■ ■■^,'/-V
' '- -i
,*
V
/
A
E
Y
TO THE
COMPLETE PRACTICAL
ARITHMETICIAN.
CONTAINING
ANSWERS TO ALL THE QIJESTIONS IN THAT
WORK,
With the Solutions at full Length, wherever there is the
faialleft Appearance of Labour or Difficulty;
THE WHOLE INTERSPERSED WITH
Several ufeful Notes and Obfervations.
TO WHICH IS A D 1) E r-j
A N
APPENDIX,
CONTAINING,
A Synopiis oPLogarichiTiical Arithmetic,
Sliev/ing chcir Natu.c, Couilrrudion, and Uie, in the piaineft. Manner
poifiblc.
Tables of Comoo^ind Intereft and Annuities,
Kxlending froiu One to Foity Vears.
Alfj, Gciicral and univcynl Demon i>iatioiis of the principal
RULES IN TK^ COMPLETE PRACTICAL
ART r H M E 1' I C I A N.
The whole tog-etlier forming the moil: Complete Sydem of
Arithmetic extant, both in Thec--^- •^■■'•A V-^'i^^t-i^^e.
:riE-Xafy Inrro-
B V T_H O -M A S
Teichcr of the Mathematics, Author cr
^ dudlon to ti:e Sc'.cnc- of Cii.oGR avh y, the
.^RJTHMETICIAN, ic<.
^h '-V- 1— 4^..-^^-
-i:
/,
LONDON:
FRINTEO BY JOHN CROWDER,
F 'J K r. LAW, A V E - M A R I A - L A N E,
4qo.
PREFACE.
A S this work makes its appearance with
■*• ^ the approbation, and by the particular
defire of feveral eminent Mathematicians and
Schoolmafters, it would be fuperfluous for me
to fay any thing farther in its commendation,
than^ that I have paid every attention to
corredlnefs and brevity, confident with per-
fpicuity.— 'In working the feveral queftions,
I have, in general, made ufe of the Ihorteft,
and mofl fimple method of folution I could
devife, and where neceffary, have explained
the feveral fleps thereof — I have fometimcs
folved the queftions by the rules, and fome-
times by the notes, in order to exemplify
both ; this will be of cqnfiderable advantage
to thofe who are not proficients in Arith-
metic, and it can be no detriment to the
a 2 ingenious
f^JL ^-.v O. 'O / V/ 'if
IV r R E F A c r.
ingenious Teacher, who is at liberty to ufc
what inethod of folution he pleafes, for I
have not the vanity to luppofe that my ib-
lutions are fo complete as not to admit of
improvement.
The Appendix commences with a Synopfis
of Logarithmical Arithmetic, which I have
had by me in manufcript above feven years.
This fynopfis I intended to prefix to a fmall
fet of rabies, on the plan of thofe publiHicd
by M. VAhhe de la Caille, The utility of fuch
a work was firft pointed out to me by Mr.
Landmann, ProfeiTor of Fortification and Ar-
tillery, in the Royal Military Academy at
IVoolwich, For, at that time, Sherwins Tables,
publifbed in 1705 and 17065 and Gardiner's
editions of the fame book, in 1741 and 1742,
were the only tables to be depended upon.
On the appearance of Dr, Hutton\ Tables,
which 1 believe are exceedingly corred, I
laid afide my defign.
In the year 1789 I publiihed the fub{l:ance
o{ xkixs fynopfis y in the fixft number of a pe-
riodical work, but by the unexpecled death
of
l^REFACE. V
of the ingenious condudtor thereof (Mr, J,
Davi/cnJ, the fecond number was never pub-
lifhed. At the inftance of fcveral refpe^'^able
contributors to that work, xhxs fynopfts is now
republifhed, as containing the plained r.nd
mofl comprehenfive indru^lions extant, for
performing Multiplication, Divifion, Involu^
tion, Evolution, &c. — Then follows, in order,
a complete fet of Tables of Compound Interefb
and Annuities, accurately calculated from i to
40 years. This was hinted to me as a mate-
rial improvement, by Mr. Hardy, an eminent
teacher of the Mathematics, at CottingbamyU^diV
Hull
I thought it quite unnecefiary to dem.on-
flrate every rule in the Complete Practical
Arithmetician, becaufe many of the opera-
tions carry their rationale along with them,
and an attempt to demonftrate a propofuion,
which is nearly felf-evident, fom.etimes occa-
fions obfcurity.
I have nothing further to add, than my
fincere thanks to thofe gentlemen, from whom
1 have received fuch liberal encouragement ;
a 3 and
VI
Preface
and more particularly to thofe, who have fa-
voured me with remarks, tending to render
this work as complete and perfedl as pof-
fiblc.
THOMAS KEITH.
Londeriy 21th September^ ^79^'
The C O N T E N T S.
The Contents of Part I. and II. follow in
the fame manner as in the COMPLETE
PRACTICAL ARITHiMETIClAN.-
THE CONTENTS OF THE APPENDIX^'-
THE nature and formation of the Logarithni^
page i:
A Jhort and eafy rule for the conftruSlwn of Logarith?ns^ .
either common or hyperbolic^ tllujt rated by examples 4
The ufe of a table of Logarithnn — """ 7
To find the logarithm of a FraB'ion — ——8
To find the logarithm of a Circulating Deci?nal ib
Two or more nmnbers being giveny to find their produ^
by logarithms — — — 9
T^o find the quotient of one number by another^ by loga^
rithms — — — — 1 1
To involve a giv£n numd>er to any power — ] 5
To extracl any root of any given power — jg
The application of logarithms to Compoufid Inter ejl 22
The conjiru^ion of Table I. being the amount of il. from
'I to 40 years^ at 3, 3f, 4, 4^5 and 5 per cent. ib.
The conjirutlion of Table ll. being the amount of ll. for
The conftruSlion of Table III. being the prefent worth of
ll. for years — — — ib.
The conJtru5}ion of Table IV . being the amount of I /. per
annum^ or annuities for years — 23
The
\\\i The Conten-ts of the Appentdix.
7 he conf.ni5iiQn of Table V. hc'ing the prefent ivorth of
\L per annum-t or annuities for years — 23
The conJlrvMion of Table VI, being the annuity which li.'
will purchafe for any number of years — ib.
ihe life of Table I. II. and HI. — — 33
The uf of Table IV, and V. — ^ '31
The tfe of Table VI. - — _ 32
I. De?nonJ}ration of the method of proving j^ddition^ hy
cajiing out the 7jines — — — 34,
II. DernoYiflration of the method of proving Aiultiplicatiouy
by cajVing out the nines — — 35.
III. Demon/haticn of the Rule of Three — ib.
IV. Demonp.raticn of the Rule of Five ^ i^c. 36
V. General princiyies and theory of FraBions 37
VI. ^emonjl rations of the rules of Circulating Decijtiah 40
VII. Demonfiratian of the ccnimon rule of Equation of
Payments^ on Cocker's, Hatton's, and Moreland*s,
or Burrow's, principle — — 42, 43
VIII. Demorifiration of the Rules of Fellowjhip^ by Air.
Adamfori — — — 44
IX. Demorflration cf the Rules of Lrfs and Gain 47
X. Demonj'iration of the Rules of Alligation -. 50
XL Dejmnjlration of the rulei of Pofition — 51
XII. Deimnjlraticn of the rules in Arithmetical Pro-
greffon — . — -- _« 53
XI II. Demoiifl ration of the rides m Geometrical Prc^
grejpon — . — ^ ^6
XIV. Demonjiration of the rules of Variations 61
XV. Demorijlration of the rules of Combinations 62
XVI. Demonjiration of the rules of Siinple Inter ejl^ by
Dicimals — — — ^3
XVII. Demonjiration ^Malcolm's rule of Equation of
Payments — — — ib.
XV III. Demonjiration of the ru'cs of Compound Inter eji^
hy Decimals — — •— 64
xix; J
The Contents of the Appen"dix, ix
XIX. J general Demonftratlon of the rule of Equation of
payments^ at Compound Interefi^ on Moreliind's or
Burrow's principle \ alfo on Kerfey's, Malcolm's,
Cocker's, and Hatton's — — 64
XX. Demonjlration of the rules for Annuities in Arrears^
at Simple Inter ejl — — — 65
XXI. Demonjlration of the rules for the prefent worth of
Annuities^ at Simple Interejl — — ib.
XXII. Demonftration of the rules for Annuities in Ar-
rears^ at Compound Intereji — — 66
XXill. Demonjlration of the rules for the prefent worth
of Aunuitiss in Arrears^ at Compound Interejl ib*
BOOKS printed for and fold by B. LA\\''^
At No. 13, Ave-Maria-Lane,
Price 3s. neatly bound,
THE
COMPLETE PRACTICAL
A R IT H M E r I C I A N.
CONTAl NING
Several new and ufeful Improvements,
ADAFTEB T© tHi V%t OP leHOQLI ANB r&IVATll
TUITION.
By THOMAS KEITH,
Teacher of the Mathematics*
* This work, which was underta'.cn with a vi6w to fur^
nilTi a complete fyltem of Pradicai Arithmetic, for the ufe
of Schools, contains more ufeful rules and obferfationa
than are to be met with in any fyftem of Arithmetic ex-
tant, of double the fize and price ; and thefc rules are
fucceeded by near two thou (and ufeful and inftrud\ive
Examples; bcfides a variety of Bills of Parcels, Invoices,
Bills of Exchange, &c. &c. — The nature and praftice
of Circulating Decimals ; and the rules of Lofb and Gain,,
Fello".\fliip, Exchange, Sec. are here thoroughly confi-
dered, and treated of in a different manner to what tluy
have hitherto been.
Books printed for B. Law,
Price IS. 6d. bound,
A
SHORT AND EASY
INTRODUCTION
TO THE
SCIENCE
O F
GEOGRAPHY.
CONTAINING
A concife and accurate defcription of the fituation, extent,
boundaries, divifion, chief cities, &c. of the feveral^ em-
pires, kingdoms, ftates, and countries, in the known v.'orld :
with the ufe of the terreftrial globe, and geographical
liiaps. Sec,
DESIGNED FOR THE USE OF SCHOOLS AND PRIVATE
TUITION.
llluftrated with the neceiTary engravings, and an accurate
map of the world, including the modern difcoveries.
The SECOND EDITION, correded and improved.
By THOMAS KEITH,
Teacher of the Mathematics, Sec,
^^ The favourable reception which this little treatife has
met with from the public, has induced the author to re-
vife the whole with the greateft care, and to make fuch
alterations and additions as h,e conceives will be a mean
of rendering this edition more extenfivelv ufeful than the
former.
^ OK S PRINTED F R B. L A W,
THE ACCIDENCE, or FIRST RUDIMENTS OF
INGUSH GRAMiMAR, defigncd for the ufe of young
Xadies.
By ELLIN DEVIS.
The Sixth Edition, ^with confiderable Additions. Price
AS. 6d.
MISCELLANEOUS LESSONS on fynonimous Exprcf-
?fions, defigned for the ufe of young Ladies.
By ELLIN DEVIS.
Price 2s. 6d.
A PRACTICAL ENGLISH GRAMMAR, for the
•ufe of Schools and private Gentlemen and Ladies ; with
Exercifes of falfe Orthography, and Syntax at large.
By the late Rev. Mr. HODGSON, Mailer of the
Grammar School in Southampton.
JFifth Editioa, with Improvements. Price is. 6d.
A NEW SYSTEM OF READING : or, THE ART
OF READING ENGLISH, praaicrlly exemplified in
almoft every word in ufe : and farther illuftrated from the
beauties of tl:ie whole Bible, arranged under different heads,
according to the moral virtues therein recommended, or
vices reproved ; with every word accented, and rules for
placing the accent. A f)ltematical arrangement, on a plan
fo entirely new as not to bear the lealt refemblance to any
thing of the kind hitherto atiemptcd by other Grammarians :
and, by the help of whic , pupils, whether F^nglifh or
F'oreigners, may be taught to read Englifh in one-tenth
part of the time ufually devoted to that purpofe.
By Mr. DU MITAND, Author of a fimilar Syftem for
reading French and oth«M- grammatical works ; Teacher of
Greek and Latin, anA of the tciip 'ncipal turopcai living
tongues. Pxicc 2s,
K E
TO THE
COMPLETE PRACTICAL
\ R I T H M E T I C I A N
PART
NUMERATION.
FORTY-NINE.—Seventy-five.—One thoufand and
feventy-five. — Three hundred and feventy-eight. — Four
hundred and thirty- fevcn. — T hree hundred and live. — One
thoufand and eighty -feven. — Forty-feven thoufand, three
hundred and eighteen. — Seventeen thoufand, three hundred
and forty-nine. — Ten thoufand, eight hundred and feven. —
Three hundred and fourteen thoufand, eight hundred and
fifteen. — One hundred and feven thoufand, and forty-eight.
— One hundred and forty-nine thoufand, three hundred and
eighty- feven. — One million, feventy-eight thoufand, four
hundred. — Thirty million, one hundred and eighty thoufand,
and feventy. — One hundred and eight million, three hundred
and feventy-four ihoufand, one hundred and eight.
(2.)
89. 750. 5,007. 10,087. 20,005. 685, .^60.
1,500,001, 27,365,000. 585,748,305. IT,000-f-I,J03
fji = 12,111, 50,000,0004-50,000-1- 5,coo + 50 =^
50,05^,050.
B
Simple Addition, Subtraction, kc.
SIMPLE ADDITION.
NL>t. Answers.
(I.) 26038
(2.) 180212
ij.) 1673039
{4.) 86S6587
(5.) 29S295769
NiTM. Answers.
(6.) 6898970405
(7.) 2079^-501
(S.) 81 Peterborough, 132
Lincoln, and 173 Hull,
SIMPLE SUBTRACTION.
KuM. Answers.
(I.) 8087801
(2.) 280007
(3.) 299S86 .
(4..) 212201
Num. Answers,
(5-) 33892
(6.) 45396
(7.) 4835656
(8.) 1344138
(9-)
Chriil: came into the world 4C00 years after the creation ;
hence 4000+ il^9=S'1^9 Y^^^^ ^^"^^ ^^^ creation.
(10.) 89 years old, 62 years I (11.) 34 years, and 46 years,
fincehedied. — 1789. |
(12.)
A + B + C won 97
B + C won 6q
A won
37
B -j- C won
B won
A + B + C won 97
A f- C won 62
B won
33
C won
60
35
25
SIMPLE MULTIPLICATION
KuM. Products,
(1.) 942694650
(2.) I ' 144SC221
(3.) 1R8194972
(4.) 28,^674870
(5.) 223C84518
(6.) 343353-^oo43
(7.) 297r949749<^
Num. Products,
(8.) 42378429339
(9-) 5;H937H3o
(10.) 40872864296
(it.) 179245888S -
(12.) 2639559272
(13.) 989830464
(14.) 8172S6228
Part I.
Simple Division.
Num. Products.
Num. Products.
(15.) 1132497912
(57.) 1 8765700000c oco
{16J 472087728
{38.) 191 69700000
(17.) 302020848
(39.) 15463500000
(18.) 244512951
(40.) 28610266500000
(19.) 3276600516
(41.) 93807952920000
(20.) 684S734752
(42.) 23190200250000®
(21.) 62244248^:^8
(43-) 330043
(22.) 10139541C480
(44.) 4459128
(23.) 2072213S22625
(45.) 8235460800
(24.) 320021 195962
(46.) 26156958360000
(25.) 556321146764
(47.) 2400355079025000
(26.) 17681 97301755
(48.) 137829543416436
(27.) 269510713983
(49.) 152415787501905210
(28.) 17529177479355
(50.) 1 21932631 11 26352690
(29.) 17464274403363
{ 51 ) 86439657760572480000
(30.) 26885596435656
(52.) 777'5
(31.) 260232989070535
{53.) lolhort
{33.) 287523274925880
(54.) 160010532
i33') 2734390884446865
{^^.) 40 difference
(34.) 2r20503567i869797
{56.) 17624 fum, 76945744
(35.) 16651582419409925
produd
(36.) 21153742978114592
SIMPLE DIVISION.
Num. Qjjot, Rem.
Num. Quot. Rem.
