINATE NUMBERS.
Rule. — S)et the niultipUer under the lowest denomination
of the multipticand, and muUljjhj each denomination in
succession, ohservincf to redtice each -product to the next
higher denomination. Write the remainder, if any, from
each reduction, and carry the quotient to the next product.
1. Multiply 27 17 5^
by 6
EXERCISES.
167 4 7^
6 farthinp;8 are Hd.,
write ^ and carry Id.
3 Id. are 2s. 7d., write
7d. and carry 2s. 104s.
are £5 4s., write 4s., and
carry £5, &c.
I'
): I
il
84
•
ARITHMETIC.
(2)
£ 8. d.
fi4 11 dh
3"
£ 8. (1.
78 5 llf
9
(4)
£ n. (1-
147 12 U
12
29 6 45
(8)
lbs. oz. drs. 8or. gn.
3 7 6 1 15
11
40 2 1 5
106 6 14
(9)
mllra. fur. rods.
5 7 15
8
47 3
193 15 4^ 704 13 9f 1771 5 3
(5)
(6)
(7)
ana. cwt. His.
« Ms. oz. dm.
n>H. or. dwt. gn
5 17 29
17 11 13
7 4 15 21
5
6
7
51 9 11 »
ao)
9 1 » I
10
93 2 K
s: I.
(11)
5 31 42
4
22 6 48
(12)
h. III. 800.
7 12 55
, 7
50 30 25
h. m. Bee,
4 56 28
5
24 42 20
When the multiplier is more than 12 it is usual to multi-
ply by factors. Thus,
14. Multiply 24 18 101 by 28. 28=7x4.
7
174 12 3^
4
698 9 1 product by 28.
=#
DENOMINATE NUMBERS.
96
acres.
;15. Multiply 15
, 15
roodd
3
roda.
25 by 243
25x3
10
Product by 10, 159
10x4
10
If " 100,1590
2
20
2
j " 200,3181
} « 40, 636
<« « 3, 47
1
1
2
35
^-
3865 35 Product by 243.
16. Multiply 18 tons, 12 cwt., 61 lbs. by 84.
Ans. 1564 tons, 19 cwt., 24 fcs.
17. Multiply 27 acres, 2 roods, 29 rods by 72.
Ans. 1993 a., roods, 8 rods.
18. Multiply 11 yds., 2 ft., 7 in. by 150.
Ans. 1779 yds., ft., 6 in.
19. Multiply 49 lbs., 11 oz., 12 drs. by 67.
Ans. 1 ton, 13 cwt., 32 fl)s., 3 oz,, 4 drs.
20. Bought 7 loads of Hay, each weighing 1 ton, 3 cwt.,
87 BJE.; what was the weight of the whole ?
Ans. 8 tons, 7 cwt., 9 I^s.
21. If a man can reap 3 acres, 35 rods per day, how much
will he reap in 30 days ? Ans. 96 acres, 90 rods.
22. If a man saw a cord of wood in 8 hours, 45 min., 50
sec, how Ion'' will he be sawing 1 1 cords ?
Ans. 96 h , 24 m., 10 s.
23. If 12 gals., 3 qrts., 1 pint of molasses be used in a
hotel in a week, how much would be used in a year at the
same rate? Ans. 669 gals., 2 qrts.
24. If 13 wagons carry 3 tons, 15 cwt., 40 lbs. each, how
i much do they all carry ? Ans. 49 tons, cwt., 20 flis.
f^.
iti'X^^-'.j,,,
ARITHMETIC.
DIVISION OP DENOMINATE NUMBERS.
Rule. — Begin with the highest denomination, and divide
each in succession, writing the quotient beneath. When a
remainder occurs, reduce it to the next lower denomination,
adding in the number of that denomination, and use the
sum as tite next dividend. So jproceed to the end.
Divide £47 13 8^ by 7.
£ a. d.
7 ) 47 13 8^
/
, £' 6 16 2f |. Ans.
7 into 47, 6 times and £5 over ; write 6, and reduce £5 to
shillings, thus, 5x20=100, add 13=113; 7 into 113, 16
times and 1 shilling over ; reduce the 1 shilliiig to pence, and
add 8=20 pence; divide by 7, twice and 6 pence over; re-
duce 6 pence to farthings, and add 2=26; divide by 7, =3
times and 5 over, which divided by 7=5.
Ans. £6 16 2| |.
EXERCISES.
2. Divide £476 19 5 by 5. Ans. £95 7 10^ f
3. What is the ^ of £927 4 IH? Ans. £115 18 \\ f.
4. Find the -^ of £1728 1 3|. Ans. £192 If.
5. Find the ^ of 27 ton% 16 cwt., 56 fcs.
Ans. 2 tons, 6 cwt., 38 lbs.
6. Find the ^ of 147 Jbs., 14 oz., 6 drs.
Ans. 13 fcs., 7 oz., 2 drs.
7. What is the \ of 62 ibs., 5 oz., 16 dwts., 1 gr.? »
Ans. 8 fcs., 11 oz., 2 dwt., 7 grs.
8. Divide 483 acres, 3 roods, 35 rods by 10.
Ans. 48a., Ir., 23^ rods.
When the divisor is more than 12, ive may either di/vide
by factors, or employ the process of long divisign.
9. Divide £7629 14 2 by 28.
- ^-. i v---.:^-:-
p:rst mktuud.
£ 1. d.
4 ) 7629 14 2
7 ) 1907 8 6^.
272 1) 9^. A»3..
PENOMINATK NUMBERS.
87
8KC0ND METHOD.
JB s. d. £ s,
28 ) 7629 14 2 (272 9
. . 56
d.
Ans.
.';r
.."sr
202
196
69
56
-
13
20
«
274
252
22
12
'
266
252
14
4
. .
56
56
-
10. Divide 1564 tons, 19 cwt, 24 lbs. by 84.
Ans. 18 tons, 12 cwt., 61 fi)B.
11. Divide 1993 acres, roods, 8 rods by 72.
Ans. 27a., 2r., 29 rods.
12. Divide 1 ton, 13 cwt., 32 flm., 3 oz., 4 dis. by ()7.
Ans. 49 lbs., 11 oz., 12 drs.
13. 7 loads of hay woi<i;he(l H tons, 3 rwt., ^7 ltin. in the
aggregate; what was the wuight ot'cin'h h»:ul un iiii average?
Ans. 1 ton, 3 cwt., 41 ftjB.
14. A silversmith mad<' half-a-dox.en bjkxiiih, wcif^hhig 8
Ifc^., 8 oz., 10 dwt.; what was the weight ol each ?
Ans. 5 ()/., H dwt., H gra.
15. If 45 wagons (;arry 685 bushels, 2 peckB, 4 auuilHi i\QW
much does each carry on an (n\n\\\ diHiriliutiou?
.' Ans. 1,1 bush., 7| qrt8.
iV
88
ARITHMETIC.
H
»
I
I
16. If a steamer occupies 48 days, 17 hours, and 40
minutes, in making 121 trips; what is the average time?
Anfe. 9 h., 40 min.
17. If 98 bushels, 3 pecks, and 2 quarts of grain, can be
packed in 37 e(iual-8ized barrels ; how much will there be in
each ? Ans. 2 bush., 2 pecks, 6^ qts.
18. In a coal mine, 97 tons, 13 cwt., 2 qrs., 8 lbs. were
raised in 97 days ; how much was that per day on an average?
Ans. 1 ton, cwt., qrs., 14 fl)s.
19. If $15.50 be the value of 1 B). of silver, what will be
the weight of $500000 worth ?
Ans. 32258 fl)s., oz., 15 dwts., U^ grs.
20. If 13 hogsheads of sugar weigh 6 tons, 8 cwts., 2 qrs.,
7 ttls., what is the weight of each ? Ans. 9 cwt., 3 qrs., 14 BDs.
21. What is the twenty-third part of 137 fos., 9 oz., 18
dwts., 22 gT-8. ? Ans. 5 lbs., 1 1 oz., 18 dwts., 5^\ grs.
22. A shinmont of Kugar ninHJHtod of 8003 tons, 17 cwt., 1
qr., 12 II)S., 10 oz, net weight; it was to be shared equally
by 451 grocers; how mucli did each get?
Ans. 17 liiiirf, 14 cwl., B »ji^8.. l8 lljs. 14 oz.
23. If a horse runs 174 miles, 26 rods, iu 14 hours, wJiat
is his speed per hour? Ann. 12 miles, 3 fur., 19 rods.
24. A farmer divided liis farm containing 322 acres, 2
roods, 10 rods, (.'((ually among his seven sons and six sons-in-
law ; what was the share of each ?
Ans, 24 acres, 5 r,Q.o4s, 10 ro^s.
THE CENTAL.
A "(\inlal" moiinH 10011)8. avoirdupois, and as the
"Weights and Pleasures Act" of 1873 provides that "from
Bhall \
ind atlHl (III lii I -lay of JaUllUty, lMT4," all grains, vege-
tables, ^tf!., " wjinn honglit or sijM py wpjghtj sliall ue
BpeciHed by the ('ental and parts of a iJental ; ana as this
pratilimi Hct'ins likely lo Iiiu'diiiA univalent, some diflRculty
may lin jmiitu ipni'tul In liinling what price tier ceiilid yi\\\ Did
respond with a given price per biisluil, ana vmvevfV The
following Itules aiv'? jafiven to meet any (lllticjulties likely to
iiiie unuiu' iltlM huuUi
THE CENTAL.
89
To find the price per cental to correspond with a given
price per bushel
Rule. — Multi'ply the given price per bushel by 100, and
divide by the weujht of a bushel in pounds,
1. What is the price per cental of wheat, when the price
per bushel is $2.10?
6,0)210.00
$3.50 price per cental.
2. What is the price per cental of wheat, when the rate per
bushel is $1.80 ? Ans. $3.00.
3. When clover seed is $4.20 per bushel, what should be
the price per cental ? Ans. $7.00.
4. When Indian corn is worth $1.12 per bushel, what
should be the price per cental ? Ans. $2.00.
5. Wlicn rye is $1.40 per bushel, what should be the
price per cental ? Ans. $2.50.
G. When outs ;ire 45 cents per bushel, what is {\\v. price
percental? Ans. $1.32.
7. Whon poi'itoes are 90 (.'(Uits per bushel, whiil [h tne
corresponding price per cental ? Ans. $1.50.
To find the price per bushel to correspond with a given
price per cental,
KuLK. — Multlpln the (jlvp.li prion per cental by the num-
ber of pounds to the bushel of the co/fi/fnodity mentioned^
and divide IJie product by 100.
1. If wheat is worth $3.25 per cental, what ssfe/>*/f/} be th«
price per bushel ?
$3.25
60
IU5.00
$1.95 price per bushel.
, 2. Wiien oats are $1.30 per cental, what should they be
jjet bushel. Ans. 44^ cents.
3. If Timothy seed sells for $10 per cental, what is the
price per bushel ? Ans. $4.80.
,' i. When clover seed is $1U per ceulal, what is the price
Jiui lii|H he) ? Ans. $7.20.
5. Wneh rye ia $9.y4 per ohiiIuI, what in tl»e price per
WM'i Ans. $1.^
90
ARITHMETIC.
LONGITUDE AND TIME.
Given the difference of longitude of two places to find the
differ* !»ce of time.
•
Since the earth makes a complete revolution of 3(iO de-
grees in 24 liours, the Hun appears to pass over tlie earth at
that rate, which is lo dejT^rees per hour. Therefore if the
number of def>;rons of longitude between two places bo divided
by 15, the (juotient will represent tlio number of hours occu-
pied by the isun in passing from the meridian of one of the
places to the meridian of the t)ther ; and since the ratio of
degn^CH, (") m ill I ites (') and seconds ( '^) to one another is the
Eume as that of lio.ns, minutes ;md seconds, if any ditterence of
longitude, expressed in degrees, minutes and seconds, be
divided by 15, the »[Uotient will express the number hours,
minutes and seconds in the diflference of time.
What is the difference of time between two places whose
difference of longitude is 56^* 28 ^ ?
15)56'' 28 '(3
45
mm.
45
sec.
52.
Ans.
'1
60
688
60
I
60
■('
780
75
30
LON(HT[^DP! ^ND TIME.
9T
A,i(.'iin, sinco (50 is PxacHy 4 fiincs 15, if anv .|u;nitity ho
multiplied l)y 4, uiid the product l)o divided by (|(), tlio residt
will 1)0 the Biinio jis dividiii;^ l)y l^. Ai.d tliis is tho more
convenient here, hec:ms(! (10 is the ratio of tlw; table, which
rediicea the process to simply niiiltiplyiu;^ 1)y 4.
The above ([iiestion will therefore be solved thus : —
J Difference of lonijf it I ide, CA't^ 28'
4
Difference of time,
3h. 4")m. 528ec. Ans.
Rule. — Multi'pljj tha diferrncc of hmuUvde hy four ; oh'
8c.rvln<i that the product of tha vt!uuteH (') w beconds, and
the product of the degrees f) is minutci^.
EXERCISES.
' 1. The lonj?itude of D-iblin in about 7° 20' W., and of St.
John's, Newfoundland, about o2" 41 ' \V. ; what is the difffr-
ence of time? Ans. 3h. Ira. 24sec.
2. What is the difference of time between St. John's,
Newfoundland, in longitude 5LJ'-' 41 ' \V., and Toronto, Onta-
rio, in lon;Tjitude 79^ 30' W. Ans. Ih. 47m. 16h«c.
3. What is the difference of time between Halifax, N. S.,
and St. John, N. B., — the lonfi;itude of Halifax beinj? 03*
34 ' W., and of St. John, ()()'' 0^ W. ? Ans. 9m' 448ec.
Since the apparent motion of the sun is toward the West,
of two places that which is farther East will have the sun on
its meridian first, and consequently its time will be the faster.
4. London, England, is nearly on the first meridian, that
is, its lonj^itude is nearly nothing, what time is at Halifax
in longitude 63^* 34 ' W., when it is noon at London ?
Ans. 7h. 45m. 44sec., A. M.
5. What time should it be in Montreal in longitude 73*
44 ' W., when it is noon at Fredericton, N. B., in longitude
66" 43 ' W. ? Ans. Uh. 31m. 56sec., A. M.
6. What time should it be in Pictou, N. S., when the
noon gun sounds at Halifax, — longitude of Pictou 62* 42 ' W.,
and that of Halifax, 63* 34 ' W. ? Ans. 3 m. 28sec. past noon.
7. Yarmouth, N. S., is in about 66* 7' W. longitude, and
Quebec in about 71* 24' W. longitude ; what time should it
be at Yarmouth when it is noon at Quebec ?
Ans. 21m. Ssec. past noon.
.ISi
IMAGE EVALUATION
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Sdaices
Corporation
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WIBSTIR,N.Y. 14SS0
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92
ABITHMEIIC.
8. Greenwich, England, is on the first meridian; what
time should be shown by a ship's chronometer, showing
Greenwich time, when the ship is in longitude 74** W., ar^d
her correct time, 9h. 30m. A. M. ? Ans. 4h. 34m.
Given the difference of time between two places, to find
the difference of longitude. , . , , , , ^ ,
This is the converse of the last case.
Rule. — Multiply the difereTice of time by 15. Or, multi-
ply the hours by 60, add in the minutes, and divide the sum
and seconds by 4.
1. The difference of time between two places is found to
be 3 hoiu-8, 45 min., 52 sec. ; required the difference of longi-
tude. ■.,.;■■■ ,;.i ;;:■ ■; -: ;^.
SKCONB METHOD,
h. m. a.
3 45 52
.60
FIRST METHOD.
h.
m.
-
3
45
52
3
11
17
36
5
4)225 52
56'' 28/ Ans.
Se** 23' 00" Ans.
2. The difference of time betweei Halifax and Fredericton,
N. B., is 12 minutes, 36 seconds ; required the difference of
longitude? Ans. 3«» 9'
3. When it is noon at Yarmouth, N. S., it is llh., 43mi.n.,
24sec., A. H., at Portland, Me. ; what is the difference of
longitude ? Ans. 4«> 9'
4. When it is noon at Greenwich, England, it is 7 hours,
36 min., A. M., at St. John, N. B. ; what is the longitude
of St. John? Ans. eO** 0' W.
5. What is the longitude of Montreal, if when it is noon at
Greenwich, it is 7 hours, 5 min., 4 sec. A. M., at Montreal ?
Ans. 73^ 44/ W.
6. What is the longitude of a ship whose correct time is
found to be 5 hoiurs, 35 min., 40 sec. faster than the time at
Greenwich? Ans. 83® 55/ E.
7. Whet is the longitude of a ship whose correct time is
8 hours 43 min., slower than the time at Greenwich ?
Ans. 130« 45' W
I
*f
^.•*-
BATIO AND PROPORTION.
93
■4 -%yt ):
K\
RATIO AND PROPORTION.
Katio is the relation which one quantity boars to another
of the same kind with respect to magnitude.
Thus, the ratio of 2 to 6 is tlie relation which 2 bears to 6
in respect to the quantity expressed by «ach ; and since 2 is
the \ of 6, tliis ratio is equal to ^.
Hence the ratio of one number or quantity to another is
measured by the tjuotient obtained by dividing the first by
the second.
Thus the ratio of 4 to 8 is * ; of 5 to 20, \ ; of 12 to 6, 2 ;
of 27 to 3, 9.
Ratio is generally expressed by the sign (:) placed between
the quantities, thus 3 : 12 expresses the ratio of 3 to 12,
and is equal to \.
The two numbers or quantities of a ratio are called its
terms.
The first term is called the antecedent ; the second the
consequent.
A Simple Katio is an expression of the .relation of two
quantities only, as 7 : 21.
A Compound Ratio is a combination of two or more simple
ratios, as J 3 : 6. 1
\2:3.j
A compound ratio is reduced to a simple one by multipli-
cation.
Thus, 3 : 6| ^6 . 18, or |. X f = A=6 : 18. ..
:81 '
i 5 ^ =120 : 120, or, | x^ x-|=H>=120 : 120.
:3j
Also, 5 : 8"
4
6
A Ratio of Equality is one in which the antecedent is
equal to the consequent, as 7:7.
' A Ratio of Majority is one in which the antecedent is
greater than the consequent, as 12 : 8.
A Ratio of Minority is one in which the antecedent is less
than the consequent, as 8 : 24.
Note. — The English method makes the antecedent the
numerator, and the conse(i[uent the denominator of the frac-
tion. The French method makes the consequent the nmne-
lator, and the antecedent the denominator ; thus, 3 to 7, by
the English, f , by the French, |.
94
ARITHMETIC.
Proportion is an expression of two or more ratios equal
to one another.
A proportion or Analogy is an expression of the equality
of two ratios.
A Simple Proportion expresses the equality of two simple
ratios, usually by means of the sign^ (: :) Thus 2 : 4 : : 7 : 14,
which indicates that the ratio of 2 to 4 is equal to the ratio
of 7 to 14, and is read, 2 is to 4 as 7 to 14.
The four quantities of a simple proportion are called its
terms.
The first and fourth terms are called the Extremes ; the
second and third, the Means.
In every proportion the product of the Extremes is equal
to the product of the Means.
The fourth term is generally known as the Fourth Pro-
portional. ^ ; , .: '
To find a fourth proportional, the first three terms being
given. r
What is the fourth proportional to 3, 21 and 10 ?
Multiply the means together, — 21 x 10=210. Now since
the product of the means is the same as the product of the
extremes, the number, 210, is the product of two factors, one
of which is 3. Therefore, if 210 be divided by 3, the
quotient will be the other extreme, or fourth proportional.
210^3=70, Ans.
Rule. — Multiply the second and third terms together,
and divide theproduct by thefirst. { •.
>v EXEKCISES. ' !
1. Find the fourth proportional to 5, 15 and 24.
' '!a. Find the fourth proportional to 17, 34 and 19. ^
3. What is the fourth proportional to 9, 36 and 48 ? >
■ 4. What is the fourth proportional to 8, 48 and 72 ? *
Whenever the first terra, or any factor of it, is a factor of
one of the others, the operation may be shortened by can-
<!elling. /T
VSib
'f4»: y^*
t->
RATIO AND PROPORTION. <'95
Thus, in the last question, i
0:^0::72 or, $:4S::'^ji
6 6 9 9
' ^ ^1, ' 4S2, Ans. 432, Ana.
5. Find the fourth proportional to 27, 72 and 31.
Jit : :t;i : : 31 8 is a factor of the first
3 8 8 and second terms.
3)248
82f, Ans.
^^ Find the fourth proportional to 16, 27 and 56.
i$ : 27 : : ^0 8 is a factor of the first
2 7 7 and third terms.
, , , • 2)189
94^, Ans.
7. Find the fourth proportional to 14, 21, 32. Ans. 48.
. 8. Find the fourth proportional to 22, 37, 363.
'^ Ans. 610^.
9 "What is the fourth proportional to 9, 19, 99 ?
Ans. 209.
The following principles will be found useful to the learner :
In the following or any other proportion : —
8 : 6 : : 12 : 9.
By inversion, the second is to the first, as the fourth is to
the third, thus, .; ■ ' •' ' ■ ,
tr ., i^./v.: y;-; 6 : 8 : • 9 : '12. ' 7--;
By alternation, the first is to the third, as the second is to
the fourth,^ thus,
8 : 12 : : 6 : 9.
By composition, the sum of the first and second, is to the
second, as the sum of the third and fourth is to the fourth,
thus, :- I * '
14 : 6 : : 21 : 9.
91
96
ARITHMETIC.
I'r
i
By additMn, the first is to the sum of the first and second
as the third is to the sum of the third and fourth, thus,
8 : 16 : : 12 : 21.
By division, the difference between the first and second is
to the second, as the diffe'^ence between the third and fourth
is to the fourth, thus,
2 : 6 : : 3 : 9.
By conversion, the first is to the difference between the
firot and second, as the third is to the difference between the
third and fourth, thus,
8 : 2 : : 12 : 3.
By mixing, the sum of the first and second is to their
difference, as the sum of the third and fourth is to their dif-
ference, thus,
14 : 2 : : 21 : 3.
f f
SOLUTION OF QUESTIONS BY SIMPLE PROPORTION.
Questions to be solved by Simple Proportion contain, or
indicate three terms, two of which are alike, and are to be
taken as the terms of one ratio ; and the third is of the same
kind as the required answer, and between which and the
answer there exists, by the nature of things, the same ratio
as between the first two.
If 3 barrels of apples cost $7, what will be the price of
12 barrels.
Now in this question, the two terms, 3 barrels and 12 bar-
rels, are of the same kind, — let them be taken as the terms of
a ratio, thus 3: 12. This ratio is evidently equal to that of
the price of 3 barrelb, $7, to the price of 12 barrels, which is
the required answer. We may, therefore, state the question
in the form of a proportion, the fourth term of which is to be
found. Thus,
3 : 12 : : 7: the fourth proportional which is obtained
by the rule already given.
The completed proportion will be ~ .
f r
3
bbla.
12
f
7
28.
RATIO AND rROPORTION.
&7
* By examining the oreviouB examples of Proportion,
it will be seen that whenever the fourth term is greater
than the third, the second is greater than the first ; and when-
ever the fourth term is less than the third the second is less
than the fiist. Therefore,
To state questions in Simple Proportion :
EuLE. — Place tlie term, or qvantity which is of the same
kvnd as the required answer in the third place. Then, when
the answer, or fourth term, is to he greater tJtan this third
term, make tJie greater of the other two the second term, and
the less the first ; hut wlten the answer is to he less than the
third term, Tnake tJie less of the other two the second term,
and the greater the first
If 14 reams of paper cost $44.10, what will 3G reams cost ?
14 : 3G : : $44.10
36
26460
13230
14)158760($113.40, Ana.
^BXSBOIBSS.
1. If 6 barrels of flour cost $32, what will 75 barrels cost ?
Ans. $400.
2. If 18 yards of cloth cost $21, what must bo paid for 12
yards ? Aua. $14.
3. How much must be paid for 15 tons of coal, if 2 tons can be
purchased for $15 ? Ans. $112.50.
4. If you can walk 84 miles in 28 hours, how many minutes
will you require to walk 1 mile ? » Ans. 20,
5. What will 14 horses cost, if 3 of the average value can be
bought for $270 ? Ans. $1260.
6. What must be paid for a certain piece of cloth, if § of it cost
$9. Ans. $13.50.
^ 7. If 5 men are required to build a wall in 5 days, how many
men will do the same in 2^ days ? Ans. 10.
8. If 16 sheep are f of a flock, how many are there in the same ?
Ans. 24.
9. tVliat must be pud for 4^ cords of wood, if the cost of 3 cords
iaZlO? Ans. $15.
^8
ABirmCETIO.
10. What is the height of a tree which oasts a shadow of 125
feet, if a stake 6 feet high produees a shadow of 8 feet ? Ans. 93f •
11. ilow lung will it take a train to ran from Syracuse to Os-
wego (a distance of 40 miles), at ihe rate of 5 miles in 15/., minutes ?
12. If 15 men can build a bridge in 10 days, how many men
will bo repaired to erect iJiroo of the same dimensions in ^ the time ?
Ans. 90.
Va. If a man receive ^-1.50 for 3 day.s* work, how many days
ought lio to remain in his place for $25? Ans. 1G§ days.
14. How much may a porsoti spend in 01 days, if he wishes to
save $73, .50 out of a salary of $500 per annum ? Ans. $109. H4.
15. If n cwt., 3 (!;•»., 14 Ibd. of sugar cost $36.50, what will
2 .jrs , 2 lbs. cost ? Ans. $4.879-f-.
ll). 5 men are employed to do a piece of work in 5 days, but
alter workinH; 4 days tl>oy find it impossible to complete the job in
Khs thiiii 3 day.s mojv, how many additional men must bo employed
to do the work in the time agreed upon at first ? Ans. 10.
17. A watch is 10 minutes too fast at 12 o'clock (noon) on Mon-
day, and it gains 3 minutes 10 seconds ai day, what will be the time
by the watch at a quarter past 10 3'cloek, A. M., on the following
l^aturday ? Ans. 10 h. 40 m. 36-/g s.
18. A bankrupt owes $972, and his property, amounting to
$007.50, is dirttributed among his creditors ; what does one receive
whose denuMitl is $11.33^ ? Ans. $7,083-}-.
19. What is the value of .15 of a hhd. of lime, at $2.39 per
hhd.? Ans. $.3585.
20. A garrison of 1200 men has provisions for | of a year, at
the rate of I of a pound per day ; how long will the provisions last
at the same allowance if the garrison be reinforced by 400 men 'f
Ans. (jjl mouths.
21. If a piece of land 40 rods in length and 4 in breadth make
an acre, how long must it be when it is 5 rods 5^ feet wide ?
Ans. 30 rods.
22. A borrowed of B $745, for 90 days, and afterwards would
return the favor by lending B $1341 ; for how long should he . lend
it? /; r ,^ Ans. 50 days.
V. 23. If a man can walk 300 miles in G successive days, how
many miles has he to walk at tlio end of 5 days ? Ans. 50c
RATIO AND PROPORTION. 99
24. If 495 gallons of ivine cost $394 ; how much will $72 pay
.for ? Ads. 90 gal+.
25. If 112 head of oattlo consume a certain quantity of hay in 9
' days ; how long will the sutue quantity last 84 head? Ans. 12 da^s.
2C. If 171 men can bui' I a house in 168 days; in what time
will 108 men build a similar house ? , Ans. 266 days.
27. It has been proved that the diameter of every circle is to the
rircuniicrence as 113: IJ55; what then is the circumlcreiico of the
moon's orbit, the diameter being, in round numbers, 480,000 miles ?
Ans. 1,507,964 .Ya m.
28. A round table is 12 ft. in circuu'ference ; what is its diameter ?
Ans. 3 ft. 9!|nin.
29. A was sent with a wanant ; after he had ridden 65 miles, B
was sent after hitii to stop the execution, and for every 16 miles that
A rode, B rode 21 ; How fur had each ridden when B overtook A?
Ans. 273 miles.
.30, Find a fourtli proportional to 9, 19 and 99. Ans. 209.
" 31. A detective chu.sed a culprit for 200 miles, travelling at the
rate of 8 miles an hour, but the culprit had a start of 75 miles ; at
what rate did the latter travel ? Ans. 5 miles an hour.
3^ IJ', w much rum may bo bought for §119.50, if 111 gallons
cost :j89.625 ? Ans. 148 gallenr-,
33. If 110 yards of cloth cost $18 ; what wUl $63 pay for ?
Ans. 385 yards.
34. If a man walk from Rochester to Auburn, a distance < 1 (say)
79 miles in 27 hours, 54 minutes ; in what time will he wulii. at the
same rate from Syracuse to Albany, supposing the distance to he
152 miles ? Ans. 53 h. 41 m. nearly.
35. A butcher used a false weigh i ' l.J oz., instead of 16 oz. k/i*
a pound, of how many lbs. did he defraud a customer who bought
112 just lbs. horn him? Ans. 9|^ lbs.
, 36. If 123 yards of muslin cost $205 ; how much will 51 yards
cost? Ans. $85.
37. In a copy of Milton's Paradise Lost, containing 304 p""-"?,
the combat c^ Michael and Satan commences at the 139th page ; at
what page may it he expected to commence in a copy containing 328
pages? , '
Ans. The "ourth proportional is 149|| ; and hence the papsage
will commence at the foot of page 150
38. Suppose a man, by travelling 10 hours a day, performs •
100
AniTHMEno.
journey in four weeks without desecrating tho Sabbath ; now many
weeks would it take him to perform the some journey, providou ho
travels only 8 hours per day, aud pays no regard to tho Sabbath ?
Ans. 4 weeks, 2 days.
39. A cubic foot of pure fresh water wciglis lOOO oz., avoirdu-
pois ; find the weight of a vessel of water containing 217^ cubic in.
Ans. 7 1bH., 131.1;] oz.
40. Suppose a certain pasture, in which arc 20 cows, is sufficient
to keep them G weeks ; how many must be turned out, that the same
pasture may keep the rest G months ? Ans. 15.
41. A wedge of gold weighing 14 lbs., 3 oz., 8 dwt., is valued at
£514 48. ; what is tho value of an ounce ?
Ans. £,?i.
42. A mason was engaged in building a wall, when another cumo
up and asked him how many feet he had laid ; ho repli -l, that tho
part ho had finished bore the same proportion to one league which
^y does to 87 ; how many feet had he laid ? Ans. 3^4,%^'
43. A farmer, by his will, divides his farm, consisting of 97
acres, 3 roods, 5 rods, between his two sons so that the share of tho
younger shall be J- the share of the elder; required the shares.
Here the ratio of the shares is 4 : 3, and we have showiythat if
four magnitudes arc proportionals, the first term increased by tho
second is to the second as tho third increased by the fourth is to tho
fourth. Now, 97 acres, 3 roods, 5 rods, being the sum of tho shares,
we must take the sum of 4 and 3 for first term, and either 4 or 3 for
the second, and therefore 7 : 4 : : 97 acres, 3 roods, 5 rods : F.P., i. e.,
the sum of the numbers denoting the ratio of the shares is to one of
them as tho sum of the shares is to one of them. This gives for tho
elder brother's share, 55 acres, 3 roods, 20 rods, and the younger's
share is found either by repeating the operation, or by subtracting
the share thus found from the whole, giving 41 acres, 3 roods, 25
rods. , : . .
44. A legacy of $398 is to be divided among three orphans, in
parts which shall be as the numbers 5, 7, 11, the eldest receiving tho
largest share ; required the parts ?
23 : 5 : : 398 : 86^*, the share of the youngest.
' 23 : 7 : : 398 : 121^, the share of the second. = J
.:_ 23 : 11 : : 398 : 190j«3, the share of tho eldest. , , :. " t
45. Three sureties on $5000 are to be given by A, B and C, so
that B's share may be one-half greater than A's, and G's one-half
greater than B's ; required the amount of , the security of each ?
COHPOUSn) PBOPOBTION.
101
Ans. A'8flhare, $106? 63^=^ ; B's, 11578.94}^ ; C's, $2368.42^.
46. Suppose that A starts from Washington and walks 4 miles
an hour, and B at the same time starts from Boston, to meet him, ut
the rate of 3 miles an hour, how far from Washington will they
meet, the whole distance hcing 432 miles ? Ana. 246f miles.
47. A certain number of dollars is to be divided between two
persons, the less share being § of the greater, and the difference of
the Hliares $800 , what are the shares, and what is the whole sura to
be divided ? Ans. Less share, $1600 ; greater, $2400 ; total, $4000.
48. A certain number of acres of land are to bo divided into
two parts, such that the one shall be i} of the other ; required the
parts and the whole, the difference of the parts being 716 acres ?
Aus. the less part 537 acres; the greater, 1253 acres ; the whole,
1790.
49. A mixture is made of copper and tin, the tin being ^ of the
copper, the difference of the parts being 75 ; required the parts and
the whole mixture ? Ans. tin, 37^ ; copper, 112^ ; the whole. 150.
50. Pure water consists of two gasses, oxygen and hydrogen ; the
hydrogen is about -f^ of the oxygen; how maiy ounces of water will
there be when there are 764 J | oz. of oxygea more than of hydrogen ?
I r . , ^^- 1000 oz.
COMPOUND PROPORTION.
■1 ■
Proportion is called simple when the question involves only one
condition, and compound when the question involves more conditions
than one. As each condition implies a ratio, simple proportion is
expressed, when the required term is found, by two ratios, and com-
pound, by more than two. Thus, if the question be, How many
men would be required to reap 65 acres in a given time, if 96 men,
working equally, can reap 40 acres in the same time ? Here there
lis but one condition, viz., that 96 men can reap 40 acres in the
given time, which implies but one ratio, and ^hen the question has
been stated 40 : 65 : : 96 : F.P., and the required term is found to
be 156, and the proportion 40 : 65 : : 96 : 156, we have the propor>
tion, czprc sed by two ratios. But, suppose the question were, If
a man walking 12 hours a day, can accomplish a journey of 250
miles in 9 days, how many days would he require walking at the
102
AltrrRMETIO.
I
N
Mone rate, 10 houra eaoh day, to travel 400 miles ? Hero there arc
two conditions, vii. : jirtt^ that, in the one oaso, ho travoli> 12 honra
a day, und in the other 10 houra ; and, iecondlj/, that the distanoefl
are 250 and 400 miles. The statement, as we shall presently show,
would be 10 : 12 ) . . g . 17 7 ^^^^° ^^^^ condition im-
250 : 400 J ■ ' " ^*' pUes one ratio, 10 : 12 and
250 : 400, and when the required term, which in IT^^, is found,
there are four rntioH, viz., the two already noted, and 9 : IT-^^, gives
two more, one in relation to 10 : 12, and one in relation to
250 : 400. Thiti will bo evident, when wo have shown the method
of statement and operation.
EXPLANATORY STATEMXMT
AND OPERATION.
11
1
33;
3
12
12
F.P.
36
PRACTICAL STATEMENT
AND OPERATION.
11:33
18: 5
}■■■
12 : F. P.
18
, 1
6
5
36
2
F.P.
10.
1
3
3
5
2 : F. P.
}: I}:: 2:10.
Let the question be, How many men would be required to reap
33 acres in 18 days, if 12 men, working equally, can reap 11 acres
in 5 days ?
Wo firet proceed, as on the left mai^n, as if there were only one
condition in the question ; or, in other words, as if the number of
days were the same in both cases, and the question were — If 12
men can reap 11 acres in a given time, how many men will be re-
quired to reap 33 acres in the same time. This, then, is a question
in simple proportion, and by that rule we have the statement —
11 : 33 : : 12 : F. P., which, by contraction, becomes 1 : 3 : : 12 : F.
P. ; and thus, we find F. P. to be 36, the number of men required,
if the time were the same in both cases. The question is now
resolved into this : How many men will be required to reap, in 18
days, the same quantity of crop that 36 men can reap in 5 days ?
This is obviously a case of inverse proportion, for the longer the
time allowed the less will be the number of men required, and hence
the statement, 18 : 5 : 36 : F. P., which, by contraction, becomes
1 : 5 : : 2 : F. P., which gives 10 for the number of men. The
work may be shortened by making the two statements at once, as on
the right mar^. We firet notice that the last term is to represent »
COMTOUND r'^OPORTlON.
108
11X18:33X5
198
eortain number of men, und, tliorcforo, wo place 12 in the third
plaoo; next, wo boo that, other thinga buittg equals it will take more
mon to roap :i3 than to reap 1 1 acrcH, and that, therefore, oo far as
that M concerned, tho fourth term will be greater than the third, and
so wo put 1 1 in the fir^t place, and WW in the second. A^ain wo see
that, olhQr thimjn bving iqmtl, u le«3 nunibiir of men will be required
when 18 days are allowed tor doin*^ the work, than when it is re-
quired to be dono iit 5 days, and that therefore the fourth term, oh
far mi that in concerned, will be Ichh than tho third, iiiiJ theruforo wu
write 18 : 5 below the other ratio as on the margin. Then by cou-
traotion wo got .^ ; |, i : : 2 : F. P. Now, uh 3 in the first term is to
bo a multiplior, and 3 in the second u divisor, we may omit those
also, and we obtain . ' . |- : : 2 : 10, the answer as before.
The full unoontractcd operation
would bo to multiply 18 by 11, which
gives 198, then to multiply 33 by 6,
which gives 105, then multiply 165,
tho product of tho two second terms,
by 12, and divide the result, 1980,
by 198, tho product of tho two first
terms, which gives 10 as before.
Bocause in the analogy 198 : 165 : : 12 : 10, the first two terras
aio products, this kind of proportion has been called compound^
and the ratio of 19 to 165 is called a compound ratio. Wo can show
the strict and original meaning of tho term compound ratio more
easily by an example, than by any explanation in words. Lot us
take any scries of numbers, wl Ac, fractional or mixed, say 5, ^,
1, 19, 12, 1, 17, 11, 15, 25, then the ratio of tho first to tho last is
said to bo compounded of tho ratio of tho first to the second, the
second to the third, the third to the fourth, &c., &o., &c., to the end.
Now the ratio of 5 to 25 is ^/^=5, and the several ratios are in this
order,ix|xi-Xl'5XT3X-'r''-X!4xifx||^^»o'» ^^'^^^"S fio'^^J
''/=5 OS before. If we took them in reverse order, viz., \=^, it
is obvious that all therein could be cancelled, as each would in suo-
oossion be a multiplier and a divisor.
Wc would also remark ihat compound proportion is nothing else
than a number of questions in simple proportion solved by one opentr
165::*12:F. P.
166X12:^10
198
104
ABITHMETEO.
tion. This will bo evident from our second example by oomparing
the two operations on the opposite margins. Again, we remarked
that every condition implies a ratio, and that therefore the third and
fourth terms of our first example really involve two ratios, one in
relation to each of the preceding. Hence v 'vcrsally the number of
ratios, expressed and implied, must always ^o double the number of
conditions, and therefore always even. As the third ratio is only
written once, the number of ratios appears to be odd, but is in reality
even.
RULS:
Place, as in simple jproportion, in the third place the term that
is the same as the required tern . Then conrider each condition
separately to see which viust he placed Jirst, and which second^ other
things being equal.
EXAMPLE .
1. If $35.10 pay 27 men for 24 days; how much will pay 16
men 18 days ? Hero we first observe that the
answer will be money, and therefore $35.10
must be in the third place. Again, it will
take less money to pay 16 men than 2? men,
and therefore, other things being equal, the
ant>wer, as far as this is concerned, will be lesa
than $35.10, and therefore we put the less
quantity, 16, in the second pkce. So also
because it will take less to pay any given num-
ber of men for 18 days than for 24 days,
therefore we put the less quantity in the second
place, which the statement shows*in the margin.
27
24
16:: $35.20
18
3:
3:
. 1
2
2 ^
9:
4:: $35.10
4
•
9)140.40
Ans. $15.60
Vf---
BXEBCISES.
1. If 15 men, working 12 hours a day, can reap 60 acres in 16
days ; in what time would 20 boya, working 10 hours a day, reap 98
acres, if 7 men can do as much as 8 boys in the same tkne ?
#t: , •^^'*' 26|| days.
2. If 15 men, by working 6f hours a day, l»n dig a trench 48
ftot long, 8 feet broad, and 5 feet deep, in 12 days; how many houn
ft day must 25 men work in order to dig a trench 36 feet long, 12
fti|( broad, and 3 feet deep, in 9 days? Ans. 3{.
COMPOUND PROPORTION.
105
:t, 3. If 48 incr can build a wall 864 feet long, 6 feet high, and Tl
feet wide, in 36 days ; how muuy men will be required io build a
wall 36 feet long, 8 feet high, and 4 feet wide, in 4 days? Ana. 32.
4. In what time would 23 men weed a quantity of potato ground
which 40 women would weed in 6 days, if 7 men can do as much as
9 women ? Ans. 8g^j, days.
5. Suppose that 50 men can dig in 27 Hays, working 5 hours a
day, 18 cellars which are each 48 feet long, 28 feet wide, and 15 feel*
de(p; how many days will 50 men require, working 3 hours each
day, to dig 24 cellai-s which are each 36 feet long, 21 feet wide, and
20 feet deep? Ans. 45 days.
^- 6. If 15 bars of iron, each 6 ft. 6 in. long, 4 in. broad, and 3 in.
thick weigh 20 cwt., 3 qrs., (28 lbs.) 16 lbs. ; how much will 6 bars
4 ft. long, 3 in. broad, and 2 in. thick, weigh ?
. r Ans. 2 cwt., 2 qrs., 8 lbs.
7. If tl2 men can seed 460 acres, 3 roods, 8 rods, in 6 days ;
how many men will be required to seed 72 acres in 5 days ?
Ans. 21.
8. If the freight by railway of 3 cwt. for 65 miles be $11.25 j
how far should 35/j cwt. bo carried for $18.75 ? Ans. 9^5
9. If a family of persons can live comfortably in Philadelphia
for $2500 a year j what will it cost a family of 8 to live in Chicago,
all in the same style, for seven months, prices supposed to bo | of
what they would be in Philadelphia ? a -^ . Ans. $1037.04
. ,i 10. If 126 lbs. of tea cost $173.25 ; what will 63 lbs. of a differ-
ent quality cost, 9 lbs. of the former being equal in value to 10 lbs.
of the latter? Ans 884.15.
11. If 120 yards of carpeting, 5 quarters wide, cost $60; what
will bo the price of 36 yards of the same quality, but 7 quarters
wide? Ans. $25.20.
12; If 48 men, in 5 days of ^ hours each, can dig a canal 139f
yards long, 4| yards wide, and 2 J yards deep ; how many hours per
day must 90 men work for 42 days to dig 491 -^ yards long, 4|
yards wide, and 3^ yards deep ? Ans. S§^|-.
13. A, standing on the bank of a river, discharges a cannon, and
B, on the oppo<iite bank, counts six pulsations at his wrist between
the flash and the report ; now, if sound traveb 1142 feet per secondi
106
iiaiTHiCEno.
and tho pulse of a person in health beats 75 ttroces in a minute,
vluit is Uie breadth of the river? Ans. 1 mile, 201 1 feet.
14. If 264 men, working 12 hours a day, can make 240
yards of a canal, 3 yards wide, and 12 yards deep, in ^ days ; how
long will it take 24 men, working 9 hours a day, to make another
portion 420 yards long, 5 yards wide, and 3 yards deep ?
Ans. 53^^.
• 15. If ine charge per freiglit train for 10800 lbs. of flour be $16
for 20 miles; how much will it be for 12500 lbs. for 100 miles?
. ' - ■ .1 '■, , - - ;. .- . . \ :i Ans. $92^§.
16. If $42 keep a family of 8 persons for 16 days ; how long, at
that rate, will $100 keep a family of 6 persons ? Ans. 50^2 days.
17. If a mixture of wine and water, measuring 63 gallons, con-^
sist of four parts wine, and one of water, and be wcrth $138.60 ; ;what
would 85 gallons of the same wine in its purity be worth ?
Aas. $233.75.
18. If I pay 16 men $62.40 for 18 days work ; how much must
I pay 27 men at the same rate ? Ans. $105.30»
19. If 60 men can build a wall 300 feet long, 8 feet high, and
6 feet thick, in 120 days, when the days are 8 hours long ; in what
time would 12 men build a wall 30 feet long, 6 feet high, and 3 feet
thick, when the days are 12 Liours long ? Ans. 15 days.
20. If 24 men, in 132 days, of 9 hours each, dig a trench of four
d^ees of hardness, 337^ feet long, 5| feet wide, and 3^ feet deep ;
in how many days, of 11 hours each, will 496 men dig a trench of 7
d^ees of hardness, 465 feet long, 3§ feet wide, and 2| feet deep ?
Ans. 5^;
21. If 50 men, by working 3 hours each day, can dig, in 45 days,
24 cellars, which are each 36 feet long, 21 feet wide, and 20 feet
deep ; how many men would be required to dig, in 27 days, working
5 hours each day, 18 cellars, which are each 48 feet long, 28 feet
wide, and 15 feet deep ? Ans. SO.
22. If 15 men, 12 won:on, and 9 boys, can complete a certain
piece of work in 50 days ; t aat time would 9 men, 15 women, and
18 boys, require to do twice as much, the parts performed by eaoh»
in the same time, being as the numbers 3, 2 and 1 ? Ans. 104 days.
23. If 12 oxen and 35 sheep eat 12 tons, 12 owt. of hay, in 8
flays ; how much will it cost per month (of 28 days,) to feed 9 oxen
•nd 12 sheep, the price of hay being $40 per ton, and 3 oxen being
supposed to cat as much as 7 sheep ? Ans. $924.
inSCELLANEOXTS EXERCISES.
lOT
24. A vessel, whose speed was 9^ miles per hour, left Belleville
at 8 o'clock, a. m., for Oananoque, a distance of 74 miles. A second
vessel, whose speed was to thut of the first as 8 is to 5, stHrting from
the same place, arrived 5 minutes before the first ; what time did the
second vessel leave Belleville ? Ans. 55 ni"n. past 10 o'clock, a. m,
25. If 9 con:positors, in 12 days, working 10 hours each day, can
compoce 36 sheets of 16 pages to a sheet, 50 lines to a page, and 46
letters in a line ; in how many days, each 1 1 hours long, can 5 com.
positors compose a volume, consisting of 25 sheets, of 24 pages in a
sheet, 44 lines in a page, and 40 letters in a line ? Ans. 16 days.
MISCELLANEOUS EXERCISES ON THE PRECEDING RULES.
I. What is the value of .7525 of a mile?
Ans. 6 fur., rd, 4 yds, 1 ft., 2| in.
2: What is the value of .25 of a score ? Ans. 5.
3. Reduce 1 ft. 6 in. to the decimal of a yard. Ans. .5.
4. What is the value of 14 yards of cloth, at $3,375 per yard ?
Ans. $47.25.
' 5. What part of 2 weeks is j\ of a day ?
* 6. What part of £1 is 13s. 4d?
7. Reduce ijP^ of a day to hours, minutes and seconds.
Ans. 2 hours, 52 min., 48 seo.
8. Add I of a furlong to f of a mile.
Ans. 7 fur., 31 rds, yd., 1 ft., 10 in.
' 9. What is the value of .857^ of a bushel of rye ?
■v--:-;^'.!,';^ ■: v-'-^ ;;iv-.-?T ,.-:■_ ,,=.-,,;■■ Ans. 48 pounds...
' 10. Reduce 47 pounds of wheat to the decimal of a bushel.
Ans. .783J.
II. Reduce 9 dozen to the decimal of a gross. Ans. .75.
12. Add y'j of a cwt. to ^ of a quarter. Ans. 3 qrs., 10 lbs.
13. Subtract 1 1
Ans. tIj.
Ans. f.
? a day from ^ of a we<
14. From II of 5 tons take 3 of 9 cwt.
days
TS
Ans. 2 tons, 17 cwt., 1 qr., §§ lbs.
;* 15. Bow many yards of cloth, at $3J a yard, can be bought for
U^?
Ans. 13^^ yards.
16. A-roan bought | of a yard of cloth for $2.80 j what was ihor
rate per yard ?
Ans. $3.20.
17. How many tons of hay, at $16^ per ton, can be bought for
11964 7
Ans. 11|| tons.
108
ABITHMRTIO.
;i.
i;
18. At $17f per week, how many weeks can a family board for
t765f ? Ads. 43^ weeks.
19. What nnmber must be added to 26|, and the sum multipli-
ed by 7^, that the product may be 496 ? Ans. 37|.
20. A man owns f of an oil well. He sells § of his share for
$3C00 ; what port of his share in the well has he still, and what is it
Wf'rth at the same rate ? ' Ans. $1750.
21. How long will 119^ hhds. of water last a company of 30
men, allowing each man § of a gallon a day ? Ans. 627 days.
22. Reduce § of 2*, j% of Ig, and 3^ of 2^, to equivalent frao-
tions having the least common denominator. Ans. |f, f !> W^.
23. From f of 2* of 4, take VV of 6| of ^. Ans. 2|.
24. What is the sum of J, J, J, {, J, 4, ^, and ^? Ans. l^U-
25. What is the sum of | ^ of S^-^ J of 85 ? Ans. 22-S ^SJ.
26. How long will it take a person to travel 442 miles, if he
travels 3^ miles per hour, and 8^ hours a day ? Ans. 16 days.
. 27. Find the sum of 2J of /^, 3J of | of ^% of 4J and J.
Ans. 6^^.
28. A has 2] times 8| dollars, and B 6^ times 9f dollars ; how
mnoh more has B than A? ^ Ans. $44||.
S, 29. If I sell hay at $1.75 per owt. ; what should I give for 9|
tons, that I may make $7 on my bargain. Ans. $329.
30. If 7 horses cat 93^ bushels of oats in 60 days ; how many
'bu' hols will one horse eat in 87f days ? Ans. 19|.
31. Bought 14/q yards of broadcloth for $102.90 ; what was tho
^alue of 87| yards of the same cloth ? Ans. $612.
32. How many bushels of wheat, at $2f per bushel, will it re-
•quire to purchase 168^^9 bushels of corn worth 75 cents per bushel?
Ads. 47y\.
• '-' 33. If in 82j^ feet there are 5 rods ; how many rods in one mile ?
Ans. 320.
34. Suppose I pay $55 for g of an acre of land ; what is that per
«ore ? Ans. $88.
•^ 35. If I of a pound of tea cost $1.66^ ; what will ^ of a pound
«ost? Ans.$1.55||.
36. Subtract the 2ium of 2} and 1 /j, from the sum of f , 7^ and
3, and multiply the remainder by 3^.
37. If i lb. oost 23^ cents i what wiU 2}^ cost ?
Aaa. 71^ oeats*
Ans. 24|f
MISCELLANEOUS EXEBaSES.
109
38. What is the difference between 2^X3^ and 2JX3,'g ?
Ana. ^^.
39. If I lb. cost $3 ; what will | ^ lb. cost ? Ans. 3C^ centa.
40. What is thd difference between J of J+H-^X.', and
tH+l ? Ans. |2§.
41. If 4|^,- yards cost Sl^'j , what will 2^ yards cost?
' . - ; ^ Ans. 475 cents.
42. Bought 5 of 2000 yards of ribbon, and sold 3 of it; how
muoh remains? Ans. 285!) yards.
43. Divide the sum of ^, f,
h h
1111
J. i^»^i»^(l5,n2by«iesumofJ,
, and divide the quotient by t»yX«, and multiply
the result by f of ^. Ans. g.
V 44. I bought g of a lot of wood land, consisting of 47 acres, 3
roods, 20 rods, and have cleared ^ of it ; how much remains to be
cleared ? Ans. 20 acres, 3 roods, 31^ rods.
45. What is the differon«e between If^^ and Igg ? Ans. |gj.
46. If $[2 pay for a IJ st. of flour; for how much will $| pay ?
Ans.
h\ St.
47. Mount 31anc, the highest mountain in Europe, is 15,872
feet above the level of the sea ; how far above the sea level is a clim-
ber who is y'j of the whole height from the top, i, c, ,'5 of perpen-
dicular bight ? Ans. 12896 feet.
^48. What will 45.94375 tons cost if 12.796875 tons cost $54.64 ?
Ans- $196.17.
■ " 49. If I gain 837.515625 by selling goods worth $324.53125 j
what shall I gain by selling a similar lot for $520.6635416. ?
/ . Ans. $60.1884.
50. If 52.815 cwt. cost $22.345 ; what will 192.664 cwt. cost at
the same rate ? Ans. $81.512-|-
51. Kcquired, the sum of the surfaces of 5 boxes, each of which
is 5^ feet long, 2^ feet high, and 3^ feet wide, and also the number
of cubic feet contained in each box. The box /supposed to be made
from inch lumber 7 Ans. 369^ Superficial ft.
52. If I pay $j^^ for sawing into three pieces wood that is 4 ft.'
long ; how muoh more should I pay, per cord, for sawing into pieces
of the same length, wood that is 8 feet long ? Ans. 22^ cents.
53. A sets out from Osw^, un a journey, and travels at the '
rate of 20 miles a day ; 4 days citer, B sets out from the same place,
and travels the same road, at the rate of 25 miles per day ; how many
4ays before B will overtake A ? Ans. 16^
110
ABITHMETIO.
54. A farmer having 56^ tons of hay, gold | of it at $10§ per
ton, and the remainder at $9.75 per ton ; how much did he receive
for his hay? Ans. S5804g.
55. If the sum of 87}^ and 117^j^ is divided by their difference ;
what will be the quotient ? Ans. 6^5 J.
56. If 8f yards of silk make a dress, and 9 dresses be made from
a piece containing 80 yards ; what will be the remnant left ?
Ans. IJ yards.
57. A merchant expended $840 for dry goods, and then had re-
maining only -^l as much money as he had at first ; how much money
had he at first ? Ans. $3430.
58. If a person travel a certain distance in 8 days and 9 hours,
by travelling 12 hours a day; how long will it take him to perform
the same journey, by traveling 8f hours a day ? Ans. 12 days.
59. If 15 horses, in 4 days, consume 87 bushels, 6 qrts. of oats ;
how many horses will 610 bushels, 1 peck, 2 qrts, keep for the same
time? Ans. 105.
60. Beduce 1 pound troy, to the fraction of one pound avoirdu-
pois. ' Ans
I 44
61. Beduce
to a Himple fraction.
Ans. ^.
' 62. What will be tho cost of 8 cwt., 3 qrs., 12J lbs. of beef, if 4
cwt. cost $34 ? Ans. $75y'g.
63. If 4 men, working 8 hours a day, can do a certain piece of
work in 15 days ; how long would it take one man, working 10 hours
a day, to do the same piece of work ? Ans. 48 days.
64. Divide $1728 among 17 boys and 15 girls, and give each boy
■Pj as much as a girl ; what sum will each receive ?
Ans. Each girl, $66§f j each boy, $42|f .
65. If A can cut 2 cords of wood in 12^ hours, and B can cut 3
cords iu 17^ hours ; how many cords can they both cut in 24^ hours?
' ■ ^«-".- •--r-^ ■;>-; ^^":r- :■■:■ "-f'-'-^ Ans. 8,^.
66. If it requires 30 yards of carpeting, which is f of a yard
wide, to cover a floor ; how many yards, which is 1 J yards wide, will
be necessary to cover the same flooi ? Ans. 18.
67. A person bought 1000 gallons of spirits for $1500 ; but 140
gallons leaked out ; at what rate per gallon must he sell the remain-
der so as to make $200 by his bargain ? Ans. $1.98 nearly.
68. What must be the breadth of u piece of land whose length is
40^ yards, in order that it may be twice as groat aa another piece of
ANALYSIS AND SYNTHESIS.
m.
14'
land whoso length m 14§ yards, and whose breadth is ISy-, yards?
Ans. 9^ yards.
69. If 7 men can reap a rectangular field whose length is 1,800
feet, and breadth 960 feet, in 9 days of 12 hours each ; how long
will it take 5 men, working 14 hours a day, to reap a field whose
length is 800 foet, and breadth 700 feet ? Ans. .3^ days.
70. 124 men dug a trench 110 yards long, 3 feet wide, and 4
feet deep, in 5 days of 11 hours each ; another trench was dug by
one-half the number of men in 7 days of 9 hours each ; how many
feet of water wiw it capable of holding ? Ans. 2268 cubic feet.
71. If 100 men, by working 6 hours each day, can, in 27 days,
dig 18 cellars, each .40 feet long, 36 feet wide, and 12 feet deep ;
how many cellars, t^at are each 24 feet long, 27 feet wide, and 18
feet deep, can 240 men dig in 81 days, by working 8 hours a day ?
Ans. 256.
72. A gentleman left his son a fortune, ^ of which he spent in
2 months, ^ of the remainder lasted him 3 months longer, and f of
what then remained lasted him 5 months longer, when he had only
$895.50 left; how much did his father leave him ? Ans. $4477.50.
73. A farmer having sheep in two different fields, sold ^ of the
number from each field, and had only 102 sheep remaining. Now
12 sheep jumped from the first field into the second ; then the num-
ber remaining in the first field, was to the number in the second
field as 8 to 9 ; how many sheep were there in each field at first ?
Ans. 80 in first field ; 56 in second.
74. A and B paid $120 for 12 acres of pasture for 8 weeks, with
an understanding that A should have th : grass that was then on the
field, and B what grew during the time they were grazing; how
many oxen, in equity, can each turn into the pasture, and how much
should each pay, providing 4 acres of pasture, together with what
grew during the time they were grazing, will keep 12 oxen 6 weeks,
and in similar manner, 5 seizes will keep 35 oxen 2 weeks ?
A should turn into the field 18 oxen, and pay $72.
B should turn into the field 12 oxen, and pay $48.
Ans. -j
-It*"'
ANALYSIS AND SYNTHESIS.
' Analysis is the act of separating and comparing all the different
parts of any compound, and showing their connection with each
other, and thereby exhibiting all its elementary principles.
»»i
112
ABirHUxno.
The oonvorae of Analysis is Synthesis. The meaning and nsa of
these terms will probably be most readily comprehended by referonoe
to thoir derivation.
Thoy are both pure Greek words. Analysis means looking up.
The general reader woald here probably expect looting down, as
employed in most popular definitions; but we may illustrate the
Greek term, loosing up, by our own everyday phrase, tearing up,
which means rending into shreds, the English up conveying the
same idea here as the Greek ana in analysis. The Greek synthesis
means literally placing together ; that is^ the component parts being
known, the word synthesis indicates the act of combining them into
one. We might give many illustrations, but ode will suffice, and we
choose the one which will be most generally understood. When we
analyse a sentence, we loose it up, or tear it up, into its component
parts, and by synthesis v^e write or compose, i. e., put together the
parts, which, by analysis, we have found it to consist of.
When we commence to analyse a problem we reason from a given
quantity to its unit, and then from this unit to the required quan-
tity ; hence, all our deductions are self-evident, and we therefore
require no rule to solve a problem by analysis.
, Although this part of arithmetic is usually called analysis, yet,
as it is really both analysis and synthesis, we have given it a title in
aocordoiice with the principles now laid down.
EXAMPLE.
i 1. If 12 pounds of sugar cost $1.80, what will 7 pounds cost ?
* ' ■• - t SOLUTION.
12)1.80
.15
$1.05
If 12 lbs. cost $1.80, one pound will cost the
j^2 of $1.80=15 cents. Now, if 1 lb. cost 15
cents, 7 lbs. will cost 7 times 15 cents=to $1.05.
Therefore, 7 lbs. of sugar will cost $1.05, if 12
lbs. cost $1.80.
Note. — ^The work may be somewhat shortened, especially in long ques-
tions, by arranging it in tlie following, manner, so as to admit of canceUinn^
if possible : —
■ 15
1 . ,1M. .7 105_ft, ns Ans.
M
U^ 1 ^1 1
2. If 5 bushels of pease cost $5.50, for what can you purchase
19 bushels? Ans. $20.90.
ANALTOIS AND SYNTHESIS.
118
3. If 9 men can perform a certain piece of labor in 17 dojfl, how
long ivill it take 3 uen to do it ? Ana. 51 days.
4. How many pigs, at $2 each, must bo given for 7 sheep, worth
$4 a bead ? Ans. 14.
5. If $100 gain $G in 12 months, how much would it gain in 40
months ? Ans. $20.
6. If 4j bushels of apples cost $3^, what will be the cost of 7^
bushels ? ,
SOLUTION.
In the first place, 4'1 bu8hela=V bushels, and 83J=$-%9.
Now, since -'j'- bushels cost S^, one bushel will cost p^-~-i^^=
^Xi\=9ij and 7i or '/- bushels will cost J^- times $'-=l X-^jt'=
$5, the value of 7^ bushels of apples, if ^ bushels arc worth $3^.
OPERATION
5
181
9
5
1'
7. Iff of 3f lbs. of tea cost $11 ; what will bo the cost of 5^
pounds ? Ans. $4.12^.
8. 100 is f of what number? ' r - r Ans. 150.
9. If ^ of a mine cost $2800 ; what is the value* of § of it ?
. Ans. $4200.
10. ^ of 24 is If times what number ? Ans. 10.
11. I of 40 is y»3 of how many times J of | of 20 ? Ans. 9.
12. A is 16 years old, and his age is § tim6s 'j of his father's
ige ; how old is his father? Ans. 36.
13. A and B were playing cards ; A lost $10, which was I times
} as much as B then had ; and when they commenced f^ of A'a
money was equal to f of B's ; how much had each when he begMi
to play? Ans. A $45; B $40.
14. A man willed to his daughter $560, which was I of f of
what he bequeathed to his son ; and 4 times the son's portion was |
Ihe value of the father's estate ; what was the value of the estate ?
Ans. $13,440.
15. A gentleman spent ^ of his life in St. Louis, ^ of it in Bos*
Ion, and the remainder of it, which was 25 years, in WashingtSi;
what age was he when he died ? \ns. 60;
!
114
ARITHICSTIO.
16. A owt.H ^, and B j^ of a ship ; A's part is worth $650 more
than B'h ; what is the value of the ship ? Ans. $16,600.
17. A post stands ^ in the mud, ^ in tho water, and 15 feel
above the water ; what is the length of tho post ? Ans. 36 feet.
18. A grocer bought a firkin of butter containing 56 pounds, for
$11.20, and sold ^ of it for $8'| ; how much did ho get a pound ?
Ans. 20 cents.
10. Tho head of a fish is 4 feet long, tho tail as long ns tho head
and ^ the length of ho body, and the body is as long as the head
and tail ; what is tho length of tho fish ? Ans. 32 feet.
20. A and B have tho sumo income ; A saves \ of his ; B, by
spending $65 a year more than A, finds himself $25 in debt at tho
end of 5 yours ; what did B spend each year ? Ans. $425.
21. A can do a certain piece of work in 8 days, and B can do
the same in 6 days ; A com&icnced and worked alone for 3 days,
when B assisted him to complete tho job ; how long did it take them
to finish tho work ?
SOLUTION.
If A can do the work in 8 days, in one day ho can do the ^ of it,
and if B can do the work in 6 days, in one day he can do tho ^ of it,
and if they work together, they would do ^-\-i=^i of the work in
one day. But A works alono for 3 days, and in one day ho can do |f
of tho work, in 3 days he would do 3 times |r=f o^ ^^^ work, and as
the whole work is equal to g of itself, there would be | — '^=^ of the
work yet to be completed by A and B, who, according to Ihe con-
ditions of the question, labour together to finish the work. Now A
and B working together for one day can do g'^j of the entire job, and
it will take them as many days to do the balance f as ^^ is contain-
ed in f, which is equal |X V^^^l ^^J^-
"22. A and B can build a boat in 18 days, but if C assists them,
they can do it in 8 days ; how long would it take C to do it alone?
Ani, 14| days.
23. A certain polo was 25^ feet high, and during a storm it was
broken, when ^ of what was broken ofi^, equalled § of what remained ;
how much was broken off, and how much remained ?
• ■ Ans. 12 feet broken off, and 13^ remained.
24. There are 3 pipes leading into a certain cistern ; the first will
fill it in 15 minutes, the second in 30 minutes, and the third in one
hour ; in what time will they all fill it together ?
Ans. 8 min., 34| see.
ANALYSIS AND SYNTHESIS.
116
A^i
25. A. nnd B. start top;cther by railway train from Buffalo to
£rio a distanco of (»uy) 100 inilRs. A •;oc.s by rroii^iit train, at the
rate of 12 luilcH per hour, anil B by mixed train, at tliu rato of 1^
milcH |K!r hour, C loaves Erie r«ir BufTulu at the nanio time by ex
press train, wliieli runs at (ho rato uf 22 uiiloM per hour, how far from
Buffalo will A and B each bo when C moots thorn. A,?).')^^ B, 45.
20. A cistern ban two pipoH, ono will Gil it in 48 minutos, and
the other will empty it in 72 minutes ; what time will it rccjuiro to
fill the cistern whon both aio running? Ann. 2 hours, 24 min.
27. If u man spends ,'\ of his time in working, J in sloopitiir, /^
in eating, and 1^ hours each day in reading; how much tini<i will be
left? Ans. 3 hours.
28. A wall, which was to be built 32 feet high, was raised 8 feet
by 6 men in 12 days ; how many men must bo employed to finish
the wall in G days ? Ans. 30 men.
29. A and B can perform a piece of work in 6/, days ; B and
in 6§ days ; and A and C in 6 days ; in what time would each of
them perform the work alono, and how long would it take them to do
the work together ?
Ans. A, 10 days; B, 12 days; C, 15 days; and A. B, and C,
together, in 4 days.
; 30. My tailor informs me that it will take 10|- square yards of
cloth to make mo a full suit of clothes. The cloth I am about to
purchase is 1^ yards wide, and on sponging it will shrink V^ in
width and length ; how many yards of this cloth must I purchase for
my " new suit ?"
Ans.
a (I'i
yards.
31. If A can do § of a certain piece of work in 4 hours, and B \y
can do f of the remainder in 1 hour, and C can finish it in 20 min. ; ^\^
in what time will they do it all working together ? .
Ans. 1 hour, 30 min.
32. A certain tailor in the City of Brooklyn bought 40 yards of
broadcloth, 2^ yds wide ; but on sponging, it shrunk in len;.:th upon
every 2 yards, ^'g of a yard, and in width, 1^ sixteenths upon every
1| yards. To line this cloth, he bought flannel 1^ yards wide
which, when wet, shrunk ^ the width on every 10 yards in length,
and in width it shrunk ^ of a sixteenth of a yard ; how many yarda
of flannel had the tailor to buy to line his broadcloth ?
Ans. 71,'^^ yards.
33. If 6 bushels of wheat are equal in value to 9 bushels of bar-
ley, and 5 bushels of barley to 7 bushels of oats, and 12 bushels oC
116
ARITBIIETIO.
oats to 10 bnihelfl of peaao, and 13 bnahels of pease to ^ ton of hay,
und 1 ton of h&y to 2 tons of ooal, how many tons of oool are eqiui
ill value to 80 bushsls of wheat ?
SOLUTION.
If 6 bushels of wheat are equal in value to 9 busliela of barley,
or 9 bushels of barley to 6 busiiels of wheat, one bushel of barley
would be equal to ^ of 6 bushels of wheat, equal to 'j, or ^ of u
bushel of wheat, and 5 bushels of barley would bo equal to 5 times
I of a bushel of wheat, equal to §xB"Y--^i bushels of wheat.
But 5 bushels of barley are equal to seven bushels of oats ; hence, 7
bushels of oats are equal to 3| bushels of wheat, and one bushel of
oats would be equal to 3^-i-7=:^7 bushels of wheat, and 12 bushels
of oats would bo equal to 12 times .^^f=:}.^p.=^ bushels of wheat.
But 12 bushels of oats ore equal in value to 10 bushels of pease,
hence, 10 bushels of pease are equal to 5!^ bushels of wheat, and one
bushel of pease would equal 5!^-;-10=:} of a bushel of wheat, and 13
bushels of pease would equal :}X 13=^7^=71^ bushels of wheat.
But 13 bushels of pease equal in value ?j ton of hay, henee, ^ ton of
bay equals 7^ bushels of wheat, and one ton would equal 7|X2=
14^ bushels of wheat. But one ton of boy equals 2 tons of ooal,
henoe, 2 tons of ooal are equal in value to 141} bushels of wheat, and
one ton would equal 141^-^2=7]^ bushels of wheat. Lostly, if 7i}
bushels of wheat be equal in value to one ton of coal, it would take
as many tons of ooal to equal 80 bushels of wheat, as 7f is contained
la 80, which gives 10{§ tons of coal.
NoTH. — This qnestion belongs to that port of arithmetic usually called
''.' Djoined Proportion, or, by some, the " Chain Rule," which has each ante-
jedeat of a compound ratio equal in value to its consequent. We have
thoaght it best not to introduce such questions under a head by themselves,
on accuant of their theory being more cosily understood when exhibited by
Analysis than by Proportiou. Questions that do occur like this will most
probably relate to Arbitration of Exchange. Although they may all be
worked by Compound Proportion as well as by Analysis, yet ibo most expe>
ditious plan, and the one generaly adopted, is by the following
BULB.
Place the antecedents in one column and the oonsequ,cnt$ in
another, on the right, with the sign of equality between them. Di-
vide the continued product of the terms in the column containing the
odd term hg the continued product of the other column, and the
guotient wiU he the amwer.
ANALYSIS AND 8YNTHESIH.
117
Let us now take oor last oxhoidIo (No. 33), and sohe it by thii
G bushels of whoat=9 btubels of barley.
