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DABOLL's
schoolwiastER's Assistant.
IMPROVED AND ENLaitor, D.
B-IN- a
PLAIN PRACTICAL SYSTEM
A R T H M ET 1 c K.
- ---------- º
Tiir. UNITED STATEs.
-
B Y WAT IIA N 19.4 ± 0.1, L.
----------------------
FARMERS" AND MECHANICKS 19EST
METHOD OF BOOK-KEEPLNG.
-----------
COMPANION TU DABOLL's ARITH METICE.
BY SAMUEL GREEN.
It hac A. N."y.
ANDRUS, wooDRUFF. & GAUNTLETT
-
1843.
- - -
---
- - -
Entered according to Act of Congress, in the year 1839,
by Mack, Asprus, & Woodhury, in the Clerk's office of
the Northern District of New York.
RECOMMENDATIONS.
-
Yale College, Nov. 27, 1790.
I have read Daboli's SchoolMaster's Assistant.
The arrangement of the different branches of Arithmetic
is judicious and perspicuous. The author has well ex-
plained Decimal Arithmetic, and has applied it in a plain
and elegant manner in the solution of various questions,
and especially to those relative to the Federal Computation
of money. I think it will be a very useful book to School-
masters and their pupils.
JOSIAH MEIGS, Professor of Mathematics
and Natural Philosophy.
[Now Surveyor-General of the United States.]
! have given some attention to the work above men-
ºtiºned, and concur with Mr. Professor Meigs in his opinion
ºf its merit. NOAH WEESTER.
New-Haven Dec. 12, 1799.
-
Rhode-Island College, Nov.30, 1799.
I have run through Mr. Dabout.’s SchoolMaster's
Assistant, and have formed of it a very favourable opinion.
According to its original design, I think it well “calculated
to furnish Schools in general with a methodical, easy, and
comprehensive System of Practical Arithmetic.” I there-
fore hope it may find a generous patronage, and have an
ºxtensive spread.
ASA MESSER, Professor ºf the Learned Languages,
and teacher of Mathematics.
(Nºw President ºf that Institutinn.]
-----------T-------
Plainſteld Academy, April 20, 1802
I make use of Daboli's Schoolmasten's Assistant
in teaching common Arithmetic, and think it the best cal
culated for that purpose of any which has fallen within my
observation. JOHN ADAMS,
Rector of Plainfield Academy.
[Now Principal of Philips' Academy, Andover, Mass.]
-
Billerica Academy, (Mass.) Dec. 10, 1807.
Having examined Mr. Danoll’s System of Arithmetic,
I am pleased with the judgment displayed in his method,
and the perspicuity of his explanations, and thinking it as
easy and comprehensive a system as any with which I am
acquainted, can cheerfully recommend it to the patronage
ºf Instructºrs. SAMUEL WHITING,
Teacher of Mathematics.
-
fºrum Mr. Kennedy, Teacher of Mathematics.
1 became acquainted with Damodt's SchoolMaster's
As ustant, in the year 1802, and on examining it atten-
tively, gave it my decisive preference to any other system
extant, and immediately adopted it for the pupils under my
harge; and since that time have used it exclusively in
elementary tuition, to the ſº advantage and improve-
ºnent of the student, as well as the ease and assistance of
he preceptor. I also deem it equally well calculated for
the benefit of individuals in private instruction; and think
it my duty to give the labour and ingenuity of the author
the tribute of my hearty approval and recommendation.
ROGER KENNEDY
New-York, March 20 1811.
PIR. E. F.A.C.E. . .
-
The design of this work is to furnish the schools of the
lºnited States with a methodical and comprehensive system
of Practical Arithmetic, in which I have endeavoured,
through the whole, to have the rules as concise and fami-
liar as the nature of the subject will permit.
During the long period which I have devoted to the in-
struction of youth in Arithmetic, I have made use of various
systems which have just claims to scientific merit; but the
authors appear to have been deficient in an important
point—the practical teacher's experience. They have been
too spuring of examples, especially in the first rudiments;
in consequence of which, the young pupil is hurried through
the ground rules too fast for his capacity. This objection
I have endeavoured to obviate in the following treatise.
In teaching the first rules, I have found it best to en-
tourage the attention of scholars by a variety of easy and
familiar questions, which might serve to strengthen their
minds as their studies grew more arduous.
The rules are arranged in such order as to introduce the
most simple and necessary parts, previous to those which
are more abstruse and difficult.
To enter into a detail of the whole work would be te-
duous; I shall therefore notice only a few particulars, and
refer the reader to the contents.
Although the Federal Coin is purely decimal, it is so
nearly allied to whole numbers, and so absolutely necessary
to be understood by every one, that I have introduced it
immediately after addition of whole numbers, and also
shown how to find the value of goods therein, immediately
after simple multiplication; which may be of great advan-
tage to many, who perhaps will not have an opportunity of
learning fractions.
In the arrangement of fractions, I have taken ºr tº:
new method, the advantages and facility ºf wºr" ºf
**iciently apºlogize f its not heing w riºt a rº
a 2
-- ºn------
systems. As decimal fructions may be learned much eashes
than vulgar, and are more simple, useful, and necessary,
and soonest wanted in more useful branches of Arithmetic,
they ought to be learned first, and Vulgar Fractions omitted,
until further progress in the science shall make them ne-
cessary. It may be well to obtain a general idea of them,
and to attend to two or three easy problems therein; after
which, the scholar may learn decimals, which will be ne-
cessary in the reduction of currencies, computing interest,
and many other branches.
Besides, to obtain a thorough knowledge of Vulgar Frac.
tions, is generally a task too hard for young scholars who
have made no further progress in Arithmetic than Reduc.
tion, and often discourages them.
I have therefore placed a few problems in Fractions, ac
cording to the method above hinted; and after going through
the principal mercantile rules have treated upon Vulga
Fractions at large, the scholar being now capable of going
through them with advantage and ease.
In Simple Interest, in Federal Money, I have given seve-
rul new and concise rules; some of which are particularly
designed for the use of the compting-house.
The Appendir contains a variety of rules for casting
Interest, Rebate, &c. together with a number of the most
easy and useful problems, for measuring superficies and
solids, examples of forms commonly used in transacting
business, useful tables, &c. which are designed as aids ir
the common business of life.
Perfect accuracy, in a work of this nature, can hardly
be expected; errors of the press, or perhaps of the author
may have escaped correction. If any such are pointer
ºut, it will be considered as a mark of friendship and ſº
tour, by
The public's most humble
and obedient Servant,
MATHAN DABOLL.
- - -
annition, simple, - - - - - - - - 17
- of Federal Money, - - - - - - -1
- Compound, - - - - - - - 33
Alligation, - - - - - - - - - 177
Annuities or Pensions, at Compound Interest, - - - 19-
Arithmetical Progression, - - - - - - 182
Barter, - - - - - - - - - - 126
Brokerage, - - - - - - - - 113
Characters. Explanation of - - - - - - 14
Commission, - - - - - - - - 112
Conjoined Proportion. - - - - - - - 137
Coins of the United States, weights of - - - - 220
Division of whole Numbers, - - - - - - º-
-Contractions in. - - - - - - º
Compound, - - - - - - - 53
Discount, - - - - - - - - 123
Duodecimals, - - - - - - - - 216
Ensurance, --- - - - - - - - 114
Equation of Payments, - - - - - - - 125
Evolution or Extraction of Root-, - - - - - 167
Exchange, - - - - - - - - - 139
Federal Money, - --- - - - - - 21
- subtraction of - - - - - 25
Fellowship, - - - - - - - - - 1-2
Compound, - - - - - - - 134
Fractions, Vulgar and Decimal, - - - - -59-143
Interest. Simple, - - - - - - - 104.
- by Decimal-, - - - - - - 157
- Compound, - - - - - - - 122
- by Decimals, - - - - - - 155
Inverse Proportion, - - - - - - - 155
involution, - - - - - - - - - 166
Loss and Gain, - - - - - - - - 128
Multiplication, Simple, - - - - * - - -7
- Application and Use of - - . - -0
Supplement to, - - - - - 37
Compound, - - - - - - 48
Numeration, * * - - - - - - - 15
Practice, - . - - - - - - - ºg
Position, - . - - - - - -- º
Permutatiºn of Quantities - - - - . 19-
º -a-Lº - P -o-TENTs.
Questions for Exercise, - - - - - - -
Reduction, - - - - - - - - -
— of Currencies, do. of Coin, - - - -
Rule of Three Direct, do. Inverse, - - - - -
Double, - - - - - - -
Rules for reducing the different currencies of the several United
States, also Canada and Nova Scotia, each to the par
of all others, - - - - - - -
– Application of the preceding, - - - - -
– Short Practical, for calculating Interest, - - -
for casting Interest at 6 percent, - - - -
- for finding the contents of Superfices and Solids -
- to reduce the currencies of the different States to Fede-
ral Money, - - - - - - - -
Rebate, a short method of finding the, of any given sum, for
months and days, - - - - - - -
Subtraction, Simple, - - - - - -
Compound, - - - - - -
Table, Numeration and Pence, - -
- Addition, Subtraction, and Multiplication,
– of weight and Measure, - - -
– of Time and Motion, - - - - - -
– showing the number of days from any day of one month
to the same day in any other month, - - -
– showing the amount of 11, or 1 dollar, at 5 and 6 per
cent. Compound Interest, for 20 years, - -
– showing the amount of 11. annuity, forburne for 31 years
or under, at 5 and 6 percent. Compound Interest, -
— showing the present worth of 11. annuity, for 31 years,
at 5 and 6 percent. Compound Interest, -
– of Cents, answering to the currencies of the United
States, with Sterling, &c. - - - - -
showing the value of Federal Money in other currencies,
Tare and Tret, - - - - - - -
Useful Forms in transacting business, - - - -
Weights of several pieces of English, Portuguese, and French
Gold Coins, in dollars, cents, and mills, - - -
– of English and Portuguese Gold, do. do. -
— of French and Spanish Gold, do. do.
191
5.
82, ºr
90, 97
13:
º:
º
11-
20-
-uſ.
200
20.
--
4:
º
1.
11
1.
tºr
--t
--
2-1
-25
103
---
DABOLL's
SCHOOLMASTER’s ASSISTANT.
ARITHMETICAL, TABLES.
Numeration Table. Pence Table.
- al. s. d. d. s.
- H 20 is 1 8 || 12 is 1
- º 30 2 6-1 24 2
* : 3 = 40 3 4 || 36 .3
F = F : 50-4 2-1 48 4
* = ‘s = - 60 5 0 || 60 5.
* = . * * * * 70 5 10 || 72 tº
E = 3 E = 3 # 80 tº 8 || 84 7
E = 2 = z = E = 3 90 7 6 || 96 8
= E = E = E = E = | 100 8 4 || 108 9
* . . . . . . . . . . ; ; ; ;
120 10 0 || 132 11
9 8 7 6 5 4 3 2
-9 8 7 6 5 4 3
* * * * * * * * - -
9 8 7 6 5
- 9 8 7 6 make
9 8 7 -
4 farthings 1 penny, d.
- 9 8 12 pence 1 shillin ſ -
9 20 shillings, º +-l.
-n ARTI-MET--AL -at-LE-
ADDITION AND SUBTRACTION TABLE
1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 |10||1||12
2 || 4 || 5 || 6 || 7 || 8 || 9 || 10111112 |13114
3 || 5 || 6 || 7 || 8 || 9 || 10 || 1 |12||13114 15
4 || 6 || 7 || 8 || 9 || 10 || 11 |12 || 13 || 14 15 16
5 || 7 || 8 || 9 |10|1|12||1311||15 ITGT17.
GTSTJTIOTITIET13 IIITISTIGI17 TIS
7| 9 |10|11 || 12 1814 ||5||16 17 |18 |19
| 8 || 10 || 11 |12 || 13 || 14-15 16 || 17 |18 |19| 20
ſº IIITIETI3T IIII: TIGI17 TIs IIITZUT2)
|IGI12 BIII.ii.5IIGilzi is III:02.1122
MULTIPLICATION TABLE.
l
| Tº
GT7TSTUTIOT 1
12 14 | 16 || 18- 20 22
ISTEITETI 27 Tº
24.128 13:2 .
| 48 | 72 ISUISSIſ
15 T54 TG3 Tº TSI Tºol ºf 108
| | |
1-20
~ I
To learn this Table: Find your multiplier in the let
hand column, and the multiplicanda-top, and in the com-
mon angle of meeting, or against your multiplier, alºng a-
the right hand, and under your multiplicand, you will fin
the product, or answer.
- 12 ounces I und
- -
anti-METICAL TABLEs.
2. Troy Weight.
24 grains (gr.) make 1 penny-weight, marked
80 penny-weights, 1 ounce,
3. Avoirdupois Weight.
16 drams (dr.) make 1 ounce, º-
16 ounces, 1 pound, lb.
| 28 pounds, 1 quarter of a hundred weight, ºr.
| 4 quarters, I hundred weight, cunt.
| 20 hundred weight, 1 tun. T.
By this weight are weighed all coarse and drossy goods,
| grocery wares, and all metals except gold and silver.
4. Apothecaries Weight.
to grains (gr.) make I scruple, B
3 scruples, 1 dram, º
8 drams, - 1 ounce, w
| 12 ounces, 1 pound, º
Apothecaries use this weight in compounding their me-
icines.
5. Cloth Measure.
4 nails (na.) make I quarter of a yard, qr.
4 quarters, - 1 yard, yº.
3 quarters, 1 Ell Flemish, E. rºl.
5 quarters, 1 El English, E. E.
6 quarters, 1. Ell French, E. Fºr
6. Dry Measure.
2 pints, (pt) make 1 guart, 7t.
8 quarts, I peck, pk.
4 peeks, 1 bushel, bu.
This measure is applied to grain, beans, flax-seed, salt
was, oysters, coal, &c.
12 -RITH-ETICAL Tablºs.
7. Wine Measure.
4. gills (gi.) make 1 pint, pt.
2 pints, 1 quart, gt.
4 quarts, 1 gallon, gal
314 gallons, 1 barrel, bl.
42 gallons, 1 tierce, tier.
63 gallons, 1 hogshead, hhd
2 hogsheads, 1 pipe, P.
2 pipes, 1 tun, T.
All brandies, spirits, mead, vinegar, oil, &c. are measu
ed by wine measure. Note. 231 solid inches, make a gal
lon.
8. Long Measure.
3 barley corns (b. c.) make 1 inch, marked in.
12 inches, I foot, ft.
3 feet, 1 yard, d.
54 yards, 1 rod, pole, or perch, ra.
40 rods, 1 furlong, fur.
8 furlongs, 1 mile, ---
3 miles, I league, lea.
69) statute miles, 1 degree, on the earth.
360 degrees, the circumference of the earth.
The use of long measure is to measure the distance of
places, or any other thing, where length is considered, with-
out regard to breadth.
N. B. In measuring the height of horses, 4 inches make
hand. In measuring depths, 6 feet make 1 fathom or
French toise. Distances are measured by a chain, four
rods long, containing one hundred links.
an-TH-METICAL TABLEs. 1-
9. Land, or Square Measure.
144 square inches make square foot.
4 square roods,
640 square acres,
10. Solid, or Cubic Measure.
1728 solid inches make 1 solid foot.
40-feet of round timber, or
50 feet of hewn timber, }
* solid feet, or 8 feetlong, } 1 cord of wood.
wide, and 4 high,
All solids, or things that have length, breadth, and depth,
ºre measured by this measure. N. B. The wine gallon
ºntains 231 solidor cubic inches, and the beer gallon, 282.
* bushel contains 2150,42 solid inches.
square acre,
square mile.
9 square feet, 1 square yard
30-square yards, or
*724 square feet, } 1 square rod.
40 square rods, 1 square rood.
1.
1.
1 tun or load.
11. Time.
60 seconds (S.) make 1 minute, marked M.
60 minutes, 1 hour, k.
24 hours, 1 day, d.
7 days, 1 week, -º-
4 weeks, 1 month, ºn-
13 months, 1 day and 6 hours, 1 Julian year, yr-
Thirty days hath September, April, June, and November
February twenty-eight alone, all the rest have thirty-one.
N. B. In Bissextile, or leap year, February hath 29 days.
12. Circular Motion.
60 seconds (") make 1 minute, ,
60 minutes, I degree, -
30 degrees, 1 si
- S.
13 signs, or 360 degrees, the whoſe great circle of the
| Zodiack.
14 CHARACT-R-.
Explanation of Characters used in this Book.
-
= Equal to, as 12d. = 1s, signifies that 12 pence are equal
to 1 shilling.
+ More, the sign of Addition; as, 5+7=12, signifies that
5 and 7 added together, are equal to 12.
– Minus, or less, the sign of Subtraction; as, 6–2=4, sig-
nifies that 2 subtracted from 6, leaves 4.
× Multiply, or with, the sign of Multiplication; as,
4×3=12, signifies that 4 multiplied by 3, is equal to 12.
+ The sign of Division; as, 8+2=4, signifies that 8 diº
vided by 2, is equal to 4; or thus, 3–4, each of which
signify the same thing.
* : Four points set in the middle of four numbers, denote
them to be proportional to one another, by the rule of
three; as 2:4:: 8: 16; that is, as 2 to 4, so is 8 to 16.
v. Prefixed to any number, supposes that the square root of
that number is required.
*/ Prefixed to any number, supposes the cube root of that
number is required.
v Denotes the biquadrate root, or fourth power, &c.
ARITHMETIC,
ARITHMETIC is the art of computing by numbers,
and has five principal rules for its operation, viz. Numera-
tion, Addition, Subtraction, Multiplication, and Division.
NUMERATION.
Numeration is the art of numbering. It teaches to ex-
press the value of any proposed number by the following
characters, or figures:
-1, 2, 3, 4, 5, 6, 7, 8, 9, 0–or cipher.
Besides the simple value of figures, each has a local
ralue, which depend upon the place it stands in, viz. any
igure in the place of units, represents only its simple value,
or so many ones; but in the second place, or place oftens, it
becomes so many tens, or ten times its simple value; and in
the third place, or place of hundreds, it becomes a hundred
times its simple value, and soon, as in the following
Nºte:-Although a cipher standiº alone signiſes nothing; yet when it
is placed on the right hard uſ figures, it increases their value in a tenſold
prºportion, by throwing them into higher places. Thus, 2 with a cipheran-
nexed tº it, becomes 20, twenty, and with twº ciphers, thus, 200,two hundred.
º: when numbers consisting of many figures, are given to be read, it
will be ſound convenient to divide them into as many periods as we can, of
six figures each, reckoning from the riºt hand towards the leſ, calling the
ºr-tº- riod of units, the secºnd that of millions, the third billions, the
four-n trillions, &c. as in the following number:
5 D7 - 5 - 5 - 5 - 7 - 9 0 1 2 5 0 g º q +
4. Period ºf 3. Period ºf 2. Periºd ºf 11. Period of
Trillion. Billiºns. -Millions. Uniº.
- - - -
ºn- tº Tººl- 500-792
The fore roung number is read thus-Eight thousand and seventy-three
trilliºns six hundred and twenty-five thousand, ſour hundred and sixty-
two billions; seven hundred and eighty-nine thousand and twelve millions;
ºve hundred and six thousand seven hundred and ninety-two.
N. B. Billiºn-i-substituted for millions of milliºns.
Trillions ſº millions of millions of millions.
Quatrillions for millions of milliºus aſ milliºn of millions, ºc.
-6 NUMERATI-N-
TABLE.
*****==35
c - E - - E E = =
--> --> : -* →
==# ==# 3. . .
== 3 = + , ,
É É # = * - I -One
5-5. . . 2 1 -Twenty-one.
. . . * * * 3 2 1 -Three hundred twenty-one.
* * * * * 4 3 2 1 -Four thousand 321.
* * * * 5 4 3 2 1 -Fifty-fourthousand 321.
* - 6 5 4 3 2 1 -654 thousand 321.
* - 7 6 5 4 3 2 1 -7 million 654 thousand 321.
8 7 6 5 4 3 2 1 -87 million 651 thousand 321.
9 8 7 6 5 4 3 2 1 -98.7 million 654 thousand 321
2 3 4 5 tº 7 S 9 -123 million 456 thousand 789.
9 8 7 6 5 4 3 + 8 -98.7 million 654 thousand 348.
To know the value of any number of figures:
Rule.-1. Numerate from the right to the loſt hand, each º:
its proper place, by saying, units, tens, hundreds, &c. as in the Num
ration Table.
2. To the simple value of each figure, join the name of its plac-
beginning at the left hand, and reading to the right.
- Ex-MPLES.
Read the following numbers.
365, Three hundred and sixty-five.
5461, Five thousand four hundred and sixty-one.
1234, One thousand two hundred and thirty-four.
540:26, Fifty-fourthousand and twenty-six.
123461, One hundred and twenty-three thousand foul
hundred and sixty-one.
4666.240, Four millions, six hundred and sixty-six thou-
sand two hundred and forty.
Note. For convenience in eading large numbers, they
may be divided into periods of three figures each, as follows
987, Nine hundred and eighty-seven.
987 000, Nine hundred and eighty-seven thousand.
987 000 000, Nine hundred and eighty-seven million.
987 654 321, Nine hundred and eighty-seven million, sº
hundred and fifty-four thousand, three hun
drel and twenty-one.
--PLE ADDITI-N. --
To write numbers.
Raº – degin on the right hand, write units in the units place,
ºns ºn he tens place, hundreds in the hundreds place, and so on,
towards the left hand, writing each figure according to its proper value
in numeration; taking care to supply those places of the natural
urder with ciphers which are omitted in the question
Ex-PLEs.
Write down in proper figures the following numbers:
Thirty-six. - -
Two hundred and seventy-nine.
Thirty-seven thousand, five hundred and fourteen.
Nine millions, seventy-two thousand and two hundred.
Eight hundred millious, forty-fourthousand and fifty-five.
-SIMPLE ADDITION.
Is putting together several smaller numbers, of the same
lenomination, into one larger, equal to the whole or sum
ºutal; as 4 dollars and tº dºlars in one sum is 10 dollars.
Ruu-Having placed units under units, tens under tens, &c. draw
- line underneath, and begin with the units; after adding up every
figure in that column, consider hºw many tens are contained in their
muni set down tº remainder under the units, and carry so many as
rou have tºns, to the next column of tºns; proceed in the same man-
her through every column or row, and set down the whole amount
ºf the last row.
F-----L-->.
(1.) (*) (3) 4.
(
-
)
- - - -
º # * 3 = + 2
* E . . * = . . FF # 3. . .
- - - --- -- "t
# 5 # = E = ** *** = ##
-- --- - E -- ~ * = E = 3
4-2 4 1-1 7 º' tº 5 º 2 (; 2
5 3 2 ſº I 0 1 2 - º 4 tº Q 7 7
5 * 8 º . 9 - 7 - + 1 + 3 + 4)
I º 1 5-2 tº G-d 3 2 1 0 1 2.
S 9 G 9 - 7 - 2 - - 7 tº 5 4 :
-
-
-
simple addition.
--------- ----
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golº do ºr + e- + ac
---- - - - - ….…
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····---···---··· -
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|-
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----------
----+ .：
----
begin at the top of the sum, and reckon
he same manner as they were added up
In To prove Addition,
the figures downwards in
---LE ADDITION. tº
wards, and if it be right, this sum total will be equal to the first: Or
cut off the upper line of figures, and find the amount of the rust; then
if the amount and upper line, when added, be equal to the total, the
work is supposed to be right.
2. There is another method of proof, as follows:–
Reject or cast out the nines in each row exampur.
or sum of figures, and set down the re- 37 S 2 | f 2
mainders, each directly even with the figures 5 7 6 6 º 6
in its row ; find the sum of these remain- 8 7 6 5 37
ders; then if the excess of nines in the - 2 –
sum found as before, is equal to the excess 18 30 31.36
-
º -
of nines in the sum total, the work is sup- -
posed to be right.
15. Add 8535, 2194,7421, 5053, 2106, and 1245, to-
ºther. - Ans. 25754.
16. Find the sum of 31-2, 7S3515, 318, 7530, and
Mºº-0-15. - Ans. 1047.3020.
17. Find the sum total of 501, 4680, 98,64, and 54.
- Ams. Fifty-five hundred.
ls. What is the sum total of 24074, 16742, 34678, 10467,
and 1:24:07 Mns. One hundred thousand.
19. Add 1021, 3189,26703, 2-0, and titºs, together.
ſus. Forty thousand.
20. What is the sum total of the following numbers, viz.
2:340, 1055, 3700, and 400.5 ! Ans. 11111.
21. What is the sum total of the following numbers, viz.
Nine hundred and forty-seven,
Seventºusand six hundred and five,
Forty-five thousand six hundred,
Three hundred and eleven thousand,
Nine millions, and twenty-five,
Fifty-two millions, and nine thousand 1
Answer, 61374.177
-2. Required the sum of the following numbers viz.
Five hundred and sixty-eight,
Eight thºusand eight hundred and five
Seventy-nine thousand six hundred,
-0 --L---L------
Nine hundred and eleven thousand,
Nine millions and twenty-six.
Answer, 9999999
QUESTIONs.
1. What number of dollars are in six bags, containing
each 375.42 dollars? Ans. 22525-2.
2. If one quarter of a ship's cargo be worth eleven thou-
sand and ninety-nine dollars, how many dollars is the whole
cargo worth 1 Ans. 44396 dols.
3. Money was first made of gold and silver at Argos,
eight hundred and ninety-four years before Christ; how
long has money been in use at this date, 1814
Ans. 2708 years.
4. The distance from Portland in the Province of Main,
to Boston, is 125 miles; from Boston to New-Haven, 16.
miles; from thence to New-York, SS ; from thence tº
Philadelphia, 95; from thence to Baltimore, 102; from
thence to Charleston, South Carolina, 716; and from thence
to Savannah, 119 miles—What is the whole distance from
Portland to Savannah Ans. 1407 miles.
5. John, Thomas, and Harry, after counting their priz.
money, John had one thousand three hundred and seventy
five dollars; Thomas had just three times as many as John
and Harry had just as many as John and Thomas both—
Pray how many dollars had Harry? Ans. 5500 dollars.
- - -
FEDERAL MONEY.
NEXT in point of simplicity, and the nearest allied tº
whole numbers, is the coin of the United States, or
FEDERAL MONEY.
This is the most simple and easy of all money—it in
creases in a tenfold proportion, like whole numbers.
10 mills, (m.) make 1 cent, marked t-
10 rººts, I dine, d.
10 lºes, I dollar, 3.
E.
10 ars, I engº,
addition or PEDERAL MONEY. 2.
Dollar is the money unit; all other denominations being
ralued according to their place from the dollar's place-
A point or comma, called a separatriz, may be placed after
the dollars to separate them from the inferior denominations;
then the first figure at the right of this separatrix is dimes,
the second figure cents, and the third mills."
ADDITION OF FEDERAL MONEY.
Rule.-1. Place the numbers according to their value; that is,
dollars under dollars, dimes under dimes, conts under cents, &c. and
proceed exactly as in whole numbers; then place the separatrix in
the sum total, directly under the separating points above.
examples.
-º-d. c. m. $. d. c. m. $. d. c. m.
365, 5 4 1 439, 3 0-4 136, 5 I 4
487, 0-6-0 416, 3-9 0 125, 0 90
94, 6-7 9 168, 9 3-4 200, 9 0-9
439, 0 & 9 239, 0-6 0 304, 0-0 6
742, 5-0 0 143, 0-0 5. 111, 19 1
212°, 8 G 0
2. When accounts are kept in dollars and cents, and no other de-
ominations are mentioned, which is the usual mode in common reck-
-ning, then the first two figures at the right of the separatrix or point,
any be called so many cents instead of dinnes and cents : for the
lace of dinºsis only the ten's place in cents; because ten cents make
a dime; for example, 48.75, forty-eight dollars, seven dimes, five cents,
may be read forty-eight dollars and seventy-five cente.
If the cents are less than ten, place a cipher in the ten's place, or
pace of dimes-Erample, write down four dollars and 7 cents.
Thus, s:4-07 eſs.
* It may be observed, that all the figures at the left hand of the separatrix
are dºllars; or you may call the first figure dollarº, and the ºther cagº,
* Thus any sum of this money may be read differently, either whoſſy in
the lowest denomination, or party in the higher, and partly in the lowest;
or example, 3754, may be either read 37.54 centºrº lines and 4 cents
-$7 gallars 5 dimes and 4 cents, or 3 eagles 7 dollar-5 dimes and 4 cents.
-- ADDITION OF FE-DERAL-Mow-
EXAMPLES.
1. Find the sum of 304 dollars, 39 cents; 291 dollars, 9
cents; 136 dollars, 99 cents; 12 dollars and 10 cents.
304, 39
291, 09
Thus, 136, 99
12, 10
Sum, 744, 57. Seven hundred forty-four dol
lars and fifty-seven cents.
(2.) (3.) (4.)
3. cts. $. cts. $. cts.
0, 99 364, 00 3287, 80
0, 50 21, 50 1729, 19
0, 25 8, 09 4219, 99
0, 75 0, 99 140, 01
(5.) (6.) (7.)
$. $. cts. $. cts.
- 246S 124, 50 16}
1900 9, 07 , 99
246 0, 60 , 864
146 231, 01 ,-17
157 0, 75 , 67:
46 24, 00 , 72
19 9, 44 , 99
8 0, 95 , 09
- -
8. What is the sum total of 127 dols. 19 cents, 278 dols
19 cents, 34 dols. 7 cents, 5 dols. 10 cents, and 1 dol. 99
cents? Ans. $445, 54 cts.
9. What is the sum of 378 dols. I ct-, 135 dols, 91 cts.
344 dols. Scts., and 365 dos. 7 Ans. $12:24.
10. What is the sum of 46 cents, 52 cents, 92 cents, an
10 cents 1 Ans. $2.
ll. What is the sum of 9 dimes, 8 dimes, and 80 cents
Ans. $24.
simple & Up-Tº-ACTION. ºs
12. I received of A, B, and C, a sum of money; A paid
me 95 dols. 43 cts., B paid me just three times as much as
A, and C paid me just as much as A and B both : can you
tell mehow much money C paid me? Ans. $381,72cts.
13. There is an excellent well built ship just returned
from the Indies. The ship only is valued at 12145 dols.
86 cents; and one quarter of her cargo is worth 25411 dols.
65 cents. Pray what is the value of the whole ship and
cargo? Ans. 113792, 46 cts.
-
A TAILOR'S BILL.
Mr. James Paywell,
To Timothy Taylor, Dr.
1814, 3. cts. º-cºs.
pril 15. To 2 yds. of Cloth, at 6, 50 per yd. 16 25
To 4 yds. Shalloon, 75 3 00
To making your Coat, 2 50
To l silk West pattern, 4 10
To making your West, 1 50
To Silk, º &c. for West, 0 45
Sum, 827 80
ºr By an act of Congress, all the accounts of the United States,
the salaries of all officers, the revenues, &c. are to be reckoned in
federal money; which mode of reckoning is so simple, easy, and con-
renient, that it will soon come into common practice throughout al.
the state.
SIMPLE SUBTRACTION.
-
Subtraction of whole Numbers,
TEACHETH to take a less number from . er, of
he same denomination, and thereby shows the difference,
ºr remainder: as 4 dollars subtracted from 6 dollars, the re-
mainder is 2 dollars. -
Ruiz-Place the least number under the greatest, so that units
º stand under units, ten- under tens, &c, and draw "a under
--
-
24 simple SUBTRACTION.
2. Begin at the right hand, and take each figure in the lower line
from the figure above it, and set down the remainder.
3. If the lower figure is greater than that above it, add ten to the
upper figure; from which number so increased, take the lower and
set down the remainder, carrying one to the next lower number, with
which proceed as before, and soon till the whole is finished.
Proof.
be equal to the greatest, the work is right.
Add the remainder to the least number, and if the sum
Exa-PLEs.
(1.) (2.) (3.)
Greatest number, 2 4 6 S 6 2 1 57 8 7-9 G 4 7 5
Least number, 1. 3 4 5 - 2 1 + 8 + 6-4 5-4 & 9
Difference, - -
Proof, - -
(4.) (5.) (6.)
From 41678839 91-75-1520 6543.2167800
Take 31542.999 912-19806 12345607098
Rem. -
- (8.)
--
From 917-114043005
356,2176255.002
Take 40600Sºlt;4 12:35:27-10-2105.
Rem. -
(9.) (10.) (11.) (12.)
From 100000 2521GG5 200000 10000
Take 65321 2000,000 99.999 I
Diſ. - -
13. From 360418, take 293752. Ans. 66666.
14. From 765410, take 34747. Ans. 730GG-3.
15. From 341209, take 198765. Ans. 142444.
16.
17.
From 1000-16, take 10009.
From 2637804, take 2376982.
Ans. 90037.
Ans. 260822.
18. From ninety thousand, five hundred and forty-six
take forty-two thousand, one hundred and nine.
Ans, 48437.
19. From fifty-four thousand and twenty-six, take nine
thousand two hundred and fifty-four.
Ans. 41772.
NLY. 25
20. From one million, take nine hundred and ninety-nine
thousand. Ans. One thousand.
21. From nine hundred and eighty-seven millions, take
nine hundred and eighty-seven thousand.
Ans. 98.6013000.
22. Subtract one from a million, and show the remainder.
Ans, 999999.
-UESTIONs.
1. How much is six hundred and sixty-seven greater
man three hundred and ninety-five? Ans. 272.
2. What is the difference between twice twenty-seven,
and three times forty-five 1 Ans, 81.
3. How much is 1200 greater than 365 and 721 added
together? - Ans. 114.
4. From New-London to Philadelphia is 240 miles. Now
ºf a man should travel five days from New-London towards
Philadelphia, at the rate of 39 miles each day, how far
would he then be from Philadelphia. Ans. 45 miles.
5. What other number with these four, viz. 21, 32, 16,
and 12, will make 1001 Ans. 19.
6. A wine merchant bought 721 pipes of wine for 90846
dollars, and sold 543 pipes thereof for 80049 dollars; how
many pipes has he remaining or unsold, and what do they
stand him in 1
Ans. 178 pipes ansold, and they stand him in $1797.
-
subtraction of Federal, Money.
Ruus-Place the numbers according to their value; that is, dollars
under dollars, dimes under dimes, cents under cents, &c. and subtract
is in whole numbers.
E-A-M-E--
$. d. c. m.
From 45, 4 7.5
Take 43, 4 85
Rem. §t, 99.0 one dollar, nine dimes, and nine cents
or one dollar and ninety-nine cents
-
-J -UBT mat-Titon ()- PELErual, Mon---
8. d. c. $. d. c. m. d. c. m.
From 45, 7 4 46. 2 4 6 211, 1 1-0
Take 13, 8 9 36, 1 6 4 111, 1 1 4
Rem. -
8. S. cits. 8. cfs.
From 4 2-8-4 411, 24 960, 00
Take 1 99 3 13, 09 136, 41
Rem -
8, cts. 3. cis- 3. cts.
Frem 4-106, 71 1901, 08 365, 00
Take 221, 69 864, 09 100, 01
Rem. - -
11. From 125 dollars, take 9 dollars 9 cents.
Ans. 115 dolls, 91 cte
12. From 127 dollars 1 cent, take 41 dollars 10 cents.
Ans. 85 dolls. 91 cts.
13. From 365 dollars 90 cents, take 168 dols. 99 cents
Ans. $196, 91 cts.
14. From 249 dollars 45 cents, take 180 dollars.
Ans. $69, 45 cts.
15. From 100 dollars, take 45 cts. Ans. $99, 55 cts.
16. From ninety dollars and ten cents, take forty dollan
and nineteen cents. Ans, 849, 91 cts.
17. From forty-one dollars eight cents, take one dolla.
nine cents. Ans. $39, 99 cts.
18. From 3 dols, take 7 cts. Ans. 82, 93 cts.
19 From ninety-nine dollars, take ninety-nine cents.
Ans. $98, 1 ct.
20. From twenty dols, take twenty cents and one mill.
Ans. 819, 79 cts. 9 mills.
21. From three dollars, take one hundred and ninety-nine
------- Ans. $1, 1 ct.
ºl. From 20 dols, take 1 dime. Ans. $19,90cts.
*3. From ºue dollars and ninety cents, take ninety-nine
-une Ans, 0 remains.
* Ja- wºre money was 219 dollars, and Thomas
st-PLE MULTIPLICATIo- -7
received just twice as much, lacking 45 cents. How Much
money did Thomas receive? Ans. 3437, 55 cts.
25. Joe Careless received prize money to the amount of
1000 dollars; after which he lays out 411 dolls. 41 cents
for a span of fine horses; and 123 dollars 40 cents for a
gold watch and a suit of new clothes; besides 359 dols.
and 50 cents he lost in gambling. How much will he have
left after paying his landlord's bill, which amounts to 85
dols, and 11 cents? Ans. $20, 58 cts.
SIMPLE MULTIPLICATION
TEACHETH to increase or repeat the greater of two
numbers given, as often as there are units in the less, or
multiplying number; hence it performs the work of many
additions in the most compendious manner.
The number to be multiplied is called the multiplicand.
The number you multiply by, is called the multiplier.
The number found from the operation, is called the pro-
duct. -
Note. Both multiplier and multiplicand are in general
called factors, or terms.
CASE 1. -
When the multiplier is not more than twelve.
Rule.-Multiply each figure in the multiplicand by the multiplier.
carry one for every ten, (as in addition of whole numbers,) and you
will have the productor answer.
Proop—Multiply the multiplier by the multiplicand.”
Ex-MPLEs.
What number is equal to 3 times;865?
Thus, 365 multiplicand.
3 multiplier.
Ans. 1095 product.
* Multiplication may also be proved by casting ºut the as in the two
actors, and setting ºn the remainders; then multiplying the two re-
-inders together; if the excess of 9's in their product is equal to the ex-
re-afºe in the total product, the work is ºut-tº-ed to be right-
28 sniple witnºrºrlivartoº."
Multiplican.” 7.4635 5.432 2345 907.
Multiplier. 3. 4 5 0.
Product,
47094 no.34 siegi tºo
7. * 9 10
1432016 *240613 1684.114
11 12 12
CA*E. 11.
When the ºn...tiplier cºnsists of several figures.
Rule.—The mulºpºer being p-ced under the multiplicanã, unlu
under units, tens under tens, &c. multiply by each significant figure
in the multiplier sepºinºly, racing the first figure in each product
exactly under its mulºlºr: tºn add the several products togethe
in the same order as they stent, and their sum will be the total product
-----PLEs.
What number is equa; to 47 times 365?
Multiplicand, 3 65
Multiplier, 1 7
2 5 5-7
4 6-0
Ans. 1 7-1 5-5 product
Multiplicand, 37864 34.293 47042
Multiplier, 209 74. 91
340776 -
75728
Product, 791.357.6 25.37682 4280822.
25.3 25203 2193 game
26 4025 4072 9105
esºs toº; sº sº; gºssºs
- - -
-------L-L-T-------T-O- no
2-u 81 2619.8% 40634
+-vº 76:2- 1206S
ºssº 20010-1906S 1709:591112
13-1092 91s273515
87362 100.3245
117-145.45304 92.1253442.978025
14. Multiply 760483 by 9.152. Ans. 69509-10-116.
15. What is the total product of 7603 times 365.432.
Ans. 2780-20ttºº.
16. What number is equal to 40003 times 48975-5.
Ans, 195922003055,
- CASE III.
When there are ciphers on the right hand of either or
both of the factors, neglect those ciphers; then place the
significant figures under one another, and multiply by them
ºnly, and to the right had of the product, place as m
*phers as were omitted in both the factors.
ExAMPLEs.
21200 3.1800 81600
70 36 34000
1484,000 1144800 2-76.00000
35926000 82530
3040 9-260000
1092.14040000 810:0:297-00000
7065000 x 8700-01465500000
7-1964:3000 x 695000–52 1001885,000unt
300000×1200,000–132000000000
CASE IV.
When the multiplier is a composite number, that is, when
it is produced by multiplying any two numbers in the ºve
together, multiply first by one of those figures, ºn the
c 2
-- a-PLE MULTI-Litº Ation.
product by the other, and the last product will be the total
required.
Exam-PLEs.
Multiply 41364 by 35
* x 5–35. 7
289548 Product of 7.
5
1447740 Product of 35
2. Multiply 764131 by 48. Ans. 3667.82-8
3. Multiply 342516 by 56. Ans. 19180896
4. Multiply 209102 by 72. Ans. 15076914.
5. Multiply 91738 by 81. Ans. 7430778,
6. Multiply 34462 by 108. Ans. 37:21800.
7. Multiply 6152.43 by 144. Ans. SS59-1992
- CASE v.
To multiply by 10, 100, 1000, &c. *nex to the muli
plicand all the ciphers in the multiplier, and it will make
the product required
-xAMPLEs.
1. Multiply 365 by 10. Ans, 3650.
2. Multiply 4657 by 100. Ans. 465700.
3. Multiply 52.24 by 1000. Ans. 5224000.
4. Multiply 26460 by 10000. Ans. 264600000.
ExAMPLES -Gº - XER-SE.
1. Multiply 1203450 by 9004. Ans. 108.35863300.
2. Multiply 9087061 by 56708. Ans. 515309,055.188.
3. Multiply 8706544 by 67089. Ans, 58.4113330.416.
4. Multiply 4321209 by 123409. Ans. 53327.6081481.
5. Multiply 3456789 by 567090. .1ms. 1960.310474010.
6. Multiply 84964.27 by 874359. Ans. 74289.27415293.
98763542 x98763542=875-1237,228.385764.
-
Application and Use of Multiplication.
ºn making out bills of parcels, and in finding the value tº
goods; when the price of one ward, pound, &c. is given (in
Federal Money) to find the value of the whole quantity,
º-PL--U-I-T-I-L-L-ATION. 31
Rulz.-Multiply the given price and quantity together, as in
whole numbers, and the separatrix will be as many figures from the
ight hand in the product as in the given price.
Ex-M.--L-->.
1. What will 35 yards of broad- } $. d. c. m.
cloth come to, at 3, 4 9 6 per yard?
3 5
17 4-8 t)
10.4 S 8
Ans. $122, 3 60=122 dol-
[lars, 36 cents.
2. What cost 35 lb. cheese at 8 cents per lb. ?
,08
-
Arts. 32, 80–2 dollars 80 cents.
J. What is the value of 29 pairs of men's shoes, at 1 dol-
w-51 cents per pair? Ans. $43, 79 cents.
4. What cost 131 yards of Irish linen, at 38 cents per
yard 7 Ans. $49,78 cents.
5. What cost 140 reams of paper, at 2 dollars 35 cents
aer ream 7 Ans. $329.
6. What cost 144 lb. of hyson tea, at 3 dollars 51 cents
per lb. ? Ans. $505, 44 cents.
7. What cost 94 bushels of oats, at 33 cents per bushelt
Ans. $31, 2 cents.
8. What do 50 firkins of butter come to, at 7 dollars 1:
cents per firkin Ans. $357.
9. What cost 12 cwt. of Malaga raisins, at 7 dollars 31
cents per cwt. 2 Ans. 887, 72 cents.
10. Bought 37 horses for shipping, at 52 dollars perhead:
what do they come to? Ans. $1924.
11. What is the amount of 500 lbs. of hog's-lard, at 15
cents per lb. ? Ans. $75
12. What is the value of 75 yards of satin, at 3 dollars
75 cents per yard? - Ans. $281, 25.
13. What cost 307 acres of land, at 14 dols. 67 cents
per acre? Aºns. $5383, 80 cents.
32 D-15-10- ºr witu. E. NUM-nºns.
14. What does 857 bls, pork come to, at 18 dols. 9º
cents per bl. 1 Ans. $162:23, 1 cent.
15. What does 15 tuns of hay come to, at 20 dols, 7:
cts. per tun? Ans. $311, 70 cents.
16. Find the amount of the following
--------------
New-London, March 9, 1814.
Mr. James Paywell, Bought of William Merchant
S. cts.
28 lb. of Green Tea at 2, 15 per tº
41 lb. of Coffee, at 0, 21
34 lb. of Loaf Sugar, at 0, 19
13 cwt. of Malaga Raisins at 7, 31 per cwt.
35 firkins of Butter, at 7, 14 perfºr.
27 pairs of worsted Hose, at 1, 0.4 per pair.
94 bushels of Oats, at 0, 33 per bush.
29 pairs of men's Shoes, at 1, 12 per pair.
Amount, $511, 78.
Received payment in full, William Merciann
A SHORT RULE.
Note. The value of 100lbs. of any article will be jua
as many dollars as the article is cents a pound.
For 100 lb. at 1 cent per lb.-100 cents=1 dollar.
100 lb. of beef at 4 cents a lb. comes to 400 cents-
dollars, &c.
DIVISION OF WHOLE NUMBERS.
SIMPLE DIVISION teaches to find how many time.
one whole number is contained in another ; and also wha
remains; and is a concise way of performing several sub
tractions.
Four principal parts are to be noticed in Division:
1. The Dividend, or number given to be divided.
2. The Divisor, or number given to divide by.
3. The Quotient, or answer to the question, which shows
how many times the divisor is contained in the dividend.
4. The Remainder, which is always less than the divisor
and of the same name with the Dividend.
to-vision or wildle NUM-ERs. 33
Rºle.—First, seek how many times the divisor is contained in as
many of the left hand figures of the dividend as are just necessary :
(that is, find the greatest figure that the divisor can be multiplied by,
so as to produce a product that shall not exceed the part of the divi-
send used:) when found, place the figure in the quotient; multiply
the divisor by this quotient figure; place the product under that part
ºf the dividend used; then subtract it therefrom, and bring down the
next figure of the dividend to the right hand of the remainder; after
which, you must seek, multiply and subtract, till you have brought
down every figure of the dividend.
Paoor. Multiply the divisor and quotient together, and and the
emainder, if there be any, to the product; if the work be right, the
sum will be equal to the dividend.”
Ex-MPLEs.
1. How many times is 4 2. Divide 3656 dollars
contained in 9391 7 equally among 8 men.
lººr-Div. Quotient. Divisor, Div. Quotient.
4).939.1(2347 8)36.56(457
8 4. -
13. 9388 - 45
12 +3 Rem. 40
tº 9391 Proof. 56
16 56
31 3656 Proof by
28 addition.
3 Remainder.
* Another method which some make use of to prove division is as ſol-
lows: viz. Add the remainder and all the products of the several quotient
figures multiplied by the divisor together, according to the order in which
they stand in the work; and this sum, when the work is right, will be equal
to the dividend.
A third method of proof by excess of nines is as follows, viz.
1. Cast the nines out of tre divisor, and place the excess on the left hand.
2. Do the same with the quotient, and place it on the right hand.
3. Multiply these two figures toºether, and add their product to the re-
mainder, and reject the nines, and place the excess attop.
4. Cast the nine-ºut ºf the º and place the excess at bottom.
Note. If the sum is rºut, the top and Lottom ſºurce -ll be alike
34 division of whole numbens.
Divisor. Div. Quotient.
29)19359(529 365,49640(136
145 365
Proof by - -
excess of 9’s. 85 1314
5 5S 1095
2X7 279 2190
5 261 2190
Remains 18 0. Rem.
Divisor. Div. Quotient. 95(S5595(901
61)2S609(469 736)S63255(1172
472)251.104(532 there remains 664.
9. Divide 1893.312 by 912. Ans. 207t
10. Divide 1893.312 by 2076. Ans. 912
11. Divide 47254149 by 4674. Ans. 10110 rººt.
12. What is the quotient of 330008048 divided by 420
Ans. 78104.