(1.) 874671172—1
(18.) 35685129—126
(2.) 157116523—2
(19.) 10823637 — 12
{3.) 101776234— I
(20.) 4760566—38
(4.) 1408142S— 3
(21.) 364993—43
(5.) 28341563—
(22.) 407294 — 1080
(6.) 70534296—3
(23-) I3I95133--1S42
(7.) 38588S1342-7
(24.) 125139201 I3CIO
(8-) 4597301659—6
(25.) 269577255S82 — 5561
(9.V 7 100057 147—9
(26.) 14243757748— 35411
(10.) 3400653124—10
(27-) 15395919— I22I4
(ii-) 3925476122— II
(28.) 3COOICOO 6347
{It.) 2859/3914-9
(29.) 131809655 104990
(13.) 272202735—12
(30.) 300335575 — 273118
(14.) 5203802—61
{31.) 9948157977—81605
(15.) 11805558—53
(32.) 101489 62899
{16.) 39096821—64
(3 3-) 15655794—9075
{'7') 34297^82—58
(34.) 2667376—42700
B z
S I XI r L E Division.
Num. Quot. Rem.
iSS-} 16871651—2944
(36.) 4C24416— 263149
(37-) 8909748—722934
(38.} 7264348958— 125715
{39.) 93191497743— 95257
{40.) 4087692937— 381715
(41.) 7943859—39
{42.) 1 19092— 12348
i)348
97
(5^0
2)194
97
471 greater J
I by
77 lers 5
note 6tli.
Num. Quot. Rem.
(43.) 59085714—84
(44.) 1258127— 1578— s
{45-) 123456789
(46.) 123456789
(47.) 714007 14374
(48.) 937143718740
149-) 1924530—652547
{50.) 119191753— 90107
(52.) 865
,5J-) 942
(14.)
8117119^^865^ Tto'niJ+Tr4'-i^5-=4^4 Anf.
ider.
Then 5889454-237 = 589182 A
I '<^5 4 c = 237 remainder, 1185 X 497 ^ 58694I
niwer.
^56.) i^
I 2
2)144 a dozen dozen
6
S64 fix dozen dozen
7 2 half a dozen dozen
792 difference.
(57O 17^493745-^759 gi^'es
2:5946 for the quo-
tient, and 731 re-
mainder ; hence 731
—5001=231 exceeds,
Anfvver.
Parti. Compound Addition 8c Surtr action.
KuM Answkrs,
COMPOUND ADDITION,
M. Answers.
509 a. 2 1. 1 8 p.
4797 a. 2 r. II p.
403 a. I r. I p.
12.)
(3.)
(4.)
(5-)
(6.)
(7-)
(9.)
(10.)
(II.)
(12.)
US')
(»4.)
('5.)
(16.)
(17.)
(18.)
(19-)
(20.)
(21.)
(22.)
(23.)
(24.)
(25.)
(26.)
(27.)
(28.)
£'
779
976
1709
567
684
205
223
240
d.
H
o
10
4
5
3
12
6
» I
10
si
8i
23821b. 1 oz. i6d\vt.
41290Z. i9dwt. 20gr.
.4621b. 907.. i4dwt.
407 oz. i4dvvt. 5gr.
51 1 lb. II oz. 3 dr.
35280Z. 6 dr, I fcr.
3J02 dr. I fcr. 18 gr.
464 lb. 5 dr.
3833 ton. 7cwt. iqr.
3030 cwt. 1 qr. 27 lb.
370 qr. 151b. looz.
347 lb. 7 oz. 6 dr.
471 yd. 2 qr. 1 n.
484611. Eng. 2 qr.
3821 ell. Fr.
4768 cll. Fl. 2n.
489 lea. I m. 6 f.
4487 fur. 35P- 5y^^s.
47 08 p. I yd.
644 f t . lb. c.
Nu
(29
(30
(3'
(32
(33
(34
(3S
(36
(37'
(38,
(39'
(4a
(41
(42
(43
(44
(45
(46
(47
(4B
(49
(50
(5^
(52
(53
(54
(55
(56
m
5P-
3209 tuns. 27 gall.
5422 p. 57g^^l- 2qt.
460 tier. 29 gall. I qr.
297 gall. 2 qt.
249 B.B. I fr. 6 gall.
323 A.B. 2 fr.
3i(foAhhd. 46gal. iqt
522oBhhd. 4gal. 2 qt.
529 ch. I b. 2 p.
3200 ch. iqr. 7 b.
3842 qr. 6 b. 2 p.
409 f. i2ch. 19 b.
4299 yrs. 8 m. 2 vv.
525 m. 4d.
ii27d. 2 hrs. 50 m.
4444 hrs. 23 m. 50 f.
X.2490 16 si
£' 397 I 4t
/•1729 19 3
>C- 551 15 3
£•^^^1 7 9
1241b. 10 oz. 3d\vt.
37 cwt. oqr. 17:5:1b.
255 yds. 1 qr. i n.
COMPOUND SUBTRACTION
Ni^M. Answers.
£' s. d,
(r.) 800 18 9^
(2.) 3254 19 ii,i
(3-) 75 o 8f
(4.) 497 ^o 71-
Num. Answers.
(5.) 26 17 c|
(6.) 30 II IOy
(7-)^ 34 13 9i
(8.)* 32 13 'i
In this Example, fo. ii^-j £, read 747/.
B3
Compound Subtraction,
Num. Answers,
(9
(iO
2-)
d.
oi
3i
£■ ■'■
lOJ I
474 J
9524 8
in all.
62223 3
mains to pa)
8073 19 4|
out in all.
paid
7i
re-
laid
63365
12
Lr re-
mains in hand.
(15.)
(16.)
(17O
(18.)
(19.)
(20.)
(21.)
(22.)
(24
(25
(26
{28
(29
(30
2464 16
lance.
3 lb. ooz. i^dvvt.
90Z. 17 dwt. 20 gr.
1 5 lb. 3 oz. 1 6 dwt.
20Z. 18 dwt. 21 gr.
791b. I ooz. 6 dr.
1 2 oz. 4 dr. 2 fcr.
13 dr. 15 gr.
81b. looz. 7dr.
.) 12 ton, I7cwt. 3 qr.
.) 2 cwt. 2 qr. 261b.
.) 69 qr. 2 lb. 140Z.
.) 1341b. i4'Oz. 13 dr.
.) 134 yd. 2qr. 3n.
.) i24cllEng. 3qr. 3 n.
.) 96 ell Fr. 2 qr. i n.
ell Fl. I qr. 2 n.
Nu
(3'.
(32-
(33-
(34-
i3S'
{36.
(37-
(3«.
(39-
(40.
(41.
(42.
(43-
(44-
(46.
(47.
(48.
(49-
(90.
(51-
(52-
iS3'
(54.
(55-
(56.
(58.
(59-
(60.)
M. Answers.
1 7 lea. 2 m. 6 f .
I f. 34 p. 5 yds.
4 p. 3 1 yd. 2 ft.
oin. 1 b. c.
2 r. 1 8 p.
2r. 34 p.
I r. 26 p.
2r. 39 p.
;oa.
37 a.
I a.
4a.
7t. 2hhd. 55 g.
67 pun. i4g. 3qt.
I tier. I g. 3 qt.
6 gal. 2 qt. I pt.
1 A.B. 3f. 6g.
1 07 B. B. I f. 4 g.
221 A. hhd. 1 g. 3 qt.
63 B.hhd. 2 g. 3 qt.
* 27 ch. ob. I p.
2ch. 3 qr. 2 b.
52 qr. 6 b. 3 p.
32 fcore, I ch. 13 b.
2 yrs. 1 1 m. 3 w.
127 m. 3 w. 6d.
147 d. 21 h. ^6 m.
79 h. 50 m. 5:4 fee.
3I. 9s.
I21I. 17s. 0;^d.
1500I. 2S. 6d. Dr.
520I. OS. 5d. Cr.
980I. 2S. id. ba-
lance.
1 80 1. II cwt. 1 2lb.
J y. II m. o w. 6 d.
8 h. 22 m.
difF.lat. 34° 35^ long.
165° i8^
* In. this Example the 74 ch. ought to be uppcrmoft.
Parti. Compound Multiplication, See.
COMPOUND MULTIPLICATION,
Num. Answers.
(l.) I 10 2
(2.) I 9 2
(3-) 2 o 5
(4-) -2 ' Si
(5-) 3 2 8
(6.) 5 6 Hi
(?•) 8 3 9
{8.j i6 8 7-i
(9.) 22 13 o
(lo.j 133 y^«' 3qr- 2n.
(ir.) 138 oz. 9dwt. i2gr.
(12.)
(13-) ^-3 19 4z-
(14.) 59 10 8
(15:.) 8 19 o
(16.) 60 14 pi
(17.) 208 13 9
(/8.) 154 12 3
(19.) 42 I 6
(20.) 123 17 9
fzi.) 819 6 o
(22.) 82 lb. 8 oz. iS dwt.
i6gr.
(23O 75a. 3^- 39P-
(24.) X.16 19 3
(25.J 19 10 81
Num. Answers.
/. s. d,
(26.) 12 18 A,k
14
8
(27-) 33 3
(28.J 83 2
(29-) ^37 7 3
(30.) 698 2 o
(31. j 743m. i^d. 6Ii.
(32.) 222 cwt. 1 8 lb.
(33.} 15 cwt. 271b. II oz,
7 dr.
(34-) ;^-343 18 7
(3S.) 2684 18 9
(36.) Z<^i 2 7i
(37.) 257 12 4
(38.) 62 I 7i
(39.) 15299 18 4
(40.) ^i 7 2f
(41.} 344 o 6
(42.) 1364 o o
(43-J 98 7 4f
(44-) I 14 2
(45-) ?66 5 3f
(46.) 17038 10 iii-|
(47.) 12422 2 7^-J.
(48.) 2875 o 7i
(49.) 1292 I 6|.i
(50.} 658 o \iy.x
COMPOUND DIVISION,
Num. Ans. Rem.
^. .. d.
(I.) '3 9 I
(2.) 15 19 9^—1
(3.) I II 5^—16
(4.) o 15 i|— 15
(5.) o 4 7I-- 122
(6.) 65 18 7i-,3_
(7.) 41 5 2 .>^
(8.) 9 ,8 3I--92
Num. Ans. Rem.
(9.) 9 yds. 2 qr. in — I
(ic.j 41 a. 3r. I p.— 5
(11.) 8 lb. 7 dwt. 15 gr. — 45
(12.) 7 cwt. 3 qr. 6 lb. — 30
(13.) 2Eng. ell, 3qr.— 107
(14.} 8ch. 13 b. ip.-||ij
(15.) ioz.7dwt.9gr.~-|J|
(16.JX1 17 6
(n-) o o 6^-^
8
Num. Answers
£.s. d.
(18.) o 17 III
(19.) 024
(20.) 7 j6 8|
(21.) O O 5I:
(22.) o o loj
(23.J o 4 9i
Compound D i v 1 s i o -x.
Num. Answers.
(24.) 1050— 175 -7- 150 =
X.516 8
(25.) 10 oz. ijdwt. 4gr.-i^V
(2f'.) I7cwt. 2qr. lilb. -^|
(27.) 36 fl. ell, I qr. 311. -/j
(2S.) 2a. or, i7i\-p.
(-9)
lb. oz. dw. gr.
2^)34 3 J I Hfilb. 30Z. i6dwt. 14^53 gr. wt. of each
— . wedge.
12
26)99(3 oz.
21
20
26)43 1( 16 dwt.
lb. oz. dw. gr. 02. dwt. gr.
34 3 II 14= 411 II 14
3f
171
24
26)374(i4gr.
114
m I
10 rem.
1234 14' 18
308 13 16I:
Guineas 1 543 8 lof
A nfwer 1 5:43 guineas, and 2 dwt. 6gr.
over ; or 1 54314^ guineas. — For if wc
divide 8 dwt. lOj gr. by 3I the quo-
tient will be 2 dwt, 6gr. = H guin.
(3C.)
Admit the weight of the drofs and hhd. to be t, then the
weight of pure fu^ar will be 13, and both together 14:
cwt. qr. lb.
Hence 14)7 3 14 whole weight
027 wt. of the drofs and hhd.
7 I 7 wt. of purefuL:^.r.
Part I.
Reduction,
. (3'-)
il. 3s. lo^^gd. the value of a fhekel, and i^6oz. lydwt.
1 2 o-r. wt. of a talent.
(32-)
AnAver
REDUCTION.
Num. Answirs.
(1.) 1055. i26od. 5:04c f.
(2.) I200d. IOCS. 5I.
(3.) 380s. 4.56od. 1 8240 f.
(4.) 1 1 55s. r386od. ^544of.
(5.) 524.0J farthings
(6.) 37249 halfpence
(7.; 22736d.
f8.) ^726 twopences
(9.) ^767 threep. SjOid.
(lo.) 2879 groau, n|i6d.
4006^ f,
(11,) 4343 fixpences
III,) 216 crowns, 432 half=
crowns, loS'o Ihill.
ai^ofijtp. i296od.
(13.) 1493 s. 5975 ihreep.
7 1 700 f.
(14.) 288od. 24CS. 12I.
(15.; 74I. I OS.
(16.) 100 guineas
(17.) 105I.
(18.) 2410 cr. 602I. los.
(19.) 77760 groats, 25920s.
5184 crowns, 1296I.
(20.) 282 of each
{2f.} loi 1 69groats, 337235.
134H9J halt" crowns,
6744I cr. i6S6i^l
(l2.) 9968 farthings
(23.) 100320 grs."
(24.) 14 oz.
(25.) 184800 grs.
(26.) 13841b.
Num. Answebs.
(27.) 23 pints, and 3 oz.
i4dwt. over
(23.) 42 tea-fpoons
(29.) 7200 fcr* i4400ogrs.
(30.) 204 oz. 17 lb.
(31.) 86962 grs.
(32.) 3cilb. 40Z. 3drs. iicr.
(jj.) 69 of each, and jjgf.
over
(34.) 368801b.
14 tons
lb.
30 t. 13 cwt. I qr.
312 lb. 4OZ,
131 cwt. 2 qr, 141b*
14625 cwt.
168 parcels -
853 common lb.
350 great lb.
105840 lb.
(36.) 4207
(37
(38.)
(39-)
(40.)
(41.)
(42.)
(45.)
(4+.)
(45-)
(46.)
(47.)
(48.)
(49-)
(5^0
(530
27 II parcels
5024 nails
864 yards
78 f fing. ells
23940 nails
9996 yards
9160^- yards
36516! yards
3768 f. 1 507 20 poles
70 miles
88000 yd. 264000ft.
3 1 68000 in 9504000b. c.
i« Reduction.
NvM. Answers. Num. Answers.
(5^0
m. f. p. f.
37 z 57 5
298 f.
40
1 1957 poles
16k
71747^6
^^957
f978 6
197296 feet
C56.) i74of. 2 1 71 miles
(57-) 4755801600 barley.
corns, or 1 008 qr. 1 6 cpt,
5268 b. c.
(58.
(59.
(60.
(61.
(62.
(64.
(65.
{66,
(67.
(68.
(69.
(70.
(71.
(72-
(73.
(7+.
(75-
(76,
(77r
126720 times
12374 perches
108 acres
5726 perches
19 tenements
8064 quarts
13 tuns
4725 g'*^- 37800 pts.
970I half anchois
8 of each
14592 pints
15504. pints
297216 half pints
17Q0 gallons
1408 pecks
12 lafts
7200 pecks
2320 facks
31556935 feconds
30617316^ feconds
(78-)
1789
4004
5793 years fince the
365-^ creation.
2115893^ days
24
50781438 hours
60
3046886280 minutes
60
182813176800 feconds
(79*) 25395102 hours
y. d.
1815 o
1788 89
(8c.)
10 Feb.
31 Mar.
30 Apr.
1 8 May
89 days
(81.)
26 276
977 2| days
ft. lb.
234 4
8
908
22
23)41976(341
507
156
lb.
26
22
30
16
'5
23
33
341 of each, and 331b. over.
Part I. Reduction,
(82.)
For 5s. 4d. (as printed) read 5s. 4fd.
II
I. s.
58 14
d.
6
a each
s. d.
5 4i
4
' ■ s. d.
I
8
6
4
57 12
20
1152
12
6
ople 11
I 1 6
parifii.