6 biuhelH of barley--? buahela of oats.
12 bushels of oat8=::=10 bosbels of pease.
13 bushels of pease=^ ton of hay.
1 ton of hay:-2 tons of ooal.
— tons of ooal:=80 bushels of wheat.
Ans.
9. %, ni^
%, ^, %,
?.
34. If 12 bushels of wheat in Boston are equal in value to 12|
bushels in Albany, and 14 bushels in Albany are worth 14^ bushels
in Syracuse ; and 12 bushels in Syracuse are worth 12| bushels in
Oswego ; and 25 bushels in Oswego are worth 28 bushels in CIeye<
land ; how many bushels in Cleveland ore worth 60 bushels in
Boston ? Ans. 75|g.
35. If 12 shillings in Massachusetts are worth 16 shillings in
New York, and 24 shillings in New York are worth 22| shillings in
Pennsylvania, and 7^ shillings in Pennsylvania ore worth 6 shillings
in Canada ; how many shillings in Canada are worth 50 shillings in
Massachusetts ? Ans. 41 f .
36. If 6 men can build 125 rods of fencing in 4 days, how many
days would seven men require to build 210 rods ?
f
SOLUTION. ,j. •
If 6 men can build 120 rods oi fencing in 4 days, one man
could do I of 120 rods in the same time ; and ^ of 120 rods is 20
«ods. Now, if one man can build 20 rods in 4 days, in one day he
would build ^ of 20 rods, and ^ of 20 rods is 5 rods. Now, if one
man can build 5 rods in ono day, 7 men would build 7 times 5 rods
In one day, and 7 times 5 rods=35 rods. Lastly, if 7 men can
l)uild 35 rods in one day, it would take them as many days to build
210 rods as 35 is contained in 210, which is 6 ; therefore, if 6 men
ean build 120 rods of fencing in 4 days, 7 men would require 6 days
to build 210 rods.
37. If 12 men, in 36 days, of 10 hours each, build a wall 24
feet long, 16 feet high, and 3 feet thick; in how many daysi of 8
118
ABITBMETIO.
( )■
honrs eaofa, woald the same lot (^ men bnild a wall 20 feet long, 12
feet high, and 2^ fedt thick ? Aiis. 23/b.
38. If 5 men can perform a piece of work in 12 days of 1^ hours
each; how many men wiH perform a piece of work four times as
large, in a fifth part of tK time, if they work the same number of
hours in a day, supposing that 2 of the second set can do as much
work in an hour as '6 of the first set ? Ans. 66§ men.
No^K. — Such questions ns this, where the nnswer involves • frnction, may
frequently occur, and it may be aaked how } of a man can do any work. Th(«
answer is simply this, that it requires 60 men to do the work, and one man
to continue on workinj[r } of a duy more.
39. Suppose that a wolf was observed to devour a sheep in ^ of
an hour, and a bear in f of an hour ; how long would it take them
together to eat what remained of a sheep after the wolf had been
eating ^ an hour ? Ans. 10, ^^m in.
40. Find the fortunes of A, B, C, D, E, and F, by knowing that
A is worth $20, which is ^ as much as B and C are worth, and that
is worth ^ as much as A and B, and also that if 19 times the sum
of A, B and C's fortune was divided in the proportion of f , ^ and ^,
it would respectively give f of D's, J of E's, and J of F's fortune.
f Ans. A, 20 ; 3, 55 ; C, 25 ; and D, E and F. 1200 each.
, ' 41. A and B set out from the same place, and in the same direc-
tion. A travels uniformly 18 miles per day, and after 9 days turns
and goes back as far as B has travelled during those 9 days ; he thea
turns again, and pursuing his journey, overtakes B 22^ days after
the time they first set out. It is required to find the rate at which
B uniformly travelled. Ans. 10 miles per day.
42. A hare starts 40 yards before a greyhound, and is not per-
ceived by him until she has been running 40 seconds, she scuds
away at the rate of 10 miles an hour, and the dog pursues her at the
irate of 18 miles an hour ; how long will the chase last, and what dis-
tance will the hare Jiave run ? Ans. 60^^ sec. ; 490 yards.
43. A can do a certain piece of work in 9 days, and B can do
the same in 12 days ; they work together for 3 days, when A ia>
taken sick and leaves, B continues on working alone, and after 2.
days he is joined by 0, and they finish it together in 1^ days^ how
long would be doing it alone ? Ans. 12 days.
44. A, in a scufiSe, seized on f of a parcel of sugar plums ; B
caught § of it out of his hands, and laid hold on -f^ more ; D ran
off with all A had left, exc^t 4 which E afterwards secured dyly for
himself; then A and C jointly set upon B, who, in the oonflioti leC
I'lULCnCE.
m
fall i he liud, which were equally picked up by D and E, 3iFl}Q..la]^
perdu. B then kicked down C's hat, and to work they all . w^nt'.
anew for what it contained ; of which A got :^, B ^, D ^, und C acfd
E equal shares of what was left of that stock. D then struck 'f of
what A and B lust acquired, out of their hands ; they, with somo
difficulty, recovered f of it in equal shares again, but tho other three
carried off f a piece of the same. Upon thia, they called d truce,
and agreed that tho ^ of the who\e left by A at -first, sb.ould be
equally divided among them ; how many plums, after this distribu'
tion, had each of the competitors ?
Ans. A had 2863 ; B, 6335 ; C, 2438 ; D, 10294 and E, 4950.
.1^ '» • * '
PRACTICE
'" The rule which is called Practice is nothing else man a partica«
far case of simple proportion, viz., when the first term is unity.
Thus : if it is required to find tho price of 28 tons of coal, at $7 a
ton — as a question in proportion, it would be, if 1 ton of coal costs
$7, what will 28 tons cost? and the statement would be 1 : 28;^;>7 :
F. P. Here the first term being 1, the question becomes one of
simple multiplication, but the answer, $196, is really
the fourth term of an analogy. ^
Again, to find the price of 46 barrels of flour, at
$7.62J per barrel, we have only to multiply $7.62j^
by 46. In many cases, however, it is more conveni-
ent to multiply the 46 by 7, which will give the price
of 46 barrels at $7 each. Now, 50 cents being half
a dollar, the pricoi of 46, at 50 cents, will bo $23,
aind 12^ cents being \ of, 50 cents, the price at 12^
cents will be the fourth of that at 50 cents, or
$5.75, and the whole comes to $350.75.
T(r find the price of 36 cwt., 2 qrs., 15 lbs.,
at $4.87|f. Hero the question stated afc
length would be, if 1 cwt. cost $4.87|^, whai^v
^ill 36 cwt., 2 qrs., 15 lbs. cost? Th#<
statement would be 1 : 36., 2., 15 : : $4.87^:
$350.75 ^' ^' ^^^ becoces a question of multi*
$7.62^
46
23
4572
3048
$350.75
SO
I2i
\
46
7
322
23
6.75
120
jaaimcEno.
plioation beoanse the first term is unity, and divided 1^ 1 wonld not
alter the prodnot of the other two terms. Thus :
2 qrs.
I of 1 owt.
I
10 lbs.
5 *^
j^ of 2 qrs.
^ of 10 lbs.
4.87^
36
18
2922
1461
V,-. 'V.
' ■■>.
!■< ' ;?
^li
175.50 == pnoe of 3 owt., (^ 94.87} per owt.
2.437= « 2 qrs. « « "
.487= « 10 lbs. « « «
.243= « 5 " " " **
$178,667= " 36owt., 2qr8., 151b8.
We would call the learner's special attention to the following
direction, as the neglect of it is a fertile source of error. Whenever
yon take any quantity as an aliquot part of a higher to find the
price of the former, he sure you divide the line which is the price at
the rate of thai higher denomination.
To find the rent of 189 acres, 2 roods, 32 rods, at $4.20 per
acre.
2 roods=]^ of 1 acre,
20 rods=^ of 2 roods,
10 rod8=^ of 20 rods,
2 rods=^ of 10 rods,
Since the rent of 1 acre is
$4.20, the half of it, $2.10,
will be the rent of 2 roods,
the rent of 20 rods will be
.525, the ^ of the rent of 2
roods, the half of that, . 2625,
will be the rent of 10 rods,
and, lastly, .0525 will be
the rent of 2 rods, which is
the ^ of 10 rods. We then
multiply by 189, and set the
figures of the product in the
usual order, so that the first figure of the product by 9 shall be under
the units of cents, &c., and then adding all the partial resi^lts, we
fibd the final answer, $796.74, the rent of 189 acres, 2 roods and
32 perches.
XXXBOISES.
1. What is the price of 187 owt. at $5.37^ per ewt. ?
Ads. $lt)0&.l!2|.
4.20
189
210
525
.y. 2625
525
3780
3360
420
$696.74
SBAOTXCE.
121
2. What is the value of 1857 lbs., at 13.87^ per lb. ?
Ana. $7195.871
3. What will 4796 tons amount to at (4.50 per ton ?
"'■■''- -■'--[ .^ x-:,'^i-. Ans. $21582.
I. What is the price of 29 score of sheep; at $7.62^ each ?
Ans. $4422.5&-
. 5. Sold to a cattle dealer 196 head of cattle at $18.75 each, fin^
the ar unt. Ans. $3676
u. Sold to a dealer 97 head of cattle, at $1G.12| each, on th«
average ; find the price of all. Ans. $ L564.12|.
7. What Is the price of 16 tons, 17 cwt., 2 qrs. of coal, at
$8.62^ per ton ? ^ Ans. $145.54.
8. What is the yearly rent of 97 acres, 3 roods, 20 rods, at
$4.37i per acre ? Ans. $428.19.
9. If a man has $12.50 per week ; how much has he per year ?
Ans. $650.
10. If a clerk has $2.12| salary for every working day in the
year ; what is his yearly income ? Ans. $665.12J.
II. If a tradesman earn $1.64 per day ; how much does he earn
in the year, the Sabbaths not being reckoned ? Ans. $513.32.
12. If an officer's pay is a guinea and a half per day ; how much
has he a year ? Ans. £574 17s. 6d.
13. What is the price of 479 cwt. of sugar, at $17.90 per cwt.
,.^^., . Ans.$8574.10.
i. 14. Find the price of 879 articles, at $1.19 each.
15. Find the cost of 1793 tons of coal, at $7.8T| per ton.
16. What is the value of 2781 tons of hay, at $S.62J per ton ?
17. What is the rent of 189 acres, 2 roods, 32 rods, at $4 20
per acre? Ans. $796.74.
18. What is the price of 879 hogs, at $4.25 each ?
:ri^- Ans. $3735.75.
19. What will 366 tons of coal come to at $8.12^ per ton ?
I -.; Ans. $2973.75.
20. What ia the price of 118 acresi 3 roods and 20 rods
of cleared land, at $36.75 per acre ? Ans. $4368.66.
21. What is the price of 286 acres, 1 rood, 24 rods of uncleared
land, at $7.25 per acre ? Ans. $2076.40.
22. A has 84 acres, 2 roods, 36 rods of cleared land, worth
t24i60 an ai»e; B has 298 acres, 3 roods, 24 rods of uncleared
land, worth $4.40 an acre — they exchange, the difference of value to
be paid in cash ; which has to pay, and bow much ? Ans. B $989.08.
222 ARITHMETIC.
ACCOUNTS AND INVOICES. >
AccocNTS are atatemento from merchants to customers that baro pW
obaeed 'goods on credit, and ure generally made out. periodically, ualefir
specially called for.
An mvoice is simplv a statement rendered by the noller to the buyer, «
time of purchase, showing the articles bought, and the prices of each.
1. New YoKK, July Ist, 18Ga.
Mb. James Axdebson,
lo Pbesch. WnrrE & Co., 2)r. . ;^ j i; > , ;
1866.
2.«« !.»» 2.00
Jany. 4, To 2 Ite tea, l.^*^-^ : 8 lbs. coffee, 45c. ; 20 lbs. rice, lOc. . .
4.iT>i 2.»'s
" 29, " 2i yds. Amer. tweed, !.•» ; 1 vest
Feb. 10, " 14 lbs. Mus. sugar, 12Ac. ; 10 lbs. cms. white sugar, 20c. . .
60c. 2fip. l."* •
<' 22, " lib. bk. soda, ; 1 lb. car. soda, ; 4 lbs. coffee, 45c.. .
3.«» 87Jc.
Mar. 11, " 10 yds. print, 30c. ; trimming, &c., per bill
I."" 85c. 2.«»
" 19, " 2 lbs. tobacco, 90c. ; 1 gal coal oil, : 2 gals, syrup, l.»«> .
I.''* 1.*^^
Apr! 12, " i yd. blk. silk, 3.«o ; i yd. blk. velvet, Gfi»h ; : , '
3.«« 40c. 60c. t
May 6, " 2 lbs. toa, 1.««'4 ; 1 bottle pickles, ; 1 lb. pepper, ..
35c. l.o» !.«<»
« 20, " 1 bag salt, ; 10 lbs. sugar, lOo. ; 3 lbs. raisins, 50c. ...
75c. 2.«o
" 31, " 3 lbs. currants, 25o. ; 10 tbs. white sugar, 25c
l.»» 12jc. 2.«>»
June 10, " 2 lbs. tobacco, 75c. ; i lb. B. soda, 25c. : 20 lbs. rice, 10c. .
40c. 10c. 30c. l.T*
« 17, '* 1 lb.cloTes, ; \ lb.Butmegs, ; } cinnamcn, ; 1 Ib.tea,
'• . ■ * ,-..' :■'■■'. ■ ,.,:.. ;:W:..: $47.61
t. ^^ • '■-'■ JitsamQiRa, Oct. Ist, 1866.
Mb. Wmxui Fattcbsok,
To Moffat k Mubrat, Br.
1866. Ai _
July 3, To 14 yds. fancy print, 20c. ; 12 yds. ool'd silk, 2.f^
« 14, " 2 ladies' feli hats, 2.oo; 2 prs. kid gloves, l.«»
« 22, " 4 prs. cotton hose, 40c. j 3 yds. red flannel, 8O0
Aug. 19, " 2^ yds. blk. cassimere, S.*"' ; 2( yds. cotton, 20c
" 27, " l| yds. white flannel, 75c. ; buttons, 10c. ; twist, 15c.. ..
Sept 1, " 2 suits boys' clothes, 9.«o ; 2 felt hats, l.» »
•* 8," 2 prs gloves 80c. ; 2 neckties, 62Ao
«♦ 22, " I doz. prs. cotton hiose, 7.«« ; \ doa. shirts, 26.««>
Ocyidra. Or.
20.00 15.00
Aug. 18, By Cash, ;27,Cash, |35.00
u 25. " firkin butter, 95 lbs., at 22c 20.90 55.90
Balance due |41.71i
Aeoeived payment in full,
MOFFAT &MUBBAT.
AOOOUMTS AND INVOIOES. 128
-!):»1P>9* ' ' RocHESTEB, Jan. 2nd, 186Si
lb. John Deans.
To Wood & FuoGEU, jDr. ' ' ' '• ' ^^
1866. -*•;'.' ■--:'• ':'^- ' ' ■■ V." r, , ..'i' ^ .v. l-
July 4, To 12 Ibx sugar, 10c. ; 3 lbs. tea, 1*' ; 2 lbs. tobacco, 87Jo.
" 11, " 1 bbl. salt, 2»» ; 2 lbs. indigo, 25c. ; 11 lbs. pepper, 30c.
" 18, " 2 prs. socks, 45c. ; 1 ncclc-tie, 75c. ; 2 scarfe, 25c
" 25, " 10 lbs. sugir, lie; 20 lbs. dr'd apples, 10c. ; 2 Iba. coffee 28o
" " " 18 lbs. dried peaches, 12Jc.; 1 bush.. onions, li«^«.
Aug. 4, *' 12 lbs. rice, 7c. ; 2 gals, syrup, 75c. ; 14 lbs. BUgar^ 12c...
" " " 13 lbs. mackerel, 12c. ; 2 lbs. ginger, 20c. ; 2 lbs. tea, l.»*
" 21. " 2 prs. kid gloves, 1«« ; 2 boxes collars, 37^0
Sept. 12, "10 lbs. sugar, 15c. ; 2 lbs. coffee, 35c. ; 1 lb. chocolate, 40c.
Oct 4, " 2 felt bats, 1><<; shoe blacking. 25c
" 21. " 2 lbs. pepper, 15c. ; soda, 40c. ; salpetre, 30c. ; salt, 75c.
Contra. Or.
10.OP 6.0*
Sept. 14. By Cash, ; Oct. 4, Cash
Oct 17. " 2 bbls. winter apples, 2«
Boston, Nov. 1st, 1866,
Mb. Wm.
To
Aug. 4,
" 17.
Sept 4,
" 26.
Oct 11,
" 22,
•« 27,
" 30.
Reid.
Camfbell, Linn & Co., Lr.
To 2 prs. kip boots, S** ; 2 prs. cobourgs, 2«* ,
" 7 yds. fancy tweed, 2*> ; trimmings, loo . buttons 25c..
" 2 prs. gloves, 75u. ; 3 prs. socks, 35c. ; 2 straw hats, 40c,
" 10 yds. print 35c. ; trimmings. 1"* j ribbons, 75c.
" 3 neck-ties, 62 Jc; 2 prs. boyj' gaiters, if"; shoe tie8,12Jc.
" 1 business coat, 14oo ; 2 felt hats, 1^^ ,• 1 umbrella, 2^"
" 2 flannel shirts, 4as ; 1 pr. pants, 8»o ; over-coat, 160».
" 2 lace scarfs, 2^" ; 3 prs. woollen mits 75c. ; pins, 25o.
/ i
I
Contra.
Sept 12.
Oct 24.
a-.
10o» 800 „, ^ .,,^ ^ ..
ByCash ; Oct 4, Cash, ,«
•• . 300 lbs. cheese, 10c. ; 76 Ibo. butter, 25c
Balance due $37.60
*^ " Reeeivetf payment
CAMPBELL, LINN ft Oa
124 '■'''-■ ''''"^AtOTBxn^ '^''
AtBUBN, Sept. Ist, 1868.
Mb. S. Bhob
3b WiuoN, Rat & Co., J)r. ...^„_
1866.
■Jan. 15, To 6 yds B. cloth, 4/o; 2 doz. bnttons, 30c.; 9 ozs. thread,lSo. ''-^'^
" 20, « 40 ydfl. &c. cot, ICc. ; 7 spools cot, 4c. ; 12 yds. rib., S5a ^''*
" 30. '< 16 yds. B. sUk, 2.>o^ 16yds. lining, ISc. ; Ssilk spools, Ilo. "*
Feb 20. " 3yds.drill,Slc. ; 5yds.cob'rg,34o. ; 2papei<)need.l8c.
liar. 18, " 9 yds. coating, (i.^o; l^ydiB. vesting, l.*o; 6 pr. hose, 40o. *
" 31. « 21 yds. print, 20c.; 19} yds. muslin, 30c.; 2 prs. gloves, lA*
Apr.l5, " I prs. gloves, l.^o; 16 yds. ribbon, 18o. ; 6 hand'Ic. 36c.' '^^'^
" 25. " 3prs.blanket8 6.3o. 4 counterpanes, 3.*o; 15 yds. cot.,25o. '
May29, " 2 summer hats, l."; 6 yds. ribbon, 40o. ; 2 feathers, 2**
June 5, " 4 prs. slippers, l.*"] 4 prs. hose, 60c. ; 3 [prs. hose, 40c
" 15. " 3 wool shawls, 6.3 o; i b. suit, S0.«»; 9 ozb. thread, 18o.
July 6. " 40 yds. cotton, 30c. ; 3 spools, 12c. ; 2 spools, 10c
Aug. 10. " 13 yds. flannel, 75c. ; 4 hand'ks., 35o. ; 12 yds. tape, 13c.
COnira. . Or.
15.««> 10.««
Jan. 15. By Cash, ; 22. Cash, \ {-
Feb. 20. " 60 lbs. butter, 40c. ; 6 cwt. pork, 10*«
Hay 15. " 6 geese, 80c. ; 14 fowls, 40c
Junes. " 60 lbs. wool, 60c. ; 16 lbs. wool, 60o.
30.00 r lo.oo " ^.
July 6. "Cash, ; Aug. 10, Cash,
Bbooklin, July 15th, 1866ii
Ub. R. R. Hilus.
2b J. WiLLUMS, Dr. , -,/.
1866. )^'^' h '^^'' '■■'
Jan. 10, To lblb8.H.8ugar,15c.; 16ibB.W.sngar,20c.; 121bs.Cj9ugar,18c.
" 30. " 151bs. raisins, 16c.; 13 lbs. raisins,16c.; 10 lbs. raisins,18o.
Feb. 12. " 9 lbs. oar'nts, 13c.; 12 lbs. cur'nts, 14c.; 6 lbs. cur'nts,20o.
Har. 30. ** 60 lbs. salt, 2 ; 2 lbs. wash, soda, 23c.; 1 lb. bak. Boda,25o.
Apr. 6, " 61bs.D.apple8,12c.;101bs.bi8o'ts, 17c.;61bs.bisc'ts,21o.
<* 25. " .3 cwt flour, 4.«o;^ 2 cwt C. meal, 2.30; 3 lbs. butter, 26o. r
May 1. " 161bs.pork, 20c. ; 19 lb«. cheese, 10c. ; 14 lbs. 8ugar,16o.
Jnnel5. " 6 lbs. tea, 1."; 9 gals, molasses, 40c. ; 6 doz. eggs, 12c.
Jalyl2, <' 6 lbs. sngar,16c.;9| lbs. raisins, 16c.; 10 lbs.Gur'nt8,12}o.
*' 29, <• 14 lbs. bacon, 12c. ; 5 lbs. cheese, 16c. ; 4 lbs. butter,25o.
** 31, '< 4 lbs. tea, 1.* o; 2 lbs. tea, l.a o; 6 lbs. coffee, 35o. ^ ,
MM « 40 lbs. salt, lie. ; 3 lbs. indigo, 90o. ; If lbs. blue, 30e.
**<*** 31be.8altpetre,36o.;4doz.egg8,12|o.;61b8.bntter,15c.
liiiiit
Xeoeived payment
4^> J. WlLUAMS.
iOCOUKTB AND XNYOIG£S. 125
,^n i AlbaaT, Dec. 1, 1868
li;.*' ■'■
Er. Geo. Smfsoi?,
To Tatlor A Grant, i9r. i
1866. T
July 'i, To 12 lbs. sugar, 15c. ; 2 lbs. tea, l.«* ; 3 lbs. coffee, 35c. .. .
" 12, " 21b8.tobacco,87)c.;31b8.rateins,30c.;121bs.cnnanta,16c.
« 2't " 3 lbs. gunpowder, 62)0.; Clbs. shot, 18c. ; 2 lb8.glne,26o.
Aug. 4, '* 12 lbs. washing soda, l&c. ; 4 lbs. baking soda, 25c
" 12, " Iboxmustard,l.«o;21b8.fllbert8,30c.;21b8.alm'd8,35c.
Sept.21, " 8 lbs. sugar, 14o. ; 1 *b. tea, l.^»^ ; 3 lbs. chocolate, 40o.
Oct. 12, " 4 lbs. figs, 15c. ;.2 lbs. orange peel, 30c. ; spices 40c
** 20, " 2 lbs. but. blue, i8o.; 2 lbs. sulphur, 20c.; 3 lbs. soda, 35o.
18.«»
Not. 4, " 2 lbs. smok. tobacco, 90c.; 2 lbs. snuff, 20o.;l business suit,
Contra.
x:i't .'
Or.
8.00 6.00
Aug. 12, By Cash, ; Sept 21, Cash, ;.....
Oct 20, « 100 lbs. dried apples, 15c. ; 60 lbs. peaches, 20c.
I
■
ii
Balance due.
$7.01
ISPi^
■?v«.
■»■•..« jy ■^'
Detroit, bept. 30th, 1866.
MR.S.S1IITH,
f - - — .■ t- ' ■
3bBAT,Hiu.&Co.,J9r., -^ •• ,-)■ w j:,,jA /y ' i , v
1866. ,:..-,...:..»-.,..„.*.;.v..^:i;..v.^ ^ ■& ^^ ■
1, To 6 lbs. tea, l.'o ; 15 lbs. sugar, 15c.; 1} lbs. cinnamon, 2."o.
10, " 18 lbs. rtce, 10c. ; 16 lbs. salt, 4c. ; 34 lbs. oat meal, 6c.. . ; [,
13, " 12 lbs. raisins, 18c. ; 3 lbs. tobacco, 58c. ; } lb. snuff, 34c..
2, " 10 lbs. cur'nts, 17c.; 10 lbs. ginger, 41c.; 5 lbs. mustard, 42c.
8, " 6 lbs. sugar, 18c.; 13 lbs. rice, 8c.; 21 lbs. dr'd apples, 16c.
13, " 25 lbs. raisins, 18c. ; ^ lb. B. fioda, 30c. ; f lb. nutmegs, 22c.
4, " 12 lbs. coffee, 36c.; 6 lbs. M. r^ngar, 15c.; 4 lbs.W. sugar, 20e.
" 15," 41bB.mn8tard,30c.;31bs.tobacoo,30o.;121bs.ginger,27c.
A:^7l. 6, " 2 lbs. currants, 20c. ; 14 lbs. rice, 8c. ; 9 lbs. tur. seed, 45c.
** 14, " 1| lbs. cin'mon, 70c.; 12 lbs. sago, 31c.; 14 lbs. sugar, 21c.
Hay 10, '< 16 lbs. salt, So. ; 2 lbs. indigo, 90c. ; 61 lbs. com starch. 14c.
June 12, » 40 lbs. floor, 4c. ; SO lbs. com meal, 3c. ; 25 lbs. coffee, 88c.
Jan.
«
Feb.
«
«
Mar.
$88.41
126 i'M • ABITHIIBIIO.
31. Chioaoo, Jan. 4th, 1866.
Mr. EUAS 0. CONKLIN, > '. J*
Bought of J. BuNTiN & Co.,
12 reams of foolscap paper „.@ $3.25 ^"'^^
15 dozen school books @ 4.50
23 slates @ 1.30
7 " photograph albums @ 15.00
3 " Bullion's grammar @ 7.00 : :
8 " fifth reader @ 3.50
5 gallons of black ink 1.10
4 doBcn American Commercial Arithmetic @ 18.00 ..
■■- '■■Wv>.' ' $367.90
' ' Bcoeiyed payment,
.,^ ' J. BUNTIN & Co.
32 ToBOKTO, Jan. 12tii, 1866.
Mr. James H. BuaniTT,
^ou^A< o/MoBBisoN, Tatlob & Co.,
15 owt. of cheese @ $9.00
4cwt. offlour @ 4.25
120 pounds of bacon @ 0.14 '■^'.
7 bushels of corn meal @ 0.75
12 firkins of butter @ 13.50
20 bushels of dried apples @ 2.25
13 " " peaches @ 4.00
11 cwt. of buck-wheat flour @ 5.50
15 owt. maple sugar @ 8.00
25 bags of common salt @ 1.15
57 barrels of mess pork @ 13.00
^8 " beef @ 9.75 ^
13 bushels of clover seed @ 7.50* " ?
V $2143.80
Beoeived payment by note at 30 days.
Fob MORRISON. TATLOR&Cc,
I
BILLB OF PABOELS. ( 127
33 "*• ' ^ '-'«c; Hamilton, January 2nd, 1866.
Mr. M. MoCuLLOcn,
To Joseph Light, Stationer, Dr.
For 500 French envelopes @ $3.00 per thousand.
" 12 doz. ^British American copy books.. .@ 1.15
" 6 " B. B. lead pencils @ .50 .
" 5 gross mourn in<; envelopes @ 1.06
" 2 reams mourninj; note paper ,@ 3.15
" 4 " tinted note paper @ 3.15
" 2J " Foreign note paper @ 1.40 ^
" 3 '* *' letter paper @ 3.00 . . -
" 1 doz. First Books @ .15
" 5 boxes Gillott's No. 303 pens @ .90 'i
« 5 doz. Third Books @ 1.62J -
" 10 quires blank books, half bound @ .35
** 2 packs visiting cards @ .37^
$71.98
NofE.— Bills should not be signed until settled.
I
t . ' '
34. Brookvillb, Jan. 5th, 1866.
N. D. GALBREAITa,
7b R. FiTZSiHMONS & Co., 2>r. /'
For 24 lbs. Mackerel @ (i^o,
" 3 gallons Molnsses @ 45
" 13 lbs. Young Hyson Tea @ 87^
" 13 lbs. brown Sugar @ 11
" 15 bushels of Potatoes @ 45
; vy- ■ V . . _ .:>,..,^ .■ , $22.23
v. "..' "'' Cr. "r-i* ■ :■ -'^
I'or 10 lbs. Butter @ 17o. -.
" 5doz.EggB @ 12J
" 3 gallons Maple Molasses @ &5
" Note at 20 days, to balance 17.05
$22.23
BFITZSDiMONS&Co.
IfoTB.— Such a Bill as this would be termed a Barter Bill.
128 ABIXBMSTIO. ^
85 > ,^ ,v. . V . . KmoiTON, Jan. 2nd, 1866.
James Thompson, Esq., ;. ^4^
To A. Jabdinx & Co., Dr. ;,
For 3dos. Buttons @ $0.12 ..^„
" 5^ yards of block Broadoloth @ 5.50
*< 20 yards Sheeting % .15
" 1 chest Y. H. Tea, 83 lbs @ .95
" 18 yards French Print @ .20
" 2 skeins of Silk Thread @ .09
" 6 yards black Silk Velvet % 3.60
*' 20 lbs. Loaf Sugar @ .18
" 2 gallons Molasses @ .40
« 1 bog of common Salt @ 1.15
"25 lbs. Rice @ .09
« 3 saoks Coffee, 70 lbs. oooh % .12
Ob. #166.74
By Cash 50.00
Balance duo -. $116.7«i
• c
,*>■; :.■■■.''•'•'•■•.■• ;•
36 Algonqitiit, Jan« 15th| 1865.
W. FMMINO' & Co., ' ,^^-.. ^>'.^s'V--VfV'>.i''- ; ./i;v'*, , ^
Bought o/J, & A. Wbight, : » ; ;< > , .jrt ; * u-M^-
1500 lbs. Canadian Cheese (^$.09
300 bushels Fall Wheat... @ 1.25
9 brls Pot Ash, net 7056 lbs @ 5.75 per owt.
150 bushels Spring Wheat @ 1.15
: 200 " Potatoes @ .45
600 " Oats @ .37J '
150 " Pease @ .65 'J-i
60 " Indian Com @ .50 •
60 " Apples @ .60' *
3 kegs Butter, 110 lbs. each @ .18 .
, 60 bushels Eye @ .70 _n:
* 40 " Barley. @ .80
^m^.:/:rr $1688.12:
J.&A.WBIGHT.
■fHf,^'*
PBBdMTAOS.
•••^^ ••* PEROENTAOE.
129
I
1^.— PiBOBNTAOX is an allowance, or redaction, or estimate ol
a certain portion of each 100 of the units that enter into any given
eakmlation. The term is a contraction of the Latin expression for
one hundred, and means literally hy the hundred. In calculating
dollars and cents, ti per cent, means 6 dollars for every 100 dollars,
or 6 cents for every $1, or 100 cents. If we are estimating the rate
of yearly increase of the population of a rising village, and Ond that
at the end of a certain year it was 100, and at the ond of the next
it was lOG, wo say it has increased 6 per cent. i. "J., G persons have
been added to the 100. So, also, if a largo city has a population of
100,000 at the end of a certain year, and it is found that it has
106,000 at the end of the following year, we say it has increased 6
per cent., which means that if wo count the population by hundreds
wo shall find that for every 100 at the end of the one year, there
are 106 at the end of the next ; because one hundred tliousnnds is
the same as one thousand hundreds, and we have supposed the increase
in every 100 to be 6, the total increase will be one thousand sixes or
6,000, giving a total population of 106,000 as above, or an increase
at the rate of 6 per cent. A decrease would be estimated in the
same manner. Thus, a falling ofif in the population of 6 persons in
tho hundred would be denoted by 100 — 6=94, as an increase of 6
in the hundred would be denoted by 100-j-6=106. So, also, in our
first example, a deduetion of $6 in $100 would be $100— 6=:$94,
ond a gain would bo $1004-$6=^106.
The portion of 100 so allowed or estimated, is colled the rate per
cant, as in the examples given, 6 denotes the rate per cent., or the
allowance or estimate on every 100. Should the sum on which the
estimate is made not reach 100, we can, nevertheless, cstimato what
is to be allowed on it at the same rate. Thus, if 6 is to be allowed
for 100, then 3 must be allowed for 50, and 1^ for 25, &c.
The number on which the perc^tage is estimated is called the
basis. Thus, in the example given regarding the population of a
city, 100,000 is the basis.
When the basis and percentage are combined into one, the result
is called the amount. If the rate per ecnt. be an increase or gain, it
is to be added to the basis to get the amount, and if it is a decrease,
or loss, it is to be subtracted from the basis to get the amonnt.
This latter result is sometimes called the remainder.
i
I
i
.1
180
ABnHMITXO.
From what has b«6n laid, it if plain that peroentige ii noiUn^
else than taking 100 as a standard unit of measure — (See Art, 1)—
and malting the rate a fraction of that unit, so that 6 per cent, if
-)'Jo=(Art. 15, V.) .06. Wo may obtain the same result bj thf
rule of proportion. Thus, in our illustrative example of an inoreoso
of 6 pcTbons fur every 100 on a population of 100,000, the analogy
will bo 100 porsoHH : 100,000 persons : : 6 (the increase on 100) :
6,000, the inorooso on 100,000. It is manliest that the same result
will be obtained whether we multiply the third by the second, and
divide by the first, or whether we divide the third b> the first, and
multiply the result by the second ; or, which is the same thing, mul«
(iply tlio second by the result. Now, we already found that
6-^-100— J J{^— .06, the same as before. So also, 7 per cent, of any
loss is seven one-hundredths of it, t. e., ^ Jg=.07. It should bo
earefully observed that such decimals represent, not the rate per cent.,
iut the rate per unit.
Though this is easily comprehended, yet wo know by experience
that learners arc constantly liable to commit errors by neglecting to
place the decimal point correctly. We would therefore direct parti-
eular attention to the above caution, which, with the rule already
laid down, under the head of decimal fractions, should be sufficient to
guide any one who takes even moderate pains.
EXERCISES ON FINDING THE BATE FEB UNIT.
At ^ per cent., what is the rate, per unit ? .,^.
At ^ per cent., what is the rate per unit ? ,
At 1 per cent., what is the rate per unit ?
At 2 per cent., what is tlie rate per unit ?
At 4 per cent., what is the rate per unit ?
At 7^ per cent., what is the rate psr unit ?
At 10 per cent., what is the rate );/er unit ?
At 12^ per cent., what is the rate per unit ?
At 17 per cent., what is the rr^te per unit ?
At 25 per cent., what is the rate per unit ?
At 33^ per cent., what is the r&te per unit ?
At 66f per cent., what is the rate per unit ?
At 75 per cc: .t., what is the rate per unit ?
At 100 per cent., what is the t«te per unit ?
At 112^ per cent., what is the rate per unit ?
At 150 per cent., what is the i-ftte per unit?
At 200 per cent., what is the rate per unit ? ^
Ans. .00^.
Ans. .00|.
Ans. .01.
Ans. .02.
Ans. .04.
* Ans. .07J.
Ans. .10.
Ans. .12^.
Ans. .17.
Ans. .25. ^
Ans. .33^.
Ans. .66f .
Ans. .75.
Ans. 1.00.
Anf. 1.12|.
Ans. 1.50.
Ans. 2.00»
FEBOENTAOB.
181
I. To find the {Mroontage on any given qnantity at a giTcn
fate:
On the priooiples of proportion, we have as 100 : given qaan^
iity : : rate : porceutago, and as tho third .term, divided by the fint,
gives tho ruto per unit, we have tho Himple
rulk:
Multiply the given quantity by the rate per unitf and theproduet
will he the percentaye.
EXAHPLB8.
To find how much 6 per cent, is on 720 bushels of wheat, wo
have 6-;-100— .06, tho rate per unit, and 720X.06=431 bushels,
the porcontago.
To find 8 per cent, of $7963-75, in like monnor, we have .08,
the rate per unit, and $7963.75 X 08 gives $637.10, the percentage.
Instead o{per cent the mark ("/„) is now commonly used.
XXXaOISKS ON THK RUTiE.
I. What does 6 per cent, of 450 tons of hay amount to ?
Ans. 27.
' 2. What is 10 per cent, of $879.62^ ? Ans. $87.06.
3. If 12 per cent, of an army of 47,800 men be lost in killed and jg^-
-wounded ; how many remain ? Ans. 42,064.
4. What is 5 per cent, of 187 bushels of potatoes ? Ans. 9.35.
''■ 6. What is 2^ per cent, of a note for $870 ? Ans. 21.75.
6. Find 12^ per cent, of 97 hogsheads ? Ans. 12.12^.
II. To find what rate per cent, one number is of another given
number : — Let us take as an example, to find what per cent. 24 is of
96. Hero the basis is 96, and we take 100 as a standard basis, and
these are magnitudes of the same kind, and 24 is a certain rate on
96, and wo wish to find what rate it is on 100, and by the rule of
proportion, we have the statement 96 : 100 : : 24 : F. P.=l^git=25.
Therefore 24 is 25 per cent, of 96.
From this we can deduce the simple *" ' * " " ~ — y- *
BULE.
Annex two cipher$ to the given percentagef and divide that by the
ioiiSf the quotient will be the rate per cent.
7. What per cent, of 150 is 15 7 Ans. 10.
8. What per cent, of 240 is 36 ? ' Ans. 15^
f
5
i!
11
in2
ABTTHMETIO.
• 9. What per cent, of 18 is 2 ? ^ Ans. ll^^V
10. V/hp*. per cent, of 72 is 48 ? '' ' ' Ans. 66§.
11. What per cent, of 576 is 18? Aus. 3J.
12. What per cent, is 12 of 480 ? ' ' ' Ans. 2J. j(
13. Bought a block of buildings in King street for $1719, and
sold it at a gain of 18 per cent. ; what vras the gain ? '^ ' ' .'* "' y^
-• :'iv,:i^ ■; Ans. $309.42;"
14. Vested $325 in an oil well speculation, and lost 8 per cent. ;
what was the loss ? Ans. $26.00.
15. In 1841 the population of Cleveland was about 15,000, it is
now about 50,000 ; what is the rate of increase ? Ans. 233|.
16. An estate worth $4,500 was sold ; A bought 30 per cent, of
it ; B, 25 per coot. ; C, 20 per cent. ; and D purchased the remain- •'f
der ; what per cent, qf the whole was D's share ? Ans. 25.
17. If a man walk at the rate of 4 miles an hour ; what per cent.
is that of a journey of 32 miles ? Ans. 12j^.
18. What is the percentage on $1370 at 2| per cent. ?
Ana. 37.C7J.
III. Given, a number, and the rate per cent, which it is of
another number, to find that other number, .400 is 40 per cent, of a
certain number, to fi id that number. As 40 : 100 : : 400 : F. P.=: .
4M^M=1,000. Hence we derive the
• ' Bn L£ .
Annex two ciphers to the given numhery and divide by the rate
per cent.
E X X R 1 8 E S .
1. A bankrupt can pay $2600, which is 80 per cent of his debts ^ W
how much does he owe ? Ans. $8250.
2. A clerk pays $8 a month for rent, which is 16 per cent, of w
hb salary ; what is his yearly salary ? Ans. $*)00.
3. In a manufacturing district in England, 40,000 persons died
of cholera in 1832, this was 25 per cent, of the population ; what was
the population ? , j Ans. 160,000.
4. Bought a certain number of bags of flour, and sold 124 of
ihcm, which is 12^ per cent, of the whole. Bequired, the number/*
of bagf] purch^jcd. Aus. 992.
5. In a shipwreck 480 tons are lost, and this amount is 15 per
cent, of the whole cargo. Find the caigo. An!]. 3210 tons*
PEBOENTAOE.
138
6. A firm lost $1770 by tho failure of another firm ; the loss was ^
30 per cent, of their oapitul ; what was their capital ? Ans. $5900. ^>
IV. To find the basis when the amount and rate are given : —
Suppose a man bays a piece of land for a certain sum, and by selling
it for 0300, gains 25 per cent. ; what did he pay for it at first ? —
Here it is plain that for every dollar of the co^t, 25 cents are gained
by tho sale, i. e., 125 cents for every 100, which gives us tho analo- '
gy, 125 : ICO : : 300 : F. P. ; or, dividing the two terms by 100,
1.25 : 1.00 : : 300 :,F. P., which by the rules for the multiplication
and division of decimals, giyes -^f |^^=$240, the original cost.
Again, suppose the farm had been sold at a loss of 25 per cent.
This being a loss, we subtract 25 from 100, and say, as 75 : 100 : :
300 : P. P.=^^o^=$^0, the prime cost in this case.
, . Hence we derive the
. RULE.
Divide the given amount hy one increcued or diminished hy the
given rate per unit^ according as the question implies increase or
decrease, gain or loss.
,^;,:.i .,.,- .^ ■M.-i EXERCISES. ; ■..,
1. Given the amount $198, and the rate of increase 20 per cent.
to find the number yielding that percentage. Ans. $165.
2. A fiel(^ yields 840 bushels of w)ieat, which is 250 per cent, on
the seed ; how many bushels of seed were sown ? Ans. 336 bu.
3. At 5 per cent, gain ; what is the basis if the amount be $126 ?
''':■, ^- ;::■'::;:"■ > Ans. $120. '
4. At 10 per cent. loss; what is the basis, the amount being
$328.5? Ans. $365. ;
• 5. A ship is sold for $1*2045, which is' a gain of f per cent, on
the sum originally paid for it ; for how much was it bought at first ? /%,
Ans. $12000.
"^ 6. A gambler lost 10 per cent, of his money bv a venture, and
had $279 left ; how mucu had he at first, and how much did he
lose ? - Ans. He lost $31, and had $310 at first.
7. A grocer bought a lot of fiour, and having lost 20 per cent, of
tile whole, had 160 bags remaining; how many bags did he buy ?
Ans. 200.
8. A merchant lost 12 per cent, of his capital by a bankruptcy,
«Dd had still $2200 left ; what was his whole capital ? Ans. $2500.
Ill
y
i
184
ABTTHMETIO.
. 9. Sold a sheep for |5, and gained 25 per cent. ; vrhat did I pay
/^rit? Ans. $4.
10. Lost $12000 on an investment, which was 30 per cent, of the
whole ; what was the investment ? Ans. $40000.
.b . :,-.::;: ,„,,:. _ INTEREST. ,:;„-: _ \^^^i_
From a transition common in language, the word interest has
been inappropriately applied to the mm paid for the use of money,
but its original and true meaning is simply the Mse of money. To
illustrate this, we will suppose that A borrows of B $100 for one
year, and at the end of the year, when A wishes to settle the account,
he gives B $107. Wore we to ask tl\p question of almost any per-
son except an accountant, whether A or B received the interest, we
should undoubtedly receive for an answer that B received it. But
such is not the case. A having had the nse of that money for one
year, paid B $7 for that use or interest ; hence A received the inter-
terest or use of that money, and B received $7 in cash for the same.
It is only by considering this subject in its true light that account-^
ants are able to determine upon the proper debits and credit^ that
arise from a transaction where interest is involved. If an individual
borrows money, he receives the use of that money, and when ho pays
for that use or interest, he places the sum so paid to that side of his
"interest account" which represents interest received, and if he lends
money, he lias parted with the use of that money, and when he re-
ceives value for that use or interest, he places ti.e sum so received
to that side of his " interest account" which represents interest de>
livered. • . -;bi* ■;«.-■ -'-i
We think that this explanation is sufficiently clear to illustrate
the difference between interest and the value received or paid for it.
It will also be noticed that we have given many of the exercises
in the usual form, e. g., we say what is the interest on $100 for one
year, instead of saying what must be paid /or the interest of $100 for
one year, but we have done this more in accordance with custom
than from any intention to deviate from the true meaning of the
word interest.
Interest is reckoned en a scale of so many units on every $100
for one year, and hence it is called so much per cent, per annvrnj
from the Latin per centum, by the hundred, and pa^ annum, by the
year. Thus, $6 a year for eyery $100, is called six per cmUper
IKTEBEST.
135
annum. The term is also extended to desigmite the return aooruini
from any investment, such as shares in a joint stock company.
To show the object and use of such transactions, we may su'ppose
ft case or two.
A person feels himself cramped or embarrassed in his circum-
stances and operations, and he applies to some friendly party thai
lends him $100 for a year, on the condition that at the expiration of
the year he is to receive $106, that is, the $100 lent, and ^6 more
a» a return for the use of the $100 ; or, if the Borrower gets $G00,
he pays at the and of the stipulated time not only the $600, but nls«
$36 ($6 for each $100) in return for the use of the $600. By thii
means the borrower gets dear of hvn difficulty, and maintains hit
credit at a small sacrifice. i:p;\':;--^.^ix. «<--%- -;t :■;/,- -■^)=; -^■i^r ;. :.:k--^j^
The ,5m on which interest is paid is called the principal. "^ f
The sum paid for the use of money is called the interest.
The sum paid on each $100 is called the rate.
The sum of the principal and interest is called the amount.
When interest is charged on the principal only, it is called simpk
interest. "-■'■' •■'"'■'''■
When interest is charged on the amount, it is called compound
interest.
When a certain rate per cetti. is established by law, it is oalied
legal interest.
When a higher rate per cent, is charged than is allowed by law,
H is called usury.
The legal rsk; per cent, differs in different States and in different
oonntries, so r.';' iojs the mode of calculation differ. In some
States it is consiocr^d legal, to reckon the month as consisting of 30
daySj in the calculaiii-.g of interest on any sum for a short period, in
others it is considered illegal. We have given the different modet
of calculation in order to make the work applicable to all the States.
For the legal rate per oent. of each State, see ".Laws of the States,'*
at the end of this work. ^ ^'-^^^ y--w*.-H^?-M«i^^^..5J.>^v.",, -^^^v;: ^t;
fri
i
m
U M
ill ,^
u
I
SIMPLE INTEREST.
As sibtple interest, when calculated for one year, differs in no
way from a percentage on a given sum, we have only four things to
consider, vis., the principal, the Mte (100 being tb«> hms), the inters
136
t
ABkTHMETIO.
ost, and tho time, any three of which being known, the fourth oan bo
found. The finding of the interest indudes by far the greatest
number of oases.
We shall first show the general principle, and from it deduce an
easy practical rule.
Let it be required to find the interest on |468 for one year, at €
per cent.
As 100 is taken as the basis principal in relation to which all
calculations are madC) it is plain that 100 will have the same ratio
to any given principal that the rate, which is the interest on 100,
has to the interest on the ^ven principal. Hence, in the question
proposed, we have as $100 : $468: : $6 : interest=$468XTBQ=
$468x.06=$28.08. Now .06 is the rate per unitj and from this
we can deduce rules for all cases. : : t
' ■ ' .-^■^••i>'---v>k'S.-v).r-:„. CASE l.:>-''&^'i'-^'^''-'^i-^-^- '■'•^'i'^-'^^-
To find tho interest of any sum of money fbr one year, at any
f'lven rate per cent.
■. "" ' RULE. -A^'-U:-
MuUiply the principal by the rate per unit.
EXEBOISES.
1. What is the interest on $15, for 1 year, at 3 per cent. ?
An«. $0.45.
2. What is the interest on $35, for 1 year, at 5 per cent. ?
Ans. $1.75.
3. What is the interest on $100, for 1 year, at 7 per cent. ?
Ans. $7.00.
4. What is the interest on $2.25, fer 1 year, at 8 per cent. ?
Ans. $0.18.
5. What is the interest on $6.40, for 1 year, at 8]^ per cent. ?
Ans. $0.54.
6. What is the interest on $250, for 1 year, at 9^ per cent. ?
Ans. $23.75.
7. What is the interest on $760.40, for 1 year, at 7^ per cent.?
Ans. $57.03.
8. What is the interest on $964.50, for 1 year, at 6^ per cent. ?
Ans. $62.69.
9. What is the interest on t568.75, for 1 year, at 7| per cent. ?
Ans. $41.23.
an;
• v>:j
IKTEBESP.
OABE II.
187
M^t
To find the interest of any sum of moneyi for any number of
years, at a given rate per cent. .:
BULE.
^ -*Find the intereit for one year, and muUiply by the ntimier oj
years.
'• '^; 10. What is the interest of $4.60, for 3 years, at 6 per cent ?
Ans. $0.83.
11. What is the interest of $570, for 5 years, at 7| per cent. ?
An's. $213.75.
•^^ 12. What is the interest of $460.50, for 3 years, at 6J per cent, f
Ans. $86.34.
13. What is the interest of $17.40, for 3 years, at 8^ per cent. ?
;,.: •.■■:-:;■'. . •"■": Ans. $4.35.
14. What is the interest of $321.05, for 8 years, at 5f per cent. ?
Ans. $147.68.
15. What is the interest of $1650.45, for 2 years, at 9 per cent. ?
Ans. $297.08.
16. What is the interest of $964.75, for 4 years, at 10 per cent. ?
Ans. $385.90.
',:,.. 17. What is the interest of $1674.50, for 3 years, at.lOJ per
«tot ? Ans. $527.47.
, . 18. What is the interest of $640.80, for 5 years, at 4f per cent. ?
. Ans. $152.19.
19. What is the interest of $965.50, for 7 years, at 5^ per cent. ?
'''' Ans. $371.72.
20. What is the interest of $2460.20, for 4 years, at 7 per cent. 7
, Ans. $688.86.
CASE III.
To find the interest on any sum of money foi' any number of
months, at a given rate per ce&t.
R U^L B .
Find the interest fw one year, and take oXiqwit j^rte for thit
fMiith* ; or,
Find the interest for one yeaTf divide hy 12, oa^ mviUiply by th/e
ktmtber of months. .-],.,
'< iit^
I ;
* i
m
m
I I I'll
5! :!
1 'I I
I
i
188
ABITHMBTIO.
V,
^•, RXBB0I8BS. ,..v - . /: -:.!. ..vVy.,...
21. What is the interest on $684.20, for 4 months, at 6 per centt
Ans. $13.68.
^ 22. What is the interest on $760.50, for 5 months, at 7 per cent. 1
^ Ans. $22.18.
23. What is the interest on $899.99, for 2 months, at 8 per cent. ?
Ans. $12.00.
24. What is the interest on $964.50, for 4 months, at 9 per cent. ?
Ans. $28.94.
25. What is the interest on $1500, for 7 'Jionths, ut 10 per cent. ?
Ans. $87.50.
26. What is the fnterest on $1560, for 11 months, at 7^ pet
oent.? Ans. $107.25.
'"^ 27. What is the interest on $1575.54, for 8 months, at 6^ per
cent. ? Ans. $65.65.
28. What is the interest on $1728.28, for 9 months, at Sji per
eent. ? Ana. $110.18.
29. What is the interest on $268.25, for 13 months, at 7 per
eent. ? Ans. $20.34.
30. What is the interest on $1569.45, for 1 year, 3 months, at
^ 8 per cent. ? Ans. $156.95.
31. What is the interest on $642.9l, for 1 .year, 5 months, at
% 10 per cent. ? Ans. $91.09.
32. What is the interest on $560.45, for 1 year, 6 months, at 9^
per cent. ? Ans. $79.86.
33. What is the interest on $48.50, for 3 years, 9 months, ac 10|
percent.? : /""' '• Ans. $19.10.
34. What is the interest on $560.80, for 2 years, 8 months, at
llf per cent. ? Ans. $175.72.
35. What is the interest on $2360.40, for 19 months, at 12 per
cent. 7 Ans. $448.48.
CASE IV.
To find the interest on any sum of money, for any number of
months and days, at a given rate per c«nt. a. v>, ,%
. ; ; :- ^ V ,,';.■ V.-;K5>:>?:>^-„.-v.-iI,S'':i?..i;--. RULE. ■?i;;^U.v-i :.::;;;;,;:*;,: y.i-
Find the interest /or the fnontha, and take aliquot jpartt for the
dayt, reckoning the month at conmting o^SO dag$,
EXAMPLE.
36. What is the interest on $875.50, for 8 months, 18 days, tft
11 per oent. ?
iT.:««
SDCFIiB INTEBE8T. 139
V • ' SOLUTION. I , ,' }.
Principal $876.60
Rate per unit .11
Interest for 1 year 96.3050
Interest for 6 months ; or, ^ of interest for 1 year......!.. 48.1525
Interest for 2 months ; or, ^ of interest for 6 months 16.0508
Interest for 15 days ; or, ^ of interest for 2 months 4.0127
Interest for 3 days ; or, ^ of interest for 15 days 8025
Interest for 8 months, 18 days., $69.0185
We find the intereF:t for 1 year to be $96,305, and as 6 months
»re the ^ of 1 year, the interest for 6 months will be the ^ of
the interest for 1 year ; likewise the interest for 2 months will be
the § of the interest for 6 months, and as 15 days are the |- of 2
months or 60 days, the interest for 15 days will be the ^ of the in-
terest for 2 months, and likewise the interest for 3 days, will be the
I of the interest for 15 days. Adding the interest for the months
and days together, we obtain $69.02, the sum to be paid for the uso
of $875.50, for 8 months, 18 days, at 11 per cent.
EXERCISES.
37. What is the interest on $468.75, for 4 months, 15 days, at
7 per cent. ? Ans. $12.30.
38. What is the interest on $1654.40, lor 3 months, 8 days, at 5
per cent. ? Ans. $22.52.
39. What is the intercst'on $345.65, for 11 months, 25 days, at
6 per cent. ? Ans. $20.45.
40. What is the interest on $ 14.85, for 5 months, 22 days, at 9
per cent. ? Ans. $3.22.
41. What is the interest on $673.75, for 8 months, 19 days, at
7J percent.? i' • . .^< ^-^ —tv . Ans. $36.35.
42. AVhat is the interest on $57.45, for 1 year, 2 months, 12
days, at 6 per cent. ? Ans. $4.14.
43. What is the interest on $2647, for 1 year, 5 months, 18 days^
at 61 per cent. ? Ans. $242.64.
44. What is the interest on $268.40, for 2 years, 1 month, 1 day,
at 8 per cent. ? Ans. $44.79.
45. What is the interest on $2345.50, for 3 years, 7 months, 20
days, at 10 per cent. ? : ,^ r , . Ansi. $853.50.
I
;si
140
ABEFBMBnO.
46. What 18 the interest on $4268.45, for 4 yean, 11 months,
11 days, at llf per cent. ? Ana. $2481.24.
47. What is the interest of $642.20, for 2 years, 7 months, 24
HojB, at 12 per cent. ? Ads. $204.22.
48. What is the interest of $64.50, for 2 years, 11 months, 2
days, at 7 per cent. ? Ans. $13.19.
49. What is the amount of $746.25, for 1 year, 10 mooths, 12
days, at 5 per cent. ? Ans $69.65.
50. What is the interest of $680, for 4 years, 1 month, 15 days,
at 6 percent.? Ans. $138.30.
A 8 E V .
To find the interest on any sum of money, for any number of
days, at a given rate per cent."'
BULE.
Find the interest for one year, and iay, as one year (365 days,)
u to the given number of days, so is the interest for one year to the
interest required ; or,
Having found the interest for one year^ muUiply it by the given
Humher of days, and divide by 365.
EXEBOISES.
51. What is the interest on $464, for 15 days, at 6 per cent. ?
•^ Ans. $1.14.
52. What is the interest on $364, for 12 days, at 7 per cent. ?
Ans. 84 cents.
53. What is the interest on $56.82, for 14 days, at- 8 per cent. ?
Ans. 17 cents.
> * To find' how many years elapse between any two dates, wo have only
to subtract the earlier from the later date. Thus, the number of years from
1814 to 1865 is 51 years. To find months, we must reckon from the given
date in the first named month, to the same date in each successive month.
Thus, five months from the 10th of March brings us on to the 10th of August.
To find days, we reqidre to count how many days each month contains, for
to consider every month as consisting of 80 days, in the calculation of inter-
est, is not strictly correct, although for portions of a single month it causes
ne serious error. Thus, the correct time from March 2nd to June 14th, would
l>e 104 days, viz., 29 for March, 30 for April, 31 for May, and 14 for June. A
rery convenient plan for reckoning time between two given dates is to cooot
the number of months and odd days that intervene. • Thns, from June 14th
to November 20th, woold be 6 moBths and 6 days.
SIMPLE VXTSBMffT,
141
64. What is the interest on $75.50, for 18 days, at 8^ per cent. ?
Ans. 32 cents.
55. What is the interest on $125.25, for 20 days, at 5 per cent. ? '
Ans. 34 cents.
56. What is the interest on $150.40, for 33 days, nt 6 per cent. ?
Ans. 82 cents.
57. What is the interest on $56.48, for 45 days, at 6^ per cent. ?
Ans. 45 cents.
58. What is the interest on $75.75, for 65 days, at 7 per cent. ?
Ana. 94 cents.
59. What is the interest on $268.40, for 70 days, at 7^ per cent. ?
Ans. $3.86.
60. What is the interest on $464.45. for 80 days, at 8 per cent. ?
Ans. $8.14.
61. What is the interest on $15.84, for 120 days, at 9 per cent. ?
' '-*'■■ Ans. 47 cents.
62. What is the Interest on $240, for 135 days, at 9^ per cent. ?
Ans. $8.43.
63. What is the interest on $2460, for 145 days, at 10 per cent. ?
Ans. $97.73.
64. What is the interest on $1568, for 170 days, at 11 per cent. ?
Ans. $80.33.
f.5. What is the interest oi $2688, for 235 days, at llf per
sent. ? Ans. $203.35. ,
66. Wliat is the amount of $364.80, for 320 days, at 11^ per '
^nt? Ans $401.58.
111
If]
ik
OASB YI.
To find the interest on any sum of money, for any time, at 6 per
cent.
Since .06 would be the rate per unit, or the interest of $1 for 1
year, it follows that the interest for one month would be the j\i of
.06, or yij of a cent, equal to ^ cent or .005, and for 2 months it
would equal ^ cent, or .005x2=.01. Therefore, when interest is
at the rate of 6 per cent., the interest pf $1, for every 2 months, is
one cent. Again, if the interest of $1, for ane month, or 30 days, is
^ cent or .005, it fellows that the interest for 6 days will be the | of
.005 or .001. Therefore, when interest is at the rate of 6 per oent.^
the interest of $1 for every 6 days is one mil. Hence tho
'■I
if
I
142 • ABXTHXEIXO.
BULL
Find the irUerett of $1 for the given time hy reckoning 6 cwte
for every year, 1 cent for every 2 monthe, and 1 miU/or every 6 dayt;
then multiply the given principal by the number denoting tJuit t'n-
tereet, and the prodv.ct will be the interest required.
Note.— This method can be adopted for any rate per cent by first finding
the Interest at G per cent., then adding to, or subtracting from the interest so
found, such a part or parts of it, as the given rate exceeds, or is less than 6 per
cent.
This method, although adopted by some, Is not exactly correct as tho
year is considered as consisting of 360 days, Instead of 365 ; so that the in-
terest, obtained in this manner, is too large by g{y or i^, which for every
$73 interest, is $1 too much, and must therefore be subtracted if the exact
amount be required.
BXAMPLS.
67. What is the interest of $24, for 4 months, 8 days, at 6 per
cent. ? ,
;,.? V . • SOLUTION.
The interest of |1, for 4 months, is 02
The interest of $1, for 8 days, is ,.001J
Hence the interest of $1, for 4 months, 8 days, is 021^
Now, if the interest of $1, for the ^ven time, is .021^, the inter*
«st of 124 will be 24 times .021^, which is $.512. ; ,,,
■''•"■' "' '' SXBBOISKS. '"'
68. What is the interest on $171, for 24 days, at 6 per cent. ?
Ans. 68 cents.
69. What is the interest on $112, for 118 days, at 6 per cent. ?
"'■'•^<"- • .-■ '"■■ - Ans. $2.20.
70. What is the interest on $11, for 112 days, at 6 per cent. ?
Ans. 21 cents.
71. What is the interest on 50 cents, for 360 days, at 6 per
•ent. ? Ans. 3 cents.
72. What is the interest on $75.00, for 236 days, at 6 per cent. ?
'^W -^ . Ans. $2.95.
73. What is the interest on $111.50, for 54 days, at 6 per cent. ?
Ans. $1.00.
74. What is the interest on $15.50, for 314 days, at 6 per cent. ?
,„ V ,, . * Ans. 81 cents.
■and
SIMPLE INTEBE8T.
148
75. What is the iDterest on $174.25, for 42 days, at 6 per cent. 7
Ads. $1.22.
76. What
cent.
77. What
78. What
per oont. ?
79. What
. 80. What
at 7 per cent. ?
81. What
82. What
83. What
at 10 per cent
84. What
per cent. ?
85. What
7 per cent. ?
86. What
per cent. ?
87. What
8 per cent. ?
88. What
per cent. ?
89. What
cent. ?
90. What
per cent. ?
s tho interest on $10, for 1 month, 18 days, at 6 per
, ' * Ans. 8 cents.
8 tho interest on $154, for 3 months, at 6 per cent. ?
Ans. $2.31.
s the interest on $172, for 2 months, 15 days, at 6
Ans. $2.15.
s the interest on $25, for 4 months, at 6 per cent. ?
Ans. 50 coats.
H the interest on $36, for 1 year, 3 months, 11 days,
Ans. $3.23.
8 tho interest on $500, for 160 days, at G per cent. ?
Ans. $13.33.
s the interest on $92.30, for 78 days, at 5 per cent. ?
Ans. $1.00.
s the interest on $125, for 3 years, 5 months, 15 days,
Ans. $43.23.
s the amount of $200, for 9 months, 27 days, at 6
Ans. $209.90.
s the interest on $125.75, for 5 months, 17 days, at
Ans. $4.08.
a the interest on $84.50, for 1 month, 20 days, at 5
Ans. 59 cents,
s the amount of $45, for 1 year, 1 month, 1 day, at
Ans. $48.91.
s tho interest on $175, for 7 months, 6 days, at 5^
Ans. $5.78.
s the interest on $225, for 3 months, 3 days, at 9 per
. Ans. $5.23.
s the interest on $212.60, for 9 months, 8 days, at 8^
Ajis. $13.95.
OASE VII
To find the interest on any sum of money, in pounds, shillings,
and pence, for any time, at a given fate per cent ,
BULB. ^j
MuUiphf the principal by the rate per cent., and divide hjf 100.
ill!
m
m
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1 »
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144
ABXIHICBIZC.
IZAMPLI.
91. What is the interest of £47 15s. 9d., for 1 year, 9 monthly
15 days, at 6 per cent. ? . . , . . v
SOLUTION.
£ 8. D. £ S. D.
Interest for 1 year...: 2 17 4 47 15 9
Interest for G mos., or ^ of int. for 1 year, 18 8 6
Interest for 3 mos., or J of int. for mos., 14 4
Interest for 15 days, or i of int. for 3 mo8., 2 ^ 2^86 14 6
20
Interest for 1 yoarj 9 mouths, 15 days.... £5 2 S^
12
4;i4
92. Wliat is the intcreat of £25, for 1 year, 9 months, at 5 per
cent. ? Ans. £2 3s. 9d.
93. What is the interest of £75 12a. 6d., for 7 months, 12 days,
at 8 per cent. ? Ans. £3 14s. l^d.
94. What is the amount of £64 lOs. 3d., for 3 months, 3 duys,
at 7 per cent. ? Ans. £65 13s. 7d.
95. What is the interest of £35 4s. 8d., for 6 months, at 10 per
cent.? Ans. £1 15s. 2|d.
96. What is the junonnt of £18 12s., i'oi 10 months and 3 days,
at 6 per cent.? ^/^'^'^^i
CASE VIII.
To find the pkincipal, the interest, the time, and the rate per
cent, being given. . ,. ,
EXAMPLE.
't.
•'..^
97. What principal will produce $4.50 ii)' .ost in 1 year, 3
months, at 6 per cent. ?
If a principal of $1 is put on interest for 1 year, 3 months, at 6
per cent., it will produce .075 interest. Now, if in this example, .075
be the interest on SI, the number of dollars required to produce
$4.50, will be represented by the number of times that .075 is con-
tained in $4.50, wMoh is 60 times. Therefore, $60 will produce
$4.50 interest in 1 year, 3 months, at 6 per cent. Hence the
SIMPLE INTEBE8T.
RULI.
145
Divide the given interest hy the intereet o/$l for the given (tW,
at the given rate per cent.
KXER0I8K8. , '■
98. What principal will produce 77 couts interest in 3 months, 9 j^
days, at 7 per cent. ? . Ana. 940.
99. Whiit principal will produce $10.71 interest in 8 months, 12
dayc, at 7| per cent. ? Ans. $204.
100. What principal will produce $31.60 interest in 4 years, at
3^ per cent. ? Ans. $225.
101. What sum of money will produce $79.30 interest in 2 years, v^
6 months, 15 dovs, at 6^ per cent. ? Ans. $480. ^
102. What sum of money is suffioio' .> produce $290 interest
in 2 ye» and 6 months, at 7|- per c^ . 't Ans. $1600.
' CASK IX.
To find the rate per cent., the principal, the interest, and tho
time being given.
E X A M P T. E .
103: If $3 be the interest of $60 fo. 1 j car, what is the rate per
■ * . SOLUTION. ' • •'■
If the interest of $60 for 1 year, at 1 per cent, is .60, the re-
quired rate per cent, will bo represented by the number of times that
.60 is contained in 3.00, which is 5 times. Therefore, if $3 is tho
interest of $60 for 1 year, the rate per cent, is 5. Hence the
RULE.
^••■/v
Divide the given interest hy the interest of the given principal at
1 per cent, for tJie given time.
,._.. EXERCISES. . ^ %'%®it'.'' "
104. If the interest of $40, for 2 years, 9 t , aths, 12 days, is*
$18.36 ; what is the rate per cent. ? Ans. 12.
105. If I borrow $75 for 2 months, and pay $1 interest ; what is
the rate per cent. ? , AniJ;8.
.1
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146
ABITHICETIO..
106. If I give $2.25 for the use of $30 for 9 months ; what rat*
per oont. am I paying ? Ana. 10.
107. At what rate per cent, will $150 amount to $165.75, in 1
year, 4 months, 24 days ? . Ans. 7J.
lOS. At what rate per cent, must $1, or any sum of money, be
on interest to double itself in 12 years ? ^ Ans. Ans. 3J.
109. At what rate per cent, must $425 be lent to gain $11.73
/\ in 3 months, 18 days ? Ans. 9|.
S^ 110. At what rate per cent, will any sum of money amount to
^hree times itself in 25 years ? Ans. 8.
111. If I give $14 for the interest of $125 for 1 year, 7 montha,
^K,. 6 days ; what rate per cent am I paying ? Ans. 7.
CASE X.
To find the time, the principal, the interest, and the rate per
cent, being given.
EXAMX'LE.
112. How long must $75 be at interest, at 8 per cent., to gain
$12?
SOLUTION.
The interest for $75, for 1 year, at 8 per cent., is $6. Now, if
$75 require to be on interest for 1 year to produce $6, it is evident
that the number of years required to produce $12 interest, will be
represcnte 1 by the number of times that 6 is contained in 12, which
is 2. Therefore, $75 will have to be at interest for 2 wjan to gain
$12. II once the
RULE.
Divide the given interest ty the interest of the principal for one
year, at the given rate per cent.
EXEROIEES.
113. In what time will $12 produce $2.88 interest, at 8 per
cent?- ^, - r : . Ans. Syears.
•f-, 114. In what time will $25 produce 50 cents interest, at 6 per
cent.? '.H< I Ans. 4mopth8.
. ^; 115. In wh^ time will %¥^ produce \ ^ cents interest, at 6| per
cent ? ^ Alls. 3 montlis, 18 days.
BIMFLE INTEREST.
147
116. In viheA time will any sum of money double itself, at 6 per
cent. ? Ans. 16 years, 8 months.
117. In what time will any mm of mon^y quadruple itself "at 9
per cent. ? Ans. 33 years, 4 months.
118. In what time will $125 amount to $138.75, at 8 per cent.?
Ans. 1 year, 4 months, 15 days.
119. Borrowed, January 1, 1865, $60, at 6 per cent, to bo paid )f
as soon as the interest amounted to one-half the principal. When is
it due ? Ans. May 1, 1873.
'A
1' \
u
b
If
120. A^^n^rchant borrowed a certain sum of money on January ^^
2, 1856, at 9 per cent , agreeing to settle thj account when the in
terest equalled the principal. When should he pay the same ?
A. y Ans. Feb. 12, 1867.
merchants' table VV^ ' 'V . ^ '
•\
K
For showing in, what time any sum of money will double itself, at
any rate per cent., from one to twenty, simple interest.
\'
h,
Per cent.
Years.
Per cent.
Years.
Per cent.
Years.
Per cent.
Years.
1
100
6
16f
11
?r
16
^
2
50
7
;4|
i2i
12
17
5|4
3
m
8
13
7-^H
18
5i
4
25
9
in
14
"??
19
h%
5
20
10
10
15
4
20
5
A"
7 <^^
'/ f4
A.-"' ^
s ,
MIXEDEXEECI8E.S.
121. What is the interest on $64.2.- lor 3 years, at 7 per cent.?
' Ans. $13.49.
