7^^ ^, £^^
IN MEMORIAM
FLORIAN CAJORl
d
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.ah'tnson'B ^\iaxitx Course.
THE
COMPLETE
AEITHMETIC.
ORAL AND WRITTEl^.
SECOND PART.
By DANIEL W. FISH, A.M.,
▲UTHOB OP BOBINSON'S 8BEIES OP PBOGRBSSIVE ABITHlfETICB
IVISON, BLAKEMAN, TAYLOR & CO.,
- NEW YORK AND CHICAGO
1881.
EOBINSON'S
Shorter Course.
FIJ^ST BOOK IN- ARITHMETIC, Primary,
COMPLETE ARITHMETIC. In One volume,^
COMPLETE ALGEBRA.
ARITHMETICAL PROBLEMS, Oral and Written.
ALGEBRAIC PROBLEMS.
KE YS to Complete Arithmetic and Problems, and
to Complete Algebra and Problems,
in separate volumes, for Teachers,
Arithmetic, ouaz and wkitten, usually taught in
THREE hooks, is now offered, complete and thorough,
in ONE booh, " the complete arithmetic:'
* This Complete Arithmetic is also published in two volumes. PAMT Xm
and TA:RT II, are each bound separately^ and in cloth.
Copyright, 1874, by DANIEL W. FISH.
JDlectrotyped by Smith & McDougal, 82 Beekmau St., N. Y.
PREFACE
rpHE design of the author, in the preparation of the Completb
-^ Arithmetic, has been to furnish a text-book on the subject of
arithmetic, complete not only as a treatise, but as a comprehensive
manual for the class-room, and, therefore, embodying every necessary
form of illustration and exercise, both oral and written. Usually,
this subject has been treated in such a way as to form the contents
of three or more graded text-books, the oral exercises being placed in
a separate volume. In the present treatise, however, the whole
subject is presented in all its different grades ; and the oral, or men-
tal, arithmetic, so called, has been inserted, where it logically and
properly belongs, either as introductory to the enunciation of prin-
ciples or to the statement of practical rules — the treatment of every
topic from the beginning to the end of the book being thoroughly
inductive.
In this way, and by carefully constructed analyses, applied to all
the various processes of mental arithmetic, the pupiFs mind cannot
fail to become thoroughly imbued with clear and accurate ideas in
respect to each particular topic before he is required to learn, or
apply to written examples, any set rule whatever. The intellect of
the pupil is thus addressed at every step ; and every part of the
instruction is made the means of effecting that mental development
which constitutes the highest aim, as well as the most important
result, of every branch of education.
This mode of treatment has not only the advantage of logically
training the pupil's mind, and cultivating his powers of calculation,
but must also prove a source of economy, both of time and money,
inasmuch" as it is the means of substituting a single wlume for an
entire aeries of text-hooks.
IV PREFACE.
As the time of many pupils will not permit them to pursue this
study through all of its departments, the work is issued in two parts,
as well as in a single 'colume. This will, it is thought, be also con-
venient for graded schools^ in supplying a separate book for classes
of the higher and lower grades respectively^ without requiring any
unnecessary repetition or review.
In this, the Second Part, all the higher departments of arith-
metic including Mensuration are presented, commencing with Per-
centage, the study of which can be taken up by the pupil imme-
diately on completing the First Part. This part of the subject
has been treated in a comprehensive manner, and is, in all respects,
adapted to the wants of the present time, recognizing and explain-
ing all the recent changes in Custom-house Business, Exchange, etc.
An Appendix of forty- eight pages of valuable Tables and Problems
has been added to this pa/rt of the work, containing much useful
and practical information, fresh and important^ obtained by much
labor of research and inquiry, which, with many other improve-
ments, particularly adapt this work to the wants of the student
qualifying for business, and of graduating classes in High Schools
and Academies, as well as of Mercantile and Commercial colleges.
The Beviews interspersed throughout the book will be found to
be just what is needed by the student to make his progress sure
at each step, and to give him comprehensive ideas of the subject
as he advances. Carefully constructed Synopses have also been
inserted, with the view to afford to both teacher and pupil a ready
means of drill and examination, as well as to present, in a clear,
concise, and logical manner, the relations of all the different depart-
ments of the subject, with their respective sub4opics, definitions,
principles, and rules.
It is confidently believed that, on examination, the work as a
whole, as well as in its separate parts, will commend itself to teach-
ers and others, by the careful grading of its topics ; the clearness
and conciseness of its definitions and rides ; its improved methods
of analysis and operation; the great number and variety of its
examples, both oral and written, embodying and elucidating all the
ordinary business transactions ; and in the omission of all obsolete
PREFACE. V
teTfns and discarded usages, as well as in the introduction of many
novel features favorable both to clearness and brevity.
Great oains have also been taken to make this work superior to
all others in its typographical arrangement and finish, and in the
general tastef ulness of its mechanical execution.
The author takes pleasure in acknowledging his indebtedness for
many valuable suggestions received from teachers of experience
and others interested in the work of education ; especially to Joseph
Ficklin, Ph. D., Professor of Mathematics in the University of
Missouri, by whom chiefly the sections upon Involution, Evolution,
Progressions, and Annuities have been prepared ; as well as to
Henry Kiddle, A. M. , Superintendent of Schools in the city of New
York, for valuable assistance, especially in the higher departments
of Percentage, and for important suggestions in relation to other
parts of the work.
D. W. F.
Bbookltn, January^ 187S.
"TN order to teach any subject with the best success, the instruc-
-^ tor should not only fully understand it, in all its principles and
details, but should also clearly perceive what particular faculties
of the mind are concerned in its acquisition and use.
Arithmetic is pre-eminently a subject of practical value ; that is,
it is one to be constantly applied to the practical affairs of life.
But this is true only in a limited sense. Very few ever need to
apply to any of the purposes of business more than a small part of
the principles and rules of calculation taught in the text-books.
Every branch of business has its own requirements in this respect,
and these are all confined within very narrow limits.
The teaching of arithmetic must, therefore, to a great extent, be
considered as disciplinary, — as training and developing certain
faculties of the mind, and thus enabling it to perform its functions
with accuracy and dispatch. The following svggestions, having
reference to this twofold object of arithmetical instruction are pre-
sented to the teacher, as a partial guide, not only in the use of this
text-book, but in the treatment of the subject as a branch of
education.
Seek to cultivate in the pupil the habit of self-reliance. Avoid
doing for him anything which, either with or without assistance,
he should be able to do for himself. Encourage and stimulate his
exertions, but do not supersede them.
Never permit him to accept any statement as true which he does
not understand. Let him learn not by authority but by demonstra-
tion addressed to his own intelligence. Encourage him to ask
questions and to interpose objections. Thus he will acquire that
most important of all mental habits, that of thinking for himself.
SUGGESTIOJS^S TO TEACHERS, VU
Carefully discriminate, in the instruction and exercises, as to
wliich faculty is addressed, — whether that of analysu or reasoning,
or that of calculation. Each of these requires peculiar culture, and
each has its appropriate period of development. In the first stage
of arithmetical instruction, calculation should be chiefly addressed,
and analysis or reasoning employed only after some progress has
been made, and then very slowly and progressively. A young
child will perform many operations in calculation which are far
beyond its powers of analysis to explain thoroughly.
In the exercise of the calculating faculty, the examples should be
rapidly performed, without pause for explanation or analysis ; and
they should have very great variety, and be carefully arranged
so as to advance from the simple and rudimental to the complicated
and diflBlcult.
In the exercise of the analytic faculty, great care should be taken
that the processes do not degenerate into the mere repetition of
formulcB, These forms of expression should be as simple and con*
cise as possible, and should be, as far as practicable, expressed in
the pupil's own language. Certain necessary points being attended
to, the precise form of expression is of no more consequence than
any particular letters or diagrams in the demonstiation of geomet-
rical theorems. Of course, the teacher should carefully criticise
the logic or reasoning, not so as to discourage, but still insisting
upon perfect accuracy/ from the first.
The oral or mental arithmetic should go hand in hand with the
written. The pupil should be made to perceive that, except for the
difficulty in retaining long processes in the mind, all arithmetic
ought to be oral, and that the slate is only to be called into requi-
sition to aid the mind in retaining intermediate processes and
results. The arrangement of this text book is particularly favora-
ble for this purpose.
BeflnitioTis and principles should be carefully committed to
memory. No slovenliness in this respect should be permitted. A
definition is a basis for thought and reasoning, and every word
which it contains is necessary to its integrity. A child should not
be expected to frame a good definition. Of course, the pupil should
Tlli SUGGESTIOIJ^S TO TEACHERS.
be required to examine and criticise the definitions given, since this
will conduce to a better understanding of their full meaning.
In conducting recitations, the teacher should use every means
that will tend to awaken thought. Hence, there should be great
variety in the examples, both as to their construction and phrase-
ology, so as to prevent all mechanical ciphering according to fixed
methods and rules.
The Rules and FormulcB given in this book are to be regarded as
summaries to enable the pupil to retain processes previously ana-
lyzed and demonstrated. They need not be committed to memory,
since the pupil will have acquired a sufficient knowledge of the
principles involved to be able, at any time, to construct rules, if he
has properly learned what precedes them.
In the higher department of arithmetic, the chief difficulty con-
Bists in giving the pupil a clear idea of the nature of the business
transactions involved. The teacher should, therefore, strive by
careful elucidation, to impart clear ideas of these transactions before
requiring any arithmetical examples involving them to be per-
formed. When the exact nature of the transaction is understood,
the pupil's knowledge of abstract arithmetic will often be sufficient
to enable hiTn to solve the problem without any special rule.
The teacher should be careful not to advance too rapidly. The
mind needs time to grasp and hold firmly every new case, and then
additional time to bring its new acquisition into relation with those
preceding it. Hence the need of frequent reviews, in order to give
the pupil a comprehensive as well as an accurate and permanent
knowledge of this subject.
The Synopses for Beview interspersed through this work, are de-
signed to afford assistance to the teacher in accomplishing this object.
Each of these Synopses exhibits a brief, but definite, summary of all
that is treated under the particular topic referred to, systematically
and logically arranged, showing not only the different sub-topics, and
their relations to each other and the general subject, but also the
necessary preliminary definitions. Thus the teacher wiU be able
readily to ask an exhaustive series of questions, without having
recourse to every paragraph and page preceding.
SUGGESTIOKS TO TEACHERS. IX
Various useful exercises may be based upon these synopses.
After the pupil has become familiar with their mode of construc-
tion, he may be required to write out, from memory, an outline
synopsis of each section that he has studied, so as to show whether
or not he has comprehended the relations of the various parts of the
subject which he has passed over. Or the whole or a part of a
Synopsis, embracing one or more topics, may be placed upon the
blackboard, and the pupil required to give briefly but accurately
the sub-divisions, definitions, principles, etc., involved in each. By
this means, if further tested by questions, a thorough and well-
classified knowledge of the whole subject will be permanently
impressed upon his mind-
Editions of this book are bound with and witJiout answers. Those
with answers will be sent, unless otherwise ordered.
artfiV'^abk'iitraB
PAGB
Percentage. 1
Profit and Loss 13
Commission 20
Synopsis 28
Interest 29
Problems in Interest. .... 40
Compound Interest 45
Annual Interest 48
Partial Payments 50
Discount 54
Bank Discount 57
Savings Bank Accounts . . 62
Synopsis 65
Stocks 66
Insurance 76
Life Insurance 80
Taxes 84
Synopsis 88
Exchange 89
Foreign Exchange 93
Arbitration of Exchange. . 98
Custom-house Business. . . 102
Equation of Payments 105
Averaging Accounts 110
Synopsis 118
Ratio 119
Proportion 123
Simple Proportion 126
Compound Proportion. ... 131
PAG 1
Partnership 137
Alligation 143
Synopsis 149
Test Problems. 150
Involution 155
EVOLLT^ION, 161
Cube Root 168
Roots of Higher Degree. . 174
Arithmetical Pr(^ression. 175
Geometrical Progression.. 180
Annuities 185
Synopsis 189
Mensuration. 190
Triangles. 191
Quadrilaterals 195
Circles. 197
Similar Plane Figures 199
Solids 203
Prisms. 203
Pyramids and Cones 205
Spheres 208
Similar Solids 209
Gauging 210
Synopsis 212
Metric System. 213
Vermont Partial Pay-
ments 227
Vermont Taxes. . . , 231
Measures and Tables 233
•^¥,
®P^1^g^^^^5"^^5^c75-
O B AT^ EXERCISES,
495. 1. What is ^0 of $100 ? ^f^? t¥o? tVo?
2. What is yf « of $500 ? Of $700 ? Of $1000 ?
3. What is ^lo of $600? ^^? ^? ^?
4. How many hundredths of $100 are $5 ? $7 ? $18 ?
5. How many hundredths of $500 are $25 ? $35 ? $50 ?
496. Percentage is a term applied to computations
in which 100 is employed as afxed measure, or standard.
497. Per Cent, is an abbreviation of the Latin
phrase per centum, which signifies by the hundred.
Thus, 5 per cent, means 5 of every 100, or yf ^, the 5 standing for
the numerator, and the words ''per cent.'' for the denominator 100.
Thus, 25 per cent. = ^^^ or .25.
498. The Sign of Per Cent, is %. It is read per
cent. Thus Q% is read 6 per cent.
2
PERCENTAGE.
What per cent, of a number is -^ of it ? -^ ? .08 ?
.12X? ^? ^? .025? .OOi? .04|? .375? .0325?
499. What per cent, of a number is ^ of it ?
Ai^ALYSis. — Since the whole of any number is ^§, J of the
33-
same is \ of ^g, or :^, equal to 33J%. Hence, etc.
What ^ of a number is i of it ? }? \'i |? f? f?
I? f? ^? if? V A? f*?
500. ^hdii fractional part of a number is 12^^ of it?
Analysis.— 12^% is — ^, or ^%^^, equal to ^. Hence, etc.
What part of a number is 8^^ of it? 16|^? 15^?
20^? 37i^? 7i^? 6i^? 25^? 66f? 75^?
501. What part of a number is -J-^ of it ?
1
Analysis. — \% is -^, equal to ^. Hence, etc.
100
What is i^ of a number? \%1 i%? ^%? i%?
503. Any per cent, may be expressed either as a deci'
mal or as a fraction, as shown in the following
Table.
*er cent.
Decimal. Fraction.
Per cent.
Decimal
Fraction.
1^
.01 Tftr-
Wo
.75
1
2%
.02 ^
100^
1.00
^%
.04 ^
125^
1.25
li
Q%
.06 ^
i%
.005
¥^17
Wo
.10, or .1 r^
i%
.0075
zis
Wo
.20, or .2 i
Wo
.08i
^
25%
.25 J
m%
.125
i
50%
.50 J
m%
.1625
M
PERCENTAGE. 3
WJtITTEN EXEnCISES,
603. Change to expressions having the per cent. sign.
1. .15; .085; .33^; .375; .00| ; H; H; .75f.
^. ^i; A; .oof; h; I; iV; .00125; |; 2|.
Change to the form of decimals,
3. 5i^; 9i^; 20^^; 3^^; i^; 3^^; If^; 112^^.
Change to the form of fractions,
4. 24^; 1^; 6^^ ; 37^^; |^ ; 3^^; 120^; 75^.
504. In the applications of percentage, at least three
elements are considered, viz. : the Rate, the Base, and the
Percentage, Any two being given, the other can be found.
505. The Rate is the number per cent, or the num-
ber of hundredths. Thus, in b%, .05 is the rate. Hence,
Bate per cent, is the decimal which denotes how many hundredths
of a number are to be taken or expressed.
506. The Rase is the number of which the per cent,
is taken.
Thus, in the expression, 5^ of $15, the 'base is $15.
507. The Percentage is the result obtained by
taking a certain per cent, of the base.
Thus, in the statement, 6 % of $50 is $3, the rate is .06, the base
$50, and the 'percentage is $3.
508. The Amount is 'the sum of the base and the
percentage.
Thus, if the base is $80, and the percentage $5, the amount is
$80 + $5 = $85.
509. The Differ ence is the remainder found by
subtracting the percentage from the base.
Thus, if the base is $80, and the percentage $5, the difference is
$80 - $5 = $75.
4 PERCENTAGE.
510. The base and rate being given to find the
percentage.
OMAL EXEItCISES.
1. What is 10% of 140 ?
Analysis.— 10^ is y^^ = ^, and f^ of 140 is 14 Hence 10%
of 140 is 14.
What is
2. 5^ of $80?
3. 7^ of 200 lb. ?
4. 6^ of 150 men ?
5. 25^ of 120 mi. ?
Find the amount
10. Of 100 A. +27^.
11. Of $75 + 5^.
12. Of 32doz. + 12J^.
How much is
6. 12^^ of 72 gal. ?
7. 40^ of 60 sheep ?
8. S% of 50 bu. ?
9. 50^ of $240 ?
Find the difference
13. Of 90 hhd. — 10^.
14. Of 63 Cd. - 33^^.
15. Of $200 — 2i^.
16. A farmer had 150 sheep, and sold 20^ of thera.
How many had he left ?
17. A mechanic who received $20 a week had his sal-
ary increased %%. What were his daily wages then ?
18. From a hhd. of molasses containing 63 gal. 33^^
was drawn. How many gallons remained ?
19. A grocer bought 150 dozen eggs, and found 16f^
of them bad or broken. How many were salable ?
20. A train of cars running 25 miles an hour increases
its speed 12^^. How far does it then run in an hour ?
511. Prikciple. — The percentage of any number is
the same part of that iiumher as the given rate is of 100^.
PERCENTAGE.
WRITTEN JEXBRCIS ES.
513. 1. What is 17^ of $4957 ?
OPERATION.
$4957
$842.69
Analysis. — Since 17% is .17, the required
percentage is .17 of $4957, or $4957 x .17, which
ill is $842.69.
What is
2. 35^ of 695 lb. ?
3. 75^ of $8428 ?
4. 12^^ of £2105 ?
Rule. — Multiply the iase iy the rate,
^art of the base as the rate is of 100^.
This rule may be briefly expressed by the following
Formula. — Percentage = Base x Bate.
Eind
5. 33^% of 8736 bu.
6. ^% of $35000.
7. 120^ of $171.24.
Or, take such a
What
Find
8. Is4|^of 312.8rd.?
13.
84^ of 354 bu.
9. Is 105^ of $5728?
14.
85^ of -J of a ton.
10. Is $3140.75 + 11^?
15.
ifc of 16400 men.
11. Is2|mi. + 7i^?
16.
f ^ of 1 of a year.
13. Is 400 ft. -3i^?
17.
f ^ of if of a hhd.
18. The bread made from a barrel of flour weighs 35^
more than the flour. What is the weight of the bread?
19. A man having a yearly income of $4550 spends 20;^
of it the first year, 25^ of it the second year, and dll^% of
it the third year. How much does he save in 3 years ?
20. A man receives a salary of $1600 a year. He pays
18^ of it for board, S^% for clothing, and 16^ for inci-
dentals. What are his yearly expenses, and what does he
save ?
6 PEECEKTAGE.
21. A man owning |^ of a cotton-mill, sold d6% of his
share for $24640. What part of the whole mill did he
still own, and what was its value ?
22. Smith had $5420 in bank. He drew out 16% of it,
then 20^ of the remainder, and afterward deposited 12|-^
of what he had drawn. How much had he then in bank ?
513. The base and percentage being given to find
the rate.
omatj exercises.
1. What per cent, of 25 is 3 ?
Analysis— Since 3 is ^^ of 25, it is /^ of 100% , or 12^ . Hence,
Sis 12% of 25.
What per cent.
2. Of 24 is 18 ?
3. Of $16 are $4 ?
4. Of200 figs are 20 figs?
5. Of 40 lb. are 15 lb. ?
6. Of 12i bu. are 2|^ bu. ?
7. Of 2 A. are 80 sq. rd. ?
8. Of 1 da. are 16 hr.?
16. f of an acre is what per cent, of it ?
17. f of a cargo is what per cent, of it ?
18. 2^ times a number is what per cent, of it ?
19. If $6 are paid for the use of $30 for a year, what is
the rate per cent. ? ^ '
20. If a milkman adds 1 pint of water to every gallon
of milk he sells, what per cent, does he add ?
514. Principle. — The rate is the number of hundredths
which the percentage is of the base.
What per cent.
' 9. Are 6^ mi. of 12imi.?
10. Are 18 qt. of 30 qt. ?
11. Are 16f cents of $1 ?
^12. Is $i of $25 ?
'"l3. Isf of f?
14. Isf of2i?
15. Is I of 3|?
PERCENTAGE.
WBITTEH^ EXEltCISES.
515. 1. What per cent, of 72 is 48 ?
OPERATION.
72=:.66f = 66f^
48
Analysis.— Since the per-
centage is the product of
the base and rate, the rate
Or, f I = f ; 100^ X f = ^^% is the quotient found by di-
viding the percentage by
the base ; and 48 divided by 72 is f f = | = .66| ; hence the rate is '
66|%. Or,
Since 48, the percentage, is f of the base, the rate is f of 100 fo,
or66|%.
What per cent.
2. Of 300 is 75?
3. Of 66 is 16i ?
4. Of $20 are 121.60 ?
What per cent.
5. Of $18 are 90 cents ?
6. Of 560 lb. are 80 lb. ?
7. Of 980 mi. are 49 mi. ?
EuLE. — Divide the percentage hy the base. Or, take
such a part of 100^ as the percentage is of the base.
Formula. — Bate — Percentage -^ Base.
* What per cent.
8. Of $480 are $26.40 ?
9. Of 192 A. are 120 A. ?
10. Of 15 mi. are 10.99 mi. ?
11. Of 46 gal. are 5 gal. 3 qt.?
12. Of $4 are 30 cents ?
13. Of 6 bu. 1 pk. are 4 bu.
2pk. 6qt. ?
What per cent,
14. Are 448 da. of 5600 da.?
15. Are 5 lb. 10 oz. of 15 lb.
Avoir. ?
16. Is 13.5 of 225 ?
17. Isfiof^^?
18. Is 3f of 181 ?
19. Is 22| of 182.4 ?
20. A grocer sold from a hogshead containing 600 lb.
of sugar, \ of it at one time, and \ of the remainder at
another time. What per cent, of the whole remained ?
21. A merchant owes $15120, and his assets are $9828.
What per cent, of his debts can he pay ?
5 PERCENTAGE.
516. Tlie rate and percentage being given to find
the base.
OMJLIj jexjemcisjes.
1. 18 is 3^ of what number ?
Analysis. — Since 3%, or y^^, of a certain number is 18, jj^ is J
of 18, or 6, and -JgJ is 600. Hence 18 is 3% of 600.
Of what number
2. Is 15 26% ?
3. Is 24 75^ ?
4. Is 48 8^?
6. Is 1.2 e%?
Of what are
6. 30 1b. 20^? 25^?
7. $84 12^ ? 21^ ?
8. 15bu. 30^? 50^?
9. 16Aoz.l2i%? S^%?
10. 12^^ of 96 is 33|^ of what number ?
517. Principle. — The base is as many times the per-
eentage as 100^ is times the rate.
WBJTTJEN XJXEMCISilS*
518* 1. 144 is 75^ of what number ?
OPERATION. Analysis. — Since the percent-
244 -=- »7»5 =: 192 *^S^ *^ i^e product of the base by
the rate, the base is equal to the
Or, 100 -7- 75 =: 4=1^ =: -| percentage divided by the rate ;
144 X I = 192 and 144 ^ .75 is 192. Or,
Since the rate is .75, the per-
eentage is j^^^^, or f of the base ; hence the base is | of the percent-
age, and I of 144 is 192.
2. $54 are 15^ of what ? r4. 4.56 A. are 6% of what ?
3. $18.75 are 2^% of what ? h's. 39.6 lb. are 1^% of what ?
EuLE. — Divide the percentage ly tJie rate. Or, take a&
many times the percentage as 100^ is times the rate.
Formula. — Base = Percentage -^ Rate.
PERCENTAGE.
9
Of what number
Of what
6. Is 828 120^ ?
10. Are $281.25
37^^?
7. Is 6119 105^^?
11. Are $4578
84^?
- 8. Is .43- 71f i ?
12. Are 37^ bu.
6i^?
9. Is3H ^H%?
13. Are 1260 bbl.
IH%?
- 14. 25% of 800 bu. is 2^% of how many bushels ?
15. A farmer sold 3150 bushels of grain and had 30%
of his entire crop left. What was his entire crop ?
16. A man drew 25% of his bank deposits, and expended
33^% of the money thus drawn in the purchase of a horse
worth 1250. How much money had he in bank at first ?
17. If a man owning 4:6% of a steamboat sells 16|^ of
his share for $5860, what is the value of the whole boat ?
18. If $295,12 are 13^^ of A's money, and 4|^ of A's
money is 8% of B's, how much more money has A than B ?
519. The amount, or the difference, and the rate
being given to find the base.
OHjLZ jexjemcis bs.
1. What number increased by 25^ of itself amounts
to 60 ?
Analysis. — Since 60 is the number increased by 25 % of itself,
it is Iff, or f of the number ; and if f of the number is 60, the
number itself is 4 times J of 60, or 48.
2. What number increased by 8J^ of itself is 130 ?
3. $70 are 40^ more than what sum ?
4. A man sold a saddle for $18, which was 12|-^ more
than it cost him. What did it cost him ?
5. A grocer sold flour for $8.40 a barrel, which was 16f ^
more than he paid for it. What did he pay for it ?
10 PERCENTAGE.
6. What number diminished by 20^ of itself is 40 ?
Analysis. — Since 40 is tlie number diminished by 20% of itself,
it is -f-^Qy or f of the number ; and if f of the number is 40, the
number itself is 5 times i of 40, or 50.
7. What number diminished by 6% of itself is 38 ?
8. What sum diminished by 50% of itself Is 120.50 ?
9. 68 yd. are 15% less than what number ?
10. A tailor, after using 75% of a piece of cloth, had 9|
yards left. How many yards in the whole piece ?
11. A sells tea at $.90 a pound, which is 10% less than
he paid for it. What did he pay for it ?
WBITTEN EXERCISES,
520. 1. What sum increased by 37% of itself is $2055?
OPERATION. Analysis.— Since
1 + .37 = 1.37 the number is in-
$2055-^1.37z3$1500 ^^^^^^^ ^'^^^^ ^^ ^^
.37 of itself, $2055
^^^ is 137%, or 1.37 the
Iff of $2055 = $2055-r-137xl00=:$1500 number. Hence
$2055 divided by
1.37, is the base or required number. Or,
Since $2055, the amount, is UJ of the base, 100 times j^y of
$2055, or $1500, is the base.
2. What number increased by 18% of itself equals 2950 ?
3. What sum increased by 15% of itself is $6900?
4. What number diminished by 12% of itself is 2640 ?
OPEKATiON. Analysis.— Since the number
1 —.12 =3 .88 is diminished 12%, or by .12 of
2640 — 88 = 3000 ^*^^^^' ^^^^ ^^ ^^^* ^^ '^^ ^^ *^®
* number. Hence 2640 divided by
Or, 2640-^22 X 25 = 3000 .88 is the base or required num-
ber. Or,
Since 2640, the difference, is j^o or || of the base, 25 times -^ of
2640, or 3000, is the base.
PERCENTAGE. 11
5. If the difference is $1000 and the rate 20^, what is
the base ?
6. What sum diminished by 36% of itself equals $4810 ?
EuLE. — Divide the amount ly 1 plus the rate; or,
divide the difference iy 1 r)iinus the rate,
^ D . _ i ^ynount -~ (1 + Rate).
~~ ( Difference — (1 — Rate).
What number increased
7. Byl2^ofitself is3800:
8. By 10^ is 39600 ?
9. By 15^ is $2616.25?
10. By 22^ is 1098 bu. ?
What number diminished
11. By7i% of itself is 740?
12. By 4.% is 312 acres ?
13. By 8^ is $2281.60?
14. By 37i^ is $234,625?
15. A man sold 160 acres of land for $4563.20, which
was 8^ less than it cost. What did it cost an acre ?
16. A speculator bought 48 bales of cotton, and after-
ward sold the whole for $2008.80, losing 7^. What was
the cost of each bale ?
17. A dealer bought a quantity of grain by measure and
sold it by weight, thereby gaining 1^% in the number of
bushels. He sold at 10^ above the purchase price, and
received $4910.976 for the grain. Eequired the cost.
18^ A merchant, after paying 60;^ of his debts, found
that $3500 would discharge the remainder. What was
his whole indebtedness ?
19. The net profits of a mill in two years were $6970,
and the profits the second year were b% greater than the
profits the first year. What were the profits each year?
20. A man sold two houses at $2500 each ; for one he'
received 20^ more than its value and for the other 20^
less. Eequired his loss.
13 P E R C E K T A G E .
^ APPLICATIONS OF PERCENTAGE.
521. The applications of percentage are those which
are independent of time, as, Profit and Loss, Commission,
Stocks, etc. ; and those in which time is considered, as.
Interest, Discount, Exchange, etc.
Since some one of the four formulas of percentage
already considered will apply to any of these applications,
the following will serve as a general
EuLE. — Note ivTiat elements of Percentage are given in
the problem, a7id what element is required, and then apply
the special rule or formula for the corresponding case.
PEOFIT AI^D LOSS.
522. Profit and Loss are terms used to express
the gain or loss in business transactions.
523. Gains and losses are usually estimated at a 7*ate
per cent, on the cost, or the money or capital invested.
524. The operations involve the same principles as
those of Percentage.
525. The corresponding terms are the following :
1. The Base is the Cost, or capital invested.
2. The Mate is the per cent, of profit or loss.
3. The Percentage is prQfit or loss.
4. The Amount is the cost p)lus the profit, or the
Selling Price.
5. The Difference is the cost minus the loss, or the
Selling Price,
PRO FITAKD LOSS. 13
OnAL BXERCISES.
536. 1. A horse that cost $200 was sold at a gain of
12^. What was the gain, and the selling price ?
Analysis. — Since the gain was 12 ^ , it was -^^^ of $200, which is
$24 ; and the selling price was $200 + $24 = $224. Hence, etc. (510.)
2. A saddle that cost $25 sold at a loss of 10^. What
was the loss, and the selling price ?
3. A tailor bought cloth at $6 a yard, and wished to
sell it at a gain of 25^. At what price must he sell it ?
4. For how much must a grocer sell tea that cost $.60
a pound, to gain 30^ ?
5. A merchant buys gloves at $.75 a pair, and sells them
at a profit of 33 J^;^. For how much does he sell them ?
6. Bought a carriage for $160, and, after paying 10^
for repairs, sold it at 12^^ profit. What was the gain,
and the selling price ?
7. If butter bought at 36 cents a pound is sold at a loss
of 16f^, what is the selling price?
8. What must be the selling price of coffee that cost
25 cents a pound, in order to gain 20^?
9. At what price must an article that cost $5 be sold,
to gain 100^? 120^? 150^? 200^?
537. 1. A merchant bought cloth at $5 a yard, and
sold it at $6 a yard. What was the gain per cent. ?
Analysis. — The whole gain is the difference between $6 and $5,
which is $1. Since $5 gain $1, or | of itself, the gain per cent, is
J of 100% or 20%. Hence, etc. (513.)
2. What is gained per cent, by selling coal at $7 a tott,
that cost $6 a ton ?
3. Sold a piano for $300. which was f of what it cost.
What was the loss per cent ?
14: PERCENTAGE.
4. Sold melons for $.75 that cost $.50. What was the
gain per cent. ?
5. What is gained per cent, by selling pine-apples at 30
cents each, that cost $15 a hundred ?
6. Sold a sewing machine at a loss of -J of what it cost.
What was the loss per cent. ?
7. What % is gained on goods sold at double the cost ?
8. What % is lost on goods sold. at one-Jialf i\\Q cost?
9. What per cent, profit does a grocer make who buys
sugar at 10 cents and sells it at 12 cents ?
10. What per cent, is gained on an article bought at $3
and sold at $5 ?
■* 538. 1. A dealer sold flour at a profit of $2 a barrel,
and gained 25^. What was the cost ?
Analysis.— Since the gain was 25% = ^^^, or J, $2 is J of the
cost ; $2 is i of 4 times $2, or $8. Hence, etc. (516.)
3. Sold hats for $1 less than cost, and lost 16f ;^. What
did they cost ?
3. A merchant sells silk at a profit of $1 1^ a yard, which
is 40^ gain. What did it cost, and what is the selling
price ?
4. If com selling for 21 cents a bushel more than cost
gives a profit of 30^, what did it cost ?
5. Sold sheep at $2^ more than cost, which was a profit
of 50^. What did they cost, and what is the selling price ?
6. Shoes sold at $.50 above cost giye a profit of 12|^^.
What did they cost ?
7. A farmer, by selling a cow for $12 less than she
cost, lost 33^^. ' What did she cost ?
8. A grocer sells a certain kind of tea for 6 cents a
pound more than cost and gains 6%, What did it cost ?
PEOFITAi^DLOSS. 15
539. 1. A watch was sold for $120, at a gain of 20^.
What was the cost ?
Analysis.— Since the gain was 20%, or i, of the cost, $120, the
selling price, is | of the cost. J of $120, or $20, is J of the cost, and
I, or the cost itself, is 5 times $20, or $100. Hence, etc. (518.)
2. Sold tea at $. 90 a pound, and gained 26%. What
did it cost ?
3. A newsboy, by selling his papers at 4 cents each,
gains 33^%, What do they cost him ?
4. A man sold a horse and harness for $330, which was
10^ more than they cost. What was their cost ?
5. If 20^ is lost by selling wheat at $1.60 a bushel,
what would be gained if sold at 20^ above cost ?
6. John Eice lost 40^ on a reaper, by selling it for $60.
For what should he have sold it to gain 40^ ?
7. If, by selling books at $2 a volume, there is a gain
of 25%, at what price must they be sold to lose 15^?
8. Two pictures were sold for $99 each ; on one there
was a gain of 10^, on the other a loss of 10^. Was there
a gain or loss on the sale of both, and how much ?
^ WRITTEN EXJEHCISJES.
530. 1. A hogshead of sugar bought for $108.80 was
sold at a profit of 12^^. What was the gain ?
OPERA.TION.— $108.80 X .12i = $13.60. (512.)
Formula. — Profit or Loss = Cost x Bate %.
Find the Profit or Loss,
2. On land that cost $1745, and was sold at a gain of 20^.
3. On goods that cost $3120, and were sold at 27'<^ gain.
4. On a boat bought for $2545|^, and sold at 25^ loss.
16 PERCENTAGE.
5. On goods bought for $2560.75, and sold at S% loss.
6. On 25 tons of iron rails bought at %bS a ton, and
sold at an advance of 11 \%.
7. A merchant pays $6840 for a stock of spring goods,
and sells them at an advance of 26^^ on the purchase
price. After deducting $3 75 for expenses, what is his gain ?
- 8. A miller bought 1000 bushels of wheat at $1.84 a
bushel, and sold the flour at 16f^ advance on the cost of
the wheat. What was his profit ?
9. Bought 128 tons of coal at %b,l^ a ton, and sold it
at a gain of 22^. What was the entire profit ?
10. A ship, loaded with 3840 bbl. of flour, being over-
taken by a storm, found it necessary to throw 37^^ of her
cargo overboard. What was the loss at $7.65 a bbl. ?
11. A man bought a pair of horses for $450, which was
25^ less than their real value, and sold them for 25^ more
than their real value ; what was his gain ?
531. 1. Bought a house for $4380. For what must it
be sold to gain 14|^?
Operation.— $4380 x (1 + .14^) or 1.145 = $5015.10. (512.)
2. At what price must pork, bought at $18.40 a barrel,
be sold, to lose 15^?
Operation.— $18.40 x (1 - .15), or .85 = $15.64. (512.)
^ CI IT -n • ( Cost X (1 + Rate % of Gain).
FoKMULA.—Sellmg Prices ] ^ ^ /-,-r»x^i.-r ^
^ ^^ ( Cost X (1— Eate % of Loss).
Find the Selling Price,
3. Of goods bought at $187.50, and sold at 11^^ gain.
4. Of beef bought at $12| a barrel, and sold at 9^^ loss.
5. Of cotton bought at $.14, and sold at a gain of 21|^.
o. Of cloth that cost $5^ a yard, and was sold at a
profit of 18^^ ?
PROFIT AND LOSS. 17
7. At what price must goods that cost $3^ a yard be
marked, to gain 26% ? To lose 20^ ?
8. Sold a lot of damaged goods at a loss of 16%. What
was the selling price of those that cost $.62^ ? $1.25 ?
9. Bought a hogshead of sugar containing 9 cwt. 56 lb.
for $86.04, and paid $4.78 freight and cartage. At what
price per pound must it be sold to gain 20^ ?
^ 533. 1. Bought wool at $.48 a pound, and sold it at
$. 60 a pound. What per c§nt. was gained ?
Operation.— $.60 - $.48 = $. 12 ; and $.12 -f- $.48 =r .25 = 25% .
(515.)
2. Sold for $10.02 an article that cost $12. What was
the loss per cent. ?
Operation.— $12-$10.02=:$1.98; and$l 98^$12=.16|=16i%.
Formula.— i^^^fe % =. Profit or Loss -r- Cost.
^ Find the rate per cent, of profit or loss,
3. On sugar bought at 8 cents and sold at 9^ cents.
4. On tea bought at $1, and sold at $.87|-.
5. On goods that cost $275, and were sold for $330.
6. On grain bought for $1.25 a bushel, and sold for
$1.60 a bushel.
7. On a sewing-machine sold for $72.96, at again of
$9.12.
8. On goods sold for $425.98, at a loss of $134.52.
9. Bought paper at $3 a ream, and sold it at 25 cents
a quire. What was the gain per cent.?
10. A dealer bought 108 bbl. of apples at $4.62^, and
sold them so as to gain $114. 88|. What was his gain ^?
11. If 1^ of an acre of land is sold for f the cost of an
acre, what is the gain per cent. ?
18 PERCEKTAGE.
12. If f of an acre of land is sold for the cost of ^ of
an acre, what is the loss per cent. ?
13. If I of a chest of tea is sold for what the whole
chest cost, what is the gain per cent, on the part sold ?
533. 1. A speculator sold grain at a profit of 33^%, by
which he made 25 cents on a bushel. What did it cost ?
Operation.— $.25-5-.33i=$.75. Or, $.25-4-1= $.75. (518.)
2. Lost $45. 75 on the sale of a horse, which was 20%
of the cost. What was the cost ?
Operation.— $45.75h-.20=$228. 75. Or $45.75 -5- J =$228. 75.
Formula. — Cost = Profit or Loss -f- Bate %.
Find the Cost,
3. Of goods sold at $1500 profit, or a gain of 16^.
4 Of fiour sold at a loss of $.88, or 10^, on a barrel.
5. Of wheat sold at a loss of 6 cents, or 4^, on a bu. ?
6. Of lumber sold at an advance of $4.95 per M., or
35^ gain.
7. If a grocer sells his stock at a profit of 15^, what
amount must he sell to clear $2500 ?
8. A and B engage in speculation. A gains $2000,
which is 12|^^ of his capital, and B loses $500, which is
6% of his capital. What sum did each invest?
534. .1. A furniture dealer sold two parlor sets for
$450 each ; on one he made 15^, on the other he lost 15^.
What did each cost him ?
Operation ($450^(1 + .15)=$391. 30 + , cost of one.
UPERATION. j $45o^^i_.i5)::,|529.41 + , costof theother. (520.)
T? ^^ 4 cr77-"'T>- . {{^ + R(^ie% of gain.)
■FouMVLA.-Cost=Selhng Price-^ | ^i_^,,^^f fo,,.)
PROFIT AKD LOSS. 19
Find the Cost^
2. Of coal sold at $6, being at a loss of 12J^.
3. Of grain sold at 1.96 a bushel, at a gain of 28%.
L Of silk sold for $5.40 a yard, at a profit of 10^.
5. Of hops sold at 16 cents a pound, at a loss of 20^.
6. Of fruit sold for $207.48, at a loss of lb%.
,^ 7. Having used a carriage 1 year, I sold it for $125,
which was 25^ below cost. What should I have received
had I sold it for 10^ above cost ?
8. B sold a span of horses to C and gained 12|^^ ;
sold them to D for $550, and lost 16f ^. What did the
horses cost B ?
9. If a piece of property increases in value each year at
the rate of 25^ on the value of the previous year, for 4
years, and then is worth $16000, what did it cost ?
535. 1. Bought cloth at $3.60 a yard. At what price
must it be marked that 12|^ may be abated from the
asking price, and still a profit made of 16f ^ ?\.
Operation ~ \ ^^^^""^ ^^^'^^ =^^-^^ ^ (1 + •l6|)-$4.20.
(if«r^^/^5rPnce=$4.30-^(l-.12l)=:$4.80. (519.)
2. At what price must shovels that cost $1.12 each be
marked in order to abate h%, and yet make 25^^ profit ?
3. How must a watch be marked, that cost $120, so
that 4^ may be deducted and a profit of 20^ be made ?
4. A merchant, on opening a case of goods that cost
$.80 a yard, finds them slightly damaged. How must he
mark them, to fall 25^ in his asking price, and sell at cost?
5. Bought land at $60 an acre ; how much must I ask
an acre, that I may deduct 25^ from my asking price, and
still make 20^ on the purchase price ?
20 PEBCEKTAGE.
COMMISSIOK
536. An Agent or Commission Mer'chant
is a person who buys or sells merchandise^ or transacts
other business for another, called the Pmicipal
537. Commission is the fee, or compensation,
allowed an agent or commission merchant for transacting
business, and is usually computed at a certain rate per
cent, of the money involved in the transaction.
538. A Consignment is a quantity of goods sent
to a commission merchant to be sold.
539. The Consignor is the person who sends the
goods for sale. A consignor is sometimes called a Shipper.
540. The Consignee is the person to whom the
goods are sent. He is sometimes called a Correspondent.
541. The Net Proceeds of a sale or other transac-
tion is the sum of money that remains after all expenses
of commission, etc., are paid.
543. A Guaranty is security given by a commis-
sion merchant to his principal for the payment of goods
sold by him on credit.
543. An Account Sales is a written statement
made by a commission merchant to his principal, contain-
ing an account of goods sold, their price, the expenses^
and the net proceeds.
544. A JBroker is a person who buys or sell stocks,
bills of exchange, real estate, etc., for a commission,
which is called BroJcerage.
COMMISSION. 21
545* The principles and operations of Percentage in-
volved in Commission and Brokerage are the same as
those already treated.
54:6. The following are the corresponding terms :
1. The Base is the amount of sales, money invested,
or collected.
2. The Rate is the per cent, allowed for services.
3. The JPercentage is the Commission or Broker-
age.
4. The Amount or Difference is the amount of
sales, plus or minus the commission.
WRITTEN EXERCISES.
547. Find the Commission or BroTcerage^
1. On a sale of flour for $2575, at %\%.
Operation.— $2575 x .025 = $64.37i (512.)
Formula. — Amount of Sales x Rate % = Commission.
2. On the purchase of a farm for $13750, at 2|^.
3. On the sale of a mill for $9384, at |^.
4. On the sale of $21680 worth of wool, at 1^%.
5. On the sale of 250 bales of cotton, averaging 520 lb.,
at 14| cents a pound ; commission 1^%.
6. On the sale of 175 shares of stock, at $92| a share ;
brokerage, ^%.
7. On the sale at auction of a house and the furniture
for $9346.80, at 6}^.
8. A commission merchant sells 225 bbl. of potatoes
at $3.25 per bbl., and 316 bbl. of apples at $4^ per bbl.
What is his commission at 4^^ ?
22 PERCE2!^TAGE.
548. Find the rate of commission or brokeiBge,
1. When $89 commission is paid for selling goods for
$3560.
Operation.— 89 -^ 3560 =: m\ = 2J-^. (515.)
Formula. — Com7nission -^ A^nount of Sales = Rate %.
2. When $165 com. is paid for selling goods for $4950.
3. When $63 is paid for collecting a debt of $1260.
4. When $117.75 is paid for selling a house for $7850.
5. When 1235.40 is paid for buying 26750 lb. of wool
at 32 cents a pound.
6. When $125 is paid for the guaranty and sale of goods
for $2500.
7. Paid my N. 0. agent $74.25 for buying 26400 lb. of
rice, at 4|^ ct. a lb. What was the rate of his commission ?
549. Find the Amount of Sales,
1. Wlien a commission of $147 is charged at Z^%.
Operation.— $147 -^ .035 = $4200. (517.)
Formula. — Commission -r- Rate % = Amount of Sales.
2. When $92.80 commission is paid at 3^%.
3. When $210 commission is charged at 6%.
4. When $24 brokerage is paid at i%.
5. When $135 commission is charged at 1^%.
6. Paid an attorney $72.03 for collecting a note, which
was a commission of '7^%. What was the face of the note ?
"^ 550. Find the Amount of Sales,
1. When the net proceeds are $4875, commission 2^%.
Operation.— $4875 ^ .975 = $5000. (519.)
Formula. — Net proceeds -^-{1—- Rate %)=zAmt. of Sales.
2, When the net proceeds are $3281.25, commission 12^^.
COMMISSION. 23
3. When the net proceeds are $560, and the com. 4^.
4. After deducting 6^% commission and $132 for
storage, my correspondent sends me $23654.25 as the net
proceeds of a consignment of pork and flour. What was
the gross amount of the sale ?
551. Find the amount to be invested ,
1. If $9500 is remitted to a correspondent to be invest-
ed in woolen goods, after deducting 6% commission.
Operation.- $9500 -f- 1.05 == $9047.62. (519.)
¥oRM.VLA,— Amount Remitted-^ (1 + Rate %) ■= Sum
Invested.
2. If $4908 be remitted, deducting ^\% commission.
3. If $3246.20 be remitted, deducting %% commission.
4. If $1511.25 be remitted, deducting ^% commission.
5. If $10701.24 be remitted, deducting ^% brokerage.
6. A dealer sends his agent in Havana $6720.80, with
which to purchase oranges and other fruits, after deduct-
ing his commission of b%. What sum did the agent invest,
and what was the amount of his commission ?
^ 7. What amount of sugar can be bought at 8 cents a
pound, for $2523.40, ?ifter deducting a commission of 1^%,
8. Eemitted to a stockbroker $10650, to be invested in
stocks, after deducting \% brokerage. What amount of
. stock did he purchase ?
9. A broker received $45337.50 to invest in bond and
mortgage, after deducting a commission of ^\%, What
amount did he invest, and what was his commission ?
^ 10. Sent $250.92 to my agent in Boston, to be invested
in prints at 15 cents a yard, after taking out his commis-
sion of ^%, How many yards ought I to receive ?
24 PERCEII^TAGE.
REVIEW.
OMAZ EXEHCISES.
553. 1. If stoves bought at $36 each are sold at a
profit of ^\%, what is the gain?
2. What will be the expense of collecting a tax of $1000,
allowing b% ?
3. What will a broker receive for selling $600 worth of
stock, at ^% brokerage ?
4. A man having $250 spent $80. What per cent, of
nis money had he left ?
5. If a man sells a building lot that cost $300, at an
advance of 166f^, what is his gain ?
6. I of 30^ is what per cent, of 72^ ? Of 144^? Ofv
180^? 240^?
7. Bought a horse for 20^ less than $200, and sold him
for 10^ more than $200. What per cent, was gained ?
8. How many bushels of wheat at $2 a bushel can an
agent buy for $2040, and retain 2% on what he expends
as his commission ?
9. If by selling land at $150 an acre I lose 25^, how
must I sell it to gain 40^ ?
10. A boy bought bananas for $3 a hundred, and sold
them for 5 cents each. What per cent, did he gain ?
11. Bought cannel coal at $19 a ton, which was ^% less
than the market price. What was the market price ?
12. Paid an agent $150, or a commission of 1\%, for
selling my house. For what sum was the house sold ?
13. If an article is sold so as to gain f as much as it
cost, what per cent, is gained ?
REVIEW. 25
14. A merchant tailor sold some linen coats at $1.80
each, which was 33^% below the marked price. What
was the marked price ?
15. A grocer bought 40 gal. of maple syrup at the rate
of 4 gal. for $6, and sold it at the rate of 5 gal. for $8.
What was his whole gain, and his gain per cent. ?
16. How much wheat must a farmer take to mill that
he may bring away the flour of 4| bushels, after the miller
takes his toll of 10^?
WRITTEN EXERCISES.
553. 1. After taking out 15^ of the grain in a bin,
there remained 40 bu. 3| pk. How many bushels were
tl^e at first ?
/^2. The net profits of a farm in 2 years were $3485, and
,, the profits the second year were b% greater than the
profits the first year. What were the profits each year ?
3. A has 32^ more money than B ; what per cent, less
is B's money than A's ?
4. Bought 450 bushels of wheat at $1.25 per bushel, and
sold it at $1.40 per bushel. What was the whole gain,
and the gain per cent. ?
5. A man drew out of the bank | of his money, and ex<
pended 30^^ of 50^ of this for 728 bu. of wheat, at $1.12^
a bushel. What sum had he left in bank ?
6. Sold goods to the amount of $47649, at a profit of
16f^. Eequired the cost and the total gain.
7. A broker received $37.50 for selling some uncurrent
money, charging \% brokerage. How much did he sell ?
8. If f of a farm is sold for what | of it cost, what is
the gain per cent. ?
26 PERCEKTAGE.
9. An architect charged ^% for plans and specifications,
and If ^ for superintending a building that cost $25000.
What was the amount of his fee ?
10. If a stationer marks his goods 50^ above cost, and
then deducts 50^, what per cent, does he make or lose?
11. Sold a farm for $14700, and lost 12^. What per
cent, should I have gained by selling it for $21000 ?
12. If an article bought at 20^ below the asking price
is sold at 16% below that price, what is the rate of gain ?
13. A commission merchant sold a consignment of
goods for $5250, and charged 3^% commission, and 2j^%
for a guaranty. Find the net proceeds.
14. Smith & Jones bought a stock of groceries for
$13680.^ They sold ^ of the entire stock at 15^ profit, ^
at 18|-^, ^ at 20^, and the remainder at 33^^ profit. What
was the whole gain, and the average gain per cent. ?
15. Give the marking prices at 25^ advance, of the
following bill of goods, and the amount when sold at a
reduction of 10^ from those prices :
1 Case of Prints, 450 yd., @ $.12
3 Pieces Cassimeres, 65 '' % 3.25
1 Bale Ticking, 244 '' ® .20
25 Dress Shawls, @ 7.36
1 Gr. gross Clark's Thread, 144 doz., @ .70
50 Gross Buttons, @ 1 . 00
16. How much would the above bill of goods amoun;
to if sold at 5|^^ below a marking price of 15^ above cost ?
17. What would be the net proceeds of a sale of 18 cwt.
75 lb. of sugar, at $9f per cwt., allowing 2^;^ commission,
and $1,6| for other charges?
COMMISSION. 27
18. A broker receives $7125 to invest in cotton, at 11 J
cents a pound. If his commission is 2^%, how many
pounds of cotton can he buy?
19. If the sale of potatoes at $.75 a barrel above cost
gives a profit of 18|^, how much must be added to this
price to realize a profit of 31^% ?
20. An agent in Chicago purchases 1000 bbl. of flour
at $6. 80, and pays 5 cents a barrel storage for 30 days ;
also, 3000 bu. of wheat at $1.20. He charges a commis-
sion of 1^% on the flour, and 1 cent a bushel on the wheat.
What sum of money will balance the account, and what is
the amount of his commission ?
21. An agent in Boston received 28000 lb. of Texas
cotton, which he sold at $.12|^ a pound. He paid $45.86
freight and cartage, and after retaining his commission,
he remits his principal $3252.89 as the net proceeds of the
gale. What was the rate of his commission?
22. The following bill of goods was sold at auction :
IJ bbl. A Sugar, 312 lb., @ $.12|^ that cost $.11^
I '' Pulv. '' 96 '' % .\\\ " " .14
1 Chest Y. H. Tea, 84^^ @ 1.10 " '' 1.12^
1 Box Soap, 60 " % .13 " " .10|
1^ Sacks Java Coffee, 110 ^^ % .22^ " " .24|
184 lb. Codfish, @ .07i " " .08|
Allowing a commission of 4^^ for selling, find the entire
profit or loss, and the gain or loss per cent, on the whole.
23. A merchant in New York imported 2400 yd. of
English cloth, for which he paid in London 10s. sterling
a yard, and the total expenses were $255. He sold the
cloth for $3.81 a yard, U. S. money. What was his whole
gain, and his gain per cent. ?
28
PERCENTAGE.
554.
SYNOPbiS FOR EEVIEW.
1. Definitions.
2. Elements.
'' 1. Percentage. 2. Per Cent. 3. Sign
of Per Cent. 4. Rate, or Rate ^ .
5. Base. 6. Percentage. 7. Amount.
8. Difference.
( 1. How many considered.
( 2. How many must be given.
3. 510. 1. Principle. 2. Rule. 3. Formula.
4. 513. 1. Principle. 2. Rule. 3. Formula.
5. 516. 1. Principle. 2. Rule. 3. Formula.
6. 519. 1. Principle. 2. Rule. 3. Formula.
Afplications of
Percentage.
8. Profit and Loss.
9. Commission.
( 1. Diff't kinds. -1 ^- "^^'^^'^^^ '^'^'
\ I 2. With Time.
( 2. General Rule.
C 1. Definition.
J 2. To estimate gains and losses.
1. Base.
3
Correspond-
ing terms.
1. Definitions. <
2. Rate.
3. Percentage.
4. Am't and Biff.
1. Agent, or Com-
mission Merchant,
2. Commission.
3. Consignment.
4. Consignor.
5. Consignee.
6. Net Proceeds.
7. Guaranty.
8. Account Sales.
9. Broker.
2. Prin. and Operations Involved.
1. Base.
2. Rate.
3. Percentage.
4. Am't and Biff,
3. Correspond-
ing terms.
OUAZ, EXERCISES,
555. 1. When h% is charged for the use of money,
how many dollars should be paid for the use of $100 ?
For the use of $200 ? Of 1500 ? Of $50 ?
2. At 1% a year, what should be paid for the use of
$100 for 2 years ? Of $200 for 3 years ?
3. If $500 is loaned for 3 years, what should be paid
for its use, at b% a year ? At 6^ a year ?
4. If I borrow $250, and agree to pay ^% a year for its
use, how much will be due the lender in 5 years ?
5. If $7 is paid for the use of $100 for 1 year, what is
the per cent, ?
6. If $50 is paid for the use of $100 for 5 years, what
is the per cent. ?
7. If $14 is paid for the use of $200 for 1 year, what is
the per cent. ?
8. At 6^, what decimal part of the money borrowed is
equal to the money paid for its use ? At 7^ ? 8^ ? 9^ ?
DEFINITIONS.
556. Interest is a sum paid for the use of money.
557. The JPrincipal is the sum for the use of
which interest is paid.
558. The Rate of Interest is the per cent., or
number of hundredths, of the principal, paid for its use
for one year.
30
PEECEKTAGE.
559. The Amount is the sum of the principal and
the interest.
560. Legal Interest is the interest according to
the rate per cent, fixed iy law.
561. Usury is a higher rate of interest than is al-
lowed by law.
563. The legal rates of interest in the different States
are as follows :
Name of State.
Alabama
Arkansas''^
Arizona
California*
Canada and Ireland
Connecticut ...
Colorado*
Dakota
Delaware
Dist. Columbia. .
England and France
Florida*
Georgia
Idaho
Illinois
Indiana
Iowa
Kansas
Kentucky
Louisiana
Maine*
Maryland
Massachusetts*. .
Michigan
Kate.
8%
6%
10%
10%
6%
7%
10%
7%
0%
0%
5%
B%
7%
10%
6%
6%
6%
7%
6%
5%
6%
6%
6%
7%
Any.
Any. I
Any.
Any.
Any.
10%
Any.
10%
10%
10%
12%
10%
8%
Any.
Any.
10%
Name of State.
Minnesota ,
Mississippi ,
Missouri
Montana
New Hampshire,
New Jersey
New York
North Carolina. .
Nebraska
Nevada*
Ohio
Oregon
Pennsylvania . . .
Rhode Island*. .
South Carolina*.
Tennessee
Texas
Utah*
Vermont
Virginia
West Virginia. .
Washington T.*
Wisconsin
Wyoming
Rate.
7%
.6%
6%
10%
6%
6%
6%
6%
10%
10%
6%
10%
6%
6%
7%
6%
8%
10%
6%
6%
6%
10%
7%
12%
12%
10%
10%
8%
15%
Any.
8%
12%
Any.
Any.
10%
12%
Any.
12%
Any.
10%
1. When the rate per cent, is not specified in accounts, notes,
mortgages, contracts, etc., the legal rate is always understood.
2. Where two rates are specified, any rate above the lower, and
not exceeding the higher, is allowed, if stipulated in writing.
3. In the States marked thus (*) the rate per cent, is unlimited if
agreed upon by the parties in writing.
IKTEKEST. 31
563. In the operations of interest there avefive parts,
or elements, namely :
The Princijjal ; the Rate per Cent, per Annum (for one
year) ; the Interest j the Time for which the principal is
lent ; and the Amount, or sum of the Prin. and Int.
564. These terms correspond respectively to Base,
Rate, Percentage, and Amount in Percentage, excluding
Time, which is an additional element in Interest.
OBAL EXEItCISES,
565. 1. At %%, for 1 yr., what decimal part of the prin-
cipal equals the interest ? At 5^ ? At 8% ? At 12 1^ ?
2. What is the interest of 120 for 1 year at b% ?
Analysis. — Since the interest of any sum at 5% for 1 yr. is .05
of the principal, the interest of $20 for 1 yr. at 5^ is .05 of $20, or $1.
3. What is the interest of $50 for 1 yr. at 5^ ? 6^ ? 7^ ?
4. What is the interest of $80 for 1 yr. at 7^ ? 8%? 10^ ?
5. At 7^ for 5 yr., what decimal part of the principal
equals the interest ?
Analysis.— Since the interest at 7% for 1 yr. is .07 of the prin-
cipal, the interest for 5 yr. is 5 times .07, or .35 of the principal.
Or, it is 5 times the interest for 1 year.
6. At 6^ for 3 yr., what decimal or fractional part of
the principal equals the interest ? At 1% for 6 yr. ? At
h% for 5 yr.? At ^% for 2 yr. ? At 10^ for 4 yr.?
7. Find the interest of $30 for 3 yr. at b%.
Analysis.— Since the interest of any sum at 5% for 1 yr. is .05
of the principal, for 3 yr. it is .15, and .15 of $30 is $4.50. Or, the
interest for 1 yr. is .05 of $30, or $1.50, and for 3 yr. it is 3 times as
much, or $4.50.
8. Find the int. at 6^ of $20 for 2 yr. Of $40 for 3 yr.
9. Find the int. at 8^ of $5 for 5 yr. Of $10 for 1 yr.
32 PERCENTAGE.
10. At S% for 2 yr. 6 mo., what decimal part of the
principal equals the interest ?
Analysis. — Since the interest of any sum for 1 yr. at 8% is .08 of
the principal, the interest on the same for 2 yr. 6 mo. is 2i times .08,
or .20 of the principal. Or, it is 2^ times the interest for 1 year.
11. At 6% for 3 yr. 3 mo., what decimal part of the
principal equals the interest? At 9% for 3 yr. 3 mo. ?
12. Find the int. of $9 for 2 yr. 4 mo. at 7%. At 8%.
13. What is the int. of $1000 for 2 yr. 3 mo. at 10^?
For 4 yr. 6 mo. ? For 5 yr. 3 mo. ? For 8 mo. ?