13. What is the quotient of 76.1858465'ivided by S40;
Ans. 90001.
14. How often does 76.1858465 contain 90001 .
Ans, 8465.
15. How many times 38.473 can you have in 119184693
Ans. 3097 ºf .
16. Divide 280208122081 by 912314.
Quotient, 3071.40; ºri.
MORE EXAMPLES FOR EXERCISE.
Divisor. Dividend. Remainder.
234063)5906249.22( º
47614)327879186( 91S2
987654(98864.1654 )---0 -
CASE II.
When there are ciphers at the right hand of the divisºr
cut off the ciphers in the divisor, and the same number of
figures from the right hand of the dividend; then divide the
remaining ones as usual, and to the remainder (if any) an:
nex those figures cut off from the dividend, and you will
have the true remainder
D-10M ºr wil-L-E NUMBERs. 3-
ENAMI'l-Es.
1. Divide 4673625 by 21400.
***(00)46736)25(218 ºn true quotient by Restitution
428--
303
214
1796
1712
-
8-125 true rem.
2. Divide 379132675 by 6500. Ans. 58374; 33.
3. Divide 421400000 by 49000. Ans. 8000.
4. Divide 11659112 by 89000. Ans. 131 = }}:s.
5. Divide 9187042 by 9170000. Ans. 1. Hºs.
Mºnº. Ex-MPLEs.
Divisor. Dividend. Remains.
125000)4362.50000 Quotient. ) 0.
120000)14959647S( ) 76.478
00:0000543.17230, )??1230
720,000 987654000ſ )534000
CASE III.
short Division is when the Divisor does not exceed 12.
Rule.-Consider huw many times the divisor is contained in the
ºrst figure or figures of the dividend, put the result under, and carry
* many tens to the next figure as there nº ones over.
Divide every figure in the same manner till the whole is finished.
Ex-MPLEs.
Divisor. Dividend.
2) 13415 3)854.94 !)39,107 5)94879
Quotient, 56.707–1
5)120616 7)1527.15 8)96872 9)118724
1. *0-6107 12)14814096 12)57.01963S2
-
- - -- -
-5 -ONTRA-" -- IN L, 1-1-1 on.
Contractions in Division.
When the divisor is such a number, that any two figures
in the Table, being multiplied together, will produce it, di-
vide the given dividend by one of those figures; the quo-
tient thence arising by the other; and the last quotient will
be the answer.
Note. The total remainder is found by multiplying the
last remainder by the first divisor, and adding in the firr
-remainder.
Examples.
Divide 162641 by 72
9)162641 or 8)162641 last rem. 7
8)18071–2 9)20330–1 x0
2258–7 2258–8 65,
- - first rem. -->
True Quotient 225844. -
True rem. 65
2. Divide 178464 by 16. Ans. 11154.
3. Divide 407412 by 24. Ans. 1947.5%:.
4. Divide 942341 by 35. Ans. 26924.
5. Divide 70638 by 36. Ans. 2212.
G. Divide 144872 by 48. Ans. 3018; 4.
7. Divide 93.7387 by 54. Ans. 17359sºr.
8. Divide 93.975 by 84. Ans. 11184;.
9. Divide 145260 by 108. Ans. 1345.
10. Divide 1575360 by 144. Ans, 10940.
2. To divide by 10, 100, 1000, &c.
Rule.—Cutoff as many figures from the right hand of the dividen
as there are ciphers in the divisor, and these figures so cut off are th
remainder; and the other figures of the dividend are the quotient.
-xAMPLEs.
1. Divide 365 by 10. Ans. 36 and 5 remains
2. Divide 5762 by 100. Ans. 57–62 rem.
3. Divide 763753 by 1000. Ans. 763 - 753 rem.
supplement. To MULTIPLICATION. 37
SUPPLEMENT TO MULTIPLICATION.
To multiply by a mixt number; that is, a whole number
joined with a fraction, as 84, 5}, 6, &c.
Rule.—Multiply by the whole number, and take 4, 5, 3, &c. of
the multiplicand, and add it to the product.
Exampl.-S.
Multiply 37 by 234. Multiply 48 by 2.
2)37 48
23; 2:
111 12=1
74 96
869, answer. 132 Ams.
3 Multiply 211 by 50}. Ans. 106554.
4. Multiply 2464 by 84. Ans. 20533.
5. Multiply 345 by 19!. Ans. 6598).
6. Multiply 6497 by 5}. Ans. 334134.
Questions to exercise Multiplication and Division.
1. What win; Qi tuns of hay come to, at 14 dollars a
unº Ans. $1364.
2. If it take 820 rods to make a mile, and every rod
Pontains 54 yards; how many yards are there in a mile !
Ans. 1760.
3. Sold a ship for 11516 dollars, and I owned of her;
what was my part of the money? Ans. $8637.
4. In 276 barrels of raisins, each 3, cvt. how many
hundred weight 1 Ans. 966 cwt.
5. In 36 pieces of cloth, each piece containing 24.
yards; how many yards in the whole? Ans. 873 yds.
6. What is the product of 161 multiplied by itself?
Ans, 25921.
7. If a man spend 492 dollars a year, what is that per
-elendar month? Ans. $41.
8. A privateer of 65 men took a prize, which being
equally divided among them, amounted to 1191 per man;
what is the value of the prize? Ans, jºº.
-
38 ComPol-L- a LD-Tow.
9. What number multiplied by 9, will make 225?
Ans. 25.
10. The quotient of a certain number is 457, and the
divisor 8; what is the dividend ? Ans, 3656.
11. What cost 9 yards of cloth, at 3s. per yard?
Ans. 27s.
12. What cost 45 oxen, at Sl. perhead? Ans. C360.
13. What cost 144 lb. of indigo, at 2 dols. 50 cts, or
250 cents per lb. Ans. $360.
14. Write down four thousand six hundred and seven-
teº, multiply it by twelve, divide the product by nine, and
add 365 to the quotient, then from that sum subtract five
thousand five hundred and twenty-one, and the remainder
will be just 1000. Try it and see.
- -
COMPOUND ADDITION,
IS the adding of several numbers together, having dif
ferent denominations, but of the same generic kind, tu
pounds, shillings and pence, &c. Tuns, hundreds, quar-
ters, &c.
Rule.—1. Place the numbers so that those of the same denomina
tion may stand directly under each other.
2. Add the first column or denomination together, as in whole num
bers; then divide the sum by as many of the same denomination as
make one of the next greater; setting down the remainder under the
column added, and carry the quotient to the next superior denomina
tion, continuing the same to the last, which add, as in simple addition."
1. STERLING MONEY,
Is the money of accountin Great-Britain, and is reckon-
ed in Pounds, Shillings, Pence and Farthings. See the
Pence Tables.
-
- The reason of this rule is evident: For, addition of this money, as 1
in the pence is equal to 4 in the farthingº.1 in the shillings, to 12 in the
pence; and 1 in * pounds, to 20 in the shillings; therefore carrying as di
rected, is the arranging the monº arising from each column, properly in
the scale of denominations: and this reasoning will hold good in the ad.
diuono compound numbers of any denomination whatever,
co-Poux D Ann-Tion.
---------- C. s. d.
What is the sum total of 471. 13s. 47 13 tº
ºd.-19. 2s. 9|a.—141. 10s. I 11 d. Thus 19 2 9.
and 121. 9s. 14. 14 10-11.
12 ºn 1.
Answer, E. 93 16 4.
(2.) (3.) (4.)
E. s. d. ... s. d. ar. £. s. d. ºr,
17 13 11 84 17. 5 3 30 II 1 -
13 10 2 75 tº 4-3 15, 10 0 1
10, 17 3 50 17 - 2 I 0 1 1
8 7 20 10 10 1 3 9 & 3
* 3 1. 16 5 0. 4. 6 x 1
(5.) (6.) (7.)
E. s. d. ºr E. s. d. ºr £. s. d, ºr,
17 17 tº 2 7 17 10 x 5-11 0 0 tº
: Q 10 3 tº tº 8 () 711 to 8 .
59 17 11 2 7 14 11 2 918 tº 9 a.
17 tº 9 3 18, 19 9 3 140 15 10 |
tº 19 10 1. 91 15 8 2 300 19 11 3
107 17 g 2 18 17 10 × 48-10 7 :
1 10 9 to 5 0 1 2 tº 14 ºn 3
(8.) (9.) (10.)
E. s. d. E. s. d. E. s. 1.
105 17 tº 9-10 10 7 97 11 tº
193 10 11 ºt; 9 || 20 0 1
901 13 tº 11 - 10 144 10
319 19 7 141 10 tº 17 11 9
18 17. 4 120 14 0. 9 16 101
101 11 9. 10+ 19 7 0 19 9.
96 its 7 160 10 tº 19 9 4
111 9 9 100 () () 234 11 lu.
975 to 10 0 0 9 180 14 tº
449 12 tº 0 19 tº 421 10 31
29, 10 4 120 U 8 3.41 10 4
T-
11. Find the amount of the following £. s. d.
sums, viz. 4:21, 13s. 5d.-111. 10s.-il.
17s. 8d.-13t, ºs. 7d.-19s. 44-271.
and 151, tis.
-Ins. C. 11, 7
- --
40 co-found -2dition.
12. Add 3041, 5s. and 0}d.—341, 19s. 7d.—71. 18s. 5d
–247.0s. 11d.-19s. 6d. 14r, and 451, together.
Ans. E. 640 3s. 5; d.
13. Find the sum total of 141, 19s. 6d.-111. 4s. 0d.-
25l. 10s.-41. 0s. 6d.-31. 5s. 8d.-19s. 6d. and 0s. 6d.
Ans. E. 600s. 5d.
14. Find the amount of the following sums, viz.
Forty pounds, nine shillings, - - - - - £. s. d
Sixty-four pounds and nine pence, - - -
Ninety-five pounds, nineteen shillings, - -
Seventeen shillings and 4d. - - - - -
Ans. E. 201 6s. Ila
15. How much is the sum of
Thirty-seven shillings and sixpence, -
Thirty-nine shillings and 4d. - - - -
Forty-four shillings and nine pence, -
Twenty-nine shillings and three pence,
Fifty shillings, - - - - - - - - - - -
Ans. E. 10 0s. 1044.
16. Bought a quantity of goods for 125l. 10s. ; paid for
truckage, forty-five shillings, for freight, seventy-nine shil
tings and sixpence, for duties, thirty-five shillings and ten
pence, and my expenses were fifty-three shillings and mn
pence; what did the goods stand me in? -
Ans. E. 136 4s. 1d.
17. Six men took a prize, and having divided it equally
amongst them, each man shared two hundred and forty
pounds, thirteen shillings and seven pence; how much
money did the whole prize amount to 1
Ans. E. 1444 is. 6d
2. troy weight.
lb. oz. punt. gr. lb. oz. plot. gr.
16 11 19 § 8 11 º 1.
4 + 15 21 G 10 16 8
8 8, 19 14 7 8 17 21
6 Q 11 17 1 tº 8 23
+ 7 10. 7 9 7 14, 17
0 7 11 12 7. 9 13 10
-
-- ---
--
-
t
r
l
l,
#
i
1 17
0 15
5 2. 12
6 10
d-or-na.
}; º 3.
13 2 1
10 0 1
12 a 3
57 2 2
19 2 2
*
2
1.
19 1 + 2
8 0 0 3
10 2 1 1
Cº-Protº-N-1). A DL-ITION.
3. avoiadupots weight.
lb. or.
24, 13
17. 12
26, 12
1G 8
24 10
11 12
dr.
14
11
15
7.
12
-
12
3 3 B gr.
10 7.
G 3
7 tº
9 5.
5.
_9_3
T. cwt. ºr lb. oz. dr
91 17 2 24 13. 14
19 9 to 17 10 12
1-1 13 2 01 0 11
47 11 3 19 11 5
69 Dº I 00 12
77
5. cloth measure.
E. E.
44 º
49 4
06
84 4.
07 0
61-2-1
".
º
2
---
2
º
º
I
0.
6, only measure.
* ºt-
-
5
7.
g
4.
t;
7
Y. wine measure.
*hd.gal. qu. pt.
42
27
9.
()
16
tºl
39
14
º
24
5 00 3
nº
3.
|
& 3 B f;
11 6 1 5
ºn 7 0 12
10 1 2 16
8 2 19
0 0 1 10
9 2. 1 tº
E. F. gr. na.
§: "º I
07. 1 3
76 0 2
52 2 3
5- 2 º'
09 º
bu. p.k. it. pt.
25 º º I
tº 2 G 1
43 0 + ()
52 : 5. I
9. 3 tº
54 7 0.
tun.hhu ºral. at
†"ſº
10 : 50 I
28, 2 2 1
19 () ºr 2
37 º 11
0 1 0.
:
!
42
in
b
1.I
COMPOUND addition.
8. Long measurae.
tº fur tº:
le.
85
52
tº-1
7:
7.
28
9. land on squarte measure.
sº-ſt-sº-ºn
**i;
6 129
134
1-13
31
ches
144t.
acres, roods, rods. acres, roods, rods.
478 3. 31 2 18
816 2 17 19 3. 00
49 1 27 9 1. 39 8
63 3. 34 l 3 00 0.
9 3. 37 0. º 27 4.
10, solid measure.
T. - cords, feet. feet. in
41 4; 3. 122 13
12 4. 4. 111 15
49 6 7 83 3.
4. 2 10 127 14
11. time.
P. m. tr. da. Yr. da. h. m. sec.
57 11 3: 6 24 363 23 54 34
3 Q 2 3 21 40 12 40 24
20 & 2 5 13 112 14 00 17
45 10 2 4 14 9 11 18 14
10 7. 1 2 8 24 8 16 13
12, circular Motion.
S. - - - S. - - -
3 29, 17 14 11 29 59 50
1 6 10-17 0 00 40 10
4 18 17. 11 9 4 10 49
5 14 18 10 4 11 6 10
ro-pound sub-aaction.
COMPOUND SUBTRACTION,
TEACHES to find the difference, inequality, or excess,
between any two sums of diverse denominations.
--
Rule.-Place those numbers under each other, which are of the
same denomination, the less being below the greater; begin with the
least denomination, and if it exceed the figure over it, borrow as many
units as make one of the next greater; subtract it therefrom; and to
the difference add the upper figure, remembering always to add one
to the next superior denomination for that which you borrowed.
Note.
1. Sterling Money.
The method of proof is the same as in simple subtraction.
Ex-MPLEs.
(1.) (2.) (3.)
£. s. d, gr. £. s. d. ar. £. s. d.
From 340 16 5. 3 14 14 tº 2 94. 11 6
Take 128 17 2 10 19 6 3 30, 14 8
Rem. 217 19 1 1 -
(4.) (5.)
£. s. d. £. s. d. gr.
Borrowed 44 10-2 Lent 36 0 & 2
Paid 36 11 8 Received 18 10 7 3
Remains Due to me
unpaid
(6.) (7.) (8.)
E. s. d. £. s. d qr. £. s. d.ºr
From 5 0 0 7 11 - 2 476 10-9. I
Tave 4 19 11 4 17 3. 1 277-17 7 :
Rem.
(0.) (10.) (11.)
£. s. d. ar. £. s. d £. s. d. ºr,
From 141 14 9 2 125 01 8 10 13 7 1
Take 19 13 10-2 0 0 6 3
Rem.
124 19 8
44 COMPOUND SU BTRACTION.
12. Borrowed 271. 1 1s. and paid 191. 17s.6d. how much
remains due 7 Ams. £7 13s. 6d.
13. How much does 3471. 6s. exceed 178l. 18s. 5; d. 7
-- Ans. £1387 s. 6; d.
14. From eleven pounds take eleven pence.
Ams. £10 19s. 1d.
15. From seven thousand two hundred pounds, take 18l.
17s. 64d. Ans. £7181 2s. 5}d.
16. How much does seven hundred and eight pounds,
exceed thirty-nine pounds, fifteen shillings and ten pence
halfpenny ? Ans. £668 4s. i d.
17. From one hundred pounds, take four pence half
penny. Ans. £99. 19s. 74d.
18. Received of four men the following sums of money,
viz. The first paid me 371. 11s. 4d. the second 25l. 16s.
7d. the third 19l. 14s. 6d. and the fourth as much as all
the other three, lacking 19s. 6d. I demand the whole sum :
received 3 Ams. £165 5s. 4d.
2. TROY WEIGHT.
lb. oz. put. Oz. put. gr. lb. oz. put. g."
From 6 11 14 4 19 21 44 9 6 lº
Take 2 3 16 2 14 23 17 3. 16 18
Rem.
lb. oz. plot. gr. lb. oz. put.gr.
654 3 "iO i* 942 2 ()
683 * 1 9 13 892 9 2 3
3. AVOIRDUPOIS WEIGHT.
lb. oz. dr. cwt. qr. lb. T. cwt. gr. lb. oz dr.
7. 9 12 7 3 13 7 10 17 5 12
3 12 9 5 1 15 3 12 I 19 10 9
T. ºpt. gr. lb. oz. dr. T. cwt. gr. lb. or d;
810 II 20 10 II 3.17 12 H 12 9 12
193 17 1 20 12 14 180 12 || || 4 || 0 || 4
t- - -
co
45
MPOUND Suetºn ACTION.
4. apothecatues" weight.
3 B gr. th & 3 B gr.
4 1 17 35 7 3. 1 14
1 2 15 17 10. 6 1 18
allºt. pt. gi.
5. 2 5 *
14
2. 1 &
ma.
G12
75
§
37
yd...ft. in. b.c.
+ 2 11 ()
2 2 11 1
le, m. ºn.
# Tº
19 2 4 3)
--
5. cloth measure.
E.E. gr. na. E. Fl. ºr na
4.67 + 1 765. 1 3
291 a 2 149 2. '
E.E. gr. na. E.F. ºr nºt.
615 % I 845 1 1
225, 2 2. 57t; 2 3
6. day measure.
bu.pk. 4t. bu.pk. Qt-pt
8, 1 º 17 2 º: ()
3. 1 tº t; 2 tº 1
7. wine measune.
hhd. gal. qt. pt. T. h.hd. ºral. at ut
13 0. | 0. 2 3 º § º
10 60 & 1 1 2 27 0 ()
-at- nº. hhd. gal. º: pt. -
º § 521, 1+ 2 1
1 1 250 25 + ()
8. Long Measure.
m. fur-po. le. m. ſurpo.
41 tº 22 86. 2 6 º'
10 (; 23 24. 1 7 31
le. m. ºr. - le, m, fur.
16 5" º 9. #ſº º
10 1 - 5 1 1 1 8
66 - COMPOUND SI BTRACTION.
9. LAND OR Squarp, MEASURE.
A. roods. rods. A. r. po- - sq.ft. sq., tº
29 | 10 29 2 17 399 13]
24, 1 25 |7 || 36 19 13%
A. gr. rods. A. gr. rods. sq. ft. sº in
540 () 25 - 130 1 10 8t;{} S4
I 19 1 27 - 49 1 11 143 125
10. solid MEASURE.
tums. Jī; cords. ft. tuns. ft. in
116 24 72 il4 45 18 I 44;
109 39 41 120 - | 6 || 4 || 4:
11. TIME.
yrs. mo. w. da. 3/7's, days. h. min. Sec.
54 II 3 1 24, 352 20 41 20
T 43 II 3 5 14 356 20 49 19
º, a nºmin, sec. w, d. h. min. Sec.
472 2 13 18 42. 781 i 8 23 24
218 4 16 29 54 197 3 12 42 53
12. CIRCULAR MOTION.
S. O f // S. O / fe
9 23 45 54 - 9 29 34 54
3 7 40 56 7 29 40 36
QUESTIONS,
Shewing the use ºf Compound Addition and Subtraction
NEw-York, MARCH 22, 1814.
1. Bought of George Grocer,
12 C. 2 qrs. of Sugar, at 52s. per cwt. £32 1 0 &
28 lbs. of Rice, at 3d. per lb. {} 7 (!
3 loaves of Sugar, wt. 35 lb. at 1s. 1d. per lb. I 17 11
3 C. 2 qes. 14 lb. of Raisins, at 36s. per cwt. 6 10 G
- Ans. 41 5 5
*
\
questions, &c. - 47
2. what sum added to 171. 11s. 8d. will make 100l.”
Ans, 821. 8s. 3d, 34".
3. Borrowed 50l. 10s. paid again at one time 171. Ils.
ºd, and at another time, 91.4s. Sd. at arºther time 17t. 9s.
jd, and at another time 19s. 6d. how much remains un-
paid : Ans. C4 4s. 9d.
4. Borrowed 100l. and paid in part as follows, viz. at one
time 21, 11s. 6d. at another time 1.9l 17s. 4d. at another
time 10 dollars at 6s, each, and at another time two English
guineas at 28s. each, and two pistareens, at 14:1. each;
how much remains due, or unpaid Ans. E52 12s. 8d.
5. A, B, and C, drew their prize money as follows, viz.
A had 75l. 15s. 4d. B had three times as much as A.
lacking 15s. 6d. and C, had just as much as A and B both;
tray how much had Cº. Ans, ºtº 5s. 10d.
6. I lent Peter Trusty 1000 dols, and a serwards lent
him 26 dols. 45 cts, more. He has paid me at one time
151 dols. 40 cts. and at another time 416 dols. 09 cents,
*esides a note which he gave me upon James Paywell, for
| 13 dols, 90 cts. ; how stands the balance between us?
Ans. The balance is $105-06 cts, due to me.
7. Paid A B, in full for E Fºs bill on me, for 1051. 10s.
riz. I gave him Richard Drawer's note for 151. 14s 9d.
Peter Johnson's do. for 30l. 0s. 6d. an order on Robert
Dealer for 391. 11s. the rest I make up in cash. I want to
know what sum will make up the deficiency?
- Ans. C20 8s. 9d.
8. A merchant had six debtors, who together owed ºut.
29.171. 10s. 6d. A, B, C, D, and E, owed him 1675l. 1:2.
9d. of it; what was I's debt 7 Ans. C1241 16s. 9d.
9. A merchant bought 17 C. º qºs. 14 lb. of sugar, of
which he sells 9 C. 3 urs. 25 lb., how much of it remains un-
told 7 Ans. 7 C, 2 grs. 17 lb.
10. From a fashionable piece of cloth which containe,
52 yds. 2 na: a tailor was ordered to take three suits, each
5 yds. 2 qºs, how much remains of the piece?
Ams. 32 yds. 2 ºrs. 2 na,
11. The war between England and America commenced
48 COMPOUND MULTIPLICATION.
April 19, 1775, and a general peace took place January
20th, 1783; how long did the war continue?
Ans. 7 yrs. 9mo. 1 d.
compound Multiplication.
COMPOUND Multiplication is when the Multiplicand
consists of several denominations, &c.
1. To Multiply Federal Money.
Rule.—Multiply as in whole numbers, and place the separatrix a-
many figures from the right hand in the product, as it is in the mul
tiplicand, or given sum.
Exa-PL-8.
$ cts. 3 d. c. m. -
1. Multiply 35 09 by 25. 2. Multiply 4900 5 by 97.
25 97.
17545 34.3035
701S 44.1045
- -
Prod: $877, 25 $4753, 4 S 5
$. cts.
3. Multiply 1 dol. 4 cts. by 305 Ans. 317, 20
4. Multiply 41 cts. 5 mills by 150 Ans. 62, 25
5. Multiply 9 dollars by 50 Ans. 450, 00
6. Multiply 9 cents by 50 Ans. 4, 50
7. Multiply 9 mills by 50 Ans, 0, 45
8. There were forty-one men concerned in the payment
of a sum of money, and each paid 3 dollars and 9 mills;
how much was paid in all? Ans. $123 ºcts. 9 mills.
9. The number of inhabitants in the United States is
five millions; now suppose each should pay the trifling
sum of 5 cents a year, for the term of 12 years, towards
a continental tax; how many dollars would be raised there-
by Ans. Three millions Dollars.
2. To Multiply the denominations of Sterling Money
Weights, Measures, &c.
Rule.-write down the Multiplicand, and place the quantity un-
derneath the least denomination, for the Multiplier, and in multiply-
co-i-o-L-L--ULT-P-CATI-N. 49
ing by it, observe the same rules for carrying from one denomination
to another, as in compound Addition.*
---------------------------
£. s. d. 4. s. d.
Multiply I 11 6 2 by 5. How much is 3 times 11 º'
5 º
erud, eſſ tº sº. £1 tº 3
E. s. a ... s. d. £. s. al-
15, 10 8 2-1 12 tº 21 15 º
2 3.
tº it to 10 tº 4 31 to 9.
5. t; 7.
in 15 s 12 17 10 iſ to 7,
8 9 1-
ºf 12 to tº 19 25 s in
11 12 12.
IT_- T L
Practical Questions.
What cost nine yards of cloth at 5s. 5d. per yard 1
£0.5 6 price of one yard.
Multiply by 9 yards.
Ans. E2 9 5 price of nine yards.
-UESTIONS. ANswers.
£. s. d. £. s. d.
4 gallons of wine, at 0 8 7 per gallon. I 14 4
5 C. Malaga Raisius, at 1 2 3 per cwt. 5 11 º'
7 reams of Paper, at 0-17 91 per ream. 6 4-6,
* When accounts are kept in pounds, shillings, and pence, this kind of mu-
ºpºcation is a concise º mºnº of finding the value of goods, at
sº much per yard, lb. ºc, the general rule being to multiply the given price
by the quantity.
50
----------
8 yds. of broadcloth, at
9 lb. of cinnamon,
II tuns of hay,
12 bushels of apples,
12 bushels of wheat,
2. When the multiplier, that is, the quantity, is a com-
posite number, and greater than 12, take any two such
numbers as when multiplied together, will exactly produce
the given quantity, and multiply first by one of those
figures, and that product by the other; and the last pro lºci
will be the answer.
Multiply by
Produces
Multiply b
al
at
al
al
--L---------T--
7 91 per yard.
I
U 11
2 I 10
0 1 0.
0 0 10
Ex-M-LEs.
What cost 28 yards of cloth, at 6s. 10d. per yard?
£. s. d.
0 5 10 price of one yard.
7
4. per lb.
per tun.
per bush.
per bush.
27 10 price of 7 yards.
4.
Answer, E911 4 price of 28 yards.
- UESTIONS.
24 yards at
27 – at
44 – at
55 — at
72 - at
20 – at
84 – at
96 – at
63 – at £1
-44 – at 1
18
11
17
4.
d.
º
-
per yard,
ANSWERs
£.-s. d.
8, 17 t,
13 5 tº
27 4 6
22 1.
71 14
3 10 it
77 ºr 6
56 S ()
* 18 2 tº
171 0 0
11 2
5 2.
2:3 u
| 1
5 is
0
3 When no two numbers multiplied together will exactly
make the multiplier, you must multiply by any two whose
product will come the nearest; then multiply the upper
itne by what remained; which, added to the last product,
- the answer
--------------T----------N -
Ex-MPLEs.
What will 47 yds, of cloth come to at 17s. 9d. per yd. 7
£. s. d.
0 17 9 price of 1 yard.
Multiply by 5.
produces i s 9 price of 5 yards.
Multiply by 9
Produces 39 IS 9 price of 45 yards.
1 15 6 price of 2 yards.
Antser, E41 14 3 price of 47 yards.
-UESTIONs. ANswººns.
£. s. d. £. s. d.
23 ells of linen, at 0 3 61 per ell. 4 1 54
17 ells of dowlas, at 0 1 61 per ell. 6 2.
39 cwt. of sugar, at 3 10 tº per cwt. 137 9 tº
52 yds. of cloth, at 0 5 9 per yd. 14 19 ()
19 lbs. of indigo, at 0 11 6 per lb. 10 18 tº
*9 yds. of cambric, at 0 13 7 per yd. 19 13 11
| 11 yds. broadcloth, at 1 2 5 per yd. 124 17 6
94 beaver hats, at 1 9 4 a piece. 137 17 4
4. To find the value of a hundred weight, by having the
ºrice of one pound.
If the price be farthings, multiply 2s. 4d. by the farthings
* the price of one lb.-Or, if the price be pence, multiply
is. 4d. by the pence in the price of one lb. and in either
use the product will be the answer.
Ex-MPLEs.
1. What will 1 cwt. of rice come to, at 24d. per lb. ?
s, al.
112 farthings=2 4 price of 1 cwt. at ad. perlb.
9 farthings in the price of 1 lb.
Ans. E! I tº puce of 1 cwt. at 21 d. per h.
COMPOUND MUET i P1.10 Årson 5?
EXAMPLES.
What will 47 yds. of cloth come to at 17s. 9d. per yd. 1
£. s. d. -
0 17 9 price of 1 yard.
Multiply by 5
Produces a s 6 price of 5 yards.
Multiply by 9
Produces 39 18 9 price of 45 yards.
I 15 6 price of 2 yards.
---
Amwser, É41 14 3 price of 47 yards.
QUESTIONS. - ANSWERS.
£. s. d. £. s. d.
23 ells of linen, at 0 3 64 per ell. 4 l 5%
17 ells of dowlas, at 0 || 64 per ell. | 6 24
39 cwt. of sugar, at 3 10 per cwt. 137 9 6
52 yds. of cloth, at 0 5 per yd. 14 19 ()
19 lbs. of indigo, at 0 1 1 per lb. 10 18 6
29 yds. of cambric, at 0 13 per yd. 19 13 II
| | | yds. broadcloth, at I 2 6 per yd. 124 17 6
94 beaver hats, at I 9 4 a piece. 137 17 4
4. To find the value of a hundred weight, by having the
price of one pound.
If the price be farthings, multiply 2s. 4d. by the farthings
in the price of one lb.-Or, if the price be pence, multiply
9s. 4d. by the pence in the price of one lb. and in either
case the product will be the answer.
l
2
l
2
:
EXAMPLES.
1. What will 1 cwt. of rice come to, at 24d. per lb. ?
s. d.
112 farthings=2 4 price of 1 cwt. at #d. per lb.
9 farthings in the price of 1 lb.
Ans. £1 1 0 price of 1 cwt. at 2; d. per lb.
52 tº-1-0u-L MULTIPLICATION.
2. What will 1 cwt. of lead come to at 7d. per lb. ?
s, d.
9 4
7
Ans. E3 5-4
Questions. Answers.
1 cwt. at 24. per lb. = £1 3 4
1 ditto, at 2:4. — = 1 5-8
I ditto, at 3d. — = 1 & 0
I ditto, at 2d. – = 0 18 8
I ditto, at 3}d. – = 1 12 8
-
Examples of Weights, Measures, &c.
1. How much is 5 times 7 cwt. 3 qºs. 15 lb. ?
Cwt. ºrs, lb.
7-3 15
5
Ans. Cwt. 39 1 19
lb. ox. piet.gr. cwt. ºr lb oz
2. Multiply-20 2-7 13 by 4. (3) 27-1 13 12
4 t;
- -
Product lb. S0 9-10 4 lb. 164 () 26 8
questions. ANswº-as.
yds, ºr na. yds, qr,
4. Multiply 14 3 2 by 11 163. 2 º'
hhd.g. at pt. hhd.g. qt pt.
5. Multiply 21 152 by 12 254 G1 2
le. m. fur-po. le. m. ſurpo
6. Multiply S1 2 5 21 by 8 655 1 4 s
a. *-*. - -
7. Multiply 41 2 11 by 18 748 to 3-
yr. m. tº d. yr. m. to d
5. Multiply 20 5 & 6 by 14 286 5, 2 tº
S. * - S. º
9. Multiply 1548 ºn by * 7 19 2 -
= -
compounty Division. 5-
cds. ft. cds. ft.
10. Multiply 3 87 by 8 20 ºt;
Practical Questions in
WEIGHTS AND MEASURES.
1. What is the weight of 7 hlids, of sugar, each weigh
ng 9 cwt. 3 qrs. 12 lb. ? Ans. 69 cwt.
2. What is the weight of 6 chests of tea, each weighing
3 cwt. 2 qºs. 5 lb. ? Ans. 21 cwt. 1 ºr 26 lb.
3. How much brandy in 9 casks, each containing 41
gals, 3 qts. pt. 2 Ams. 376 gals, 3 qts. I pt.
4. In 35 pieces of cloth, each measuring 27 yards, how
many yards? Ans. 971 yds. I qr.
5. In 9 fields, each containing 14 acres, I rood, and 25
poles, how many acres? Ans. 129 a. 2 ºrs. 25 rods.
6. In 6 parcels of wood, each containing 5 cords and 96
feet, how many cords? Ams. 34 cards.
7. A gentleman is possessed of 11 dozen of silver spoons,
each weighing 2 oz. 15pwt. 11 grs. 2 dozen of tea-spoons,
each weighing 10 pºwt. 14 grs. and 2 silver tankards, each
21 oz. 15pwt. Pray what is the weight of the whole?
Ans, 8 lb. 10 oz. 2put. 6 grs.
- - - -
COMPOUND DIVISION,
TEACHES to find how often one number is contained
in another of different denominations.
Division or PEDERAL MONEY.
ſº-Any sum in Federal Money may be divided as a
whole number; for, if dollars and cents be written down as
a simple number, the whole will be cents; and if the sum
consists of dollars only, annex two ciphers to the dollars,
and the whole will be cents; hence the following
Genenal. Rule.-Write down the given sum in cents, and divide
-s in whole numbers; the quotient will be the answer in cºnts.
Nore. If the cents in the given sum are ºr than 11, fou must
always place a cipher on their left, or in tº º ºn tº the cert:
before you write them down
- 2
§§ ºC Yi POUND DIV (SIONº.
ExAMPLEs.
1. Divide 35 dollars 68 cents, by 41. - º:
41)3568(87 the quotient in cents ; and when ther,
328 is any considerable remainder, you may
- annex a cipher to it, if you please, and
28S divide it again, and you will have thº
287 mills, &c.
Rem. I
2. Divide 21 dollars, 5 cents, by 14.
14)2105(150 cents= 1 dol. 50 cts. but to bring cent,
14 into dollars, you need only point off two
-- figures to the right hand for cents, and
70 the rest will be dollars, &c.
70
5
. Divide 4 dols. 9 cts, or 409 cts. by 6. Ans. 68 cts. A
. Divide 9 dols. 24 cts. by 12. Ans. 77 cts.
. Divide 97 dols. 43 cts, by 85. Ans. $1 14 cts. 6m.
. Divide 248 dols. 54 cts. by 125.
Ams. 198 cts. Sm. =$1 98 cts. 8m.
:
7. Divide 24 dols. 65 cts. by 248. Ans. 9...cts. 9m.
8. Divide 10 dols. or 1000 cts. by 25. Ans. 40 cts.
9. Divide 125 dols. by 500. Ans. 25 cts.
10. Divide I dollar into 33 equal parts. Ans. 3 cfs.-:
PRACTICAL QUESTIONS. -
1. Bought 25 lb. of coffee for 5 dollars; what is that "
pound 3 Ans. 20 cłs.
2. If 131 yards of Irish linen cost 49 dols. 78 cts. what
is that per yard 7 Ams. 38 cits.
3. If a cwt. of sugar cost 8 dols. 96 cts. what is that per
pound ! - Ams. 8 cts.
4. If 140 reams of paper cost 329 dols. what is that
per ream 3 Ans. $2.35 cts.
5. If a reckoning of 25 dols. 41 cts. be paid equally among
14 persons, what do they pay apiece? Ans. $181; cts.
6. If a man's wages are 235 dols. S0 cts, a year, what is
that a calendar month 1 Ans, $19 65 cts.
|
compoux D Divišov. 55
7. The salary of the President of the United States, is
twenty-five thousand dollars a year; what is that a day ?
Ans. $68.49 cts.
To divide the denominations of Sterling Money
Weights, Measures, &c.
RULE.—Begin with the highest denomination as in simple divisiv.;;
"d if any thing remains, find how many of the next lower denomi-
e-Ation this remainder is equal to ; which add to the next denomina-
tion : then divide again, carrying the remainder, if any, as before ;
and so on till the whole is finished.
-Proof. The same as in simple Division.
EXAMPLES.
- £ s. d. qr. 36 s. d.
Divide 97 3 II 2 by 5 8)27 18 6
Quo’t. £19 8 9 2 #3 9 9;
36 s. d. - £ s. d.
3. Divide 31 II 6 by 2 Ams. 15 15 9
4. Divide 22 3 9 by 3 7 7 || 1
5. Divide 70 10 4 by 4 17 12 7
6. Divide 56 11 5} by 5 11 6 3}
7. Divide 61 14 8 by 6 10 5 9.
8. Divide 24 15 6 by 7 3 10 94
9. Divide 185 17 6 by 8 23 4 84
(). Divide IS2 16 8 by 9 20 6 34
11. Divide 16 I II by 10 1 12 24
12. Divide I 19 8 by 11 0 3 74
13. Divide 6 6 6 by 12 0 10 64
14. Divide I 2 6 by 9 0 2 6
15. Divide 948 11 6 by 12 79 0 1 1,
2. When the divisor exceeds 12, and is the product of two
or more numbers in the table multiplied together.
RULE.—Divide by one of those numbers first, and the quotient by
ºne other, and the last quotient will be the answer.
EXAMPLES.
36 s. d. £ s. d.
1. Divide 29, 15 0 by 21 Ams. I 8 4
2. Divide 27, 16 0 by 32 0 17 4
3. Divide 67 9 4 by 44 | H 0 8
ºn
56 rºxtºn UNL) Divisiºn.
£ s. d.
4. Divide 24 15 6 by 36
5. Divide 128 9 0 by 42
6. Divide 269 12 4 by 56
7. Divide 248 10 S by 64
S. Divide 65 14-0 by 72
9. Divide 5 10 3 by 81
10. Divide 115 10-0 by 90
11. Divide 136 16 G by 108
12. Divide 202 13 6 by 12
13. Divide 34 4 0 by 144
3. When the divisor is large, and not a composite num
her, you may divide by the whole divisor at once, after man
ner of long division, as follows, viz.
Ex-MPLEs.
1. Divide 1281. 13s. 8d. by 47.
£ s. d. E. s. d.
47)128 13.3(214-9 quotient.
94
17
34 pounds remaining.
Multiply by 20 and add in the 13s.
produces 693 shillings, which divided by 47, give
47 [14s. in the quotient.
223
ISS
35 shillings remaining.
Multiply by 12 and add in the 3d.
produces 423 pence, which, divided as abºve
423 [gives 9d. in the quotient.
£ s. d. £ s. d
2. Divide 113 13. 4 by 31 Ans. 3-13 4
3. Divide 85-6 3 by 75 1 2 ºn
4. Divide 315 3 10, by 365 0-17 3
5. Divide 132 0 8 by 68 I 1-10
5. Divide 740 16 S by 100 * 8. 2
7. Divide SSS IS 10 by 95 9 7 11
compound L-1510N. 57
Examples of Weights, Measures, &c.
1. Divide 14 cwt. I qr. Slb. of sugar equally among 8 men.
C. qr, lb. oz.
8)14 1 8 0
1. 3 4 8. Quotient.
8
14. 1 8 0 Proof.
2. Divide 6 T. 11 cwt. 3 qrs. 19 lb. by 4.
Ans. 1 T. 12 cwt. 3 qrs. 25 lb. 12 oz.
3. Divide 14 cwt. I qr, 12 lb. by 5.
Ans. 2 cwt. 3 q’s 13 lb. 9 oz. 9 dr.--
4. Divide 101b. 13 oz. 10 dr. by 6. Ans.21). 12 oz. 15 dr.
5. Divide 56 lb. 6 oz. 17 pºwt. of silver into 9 equal
parts. Ans. 5 lb. 3 oz. Spºrt. 13 grs.--
6. Divide 26 lb. 1 oz. 5 ºwt, by 24.
Ans. 1 lb. 1 oz. 1 punt. I gr.
7. Divide 9 hlids. 28 gals. 2 qts, by 12.
Ans. 0.hhd. 49 gals. 24ts. 1 pt.
8. Divide 168 bu. I ple. 6 qts, by 35.
Ans. 4 bu. 3 p.ks. 2 qts.
9. Divide 17 lea. 1 m. 4 fur. 21 po by 21.
Ans. 2 m, 4 fur. 1 po.
10. Divide 43 yds. I qr, 1 na: by 11.
Ans. 3 yds. 3 ºrs, 3 na.
11. Divide 97 E. E. 4 qrs. 1 na: by 5.
Ans. 19 yds. 2 qrs. 3 na-
12. Divide 4 gallons of brandy equally among 144
soldiers. Ans. I gill apiece.
13. Bought a dozen of silver spoons, which together
weighed 3 lb. 2 oz. 13 pºwt. 12 grs, how much silver did
each spoon contain? Ams. 3 oz. 4 plot. 11 gº.
14. Bought 17 cwt. 3 qrs. 19 lb. of sugar, and sold out
one third of it; how much remains unsold?
Ans. 11 cipt. 3 ºrs. 22 lb.
15. From a piece of cloth containing 64 yards 2 na, a
tumor was ordered to make 9 soldiers' conts, which took one
third of the whole piece; how many yards did each coat
cºntain? Ans. 2 yds. 1 ºr 2 ha.
5 ſº -º-Mºo-NL Divisi-)-.
PRACTICAL QUESTIONs.
1. If 9 yards of cloth cost 41. 3s. 7] d. what is that
per yard?
£ s. d. ºr,
9)4 3 7 2
9 3 2 Answer.
2. If 11 tons of hay cost 231. 0s. 2d. what is that pe.
tun? Ans. Cº 1s. 10d.
3. If 12 gallons of brandy cost 4, 15s. 6d. what is
that per gallon? Ans. 7s. 11d. 24rs.
4. If 84 lbs. of cheese cost 11, 16s. 9d. what is that
per pound? Ans. 5; d.
5. Bought 48 pairs of stockings for 11ſ. 2s, how muc
a pair do they stand me in? - Ans. 4s. 7d.
6. If a reckoning of 51. 8s. 10 d. be paid equally among
13 persons, what do they pay apiece? Ans. 8s. 4d.
7. A piece of cloth containing 24 yards, cost 18, 6.
what did it cost per yard? Ans. 15s. 3d
S. If a hogshead of wine cost 331. 12s. what is it a gu.
lon? Ans, 10s. 8d.
9. If I cwt. of sugar cost 31. 10s. what is it per pound
Ans. 7d.
10. If a man spend 711. 14s. 6d. a year, what is that
per calendar month? Ans. £5 10s. Gºd.
11. The Prince of Wales' salary is 150,000l. a year
what is that a day? Ans. C110 19s. 2d.
12. A privateertakes a prizeworth 12465 dollars, of whicl
the owner takes one half, the officers one fourth, and the re-
mainder is equally divided among the sailors, who are 125 it
number; how much is each sailor's part? Ans. $2493cts.
13. Three merchants A, B, and C, have a ship in com
pany. A hath #, Bi, and C+, and they receive for freigh
2281. 16s. 8d. It is required to divide it among the own
ers according to their respective shares.
Ans. A's share E143 0s. 5d. B's share £57 4s. 2d
rº's share C28 12s. 1d.
A privateer having taken a prize worth $6850, it
R-L-L-------. 59
divided into one hundred shares; of which the captain is to
have 11; 2 lieutenants, each 5; tº midsipmen, each 2;
and the remainder is to be divided equally among the
sailors, who are 105 in number.
Ans. Captain's share $753 50cts. ; lieutenant's, $342
50 cºs.; a midshipman's, $137, and a sailor's, $35 SS.
_
- - -
REDUCTION,
TEACHES to bring or change numbers from one name
to another, without altering their value.
Reduction is either Descending or Ascending.
Descending, is when great names are brought into small,
as pounds into shillings, days into hours, &c.—This is done
by Multiplication.
Ascending, is when small names are brought into great,
is shillings into pounds, hours into days, &c. This is per-
ºrmed by Division.
REDUCTION DESCENDING.
Rule-Multiply the highest denomination ºven by so many of
-- nºt lºss as make one of that greater, and thus continue till you
ave brought it down as low as your question requires.
Proof. Change the order of the question, and divide your last
ºduct by the last multiplier, and so on.
Ex-M-LEs.
I. In 25l. 15s. 9d. 24rs, how many farthings?
£ s. d. ºrs.
25 15 9 2 Proof.
~0 4)247.58 Ans.-24758.
515 shillings. 12)61892 qrs.
12
- 20).51594.
G189 pence. -
4. C25 1592
24798 farthings.
Note. In multiplying by 20, I added in the 15s.-by 12
neºd—and by 4 the 24rs, which must always be done in
-----------
2. In 311 11s. 10d. Iqr, how many farthings?
Ans. 30329.
60 RELUCT-U-.
3. In 46l. 5s. 11d. 34 rs, how many º
Ans. 44447.
4. In 611.12s. how many shillings, pence, and farthing-
Ans. 1232s. 14784d. 591364rs.
5. In 841. how many shillings and pence?
Ans. 1680s. 201604
6. In 18s. 9d, how many pence and farthings?
Ans. 225d. 9004's.
7. In 3121.8s. 5d. how many half-pence? Ans. 149962.
8. In 846 dollars, at 6s. each, how many farthings?
Ans. 243648.
9. In 41 guineas, at 28s. each, how many pence?
Ans. 13776.
10. In 59 pistoles, at 22s, how many shillings, pence
and farthings? Ans. 1298s. 15576d. 62304 qrs.
11. In 37 half-johannes, at 48s, how many shillings, six
pences, and three-pences?
Ans. 1776s. 8552 six-pences, 7104 three-pences.
12. In 121 French crowns, at 6s. 8d. each, how many
pence and farthings? Ans. 9680d. 38720grs.
-
REDUCTION ASCENDING.
Rule.-Divide the lowest denomination given, by so many of than
name as make one of the next higher, and so on through all the de
nominations, as far as your question requires.
Proof. Multiply ºnversely by the several divisors.
exa-PLEs.
1. In 224765 farthings, how many pence, shillings and
pounds?
Farthings in a penny-4)224765
Pence in a shilling =12)5619.1 1 qr.
Shillings in a pound =20)468.27a.
£234 2s. 7d. I gr.
Ans. 56191d. 4682s. 234.
Note. The remainder is always of the same name as
the dividend.
2. Bring 30329 farthings into pounds. -
Ans. £31 11s. 16d. Iqr.
REDUCTION. G1
3. In 44447 farthings, how many pounds?
Ans. £46 5s. 11d. 34 rs.
4. In 59136 farthings, how many pence, shillings, and
rounds? Ans. 14784d. 1232s. e61 12s.
5. In 20160 pence, how many shillings and pounds?
Ans. 1680s, or £84.
6. In 900 farthings, how many pounds?
Ans. CO 18s. 9d.
7. Bring 74981 half-pence intopounds. Ans. E1564s.2d.
8. In 243648 farthings, how many dollars at 6s. each?
- Ans. $846.
9. Reduce 18776 pence to guineas, at 28s. per guinea.
Ans. 41.
10. In 62304 farthings, how many pistoles, at 22s. each?
- Ans. 59.
11. In 7104 three-pences, how many half-johannes, at
SS. 7 Ans. 37.
12. In 38720 farthings, how many French crowns, at
is. 8d.: Ans. 121.
- -
Reduction Ascending and Descending.
1. money.
1. In 1211. 0s. 9d. how many half-pence? Ans. 58099.
2. In 58099 half-pence, how many pounds?
Ans. 1211. 0s. 9d.
3. Bring 23760 half-pence into pounds. Ans. E4910s.
4. In 214. 1s 8d, how many shillings, six-pences, three-
lences, and farthings? Ans. 4281s. 8562 six-pences,
17125 three-pences, and 205500 farthings.