122. What is the interest on $125.4t; for 6 months, at 6 per
cent. ?* Ans. 3.76.
123. What is the amount of S369.29 for 2 years, 3 months, 1 »
day, at 9 per cent. ? Ans. $444,16. ^^
124. What milst be paid for the use of 75 cents for 6 years, 9 \J^
months, 3 days, at 10 per cent. ? Ans. 51 cents.
125. What will $54 amount to in 254 days, at 10 per cent. ?* ^ ^
Ans. $57.81. ^C
• This and the following exercises (marked with a •) are to be worked by
Case VI.
i.i,,Vv.:a.(v''*,',' ««t
I.' I
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148
ARITHMETIC.
'■i
126. What must be paid for the interest of $45 for 72 days, at
9 percent.?* ^ Ans. 81 ceiks.
v^ 127. Wliat is the interest of $240 from January 1, 1866, to
/\ June 4, 1866, at 7 per cent. ? Ans. $7.14.
128. What will $140.40 amount to from August 29, 1865. to
y November 29, 1866, at 6J per cent. ? Ans. $151.8 .
129. What principal will give $4.40 interest in 1 year, 4 months,
15 days, at 8 per cent. ? ' Ans. $40.
130. In what time will $40 amount to $44.40, at 8 per cent/;'
Ans. 1 yr., 4 raos., 15 days.
131. At what rate per cent, will $40 produce in 1 yr., 4 mos., 15
%^ days, $4.40 interest ? Ans. 8.
132. What must bo paid for the interest of $145.50 for 240
days, at 9A per cent. ?* Ans. $9.22.
133. What will $160 amount to in 175 days, at 6 per cent. ?*
Ans. $164.67.
134. At what rate per cent, must any sum of mondy be on
^>ii^^ interest to quadruple itself in 33 years and 4 months ? Ans. 9.
135. In what time will any sum of money double itself, at 10
per cent. ? Ans. 10 years.
CASE X I.
To find the interest on bonds, notes, or other documents draw-
ing 7 ,",j per cent, interest.
Since .07-,^g or, .073 would be the rate per unit, or the interest of
$1 for I year or 365 days, it follows that the interest for 1 day
would be the ^ J^ part of .073 which is .0002, equal to two tenths
of a mill, hence the
BULL .
3Iultlplii the principal hy the number of days, and the product
hy two tenths of a mill the result will be the answer in mills.
EXAMPLE. . > ;
What must be paid for the use of $75 for 36 days at 7/jj per
jeut. ?
SOLUTION.
, The interest on $75 for 36 days would be the same as the inter-
est on $75x36=r$2700 for 1 day, and at ~-q of a mill per day
would be $2700 X. 0002=54 cents. "^-^ Vfe:^^' '^^^ %**iW
•i 2. What would be the interest on $118.30 for 42 days at 7/^
per cent. ■ ' Ans. 99cts,
X
COMMEBCIAL PAPER.
140
COMMERCIAL PAPER.
OoMMEBOiAL paper is divided into two classes^— NEGOTIABLE
and NON-NEGOTIABLE.
NEGOTIABLE COMMERCIAL PAPER.
Negotiable commercial paper is that which may be freely trans*
ferred from one owner to another, so as to pass the right of action to
the holder, without being subject to any set-ofis, or legal or equitable
defences existing between the original parties, if transferred for a
valuable consideration before maturity, and received without any
defect therein.
Negotiable paper is made payable to the payee therein named, or
to his order, or to the payee or bearer, or to bearer; or some similar
term is used ; showing that the marker intends to give the payee
authority to tiansfer it to a third party, Irce from jjl set-offs, or
equitable or legal defences existing between himself and the payee.
NON-NEGOTIABLE COMMERCIAL PAPER.
Nbn-ncgotiahle commercial paper is that wfcich is made pajrable
to the pnyee therein named, without authority to transfer it to a
third party. It may be passed from one owner to another by assign-
ment, 04: by indorsement, but it passes subject to all set-offs, and
legal or equitable defences existing between the original parties.
now THE TITLE PASSE>i.
The title to negotiable paper passes from one owner to another by
delivery, if made payable to payee or bearer, or to bearer. It passes
by iudorsement and delivery, if made payable to payee or order.
The title to non-negotiable paper passes by a mere verbal assignment
and delivery, or by indorsement and delivery.
PRIMARY DEBTOR.
In a promissory note there are two original parties — the maker
and the payee. The obligation of the maker is absolute, and con-
tinues until the note is presumed to have been paid under the Statute
of Limitations. The maker is the primary debtor. In a bill of
exchange there are three parties. When the drawee accepts the
bill, he becomes the primary debtor upon the bill of exchange.
PROMISSORY NOTE NOT PAYABLE IN MONEY.
When a promissory note is payable in anything but money, it
does not come within the Statute. There is no presumption that it
Is founded upon a valuable consideration. A consideration must be
^■Q
tall
r^%
Ml
!;^>!'
'IS
Si ■,
150
▲BITHIIETIO.
alleged in the complaint, nnd proved on the trial. The aoknowledg*
mentof a consideration in such promissory note, by inserting the
words " value recrived" is sufficient to cast upon tho defendant the
burden of proof that there was no consideration. Tho acknowledg-
ment of '* value received," raises tho presumption that tho note was
given for value ; but this presumption may bo rebutted by the de-
fendint.
A negotiable instrument is a written promise or request for the
payment of a certain sum of money to order or bearer.
A negotiable instrument must be made payable in money only,
and without any condition not certain of fulfillment.
The person, to whoso order a negotiable instrument is made
payable, must lio as^ccrtainable at the time tho instrument is made.
A negotiable instrument may give to the payee an option between
the payment of the sum specified therein, and the performance of
another act.
A negotiable instrument may be with or withput date; with or
without seal ; and with or without designation of the time or place of
payment.
A negotiable instrument may contain a pledge of collateral secu-
rity, with authority to dispose thereof.
A negotiable instrument must not contjiin any other contract
than such as is specified. Two different contracts cannot bo ad-
mitted.
Any date may be inserted by the maker ef a negotiable instru-
ment, whether past, present, or future, and the instrument is not
invalidated by his death or incapacity at the time of the nominal
date.
There are several classes of negotiable instruments, namely : —
1. Bills of Exchange; 2. Promissory Notes; 3. Bank Notes;
4. Cheques on Banks and Bankers ; 5. Coupon Bonds ; 6. Certifi-
cates of Deposit; 7. Letters of Credit.
A negotiable instrument that doea ;iot specify the time of pay-
ment, is payable immediately.
A negotiable instrument which docs not specify a place of pay-
ment, is payable wherever it is held at its maturity.
An instrument, otherwise negotifiblc in form, payable to a person
named, but adding the words, " or to his order," or " to bearer,"
or equivalent thereto, is in the former case rayablo to the written
order of such person, bnd in the latter case, payable to the bearer.
A negotiable instrument, made payable to the order of tho maker,
or of a fictitious person, if issued by the maker for a valid considera-
tion, without indorsement, has the same effect against him and all
other persons having notice of the facts', as if payable to the bearer.
A negotiable instrument, mado payable to the order of a person
obviously fictitious, is payable to the bearer.
The signature of every drawer, acceptor and indorser of a aego-
aiM»»MaaaMaitgai»i n
COilMEKCIAL r.\rER.
151
tiable instwimcnt, is presumed to. have been niaoc for a valnabla
consideration, before the maturity of the instnimcnf, and in thtj
ordinary course of business, and the words '• value received,"
noknowledgo a consideration.
Ono who writes his name upon a negotiable instrument, otlierviise
than as a, maker or acceptor, and delivers it, with lii.s name thetoon,
to another person, is called an itidorser, and his act is culled ab
indorsement.
One who agrees to indorse a negotiable instrument is bound to
write his .signature upon the back of the instrument, if there ia
suJEcient space thereon for that [)urpose.
When there is not room lor a signature upon thr h;ick of a nego-
tiable instrument, a signature equivalent to an indorsement thereof
may bo made upon a paper annexed thereto.
An indorsement may be general or .'•pecial.
A general indorsement is one by which no indorser is named. A
special indorsement (jpeciiies the indorsee.
A negotiable iiistr'.^ient bearing a general indor.'-ement cannot
be afterwards specially indorsed ; but any lawful holder may turn a
p;eueral indorsement into a special one, by writing abovo it a direction
for payment to a particular per.son.
A special indorsement may, by express words Ibr tliat i)urpo.sc,
but not olherwine, be so made as to render the instrument not negoti-
able.
Every indorser of a negotiable instrument warrants to every subs^
quent holder thereof, who is not liable thereon to him :
1. That it is in all respects wrmt it purports to be;
2. That he has a good title to it;
3. That the signatures of all prior parties are binding upon them ;
4. That if the instrument is dishonored, the indorser will, upon
notice thereof duly given unto him, or without notice, wlioro it ia
excused by law, pay so much of the same as the holder paid therefor,
with interest.
One who indorses a negotiable instrunjent before it is delivered
to the payee, is liable to the payee thereon, as an indovf-cr.
An indorser may qualify his indorsement with tho words, " with-
out recourse," or e([uivalent words ; and upon such indorsement,
he is responsible only to the same extent as in the case of a (ran:- Per
without indorsement.
Except as otherwise prescribed by the last section, an indorse-
ment " without recourse" has the same effect as any other indoise-
ment.
An indorsee of a negotiable instrument has tlic same right
against every prior party thereto, that he would have had if tho
contract had been made directly between them in the first instance.
An indorser has all the rights of a guarantor, and is exonerated
from liability in like manner.
■:|
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III;
n
152
AlOTHMEKO.
One who Indorses a negotiable instrument, at the request, and for
the " accommodation" of another party to the instrument, has all the
rights of a surety, and is exonerated in like manner, in respect to
Kvery one having notice of the facts, except that he is not entitled to
contribution from subsequent indorsers.
The wiint of consideration for the undertaking of a maker,
Acceptor, or indorser of a negotiable instrument, does not exonerate
him from liability thereon, to an indorsee in good faith for a consid-
eration.
An indorsee in due course is one who in good faith, in the ordi-
nary course of business, and for value, before its apparent maturity
or presumptive dishonor, acquires a negotiable instruuMint duly
indorsed to him, or indorsed generally, or payable to the bearer.
An indorser of a negotiable instrument, in due course, acquires
an absolute title thereto, so that it is valid in his hands, notwith-
standing any provision of law making it generolly void or voidable,
and notwithstanding any defect in the title of the person from whom
he acquired it.
One who makes himself a party to an instrument intended to be
negotiable, but whicli is left wholly or partly in blank, for the pur-
pose of filling ul'tcrwards, is liable upon the instrument to an indorsee
thereof iw du(j course, in whatever manner, and at whatever time it
may bo filled, so long ds it remains negotiable in form.
It is not necessary to make a demand of payment upon the
principal debtor in a negotiable instrument in order to charge him ;
but if the instrument is by its terms payable at a specified place, and
he is able and willing to pay it there at maturity, such ability and
willingness are equivalent to an oiFer of payment upon his part.
Presentment of a negotiable instrument for payment, when
necessary, nmst be made as follows, as nearly as by reasonable dili-
gence it is practicable : .
1. The instrument must be presented by the holder, or his
authorized agent.
2. The instrument must be presented to the principal debtor, if
he can be found at the place "where presentment should be made, and
if not, then it must be presented to some other person of discretion,
if one can be found there, and if not, then it must be presented to
some other person of discretion, if one can be found there, and if not,
then it must be presented to a notary public within the State ;
3. An instrument which specifies a place for its payment, must
be presented there, and if the place specified includes more than one
house, then at the place of residence or business of the principal
debtor, if it can be found therein ;
4. An instrument whioli does not specify a place for its payment,
must be presented at the place of residence or business of the prin-
cipal debtor, or wherever he mav be found, at the option of the
presenter ; and,
COMMERCIAL PAPER.
153
5. Tho instrument must bo presented upon the day of its appar*
ent maturity, or, if it is payable on demand, at any time before its
apparent maturity, within reasonable hours, and, if it is payable at a
baukin<; house, within the usual banking hours of the vicinity ; but,
by the consent of the person to whom it should be presented, rt may
be presented at any hour of the day.
The apparent maturity of a netj^otiable instrument, payable at a
particular time, is tho day «n which by its terms it becomes due ; or,
when that is a holiday, it should be paid the previous day.
A bill of exchancje, payable at a specified time after sight, which
is not accepted within ten days after its date, in addition to the
time which would suffice, witli ordinary diligence, to forward it for
acceptance, is presumed to have been dishonored.
The apparent maturity of a bill of exchange, payable at sight or
©n demand, is :
1. If it bears interest, one year after its date ; or,
2. If it docs not bear interest, ten days after its date, in addition
to the time which would' suffice, with ordinary diligence, to forward
it for acceptance.
The apparent maturity of a promissory note, payable at eight or
on demanc., is :
1. If it bears interest one year after its date ; or,
2. If it docs not bear interest, six months after its date.
When a promissory note is payable at a certain time after sight
or demand, such time is to bo added to tho periods mentioned in the
last paragraph.
A party to a negotiable instrument may require, as a condition
concurred to its payment by him :
1. Tluit the instrument be surrendered to him, unless it is lost
or destroyed, or the holder has other claims upon it ; or,
2. If the holder has a right to retain the instrument, and does
not retain it, then that a receipt for the amount paid, or an exonera-
tion of tho party paying, be written thereon ; or,
3. If the instrument is lost, then that the holder give to him a
bond, executed by himself and two sufficient sureties, to indemnify
him against ajiy lawful claim thereon ; or
4. If the instrument is destroyed, then that proof of its destruc-
tion be given to him.
A negotiable instrument is dishonored when it is either not
paid, or not accepted, according to its tenor, or presentment for tho
purpose, or without presentment, where that is excused.
Notice of the dishonor or protest of a negotiable instrument may
be given :
1. By a holder thereof; or,
2. By a party to the instrument who might be compelled to pay
it to the holder, and who would, upon taking it up, have a right to
reimbursement from the party to whom the notice is given.
) •■
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154
ABn^miETic.
A notice of dishonor may bo given in any form which dcscribe»
the instrunicnt nvith reasonable certainty, and substaiitiully informs
the puity receiving it that the instrument has been dishonored.
A notice of dishonor may be given :
1. By delivering it to the party to bo chained, ])cr.sonally, at
any place ; or,
2. By delivering it to some person of discretion at tlie placo of
residence or business of such party, apparently acting f u' liini ; or,
ii. By properly folding the notice, directing it to the party to be
charged, at his place of resilience, according to the hos-t information
that the person giving the notice can. obtain, depositing it in the
post-office most conveniently accessible from the place where the
presentment was made, and paying the postage thereon.
In case of the death of a party to whonj notice of dishonor should
otherwise be given, the notice must be given to one «!' liis personal
representatives; or, if there are none, then to any member of hiS
family who resided with liim at his death , or, if then; is none, thea
it must be mailed to his last place of nsidence, as prescribed by
subdivision 3 of the last paragraph.
A notice of dishonor sent to a pnrty after his death, but in igno-
ranee thereof, and in good faith, is valid.
Notice of dishonor, when given by the holder of an instrument,
or his agent, otherwise than by mail, must be given on the day of
dishonor, or on the next business day thereafter.
When notice of dishonor is given by mail, it must bo deposited
in the post-office in time for the first mail which closes alter neon of
the first business day succeeding this dishonor, and which leaves tho
place where tlie instrument was dishonored, for tho place to which
the notice should be sent.
When the holder of a negotiable instrument, at the time of its
dishonor, is a mere agent for the owner, it is sufficient for him to
give notice to his principal in the same manner as to an indorser, and
his principal may give notice to any other party to be charged, as if
he were himself an indorser. And if an agelit of the owner em-
ployes a sub-agent, it is sufficient for each successive agent or sub-
agent to give notice in like manner to his own principals.
Every party to a negotiable instrument, receiving notice of ita
dishonor, has the like time thereafter to give similar notice to prior
parties, as the original holder had after its dishonor. But this addi-
tional time is available only to tho particular party entitled thereto.
A notice of the dishonor of a negotiable instrument, if valid in
favor of tho party giving it, inures to the benefit of a)l other parties
thereto, whose right to give tho like notice hxs cot then been lost.
i
COMMERCIAL PAPER
ir)/>
18
"^ ' FORMS OF FOBEiay BILLS OF EXOHANOE.
FiiEXcn.
mie, le28 Septemfjre. 1 84 8. Bon pour £ 1 58 9 Sterllnga.
Ah vinrjt cinq IXrevihre prochaln, 11 vovn plnira pnytr par ce mamlat d
Vordre de iioiisin^ities la sirmrnc de cent einr/uante huil livrea aterlings U ahelUnga
valeur en ii<nmni£inen d que ptmaercz suivant I'avia de
4 JHesuwirs' '
u Londres, ,• i.
Gkkman.
Nitrnherf/. den 'JS Oc/oher, 1R:58. Pro X10(» Sterlhuj.
Zicci moiiute nucU d'do zulUeu Sic fjfrjm dle/ien Prhna Wrrhsd an die <)rdr«
des Ileirn Ein Ilundirt Vfund ISkrlimj den it'-""'
erkalten. A'l'e briiujen aotclie uuf licchnurifj hut Bcricht von
lleirn
Werth
Lohduu.
iTAiJAV. **
Livorno, le 25 Seliembre.. 1848. Per .<:.')00 Sterling.
A Tre mesi data pagate per quesia prima de Cumhio (una sol voliai alV ordine
Jasomma di Lire einqne cento tiledine valuta cam'
biata, c jX'nete in conto M. ^'. seconrio Vavviao Addio.
AL
Lotulra.
Malaga, a 20 do Seib" de 1848.
Spanish.
Son£'iQO.
A noveidd diasfecha se serviran K* mandar pagar por esta nrimera de cam-
bio (i la orden de /(w 8"': 'ires cicntus libra h 3iterllna9
m oro pluta valor recibido de dhoa ii'" que anotaran valor en cuenia aegun
aviso de
A los H'-"
lAmdrea.
POBTfOUKSK.
£()()() Estcrlivatt. Lisbon, aos 8 dc Daemhro de 1848.
A f)e,ssenla dia.i de vista prtcizoa pngnra V por
eata nossa %inica via de Letra tiegnra. a iio.ssa Ordcm a qu4ntia acima de oeia
Ctntas Livras Ksterlinas valor de 7103 recibido em Fuzcndas, que pasaera em
(jomia segundo aviso de
Ao i^t'u^
Londres. b
EiUi OK Excuanob on London.
£317 lO.s-. l"d. '**»9rhiladelphia. Oct. 25th, 1866.
Sixty days after sight of this, my first Bill of Exchange [seeond and third of
the same dide, and tenor unpaid), pay to the order of Wdliams & Mann, Three
Ilundrt:<l and Fhrty-seven Pounds, JSineteen Shillings and Ten peuce. Sterling,
value received, with or vcithoid further advice.
KERR, BROWN & Co.
To K. XL Gladstone, Banker,
London,
Inland Duait.
$97 1 ^J{f Chicago. Sept. 10th, 186C.
Ninety days after sight, pay to the order of Manning and Muriaon, Nine
Hundred and Sevejity-one and ^^ Dollara, value received, and charge ike samt
to our acamnt.
SMITH & EVANS.
To Sa.mi;kl Small A Co.,
BaUimore, Mi.
11
156
ARITHMETIO.
Bills of Bzohango are iho highest oioss of commeroial paper
known to the law, and it Ims never been the cherished object of the
law merchant, — which has been permitted by the EnpUsh courts to
insinuate itself into the common law, till it now forms a part of that
code, — to uphold them inviolate, as far as possible. While the lex-
mercatoria (or mercantile law) is deeply impregnated with the
principles of equity, those principles have been chiefly marked to
enable courts of law to 'inforce equitable rights, and upon this
principle was the negotiability of bills of exchange insisted upon and
finally maintained at the common law ; but when equitable principles
have been invoked for the purpose of dcstroving the validity and
security of bills of exchange, they have been Usteued to with great
disfavor and only admitted as exceptional cases
CHECKS.
1. A check is substantially the same as an inland bill of exchange :
it passes by delivery, when payable to bearer, and the rules as to
presentment, diligence of the holder, &o., which are applicable to the
one, are generally applicable to the other.
2. A check is an appropriation of the drawer's funds, in the
hands of the banker, to the amount thereof, and, consequently, the
drawee has no right to withdraw them before the cheek is paid.
3. The eharocteristios which distinguish cheeks from bills of
exchange arc, that checks are always drawn on a bank or banker ;
that they are payable immediately upon presentment, and without
days of grace ; and that they are not presentable for acceptance,
but only for payment. The want of due presentment of a check,
and notice of the non-payment thereof only exonerates the drawer in
BO far as actual damages have thereby resulted to him.
LETTERS OF CREDIT.
In addition to the commercial paper before mentioned, there is
an extensive business done by the issue of " Letters of Credit.^'
These are issued by prominent bankers in London, Paris, New York
and other cities, to travellers who are about to visit foreign countries,
and who are thus saved tlie risk and expense of carrying any large
amount of cash about them.
These Letters of Crfdit are addressed by the banker to his
oorrespondents abroad, authorizing any one or more of them to pay
to jthe person named, any portion of the sum mentioned in the letter.
Thus a person leaving New York for the Pacific Ports, South
America or Arctic Ports, or any city or place in Europe or other
portions of the world, need carry very little cash. At the first port
of arrival he is able to realize such funds as may be necessary to pay
COMMEBCIAIi PAPER.
167
hiH cxpcnHes to u further port by using his Letters ot Credit. A
traveller nioy fio round the vrorld, with the aid of Kuch a Crkdit,
und never have ujoro than one hundred ddlurs in his pookut. No
loss from oxohango need occur, in such ca.seM : bills un London being
in deniund throughout the civilizcl world.
The u.suul charge by the bunkers for such *' Letters of Credit,"
is one per rait, where the trader does not pay thu amount of the
Letter in iidvance. Whore he pays in udvanco, no charge is made ;
the use of tlio money in the banker'n hands boing an equivalent for
the oortt of the credit.
Letters of Credit are also extensively used hy importers when
travelling abroad for thu purchase of goods ; also ))y supercargoes
and captains of vessels for the purchase of cargoes in ibreign ports ;
also as remittances to distanc ports in Asia, Australia, i&c. for the
purchase of cargoes of foreign goods. Before Letters of Credit werr-
adopted or in circulation, it was the practice among American and
other merchants to remit specie to remote parts for investment in
foreign merchandize
DAYS OF GRACE.
1. In most countries, when a bill or noto in payauie ar a certain
time after date, or after sight, or after demand, it is not payuljlc the
precise tinie mentioned in the bill or note, but days of grace are
allowed.
2. The days of grace are so called, bccau.se they ^voro formerly
gratuitous, and not to be claimed as a right by the person on whom
it was incumbent to pay the bill, and were dependant on the inclina-
tion of the holder ; they still retain the name of days of grace, though
the custom of merchants, recognized by law, has long reduced them
to a certainty, and established them as a right.
3. In England, Scotland, Wales, and Ireland, three days grace
are allowed ; in other countries they vary from three tn twelve days.
4. The days of grace as allowed in England, are generally
allowed in the United States, at least no traces can be f iun<l of a
contrary decision, except in the State of Massachusetts, whore it has
been held that no days of grace are allowable, unless stipulated in
the contract itself.
It is probable that a I'll of exchange was, in its original, nothing
more than a letter of crcd.t from a merchant in one country, to his
debtor, a merchant in another, requesting him to pay the debt to a
third jKJrson, who carried the letter, and liappened to be trnvciling
to the place where the debtor resided. It was discovered, by oxpcri-
cnce, that this mode of making payments was extremely convenient
to all parties : —to the creditor, for ho could thus receive his debt
without trouble, risk or expense — to the debtor, for the I'acility of
i „ f *
f ■:
158
ARTTHKETIO.
piiyniont wan an oqual aooomniodation to him, and porliipa drew
ufter it fiiciijty of credit to Uio bonror of tlio letter, vrhn loiind liiniHolf
in fundH in u for(ii|i;n country, without the duiij^cr and incuinbrunco
of carrying H|x!cio. At Brut, perhaps, the letter coiiiainrd many
other thiujiM bi'sidoa tlio order to give credit. Hut it wmh tiuiiid tluit
the original bean'r nii^ht often, with udvanta^'c, triitihici- il. to
another. The letter was thou iliRoneunibercd of all other matter ; it
was opened and not aeuled, und the page on which it waa written,
gradually Mhrunk to the slip now in u.se. The aHsi^'iieo w;i.h, perlia[»«,
dcairoufl to know beforehand whether the party to whom it was
uddreHNcd would pay, and hoiuetiuics 8howed it to him lor that pur-
pose; his promise to pay w«8 the origin of aeeeptaiieoM. TlieMe
letters or hills, the representatives of debts duo in a l'i)reij;ii country,
were sometimes more, sometimes less, in demand ; tliey became, by
degrees, articles of traffic ; and the present complicatod und abstruse
practice and theory of ezehango was gradually formed.
PARTIAL PAYMENTS
Partial payments, as the term indicates, arc the part payments
'of promissory notes, bonds, oi: other obligations.
When these payments are made the creditor specifies in writing,
on the hack of the note, or other instrument, the sum paid, and the
time when it is paid, and acknowledges it by signing his name.
The method approved of by the Supreme Court of t; United
States, for casting interest upon bonds, notes, or other obligations,
upon which partial payments have been made, is to apply the pay-
ment, in the iirst place, to the discharge of the interest then due. If
the payment exceeds the interest, the surplus goes towards discharg-
ing the principal, and the subsequent interest is to be computed on
the balance of the principal remaining due. If the payment be less
than the interest, the surplus of interest must not be taken to aug-
ment the principal, but interest continues on the former principal
until the time when the payments, taken together, exceed the in-
terest due, and then the surplus is to be applied towards discharging
the principal.
nULE.
Find the amount of the principal to the time of the first pay-
ment ; subtract the jjnyment from the amount, and then find the
amount of the remainder to the time of the second payment ; deduct
the payment att before ; and so on to the time of settlement.
But if any payment is less than the interest then due, find the
amou7U of the sum due to the time when the payments, added to-
getJicr, shall be equal, at lea^t, to the interest already due; then find
the balance, and proceed as before. ; . . .;-J^ •: /t j,; sirj. ),r
FABTZAL PAIMEMTS. 169
• .'<"-' BXAMPLK. • ' *';tv ■!
1. On the 4th of January, 18G5, a note was given for $800, ^^
payable on demand, with interest at G per eont. The following pay-
montfl wore, receipted on the back of the note :
February 7th, 18G5, received $150
April IGth, " « 100
Sopt.,:iOth, " " 180
January 4th, 1866. " 170
March L'4th, " " 100
Juno 12th, " " 60
Settled July 1st, 1SG7. IIow much was due ?
SOLUTION:
Faro of the note, or principal $800.00
Interest on the saiuo to February 7th, 1865 (1 month, 3
days) 4.40
Amount due ut tiuio of 1st payment 804.40
First payment to bo taken from this amount 150.00
Balance remaining duo February 7th, 1865 6l>i,40 -. ■ ^ .
Interest on tho name from February 7*H, *i3G<i,'to April
16th, 18(J5 7.625
Amount duo at time of 2nd payment 661.925
Second payment to be taken from this amount 100.000
Balance remaining due April 16th, 1865 561.925
Interest on the same from April 16th, 1865, to September
30th, 1865 15.359
Amount duo at time of 3rd payment 577.284
Third payment to be taken from this amount 180.000
Balance remaining due Sept. 30th, 1865 397.284
Interest on the same from Sept. 30th, 1865, to January
4th, 1866 6.290
Amount duo at time of 4th payment '. 403.b74
Fourth payment to bo'taken from this amoun|) 170.000
Balance remaining due January 4th, 1866 233.574
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Interest on the same from Jan. 4th, 1866, to March 24th,
1866 3.114
Amount due at ttbeof 5th payment 236.688
Fifth payment to bo taken from this amount 100.000
Balance remaining due, March 24th, 1866 136.688
Interest on the same from March 24th, 1866, to June
12th, 1866 1.799
Amount due at time of 6th payment 138.487
Sixth payment to bo taken from this amount 50.000
Balance remaining due June 12th, 1866.. 88.487
Interest on the same from June 12th, 1866, to July 1st,
,1867 5.589
Amount due on settlement 94.076
2. $1600. Charleston, February 16th, 1865.
On demand, I promise to pay Jacob A;iderson, or order, one
thoiiaand six hur^'^ed dollars, u'.th interest, at 7 per cent.
John Fortune Jr.
There was paid on this note,
April 19th, 1865 $460
July 22nd " 150
August 25th, 1866 50
Sept. 12th, '' 100
Dec. 24th. " 700
Hov much was due December 31st, 1866 ?
SOLUTION.
Face of the note or principal $1600.00
Interest on the same from Feb. 16th, 1865, to April 19th,
1865 , 19.60
Amount due at time of 1st payment 1619.60
First payment to be .taken from this amount*. 460.00
Balance remaining due, April 19th. 1865 1159.60
,, . ■«
PABTIAL PAYMENTS. 16X
Interest on Ihe same from April 19th, 1865, to July 22ad,
1865 20.969
Amount due at time of 2nd payment..... 1180.569
Second payment to be taken from this amount 150.000
Balance remaining due, July 22nd, 1865 1030.569
Interest on the same from July 22nd, 1865, to Aug. 25th,
1866, greater than 3rd payment,*
Interest on the same frou July 22nd, 1865, to Sept. 12th,
1866 82.359
Amount due at time of 4th payment 1112.923
Third and fonrth payments to bo taken from this amount, 150.000
Balance remaining due Sept. 12th, 1866 962.928
Interest on the same from Sept. 12th, 1866, to Dec. 24th,
1866 19.098
Amount due at time of last payment .*. 982.026
Last payment to be taken from this amount 700.000
Balance remaining due Dec. 24th, 1866 282.026
Interest on the same from Dec. 24th, 1866, to Dec. 31st,
1866 , 382
Amount due at time of settlement, Dec. 31st, 1866 $282.20.
3. $350. ' BosTONjMay 1st, 1864.
On demand I promise to pay William BrowUy or order j
three hundred andji/ty dollars, toith interest, at 6 per cent.
James Weston.
There was paid on this note,
» December 25th, 1864 $50
V June 30th, 1865 5
* The interest on $1030.569, from Jnlj 22Dd, 1865, to August 25th, 1866,
is $78,752, and the payment made at this date, is only $50, not euoagh to
pay the interest, so if we proceeded, op in the former case, to add the interest
to the principal, and subtract the payment from the amount obtained, we
would be taking interest, until the next payment, on the excess of the interr
est, $78,762, over the payment, $50, which would be in effect interest upon
interest or compound interest which the law does not allow.
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Anga8t22Dd, 1866 IS ^ i.-r/r:^^^.
Juno 4th, 1867 100
How much was due April 5th, 1868 ? Ans. $251.67.
4. $609.65. Bbantfobd, June 8th, 1861.
Six months after date, we jointly and severally promist
to pay John Anderson, or order, six hundred and nine -^^ dollars,
at the Royal Canadian Bank in Toronto, with interest at 6 per cent,
after maturity.
Samuel Graham.
T. B. Bearman.
There was paid on this note,
October 4th, 1862 $25.00
March 15th, 1863 16.25
August 24th, 1864 36.56
What was due December 19th, 1865 ? Ans. 679.27.
6. $874.95. Kingston, May 9th, 1863.
Tliree months after date, I promise to pay Harmon
Cummings, or order, eight hundred and seventy four -y^^ dollars,
with interest after maturity at 6 per cent.
Thomas Gooppat.
There was paid on this note,
April 12th, 1864 $56.30
July 14th, 1865 24.80
Sept. 18th, 1866 240.60
- What was due February 9th, 1868? Ans. $772.94.
When the interest accruing on a note is to be paid annually
adopt the following
RULE.*
Compute the interest on the principal to the time of settlement,
and on each yearns interest after it is due, then add the sum of the
* When notes, bonds, or other obligations, are given, " with interest
payable annually," the interest is due at the end of each year, and may be
collected, but if not collected at that time, the interest due draws only simple
irUeresty and the original principal must not be increased by any addition of
yearly interest If nothing has been paid until maturity on a note drawing
annual interest, the amount due consists of the principal, the total annual
interest, or the simple interest, and the simple interest on each item of annoal
interest from the time it became due until paid. ^ !>:.'-" i < ,..^-,er.a,'
•■ ■■ "
PARTIAL PAYMENTO. 163^
interests on the annttnl interests to the amovmt of the principal^ ana
from this amount take the payments, and the interest on each, from
the time they were paid to the time of settlement, the remainder will
he the amount due.
6. $500. Prescott, May 1st, 1864,
One yewr after date, for value received, I promise to pay
Musgrovc do Wright, or order, Five Hundred Dollars, at their office^
in the city of Toronto, with interest at Q per cent., payable annually,
James Manning.
There was paid on this note :
May 4th, 1865 $150
Dec. 18th, " 300
How much was due Juno 1st, 1866 ?
SOLUTION.
Face of note, or principal $500.00
Interest on the same from May 1st 1864, to June 1st,
1866 62.50
Amount of the principal at time of settlement 562.50
First year's interest on principal $30
Interest on the same from May 1st, 1865, to June
1st, 1866 $1.95
Second year's interest on principal $30
Interest on the same from May 1st, 1866, to Juno
1st, 1866 15
Amount of interest upon annual intere. 2.10
Total amount of principal $564.4>0
First payment, May 4th, 1865 $150.00
Interest on the same from May 4th, 1805, to
June 1st, 1866 ^ 9.70
Second payment, December 18th, 1865 300.00
Interest on the same from December 18th, 1865,
to June 1st, 1866 8.20
Payments and interest on the same 467.90
Amount due June 1st, 1866 $96.70
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164 ABTTHMETIC.
7. $700. Cincinnati, January 2nd, 1863.
Eighteen months after date, I promise to pay to tht
order of J. H. Wilson, Seven Hundred Dollars, for value received,
with interest at 6 per cent., payable annually.
•
Thos. a. Bryoe.
There was paid on this note :
January 15th, 1864 $350
July 2nd, 1864 300
What amount was duo January 2nd, 1865 ? Ans. $107.22,
8. $950. . Indianapolis, Jan.3rd, 1863.
Two years after date, I promise to pay A. R. Tennison,
or order, Nine hundred and Fifty Dollars, loith interest at 9 per
cent., payable annually, value received.
James S. Parmenter.
The following payments were receipted on the back of this note :
February 1st, 1864, received $500
May 14th, « " 100
• January 12, 1865, " 300
What was due May 6th, 1865 ? Ans. $188.94.
9. $250. Mobile, January 2nd, 1863.
J Three years from, date, for value received, I promise to
pay Michael Wright, or order, Two Hundred and Fifty Dollars^
vnth interest, payable annually, at Qper cent.
Calvin W. Pearsons.
At First National Bank here.
What was the amount of this note at maturity ? Ans. $297.70.
CONNECTICUT RULE.
The Supreme Court of the State of Connecticut has adopted the
following
RULE.
Compute the interest on the principal to the time of the first pay-
ment ; if that be one year or m^jre from the time the interest com-
menced, add it to tJie principal, and deduct the payment from the
sum total. If there be afterpayments made, compute the interest on
the balarice due to the next payment, and then deduct the payment as
dbvve, and in like manner from one payment tf^ apother. t'V oV thr
TARTIAL PAYMENTS. -: ''^■'^
payments are absorbed, provided tJui time between one payment and
another he one year or more.
If any payments be made he/ore one year's interest hasaccruedy
then compute the interest on the principal sum due on the obligation
for one year, add it to the principal, and compute the interest on the
sum paid, from the time it loas paid, tip to the end of the year ; add
it to the sum paid, and deduct that mm from the principal and
interest, added as above. ^ ' •
If any payments be made, of a less sum than the interest arising
at the time of such payment, no interest is to be comjmted, but only
on the principal sum for any period.
Note.— If a year extends beyond the time of settlement, 6nd the amount of
tho remaining principal to the time of settlement ; find also the amount of the
payment or payments, if any, from tho time they were paid to the time of
settlemunt, and subtract their sum from the amount of the principal.
K>:A MPLES.
10. $900. Kingston, June 1st, 1862.
On demand wc promise to pay J. R. Smith & Co., or
order, nine hundred dollars, for value received, with interest from
date, at G per cent.
Jones & VViiiaHT.
On the back of this note were receipted tho following payments :
June 16th, 1863, received $200
August 1st, 1864, " 160
Nov. 16th, 1864, " 75
Feby. 1st, 1866, " 220
Wlaat amount was due August 1st, 1866 ?
SOLUTION .
Face of note or principal $900.00
Interest on the same from June 1st, 1862, to June 16tli,
1863 56.25
Amount of principal and interest, June 16th, 1863 956.25
First payment to be taken from this amount 200.00
Balance due 756.25
Interest on the same from June 16th, 1863, to August
1st, 1864 51.046
Amount due August 1st, 1864 807.296
" il
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166 ABTTHMETIO.
Second payment to be taken from this amount 160.000
Balance due 647.296
Interest on the same for one year 38.837
Amount duo August 1st, 1865 686.133
Amount of 3rd payment from Nov. 16th, 1864, to August
1st, 1865 78.187
Balance due -. 607.946
Interest on the same from August 1st, 1865, to August
1st, 1866 36.476
Amount due August 1st, 1866 644.422
Amount of 4th payment from February 1st, 1866, to
August 1st, 1866 226.600
Balance duo August Ist, 1866 6417.822
MEBOHANTS' RULE.
It is customary among merchants and others, when partial pay-
ments of notes or other debts are made, when the note or debt is
settled within a year after becoming due, to adopt the following
RULE.
Find the amount of the principal from the time it became due
until the time of settlement. Then find the amount of each payment
from the time it was paid until settlementf and subtract their sum
from the amount of the principal.
EXAMPLE.
11. 6400. Maitland, January 1st, 1865.
For value received, I promise to pay J. B. Smith & Co.j
or order, on demand, four hundred dollars, with interest at 6 per
cent. A. B. Cassels.
The following payments were receipted on the back of this note :
February 4th, 1865, received $100
May 16th, " " 75
August 28th " « . 100
November 25th, " « 80 ..
What was due at time of settlement, which was December 28th,
1865?
PARTIAL rAYMENTS. ^g^
SOLUTION.
Principal or face of note $400.00
Interest on the same from Jan. 1st, 1865, to Dec. 28th,
1865 23.80
Amount of principal at settlement 423.80
First payment $100.00
Interest on the same from Feb. 4th, 1805, to
Dec. 28th, 1865 5.40
Second payment 75.00
Interest on the same from May 10th, 1865, to
Dec. 28th, 18G5 2.77J
Third payment 100.00
Interest on the same from August 28th, 1865,
to Dec. 25th, 1865 2.00
Fourth payment 80.00
Interest on the same from Nov. 25th, 1865, to
Dec. 28th, 1865 44
Amount of payments to be taken from amount
of principal 365.61^
Balance due, December r8th, 1865 '. $58.18^
12. $500. Cleveland, January 1st, 1865.
Three months after date, I promise to pay James Man-
nine/, or order, Jive hundred dollars, /or value received, at the First
National Bank of Buffalo. Cyrus King.
Mr. King paid on this note, July 1st 1865, $200.
What was due April 1st, 1866, the rate of interest being 7 per
cent ? Ans. $324.50.
13. $240. Philadelphia, May 4th, 1865.
On demand, I promise to pay A. K. Frost & Co., or
order, two hundred and forty dollars, for vfllue received, with in-
terest at 6 per cent. David Flock.
The following payments were receipted on the back of this note :
September 10th, 1865, received $60
January 16th, 1866, *' 90
What T?«« <l"ft at the time of settlement, which was May 4th,
1866? Ans. $100.44.
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II
168
ABITHMETIO.
:l
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14. $340. Lowell, Juno l<)th, 1864.
Three months after date, I promise to pry Thomas
Culverwell, or order, three hundred and forty dollars, ivi'h interest,
at 6 per cent. William Mavnino.
On this note were rocciptcd the following payments :
October 14th, 18G4, received $86
February 12th, 1865, " 40
What was duo at time of settlement, Aug. 10, 1865? Ans. $232.96.
COMPOUND INTEREST.
When interest is unpaid at the end of the year, it may, by special
agreement, be added to the principal, and in its turn bear interest,
and so on from year to year. Wlien added to the principal in this
way, it is said to be compound.
A person may take compound interest and not be liable to the
charge of usury, provided the person to whom he lends money
chooses to pay compound interest, but he cannot legally collect it
unless there lias been a previous agreement to that effect.
E X A 31 P L E .
1. What is the compound interest of $60, for 4 years, at 7 per
cent. ?
SOLUTION.
Principal $60.00
Interest on the same for one year 4.20
New principal for 2nd year G4.20
Interest on the same for one year 4.494
New principal for 3rd year 68.694
Interest on the same for one year 4.808
New principal for 4th j'ear... 73.502
Interest on the same for one year 5.145
Amount for 4 j-ears 78.647
Principal to be taken from same '. 60.000
Compound interest for 4 years $18647
The method of finding compound interest is usually much short-
ened by the following table, which shows the amount of 81 or £1
for any number of years not exceeding 50, at 3, 3^, 4, 5, 6 and 7
Ser cent. The amount of $1 or £1 thus obtained, being multiplied
y the given principal, will give the required amount, IVom which, if
the principal be taken, the remainder will bo compound interest :
•BO
b
G
7
8
9
10
11
12
13
14
15
IG
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
COMPOUND INTEREST. 169
TABLE,
•Hoimo m Amnrrr or (ms dollar at coin>ouirt> nrrauiT for akt RmtBiR of tram
NOT RxouniNO rirrr.
No.
T
3 per cent.
3}i per cont.
4 per cent
6 por cont
por cent.
7 por cont
1.030 000
1.035 000
1.040 000
1.060 000
l.OoO 000
1.070 000
2
l.OliO 900
1.071 225
1.081 600
1.102 600
1.123 600
1.144 900
3
1.092 727
1.108 718
1.124 864
1.157 625
1.191 016
1.225 043
4
1.125 609
1.147 623
1.169 859
1.215 606
1.262 477
1.310 796
6
1.159 274
1.187 686
1.216 653
1.276 282
1.338 226
1.402 652
C
1.194 052
1.229 255
1.265 319
1.340 096
1.418 519
1.500 730
7
1.229 874
1.272 279
1.315 932
1.407 100
1.503 630
1.605 781
8
1.2C6 770
1.316 809
1.368 669
1.477 455
1.693 848
1.718 186
9
1.304 773
1.362 897
1.423 312
1.551 328
1.C89 479
1.838 45a
10
1.343 916
1.410 599
1.480 244
1.628 895
1.790 848
1.967 161
11
1.384 234
1.459 970
1.539 454
1.710 339
1.898 299
2.104 852
12
1.425 761
1.511 069
1.601 032
1.795 856
2.012 196
2.252 192
13
1.468 534
1.563 956
1.665 074
1.885 649
2.132 928
2.409 845
14
1.512 590
1.618 694
1.731 676
1.979 932
2.260 904
2.578 634
15
1.557 967
1.675 349
1.800 944
2.078 928
2.396 658
2.759 032
16
1.604 706
1.733 986
1.872 981
2.182 875
2.640 352
2.952 164
17
1.652 848
1.794 675
1.047 901
2.292 018
2.692 773
3.158 815
18
1.702 433
1.857 489
2.025 817
2.406 619
2.8.J4 339
3.379 932
19
1.753 506
1.922 501
2.106 849
2.626 950
3.025 600
3.616 626
20
1.806 111
1.989 789
2.191 123
2.653 298
3.207 135
3.869 684
21
1.860 295
2.059 431
2.278 768
2.785 963
3.399 664
4.140 562
22
1.916 103
2.131 612
2.369 919
2.925 261
3.603 637
4.430 402
23
1.973 587
2.206 114
2.464 716
3.071 524
3.819 750
4.740 630
24
2.032 794
2.283 328
2.563 304
3.225 100
4.048 935
6.072 367
25
2.093 778
2.363 215
2.665 836
3.386 355
4.291 871
5.427 433
26
2.156 591
2.445 959
2.772 470
3.555 673
4.549 383
5.807 353
27
2.221 289
2.631 567
2.883 369
3.733 456
4.822 346
6.213 868
28
2.287 928
2.620 177
2.998 703
3.920 129
5.111 687
6.648 838
29
2.356 566
2.711 878
3.118 651
4.116 136
5.418 388
7.114 257
30
2.427 262
2.806 794
3.243 398
4.321 942
5.743 491
7.612 255
31
2.500 080
2.905 031
3.373 133
4.538 039
6.088 101
8.145 113
32
2.575 083
3.006 708
3.508 059
4.764 941
6.453 387
8.715 271
33
2.652 335
3.111 942
3.648 381
5.003 189
6.840 690
9.325 340
34
2.731 905
3.220 860
3.794 316
5.253 348
7.251 025
9.978 114
35
2.813 862
3.333 690
3.946 089
5.516 015
7.686 087
10.676 581
36
2.890 278
3.450 266
4.103 933
6.791 816
8.147 252
11.423 942
37
2.985 227
3.671 025
4.268 090
6.081 407
8.636 087
12.223 618
38
3.074 783
3.696 Oil
4.438 813
6.385 477
9.154 252
13.079 271
39
3.167 027
3.825 372
4.616 366
6.704 751
9.703 607
13.994 820
40
3.262 038
3.959 260
4.801 021
7.039 989
10.285 718
14.974 458
41
3.359 899
4.097 834
4.993 061
7.391 988
10.902 861
16.022 670
42
3.460 696
4.241 258
5.192 784
7.761 688
11.557 033
17.144 257
43
3.564 617
4.389 702
6.400 495
8.149 667
12.250 455
18.344 355
44
3.671 452
4.643 342
6.616 616
8.557 150
12.985 482
19.628 460
45
3.781 696
4.702 368
6.841 176
8.985 003
13.764 611
21.002 462
46
3.895 044
4.866 941
6.074 823
9.434 258
14.590 487
22.472 623
47
4.011 895
5.037 284
6.317 816
9.905 971
15.465 917
24.045 707
48
4.132 262
5.213 689
6.670 528
10.401 270
16.393 872
25.728 907
49
4.256 219
6.396 066
6.833 349
10.921 333
17.377 604
27.529 930
60
4.383 906
6.684 927
7.106 683
11.467 400
18.420 154
29.467 025
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Noa.->lf each of the namberB In the table bo dimiaished by 1, the reouinder will denot*
the interoat of $1, inatead of ita amount.
170 ABITHMBTia
ISXBROISIB.
2. What is the oompoimd intorost on $75, for 3 yoars, at 7 p«r
coDt. ? Ans. $10.87.
3. What will $50 amount to io 3 years, at 6 per cent., compound
interest ? Ans. $59.55.
4. What is the compound interest on $600, for 2 years, at G per
cent., payable half-yearly ? Ans. $75.31.
6. What will $320 amount to in 2J- years, at 7 per cent, com-
pound interest ? Ans. $379.19.
C. What is the compound interest of $150, for 3 years, at 9 per
cent. ? Ans. $M.25.
7. What is the compound interest on $1,000, for 2 years, at 3|f
per cent, payable qunrterly ? Ans. $72.18.
8. What will $460 amount to in 3 years, 4 months, 10 days, at
6 per cent., compound interest ? Ans. $559.74.
9. What is the compound interest on $1860, for 8 yc; at 7 per
cent.? Ans. ^1335.83.
10. What will be the compound interest on $75.20, for 20 years^
at 3J j^r cent. ? Ans. $74.43.
11. How much more will $500 amount to at compound than
simple interest, for 20 years, 3 months, 15 days, at 7 per cent.?
Ans. $764.14.
12. What sum will $50, deposited in a savings bank, amount to
at compound interest, for 21 years, at 3 per cent, payable half yearly ?
Ans. $173.03.
13. If a note of $60.60, dated October 25th, 1856, with tho
interest payable yearly, at 6 per cent., be paid October 25th, 1860;
what will it amount to at compound interest ? Ans. $76.51.
14. What remains due on the following note, April 1st, 1863, at
7 per cent, compound interest?
$1,000. Cleveland, January 1, 1858.
For value received, I promise to pay A. B. Smith & Co., or
vrder, one thousand dollars on demand, icith interest at 7 per cent,
J. D. Foster.
On the back of this note were receipted the following payments:
June 10, 1858, received $70
Sept. 25^ 1859, « 80 •
July 4, 1860, « 100 ,
DISCOUNT AND rBESENT WOIITH.
171
Nov. 11, 18(J1,
Juno 5, 1862,
u
«(
30
50
Ans. $1022.34.
DISCOUNT AND PRESENT WORTH.
Discount boing of the same nature as interest, i.s, strictly speak-
ing, the UHc of money bofuro it is due. Tho term is applied, however,
to a deduction of .so mucli per cent, from the fucc of a bill, or the
dcductin,!^ of interest from ;i note before any interest lias accrued.
This i.s tho practice followed in our Bank-s, and is therefore called
Bank discount, in order to distinguish it from true discount.
The method of computini; bank discount differs in no way from
that of computing simple interest, but the method of finding true
discount is quite difforeut, e. g., a debt of $107, duo one year hence,
is consider'! to bo worth $100 now, for tlie reason that $100 lot out
at interest now, at 7 per cent., would amount to 8107 at the end of
a year.
In calculating interest, the sum on which interest is to be paid is
known, hut in computing discount we have to find what sum must
be placed at interest so that that sum, toc!;ether with its interest, will
amount to tho <iiven principal. Tho sum thus found is called tho
" Present Worth."
We have already seen that $1.00 is tho present worth of $1.07-
due one year hence, at 7 per cent., therefore, to get the present worth
of any sura due ono year hence, at 7 per ccvil., it is only necessary to
find how many times $1.07 is contained in tho given sum, and wo
havf^ the present worth ; henco ,
To find tho present worth of any sum, and the discount for any
time, at any rate percent., we have the following
RULE.
Divide the given sum hy the amount of $1 for the given time and
rate, and the quotient will he the present worth.
Fro,. I the given sum suhtmct the present worthy and the remainder
will be the discount.
£X£11CISE£).
1. What is the present worth of $224, due 2 years hence, at 6
per cent. ? ^^ Ans. $200.
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172
jkBiTHiarno.
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2. What ii tho diBoount on $670, due 1 year and 8 months honoe,
at 7 por oont. ? Ans. $70.
3. What is tho discount on $&01, duo 1 year and 5 months honco,
at 8 por coDt. ? Ans. $51.
4. What is tho present value of a debt of $678.75, duo 3 years
and 7 months honoc, at 7^ por cent. ? Ans. $534.97^.
6. What is tho discount on $88.16, duo 1 year, 8 months, and
12 days honoc, at 6 por oont. ? Ans. $8.16.
6. If tho discount on $1060, for 1 year, at 6 por cent., is $60;
what is tho discount on tho same sum for ono-half tho timo ?
Ans. $30.87.
7. How much cosh will discharge a dobt of $145.50, duo 2 years,
6 months and 12 days henco, at por cent. ? Ans. $126.30.
8. If I am offered a certain quantity of goods for $2500 oash, or
for $2821.50, on 9 months credit; which is tho best offer, and by
how much ? Ans. Cash by $200.
9. What is tho differonoo between tho interest and discount of
$46.16, duo at the end of 2 years, 6 months, and 24 days, at 6 per
cent. ? Ans. 95 cents.
10. A merchant sold goods to tho amount of $1500, ono-half to
be paid in 6 months, and the balance in 9 months ; how much cosh
ought he to receive for them after deducting 1^ por cent, a month ?
Ans. $1331.25.
11. Suppose a merchant contracts a debt of $24000, to bo paid
in four instalments, as follows: one-fifth in 4 months; one-fourth in
9 months ; one-sixth in 1 year and 2 months, and the rest in 1 year
and 7 months ; how much cosh must ho give at once to discharge tho
debt, money being worth 6 per cent. ? Ans. 22587.65.
12. Bought goods to the amount of $840, on 9 months credit;
how much money would discharge tho debt at the timo of purchasing
tho goods, interest being 8 per cent. ? Ans. $792.45.
13. A bookseller marks two prices in a book, one for ready
money, and the other for one year's credit, allowing discount at 5 per
cent. If the credit price be marked $9.80 ; what ought to be the
price marked for oash ? Ans. $9.33.
14. A man having a horse for sale, offered it for $225, oash ; or,
$230 at 9 months credit ; the buyer chose the latter ; did the seller
lose or make by his bargain, and how much, supposing money to be
worth 7 per cent. ? Ans. He lost $6.47.
15. A. B. Smith owes John Manning as follows : — $365.87, to
: i
DANEB AND RANKINO.
173
bo paid December 10th, 1863; $1(11.15, tobopaid.Tnlv 16th, 1864 ;
$112.50, to bo puid Juno 2:Jrd, 1862] !?!)6.H1, to bo paid April 19th,
1866, allowiii<{ discount tit 6 per cent. ; how much cuith Hhould
M'*nning roccivo ns iin ccjuivulcnt, Junuury Ut, 1862?
Ans. $6 Jo. 40.
16. I buy IV bill of goods nntountinfi to $2500 on n\x months'
credit, and can got 5 per cent, off by piiyin;? cnph ; how much would
1 frairi by payinj; tlio bill now, provided I huvo to borrow the money,
and pay 6 pur cent, a your ibr U '( Aau, $5U.75.
'ft
|1 ' l;«
BANKS AND BANKING.
Grnernl Principles of Banking. — Btinka arc commonly divided
into tlio two great cla.'jscs of banks of deposit and bauka of issue.
This, howover, appears at first flight to be ratiior an imperfect classi-
fication, inasmuch as almost all banks of deposit are at tho same time
banks of issue, and almost all banks of issue also banks of deposit.
But there is in reality no antbiguity; for by bunks of deposit aro
meant banks for tlio custody and employnictitof tiie money deposited
with tlieia or entrusted to their caro by tlieir customers, or by tho
public ; while by banks of issue aro meant banks which, besides
employing or issuing tho money entrusted to them by (tthcrs, issue
money of their own, or notes payable on demand. The Bunk of
England is principally a bank of issue ; but it, as well as the other
banks in the different parts of the empire that issue notes, is also a
. great bank of deposit. The private banking companies of London,
and tho various provincial banks, that do not issue notes of their
own, arc strictly banks of deposit. Banking business may bo con-
ducted indifferently by individuals, by private companies, or by joint
fitock companies or associations.
Utility and Functions of BanJes of Deposit. — Bunks of this class
execute all that is properly understood by banking business ; and
their establishment lias contributed in no ordinary degree to give
security and facility to commercial transactions. They afford, when
properly conducted, safe and convenient places of deposit for the
money that w(<uld otherwise have to be kept, at a considerablo risk,
in private houses. They also prevent, in a great measure, the
necessity of carrying money from place to place to make payments,
and enable them to be made in tho most convenient and least expen-
sive manner.
The objects of hanking. — Correct sentiments beget correct con-
duct. A banker ought, therefore, to apprehend correctly, the
objects of banking. They coi list in making pecuniary gains for
the stockholders Vy legal operations. Tho business is eminently
7
174
ABTTHMETIO.
t' 1
1;
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beneficial to society; but some bankers have deemed the good of
Bociety so much more worthy of regard than the private pood of
stockholders, that they havo supposed all loans should be di;jiietised
with direct reference to the beneficial effect of ihe loans on society,
irrespective, in some degree, of the pecuniary interests of the disjien-
sing bank. Such a bunker will lend to bu'lders, that houses or (<! ips
may be multiplied; to manufacturers, that useful fabrics niny he
increased ; and to merchants, that goods may bo seasonably replen-
ished. Do deems hiniseli', ex-officioy the patron of all interesti^ tluit
concern his nci;,dibourhood. and regulates his loans to these interests
by the urgeney of their necessities, rather than by the pecuniary
profits of the operations to the bank, or the ability of the bank to
sustain such demands. The late Bank of the United States is a
remarkable illustration of these errors. Its manager seemed to
believe that his dutes comprehended the equalization of foreign and
domestic exchanges, the regulation of the price of cotton, the up-
holding of iStuto credit, and the control, in some particulars, of
Congress and the President — all vicious perversions of banking to
an imagined paramount end.
When ^ve perform well the direct duties of our station, v.c need
not curiously trouble ourselves to effect, indirectly, some remote
duty, llcsults belong to Providence, and by the natural catenation
of events (a system admirably adapted to our restricted foresight),
a man can usually in no way so efiiciently promote the general wel-
fare, as by vigilantly guarding the peculiar interests tCommitted to
his care. If, lor instance, his bank is situated in a region dependent
for its prosperity in the business of lumbering, the dealers in lumber
will naturally constitute his most profitable customers; hence, in
promoting his own interest out of their wants, he will, legitimately,
benefit them as well as himself, and benefit them more permanently
than by a vicious subordination of his interests to theirs.
Men will not engage permanently in any business that is not
pecuniarily beneficial to them personally ; hence, a banker becomes
recreant to even the manufacturing and other interests that he would
protect, if he so manage his bank as to make its stockholders unwill-
ing to continue the employment of their capital in banking. This
principle, also, is illustrated by the late United States Bank, for the
stupendous temporary injuries which its mismanagement inflicted on
society, are a smaller evil than the pernicment barrier its mismanage-
ment has probably produced against the creation of any similar
institution.
Banh of England Notes Legal Tender. — According to the law as
it stood previously to 1834, all descriptions of notes were legally
payable at the pleasure of the holder in coin of the standard weight
and purity. But the policy of such a regulation was very question-
able ; and we regard the enactment of the Stats. 3 & 4, Will. 4, c.
99, which makes Bank of England notes legal tender, everywhare
^
BANES AM> BANKINO.
m
except at the Bank and its branches, for all suras above £5, as a
great improvement.
Savings Banks have been in use in Europe over fifty year?, and in
Canntia and the United States, almost as long. They aie established
for tlie purpose of receiving from people in moderate ciroumstanccs,
small sums of money on interest. In England the deposits are held
by the Government, and invested in the three per cent, funds. In
Mew England, New York and other States, the deposits are generally
loaned on bond and mortgage tii six or seven per cent, interest.
Ft'imdbj Societies. — Friendly Societies are a.ssoci!ilions, mostly
in England, of persons chiefly in the humblest classes for the pur-
pose of making provision by mutual contribution against those '.'jn-
tingencies in hum:m life, the occurrence of which can be calculated
by way of average. Tlic principal objects contemplated by such
societies are the following : The insurance of a sum of money to be
paid on the birth of a member's child, or on the death of a member
or any of his family; the maintenance of members in old age and
widowhood ; the administration of relief to members incapacitated
for labor by sickness or accident ; and the endowment of members
or their nominees. Friendly Societies are, therefore, associations
for mutual assurance, but arc distingushed from assurance societies,
properly so called, by the circumstance that the sums of money
which they insure are comparatively small.
BANK DISCOUNT.
The Bank Discount of a note is the simple interest tu the sum
for which it is given from the time it is discovntcd to i\\<^ time it
becomes due, including three days of grace.
Suppose, for example, in getting a note of $200 discounted at a
bank I am charged 812 for discount, which being deducted, I receive
but $188, so that I pay interest on $12 which I did not receive.
From this it is clear that I am paying a higher rate of interest in
discounting a note at a bank, than I would pay were I to borrow
money at the same rate. As bank discount is the saoae as interest,
we derive the following
RULE.
Find the interest on the sum specified in the note at the given
rate, and for the given time, including three days of grace, and thi'
will he the BANK UISOOUNT.
Subtract the discount from the/ace of the note, and the remain
der will he the PROCEEDS OR PRESENT WORTH.
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176
ARITHMETia
EXER OISES
1 . What is the bank discount on a note, given for 60 days, for
$350, at per cent.?* Ans. $3,62.
2. What is the bank discount on a note of $495, for 2 months,
at 5 per cent. ? Ans. 4.33.
3. What is the present value ci a note of $7840 discountod at a
bank for 4 months and 15 days, at o per cent. ? Ans. $7059 68.
4. How much money should bo received on a note for $125,
payable at the end of 1 year, 3 months, and 15 days, if discounted
at a bank at 8 per cent.? Ans. $112.
5. A note, dated December 3rd, 1860, for $160.40, and having
6 months to run, was discounted at a bank, April 3rd, 1861, at 6
per cent. ; how long had it to run, and what were the proceeds ?
Ans. 64 days ; proceeds $158.71.
6. On the first day of January, 1866, 1 received a note for $2405
at 60 days, and on the 12th of the same month had it discounted at
a bank at 7 per cent. ; how much did I realize upon it.
Ans. $2381.02.
7. A merchant -sold 240 bales of cotton, each weighing 280
pounds, for 12J cents per pound, which cost him, the same day, 10
cents per pound ; he received in payment a good note, for 4 months'
time, which he discounted immediately at a bank at 7 per cent, j
what will be his profits? Ans. $1479 10.
8. I hold a note against Clemes, Rice & Co., to the amount of
$327.40 dated April 11th, 1806, having six months to run after
date, and drawing interest at the rate of 6 per cent, per annum.
What are the proceeds if discounted at the Girard Bank on the 10th
of August, at 7 /„ per cent. ? Ans. $332.86.
Note. When a note drawing interest, is discounted at a bank, the interest
is calculated on the face of the note from its date to the time of maturity,
and added to the face of the note, and this amount discounted for the length
of time the note has still to run.
9. What will be the discount on the following note if discounted
at the City Bank on the 17th of Nouember, at 6 per cent.
Ans. $4.34.
* Throughout all the exercises, unless otherwise specified, the year is to
be considered as consisting of 3G5 days. Since it is cuHtomaiy in business
when a fraction of a cent occurs in and result to reject it. if less than half a
cent, and f not less, to call it a cent, we have adopted this principal through-
out thebcoV-
RANK DISCOUNT.
177
k34.
$527.-,V{ji Oberlin, Oct. 4, .866.
Ninety aays after date for value rcceicnl, icc promist
to pay to the order of Smith, Warren,ii' Co., Jivr humlrvd twenty-
seven and ,"„'^, dollars at the City Bank, Obrrlin, with iiit'test al
eight per neiit. Thompson k BuiiNs.
10. AVhrit will be the discount at 7,-'^, per cent, on ii note for
$227.41, drawing interest ut 8 per cent., dated iMay l.st, 1SG5, at
1 year after date, if discounted on March 7th, i8(JG? An,«. .$2.<sr».
11. What amount of money will I receive on the following note,
if discounted at the First I^ational Bank of Detroit on June 21st,
at 9 percent.? , , Ans $47;1G6.
$473.80. Detroit, May 17, 1866.
Three months afterdate Ipromitye to jxiy to the order oj
J. R. Sing <& Co., four hundred and seventy-three and f^^^^ Dollars^
at the First National Banh, Detroit, for value r<veiied icith interest
at1-^\pcr cent. lllcilAUl) DuNN.
12. Vvhat must I pay for the following note on August ISth,
1866, so a.s to make at the rate of 30 per tent, interest per annum
on the money I pay for it? Ans. $758.14.
$746.75. Adrian, January 19, 1866.
One year from date, for value received, we j>romise to
pay James Ames, or order, seven hundred and forty-six ^\\^ dollars,
at the Commercial Bank, Adrian, with interest at 7 ,-'y per cent, per
annum. . "Wilson & Cummixgs.
13. A holds a note against B to the amount of lj.478.02, dated May
10th, 1805 at 1 year after date drawing 7, •'„ per cent, interest. I
purchase this note from A. on August 18th, paying ibr it huch a
sum that will allow nic 20 per cent, interest on my money. What
shall I pay for it? Ans. 8 148.33.
14. 1 got my note for §2000 discounted at a bank. May 20, 1862,
for 2 months, and immediately invesled the sum received in flour
June 7, 18G2,' I sold half the flour at 10 per cent. lcs.s than cost, and
put the money on interest at percent. August 13, 1862, I sold
the remainder of the flour at 18 per cent, advance, and expended tho
money for cloth at $1 per yard ; 12 days after T sold the cloth at
$1.1 6 j per yard, receiving half the pay in cash, which 1 lent oa
interest at 7^ per cent, and a note for the other half, to be on inter-
N'
k--
;' 1
iiir
■ ■ ■ 1m'
!;:■■ •kM.;
IS
ir
178 Ajm^nMETic
est frnm October 4, 1802, at J per cent. When my note at tlio
biiiilc became duo 1 renewed it ibr 5 months, and when this noto
biciuue due I renewed it for 2 months, and when this note became
due T V'.newed it ibr such a time that it becunie duo July 20, 18(].'»,
at wl.K'h tin#r T collected the amount due me, and paid my note at
tlic bank, llequlred the loss or gain by the transaction. , , , ,
Ans. #29!).i,l
It i.s !-omotimcs nL(!cssary to know tlie amount for which a noto
must be <;iveii, in order that it hliail produce a given sum when dis-
counted at a bank.
E X A M P L F, .
1. Suppose wc require to obtain lj'2:]6.22 from a bank, and Hint
we. are to give our noto, due in two months ; for what amount must
wo draw the note, supposing thai, money is worth 9 per cent. ?
SOL i: T ION.
Vvom tlio nature of tliis example wo can readily poroei'/^ <hat
suih a sum must }k> jiut on the. i'ace ol' the note, that wben dis-
ciiutited the jivooeeds wiil \h- exactly ^2o(.).22. If we were iii ( ikdtl
iii'.c (l<iil(ir iioic and di^^count it .-it a bank for tlie given time, all/l fti
the given rate, tlie ]/roceeds would be .'.)cS-12r). Ifenco, for cveri/ do/-
/ill- wo put upon the faci' of the note we receive .9842r», and \(i n-
ceivo §2P>(>.22 wv- would have to put as iii<u»y dollars on the fac^ of
the note as are represented by the number of times that .98-125 is
contained in fi2l>().22, which is 240. Therefore, we must put $240
on the lace of a I'.ole due at the end of two montlis to produce
$230.22 when discounted at a bank at 9 per cent. From this wo
deduce ilio following
n i: L E .
.Dc(7Hrf ih<: hanJc (JiscoiDit on 'ii<^ . Jo)' the giroi tivic and rate,
from ^'i, atal duldr the drsirrd amount Inj the remainder. The
quotient v:Ul he the face of the. vote required.
2. For what sum must a note be given, having 4 months to run,
that .^' ili ]iroduce $19r)0, if discounted at a bank at 7 per cent. ?
A:is. $1997.78.
0. What must be the face of a note, so that when discounted for
5 months and 21 days, at 7 per cent,, it will produce $57.97, cash?
Ans. S'k>.
i'l
B.VNK DISCOUNT.
179
'(, I ill
4-. Suppose your note for months is discounted at a bank nt 6
per cont., and $48-1.75 j<laccd to your credit, what must liuvo been
Hh; fiico of the note ? Ans. $500.
5. A merchant bought a quantity of goods for SCOO. For wl.at
sum uiust lie write his note, to be (liecouvjtcd at a bank fi>r fi months,
ut 6 j)er cent. ? Ans. .^G18.S.S\
p. A fiirmer boui^ht a farm for $5000 cas^h, and having only one-
half (if [jlc Slim on haiui. he wishes to obtain the balance from the
baii|c. For what sum must "he give his note, to be discounted for 9
mouljiH, iii II put f!L»((f. ? Ans. $2619.17.
7. II' II uicrcliaut wit,hcs (o obtain $550 of a bank, for what sum
must he give his note, ptiyablc in GO days, allowing it to be dis-
counted at ^ per cent, per month ?
Ans. $555.75.
8. 1 sold A. Mills, merchandize valued at $91S.1G, for which he
was to pay uic cash, but being dinappuintcd in receiving money ex-
pected, he gave me his endorsed uote at 00 days, for such an amount
(Jiat when discounted at the bank at 7 per cent, it would produce
the price of the raercliandize. AVliat was tiic face of the note ?
Ans. $!J.'J4.82.
9. I am owing T?. Harrington on accouut, now due, $1G8.45 ; liC
also holds a note against mo for 8210, duo in 34 days, including
days of grace ; he allows a discount of 8 per cent, on the note, and if I
give him my note at GO days for an amount that will be sufficient if
disccmntud at G per cent., to produce the amount of account and
note. VV hat will be the face ol new note ? Ans. $380. <S3.
10. t-^amuel Johnson has been owing mc $274.48 for 84 days. I
charge lum interest at G per cent, per annum for this time, and lie
gives nie his note at 90 days lor such an amount that when dis-
co'iiitvd at the Cirard Bank, at 8 per cent., the proceeds will ecual
the amount now due. AV liat is the face of the note ?
Ans. 8284.05.
From the many dealings business men have, in regard to dis-
count and interest, it is irequently required to know what rate of
interest corresponds to a given rate of bank discount.
E X A Jl P L E .
;. V»";i;itj..v of interest is paid when a note, payable in 362
day ', i.-. (., ; L AU.U'd at 10 per cent. ?
I
via
at
r
180
ABirHMETIC.
SOLUTION.
If WO discount $1 for the given time, and at tho given rate, the
proceeds will bo .90, or 90 cents. Hence, tho discount being 10
cents, wo are paying 10 cents for tho use of 90 cents. Now, if we
pay 10 cents for tho use of 90, for the use of 1 cent we must pay
t/u of 10 cents, or J of a cent, and for $1, or 100 cents, we must pay
100 times J of ft cent, or ■i'{^=.ll J, and for $100, $11 J, or 11 J per
cent. Therefore, to find tho rate of interest corresponding to a
given rate of bank discount, wo deduce the following
!