566. PRiisrciPLE. — The interest is the product of three
factors ; namely, the principal, rate per annum, and time
{expressed in years or parts of a year).
WRITTEN EXERCISES.
56*7. To find the interest or amount of any sum,
at any rate per cent., for years and months.
1. Find the amount of $97.50, at 7^, for 2 yr. 6 mo.
OPERATION. Analysis.— Since the interest of
$97.50 any sum at 7% for 1 yr. is .07 of
Q,v the principal, the interest of $97.50
'- — at 7% for 1 yr. is .07 of $97.50, or
$6.8250 Intforlyr. $6,825; and the interest for 2 yr.
2|- 6 mo. is 2 1 times the interest for 1
i7n^ T . , o a y^'^ '''' $17,061, and $17.06H $97.50
17.06^5 Int. for 2 yr. 6 mo. '' L^^/^^i ,; . 4 ^
= $114,561, *he Amount.
97.50 Principal.
$1145625 Amount.
Find the interest and the amount,
2. Of $450 for 3 yr. 9 mo. at 6%. For 8 mo. at 7%.
3. Of $247 for 5 yr. 3 mo. at 6^%- For 10 mo. at 8^
4. Of $500 for 4 yr. 2 mo. at 10^. For llmo. at 5^.
INTEREST. 33
EuLE. — I. Multiply the principal hy the rate, and the
product is the interest for 1 year.
II. Multiply the interest for 1 year by the time in years,
and the fraction of a year j the product is the required
interest.
III. Add the pri7icipal to the interest for the amount.
Formula. — Interest = Principal x Rate x Time.
Find the interest,
5. Of $36.40 for 1 yr. 7 mo. at 6^. At 7^. At ^%.
6. Of $750.50 for 3 yr. 1 mo. at b%. At 8^. At 9^.
7. Of $1346.84 for 2 yr. 4 mo. at 6J^. At 1^%.
8. Of $138.75 for 4 yr. 3 mo. at 10^. At n\%.
9. Find the amount of $640 for 5 yr. 6 mo. at 1%.
10. Find the amount of $56.64 at S% for 3 yr. 3 mo.
11. Made a loan of $1040 for 1 yr. 9 mo. at ^%. How
much is due at the end of the time ?
12. If a note for $375, on interest at %%, dated June 10,
1874, be paid Sept. 10, 1876, what amount will be due ?
568. To find the interest on any sum of money,
for any time, at any rate per cent.
Obvious EELATioiiirs betweek Time akd Iktekest.
I. The interest on any sum for 1 year at 1% is .01 of
the principal.
It is therefore equal to the principal with the decimal point re-
moved tiDO places to the left.
II. The interest for 1 mo. is -^ of the interest for 1 yr.
III. The interest for 3 days is -jV, or ^, of the interest
for 1 month ; hence any number of days may readily be
reduced to tenths of a month by dividing by 3.
34 PERCENTAGE.
IV. The interest on any sum for 1 month, multiplied
by the number of months and tenths of a month in the
given time, and the product by the number expressing
the rate, will be the required interest.
569. 1. Find the int. of $361.20 for 1 yr. 3 mo.24 da. at 7^.
OPERATION.
$3,612 (01 of the Prin.) Int. for 1 yr. at 1 % (568, I).
.301 Int. for 1 mo. at 1% (568, II).
15.8 Number of months and tenths (568, III).
$4.7558 Int. for 1 yr. 3 mo. 24 da. at IJ
7
$33.2906 Int. for 1 yr. 3 mo. 24 da. at 1% (568, IV).
What is the interest,
2. Of $137.25 for 1 yr. 6 mo. 10 da. at 6^ ? At 4^ ?
3. Of $510.50 for 3 yr. 7 mo. 15 da. at b% ? At 8^?
4. Of $1297. 60 for 2 yr. 11 mo. 18 da. at t% ? At 7^^?
EuLE. — I. To find the interest for 1 yr. at \%,
Remove the decimal point in the given principal two
places to the left.
II. To find the interest for 1 mo. at 1%.
Divide the interest for 1 year ly 12.
III. To find the interest for any time at 1%.
Multiply the interest for 1 month hy the number of
months and tenths of a month in the given time,
IV. To find the interest at any rate %,
Multiply the interest at l%for the given time ly the num-
ler expressi^ig the given rate,
5. Find the int. of $781.90 for 1 yr. 1 mo. 12 da, at 7%.
6. Find the int. of $3000 for 11 mo. 21 da. at 10^.
IlfTEKEST. 35
7. What is the ami of $1049 for 2 yr. 3 mo. 9 da. at (j^% ?
8. What is the amt. of $216.75 for 3 yr. 5 mo. 11 da. at S% ?
9. Eequired the int. of $250 from Jan. 1, 1873, to
May 10, 1875, at 7^?
10. Eequired the amount of $408.60 from Aug. 20 to
Dec. 18, 1876, at 10^?
11. What is the interest on a note for $515.62, dated
March 1, 1873, and payable July 16, 1875, at 7%?
12. A man sold his house and lot for $12500 ; the
terms were, $4000 in cash on delivery, $3500 in 9 mo.,
$2600 in 1 yr. 6 mo., and the balance in 2 yr. 4 mo., with
6% interest. What was the whole amount paid ?
570 • SIX PER CENT METHOD.
At 6% per annum, the interest of $1
For 12 mo is 6 cents, or .06 of the principal.
'' 2 '^ or ^ of 12 mo., 'a cent, ^^.01 '' ''
" 1 " "i^"Vl " "\ " " .005 "
'' 6da.'^^'' 1 " "^^ " ^^.001 " "
'' \ " " \" 6 da. " .000^'^
571. Principles. — 1. Tlie interest of any sum at 6%
is ONE-HALF as many hundredths of the principal as
there are months in the given time,
2. The interest of any sum at 6% is one-sixth as
many thousandths of the principal as there are days in
the given time.
Thus, the interest on any sum at 6% for 1 yr. 3 mo., or 15 mo.,
is J of .15, or .075, of the principal ; and for 18 da. it is -J of .018,
or .003, of the principal. Hence, for 1 yr. 3 mo. 18 da., it is .075
+ .003 = .078 of the principal.
It is evident that an odd month is ^ of .01, or .005; and that
any number of days less than 6 is such a fractional part of .001 as
the days are of 6 days.
36 PERCENTAGE.
oraij exercises.
573. What is the interest,
1. Of $1 at Q% for i year ? 2 years ? 3 years ? 5 years ?
8 years ? 12 years ?
2. Of $1 at 6^ for 1 month ? 2 mo. ? 3 mo. ? 4 mo. ?
5mo. ? 7mo. ? 9 mo. ? 10 mo.? 15mo.?18mo. ?
At 6^, what is the interest,
3. Of $1 for 1 yr. 4 mo. ? 1 yr. 7 mo. ? 2 yr. 2 mo. ?
4. Of $1 for 1 day ? 6 da. ? 12 da. ? 19 da. ? 24 da. ?
33 da.? 36 da.? 45 da.? 63 da.?
5. Of $1 for 1 mo. 12 da. ? For 3 mo. 15 da. ? For
6 mo. 25 da. ? For 7 mo. 11 da. ? For 11 mo. ]8 da. ?
Find the interest,
6. Of $1, at Q%, for 1 yr. 3 mo. 6 da. For 1 yr. 9 mo.
18 da. For 1 yr. 5 mo. 19 da.
7. Of $1 at %% for 2 yr. 1 mo. 9 da. For 3 yr. 24 da.
8. Of $1 at 6^ for 5 yr. 5 mo. 5 da. For 4 yr. 7 mo. 10 da.
At 6^, find the interest,
9. Of $1 for 2 yr. 6 mo. Of $2. Of $3. Of $5.
10. Of $1 for 4 yr. 2 mo. Of $10. Of $20. Of $30.
11. Of $5 for 1 yr. 4 mo. For 2 yr. For 2 yr. 8 mo.
12. Of $1 for 33 da. For 63 da. For 93 da. For 123 da.
13. Of $6 for 33 da. Of $4 for 63 da. Of $2 for 93 da.
14. If the interest of a certain principal at 6^ is $18,
what would the interest be at b% ? 7;^ ? 8^ ? 9^ ?.
5% is I less than 6% ; 7% is J more than 6% ; 8% is J more, etc.
15. If the interest of a certain principal is $16, what
tvould the int. be at Z% ? ^%? 6%? U% ? 8^ ? 12^ ?
16. If the interest of a certain principal is $30, what
would the int. be at 2^? 4=%? 7^? 8^? 10^? 14^?
IKTEKEST. 37
WRITTEN EXEItCISES.
573. 1. What is the int. of $427.20 at 6^ for 2 yr. 5 mo.
27 da.
OPERATION. Analysis.— Since the in-
2 yr. 5 mo. =. 29 mo. $427.20 *^^^»* «^ ^^ ^^^ ^ y^- ^ ^^•
/ _ ^ ^'^ ^^- ^^ $149|, or of any
^ of .29 = .145 AA:^ g^j^ jg 1491 ^f ^^^ princi-
1^ of .027 = .004^ 163.8664 pal (571), $427.20 x .149J-
Int. =.149i0fthePrin. =163.866+ is the required
interest.
Find the interest at 6^ of
2. $597.25 for 7 mo. 18 da.
3. $418.75 for 1 mo. 25 da.
4. $309.18 for 2 yr. 24 da.
5. $1298 for 3 yr. 1 mo. 13 da.
6. $2000 for 2 yr. 7 mo. 24 da.
7. $4010 for 1 yr. 1 mo. 13 da.
EuLE. — MuUijjly the given principal by the decimal ex-
pressing the i?iterest of $1 ; or by the decimal expressing
one-half as many hundredths as there are months, and one-
sixth as many thousandths as there are days, in the given
time, and the product will be the required interest.
To find the interest at any other per cent, by this method, increase
or diminish the interest at 6 % by such part of itself as the given
rate is greater or less than 6%.
574. To compute Accurate Interest^ that is,
reckoning 365 da. to the year, use the following
EuLE. — Find the interest for years and aliquot parts of
a year by the common method, and for days take such part
ofl yearns interest as the number of days is o/'365. Or,
When the time is in days and less than 1 yea.r,find the
interest by the common method and then subtract -^ part
of itself for the common year, or -^^ if it be a leap year*
38 PEKCENTAGE.
1. Find the accurate interest of $1560 for 45 da. at 1%.
The exact int. of $1560 for 45 da. at 7 fc = $109^x_45 ^ ^^^ ^ ^
Or, It is $13.65 - ^'-^'t^ "" ^ = $13.46 +.
2.^ Find the exact int. of $1600 for 1 yr. 3 mo. at 6%,
3. What is the difference between the exact interest of
$648.40 at S% for 1 yr. 3 mo. 20 da. and the interest
reckoned by the 6^ method?
4. Find the exact interest of $875.60 at 7% for 63 da.
5. Eequired the exact interest on three U. S. Bonds of
$1000 each, at 6%, from May 1 to Oct. 15.
6. What is the exact interest on a $500 U. S. Bond, at
5%, from Nov. 1 to April 10 following ?
575. Find the interest, by any of the ordinary methods,
1. Of $721.56 for 1 yr. 4 mo. 10 da. at 6^.
2. Of $54.75 for 3 yr. 24 da. at 5%.
3. Of $1000 for 11 mo. 18 da. at 7^.
4. Of $3046 for 7 mo. 26 da. at S%.
5. Of $1821.50 from April 1 to Nov. 12 at 6%.
6. Of $700 from Jan. 15 to Aug. 1 at 10^.
7. Of $316.84 from Oct. 20 to March 10 at 11%.
What is the amount
8. Of $3146 for 2 yr. 3 mo. 10 da. at 1%?
9. Of $96.85 for 3 yr. 1 mo. 27 da. at 6% ?
10. Of $1008.S0 for 10 mo. 16 da. at Qi% ?
11. Of $2000 for 15 da. at 12^^ ?
12. Of $137.60 for 127 da. at 10^?
^13. If $1671.64 be placed at interest June 1, 1874, what
amount will be due April 1, 1876, at 11% ?
INTEREST. 33
14. How much is the interest on a note for $600, dated
Feb. 1, 1872, and payable Sept. 25, 1875, at 6^ ?
15. If a man borrow $9700 in New York, and loan it
in Colorado, what will it gain at legal int. in a year r
16. Eequired the interest of $127.36 from Dec. 12^ 1873,
to July 3, 1875, at 4^^.
17. A note of $250, dated June 5, 1874, was paid Feb.
14, 1875, with interest at S%. What was the amount ^^
18. A note for $710.50, with interest after 3 mo., at 7^,
was given Jan. 1, 1874, and paid Aug. 12, 1876. What
was the amount due ?
19. A man engaged in business was making 12^% an-
nually on his capital of $16840. He quit his business
and loaned his money at 7^%. What did he lose in 2 yr.
3 mo. 18 da. by the change ?
20. A man borrows $2876.75, which belongs to a minor
who is 16 yr. 5 mo. 10 da. old, and he is to retain it until
the owner is 21 years old. What will then be due at 8%
simple interest ?
21. A speculator borrowed $9675, at 6%, April 15, 1874,
with which he purchased flour at $6.35. May 10, 1875,
he sold the flour at $7f a barrel, cash. What did he gain
by the transaction ?
■^ 22. A man borrows $10000 in Boston at 6^, reckoning
360 da. to the year, and lends it in Ohio at 8^, reckoning
365 da. to the year. What will be his gain in 146 days?
23. A tract of land containing 450 acres was bought at
$36 an acre, the money paid for it being loaned at 6^%.
At the end of 3 jr. 8 mo. 24 da., f of the land was sold
at $40 an acre, and the remainder at $38|^ an acre. What
was gained or lost by the transaction ?
4D PERCENTAGE.
^^ PKOBLEMS IN INTEEEST.
576« Interest, time, and rate given, to find the
principal.
OBAZ EXJERCISES,
1. What sum of money will gain $10 in 1 yr. at b% ?
Analysis. — The interest of $1 for 1 yr. at 5% is .05 of the prin-
dpal, and therefore $10 -r- .05, or $200, is the required sum. Or,
Since $.05 is the interest of $1, $10 is the interest of as many-
dollars as $.05 is contained times in $10, or 200 times. Hence, etc.
What sum of money will gain.
2. $20 int. in 2 yr. at b% ?
3. $25 int. in 5 yr. at b% ?
4. $60 int. in 2 yr. at 6%?
5. $84 int. in 2 yr. at 1% ?
6. $50 int. in 6 mo. at 10^?
7. $30int. inSmo. at 8;^?
WRITTEN EXERCISES,
577. 1. What sum of money, put at interest 3| yr. at
&%, will gain $346.50?
OPERATION.
Int. of $1 for 3| yr. at 6% = $.21. Analysis.— Same as in
$346.50 -^ $.21 = 1650 times ; ^^^^ exercises. (576.)
$1 X 1650 =: $1650.
What principal
2. Will gain $49.50 in 1 yr. 3 mo. at 6% ? At 5^ ?
3. Will gain $153.75 in 3 mo. 24 da. at 7^ ? At 8^?
EuLE. — Divide the given interest ly the interest of $1
for the given time, at the given rate.
Formula. — Principal = Interest -r- {Bate x Time).
What sum of money
^ 4. Will gain $213 in 5 yr. 10 mo. 20 da. at 7% ?
5. Willgain$173.97in4yr.4mo. at6^? At 12^?
^
IKTEREST. 41
6.* A man receives semi-annually $350 int. on a mort-
gage at 1%, What is the amount of the mortgage ?
578. Amount, rate, and time given, to find the
principal.
OltATj EXEHCIS ES.
1. What sum of money will amount to $107 in 1 yr.
at 7^?
Analysis. — Since the interest is .07 of the principal, the amount
is 1.07, or igj, of it. If $107 is igj of the principal, yi^ of the prin-
cipal is y^7 of $107, or $1 ; and ^gg, or the principal itself, is $100. Or,
Since $1.07 is the amount of $1, $107 is the amount of as many
dollars as $1.07 is contained times in $107, or $100.
What sum of money will amount to
2. $130 in 5 yr. at 6^?
3. $228 in 2 yr. at 1% ?
4. $412 in 6 mo. at 6^?
5. $250 in 10 yr. at 10^?
6. $350 in 15 yr. at 5^?
7. $260 in 3 yr. 9 mo. at 8^?
WJtITTEN EXEBCISES ,
579. 1. What sum will amount to $337.50 in 5 yr.
at 7^?
OPERATION.
Am't of $1 for 5 yr. at 7% = $1.35. Analysis. — Same as
$337.50 -^ $1.35 :r3 250 times ; in oral exercises. (578.)
$1 X 250 = $250.
What principal
2. Will amount to $1028 in 4 mo. 24 da. at 7%?
3. Will amount to $1596 in 2 yr. 6 mo. at 6^% ?
^4. Will amount to $1531.50 in 3 mo. 18 da. at 7%?
Rule. — Divide the given amount hy the amou7it of $1
for the given time, at the given rate.
Formula. — Prin. = Amt. -r- (1 + Bate x Time).
43 PERCENTAGE.
5. What is the principal which in 217 days, at 6^%,
amounts to $918.73 ?
6. What principal in 3 yr. 4 mo. 24 da. will amount
to $761.44 at 5^?
580. Principal, interest, and time given, to find
the rate.
ORAL bxehcises,
1. At what rate will $100 gain $14 in 2 years?
Analysis. — Since the interest of $100 is $14 for 2 yr., it is $7 for
1 yr., and $7 is .07 of $100, the principal. Hence the rate is 7 % . Or,
Since the interest of $100 for 2 yr. at 1 % is $2, $14 is as many
per cent, as $2 is contained times in $14, or 1%.
At what rate will
.2. $300 gain $60 in 4 yr. ?
3. $500 gain $100 in 5 yr. r
4. $400 gain $84 in 3 yr. ?
5. $5 gain $1 in 3 yr. ?
6. $120 gain $60 in 10 yr.?
7. $150 double itself in 10 yr.?
WRITTEN EXERCISES,
581. 1. At what rate per cent, will $1600 gain $280
interest in '^ years ? .
OPERATION.
Int. of $1600 at 1% for ^ yr. = $40. ANALYsis.-Same as
$280 - $40 =. 7 tiihes^; 1% x 7=7^. '(^^oT '^''''''''
At what rate per cent
2. Will $2085 gain $68.11 in 5 mo. 18 da. ?
3. Will $1500 gain $252 in 2 yr. 4 mo. 24 da. ?
EuLE. — Divide the given interest hy the interest of the
^iven principal, for the given time, at 1%.
Formula. — Rate — Int, -r- {Prin. x 1% x Time).
INTEREST. 43
4. A house that cost 114500 rents for $1189. What per
cent, does it pay on the investment ?
^^ 5. At what rate will $1500 amount to $1684.50 in
2 yr. 18 da. ?
6. At what rate per fnonth will $2000 gain $120 in
90 da. ?
7. A man invests $15600, which gives him an annual
income of $1620. What rate of interest does he receive ?
8. At what rate per annum will any sum double itself
in 4, 6, 8, and 10 years, respectively ?
At 1 % , any sum will double itself in 100 yr. ; hence, to double
itself in 4 yr., the rate will be as many times 1^ as 4 yr. are con-
tained times in 100 yr , or 25%, etc.
9. At what rate per annum will any sum triple itself
in 2, 5, 7, 12, and 20 years, respectively ?
10. I invest $49500 in a business that pays me $297 a
month. What annual rate of interest do I receive ?
11. Which is tiie better investment, and how much,
one of $4200, yielding $168 semi-annually, or one of
$7500, producing $712^ annually ?
■■' ^
583. Principal, interest, and rate g^iven, to find
the time.
OJRAIj JEXER CIS es,
1. In what time will $200 gain $56 at 7^ ?
Analysis.— The given interest, $56, is -f^^, or .28, of the princi-
pal, $200; therefore, the time is as many years as .07, the given
rate, is contained times in .28, or 4 times. Hence, etc.
Or, the interest of $200 at 7% for 1 yr. is $14; therefore, the
time is as many years as $14 are contained times in the given inter-
est, $50, or 4 years. Hence, etc.
44 PERCENTAGE.
In what time will
2. $40 gain $10 at 6% ?
3. $500 gain $100 at 4^?
4. $25 gain $20 at 6^?
5. $1000 gain $250 at 6% ?
6. $5 gain 90 cents at 6% ?
7. $50 gain $12| at 10^ ?
WRITTEN EXEnc IS i:s,
583. L In what time will $840 gain $78.12 at 6^?
OPERATION.
$840 X .06 z= $50.40 Int. for 1 yr. Analysis.— Same as in tb^
$78.12-T-$50.40i=1.55. oral exercises. (582.)
1 yr. X 1.55 = 1 yr. 6 mo. 18 da.
In what time
2. Will $175.12 gain $6.43 at 6^?
3. Will $1000 amount to $1500 at 7|-^?
KuLE. — Divide the given interest hy the interest of th$
given principal, at the given rate for 1 year.
Formula. — Time =z Literest — {Prin. x Rate),
4 In what time will $8750 gain $1260 at 2% a month?
V 5. How long must $1301,64 be on interest to amount
to $1522.92 at 5^?
6. How long will it take any sum of money to double
itself at ^%, 6fc, 6%, 1^%, and 10^, respectively?
At 100 % , any sum of money will double itself in 1 year ; hence
to double itself at 10%, it will require as many years as 10% is
contained times in 100%, or 10 yr.
7. How long will it take any sunju to triple itself at
^%, ^%y 7^, ^%, and l^%, respectively ?
8. In what time will the interest of $120, at %%, equal
the priacipal ? Equal half the principal ? Equal twice
the principal ?
I2!^^TEREST. 45
/; COMPOUND INTEREST.
584. Cofnpound Interest is interest not only on
the principal, but on the interest added to the principal
when it becomes due ?
ORAL EXERCISES,
585. 1. What is the comp. int. of $500 in 2 yr. at 6^ ?
Analysis. — The simple interest of $500 for 2 yr. is $60 ; the in-
terest of the first year's interest, $30, for the second year is $1.80,
which, added to $60, gives $61.80, the compound interest. Or,
The interest of $500 for 1 yr. at 6% is $30, and the amount is
$530, which is the principal for the second year ; the interest of $530
for 1 yr. at 6% is $31.80, which added to $530 gives $561.80, the
final amount ; and deducting $500, the original principal, gives
$61.80, the compound interest.
What is the compotmd interest
2. Of $600 for 2 yr. at b% ? 4. Of $300 for 2 yr. at 10^?
3. Of $100 for 2 yr. at 1% ?
5. Of $1000 for 2 yr. at 5^?
What is the amount at compound interest.
6. Of $800 for 2 yr. at h% ?
8. Of $400 for 2 yr. at 4^?
7. Of $2000 for 2 yr. at 10^? 9. Of $500 for 2 yr. at 8^ ?
WRITTEN EXAMPLES,
586. 1. What is the comp. int. of $750 for 2 yr. at 6^?
OPERATION. Analysis.— Since the amount is 1.06
$750 Prin. for let yr. of the principal, the amount at the end
1.06 ®^ *^^ fi^s* y^^^ ^s $795, which is the
'ZZZZ , principal for the 2d year, and the amount
$795 Prm.for 2dyr. ^^ ^j^^ ^^^ ^^ j,^^ 2^ ^^^^ .^ ^^3 ^(,
ll_l^ Hence, by subtracting the given princi-
$842.70 Total amount. V^\ $750, the result is the compound
.j^^Q t- interest, $92.70.
$92.70 Compound int.
46 PEKCEKTAGE.
2. What will $350 amt. to in 3 yr. at 7^, comp. int. ?
3. What is the compound int. of $1200 for 3 yr. at 6% ?
KuLE. — I. Find the amount of the given principal for
the first period of time at the end of ivhich interest is due,
and make it the principal for the second period,
II. Find the amount of this principal for the next period;
and so continue till the end of the given time.
III. SuMract the given principal from the last amounty
and the remainder will be the compound interest.
When the time contains months and days, less than a single
period, find the amount up to the end of the last period, and com-
pute the simple interest upon that amount for the remaining months
and days, which add to find the total amount.
4. What will $864.50 amount to in 4 yr. at 8%^ com-
pound interest ?
5. What is the compound interest of $680 for 2 yr. at
7^, interest being payable semi-annually ?
6. What is the compound interest of $460 for 1 yr.
5 mo. 18 da. at 6%, interest payable quarterly ?
7. What will be the amount of $1250 in 3 yr. 7 mo.
18 da. at 5%, interest being semi-annual?
8. Find the compound interest of $790 for 9 mo. 27 da.
at S%, payable quarterly.
The computation of compound interest may be abridged by
1 sing the following table.
To use the table, multiply the given principal by the number in
the table corresponding to the given number of years and the given
rate. If the interest is not annual, reduce the time to periods, and
the rate proportionally. Thus, 2 yr. 6 mo. , by semi-annual payments,
at 7%, is the same as 5 yr. at 3.^% ; and 1 yr. 9 mo., quarterly
payments, at 8^, the same as 7 yr. at 25^.
INTEREST.
47
587. Table showing the arnt. of $1, at 2^, 3, 3^, 4, 5, 6, 7,
8, 9, 10, 11, and 12%, compound int,,fro7n 1 to 20 years.
Yrs.
2i percent.
3 per cent.
3i per cent.
4 per cent.
5 per cent.
6 per cent.
1
1.025000
1.030000
1.035000
1.040000
1.050000
1.060000
2
1.050625
1.060900
1.071225
1.081600
1.102500
1.123600
3
1.076891
1.092727
1.108718
1.124864
1.157625
1.191016
4
1.103813
1.125509
1.147523
1.169859
1.215506
1.262477
5
1.131408
1.159274
1.187686
1.216653
1.276282
1.338226
6
1.159693
1.194052
1.229255
1.265319
1.340096
1.418519
7
1.188686
1.229874
1.272279
1.315932
1.407100
1.503630
8
1.218403
1.266770
1.316809
1.368569
1.477455
1.59S848
9
1.248863
1.304773
1.362897
1.423312
1.551328
1.689479
10
1.280085
1.343916
J.410599
1.480244
1.628895
1.790848
11
1.312087
1.384234
1.459970
1.539454
1.710339
1.898299
12
1344889
1.425761
1.511069
l.n01032.
1.795856
2.012197
13
1.378511
1.468534
1.563956
1.66^074
1.885649
2.132928
14
1.412974
1.512590
1.618695
1.731676
1.9';9932
2200904
15
1.448298
1.557967
1.675349
1.800944
2.078^.28
2.396558
16
1.484506
1.604703
1.733986
1.872981
2.182875
2.540852
17
1.521618
1.652848
1 794676
1.947901
2.292018
2.692773
18
1 559659
1.702433
1.857489
2.025817
2.4C6619
2.854339
19
1.598650
1.753506
1922501
2.106849
2.526950
3.025600
20
1.638616
1.806111
1.989789
2.191123
2.653298
3.207136
Yrs.
7 per cent.
8 per cent.
9 per cent.
10 per cent.
11 per cent.
12 per cent.
1
1.070000
1.080000
1. Of. 0000
I.IOCOOO
I.IICOCO
1.120000
2
1.144900
1.166400
1.188100
1.210000
1.232100
1.254400
3
1.225043
1.259712
1.295029
1.3310C0
1.867631
1.404908
4
1 310796
1.360489
1.411582
1.464100
1.516070
1.573519
5
1.402552
1.469328
1.538624
1.610510
1.685058
1.762342
6
1.500730
1.586874
1 677100
1.771561
1.870414
1.973822
7
1.605781
1.713824
1.828039
1.948717
2.076160
2.210681
8
1.718186
1.850930
1.992563
2.143589
2.804537
2.47^963
9
1.838459
1.999005
2.171893
2.357948
2.558036
2.773078
10
1.967151
2.158925
2.367364
2.593742
2.839420
3.105848
11
2.104852
2.331639
2.580426
2.853117
3.151757
3 478549
12
2.252192
2.518170
2.812665
3.138428
3.498450
3 895975
13
2.409845
2.719624
3065805
3.452271
3.888279
4.363492
14
2.578534
2.937194
3.341727
3.797498
4.310440
4.887111
15
2.759031
3.172169
3.642482
4.177248
4.784588
5.473565
16
2.952164
3.425943
3.970300
4.594973
5.310893
6.130392
17
3.158815
3.700018
4.327633
5.054470
5.895091
6.666040
18
3.379932
3.996019
4.717120
5.559917
6.54;551
7.689964
19
3.616527
4.315701
5.141661
6.115909
7.263342
8.612760
20
3.869684
4.660957
5.604411
6.727500
8.062309
9.646291
48 PERCENTAGE.
9. Find by the table the compound interest of $950 for
1 yr. 5 mo. 24 da., at 10^, interest payable quarterly.
OPERATION.
1 yr. 5 mo. 24 da. = 5 quarters of a year + 2 mo. 24 da.
10% per annum = 2| % per quarter.
Amount for 5 yr. at 2^% = 1.131408 of principal.
$950 X 1.131408 = $1074.837, amount for 1 yr. 3 mo.
Interest of $1074.837 for 2 mo. 24 da. at 10 fo = $25,079.
$1074.837 + $25,079 = $1099.916, total amount.
$1099.916 — $950 = $149,916, compound interest.
10. Find the amount, at compound interest, of $749.25
for 10 yr. 4 mo., at 7%, interest payable semi-annually.
11. What sum placed at simple interest for 3 yr. lOmo.
18 da., at 11%, will amount to the same as $1500 placed at
compound interest for the same time, and at the same
rate, payable semi-annually ?
rl2. At S%, interest compounded quarterly, how much
will $850 amount to in 1 yr. 10 mo. 20 da. ?.
13. What will $500 amount to in 20 yr. at 7^, comp. int.?
14. A father at his death left $12500 for the benefit of
his only son, 14 yr. 8 mo. 12 da. old, the money to be paid
him when he should be 21 years of age, with 6% interest
compounded semi-annually. What did he receive ?
ANI^UAL INTEREST.
588. Annual Interest is interest on the principal
and on each year's interest remaining unpaid, but so
computed as not to increase the original principal.
It is allowed in the case of promissory notes and other contracts
which contain the words, " with interest payable annually," or with
" compound interest. " In such cases, the interest is not compounded
beyond the second year.
INTEREST. 49
WMITTEN EXERCISES,
589. 1. Find the annual interest and amount of $8000
for 5 yr., at 6^ per annum.
OPERATION. Analysis.— The in-
Int. of $8000 for 5 yr. at 6^=:$2400. t^rest on $8000 for i
- - $480 for 10 yr. at %% = $288. f ' f ^^^. ^^Jf^^> ^^^
•^ ^ for 5 yr. is $2400.
$2400 + $288==$2688, Annual int. The interest for the
$8000 + $2688=1 $10688, Amount. first year, remaining
unpaid, draws interest
for 4 yr. ; that for the second year, for 3 yr. ; that for the third year,
for 2 yr. ; and that for the fourth year, for 1 yr., the sum of which
is equal to the interest of $480 for 4 yr. + 3 yr. + 2 yr. + 1 yr. = 10 yr. ;
and the interest of $480 at 6% for 10 yr. is $288. Hence the total
amount of interest is $2400 + $288, or $2688, and the amt. is $10688.
2. What is the annual interest of $1500 for 4 yr. at 7^?
EuLE. — Compute the interest on the princij)dl for the
given time and rate, to which add the interest on each
yearns interest for the time it has remained unpaid.
To obtain the latter, when the interest has remained
impaid for a number of years, multiply the interest for
one year by the product of the number of years and half
that number diminished by one.
Thus, if the time is 9 yr., the interest for 1 yr. should be multi-
plied by 9 X (9 — 1) -^ 2, or 9 x 4 = 86. Since the interest for
the first year draws 8 years' interest, that for the second year 7
years' interest, etc., and the sum of the series 8 + 7 + 6 + 5 + 4 + 3 + 3
+ lis86.
3. What will $3500 amt. to in 10 yr., annual int., at 8^?
4. What is the difference between the annual interest
and the compound interest of $2500 for 6 yr. at 6^?
5. Find the amt. of $575, at %% annual int., for 9|- yr.
50 PERCEIbf TAGE.
6. $800. Macon, June 15, ia72.
Four years after date, for value received, I promise to pay
Robert E. Park, or order, eight hundred dollars, with in-
terest at seven per cent,, payable annually.
J. W. Burke.
What amount is due on this note at maturity, no in-
terest haying been paid ?
PARTIAL PAYMEl!TTS.
590. Partial Faj/iuents are payments in part of
the amount of a note, bond, or other obligation.
591. Indorsenients are the acknowledgment of
such payments, written on the back of the note, bond,
etc., stating the time and amount of the same.
593. A JProniissof^y Note is a written promise to
pay a certain sum of money, on demand or at a specified
time.
593. The Maker or Drawer of the note is the
person who signs it.
594. The Payee is the person to whom, or to
whose order, the money is paid.
595. An Indorser is a person who, by signing
his name on the back of the note, makes himself respon-
sible for its payment.
596. The Face of a note is the sum of money made
payable by the note.
597. A Negotiable Note is one made payal^le to
bearer, or to any person's order. When so made, it can
be sold or transferred.
PARTIAL PAYMENTS. 51
WRITTEN EXERCISES.
1. $800. New York, Jan. 1st, 1874.
One year after date, I proynise to pay Caleb Barlow, or
order, eight hundred dollars, for value received, with in-
terest. James Dunlap.
Indorsed as follows : April 1, 1874, $10 ; July 1, 1874,
$35 ; Not. 1, 1874, $100. What was there due Jan. 1, 1875 ?
Analysis— The interest of $800 for 3 mo., from Jan. 1 to April 1,
at 7%, is $14; am% $814. Since the payment is less than the in-
terest, it cannot be deducted for a new principal without com-
pounding the int., which is illegal ; hence, find the int. of $800 to
the time of the next payment, 3 mo , which is $14, and the amt. to
that time, $828, from which deduct the sum of the two payments,
or $45, leaving $783, a new principal. The int. of $783 for 4 mo.,
to Nov. 1, is $18.27; amt., $801.27; from which deduct the third
payment, $100, leaving $701.27, the next principal, the amt. of
which for 2 mo., to Jan. 1, 1874, is $709.45, sum due at that time.
Peinciple. — The principal must not he increased hy the
addition of interest due at the time of any payment, so as
to compound the interest.
Upon this principle is based the rule prescribed by the
Supreme Court of the United States :
U. S. EuLE. — I. Find the amount of the given princi-
pal to the time of the first payment, and if this payment
equals or exceeds the interest then due, subtract it from the
amt. obtained, and treat the remainder as a new principal.
II. If the interest exceed the payment, find the amou7it
of the same principal to a time lohen the sum of the pay-
ments equals or exceeds the interest then due, and subtract
the sum of the payments from that amount.
III. Proceed in the same manner loith the remai7iing
payments.
52 PERCEK"TAGE,
$500. Philadelphia, Feb. 1, 1875.
2. Three months after date, I promise to pay to J. B
Lippincott & Co., or order, five hundred dollars, with
interest, without defalcation. Value received,
James Mo^^roe.
Indorsed as follows : May 1, 1875, $40 ; Nov. 14, 1875,
$8; April 1, 1876, $18; May 1, 1876, 130. What was
due Sept. 16, 1876 ?
OPERATION.
Face of note, or principal $500.00
Interest to May 1, 1875, 3 mo., at 6% 7.50
Amount 507.50
Payment, to be subtracted 40.00
2cl principal 467.50
Int. of $467.50 to Nov. 14, 1875, 6 mo. 13 da. . . $15.04
Int. of $467.50 to April 1, 1876, 4 mo. 17 da. . . 10.67 25.71
Amount 493.21
Sum of payments, to be subtracted 26.00
3d principal . 467.21
Int. to May 1, 1876, 1 mo 2.34
Amount 469.55
Payment, to be subtracted 30.00
4tli principal 439.55
Int. to Sept. 16, 1876, 4 mo. 15 da., . T 9.89
Amount due $449.44
3. What was the amount due October 25, 1873, upon a
note for $1500, dated New Orleans, April 1, 1872, and
on which the following payments were endorsed : June 5,
1872, $300 ; Oct. 15, 1872, $37.75 ; May 1, 1873, $97.25 ;
Aug. 6, 1873, $495?
PARTIAL PAYMENTS. 53
$700. Detroit, Nov. 1, 1873.
4. On demandy 1 promise to pay Charles Smith, or
order, seven hundred dollars, with interest. Value re-
ceived, Abraham Isaacs.
Indorsed as follows : Dec. 5, 1873, $75 ; Jan. 10, 1874,
$350; April 11, 1874, $11.25; May 15,^874, $250.
What was due Sept. 1, 1874?
$4 97^A' Chicago, March 15, 1874.
5. Three months after date, tve promise to pay James
Kelly, or order, four hundred and ninety-seven ^^^ dollars,
with interest at 6%, Value received.
Brown, Nichols & Co.
Indorsed as follows : Nov. 3, 1874, $57.50 ; June 15,
1875, $22.25 ; Aug. 1, 1875, $125 ; Sept. 15, 1875, $175.
What was due Jan. 1, 1876?
598. The following method of computation is often
used by merchants in the settlement of notes and of in^
terest accounts running a year or less ; hence called the
Mercantile Eule:
I. Find the amount of the note or deU from its date
to the time of settlement.
II. Find the amount of each payment frmn its date
to the time of settlement.
III. Subtract the sum of the amounts of payments from
the amount of the note or debt.
An accurate application of this rule requires that the time should be reduced
to days, and that the interest should be computed by the rule for days (574).
For the Vermont State method of computation, and also of assessing taxes,
see pages 227-231.
54 PERCENTAGE,
1. On a note for $600 at 1%, dated Feb. 15, 1874, were
the following indorsements : March 25, 1874, $150 ; June
1, 1874, $75 ; Oct. 10, 1874, $100. What was due Dec. 31,
1874 ?
OPERATION.
Am't of $600 from Feb. 15 to Dec. 31, 319 da., $636.71
'' '' $150 '' Mar. 25 '' " 281 da., $158.08
" '' $75 " June 1 '* *' 213 da., 78.06
** *' $100 " Oct. 10 *' '* 82 da,, 101.57 837.71
Balance due Dec. 31, 1874, $299.00
2. A note for $950, dated Jan. 25, 1876, payable in
9 mo., at 7^ interest, had the following indorsements :
March 2, 1876, $225 ; May 5, 1876, $174.19 ; June 29,
1876, $187.50; Aug. 1, 1876, $79.15. What was the
balance due at the time of its maturity ?
3. Payments were made on a debt of $1750, due April 5,
1875, as follows : May 10, 1875, $190 ; July 1, 1875,
$230 ; Aug. 5, 1875, $645 ; Oct. 1, 1875, $372. What
was due Dec. 31, 1875, interest at 6^ ?
DISOOUJ^fT.
599. Discount is a certain percent deducted from
the price-list of goods, or an allowance made for the pay-
ment of a debt or other obligation before it is due.
600. The Present Worth of a debt payable at a
future time without interest, is such a sum as, being put
at legal interest, will amount to the debt when it becomes
due.
601. The True Discount is the difference between
the whole debt and the present worth.
DISCOUKT. 55
OJB^X EXERCISES,
603. 1. What is the present worth of a debt of $224,
to be pdid in 2 yr., at 6^ ?
Analysis. — Since in 2 yr., at 6^, the int, is .12 of the principal,
the amt. is 1.12 of it ; therefore, $234, the debt, is 1.12, or ^^f of
the present worth, and jgg, or the present worth itself, is $200.
Or, since $1.12 is the amt, of $1, $224 is the amt. of as many dol-
lars as $1.12 is contained times in $224, or $200. (578.)
What is the present worth
2. Of $315, due in 10 mo., at 6^?
3. Of $570, due in 2 yr., at 7^?
4. Of $408, due in 3 mo., at 8^ ? *
5. Of $51, due in 4 mo., at H ?
6. Of $440, due in 2 yr., at b% ?
Find the true discount at 6^,
7. Of $1019, due in 3 mo. 24 da.
8. Of $102.20, due in 4 mo. 12 da.
9. Of $5035, due in 1 mo. 12 da.
WRITTEN EXERCISES,
603. 1. What are the present worth and the true dis*
count, of $362.95, payable in 7 mo. 12 da., at 6^ ?
OPERATION.
Amt. of $1, for 7 mo. 12^da., at 6^ = $1,037
$362.95 -T- $1,037 = 350*times.
$1 X 350 = $350, Present Worth.
$362.95 — $350 = $12.95, True Discount.
Analysts.— Since the amount of $1 for 7 mo. 12 da. at 6% is
$1,037 (579), $362.95 is the amount of as many dollars as $1,037
is contained times in $362.95, or 350 times. Hence the present
worth is $350 ; and the true discount is $362.95 — $350, or $12,95.
56 PERCENTAGE.
^'
2. What is the present worth of a debt of $287.'}'5 to be
id in 3 mo. 18 da. at 11% ?
3. What is the true discount on a debt of $2202.90 due
in 8 mo. 12 da. at 7% ?
KuLE. — I. Divide the debt iy the amount of $1 for the
given rate and time, and the quotie7it is the present worth.
11. Subtract the j^f'esent worth from the debt, and the
remainder is the true discount.
Formula. — Present Worth = Debt -r- Amt. of $1.
Hence the present worth is the principal of which the true dis-
count is the interest, anij the whole debt the amount.
4. Bought a house and lot for $19500 cash, and sold
them for $22000, payable one-fourth in cash and the re-
mainder in 1 yr. 6 mo. How much ready money did I
gain, computing discount at 6%?
5. A merchant buys goods for $4200 on 4 mo. credit,
but is offered a discount of 3% for cash. If money is
worth ^% a month, what is the difference ?
6. Bought a bill of lumber amounting to $3500, on
6 mo. credit ; 2 months afterward paid on account $1500,
and 1 month later, $1000. Find the present worth of
the balance, at the time of the second payment, int. at 7%.
7. A merchant holds two notes, one for $356.25 due
Dec. 1, 1875, and the otheii' for $497.50, due Feb. 1, 1876.
What would be due him in cash on both notes Sept. 15^
1875, at 6% ?
8. A bookseller bought $300 worth of books at a dis-
count of 33^^ from list prices, and sold them at the reg-
ular retail price, on 6 mo. time. Money being worth 6^,
what per cent, profit did he make ?
DISCOUKT. 57
9. A speculator bought 230 bales of cotton, each bale
eontainiug 470 lb., at llf cents a pounds on a credit of
9 mo. He at once sold the cotton for $13000 cash, and
paid the pres. worth of the debt at 1%. What was his gain ?
10. Which is the more profitable, to buy flour at $8.75 a
barrel on 6 mo. credit, or at $8.60 on 2 mo., money being
worth 7^?
11. A person sold goods to the amount of $3750, 15^
payable in cash, 25^ in 3 mo., 20^ in 4 mo., and the re-
mainder in 6 mo. What ready money would discharge
the whole debt, money being worth Q% ?
r BANK DISCOUNT. ^
604. A Bank is a corporation chartered by law for
the safe-keeping and loaning of money, or the issuing of
bills for circulation as money.
605. Bank Bills or Notes are promissory notes
issued by banks, and payable on demand.
A bank which, issues notes to circulate as money is called a Bank
of Issue ; one which lends money by discounting notes, a Bank of
Discount ; and one which takes charge of money belonging to other
parties, called depositors, a Savings Bank, or Bank of Deposit,
Some banks perform two and others all of these duties.
606. Bank Discount is a deduction made for
interest in advancing money upon a note not due, or pay-
ment by a borrower, in advance, of interest upon money
loaned to him. It is equal to the interest at the given
rate for the given time (including the days »of grace) on
the whole sum specified to be paid.
607. Bays of Grace are the three days allowed
by law for the payment of a note after the expiration of
the time specified in the note. They are counted in by
bankers in discounting notes.
58 ^ PERCENTAGE.
608. The Maturity of a note is the expiration of
the whole time, including the days of grace.
609. The Term of Discount is the time from the
discount of a note to its maturity.
610. A JBank Chech is a written order for money
by a depositor, upon a bank.
611. The Proceeds or Avails of a note is the
sum received for it when discounted, that is, the face of
the note less the discount.
613. A Protest is a formal declaration in writing,
made by a Notary-Public, at the request of the holder of
a note, to give legal notice to the maker and the indorsers
of its non-payment.
1. The failure to protest a note on the third day of grace releases
the indorsers from all obligation to pay it.
2. If the third day of grace or the maturity of a note occurs on
Sunday or a legal holiday, it must be paid on the day previous.
3. The transaction of borrowing money at a bank is conducted
as follows : The borrower presents a note, either made or indorsed
by himself, payable at a specified time, and receives for it a sum
equal to the face less the interest for the time it has to run, in-
cluding the days of grace. A note for discount at a bank must be
made payable to the order of some person, by whom it must be
indorsed. When the note bears interest, the discount is computed
on its face plus the interest for the time it has to run.
613. Bank discount being simple interest, the follow-
ing are corresponding terms :
The Face of the Note is the principal.
The Term of Discount is the time.
The Bank Discount is the interest.
The Proceeds is the principal less the interest.
DISCOUiTT. 59
614. To find the bank discount and proceeds of
a note.
OJtA.L MXEJiClSBS,
1. What is the bank discount on a note for $2000 due
in 2 mo. 15 da. at Q%, and the proceeds ?
Analysis. — After adding 3 da., tlie time is 2 mo. 18 da. ; the in-
terest for which at 6% is .013 of the principal ; .013 of $2000 is $26,
the hank disco ant, and $2000 — $26 equals $1974, or the proceeds.
What are the l)a7iJc discount and the proceeds of a note
2. Of $80 for 5 mo. 27 da., at 7^?
3. Of $100 for 2 mo. 21 da., at 6^ ?
4. Of $200 for 8 mo. 9 da., at 1% ?
5. Of $150 for 4 mo. 21 da., at b% ?
6. Of $100 for 30 da., at Q>% ?
WniTTBN JEX EB CISES.
615. 1. Eequired the bank discount and proceeds of
a note for $1250 due in 90 days, at 7^.
OPERATION.
$1250 X. 07 ^ gg ^ ^^^ g^^ g^^^ Discount.
(565
$1250 — $22.32 = $1227.68, Proceeds.
Analysis. — The interest of $1^50 for 93 da., at 7%, reckoning
865 da. to the year, is $33.32, which is the bank discount. If 360 da.
are reckoned to the year, the bank disc't is $33,604. Deducting the
bank disc't from the face of the note, the remainder is the proceeds.
Rule. — I. Compute the interest on the face of the note
(or if it hears interest, on its amount at maturity), for
three days more than the specified time, and the result ns
the hank discount.
II. Suhtract the discount from the face of the note, or '
its amount at maturity, and the remainder is tJ^e proceeds.
60 PERCEISTTAGE.
2. Wkat is the bank discount, and what is the pro-
ceeds of a note for $597.50, due in 60 da., at (j% ?
3. What will be the proceeds of a note for $1G15, due
in 90 da. with interest at 1%, discounted at the Nassau
Bank in New York ?
4. Sold a farm, containing 173 A. 95 P., for %Q^ an
acre, and received payment as follows : $2000 cash, and
the balance in a note payable in 5 mo. 18 da. at 1% inter-
est, which was discounted at a bank. How much ready
money did the farm bring ?
Find the date of maturity, the term of discount , and
the proceeds of the following :
$957^. Chicago, July 27, 1875.
5. Three months after date, I promise to pay to the
order of D. L. Moody, nine hundred fifty-seven and -^^
dollars, for value received.
* Discounted Aug. 10, at S%. William Thomson".
$916^. San Francisco, Feb. 5, 1874.
6. Two months after date, we jointly and severally
agree to pay 0. H. Thomas^, or order, nine hundred six-
teen and y^ dollars luith interest at 8^, value received.
Discounted at Marine Bank, James Barnes.
Feb. 21, at 10^. George Childs.
$1315y^. New York, May 1, 1875.
J. Ninety days after date, I promise to pay to the
order of Ivison, Blakeman, Taylor & Co., one thousand
three hundred fifteen and ^^ dollars, for value received.
Discounted May 15, at 7^. William Hewson".
* Banks usually count the actual number of days in the given time, and
"ijS days to the year.
DISCOUNl, 61
$1250. Boston, June 12, 1876.
8. Six months after date, I promise to pay Knight,
Adams & Co., or order, twelve hundred fifty dollars, with
interest at 6 per cent., value received.
Discounted at a broker's, Geo. B. Damok.
Nov. 15, at 6%.
616. The proceeds and time of a note given, to
find the face.
OMAJj EXEB,C IS ES .
1. For what sum must a note be drawn, at 2 mo. 15 da.,
• at 6^, so that the proceeds when discounted may be $987 ?
Analysis. — The bank discount for 3 mo. 18 da. at 6% is .013 of
the face of the note, and the proceeds must therefore be 1 — .013,
or .987 of the face ; and if .987 of the face is $987, the whole face
of the note is $1000.
Required the face of a note, so that the proceeds maybe
2. $972, for 4 mo. 21 da. at 7^.
3. $194, for 5 mo. 27 da. at 6^.
4. $97.60, for 3 mo. 15 da. at %%.
5. $980, for 4 mo. 21 da. at b%.
6. $184, for 9 mo. 15 da. at 10^.
wit ITT EN EXEHCISES.
617. 1. What must be the face of a note at 9 mo.
27 da., interest S%, so that the proceeds may be $448 ?
OPERATION.
The bank discount of $1 for 10 mo. at 8% is $.066|.
The proceeds of $1 = $1 - $.066f or $.933^.
Hence $448 -7- .933^ = $480, the face of the note.
2. What is the face of a note at 30 da., the proceeds of
which, when discounted at bank, at 7^, are $1425 ?
62 PEECENTAGE.
KuLE. — Divide the given proceeds dy the j)TOceeds of |1
for the time and rate given ; the quotient is the face of
the note.
Formula, — Face = Proceeds — (1 — Bate x Time).
3. Find the face of a 3 mo. note the proceeds of which,
discounted at 2% a month, is $675.
4. The proceeds of a note are $1915.75, the time 3 mo.,
and the rate of interest 7% ; what is the face of the note ? "
5. Bought merchandise for $2250, cash ; for what sum
.must I draw my note at 3 mo., so as to obtain that sum
at the bank, interest at 7^ ?
6. The avails of a 3 months note, when discounted at
tl^%, were $315.23 ; what was the face of the note ?
7. For what sum must a note dated April 5, for 90 da.,
be drawn, so that when discounted at 7^, on Ajjril 21,
the proceeds may be $650 ?
8. For how much must I draw my note at 90 da., in
order that when discounted at a bank, at 7^, its avails
will pay for 137i J^- <>* cloth at $2| a yard?
SAVINGS-BANK ACCOUNTS.
618. A Savings-Bcmk is designed chiefly to ac-
commodate depositors of small sums of money.
Interest is allowed semi-annually on all sums that have been on
deposit for a certain time, if not drawn out before the regular daj
of paying interest — generally on the 1st of January and of July.
Savings-banks generally allow interest only from the commence-
ment of each quarter ; but in some banks money deposited pre-
vious to the 1st day of any month draws interest from that date to
the day of declaring interest dividends, provided it has not been
previously withdrawn.
DISCOUNT.
63
WniTTEN JEXEB CIS ES.
619. 1. A person had on deposit Jan. 1, 1874, $150.
His subsequent deposits were, Feb. 3, $35 ; March 29,
$20 ; April 10, $43 ; May 15, $26. His drafts during the
same time were, Jan. 15, $50 ; Feb. 27, $15 ; April 19,
$45. What interest was due July 1st, at Q% ?
OPERATION.
Date of
Balance
SmaUestBal.
Interest
SmaUestBal.
Interest for
Int. paym'ts.
1st of month.
during mo.
for 1 month.
dur'g Q'rter.
1 Quarter.
Jan. 1
$150
Feb. 1
100
$100
$.50
Mar. 1
120
100
.50
Apr. 1
140
120
.60
$100
$1.50
May 1
138
138
.69
June 1
164
138
.69
July 1
164
164
.82
138
2.07
$3.80
$3.57
Balance due, with int. by monthly periods, $167.80.
** ** " quarterly *' $167.57.
Analysis. — At the end of January, tbe balance due is $100, which
having been on deposit for the month, draws interest for 1 mo. ; at
the end of February, the balance is $120 ; but the smallest balance
during the month is $100 ; hence interest is allowed only on that
sum. The same principle applies to the other balances. If only
quarterly periods of interest are allowed, the interest is calculated
at the end of each quarter on the smallest balance during the quar-
ter, or, in this case, on $100, April 1, and $138, July 1.
2. Find the balance, due July 1, on the following
account : Deposits, Jan. 15, $175 ; April 10, $60 ; May 31,
$110. Drafts, March 5, $75; May 1, $35 ; June 10, $50.
Interest at 6^, from the 1st day of each months
64
PERCENTAGE.
3. A person deposits in a savings-bank the following
sums : Jan. 1, $350 ; Feb. 5, $150 ; March 15, $75 ;
May 10, $30 ; June 15, $100. During the same time he
draws, Jan. 15, $150 ; Feb. 10, $200 ; March 31, $50 ;
June 1, $75. What interest at 6%, payable from the 1st
of each month, must be added to the account July 1 ?
4. Balance the following, Jan. 1, 1875 : Balance due to
Margaret Brown, July 1, 1874, $275. Deposits received
as follows : Aug. 1, $125 ; Sept. 15, $57 ; Oct. 10, $350.
Drafts paid : July 15, $100 ; Sept. 1, $150 ; Nov. 15,
$6S ; Dec. 15, $125. Interest at 6%, from the 1st of each
quarter, July 1 and Oct. 1.
KuLE. — At the end of each term complete the interest for
the term on the smallest balance on deposit at any time
during the quarter ; and at the end of each period of six
months add to the balance of principal the whole amount of
interest due, and the sum will he the principal at the com-
mencement of the next six inonths.
5. How much was due Jan. 1, 1876, on the following
account, allowing interest, computed from the 1st of each
quarter, Jan. 1 to July 1, at 6% per annum ?
Br. Greenwich Savings Bank, in acct. with Mary Williams. Or.
1874.
1874.
Jan. 1
To Cash .
$136
00
Sept. 15
By Check
$75
00
Mar. 17
(( u
25
00
1875.
Aug. 1
ii St
87
50
Jan. 20
t( (t
37
50
1875.
Mar. 3
it (I
50
00
June 11
(t a
150
00
Nov. 17
c( a
72
00
REVIEW,
65
630.
SYNOPSIS FOR KEVIEW.
10. Interest.
1. Defs,
j 1. Interest. 2. Principal. 3. Rate.
• ( 4. A
6% Method
■U:
11. COMPOUN
Interest,
12. Annual
Interest. ) 2,
S'D j 1,
iT. ( 2,
r. ( 2,
13. Partial
Payments.
14. Discount.
i::
15. Bank Dis-
count.
16. Savings-Bank
Amt. 5. Legal Int. 6. Usury ^
Corresponding Elements.
1. Principle. 2. Rule, I, II, III.
Relations between Time ) j jj jjj j^
and Interest. ) ' ' '
569. Rule, I, II, III, IV.
Principles, 1, 2.
Rule.
Accurate Interest. Rule.
r 576. 1. Bule. 2. Formula.
Problems J ^78. 1. i?i/^^. 2. Formula.
1 580. 1. Rule. 2. Formula.
t 582. 1. i??/^6. 2. Formula.
Definitions — Compound Interest.
Rule, I, II, III.
Definitions.
Rule, I, II.
1. Part. Pay'ts. 2. Indorsem'ts.
3. Promissory Note. 4. Maker
or Drawer. 5. Payee. 6. In-
dorser. 7. Face of a Note.
8. Negotiable Note.
U. S. Rule, I. TI. Merc. Rule.
Discount. 2. Present Worth.
True Discount.
Rule, I, II.
1. Bank. 2. Bank Bills or Notes.
3. Bank Discount. 4. Days of
Grace. 5. Maturity of Note.
6. Term of Discount. 7. Bank
Check. S. Proceeds or J vails i
9. Protest.
Corresponding Terms.
614. Rule, I, II.
616. Rule ,
Accounts— Rule.
1. Defs.
2. Principle.
Defs. j o*
Defs. <
/y
631. A Corporation is an association of indi-
viduals authorized by law to transact business as a single
person.
633. A Charter is the legal act of incorporation
defining the powers and obligations of the body incor-
porated.
633. The Capital Stock of a corporation is the
capital or money contributed, or subscribed to carry on
the business of the company.
634. Certificates of Stock or Scrip are the
papers or documents issued by a corporation, specifying
the number of shares of the joint capital which the
holders own.
635. A Share is one of the equal parts into which
capital stock is diyided.
The value of a share in the original contribution of capital varies
in different companies. In bank, insurance, and railroad comj)^-
nies, it is usually $100.
636. Stocks is a general term applied to shares of
stock of various kinds, Government and State bonds, etc.
Stockholders are the owners of stock, either by original title or by
subsequent purchase. The stockholders constitute the company.
637. The Far Value of stock is the sum for which
the scrip or certificate was issued.
638. The Market Value of stock is the sum for
which it can be sold. «
STOCKS. 67
Stock is at par when it can be sold for its original or face value,
or 100^ ; it is above par, or at a premium, when it will bring more
than its face value ; and it is below par, or at a discount, when it
sells for less than its face value. Thus, when stock is at par, it is
quoted at 100 ; when it is 5% above par, at 105 ; and when His 5%
below par, at 95.
639. JPremium^ Discount^ and JBrokerage
are each 2i percentage computed upon the par value of the
stock as the base.
630. A Stock Broker is a person who buys and
sells stocks, either for himself, or as the agent of another.
631. Stock-jobbing is the buying and selling of
stocks with the view to realize gain from their rise and
fall in the market.
633. An Installment is a portion of the capital
stock required of the stockholders as a payment on their
subscription.
633. An Assessment is a sum required of stock-
holders, to meet the losses, or to pay the business expenses
of the company.
634. A Dividend is a sum paid to the stockholders
from the profits of the business.
Dividends and assessments are a percentage computed upon the
par value of the stock as the base.
635. Net Earnings are the moneys left from the
profits of a business after paying expenses, losses, and the
interest upon the bonds.
636. A JBond is a written instrument securing the
payment of a sum of money at or before a specified time.
The principal bonds dealt in by brokers are Government, State,
City, and Railroad bonds.
ffS PERCEKTAGE.
637. Z7. S. Sands are of two kinds ; viz., these
which are payable at a fixed date, and those which, while
payable at a fixed date, may be paid at an earlier specified
time, as the Government may elect.
1. The former are quoted in commercial transactions by the rate
of interest which they bear ; thus, United States bonds bearing 6%
interest are quoted U, S. 6's. The latter are quoted in commercial
transactions by a combination of the two dates ; thus, U. 8. 5-^0's,
or U. 8. 6*8 5-20 y means bonds of U. S. bearing 6 % interest, and pay-
able at any time from 5 to 20 years, as the Government may choose.
2. When it is necessary to distinguish different issues bearing the
game rate of interest, the year at which they become due is also
mentioned ; thus, U. 8. S's of '71; JJ. 8. S's of '74; U. 8. 6'8, 5-20,
of '84; U. 8. 6' 8, 5-20, of '85,
3. The 5-20's were issued in 1862, 'G4, '65, '67, and 70. They
bear interest at 6 % , paid semi-annually in gold, except the issue of
1870, called 5's of '81-, which bear int. at 5%, paid quarterly in gold.
4. Bonds issued by States, cities, etc., are quoted in a similar
manner. Thus, 8. G. 6'8 are bonds bearing 6% interest, issued by
the State of South Carolina.