2 6=30 d*
013832
461 pe^
4
1S44 people, in all, partook of the charity.
(83-)
lb. oz. dw.gr, lb. oz, dw. gr.
371421X18 — 657718
From which fubtradl the drofs= o 7 i 14
oz. dw. gr. 65 o 6 4 =: 374548 gr.
18 14 10 ' ■
19 15 II
24 10 14 Then 374548^43297=811411 of
412 o each, or 8 of each, and 58 oz,
22 II 14 I3dwt. 20 gr.
90 4 i=43297|;rs.
(84.)
365 d. 6 hrs. = 525960 minutes in a year.
500 X 100 X 525960 = ^.26298000000 the 500
men would count in i year.
2629 8000000 ) I, ooo,ooo.ooo>oQo [sS^ll^j years, Anfwcr.
Rem. 676
12
R
EDUCTION,
(85.)
d.
h. / //
d. h.
365
600
5 48 55
2)365 6
1S2 15
II 5
60
66^
24
4383 hours
60
262980 minutes
60
665) 1 5:77880c (23727111 years
Rem. 34;
(86.)
oz.
dw. gr.
lb.
07.
5)14
II 15! =
: I i
ivoird. 44-^ ) 1 2 in a lb. troy. .
6 ijf
wt.
of a halfpenny 5^w. lo^yygr.
wt. of a guinea
15KI
rs. difference
(87.)
I. s. d.
500 o o he left in cafh
/. 84 10 6 y. S =i 422 12 6 in bills
92212 6 he left in all
120 + 20 == 140 o o his debts
4) 782 12 6 the refidue
1 95 1 3 1 1 the eldeil fon's (hare
4 )586 19 4 1
1 46 14 10 each of the other's fharc
rsrtl. The Role OF Three Direct. 13
The rule of THREE DIRECT.
If 2cwt. 3qr. 141b. : 61. 14s. 2d. : : I2cwt. 3qr. : 29I. ijs.
(2.)
If I2cwt. 3qr. : 29I. 15s. : : 2 cv/t. 3 qr. 141b. : 61. 148. id.
(3-)
If 61. 14s. 2d. '. 2cwt. 3qr. 141b. : : 29I. 1 js. : I2cwt. 34r.
(+•)
If 29I. 15s. : I2cwt 3qr. :: 61. 145,24!. : 2c\vt. 3qr. 14 lb,
(S-)
If 1 12 lb. : gl. i6s. :: i lb. : is. 9J.
(6.)
If lib. : jjd. : : 561b. : il. 63. lod,
(7-)
]f jlyds. : 2I. i6s. ^d, :: 28Jyds. : 23I. 2S. oid. |,
If40guln. : 56 yds,:: 1135-1. los.; 1514yds. p 201 SFIcin,
eils, 2qr.
(9-)
From the 14th of May, 1 780, to the 1 1 th of December, 1783,
are 1 306 days.
If 28 days : 25s. :: 1306 days : 5SI. 65. o|d. ^
(10.)
If IS. 6d. : I w, : : lool. : 1333^ ^* = 25 yrs. 8 m. i\ w.
(II.)
If III-. 5s. : 22oqrs. (ir 55yds.) : : looguin. : 2053jqri
==410 Eng. ells, 3jqrs.
(12.)
1. s. d.
30 quarters coft 76 17 6
150 ^ 361 II 8
gain 20
!lfi8oqrs. ; 458 9 2 : : i bu. : 63. 4id. J|*
14 The Ruli or Three Direct.
(.3-)
If I ell : 7s. 6d. : : 27 pieces each 34 ells : 344I. 51.
(14O
If il. : los. 6d. :: 475I. los. : 249I. 12s. pd.
(15.)
If 53cwt. 2qr. 51b. : 18I. 143. 9|d. :: i lb. : -'d.
(16.)
If 2547I. 14s. 9d. : loooiguin. :: il. : 8s. 2jd. 4^||.f|
(I7-)
If47i4l. IIS. icd. : 117]. 17s. 3id. :: il. : 5H ||HH
(IB.)
If 4714I. IIS. icd. : 235I. 14s. 7d. :: 350I. 14?. : 17I. los,
(19-)
Ifil. : 9d. :: 350I. 14s. : 13I. 3s. ojd.-j-
(20.)
If I cwt. : 2I. 14s. 9d. :: i4hhds, each 17 cwt. i qr.
141b, ; 282I. 1 25. ii^d.
(2..)
If 1 todd : 17s. 6d. : : 2 cwt. 2qr. i4ib. : 9I. 3s. 9d,
(22.)
If I cwt. : 5I. 5s. 4 : 157 foth. each 19^ cwt. : 16072I.
17s. 6d. to which add 5I. 5s. the fum is 16078I. 2S. 6d.
the whole expence of the lead. Again,
If 1 57 foth. : 1607SI. 2S. 6d. : : I lb. : i i^d. :rtKi Anf.
(23.)
If I oz. : 3I. : : 14 ing. each 31b. 11 oz. 15 dwt.
i3gr. : 2C06I. I2S. 9d.
(24.)
If4s. gd. : I yd. :: 722I. : 3040 yds. = 2432 ells Engliih,
which divided by 76, gives 32 ells Eng. in each piece.
(2v)
If 4 tons lefs 64 gall. : 247I. i;§, ;; j gall, : 5s. z^d. f*.
Part I. The Rule of Three Direct, ij
(26.)
If I yd. : 17s. 9^. :: ii'ji yds. : 104I. 5s. -jld. value of
the broad cloih, which deduced from i 24I. gives 19I. 14s.
4td. value of the baize.
If 5 yds. : liyd. : : 11 7-'- yds. : 35! yds. of baize he bought.
If 35|yds. : 19I. 14s. 47^.' :: i yd. : lis. z^d. ^'y the vaitie
of a yard of baize.
I. s.
If I hhd, ; 12 guin. : : 59 tuns : 2973 12
The freight - - 47 i «>
Loading and unloading 7 10
Cudom - - 24 o
Charges of the cellar 3 3
Prime coft of the wine 3055 15
The gain 150 o
3205 j;
If 59 tuns : 3055I. 15s. : : 1 gall. : 4s. i *:d. -jV/^ prime coft
of 1 gallon.
If 59 tuns : 3205I. 15s. : : i gall. : 4s. 3id. ||f|hemuft fell
it per gallon.
(28.)
Firfl 1 5 ells Eng. zqr. 3n. = 3iin. 311 X 15X7X5 =
163275 nails.
If ^yds. : 4I. 7s. 9d. : : 163275 nails : 8954I. 12s. 3i'^d.
(29.)
If 2^ hrs. 56m. : 36odeg. each 691 miles : : i hr. : 1045
miles, 3 ^y^ furlongs, the inhabitants upon the equator are
carried per hour. Again,
If '23 hrs. 56 m. : 360 deg. each 37 m. 2 f. 37 p.
5t ft. : : 1 hr. : 562 ra. 18 p. 8 yds. lift.^Y* ^^^ v^^-
bitants oi Londofi are carried per hour,
{30.)
^^ 3 + 3 + 3 + 2 =z II.
If II : 225I. los. : : 3 : 61I. ics. A, B, and C each
If 3d. : 61I. ics. :: 2d. : 41I. Dpaid.
C 2
%6 The Rule or Three Direct,
d. '^-^
I 20 -f- 3 = 40
1 20 -7- 2 ~ 60 If I ood. : 240 eggs : : ^d. : 12 eggs. Anf.
Tlie eggs cofl 1 00
Fir/?, While the hour-hand gees once round, the minute-
hand goes 12 times round, therefore the minute-hand gains
J I rounds eery 12 hours.
If II r. : izhrs. : : 2 r. : 2 hrs. loj-ymin. hence the hands
will be together loj^ minutes paft 2 o'clock.
SecoAaJj, While the minute-hand moves 5 minutes, the
hour-hand moves -j^^j of a minute, then 5 — ^K- = 4,\ minutes,
the minute-hand gains of the hour-hand every 5 minutes.
If4/jm. : 5m. :: I5ra. : 2 7y\min. hence the hands
Vrijl be 1 5 minutes (or 90 degrees) apart at 27-j\. minutes pafl
2 o'clock.
77?/>i//y, If J I r. : 12 hrs. :: 3 r. : 3 hrs. i6/,-min. hence
the hands will be together a fecond time at 1 6/1 minutes
paft 3 o'clock.
(33-)
IfzG.lps. : 3H. Ips. :: 3G.Ips. : 4l:H. Ips.
Here 3 greyhounds' leaps are equal, in diftance, 104!: hare's
leaps ; hence every 3 leaps the greyhound took he gained ^ a
hare's lenp, for while the greyhound took 3 leaps, the hare
took 4.
IfiH.Ip.
: 3 dps.
:: J 44 H. Ips.
: 864 G. Ips. Anf.
s. d.
(34-)
I. s. d.
If I p. :
If I yd. :
4 S ••
3 7 •
: i7sop.
: i749y^s.
: 3S6 9 2
' 3^3 1 3
Shipped for Jamaica goods to the value of 699 1 6 5
I. s. d. L s. d.
trig. : o 6 9 :: 47f g. : 161 6 o^
Ificw. : 3 15 7 :: 2-hd, each 7CW. 3qr. 151b. : S04 9 Jt-i^
Received frcm y<7'/rff;V^ - - - 9^5^52 A
699 16 5
Balance 265 18 9 v'j
Part I. The Rule of Three Direct. 17
(35.)
Ifiw. : 32I. 15s. :: 52 w. : 1703I. yearly expences.
3780 -T- 9 = 420I. land-tax.
3780 — 1703 -f- 420 divided by 20 = 82I. 17s. charit. dona.
If I d. : il. IIS. 6d. : : 365 d. : 574I. 17s. 6d. pock. ex.
Whole expences 2780!. 14s. 6d. this de-
duced from 3780!, leaves 999I, 5s. 6d. he lays up at
the year's end.
(36.)
s. d. g. I. s. d. t. hd. g.
If 3 7 : I :: 571 i 8 : 12 2 37|| quan. of wine bought
If 7 6 : I :: 419 no: 41 47^^ quantity of wine fold
8 o S'^zl^T quality of wine loft.
(37-)
cwt. 1. s. d. cwt. I. s. d.
If I : 14 19 7 : : 22i : 337 o 7^
90
cwt. Is, d. cwt.
If I : 12 17 6 ; : 17-;-
;f-427
7^
: 222
90
I 10^
£-312
I lOj
cwt.
If22i- :
If 17;- :
1. s. d. oz.
427 7I : : I
312 I o| : : I
d.
9I. 675 T
s. d.
If 5 6 :
(38.)
g. 1. s. d.
; I : : 41 14 6 :
gall.
1 5 i-f\ of rum and water.
84 of rum.
^Vtt of water put in.
C )
I 5 1 K V 1 » £ E P R O F O R T 1 O v.
(39-)
guin.
If 47p. : 21 X 2 : : 62 p. : 55^9 guinea! worth at 17I
guineas per hhd.
guin.
I^ 55tU- : i7t X 2 : : 65g. : 43I. 2». 3id.i°. Anf.
(40.)
If i7cwt. 3qr. lolb. : i6cwt. 141b. :: ilb. : 140Z.
7/,/^5^dr. which fubtraded from ilb. leave loz. SJlf f dr.
loit in every pound.
If lib. :'Sid.-f i^d. :: I7cwt. 3 qr. lolb. : 81I. 3s. 4id.
the tobacco ftood him in.
If i6cwr. 14.1b. : 81I. 3s.41d.-j- lol. los. :: ilb. : is.cd.
-Jiilhemulllellitpsrlb.
INVERSE PROPORTION.
(2.)
If 20m. ; 6d. : : icm. : i2d.
(3.)
If2S. : Soz. :: 2S. 6d. : 6oz. 6]^ dr.
(+■)
If 3 m. : i8oof. : : i m. : ^400 f.
Then 5400 — 1 8co =: 3600 foldiers. Anfwer.
If I yd. ^qr. : jyds. 2qr. :: :Jyd. : 24iyds.
Then a+^yds. + :| yd. = 25iyds. Anfwer.
(6.)
If 20cl. : 12 m. :: 150!. : 16 m.
(7-)
If 22 yds. : 220 yds. :: 40yds. : 121yds.
If 1 m. : 72cm. :: 5 ra. : 144 mCD, which dedud^ed
from 720, leaves 576 men, Anfwer,
Part I. T H E R u L E OF F I V s. 19
(9)
If (^ox. ; 7 colts. : : 2 ox. : 5-f colts.
If 7 colts. ; 87 d. :: 5-^ colts. : 105 days. Anfwer.
(10.)
yds.
If i;Jyd. : 1000X2!: :: |yd. : 4166yds. 2* qrs.
(II.)
From II Dec. 1786, to May 10, 1787, are i^o days.
From Sept. 5, 1788, to Chriftmas, 1789, are 478 days.
If 91 g. : 150 d. . : 661, 13s. 4d. : 214^! days.
478 — 2i4|§ = 263^Vdays. .1
If 661. 15s. 4d. : 2633-'od.. :: 4ol.^^438|-J days, hence
A is the perfon obliged , and mud lend Bij.ol. for438i-5-days.
(12.)
IfiSIb. : 100 ft. :: z+lb. : 75 ft.
The rule op FIVE,
(3-)
If 7 m. : 1 2d. : 126 a. 7X12X72 = 6048 dividend.
16 m. : * : 72a. 16 X i zS = 2016 divifor.
Then 6048 -7- 2016 nj days. Anfwer,
(4-)
If4it, : iSm. : 15I. 12s. (=3123.) 4JxiSx42o=:34oio dividend.
* t. : 72 m. ; 21I. os. (—420s.) 72 x 312 — 22464 divifor.
Then 34020 divided by 22464— I ^i^^ll ton. rr It. locwt. iqr. 4lb.-|-'^
(;•)
Ifiool. : 12 m. : 4I. 100 X 12 X 20 = 24000 dividend.
* : 19m. : 20I. 19 X 4 = 76 divifor.
Then 24000—76=315^51. = 315!. 15s. 9id.-j|- Anfwer,
(6.)
1 1 cwt. 2 qr. = 1 288 lb. 15 cvvt. I qr. 22 lb. = 1730 lb.
and 61. 14s. 2d. = i6i6d.
If r 2881b. : 150 m. : i6i6d.
17301b. : 64m. : *d.
1730 X 64 X 1616 z= 178923520 dividend.
1288 X 150 1= 193200 divifor.
Thenr78923520-rj93200=926TVAV=3l»i7S'2iVTV'Anf
20 Uniffrsal Proportiok,
(7-)
IfiSySfol. : 336 d. : 702 qrs. a2 536 x 112x702=: 1771S70464 div,
22536 fol. : ii2d. : *4rs. 1878x336 iz 63100S divifor.
Then 1771870464-a- 63 1008 :r 2808 fcldiers. Anfwcr.
100!. — 240c d 144I. 14s. 9d. — 34737 d.
If 2400 d. : 365 d. : 5I. 34737 x 495 X 5 r: 859740-5 dividend.
34737 : 495 d. : *1. 2400 X 365 rr: 8760)000 divifor.
Then S5974075-T-876C0001Z97 j^£^.7^1. - 9I. i6s. 3id.-f|f Anf.
(9.)
Ifiztaylors : 7 d. : 13 fuits.
* taylors : 19 d. : 494 faits.
12 X 7 X 494 =: 41496 dividend.
19 X 13 = 247 divifor.
Then 41496 -r- 247 = 168 taylors. Anfwer.
(ic.)
If ICO m. : 2s. 6d, : 20I.
* m. : IS. 9d. : 7I.
100 X 30 X 7 = 21000 dividend.
21 X 20 r:: 420 divifor.
Then 210CO — 420 zz 50 men, Anfwer.
UNIVERSAL PROPORTION.
Ifi261K : ioorr..Vs\^ 6s. 126 X 100X21 = 264600 dividerul
*lb. : 750 m. rN 21s. 750 X 6 = 4500 divifor.
Then 264600 -^ 4500 = 5S4lb. Anfwer.
(3.)
feive dtviillhd.
If24mea. : 3s.4d,Ly(i6m. t 6d. 24x40 (=3^- 4«I0 X48X4=i8432o
* mea..:2s.8d.^^48m. : 4d. 32(^=23. 8d.) X i6x6=::307zdivifo^>
• ferve
Then 18432c -7- 3072 =60 mcafurcs, Anfwer.