RULE.
Divide the given rate per cent., expreMed decimally, or tlic rate
per unit, hy the number denoting the proceeds of $1 for tlie given time
and rate. The quotient tcill be the rate of interest required.
EXERCISES.
2. What rate of interest is paid when a note, payable in 60 days,
is discounted at 7 per cent. ? Ans. l-§4-iv
3. What rate of interest is paid when a note, payable in 3
months, is discounted at 6 per cent. ? Ans. Cj'g^J'y.
4. A note, payable in 6 months, is discounted at 1 per cent, a
month ; what rate of interest is paid ? Aus. V<i>l\'l,
5. What rate of interest is paid, when a note of $200, payable in
70 days, is discounted at f per cent, a month ? Ans, ^f,y^-
6. When a note of $45, payable in G5 days, is discounted at 7
per cent., to what rate of interest does the bank discount correspond?
iins. <goui;*
7. A bank, by discounting a note at 6 per cent., receives for its
money a discount equivalent to 6^ per cent, interest j how long must
the note have been discounted before it was due ?
Ans. 1 yr., 3 mos., liii
C0MMI8SI0N.
Commission is tho tcrnj applied to money paid to an agent to
remunerate hlia for his trouble in buyiui^, selling, valuing, or for |
forwarding merchandise or other property.
The goods sent to a commission merchant oi agent, to be sold on j
account and ri&k of anothor, arc termed a consiijnment.
COMMISSION.
181
Tlic person to whonj tlicse goods arc consigned is called the coiV'
tlp*^'jc or I'orrcspondciif.
Thfc term shipineiit is Honictiincs used instead of consignment.
E X A 51 P L K .
A oiinnission merchant soils lor mo goods worth $1200, and
'C*.iar[i;cff t per cent. ; what have T to pay him ?
SOLUTION.
4 per cent, of $1200 is cfjual to $1200X.0,l^$48. Ilcnco I
•would have to pay S-18, and I'rom this we dodueo the following
RULE.
Find (he, pctrcntage on the given sum at the giviu rate, whieh
iKiU be the commission.
K X E n C I S E s .
1. Consigned to A. K. Boomer, Es(|., Syracuse, hy the Troy, N.l .,
foundry, agricultural implements which arc sold for $1875.75, what
is the agent's commission at 2\ per cent. ? Ans. 6-10.81).
2. Uought in Boston 12 chests of tea, containing 04 lbs. each, at
81.12i per lb., on a commission of \'{ per cent. ; what was my com-
mission ? Ans. $15.12.
3. My Tokdo correspondent has bought for me 2768 lbs. of
bacon, at Vl\ cts. a pound ; what is his commission at 3.^ i)cr cent.?
$11.25.
4. Bought a carriage and pair of liorses, per the order of S.
Williams, Portland ; paid for the liorses $240, and charged 4J per
cent, and paidlbr tiie carriage $100, and charged li percent. ; how
much did 1 earn ? Ans. $13.20.
5. A commission atrent in a Southern State bought cotton worth
$2284 tor an English manufacturer, and charged hh per cent.; what
is his counuj^sion ? Aua. $125,02.
0. On another occasion tlio mnnui'aeturcr gave the commission
merchant ^',105 78, lor purchasing ibr him ootton worth l|;ill84 \ what
was the rate per cent ? Ans. 4^
7. All Kuglisli \«»mmlsslon merchant buys for a Porlland house,
,1.5T0 lOs. Od. worth of provisions, and charges l| per cent. ; what
is his commissi^m ? Ans. £25 18s. 10 ^d.
8. A New York provision merchant instructs a Belfast (Ireland)
connnission nierclii>.i)t to purchase ibr him £534 4s. Od. wort]\ of
li
i H
m
1
1
I:,;
1
i
182
ABixmrnTio.
''
I: ■
bacon and hams, and offers him ^\ per cent. ; what does the agent
get ? Ans. iE38 I4s. 7d.
9. A book apicnt in Cincinatti, sells $487.50 worth of books for
Day & Co., of Montreal, and receives $72.05 for his trouble; at
what rate per cent, was he paid ? Ans. 15 nearly.
10. An agent sells 84 sewing machines at $25 each, and his
commission amounts to $262.G0 ; what is the rate ? Ans. 12^.
1
When a sum has to be sent to a commission agent, such that it
will bo equal botli to the sum to bo invested, and the agent's com-
mission, it is plain, as already noted, that this is merely a case of
percentage. It is the same as the first part of case IV., and wc will
have the corresponding
RT LE.
Divide the given amount hy 1, increased hi/ tJic given rate per
unit, and the quotient will be the sum to be invested ; subtract this
from the given amount, and the remainder will be the commission.
EXAMPLE.
If I send $1890 to a commission merchant, and instruct him to
buy merchandise with what is left after his commission at 5 per cent,
is deducted ; what will be the sum invested, awd the agent's com-
niission ?
SOLUTION.
It is plain that for every dollar of the proposed investment I
must remit 105 cents, 100 towards the investment, and 5 towards
the commission, and hence the number of dollars which can be In-
vested from the sum remitted will be the same as \\\o number oE
times that 1.05 is contained in 1800. Mow, $l890-:4.05 gives
$1800, the sum to be invested, and tliifl subtraettid from $1890,
loaves $90, the commisBlou to which the agent is entitled.
KX WHO IH Hh.
1. lUMulltud to A. U., St. I'lmb, ^088 to purchase flour for me
with the balanoti llmt ii'.umliis uitcr deduellug his commission at 4
pov cent. J required the purchase raon^y and percentage ?
Ans. $950 and $;J8.
2. Heeeived a commission to buy wheat with $779, less by my
commission at 2J per cent. ; re (ulred the price of the wheat and my
oommission. Ans. $700, and $11).
BKOICEliAGi:.
183
8. Ilrmittcd to my cont.^iM/.ulcnt to Au^'usta S2(»r».70, to pay
for luinlx!!' which ho purchased for iiii-, and to pay liis own ((ttiiiiiis-
tihn at 4 per cent. ; what was tho price of the hiiubir. Jintl what tiio
(onuiiissioti ? ' Ans. liJ25l».Jil», and 610.2(1.
4. John -Tones, Ncwniarkot, cop^niis.sions W. Orr, Portland, to
procure for liim a (juantlty tif liiu! liour, and remits !i:!»17.<Il ; how
niucli flour can he have, after allo\vin;j: -1^' jx^r cent., and what, will
the commission amount to ? Ans. $^~i'\ and $M.I)1.
5. John Stalker. Jjondon. commissions J. Fleming' New York, to
pnrcliasc for hin) as much butter as ho can procure for tho balance
between ^""0.52, and his own commission at 1^ per cent. ; how many
pounds butter did he j:;ct at 25 cents per lb. ; what the whole jirice,
and what was th(! commission ? Ans. .■5072 lbs., 87(jS, and $1 1.52.
(i. Dr. (lallipot is about !(» remove to ]']ni;'land, and sends to a
London cabinet maker ;• -1005. 15 towards getting bis bouse furnished,
he is charged '.t\ \)cy cent, over and abovo the price of the furniture,
for time and bdjour, what dons tho I'urniture cost? Attn. ^;1870.
7. (Jraliam iJros., of Newbury, Kofid lo U. AVhite, Cliarles|on,
bacon and liams worth SI 500, they charge 5 A per oellt- OOUUUIhhIdU,
aud the charge for lading is §75.15; how much do(.'H 11. White owo
them? Ans. 11720.95.
8. P. Robson, commission merchant, Albany, buys fur T. Black
& Co., Baltimore, groceries, tho price of which, together with tlirir
commission at 4 per cent. con)e.s to §475.02; what was iim i^f/vce of
the goods, and what was the amount of the commission ?
' Ans. $45(3.75, and $18.27.
Mi
184
ARITHMETia
BROKERAO-E
Brokeraof. is a per ccDtage paid to an ogont for ncgooiating bills,
oxchangiDg ; .oncy, buying and selling railroad, bank, and building
society stocks, Government bonds and gold. Such an agent is called
a broker. A. nmullcr per centago is usually allowed to a broker than
to a commission merchant, because the work ho has to do requires
less time and labor.
Brokers charge, generally, one-eighth of one per cent, for buying
or selling stocks, bonds, gold, &c., and it is always reckoned on tho par,
or face value. For instance, if a broker wore to purchase lor you a
shiiro 111 N. Y. C. 11. U. stuck at 112J, or 12J per cent, premium, tho
brokerage would be ^ per cent, on $100, and not on tho $112.50.
Tho charge would still be the same if purchased at 85, or 15 per c t.
discount. In gold operations, (he brokerage is calculated on ..he
gold, althougli the brokerage itself is taken in currency. For in-
Htance, it' a broker purchases $10,000 in gold for a customer, tho
charge would be ^ per cent, on $10,000, viz. : $12.50 in currency.
A great many of tho transactions made by brokers consist in the
buying and selling of Government Bonds, called " Five-Twenties,"
" Ten-Forties," and " Sevpn-Thirties." The " Five-Twenties" are
so called because they are payable, at the option of the Government,
at five years after their date, or at tho end of twenty years. The
•' Ten-Forties" are payable, at the option of the Government, ten
years after their date, or forty. Tho " Seven-Thirties" are so called
because they bear interest at the rate of seven and three-tenths per
cent, per annum — (7,^„",).
In buying or selling those bonds, the seller of a " Seven-Thirty"
alwayg receives the interest that has accrued on it, from the time of
last payment of interest by Government, until the time of sale ; but
in all other bonds the buyer has the benefit of the interest. The
only reason that can bo assigned why the " Seven-Thirties" should
bo an exception to the general rule, is, that the interest is so easily
calculated, being just one cent per day on every fifty dollars.
In reality Ihe result is the same, because, if in the ** Seven-Thir-
ties" the buyer received the interest, the (quotation or market value
of them would be greater.
BROKERAGE.
185
Brokers frequently, ninong each other, buy and sell bonds, |^Id,
&c., at .'»() daya, usln^ tlie terms " seller IIO" or " buyer 30," which
slgnifieH, if the term " seller 'JO" Ih used, that the seller of the stock
can, any time during the thirty days, deli\ r tho Mtoek to the buyer,
and receive his luoney ; if " buyer 30" is used, the buyer has the
privilege of calling in the stock bought, any time during the 30
days.
Tliis practice, as will be scon at u glance, gives a great range for
speculation. To illustrate : Suppose that A sells to li .^>00 shares
of Krio 11. 11. stock at G4 or SG per cent, dis., " hcllcr 30," now it
is not at all probable that A has this stock on hand that he has sold
to 13, but expects to bo able to purchase it before thu expiration of
the 30 days, at something less than M. This is called celling
"short." It not unfrcquently occurs, in transaclions such as just
mentioned, that A may not mtih, or B require, the delivery of tho
stock when tho time arwvcs; if tliis is the cafe, A simply pays, or
receives from, B the differenc* between what the stock was sold at,
and what it is worth at tho time of settlement. In purchases liko
the above, unless the parties are known to bo reliable men, a certain
amount of money must bo put tip, termed u "margin," that may bo
considered suflficicnt to cover flueluations in the value of the stock.
If either party is unable to meet his part of Hie contract, tlie term
" Lame Duck" is applied to him. Any person may buy or sell stock
through a broker at " buyer 30" or " seller 30" by putting up what
the broker may consider to bo a sufficient " margin." Interest is
generally allowed on this margin.
Where there are a number of brokers operating in any one stock,
they are, according to brokers' phrases, divided into two classes,
called " Bulls" and '• Bears." The " Bear" is always tho seller, while
the buyer is always a '• Bull." If A sells to B stock at 97, " seller
30," he is evidently a " Bear," as it is to his interest to constantly
bear down the price of the stock he has sold, so that he may be able
to purchase at a price less than 97, while it is always the interest of
the buyer to "Bull" or raise its value.
The par value of stock in the following examples is considered to
be $100, and tho brokerage ^ per cent., unless otherwise mentioned^
Some of the answers requested may not belong, legitimately, to ques-
tions in brokerage ; but the teacher or learner may ask or give but
•ill
^i:
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ARITHMETIO.
one answer. Like commission, brokerage is merely a particular oasa
of per ccntage, and hence the
RULE.
To find the brokerage on any sum, find the per ccntage on the
aiven sum at the given rate, which will he the brokerage.
1. I purchased for C. R. Sing 10 shares of Hudson lliver R. R.
stock at 103; what is the broktragc ? Ans. $1.25.
2. Sold the alovo stock for the same person at 103|^ ; what is
the brokerage? Ans. $1.25.
3. Bought for J. C. Baylies 50 shares of N. Y. C. R. R. stock
at 107^ ; what i3 n»y brokerage ? Ans. $6.25.
4. Sold for Kimball & Co. $5000 in gold, at 137^- j what is my
brokerage at § per cent. ? Ans. $18.75.
5. Purchased through my broker 100 shares Ilarlem R. R. stock
at 109J, •' buyer 30 ;" at the expiration of the 30 days he sold tho
same, per my order, at 110?r; what was my gain, and what the bro-
kerage? Gain, $100 ; brokerago, $25»
6. Paid a broker f per cent, for exchanging $245 fractional cur-
rency for bills of a larger denomination ; what is the brokerage ?
Ans. $1.5C.
7. My broker has purchased forme a "Five-Twenty Bond "
for $4500 at 108|^ ; what is the brokerage at |- per cent., and
what does it cost me ? Ans. Cost $4916.25 ; B. $38.75.
8. Instructed a broker to purchase Seven-Thirties to tho
amount of $7500, which he did on March 8th, 1867, at 107f ;
interest on this bond payable on the 1st Jan. and July ; what is
the brokerage, and its cost ? Ans. Br. $9.37-|^ ; Cost $8181.751
9. I purchased through a broker $15,000 gold, at 134^^, ho
sells it for me at $131^ ; brokerage on purchase \ per cent., on
Belling ^ ; what is the brokerage, and what my loss ?
i' "...a-,,-; ■.'.J.-/. Loss $571.87^.
10. A broker purchased for me 150 shares of Michigan Cen-
tral stock, at 87| — brokerage | per cent. ; 80 shares Reading
R. R. stock, at 102^ — brokerage \ per cent., and $8000 of Ten-
Forties, at lOlf ; what is the brokerage and full cost ?
Ans. Brokerage, $80 ; Cost $24457.50.
MISCELLANEOUS EXAMPLES.
187
MISCELLANEOUS EXAMPLES.
1. A commission merchant has purchased for me 5G48 lbs. lone;
cut ham?, at 14J cts. per ib. ; what is his commission at 1 J per
cent.? Ans. $10.0G.
2. My agent in Richmond has purchased cotton for nic to the
amount of ^1785.80 and charges me a commission of § per cent. ;
hovr much have I to remit him to pay for the cotton and commis-
sion ? Ans. $1801.42-3.-.
3. I remit J. Purdy, com. mcr., New Orleans, $1142.40, in-
structing him to invest it in cotton at 32 cts. per lb. ; after deduct-
ing his commission at 2 per cent., how many lbs. of cotton do I re-
ceive ? Ans. 3500 lbs.
4. Morrison & Thompson have sold for me 112 bbls. white fish,
at $9.50 per bbl., and 85 bbls. flour at $12.40 — commission 2^ per
cent. I have instructed them to invest the proceeds in bacon, nt
13-^- cts. per lb., after deducting their commission at 1 J per cent. ;
how much is the commission, and how many lbs. of bacon do I re.
ceive? Ans. Bacon. 15070^ lbs. Comn. $83.47.
5. A purchased, per the order of Andrew Campbell & Co., Nash
ville, Tenn., 14872 lbs. C. C. bacon at 13|- cts. per lb., charing a
commission of 1^ per cent. A wishes to draw on them for reim-
bursement ; what must be the face of the draft if it cost ^ per cent,
to get it cashed, and what is the com.mission on purchase ?
Ans. Face of dft. $2010.15; Commission $29.56
6. I have received, from a correspondent in Troy, $4781.25^
with iiistructions to invest the same in Five-twenties, at 105J, first
deducting my <3ommission at f per cent. ; what is the brokerage, and
what amount of Five-twenties can I purchase ?
Ans. Brokerage $33.75 ; invested in Five-twenties $4500.
7. An accountant is entrusted to make schedules of the debts
ind assets of a bankrupt ; he charges only 2^ per cent, on the debts,
jn the principle that he will have little trouble in getting the accounts
due by the bankrupt sent in ; but as he knows very well that he will
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ABITHMETIO.
have trouble in getting correct statements sent in ot accounts due to
iuie bankrupt, he stipulates for 5^ per cent, on these ; how much does
he get altogether, the debts being $2786, and the assets $618 ?
Ans. $103.64.
8. I have sold for Walker & Smith, Cinoinnatti, a consignment
•of 100 bbls. of pork, at 27f|/*5 per bbl. I ha^o paid out for charges
$31.40 — my commission is 2^ per cent. I remit them their net
proceeds by draft on Cincinnati, purchased at f p^r cent, discount,
charging ^ per cent. com. on face of drnft; what commission do I
receive, and what is the face of dft. that I remit them ?
Ans. Commission $72.08 ; Face of dft. $2666.52.
9. On the 14th of March, 1867, a broker purchased for B, 100
shares of Erie R. R. stock, at 71 ; 50 shares C. and R. I. R. R.
stock, at y5f ; 200 shares N. Y. C. J?. R. stock, at 103J, and a Seven-
thirty bond for $6000, interest payable Dec. and June, at 106J.
They were sold on April 12th at 68|, l)7f, 103J, and 106J, respec-
tively ; what is the brokerage at J per cent, for baying, and ^ for
•fielling; and B's g^ain or loss on the transaction ?
Ans. Brokerage $153.75; Loss, $240.20.
10. I sent to Taylor & Morrison, com. merchants, New York,
'250 firkins butter, containing on an average 56 lbs. each, at 15 cts.
per lb. They sold at an advance of 10 per cent. ; freight, &c., de-
ducted $10.45, commission 2^ per cent. They have remitted me a
sight draft for net proceeds, which they purchased at f per cent,
premium, charging ^ per cent, commission on face of draft. What
amount of draft did I receive, and what amount of commission
charged? % Ans. Com $63.32; Face of dft $2227.88.
\ ;
f^rt;n^-*^^i ifyr? -;»:;* -r.-ry^fiA Hii.« i,- j£fiM ^ •J^-.tiJ-riy-Sf.SS'i,..^,--^.;;^,
m' '-»
"-CVAi
INSUBANCE.
INSURANCE.
189
Insurance is an engagement by wiiich one party ia bound, in.
eonsideration of receiving a certain sum, to indemnify another for
something in case it should in any way be lost. The party under-
taking the risk is seldom, if ever, an individual, but a joint stock
company, represented by an agent or agents, and doing business
under the title of an " Inturance Company,^' or " Assurance Com-
pany,^^ such as the "Globe Insurance ""ompany," the "Mutual
Insurance Company."
Some companies arc formed on the principle that each individual
shareholder is insured, and shares in the profits, and bearc his portion
of the losses. Such a company is usually called a Mutual Insurance
Company.
The sum paid to the party taking the risk is called the Premium
o/ Insurance, or Biinpij the Premium. -.
The document binding the parties to the contract, is called the
Policy 0/ Insurance, or simply the Policy.
The party that undertakes to indemnify is called the Tnsurer, or
underwriter after he has written his name at the foot of the policy.
The person or party guaranteed is called the Insured.
As there are many different kinds of things that may be at stake
or risked, so there are different kinds of insurance which may be
classified under three heads.
Fire Insurance, including a!l cases on land where property is ex-
posed to the risk of being destroyed by fire, such as dwelling houses,,
stores and factories.
Marine Insurance. — This includes all insurances on ships and
cargoes. Such an insurance may be made on the ship alone, and in
that case it is sometimes called hull insurance, and sometimes bot-
tomry, the ship's bottom representing the whole ship, just as we say
fifty sail for fifty ships. The insurance may be made on the cargo
alone, and is then usually called Cargo Insurance. It may be made
on both ship and cargo, in which case the general term Marine Ii-
surance will be applicable. This kind, as the name implies, insures
against all accidents by sea.
Life Insurance. — This is an agreement between two parties, that
in case the one insured should die within a certain stated time, the
other shall, in consideration of having received a stipulated sum
aimaally, pay ,to the lawful heir of the deceased, or some one men-
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ABTTHMETIO.
fi!
tioned in his vrill, or some other party entitled thereto, the amount
recorded in the policy. " "
For instance, a man may, on the occasion of his marriage, insure
his life for a certain sum, so thnt should he die within a certain time,
his wHow or children shall be paid that sum by the other party.
Again, a father may insure the life of his child, so that in cose of the
child's death within a specified time, he shall bo paid the sum agreed
upon, or that the child, if it lives to a certain »ge, shall be entitled
to that sum. One person may insure the life of another. Supposing
that A owes B a certain sum, there is the risk that A may die before
he is able to pay B ; another party engages, for a certain yaarly sum^
to pay B in case A should fail to do so during his life time.
In some instances, insurances are effected to gain a support in
case of sickness. Such a contract is called a Health Insurance. In-
surances are now also effected ior compensation in case of railway
accidents. These we may call Railway Accident Insurances.
A policy is often transferred from one party to another, especi-
ally as collateral security for debu or some analogous obligation. If
the payments agreed upon are not regularly kept up, the policy
lapses, that is, becomes null and void, so that the holder of it forfeits
not only his claim to the sum insured, but also the instalments pre-
viously paid. In many companies a person can insure in such a
■way as to be entitled to have a share of the profits.
The date at which the system of insurance began cannot be
clearly ascertained ; but, whatever its dat/a, its origin seems to have
been protection against the perils of the sea. We know that it was
practised, in a certain way, by the ancient Greeks and Eomans. If
a Roman merchant sent a cargo to a distant port, he made a contract
rvith some one engaged in such business, that he would advance
a certain sum, to be repaid with interest, if the vessel reached her
destination iivsafetj, but should the vessel or cargo, or both be lost,
the lender was to bear the loss. This was termed respondentia, (a
respondence) a term corresponding pretty nearly to the English
word repayment. It was lawful to charge interest in such cases,
above the legal interest in ordinary cases, on account of the great-
ness of the risk. The lender of the money usually sent an agent of
his own on board the vessel to look after the cargo, and receive the
repayment on the safe delivery of the goods. This agent comS'
poQfled pretty nearly to our more modern aupercargo. As the art
•of navigation advanced, and the soourities afforded by law beoune
inbuhance.
191
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(a
more Biringent, and also facilities of communication increased, this
Bystem gradually gave way, and lias eventually been supplanted by
communications by post, and telegraphic uefiSages to agents at the
ports of destination.
With regard to the equitableness of insurances, and their utility
in promoting commorcial cxterprisc, wo may remark that they make
the interest of every merchant, the interest of every other. To show
this, we may c Draparo an insurance office to a club. Suppose the
merchants of a town lo form a club, and establish a fund, out of
which every member, if a loser, was to be indemnified, it is plain
that no loss would fall on the individual, except his share as a mem-
ber of the club. Even so the insurance system causes that each
ppeculator, by insuring his own stake, contributes so much to the
funds of a company, which is bound to indemnify eacli loser. Ca
the other hand, the insurer or insuring company, gains in this way,
that the profits accruing from cases where no lo is is sustained, far
exceed the cases where loss is sustained, and the trifling expense of
insuring is of no moment to the insured, in comparison with the
<lamage of a disastrous voyage, or consuming conflagration. By the
insurance system, loss is virtually distributed over a large commu-
nity, and therefore i'alls heavily on no individual, from which we
draw our conclusion, that it is equivalent to a mutual mercantile
indemnification club.
Wo must now show the rules of the clvh, and principles on which
its calculations arc made.
The principal thing to be taken into account, in all insurances, is
the amount of risk. For example, a store, where nothing but iron
is kept, would be considered safe; a factory, where fire is used,
would be accounted hazardous, . and one where inflammable sub-
stances are used would be designated extra hazardous, and the rates
would be higher in proportion to the increased risks. As, however,
the degrees of risk are so very varied, only a rough scale can be
made, and hence the estimate is nothing more than a calculation of
probabilities. In life insurances, the rates are regulated chiefly by
the age, and general health of the individual, and also by the gen-
eral health of the family relations. Connected with this is the cal-
culation of thrt average length of human life.
Almost all the calculations in insurance come under two heads.
FiKST, to find the premium of insurance on a given amount, and at
■a given rate ; and, secondly, to find how much must be insured at a
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ABITHMETIO.
given rate, so that in (Jase of loss, both the principal and premionk
may be reoove^d. «' ' ■ -, : C
i' T As the premium is reckoned as so much bj the hundred, insur-
ance is merely a particular case of percentage. Hence to find the
premium of insurance on any given amount, at a given raie per cent.,
yfQ deduce the following
BULK.
Multiply the given amount hy the rate per unit.*
EXAMPLES.
1. To find the cost of insuring a block of buildings valued at
$2688, at 6 per cent. ? Here we have .06 fcr the rate per unit, and
$2688 X. 06=0161.28, the answer.
2. What will bo the cost of insuring a cargo worth $3679, at 3
per cent.? The rate per unit is .03, and $3679X.03=$110.37,
the answer.
3. A gentleman employed a broker to insure his residence and
outhous«>9, valued st $2760, the rate being 8 per cent., and the bro-
ker'.^ charge \\ per cent, ; how much had he to pay ? The cost of
insurance is $2760X •08=$220.80, and the brokerage $41.40, which
Added to $220.80, will give $262.20, the answer.
EXERCISES.
What will be the premium of insurance on goods worth $1280,
at 5J per cent. ? Ans. $70.40.
2. A ship and cargo, valued at $85,000, is insured at 2\ per
, cent. ; what is the premium ? Ans. $1912.50.
' 3. A ship worth $35,000, is insured at 1^ per cent., and her
cargo, worth $55,000, at 2^ per cent. ; what is the whole cost ? ^
Ans. $1900.00.
4. What will be the cost of insuring a building valued at $58,000,
♦at 2J per cent. ? Ans. $1450.00.
T(
at 4
$100^
requii
This
* It is plain that the rate can be found, if the amount and premium are
given, and the amount can be found if the rate and preniUiin are given. In
the case of insuring property, a pvofessional surveyor is often employed to
value it, and likewise in the case of life insurance, a medical certificate i»
veqniredj and hveaeh case the fee must be paid by the person insured. A»
100, the basis of percentage, is a constant quantity, when any tvro of thv
ether quantities are given, the third can be foond.
m
INSUBANOB.
198
5. What must I pay to insure a house rained at $898.50, at }
per cent. ? ^ Ans. $6.74.
6. A village store was valued at $1130 ; the proprietor insured
it for six years ; the rate for the first year was 3^ per cent., with
ft reduction of ^ each succeeding year ; the stock maintained an aver-
age value of $1568, and was insured each of the six years, at 2^ per
eeut. ; how much did the proprietor pay for insurance during the six
years ? Ansv $397.53.
7. A store and yard were valued at $1280, and insured at 1^ ^r
cent. ; the policy and surveyor's fee came to $2,25 ; what was the
"whole cost of insuring ? Ans. $16.65.
8. W. Smith, Port Huron, requests B. Tomlinson, Toronto, to
insure for him a building valued at $976 ; R. Tomlinson effects the
insurance at 4| per cent., and charges | per cent commission ; how
much has W. Smith to remit to R. Tomlinson, the latter having paid
*ho premium ? Ans. $46.36
9. The cost of insuring a factory, valued at $25,000,- is $125
what is the rate per cent. ? Ans. ^
10. A 1|- per cent, insuring my dwelling house cost me $50
what is the value of the house ? ' Ans. $4000.00,
To find how much must bo insured for, so that in case of loss,
both principal and premium may be recovered.
Here it. is obvious that the sum insured for must exceed the
value of the property in the same ratio that 100 exceeds tho rate.
■:<k ■-■■■' ■ ■ t . EXAMPLE*
To find what sum must be insured for on property worth $600,
at 4 per cent., to secure both property and premium, wo have as
$100^=$96 : $100 : : $600 : F. P.=^'>-<y<gU>i>=:=$625, the sum
required. Taking the rate per unit we find ^^^Q^=:-^^\i=^ . 96.
This gives the »
RULE.
Divide the value of the pfoperhj by 1, diminished hy the rate per
unity and the quotient will be tJie sum required.
EXAMPLEJ.
1. A foundry is valued at $874 : for what sum at 8 per cent,
most it be insured to secure both the value of the property and the
prequum ? One mimia the rate or 1.00— .08=.92, and $874-j-.92
=$950, the answer.
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AiirnDtL.''UT.
*• The promises of a gunsmith, who sells gunpowder, arc valued at
112618.85: for how much, at 15 per cent., must they bo insured in
order to recover the value of the property and also the premium of
insurance ? Subtract .15, the rate per unit, from 1, and the remain-
dor is .85 and $2618.85-i-.85 gives $3081, the sum required.
,■.•■-••' , ■
EXERCISES ,
' 1. A chemist's laboratory and appurtenances are valuea ac
$26,250, for what sum should ho insure them at 6^ per cent., to
.secure both property and premium ? $28,000.
2. A New York merchan. sent goods worth $1,186 by water
conveyance to Chicago ; ho insured them from New York to Buffalo
at 1 J per cent., and from Buffalo to Chicago at 2^ per cent., and in
both cases so as to secure the premium as well as the cargo; how
much did the insurance cost him ? Ans. $45.42.
3. A person owned a flour mill, valued at $1846.05, which he
in^ul'ed at 1^ per cent. He also owned a flax mill, valued at
$846.30, which ho insured at 2^ per cent., and in both eases at such
n ibum as to secure both property and premium. Which cost him
most, and how much more ?
Ans. The .flour mill cost him $1.67 more than the other.
4. Collins & Co., of Philadelphia, ordered a quantity of pork
from G. S. Coates & Son, Cincinnati, which amounts to $2423.10.
They insure it to Pittsburg at ^ per cent., and from PittsbuVg to
Philadelphia at 3 per cent., and in all cases so as to secure both the
price and premium. How much does the whole insurance come to ?
.•...,».,.,■-■:.:.. .■ .-u. .T•.^-.v -^iv^. ■■..;>-. ;i AhH. $87.12.
5. In order to secure both the value of goods shipped and the
premium, at 1^ per cent., an insurj^ncc is effected on $1526.72.
What is the value of the goods ? Ans. $1500.00.
6. The Mechanics' Institute is valued at $18,000 ; it is insured
. at IJ per cent., so that in ease of fire, the property and premium may
both be recovered. For how much is it insured ?
-:■ • / :■:■-."•'- '-./::'-■ ^r--'-' ■■^•-v;-^: -rw k ■ Ans. $18,227.85.
7. How much must be insured on a cargo worth $40,000, at J
per cent., to secure both the value of the cargo and the cost of
insurance? ' Ans. $40,201.00.
LIFE INSURANCE.
195
8. Tho RosBin House, King-street, Toronto, is valued at, say,
$160,000, and is insured at l.jj per cent, so that in case of another
oonfln'gralion, both tho value of the property and the premium of
insurance may be recovered. For how much must it bo insured ?
Ans. $152,(571.70, nearly.
9. A jail and court-house, adjoining chemical works, and there-
fore deemed hazardous, will not bo insured under 2^^ per cent.
How much will secure both property and premium, the valuation
being .$17,550,00 ? ^ Ans. $18,000.00.
•' 10. A cotton mill is insured for $12,000, at 4 per cent., to secure
both premium and property. What is the value of tho property ?
Ans. $11538.46.
11. What sum must bo insured on a vessel and cargo valued at
, $40,000, at 5^ per cent., in order to secure both the premium and
property ? Ans. $42,328.04.
12. How much must be insured on property worth $70,000, at
4^ per cent., to secure both premium and property, a commission of
f per cent, having been charged ? Ans. $73,848.17,
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LIFE INSURANCE.
A Life Insurance may be eflFected either for a term of years or
for the whole period of life. The former is called a Temporary
Insurance f and binds tho insurer to pay the amount to the legal
heir or legatee or creditor, if the insured should die within the
specified time. The latter is called a Life Insurance, because it is
demandable at death, no matter how long the insured may live.
The rate per annum that the insured is to pay is reckoned from
tables constructed on a calcutation of the average duration of life
beyond different ages. This calculation is made from statistical
re^rns called Bills op Moutality, and the result is called The
Expectation op Lipe.
The annual preipium is fixed at such a rate as would, at the
end of the expectation of li^'e, amount to the sum insured. From
tables of the expectation of life other tables are constructed, show-
ing the premium on $100 for one year, calculated on tho supposi-
tion that it is^to bo paid annually in advance.
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lt>'3 AWTHMETia
LIFK INBITRANOI T A B L Jl
A|{n tt«xt
BlrtbilMy.
1 yi>ar.
•*
7 yiMii
1
VoT Utt.
Ago next
nirth(l;iy.
1 yfur.
7 ynoni.
For Lir<L
15
.83
.85
1.44
38
1.19
1.28
2.75
16
.84
.86
1.47
39
1.22
1.31
2.85
17
.85
.87
1.51
40
1.24
1.36
2.95
18
.86
.88
1.54
41
1.27
1.41
3.07
19
.87
.90
1.58
42
1.31
1.47
3.19
20
.88
.91
1.02
43
1.35
1.54
3.32
21
.89
.92
1.6G
44
1.40
1.62
3.45
22
.90
.93
1.70
45
1.47
1.71
3.60
23
.91
.95
1.74
46
1.54
1.80
3.76
24
.92
.96
1.79
47
1.62
1.90
3.92
25
.93
.98
1.84
48
1.71
2.02
4.09
26
.95
.99
1.89
49
1.81
2.14
4.27
27
.96
1.01
1.94
50
1.91
2.28
4.46
28
.98
1.03
2.00
51
2.03
2.42
4.67
29
.99
1.05
2.06
52
2.15
2.59
4.89
SO
1.01
1.07
2.12
53
2.29
2.76
612
81
1.03
1.09
2.18
54
2.44
2.95
6.36
. 32
1.05
1.11
2.25
55
2.60
3.15
5.62
33
1.07
1.14
2.32
56
2.78
3.38
5.89
34
1.09
1.16
2.40
67
2.96
3.62
6.19
35
1.11
1.19
2.48
58
3.17
3.87
6.50
36
1.14
1.21
2.56
59
3.39
4.17
6.83
37
1.16
1.24
2.65
60
3.64
4.50
7.18
EXAMPLES.
Supposing a young man, on coming of age, vrishes to effect an
insurance for $3000 for the wJiole period of his life. To find tho
annual premium which ho must pay, we look for 21 in the left hand
column, and opposite that, in the column headed FOR life, wo find
the number 1.66, which is the premium for one year on $100, itpd
|g||=:^.0166 is tho premium on 01 for 1 year, and hence $3000 X
.0166=$49.80, is the whole annual premium.
If the insurance is to last /br seven years only, we find under that
heading .92, an^ f^^=.0d2, and $3000X. 092=^327.60, the annual
premium.
If the insurance is to be for one year only, we find .89 und^r that
head, and $3000X. 089=026.70, the premium.
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LIFE INSURANCE.
197
From thoao ozplanations wo con now derive a raid for finding the
nnnnal premium, when the ago of tho individual and tho sum to be
insured for nro known.
RU L 1.
Fitul the age in the left hand column of the talk, and oppotite
thi» in the vertical column for the given period will be found the
premium on $100 for one year, and thin divided hg 100 wilt give
the premium on Z^ for one year, and the given turn multiplied by
this will be tho wliole annual premium.
•
S X K R I B B S .
1. What will bo tho annual premium for insuring a person's
life, who is 18 years old, for $1000 for 7 years ? Ans. $8.80.
2. What amount of annual premium must bo paid by A. B.
Smith, who wishes to insure his life for 7 years for $2000, his ago
being 25 years ? Ans. $19.G0.
'd. John Jones, 35 years of ago, wishes to effect an insurance for
life for $1500. What amount of annual premium must ho pay ?
Ans. $37.20.
4. A gentleman in Chicago, 32 years of age, being about to start
for Australia, and wishing to provide for his family in ease of his
death, obtains an insurance for seven years for $3000. What
unoUnt of annual premium must he pay ? Ans. $33.30.
5. Amos Fairplay, 48 years of ago, being bound on a dangerous
•voyage, and wishing to provide for tho support of his widowed
mother, in case of accident to himself, insures his life for 1 year for
$2500. What amount of premium must he pay ? Ans. $42.75.
6. 1^ gentleman, 50 years of age, gets his life insured for $3000,
t)y paying an annual premium of $4.4ti on each $100 insured ; if ho
should die at tho ago of 75 years, how much less will be the amount
of insurance than tho payments, allowing the latter to be without
interest? Ans. $345,
7. A gentleman, 45 years of age, gets his life insured for $5000,
for which he pays an annual premium of $180, and dies at tho age
of 50 years. Suppose we reckon simple interest at 7 per cent, on
iiis payments, what is gained by the insurance ? . Ana, $391.X,
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ABITHSEETIO.
PROFIT AND LOSS.
•fii'
In the language of arithmetic, the expression Profit and Los" is
usually applied to something gained or something lost in mercantile
transactions, and lO most important rule relating to it directs how
to find at what increased rate above the cost price goods must bo
sold to produce a fair remuneration for time, labour and expendi*
ture ; or, in case of loss by unforeseen circumstanees, to estimate the
amount of that loss as a guide in future transactions.
There are other cases, however, which we shall consider in
detail. /
CASE I.
When the prime cost nnd selling price are known, to find the gain
or loss.
■ BULB .'■'
Pind, h^i tTiQ rule of practice, the price at the difference between
the prime cost and telling price, which will be the gain or loss ac-
cording as the selling price is greater or less than the prime cost; or.
Find the price at each rate, and take the difference.
"' ■ ■' , -. • ■ .
EXAMPLES.
To find what is gained by selling 4 owt. of sugar, which cost 12\
cents per lb., at 15 centt! per lb. %«: .t^/S ,
Here the difference between the two prices is 2^ cents per lb.,
and 400 lbs., at 2^ cents pei lb., will give $10. Also, 400 lbs. at
15 cents per lb.=$60, and at 12^ cents=$50, and $60— $50=|10.
Again, if 120 lbs. of tobacco be bought at 92 cts. per lb., and, being
damaged, is sold at 75 cents per lb., the loss will be a lo3s of 17 cents
in the pound, and 120 lbs., at 17 oenia per lb., is $20.40 ; or, 120
lbs., at 92 cents, will come to $110.40, and at 75 cents, to $90, and
$110.40— $90=$20.40. -y-<i. ,
EXERCISES.
1. If 224 lbs. of tea be bought at 60 cents per lb., and sold at 9&
cents per lb. ; how much is gained ? Ana. $78.40.
2. A 2P'ocer bought 24 barrels of flour, at $5.80 per barrel, aud.
sold 12 barrels of it at $6.10 per barrel, 9 barrels at $6.20 per bar*
rel, and the rest ut $6.25 ; how much did he gain ? Ana. $8.56.
3. If a person is obliged to sell 216 yards of flannel, which cost
him $86.40, at 37|- cents per yard ; how much does he lose 7
Ans.$5.4a.
PBOSlt AND LOSS.
199
i. If a dealer buys 78 bu&hels of potatoes, at 62^ cents per
bushel, and retails them at 87} cents per bushel ; how much does he
gain? ' Ans. $19.60.
5. A wine merchant bought 374 gallons of wine, at $3.20 per
gallon, and sold it at $3.35 per gallon ; how much did he gain ?
Ans. $56.10.
CASE II.
To find at what price any article must be sold, to gain a cert(un
rate per cent., the cost price, and the gain or loss per cent, being
known.
BULB.
Multiply tu9 cott price hy tplus the gain, or 1 minus the lots,
. . EXAMPLE.
If a quantity of linen be bought for 75 cents a yard ; at what
price must it be sold to g^in 16 per cent. ?
Since 16 per cent, is 16 cents for every dollar, each dollar in tho
cost price would bring $1.16 in the selling price, so that we havo
$1.16X.75=.87, or 87cents.
EXEBOISES.
1. Railroad shares being purchased for $2500, and sold at a gain
of 20 per cent. ; for what amount were they sold ? Ans. $3000.
2. A property having been bought for $2000 was sold at a gain '
of 10 per cent. For what was it sold ? Ans. $2200.
3. A horse was bought for $50, but, proving lame, was sold at a
loss of 15 per cent. At what price was he sold ? Ans. $42.50.
4. Bought a horse for $897 and sold it at a loss of 11 per oeni;
for What sum was it sold ? Ans. $798.83.
5. A merchant buys dry goods for $1562 and sells them at a
profit ' " ~2 per cent. For what were they sold ? Ans. $1905.64.
' ^ CASS III.
To find the cost when the selling price and tho gain percent, are
known.
'H
RULX,
-r.*'
Dunde the sdling price by 1 j^dw the gain, or 1 minus the 2(KV.
To find what was the first cost of a quantity of 4oiur If hioi.^
prodttoed 8 per cent, profit by being sold for $127.44. ^^^ - >'f
200
ABITHMEIIO.
iince the gain ib 8 per cent, of the eost, it follows that oaeh
dollar laid out has brought in a return of $1.08, and therefore the
eost must have been as many dollars as the number of times that
1.08 is contained in 127.44, which is 118, and therefore the first
cost must have been $118. * - , " :^ ;
■ ""^ BXEBCIBES.
1. If flaxseed is sold at $17.40 per bushel, and 13 per cent
lost, what was the first cost ? Ans. $20.00.
2. A dealer bought 116 hogs for $580, and sold them at a gain of
25 per cent. ; at what price did he sell e&oh on an average ? $6.25.
3. If 13 sheep be sold for $52.65, and 25 per cent, gained on
the first cost, how much was paid for each at first ? Ans. $3.24.
4. If 16f per cent, be lost on the sale of linen at $1.25, what
was the first cost ? Ans. $1.50.
5. If a quantity of glass be sold for $4, and 10 per cent, gained,
for what sum was it bought ? Ans. $3.64, nearly.
u""
CASE IV.
To find the gain or loss per cent, when the first cost and soiling
price are known.
.';,:. ■''... (. . ■ ^ BULB. ^
Divide the gain or lou by the fint oott. '
EXAMPLE.
'■ i
If a web of linen be bought for $20 and sold for $25, what is
the gain per cent?
* Here $5 are gained on $20, and $20 is I of $100, therefore $25
will be gained on $100, t. e;, 25 per cent. • .
iV. vt
EXERCISES.
.-!-■!' I
■fV-.^-
1. If a quantity of goods be bought for'$318.60, and sold for
$299.39, how much per cent, is lost ? Ans. 6 per cent.
2. If two houses are bought, the one for $150 and the other for
$250 ; and the first sold again for $100 and the latter for $350,
what per cent/is gained on the whole ? Ans. 12^.
A grocer buys butter at 24 cents per lb. and seUs it at 30 cento
per lb., what does he gaia per cent ? Ans. 25.
m
FBOFIT AND LOSS.
201
j'i:?
4. If a oattlfl dealer buys 20 cows, at an average pric* of $20,
and pays 50 cents for the freight of each per railroad, what per
eent. does he gain by selling them at $25.62} each ? Ans. 25.
b. A tobacconist bought a quantity 4)f tobacco for $75, but a
part of it being h^st, he sold the remainder for $60 : what per cent,
did he lose 7 Ans. 20.
A B E V . ♦
Given the gain or loss per cent, resulting from the sale of goods
at one price, to find the gain or loss per <^nt. by selling the same a(i
anot^ price. v '
« BULK.
t
Find hy case in. the first cost, and then hy case IV. the gain or
lass per cent, on that cost at the second selling price. >
■ . . EXAMPLE.
If a farmer sells his hogs at $5 each, and realizes 25 per cent. ;
what per cent, would he realize by selling them at $7 each.
We find by case III., that the cost was $4, and then by case IV.
what the gain per cent, would be on the second supposition, that is
$3-h4=.75, or 75 per cent.
EXERCISES.
1. If a grocer sells rum at 90 cents per bottle, vcui gains 20 pjier
eent. ; what per cent, would he gain by selling it at $1.00 per bottle?
Ans 33|.
2. If a hatter sells hats at $1.25 each, and loses 25 per cent. ;
whal per cent, would he lose by selling them at $1.60 each ?
-, ^•/'-- .;•, ,-: Ans. 4.
3. If a storekeeper sells cloth at $1.25, and loses 15 per cent. ;
would he gain or lose, and how much, by selling at $1.65 ?
Ans. He would gain 12 per cent, nearly.
4. A milliner sold bonnets at $1.25, and thereby loRt 25 per
eent. ; wquld she have gained or lost by selling them at $1.40 ?
Ans. She would have lv)st 16 per cent.
5. A merchant sold a lot of goods for $480, and lost 20 per cent. ;
would he have gained or lost b]^ selling them for $720, and how
much ? Ans. He would have gained 20 per cent.
6. A quantity of grain was sold for $90, which was 10 per cent,
less than the cost ;-what woi|ld bave bew the gala per cent, if it had
been sold for $160? " * f : £ . ^ Ass. 60.
J!?
'}''} i|*j
II
lltr,
202
ABTTHMETIO.
7. A grocer sold tea at 45 cents per pound, and thereby guned
12]^ per cent. ; what would he have gamed per cent, if he had sold
the tea at 54 cents per pound ? Ans. 35.
8. A fanner sold com at 65 cents per bushel, and gained 5 per
cent. ; what per cent, would he have gained if he had sold the com
at 70 cents per bushel ? Ans. 13j^.
!i'
MISCELLANEOUS EXBBOISBS.
1. If I buy goods amounting to $465, and nell them at a gain of
15 per cent. ; what arc my profits ? Ans. $69.75.
2. Suppose I buy 400^ barrels of flour, at $16.75 a barrel, and
Bell it at an advance >f | per cent. ; how much do I gain?
Ans. $25.14.
3. If I buy 220 bushels of wheat, at $1.15 per bushel, and wish
to gain 15 per cent, in soiling it ; what must I ask a bushel. $1.32^.
4. A grocer bought molasses for 24 cents a gallon, which he sold
for 30 cents ; what was his gain per cent. ? Ans. 25.
5. A man bought a horse for $150, and a chaise for $250, and
sold the chaise for $350, and the horse for 100 ; what was his gain
per cent. ? Ans. 12|.
6. A gentleman sold a horse for $180, and thereby gained 20
per cent. ; how much did the horse cost him ?
Ans. $150,
f 7. In one year the principal and interest* of a certain note
amounted to $810, at 8 per cent. ; what was the face of the note ?
Ans. $750,
8. A carpenter built a house for $990, which was 10 per cent,
less than what it was worth ; how much should he have received for
U so as to have made 40 per cent. 7 Ans. $1040.
9. A broker bought stocks at $96 per share, and sold them at
$102 per share ; what was his gain per cent. ? Ans. 6^»
10. A merchant sold sugar at 6^ cents a pound, which was 10
per cent, less than it cost him -, what was the cost price ? .
Ans. 7| cents per pound.
11. A merchant sold broadcloth at $4.75 per yard, and gained
12^ per cent. ; what would he llave gained per cent, if he had sold
it at $5.25 per yard ? Ans. 24||.
12. I sold a horsJB for $75, and by so doing, I lost 25 per cent. ;
whereas, I ought to have gained 30 per cent. ; how much was he sold
for under his real value? Ans. $55.
m
PROFIT AND LOSS. ' 208
13. A watch which cost mo $30 I have sold fbr $35, on a credit
of 8 months ; what did I gain bj my bargain, allowing money to bo
worth per cent.? . , . Akj $3.65.
14. Bought 84 yai-dfl of broadcloth, at $5.06 per yard ; what
must be my asking price in order to fall 10 per cent., ^nd still make
10 per cent, on the cost ? Ans. $6.11^.
15. A farmer sold land at 5 cents per foot, and gained 25 per
cent, more than it cost him ; what would have been his gain or loss
per cent, if he had sold it at 3^*ccnts per foot ? ' '
" ~" '■ ' " Ans. 12J^ per cent. loss.
16. What must I ask per yard for cloth that cost $3.52, so that
I may fall 8 pe** cent., and still make 15 per cent., allowing 12 per
cent, of sales to bo in bad debts ? ■ Ans. $5.
17. A merchant sold two bales of cotton at $240 each ; for one
he received 60 per cent, more than its cost, and for the other 60 per
cent, less than its cost. Did he gain or lose by the operation, and
how much? , ^ - , Ans. loss $270.
18. Bought 2688 yards of cloth at $2.16 per yard, and sold
oncrfourth of it at $2.54 per yard ; one-third of it at $2.75 per
yard, and the remainder at $2,90 per yard. Find the whole gain,
and the gain per cent. Ans. $1612.80 and 27^ per cent.
19. A flour merchant bought the following lots : —
118 barrels at.... .$9.25 per barreL
212 « 9.50 "
315 " 9.12^ « ^
400 " 10.00 «
The expenses amounted to $29.50, besides insurance at ^ per cent.
At what price must he sell it per barrel to gain 15 per cent ?
Ans., $11.05.
20. Bought 100 sheep at $5 each ; having resold them at once
and received a note at six months for the amount ; having got the
note discounte 1 at the Fourth National Bank, at six per cent., I
found I htui gained 20 per cent, by the transaction. What was the
Belling price of each aheep? • Ans.. $6.19.
'- . - , •. " ., - '
-. '' " . ■ ■ >y ■'■- : -.'. 'i.:.;:. ,,:■ r^mxr':' ''■'^'■^
y .-■•-■■'■'■.
I
I
204 ABriHMEno.
■r," STORAGE. ; ■-■- -,- ;-v,-^^
Whan ft charge is njtde for the aooommodation of having goodi
kept in store, it is called storage.
Accounts of storage contain the entries showing when the goods
were l^eceivcd and when delivered, with the number, the description
of the articles, the sum charged on each for a certain time, and the
total amount charged for storage, iii;}iioh is generally determined by
an averago reckoned for some specified time, usually one month (30
EZAMPLX8.
1. What will be the cost of storing wheat at 3 cents per bushel
. per month, which was received and delivered as Mows : — Received,
August 3rd> 18G5, 800 bushels ; August 12th, 600 bushels. De-
livered, August 9th, 250 bushels ; September 12th, 350 bushels ;
September 15tb, 400 buuheb, and Ootolor Ist^ the balance.
SOLUTION. . i
18S5. Bush. Days. Bnah.
August 3. Beceived 80C ' X 6 = 4800 in store for one day.
« 9. Delivered 250
Balance 550 X 3 = 1650 in store for one day.
" 12. Beceived 600
Balance 1150 X 31 =35660 b store for one day.
Sept. 12. Delivered 350 - , _ /,
^ ■ '^ Balance 800 X 3 = 2400 in ato^ for one day.
" 15. Delivered 400 * :^ ?
Balance 400 X 16 = 6400 in store for one day.
OotI 1. Delivered 400
Total....: :..... 50900 in store for one day.
50,900 bushels ita store for one day would be the same as
B0900-^30=1696f busheb in store for one month of 30 dayt, and
the storage of J. 697 bushels for one month, at 3 cents per month,
would equal 1697X.03=$50.91.
It u customary, in business, when the number of .articles upon
wbioh storage is to be charged, as found, contains a fraction leu
stohaoe.
205
than a half, to reject the fraction ; but it* it is more than a haJ/, to
regard it &s an cntiro article.
From the solution of the foregoing example, yre deduce the foU
lowing
Multiply the number of hunJicls, harrch, vr other articles, by the
number of dai/n they (ire. in store, and divide the sum of the pro-
Juct.t by .*>(>, or the number of days in any term agreed upon. The
quotient will give the number of bushels, barrels, or other articles on
which storage is to he. charged for that term, «.
2. What will be the cost of storing salt at 3 <!cnts a barrel per
month, which was put in store and taken out as Ibllows : — Put in^
.January 2, 180G, 450 barrels ; January 3, 75 barrels ; January 18,
300 barrels; January 27, 200 barrels; February 2, 75 barrels.
Taken out, January 10, tiO barrels; January 30, 150 barrels;,
February 10, 190 barrels ; February 20, 3()0 barrels; March 1, 250
barrels; said on March 12, the lialancc, 150 barrels? Ans. $39.44.
3. Kccelvcd and delivered, on account of T. C. Musgrove,
sundry bales of cotton, as Ibllows: — llcceivcd January 1, 1866,
2310 bales; January 16, 120 bales; February 1, 300 bales. Deli-
vered February 12, 1000 bales; March 1, 600 bales; April's, 400
bales ; April 10, 312 bales ; May 10, 200 bales. Required the num-
ber oi bales remaining in store on June 1, and the cost of storage
up to that date, ut the rate of 5 cents a bale per month.
Ans. 218 bales in store ; ^321.18 cost of storage.
4. W. T. Leeming & Co., Comniission Merchants, Albany, in
account with A. B. Smith & Co., Oswego, for storage of salt and
gunpowder, received and delivered as fbllows :
Keceivcd, January 18, 1866, 400 kegs of gunpowder and 50
barrels of salt; January 25, 250 barrels of salt; February 4, 150
barrels of salt, and 50 kegs of gunpowder; February 15, 100 kegs
of gunpowder; March 5, 64 kegs of gunpowder; April 15, 50 kegs
of gunpowder, and 75 barrels of salt. Delivered, February 25, 15
legs of gunpowder," and 40 barrels of salt ; March 10, 150 kegs of
gunpowder, and 285 barrels of salt; April 20, 200 kegs of gunpow-
der; April 25, 150 barrels of salt, ond 200 kegs of gunpowder.
Required the number of barrels of suit and kegs of gunpowder in
■tore May 1, and the bill of storage up to that date. The rate of
i
II!,
206
ABITHMETIO.
Storage for salt being 3 centti a barrel per month, and for gunpowder
12 ocnts a keg per month.
Ans. Id store, 150 barrels of salt and 99 kegs of gunpowder; bill
of storage, 8206.01.
O-ENERAL AVERAGE,
Tins is the term used to denote the contribution of all persons
interested in a ship, freight, or cargo, towards the loss or damage
incurred by any particular part of a ship, or cargo, for the preserva-
tion of the rest. This sacrifice of property, is caWod jettison, from
the goods being cast into the sea to save the vessel ; although not
only property destroyed in that way is the subject of general average,
but also any damages or expenses voluntarily incurred for the good of
all. For pxamplc, the expense of unloading the cargo that the ship
may be repaired ; masts or sails cut away and abandoned to save the
The only articles exempt from contribution are provisions, wear-
ing apparel of passengers, and wages of the seamen.
The owners contribute according to the clear value of the ship
and freight at the end of the voyage, after deducting the wages of
the crew and other expenses.
In New York ^, and in other States i^ of gross freight is some-
times deducted for seamen's wages; but as a general custom the
exact amount is ascertained and deducted.
Goods that have been subject to jettison, and are lost, are valued,
when the average is calculated at the place of the ship's destination,
at the price they could have sold for there ; but when the averaiie is
to be ascertained at the port of lading, the invoice price is the
standard of value. v ' • 't}!^ ' *
In making an account of the articles which are to contribute, the
property lost or sacrificed must be included, and its owners must
sufier the same proportionate loss as the rest. The losses to the dif-
ferent parties interested in the vessel, freight and cargo, are paid by
their insu'-crs.
When repairs have to be made to a ship — new sails, masts, or
rigging, ibr example — one-third of the expense is deducted on account
of mcUoralion, or the improved condition of the ship by these repairs.
When the ship is now, and on her first voyage, the full amount of
«thc cspeusc of repairs is allowed in computation of the loss.
OENERAL AVERAGE.
EXAMPLE.
207
On the 26th Judo, 1865, tho utoamcr Cuba left New York for
Liverpool with a cargo, ua follows : —Shipped by T. A. CoUiqs,
$7480 ; R. Evans & Co., $5305 ; H. C. WrFght, $9218 ; W. Man-
ning & Co., 011428 ; E, Carpenter, $7559. When off Sundy Hook
a houvy galo was experienced, during which cargo to tho value of
$3498 was thrown overboard; of tlii $1123.40 belonged to R.
Evans & Co., aiid tho balance to E. Carpenter. The necessary
repairs of tho steamer cost $878, and the expenses in port, while
getting repaired, were $253. The steamer was valued at $100,000 ;
gross freight, $4310. Tho seamen's wages were $860. What waa
tho loss per cent., and what was the lo^s of each contributory in-
terest? ... , , ...,:, , . ir.^'.'V ■>,..- . , . --■ i
* ' ■ ' ' ' . • " ';■"'-''."
V - ; SOLUTION. • ,
Loss for general henefit. Contributory tnterats. ' '
Cargo thrown overboard,$3498 Value of steamer i $100,000
.Repairs to steamer less J 584 Invoice price of cargo.... 41,050
Expenses in port 253 Fr'ght, less seamen's wages 3,460"
Total loss $4,335 Total contrib. int.... $144,500
$4835-v-144,500=:.03 loss per unit, or 3 per cent. ,y j;;^ «,v
$100,000 X.03=$3000.00, steamer's share of loss. ;. ,
7,480 X -03== 224.40, T.A.Collins' share of loss. ( ;, ; .
5,365 X. 03=:. 160.95, R. Evan & Co.'s share of loss. •;
.,. ' 9,218 X. 03= 276.54, H. C. Wright's share of loss.
11,428X.03= 342.84, W. Manning & Co.'s share of loss.
7,559 X. 03= 220.77, E. Carpenter's share of loss. •.',;,.
3,450 X .03= 103,50, Freight's share of loss.
$3000.00-
1123.40-
2374.60-
$4335.00, Total loss. • ' ' *^
-837.00=$2163.00, balance payable by steamer. ' ,
-160.95^ $962.45, balance receivable by R. Evans & Co.
-226.77= 2147.83, balance rcocivablc by E. Carpenter. ,
NoTK.— It is evident tfaat since the steamur lost $8:!7 ($581 by repairs,
and $253 by uxpenses), — that tliu net amount required I'rom the steamer will
be $3000— 837=$2163. R. Evans & Co. having lost by merchandize being
thrown overboard $1123.46, a sum greater than their share of tho general
loss, 80 that there must be duo them $1123.40 -160.95=$962.45 ; m also the
amount of £. Carpenter's share of the general loss must be deducted from bis
iadlTidual loss in order to find the balance due him.
4
■'0
m
in
208
ABirmcEno.
R n L 1 . Ji»
Find the rate per unit of lo»$, hy which multiply the value 0/
saih contributory intereit, and the product will he the §har& of \ou
to be iuatained by each.
KXBBOI8I8.
( <;
1. Tho steamship Ocean Queen on her trip finom PHiladelphia to
Liverpool, wns crippled in a storm, in consequence of which the
captain had to throw overboard a portion of the cargo, amounting in
vnluo to $4465.50, and tho necessary repairs of tho vessel cost $423.
Tho contributory interests were as follows: — Vessel, $30,000; gross
freight, $6225 ; oar^o shipped by J. Jones & Co., $3650 ; by Henry
Anderson, $0500 ; by George Millan, $2000 ; by J. Foster & Son,
$550 ; by Brown Brothers, $5450 ; and by Wilson &, Carter, $8500.
Of tho cargo thrown overboard, thcro belonged to Henry Anderson
tho value of $3000, and to Brown Brothers tho remainder, $1465.50.
The cost of detention in port, in consequence of repairs, was $116.50;
seaman's wages, $2075. How ought the loss to bo shared among
the contributory interests ? Ans. 8 per cent.
2. .The steamer Persia left Boston for Halifax, June 30th,
loaded with 7210 buchels of spring wheat, shipped by J. M. Mus-
grove, and invoiced at 95 cents per bushel; 4815 bushels of corn,
shipped by Thomas A Bryco & Co., and invoiced at 60 cents per
bushel ; 2180 barrels of flour, shipped by A. B. Smith & Co., and
invoiced at $5.50 per barrel. When near Halifax, the steamer
collided with the Bay State, and the captain found it necessary to
throw overboard 1600 bushels of wheat, 1280 bushels of corn, and
720 barrels of flour. On estimating the proportionate loss, it was
allowed that the wheat would have sold in Halifax at an advance of
10 per cent., the corn at an advance of 15 per cent., and tho flour
for $5 per barrel. The contributory interests were: — Steamer,
$95,000; cargo, $ ; gross freight, ^2361. 20. The cost of
fcpairs to steamer was $2198.15; cost arising from detention during
repairs, $318; seamen's wages, $1252.50. How much of the loss
had each contributory interest to bear ?
3. Tho eteamer Edith left Baltimore for New Orleans with Y600
bushels of wheat, valued at $1.25 per bushel, shipped by Dunn,
Lloyd & Co., and insured in the Hartford Insurance Company at If
per cent., 9200 bushels of com, valued at* 75 cents per bushel,
4
TAXES AND CUSTOM DUTIES.
too
ihipped by J. W. Roo, and insured in tho JRtna, Insurance Company
at 1'^ per cent.; 14,800 busheln of oatii, valued at .'}7^ conts per
bnahel, shipped by Morriit, Wright & Co., and insured in the Mutual
Insurance Company at 1^ per cent. ; 1,800 barrels o? flour, valued
tt $6.25 per barrel, shipped by Smith & Worth, and imiurcd in the
Beaver Insurance Company at l^ per cent. In consequence of a
violent gale in the Gulf of Mexico, it was found ncoesaary to throw
overboard the flour, 4,000 bushels of oats, and 3,100 bushels oi
wheat. Tho propeller was valued at $45,000, and insured in the
Beaver Insurance Company for $12,000, at 2 per cent., and in the
Western for $25,000, at 2^ per cent. The gross freight was $4950 ;
seamen's wages, $340, and repairs to tho boat, $3953.75 ; what was
tho loss sustained by each of the contributory interests, the propeller
being on her first trip ?
TAXES AND CUSTOMS DUTIES.
A tax is a money payment levied upon tho subjects of a State
or tho members of any community, for the support of tho govern-
mcnt. : /,v;.; v; > ■• '/^, .^^ ,.
A tax is cither levied upon tho property or tho persons of indi-
viduals. When levied upon the person, it is called npoll tax.
It may be either direct or indirect. When direct, it is levied
from the individuals, or the property in the )> mJs of tlio ultimate
owners. When indirect, it is in the nature ' .' a cusioms' or excite
duty, which is levied upon imports, or i iuiiuiacturci', bd'orc they
reach tho consumer, although in tho end thoy aro pal J liy tho latter.
Customs^ duties are paid by the iinportcr cl' gooJ- at tho port of
entry, wliero a custom-house is stationed, with government employees
called custom-Jiouse ojfficers, to collect these dues.
Excise duties are those levied upon articles manufactured in the
country.
An invoice is a complete list of the particulars and prices of
goods sent from one place to another.
A Specific duty is a certain sum paid on a ton, hundred WAic^ht.
yard; gallon, &c., without regard to the cost of the article
An ad valorem duty is a percontogo levied on tho actual com,, or
fair market value of the coods in the country from which th*^, v«
imported. ,>^
,
210 ' ARn'HMETlC.
OroM tcelffht is tho vrcight of good;*, upon which a upeoifio duty
iit to be loviod, bolbro any allowunoen uru doduotod.
Xet weight M tho weight of tho goods after all allowanoos aro
doduottid.
Among tho ullowancoH made nro tho following:
Breakage — iin allowance on fluids contained in bottles or break-
able vessels.
Draft — thi! allowonco for waBtc.
Lotkagr — an allowance for waste by leaking.
Tare and (ret are the deductions lundo for tho weight of the ease
or barrel which contuinH tho ;;c)ods.
When goods, invoiced at gold value, upon which duty is payable,
aro imported Mito this country from any foreign country, tho custom
houHc duticH are payable in gold, for cIho manifest injustice might be
done. Jf tho duty were payable in greenbacks, it would bo nccce
sary, in order to obtain uniformity, cither to incroaso or decrease the
rato pep cent, of duty, an greenbacks fluctuated in value, compared
with gold (tho invoice price ol'tho goods), or cIjo tho goods imported
would rciiuire to bo retJucod to their vuluo in greenbacks at time of
delivery. To avoid all this troublo and confusion, goods that are
invoiced at their gold value, tho duties arc uiidc payable in tho same
currency.
When goods aro imported from any country which has a doprc-
oiated currency, a note is attached to tho invoice, certifying tho
amount of depreciation. This is tho duty of tho Consul represent-
ing tho country to which tho goods aro exported, and residing at the
port /roM which they aro exported.
EXAMPLES.
^ To find the specific duty on any quantity of goods.
Suppo.so an Albany Provision Merchant imports from Ireland 59
casks of butter, each weighing G8 lbs., and that 12 lbs. tare is allo^ved
un each cask, and 2 cents per lb. duty on the net weight.
- We find the grcss is 5I]|XCSrr^4012 lbs.
" tare is....; 59X12^- 708 ibs.
Hence tI/6 net weight is 3304 lbs
The duty is 2 cents per lb..... 2 '"'"'
The duty, therefore, is 866.08
TAXES AND CU8T0M DUTIES.
211
To find the ad valorem dvtty od any quantity of goods.
Suppoio a Troy dry y: odtt niurobant to import from Montreal
430 yards of nilk, at $1.75 per yard, and that Ii5 per oont. dnty ig
chur;!;od on them.
Hero wo find the wholo price by tho rulo of Practloo to bo
$7C3, tlion thu rest of tho operation in a direct case of percentage,
and thoroforo wo multiply $703 by .liO, which gives $207.05, the
amount of duty on tliu wholo. ;,. " .,
Uonoe wo have tho following
>rv
fill] ■
ii.' I.'
RULE FOR 8PK0IFI0 DUTY.
Subtract the tare, or other allowance, and multiply r/i5 remain'
dtr by the rate of duty per box, yallnn, dkc.
nULBFORADVALOREMDUTT.
Multiply the amount of the invoice by the rate per unit,
« EXEROIBES.
1. Find the specific duty on 5120 lbs. of sugar, the tare being 14
per cent., and the duty 2^ cents per lb. Ans. $121.09.
2. What is tho ad valorem duty on a quantity of silks, the
amount of the invoice being $95,800, and the duty 02^ per cent ?
^ ^ ^, , w/ Ans. $59,875.
3. At 30 per cent., what is the ad valorem duty on an importa-
tion of china worth $1200. ? Ans. $378.
3. What is the specific duty, at 10 cents per lb., on 45 chests of
tea, each weighing 120 lbs., the tare being 10 per cent. ? Ans. $486.
5. What is the ad valorem duty on a shipment of fruit invoiced
at $4560, the duty being 40 per cent.? Ans. $1824.
6. What is tho specific duty on 950 bags of coffee, each weighing
200 lbs., the duty being 2 cents per lb., and the tare 2 per cent?
Ans. $3724.
7. What is the ad valorem duty on 20 casks of wine, each con-
taining 75 gallons, at 18 cents a gallon ? Ans. $270.
8. A. B. shipped irom Oswego 24 pipes of molasses, each con- ,*
taining 90 gallons; 2 percent, was deducted for leakage, and 12
cents duty per gallon charged on the remainder ; how much was the
duty? Ans. $270.95.
212
AI5ITHBIETIC.
9. Peter Smith & Co., Brooklin, import from Cadiz, 80 baskets
of port wino, ut 70 francs ror basket ; 42 boskets of sherry wine, at
35 franes pcir basket ; CO casks of champagne, containing 31 gallons
each, at 4 I'rancs per gallon. Tho waste of the wine in the casks
was reckoned at u gallon each cask, and the allowance for breakage:
'n the baskets was 5 per cent. ; what was the duty at 30 per cent.,
18jJ cents being taken as equal to 1 franc? Ans. $776.54.
10. J, John.son & Co., of Boston, import from Liverpool 10
pieces of Brussels carpeting, 40 yards each, purchased at 5s. per
yard, duty 24 per cent. ; 200 yards of hair cloth, at 4.'- per yard,
duty 19 cwt. ; 100 woollen blankets, at 2a. (id., duty 10 per cent. ;
and shoe-lasting to the cost of £60, duty 4 per cent. Kequired the
whole amount of duty, allowing the value of the pound sterling to
bo 94.84. Ans. $174.24.
11. John McMaster & Co., of CoUingwood, Canada West.,
bought of A. M. Smith, of Buffalo, N. Y., goods invoiced at
$5430.50, which should have passed through tho custom-house dur-
ing, the first week in May, when the discount oa American invoices
was 43|^ per cent., but they were not passed until the fourth week
in May, when the discount was 36f per cent. Tho duty in both
cases being 20 per cent. ; what was the loss sustained by McMaster
& Co. on account of their goods being delayed ? Ans. $70.72.
STOCKS AND BONDS.
Capital is a term generally applied to tho property accumulated
by incividuals, and invested in trade, manufactures, railroads, build-
ings, government securities, banking, &c. The capital of incorpo-
rated companies is generally termed its " capital stock," and is
divided into shares ; the persons owning ono or more of these shares,
being called stockholders. The shares in England, are usually
£100. £50, or £10 each. In the United States they are generally
$100, 850, or 810 each.
The management of incorporated -companies is generally vested
in officers and directors, as provided in the law or laws, wh.* are
elected by the stockholders or shareholders ; each stockholder, in
most cases, being entitled to as many votes as tho number of shares
he holds ; but sometimes the holder of a few shares votes in a larger
proportion than the holder of many.
The a«»pumulating profits which are distributed among the stork-
holders, once or twice a year, are called " dividends," and when
** declared," are a certain percentage of the par value of the shares.
In mining, and some other companies, where the shares are only a
STOCKS AND BOIJDS.