638. A Coupon is a certificate of interest attached
to a bond, to be cut off and presented for payment when
the interest is due.
639. Currency is a term used to denote the circu-
lating medium employed as a substitute for gold and
silver. It consists, at present, in the United States, of
U. S. Legal-tender Notes, or ^^ Greenbacks," and the
Bills issued by the Nat. Banks, and secured by U. S. Bonds.
If from any cause the paper medium depreciates in value, gold
becomes an object of investment, the same as stocks. Gold being
of fixed standard value, its fluctuations in price indicate changes
in the value of the currency. Hence, when gold is said to be at
a premium, currency is virtually below par, or at a discount.
STOCKS. d9
ORAZ JSXJUBCISJSS.
640. 1. Find the cost of 100 shares of Chicago and
Kock Island Eaih'oad stock at 90 ; brokerage ^%.
Analysis. — Since the cost of one share is 90% of $100, or $90,
the cost of 100 shares is 100 times $90, or $9000, to which add the
brokerage, i% of $10000, or $12|, and the sum $9012^, is the entire
cost of the stock.
2. What cost 50 shares of N. Y. Central R. R. Stock,
at par ; brokerage, ^% ?
, 3. Find the cost of 10 shares of Bank Stock at 104 ;
brokerage ^%\
4. What is the cost of $2000 U. S. 6's 5-20, at 112 ;
brokerage ^% ?
641, 1. A broker has $5010 to invest in bank stock at
25^ premium ; how many shares can he buy, charging ^%
for brokerage ?
Analysis. — Since the stock sells at 25% premium, each share
with brokerage will cost $125| ; hence he can buy as many shares
as $125J are contained times in $5010, or 40 shares.
2. A speculator invested $52000 in Ohio and Missis-
sippi E. R. stock at 25|, allowing ^% brokerage ; how
many shares did he buy ?
3. If I invest $2350 in U. S. 6's, '81, at n7|, broker-
age ^%, how many $1000 bonds do I receive ?
643. 1. A man bought a number of shares of mining
stock at 60, and sold the same at 68, and gained $800
by the transaction. How many shares did he buy ?
Analysis. — Since he bought at 60% and sold at 68%, he gained
8% of the pa'r value ; hence $800 is 8% of $10000, the par value,
and the number of shares at $100 each is 100.
70 PERCENTAGE.
2. Bought K. R. stock at 90, and sold at par, gaining
$1000. Required the number of shares.
3. I purchased stock at 110 and sold at 98, losing
$1200. How many shares did I buy ?
4. A broker bought some stock at par, and sold it
at 95, losing $2000. How many shares did he buy ?
643. 1. What sum must be inyested in California 7's,^
at 110, to obtain therefrom an annual income of $1400 ?
Analysis. — Since the annual income is |7 on each share, the
number of shares must be equal to $1400 -i- $7, or 200 shares ; and
200 shares at $110 amount to $22000, the required investment.
2. What sum must I invest in stock at 115, paying
10^ yearly dividends, to realize an income of $2000 ?
3. What sum must be invested in N. Y. 7's at 103|-,
in order to receive therefrom an annual income of $2100 ?
644. 1. What per cent, does money yield which is
invested in S% stock at 120 ?
Analysis. — Since each share costs $120, and pays $8 income, the
per cent, will be yf^, or ^^ of 100%, equal to 6f %.
2. What per cent, does stock yield when bought at
90, paying 6% dividends ? When bought at 75 ? At 120 ?
3. What per cent, of interest does stock yield, which
pays 6% semi-annual dividends, if bought at 150 ? At .
140 ? At 120 ?
645. 1. What should be paid for stock yielding C)%
dividends, in order to realize an annual interest on the
investment of 8% ?
Analysis. — Since the annual dividend on each share is $6, this
must be 8% of the sum required ; and if 8% is $6, 1% is $f, and
100 % is $75. Hence the stock must be bought for 75.
STOCKS. 71
2. For what must stock that pays 7^ dividends be
bought to realize 10^ interest ? 9^ ? S%?
3. For what should Missouri 6's be bought to pay 6%
interest? 6^%? 6^%? S%?
646. 1. How much currency can be bought for $500
in gold, when the latter is at a premium of 10^ ?
Analysis.— Since $1 in gold is worth |1,10 in currency, $500 in
gold is worth 500 times $1.10, or $550. Hence, etc.
2. How much currency can be bought for $200 in
gold, when the latter is at a premium of d% ?
3. What is $1000 in gold worth in currency, when
the former is at a premium of 12^ ? Of 9^% ? Of 10^^ ?
64*7. 1. How much gold can be bought for $440 in
currency, when the former is at a premium of 10%'?
Analysis.— Since $1 in gold is worth $1.10 in currency, $440
will buy as many dollars in gold as $1.10 is contained times la
$440, or $400 in gold. Hence, etc
2. How much gold selling at 9% premium will $1090
in currency buy ? $218 ? $654 ?
3. How much gold at 11^ premium will $444 buy ?
' WRITTEN EX EKCISJES.
648. Find the cost
1. Of 220 shares of bank stock, the market value of
which is 103J, brokerage J^
Operation.— (103f% + J%)of $100 = $104, cost of 1 share.
$104 X 220 = $22880, cost of 220 shares. (640.)
Formula. — Entire Cost = {Marhet Value of 1 Share
+ Brokerage) x No. of Shares.
2. Find the cost of 350 shares of Western Union Tele-
graph stock, market value 97f , brokerage i%.
72 PEECENTAGE.
3. A broker bought for me 15 one-thousand-doUar TJ. S.
5-20 bonds at 112J, brokerage ^%. What was their cost ?
4. My broker sells for me 125 shares of stock at 127^.
What should I receive, the brokerage being J^ ?
649. Find the nuniber of shares
1. Of bank stock at 105, that can be bought for $25260,
including brokerage at \% ?
Operation.— (105% +\%)oi $100 = $105|^, cost of 1 share.
$25260 -^ $1051 == 240, No. of shares. (641.)
Formula. — No. of Shares = Investment -r- Cost of 1
Share.
2. How many shares of N. J. Central R. E. stock at
107|, brokerage ^%, can be bought for $27000?
3. How many shares of Mo. 6's at 97|, brokerage J^,
will $21560 purchase?
4. Bought Pacific Mail at 29|^, and sold at 31J, paying
\% brokerage each way. How many shares will gain $330 ?
Opekation.— (31J% -29lfc)-i% =H%, gain.
-^ $1.50 = 220, No. of shares. (642.)
Formula. — No. of Shares = Wliole Gain or Loss -j-
Gain or Loss per Share.
5. How many shares of stock bought at 97|^ and sold
at 102}, brokerage \% each way, will gain $990 ?
6. Lost $1680 by selling N. Y. Central at 101 that cost
104. Brokerage being \% each way, how many shares did
I sell?
7. How many shares of the Bank of Commerce bought
at 110| and sold at 116J, brokerage \% on the purchase
and the sale, will gain $1200 ?
^ STOCKS. 73
650. Find the amount of investment
1. In U. S. 5's, of '81, at 111, so as to realize therefrom
an annual income of $2500 ?
Operation. — $2500 -^^ $5, income on 1 share = 500, No. of shares.
$111, price of 1 share x 500 = $55500, investment. (643.)
FoEMULA. — Investment = Price of 1 Share x iVb. of
Shares,
2. What sum must be invested in Tennessee 6's at 85,
to yield an annual income of $1800 ?
3. How much money must be invested in any stock at
105|^, which pays 6% semi-annual dividends, to realize an
annual income of $2000 ?
4. What sum invested in stock at $63 per share, will
yield an income of $550, the par value of each share being
$50, and the stock paying 10^ annual dividends?
651. Find the rate per cejit, of income, realized
1. From bonds paying 8^ interest, bought at 110.
Operation. — $8, interest per share -^ $110, cost per share =
.073?3:, or7i\%. (644.)
Formula. — Rate % of Income = Interest per Share -r-
Cost per Share.
2. If stock paying 10^ dividends is at a premium of
12^^, what per cent, of income will be realized on an in-
vestment in it ?
3. Which will yield the better income, S% bonds at 110,
or 5's at 75 ?
4. Which is the more profitable, and how much, to buy
New York 7's at 105, or 6 per cent, bonds at 84?
74 PERCEKTAGE.
5. What per cent, of income does stock paying ].0^
dividends yield, if bought at 106 ?
6. What per cent, will stock which pays 6% dividends
yield, if bought at a discount of 16% ?
7. What rate per cent, of income shall I receive, if I
buy U. S. 5's at a premium of 10^, and receive payment
at par in 15 years ?
'653. Find at what price stock must he bought
1. That pays 6% dividends, so as to realize an income
of 7^% on the investment.
Operation.— .06 -f- .075 = .80 or 80%, price of stock. (045.)
Formula. — Pince of Stock = Dividend -^ Rate of hi-
come,
2. What must be paid for h% bonds, that the invest-
ment may yield 8^?
• 3. How much premium maybe paid on stock that pays
10% dividends, so as to realize 7\% on the investment ?
, 4. What must I pay for Government 5's of '81, that my
investment may yield 1% ?
* 5. At what price must stock, of the par value of $50 a
share, and that pays 6% dividends, be bought, to yield an
income of 7^% ?
* 6. At what price must 6% stock be bought, to pay as
' good an income as 8% stock bought at par ? As 9% stock ?
653. Find the value in currency,
1. Of $3750 in gold, quoted at llOf
Opekation.— $1.10| X 3750= $4143. 75, value in currency. (646.)
Formula. — Total Value in Currency = Value of $1 in
Currency x No. of Dollars in Gold,
STOCKS. 75
2. Find the value of $4975 in gold, at a premium of
" 3. What is the semi-annual interest of 18000 6% gold-
bearing bonds worth in currency, when gold is at lllf ?
• 4. A merchant bought a bill of goods, for which he
was to pay l>7000 in currency, or $6625 in gold. Gold
being at 109|, which is the better proposition, and how
much in currency ?
654. Find the value in gold,
1. Of 12150 in currency, when gold is at a premium of
Operation.— $2150 h- 1.105 = $1945.70, value in gold. (G47.)
Formula. — Total VaUie in Gold = AmL of Currency
-r- (1 + Premium),
2. What is $4500 in currency worth in gold, when the
latter is at a premium of 12|;^ ? At \\\% ? At 9^^ ?
3. How much money must be invested in U. S. 6's at
111, when gold is quoted at llOf, in order to obtain a
semi-annual income of $2210 in currency?
4. The Mechanics Bank of New York having $109737.50
to distribute to the stockholders, declares a dividend of
^\% ; what is the amount of its capital ?
5. A man owns a house which rents for $1450, and the
tax on which is 2|^ on a valuation of $8500. He sella
for $15300, and invests in stock at 90, that pays 7;^ divi-
dends. Is his yearly income increased or diminished,
and how much ?
6. If I have $36500 to invest, and can buy N. Y. Cen-
tral 6's at 85, or K Y. Central 7's at 95, how much more
profitable will the latter be than the former ?
y
7fi PEKCENTAGE.
7. Which is the better investment, a mortgage for 3 yr.
of $5000, paying 1% interest, and purchased at a discount
of b%, or 50 shares of stock at 95, paying S% dividends,
and sold at the expiration of 3 years at 98 ?
8. Henry Ivison, through his broker, invested a certain
sum of money in New York State 6's at 107|^, and twice
as much in U. S. 5's, of ^81, at 98J, brokerage in each
case ^%. The annual income from both investments was
$3348. How much did he invest in each kind of stock ?
9. A gentleman invested $12480 current funds in
U. S. 5-20's of ^85, at 104. What will be his annual
income in currency when gold is 1 10 ?
I]:^SUBAI^CE.
655. Insurance is a contract of indemnity against
loss or damage. It is of two kinds : insurance on prop-
erty, and insurance on life.
656. The Insurer or Underwriter is the party
who takes the risk or makes the contract.
657. The Policy is the written contract between the
parties.
658. The Premium is the sum paid for insurance,
and is a certain per cent, of the sum insured.
659. Insurance business is generally conducted by Companies,
wliich are either Joint-stock Companies, or Mutual Companies,
A StocJc Insurance Company is one in which the capi-
tal is owned by individuals called stockholders. They alone share
the profits, and are liable for the losses.
A 3£utnal Insurance Cornj)anif is one in which the profits
and losses are divided among those who are insured.
Some companies are conducted upon the Stock and Mutual plans
combined and are called Mixed Companies.
Insurance on property is principally of two kinds: Fire
Insurance, and Marine and Inland Insurance.
660. Fire Insurance is indemnity for loss of
property by fire.
661. Marine and Inland Insurance is in-
demnity for loss of vessel or cargo, by casualties of navi-
gation on the ocean, or on inland waters.
Transit Insurance refers to risks of transportation by land only,
or partly by land and partly by water. The same policy may cover
both Marine and Transit Insurance.
Stock Insurance is indemnity for the loss of cattle, horses, etc.
Most insurance companies will not take risks to exceed two-thirds
or three-fourths the appraised value of the property insured.
When only a part of the property insured is destroyed or dam-
aged, the insurers are required to pay only the estimated loss ; and
sometimes the claim is adjusted by repairing or replacing the
property, instead of paying the amount claimed.
662. The operations are based on the principles of
Percentage, the corresponding terms being as follows :
1. The Base is the amount of insurance.
2. The Mate is the per cent, of premium.
3. The Percentage is the premium.
ORAL EXEMC IS ES.
663. 1. How much must be paid for insuring a house
and furniture for $4000, at 1\% premium ?
Analysis. — Since the premium is 1}%, or ^^, equal to /^ of
the sum insured, the premium on $4000 will be ^^^f ^^ $4000, or
$50. Hence, etc. (510.)
2. What will be the annual premium of insurance, at
i%, on a building valued at $8000 ?
78 PERCENTAGE.
3. What will be the cost of insuring a quantity of flour,
valued at $1500, at ^% ?
4. What must be paid for insuring a case of merchan-
dise, worth $640, at 2^% ?
X 5. A man owns f of a boat-load of corn valued at
$1800, and insures his interest at ' If ^. What premium
does he pay ?
6. Paid $6 for insuring $300 ; what was the rate ?
Analysis. — Since the premium on $300 is $6, the premium on
$1 is ^ of $6, or $.02, equal to 2% . Hence, etc. (513.)
7. Paid $12 for an insurance of $800 ; find the rate.
^**v8. Paid $24 for an insurance of $1000 ; find the rate.
9. At 2%, what amount of insurance can be obtained
for $30 premium ?
Analysis. — Since 2% is yf ^ or ^^ of the amount insured, $30, the
given premium, is ^^ of the amount insured ; and $30 is -^j^ of 50
times $30, or $1500. Hence, etc. (516.)
What amount of insurance can be obtained,
10. On a house, for $75, at 3% premium ?
11. On a boat load of flour, for $150, at l%r
12. On a car load of horses, for $90, at ^% ?
13. On a store and its contents, for $105, at 1^% ?
W MITTBN EXJEJiCISJES,
664. Find the Premium
1. For insuring a building for $14500, at 1^%.
Operation.— $14500 x .015 = $217,50. (512.) ^ ,,^
Formula. — Premium = Amount Insured x Rate.
Find the premium for insuring
2. A house valued at $5700, at f^.
3. Merchandise for $2750, at 1%.
INSUBAKCE. 79
4. A fishing craft, for $15000, at 1^%.
5. If I take a risk of $25000, at If ^, and re-insure ^
of it at 21%, what is my balance of the premium ?
665. Find the Rate of Insurance,
1. If $36 is paid for an insurance of $2400.
Operation.— $36 -h $2400 = .015, or 1^% . (5 15.)
Formula. — Rate of Insurance = Premium -r- Sum
hisured.
What is the rate of insurance,
2. If $280 is paid for an insurance of $16000 ?
3. If $4.30 is paid for an insurance of $860 ?
4. A tea merchant gets his vessel insured for $20000
in the Eoyal Company, at f^, and for $30000 in the
Globe Company, at ^%, What rate of premium does he
pay on the whole insurance ?
" 666. To find the Amount of Insurance.
1. A speculator paid $262.50 for the insurance of a
cargo of corn, at 1^%. For what amount was the com
insured ?
Operation.— $262.50 ^ .015 := $17500, the sum insured. (518.)
Formula. — Su7n Insured = Premium -V Rate.
^^. If it cost $93.50 to insure a store for one-half of its
value, at \\%, what is the store worth ?
^ 3. Paid $245 insurance at 4f ^ on a shipment of pork,
to cover | of its value. What was its total value ?
4. A merchant shipped a cargo of flour worth $3597,
from New York to Liverpool. For what must he insure
it at 3 J^, to cover the value of the flour and premium ?
Operation.— $3597 h- (1 - .03J) or ,9675 = $3717.829. (520.)
80 PEBCEl^TAGE.
5. An underwriter agrees to insure some property for
enough more than its value to cover the premium, at the
rate of 26 cents per $100. If the property is worth
$22163, what should be the amount of the policy ?
6. For what sum must a policy be issued to insure a
dwelling-house, valued at $35000, at ^%, a carriage-house
worth $9500, at |^, and furniture worth $4500, at |^,
10^ being deducted from the premium, which is to be
covered by the policy ?
7. A person insured his house for f of its value at
40 cents per $100, paying a premium of $73.50. What
was the value of the house ?
8. A dealer shipped a cargo of lumber from Portland
to New York ; the amount of insurance, including the
value of the lumber and the premium paid, at If ^, was
$25200. What was the value of the lumber.?*
9. A merchant had 500 bbl. of flour insured for 80^ of
their cost, at 3^%, paying $107.25 premium. At what
price per barrel must he sell the flour to gain
LIFE INSUKANCE.*
667. Life Insurance is a contract by which a
company agrees to pay a certain sum, in case of the death
of the insured during the continuance of the policy.
668. A Term Life Policy is an assurance for one
or more years specified.
669t A Wlpole Life Policy continues during the
life of the insured.
* See note at bottom of page 82.
INSURANCE. 81
•
Premmms may be paid annually for life, or in 5, 10, or more
installments (called 5-payment, 10-payment policies, etc.), or the
entire premium may be paid in one sum in advance.
The premium is computed at a certain sum or rate per $1000
insured, the rate varying with the age of the insured at the time
the policy is issued.
A policy of endowment is not in all respects an insurance policy,
but is rather a covenant to pay a stipulated sum at the end of a
certain period to the person named if living.
Most companies issue a form of policy that combines the princi-
ples of Term Life Assurance and Simple Endowment, called for
brevity Endowment Policy. Hence,
670. An Undow^nent Policy is one in which
the assurance is payable to the person insured at the end
of a certain number of years named, or to his heirs if
he die sooner.
An endowment policy is really two policies in one, and the
assured pays the premiums of both.
671. A Dividend is a share of the premiums or
profits returned to a policy-holder in a mutual life in-
surance company.
673. A Table of Mortality shows how many per-
sons per 1000 at each age are expected to die per annum.
673. A Table of Hates shows the premium to be
charged for $1000 assurance at the different ages.
Such a table is based upon the table of mortality, and the proba-
ble rates of interest for money invested, with a margin or loading
for expenses.
674. The following condensed table gives data from
the American Experience Table of mortality, and the
annual premium on the kinds of policies most in use.
82
PERCENTAGE.
American Experience Table— Mortality and Premiums.
3
^
ANNUAL PREMIUM PER $1000.
Life Table.
Endow-
AGE.
1
One
Whole Life,
ment
(AND
Year
Term
Term
1
Payments
Payment
Payment
Single
Payment.
Life).
10
years.
^
{Net).
during
life.
for 10 yr.
only.
for 5 yr.
only.
25
8.1
7.75
$19.89
$42.56
$73.87
$326.58
$103.91
26
8.1
7.82
20.40
43.37
75.25
332.58
104.03
27
8.2
7.83
29.93
44.22
76.69
338.83
104.16
28
8.3
7 95
21.48
45.10
78.18
345.31
104.29
29
8.3
8.02
22 07
46.02
79.74
352.05
104.43
30
8.4
8.10
22.70
46.97
81.36
359.05
104.58
31
8.5
8.18
23.35
47.98
83.05
366.33
104 75
32
8.6
8.28
24.05
49.02
84.80
373.89
104.92
as
8.7
8.33
24.78
50.10
86.62
381.73
105.11
34
8.8
8.49
25.56
51.22
88.52
389.88
105.31
35
8.9
860
26.38
52.40
90.49
398.34
105.53
40
9.8
9.42
31.30
59.09
101.58
445.55
106.90
45
11.2
10.73
37.97
67.37
115.02
501.69
109.07
50
13.8
13.25
47.18
77.77
131.21
567.13
112.68
The actual net cost of insurance for a single year at each age
given in the table, on the mortality assumed, is as many dollars and
tenths of a dollar as there are deaths, but discounted for 1 year.
Thus, at age 25, deaths 8.1 per 1000, net cost, which is $8.10, dis-
counted at 4|-% by the insurance law, $7.75. If this sum, $7.75,
is loaded for expenses at, say 25%, the total premium for 1 year
is $9.69, if at 40%, then it would be $10,85.
In a Term Life Policy the premium may vary, increasing slightly
each year of the term, according to the assumed increasing liability
to decease, or it may be averaged for the term so as to be the same
each year.
Note. — As there is no uniformity in the Tables and Methods used
by different Life Insurance Companies, the pupil may very properly
omit this subject
I2S"SURANCE. 83
WRITTEN EXERCISES.
675. To find the amount of premium
1. For a life policy of $5000 issued to a person 30
years old.
Operation.— $22.70 x 5 = $113.50.
2. For a life policy of $7500, age being 45.
'RvL^,— Multiply the premium for $1000 assurance by
the number of thousands.
Formula. — Premium — Bate jper $1000 x iVb. of thou-
sands,
3. Find the annual premium for an endowment policy
of $10000, payable in 10 years, age 35.
4. What premium must a man aged 30 pay annually
for life, for a life policy of $5000 ?
What premium annually for 10 years ?
What premium annually for 5 years ?
What premium in a single payment ?
OPERATION. Analysis.— Multiply the rate
$22.70x5000= $113.50 P^^ thousand dollars, found in
»«.9rx6«oo= .m85 L7.S';ieZ£li:S
$81.36 X 5000 = $406.80 pressing the hundreds, tens, and
$359. 05 X 5000 = $1795. 25 units decimally.
5. What annual premium will a man aged 35 years pay
to secure an endowment policy for $5000, payable to him-
self in 10 years, or to his heirs, if death occurs before ?
6. If he dies at the beginning of the ninth year, how much
will the assurance cost, reckoning simple interest at 6^ ?
7. How much less would he have paid in the whole life
(annual payment) plan, interest included ?
84 PERCEKTAGE.
8. A man aged 45 insures his life for $7500 on the sin-
gle-payment plan, and dies 3 yr. 5 mo. afterward. How
much less would his insurance have cost him had he in-
sured on the annual payment plan, reckoning int. at 6^ ?
9. A person aged 27 takes out a 10-year endowment
policy for $5000 ; the dividends reduce his annual pre-
miums 15^ on the average. Computing annual interest at
1% on his premiums, does he gain or lose, and how much ?
10. A man aged 35 years took out a life policy for
$12000, on the 5-payment plan, and died 3 yr. 6 mo.
afterward. What was gained to his estate by insuring,
computing compound interest on his payments at 7^,
also adding two dividends of $95 each ?
P TAXES.
676. A Tax is a sum of money assessed on the per-
son, property, or income of an individual, for any public
purpose.
677. A Poll Tax or Capitation Tax is a cer-
tain sum assessed on every male citizen liable to taxation.
Each person so taxed is called a poll,
678. A JProperty Tax is a tax assessed on prop-
erty, according to its estimated, or assessed, value.
Property is of two kinds : Real Property, or Real Es-
tate, and Personal Property.
679. Real Estate is fixed property ; such as houses
and lands.
680. Personal Property is of a movable nature ;
such as furniture, merchandise, ships, cash, notes, mort-
gages, stock, etc.
TAXES. 85
681* An Assessor is an officer appointed to deter-
mine the taxable value of property, prepare the assess-
ment roUs, and apportion the taxes.
683. A Collector is an officer appointed to receive
the taxes.
683. An Assessment Moll is a schedule, or list,
containing the names of all the persons liable to taxation
in the district or company to be assessed, and the valua-
tion of each person's taxable property.
684. The Hate of JP^'Ojjerty Tax is the rate per
cent, on the valuation of the property of a city, town,
or district, required to raise a specific tax.
WRITTEN EXEMCISES.
685. 1. What sum must be assessed to raise $836000
net, after deducting the cost of collection at 5% ?
Operation.— $836000 ^ .95 = $880000. (519.)
FoKMULA. — Sum to be raised -— (1 — Bate of Collection)
= Sum to be Assessed.
2. What sum must be assessed to raise a net amount
of $11123, and pay the cost of collecting at 2% ?
3. In a certain district, a school-house is to be built at
a cost of $1 8500. What amount must be assessed to cover
this and the collector's fees at 3% ?
4. The expense of building a public bridge was $1260.52,
which was defrayed by a tax upon the property of the
town. The rate of taxation was 3^ mills on a dollar,
and the collector's commission was 3^%. What was the
valuation of the property ?
86
PERCEITTAGE.
5. In a certain town a tax of $5000 is to be assessed.
There are 500 polls, each assessed 75 cents, and the
valuation of the taxable property is $370000. What will
be the rate of property tax, and how much will be A's tax,
whose property is valued at $7500, and who pays for 2 polls ?
Operation.— $.75 x 500 = $375, amt. on polls.
$5000 - $375 = '* '* property.
$4635 -r- $370000 = .0125, rate of taxation.
$7500 X .0125 = $93.75, A's property tax.
$93.75 + $1.50 = $95.25, A's whole tax.
EuLE. — I. Find the amount of poll tax, if any, and
subtract it from the whole amount to be assessed ; the
remainder is the property tax.
II. Divide the property tax by the whole amount of
taxable property ; the quotient is the rate of taxation.
III. Multiply each marCs taxable property by the rate
of taxation, and to the product add his poll tax, if any ;
the result is the lohole amount of his tax.
A table such as the following is a great aid in calculating the
amount of each person's tax, according to the ascertained rate.
Assessor's Table.
{Rate .
0087.)
Prop.
Tax.
Prop.
Tax.
Prop.
Tax.
Prop.
Tax.
$1
$.0087
$9
$.0783
$ 80
$ .696 '
$ 700
$ 6.09
2
.0174
10
.087
90
.783
800
6.96
3
.0261
20
.174
100
.87
900
7.83
4
.0348
30
.261
200
1.74
1000
8.70
5
.0435
40
.348
300
2.61 i
2000
17.40
6
.0522
50
.435
400
3.48 j
8000
26.10
7
.0609
60
.522
500
4.35 !
4000
34.80
8
.0696
70
.609
600
5.22 1
5000
43.50
T A X E 8 . 87
6. Find by the table the tax of a person whose property
is valued at $3475, the rate being .0087,
Opebation.— Tax on $3000 = $26.10
*' " 400 = 3.48
*' " 70 = .609
*' " 5 = . 0435
" " $3475 = $30.2325, or $30.2a
Find by the table the tax of a person whose property
^ 7. Is $2596, and who ;^ays for 5 polls at $.50.
--8. Is $9785, polls 3 at $.75.
,- 9. Is $12356, polls 4 at $1.25.
\L 10. Is $25489, polls 5 at $.95.
^ 11. A tax of $11384, besides cost of collection at S^%,
is to be raised in a certain town. There are 760 polls
assessed at $1.25 each, and the personal property is
valued at $124000, and the real estate at $350000. Find
the tax rate,^make an assessor's table for that rate,' and
find a person's tax, whose real estate is valued at $6750,
personal property at $2500, and who pays for 3 polls.
12. In the above town, how much is B's tax on $15000
real estate, $2750 personal property, and 5 polls ?
13. What is C's tax on $9786 and 1 poll ?
14. How much tax will a person pay whose property is
^^assessed at $7500, if he pays If ^ village tax, ^% State tax,
and 1 J mills on a dollar school tax ?
"^-.^ 15. The expense of constructing a bridge was $916.65,
which was defrayed by a tax upon the property of the
town. The rate of taxation was 2J mills on a dollar,
and the commission for collecting 3% ; what was the
assessed valuation of the property of the town ?
Note. — Amt. to be raised -5- by rate = valuation.
88
PERCEKTAGE.
686.
o
SYNOPSIS FOE KEVIEW.
" 1. Corporation. 2. Charter. 3. Capital Stock,
4. Certificate of Stock, or Scrip. 5. Share.
6. Stocks, 11. Stockholders. 8. P«r Value,
9. Market Value. 10. Premium, Discount,
1. Defs. -^ Brokerage. 11. >8^^A; Broker. 12. ^ocA:-
jobbing. 13. Installment. 14. Assessment.
15. Dividend. 16. J^ef Earnings. 17. Bond.
18. Dif. ^m^Z* <?/ CT. >8^. J?(wc?«. 19. Cbw-
^n. 20. Currency.
2. CM8. 1 r C6?«^.
3. 64:9. iV^<?. of Sf tares.
4. 050. ) ^w*. of Investment.
5. 651. > To find K i?af^ % Income.
6. 652. Pwe fo j?ay Income.
7. 653. FaZ2/e 0/ G^o^ iri (7?/r.
8. 654. J I Fa^M6 o/Owr. iw (?oZd
mula.
H
fe
!1. Insurance. 2. Insurer or Underwriter,
3. Policy. 4. Premium. 5. i^*>6 Insurance.
6. Marine or Inland Insurance.
J 2. Corresponding Terms in Percentage.
3. 664r. i ( Premium. ^
4. 665. f To find ^ -Ba^6 <?/ Insurance. V Formula.
^ 5. 666. ) ( ^w^. of Insurance. )
^ 1. Life Insurance. 2. Term Life Policy. 3.
Tfi^^6 X^/(3 Policy. 4. Endowment Policy.
1. Defs. ^ ^,Dimdend. 6. Table of Mortality. 7. Table
of Bates.
2. 675. Rule. Formula.
1. Defs.
2. 685
L 3. 686.
u
1, r^aj. 2. Poll Tax. 3. Property Tax. 4.
ii?6aZ Estate. 5. Personal Property. 6.
Assessor. 7. Collector. S. Assessment Boll
9. JSa^6 ^/ Property Tax.
T fi r1 ^ /Swm ^(? &6 raised. Formula.
I ^m^. ^/ Taa;. Rule, I, II, III.
t^^
^^v ^^^^ ^
687. Exchange is the giving or receiving of any
sum in one currency for its value in another.
By means of exchange, payments are made to persons at a dis-
tance by written orders, called Bills of Exchange.
688. Exchange is of two kinds. Domestic, or In-
land, and Foreign.
689. Domestic or Inland Exchange relates
to remittances made between different places in the same
country.
690. Foreign Exchange relates to remittances
made between different countries.
691. A Bill of Exchange is a written request, or
order, upon one person to pay a certain sum to another
person, or to his order, at a specified time. An inland
bill of exchange is usually called a Draft.
693. A Set of Exchange is a bill drawn in dupli-
cate or triplicate, each copy being valid, until the amount
of the bill is paid. These copies are sent by different
conveyances, to provide against mis6arriage.
693. A Sight Draft or Bill is one which requires
payment to be made ^^at sight, '^ that is, at the time it is
presented to the person wjio is to pay it.
90 • PERCENTAGE.
694. A Time Draft or Bill is one that requires
payment to be made at a cerlain specified time after date,
or after sight.
695. The Buyer or Meniitter^ of a bill is the
person who purchases it. The buyer and payee may be
the same person.
696. The Acceptance of a bill or draft is the agree-
ment by the drawee to pay it at maturity. The drawee
thus becomes the acceptor, and the bill or draft, an
acceptance.
1. The drawee accepts by writing the word " accepted " across
the face of the biU, and signing it.
2. Three days of grace are usuaUy aUowed on bills of exchange,
as well as on notes. When a bill is protested for non-acceptance,
the drawer is bound to pay it immediately.
697. The Par of JExchange is the estimated value
of the coins of one country as compared with those of
another. It is either intrinsic or commercial,
1. The Intrinsic Par of Exchange is the comparative value of the
coins of different countries, according to their weight and purity.
2. The Commercial Par of Exchange is the comparative value of
the coins of different countries, according to their market price.
698. The Course or Bate of JExchange is the
current price paid in one place for bills of exchange on
another place.
This price varies according to the relative conditions of trade and
commercial credit at the two places between which the exchange is
made. Thus, if New York is largely indebted to London, bills of
exchange on London will bear a high price in New York.
EXCSAKaE.: "91
699. FORMS OF DRAFTS AND BILLS.
A SIGHT DRAFT.
$500. New York, Jvly 1, 1874.
At sight, pay to the order of William Thompson, five
hundred dollars, value received, and charge to the acct, of
He:n^ry J. Carpenter.
To Harris, Jones & Co.,
Cincinnati, 0.
Other drafts have the same form as. the aDove, except that in-
stead of the words *^ at sight," " days after sight," or **
days after date/' are used. When the time is after sights it meana
after acceptance.
SET OF EXCHANGE.
i;700. New York, ^w^ws« 1,1874.
At sight of this First of Exchange (Second and Third
of the same tenor and date unpaid), pay to the order of
Samuel Monmouth, Seven Hundred Pounds Sterling, for
value received, and charge the same to the account of
Morton, Bliss & Co.
Morton, Eose & Co., London.
The above is the form of ilie first bill ; the second requires only
the change of ''First" into "Second," and instead of "Second
and Third of the same tenor," etc., '' First and Third." The Third
Bill varies similarly.
DOMESTIC OR INLAND EXCHANGE.
The course of exchange for inland bills, or drafts, is always ex-
pressed by the rate of premium or discount. Time drafts, however,
are subject to bank discount, like promissory notes, for the term
of credit given. Hence, their cost is affected by both the course of
exchange and the rate of discount for the time.
92 PERCENTAGE.
WRITTEN EXERCISES,
700. What is the cost
1. Of a sight draft on New Orleans for $1750, at l\%
premium ?
Operation.— $1750 x LOIJ = $1771.871. (512.)
^ ^ . 7^ S 1 + Rcite of Premium,
Formula. — Cost = Face x i ^ t> . /. r^-
( 1 — Rate of Discount.
2. Of a sight draft on Troy for 11590, at 1^% discount ?
3. Of a draft on Boston for $1650, payable in 60 days
after sight, exchange being at a premium of If ^ ?
Operation. — $1.0175 = Course of Exchange.
$.0105 = Bank Dig. on $1, for 63 da.
$1,007 = Cost of Exchange, for $1.
$1,007 X 1650 = $1661.55, value of Draft.
w- 4. Of a draft on New York at 30 da. for $4720, at l^%
premium ?
5. Of a draft on New Orleans, at 90 da., for $5275, int.
being 1%, and exchange Y/c discount ?
\v 6. Find the cost in Philadelphia of a draft on Denver,
at 90 da., for $6400, the course of exchange being lOlf?
\^ 7. What must be paid in New York for a draft on
San Francisco, at 90 da., for $5600, the course of ex-
change being 102^^ ?
701. Find the Face
1. Of a draft on St. Louis, at 90 da., purchased for
$4500, exchange being at 101^^.
Operation. — $1,015 = Course of Exchange.
$.0155 = Bank Dis. of $1, for 93 da., at 6%.
$.9995 = Cost of Exchange of $1.
$4500 -r- .9995 = $4502.25. (520.)
EXCHANGE. 93
_^2. Of a draft on Richmond at 60 da. sight, purchased
for $797.50, interest 7^, premium 2j^%.
-—3. Of a sight draft bought for $711.90, discount 1^%,
4. A commission merchant sold 2780 lb. of cotton at
/11|^ cents a pound. If his commission is 2^%, and the
course of exchange 9S^%, how large a draft can he buy to
remit to his consignor ?
5. The Broadway Bank of New York having declared
a dividend of 5^, a stockholder in Chicago drew on the
bank for the sum due him, and sold the draft at a pre-
mium of H%y thus realizing |2283.18f from his dividend-
How many shares did he own ?
tJ3. A man in Rochester purchased a draft on Louisville,
y., for $5320, drawn at 60 days, paying $5151.09. What
was the course of exchange ?
\i 7. Received from Savannah 250 bales of cotton, each
weighing 520 pounds, and invoiced at 12| cents a pound.
Sold it at an advance of 25^, commission 1^%, and
remitted the proceeds by draft. What was the fece of
the draft, exchange being ^% discount ?
^ FOREIGN EXCHANGE.
703. Money of Account consists of the denomi-
nations or divisions of money of any particular country,
in which accounts are kept.
The Act of March 3, 1873, provides that *' the -ralue of foreign
coin, as expressed in th« moniBj of account of the United States,
shall be that of the pure metal of su«h coin of standard value ; and
the values of the standard coins in circulation, of the various na-
tions of the world, shaU be estimated annually/ hj the Director of
the Mint, and be proclaimed on the first daj of Januaiy hj the
Secretary of the Treasury."
u
PERCEIifTAGE.
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.2
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^.-a
EXCHANGE. 95
704. Sterling Bills or Sterling Exchange
are bills on England, Ireland, or Scotland. Such bills
are negotiated at a rate fixed without reference to the par
of exchange.
Formerly such bills were quoted at a certain rate fc above the
old par value of a pound sterling, which was $4.44f . As this was
entirely a fictitious value, and always about 9% below the real
value, the course of exchange always appeared to be heavily against
this country, and thus tended to impair its credit. By the Act of
March, 1873, ** all contracts made after the first day of January,
1874, based on an assumed par of exchange with Great Britain of
fifty-four pence to the dollar, or $4.44f to the sovereign or pound
sterling," are declared nuU and void. The par of exchange between
Great Britain and the United States is fixed at $4.8665.
705. Exchanges with Europe are effected
chiefly through the following prominent financial circles:
London, Paris, Antwerp, Amsterdam, Hamburg, Frank-
fort, Bremen, and Berlin.
In exchange on Paris, Antwerp, and Switzerland, the unit is
the franc, and the quotation shows the number of francs and
centimes to the dollar. Federal Money. In exchange on Amster-
dam, the unit is the guilder, quoted at its value in cents ; on Ham-
burg, Frankfort, Bremen, and Berlin, the quotation shows the value
of fottr reichsmarks (marks) in cents,
WJRITTEN EXAMJPJ^ES.
'706. Find the «>5/
1. Of a bill of exchange on London at 3 days' sight,
for £393 15s, 6d, exchange being quoted at 4.89^, and
gold at 1.10^.
OPERATION.
£393 15s. ^di, = £393.775.
$4,895 X 393 775 = $1927.529, gold value of bOL
$1927.529 X 1.10| = $2122.69, value in mirrene^.
rBKCCI^T ACB.
S.Ofaia
EXCHANaB, 97
13, What will it cost to remit directly from Boston to
Amsterdam, 12560 guilders, at 41 J?
14. What will be the cost of remitting 13550 marks
from New York to Frankfort, exchange selling at 94^,
and gold at lOOJ ; brokerage, ^% ?
707. What will be the /ace
1. Of a bill of exchange on London that can be bought
for $5500, in currency, exchange selling at 4.86, and gold
at 1.10?
Operation.— $5500 currency -*- 1.10 = $5000, gold. (519.)
$5000 -J- $-^.80 = 1028.806 -h .
£1028.806 = £1028 168, li<L
2. Of a bill on Manche^ter^ England, that can be
bought for $7500, gold ; rate of exchange, 4.86 ?
3. Of a bill on Berlin that cost $4000 in gold, ex-
change 93|^.
Operation.— <$1000 -^ $.9375) x 4 = 17066| marks.
Analysis. — Since $.93} will buy 4 marks, $4000 \^ill buy 4 times
as many marks as $.93} is contained times in $4000, or ITOOOf marks.
4. Of a bill on Hamburg that cost $550 in gold, ex-
change 94| ?
5. Of a bill on Frankfort that cost $395.75 in gold,
exchange 95^?
6. Of a bill on Geneva, Switzerland, that cost $325 in
gold, exchange at 5.17?
Operation.— 5.17 fr. x 325 = 1680.25 francs.
Analysis. —If $1 \^ill buy 5.17 francs, $325 will buy 325 times
5.17 francs, or 1680.25 francs.
7. A merchant in New Orleans gave $6186, currency,
for a bill on Paris, at 5.15^. What was its face ?
8. What is the face of a bill on Antwerp, that may be
purchased in New York for $2500, exchange at 5.16i?
t)9 PERCENTAGE.
ARBITEATION OF EXCHANGE.
108. Arbitration of Exchange is the process of
computing the cost of exchange between two places by
means of one or more intermediate exchanges. Such ex-
change is said to be indirect, or circuitous.
By this computation the relative cost of direct and indirect ex-
change is ascertained. Sometimes, owing to the course of exchange
between different places, it is more advantageous to remit by the
latter than by the former.
Arbitration is either simple or compound.
709. Simple Arbitration is that in which there
is but one intermediate place.
710. Compound Arbitration is that in which
there are several intermediate places.
WniTTEN EXERCISES.
711. 1. I owe 1500 marks to a merchant in Frankfort.
Should I remit directly from New York, or through Lon-
don, exchange on Fra-nkfort being 94, on London 4.87|^,
and in the latter place on Frankfort 20. 75 marks to the
pound, and the London brokerage \% ?
Operation.— $.94 x 1500-^4= $352.50, cost of direct exchange.
1500 marks -f- 20.75 marks = £72.29.
£72.29 + 4% =£72.38.
$4.87i X 72.38 = $352.85.
$352.85 — $352.50 = $.35, loss by ind. exchange,
2. What will it cost to remit from Boston to Berlin
750 marks, by indirect exchange, through Paris, exchange
in New York on Paris being at 5.15, and 4 marks
at Paris being worth 4.91 francs, the brokerage being
ati^?
EXCHANGE. Cd9
3. What will it cost to remit 2500 guilders from New
York to Amsterdam, through London and Paris, the rates
of exchange being as follows : at New York on London
4.83, at London on Paris 24.75 francs to the pound, and
at Paris on Amsterdam 2.09 francs to the guilder, broker-
age at London and Paris i% each ?
OPERATION.
$ X = 2500 guilders.
1 guilder = 2.09 francs.
1 franc (net) = 1 .00|^ (with brokerage).
24.75 francs = £1.
£1 (net) = £1.00^ (with brokerage).
£1 = $4.83.
2500x2.09x1.001^x1.001x4.83
Hence, -, ^^ ^— , or
By cancenatio«, 100x19 xl.OOj-xl.OOjx 1.61 ^ ^^^^^^^
o
Analysis. — Since the members of each equation are equal, the
product of the corresponding members of any number of equations
are equal ; hence, the product of all the second members divided by
the product of all the first members except one, must give that
member, which is the value required.
4. A merchant in St. Louis directs his agent in New
York to draw upon Philadelphia at 1% discount, for
$1500 due from the sale of mdse. ; he then draws upon
the New York agent, at 2% premium, for the proceeds,
after allowing the agent to reserve ^% commission. What
BvCm does he realize from his mdse. ?
OPERATION.
( a; ) St. L. = 1500 Philadelphia.
100 Phil. = 99 N.York.
100 N. Y. = 102 St. Louis.
1 = .995 (net proceeds).
By cancellation, . 15 x 99 x 102 x .995 =$1507. 13.
100 PERCEKTAGE.
Analysis.— $100 on Philadelphia = $99 on N. Y., and $100 on
N. Y. = $102 on St. Louis ; and since the agent reserves |% com-
mission, $1 realized = $.995 net proceeds. Arranging, canceling,
and multiplying, we find the result to be $1507.13.
EuLE. — I. Represent the required sum iy {x), tvith the
proper unit of currency affixed, and place it equal to the
given sum on the right.
II. Arrange the given rates of exchange so that in any
two consecutive equations the same unit of currency shall
stand on opposite sides.
III. When there iscomrnissionfor drawing, place 1 minus
the rate on the left if the cost of exchange is required, and
on the right if proceeds are required ; and when there is
commission for remitting, place 1 plus the rate on the
right, if cost is required, and on the left, if proceeds are
required.
IV. Divide the product of the numbers on the right iy
the product of the numbers on the left, canceling equal fac-
tors, and the result will be the required sum.
Commission for drawing is commission on the sale of a draft ;
commission for remitting is commission on the purchase price of a
draft.
The above method of operation is sometimes called the Chain Rule.
5. If at New York exchange on London is 4.84|^, and
at London on Paris it is 25.73 francs to the £, what is
the arbitrated course of exchange between New York and
Paris?
6. If in London exchange on Paris is 25.71, and in
New York on Paris it is 5.15^, what is the arbitrated
course of exchange between New York and London ?
EXCHA:efGE. 101
7. A banker in New York remits $5000 to Liverpool
by indirect exchange, through Paris, Hamburg, and Am-
sterdam, the rates being as follows : in New York on
Paris 5.18 fr. to the dollar, in Paris on Hamburg 1.22 fr.
to the mark, in Hamburg on Amsterdam 1.70 mark to the
guilder, and in Amsterdam 11.83 guilders to the pound
sterling. How much sterling will he have in bank at
Liverpool, and how much does he gain by indirect ex-
change, sterling being worth in New York 4.83^ ?
8. A merchant in Philadelphia owes a correspondent
in Paris 35000 francs. Direct exchange on Paris is 5.15 ;
but exchange on London is 4.83, and London exchange
on Paris is 25. 12. Allowing ^% commission for brokerage
at London, which is the more advantageous way to remit,
and by how much ?
9. An American resident at Amsterdam wishing to
obtain funds from the U. S. to the amount of 14500,
directs his agent in London to draw on Philadelphia, and
remit the proceeds to him in a draft on Amsterdam, ex-
change on London in Phil, selling at 4.87^, and in Lon-
don on Amsterdam 11.17^ guilders to the pound sterling.
If the agent charges commission at ^% both for drawing
and remitting, how much better is this arbitration than
to draw directly on the U. S. at 41|^ cents per guilder ?
10. A speculator residing in Cincinnati, having pur-
chased 165 shares of railroad stock in New Orleans, at
75^, remits to his agent in N. York a draft purchased at
2% premium, directing the agent to remit the sum due on
N. Orleans. Now, if exchange on N. Orleans is at f ^ dis-
count in N. Y., and the agent's commission for remitting
is ^%, how much does the stock cost in Cincinnati ?
102 PERCENTAGE.
^ CUSTOM-HOUSE BUSIK^ESS.
713. A Custom^House is an office established by
government for the transaction of business relating to the
collection of customs or duties, and the entry and clear-
ance of vessels.
713. A Port of Entry is a seaport town in which
a custom-house is established.
714. The Collector of the JPort is the officer ap-
pointed by government to attend to the collection of
duties and to other custom-house business.
715. A Clearance is a certificate given by the Col-
lector of the port, that a vessel has been entered and
cleared according to law.
By the entry of a vessel is meant the lodgment of its papers in
the custom-house, on its arrival at the port.
716. A Manifest is a detailed statement, or invoice,
of a ship's cargo.
No goods, wares, or merchandise can be brought into the United
States by any vessel, unless the master has on board a full mani-
fest, showing in detail the several items of the cargo, the place
where it was shipped, the names of the consignees, etc.
717. Duties or Customs are taxes levied on im-
ported goods.
The general object of such taxes is the support of government,
but they are also designed sometimes to protect the manufacturing
industry of a country against foreign competition.
718. A Tariff is a schedule showing the rates of
duties fixed by law on all kinds of imported merchandise.
Duties are of two kinds. Specific and Ad Valorem.
CUSTOM-HOUSE BUSIl^ESS. 103
719. A Specific Duty is a fixed sum imposed on
articles according to their weight or measure, but without
regard to their value.
*730. An Ad Valorem Duty is an import duty
assessed by a percentage of the value of the goods in the
country from which they are brought.
Before computing specific duties, certain deductions, or allow- '
ances, are made, called Tare, Leakage, Breakage, etc.
721. Tare is an allowance for the weight of the box, cask,
bag, etc., that contains the merchandise.
722. Leakage is an allowance for waste of liquors Imported
in casks or barrels.
723. breakage is an allowance for loss of liquors imported
in bottles.
734. Gross Weight or Value is the weight or
value of the goods before any allowance is made?.
735. N'et Weight or Value is the weight or value
of the goods after all allowances have been deducted
WniTTEN JEXEMC IS ISS.
736. Find the Duty
1. On 355 yds. of carpeting, invoiced at lis. 6d. per
yd., the duty being 50%.
Operation.— lis. 6d. = £.575.
£.575 X 355 = £204.125.
$4.8665 (par value of £1) x 204125 =$993.37.
$993.37 X JO = $496.68, duty. (510:)
2. On 50 hhd. of sugar, each containing 500 lb., at 5|
cts. per lb. ; duty If ets. per lb.
3. On 350 boxes of cigars, each containing 100 cigars,
invoiced at 17.50 per box,; weight, 12 lb. per 1000 ; duty,
$2.50 per lb., and 25;^ ad valorem.
104 PEECBNTAGE.
4. A wine merchant in New York imported from Havre
100 doz. quart bottles of champagne, at $13 per doz., and
25 casks of sherry wine, each containing 30 gals., at $2.50
per gal. What is the duty, the rate on the champagne
being $6 per dozen,, and on the sherry 60 cents per gal.,
and 25^ ad valorem ?
5. Imported from Geneva 25 watches invoiced at $125
each, and 15 clocks, at $37.50. What was the duty, the
rate being on clocks 25^, and on watches, 35^ ad valorem?
6. A liquor dealer receives an invoice of 120 dozen pint
bottles of porter, rated at $.75 per dozen. If 2^% of the
bottles are found broken, what will be the duty at 36 cts.
per gallon ?
7. H. B. Claflin & Co. imported 20 cases of bleached
muslins, each case containing 175 pieces of 24 yards
ea<^h, IJ yards wide. What was the duty at 5^ cts. per
square yard ?
8. What was the duty on 10 cases of shawls, average
weight of eaeh case 213|^ lb., invoiced at 19375 francs ;
rate of duty, 50 cts. per lb. and 35^ ad valorem ? If I pay
fox tha invoice with a bill of exchange bought at 5.15^,
and pay charges amounting to $67.50 currency, what do
the shawls cost me in currency, gold selling at 1.10 ?
9. Olmsted & Taylor, of New York, import from
Switzerland 1 ease of watches, invoiced at 7125 francs;
duty, 25^; charges, 13.50 francs; comniissions, 2^%,
What was the cost of the watches in U. S. gold ?
10. Imported from England 5 cases of cloths and cassi-
meres, net weight, 695 lb. ; value as per invoice, £375
10s. What was the duty in American gold, the rate
being 50 cts. per lb. and 35^ ad valorem ?
EQUATIOK OF PAYMENTS. 105
EQUATIOJN" OF PAYMENTS.
737. Equation of Payments is the process of
finding the average time for the payment of several sums
of money due at different times, without loss to debtor
or creditor.
738. The Equated Time is the date at which the
several debts may be discharged by one payment.
739. The Term of Credit is the time at the
expiration of which a debt becomes due.
730. The Avei^age Term of Credit is the time
at the end of which the several debts due at diff'erent
dates, may all be paid at once, without loss to debtor or
creditor.
ORAJL EXERCISES,
731. 1. The interest of $100 for 3 mo. equals the
interest of $50 for how many months ?
Analysis. — At the same rate, the interest of $100 equals the
interest of $50, or one-half of $100, for twice the time, or 6 mo.
2. The interest of $20 for 4 mo. equals the interest of
$10 for how many mo. ? Equals the interest of $5 for
how many mo. ? Of $1 ? Of $40 ? Of $100 ?
3. The interest of $25 for 6 mo. equals the interest of
$5 for how many mo. ? Of $10 ? Of $1 ?
4. The interest of $10 for 6 mo., and of $100 for 2 mo.,
taken together, equals the interest of $1 for how many
months? r.
106 PERCENTAGE,
5. If I borrow $50 for 3 mo., for how many months
should I lend $100 to repay an equal amount of interest ?
Analysis. — The interest of $50 for 3 mo. is the same as the
interest of $1 for 50 times 3 mo., or 150 mo. ; and the interest of $1
for 150 mo. is the same as the interest of $100 for j^^ of 150 mo.,
or 1| mo.
6. If I lend $200 for 3 mo., for how long a time should
I have the use of $50 to balance the favor ?
7. If A borrows of B $1000 for 3 mo., what sum
should A lend B for 9 mo. to discharge the obligation ?
*733. Peinciple. — The interest arid rate remaining the
samey the greater the principal the less the time, and the
less the principal the greater the time.
wit ITT EK EXDItCISBS,
733. Find the average term of credit
1. Of $300 due in cash, $500 due in 3 mo., $750 due
in 8 mo., and $950 due in 10 mo.
OPERATION Analysis— On $300, the first
3 X = payment, there is no interest,
^^ ^ i^^nn since it is due in cash ; the int.
b^^ X d_15U0 ^^ ^5()Q ^^^ 3 ^^^ .g ^j^^ ^^^ ^^
75 X 8 = 6000 the int. of $1 for 1500 mo.; the
950 Xl0 = 9500 int. of $750 for 8 mo. is the same
^Kr\r\ vTtOOO ^^ *^^^* of $1 for 6000 mo. ; and
^ the int. of $950 for 10 mo. is the
6 1- mo. same as the int. of $1 for 9500 mo.
Therefore, the whole amt. of int.
is that of $1 for 1500 mo. 4- 6000 mo. -f 9500 mo., or 17000 mo. ; hut
the whole debt is $2500 ; and the int. of $1 for 17000 mo. is equal
to the int. of $2500 for ^^tttf of 17000 mo., or ^ mo.
2. Find the average term of credit of $800 due in 1 mo.,
$750 due in 4 mo.^ and $1000 due in 6 mo.
EQUATION OF PAYMENTS. 107
Rule. — I. Multiply each payment ly its term of credit,
and divide the sum of the products iy the sura of the pay-
ments; the quotient is the average term of credit.
II. (To find the equated time of payment,) Add the
average term of credit to the date at which the several
credits begin.
3. On the first day of December, 1876, a man gave
3 notes, the first for $500 payable in 3 mo. ; the second
for $750 payable in 6 mo. ; and the third for $1200 paya-
ble in 9 mo. What was the average term of credit, and
the equated time of payment ?
^ 4. Bought merchandise Jan. 1, 1875, as follows : $350
on 2 mo., $500 on 3 mo., $700 on 6 mo. What is the
equated time of payment ?
. 5. A person owes a debt of $1680 due in 8 months, of
which he pays -J^ in 3 mo., J in 5 mo., ^ in 6 mo., and
■J^ in 7 mo. When is the remainder due ?
6. Bought a bill of goods, amounting to $1500 on 6
months' credit. At the end of 2 mo., I paid $300 on
account, and 2 mo. afterward, paid $400 on account, at
the same time giving my note for the balance. For what
time was the note drawn ?
OPEKATiON. Analysis. — $300 paid
300x4 = 1200 ^ ^^- l>^fc)re it is due, and
/lAAv. o Q(\f\ $400, 2 mo. before it is
, due, are equivalent to the
800 ' )2000 use of $1 for 2000 months,
21 or the use of $800 (the
,^ , . ^^ ,^ balance) for 2^ mo. bevond
(6 mo. -4 mo.) + 2* mo.=4i mo. ^he original time. H;nce.
the note was drawn for
4i mo. after the second payment.
108 PERCENTAGE.
7. On a debt of $2500 due in 8 mo. from Feb. 1, the
following payments were made : May 1, $250, July 1,
$300, and Sept. 1, $500. When is the balance due ?
8. Find the average term of credit, and the equated
time of payment from Dec. 15, of $225 due in 35 da.,
$350 due in 60 da., and $750 due in 90 da.
9. Dec. 1, 1874, purchased goods to the amount of
$1200, on the following terms : 25^ payable in cash,
30^ in 3 mo., 20^ in 4 mo., and the balance in 6 mo.
Find the equated time of payment, and the cash value of
the goods, computing discount at 1%.
734. To find the equated time when the terms of
credit begin at different dates.
1. J. Prince bought goods of W. Sloan as follows :
June 1, 1874, amounting to $350 on 2 mo. credit ; July 15,
1874, $400, on 3 mo. credit ; Aug. 10, $450, on 4 mo.
credit; Sept. 12, $600, on 6 mo. credit. What is the
equated time of payment?
OPERATION.
$350
due
Aug. 1,
350 X =
400
«*
Oct. 15,
400 X 75 = 30000
450
<(
Dec. 10,
450 X 181 = 58950
600
((
Mar. 12,
600 X 223 = 138800
1800 1800)222750
128f
Hence the equated time is 124 da. from Aug 1, or- Dec. 3.
Analysis. — Computing the terms of credit from Aug. 1, the
earliest date at which any of the debts become due, we find the
terms of credit to be from Aug. 1 to Oct. 15, 75 da. ; to Dec. 10,
181 da., and to March 12, 228 da. The average term of credit is
therefore 124 da. from Aug. 1, and the equated time Dec. 3.
EQUATION OF PAYMENTS. 109
Proof. — Assume as the standard time the latest date, March 13,
The operation will then be as follows :
350 X 223 = 78050
400 X 148 = 69200
450 X 92 = 41400
600 X =
1800 )178650
99i
Hence, the equated time is 99 da. previous to March 12, or Dec. S.
2. Peake & Co. sell to Wm. Jones the following bills
of goods : March 1, 1875, on 60 da., $800 ; April 15, on
30 da., $350 ; May 20, on 4 mo., $3800.
What is the equated time for settlement ?
Rule,— I. Find tJie date at which each deit becomes due.
II. From the earliest of these dates as a standard com^
pute the time to each of the others,
III. Then find the average term of credit and equated
time as in (733).
Proof. — Compute the terms of credit backward from the
latest datCy and subtract the average time from that date
for the equated time.
If the earliest date is not the first of the month, it is more con-
venient to assume the first of the month as the standard date.
3. Bought mdse. as follows : Jan. 15, 1876, on 4 mo.,
$375 ; Feb. 3, on 60 da., $550 ; March 25, on 4 mo.,
$1100 ; April 2, on 30 da., $250. Find the equated time.
4. Ira Blunt, of Gadsden, Ala., bought of Opdyke &
Co. the following bills of goods on 4 months' credit :
Jan. 1, 1874, $650 ; Feb. 10, $380 ; March 12, $900 ;
March 18, $350 ; April 3, $600.
April 5, he discounted his bills at 2^ per month. Find
the equated time of payment, and the discount.
110
PERi
DENTAGE.
5. James Smith
to
Thomas Browk, Dr.
March 10, 1874.
To mdse.
$835.
" 18, ''
a
((
330.
" 26, ''
a
a
475.
April 5, ''
(C
66
600.
" 12, ''
cc
66
350.
Allowing 30 days' credit on each of the bills, what i&
the equated time of payment ?
6. Purchased goods as follows :
Sept. 15, 1875, a bill of $275, on 3 mos.
Oct. 10, '' " 351.50, '' 60 da.
" 28, '' '' 415.75, " 30 da.
NoY. 3, " " 500, " 4 mos.
Dec. 15, " " 710, '' 3 mos.
What was due on this account Aug. 10, 1876, com-
puting interest at 7^ ?
7. I have four notes, as follows : the first for $425, due
April 1, 1875 ; the second for 1615, due May 10, 1875 ;
the third for $1500, due May 28, 1875 ; and the fourth
for $750, due June 10, 1875.
At what date should a single note be made payable, to
be given in exchange for the four notes ?
ayeragi:ng accounts.
735. An Account is a written statement of debit
and credit transactions, with their respective dates.