Part I. Universal Proportion. 21
(4-)
eat
If 3600m. : 35 d. : 240Z. k^ I quantity.
* m. : 43 d. : 140Z. rS 2 quantity.
eat
3600 X 3^ X 24 X 2=:6o48ooodividend, 45 X r4~63odivifor
'ihen 6048000 -f- 630 zz 9600 men. Anfwer.
If36coom. : 24 oz. : 35d. k>^ i quantity of bread.
4800m. : * oz. : 45 d. rN i (the fame) quantity.
eat
-3600 X 24 X 3 5 =3024000 div. 4800 X 4? == 2 1 6000 divifor.
Then 3024000 -r 216000 z: 14 oz, Anfwcf.
(6.)
coft
If 150ft. : 3ft. : 40m. Kr>3l. 5:4X 8 X 25 X 3=32400 divid,
54 ft. ; 8ft. : 25m. rS*l. 150X3X40=18000 divifor,
coji
Then 32400 -r 18000 = i\^%\*zz il. i6s. Anfwer,
(7.)
If lool. : 365 d. y^ 5I.
1627I. 103.: zijd. ^'^ *\,
gain
32550 (=31627!. los.) X2I9X5 = 35642250 dividcnJ.
20C0 ( — locl.) X 365 =r 730000 divifor.
Then 3 5642250 -r 730000 = 48 ——-1. zz 48I. i6s. 6d. Anfwer.
73000
(S.)
3yv.l3. -^qr. 11 I5qrs. Ss. gd. n 105 d. 257yds. 2 qr. rr l03oqrs.
• c.Ji
1? 15 qrs. ^^^^ I05d. 1030x105 — 108150 dividend.
I03cf qrs. ^^S *d. Then 108150 -i- 15 .-=: 72iod. — 30I. 8:, 4d.
cofl
(90
drink
If 336 m. : 4d. : i pt. : 30]. k^ 20I.
' 250 m. : *d. : i^:pt. : 24I. rS 500I.
drifik
336x4X30X500 zz 20160000 dividend.
2 50 X I i: X 24 X 20 =: 1 80000 divifor.
Then 20160000 ^ 1 80C00 zz 1 12 days. Anfwer.
2 2 Universal P r o r o r t i o n»
(10.)
- (iig a trench
If^-^Sm. : 5d. : johrs. k^ 5 deg. : 70yds. : 3 w. : 2d.
s4om. : 9 d. : lahrs. j^^ 6 dcg. : * yds. ; 5 w, : 3d.
dig a trer.cb
24CX9X I2X 5X''CX3x"- — 544320CO dividend.
■ 55<^X5Xiox6x5X3 = i5i2cooaivifor.
Then 5443204,0 -j- 1512CCO zr 36 yards. Anfwcr.
(II.)
huild
If 96m. : 48 d. k>( 1 wall.
*m. : 384 d. KS I wall.
huild
96 X 48 rr 4608 dividend.
Then 460S -T- 384 zz \i men. Anfwer.
(12.)
I cwt. = r 1 2 lb. il. 17s. 4d. = 448d,
ccji
If 1 12 lb. k^ 448d.
J lb. rS *
cofi
448 -^- 1 12 z: 4d. Anfwer.
(I3-)
hatter domon
Ifi2C. : iSpo. : 1 hr. k^ i caftle.
9C. : 24 po. : *hr. rN i caftle.
halter doixin
12x18 = 216 dividend, 9 x 24 — 216 divifor.
Hence tl:e nine 24 pounders would batter down the caftle
in the fame time.
(140
eat
If 1 2 ox. : 4 w. k^ 3 J a.
21 ox. : 9 w. ri * a.
eat
21 X 9 X 3-] =630 dividend, 12 X 4 = 48 divifor.
Then 630 -^-48 = 13^ acres, 21 oxen will eat in 9 weeks,
admitting the grafs does not grow ; but by the queftion 21
oxen will eat only 10 acres in 9 weeks; therefore, in
(9 — 4 = ) 5 weeks, 10 acres by the growth of the grafs are
ecjuivalcnt to 13I acres; hence 131 — 10 :=l ^\ acres, the
Parti. Reduction op Vulgar Fractions.
increafe upon lo acres in 5- weeks. Now the real quantity
which 12 oxen eat in 4 weeks without any increafe was {^
acres ; hence the /mvc" of increafe upon 24 acres is only
If 10 a.
24 a.
(18 — 4zr)i4weeks. Again,
h/crea/e
pv. k^ 3ia.
4W. rS * a.
24 X 1 4 X 3i = 1 050 dividend, 10X5 — 50 divifor.
Then 1050 -^- 50 =1 21 acres, the increafe on 24 acres in
14 weeks; hence 24 -f- 2: =45 acres, the whole quantity
of grafs which is to feed the required number of oxen 18
weeks. Therefore,
eat
If 12 ox. : 4 w. k^ 3|a.
* ox. : 18 w. rS 45 a.
eai
12 X 4 X 45 rr 2160 dividend, i8 X 33-21:60 divifor.
Then 2i6o-r-6o = 36 oxen. Anfvver.
Note. This queftion may be folved by feveral different methods, but
I apprehend the above Iblution to be the moft intelligible that can be
^iven, for thofe who uuderftaiid Arithmetic only.
REDUCTION OF VULGAR FRACTIONS,
(^0
(3-)
(4.)
(5.)
(6.)
(7.)
(8.)
(9-)
(10.)
Common meafure 3
Common meafure 6
Common meafure 9
Common meafure 375
. ai6 ^
Anfwer.
1080 540*
410 41
Anfwer.
10)-^ — ZZ-^. Anfwer.
'510 51
5) rr — . Anfw.
^745 349
i?) zz--^, Anfw.
9747 361
•^'7143 2381
T26
(la.) 14 ir — . Anlvv'cr.
9
(13.) 15 = ^^' Anfwer.
3094
(14.) 34ir . Anfwert
3
(15.) ^-5^ = -^
ro3
Anfwer.
(16.) 149 J — . Anfwer.
24 Reduction op Vulgar Fractions.
94^ 375'^- -375+ 94 _o7:^^ Anfwer.
y9 99 99
(i8.)
543 _ i74;4^^ccc — 174^4 + '-^1 _ 174938^^49 ..,
99999 5^999 99999
(190
473... ^- 9.
(20.)
1 6106 - -
1 - 80 ^ = . Ani wer.
' ^' 9
(21.)
^ = I. Therefore 7 7 -J = 78. Anf^ver.
(22.)
97vV Example is ivorked*
(23-)
^5 )4790 I ^9^U = '9^1* Anfwer.
(24.)
)o8 } 1 5 12 { 14. Anuver.
999)37S9+i( 376 -9W Anfwer.
(26.)
349 ) 374517+ ( ^^ll^i^y' Anfwer.
(27.)
T'i'/V Example is fJJorheJ,
* Should the reader not immediately comprehend the manner in wliich
this and the following Example are worked, he may confult Note 6th,
p=gc 40th, of the Complete PraQlcal Arithmetician.
Part I. Reduction- of Vulgar Fractions. 25
(28.}
(29.)
^t_ = --S.= -i^^^ := ^\ =-^^-. Anrwcr.
(50.)
; " I 5 5
(31.)
1789 'V^ SHXi7^9 9^9546
(32.)
394^^ _ 'IV! __^9^!i>l7i9_ 2 80985^0
894)!^-"' VtV 643333x^9^^89967'
(33-)
TZv'j Ex a '■/■pic is njoorkciL
(3+-)
4 9 lOI 13 4;. 2 ;< 101X13 IC5C4
41 ^ 41 X 4^ ''41 l6Sr ^ ^
-— X— X~T. X -'^rr : ~ — -— . Anf.
no 19 /p'8 7 110x19X12X7 175560
12
(36.)
l^X ^ y, -J-^- X ^^z^ "9 X j; X3 X 7 _ 50J
S S^ ^c<X- 1 8x43
43
D
26 Reduction of Vulgar Fraction's.
(37-)
^Jx^x ^'-^ X -^^iz iiilIii-LiH!7= 970785: p^^^^
1 0^ /3^ ^ 7x19x12 2261
^9 34
17
(38.)
54_-^055:
37I 150+
59 /^^ ^^4 I 59X376 22184
376
(39-)
TT-zV ExafHple is ixjorkeJ,
(40.
1 X 9 X 8 X 16 = 1152!
4 X 5 X 8 X 16 =z 2960 Kt
5X5X9X16ZZ 3600 f
II X 8 X 9 X 5 = 3960 J
ew numerators.
CX9X8X16ZZ 5760 Common denominatorji
1 152 2560 3600 3960
The Fradions are • — 7-, —r-y TTTl^ T^-^
5760 5760 5760 5,00
(41O
238+8 4. 7 ^ zb
5 9
HcHce^he new Fraaions are ^, and --.
Part I. Reduction of Vulgar Fractions,
r^ — ^i ^5 — 26 y.^- ~r 3 7 AS
a *
8x7X9X8—. 4032. Common denominator.
8)4032 7)4C52 9)4032 8(4032
504, 576, 448, 504,
41 26 37 53
20664 num. 14976 num. 16576 num. 26712 num..
« „. %co6a. 1AQ76 16576 , 26712
Hence the Jradtioas are ^> < — - — , — — » and ^ .
4032 4032 4032 4032
(43.)
J. —A. 11j —III fi_ _il^ i 5J" _^
15 "105' 94 ""470* 89|~7i7' 5f~~93*
105 . 470 . 717 . 93 denominators.
94 • 717 • 93
94 • 239 . 3;
5X3x7X94x239x311^73126830 the leafl: common denominator*
105)73126830 470)73126830 717)73126830 93)73126830
696446 1555S9 101990 786310
4 171 512 128
278 5784 num. 26605719 num. 5221S880 num. 100647680 num
Hence the new Fraftions in their lowefl: terms are,
7857S4 26605719 52218880 1006476.80
■ and
73126830 73126830' 73126830 73126830
Had I not reduced thefe Fradlions, the refult of each numerator, Scc»
would have been 45 times greater than it is.
* To find the Icaft common multiple of two or more numbers, or to
find the leaft number that can be divided by two or mo.e given numbers
without a remainder.
Rule. Divide the given numbers by any number that will divide two
or more of them v/ithout a remainder, and fet the quotients, together
with the undivided numbers, in a line underneath j divide this fecond
line as before, ahd {0 on till there are no two numbers that can be divided,
then the continual products of the divifois and quotients will give the
multiple required*
D a x.
28
Reduction of Vulgar Fractions.
(4+0
3X3X7X2 = 1 26
2X5X7x2 = 140
1X5X3x2= 30
1x7x3X5 = 105 J
New numerators.
5X3X7X2 = 210 Common denominator.
"5
Hence t!ie Fraftions are — -, — -
210 2IO
- — , and
10 210
(45-)
4 9 _36 8 _2SS 5 7 _35 8 _28o_
7 9 03 8 504. 9 7 63 8 504
._X^:rii25 7^63^44^ 524^9^76 Hence th.
S 03 504 8 03 504' ^ 504 504
Frafljons in their loweft terms are
0576
7^4"
1S9 2S8 2S0 i^4i
5C4' 5^* 5-4' 5^4*
{46.)
This Example is nvorhd,
(47-)
This Ex an: tic is <v:or'ked.
ExAMTLE. Find the leaft number that can be divided by the n'fie
• ,lti without a remainder.
I
-
3
• -4
.5.6.
7
• S
• 9
'
■ I
3
2
• 5 • 3 •
-
-t
• 9
I
• I
5
I
5 • 3 •
7
* ^
• 9
• I
I
^
5 • I .
7
• 2
• 3
Then 2X2X 3X5 X7X 2#< 3
Anfw,
I have given this Ru!e and Example hc:e, becaufe th<gr maybe occa-
fionaMy ufeful in reducing fraf^.'ons to a coniinon denominatcr (as in
Example 43d). For by this Rule the ieaft common denominator may be
i MvA\ and thea the fevcral numerators, by the latter plrt cf the Ruie
tj Prop. Stn, page 29th, in the Ctirficte PraEilcal ArittiKetician.
Parti. RfiDUCTioN of Vulgar Fractions.
29
Num. Answers.
(48.) 6H.-!-
(49.) 70Z. i^dr.
{^o } 70Z. 4d'vvt.
(51.) im. 6 f. 16 p.
(^2.) I r. 20 p.
(93.) 8yds. I qr. ifn.
()4-) 3^1^^- 7 gall.
(55.) 29 gall. iqt. i/-j-pt.
Num. Ans\^ers,
(5:6.] 14 weeks.
(5:7.) 20 bufhels.
(58.) 5s. 4d.
(99.) 6cwt. 3qr. 61b.
(60.) 10 ft. 216 in,
(61.) 30 gallons.
(62.) This Example is njoorhed^
i^S')
15s. lid. =r iQidr Then
191
= — ^I. Anfwer.
240
12 X 20
(64.)
5^d. =z ^M. Then -^ =i:|s. = i^s. Anfwer.
-'^ 9x12 108 54
I cwt. 2 qr. 61b. 3 oz. 8| dr. = 44600I z= *°^/°8 jj.^
401408
4014.08 14
^^^ — -^— —— cwu
9X16x10x28X4 25S048 9
(66.)
^ ^^ ^ 5x16x16
6999 1- I8IQ87
40 5760
416
izSo
40
■lb. avoird.
Hence -^ x
^ ,, -lb. tro5'. Anfwer,
460800
(^7-)
3qr. 3^n. = 15^ n.
Tten — i^ zz-if^ -^ cUs Eng. Anfwer.
9 A 4 A J I Eg 45 °
(68.)
.47 d. . jh. = 3;4.3 h. Then ^iil ~'J^ yrs.
8766 2920
155. 5-|d, =: 579 halfpence. Then
D 1
AnAv,
X I 2 X 20 400
Reduction of \''uLGiR Fractions.
(70.)
V of a groat. Anfwer.
(-■•)
locwt. 1 qr. 12 lb. rr 296960 drams.
Sc'.vt. I qr. 25 lb. i oz. 7-A-dr. rz 24.2967 ^\ drams.
242967YV 2672640 9
296960 3266560 II
1 , 128 ^, 128 128 4 , .
4b. 2fp. = p. Then ir — =-qrs. Anf.
(73.) V'^ yd. Anfwer. I {-j^.) This Example is -ivirkcd.
{74.} -ir| a. Anfwer. j {'-jS.) This Example is ^Morked,
(77-)
— X— X-r, = = — -X Anfwer.
7 12 /p{ 7x12x4 336
4
(7?.)
— ;- X — X-- rz — of apennv. Anfwer.
£?;'P 11 +
4
(79-)
^Tds. troy. Anfwer.
6 ii:5 "^12-
I I
X4X12 2SS
4
(So.)
I xi U 4 ^
X— X— =~awt. Amwcr.
;<p.o' I I 5
-^ X — X— = =: cwt. Anfwer.
5 2d 4 5 X 23 140
^ X^ X^=^lb. Anfwer
^l/^ I I 7
7
Parti. Addition of Vulgar Fractions. 31
(33.) ' >
7 7 24 60 6o ^233600. , . ^
-^ X-^ X— X-- X—zz:^^ feconds. Anfwcr.
71 I I 1 I 71 -
(84.)
— X -7- zi: of a hhd. Anfvver.
13 63 bi9
ADDITION OF VULGAR FRACTIONS,
T'Z/y Example is <worked,
587 280 2S0
(3.}
5 II 55 -8 8
Then -L +4|. ^2389^ i£l, ^^^^^
55 8 4+0 44.0
(4.)
^8 "8' 94.7_.~'ic4i' 314.1 "14157* ^" 9 "^ S' "^
405 2140 __ 43^^^''^4S 4- ^9065^^10 + 186872404- i;94o64o ^
1041 14157 ~ 3S'^'y9333 ""
i> -•« . Anfwer,
39299b3z
4of-^=-i., JorIl=^L, 4of^=^-l.
£^ 5 15 4 I 4 $ I 2
3 2
2 4 15 4 ^5 60 60
Addition of Vulgar Fractioj:*.
(6.)