213
few dollars each, the dividend is usually a fixea sum "per share."
Certificates of stock are issued by every company, signed by the
proper ofiicers, indicating the number of shares each stockholder is
entitled to, and us an evidence of ownership ; these are transferable,
and may be bought ;^nd sold like any other property. When the
market value equals their nomiaal value they are said to be " at
par J" When they sell for more than their nominal value, or face,
they are said to be above par, or at a " premium" ; when for less,
they are below par, or at a " discount." Quotations of the market
value are generally made by a percentage of their par value. Thus,
a share which is $25 at par, and sells at $28, is quoted at twelve
per cent, premium, or 112 per cent.
When states, cities, counties, railroad companies, and other
corporations, borrow largo amounts of money, for the prosecution of
their objects, instead of giving common promissory notes, as with
the mercantile community, they issue bonds, in denominations of
convenient size, payable at a specified number of years, the interest
usually payable semi-annually at some well known place. These are
usually payable to "bearer," and sometimes to the "order" of the
owner or holder. When issued by Governments or States, these
bonds are frequently called Government stocks or State stocks,
under authority of law. To these bonds are attached, what are
called " coupons," or certificates of interest, each of which is a due
bill for the annual or semi-annual interest on the bond to which it is
attached, representing the amount of the periodical dividend or
interest; which coupons were usually cut off, and presented for
payment as they become due. These bonds and coupons are signed
by the proper officers, and like certificates of capital stock, are nego-
tiable by delivery. The loan is obtained by the sale of the bonds,
with coupons attached, but they are sometimes negotiated at par.
Their market value depends unon the degree of confidence felt by
capitalists of their being paid at maturity, and the rate of interest
compared with the rate in the market.
Treasury notes are issued by the United States Government,
for the purpose of effecting temporary loans, and for the payment
of vfontracts and salaries, which resemble bank notes, and are made
payable without interest generally. Recently such notes have been
issued bearing one year or three years' interest.
"Consols" is a term abbreviated from the expression " consoli-
dated," the British Government having at various times borrowed
money at different rates of interest and payable at different times,
" consolidated" the debt or bonds thus issued, by issuing new stock,
drawing interest at three percent, per annum, payable semi-f.nnually,
and redeemable only at the option of the Government, becoming
practically perpetual annuities. With the proceeds of this, the old
Btodc was redeemed. The quotations of these three per cent, per-
petual annuities, or " consols," iodicate ordinarily the state of the
:» .:
r^\
( ?;■•
214
AUITHMETIO.
1
money market, as they form a large portion of the Briti&h public
debt.
" Mortgage Bonds*' are frequently issued by owners of real
property, with coupons attached, which renijl^r the bonds more
saleable as well as more conveDient for the collection of interest.
" Coupon Bonds," being negotiable by delivery, are payable to
the liolder ; and in case of loss or theft, the amoiint cannot be
recovered from the government or corporation issu ug them, unless
ample notice is given of the loss.
" Begistered Bonds" arc those payable only to the " order" of
the holder or owner, and are more safe for investment.
By law, stockholders are liable for the whole debts of the corpo-
ration, in case of failure. In some States the law provides that they
are liable only to an amount equal to their stock. In England the
statute provides for " Limited" liability, by an Act passed in 1862
termed the "Limited Act."
CASE I. ' '''''-' -■"*■■'■■•
The premium or discount being known, to find the market value
of any amount of stock.
'••^••'- 'I.:' ■.-•■. .^' /,;■■. BXA-MPLES. ■■l-\ ,/'■
k If G. W. R. shares are at 7 per cent, premium, to find the value
of 30 shares of $100.
■ ; Here it is plain that each $100 will bring $107, and that each
$1 will bring $1.07, and as the par value is $3000, the advanced
value will be 3000 times 1.07, which gives $3210, the market value,
and $3210— $3000=$210, the gain.
Again, if the same arc sold at a discount of 7 per cent., it is plain
that each $100 would bring only $93, and therefore each $1 would
bring only $0.93, and therefore as the par value is $3000, the de-
preciated value will be 3000 times .93, which gives $2790, and
therefore the loss would be $3000—2790=210.
From this we derive the
nULK.
^. ■;>'-
Multipltf the par value hy 1 plut or mintu tne rate per wnitp
according as the shares are at a premium or a discount
STOCKS AKD B02n)S
2lSr
KXKB0ISK8. .'. V
1. What is the market value of $450 stock, at 8^ per cofit/'dlfi!-
oonnt ? Ans. $411.75.
2. What is the value of 29 shares of $50 each, when the shares
are 11 per cent, below par ? Ans. $129p.50.
'6. A man purchased 60 shares oi^ $5 each, from an oil \ireii
company, when the shares wore at a discount of 8 per cent!,, and
sold them when they were at a premium of 10 per cent; how much
did ho gain ? Ans. $54 •
4. A man purcliascd $10,000 stock when it was at an advance of
8 per cent., and sold when it was at a discount of 8 per cent. ; how
much did he lose ? Ans. $1600.
5. If a man buys 15 shares of $100 each, when the shares are
at a premium of 5 per cent., and sells when they have advanced to
12 per cent., how much docs he gain ? Ans. $105.
■ -' ) , CASE II. „, ...
To find how much stock a given sum will purchase at a given
premium or discount. .
Let it be required to find how much stock can be purchased for
$21,600 when at a premium of 8 per cent.
In this case it will require $108 to purchase $100 stock, and
therefore $1.08 to purchase $1 stock, and hence the amount that
can be purchased for $21600 will be represented by the number of
times that $1.08 is oontained in 21600, which gives $20000.
Again : Let it be required to find how much stock can be pur-
chased for $5520, when at a disc( unt of 8 per cent. When stocks
are 8 per cent, below par, $92 will purchase $100 stock, and there-
fore $0.92 will purchase $1, and hence the amount that can be pur-
chased for $5520 will ^be represented by the number of tisies that
.92 is contained in 5520, which gives $6000 stock.
Hence we derive the
RULE. ■•■■'"■'''' ' ■'■"'""
Divide the given sum hy 1 plu& or minua the rate per unit, accord^
tuff as the shares are at a premium or a discount.
- A*' :
SXEB0I3ES.
6. When stocks arc at a premium of 12 per cent., how mooh oanr
be purchased for $8064 ? Ans. $7200.
216
ABUHXEITOt
7. When stocks are at a disconnt of 9 per cent., how much can
he bought for $3640 ? Ans. $4000.
8. When G. T. R. stock is at 18 per cent, below par, how much
can be bought for $42,640. Ans. $52000.
9. When Gt. W. B. stock is at a premium of 9 per cent., how
much will $4578 purchase ? Ans. $4200.
10. When government stock is selling at 92^, what amount of
stock will $28,G75 purchase, and to what will it amount with broker-
age at J per cent. ? * r Ans. $31077.50.
CASK III. '' ' " ■^--' '■■'*'■'-■"
The premium or discount being known, to find the par value.
To find tTiA par value of $1,296, when stock is at a premium of
8 per cent.
At 8 per cent, premium, each $1 brings $1.08, hence the par
value will be represented by the number of times 1.08 is contained
in 1296, which ^ives $1200 for the par value.
To find the par value of $1104, when stock is at a discount of 8
per cent.
Each $1 will bring $0.92, and therefore the par value w:'l be
represented by the number of times that .92 is contained in 1104,
which gives $1200, the par value. Hence the ., >
y .1 3^- ■ 'U..; --^ ■ - •,.- RULE.
IQjivide the market value hy 1 plu» or minui the rate per untV,
according as the stocks are selling above or below par.
S X B R I M E S .
11. What is the par value of $24420, when stock is 11 per cent,
above par ? Ans. $22000.
12. What is the par value of $10800, when stocks are at a dis-
oount of 4 per cent. ? Ans. $11250.
13. When government stocks are at 6 per cent, premium ; how
much w.U $20246 purchase at par value ? Ans. $19100.
14. The shares in a canal company are at 15 per cent, discount;
how many shares of $100 will $11390 purchase ? Ans. 134.
15. The shares of a British gas company were selling in 1848,
at a discount of 12 per cent. ; a speculator purchased a certain num-
ber of shares for £792 ; the value of tho shares suddenly rose to par ;
how many shares did ho purchase, and how much did he gain ?
. Ans. 9 shares; £108 gain.
STOOES AMD IVOMDS.
217
i
' OASBIV.
To find to what rate of interest a given dividend corresponds.
If a person receives a dividend of 12 per cent, on an investment
made at 20 per cent, above par, the corresponding interest may be
calculated thus :
As the stock was bought at 20 per cent., or .20 above par, $1.20
of market value corresponds to $1 of par value, and as every $1 of
par value corresponds to 12 per cent, interest, or .12, it follows that
the per cent, which was invested will be represented by the num-
ber of times that 1.20 is contained in .12, which is .10 or 10
percent Hence tho , ^ , v ..
Divide the rate per unit of dividend hy 1 phu or minus the rait
per cent, premium or discount, according as the stocks are above or
helov) par.^ -^' ".•'«•.".■;;''•,;■ ■'■^»";''' ,•',.,..■, '^',
■ EXERCISES. ;>*.•■-■*'''.:'.'',
16. If a dividend of 10 per cent, be declared on stock vested at
25 per cent, advance ; what is the corresponding interest ?
Ans. 8 per cent.
17. If a dividend of 4 per cent, be declared on stock invested at
12 per cent, below par, what is tho corresponding interest ?
Ans. 4j^.
18. If money invested at 24 per cent, yields a dividend of 15
per cent., what is the rate of interest ? Ans. 123^.
19. If railroad stock is invested at 18 per cent, above par, and a
dividt^nd of 6 per cent, bu declared, what is the rate of interest ?
Ans. 5^^.
20. If bank stock be invested at 15 per cent, below par, and a
dividend of 10 per cent, declared, what is the rate of interest ?
Ans. llif.
UISCBLIiANEOUS EXEB0I8BS.
1. What must be paid for 20 shares of railway stock, at 5 per
cent, premium, the shares being $100 each? Ans. $2100.
* To find at what price stock paying a given rate per cent dividend can be
purchased, so that th<' noney invested shall produce a given rate of interest^
divide fAe rcrfe per uniA of ditulend ^y <A« rate per untf of Mttrest.
ii''
V
%•• Si
VI
m
i:
218 ;i?
.n ABJTHMXnO.
/
/
2. What is tho par value of bank stock worth $8740, at a pro>
mium of 15 per cent. ? Ans. $7600.
3. Bailway stock was bought at 15f below par, for $I895.62| ;
how many shares were there, each share being $150 ?
Ans. 16 shares.
4. If 6 per cent, stock yields 8 per cent, on an investment, at
what per "'"t. discount was it bought? Ans. 25.
5. If bank stock which pays 11 per c^int. dividend, is 10 per
cent, above par, what is tho corresponding rate of interest on any
investment? ■ ' V' - Ans. 10.
6. When 4 per cent, stocks were at 17J discount, A bought
$1000 ; how much did he pay, and how much did he gain by selling
when stock had risen to 86^ ? Ans. $821.25, and $41.25.
7. What will $850 bank stock cost at a discount of 9f per
cent., ^ per cent, being charged for brokerage ? Ans. $771.38.
8. On the data of the last example, how much would be lost by
selling out at 10^ per cent. ? Ans. $10.^3. ^^
9. What income should I get by laying out $1620 in the pur-
chase of 3 per cent, stock ut 81 ? Ans. $60.
10. What sum must be invested m the 4 per cent, stocks at 84,
to yield an income of $280 ? Ans. $5880.
11. What rate of interest will a person receive by investing in
the 4^ per cent, stocks at 90 ? Ans. 5 per ccnt»
12. A person transfers his capital from the 3^ per cent, stocks at
77, to the 4 per cent, at 117^, what is the increase or decrease per
cent, in his income ? Ans. Decrease 25.
~ 13. A person sells out of the 3 per cent, stock at 96, and invests
his money in railway 5 per cent, stock at par ; how much per cent.
is his income increased ? Ans. 60.
14. What must be the market valne of 5]^ per cent, stock, so
that after deducting an income tax of 2 cents on the doUar, It may
produce 5 per cent, interest ? Ans. 107|.
15. A gentleman invested $7560 in tlie 3} per cent, stocks at
94^, and on their rising to 95 sold out, and purchased G. T. B. 4
per cent, stock at par; what increase did he make in his annual
income ? Ans. $24.
16. How much more may a person increase his annual income
by lending $3800, at 6 per cent., than by pahi'i9siDg railway 5 per
«eiit stock at 95 ? Ans. $28.
STOCKS AND BONDS.
219
17. A person sells $4200 railway stock which pays C per cent,
at 116, and iavcsts one-third of the proceeds in the 3 per cent. oon.
60I.4 at 80^, and the balance in saviD<;s bank stock, which pays 9 per
cent, at par \ what is the decrease or increase of his annual income ?
Ans. Increase $97.80.
18. A person having $10,000 consols, sella $5000 at 94|, and
on their rising to 98f he sells $5000 nioro ; on their again rising ho
buys back the whole at 96 ; how much docs ho gain ? Ans. $75.
19. The sum of $4004 was laid out in purchasing .3 per cent,
stocks at 89^, and a whole year's dividend having been rescivod upon
it, it was sold out, the whole increase of capita^ being $302.40; at
what price was it sold out ? 1 • ■-.■.,),' Ans. 93|f.
20. Suppose a person to have been an Original subscriber for 500
shares of $50 each, in the First National 'Jank, payable by instal-
ments, as follows : — J in three months, which he sold for 5J per
cent, advance : § in 6 months, which brought him 0^ per cent, ad-
vance, and the balance in nine months, which he was compelled to
sell at 8f per cent, riisoount ; what did he gain by the whole transac-
tion ? Ans. $808.33.
¥
pr
f
-f
21. A gentleman purchased $5000 of Fivs-twenties (gold 6 per
cents) at 108 ; gold at time of purchase was at 35 per cent, pre- -^
mium ; if it remained so when the interest was payable, what was
the riLte per cent, of interest on amount invested ?
Ans. 7J per cent.
22. From which would be derived the greater income, Seven-
thirties purchased at 104, or Five-twenties (C per cent, gold) at
109^, interest on both bonds payable at the same time, and gold,
quoted at 140 ? Ans. From the Five-twenties.
23. On Jan. Ist I wish to make an investment of money that
will allow me 7^ per cent, interest on the investment ; what can I
afford to pay for !f en-forties (interest payable in gold at G per cent.)
and what for Seven-thirties^ calculating gold at 35 per cent. prem. ?
Ans. For Ten-forties 108 ; Seven-thirties 97^.
24. In the above example, what could I give for the Ten-forties
Ans.9&.
K
if gold wereoalcnlated at 20 per cient. prem. ?
U i*''*.-*
.'iCfc,, «*««# *%*
!!i!:.
(
r^
220
AlUTHMETIO.
25. On May 2l8t, a broker purchased for me » Seven-thirty
bond to the amount of $12,000 at 104| ; the interest on this bond
is payable on tho Ist Feb. and August ; what does the bond cost me,
the brokerage being ^ per cent. ? Ans. $12861.60.
26. After receiving the interest, on Aug. 1st, on tho bond men-
tioned in last question, the broker immediately sold it for me at 103f
oharging ^ per cent, for selling ; did I gain or loso by the transac-
tion, and how much, money being worth 6 per cent. ?
, Ans. Lost $157.94:
27. A gentleman subscribed $15,000 in a railroad company,
having a paid-up capital of $^50,000 ; but only 40 per cent, of sub-
scribed capital paid in. A cash dividend of 3^ per cent, on the par
value is declared ; what rate per cent, does he receive on his invest-
ment? Ans. 8f per cent
28. The capital of the " First National Bank," of Cincinnati, is
$1,500,000, of which A has subscribed $7,500. There has been
25 per cent, called in. A cash dividend of 4 per cent, on the paid-
up capital is declared, and 10 per cent, on paid-up capital carried to
the credit of the stockholders ; how much money does A receive as
« dividend, what per cent, on subscribed capital is carried to credit
of stockholders, and what has A still to pay on his stock ?
Ans. A receives $75 ; to credit of stockholders 2^ per cent. A
lias still to pay $5437.50.
29. A having $25,000 for investment on May 1st, placed it in
the hands of B, a broker, advising him to spc oulate in buying and
fielling stocks, bonds, and gold, for 60 days, and then return what
the money produced, after deducting brokerage of ^ per cent, on the
actual sales and purchases. B immediately purchased 200 shares of
Erie R. B. stock at 59 " buyer 30," no margin required, and sold
400 shares Reading R. R. stock, at 102| >< buyer 15." Five days
after, B called in the Erie R. R. stock and sold it to C at 61|^ ;
May 8th he bought $20,000 of Five-twenties at 109|, at the same
time the person to whom the Reading R. R. stock was sold, called it
in ; B paid the difference, the stock being valued at 102^ ; May
27th, gold Ifaving appreciated in value as compared with Greenbacks,
B sold the Five-twenties at 110^ cash, and at the same time made a
farther sale of $15,000 in the same kind of bonds at 110^ " seller
ed it in
ing and
•n what
on the
lares of
ind sold
ive days
at6l|;
he same
called it
i; May
snbaoks,
made a
" seller
STOCKtt AND BONDS.
221
30/' which ho was able to purchase and deliver in ton days at 109}.
June 20th, B sold $20,000 in gold at 137^ "seller 10," which was
not delivered at tho expiration of the ten days, but settled at 131^ ;
hoyi much money is A to receive from B ? Ans. $25,475.
L
PARTNERSHIP,
,V:tel
Partnership has been defined to bo tho rcsiilt of a contract, under
which two or more persons ngroo to combine property, or labour, for
the purpose of a common undertaking, and tho acquisition of a com-
mon profit.
A dormant, or sleeping partner, is one who shares in tho concern^
but does not appear to tho world as such.
A nominal partner is one who lends his name and credit to %
firm, without having any real interest in the profits.
AH the partners may contribute equally to the business ; or the
capital may be contributed by some or one, and tho skill and labour
by the other ; or, unequal proportions may be furnished by each.
The contract need not be in writing, but all parties to be bound
must assent to it, and it is usually contained in an instrument called
" Articles of Partner»htj>"
Too much pains cannot be taken to have this agreement so plain
and explicit in regard to particulars, that it cannot possibly be mis.
understood. A grcht deal of litigation has arisen from carelessness
in this respect.
These Articles of Partnership should particularly specify the
amount of investment by each partner, whether the personal atten-
tion of thf) partners is required to tho business, duration of partner-
ship, and sometimes an agreement with regard to the vrithdrawal of
money from the business.
A dissolution can take place at any time by mutual consent. '
A partnership at will is one in which there is no limited time-
affixed for its oontlnuanoe, and the whole firm may be dissolved
M
«i
SI
222
ABTTHMITIO*
by any of its members at a moment's notice. A dooumont is, how*
ever, gonorally drawn up and signed upon a dissolution, called a
tettlement, which contaius ti statement of the mode of adjustment of
the accounts, and the apportionment of profits or losses.
-KXAMPLK.
Two persons, A. and B., enter into nartnership. A. invests
$300 and B. $400. Thoy gain during one year $210; what is
each man's share of the profit ?
SOLUTION BT P It PC ATI N.
^ ' A.'s Stock, $300 , '■
B.'b " 400 ' '
Entire stock $700 : 300: : $210 : $90 A.'s gain. ) ;' ^
« " 700:400::$210:120 B.'s "
'. :;'• ■.■••;'■
« SOLUTION BY PS BCENTAOX.
Since the entire amount invested is $700, and the gain $210,
the gain on every $1 of investment will be represented by the num-
ber of times that 700 is contained in $210, which is .30 or 30 cents
on the dollar. Now if each man's stock be multiplied by .30 it will
represent his share of the gain thus :
$300x.30=$ 90 A'sgoin.
■ ^j^ /. 400x.30=: 120 B.'s
((
v' ^
Entire stock 700 210 Entire gain.
Henpe, — To find each partner's share of the profit or loss, when
there is no reference to time, we have the following
RULE.
As the whole stock is to each partner's stock, so is the whole gain
or loss to each partner's gain or loss; or, divide the whole gain or
loss hy the number denoting the entire stock, and the quotient will he
the gain or loss on each dollar of stock ; which multiplied hy the
. number denoting each partner's share of the entire stock, will give
hii share of the entire gain or loss.
EXEBOISES.
1. ThMo persons, A., B., and 0., enter into partnership. A.
advances $500, B. $550, and 0. $600 ; they gain by trade $412.50.
What is each partner's share of the profit ?
Ans. A.'s $125 ; B.'s $137.50 ; O.'s $100.
PABTNEBSinP.
228
2. A, B, and D purohoso an oil well. A payg for G shares, B
for 6, for 7, and D for 8. Their not profits ut tho end of three
months have aniountod to $7800 ; what sum ought each to receive ?
Ans. A, 81800 ; B, $1500 ; C, $2100 ; D, $2400.
3. A und B purchased u lot of land for $4500. A paid ^ of tho
price, und \i tlio roniuindcr ; they gained by the sub of it 20 per
cent. ] what wau each man's aharc of the profit ?
Ans. A, $300^ B, $600.
4. A captain, mate, and 12 sailors, won u prize of $2240, of
which tho captain took 14 sharcHi, tho mate 0, and tho remainder
was cciuully divided among tho sailors ; how much did each receive ?
Ans. Tho captain, $080 ; tho mate, $420 ; each sailor, $70.
5. A und B invest equal sums in trade, and clear $220, of which
A is to have 8 shares on account of transacting the bu.sinuss, and B
only 3 shares ; what is each man's gain, and what allowance is made
A for his time ? Ans. Each n)an'8 gain $60 ; A $100 for his time.
6. A, B, C und D enter into partnership with u joint capital of
$4000, of which A furnishes $1U00 ; B $800 ; C $1300, and D tho
balance ; at tho end of nine months their net profits amount to
$1700 ; what is each partner's share of tho gain, supposing B to re-
ceive $100 for extra services ?
Ans. A, $400 ; B, $320 ; C, $520 ; D, $360.
7. Six persons, A, B, C, D, E and F, enter into partnership, and
gun $7000, which is to be divided among them in tho following
I'Qanner: — A to have I; B, ^ ; C, J as much as A and B, and tho
remainder to bo divided between D, E and F, in tho proportion of
2, 2^ and 3^ ; how much does each partner receive ?
Ans. A, $1400 ; B, $1000 ; C, $800 ; D, $950 j E, $1187.50 ;
F, ^1662.50. V
8. A, B and C enter into partnership with a joint stock of
$30,000, of which A furnished an unknown sum; B furnished IJ,
and C 1^ times as much. At the end of six months their profits
were 25 per cert, of the investment ; what was eaeh man's share of
tho gain ? Ans. A's, $2000 ; B's, $3000 ; and C's, $2500.
9. A, B, C and D trade in company with a joint capital of $3000 ;
on dividing the profits, it is found that A's share is $120 ; B's, $255 ;
C's, $225 ; and D's, $300 ; what was each partner's stock ?
Ans. A's, $400; B's, $850; C's, $750; and D's, $1000.
10. Three labouring men. A, B and C, join together to reap a
ccrudn field of wheat, for which they agree to take the sum of
III
i
y '» .•
224
ABITHiaEnO.
$19.84; A ind B oftlouUte that they oan do | of tho work ; A and
f ; B and { of it ; how mnch ahould oooh reoeiye aooording to
these ostimAtes ? Ana. A, $8.32 ; B, $7.04 ; and 0, •4.48.
I, r
To find each partner's share of the gain or loss, when the capital
is invested for different periods.
IXAMPLE. ■*
Two morohants, A and B, enter into partnership. A invests
$700 for 15 months, and B $800 for 12 months; they gain 1603 ;
what is each man's shore of tlie profits ? ^
SOLUTION.
$700X15==$10500
$800X12= 9600
20100 : 10500 : : $603 : $315 A's gain.
20100 : 9600 : : $603 : $288 B's gain. •
The reason for multiplying each partner's stock by the time it
was in trade, is evident from the consideration that $700 invested
for 15 months would be equivalent to $700x15 equal to $10500 for
one month, that is $10500 would yield, in one month, the same in-
terest that 1(700 would in fifteen montha. Likewise $800 invested
for 12 mctiths would be the same as $9600 for one pionth; henoo
the question becomes one of the previous case, that is, their invest-
ments are tho same as if they had invested respectively $10500 and
$9600 for equal times ; hence the
. ^- BULL " ■"" "■ ■*' '■^■^--'f
Multiply each man^$ atock hy the time he continuei it in trade;
then say, as the mm of the products it to each particular prodiict, to
it the whole gain or lott to each man't thare of the gain or htt,
EXEBOISXS.
11. A, B and are associated in trade. A furnished $300 for
6 months ;'B, $350 for 7 months, and C, $400 for 8 mopths. Their
profits amouTited to $1490 at the time of dissolution ; what was tho
profit belon{i;ing to each partner ?
Ans. A, $360 ; B, $490 ; 0, $640.
PABTMIBSHIP.
225
12. A, B uifl eontnot to perform a certain pieoo of work ; A
cmployi 40 meo for 4j^ months ; B 46 men for 3^ months, 4nd 60
aien for 2^ months. Their profits, after paying all oxponscs,. art
$860 ; how mueh of this belongs to each ?
Ans. A, $340 ; B, $297.60 ; 0, $212.60.
13. Four men. A, B, and D, hired a pasture for $27.80 ; A .
pats in 18 sheep for 4 months ; B, 24 for 3 months ; C, 22 for 3
months ; and D, 30 for 3 months ; how much ought oaoh to pay ?
Ans. A and B each, $7.20 ; (.\ 94.40 ; D, $9.
14. On the first day of January A began business with a capital ,
of $760, and on the first of February following ho took in B, who
invested $640 ; and on the first of June following they took in 0,
who put into the business $800. At the end of the year they found
they had gained $872 ; how much of this was each man entitled to ?
Ans. A, $384.93 ; B, $250.71 ; C, $236.36.
16. Three merchants, A, B and 0, entered into partnership with
a joint capital of $6876, A investing his stock for 6 months, B his
for 8 months, and his for 10 months ; of the profits each partner
took an equal sbnro ; how much of the eopital did each invest ?
Ans. A, $2600; B, $1875; C, $1600.
16. Two merchants, A and B, entered into partnership ibr two >'
years ; A at first furnished $800, and at the end of one year, $600
more ; B furnished at first $1000, at the end of 6 months, $600
more, and after they had been in business one year, he was compelled
to withdraw $600 At the expiration of the partnership their net
profits were $2560 ; how much must A pay B who wishes to retire
from the business ? Ans. $2190.
17. Three persons, A, B and C, form a partnership for one year,
commencing January 1st, 1866 ; A puts in $4000 ; B, $3000 ; and 0,
$2600; April 1st, A withdraws $500, and B withdraws $600 ; June
1st, puts in $800 more ; September 1st, A furnishes $700 more,
and B $400 more. At the end of the year they find they have
guned $1600 ; what is each partner's share of it?
Ans. A, $608.68; B, $423.31 ; C, $468.01.
18. John Adams commenced business January first, 1866, with .
ft capital of $10000, and after some time formed a partnership with
William Hickman, who contributed to the joint stock the sum of
$2800 cash. In course of time they admitted into the firm Joseph
Williams, with a stock worth $3600. On making a settlement
January first, 1866, it was found that Adams had gained $2260;
. ^ i
tarn
■ ■■ ''I
S
w
I
i
226
ARirmisTio.
Hickman, 8120 ; and Williams, $405 ; how loQg had Hickman's and
Williams' money heen employed in the business, and what rate of
interest per annum had each of the partners gained on their btook ?
Ads. Hickman's 8 months ; Williams' 6 months. Gain, 22^
per cent, interest.
BANKRUPTCY.
_ «
When any person is unaoic to meet his liabilities, he makes an
assignment of his property to some other person or persons, called
official assignee or assignees, whose office it is to distribute the avail*
able property, after paying expenses, rateably among the creditors.
An allowance for maintenance is generally made to the insolvent, but
sometimes lie is compelled to surrender all his estate, but only in
case of iKunifest /rawe?, which the word hanhmpt originally implied,
though now it is used as nearly 'ynonimous with insolvent. The
property to bo divided is called the assets. The shares of the pro-
perty which are divided among creditors, are called dividends.
EXAMPLE.
A bankrupt owes A $400 ; B. $350, and C, $600 ; his net assets
amount to $810 cash; Jow much is he able to pay on the $1, and
how much will each creditor receive ?
SOLUTION.
$400-[-$3504-$600=$1350, total liabilities. JNow, if he nas
$1350 to pay, and only $810 to pay it with, he will only be able to
pay $810-:-1350=.60or eOcents on the $1. Therefore, A will
receive $400X.60=$240; B,$350X.60=$210,and d, $600 X. 60
=$360. Hence the
RULE.
Divide the net assets hy the number denoting the total amount of
the debts, and the quotient will be the sum to bepaia on each dollary
then multiply each m^n's claim by the sum paid on the dollar^ and
the product will be the amount he is to receive. f ' f — :
BANKBUPTCY.
227
^ EXERCISES.
1. A becomes bankrupt. He owes B, $800 ; G, $500 ; D,
$1100, and E, $600. The assets amount to $1110 ; how much can
he pay on the dollar, and how much does each creditor receive ?
Ads. He can pay 37 cents on the dollar, and B receives $296 ;
0, $185 ; D, $407 ; and E, $222.
2. A house becomes bankrupt; its liabilities are $17940; its
assets arc $8970 ; what is the dividend, and what is t^ac shase of the
chief creditor to whom $1282 are due ?
Ans. The dividend is 50 cents on the dollar, and the principal
^ , creditor gets $641.
3. A shipbuilder becomes bankrupt, and his liabilities are
1303000 ; the premises, building and stock are worth $220000, and
i>8 has in cush hsd notes $12875 ; the creditors allow him $3000
for maintenance of his family; the costs are 3^ per cent, of the
amount available for the creditors; what is the dividend, and how
much does a creditor get to whom $1360.60 are due ?
.,^; Ans. Dividend, 73^ percent. Creditor gets, $995.95.
iit' I
!i
i
I'
I?
iH
4. Foster & Co. fail. They owe in Albany, $22000; in Balti-
more, 818000; in Philadelphia, $1/100; in Charleston, $16000;
in Bojton, $4400, and in Newark, $4200. Their assets are : house
prop rty, $14000 ; farms, $2200 ; cash in bank, $4400 ; railway
stock, $4200 ; sundry sums due to them, $20135 ; what is the divi-
dend, and how much goes to each city ?
Ans. Dividend, 55 cents on the dollar; to be paid in Albany,
$12100; in Baltimore, $9900; in Philadelphia, $9405; in
rT ; Charleston, $8800 , in Boston, $2420; in Newark, $2310.
5. The firm of Reuben Ring & Nephews becomes bankrupt. It
owes to Buchanan & Ramsay, $1080 ; to Kinneburgh & McNabb,
$850 ; to Collier Bros., $1720 ; to David Bryce & Son, $1580 ; to
Sinclair & Boyd, $970. The assets are : house and store, valued at
$848 ; merchandise in stock, $420 ; sundry debts, $220. What can
the estate pay, and what is the share of each creditor?
Ans. The estate payr. 24 cents on the dollar, and the payments
are: to Buchanan & Ramsay, $259.20; to Kinneburgh &
McNabb, $204; to Collier Bros , $412.80 ; to David Bryco ,
& Son, $379.20 ; to Sinclair & Boyd, $231.80.
m\
! !•■(
,, w5,^f ;-'
228 . ABETHXETIO.
EQUATION OF PAYMENTS.
Equation of Payments is the process of finding the average or
mean time at which the payment of several sums, duo at different
times, may all be made at one time, so that neither the debtor nor
creditor shall be at any loss.
The date to be found is called the equated time^
The mode of finding equated time almost universally adopted is
very simple, though, as vrc shall show in the sequel, not altogether
correct. It is known as tJie mercantile rule.
Let us observe, in the first place, that t'no stftPv'-'Td by which
men of business re(jcon the advantage that accrues to them from
receiving money beibre the time fiked for its payment, and the loss
they sustain by the payment being deferred beyond the appointed
time, is the interest of money for each such period. Thus, if $50
be a year overdue, the loss is $3", at 6 per cent. ; and, if $50 be paid
a year in advance of the time agreed upon, the gain to the payee is
$3, at the same rate. In the former case, the person receiving the
money charges the payer $3 interest for the inconvenience of lying
out of his money, but, in the latter case, ho deducts $3 from the
debt, for the advantage of having the money in hand. If, on the 1st
May, A gives B two notes, one for $50, at a term of three months, and
the other for $80, at a term of seven months, the first will V; hjgally
due on the 1st August, and the 2nd on the 1st December; i>;a / is
not able to meet the first at August, and it is held over till 'h.> ** t
November, when A finds himself in a position to pay both at c^ce.
The first is then three months over-due, and accordingly B claims
interest for that time, which, at 6 per cent., is 75 cents, but as A
tenders payment of the whole debt at once, and the second note will
not be due for another month, A claims a deduction of one month's
interest, which, at the same rate, is 40 cents, and accordingly A, in
addition to the debt, pays B 35 cents. ? '^' ;'; t <
Let us now suppose another case. A owes B $130, as before, and
he gives B two notes — one for $50, on 1st May, at 3 months, and
another, on the 6th May, for $80, at 8 months. The first falls due
on Ist Aiigust, and the other on the 6th January, but A and B
agree to settle at such a time that neither shall have interest to pay,
but that A shall simply have to pay the principal. Supposing thtft
a settlement is made on 6th November, we find that the 1st note is
EQUATION OF PAYMENTS.
229
3 months and 6 days over due, and the interest on it for that period
is 80 cents, while the second will not he due for 2 months, and the
interest on it for that period is also 80 cents ; consequently, the
interest that A should pay, and that which B should allow being
equal, they balance each other, and the principal only has to be paid.
There arc, then, three methods for the payment of several debtS}
or a debt to be paid by instalments. The first is to pay each instal*
ment as it becomes due. This needs no elucidation, nor is it often
practised, except in the case of small debts, due by persons of con*
tracted means.
The second is what has been illustrated above by the first exam*
plo, viz., that interest is added for overdue money, and deducted for
sums paid in advance of the stipulated time.
The third has been illustrated by the second example, viz., to fix
on such a time that the interests on the overdue and underdue sums
shall be equal, so that the debtor has only to give the principal to
the creditor. If, in this last case, the time should come out as a
mixed number, the fraction must be taken as another day, or thrown
off, making the payment fall due a day earlier. The principle on
which all such settlements are made is, that the interest of any sum
paid in advance of a stipulated time is equivalent to the interest of
the same sum overdue for a like time.
With these explanations we are now ready to investigate a rule
for the Equation of Payments. For this purpose let us suppose a
ease. R. Evans owes J. Jones $200, which he undertakes to pay
by two instalments of $100 each, (basis of interest 6 per cent.,) the
first payment to be made at once, and the second at the expiration of
two years. But the first payment is not made till the end of the
first year, at which time B. E. tenders payment of the whole amount.
For the accommodation of having the first payment deferred for one
year he is to pay $6, i. e., $1 06 in all, and in return for making the
second payment a year before it is due, he claims a discount at the
same rate, which gives $6. He has therefore, by the mercantile
rule, to pay $106-}^94=$200, so that the $6. in the latter case
balances the $6 in the former. This takes one year as the equated
time, and is the mode usually adopted on account of its simplicity,
though not strictly accurate.
To find the equated time when there are several payments to bo
made at difierent dates. ^ ^ ^
l|V-
230
ARITHMETIC.
If A owes B $1200, due by instalments as follows : — $300
in 4 montHs, $500 in 6. months, and $400 in 10^ months,
what is the equated time for the payment of the whole ?
$300 X 4 = 1200
500x 6 = 3000
400x101= 4200
$1200 )
8400 ( 7 months
8400 (equated time.
The interest of $300 for 4 months is the same as the in-
terest of $1 for 1200 months ; the interest of $500 for 6
months is the same as the interest of $1 for 3000 months ;
and the interest of $400 for 10^ months is the snme as the
interest of $1 for 4200 months. The sura of all these is
8400 months ; therefore the interest of the whole is the same
as the interest of $1 for 8400 months. Now, if $1 require
8400 months to produce a certain interest, the whole debt,
$1200, will require only ^nj^g^ part of that time to produce
the same interest ; and 8400 -j- 1200 =7. Hence the equated
time is 7 months. ; ; - > ,; '
Rule 1. — Multiply each payment by its time, and divide
the 8um of the products by tJie sum of the payments.
Another method of producing the same result is the fol"
lowing : —
Interest of $300 for 4 months= $12.00 @ 12 per cent.
Interest of 500 for 6 months = 30.00 '@ «
Interest of 400 for 10^ months= 42.00 @ ««
Interest of $1200 for 1 month =12)84.00 ( 7 months.
Rule 2. — Fimd the interest of each instalment for its
time, at any convenient rate, and divide the sum of the
interests by the interest of the ivhole debt, at the same rate
for one m,onth.
Note. — 12 percent, is a vety convewient rate, because the interest is flo
easily found, being 1 per cent, a month, and consequently the hundredth
part of the principal f err I month, The interest is therefore found by
simply multiplying the principal by the number of months, and pointing
off two places of decimals.
The process by Eule 2 becomes identical with that by Eule 1 by reckon-
ing the interest at 1 200 per cent.
EXERCISES.
i., ■
' 1. Find the ecpiated time for tlie payment of three debts,
tlie first lor $45, due at the end of 6 months ; the second for
§300
ntbs,
he in-
for 6
mths ;
IS the
lese is
3 same
•equire
e debt,
irocluce
quated
divide
he fol-
ceut.
((
•I VJ
18.
^or its
of the
hne rcbt&
[rest is so
lundredth
found by
pointing
ly reckon-
^e debts,
2ond for
EQUATION OP PAYMENTS.
231
$70, due at the end of 1 1 months ; and the third for $75j
due at the end of 13 months. . ■ - ' '. •
$45 X ()=270 ' < . ' •
70x11=770 '
75x13=975 .
$190 ) 2015 (
10.61
80
10 mqs. 18'days. Ans.
18.30
"When the division is not exact, continue it to two phices
of decimals, and reduce to days.
2. If a person owes $1200, to be paid in four instalments,
$100 in 3 months ; $200 in 10 months ; 300 in 15 montlis,
and $600 in 18 montivs, in what time should he pay the
whole sum at once ?
In this and similar questions, the work may be somewhat
shortened by counting no time for the first payment, and
deducting its time from that of each of the others. Thus :
$100 X 0=
200 X 7= 1400 . '
300x12= 3600
. ^ 600x15= 9000
$1200 ) 14000 ( 11§, to which add 3 months,
and we have for the equated time 14§ months.
3. J. Smith owes R. Evans $1300, of which $700 are to
be paid at the end of 3 months, $100 at the end of 4 months,
and the balance at the end of 8 months. Required the
equated time for the payment of the whole ? Ans. 5 mos.
4. T. C. Musgrove owes H. W. Field $900, of which $300
are due in 4 months, $400 in 6 months, and $200 in 9
months ; what is the equated time for the payment of the
whole amount ? Ans. months.
5. A. & W. MoKinlay have in their possession five notes
drawn by G. W. Armstrong, all dated 1st January, 1873 ;
the first is drawn at 4 months, for $45 ; the second at 8
months, for $120 ; the tliird at 10 months for $75 ; the fourth
at 11 months, for $60 ; and the fifth at 15 months, for $90 :
for what len<»th of time must a sinirle note be drawn, dated
1st May, 1873, so that it may fall due at the properly equa-
ted time ? Ans. 6 months.
6. A gentleman left his son $1500, to be paid as follows :
I in 3 months, | in 4 months, ^ in G months, and the remain-
ll
;;il.
Li,':,
IliiS ,'
hi!;;;:;
m
W,
M
232
ABITHMETIC.
der in 8 monthe ; at what time ought the whole to be paid
at once? Ans. 4 months 15 days.
7. A merchant bought goods amounting to $6,000. He
agrees to pay $500 down, $600 in 6 months, $1500 in 9
months, and the remainder in 10 months; at what time
ought he to pay the whole in one payment ?
Ans. 8 mos. 16 days.
8. A r^rocer sold 484 bbls. of rosin, as follows : February
6th, 35 bbls. @ $3.12^ on 4 months time ; March 12th, 38
bbls. @ $3.00 on 4 months time; March 12th, 411 bbls.
@ 2.62^ on 4 months time. What is the equated time for
the payment of the whole ?
Febr'y 6, 109 xo' 6= 22 /
March 12, 1193x1 12=1193
476 ■• '• •■' :■■;■'
1302 )
1.691
1302
(
m.
1—3
m
,f
3890 ' 9.0 days.
1 month, 9 days, not reckoning the credit of 4 months on
which the whole was bought. Add 4 months to this time,
and »the result is & months, 9 days to be counted forward
from the beginning of February — making July 9th the date
on which the whole should be paid.
In the above example we have taken the beginning of Feb-
ruary as a convenient point from which to reckon the time
on each item. From that point the time on the first item
(from which we have omitted the cents .is of no consequence in
the calculation) is 6 days, .and on the second 1 month, 12 days.
When the time is expressed in months, wa have simply to
multiply by the months. When there are days, multiply ^
the principal by ^ the number of days, for -^ the principal
will be the product by -j^ of a month, or 3 days, and as 6
days are twice 3 days, the product to be carried out is twice
^ the principal, -^ the principal is obtained by simply cut-
ting off the right hand figure, which if 5 or more should be
considered as 1, and added to the remaining figures. Thus,
in the above example, we have carried out t\«ice 11 for 6
days, and 4 times 119 for 12 days.
When the number of days does not contain 3 an exact
nimiber of times, the nearest niunber that does may be taken
first, and then the odd days over, which must be either ^ or
EQUATION OP rAYMENTS.
233
days.
3ruary
th, 38
LbblB.
no for
f of 3 days. Work 5 days as -J- of a month, 10 days as ^ of
a month, 15 days as ^ a month, &c., when more convenient.
9. Purchased Goods of J. R. Worthington & Co,, at differ-
ent times, and on various terms of credit, as by the following
statement : —
March Ist, 1872, a bill of $675.25, on 3 months.
t(
,y-
nths on
^1 time,
brward
le date
ofFeb-
le time
rst item
aence in
12 days,
mply to
tiply A
rincipal
nd as 6
is twice
ply cut-
lould be
Thus,
11 for 6
,n exact
betaken
her i or
July 4th,
Sept. 25th, «
Oct. 1st, •«
Jan'ry Ist, 1873,
Feb'y 10th, «
Mar. 12th, «
April 15tl, «
c;
((
(i
((
it
u
376.18
821.75
961.25
144.50
811.30
567.70
369.80
4
2
8
3
6
5
4
((
((
What is the equated time for the payment of the whole ?
m. d.
1 =
3, March 1,1872, 675x3
4, July 4,
2, Sept. 25,
376x8 4
822 X 8 25 =
8, Oct
1,
96i X 15
3, Jan'ry, 1,1873, 145x13
1
1
6, Feb'y, 10,
5, March,! 2,
4, April 1.%
811x17 10
568x17 12
370x17 15
4728
)
2025 for 3 months.
23 « 1 day.
3008 « 8 months.
38 « 3 days.
13 « 1 "
6576 « 8 months.
656 « 24 days.
27 " 1 «
14415 « 15 months.
32 « 1 day.
1885 « 13 months.
5 " 1 day.
13787 « 17 months.
270 « 10 days.
9656 " 17 months.
228 "12 days.
6290 « 17 months.
185 « 15 days.
m.
59119 (12.50
4728 30
4- .■-,
11839
9456
,, 23830
.23640
L 1900
15.00dayB.
'i:
Mil.,
Ik
Iff
1 1
lli
iWII
234
ABITHMETIC.
12 months, 15 days, from the beginning of March, 1872,
gives March 15th, 1873. Ans.
3 months' credit on the first bill, and 1 day in March gives
the time on the first bill ; 4 months from March to July and
4 months' credit with 4 days in July gives the time on the
second bill ; 6 months from March to September, and 2
months' credit with 25 days in September, gives the time on
the third bill, &c.
To carry out the products, — Ist, multiply the first bill by
the months ; — for the one day, take ^ of 68. 2nd, multiply
the second bill by the months, throw off the 6 and take the
remaining figures of the principal, plus 1, for 3 days, — take
^ of that for 1 day. 3rd, multiply 822 by 8, for 8 months, —
multiply 82 by 8, for 24 days, — ta)ce ^ of 82 for 1 day, &c.
10. Bought of A. & W. Smith, 1650 barrels of flour, at
different times and on various terms of credit, as by the fol-
lowing statement : —
May 6th, 150 barrels @ $4.50, on 3 months' credit.
May 20th, 400 " « 4.'75, on 4 « «
July 10th, 500 " « 5.00, on 5 " « .
August 4th, 600 " " 4.25, on 4 « «
What is the equated time for the payment of the whole ?
Ans. November 7th.
11. J. B. Smith & Co. bought of A. Hamilton & Son 576
barrels of rosin, as follows : —
May 3, 62 bbls. @ $2.50, on 6 months.
May 10, 100 « « 2.50, on 6 months.
May 18, 10 « " 2.50, as cash.
May 26, 50 « " 2.75, on 30 days. - v '
> May 26, 345 « « 2.50, on 6 months.
:i? May 26, 9 " " 2.00, on 6 months. ii^U^
What is the equated time for the payment of the whole ?
Ans. November 3rd.
12. T. B. Jones & Co. sold goods on 3 months' credit, as
follows : —
^ ^ ^ . - May 9, a bill of $435.60.
« 30,
(( ((
75.30.
July 17,
(( ((
183#75.
Aug. 28,
(( ((
239.18.
Sep. 21,
t( . ((
' 82.10.
Oct. 23,
(t (C
39.85.
Nov. 30,
(( (t
390.67
235
AVERAOINO ACCOUNTS.
a-r-f.
W.
When, in equity, ought they to have received the whole in
one sum, and, allowinfr money worth 6 per cent, what sum
ought they to have received at the date of the last sale ?
Ana. Equated time, Nov. 13th ; cadh to settle
Nov. 30th, $1450.49.
13. Bought of T. & E. Kenny, on 6 months' credit, goods as
follows: »
January 3, to the amount of $250.00
February 6,
Maich 9,
April 12,
May 15,
June 18,
July 21,
August 24,
Sept'br. 27,
October 30,
Nov'br. 29,
Dec'br. 11.
tt
((
((
((
317.40
171.70
88.12
623.50
49.04
73.90
218.75
8.15
55.84
398.00
191.25
What *3 the equated time of settlement, and allowing in-
terest at 7 per cent., if payment be delayed till Februaey Ist,
1874, how much will then be due? Ans. Equated time,
Dec. 16th, 1873. Du? Feb. 1st, 1874, $2467.69.
AVERAGING ACCOUNTS.
When one merchan c trades with another, exchanging mer-
chandise, or giving aud receiving cash, the memorandum of
the transactions is called an Account Current. The fixing
on a time when the account may be settled by simply pay-
ing the balance without interest against either party, is
called Averaging the Account.
A merchant sold goods amounting to $4000 on 8 months'
credit. The purchaser paid ^ down, and ^ in 3 months;
what time should be allowed him for for the payment of the
remainder ?
$4000 X 8 = 32000
2000 X =
1000 X 3 = 3000
'
3000 , ,3000
subtract from 320(
1000 :- 4^ w 29000
( 29 months =
2 years, 5 mouths
''„*
^1
286
ARTrHMETIC.
The buyer, by the terms of the purchase, is entitled to the
use or interest of $4000 for 8 montha, which is the same as
the interest of $1 for 32000 months. He lias received on the
first $2000 no credit, and on the $1000 paid, only 3 months,
which is equal to the interest of $1 for 3000 months. He
has therefore to receive on the remaining $1000 what is equal
'to the use of $1 for 29000 months. But the interest of $1
for 29000 months is the same as the interest of $1000 for
the Y^jfj^ part of 29000 months, which is 29 months, or 2
years, 5 months.
A merchant sold W. M. Brown, Esq., goods to the amount
of $3051, on a credit of 6 months from September 25th,
1873. October 4th, Mr. Brown paid $476 ; Nov. 12th, $375 ;
December 5th, $800; and on January Ist, 1874, $200.
When, in equity, ought the merchant to receive the balance ?
6, Sept. 25, 3051 x 6-25=18306 • ^ ,
" -m. . - ;^>*^'v,:- : . 2440 ■;-
102
20848
Oct 4,
- - .
Nov. 12,
Dec. 5,
Jan'y 1,
476x1-4 = 476
•' 16 - ;
375x2-12= 750 ,
152
800x3- 5= 2400
183
200x4- 1= 800
7
1851 4782
1200 ) 16066(13.39 mo«.
1200 30
" ' 4066 11.70 d
' 3600
; ^ 4660
- :;: 3600
^ ':i
10600
13 months, 12 days, after the 31st of August, 1873, which
will be October 12th, 1874. v .
AVKRAOINO ACCOUNTS.
237
The intcreut on the debtor Hide from Aiip;. 31, 1873, is equal
to the interest of ^1 for 20848 months. The iutorest on the
credit side from the same date is e({iial to the interest of $1
for 4782, which leaves a difference, in favor of the credit
side, of the interest of ^l for KiOGG months, that is the in-
terest of the balance, 1^1200 for thn Tnj'jnr P^rt of 1 6066 months,
or 13 months, 12 days. Thereforotho merchant should allow
Mr. Brown the use of the balance of the account 13 months,
12 days, from August 31, 1873, or till October 12th, 1874.
When did the 1>alance of the followinjif account fall due,
the merchandize items being on C months credit.
Dr.
McDonald Bros.
1872.
May 15.
July 20.
Sept. 27.
1872.
To Mdse.
$350.75
Juno 9.
(( ((
185.10
1873.
U i(
431.73
Feb. 18.
Mar. 8.
By Mdse.
By cash...
By Mdse.
6. May 15, 351 x 6.15 = 2106
175
6, July 20, 185 X 8.20 = 1480
114
6, Sept. 27, 432x10.27 = 4320 '
387
968
8594
Cr.
$200.20
800.00
290.00
I
ii
r
4.1V 1
•-.
i
(, which
6, June 9, 200 x 7. 9 = 1400
60
Feb'y 18, 300 x 9.18 = 2700
, 180
6, March 8, 290 x 16.8 = 4640
58
■■'i:i--y
790
178
)
■Jy
9057
463 (
356
1070
1068
20
2.60 mos.
30
18.00 days.
W
((1^:
238
ARITHMETIC.
2 montbH, 18 davHi to
finning of May, 1872, wh
be counted backward from the ])e-
l^nninjf of May, 1872, which gives February 11th, 1872, the
time from wJiich interest is to be charged on the balance.
The interest of the debit side, from April 30, 1872, is enual
to the interest of $1 for 8594 months, while the interest on
the credit side from the same date is e(iual to the interrst of
$1 for 9057 months, which gives a difierenco in favor of tlic
debit Hide, of the interest of $1 for 4G3 months, equal to the
interest of the balance, $178, for j [^ part of 403 montliH,
that is, 2 months, 18 days.
From the examples given wo may deduce the following:
Rule. — Proceed with each sloe of the account as in
Equation of Payments^ counting the time for each aide,
from the hefjinning of the month of the earlic8„ date in the
account.
Take tJce difference between the aumr ' the products of
the tivo 8idc8j and divide it by the bal of the account.
Count the quotient 'inonths, and carry it to tivo places of
decimals. Reduce the decimals to days.
WJien the sum of the products of the larger side is
greater than the sum of the products of the smaller side,
reckon tlie time denoted by the quotient forward, but when
the opposite of this is the case^ reckon backward from the
date from which all tlie tlTne has been reckoned.
EXERCISES.
Find the times at which the balances of the following
accounts became due, or subject to interest : —
1. Dr. > J. S. Peckham. Cr.
May 16, 1872 $724.45 | July 29th, 1872 $486.80.
Ans. December 16th, 1871.
2. Dr. T. B. Reagh. Cr.
November 19, 1873,... .$635. | December 12, 1873 $950.
Ans. January 28th, 1 874.
3. Dr. * Jno. T. Lithgow & Co. Cr.
February 24, 1873... .$512.25 | June 10, 18^2 $309.70.
Ans. February 24th, 1874.
4. Dr. ' T. J. Golden & Co. Cr.
March 17, 1873 $145 | January 15, 1873... .$695.60.
; . ^y^*'. ^ : Ans. December 30th, 1872.
AVEMQINQ ACCOUNTS,
239
itcta of
:Gouiit.
aces of
nde is
:r aide,
d when
om the
llowing
Or.
;486.80.
1871.
6. Dr. S. E. WiiisTON. Cr.
August 27, 1873 $341. | November 7, 1873 $247.
Ana. Februury 2l8t, 1873.
6. Dr. L. C. Eaton. Or.
July 20, 1873,... $711. | April 14, 1873, $1260.
Aus. Docoraber Uth, 1872.
1. Dr. Gordon & Keith. Or.
June 24, 1872....! $1418. | September 7, 1873 $2346.
An8. July 9tli, 1875.
8. Dr, Gi:o. W. Jones. O?'.
December 2, 1873!.. $1040.80. | Augunt 13, 1873,...$1112.40.
A , Au8. March 0th, 1869.
9. Required the time wlicn the Imlance of the following
account becomes subject to interest, allowing the merchan-
dise items to havtj been on 8 months' credit ?
Dr.
S. T. Hall & Co.
C^.
1872.
1873.
May 1.
To Mdse,.
$300.00.
.Fan. 1.
July 7.
(( i(
7r)9.06.
Feb. 18.
Sept. 11.
•
417.20.
Mar. 19.
Nov. 25.
(( ((
287.70.
April 1.
Dec. 20.
(( ((
571.10.
May 25.
By Cash,....
« Mdse,...
" Cash,....
« Draft....
" Cash,....
$500.00.
481.75.
750.25.
210.00.
100.00.
;,, • ' #;. Ans. August 7th, 1873.
10. When will the balance of the following account fall
due, the merchandise items being on G months' credit ?
Dr. ' Barnes & Co. CV.
1873.
May 1.
May 23.
June 12.
July 29.
Aug. 4.
Sept. 18.
To Mdse
"Cashpd.dft.
"Mdse
« Cash
$312.40
85.70
105.00
243.80
92.10
50.00
Ans. January 11th, 1874.
11. When does the balance of the following account be-
come subject to interest ?
1873.
June 14,
July 30,
Aug. 10,
Aug. 21,
Sept. 28,
By Cash,.
" Mdse.
" Cash...
" Mdse,
$200.00
185.90
100.00
58.00
45.10
ill
!l!*'
!-i(
'W!!
I
240
] i
Dr.
AKITHMETIC.
Beard & Venning.
Cr.
1873.
Aug. 10,
Aug. 17,
Sept. 21,
Oct. 13,
Nov. 25,
Nov. 30,
Dec. 18,
1874.
Jan. 31.
To Mdse 4 mos.
^' " 60 days
« Cashpd. dft.
" Mdse 6 mos.
« " 90 days
" " 2 mos.
((
$285.30
192 60
256.80
190.00
432.20
215.25
68.90
1873.
Oct. 13
Oct. 26
Dec. 15
Dec. 30
1874.
Jan. 4
Jan. 21
By Cash.., $400.00
150.00
«Mse2m 345.80
« « 4m 230.40
« Cash....
340.30
180.00
Cash 100.00
Ans. August IQtb, 1»74.
12. In the following account, when did the balance be-
come dufc, the merchandise articles being on 6 months' credit ?
Dr. S. Kerr in account with T. E. Jones & Co. Cr.
To Mdse.
((
cash paid drft.
Mdse
cash paid drft.
Mdse
$240.00
48.88
50.00
1873.
Jan. 4,
Jan. 18,
Feb. 4,
Feb. 4,
Feb. 9,
Mar. 3,
Mar. 24,
April 9,
May 15
May 21,
Ans. December 20th, 1873.
When, in equity, should the balance of the following
account be payable?
Dr. Danei & Boyd. Cr.
u
it
((
((
1873.
$ 96.57
Jan. 30
By Cash,
57.37
April 3
(( ((
80.00
May 22
(( i(
3§.96
50.26
15446
42.30
23.60
28.46
177.19
1873.
Jan. 3,
Jan. 31,
Feb. 8,
Feb. 21,
Mar. 10,
Mar. 24,
Apr. 12,
June 1,
Ju 20,
Ju 4^
Sep-.. 27,
Dec. 9j
?o Cash,.
$200
(( u
300
(( ((
75
U li
100
(( (C
350
U ((
25
(( ((
40
(( ((
80
(( ((
125
(( u
268
(C (t
250
(( ((
looj
1872.
Sep. 20,
Oct. 27,
Dec. 5,
1873.
Jan. 18,
Feb. 26,
Apr. 15,
June 12,
Sep. 21,
Dec. 29,
By Mdse, 6 mos.
« 4 « .
a
a
« 60 days...
" 6 mos...
U A it
^C • • •
« 2 « ...
« 6 " ...
« 6 « ...
$583.17
321.00
137.00
98.75
53.98
634.00
97.23
84.00
132.14
Ans. October 16th, 1847.
}'
CASH BALANCE.
CASH BALANCE.
24l
JM
EXAMPLE,
What is the balance of the following account on January
19th, 1873, a credit of three months l)eing allowed on the
merchandise, money being worth 6 per cent ?
Dr. KisRu & Thorn. CV.
1872.
Mar. 12,
Apr. 21,
May 6,
May 27,
July 16,
Sep. 10,
Oct. 19,
To Mdse,.
((
(t
" cash pd. dft.
« Mdso
« Cash
" Miise
((
((
$340.00
150.00
165.00
215.00
100.00
310.00
120.00
1872.
Apr. 20,
May 4,
June 15,
Aug. 10,
Sep. 23,
Nov. 12,
Dec. 15,
By Mdse.
" Cash..
" Mdse..
« Cash..
$200.00
110.00
230.00
180.00
50.oO
50.00
100.00
FIRST METHOD.
If the above account bo averaged by the method already f;iven, it will
be found that the balance fell duo on September 3rd, lb72. If, therefore,
iitbe not paid till January 19th, 1873, the amount due will consist of
9480, the balance, and the interest on it from Se))tember 3rd, 1872, to
January 19th, 1873. Now the interest of $480 for 130 days, the intorral
referred to, i8S10.89, and $480 + 10.89=8490.89, the sum due Jan. 19, 187S.
SECOND METHOD.
EuLE. — Find the interest on eich item, from the time it
falls due to the time of settlement^ and thence the balance
of interest. Add the balance of interest to that side of the
account which 'produces the larger amount of interest. The
balance oftJie account ivill then be the Cash Balance at the
date to which the interest of the several items has been
reckoned.
This method is illustrated in the following
ACCOUNT CURRENT AND INTEREST ACCOUNT.
MnigroTC * Wrigbt lu aecoxint current and lotcrcat account to Jan. IBth, 1873, with Eaton * PiMee.
DATm
iTCMf.
raixci-
PAU
WHtK
Uva.
TlHI.
IKT.
.DlTB.
iTixa
I'aiKci-
P4U
WUJM
DlB.
Tma
Im.
I873L
1873.
1872.
1872.
Mar. 19
To Mdic. 3 moa.
S34O00
7un<-ll
221 d'K
ai23i
Apr. 30
VCjlt 4
njr Md««,. 1 mod.
S'WOO
July 20
ind'i
flea
Apr. 21
Hat a
•• " 3 "
15000
JuItZI
18;; •'
449
■• Cash
.1000
May 4
200"
"to
"C««h, paid draft,
16S00
May 6
2SS "
7 00
Junp i
It 4t
23000
JunelS
218"
834
• n
" MdM. 3 mo*.
315 00
Auit.W
145 •'
512
KugAO
'■ Mdie.,lm(M.
ISO 00
Noy.io
70"
SOT
Juir IS
" Cash,
10000
juiy le
187 "
8 07
Si'pt.33
" Caih
50 CO
Sept23
Ul»"
007
eeptlO
"Mdiir. Smoc.
31000
Dee. 10
40 "
304
NOT.12
■1 «,
50 »)
Not. 12
89 "
OM
Oct. 10
.. .. 3 ..
13000
1873.
Dec. 13
<« 4(
100 no
Deals
33 "
oaa
187SL
Jaal9
"BaLofinterMt,
10 B
Jaa.M
" Bal. of mtrrett,
loss
1873.
Jan. U
" Balance,
iSOM
SU10V3
fl340T
. )
gUlODQ
MOT
—
The difforenctt observable in the rrault of the tTro metboda is due to the different modm of
counting the time. The latter answer i.t tb» more ancurate. The difference, however, is iasi^
idfleuit, amounting to only about bait a day's interest on the baianoo of account. •
i
ill I
11; I
Itlk
142
ABITHXETIO.
2. The following account was settled in full on December Ist, 1865 ;
what amount was paid, intor'^t 6 per cent. ?
0. F. Muds in account cnrrent su< aterest account to Dec. Ist, 1865, with T. R BrowB.
Dati.
ITEMS.
rr.ixc:-
PAL.
wnfM
DUK.
TWl.
INT.
DATZ.
I1EII8.
PBIirci.
PAL.
WHKN
Due.
TIMB.
Iirr.
San. 1
To Ifdsi'., 6 inoJ.
" Cash pd. dft.
" UdM., 4 mos.
.. .. 4 ..
" Cash pd. dft.
" Udie.. e mm.
(15C lo
100 00
31S»u
103 0(1
100 0(.
313 00
F«b. 1
Mar. 30
Mlj 1
July 1
Sep. 10
Br C.vih
*' Mdne., 4 inn3.
•• « "
.. .. 4 ..
" •' 4 "
•130 i»
430 18
auooo
60 00
»»84
Veb. S
Kw.ao
*
" 90
■Uyis
Au|t.30
Ans. 961.36.
t;
3. What will be the cash balance of the following account if settled
<m January 1, 1865, allowing interest at 8 per cent, on each item after
it is duo? Ans., $110 86.
J. Smith Homans in'acct. current and interest account to'Jan. 1st, 1865, wiA T. C. Musgrove.
DATS.
Items.
PRINCI-
PAL.
WOTN
Duii.
Time
1ST.
DATE.
ITEMS.
PKtNCI-
PAL.
WHEN
Due.
Time.
IKT.
US4.
}unel1
To Uise., 4 mos.
•• " 6 ••
" Cash pd. dft.
" Cash :
" Milio., 3 mo3.
•'■ " 1 mo.
;| CKh
" Uine. an Casli
•315 00
18» OU
200 00
75 00
50 00
100 00
RJOO
150 00
.TOO 00
1854.
.\pr. 15
May 10
JUIl£l2
•••so
July If,
•' 37
.Vug.
" 20
•• 80
By Mdsa., 3 mnsi.
.. 4 ..
.. g ..
" Cash
it 4i ' ^
" KAnc. as Cash
" Caih
" MdRC., 3 mon.
(350 00
130 00
340 00
100 00
90 00
SO 00
100 00
175 00
76 00
" 29
fnlylA
1
.
1
i
•flrt.^
1
Oct 14
" 19
4. What will be the amount due on the following account, July 1st
1867, interest calculated al 7-^% per cent. ?
B. G. Conklin in account current and interest account to .July Ist, 1867, with J. B. Hams.
DAIU.
ITEMS.
Pnisci-
I'AL.
WHEN
DUE.
TIME.
INT. 1
Date.
Items.
PRINCI-
PAL.
WHES
DOE.
TIME.
Int.
186S.
July 15
Auk. 30
ToHd'e.(a.'!0d5'!i.
•' 3 moi.
" Co dya.
" 4 mo!.
SOdys.
" 4 mos.
2 mo3.
30 dya.
(,',ot m
3;i8 V2
IM '^7
1248 .Oil
110 8M
428 30
1011 no
2.'.7 V,
1
IStifl.
.Vug. 14
1 " 31
Oot. 18
Not. 1
Doc. 20
Jan. 30
Mar. 4
JiiiieK
nj Cvh
•• Md?c.V.'ib'iiy.«
" Cash
" Draft® CO dya
" Cash
" Onl.ouJ.JoD':?
" Mdso.. CO dya
(400 00
104 50
200 00
185 14
300 00
350 OU
600(10
475 00
312 00
!
Bop. 19
Kor.35
Doc. 20
ltlli7.
1
Jan. 13
Mar. 19
May 30
5. What will be the balance of the following account on March 25,
1865, each item drawing 7 per cent, interest from its date ?
, * ' Ans. $50.64.
,f^.'
-..■?.'
;i'
CASH BALANCE.
248
J. C. Baylies in acconnt current and interest account to March 25tb, 1865, with E. E. Bryan.
tn.
mr.
riME.
IKT.
".'.!!!'.
TIMS.
INT.
Date,
iTEiia.
Princi-
rue
WnF.s
UUK.
Time.
Int.
Idate.
ITEMS.
PniKci-
PAL,
wmjji
UUI.
Tnn.
iin.
ISftt.
July 4
Hvpt. 8
ToMcrchMUl'so..
t« ««
•1 *•
•• <>
«* «,
(200 00
30U0U
250 00
600 00
400 00
fiOO'K)
100 00
moo
1864.
J Illy 20
Aug. 15
Snpt, 1
Nov. 1
Dec. 6
" 20
1865.
Feb, 1
" 28
By Cash
if
By Merchandise.!
By CuKh
ByUcrebaBdiMii
(300 00
450 00
400 00
820 00
600 00
lOOOO
200 00
1W08
Octr .1
Nut. 20
J)cc. 12
1865.
Jan. 15
Jfaur.U
!■••••
ACOOUHT OF SALES.
Aa account of sales is a statement made by the consignee (gen-
erally a commission merchant) to the consignor, the person from
whom the merchandise was received tp sell, showing the parsons
to whom sold, the price, time, charges, and net proceeds.
• The net proceeds is the :; mount due the consignor, from proceeds
of sales, after all charges are deducted, and are due to the consignor
at the avoragc time of sales.
Commission merchants often become interested in the merchan-
dise consigned to them for sale, by accepting a certain share and sell-
ing oa joint account of themselves and the consignor ; when this is
the case, the gain or loss is shared according to the way in which the
, merchandise was originally divided.
When the commission merchant accepts the merchandise to sell
on joint account, the terms upon which he becomes responsible for
his share should be known, whether payable as cash, some definite
term of credit, or at average time of sales.
In the following account sales of merchandise sold for A. R.
Eastman, at what time will his net proceeds be due, as cash, and
what will be the amount due A. R. E. on May 14, 1867, discount 6
per cent. ?
Ans. Net proceeds due May 18, 1867.
Due i. R. E.. 82371.36.
rltt
lii m
i
I 4
w ri
I
244
ABITHMETIC.
Account sales of 8745 lbs. bacon, 2976 lbs. choese, and 1245 lbs.
batter, for account and risk of A. R. Eastman, Cbioago, 111.
1867.
Mar. 10
Apr. 20
^ay 14
March 1
« 9
May 14
« ((
Sold to R. White, at 30 days—
4000 lbs. bacon, at IGc...
500 lbs. butter, at 40c..
Sold to J. B. Harris, for cash —
4745 lbs. bacon, at 15|c..
Sold to J. C. Parsons, at 60 days-
2976 lbs. cheese, at 22c..
745 lbs. butter, at 41c..
CHABGES.—
$640 00
200 00
Paid freight in cash »
Paid for labor resalting biieon
Storage :
Commission on $2547.51 at 2^ per ct.
Net proceeds due per average May 21
654 72
305 45
97 40
8 50
5 00
63 69
$840 oa
747 34
960 17
$2547 51
174 59
2372 92
Cleveland, 0., E. and 0. E.
May 14, 1867. R. Felton & Co.
September 4, 1866, ve received from W. Cummings, Cincinnati,
a consignment of 120 brls. of mess pork at $25.00 per bbl., and 742
bushels clover seed at $8.30 per bushel, to be sold on joint account
of consignor and consignees, each one-half; consignees' half subject
to average sales. The same date we cashed his demand draft in favor
of Third National Bank, for $3450. The following is the account
sales. At what time are the net proceeds due as cash, and what
amount in cash will settle our account with W. Cummings on Jan.
1st, 1867, interest 6 per cent ?
. Ans. To last question, $6140.67.
Account sales of 120 brls. mess pork, and 742 bushels clover seed, on
joint account of W. Cuuunings, Cincinnati, and ourselves (each
one-half.)
Sept. 12
« 30
Sold to C K. Sing, for cash —
250 bshls. clover seed, at $8.95
Sold to M. HoUingsworth, at 60 dys. —
25 brls. mess pork, at $32
80 bshls. clover seed, at $9.25...
$2237 50
1540 00
'M.\i
CASH BALANCE.
245
15 lbs.
m.
?47 34
960 IT
'Oot> 18
Novr. 2
« 15
Sept. 4
<( «
Octr. 8
Nov. 15
Sold to T. M. Ames, at 30 days—
200 bshls. clover seed, at $9.25...
10 brls. mess pork, at $32.50...
Sold to T. B. Brown, for cash—
200 bshls. clover seed, at $9.30.
Sold to A. W. Purdy, at 6 months—
85 brls. mess pork, at $33.......
12 bshls. clover seed, at $9.30 .
CHAROES.-
Paid frcii;ht and cartage in cash
Paid insurance on $9500, at 1^ p. et.
Paid for storage, cooperage, and labor
Commission on $10,729.10, at2j^p.c.