B^it means wliat is owed by the person with whom the account
is kept ; credit, what is due to him from the person keeping the
account.
736. To Average an Account is to lind, either
AVERAGIlJfG ACCOUNTS.
Ill
the equated time of paying the balance, or the cash balance
at any given time.
Each item of a book account should draw interest from the time
it becomes due.
WRITTEN EX EMCISES .
737. 1. Find the equated time of paying the balance
of the following account.
Dr. William Sampson. Cr.
1875.
1875.
Jan. 11
To mdse. , . .
$750
Feb. 10
By draft at 60 da.
$500
Feb. 1
** " at 3 mo.
600
Mar. 3
*' cash . . .
700
Mar. 15
** ** at 6 mo.
1500
Apr. 15
it ii
300
May 3
** *' at 4 mo.
900
Operation I. {Method hy Products)
Due.
Jan. 11.
May 1.
Sept. 15.
** 3.
Amt. Days. Product. Paid.
750 X 10 = 7500
600 X 120 = 72000
1500 X 257 = 3^5500
900 X 245 = 220500
3750 685500
1500 125400
Apr. 14.
Mar. 3.
Apr. 15.
Amt.
500 :
700 :
300 :
1500
103 = 51500
61 = 42700
104 = 31200
125400
2250
) 560100
248ff , or 249 da.
Balance due 249 da. from Jan. 1, or Sept, 7.
Analysis.— Assuming for convenience Jan. 1 as the standard
date, we find as in 734 the term of credit of each debit amount ;
and, reckoning from the same date, the time to each credit amount.
Multiplying each amount by its time in days, and adding the debit
and credit products, we find the number of days' interest of $1 due
to the debtor, and the number of days' interest of $1 he has already
received. The difference, 560100, shows the number of days' inter-
est of $1 still due, and as the balance is $2250, the time must be
^^ of 560100 da., or 249 da. Hence, the equated time is 249 da.
from Jan. 1, or. Sept, 7.
il2
PERCENTAGE.
Operation II. {^Method by Interest.)
Dt.
$750 to Jan, 11 (from Jan. 1)=
600 '' Feb. 1 + 3 mo. , = 4 mo.
1500 '' Mar. 15 + 6 mo. =8 mo. 14 da.,
900 '' May 3 + 4 mo. = 8 mo. 2 da.,
$3750
Gr,
$500 to Feb. 10 + 63 da. = 3 mo. 13 da., int. at 1% per mo. $17.17
10 da., int. at 1% per mo. $2.50
24.00
127.00
" 72.60
$226.10
700 '' Mar. 3
300 *' Apr.15
= 2 mo. 2 da.,
= 3 mo. 14 da..
14.47
10.40
$42.04
$1500
$226.10 - $42.04 = $184.06, int. at 1% per mo. due.
Int. of balance, $2250, for 1 mo., at 1^ = $22.50.
Hence, $184.06 -i- $22.50 = 8.18+ mo., or 8 mo. 6 da.
8 mo. 6 da. from Jan. 1, or ISept. 7, Equated Time.
In this operation, 12 % per annum or 1 % per mo. is assumed for
convenience ; since the int. at 1 ^/o per mo. is as many hundredths
as there are months, and one-third as many thousandths as there
are days. Thus, the int. of $249 for 2 mo. 9 da. is $498 + $.747
= $5,727(571).
2. Find the equated time of the following ;
Dr, William Simpsok.
(7n
1874.
1874.
Aug. 5
To mdse. at 3 mo.
$720
Oct. 10
By cash . . .
$500
Sept. 10
,. - - 2 **
850
Dec. 15
" draft at 60 da.
450
Nov. 3
<( f(
1200
'' 25
'' cash . . .
900
1875.
1875.
Jan. 20
'' sundr's at 5 mo.
620
Jan. 3
(t <(
250
EuLE 1. — I. Find the date at which each debit item is
due, and each credit item is paid or due.
II. Tahe the first day of the month in the earliest date
on either side of the account as a standard date, and 7nul'
AVEBAGIKG ACCOUKTS.
113
tiply each sum due or paid hy the number of days between
its time and the standard date.
III. Add the products, and their difference divided by
the balance due will give the number of days between the
standard date and the equated time. Or,
Rule 2. — Find the time of each item from the standard
date as before^ and compute the interest on each at 1% a
month. The differ e7ice between the amount of interest on
each side divided by the interest of the balance at l%for
one month will be the equated time.
When the terms of credit are long, Rule 2. gives the shorter
method.
3. Find the equated time of the following, allowing
^60 da. credit on each debit item :
Dr. John Deiscoll. Cr.
1877.
1877.
June 1
To mdse. . .
$950
Aug. 1
By cash . .
$700
July 6
(( i(
300
Sept.20
(( K
1000
Sept. 8
it t(
1900
Nov. 1
it it
1200
Oct. 20
*' '* . .
2600
H
4. What is the equated time for the payment of the
balance of the following account, allowing 4 months'
credit on all the debit items ?
Dr. DoDD, Brown & Co. Cr.
1878.
1878.
Jan. 20
To mdse. . .
$570
Feb. 14
By mdse. . .
$490
'* 28
300
Mar. 1
*' cash . .
1000
Feb. 11
720
Apr. 2
(( tt
1800
'' 26
835
Mar. 10
1150
'' 28
930
Apr. 15
475
114
PERCEKTAGE.
738. 1. Find the cash halance of the following account
on the 22d of August, allowing interest at 6^ :
Dr. George Hammond. Cr,
1875.
1875.
Mar. 15
Tomdse.,at3mo.
$600
May 10
By cash . .
$300
Apr. 3
<t " *'4ino.
700
July 1
ti ti
400
May 10
^' '* '^6mo.
1000
Aug.l5
{( it
500
Operation. — By averaging the account, the equated time for
paying the balance, $1100, is found to be ^ov. 4. (734.)
True present worth of $1100 for 74 da. (from Aug. 22 to Nov. 4)
is $1086.60, or cash balance Aug. 22.
Or, by Interest Method, as follows :
Dr.
Int. of $600, from June 15 to Aug. 22, 68 da., $6.71 (574.;
** ** 700, '' Aug. 3 '' " 19 da., 2.1 9
$8.90
Cr.
Int. of $1000, from Aug. 22 to Nov. 10, 80 da., $13.15
300, *' May 10 '' Aug. 22, 104 da ,
400, '' July 1'* " 52 da.,
500, " Aug. 15 " ** 7 da..
513
3.42
58
$22.28
8.90
Balance of interest due Hammond, $13.38
$1100 - $13.38 = $1086.62, Cash Balance, Aug. 22.
Analysis. — Charge Hammond with interest on each debit item
from the time it is due to date of settlement, and credit him with
interest on each sum paid from the date of payment to date of set-
tlement, also on each debit item which becomes due after the date
of settlement. Hence, he is entitled to interest on $1000 from
Aug. 22 to Nov. 10. As the balance of interest is in favor of Ham-
mond, it must be deducted from the balance of the account, to ob-
tain the cash balance. There is a slight difference in the results,
but the interest method is the more accurate. By the use of Inter-
est Tables, it is also the shorter of the two methods.
AVERAGING ACCOUNTS,
115
EuLE 1. — I. Average the account^ and find the equated
time of payment of the balance,
11. If the date of settlement is prior to the equated time,
find the present worth of the balance of account for the
cash balance ; if later, find the interest of the balance of
account for the intervening time, and add it to find the
cash balance, Ov,
B>VL^ 2.— Find the interest on each debit and credit
item, from the time it is due or paid to the date of settle-
me7it, placing on the same side of the account the interest
on each item due prior to the date of settlement, and on the
opposite side the interest on each item due after the date
of settlement. If the balance of interest is on the same side
as the balance of the account, add it, if on the other side
subtract it ; and the result ivill be the cash balance at the
date of settlement,
2. I owe $1500 duo May 1, and $750 due Aug. 15. If
I give my note at 30 da. for $450, June 1, and pay $370
in cash July 15, what is the equated time for paying the
balance ; and what would be due in cash Dec. 10, allow-
ing interest at 7^?
3. When is the balance of the following account due
per average ?
Dr. 0. B. TiMPSOiT. Cr.
1875.
1875.
Aug. 10
To mdse. @ 60 da. .
$751.35
Oct. 3
By cash
$300.00
Sept. 5
" " @30da. .
425.00
Nov. 15
" note @ 90 da. .
450.00
Nov. 1
" '' @90da. .
927.83
Dec. 20
'* cash ....
500.00
Dec. 5
" " @30da. .
1200.00
116
PERCENTAGE.
4. What IS the cash balance of the aboye account Jan. 1,
1876, allowing interest at 10^?
5. Find the equated time, and cash balance July 1, of
the following, allowing 7^ interest :
Dr.
Thomas Smith.
Cr.
Jan. 4
To mdse. @ 4 mo.
$1600
Feb. 1
By mdse. @ 4 mo.
$500
" 6
" " @3mo.
1500
Mar. 2
" cash . . .
2000
Apr. 10
" " @60da.
3000
" 25
(( (I
3150
" 28
" " @30da.
2500
Apr. 16
ti. ((
800
6. Average the following account, and find for what
amount a note at 60 days should be given Aug. 1, to pay
the balance, interest at 6^.
Dr. Orson Hinman. Or.
1875.
1875.
«♦
Apr. 2
To charges
$87.25
Feb. 25
By mdse. @ 8 mo.
$600
May 15
(( (i
35.75
Mar. 3
*' « @6 ''
300
Apr. 1
'' " @6 ''
500
739. 1. Average the following Account Sales, and find
when the net proceeds are due. (543.)
Account Sales of 1200 iils. of flour received from
SmitJi, Tyler & Co., Cincinnati.
Date.
Buyer.
Quantity.
Price.
Amount.
1876.
May 1
June 5
'* 15
July 1
J. Brooke
W. Long
A, Bruce
W. Case
300 bbl.
450 ''
250 ''
200 "
% $5.50, 3 mo.
% 6.20, 4 mo.
% 6.50, 6 mo.
@ 5.75, 2 mo.
$1650.00
2790.00
1625.00
1150.00
$721500
AVERAGIKG ACCOUKTS. 117
Charges.
Apr. 28. Freight . $674.50
*' " Cartage 37.50
May 1. Storage 191.00
Commission on $7215, @ 2} fc . . . 162.34
Total charges $1065.3 4
Net proceeds due per average $6149.66
OPERATION.
' -Average of sales, found by the method of Equation of Payments,
Oct. ly which is the date at which the commission is due.
Average of charges, including commission (Oct. 1), May 22.
Equated time of $7215 due Oct. 1, and $1065.34 due May 22, 0cL2Ji,,
date when the net proceeds are due.
EuLE. — I. Average the sales alone^ and the result will
ie the date to be given to the commission and guaranty.
II. Make the sales the credits and the charges the debits,
and find the equated time for paying the balance.
2. Make an account sales, and find the net proceeds
and the time the balance is due :
Wm. Brown, of N. York, sold on acct. of J. Berry, of Chi-
cago, June 1, 350 bu. Winter Wheat, @ $1.35, at 60 da. ;
June 15, 275 bu. Spring Wheat, @ $1. 75, at 90 da. ; July 3,
1260 bu. Indian Corn, @ $.79, at 6 mo.; and July 10,
375 bu. Eye, @ $1.02, at 3 mo. Paid freight, May 28,
$567.50; cartage. May 30, $22.50; insurance, June 5,
$56.25 ; and charged com. at 3^%, and 1^% for guaranty.
3. Sold on account of Brown, Sampson & Co., at 6
mo. : Oct. 1, 1874, 13 hhd. sugar, averaging 1520 lb., (^
$.12^ ; Oct. 5, 15 chests Hyson Tea, each 95 lb., @ $1j05.
Paid charges : Oct. 3, Insurance, $85 ; Oct. 10, Cooper-
age, etc., $24.50 ; Oct. 20, Cartage, $125. Charged com-
mission and guaranty, 4:^%. Make an account sales, and
find the equated time for paying the net proceeds.
118
PERCENTAGE.
740.
SYNOPSIS FOR REVIEW.
Exchange, 3. Domestic Exchange, 3. i^(?r-
ej^^ Exchange. 4. ^2^^ <?/ Exchange. 5.
/S'^^ <?/ Exchange. 6. /Sji^A^ l>ra/if <?r -Si7^.
7. 2Vm^ Draft or Bill. 8. Buyer or Be-
mitter. ^.Acceptance. 10. Par of Exchange,
11. Course or Bate of Exchange.
A Sight Draft. 2. Set of Exchange.
TOO. i rp fi ^ t Cost of Draft. Formvla.
701. i ^ \ Face of Draft.
1.
Defs. .
2.
Forms.
3.
Inland
Exch.
j 1. TOO. )
( 2. 701. )
4 Foreign
Exch'ge.
5. Arbitra-
tion of
Exch'ge.
22. Custom-
house
Business.
23. Equation
OF Paym'ts.
24 AVEKAGINa
Accounts.
j 1. Money of Account,
1. Defs. .
Sterling Bills, or Exchange.
2. Exchange with Europe — how effected.
f:l
To find
j CostofBiU.
\ Face of Bill.
1. Defs.
706.
707.
1. Arbitration of ExcJiange.
2. Simple Arbitration.
I 3. Compound Arbitration.
Rule, I, II, m, IV.
1. Custom Rouse. 2. Port of
Entry. 3. Collector. 4.
Clearance. 5. Manifest. 6.
Duties or Customs. 7. Tariff.
S. Specific Duty. 9. Ad Val-
orem Duly. 10. Gross Wght.
11. Net Weight-
To find the Duty.
1. Equation of Payments. 2.
Equated Time. 3. Term of
Credit. 4 Average Term of
Credit.
2. Principle.
3. 733. Rule, I, IL
4 734. Rule, I, II, III. Proof.
1. Defs. 1. Account. 2. To Average an AceL
2. 737. Rule 1, I, II, III. Rule 2.
3. 738. Rule 1, 1, II. Rule 2.
4 739. Rule, I. II.
1. Defa. .
2. 726.
1. Defs. <
ORJLL EXERCISES.
741. 1. A father is 30 years old, and his son 6 ; ho\v
many times as old as the son is the father ?
2. 30 are how many times 6 ? 30 ~- 6 == ?
3. What part of $30 are $6 ? Of 20 cents are 5 cents ?
4. What is the relation of 8 to 2 ? Of 40 rd. to 4 rd. ?
5. What relation has 12 to 3 ? 60 lb. to 20 lb. ?
Compai-e the following, and give their relative values.
6. 75 with 5.
7. 25 with Gf
8. 1 with 7.
9. \ with 7.
10. ^ with 3f
11. .9 with .3.
12. $.G with $.2.
13. .42 with .3.
14. f with f .
DEFINITIONS.
743. Ratio is the relation between two numbers of
the same unit value, expressed by the quotient of the first
divided by the second. Thus the ratio of 12 to 4 is
12 ^ 4 = 3.
743. The Sign of ratio is the colon ( : ), or the sign
of division with the line omitted.
Thus, the ratio of 9 to 3 is expressed 9:3, or 9^3, or in the form
of a fraction |, and is read, the ratio of 9 to 3, or 9 divided by 3.
744. The Terms of a ratio are the two numbers
compared.
745. The Antecedent is the first term, or dividend.
746. The Consequent is the second term, or divisor.
120 RATIO.
747. The Value of a ratio is the quotient of the antece-
dent divided by the consequent, and is an abstract number.
Thus, in the ratio $18 : $6, $18 and $6 are the terms of the ratio ;
$18 is the antecedent ; $6 is the consequent ; and 3, the quotient of
$18 H- $6, is the value of the ratio.
748. A Simple Ratio is the ratio of two numbers ;
as 10 : 5.
749. A Compound Ratio is the ratio of tho
products of the corresponding terms of two or more sim^
pie ratios.
Thus the ratio compounded of the simple ratios,
I ; ^^ f may be expressed { ^^^ ^^ \^^ ": J'j /|^ } =72 : 48 ;
Or, f X V^ = I = 3 : 2.
When the multiplication is performed the result is a simple ratio.
750. The Reciprocal of a ratio is 1 divided by the
ratio (196), or it is the consequent divided by the ante-
cedent. Thus the ratio of 8 to 9 is 8 : 9, or f , and its
reciprocal is f.
The ratio of two fractions is obtained by reducing them to a
common denominator, when they are to each other as their nume-
rators (241).
If the terms of a ratio are denominate numbers, they must b«
reduced to the same unit value.
751. From the preceding definitions and illustrations
are deduced the following
Formulas. — 1. The Ratio = Antecedent -r- Consequent.
2. The Consequent == Antecedent-^ Ratio.
3. The Afitecedent = Consequent x Ratio.
RATIO. 121
753. Since the antecedent is a dividend, and the con-
sequent a divisor, any change in either or both of the
terms of a ratio will aflfect its value according to the laws
of division or of fractions (200), which laws become the
Gei^eral Principles of Eatio.
1. Multiply inq the antecedent y or ),..,,. ,. ,, ,.
^. .;. ,^ ^ } Multiplies the ratio.
Dividing the consequent, )
2. Dividinq the antecedent, or ) r^ • • 7 ,i ,-
,^ ^,. /. ,^ ", } Divides the ratio.
Multiplying the consequent, )
3. Multiplyinq or dividing loth ] ^ , , ,,
^ ^ -, ^ , -, _Li [ Does not chanqe the
antecedent and consequent by Y , .
the same number, )
753. These principles may be embraced in one
GENERAL LAW.
A change in the antecedent produces a like change in
the ratio ; hut a change in the consequent produces an
OPPOSITE change iii the ratio.
JSXERCISES.
754. 1. Express the ratio of 11 to 4 ; of 16 to 2 ; of 20
lo 6| ; of $36 to 112 ; of 9 lb. to 27 lb. ; of ^ bu. to 9 bu.
2. Can you express the ratio between $15 and 5 lb. ?
Why not ?
3. Indicate the ratio of 18 to 20 in two forms. What
are the terms of the ratio ? The antecedent ? The co^^-
sequent ? The hind of ratio ? The value of the ratio.
In like manner express, analyze, and give the value,
4. Of 80 to 120 ; of 12^ to 37i ; of l^ to |.
2x27x42
5. Of 5.2 to 1.3; of f to^; of
12x4x126'
122 RATIO.
6. The antecedents of a ratio are 7 and 10, and the
consequents, 5 and 4. What is the value of the ratio ?
7. The first terms of a ratio are 18, 12, and 30, the
second, 54, 6, and 15. What is the kind of ratio ? Ex-
press in three forms. Find its value in the lowest terms.
Solve, and state the formula applied to the following :
8. The consequent is 3;^, the antecedent -^f ; what is
the ratio ?
9. The antecedent is 60, the ratio 7 ; what is the con-
sequent ?
10. The consequent is $6.12^, the ratio ^ ; what is
the antecedent ?
11. The ratio is 2f, the antecedent i^ of | ; what is the
consequent ?
12. The ratio is 6, the consequent 1 wk. 3 da. 12 hr. ;
what is the antecedent?
13. Express the ratio of 120 to 80, and give its value
in the lowest terms.
14. Make such changes in the last example as will
illustrate Prik. 1.
15. With the same example, illustrate Prin. 2.
16. Illustrate by the same example Prin. 3.
17. Find the reciprocal of the ratio of 75 to 15.
18. Find the reciprocal of the ratio of 2 qt. 1 pt. to
4 gal. 1 qt. 1 pt.
What is the ratio
19. Of 40 bu. 4.5 pk. to 25 bu. 2 pk. 1 qt.
20. Of 6 A. 110 P. to 10 A. 60 P.
21. Of 25 lb. 11 oz. 4 pwt. to 19 lb. 5 oz. 8 pwt.
2.. Ofl?itoi^a
ORAL EXBMC IS E8,
755. 1. What is the ratio of 4 to 2 ? Of 6 to 1 ? Of
14 to 7 ? Of 21 to 3 ?
2. Find two numbers that have the same quotient as
8^2. As 27 -r- 3. As 16 -r- 4. As 30 -^ 6. As 4-^|.
3. Express in the form of a fraction the ratio of 26 to
13. Of 32 to 8.
4. Express in both forms the ratio of two other num-
bers equal to the ratio of 10 to 2. Of 15 to 5. Of 12 to 3.
5. If 4 stamps cost 12 cents, what will 20 stamps cost
at the same rate ?
6. What number divided by 12, gives the same quo-
tient as 20 -^ 4 ?
7. What number has the same ratio to 12, that 20 has
to 4?
8. To what number has 48 the same ratio that 80 has
to 5 ? That 24 has to 3 ?
9. The ratio of 20 to 5 is the same as the ratio of what
number to 4 ? To 6 ? To 5^ ? To 6^ ?
10. The ratio of 45 to 9 is the same as the ratio of 15
to what number ? Of 30 to what number ?
11. 28 is to 7 as 8 is to what number ?
12. 56 is to 8 as what number is to 5 ?
13. 63 -T- what number equals the ratio of 36 to 4?
124 PKOPORTIOK.
DEFINITIONS.
756. A Proportion is an equation in which each
member is a ratio ; or it is an equality of ratios.
757. The equality of the two ratios may be indicated
by the sign = or by the double colon : :
Thus, we may indicate that the ratio of 8 to 4 is equal to that of
6 to 3, in any of the following ways :
8:4rr6:3, 8:4::6:3, | = | 8^-4 = 6^3.
This proportion, in any of its forms, is read. The ratio of 8 to 4 is
equal to the ratio of 6 to 3, or, 8 is to 4 as 6 is to 3,
758. Since each ratio consists of two terms, every pro-
portion must consist of at least /o^^r terms. Each ratio is
called a Couplet, and each term is called a Proportional.
759. The Antecedents of a proportion are the first
and third terms, that is, the antecedents of its ratios.
760. The Consequents are the second and fourth
terms, or the consequents of its ratios.
761. The JExtremes are the first and fourth terms.
763. The Means are the second and third terms.
In the proportion 8 : 4 : : 10 : 5, 8, 4, 10, and 5 are the propor-
tionals; 8 : 4 is the first couplet, 10 : 5 the second couplet ; 8 and 10
are the antecedents, 4 and 5 are the consequents; 8 and 5 are the ex-
tremes, 4 and 10 are the means.
Three numbers are proportional, when the ratio of the first to the
second is equal to the ratio of the second to the third. Thus the
numbers 4, 10, and 25 are proportional, since 4 : 10 = 10 : 25, the
ratio of each couplet being f ,
When three numbers are proportional, the second tenn is called
a Mean Proportional between the other two.
PROPORTION. 125
The proportion 8 : 4 : : 10 : 5 may be expressed thus, i=^
(757). Reducing these fractions to equivalent ones having a com-
. ^ 8x5 10x4
mon denominator, -— — = .
Since these fractions are equal, and have a common denominator,
their numerators are equal, or 8 x 5 = 10 x 4.
763. Principles. — 1. The product of the extremes of
' a proportion is equal to the product of the means,
2. The product of the extremes divided iy either mean
will give the other mean,
3. Tlie product of the means divided iy either extreme
will give the other extreme.
EXJERCISJES.
764. 1. The ratio of 4 to 10 is equal to the ratio of 6
to 15. Express the proportion in all its forms (757).
Brill Exercise. — How many terms has a proportion ? What are
they called ? How many ratios ? What are they called ?
Name the proportionals in example 1 ; the couplets ; the ante-
cedents ; the consequents ; the extremes ; the means. What is the
product of the extremes ? Of the means ? What is the dividend
of the first ratio ? The divisor of the second ratio ? The divisor
of the first ratio ? The dividend of the second ratio ? In the frac-
tional form what are the numerators ? The denominators ?
2. The ratio of 6 to 15 equals the ratio of 8 to 20.
3. The ratio of 4|^ to 18 equals the ratio of 6 to 24.
Change to the form of equations by Prin. 1 :
4. 12 : 1728 : : 1 : 144.
5. 2| : 17 : : 20 : 143^.
6. 27.03 : 9.01 : : 16.05 : 5.35.
7. f :f ::|:^.
8. The extremes are 15 and 48, and one of the means
is 10. Find the other mean.
9. The means are 25 and 75, and one of the extremes
is 12^. Find the other extreme.
126
p R o p o R T I o :n^ .
The required or omitted term in a proportion, or in an operation,
will hereafter be represented by x.
Find the term omitted in each of the following pro-
portions :
IT. 4|yd.:cryd.::$9|: $27.25.
18. x: 0.01 :: 16.05: 5.35.
19. |yd.::?ryd.::$|: $59.0625.
20. -^:|::^:|.
21. .r:38i::8|:76f
22. 7.5:18::a:oz. : 7^V ^z-
11. 8:52::20:ir.
12. 12 :ic:: 1:144
13. a;: 20:: 120: 50.
14. $80: $4::^: 4.
15. 2.5:62.5::5:ir.
16. $175.35: $2: ::i:f.
SIMPLE PROPOETION.
765. A Simple JProjfortion is an expression of
equality between two simple ratios. It is used to solye
problems of which three terms are given, and the fourth
is required.
Of the three given numbers, two mnst always be of the same
kind ; and the third y of the same kind as the required term.
766. A Statement is the arrangement of these
terms in the form of a proportion.
WRITTEN EXEItCISTSS.
767. 1. If 4 tons of coal cost $24, what will be the
cost of 12 tons at the same rate ?
STATEMENT.
4T.: 12 T. :: $24:$a;
OPERATION.
12 X 24-^4=r$72
Or By Cancellation.
12 X t^'
>x^-
$72
Analysis. — Since 4 tons and 12
tons have the same unit value, they
can be compared, and will form one
couplet of the proportion.
For the same reason $24 the cost
of 4 tons, and %x the cost of 12 tons,
will form the other couplet.
Then by Prin. 3, $a;= 24 x 12 -5-4
z=$72.
PROPORTION. 127
Proof. — 4 x 72=12 x 24. (763, Prin. 1.) In practice, that number
which is of the same unit value as the required term, is generally
made the antecedent of the second couplet or third term of the pro-
portion, and the required term, ic,the fourth term. The terms of the
first couplet are so arranged as to have the same ratio to each other,
as the terms of the second couplet, have to each other, which is
easily determined by inspection. The product of the means 12 and
24, divided by the given extreme 4, gives the other extreme, or
required term, $72. (763, Prin. 3.)
Drill exercises like the following, will soon make the pupil
familiar with the principles and operations of proportion.
2. If 4 horses eat 12 bushels of oats in a given time,
how many bushels will 20 horses eat in the same time ?
In this example, what two numbers have the same unit value ?
What do they form ? What is the denomination of the third term ?
Of the required term ? What is the antecedent of the second «
couplet ? From the conditions of the question, is the consequent
of the second couplet or required term, greater or less than the
antecedent ? If greater, how must the antecedent and consequent
of the first couplet compare with each other ? If less, how com-
pare ? What is the ratio of the first couplet ? Why not 20 to 4 ?
Make the statement. How is the required term found ?
3. If 96 cords of wood cost 1240, what will 40 cords cost ?
4. If 20 lb. of sugar cost $1.80, find the cost of 45 lb.
5. If 18 bu. of wheat make 4 barrels of flour, how many
barrels will 200 bu. make ?
EuLE. — I. Mahe the statement so that two of the given
numbers which are of the same unit value, shall form the
first couplet of the proportion, and have a ratio equal to
the ratio of the third given term to the required term.
II. Divide the product of the means iy the given extreme^
and the quotient will he the number required.
128 PROPORTION.
CAUSE AND EFFECT.
768. The terms of a proportion have not only the
relations of magnitude^ but also the relations of cause
and effect
Every problem in proportion may be considered as a
comparison of two causes and two effects.
Thus, if 4 tons as a cause will bring when sold, $24 as an effect,
12 tons as a cause will bring $72 as an effe^^t. Or, if 6 horses as a
cause draw 10 tons as an effect, 9 horses as a cause will draw 15
tons as an effect,
769. Since like causes produce like effects, the ratio
of two like causes equals the ratio of two like effects pro-
duced by these causes. Hence,
1st cause : 2d cause : : 1st effect : 2d effect.
WRITTEN BXEUCISES.
770. 1. If 8 men earn $32 in one week, how much will
15 men earn at the same rate, in the same time ?
STATEMENT. ANALYSIS. — In this ex-
ist cause. 2d cause. 1st effect 2d effect ample an eJf(3C^ is required.
8 men : 15 men :: $32 : ^x The first cause is 8 men,
the second cause 15 men,
OPERATION. , . ^, T.,
and since they are like
%X z=:15 Xt3/^-r-o^^=^bO causes they can be com-
pared.
The effect of the first cause is $32 earned, the effect of the second
cause is %x earned, or the required term. Since like effects have
the same ratio as their causes (769), the causes may form the
first couplet, and the effects the second couplet of the proportion.
The required term is readily obtained by (7C>3, 3).
2. If 20 bushels of wheat produce 6 barrels of flour,
how many bushels will be required to produce 24 barrels ?
PROPORTION. 129
STATEMENT. ANALY8IS.~In this ex-
ist cause. 2d cause. 1st effect. 2d effect. ample a cause is required.
20 bu. : iiJ bu : : 6 bbl. : 24 bbl. The first cause is 20 bu.,
the second cause is x bu.
OPERATION. ,, . ,
or the required term.
a; bu. = 20 X 24 -r- 6 = 80 bu. The effect of the first
cause is bbl. of flour,
the effect of the second cause is 24 bbl. of flour. Since like causes
have the same ratio as their effects (709), the statement is made
as in Ex. 1, and the required term found by (703, 2).
3. If 5 horses consume 10 tons of hay in 8 mo., how
many horses will consume 18 tons in the same time ?
Drill Exercise. — In this example, what is the first cause ? The
second cause ? The first effect ? The second effect ? Is the re-
quired term a cause or an effect ? A mean or an extreme ? What
is the first couplet ? What, the second ? Make the statement.
How is the required term found ?
4. If 8 yards of cloth cost $6, how many yards can be
bought for $75 ?
5. How many men will be required to build 32 rods of
wall in the same time that 5 men can build 10 rods ?
Rule. — I. Arrange the terms in the statement so that
the ratio of the causes which form the first couplet, shall
equal the ratio of the effects which form the second couplet y
putting X in the place of the required term.
II. If the required term^ te an extreme, divide the pro-
duct of the means by the given extreme ; if the required
term he a mean, divide the product of the extremes hy the
given mean.
To shorten the operation, equal factors in the first and second, or
in the first and third terms may be canceled.
Solve the following by either of the foregoing methods :
6. If 5 sheep can be bought for $20.75, how manj
sheep can be bought for $398.40 ?
130 PEOPORTION.
7. When 10 barrels of flour cost $112.50, what will be
the cost of 476 barrels of flour Z
8. If a railroad train run 30 miles in 50 min., in what
time will it run 260 miles ?
9. How many bushels of peaches can be purchased for
$454.40, if 8 bushels cost $10.24 ?
10. If a horse travel 12 miles in 1 hr. 36 min., how far,
at the same rate, will he travel in 15 hours ?
11. How many days will 12 men require to do a piece
of work, that 95 men can do in 7^ days ?
12. If f of an acre of land cost $60, what will 45|- acres
cost?
13. At the rate of 72 yards for £44 16s., how many
yards of cloth can be bought for £5 12s. ?
14. If ^ of a^barrel of cider cost $1^, what is the cost
of I of a barrel?
15. If the annual rent of 35 A. 90 P. is $284.50, how
much land can be rented for $374.70 ?
16. What will 87.5 yd. of cloth cost, if If yd. cost $1.26 ?
17. If by selling $5000 worth of dry goods, a merchant
gains $456.25, what amount must he sell to gain $1000 ?
18. Bought coal at $4.48 per long ton, and sold it at
$7.25 per short ton. What was the gain per. ton ?
19. What will be the cost of a pile of wood 80 ft. long,
4 ft. wide, 4 ft. high, if a pile 18 ft. long, 4 ft. wide, 6 ft.
high cost $30.24?
20. If 36 bu. of wheat are bought for $44.50, and sold
for $53.50, what is gained on 480 bu. at the same rate ?
21. If a business yield $700 net profits in 1 yr. 8 mo., in
what time will the same business yield $1050 profits ?
PROPORTION.
131
COMPOUND PROPOETION.
771. A Compound Proportion is an expression
of equality between two ratios, one or both of which are
compound.
AH the terms of every problem in compound proportion appear
in couplets, except one, and this is always of the same unit value as
the required term.
The order of the ratios, and of the terms composing the ratios, is
the same as in simple proportion.
WRITTEN EXERCISES,
773. 1. If 18 men build 126 rd. of wall in 60 da.,
working 10 hr. a day, how many rods will 6 men build in
110 da., working 12 hr. a day?
STATEMENT.
r.
18 men : Omen ^ ^o'ds. rods.
X
60 days : 110 days V : : 126 : x
^U
11011
10 hours : 12 hours )
5 00
U
nPT?,!? ATTON"
i0
m^^
11 42
5
463
r<^B:_0xM0Xl^xW_,«,_9,„ ,
93|=a;rd.
^ rA^tiiAs^xA - i y-iiva.
$ 5
Analysis. — All the terms in this example appear in couplets, ex-
cept 126 rods, which is of the same unit value as the required term,
and is made the third term of the proportion, and x rods, the fourth.
The required number of rods depends upon ^^7*66 conditions : 1st,
the number of men employed ; 2d, the number of days they work ;
and 3d, the number of hours they work each day.
Consider each condition separately, and arrange the terms of the
same unit value in couplets, and make the statement as in simple
proportion (767). Then find the required term by (763, 3),
132 PROPORTION.
2. If 20 horses consume 36 tons of hay in 9 mo., how
many tons will 12 horses consume in 18 months ?
Drill Exercise. — In this example, what is the denomination of
the required term ? What given number has the same unit value ?
What will be the third term of the proportion? The fourth?
How many couplets are there ? Give them. What kind of a ratio
do they form ? How is the antecedent and consequent of each
couplet determined ? How is a compound ratio reduced to a simple
one ? Make the statement. Is the required term a mean or an
extreme? How is it found? (763, 3.)
3. If $320 will pay the board of 4 persons for 8 weeks,
for how many weeks will $800 pay the board of 15
persons ?
4. If a man walk 192 miles in 6 days, walking 8 hr. a
day, how far can he walk in 18 days, walking 6 hr. a day ?
5. If 6 laborers can dig a ditch 34 yards long in 10
days, how many days wiB 20 laborers require to dig a
ditch 170 yards long?
KuLE. — I. Form each couplet of the compound ratio
from the numbers given^ iy comparing those lohich are of
the same unit value, arranging the terms of each in respect
to the third term of the proportion, as if it were the first
couplet of a simple proportion. (767.)
II. Divide the product of the second and third terms hy
the product of the first terms, the quotient will he the num-
ber required.
The same preparation of the terms by reduction is to be observed
as in simple proportion.
When possible, shorten the operation by cancellation. When
the vertical line is used, write the factors of the dividend on the
right, and the factors of the divisor with x on the left.
PBOPORTION.
133
CAUSE AND EFFECT.
773. If we regard the conditions of each problem as
the comparison of two causes and two effects, the com-
pound proportion will consist of two ratios, one or both
of which may be compound, and the required term will
be either a simple cause, or effect, or a single element of a
compound cause, or effect.
WRITTEN EXEBC IS ES.
774. 1. If 8 men earn $320 in 8 days, how much will
12 men earn in 4 days ?
1st cause.
8 men :
8 days
STATEMENT.
2d. cause.
: 12 men [ ^ ^
• 4 days \
OPERATION.
12x4x^ ^0^
^x^
Or,
$320 : fe ^
= $240
12
4
$240
Analysis. —
In this example
the first cause is
8 men at work 8
days, the second
cause is 13 men
at work 4 days ;
the two form a
compound ratio.
The effect of the first cause is $320 earned, the effect of the sec-
ond cause is $x earned, and is the required term ; the two effects
form a simple ratio.
The value of the required term depends upon two conditions :
1st, the number of men at work ; 2d, the number of days they work.
Consider each condition separately, and arrange the terms of the
same unit value in couplets, and make a statement in the same man-
ner as in simple proportion. The required term being an extreme,
is found by (763, 3).
2. If it cost $41.25 to pave a sidewalk 5 ft. wide and
75 ft. long, what will it cost to pave a similar walk 8 ft.
wide and 566 ft. long ?
134 PROPORTION.
3. How many days will 21 men require to dig a ditch
80 ft. long, 3 ft. wide, and 8 ft. deep, if 7 men can dig a
ditch 60 ft. long, 8 ft. wide, and 6 ft. deep, in 12 days ?
Or,
80 X
3 u n
8 ^m ^(^8
8
OPERATION.
8
;?xlgx^0x$x^ _8
"^^ ^1x00x^x0 -3-^t^a-
3 X— 2| da.
Analysis. — In this example the causes and the effects each form
a compound ratio. The required term is an element of the second
cause and a mean. Hence divide the product of the extremes by
the product of the given means, and the quotient is the required
factor or term, 3| da. (763, 2).
4. If 4 horses consume 48 bushels of oats in 12 days,
how many bushels will 20 horses consume in 8 weeks ?
EuLE. — I. Of the given numbers, select those which con-
stitute the causes, and those which constitute the effects,
and arrange them in couplets as in simple cause and effect,
putting X in the place of the required term.
II. If the required term, x, he an extreme, divide the
product of the means by the product of the given extremes ;
if X be a mean, divide the product of the extremes by the
product of the given means ; the quotient will be the re-
quired term.
Solve the following by either of the foregoing methods :
5. What sum of money will produce $300 in 8 mo., if
$800 produce $70 in 15 months ?
PROPOETION. 135
6. If 20 reams of paper are required to print 800 copies
of a book containing 230 pages each, 40 lines on a page,
how many reams are required to print 3000 copies of
400 pages each, 35 lines on a page ?
7. If 10 men can cut 46 cords of wood in 18 da., work-
ing 10 hr. a day, how many cords can 40 men cut in
24 da., working 9 hr. a day ?
8. What is the cost 36^ yards of cloth 1| yi wide,
if 2i yards If yd. wide, cost $3-37^ ?
9. A contractor employs 45 men to complete a work
in 3 months ; what additional number of men must he
employ, to complete the work in 2^ months?
10. If a vat 16 ft. long, 7 ft wide, and 15 ft. deep
holds 384 barrels, how many barrels will a vat 17J ft.
long, lOJ^ ft. wide, and 13 ft. deep hold ?
11. What is the weight of a block of granite 8 ft. long,
4 ft. wide, and 10 in. thick, if a similar block 10 ft. long,
5 ft. wide, and 16 in. thick, weigh 5200 pounds ?
12. If it cost $15 to carry 20 tons 1^ miles, what will
it cost to carry 400 tons ^ of a mile ?
13. If it take 13500 bricks to build a wall 200 ft. long,
20 ft. high, and 16 in. thick, each brick being 8 in. long,
4 in. wide, and 2 in. thick, how many bricks 10 in. long,
5 in. wide, S^ in. thick, will be required to build a wall
600 ft, long, 24 ft. high, and 20 ft. thick ?
14. What will 15 hogsheads of molasses cost, if 28^
gallons cost $7^ ?
15. At 6^d. for If yards of cotton cloth, how many
yards can be bought for £10 6s. 8d. ?
16. If $750 gain $202.50 in 4 yr. 6 mo.^ what sum will
gain $155.52 in 1 yr. 6 mo. ?
136 PROPORTIOIf.
17. In what time can 60 men do a piece of work that
15 men can do in 20 days ?
18. If 2^ yd. of cloth 6 quarters wide can be made from
1 lb. 12 oz. of wool, how many yards of cloth 4 quarters
wide can be made from 70 lb. of wool ?
19. If the use of $300 for 1 yr. 8 mo. is worth $30, how
long, at the same rate, may $210.25 be retained to be
worth $42,891 ?
20. A farmer has hay worth $18 a ton, and a merchant
has flour worth $10 a barrel. If the farmer ask $21 for
his hay, what should the merchant ask for his flour ?
21. How many men will be required to dig a cellar
45 ft. long, 34.6 ft. wide, and 12.3 ft. deep, in 12 da. of
8.2 hr. each, if 6 men can dig a similar one 22.5 ft. long,
17.3 ft. wide, and 10.25 ft. deep, in 3 da. of 10.25 hr. each?
22. If a bin 8 ft. long, 4^ ft. wide, and 2^ ft. deep,
hold 67|^ bu., how deep must another bin be made, that is
18 ft. long and 3|- ft. wide, to hold 450 bu. ?
23. What wiU 120 lb. of coffee cost, if 10 lb. of sugar
cost $1.25, and 16 lb. of sugar are worth 5 lb. of coffee ?
24. Two men haye each a farm. A's farm is worth
$48.75, and B's $43| ; but in trading A values his at $60
an acre. What value should B put upon his ?
25. If 6 men in 4 mo., working 26 da. for a month,
and 12 hr. a day, can set the type for 24 books of 300 pp.
each, 60 lines to the page, 12 words to the line, and an
average of 6 letters to the word, in how many months of
24 da. each, and 10 hr. a day, can 8 men and 4 boys set
tlie type for 10 books of 240 pp. each, 52 lines to the
page, 16 words to the line, and 8 letters to the word, 2
boys doing as much as 1 man ?
ORAL EXEMCISES.
775. 1. If John has 10 marbles, William 15 marbles,
and Charles 25 marbles, what part of the whole has each ?
2. Two men bought a barrel of flour for $9, the first
paying $4 and the second $5. What part of the flour
belongs to each ?
3. Three men bought 108 sheep, and as often as the
first paid $3, the second paid $4, and the third $5. How
many sheep should each receive ?
4. If $45 be divided between two persons, so that of
every $5, one receives $2, and the other $3, how many
dollars does each receive ?
5. Two men hired a pasture for $36 ; one put in 2
horses for 3 weeks, the other 3 horses for 4 weeks. What
should each pay ?
DEFINITIONS.
776. Partner ship is the association of two or more
persons under a certain name, for the transaction of busi-
ness with an agreement to share the gains and losses.
777. A Firnif Company or Mouse is any par-
ticular partnership association.
778. The Capital i& the money or property invested
by the partners, called also Investment, or Joint- Stoch.
138 PARTNERSHIP.
imQ. The Resources of a firm are the amounts due
it, together with the property of all kinds belonging to
it ; called also Assets, or Effects,
HSO. The lAahilities of a firm are its debts,
781. The Net Capital is the excess of resources
over liabilities.
WJtXTTMN EXERCISES,
783. To apportion grains or losses according to
capital invested.
1. A and B engage in trade ; A furnishes $400 capital,
B 1600. They gain $250 ; what is the profit of each ?
1st operation. {By Fractions.)
$400, A/s investmeoit = y^^TT = i of ^^^ whole capital.
600, B.'s - = A'A = f "
$1000, whole "
$250 X f rz: $100, A/s share of the gain.
$250 X f = $150, B.'s '*
2d operation. {By Proportion.)
$1000 (whole cap.) : $400 (A/s in v.) : : $250 (whole gain) : A/s share.
$1000 (whole cap.) : $600 (B.'s lav.) : : $250 (whole gain) : B/s share.
3d operation. {By Percentage.)
$250 gain is y%V^ :=: 25% of the whole capital.
$400 X .25 = $100, A/s gain ; $600 x .25 = $150, B.'s gain.
Analysis. — {1st Method.) Since $400, A.'s investment, is -^-^y
or f , of the whole capital, he is entitled to f of the gain, or $lt)0 ;
and B is entitled to f of the gain, or $150.
M Method. The ratio of $1000, the whole capital, to $400, A.'s
investment, is equal to the ratio of $250, the whole gain, to A.'s
share of the gain. Hence the proportions, etc.
Sd Method. Since the gain is 25 % of the whole capital, each
partner is entitled to 25 % of his investment as his share of the gain.
The third method (hy dividend) is that generally adopted by joints
stock companies having numerous shareholders.
PARTNERSHIP. 139
2. At the end of the year, Norton, Smith & Co. take
an account of stock, and find the amount of merchandise,
as per inventory, to be $8400 ; cash on hand, $4850 ; due
from sundry persons, $5273. Their debts are found to
amount to $4223. S. Norton's investment in the busi-
ness is $5000 ; E. Smith's, $4000 ; and C. Woodward's,
$2000. Make a statement showing the resources, lia-
bihties, net capital, and net gain ; and find each part-
ner's share of the gain.
OPERATION.
Mesources.
Mdse. as per inventory, $8400
Cash on hand, 4850
Debts due the firm, • 5273
$18523
lA abilities.
Debts due to sundry persons, 4223
Net capital, .... $14300
Investments,
S. Norton, . $5000
R. Smith, 4000
C. Woodward, 2000
Total investments ..... $11000
Net gain, $3300
S. Norton's fractional part, ^\^^% = A o^ $3^^^ = $1^^0-
R. Smith's '' '' Am = XT of $3300 = $1300.
C. Woodward's ** '' A^ = A of $3300 = $ 600.
Proof.— $1500 + $1200 + $600 = $3300, total gain.
140 PAKTKERSHIP.
Eul:e 1. Find what fractional part each partner^ s in-
vestment is of tJie whole capital^ and take such part of the
tvhole gain or loss for his share of the gain or loss. Or,
2. State hy proportion, as the whole capital is to each
partner's investment t, respectively, so is the whole gain or
loss to each partner's share of the gain or loss. Or,
3. Find what per cent, the gain or loss is of the whole
capital, and taTce that per cent, of each partner's invest-
ment for his share of the gain or loss, respectively,
3. A furnishes $4000, B, $2700, and C, $2300, to pur-
chase a house, which they rent for $720. What is each
one's share of the rent ?
4. Four persons rent a farm of 230 A. 64 P. at $7|^ an
acre. A puts in 288 sheep, B, 320 sheep, C, 384 sheep,
and D, 648 sheep ; what rent ought each to pay ?
5. Prime & Co. fail in business ; their liabilities
amount to $22000 ; their available resources to $8800.
They owe A $4275, and B $2175.50 : what will each of
these creditors receive ?
6. Four persons engage in manufacturing, and invest
jointly $22500. At the expiration of a certain time, A's
share of the gain is $2000, B's $2800.75, C's $1685.25,
and D's $1014. How much capital did each put in ?
7. An estate worth $10927.60 is divided between two
heirs so that one receives ^ more than the other. What
does each receive ?
8. Three persons engage in the lumber trade with a
joint capital of $37680. A puts in $6 as often as B puts
in $10, and as often as puts in $14. Their annual gain
is equal to O's stock. What is each partner's gain ?
PARTKEESHIP. 141
9. Ames, Lyon & Co. close business in the following
condition : notes due the firm to the amount of $24843.75,
cash in hand, $42375.80, due on account, $26500, mer-
chandise per inventory, $175840. Notes against the
firm, $14058.75, due from the firm on account, $12375.80.
Ames invested $60000, Lyon, $40000, and Clark $25000.
Make a statement showing the total amount of resources,
liabilities, investments, net capital, net gain, and each
partner's share of the gain.
783. To apportion ^ains or losses according to
amount of capital invested, and time it is employed.
1. Three partners. A, B, and C, furnish capital as fol-
lows : A, $500 for 2 mo.; B, $400 for 3 mo.; C, $200 for
4 mo. They gain $600 ; what is each partner's share ?
OPERATION.
500 X 2 rr 1000 = UU = i X ) ( $200, A's share.
400 X 3 r= 1200 = iUi = f X >- $600 = J $240, B's "
200 X 4 =_800 = ^%% =^\x ) I $160, C's "
3000
Analysis.— The use of $500 for 2 mo. is the same as the use of
$1000 for 1 mo. ; the use of $400 for 3 mo. is the same as that of
$1200 for 1 mo. ; and the use of $200 for 4 mo. is the same as that
of $800 for 1 mo. Therefore the whole capital is the use of $3000
for 1 mo. ; and as A*s investment is $1000 for 1 mo., it is J of the
capital, and hence he should receive J of the gain, or $200. For
the same reason, B should receive f, and C y% of the gain, or $240
and $160, respectively.
The other methods of operation may be applied by considering
the products of investment and time as shares of the capital. Thus,
$600 is 20% of $3000; and 20% of $1000, $1200, and $800 will
give $200, $240, and $160, respectively, the shares of gain required.
142 PABTIS^EESHIP.
2. Barr, Banks & Co. gain in trade $8000. Barr fur-
nished $12000 for 6 mo.. Banks, $10000 for 8 mo., and
Butts 18000 for 11 mo. Apportion the gain ?
EuLE 1. — Multiply each partners capital iy the time
it is invested, and divide the whole gain or loss among the
partners in the ratio of these products. Or,
2. State hy projjortion: The sum of the products is to
each product, as the whole gain or loss is to each partner's
gain or loss.
3. Jan. ], 1876, three persons began business with
$1300 capital furnished by A ; March 1, B put in $1000 ;
Aug. 1, C put in $900. The profits at the end of the
year were $750. Apportion it.
4. In a partnership for 2 years, A furnished at first
$2000, and 10 mo. after withdi^ew $400 for 4 mo., and
then returned it ; B at first put in $3000, and at the end
of 4 mo. 1500 more, but drew out $1500 at the end of 16
mo. The whole gain was $3372. Find the share of each.
5. The joint capital of a company was $5400, which
was doubled at the end of the year. A put in ^ for 9 mo.,
B f for 6 mo., and C the remainder for ] year. What is
each one's share of the stock at the end of the year ?
6. Crane, Child & Coe, forming a partnership Jan. 1,
1875, invested and drew out as follows: Crane invested
$2000, 4 mo. after $1000 more, and at the end of 9 mo.
drew out $600. Child invested $5000, 6 mo. after $1200
more, and at the end of 11 mo. put in $2000 more. Coe
put in $6000, 4 mo. after drew out $4000, and at the
end of 8 mo. drew out $1000 more. The net profits for
the year were $7570. Find the share of each.
784. Alligation treats of mixing or compounding
two or more ingredients of different values or qualities.
785. Alligation Medial is the process of finding
the mean or average yalue or quality of several ingredients.
786. Alligation Alternate is the process of find-
ing the proportional quantities to be used in any required
mixture.
WRITTEN EX A MPZJES,
787. 1. If a grocer mix 8 lb. of tea worth $.60 a pound
with 6 lb. at $.70, 2 lb. at $1.10, and 4 lb. at $1.20, what
is 1 lb. of the mixture worth ?
Analysis. —Since 8 lb. of tea at $.60 is
worth $4.80, and 6 lb. at $.70 is worth
$4.20, and 2 lb. at $1.10 is worth $2.20,
and 4 lb. at $1.20 is worth $4.80, the mix-
ture of 20 lb. is worth $16. Hence 1 lb. is
worth ^\ of $16, or $16 -5- 20 = $.80.
2. If 20 lb. of sugar at 8 cents be mixed with 24 lb. at
9 cents, and 32 lb. at 11 cents, and the mixture is sold
at 10 cents a pound, what is the gain or loss on the whole ?
'RvL^.—Find the entire cost or value of the ingredients^
and divide it iy the sum of the simples.
OPERATION.
$.60
X8 =
$4.80
.70
X6 =
4.20
1.10
X3 =
2.20
1.20
x4 =
4.80
20 )
$16.00
144 , ALLIGATION.
3. A miller mixes 18 bu. of wheat at $1.44 with 6 bu.
at $1.32, 6 bu. at $1.08, and 12 bu. at $.84. What will be
his gain per bushel if he sell the mixture at $1.50 ?
4. Bought 24 cheeses, each weighing 25 lb., at If a
pound ; 10, weighing 40 lb. each, at IW ; and 4, weigh-
ing 50 lb. each, at 13^ ; sold the whole at an average
price of ^^f a pound. What was the whole gain ?
5. A drover bought 84 sheep at $5 a head ; 96 at $4.75 ;
and 130 at $5^. At what average price per head must
he sell them to gain 20^ ?
188. To find the proportional parts to be used,
when the mean price of a mixture and the prices of
the simples are given.
1. What relative quantities of timothy seed worth $2 a
bushel, and clover seed worth $7 a bushel, must be used
to form a mixture worth $5 a bushel ?
i
OPERATION. ANAiiYSis. — Since on evAry ingredient usea
2 ) whose price or quality is less than the mean
• Ans. rate there will be a gain, and on every ingre-
dient whose price or quality is greater than
the mean rate there will be a loss^ and sincp the gains and losses
must be exactly equal, the relative quantities used of each should
be such as represent the unit of value. By selling one bushel of
timothy seed worth $2, for $5, there is a gain of $3 ; and to gain $1
would require i of a bushel, which is placed opposite the 2. By
selling one bushel of clover seed worth $7, for $5, there is a loss
of $2 ; and to lose $1 would require J of a bushel, which is placed
opposite the 7.
In every case, to find the unit of value, divide $1 by the gain or
loss per bushel or pound, etc. Hence, if every time J of a bushel
of timothy seed is taken, J of a bushel of clover seed is taken, the
gain and loss will be exactly equal, and i and J will be the propor-
tional quantities required.
ALLIGATION.
145
OPERATION.
6
f
1
%
3
4
5
3
i
4
4
4
\
1
1
7
1
%
2
.10
i
3
3
To express the proportional numbers in integers, reduce these
fractions to a common denominator, and use their numerators, since
fractions having a common denominator are to each other as their
numerators (241) ; thus, \ and \ are equal to f and |, and the pro-
portional quantities are 2 bu. of timothy seed to 3 bu. of clover seed.
2. What proportions of teas worth respectively 3, 4, 7,
and 10 shillings a pound, must be taken to form a mix-
ture worth 6 shillings a pound ?
Analysis. — To preserve the equality
of gains and losses, always compare
two prices or simples, one greater and
one less than the mean rate, and treat
each pair or couplet as a separate ex-
ample. In the given example form two
couplets, and compare either 3 and 10,
4 and 7, or 3 and 7, 4 and 10.
We find that J of a lb. at 3s. must be
taken to qain 1 shilling, and \ of a lb.
at 10s. to lose 1 shilling ; also ^ of a lb. at 4s. to gain 1 shilling, and
1 lb. at 7s. to lose 1 shilling. These proportional numbers, obtained
by comparing the two couplets, are placed in columns 1 and 2. If,
now, the numbers in columns 1 and 2 are reduced to a common de-
nominator, and their numerators used, the integral numbers in
columns 3 and 4 are obtained, which, being arranged in column 5,
give the proportional quantities to be taken of each.
It will be seen that in comparing the simples of any couplet, one
of which is greater, and the other less than the mean rate, the pro-
portional number finally obtained for either term is the difference
between the mean rate and the other term. Thus, in comparing 3
and 10, the proportional number of the former is 4, which is the
difference between 10 and the mean rate 6 ; and the proportional
number of the latter is 3, which is the difference between 3 and the
mean rate. The same is true of every other couplet. Hence, when
the simples and the mean rate are integers, the intermediate steps
taken to obtain the final proportional numbers as in columns 1 , 2, 3,
and 4, may be omitted, and the same results readily found by taking
the difference between each simple and the mean rate, and placing
it opposite the one with which it is compared.
146 ALLIGATIOK.
3. In what proportions must sugars worth 10 cents,
11 cents, and 14 cents a pound be used, to form a mix-
ture worth 12 cents a pound ?
4. A farmer has sheep worth $4, $5, $6, and $8 per
head. What number may he sell of each and realize an
average price of $5^ per head ?
EuLE. — I. Write the several prices or qualities in a
column^ and the mean price or qicality of the mixture at
the left.
II. Forra couplets hy comparing any price or quality
less, with one that is greater than the mean rate, placing
the part luhich must be used to gain 1 of the mean rate
opposite the less simple, and the part that must be used to
lose 1 opposite the greater simple, and do the same for each
simple in every couplet.
III. If the proportional numbers are fractional, they
may be rediiced to integers, and if two or more stand in
the same horizontal line, they must be added ; the final re-
sults will be the proportional quantities required.
1. If the numbers in any couplet or column liave a common fac-
tor, it may be rejected.
2. We may also multiply the numbers in any couplet or column
by any multiplier we choose, without affecting the equality of the
gains and losses, and thus obtain an indefinite number of results,
any one of which being taken will give a correct final result.
5. What amount of flour worth $5^, $6, and $?| per
barrel, must be sold to realize an average price of $6^ per
barrel ?
6. In what proportions can wine worth $1.20, $1.80,
and $2.30 per gallon be mixed with water so as to form a
mixture worth $1.50 per gallon ?
ALLIGATION. 147
789. When the quantity of one of the simples is
limited.
1. A farmer has oats worth $.30, com worth $.45, and
barley worth $. 84 a bushel. To make a mixture worth
$.60 a bushel, and which shall contain 48 bu. of corn,
how many bushels of oats and barley must he use ?
OPERATION. Analysis. — By tlie
^30 JL 4 4 24^ same process as in
^^ .^ . ^ ,^ .r.\ J (788), the proportional
60^5 ^ 8 S ^slAns. quantities of each are
.84: ^ ^ 5 5 10 60 J found to be 4 bu. of
oats, 8 of corn, and 10
of barley. But since 48 bu. of corn is 6 times the proportional num-
ber 8, to preserve the equality of gain and loss take 6 times the
proportional quantity of each of the other simples, or 6 x 4 = 24 bu.
of oats, and 6 x 10 =: 60 bu. of barley. Hence, etc.
2. A dairyman bought 10 cows at $40 a head. How
many must he buy at $32, $36, and $48 a head, so that
the whole may average $44 a head ?
EuLE. — Find the proportional quajitities as in (788).
Divide the given quantity hy the proportional qua7itity of
the same ingredient, and multiply each of the other propor-
tional quantities iy the quotient thus ohtained,
3. A grocer having teas worth $.80, $1.20, $1.50, and
$1.80 per pound, wishes to form a mixture worth $1.60 a
pound, and use 20 lb. of that worth $1.50 a pound.
4. Bought 12 yd. of cloth for $30. How many yards
must I buy at $3^ and $1| a yard, that the average price
of the whole may be $2| a yard ?
5. How many acres of land worth $70 an acre must be
added to a farm of 75 A., worth $100 an acre, that the
average value may be $80 an acre.'^
148
ALLIGATION.
OPEBATION
r 6
i
3
3
37
10 <
7
13
i
3
3
3
3
18
37
.13
i
4
4
36
13
108
790. When the quantity of the whole compound
is limited.
1. A grocer has sugars worth 6 cents, 7 cents, 12 cents,
and 13 cents per pound. He wishes to make a mixture
of 108 pounds, worth 10 cents a pound ; how many
pounds of each kind must be use ?
Analysis. — The proportion-
al quantities of each simple
found by (788) are 3 lb. at
6 cts., 2 lb. at 7 cts., 3 lb. at 12
cts., and 4 lb. at 13 cts. Add-
ing the proportional quantities,
the mixture is but 12 lb ,
while the required mixture is
108, or 9 times 12. If the
whole mixture is to be 9 times as much as the sum of the propor-
tional quantities, then the quantity of each simple used must be 9
times as much as its respective proportional, or 27 lb. at 6 cts.,
18 lb. at 7 cts., 27 lb. at 12 cts.. and 36 lb. at 13 cts.
2. A man paid $330 per week to 55 laborers, consisting
of men, women, and boys; to the men he paid $10 a
week, to the women $2 a week, and to the boys $1 a week ;
how many were there of each ?
EuLE. — Find the proportional numbers as in (788).
Divide the given quantity ly the sum of the proportional
quantities^ and multiply each of the proportional quanti-
ties ly the quotient thus obtained.
3. How much water must be mixed with wine worth
$.90 a gallon, to make 100 gal. worth $.60 a gallon ?
4. One man and 3 boys received $84 for 56 days' labor ;
the man received $3 per day, and the boys %\, $|, and
$lf respectively ; how many days did each labor ?