5. In 1371, how many pence, and English or French
rowns, at 6s. 8d.: each? Ans. 32880d. 411 crowns.
6. In 249 English half-crowns, how many pence and
pounds? Ans. 9960.d. and £41 10s.
7. In 346 guineas, at 21s, each, how many shillings,
groats, and pence? Ans. 7266s. 21798 grºts, and 87.192d.
8. In 48 guineas, at 28s, each, how many 4d. pieces?
Ans. 358.
9. In 81 guineas, at 27s. 4d. each, how many pounds?
Ans. E110 14s.
62 R-T------
10. In 24396 pence, how many shillings, pounds, a 1d
pistoles? Ans. 2033s. E101.13s. and 92 pistoles. 9s. over.
11. In 252 moidores, at 36s, each, how many guineas a
º-s, each 1 Ans. 324.
12. In 1680 Dutch guilders, at 2s. 4d. each, how many
pistoles at 22s. each? Ans. 178 pistoles, 4s.
13. Borrowed 1248 English crowns, at 6s. 8d. each, how
many pistareens, at 14, d. each, will pay the debt?
Ans. 6885 pistareens, and 73d. -
14. In 50l., how many shillings, nine-pences, six-pences,
four-pences, and pence, and of each, an equal number?
12d.--9d.--6d.--4d.--1d.-32d. and £50=
12000d.--32–375 Ans.
Examples in Reduction of Federal Money.
1. Reduce 2745 dollars into cents.
2745 dollars Here I multiply by 100, the cents in
100 a dollar; but dollars are readily brought
into cents by annexing two ciphers,
Ans. 27.4500 and into mills by annexing three ci.
phers. Also, any sum in Federal money
may be written down as a whole number, and expressed in
its lowest denomination; for, when dollars and cents are
joined together as a whole number, without a separatrix,
they will show how many cents the given sum contains;
and when dollars, cents, and mills, are so joined together,
they will show the whole number of mills in the given
sum-Hence, properly speaking, there is no reduction of
this money; for cents are readily turned into dollars by cut-
ting off the two right hand figures, and mills by pointing
off three figures with a dot; the figures to the left hand of
the dot, are dollars; and the figures cut off are cents, or
tents and mills.
2. In 345 dollars, how many cents, and mills?
Ans. 34500 cºs. 345000 mills.
3. Reduce 48 dols. 78cts. into cents. Ans. 4878
4. Reduce 25 dols. Scts, into cents. Ans. 2508
5. Reduce 54 dols. 36 cts. 5 m. into mills. Ans. 54.365
ti. Reduce 9 dols, 9 cts, 9 m, into mills. Ans, 9099
-
REDUCTION. tº 3
s cºs.
7. Reduce 41925 cents into dollars - Ans. 419 25
8, Change 4896 cents into dollars. 48 96
9. Change 45009 cents into dollars. 450 00
10. Bring 4625 mills into dollars. 1 tº 5
-
2. traoy weight.
1. How many grains in a silver tankard, that weighs
Ib. 11 oz. 15 pºvt.
lb. oz. purt.
1 11 15
12 ounces in a pound. -
23 ounces.
20 pennyweights in one ounce
475 pennyweights.
24 grains in one pernyweight.
1900
950
Proof. 24)11400 grains. Ans.
2,0)47,5
12)23 15 pyt.
1 lb. 11 oz. 15 pºwt.
* In 246 oz. how many pºwts, and grains?
Ans. 4920 pupt. 118080 grº.
* Bring 46080 grs. into pounds. Ans. 8.
4. In 97.397 grains of gold, how many pounds?
Ans. 16 lb. 10 oz. 18 put. 5 grº.
5. In 15 ingots of gold, each weighing 9 oz. 5pwt. how
any grains? Ans. 66600.
6. In 4 lb. 1 oz. 1 pºwt. of silver, how many table-spoons,
weighing 23 pºvt. each, and tea-spoons, 4 pºwt. 6 grs, each,
tan be made, and an equal number of each sort?
23pwt.*4pwt. Gºrs.-654grs, the divisor; and 4th. 1 oz.
Iput.-23544 grs, the dividend. Therefore 23544-654–
30 Auswer.
64 REDUCTION.
3. avoiadurois weight.
In 89 cwt. 3 qrs. 14 lb. 12 oz. how many ounces?
4.
359 quarters Proof.
--- 16)161068
2-76 28)10066 12 oz.
719
- 4)359 14 lb.
10066 pounds
16 Ans. 89 cwt. 3 qrs. 14 lb. 12 oz.
60398
1006.7
-
161068 ounces. Answer.
2. In 19 lb. 14 oz. 11 dr. how many drams? Ans. 5099.
3. In 1 tun, how many drams? Ans. 578440.
4. In 24 tuns, 17 cwt. 3 qrs. 17 lbs. 5 oz. how mas.
ounces? Ans. 892.245.
5. Bring 5099 drams into pounds. Ans. 191b. 14oz. 11 dr.
6. Bring 573440 drams into tuns. Ans. 1.
7. Bring 892:24.5 ounces into tuns.
Ans, 24 tuns, 17 cwt. 3 grs. 17 lb. 5 oz.
8. In 12 hhds of sugar, each 11 cwt. 25 lb. how man,
pounds? Ans, 15084.
9. In 42 pigs of lead, each weighing 4 cwt. 3 qrs, how
many fother, at 19 cwt. 2 qrs.” Ans. 10 fother, 4} cut.
10. A gentleman has 20 hlids, of tobacco, each 8 cwt
3 qrs. 14 lb. and wishes to put it into boxes containing 78
lb. each, I demand the number of boxes he must get?
Ans. 284.
-
4. apothecanies' weight.
1. In 91583 13 2B. 19 grs, how many grains?
Ans. 55799
2. In 55799 grains, how many pounds?
Ans. 9th 83 132B 19.g.
REDUCTION 65
5. cloth measure.
1. In 95 yards, how many quarters and nails?
Ans. 380 qrs. 1520 na.
. In 341 yards, 3 qrs. 1 na: how many nails?
Ans. 5469.
. In 3783 nails, how many yards?
Ans. 236 yds. 1 ºr 3 na.
. In 61 Ells English, how many quarters and nails?
Ans. 305 ºrs. 1220 na.
. In 56 Ells Flemish, how many quarters and nails?
Ams. 168 ºrs. 072 na.
. In 148 Ells English, how many Ells Flemish
Ans. 246 E. F. 2 qrs.
7. In 1920 nails, how many yards, Ells Flemish, and
Ells English º Ans. 120 yds. 160 E. F. and 96 E. E.
8. How many coats can be made out of 35 yards of
wroadcloth, allowing 11 yards to a coat? Ans. 21.
-
9, day measure.
1. In 136 bushels, how many pecks, quarts and pints?
Ans. 544.pks. 4352 7ts. 8701 pts.
. In 49bush. 3pks. 54ts, how many quarts? Ans. 1597.
. In 8704 pints, how many bushels? Ans. 126.
. In 1597 quarts, how many bushels?
Ans. 49 bush. 3 pºs. 57ts.
5. A man would ship 720 bushels of corn m barrels,
which hold 3 bushels 3 pecks each, how many barrels
must he get? Ans, 192.
-
7. wine measure.
1. In 9 tuns of wine, how many hogsheads, gallons and
marts? Ans. 36 hºlds. 2268 gals. 90724ts.
2. In 24 hlids. 18 gals. 2 que. how many pints?
Ans. 12244.
3. In 9072 quarts how many tuns? Ans. 9.
4. In 1905 pints of wine, how many hogsheads?
Ans. 3 hºds. 49 gals. 1 pt.
5. In 1789 quarts of cider, how many barrels?
Ans, 14 his 25 qts.
:
-2
66 ſºul-CTION.
6. What number of bottles, containing a pint and a half-
each, can be filled with a barrel of cider? Ans. 168.
7. How many pints, quarts, and two quarts, each aſ
equal number, may be filled from a pipe of wine! Ans. 144.
-
8. Loxo measune.
1. In 51 miles, how many furlongs and poles?
Ans. 408 fur. 10320 poles.
2. In 49 yards, how many feet, inches, and barley-corns
Ans. 147 ft. 1764 inch. 5292 b. c.
3. How many inches from Boston to New-York, it being
248 miles? Ans. 1571.3280 inch.
4. In 4352 inches, how many yards?
Ans. 120 yds. 2.ft. Sin.
5. In 682 yards, how many rods?
Ans. GS2X2-11–124 rods.
6. In 15840 yards, how many miles and leagues?
Ans, 9 m. 3 lea.
7. How many times will a carriage wheel, 15 feet and
inches in circumference, turn round in going from New
York to Philadelphia; it being 96 miles?
Ans. 30261 times, and 8 feet over.
8. How many barley-corns will reach round the globe
it being 360 degrees? Ans. 4755S01000.
9. Laxd on square measure.
I. In 241 acres, 3 roods, and 25 poles, how many square
rods or perches? Ans. 38705 perches.
2. In 20692 square poles, how many acres? -
Ans. 129 a. 1 r. 12po.
3. If a piece of land contain 24 acres, and an enclosure
of 17 acres, 3 roods, and 20 rods, be taken out of it, how
many perches are there in the remainder?
Ans. 980 perches.
4. Three fields contain, the first 7 acres, the second it
acres, the third 12 acres, 1 rood; how many shares can
they be divided into, each share to contain 76 rods
Ans, 61 shares and 44 rods oner.
Reduction.
10. solid measure.
1. In 14 tons of hewn timber, how many solid inches?
Ans. 1.4 × 50 x 1728–1209600.
2 in 19 tons of round timber, how many inches?
Ans. 1:31:32.80.
3. In 21 cords of wood, how many solid feet?
Ans. 21 x 128–2688.
4. In 12 cords of wood, how many solid feet and inches?
Ans. 1536 ft, and 2654.208 inch.
5. In 4608 solid feet of wood, how many cords?
Ans, 36 cas.
-
11. time.
1. In 41 weeks, how many days, hours, minutes, and
seconds? Ans. 287 d. 6888.h. 41:5280 min, and 24796800 sec.
2. In 214 d. 15 h. 31 m. 25 sec. how many seconds?
Ans. 18545.485 sec.
3. In 24796800 seconds, how many weeks? Ans. 41 whºs.
4. In 184009 minutes, how many days?
Ans. 1:27 d. 18 h. 40 min.
5. How many days from the birth of Christ, to Christ-
mas, 1797, allowing the year to contain 365 days, 6 hours 1
Ans, 656354 d. 6 h.
6. Suppose your age to be 16 years and 20 days, how
many seconds old are you, allowing 365 days and 6 hours
to the year? Ans. 5066-19600 sec.
7. From March 2d., to November 19th following, inclu-
sive, how many days? Ans. 262.
-
12, circular Motion.
1. In 7 signs, 15° 21' 40", how many degrees, minutes,
and seconds? Ams. 225° 13-24' and slitsu”.
2. Bring 1020300 seconds into signs.
Ans. 9 signs, 13° 25'.
-
Questions to erercise Reduction.
! In 1259 groats, how many farthings, pence, shillings,
and guineas, at 28s. 1 Ans. 20144.jps. 5036/1.4.19s. 8d.
and 14 guineas. 27s. 8d.
º RELUCTION.
2. Borrowed 10 English guineas at 28s, each, and 24
English crowns at 6s. and Sd. each; how many pistoies at
22s. each, will pay the debt? Ans. 20.
3. Four men brought each 171.10s, sterling value in gold
into the mint, how many guineas at 21s, each must they
receive in return? Ans. 66 guin. 14s.
4. A silversmith received three ingots of silver, each
weighing 27 ounces, with directions to make them into
spoons of 2 oz., cups of 5 oz., salts of 1 oz., and snuff-boxes
of 2 oz., and deliver an equal number of each; what was
the number 1 Ans. 8 of each, and 1 oz. over.
5. Admit a ship's cargo from Bordeaux to be 250 pipes,
130 hlids, and 150 quarter casks, [] hºlds.] how many gal-
lons in all; allowing every pint to be a pound, what burden
was the ship of 1 Ans. 44.415 gals. and the ship's burden
was 158 tons, 12 cwt. 2 qrs.
6. In 15 pieces of cloth, each piece 20 yards, how many
French Ells? - Ans. 200.
7. In 10 bales of cloth, each bale 12 pieces, and each
piece 25 Flemish Ells, how many yards? Ans. 2250.
8. The forward wheels of a wagon are 14 feet in cit.
crumference, and the hind wheels 15 feet and 9 inches; how
many more times will the forward wheels turn round than
the hind wheels, in running from Boston to New-York, it
being 248 miles? Ans. 7167.
9. How many times will a ship 97 feet 6 inches long,
sail her length in the distance of 12800 leagues and ten
yards? Ans. 20795.08.
10. The sun is 95,000,000 of miles from the earth, and
a cannon ball at its first discharge flies about a mile in 71
seconds; how long would a cannon ball be, at that rate in
flying from here to the sun ? Ans. 22 yr. 216 d. 12 h. 40m.
11. The sun travels through 6 signs of the zodiac in
half a year; how many degrees, minutes, and seconds?
Ans. 180 deg. 10800 min. 648000 sec.
12. How many strokes does a regular clock strike in 365
days, or a year? ans. 56940.
13. How long will it take to count a million, at the rate of
50 a minute? Ans. 33.3 h. 20 m. or 13 d. 21 h. 20 m.
-
PRACTIONS. cº
14. The national debt of England amounts to about 279
millions of pounds sterling; how long would it take to count
this debt in dollars (4s. 6d. sterling) reckoning without in-
termission twelve hours a day at the rate of 50 dols, a mi-
nute, and 365 days to the year?
Ans, 94 years, 134 days, 5 hours, 20 min.
FRACTIONS.
FRACTIONS, or broken numbers, are expressions for
any assignable part of a unit or whole number, and (in
general) are of two kinds, viz.
WULGAR AND DECIMAL.
A Vulgar Fraction, is represented by two numbers placed
one above another, with a line drawn between them, thus,
&c. signifies three fourths, five eighths, &c.
The figure above the line, is called the numerator, and
at below it, the denominator;
5 Numerator.
Thus, | -
s Denominator.
The denominator (which is the divisor in division) shows
how many parts the integer is divided into; and the nume
rator (which is the remainder after division) shows howma
ny of those parts are meant by the fraction.
A fraction is said to be in its least or lowest terms, when
it is expressed by the least numbers possible, as : when re-
duced to its lowest terms will be 1, and ºr is equal to 1, &c.
-------------
To abbreviate or reduce fractions to their lowest terms.
Rule.-Divide the terms of the given fºrtion by any number which
will divide them without a remainder, and the quotients again in the
ame manner; and so on, till it appears that there is no number
greatºr than 1, which will divide them, and the fraction will be in its
--torns.
Ex-MPLEs.
1. Reduce ºf to its lowest terms.
(3) (2)
8); # =#-º-; the Answer.
2. Reduce 4; to its lowest terms. Ans. 1
3. Reduce is to its lowest terms. Ans. A
1. Reduce ºr to its lowest terms. Ans.
70 PRACTIONS.
5. Abbreviate #4 as much as possible. Ans. H
6. Reduce ºf to its lowest terms. Ans. #
7. Reduce ### to its lowest terms. Ans. :
8. Iteduce ºn to its lowest terms. Ans. .
9. Reduce H* to its lowest terms. Ans. };
10. Reduce #4:4 to its lowest terms. Ans. .
-------------
To find the value of a fraction in the known parts of the
integer, as to coin, weight, measure, &c.
Rule-Multiply the numerator by the common parts of the integer
and divide by the denominator, &c.
Exam-PLEs.
What is the value of 3 of a pound sterling?
Numer. 2
20 shillings in a pound.
Denom. 30(13s. 4d. Ans.
3.
10
9
I
12
3,124
12
What is the value of #4 of a pound sterling
Ans. 18s. 5d. 2's qrl.
- Reduce of a shilling to its proper quantity. Ans. 4d
- What is the value .#
What is the value of +3 of a pound troy? Ans. 9oz.
. How much is ºr of a hundred weight?
of a shilling? Ans. 4d.
Ans. 3 ºrs. 71b. 10P, oz.
. What is the value of of a mile?
Ans. 6 fur. 26 po. 11 fi
. How much is ; of a cwt.” Ans. 34 rs. 3 lb. 1 oz. 12; d.
Reduce 5 of an Ell English to its proper quantity.
Ans. 2 qrs. 34 na
. How much is of a hind. of wine? Ans. 54 gºd
-]
-na-TION5. 7.
tl. What is the value of tº of a day?
Ans. 16 h. 36 min. 55, a sec.
Pro-LEM III.
To reduce any given quantity to the fraction of any
greater denomination of the same kind. -
Rule.—Reduce the given quantity to the lowest term inentioned
for a numerator; then reduce the integral part to the same term, for
denominator; which will be the fraction required.
Exa-P-Es-
1. Reduce 13s.6d. 24rs. to the fraction of a pound.
20 integral part 13 62 given sum.
12
240 162
4 1.
960 Denominator. 650 Num. Ans. *******.
2. What part of a hundred weight is 3 qºs. 14 lb. ?
3 grs. 14 lb.-98 lb. Ans, "ºº-
What part of a yard is 3 qºs. 3 na. ? Ans. H.
4. What part of a pound sterling is 13s. 4d.” Ans. 4
5. What part of a civil year is 3 weeks, 4 days?
3.
---
G. What part of a mile is 6 fur. 26 po 3 yds. 2.ft. 1
Jºur. po- yds. ft. feet.
6 26 & 2–4400 Num.
a mile =5280 Denom. Ans. #33-#
7. Reduce 7 oz-4pwt to the fraction of a pound troy.
Ans.
8. What part of an acre is 2 roods, 20 poles? Ans.
9. Reduce 54 gallons to the fraction of a hogshead ol
vine. Ans. *
| 10. What part of a hogshead is 9 gallons? Ans.
| 11. What part of a pound troy is 10oz. 10 pºt. 10 grs.
Ans. #
- DECIMAL FRACTIONS.
A Decimal *raction is that whose denominator is a unit,
with a cipher, or ciphers annexed to it, Thus, , , , , , , , º,
&c. &c.
72 -na-TION--
The integer is always divided either into 10, 100, 1000
&c. equal parts; consequently the denominator of the frac
tion will always be either 10, 100, 1000, or 10000, &c. which
being understood, need not be expressed; for the true value
of the fraction may be expressed by writing the numerator
only with a point before it on the left hand thus, ºr is writ-
ten ,5; ºr ,45; Fºr ,725, &c.
But if the numerator has not so many places as the de-
nominator has ciphers, put so many ciphers before it, viz.
at the left hand, as will make up the defect; so write rº,
thus, ,05; and rººm thus, ,006, &c.
Note. The point prefixed is called the separatrix.
Decimals are counted from the left towards the right
hand, and each figure takes its value by its distance from
the unit's place; if it be in the first place after units, (orse-
parating point) it signifies tenths; if in the second, hun
dredths, &c. decreasing in each place in a tenfold propor
tion, as in the following
NUMERATION T-a-º.
* *
+ º-
- - -
3 : * =
- - - - - -
* ºf E = 3 E
E = . #####
- * ºr . #2 = E = 5.
º, º - - = E = E = -
* = E = nº E-TF = 3 -
= E = 5 & 5 : E = 3
- - - +E → -- -
=F = E = #3 # E = − ==
- : , ; F = ~...~ - - - - * -
7 6 5 4 3 2 1 2 3 4 5 6 7
Whole numbers. Decimals.
Ciphers placed at the right hand of a decimal fraction
do not alter its value, since every significant figure conti-
nues to possess the same place: so ,5,50 and 500 are all
the same value, and equal to ſº, or .
But ciphers placed at the left hand of decimals, decrease
their value in a tenfold proportion, by removing them fur-
ther from the decimal point. Thus, 5 ,05 ,005, &c. are
five tenth parts, five hundredth parts, five thousandth parts.
&c. respectively. It is therefore evident that the magnitude
DE-1-1AL FRACTIONs. - 73
of a decimal fraction, compared with another, does not de
end upon the number of its figures, but upon the value of
its first left hand figure: for instance, a fraction beginnin
with any figure less than 9 such as 899.229, &c. if .#
ed to an infinite number of figures, will not equal 9.
ADDITION OF DECIMALS.
Rule-1, Place the numbers, whether mixed or pure decimals, un-
wer each other, according to the value of their places.
2. Find their sum as in whole numbers, and point off so many places
or the decimals, as are equal to the greatest number of decimal parts
in any of the given numbers. -
----------
1. Find the sum of 41,653+36,05+24,009+1,6
41,653
36,05
Thus, 24,000
1,5
Sum, 103,312, which is 103 integers, and ſº, parts of
unit. Or, it is 103 units, and 3 tenth parts, 1 hundredth
part, and 2 thousandth parts of a unit, or 1.
Hence we may observe, that decimals, and FEDERAL
Moxey, are subject to one and the same law of notation,
and consequently of operation.
For since º is the money unit; and a dime being
the tenth, a cent the hundredth, and a mill the thousandth
º of a dollar, or unit, it is evident that any number of
ollars, dimes, cents and mills, is simply the expression of
jollars, and decimal parts of a dollar. Thus, 11 dollars, 6
dimes, 5 cents, e1165 or 11 ºr dol. &c.
2. Add the following mixed numbers together.
(2) (3) (4)
Yards. Ounces. Dollars.
46,23456 12,3456 48,9108
24,90400 7,891 1,819.1
17,00411 2,34 3,1030
3,01111 5,6 ,7012
- -
74 LE-MAIL FRA-T-I-ONº.
5. Add the following sums of Dollars together, viz.
$12,34565+7,891+2,34+14,--,00.11
Ans. $36,57775, or $36, 5di. 7cts. 7 ºr mills.
6. Add the following parts of an acre together, viz.
,7569+,25+,654+,199. Ans. 1,8599 acres.
7. Add 72,5+32,071-1-2,1574-H 371,4+2,75.
Ans. 480,8784
8. Add 30,07+200,71+59,4+3207,1. Ans. 3107,28
9. Add 71,467—H27,94+16,084+98,000+86,5. Ans.300
10. Add,75094-,00.74+,69+,840s-H,6109. Ans, 2,9
11. Add,5+,099-H,37+,905+,026. Ans. 2
12. To 9,999999 add one millionth part of a unit, and
the sum will be 10.
13. Find the sum of
Twenty-five hundredths, - - - - - -
Three hundred and sixty-five thousandths,
Six tenths, and nine millionths, - - - -
Ans. 1,215009
-
SUBTRACTION OF DECIMALS.
Rule.—Place the numbers according to their value; then subtrº-
s in whole numbers, and point off the decimals as in Addition.
Exa-ºl-Es.
- Dollars. - Inches
1. From 125,64 2. From 14,674
Take - 95,587.56 Take 5,91
5. From 761,8109 719,100.09 27,15
Take 18,9113 7,121 1,51679
d. From 480 take 245,0075 Ans. 234,9925
7. From 236 dols, take, 540 dols. Ans. $235,451
8. From 145 take ,09684 Ans. ,04816
9. From .2754 take 237. Ans. ,0383
10. From 271 take 215.7 Ans. 55,3
11. From 270.2 take 75 'º Ans. 194,7925
*. From tº tase ODU’ Anº 106,0993
u-t-Ma-Faa-TIONs. 7-
13. From a unit, or 1, subtract the millionth part of it-
welf. Ans, 999999
-
MULTIPLICATION OF DECIMALS.
Rule-1, whether they be mixed numbers, or pure decimals, place
the factors and multiply them as in whole numbers.
2. Point of so many figures from the product as there are decimal
places in both the factors; and if there be not so many places in the
product, supply the defect by prefixing ciphers to the left hand.
Ex-MPLEs.
I. Multiply 5,236 2. Multiply 3,024
by 00s by 2,23
Product, ,041SSS 6,74352
3. Multiply 25,238 by 12,17. Answers. 307,14646
4. Multiply 2461 by ,0529. 130, 1869
5. Multiply 7853 by 3,5. 274S5.5
6. Multiply ,007853 by ,035. ,00027-1855
7. Multiply 004 by .004. ,000016
8. What cost 6, 21 yards of cloth, at 2 dols. 32 cents, 5
mills, per yard? Ans. $14,4d. 3c. Sºm.
9. Multiply 7,02 dollars by 5,27 dollars.
Ams. 36,995.4 dols, or $36 99 cts, 5*m.
10. Multiply 41 dols. 25cts, by 120 dollars. Ans. $4950
11. Multiply 3 dols. 45 cts. by 16 cts.
Ans. $0,5520–55 cts, 2 mills.
12. Multiply 65 cents, by ,09 or 9 cents.
Ans. 80,0585=5 cts, 84 mills.
13. Multiply 10 dols, by 10 cts. Ans. $1
14. Multiply 341,45-dols, by ,007 or 7 mills. Ans. $2,39
To multiply by 10, 100, 1000, &c. remove the separating
pºint so many places to the right hand, as the multiplier
has ciphers.
(Multiplied by 10, makes 4,25
So ,425 || – by 100, makes 42,5
l by 1000, is .425
For ,425 x 10 is 4,250, &c.
DIVISION OF DECIMALS.
Rutº-1. The places of the decimal parts of the divisor and quo-
uent counted together, must always be equal to those in the dividend.
70 ------L--R-T-Ns.
therefore divide as in whole numbers, and from the right hand of the
quotient point off so many places for decimals, as the decimal places
in the dividend exceed those in the divisor.
2. If the places in the quotient be not so many as the rule requires,
supply the defect by prefixing ciphers to the left hand of said quotient.
Nore.-If the decimal places in the divisor be more than
those in the dividend, annex as many ciphers to the divi-
dend as you please, so as to make it equal, (at least,) to the
divisor. Or, if there be a remainder, you may annex ciphers
to it, and carry on the quotient to any degree of exactness
Ex-MPLEs.
9,51)77,4114(8,14 3,8), 21318(,0561
76,08 190
1,331 231
951 228
3S04 38
3S04 38
00 00
3. Divide 780,517 by 24,3. Answers. 32.1%
4. Divide 4,18 by , 1812. 23068+
5. Divide 7,25496 by 957. ,00758
6. Divide,0007S759 by ,525. ,00150+
7. Divide 14 by 365. ,038356.--
8. Divide $246,1476 by $604,25. ,407:364-
9. Divide $1-5513,239 by $304,81. 611,94.
10. Divide $1,28 by $8,31 ,154+
11. Divide 56 cts, by 1 dol. 12 cts. 5
12. Divide 1 dollar by 12 cents. 8,333+
13. If 21; or 21,75 yards of cloth cost 34,317 dollar.
what will one yard cost?
Nore...—When decimals, or whole numbers, are to be d
vided by 10, 100, 1000, &c. (viz. unity with ciphers,) it i.
$1,577+
performed by removing the separatrix in the dividend, sº
many places towards the left hand as there are ciphers in
the divisor
nec-1-AL -n. A tº T-in-5 7-
ExA MI-LEs.
10, the quotient, is 57,2
572 divided by |: - - - - 5,72
1000, - - - - ,572
-
REDUCTION OF DECIMALS.
CASE 1.
To duce a Vulgar Fraction to its equivalent Decimal.
Rule--Annex ciphers to the numerator, and divide by the deno-
minator, and the quotient will be the decimal required.
Nota—So many ciphers as you annex to the given nu-
merator, so many places must be pointed in the quotient:
and if there be not so many places of figures in the quotient
make up the deficiency by placing ciphers to the left hand
of the said quotient.
ExAMPLEs.
1. Reduce to a decimal. S)1,000
- Arts., 1:25
2. What decimal is equal to #1 Answers. ,5
3. What decimal is equal to #1 - - - - - ,75
4. Reduce : to a decimal. - - - - - - -2
5. Reduce H to a decimal. - - - - - ,6875
6. Reduce #1 to a decimal. - - - - - - ,85
7. Bring sº, to a decimal. - - - - - ,09375
8. What decimal is equal to ºr 1 - - - ,037037-
9. Reduce to a decimal. - - - - ,333333+
10. Reduce fºr to its equivalent decimal. - - ,008
11. Reduce ºr to a decimal. - - - , 1923076+
CASE
To reduce quantities of several denominations to a Decimal.
Rºut.-1. Bring the given denominations first to a vulgar fraction by
Fº 111 page 71; and reduce said vulgar fraction to its equivalent
--------- -
º: Place the several denominations above each other letting the
highest denomination stand at the bºttom; ther divide each enomina-
ºn (beginning at the top) by its º the nextd enominatº
last quotient will give the decimal reºu
- -
76 DE-1-L -aa-t-ºn-
Exam-PLEs.
1. Iteduce 12 s. 6d. 3 qºs. to the decimal of a pound
12
150
4.
900)603,000000(,028125. Answer
5760
2700 By Rule 2.
1920 4 3,
7sº 12, 6,75
76St.
20, 12,5625
1200 -
950 ,028125
G
2400
1920
4800
4800
2. Reduce 15s. 9d. 34rs to the decimal of a pound.
Ans. ,790025
3. Reduce 9d. 3 qrs, to the decimal of a shilling.
Ans. ,81:25
4. Reduce 3 farthings to the decimal of a shilling.
Ans. ,0625
5. Reduce 3s. 4d. New-England currency, to the deci
ºnal of a dollar. Ans. 555555-4-
5. Reduce 12s, to the decimal of a pound. Ans. .6
Note.—when the shillings are even, half the number
with a point prefixed, is their decimal expression; but if the
number be odd, annex a cipher to the shillings, and then
by halving them, you will have their decimal expression.
7. Reduce 1, 2, 4, 9, 16 and 19 shillings to decimals.
Shillings 1 2 - 4 º 16 19
Answers. .05 .. .2 .45 s .95
0.
10.
11.
12.
|->.
11.
15.
Its.
17.
DE-1.-1.-1. --R-T-Ns. 73
... What is the decimal expression of 41.19s. 6] d. 1
Ans. E4,977.08+
Bring 341. 16s. 73d. into a decimal expression.
Ans. C34,8322916+
Reduce 25l. 19s. 5; d. to a decimal.
Ans. E25,972916+
Reduce 3 ars, 2na to the decimal of a yard. Ans. S75
Reduce 1 gallon to the decimal of a hogshead.
Ans. ,015873+
Reduce 7 oz. 19pwt. to the decimal of a lb. troy.
Ans. ,6625
Reducesqrs. 21 lb. avoirdupois, to the decimal of acwt.
Ans. 9.375
Reduce 2 roods, 16 perches, to the decimal of an acre.
Ans. 6
Reduce 2 feet 6inches to the decimal of a yard.
Ans. S33333+
Reduce 5 fur. 16po. to the decimal of a mile. Ans.,675
18. Reduce 4 calendar months to the decimal of a year.
Ans. .375
-
CASE III.
to find the ralue of a Decimal in the known parts of the In-
teger.
Rule.-1. Multiply the decimal by the number of parts in the next
cºs denomination, and cut of so many places for a remainder, to the
right hand, as there are places in the given decimal.
2. Multiply the remainder by the next inferior denomination, and
ºut of a remainder as before; and so on through all the parts of the
integer, and the several denominations standing on the left hand
make the answer.
----------
What is the value of 5724 of a pound sterling?
£,5724
20
11,4480
12
5,3760 [Carried up.,
60 decimal, Pnactions.
5,3760
4.
1,5040 Ans. 11s. 5d. 1,5 qrs
. What is the value of .75 of a pound? Ans. 15s
... What is the value of ,85251 of a pound?
Ans. 17s. 0d. 2,4 qrs.
... What is the value of,040625 of a pound? Ans. 9d.
. Find the value of 8125 of a shilling. Ans. 9ºd.
... What is the value of,617 of a cwt. 1
Ans. 24rs. 13 lb. 1 oz. 10,6 dr.
7. Find the value of ,76442 of a pound troy.
Ans. 9 oz. 3 pict. 11g
8. What is the value of,875 of a yd. Ans, 3 grs. 2nd
9. What is the value of ,875 of a hind. of wine?
Ans. 55 gals. 04t. 1 p.
10. Find the proper quantity of,089 of a mile.
Ans. 28po. 2 yds. Ift. 11,04 in.
11. Find the proper quantity of,9075 of an acre.
Ans. 3r. 25,2po.
12. What is the value of ,569 of a year of 365 days?
Ans. 207 d. 16 A 26 m. 2-1 sec.
13. What is the proper quantity of .002084 of a poundtra,
Ans. 12,00384 gr.
14. What is the value of,046.875 of a pound avoirdupois?
Ans. 12 dr.
15. What is the value of,712 of a furlong?
Ans. 28po. 2 yds. 1.ft. 11,04 in.
16. What is the proper quantity of 14:2465 of a year?
Ans. 51,909725 days.
:
;
-
CONTRACTIONS IN DECIMALS.
------------
A CONCISE and easy method to find the decimal of any
number of shillings, pence and farthings, (to three places)
by Inspection.
Rule-1, write half the greatest even number of shillings for the
first decimal figure.
* Let the ºrthings in the given pence and farthings possess the
second and third places; observing to increase the second place -
u-Mal. FRACTIONs. º
place of hundredths, by 5, if the shillings be odd; and the third place
iy 1 when the farthings exceed 12, and by 2 when they exceed 36.
ExAMPLEs.
1. Find the decimal of 7s. 9; d. by inspection.
3 = 6s.
5 for the odd shillings.
39–the farthings in 9;d.
2 for the excess of 36.
£. .391=decimal required.
2. Find the decimal expression of 16s. 4d. and 17s. 8d.
Ans. E. ,819, and £,885
3. Write down £47 1810, in a decimal expression.
Ans. E47,943
1. Reduce £18s. 2d. to an equivalent decimal.
Ans. El 40
-
------------
ºf ºrt and easy method to find the value of any decimal of
a pound by inspection.
Rut---Double the first figure, or place of tenths, for shillings, and
ºf the second figure bes, or more than 5, reckon another shilling; then,
after this 5 is deducted, enll the figures in the second and third places
so many farthings, abating 1 when they are above 12, and 2 when
above 36, and the result will be the answer.
Note:- When the decimal has but 2 figures, if anything
remains after the shillings are taken out, a cipher must be
annexed to the left hand, or supposed to be so.
----------
1. Find the value of C. ,679 by inspection.
lºs-double of 6
I for the 5 in the second place which is to be
[deducted out of 7
Ald 7, d.-29 farthings remain to be added.
Deduct d. for the excess of 12.
ins. 13s. 7d.
2. Find the value of £.,876 by inspection. Ans. 17s.6d.
3. Find the value of £, S42 by inspection. Ans. 16s. 10d.
4. Find the value of 4 097 by inspection. Ans: 1s. 111.d.
T
82 REDUCTION OF (-URRENCIE-
REDUCTION OF CURRENCIES.
Rules for reducing the Currencies of the several United
States" into Federal Money.
CASE I.
To reduce the currencies of the different states, where a
dollaris an even number of shillings, to Federal Money.
They are
New-England, New-York, and 1
Virginia, North Carolina.
Kentucky, and
Tennessee.
Rule.-1. When the sum consists of pounds only, annex a cupne.
to the pounds, and divide by half the number of shillings in a dollar,
the quotient will be dollars.t
2. But if the sum consists of pounds, shillings, pence, &c., bring the
given sum into shillings, and reduce the pence and farthings to a de-
cimal of a shilling; annex said decimal to the shillings, with a decimal
point between, then divide the whole by the number of shillings con
tained in a dollar, and the quotient will be dollars, cents, mills, &c.
Ex-MPLES-
1. Reduce 731. New-England and Virginia currency, t
Federal money. 3)730
& rºs.
sº-2 is 33
2. Reduce 451.15s. 744. New-England currency, to fed
20
[ral money.
- d.
A dollar–6)915,625 12)7,500
$152,604+ Ans. ,525 decimal.
* Formerly the pound was of the same sterling value in all the colonies
as in Great-Britain, and a Spanish Dollar worth-4s. 6d.-but the legisla
tures of the different colonies emitted bills of credit, which aſte -de-
preciated in their value, in some states more, in others less, &c.
Thus a dollar is reckoned in
fººd l Žº cº
irginia *ennsultania - arolina
kº, and y - fº. * ! is. 6d.” & 4s. su.
Tennessee ºfºrºd, Georgia,
-Meu- ºrk, and s
orth carºlina, "
+ Adding a cipher to the pounds, multiplies the whole by 10, bringing
them into tenths of a pound; then because a dollaris just three tenths of
* N. E. currency, dividing those tenths by s, brings them into dolla-
º sº. Note, puge 78.
reduction or tº unnencies 83
2 – = ,50 divide by 12, you will have the
3 – = .75) decimal required.
3. Reduce 3451. 10s. 11+d. New-Hampshire, &c. curren-
cy, to Spanish milled dollars, or federal money.
£345 10 114
Nore. I farthing is 25 | which annex to the pence, and
-
20 d.
– 12)11,2500
6)6910,9375 -
- ,9375 decimal.
$1151,8229-1- Ans.
4. Reduce 1051. 14s 3d. New-York and North-Caroli-
na currency, to federal money.
£105 14 3: al
20 12)3,7500
A dollar–S)2114,3125 ,3125 decimal.
$264,289 06 Ans.
Or 3 dom. tº
5. Reduce 4311. New-York currency to federal money.
this being pounds only,”– 4)4310
- º
-
Ans. $1077–1077,50
6. Reduce 281. 11s. 6d. New-England and Virginia cur-
tency, to federal money. Ans. 895, 25 cts.
7. Change 4631. 10s. 8d. New-England, &c. currency,
to federal money. Ans. $1545, 11cts. Im.--
8. Reduce 35l. 19s. Virginia, &c. currency, to federal
money. Ans. $119, 83 ats. 3m.--
9. Reduce 2141 10s. 74d. New-York, &c. currency, to
federal money. Ans. $536, 32 cts. 8 m.--
10. Reduce 304. 11s. 5d. North-Carolina, &c. currency,
to federal money. Ans. 8761 42 cts. 7 m.--
11. Change 2191 11s. 744. New-England and Virginia
currency, to federal money. Ans. $731. 94 cts.--
* A dollar is 8s. in this currency-,4=4-10 of a pound; therefore, multi-
ply Hy 10, and divide by 4, brings the pounds into dollars, &c.
B1 ar. DUCTION UF - Unliu-NC-Es.
12. Change 2411. New-England, &c. currency, into fe
deral money. Ans. $803, 33 cts.
13. Bring 201. 18s. 5; d. New-England currency, intº
dullars. Ans. $69, 74 cts. 64 m.-H.
14. Reduce 4681. New-York currency to federal money
Ans. $1170
15. Reduce 17s. 9ºd. New-York, &c. currency, to dol-
lars, &c. Ans. $2, 22 cts. 6,5 m.-H.
16. Borrowed 10 English crowns, at 6s. 8d. each, how
many dollars, at 6s. each, will pay the debt?
Ans. $11, 11 cts. I m.
Nore...—There are several short practical methods of re-
ducing New-England and New-York currencies to Federal
Money, for which see the Appendix. -
CASE II.
To reduce the currency of New-Jersey, Pennsylvania,
Delaware, and Maryland, to Federal Money. -
Rule.--Multiply the given sum by 8, and divide the product by 3.
and the quotient will be dollars, &c." -
Ex------->.
1. Reduce 2451. New-Jersey, &c. currency, to federu
money. -
£215x8–1960, and 1900+3=$653–$653, 33}cts.
Nºne.--When there are shillings, pence, &c. in the given
sum, reduce them to the decimal of a pound, then multin ly
and divide as above, &c.
2. Reduce 361. 11s. Sld. New-Jersey, &c. currency, tº
federal money. £36,5854 decimal value.
S
- >
3)292,6832(97,56106 Ams. Answers.
… s. d. * cºs. m.
3. Reduce 240 0 0 to federal money 640 00
4. Ireduce 125 S () º: 40
5. Reduce 99 7 64 - 265 00 5-H
*... ºceduce 100 0 0 - 266 66 6+
7. Reduce 25 3 7 - 07 14 +
8. Reduce 0 17 9 * 36 6,6
*Adollarists. 5d.-90d-in this currency-30-340-3-8 of a pound; ther-
fore multiplying by 8, and dividing by s, gives the dollars, cents, &c.
T
REDUCTION OF CURRENCIES. 85
CASE III.
To reduce the currency of South-Carolina and Georgia,
to Federal Money.
Rule.—Multiply the given sum by 30, and divide the product by
7, the quotient will be the dollars, cents, &c."
Ex-MPLE.S.
1. Reduce 100l. South-Carolina and Georgia currency,
ofederal money.
1001 x 30– 000; 3000+7=$428,5714 Ans.
2. Reduce 541. 16s 3d. Georgia currency, to federal
money. 54,8406 decimal expression.
- 30
7)1645,2180
Ans. 235,0311 ANswers.
- +. s, d. & cts. In
3. Reduce 94 14 8 to federal money, 405 99 8+
4. Reduce 19 17 6. - 85 IS 7-i-
5. Reduce 4.17 14 6 - 1790. 25
tº. Reduce 140 10 0 - 602 14 2-H
7. Reduce 160 0 0 - 685 71 4
8. Reduce 0 11 6 2 46 4+
9. Reduce 11 17 9 - 17951 4tº,
-
CASE IV.
To reduce the currency of Canada and Nova-Scotia to
Federal Money.
Rule.—Multiply the given sum by 4, the product will be dollars.
Note—Five shillings of this currency are equal to a
dollar; consequently 4 dollars make one pound.
Ex--------
1. Reduce 125l. Canada and Nova-Scotia currency, to
ederal money. 125
- - 4.
Ans. $500
- is, sº or 56d to the dollar–º =* of a pound;
herefore ×30+7.
-
86 RELuction or coln.
2. Reduce 551. 10s. 6d. Nova-Scotia currency, to dollars
55,525 decimal value.
4.
- * cts.
Ans. $222, 100-222 10 ------>
3. Reduce 2411. 18s. 9d. to federal money, sø67 75
4. Reduce 58 13 6, - 234 70
5. Reduce 528 17 8 2115 53
6. Reduce 1 2 6 - - 4 50
7. Reduce 224 19 0 - S99 So
8. Reduce 0 13 11: - 2 79
- -
REDUCTION OF COIN.
Rules for reducing the Federal Money to the currencies
of the several United States.
To reduce Federal Money to the currency of
}: Rule.—Multiply the given sum by 3, and the
1.
Pirginia, - -
product will be pounds, and decimals of a
Kentucky, and pound.
Tennessee.
Neur-York, and - º
- }º: º will be pounds, and decimals of 1
}: t Rule.—Multiply the given sum by 3, and dº-
3.
2
Pennsylvania, vide the product by 8, and the quºtientwº
Delaware, and
Maryland. be pounds, and decimals of a pound.
- - Rule.-Multiply the given sum by 7 and
4. jº- divide by 3, the quotient will be the
Georgia answer in pounds, and decimals of a
gua. pound.
Examples in the foregoing Rules.
1. Reduce $152, 60 cts, to New-England currency.
º
-
£45, 780 Ans.—£45 15s. 7,2d.
20 But the value of any decimal of
15, 600 tion. See Problem II. page 81.
12
7, 200
! Rule.—Multiply the given sum by 4, and the
- a pound, may be found by inspec-
REDUCTION OF COLN. 87
2. In $196, how many pounds, N. England currency?
--
£58,8 Ans.--C5S 16
3. Reduce #629 into New-York, &c. currency.
,4
£251,6 Ans.—C251 12
4. Bring silo, 51cts. 1 m. into New-Jersey, &c. currency.
sl 10,511
3 Double 4 makes 8s. Then 30 farthings
8)331,534 are 9d. 34rs. See Problem II. page 81.
£11,441 Ans.—£21 5s. 9|al. by Inspection.
5. Bring 565, 36 cts into South-Carolina, &c. currency.
3),45,752
£15,250=E15 5s. Ans. ---------
* cºs. E. s. d.
6. Reduce 425,07 to N. E. &c. currency. 127 10 5 +
7. Reduce 36,11 to N. Y. &c. currency. 14 8 10}-H
8. Reduce 315,44 to N. J. &c. currency, 118 5 95-H
9. Reduce 600,45 to S. C. &c. currency. 161 2 1,2
-
To reduce Federal Money to Canada and Nºora-Scotia currency.
Rule-Divide the dollars, &c. by 4, the quotient will be pounds,
and decimals of a pound.
- -----------
1. Reduce º into Canada and Nova-Scotia currency.
rts.
4)741,00
£is3.25–E185 5s.
2 Bring $311, 75cts, into Nova-Scotia currency.
* cºs.
4)311,750
£77,9375–677, 18s. 9d.
3. Bring $2007, 56 cts, into Nova-Scotia currency.
Ans. E726 17s. 94a.
Reduce sºl14, 50cts, into Canada currency.
Ams. E528 12s. 6d.
88 a ULes Poa ar, LLCIN-C-Un-R-NC-Es.
RULES for º's the Currencies ºf the several United States, al-
Canada, Nova Scotia, and Sterling, to the par of all the others.
it P. See the given currency in the left hand column, and then cast yout
eye to the right-hand, till you come under the required currency, and yo-
will have the rule.
wº-ºº-º-º-º-º-º-ey.
land, ºr- Pennsylva--wººn-York. Sout-tº-1 canada.
ºua, ºn--1-1-and-wºrth-rºund, and --- Sterlin-
tucº-a-dl war-a-ul-ul- lººr- -Yºu-cotia
Tennes-l-Maruland.
--ºn- Adº one. Add ºn-Multiply the Multiply the D-ºn-on-
---, --- fºurth to the third to the º º wºn sumºurth from
-inia, ºn -wºn-un aven sun, by 7, and dº º -------- -
turku, and º º -- º ---
Tºur- duct by 9 uu-by-º.
wººl neaua ºne Add one Multiply the 10-ton-Muluuy the
Pennsyleº-iſºn from ºff-th ---n - nºn- º ---
- 10-1l-- - - ºne ºnly ºundue ºvenly 3, and in
ware. --un- --- divide ºn-aum. -------
Maryland. * by duet by-
--
Dºlution- Dºduct one Multiply the Multiply the Multiply--
ºrk, ſºuth ºth. -- ----------- -------- ---
and Mºrtº-New-ºn the N yº, and di- º and by tº and di
Carolina. York-- York. --- º ------- º
duet by 1-uu-ºy duet by Jº.
|
Multiply the Multiply the Multiply the |Multiply the From the
sºuth-Ca-Lºivº -um º --- º --- -- ----------- --
rºlina, and by 9, and di-y 45, and by Iº, and º 15, and d-duet on-
--- wide the pro-divide the divide tº divide the twenty-
duct by 7- ºduct byprºduct by 7. - by-hui.
- 1.
- - - º: Multiply the Leduction- Deduction-
Canada, fiſh to the half to ºven sum ºn-nth ºnth from
und Canada.--Canada -- ººm th--- the rive-
Nºn-cºtia --- * º: ----------- --- -
--
Tº the E-Multiply the Mulºy the Tº the Eurº Ada one
--- -lish -un º º º º non-yninth to the
- --- - --
- *. - --- º th- º ºnly º ------
product by-prºductººl-ºuth.
- --
n-En-T-low or co-N. 99
Application of the Rules contained in the foregoing Table.
Ex-M-Lºs.
1. Reduce 461. 10s. 6d. of the currency of New-Hamp-
shire, into that of New-Jersey, Pennsylvania, &c.
... s. d.