9 ^ i ^UL —112]
ffi ^11 "* 8 -176-
2
'76
{7-)
5 176 bSo Sto
s. d.
41. = II ii-/f
is. =: o 4T
-id. — o ot-il
Sum 1 1 6i;-i\
(8.)
I. s. d.
^ of ll. lOS. s= I 2 6
~ of 5I. los. = 017 6
3^0 of icoguin. = 220
Sum 420
(9-)
oz. dut.
f lb. tr. = 2 8
^ oz. tr. zz o 2 J
Sum 2 lOv-
(10.)
5 I 80—335 p«Tf —
9cwt. 1 qr. 6 J lb. AnC
-^ X-^ =-^ells Eng. Jj x-^=: -ells Eng.
2 3
Then -^ -f - =-y- ells Ene. = 2 ells Eng. 4—- qr;
236" 6
(12.)
I X T7oo = 450 mile. } X ^,Vo = Tjy^c mile.
-A + :i':c + izhc = llMlmile = 3^"^- 25?. 37yo->-ds.
rof¥=vf<- =
r. p. ft. in.
2 20
3
7
Sum,
2 20 U yi^T
Fart I. Subtraction of Vulgar Fractions. ^j
(hO
Jx-^-3 hhds. Then 3 + - = 3-^ hhd. =:
4 ' 5 5
226— gallons. Anfvver.
4
^-^ X— = 20 budi. to which add -^ b. and the anfw^r i
' 7
20 -=^ bufliels*
7
(16.)
1 7 _ 7 J ' ' — ' J
4 I 4.5 24 ~" 120 '
J. 4- -j — ~-i = 2d, 2 hrs. Anfwer.
4. 120 3 12
(17.)
JLx-ixiLii2d.-f x-^=id.
^41 4 8 gf 24
852 //^ 6^0
8
21Q ? 273 106339 1 iQi . /.
4 24 640 1920 "^80
SUBTRACTION of VUL.GAR
FRACTIONS.
(I.)
T'^is Example is ^doorkcdt
(2.)
■^-^4:=f' Anfwer.
34 Subtraction of Vulgar Fractions.
(3.)
3 8 24* ^ 8 8 "-24*
Then -7-^^=12 =3!. Anfwer.
24 24 24 24
97 776* H5tt 799<^*
Theni^-ii^=i±llii. AnAver.
776 7990 3100120
(J-) (6.)
II
^8 ^^8 ^^4 I 713 713
(7-)
S6^ = 3^4— • Anfweib
15 ^15
(8.)
m, 288 2 C18 ? A r
Then — -^ — = cn-^. Anfwer*
5 45 9 9
(9-)
-J- zz—. Anfwer.
8 5 40
(10.)
^ 7-5 — ' 7tt
^i — _ I o
;t — 7.T
9i-*- Anfwer*
lof^zzl, Jof4:=i-^
II 9 99 5 8 40
rru 5 21 33761 „2o8r . -
Then -^- + 9 =^^i-^ == 8 — -- . Anfwer.
99 40 39^0 3960
Parti. Subtraction OF V^ULGAR Fractions, ^p
(12.)
s. d.
■II ~ 15 o
4s. -
o SH
Anfwer 14 3^-^
oz. dvvt.
4 lb. rz 10 o
1 oz. n — 1 2 I-
Anfwcr 9 - ^-
(14.)
— of -^ =z— lb. — ton. = 7 cwt, 2 qr. from which take !- lb.
;S 4 2 S
2
the remainder will be 7 cwt. i qr. 27-i-lb. Anfwer.
Cv)
2 10
pt.
gal. pt.
|hhd. = 2S 1}
i of -"- ~ -5-
Anfwef 25 I
(.6.)
i X^ =-2 miles. -2 _ J =11 = , 1 miles. Anf^r.
J 1 5 5 8 40 40
(•7-)
-I of ~I— =— !^ days, -^of ^ of-r-, =z^ days.
^ 4 12 ^ 7 16 /^ 896 ^
r-T = — 7T^ = 202—7—0. =: 202 days,
12 896 2638 2688
21 hrs. 45 m. 32y fee.
^6 Multiplication of Vulgar Fractions.
MULTIPLICATION of VULGAR
FRACTIONS.
T'.Js JixaTnjle is ivcrhd^ '
(2.)
~ X — = — . Ajilvver.
7 A^ 35
5
^ xj^=^ - 252 4. AnAvcr.
I ^^ 6 ^6
'6
111 il =1jS —111 — j^l Anfwer
27
'31 3 13
(6.)
Anfwer,
J X^ X-^ X^; xi- xi =^. Anfwer.
$ 7 ^ / II I 308
4
(7.)
4__._3. 7j'-_!i.. 35s __ 424? ^"
si ""4* 15 ""25' i29-j\ 15512*
^hen-lx^X-^=:-^. Anfwer.
? 3878
-^ X 1- X^' x^^ x^,^ =ii^^= 62-li. Anfw
5 II I ^^ $ 220 220
4
er.
Part I. Division of Vulgar Fractions. :
(9.)
7ft. 9 in. = 2^ ft. 3 ft. II in, = ^, eft. 3 in. = — .
4 II ^ -^ 4
31 xi-7 x^'zzl^ = i59r'- Anfwer.
4
(10.)
12 ft. gin. = ~, eft. 7 in. = — .
^ 4 -' ' 12
il^p-im=.,l. Anfwer.
4 - ^/ 16 '16
4
(ir.)
r . • 160 „ . 30
1 7 ft. q 1^ in. = . Q ft. Q in. = ^.
' ^' 9 4
if xi'=i^=i73irt. Anfwer.
3
DIVISION OF VULGAR FRACTIONS,
(I.)
T/v's Example is ivorkcdt
(2.)
16 7 112 cc . _
^9 3 57 57
(3.)
3
CJ ; X -; = - . Anfwer.
~ ; X — =: — -> Anfwer.
37 7 259
E
3.3 Division of Vulcax FiACTiow*.
I?; X ^=z^ = .-. Anfwer.
(6.)
2
7/'
X 1^=^.
7 7
An{v,tT
(7.)
•
■^X-^;xi:
7 9 $
16 F28 . .
X— =-T~' Anfwer.
3
(8.)
20
9 '
8 II
X -: X —
^ 3
__i76o
(9.)
= 6s-^^ Ar
"! = ■?
1 54I
' 93A
«.4763
8224*
239
7 '
8224
X — z^ —
4763
1965536
33341
(10.)
- = 58^'^^'.
33341
?'A
_ S6z
7' _
568
95 ^045 H9f ^^95
281 239
Then i^i . yr^M. -tll31 - , ^!?I. Anfwer
209 284
(II.)
r^Jx^i'; xix^x-l=i^=,-il. ;tnr.
$ ^ gf ' I I 19 114 114
^ 3
3
Parti, The Rule of Three Direct in, &c, fg
(12.)
21
tc-i z=J, Tlien &; X -^ = 21. Anfwer.
4 4 4 3
(13.)
•^ ; X — ^ = — ^. ' Anfwer.
12 108 1296
(14.)
-~ ; X ~ X — X = -%-. Anfwer.
RULE OF THREE DIRECT im
VULGAR FRACTIONS.
Tlis Example is <wdrhd*
yds.
If I'V X I J 'OS* 2d.-TV : : I : 88. Anfwer.
(3.)
If|lb. : 7s. 9d. : : ^fMb. ; j^l. 5s. 6|d.-f, Anfwer,
(4.)
If A of |xV X| : 2Vt1. : : |x 'i^ ; 68lr 13s. lo^d.^.
Anfwer.
{$■)
If -fell. : 5s. 3fd. : : 5t X 35^ : J2l. os. 4|d.-t. Anf.
(6.)
quarts. -
If iqt. : 3s.3{d. : : i4fVX-ioo8 : 2405I. iis. iiid.-s\*
Anfwer,
E 2
40 The Rule of Three Inverse, and
(7.)
ells. I.
IfjXjXl : |X| :: '^^dls. ; 199I. 153. 6id.f Anf.
(8.)
If I lb. : *jld. :: II hhds, each 4cwt. 3qr. 15-Jlb. ; 194L
los. 53.'^d. Anfwer.
The rule of THREE INVERSE
IN VULGAR FRACTIONS.
This Example is ^worked*
(2.)
U j^ift.br. : 27|ft. Ion. : : ^ft. br. : |73if. =:
1 9 1 yds. i ft. Anfwer,
M
If 4s. :d. : 57|yds. :; js. 8d. : i^j^^^yis. Anfwer.
(4-)
If I J yd. : n^ydr. :: lyd. : 39iyds, Anfwer.
(5-)
]f jjcwt. : ijtm. :: 9i-cwt. : 6-y'^ m. Anfwer.
The RULE of FIVE in VULGAR
FRACTIONS.
(1.)
This Example is fworked,
(2.)
gain
If I col. : 1 vr. K^ 4I-I.
4S011. : 7iyr. /\ * 1.
gam
12^ X^Jx -2,X-L =m^l. = ,7,I.2S.„ld..^. Anf
442 100 3200^ ' *
F^artl. RuLe OF Five in Vulgar Fragtioks, 41
(3-)
tra'vel
IfiF. : 7|d. : lalTirs. k^ *|+ m.
I F. : *d. : lojhrs. rN i47|m.
trauel
13 X ^^ X 1^7. ^i. X _L= ^ = 4lif^d. Ann
^2 8 32 294 150528 i^o^-21%
equal
jf5o_oo^gals . i_5fj^ . 2iin. k> I 1 quantity
* deals : V f^* • i ^"- *^ i J wood.
of
200
c'0'p;:^ K 7 5^ 4 6ccoo „ ,^ . -
V^T^7' X ^^X~ = -^=857^?ft. Anf.
(5.)
coji
If 1 3I ells : f yd. k> 5I guin,
33*: X| ells ; I X|yd. rN *guin.-
cojl
-ii-X-^X-^X-^X-7; X -^ X-^ = -^guin, 1=
4 i 3 4 i 27 3 486 ^
2
15I. 1 6s. o|d.-||, Anfwer.
(6 )
'dig a tre7tch
lf*4«m. : 5id. : iihrs. k^ 7^. : 232iyd. :3jyd.:2ivd.
ym. : *d. ; 9^^^- ^ 4^1. : 337iyd. rsly^- = 3iyd.
dig a treJ2ch
X^ J^Zt 4
i^^X-rX— X — X^Xi3^x-^; X-7-. X — X
I i I \ i t i a ^
E3
4* Promiscuous Questions in
PROMISCUOUS QUESTIONS in
VULGAR FRACTIONS.
3ofi=-d. Theni-r5=i. Anfwer.
0^3 3*9
3
(2.)
s. d.
17 o
I of Vs. = 14 2
3 of Vs. = 129
jf . 2 511 This fum multi-
plied by 7 J gives 16I. 16s. 8Jd.-j. Anfwer.
(3.)
47A
141 = Hll
62t'^ Anfwer.
(4.)
1 1 200I. Anfwer.
i!£ 2I_ = i?l the part I have left.
16 2816 2816 '^
80c 16000 2012CO, , ■ J 1 .V
2010 1 44
yaluc of the part I have left.
Parti. Vulgar Fractioks.
(6.)
loa =11. io(L =2i-.
8 7 56 II 19 209
Then i|+li=^, and ii|5_ ±L= J^llii = 45 2211
56 1 56 56 209 XI7O4 11704
43
.Anf
As 6— d. »
2
i-td.
(7-)
work : ; id. ; — of the work A can do in x day.
13
work : : id. : — ■
J?
± X 'I
3 — d. : — work : : id. ; —
Then j- A + -1. — _Li of the work they all can do In i day
13 17 10 2210 '
working together.
Tr '5^3 , 1 , J , 2210 687 , , .
If work : — day : : — work : — ZZ i days. Anf.
2210 I I 1523 1523
(8.)
In the fourth line of this Example, /or A, B, C, D, and E, In ii days,
read A, B, C, D, and E, in 12 days.
days, work dayi
19
13 :
IS .. —
l» : — : : X ;
I :-l 1
13
I :
Then,
"a+b+c+d+*
A + B + C+»+E
•* ' ■ 19
— yf^ A + B + * + D + E^
A + * + C + D + E
14 : — : : j : —
J4
* +B+C + D + E
>
44
Promiscuous Questions ii^
Then — ^ H 1 +• — z; — ^— ^ part of the work they
13 15 la 19 14 311220
would do in 4 * days all working together, tonfequently they would do
1 109233 109233
— of ^ or -— - part of the work b 1 dav.
4 311220 1244880
109233 1 I 1244880 43317
As — ^-—- work : — day : : — work : -^^ = ii
1244880 I I 109233 109233
days, the time in which they would all finifh the work.
Again,
109233
20313
I244SS0
14
~
I244SS0 "^
109233
1
-
43713
1244880
1244880
1C9233
1
12
=
5493
1244880
12448S0
109233
1244880
1
=
26241
1244880
109233
I
__
13473
1244880 13
1244880
part of the work A could do In i day.
B
C
D ^
E
Hence it appears that B would finifli the work the fooneft if left to
hixnfelf, the fra<aion oppofite B being the greateft.
• If the reader fliould not, at firft fight, be able to comprehend
whence thefe four days are got, he may caft his eye to the right hand
of the foregoing page, where he will ^txctivt four A^s, four Wsyfour C's,
four D's, iJidftur E's J confequently, they had each worked four da)'S.
(9-)
hrs. referv. hr.
As 6 : — : : I
8 : -^ : : 1
JO :
8
< B
C
Hence — + -^ + — = — of the refervoir will be filled in one
6 8 10 120
hour, if the filing cocks A, B, and C; are fet open together.
Part I.
Vulgar Fractions.
Agaiil,
4i:
hrs. rei'erv. hr
As 9 :
13 : --
£"2 1 Is
Hence — + — H — "^^ of *hc refervolr will be emptied i;
1287
tig cocks D, E, and F, are fet open*
9 I
2 hour, if the difcbarg
Then -i^ ^^ = iZi^ part of the rcfervoir will be filled
480 1287 154440
in I hour, if all the cocks A, B,.C, D, E, and F, are fet open together.
A, iliSi refcrv. : ^ hr. : , -1 referv. = iiiii2 hr. = 8 hrs.
154440 I I 17409
Ki m. 16 ^ f. Anfwer.
^ 5803
* Gentlemen acquainted with the doftrine of fuiJi will readily per-
ceive, that this folution is on fuppofition that the weight of the column
of water in the refervoir, and the prelTure of the atmofphere, are
uniform during the influx and efflux of the water j if thefe were allowed
for, and the dimenfions of the ciftern given, the folution would be dif-
ferent, but thefe minutiae are ufelefs to one acquainted only with
Arithmetic.
(10.)
98. 9jd.-^. Anfwer.
Firft -of-? of c4 =
25 "^ 8
14 _ 112
95I" """763
(II.)
80 '
Hi
94-
86
4.70
^^T=f'
Thenii9^
43 U ^1
^ ^0 .. Z/;^^^ $^__ i2QX43Xt7 _ 94299
^0'^^^fi ^0^ $ 20X94X109 20+920
40 94 109
20
the numerator of the new fradion.
46 Promiscuous Questions im
Again,
^71 792' i"~ 7 ' i5i""34i' ^8 » '
7 5*9
_^55^^^^/^ ^^ 4^/J_ ^^^X7Xj)9 _ 2705'?
^ 341 ^ 22x341 X 4 3000S
22 4
the denominator of the new fraftion.
3751
Hence '^^^^'^^'^ = 94299X3^^ _ 3?37'5549 ^he
VoVoV 2^ops>^iM^ii^ 6929241725
25615
fra^on required.
(12.)
Flrft,
Let — reprefcnt the number of marbles in the ring*
— of— = — A fnatchedoutofthe ring*
515
-I of -^ =: — B fnatchcd from A.
« 5 ao
at T
— 1— — z: — rcmamdcr in A's hands*
5 20 4
•1 of -i =: i. C got from A.
5 4 20
J. — -^ =: — A had then left,
4 20 20
123
— — — n: — A left in the ring.
1 5 5
— of ~ := — E got.
12 5 20
3 1 II ^ «. . ,
— — — . — — D ran oft with.