REVIEW.
149
791.
RATIO.
PROPOR-
TION.
SYNOPSIS FOR EEVIEW.
1. Ratio. 2. Sign of Ratio. 3. Terms.
4. Antecedent. 5. Consequent. 6.
1. Defs. ^ Value of a Ratio. 7. Simple Ratio.
8. Compound Ratio. 9. Reciprocal of
a Ratio.
2. Formulas, 1, 2, 3.
3. General Principles, 1, 2, 3.
4. General Law.
1. Defs. <
'' 1. Proportion. 2. Sign. 3. Couplet.
4 Proportional. 5. Antecedents. C.
Consequents. 7. Extremes. 8. Means,
9. Mean Proportional.
2. Principles, 1, 2, 3, 4.
3. Simple Pro-
portion.
4. Compound
Proportion, i
.■U:
... Simple Proportion,
' " Statement,
2. Rule, I, II.
3. Cause and Effect,
t 4. Rule, I, II.
" 1. Def. Compound Proportion.
2. Rule, I, II.
3. Cause and Effect.
4. Rule, I, II.
PARTNER.
SHIP.
ALLIGA-
TION. ^
{1. Partnership. 2. Firm, Company, or
House. 3. Capital. 4. Resources.
5. Liabilities. 6. Net Capital.
2. 782. Rule, 1, 2, 3.
3. 783. Rule, 1, 2.
1. Defs.
■]'■
Alligation. 2. Alligation MediaL
3. Alligation Alternate.
2. 787. Rule.
3. 788. Rule, I, II, III.
4. 789. Rule.
5. 790. Rule.
150 GENERAL REVIEW.
TEST PROBLEMS.
792. 1. The sum of two numbers is 120, and their dif-
ference is equal to ^ of the greater. Find the numbers.
2. E's age is H times the age of D, and F's age is 2^
times the age of both, and the sum of their ages is 124.
What is the age of each ?
3. If 7 bu. of wheat are worth 10 bu. of rye, and 5 bu.
of rye are worth 14 bu. of oats, and 6 bu. of oats are
worth $6, how many bushels of wheat will $60 buy?
4. A mechanic was engaged to labor 20 days, on the
conditions that he was to receive 15 a day for every day
he worked, and to forfeit $2 a day for every day he was
idle ; at the end of the time he received $86. How many
days did he labor?
5. One man can build a fence in 18 da., working 10 lir.
a day ; another can build it in 9 da., working 8 hr. a day.
In how many days can both together build it, if they
work 6 hours a day ?
6. If 6 boxes of starch and 7 boxes of soap cost $33,
and 12 boxes of starch and 10 boxes of soap cost $54, what
is the price of 1 box of each?
7. Three men agree to build a bam for $540. The first
and second can do the work in 16 da., the second and
third in 13 J^ da., and the first and third in llf da. In
how many days can all do it working together ? In how
many days can each do it alone ? What part of the pay
should each receive ?
8. A dealer paid $182 for 20 barrels of flour, giving $10
for first quality, and $7 for second quality. How many
barrels were there of each ?
GENERAL REVIEW. 151
9. The hour and minute hands of a clock are together
at 12 M. When will they be exactly together the third
time after this ?
10. Bought 15 bu. of wheat and 30 bu. of oats for $35,
and 9 bu. of wheat and 6 bu. of oats for $15. What was
the price per bushel of each ?
11. If Ames can do as much work in 3 days as Jones
can do in 4^ days, and Jones can do as much in 9 days as
Smith can do in 12 days, and Smith as much in 10 days
as Eay in 8 days, how many days' work done by Kay are
equal to 5 days done by Ames ?
12. A merchant bought 40 pieces of cloth, each piece
containing 25 yd. , at $4f per yard, on 9 mo. credit, and
sold the same at $4f per yard, on 4 mo. credit. Find his
net cash gain, money being worth 6%.
13. There are 70 bu. of grain in 2 bins, and in 1 bin
are 10 bu. less than f as much as there is in the other.
How many bushels in the larger bin ?
, 14. Three men can perform a piece of work in 12 hr. ;
A and B can do it in 16 hr., A and C in 18 hr. What
part of the work can B and C do in 9\ hours ?
15. What per cent, in advance of the cost must a mer-
chant mark his goods, so that after allowing 6% of his
sales for bad debts, an average credit of 6 mo., and 4:% of
the cost of the goods for his expenses, he may make a
clear gain of 12|^^ on the first cost of the goods, money
being worth 7% ?
16. An elder brother's fortune is 1^ times his younger
brother's ; the interest of J of the elder brother's fortune
and -J- of the younger's for 5 years, at 6^, is $2400. What
is the fortune of each ?
152 GENERAL REVIEW.
17. The top of Trinity Church steeple in New York is
368 ft. from the ground; f the height of the steeple
above the church plus 12 ft. is equal to the height of the
church. Find the height of the steeple above the church ?
18. Two persons have the same income : A saves J of
his, but B by spending $300 a year more than A, at the
end of 2 years is $200 in debt. What is their income ?
19. Divide $2520 among 3 persons, so that the second
shall have f as much as the first, and the third ^ as much
as the other two. What is the share of each ?
20. A man owes a debt to be paid in 4 equal install-
ments of 4, 9, 12, and 20 months respectively ; a discount
of 6% being allowed, he finds that $1500 ready money will
pay the debt. What is the amount of the debt ?
21. A quantity of flour is to be distributed among some
poor families ; if 50 lb. are given to each family, there
will be 6 lb. left ; if 51 lb. are given to each, there will be
wanting 4 lb. What is the quantity of flour ?
22. 1 have three notes payable as follows : one for $400,
due Jan. 1, 1875 ; another for $700, due Sept. 1, 1875 ;
and another for $1000, due April 1, 1876. What is the
average of maturity ?
23. An estate worth $123251.82 is left to four sons,
whose ages are 19, 17, 13, and 11 years, respectively, and
is to be so divided that each part being put out at 7%
simple interest, the amounts shall be equal when they
become 21 years of age. What are the parts ?
24. If a piece of silk cost $1 .20 a yard, at what price
must it be marked that it may be sold at 10^ less than
the marked price, and still make a profit of 20^ ?
25. A farmer sold 100 geese and turkeys ; lor the geese
GENERAL EEVIEW. 153
he received $.75 apiece, and for the turkeys $1.25 apiece,
and for the whole $104. What was the number of each ?
26. A man left his property to three sons ; to A ^ want-
ing $180, to B i, and to C the rest, which was $590 less
than A and B received. What was the whole estate ?
27. What is the simple interest and the amount ; the
compound interest and amount ; the present worth and
the true discount ; the bank discount and the proceeds
of $1920, for 2 yr. 5 mo. 12 da., at 6% ? Also the face of
the note, which when discounted at a bank for the same
time, and at the same rate, will produce the same sum ?
28. Divide $1500 among 3 persons, so that the share
of the second may be ^ greater than that of the first, and
the share of the third ^ greater than that of the second.
29. A merchant owes for three bills of goods as follows :
$500 due March 1, $800 due June 1, and $600 due Aug. 1.
He wishes to give two notes for the amount, one for $1000,
payable April 1 ; what must be the face, and when the
maturity, of the other ?
30. A man in New York purchased a draft on Chicago
for $10640, drawn at 60 da., $10302.18. What was the
course of exchange ?
31. B. B. Northrop, through his broker, invested a
certain sum in U. S. 6's, 5-20, at 107|^, and twice as much
in U. S. 5's of '81, at 98|, brokerage on each, ^%. His
income from both investments is $1674. How much did
he invest in each kind of stock ?
32. A, B, and C are under a joint contract to furnish
6000 bu. of corn, at $.48 a bushel ; A's corn is worth $.45,
B's $.51, and C's $.54 ; how many bushels must each put
into the mixture that the contract mav be fulfilled ?
154 GEKERAL REVIEW.
33. A cask contains 42J U. S. gallons of wine, worth
$4|- per gallon. How much less will it cost in U. S.
money, at the rate of £1 2s. per the Imperial gallon ?
34. A garden 400 ft. long and 300 ft. wide has a walk
20 ft. wide laid off from each of its two sides. What is
the ratio between the area of the walk and the area of
what remains ?
35. A commission merchant in Charleston received into
his store on May 1, 1875, 1000 bbl. of flour, paying as
charges on the same day, freight $175.48, cartage $56.25,
and cooperage $8.37. He sold out the shipment as fol-
lows: June 3, 200 bbl. @ $6.25 ; June 30, 350 bbl. @
$6.50 : July 29, 400 bbl. @ $6.12|^ ; Aug. 6, 50 bbl. @ $6.
Eequired, the net proceeds, and the date when they shall
be accredited to the owner, allowing commission at 3^%,
and storage at 2 cents per week per bbl.
"^ 36. Three men engage in manufacturing. L puts in
$3840 for 6 mo. ; M, a sum not specified for 12 mo. ; and
N, $2560 for a time not specified. L received $4800 for
his capital and profits ; M, $9600 for his ; and N, $4160
for his. Eequired, M's capital and N's time.
37. My expenditures in building a house, in the year
1874, were as follows : Jan. 16, $536.78 ; Feb. 20, $425.36 ;
March 4, $259.25 ; April 24, $786.36. At the last date I
sold the house for exactly what it cost, interest at 6 per
cent, on the money expended added, and took the
purchaser's note for the amount. What was the face of
the note ?
38. A man bought a farm for $6000, and agreed to pay
principal and interest in 3 equal annual installments.
What was the annual payment, interest being 6% ?
ORAL BXEMCISES .
793. 1. What is the product of 3 used twice as a
factor ?
2. What is the product of 3 used 3 times as factor .
3. What is the product of 4 used 3 times as a factor ?
4. What is the result of using 5 twice as a factor ?
5. What is the product of -^ used twice as a factor ?
6. What is the result of using f twice as a factor?
f , three times as a factor ?
7. What number will be produced by using .3 twice as
a factor? .7, twice ? A, three times ? .05, twice ?
8. A room is 9 ft. on each side ; how many square feet
in the floor ?
9. A cubical block of stone is 4 ft. on each edge ; how
many cubic feet does it contain ?
DEFINITIONS.
794. A Power of a number is the product of factors,
each of which is equal to that number. Thus, 27 is the
third power of 3, since 27 = 3x3x3.
795. Involution is the process of finding any power
of a number.
156
INVOLUTION.
796. The JBase or Root of a power is one of the
equal factors of the power. Thus, 27 is the third power
of 3, and 3 is the base, or root, of that power.
797. The ExiJOiient of a power is a number placed
at the right of the base and a little above it, to show how
many times it is used as a factor to produce the power.
It also denotes the degree of the power. Thus,
3^ or 3 = 3, the^r^^ power of 3.
32 z= 3 X 3 =9, the second power of 3.
33 zzz 3 X 3 X 3 = 27, the third power of 3.
34 1= 3 X 3 X 3 x3 = 81, the/o^^r^Apowerof3.
3-^ = 3x3 = 9
^
^^I
\ -^ -^^5^
^
iii;
,:',.;r|!'ll
III
'itll'iiiSi
ililHili^illMili
ii!|iV"'.; "
miiiiyiiiumimi
8» = 3x8x3 = 27
798. The Square of a num-
ber is its second power, so called
because the number of superficial
units in a square is equal to the
second power of the number of
linear units in one of its sides.
799. The O^ft^ of a num-
ber is its third power, so
called because the number of
units of Yolume in a cube is
equal to the third power of
the number of linear units
in one of its edges.
800. A Perfect Power is a number which can be
resolved into equal factors. Thus, 25 is a perfect power
of the second degree, and 27 is a perfect power of the
third degree.
IN^VOLUTION. 157
801, Pkin^ciple. — The sum of the exponents of two
powers of the same number is equal to the exponent of the
product of those powers. Thus, 2^ x 2^=25 ; for 22=:2 x 2,
and 2^=2 x 2 x 2 ; hence 2^ x 23=2 x 2 x 2 x 2 x 2=25.
WRITTEN JEXEHCISES.
803. To find any power of a number.
1. Find the third power of 35.
OPERATION. Anai^tsis. — Since using
35 = 351 ; 35 X 35 = 35^ =z 1225 any number three times
1225 X 35 = 353 = 42875 Z \ ^^^"^^ ^jl^T' *^^
third power of that num-
ber (797), 35 X 35 X 35 :rr 353 = 42875.
2. Find the square of 37. Of 42.' Of 56. Of 75.
3. Find the cube of 15. Of 18. Of 42. Of 54.
4. What is the value of 63^ ? of 48^ ? of 32^ ? of 12^ ?
EuLE. — Find the product of as many factors^ each
equal to the given number y as there are units in the expo-
nent of the required power.
5. What is the third power of f ?
r^ /^x2 . ^ d 4x4x4 43 64
Operation.— (If = f x f x | = ^—^—z = -^ = :r~,
^^ ^ * * 5x5x5 5^ 125
EuLE. — A fraction may be raised to any power by in*
volving each of its terms separately to the required power.
6. What is the square of ^^^ ? The cube of |f ?
7. Eaise ^ to the 4th power. 2|^ to the 5th power.
Find the required power of the following :
8.
35.42.
13.
.03438.
16.
(182i)2.
9.
1063.
13.
.5«.
17.
(4.07i)2,
10.
(44i)2.
14.
36.03».
18.
(1t^)=-
11.
(H)*.
15.
.403163
19.
.00638.
158.
INVOLUTION.
Find the value of each of the following expressions ;
20. 4.63 X 253.
21. 6.754 -(7^)2.
22. -Jof(i)3x(3|)2.
26. (43x56x123)-^
23. 8« -f- .4096.
24. 2.53x(12|)2.
25. (7.5)3 _^ (1^)3.
(42x104x32).
FORMATION OF SQUARES AND CUBES BY THE ANALTT
ICAL METHOD.
803. To find the square of a number in terms of
its tens and units.
1. Find the square of 27 in terms of its tens and units.
OPERATION.
27 =
27 =
20 + 7
20 + 7
189= (20x7)+ 72
540= 202 +(20x7)
729 = 202+(2x20x7)+72
Analysis. — The product of 20
+ 7 by 7 is 20 X 7 + 7\ and the
product of 20 + 7 by 20 is 20^ + (20
x7); hence 202 + (2x20x7) + 72,
which is the sum of these partial
products, is the square of 20 + 7
or 27.
Principle. — The square of a number consisting of tens
and units is equal to the sum of the squares of the tens
and units increased iy twice their product.
Geometrical Illustration.
Let ABCD be a square, each side
of which is 27 feet, and let lines be
drawn as represented in the figure. It
is evident that the square ABCD (27^)
is equal to the sum of two squares, one
of which is the square of tens (20^), the
other the square of the units (7^), to-
gether with two rectangles each of
whose areas is 20 x 7.
INVOLUTION. 159
2. What is the square of 37 ?
2 X 30 X 7 = 420
7'=r 49
372 = 1369 (803, Prin.)
3. Find the square of 42 in terms of its tens and units.
In like manner find the square
4. Of 48.
5. Of 56.
6. Of 98.
7. Of 125.
8. Of 105.
9. Of 225.
10. Of 197.
11. Of 342.
804. To find the cube of a number in terms of
its tens and units.
1. Find the cube of 25 in terms of its tens and units.
OPERATION.
252= 202+ (2x20x5) + 58
25 = 20 + 5
252x 5 = (202x5) + (2x20x52) + 53
25^x20 zir203 + (2x20^x5)+ (20x5^)
253 = 203+{3 X 202 x5) + (3x 20x52) + 53
Analysis. —The square of 35 is 20^ 4- (2 x 20 x 5) + 5^. (803, Prin.)
Multiplying this by 20 -»- 5 gives the cvbe of 25.
2. Find the cube of 34 in terms of its tens and units.
Principle. — The cube of a number consisting of tens
and units is equal to the cube of the tens, plus three times
the product of the square of the tens by the units, plus
three times the product of the tens by the square of the
units, plus the cube of the units.
160
INVOLUTIOIT.
Geometrical Illustration.
Fig. 1.
The volume of the
cube marked A, Fig. 1 ,
is 20^ ; the volume of
each of the rectangu-
lar solids marked B is
20 X 20 X 5, or 20-^ x 5 ;
the volume of each of
the rectangular solids
marked C, in Fig. 2, is
B 20 X 5 X 5, or 20 x 5^ ;
|B and the volume of the
small cube marked D
is 5^. It is evident,
that if all these solids
are put together ais
represented in Fig. 3,
a cube will be formed,
each edge of which
is 25.
3. Find the cube
of 46?
OPERATION.
403= 64000
402x6x3 = 28800
40x62x3= 4320
63= 216
463=97336
In like manner
find the cube
4. Of 48.
5. Of 64.
6. Of 95.
7. Of 125.
805. 1. What are the two equal factors of 25 ? 36 ?
2. What are the three equal factors of 27 ? 64 ? 125 ?
3. What are the four equal factors of 16 ? 81 ? 256 ?
4. Of what is 81 the 2d power ? The 4th power ?
DBFUnTITIONS.
806. The Square Hoot of a number is one of the
two equal factors of that number ; the Cube Root is
one of the three equal factors of that number, etc.
Thus, 3 is the square root of 9, 2 is the cube root of 8, etc.
807. Evolution is the process of finding the root
of any power of a number.
808. The Madical Sign is V. When prefixed to
a number, it indicates that some root of it is to be found.
809. The Index of the root is a small figure placed
aboye the radical sign to denote what root is to be found.
When no index is written, the index 2 is understood.
Thus, /\/T00 denotes the square root of 100 ; \/\2^ denotes the
cube root of 125 ; v^256 denotes the fourtJi root of 256 ; and so on,
Evolution, or both involution and evolution, may be indicated in
the same expression by a fractional exponent, the numerator de-
noting the required power of the given number, and the denomina-
tor the root of that power of the number. Thus,
Oi is equivalent to y^O ; 643 , to /^64 ; and 8f , to the cube root
of the second power of 8, equivalent to >y/8^, etc.
163 EVOLUTION".
EVOLUTION BY FACTORING.
WJRITTJSN EXERCISES,
810. To find any root of a number by factoring.
1. Find the cube root of 1728.
OPERATION.
3)1728
Q \ K 7 «
-^ Analysis. — A number that is a perfect cube, is
3)192 composed of three equal factors, and one of them
TTTT is the cube root of that number.
^y^ The prime factors of 1728 are 3, 3, 3, 2, 2, 2,
2)3 2 2, 2, 2 ; hence 1728 = (3 x 2 x 2) x (3 x 2 x 2) x
aTTfi (3x2x2); therefore the cube root of 1728 is
<-— (3 X 2 X 2), or 12.
2)8
2)4
2
EuLE. — Resolve the given number into its prime factors ;
then, to produce the square root, take one of every two equal
factors ; to produce the cube root take one of every three
equal factors ; and so on,
2. Find the square root of 64. Of 256. Of 576. Of 6561.
3. Find the cube root of 729. Of 2744. Of 9261. Of 3375.
GENERAL METHOD OF SQUARE ROOT.
811. A Perfect Square is a number which has
an exact square root.
813. Pkinciples. — 1. The square of a number ex-
pressed by a single figure contains no figure of a higher
order than tens,
2. The square of tens contains no significant figure of a
lower order than hundreds, nor of a higher order than
thousands.
SQUARE ROOT. 163
3. The square of a number contains twice as many
figures as the number, or twice as many less one. Thus,
12 = 1, 102 =: 100,
92 = 81, 1002 = 10000,
992 = 9801, 10002 z= 1000000.
Hence,
4. If any perfect square be separated into periods of two
figiores each, beginning with units^ place, the number of
periods will be equal to the number of figures in the square
root of that number.
If the number of figures in the number is odd^ the left-hand
period will contain only one figure.
WniTTEN X:XEItCISE8.
813. To find the square root of a number.
1. Find the square root of 4356.
OPERATION. Analysis.— Since 4356 con-
43,56(60 + 6 = 66 si^ts of two periods, its
gQ2 3600 square root will consist of
two figures (812, Prin. 4).
130 + 6 = 126 ) 756 Since 56 cannot be a part of
756 tlie square of the tens (812,
Prin. 2), the tens of the root
must be found from the first period 43.
The greatest number of tens whose square is contained in 4300
is 6. Subtracting 3600, which is the square of 6 tens, from the
given number, the remainder is 756. This remainder is composed
of twice the product of the tens by the units, and the square of the
units (803, Prin.). But the product of tens by units cannot
be of a lower order than tens ; hence the last figure 6 cannot be a
part of twice the product of the tens by the units ; this double
product must therefore be found in the part 750.
Now, if we double the tens of the root and divide 750 by the
result, the quotient 6 will be the units' figure of the root, or a
164 EVOLUTION.
figure greater thaD the units' figure. This quotient cannot be too
small, for the part 750 is at least equal to twice the product of the
tens by the units ; but it may be too large, for the part 750, be-
sides the double product of the tens by the units, may contain tens
arising from the square of the units. (812, Prin. 1.) Subtracting
6 X 120 + 6* or 6 x 120 + 6 from 756, nothing remains. Hence 66 is
the required root.
1. In this example, 120 is a partial or trial divisor, and 126 is a
complete divisor.
2. If the root contains more than two figures, it may be found by
a similar process, as in the following example, where it will be
seen that the partial divisor at each step is obtained by doubling
that part of the root already found.
2. Find the square root of 186634.
OPERATION.
18,66,24(400 + 30 + 2=432
16 00 00 nru ' X. *!, . W
The ciphers on the right
400 X 2 + 30 = 830 ) 2 Q^ 24 are usually omitted for the
2 49 00 sake of brevity. Thus,
400x2 + 30x2 + 2=862)1724 18,66,24(432
1724 16
83)266, etc.
3. Find the square root of 7225.
4. What is the square root of 58564.
Rule. — I. Separate the given number into periods of two
figures each, beginning at the units' place.
II. Find the greatest number whose square is contained
in the period on the left ; this will be the first figure in the
root. Subtract the square of this figure from the period on
the left, and to the remainder annex the next period to form
a dividend.
SQUARE ROOT.
165
III. Divide this dividend, omitting the figure on the
right, iy double the part of the root already found, and
annex the quotient to that part, and also to the divisor ;
then multiply the divisor thus completed ly the figure of
the root last obtained, and subtract the product from the
dividend.
IV. // there are more periods to be brought down, con-
tinue the operation in the same manner as before.
1. If a cipher occur in the root, annex a cipher to the trial divi-
sor, and another period to the dividend, and proceed as before.
2. If there is a remainder after the root of the last period is
found, annex periods of ciphers and continue the root to as many
decimal places as are required.
Find the square root
5. Of 9G04.
7. Of 11881.
9. Of 2050624.
6. Of 13225.
8. Of 994009.
10. Of 29855296.
11. Find the square root of IfJ.
Operation. — \/Hi
-_vioo_,„
V121
EuLE. — Tlie square root of a fraction may be found by
extracting the square root of the numerator and denomina-
tor separately.
Mixed numbers may be reduced to the decimal form before ex-
tracting the root ; or, if the denominator of the fraction is a perfect
square, to an improper fraction.
In extracting the square root of a number containing a decimal^
begin at the units' place, and proceed both toward the left and the
right to separate into periods, then proceed as in the extraction of
the square root of integers.
Extract the square root
12.
Of iff.
15.
Of .001225.
18.
Of 58.1406|»
13.
Ofi^T-
16.
Of 196.1369.
19.
Of 17f.
14.
OfyHir-
17.
Of 2.251521.
20.
Of 10795.21.
L66
EVOLUTIOK.
21. What is the square root of 3486784401 ?
22. What is the square root of 9.0000994009 ?
23. Find the value of 32^ to 6 decimal places.
24. Find the square root of 2f to 4 decimal places.
25. Find the square root of f to 5 decimal places.
26. Find the value of .1254 to 5 decimal places.
Find the second member of the following equations :
27. a/3369 + Vi296= ? | 28. (36^)^ x a/."25^= ?
29. 2.83 -^ a/. 11 7649 zz:?
30. vnu-^m^-^^=?
31.
9
,i
1= X
\/32 a/92
32. 'v/^:6896 + .3729 x| of a/.256=: ?
33. (7.2 - A/277or)' -r- (|f = ?
34. (a/8T— 16*) X (a/T69 + 25*) = ?
35. a/2642 X 4.41 -t- (5.3361)* - (2.3^ x
Geometrical Explanation of Square Eoot.
814. What is the length of one side of a square whose
area is 729 square feet ?
Fia. 1. Let Fig. 1 represent a square whose area
is 729 square feet. It is required to find the
length of one side of this square.
Since the area of a square is equal to the
square of one of its sides, a side may be
found by extracting the square root of the
area.
Since 729 consists of two periods, its square
root will consist of two figures. The great-
est number of tens whose square is con-
tained in 700 is 2. Hence the length of the side of the square ia
20 feet plus the units' figure of the root.
Fig. 2.
^^^^^^1
B
jiiiii
§H
SQUARE ROOT. 167
Removing the square whose side is 20 feet and whose area is 400
square feet, there remains a surface whose
area is 329 square feet (Fig. 2). ^i'his re-
mainder consists of two equal rectangles,
each of which is 20 feet long, and a square
whose side is equal to the width of each
rectangle.. The units' figure of the root
is equal to the width of one of these
rectangles.
The area of a rectangle is equal to the
product of its length and width (462) ;
hence, if the area be divided by the length,
the quotient will be the width. Now, since the two rectangles
contain the greater portion of the 329 square feet, 2 x 20 or 40,
the length of the two rectangles, may be used as a trial divisor to
find the width. Dividing 329 by 40, the quotient is 8. But this
quotient is too large for the width of the rectangles, for if 8 feet is
the width, the area of Fig. 2 will be 40 x 8 + 8^ or 384 square feet.
Taking 7 feet for the width of the rectangles, the area of Fig. 2 is
40 X 7 + 72 or 329 square feet. Hence 20 + 7 or 27 feet is the length
of a side of the square whose area is 729 square feet.
PROBIjEMS,
815. 1. A square field contains 1016064 square feet.
What is the length of each side ?
2. A square farm contains 361 A. Find the length of
one side.
3. A field is 208 rd. long and 13 rd. wide. What is the
length of the side of a square containing an equal area ?
4. If 251 A. 65 P. of land are laid out in the form of a
square, what will be the length of each of its sides ?
5. A circular island contains 21170.25 P. of land. What
is the length of the side of a square field of equal area ?
6. If it cost $312 to enclose a field 216 rd. long and
24 rd. wide, what will it cost to enclose a square field
of equal area with the same kind of fence ?
168 EVOLUTION.
CUBE ROOT.
816. A Perfect Cube is a number which has an
exact cube root.
817. The Cube Hoot of a numbei* is one of the three
equal factors of that number. Thus^ the cube root of 125
is 5, since 5x5x5 = 125.
818. Pkii^ciples. — 1. The cuie of a mimier expressed
by a single figure contains no figure of a higher order than
hundreds.
2. The cube of tens contains no significant figure of a
tower order than thousands, nor of a higher order than
hundred thousands.
3. Tlie cube of a number contains three time» as many
figures as the number, or three times as many, less one or
two. Thus,
Pzzi 1 103= 1^000
3» = 27 lOQS = 1,000,000
93 = 729 10003 = 1,000,000,000
99s = 907,299 100003 = 1,000,000,000,000
4. If any perfect cube be separated into periods of three
figures each, beginning with units' place, the number of
periods will be equal to the number of figures in the cube
root cf that number.
WRITTEN EXJEHC I S^S,
819. To find the cube root of a number.
1. Find the cube root of 405224.
OFERATIOKT.
405,224 ( 70 + 4 ^ 74, cube root.
70^ = 343 000
702 X 3 = 14700 ) 62 224
743= 405 224
CUBE ROOT. 169
Analysis. — Since 405224 consists of two periods, its cube root
will consist of two figures (818, Prin. 4), Since 224 cannot be a
part of the cube of the tens of the root (818, Prin. 2), the first
figure of the root must be found from the first period, 405. The
greatest number of tens whose cube is contained in 405000 is 7.
Subtracting the cube of 7 tens from the given number, the remain-
der is 62224. This remainder is equal to the product of three timea
the square of the tens of the root by the units, plus three times the
product of the tens by the square of the units, plus the cube of the
units (804, Prin.) But the product of the square of tens by units
cannot be of a lower order than hundreds (818, Prin. 2) ; hence
the number represented by the last t\vo figures, 24, cannot be a part
of three times the product of the square of the tens of the root by
the units ; the triple product must therefore be found in the part
62200. Hence, if 62200 be divided by 3 x 70^ the quotient, which
is 4, will be the units' figure of the root or a figure greater than the
units' figure. Subtracting 74^ from the given number, the result
is ; hence 74 is the required root.
Instead of cubing 74, the parts which make up the remainder
62224 may be formed and added thus :
3 X 702 X 4 = 58goO
3 X 70 X 42 = 3360
48 = 64
62224/
Or, since 4 is a common factor in the three parts which make up
the remainder, these parts may be combined thus :
3 X 702 = 14700
3 X 70 X 4 = 840
42= 16
15556x4=62224.
1 In this example, 14700 is a partial or trial divisor, and 15556 is
a complete divisor,
2. If the cube root contains more than two figures, it may be
found by a similar process, as in the following example, where it
will be seen that the partial divisor at each step is equal to thre©
times the square of that part of the root already found.
170
EVOLUTION.
2. Find the cube root of 12812904.
OPERATION.
2003==
1st par. divisor 3 x 200' == 120000
3x200x30^ 18000
30^ =r 900
1st complete divisor 138900
3d par. divisor 3 x 280^ = 1 58700
3 X 230 X 4.= 2760
42= 16
Cube Root
12,812,904 ( 200 + 30 + 4==234
8 000 000
4 812 904
4167000
645904
645904
2d comilete divisor 161476
The operation may be abridged as follows :
12,812,904(234
2^= 8
1st partial divisor 3 X 20^ =1200
3x20x3= 180
3'^= 9
1st complete divisor
1389
2d par. divisor
3x230* =158700
3x230x4= 2760
42= 16
2d complete divisor
161476
4812
4167
645904
645904
EuLE. — I. Separate the given number into periods of
three figures each, beginning at the units'^ place.
II. Find the greatest number whose cube is contained in
the period on the left ; this will be the first figure in the
root. Subtract the cube of this figure from the period on
the left, and to the remainder annex the next period to
form a dividend,
III. Divide this dividend by the partial divisor, ivhich
is 3 times the square of the root already found, considered
as tens ; the quotient is the second figure of the root.
CUBE ROOT. 171
IV. To the partial divisor add 3 times the product of the
second figure of the root by the first considered as tens, also
the square of the second figure, the result will be the com-
plete divisor.
V. Multiply the complete divisor by the second figure of
the root and subtract the product from the dividend,
VI. // there are more periods to be brought down, pro*
ceed as before, using the part of the root already found,
the same as the first figure in the previous process.
1. If a cipher occur in the root, annex two ciphers to the trial
divisor, and another period to the dividend ; then proceed as before,
annexing both cipher and trial figure to the root.
2. If there is a remainder after the root of the last period is found,
annex periods of ciphers and proceed as before. The figures of the
root thus obtained will be decimals.
What is the cube root
3. Of 15625 ?
4. Of 166375 ?
5. Of 1030301 ?
6. Of 4492125 ?
7. Of 1045678375 ?
8. Of 4080659192 ?
9. Find the cube root of ^.
Operation.— ^A = J^-A = |.
EuLE. — The cube root of a fraction may be found by
extracting the cube root of the numerator and denominator.
In extracting the cube root of decimal numbers, begin at the
units' place and proceed both toward the left and the right, to
separate into periods of three figures each.
Extract the cube root
14. Of .091125.
15. Of 12.812904.
10. Of if^f 12. Of ^.
11. Of flfff. 13. Of 39304.
16. What is the cube root of 98867482624 ?
17. What is the cube root of .000529475129 ?
18. Find the cube root of ^ correct to 4 decimal places,
172
EVOLUTION.
Find the second member of the following equations
19. 1.443+2.53==? 21.
22.
^0. ^iifix^ifi==?
V.4096 — .2368 = ?
^^'54:872 — (21.952)*=?
23. 24.8 + v^l03.823 x (.125)i = ?
U. V^166 -r- \/6i - (4 X ^^."512) = ?
Geometrical Explanation of Cube Root.
830. What is the length of the edge of a cube whose
volume is 15625 cubic feet ?
^i^- 1- Let Fig. 1 represent a
cube whose volume is
15625 cubic feet. It is
required to find the length
of the edge of this cube.
Since the volume of a
cube is equal to the cube
of one of its edges, an
edge may be found by
extracting the cube root
of the volume.
Since 15625 consists of
two periods, its cube root
will consist of two figures. The greatest number of tens whose
cube is contained in 15000 is 2. Hence, the length of the edge of
the cube is 20 feet plus the units' figure of the root. Removing the
cube whose edge is 20 feet and whose volume is 8000 cubic feet,
there remains a solid whose volume is 7625 cubic feet (Fig. 2).
This remainder consists of solids similar to those marked B, C, and
D, in Fig. 1 and Fig. 2 of Art. 804.
15,625(25
2^= 8
8 X 202 = 1200
3 X 20 X 5 r= 300
52= 25
7625
1525 7625
CUBE ROOT.
173
The volume of a rectan- FiG. 2.
gular solid is equal to the
product of the area of its
base by its height or thick-
ness (472) ; hence, if the
volume be divided by the
area of the base the quo-
tient will be the thickness.
Now, since the three equal
rectangular solids, each
of which is 20 feet square
and whose thickness is the
units' figure of the root,
contain the greater por-
tion of the 7625 cubic feet, 3 x 20^ or 300 x 2^ may be used as a trial
divisor to find the thickness. Dividing 7625 by 1200 the quotient
is 6. But this quotient is too large, for if 6 feet is the thickness,
the volume of Fig. 2 will be 3x202x6 + 3x20x62 + 63, or 9576
cubic feet. Taking 5 feet for the thickness, the volume of Fig. 2
is 7625 cubic feet, for 3 x202 x 5 + 3 x 20 x52 + 5'=:(300x 2^ + 30x2
X 5 + 52) 5 = 1525 X 5=7625. Hence, 25 feet is the length of the edge
of a cube whose volume is 15625 cubic feet.
PJtOBTjEMS,
831. 1. What is the length of the edge of a cubical
box that contains 46656 cu. inches ?
2. What must be the length of the edge of a cubical
bin that shall contain the same volume as one that is
16 ft. long, 8 ft. wide, and 4 ft. deep ?
3. What are the dimensions of a cube that has the
same volume as a box 2 ft. 8 in. long, 2 ft. 3 in. wide, and
1 ft. 4 in. deep ?
4. How many square feet in the surface of a cube
whose volume is 91125 cubic feet ?
5. What is the length of the inner edge of a cubical
bin that contains 150 bushels ?
174 EVOLUTION.
6. What is the depth of a cubical cistern that holds
200 barrels of water ?
7. Find the length of a cubical vessel that will hold
4000 gallons of water.
ROOTS OF HIGHER DEGREE.
833. Any root whose index contains no other factors
than 2, or 3, may be extracted by means of the square and
cube roots.
If any power of a given number is raised to any required power,
the result is that power of the given number denoted by the pro-
duct of the two exponents. (801.) Conversely, if two or more
roots of a given number are extracted, successively, the result is
that root of the given number denoted by the product of the indices.
1. What is the 6th root of 2176782336 ?
OPERATION. Analysis. — The index of the re-
V^2176782336 = 46656 ^^^^^^ ^^^* is 6 = 2 x 3 ; hence ex-
3 tract the square root of the given
V 46656 = 36 number, and the cube root of this
Or, result, which gives 36 as the 6th or
-^2176783336 = 1396 '^^li^ed "^t-^ O^, first find the
cube root of the given number, and
V 1296 = 36 then the square root of the result.
KuLE. — Separate the index of the required root into its
prime factors, and extract successively the roots indicated
hy the several factors obtained; the final result will ie the
required root.
2. What is the 4th root of 5636405776 ?
3. What is the 8th root of 1099511627776 ?
4. What is the 6th root of 25632972850442049?
5. What is the 9th root of 1.577635 ?
For further practical applications of Involution and Evolution,
Bee ** Mensuration."
833. An Arithmetical Progression is a suc-
cession of numbers, each of which is greater or less than
the preceding one by a constant difference.
Thus, 5, 7, 9, 11, 13, 15, is an arithmetical progression.
834. The Terms of an arithmetical progression are
the numbers of which it consists. The first and last terms
are called the Extremes^ and the other terms the Means.
835. The Common Difference is the difference
between any two consecutive terms of the progression.
836. An Increasing Arithmetical Frogres-
sion is one in which each term is greater than the pre-
ceding one.
Thus, 1, 3, 5, 7, 9, 11, is an increasing progression.
837. A Decreasing Arithfuetical Progres-
sion is one in which each term is less than the preced-
ing one.
Thus, 15, 13, 11, 9, 7, 5, 3, 1, is a decreasing progression.
838. The following are the quantities considered in
arithmetical progression and the abbreviations used for
them:
1. The first term, (a).
2. The last term, (I).
3. The common diflFerence, (d).
4. The number of terms, (n).
5. The sum of all the terms, («).
176 PROGRESSIONS.
WMITTE N EXEJRCISES,
839. To find one of the extremes, when the other
extreme, the common difference, and the number
of terms are given.
1. The first term of an increasing progression is 8^ the
common difference 5, and the number of terms 20 ; what
is the last term ? • u»
OPERATION. Analysis. — The 2d term is 8 + 5;
20 1 = 19 the 3d term is 8 + (5 x 2) the 4th term
r-Q r^ ^ ^ ^o J is 8 + (5 X 3) ; and so on. Hence 8 +
ly X D + b — iOd _ ^. ^^g ^ g^ ^j. ^^3 .g ^^^ 2Q^^ ^^ ^^^^^ ^^^^
2. The last term of an increasing progression is 103,
the common difference 5, and the number of terms 20 ;
what is the first term ?
OPERATION. Analysis.— The 1st term must be
- a number to which, if 19 x 5 be added,
"^ the sum shall be 103 ; hence, if 19 x 5
103 — 19x5=:8=:6]^ is subtracted from 103, the remainder
is the first term.
3. The first term of a decreasing progression is 203,
the common difference 5, and the number of terms 40 ;
what is the last term ?
4. The last term of a decreasing progression is 1, the
common difference 2, and the number of terms 9 ; what
is the first term ?
EuLE.— I. If the given extreme is the less, add to it the
product of the common difference by the number of terms
less one.
II. If the given extreme is the greater, subtract from it
the product of the common difference by the number of
terms less one,
^ il z=za + (n — 1) X d.
Formula.— i , ; ^i j
* a=.l — {n — 1) X d.
PROGRESSIONS. 177
5. The first term of an increasing progression is 5, the
common difference 4, and the number of terms 8 ; what
is the last term ?
6. The first term of an increasing progression is 2, and
the common difference 3 ; what is the 50th term ?
7. The first term of a decreasing progression is 100,
and the common difference 7 ; what is the 13th term ?
8. The first term of an increasing progression is f , the
common difference |, and the number of terms 20 ; what
is the last term ?
830. To find the common difference, when the
extremes and number of terms are given.
1. The extremes of a progression are 8 and 103, and
the number of terms 20 ; what is the common difference ?
OPERATION. Analysis.— The difference between
I QQ Q _^ i Q __ K __ ^ the extremes is equal to the product
of the common difference by the
number of terms less one (829) ; hence the common difference is
ff,or5.
2. The extremes of a progression are 1 and 17, and the
number of terms 9 ; what is the common difference ?
EuLE. — Divide the difference between the extremes by
the number of terms less one.
Formula. — d = -^^? .
n — 1
3. The extremes are 3 and 15, and the number of terras
7 ; what is the common difference ?
4. The extremes are 1 and 51, and the number of terms
76 ; what is the common difference ?
178 PROGRESSIONS.
5. The youngest of ten children is 8^ and the eldest 44
years old ; their ages are in arithmetical progression.
What is the common difference of their ages ?
6. The amount of $800 for 60 years, at simple interest,
is $4160. What is the rate per cent. ?
7. The extremes are and 2^, and the number of terms
18 ; what is the common difference ?
831. To find the humber of terms, when the ex-
tremes and common difference are given.
1. The extremes of a progression are 8 and 103, and
the common difference 5 ; what is the number of terms ?
OPERATION. Analysis. — The difference between the
-1 no Q _i_ K ___ 1 Q extremes is equal to the product of the
1 * OA common difference by the number of
■^^ + ^^^^~^ terms less one (830) ; hence the number
of terms less one is equal to -^/ or 19 ;
therefore 19 + 1 or 20 is the number of terms.
2. The extremes of a progression are 1 and 17, and the
common difference 2 ; what is the number of terms ?
EuLE. — Divide the difference between the ecctremes by
the common difference, and add one to the quotient.
FOEMULA. — 71 = ~ — h 1.
3. The extremes are 5 and 75, and the common differ-
ence is 5 ; what is the number of terms ?
4. The extremes are \ and 20, and the common differ-
ence is 6|- ; what is the number of terms ?
5. A laborer received 50 cents the first day, 54 cents
the second, 58 cents the third, and so on, until his wages
were $1.54 a day ; how many days did he work ?
6. In what time will $500, at 7 per cent, simple inter-
est, amount to $885 ?
PEOGRESSIONS. 179
832. To find the sum of all the terms, when the
extremes and the number of terms are given.
1. The extremes of an arithmetical progression are 2
and 14, and the number of terms is 5 ; what is the sum
of all the terms ?
Analysis. — The common dif-
OPEKATION. erence is found to be 3 (830) ;
2_L. 5_L «_l11j_14 hence the required sum is
""i^TuT o T K .. ^q^^l *o 2 + 5 + 8 + 11 + 14, or
,^ 14 + 11+ 8+ 5+ 2 i4^^i^s^5^3^ Addingthe
2 S =16 + 16 + 16 + 16 + 16 corresponding terms of these
2 5 = 16 X 5 z= (2 + 14) X 5 ^^o progressions, we have 2
o I 1 4^ times the sum — 16 x 5 = (2 +
S =■ X 5 = 40. 14) X 5 ; hence the sum is
2
^ + 14 K Al^
X 5 = 40.
2
2. The extremes of an arithmetical progression are 5
and 75, and the number of terms is 15 ; what is the sum
of all the terms ?
KuLE. — Multiply the sum of the extremes by half the
number of terms.
Formula. — s = - x {a + I).
3. The extremes are 4 and 40, and the number of terms
is 7 ; what is the sum of all the terms ?
4. The extremes are and 250, and the number of
terms is 1000 ; what is the sum of all the terms ?
5. How many strokes, beginning at 1 o'clock, does the
hammer of a common clock strike in 12 hours ?
6. A body will fall 16^ ft. in the first second of its
fall, 48|^ ft, in the second second, 80^^^ ft. in the third
second, and so on ; how far will it fall in one minute ?
180 PROGRESSIONS.
833. A Geometrleal Progression is a succes-
sion of numbers, each of which is greater or less than the
preceding one in a constant rafio.
Thus, 1, 3, 9,27, 81, etc., is a geometrical progression.
834. The Terms of a geometrical progression are
the numbers of which the progression consists. The first
and last terms are called the Extremes^ and the other
terms the Means.
835. The Ratio of a geometrical progression is the
quotient obtained by dividing any term by the preceding
one.
836. An Increasing Geometrical Frogres'
sion is one in which the ratio is greater than 1.
Thus, 1, 3, 4, 8, 16, etc., is an increasing progression.
837. A Decreasing Geometrical JProgres-^
sioni& one in which the ratio is less than 1.
Thus, 1, J, J, J, Y^^, etc., is a decreasing progression.
838. An Infinite Decreasing Geometrical
Progression is one in which the ratio is less than 1,
and the number of terms infi7iite.
Thus, 1, i, J, J, ^^, 7^, ^, and so on is an infinite decreasing
progression.
839. The following are the quantities considered in
geometrical progression :
1. The first term {a),
2. The last term (Z).
3. The ratio (r).
4. The number of terms (w).
5. The sum of all the terms {s).
PROGRESSIONS. 181
WRITTJEN EXEJRC IS 1SS.
840. To find one of the extremes, when the other
extreme, the ratio, and the number of terms are
given.
1. The first term of a progression is 2, the ratio 3, and
the number of terms 10 ; what is the last term ?
Analysis.— The 2d term is 3 x 3 ; the third
3'*^ =19683 term is 2x3x3 or 2x3'^; the 4tli term is
2 2x3^; and so on. Hence the 10th or last
oaoaa 7 ^^^'^ is 2 x 3® or 39366.
2. The last term of a progression is 39366, the ratio 3,
and the number of terms 10 ; what is the first term ?
OPERATION. Analysis. — The first term must be a num-
39355 ber, by which if 3^ be multiplied the product
—39— = 2 = a • siiall be 39366 ; hence, if 39366 be divided by
3®, the quotient will be the first term.
3. The first term of a progression is 1, the ratio |, and
the number of terms 9 ; what is the last term ?
EuLE. — I. If the given extreme is the first term, rmdti-
ply it hy that power of the ratio whose exponent is one less
than the number of terms.
II. If the given extreme is the last term, divide it hy
that power of the ratio whose exponent is one less than the
number of terms.
FoRMULiE. — I =: ar"^'^ ; a = — — : .
4. The first term of a geometrical progression is 6, the
ratio 4, the number of terms 6 ; what is the last term ?
5. The last term is 192, the ratio 2, and the number of
terms 7 ; what is the first term ?
182 PROGRESSIOl^S.
6. A drover bought 20 cows, agreeing to pay $1 for the
first, $2 for the second, $4 for the third, and so on ; how
much did he pay for the last cow ?
7. Find the amount of $250 for 4 years at 6 per cent,
compound interest.
The first term is 350, the ratio 1,06, and the number of terms 5.
8. If 1 cent had been put at interest in 1634, what
would it have amounted to in the year 1874, if it had
doubled its value every 12 years ?
841. To find the ratio, when the extremes and
the number of terms are given,
1. The first term is 2, the last term 512, and the num-
ber of terms 5 ; what is the ratio ?
OPEKATiON. ANAiiYBTS. — ^If the 4th power of the
.gig =256 ratio be multiplied by 2, the product will
4/^— _ . _ be 512 (840); henc6, if 512 be divided
Vx55b — 4 — r ^^ ^ ^^^ quotient, 256, will be the 4th
power of the ratio. Hence the ratio is the 4th root of 256, or 4.
2. The first term is 1, the last term ^-J-g-, and the num-
ber of terms 9 ; what is the ratio ?
EuLE. — Divide the last term by the first, and extract
that root of the quotient whose index is one less than the
number of terms.
Formula* — r= V --
^ a
3. The first term is 8, the last term 5000, and the num-
ber of terms 5 ; what is the ratio ?
4. The first term is .0112, the last term 7, and the
number of terms 5 ; what is the ratio ?
5. The first term is ^, the last term 15^, and the
number of terms 7 ; what is the ratio ?
PROGBESSIOKS. 183
84:3« To find the number of terms, when the
extremes and the ratio are ^ven,
1. The extremes are 2 and 512> and the ratio is 4 ; what
is the number of terms ?
OPERATION. Analysis. — If 512 be divided by 2, the quotient,
2 ) 512 256, will be that power of the ratio whose exponent
~Tr^ is one less than the number of terms (841). But
256 is the 4th power of the ratio 4 : hence the nuTn-
4 = 2o6 her of terms S^^.
2. The extremes are 1 and ^^, and the ratio is ^ ; what
is the number of terms ?
Rule. — Divide the last term iy the first ; then the expo-
nent of the power to which the ratio must ie raised to pro-
duce the quotient is one less than the number of terms.
. I
Formula. — r"^'^ = - .
a
3. The extremes are 2 and 1458, and the ratio is 3 ;
what is the number of terms ?
4. The extremes are -^ and -^, and the ratio 2 ; what
is the number of terms ?
843. To find the sum of all the terms, when the
extremes and the ratio are given.
1, The extremes are 2 and 128, and the ratio is 4 ; what
is the sum of all the terms ?
OPERATION.
(128x4)^2 510
4-1 ~ -3- - 17U ^ 5
4: 8 = 8 + 32 + 128 + 512 Analysis.— Subtract the sum from 4
8 = 2 + 8 + 32 + 128 times the sum, and 510 remains, which
3 « = 512 — 2 = 510 is 3 times the sum ; hence, ^^yOT 170,
510 _ 170 _ g. is the sum.
184 PROGBESSIONS.
2. The extremes are 1 and -^, and the ratio is | : what
IS the sum of all the terms ?
is= i + i + i + iV + A
B,VL^,— Multiply the last term by the ratio^ and divide
the difference between the product and the first term by the
difference between 1 and the ratio.
Formula. — s = ;-.
r — 1
3. The extremes are 3 and 384, and the ratio is 2 ; what
is the sum of all the terms ?
4. The extremes are 4f and ;^, and the ratio is | ;
what is the sum of all the terms ?
5. What is the sum of all the terms of the infinite pro-
gression 8, 4, 2, 1, 1^, i, .... ?
The last term of this progression may be conceived as 0.
6. What is the sum of all the terms of the infinite pro-
gression 1, I, :^, ^, -gL., . . . . ?
7. What is the sum otl+^ + ^ + ^, etc., to infinity?
8. The first is 7, the ratio 3, and the number of terms
4 ; what is the sum of all the terms ?
First find tlie last term by Art. 840.
9. A drover bought 10 cows, agreeing to pay $1 for the
first, $2 for the second, S4 for the third, and so on ; what
did he pay for the 10 cows ?
10. If a man were to buy 3 2 horses, paying 2 cents for
the first horse, 6 cents for the second, and so on, what
would they cost him ?
844. An Annuity is a sum of money payable an-
nually. The term is also applied to a sum of money
payable at any equal intervals of time.
845. A Certain Annuity is one wbich continues
for a definite period of time.
846. A Perpetual Annuity or Fe7^j)etuity
is one which continues forever.
847. A Contingent Annuity is one which begins
or ends, or both begins and ends, on the occurrence of
some specified future event or events.
848. An Annuity Forborne or in A7*rears
is one the payments of which were not made when due.
849. The Amount or Final Value of an an-
nuity is the sum of all the payments increased by the
interest of each payment^ from the time it becomes due
until the annuity ceases.
850. The Present Worth of an annuity is such a
sum of money as will, in the given time, and at the given
rate per cent., amount to the final value.
851i An annuity is said to be deferred when it does
not begin until after a certain period of time ; it is said
to be reversionary when it does not begin until after the
occurrence of some specified future event, as the death
of a certain person ; and it is said to be in possession
when it has begun, or begins immediately.
186 AKKUITIES.
ANNUITIES AT SIMPLE INTEREST.
853. All problems in annuities at simple interest may
be solved by combining the rules in Arithmetical Pro-
gression with those in Simple Interest.
WMITTJEN EXJSRC ISBS,
853. 1. What is the amount of an annuity of $300 for
5 years, at 6 per cent, simple interest ?
OPERATION.
300 + 372 Analysis.— At the end of the 5th
H X 5 = 1680 year the following sums were due :
The 5th year's payment = $300,
The 4tli year's payment == $300 + $18 =
The 3d year's payment = $300 + $36 =
The 2d.year's payment = $300 + $54 = $354,
The 1st year's payment = $300 + $72 =: $373.
These sums form an arithmetical progression, in which the first
term is the annuity, $300, the common difference is the interest of
the annuity for 1 year, and the number of terms is the number of
years. The sum of all the terms of this progression is $1680 (882),
which is the amount of the annuity.
2. A father deposits annually^for the benefit of his son,
beginning with his tenth birthday, such a sum that on
his 21st birthday the first deposit, at simple int., amounts
to $210, and the sum due his son is $1860. Find the
annual deposit, and at what rate per cent, it is deposited.
OPERATION.
6 X (1st term + 210) = 1860. (833.)
Hence, 1st term = 310 — 210 = 100 — a.
(210 - 100) ^ {1^ -1) =:i^ = 10 ^ d. (830.)
Analysis. — Here $210, the first deposit, is the last term ; 12, the
number of deposits, is the number of terms ;
AIS^NUITIES. 187
and, $1860, the final value of the annuity, is the sum of all thsy
terms. Using the principle of 832, we find the first term to bv.
$100, which is the annual deposit. By 880, the common dif
ference is found to be $10 ; hence 10 per cent, is the required rate.
3. What is the amount of an annuity of $150 for b\
years, payable quarterly, at 1-| per cent, per quarter ?
4. What is the present worth of an annuity of $300
for 5 years, at 6 per cent. ?
5. What is the present worth of an annuity of $500
for 10 years, at 10 per cent. ?
6. In what time will an annual pension of $500 amount
to $3450, at 6 per cent, simple interest ?
7. Find the rate per cent, at which an annuity of $6000
will amount to $59760 in 8 years, at simple interest.
8. A man works for a farmer 1 yr. 6 mo., at $20 per
month, payable monthly ; and these wages remain unpaid
until the expiration of the whole term of service. What
is due the workman, allowing simple interest at 6 per
cent, per annum ?
ANNUITIES AT COMPOUND INTEREST.
854. All problems in annuities at compound interest
may be solved by combining the rules in Geometrical
Progression with those in Compound Interest.
WBITTEN EXBItCIS ES,
1. What is the amount of an annuity of $300 for 5
years, at 6 per cent, compound interest ^
OPERATION. Analysis.— At the end of the
300x1.065—300 -,^0110 5th year the following sums
' ttt: "^^^ ioyi.it) J
.06 8^6 <1^G :
188 AKKUITIES.
The 5tli year's payment = |300,
The 4th year's payment + interest for 1 year = $300 x 1.06,
The 3d year's payment + compound int. for 2 years ~ $300 x 1.06^,
The 2d year's payment + compound int. for 3 years = $oOO x 1.06^
The 1st year's payment + compound int. for 4 years = $300 x 1.06*.
These sums form a geometrical progression, in which the first
term is the annuity, $300, the ratio is the amount of $1 for 1 year,
and the number of terms is the number of years. The sum of all
the terms of this progression is $1691.13 (843), which is the
amount of the annuity.
2. What is the present worth of an annuity of $300 for
5 years, at 6 per cent, compound interest ?
OPERATION.. Analysis. — The amount of this an-
1691.13 nuity is $1691.13. The amount of $1 for
= 1363.71 5 years, at 6 percent, compound interest,
is $1.338226 (587). Hence the present
1.338226
worth of the annuity is -f^^' or $1263.71.
3. Find the annuity whose amount for 25 years, at 6
per cent, compound interest, is $16459.35.
4. What is the present worth of an annuity of $700
for 7 years, at 6 per cent, compound interest ?
5. An annuity of $200 for 12 years is in reversion 6
years. What is its present worth, compound interest
at 6^?
6. A man bought a tract of land for $4800, which was
to be paid in installments of $600 a year ; how much
money, at 6 per cent, compound interest, would discharge
the debt at the time of the purchase ?
7. What is the present value of a reversionary lease of
$100, commencing 14 years hence, and to continue 20
years, compound interest at 5 per cent. ?
REVIEW. 189
855. SYNOPSIS FOE EEVIEW.
[ Defs i ^' ^Pow^^- 2- Involution. 3. Base, or Root. 4. Ex-
I pouent. 5. Square. 6. Cube. 7. Perfect Power.
2. Principle.
3. 802. Rule. 1. For Integers. 2. For Fractions.
4. 803, 1- Principle. 2. Geometrical Illustration.
5. 804. 1 . Principle. 2. Geometrical Illustration.
1 Defs i ^' ^^^^^^ Root. 2. Cube Root, etc. 3. Evolution.
( 4. Radical Sign. 5. Index.
2. 810. Rule.
3. 812. Principles, 1, 2, 3, 4
4. 813. Rule, I, II, III. For Fractions.
5. 814:. Geometrical Illustration.
6. 818. Principles, 1, 2, 3, -4.
7. 819. Rule, I, II, III, IV, V, VI. For Fractions.
8. 820. Geometiical Illustration.
9. 822. Roots of a Higher Degree. Bide.
{ 1. Arithmetical Progression. 2. Terms. 3. Common
^ 1. Defs. -j Difference, 4. Increasing Arithmetical Progression.
i 5. Decreasing Arithmetical Progression,
2. Quantities considered.
3. 829. Rule, I, II. Formulm.
4. 830. Rule. Formula.
5. 831. Rule. Formvla.
6. 832. Rule. Formula.
( 1. Geometii;Cal Progression. 2. Terms. 3. Batio.
1. Defs. ■< 4. Increasing Geom. Prog. 5. DecreoMng Oeom,
\ Prog. 6. Infinite Decreasing Geom. Prog.
2. Quantities considered.
3. 840. Rule, I, II. Formulm,
4. 841. Rule. Formula.
5. 842. Rule. Formula.
6. 843. Rule. Formula.
1. Annuity. 2. Certain Annuity. 3. Perpetuity.
4. Contingent Annuity. 5. Annuity in Arrears.
6. Amount. 7. Present Wortli of an Annuity.
8. Deferred Annuity. 9. Reversionary Annuity.
10. Annuity in Possession.
3. Annuities at Simple Interest. ) „ , , , , .
^ 2. Annuities at Comf. Interest. | ^^^1^°^' how solved.
1. Defs. <
85G. Mensuration is the process of finding the number of
units in extension.
LINES.
857. A Straight Line is a line that
does not change its direction. It is the short-
est distance between two points.
858. A* Curved lAne changes its direc-
tion at every point.
859. JFarallel Lines have the same
direction ; and being in the same plane and
equally distant from each other, they can never
meet.
860. A HoHzonfal Line is a line par'
allel either to the horizon or water level.
801. A JPerpendirtilar Line is a
straight line drawn to meet another straight
line, so as to incline no more to the one side
than to the other.
A perpendicular to a horizontal line is called a vertir
ca^ lina
ANGLES.
862. An Angle is the difference in the
direction of two lines proceeding from a com-
mon point, called the vertex.
An^es are measured by degrees. (301.)
863. A Might Angle is an angle formed
by two lines perpendicular to each other.
864:. An Obtuse Angle is greater than
a right angle.
865. An Acute Angle is less than a
right angle,
except rigbt angles are caQed obHgtte (mgke.
Horizontal.
TKIAKGLES.
191
PLANE FIGURES.
866. A Platte Figure is a portion of a plane surface bounded
by straight or curved lines.
867. A Polygon is a plane figure bounded by straight lines.
868. The Perhneter of a polygon is the sum of its sides.
869. The Area of a plane figure is the surface included
within the lines which bound it. (460.)
A regular polygon has all its sides and all its angles equal.
The altitude of a polygon is the perpendicular distance between its hose and ^
side or angle opposite.
A polygon of three sides is called a trigon, or triangle ; of four sides» a tetror
gon^ or quadrilateral; of five sides, & pentagon^ etc.
Pentagon. Hexagon. Heptagon. Octagon. Nonagon. Decagon.
TRIANGLES.
870. A Triangle is a plane figure bounded by three sides,
and having three angles.
871. A night- Angled Triangle
is a triangle having one right angle.
872. The Hypothenuse oi2iTLg\ii'
angled triangle is the side opposite the
right angle.
873. The Base of a triangle, or of
any plane figure, is the side on which it may be supposed to stand.
874. The Perpendicular of a right-angled triangle is the
side which forms a right angle with the base.
875. The Altitude of a triangle is a line drawn from the angle
opposite perpendicular to the base.