See the rule 4)46 10 6
in the table. +11 12 71
Ans. E. 58 3 11
2. Reduce 25l. 13s. 9d. Connecticut currency, to New-
York currency. £. s. d.
3)25 13.9
By the table, -H, &c. --8 11 3
Ans. Cº. 5 ()
3. Reduce 1251. 10s. 4d. New-York, &c. currency, to
south-Carolina currency. £. s. d.
Rule by the table, 125 10-4
×7,--by 12, &c. 7
|2)878 12-4
Ans. E73 4-4
4. Reduce 461. 11s. 8d. New-York and N. Carolina cur-
tency to sterling or English money. £ s. d.
4t; 11 S.
9
See the table. , 16–4x4)419 50
× given sum º 4)104 16 3
9,--by 16, &c. -
Ans. E26 4 0.
To reduce any of the different currencies of the several
States into each other, at par; you may consult the prece-
ding table, which will give you the rules.
Mortº. Ex-ML-L--> -------------
5. Reduce 841. 10s. 8d. New-Hampshire, &c. currency,
into New-Jersey currency. Ans. E105 13s. 4d.
6. Reduce 120l. 8s. 3d. Connecticut currency, into New-
York currency. Ans. Eigo 11s. 0d.
u-2
90 RULE OF THI---L--RECT-
7. Reduce 120l. 10s. Massachusetts currency, into South
Carolina and Georgia currency. Ans. C93. 14s. 5d.
8. Reduce 4101. 18s. 11d. Rhode-Island currency, int.
Canada and Nova-Scotia currency. Ans. 4:34:29s. 1d.
9. Reduce 5241.8s. 4d. Virginia, &c. currency, into ster
ling money. Ans. C393 tºs. 3d.
10. Reduce 2141. 9s. 2d. New-Jersey, &c. currency, intº
N. Hamp. Massachusetts, &c. currency. Ans. 1711.11s. 4d.
11. Red+ce 100l. New-Jersey, &c. currency, into New
York and North-Carolina currency. Ans. 10.6l. 13s. 4d.
12. Reduce 100l. Delaware and Maryland currency intº
sterling money. Ans. 601.
13. Reduce 1161.10s. New-York currency, into Connec.
ticut currencv. Ans. 871. 7s. tºº.
14. Reduce 1121.7s. 3d. S. Carolina and Georgia curren-
cy, into Connecticut, &c. currency. Ans. 1441. 9s. 3; d.
15. Reduce 100l. Canada and Nova-Scotiacurrency, int.
Connecticut currency. Ans. 120l.
16. Reduce 1161. 14s. 0d. sterling money, into Connec
ticut currency. Ans. 1551. 13s.
17. Reduce 1041. 10s. Canada and Nowa-Scotia curren
cy, into New-York currency. Ans. 1671. 4s.
18. Reduce 100l. Nowa-Scotia currency, into New-Jer
sey, &c. currency. - Ans. 1501.
RULE OF THREE Dinº CT.
THE Rule of Three Direct teaches, by having three
numbers given to find a fourth, which shall have the same
proportion to the third, as the second has to the first.
1. Observe that two of the given numbers in your ques-
tion are always of the same name or kind; one of which
must be the first number in stating, and the other the third
number; consequently the first and third numbers must al
ways be of the same name, or kind; and the other number,
which is of the same kind with the answer, or thing sought,
will always possess the second or middle place.
2. The third term is a demand; and may be known by
these or the like words before it, viz. What will? What cost!
How many? How far? How long? or, How much? &c.
RULE or Tunes D1- CT. al-
Rule.-1. State the question; that is, place the numbers so that
the first and third terms may be of the same kind; and the second
term of the same kind with the answer, or thing sought.
2. Bring the first and third terms to the same denomination, and
educe the second term to the lowest name mentioned in it.
3. Multiply the second and third terms together, and divide their
product by the first term; and the quotient will be the answer to the
question, in the same denomination you left the second term in, which
may be brought into any other denomination required.
The method of proof is by inverting the question.
[NOTE.-The following methods of operation, when they can be used,
perform the work in a much shorter manner than the Aeneral rule.
1. Divide the second term by the first; multiply the quotient into the third,
and the product will be the answer. Ör,
2. Divide the third term by the first; multiply the quotientinto the second,
and the product will be the answer. Or,
3. Divide the first term by the second, and the third by that quotient, and
the last quotient will be the answer: , or,
4. Divide the first term by the third, and the second by that quotient, and
the last quotient will be the answer.]
Ex-A-Lºs.
1. If 6 yards of cloth cost 9 dollars, what will 20 yards
cost at the same rate? Yds. 8 ds.
Here 20 yards, which moves the 6 - 9 : : 20
question, is the third term; 6 yds. 9.
the same kind, is the first, and 9 -
dollars the second. 6)180
Ans. $30
2. If 20 yards cost 30 dols. 3. If 9 dollars will buy 6
what cost 6 yards? yards, how many yards will
Yds. & Yºs. 30dols, buy? § yds. *
20 - 30 : : 6 9 : tº : : :º)
_6 t;
2,0)18,0 9)180
Ans. $9 Ans. 20/4.
4. If 3 cwt. of sugar cost ºl. 8s. what will 11 cwt. I tr.
24 lb, cost?
3 cwt, Sl. 8s. C ºr lb. 10. -
112 20 11 : 24. As ºf : 168 - 128-110.
º, T- 4. tº-
º6 lb. 168s. - -
45 [Carried up.) trºº
97. at LE OF THRE. Din EcºT.
45 10272
28 77.04
- 1284
j4. –(2,0)
92 336): 15712(64.2
2010 –
1284 –521.2s
1411 Ans
1344
672
672
5. If one pair of stockings cost 4s. 6d. what will 19 do
zen pair cost? Ans. C51 6s.
6. If 19 dozen pair of shoes cost 511. bs. what will cºns
pair cost? Ans. 4s. 6d.
7. At 10] d. per pound, what is the value of a firrin ºf
butter, weight 56 pounds? Ans. E29s.
8. How much sugar can you buy for 23, 2s. at 9d pe.
pound? Ans. 5 C. 2 ºrs.
9. Bought Schests of sugar, each 9 cwt. 2 qrs, what du
they come to at 21, 5s. per cwt. Ans. C171.
10. If a man's wages be 75l. 10s, a year, what is that a
calendar month? - - Ans. EG 5s. 10d.
11. If 4A tuns of hay will keep 3 cattle over the winter;
how many tuns will it take to keep 25 cattle the same time
Ans. 374 tuns,
12. If a man's yearly income be 20s! is: what is that a
day? - Ans. Its. 4d. 3 ºr qrs.
13. If a man spend 3s. 4d. per day, how tºuch is that a
year? Ans. E60 16s. 8d.
14. Boarding at 12s. 6d. per week, how long will 321
10s, last me? Ans. 1 year.
15. A owes B 3475l. at B compounds with him for 13s
4d. on the pound; pray what must he receive for his debt?
Ans. E2316 13s. 4d.
16. A goldsmith sold a tankard fºr Sl. 12s. at 5s. 4d. per
oz.what wastneweight of the tankard! Ans. 21. Soz. 5punt.
17. If 2 cwt. 3 ºrs. 21 lb. of sugar cost tº 1s. Sd. what
cost 35 cwt. 7 Ans. L72.
º
-
RULE OF THREE, DLRE-T- 93
18. Bought 10 pieces of cloth, each piece containing 9.
yards, at 11s. 4d. per yard; what did the whole come to?
Ans. E559s. 0d.
FEDERAL MONEY.
Note 1. You must state the question, as taught in the
Rules foregoing, and after reducing the first and third terms
to the same name, &c. yournay multiply and divide accord-
ing to the rules in decimals; or by the rules for multiplying
and dividing Federal Money.
Exam-PLEs.
19. If 7 yds. of cloth cost 15 dollars 47 cents, what will
12 yds cost? Yds. $ cts, yds.
7 : 15,47 : : 12
12
7)IS5,64
Ans. 26,52=$26, 52 cts.
But any sum in dollars and cents may be written down
as a whole number, and expressed in its lowest denomina-
tion, as in the following example: (See Reduction of Fede-
ral Money, page 62.)
20. What will 1 qr, 9 lb. sugar come to, at 0 dollars 45
tºs. per cwt. 1 qr, lb. lb. cts. lb.
1 9. As 112 : 615 : : 37
28 37
37 lb. 1515
1935
rt.
112)23865(213+Ans=$2,13.
224
146
112
345
3:56
-
9.
94 RULE or THREE Duke CT.
Note 2. When the first and third numbers are federal
money, you may annex ciphers, (if necessary,) until you
make their decimal places or figures at the right hand of
the separatrix, equal: which will reduce them to a like de-
nomination. Then you may multiply and divide, as in whole
numbers, and the quotient will express the answer in the
least denomination mentioned in the second, or middle term. -
----------
21. If 3 dols, will buy 7 yds. of cloth, how many yds. can
buy for 120 dols. 75 cts.” cºs. vºls. cts.
As 300 : 7 : : 12075
7
Jus. -
300)84525(281, Ans
22. If 12 lb. of tea cost 6 dols. 600
78cts, and 9 mills, what will 5 lb. -
cost at the same rate? 24.52
15. mills. Ib. *2100
As 12 : 6789 : : 5
5 525
300
12)339.45 -
$cts.m. 2:25
Ans. 2828+ mills.-2,82,8. 1.
900(3 yrs.
900
§ ets.
23. If a man lay out 121, 23 in merchandise, and thereby
gains $39.51 cts, how much will he gain by laying out $13
at the same rate? Cents. Cents. Cents.
As 12128 - 395.1 : : 1200
1200
–cts. - rts.
121:23)4741200(391=3,91 Ans.
35369
110 ºn
109107
- [Carried up.
RULE OF THREE DIRECT- 95
13230
121:23
1107
24. If the wages of 15 weeks come to $64 19 cts what is
a year's wages at that rate? Ans. $222, 52 cts. 5m.
25. A man bought sheep at $1 11 cts. per head, to the
amount of 51 dols. 6 cts. ; how many sheep did he buy?
Ans, 46.
26. Bought 4 pieces of cloth, each piece containing 31
yards, at 16s. 6d. per yard, (New-England currency;) what
does the whole amount to in federal money? Ans. $341.
27. When a tun of wine cost 140 dollars, what cost a
quart? Ans. 13 cts. Sºm.
28. A merchant agreed with his debtor, that if he would
pay him down 65 cts on a dollar, he would give him up a
note of hand of 219 dols. SS cts. I demand what the debtor
taust pay for his note? Ans. $162.42 cts. 2m.
29. If 12 horses eat upºº bush. of oatsin a week, how many
ºushels will serve 45 horses the same time? Ans. 112 bush.
30. Bought a piece of cloth for $48.27 cts, at $1 19ets, per
| 1 , how many yds. did it contain? Ans.40 yds. 24rs.”.
31. Bought 3 hlids of sugar, each weighing 8 cwt. I qr.
12 lb. at $7.25cts. per cwt. what come they to ?
Ans. $182 1 ct. 8 m.
32. What is the price of 4 pieces of cloth, the first piece
containing 21, the second 23, the third 24, and the fourth
17 yards, at 1 dollar 43 cents per yard?
Ans. $13585cts. 21+23+24+27–95 ya's.
33. Bought 3 hlids of brandy, containing 61, 62, 62
ºlons, at I dollar 38 cts. per gallon, I demand how much
they amount to? Ans. $255 99 cºs.
34. Suppose a gentleman's income is $1836 a year, and
he spends $3.49cts, a day, one day with another, how much
will he have saved at the year's end? Ans.3562, 15 cts.
35. If my horse stand me in 20 cts, per day keeping,
what will be the charge of 11 horses for the year, at that
rate? Ans. $803.
96 RULE or Tukee direct.
36. A merchant bought 14 pipes of wine, and is allowed
6 months credit, but for ready money gets it 8 cts, a gallon
cheaper; how much did he save by paying ready money?
Ans. $141, 12 cts.
Examples promiscuously placed.
37. Sold a ship for 5371 and I owned of her; what
was my part of the money? Ans. E201 7.s. 6d.
38. If nº of a ship cost 781 dollars 25 cents, what is the
whole worth 2 As 5:781,25: ; 16 : $2500 Ans.
39. If I buy 54 yards of cloth for 311 10s. what did it
cost per Ell English Ans. 14s. 7d.
40. Bought of Mr. Grocer, 11 cwt. 3 qrs. of sugar, at 8
dollars 12 cents per cwt. and gave him James Paywell's
note for 191.7s. (New-England currency) the rest I pay in
cash; tell me how many dols. will make up the balance?
Ans. $30, 91 cts.
41. If a staff 5 feet long cast a shade on level ground 8
feet, what is the height of that steeple whose shade at the
same time measures 181 feet? Ans. 1131 ft.
42. If a gentleman have an income of 300 English gui-
neas a year, how much may he spend, one day with ano.
ther, to lay up $500 at the year's end? Ans. $2,46cts. 5m.
43. Bought 50 pieces of kerseys, each 34 Ells Flemish, at
8s. 4d. per Ell English; what did the whole cost? Ans. E425.
44. Bought 200 yards of cambrick for 901, but being da-
maged, I am willing to lose 71. 10s. by the sale of it; what
must 1 demand per Ell English? Ans. 10s. 33d.
45. How many pieces of Holland, each 20 Ells Flemish,
may I have for 23.8s, at 6s.[5d. per Ell English? Ans. 6pcs.
46. A merchant bought a bale of cloth containing 240 yds
at the rate of 57* for 5 yds. and sold it again at the rate of
$114 for 7 yards; did he gain or lose by the bargain, and how
much Ans. He gained $25,71 cts, 4 m.--
47. Bought a pipe of wine for 84 dollars, and found it had
leaked out 12 gals. ; I sold the remainder at 12}cts, a pint;
what did I gain or lose? Ans. I gained $30.
48. A gentleman bought 18 pipes of wine at 12s. 6d.
(New-Jersey currency) per gallon; how many dollars will
pay the purchase ? Ans. $3780.
ºut------ T-1-1----N-Euse. 27
49. Bought a quantity of plate, weighing 15 lb. 11 oz. 13
pwt. 17 gr: how many dos. will pay for it, at the rate of 12s.
7d. New-York currency, per ox.” –ins. $301, 50, cts. 2, ºm.
50. A factor bought a certain quantity of broadcloth and
drugget, which together cost Sil, the quantity of broadcloth
was 50 yds., at 18s, per yd., and for every 5 yds. of broad
cloth he had 9 yards of drugget; I demand how many yds.
ºf drugget he had, and what it cost him per yard?
Ans. 90 yds. at 8s, per yd.
51. If I give Leagle, 2 dols. 8dimes, 2 cts and 5m. for 675
tops, how many tops will 19 mills buy? Ans. I top.
52. Whereas an eagle and a cent just threescore yards
did buy,
How many yards of that same cloth for 15 dimes had I?
Ans. 8 yds. 3 ºrs. 3 na.--
53. If the legislature of a state grant a tax of 8 mills on
the dollar, how much must that man pay who is 319 dols.
*5 cents on the list? Ans. $2, 55 cts, 8 m.
54. If 100 dols, gain 6 dols. interest in a year, how much
will 49 dols. gain in the same time? Ans. $2, 94 cts.
55. If 60 gallons of water, in one hour, fall into a cistern
containing 300 gallons, and by a pipe in the cistern 35 gal-
ons run out in an hour; in what time will it be filled 2
Ans. in 12 hours.
56. A and B depart from the same place and travel the
same road; but A goes 5 days before B, at the rate of 15
miles a day; B follows at the rate of 20 mile a day; what
distance must he travel to overtake A1 Ans. 300 miles.
THE Rule of Three Inverse, teaches by having three
numbers given to find a fourth, which shall have the same
proportion to the second, as the first has to the third.
If more requires more, or less requires less, the question
belongs to the Rule of Three Direct.
But if more requires less, or less requires more, the ques-
tion belongs to the Rule of Three Inverse; which may al-
ways be known from the nature and tenor of the question.
For example:
º RULE ºr tº R-E-Nº-Erase
If 2 men can mow a field in 4 days, how many days will
it require 4 men to mow it?
--- da ---
1. If 2 require how much time will 4 require 1
Answer, 2 days. Here more requires less, viz. the more
men the less time is required.
ºn-ºn- days ---
2. If 4 require 2 how much time will 2 require?
Answer, 4 days. Here less requires more, viz. the less the
number of men are, the more days are required—therefore
the question belongs to Inverse Proportion.
Rule.-1. State and reduce the terms as in the Rule of Three Di-
rect.
2. Multiply the first and second terms together, and divide the pro-
duct by the third; the quotient will be the answer in the same deno.
ºuination as the middle term was reduced into.
Ex--------
1. If 12 men can build a wall in 20 days, how many men
can do the same in 8 days? Ans. 30 men
2. If a man perform a journey in 5 days, when the day
is 12 hours long, in how many days will he perform it when
the day is but 10 hours long? Ans, 6 days.
3. What length of board 71 inches wide, will make a
square foot? Ans. 104 inches.
4. If five dollars will pay for the carriage of 2 cwt. 150
miles, how far may 15 cwt. be carried for the same money!
Ans. 20 miles.
5. If when wheat is 7s. 6d. the bushel, the penny loaf
will weigh 9 oz. what ought it to weigh when wheat is 6s.
per bushel? Ans. 11 oz. 5 punt.
6. If 30 bushels of grain, at 50 cts. per bushel, will pay
a debt, how many bushels at 75 cents per bushel, will pay
the same? Ans. 20 bushals.
7. If 100l. in 12 months gain Gl. interest, what principal
will gain the same in 8 months? Ans. C150.
8. If 11 men can build a house in 5 months, by working
12 hours per day—in what time will the same number of
men do it, when they work only 8 hours per day?
Ans. 74 months.
* Whº number of men must be employed to finish in 5
dº whº is men would be 20 days about? Ans. 60 men.
--- - º
nº-
10. Suppose 650 men are in a garrison, and their provi-
sions calculated to last but 2 months, how many men must
leave the garrison that the same provisions may be suſh-
tient for those who remain 5 months? Ans. 390 men.
11. A regiment of soldiers consisting of 850 men are to
be clothed, each suit to contain 34 yards of cloth, which is
1: yds. wide, and lined with shalloon : yd. wide; how nº
ny yards of shalloon will complete the lining?
Ans. 6941 yds. 2 qrs. 23 na.
-
- PRACTICE.
PRACTICE is a contraction of the Rule of Three Direc.
when the first term happens to be a unit or one, and is a
concise method of resolving most questions that occur in
trade or business where money is reckoned in pounds, shil-
ings and pence; but reckoning in federal money will ren-
der this rule almost useless: for which reason I shall no
ºnlarge so much on the subject as many other writers have
done.
Paa------
Tables of Aliquot, or Even Parts.
Parts of a shilling. Parts of a pound. Parts of a cwt.
d. -- s, d. £ lb. twº.
6 is 1 10 0 is . 56 is 4.
4 = + 68 – } 28 = 1
3. . 5 0 ! 16 +
2 4. () ! 14 !
14. ...} 3 + 4 7 º’s
Parts of 2 shillings. -2 6 º
ls. is . 1 8 *
8d. = } The aliquot part of any number is
6d. . such a part of it, as being taken a cer-
44. º tain number of times, exactly makes
3d. that number. -
2d. i’s
CASE I.
When the price of one yard, pound, &c. is an even part
ºf one shilling–Fund the value of the given quantity at
is a yard, pound, &c. and divide it by that even part, and
the quotient will he the answer in shillings, &c.
-
--
:
-
-
-
()0 1-tº-1---
º
Or find the value of the given quantity at 2s. per yd. &
and divide said value by the even part which the given
price is of 2s. and the quotient will be the answer in shil-
lings, &c. which reduce to pounds.
N. B. To find the value of any quantity at 2s. you need
only double the unit figure for shillings; the other figures
will be pºunds.
!
----------
1. What will 4614 yds. of tape come to at lºd. per vd 1
d
-- -
11 d. | 1 || 461 6 value of 461 yds. at 1s. per yd
5,7 8.
º 17s. Sºd. value at 14d.
2. What cost 256 lb. of cheese at 8d. per pound?
Sd. £25 12s, value of 256 lb. at 2s. per lb.
£810s. 8d. value at 8d. per pound.
s. d
Yards, per yard. £. s. d.
486; at Id Answers. 2 0 64
862 at 2d. 7 3-8
911 at ºd. 11 7 9
749 at 4d. 12 9 8
113 at 6d. - - 2 16 tº
899 at Sd. 29 19 4
CASE II.
When the price is an even part of a pound—Find the
value of the given quantity at one pound per yard, &c. and
divide it by that even part, and the quotient will be the an
swer in pounds.
k
Ex-M-LEs.
What will 1291 yards cost at 2s. 6d. per yard?
s. d. £. s. £.
2 G | | | 129 10 value at 1 per yard.
Ans, £163s. 9d, value at 2s. 6d. per yard.
Yals. s. d. £. s. d.
123 at 100 per yard Answers. 61 10-0
687 at 5 0 - 171 tº 6
tº º
- -
- - -
- - - -
PRACT-ce. 11)
Pºds. s. d. E. s. d.
211+ at 40 per yard. 42 5 0
543 at 6 8 – 181 0 0
127 at 3 4 – 21 & 4
461 at 1 8 – 38 S 4
Note.—When the price is pounds only, the given quan-
ity multiplied thereby, will be the answer.
Example.—11 tuns of hay at 41 per tun. Thus, 11
Ans. C44
CASE III.
When the given price is any number of shillings un-
der 20.
1. When the shillings are an even number, multiply the
quantity by half the number of shillings, and double the
first figure of the product for shillings; and the rest of the
product will be pounds.
2. If the shillings be odd, multiply the quantity by the
whole number of shillings, and the product will be the an
swer in shillings, which reduce to pounds.
----------
1st–124 yds. at 8s. 2d.—132 yds. at 7s. per yd
4. 7
£49 12s. Ans. 2,092,4
£46,4 Ans.
ºds. £. s. Yals. £. s.
562 at 4s. Ans. 112 8 || 372 at 11s. Ans. 204 12
378 at 2s. 37 16 || 264 at 9s. 11- tº
913 at 14s. 639 2 250 at 16s. 200 tº
CASE IV.
When the given price is pence, or pence and furthings,
and not an even part of a shilling—Find the value of he
given quantity at is per yd. &c. which divide by the great-
est even part of a shilling contained in the given price, and
take parts of the quotient for the remainder tº free,
and the sum of these several quotients will ----
in shillings, &c. which reduce to pounds.
tº
- - - T
102 I-n-A-T-I-e.
ExAMPLEs. ſ
What will 245 lb. of raisins come to, at ºd. per lb. ?
-- - -
6d. 245 0 value of 245 lb. at 1s. per pound.
3d. |}| 122 6 value of do. at 6d. per lb.
id. | 1 || 61 3 value of do. at 3d. per lb. *
15 3; value of do. at ºd. per lb.
2,019,9 0.
Ans. E9 19 0 value of the whole at 9; d. per lb.
+.
ll. d. E. s. d. ll. d. - - -
372 at 1: Ans, 2 14 3 576 at 74 Ams. 18 0 6
325 at 2, 3 0 111 || 541 at 91 20 17-0,
827 at 4. 15 10 11 G72 at 11: 3-2 18 ()
CASE v.
When the price is shillings, pence and furthings, and nº
the aliquot part of a pound—Multiply the given quantitr
by the shillings, and take parts for the pence and farthings,
as in the foregoing cases, and add them together; the sum.
will be the answer in shillings.
-----------
1. What will 246 yds. of velvet come to, at 7s. 8d. per yd."
d
--
3d, 1 ; 1246 0 value of 246 yards at 1s. per yd.
7
Tº 0 value of do. at 7s. per yard.
G1 6 value of do. at 3d. per yard.
-
-
2,0)iºs, 3 6
Ans. ES9 36-value of do. at 7s. 8d. per yard.
ANswers.
- d. -E. s. d.
2. What cost 139 yds. at 9 10 per yd. 68 6 10
3. What cost 146 yds. at 14 9 per yd. * 107 13 6
4. What cost 120 cwt. at 11 3 per cwt.” G7 10 -
5. What cost 127 yds. at 9 84 per yd. 61 12 11,
6. What cost 491 lbs. at 3 113 per lb. ? 9 15 11.
Tane axi, Tae- 103
CASE WI.
When the price and quantity given are of several deno-
a.inations—Multiply the price by the integers in the given
quantity, and take parts for the rest from the price of an in-
teger; which, added together, will be the answer. This is
applicable to federal money.
----------
1. What cost 5 cwt. 3 qrs. 2. What cost 9 cwt. 1 qr.
14 lb. of raisins, at 21, 11s. |8 lb. of sugar, at 8 dollars,
*d. per cwt. 1 65 cts. per cwt.”
£. s. d. $ cºs.
2 qrs. |}| 2 11 8 I q1 || 8,65
5 9
12 18 4 77,85
1 qr, 1 5 10 7 lb. 2,1625
| 1.4 lb. 12 11 1 lb. ,5406
6 5. ,772
Ans. E15 3-6, Ans. $80,6303
C. ºrs. Ib. --------
7 3. 16 at $9, 58 cts, per cwt. $75, 61 cts. 3m.
5 1 0 at 21, 17s. per cwt. £14 19s. 3d.
14 3 7 at 0l. 13s. 8d. per cwt. £10-2s. 541.
12 0 7 at $6, 34 cts. per cwt. $76, 47 cts. 6 m.
0 0 24 at $11, 91 cts. per cwt. 52.55 cts. 2's m.
TARE AND TRET.
TARE and Tret are practical rules for deducting cer-
tain allowances which are made by merchants, in buying
and selling goods, &c. by weight; in which are noticed the
following particulars:
1. Gross Weight, which is the whole weight of any sort
of goods, together with the box, cask, or bag, &c. which
contains them.
2. Tare, which is an allowance made to the buyer, for
the weight of the box, cask, or bag, &c. which contains the
goods bought, and is either at so much per box, &c. or at
ºr much per cwt. or at so much in the whole gross weight.
104 TARE ANL TRET.
3. Tret, which is an allowance of 4 lb. on every 104 m.
for waste, dust, &c.
4. Cloff, which is an allowance made of 2 lb. upon every
3 cwt.
5. Suttle, is what remains after one or two allowances
have been deducted.
CASE I.
When the question is an Invoice–Add the grossweight,
into one sum and the tares into another; then subtract the
total tare from the whole gross, and the remainder will bº
the meat weight.
----------
1. What is the meat weight of 4 hogsheads of Tobaccº
marked with the gross weight as follows:
C. º lb. lb.
No. 1 – 9 0. 12 Tare 100
2–8 3. 4 – 95.
3 – 7 I 0 – Sº
4 – 6 3. 25 — S1
Whole gross 32 0. 13 359 total tare.
Tare 359 lb. = 3 3 23
Ans. 28 3 Tis meat.
2. What is the neat weight of 4 barrels of Indigo, No
and weight as follows: C. ºr lb. lb.
No. 1 – 4 10 Tare 36
2 – 3 3 02 – 29
3 – 4 0 19 – 32 cut-qr. In
4 – 4 0 0 – 35 W. Ans. 15 0 11
CASE II.
When the tare is at so much per box, cask, bag, &c.-
Multiply the tare of 1 by the number of bags, bales, &e
the product is the wholetare, which subtract from the gross,
and the remainder will be the meat weight.
ºx--------
1. In 4.hhds of sugar, each weighing 10 cwt. I qr, 15 ſh
gross; tare 75 lb. per hlid, how much neat?
Curt. ºrs. Ibs.
10 1 15 gross weight of one had.
4. [ºarried up.]
Tarur, A-1) TRET. 105
41 2 4 gross weight of the whole.
15x4-2 2 20 whole tare.
Ans. 3s 3 tº meat.
2. What is the meat weight of 7 tierces of rice, each
*eghing 4 cwt. I q. 9 lb. gross, tare pertierce 34 lb. ?
Ans. 28 C. 0 qr, 21 lb.
3. In 9 firkins of butter, each weighing 2 qrs. 12 lo: gross,
tare 11-lb. perfirkin, how much meat? Ans. 4 C. 24rs. 91b.
4. If 241 bls of figs, each 3 qºs. 19 lb. gross, tare 10 lb.
per barrel; how many pounds neat? Ans. 22413.
5. In 16 bags of pepper, each 85 lb. 4 oz. gross, tare per
bag, 3 lb. 5 oz. ; how many pounds ment Ans. 1311.
6. In 75 barrels of figs, each 2 qºs. 27 lb. gross, tare in the
whole 597 lb.; how much ment weight? Ans. 50 C. 1 ºr.
7. What is the meat weight of 15 hlids of Tobacco, each
weighing 7 cwt. 1 ºr 13 lb. tare 100lb. perhbd.”
Ans. 97 C. 04r, 11 lb.
CASE III.
When the tare is at so much per cºvt-Divide the gross
weight by the aliquot part of a cwt. for the tare, which sub-
tract from the gross, and the remainder will be meat weight.
----------
1. What is the meat weight of 44 cwt. 3 qrs. 16 lb. gross,
unre 14 lb. per cwt. 2 C. qrs. lb.
| 1.4 lb. 44 3 16 gross.
- 5, 2 12+ tare.
Ans. 39 1 34 meat.
2. What is the meat weight of 9 hlids. of Tobacco, each
weighing gross 8 cwt. 3 qrs. 14 lb. tare 16 lb. per cwt.”
Ans. 68 C. I q. 24 lb.
3. What is the neat weight of 7 bls, of potash, each weighing
2011b. gross, tare 10 lb. per cwt.” Ans. 1281 lb. 6 oz.
4. In 25 bls, of figs, each 2 cwt. I qr. gross, tare per cwt.
its b-; how much meat weight? Ans. 48 cwt. 24 lb.
5. In 83 cwt. 3 qrs, gross, tare 20 lb. per cwt. what meat
-eight? Ans. 68 cwt. 3 ºrs. 5 lb.
6. In 45 cwt. 3 qrs. 21 lb. gross, tare 8 lb. per cwt. how
such ment weight? Aus. 42 cwt. 2 qrs. 173 lb.
7. What is the value of the meat weight of Shlids, of su-
105 -A-R-AND Ta-T.
gar, at $9,54 cts. per cwt. eachweighing 10 cwt. I q. 14 lb
gross, tare 14 lb. per cwt. Ans. $592, 84 cts. 24 m.
CASE IV.
When Tret is allowed with the Tare.
1. Find the tare, which subtract from the gross, and call
the remainder suttle.
2. Divide the suttle by 26, and the quotient will be the
tret, which subtract from the suttle, and the remainder will
be the meat weight.
EXAMPLES.
1. In a hogshead of sugar, weighing 10 cwt. 1 qr. 12 lb.
gross, tare 14 lb. per cwt., tret 4 lb. per 104 lb.," how much
neat weight? Or thus,
curt. gr. lb. curt, ar. lb.
10 1 12 14=})10 1 12 gross.
_4 l l 5 tare.
41 26).9 0 7 suttle
28 1 11 tret.
3:30 Ans. S 2 24 meat.
S3
14=})1160 gross.
145 tare.
26)1015 suttle.
39 tret.
Ans. 976 lb. meat.
2. In 9 cwt. 2 qrs. 17 lb. gross, tare 41 lb., tret 4 lb. per
104 lb., how much meat 2 Ans. 8 cwt. 3 qrs. 20 lb.
3. In 15 chests of sugar, weighing 117 cwt. 21 lb. gross,
tare 173 lb., tret 4 lb. per 104, how many cwt. neat?
Ans. 111 cwt. 22 10.
4. What is the meat weight of 3 tierces of rice, eachweigh-
ing 4 cwt. 3 qrs. 14 lb gross, tare 16 lb. per cwt., and allow-
wng tret as usual? Ans. 12 cwt. 0 ºrs. 610.
5. In 25 bls, of figs, each 84 lb. gross, tare 12 lb. per cwt.
tret Alb. per 104 lb.; how many pounds meat? Ans. 1803+
* This is the tret allowed in London. Thereason of divividing by 26,
because 4 lb. is 1-26 of -04 lb. but iſ the tret is at any other rate, other put
must be taken, according to the rate proposed, &c.
tane and thet. 107
6. What is the value of the neat weight of 4 barrels of
Spanish tobacco; numbers, weights, and allowances as fol-
ows, at 9, d. per pound?
cwt. ºrs, lb.
No. ! Gross ! ; º Tare 16 lb. per cwt.
* } Tret 4 lb. per 104 lb.
3 1 0 09
4. 0 & 21 Ans. E17. 16s. 8d.
CASE W.
When Tare, Tret, and Cloff, are allowed:
Deduct the tare and tret as before, and divide the suttle
by 168 (because:2 lb. is the ºr of 3 cwt.) the quotient will
be the cloff, which subtract from the suttle, and the remain-
der will be the neat weight.
-----------
1. In 3 hogsheads of tobacco, each weighing 13 cwt. 34rs.
ºlb. gross, tare 1071b, per hlid., tret 4 lb. per 104 lb., and
aloff 2 lb. per 3 cwt., as usual; how much meat?
cut-ºrs, lb.
- 13 3 23
4.
55
28
443
112
1563 lb. gross of 1 had.
3.
4689 whole -
107-X3–321 tare. gross
26)435s suttle.
168 tret.
168)4200 suttle.
25 cloff.
Ans. 4175 meat weight.
2. What is the meat weight of 26 cwt. 3 qrs. 201b. gross,
are 52 lb., the allowance of tret and cloff as usual?
Ans, meat 25 cwt. 1 ºr 5 lb. 1 oz. nearly; omitting fur-
ther fractiºns.
100 --Tº-n-Est.
INTEREST.
INTEREST is of two kinds; Simple and Compound
SIMPLE INTEREST.
Simple Interest is the sum paid by the borrower to the
lender for the use of money lent; and is generally at a cer-
tain rate per cent. per annum, which in several of the Uni-
ted States is fixed by law at 6 percent. per annum; that is,
6l. for the use of 100l. or 6 dollars for the use of 100 dol
lars for one year, &c.
Principal, is the sum lent.
Rate, is the sum per cent. agreed on.
Amrºt is the principal and interest added together.
CASE 1.
To find the interest of any given sum for one year.
Rule.—Multiply the principal by the rate per cent. and divide the
product by 100; the quotient will be the answer.
Ex-ML-L---
1. What is the interest of 391. 11s. 84d. for one year *
61 percent. per annum ? -
£. s. d.
39 11.8;
6
237 103
20
750
12
gos
4.
012 Ans. £2.7s.6d.rºr.
2. What is the interest of 236. 10s. 4d. for a year, at 5
per cent? Ans. # 11 16s. 8d.
º-PLE INTER-E-T. lon
3. What is the interest of 5711. 13s. 9d. for one year, at
Gl, per cent.” Ans. E346s, 0, d.
4. What is the interest of 21. 12s. 9ºd. for a year, at 6l.
percent, 7 Ans. EO 3s. 2d.
-
FEDERAL MONEY.
5. What is the interest of 468 dols. 45 cts. for one year,
at 6 per cent.” 3 cts.
468, 45
G
Ans. 2810, 70–$28, 10cts. 7 m.
Here I cut of the two right hand integers, which divide
by 100: but to divide federal money by 100, you need only
ball the dollars so many cents, and the inferior denomina-
ions decimals of a cent, and it is done.
Therefore you may multiply the principal by the rate,
and place the separatrix in the product, as in multiplication
affederal money, and all the figures at the left of the sepa-
ºatrix, will be the interest in cents, and the first figure on
'e right will be mills, and the others decimals of a mill, as
ºn the following
----------
6. Required the interest of 135 dols, 25cts, for a year at
5 per cent? $ cts.
135, 25
d
Ans. 811, 50–38, 11 cts. 5 m.
7. What is the interest of 19 dols. 51 cts, for one year, at
5 per cent.” 3 cts.
19, 51
5.
Ans. 97, 55–97 cts. 54m.
8. What is the interest of 436 dols, for one year, at 6per
cent.” º
Ans, 2616 cts.-826, 16 cts.
-
10 s-M1-1-1. IN 1-ºn-Est.
ANOTHER METHOD,
Write down the given principal in cents, which multiply
by the rate, and divide by 100 as before, and you will have
the interest for a year, in cents, and decimals of a cent, as
follows:
9. What is the interest of $73, 65 cents for a year, at 6
per cent. 1
Principal 7365 cents.
6
Ans. 441,90=441*, cts, or $4,41 cts, 9m.
10. Required the interest of $85,45 cts. for a year, at "
per cent.”
Cents.
Principal 8545
7
Ans. 598, 15 cents, 35,98cts. 13rm.
CASE II
To find the simple interest of any sum of money, for ary
number of years, and parts of a year.
GENERAL Rule-1st. Find the interest of the given sum for one
year.
2d. Multiply the interest of one yearby the given number of years,
and the product will be the answer for that time.
3d. If there be parts of a year, as months and days, work for the *
months by the aliquot parts of a year, and for the days by the Rule of
Three Direct, or by allowing 30 days to the month, and taking aliquot
parts of the same.*
* By allowing the month to be 30 days, and taking aliquot parts thereoſ,
you will have the interest of any ordinary sum sufficiently exact for common
use; but if the sum bevery large, yºumaysay,
As 365 days : is to the interest of one year ; ; so is the given number ol
days: to the interest required
six-tº-Lº. INTER-Es- 111
Ex-M1-Lºs.
1. What is the interest of 75l. Ss. 4d. for 5 years and 2
months, at 6l. per cent. per annum ?
£. s. d.
75 S. 4 £. s. d.
6 || 2 mo.-4)4 10 6 Interest for 1 year.
->
452 10 0
20 22 12 6 do. 5 years.
- 0 15 I do. for two months.
10|50 -
12 E23 7 7 Mns.
at)0
2. What is the interest of 64 dollars 58 cents for 3 years,
months, and 10 days, at 5 per cent.”
$64,58
5
322,90 nterest for 1 year in cents, per
3. [Case I.
968,70 do. for 3 years.
4 mo. 3 || 107,63 de for 4 months.
-
1 mo. 26,90 do. for 1 month.
10 days, 8,96 do. for 10 days.
Ans. 1112,19–1112cts, or § 11, 12c. 11", m.
3. What is the interest of 789 dollars for 2 years, at 6
per cent.” Ans. $94, 68 cºs.
4. Of 37 dollars 50 cents for 4 years, at 6 per cent. per
annum ? Ans. 900 cts, or $9.
5. Of 325 dollars 41 cts, for 3 years and 4 months, at 5
percent.” Ans. $54, 23 cºs. 5 m.
6. Of "51. 12s. 8d. for five years, at 6 per cent.”
Ans. C97 13s. 8d.
7. Of 1.41 10s. 6d. for 3 and a half years, at 6 percent.”
Ans. E30 13s.
8. Of 150l. 16s. 8d. fºr 4 years and 7 months, at 6 per
tent.” Ans. E4 ºs. 7d.
-
112 . COMM-15-1-N.
9. Of I dollar for 12 years, at 5 per cent.”
Ans. 60 cas.
10. Of 215 dollars 34 cts, for 4 and a half years, at 4
and a half per cent. Mns. $33, 91 cts, 6m.
11. What is the amount of 324 dollars (51 cents for a
years and 5 months, at 6 per cent.”
Ans. $430, 10cts. Sºm.
12. What will 3000l. amount to in 12 years and 10
months, at 6 per cent.” Ans. C5310.
13. What is the interest of 2571. 5s. 1d. for 1 year and
3 quarters, at 4 per cent. 1 Ans. E18 0s. 1d. 34ps.
14. What is the interest of 279 dollars 87 cents for 2
years and a half, at 7 per cent per annum
Ans. $48, 97cts. 7am.
15. What will 2791. 13s. Sd. amount to in 3 years and a
half, at 5% per cent per annum?
- Ans. -Cº-1 1s. 6d.
16. What is the amount of 341 dols. 60cts for 5 year.
and 3 quarters, at 7 and a half per cent. per annum ?
- Ans. $4SS, 911 cts.
17. What will 780 dols, amount to at 6 per cent. in 5
years, 7 months, and 12 days, or ºs of a year?
Ans. $975, 99 cts.
18. What is the interest of 1825l. at 5 per cent, per an:
num, from March 4th, 1796, to March 29th, 1799, (allow
ing the year to contain 365 days?)
Ans. C280.
Note.—The Rules for Simple Interest serve also to cal-
culate Commission, Brokerage, Ensurance, or anything
else estimated at a rate per cent.
-
COMMISSION,
IS an allowance of so much per cent, to a factor or cor-
respondent abroad, for buying and selling goods for his em-
ployer.
-----------
1. What will the commission of 8431. 10s, come to at tº
per-cent. 1
ºn-or-E-A-Gº- -13
E. s. Or thus,
843. 10 £. s.
5 £5 is *)843 10
12, 17 10 Ans. E42 3 6
20 -
350
12
600 £423s.6d.
* Required the commission on 964 dols. 90 cts. at 21.
per-ent.” Ans. $21, 71 cts.
3. What may a factor demand on 1; per cent. commis-
sion for laying out 3568 dollars? Ans. $62, 44 cts
BROKERAGE,
IS an allowance of so much per cent to persons assist-
hig merchants, or factors, in purchasing or selling goods.
----------
1. yº is the brokerage of 750l. 8s. 4d. at 6s. 8d. per
cent.
£ s. d.
750 S 4 Here I first find the brokerage at 1 pound
I per cent. and then for the given rate,
- which is of a pound.
7,50 tº 4
20 s, d. E. s. d. ºrs.
6 8–1)7 10 1 0
10,08
12 Ans. £2 10 0 14
1,00
2. What is the brokerage upon 4125 dols at , or 75 cents
cent. 1 Ans. $30, 93 cºs. 71 m.
3. If a broker sell goods to the amount of 5000 dollars,
what is his demand at 65 cts, percent.”
Ans. $32, 50cts.
- 2
114 ---------.
4. What may a broker demand, when he sells goods tº
the value of 5081. 17s. 10d, and I allow him 11 per cent. 1
Ans. E7 12s. 8d.
ENSURANCE, |
IS a premium at so much per cent, allowed to persons
and offices, for making good the loss of ships, houses, mer.
chandise, &c. which may happen from storms, fire, &c.
-----------
1. What is the ensurance of 7:251. Ss. 10d. at 12 pe.
cent.” Ans. E90 13s. 71a.
2. What is the ensurance of an East-India ship and cas.
go, valued at 12:34:25 dollars, at 15 per cent.”
Ans. $19180, 87 cºs. 5 m.
3. A man's house estimated at 3500 dols., was ensurell
against fire, for I; per cent, a year: what ensurance did
he annually pay? Ans. $51, 25 cts.
-
Short Practical Rules for calculating Interest at 6 percent.
either for months, or months and days.
1. FOR STERLING MONEY.
Rule.-1. If the principal consist of pounds only, cut of the unit
figure, and as it then stands it will be the interest for one month, in
shillings and decimal parts.
2. If the principal consist of pounds, shillings, &c. reduce it to its
decimal value; then remove the decimal point one place, or figure,
further towards the left hand, and as the decimal then stands, it will
show the interest for one month in shillings and decimals of a shil.
ling.
----------
1. Required the interest of 541, for seven months and ten
days, at 6 per cent.
st-ºn-T ºn A-T-I-A1, Ru-LES. I-15
-
days=y)5,4 Interest for one month.
7
37s ditto for 7 months.
1,8 ditto for 10 days.
-
Ans. 39,6 shillings=E1 19s. 7,2d.
12
7,2
2. What is the interest of 421. 10s. for 11 months, at 6
per cent."
E. s. E.
42 10 = 42.5 decimal value.
Therefore 4.25 shillings interest for 1 month.
11
- £. s. d.
Ans. 46,75. Interest for 11 mo. - 2 6 9
3. Required the interest of 941. 7s 6d. for one year,
five months and a half, at 6 per cent. per annum ?
Ans. ES 5s. 1d. 3,5prs.
4. What is the interest of 121, 18s. for one third of a
month, at 6 per cent.” Ans. 5,161.
II. FOR FEDERAL MONEY.
Rule-1. Divide the principal by 2-placing the separatrix as usual,
ind the quotient will be the interest for one month in cents, and deci-
mals of a cºnt; that is, the figures at the lºſt of the separatrix will be
cents, and those on the right, decimals of a cent.
2. Multiply the interest of one month by the given number of
months, or months and decimal parts thereof, or for the days take the
even parts of a month, ºr
11ſ. sº-on-T PR-A-T-L RULES
-----------
1. What is the interest of 341 dols. 52 cts for 74 months.1
2)341,52
- Or thus, 170,76 Int. for 1 month.
170,76 Int, for 1 month. x7.5 months.
73. -
- 85.380
1195,32 do. for 7 mo. 119.532
85,38 do. for mo. - $ cºs. m.
- 1280,700cts. - 12,807
1280,70. Ans, 1280,7cts.-S12, 80cts. 7m.
2. Required the interest of 10 dols. 44 cts, for 3 yea-
5 months, and 10 days.
2)10,44
10 days=}) 5,22 interest for 1 month.
41 months.
5,22
-
208,8
214,02 ditto for 41 months.
1,74 ditto for 10 days.
215,76cts. Ans. =$2, 15 cts. 7 m.--
3. What is the interest of 342 dollars for 11 months?
The is 171 interest for one month.
11
Ans. ISSI cts.-818, 81 cts.
Note:-To find the interest of any sum for two months,
at 6 per cent, you need only call the dollars so many cents,
and the inferior denominations decimals of a cent, and it is
done: "Thus, the interest of 100 dollars for two months, is
100 cents, or one dollar; and 325, 40 cis- is 25 cts, 4 m.
&c. which gives the following
Rule II.-Multiply the principal by half the number of months,
and the product will show the interest of the given time, in cents and
docimals of a cent, as above.
- -
Fon tº A-L-L-L-T-N- INTEREST. 17
ºxa-PLEs.
1. Required the interest of 316 dollars for 1 year and 10
taonths. 1–1 the number of mo.
Ans. 3476 cts. -$34, 76 cts.
2. What is the interest of 364 dols. 25 cts, for 4 months
$ cºs.
364, 25
2 half the months.
728, 50cts. Ans.-37, 28 cts. 5 m.
III. When the principal is given in federal money, at 6
wer cent to find how much the monthly interest will be in
New-England, &c. currency.
Rule-Multiply the given principal by ,03, and the product will be
he interest for one month, in shillings and decimal parts of a shilling.
-----------
1. What is the interest of 325 dols, for 11 months?
,03
9,75 shil. int. for one month
× 11 months.
Ans. 107,25s.-E5 7s. 8d.
1. What is he interest in New-England currency of 31
dols, 68 cus, for 5 months?
Principal 31,68 dols.
". ,03
9504 Interest for one month.
5.
Ans. 4,7520s.-1s. 0d.
12
9, tº 10
110 *11-I-T-I-RACTICAL RULES
IV. When the principal is given in pounds, shillings, &c.
New-England currency, at 6 percent. to find how much the
menthly interest will be in federal money.
Rule.—Multiply the pounds, &c. by 5, and divide that product by
3, the quotient will be the interest for one month, in cents, and deci-
mals of a cent, &c.
-----------
1. A note for £411 New-England currency has been on
interest one month; how much is the interest thereof infe
deral money? --
411
-> -
3)2055
Ans. 685 cts.-S6, S5 cts.
2. Required the interest of 391. 18s. N. E. currency, for
7 months? £
-
39.9 decimal value.
5
Interest for 1 mo. 66,5 cents.