5 20 20
2
Hence at the end of this fcuffle there remained with A —-, with B
20
— , with C — , with D — , and with E —
20 20 20 ao
J*art I. Vulgar Fractions. 47
Second,
-Z- of — =: — A and C fnatched from Dt
II 20 20
20 20 20 100
-i of -Z = -Z-. B got.
5 20 100
2. ^
20
7 28
-^ = E got.
100 100
^ +
_L zr -^ B had in aU.
100 100
33
100
II 26
= C had in alt.
100 100
i-of
II
,22 2 ^
100 100
2 2 12 .
-\ — A had in all.
100 20 100
— — — 13 B had in all.
100 100 100
Hence at the end of this fcuffle there remained with A , with
100
^20 . , « 26 . , ^ 20 , . , ^ 22
B , with C — — , with D , and with E ♦
100 100 100 lob
Third,
J , 22 2 ^, 22 2 20
— of = Then = E had left.
J I 100 100 100 100 100
12 2 14 1
— — A rr ■ A had, after receiving — of E's.
100 100 ICO II
3 ^ 26 6 _. 26 6 20
J. of = Then = C had left.
13 100 100 100 100 100
^14 6 20 5
'■ ■ ■ + — ZI ■ A had, after receiving ~ of C*s.
100 100 100 *' 13
Heace after their difpute was thus fettled, there remained with A -
100
with B , with C — , with D , and with E : fo that
100 100 100 100
there might be loo marbles in the ring at firft, and if fo, each boy put
in 20. This queftion is unlimited, but the above folution is the molt
natural that can be given, at leaft in tiiis place.
48
Addition op Decimals.
('3.) ,
11 of — =: 1^ left the elder fon ; -i— li ^^ — the remainder.
79 I 79 I 79 79
35
44 _ J540
of:^ =
79 79
6241
left the younger fon.
3S r542 -.|l£i left to both the fons.
79 624.1 6241
-i «- 12£i — iiL rcfidue allotted to the Vfldow.
I 6241 6241
U ^ iil2 = iii2 of the Eftatc worth 500I.
79 6241 0241
If IliiE. : 5£?1. f. I22!e. : 7901- 4S. oJd..ti what was
6241 I 6241 * 49
allotted to the widow.
If 4^'e. : i^l. : : -^E. : 25471. 6s. iiW.-l the value
6241 I I •' 49
ef the whole Eftatc.
If 4w. : 57-l-yds. : 12/^ ft. : ifb.
ccj}
4w. : 57-l-yds. : 12/^ft. : if b. L^ 342II. .
4w. : 34|yds. : ii^ft. : 2^b. KS *l
cojl
By working this ftating, the Anfwer will be found
308I. 4s. 2|d..|||f
DECIMAL FRACTIONS,
ADDITION OF DECIMALS.
Num. Answers.
(i.) This is do7ie at length,
(2.) 27-2087
(3.) 88-76257
X
Num. Answers.
(5-) '835-599
(6.) 397*547
{7.) 31-02464
Prirt I. Subtraction, &c. of Decimals.
49
SUBTRACTION of DECIMALS.
Num. Answers.
( [ . ) This is done*
(2.) 51-722
(3.) 2-7696
(4.) 1571-8;
(5.) -6946
Num. Answers,
(6.) -89575
(7.) 603-925
(8.) i379'25922
(g.j 99'7o5
(10.) 17-949
MULTIPLICATION of DECIMALS.
Num. Answers.
( r . ) This is -ojorkcd at length,
(2.) Ditto.
(3.) 26-99178
(4.) 10376-283913
(5-i 2'7S?39^"^65
Num. Answers.
(6.) '020621 1250
(7.) -28033797099
(8.) 175-26788396
(9.) -00043204577
(10.; 215-67436625
CONTRACTED MULTIPLICATION of
DECIMALS.
(I.)
This Example is ivorkcd at length.
(2.)
(3.)
54*7494367
475"7io5<54
357427-4
4946143-
21899775:
142713
3S32460
19028
109499
476
21900
285-
274
^9
16
4
2^8*67756
62-525
5^
Division of Decimals, &c.
3754-+078
^75437'
262S08546
11263223
1501763
187720
26281
2253
2757-89786
(v)
4745*679
9454-157
332^975
23728+
4746
1898
237
19
4
3^66163.
DIVISION OF DECIMALS.
(I.)
T^is Example is 'worked.
Num. Quot. Rem.
KuM. Quot. Rem.
(2.) 487-74—37378
(3-) •13956—583
(4.) 1918;!. 528—696
(5.) -004735—0
(6.) '01666, &c. — 640
(7.) -006944, &c.
680
(8.) i-36832?i7— 3425
(9.) 12976*8062 — 474
(10.) '004958x31 — 69466
CONTRACTED DIVISION of
DECIMALS.
This Example is inorkai at length.
(i-
(3-:
(3 80C06 .
8864 495'783i69)i7493«407704962(35'2843
14-7 261990
662 14008
65 41S2
5 216
18
3
Part I. Reduction of Decimals, (j i
(4.)
5-476 56 i8)98'i874^7poco( 1 1 '5834036625
i34aiS-lL^
494525720
(5.)
706976300
28S5I3560
14-7349
5)47i94-375457(3202-8S6a
2989529
4^539
13069
lie I
ICZ
2
34216706
310459
56161
5303
ai;
47
IlEDUCTION or DECIMALS.
T/:is ExaiTiple is ivothd.
NwM. QuoT. Rem.
(2
(3
(4
(9
(6
(7
{«
(9
(10
; .•57142^—4
) '0041152263 — 91
) 'IS
) '20835, &^c.— 8
) 15-38461—7
) •^241379— 9
) -026178010471 — 79
) This Example is do?ie,
) -37291666, &C. 1.
Num. Quot.
(ii.j -0062^1.
(12.)
ii'9-
12 7-818181, &c.
20 4-65 1 51 5, &c,
•2325757, &c. I
(13.) •12968751b,
(14.} -05 oz.
(15-
50Z. 4 dr. = •3281251b. avoirdupois. Now i lb. avoir-
dupois = 6999-5 gr» troy.
_. "■ 6qqq*c •328r2c „ „
Hence -^-^ H^ ^= '39873453 lb. troy. Anf.
5760
(16.)
•168751011. Aafwer,
(17.)
-•6339285714 cwt. Anfwer.
F a
52
Reduction of Decimals.
(i8.)
51b. I oz, 3dw-t. i3grs. =: 33685 grains.
Then 33685 -^ 6999*5 = 4'8i 24866061 86 lbs. avoirdu-
pois, which divided by 112 lb. give '04296863 cwt. the
decimal required.
Nu
(19.
(20.
(21.
(22.
(23-
(24.
(25-
(26.
(27-
(28.
(29.
(30.
i32'
M. Answers,
Num. Answers.
) -3125 yards.
) '55 ellsEng.
) '0084359217 miles.
) '48125 acres.
) 'C099206349 hhds.
^33') I qr- 23 lb. 3 oz,
13-44 drs.
(34.) 3 cwt. 2 qrs.
{3S') 15*04 drs.
[36.) 6dwt. 4-32 grs.
) '1 171 875 chaldron.
(37.) 1-2 n.
} '07472051165 hrs.
(38.) 2qr. i-5n.
) -12251.
(39.) i4p. 2 yds. 2 ft. 5 in.
) -2685 acres.
i'i2S b. c.
) -10:051136 miles.
(40.) 2r. 22 p.
) This Example is nMorked.
) 15s. I|d, -2
} 9-o522d.
(41.) 30 gall. 3qt. i-968pt.
(42.) 156 d. 12 hrs. i-^ m,
5 if. 36 thirds.
) 7s. loid.
(430
'475
20
f. 9-500
*375
add
.b. 9-875
12
d. 10-500
4
f. 2-000
9s. lo^d.
A
(44-)
•75'
£
ft. 2-255
3*036
•573 add.
. in. 3-609
3
b. c. 1-827
. V
2 ft. 3 in. i-82 7b. c. Anii
Part I. C I » c o L A T I N o Decimals. jj
'AS-)
3 fur. 2j p.
(46.)
4cwt. 3qr. 41b. 1 1 cz. 4*224drs,
(47.)
3Cwt. 2qr. 261b. 12*8 drs.
- (48.J
70Z. i4dvv't. 2i'6grs.
CIRCULATING DECIMALS,
___ 3 I 135 5 ~"
_ ^ _ ^ , an I ^ _ ^^^^ _ ^^.
.^ 6 2 , 162 6 , 769230 10
9 3 999 37 999999 ^3
999 39 99 i^
(3.)
' 999999 ^43 ' 99 ^^
142857 1
(4.)
T^i^/V Example is luorhd,
,,38. =IiizLi2=Iii=±; 7.54,3.= mirZi -
■ 900 (;oo 37 990
746S_ 3734 — 7 ll2; 043'54' = ^^^^4 — 4 _. 435° _.
990 495 495' '^ c^yc^GO 99900
^9 . .„..,> ^ 3754—371 _ 3379 _ 49
666 ' ^7 i4 — ^^ — ^^^ .— 37 ■^- i
990 990 330' ^^^^^—999990 333330
F 3
^4- Reduction op
(6.)
Tor '7' 5', read '75'; and/c/r '4'5', read '45'.
Th^ . / — ILZLl— ^— 11. .4V8' = lli^=^=:—
" '^ 90 90 45' 990 990
217 , 93 — 9 S4 7 47543 — 47
— ; •093' — ^zr rr — ; 4*75'43' n: ■
495 9C0 9-0 75 ^ '^^^ 9990
^74 134 .^^.o,., _ 9S7--9 _ j6?_.
~ — r- rr 4—7- j •oc9b'7' zr rr -7 ;
1S5 185 99000 16500
90 so
(7-)
^his Example is ivorkedm
(8.)
)2I0 3 ~2-f-3-T-2-r2
771^-76' ^^'"^ 16 =rS =:4 =2 = 1. Here the
denominator vanifhes after four divifions ; hence the decimal will be
21C
£nite, and conf.ft of four places, viz. — iz •1S75. ^This Ix-
1120
ample is the fame as the firft in Mr, Bcnrycajile'^s Arithmetic, page 154,
third ed tion, where, by inadveitency, he makes the decimal infinite,
and to confift of fix circulating fijjures.
4')-4- =— i tlien
/iibo 290'
{9-)
-7- 10
290'^ 290 — 29.
29 ) 9999» &^c. to 28 nines. Here 28 nines are made ufe of be-
fore o remains, and the denominator has been abridged once 5 hence the
tlcc'mal wiil c.nfift of one finite decimal, and 2S pure repetends, viz,
-It- rr •co'344Sz75S62o689655i724i3793i/
IIOO
(10.)
\ 12 T
Firft, 12 — =-.
/132 II
J I ) 99 ( Here are two nines made ufe of } hence tlie decimal will b&
J2
infi.iite, and confiii of two pure repetends, viz. — - r: •c'9'.
Parti. Circulating Decimals, ^^
Secondly,
> 80 _ lb
V 1T5 "" 27'
'^7 ) 999 ( Here are three nines made ufe of before o remains j
hence the decimal will be infinite, and confift of three pure repetend>,
80
VIZ. m s'92'.
135
Thirdly, 9") — — — . Then
yi35 15
8 a^u„_ -^ 5
15 = 3
3 ) 9 ( Here is one 9 made ufe of, and the denominator has been
abridged once, therefore the decimal will contain two places, one finite,
, .... . 72
the otner mnnite, viz. zz "\V»
135
Fourthly, -^. Then f ^ "^"^ ^^ '^.! "^*
8544 8544 — 4272 — 2136 — 1068 zz 534
= 267.
267 ) 9999, &c. to 44 nines. ( Here the decimal will confift of 5
finite decimals, and 44 pure repetends.
Fifthly, 3)'-^=^; thenV "' "J
■'/792 264' 264 ~ 132 ~ 66 — 33.
33 ) 99 ( Here the decimal will confift of five figures, three finite,
255
and two infinite, viz. zn "12JQ'6'%
792
ADDITION OF CIRCULATING
DECIMALS.
('.)
77'/V Example h nuorked^
(a.)
D'ljfimilart S'lm'ilar. Similar and Conterminous*
67-34'5' — 67-34'5/ = 67-34'54545/ 4545
9«6'5i' =: 9'65'i6' rr 9'65'i65i6' 5165
•2'5' — •25'2' — •25/25252 ..... 5252
17*47' = 17-47' = I7*47'77777' 7777
•5' =z -55' — -SSS'SSSS' 5555
I'he corrcB fum gs'^^' '^9^4-7' two to carry,
55 A D D I T r o N o p
(3-)
DiJfi>nUar, Sitr.'ilar. Similar and Cc7iternnnous,
•4'75' = •47547' =: •47547547' — • 5475^
3-754'3' — 3'754'3' = - 3*754'34343' •— 4343
64-7'5' n 64-757'5' — H"JSYS1^1S' •— 7575
•5'7' = .:75y = 'SlS'-^SlSl" •— 5757
i7'88' = •i78'87' =: •i78'S7887' .... SS7S
Sum 6q'742'o3ii2' three to carry^
(+•)
DlJfrrMar.
5
nilar.
Similar ar.d C'^v.tcrmimui.
•5' =
•55'
= •55'55555555555' •— 555^
4-37' , =
4-37'
= ^-hi'iiiiniiiii' '• in
49*45 7 =
49 •457' .
= 49'45'7 57 5757 575?' — • 575
•49' 54' =
•49 54
— -49-54954954954' .... 954
•7-345' =
•73'457'
= •73'45734573457' .— 345
55-62'0978c43-^5C-3' three to carry
is-)
DiJltr.Uar.
Similar. Similar and C»nt€rrr.irBUS.
•l'75'
~
•1/51
= •I7'5i75»' •••• 7517
4*-5'7'
\ —
42-57'5'
= 42-57'57575' — • 7575
•37'53'
—
•57'5:
s' = •??'53753' •— 7537
•59'45'
—
•59%5' = •59'45945' •••• S459
3 75V
3-75 4'
= 3*75 45454 •— 5454
Sum. 47'4-''544Si' three te carry^
(6.)
Dijfnilar.
Similar,
1 65- 1 'C- 4'
ri i65'i6'4i'
147 -oV
— 1 47 -040'
4-9'5'
= 4-95'9'
S4-37'
— 94-37'
Similar end Conterminous,
n i65«i6'4i64i6, See*
zz I47«c4'c40404, See.
— 4-95'959595i- ^^-
— 94-37'777777, -^'C
4-7'i23456' — 4-7i'234567' — 4-7i'234567', &c.
•"aw 4i6'25'42S76i, &c.
* If thefe Girculatirg Figures were made conterminous, they ;v:.uld
.■^un cut to 42 p'aces of pare repetcr.d. 5 for the left common mukiplc
of th - number of figures contained in each circulating decimal (by the
Ruk, page 27th, o.'thisKeyj is 42. Thus,
The iff, 2d, 3d, 4th, 5th, Ciicu'aring decimal
Confiftsof2)3 ,2,2,1,7 piacs of pure repeteads*
Then 2 X 3 X 7 — 42,—— The aLove f^iution is carried far enough
for acy pra(^ical p urpofe»
Part I. Circulating Decimals, ^-j
SUBTRACTION of CIRCULATING
DECIMALS,
(I.)
^'^is Example is njoorked,
(2.)
Di[Jlm'dar. Similar, Similar and Contcrmlnsus*
47-53' = 47-53' = 47*53'33' .— 333
i-7'57' =: i7S'77' = i-75'77' —• 577
■D'ff' 45*77'55' one to carry*
(3-)
ViJJimtlar, Similar, Similar and Conterminous,
i7-5'73' = i7-57'35' = i7-57'35' •••• 735
34-57' = H-57' = i4-57'77' •••• 777
Diff* 2*99'57' one to carry*
■ ■! ■
(4-)
^ijjlmilar. Similar and Conterminous*
17*43' = 17*433' •••• 33?-
iz-345' = 12*345' •••• 555
Diff, 5-087' one to carry*
Is'-)
Dijffimilar, Similar, Similar and Conterminous*
l-i27'S4' = i*i27'54' = i*i27'54754754754' ••• 73^
•47'3S4' = •473'247' = *473'S473S473S47' — 3^4
•653'70oi62So907'
(6.)