Net proceeds of sales.
Your ^ net proceeds, due as per aver.
1850 00
325 00
2805 00
111 60
210 75
118 75
15 00
268 23
2175 00
1860 00
2916 60
10729 10
612 73
10116 37
5058 18
Columbus,
Nov. 15, 1866.
E. and 0. E.
J. O. Denison & Co.
January 2, 1867 — Beceived from D. M. Haiman, Owosso, Mich.
200 brls. pork, invoiced at $18 per brl. ; 3750 lbs. cheese, at 10c.
per lb., and 100 firkins of butter, each 80 lbs., at 16c. per lb., to be
sold on joint account, of shippers f , and ourselves ^ ; our ^ of invoice
due as cash.
January 21 — Cashed I). M. Harman's sight draft in favor of
First National Bank, Cleveland, for $1264.50.
February 14 — Accepted D. M. Harman's one month sight draft
in favor of Thos. L. Elliot, Owosso, for mutual accommodation, for
$864.
February 28 — Cashed D. M. Ilarman's demand draft in favor of
Third National Bank, Detroit, for $1174.75.
Find equated time of sales in the following account ; equated time
of D. M. Harman's account ; also, give the amount that would bal-
ance his account on May 14th, 1867, reckoning interest at the rato
of 7 per cent, per annum.
Ans. Equated time of sales, April 8, 1867.
^ ^ ;; Equated time of Harman's acct., April 7th.
Balance of account on May 14, $2458.56.
'in
! .
iii>i
246
ABITHMETIO.
Account sales of 200 brls. pork, 3750 lbs. clioese, and 100 firkins
butter, on joint account of D. M. Harman §, and ourselves J.
tfan. 16
Sold A. h5. Morrison, on 90 days —
40 brls. pork, at $18.75* per brl.
50 firkins butter, tiO lbs. each, at
24c per lb
$750 00
960 00
1162 50
149 50
$1710 00
Feby. 9
Sold W. E. Glennie, on 2 months—
60 brls. pork, at $19.37^ per brl.
1150 lbs. cheese, at 13o per lb
Sold A. n. Peatman, on his note at 2
months —
50 firkins butter, 80^ lbs. each,
at 24c oer lb
1312 00
« 27
966 00
265 30
1895 lbs. cheese at 14c.
1231 30
Sold 11. D. Wright, for cash—
75 brls. pork, at $19.25 per brl.
Sold A. B. May, on 30 days—
25 brls. pork, at $19.87^ per brl.
705 lbs. cheese, at 15c per lb....
March?
1443 75
" 24
496 88
105 75
602 63
122 75
6 50
12 46
11 25
157 49
6299 68
Jan'y. 2
Paid for freight and cartage by cash...
Paid for cooperage and extra labor by
cash
" 15
Paid Insurance at IJ per cent
Mar. 24
" 24
Charges for storing in storehouse
Com. at 2^ per cent, on $6299.68 is.
•
Net proceeds of sales
' 310 45
5989 23
Your § of N. P. due, as per average,
April 12th
■
t.
•
3992 83
.E.J
md 0. E. E. Geo. (
1
JONKLIN &
Clevel
5 Co.
and, 0..
tt ■
Dec: Ist, 1866--We received from Messrs. Gillespie, Moffatt &
Co., Boston, 27 cases Mackinaw blankets, 340 prs. at $3.90; 2 cases
? ^)
CASH D.ILAKCE.
247
^rkins
9 J.
no 00
312 00
1231 30
1443 75
602 63
6299 68
• 310 45
5989 23
3^2 82
Co.
xnd, 0.
Moffatt &
)0; 2 case*
chintz cotton, 987 yds. at 7^ ; 20 pea. table oil cloth, at $3.70 ; 4
pes. do., at $5.62]^ ; 7 pes. West England broad cloth, 126 yds. at
$3.70 -, 7 bales cotton batts, at $6.20 ; to be sold on joint account and
risk of consignor and consignee, each one-half, our one-half as cash*
Dee. 5th— Wo cashed their demand draft for $1200.
Dec. 17 — Accepted their draft on us at 30 days' sight, for $984.
Jan. 14 — Cashed their draft on demand, for $500.
Sales of merchandise as per account sales annexed.
At what date arc not proceeds due as cash ? A. Feb. 22, '67.
What is cquatea time of 6. & M.'s account ? A. June 10, '64,
What is the cash balance March 24, 1867 ? A. $76.54.
Account sales of 27 cases Mackinaw blankets, 2 cases chintz cotton,
24 pes. table oil cloth, 7 pes. West England broad cloth, and 7
bales cotton batts, on joint account of Gillespie, Mofifatt & Co.,
Boston, and ourselves (each one-half).
i f
Deer. 6 Sold John McDonald & Co., ^ cash, ^
on account 30 days —
13 cases blankets, 260 prs. at
$4.20
1 case chintz cotton, 425 yds. at
9c ...;
7 pes. table oil cloth, at $4.50 ...
<«
u 17
9 Sold K. Chisholm & Co., on note at 6
months —
7 cases blankets, 140 prs. at $4.50
3 pes. W. E. broad, 54 yds., at
$4.20
4 pes. table oil, at $6
** 14|Sold Thomas & Arthurs, for cash —
1 case chintz cotton, 562 yds. at
9o
13 pes. table oil, at $4.40...
3 pes. W. E. broad, 54 yds., at
$4
Sold James A. Dobbie & Co., note at
90 days
2 bales cotton batts, at $7
4 cases Mackinaw blankets, 80
prs., at $6.70
1 pc. W. E. broad, 18 yds., at $5
w
■'i
till
ii
I
948
♦' 28
Beer. 1
1
1
ABXTHlOEnO.
Sold Thiotnu Speneo & Co., | cash,
bolanoo on aoot. at 30 dajs —
3 cases M. blankets, 60 pn., at
•6.76
6 bales cotton batts, at $7.25. ...
OHAROES.
Jany. 1
Paid freight and expenses from depot,
cash
Storage
Com. at 2| per cent, on sales
Net proceeds ^
Your ^ of net proceeds, due as per
average
$94 75
34 48
Milwaukee, Wis.,
January Ist, 1867.
£. and 0. E.
J. 0. Spencer & Co.
ALLIGATION.
Alligation is the method of making calculations regarding the
compounding of articles of different kinds or different values. It is
a Latin word, which means binding to, or binding together.
It is usual to distinguish alligation as being of two kinds, medial
^and alternate,
ALLiaATION MEDIAL.
Alligation medial relates to the average value of articles com-
ponndei, when the actual quantities and rates are given. 7"" "
■ X AMPLE.
A miller mixes three kinds of grain: 10 bushels, at 40 cents a
bushel ; 15 bushels, at 50 cents a bushel ; and 25 bushels, at 70 cents
a bushel; it is required to find the value of the mixture. j^
iUIOiXIOBf.
249
10 bushels, at 40 cents a bushel, will be worth 400 cents.,
^■,,, 16 bushels, at 50 cents a bushel, will be worth 750 cents.,
25 bushels, at 70 cents a bushel, will be worth 1750 cents.,
giving a total of 50 bushels and 2900 cents, and hence the mixtoro
is 2900-f 50=58 cents, the price of the mixture per bushel. Hence
the
BULB.
Find the value of each of tlie articles, and divide the sum of
their values hy the number denoting the sum of the articleSy and tht
quotient will he the price of the mixture,
EXEBO ISES.
1. A farmer mixes 20 bushels of wheat, worth $2.00 per bushel,
with 40 bushels of oats, worth 50 cents per bushel ; what is the
price of one bushel of the mixture ? Ans. $1.
2. A grocer mixes 10 pounds of tea, at 40 cents per pound; 20
pounds, at 45 cents per pound, and 30 pounds, at 50 cents per
pound ; what is a pound of this mixture worth ? Ans. 46$ cents.
3. A liquor merchant mixed together 40 gallons of wine, worth
80 cents a gallon ; 25 gallons of brandy, worth 70 cents a gallon ;
and 15 gallons of wine, worth $1.50 a gallon ; what was a gallon of
this mixture worth ? Ans. 90 cents.
4. A farmer mixed together 30 bushels of • wheat, worth $1 per
bushel; 72 bushels of rye, worth 60 cents per bushel; and 60
bushels of barley, worth 40 cents per bushel ; what was the value of
2^ bushels of the mixture ? Ans. $1.50,
5. A goldsmith mixes together 4 pounds of gold, of 18 carats
fine ; 2 pounds, of 20 carats fine ; 5 pounds, of 16 carats fine ; and
3 pounds, of 22 carats fine ; how many carats fine is one pound of
the mixture ? Ans. 18f .'
ALLIGATION ALTERNATE. ^
Alligation alternate is the method of finding how much of seve*
Tal ingredients, the quantity or value (if which is known, must be
combined to make a compound of a given value.
CASK I.
Given, thQ value of several ingredieuts, to make a oompoand of
uneven Tahio. ._
I
250
^srrBMKno.
.^
,Mi: y ty^^'.m
■ XAMPLB ^ ;';
"■'^ •.'■a ''4
How mnoh Bugor that is worth 6 oents, 10 oents, and 13 oenta
per pound, must bo mixed together, so that the mixture may bo
worth 12 oents per pound ?
SOLUTION.
12 ccntfl.
' 1 lb., at G cents, is a gain of 6 cents. ) Gain.
1 lb., at 10 cents, is a gain of 2 cents. J 8
1 lb., at 13 oents, rs a loss of 1 cent.
1 lbs. more, at 13 cents, is a loss of.
Loss.
1
7
Gain 8 Loss 8
It is evident, in forming a mixture of sugar worth 6, 10 and 131
oents per pound so as to bo worth 12 cents, that tho gains obtained^
in putting in sugar of has value than tho average price must exactly
balance the losses sustained in putting in sugrxr of (^rca^er valuo than
the average price. Hence in our example, sugar that is worth 6
cents per pound when put in the mixture will sell for 12, thereby
giving a gain of 6 cents on every pound of this sugar put in tho
mixture. So also sugar that is worth 10 cents pur pound, when ioi
the mixture will bring 12, so that a gain of 2 cents is obtained on
every pound of this sugar used in the compound. Again, sugar that
is worth 13 cents per pound, on being put into the mixture will cell
for only 12 cents, consequently a loss of 1 cent is sustained on every
pound of this sugar used in forming the mixture. In this manner
we find that in taking one pound of each of the different qualities of
sugar there is a gain of 8 cents, and a loss of only 1 cent. Now, our
losses must equal our gains, and therefore we have yet to lose 7 cents*
and as there is only one quality of sugar in the mixture by which
we can lose, it is plain that we must take as much more sugar at 131
oenta as will make up the loss, and that will, require 7 pounds.
Therefore, to form a mixture of sugar worth 6, 10 and 13 cents per
pound, so as to be worth 12 cents per pound, we will require 1
pound at 6 cents, 1 pound at 10 cents, and 1 pound at the 13
cents-f-7 pounds of tho same, which must be ^taken to make the loss
equal to the gain. .,..:.,; ^ i»^.
By making a mixture of any number of times these answers, it
will be observed, that the compound will be^oorrectly fortned. Hence
we con readily perceive that any number of answers may be obtained
ALLIGATION ALTERNATE.
251
to all ezoroises of this kind,
tho Mowing < .;
From whnt has boea said wo dodttoo
RULE.
Find how much i$ gained or lost by taking one of each kind of
the proposed ingredients. Then take one or more of the ingredient^
or »uch parts of them as will make the gains and losses equal.
KXEROISES.
1. A grocer wiahea to mix together tea worth 80 cents, $1.20,
$1.80 und $2.40 per pound, so as to make a mixture worth $1.C0
per pound ; how many pou ids of each sort must ho take ?
Ans. 1 lb. at 80 cents; 1 lb. at $1.20; 2 lbs. at $1.80, and 1 lb.
at $2.40.
2. How muoli corn, at 42 cents, GO cents, 67 cents, and 78 cents
per bushel, must bo mixed together that tho compound may bo worth
64 cents per bushel ? .
Ads. 1 bush, at 42 cts. ; 1 bush, at CO cts. ; 4 bush, at 67 cts. ;
and 1 bush, at 78 cts.
3. It is required to mix wine, worth 60 cents, 80 cents, and
$1.20 per gallon, with water, that tho mixture may be worth 75 cts.
per gallon ; how much of each sort must be taken ?
Ans. 1 gal. of water ; 1 gal. of wine at 60 cts. ; 9 gal. at 80 cts.;
and 1 gal at $1.20.
4. In what proportion must grain, valued at 50 cents, 56 cents,
62 cents, and 75 cents per bushel, be mixed together, that the com-
pound may be 62 cents per bushel ?
Give, at least three answers, and prove tho work to be correct.
5. A produce dealer mixed together corn, wortfi 75 cents per
bushel ; oats, worth 40 cents per bushel ; rye, worth 65 cents per
bushel, and wheat, worth $1 per bushel, so that the mixture waB
worth 80 cents per bushel ; what quantity of each did he take ?
Givo four answers, and prove the work to be correctly done in^
each case.
CASE II.
When one or'more of the ingredients are limited in quantity, to
find the other ingredients.
BXAMFLS.
"■'I
I!'
! ^ How much barley, at 40 cents ; oats, at 30 cents, and ooni, at 60
jii
,S5S ARrrBMBTIO.
«wti per bttiliel, must bo mixed with 20 bnehels of rye. At 65 eflnte
per bushel, so that the mixture may be worth 60 ceuts per bmlial tli
SOLUTION.
Buih. Cents. ^ Gain. Loai.
1 at 40, gives 20 ...
1 at 30, givos 30
1 at 60, gives 00 .00
20 at 85, gives 6.00
.60 6.00
Oat 40, gives 1.80
9 at 30, gives 2.70 ... v
$5.00 $5.00
By taking 1 bushel of barley, at 40 cents, 1 bushel of oats at 30
oents, and 1 boabcl of com at 60, in connection with 20 bushels of
rye at 85 cents per bushel, wo observe that our gains amount to 60
oents and our losses to $5.00. Now, to make the gains equal the
losses, wo have to tako 9 bushels more at 40 cents, and 9 bushels
more at 30 oents. This gives us for the answer 1 bushol-f 9=10
bushels of barley, 1 bushel-{-9=10 bushels of oats, and 1 bushel of
corn. From tl- we deduce the -^
RULB.
JFind how much is gained or loit, hy taking one of each of the
proposed ingredients, in connection with the ingredient which is
limitedf and if the gain and loss he not equal, take such of the pro-
posed ingredients, or such parts of them,, as will make the gain and
loss equal. ^ . > i < .
EXERCIB18.
6. How much gold, of 16 and 18 carats fine, must be mind
with 90 ounces, of 22 carats fine, that the compound may be 20
carats fine ?
Ans. 41 ounces of 16 carats fine, and 8 of 18 carats fine.
7. A grocer mixes teas worth $1.20, $1, and 60 cents per pound,
wUh 20 pounds, at 40 cents per jraUnd ; how much of each sort must
he take to m^ke the composition worth 80 cents per pound? '^ ■■>
8. Bow much barley, at 60 cents per bushel, and at 60«cent8
per bushel, must be mixed with tra bsshels pf itease, worth 80 oenta
ALUOATION ALTKBNATB. 268
P«r Imihel, and 6 bushola of ryo, worth 85 cents per bnshel, to make
A mizturo worth 75 oentM por bushol ?
Am. !t buBholfl, at 50 oontH ; 2^ bushels, at 60 oonts.
9. How many poundH of Hugar, at 8, 14, and 13 oonts pe
pound, must bo mixed with 3 poundn, worth 0\ oontH per pound ; 4
pounds, worth 10^ cents per pound; and (i pounds, worth 13j^ cents
per pound, so thut tho mixture may bo worth 12^ cents por pound ?
Ans. 1 lb., at 8 ots. ; 9 lbs., at 14 ots. ; and 5^ Iks., at 13 eta
0A8E III.
To find tho quantity of each ingredient, when tho sum of tho
iogrodients and the average price are given.
■'''' KXAMPLE.
A grocer has sugar worth 8, 10, 12 and 14 oonts per pound, and
ho wishes to make a mixture of 240 pounds, worth 11 oonts ptK
pound ; how much 5f each sort mu,st he take ?
^ \ 8 L u T I N .
Gala. Loss.
1 lb., at 8 cents, gives 3 •
1 lb., at 10 cents, gives 1 •
\ 1 lb., at 12 cents, gives 1
1 lb., at 14 cents, gives 3
41bs, ^ 4 4
240 lb8.-i-4=:60 lbs. of each sort
By taking 60 lbs. of each sort we have Che required quantity,
and it will be observed that the gains will exactly balance the losses,
oonsequently tho work is correct. Hence the
RULE.
J,
Find the least quantity of each ingredient by Case I., Then
divide the given amount by the sum of the ingredients already /ound^
and multiply the quotient by (lie quantities found for the propor-
tional quantities. ., ; .
10. What quantity of three different kinds of raisins, worth 15
cents, 18 cents, and 25 cents per pound, must be mixed together to
tHk a box containing 680 lbs., and to be worth 20 cents per pound ?
Ans. 200 lbs., at 15 cents ; 200 lbs., at 18 cents ; and 280 lb0.»
at 25cents. , ^ -.
ii.
254
ARITHMETIC.
11. How much sugar, at 6 cents, 8 cents, 10 cents, and 12 cents
per pound, must be mixed together, 80 as to form a compound of 200
pounds, worth 9 cents per pound ? r. Ans. 50 lbs. of each.
12. How much water must be mixed with wine, worth 80 cents
per gallon, so as to fill a vessel of 90 gallons, which may be offered
•at 50 cents per gallon ? Ans. 5G^ gals, wine, and o3|; gals, water.
13. A wine merchant has wines worth $1, $1.25, $1.50, $1.75,and
$2. per gallon, and ho wishes to form a compound to fill a 150 gallon-
cask that will sell at $ I. -10 per gallon; how many gallons of each
i^ort must ho take ? Ans. 54 of SI, and 24 of each of the others.
14. A grocer has sugars worth 8 cents, 10 cents 12 cents, and 20
cents per pound ; with these he wishes to fill a hogshead that would
contain 200 pounds.; how much of each kind must he take, so that
the mixluro may be worth 1 5 cents per polmd ?
Ans. 33] lbs. of 8, 10, and 12 cents, and 100 lbs. of 20 cents,
15. A grocer requires to mix 240 pounds fi? different kinds of
raisins, worth 8 cents, 12 cents,' 18 cents, and 24 cents per lb., so
that the mixture shall be worth 10 cents per pound ; how much must
be taken of each kind ?
Aus. 192 lbs. of 8 cents, and 16 lbs. of each of the other kinds,
MONEY.; ITS NATURE AND VALUE.
Money is the medium through which the incomes of the different
members of the community arc distributed to them, and the measure
by which they estimate their possessions.
The precious metals have, amongst almost all nations, been the
standard of value I'rom the earliest lime. Except in the very rudest
state of society, men have felt the necessity of having some article,
of more or less intrinsic value, Jthat can at any time be exchanged
for different commodities. No other substances were so suitable for
this purpose as gold and silver. They arc easily divisible, portable,
and among the least imperishable of all substances. The work of
dividing the precious metals, and marking or coining them, is
generally undertaken by the Government of the country.
Money is a commodity, and its value is determined, like that of
Other commodities, by demand • and supply, and cost of production.
When there is a large supply of money it becomes cheap ; in other
words, more of it is required to purchase other articles. If all tho
quad
islij
Willi
silvel
currj
issud
will
papel
pareq
MONEY: ITS NATUBE AND VALUE.
255
money in circulation wore doubled, prices would be doubled. The
usefulness of money depends a great deal upon the rapidity of its
circulation. A ten-dollar bill that changes hands ton times in a
month, purchasas, during that time, a hundred dollars' worth of
goods. A small amount of money, kept in rapid circulation, does
the same work aa a far larger sum used more gradually. Thcrefr .-< ,
whatever may bo the quantity of money in a country, only that pan
of it will ciFect prices which goes int9 circulation, and Is act slly
exchanged for goods.
Money hoarded, or kept in reserve by individuals, does not act
upon prices. An increase in the circulating medium, conforma^^o
in duration and extent to a temporary activity in business, does not
raise prices, it merely prevents the fall that would otherwise enauo
from its temporary scarcity.
. PAPER CURRENCY.
Paper Currency may be of two kinds — convertible and incon-
vertible. When it is issued to represent gold, and can at any timo
be exchanged for gold, it is called convertible. When it is issued by
the sovereign power in a State, and is made to pass for money, by
merely calling it money, and from the fact that it is received in pay-
ment of taxes, and made a legal tender, it is known as an inconver-
tible currency. Nothing more is needful to make a person accept
anytliing as money, than the persuasion that it will be taken from
him on the «ame terms by others. That alone would ensure its
currency, but would '.lot regulate its value. This evidently cannot
depend, as in the cade of gold and silver, upon the cost of production,
for that is very trifling. It depends, then, upon the supply or tho
quantity in circulation. While the issue of inconvertible currency
is limited to something under the amount of bullion in circulation, it
will on the whole maintain a par value. But as soon as gold and
silver are driven out of circulation by the flood of inconvertible
currency, prices begin to rise, and got higher with every additional
issue. Among other commodities the price of gold and silver articles
will rise, and the coinage will rise in value as mere bullion. Tho
paper currency will then become proportion ably depreciated, as com-
pared with the metallic Qurrcncy of other countries. It would be
■I
256
ABirSBCETIC.
quite impossible for these results to follow the issue of convertible
paper for which gold could at any time be obtained.
All variations in the value of the ciroulating medium are mis-
chievous ; they disturb existing contracts and expectations, and the
liability to such disturbing influences renders every pecuniary
engagement of long date entirely precarious.
A convertible paper currency is, in many respects, beneficial.
It is a nA>re convenient medium of circulation. It ib clearly a gain
to the issuers, who, until the notes are returned for payihent, obtain
the use of them as if they were a real capital, and that, without any
loss to the community.
i*
due
is a
eannc
persoj
make I
that
ehan^
from
have
very
mcrcl
bougl
BXOBANOI.
26T
: f
EXCHANaK
It often becomes necessary to send money from one town oi
country to another for various purposes, generally in payment for
goods. The usual mode of making and receiving payments between
distant places is by bills of exchange. A merchant in Liverpool,
whom we shall call A. B., has recdlved a consignment of flour from
C. D., of Chic. go; and another man, E. F., in Liverpool, has
Bhipped a quantity of cloth, in value equal to the flour, to 6. H. in
Chicago. There arises, in this transaction, an indebtedness to Chi<
cage for the flour, as well as an iudebtedness from Chicago for the
doth. It is evidently unnecessary that A. B,,-in Liverpool, should
Bcnd money to C. D. in Chicago, and that G. II., in Chicago,
should send an equal sum to E. F. in Liverpool. The one debt
may be applied in paymo'-t of tht other, and by this plan the expense
and risk attending the double transmission of the money may bo
saved. C. B. draws on A. B. for the amount which he owes to him;
and G. H. having an equal amount to pay in Liverpool, buys this
bill from C. D., and sends it to E. F., who, at the maturity of the
bill, presents it to A. B. for payment. In this way the debt due
from Chicago to Liverpool, and the debt due from Liverpool to
Chicago are both paid without any coin passing from one place to
the other, ,' : •
An iurangement of this kind can always be made when the debts
due between the different places are equal in amount. But if there
is a greater sum due from one place than from the other, the debts
cannot be simply written off againut one another. Indeed, when a
person desires to make a remittance to a foreign country, he does not
make a personal search for some one who has money to receive from
that country, and ask him for a bill of exchange. There arc ex-
change brokers and bankers whose business this is. They buy bills
from those who have money to receive, and sell bills to those who
have money to pay. A person going to a broker to buy a bill may
very likely receive one that has been bought the same day from a
merchant. If the broker has not on hand any exchange that he has
bought, he will often give a bill on his own foreign correspondent ;
and to place his correspondent in funds to meet it, he will remit to
him all the exchange which he has bought and not re-sold. .
i.
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258
AiiniQiBTXi
When broken find that they are asked for more bilk than are
offered to them, they do not absolutely refuse to give them. To
enable their oorrcspondents to meet the bills at maturity, as they
have no exchange to send, they have to remit funds in gold and
silver. There are the expenses of freight and insurance upon the
specie, besides the occupation of a certain amount of capital involved
in this ; and an increased price, or premium, is charged upon the
exchange to cover all. •
The reverse of this happens when brokers find that more bills
are offered to them than they can sell or find use for. Exchange on
the foreign country then falls to a discount, and can be purchased
at a lower rate by those who require to make payments.
There are other influences that disturb the exchange between
different countries. Expectations of receiving large payments from
a foreign country will have one effect, and the fear of having to
make larger payments will have the opposite effect.
AMEBICAN EXCHANQE
Exchange between Canada and the United States, especially the
northern, is a matter of every day occurrence on account of the
proximity of the two countries, and the incessant intercourse between
them, both of a social and commercial character. The exigencies of
the Northern States arising from the late war, compelled them to
issue, to an enormous extent, an inconvertible paper currency, known by
the name of " Greenbacks." As the value of these depended mainly
on the stability of the government and the issue of the war, public con-
fidence wavered, and in consequence, the value of this issue sunk
materially. This caused a gradual rise in the value of gold until it
reached the enormous premium of nearly two hundred per cent., or
• quotation of nearly three hundred per cent., that is, it took nearly
three hundred dollars in Greenbacks to purchase one hundred dollars
in gold. It is to be hoped and expected, however, that as peace is
nom restored, matters will soon find their former level.
'^ It has been deemed essential that this should be distinctly ez>
plained, as it has brought about a necessity for a constant calculation
AMEBIOAN EXCHANGE.
259
of the relative values of gold and greenbacks, and has generated an
extensive business in that species of exchange.
When the term " American currency" is used in the following
exercises it is understood to bo Greenbacks.
A s E I .
To find the value of $1, American currency, when gold is at a
premium.
EXAMPLE.
When gold is quoted at 140, or 40 per cent, premium, what is
the value of $1, American currency ?
SOLUTION.
Since gold is at a premium of 40 per cent., it requires 140 cdhts
of American funds to equal in value $1, or 100 cents in gold. Hence
the value of $1, American money, will be represented by the number
of times 140 is contained in 100, which is .71| or 71| cents.
Hence to find the value of $1 of any depreciated currency reckoned
in dollars and cents, we deduce the following
RULE.
Divide 100 cents hy 100 ^Zms the rate of premium, on gold, and
the quotient will he the value o/ $1.
Subtract this from $1, and the remainder will be the rate of
discount on the given currency.
CASE II.
To find the value of any given sum of American currency when
gold is at a premium. %
EXAMPLES.
What is the value of $280, American money, when gold is quoted
at 140, or 40 per cent, premium ?
SOLUTION.
We find by Case I. the value of $1 to be 71 1 cents. Now, it is
evident that if 71 f cents be the value of $1, the value of $280 will
be 280 times 71 f cents, which is $200, or $280H-l.40=28000-i-
140=$200. Hence we have the following ''^^ '"' "^ '^ "'
4«ii
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260
abuhmeiio.
r
\
RULE.
Multiply the value of $1 2iy the number denoting the given
amount of American money, and the product will he the gold value;
or, . '
Divide the given sum of American money hy 100 (the number of
cents in $1,) plus the premium, and the quotient will he the value in
gold.
CASE III.
To find tbo premium on gold when American money is quoted
at a certain rate per cent, discount.
EXAStPLE. «
When tbe discount on American money is 40 per cent., what is
the premium on gold ?
*' * SOLUTION.
If American money is at a discount of 40 per cent., the discount
on $1 would be 40 cents, and consequently the value of $1 would bo
equal to $1.00 — 40 cents, equal to 60 cents. Now, if 60 cents in
gold be worth $1 in American currency, $1 or 100 cents in gold
would bo worth 100 times g'g of $1, which is $1.'^6§, frc aa which if
we subtract $1, the remainder will bo the premium. Iherefore, if
American currency be at a discount of 40 per cent., the premium on
gold would be 66§ per cent. Hence we deduce the following
BULE.
Divide 100 cents by tJie number denoting the gold value of $1,
American currency, and the quotient will he the value, in American
currency, of ^1 in gold, from which subtract $1, and the remainder
will he the premium. ^ ' > '
CASE IV. • - , 'y-,y '■'■'
To find the value in American currency of any given amount of
gold. ^ ;.
EXAMPLE.,
What is the value of $200 of gold, in American currency, gold
( being quoted at 150 ?
SOLUTION.
\ When gold is quoted at 150, it requires 150 cents, in American
currency, to equal in value $1 in gold. Now, if $1 in gold bo worth
91.50 in American currency, $200 will be worth 200 times $1.50,
which is $300. Hence the <
AMERICAN EXCHAKOE
261
<
BULE.
Multiply ilie value of %\ hy the number denoting the amount of
gold to he changed, and theproditct will he the value in American
currency; or
To the given turn add the premium on itself at tJie given rate,
and the result will he the value in American currency.
EXERCISES. ■
1 If American currency ia at a discou^" > of 00 per cent, what ia
the value of $450 ? Ans. $225.
2. The quotation of gold is 140, what is the discount on Ameri*
can currency? Ans. 28 ij percent.
•3. A person exchanged $750, American money, at a discount of
35 per cent, for gold ; how much did he receive ? Ang. $487.50.
4. Purchased a draft on Montreal, Canada East, for $1500 at a
premium of 64J^ per cent. ; what did it cost rac ? Ans. $2473.12.
5. If Amfcrican currency is quoted at 33i^ per cent, discount ;
what is the premium on gold ? Ans. 50 per cent.
6. Purchased a suit of clothes in Toronto, Canada West, for $35,
but on paying for the same in American funds, the tailor charged ^
me 32 per cent, discount ; how much had I to pay him ?
Ans. $51.47.
7. What would be the difference between the quotations of gold,
if greenbacks were selling at 40 and 60 per cent, discount ? V "^v^/
Ans. 83^ per cent.
8. P. Y. Smith borrowed from C. 11. King, $27 in gold, and
wished to repay him in American currency, at a discount of 3S
per eent. ; how much did it require ? Ans. $43.55.
9. J. E. Pekham bought of Sidney Leonard a horse and cutter
for $315.50, American currency, but only having $200 of this sum, 'V^
be paid the balance in gold, at a premium of C5 per cent. ; how much
did it require ? Ansi $70.
10.^ A cattle drover purchased of a farmer a yoke of oxen valued
at $135 in gold, but paid him $112 in American currency, at a y^
discount of 27 j per cent. ; how much gold did it require to pay the
balance ? Ans. $53.80.
11. W. H. Hounsfield & Co., of Toronto, Canada West, purchased
in New York City, merchandise amounting in value to $4798.40, on
3 months' credit, premium on gold being 79f per cent. At tho
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262 ARITHMETIC.
expiration ol' the three months they purchased a draft on Adams,
KimbuU nnd Mooro, of Now York, for tho amount duo, at a discount
of 57J per cent. ; what was tho gain by exchange ? Ans. $G47.75.
12. A makes an cxchanp;e of a horso for a carriage with B ; tho
horse being valued at $127.50, in gold, and the carriage at S210,
American currency. Gold being at a premium of 05 per cent.;
what was tho diftbrcncc, and by whom payable ?
Ans. B pays A 2.^ cents in gold, or 37 cents in greenbacks.
13. A merchant takes 803 in American silver to a broker, and
wishes to obtain for tho same greenbacks which arc selling at a dis-
count of 30 per cent. The broker takes the silver at 3i per cent,
discount ; what amount of American currency does the merchant
receive ? Ans. $80.85.
14. I bought tho following goods, as per invoice, from John
McDonald & Co., of 3Iontreal, Canada East, on a credit of 3 months :
1120 J yards Canadian Tweed at 95 cents per yard.
2190 " long-wool red flannel at GO *• " «
3400 " " white flannel at 55 " " «
Paid custom house duties, 30 per cent. ; also paid for freight,
$37.40. Gold at time of purchase was at a premium of G3£ per
cent. ; what shall I mark each piece at per yard to make a net gain
of 20 per cent, on first cost ?
« Ans. C. tweed, $2.44 ; red flannel, $1.54 ; white flannel, $1.41.
15. A merchant left Toronto, Canada Wegt, for New York City
to purchase his stock of spring goods, taking with him to defray
expenses $95 in gold. After purchasing his ticket to the Suspension
Bridge for $2.40, he expended tho balance in greenbacks, which
were at a discount of 41^ per cent. When in New York he drew
from this amount $23.85 to ''square" an old account then past due.
On arriving home he found that he still had in greenbacks $16.40,
which he disposed of at a discount of 43f per cent., receiving in
payment American silver at a discount of 3^ per cent., which he passed
,oflf at 2 J per cent, discount for gold. What were his expenses in
gold ; the actual amount in greenbacks paid for expenses, and the
amount of silver received ?
1 Ads. Total expenses in gold, $71.76; expenses in greenbacks,
$118.04 ; silver received, $9.53. '(-;r-i
EXCHANQE WITU OHEAT BRITAIN.
. ■ EXCHANGE "WITH GREAT BRITAIN^; l. ;,.,i.
In Britain money is reckoned by pounds, shillings and )^rAh,
and fractions of a penny, and is called Sterling mon^-y,. the gold
sovereijrn or the pound sterling, consisting of 22 parts gold and 2
alloy, being the standard, and the sliilling, one-twentieth part of tho
pound, a silver coin of 37 parts ailvcr and 3 copper, and the penny,
one-twelfth part of the shilling, a copper coin, tho ingrcdicuts and
size cf which have frequently been altered.
Tho comparative value of the gold sovereign in the United States
previous to the year 1834 was $4.44 .J, but by Act of Congress
passed in tliat year it was made a legal tender at the rate of ^^f^
cents per pennyweight, because the old standard was less than the
intriLsic value and also because the commercial value, though fluc-
tuating, was always considerably higher. Hence, the full weight of
the sovereign being 5 dwts. 3.274 grs., it was made equivalent to
4 dollars and 86<| cents. The increase in the standard value was,
therefore, equal to 9^ per cent, of its nominal value.
The real par of exchange between two countries is that by which
an ounce of gold in one country can be replaced by an ounce of
gold of equal fineness in the oth'^r country.
If the course of exchange at New York on London were 108^
per cent. ] and the par of exchange between England and America
109|per cent., it follows that the exchange is 100 percent, against
England ; but the quoted exchange at New York being for bills at
60 days sight, the interest must be deducted from the above differ-
ence.
The general form for the quotation of exchange with England
is: 108, 108J, 109, 109i &c., which indicates that it is at 8, 8^, 9,
or 9^ per cent, premium on its nominal value.
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What amount of decimal money will be required to purchase a
draft on London for JE648 17s. 6d. ?— exchange 108. ., . -
Tho old par value or itominai value is $4.44J=^^®=^ of $40
H
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264
RITHMETia
by reaucbg to an improper fraction. Now, tio quotation in 108,
or 8 per cent, above *ho nonunal vuluc, vc find the premium on $40
at 8 per cent., which itt $3.20, which added to $40 will gwo $43.20,
and $43.20+9=$4.80 to be rcniilUid for every pound sterling, and
therefore £648 178. 6d. multiplied by 4.80 or 4.8 will be the value
in our mooej. ITs. 6d.=:.875 of a pound, and the operation is ai
follows :
£648,875
4.8
6191000
2595500
$3114.6000
auLB.
Po $40 €tdd the premium on tttel/ at the quoted rate, multipijf
the sum hy the number repre$eixting the amount of iterling moneys
and divide the rctult hy 9, the quotient will be the equivalent of the
aterling money in dollars and ceiits.
Note.— If there be shillings, pence, &c., in the Btcrlicg money, tbej are
to be reduced to the decimal of £1.
To find the value of decimal money in sterling money, at any
given rate above par.
Lot it be required to find the value of $465 in sterling money, at
8 per cent above its nominal value. Here wo have exactly tho
converse of tho last problem, and therefore, havinj; found the value
of £1 sterling, we divide the given sura instead of multiplying ;
thus the premium on $40, at 8 per cent., is $3.20, which added to $40
makes $43.20, and 43.20->-9r^4.80, and $465-r4.80=£96.17.6.
■- - ♦
a u L 1 . .
Divide the given sum hy the nvmber denoting the value of on^
pound sterling at the given rate abo, fpar, and if there be a decimal
remaining reduce it to shillings and pence.
^. • * IXIRC ISBS.
1. When sterling exchange is quoted at 108. what is the valua
of £17 Ai)5.i4.80»
ESCHANQE WTTn GREAT DBITAIN.
266
3. If £1 sterling bo worth $4.84^, what is the promium of cx«
y change botweeti Londua and America. Ana. 9 per cent.
3. At 10 per cent, above itn nominal value, what is the worth of
£60 sterling, in decimal currency ? Ans. $244.44.
4. When sterling ezohangc is quoted at 9\ per cent, premium,
V what is the valuo of $1000 ? Ans. £205 18s. llfd.
5. At 12 per cent, above its nominal value, what will a bill for
£1800 cost in dollars and cents ? Ans. $8960.
6. A merchant sold u bill of exchange on London for £7000, at
an advance of 11 per cent ; what did he rooeive for it more than its
real value? Ans. $466.66| .
7. Bought a bill on London for £1266 15s. at 9^ per cent, pre-
mium ; what nhall I have to pay fur it ? Ans. $6164.85.
8. A merchant sells a bill on London for £4000, at 8 per cent,
above its nominal value, instead of importing specie at an expense of
2 per cent. ; what does ho save ? Ans. $122.66j.
9. A merchant in Kingsto i paid $7300 for a draft of £1500 on
Liverpool at what per cent, of premium was it purchaaod ?
Ans. 9J.
10. Exchange on London can be purchased in Detroit at 108J ;
^ in New York at 108 J. At which place would it be the most advan-
tageous to purchase a bill for £;]58 148. Dd., supposing the N.Y.
broker charges ^ per cent, commission for investing and gold drafts
Dn New York are at a premium of j| per cent.
Ans., Detroit by $6.82.
11. A broker sold a bill of exchange for X2000, on commission,
it 10 per cent, above its nominal value receiving a commission of
,'0 per cent, on the real value, and 5 per cent, ou what he obtained
for the bill above its real value ; what was his commission ?
Ans. $11,955.
12. I owe A. N. McDonald & Co., of Liverpool, $7218, net pro-
ceeds of sales of merchandise c£fected for them, which I am to remit
them in a bill of exchange on London for such amount as will close
the transaction, less \ per cent, on the face of the bill for my com-
mission for investing. Bills on London are at 8 per cent, premium.
Required the amount of the bill, in sterling money, to be remitted.
18 Ans. £1500.
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. , ABrrnMsno. . > . . «
TADLB or rORHION UONBTB.
CiTUM AKD Conmiia
London, Liverpool, &c
DRiraKt.<UTiom or Moobt.
raris, Havre, &o
Amsterdam, Ilaguo, &o
Bremen
Hamburg, Lubeo, &o...
Berlin, Dantzic
Belgium
St. Peteruburg.
Stockholm
Oopenbagcn .
Vienna, Trieste, &c....
Naples
Venice, Milan, &c
Florence, Leghorn, &o.
Genoa, Turin, &e.
Sicily
Portugal
Spain
Oonstantinoplo
British India..
Canton ,
Mexico
Monte Video.
Brazil.
Cuba..
Turkev........
United States.
Kew Brunswick.
Now Scotia
Newfoundland....
12 pence - 1 shilling; 20 sbillingf
-1 pound --^
100 centimes=:-:l ftronc =
lUO cents—l guilder or florin. ..=;
5 swares—l groto; 72 grot08=l
rit clollar =
12 pfennings=:l schilling; 168.=
1 iwtrli banco =
12 pfonningsr^-l groschcn; 30gro.
^=1 thaler =
KM) centime8--l franc =
100 kopecks -^1 ruble =
12 rund8tyok.s:=^16 skillings; 48s.
r-:l rlx dollar gpecie =:^-
10 8killing3=l nmrk ; G m.=l rix
dollar =
Go krcutzer.s;:::=l florin ==
10 graui:r^l oarlino ; 10 car.=l
ducat ....=
100 ccntcsitni=:l lira =
100 eentcsimini^l lira =
100 cente8imi::=l lira =
20 grani=l tare ; 30 tari=l oz.=:
1000 reas=l millrea =
i34 maravedis=l real vellon=
68 maravedis=l real plate. . =
)0 asper8=:l jpta«<er. =
12 piee=l anna; 16 anna8=l
rupee =
100 oandarines=l mace ; 10 m.:=
1 tael. ==
8 rial8=l dollar =
100 oentesimas=l rial ; 8 riab=:l
dollar =
1000 reas=l milrea =
8 reals plate or 20 reals vellon=l
dollar =
100 aspers=l piaster =
10 mills=:=4 cent ; 10 oent8=l
dime ; 10 dimes=l dollar.... =
}4 farthings=l penny ; 12 pence
=1 shilling ; 20 8hilling8=:l
poimd =
Vawi.
$4.8
.18
.40
b1
.78|
.86
.G9
.18§
.76
1.06
1.06
.48i
.80
.16
.16
.181
2.40
1.12
.06
.10
.06
1.48
1.00
.83V»^
.821
1.00
.06
variable.
4.00
ABBrnunoM ov uciiamqe.
267
ARBITRATION OF EXOHANQE.
Arbitration of Exchange is tho method of finding iho rato of
ozohango butwoon two countries through tho intervention of ono or
moro other countries. Tho object of this is to osoertuia what is tho
most advantageous channel through wliich to remit money to a foreign
oountry.
Three things have here to bo considered. First, what is the
most secure channel ; secondly, what is tlie least expensive, and
thirdly, tho comparative value of the currcueies of the different
countries. Regarding the two first considerations no general rulo
can be given, as there must necessarily be a continual fluctuation
arising from political and other causes. We arc therefore compelled
to confine our calculation to tho third, viz., the comparative value of
tho coin current of different countries.
For this purpose we shall investigate a rule, and append tables.
Let us suppose an English merchant in London wishes to remii
money to Paris, and finds that owing to certain international rela-
tions, he can best do it through Hamburg and Amsterdam, and that
the exchange of London on Hamburg is 13^ marcs per pound ster-
ling ; that of Hamburg on Amsterdam, 40 marcs for 3C^ florins, and
that of Amsterdam on Paris, 56| florins for 120 francs, and thus
the question is to fiind tho rate of exchange between London and
Paris.
solution:
We write down the equivalents in ranks, the equivalent of the
first term being placed to the right of it, and the other pairs below
them in a similar order. Henco the first term of any pair will be of
the same kind as the second term of the preceding pair. As tho
answer is to be the equivalent of tho first term, the first term in tho
last rank corresponds to the third term of an analogy, and is there-
fore a multiplier, it must be placed below the second rank. The
•A
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268
ABITHMETia
i
terms being thus arranged, we divide the product of the seoond tank
by that of the first, and the quotient will be the equivalent, a* exhi-
bited below :
£1 stcrling= 13J marcs.
40 marcs = 36|- florins.
56 J florins =120 francs
£1 StR.
As it is most convenient to express the fractions decimally^ wa
Itavc
1 3.fiX3B.2f)Xl00Xl
=25.87^franc3.
1X4 0X3C.76
The foregoing explanations may be Condensed into the form of ft
BULB.
Write (hicn the first term, and its equivalent to the right of it,
<ina the other pairs in the same order, the odd term being placed
vnder the second rank, and then divide the product of the second
rank hy the product of the first, the quotient will he the required
equivalent.
Note.— The Inie principle on whicli tliis operation is founded, iatbat each
j)air consists of the aDtecedcnt aud consequeut which are to each other iu
the ratio ot equality in point op nn'niNSic vaixe, though not in regard to
TBE NUMBERS Bv 'wuicu TUEY AKE EXPREssEP, and therel'oro the required term
and its equivalent must have the same relation to each other, that is, they
■will be an antecedent and a consequent in the ratio of equality as regards
their value, but not aa regards the numbers by which they are expressed.
EXERCISES.
1. If the exchange of London on Paris is 28 irancs per pound
sterling, and that of America on Paris 18 cents per franc ; what is
the rate of exchange of America on London, through Paris?
Ans. $5.04 per £ sterling.
2. If exchange between New York and London is at 8 per cent,
premium, and between London and Paris 25 J^ francs per pound
sterling ; what sum in New York is equal to 7000 IVancs in Paris ?
3. When exchange between Portland and Hamburg is at 34 cents
per mark banco, and between Hamburg and St. Petersburg is 2
marks, 8 scliillings per rub'j ; how much muEt be paid in St. Peters-
burg for a draft on Portland for $650 ?
Ans. 764 rubles, 70} 2 kopecks.
EXOHANOE.
2Ga
4. If a merchant buys a bill in London, drawn on Parrs, at the
rate of 25.87 francs per pound sterling, and if this bill bo sold in
Amiiterdam at 120 francs for 5Gf florins, and the proceeds be in-
vested in a bill on Hamburg, at the rate of 3GJ- florins for 40 marcs ;
what is the rate of cxahange between London and Hamburg, or
what is £1 sterling worth in Hamburg ? Ans. 13.499-fmarc8.
5. A merchant of St. Louis wishes to pay a debt of $5000 in
New York; the direct exchange is It per cent, in favour of New
York, but oo New Orleans it is A- per cent, discount, and between
New Orleans and New York at a J per cent, premium ; how much
would be saved by the circular exchange coinjiarcd with the direct?
Ans. $87.56-
6. A merchant in Detroit wishes to remit to J. B. Gladstone &
Co., of London, £3G00 sterling. Exchange on London, in Detroit,
is at a premium of 10 per cent. Excliangc on London can bo
obtained at New York for per cent, premium. If Detroit bills on
New York are at a discount of ^ per cent., and the merchant remits
a draft to New York, and pays lii.s agent i per cent, for investing it
in bills on London ; what will he gain over the direct exchange ?
Ans. 6123.80.
7. A merchant in London remits to Amsterdam £1000, at the
rate of 18 pence per guilder, directing his correspondent at Amster-
dam to remit the same to I'aris at 2 francs, 10 centimes per guilder,
less -^ pur cent, for his commission ; but the exchange between Ams-
terdam and Paris happened to l)e, at the time the order was received,
at 2 francs, 20 centimes per guilder. The merchant at London,
not apprised of this, drew upon Paris at 25 francs per pound ster-
ling. Lid he gain or lose, and how much per cent. ?
Ans. lG:i|? percent, gain.
reasUHHe
• 2. Bar
MIXED E X E R C I S K S IN EXCHANGE.
ien gold is <jUoted at 150 per cent, premium ; what is tho
^enbaeks are not at a uiseount of 50 per cent. ?
,r gold in London is 77s. Od. per oujiee standard; required,"
the arbitrated rate of exchange produced by its import to this coun-
try for coinage, at the rate of 23 2 ^ grains of fine gold for the eagle
of 10 dollars.
3. What sum in decimal money must I pay for a bill on
London of X76 14s. Id., exchange beings 9-^ per cent, pre-
mium, and the broker's commission for negotiating the bill
being ^ per cent ? Ans. $3.75.
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270
ARITHMETIC.
4. A mcrchiint shipped 2560 barrels of flour to liis agent in
Liverpool, wlio si Id it at £1 8s. Gd. per barrel, and charged 2 per
cent, commission ; what was the net amount of the flour in decimal
money, allowing exchange to be at a premium of 8 per cent. ?
Ans. $17160.19.
5. What is the cost of a 30 days' bill on Montreal, at ^ per cent.
premium, the f#cc of the bill being $1500 ?
Ans. $1507.50.
6. What must be the face of a 60 days' draft on Now Orleans
to yield $1641.75, when sold at a discount ofh per cent. ?
Ans. $1650.
7. What is the cost of a 30 days' bill on Chicago, at f per cent,
premium, and interest off at (5 per cent. ; the face of the bill being
S9256.40 ?* Ans. $9240.20.
8. A merchant paid $14400.12 for a bill on Havre for $79000
francs ; how much was exchange below par ? Ans. 2 per cent.
9. I have in possession the net proceeds of a sale of cotton
amounting to $3765, which my correspondent desires mc to remit to
him in New Orleans ; exchange on New Orleans is at a discount of
2J per cent., and I invest the whole in a draft at that rate, which I
remit to him; what is the face of the draft? Ans. $3861.54.
10. The proceeds of a sale of goods, consigned to me from
Bremen, is $2764.67, on which I am to charge a commission of 10
per cent., and remit the balance to my consignor in such a way as
shall be most advantageous to him. Exchange on Paris can be had
at 92 cents per 5 francs, and in Paris exchange on Bremen is 17
francs to 4 thalers. Exchange on Liverpool can be had a 9 per
cent, premium, and in Liverpool exchange on Bremen is 6 thalers to
the pound sterling. Direct exchange is 80J cents per thaler. Which
course will be the best, allowing i per cent, brokerage to correspon-
dents both in Liverpool and Paris ? Ans. By way of Paris.
11. A, of Buffalo, sent articles to the World's Fair in London,
which were afterwards sold by B, of London, on A's account, net
proceeds £1266 15s. sterling. B was instructed to invest this
amount in bills on New York, and remit to A, which was accordingly
done. B charged J per cent, brokerage on the face of the bills for
iwcsting, and purchased the bills at 7 per cent, discount. Required
*When then.' is interest to bo computed, it must be reckoned ou the face
of tho bill or draft. When other than the value or coat of the bill is to he
found, proceed aa in perceut&ge.
ber,
the
60
EXCHANaB.
271
ihe amount of the bill A must receive in dollars and cents to close
the transaction. Ans. $6037.58 nearly.
12. A merchant in Boston having to remit £434 153, to Liver-
pool, wishes to know which' is the most profitable, to buy a set of
exchange on Liverpool at 10^ per cent, premium, or send it by way
of France ; exchange on tlie latter place being 19f cents per franc,
and exchange on Liverpool can bo bought in France at the rate of 24^
francs per pound sterling, and ho has to pay his correspondent in
France f of 1 per cent, for purchasing the bill on Liverpool.
Ans. By way of France, $15.69.
f^ 13. John DcDonald & Co., of Toronto, Canada AVest, wish to
remit to a creditor in London £1241 15s. 9d. Exchange on
London can be bought in Toronto at 109|, but Exchange on Lon-
don can bo purchased in New York for gold at 108^. In New
York it takes $1.85 greenbacks to equal $1 in gold. The broker in
New York charges |- per cent, on the greenback value for investing.
If Exchange on New York is at 47 per cent, discount, at which
place would il bo the most advantageous to purchase, and how much
gain, and if the remittance be made by way of New York, what
would be the face of the draft ?
Ans. New York by $141.72; face of draft, $11161.21.
14. Find the arbitrated rate of exchange between London and
Amsterdam when the exchange of London on Madrid is 37 pence
for one dollar of plate, and that of Amsterdam on Madrid is 100
florins, 75 cents, for 40 ducats of plate.
15. Hughes Bros. & Co., purchase of E. ChaflFey & Co., a ster-
ling bill at 60 days on Gladstone & Hart, of London, for £3956 lOs.
Thoy remit this bill to James Alder, in London, where it is accepted
by Gladstone & Hart, and falls due on the 20th November, at which
time it is protested causing an expense of £2 19s. Gladstone & Hart
having failed, E. Chaffey & Co.'s agent in London pays James Alder
on the 20th November, £2000 on account. How much must E.
Chaffey & Co., pay to Hughes, Brothers & Co., on the 24th Decem-
ber, to cover the amount still due in London, allowing interest at
the rate of 10 per cent, from November 2Gth, to the maturity of a
60 days' bill at date of 24th December, and ^ of 1 per cent, commis*
sion for their trouble iu negociating a now bill? Ans. $9815.91.
4'
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ABITHMETIO.
INVOLUTION.
of flodli
Involution is tho process of finding a given power of a givea
number.
We luive noted already, under tlie |ica4 of multiplication, that the
product of any number of e(|iial fiiotors is called the second, third,
fourth, &c., power of the number, uoiuirding as the factor is taken
two, throe, four, &o., times. Thus: d^;\/,\i \^ tlie second power of
3; 27=^:3X3X3 is the third power of three; 81=3x3x3X3 is
the fourth power of ;\ These are often written thus : 3'^, 3^, 3"*,
&o. The small figures, 2, 3, 4, indicate the number of factors, and
therefore each is called the index or exponent of the power. I|!epco
to find any required power of a given quantity, wo have the
il. - '
BULE.
Multiply the quantity continually by itself until it has been used
as a factor as often as there are units in the index.
Since the first mnltipUoatjon exhausts two factors, the number of
opot'llllllllH will be iHiH |i Bd iiljili lllii nuinbor of factors.
Itlvulullou, then, is nothing more than multiplication, and for
any power above the second, it is a case of continual multiplication.
For the sake of uniformity the original quantity is called the first
power, and also the root in relation to higher powers. Again, if we
multiply 3X3 by 3X3X3, we have five factors, or 3X3X3X3X3,
but this being an inconvenient form, it is written briefly 3^, the 5
indicating the number of times that 3 is to be repeated as a factor.
Hence, since 3x3 is written 3^, and 3x3x3 is wiicten 3^, it fol-
lows that 3^X3 ^:=3'', and therefore we may multiply quantities so
expressed by adding their indices, and so also we may divide such
• quantities by subtracting the index of the divisor from that of the
dividend. For example 3 3 ~-33 =3 or 3 ' . If we divide 3 » by 3 »
by subtracting the index of the divisor from that of the dividend, wo
obtain 3*^, but 3 or 3* dividiid ijy ;* or 3* is equrl to 1, and there-
fore any quantity with aa ir.c<rK ..ero b oij'.;al to unity.
When high powers arc t > • t fcimi, the operation may be short-
ened in the foUowir 5 manner - L jt i" be required to find the six-
teenth power of 2. We first £nd tho second power of 2, which is 4,
INVOLUTION,
273
then 4X4=16, which is the fourth power, and 16x16=256, th«
eighth power, and 256x256= 65536, the sixteenth power. If wo
wished to find the nineteenth power, we should only have to multiply
the last result by 8, which is thf) third power of 2, for 2' « X23=2» ».
£X£BCISES.
1. Find the second power of 697.
2. What is the third power of 854 ?
3. What is the second power of 4.367 ?
Find the fourth power of 75.
What is the sixth power of 1.12?
What is the second power .7, correct to six places ?
Ans. .060893+
Ans.'485809.
Ans. 622835864.
Ans. 19.070689.
Ans. 31640625.
Ans. 1.9738+.
r.sfii
OQ4 9
7. What is the fifth power of 4 ?
8. Find the third power of .3 to three places ?
9. What is the third power of I ?
10. What is the fifteenth power of 1.04 ?*
11. Raise 1.05 to the thirty-first power.
12. What is the eighth power of | ?
13. What is the second power of 4| ?
14. Expand the expression 6''*.
15. What is the second power of 5J ?
16. What part of 8^ is 2" ?
17. What is the difference between 5^ and 4° ? Ans, 11529.
18. Expand 3^ X2^ Ans. 3888.
19. Express, with a single index, 47^ X47^ X47<' ? Ans. 47* "*.
20. How many acres are in a square lot, each side of which is
135 rods ? Ans. 113 acres, 3 roods, 25 rods.
Ans. 1024.
Ans. .027.
Ans. »^^.
Ans. 1.800943.
Ans. 4.538039.
Ans.
Ans
Ans. .7776.
Ans. -4^=30 J.
Ans. |«
21. What is the sixth power of .1 ?
22. What is the fourth power of .03 ?
23. What is the fifth power of 1.05 ?
24. What is the third power of .001 ?
25. What is the second power of .0044 ?
Ans. .000001.
Ans. .00000081.
Ans. 1.2762815625.
Ans. .000000001.
Ans. .00001936+.
The second power of any number ending with the digit 6 may
be readily found by taking all the figures except the 5, and multi-
• This exercise will be most readily worked by finding the sixteenth
power, and dividing by 1.04. go in the next exercise, find the thirty-second
power, and divide by 1.05. A still more easy mode of working such ques*
ttons will be found under the head of logarithms.
! t «
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2T4'
ABIIBMETIO.
;4ying that by itself, increased by a unit, and annexing 25 to tht
result^ I
Thus, to find the second power of 15, cut off the 5, and 1 remains,
and this increased by 1 gives 2, and 2X1=^2, and 25 annexed will
give 225, the second power of 15. So also,
625
3.5
4
1225
6.5
7
4225
10.5
11
11025
21.5
22
46225
57.5
58
330625
EXERCISES ON THIS METHOD.
26., What is the second power of 135 ?
27. What ia the second power of 205 ?
28. What is the second power of 335 ?
29. What is the second power of 455 ?
30. What is the second power of 585 ?
31. What is the second power of 795 ?
Ans. 18225.
Ans. 42025.
Ans. 112225.
Ans. 207025.
Ans. 342225.
Ans. 632025.
Note.— The square root of an7 quantity ending in 9, must end in either 3
or 7.
No second power can end in 8, 7, 3 or 2.
The second root of any quantity ending in (5, must .^nd in 4 or 6.
The second root of any quantity ending in 5, nyist end also in 5.
The second root of any quantity ending in 4, must end cither in 8 or 3.
The second root of any quantity ending in 1, must end either in 1 or 9.
The second rout of any quantity ending in 0, must also end in 0.
EVOLUTION.
The root of any quantity is a number such that when repeated,
to a factor, the specified miinbor of times, will prodiioo that quantity.
Thu.'5, -5 rcpcutod twice hh u I'aotur gived 5), and Hidrofon^ 3 is oiiliod
the second root nf 9, whilo 3 taken three times as a faotor will givo
27, and therefore 3 is called tlio thini root of 27, Utl4 so also it ia
oallod the /(n/r<^ roo< of 81,
Thcro aro two waya of liulioatlng this. First, by the mark ;/
which is merely a modified form of the letter r, the initial letter of
the English word root, and the Latin word radix (root). When no
marlc i; attached, the Hiniplo quantity nr fitnt rort In liullcated.
When Il» • «ntn»il ivut la meant, the mark y alone is placed before
the quantity, bu'. if the third, fourth, &c., rouia are tu be iudioated.
SECOND OB 8QUABE BOOT. 275
the figures 3, 4, &o., are written in the angular space. Thus:
3=|/9=^ 27=^^81 =^243, &c., &c. The other method is to
write the index as a fraction. Thus, 9^ means the second root of
the first power of 9, i. e. 3. So also, 27^ is the third root of the
first power of 27. In the same manner G4* means the third root of
the second power of 64, or the second power of the third root of 64.
Now the third root of 64 is 4, and the second power of 4 is 16, or
the second power of 64 is 4096, and the third root of 4096 is 16, so
that both views give the same result.
Evolution is the process of finding any required root of a given
quantity.
SECOND OR SQUARE ROOT.
Jf
Extracting the square or second root of any number, is the find-
ing of a number which, when multiplied by itself, will produce that
number.
To find the second root, or square root of any quantity.
By inspecting the tabic of second powers, it will bo found that
the second power of any whole number less than 10, consists of either
one or two digits ; the second power of any number greater than 9,
and less than 100, will in like manner be found to consist of three or
four digits ; and, universally, the second power of any number will
consist of either twice the number tC Mgits, or one less than twice
the number of digits that the root i^^olf consists of Hence, if wo
begin lit Uiii nulls' figure, and mark oil" lliu (flvtiii niimbiM' In porloda
of two figures each, We "hall find that the number of digits contained
in the root will |)o tlio same as tliti number iil' purloilti. H' llio nuni
bet of digits is even, each period will consist of two figures, but if
the number of digits be o4d, the last period to the lol't will consist
of only one figure.
Let it now bo rnf|nirod to find tho Huound root of 141. Wo
know by the rule of involution that 144 is the second pou'er of 12.
Now 12 may bo resolved into one ten and tico units, or 10-|-2, and
10-f 2 multiplied by itself, as in the mi'Tgin, gives 100-|-404-4, ancl
uinm 100 is the second pnwer of 10, and \ the mmn] power of 3,
and 40 is twleo the product of 10 and 2, wc conclude that the second
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276
AIUTHMETIO.
I ti
I' I'
10+2
10+2
100+20
20+4
100+40+4
power of any number thus resolved is equal to the sum of the second :
powers of the parts, plus twioo the product of the
parts. Hence to find the second root of 144, let us
resolve it into the three parts 100+40+4, and wo
find that the second root of the first part is 10, and
since 40 is twice the product of tho parts, 40
divided by twice 10 or 20 will give the other part
2, and 10+2=12, tho second root of 144. We
should find tho same result by resolving 12 into
11+1, or 9+3, or 8+4, or 7+5, or G+6, but
the most convenient modo is to resolve into the
tens and the units. In tho same manner, if it bo required to
find the second root of 1369, we have by resolution 900+420+49,
of which 900 ia the second power of 30, and 30x2=00, and
420-j-60=7, the second part of the root, and 30+7—37, the whole
TOOt.
Again, iel U. Idc required to find the second root of 15129. This
may be resolved as below :
10000 is the second power of 100.
400 is the second power of 20.
9 is the second power of 3.
4000 is twice the product of 20 and 100.
600 is twice the product of 100 and 3.
120 is twice the product < f 20 and 3.
I :
15129 is the sum of all, and hence 1 is the root of the hundreds,
2 the root of the tens, and 3 tho root of the units.
Gietleralizing these investigations, we find that the second power
of a number consisting of units alone is the product of that number
by itself; that the second power of a number conslsliug of (ens and
units is the second power of the tens, plus the secon(^ power of the
units, plus twice the product of the tens and units ; that the seoond
power of a number, consisting of hundreds, tens and units, is thti
fcum of the squares of the hundreds, tho tens, and the units, plus
twice the product of each pair. Now since the ooinjiioniont of tho
full second power, to the sum of tho seoniir] powers of the parts, is
twice the product of the parts, it follows that, when the first figure
of tlie root has been found, it must be doubled before used as a divi-
sor to find the sccuud Lurm, and for tho Hiunu ronson each figure,
whnn fbuiid, must |)p !|Qu|j{e(| to gjyo eorreotjy tho next divisor,
ileuoe the r ■
SECOIO) OR 8QUABE ROOT.
277
RULE.
Beginning at the v,nit»^ figure, mark off the whole line inpmod»
of two figures each ; find the greatest power contained in the left hand
period, and subtract it from that period; to the remainder annex the
next period; for a new dividend, place the figure thus obtained as a
quotient, and its double as a divisor, and find how often that quantity
is contained in the second partial dividend, omitting the last figure ;
annex the figure thus found to both divisor and quotient, multiply
and subtract as in common division, and to tJie remainder annex the
next period; double the last obtained figure of the divisor, and proceed
as before till all the periods arc exha'ustcd,r-if there be a remainder^
annex to it two ciphers, and the figure thence obtained will be a
decimal, as will every figure thereafter obtained.
EXAMPLES.
1. To find the secon(\ root of 7l)744!V
First, comiuiiiuilng with tlio units' figure, wo divide tho line into
perigilit, viz., 49, 74 and 79,— wo then note that tho Rrcatofit squaro
ooulaiued In 79 is 6-4, — this we subtract
from 79, and find 15 remaining, to which
803 we annex the next period 74, and placa
8, the second ropt of 1)4, in the quotient,
and ita double IG as a divisor, and try
how often 16 is contained in 157, which
we find to be 9 times ; placing the 9 in
both divisor and quotient, we multiply
and subtract as in common division, and
find a remainder of 53, to which wo annex
tho last period 49, and proceeding as
before, we find 3, tho last figure of the root, without remainder, and
now wo have tho complete root 893.
2. This operation may bo illustrated aa follows :
To find the second root of 273529.
ir
6
169
1783
797449
64
1574
1521
5349
5349
600
500x2=1000+20, or
1020 ' ,
10004-2X20+3=1043
273529
250000
3129
3129
3=523
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6
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278 AIOTHMETIO.
3. To find tho second root of 153687.
Hero wo obtaia, by tho same process as in the lost example, thu
whole nuiubor 392, with a romaiadur of 23, whioh can produoo only
a fraction.
69
782
78402
784049
392.029+
230000
156804
7319600
7056441
263159
Wo now annex two ciphers,
placing tho decimal point after
the root already found, but oa
the divisor is not contained in
this Dew dividend, wo place a
cipher in both quotient and di-
visor, and annex two ciphers
more to the dividend, and by
continuing this process wo find
the decimal part of tho root., and
the whole root is 392.029+.
Ana. 529.
Ans. 8642.
Ans. 678.
Ans. 28.01785+.
Ans. 41.569219+.
Ans. 25.8069+.
EXEROISBB.
1. What is the second root of 279841 ?
2. What is tho second root of 74684164?
3. What is tho second root of 459684 ?
4. What is tho second root of 785 ?
5. What is tho sojond root of 1728 ?
6. What is the second root of 666 ?
7. What is the second root of 123456789 ?
Ans. 11111.11106+.
8. What is the second root of 5 to three places ? Ans. 2.236.
9. What is the side of a square whose area is 19044 square
feet ? Ans. 138 feet.
10. What is tho length of each side of a square field containing
893025 square rods ? Ans. 945 linear rods.
The second root of a fraction is found by extracting the roots of
its terms, for it=^X| and therefore |/^«=|/|x!=5- So
also, i/i^i=l' Again, since i/j%^q=j%=.09, and .3X.3=.09, the
second root of .09 is .3. This follows from the rules laid down for
the multiplication of decimals.
To find the second root of a deoimai or of a whole number and a
decimal:
SECOND OB SQUARE ROOT.
w
Point off period* of two figurea tack from the decimal point
towards the right and left, adding a cipher, or a repetend, if th§
number of figurea be odd. ,
From what has boon said, it is plain that ovcry period, except
tho first on the left, must, consiat of two dij^its, and every decimal
prcauppoaoH soiuethint^ Koi'^f^ before, for .5 indicates tho h.ilf of somo
unit under consideration, an 1 .5 is cijuivalont to .50, and not to .05,
from which it is cbviour? that the second root of .5 is not tho root of
.05, but of .50, and tlioroforo tho secoJid root of .5 is not .2-|-, af
the beginner would naturally suppose, but ,7-f-, for .2+ is th«
approximate root of .05.
ADDITIONAL EX£HCI8XS.
11. What is the second root of .7 to five places of decimals ?
, . Ans. .83666.
12. Find the second root of .07 to six plaoes.
13. What is the second root of .05 ?
14. What is tho second root of .7 ?
15. Find the second root of .5.
16. What is the second root of .1 ?
17. What is the second root of .1 ?
18. What is tho second root of 1.375 ?
19. What is the second root of .375 ?
20. What is the second root of 6.4 ?
21. Find to four decimal places v'^^j'j.
22. Find |/2 to four decimal places.
23. Find the value of v/3271.4207.
24. Find the second root of .005 to five places. Ans. 07071.
25. Find the square root of 4.372594. Ans. 2.09107-|-.
26. What is the second root of .01 ?
27. What is the second root of .001 ?
28. "What is the square root of .0001 ?
29. What is the second root of .000001 ?
30. What is the second root of 19.0968 ?
Ans. 264575.
Ans. .2236-f .
Ans. .8919-f .
Ans. .74535+.
Ans. .31622774-.
•
Ans. .3.
Ans. 1.1726, &c.*
Ans. 61237, &c.*
Ans. 2.52982-}-.
Ans. 1.774a
Ans. 1.4142.
Ans. 57.1964-.
Ans. .L
Ans. 031624-.
Ans. .01.
Ans. .001.
* The young student would naturally expect that the decimal figures of
^1.376 and y/.^JS would be the same, but it is not so. If it were so, 1/14-
y.375 would be equal to y/ 1.376. That such ia not the case, may be showo
by a very simple example. i/16-{-y'9z=i-^Z—7, but y'Ki-f 9=^/25=5.
Let it be carefully observed, therefore, that the sum of the second roots is not
the same aa the second root qf the awn.
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IMAGE EVALUATION
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Hiotographic
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23 WIST MAIN STRffT
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280
ABUHScEna
OPBRATXON
4
i9.0966
16
4.37 trial
83
4.36 troo.
Trial 867
309
249
/
?oo great by 1
6068
6069
True 866
6068
5196
■
^''
872
Fore trd find the remtunder, 872, is greater than the diyiaor,
866, which seems inconsistent with ordinary rules ; but it mnst b«
observed that we are not seeking an exact root, bat only the closest
possible approximation to it. If the given quantity had been
19.0969, we should have found an exact root 4.37. The remainder
872 being greater than the divisor, shows that the last figure of the
root is too small by j^^g, whereas 7 would be too great by y^^, and
that 866 is not a correct divisor but an approximate one, and that
the true root lies between 4.36 and 4.37. i • v . -
When the root of any quantity can be found exMStly, it is called
% perfect power or rational qtiantiti/, but if the root cannot be found
exactly, the quantity is called irrational or iurd.
A number may be rational in regard to one root, and irrational
in regard to another. Thus, 64 is rational as regards |/64=8,
]^64=4 and |/64=2, but it is irrational regarding any other root
expressed by a whole number. But 64, with the fractional index f,
i. e., 64^, is rational, because it has an even root as already shown.
We may call 64^ either the second power of the third root of 64, or the
third root of the second power. In tho former view, the third root
of 64 is 4, and the second power of 4 is 16, and according to tha
aeoond view, 64^ is 4096, and thd third root of 4096 is 16, the same
aa before. |/81=:3 is rational, and |/81=9 is rational, but 81 is
oot rational regarding any other root ; while y'25 is rational only
regarding the second root, and ^8:=2 only regarding the third root.