7
Ditto for 7 mo. 465,5cts.-$4, 65 cts. 5 m. Ans.
V. When the principal is given in New-England and Vir
ginia currency, at 6 per cent to find the interest for a year,
in dollars, cents, and mills, by inspection.
Rule.-Since the interest of a year will be just so intºny cents as
the given principal contains shillings, therefºre, write down the shil.
lings and call them cents, and the pence in the principal made less by
1 if they exceed 3, or by 2 when they exceed 9, will be the mills, very
nearly.
ºn tº AL-U-LATING INTEREST. 119
----------
1. What is the interest of 21, 5s, for a year, at 6 per ct. 1
£25s-15s. Interest 45 cts, the Answer.
2. Required the interest of 100l. for a year, at 6 per et. "
£100–2000s. Interest 2000 cts.-sºo Ans.
3. Of 27s.6d. for a year?
Ans. 27s. is 27 cts, and Gd. is 5 m.
4. Required the interest of 51. 10s. 11d. for a year?
£5 10s.-110s. Interest 110 cºs.-31, 10cts. 0 m.
11 pence.—2 per rule leaves 9– 9
Ans, si, to 9
-
VI. To compute the interest on any note or obligation,
when there are payments in part, or endorsements.
Rule.-1. Find the amount of the whole principal for the whole
ine.
2. Cast the interest on the several payments, from the time they
were paid, to the time of settlement, and find their amount; and lastly
educt the amount of the several payments from the amount of the
rincipal. -
----------
Suppose a bond or note dated April 17, 1793, was given
for 675 dollars, interest at 6 per cent. and there were pay-
ments endorsed upon it as follows, viz.
First payment, 148 dollars, May 7, 1794.
Second payment, 341 dols. August 17, 1796.
Third payment, 99 dols. Jan. 2, 1798. I demand how
nuch remains due on said note, the 17th June, 1798?
$ cts.
148, 00 first payment, May 7, 1791. Mr. mo.
36, 50 interest up to-June 17, 1798–4 14
184, 50 amount -
341, 00 second payment, Aug. 17, 1796. Yr. mo.
37, 51 interest to—June 17, 1798 –1 10
378, 51 amount.
[Carried over.]
120 -11-R-T ºn A-T-CAL I-ULE
8 rºs.
90, 00 third payment, January 2, 1798.
2, 72 interest to—June 17, 1798–54 mo,
101, 72 amount.
184, 50
378, 51 | several amounts.
101, 72
664, 73 total amount of payments.
575, 00 note dated April 17, 1793. Yr. mo
209, 25 interest to—June 17, 1798. =5 2
884, 25 amount of the note.
664, 73 amount of payments.
$219, 52 remains due on the note, June 17, 1798.
2. On the 16th January, 1795, Ilent James Paywell 50°
dollars, on interest at 6 per cent, which I received back in
the following partial payments, as under, viz.
1st of April, 1796 - - - - $ 50
16th of July, 1797 - - - - 400
1st of Sept. 1798 - - - - 60
How stands the balance between us, on the 16th Novem
ber, 1800? Ans, due to me, $63, 18 cºs.
3. A promissony Note, viz.
c62 10s. New-London, April 4, 1797.
On demand, I promise to pay Timothy Careful, sixty-two
pounds, ten shillings, and interest at 6 percent. per annum,
till paid; value received.
John Stawny, PETER PAY WELL.
Richand Testis.
Endorsements. £. s.
1st. Received in part of the above note,
September 4, 1799, 50 0
And payment June 4, 1800, 12 10
How much remains due on said note, the 4th day of De
celnber, 1800. £. s. d.
Ans, ºn 12 tº
run ------ LAT-Mº, NTE-F-T. I-1
Note:-The preceding Rule, by custom, is rendered so
popular, and so much practised and esteemed by many on
ºccount of its being simple and concise, that I have given it
a place: it may answer for short periods of time, but in
a long course of years, it will be found to be very errone-
----- -
Although this method seems at first view to be upon the
ground of simple interest, yet upon a little attention the
following objection will be found most clearly to lie against
it, viz. that the interest will, in a course of years, complete-
ly expunge, or as it may be said, eat up the debt. For an
explanation of this, take the following
--------
A lends B 100 dollars, at 6 per cent. interest, and takes
His note of hand; B does no more than pay A at every
rear's end 6 dollars, (which is then justly due to B for the
*se of his money) and has it endorsed on his note. At the
and of 10 years B takes up his note, and the sum he has to
lay is reckoned thus: The principal 100 dollars, on inte
rest 10 years amounts to 160 dollars; there are nine en-
dorsements of 6 dollars each, upon which the debtor claims
interest; one for nine years, the second for 8 years, the
third for 7 years, and so down to the time of settlement;
the whole amount of the several endorsements and their in-
terest, (as any one can see by casting it) is $70, 20cts, this
subtracted from 160 dols, the amount of the debt, leaves in
favour of the creditor, $89, 40 cts. or 810, 20cts, less than
the original principal, of which he has not received a cent,
but only its annual interest.
If the same note should lie 20 years in the same way, B
would owe but 37 dols. 60 cts, without paying the least
fraction of the 100 dollars borrowed.
Extend it to 28 years, and A the creditor would fall in
debt to B, without receiving a cent of the 100 dols, which
he lent him. See a better Rule in Simple Interest by de-
timals, page 175.
-
22 -------------------T
COMPOUND INTEREST,
1S when the interest is added to the principal, at the
end of the year, and on that amont the interest cast for ano-
ther year, and added again, and so on: this is called inte-
rest upon interest.
Rule.—Find the interest for a year, and add it to the princi-
pal, which call the amount for the first year; find the interest
of this amount, which add as before, for the amount of the se-
cond, and so on for any number of years required. Subtract the
original principal from the last amount, and the remainder will be
the Compound Interest for the whole time.
-----------
1. Required the amount of 100 dollars for 3 years at L
percent. per annum, compound interest!
$ cas. s rºs.
1st Principal 100,00 Amount 106,00 for 1 year.
2d Principal 106,00 Amount 112,36 for 2 years.
3d Principal 112,36 Amount 19,1016 for 3 yrs. Ans.
2. What is the amount of 425 dollars, for 4 years, at a
per cent. per annum, compound interest?
Ans. $516, 59cts.
3. What will 400l. amount to, in four years, at 6 per
cent. per annum, compound interest?
Ans. E504 19s. 9d.
4. What is the compound interest of 1501. 10s. for 3
years, at 6 per cent. per annum? Ans. £28, 14s. 11+d.--
5. What is the compound interest of 500 dollars for 4
years, at 6 per cent. per annum Ans. $131,238+
6. What will 1000 dollars as "ºnt to in 4 years, at 7 per
cent. per annum, compound interest? -
Ans. $1310, 79 cts. G. m. --
7. What is the amount of 750 dollars for 4 years, at 6
per cent. per annum, compound interest?
Ans. $946, 85 cts. 7,72 m.
8. What is the compound interest of 876 dols, 90 cents
or 8 years, at 6 per cent per annum !
Ans, slº, S3 cts.--
-------NT 113
DISCOUNT,
Is an allowance made for the payment of any sum of
money before it becomes due; or upon advancing ready
money for notes, bills, &c. which are payable at a future
day. What remains after the discount is deducted, is the
present worth, or such a sum as, if put to interest, would
at the given rate and time, amount to the given sum or
debt.
Rule.—As the amount of 100t, or 100 dollars, at the given rate
and time: is to the interest of 100, at the same rate and time: ; so is
the given sum : to the discount.
subtract the discount from the given sum, and the remainder is the
present worth.
Or—as the amount of 100: is to 100: ; so is the given sum or
debt i to the present worth.
Paoon.-Find the amount of the present worth, at the giver
rate and time, and if the work is right, that will be equal to the
given sum.
---------
1. What must be discounted for the ready payment of
100 dollars, due a year hence at 6 percent, a year?
3 & $ 8 cts.
As 106 : 6 : : 100 : 5 (56 the answer.
100,00 given sum.
5,66 discount.
sø1,34 the present worth.
2. What sum in ready money will discharge a debt of
925, due 1 year and 8 months hence, at 6 percent.
£100
10 interest for 20 months.
110 Am"t. C. £. £. £. a. d.
As 110 : 100 : : 925 - 840 18 2+ Ans.
3. What is the present worth of 600 dollars, due 4 years
hence, at 5 percent.” Ans, 8500.
4. What is the discount of 275l. os. for 10 months, at
5 per cent. per annum ? 4ns. E1.32s. 4d.
2-1 -----T-I-R
5. Bought goods amounting to 615 dols. 75 cents, at 7
months credit; how much ready money must 1 pay, dis-
count at 41 per cent. per annum ? Ans. $600.
6. What sum of ready money must be received for a bill
of 900 dollars, due 73 days hence, discount at 6 per cent.
per annum ? Ans. $889, 32 cts. Sm.
Note.—When sundry sums are to be paid at different
times, find the Rebate or present worth of each particular
payment separately, and when so found, add them into one
-------
-----------
7. What is the discount of 7561, the one half payable in
six months, and the other half in six months after that, at 7
per cent.” Ans. C37 10s. 21d.
S. If a legacy is left me of 2000 dollars, of which 500
dols, are payable in 6 months, 800 dols, payable in 1 year,
and the rest at the end of 3 years; how much ready morey
ought I to receive for said legacy, allowing 6 per cent. dis-
count? Ans. $1833, 37 cits. 4 m.
-
ANNUITIES.
AN Annuity is a sum of money, payable every year, ºr
for a certain number of years, or for ever.
When the debtor keeps the annuity in his own hands
beyond the time of payment, it is said to be in arrears.
The sum of all the annuities for the time they have been
foreborne, together with the interest due on each, is called
the amount.
If an annuity is bought off, or paid all at once at the
beginning of the first year, the price which is paid for it is
called the present worth.
To find the amount of an annuity at simple interest.
Rule.-1. Find the interest of the given annuity for 1 year.
2. And then for 2, 3, &c. years, up to the given time, less 1.
3. Multiply the annuity by the number of years given, and add-
the product to the whole interest, and the sum will be the amrust
sought.
ANNU-T-Fº. 125
Ex--------
! If an annuity of 701. be forborne 5 years, what will
be due for the principal and interest at the end of said
term, simple interest being computed at 5 per cent. per
- -
annum 2 E. s.
1st. Interest of 701 at 5 per cent, for 1-3 10
2– 7 0.
8–10 10
1–14 tº
2d. And 5 yrs. annuity, at 701 per yr. is 350 ()
Ans. E335 0
2. A house being let upon a lease of 7 years, at 400
dollars per anuum, and the rent being in arreur for the
whole term, I demand the sum due at the end of the term,
simple interest being allowed at 61 per cent, per annum?
Ans. E3304.
-
To find the present worth of an annuity at simple
interest.
Rule.—Find the present worth of each year by itself, discounting
from the time it falls due, and the sum of all these present worth-
will be the present worth required.
---------
1. What is the present worth of 400 dols, per annum,
to continue 4 years, at 6 percent, per annum?
10t, 377,35849 = Pres, worth of 1st yr.
112 357.14:285 – – 2d yr.
* -- intº -- - - y
is ?: "****ś30; 3d yr.
124 322,58064 = – 4th yr.
Ans. $1396,0.650.4 = $1340, ºcts, 5m.
2. How much present money is equivalent to an annuity
of 100 dollars, to continue 3 years; rebate being made at
6 per cent.” Ans. $268, 37 cº,
3. What is 80 yearly rent, to continue 5 years, worht
in ready mºney, at 61 percent. 4ns, £340, 5s 3d.
2
lºſ, Equatiºn of raw Mºnt-
EQUATION OF PAYMENTS,
1S finding the equated time to pay at once, several debts
nue at different periods of time, so that no loss shall be
sustained by either party.
Rule.—Multiply each payment by its time, and divide the sum ºf
the several products by the whole debt, and the quotient will be the
equated time for the payment of the whole.
Exami-LEs.
1. A owes B 380 dollars, to be paid as follows—viz. 100
dollars in 6 months, 120 dollars in 7 months, and 160 dol-
lars in 10 months: What is the equated time for the pay.
ment of the whole debt?
100 x G = G00
120 × 7 = 840
100 x 10 - 1500
3S0 )3040(8 months. Ans.
2. A merchant hath owing him 300l. to be paid as fol
lows: 50l. at 2 months, 100l. at 5 months, and the rest ta
8 months; and it is agreed to make one payment of the
whole: I demand the equated time? Ans, 6 months.
3. F owes H 1000 dollars, whereof 200 dollars is to be
paid present, 400 dollars at 5 months, and the rest at 15
months, but they agree to make one payment of the whole;
I demand when that time must be? Ans. S months.
4. A merchant has due to him a certain sum of money,
to be paid one sixth at 2 months, one third at 3 months,
and the rest at 6 months; what is the equated time for the
payment of the whole 1 Ans. 41 months.
__
- -
BARTER,
1S the exchanging of one commodity for another, and
directs merchants and traders how to make the exchange
without loss to either party.
Rule:-Find the value of the commodity whose quantity is given;
then find what quantity of the other at the proposed rate can be
*ght for the same money, and it gives the answer.
man ren. -7
-----------
1. What quantity of flax at 9 cts. per b. must be given
in barter for 12 lb. of indigo, at 2 dols. 19 cents per lb. ?
12 lb. of indigo at 2 dols. 19 cts. per lb. comes to 26
dols. 28 cts.-therefore, As 9 cts. : 1 lb.: ; 26.28 cts. :
292 the answer.
2. How much wheat at I dol. 25 cts, a bushel, must be
given in barter for 50 bushes of rye, at 70 cts a bushelt
Ans. 28 bushels.
3. How much rice at 28s, per cwt. must be bartered for
3 cwt. of raisins, at 5d. per lb. ?
Ans. 5 cwt. 3 qrs. 9113th.
4. How much tea at 4s. 0d. per lb. must be given in
barter for 78 gallons of brandy, at lºs. 3}d. per gallon 1
Ans. 201 lb. 133102.
5. A and B bartered: A Jºud 8 cwt. of sugar at 12 cts.
per lb. for which 3 gave him (S. cwt. of flour; what was
he flour rated a per lb. Ans. 54 cts.
G. B. delivered 3 hlids of brandy, at 6s. 8d. per gallon,
to C, for 126 yds of cloth, what was the cloth per yard?
Ans. 10s.
7. D gives E 250 yards of drugget, at 30 cts. per yd.
ºor 319 lbs. of pepper; what does the pepper stand him in
per lb. ? Ans. 23 cts. 5,'ºm.
8. A and B bartered: A had 41 cwt. of rice, at 21s.
per cwt. for which B gave him 201 in money, and the
rest in sugar at 8d. per lb., I demand how much sugar B
gave A besides the 201.7 Ans. 6 cwt. 0 qrs. 1941b.
9. Two farmers bartered: A had 120 bushels of wheat
at 11 dols, per bushel, for which B gave him 100 bushels
of barley, worth 65 cts, per bushel, and the balance in oats
at 40 cts. per bushel; what quantity of oats did A receive
from ºt Ans. 287 bushels.
10. A hath linen cloth worth 200. en ell ready money;
but in barter he will have 2s. B hath broadcloth worth 14s.
5d. per yard ready money; at what price ought B to rate
his broadcloth in barter, so as to be equivalent to A's bar-
tering price? Ams. 17s. 4d. 3 ºrs.
170 ---- a-i, u A-w.
11. A and B barter: A hath 145 gallons of brandy nº
| dol. 20 cts, per gallon ready money, but in barter he
will have I dol. 35 cts. per gallon: B has linen at 58 cts
per yard ready money; how must B sell his linen per
yard in proportion to A's bartering price, and how many
yards are equal to A's brandy?
Ans. Barter price of B's linen is 65 cts. 21 m. and he
must give A 300 yds. for his brandy.
12. A has 225 yds. of shalloon, at 2s. ready money pe.
yard, which he barters with B at 2s. 5d. per yard, taking
indigo at 12s. 6d. per lb. which is worth but 10s. how
much indigo will pay for the shalloon; and who gets the
best bargain?
Ans. 4341b. at barter price will pay for the shalloon, and
B has the advantage in barter.
Value of A's cloth, at cash price, is £22. It
Value of 434th. of indigo, at 10s. per lb. 21, 15.
B gets the best bargain by £0 li
LOSS AND GAIN,
IS a rule by which merchants and traders discover their
profit or loss in buying and selling their goods: it also in
structs them how to rise or fall in the price of their good",
so as to gain or lose so much per cent or otherwise.
Questions in this rule are answered by the Rule of Three.
- ----------
1. Bought a piece of cloth containing S5 yards, for 19t
dols. 25 cts, and sold the same at 2 dols. S1 cts. per yard;
what is the profit upon the whole piece?
Ans. $47, 60 cts.
2. Bought 12 cwt. of rice, at 3 dols. 45 cts, a cwt. and
sold it again at 4 cts, a pound; what was the whole gain
Ans. 312, 87 cts. 5m.
3. Bought 11 cwt. of sugar, at Gºd. per lb. tº it could not
sell it again for any more than 2, 16s. per cwt. ; did I gau
or lose by my bargain? Ans. Lost, C2 11s 4d.
4. Bought 44 lb. often for tºl. 12s, and sold it again for
ºl. 10s. 6d. what was the profit on each pound 1
Ans. 10-d
Luss ANL) tº A1x. 120
5. Bougºut a hlid. of molasses containing 119 gallons,
at 52 cents per gallon; paid for carting the same I dollar
25 cents, and by accident 9 gallons leaked out; at what
rate must I sell the remainder per gallon, to gain 13 dol-
lars in the whole? Ans. 69 cts. 2 m.-H.
-
II. To know what is gained or lost per cent.
Rule.—First see what the gain or loss is by subtractiºn; then, As
the price it cost: is to the gain or loss :: so is 100', or $100, to the
gain or loss per cent.
EXAMPLES-
1. If I buy Irish linen at 2s. per yard, and sell it again
at 2s. 8d. per yard; what do I gain per cent, or in laying
out 100t. . As : 2s. 8d. : : 1001 : £33 6s. 8d. Ans.
2. If I buy broadcloth at 3 dols. 44 cts. per yard, and sell
it again at 4 dols. 30 cts. per yard: what do I gain per ct.
ut in laying out 100 dollars?
* cºs.
sold for 4, 30 3 cºs. cfs. s 8
Cost 3, 44 As 3 44 - 86 : : 100 - 25
Ans. 25 per cent.
Gained per yd. 86
3. If I buy a cwt. of cotton for 34 dols. 86 cts, and sell it
again at 414 cts. per lb. what do I gain or lose, and what
per cent.” $ cºs.
1 cwt. at 411 cts, per lb. comes to 46,48
Prime cost 34,86
Gained in the gross, $11,61
As 34.86 : 11,62 : : 100 : 33). Ans. 33 per cent.
* Bought sugar at Sºd. per Ib, and sold it again at 41.
17s. per cwt. what did I gain percent.
Ans. E25 19s. 5; d.
5. If I buy 12 hlids of wine for 2041, and sell the same
again at 14, 17s.6d. per hld. do I gain or lose, and what
per cent. 1 Ans. I lose 121 percent.
6. At Hºl. profit in a shilling, how much per cent.”
- Ans. E12 10s.
ºt) Lºss ANL). GAIN
7. At 25 cts, profit in a dollar, how no ºn per cent.”
Ans. 25 per cent.
Note:-When goods are bought or sold on credit, you
must calculate (by discount) the present worth of their
price, in order to find your true gain or loss, &c.
Ex-MPLEs.
1. Bought 164 yards of broadcloth, at 14s. 6d. per yard
ready money, and sold the same again for 1541. 10s. on 6
months credit; what did I gain by the whole; allowing
discount at 6 per cent a year?
- +. £. £. . s. £. s.
As 103 : 100 : : 154 10 : 150 0 ºresent worth.
118 18 prime cost.
Gained £31 a 41-seen.
2. If I buy cloth at 4 dols. 16 ct, per yard, on eigh
months credit, and sell it again at 3 dols. 90 cts. per yº
ready money, what do I lose per cent, allowing 6 per cen
discount on the purchase price? Ans. 24 per cent.
-
III. To know how a commodity must be sold, to gau,
or lose so much per cent.
Rule.—As 100 : is to the purchase price : ; so is 1001, or 101
dollars, with the profit added, or loss subtracted : to the selling
price.
Ex-MPLEs.
1. If I buy Irish linen at 2s. 8d. per yard; how must 1
sell it per yard to gain 25 per cent.” -
As 100. : 2s. 8d. : : 125l. to 2s. 9d. 3 qrs. Ans.
2. If I buy rum at I dol. 5 cts, per gallon; how must 1
sellit per gallon to gain 30 per cent.”
As $100 : $1,05 : : $130 : $1,364 cts. Ans.
3. If tea cost 54 cts. per lb.; how must it be sold per lb
to lose 121 per cent. 1
As $100: 54 cts. : : $87, 50cts. : 47 cts. 24 m. Ans.
4. Bought cloth at 17s.6d. per yard, which not provina
so º as I expected, I am obliged to lose 15 percent. *-
it; how must I sell it per yard? Ans. 14s. 10; d.
Lºss AN-u-N. 131
5. If 11 cwt. I qr, 25 lb. of sugar cost 126 dols, 50cts.
how must it be sold per lb. to gain 30 per cent. ”
Ans. 12 cts, 8m.
6. Bought 90 gallons of wine at 1 dol. 20 cts. per gal:
but by accident 10 gallons leaked out; at what rate must I
sell the remainder per gallon to gain upon the whole prime
cost, at the rate of 124 per cent.” Ans 81, 51 cts. Sºm.
- -
IV. When there is gained or lost per cent to know
what the commodity cost.
Rule.—As 100, or 100 dols, with the gain percent, added, or loss
percent, subtracted, is to the price, so is 100 to the prime cost.
----------
1. If a yard of cloth be sold at 14s. 7d. and there is gain-
* 16t. 13s. 4d. percent. ; what did the yard cost?
£. s. d. s. d. E.
As 116 13 4: 14 7: : 100 to 12s. 6d. Ans.
2. By selling broadcloth at 3 dols. 25 cts. per yard, I
use at the rate of 20 per cent. ; what is the prime cost of
raid cloth per yard? Ans. $4,06 cts. 21 m.
3. If 40 lb. of chocolate be sold at 25 cts, per lb, and I
gain 9 percent.; what did the whole cost me?
Ans. $9, 17 cºs. 4m.--
4. Bought 5 cwt. of sugar, and sold it again at 12 cents
wer lb. by which I gained at the rate of 25 per cent. ;
what did the sugar cost me per cwt. 1
Ans. $10,70cts. 9.m.-H.
-
W. If by wares sold at a given rate, there is so much
f. or lost per cent to know what would be gained or
ost per cent. if sold at another rate.
Rule.--As the first price: is to 1001, or 100 dols, with the profit
percent added, or loss percent subtracted :: so is the other price: to
the gain or loss percent. at the other rate.
N. B. If your answer exceed 100l. or 100 dols, the
excess is your gain per cent.; but if it be less than 100,
that defriency is the loss percent.
132 --------------
Ex-M-L-->. -
1. If I sell cloth at 5s. per yd. and thereby gain 15 per
cent what shall I gain per cent. if I sell it at 5s. per yd: 1
s. £ s. E. -
As 5 : 115 :: 6 : 138 Aus. gained 38 per cent.
2. If I retail rum at I dollar 50 cents per gallon, ºnal
thereby gain 25 percent what shall I gain or lose per cent
if I sell it at I dol. Scts. per gallon 1
8 cts. 8. § cts. 8
1,50: 125:: 1,08 : 90 Ans. I shall lose 10 per cent.
3. If I sell a cwt. of sugar for 8 dollars, and thereby
ose 12 per cent. what shall I gain or lose percent. if I sell
4 cwt. of the same sugar for 36 dollars?
Ans. I lose only 1 per cent.
4. I sold a watch for 171. Is. 5d. and by so doing lost
15 per cent, whereas I ought in trading to have cleared
20 per cent. ; how much was it sold under its real value?
£. E. s. d. E. E. s. 1.
As 85 : 17-1 5 : : 100 : 20 1 s the prime cost.
100 : 20 1 S : : 1:20 : 24 2 0 the real value.
Sold for 17 - 5
-C7 0 7 Answer.
- - - - -
FELLOWSHIP,
IS a rule by which the accounts of several merchants
or other persons trading in partnership, are so adjusted,
that each may have his share of the gain, or sustain his
share of the loss, in proportion to his share of the joint
stock-Also, by this Rule a bankrupt's estate may be di-
vided among his creditors, &c. -
SINGLE FELLOWSHIP,
Is when the several shares cf stock are continued in
trade an equal term of time.
Rule-As the whole stock is to the whole gain or loss: so is each
man's particular stºck, to his particular share of the gain or loss.
FELL-w's H- 133
Paoor-Add all the particular shares of the gain or loss to-
cather, and if it be right, the sum will be equal to the whole
gain or loss
---------
1. Two partners, A and B, join their stock and buy
a quantity of merchandise, to the amount of 820 dollars;
in the purchase of which Alaid out 350 dollars, and B-470
dollars; the commodity being sold, they find their clean
gain amounts to 250 dollaº. What is each person's share
of the gain?
A put in 350
B 470
As 820 - 250 : :
350 : 106,7073+A's share.
470 : 143,2926-1-B's share.
Proof 249,9999-–$25
2. Three merchants make a joint stock of 1200I. of
which A put in 240l. B 860l. and C. 600l. ; and by trading
hey gain 325l. what is each one's part of the gain?
Ans. A's part £65, B's £97 10s. C's £162 10s.
3. Three partners, A, B, and C, shipped 108 mules for
Ahe West-Indies; of which A owned 48, B 36, and C 24;
But in stress of weather, the mariners were obliged to
throw 45 of them overboard; I demand how much of
the loss each owner must sustain?
Ans. A 20, B 15, and C 10.
4. Four men traded with a stock of 800 dollars, by
which they gained 307 dols. A's stock was 140 dols.
B's 260 dols. C's 300 dols. I demand D's stock, and
what each man gained by trading 1
Ans. D's stock was $100, and Agained $53,72cts, 5 m.
B $99,774 cts. C. §115, 12 cts, and D $38,374 cts.
5. A bankrupt is indebted to A 21-11 to B 300l. and to
C 891. and his whole estate amounts only to 675l. 10s.
which he gives up to those creditors; how much must each
have in proportion to his debt?
Ans. A must have £158 0s. 344. B C224 13s. 4d. and
C £292 16s 3d.
134 CºMPOUNL PELLOWSHIP
6. A captain, mate, and 20 seamen, took a prize worth
3501 dols. of which the captain takes 11 shares, and the
mate 5 shares; the remainder of the prize is equally di-
vided among the sailors; how much did each man receive!
8 cts.
Ans. The captain received 1069, 75
The mate 486, 25 -
Each sailor 97, 25
7. Divide the number of 360 into 3 parts, which shall be
to each other as 2, 3 and 4. Ans. 80, 120 and 160.
8. Two merchants have gained 450l. of which A is to
have three times as much as B; how much is each to have?
Ans. A £337 10s. and B.E.1.12 10s.-1.-H.3–4 : 450 : :
3 : E337 10s. A's share.
9. Three persons are to share 600l. A is to have a cer-
tain sum, B as much again as A, and C three times as
much as B. I demand each man's part 1
Ans. A £666, B.E.1333, and C++00.
10. A and B traded together and gained 100 dols. A pu!
in 640dols. B put in so much that he must receive 60 dols.
of the gain; I demand B's stock 2 Ans. $960.
11. A, B and C traded in company: A put in 140 dols.
B 250 dols, and C put in 120 yds. of cloth, at cash price;
they gained 230 dols, of which C took 100 dols. for hº
share of the gain; how did C value his cloth per yard in
common stock, and what was A and B's part of the gain!
Ans. Cput in the cloth at $2 per yard. A gained $46,
67 cts. 6 m.--and B SS3, 33 cts, 3 m.--
~ - -
COMPOUND FELLOWSHIP,
OR Fellowship with time, is occasioned by several shares
of partners being continued in trade an unequal term of
time.
Rule.-Multiply each man's stock, or share, by the time it was
eontinued in trade: then, -
As the sum of the several products,
Is to the whole gain or loss:
So is each man's partic-lar product,
To his particular share of the gain or loss.
co-Poux in FELLuwsuitº 135
examples.
A, B and C hold a pasture in common, for which they
pay 191 per annum. A put in Soxen for 6 weeks; B 12
oxen for 8 weeks; and C 12 oxen for 12 weeks; what must
each pay of the rent?
£. s. d.
8× 6– 48 48 : 3 3 4 A's part.
12 x 8– 96 96 : 6 (; 8 B's –
12×12–144 - As2S8: 191. :: * 144 - 9 10 0 C's –
Sum 288 . Proof 19 0 ()
2. Two merchants traded in company; A put in 215
dols, for 6 months, and B 300 dols. for 9 months, but by
misformane they lose 200 dols.; how must they share the
loss? Ans. A's loss $53, 75cts. B's $146,25cts.
3. Three persons had received 665 dols, interest: A had
put in 4000 dollars for 12 months, B 3000 dollars for 15
months, and C 5000 dollars for 8 months; how much is
each man's part of the interest?
Ans. A s?40, B-S2:25, and C. §200.
4. Two partners gained by trading 1101.12s. : A's stock
was 120l. 10s. for 4 months, and B's 2001, for 6 months;
what is each man's part of the gain? -
Ans. A's part £29.18s. 31.1.hº. B's E80 13s. 8d.º.
5. Two merchants enter into partnership for 18 months.
A at first put into stock 500 dollars, and at the end of 8
months he put in 100 dollars more; B at first put in 800
dollars, and at 4 months’ end took out 200 dols. At the
expiration of the time they find they have gained 700 dol-
lars; what is each man's share of the gain?
Mns | S324,07 4-4-4's share.
" .. 8375,92 5-H B's do.
6. A and B companied: A put in the first of January,
1000 dollars; but B could not put in any till the first of
May; what did he then put in to have an equal share with
A at the year's end ? -
Mo. s Mo. º
As 12 : 1000 : : 8 - 1000×12–1500 Ans
º
lºſ, dot-ELE RULE - F THREE.
DOLIBLE IRULE OF THREE.
THE Double Rule of Three teaches to resolve at once
such questions as require two or more statings in simple
proportion, whether director inverse.
In this rule there are always five terms given to find a
sixth; the first three terms of which are a supposition, the
last two a demand.
Rule.—in stating the question, place the terms of the supposi
tion so that the principal cause of loss, gain, or action, possess
the first place; that which signifies time, distance of place, &c.
in the second place; and the remaining term in the third place.
Place the terms of demand, under those of the same kind-in
the supposition. If the blank place, or term sought, fall un-
der the third term, the proportion is direct; then multiply the
first and second terms together for a divisor, and the other thren
for a dividend - but if the blank fall under the first or second
term, the proportion is inverse; then multiply the third-and
fourth terms together for a divisor, and the other three for a di
vidend, and the quotient will be the answer.
Ex-MPLEs.
1. If 7 men can build 36 rods of wall in 3 days; how
many rods can 20 men build in 14 days?
7 : 3 : : 36 Terms of supposition.
20 : 14 Terms of dºsaand.
36
84
42
504
20
7x3–21)10080(480 rods. An-
2. If 100l. principal will gain 6l. interºst in 12 months
what will 400l. gain in 7 months? -
Principal 1001 : 12 mo. . . ºr interest.
400 - 7 Ana. -11.
–
co-oº-º-n PROPORTION. 1.37
3. If 100. will gain G. a year; in what time will 400.
gain 141. E. mo. E.
It)0 : 12 : : 6
400 : : : 14 Ans. 7 months.
4. If 400. gain 14 in 7 months: what is the rate per
tent. per annum? E. mo. Int.
400 : 7 : : 11
100 : 12 Ans. E6.
3. What principal at 61 per cent per annum, will give
141, in 7 months? E. mo. Int.
100 : 12 : : 6
7 : : 14 Ans, e-100.
6. An usurer put outsºl to receive interest for the same:
and when it had continued 8 months, he received principal
and interest, SSl. 17s. 4d. ; I demand at what rate per ct-
per ann. he received interest? Ans. 5 per cent.
7. If 20 bushels of wheat are sufficient for a family of
8 persons 5 months, how much will be sufficient for 4 per-
tons tº months? Ans. 24 bushels.
8. If 30 men perform a piece of work in 20 days; how
many men will accomplish another piece of work 4 times
as large in a fifth part of the time !
S0 - 20 : : 1
4 : -4 Ans. 600.
9. If the carriage of 5 cwt. 3 qºs. 150 miles, cost 24
dollars 58 cents; what must be paid for the carriage of 7
ewt. 2 qrs. 25 lb. 64 miles, at the same rate?
Ans. s.14,08 cºs. Gm.--
10. If 8 men can build a wall 20 feet long, 6 feet high,
and 4 feet thick, in 12 days; in what time will 24 men
build one 200 feet long, 8 feet high, and 6 feet thick?
8 : 12 : : 20 × 6-4
2-1 : 200x8x6 80 days. Ans.
CONJOINED PROPORTION,
IS when the coins, weights or measures of several coun-
ries are compared in the same question; or it is joinin
many proportions together, and by the relation º
m 2
138 CON-INEL Prº-1-1-1-1-N.
-
several antecedents have to their consequents, the propor
tion between the first antecedent and the last consequent is
discovered, as well as the proportion between the others in
their several respects.
Note:-This rule may generally be abridged by can-
celling equal quantities, or terms that happen to be the
same in both columns: and it may be proved by as many
statings in the Single Rule of Three as the nature of the
question may require.
CASE I. -
When it is required to find how many of the first sort
of coin, weight or measure, mentioned in the question, are
equal to a given quantity of the last.
Rule.—Place the numbers alternately, beginning at the left hand,
and let the last number stand on the left hand column; then multi-
ply the left hand column continually for a dividend, and the right
hand for a divisor, and the quotient will be the unswer.
1-xAMI-Lºs.
1. If 100 lb. English make 95 lb. Flemish, and 19 lb.
Flemish 25 lb. at Bologna; how many pounds English are
equal to 50 lb. at Bologna?
lb.
100 Eng. =95 Flemish.
19 Fle. =25 Bologna.
50 Bologna. Then 95 x 25–2375 the divisor,
95000 dividend, and 2375).95000(40 Ans. -
2. If 40 lb. at New-York make 48 lb. at Antwerp, and
30 lb. at Antwerp make 36 lb. at Leghorn; how many lb.
at New-York are equal to 144 lb. at Leghorn?
3. If 70 braces at Venice be equal to 75 braces at Leg-
horn, and 7 braces at Leghorn be equal to 4 American
yards; how many braces at Venice are equal to 64 Ameri-
can yards? Ans. 104*.
CASE II. -
When it is required to find how many of the last sort of
coin, weight or measure, mentioned in the question, are
equal to a given quantity of the first.
Ans. 100 lb,
Ex-11AN-E. 130
Rule.—Place une numbers alternately, oeginning at the left hand,
and let the last number stand on the right hand; then multiply the
first row for a divisor, and the second for a dividend
ExAMPLEs.
1. if 24 lb. at New-London make 20 lb. at Amsterdam,
and 50 lb. at Amsterdam 60 lb. at Paris; how many at
Paris are equal to 40 at New-London?
Lºft. Right.
2-1 … 20 20 × 60 × 40 = 48000
50 - 00 –= 40 Ans.
10 21 × 50 = 1200
2. If bo whº at New-York make 45 at Amsterdam, and
S0 lb. at Aºsterdam make 103 at Dantzic ; how many lb.
at Dantzic are equal to 240 at N. York? Ans. 27°º,
3. If 20 braces at Leghorn be equal to 11 vares at Lis-
bon, and 40 vares at Lisbon to 80 braces at Lucca; how
many braces at Lucca are equal to 100 braces at Leghorn?
- Ans. 110.
- -
EXCHANGE.
BY this rule merchants know what sum of money ought
to be received in one country, for any sum of different spe-
cie paid in another, according to the given course of ex-
change.
To reduce the moneys of foreign nations to that of the
nited States, you may cºnsult the following
TABLE:
owing the value of the moneys of account, of foreign
nations, estimated in Federal money." § cts.
Pound Sterling of Great Britain, 4 44
Pound Sterling of Ireland, 4. It)
Livre of France, 0 18-
Guilder ºr Florin of the U. Netherlands, 0-30
Mark Banco of Hamburgh, 0.334
Rix Dollar of Denmark, I ()
* Law. J. S. A.
*40 FYCHANGE.
Rial Plate of Spain, 0 in
Milrea of Portuga, 1 24
Tale of China, 1 48
Pagoda of India, 1 94 -
Rupee of Bengal, 0. 55
p "ºf GREAT BRITAIN. 554
Ex--------- - º
1. In 451. 10s, sterling, how many dollars and cents :
A pound sterling being=444 cents,
Therefore–As il. : 444 cts. : : 45,51. : 20202 cts. Amr,
2. In 500 dollars how many pounds sterling?
As 444 cts. : 11. : : 50000 cts. : 1121. 12s. 8d.-- Ans.
II.-OF. IRELAND.
ExAM-LEs.
1. In 901. 10s. 6d. Irish money, how many cents?
11. Irish–110 cas.
£, cts. £. cts. $ cts
Therefore–As I : 410 : : 00,525 : 37.115–371, 1
2. In 168 dols. 10cts, how many pounds Irish?
As 410 cts. : 11. : : 16810 cts. : £41 Irish. Ans.
III.-OF FIRANCE.
Accounts are kept in livres, sols and deniers.
| 12 deniers, or pence, make 1 sol, or shilling.
20 sols, or shillings, - I livre, or pound.
Exa-PL)-5.
1. In 250 livres, 8 sols, how many dollars and cents.
1 livre of France=18, cts, or 185 mills.
£. m. £. -n. & cºs. m.
As I : 185: : 250,4 : 463:24 = 4G 32 4 Ans.
2. Reduce S7 dols. 45 cts. 7 m. into livres of France.
mills. Jin. mills, lin. so. den.
-
As 185 : 1 : : 87,457 : 472 14 9+ Ans.
IV.-OF THE U. NETHERLANDS.
Accounts are kept here in guilders, stivers, groats and
hennings.
S phennings make i groat.
| 2 groats - I stiver.
20 stivers - I guilder or florin.
A guilder isº-39 cents, or 300 mills.
Exº-Ha!--- I-1
ºx---------
Reduce 124 guilders, 14 stivers, into federal money.
Guil, cts. Guil. $ d. c. m.
As I : 39 : : 124,7 : 48, G 3 3 Ans.
mills. G. mills. G.
As 390 : 1 : : 48.633 : 124,7 Proof.
W.-OF HAMBURGH, IN GERMANY.
ºccounts are kept in Hamburgh in marks, sous and de-
re-lºbs, and by some in rix dollars.
12 deniers-lubs make 1 sous-lubs.
| 16 sous-lubs, - 1 mark-lubs.
3 mark-lubs, - I rix dollar.
lºote.--A mark is −33 cts, or just of a dollar.
Autº-Lºvide the marks by 3, the quotient will be dollars.
-----------
Reduce 641 marks, 8 sous, to federal money.
3)641,5
$213,833 Ans.
But to reduce federal money into marks, multiply the
liven sum by 3, &c.
ExA-Lºs.
Reduce 121 dollars, 90 cts. into marks banco.
12,90 -
3
365,70–365 marks, 11 sons, 2,4 den. Ans.
WI.-Or SpAIN.
Accounts are kept in Spain in piastres, rials, and mar-
wadies.
| 34 marvadies of plate make 1 rial of plate.
8 rials of plate - I piastre or piece of 8.
To reducerials of plate to federal money.
Since a rial of prºte is - 10 cents or 1 dime, you need
only call the rials so many dimes, and it is done.
-----------
485 rials-485 dunes--is los. 50 cts. &c.
-12 ºx------.
But to reduce cents into rins of plate, divide by 10,
Thus, 845 cents-10–84,5–84 rials, 17 marvadies, &c.
VII.-O." PORTUGAL.
Accounts are kept throughout this kingdom in my s,
and reas, reckoning 1000 reas to a milrea.
Note.—A milrea is – 124 cents; therefore to r ice
milreas into federal money, multiply by 124, and th pro-
duct will be cents, and decimals of a cent.
---------
1. In 340 milreas how many cents?
340 x 124–42160 cents=$421, 60ct Ins.
2. In 211 milreas, 48 reas, how many cents? -
Note.—When the reas are less than 100, place a cipher
before them.–Thus, 211,048 x 124=26169,952 cu. or 261
dols. 69 cts. 9 mills.--Ans.
But to reduce cents into milreas, divide them by 124;
and if decimals arise you must carry on the quotient as far
as three decimal places; then the whole numbers thereof
will be the milreas, and the decimals will be the reas.
-----------
1. In 4195 cents, how many milreas?
4195-124=33,830+or 33 milreas, 830 reas. Ans.
2. In 24 dols. 92 cents, how many milreas of Portual?
Ans. 20 milreas, 096 reas.
VIII.-EAST-INDIA MONEY.
To reduce India Money to Federal, viz.
Tales of China, multiply with 148
| Pagodas of India, 194
Rupee of Bengal, 55} -
Ex-MPLEs.
1. In 641 Tales of China, how many cents?
Ans. 9.1868
2. In 50 Pagodas of India, how many cents?
Ans. 9700
3. In 98 Rupees of Bengal, how many cents?
Ans. 54:39
Wu L-AR -RACTIONs
WULGAR FRACTIONS.
HAVING briefly introduced Vulgar Fractions imme-
diately after reduction of whole numbers, and given some
general definitions, and a few such problems therein as
were necessary to prepare and lead the scholar immediate-
ly to decimals; the learner is therefore requested to read
those general definitions in page 69.
Vulgar Fractions are either proper, improper, single,
compound, or mixed.
1. A single, simple, or proper fraction, is when the nu-
merator is less than the denominator, as , , 3, 4, &c.
2. An Improper Fraction, is when the numerator ex
*eeds the denominator, as , , ,”, &c.
3. A Compound Fraction, is the fraction of a fraction,
oupled by the woºd of, thus, of ſº, of 3 of , &c.
4. A Mired Number, is composed of a whole number and
a fraction, thus, Sł, 14", &c.
5. Any whole number may be expressed like a fraction
by drawing a line under it, and putting 1 for denominator,
thus, 8–1, and 12 thus, a, &c.
6. The common measure of two or more numbers, is
that number which will divide each of them without a re-
mainder; thus, 3 is the common measure of 12, 24, and 30;
and the greatest number which will do this is called the
greatest common measure.
7. A number, which can be measured by two or more
numbers, is called their common multiple: and if it be the
least number that can be so measured, it is called the leas
common multiple: thus 24 is the common multiple 2, 3 ane
4; but their least common multiple is 12.
To find the least common multiple of two or more num-
bers.
Rule.-1. Divide by any number that will divide two or more of
the given numbers without a remainder, and set the quotients, toge-
her with the undivided numbers, in a line beneath.
2. Divide the second lines as before, and so on till there are no two
numbers that can be divided; then the continued product of the di-
visors and quotients, will give the multiple required.
| 44 nEDUCTION OF vºlcan Faactions.
Ex--------
1. What is the least common multiple of 4,5,6 and 101
Operation, ×5)4-5-6 10
×2)4 1 6 2
x2 1 × 3. 1
5x2x2 x3–60 Ans.
2. What is the common multiple of 6 and 8?
Ans. 24.
3 What is the least number that 3, 5, 8 and 12 wil
measure? Ans. 120.
4. What is the least number that can be divided by the
9 digits separately, without a remainder? Ans. 2520.
REDUCTION OF WULGAR FRACTIONS,
IS the bringing them out of one form into another, in or
der to prepare them for the operation of Addition, Sul,
traction, &c.
CASE I.
To abbreviate or reduce fractions to their lowest terms.
Rule.-1. Find a common measure, by dividing the greater term
by the less, and this divisor by the remainder, and so on, always di-
viding the last divisor by the last remainder, till nothing remains:
the last divisor is the common measure.”
2. Divide both of the terms of the fraction by the common mea-
sure, and the quotients will make the fraction required.
* To find the greatest common measure of more than two numbers, you
must find the greatest common measure of two of them as per rule above;
then, of that common measure and one of the other unmbers, and so on
through all the numbers to the last; then will the greatest common mea
sure lºst found be the answer
REL-U-T-I-M OF --L-A-I- FR-CT-DN-5. 145
ºr, if you choose, you may take that easy method in Problem I.
ºp-gº tºº.)
----------
1. Reduce #: to its lowest terms.
s)+(, Operation.
**): - common measure, 8)+2=4 Ans.
* Rem.
2. Reduce : to its owest terms. Ans. **
3. Reduce Hºº to its lowest terms Ans. Hº
4. Reduce #### to its lowest terms. Ans. }
CASE II.
To reduce a mixed number to its equivalent improper
fraction.
Rule.—Multiply the whole number by the denominator of the gi-
ºn fraction, and to the product add the numerator, this sum written
dove the denominator will form the fraction required
----------
1. Reduce 45 to its equivalent improper fraction
45×8+7="#" Ans.
2. Reduce 1914 to its equivalent improper fraction.
Ans. *
3. Reduce 16,'º', to an improper fraction.
Ans. *
4 Reduce 6144: to its equivalent improper fraction.
Ans. *****
CASE III.
To find the value of an º fraction.
Rºle-Divide the numerator by the denominator, and the quo
went will be the value sought.
EXAMPLES
---------
1. Find the value of * 5)48(9.
2. Find the value of * 1914
3. Find the value of * 84)",
4. Find the value of ***** 6144:
5 Find the value of ºf
145 Re-LUCTION or vuluan Fra A.C.T.u.Ns.
CASE IV.
To reduce a whole number to an equivalent fraction, hav
ing a given denominator.
Rule.—Multiply the whole number by the given denominator
place the product over the said denominator, and it will form the
fraction required.
Ex---L-->.
1. Reduce 7 to a fraction whose denominator will be 9.
Thus, 7×9–63, and * the Ans.
2. Reduce 18 to a fraction whose denominator shall be
12. Ans. **
3. Reduce 100 to its equivalent fraction, having 90 for a
denominator. Ans. ****=***=" tº
CASE W.
To reduce a compound fraction to a simple one of equal
value.
Rule.-1. Reduce all whole and mixed numbers to their equiva
lent fractions.
2. Multiply all the numerators together for a new numerator. and
all the denominators for a new denominator; and they will form *
ſinction required.
Ex-MPLEs.
1. Reduce of 3 of , of ºn to a simple fraction
1 x2 x3 × 4
—=*-* Ans.
2x3 × 4 × 10
2. Reduce 5 of of to a single fraction. Ans. *
3. Reduce of of 18 to a single fraction.
Ans, tºº,
4 Reduce of ; of 8 to a simple fraction.
Ans. *=34
5. Reduce : of 13 of 42+ to a simple fraction.
Ans. *****=21 tº
*** -—If the denominator of any member of a com
* * *ction be equal tº the numerator of another mem.
--Luction or vu-uan FRACTION-- 141
per thereof, they may both be expunged, and the other
members continually multiplied (as by the rule) will pro-
duce the fraction required in lower terms.
6. Reduce 3 of , of to a simple fraction.
Thus 2 × 5
–=#|-º, Ans.
4 × 7
7. Reduce of ; off of 4 to a simple fraction.
Ans. }}=}}
CASE WI.
to reduce fractions of different denominations to equiva
lent fractions having a common denominator.
RULE. I.