1. The dotted lines in the following figures represent the altitude.
2. Triangles are named from the relation both of their sides and angles.
192 MENSURATION.
876. An Equilateral Triangle has its three sides equal.
877. An Isosceles Tri'angr^<? has only two of its sides equal.
878. A Scalene Triangle has all of its sides unequal.
Fig. 1. FiQ. 2. Fio. 3.
Equilateral. Isosceles. Scalene.
879. An Equiangular* Triangle has three equal angles
(Fig. 1.)
880. An Acute-angled Triangle has three acute anglea
(Fig. 2.)
881. An Obtuse-angled Triangle has one obtuse angle.
(Fig. 3.)
SS^2. The base and altitude of a triangle being
given to find its area,
1. Find the area of a triangle whose base is 26 ft. and altitude
14.5 feet.
145
Operation.— 14.5 x 26^2=rl88Jsq.ft. Or, 26 x -^=188^square
feet, area.
2. What is the area of a triangle whose altitude is 10 yards and
base 40 feet ?
Rule.— 1. Divide the product of the base and altitude hy 2. Or,
2. Multiply the base by ona-half the altitude.
Find the area of a triangle
3. Whose base is 13 ft. 6 in. and altitude 6 ft. 9 in.
4. Whose base is 25.01 chains and altitude 18.14 chains.
5. What is the cost of a triangular piece of land whose base is
15.48 ch. and altitude 9.67 ch., at $60 an acre?
6. At $.40* a square yard, find tho cost of paving a triangular
court, its base being 105 feet, and its altitude 21 yards ?
7. Find the area of the gable end of a house that is 28 ft. wide,
and the ridge of the roof 15 ft. higher than the foot of the rafters.
TRIANGLES. 193
883. The area and one dimension being given to
find the other dimension.
1. What is the base of a triangle whose area is 189 square feet
and altitude 14 feet ?
Operation.— (189 sq. ft. x 2)-t-14 = 27 ft., hose,
2. Find the altitude of a triangle whose area is 20J square feet
and base 3 yards.
Rule. — Double the area, then divide hy the given dimension.
Find the other dimension of the triangle
3. When the area is 65 sq. in. and the altitude 10 inches.
4. When the base is 42 rods and the area 588 sq. rods.
5. When the area is 6 J acres and the altitude 17 yards.
6. When the base is 12.25 chains and the area 5 A. 33 P.
7. Paid $1050 for a piece of land in the form of a triangle, at the
rate of $5 J per square rod. If the base is 8 rd., what is its altitude ?
884. The three sides of a triangle being given to
find its area.
1. Find the area of a triangle whose sides are 30, 40, and 50 ft.
Operation.— (30 + 40 + 50)-j-2 = 60 ; 60-30 = 30 ; 60-40 = 20 ;
60-50 = 10. /v/60x 30x20x10 = 600 ft., area.
2. What is the area of an isosceles triangle whose base is 20 ft.,
and each of its equal sides 15 feet ?
BjjuR.—From half the svm of the three sides, subtract each side
separately ; multiply the half-%um and the three remainders together;
the square root of the product is the a/rea,
3. Find the area of a triangle whose sides are 25, 36, and 49 in.
4. How many acres in a field in the form of an equilateral tri-
angle whose sides each measure 70 rods ?
ff. The roof of a house 30 ft. wide has the rafters on one side
20 ft. long, and on the other 18 ft. long. How many square feet of
boards will be required to board up both gable ends ?
194
MEKSUEATIOK.
885. The following principles relating to right-angled triangles
have been established by Geometry ;
Principles. — 1. The square of the
hypothenuae of a right-angled triangle
is equal to the sum of the squares of
the other two sides.
2. 2^he square of the base, or of the
perpendicular, of a right-angled tri-
angle is equul to the square of the
hypothenuse diminished by the square
of the other side,
886. To find the hypothenuse.
1 . The base of a right-angled triangle is 13, and the perpendicu-
lar 16. What is the length of the hypothenuse ?
Operation.— 122 + 16^ = 400 (Prin. 1). ^^ = ^^^ hypotJienuae,
2. The foot of a ladder is 15 feet from the base of a building, and
the top reaches a window 36 feet above the base. What is the
length of the ladder ?
Rule. — Extract the square root of the sum of the squares of the
base and the perpendicular ; the result is the hypothenuse,
3. If the gable end of a house 40 ft. wide is 16 ft. high, what is
the length of the rafters ?
4. A park 25 chains long and 23 chains wide has a walk running
through it from opposite corners in a straight line. What is the
length of the walk ?
5. A room is 20 ft. long, 16 ft. wide, and 12 ft. high. What is the
distance from one of the lower corners to the opposite upper corner t
887. To find the base or perpendicular,
1. The hypothenuse of a right-angled triangle is 35 feet, and the
perpendicular 28 feet. Find the base.
Operation.— 352 - 28" = 441 (Prin. 2). y'Sl = 21 ft., base.
QUADRILATERALS.
195
2. The hypothenuse of a right-angled triangle is 53 yards and
the base 84 feet. Find the perpendicular.
Rule. — Extract the square root of the difference between the square
of the hypothenuse and the square of the given side; the result is the
required side.
3. Find the width of a house, whose rafters are 13 ft. and 15 ft.
long, and that form a right angle at the point in which they meet.
4. A line reaching from the top of a precipice 120 feet high, on
the bank of a river, to the opposite side is 380 feet long. How
wide is the river ?
5. A ladder 52 ft. long stands against the side of a building.
How many feet must it be drawn out at the bottom that the top
may be lowered 4 feet ?
QUADRILATERALS.
888. A Quadrilateral is a plane figure bounded by four
straight lines.
There are three kinds of quadrilaterals, the ParaUdogram, Trapezoid^ and
Trapezium.
889. A Parallelogram is a quadrilateral which has its
opposite sides parallel.
There are four kinds of parallelograms, the Square, Bectangle, Bhombcid, and
Bhomlus,
890. A Hectangle is any parallelogram having its angles
right angles.
891. A Square is a rectangle whose sides are equal.
892. A Rhomboid is a parallelogram whose opposite sides
only are equal, and whose angles are not right angles.
893. A Hhombtis is a parallelogram whose sides are all
equal, but whose angles are not right angles.
Square.
Rectangle.
Rhomboid.
Rhombus.
196
M E i^ S U R A T I N .
894. A Trapezoid is a quadrilateral, two of whose sides are
parallel.
895. A Trapezium is a quadrilateral having no two sides
parallel.
896. The Altitude of a parallelogram or trapezoid is the per
pendicular distance between its parallel sides.
The dotted vertical lines in the figure represent the altitude.
897. A lyiagonal of a plane figure is a straight line joining
the vertices of two angles not adjacent.
Parallelogram.
Trapezoid.
Trapezium.
1*12 O B L EMS .
898. To find the area of any parallelogram.
1. Find the area of a parallelogram whose base is 16.25 feet an'*
altitude 7.5 feet.
Operation.— 16.25 ft. x 7.5 = 121.875 sq. feet, area.
2. The base of a rhombus is 10 feet 6 inches, and its altitude
8 feet. What is its area ?
Rule. — Multiply the base by the altitude.
3. How many acres in a piece of land in the form of a rhomboid,
the base being 8.75 ch. and altitude 6 chains ?
899. To find the area of a trapezoid.
1. Find the area of a trapezoid whose parallel sides are 23 and
11 feet, and the altitude 9 feet.
Operation.— 23 ft. + 11 ft. -^2=17 ft. ; 17 ft. x 9=153 sq. ft., area.
2. Required the area of a trapezoid whose parallel sides are 178
and 146 feet, and the altitude 69 feet.
Rule, — Multiply one- half the sum of the parallel sides by the
altitude.
0IECLES. 19?
3. How many square feet in a board 16 ft. long, 18 inches wide
at one end and 25 inches wide at the other end ?
4. One side of a quadrilateral field measures 38 rods ; the side
opposite and parallel to it measures 26 rods, and the distance be-
tween the two sides is 10 rods. Find the area.
900. To find the area of a trapezium.
1. Find the area of a trapezium whose
diagonal is 42 feet and perpendiculars to this
diagonal, as in the diagram, are 16 feet and
18 feet.
Operation.— (18 ft. + 16 ft. -^2) x 42 = 714 sq. feet, area,
2. Find the area of a trapezium whose diagonal is 35 ft. 6 in., and
the perpendiculars to this diagonal 9 feet and 3 feet.
Rule. — Multiply the diagonal hy half the sum of the perpendicu-
lars drawn to it from ihe vertices of opposite angles,
3. How many acres in a quadrilateral field whose diagonal is
80 rd. and the perpendiculars to this diagonal 20.453 and 50.832 rd. ?
To find the area of any regular polygon, multiply its perimeter, or the Bnm of
ita sides, by one-half the perpendicular falling from its center to one of its sides.
To find the area of an irregular polygon, divide the figure into triangles and
trapeziums, and find the area of each separately. The sum of these areas will
be the area of the whole polygon.
THE CIRCLE.
901. A Circle is a plane figure bounded by a curved line,
called the circumference ^ every point of which is
equally distant from a point within called the
center,
902. The Diameter of a circle is a line
passing through its center, and terminated at both
ends by the circumference.
903. The Ttaditis of a circle is a line extending from its cen-
ter to any point in the circumference. It is one-half the diameter.
198 MENS U RATIO K.
rS OB LE M S »
904. Wlien either the diameter or the circum-
ference of a circle is given, to find the other di-
mension of it.
1. Find the circumference of a circle wliose diameter is 20 inches.
Operation.— 20 in. x 3.1416 = 62.832 in. = 5 ft. 2.832 in., ci/rcum.
2. Find the diameter of a circle whose circumference is 62.832 ft.
Operation.— 62.832 ft.-f-3.1416 = 20 ft., diameter,
3. Find the diameter of a wheel whose circumference is 50 feet.
Rule. — 1. Multiply the diameter by 3.1416 ; the product is the oir-
cumference.
2. Bimde the circumferenee by 3.1416 ; the quotient is the diameter.
4. What is the diameter of a tree whose girt is 18 ft. 6 in. ?
5. What is the radius of a circle whose circumference is 31.416 ft.?
6. Find the circumference of the greatest circle that can be
drawn with a string 14 inches long, used as a radius.
905. To find the area of a circle, wlien both its
diameter and circumference are given, or when
eitlier is given.
1. What is the area of a circle whose diameter is 10 feet and cir-
cumference 31.416 feet? t
Operation.— 31.416 ft. x IOh-4 = 78.54 sq. ft., area,
2. Find the area of a circle whose diameter is 10 feet.
Operation.— 10 ft.^ x .7854 = 78.54 sq. feet, a/rea,
3. Find the area of a circle whose circumference is 31.416 feet.
Operation.— 31.416 ft.-^3.l416=rlO ft., diam,; (10 ft.j^x .7854=
78.54 sq. feet, a^ea.
Rules. — To find the area of a circle :
1. Multiply i of its diameter by the circumference,
2. Multiply the square of its diameter by .7854.
4. What is the area of a circular pond whose circumference is
200 chains
5. The distance around a circular park is 1 J miles. How many
acres does it contain ?
CIRCLES. 199
906. To find the diameter or the circumference
of a circle, when the area is g^iven.
1. What is tlie diameter of a circle whose area is 1319.472 ?
Operation.— 1319.472^.7854 =z 1689 ; \/lQSO = 40.987 + , diam-
eter.
2. What is the circumference of a circle whose area is 19.635 ?
Opekation— 19,635 -i- 3.1416 = 6,25 ; ^"6^5=12.5, racUiis; 2.5 x
2 X 3.1416 = 15.708, circumference.
Rule.— 1. Divide the area by .7854 and extract the square root of
the quotient ; the resuU is the diameter,
2. Divide the area hy 3.1416 and extract the square root of ths
quotient ; the result is the radius^ The cireumference is obtained by
Art. 904, 1. Or,
8. Divide the area by .07958 and find the square root of the quotient.
3. The area of a circular lot is 38.4846 square rods. What is its
diameter ?
4. The area of a circle is 286.488 equare feet. Required the
diameter and the circumference.
907. To find the side of an inscribed square when
the diameter of the circle is known.
1. What is the side of a square inscribed in a
circle whose diameter is 6 rods ?
Operation. — 6^ -f- 2 = 18 ; ,y^l8— 4.24 rods, side
ofsquAire.
2. The diameter of a circle is 200 feet. Find
the side of the inscribed square.
Rule.— 1. Extract the square root of half the square of tlie diam |
t^er. Or,
2. Multiply the diameter by .7071.
3. The circumference of a circle is 104 yards. Elnd the side of
the inscribed square.
4. The area of a circle is 78.54 square feet Find the side of the
inscribed square.
200 MEKSURATIOK.
908. To find the area of a circular ring formed
by two concentric circles.
1. Find the area of a circular ring, when
the diameters of the circles are 20 and 80 feet.
Opekation.— (30 + 20 X 30 - 20) x .7854 =
392.7 sq. ft., area,
2. Find the area of a circular ring formed
by two concentric circles, whose diameters are
7 ft. 9 in. and 4 ft. 3 in.
Rule. — MulMpiy the mm of the two diameters by their difference,
and the product by .7854 ; the result is the area.
3. Two diameters are 35.75 and 16.25 ft. ; find the area of the ring.
4. The area of a circle is 1 A. 154.16 P. In the center is a pond of
water 10 rd. in diameter ; find the area of the land and of the water.
909. To find a mean proportional between two
numbers.
1. What is a mean proportional between 3 and 12 ?
Operation. — '\/l2 x 3 = 6, the mean proportional.
When three nnmhers are proportional^ the product of the extremes is equal to
the square of the mean.
EuLK — Extract the square root of the product ofth^e two numbers.
Find a mean proportional between
2. 42 and 168. | 3. 64 and 12.25. | 4. |f and /y.
5. A tub of butter weighed 36 lb/ by the grocer's scales ; but
weighing it in the other scale of the balance, it weighed only 30
pounds. What was the true weight of the butter ?
SIMILAR PLANE FIGURES.
910. Similar I^lane Fignres are such as have the same
form J viz., equal angles, and their like dimensions proportional.
All circles, squares, equiangular triangles, and regular polygons of the same
number of sides are similar fignres.
The like dimensions of circles are their radii, diameters, and circumferences.
Principles. — 1. The like diwjensions of similar plane figures are
proportional.
SIMILAR PLANE FIGURES. 201
2. The areas of dmilar plane figures are to each other as the squa/res
of their like dimensions. And conversely,
3. The like dimensions of similar plane figures are to each other aa
the square roots of their areas.
The same principles apply also to the surfaces of all similar figures, such as
triangles, rectangles, etc. ; the surfaces of similar so/erfs, as cubes, pyramids, etc.;
and to similar curved surfaces, as of cylinders, cones, and spheres. Hence,
4. The surfaces of all similar figures are to each other as the squa/res
of their like dimensions. And conversely,
5. Their dimensions are as the square roots of their surfaces.
rR OBLEM S,
1. A triangular field whose base is 12 ch. contains 2 A. 80 P.
Find the area of a field of similar form whose base is 48 chains.
Operation.— 122 : 48^ : : 2 A. 80 P. : a? P. =6400 P. = 40 A., area.
(Prin. 2.)
2. The side of a square field containing 18 acres is 60 rods long.
Find the side of a similar field that contains J as many acres.
Operation.— 18 A. : 6 A. : : 60"^ : ic^ =1200 ; ^1200 = 34.64 rd. + ,
side, (Prin. 3.)
3. Two circles are to each other as 9 to 16 ; the diameter of the
less being 112 feet, what is the diameter of the greater?
Operation— 9 : 16 : : 112^ : a;^ = 3 : 4 : : 112 : a; = 149 ft. 4 in.,
diameter. (Prin. 2.)
4. A peach orchard contains 720 square rods, and its length is to
its breadth as 5 to 4 ; what are its dimensions ?
Operation. — The area of a rectangle 5 by 4 equals 20 (898).
20 : 720 : : 52 : aj2 rr: 900 ; ^900 = 30 rd., length.
20 : 720 : : 42 : a;2 = 576 ; ^/^= 24 rd., width.
5. It is required to lay out 283 A. 107 P. of land in the form of
a rectangle, so that the length shall be 3 times the width. Find
the dimensions.
6. A pipe 1.5 in. in diameter fills a cistern in 5 hours ; find the
diameter of a pipe that will fill the same cistern in 55 min. 6 sec.
7. The area of a triangle is 24276 sq. ft., and its sides in proportion
to the numbers 13, 14, and 15. Find the length of its sides in feet
202 MENSURATION.
8. If it cost $167.70 to enclose a circular pond containing 17 A.
110 P., what will it cost to enclose another i as large ?
9. If 63.39 rods of fence will enclose a circular field containing
2 acres, what length will enclose 8 acres in circular form ?
REVIEW OF PLANE FIGURES.
l*IiO B LJE JI S ,
911. 1. How much less will the fencing of 20 acres cost in the
square form than in the form of a rectangle whose breadth is J the
length, the price being $2.40 per rod ?
2. A house that is 50 feet long and 40 feet wide has a square or
pyramidal roof, whose height is 15 ft. Find the length of a rafter
reaching from a corner of the building to the vertex of the roof.
3. Find the diameter of a circular island containing IJ sq. miles.
4. What is the value of a farm, at $75 an acre, its form being a
quadrilateral, with two of its opposite sides parallel, one 40 ch.
and the other 22 ch. long, and the perpendicular distance between
them 25 chains ?
5. Find the cost, at 18 cents a square foot, of paving a space in
the form of a rhombus, the sides of which are 15 feet, and a per-
pendicular drawn from one oblique angle will meet the opposite
side 9 feet from the adjacent angle.
6. A goat is fastened to the top of a post 4 ft. high by a rope 50 ft.
long. Find the area of the greatest circle over which he can graze.
7. How much larger is a square circumscribing a circle 40 rods
in diameter, than a square inscribed in the same circle ?
8. What is the value of a piece of land in the form of a triangle,
whose sides are 40, 48, and 54 rods, respectively, at the rate of
$125 an acre ?
9 . The radius of a circle is 5 feet ; find the diameter of another
circle containing 4 times the area of the first.
10. How many acres in a semi-circular farm, whose radius is
100 rods ?
11. What must be the width of a walk extending around a gar-
den 100 feet square, to occupy one-half the ground?
12. An irregular piece of land, containing 540 A. 36 P. is ex-
changed for a square piece of the same area ; find the length of one
of its sides ? If divided into 42 equal squares, what is the length
of the side of each ?
SOLIDS.
303
13. A field containing 15 A. is 30 rd. wide, and is a plane inclining
in the direction of its length, one end being 120 ft. higher than the
other. Find how many acres of horizontal surface it contains.
14. If a pipe 3 inches in diameter discharges 12 hogsheads of
water in a certain time, what must be the diameter of a pipe which
will discharge 48 hogsheads in the same time ?
SOLIDS.
912 . A Solid or Sody has three dimensions, length, breadth,
and thickness.
The planes which bouud it are called its faces^ and their intersections, its
913. A Prism is a solid whose ends are equal and parallel,
similar polygons, and its sides parallelograms.
Prisms take their names from the form of their bases, as triangular^ quad-
rangular^ pentagonal^ etc.
914. The Altitude oi a prism is the
perpendicular distance between its bases.
915. A Parallelopipedtni is a
prism bounded by six parallelograms, the
opposite ones being parallel
916. A Cube is a parallelopipedon
whose faces are all equal squares.
917. A Cf/llnder is a body bounded
by a uniformly curved surface, its' ends being equal and parallel
circles.
1. A cylinder is conceived to be generated by the revolution of a rectangle
about one of its sides as an axis.
2. The line joining the centers of the bases, or ends, of the cylinder is its aUi'
tude^ or axis.
Cube.
Triangular
Prism.
Quad ran 2:ular
Prism.
Pentagonal
Prism.
Cylinder.
204
MENSURATION.
I^ltOBLEMS.
918. To find the convex surlace of a prism or
cylinder.
1. Find the area of the convex sur-
face of a prism whose altitude is 7 ft.,
and its base a pentagon, each side of
whicli is 4 feet.
Operation. — 4 ft. x 5 = 20 ft., pert-
metcT.
20 ft. X 7=140 sq. ft., convex surface.
2. Find the area of the convex sur-
face of a triangular prism, whose alti-
tude is 8 J feet, and the sides of its base
4, 5, and 6 feet, respectively.
Operation. —4 ft. + 5 f t. + 6 ft. =
15 ft., perimeter.
15 ft. X 8J=127J sq. ft., contex surface.
3. Find the area of the convex surf qice pf a cylinder whose altitude
is 2 ft. 5 in. and the circumference of
its base 4 ftr^ jn.
Operation.— 2 ft. 5 in. = 29 m. ; 4 ft.
9 in, = 57 in,
57 in. X 29 = 1653 sq. in. = 11 sq. ft.
?!^^ 69 sq. inches, convex surface.
Rule. — Multiply the perimeter of the hose l)y the altitude.
To find the entire surface, add the area of the bases or ends.
4. If a gate 8 ft. high and 6 ft. wide revolves upon a point In its
center, what is the entire surface of the cylinder described by it ?
5. Find the superficial contents, or entire surface of a parallelo-
pipedon 8 ft. 9 in. long, 4 ft. 8 in, wide, and 3 ft. 3 in. high.
6. What is the entire surface of a cylinder formed by the revo-
lution about one of its sides of a rectangle that is 6 ft. 6 in. long
and 4 ft. wide ?
7. Find the entire surface of a prism whose base is an equilateral
triangle, the perimeter being 18 ft., and the altitude 15 ft.
PYRAMIDS AKD COJSTES,
205
919. To find the volume of any prism or cylinder.
1. Find the volume of a triangular prism, whose altitude is 20 ft.,
and each side of the base 4 feet.
Opekation.— The area of the base is 6.928 sq. ft. (882>
6.928 sq. ft. x 20 = 138.56 cu. ft., mlume.
2. Find the volume of a cylinder whose altitude is 8 ft. 6 in., an<
the diameter of its base 3 feet.
Operation.— 32 x .7854 = 7.0686 square feet, area of base (905).
7.0686 sq. ft. x 8.5 = 60.083 cubic feet, wlume.
Rule. — Multiply the area of the base by the altitude,
3. Find the solid contents of a cube whose edges are 6 ft. 6 in.
4. Find the cost of a piece of timber 18 in. square and 40 ft. long,
at $.30 a cubic foot,
5. Required the solid contents of a cylinder whose altitude is
15 ft. and its radius 1 ft. 3 in.
6. What is the value of a log 24 ft. long, of the average circum-
ference of 7.9 ft., at $.45 a cubic foot ?
PYRAMIDS AND CONES. .
920. A JPyramid is a body having for its base a polygon,
and for its other faces three or more triangles, which terminate in
a common point called the mrtex.
Pyramids, like prisms, take their names from their baees, and are called tri'
angular^ square^ or quadrangular^ pentagonal^ etc.
Frustum.
Cone.
Frustum.
Pyramid.
921. A Cone is a body having a circular base, and whose con-
vex surface tapers uniformly to the tertex.
It is a body conceived to be formed by the revolution of a right-angled triangle
about one of its sides containing the right angle, as an immovable axis.
922. The Altitude of a pyramid or of a cone is the perpendic-
ular distance from its vertex to the plane of its base.
306 MENSURATION.
923. The Slant Height of 2i pyramid is the perpendicular dis.
tance from its vertex to one of the sides of the base ; of a cone, is a
straight line from the vertex to the circumference of the base.
924. The Frustum of a pyramid or cone is that part which
remains after cutting off the top by a plane parallel to the base.
l^JiOBljJEMS.
925. To find the convex surface of a pyramid or
cone.
1. Find the convex surface of a triangular pyramid, the slant
height being 16 ft., and each side of the base 5 feet.
Operation.— (5 ft. + 5 ft. + 5 ft.) x 16^2 = 120 sq. ft., com. surf.
2. Find the convex/Surface of a cone whose diameter is 17 ft. 6 in.,
and the slant height 30 feet.
Rule. — Multiply the perimeter or circumference of the base hy one-
half the slant height.
To find the entire surface, add to this product the area of the base.
3. Find the entire surface of a square pyramid whose base is 8 ft.
6 in. square, and its slant height 21 feet.
4. Find the entire surface of a cone the diameter of whose base
is 6 ft. 9 in. and the slant height 45 ft.
5. Find the cost of painting a church spire, at $.25 a sq. yd., whose
base is a hexagon 5 ft. on each side, and the slant height 60 feet.
926. To find the volume of a pyramid or of a cone.
1. What is the volume, or solid contents, of a square pyramid
whose base is 6 feet on each side, and its altitude 12 feet.
Opekation.— 6 X 6 X 12 -r- 3 — 144 cu. ft., wlume.
2. Find the volume of a cone, the diameter of whose base is 5 ft.
and its altitude lOJ feet.
Operation.— (52 ft. x .7854) x lOJ-f-3 = 68.721^ cu. ft., 'oolume.
Rule. — Multiply the area of the lose hy one-third the altitude.
3. Find the solid contents of a cone whose altitude is 24 ft., and
the diameter of its base 30 inches.
4. What is the cost of a triangular pyramid of marble, whose
altitude is 9 ft., each side of the base being 3 ft., at $2^ per cu. foot ?
5. Find the volume and the entire surface of a pyramid whose
base is a rectangle 80 feet by 60 feet, and the edges which meet at
the vertex are 130 feet.
PYRAMIDS AND CONES. 207
927. To find the convex surface of a frustum of a
pyramid or of a cone.
1. What is the convex surface of a frustum of a square pyramid,
whose slant height is 7 feet, each side of the greater base 4 feet, and
of the less base 18 inches?
Operation. — The perimeter of the greater base is 16 ft., of the less
6 feet.
16 ft. + 6 ft. X 7-5-2 = 77 sq. ft., co7ivex surface.
3. Find the convex surface of a frustum of a cone whose slant
height is 15 feet, the circumference of the lower base BO feet, and
of the upper base 16 feet.
Rule. — Multiply the sum of the perimeters, or of the circumfer-
ences, by one-half the slant height.
To find the entire surface, add to this prodnct the area of both ends, or bases.
8. How many square yards in the convex surface of a frustum
of a pyramid, whose bases are heptagons, each side of the lower
base being 8 feet, and of the upper base 4 feet, and the slant height
55 feet?
928. To find the volume of a frustum of a pyramid
or cone.
1. Find the volume of the frustum of a square pyramid whose
altitude is 10 feet, each side of the lower base 12 feet, and of the
upper base 9 feet.
Operation.— 12' + 9' = 225 ; (225+ ^144x81) x 10-^3=1110 cu.
feet, volume.
2. How many cubic feet in the frustum of a cone whose altitude
is 6 ft. and the diameters of its bases 4 ft. and 3 feet ?
Rule. — To the sum of the areas of loth bases add the square root
of the product, and multiply this sum by one-third of the altitude,
3. How many cubic feet in a piece of timber 30 ft. long, the
greater end being 15 inches square, and that of the less 12 inches ?
4. How many cubic feet in the mast of a ship, its height being
50 ft., the circumference at one end 5 feet and at the other 8 feet If
208 MElf SURATION.
THE SPHERE.
929. A Sphere is a body bounded by a uniformly curved sur-
face, all the points of which are equally distant
from a point within called the center.
930. The Diameter of a sphere is a
straight line passing through the center of the
sphere, and terminated at both ends by its
surface.
931. The Radius of a sphere is a straight line drawn from
the center to any point in the surface.
932. To find the surface of a sphere.
1. Find the surface of a sphere whose diameter is 9 in.
Operation.— 9 in. x 3.1416 = 28.2744 in., circumference,
28.2744 in. X 9 = 254.4696 sq. in., surface.
Rule. — Multiply the diameter hy the circumference of a great circle
of the sphere.
2. What is the surface of a globe 3 feet in diameter ?
3. Find the surface of a globe whose riadiua is 1 foot.
933. To find the volume of a sphere.
1. Find the volume of a sphere whose diameter is 18 inches.
Operation. — 18 in. x 3.1416 = 56.5488 in., circumference.
56.5488 in. x 18 = 1017.8784 sq. in., mrface,
1017.8784 sq. in. x 18-^6=3053.6352 cu. in., mlume,
'Rule.— Multiply the surface by \ of the diameter, or \ of the radius.
2. Find the volume of a globe whose diameter is 30 in.
3. Find the solid contents of a globe whose radius is 5 yards.
934. To find the three dimensions of a rectangu-
lar solid, the volume and the ratio of the dimensions
being given.
1. What are the dimensions of a rectangular solid,, whose volume
is 4480 cu. ft., and its dimensions are to each other as 2, 5, and 7 ?
Operation.— <v/4480 ^ (2 x 5 x 7) = 4 ; 4 ft. x 2 = 8 ft., height .
i ft. X 5 = 20 ft., yyidth; 4 ft. x 7 = 28 ft., length.
REVIEW OF SOLIDS. 209
RuLE.—I. Divide the wlume hy the product of the terms proportional
to the three dimensions, and extract the cube root of the quotient.
II. Multiply the root thus obtained by each proportional term ; the
products will be the corresponding sides.
2. What are the dimensions of a rectangular box whose volume
is 3000 cu. ft., and its dimensions are to each other as 2, 3, and 4 ?
3. A pile of bricks in the fonii of a parallelopiped contains 30720
cu. feet, and the length, breadth, and height are to each other as 3,
4, and 5. What are the dimensions of the pile ?
SIMILAR SOLIDS.
935. Similar Solids are such as have the same form, and
diifer from each other only in volume.
Principles. — 1. The volumes of similar solids a/re to each other as
the cubes of their like dimensions.
1. If the volume of a cube 3 inches on each side is 27 cu. in.,
what is the volume of one 7 inches on each side.
Operation.— 33 : 7^ : : 27 cu. in. : a; = 343 cu. in., volume.
2. The like dimensions of similar solids are to each other as the cube
roots of their volumes.
3. If the diameter of a ball whose volume is 27 cu. in. is 3 in.,
what is the volume of one 7 inches on each side ?
Operation.— ^27 : -^343 :: 3 : aj = 7 in. diameter.
REVIEW OP SOLIDS.
mOBLEMS,
936. 1. What is the edge of a cube whose entire surface is
1050 sq. feet, and what is its volume ?
2. What must be the inner edge of a cubical bin to hold 1250 bu.
of wheat ?
3. How many gallons will a cistern hold, whose depth is 7 ft.,
the bottom being a circle 7 feet in diameter and the top 5 feet in
diameter ?
4. What is the value of a stick of timber 24 ft. long, the larger
end being 15 in. square, and the less 6 in., at 28 cents a cubic foot ?
310 MENSURATION.
5. If a cubic foot of iron were formed into a bar ^ an inch square,
without waste, what would be its length ?
6. If a marble column 10 in. in diameter contains 27 cu. ft., what
is the diameter of a column of equal length that contains 81 cu. ft.?
7. How many board feet in a post 11 ft. long, 9 in. square at the
bottom, and 4 in. square at the top ?
8. The surface of a sphere is the same as that of a cube, the edge
of which is 12 in. Find the volume of each.
9. A ball 4.5 in. in diameter weighs 18 oz. Avoir. ; what is the
weight of another ball of the same density, that is 9 in. in diameter ?
10. In what time will a pipe supplying 6 gal. of water a minute
fill a tank in the form of a hemisphere, that is 10 ft. in diameter?
11. The diameter of a cistern is 8 feet ; what must be its depth
to contain 75 hhd. of water?
12. How many bushels in a heap of grain in the form of a cone,
whose base is 8 ft. in diameter and altitude 4 feet ?
GAUGING.
937 . Gauging is the process of finding the capacity or volume
of casks and other vessels.
A cask is equivalent to a cylinder having the same
length and a diameter equal to the mean diameter oi the
cask.
To find the mean diameter of a cask {yearly).
Add to the head diameter f , or, if the staves
are hut little curved, .6, of the difference between the head and hung
diameters.
To find the volume of a cask in gallons.
Multiply the square of the mean diameter hy the length {both in
inches) and this product hy .0034.
1. How many gallons in a cask whose head diameter is 24 inches,
bung diameter 30 in., and its length 34 inches ?
Operation. — 24 + (30 — 24 x |) =r 28 in., mean diameter,
28^ X 34 X .0034 - 90.63 gal., capacity.
2. What is the volume of a cask whose length is 40 inches, the
diameters 21 and 30 in.,. respectively.
3. How many gallons in a cask of slight curvature, 3 ft. 6 in. long,
the head diameter being 26 in., the bung diameter 31 in. ?
FORMULAS.
811
938.
1. The Diameter s
The Circum-
ference
3. The Area
939.
1. The Surface
2. The Volume
3
CIRCLES.
y = the circumference,
c = the side of an equal square.
\ = the side of an inscribed equi-
\ lateral triangle.
^ = the sideof an inscribed square,
j- = the diameter.
j- = the side of an equal square.
\ = the side of an inscribed equi-
\ lateral triangle.
\ — the side of an inscribed square,
\ — the radius.
— the square of the radius,
\ - the square of the diameter.
i !07958 [ ~ the sq're of the circumference.
SPHERES.
{Circumference x itsdiam,
Radius^ x 12.5664.
IHametef^ x 3.1416.
Circumference^ x .3183.
(Surface x J its diameter,
Badius^ x 4.1888.
Diameter^ x .5286.
Circumference^ x .0169.
X 3.1416
-f- .3183
X .8862
-i- 1.1284
X .8660
-f- .1547
X .7070
-^ 1.4142
X .3183
-i- 3.1416
X .2821
-f- 3.5450
X .2756
-^ 3.6276
X .2251
-i- 4.4428
X .15915
-f- 6.28318
r -^ 3.1416
X 1.2732
■i- .7854
The Diatneter
4. The Circumference
5. The Madius
6. The Side of Inscribed Cube ■•
I \^ 0f surf ace x .5642.
^ Of volume X 1.2407.
I ^ Ofsu^if^e X 1.77255.
^ Of volume x 3.8978.
j ^J Of surfac e x .2821.
\ ^ Of volume X .6204.
j Radius x 1.1547.
\ Diameter x .5774.
213
BEVIEW.
940.
SYNOPSIS FOR EEVIEW.
r 1. Definition. 3. Lines. 3. Angles. 4. Plane Figubes.
1. Defs.
o
H
12;
5. Tri-
angles.
2. Prob-
lems.
Triangle. 2. Right-angled Tri. 3. .
4. ^056. 5. Perpendicular. 6. Altitude, 7. Equi-
lateral Triangle. 8. Isosceles Triangle. 9. Scalene
Triangle. 10. Equiangular Triangle. 11. ^cw^e-
angled Triangle. 12. Obtuse-angled Triangle,
f Area of Triangle.
Either Dimension.
Area of a Triangle.
The Hypothenuse.
The Base or Perp.
882.
883.
884.
886.
887.
To find
y Rule.
, Quad-
rilat-
erals.
7. Circle.
1. Defs.
2. Prob
lems.
I"
( 898.)
( 900. )
Quadrilateral. 2. Parallelogram. 3. Rectangle,
4. Square. 5. Rhornboid. 6. PJioml)us. 7. Trape-
zoid, 8. Trapezium. 9. Altitude,
Parallelogram. \
Trapezoid. > Rule.
Trapezium. )
To find
area of
1. Defs. 1. Circle. 8. Diameter, 3. Radius.
2. Prob-
lems.
904.
905.
906.
907.
908.
909.
■To find
Diam. or Circum. Rule, 1, 2.
^/•ea. Rule, 1, 2.
Diam. or Circ. Rule, 1, 2, 3.
Side of Ins. Square. Rule, 1, 2.
J.rea q/" Circular Ring. Rule.
^ JlfeaTi Proportional. Rule.
Similar Plane Figures. 1. Defs. 2. Prin. 1, 2, 3, 4, 5.
9. SOUDS. -
1. Defs.
. Prob-
lems.
( 1. Solid or Body. 2. Prism. 3. Altitude. 4. Par-
alldopipedon. 5. Cwde. 6. Cylinder. 7. Pyra-
mid. 8. Cone. 9. Altitude of Pyramid or Cone.
10. Slant Height. 11. Frustum. 12.
13. Diameter. 14. Radius.
918.
919.
925.
926.
927. }^ To find
928.
932.
933.
934. J
3. Similar Solids.
10. GA.UGING. 1. Definitions.
Conv.Surf. of Prism or Cyl. Rule.
Volume " " Rule.^
Com.Surf.ofPyr. or Cone. Rule.
Volume " " Rule.
Cbzzv. Surf, of Frustum. Rule.
Fo^^^me " " Rule.
Surface of Sphere. Rule.
Fo/Mm6 " " Rule.
^ Dim. of Rectang. Solid. Rule.
1 Defs. 2. Principles^ 1, 2.
2. Rules.
<
The edges of this cube are each 1 Me^ter^ or 10 Dec/i-me^ters, or 100 Cen^ti-
me'ters^ m length.
ScAiiE, ^ of the Exact Size*
94:1. The MetTic System of weights and measures is based
upon the decimal notation, and is so called because its primary unit
is the Metier.
942. The Me'ter (m.) is the base of the system, and is the
one ten -millionth part of the distance on the earth's surface from the
equator to either pole, or 39.37079 inches.
Me^ter means measure ; and the three principal units are units of lengthy
capacity or wlume^ and weight.
21'i METRIC SYSTEM.
943. The Multiple UnifSf or higher denominations, are
named by prefixing to the name of the primary units the Greek
numerals, Dek'a (10), Hek'to (100), KU'o (1000), and Myr'ia (10000).
Thus, 1 dek'a-me'ter (Dm.) denotes 10 me'ters (m.) ; 1 hek'to-me'ter (Hm.)^
100 me'ters ; 1 kil'o-me'ter (Km.), 1000 me'ters ; and 1 myr'ia-me'ter (Jfm.),
10000 meters.
944:. The Snb-fnultiple Units^ or lower denominations,
are named by prefixing to the names of the primary units the Latin
ordinals, Dec'i (y^), Cen'ti (j^J^), Mil'li (ywff)-
Thus, 1 dec'i-me'ter (dm.) denotes ^, or .1 of a me'ter ; 1 cen'ti-me'ter (cm.),
tJs, or .01 of a me'ter; 1 milli-me'ter {mm\ t^i or .001 of a me^ter.
Hence, it is apparent from the ruwie of a unit whether it is greater ot less than
the standard unit, and also how m^my Um£S.
945. The Metric System being based upon the decimal scale y the
denominations correspond to the orders of the Arabic Notation ; and
hence are written like United States Money, the lowest denomina-
tion at the right. Thus,
g
Ob
^
s
0)
o
1-i
OQ
3
'2
CD
J
E-<
H
»
^
6
7
1
1
««
i
i
^
^
ft
fii
5 '2
P f^ Eh w h
*«
§
I
The number is read, 67015.638 me'ters. It may be expressed in
other denominations by placing the decimal point at the right of the
required denomination, and writing the name or abbreviation after
the figures.
Thus, the above may be read, 670.15638Hm. ; or 67.015638 Km. ;
or 670156.38 dm. ; or 6701563.8 cm. ; or it may be read,
6 Mm. 7 Km. Hm. 1 Dm. 5 m. 6 dm. 3 cm. 8 mm.
Write 3672.045 me'ters^ and read it in the several orders ; read it
in kil'o-me'ters ; in hek'to-me'ters ; in dek'a-me'ters ; in dec'i-
me'ters ; in cen'ti-me'ters.
The names miU, cent^ dim£, used in United States Money, correspond to
mil^li, cent% de&U in the Metric Systeim Hence the eagle might be called the
dek'a-doUar^ since it is 10 dollars ; the dime, a decfirdoUar, eince it is xV of a
dollar, etc.
METRIC SYSTEM.
215
MEASURES OP LENGTH.
946. The Mefter is the unit of length, and is equal to 89.37 in.
or, 1.0936 yd. +.
Metric Denominations. U. S. Value.
1 Mil'li-me'ter = .08937 in.
10 Mil'li-me'ters, mm. = 1 Cen'ti-me'ter — .8937 in.
10 Cen'ti-me'ters, cm. = 1 Dec'i-me'ter = 3.937 in.
10 Dec'i-me'ters, dm. = 1 Meter = 39.37 in.
10 Me'ters, m, =1 Dek'a-me'ter = 32.809 ft.
10 Dek'a-me'ters, Dm. = 1 Hek'to-me'ter=:19.8842 rd.
10 Hek'to-me'ters,J3m. = 1 Kiro-me'ter = .6213 mi.
10 Kil'o-me'ters, Km. = 1 Myr'ia-me'ter=: 6.2138 mi.
Units of long measure form a scale of tens;
hence, in writing numbers expressing length, one
decimal place must be allowed for each denomina-
tion.
Thus, 9652 mm. may be written 965.2 cm., or
96.52 dm., or 9.652 m , or .9652 Dm.
1. The Metier is used in measuring cloths and short dis-
tances.
2. The KU'o-me'ier is commonly used for measuring long
distances, and is about | of a common mile.
3. The Cent'i-me'ter and MU'li-ine'ter are used by mechanics
and others for minute lengths.
4. In business, Dgc'i-me'^ers are usually expressed in CenVi-
me'ters.
5. The BeWa-me'ter, Bek'to-rm'ter. and Myr'ia-me'ter are
seldom used, but their values are expressed as EM'o-me'ters.
EXERCISES.
Read the following :
8.9 m. .
36 dm.
428 cm.
6.57 dm.
346 Dm.
57.9 Hm.
479.6 m.
36.75 mm.
451 Hm.
593.7 Km.
105.6 Dm.
6000 Km.
4 in, 1 ffm
ii:
II
cokl
. ?
13.043 Km.
500.032 m.
31045.7 cm.
216 METRIC SYSTEM.
Change the following to metiers :
327 Dm. 947 cm. 0.72 Km. 30674 mm.
28 Hm. 236 dm. 1.73 Hm. 83.062 cm.
16.8 Km. 43.5 cm. 35.4 Dm. 4000.5 dm.
1. Write 6 kilometers 6 dekameters 6 meters 6 decimeters 6 centi-
meters. Ans. 6.06666 Km., or m.^m^ Hm., or 606.666 Dm., etc.
Write the following, expressing each in three denominations]:
2. 24379 dm.; 15032036 cm.: 2475064 mm.; 30471 Dm.
3. 6704 Hm. ; 85 Km. ; 120000 m. ; 780109 cm. ; 75 m.
Similar examples should be given, until the pupil is familiar with the reduc-
tion of higher to lower, and of lower to higher denominations, by changing the
place of the decimal point and using the proper abbreviations.
947. To add, subtract, multiply, and divide
Metric Denoniiuations.
1. What is the sum of 314.217 m., 53.062 Hm., and 225 cm. ?
Operation. 314.217 m. + 5306.2 m. + 2.25 m. = 5622 667 m., Ans.
2. Find the difference between 4.37 Km. and 1246 m.
Opbration. 4.37 Km. — 1.242 Km. = 3.128 Km., Ans.
3. How much cloth in 8 J pieces, each containing 43.65 m. ?
Operation. 43.65 m. x 8.25 = 384.8625 m., Am.
4. How many garments, each containing 3.5 m., can be made
from a piece of cloth containing 43.75 Dm. ?
Operation. 437.5 m. -s- 3.5 m. = 125 times; hence, 125 garments, Ans.
Rule. — Reduce the given numbers to the same denominations,
when necessary ; then ^proceed as in the corresponding operations with
whole numbers and decimals.
EXEItCISES,
1. Add 7.6 m., 36.07 m., 125.8 m., and 9.127 m.
2. Express as meters and add 475 dm., 3241 cm., and 725 mm.
3. Add 56.07 m., 1058.2 dm., 430765 cm., 6034.58 m., and express
the result in kilometers.
4. From 8.125 Km. take 3276.4 m. Ans. 4.8486 Km.
METRIC SYSa:EM. 217
5. The distance around a certain square is 3.15 Krn. How many
meters will a man travel who walks around it 4 times?
6. How many meters of ribbon will be required to make 32 badges,
each containing 40 centimeters ? Ana. 12.8 m.
7. What will be its cost, at 15 cents a meter?
8. Find the difference between 25.3 Km. and 425.25 m.
9. If an engine runs 36.8 Km. in an hour, how far does it run
between 8 o'clock and 12 o'clock ?
10. In what time will a train fun from Boston to Albany, at the
rate of 46.55 Km, per hour, the distance being about 325.85 Km. ?
11. From a piece of cloth containing 45.75 m., a tailor cut 5 suits,
each containing 7.5 m. How much remained ?
12. A wheel is 3.6 m. around. How many times will it revolve in
rolling a distance of 1.08 Km.? Arts, 300.
MEASURES OF SURFACE.
948. The units of square measure are
squares, the sides of which are equal to a unit
of long measure. 1 sq. cm., Exact Size,
lOOSq.Decl-me'ters ^W^''^'' ^ i =\f^
( 1 Centar {ca.) ) (1.1
lOOSq.Me'ters = -} ! 1' ^f"*. Ui'
100 Sq. Mirii-me'ters(«g.?wm.) = 1 sq. cm, = 0.155 sq. in.
100 Sq. Cen'ti-me'ters = 1 sq. dm. = 15.5 sq. in.
[10.764 sq.ft.
.196sq.yd.
\ 3.954 sq. rd.
.0247 acre.
100 Sq. Dek^a-me'ters = \] l^' f/^' . „ x !■ = 2.471 acres.
( 1 Hektar (Ea.) )
100 Sq. Hek'to-me'ters = 1 sq. Km. = .3861 sq. mi.
Units of square measure form a scale of hundreds; hence, in
writing numbers expressing surface, two decimal places must be
allowed for each denomination.
Thus, 36 sq. m. 4 sq. dm. 27 sq. cm. are written 36.0427 sq. m. ;
and 6 Ha. 5 a. ^ ca. are written 6.0503 Ha., or 605.03 a., etc.
1. The Square Me'teris the unit for measuring ordinary surfaces of small
extent, as floors, ceilings, etc.
2. The Ar, or Square Bek'a-me'ter, is the unit of land measure, and is equal
to 119.6 sq. yd., or 3 954 sq. rd., or .0247 acre.
218 METRIC SYSTEM.
EXEMCTS ES.
1. Read 36145 sq. m., naming each denomination.
Ans. 3 sq. Hm. 61 sq. Dm. 45 sq. m.
2. Write in one number 4 of each denomination from sq. Hm. to
sq. mm., expressed in sq. Hm. Ans. 4.0404040404 sq. Hm.
3. Express the following, each in three denominations ;
6 sq. Km. 6 sq. Hm. 24 sq. Dm. 5 sq. m. ;
16 sq. Dm. 8 sq. m. 4 sq. dm. 15 sq. cm.
4. In 15 sq. Hm. how many square meters?
5. What is the surface of a floor 12 m. long and 7 m. wide ?
6. Add 8 times 4 Ha., 7 times 9 a., and 12 times 14 ca.
7. What is the area of a piece of land 42 Dm. long and 36 Dm.
wide? Ans. 1512 sq. Dm., or 15.12 Ha.
8. Divide 125000 ca. into 8 equal parts.
9. How many times is 2.50 sq. m. contained in 5 Ha. ?
10. How many meters of carpeting 0.6 m. wide will cover a floor
8 m. long and 5.7 m. wide? Ans. 76 m.
11. At 15 cents a sq. m., what is the cost of painting a surface
20.5 m. long and 6.8 m. wide? Ans. $20.91.
12. A man having 5 Ha 8 a. 7 ca. of land, sold .3 of it, at $25 an
ar. What did he receive for what he sold ?
MEASURES OP VOLUME.
949. The units of cubic measure are cubes,
the edfices of which are equal to a unit of long ^ _
^ ^ ° leu. cm., Exact Size.
measure.
1000 Cu. MiVli-meHers {cu. mm.) = 1 cu. cm. = .061 cu. in.
.^^^ ^ r. ,.. ,. \1 cu. dm. ) S .0353 <
1000 Cu. Cen'ti-me'ters = -j ^ ^.,^^^^^^^ f ^^-j ^^^.^^
.... ^ T. ,. ,. \\eu.7n. 35.316
1000 Cu. Dec'i-me'ters == -j ^ g^^^ ^^^ J --j ^^^
L.0567 li. qt.
;.3165cu.ft.
759 cord.
Units of cubic measure form a scale of thousands; hence, in
writing numbers expressing volume, three decimal places must be
allowed for each denomination.
Thus, 42 cu. m. 31 cu. dm. 5 cu. cm. are written 42.031005 cu. m.
The cubic dec^lmeUer, wheu used as a unit of liquid or dry measure, is called
a Wter.
METRIC SYSTEM. 219
WOOD MEASURE..
1000 Cu. Dec'i-me^ters {cu, dm.) ) _ \1 cu. m. | _ j .2759 cord.
10 Dec^-sters {ds.) ) ~" \\Ster,8. \ " (85.3165 cu. ft.
10 Sters = 1 Dek'a-ster, Da, = 2.759 cord.
Units of wood measure form a scale of tens ; hence, but one deci-
mal is required for each denomination.
Thus, 9 Ds. 4 s. 7 ds. are written 94.7 s. ; or 9.47 Ds.
1. The Cubic Metier is the unit for measuring ordinary solids ; as excavations,
embankments, etc.
2. Cubic Cen'ti-me'ters and MiVli-me'ters are used for measuring minute
bodies.
3. The CvMc Me'ter when used as a unit of measure for wood or stone is
called a 8ter.
4. The common Cord is about the same as 3.6 sters ^ or 36 de&i-sters.
EXERCISES,
1. Write 30 Ds. 6 s. 8 ds. Ans. 30.68 Ds.
2. Express in cu. m., 3 cu. m. 3 cu. dm. 3 cu. cm. 3 cu. mm.
Ans. 3.003003008 cu. m.
3. Write and read the following, each in cu. dm., in cu. cm., and
in cu. mm. :
16 cu. m. 275 cu. dm. ; 204 cu. m. .016 cu. dm. .024 cu. cm. ;
10 cu. m. 324 cu. dm. .016 cu. cm. 3244 cu. cm.
4. Express in cu. meters and add : 7 cu.m., 55 cu. dm., 12 cu. m.,
6 cu. dm., 15 cu. cm., 10532 cu. cm. Ans. 19.071547 m.
5. From 36 cu. m. subtract 8 times 42 cu. dm. Ans. 35.664 m.
6. How many cubic meters of brick in a wall 16 m. long, 3 m.
high, and 8 dm. thick? Ans. 38.4 cu. m.
7. How many cu. meters of earth niust be removed in digging a
cellar 16.5 m. long, 8.2 m. wide, and 3.2 m. deep?
8. In a pile of wood 9.3 m. long, 2.8 m. high, and 1.5 m. wide,
how many sters ? Ans. 39.06 s.
9. At $2.25 a ster, what would be the cost of a pile of wood 5.6 m.
long, 3.4 m. wide, and 2.5 m. high ?
10. If a cu. centimeter of silver is worth $.75, what is the value
of a brick of silver 12.4 cm. long, 3.6 cm. wide, and 2.5 cm. thick?
220
METRIC SYSTEM.
MEASURES OF CAPACITY.
950. The Li'ter is the unit of ca-
pacity, both of Liquid and of Dry
Measures, and is equal in volume to one
cu. dtci-me'ter, equal to 1.0567 qt.Liquid
Measure, or .908 qt. Dry Measure.
lOMirii-li^ters, m?.=l Cen'ti-li'ter -
10 Cen^ti-U'ters, c?. =1 Dec'i-li'ter -
10 Dec'i-li'ters, dl. -1 Ijiter
IOLi'terb, /. =1 Dek'a-li^ter -
10 Dek'a-li'ters, J)l = \ Hek'to-li^ter
10 HekHo-li'ters,5?.=l Kil'o-li'teror Stem
10 Ril'o-li'ters, Kl.=\ Myr'ia-li'ter (Ml.)^
Dry M,
.61 cu. in.
: 6.10 " "
.908 qt.
r 9.081 *' :
: 2.837 bu. :
(28.37bu. )
'jl.BOScu.ydJ
:283.73bu. :
Liquid M.
=.338fl'doz.
= .845 gi.
=1.0567 qt.
=2.64175 gal.
=26.4175 "
=264.175 "
=2641.75 "
1. The Li'ter is used in measuring liquids in moderate quantities.
2. The Hek'to-Wter is used for measuring grain, fruit, roots, etc., in large
quantities, also wine in casks.
3. Instead of the KiVo-Wter and MiVli-me'ter^ the Cubic Me'ter and Cuby:
Cen'ti-me'tery which are their equals, may be used.
ext:ti c isljs,
1. Write 5 kiloliters 5 liters 5 deciliters 5 centiliters.
Ans. 5.00555 Kl., or 5005.551.
2. Read, naming each denomination, the following :
45624 cl. ; 306721 ml. ; 76031 dl. ; 89764 i.
3. In 3846 1. how many cl. ? How many Dl. ? Kl. ? dl. ? ml. ?
' 4. Find the sum of 175 1., 25 HI., 42 cl., and 16 dl.
5. From 6 times 25 HI. take 15 times 36 I.
6. Divide 5 HI. of corn equally among 25 persons. Ans. 20 1.
7. From a cask of wine containing 2 HI. of wine, 125 1. were
drawn out. How much remained ?
8. How many HI. of wheat can be put into a bin 3 m. long, 2 m.
wide, and 1.5 m. deep? Ans. 90 HI.
9. What must be the length of a bin 1.5 m. wide, 1 m. deep, to
contain 7500 liters of grain ? Ans. 5 m.
METRIC SYSTEM. 221
MEASURES OF WEIGHT.
951. The Gram is the unit of weighty and is equal to the
weight of a cu, ceii^ti-me' ter of distilled water.
A Gram is equal to 15.432 gr. Troy, or .03527 oz. Avoir.
10 Mirii-grams, mg, = 1 Cen'ti-gram = .1543 + gr. Tr.
10 Cen'ti-grams, eg, = 1 Dec' i -gram = 1.5432+ ** "
10 Dec'i-grams, dg, = 1 Gram ^-l^^fflJ"" 1
\ .03527+ oz.Av.
IOGkams, g. = 1 Dek'a-gram - .3527+" "
10 Dek'a-grams, Dg, = 1 Hek'to-gram = 3.5274+ " '*
-I A XT 1 /+ Tj -, ( Kiro-gram, ) ( 2.6792 lb. Tr.
10 Hek'to-grams, Hg. = 1< ^?.„ >•= i ^ ^^.^ „ .
^ ^ j or KWo ) i 2.2046+ lb. Av.
10 Kiro-grams, Kg. = 1 Myr'ia-gram ■= 22.046 +
10 Myr'ia-grams, IJg., or \
100 Kil'os,
lOQuin'tals, Q., or
1000 Kilos, K f ~ "] or Ton {~ I 1.1023 + tons.
•1 = 1 Quin'tal = 220.46 +
[ _ . j Tonneau, | __ j 22
) ( orTouf~U.:
1. The Gram is used for weighing letters, gold, silver, medicines, and all
small, or costly articles.
2. The KWo-gram or KiVo is the weight of a cu. dm. of water, and is the unit
of common weight in trade, being a trifle less than 2| lb. Avoir.
3. The Ton is the weight of a cu. m. of water, and is used for weighing very-
heavy articles, being about 294^ lb. more than a common ton.
4. The Avoir, oz. is about 28 g. ; the pound is a little less than ^ a kilo.
BXEBCISBS.
1. Read 340642 eg. in grams ; in hectograms; in kilograms.
2. Change 16.5 T. to kilos ; to grams ; to decigrams.
3. If coffee is $.80 a kilo, what will 5 quintals cost ?
4. How many boxes containing 1 gram each, will be required to
liold 1 kilo of quinine ? Ans. 1000.
5. If a letter weighs 3.5 g., how many such letters will weigli
1.015 Kg.? - Ans. 2^0.
6. A car weighing 6.577 T. contains 125 barrels of salt, each
weighing 102.15 K. What is the weight of the car and contents ?
7. Find the difference in the weight of the car and its contents ?
222 METKIC SYSTEM.
952» To change the Metric to the Common Sys-
tem,
1. In 3.6 Km., how many feet?
OPERATION. Analysis.— The meter is
3.6 Km. X 1000 = 3600 m. the principal unit ofthe table;
,^^^^ , hence, reduce the kilometers
39.37 in. x 3600 = 1417^2 in. ^^ ^^^^^8. Since there are
141732 in. -f- 12 = 11811 ft., An^, 39.37 inches in 1 meter, in
3600 m. there are 3600 times
39.37 in., or 141732 in. = 11811 ft. Therefore, 8.6 Km. are equal to 11811 ft.
Rule. — Beduce the metric nnmber to the denomination of the
principal unit of the table ; then multiply by the equivalent , and
reduce the product to the required denomination.
jjxERC is:es.
2- How many feet in 472 centimeters? Ans. 15.485 ft
3. How many cubic feet in 2000 sters ?
4. How many gallons, liquid measure, in 325 deciliters ?
5. How many gallons in 108.24 liters? Ans. 28 gal. 2.77 qt.
6. How many bushels in 3262 kiloliters ?
7. How many acres in 436 ats? Avs. 10.774 A.
8. In 942325 centiliters, bow many bushels?
9. In 456 kilograms, how many pounds ? Ans, 1005.024 lb.
10. In 42 ars, bow many square rods ?
11. Change 75.5 hektars to acres. Ans, 186.56 A.
12. How many gallons in 24J liters of wine ?
13. How many pounds of butter in 124 kilos?
14. In 28 sters, how many cords? An^. 7.725 C.
15. In 72 kilometers, how many miles ?
16. Change 148 grams to ounces Avoirdupois. Ans, 5.22 oz.
17. Change 150.75 kilos to pounds.
18. How many sq. rods in 5 a. 85 ea. ? Ans 23.13 sq, rd.
19. What is the weight of 24 cu. dm. 148 cu. cm. of silver, if a
cu. centimeter weighs 11.4 g.? Ans. 737.553 lb. Tr.
METRIC SYSTEM. 223
953. To change the Common to the Metric Sys-
tem.
I. In 10 lb. 4 oz. Troy, how many kilograms?
OPERATION. Analysis.— The gram,
10 lb. 4 oz. — 10.25 lb. the principal unit of the
10.25 lb. X 5760 = 59040 gr. *^*^1^' ^^ expressed in
erains : hence, reduce the
59040 gr.-15.432gr. = 3825.75 g. %^,^ ,,^ '^^,,es to
3825.75 g. -^ 1000 == 3.82575 Kg., Ans, grains. Since 15.433 gr.
make 1 gram, there are
as many grams in 59040 gr. as 15.432 gr. is contained times in 59040 gr., or
3825.75 g. And since there are 1000 grams in a kilogram, dividing 3885.75 g. hy
1000 g., the quotient is 3.82575. Therefore, there are 3.82575 Kg. in 10 lb. 4 oz.
Rule. — Reduce the given quaTvtity to the denomination in which
the equivalent of the principal unit of the metric table is expressed ;
divide by this equivalent, and reduce the quotient to the required
denomination.
EXEItCISES.
2. In 6172.9 lb av., how many kilograms ? Ans. 2800.009 Kg.
3. How many ars in a square mile ?
4. How many cu. decimeters in 1892 cu. feet ?
5. In 892 gr., how many grams? Ans. 57.8 g.
6. In 2 mi. 272 rd. 5 yd., how many kilometers? Ans. 4.59 Km.
7. How many sters in 264.4 cu. feet ?
8. How many liters in 3 bu. 1 pk. ? Ans. 114.5 1.
9. How many grams in 6 lb. Troy ? In 6 lb. Avoir. ?
10. How many meters in 3 mi. 272 rd. ?
II. In 1828 cu. yd. how many cu. meters? Ans. 1397.52 cu. m.
12. In 3588 sq. yards, how many sq. meters?
13. Bought 454 bu. of wheat, at $3 a bushel, and sold the same
at 18.75 per hektoliter ; how many hektoliters did I sell ? Did I
gain or lose, and how much ? Ans. 160 HI. ; gain, $38.