The second root of an even square may be readily found by r0>
•olrii^ the number into its prime faotorsi and taking each of thost
THIBD BOOT OB CUBE BOOT.
281
AetoM once,— the product will be the root. Thus, 441 is 3x3X7X7
and oaoh factor taken once is 3x7=21, the second root. Here let
it be observed, that if we used eauh factor twice we should obtain
the tecond prywer, but if we use each factor half the number of times
that it occurs, , we shall have the second root of that power. 64 is
2X2X2X2X2X2=26, t. e., 2 repeated sis timr^a as a factor gives
the number 64, and therefore half the number of those factors will
give the second root of 64, or 2x2x2=8, and 2X2X2 multiplied
by 2X2X2=8X8=64.
As this cannot be considered more than a trial method, though
often expeditious, we would observe that the smallest possible divisois
should be used in every case, and that If the number cannot be thus
xesolved into factors, it has no even root, and must be carried out
into a line of decimals, or those decimals may be reduced to common
firaotions.
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THIBD ROOT OR CTJBE ROOT.
As extracting the second root of any quantity is the finding of
irhat two equal factors will produce that quantity, so extracting the
(third root is the finding of what three equal factors will produce tho
quantity. ^
By inspecting the table of third powers, it will be seen that no
third power has more than three digits for each digit of the first
power, nor fewer than two less than three times the number of
digits. Hence, if the given quantity be marked off in periods of
three digits each, there will be one digit in the first power for each
'period in the third power. The left hand period may contain only
one digit.
From the mode of finding the third power from the first, we esn
deduce, by the converse process, a rule for finding the first power
II
I :
ii
i;
282 ABxrmana
from the third. We know by the mle of involation that the third
power of 25 is 15625. If we resolve 25 into
20+5, and perform the multiplication in that fbnn.
we have 20-J-5
400+100 ^ ; .
100+26
i; iv : 400+200+25=(20+5)a
20+5 ' '
8000+4000+500
2000+1000+125 . • ' "^
8000+6000+1500+125=(20+5) '=15625
Now, 8000 is the third power of 20, and 125 is the third power
•f 6 ; also, 6000 is three times the product of 5, and the second
power of 20, and 1500 is three times the product of 20, and the
second power of 5. Let a represent 20 and b represent 5, then
a»=20» = 3000
3aa6=3X20«x5 = 6000
3 a 6a;=3x20x5« = 1500
. . 6»=5» = 125
15625 *
By using these symbols we obtain the simplest possible method
of extracting the third «oot of any quantity, as exhibited by the
fubjoined scheme :
^ Given quantity. 15625
.. . > a»=20»=:20x20x20 = 8000 ^
>'-■ ^'-' ■'-;■ II. - v.;::'r
' . ^. Bemainder 7625
U; , w 3 oa 6=3X202X5 = 6000
* *^* Bemainder 1625
3 o 62=3X20X5* = 1500
Bemainder 125
6»=5»=5X5X'^ = 125
From this and similar examples we see that a number demoted
bj more than one digit may be resolved into tens and units. Tluu,
25 is 2 tens and 5 units, 123 is 12 tens and 3 unite, and so of all
aniBben. ■ '
of A
of A
THIBD BOOT OB CUBE BOOT.
283
power
seoond
md. tha
len
method
1 by the
denoted
. Thii8,
80 of all
To find the third root of 1860867 : ''
As this number consists of three periods, the root will consist of
three digits, and the first period from the left will give hundrads, thft
second tens, and the third units, and so also in ease of remainder,
each period to the right will give one decimal place, the first being
tenths, the second hundredths, &o., &c.
We may denote the digits by a, b and e.
a=100
a»=1003=
and 30000x20=
3 a 6»=3X 100X400=
1860867(100+20+3—123
1000000
860867 remainder.
600000
Si60867 remainder.
120000
i3=203=
140867 remainder.
8000
Now (a+6)=120 . • . 3 (a+6)2=132867 remainder.
43200, which is contained 3 times+ „ r
in 132867, . • . c=3, and 3 (a+bW
=3X120^X3= 129600
And 3 (a+i) c2=3X 120X9=
And lastly, c»=39=
8267 remainder.
3240
27
27
no remainder*
BULB.
Mark off the given nwnher in periods of thra figures each.
Find the highest third power contained in the left hand periodf
and subtract it from that period. Divide the remainder and next
period by three times the second power of the root thus found, and the
guotient will be the second term of the root.
From the first remainder subtract three times the product of the,
second term, and the square of the first, PLUS three times the product
of the first term, and the square of the second, plus the thirt vower
of the second.
Divide the remainder by three times the square of the *um of Out
first and uoond terms, and the auotient will be the third term.
^ ■ !
< i
li'l
284
xs:m>i
ABITHMZnO.
From the Icut remainder subtract three timee the product of th^
term kut found, and the square of the SUM of the preceding ternu^
PLUS the product of the square of the last found term hy the snic
of the preceding ones, flub the third power of the last found termf
ondsoon.
BXBR0I8X8.
1. What is the third root of 46656 ? Ana. 36.
2. What is the third root of 250047 ? Ana. 63.
3. What is the third root of 20Q0576 ? Ans. 126.
4. What is the third root of 5545233 ? Ans. 177.
6. What is the third root of 10077696 ? Ans. 216.
6. What is tLe third root of 46268279 ? ' Ans. 359.
7. What is the third root of 85766121 ? Ans. 441.
8. What is the third root of 125751501 ? Ans. 501.
9. What is t'le third root of 153990656 ? Ans. 536.
10. What is the third root of 250047000 ? Ans. 630.
11. What is each side of a square box, the solid content of which
Is 69319 ? Ans. 39 inches.
12. Whal is the third root of 926859375 ? Ans. 975.
13. Fmd the third root of 44.6. < Ans. 3.456-f-.
14. What is the third root of 9 ? Ans. 2.08008+.
15. What is the length of each side of a cubic vessel whose solid
DOiitent is 2936.493568 cubic feet 7 Ans. 1432 feet
16. Find the third root of 6. Ans, 1.7099.
17. A store has its length, breadth and height all equal ; it can
hold 185193 cubic feet of goods ; what is each dimension ?
Ans. 57 feet.
18. How many linear inches must each dimennon of a cubic
vessel be which can hold 997002999 cubic inches of water ?
Ans. 999 inches.
> 19. What is the third root of 1 ? Ans. 1.
20. What is the third root of 144 ? , -^ Ans. 6.241483.
The third root of a fraction is found by extracting the third root
of the ter.ms. The result may be expressed either as a common
fraction, or aa a decimal, or the given fraction may be reduced to ft
decimal, and the root extracted under that form.
in
the
are
tiou.
TBntD BOOT OB CUBE BOOT.
28S
S^-,i> V'l J-
Aqs. f =.75.
1XBB0I8I8.
1. What is Ihe third root of || ? '^
,,, V Otherwise:
• g3=.421875. To find tho third root of
this we have .42i875(.70+.05=u:.76
703=:
3x702x5 =73500 ■)
3X70 X52= 5250 f-
5»= 125) =
343000
78875 remainder.
78876
no remainder.
The third root of a mixed quantity Till lo most readily found by
reducing the fractional part to the dec imal form, and applying the
general rule.
It has been already explained that the second root of an even
power may be obtained by dividing the given number by the smallest
possible divisors in succession, and taking half tho number of thosa
divisors as factors. The same principle will apply to any root. If
the given quantity is not an even power, it may yet be found approx-
imately. If we take the number 46656, we notice that as the last
figure is an even number, it is divisible by 2, and by pursuing the
same principle of operation we find six twos as factors, and afterwards
six threes; and, as in the case of the second root, we take each factor
half the number of times it occurs, so in the case of the third root,
we take each factor one-third the number of times it occurs.
The same principle on which the extraction of the second and
third depends may be applied to any root, the line of figures being
divided into periods, consisting of as many figures as there are units
in the index ; for the fourth root, periods of four figures each ; for
the fifth, five, &c., &o. We may remark, however, that these modes
are now superseded by the grand discovery of Logarithmic Computa--
■tiou. '
.'1'
•m
k
•ft
^■)f.l
'■.ill
M
I
'I
■V-
V'
■-•jiti ■ -i '
-Ji,'*»%:
s
1
283 ABiiBMina
, . PROGRESSION.
A «ene« is a saocession of quantities increasing or decreasing by a
Common Difference, or a Common Ratio.
Progression hy a Common Difference forms a series by the addi-
tion or subtraction of the same quantity. Thus 3, 7, 11, 15, 19, 23
forms a series increasing by the constant quantity 4, and 28, 21, 14,
7, forms a series decreasing by the constant quantity 7. Such a
progression is also called an equidifferent series.^
Progression hy a Common Ratio forms a series increasing or
decreasing by multiplying or dividing by the same quantity. Thus,
3, 9, 27, 81, 243, is a series increasing by a constant multiplier 3,
and 64, 32, 16, 8, 4, 2, is a series decreasing by a constant divisor 2.
The quantities forming such a progression are also called Con-
tinual Proportionals,^ because the ratio of 3 to 9 is the same as the
xatio of !) to 27, &c., &c. From this it is plain that in a progression
by ratio, each term is a mean proportional between the two adjacent
ones, and also between any two that are equally distant from it.
The first and last terms are called the Extremas^ and all betweea
them the Means, ,
PROGRESSION BY A COMMON DIPPERENOE.
In a series incKasing or decreasing by a common difference, the
sum of the extremes is always equal to the sum of any two that are
equally distant from them. Thus, in the first example 3-|-23=7-|-
19=11-1-15=26, and in the second 28-1-7=21 -|- 14=35.
^ '' If the number of terms be odd, the sum of the extremes is equal
to twice the middle term. Thus in the series 3, 7, 11, ?5, 19,
3-{-19=2X 11=22, and hence the middle term is half the sum of
the extremes.
" The names ArUhmeiicdl Progression and OtotMtrical ProgrissUm are
often applied to quantities so related, but these terms are altogether inapprc- ,
priate, oa they would indicate that the one kind belonged solely to arithmetic,
and the other solely to geometry, whereas, in reali^, »ach belongs to botb
these branches of science. '
PBOORESSION B7 A COMMON DIFFEBENOE.
287
In treating of progressions by difference or cquidifferent seriefl,
there are five things to be con'iidercd, viz., the first term, the last
term, the common difference, the number of terms, and the sum of
the series. These are so related to each other that wlion any three
of them are known we can find the other two.
Given tho first term of a scries, and the common difference, to
find any other term.
Suppose it is required to find tho seventh term of tho scries 2, 5^
8, &c. Here, as the first term is given, no addition is required to
find it, and therefore six additions of tho common difference will
complete the series on to seven terms. In other words, the common
difference is to be added to the first term as often as there are units
in the number of terms diminished by 1. This gives 7 — 1=6, and
6X3=18, which added to the first term 2 gives 20 for the seventh
term. If we had taken the series on the descending scale, 20, 17,
14, &c., we should have liad to subtract the 18 from tho first term
29 to find the seventh term 2. The term thus found is usually
designated tJie last term, not beoausc the series terminates there, for
it does not, but simply because it is the last term considered in each
question proposed. From these illustrations we derive the
RULE (1.)
Svibtract 1 from the number of terms, and multiply the remainder
by the common difference; then if the series be an increasing ontf
add the result to the first term, and if the series be a decreasing one,
tubtract it.
EXAMPLES.
To find the fifty-fourth term of the increasing series, the first
term of which is 33f, and the common difference 1^. Here
54—1=53, and 53X11=66^, and 66J+33f=100, the fifty-fourth
term. ": ■^■.> ■'■'" ■'■' "-' '"
Given 64 the first term of a decreasing series, and 7 the common
difference, to find the eighth term. Here 8— 1=:7, and 7x7=49,
and 64— 49=:15, the eighth term. , »
EXERCISES.
1. JB'ind the eleventh term uf the decreasing series, the first term
of which is 248f, and the common difference 3^. Ans. 216^.
2. The hundredth term of a decreasing series is 392|, and the
common difference :1s 3|, what is the bst term ? Ans; 36.
I''
'i
II
I
m
s
m
I
!I88
ABmnntria
3. What is the one-thonauidth torm of the ieriea of the odd
figures? AnR. 1999.
4. What is the fiye-handredth term of the Miies of even digits ?
Ads. 1000.
5. What is the sixteenth term of the deoreasing series, 100, 96,
92, &c. ? * Ans. 40.
To find the smn of any eqnidifforent series, when the number
of terms, and either the middle term or the extremes, or two tenns
equidistant from them, are given.
Wo have seen -already that in any such series the sum of the
extremes is equal to the sum of any two terms that are equidistant
from them, and when the number of terms is odd, to twice the mid-
dle term. Hence the middle term, or half the sum of any two terms
equi-distant from the extremes, will be equal to half the sum of those
extremes. Thus, in the series 24-7+12+17+22-1-27+32, we
have ^21=2111=17, the middle term. It is plain, therefore, that
if we take the middle term and half the sum of each equi-distant
pair, the series will be equivalent to 17+17+17+17+17+17+17,
or 7 times 17, which will give 119, the some as would be found by
adding together the original quantities. The same result woj^ld be
arrived at when the number of terms is even, by taking half the sum
of the extremes, or of any two terms that are equi-distant from them.
From these explanations we deduce the
RULE (2.)
Multiply the middle term, or half the «um of the extreme*, or of
any two term* that are equidistant from iher.i, hy the number of
temu, , . . : ■'■'■•'' ^
NoTK.— If the sum of the two terms be an odd number, it is generally
more convenient to multiply b7 the number of terms before dividing by 2. '
EXAMPLES.
■',^h'*>*i^;^ri^y'^^r\f-^ft
Given 23, the middle term of a series of 11 numbers, to find the
earn. Here we have onlyto multiply 23 by 11, and we find at once
the sum of the series to be 253.
Qiven 7 and 73, the extremes of an increasing series of 12 nam-
ben, to find the sum. The sum of the extremes is 80, the half of
wbiol) ia 40.and 40x12=480. the sum required.
PBOOBESSION BT A OOIOCON DIFFEBENCE.
289
.;/lT.l?a
. 'ira
Two equidistant terms of a series, 35 and 70, arc given in a
leries of 20 numbers, to find the sum of tho series. In this eaie, W0
have 96+70=105, and 105x20=2100, and 2100-^2=1050, tho
■am required.
' CXER0I8I8.
1. Find the sum of the series, consisting of 200 terms, the first
term being 1 and the last 200. Ahh 20100.
2. What is the sum of the scries whoso first term is 2, and
twenty-first 62 ? Ans. 672.
'' 3. What is the sum of 14 terms of the scries, the first term of
which is }t and tho last 7 ? Ans. 52^.
4. Find the sum to 10 terms of tho decreasing series, tho fint
term of which is 60 and the ninth 12. Ans. 360.
6. A canvasser was only able to cam $G during the first month
ho was in tho business, but at the eud of two ycar.s was ablo to earn
$98 a month ; how much did ho earn during the two years, supposing
the increase to have been at a constant monthly rate ? An^. $1248.
6. If a man begins on the first of January by saving a cent oo
the first, two on the second, three on tho third, four on the fourth,
&o., &c., how much will he have saved at the end of the year, not
counting the Sabbarhs ? Ans. $490.41.
7. How many strokes docs a clock strike in 13 weeks ?
Ans. 14196.
8. If 8£ is the fourth part of tho middle term of a series of 99
numbers, what is the sum? Ans. 3465.
9. In a series of 17 numbers, 53 and 33 are equidistant from
the extremes ; what is the sum of the series? Ans. 731.
10. In a series of 13 numbers, 33 is the middle term ; what is tho
turn ? Ans. 429.
To find tho number of terms when the extremes and common
difference are given :
As in the rule (1), we found the difference of the cxtrcma bj
multiplying by one leas than the numb&r of terms, and added tho fini
term to the result, so now we reverse the operation and find the
,-»*■ ■ i .
RULE (3.)
1/ivide the difference of the ex*remet bjf the common diffi
and add 1 to the retult. *r:5T^^ •
Ui
''':.>]
n
t-M.j'f'it.
:'?*T
990
AIUTBXETIO.
I
BX AM PLit.
, Given the extremes 7 and 109, and the oommon diffieronee, 8, to
find t!io nuiubor of tcrnia.
In this case wo have 109—7:^102, and 102-i-3=:34, and
344-1—3!), thu number of terms.
K X R R I B E 8 .
I .■
1. Wliat iH the number of terms wboft the extremes ore 35 and
333, und tlio common difference 2 ? Ans. 150.
2. Two C(|uidi8t:int turniH are 31 and 329, and the common dif-
ffercnco 2 ; what is the number of terms ? Ans. 150.
3. Thu fifi^t term of n series is 7, nnd the lost 142, and the com-
mon dilTorcuco \ ; what is the number of terms ? Ans, 541.
4. Tiic first and lust terms of u series are 2^ and 35^, and the
•oniiuon difference J ; what is the number of terms ? Ans. 100.
5. The first term of u series is ^ and last 12J, and the common
diffen.'nce ^ ; what is the number ot terms ? Ans. 25.
Given one extreme, the sum of the series and the number of
tcrmn, to find the other extreme.
This case may be solved by reversing llulo (2), for in it tho
data arc tho same, except that there the second extreme was
given to findihe sum, and now the sum is given, to find the second
extreme. Therefore, as in that rule we multiplied the sum of the
extren.es by tho number of terms and halved the product, so how we
must double, tiie sum of the series and divide by the number of
terms to find the sum of the extremes, and from this subtract the
given extreme, and the remainder will bo the required extreme.
This will illustrate the
r ■»* *■
BULE (4.)
Divide twice the turn of the series hy the numher of terms, and
from the quotient subtract the given extreme, and the remainder will
be the required extreme.
EXAMPLE. ^ . ™
Given 5050, the sum of a series, 1 the first term, and 100 th«
Onmbcr of terms, to find tho other extreme.
Twice the sum is 10100, which, divided by 100, gives 101, and
101 — 1=:^100, the number of terms.
PRoamEssioN by a common difference.
291
■■■< H-./.a
\:^ KXRnOIHES.
1. Given SO, tho greater cxtromo of n decreasing icrios, 442, tha
aum, and 17 tho number of terms, to Gnd the other extreme.
Ans. 2.
2. If 1212G8 bo tho Hura ofu Norics, 8 tho loss cstrcmc, and 142
tho number of terms ; what is tho ^Toutor oxtrcrao ? Ans. 1700.
3. Tho sum of a scries of 7 terms is 105, tho greater extreme is
21, and tho number of tcniis 7 ; what is tho less extreme ? Ans. 9.
4. Tho sum of a scries is 570, tho numlior of terms 24, and the
greater extreme is 47 ; what is tho loss extreme ? Ans. 1.
5. Tho sum of a scries is 30204^, tho greater extreme 312, and
tho number of terms 193 ; what is tho loss extreme ? Ans. 1.
Qiven tho extremes and nuuiber of terms, to find tho common
difibrcnce.
As explained in tho introduction to Rule (1), tho number of
common differeneos must bo one hss than tho number of terms. It if
obvious also, that tho sum of these differences constitutes the differ-
cnco between tho cxtrotiics, and that therefore the sum of tho differ-
ences is tho same as 1 less than tlic numKcr of terms. Therefore the
difforcnco of tho extremes, divided by the sum of the differences, will
give one difference, i. c, tho common difference. This gives us tho
nuLE (5.)
Subtract 1 from the number of terms, and divide the difference
of the extremes by the remainder. *
;',^). .V'.^,.^w:....,,\.< .,. EXAMPLE.
If the extremes of an increasing series be 1 and 47, and the
number of terms 24, wo can find the common difference thus: —
47 — 1=46, and 46-7-23^=^2, tho common difference.
EXERCISES.
1. If the extremes are 2 and 36, and the number of terms 18;
vhat the common difference ? Ans. 2.
2. What is the com non difference if the extremes arc 58 and 3,
and the number of terms 12 ? .y:%^'\^
■« I ■
Ans. 5.
3. In a decreasing series given 1000 the less extreme, and
1793 the greater, and 367 the number of terms, to find the common
difference. Ans. 2^.
11
•n H
I
292
abuhhetio.
4. If 6 and 60 are the extremes in a series of 10 numbers, what
is the common diffurenoe ? Ans. 6.
'^' 5. What is the common difference in a decreasing series of 42
tenr i^he extremes of which are 9 and 50 ? Ans. 1.
There are fifteen other cases, but they may all be deduoei from
the five here given.
We subjoin the ^Jgebraio form as it is more satisfaotoiy and
complete, and also more easy to persons acquai ited with the symbcla
of that science.
Let a bo the first term, d the common difference, n the number
of terms, 5 the sum of the series ; the series will be represented by
a+(a+d)+(o+2d)+(a-}-3<i)+&c., to | o+(n— l)rf. | By in-
apeoting this series it will bo seen that the co-efficient of d is always 1
less than the number of terms, for in the second term where d first
appears, its co-efficient is 1, in the third it is 2, and therefore since
n represents Ihe number of terms, the co-efficient of d in the last
term is n — 1, and that term therefore is a-\-(n—l)d. If the series
Wf^re a decreasing one, that is, one formed by a succession of sub-
iTaotions, the last term would be a — (n — l)d. , v
'. To find the sum of an equidifferent series. . i
y . ■ -'b
We have here 8=a-\-(a-{-d)-{-(a-\-2d)+(a-{-3d)+ &o.........
-\-\ a-\-(n — l)d. I" Since a-{-(n — l)d is the last term, the last
but one will be a-f- (n — 2)d, and the last but two will be a-}- (n — S^d,
&o., &o. But the sum of any number of quantities is the same in
whatever order they may be written. Let us therefore write this
series both as above, and also in reversed order :
«=o+(a+<i)+(a+2d)+(a+3rf)+(a-f4cf)-i-&c ..^. ..
+a-|-(«— 3)d+a+(n— 2)<?-f-a-|-(n— l)c?. ^
«=a-j-(n— l)(£+a-f(»— 2)c?-}-o+(n— 3)i+&c
(a+4(i)+(a-f3d)+(a-{-2rf)+(a-f-d)4-a.
Adding tho two members of the second to those of the first,
we ob t a i n 2»= 1 2a-\-(n—l)d | + 1 2a-{-(nr-'l')d I + 1 2a+
(i^-l)<l 1 + 1 2aH-(n— 1)<« J +&c., to n terms.
PB0OBE8SI0N BT A COIQCON DIFFEBENOE.
293
In tlie last expression all the terms are the same, bat thero are
n terms, and therefore the trhole will be
2«=n -j 2a-j-(n — l)dl and therefore
•=^1 2a+(«-l)i} (1.) , . ,^4
As we have used no single symbol to represent the last term, vo
must now show how it may be obtained from the other data. Wo
iiave seen that the last term is o-\-(n — l)d, which we may donoto
by If which will ^vo us the formula
l=za-{-(n — l)<f.
This formuiti, in the case of a decreasing series, will beoomo
^ : l=a — (n — l)(f, and generally
\ , l=a±ln—l)d.(2.)e '
This formula is the same as Eule (1.) .
We may modify (1) by (2) by substituting I for a+(»— l)dL
■Thus: ,.:.;.,,:^^. ^ ..'v-v..,. ,,-..:-:0->. . , ,V
«=^(a+0.(3.)
This is a convenient form when the last term is given. Using I
for the last term, we have five quantities to consider, viz., a, I, d,
n, «, and, as already stated, any three of these being ^ven. the other
two can be found from (1) and (2.) - "^ f * '
To find (2 when a, Z, n are given :
By (2.) I=a-\-(nr-l)d
J i- . • . I — a=(« — l)df
j _ I— a . \
This finds the common difference, when the extremes and nam-
ber of terms are given, and corresponds to Rule (5.") ^ „ . ,.^~ , ...
If a, n, s are given, we have
J3y(l.) ,=||2a+(n-l)d|.
. 2»=2an+n (n— 1)<?
• . dn (n — ^1)=2 (« — an)
,_2 (»-<in)
• • ^ »(n-l)-
(
... ^] - .,. ..■;,?^%-.,;.4-.
1
¥:...: v:r.| .j.n\-^4^,-. b
, !
'' --t': ■■■r-^i.- -v^ .
■■ . :. - ., - ., ,-..-^t;,i&''.
1
■m, •
■■-<i
t
«
294
ABITHMETIO.
'- If n b to be found from a, d, s, wc hate
1)7(1.) '" »=|{2a-h(n-l)d}
v' , C . • . 2s=2an=r:dn^ — dn
. • . <in24-n(2a— d)=2»
And by Bolving this quadratic equation, we find
..s/^:: .'■;(:•;
d—2a±
l/|8<&+;2a— d)a j
2d
EXAMPLES.
Given a==6, dl=4, n— 20, to find t.
First by (2) l—a-^(n—l)d
:./.<«3vM';.f >v;:i-r.:^-?...;^'.'V =6+(20~l)4 ■,
=82
20 *
tnd hence by (3) »=--.(6+82) j.
=880. •*
k ; X ' V ; . Giyen 0=3, 2=300, n=33, to find d
By (4) cfc=i=?
297
>j^-:-; n
32— »U2-
MIXED 2XER0ISES.
1. Given 70, the less extreme, 10 the common difference, and
44 the number of terms, to find the sum. Ans. 12540.
2. What is tl^e less extreme when the grealer is 579, the common
difference 9, and the sum of the series 18915 ? Ans. 3.
3. What is the series when *=143, d=2, n=ll ?
• Ans. 3, 5, 7, 9, 11, 13, 15, &o.
4. Given 4 and 49, the extremeti, and 6 the number of terms, to
find the series. Ans. 4, 13, 22, 31, 40, &c
„ -li 5. K 120 stones are laid in a straight line, on level ground, at a
K^lar distance of a yard and a quarter, how far must a person
travel to pick them all up one by one and carry them singly and
place them in a heap at the distance of 6 yards from the first, and in
the same line with the stones ? Ans. 10 m. 7 fur., 27 rds., 1^ yds.
• 6. Insert three means between the extremes 117 and 477.
/ Ans. 207, 297, and 387.
The other variations are left as exercises for the student
FBOaBESSIOMS BY RATIO.
595
7. A courier agreed to ride 100 miles on condition of being pai^
1 cent for the first mile, 5 for the second, 9 for the third, and so oqf;
how much did he get per mile on an average, how much foV tjh? la^t
mils and liow much altogether? .|
Ans. $1.99 per mile, $3.97 for the last, and $199 for all.
8. A man performed a journey in 11 days on horsebackT-t^jB
first day ho rode 45 miles, but, his horse getting lame, he was forced
to slacken the puce at a certain rate per day, so that ct the last day
he made only five miles ; what was the length of the journey, m^d iat
what rate did h« slacken his speed ?
Ans. The journey was 275 miles, and the slackening of speed 4
m. per day.
9. Find the series of which 72 is the sum, 17 the first term, and
number of terms 6. ' Ans. 17, 15, 13, 11, 9, 7.
10. The Venetian clocks strike the hours for the whole day ;
how many strokes will one of these strike in a year. Ans. 109500.
11. An Eastern monarch being threatened with invasion, offered
his commander-in-chief a reward equivalent to a mill for the first
soldier he would enlist within a month, two for the second, three for
the third, and so on ; the officer enlisted 999,999 men ; what was
his reward equal to in our money. Ans. $499,999,500.
12. One hundred sailors wore drawn up in line at a distance from
each other of 2 yards, including the breadth of tho body — the pay-
master, seated a distance of two yards from the fir«t, sent a lieutenant
to hand to the first a sum of prize money, then back again to the
Becondj and so on to each sugly ; how far had the lieutenant to walk?
Ans. 11 miles, 3 fur., 32 rods, 4 yds.
. 1 ■■:•
PROGRESSIONS BY RATIO.
i-1 ■
There are in progression by ratio, as in progression by difference,
the same five quantities to be considered, except that in place of a com-
mon difference we have a common ratio ; that is, instead of increase
or decrease by addition and subtraction, we have increase or decrease
hy multiplication «r division. If any three of these are known the
other two can be found.
We have noticed already that if any quantity, 2, bo multiplied by
itself, the product, 4, is called the square, or second power of that
206
ABIFHMETIO.
quantity ; if this bo again multipUed by 2, tho prodnot, 8, is ealled
the cube, or third power of that quantity ; if this again be multi-
pUed by 2, tho product is called the fourth power of that quan-
tity, and 80 on to the fifth, sixth, &o., powers. To show the
short mode of indicating this, let us take 3x3x3x3x3=243.
For brevity this is written 3^, which means that there are 5 factors,
all 3, to be continually multiplied together, and 5 is called the index,
because it indicates the number of equal factors.
Given the first term and the common ratio to find the last pro-
posed term.
Let it be required to find the sixtn term of tho increasing series,
of which the first term is 3 and the ratio 4.
This may obviously be found by successive multiplications of tho
first term, 3, by the ratio, 4, — thus : — . , . ,,^. ,
, „ ,• , . 3=l8t term. ,. • „ '■-' :.'^-Jv
»X4= 12=2ndterm. V ^ .
12X4= 48=3rdterm. ' ^r
. 48X4= 192=4th term. ■ .:.■ ''^^■;ir.,':^/^■'
...■'■■':^ 192X4= 768=5th term. ' ' : ' "^^
f V fc* ^ 768X4=3072=6th term. <: -
The series, therefore, is 3, 12, 48, 192, 768, 3072. From this, it
is plain, that as to find the last of 6 terms, only 5 multiplications bf
the first are required, in all cases the number of multiplications will
be one less than the number of terms. But to multiply five times
by 4 is the same as to multiply by 1024, the fifth power of 4, for
4X4X4X4X4=1024, and 1024X3=3072.*
This gives us tho general
RULE (1.)
Multiply the first term by that power of the given ratio which
M a unit less than the number of terms.
If the series be a decreasing one, divide instead of multiplying.
BX AMP LBS. ^
Given in a sf^ries of 12 cumbers, the first term 4 and the ratio 2,
to find the last term.
Since 11 is one less than the number of terms, we find the lltli
'poprer of 4, which is 2048, and this, multiplied by the first term, 4,
glres 8192 for the twelfth term.
^
* For the abbreviated mode see Involution.
PBOGBES8I0NS BT BATIO.
29T
Given the ninth term of a decreasing series, 39366, and the
ratio 3, to find the first term.
As there are 9 terms, we take . the 8th power of the ratio, 3,
which we find to be 6561, and the first term 39366—6561 =6, the
first term. -,..,-■-■'- v-v ^>,v^,--,^; .■■.'-.' .c ^ .:.;,-'■.--'
EXERCISES.
1. What is the ninth term of the increasing series of which 5 is
the first term and 4 the ratio ? Ans. 327680.
2. What is the twelfth term of the increasing series, the first term
of which is 1 and the ratio 3 ? Ans. 177147.
3. In a decreasiDg series the first term is 78732, the ratio 3, and
the number of terms, 10 ; what is the last term ? Ans. 4.
4. What is the 20tk term of an increasing scries, the first of
rhich is 1.06, and also the ratio 1.06 ? Ans. 3.207135.
0. In a decreasing series the first term is 126.2477, the ratio
1.06 ; what is the last of 5 terms ? Ans. 100.
Given the extremes and ratio, to find the sum of the series.
It is not easy to give a direct proof of this rule without the aid
of Algebra, but the following illustration may be found satisfactory,
■and, in some sort, be accounted a proof.
Let it be required to find the sum of a series of continual pro*
portions, of which the first term is 5, the ratio 3, and the number of
terms 4.
Since 3 is the common ratio, we can easily find the terms of the
series by a succession of multiplications. These are —
5+154-45-f 13j>, and the sum is 200
15-^45-{-135-|.405
400 ' - '
Let us now multiply each term by the ratio, 3, and, for oonye-
nience and clearness, place each term of the second line below that
one of the first to which it is equal. Let us now subtract the upper
from the lower line, and we find that there is no remainder, except
the difference of the two extreme quantities, viz., 400. Now, it will
bo seen that this remainder is 'jzaotly double of the sum of the
aeries, 200, and consequently 400 divided by 2, will give the som
200. Also, 405 is the product of the last term by the ratio, and 400
is the diffei'cnoe between that product and the first term, and t|M
divisor, 2, is a unit less than the ratio, 3. Henoe the
20
298
ABITHMETIO.
BULK (2.)
Multtplj/, the last term hy the ratio, from this product suhtrad
the first term, and divide the remainder by the ratio, diminished by
unity, > • .^'r ^«!.,,vr
EXAMPLE.
Given the first tenn of an increasing series, equal 4, the ratio 3,
and the number of terms G, to find the sum of the series.
By the former rule we find the las* term to bo 972. This, mut
tiplied by the ratio, gives 2916, and the first extreme, 4, subtracted
from this, leaves 2912, and this divided by 2, which ial less than tho
ratio, gives 1456, the sum of the series. ' ''j ' ,,
'' "'-'' '■ '' ' ''■■'^■' EXERCISES. '
1. What is the sum of the series, of which the less extreme is 4,
the ratio 3, and the number of terms 10? Ans. 118096.
2. What is the sum of the series, of which 1 is the less extreme,
2 the ratio, and 1 4 the number of terms ? Ans. 16383.
.3. What is the sum of the series, of which the greater cxtremo
is 18.42015, tho less 1, and the ratio 1.06 ? Ans. 308.755983.
4. A cattle dealer offered a farmer 10 sheep, at the rate of a mill
for the first, a cent fok the second, a dime for the third, a dollar for
the fourth, &c., &c. ; in what amount was he '* taken in." supposing
that each sheep was worth $11,111? Ans. ^11111 00.00,
5. What is the sum of six terms of the series, of which tho
ijreater extreme is J and the ratio | ? Ans. 1 735, or IgVVs-
To find the ratio when the extremes and number of terms are
given :
Let it be required to find the ratio when the extremes are 3 and
192, and the number of terms 7. This is effected by simply
reversing the first rule, and Iherefore we divide 192 by 3 and finol
64, and take the 6th root of 64, which is 2, the ratio. Hence the
RULE (3.) "^^
Divide th^ greater extreme hy the less, and find that root o/thf
quotient, the index 0/ which is one less tJum the number of terms,
EXAMPLE.
If the greater extreme is 1024, and^the less 2, and the numbet
of terms 10, we divide 1024 by 2, and find 512, and then by
•ztraoting the ninth root of 512, we find the ratio, 2.
FBOOBESSIONS BY BATIO.
299
EXEROISEB.
1. If the first yearly dividend of a joint stock company bo $1,
and the dividends increase yearly, so as to form a series of continual
proportionals, vrhat will all amount to in 12 years, the last dividend
being $2048, and what will bo the ratio of the increase ?
Ans. ratio, 2 ; sum, $4095.
2. What is the ratio, in the series of which the less extreme is 3
and the greater 98034, and the number of tcriiis 16. Ans. .10GG05.
3. What is the ratio of a series, the extremes of which are 4 iod
324, and the number of terms 5 ? Ans. 3.
4. What is the ratio of a series, the number of terms being 7
and the extremes ?> and 12288 ? Ans. 4.
o. In a series of 23 terms the extremes are 2 and 8388G08 ;
what is the ratio? Ans. 2.
To insert any number.of means between two given extremes :
Find the ratio hy Rule (3), and multiply the first extreme by
thin ratio', and the second tciU be obtained, and divide the last by the
ratio, and the last but one will be obtained ; continue this operation
until the required term or terms be procured.
Note.— A mean proportional is found by taking the sguare root of the yrih
duct of the extremes.
\,..U-Vn:'>'-V'' ■-^■~. !:.■::■ \.''-^ EXAMPLE. '.'u''
'■' Let it be required to insert between the extremes 5 and 1280
three terms, so that the numbers constituting the series shall bo oon«
tinual proportionals.
The number of terms here is 5, and henc^, by Rule (3), we find
the ratio to be 4, and 5 multiplied by this will give the second term,
20, and that again multiplied by 4 will give 80, the third, and that
again multiplied by 4 will give the fourth term, 320, so that the
full series is found to be 5, 20, 80, 320, 1280. The same result
would be found by dividing the greater extreme by 4, and so on
downwards, thus: 1280, 320, 80, 20, 5.
'iili
•«■>• *
EXERCISES
1. Between 5 and 405 insert three terms, which shall make the
irhole a series of continual proportionals Ans. 5, 15, 45, 135, 405.
2. Insert between ^ and 27 four terms to form a scries, and give
Ihe ratio. Ratio, 3 ; series, I, |, 1, 3, 9, 27.
500 ABITHMETIO.
3. What tluroe numbers inserted between 7 and 4376 will form
a lerieK of continual proportionals? Ans. 35, 176, 875.
4. What is the mean proportional between 23 and 8464 ?
Ans. 441.2164+.
5. Find a mean proportional between |l} and |. Ans. {.
ALOKBRAIO FORM.
Let a represent the first term, I the last, r the ratio, n the num-
lier of terms, and t the sum.
Then $=a-{-ar-{-ar"-\-ar^-\'ar*-\-8K oi*-^-!"****"'*
Multiplying the whole equation by r, wo obtain
r$=ar-\-ar^-^ar^'{-ar*-{-ar'^-{-&o a»*~*+or*.
'Bxkt$=a-\-ar-^ar^-{-ar^-{-ar*-{-ar'^-{-&c ai*~*.
Subtracting, we obtain
n — 8==*(r — l)=ar"— a, and therefore
. But we found the last term of the series to bo at^^, calling this
i, we have from (1.) »=^ (2.)
If r is a fraction, r^ and ar^ decrease as n increases, as already
shown under the head of fractions, so that if n become indefinitely
great, at* will become unassignably small, compared with any finite
quantity, and may be reckoned as nothing. In this case (1) will
heaome 9=^z=r~ (3.)
By this formula ^e can find the sum of any infinite series so
closely as to diffu from the actual sum by an amount less than any
assignable quantity. This is called the limitj an expression more
strictly correct than the turn.
From the formula *==^—{i any three of the quantities a, r, Z, t
being ^ven, the fourth can be found.
Let it be required to find the sum of the serira l+i+l+i+
isc., to infinity.
Here a=l and r=J . • . »=1— J=-|- =1 X2=2. Therefore, 2
is the number to which the sum of the series continually approaches,
by the increase of the number of its terms, and is the limit from
whibh it may be made to differ by a quantity less than any assignable
quantity, and is also the limit beyond which it can never pass.
PB00BES8I0NS BT RATIO.
801
Bj adding the first two terms, we find 1-f ^=^=2 — }=1|.
By adding the first three terms, we find ^-f-J— ]=2— J=l^.
*' By adding the first four terms, we find |-f |=Y— 2 — ^=lf.
By adding the first five terms, we find Y-+ A=ii=2 — j'g=
By adding the first six terms, we find ?a+a3=Si=2— g'j=
It will bo observed hero that the difference from 2 is continually
decreasing. The next term would differ from 2 by g'^ , and the next
by j^g, &o., &c. Thus, when the series is carried to infinity, 2 may
be taken as the sum, because it differs from the actual sum by a
quantity less than any assignable quantity.
EXAMPLES.
To find the sum of the first twelve terms of the series l-f-3-f9-f-
27-1-&0.:
Herea=l, r='6, '
And ,=;:t-_3_^^^=::,2m2JjLTz:l_265720.
'* To find the sum of the series 1, — 3, 9, — 27, &o., to twelve terms,
II
■iX-» -I
•lX-177147— I
=—132860.
1
-3-1 ~" , r' . . .
In the case of infinite series, if a is sought, « and r being given,
we have from (3) a— « (1 — r), and if r is sought, a and « being
♦
given, wc have »*=— or 1 — -.
EXERCISES.
1. Find the sum of the series 2, 6, 18, 54, &c., to 8 terms.
Ans. 6560.
2. Find the sum of the infinite series \ — ^+t3 — 3i> Observe
here r= — \. Ans. §.
3. What is the sum of the series 1, ^, |, &o., to infinity ?
*■ Ans. -^.
- 4. Find the sum of the infinite series 1 — %-\-% — ^^-\-ka.
Ans. 3.
5. What is the sum of nine terms of the series 5, 20, 80, &o. ?
Ans. 436905.
6. Find the sum of i/^+l+l/i+^o-) to infinity.
Ans. i/J-^l.
7. What is the limit to which the sum of the infinite series f, ^,
\% \i ^'i continually approaches ? Am. |.
if
I)
802'
ABITHMEnO.
8. What is tho sum often terini of tho series 4, 12, 36, &o. ?
Ana. 118096.
9. Insert three terms between 39 and 3159, so that tho wholo
shall be a series of continual proportionals.
. - , Ans. 117, .351 and 1053.
10. Insert four terms between ^ and 27, so that the wholo shall
form n scries of euntinual proportionals. Ans. ^, 1, 3, 9.
11. The sum of a scries of continual proportionals is 10^, the
first term 3^ ; whut is the ratio ? Ans. §.
12. Tho limit of an infinite scries is 70, the ratio | ; what is the
first term? Ans. 40.
ANNUITIES.
Ti.
The word Annuity originally denoted a sum paid annually^ and
though such payments are often made half-yearly, quarterly, &o., still
tho term is applied, and quite properly, because the calculations nro
made for tho year, at what iimc soever tho disbursements may be
made.
By the term annuities certain is indicated such as have a fixed
time for their commencement and termination. a "' '
By the term annuities contingent is meant annuities, the com-
menceiiient or termination of which depends on some contingent
event, most commonly the death of some individual or individuals.
By tho term annuity in reversion or deferred, is meant that the
person entitled to it cannot enter on the enjoyment of it till after the
lapse of some specified time, or the occurrence of some event, geher-
ally tho death of some person or persons.
An annuity in perpetuity is one that ** lasts for ever," and thore>
fore is a species of hereditary property.
An annuity forborne is one the payments of which have not
been made when due, bt:t have been allowed to accumulate.
By the amount of an annuity is meant the sum that the principal
and compound interest will amount to in a given time.
The present worth of an annuity is the sum to which it would
amount, at compound interest, at the end of a given time, if forborne
jbr that time.
Tables have been constructed showing the present and final
values per unit for different periods, by which the value of any
annuity may be found according to the following
ANNTTTmS.
308
BULKS.
To flrd either tiie amottDt or the present value of ao annoity,—
Multiply the value of the unit, as fmnd in the tablet, by ih$
number denoting the annuity.
If the annuity bo in perpetuity, —
Divide the annuity by the number denoting the »'«/««< 0/ th§
unit for ohe year.
If the annuity bo in reversion, —
Find the value of the unit up to the date of commencement, and
alto to the date of termination, and multiply their difference by the
number denoting the annuity. * ;,. '. .. ,
To find the annuity, the time, rate and present worth being
given. ; ,
Divide the present worth by the worth of the unit.
Tables are appended varyini^ from 20 to 50 vcars.
EXAMPLES.
To find what an annuity of 8400 will amount to in 30 years, at
3^ per eent.
We find by the tables the amount of $1, for 80 years, to be
$51.G22G77, whieh multiplied by 400 gives $20649.07 nearly.
To find the present worth of an annuity of $100 for 45 years, at
3 per cent.
By the table we find 024.518713, and this multiplied by 100
gives- $2451.88.
To find the present worth of a property on lease for ever, yielding
$600, at 3^ per cent.
The rate per unit for one year is .035, and 600 divided by this
gives $17142.86.
To find the present worth of an annuity on a lease in reversion, to
oommenoe at the end of three years and to last for 5, at 3^ per cent.
By the table we find the rate per unit ibr 3 years to be $2.801637,
and for 8 years, the time the lease expires, $6.873956 ; the differ-
ence is $4.072319, which, multiplied by 300, gives $1221.70.
Qiven $207.90, the present worth of an annuity continued for 4
years, at 3 per cent., to find the annuity.
By the tables, the valuo for $1 is $3.717098. and $207.90,
divide] by tiilii, [jivcd ?J5.9U.
'!■:
•1!
804
AMTHlCEnO.
TABLB,
nowivo m amocmt or ak Amnrrrr or otn doluk rcB amkum, niraom
AT COMTOCND LiTnUeflT VOR ANY NVMBBR OF TRAIU NOT KZORRDrNU niTT.
3 per cent. '3) percent.
1
2
3
4
fi
6
7
8
P
10
11
12
13
14
15
16
17
18
10
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
60
1.000 000
2.03U 000
3.090 UOO
4.1 83 627
5.309 136
6.468 410
7.662 462
8.892 336
10.169 106
ll.4fi:JM79
12.807 796
14.192 030
15.617 790
17.086 324
18.598 914
2) 156 881
21.761 688
23.414 435
23.116 868
26.870 374
28.676 486
30.536 780
32.452 884
34.426 470
36.459 264
38.553 042
40.700 634
42.930 923
46.218 850
47.575 416
50.002 678
62.502 759
55.077 841
67.730 177
60.462 082
63.271 944
66.174 223
09.159 449
72.234 233
75.401 260
78.663 298
82.023 196
85.483 892
89.048 409
92.719 861
96.501 457
100.396 501
104.408 896
108JS40 648
112.796 867
4 per cent.
5 per cent.
1.000 000
2.035 000
3.106 J^S
4.214 9 13
6.:i62 466
6.550 162
7.779 408
9.051 687
10.308 496
11.731 393
13.141992
14.601 962
16.113 030
17.676 98(1
19.295 681
20.971 030
22.705 016
24.499 691
26.3.'»7 180
28.279 682
30.269 471
32.328 902
34.460 414
36,666 628
38.949 857
41.313 102
42.759 060
46.290 627
48.910 799
51.622 677
54.429 471
57.334 602
60.341 210
63.453 152
66.674 013
70.007 C03
73.467 869
77.028 895
80.724 906
84.550 278
88.509 637
92.607 371
96.848 629
101.288 831
105.781 673
110.484 031
115.350 973
120.388 297
125.601 846
130.999 910
1.000 000
2.040 000
3.121 600
4.246 464
5.416 323
6.r.:{2 976
7.898 204
9.214 226
10..')82 795
12.006 107
13.486 351
15.02.'^ 805
16.626 838
18.291 911
20.023 588
21.824 531
23.697 512
25.645 413
27.671 229
29.778 079
31.9G9 202
34.247 970
36.617 889
39,082 604
41.615 908
44.311 745
47.084 214
49.967 683
62.966 286
56.084 938
69.328 335
62.701 469
66.209 627
69.857 909
73.652 225
77.698 314
81.702 246
85.970 336
90.409 150
95.026 616
99.826 636
104.819 698
110.012 382
115.412 877
121.029 392
126.870 668
182.946 390
139.263 206
146.83) 734
6 per cent.
1.000 000
2.050 000
3.152 500
4.310 125
6.625 631
6.801 913
8.11.42 008
9.649 109
11.026 564
12.577 893
14.206 787
16.917 127
17.712 983
19.598 6:i2
21.578 664
23.657 492
25.840 366
28.132 385
30.539 004
33.065 954
35.719 252
38.606 214
41.430 475
44.501 999
47.727 099
51.113 454
54.669 126
68.402 583
62.322 712
66.438 848
70.760 790
75.298 829
80.063 771
85.066 959
90.320 307
95.836 323
101.628 139
107.709 546
114.095 023
120.799 774
127.839 763
135.231 751
142.993 339
151.143 006
159.700 156
168.685 164
178.119 422
188.026 393
198.426 663
7 por cent.
162.667 0841209.347 976
1.000 000
2.060 000
3.183 600
4.:.74 616
5.637 093
6.975 319
8.893 838
0.897 468
11.491 316
13.180 795
14.971 64:t
16.869 94)
18.882 138
21,015 060
23.275 970
25.670 528
28.212 880
30.905 653
33.759 992
36.785 691
39.992 727
43.392 290
46.995 828
60.815 577
64.864 51*2
59.156 383
63.705 766
68.528 112
73.639 798
79.058 18(i
84.801 677
90.889 778
97.343 165
104.183 755
111.434 780
119.120 867
127.268 119
135.904 206
145.058 458
154.761 966
165.047 684
175.950 645
187.607 677
199.768 032
212.74.<\ 614
226Ji08 126
241.098 612
266.664 629
272.968 401
290.33fi 905
1.000 000
2.070 000
3.214 900
4.439 943
6.750 739
7.153 291
8.651 021
10.259 803
11.977 C89
13.816 448
1.5.783 599
17.888 451
20.140 643
22.550 488
25.129 022
27.888 054
30.840 217
33.999 033
37.378 965
40.005 492
44.865 177
49.005 739
53.436 141
68.176 671
63.249 030
68.676 470
74.483 823
80.697 691
87.346 629
94.460 786
102.073 041
110.218 164
118.933 426
128.258 766
138.236 878
148.913 460
160.337 400
172.661 020
IFAMO 292
199.635 112
214.609 670
280.632 240
247.776 496
266.120 861
285.749 311
306.751 763
329.224 386
363.270 093
378.999 000
,406ii28 9 2ft
AMJIU1T1JE8.
806
TABLB,
nownio nil nmtm woktr or an akktitt of otni dollau per annum, to
ooNTtNUi ron > vr nvmbbr or tbarh not exchidino wtnx.
1
1
3 per cent.
■
3| per cent
4 per cent,
5 pur cent
6 percent.
7 per cent, g
0.970 «74
0.966 184
0.961 538
0.962 381
0.943 396
0.934 579
1
2
1.013 470
1.899 694
1.8.^6 095
1.859 41U
1.833 393
l.h08 017
2
3
2.H28 611
2.801 637
2.775 091
2.723 248
2.673 012
2.624 314
3
4
3.717 (198
3.673 079
3.629 895
3.545 951
3.4(i5 106
3.387 209
4
6
4.579 707
4.515 052
4.451 822
4.329 477
4.212 364
4.100 195
6
6
5.417 101
5.328 553
6.242 137
5.075 692
4.917 321
4.766 537
6
7
6.-2:i() 2H3
6.114 644
6.002 055
6.786 373
5.582 381
5.389 286
7
8
7 019 092
6.873 95(i
6.732 745
6.463 213
6.203 744
6.971 295
8
7.786 109
7.607 687
7.435 33k
7.107 822
6.801 692
6.515 228
9
10
8.530 203
8.3 1<* 605
8.110 89(1
7.721 735
7.360 087
7.023 577 1
11
9.252 62 1
9.001 551
8.760 477
8.306 414
7.886 875
7.498 669 1
1
12
9 954 004
9.663 334
9.385 074
sMy^ 252
8.383 844
7.942 671 12
13
10.634 955
10.302 738
9.985 648
9.393 573
8.852 68H
8.:ij7 635 13
14
11.2!l« 073
10.920 520
10.563 12:{
9.898 641
9.294 984
8.745 452 14 .
15
11.U..7 935
11.517 411
11.118 387
10.379 658
9.712 249
9.107 898 15
16
12.561 102
12.094 117
11.6.52 29(i
10.837 770
10.105 895
9.446 632 1
6
17
13.166 118
12.651 321
12.165 66*J
11.274 066
10.477 260
9.763 206 1
7
18
13.753 513
13.189 682
12.659 297
11.689 587
10.827 603
10.059 070 1
8
1!)
14.323 799
13.709 837
13.133 939
12.085 321
11.158 11(1
10.335 578 1
9
20
14.S77 475
14.212 403
13.590 326
12.462 21U
11.469 421
10..''.93 997 2
21
15.416 024
14.697 974
14.029 160
12.821 153
11.764 077
10.835 527 2
1
22
15.936 917
15.167 125
14.451 115
13.163 003
12.041 582
11.061 241 2
2
23
16.443 608
15.620 410
I4.8.-.6 842
13.488 574
12.303 379
11.272 187 I
3
24
16.935 542
16.058 368
15.246 963
13.798 642
12.550 358
11.469 334 2
4
25
17.413 148
lo.481 515
15.622 080
14.093 945
12.783 356
11.653 683 2
6
26
17.876 842
10.890 352
15.982 7(>S
14.275 185
13003 166
11.825 779 2
6
27
18.327 031
17.285 365
16.329 58l>
14.643 034
13.210 534
11.986 709 2
7
28
18.764 108
17.067 019
16.663 063
14.898 127
13.406 164
12.137 111 2
8
29
19.188 455
18.035 767
16.983 715
15.141 074
13.590 721
12.277 674 2
9
30
19.600 441
18.392 045
17.292 03:;
15.372 451
1.3.764 831
12.409 041 3
31
20.000 428
18.736 276
17,588 49'J
15.592 811
13.929 086
12.531 814 3
1
32
20.338 766
19.068 865
17.873 552
15.802 677
14.084 043
12.646 655 3
2
33
20.765 792
19.390 208
18.147 646
16.002 549
14.230 230
12.758 790 3
3
34
21.131 837
19.700 684
18.411 198
16.192 204
14.368 141
12.854 009 3
4
35
21.487 220
20.000 661
18.664 613
16.374 194
14.498 246
14.n20 987
12.947 672 3
5
36
21.832 252
20.290 494
18.908 282
16.546 852
13.035 208 3
6
37
22.167 235
20.570 525
19.142 679
16.711 287
14.736 780
13.117 017 3
7
38
22.492 462
20.841 087
19.367 864
16.867 893
14.846 019
13.198 473 3
8
39
22.808 215
21.102 600
19.584 485
17.017 041
14.949 075
13.264 928 3
9
40
23.114 772
21.355 072
19.792 774
17.159 086
15.046 297
13.331 709 4
41
2.3.412 400
21.599 104
19.993 052
17.294 368
15.138 016
13.394 120 4
1
42
23.701 359
21.834 883
20.186 627
17.423 208
16.224 643
13.452 449 4
2
43
23.981 902
22.062 689
20.370 795
17.646 912
16.306 173
13.506 9C2 4
3
44
24.254 274
22.282 791
20.648 841
17.662 773
16.383 182
13J67 908 4
4
45
24.518 713
22.495 460
20.720 040
17.774 070
16.455 832
13.605 522 4
5
46
24.775 449
22.700 918
20.884 654
17.880 067
16.624 370
13.660 020 4
6
47
25.024 708
22.899 438
21.042 936
17.981 016
16.689 028
13.691 608 4
7
48
25.266 707
23.091 244
21.196 181
18.077 168
15.660 027
13.730 474 4
8
49
25.601 667
23.276 664
21.341 472
18.168 722
15.707 672
13.76C 799 4
9
|80
26.729 764
23.465 618
21.482 186
18.2.5 925
15.761 861
13.800 746 6
306
■;:'«?'
PABTNFKSHIP SETTLEMENTS.
.The oireumstances under wbbh partnerships are formed, tha
oonditions on which they are made, and the causes that lead to their
dissolution, arc so varied that it is impossible to do more than give
general dircoUons deduced from thd oases of most common occur-
rence. In forming a partnership, the great requisite is to have the
terms of agreement expressed in the most clear and yet concise lan<-
guage possible, setting forth the sum invested by each, the duration
of partnership, the share of gains or losses that fall to each, the
sum that each may draw from time to time for private purposes, and
any other circumstances arising out of the peculiarities of each case.
The ease and satisfaction of making an equitable settlement, in case
of dissolution, depends mainly on the clearjess of the original' agree-
ment, and hence the necessity for its being distinct and explicit.
Even when no dissolution is contemplated, settlements should be
frequently made, in-order that the parties may know how they stand
to each other, and how the business is succeeding. This is of great
importance in preserving unanimity and securing vigour and regu-
larity in all the transactions of a mercantile Houbq.
A dissolution may take place from various causes. If the part-
nership is formed for a term of years, the expiration of thosb years
necessarily involves either a dissolution or a new agreement. The
death of one of the partners may or may not cause dissolution, for
the deceased partner may have, by his will, left his share in the
business to his son, or some other relative or friend. In no case,
bowever, can an equitable settlement be made, except by the mutual
consent of the parties, or else in exact accordance with the terms of
agreement. It is also necessary that when a dissolution takes place
public notice should be given thereof, in order that all parties having
dealings with the firm may be apprized of the change, and hu,ve their
accounts arranged. For the same reason, it is necessary that Bome
one of the partners, or some trustworthy accountant appointed by
them, should be authorized to oolleot all debts due to the firm, and
pay all accounts owing by it.
» Partnerships are sometimes formed for a specific speoulation, and
tj^erefore^of course, oease with the completion of the transaction, and
s! settlement must nec^sarily be then made. No matter for what
PABTNEBSHIP SITnUSMENTS.
SOT
by
time tho partnership has been made, .any partner is at liberty, at any
time, to withdraw, on showing sufficient cause and giving proper
notice. This is a just provision, for the circumstances of any part-
ner may so change, from various causes, as to make it undesirable
for him to remain in the business. If one partner is deputed to
ce'wtle the accounts of the house, it would be reckoned fraudulent for
any other partner to collect any moneys due, except that on receipt
of them he hands them directly over to the person so deputed.
The resources and liabilities, with the net investment on com-
mencing business, being given, to find the net gain or loss.
1. W. Smith and B. Evans ire partners intbusiness, and invested
when commencing $1000 each. On dissolving the partnership, the
assets and liabilities are &a follows : — Merchandise valusd at $1295 ;
cash, $344 ; notes against sundry individuals, $790 ; W. H. Monroe
owes on account $86.40 ; E. B. Carpenter owes $132.85, and C. F.
Musgrove owes $67.50. They owe on sundry notes, as per bill book,
9212,40 ; E. G. Conklin, on account, $29.45, and H. G. Wright, on
account, $41.30. What has been the net gain ?
SOLUTION.
Assets.
Merchandise on hand... $1295.00
Cash on hand .1 344.00
Bills Receivable 790.00
Amt. due from W. H.
Monroe 86.40
Amt. due from E. B.
; Carpenter 132.85
Amt. due from C. F.
Musgrove 67.50
• it
LiaMKties,
Total amount Assets.... $2715.75
'« '* LiabiUties, 2283.15
Bills Payable $212.40
Amt. due £. G. Conklin. 29.45
Amt. due H. C. Wright. 41.30
W. Smith's investment... 1000.00
B. Evan's investment.... 1000.00
^:.>i.f,< -
'■■^ ■]
„ S2283.15
-.:: •,:v.V.i • ': •-
■,J'i'
- •':.■' ,.('i>!4 ^I'iV
.^j / ■-*> ' 'S
:': ,'^>' ,'i ' it
'.:':? . -O-. .»■*
ilj •;' ,: '-t-^^.H-V
H, J';'. •"
. •#- f»\ .■-:i:ji
i<h -vc
Wi^
■«..;- J.
Net gain $432.60
BULB* J-*:"'^:'^ ' .i;, ,». ►.i , S >,^iiv.- .i^v.Ai -
Find the sum of the assets and UabiUties ;fnm the assets subtract
ike liahilities, (including the net amaunt invested) and the differ-
ence toill be the net gain ; or, if the tiabiKties be the larger, subtract
ih€ assets from the liabilities^ and the difference will be the net loss.
I
iiii
808
ARITHMBTIC.
2. Harvey Miller and James Oarcy are partners in a dry good»
business '/Karvoy Miller investing $1400, and James Carey $1250.
When closing the books, they have un hand-'-^ash, $1125.30; mev-
chandiso as per inventory book, $1855.75 ; amount deposited in First
National Bank, $1200 ; amount invested in oil lands, $963 ; a site
of land for building purposes, valued at $1600 ; Adam Dudgeon owes
them, on aooount, ^104.92 ; William Fleming '' iA $216.80 , a note
against Afl^rod Mills for $69. 43, and a due bill for $33^ drawn by
James Laing. They owe W. S. Hope & Co., on account, $849.21 ;
B. J. King & Co., $603.12, and on notes, $1326.14. What has
been the net gain or loss ? Ans. $1761.73 gain.
3. James Henning and Adam Manning have formed a co-part-
nership for the purpose of conducting a general dry goods and
grocery business, each to share gains or losses equally. ^ ^,. '^
At the end of one year they close the books, having $1280 wortb.
of merchandise on hand ; casli, $714.27 ; Girard Bank stock, $500 ;
deposited in Merchants' Bank, $320.60 ; store and fixtures valued at
$3100; amount due on notes and book accounts, $3471.49. Tho
firm owes on notes $3400, and on open accounts $747.10.
James Henning invested $1200, and Adam Manning, $1000 i.
what is each partner's interest in the business at closing ? ^
Ans. James Henning's interest, $2719.63. Adam Manning's
• \ interest, $2519.63.
Note. — Where the interest of each partner at closing is required, tho gam
or loss is first found, as in t'ormer examples, then the share of gain or loss
ia added to or subtracted from each partner's investment, and the sum, or
difference, is the interest of each partner. If a partner has withdrawn any-
thing from the business, the amoiint thus withdrawn must bo deducted trom
the sum of his investment, p{u.<r his share of the gain, or minus his share of
the loss, and the remainder will bo his net capital ir interest. :};
4. F. A. Clarke, W. H. Marsden, and J. M. Musgrove, are con-
ducting business in partnership ; F. A. Clarke is to be ^ gain or
loss, W. H. Marsden and J. M. Musgrove, each ^.
On dissolving the partnership, they have cash on hand $712.90 ;
merchandise as per Inventory Book, $4360 ; bills receivable, as per
Bill Book, $1450.75 ; amount deposited in Third National Bank of
Syracuse $3475 ; merchandise shipped to Richmond, to be sold on
ojrn account and risk, valued at $995; debts due from individ-
uals on book account, $2644.67. They owe on notes $3760, and to
Manning and Munson, $1312.60.
; vrP-'T ■^r.j-.j~:f^^~
PARTNERSHIP SETTLE3CENTS.
809
90;
kid-
d to
5. A, and aro partners.
shared as follows : A, f, ; B, Jj ; and C, yj.
■^^ P. A. Clarke invested $5760, and has drawn out $875 ; W. H.
Marsden invested $2500, and has drawn oat $500 ; J. M. Musgrove
invested $3000, and has drawn out $750. What has been the net
gain or loss, and what is each partner's interest in the basiness ?
Ans. Net loss, $559.28 ; F. A. Clarke's interest, $4595.36 ; W.
K!. II. Marsden's interest, $1860.18 ; J. M. Musgrove's inter-
..i..., est, $2110.18.
Note. — la this and succeeding examples, no interest is to be allowed oa
Investment, or charged on amounts withdrawn, unless so specified.
The gains and lo;:93S are to be
^•j. A invested $3000,
and has withdrawn $2500, with the consent of B. and C, upon
which no interest is to bo charged; B invested $2700, and has
withdrawn $1150; C invested $2500, and has withdrawn $420.
After doing basiness 14 months, retires. Their assets consist of
bills receivable, $2937.20 , merchandise, $1970 ; cash, $1240.80 ; 50
shares of the Chicago Permanent Bailding and Savings' Society
Stock, the par valae Qf which is $50 per share; cash deposited in the
Third National Bank, $1850 ; store and farnitare, $3130 ; amoant
daefrom W. Smith, $360.80; G. S. Brown, $246.40; and E. R.
Carpenter, $97.12. Their liabilities are as follows : Amoant dae
Samuel Harris, $1675 ; unpaid on store and furniture, $935 ; and
notes unredeemed, $3388.76. The Savings' Society stock is valued
at 10 per cent, premium, and C in retiring takes it as part payment*
What is the amount due C, and what is A's, and what is B's interest
in the busitess ?
Aus. Due C, $815.32 ; A's interest, $2356.90 ; B's interest,
$2064.14.
6. E, F, Q and H are partners in business, each to share ^ of
profit and losses. The business is carried on for one year, when E
and F purchase from O and H their interest in the business, allow*
ing each $100 for his good will. Upon examination, their resources
are found to bo as follows : Cash deposited in Girard Bank, $3645 ;
cash on hand, $1422 ; bills receivable, $1685 ; bonds and mortgages,
$2746, upon which there is interest due $106 , Metropolitan Bank
stock, $1000; Girard Bank stock, $500; store and fixtures,
$3500; house and lot, $1800; span of horses, carriages, harness,
ico., $495; outstanding book debts due the firm. $4780. Tfabir
liabilities ure : Notes payable. $2345 ; upon which there is interest
due, $57 ; duo on book debts. $1560. E invested $5000 ; F $4500 ;
310
ABITHMETIO.
CK $4000 : and H, $3000. E hcs drawn from the business $1200,
upon whi(m he owes interest $32 ; F has drawn $1000— owes interest
$24.50 ; G has drawn $950— owes interest $12 ; and H has drawn
nothing. In the settlement a discount of 10 per cent., for bad debts,
is allowed, on the book debts due the firm and on the bills- receivable.
G takes the Metropolitan Bank stock, allowing on the same a pre-
mium of 5 per cent. ; and H takes the Girard Bank stock, at a
premium of 8 per cent. ; E and F take the* assets and assume the
liabilities, as above stated. What has been the net gain or loss, tho
balances duo G and H, and what are E and F eaca worth after tho
settlement?
Ans. Due G, $3057.75 ; due H. $3529.75 ; E's net capital,
$4637.75; F's net capital, $4345.25.
'vr 7. H. C. Wright, W. S. Samuels, and E. P. Hall, are doins
business together — H. C. W. to have J- gain or loss ; W. S. S.
and E. P. H. each J. After doing business one year, W. S, S.
and E. P. H. retire from the firm. On closing the books and taking
stock, the following is found to be the result : merchandise on hand,
$3216.50; cash deposited in Sixth National Bank, $162V,35; oush
in till $134.16 ; bills receivable, $940.60 ; G. Brown owes, on ac-
count, $112.40; Thos. A. Bryce owes $94.12; W. McKeo owes
$143.95; J. Anderson owes $54,20 ; B. H. Hill owes $43.60 ; and
S. Graham owes $260.13. They owe on notes not redeemed $1864 ;
H. T. Collins, on account, $124 45; and W. F. Curtis, $79.40.
H. C. Wright invested $3200, and has drawn from the business $350.
W. S. Samuels invested $2455, and has drawn $140 ; E. P. Hall
invested $2100, and has drawn $2000. A di.scount of 10 per cent,
is to be allowed on the bills receivable and book accounts due
the firm for bad debts. H. C. Wright takes the assets and assumes
tho liabilities as above stated. What has been the net gain
or loss, and what does H. C. Wright pay W. S. Samuels and E. P..
Ha'l on retiring ?
8. T. P. Wolfe, J. P. Towlcr and E. B. Carpenter have been
doing business in partnerchip, sharing the gains and losses equally.
After dissolution and settlement of all their liabilities they make a
divbion of the remaining effects without regard to the proper pro-
poytion each should take. The following is the result according to
their ledger :— T. P. Wolfe invested $3495, and has drawn $2941 ;
J. P. Towler invested $2900, and has drawn $2200 ; E. B. Carpentck)
FABTKi92k8HIP SEITLEMENTEk
sn
ih
inyested $3150, and hbs drawn $3000. How will tfleiparfBOfs
fettle with each other? . . , i'. •..» t \:\;
.^ Ans. E. R. Carpenter pays T. P. Wolfo $86, and J.P^ Towlec<$232.
9. I, J, K, L and M havo entered iato oo-partnor^hlp, {peeing
to share the gains and looses in the following proportion :^-I, j\ ; J,
1^5; K,
To >
H .
T5»
and M, /j. Wiien dissolving the* partnership
the resources consisted of cash $4700 ; merchandise, $0855 ;■ 'botes
on hand $7G80 ; debentures of tlie city of Albany valued at $$780,
on which thuro is interest duo, $123 ; horses, waggons, &o., $1280';
Merchant's bank stock, $5000 ; First National bank stock, $5to;
mortgages and bonds, $3600; interest due on mortgages, $345.80 -;
store and fixtures, $3000; amount due from W. P. Campbell & Co.,
$2418; due from R. B.Smith, $712.60; due from J. W.Jones,
$1000. The liabilities are : — Mortgage on store and fixtures, $5000 ;
Interest due on the same, $212.25 ; due the estate of R. M. Evans,
$14675 ; notes and acceptances, $11940, on which interest is due,
$85 ; sundry other book debts, $7500 ; I invested $7800, interest
on his investment to date of dissolution, $702 ; J invested $6400^
interest on investment, $576 ; K invested $6100, interest on invest-
ment, $549 ; L invested $5800, interest on investment, $522 ; M
invested $5000, interest on investment, $450. I has withdrawn
from the firm at different times, $2425, upon which the interest calcu-
lated to time of dissolution is $183.40 ; J has drawn $2960, interest,
$267.85; K has drawn $1850, interest $87.30; L has drawn
$3000, interest, $460 ; M has drawn $895, interest, $63.45. What
is the net gain or loss of each partner, and what is the net capital of
eaoh partner ?
I Anfl. I's net loss, $1233^29 ; I's net capital, $4660.31. J's net
^' loss, $924.97; J's net capital^ $2823.18. K's net loss,
> ' $616.65 ; K'l) net capital, $4095.05. L's net loss, $1541.62 ;
' f L's net capiUvl, $1320.38. M's net loss, $308.32 ; M's net
'^ ' capital, $4183.23.
" 10. A, B, C and D are partners. At the time of dissolution,
and after the liabilities are all cancelled, they make a division of the
effects, and upon examination of their ledger it shows tho f'^Dcwing
result : — A has drawn from the business $3465, and invested on
eonimenoement of business, $4240 ; B has drawn $4595, and invested
$3800: C has drawn $5000, and invested $3200; D has drawn
$^00, «nd iuvested $2800. The profit or loss was to be divided in
312
ABnmano:
proportion to the ori^pnal inTestment. What has heen each partner*!
gain or loss, and how do the partners settle with each other?
Ans. A's.net gain, $368.43; B's net gain, $330.20; G's net
gain, $278.06; D's net gaia, $243.31. B has to pay in
$464.80 ; has to pay in $1521.94. A receives $1143.43 ;
D receives $843.31.