DiJJlmilar, Similar and Conterminous,
4-75 = 4750' •••• 000
•375' = '375' •— 555
DiJ^, 4*374' one to carry.
58 Multiplication or
(7-)
VifftniUar* Simlar and Conterminous^
4-704 rr 4*794o'oo' .... 000
.J7'44' ~ •i744'74' •— 474
Pif. 4'6i95'25' one to carry^
DljfitnUar, Sml'ar amd'Corterminout*
J-457' = 1*45777' •••• 777
•3754 = '3754° •►" ^^^
t>lg. i-og2<;7'
(9.)
D'ljfmilaf* Similar. Similar and Cttitermncuu
J-49'S7' = i-49379'37' = i-49379'37' .••• 93
•J47S zz '14750' "zz .147 5000' .... 00
Dlff. 1-3462/37' .-. 9S
MULTIPLICATION of CIRCULATING
DECIMALS.
{•■■)
This
Example is
^jL'Gfked,
(^->
(.3-T
•3754'
4-753'
^4*75
7-437
18772'
... 2Z
33273' ..• 35
26z8i'i
... TI
14260'© ... 00
i5oi7'77
... 77
i90i3'33 .. 33
3754'444
... 44
sne tc carry^
33273'333 — 33
5-537S05'
3 5 '3 50540' ^r.etoi
Part I.
Circulating
(4.)
66
88
2Z
92 1 1 6'
2i4938'8
12282222
prod, '014523727' one ^0 curry.
(5.)
-14752'
•1497
303125 ... 55
i3259o'o ... 00
58928'88 ... 88
14732'222 ... 2'2
Prcd- 022054136' one to carry.
(6-:
337890' ..
i5oi73'3 ••
37543'33 ••
262So3'333 ..
00
33
33
33
•268397290' one to carry*
Th^ next Example is iccrked.
IS printed No. 6, by niijlake,
(7.)
zdpericd,
•37'54' — 754
17-43
ii2'64' ... 264
i5o'i9'o ... 190
262'83'28 ... 32S
37'54'754 — 754
^•5445'37' one to carry.
Decimals.
(8.)
59
zd period,
4*73'5' "• 35
7-349
42 6 1 '8'
iS94'i'4
i42o'6'o6
33i4'7'474
.. 181S
.. 1414
.. 0606
•• 7474
34«8ooii'3' one to c-Hrry,
(90
id period.
4.i'42857' ... 142857
• 1797
29o'ooooo'*
372'857i4'2
290'cooco'oo
4i'42S57'i42
000
857
000
S57
'74447'i4-iS5' tne to carry*
(10.)
zd period,
7'H'93' ••• 493
5-43
2144'So' ... 4S0
2S59'73'9 — 739
3574'67'46 ... 74^
38-8209'66' one to carry.
(II.)
zd period,
.40'705' ... 0705
7-345
203 525'
l62'820'2
35^
820
I22'lI5'2I ... 152
284'935'493 «.• 549
2.98978'742' one to carry.
(12.) Thii Example is ivorked.
* See the note to the rule, p. 42.
6o
M u
(130
1-475
1-754'
LTI
P L I C A T]
[ON OF
(MO
173-715 ■
3-7545' •
9)5900
9)86.8575
6555'
7375
10325
1475
2.5S7805'
9650S3'
694860
86S575
1216005
521145
652«2226i83'
(»50
•37504
•7153'
9)112512
i25ei3'
187520
37504
262528
Prod. '26827861 3'
(16,]
5-" 534
•^735'
9)287670
319633'
172602
402738
57534
Prcd. -99553453'
(170
(20O
9)262451
291612'
187465
262451
Prod* '28411362'
54;
(18.)
^I'li Exatrfle is iccrked*
(19O
*Tl''is Example is worked*
4-7157
3-754
3-7 540;rfuWJ</. 18862S
— 2^5785
330099
141471
17*7027378
17702
I
Prod. i7'7045'o82
•7378
. 770 &-C
ore tt/
carry.
Part L Circulating Decimals.
6x
■007 5
00
(21.)
47-1937
•OC75
(".)
'•7 543S
I
4-37595
1-75434
'OOl ^neivmuh 2359685
3303559
•35395275
353952
75
3539 — 52j &«
35 — 39. &c
... 35» ^^'
'•75434«^ww«//. 1750380
. 1312785
1750380
2187975
3063165
437595
Prod. •357528o'3' tivl? to
• • • ■- ■ — carry.
(230
371473
•7'53H'
1485892
371473
II 14419
1^5^365
26003 1 1
279771.17522
279771, Sec.
2, &c.
7'676(,t)4i23o
767690
7
(16.)
3-745
1.47'
9)262i8'8S8888S8
2913*20987654'
149S2'22222222
3745'555555555
4121
67 &c.
320
222
555
Prod. 279773-9'7295'
Pred. S'SIS°'9^7^5^2^' ore to carry
(24.) 7his Example is worked.
(25.)
4-57
2-45'
9)2288'8888S8S8
25432098765'
i83i'iiiiiiii
9i5'555555555
Prod.
432
III
555
(27.)
5-7195
1-788'
9)45756444444444
50840^49 3 827 16'
4 575^5 444444444
40O368'888888888
57i95'5555555555
ii-24o'98765432' ove to
■ ' ■» carry
Prod. 10-23 1649'3827 1604'
,.. 049
.. 444.
,.. 88S
- 555
ore to
carry%
Bt Multiplication or
(28.) {29.)
3-7S3*' 7H-32^'
•3425' 3-456'
9)i8766i'iiiiiiii 9)4^8593*33
2oS5i'2345679o' ... 123 4762i'48' ... 14.3
75o64'44444444 — 444 357i6i'ii ... iii
i5oi28'8888SSS8 ... 888 285728'888 ... 888
ii2596'6666666666 ...666 2i4296'6666 ... 666
Prod. i'2556S7i'2345679o' two to Prai* 2469«i748'i4' one io.carry,
(30.) Tb'ii Example is ivorked* (32O
zd period*
(31.) •4i'32.' ..13a
td period t 1*432'
■573 4 — 34
1-57.3' 9)S2'64'264264
9)i72o'3'o303 9'i825T584' ... 91
— , i2396'396396 ... 39
i9i'i4478' ... II i65'28'5285285 ... 28
4oi4'o'4040 ... 40 4i'32'i32i32i3 ... 21
a867'i'7i7i7 ... 17
573'4'343434 ••• 34 Prod* •59i8'i3i46479' onetocarry.
Prod* •9022o'3367o' cm to carry
(330
2dper:od»
(34-) 3*7'534' — 7534
^d period* -3757'
7*00430 ... 0^30
4.7005' 9)262'743' 2743, &c.
9)35o2'i5o' 21, &c.* 29'i93, &c. ro4ijfgtfrM
i87'673', &c.
3891278, Sec. 262'743'2, &c.
49C3'oio'3oi, &c. ii2'6o4'26, &c.
28oi'720'l720, &c.
Prod. I"4I04'7 circu!. conftjis of "ifi
Trod. 32 '9241 • • • • &c. fgurei.
* This and the following Examples would have taken up too much
room, had I continued them far enough to recur ; I have therefore cun-
tirmed them no farther than is neceflary in pradice, prefuming, that
when the youug itudent has attentively perufed what I have written on
this fubjed, he will be enabled to work any kind of circulating decimals,
with as much kwlJty as common or finite deciihals, without the help of
a tutor.
Part I. Circulating Decimals, 6^^
(350
2d pcn:d, (S^O '^'^i^ Example is vuorked,
5'437'i5' ... 715
• 37^5 3' (37' ) *I'hh Example is 'worked.
9)16311-47' 14, &c.
(38.)
1812385, &c.
27i85'78'5, &c. For I'lis' read I'l'i' s''
38060' io'oic,&c..
j63ii'47'i47i, &c. i'73'5'
-47053' 17
Frod, a'oi4647 . . . i;c. i"7i8 - -
—————— . 1*710 netu
376426' ... 66 "^ — muU
(39-) 47053'3 • • 33 I
54-7'53' 3'-9373'33— Zl Unetocarry,
3-4573' 5+ 47C53'333— 33 J
54.599 ■
■ 54*699 ruio •80837626'
3iii6o'..oo i ■ mul. 808376, &c.
3iii6o'o..oo I
ao744o'oo..oo i
i38293'333-33 \oneto
172866'6666..66-'
8083, &c.
^)0, &c.
carry.
72866'6666.. 66-' trod. -81654 &c.
j89'ii2676o'
1891126, &c«
1891, &c. (40O ^'^'^ Examplt it worked*
Prt* i89'3oi97 • • Sec, (41.) Tiiis Example is worked.
(42.)
2d per lid,
4-57l'37' ..• J37
•148078
'M957'3'
3657o'97' ... 097 1495
3i999'59'9 ... 599 I
3657o'97'o97 ... 097 I. '148078 ntwmult,
i8285'48'5485 ... 485 V — _
457J'37'i37i3 ••• lil-^ one to carry.
•6769i9529'92'
67691, &c.
676, &c.
6, &c.
P;W. -6837571 . . . ; &c.
G 2
6+ Division
(43r)
2«/ period*
5-7U9Y — 93
4-7488 4*75'35'
47
457195 I ... 51
457i95'i'5 — 15 / 4-748« kcw ps'J^.
228597'5'75 .,. 75 L
4ooc45'7'575 ... ti^X two ts carry.
^^8597'5'7575 ... 7$''
27-13910419 3
2713, &c.
2, &c.
Prorf. 27-16627 • • . • ii04
DIVISION OF CIRCULATING
DECIMALS.
Tlis Example is fworked^
^his ExampU is ^worked* ■
(3.)
For 7-54'o3, r^/7^7-54'o3'.
Then 14*25 ) l'S^'03%03^03, &c. ( -52914669, &c.
Rem. 7 8
(4.)
•075 ) 4-3i73'33* &c. ( 57-564'
Rem. 33
(5.)
178? ) I4-937V7575* ^'c C 8-3543+» &^-
Rem. 1583
Part I. Circulating De-cimals. 65
(6.)
•00456 ) 43-s'75'57557^' &c. ( 955^*'^472* ^<^'
Rem. 343
(7.)
Th's Example is ixorhd,
(8.)
This Example is fworked*
M
quotient
i'7'59' ) 47*34Jooo ( 26-90424, &c.
•J 4.7345
neTrdivifor 1*758 ) 47-297655 new dividend
Rem. 1 1 8
(10.)
quotient
3*7S'3') -35^780^ ( •0937> ^^'
37 35^8
new divifor 5-716 ) "3482622 new dividend
Rem. 730
(...)
quotient
•4'5678' ) 17*34200000 ( 37*965, kc.
17342
new divifor '45678 ) i7'34i82658 new dividend
Rem. 17388
G 3
65 Division or. Sec,
(12.)
quotient
1 V^j*;' ) '374530000 ( -26309, &c.
' 37453
new divifor 1*4234 ) '374492547 new dividend
Rem. 10241
(13.)
.- this Example, for I3*5'j69533', read iS'S^'^S'iSS'y '^'^
i^e Example is ^worked right.
^his Example is ^worked.
This Example is <^csrked.
(16.)
This Example is ivorked*
(17.)
For 357V, read'SSl'i'-
Then -357'-*'-= •357'^72'* and 49'5'735'. = 49'573'557
quotient
49'573'557' ) 'SSl'^l^' ( -0072, &c.
4957 35
new divifor 49-568600 ) 'S'il^}'! new dividend
Rem. 34308
(18.)
S — ^ SS' ,
'555 ) '1 5400000 ( 'i-]7AT quotient
Rem. 41 5
Part I. Practice.
"3' ^ 'Si's' quotient
5 3
new divifor '^O '33° new dividend
Rem. 1 9 i
6^
(20.)
The divifor nxjants a point before it. Then '7' = •7^7 77'
quotient
•7'777' ) 4'5'732' ( 5'8799» &c.
4
new divifor '7777 ) 4*5728 new dividend
Rem. 177
PRACTICE.
This Example is nvorked.
Num. Answers.
d.
U
(3
(+
(5
(6,
(7
(B
<9
(la
(II
(12
('3
o
o
17
7
8^
TT'M /f ivorked,
IS 12 5
3
31
7
29
43 .
77»;V is ^worked,
29
5 2t
Num. Answ
(14.)
I
(I?.)
16
7
(16.)
I
7
(17.)
66
7
(18.)
23
13
(I9-)
8
7
(20.)
27
6
(21.)
13
S
(22.)
32
^5
{23.)
74
3
(24.)
3
12
(25.)
3
3
ERS.
1 1
n
li
lot
1
lol-
68
P R A
Num. Answers.
NUM
£. s, d.
(26.) 85 14 7i
(68.
{^7-) 35 15 4j
(69.
(28.) 19 iO Ji
(70.
(29.J 47 3 Oi
(71-
(30.) 9 16 33.
(72-
(3I-) '5 19 '
(73.
(32.) 11 5 Hi
74.
{33.) 171 II lO^
(75-
(34-) 53 4 iij
(76.
(35.) 49 18 8
(77-
(36.) 25- r4 II J
(78.
(37-) 167 2 7i
(79.
(38O 63 13 lOX
(80.
(39.) ,85 15 6
(81.
(40.) 275 6 oi
. (82.
(41.) 19 II I
(83.
(42.) 15 3 loi:
(84.
(43.) 19 12 6
^ll'
(44.) 160 3 ii|
. (86.
(45.) 217 6 il
(87.
(46.) 227 16 9
(88.
(47-) 283 13 3t^
(89.
(48.) * This is ivaried.
(90.
(49.) 16 H 4l
(9i->
(50.) IS 8 Hi
(92-
(51.) 31 J ^o.
(93-
(52-) 37 5 3f
(94-
(53-) 278 7 7i
■ (99-
(54.) 32 17 8i
(96.
(55.) 28 17 6
■ (97-
(S6.) 339 5 4f
(9^-
(57-) 22 8 3i:
(99-
(S8.) 289 14 3i
(100.
(59-) 232 8 9,
(lOI.
(60.) II I li
(102.
(61.) 306 14 it
(103.;
(62.) 24 10 lOi
(104.
{6i.) 33 4
(105:.
(64.) 251 9 4i
(106.J
(65.) 392 ^6 9,
(107.J
(66.) 34 9 6a
(108.J
(67.) 265 13 »i
(109.J
* lo-tbis Example, ^
'or 4576,
Answers,
£' s, d.
10 I
6
10
II
4
12
o
18
15
S
11
5
12
3
35
27
428
371
28
44
293
37
29
38
47
39
42
32
455
33
33S
44
545
549
40
,51
689
532
69
493
483
367
lOt
3t
Ik
I I
J.
II
O
II
11
15
16
4
2
17
2
17
10
5
8
4i
2+
10
oi
o
t
si
3
2
4i
T
10
18
4
4
^ , 10 ,_
Ti6;> /V 'worked*
47 10 o
56 17
350 16
447 o
536 14
144 J 8
2165
3237
172
436
.256
06
4
6
o
14
4
^7
read ^7 $6*
Part I.
P R A
T I C E.
69
NUM
. Answers.
, , /•
J. d.
(no.) 2^9
14
(IM) 371
5
{n2.) 2992
16
(1 13.) 674
18
(m4.) 444
12
([15-.) 35:2
{116.) This isiuoyked.
[W].) This is rworked.
(n8.) 82
10
(n9.) 160
4 10
(120.) 616
(lii.j 237
6 8
(122.) 192
(I23-) 495
11
(124.) 841
14 6
(125.) 78
I 2
(126.) 213
9
(127.) This is
ivorked.
(128.J This is ^worked.
(129.) 3091
8
(130.) 1910
13
(131. j 1370
16
('32.) 3833
10
(I33-) 2349
6
(^340 J82
18
(135.) 2764
16
(136.J 1043
(163.)
1. s.
d.
4 'I
9
6
, 27 10
6
3
qr. lb. 82 n
6
2
^ 4 II
9,
I -
r 2 5
10^
^ 7 ,
\ I 2
I'i
° 4,
^ 5
81 -2^
3
3i -28^
91 I
• o| -53^
Num. Answers,
^. - ^.
(I3-.)
900
(138.)
877 4
(I39-)
691 4
(140.)
1632 3
(141.)
2348 8
(142.)