14. In 13 gal. 3 qt. 2 pt. 3 gi., how many liters?
Ans. 53.351.+.
15. Sow many sq. meters of plastering in a room" 18 ft. 6 in.
long, 14 ft. wide, and 9 ft. 6 in. high? Ans. 55.367 sq. in. +.
224 METRIC SYSTEM.
TEST PEOBLEMS.
954. 1. Find the weight of a barrel of flour (196 lb.) in Kg. ?
2. What is the cost of a carpet for a room 10.5 m. long, and 8.4 m.
wide, if the carpet is 84 cm. wide and costs $2.75 a meter?
Ans. $288.75.
3. A farmer sold 540 HI. of wheat, at $2 a bushel, and invested
the proceeds in coal at $7 per ton. How many tons did he buy ?
Ans. 437.785 T. + .
4 What is the cost of a building lot 75 m. long and 63 m. wide,
at $40 an ar ? Ans, $1860.
5. A bushel of wheat weighs 60 lb. What is the weight of 5 HI.
of wheat, in kilograms ? Ans. 386.05 Kg.
6. What will be the cost of a pile of wood 15. 7 m. long, 3 m.
high, and 7.53 m. wide, at $1.50 a ster?
7. The new silver dollar weighs 412 J gr. Troy. How many
grams does it weigh ? A7is. 26.73 g.
8. How many acres of land in 24.6 Km. of a highway, which is
20 m. wide ? Ans. 121.573 A.
9. A bin is 4.2 m. long, 2.8 m. wide, and 1.5 m. deep. What will
be the cost of filling it with charcoal, at 25 cts. a hektoliter ?
10. A merchant bought 300 m. of silk in Lyons, at 12.5 francs a
meter ; he paid 75 cents a yard for duty and freight, and sold it in
New York at $5 a yard. What was his gain ? Ans. $670.61.
11. What price per pound is equivalent to $2.50 per Hg. ?
13. If a man buys 5000 g. of jewels, at 35 francs a gram, and sells
them at $15 a pennyweight, what was his gain or loss ?
13. If a field produces 40 HI. of oats to the hektar, how many
bushels is that to the acre? Ans. 45.93 bu.
14. What price per peck is equivalent to 80 cts. a dekaliter ?
15. What will be the cost of excavating a cellar 18.3 m. long,
10.73 m. wide, and 3.4 m. deep, at 20 cents per ster ?
16. How many pounds Avoir, are there in 96.4 kilos of salt ?
17. How many liters will a cistern hold that measures on the
inside 5.5 ft. long, 4 ft. 6 in wide, and 4 ft. djep ? Ans. 3803.383 I.
METRIC SYSTEM. 225
18. How many meters of lining that is 60 cm. wide will line
15 m. of silk that is 75 cm. wide ? Ans. 18,75 cm.
19. A lady bought 40.5 m. of silk in Paris. What would be its
value in Boston, at $4 75 per yard ?
20. A bin is 4 m. long, 2.3 m. wide. How deep must it be to
contain 40 HI. of grain? Ans. 4.347 + dm.
21. How many sters of wood can be piled in a shed 8.5 m. loag,
5.8 m. wide, and 4.2 m. high ? What would be its value at $3.25 a
cord? ^;is. 207.03 8.; $185,665.
22. A dray is loaded with 60 bags of grain, each bag holding
8 Dl. ; allowing 75 K. of grain to the hectoliter, what is the weight
of the load in metric tons ? Ans. 3.6 T.
23. How many meters of shirting, at $.18 per meter, must be
given in exchange for 250 HI. of oats, at $1.20 per hectoliter?
24. A merchant shipped to France 50 barrels of sugar, each con-
taining 250 lb., paying $2 per cwt. for transportation. He sold the
sugar at $.34 per kilogram, and invested the proceeds in broadcloth,
at $4 per meter. How many yards did he purchase ?
25. A cu. decimeter of copper weighs 8.8 Kg. What is the value
of a bar of the same metal 15 dm. long, 9.6 cm. broad, and 6.4 cm.
thick, at $1.30 a kilogram? Ans. $105.43.
26. How many bricks, each 20 cm. lon^ and 10 cm. wide, will
pave a walk 95.4 m. long and 2.1 m. wide; and what will they
cost, at $1.75 per hundred ? Ans. 10017 bricks ; $175,297.
27. What is the value of a pile of wood 40 ft. 6 in. long, 4 ft.
broad, and 6 ft. 6 in. high, at $6.50 per dekastere ?
28. What will be the cost of building a wall 96 Dm. 6 m. 8 dm.
long, 1 m. 6 dm. thick, and 2 m. 4 cm. high, at $6.75 a cu. metiV?
29. A wine merchant imported to Boston 1000 dekaliters of wine,
at a cost of $.75 a liter, delivered. At what price per gallon must
he sell the same to clear $2000 on the shipment ? Ans. $3,596.
30. How many gallons of water will a cistern contain that is 3 m.
deep, 2 m. long, and 1.5 m. wide; and what will be its weight in
metric tons ? Ans. 2377.575 gals. ; 9 T.
226
METRIC SYSTEM.
TABLE OF EQUIVALENTS.
955. The equivalents here given agree with those that have
been established by Act of Congress for use in legal proceedings and
in the interpretation of contracts.
1 inch = 2.540 centimeters.
1 foot = 3.048 decimeters.
1 yard =: 0.9144 meter.
1 rod — 0.5029 dekameter.
1 mile = 1.6093 kilometers.
1 sq. in. = 6.452 sq. centimeters.
1 sq. ft. =: 9.2903 sq. decimeters.
1 sq. yard = 0.8361 sq. meter.
1 sq. rd. = 25.293 sq. meters.
1 acre = 0.4047 hektar.
1 sq. mile =: 2.590 sq. kilometers.
1 cu. in. = 16.387 cu. centimeters.
1 cu. ft. r= 28.317 cu. decimeters.
1 cu. yard = 0.7645 cu. meter.
1 cord = 3.624 sters.
1 liquid quart = 0.9463 liter.
1 gallon = 0.3785 dekaliter.
1 4ry quart = 1.101 liters.
1 peck = 0.881 dekaliter.
1 bushel = 3.524 dekaliters.
1 ounce av. = 28.35 grams.
1 pound av. = 0.4536 kilogram.
1 T. (2000 lbs.) = 0.9072 met. ton.
1 grain Troy = 0.0648 gram.
1 ounce Troy = 31.1035 grams
1 pound Troy = 0.3732 kilogram.
1 centimeter = 0.3937 inch.
1 decimeter = 0.328 foot.
1 meter = 1.0936 yds. - 39.37 in.
1 dekameter = 1.9884 rods.
1 kilometer = 0.62137 mile.
1 sq. centimeter = 0.1550 sq. in.
1 sq. decimeter = 0. 1076 sq. ft.
1 sq. meter — 1.196 sq. yards.
1 ar =r 3.954 sq. rods.
1 hektar = 2.471 acres.
1 sq. kilometer = 0.3861 sq. mi.
1 cu. centimeter — 0.0610 cu. in.
1 cu. decimeter = 0.0353 cu. ft.
1 cu. meter = 1.308 cu. yards.
1 ster = 0.2759 cord.
1 liter = 1.0567 liquid quarts.
1 dekaliter = 2.6417 gallons.
1 liter = 0.908 dry quart.
1 dekaliter = 1.135 pecks.
1 hectoliter = 2.8375 bushels.
1 gram = 0,03527 ounce Av.
1 kilogram = 2.2046 pounds Av.
1 metric ton =: 1 .1023 tons.
1 gram = 15.432 grains Troy.
1 gram = 0.03215 ounce Troy.
1 kilogram = 2.679 pounds Troy.
PARTIAL PAYMENTS. 227
VERMONT RULE EOE PARTIAL PAYMENTS.
956. The General Statutes of Vermont provide the following
HuLE for computing interest on notes, when partial payments have
been made :
*' On all notes y hills, or other similar obligations, whether made
payable on demand or at a specified time, with interest, when
payments are made, such payments shall be applied : first, to liqui-
date the interest that has accrued at the time of such payments ;
and, secondly, to the extinguishment of the principal.
*' On all notes, bills, or other similar obligations, whether made
payable on demand or at a specified time, with interest annu-
ally, the annual interests that remain unpaid shall be subject to
simple interest, from the time they become due to the time of final
settlement ; but if in any year, reckoning from the time such annual
interest began to accrue, payments have been made, the amount of
such payments at the end of such year, with interest thereon from the
date of payment, shall be applied : first, to liquidate the simple inter-
est that has accrued upon the unpaid annual interests ; secondly, to
liquidate the annual interests that have become due; and thirdly, to
the extinguishment of the principal.'*
EXERCISES.
$3458. Bradford, Vt., Sept. 13, 1869.
1. For value received, I promise to pay E. W. Colby or order three
thousand four hundred and fifty-eight dollars, on or before the first
day of January, 1878, with interest. Samuel S. Green.
Indorsed as follows: Dec. 16, 1870, $100; May 1, 1871, $1000;
Jan. 13, 1874, $85 ; April 13, 1876, $450.75.
What was due Jan. 1, 1878? Ans. $3239.90.
%^'^^' St. Johnsbury, Vt., Nov. 22, 1868.
^ 2. For value received, I promise to pay James Ferguson or order
eight hundred and seventy two dollars, on demand, with interest
annually. Sylyanus E. Boyle.
Indorsed as follows : April 4, 1869, $28 ; July 10, 1872, $94.40 •
Dec. 10, 1874, $6.72 ; Jan. 14, 1877,
What was due Dec. 28, 1878 ?
228 PARTIAL PAYMENTS.
OPERATION.
Int. oil Yearly
Int. Int. Prin,
int. of prin. to Nov. 22, 1869 . , . . . $52.32
Am't of 1st payment 29.06
Bal. of unpaid yearly int. ...... 23.26
Int. of prin. to Nov. 22, 1872 156.96
Int. on 1 year's int. 3. years $9.42
int. on bal. of unpaid yearly int. 3 years . 4.1J^ 13.61
193^83
Am't of 2d payment 96.48
Bal. of unpaid yearly int 97.35
Int. of prin. to Nov. 22, 1875 156.96
Int. on 1 year's int. 3 years 9.42
Int. on bal. of unpaid yearly int. 3 years .17.52
26.94 25431
Am't of 3d payment 7.10
Bal. of int. on int 19.84
Int of prin. to Nov. 22, 1877 104.64
Int. on 1 year's int 1 year 3.14
Int. on bal. of unpaid yearly int. 2 years . 30.52 53.50 412.45
1284.45
Am't of 4tli payment 416.33
New principal 868.12
Int. of new prin. to Dec. 28, 1878 57.30
Int. on 1 year's int. 1 mo. 6 d .31
Due, Dec. 28, 1878 $925.73
ExPL\NATiON.— We compute the interest for one year from the date of the
note, as a payment is made within that year, and deduct the amount of the pay-
ment at the end of the year from the interest due. The balance of interest hears
interest till Nov. 22, 1872. The amount of the payment at the end of this year
exceeds the interest on interest due. We therefore deduct the amount of the
payment from the total interest due, and have a balance of unpaid yearly inter-
est, $97.35, which bears simple interest till Nov. 22, 1875. At this date the
amount of the payment is less than the interest on interest due. We there-
fore deduct the amount of the payment from the amount of interest on interest,
and have a remainder of $19.84, which is without interest. The amount of un-
paid yearly interest at this date bears simple interest till the next balance.
PARTIAL PAYMENTS. 220
The amount of the fourth payment, Nov. 22, 1877, exceeds the total interest
due. We therefore deduct it from the sum of the interest and principal. The
remainder forms a new principal, which bears simple interest to the settlement
of the note, Dec. 28, 1878, and one year's interest on the same bears interest from
Nov. 22, 1878, to Dec. 28, 1878, which interest, added to the new principal, gives
the amount due Dec. 28, 1818— $925.73.
In cases of annual interest with partial payments, like the above
example, obssrve the following notes ;
1. To avoid compounding interest, keep the principal, unpaid yearly inter-
ests, and interest on yearly interet^t, in separate columns.
2. Deduct the amount of the payment or payments at the end of the year
from the interest on the unpaid yearly interest, when it does not exceed this
interest. The remainder never draws interest, but is liquidated by the first pay-
ment that equals or exceeds it.
3. Deduct the amount of the payment or payments at the end of the year
from the sum of the unpaid yearly interests and the interest on the unpaid
yearly interests, when this amount exceeds the interest on the interest, but is
less than such sum. The remainder is a balance of unpaid yearly interest which
draws simple interest imtil canceled by a payment.
4. Deduct the amount of the payment or payments at the end of the year
from the sum of the total interest due and the principal, when it exceeds the
total interest due. The remainder forms a new principal, with which proceed
as with the original principal.
$5000. Newport, Vt., Oct. 19, 1862.
3. For "oalue received, we jointly and severally promise to pay John
Smith or hearer five thousand dollars, sixteen years after date, with
interest annually, Geo. S. Leazer.
E. D. Crawford.
Indorsed as follows : Jan. 13, 1866, |393 ; Sept. 24, 1866, $48 ;
July 10, 1869, $493.47; Oct. 14, 1873, $100; Dec. 12, 1877, $3200;
April 15, 1878, $65.
What was due Oct. 19, 1878? Ans, $7056.17.
Burlington, Yt., March 23, 1872.
^. For value received, I promise to pay Jas. B. Vinton or order
four hundred and tweiity dollars, six years from date, with interest
annually. Geo. A. Bancroft.
Indorsed as follows ; Oct. 3, 1873, $40.23 ; March 1, 1874, $8 :
Sept. 13, 1875, $33.38.
What was due March 23, 1878 ? Ans. $494.62.
230
PARTIAL PAYMENTS.
Barton, Vt. Aug. 20, 1873.
5. For value receivedy I promise to pay E. J. Baxter or order six
hundred and thirty-nine dolla/rSj on demand, with interest annually,
Samuel Macomber.
Indorsed as follows : Oct. 14, 1877, $10 ; Dae. 24, 1878, $20.
What was due March 30, 1879 ? Ans, $904.58. "
TABLE.
Showing amount of $1.00 from 1 to 20 years, at ^ 5, 6, 7 and 8 per
cent.y Annual Interests
Years.
4 per cent.
5 per cent.
6 per cent,
7 per cent.
8 per cent.
Years.
1 .
$1 0400
$1.0500
$1.0600
$1.0700
$1.0800
. 1
2 .
1.0816
1.1025
1.1236
1.1449
1.1664
. 2
3 .
1.1248
1.1575
1.1908
1.2247
1.2592
. 3
4 .
1.1696
1.2150
1.2816
1.3094
1,3584
. 4
5 .
1.2160
1.2750
1.3360
1.3990
1.4640
. 5
6 .
1.2640
1.3375
14140
1.4935
1.5760
. 6
7 .
1.3136
1.4025
1.4956
1.5929
1.6944
. 7
8 .
1.3648
1.4700
1.5808
1.6972
1.8192
. 8
9 .
1.4176
1.5400
1.6696
1.8064
1.9504
. 9
10 .
1.4720
1.6125
1.7620
1.9205
2.0880
. 10
11 .
1.5?80
1.6875
1.8580
2.0395
2.2320
. 11
12 .
1.5856
1.7650
1.9576
2.1634
2.3824
. 12
13 .
1.6448
1.8450
2.0608
2.2922
2 5392
. 13
14 .
1.7056
1.9275
2.1676
2.4259
2.7024
. 14
15 .
1.7680
1.0125
2.2780
2.5645
2.8720
. 15
16 .
1.8320
2.1000
2.3920
2.7080
3.0480
. 16
17 .
1.8976
2.1900
2.5096
2.8564
3.2304
. 17
18 .
1.9648
2.2825
2.630S
3.0097
3.4192
. 18
19 .
2.0336
2.3775
2.7556
3.1679
3.6144
. 19
20 .
2.1040
2.4750
2.8840
3.3100
3.8160
20
ASSESSMEiq^T OF TAXES. 231
VEEMONT METHOD OP ASSESSING TAXES.
957. The Grand List is the base on which all taxes are assessed ;
it is Ifo of the appraised value of the real estate and personal
property, together with the poll list.
The Poll List is $2.00 for every male inhabitant, from 21 to 70
years of age, except such as are specially exempt by law.
The General Statutes of Vermont provide that the listers in each
town shall make a list of all the real estate and personal property,
and the number of taxable polls in such town, and that the said
list shall contain the following particulars :
*' First. The name of each taxable person.
*' Second, The number of polls and the amount at which the same are set in
the list.
*' TMrd, The quantity of real estate owned or occupied by such person.
" Fourth, The value of such real estate.
" Fifth. In the fifth column the full value of all taxable personal estate owned
by such person.
" Sixth. In the sixth column shall be set the one per centum on the value of
all personal and real estate, tojjether with the amount of the polls, which sum
shall be the amount on which all taxes shall be made or assessed.
The State and County Taxes are assessed by the Legislature.
The minimum of the State School and Highway Taxes is fixed by
law, and a higher rate left optional with the town.
A Town Tax is assessed by vote of the town, a Village Tax by
vote of the village, and a School District Tax by vote of the district,
EXERCISES.
1. The town of Montpelier voted a town tax of $2.60 on each
dollar of the grand list. The appraised value of the real estate was
$702727, and of the personal property $309987, and there were
740 taxable polls. What was the grand list of the town? How
much money wfis raised by this vote ? Wh^t was John Hammond's
town tax, who was 30 years of age, and whose property was ap-
praised at $8927.75?
232 ASSESSMENT OF TAXES.
OPERATION.
$702727 + $309987= $1012714, assessed value of the property.
$1012714 X .01 =$10127.14, 1 % of the assessed value.
$2.00 X 740= $1480, the poll list.
$10127.14-4- $1480=111607.14, the grand list.
$2.60 X 11607.14=$30178.56, amount of money raised.
$8927. 75 X. 01 = $89.28,1% of the assessed value of John Ham-
mond's property.
$89.28 + $2.00, his poll list = $91.28, John Hammond's grand list.
$2.60 X 91 .28 =$237. 33, John Hammond's town tax.
2. The appraised value of property, both real and personal, in
the town of Rutland, for the year 1878, was $3415264. The num-
ber of taxable polls was 2066. The town voted to raise a tax of
$28713.48. What was the tax on a dollar of the grand list ?
Ans, $0.75.
3. The appraised value of the real estate in the city of Burling^
ton was $2542373; of the personal property, $399937. There
were 2040 taxable polls. The city voted to raise $60305.58 city
tax. What was the amount of Henry Cook's tax, a resident, who
was 73 years of age, and whose real estate was appraised at $750,
and his personal property at $475.50 ? Ans. $22.06.
4. The grand list in the town of Chelsea was $4403.74. The ap-
praised valug of all the property was $368774. How many taxable
polls were there in that town ? Ans. 358.
5. The estimated cost of schools in school district No. 8, in the
town of Cabot, for one year, was $765. The amount of public
money received from the town was $71.50. The appraised value of
the real estate in the district was $48545 ; of the personal estate
$15428.75 ; the number of taxable polls in the district 103. How
much tax on a dollar of the grand list must the district vote, to pay
its expenses ? Ans. $0.82.
6. James Bell resides in Hardwick ; he is 44 years of ag-e ; his
property, both real estate and personal, is appraised at $8975.50.
Hardwick voted a town tax of $1.60 on a dollar of the grand list.
The highTvay tax is $0.40 : the state tax is $0.45 ; the state school
tax is $0.09 ; the school tax is $0.86 ; and the county tax $0.01, on
the dollar. What is the amount of his taxes ? Ans. $315.64.
.^x^IZg^^ag^Cv-. g-^^^
MEASTJUE S E^
1. A Measure is a standard unity established by
law or custom, by which quantity, as extent, dimension,
capacity, amount, or value is measured or estimated.
Thus, the standard unit of Measures of Extension is the yard ;
of Liquid Measure, the wine gallon ; of Dry Measure, the Winches-
ter hushel; of Weight, the Troy 'pound , etc. Hence the length of a
piece of cloth is ascertained by applying the yard measure ; the
capacity of a cask, by the use of the gallon measure ; of a bin, by
the use of the hushel measure ; the weight of a body, by the pound
weight, etc.
3* Measures may be classified into six hinds :
4. Time.
1. Extension.
2. Capacity.
3. Weight.
5. Angles or Arcs.
6. Money or Value.
MEASUEES OF EXTEITSIOK
3. Extension has length, Ireadth, and thichness.
4t A Line has length only.
5. A Surface or Area has length and breadth.
6. A Solid has length, Ireadth, and thichness.
^34 EXTEiq^SIOiq'.
LINEAR MEASURE.
'7. Linear Measure^ also called Long MeaS'
ure IS used in measuring lines and distances.
Table.
13 Inches (in.) = 1 Foot .... /if.
3 Feet = 1 Yard . . , . pd.
5i Yards, or) ^ ^ ^^^ . . . , rd.
\U Feet )
320 Rods - 1 Mile .... mi.
1 ML =
63360 in,
5280 /If.
1760 yd.
320 rd.
8. Cloth Measure is practically out of use. In
measuring goods sold by the yard, the yard is divided
into halves, fourths, eighths, and sixteenths,
2i Inches = 1 Sixteenth, -jV yd.
2 Sixteenths, (4^ in.) = 1 Eighth. ^ yd.
2 Eighths, (9 in.) = 1 Quarter, \ yd.
4 Quarters = 1 Yard, 1 yd.
At U. S. Custom-Houses, in estimating duties, the yard is divided
into tenths and hundredtJis.
9. Mariners use the following denominations :
9 Inches = 1 Span, Sp.
8 Spans or 6 Ft. = 1 Fathom, fath.
120 Fathoms = 1 Cable's Length, c. I.
74 C. Length = 1 Nautical Mile (or Knot), mi.
3 Miles = 1 League, lea.
10. In geographical and astronomical calculations :
1 Geographic Mile = 1.152| Statute Miles.
3 " *' = 1 League.
60 " " or ) _ ^ j^ j of Latitude on a Meridian,
69.16 Statute " ) (or of Long, on the Equator
360 Degrees = the Circumference of the Earth.
MEASURES OF EXTENSION.
235
11. The following are sometimes used :
3 Barley-corns, or Sizes = 1 Inch. Used by slioemakers.
4 Inches = 1 Hand.
, J j to measure the height of
( horses at the shoulder.
8-1% Feet
3 Inches
21.888 Inches
6 Points
12 Lines
= 1 Pace.
= 1 Palm.
=z 1 Sacred Cubit.
= 1 Line. )
= 1 Inch. )
Used in clock-making.
1. The nautical mile (or knot) is the same as the geographical
mile, and is used in measuring the speed of vessels.
2. The geographical mile is (3V of 3^77 or j-rhoo of the distance
round the center of the earth. It is a small fraction more than 1.15
statute miles.
3. The length of a degree of latitude varies, being 68.72 miles at
the equator, 68.9 to 69.05 miles in middle latitudes, and 69.30 to
69.34 miles in the polar regions. The mean or average length,
69.16, is the standard recently adopted by the U. S. Coast Survey.
A degree of longitude is greatest at the equator, where it is 69.16
miles, and it gradually decreases toward the poles, where it is 0.
12. Surveyors^ Linear Measure is used by
land suryeyors in measuring roads and boundaries of
land.
Table.
7.92 Inches r== 1 Link . .
25 Links = 1 Rod . .
4 Rods = 1 Chain . .
80 Chains = 1 Mile . .
1. A Gtmter^s Chain is the unit of measure, and is 4 rods, or
66 feet long, and consists of 100 links.
2. Engineers commonly use a chain or measuring tape, 100 feet
long, each foot divided into tenths.
3. Measurements are recorded in chains and hundredths.
I
' 63330 in.
rd.
ch.
1 Mi. = ^
8000 I.
320 rd.
mi.
'&<Sch.
236
MEASURES OF EXTEKSION.
13.
COMPARISON OF DISTANCES.
Country.
Distance. U.
S.mile.
Country.
Distance.
U. S. mile.
England,
1 Mile =
1
Russia,
1 Verst
= .66
France,
1 Km. =
.62
Turkey,
1 Berri
= 1.04
Spain,
1 League =
4.15
Portugal,
IMilha
= 1.28
Prussia,
1 Meile =
4.93
Persia,
1 Farsang
= 4.17
Austria,
1 Meile =
4.98
China,
ILi
= .35
Sweden,
IMil =
QM
Egypt,
1 Mill
= 1.15
Switzerland,
1 Lieue =
2.98
East Indies,
1 Coss
= 1.14
Mexico,
1 SiUo =
6.76
Japan.
IRi
=2.562
SURFACE OR SQUARE MEASURE.
14. Surface or Square Measure is used in
computing areas or surfaces ; as of land, boards, paint-
ing, plastering, paving, etc.
Table.
144 Square Inches (Si
9 Square Feet
30 J: Square Yards
160 Square Rods
640 Acres
sq. mi. A. sq, rd.
sq.ft.
, in.) = 1 Square Foot
= 1 Square Yard . . sq. yd.
= 1 Sq. Rod or Perch sq. rd.; P.
= 1 Acre A.
= 1 Square Mile . . sq. mi.
sq. yd. sq ft. sq. in.
1 = 640 = 102400 = 3097600 = 27878400 = 4014489600
1 = 160 =^ 4840 = 43560 =^ 6272640
1 = 30i= 272J:= 39204
1 = 9 = 1296
1 = 144
15. Artificers estimate their work as follows :
By the square foot ; as in glazing, stone-cutting, etc.
By the square yard, or by the square of 100 square feet ; as in
plastering, flooring, roofing, paving, etc.
One thousand shingles, averaging 4 in. wide, and laid 5 in. to the
weather, are estimated to be a squa/re.
MEASURES OF EXTEKSIOK. 237
16. Surveyors^ Square Measure is used by
surveyors in computing the area or contents of land.
Table.
625 Square Links {sq, I.) =1 Pole P.
sq. ch.
16 Poles = 1 Square Chain
10 Square Chains = 1 Acre . . .
. A.
. sq. mi,
. Tp.
l.
640 Acres = 1 Square Mile .
36 Square Miles (6 miles square) = 1 Township .
Tp. sq. mi. A. sq. ch. P. sq.
1 = 36 = 23040 = 230400 = 3686400 = 2304000000
1 = 640 = 6400 = 102400 = 6400000
1 = 10 = 100 == 100000
1. The A^cve is the unit of land measure.
2. Government lands are divided into Townships, by parallels and
meridians, each containing 36 square miles or Sections. Each sec-
tion contains 640 acres (1 sq. mile), and is subdivided into half-sec-
tionSy qua/rier -sections, etc.
3. Measurements of land are commonly recorded in square miles,
acreSy and hundredth's of an acre. The rood is no longer used.
CUBIC OR SOLID MEASURE.
17. Cubic or Solid Measure is used in com-
puting the contents of solids ; as timber, wood, stone,
boxes of goods, the capacity of rooms, etc.
Table.
1728 Cubic la (C2^.m.)=l Cubic Ft., cu.fi. L '^^ ^_ j 46656 cw.iX
27 Cubic Ft. =1 CnhicYd., cu.yd.] ( %1cu.fi.
1. A Begist&r Ton, used in measuring the entire internal capacity
or tonnage of vessels, is 100 cubic feet.
2. A Shipping Ton, used in measuring ca/rgoes, is 40 cubic feet in
the U. S.^ and in England 42 cubic feet.
238 MEASURES OF EXTENSION.
18. Wood Measure is used to measure wood and
rough stone.
Table.
16 Cubic Feet = 1 Cord Foot .... cd.ft.
^\ r= ICord Cd.
128 Cubic Feet )
% Cord Feet, or)
18 Cubic Feet )
24^ Cubic Feet = 1 \ ^^^^^ °^ ^^^^^^ I Pch,
I or of Masonry. )
or of Masonry.
A Cord of wood is a pile 8 ft. long, 4 ft. wide, and 4 ft. high.
A cord-foot is 1 ft. in lengtb of such a pile ; that is, 1 ft. long,
4 ft. wide, and 4 ft. Mgh.
A Perch of stone or of masonry is 16^ ft. long, \\ ft. wide, and
1 ft. high. Stone-masons usually call 25 cu. ft. a perch.
19. Duodecimals are the parts of a unit resulting
from continually dividing by 12, and are sometimes used
in measuring surfaces and solids.
Table.
12 Fourths ("") = ! Third
12 Thirds ~ 1 Second
12 Seconds = 1 Prime
12 Primes = 1 Foot .
The marks ', ", " ', " ", are called indices.
Railroad and transportation companies estimate light freight by
the space it occupies in cubic feet, but heavy freight by weight.
Masonry is estimated by the cubic foot, and perch ; also by the
square foot and square yard.
Materials are usually estimated by cubic measure ; the uork by
cubic, or by square measure.
Engineers, in making estimates for excavations and eTrtbanTcmentSy
take the dimensions with a line or measure divided into feet and
decimals of a foot. The computations are made in feet and deci-
mals, and the results are reduced to cubic yards. In civil engineer-
ing, the cubic yard is the unit to which estimates for excavations
and embankments are finally reduced.
V"
^ 20736'
1"
V
lFt. = ^
1728'
144'
Ft.
12'
MEASURES OF EXTENSION. 239
A cubic yard of common earth is called a load.
Brickwork is generally estimated by the 1000 bricks, sometimes in
cubic feet.
Bricks are of various dimensions. The average size of a common
brick is 8 in. long, 4 in. wide, and 2 in. thick.
Philadelphia or Baltimore front bricks are 8^: x 4^ x 2| inches ;
North Eiver bricks, 8 x 3J^ x 2 J inches ; Maine bricks, 7^ x 3J x2|;
and Milwaukee bricks, 8^ x 4^ x 2| inches.
A cubic foot is estimated to contain 27 bricks laid dry. Laid in
mortar, an allowance is made of from -,^j to i for the mortar.
Five courses of bricks in the height of a wall are called a foot.
A brick wall which is a brick and a half thick is said to be of the
standard thickness.
In estimating material, allowance is made for doors, windows,
and cornices.
In estimating the work, masons measure each wall on the outside.
Ordinarily, no allowance is made for doors, windows, and cornices,
but sometimes an allowance of one-Mlf is made, this being a matter
of contract.
In scaling or measuring timber for shipping or freighting, I of
the solid contents of round timber is deducted for waste in hewing
or sawing. Thus, a log that will make 40 feet of hewn or sawed
timber, actually contains 50 cubic feet by measurement ; but its
market value is only equal to 40 cubic feet of hewn or sawed
timber.
Sawed timber, joists, plank, and scantlings are geijerally bought
and sold by what is called board measure. Hewn and round timber
by cvMc measure.
In board and lumber measure, all estimates are made on 1 inch
in thickness ; in buying and selling lumber, one-fourth the price is
added for every \ inch thickness over an inch.
In Board Measure all boards are assumed to be 1 in. thick.
A board foot is 1 ft. long, 1 ft. wide, and 1 inch thick ; hence 12
board feet make 1 cubic foot.
Board feet are changed to cubic feet by dividing by 12.
Cubic feet are changed to board feet by multiplying by 12.
C 32 gi.
1 Gal, = ■} Spt,
24fO MEASURES OF CAPACITY.
MEASUEES OF CAPACITY.
30. Capacity signifies extent of room or space.
31. Measures of capacity are divided into two classes ;
Measures of Liquids and Measures of Dry Substances.
33. The Units of Capacity are the Gallon for Liquid,
and the Bushel for Dry Measure.
LIQUID MEASURE.
33. Liquid Measure is used in measuring liquids ;
as spirituous liquors, oil, molasses, milk, water, etc.
Table.
4 Gills (gi.) = 1 Pint . . . pt.
2 Pints = 1 Quart . . . qt.
4 Quarts = 1 Gallon . . gcd.
The Standard Liquid Gallon of the United States contains 231
cubic inches, and is equal to about 8^ lb. Avoir, of pure water.
The Imperial Gallon of Great Britain contains 277.274 cubic
inches, and is equal to about 1.2 U. S. liquid gallons.
The Old Ale or Beer Measure is out of use. The gallon contained
282 cubic inches.
34. In estimating the capacity of cisterns, reservoirs,
etc.:
31i Gallons make 1 Barrel . . . Hbl.
63 Gallons " 1 Hogshead . . hhd.
1. The barrel and hogshead are not jfixed measures, but vary
-when used for commercial purposes, the former containing from 28
to 45 gallons, the latter from 60 to 125 gallons.
2. In some of the New England States the barrel is estimated at
32 gallons ; in some States 31^ gallons, and in others from 28 to 32.
3. The tierce, hogshead, pipe, puncheon, butt and tun are the
name of casks, and do not express any fixed or definite measures.
They are usually gauged, and have their capacities in gallons
marked on them.
MEASURES OF CAPACITY.
241
35. Apothecaries^ Fluid Measure is used by
physicians and apothecaries in prescribing and com-
pounding liquid medicines.
Table.
60 Minims, or drops (TTL or gtt, ) = 1 Fluidraclim ,. . /3 .
8 Fluidraclims = 1 Fluidounce . . /I .
16 Fluidounces = 1 Pint 0.
8 Pints = 1 Gallon .... Gong,
Cong. 1 = 0. 8 =/f 128 =fl 1024 = Itl 61440.
0. is an abbreviation of octans, tlie Latin for one-eighth ; Gong, for
congiarium, the Latin for gallon.
A common teaspoonful, or 45 drops, makes about one fluidrachm.
A common teacup holds about 4 fluidounces ; a common tablespoon,
about half a fluidounce ; a pint of water weighs a pound.
![^is an abbreviation for recipe , or take ; a., aa., for equal quanti-
ties; j. for 1 ; ij. for 2 ; ss. for senfii, or half; gr. for grain; P. for
a little part ; P. aeq. for equal parts ; q. p., as much as you please.
36. COMPARISON OF
Country.
England,
France,
Prussia,
Austria,
Sweden,
Measure. U. S. gal,
1 .Gal. =1.2
1 Dl. = 2.64
1 Quart = .30
1 Maas =r .37
LIQUID MEASURES.
Country.
Measure.
U.S. gal.
Switzerland,
1 pot
= .40
Turkey,
Almud
= 1.38
Mexico,
1 Fasco
= .63
Brazil,
1 Medida
= .74
Cuba,
1 Arroba
= 4.01
South Spain
1 Arroba
=: 4.25
1 Kanna = .69
Denmark, 1 Kande = .51
DRY MEASURE,
37. I>ry Measure is used in measuring articles
not liquid ; as grain, fruit, salt, roots, etc.
Table.
2 Pints (pt) = 1 Quart . .
. qt
{ 64: pi.
8 Quarts = 1 Peck . .
. pk.
IBu. =
• S2qt.
4 Pecks = 1 Bushel .
. bu.
i ^Pk.
242 MEASUKES OF CAPACITY.
The 8tanda/rd Bushel of the United States contains 2150.42 cubic
inches, and is a cylindrical measure 18^ inches in diameter and 8
. inches deep.
The half-peck, or dry gallon, contains 268.8 cubic inches. Six
quarts dry measure are equal, to nearly 7 quarts, liquid meas-
ure.
The Imperial Bushel of Great Britain contains 2218.192 cu. in.
The English Quarter contains 8 Imperial bushels, or 8^ U. S,
bushels.
Grain is shipped from New York by the Quarter of 480 lb.
(8 U. S. bu.), or by the Ton of 33i U. S. bushels.
The bushel, heap measure, is the Winchester bushel, heaped in
the form of a cone, not less than 6 inches high and 19^ inches in
diameter, equal to the outside diameter of the standard bushel
measure, and equal to 2747.715 cu. in.
Grain, seeds, and small fruits are sold by stricken measure, or the
measure must be even full.
Corn in the ear, potatoes, coal, large fruits, coarse vegetables
and other bulky articles, are sold by heap measure.
It is suflBiciently accurate in practice to call 5 stricken measures
equal to 4 heaped measures.
The value of many kinds of grain, seeds, fruit, and other articles,
are often determined by weight instead of by bulk.
American coal is bought and sold, in large quantities, by the ton;
in small quantities, by the bushel.
The liquid and dry measures of the same denomination are of
different capacities. The exact and the relative size of each may be
readily seen by the following
38. COMPARATIVE TABLE OF MEASURES OF
CAPACITY.
Cubic in. in
Cubic in. in
Cubic in. in
Cubic in. in.
one gallon.
one quart.
one pint.
one gill.
Liquid measure . . v
. . 231
mi-
385
rh
Dry measure (J pk.)
. . 2681
67J
331
81
A cubic foot of pure water weighs 1000 oz , 62 V lb. Avoir.
MEASUEES OF CAPACITY.
243
39.
COMPARISON OF GRAIN MEASURES
Country.
Measure. U. S. bush.
Country.
Measure.
U. S. bush.
England,
1 Bushel = 1.031
Germany,
1 Schef.
= 1.5 to 3
France,
1 Hectoliter = 2.84
Persia,
1 Artaba
= 1.85
Prussia,
1 Scheffel = 1.56
Turkey,
IKilo
= 1.03
Austria,
1 Metze .= 1.75
Brazil,
IFan.
= 1.5
Russia,
1 Chetverik = .74
Mexico,
1 Alque.
= 1.13
Greece,
1 Kailon = 2.837
Madras,
1 Parah
= 1.743
ENGLISH MEASURES OF CAPACITY.
30. Wine Measure is used to measure wines and
all liquids, except malt liquors and water.
Table.
4 Gills
= 1 Pint ....
pU
2 Pints
= 1 Quart ....
gt.
4 Quarts
= 1 Gallon . , .
gal.
10 Gallons
= 1 Anker . . .
ank.
18 Gallons
= 1 Runlet . . .
run.
43 Gallons
= 1 Tierce ....
tier.
2 Tierces
= 1 Puncheon . .
pun.
63 Gallons
= 1 Hogshead . .
Mid.
2 Hogsheads
= 1 Pipe ....
pipe.
2 Pipes
= 1 Tun ... .
tun.
31. Ale and Seer Measure is
used to measure
all malt liquors and water.
Table.
2 Pints
= 1 Quart . . .
. qt.
4 Quarts
= 1 Gallon . .
. gal.
9 Gallons
= 1 Firkin. . .
. fir.
18 Gallons
= 1 Kilderkin
. kU.
86 Gallons
= 1 Barrel . . .
bar.
li Bar. or 54 gal
. = 1 Hogshead .
hhd.
2 Hogsheads
= 1 Butt . . .
butt.
2 Butts
= 1 Tun
tun.
244
MEASURES OF WEIGHT.
33. Corn or
Dry Measure
is used to measure
all dry commodities not usually heaped
r^
Table.
2 Quarts
= 1 Pottle . .
. pot.
2 Pottles
= 1 Gallon . . .
. gal.
2 Gallons
= 1 Peck . .
. ph.
4 Pecks
= 1 Bushel .
. lus.
2 Bushels
= 1 Strike . .
. str.
4 Bushels
r= 1 Coomb . .
. . coornb.
2 Coombs or 8 bu.= 1 Quarter .
. . qr.
5 Quarters
= 1 Load . .
. . load.
2 Loads or
10Qr.= lLast . .
. last.
14 Pounds = 1 Stone.
21i Stones — 1 Pig of iron or lead.
8 Pigs =: 1 Pother.
The stone varies. Legally it is 14 lb. A stone of butcher's meat
or fish is reckoned at 8 lb. ; of cheese, at 16 lb. ; of hemp, at 32 lb.
Kpig of iron or lead is 250 lb., and 8 pigs make 2i father.
MEASURES OF WEIGHT.
33. Weight is the measure of the quantity of matter
a body contains, determined by the force of gravity.
34. The Standard Unit of weight is the Tro^
Pound of the Mint, and contains 5760 grains.
TROY ^VEIGHT,
35. Troy Weight is used in weighing gold, silver,
jewels, and in philosophical experiments.
Table.
24 Grains {gr) = 1 Pennyweight . pwt. ( 5760 gr.
20 Pennyweights = 1 Ounce . . . . oz. 1 ^. = -< 240 pwt
12 Ounces = 1 Pound .... ^6. 1 { 12 oz.
measures of weight. 245
36. Table.
DIAMOND WEIGHT.
16 Parts — 1 Grain.
4 Grains = 1 Carat.
1 Carat = 3i Troy gr., nearly.
ASSAYERS* WEIGHT.
1 Carat =10 pwts.
1 Carat gr. = 2 pwts. 12 gr. or
60 Troy gr.
24 Carats = 1 Troy lb.
The term carat is also used to express the fineness of gold, each
carat meaning a twenty-fourth part.
APOTHECARIES* ^VEIGHT.
37. Apothecaries^ Weight is used by apotheca-
ries and physicians in compounding dry medicines.
Table.
20 Grains {gr, xx) = 1 Scrapie . . . . sc, or 3.
3 Scruples (3 iij) = 1 Dram dr., or 3.
8 Drams ( 3 viij) = 1 Ounce . . . . <>g., or g .
12 Ounces (.§ xij) = 1 Pound . . . . lb., or a.
rt,l = ll2 = zm=^2SS=gr. 5760.
1. Medicines are bought and sold in quantities by Avoirdupois
weight.
2. The pound, ounce and grain are the same as those of Troy
weight, the ounce being differently divided.
AVOIRDUPOIS W^EIGHT.
38. Avoirdupois Weight is used for all the ordi-
nary purposes of weighing.
Table.
16 Ounces (oz.) = 1 Pound . . . lb.
100 Pounds , =1 Hundredweight cwt.
20 cwt., or 2000 lb. = 1 Ton . . . . T.
i 32000 oz.
2000 lb.
20 ewe.
The oun£e is often divided into halves, quarters, etc.
246
MEASURES OF WEIGHT.
39. The Long or Gross ton, hundred-weight, and
quarter were formerly in common use ; but they are now
seldom used except in estimating duties at the United
States Custom-Houses, in freighting and wholesaling
coal from the Pennsylvania mines.
38 Pounds
4Qr., or 112 lb.
20 cwt. or 3240 lb.
LONO TON TABLE.
— 1 Quarter . . . qr.
= 1 Hundredweight cwt.
= 1 Ton T.
1 T.
2240 lb.
SOqr.
2t)ci^.
40. The following denominations are also used :
100 Pounds of Grain or Flour make 1 Cental.
100 Pounds of Dry Fish
100 Pounds of Nails
196 Pounds of Flour
200 Pounds of Pork or Beef
280 Pounds
41.
TABLE OF
1 QuintaL
IKeg.
1 Barrel.
1 Barrel.
1 Barrel of Salt at the
N. Y. Salt Works.
WEIGHTS.
COMPARATIVE
Troy. Apothixjaries. Avoirdupois.
1 Pound = 5760 Grains = 5760 Grains =7000 Grains.
1 Ounce = 480 '* = 480 '' = 437.5 "
175 Pounds = 175 Pounds = 144 Pounds.
42. COMPARISON
Weight. U. S.
1 KHogram r
OF COMMERCIAL ^A^EIGHTS.
Country,
France,
Germany,
Austria,
Russia,
Sweden,
Denmark,
Turkey,
Egypt,
Persia,
Madras,
1 Pfund
1 Pfund
1 Funt
IPund
IPund
1 Oka
1 Rottoli,
1 Battel,
IVis
lbs. avdp.
= 2.20
= 1.10
= 1.23
= .90
= .93
= 1.10
= 2.82
= 1.008
= 2.116
= 3.125
Weight.
Country.
Prussia, 1 Zolpf'd
Netherlands,! Pond
East Indies, 1 Seer
U. S. lbs. avdp.
l.IO
China,
Japan,
Mexico,
Brazil,
Spain,
Sicily,
Arabia,
1 Catty
IKin
1 Libra
1 Arratel
1 Libra
1 Libra
1 Maund
= 2.20
= 2.06
= 1.33
= .63
= 1.02
= 1.02
=1.016
= .7
= .8
MEASURES OF WEIGHT.
247
43. The weight of the bushel of certain grains, seeds
and vegetables has been fixed by statute in many of the
States ; and these statute weights must govern in buying
and selling, unless specific agreements to the contrary be
made.
TABLE OF AVOIRDUPOIS POUNDS IN A BUSHEL,
As prescribed hy statute in the several States named.
COMMODITIES.
Barley
Beans
Blue Grass Seed...
Buckwheat
Castor Beans
Clover Seed
Dried Apples
Dried Peaches. ...
Flax Seed
Hemp Seed
Indian Corn . .
Indian Com in ear.
Indian Corn Meal. .
Oats
Onions
Potatoes
Rye
Rye Meal
Salt
Timothy Seed
Wheat
Wheat Bran
50
40
45
32
54
60
56
56
t^
^^
48 48
60 60
14! 14
52 52
46
32
14
50
46
60
2524
60
56
44
56
68
50
|32
57 48
60 60
54 56
56
44
56
50
33>^
57
60 60
56156
50 50 50
45 45'45;45
60 60 60|60
20| |20,2a
56
60
)48
46
42
48
42
28
56
.1
50
30 32 32
52
60
50
56 56
50
60
60
48
60
14
52
4()
CO
24
3b
;56
J44
56 52
56
50
45
60,60
48'48
60 f
55 55 56
' I
56 58 56
30 32 32
I I
60 60:
56 56 56
u
'44!
0606
I I
47
56
32
56
46
46
50
50
56
60
45
42
28
56
46
60 60
In Pennsylvania 80 lbs. coarse, 70 lbs. ground, or 62 lbs. fine salt
make 1 bushel ; and in Illinois, 50 lbs. common or 55 lbs. fine salt
make 1 bushel.
In Maine 64 lbs. of ruta baga turnips or beets make 1 bushel.
A cask of lime is 240 lbs. Lime in slaking absorbs 2i times its
volume, and 2i times its weight in water.
248
MEASURES OF WEIGHT.
44, The following table will assist farmers in making
an accurate estimate of the amount of land in different
fields under cultivation.
Table.
A.
10 rods X 16
rods
= 1 A.
220
feet
X
198 feet = 1
8 " X 20
«
= 1 "
110
"
X
369 " = 1
5 " X 32
**
r= 1 "
60
a
X
726 " = 1
4 " X 40
"
, 1 ti
120
t(
X
363 " = 1
5 yds. X 96S
yds.
1 ''
200
<(
X
108.9 " = i
10 " X 484
it
1 "
100
*<
X
U5.2 '' = 4
20 " X 242
<(
= 1 "
100
"
X
108.9 " = i
40 '' X 121
"
= 1 " '
45. The following table will often be found convenient,
taking inside dimensions :
A box 24 in. x 24 in. x 147 will contain a barrel of 31 i gallons.
A box 15 in. x 14 in. x 11 in. will contain 10 gallons.
A box 8} in. x 7 in. x 4 in. will contain a gallon,
A box 4 in. x 4 in. x 3.6 in. will contain a quart.
A box 24 in. x 28 in. x 16 in. will contain 5 bushels.
A box 16 in. x 12 in. x 11.2 in. will contain a bushel,
A box 12 in. x 11.2 in. x 8 in, will contain a haZf-bu^hel,
A box 7 in. x 0.4 in. x 12 in. will contain a peck.
A box 8. 4 in. x 8 in. x 4 in. will contain a half -peck or 4 dry quarts.
A box 6 ill. by 5 J in., and 4 in. deep, will contain a half-gallon.
A box 4 in. by 4 in. and 2,\ in. deep, will contain a pint.
46. Nails are put up 100 pounds to the keg.
.a r£
«jQ
■2 aJ
Oi ,£5
^ 00
a& jD
Size.
fl
3 °3
Size.
il
?'S
Size.
§.s
3.9
^.S
5.9
^.9
T6
3.9
^.3
3cZfineblaed.
U
725
30c? com. blued.
4i
M casing
2
210
3c? com. "
u
403
40c? "
5
14
8c? ''
2i
134
Ad " "
li
300
50c? "
5i-
11
10c? *' •
3
78
M '' "
2
150
60c? "
6
8
6c? finishing
2
317
M " '*
2.^
85
6c? fence.
2
80
8c? "
2^
208
10^ " *'
3
60
8c? "
2J-
50
10c? '*
3
126
12d " "
Sj
50
10c? "
3
30
6c? clinching
2
118
16c? " "
3i
40
12c? "
31
27
8c? ''
2i
80
20cZ ''
4
20
16c? «
3i
20
10c? ''
3
45
5 lbs. of 4c? or 3J lbs. of 3c? will put on 1,000 shingles.
5f lbs. of 3c? fine will put on 1,000 lath.
MEASURES OF TIME AND ANGLES.
249
MEASURES OF TIME.
47. Time is the measure of duration.
48. The Unit is the mean solar day .
Table.
CO Seconds {sec.)
60 Minutes
24 Hours
7 Days
365 Days, or )
12 Calendar Mo. )
= 1 Minute . . min.
= 1 Hour , . . hr.
= 1 Day . , . da.
= 1 Week . . wk.
= 1 Common Year yr.
Common Year.
r 525600 min.
I Yr.= \ 8760 ;^r.
1 12 Tno.
366 Days
100 Years
= 1 Leap Year . yr.
= 1 Century . . Cen.
VW lA/lA/.
1. Every year that is exactly divisible by 4 is a leap year, the
centennial years excepted ; the other years are common years.
2. Every centennial year that is divisible by 400 is a leap year.
3. In most business transactions 30 days are called 1 month, and
12 months 1 year.
4. The civU day begins and ends at 12 o'clock, midnight. A. M.
denotes the time before noon ; M., at noon ; and P. M., afternoon.
5. The astronomical day, used by astronomers in dating events,
begins and ends at 12 o'clock, noon.
MEASURE OF ANGLES.
49. Circular or An-
gular Measure is used
in measuring angles and arcs
of circles, in determining lat-
itude and longitude, the loca-
tion of places, the motion of
the heavenly bodies, etc.
/
^ 1296000".
o
1(7. = ^
21600'
360^
a
12 8.
250 MEASUEES OF TIME AND ANGLES.
50. The Unit is the degree, which is -^ part of the
circumference of any circle.
Table,
60 Seconds (") =1 Minute .
60 Minutes = 1 Degree .
30 Degrees = 1 Sign . .
12 Signs, or 360° = 1 Circle .
A Seini-Circumf^ce is one-half of a circumference, or 180°.
A Quadrant is one-fourth of a circumference, or 90°.
A Sextant is one-sixth of a circumference, or 60°.
A Sign is one-twelfth of a circumference, or 30''.
A Degree (1°) is one-ninetieth of a right angle.
The length of a degree varies with the size of the circle ; thus,
a degree of longitude at the Equator is 69.16 statute miles, at 30°
of latitude it is 59.81 miles, at 60° of latitude it is 34.53 miles, and
at 90°, or the poles, it is nothing.
A mm-z^^g of, the earth's circumference is called a geographic mile.
LONGITUDE AND TIME.
51. Since the earth performs one complete revolution
on its axis in a day or 24 hours, the sun appears to pass
from east to west round the earth, or through 360° of
iongitude, once in every 24 hours of time. Hence the re-
lation of time to the real motion of the earth or the
apparent motion of the sun, is as follows :
Table.
For a difference of
There is a difference of
15° in
Long.
1 hr. in Time.
15' "
**
iTTiin." "
15" "
(1
1 sec. " «
1° "
(1
4min."^ "
1' "
n
4 sec. " '*
1' •*
it
A sec. - •*
MISCELLANEOUS.
251
COUNTING.
53. This measure is used in counting certain classes
of articles for market purposes.
Table.
12 Units
= 1 Dozen . .
doz.
C 1728 vnits.
12 Dozen
= 1 Gross . .
. gro.
1 G. gro. =
I lUdoz.
12 Gross
= 1 Great Gross
0. gro.
i 12 gro.
20 Units
= 1 Score . .
. sc.
Two things of a kind are often called a pair, and six things a
set; as Sipair of horses, a set of chairs, etc,
PAPER.
53. The denominations of this table are used in the
paper trade
Table.
24 Sheets
= 1 Quire .
. qr.
' 4800 Sheets.
20 Quires
2 Reams
= 1 Ream .
= 1 Bundle .
. rm.
. hurt.
1B.= ^
200 Quires.
10 Reams.
5 Bundles
- 1 Bale .
. B.
5 Bundles.
Paper is bought at wholesale by the bale, bundle, and ream ; and
at retail by the ream, quire, and sheet.
Paper may be made to order of any size, but the
greater part made up for sale is only of regular sizes.
The names generally define the sizes. Writing and Draw-
ing Papers differ in size from Printing Papers of the
same name, English sizes differ from American. How-
eyer, when English or French printing papers are made
for this country they are of American sizes.
252
MISCELLAKEOUS.
54. SIZE OF WRITING PAPERS,
FOLDED PAPERS.
Inches.
Xnches,
Billet Note ....
.6x8
Letter
10 xl6
Octavo Note . . .
.7x9
Commercial Letter
11 xl7
Commercial Note .
. 8 xlO
Packet Post. . .
lUxl8
Packet Note . . .
. 9 xll
Extra Packet Post
llixlSi
Bath Note ....
. 8ixl4
Foolscap ....
m X 16
The dimensions given above are those in most general
use. Some kinds occasionally vary a trifle.
55.
FLAT PAPERS.
Law Blank ....
Inches.
13x16
Flat Cap ....
14x17
Crown
15x19
Demy
16x21
Folio Post ....
17x22
Check Folio . . .
17x24
Double Cap . . .
17x28
Extra Size Folio . .
19x28
Inches.
Medium 18 x 23
Royal 19 x24
Super Royal . .
Imperial ....
Elephant . . .
Columbia . . .
Atlas
Double Elephant .
20 x28
22 x30
22\ X 27}
23 x33J-
26 X 33
26 X 40
Extra Size Folio is sometimes 18 x 23 inches, and
19 X 24 inches. Imperial is sometimes 93 x 31 inches.
SIZE OF PRINTINQ PAPERS.
56.
Inches.
Medium 19 x 24
Royal 20 X 25
Super Royal .... 22x28
Imperial 22 x 32
Medium-and-half . . 24x30
Small Double Medium . 24 x 36
Double Medium . .
Double Royal . . .
Double Super Royal
<( <( ((
Broad Twelves . .
Double Imperial . .
Inches.
24x38
26x40
28x43
29x43
23x41
32x46
Larger sizes and odd sizes are sometimes made, but are
not common.
MISCELLANEOUS.
253
BOOKS.
57. The terms folio, quarto, octavo, duodecimo, etc.,
indicate the number of leaves into which a sheet of paper
is folded.
When a sheet is
foldea into
2 leaves
4 "
8 '*
The book is
called
a Folio,
a Quarto or 4to,
an Octavo or 8vo. •
And 1 sheet of
paper makes
4 pp. (pages).
8 ''
16 ''
12 "
a Duodecimo or 12mo,
24 -
16 "
a 16m,o,
82 *'
18 "
24 " .
32 *'
an 18mo,
a 24mo,
a 32mo,
36 ''
48 "
64 '^
COPYING.
58. Clerks and copyists are often paid by i\\Q folio for
making copies of legal papers, records, and documents.
72 words make 1 folio, or sheet of common law.
90 " 1 '* chancery.
A folio varies in different States and countries, but usually con-
tains from 75 to 100 words.
59.
ROMAN LONG MEASURES.
Digit . . .
Uncia (inch)
Pes (foot;
Inches.
^ .72575
= .967
=r 11.604
Feet. Inches.
Cubit 1 5.408
Passus ..... 4 10.02
Mile (millarium) 4842
6^0. JEWISH LONG MEASURES.
Feet.
Cubit = 1.824
Sabbath day's journey = 3648
MUe (4000 cubits) .
Day's journey -
Feet.
. = 7296
33.164 mi.
61.
MISCELLANEOUS.
Feet.
Arabian foot . . . = 1.095
Babylonian foot . . = 1.140
FiO-vTitian finp-er . . = .06145
Feet.
Hebrew foot . . . = 1.213
cubit ... = 1.817
sacred cubit . = 2.002
254
MISCELLANEOUS.
RAILROAD FREIGHT.
Q2m When convenient to weigh them, all goods are
billed at actual loeight; but ordinarily, the articles named
below are billed, at the rates giyen in the following ^
Table.
Ale or Beer,
820 lbs
per bbl.
Highwines,
350 lbs. per bbL
Apples, green.
150
ft
«
Lime,
200
f( u
Beef,
820
tt
ff
: Nails,
108
" per keg.
Barley,
48
it
per bu.
Oil,
400
" per bbL
Beans,
60
(f
tt
Oats,
32
** per bu.
Cider,
350
it
per bbl.
Pork,
320
" per bbL
Com Meal,
220
ti
(f
Potatoes, com'n, 150
(( ti
Corn, shelled
56
t(
per bu.
Salt, fine,
300
tt tf
Corn in ear.
70
ft
tt
•* coarse.
350
ft tt
Clover Seed,
60
ft
"
" in sacks.
200
*' per sack.
Eggs,
200
ft
per bbl.
Wheat,
60
'* per bu.
Fish,
800
tc
t(
Whiskey,
350
" per bbL
Flour,
200
ft
u
2000 pounds are reckoned 1 tan^
Generally from 18000 to 20000 pounds is considered a car load.
63. Lumber and some other articles are estimated aa
follows :
Amount for
Weight. car load.
Pine, Hemlock, and Poflab, thoroughly
seasoned, per thousand feet . . . * . SOOO 6500
Black Walnut, Ash, Maple, and Cherry,
per thousand feet 4000 5000
Pine, Hemlock, and Poplab, green, per M. 4000 6000
Black Walnut, Ash, Maple, and Cherry,
green, per M 4500 4000
Oak, Hickory, and Elm, dry, per M. . . . 4000 5000
Oak, Hickory, and Elm, green, per M. . . 5000 4000
Shingles, green, per thousand 375 55 M.
Lath, per thousand 500 40 M
Brick, common, per car load 4 lbs. each. 5000
Coal, per car load 250 bu.
MISCELLANEOUS. 255
64. SCRIPTURE LONG MEASURES.
Eng.mi. Paces. Feet, Inches.
A Palm
equals
3.648
A Span
«
10.944
A Fathom
7
3.552
EzekieFs reed
10
11.328
An Arabian pole
14
7.104
A Furlong
145
4.6
.00
An Eastern mile
1
403
1.0
.00
A Day's Journey
33
172
4.0
.00
65. SCRIPTURE MEASURES OF CAPACITY.
LIQUID. DKY.
gal.
pints.
ACaph
=
.625
A Gachal
A Log
=
.83%
AKab
AKab
=
8.333
An Omer
AHin
= 1
2
A Seah
A Seah
=: 2
4
An Epah
A Bath or
Ephah
= 7
4
ALetek
A Homer
= 75
5
A Homer
ecks
. pints.
.1416
2.8333
5.1
1
1
3
3
6
32
66. MONEY MENTIONED IN SCRIPTURE.
£ s. d. $ ct8.
A Talent (gold) equal 5464 5 8 ==26592.809
A Talent (silver) ...... « 341 10 4 = 1662.0249
A Manch or Mina " 5 13 10 = 27.6990
A Pound (Mina) « 3 4 7 _ 15.7151
A Shekel (gold) ^* 1 16 5 = 8.8612
A Shekel (silver) " q 2 3 — 0.5474
A Golden Daric or Dram . , . " 1 1 10 =: 5.3127
A Piece of Silver (Stater) . . " 2 7= 0.6285
Tribute Money (Didrachm) . . " 1 3J =: 0.3142
^^B^^ah " 1 1 =r 0.2636
A Piece of Silver (Drachm) . . " 7J = 0.1571
A Penny (Denarius) " 7i = 0.1520
A^erah « 1= 0.0202
A Farthing (Assarium) .... " 0| = 0.0076
A Mite « 0A= 0.0019
256
MONEY.