1. Reduce all fractions to simple terms.
2. Multiply each numerator into all the denominators except its
own, for a new numerator; and all the denominators into each other
continually for a common denominator; this written under the seve-
ral new numerators will give the fractions required.
----------
1. Reduce {, *, *, to equivalent fractions, having a com-
non-denominator.
4 + 4 + º-24 common denominator.
1. 2 3.
x - 2. º
- - -
3. 4. 9.
× 4 4. 2
12 16 18 new numerators.
- - -
24, 24 24 denominators.
* Reduce i, º, and H, to a common denominator.
Ans. #5, ###, and #4.
3. Reduce # 3, #, and I, to a common denominator.
Ans. Hi, Hi, ###, and 4:3
18 REDUCT-N OF --L--R FRACTIONS.
-4. Reduce 4, ºr, and ſº, to a common denominator
800 300 40
- and
1000 1000 1000
5. Reduce : º, and 1:24, to a common denominator.
Ans.##, ##, º.
6. Reduce : , , and 3 of H, to a common denominator
Ans. *, *, *:::, 14:4.
The foregoing is a general rule for reducing fractions tº
a common denominator; but as it will save much labour to
keep the fractions in the lowest terms possible, the follow-
ing Rule is much preferable.
=º tº and ºt-1 ºr Ans.
- RULE II.
For reducing fractions to the least common denominator.
(By Rule, page 143) find the least common multiple of
all the denominators of the given fractions, and it will ºf
the common denominator required, in which divide encl.
particular denominator, and multiply the quotient by it:
own numerator, for a new numerator, and the new nume
rators being placed over the common denominator, will ex
press the fractions required in their lowest terms.
ExAMPLEs.
1. Reduce 3, , and #, to their least common derovaira”
4)2 4
2)2. 1 2
1 1 4x2=8 the least com. denominator.
8+2×1=4 the 1st numerator.
8-4× 3–6 the 2d numerator.
8-8-2 5–5 the 3d numerator.
These numbers placed over the denominator, give the
answer i, º, º, equal in value, and in much lower terms
than the general Rule would produce #4, #, #.
* Reduce i, º, and ſº to their least common denomina
tor. Ans. **, +, ++.
nED-CTION or wºuld An FRACTions. 149
3 Reduce # 3 and ºn to their least common denomi-
uatº - Ans. It ºr ºf ºf
4. Reduce 4 x + and ſº to their least common denomi-
nation. Ans. Tº Hº Hº tº
CASE VII.
To Reduce the fraction of one denomination to the frac-
tion of another, retaining the same value.
RULE. -
Reduce the given fraction to such a compound one, as
will express the value of the given fraction, by comparing
it with all the denominations between it and that denomi-
nation you would reduce it to ; lastly, reduce this com
round fraction to a single one, by Case V.
----------
1. Reduce of a penny to the fraction of a pound.
By comparing it, it becomes of ºr of ºr of a pound.
5 × 1 × 1 5.
- - Ans.
6 x 12 × 20. 1440
2. Reduce rºw of a pound to the fraction of a penny.
Compared thus rººm of * of ºd.
Then 5 × 20 × 12 -
- - -=}###-
1440 I I
3. Reduce of a farthing to the faction of a smilling.
Ans. ,
4. Reduce 3 of a shilling to the fraction of a ...”
Ams, ºnes'.
5. Reduce 4 of a pwt. to the fraction of a pound troy.
4ns, ºnesis
6. Reduce of a pound avoirdupois to the fraction of
--t. Ans. , is cºot.
7. What part of a pound avoirdupcis is ºr of a cºt.
Compounded thus r ºr of ; of * =#47-# 4's,
8. What part of an hour is ºn ºf a week.
Ans -
- ?
º a-Duction or Wu LGAR PRACTION--
9. Reduce of a pint to the fraction of a hind. Ans, sº
10. Reduce 4 of a pound to the fraction of a guinea.
Compounded thus, of * of ºs- Ans.
11. Express 5A furlongs in the fraction of a mile.
Thus 5–4 of 1–H Ans.
12. Reduce 1 of an English crown, at 6s. 8d. to the frac
tion of a guinea at 28s. Ans, ºr of a guinea.
- CASE VIII.
To find the value of a fraction in the known parts of the
integer, as of coin, weight, measure, &c.
RULE.
Multiply the numerator by the parts in the next inferica
denomination, and divide the product by the denominator:
and if anything remains, multiply it by the next inferior de
nomination, and divide by the denominator as before, and st
on as far as necessary, and the quotient will be the answer
Note:-This and the following Case are the same with
Problems II. and III. pages 70 and 71; but for the scho
lar's exercise, I shall give a few more examples in each.
Ex-MPLEs.
1 What is the value of ill of a pound? Ant & £44.
2. Find the value of , of a cwt. Ans. 5 ºrs, 3.5. 1 oz-12; d.
8. Find the value of 1 of 3s. 5d. Ans. 3s. 0.4.
4. How much is ºr of a pound avoirdupois?
Ans, 7 oz. 10 dr
5. Ilow much is # of a hind, of wine? Ans. 45 gale
6. What is the value of H of a dollar? Ans. 5s. 7d
* what is the value of * of a ruinea 7 Ans. 18.
add-TION or vuluan PRACTION3. 1-1
8. Required the value of lºg of a pound apothecaries.
Ans, 2 oz. 3 grs.
9. How much is of 5l. 9s. 1 Ans. E4 13s. 54a.
10. How much is of 3 of , of a hlid. of wine?
Ans. 15 gals, 37ts
CASE IX.
To reduce any given quantity to the fraction of any greater
denomination of the same kind.
[See the Rule in Problem III. page 71.]
Ex-M-LEs. Fort Ex-It-isº.
1. Reduce 12 lb. 3 oz. to the fraction of a cwt.
Ans, ºs
2. Reduce 13 cwt. 3 qrs. 20 lb. to the fraction of a ton.
Ans. #
3. Reduce 16s. to the fraction of a guinea. Ans. *
4. Reduce 1 hºld. 49 gals of wine to the fraction of a
--- Ans. *
5. What part of 4 cwt. I qr, 24 lb. is 3 cwt. 3 qrs. 171b.
3 oz. 7 Ans. *
-
ADDITION OF WULGAR FRACTIONS.
RULE.
Reduce compound fractions to single ones; mixed num-
bers to improper fractions; and all of them to their least
common denominator, (by Case VI. Rule II.) then the sum
of the numerators written over the common denominator
will be the sum of the fractions required.
Ex-MPLEs.
1. Add 5) + and 5 of 1 together.
5}=", and 3 of 1–4:
Then º', º, ºf reduced to their least common denominator
by Case VI. Rule II. will become ºn H. H.
Then 132+18+14–º-G# or 6: Ans.
15? annittox or vul-Gan Faactions.
2. Add 3, #, and together. answers. 11
3. Add +, +, and # together. - 11
4. Add 1243; and 4 together. 2011
5. Add + of 95 and 1 of 14, together. 44;
Note 1.-In adding mixed numbers that are not con
pounded with other fractions, you may first find the sum on
the fractions, to which add the whole numbers of the given
mixed numbers.
6. Find the sum of 5, 74 and 15.
I find the sum of , and # to be #-1}}
Then 111-5+7+15–28). Ans
7. Add + and 17; together. ANswers. 17”.
8. Add 25, Si and of 3 of tº 33 ºr
Nore 2–To add fractions of money, weight, &c. reduº
fractions of different integers to those of the same.
Or, if you please, you may find the value of each fraction
by Case VIII. in Reduction, and then add them in their
Łroper terms.
9. Add + of a shilling to of a pound.
1st method 2d method.
# of *=1#st. #42-7s. 6d. 0qrs.
Then rig-Hºº-ºººº. is.-0 6 33
Whole value by Case VIII.
is Ss. 0d. 34 qis. Ans. Ans. S 0 33
By Case VIII. Reduction. ;
10. Add 1 lb. Troy, to of a pwt.
Ans. 7 oz. 4 piet. 13, grº.
11. Add + of a ton, to ºr of a cwt.
Ans. 12 cwt. 1 ºr 8 lb. 12* oz.
12. Addº of a mile to ºr of a furlong. Ans.6 fºr 2Spo-
13. Add 3 of a yard, of a foot, and of a mile together
Ans. 1540 yds. 2.ft. 9 in.
14. Add + of a week, of a day, of an hour, and 1 on
a minute together. 4ns º da 2 ha. 30 min. 45 sec.
º
-
s
º
º
-
-
-
--
-º
:
subtraction or vulgan FRActions. 153
SUBTRACTION OF WULGAR FRACTIONS.
RULE.- -
prepare the fraction as in Addition, and the difference
ºf the numerators written above the common denominator,
will give the difference of the fraction required.
Ex-MPLE.S.
1. From take of
. -
5 of i-º-º: Then and nº ºr
Therefore 9–7–4–4 the Ans.
2. From 3; take # Answers. H
3. From 14 ake tº - *
1. From 14 take ºf 13 ºr
5. What is the difference of ºr and #1 słs
6. What differs tº from 41 *
7. From 14, take of 19 lº's
S. From ºf take lº 0 remains.
9. From º of a pound, take of a shilling.
of º-,+,+. Then from H.E. take rººf. Ans. ***.
Nore.—in fractions of money, weight, &c. you may, if
you please, find the value of the given fractions (by Case
VIII. in Reduction) and then subtract them in their proper
Lerºus-
10. From ºf take 3: shillings. Ans. 5s. 5d. 2; ºrs.
11. From of an oz. takeſ of apwt. Ans. 11 put. 3gr.
12. From of a cwt, take ºr of a lb.
Ans. 1 ºr 27 lb. 6 oz. 10, , dr.
13. From 35 weeks, take of a day, and of 4 of , of
an hour. Ans. 3 we 4 da. 12 ho. 19 min. 174 sec.
* In subtracting mixed numbers, when the lower fraction is greater than
the º one, you may, without reducing them to improper fractions, sub-
tract the numerator of the lower fraction from the common denominator,
and to that diſſerence add the upper numerator, carrying one to the unit's
place of the lower whole number.
Also, a fraction may be subtracted from a whole number by taking the
numerator of the fraction from its denominator, and placing the remainder
-re the denominator, then takin- one from the whole number
| 54
Multiplication, division, &c.
MULTIPLICATION OF WULGAR FRACTIONS
Rednee whole and mixed numbers to the improper frac
RULE.
tions, mixed fractions to simple ones, and those of differen:
integers to the same; then multiply all the numerators to:
gether for a new numerator, and all the denominators to
gether for a new denominator.
ExAMPLES.
1. Multiply by a Answers. 14–
2. Multiply by * :
3. Multiply 54 by * º
4. Multiply 3 of 7 by + 3++
5. Multiply #1 by tº #:
6. Multiply of 8 by of 5 13.
7. Multiply 74 by 9. 69
8. Multiply § of by of 34 #:
9. What is the continued product of , of 3, 7, 5) and
of : º Ans. 4's
- -
DIVISION OF VULGAR FRACTIONS.
RULE.
Prepare the fractions as before; then, invert the divison
and proceed exactly as in Multiplication:-The products
will be the quotient required.
ExAMPLEs.
4 x 5
1. Divideº by : Thus, -=#4 Ans.
3 x 7.
2. Divideº by 4 Answers, lºº,
3. Divideº off by *
4. What is the quotient of 17 by 41 594
5. Divide 5 by ºr 7+
6. Divide off of by of . 3.
7. Divide 4: by of 4 2*
8. Divide 71 by 127 *
9. Divide 5205; by + of 91 71.
Rule of thage unnect, inverse, &c. 1:5
RULE OF THREE DIRECT IN WULGAR
FRACTIONS.
RULE.
Prepare the fractions as before, then state your question
agreeable to the Rules already laid down in the Rule of
Three in whole numbers, and invert the first term in the
proportion; then multiply all the three terms continually
together, and the product will be the answer, in the same
name with the second or middle term.
Examples.
1. If of a yard cost of a pound, what will ºr of an Ell
English cost?
Hyd.— off of 1–4 or ; Ell English.
Ell. E. Ell. s. d. ºrs.
As : y :: *, And ºx #x º-ºf-10 3 1? Ans.
2. If of a yard cost of a pound, what will 40 yards
-me to 7 Ans. E59 8s. 64d.
3. If 50 bushels of wheat cost 1731, what is it per bush
*11 Ans. 7s. 0d. 143 ºrs.
4. If a pistareen be worth 144 pence, what are 100 pista.
reens worth 1 - Ans. E6
5. A merchant sold 54 pieces of cloth, each containing
24 yards at 9s. 1d. per yard; what did the whole amount
to 1 Ans. E60 10s. 0d. 34 q’s.
6. A person having of a vessel, sells 3 of his share for
3121. what is the whole vessel worth 7 Ans. E780
7. If I of a ship be worth 3 of her cargo, valued at 8000l.
what is the whole ship and cargo worth?
Ans. E10031 14s. 11 º'rd.
INVERSE PROPORTION.
RULE.
Prepare the fractions and state the question as before.
len invert the third term, and multiply all the three terms
together, the product will be the answer.
155 RULE OF THREE D1-E-T IN ------L-
Ex-MPLE5.
1. How much shalloon that is 3 yard wide, will line 5,
yards of cloth which is 13 yard wide?
Yds. yds. was. Yds.
As 1: ; 54 :: ; And 4 × 2 × 3–5, e164, Ans.
2. If a man perform a journey in 34 days, when the day
is 124 hours long; in how many days will he do it when
the day is but 91 hours? Ans. 4 ºr days.
3. If 13 men in 113 days, mow 21 acres, in how many
days will 8 men do the same? Ans. 18; days,
4. How much in length that is 74 inches broad, will
make a square foot? Ans. 20 inches.
5. If 254s. will pay for the carriage of a cwt. 1451 miles;
how far may 64 cwt. be carried for the same money?
Ans. 22.4% mile.
6. How many yards of baize which is 14 yards whe,
will line 18 yards of camblet yard wide?
Ans. 11 yds. I tr. 11 na.
-
RULE OF THREE DIRECT IN DECIMALS.
RULE.
Reduce your fractious to decimals, and state your ques
tion as in whole numbers; multiply the second and third to-
gether; divide by the first, and the quotient will be the an-
swer, &c.
Exº-M.--L--R-
1. If I of a yard cost ºr of a pound; what will 15; yards
come to 7 1–,875 –,583+ and i-,75
Yals. E. I'ds. E. £. s. d. ºrs.
As,875: ,583 :: 15,75: 10,494=109 10 2.24 Ans
2. If I pint of wine cost 1,2s. what cost 12,5hhds?
Ans. E378
3. If 44 yards cost 3s. 4d. what will 303 yards cost?
Ans, el 4's, 3d, 3 grº.--
simple in TEREST BY DECIMALs. 157
4. If 1,4 cwt. of sugar cost 10 dols, 9 cts., what will 9
ºwt. 3 qrs, cost at the same rate?
cult. s cuit. §
As 1,4: : 10,09 : : 9,75: 70,269–$70,26cts. 9m.--
5. If 19 yards cost 25,75 dollars, what will 4354 yards
come to ? Ans. $590, 21 cts. 7 ºr m.
6. If 345 yards of tape cost 5 dols. 17 cents, 5 m., what
will one yard cost? Ans. ,015–11 cts.
7. If a man lay out 121 dollars 23 cts, in merchandise,
and thereby gains 39.51 dollars, how much will he gain in
laying out 12 dollars at the same rate?
Ans. $3,91=$3, 91 cts.
8. How many yards of riband can I buy for 254 dols. if
* yards cost 4 dollars? Ans. 1784 yards.
. If 1784 yds. cost 25, dollars, what cost 29 yards?
Ans. $4.
10. If 1,6 cwt. of sugar cost 12 dols. 12 cts., what cost 3
whds., each 11 cwt. 3 qºs. 10, 12 lb. ?
Ans. 269,072 dols.-$269, 7 cts. 2 m.--
SIMPLE INTEREST BY DECIMALS.
A TABLE OF RATIOS.
Raº percent.T Rario. TTRare per cent. Ratio.
º ,03 5, 0.55-
4 - ,04 6 ,06
4+ ,045 6) ſº
5 _05 7 07_
Ratio is the simple interest of 11. for one year; or in fe-
deral money, of $1 for one year, at the rate percent, agreed
--- -
RULE.
Multiply the principal, ratio and time continually toge-
ther, and the last product will be the interest required.
ExAMI-LEs.
1. Required the interest of 211 dols, 45 cts, for 5 years
at 5 per cent per annum?
-
158 51-1PLE INTEREST -- DEGI-L-.
4 cts.
211,45 principal.
,05 ratio.
10,5725 interest for one year.
5 multiply by the time.
52,8625 Ans.—352, 86 cts. 24 m.
2. What is the interest of 645. 10s. for 3 years, at 5 per
cent. per annum ?
£645,5x06x3=116,190=E116 3s. 9d. 2,4 qrs. Ans.
3. What is the interest of 1211. 8s. 6d. for 4 years, at
6 per cent. per annum ? Ans. E32 15s. 8d. 1,364rs.
4. What is the amount of 536 dollars, 39 cents, for 11
years, at 6 per cent per annum ! Ans. $584,6651.
5. Required the amount of 648 dollars 50 cents for 12:
years, at 54 per cent. per annum ! Ans. $1103, 26cts.
CASE II.
The amount, time and ratio given, to find the principal.
Rule.-Multiply the ratio by the time, add unity to the product
for a divisor, by which sum divide the amount, and the quotient will
be the principal.
Ex-MPLES.
1. What principal will amount to 1235,975 dollars, in 5
years, at 6 percent. per annum 1 & &
-
,66x5+1=1,30, 1235,975(950,75 Ans.
2. What principal will amount to 873. 19s. in 9 years,
at 6 per cent. per annum? Ans. C567 10s.
3. What principal will amount to $625, 6 cents in 1:
years, at 7 percent." Ans. $340,25–$340, 25cts.
4. What principal will amount to 9561. 10s. 4,125d. in
Sº years, at 5 per cent.” Ans. EG45 15s.
CASE III. -
The amount, principal and time given, to find the ratio.
Ruis-Subtract the principal from the amount, divide the re-
mainder by the product of the time and principal, and the quotient
will be the ratio.
---------
1. At what rate percent will 950,75 dollars amount to
1230,975 dollars in 5 years?
st MPLE INTEREST ºw. DECIMALs. 1 aw
From the amount = 1235,975
Take the principal = 950,75
950,75×5–4753,75)285,2250(,06=6 per cent.
285,2250 Ans.
2. At what rate per cent. will 5671. 10s, amount to 873.
19s, in 9 years? Ans. 6 per cent.
3. At what rate percent will 340 dols. 25 cts amount to
626 dols. 6 cts, in 12 years? Ans. 7 per cent.
4. At what rate per cent will 6451.15s. amount to 956.
10s. 4,125d. in 81 years? Ans. 54 per cent.
case iv.
The amount, principa, and rate per cent, given, to find
the time.
Rule.—Subtract the principal from the amount; divide the re-
mainder by the product of the ratio and principal; and the quotient
will be the time.
------------
1. In what time will 950 dols, 75 cts, amount to 1235
Hollars, 97.5 cents, at 6 per cent per annum ?
From the amount $1235,975
Take the principal 950,75
950,75×06–57,0450).285,2250(5 years, Ans.
285,2250
2. In what time will 5671. 10s, amount to 8731. 19s. at
uper cent per annum? Ans. 9 years.
3. In what time will 340 dols. 25 cts, amount to 626
tols. 6 cts. at 7 per cent per annum? Ans. 12 years.
4. In what time will 6451. 15s, amount to 9561. 10s.
4,125d. at 54 per ct. per annum ? Ans. 8,75–8: wears.
-
To calcut--------nºt-run n---
Rute-Multiply the principal by the given number of days, and
that prºduct by the ratio; divide the last product by 365 (the num-
•er of days in a year) and it will give the interest required.
--------
1. Whatisthe interest of 3601 10s. for 146 days, at 6 prict.”
T
INTEREST BY DECIMALs.
2. What is the interest of 640dols. 60 cts. for 100 days,
SIMPLE.
160
365
Ans. $10, 53cts. +
at 6 per cent, per annum ?
„ … .
_ _ ·
*-+ e+
ſ！<
----|-
gºgº
！ 5 &
§ ¶ ¡ ¿
|-|-
s T.----
----·
-
, ， ， ***
！ ± − ×
----~
|-
----
：|-
<
----
-
|-
4. Required the interest of 481 dollars 75 cents, for
3. Required the interest of 2.
days, at 7 percent. per annum
per cent per annum ?
A TABLE, showing the number of Days from any aay or one
month, to the same day of any other month.
ſºttomi anae loa， orº
|jan||:365
| rºw || 31
mar || 59
Apºl|| 90
};120
June|151
July|181
Aug.) 21:2
sae24:3
0ct. I 27:8
Nov., 304
Dec.|| 334
|jan. № ſ'eb.liſtar.
3:34|| 306
865 837
28;8，5)
50
81||
89|| 61
1201 92
1501 122
is, iš
212， 184
ºg giá
§§246,
3，8|| 375
Apºl.
275
806
884
365
:30
61
91
122
15:
18:3
214
244
May
245
276
304
:3:35
:365
:31
61
92
12:3
15:3
184
214
June|July.
214, 184
215 215
27:3, 243
3041 274
3:34， 304
365|| 3:35
:30|| 865
611 31
91.1 62
1221 92
15:31 12:3
18:31 15:3
Aug.
15:3
122
Sept|0ct.
12:21 92
15:31 12:3
1811 151
2121 182
242 212
278 243
80s, wº
834 804
365|| 3:35
:30865
61|| 81
91|||
Waeſpec.
611 31
9:2, 62;
120 00|
1511 121||
1811 151
ſºlº isºſ
2421 212
278 243
3041 274
384|| 804
365|| 335
301 305
SIMPLE INTERF.S." -- DECIMALS- 16.1
When interest is to be calculated on cash accounts, &c.
where partial payments are made; multiply the several
balances into the days they are at interest, then multiply
the sum of these products by the rate on the dollar, and di-
vide the last product by 365, and you will have the whole
interest due on the account, &c.
Ex-MPLLs.
Lent Peter Trusty, per bill on demand, dated 1st of
June, 1800, 2000 dollars, of which I received back the 19th
of August, 400 dollars; on the 15th of October, 600
dollars; on the 11th of December, 400 dollars; on the
17th of February, 1801, 200 dollars; and on the 1st of
June 400 dollars; how much interest is due on the bill,
reckoning at 6 per cent.”
1800. dols, days, products.
June 1, Principal per bill, 2000 || 79 15S000
August 19, Received in part, 100
-
Balance, 1600 57 01:200
October 15, Received in part, Gt)0
Balance, 1000 || 57 5700ſ)
December 11, Received in part, 400
1801. Balance, 600 6S 40800
February 17, Received in part, 200
Balance, 400
June 1, Rec'd in full of principal, 400
101 11600
- 388000
Then 388600 -
,06 Ratio.
- 3 cºs. m.
365)23316,0003,879 Ans. - 63 87 94.
The following Rule for computing interest on any note,
or obligation, when there are payments in part, or endorse-
tº ents, was established by the Superior Court of the State
of Connecticut, in 1784 o 2
162 SIMPLE INTEREST BY DECIMALs.
RULE.
“Compute the interest to the time of the first payment,
if that be one year or more from the time the interest com-
menced, add it to the principal, and deduct the payment
from the sum total. If there be after payments made,
compute the interest on the balance due to the next pay-
ment, and then deduct the payment as above, and in like
manner from one payment to another, till all the payments
are absorbed; provided the time between one payment and
another be one year or more. But if any payment be made
before one year's interest hath accrued, then compute the
interest on the principal sum due on the obligation for one
-year, add it to the principal, and compute the interest or
the sum paid, from the time it was paid, up to the end of
the year: add it to the sum paid, and deduct that sum from
the principal and interest added as above."
“If any payments be made of a less sum than the interest
arisen at the time of such payment, no interest is to becom
puted but only on the principal sum for any period.”
Kirby's Reports, page 49.
Ex-M-LEs. -
A bond, or note, dated January 4th, 1797, was given for
1000 dollars, interest at 6 per cent, and there were pay
ments endorsed upon it as follows, viz. s
1st payment February 19, 1798, 200
2d payment June 29, 1799, 500
3d payment November 14, 1799, 200
I demand how much remains due on said note the 24th
of December, 1800?
1000,00 dated January 4, 1797.
67,50 interest to February 19, 1798–13) months.
1067,50 amount. [Carried up.]
* If a year does not extend beyond the time of final settlement; but if it
does, then find the amount ºf the principal sum due ºn the obligation, up to
the time of settlement, and likewise find the amount of the sum paid, from the
time it was paid, up to the time of the final settlement, and deduct this
amount from the amount of the principal. But iſ there be several payments
made within the said time, find the amount of the several payments, from
the time they were paid, to the time of settlement, and deduct their amount
from the amount of taenrincina.
simple in TEREST BY DECIMALS- 103
1067,50 amount. [Brought up.
200,00 first payment deducted.
867,50 balance due, Feb. 19, 1798.
70,845 interest to June 29, 1799–16; months.
938,345 amount.
500,000 second payment deducted.
438,345 balance due June 29, 1799.
25.30 interest for one year.
464,645 amount for one year.
269,750 amount of third payment for 74 months."
194,895 balance due June 29, 1800. mo, da.
5,6-7 interest to December 24, 1800. 5 2.5
200,579 balance due on the Note, Dec. 24, 1800.
RULE II.
* tablished by the Courts of Law in Massachusetts for
computing interest on notes, yº. on which partial pay-
ments have been endorsed.
Compute the interest on the principal sum, from the
tin-3 when the interest commenced to the first time when
a pºlyment was made, which exceeds either alone or in con-
junction with the preceding payment (if any) the interest at
that time due: add that interest to the principal, and from
the sum subtract the payment made at that time, together
with the preceding payments (if any) and the remainder
forms a new principal; on which compute ard subtract
the payments as upon the first principal, and proceed in
this manner to the time of final settlement.” -
* -t-.
*250,00third payment with its interest from the time it was paid, up to
9.75 the end of the year, or from Nov. 14, 1799, to June 29, º
which is 7 and 1-2 months.
-59.75 amount.
104 81MPLE INTEREST by DECIMALs. -
Let the foregoing example be solved by this Rule.
A note for 1000 dols, dated Jan. 4, 1797, at 6 per cent.
1st payment February 19, 1798, $200
2d payment June 29, 1799, 500
3d payment November 14, 1799, 260
How much remains due on said note the 24th of Decem
ber, 1800? & cits.
Principal, January 4, 1797, 1000,00
Interest to February 19, 1798, (134 mo.) 67,50
Amount, 1067,50
Paid February 19, 1798, 200,00
Remainder for a new principal, S67,50
Interest to June 29, 1799, (16; no.) 70,84
Amount, 938,34
Paid June 29, 1799, 500,00
Remains for a new principal, 45sº
Interest to November 14, 1799, (4 mo.) 9,86
Amount, 448,20
November 14, 1799, paid 200,00
Remains for a new principal, - 188,20
Interest to December 24, 1800, (13 mo.) 12,70
Balance due on said note, Dec. 24, 1800, 20090
& cºs.
The balance by Rule 1, 200,579
Rule II. 200,990
Difference, 0,411
Another Example in Rule II.
A bond or note, dated February 1, 1800, was given for
500 dollars, interest at 6 per cent. and there were payments
endorsed upon it as follows, viz. 8 rºs.
1st payment May 1, 1800, 40,00
2d payment November 14, 1800 8,00
co-Pou ND INTEREST BY DEL-L-L-LS. 1-5
3d payment April 1, 1801, 12,00
4th payment May 1, 1801, 30,00
How much remains due on said note the 16th of Sep
tº mber, 1801 : 8 cºs.
Principal dated February 1, 1800, 500,00
Interest to May 1, 1800, (3 mo.) 7,50
Amount 507 50
Paid May 1, 1800, a sum exceeding the interest 40,00
New principal, May 1, 1800, 467,50
Interest to May 1, 1801, (1 year,) 28,05
Paid Nov. 4, 1800, a sum less than the
interest then due, 8,00
Paid April 1, 1801, do do. 12,00
Paid May 1, 1801, a sum greater, 30,00
- 50,00
New principal May 1, 1801, 445,55
Interest to Sept. 16, 1801, (44 mo.) 10,92
Balance due on the note, Sept. 16, 1801, $455,57
ſº-The payments being applied according to this Rule,
keep down the interest, and no part of the interest ever
forms a part of the principal carrying interest.
-
COMPOUND INTEREST BY DECIMALS.
Rule.—Multiply the given principal continually by the
amount of one pound, or one dollar, for one year, at the
rate per cent. given, until the number of multiplications are
equal to the given number of years, and the product will
be the amount required.
Or, in Table I, Appendix, find the amount of one dollar,
or one pound, for the given number of years, which multiply
by the given principal, and it will give the amount as before.
166 - invol. UTION.
Ex-MPLEs.
1. What will 400l. amount to in 4 years, at 6 per cent
oer annum, compound interest?
400x1,06 x 1,06x1,06 x 1,06=E504,99-Ho
[E504 19s. 9d. 2,754rs.--Ans.
The same by Table 1.
Tabular amount of £1=1,26247
Multiply by the principal 400
Whole amount=#E504,988.00
2. Required the amount of 425 dols. 75 cts. for 3 years,
at 6 per cent compound interest? Ans. $507,74 cts. +
3. What is the compound interest of 555 dols for 1-
years at 5 per cent.” By Table I. Ans. 543,85 cts. +
4. What will 50 dollars amount to in 20 years, at 6 per
cent compound interest? Ans. $160, 35 cts. Gºm.
INVOLUTION,
IS the multiplying any number with itself, and that pro-
duct by the former multiplier; and so on; and the several
products which arise are called powers.
The number denoting the height of the power, is called
the index or exponent of that power. "
ExAMPLEs.
What is the 5th power of 8?
8 the root or 1st power.
8
6. =2d power, or square.
8
sº 3d power, or cube.
=4th power, or biquadrate.
=
32768 =5th power, or sursolid. An.
--OLUTION OR EXTRACTION O-Roc-T-- 167
what is the square of 17,11 Ans. 292,41
What is the square of,0852 Ans. ,0072:25
What is the cuba of 25,47 Ans. 16387,064
What is the hiquadrate of 12? Ans. 20736
What is the square of 71 Ans. 52*
-
EVOLUTION, OR EXTRACTION OF ROOTS,
WHEN the root of any power is required, the business
of finding it is called the Extraction of the Root.
The root is that number, which by a continued multipli
cation into itself, produces the given power.
Although there is no number but what will produce a
perfect power by involution, yet there are many numbers of
which precise roots can never be determined. But, by the
help of decimals, we can approximate towards the root to
uly assigned degree of exactness.
The roots which approximate are called surd roots, and
hose which are perfectly accurate are called rational roots.
A Table of the Squares and Cubes of the nine digits.
Rºº. III? I 3 || 4 || 5 || 5 || 7 ISIS
Sºuarº. IIITT 9 |IGI25T 35||19||5|| Si
Cº. III's 137 IGITT2531513.131513 Tº
ExTRACTION OF THE SQUARE ROOT.
Any number multiplied into itself produces a square.
To extract the square root, is only to find a number,
which being multiplied into itself shall produce the given
number.
Rule.—1. Distinguish the given number into periods of
two figures each, by putting a point over the place of units,
another over the place of hundreds, and so on; and º
there are decimals, point them in the same manner, from
units towards the right hand; which points show the num-
ber of figures the root will consist of
2. Find the greatest square number in the first, or left
mand period, place the root of it at the right hand of the
- - - -
º -----
-
-
168 Evolution, OR Extnaction of Roots.
given number, (after the manner of a quotient in division,)
for the first figure of the root, and the square number un-
der the period, and subtract it therefrom, and to the re
mainder bring down the next period, for a dividend.
3. Place the double of the root, already found, on the
left hand of the dividend, for a divisor.
4. Place such a figure at the right hand of the divisor,
and also the same figure in the root, as when multiplied
into the whole (increased divisor) the product snail be equal
to, or the next less than the dividend, and it will be the
second figure in the root.
5. Subtract the product from the dividend, and to the
remainder join the next period for a new dividend.
6. Double the figures already found in the root, for a
new divisor, and from these find the next figure in the root
as last directed, and continue the operation in the same
manner till you have brought down all the periods. -
Or, to facilitate the foregoing Rule, when you have
brought down a period, and formed a dividend in order to
find a new figure in the root, you may divide said dividend
(omitting the right hand figure thereof) by double the roof
already found, and the quotient will commonly be the
figures sought, or being made less one or two, will generally
give the next figure in the quotient.
Ex-MP-Es.
1. Required the square root of 141225,64.
1412:25,64(375,8 the root exactly without a remainder;
9 but when the periods belonging to any
- given number are exhausted, and still
67)512 leave a remainder, the operation may
169 be continued at pleasure, by annexing
- periods of ciphers, &c.
745)4325
3725
750S)00064
60064
0 remains.
-
TO
Evolutiºn, on Extraction of Roots. 109.
What is the square root of 12961 answers. 36
Of - 566441 23,8
Of - 5-1900257 2.345
Ot - 3537.2961-1 - titºl
Of - 184,21 13,57+
Of - 97.12,6938097 98,553
Of - 0,45369? ,673+
Ot - 00:2916? ,05:
Of - 45.1 6,708+
EXTRACT THE SQUARE ROOT OF WUL-
GAR FRACTIONS.
RULE.
Reduce the fraction to its lowest terms for this and all
other roots; then
1.
Extract the root of the numerator for a new numera-
or, and the root of the denominator, for a new denominator.
2.
If the fraction be a surd, reduce it to a decimal, and
-tract its root.
Ex-M-Lºs.
1. What is the square root of tº Answers. ;
2. What is the square root of ºr ? º
3. What is the square root of +441
4. What is the square root of 2011 4+
5. What is the square root of 248, 1 15:
SURDS.
6. What is the square root of ºf 1 9128-
7. What is the square root of #1 ,7745+
8. Required the square root of 364 6,0207+
APPLICATION AND USE OF THE SQUARE,
ROOT.
Paontext L-A certain general has an army of 5184
men; how many unust he place in rank and file, to form
them into a square?
-
- 10 Evolution, on Extraction of Roots.
Rult.—Extract the square root of the given number.
vºl.84=72 Ans.
Prop. II. A certain square pavement contains 20736
square stones, all of the same size; I demand how many
are contained in one of its sides? v20.736–144 Ans,
Prop. III. To find a mean proportional between twº
numbers.
Rule.—Multiply the given numbers together and extraº
the square root of the product.
Ex-MPLEs.
What is the mean proportion all between 18 and 72?
7:22, 18–1296, and v. 1296–36 Ans.
Prou. IV. To form any body of soldiers so that they may
be double, triple &c. as many in rank as in file. -
Rule.—Extract the square root of 1-2, 1-3, &c. of the
given number of men, and that will be the number of men
in file, which double, triple, &c. and the product will be the
number in rank.
Ex-MPLEs.
Let 13122 men be so formed, as that the number in rank
may be double the number in file.
13122-2–6561, and v5561=81 in file, and 81 ×2
-162 in rank. -
Pros. W. Admit 10 bhds. of water are discharged
through a leaden pipe of 24 inches in diameter, in a cer-
tain time; I demand what the diameter of another pipe
must be to discharge four times as much water in the same
time.
square by the given proportion, and the square root of the
wroduct is se answer.
2+-ºf n i 2,5×2,5–525 square.
4 given proportion.
Rutº-Square the given diameter, and multiply º
vº,00-5 inch, diam, Ans.
evoluttºx, on extraction of acots. 17.
Pros. WI. The sum of any two numbers, and their prº-
fucts being given, to find each number.
Rule.—From the square of their sum, subtract 4 times their pro-
duct, and extract the square root of the remainder, which will be the
difference of the two numbers; then half ºne said difference added to
half the sum, gives the greater of the two numbers, and the said half
difference subtracted from the half sum, gives the losser number.
-----L-->.
The sum of two numbers is 43, and their product is 442;
what are those two numbers?
The sum of the numb. 43×43-1849 square ºf do.
The product of do. 4428 4-1768 4 times he pro.
Then to the 1 sum of 21,5 [numb.
+and- 4,5 V81–9 diff of the
Greatest n amber, 26,0 4} the diff.
| Answers.
east number, 17,0
-
EXTRACTION OF THE CUBE ROOT-
A cube is any number multiplied by its square.
To extract the cube root, is to find a number, which, be-
ing multiplied into its square, shall produce the given num-
ber.
RULE.
1. Separate the given number into periods of three figures
each, by putting a pºint over the unit figure, and every
third figure from the place of units to the left, and if there
be decimals, to the right.
2. Find the greatest cube in the left hand period, and
place its root in the quotient.
3. Subtract the cube thus found, from the said period,
and to the remainder bring down the next period, calling
this the dividend.
4. Multiply the square of the quotien, by 300, calling it
the divisor.
172 EVOLUTION--R EXTRACTION C F. Roo Is.
5. Seck how often the divisor may be had in the divi
Hend, and place the result in the quotient; then multiply
the divisor by this last quotient figure, placing the product
under the dividend.
G. Multiply the former quotient figure, or figures, by the
square of the last quºtient figure, and that product by 30,
and place the product under the last; then under these two
products place the cube of the last quotient figure, and add
them together, calling their sum the subtrahend.
7. Subtract the subtrahend from the dividend, and to the
remainder bring down the next period for a new dividend;
with which proceed in the same manner, till the whole be
finished.
Note:-If the subtrahend (found by the foregoing rule)
happens to be greater than the dividend, and consequently
cannot be subtracted therefrom, you must make the last
quotient figure one less; with which find a new subtrahend.
(by the rule foregoing,) and so on until you can subtraº
the subtrahend from the dividend.
----------
1. Required the cube root of 18399,744.
18399,744(26,4 Root Ans.
8 -
2×2–18800–1200) ſº first dividend
7200
6 x 6–36 × 2-72 × 30–2160
6x6 × 6– 216
95.76 1st subtrahend.
26x26–676x300–202800)82:37:44, 2d dividend.
S11:200 - -
4×4=16 ׺-415 x 30– 124-0
4x4 ×4= t;4
sº. 2d subtrahene
evolution, on Extnatºtion of Rocts. 173
Nore.—The foregoing example gives a perfect root;
and if, when all the periods are exhausted, there happens
to be a remainder, you may annexperiods of ciphers, and
continue the operation as far as you think it necessary.
Answers
2. What is the cube root of 205379? 59
3. Of 6111257 85
4. Of 41ſt 217361 3.16
5. Of - - 146363,183? 52.7
6. Of - 29,508381? 3,00+
7. Of - S0,7631 4,324-
8. Of - ,1627.713367 ,546
9. Of - ,0006S41341 ,088+
10. Of 122615:3272.321 1968
Rule.-1. Find by trial, a cube near to the given number, and call
it the supposed cube.
2. Then, as twice the supposed cube, added to the given number, is
to twice the given number added to the supposed cube, so is the root
of the supposed cube, to the true root, or an approximation to it.
3. By taking the cube of the root thus found, for the supposed cube,
and repeating the operation, the root will be had to a greater degree
of exactness.
ExAMPLEs.
1. Let it be required to extract the cube root of 2.
Assume 1,3 as the root of the nearest cube; then—1,3×
1,3× 1,3–2,197—supposed cube.
Then, 2,197. 2,000 given number.
2 2
4,394 ºn
2,000 2,197.
As 6,394 : 6,197 : : 1,3 - 1,2500 root.
which is true to the last place of decimals; but might by re-
peating the operation be brought tº greatºr tº actress.
2. What is the cube root of 584; ****
P
- --Lutton, of Extraction of Roots.
3. Required the cube root of 7290011011
Ans. 900,000+
QUESTIONs.
Showing the use of the Cube Root.
1. The statute bushel contains 2150,425 cubic or solid
inches. I demand the side of a cubic box, which shall con-
tain that quantity?
V250,425–12,907 inch. Ans.
Note—The solid contents of similar figures are in pro-
portion to each other, as the cubes of their similar sides or
diameters.
2. If a bullet 3-inches diameter weigh 4 lb. what will a
bullet of the same metal weigh, whose diamº-er is 6 in
ches? -
3x3x3–27 tºx6x6–216. As 27: 4 in. : : 216.
ºlb. Ans. -
3. If a solid globe of silver, of 3-inche. diameter, b.
worth 150 dollars; what is the value of another globe ol
silver, whose diameter is six inches?
3 & 8 × 3–27 6-6 x 6=216, As 27 : 150 :: 216
81.200. Ans.
The side of a cube being given, to find the side of tha
cube which shall be double, triple, &c. in quantity to the
given cube. -
Rule.-Cube your given side, and multiply by the given propor
tion between the given and required cube, and the cube root of th
product will be the side sought.
----------
4. If a cube of silver, whose side is two inches, be worth
20 dollars; I demand the side of a cube of like silver whose
value shall be 8 times as much?
2x2x2–8, and 8×8–64 vº-4 inches. Ans.
5. There is a cubical vessel, whose side is 4 feet; I de
mand the side of another cubical vessel, which shall con
tain 4 times as much
4×4×4=64, and 64x4–256/355-6349+ ft. Ans.
6. A cooper having a cask tº inches long, and 32 in
-
evolution, on Exth Action or Roots. 174
ches at the bung diameter, is ordered to make another cask
of the same shape, but to hold just twice as much; wha
will be the bung diameter and length of the new cask?
10x40x40x2-12-000 then J 128000–50,34-length.
#232:32:2-65536 and wººd-4034 ºung diam.
--
A General Rule for extracting the Roots of all Powers.
RULE.
1. Prepare the given number for extraction, by pointing
off from the unit's place, as the required root directs.
2. Find the first figure of the root by trial, and subtract
is power from the left hand period of the given number.
3. To the remainder bring down the first figure in the
*ext period, and call it the dividend.
4. Involve the root to the next inferior power to that
*hich is given, and multiply it by the number denoting the
ºven power, for a divisor.
5. Find how many times the divisor may be had in the
lividend, and the quotient will be another figure of the
root.
6. Involve the whole root to the given power, and sub-
tract it (always) from as many-periods of the given number
as you have found figures in the root.
7. Bring down the first figure of the next period to the
remainder for a new dividend, to which find a new divisor
as before, and in like manner proceed till the whole be
finished.
Note—When the number to be subtracted is greater
than those periods from which it is to be taken, the last
ruotient figure must be taken less, &c.
Ex-MPLEs.
1. Required the cube root of 135796,744 by the above
general method.
wº Evolution, tº Extraction of Roc rs
185795,445, the re.
125–1st subtrahend.
5)107 dividend.
132651–2d subtrahend.
7803) 31457–2d dividend.
135796744–3d subtrahend.
*
5×5×3–75 first divisor.
51x51 x 51–132651 second subtrahend.
51 x 51 × 3–7803 second divisor.
514 × 51.4 × 514–135796744 3d subtrahen
2. Required the sursolid or 5th root of 6436343.
gigº root.
32
2x2x2x2×5–80)323 dividend.
23x23x23 x 23x 23–6.1363-13 subtrahend.
Nore.—The roots of most powers may be found by th:
square and cube roots only; therefore, when any ever
power is given, the easiest method will be (especially in t
very high power) to extract the square root of it, which re
duces it to half the given-power, then the square root o
that power reduces it to half the same power; and so on
till you come to a square or a cube.
For example: suppose a 12th power be given; the squart
root of that reduces it to a 6th power: and the square roo
of a 6th power to a cube.
Ex-MPLEs.
3. What is the biquadrate, or 4th root of 1998.71733761
Ans. 376.
4. Extract the square, cubed, or 6th root of 12230590
464. Ans. 48.
5. Extract the square guadrate, or 8th root of 72138
95.789838.336. Arts 96.
º
ALLIGATION. 177
ALLIGATION,
is the method of mixing several simples of different qua-
lities, so that the composition may be of a mean or middle
quality: It consists of two kinds, viz. Alligation Medial,
and Alligation Alternate.
ALLIGATION MEDIAL,
Is when the quantities and prices of several things are
given, to find the mean price of the mixture composed
of those materials.
RULE.
As the whole composition : is to the whole value: : so
is any part of the compºsition : to its mean price.
Exº-M'LES.
1. A farmer mixed 15 bushels of rye, at 64 cents a bush-
e, is bushels of Indian corn, at 55 cts, a bushel, and 21
bushels of oats, atº-ets, a bushel; I demand what a
bushel of this mixture is worth?
bu, cts sets. bu. $ cts, bu-
15 at 64–9,60. As 54 - 25,38 : : 1
is 55–9,90 I
21 28–5, SS cts.
- - 54)25,38(,47 Ans.
54 25,38
2. If 20 bushels of wheat at I dol. 35 cts. per bushel
be mixed with 10 bushels of rye at 90 cents per bushel,
what will a bushel of this mixture be worth 2
- Ans. $1,20 cits.
3. A tobacconist mixed 36 lb. of tobacco, at 1s. 6d.
per Ib. 12 lb. at 2s. a pound, with 12 lb. at 1s. 10d. per
lb.; what is the price of a pound of this mixture?
Ans, 1s. 8d.
4. A grocer mixed 2 C. of sugar at 56s. per C. and 1
C. at 43s. per C. and 2 C. at 50s, per C. together; I de-
mand the price of 3 cwt. of this mixture? Ans. E7 13s.
5. A wine merchant mixes 15 gallous of wine at 4s.
ºd. per gallon, with 24 gallous at 6s. 8d. and 20 gallons
at 6s. 8d.: what is a gallon of this composition worth?
4ns. 5s 10d. 24% ars,
178 a-GATION ATL/TERNATE-
6. A grocer hath several sorts of sugar, viz. one sort a
8 dols. per cwt. another sort at 9 dols, per cwt. a third soil
at 10 dols, per cwt. and a fourth sort at 12 dols, per cwt
and he would mix an equal quantity of each together; )
demand the price of 31 cwt. of this mixture?
Aus. $34 12cts. 5 m.
7. A goldsmith melted together 5 lb. of silver bullion.
of 8 oz. fine, 10 lb. of 7 oz, fine, and 15 lb. of 6 oz. fine;
pray what is the quality or fineness of this composition?
Ans. 6 oz. 13pwt. Sgr, fine.
8. Suppose 5 lb. of gold of 22 carats fine, 2 lb. of 21
carats fine, and 1 lb. of alloy be melted together; what iſ
the quality or fineness of this mass? -
Ans. 19 carats fine.
-
ALLIGATION ALTERNATE,
IS the method of finding what quantity of each of the
ingredients whose rates are given, will compose a mixture
of a given rate; so that it is the reverse of Alligation Me.
dial, and may be proved by it.
CASE I.
When the mean rate of the whole mixture, and the rate,
of all the ingredients are given, without any limited quak
tity.
RULE.
1. Place the several rates, or prices of the simples, be:
ing reduced to one denomination, in a column under each
other, and the mean price in the like name, at the left-hand
2. Connect, or link the price of each simple or ingredi
ent, which is less than that of the mean rate, with one on
any number of those, which are greater than the mean
rate, and each greater rate, or price, with one, or any num:
ber of the less.
3. Place the difference, between the mean price (or mix
ture rate) and that of each of the simples, opposite to the
rates with which they are connected.
-L-GAT-DN ALTERNATE. 179
4. Then, if only one difference stands against any rate,
it will be the quantity belonging to that rate, but if there be
several, their sum will be the quantity.
ExAMPLES.