'*'* Three mechanics, A. W. Smith, James Walker and P.
Banton, are equal partners in their business, with the understanding
that each is to be charged $1.25 per day for lost time. At the oloso
of their business, in the settlement it was found that A. W. Smith
had lost 14 days, James Walker 21 days, ahd P. Banton 30 dayi.
How shall the partners properly adjust the matter between them?
Ans. P. Banton jpays A. W. Smith, $9.58^, and James Walker,
83^ cents.
12. There are 6 mechanics on a certain piece of work in tho
following proportions :— A 'ia^%; B, ^^ ; C, 3^ ; D, /g, and E, ^jp,
A is to pay $1.25 per day for all lost time ; B, $1 ; 0, $1.50; D,
$1.75, and E> $1.62^. At settlement it is found that A has lost
24 ; B, 19 ; C, 34 ; D, 12 ; and E, 45 days. They receive in pay-
ment for their joint work, $250C. What is each partner's share of
^S 9inount according to the above regulations?
Ans; A's share, $374.12; B's, $250.41; O's, $487.83; D'a,
$787.24; E's, $600.40.
,. , 18. A. B. Smith and T. G. Musgrove commenced business in
partjoership January 1st, 1864. A. B. Smith invested, on com-
fitaaoeiment, $9000 ; May 1st, $2400; June 1st, he drew out $1800;
September 1st, $2000, and October 1st, he invested $800 more.
)Pf . C.rMu^oye; invested on commencing, $3000 ; March 1st, he
dr^w o^t $jt600; 3j[ay 1st, $1200; June 1st, he invested $1500
morej jEtnd October 1st, $8000 more. At the time of settlement, on
the 31st December, 1864, their merchandise account was — Dr.
$32000 ; Cr. $27000 ; balance of merchandise on hand, as pef
inventory, $10500 ; cash on hand, $4900 ; bills receivable, $12400;
E. l)rap&r owes on account, $2450. They owe on their notes,
$189Q, and iGi-. Boe on account, $840. Their profit and loss account
is^' jbr. $86^ ; Cr. $1520. Expense account is, Dr. $2420. Com-
mission account is. Or, $2760. Interest account is Dr. $480 ; Cr,
|^5^T T|)ek|.gaijf^, or.loss is to be divided ia proportion to eaoh
^i||bW'8^-«^^^,, and in proportion to the time it was invested.
Eeamred each partner's share of the gain or loss, the net balanot
FilBTNEBSHIP- SERLEMENTBU
818
,.?;'
due each, and a ledger specificatioa exhibiting th»«lo8in({ of ill fhs
accounts, -and the balance sheet.
4.
^ Ans..A.B. S.'s net gain, $6671.73; his net l)alancer|16071.73.
« T. G. M.'s net gain, $2748.27 ; his net balance, $1 2448.27.
14. A, B, G, and D commence business together on July Is^
1865, with the agreement that all gain or loss is to be ehared equally
by each partner, but that interest at the rate of 6 per cent, per an-
num is to be allowed on each one's investment, and the same rata
charged on all amounts withdrawn by each. A is to manage tho
business, having a salary of $2,000 per year, payable half-yearly.
Tho services of B, G, and D are not required in the business. Tho
assets are, on commencing, Gash $7440 ; Mdse. $9686 ; Bills Be-
ceivablo, estimated value $4976.C ') (face value $5237.89) invested
by G. Per. accounts Dr. $12271.40 (estimated value, 10 per cent,
discount) invested by A. Of the assets, there belongs to A
$13492.40, B, $6000, C, $5750, and to D $5000. Personal ao
counts Cr. $4292.89. Aug. 20th, A drew cash $75, B $90. Sept
4tb> D drew cash $125. Sept. 30th, A drew $200, G $80. Nov.
20th, B ilrcw $100, D, $50. Dec. 24th, A drew $150. Feb. 27th,
18C6, A drew $200, G $150, D $100. May 12th, A drew $200, D
$200. Juno 13th, B drew $150, G $100. July 1st, B made a
further investment of $1000, G, $1500, and A drew $400, D $100.
Sept. 15th, A drew $150, B $500, C, $750. Nov. 1st, A drew $100,
D $75. Dec. 31st, 1866, the books are closed and the partnership
dissolved, G and D retiring from tho business, being allowed by tho
remaining partners $150 each for their good-will in the business.
Before calculating the interest on tho partners' investments, and on
the amounts withdrawn by them, and allowing A the amount of hiiii
fjalary from time of commencement up to date, the assets and liabili-
ties are as follows: Gash $9483.50; Mdse. $14675; Bills Eeceiv-
able $6219.85, Personal accounts Dr. $7694.30. Inventory of
Mdse. consigned to "W. Smith, New York, to be sold on our account
and risk, $1265.12; 50 shares N. Y. G. B. B. stock, valued at 105;
60 shares Erie B. R stock at 69. They owe in BiUs payable
$5657.45. Personal accounts Gr. $3272.94 ; also, Samuel Zimme>
man for rent to date $1250. In the settlement a discoxint of 5 per
cent, on the B^ls BeceivaUe, 15 per cent, on tho Dr. personal ao>
21
Ji!
814
ABXTBMBTXOi
oonnts, and 10 por oont on the Mdso. shipped to New York, is lU
lowed for loss in bad debts. agrees to talce the N. Y. 0. R. R,
stock at 105, in part payment of the amount due him ; D takes th«
Erie R. R. stock at 69 ; receives cash 01200, and D $1800.
What has been the whole gain. or loss, the amount still duo and D,
the amount of cash on hand when the books are closed up, and what
are A and B each worth, the Billa Roc, persona! accounts, and con-
signmcnt to N. Y. being valiicd at par?
Ans. Gain $3370.45 ; duo C 8959.72; duo D $512.96; cash on
hand $6483.50 ; A's capital $16401.47 ; B's net capital $8183.21.
15. A book-keeper applied at our OoUege for counsel, not long
since, to settle the following accounts, between two partners, Jan.
1st, 1866. We'll call them Mr. E. and Mr. F., each \ gain or loss.
Books were kept by single entry, and tho accounts and inventory
were as follows :— Cash in Rank $5,705. Do. in office $6,000.
Bills Rcc. on file $4,921.33, upon which there was interest duo
$78.67. Mdse. unsold ^4,000. Propeller Toledo account, (techni-
cal) dr. bal. $6,210. Their shares in the boat valued at $5,000.
E drew out $1,010. F drew out $3,339, Expense account, dr. baL
$1,335, Mdse. account, dr. bal. $210. Bills payable, per B. B.,
$4,564. Rev. O. Burger, cr. bal. $200. D. C. Weed & Co. dr.
bal. $2,000. Shepard & Cottier, or. bal. $300. Jn't and Dis't ao-
count, cr. bal. $1,524. Joint account with A. L. Griffon & Co. each
^, net gains were $872. P. P. Dobbins & Son, dr. bal. $l,00a
What are B. and F. eaeh worth ?
■ ■> ■)
.■"3.'5
>«ffw»t-
is al«
QT7B8TXOKB VOB OOmCBBOXili 8TDDEMTBL
315
QX7ESTI0NS FOR COMMERCIAL STUDENTS.
1. The following questions may be found interesting and inrtruotiye
to young men preparing for the practical duties of accountants.
On the 1st of May I purchased for cash, on a commission of 2^
per cent., aud consigned to Ross, Winans & Co., commission mer-
chants, Baltimore, Md.; 380 bbls. of mess pork, at $27.50 per bbl.,
to bo sold on joint account of himself and myself, each one half.
Paid shipping expenses, $7.40. July 7th, I received from Ross,
Winana & Co. an account sales showing my net proceeds to be
$5319,79, due as per average, August 12th. August 8th, I draw
on them at sight for the full amount of their account, which I sell at
^ per cent, discount for cash ; interest 7 per cent. What amount
of money du I receive and what are the journal entries?
2. B. Empey, a merchant doing business in Montreal, Canada
East, purchased from A. T. Stewart, of New York city, on a orerUt
of three months, the following 'nvoice of goods : ■ •
Gold.
845 yds. Fancy Tweed, @ $1.90peryd. 81.99
: 1712 " Amer. black broadcloth, @ 3.85 " " 4.02
■ 423 " Blue pilot, @ 2.75 « "2.87
, 700 " Black Cassimere. @ 2.10 " "2.20
When the nbove goods were passed through the custom-house, a
discount of 27} per cent, was allowed on American invoices ; duty
25 per cent, freight charges paid in gold, $29.35. What must
each piece be marked at, per yd., to sell at a net profit of 15 per
cent, on full cost ? What would be the gain or loss by excKange, if
at the expiration of the three months B. Empey remitted A. T.
Stewart, to balance account, a draft on Adam?; Kimball & Moore,
bankers, New York city, purchased at 32^ per cent, discount, and
what are the journal entries ?
3. I purcihased for cash, per the order of J. P. Fowler, 70 boxes
0. 0. bacon, containing on an average 400 lbs. each, at ISf cents
816
ABITBICBTXa
I: i. ;.
per lb., and 140 firkini buttor, 8312 lbs., at 17} cents per id., on a
commission of 2} per cent ; paid ibipping and sundry expenses in
oash $13.40. For reimbursement I draw on J. P. Powlcr at sight,
which I sell to the bank at } per cent, discount ; what is the face of
draft, and what are the journal entries ?
Ans. Face of draft $5479.05.
4. Sept. 27th, I received from James Watson, Leeds, England*
a consignment of 1243 yards black broadcloth, inyoioed at 13s. 6d.
per yard, to be sold on joint account of consignor and myself, each
one half, my half to be as cosh, invoice dated Sept. 16th. Oct 5th,
I sold K. Duncan, for cash, 207 yards, at $6.10 per yard ; Oct 24th,
sold 317 yards to James Grant, at $6.25 per yard, on a credit of 90
days ; Nov. 18th, sold E. 0. Oonklin, for his note at 4 months, 400
yards, at $6.30 per yard ; Dec. 12th, sold the remainder to J. A.
Musgrove at $6.00 per yard, half cash and a credit of 30 days for
balance; charges for storage, advertising, &c., $13.40; my com*
mission, with guarantee of soles, 5 per cent, \yhat would be the
average time of sales ; the average time of James Watson's account ;
and what would bo the face of a sterling bill, dated Dec. 15th, at
€0 days after date, remitted James Watson to balance account
purchased at $108^^, money being worth 7 percent, and gold being 70
4)er cent, premium ?
6. Buchanan & Harris of Milwaukee, Wis., are owing W. A.
Murray & Co. of Washington, $1742.75, being proceeds of consign-
ment of tobacco sold for them, and Simpson & Oo. of Washington,
are owing Buchanan & Harris $2000 payable in Washington.
Buchanan & Harris wish to remit W. A. Murray & Co. the proceeds
of their consignment and they do so by draft on Simpson & Co., but
Washington funds are 2 per cent, premium over those of Chicago.
Kequired the face of the draft and the journal entries^
6. A. Cummings, of London, England, is owing me a certain sum,
payable there, and I am owing Charles Massey, of the same place,
$1985.42, being proceeds of consignment of broadcloth sold for
him. I remit C. Massey in full of account, after allowing him
$21,12 for inserest, my bill of exchange on A. Cummings at 60
days' sight ; exchange 109f, gold 42 per cent, premium. What is
<he fiioe of the draft, and what are the journal entries ? -^Jt.,^ , .
QUESTIONS FOB OOMMSBOUL BTUDF^TS.
317
sum,
i)lace,
for
him
xt60
kat is
7 . March 10, I shipped per stetumer VanderbiU ana oonsigaed
to Samuel Vestry, Liverpool, England, to be sold on joint account of
consigDee and consignor, each one half, (consignee's half as cosh),
27894 lbs. prime American Cheese at 1&|- cents per pound. Paid
•hipping expenses $18.30. Insurant l^ per cent, and insured for
luch an amount that, in case the oheoso was lost, tho total coat would
be recoverable. May 19, I rcooivod from Samuel Vestry an account
■ales, showing my net proceeds to be £298 14 9^, duo as per
average August 21. June 1, I drew on Samuel Vestry at the num*
ber of days after date that it would take to make tho bill fall duo at
the properly equated time of his account. Sold tho above bill to R.
Bamdsey, broker, at 108^. Required the number of days I drew
the bill at, its face, gold being at a premium of 43f per cent., the
amount of money in greenbacks I received, and tho journal entries.
8. I am doing a commission business in New York, and on Sept.
14, I received from A. J. Rice, of Hudson, to be sold on joint ao*
«ount of himself §, A. H. Peatman, of Ncwburg, ^, and myself^;
merchandise invoiced at $1262.40, paid freight $14.20. Tho same
day, I received from A. H. Peatman to bo sold on joint account of
himself §, A. J. Rice ^, and myself ^, morohnndiso invoiced at
$1102.12 ; paid freight $10.00. I also invest to be sold on joint
aooount of A. J. Rice ^, A. H. Peatman ^, and myself §, merchan-
dise valued at $745.35. The shares of each are subject to average
Bales. October 29th, I sold ^ of tho merchandise received from A.
J. Rice to S. King at an advance of 20 per cent., on a credit of 90
days. November 9tb, sold for cash one half of tho remainder at 15
per cent, advance, closed the company, and rendered account sales;
Btorago $3.50, commission 2^ per cent. November 12, sold to A.
M. Spafford, on a credit of 30 days one half of goods received from
A. H. Peatman, at an advance of 25 per cent. November 23, sold
for cash the merchandise that I invested at an advance of 15 per
eent. ; closed the company and rendered account sales ; storage $2.75,
commission 2^ per eent. December 4th, sold ^ of the remainder of
jnerohondize received from A. H. Peatman to'G. W. Wright, on a
credit of 60 days, at 33^ per cent, advance. December 12th, sold
the balance of Peatman's merchandise for cash at 25 per cent, ad-
ranoe ; dosed the company and rendered aooount sales, storage $5.00
oommission 2^ per cent. December 23rd, I wish to settle with A.
J. Bioe, and A. H. Peatman, in full ; I take to my own account,
818
ABmaOETIO
M Mih, the balanee of moralukiidiM untold at antdnneo of d per
oant.. What h the average time of aalea of each Mdae. Go., the
ETerage time of A. J. R. and A. H. P'a. accounts, the amount of
konojc I shall have to pay them on December 20, how do A. J. R.
and A. II. P. stand with each oth^r, and what are the journal entries 7
9. E. B. Carpontor, S. Northrup and Levi Williams, commonood
business togotlior as partners under the name and style of £. R.
Carpenter & Co., on January Ist, 1865, with the follo:;7ing effects :
merchandise, $7844 ; cash, $5000 ; store and fixtures, $3984 ; bills
receivable, $173:^50 ; of this amount there belonged to E. R. Car>
pcnter, as capital, $8000; S. Northrup, $6000; Levi Williams,
$4561.50. The firm assumed the liability of Levi Williams, which
was a note to tho amount of $425.80 ; this note was paid on March
10th. The loss or gain is to be shared equally by the partners, but
interest at the rate of 7 per cent, per annum is to be allowed on in*
vestments, and charged on amounts withdrawn. £. R. Carpenter
is to manage the business on a salary of $1000 a year, payable half
yearly (the time of the other partners not being required in tho
business). March 14th, S. Northrup draws cash, $300; Levi
Williams, $200; April 19th* E. R. Carpenter draws cash, $500 ; S.
Northrup, $100. On the 1st of May, they admit Qeo. Smith as a
partner, under the original agreement, with a cash capital of $4000.
The books not being closed, ho pays each partner for a participation
in the profits to this time $450, which they invest in the business.
May 14th, E. B. Carpenter draws cash, $160 ; May 24th, Levi
Williams draws cash, $100 ; Juno 12th, S. Northrup draws cash,
4250, und E. R. Carpenter, $200 ; July Ist, Levi Williams draws
cash, $300, and S. Northrup, $450 ; July 21st, Levi Williams draws
•cash, $180 ; August 1st, Levi Williams retires from the partnership,
the firm allowing him $500 for his profits and good-will in the busi-
ness, this amount, together with his capital, has been paid in cash.
Oct. 14th, Geo. Smith draws cash, $340; E. R. Carpenter, $725.
November 1st, with the consent of the firm, S. Northrup disposes of
his right, title and interest in the business to J. K. White, who is
admitted as a partner under the original agreement. J. K. White
is to allow S. Northrup $600 for his share of the profits to date, aild
his good-will in the business. J. K. White not receiving funds an-
ticipated, is unable to pay S. Northrup but $1500, the firm therefbre
iMumes the balance as a liability. December 10th, received ttdm
QUESnONH FOR OOMMIROIAL 8TUDEMT8.
819
J. K. White, and paid oTor to S. Northrup, tho full amount due him
(8. N.) to date. Dooomber Slst, the booka are olosod, and tho fol*
lowing cffoots are on hand:— Mdae, $11043.75; cash, |eil0.12;
bilb reoeivablo, 16400 ; store and flzturos, $3860 ; personal accounts
Pr. $14087.50; personal accounts Cr. $10711 ; Bills Payable nnre-
deomod, $4000. What has bocn tho not gain or loss, tho net capital
of each partner at tho end of the year, and what arc tho double
entry journal entries on commencing business, when Levi Williams
retires, when Qeo. Smith is admitted, when S. Northrup Mif\a his
interest ond right to J. K. White, for E. R. Carpenter's dalary,
tnd tho interest due from and to each partner ?
The student may also, in the above example, after finding tho
interest on tho partners investments, and on tho amounts withdrawn,
give a joomal entry that will adjust the matter of interest between
the partners without opening any profit and loss account.
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MENSURATION.*
We have already observed that no solid body can have more thaa
ikne dimensions, viz. : length, breadth, and thickness, or depth, and
that a line is length, or breadth, or depth, or it is a line or unit
repeated a certain number of times. A foot in length is a line mea-
sured by repeating the linear unit called an inch 12 times, and a
yard is the linear unit called a foot, repeated 3 times, and so on.
Thus, 1 ft. 1 ft. 1 ft. „ „ ^ T, . ,, , . . ,.
' . J =3 feet. But there may be two such Imes
drawn at right angles to each other, and each three feet long, and if
t ^ the figure be completed it is a sauare.
Also, if lines be drawn, each an inch
apart from the other, and parallel to the
two first-mentioned lines, we shall find
that thero are three small figures, each an
inch square, between the two upper hori-
zontal lines, and 3 of the same extent
uutween the two intermediate lines, and
3 between the two lower lines, making
9 in all, or 3 times 3. This is the origin of the expression that 9 is
the square of 3. Let the learner mark the difference between 3
square feet and 3 feet square, a, h and c are 3 square feet, but the
whole figure is Zfeet sqttare, and therefore three feet square must be
equal to 9 square '^set. Three feet square, then, is a square, each of
whose sides measures 3 linear feet; but 3 square feet would denote
3 squares, each side of each measuring one linear foot. The space
thus inclosed is called the area.
This is the principle on which surfaces are measured. ;
a
b
r
9
■ , '^,, _
,», •
■ T . « , -;i ;r . PROBLEM I.
To find the area of a paralellogram :
■ I,' . -ji-- .. B U II B • ,*;,
:"'*-■ .
.■i-4« .-.-** ■
Multiply the lengt\hjf the perpendicular breadth. If the figure
ie rectangular, one of the sides will be the perpendicular breadth,
* We have taken for granted that those atudying mensuration have
^earned, at least, tho elementary principles of geometry. We have, there-
fore, only gi7on th? rales, as oor space would not admit of our giving^
demonstratioDB as this woidd require a separate treatit^^
■ • . .J* •
IGSNBUfiATION.
321
■•»«i^-
Tt the figure he not reetangidar, either the perpendicular breadth
'0imt be given or data/rom which to find it.
■ XBBOISBS, ■^■'^*^'' '■ -^'' ^*'
' 1. How many aorea are there in a sauaro, each side of which ia
24 rodf ? Ana. 3 acres, 2 roods, 16 rods.
2. What b the area of a square picture frame, each side of which
is 5 ft. 9 in. ? Ans. 33 ft. 9 in.
3. How many acres are there in a rectangular field, the length of
which is 13|^ chains, and the breadth 9^ ?
Ans. 130.625 square chains, or 13 acres, roods, 10 rods.
4. What is the area of a rectangle, whose sides are 14 ft. 6 in.
and 4 ft. 9 in. ? Ans. 68 ft., 126 sq. iri.
t; > 5. What does the surface of a plaalc measure, which is 12 fC
6 in. long and 9 in. broad? Ans. 9 sq. ft. 54 sq. in,
6. What is the area of a rhomboidal field, the length of which is
10.52^hains and the perpendicular breath 7.63 chains ?
Ans. 8 acres, roods, 4.2816 rods,
7. What is the area of a rhomboidal field, the length of which 19
24 rods and the perpendicular breadth 24 rods ?
-> Ans. 3 acres, 2 roods, 16 rods.
8. What is the length of each side of a square field, the area of
which is 788544 square yards ? Ans. 888 yards.
9. The area of a rectangular garden is 1848 square yarcls, and|
one side is 56 yards ; what is the other ? Ans. 33 yards.
10. The area of a rhomboidal pavement is 205, and the length
is 20 feet ; what is the perpendicular breadth ? Ans. 10^^^ feet.
PR0BLE2E U.
To find the area of a triangle. -^
• 1. If t]ie base and perpendicular, or data to find them, be given, we
have tht ; ^;"''*''•V ':'.'■■ !'-'-'*«sji-«v!4 '■ --,' . '' ^ 'H^'
BULB.
MuUiply the base by the perpendicular^ and take half the pr(h
iiuet i or, multiply half ihe one by the other. '
2. If the three sides are given f ^ '
;'<■■': BULB.
^,, From half ihe mm of th/e «u2et tvhftmiA each tide tu^xmiv^ly,
and the tquare root of the continual product of the half wm» dmd
iheee three remainden mil be the area.
822
ABirmoEna
Expressed algebraioally this area=|/«(«— <>)(<— &)(i—c).
> XXEROISBB. '-'■'
11. What is the area of a triangle, the base of which is 17 inches,
and the altitude 12 inches? Ans. 102 square inches.
12. What is the area of a triangular garden, the length of which
is 46 rods, and the breadth 19 rods ? Ans. 437 square rods.
13. Find how many acres, &c., are in a triangular field, the
length of which is 49.75 rods, and the breadth 34 J rods.
Ans. 5 acres, 1 rood, ISj'^ rods.
14. The area of a triangular inclosuro is 150 square rods, and
the base is 30 linear rods ; what is the altitude ? Ans. 10 rods.
15. The area of a triangle is 400 roJs, ar.d the altitude 40 rods,
what is the base ? Ans. 20 rods.
16. Three trees are so planted that the lines joining them form
a right angled triangle ; the two sides containing the right angle are
33 and 56 yards, what is the area in square yards ? Ans. 924.
17. Let the position of the trees, as in
the last example, be represented by the tri-
angle ABC, and let the distance from A
to B be 50 rods, and from B to C 30 rods.
Eequired the area. — (See Euclid I. 47.)
Ans. 600 square rods.
18. In the fin;ure annexed to 17, suppose A B to represent the
pitch of a gallery in a church, inclined to the ground ai an angle of
A5' ; how many more persons will the gallery contiu.:: ih^.a if the
•eats were made on the flat B C, supposing B C Ij be 'I ' ^' at and
Ihe frontage 60 feet in length ?
2a i
J. None.
We have introduced this question and the next to correct a
common misapprehension on this point.
Jl|J■^K'lt Because the distance from B f» A is
greater than the distance from B to 0,
it is commonly supposed that more per-
sons can be accommodated on the slant
A B, than on the flat B C. By in-
specting the annexed diagram it will
be seen that the seats are not perpen-
dicular to A B, but to B C, and that
precisely the same number of seats can
be made, and the same number of per*
ions accommodated on B C as on A B.
v'^.U-Sf
1 . ■
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A
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MEKSTTBATION.
323
Hcan
per*
AB.
19. If B be half the base of a hill, and A B one of its sloping
-aides, and B G=30 yards, and A B=50 yards ; how many more
rows of trees can be planted on A B, than on B C, at 1 yard apart ?
Ans. None, because the trees being all perpendicular to the
horizon, are parallel to each other as represented by the
vertical lines in the last figure.
20. How many acres, &c., are there in a triangular field of which
the perpendicular length and breadth are 12 chains, 76 links and 9
chains, 43 links ? Ans. 6 acres, roods, 2^ rods.
21. A ship was stranded at a distance of 40 yards from the base
of a cliff 30 yards high ; what was the length of a cable which reached
from the top of the cliff to the ship ? Ans 50 yds«
22. A cable 100 yards long was passed from the bow to the stem
of a ship through the cradle of a mast placed in midships at the
height of 30 yards ; what was the length of the ship ?
Ans. 80 yards.
23. A man attempts to row a boat directly across a riyer 200
yards broad, but is carried 80 yards down the stream by the current;
through how maay yards was he carried ? Ans. 215.4-f-yards.
24. Let the three sides of a triangle be 30, 40, 20 ; to find the
area in square feet. Ans. 290.4737 square feet.
25. What is the area of an isosceles triangle, each of the equal
sides being 15 feet, and the base 20 feet ?* Ans. 111.803 sq. feet.
26. What ia the area of a triangular space, of which the base is
56, and the hypotermse 65 yards ? Ans. 924 square yards.
27. What is the area of a triangular clearing, each side of which
is 25 chains ? Ans. 27.0632 acres.
28. What is the area of a triangular clearing, of which the three
sides arc 380, 420 and 765 ? Ans. 9 acres, 37^ perches.
29. A lot of ground is represented by the three sides of a right
angled triangle, of which the hypotenuse i# 100 rods, and the base
60 rods ; what is the area? .„,..».. Ans. 15 acres.
'60. What is the area of a triangular field, of which the sides are
49, 34 and 27 rods respectively ? Ans. 2 acres, 3 roods4-.
31. What is the area of a triangular orchard, the sides of which
are 13, 14 and 15 yards ? Ans. 84 square yards.
.32. Three divisions of an army are placed so as to be represented
* This quostion, and some others may be solved by either rule, and it will
bo found a good exercise to solve by both.
324
ABITHUEnO.
M
by three sides of a triangle, 12, 18 and 24 ; how many square »iIoB
do they guard within their lines ?
Ans. Between 104 and 105 square miles.
33. A ladder, 50 feet long, was placed in a street, and reachec|
to a parapet 28 feet high, and on being turned over reached a para-
pet on the other side 30 feet high ; what was the breadth of the
street? Ans. 81.42+feet.
PBOBLBM III.
To find the area of a regular Polygon. ^ ' . ' ; ^ ;
1. When one of the equal sides, and the perpendicular on it
firom the centre, are given.
Multiply the perimeter hy the perpendicular on it frcmi its centre,
and take half the product ; or, multiply either by half the other.
2. When a side only is given.
Multiply the square of the aide hy the number found opposite the
numher of sides in the subjoined table.
Note.— This tublu shows the area when the Hide is unity ; or, which is the
aame thing, the square is the unit.
SIDES.
■'* ,!
fi
3
4
6
6
7
8
9
10
11
12
REQULAR FIOmiES.
Triangle
Square
Pentagon
Pexagon
Heptagon
Octagon
Nonagon.^.^..
Decagon
Heredecagon.
Dodecagon...
0.4330127.
1.0000000.
1.7204774.
2.5980762.
3.6339125.
4.8284272.
6.1818241.
7.6942088.
9.3656395.
11.1961524.
34. If tlie side of a pentagon is 6 feet and the perpendicular 3
feet, what is the area ? Ans. 45 feet.
35. What is the area of a r^ular polygon, each side of which is
,15 yards? Ans. 387.107325 Sq. yds.
|:j^ 36. If each side of a hexagon be 6 feet, and a lino drawn from
the centre to any orngle be 5 feet, what is the area ?
Ans. 72 8q. iMi
MENSUBAHON.
825
37. Tlio side of a decagon is 20.5 rods ; what is the area 7
Ans. 20 acres, roods, 33.5 rods, nearly.
38. A hexa^ional table has each side 60 inches, and a line firqm
die centre to any comer is 50 inches ; how raaay square feet in tho
fiorface of the table ?
39. What is ihe area of a regular heptagon, the side being 19|{
and the perpendicular 10 ? . Ans. 678.3.
40. An octagonal ennlosure has each side 6 yards, what is its
area ? . Ans. 3 acres, 2 roods, 14 rods. 19 yards.
41. Five divisions of an army guard a certain tract of country— >
each lino is 20 mile.i ; how many square miles are guarded ?
Ans. 688.2, nearly.
42. Find tho same if there are 6 divisions, and each 4ine extends
5 miles? Ans. 64.95+ miles.
. 43. The area of. a hexagonal table is 73^ feet; what is each
side? H--_-, ■ , ,. Ans. 6J feet.
PBOBLEM IV.
To find the area of tvn irregular polygon.
.Divide it into triangles hy a perpendicular on each dicyonal
from the opposite angle.
Find the area of each triangle separately ^ and the sum of these
areas will be the area of the trapezium. ., /
Note. — Either the diagoniils and perpendicultire must be given, or data
from which to find them. , ,,,'
44. The diagonal extent of a four-sided field is 65 rods, and the
perpendiculars on it from the opposite comers are 23 and 33.5 rods;
what is the area ? Ans. 1 acre, 1 rood, 22.083 rods.
45. A quadrangle having two sides pardlcl, and the one is 20.5
feet long and the other 12.25 feet, and the perpendicular dbtonce
4)etween them is 10.75 feet ; what is the area ?
i^U r . ! , . Ans. 176.03125 sq. feet.
46. Required the area of a six-sided figure, the diagonals of
which are as follows : the two extreme ones, 20.75 yards and 18.5,
and the intermediate 27.48 ; the perpendicular on the first is 8.6, on
the second 12.8, and those on the intermediate one 14.25 and 9.35 ?
Ans. 531.889 yards.
47. If tho two sides of a hexagon bo parallel, and the diagonal
panllel to them be 30.15 feet, and the perpendionlars on it from
'yr%;.VT vr
. tin^ 4 A. i
f:-x
v..
826
ABTTEMETIO.
^
the opposite angles are, on the left, 10.56, and on the right 12.24;.
and the part of the diagonal out off to the left hy the first perpendi-
cular, 8.26, and to the right by the second, 10.14 ; on the other,
nde, the perpendicular and s^;ment of the diagonal to the left are.
8.56 and 4.54, and on the right 9.26 and 3.93 ; what is the area?
Ans. 470.4155 sq. feet»^
PBOBLIK V.
To find the area of a figure, the boundaries of which are* partly
right lines and partly curves or salients. ^
Find the average breadth hy taking several perpendiculars from
the nearest and most remote points, from a fixed base, axd dividing
the sum of these by their nurnbeTj the quotient, multiplied by the
length, will be a close approximation to the area.
Let the perpendiculars 9.2, 10.5, 8.3, 9.4, 10.7, their sum is
48.1, then 48.1-4-5=9.62, and if the base is 20, we have 9.62X20=
192.4, the area. . , , .^
When practicable, as laige a portion of the space as possible
should b^ laid off, so as to form a regular figure, and the rest found
as above.
A field is to be measured, and the greater part of it can be laid
off in the form of a rectangle, the sides of which are 20.5 and 10.5,
and therefore its area is 215.25, and the offsets of the irregular part
are 10.2, 8.7, 10.9, and 8.5, the stm of which, divided by Iheir
number, is 7.66, and 7.66X20.5=157.03, the area of the irregular
part, and this, added so the area of the rectangles, gives 372.23, tho
wh6le area.
48. The length of an irregular clearing is 47 rods, and the
breadths at 6 equal distances are 5.7, 4.8, 7.5, 5.1, 8.4 and 6.5;
what is the area 7 Ans. 1 acre, 1 rood, 29.86 rods.
< PBOBLSM VI.
I' . '
To find the circumference of a circle when the diameter is known,
or the diameter when the circumference u known.*
,' The most accurate rule is the well-known theorem that tho
diameter is to the circumference in the ratio of 113 to 355, and
* In strictneas the circumiierence and diameter are not like qoantities, bat
we may Boppoae that a cord is stretched round the circumference, and then
'drawn «ut into a straight line, and its linear units compared with thos^of
tbedianeier.
MENSURATION.
827
Miueqnently the eiroumference to the diameter as 355 to 113.
Now, 855-1-113=3.1416 nearly, and for general purposes, guffioieDl
•oouracj will be attained by this
RULB.
' • . • ' f
: . • 1 1 »
To find the eircumference from a ^ven diameter, multip^^ .t^;
diameter by 3.1416 ; and to find the diameter from a given cirqom*
ference, divide by 3.1416. , ^' \ ■.'.'
49. What is the length round the equator of a 15-inch globcf?
Ana. 47.124 idcheli.'
$-•>■■ 50. If a round log has a eiroumference of 6 feet, 10 inches ; whai
is its diameter? Ans. 2 feet, 2^*0 inches nearly.
51. If we take the distance from the centre of the earth to tho
equator U. be 8979 ; what is the number of miles round the equator?
Ans. 25001 nearly*
'^ '-> ^ PROBLBK VII.
To find the area of a circle.
1. If the circumference and diameter are known, —
Multiply tJhe circumference by the diameteTf and take one-fourA
of the product.
2. If the diameter alone is given, —
" * Multiply the tquare of the diameter by .*l2Xt^.
3. If the circumference alone bo given, —
M%dtiply the tqvMre of the nun^er denoting t?te circumferenu by
.07958.
52. If the diameter of a circle is 7, and the eircumference 22 {
vhat is the area? Ans. 38^
., , 53. What k the area of a circle, the radius of which is 3§ yds?
»' Ans. 3| square yardf.
54. If a semicircular arc be denoted by 10.05 ; what is the are*
of the circle ? Ans. 289.36.
55. If the diameter of a grinding stone be 20 inches; what
•uperficial area is left when it is ground down to 15 inches diameter,
and what superficial area has been worn away ?
Ans. 176.715 sqr. inches left, and 137.445 worn away.
56. If the ehord of an arc be 24 inches, and the perpendiculat
on it from the centre 11.9 ; what is the area of the circle ? '
Ans. 2.689804.
828
ABrmimaL
MENSURATION OF SOLIDS.
To find the Bolid contents of a parallelopiped, or any r^larlj
box-shaped hody :
Lot it bo required to find tho number of cubic feet in a box 8
feet long, 4^ feet broad, and 6^ feet deep.
In tho first place, tho length being 8 feet and tho breadth 4},
the area of the base is 8X4^=36 square feet, and therefore every
foot of altitude, or depth, or thickness, will give 36 cubic feet, and
as there are 6| feet of depth, tho whole solid content will be 36
times 6|, or 243 cubic feet. Uence the
.^. ,.,;^'.^u:.;..r,.v,.iv / BULB. '■' ; . -;•
Take the continual product o/the lengthy hreadthy and depth.
Note.— Let it bo carefully observed that the unit of measure in the case
of solids is to bo talten as a cube, the base of which is a superficial unit used
in the measurement of surfoces. Tlio solid content is indicated by the repe-
tttio 1 of this unit a certain number of times. If the body is of uniform
breadth the rule needs no modification, but if it is rounded or tapering, as a
globe, cone, or pyramid, the calculation becomes virtually to find how much
the rounded or tapering body differs from the one of uniform breadth. Sup-
pose, for example, we take a piece of wood 6 feet high, in the form of a
pyramid, and having the length and breadth of tho btwe each 6 fleet, then the
area of the base is 36 ; but if, at tho height of 1 foot, the dimensions have each
diminished by 1 foot, the area is 25 ; at another foot higher it is 16 ; at the
next 9 ; at the next 4 ; at tho next 1 ; and at tho 6th 0, t. e., it has come to a
point, and the calculation is, how much remains from the solid cube after bo
much has been cut off each side as to give it this form.
This gives rise to tho following varieties : v.. , ,.^,t
I. To find the solid contents of a cone or pyramid :
Multiply the area of the bote hy the perpendicular height^ and
take one-third of the product. " "*' ' r'f;:> m*^* ! ;
II. To find tho solid contents of a cylinder or prism : ^ ' ''
Multiply the area of the hate by the perpendicular Ju^ht.
III. To find the surface of a sphere : ^^: ^ >
Multiply the square of the diameter by 3.1416. >- *' •
TV.. To find the solid contents of a globe or sphere :
MiUtiplv the third power of the diameter 2y .5236. ''- /^^'^
■■rs ,
'tn'
MZNSUIUXION or BOUDB.
329
V. To find the volume of a spheroid, the axes being g*Ten :
Mult^jf the tjuare of tht qsm ^r«^k^iof^ bjf the Jixed axitf
•wiidthejaroduUbjfJiZiQ.
l*f: '
EXER0ISB8.
57. If the diameter of the base of a cylinder be 2 feet, and its
height 5 feet, what is tho solid content ? Ans. 25.708 feet:
58. If the diameter of the base of a cone be 1 foot 6 inches, and
the altitude 15 feet, what are the solid contents ?
Ans. 8 feet, 120 inches.
59. If the diameter of the base of a cylinder be 7 feet, and the
height 5 feet, what is the solid content ? ' Ans. 245 cubic feet.
60. What are the solid contents of a hexagonal prism, each side
of the base being 16 inches and the height 15 feet?
Ans. 69.282 cubic feet:
61. A triangular pyramid is 30 feet high, and each side of the
base is 3 feet ; required tho solid contents. Ans. 39.98 cubic feet.
62. What are the solid contents of the earth, the diameter being
taken as 7918.7 mUes ? Ans. 2C '>992732079.87.
63. In a spheroid the less axis is 70 and the greater 90 ; what
are the solid contents ? Ans. 230907.6.
PILING OF BALLS AND SHELLS.
Balls are usually piled on a base which is either a triangle, or
square, or rectangle, each side of each course containing one ball
less than the one below it.
If the base is an equilateral figure, the vertex of a complete pile
will be a single ball ; but if one side of the base be greater than the
contiguous one, the vertex will be a row of balls. Hence, if the base
be an equilateral figure, the pile will be a pyramid, and as the side
of each layer contains one layer less than the oiie below it, the
number of balls in height will be the same as the number of balls ia
one side of the lowest layer. If the pile be rectangular, each layer
must also be rectangular, and the number of balls in height will bo
the same as the number in the less side of the base. If the base
be triangular, we have the
880
•*^ ABUBmriCL
BULL
.til.' :')
1'
Multiply ih0 mmher on ofM nd» of the (oMofii row Sy ittelfVLUB
one, and the product by the tame ban row FLUl <I0O, and divide ih»
HMuUhytix.
For a eomplete sqaaro pilo we have tho
'■L^i
BULS.
Multiply the nundier o/halle in one eide of the lowat eourte by
itujf FLUB onCf and thie product by double the fint muUiplier PLUS
one, and take one-tixth of the reeult. j , m
If the pile be rectangular, we have tho • '>^
BULB.
. From three time$ the number of balls in the length of the hnoett
tourte aubtract one leu than Me nun^er in the breadth of the tame
eourte ; multiply the remainder by the breadth^ and thit product by
ime-tizth the breadth plus one.
If the pile be incomplete, find what it to&uld be if complete ; find
alto what the incomplete one would be at a teparatt pile^ andnh*
tract the latter from the former. >^ii.
BXBB0I8B8.
64. In a eomplete triangular pile eaoh side of the base is 40 ;
bow many balls are there ? Ans. 11480,
65. In each side of the base of a square pile there are 20 shells ;
how many in the whole pile ? Ans. 2870.
66. In a rectangular pile there are 59 balls in the length, and 20
in the breadth of the base ; how many are in all ? Ans. 11060.
67. In an incomplete triangular pile, each side of the lowest layer
consists of 40 balls, and the side of the upper course of 20 ; what
is the number of balls 7 . „v Ans. 10150.
f riv"^.'/-* ■
Nora.— Since the upper ooune b 30. the first row In the wanting part
voold be 19.
..I f.
•i.Sffi*i^', i-^t; , *Ji;. fi'.S </-i^ .k, ^/t:^ :^«tj< :<>•.$> ,%^A«MM^-.,.i!fS?*v4;afe- .^^iJj: j^;^
^{-r
vM-;-:?iiti'.i.ji-'
■'v-'A. •
XZABUBEMZNT 07 TDCBIB.
881
wOU
MEASUREMENT OF TIMBER.
.i\
Timber is moMured sometimes by the square foot, and sometimei
by the oubio foot.
Cleared timber, such as planks, beams, &o., are usually measured
by the square foot. . „
What is called board measure is a certain length anu breadth,
and a uniform thickness of one inch.
Large quantities of round timber are often estimated by the ton.
To find either the superficial extent or board measure of a plank,
&o.
RULE.
Multiply the length in feet hy the breadth in incites, and divide
6yl2.
Note.— The ihickness being taken uniformly as one inch, tlie rule for find-
ing the content!! in square feet becomes the same as that for finding surfaoe.
If the thickness bo not an inch,—
' Multiply the hoard meature hy the thickness.
- If the board be a tapering one, take half the sum of the two *
extreme widths for the average width.
If a one-inch plank be 24 feet long, and 8 inches thick, then wo
have 8 inches equal f of a foot, and f of 24 feet=16 feet.
A board 30 feet long is 26 inches wide at the one end, and 14
inches at the other, hence 20 is the mean width, t. «., 1^ feet, and
30xlf=50 ; or, 30x20=600, and 600-j-12='50.
To find the solid contents of a round log when the girt is known.
BULK. . ■ , ;, .,,...■, ;-.;., '"'
Multiply the square of the quarter girt in inches by tfie length in
feet, and divide theproduct by 144. • ; * a^ ^^ * i 'v
' If a log is 40 inches in girt, and 30 feet long, the solid contents
will be found by taking the square of 10, the quarter girt in inches,
which is 100, and 100x30=3000, and 3000-f-144=20|.
To find the number of square feet in round timber, when Um
mean diameter is given. "^ , . . ^ i. ;
.iL'.^
'>.il
382
ABUHMma
lULB.
MuUijpljf tki diamtiet in ineha iy ka^ the diamder in inehm,
€»d theprodvct bjf the length in/eet, and divide the retuU 2y 12.
If A log is 30 feot long, and 66 inches mean diuMter, the number
of squue feet is 66 X 28 X 30-h12=:3920 feet.
To find the solid contents of a log when tho length and mean
diameter are given.
BULB.
Multiply the tquare of half the diameter in inchee ly 3.141 6, and
thie product hy the length in feet, and divide by 144.
68. How many cubic feet are there in a piece of jnbor 14x18,
and 28 feet long ? Ans. 49-f cubic feet.
69. How many cubic feet are there in a * >d log 21 inches in
diameter, and 40 feet in length ?
70. What are the solid contents of a log 24 inches in diameter,
and 34 feet in length ? Ans. 106.81 -j-cubic feet
71. How many feet, board measure, are there in a log 23 inches
in diameter, and 12 feet long ? Ans. 264^.
72. How many feet, board measure, are there in a log, the
diameter of which is 27 inches, and the length 16 feet. Ans. 486.
73. What are the solid contents of a round log 36 feet long, 18
inches diameter at one end, and 9 at the other?
74. How many feet of square timber will a round log 36 inches
in diameter and 10 feet long ield ? Ans. 640 solid feet.
76. How many solid feet are there in a board 16 feet long, 6
inches wide, and 3 inches thick ? Ana. 1/^ cubic feet^
76. What are the solid contents of a board 20 feet long, 20
inches broad, and 10 inches thick ? Ans. 27} feet,
77. What is the solid content of a piece of timber 12 feet long,
16 inches broad, and 12 inches thick ? Ans. 16 feet.
78. How many cubic feet are there in a log that is 26 inchcff in
diameter, and 32 feet long ?
79. How many feet, |)oard measure, do^ a log 28 inches in
diaiaater, and 14 feet in length contain ? Ans. 467|.
80. How many cubic feet are contained in a piece of squared
timber that is 12 by 16 inches, and 47 feet in length 7 Ans. 62|.
XEABtmZMEin* 07 TIMBER.
888
31. How many foot, board moaauro, nro there in 22 one-inob
boarda, eaeh being 13 inchea in width, and 16 feet in length ?
Ana. 381|.
BALIB, BiNa, ao.
Aa bales are UHually of tho aamc form as boxes, tbo aomo rule
applies.
82. Honoo, a balo measuring 4^ inohos in length, 33 in width,
and 3^ in doptb, is, in Holid content, 37^ fooC.
83. A orate is 5 foot long, 4| broad, and 3yT} doop, what is the
solid content ? Ans. 85:^g.
To find how many bushels are in a bin of groin :
BULB.
Find the product of the length, breadth and depth, and divide
hjf 61ft0.4.
84. A bin consists of 12 compartments ; each measures 6 feet 3
inches in length, 4 foot 8 inches in width, and 3 foot 9 inches in depth j
how many bushels of grain will it hold ? Ans. 1055, nearly.
To find how- many bushels of grain are in a oonicol heap in the
middle of a floor :
'■«■■ ^'' •
'• . " ' '-•'■' BULE . ■•''".•■
'. I ■.■■..•■'. ,
Multiply the area of the bate by one-third the height.
The base of such a pile is 8 feet diameter and 4 feet high ; what
is tho content ?
The area of tho base is 64X. 7854=83.777, and 83.777xi=-
67.02, tho number of bushels.
If it be heaped against a wall take half the above result.
If it be heaped in a corner, take one-fourth the aboye result.
^f^v^-^. .
-taf;
■;!
-;-■ ■-■' ._. .,
T (i .»i -.
/.-.t*^- e'V h:-i:'^:r--f,-' ' ,•
•-».l4 •'.:> -•.•!-•» „
•
,1
-.»,
- ■*! -^^ -
— •
- ■ --•,,-;;-= ,-7 — -
>• ..- 1
' -'Hfii ' .
r '•'!,-._.. .) -
■^. I . ^.i,v v)!' •
i-f 1J-! - .;.
: ■•»f j .<mti
..'IPa'. u\i-l]t,^:: ■ -j if
■; •■ T/<v ;
•.i.w- ;-•,..
,. ,' , T< >'Jl%i,.- ^>ij..'.
■■^^M ■
^.5«?.'"eji'
Vl^ flS
n -ir
Krfffxyy,
884
ABUHlCEnO.
MISCELLANEOUS EXEBCISEB.
\
1. What number b that f and f of which make 255 ?
Ans. 201-/^,.
2. What must be added to 217^, that the sum may be 17^ timea
19J? Ans. 118f
3. What sum of money must be lent, at 7 per cent., to accumu-
late to $455 interest in 3 months ? Ans. $26000.
4. Divide $1000 amcng A, B and C, so that A may have $156
more than B, and B $62 less than C.
Ans. A. $41 C§; B, $260§; C, $322f.
5. Where shall a pole 60 feet high be broken, that the top may
rest on the ground 20 feet from the stump ? Ans. 26§ feet.
6. A man bought a horse for $68, which was | as much again as
he sold it for, lacking $1 ; how mu«b did ho gain by the bargain ?
• > . Ans. $12.50.
7. A fox is 120 leaps before a hound, and takes 5 leaps to the
hound's 2; but 4 of the hound's leaps equal 12 of the fox's ; how
many leaps must the hound take to catch the fox? Ans. 240.
8. A, B and C can do a certain piece of work in 10 days ; how
long will it take each to do it separately, if A does i^ times as much \^
as B, and B does ^ as much as G ?
Ans. A, 30 days ; B, 45 ; C, 22^.
9. At what time between five and six o'clock, are the hour and
minute hands of a clock exactly together ?
Ans. 27 min., 16y^ sec. past 5.
10. A courier has advanced 35 miles with despatches, when a
second starts with additional instructions, and hurries to overtake
the first, travelling 25 miles for 18 that the first travels ; how far
will both have travelled when the second overtakes the first ?
• T" Ans. 125 miles.
; 11. What is the sum of the series | — ts"^^! — iVs+i^s — ^^' ^
Ans. 2*5.
12. If a man earn $2 more each month than he did the month
before **od finds at the end of 18 months that the rate of increase
will §D te him to earn the same sum in 14 months ; how much did
lie earc in the whole time ? . Ans. $4032.
13. How long would it take a body, moving at the rate of 50
mSCELLAMEOFS EZEBCISES.
885
*.■
miles an hour, to pass over a space equal to the distance of the earth
firom the sun, t. e., 95 millions of miles, a year being 365 days ?
Ans. 21C years, 326 days^ 16 hours.
14. Two soldiers start together for -a certain fort, and one travels
18 miles a day, and after travelling 9 days, turns back as far as the
second had travelled during those 9 days, he then turns, and in 22^
days from the time they started, arrives at the fort at the same time
as his comrade ; at what rate did the second travel ?
Ans. 18 miles a day.
15. What quantity must be subtracted from the square of 48, so ^^
that the remainder may be the product of 54 by 16 ? Ans. 1440. -"•"^^
16. A father gave J of his farm to his tion, the son sold § of his
share for $1260 ; what was the value of the whole farm ?
Ans. $5040.'
17. There were | of a flock of sheep stolen, and 672 were left;
how many were there in all ? Ans. 1792.
18. A boy gave 2 cents each for a number of pears, and had 42
cents left, but if he had given 5 cents for each, he would have had
nothing left. Eequiifed the number of pears. Ans. 14.
19. Simplify p-
-■ ■^"^2Ti* Ans. f.
20. A man contracted to perform a piece of work in 60 days, he
employed 30 men, and at the end of 48 days it was only half finish-
ed; how many additional hands had to be employed to finish it
in the stipulated time? : .
21. A gentleman gave his eldest daughter twice as much as his
second, and the second three times as much as the third, and the \^
third got $1573 ; how much did he give to all ? Ans. $15730. '\'^
22. The sum of two numbers is 5643, and their difference 125 ; -
what are the numbers ? Ans. 2884 and 2759.
23. How often will all the four wheels of a carriage turn round
m going 7 miles, 1 furlong, and 8 rods, the hind wheels being each \.
7 feet 6 inches in circumference, and the fore wheels 5 feet 7^ inches ?
Ans. 23716.
24. What is the area of a right angled triangular field, of which
•the hypotenuse is 100 rods and the base 60? Ans. 240'^ sq. rds.
25. Simplify 5i=5i„fli±MI of. ?1±11-
Ans. l^f.
836
ABXIHMETIO.
26. Find the value of i_|._-L_' Ans. |.
27. If § of A's age is ^ of Bs', and A is 37|, vrhA ago is B ?
Am. 40.
28. What is the excess of ■^^-
'— above «iT.-i~r«'«-r ?
TOT
U7Tf~n:uuT
Ans. -rISf ,.
29; The sum of two numbers is 5330 and their difference 1999 ;
what are the numbers ? Ans. 3664^ and 1665|.
30. A person being asked the hour of the day, replied that the
time past noon was equal to one-fifth of the time past midnight ;
what was the time? Ans. 3 P,M.
31. A snail, in getting up a pole 20 feet high, climbed up 8 feet
every day, but slipped back 4 feet every night; in what time did he
reach the top ? Ans 4 days.
32. What number is that whose jj^, ^, and ^ parts make 48?
Ans. 44^*5.
33. A merchant sold goods to a certain amount, on a commission
of 4 per cent., and, having remitted the net proceeds to the owner,
received ^ per cent, for immediate payment, which amounted to
$15.60 ; what was the amount of his commission ? Ans. $260.
34. A criminal has 40 miles the start of the detective, but the
detective makes 7 miles for 5 that the fugitive makes ; how far will
the detective have travelled before he overtakes the criminal ?
Ans. 140 miles.
35. A man sold 17 stoves for $153; for the largest size he
received $19, for the middle size $7, and for the small size $6 ; how
many did he sell of each size ?
Ans. 3 of the large size, 12 of the middle, 2 of the small.
36. A merchant bought goods to the amount of $12400 ; $4060
of which was on a credit of 3 months, $4160 on a credit of 8 months
and the remainder on a credit of 9 months ; how much ready money
would discharge the debt, money being worth 6 per cent. ?
\.,,t;,..:..v':>,..i:"r.-- ■,:.:_. ..•.. ........--.,,.....,.-.. ■Ans.$i2000.
37. If a regiment of soldiers, consisting of 1000 men, are to be
dothed, each suit to contain 3f yards of cloth that is If yards wide,
land to be lined with flannel 1| yards wide ; how many yards will it
&ke to line the whole ? Ans. 5625.
38. Taking the moon's diameter at 2180 miles, what are the
folid contents? Ans. 5424617475-f sq. miles.
MiSGELLAKEOnS EX£BCISES.
887
39. A certain island is 73 miles in circumference, and if two men
fitart out from the same point, in the same direction, the one walking
at the rate of 5 and the other at the rate of 3 miles an hour; ia
whai time will they come together ? Ans. 3U hours, 30 minutes.
40. A circular pond measures half an acre ; what length of cord
will be required to reach from the edge of tlie pond to the centre ?
Ans. 83263-j- feet
41. A gentleman has deposited $450 for the benefit of his son,
in a Savings' Bank, at compound interest at a half-jearly rate of 3^-
per cent. He is to receive the amount as soon as it becomes
$1781.60^. Allowing that the deposit was made when the son was
1 year old, what will be his age when he can come in possession of
the money ? Ans. 21 years.
42. The select men of a certain town appointed a liquor agent,
and furnished him with liquor to the amount of $825.60, and cash,
$215. The agent received cash for liquor sold, $1323.40. He paid
for liquor bought, $937 ; to the town treasurer, $300 ; sundry ex-
penses, $29 ; his own salary, §205 ; }ic delivered to indigent persons,
by order of the town, liquor to the amount of $13.50. Upon taking
stock at the end of the year, the liquor on hand amounted to
$616.50. Did the town gain or lose by the agency, and how much;
has the agent any mopey in his hands belonging to the town ; or
does the town owe the agent, and how much in either case ?
Ans. The town lost $103.20 ; the agent owes the town $7.4U.
43. A holds a note for $575 against B, dated July 13th, paytc
ble in 4 months from date. On the 9th August, A received in
advance $62 ; and on the rtth September, $45 more. According to
the terms of agreement it t;ill be due, adding 3 days of grace, on the
16th November, but o|i ih'i 3rd of October B proposes to pay a sum
which, in addition to the sums previously paid, shall extend the pay
day to forty days beyontl the IGth of November; how much must B
pay on the 3rd of October? Ans. $111.43.
44. A accepted .'>ii agency from B to buy and sell grain for him.
A received from B grain in store, valued at f 135.60, and cash,
$222.10 ; ]:c beuj^/it grain to the value of $1346.40, and sold grain
to the amcant of $1171.97. At the end of four months B wished
to close the agency, and A returned him grain unsold, valued at
$437.95 ; A was to receive for «er/ices, $48.12. Did A owe B, pr
B owe A, and how much ? Ans. B owed A 45 tHtHm
888
ABITHMETIO.
45. A general ranging his men in the form of a sqnaret had 59
men over, but having increased the side of the square by one man,
he looked 84 of completing the square ; how many men had he ?
Ans. 5100.
46. What portion, expressed as a common fraction, is a ponnd
and a half troy weight of three pounds avoirdupois ? Ans. ■^^^,
47. What would the last fraction bo if wo reckoned by the ounces
instead of grains according to the standards ? Ans. |.
48. If 4 men can reap 6^ acres of wheat in 2 J days, by working
8J hours per day, how many acres will 15 men, working equally,
reap in 3J days, working 9 hours per day ? Ans. 40 j-J days.
49. Out of a certain quantity of wheat, ^ was sold at a certain
gain per cent., ^ at twice that gain, and the remainder at three times
the gain on the first lot ; what was the gain on each, the ga:': . on the
whole being 20 per cent. ? Ans. 9|, 19 J and 28| per cent.
50. If a man by travelling 6 hours a day, and at the rate of 4^
miles an hour, can accomplish a journey of 540 miles in 20 days ;
how many days, at the rote of 4f miles an hour, will he require to
accomplish a journey of 600 miles ? Ans. 21 1.
51. Smith in Montreal,- and Jones in Toronto, agree to exchange
operations, Jones chiefly making the purchases, and Smith the sales,
the profits to be equally divided ; Smith remitted to Jones a draft
for $8000 after Jones had made purchases to the amount of
$13682.24 ; — Jones had sent merchandise to Smith, of which the
latter had made sales to the value of $9241.18 ; Jones had also made
sales to the worth of $2836.24 ; Smith has paid $364.16 and Jones
$239.14 for expenses. At the end of the year Jones has on hands
goods worth $2327.34 and Smith goods worth $3123.42. The term
of the agreement having now expired, a settlement is made, what has
been the gain or loss ? What is each partner's share of gain or
loss i '. What is the cash balance, and in favor of which partner ?
52. In a certain factory a number of men, boys and girls are
employed, the men work 12 hours a day, the boys 9 hours and the
^Is 8 hours ; for the same number of hours each man receives a
half more, than each boy, and each boy a third more than each
g^r; the' sum paid each day to all the boys is double the sum
paid to all thei girls, and for every five shillings earned by all
^e bdysj ekoh dby, twelvei shillings are earned by all the men; it
laSOELLANEOnS ISXEBCISES.
339
is required to find the number of men, the number of boys and the
number of girls, the whole number being 59.
Ans. 24 men, 20 boys and 15 girls.
53. A holds B's note for $575, jiayable at the end of 4 months
from the 13th July ; on the 9th Augmt, *A received $62 in advance,
as part payment, and on the 5th September $45 more ; according to
agreement the note will not be duo till IGth November, three days
of grace being added to tho term ; but on the 3rd October B tenders
such a sum as will, together with the payments already mide, ex-
tend time of payment forty days forward ; how much must B pay on
the 3rd of October ? Ans. $111.43.
54. If a man commence business with a capital of $5000 and
realises, above expenses, so much as to increase his capital each year
by one tenth of itself less $100, what will his capital amount to in
twenty years? - Ans. $27910.
55. A note for $100 was to come due on the 1st October, but
on the 11th of August, the acceptor proposes to pay as much in ad-
vance as will allow him GO days after the 1st of October to pay tho
balance; how much must he pay on the 11th of August ? Ans. $54.
56. A person contributed a eert:?in sum in dollars to four char*
ities ; — to one he gave one half of the whole and half a dollar ; to a
second half the remainder and half a dollar ; to a third half the re-
mainder and half a dollar; and also to the fourth half the remainder
and half a dollar, together with one dollar that was left ; how much
did he give to each ?
Ans. To the first, $16; to the second, $8; to the third, $4; to
the fourth, $3.
57. A farmer being asked how many sheep he had, replied that
he had them in four different fields, and that two-thirds of the num-
ber in the first field was equal to three-fourths of the number in tho
second field; and that two-thirds of the number in the second
field was equal to three-fourths of the number in the third field ; and
that two-thirds of the number in tho third field was equal to four-
fifths of the number in the fourth field ; also that there were thirty-
two sheep more in the third field than in the fourth ; how many
sheep were in each field and how many altogether ?
Ans. First field, 243; second field, 216; third field, 192i
fourth field, 160. Total. 811.
340
AETTHMETIC.
*.,-
68. How many hours per day must ?A7 men work lor 5 J days
to dig a trench 23J- yards long, o^ yarda vido, and 2^ deep, if 24
men working equally can dij; one 3.} J yarda long, 5} vvidj, and iih
deep, in 189 days of l-i hours caoli. Ans. IG liwurd.
59. A man bequeathed one-fourth of his property to his oldest
son ; — to the second son one-fourth of tiio remainder, and $350 be-
sides ; to tho third one-fourth of tlio remainder, togetlicr with $975 ;
to tho youngest ono-fourth of tho remainder and $1400 ; ho gives
his wife a life interest in tho remainder, and her share is found to
be one-fifth of tho whole j what was tho amount of tho property ?
■ • . Ans. $20,000.
60. Five men formed a partnership which was dissolved after
four years* continuance ; the first contributed $G0 at lirsfc and $S00
more at tho end of five months, and again $1500 at the end of a
year and eight months; tho second contributed $G00 and .<?1800
more at the end of six months; the third gave at first $!00 and
$500 every six months; the fourth did not contribute till the end of
eight months ; he then gave $900, and the same sum every six
months; tho fifth, having no capital, contributed by his labor in keep-
ing tho books at a salary of $1.25 por day; at tho expiration of tho
partnership what was the share of each, tho whole gain having been
$20000 ?
61. Four men. A, B, C, and D, bought a stack of hay containing
8 tons, for $100. A is to have 12 per cent, more of the hay than B,
B is to have 10 per cent, more than 0, and C is to have 8 par cent.
more than D. Each man is to pay in proportion to the quantity ho
receives, Tho stack is 20 i'oet high, and 12 feet squtiro at its base,
it being an exact pyramil ; and it is agreed tliat A shall take his
share first from the top of the stack, B is to take his share the next,
and then and D. IIow many foot of the perpendicular height of
the stack shall each take, and what sum shall each pay ? ,
Ans. A. takes 13.22 -[-ft., and pays $23.9:1^; B takes .^.U-j-ft.,
and pays $25.83^; takes 2.0G+ft., and pays $23.48^ ;
__.: D takes 1.58-f ft., and pays $21.71-^.
62. A merchant bought 500 bushels of wheat and sold one half
of ft at 80 cents per bushel which was 10 per cent more than it
insoELLAinsons ezzboisea
841
cost him, and 5 por oont. less than ho asked for it. Ho sold tho
remainder at 12^ por cent, more than it oost him. What was his
asking price for both lota ? What did ho rcocivo for tho last lot,
and how much did ho gain on the whole ?
63. May 1st, 1862, 1 got my noto for $2000 payable in 4 months
discounted at a bank, and immediately invested tho money received
in woodland. November 9 th, I sold the land at an advance of 15
per cent., receiving | of the price in cash, and a note for tho
remainder, payable August 10, 1864, without grace, and to be on
interest after January 1, 1864, at 7 por cent. I lent the cash re-
ceived at 6 per cent. When my noto at tho bank became due I
renewed it for tho same time as before, and at tho proper time
renewed it again, aud finally renewed it for such a time that the
note would become duo August 10, 1864. Now, if I paid 6 per cent,
on the money borrowed at the bank, and made a complete settlement
August 10, 1864, what was the amount of my gains ?
64. My agent at Mobile buys for me 500 bales of cotton, avei
aging 500 lbs. per bale, at 10 cents per pound. I pay him 1^ per
cent, on the amount paid for the cotton, and shipping charges at 60
cents from January 1 for an amount sufficient to pay for the cotton,
charges and commission including also 2 per cent, discount on the
draft. On tho receipt of the invoice, I insure for the amount of the
draft plus 10 per cent. ; I pay 1^ per cent, premium on the amount
insured, and from tho amount of the premium is discounted 1^ per
cent, for cash. On the arrival of the cotton I pay f of a cent per
pound for freight, and 5 per cent, primage to the captain on the
freight money, and also 4 cents per bale for wharfage. I sell it of.
the wharf, January 20. at $1 per bale profit, and agreed to take in
payment the note of the purchaser for 6 months from January 20.
What amount would be received on the note when discounted at a
bank at 7 per cent. ?
842
ABiisMxna
FOEEiaN GOLD COINS.
MINT TAICI.
conrriT.
Australia . ...
«
Austria
(I
ti
Belgium .....
Bolivia
Brazil
Gentr'l America
Chili
Denmaric
Equador
England
France
Germany, North
u «
« «
Germany, Sontb
Greece
HindoBtau
Italy
Japan
«<
Mexico
Naples
Netherlands....
New Granada..
« «
u u
Pern
Portugal
Prussia
Rome
Russia
Spain
8^den
Tunis
Turltey
Tuscany
BMOWXATIORI.
Pound of 1852
Sovereign 1855-60
Ducat
Souverain. ^
NewUnion Crown (assumed)
Twenty-five francs
Doubloon •
20 Milreis
Two escudos
Old doubloon
Ten Pesos
Ten thaler
Four eseiMlos
Pound or Sovereign, new .
Pound orSoverign,averago
Twenty francs, new
Twenty francs, average. . .
Ten thaler
Ten thaler, Prussian
Krone [crown]
Ducat
Twenty drachms
Mohur
20 lire
Old Cobang
New Cobang
Doubloon, average
" new
Six ducati, new.
Ten guilders
Old Doubloon, Bogota
Old Doubloon, Popayan. .
Ten pesos, new
Old doubloon
Gold crown
NewUnion Crown [assumed]
2^ scudi,iiew
Five roubles ,
100 reals
80 reals
Ducat
25 piastres
100 piastres
Seqmn
VIM'
wBonr.
MBSa
rxLvm.
Oi. Dhc.
Tiiotrs.
0.281
916.5
$5.32.37
0.256.6
916
4.85.58
0.112
986
2.28.28
0.363
900
6.75.35
0.357
900
6.64.19
0.254
899
4.72.03
0.867
870
15.59.25
0.575
917.5
10.90.57
0.209
853.5
3.68.75
0.867
870
15.69.26
0.492
900
9.15.35
0.427
895
7.90.01
0.433
844
7.55.46
0.256.7
916.5
4.86.34
0.256.2
916
4.84.92
0.207.6
899.5
3.85.83
0.207
899
3.84.69
0.427
895
7.90.01
0.427
903
7.97.07
0.357
900
6.64.20
0.112
986
2.28.28
0.185
900
3.44.19
0.374
916
7.08.18
0.207
898
3.84.26
0.362
568
4.44.0
0.289
572
3.57.6
0.867.5
866
15.52.98
0.867.6
870.5
15.61.05
0.245
996
5.04.43
0.215
899
3.99.56
0.868
870
15.61.06
0.867
858
15.37.75
0.525
891.5
9.67.51
0.867
868
15.55.67
0.308
912
6.80.66
0.357
900
6.64.19
0.140
900
2.60.47
0.210
916
3.97.64
0.268
896
4.96.39
0.215
869.5
3.86.44
0.111
975
2.23.72
0.161
900
2.99.54
0.231
915
4.36.93
0.112
099
2.31.29
Value after
Deduction.
15.29.71
4.83.16
2.27.04
6.71.98
6.60.87
4.69.67
15.5l.4G
10.85.12
3.66.91
15.61.47
9.10.78
7.86.06
7.51.69
4.83.91
4.82.60
3.88.91
3.82.77
7.86.06
7.93.09
6.60.88
2.27.14
3.42.47
7.04.64
3.82.34
4.41.8
3.55.8
15.45.22
15.53.25
5.01.91
3.97.67
15.63.26
15.30.07
9.62.68
16.47.90
6.77.76
6.60.87
2.69.17
3.95.66
4.93.91
3.84.51
2.?2.6l
2.98.05
4.34.75
2.80.14
rOBEION BILVEB COINS.
348
FOREIGN SILYEB COINS
MINT VALUB.
,01.91
.97.57
,63.26
,30.07
62.68
47.90
77.76
60.87
59.17
95.66
93.91
84.51
^2.61
98.05
34.75
80.14
COnXTBT
DnOMnfATtORS.
wiionr.
riNiinss.
TlLCk
Austria
Old rix dollar
Oz. Die.
0.902
0.836
0.451
0.397
0.596
0.8'J5
0.803
0.643
0.432
0.820
0.150
0.8G6
0.864
0.801
0.927
0.182.5
0.178
0.800
0.712
0.695
0.340
0.340
0.719
0.374
0.279
0.279
0.867.5
0.866 '
0.844
0.804
0.927
O.L 3
0.866
0.766
0.433
0.712
0.595
0.864
0.667
0.800
0.166
1.092
0..323
0.511
0.770
0.220
Tnocs.
833
902
833
900
900
838
897
903.5
667
918.5
925
850
908
900.5
877
924.5
925
900
750
900
900
900
900
916
991
890
903
901
830
944
877
896
901
909 .
650
750
900
900
875
900
899
750
899
898.5
830
S26
$1.02.27
1.02.64
(i
Old scudo
<>
Florin before 1858
New florin
New Union dollar
Maria There.sttdol'r.nsO
Five franca
51.14
«
48.63
i<
73.01
II
1.02.12
Belgium
98.04
Bolivia
New dollar
79.07
II
Half dollar
39.22
Brazil
Double Milreis
1.02.53
Canada
20 cents
18.87
Central America. . .
Dollar
1.00.19
Cliili
Old Dollar
1.06.79
II
New Dollar
98.17
Denmark
Two rigsdaler
1.10.65
England
Shilling, new
22.96
II
Shilling, average
Five franc, average
Thaler, before 1857
Now thaler
22.41
France
Germany, North...
11
98.00
72.67
72.89
Germany, South. , . .
II
Florin, before 1857
New florin fassumedl . . .
41.65
41.65
Greece
Hindostan
Five drachms
Rupee
88.08
46.62
Japan
Itzebu
37.63
II
New Itzebu
33.80
Mexico
Dollar, new
1.06.62
11
Dollar, average
Scudo
1.06.20
Naples. ...
Netherlands
95.34
2i guild
1.03.31
Norway
Specie daler
1.10.65
New Granada
Dollar of 1857
97.92
Peru
Old dollar
1.06.20
li
Dollar of 1858
94.77
14
Half dollar, 1835-38....
Thaler before 1857
New thaler
38.31
Prussia.
72.68
II
72.89
Rome . / . . .
Scudo
1.05.84
Russia
Rouble
Five lire
79.44
Sardinia
98.00
Spain
New pistareen
20.31
Sweden
Rix dollar
1.11.48
Switzerland
Two francs
39.52
Tunis.
Five piastres
Twenty piastres
Florio
62.49
Turkey
86.98
Tuscany
27.60