MEASURES OF VALUE.
67. Money is the measure of the value of things or
of services, and the medium of exchange in trade.
UNITED STATES MONEY.
68. United States Money is the legal currency
of the United States.
69. The Unit of United
States Money is the Gold
Dollar.
Table.
10 Mills (m.) — 1 Cent . . ct.
10 Cents = 1 Dime . . d,
10 Dimes = 1 Dollar . $.
10 DoUajs = 1 Eagle . E.
\E, =
10000 m.
1000 ct.
100 d.
10 $.
70. The legal Coin of the United States consists of
gold, silver, nickel, and bronze, and is as follows:
71. Gold. The double-eagle, eagle, half-eagle, quar-
ter eagle, three-dollar, and one-dollar pieces.
73. Silver. The dollar, half-dollar, quarter-dollar,
the twenty-cent, and the ten-cent pieces.
73. Nickel. The five-cent, and three-cent pieces.
74. Bronze. The one-cent piece.
1. The half -dime and three-cent pieces, the bronze two-cent, and
Hie nickel one-cent pieces are no longer coined.
MONEY. 257
2. The Trade-dollar weighs 420 grains, and is designed soleiy for
purposes of commerce and not for currency. The legal-tender dollar
weighs 412| grains.
3. The Standard Purity of the gold and silver coins is .9 pure
metal, and .1 alloy. The alloy of gold coins is silver and copper ;
the silver, by law, not to exceed -^^ of the whole alloy. The alloy
of siUer coins is pure copper.
4. The five-cent and three-cent pieces are composed of f copper
and \ nickel. The cent is composed of 95 parts of copper and 5
parts of tin and zinc.
CANADA MONEY.
75. Canada Money is the legal currency of the
Dominion of Canada. The denominations are dollars^
cents, and mills, and have tlie same nomi7ial value as the
corresponding denominations of U. S. Money.
The Currency of the Dominion of Canada was made uniform July
1st, 1871. Previous to 1858 sterling money was in use.
76« The Coin of the Dominion of Canada is silver
and bronze.
77. The Silver Coins are the fifty-cent, twenty-
five-cent, ten-cent, and five-cent pieces.
78. The JBronze Coin is the one-cent piece.
The standard silver coins consist of 925 parts (.925)
pure silver and 75 parts (.075) copper. That is, they are
.925 fine.
1. The gold coin used in Canada is the British Sovereign, worth
$4.86|, and .the Half- Sovereign.
2. The intrinsic value of the 50-cent piece in United States
money is about 46 1 cents, of the 25-cent piece 23^ cents. In ordi-
tiary business transactions, they pass the same as U. States coin.
258,
MONEY.
ENGLISH MONEY.
79. English or Sterling Money is the legal
currency of Great Britain.
80. The Vnit of
English Money is the
Sovereign, or Pound
Sterling.
The value of a Sovereign in United States Money is $4.8665.
Table.
4 Farthings (far) = 1 Penny . . . . d
13 Pence = 1 ShiUing . . . s.
1 Sovereign, or . sov.
Pound ....£.
20 Shillings
w
U. S. Value.
r .02+.
£1 = ^ .293 + .
[ $48665.
Other Denominations.
2 Shillings («.) = 1 Florin . , . fl.
5 Shillings = 1 Crown . . . cr.
U. 8. Value.
$.48665.
$1.2166 + .
81. The Coin of Great Britain in general use con-
fiists of gold^ silvery and copper , as follows :
82. Gold* The sovereign, and half-sovereign.
83. Silver. The crown, half-crown, florin, shilling,
six-penny, and three-penny piece.
84. Copper. The penny, half-penny, and farthing.
The standard gold coin contains 11 parts pure gold
and 1 part alloy ; silver coin 37 parts pure silver and 3
parts alloy.
MOKEY.
FRENCH MONEY,
259
85, French Money is the legal currency of Prance,
and is decimal
The Franc of the
Eepublic.
86. The Unit
of French Money is
the Silver Franc.
The Franc of the
Empire.
The value of a Franc in United States Money is $.193.
Table.
10 Millimes (m.) = 1 Centime . , , ct.
10 Centimes = 1 Decime , , . dc,
10 Decimes = 1 Franc . , , . fr.
20 Francs = 1 Napoleon . . Wap.
87. The Coin of France consists of gold, silver, and
bronze, as follows :
88. Gold. The 100, 40, 20, 10, and 5 franc pieces.
89. Silver. The 5, 2, and 1 franc, the 50 and the
25 centime pieces.
90. Bronze. The 10, 5, 2 and 1 centime pieces.
The standard gold and silver coins contain 9 parts of
pure metal and 1 part of alloy.
The U. S. Congress, by the Act of 1873, fixed the weight of the
silver half-dollar at 12 1 metrical grammes, so that 2 half-dollars are
precisely equivalent in value to the 5 franc silver coin of Europe.
1 N'ap. = ^
r 20000 m.
2000 ct.
200 dc.
20fr,
260
MONEY.
GERMAN MONEY.
91. The New Empire of Germany has
adopted a new and uniform system of coinage.
93. The Unit
of this new German
System of Coinage
is the Reichsmark.
The value of a Reichsmark ('* Mark'') in U. S. Money is $.2385
A pound of gold .900 fine is divided into 139| pieces, and the ^
part of this gold coin is called a "Mark," and this is subdivided
into 100 pennies (Pfennige).
The Coin of the New Empire consists of gold, sUv&t^ and nickeL
Gold* The 20, 10, and 5 mark pieces.
Silver. The 2, and 1 mark, and the 20-penny pieces.
Nickel. The 10, and the 5-penny, and pieces of less valuation.
The 10-mark piece {gold) is equal to 3J P. Thalers (old).
The l-mark {diver) is equal to 10 S. Groschen, or 1000 pennies.
The 20-penny {silver) is equal to 2 S. Groschen, or ^ of a mark.
The 10-penny {nickel) is equal to 1 S. Groschen, or ^jj of a mark.
JAPAN MONEY.
93. Japan has a new and decimal system of coinage.
94. The Unit of Japan money
is the gold Yen, valued at $.997
U. S. money.
The Coin of Japan embraces five gold coins, valued at $20, $10,
$5, $2, and $1. Also five silver coins, valued at 5, 10, 20, 50, and
100 cents, respectively.
The weight of the new trade dollar is 420 gr., and .9 pure silver.
MONEY.
261
96, The following shows the manner in
of foreign exchange are made in
as quoted Jan, 2, 1875 :
which quotations
this country, and
Sixty
4.85^4 '
4.85 (
4.84 (
London Prime Bankers' Sterl, Bills
Do Good do. do.
Do Prime Commercial do.
Paris Francs 5.17^ ©
Antwerp Francs 5.17v^ @
Switzerland Francs. ..^ 5.17^ @
Amsterdam Guilders. 41^^ @,
Hamburg* Reichsmarks 947^ @,
Fi-ankfort Reichsmarks 94% @
Bremen Reichsmarks 9i% @
Berlin Reichsmarks 94% @.
4.86
4.85X
4.85
5.I614
5.I614
5.I614
41X
951/8
95>^
951/i
95>^
TTiree Days.
4.90 © 4.90>^
4.8914 © 4.90
4.88K2
5.13X
5.13^
5.139£
41X
96
96
96
96
©
4.893/
5.12y,
5.12;^
5.121/2
41%
96X
96X
96X
963^
In the above, " Prime Bankers' Bills" are those on the most reliable banking
houses; "Good" is applied to those of somewhat inferior credit; and "Prime
Commercial" are merchants' drafts, which usually command a less price in the
market. The quotations in the Jirst column are those of 60-day bills, and in
the second column those of 3 days.
97. Rates of Exchange at London, and on London.
EXCHANGE AT LONDON, JAN. 2, 1875.
EXCHANGE ON LONDON.
ON
TIME.
BATE.
DATE.
TIME.
EATE.
Amsterdam
Antwero
short.
11.13/2@11-16)^
25.47>^@25 5214
20.78 @20.82
25.15 @25.25
25.47)^@-25.52i/2
11.37/2@11.42
20.78 @20.82
20.78 ©20.82
32i/2@ 32%
483^(1^ 48%
52%@ 525^
28.173^(??^28.22i/2
28.17Xfr?^'28.22X
28.17/2@'28.22^
Jan. 2.
short.
3 mo.
short.
11.82
2517
Hamburg
20.25
Paris
short.
3 months.
25.19
Paris
Vienna
Jan. 2.
3 mo.
short.
11050
Berlin
6.24X
Frankfort
St. Petersburg. .
my,
Cadiz
Lisbon
90 days.
3 months.
Mil m ....
Genoa
Naples
•
New Yorl?:. .
Dec. 31.
Dec. 17
60 days.
90 days.
1^4 80
Rio de Janeiro.
26% @ 2614
Buenos Ayres..
Valparaiso
Bombay
Dec. 31
Dec. 29.
Dec. 24.
Dec 25
6 mo.
Is. iokd.
Calcutta
Is. I0|;.d.
Honir Kong^ . r. .
Shanghai..
4s. 2i^d.
5s.83i£d. @,5s.9d.
96%
Alexandria.
Dec. 30.
3 mo.
262
MONEY.
98. Weight, Fineness, and Value of Foreign Gold Coins,
as determined hy United States Mint Assays,
Country.
Austria
Do
Do
Belgium
Brazil
Centr'l America
Do. do.
Chili
Colombia and
S. A. generally.
Denmark
Egypt
England
Do
France
Germany
Greece
India (British).
Italy
Japan
Do
Mexico
Do
Do
Netherlands ...
New Granada..
Peru
Portugal
Russia
Spain
Do
Do
Sweden
Do
Tunis
Turkey
Denomination.
Fourfold ducat ,
Souverain (no longer coined)
4 florins
25 francs
20 milreis
2e8cudos
4 reals
10 pesos (dollars) . .
Old doubloon
Old 10 thaler
Bedidlik (100 piasters)
Pound or Sovereign (new) . .
Pound average (worn)
20 franc (no new issues)
Old 10 thaler (Prussian) . . .
20 drachms
Mohur, or 15 rupees
20 lire (francs)
Cobang (obsolete)
New 20 ven
Old doubloon (average)
20 pesos (empire)
20 pesos (republic), new
10 guilders
10 pesos (dollars)
20 soles
Coroa (crown)
5 roubles
100 reales
80 reales
10 escudos
Ducat
Carolin (10 francs)
25 piasters
100 piasters
Weight.
0.448
0.363 ♦
0.104
0.254
0.575
0.209
0.027
0.492
0.867
0.427
0.275
0,256,8
0,S50,3
0,307
0.427
0.185
0.375
0.207
0.289
1.072
0.867
1.0P6
1.081
0.215
0.525
1.055
0.308
0.210
0.268
0.215
0.270,8
0.111
0.104
0.161
0.231
Fineness.
TJwus'dths.
986
900
9C0
899
916,5
853,5
875
870
895
875
916,5
916,5
899
903
900
910,5
899
572
900
870
875
873
809
891,5
898
912
916
896
869,5
896
975
900
900
915
Value in U.S.
gold coin.
$ cts.m.
9 13
75
93
72
4
5
4
8
48. 8
13 6
15 50 3
19 21
80
96
7 10
3 84
3 57
19 94
15 59 3
19 64 3
19 51 5
3 99 7
67 5
97 6
5
3
4
3
5
2
1 93
2 99
4 37
1. Foreign gold coins, if converted into United States coins, are
subject to a charge of one-fifth of one per cent.
2. For sU^er coins there is no fixed legal valuation, as compared
with gold. The value of the silver coins January 1, 1874, wa« com-
puted at the rate of 120 cents per ounce, 900 fine, payable in sub-
sidiary silver coin, or 113 cents in gold.
TABLE FOR INVESTORS.
99. The following liable shows the rate per cent, of Annual Income
from Bonds hearing 5, 6, 7, or 8 per cent, interest, and costing
from 40 to 125.
Purchase
Price.
5%.
6%.
7/..
8%.
Purchase
Price.
5f^.
6.02
6%.
7%.
8%.
40
12.50
15.00
17.50
20.00
83
7.22
8.43
9.63
41
12.20
14:64
17.08
19.52
84
5.95
7.14
8.33
9.52
42
11.90
14.28
16.66
19.04
8.^
5.88
7.05
8.23
9.41
43
11.63
13.95
16.28
18 61
86
5.81
6.97
8.13
9.30
44
11.36
13.63
15.90
18.18
87
5.74
6.89
8.04
9.19
45
11.11
13.32
15.56
17.78
88
5.68
6.81
7.94
9.09
46
10.86
13.04
1521
17 39
89
5.61
674
7.86
8.98
47
10.63
12.77
14.90
17.02
90
5.55
6.66
7.77
8.S8
48
10.41
12.50
1453
16.66
91
5.49
6.59
7.69
8.79
49
10.20
12.25
14.29
16.33
92
5.43
6.52
7.60
8.69
50
10.00
12.00
14.00
i6.o:)
93
5.37
6.45
7.52
8.60
51
9.80
11.73
13.73
15.68
94
5.31
6.38
7.44
8.51
52
9.61
11.53
13.46
15.38
i 95
5.26
6.31
7.36
8.42
53
9.43
11.32
13.20
15.09
! 96
5.20
6.25
7.29
8.33
54
9.25
11.11
12.96
14.81
; 97
5.15
6.18
7.21
8.24
55
9.03
10.90
12.72
14.54
! 98
5.10
6.12
7.14
8.16
56
8.92
10?70
12.50
14.28
1 99
505
6.06
7.07
8.08
57
8.77
10.52
12.27
14.03
1 100
5.00
6.00
7.00
8.00
58
8.62
10.34
12.06
13.79
101
4.95
5.94
6.93
792
59
8.47
10.16
11.86
13.55
102
4.90
5.88
6.86
7.84
60
8.33
1000
11.66
1-3.33
103
485
5.82
6.79
7.76
61
8.19
9.83
11.47
13.11
104
4.80
5.76
6.72
7.69
62
8.06
9.67
11.20
12.90
105
4.76
5.71
6.66
7.61
63
7.93
9.52
nil
12.69
106
4.71
5.66
6.60
7.54
64
7.81
9.37
10.93
12.50
107
4.67
5.60
6.54
7.47
65
7.69
9.23
10.76
12.30
108
4.62
5.55
6.48
7.40
m
7.57
9.09
10.60
12,12
109
4.58
5.50
0.42
7.33
67
7.46
8.95
10.44
11.94
no
4.54
5.45
6.36
7.27
68
7.35
8.82
10.29
11.76
111
4.50
5.40
6.30
7 20
69
7.24
8.69
10.14.
11.53
112
4.46
5.35
6.25
7.14
70
7.14
8.57
10.00
11.43
113
4.42
5.30
6.19
707
71
7.04
8.45
9.85
11.26
114
4.38
5.26
6.14
7.01
72
6.94
8.33
9.72
11.11
115
4.35
5.21
6.08
6.95
73
6.84
8.21
9.58
10.95
116
4.31
5.17
6.03
689
74
6.75
8.10
9.45
10.80
117
4.27
5.12
5.98
6 83
75
6.66
8.00
9.33
10.66
118
4.23
5.08
5.93
6.77
76
6.57
7.89
9.21
1052
119
4.20
5.04
5.88
6.'; 2
77
6.49
7.79
9.00
10.38
120
4.16
5.00
5.83
6.66
78
6.41
7.69
8.97
10.25
121
4.13
4.95
5.78
6.61
79
6.32
7.59
8.8o
10.12
122
4.09
4.91
5.73
6.55
80
6 25,
7.50
8.75
10.00
123
4.03
4.87
5.69
6 50
81
6.17
7.40
8.64
9.87
124
4.03
4.83
5.65
6.45
82
6.09
7.31
8.53
9.75
125
4.00 '
4.80
5.60
6.40
.-A
264
31 ]sr E Y .
STATUTE LIMITATIONS.
100. A forced collection of debts cannot be made after
a certain number of years specified in tlie statute of limita-
tions of the different States named in the following Table :
Name of States.
Alabama
Arkansas
California..
Connecticut
Colorado
Delaware
Dist. of Columbia.
Florida
Georgia
Illinois
Indiana
Iowa
Kentucky
Kansas
Louisiana
Maine
Maryland
Massachusetts
Michigan
*i
aj
§3
S
a
^
^
Ha
Yrs.
Yrs.
Yrs.
3
6
20
3
7
10
2
4
10
6
6
17
2
4
5
3
6
20
3
3
12
5
5
3
3
12
5
6
16
6
20
20
5
10
20
2
7
14
3
5
10
3
5
10
6
6
20
8
3
12
6
6
20
6
6
20
Name of States.
Minnesota
Mississippi
Missouri
New Hampshire.
New Jersey
New York
North Carolina..
Ohio
Oregon
Pennsylvania.
Rhode Island
South Carolina..
Tennessee
Texas ,
Utah
VennoDt
Virginia
West Virginia...
Wisconsin
^*
« B
<U
S
0,0
^
08
1
Yrs.
Yrs.
6
6
3
6
5
10
6
6
6
16
6
G
3
3
6
15
6
6
6
6
6
6
6
6
6
6
2
4
6
6
5
5
5
5
10
'
Yrs.
10
20
20
20
20
20
10
20
10
20
20
20
10
10
10
10
10
1. The above data are liable to a cbange at any time by the Leg-
islatures of the States respectively.
3. In some of the above States there are exceptions and conditions.
LEGAL TENDER.
101. All gold coins, of United States coinage, are legal
tender in payment of all amounts.
All silver coins are legal tender in sums not exceeding
Jive dollars, in any one payment.
The five-cent, three-cent, and one-cent coins are legal
tender at their nominal value, in sums not exceeding
twenty-five cents, in any one payment.
^^ GreenbacTcs " are legal tender in payment of all debts
public and private, except duties on imports, and interest
on the public debt.
MONEY.
265
103. COMPOUND INTEREST TABLE.
Amount at the end of the year, of One Dollar per annum i^paid in
advance), at Compound Interest /(?r any number of years.
Yrs.
3 per cent.
4 per cent.
5 per cent.
6 per cent.
7 per cent.
8 per cent.
1
$1.03
$1.04
$1.05
$1.06
$1.07
$1.08
2
2.09
2-12
2.15
2.18
2 21
2.25
3
3.18
3.25
3.31
3.37
3.44
3.51
4
4.31
4.42
4 53
4.64
4.75
4.87
5
547
5.63
5 80
5.98
6.15
6.34
6
^m
6.90
7.14
7.39
7.65
7.92
7
7.89
8.21
8.55
8.90
9.27
9.64
8
9.16
9.58
10.03
10.49
10.98
11.49
.9
10.46
1101
11.58
12.18
12.82
13.49
10
11.81
1249
13.21
13.97
14.78
15.65
11
13.19
14.03
14.92
15.87
16.89
1798
13
1462
15.63
16.71
17.88
19.14
20.50
13
16.09
17.29
18.60
20.02
21.55
2321
14
17.60
19.02
20 58
22.28
24.13
26.15
15
19.16
20.82
22.66
24.67
26.89
29.32
16
2076
22.70
24.84
27.21
29.84
82.75
17
22.41
24.65
27.13
30.00
33 00
36.45
18
24.12
26.67
29.54
32.76
36 38
40.45
19
25.87
23.78
32.07
35.79
40.00
44.76
20
27.68
30.97
34.72
38.99
43 87
49.42
21
29.51
33.25
3751
42.39
48.01
54.46
22
31.45
35.62
40.43
46.00
52.44
59.89
23
33.43
38.08
43.50
49.82
57.18
65.76
24
35 46
40.65
46.73
53.86
62.25
72.11
25
37.55
43 31
50.11
58.16
67.68
78.95
26
39 71
4^108
5367
62.71
73.48
86.35
27
41.93
48.97
57.40
67.53
79.70
94.34
28
44.22
51.97
61.32
72.64
86.35
102.97
29
46.58
55.08
65.44
78.06
93.46
112.28
30
49.00
58.33
69.76
83.80
10107
122.35
31
51.50
61.70
74.30
89.89
109.22
133.21
32
54.08
65.21
79.06
96.34
117.93
144 95
33
56.73
68.86
84.07
103.18
127.26
157.63
34
59.46
72.65
89.32
110.43
137.24
171.82
35
62 28
76.60
94.84
118.12
147.91
186.10
36
65.17
80.70
100.63
126.27
159.34
202.07
37
68.16
84.87
106.71
134.90
171.56
219.32
38
71.23
89.41
113.10
144.06
184.64
237.94
39
74.40
94.03
119.80
153.76
198.64
258.06
40
77.66
98.83
126.84
164.05
213.61
279.78
41
81.02
103.82
134.23
174.95
229.63
303.24
42
84.48
109.01
141.99
186.51
246.78
328 58
43
88.05
114.41
150.14
198.76
265.13
355.95
266
MONEY,
103.
CARLISLE TABLE.
the values of Annuities on Single Lives, according to
the Carlisle Table of Mortality.
Agk. 4 per cent. 5 per ct.
14.28164
16.55455
17.72616
18.71508
19.23133
19.59203
19.74502
19.79019
19.76i43
19.69114
19.58339
19.45357
19.33493
19.20937
19.08182
18.95534
18.83036
18.72111
18.G0G56
18.48649
18.36170
18.23196
18.093S6
17.95010
17.80058
17.64486
17.485S6
17.32023
17.15412
16.99683
16.85215
16.70511
16.55246
16.39072
16.21943
16.04123
15.85577
15.66586
15.47129
15.27184
15.07363
14.83314
14.69466
14.50529
14.30874
12.083
13.995
14.983
15.824
16.271
16.590
16.735
16.790
16.786
16.742
16.609
16.581
10.494
16.406
16.316
16.227
10.114
16.0G6
15.987
15.904
15.817
15.726
15.628
15.525
15.417
15.303
15.187
15.065
14.942
14.827
14.723
14.617
14.506
14.387
14.260
14.127
13.987
13.843
13.695
13.542
13.390
13.245
13.101
12.957
12.806
5 per ct.
10.439
12.078
12.925
13.652
14.042
14.325
14.400
14.518
14.526
14.500
14 448
14.384
14.321
14.257
14.191
14.126
14.007
14.012
13 956
13.897
13.805
13.769
13.697
13.621
13.541
13.456
13.308
13.275
13.182
13.096
13.020
12.942
12.860
12.771
12.675
12.573
12.465
12.354
12.239
12.120
12.002
11.890
11.779
11.608
11.551
7 per ct.
9.177
10.605
11.342
11.978
12.322
12.574
12.098
12.756
12.770
12.704
12.717
12.069
12.621
12.572
12.522
12.473
12.^129
12.309
12.31-8
12.305
12.259
12.210
12.156
12.098
12 037
11.972
11.904
11.832
11.7o9
11.693
11.636
11.578
11.516
11.448
11.374
11.295
11.211
11.124
11.033
10.939
10.845
10.757
10.671
10.585
10.494
Age.
45
4 per ct.
5 per ct.
14.10460
12.648
46
13.88928
12.480 I
47
13.66208
12.301
48
13.41914
12.107
49
13.15312
11.892
50
12.86902
11.660
51
12.56581
11.410
52
12.25793
11.154
53
11.94503
10.892
54
11.62673
10.624
55
11.29961
10.347
56
10.96607
10.063
57
10.62559
9.771
58
10.28C47
9.478
59
9.96331
9.199
60
9.66333
8.940
61
9.39309
8.712
62
9.13376
8.487
63
8.87150
8.258
64
8.59330
8.016
65
8.30719
7.765
66
8.00900
7.503
07
7.69080
7.227
68
7.37970
6.941
69
7.04881
6.643
70
6.70936
6.336
71
6.35773
6.015
72
6.02548
5.711
73
5.72465
5.435
74
5.45P12
5.190
75
5.2;3901
4.989
76
5.02399
4.792
77
4.82473
4.609
78
4.62166
4.422
79
4.39345
4.210
80
4.18289
4.015
81
3.95309
3.799
82
3.74034
3.606
83
3.53409
3.406
84
3.32856
3.211
85
3.11515
3.009
86
2.92831
2.8:30
87
2.77593
2.685
88
2.68337
2.597
89
2.57704
2.495
11.428
11.296
11.154
10.998
10.823
10.631
10.422
10.208
9.988
9.761
9.524
9.280
9.027
8.772
8.629
8.804
8.108
7.913
7.714
7.502
7.281
7.049
6.803
6.540
6.277
5.998
5.704
5.424
5.170
4.944
4.760
5.579
4.410
4.238
4.040
3.858
3.656
3.474
3.286
3.102
2.909
2.739
2.599
2.515
2.417
MONEY.
267
ANNUITY TABLE.
104. Showing the present worth of an Annuity of One Dollar
per annumfiy at Compound Interest, from 1 year to Jfiy inclusive.
i
3 per ct.
3.} per ct.
4 per ct.
5 per ct.
6 per ct.
7 per ct.
1
0.970 874
0.966 184
0.961 538
0.952 381
0.943 396
0.934 579
2
1.913 470
1.899 694
1.886 095
1.859 410
l.a33 393
1.808 017
3
2.828 611
2.801 637
2.775 091
2.723 248
2.673 012
2.624 314
4
3.717 098
3.673 079
3.629 895
3.545 951
3.4C5 106
3.387 209
5
4.579 707
4.515 052
4.451 822
4.329 477
4.212 364
4.100 195
6
5.417 191
5.328 553
5.242 137
5.075 692
4.917 324
4.766 537
7
6.230 283
6.114 544
6.002 055
5.786 373
5.582 381
5.389 286
8
7.019 692
6.873 956
6.732 745
6.463 233
6.209 744
5.971 295
9
7.783 109
7.607 C87
7.435 332
7.107 822
6.801 692
6.515 228
10
8.530 203
8.316 605
8.110 896
7.721 735
7.360 087
7.C23 577
11
9.252 624
9.001 551
8.760 477
8.306 414
7.886 875
7.498 699
12
9.951 OW
9.663 334
9.305 074
8. 863 2C2
8.383 844
7.942 671
13
10.634 955
10.302 788
9.905 648
9.393 573
8.852 683
8.&57 685
14
11.296 073
10.920 520
10.563 123
9.898 641
9.294 984
8.745 452
15
11.937 935
11.517 411
11.118 3G7
10.379 658
9.712 249
9.107 898
16
12.561 102
12.094 117
11.652 206
10.837 770
10.105 895
9.446 63^
17
13.166 118
12.051 321
12.105 6C9
11.274 0C6
10.477 260
9.763 20»
18
13.753 513
13.109 602
12.659 207
11 689 587
10.827 603
10.059 070
19
14.323 799
13.709 837
13.133 9S9
12.085 821
11.158 116
10.335 578
20
14.877 475
14.212 403
13.590 326
12.462 210
11.469 421
10.593 997
21
15.415 024
14.697 974
14.029 IGO
12.821 153
11.764.077
10.835 527
22
15.936 917
15.167 125
14.451 115
13.163 003
12.041 582
H.C61 241
23
16.443 608
15.620 410
14.856 842
13.488 574
12.303 379
11.272 187
24
16.935 542
16.058 308
15.246 963
13.798 642
12.550 358
11.469.334
25
17.413 148
16.481 415
15.622 080
14.093 945
12.783 356
11.653 5^3
26
17.876 842
16. 890 352
15.982 769
14.275 185
13,C03 166
11.825 779
27
18.327 031
17.285 365
16.329 586
14.64^ 034
13.210 534
11.986 709
28
18.764 108
17.667 019
16.663 063
14.898 127
13.406 164
12.137 111
29
19.188 455
18.035 767
16.983 715
15.141 074
13.590 721
12.277 674
30
19.600 441
18.392 045
17.292 033
15.372 451
13.764 831
12.409 041
31
20 000 428
18.736 276
17.588 494
15.592 811
13.929 086
12.531 814
32
20.338 766
19.068 865
17.873 552
15.802 677
14.084 043
12.646 555
33
20.765 792
19 390 208
18.147 646
16.002 549
14.230 280
12.753 790
ai
21.131 837
19.700 684
18.411 198
16.192 204
14.368 141
12.854 m
35
21.487 220
20.000 661
18.664 613
16.374 194
14.498 246
12,947 67^
36
21 .832 252
20.290 494
18.908 282
16.546 852
14.620 987
13.035 208
37
22.167 235,
70.570 525
19.142 579
16.711 287
14.736 780
13.117 017
33
22.492 462
20.841 087
19.367 864
16.867 893
14.846 019
13.193 473
39
22.808 215
21.102 500
19.584 485
17.017 041
14.949 075
13.264 928
40
23.114 772
21.355 072
19.792 774
17.159 086
15.046 297
13.331 709
2C8 MISCELLANEOUS.
SPECIFIC GRAVITIES.— WATER 1.
105. A Table showing the weight of each substance compared with
an equal volume of pure water, A cubic foot of rain-water weighs
1000 ounces, or 62^ lb. Avoir. To find the weight of a cubic foot
of any substance named in the table, remove the decimal point three
places toward the right, which is multiplying by 1000, and the result
icill show the number of ounces in a cubic foot.
Substances.
Acid, acetic
" nitric
" sulphuric
Air
Alcoliol, of commerce. .
" pure
Alder wood
Ale
Alum ,..,
Aluminum
Amber
Amethyst
Ammonia ,
Ash
Blood, human ,
Brass (about)
Brick
Butter
Cedar
Cherry
Cider
Coal, bituminous (about)
" anthracite
Copper
Coral
Cork
Diamond
Earth (mean of the globe)
Elm
Emerald
Fir
Glass, flint
" plate
Gold, native
" pure, cast
*' coin
Granite
Gnm Arabic
Cypsum
]loney
Ice
Iodine
Iron
'• ore
Ivory
Lard
Specific Grav
1.008
1.271
1.841 to 2.125
.001227
.835
.794
.800
1.035
1.724
2 560
1.064
2.750
.875
.800
1.054
.800
2.000
.942
.457 to '.561
.715
1.018
1.250
1.500
8.788
2.540
.240
8.530
5.210
.671
2.678
.550
2.760
2.760
15.600tol9500
19.258
17.647
2.652
1.452
2 288
1.45B
.930
4.948
7.645
4.900
1.917
.947
Substances.
Lead, cast
" white
" ore
Lignum vitae
Lime
"' stone
Mahogany
Manganese
Maple
Marble
Men (living)
Mercury, pure
Mica
Milk
Nickel
Nitre
Oil, castor
'' Unseed
Opal
Opium
Pearl
Pewter
Platinum (native)
'' wire
Poplar
Porcelain
Quartz
Rosin
Salt
Sand
Silver, cast
" coin
Slate
Steel
Stone
Sulphur, fused
Tallow
Tar
Tin
Turpentine, spirits of
Vinegar
Walnut
Water, distilled
" sea
Wax
Zinc, cast
Specific GraVo
11.350
7.235
7.250
1.333
.804
2.386
1.063
3.700
.750
2.716
.891
14.000
2.750
1.032
8.279
1.900
.970
.940
2.114
l.:337
2.510
7.471
17.000
21.041
.383
2.385
2.500
1.100
2 130
1.500 to 1800
10.474
10.534
2.110
7.816
2.000 to 2.700
1.990
.941
1.015
7.291
.870
1.013
.671
1000
1.0C8
.897
7.190
MISCELLANEOUS.
269
106. ABBREVIATIONS USED IN BUSINESS^
@
At.
Guar.
Guarantee.
% or Acc't
Account.
Gal.
Gallon.
Am't
Amount.
Hhds.
Hogsheads.
Ass'd
Assorted.
Ins.
Insurance.
BaL
Balance.
Inst.
This month.
Bbl.
Barrel
Invt.
Inventory.
Blk.
Black.
Int.
Interest.
B. L.
Bill of Lading.
Mdse.
Merchandise.
f or ct
Cents.
Mo.
Month.
%
Per cent.
Net.
Without disc't
Co.
Company.
No.
Number.
Cr.
Creditor.
Pay't
Payment.
Com.
Commission.
Pd.
Paid.
Cons't
Consignment.
Pk'gs
Packages.
Dft.
Draft.
Per
By.
Disc't
Discount.
Prem.
Premium.
Do.
The same.
Prox.
Next month.
Doz.
Dozen.
Ps.
Pieces.
Dr.
Debtor.
Rec'd
Received.
Ea.
Each.
Ship't
Shipment.
Exch.
Exchange.
Sund's
Sundries.
Exps.
Expenses.
S. S.
Steamship.
FeL
Folio.
Ult.
Last month.
Fw'd
Forward.
Yd.
Yards.
Fr't
Freight.
Yr
Year.
4 5 7
16 doz. — --, ^, — = 16 dbz., 4 of which are at $10 per doz., 5
% $13, and 7 @ $15.
9 doz., f (^ 5/ f @4/6. 3 doz. No. 4 @ 5 shillings per doz., and
6 doz. No. 5 @ 4 shillings sixpence per
doz.
8 X 10, or 8 by 10 in. 8 inches wide and 10 inches long.
270 MEASURES,
FEENCH AND SPANISH MEASURES.
958. The old French Linear ^ and Land Meas-
nre^ is still used to some extent in Louisiana,, and in
other French settlements in the United States.
Table.
12 Lines = 1 Inch. 6 Feet — 1 Toise.
12 Inches = 1 Foot. 32 Toises = 1 Arpent.
900 Square Toises = 1 Square Arpent.
The French Foot equals 12.8 inches, American, nearly.
The Arpent is the old French name for Acre^ and contains nearly
f of an English acre.
In Texas, New Mexico, and in other Spanish settle-
ments of the United States, the following denominations
are still used :
Table.
1000000 Square Varas = 1 Labor = 177.136 Acres (American).
25 Labors — 1 League = 4428.4 Acres "
The Spanish Foot = 11 11 + in. (Am.) ; 1 Vara = S3| in. (Am.);
108 Varas = 100 Yards, and 1900.8 Varas = 1 Mile.
Other Denominations in Use.
5000 Varas Square = 1 Square League.
1000 Varas Square = 1 Labor, or ^V League.
5645.376 Square Varas = 4840 Square Yards = 1 Acre.
23.76 Square Varas = 1 Square Chain = ^ -^^ Acre.
1900.8 Varas Square = 1 Section = 640 Acres.
In many answers the decimal figures following the second or third
places have been omitted, and when the first figure omitted was equal
to, or greater than 5, the last figure retained was increased by 1 .
Art. 512.
2. 2431 lb.
6d.
9.
10.
11.
12.
13
19,
3. $6321.
4. £263 2s.
5. 2912 bu.
6. $175.
7. $205.49.
8. 14.076 rd.
$6014.40.
$3180.01.
2 mi. 277 rd.
5V ft.
386| ft.
21 f bu.
U. 437i lb.
15. 123 men.
17. .004 hhd.
18. 264| lb.
$9898.25.
V. $677,331 Ex.
Savings,
$922.66|.
7. .52 ; $45760.
'^. $3903.40.
Art. 515.
2. 25%.
3. 25%.
108%.
5%.
14|%.
5%.
5|%.
9. 62iy%. '
to. 73A%.
11. m%.
12. 7i%.
4-
5.
6.
7.
8.
75%.
8%.
37i%.
13.
u>
15.
16. 6%.
17. 112i%.
18. 20%.
19. 121%.
20. 50%.
21. 65%.
Art. 518.
3. $750.
i. 91.2 A.
5. 528 11).
G. 690. <9. .6.
7. 5800. 9. 100.
i^*. $750.
11. $5450.
X?. 6r.0bu.
i<f. lOOSObbl.
U. 8G00bu.
15 4500 bu.
16. $3000.
17. $78133.33i.
18. $922.25.
Art. 520.
2. 2500.
3. $6000.
5. $1250.
6. S7400.
7. $3892.86.
8. 36000.
9. $2275.
i^. 900 bu.
11. 800.
i-^. 325 A.
13. $2480.
i^. $375.40.
15. $31 pr. A.
t?^ $4S pr. bale,
i;2:--$4398.55.
18. $8750.
19. $3400,lstyr,
$35^0,2dyr.
20. $208,331.
Art. 530.
2. $349.
3. $842.40.
4. $636,375.
5. $204.86.
6. $253.75.
$1437.60.
$306.67.
$144.32.
$11016.
8.
9.
10.
11.
Art. 531.
3. $208,125.
4. $11.31i.
5. $ 17.
6. $6.22|-,
7. $4,375 ;
$2.80.
8. $.53|-.
$1.06i.
9. $.11| per.lb
Art. 532.
3. 18f % gain.
4. 12^% loss.
5. 20%.
6. 28%.
7.
^.
5.
10.
11.
14|%,
24%.
66i%.
23%.
50%.
12.
13.
37i%.
60%.
Art. 533.
3.
4.
5.
6.
7.
8.
$9375.
$8.80.
$150.
$14.14.
$16666.66f.
A. $16000 ;
B. $10000.
Art. 534.
2.
6.86. 3. .75i
4.
5.
6.
7.
<9.
9,
$4.91.
$.20.
$244,094.
$183,331.
$586.66f.
$6553.60.
Art. 535.
2. $1.47.
3. $150.
4. $1.03|.
5. $96.
Art. 547.
2. $378,125.
3. $82.11.
4. $379.40.
5. $285.19.
272
ANSWERS.
6. $20.18.
7. $584.17^.
8. $96.90.
Art. 548.
S, 3i%. S. 2f %.
S, hfo. 6, 5%.
Art. 549.
^. $2784. 5. $9000.
:^. $3500. 6. $960.40.
4. $9600.
Art. 550.
^. $3750.
^. $583.33i
4. $25372.
Art. 551.
;^. $4696.65.
5. $3182.55.
4, $1500.
6. $10648.
^. $6400.76 Inv. ;
$320.04 Com.
7. 31000 lb.
8. $10623.44.
9. $44231.71 Inv.;
$1105.79 Com.
10. 1640 yd.
Art. 553.
1. 48 bu.
^. $1700, 1st yr.;
$1785, 2d yr.
5. 24^\%.
^. $67 50 gain;
12% gain.
5, $3640.
6, $40842 cost.
$6807 gain.
7, $30000.
5. 40|%.
5. 1468.75.
10. Loses 25%.
/i. 25f % nearly.
12. 5%.
7.1 $4948.125.
14. $2964 whole gain
215 av. gain ^.
i5. Prints @ $.15 ;
Cassim.@$4.06i;
Ticking @ $.25
Shawls @ $9.20
Thread @ $.875 .
Buttons® $1.25;
Amt. (a) $729.96.
16. $705.14.
17. $155.09.
i<?. 61788.6 lb. +
iP. $.50.
W. $10582; $132 Com.
21. 5i%.
22. $8,875; loss 4|%+.
28. $3049.20 whole
gain;
50% gain +.
Art. 507.
2. $101.25 int. ;
$551.25 amt.;
$21 int.; $471 amt.
3. $71.32 int.;
$318.32 amt.;
$16.47 int. ;
$263.47 amt.
4. $208.33 int. ;
$708.33 amt. ;
$22.92 int. ;
$522.92 amt.
5. $3.46 int. at 6%.
$4.03 int. at 7%;
$4.32 int. at 7i%;
6. $115.70 at 5%;
$185.12 int. at8%;
$208.26 int. at 9%.
7. $196.41 int. at 6^%;
$235.';0int.at7i%.
8. $58.97 int. at 10%;
$73.71 int. at 12^%
9. $888.40 amt.
10. $71.87 amt.
11. $1176.50 amt.
?. $442.50
Art. 509.
?. $12.58 int. at 6%
$8.89 at 4%.
?. $92.53(^5%;
$148.04 @ 8%.
^.
$269.47 @ 7%;
$288.72 @ 74.
$61.12 int.
$292.50 int.
$1204.12 amt.
$276.52 amt.
$41.27 Int.
$421.99 amt.
$85.72 Int.
$13227.50.
Art. 573.
$22.70 Int.
$3.84.
$38.34.
$242.94.
$318.
$269.34.
Art. 574.
$120. 5. $82.36
$.04. 6. $10.96.
$10.58.
Art. 575.
$58.93..4. $159,745.
$8.40. 5. $67.09.
$67.67. 6. $38.11
$8.63.
$3647.61.
$115.20.
$1066.36.
$2010.42.
$142.45 + .
$1886.17.
$131.40.
$263.83.
$828.07.
$1936.60.
$3925.17.
$1120.69.
$76.67.
$1931.40 lossc
Art. 577.
$660, $792.
$6936.09.
$6069.08.
ANSWERS.
273
4. $5l6;'ri.
5. $669.12 ; $334.56.
6. $10000.
Art. 579.
$1500.
$889.25.
$650.80.
Art. 581.
7%.
7%.
to.
11.
4.
5.
6.
7.
8.
10
IL
12.
IS,
2 % a month,
lOAf..
25%; 16|%; .
12i%; 10%.
100%; 40%;
28i%;16|%;10%
The 2d is J^%.^
better.
Art. nS'S.
7 mo. 10 d.
6 yr. 8 mo.
7 mo. 6 da.
3 yr. 4 mo. 24 da.
33i ; 20 ; 16| ;
13| ; 10 yr.
50; 40; 28f ;
25 ; 16 yr.
12i; 6i; 25 yr.
Art. 586.
$428.76.
$189.15.
$1 176. 14.
$100.32.
$4199 + .
$1495.77.
$53.38.
$1525.64.
$1540 79.
$987.23^'
$1934.84.
$18142.81.
10.
11.
Art. 589.
$464.10.
$7308.
$11.30.
$1161.04..
$1047.52.
Art. 597.
$659.94.
$30.14.
$162.25.
Art. 598.
$312.50.
$355.16.
Art. 603.
$281.83.
$102.90.
$1137.61.
$43.65 in favor of
dis.
$931.20.
$838.26.
45tVo%.
$931.83.
$.05 per bbl. more
profitable to buy
at $8.75 on 6 mo.
$3677.75.
Art. 615.
$6.27 Bk. dis.
$591.23 proceeds.
$1614.88.
$10839 83.
Mat. Oct. 30 ;
81 days term of
dis. ;
$940.38 proceeds.
Mat April 8 ;
46 days term of
dis. ;
$t±!r proceeds, ff^,
Mat. Aug. 2 ;
79 days term of
dis. ;
$1295.82 proceeds.
8. Mat. Dec. 15 ;
30 da. term of dis.
$1281. 77 proceeds.
Art. 617.
S. $1434.20.
3. $719.61.
$1951.03.
$2291.44.
$321.46.
$659.88.
$368.25
Art. 619.
$188.43 bal. July
1st.
$4.90.
$369.36.
$327,927.
Art. 648.
$34256.25.
$16856.25.
$15843.75.
Art. 649.
250 shares.
220 ''
220 ''
480 **
200 ''
Art. 650.
$25500. 4. $693a
$21100.
Art. 651.
8f%.
8% bonds at 110
If % better.
6% bonds at 84.
if % better.
3tt%.
Art. 652.
62J. 4- 71f.
33J%.
75 ; 66f .
5.
6.
7.
2,
3.
6.
5. $40.
274
A N S W E K S .
Art. 053.
2. $5463.28.
3. $268.20.
J^. $262.66 better to
pay in currency.
Art. 654:.
2. $4000; $4035.87;
$4109.59.
3, $74000.
Jf. $1755890.
5. Dim. $28.25.
G. $113 per annum.
7. Stock invest, is
$50 better, or
ff% yearly.
8. $21384 in N.Y.S.
6's;
$42768 U. S. 5's
of 81.
9. $792.
Art. 664.
2, $42.75.
3, $24.06.
k. $187.50.
5. $156.25.
Art. 665.
2, 1}%.
^. t%.
Art. 666.
2, $13600.
5. $22220.77.
(;. $49147.91.
7. $24500.
<S\ $24766.58. '
d, $9.90.
Art. 675.
2. $284.78.
S. $1055.30.
5. $527.65.
6. $5888.57.
7. $4416.57.
8. $3263.93.
9. $1131.12 loss.
10, $7200.
Art. 685.
2. $11350.
3. $19072.16.
J^. $401920.
7. $25.09.
8. $87.38.
5. $112.50.
10. $226.50.
11. .0228 tax rate.
$214.65.
12. $410.95.
13. $224.37.
U. $178.13.
15. $420900.
Art. 700.
2, $1566.15.
Jf. ^-4764.84.
5. $5153.24.
6. $6388 80.
7. $5632.20.
Art. 701.
2. $787.46.
3, $720.
k, $316.45.
5. 451 sliares.
<?. 97J%. 7 7 V/c
7. $20108.35..
Art. 706.
.?. $2003.25.
3. $3317.63.
5. $134.78.
G. $352.67.
8. $421.09.
9. $566.50.
11. $801.94.
12. $4621 16.
13. $5243.89.
U. $3500.40.
Art. 707.
2. £1543 4s. 2(1.
4. 2318.84 marks.
5. 1664.13 marks.
7. 31888.83 francs.
8, 12918.75 francs.
Art. 711.
2. $179.21.
5. 5.31 francs.
6. $4,987.
7. £1055 12s. 4d.;
£21 9s. 9.7d.
8. |e32.78 ind. ex.
9. 696.6 guild, loss.
10. $12617.08.
Art. 726.
2. $437.50.
3. $1703 25.
4. $1843.75.
5. $1234.88.
G. $63.18.
7. $5775.
<?. $.2376.28 duty.
$6815.75 cost in
currency.
9, $1755.89.
10. $987.08.
Art. 733.
2. 3 mo. 25 da.
S, 6 mo. 26 da. time
of Cr. ;
June 27, 77 Eq. time
Jf.. May 5, 1875.
5. 5 yr. 20 da. from
date of last paym't
7, Nov. 26, Eq. time.
8, 73 da. term of Cr.;
Feb. 26, Eq. time
9, Mar 7, Eq. time.
$1178.01 cash
value.
Art. 734.
2, Aug. 19, 1875.
Eq. time.
3. June 7. 1876.
A I^ S W E R S .
275
4. June 27, 1874 ;
Dis. $149.28.
6. Apr. 23, 1874.
6. $233711^.
7. May 20, 1875.
Art. 737.
f . Dec. 13, Eq. time.
3. Dec. 19.
4. Jan. 24, 1879.
'Art. 738.
5. May 18 ;
$1486.17 due.
8. Dec. 5, 1875.
4. $2069.59.
5. Oct. 27 ;
$:il02 58.
e. $1272.33.
Art. 739.
S. $2331.65 Sales ;
$762.83 Charges ;
$1568.82 Net pro-
ceeds ;
Bal. due, Dec. 27.
S. $3966.25 Sales ;
$412.98 Charges ;
$3553.27 Net pro-
ceeds ;
Eq. time Apr. 14,
1875.
Art. 767.
S. 60 bu.
3. $100.
4. $4.05.
5. 44| bbl.
Art. 770.
S. 9 horses.
4. 100 yd.
5. 16 men.
6. 96 sheep.
7. $5355.
8. 7 hr. 13i min.
9. 355 bu.
10. 112imi.
11. od^ da.
It $7320.
13. 9 yd.
14. $1M.
15. 46 A. 134 P.
16. $63.
17. $10958.90.
18. $3.25.
19. $89.60.
^0. $120. •
21. 2 yr. 6 mo.
Art. 772.
43i- tons.
5 1 weeks.
432 mi.
15 da.
Art. 774.
^. $498.08.
4. 1120 bu.
5. $6428.57.
6. 114^\ ream.
7. 2201 Cd.
8. $52.79.
9. 9 men.
10. 546 bbl.
ii. 2080 lb.
12. $100.
2«?. 266605f brick.
14^ $236.25.
i5. 694| yd.
16. $1728.
i7. 5 da.
18. 150 yd.
19. 3 yr. 4 mo. 24 da.
20. $11.66|.
21: 9 men.
22. 8.116 ft.
23. $48.
^4. $53.08.
^5. 1.6 mo. 4-
Art. 782.
3. A's share $320.
B's '' $316.
Cs '^ $184.
>^. A. $303.45.
B. $337.17.
C. $404.61.
D. $682.77.
5. A. $1710.
B. $870.20.
6. A. $6000.
B. $8402.25.
C. $:05575.
D. $3042.
7. $5785 20, the first;
$5142.40, the
second.
8. $3516.80 A's gain;
$5861.33^ B's ''
$8205.861 C's ''
9. $269559.55 Re-
sources ;
$26434.55 Lia-
bilities ;
$243125 Stock ;
$125000 Origi-
nal capital ;
$118125 net gain;
$56700 Ames'
share *
$37800 Lyon's
Q |"| Ck It* fit •
$23625 Clark's
share.
Art. 783.
2. $2400 Barr ;
$2666.661 Banks ;
$2933.33J Butts.
3. $388,704+ A.;
$249,169+ B.;
$112,122 C.
4. $1344.164 A.;
$2027.836 B.
5. $5700 A.;
$3760 B. ;
$1340 C.
6. $1688.434
Crane ;
$3868 862
Childs ;
$2012.708 Coe.
276
ANSWERS.
Art. 787.
^. $.32.
5. $.30 per bushel.
4. $6 gain.
6, $6.16.
Art. 788.
5. 2 lb. of first ;
2 lb. of second ;
3 lb. of third.
4. 1 at $4 ;
5 at $5
3 at $6
1 at $8,
5. 3bb
3 bbl. at $(}";
2 bbl. at $7|.
6. 3 gal. at $1.20 ;
3 gal. at $1.83 ;
15 gal. at $2.30 ;
8 gal. water.
Art. 789.
e. 10 cows at $32 ;
10 cows at $30 ;
60 cows at $48.
S. 10 lb. at $.80 ;
10 lb. at $1.20 ;
70 lb. at $1.80.
4. 12 yd. at $3^ ;
16 yd. at $li.
5. 150 acres.
Art. 790.
j^. 30 men, 5 w^omen,
20 boys.
S. 33 g^ gal. water.
4. 16, 24, 4, and 12
da. respectively.
Art. 792.
1. 72 and 48.
^. D^s age 16 ;
E's age 24 ;
F's age 84.
5. 15 bu.
4, 18 da.
6. 8| da.
Starch $2 a box ;
Soap $3.
8Ha.;
First in 26| da. ;
Second in 40 da. ;
Third in 20 da. ;
$180 share of 1st ;
$120 share of 2d ;
$240 share of 3d.
14 bbl. at $10 ;
6 bbl. at $7.
16 min. 21f'Y sec.
past 3 o'clock.
Wheat $1.33 J per
bu.;
Oats $.50 per. bu.
8 da.
$347.71.
50 bu.
27% nearly.
$7384j^3 younger ;
$11076i-f elder.
1461 ft.
$960 first ;
$ ?20 second ;
$840 third.
$1570.31.
506 lb.
Oct. 26, 1875.
^7.
$33345;
$27359.999;
$25106.82.
$1.60.
42 geese ;
58 turkeys.
$5700.
$282.24 Sim. Int. ;
$2202.24 Amt. ;
$295.56 Com. Int. ;
$2215.56 '' Amt.;
$1673.93-}- Pres-
ent Worth ;
$246.07 True Dis. ;
$283.20 Bk. Dis. ;
$1636.80 Proc'ds;
$2252.199 Face.
28. $315.79.
$473 69.
$710.52.
29. $900, July 28.
SO. $.97|.
31. $10665.80 in U. S.
6's, 5-20.
$21331.60 in U.S.
5's of '81.
32. A. 3600 bu.;
B. 1200 bu. ;
C. 1200 bu. •
33. $1.72.
34. fn-
35. $5614.27 Net
Proceeds.
July 10, Eq. time.
36. $6100 M.'s Cap.;
15 mo. N.'s time.
37. $2023.22 ; Apr. 24.
38. $2244.66.
Art. 802.
2. 1369; 1764;
3136; 5625.
3. 3375 ; 5832 ;
74088 ; 157464.
4. 3969 ; 110592 ;
1048576 ; 248832.
/? 49 . 1728
^- ¥5 6" > "sgy^T*
7 _a5iLl_ • 318S
' • 384 16 > 3 2' •
8. 645.16.
9. 1191016.
10. 1958jV.
/ 7 1 4 G 4 1
■^■^' 50625-
12. .00116964.
13. .015625.
14. 46733.803208.
15. .065528814274496.
16. 33169 If.
17. 16.6056I-.
18. 24.76099.
19. .000000250047.
20. 1520875.
21. 2023//^.
22. 5.887.
23. 640000.
24. 2540.0390625.
25. 125. 26. 1200
ANSWERS.
277
Art. 803.
3. 1764.
4. 2304.
5. 3136.
6'. 9604.
7. 15625.
<^. 11025.
9, 50625.
i^. 38809.
lU 116964.
Art. 804.
;^. 39304.
4. 110592.
5. 262144.
6. 857375.
7. 1953125.
Art. 810.
^. 8 ; 16 ; 24 ;
81.
^. 9 ; 14 ; 21 ;
15.
Art. 813.
S. 85.
4. 242.
5. 98.
^. 115.
7.109.
8, 997.
P. 1432.
10, 5464.
i^. «.
i5. If
15, .035.
m 14.0048 + .
17. 1.5005 + .
18. 7.625.
19. 4.213 + .
^^ 103.9.
;^i. 59049.
^^. 3.00001 654-.
^S. 5.656854 + .
^^. 1.5411.
^5. .91287+;
ee. .04419.
,V. 36.37.
1"^. 1.50748 + .
^9. 64.
^^. tV-
^i. 1.
S2. 1.78 + .
^«f. 72.
S4, 90.
^5. 480.8827.
Art. 815.
i. 1008 ft.
^. 240.33 rd.
S, 52 rd.
4. 200.56 rd.
5. 145ird.
6. $187.20.
Art. 819.
«^. 25.
4. 55.
5. 101.
6. 165.
7. 1015.
^. 1598.
10. a.
11' f *•
12. 1.42 + .
i<^. 34.
i.^. .45.
15. 2.34.
i^. 4624.
17. .0809.
i^. .7936.
i5>. 5.73 + .
^i. .5569.
22, 1
^<?.' 14.75.
;^.4. 60 8.
Art. 821.
1. 3 ft.
2. 8 ft.
^. 2 ft.
4. 12150 sq.ft.
5. 5 ft. 8+ in.
6. Oft. 5.3 + in
7. 8 ft. 1.4 in.
Art. 822.
2. 274.
3. 32.
4- 543.
5. 1.05 + .
Art. 829.
3. 8. ^. 149,
4. 17. 7. 16.
5. 33. 8. 7J|.
Art. 830.
2. 2. .5. 4.
^. 2. G. 7%,
4^ -|. 7. A.
Art. 831.
i*. 9.
3, 15.
4.4.
5. 27.
6. 11 yr.
Art. 832.
2.
3.
4.
5.
6,
600.
154.
125000.
78.
57900 ft.
Art. 840.
4.
5,
6.
7,
8.
6144.
3.
$524288.
$315,619 +
$10485.76.
Art. 841.
2. i. 4. 5.
3, 5. 5, 3.
Art. 842.
2. 9.
3. 7.
4. 8.
Art. 843.
3, 765.
5. 16.
7. 2.
<?. 280.
^. $1023.
10. $5314.40.
Art. 853.
3. $3819.75.
4. $1292.31.
5. $3625.
6. 6 yr.
7. 7%.
8. $375.30.
Art. 854.
«^. $300.
4. $3907.665 +
5. $1182.05 + .
6. $3725.87 + .
7. $629,426 + .
Art. 882.
2. 600 sq. ft.
3. 4:2j\ sq. ft.
4. 22 A. 6 sq.
ch. 13.45 P.
5. $449.07.
6. $147.
7. 210 sq. ft.
Art. 883.
2. 4i ft.
3. 13 in.
4. 28 rd.
5. 672 rd.
yd.
6. 8i ch.
7. 50rd.
Art. 884.
2. 111.80 sq. ft.
3. 3 sq. ft. 1.7
sq. in.
4. 13 A. 41.76
P.
5. 349.07 sq. ft
5^
278
AJq^SWERS.
Art. 886.
2. 39 ft.
8. 25 ft. 7.34 in.
J^, 33.97 ch.
5. 28 ft. 3.36 in.
Art. 887.
2. 45 yd.
3. 19 ft. 2.5 in.
Jf. 360 ft. 6f in.
6. 20 ft. .
Art. 898.
2. 84 sq. ft.
3, 5 J A.
Art. 899.
2, 11178 sq. ft.
3, 28| sq. ft.
4, 2 A.
Art. 900.
2, 213 sq. ft.
3, 17 A. 8 ch. 3.4 P.
Art. 904.
8, 15 ft. 10.98 in.
.4. 5 ft. 10.67 in.
5, 5 ft.
6, 7 ft. 3.96 in.
Art. 905.
4. 318.3 A. +
5. 114.59 A.
Art. 906.
3. 7 rd.
J^. 19.098 ft. Diam.
59.998 ft. Circum,
Art. 907.
2. 141.42 ft.
8. 23.4 yd. +
Jt-. 7.07 ft. +
Art. 908.
2. 32.98 sq. ft. 4-
3. 796.39 sq. ft.
Jt. 1 A. 75.62 P. land.
78.54 P. water.
Art. 909.
f . 84.
3' 28.
^. A-
5. 32 lb. 13.7 oz.
Art. 910.
5. 369 rd. L.;
123 rd. W.
6. 3.5 in.
7. 221 ; 238 ; and
255 ft.
8. $75.
9. 126.78 rd.
Art. 911.
1. $185.53.
^. 35.35 ft. +
3, 403.7 rd. +
.4. $5812.50.
5. $32.40.
^. 28.66 P. +
7. 5 A. ; or twice as
large.
8. $724.75. -
9. 20 ft.
iry. 98 A. 28 P.
11. 14.645 ft.
12. 294 rd.; 45.36 rd.
13. 14 A. 150.4 P.
U. 6 in.
Art. 918.
.4. 207.34 sq. ft.
5- 168| sq. ft.
e. 263.89 sq. ft.
7. 301.177 sq. ft.
Art. 919.
3. 274| cu. ft.
Jf. $27.
5. 73.63 cu. ft.
^. $53.70.
Art. 925.
2. 824 67 sq, ft.
3. 429| sq. ft.
.4. 512.9 sq. ft.
5. $25.
Art. 926.
3. 39.27 cu. ft.
4. $29.23.
5. 192000 cu. ft. vol
22284.6 sq. ft.
surface.
Art. 927.
2. 345 sq. ft.
3. 256| sq. yd.
Art. 928.
2. 58.1196 cu. ft.
3, 38i cu. ft.
It, 64.99 cu. ft.
Art. 932.
2. 28.27 sq. ft.
3, 12.57 sq. ft.
Art. 933.
2. 14137.2 cu. ft.
3, 523.6 cu. yd.
Art. 934.
2. 10 ft.; 15 ft.; and
20 ft.
3. 24 ft.; 33 ft.; and
40 ft.
Art. 936.
i. 13.228 ft. edge.
2315.03 cu. ft. vol.
2. 11 ft. 7 in.
3. 1494.257 gal.
It. $5.46.
5. 576 ft.
^. 14.42 in.
7. 40 sq. ft. 7f '.
^. 1 cu.ft. vol.of cube
Icu.ft. 659.5 cu. in.
vol. of sphere.
9. 9 lb.
10. 5 hr. 26.4 min.
11. 12 ft. 6.79 in.
12. 53.855 bu.
Art. 937.
2. 99.144 gal.
3. 120.09 gal.
RETURN TO the circulation desk of any
University of California Library
or to the
NORTHERN F
VB 17383
QA 16
PS
9
THE UNIVERSITY OF CALIFORNIA LIBRARY
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