1. A merchant has spices, some at 9d. per lb. some at 1s.
some at 2s. and some at 2s. 6d. per Ib, how much of each
tort must he mix, that he may sell the mixture at 1s. 8d.
per pound?
d. lb. al. d. lb.
9– 10 at 9 9 4. t
- 12 4 12 Gives the a liºlin in s
et) ; 8 24 Answer; or 20, 24.) ) 11 ſ :
30-~ 11-30 30- 8 \ ^
2. A grocer would mix the following qualities of sugar;
iz. at 10 cents, 13 cents, and 16 cents per lb.; what quan-
ity of each sort must be taken to make a mixture worth
º cents per pound?
-Ins. 5 lb. at 10cts. 21.b. at 13 cts, and 2 lb. at 16 cts, per lb.
3. A grocer has two sorts of tea, viz. at 9s. and at 15s.
ºr lb, how must he mix them so as to afford the composi-
ion for 12s. per lb. ?
Ans. He must mir an equal quantity of each sort.
4. A goldsmith would mix gold of 17 carats fine, with
wome of 19, 21, and 24 carats fine, so that the compound
may be 22 carats fine; what quantity of each must he take?
4ns. 2 of each of the first three sorts, and 9 of the last.
5. It is required to mix several sorts of rum, viz. at 5s.
's, and 9s. per gallon, with water at 0 per gallon, toge-
her, so that the mixture may be worth 6s, per gallon; how
nuch of each sort must the mixture consist of?
Ans, 1 gal of rum at 5s., I do. at 7s., 6 do. at 9s, and 3 gals.
water. Or, 3 gals, rum at 5s., 6 do. at 7s., 1 do. at 9s. and
gal. water.
6. A grocer hath several sorts of sugar, viz. one sort at 12
its per lb. another at 11 cts, a third at 9 cts. and a fourth
at 8 cts, per lb.; I demand how much of each sort he must
mix together, that the whole quantity may be afforded at
10 cents per pound?
100 a LT+1-NAT-ox PARTIAL.
lb. cits. lb. cits. lb. cits,
2 at 12 1 at 12 8 at 12
1 at 11 2 at 11. 2 at 11
1st Ans. 1 at 9 2d Aus. 2 at 9 3d Ans. 2 at 9
2 at S 1 at 8 3 at 8
4th Ans, 3 lb. of each sort.”
CASE II.
ALTERNATION PARTIAL,
Or, when one of the ingredients is limited to a certain
quantity, thence to find the several quantities of the rest, in
proportion to the quantity given.
RULF.
Take the differences between each price, and the mear
rate, and place them alternately as in Case 1. Then, as the
difference standing against that simple whose quantity in
given, is to that quantity: so is each of the other differ
ences, severally, to the several quantities required.
Examples.
I. A farmer would mix 10 bushels of wheat, at 70 cents
her bushel, with rye at 4Scts, corn at 36 cts, and barley a
* cts. per bushel, so that a bushel of the composition may
be sold for 38cts.; what quantity of each must be taken?
70--> S stands against the given quan
48 2 [tity
Mean rate, 38 36 10
30- 32
2 : 2; bushels of rye.
As S : 10 : : | : 121 bushels of corn.
32 : 40 bushels of barley.
- These four answers arise from as many various ways of linking the
rates of the ingredients together.
Questions in this rule admitofaninfinite variety of answers: for after the
ntities are ſound from different methods of linking; any other numbers in
the same prºportion betweenthemselves, as the numbers which composeth-
answer, win hkewise satisfy the conditiºns of the question.
al-T-RNATION PARTIAL.
2. How much water must be mixed with 100 gallons of
rum, worth 7s.6d. per gallon, to reduce it to 6s. 8d. per
gallon? Ans. 20 gallons.
3. A farmer would mix 20 bushels of rye, at 65 cents
yer bushel, with barley at 51 cts, and oats at 30 cents per
ushel; how much barley and oats must be mixed with the
20 bushels of rye, that the provender may be worth 41 cts.
per bushel?
Ans. 20 bushels of barley, and 61 ºr bushels of oats.
4. With 95 gallons of rum at 8s. per gallon, I mixed other
ram at 6s. 8d. per gallon, and some water; then I found it
stood me in 6s. 4d. per gallon; I demand how much rum
and how much water I took 2
Ans. 95 gals, rum at 6s. 8d. and 30 gals. water.
CASE III.
when the whole composition is limited to a given quantity.
RULE.
Place the difference between the mean rate, and the se-
veral prices alternately, as in Case I. ; then, As the sum of
the quantities, or difference thus determined, is to the given
quantity, or whole composition: so is the difference of each
rate, to the required quantity of each rate.
ExAMPLEs.
1. A grocer had four sorts of tea, at 1s. 3s.6s. and 10s.
er Ib, the worst would not sell, and the best were too dear;
[. therefore mixed 120 lb. and so much of each sort, as to
well it at 4s per lb.; how much of each sort did he take?
1.-- 6 6 : 60 at 1
4. #) 2 lb. lh. 2 : 20 – 3 Ib
6 J as 12 : 120 : :º) i : 10 – 6 ſº ".
10- 3 3 : 30-10
sum, ſº 120
182 -º-T-I-T-P-R------.
2. How much water at 0 per gallon, must be mixed with
wine at 90 cents per gallon, so as to fill a vessel of 100 gal
lons, which may be afforded at 60 cents per gallon 7
Ans. 33 gals, water, and 66% gals. wine.
3. A grocer having sugars at Scts. 16 cts, and 24 cts
per pound, would make a composition of 240 lb. worth 20
cts. per lb. without gain or loss; what quantity of each must
be taken?
Ans. 40 lb. at Scts, 40 lb. at 16 cts, and 160 lb. at 24 cts.
4. A goldsmith had two sorts of silver bullion, one of
10 oz. and the other of 5 oz. fine, and has a mind to mix
a pound of it so that it shall be 8 oz. fine; how much of
each sort must he take 7
Ans. 44 of 5 oz. fine, and 71 of 10 oz. fine.
5. Brandy at 3s.6d. and 5s. 9d per gallon, is to be mixed,
so that a hlid. of 63 gallons may be sold for 121. 12s. ; how
many gallons must be taken of each?
Ans. 14 gals, at 5s. 9d. and 49 gals, at 3s.6d.
- - - - - - - --
ARITHMETICAL PROGRESSION.
ANY rank of numbers more than two, increasing by
common excess, or decreasing by common difference, is
said to be in Arithmetical Progression.
So | 2,4,5,8, &c. is an ascending arithmetical series:
8,5,4,2, &c. is a descending arithmetieal series:
The numbers which form the series, are called the terms
of the progression; the first and last terms of which are
called the extremes.”
*ROHLEMI 1.
The first term, the last term, and the number of terms
being given, to find the sum of all the terms.
-
- A series in progression includes five parts, viz. the first term, last term.
number of terms, common difference, and sum of the series.
By having any three of these parts given, the other two º: ſound
which admits of a variety of Problems; but most of them are under-
stoºd by analgebraic process, and are here omº-d-
anitri-ET--AL PROGREssion |t.
Rule.—Multiply the sum of the extremes by the number
eams, and half the product will be the answer.
----------
1. The urst term of an arithmetical series is 3, the las,
term 23, and the number of terms 11 : required the sum of
the series.
23-3–26 sum of the extremes.
Then 26 x 11-2=1.43 the Answer.
2. How many strokes does the hammer of a clock strike
in 12 hours. Ans. 78.
3. A merchant sold 100 yards of cloth, viz. the first
ard for 1 ct the second for 2 cts, the third for 3 cts. &c.
demand what the cloth came to at that rate 1
Ans. $50.
4. A man bought 19 yards of linen in arithmetical
progression, for the first yard he gave Is. and for the last
wd. 11, 17s. what did the whole come toº
- An EIS is.
5. A draper sold 100 yards of broadcloth, at 5 cts, for
the first yard, 10cts. for the second, 15 for the third, &c.
increasing 5 cents for every yard; what did the whole
amount to, and what did it average per yard 7
Ans. Amount $2524, and the average price is sº, 52 cts.
5 mills per yard.
6. Suppose 144 oranges were laid 2 yards distant from
each other, in a right line, and a basket placed two yards
from the first orange, what length of ground will that boy
travel over, who gathers them up singly, returning with
them cue by one to the basket?
Ans. 23 miles, 5 furlºngs, 180 yds.
PI-OELEM II.
The first term, the last term, and the number of terms given,
to find the common difference.
Rule.—divide the difference of the extremes by the number
aſ terms less 1, and the quotiºn will be the common difference,
184 an ITH-METICAL PR-GRESSION-
Ex-MPLEs.
1. The extremes are 3 and 29, and the number of tenna
14, what is the common difference?
29
3. } Extremes.
Number of terms less 1=13)25(2 Ans.
2. A man had 9 sons, whose several ages differed alike,
the youngest was three years old, and the oldest 35; what
was the common difference of their ages?
º
Ans. 4 years.
3. A man is to travel from New-London to a certain
place in 9 days, and to go but 3 miles the first day, increa-
sing every day by an equal excess, so that the last day's
journey may be 43 miles: Required the daily increase,
and the length of the whole journey :
Ans. The daily increase is 5, and the whole journey 207
miles.
4. A debt is to be discharged at 16 different payment,
(in arithmetical progression,) the first payment is to be 14t.
the last 100l. : What is the common difference, and that
sum of the whole debt?
Ans. 5.14s. 8d, common difference, and 912. the whol.
debt.
PROBLEM III.
Given the first term, last term, and common difference, tº
find the number of terms.
Ruur-Divide the difference of the extremes by the common
difference, and the quotientincreased by 1 is the number of terms
ExA-PLEs. *
1. If the extremes be 3 and 45, and the common differ
ence 2; what is the number of terms? Ans. 22.
2. A man going a journey, travelled the first day five
miles, the last day 45 miles, and each day increased his
journey by 4 miles; how many days did he travel, and
how far?
Ans. 11 days, and the whole distance travelled 275 miles
cºunt-Trutal rºot-RESSION. le
GEOMETRICAL PROGRESSION,
is when any rank or series of numbers increase by one
common multiplier, or decrease by one common divisor
as, 1, 2, 4, 8, 16, &c. increase by the multiplier 2; and 27,
9, 3, 1, decrease by the divisor 3.
PROLLEM I.
The first term, the last term (or the extremes) and the ra-
tio given, to find the sum of the series
RULE.
Multiply the last term by the ratio, and from the pro-
duct subtract the first term; then divide the remainder by
the ratio, less by 1, and the quotient will be the sum of all
the terms. -
-----------
1. If the series be 2, 6, 18, 54, 162,486, 1458, and the
natio 3, what is its sum total!
3× 1-15S.–2
- 3–1
2. The extremes of a geometrical series are 1 and 65536,
and the ratio 4 ; what is the sum of the series?
Ans. 87:281.
=21st; the Answer.
l'Ivoir LEM1 11,
Given the first term, and the ratio, to find any other term
assigned."
CASE 1.
When the first term of the series and the ratio are equal."
-
* As the last term in a long series of numbers is very tedious to be ſound
by continual multiplications, it will be necessary for the readier nºting it out,
to have a series of number-inºrithmetical proportion, called indices, whose
commun diºrence is 1.
* When the first term of the series and the ratio are equal, the indices
must begin with the unit, and in this case, the product of any two terms is
equal to that term, signified by the sum º inutees:
a º
º GEOMETRICAL PROGilession.
1. Write down a few of the leading terms of the serien
and place their indices over them, beginning the indical
with a unit or 1.
2. Add together such indices, whose sum shall make ul
the entire index to the sum required.
3. Multiply the terms of the geometrical series belonging
to those indices together, and the product will be the term
sought.
----------
1. If the first be 2, and the ratio 2; what is the 13th
term?
1, 2, 3, 4, 5, indices. Then 5+5+3=13.
2, 4, 8, 16, 32, leading terms. 32×32XS-8192 Ans.
2. A draper sold 20 yards of superfine cloth, the first
yard for 3d., the second for 9d., the third for 27d., &c. in
triple proportion geometrical; what did the cloth come to
at that rate 7
The 20th, or last term, is 3480784401 d.
Then 3+3186784401–3
––5230170600d, the sum of all
3–1
the terms (by Prob. I.) equal to £21702402, 10s.
3. A rich miser thought 20 guineas a price too much for
12 fine horses, but agreed to give 4 cts, for the first, 16 cts.
for the second, and 64 cents for the third horse, and so
on in quadruple or fourfold proportion to the last: what
did they come to at that rate, and how much did they cost
per head one with another? -
Ans. The 12 horses came to $223696, 20 cts., and the
average price was $18641, 35 cts. per head.
Thus {} 2 3 4 5, &c. indices or arithmetical series
* \ 24 8 1632, &c. geometrical series.
Now 3+2 = 5 = the index of the fifth term, and
* 4x8 = 82 – the fifth term.
-
º
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G-O-ETRICAL PROGREssion. 187
CASE 11.
When the first term of the series and the ratio are diffe-
rent, that is, when the first term is either greater or less
than the ratio."
1. Write down a few of the leading terms of the series,
and begin the indices with a cipher: Thus, 0, 1, 2, 3, &c.
2. Add together the most convenient indices to make an
index less by 1 than the number expressing the place of the
terms sought.
3. Multiply the terms of the geometrical series together
belonging to those indices, and make the product a dividend
4. Raise the first term to a power whose index is one
less than the number of the terms multiplied, and make the
result a divisor.
5. Divide, and the quotient is the term sought.
Ex-MPLEs.
4. If the first of a geometrical series be 4, and the ratio
1, what is the 7th term 1
0, 1, 2, 3, Indices,
4, 12, 36, 108, leading terms. -
3+2+1–6, the index of the 7th term. -
108 x 3.5 x 12=46655
––2916 the 7th term required.
16
Here the number of terms multiplied are three; there-
fore the first term raised to a power less than three, is the
2d power or square of 4–16 the divisor.
* When the first term of the series and the ratio are different, their lices
must begin with a cipher, and the sum ºf the indices made choice of must
be one less than the number of terms given in the question: because 1 in
the indices stand-over the second term, and 2 in the indices over the third
term, &c. and in this case, the product of any two terms, divided by the first
is equal to that term beyond the first, signified by the sum of their indices.
Thus | 0, 1, 2, 3, 4, &c. Indices.
* - 1, 3, 9, 27, S1, &c. Geometrical series.
Here 4+3–7 the index of the 8th term.
81 × 27–21.87 the 8th term, ºr the 7th beyond the 1st.
-88 Position.
5. A Goldsmith sold 1 lb. of gold, at 2 cts, for the first
ounce, 8 cents for the second, 32 cents for the third, &c.in
a quadruple proportion geometrically: what did the whole
come to ? Ans. $11 IS-18, 10 cºs.
6. What debt can be discharged in a year, by paying 1
farthing the first month, 10 farthings, or (2}d) the second
and so on, each month in a tenfold proportion?
Ans. E115740740-14s 9d. 3 qrs.
7. A thrasher worked 20 days for a farmer, and received
for the first days work four barley-corns, for the second 12
barley corns, for the third 36 barley corns, and so on, in
triple proportion geometrically. I demand what the 20
day's labour came to supposing a pint of barley to contain
7680 corns, and the whole quantity to be sold at 2s. 6d. per
bushelf Ans. E1773 7s 6d. rejecting remainders
8. A man bought a horse, and by agreement, was to
ive a farthing for the first nail, two for the second, foul
É. the third, &c. There were four shoes, and eight nails in
each shoe; what did the horse come to at that rate 7
Ans. E44739:24 5s 3d
9. Suppose a certain body, put in motion, should mºve
the length of 1 barley-corn the first second ºf time, ºne
inch the second, and three inches the third -cond of time,
and so continue to increase its motion in triple proportion
geometrical; how many yards would the said body move
in the term of half a minute.
Ans. 953199.685623 yds, 1 ft. 1 in. Ib, which is no less
than five hundred and forty-one millions of miles.
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POSITION.
POSITION is a rule which, by false or supposed num
bers, taken at pleasure, discovers the true ones required.-
It is divided into two parts, Single or Double.
SINGLE, I-OSITION
IS when one number is required, the properties of which
are given in the question.
S-N-LE 189
Rule.—1. Take any number and perform ºne same operation
with it, as is described to be performed in the question.
Then say; as the result of the operation : is to the given
sum in the question : : so is the supposed number: to the true
one required.
The method of proof is by substituting the answer in the ques
tion.
Exº-M.--LEs.
1. A schoolmaster being asked how many scholars he
had, said, If I had as many more as I now have, half as
many, one-third, and one fourth as many, I should ther
bave 148; How many scholars had he
Suppose he had 12. As 37 : 148 : : 12 : 48 Ans.
as many = 12 48
as many = 8 24
# as many = 4 16
: as many – 3 12
Result, 37 Proof, 148
2. What number is that which being increased by 4, ,
and of itself, the sum will be 125? Ans. 60.
3. Divide 93 dollars between A, B and C, so that B's
share may be half as much as A's, and C's share three times
as much as B's.
Ans. A's share $31, B's $154, and C's $464.
4. A, B and C, joined their stock and gained 360 dols.
of which A took up a certain sum, B took 3 times as much
as A, and C took up as much as A and B both; what share
of the gain had each?
Ans. A $40, 13 $140, and C $180.
5. Delivered to a banker a certain sum of money, to re-
ceive interest for the same at til, per cent per annum, sim-
ple interest, and at the end of twelve years received 7311.
principal and interest together; what was the sum deliver-
ed to him at first 7 - Ans. E425.
6. A vessel has 3 cocks, A, B and C : A can fill it in 1
hour, B in 2 hours, and C in 4 hours; in what time will
they all fill it together? Ans 34 min. 174 sec.
90 Douei.e. Positrox.
DOUBLE POSITION,
TEACHES to resolve questions by making two suppo
sitions of false numbers.”
IRULL.
1. Take any two convenient numbers, and proceed with
each according to the conditions ºf he question.
2. Find how much the results are different from the re.
sults in the question.
3. Multiply the first position by the last error, and the las:
position by the first error.
4. If the errors are alike, divide the difference of the pro-
ducts by the difference of the errors, and the quotient will
be the answer.
5. If the errors are unlike, divide the sum of the prº-
ducts by the sum of the errors, and the quotient will be
the answer.
Note:-The errors are said to be alike when they are
both too great, or both too small; and unlike, when one
is too great, and the other too small.
Ex-Mi-Es.
1. A purse of 100 dollars is to be divided among 4 men
A, B, C and D, so that B may have four dollars more that
A, and C S dollars more than B, and D twice as many aſ
C; what is each one's share of the money?
1st. Suppose A 6 2d. Suppose A 8
B 10 B 12
C 18 C 20
* * * * *
70 So -
100 100
1st error, so 2d error, 20
* Those questions in which the results are nºt prºportional to their post
tions, belong tº this rule; such as those in which the number sought is it. .
creased or diminished by some given number, which is no known part of tº |
number required. /
Dourº-Lº. Pos-TION-- in
The errors being alike, are both too small, therefore,
Pos. Err
5 30 s
ſº 12
B 16
t; 24
D 48
8 20 Proof 100
240 120
120
10)120(12 A's part. -
2. A, B, and C, built a house which cost 500 dollars, of
which A paid a certain sum : B paid 10 dollars more than
A, and C paid as much as A and B both ; how much did
each man pay
ºns. -1 paid $120, BS130, and C$250.
3. A man bequeathed 1001 to three of his friends, after
this manner; the first must have a certain portion, the se-
cond must have twice as much as the first, wanting 81 and
the third must have three times as much as the first, want-
ing 151. I demand how much each man must have?
ºns. The first £20 10s. second 533, third, £46 10s.
4. A labourer was hired for 60 days upon this condition;
that for every day he wrought he should receive 4s. and for
every day he was idle should forfeit 2s. ; at the expiration
of the time he received 71 10s. ; how many days did he
work, and how many was he idle?
ins. He wrought 45 days, and was idle 15 days.
5. What number is that which being increased by its 1,
ls º and 18 more, will be doubled 2 1ns, 72.
A man gave to his three sons all his estate in money,
viz. to F halſ, wanting 501 to G one-third, and to H the
test, which was 101. less than the share of G: I demand
ºne sum given, and each man's part 1
ºns, the sum given was £360, whereof F had £130,
G+120, and H+110.
102 PERMUTATION OF QUANTITLEs.
7. Two men, A and B, lay out equal sums of money in
trade: A gains 1261 and B loses 871 and A's money iſ
now double to B's; what did each lay out?
- Ans. E300.
8. A farmer having driven his cattle to market, received
for them all 1301, being paid for every ox 71 for every cow
5l. and for every calf # 10s, there were twice as many
cows as oxen, and three times as many calves as cows:
how many were there of each sort?
Ans. 5 ozºn, 10 cows, and 30 calves.
9. A, B, and C, playing at cards, staked 324 crowns:
but disputing about tricks, each man took as many as he
could; A got a certain number; B as many as A and 15
more; C got a 5th part of both their sums added together;
how many did each get?
Ana. A got 1274, B142), C 54.
PERMUTATION OF QUANTITIES,
IS the showing how many different ways any given num-
ber of things may be changed.
To find the number of Permutations, or changes, that
can be made of any given number of things all different
from each other.
Rule-Multiply all the terms of the natural series of number-
from one up to the given number, continually together, and the ºx-
product will be the answer required
Ex-MPLEs.
1. How many changes can be 1 a b c
made of the first three letters of 2 a c b
the alphabet? Proof, º .
5 c h a
1 x2 x3–6. Ans. tº cab
2. How many changes may be rung on 9 bells?
Ans, 362880.
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ANNU-T-1Es un PENSIONs. 103
3. Seven gentlemen met at an inn, and were so well
pleased with their host, and with each other, that they
agreed to tarry so long as they, together with their host,
cºuld sit every day in a different position at dinner; how
long must they have staid at said inn to have fulfilled their
agreement? Ans. 110}º years.
ANNuities or pensions,
co-LTED A.T
Co-ºpou-wºn 1 NºTEREST.
CASE I.
Te find the amount of an Annuity, or Pension, in arrears,
at Compound Interest.
RULE.
1. Make 1 the first term of a geometrical progression,
and the amount of $1 or £1 for one year, at the given rate
per cent, the ratio.
2. Carry on the series up to as many terms as the given
number of years, and find its sum.
3. Multiply the sum thus found, by the given annuity,
and the product will be the amount sought.
Ex-MPLEs.
1. If 125 dols. yearly rent, or annuity, be forborne (or
anpaid)4 years; what will it amount to at 6 per cent. per
annum, compound interest?
1+1,06+1,1236+1,191016–4,374616, sum of the se-
ries.” Then, 4,374616X 125–$546,827, the amount
sought.
OR BY TABLE II.
Multiply the Tabular number under the rate, and oppo-
site to the time, by the annuity, and the product will be
the amount sought.
* The sum of the seriesthus found, is the amount of 11...or 1 dollar an-
º for the given time, which may be found in Table II. ready calcula-
ñense, either the amºunt or present worth of annuities may be readih
ſound by ta-les for that purpose.
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|- axx UITLES OR PENSI-Ns.
2. If a salary of 60 dollars per annum to be paid yearly
be forborne twenty years, at 6 per cent compound interest
what is the amount?
Under 6 per cent. and opposite 20, in Table II., you
will find,
Tabular number–36,78559
60 Annuity.
Ans, sºo-sºo, 13 cºs. 5m.4
3. Suppose an annuity of 100l be 12 years in arrears, it iſ
required to find what is now due, compound interest beint
allowed at 51 per cent per annum ?
Ans. £1591. 14s. 3,024d. (by Table II.)
4. What will a pension of 120l. per annum, payable
yearly, amount to in 3 years, at 5l. per cent compound in-
terest? Ans, cºsts.
II. To find the present worth of annuities at Compound In-
terest.
RULE.
Divide the annuity, &c. by that power of the ratio sig
nified by the number of years, and subtract the quotient
from the annuity: This remainder being divided by the ra
tio less 1, the quotient will be the present value of the an
nuity sought.
Ex-MPL-5.
1. What ready money will purchase an annuity of 50l.
to :ontinue 4 years, at 51 per cent compound interest?
4th power º =1,215506)50,000.00(41,135.13+
the ratio,
From 50
Subtract 41,13513
º-s, " ºr -ºsó487
T,297 C177 5s. 111d. Ans.
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annuities on Pensions. 19-
BY TABLE III.
Under 5 per cent. and even with 4 years,
We have 3,54595–present worth of 11 for 4 years.
Multiply by 50=Annuity.
Ans, £177,29750=present worth of the annuity.
2. What is the present worth of an annuity of 60 dols
per annum, to continue 20 years, at 6 per cent. compound
interest? Ans. $688, 194 cts.--
3. what is 30l. per annum, to continue 7 years, worth in
ready money, at 6 per cent compound interest?
Ans. £167 9s. 5d.--
III. To find the present worth of Annuities, Leases, &c. ta-
ken in Reversion at Compound Interest.
1. Divide the annuity by that power of the ratio denoted
by the time of its continuance.
2. Subtract the quotient from the annuity: Divide the
remainder by the ratio less I, and the quotient will be the
present worth to commence immediately.
3. Divide this quotient by that power of the ratio deno-
tººd by the time of Reversion, (or the time to come before
tle annuity commences) and the quotient will be the pre-
sent worth of the annuity in Reversion.
Ex-MPLEs.
1. What ready money will purchase an annuity of 50l.
payable yearly, for 4 years; but not to commence till two
years, at 5 percent.”
4th power of 1,05=1,215.506)50,000.00(41,13513
Subtract the quotient=41,13513
Divide by 1,05–1=,05)8,86487
2d power of 1,05=1,1025)177,297(160,8136=#160
16s. 8d. I q. present worth of the annuity in reversion.
OR BY TABLE III.
Find the present value of 11 at the given rate for the sum
of the time of continuance, and time in reversion added to-
gether; from which value subtract the present worth of 11.
for the time in reversion, and multiply the remainder by the
annuity; the product will be the answer.
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195 -NNU-T-I-5 OTL P-------.
Thus in Example 1
Time of continuance, 4 years.
Ditto of reversion, 2
The sum, =6 years, gives 5,075692
Time in reversion, –2 years, 1,859.410
Remaintler, 3,216282x50
Ans. E160,8141.
2. What is the present worth of 75l. yearly rent, which
is not to commence until 10 years hence, and then to con-
tinue 7 years after that time at 6 per cent.”
Ans. E233 15s. 9d.
1. What is the present worth of the reversion of a lease
of 60 dollars per annum, to continue 20 years, but not tº
commence till the end of 8 years, allowing 6 per cent tº
the purchaser? Ans. $431, 78cts. 2 ºm.
IV. To find the present worth of a Freehold Estate, or a
Annuity to continue forever, at Compound Interest.
RULE. -
As the rate per cent, is to 100l. : so is the yearly rent tº
the value required.
Ex-MPLEs. -
1. What is the worth of a freehold estate of 401 per an
num, allowing 5 per cent to the purchaser?
As £5 : £100 : : £40 : £800 Ans.
2. An estate brings in yearly 150l. what would it sell for
allowing the purchaser 6 per cent for his money?
Ans. E2500.
W. To find the present worth of a Freehold Estate, in Re-
version, at Compound Interest.
Rule.-1. Find the present value of the estate (by the foregoing
rule) as though it were to be entered on immediately, and divide the
said value by that power of the ratio denoted by the time of rever.
sion. and the quotient will be the present worth of the estate in re-
--------
Ex-MPLEs.
1. Suppose a freehold estate of 401 per annum to com
mence two years hence, he put on sale; what is its value,
allowing the purchaser 51 per cent.”
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Questions FOR Ex-Reise. lºn
As 5 : 100 : : 40 : 800–present worth if entered on
*mmediately.
Then, 1,05–1,1025).s00,00025,62858-725, 12.
544.—present worth of £800 in two years reversion. Ans.
OR BY TABLE III.
Find the present worth of the annuity, or rent, for the
time of reversion, which subtract from the value of the im-
mediate possession, and you will have the value of the es-
tate in reversion. -
Thus in the foregoing example,
1,859.410=present worth of º for 2 years.
40=annuity or rent.
74,376.400–present worth of the annuity or rent, for
[the time of reversion.
From 800,0000-value of immediate possession.
Take 74,3764–present worth of rent.
£725,6236-8725 12s. 5d. Ans.
2. Suppose an estate of 90 dollars per annum, to com-
mence 10 years hence, were to be sold, allowing the pur-
chaser 6 per cent. ; what is the worth 1
Ans. $837, 59cts. 2 m.
3. Which is the most advantageous, a term of 15 years,
in an estate of 100l. per annum; or the reversion of such
an estate forever after the said 15 years, computing at the
rate of 5 per cent per annum, compound interest?
Ans. The first term of 15 years is better than the rever-
sion forever afterwards, by £75 18s. 7] d.
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A COLLECTION OF QUESTIONS TO EXERCISE
THE FOREGOING RULES.
1. I demand the sum of 1748; added to itself?
Ans. 3-107.
2. What is the difference between 41 eagles, and 4099
dimes? Ans. 10 cts.
3. What number is that which being multiplied by 51
the product will be 1365?
a 2
193 Questions ron exercise.
4. What number is that which being divided by 19, the
quotient will be 72? Ans. 1368.
5. What number is that which being multiplied by 15,
the product will be #1 Ans. *
6. There are 7 chests of drawers, in each of which there
are 18 drawers, and in each of these there are six divisions,
in each of which is 161. 6s. 8d. ; how much money is there
in the whole? Ans. E1.2348.
7. Bought 36 pipes of wine for 4536 dollars; how must
sell it a pipe to save one for my own use, and sell the rest
for what the whºle cost? Ans. $129, 60 cts
S. Just 16 yards of German serge,
For 90 dimes had Î
How many yards of that same cloth
Will 14 eagles buy? Ans. 248 yds. 3 ºrs. 23 na.
9. A certain quantity of pasture will last 953 sheep 7
weeks, how many must be turned out that it will last the
remainder 9 weeks 1 Ans. 214.
10. A grocer bought an equal quantity of sugar, tea, and
coffee, for 740 dollars; he gave 10 cents per lb. for the su
gar, 60 cts, per lb. for the tea, and 20 cts. per lb. for the
coffee; required the quantity of each?
Ans. S22 lb. 3 oz. Sº dr.
11. Bought cloth at $1; a yard, and lost 25 per cent.
how was it sold a yard? Ans. 93; cfs.
12. The third part of an army was killed, the fourth par
taken prisoners, and 1000 fled; how many were in this ar.
my, how many killed, and how many captives?
Ans. 2400 in the army, 800 killed, and
600 taken prisoners.
13. Thomas sold 150 pine apples at 33A cents apiece, and
received as much money as Harry received for a certain
number of water-melons, which he sold at 25 cents apiece;
how much money did each receive, and how many melons
had Harry? Ans. Each rec'd $50, and Harry sold 200 melons.
14. Said John to Dick, my purse and money are worth
9.2s., but the money is twenty-five times as much as the
purse; I demand how much money was in it?
Ans, £8 15.
questions Fon exercise. 199
15. A young man received 210t, which was 4 of his el
ier brother's portion; now three times the elder brother's
portion was half the father's estate; what was the value of
he estate? Ans. E1890.
16. A hare starts 40 yards before a grey-hound, and is
not perceived by him till she has been up 40 seconds; she
ucuds away at the rate of ten miles an hour, and the dog,
or view, makes after her at the rate of 18 miles an hour:
How long will the course hold and what space will be run
over from the spot where the dog started?
Ans. 60 ºr sec. and 530 yds. space.
17. What number multiplied by 57 will produce just
what 134 multiplied by 71 will do? Ans. 16644.
18. There are two numbers whose product is 1610, the
greater is given 46; I demand the sum of their squares,
and the cube of their difference?
Ans, the sum of their squares is 3341. The cube of
heir difference is 1331.
19. Suppose there is a mast erected, so that of its
ength stands in the ground, 12 feet of it in the water, and
of its length in the air, or above water; I demand the
whole length 1 Ans. 216 feet.
20. What difference is there between the interest of 500l.
at 5 per cent for 12 years, and the discount of the same
sum at the same rate, and for the same time?
Ans. E1.12 10s.
21. A stationer sold quills at 11s. per thousand, by which
he cleared of the money, but growing scarce raised them
to 13s. 6d. per thousand; what might he clear per cent.
by the latter price? Ans. E967s. 8", d.
22 Three persons purchase a West-India sloop, towards
the payment of which A advanced 4, B #, and C 140l.
How much paid A and B, and what part of the vessel
had Cº.
Ans. A paid £267 ºr, B E305,ºr, and Cº's part of the
ressel was H.
23. What is the purchase of 12001 bank stock, at 103;
per cent.” Ans. E1243 10s.
24. Bought 27 pieces of Nankeens, each 11 yards, a
--
200 questions. For exercise.
14s. 4d. a piece, which were sold at 18d, a yard; required
the prime cost, what it sold for, and the gain.
£. s. d.
Prime cost, 19 8 1.
Ans. | Sold for, 23 5 9
Gain, 3 17 7.
25. Three partners, A, B and C, join their stock, and
buy goods to the amount of £1025,5; of which A put in
a certain sum; B put in...I know not how much, and C
the rest; they gained at the rate of 241 percent.: A's part
of the gain is , B's 1, and C's the rest. Required each
man's particular stock.
4's stock was 512,75
Ans. |; – 205,1
O's – 307,65
26. What is that number which being divided by 3, the
quotient will be 21? Ars. 157.
27. If to my age there added be,
One-half, one-third, and three times three,
Six score and ten the sum will be:
What is my age, pray show it me? Ans, 66.
28. A gentleman divided his fortune among his three
sons, giving A 91, as often as B 51 and to C but 31, as often
as B 71, and yet C’s dividend was 25841. what did the
whole estate amount toº Ans, £19166-2s. 8d.
29. A gentleman left his son a fortune, of which he
spent in three months; of the remainder lasted him 10
months longer, when he had only 2524 dollars left; pray
what did his father bequeath him 1 Ans. $5889, 33rts. +
30. In an orchard of fruit trees, of them bear apples, º
+ pears, plums, 40 of them peaches, and 10 cherries:
how many trees does the orchard contain? Ans. 600.
31. There is a certain number which being divided by 7,
the quotient resulting multiplied by 3, that product divided
by 5, from the quotient 20 being subtracted, and 30 added
to the remainder, the half sum shall make 65; can you tell
me the number 7 Ans. 1400
Questions FOR EXERCISE. 201
tº what part of 25 is of a unit? Ans als.
53. If A can do a piece of work alone in 10 days, B in
20 days, Cin 40 days, and D in 80 days; set all foul about
it together, in what time will they finish it ! Ans. 54 days.
34. A farmer being asked how many sheep he had, an-
wered, that he had them in five fields; in the first be had
of his flock, in the second +, in the third in the ourth
º, and in the fifth 450; how many had he? Ans. 1100.
35. A and B together can build a boat in 18 day, and
with the assistance of C they can do it in 11 days; in what
time would C do it alone? Ans. 284 days.
36. There are three numbers, 23, 25, and 42; what is the
difference between the sum of the squares of the first and
last, and the cube of the middlemost? Ans. 133 jº.
37. Part 1200 acres of land among A, B, and C, sº that
B may have 100 more than A, and C 64 more than B
Ans. A 312, B 412, C-47 5.
38. If 3 dozen pairs of gloves be equal in value to 2 pieces
ºf Holland, 3 pieces of Holland to 7 yards of satin, 6 yards
of satin to 2 pieces of Flanders lace, and 3 pieces of Flan-
ders lace to 81 shillings; how manv dozen pairs of gloves
may be bought for 28s. 1 Ans. 2 dozen pairs
39. A lets B have a hogshead of sugar of 18 cwt., worth
* dollars, for 7 dollars the cwt. i. of which he is to pay in
sash. B hath paper worth 2 dollars per ream, which he
gives. A for the rest of his sugar, at 24 dollars per realm;
which gained most by the bargain? Ans. A by $1920 cts.
40. A father left his two sons (the one 11 and the other
it. years old) 10,000 dollars, to be divided so that each share
*ing put to interest at 5 per cent might amount to equal
ºms when they would be respectively 21 years of age.
Required the shares? Ans. 5454. º 4545 ºr dollars.
41. Bought a certain quantity of broadeloth for 383.
202 QUESTIONS FOR Exercise.
5s, and if the number of shillings which it cost per yard
were added to the number of yards bought, the sum would
be 386; I demand the number of yards bought, and at
what price per yard? Ans. 865 yds. at 21s, per yard.
Solved by Problem VI. page 171.
42. Two partners Peter and John, bought goods to the
amount of 1000 dollars; in the purchase of which, Peter
paid more than John, and John paid.....I know not how
much: They then sold their goods for ready money, and
thereby gained at the rate of 200 per cent, on the prime
cost: they divided the gain between them in proportion to
the purchase money that each paid in buying the goods;
and Peter says to John, My part of the gain is really a
handsome sum of money; I wish I had as many such sums
as your part contains dollars, I should then have $950,000.
I demand each man's particular stock in purchasing the
goods. Ans. Peter paid $600 and Johnpaid $400.
THE Pol-Lowing questions and proposed. To surveyons:
1. Required to lay out a lot of land in form of a long
square, containing 3 acres, 2 roods and 29 rods, that º
take just 100 rods of wall to enclose, or fence it sound;
pray how many rods in length, and how many wide, mus
said lot be? Ans. 31 rods in length, and 19 in breadth.
Solved by Phoniest VI. page 171.
2. A tract of land is to be laid out in form of an equal
square, and to be enclosed with a post and rail fence, 5 rails
high; so that each rod offence shall contain 10 rails. How
large must this noble square be to contain just as many
acres as there are rails in the fence that encloses it, so that
every rail shall fence an acre?
Ans, the tract of land is 20 miles square, and contain,
256,000 acres.
Thus, 1 mile=320 rods: then 320x320-160=640
acres: and 320x4×10–12,800 rails. As 640 : 12,800 :
12,800 : 256,000, rails, which will enclose 256,000 acres-
Q0 miles square.
--
APPENDIX,
SHORT RULES,
CASTING INTEREST AND REBATE:
US EFUL RULES,
- F-L-N-G -- cox T-NT's or sur-ER-FICEs-so-Lºs. &c.
SHORT RULES,
I or CASTING INTEREST AT Six PER CENT.
[.. To find the interest of any sum of shillings for any
number of days less than a month, at 6 per cent.
RULE.
1. Multiply the shillings of the principal by the number
of days, and that product by 2, and cut of three figures to
the right hand, and all above three figures will be the interest
in pence.
2. Multiply the figures cut off by 4, still striking off
three figures to the right hand, and you will have the far-
things, very nearly.
EXAMPLEs.
1. Required the interest of 51.8s. for 25 days.
+... s.
5,8–108×25-2–5,400, and 400x4–1,600.
- Ans. 5d. 1,647s.
2. What is the interest of 211. 3s, for 29 days?
Ans ºs. 0d. 2 ºrs.
204 APT-E-L-x.
FEDERAL MONEY.
II. To find the interest of any number of cents for any
numler of days less than a month, at 6 per cent.
RULE.
Multiply the cents by the number of days, divide the pro
duct by 6, and point off two figures to the right, and all the
figures at he left hand of the dash, will be the interest in
mills, near y.
Ex-MPLEs.
Required the interest of 85 dollars, for 20 days.
3 cits. - mills.
85–8500 x20–6–2S3,33 Ans. 283 which is
28 cits. 3 mills.
2. What is the interest of 73 dollars 41 cents, or 734
cents, for 2 days, at 6 per cent.”
Ams. 330 mills, or 33 cts.
-
III. When the principal is given in pounds, shillings, &e
New-England currency, to find the interest for any num
ber of days, less than a month, in Federal Money.
RULE.
Multiply ille shillings in the principal by the number of
days, and divide the product by 36, the quotient will be the
Interest in mills, for the given time, nearly, omitting
fractions
Ex-MPLE.
Required the interest in Federal Money, of 271. 15s, for
27 days, a tiler cent.
1.
- -
Ans. 27 15–555 x 27-36–116 mills.-41 cts, 6m.
IV. When the rincipal is º in Federal Money, and
you want the interest in shillings, pence, &c. Nº.
land currency for any number of days less than a mont
APPENDI- 205
RULE.
Multiply the principal, in cents, by the number of days
and point of five figures to the right hand of the product
which will give the interest for the given time, in shillings
and decimals of a shilling, very nearly.
Ex-MPLEs.
A note for 65 dollars, 31 cents, has been on interest 25
days; how much is the interest thereof in New-England
currency? 8-cts. - s, d. ºrs.
Ans. 65,31–6531 ×25–1,63275–1 7 2.
Remarks.-In the above, and likewise in the preceding
practical Rules, (page 115) the interest is confined at 6 per
sent, which admits of a variety of short methods of cast-
ing: and when the rate of interest is 7 per cent as esta-
Elished in New-York, &c. you may first cast the interest at
per cent, and add thereto one sixth of itself, and the sum
will be the interest at 7 per ct-, which perhaps, many times
will be found more convenient than the general rule of cast
ng Interest.
EXAMPLE.
Required the interest of 75l. for 5 months, at 7 percent
-
7.5 for 1 month.
5
– E. s. d.
37.5-1176 for 5 months at 6 per cent.
++= 63
Ans. E2 39 for ditto at 7 per cent.
-
--no-T METHoº-º-o-º-º-º-º-º-º-º-º-º-º-o- any given
su-º-o-º-º-o-Tº-A-D ---
Rule-Diminish the interest of the given sum for the time by its
ºwn interest, and this gives the Rebate very nearly.
----------
1. What is the rebate of 50 dollars, for 6 months, at 6
per cent.”
200 APPENI-1-.
4 ct,
The interest of 50 dollars for 6 months, is 1 50
And, the interest of 1 dol. 50 cts, for 6 months, is 4.
Ans. Rebate, $1 46
2. What is the rebate of 150 for 7 months, at 5 per
cent.” £. s. d.
Interest of 150l. for 7 months, is 4 7 6
Interest of 41.7s.6d. for 7 months, is 2 6+
Ans. £4 4 113 nearly
By the above Rule, those who use interest tables in their
counting-houses, have only to deduct the interest of the in
terest, and the remainder is the discount.
-
A concise Rule to reduce the currencies of the different States,
where a dollar is an even number of shillings, to Federal
Money.
Rule. I.-Bring the given sum into a decimal expression by in-
spection, (as in Problem I. page 80) then divide the whole by 3in
New-England, and by 4 in New-York currency, and the quotient
will be dollars, cents, &c.
Ex-MPLEs.
1. Reduce 541. 8s. 3+d. New-England currency, to fo
leral money.
3)54,415 decimally expressed.
Ans. $181,38 cts.
2. Reduce 7s. 11; d. New-England currency, to federal
money.
7s. 11; d.-E0,399 then, 3),399
Ans 81,33
3. Reduce 5131, 16s. 10d. New-York, &c. currency, to
federal money.
,4)513,842 decimal.
Ans. $1284,604
Al-PENDIX. 207
4. Reduce 19s. 5; d. New-York, &c. currency, to Fede-
ral Money. ,4)0,974 decimal of 19s. 5; d.
$2,43; Ans.
5. Reduce 641. New-England currency, to Federal
Money. 3)64000 decimal expression.
$213,334 Ans.
Note-By the foregoing rule you may carry on the de-
cimal to any degree of exactness; but in ordinary practice,
the following Contraction may be useful.
RULE II.
To the shillings contained in the given sum, annex. 8
times the given pence, increasing the product by 2; then
divide the whole by the number of shillings contained in a
dollar, and the quotient will be cents.
Ex-MPLEs.
1. Reduce 45s. 6d. New-England currency, to Federal
Money. 6 × 8 +2 − 50 to be annexed.
6)45,50 or 6)4550
- rts.
$7,584 Ans. 75s cents.-7,58
2. Reduce 21 10s. 9d. New-York, &c. currency, to
Tederal Money.
9×8+2–74 to be annexed.
Then 8)5074 Or thus, 8)50,74
- & ºt- -
Ans. 634 cents.-6 34 $6,34 Ans.
N. B. When there are no pence in the given sum, you
must annex two ciphers to the shillings; then divide as be-
fore, &c.
3. Reduce 31 5s. New-England currency, to Federal
Morey
31 5s -65s. Then 6)6500
Ans, 1083 cents.
208 appendix.
SOME USEFUL RULES,
-on. Fix-ING THE co-ºr-nºrs of suº-º-º-c-8 -ND-50-----
SECTION I.-OF SUPERFICES.
The superfices or area of any plane surface, is compo
sed or made up of squares, either greater or less, according
to the different measures by which the dimensions of the
figure are taken or measured:—and because 12 inches in
length make 1 foot of long measure, therefore, 12×12=144
the square inches in a superficial foot, &c.
ART. I. To find the area of a square having equal sides
RULE.
Multiply the side of the square into itself and the pro-
duct will be the area, or content.
ExAMPLEs.
1. How many square feet of boards are contained in the
floor of a room which is 20 feet square?
20x20–400 feet, the Answer.
2. Suppose a square lot of land measures 26 rods on
each side, how many acres doth it contain?
Note.—160 square rods make an acre.
Therefore, 26x26–676 sq. rods, and 676-160=4 a
36 r. the Answer.
Ant. 2. To measure a parallelogram, or long square.
RULE.
Multiply the length by the breadth, and the product will
be the area, or superficial content.
Ev.AMPLES.
1. A certain ſº in form of a long square, is 96 feel
long, and 54 wide; how many square #. of ground are
contained in it? Ans. 96×54=5184 square feet.
2. A lot of land, in form of a long square, is 120 rods in
ength, and 60 rods wide; how many acres are in it?
120x60=7200 sq. rods, then º-45 acres. Ans.
3. If a board or plank be 21 feet long, and 18 inches
broad; how many square feet are contained in it?
18 inches=1,5 feet, then, 21x1,5–31,5. Ans.
-PPENDIX. 200
Or, in measuring boards, you may multiply the length in
'eet by the breadth in inches, and divide by 12, the quo-
sent will give the answer in square feet, &c.
Thus, in the foregoing example, 21 x 18-12=31,5 as
before.
4. If a board be 8 inches wide, how much in length will
make a square foot?
Rule.-Divide 144 by the breadth, thus, 8)144
Ans. 18 in.
5. If a piece of land be 5 rods wide, how many rods in
length will make an acre?
Rule.—Divide 160 by the breadth, and the quotient will be the
length required, thus,
5)160
Ans. 32 rods in length.
ART. 3.--To measure a triangle.
Definition.—A triangle is any three cornered figure which
is bounded by three right lines.”
RULE.
Multiply the base of the given triangle into half its per-
pendicular height, or half the base into the whole perpen-
dicular, and the product will be the area.
Ex-MPLEs.
1. Required the area of a triangle whose base or longest
wide is 32 inches, and the º height 14 inches.
32x T-224 square inches the Answer.
2. There is a triangular or three cornered lot of land whose
base or longest side is 514 rods; the perpendicular from the
corner opposite the base measures 44 rods; how many acres
doth it contain?
51,5×22=1133 square rods,-7 acres, 13 rods.
* A Triangle may be either right angled or oblique; in either case the
eacher can easily give the scholar a right idea of the base and perpendieu
i-, by marking it down on the slate, paper, &c.
a 2
210 APPENDIX.
TO MEASURE A CIRCLE.
ART. 4.—The diameter of a circle being given, to fin;
the circumference.
Rule.—As 7 : is to 22 : : so is the given diameter: to the circum
ference. Or, more exactly, as 113 : is to 355 : : &c. the diameter a
found inversely.
Note.—The diameter is a right line drawn across the
circle through its centre.
Ex-MPLEs.
1. What is the circumference of a wheel whose diameter
is 4 feet?—as 7 : 22 : : 4 : 12,57 the circumference.
2. What is the circumference of a circle whose diameter
is 35?–As 7 - 22 : : 35 : 110 Ans—and inversely as
22 : 7 : : 110 : 35, the diameter, &c.
Art. 5.--To find the area of a Circle.
Rule.—Multiply half the diameter by half the circumference, and
the product is the area; or if the diameter is given without the cir.
cumference, multiply the square of the diameter by ,7854, and the
product will be the area.
ExAMPLE.S.
1. Required the area of a circle whose diameter is 1:
inches, and circumference 37,7-inches.
18,85–half the circumference.
6-half the diameter.
113,10 area in square inches.
2. Required the area of a circular garden whose diame-
ter is 11 rods? ,7854
By the second method, 11×11 = 121
Ans. 95,0334 rods
SECTION 2.—OF SOLIDS.
Solids are estimated by the solid inch, solid foot, &c.
1728 of these inches, that is, 12×12×12 make 1 cubic on
solid foot.
APPENDIX. 211
Ant, 6.-To measure a Cube.
Definition.—A cube is a solid of six equal sides, each of
which is an exact square.
Rule.—Multiply the side by itself, and that product by the same
ide, and this last product will be the solid content of the cube.
Ex-MPLEs.
1. The side of a cubic block being 18 inches, or 1 foot
and 6 inches, how many solid inches doth it contain?
--- -
1. **s and 1,5×1.5× 1,5–3,375 solid feet. Ans.
Or, 18×18x18–5832 solid inches, and ###–3,375.
2. Suppose a cellar to be dug that shall contain 12 feet
every way, in length, breadth and depth; how many solid
feet of earth must be taken out to complete the same?
12×12×12–1728 sold feet, the Ans.
ART. 7-To find the content of any regular solid of three
dimensions, length, breadth and thickness, as a piece of
timber squared, whose length is more than the breadth
and º -
Rute-Multiply the breadth by the depth, or thickness, and that
product by the length, which gives the solid content.
E*AMPLEs.
1. A square piece of timber, being one foot 6 inches, or
18 inches broad, 9 inches thick, and 9 feet or 108 inches
long; how many solid feet doth it contain?
1 ft. 6 in.-1,5 foot
9 inches = .75 foot.
Prod. 1,125×9–10,125 solid feet, the Ans.
in, in in solid in.
Or 18x9x108–17496- 1728–10,125 feet.
But, in measuring timber, you may multiply the breadth
in inches, and the depth in inches, and that product by the
length in feet and divide the ast product by 144, which
will give the solid cºntent in feet, &c.
212. a PPF, NL1x. |
2. A piece of timber being 16 inches broad, 11 inches
thick, and 20 feet long, to find the content?
Breadth 16 inches.
Depth 11
Prod. 176x20–3520 then, 3520-144–24.4 feet. Ans.
3. A piece of timber 15 inches broad, 8 inches thick,
and 25 feet long; how many solid feet doth it contain?
Ans. 20,8+ feet.
ART. S.–When the breadth and thickness of a piece of
timber are given in inches, to find how much in length
will make a solid foot.
Rule.—Divide 1728 by the product of the breadth and depth, and
the quotient will be the length making a solid foot.
Ex-LEs.
1. If a piece of timber be 11 inches broad and 8 incheſ
deep, how many inches in length will make a solid foot?
11 x8=SS)1728(19.6 inches. Ans.
2. If a piece of timber be 18 inches broad and 14 inche.
deep, how many inches in length will make a solid foot?
18x14=252 divisor, then, 252)1728(6,8 inches. Ans
Arºr. 9.--To measure a Cylinder.
Definition.—A Cylinder is a round body whose bases are
circles, like around column or stick of timber, of equal big-
ness from end to end.
Rule.—Multiply the square of the diameter of the end by 7854
which gives the area of the base; then multiply the area of the base
by the length, and the product will be the solid content
EXAMPLE.
What is the solid content of a round stick of timber of
equal bigness from end to end, whose diameteris 18-inches,
and length 20 feet?
PPENDIX. 2.13
18in.-1,5 ft.
× 1,5
Square 2,25× ,7854=1,76715 area of the base.
+20 length.
Ans. 35,34300 solid content.
Or, 18 inches.
18 inches.
321 x,"854–254,4696 inches, area of the base.
20 length in feet.
144)5089,3920(35,343 solid feet. Ans.
tar. 10. To find how many solid feet a round stick of
timber, equally thick from end to end, will contain when
hewn square.
sq. RULE.
Multiply twice the square of its semi-diameter in inches
by the length in feet, then divide the product by 144, and
the quotient will be the answer.
Ex-MI-LE.
If the diameter of a round stick of timber be 22 inches
and its length 20 feet, how many solid feet will it contain
when hewn square?
11 x 11×2×20+144=33,6+ feet, the solidity when
hewn square.
Art. 11. To find how many feet of square edged boards
of a given thickness, can be sawn from a log of a given
diameter.
RULE.
Find the solid content of the log, when made square, by
the last article—Then say, As the thickness of the board
including the saw calf - is to the solid feet :: so is 12 (in-
ches) to the number of feet of boards.
Ex-MPLE.
How many feet of square edged boards, 1+ inch thick,
including the saw calf, can be sawn from a log 20 feetlong
and 24 inches diameter?
12×12×2x20+ 144-40feet, solid content.
As I i : 40 : ; 12 : 384 feet, the An-
214 APPENDIX.
being given, to find how many bushels it will contain.
RULE.
Multiply the length by the breadth, and that product by
the depth, divide the last product by 2150,425 the solid
inches in a statute bushel, and the quotient will be the an-
-e-
EXAMPLE.
There is a square box, the length of its bottom is 50
inches, breadth of ditto 40 inches, and its depth is 60
inches; how many bushels of corn will it hold 2
50×40×60+2150,425–55,84+ or 55 bushels three
pecks. Ans. -
ARt. 13. The dimensions of the walls of a brick building
being given, to find how many bricks are necessary tº
build it.
Art. 12. The length, breadth and depth of any squarebo
RULE.
From the whole circumference of the wall measure:
round on the outside, subtract four times its thickness, ther
multiply the remainder by the height, and that product by
the thickness of the wall, gives the solid content of the
whole wall; which multiplied by the number of brick.
contained in a solid foot gives the answer.
EXAMPLE.
How many bricks sinches long, 4 inches wide, and 2.
inches thick, will it take to build a house 44 feet long, 4.
feet wide, and 20 feet high, and the walls to be 1 foot thick
8×4×2,5–80 solid inches in a brick, then 1728-Su.
21,5 bricks in a solid foot.
44+40+44+40–168 feet, whole length of wall.
–4 times the thickness.
164 remains.
Multiply by 20 height.
3280 solid feet in the whole wall.
Multiply by 21,6 bricks in a solid foot.
Product, 70848 bricks. Ans.
a PPEND1- 215
Aar. 14.—To find the *. of a ship.
Rule.—Multiply the length of the keel by the breadth of the
beam, and that product by the depth of the hold, and divide the last
product by 95, and the quotient is the tonnage.
ExAMPLE.
suppose a ship 72 feet by the keel, and 24 feet by the
beam and 12 feet deep; what is the tonnage?
72×24x 12-95–218,2+tons. Ans.
RULE II.
Multiply the length of the keel by the breadth of the beam, and
hat product by half the breadth of the beam, and divide by 95.
ExAMPLE.
A ship 84 feet by the keel, 28 feet by the beam; what is
he tonnage? 84x28x14+95=350,29 tons. Ans.
lat. 15–From the proof of any cable, to find the strength
of another.
Rule.—The strength of cables, and consequently the weights of
heir anchors, are as the cube of their peripheries.
Therefore : As the cube of the periphery of any cable,
Is to the weight of its anchor;
So is the cube of the periphery of any other cable,
To the weight of its anchor.
Ex-MPLEs.
1. If a cable 6 inches about, require an anchor of 2+ cwt.
ºf what weight must an anchor be for a 12-inch cable?
As 6×6x6 : 2; cict. : : 12×12×12 : 18 cwt. Ans.
2. If a 12-inch cable require an anchor of 18 cwt. what
must the circumference of a cable be, for an anchor of 21
awt. 1
cºnt. cºnt. - in.
As 18: 12×12×12 : : 2,25: 216 V216–6 Ans.
Anº. 16.-Having the dimensions of two similar built ships
of a different capacity, with the burthen of one of them,
to find the burthen of the other.
-6 -PPENDIX.
RULE.
The burthens of similar built ships are tº each other, a
the cubes of their like dimensions.
--------
If a ship of 300 tons burthen be 75 feet long in the keel
I demand the burthen of another ship, whose keel is 10
feet long? T. cwt. ºrs, lb.
As 75×75×75:300: : 100x100×100:711, 2 0 24+
-
DUODECIMALS,
on. |
CROSS MULTIPLICATION,
IS a rule made use of by workmen and artificers in cast
ing up the contents of their work.
RULE.
1. Under the mulplicand-write the corresponding dem
minations of the multiplier.
2. Multiply each term into the multiplicand, beginning
at the lowest, by the highest denomination in the multiplier
and write the result of each under its respective-term; ol.
serving to carry an unit for every 12, from each lower dº
nomination-to-its-next superior.
3. In the same manner multiply all the multiplicand by
the inches, or second denomination, in the multiplier, and
set the result of each term one place removed to the righ
hand of those in the multiplicand.
4. Do the same with the seconds in the multiplier, set
ting the result of each term two places to the right-hand-d
those in the multiplicand, &c.
ºx-MPLES-
rº. 1. rº. I. E. J. F. I.
Multiply 7-3. 7 5 4 tº 9-7
By 4 7 8-9 5 8 97.
29-0-" 27-9-9 25-6 91 101
4-2 9 – –
a-P-N-1- ---
F. I. F. I. F. I.
Multiply 4 7 3. 8 9 7
By 5 10 7 6. 8 6 .
Product, 26 8 10 27 6 32 6 6.
F. I. F. I. F. I.
Multiply 3 11 6 5. 7 10
By 9 5 7 6. 8 11.
--
-
Product, 36 107 4S 1 6 69 10 2.
FEET, inches AND BEconos.
F. I. "
Multiply 9 8 6
By 7. 9 3
- Ltiplier.
67. 11 6 " =prod. by the feet in the mul-
7 3 4 6 "" =ditto by the inches.
2 5 1 6 =ditto by the seconds.
75 5. 3 7 & Ans.
F. I. " rº. r. "
Multiply 7 1 9 5, 6, 7
By 7 8 9 8 9-10
Product, 55 2 9 3-9 48 11 2 8 10
How many square feet in a board 16 feet 9 inches long,
and 2 feet 3 inches wide?
By Duodecimals. By Decimals.
F. I. F. I.
16 9 16 9–16.75 feet.
2 3 2 3–2,25
33 6. ' 83.75
+ 2 3 3350
3350
Anº. 37 & 8 – F. I.
An, 37.6875–87 s a
-
|
2] § APPENDIX.
TO MEASURE LOADS OF WOO!).
RULE.—Multiply the length by the breadth, and the product by the
depth or height, which will give the content in solid feet; of which 64
make half a cord, and 128 a cord.
EXAMPLE.
How many solid feet are contained in a load of wood,
7 feet 6 inches long, 4 feet 2 inches wide, and 2 feet 3
inches high 3
7 ft. 6 in.—7,5 and 4ft. 2 in. =4,167 and 2 ft. 3 in-
2.25 ; then, 7.5 × 4,167–31,2525 × 2,25=70,318.125 solid
feet, Ans. -
But loads of wood are commonly estimated by the foot,
allowing the load to be 8 feet long, 4 feet wide, and then 2
feet high will make half a cord, which is called 4 feet of
wood; but if the breadth of the load be less than 4 feet, its
height must be increased so as to make half a cord, which
is still called 4 feet of wood. -
By measuring the breadth and height of the load, the
content may be found by the following
RULE.—Multiply the breadth by the height, and half the product
will be the content in feet and inches.
EXAMPLE. -
Required the content of a load of wood which is 3 feet 9
inches wide and 2 feet 6 inches high.
By Duodecimals. By Decimals.
F. in. F.
3 9 3.75
2 6 2,5
7 6 1875
1 10 6 750
9TAT6 9,375 -
—- F. in. - º
Ans 4 8 3 4,6875=484 or half a cord and 8.
--- ºr.
inches over.
The foregoing method is concise and easy to those who are wel)
acquainted with Duodecimals, but the following table will give he
pontent of any load of wood, by inspection only, suffieiently exact for
tommon practice; which will be found ver Nonvenient.
y the
ch 64
100d.
jet 3
solid
foot,
len?
set of
et, its
Which
|, the
roduct
feelſ
A fºg, NHDíº, - 31.9
-
A TABLE of Breadth, Height, and Content.
Breadth. [Height in ſº Inches.
ſº in 1334|| | 3| 4 || 5 || 6 || 7 || 8 || 9|10|| ||
2 6 º Iſ 2| iſ 5|6||7|3|10|III2|14
7 º 1 || 3 || 4 || 5 || 6 || 8 §:
8 º; If 3 4 5 7| 8|9||1|12||1315
9 § || 3 || 4 || 6 || 7 || 8 || 9 || ||12|14}}.5
10 #: 2 3 4| 6 || 7 || 9|10|11||1314|16
II ||1835,5370|| 2 3 4 6 7 91012||1315||16
3 0 ||IS355472 23 #| G|S| 9|III.2.1415||7
|iº || 3 || || |s|{iiigiºiºi,
3 |19:5776|| 2 || 5 || SºHº 14%||7
3 || 193959|78|| 2 3' 5| 7 || 8 ||10|II #: 18
4 ||201406080 || 2 || 3 || 5 || 7 || 8|10|12||1315||7|18
5 gº 2| 3 || 5 || 7 sidiºiſilii 19
|T T6 ||3||1363.54 ji 5| 7 || 9|11||12|14|1618||19
7 ||22436486 g #| || 7 || || ||4||6|Sº
8 ||2244|66|SS|| 2 4 6 7| 9||1|1315||17|18:20
| | |##| || 3 || || || #
10|º]| | | | | jºiářižišši
| | | | | | |||}|{ij
4 0 ||34 is 7336|| 3 || 6 slidiºiáić'isºlº
TO USE THE FOREGOING TABLE.
First measure the breadth and height of your load to the nearest average
inch; then find the breadth in the left hand column of the table, then move
to the right on the same line till you come under the height in feet, and you
will have the content in inches, answering the feet, to which add the content
of the inches on the right and divide the sum by 12, and you will have the
rue content of the load in feet and inches.
JNote.—The contents answering the inches being always small, may oe
added by inspection.
EXAMPLES.
1. Admit a load of wood is 3 feet 4 inches wide, and 2 feet 10 inches nigº,
required the content.— -
hus, against 3 feet 4 inches, and under 2 feet, stands 40 inches; and un-
der 10 inches at top, stands 17 inches: then 40+17–57, true content in
inches, which divide by 12, gives 4 feet 9 inches, the answer. .
2. The breadth being 3 feet, and height 2 feet 8 inches; required the con-
tent.—
Thus, with breadth 3 feet 0 inches, and under 2 ſeet atop, stands 86
220 -PP-N-D-1-.
inches; and unders inches, stands 12-inches: now 55 and 12 make 48, twº
answer in inches; and 48+12–4 feet, or just half a cord.
3. Admit the breadth to be 3 feet 11 inches, and heights feet 9 inches,
uired the content.
Inder 3 feet attop, stands 70; and under 9 inches, is 18: 70 and 18, make
º or 7 ſt. I q.--inches, the answer.
58-12=7 feet 4 inc
Showing the amount of £1, or $1, at 5 and 6 per cent. pe
annum, Compound Interest, for 20 years.
Prºper cent.6 percent. Yrs.5 percent:6 percent,
1 | 1,05000 | 1,06000 || 11 | 1,71034 1,898.29
2 1,10250 1,12360 12 1,79585 2,012.19
3 1,157.62 1,19101 || 13 1,885.65 2,13292
4 1,21550 1,26247 14 1,97993 2,26000
5 1,27628 1,33822 15 2,078.93 2,39655
6 1,34009 || 1,41851 16 || 2,18287 2,54727
7 1,40710 1,50363 - 17 2,29201 2,69277
8 1,47745 1,59384 18 2,40661 2,85433
9 1,55132 1,68947 19 2,52695 || 3,02559
10 1,628S9 1,79084 2012,653291.3.20713
VII. The weights of the coins of the United States.
pºrt grs.
Eagles, 11 tº -
Half-Eagles, 5 15 | sº
Quarter-Eagles, 2 19: -
Dollars, 17 8
Half-Dollars, 8 16
Quarter-Dollars, * : *g."
Dimes, 1 17: -
Half-Dimes, 20:
Cents, 8 16
Half-Cents, copper
The standard for gold coin is 11 parts pure gold, and
one part alloy—the alloy to consist of silver . copper.
The standard for silver coin is 1485 parts fine to 179 puru
alloy—the alloy to be wholly copper.
---E-L---
ANNUITIES.
2-1
TABLE II.
Showing the amount ºf +1 annui-
, forburne for 31 years or un-
pºund interest. -
Yrs. 5 t;
er, at 5 and 6 per cent, com-
| TABI
5
showing the present worth
ºf +1 annuity, to conti-
nue for 31 years, at 5 and
| 6per cent. compound int.
E III.
,000000-1,000000
2,060000
3, 183600
4,374616
5,63719.3
1,859.410
2,723.248
3,545.950
4,329.477
6,9753.19
8,393838
9,549.109-9,897-168
11,02655411,491316
10 12,577.892,13,180770
5,075692
5,786278
6,463213
7,107822
7,721735
TT lºsſ 14,971643
12 15,91712616,859942
14 19,5986.3221,015066
15 21,57856423,275969
16
23,55742.25,67252:
17 25,840.366:28,212:380
is 28,132.38530,905653
19 30,539004:33,759992
20 33,065954.36,785592
21 º
38,50521443,392.291
41,430475/46,995828
44,501999.50,815578
47,727099.54,864513
51,11345.459,156.382
54,669.12663,7057.65
58,402583,68,528112
62,32271273,53979-
56,43884779,058.186
:
i
70,760790-480.1677
-
13 |17,71298.218,882138
8,305414
8,863252
9,393573
9,808611
10,379.658
10,837769
11,274.066
11,689587
12,462.210
12,821-153
13,163003
13,488574
0,9523SITU,943396
12,085321
1,833.393
2,673,012
3,465106
4,212:364
4,917.324
5,582381
6,2097.94
6,801692
7,360087
7,886875
8,383S44
8,852683
9,294.984
9,712249
10,105895
10, 177260
10,827.603
11,158116
11,46992.1
11,764077
12,041582
12,303380
- y T-
13.79sº?!?,550.357
14,09394.412,783:350
1437.5issisoºlgä
14,64308413,2105.34
14,898.127|13,400 104
15,14107313,590721
15,372.4°11′376-4831
15,59281013,929.036
-- ---------
TABLES.
-
THE three following tables are calculated agreeable tº
an Act of Congress passed in November, 1752, making
foreign Gold and Silver coins, a legal tender for the pay-
ment of all debts and demands, at the several and respec-
tive rates following, viz. The Gold Coins of Great Bri-
tain and Portugal, of their present standard, at the rate of
100 cents for every 27 grains of the actual weight there.
of-Those of France and Spain 273 grains of the actual
weight thereof—Spanish milled dollars weighing 17 put
7 gr. equal to 100 cents, and in proportion for the parts of
a dollar.—Crowns of France weighing 18 pºwt. 17 gr.
equal to 110 cents, and in proportion for the parts of
Crown.—They have enacted, that every cent shall contain
208 grains of copper, and every half-cent 104 grains
TABLE IV.
Weights of several pieces of English, Portuguese and
French Gold Coins.
TTPºt. TGr, TDaſ. Cº.T.
* - - - - - 18 TIGTOTO
Single ditto, - - - - 9 8 to U
English Guinea, - - 5. 6 4 663
Half ditto, - - - 2 15 3 #3;
French Guinea, - - - 5 6 4 59 S
ºf , ditto - - - || 3 || 3 || 3: 39 º
4. Pistoles, - - - - - 16 12 14 45 2
2 Pistoles, - - - - - 8 || 6 ; : ;
Pistole, - - - - 4 3. 3 G1 3
oidore, - 8 22 * : *
s
i
-
i
=
:
TE
|
§
ATP-N-D-1- º
º ----- ------------ ---
º:332:3:32:3:35:2-2:23: 5°3
--------------------
-----
-
:
=
º:33.
----------------------
------
º
º
=
g-º-º:
=
-- ----------------------> -----
**-->25:35:23:35:5:33:33:38.2%
s
- -
---------- -------- ---
ºte:::::::= *tºg:33: 2 tº:
--~~~~~ *-***=-2-3-2-3 -:3?
|-
--------------------- 8-----
--- ------
|- _
------------- --------- ---------
*::::::::::::::::::::::::::3RRR:::::
*********E=23::=>223:33
--- --
-
-
224 -**E*L-x.
VII. TABLE of Cents, answering to the Currence.
of the United States, with Sterling, &c.
Note—The figures on the right hand of the space
show the parts of a cent, or mills, &c.
-
6s. toss, to 7s.6d.4s. Sºl. 5s, to 4s.6d.4s. 10; d. tº
the the to the to the the to the the
Doll. Doll. Doll. Doll. Dull. Doll. Dollar.
P. cents, cents, cents, cents, cents, cents. cents.
| | | | | | | | | | | | . . . .
2, 27 2 0 2-2, 3 º' 3 3, 3 7 3 +
3, 4 1 & 1 & 3. 5.3, 5 5 5. 5 1
4 55 4 1 4-4 7 || 6 3, 7 4 tº 8
; : * * * * * * * * * * * * *
6 s 3 & 2 6 6 107 10 || 11 1 10 2
7| 9 || 7 || 7 || is 5 iſ 6' 13 g iſ 9
8, 11 1 S 3 S S 14 2. 13 3: 14 S 13 6
9 12 5 9 2 10 | 16 15 16 6 15 3
10 is sº to 4. ii il iſ slie 6 is 5, 17
gº 18
-
1, 16 G 12 5, 133 31 4 30 || 2: 2 20
2 33 3 25 **::::: 44 1 41
3, 50 37 5, 40 64 2 60 | 66 tº G1 5
4 d6 6. 50 sº a sº 7 so iss's sº
gº 8 tº 5 tº 5107 1100 111 || 102 5
6100 || 75 so 12s 5120 tº 3 tº
iii., §§ 5 ºil. i. i. º. ii., §
S133 3100 106 61714|160 1777, 164
9.150 112 5120 ſlº Slsº 300 184 tº
ºº: ºº lºº. 23" º ż żº
it is ºr siſº gº ºn ºf º 6
12200 150 160 ºf 1240 266 6, 246
13216 (1625,173 3.278 Bºº S 265 tº
14233 3.175 186 6300 280 ºil 1, 2S7
15250 187 5200 ºr 4300 lº º 307 tº
16266 Gºdſ) º 3.342. Sºo 255 5. 32.8 2.
17283 3.212 52:26 6364. 2340 377 7| 348 7
18300 225 240 385 6'360 400 369 2
1931.6 G.237 52.53 stoº 1380 # 2 389 7.
20sº 8250 256 6428-5400 4444. 410 2
arº-ND-x. 2-5
TABLE IX.
shewing the value of Federal Money in other Currencies.
New Jersey,
New Eng- New York Pennsylva-South-Car-
|Federal land, Vir- and North nia, Dela-olina, and
| Money.ginia, and Carolina ware, and Georgia
Kentuky currency. Maryland currency.
currency. | currency.
| Cents. s. d. s, d. | s, d. s. d.
I 0 0: 0 1 1 0 1 0 0}
2 0 11 0 2. 0 1: 0 1
3. 0 2+ 0 3 0 2; 0 1:
4. 0 3 0 3. 0 34 0 2.
5. 0 3. 0 4. 0 4 0 2:
6 0 41 0. 53 0. 51 0 3.
7 0. 5 0 6; 0 64 0 4
§ 5, o ż, , , ; ; ) is
9. 0 G1 0 8: 0 8 0. 5
10 0 71 0 9} 0 9 0. 5*
11 0 & 0 104 0 10 0 tº:
| 12 0 8, 0 111 0 10. 0 6-
13 0 9. 1 0} 0 11: 0 71
14 0 10 1 11 I 0} 0 7.
15 0 10, 1 2+ I 11 0 8.
16 0 111 1 31 24 0 9
17 I 0. 1 4. I 3+ 0 94
18 1 1 I 53 4. 0 10
19 1 1: I Gº || 1 5; 0 10,
20 1 24 I 7+ | 1 tº 0 111
30 I 91. 2 4: 2 3 1 41
40 2 4. 3 24 3 0 1 101
50 3 0. 4 0. 3 Q 2 4
60 3 7. 4. 9] 4 6 2 9.
70 4 24 || 5 71 || 5 3 3 3.
80 || 4 9 || 6 4; 6 o 3 81
90 5 º | 7 24 6 9 4 2
100 º s () 7_6 L 4 8
- - - -
225 --PENL)--.
A Few USEFUL FORMS IN TRANSACTING BUSINESS,
-
AN OBLIGATORY BOND.
KNOW all men by these presents, that I, C. D. ºn
in the county of am held and firmly bound to
H. W. of in the penal sum of to be paid
H. W. his certain attorney, executors, and administrators,
to which payment, well and truly to be made and done,
I bind myself, my heirs, executors, and administrators,
firmly by these presents. Signed with my hand, and
sealed with my seal. Dated at this day
of A. D.
The condition of this obligation is such, That if the
above bounden C. D. &c. [Here insert the condition.]
then this obligation to be void and of none effect; other
wise to remain in full force and virtue.
Signed, sealed, and delivered,
in the presence of | -
-
A BILL OF SALE.
KNOW all men by these presents, that I, B. A. of
for and in consideration of to me in hand paid by
D. C. of the receipt whereof I do hereby ac
knowledge, have bargained, sold, and delivered, and, by
these presents, do bargain, sell and deliver unto the said
D. C. º: specify º sold.] To have and to
houn the aforesaid bargained premises, unto the said D.C.
his executors, administrators, and assigns, forever. And
the said B.A. for myself, my executors and administrators,
shall and will warrant and defend the same against all per-
sons unto the said D. C. his executors, administrators, and
assigns, by these presents. In witness whereof, I have
hereunto set my hand and seal, this day of 1814.
In presence of
-
A SHORT WILL.
I, B.A. of &c. do make and ordain this my last will
and testament, in manner and form following, viz. I giv
a-P-N-D-x. 227
Ladbequeath to my dear brother, R. A. the sum of ten
pounds, to buy him mourning. I give and bequeath to
my son J. A. the sum of two hundred pounds. I give and
bequeath to my daughter E. E. the sum of one hundred
ands; and to my daughter A. W. the like sum of one
F. pounds. All the rest and residue of my estate,
oods and chattels, I give and bequeath to my dear be-
oved wife, E. R. whom I nominate, constitute and appoint
sole executrix of this my last will and testament, hereby
revoking all other and former wills by me at any time
heretofore made. In witness whereof, I have hereunto
set my hand and seal, the day of
in the year of our Lord
Signed, sealed, published and declared by the said tes-
ºntor, B. A. as and for his last will and testament, in the
presence of us who have subscribed our names as witnesses
thereto, in the presence of the said testator. R. A
S. D.
L. T.
Nore.-The testator, after taking off his seal, must, in
presence of the witnesses, pronounce these words: “I
publish and declare this to be my last will and testament.”
Where real estate is devised, three witnesses are ab-
solutely necessary, who must sign it in the presence of
the testator.
-
A LEASE OF A HOUSE.
KNOW all men by these presents, that I, A. B. ol
in. for and in consideration of the sum of ---
ºeived to my full satisfaction of P. V. of this
lay of in the year of our Lord have demised
and to farmlet, and do by these presents, demise and to farm let,
unto this said P. V. his heirs, executors, administrators and as:
signs, one certain piece of land, lying and being situated in said
bounded, &c. [Here describe the boundaries] with a
dwelling house thereon standing, for the term of one year from
this date. To have and to noºn to him the said P. V. his heirs,
**otº, administrators and assigns, for said term, for him the
said P. V. to use and tº "E" as to him shall seem meet and
proper. And the said A. B. doth run then covenant with the
£23 a PP-N1)--.
said P. that he hath good right to let and demise the said
letten and demised premises in manner aforesaid, and that he
the said A. during the said time will suffer the said P. quietly to
save and to Hoºp, use, occupy and enjoy said demised premises,
and that said P. shall have, º use, occupy, possess and enjo
the same, free and clear of all incumbrances, claims, rights an
itles whatsoever. In witness whereof, I the said A. B. have
nereunto set my hand and seal, this day of
Signed, sealed and delivered ;
in presence of A lº-
A NOTE PAYABLE AT A BANK.
s:00, 60] . Haarronn, May 30, 1815.
FOR value received, I promise to pay to John Merchan
3r order, Five Hundred º and Sixty Cents, at Hartfo
Bank, in sixty days from the date.
WILLIAM DISCOUNT.
AN INLAND BILL OF EXCHANGE.
[583,34 Boston, June 1, 1815.
TWENTY days after date, please to º to Thomas Good-
win or order, Eighty-Three Dollars and Thirty-Four Cents, and
place it to my account, as per advice from yºu...humble servant,
Mr. T. º”; SIMON PURSE.
New-York.
A COMMON NOTE OF HAND.
[slº New-Yonk, March 8, 1821.
FOR value received, I promise to pay to John Murray, One
Hundred and Thirty Dollars, in four months from this date, with
interest until paid. JOHN LAWRENCE.
A COMMON ORDER.
New-Yonk, June 10, 1822.
Mr. Charles Careful,
Please to deliver Mr. George Speedwell, the amount ol
Twenty-Five Dollars, in goods from your store; and charge the
same to the account of Your Obºt. Servant,
- E. WHITE.
FINIS.
THE
PRACTICAL ACCOUNTANT,
o-
FARMERS" AND MECHANICKS."
B-ST METHOD or
B O O K-K E E PIN G :
-o-T--
Easy instruction or Youth.
-----------
A COMPANION
-->
DABOLL’S ARITHMETICK
BY SAMUEL GREEN.
T. H. A. C. A., N. Y.,
PRINTED ANL PUBLISHED by MACR, ANDRus, AND wooDRUFr.
INTRODUCTION.
Scholars, male and female, after they have acquired a sufficient
knowledge of Arithmetie, especially in the fundamental rules of Addi-
tion, Subtraction, Multiplication, and Division, should be instructed
in the practice of Book Keeping. By this it is not meant to recom-
mend that the son or daughter of every farmer, mechanic. or shop
keeper, should enter deeply into the science as practised by the mer
chant engaged in extensive business, for such study would engross a
great portion of time which might be more usefully emplºyed in ae-
quiring a proper knowledge of a trade, or other employment.
Persons employed in the common business of life, who do not keep
regular accounts, are subjected to many losses and inconveniences
to avoid which, the following simple and correct plan is recommend
ed for their adoption.
Let a small book be made, or a few sheets of paper sewed toge-
ther, and ruled after the examples given in this system. In the book,
termed the Day Book, are duly to be entered, daily, all the transac-
tions of the master or mistress of the family, which require a charge
to be made, or a credit to be given to any person. N. article thus
subject to be entered, should on any consideration be deferred till
another day. Great attention should be given to write the transac-
tion in a plain hand; the entry should mention all the particulars nº-
cessary to make it fully º with the time when they took
place; and if an article be delivered, the name of the person to whom
delivered is to be mentioned. No scratching out may be suffered; be-
cause it is sometimes done for dishonest purposes, and will weaken
or destroy the authority of your accounts. But if, .. mistake,
any transaction should be wrongly entered, the error must be rectified
by a new entry; and the wrong one may be cancelled by writing the
word Error in the margin.
A book, thus fairly kept, will at all times show the exact ºute of a
persons affairs, and have great weight, should there at any time be a
necessity of producing it in a court of justice.
-
--ORM OF A L.A.W. ºor.
3.
*JEREMlAH GOODALE, Albany, January 1, 1822.
Entered. Joseph Hastings, Cr. * Ict.
1|By 3 months' wages, at $6 a month, due this date, 1800
Entered. Samuel Stacy, - Dr.
1.To 2 weeks' wages of my daughter Ann, spinning
* yarn, at 75 cents a week, ending this day, 150
- Entered. Joseph Hastings, Dr.
1.To my order for goods out of the store of Anthony
Billings, - - - - - - 1150
Entered. Anthony Billings, Cr.
1By my order in favour of Joseph Hastings, 1150
15
Eatered. Thomas Grosvenor, Dr.
1.To the frame of a house completed and raised this
day on his Glover Farm, so called, 4000 feet at 2:
cents per foot, . - - - - - 10000
19.
Entered. Edward-Jones, Cr.
1|By his team at sundry times, carrying manure on
my farm, . - - - - 554
25
Entered. Thomas Grosvenor, Dr.
1.To 48 window sashes delivered at his Glover Farm,
so-called, at $1.00 . - - $48,00
Setting 500 panes of glass by my son John,
at 14 cents, - - - - - 7,50
10 days’ work of myself finishing front room,
at $1.25 a day, - - - - 12.50
74 do. of William, my hired man, laying
the kitchen floor and hanging doors, at 6:30
84 cents a day, - 7430
Entered. Anthony Billings, Cr.
1|By 2 galls. molasses, at 36 cts, per gall. 0.72
4 yds. of India Cotton, at 18 cents, 0.74
ºflannel shirts to Joseph Hastings, 2,16
- 362
Entered. Joseph Hastings, - Dr.
1.To 2 shirts of A. Billings, - - 216
- There put the name ºf the owner of the boºk, and first date.
-On-O--DAY. Bº-º-
Albany, February 12, 1822.
ºn-red.
1.
Thomas Grosvenor, Cr.
By my order in favour of Joseph Hastings,
Entered.
1.
Joseph Hastings, Dr.
To my order on T. Grosvenor,
Entered.
16.
Thomas Grosvenor, Dr.
To 3 days’ work of myselfonyour fence at $1,25
r day, . - - - - - 3.75
3 days' do, my man Wm. on your stable and
finishing of kitchen, at 84 cts. . - 2,52
2 pr: brown yarn stockings, at 42 cts. 0.84
18-
Entered.
1.
Edward Jones, Cr.
By 4 months' hire of his son William at #10 amonth,
24.
Entered.
Edward-Jones, Dr.
1.To my draft on Thomas Grosvenor, .
Entered.
Thomas Grosvenor, Cr.
Entered.
1|By my draft in favour of E. Jones, - -
28
Thomas Grosvenor, Dr.
1.To the frame of a barn, - - - -
Entered.
Anthony Billings, Cr.
1|For the following articles,
14 lbs. muscovado sugar at #12 prewt.
1 large dish, - - - -
6 plates, - - - - -
4 cups and saucers - - - -
1 pint French Brandy, - - -
1 quart Cherry Bounce, -
Thread and tape, -
- Thimbles, - - -
1 pair Scissors, . - -
1 quire paper, - - -
Wafers, 4; ink, 6; 1 bottle, 8;
Entered.
1.
D
To a cotton Coverlet delivered Sarah Bradford, b
11
Peter Daboll, -
your written order, dated 14 Jan. - .
Port-M up a day door-
Albany, March 1, 1822.
ºntered. Thomas Grosvenor, Cr. ºct
By cash paid me this date, . - - - 7500
__
Enwered. Anthony Billings, Dr.
1.To one barrel of Cider, . - - - 31 17
1 barrel containing the same, (from Tho
mas Grosvenor,) - - - 0. 58
- 175
7
Entered. Thomas Grosvenor, Cr.
1|By 1 barrel containing Cider sold and delivered to
Anthony Billings, . - - - - 0.58
-
Entered. Anthony Billings, Dr.
1.To cash per his order to George Gilbert, 24-32
15
ºntered. Peter Daboll, Cr.
1|By amount of his Shoe account, . . 34-48
Yarn received from him for the balance of
his account, . - - . . I
- 55.
Entered. Samuel Green, Cr.
2|By amount due for 12 months New-London
Gazette, - - - - $200
4. Spelling Books, at 20 cents, for children, 0 80
1 Daboli's Arithmetic, for my son Samuel, 0.42
2-blank Writing Books, at 125 cents, . 0-25
1 quire of Letter Paper, - - - 0.34
- º
24-
Entered. Notes Payable, Dr.
2By my note of this date, endorsed by Ephraim
Dodge, at 6 months, for a yoke of Oxen bought
of Daniel Mason, at Lebanon, . - -
Entered. Jonathan Curtis, - Dr.
*To an old bay Horse, - - - - 00
A four-wheeled Wagon, and half worn
Harness, - - - - - 42 00
Catered. Samuel Green, Dr.
*To cash in full, - - - - - -
Entered.
1.
Entered.
1.
Entered.
1.
Port- or a day Box-.
Albany, April 6, 1822.
TAnthony Billings, Dr.
To 2 tons of Hay, at $11 25, - $22 50
Amount of order dated March 26, 1822,
in favour of Fanny White, paid in 1 0. 54
º stockings, - -
Hire of my wagon and horse to bring
sundry articles from Providence, 3d 3 00
of this month, . - - -
12
Thomas Grosvenor, Cr.
By his order on Theodore Barrell, New-London, for
68 dollars, - - - - - -
Anthony Billings, Dr.
To 1 hogshead Rum from Theodore Barrell,
100 gals, at 50 cents, - - -
Cash received from said Barrell for balance
due on Thomas Grosvenor's order, 18 Oo
18
Entered. Jonathan Curtis, Cr.
2By a coat $14.75, pantaloons $5,00, - -
22
Entered. Thomas Grosvenor, Dr.
1.To mending your cart by my man William, $4 00
I.
Entered.
Paid Hunt for blacksmith's work on your
By Garden Seeds of various kinds. . - 30-56
1 pair-Boots, myself. §4.00, and 1 pair for
cºurt. - - - - - - 0. 59.
Setting 6 panes of glass, and finding glass, 0-56
-5
John Rogers, Dr.
To a yoke of Oxen, at 60 days’ credit,
Anthony Billings, - cº,
John, 33.50, - - - - - 50
1 pair of thick Shoes for Joseph Hastings, 1 25
Tea, Sugar, and Lamp Oil, per bill. ... O 68
Notes Payable, Cr.
By my note to Isaac Thompson, at 6 months,
6800
580.
- -
i
Pon- or A in a y Tºor. -
Albany, May 3, 1822.
Entered. Theodore Barrell, New-London, Dr. - |ct.
2To 16 cheese, 308 lbs. at 5 cents, . ºlò 4
217 lbs. of butter, at 15.2-3cts. . . 34
24 lbs. ºf honey, at 125 cents, - 3 00
- 5240
º -
Entered. Joseph Hastings, Dr.
1.To 1 pair shoes, 29th April, from Anthony Billings, 125
12
Entered. Anthony Billings, Dr.
1.To 84 bushels of seed potatoes, at 33-1-3
cents, - - - - - ... sº 00
8 pair mittens, at 20 cents, . - - I tºo
Cash, - - - - 14 00
– 4360
15
Entered. Joseph Hastings, Cr.
1|By 4 months wages, at 7 dollars, - - 3150
20
Entered. Theodore Barrell. Cr.
2By cash in full of all demands, - - - 5240
25
tntered. Thomas Grosvenor, Cr.
1|By his acceptance of my order in favour of Anthony
Billings, - - - - - - 5-100
Entered. Anthony Billings, Dr.
1.To amount of my order on Thomas Grosvenor, 54.09
|-Sept. 24-
Entered. Notes Payable, Dr.
2.To cash paid for my note to D. Mason, - 4000
The ºregon-example-Day Book. --------------of-the-way
---------------------nºr----and-credit-r- ano-
-------hould next-prepared-accordin-to-th-ºllowin-for-terin-th-
-------L-- int-tº-º-º-º-º-º-º-tº-
D--------------------------ºr-din-ut-
------------------------------- Thu-
-------------------------- --
------------------------------ --
----------
wº--------------------L--------------
º ------------------------------------ord
ºr-or-in-twº par-------------------------
ºur---ou-ºº-º-º-º-L--------------
ºn-ºn-º-º-º-º-º-º-º-o-end of the L-ºr--alphabet-in-le-hou-
--containin-the-very per-with whom you have-count-i-º-
--it------------her-u---
Pura--- a L-L-a.
Dr. Joseph Hastings.
Tº TT | -et.
Jan'y 5Tomy order on Anthony Billings for goods, 1150
26, 2 shirts of Anthony Billings, - - - 216
Febºy 12 My order on Thomas Grosvenor, - - 350
May 18, 1 pair shoes, 29th April, from A. Billings, -
Dr. Samuel Stacy.
1822. - et,
Jan y 5To 2 weeks' wages of my daughter, at 75 cents a
week, - - - - - - 150
Dr. Anthony Billings.
1822. | s et,
March 4To 1 barrel of cider, and barrel, - - - 75
10 Cash paid your order in favour of G. Gilbert, 243.
April 16 Sundries, - - - - - - - 250-
12 ditto. - - - - - - - 6800
May 12 ditto, - - - - - - 4360
25. My order on Thomas Grosvenor, - - 54.00
Dr. - Thomas Grosvenor.
º * Ict
Jan'y 15To the frame of a house, - - - - 10000
25 Sundries, - - - - - - 7430
Febºy 16, Sundries, - - - - - 711
28. The frame of a barn, - - - - 750,
April Sundries, - - - - - - 224
Dr. Edward Jones.
Tass. T-- * ºt.
Febºy 24To my draft on Thomas Grosvenor, - - 3-00
Dr. Peter Daboll.
º; - c.
Febºy 281To sundries, - - - | 551
--- or a LEGEa.
A hired lad, Cr.
1822. | | |ct.
Jan'y 1|By 3 months' wages due this day, at #6, - - 1800
May º 4 months' wages, at $7, - - -
Farmer, Cr.
-
Merchant, Cr.
1822. 3 ºt.
Jan'y 5By my order in favour of Joseph Hastings, - 1150
261 Sundries, - - - - - - 3.02
Febºy 28 ditto, - - - - - - - 355
April ditto, - - - - - - - 999
Judge of County Court, Cr.
Febºy |12|By my order in favour of Joseph Hastings, - #350
24 My draft in favour of Edward Jones, - - 3900
March 1. Cash paid me this day, - - - - 7500
1 empty cider barrel, - - - - 58
April 12 Amount of your order on Theodore Barrell, 6500
May 25 My order in favour of Anthony Billings, - 54
Labourer, Cr.
1- - * ºt
Jan'y 18By team hire at sundry times, - - - 564
Febºy 18, 4 months' hire of his son William, at #10, - 4000
Farmer, Cr.
18- | | 4 Irt.
March 151By sundries in full, - - - - - - 551
º
-on-or a Learn
Dr. Samuel Green.
º: - - c.
March 123to cash in full of his account, - - - 319)
Dr. Notes Payable.
1822. * Let
Sept. 24. To cash paid for my note to D. Mason, - - º
| |
-
Dr. Jonathan Curtis.
1922. - - * Ict.
March 28.To a bay horse, - - - - - - 2300
A wagon and harness, - 4200
Dr. John Rogers.
1822. - * Tel
April 25 to 1 yoke of oxen at 60 days’ credit, - - 5000
-
Dr. Theodore Barrell.
* 3To 15 cheese, weigntº IDs at 5 cents, T- *:::
217 lbs, butter at 15-2-3 cents, - - - 340.
24 lbs. honey at 12 cents, - - - 3.0.
ſº
INDEx TO THE LEGER.
B. I---- H. *--
Barrell, Theodore, - - 2| Hastings, Joseph, - - I
Billings, Anthony, - - 1
J.
C. Jones, Edward, - - I
Curtis, Jonathan, - - 2 N
D. | Notes Payable, . - -
Daboll, Peter, - - - R.
Rogers, John, - - 1.
G.
Grosvenor, Thomas, - 1. S.
Glenn, Samuel, - - - Stacy, sanuel, -
------ - L-º-R.
New-London.
Tºº, |
March 15|By sundries, - - - -
- -
|
- Tºº. b. * Irt.
March By my note to Daniel Mason, at 6 months, endor-
sed by Ephraim Dodge, - - - 4800 -
April 29 Do. Isaac Thompson, at 6 months, - - 9000
-
Danbury. Cr
1822. ºr.
April 18|By a coat, - - - - - - - 1475
A pair of pantaloons, - - - 500
Hudson. Cr.
18-2. - - * Ict.
-
New-London. Cr.
18- * |ct.
May 2013 yeash in full, - - - - - 5240
5240
-
QUESTIONS TO Exercise THE STUDENT.
What is the state of the following Accounts *
Joseph Hastings, Due 'ºseph Hastings, - - 431 09
Samuel Stacy, Edward Jones, - - - 7 64
Anthony Billings, : Notes Payable, - - - 90 00
Thomas Grosvenor. - J Samuel Stacy owes, - - - 1 50
Edward-Jones, : Anthony Billings owes, - - 189 05
Notes Payable, : I Thomas Grosvenor owes, - 1957
onathan Curtis, Jonathan Curtis awes, - - 45 25
Jºhn Rogers. Jºhn Hºogen-owes, - - 60 00
12 ----U-POR--.
.4 Farmer's Bill, or Account.
Ausuax, Oct. 21, 1822.
Thomas Yates, Esq.
To John Mornington, Dr.
18-2.
April 5. To 5 barrels Cider, at $200 . - - 410.00
20 bushels Potatoes, at 0.25 . - - 5.00
55 lbs. Butter, at 0,17 . - 9.35
June 6. 1 ton of Hay, . - - - - 10,00
July 15. 40 lbs. Cheese, at 0.08 . - 3.20
2 cords of Wood, at 400 . . 8.00
Received the amount. 37.55
JOHN MORNINGTON.
N. B.-To prevent accidents, care should be taken not to receipt an
account until it is paid.
A negotiable Mote.
New-Haven, March 21, 1822.
Six months after date, I promise to pay to William Walter, or or
der, (at my house.) One Hundred Dollars, value received in two yokº
of oxen. JAMES HILLHOUSE.
Tritis best to mention where the note shall be paid, and for what
it is given. Without the words, “ or order,” a note is not negotiable
A Receipt in full.
Received, Hartford, May 22, 1822, of Theodore Barrell, Esq. Fifty
two Dollars, in full of all demands. GEO, GOOD WIN.
In If the payment be not in full, write “on account.”
N. B.-For other useful forms, see the Arithmetick.
-Nºot"E.
The affectionate Instructor, who always feels a parental solicitude
for the permanent welfare of his pupils, cannot in any way so much
contribute to their success in life, with so little trouble, as to teach
them to understand this abridged, complete and simple system of
Book Keeping. It contains all the important principles of extended
and expensive works on the science; all, in fact, that is necessary to
be known by the Farmer, Mechanic, and Shopkeeper, relating to ac-
counts; and yet with very little explanation and repeated copying and
balancing the accounts, will be so fully understood and deeply impres-
sed on the memory of scholars of common mind, as never to beforgot
ton; while their knowledge of common arithmetick and practical pen
manship will thereby be greatly improved.
Tºwns.
UNIVERSITY OF MICH IGAN
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15 O6389 4565