1023
{143J
1333 i^
('44.)
580 18
(Hi--)
894 18
(i,6.j
' 7461 6
(147O
This is ivorked*
(148.)
2702 17 o|-
(149O
10023 18 7J:
(^50.)
1906 8 it
(i?i.)
1940 ^ w^
(1,52.)
81637 15 3
(I53-)
4004^9 13 loj
(1 54-)
105668 4 4I:
(I55:-)
T^/> is ivorhed.
(156.)
13B5 9'-f
(157.)
24309 8|-i
(158.)
4125 4 8 1
(159-)
22799 4|-.'-
(r6o.)
11247 '5 lol-i
(16..)
8732 13 3 1^.
(idz.)
This is nvorked.
(.6+0
I.
s. d.
2
♦ 8
S
lb.
I
n
3
4
4
i-
44
13
4
Add 5° 3 2;--,4^,&c.
< o o 4^- '142, iScx.
Sub. 037 '28;, &'c.
Anf. 44 9 8|. -714, &c.
10
o 161^
P R
(165.)
I. 5. d.
I I
A C T 1 C fl,
(.68.r
qr. lb. 1. s. d.
2 oi II 12 5i
36
11
9
38
3
3J-I
(166.)
lb.
i6i
I. s.
I 18
9
10
19 7 6
6J:'7i4, &c.
S;^ '214, Sec,
o 6 2|:'928, &-C.
Anf. 19 I 5|: '071, &c.
(167.)
cw.qr.lb.
10 o
I. s.
14 15
4
J
oz.
5 16
2 18
' 9
o 8
o 2
O I
o o
^i
, -5
3i-5
of '62;
Oi -812, Sec.
3 '453* &c.
10 15
o^ '640, Arc.
(169.)
1. s. d.
o 4 11*
6
d\v. gr.
2 c
o 12
To
I 9
9
12
t7 17
9
o
II
51-8
•9^
18 7 6^-7-
dw
10
o 8
10
o o
33 I 9
7 7 10^
I 16 11^ 'J I
o 18 si -25
o I 10 '725
o o 7I -687, &c.
(170.)
s. d.
7
6i
5i-4S
'43
7- ^ 3 7 5i -95
7 7 •l62,&C. ■ ■ ^— «
Part I.
P> R A C T
I
C E.
(11
I.)
' ('
72.)
1.
S. d.
1.
s. d.
3
II 9iXi
2
11 9 X ;
10
10
35
17 8^X7
2i
17 6X2
10
10
'
r. p.
3S8
17 I
2
i 258
15
5
16 -
' i8ii
7
1794
5 5,
5
251
3 Hi
02 51
IS
dw.gr. 3
11 9t
I 4 12
18 9
10
JL
I
19 ioi-5
i I
5 lof
4
I
T
H 4i
52-4
16
"5
2 4; -83*
&c.
7l*o^
A
1-58,
&c.
&c. 1877
3j-i:2;
2051
13 10 -92,
10 9 -97;
(■73
(174.)
1. s.
4 10
- 1. s.
d.
10
3 II
10
8
4;
■■
5
,
28 14
8
r. p.
12
2
i
22^
4 10
2
p. 344 16
f
IO[ * ■ - - -
i?
3 ii
10
1
2
229 10
2 9
2
1
4
«0| •!
10
c
J.
4
i
I 2
r
6
9i
2
1
T
lOj •!
2
348 14
'i7
233 ^ iri
71
P R A C T
c 6 i
o 3 i
o i i
yds.
8
411 4
25 14
12 17
6 18 6
I I 5
15 6J-25
qr. n
2 o
1 o
o 2
c t.
s. d.
17 91
8 I£>|
4 5i-5
IS 61-25
458 o si '25
(176.)
qr. n.
1. s. d.
1 i.il I II 9^
12
19
I 6
10
190 15 o
7 18 III
7 i^i '^
199 I lol -5
qr. n
I 2
(1770
3 19 lit
4
15 19 4
4
63 19 o
3 19 ii|
9 iii*25
3 31*875
69
3 -125
(178.)
s. d.
10
2
1
To
-L
349
174
17
8
10
9 .
10 6
18 4i
qr. n
I
2
1
I
3
±
1
I. s. d,
III 6
I c
• 6
10 6
551
11 IO-5;
18 4i
artr.
P
R A C T I
c
E
>
yds.
47 S
4-
(»79.)
qr. n.
3 2
5)^903
I. s. d.
380
3 2
at I 14 9i
10
<:
jr. ..
I
J.
c
i
5
•7 7 8i
10
»73 »7 I
3
5-1 II 3
139 I 8
17 4i:'y
6 11^-8
661 17 3 '3
(I So.)
ells.
qr. n. s.
d.
?75
• 3i-
18
Hi
5
4)i8;6
469
9
lit
■2
3,
2
I
4
4i'87.-
2 •937>^'^^-
7 '46$, &c.
i| -281, &c
— —
«—
■'- ■ -«
s. 422
2l ,
9i'
id
d.
75
'44) I
9 9H
a.; !-
4 i| -281
44? 5 4i -281
II
Tare and Tret.
TARE AND TRET.
(I.)
This Example is njcorlied.
Num. Answers.
(2.) apcwt. 3qr. 13^ lb.
(3.) 418 cwt. 1 qr. 10 lb.
(4.) 143 cwt. iqr. 251b.
(^,) 26 cwt. oqr. 10 lb.
(6.) 6ocwt. oqr. 12 lb.
{-,) 76 cwt. 3 qr. II lb,
(■3-)
cwt. qr. lb.
I 3 18
7
4
53 2 o
I 3 18
55 I i8lb. grofs
3 3 23 '^H
I 3 25 -642
5 3 20 '928 tare
49 I 25 -071 neat
lb.
14
I
Num. Answers.
(8.) 1 15 cwt. 3qr. 141b.
(9.) M243lb.
(10.) i8Qoolb.
(^ ^O 357 c^^f* 1 8 lb.
(12.) 'This is <ivorked.
cwt. qr. lb.
97 2 15
12 O 22 '87
0313 -63
13 o 8-50 tare
84 2 6 '49 neat
,9X35=665 lb. grofs
Ha
7315,
332*
112)7647!:
68 ^V lb. tare
596 * -5 -^ lb. neat
Part I.
(i6.)
cwt. qr. lb
4 3 H
1' A R E AND Tret.
( I 7.) T^is w 'zvcrhil,
tiS.)
/!>
2 O
4
cwt. gr. lb.
12 "3 19 grof:
1 g tare
78 o o
.4 3 14
lb.
o I 26 -23 tree
82 3 i4gror3
12 oil '76 neat
10
^ I r
I I 29 -7^-
i_ l-j o 2 26 •875'
14, o 6 '625 tare
68 3 7 '375 neat
(19')
cwt. qr. lb,
7 3 19
9
7»
lb.
16
(2C.)
cwt. qr. lb.
4 i »4
9
59 ' H
2
78 3 o
4 I H
lb.
42 2 6
7 3 rg
150 I 2j grofs
i *^ 10 2 27 -78, &c.
5 113 -89
16 o 13 '67 tare
83 o 14 grofs
2 s I ' 314
I i 1 I 26 •25'
o 2 27 '125
O O 20 -781
14 I 4 '1^6 tare
26 j68 3 9 •S43 tret
2 216 '532 futtle
26)134 I II '32 futtle
5 018 '74 tret
129 o 20 '57 neat
66 o 2 I '3 1 1 neat
(zr.) T;^.- ;^ .c-;n:../.
H 2.
7C
Tare and T r e t;
(22.
lb.
(23.)
cwt.
5
41
qr. lb.
3 17
7
1 7 grofs
cwt.
7
qr. lb. 39
2 11 X5= 2
26)36
I
qr. lb.
3 II
5
27 grofs
3 27 tare
1 futtle
I i6'i5 tret
2
3 22-5
2 26-62
I 13-31
6-43 tare
4
168)34
3 1 1 -84 fee. f.
23-24 cloff
)37
I 0-56 futile
I I 20-48 tret
■168)35 3 8-07 fecond futtle
o o 23-88 cloff
35 2 12*19 ^^^^
34 2 1 6*59 neat
(z+.
cwt. qr. lb.
19 I 27
' ' 9
18 o 18
1.
s. d.
5
4
X18
m
ID.
'6;j.9o
6
14 4
I 9i
01
2 li
73
9 3
J64 ri 4i- Anfwer.
cwt
. qr. lb.
1* 3 19
grofs
1 20
• tare
12 I 26
^ neat
I. s.
d.
q^
.lb. 5 17
8
I
16
4
I
y
12
70 12
7
4.
I 9
5
3i
i
16
7
9i-S
4i
3
8 -5
73 9
3 '3
Fait r.
Tar
A* N D T R E r.
(26.)
lb.
cwt. qr. lb.
10 I 14
ib.
16
cwt. qr. lb.
4 3 n
7
54. I 7 grofs
7l-/-gji78 3 24-5' grots
II o 20*781, d'c. tare
' -'-- 4- 3 n
167 3 3-718 luttle qr. lb. o r 6-/^
6 I 22*758 tret 2 o break
315^- d ama. 5 o 2 3 ^ tare
161 I 8-960, ^jc. neat
1 i ifl 29 on f-g- futtle
' I I I5i|l^reak,
&c..
If I cwt. : il. 1 2gS. : : i6j cwt.
J qr. 8*96 lb* : 259l.cs. 5|d.
13 j 5 neat
lb.
(27-)
^wt. qr. lb.
3 3 i>
X 29
Sj,-V|ri2 I 14 grofs
8 o
tare
26)104 I II futtle
4 o It tret
If iCAr. : il. 175. 6d. :: 27 cw.
2qr. 2341 : 51I. 19s. 3d. Anf.
(28.)
1. s. d.
414 6 neat value
211 4 cuftoms^ &c,
I I 6 freight
059 factorage
8 13 I
168)100 I 10-5 fecond fut. If rcwt. : 1 1 1 lb. :: 1 1 cwt.
o 2 10*595
clofF
^99 2 27-604 neat
If icw, : il. I IS. 6d. :; 99 cw.
2qr."2 7*6o4!b, ; 157!. 2s.
Anfwer,
iqr. I 51b. : i cw. oqr. 1 5ilb.. 1
cwt. qr. lb.
I o 15V tare
lo o 27f neat
Jfiocwt. 27^ lb. : 81. i3.<;.
id. : : f cwt. : i6s. lo^d.
^ °*| the fugar flood him in.
per cwt. neat.
f 3
7S
S 1 M P L
(29.)
cwt. qr. lb.
lb.
N T E R E S T.
(30.)
87.
57..
14UJ25 1 14 grofs
m 3 o I9-J
I 2 9I-
4 2 oj tare
2C 2 13I neat
If 7ilb. : 1 gal. . . ^yj^y>^.
2qr. i3slb. : 307!^ gal. neat,
which at 5s. 4d. pergal. amount
to 82I. 2S. oid. 7^.
I049S9 g^^^- gro^^
450i\ tare
45o8y\ gall, neat
338ii-A-lb. neat
If 1 12 lb. : 5s. 3d. :: 33811/ylb.
: 79I.4S. io|d.
20CWt.
SIMPLE INTEREST.
T^is Example is n.vorled.
Num. Answer*.
(2.) 9-71. 13s. id.
(3.) 128I. 17s. 4id.
{4.) 87I. I2S. old.
(c.) 42i-+s. 5id. -75
(6.) 282I. 2S. 9d.
Num. Answers.
(7.} ic61. I IS. 6d.
(8.) J17I. OS. ii^d.
(9.) 17I. 8s. gid. -75
(10.) Sn6/> ;> (worked.
('0
As 365d. : 81. 15s. i^d. '64 :i 3i5d. : 7L 11s. ifd,
70^. X 1+.^ X 4I -^ 36^00 = 13111111. z= 13I. IIS. jii.
(13.)
As 3^500 : 4 4L iz?. i< d. X 5 1 : 29 X 7 '• J?^' i5»- i<^<^'
Cr, 7300 : 4,41. iz.» Kd. ; ; i^X 7 • 'S'* 'S* i^«^' •Ac^-
Part I.
Simple Interest.
7.9
(T4.)
As 36500 : 347I. los. y 4 : : iS X 7 ! 4I. 163. 2jcl.
From Jan. ift, to Sept. 22d, are 264 days.
As 36500(1. : 540I. los. X 4 :: 2646. : 15I. 12s. S^d,
(.6.)
(■7-)
d.
400 60
3 i
1200 I
200
14,00
2i
28
7
i
14
2
6
8
^p
9
2 5 101
35 o o
d.
3lil294-
2o)73_ 6
Anfu'er 3 13 6
Anf. 37 5 loi (18.) This is nxjorkcJ.
.20)500
4+
- 1-24)93 15
20
s. 18)75
12
d. 9)00
I9-)
m.
6
X
3
X
d.
30
i.
3
3
TO
I. s. d.
24 18 9
5
124 13 9
12 9 42:
6 4 8|:
2 I 61
145 9 4i:
o 4 i|
'5
H5 5 21 -5. Anf.
So
Simple
(20.)
INTEREST.
(21.)
From May 15th, 1787, to
Sept. 2 2d, 1789,316 2yrs. 4111.
icd.
5.15)00
I.
500^
m
4
d.
10
±
3
1
1. S.
iS 15
d.
5i
15C0
375:
57 10
ro
5
.18)75
20
Anf. 44 5
^
I.
100
4
1. 4)00
ID.
2
d.
IC
6
J
1. s. d.
400
o 13 4
ToO , 4
i 10 O 2
ICO o
['66
rremiuni o 10 o
Anf. ici 7 jJ-33
y. m. d.
2100
^8 7 3
2 4 27
t's 10
1-43) 77 »o
20
£.15)50
12
d. 6)00
22.)
in.
I.
s.
d.
4
i
43
'5
6
2
d
30
87
II
14
II
10
jVol
3
12
i:i
1C5
'5
9f
7
3i'2
l^'S
8
5l *8
IS
10 o
980 18 5^-8
(23.)
S'ZvV Example is ivorkeJ,
Paft I. SimpleInterest. 8t
(24O
100 -f 4 X 5 = 120I.
As 120I. : lool. : : 570I. i6s. 6d. ; 47 5I. \p, gd, Anf.
100 + si X /\.i— 1 1 61. 17s. 6d.
As 1 1 61. 17s. 6d. : lool. :: 205I. iis. 7|d.-f : 175!. i8s,
Anfwer.
(26.)
100 -f 4t X 3| = 1 1 61. 17s. 6d.
As 1 1 61. 175, 6d, : igol. : ; 350I. 12s. 6d. : 300I. Anfwer,
Tiif Examph is ^worked*
(28.)
1. s. d.
465 8 3 amount
344 15 o principal
120 13 3 intereft
As ^44!. 15s. : 120I. 13s. 3d. :: tool. : 28I.
Then 28-7-7 =4^' percent, Aufwer.
(29.)
For 175I. los. read I "^^h iSs.
1. s. d.
205 II 7 1 'I amount
175 18 o principal
29 13 7^ 'I intereil
.As 175I. iBs. ; 29I. 13s. 7jd. 'I : : 100 : 1 61. 17s. 6d.
Then 16I. 17s, 6d, -J- 3.J = n-I, ics. An(\vzx,
Bx Simple Interest.
(30.)
I. s. d.
350 12 6
300 o o
50 12 6
As 300I. : 50I, I2S. 6 J. :: ico : 16I. 17s. 6d»
Then j6I. 17s. 6d. -^ 41 =1: -J. 15s. Aufwer.
1.7191 II
20
s.jSjji
12
4
1. s.
475 n
(31.)
d.
9
4
7>?7.r
is
370 16 6 amount
475 13 9 principal
I. 19!C2 1^
20
s.o'si-
12
d. 6!6o
4
9^ 2 9 intereil
As 19I. cs. 6i^. 'I : I y. : ; 05I. lu
2(40 9d. ; 5}TS.
I. s. i^cr 17 5I. I OS. rf ^^175!. I Ss,^
175 i3
4^- I. s.
■ 2C5 I! 7I •* amount
703 12 175 18 o principal
87 19 ^
29 13 tI *t infcrelt
As 7I. 78s. 3f-TVo •• ly- •• • 29I. 15s.
«iSS 7^<i- 'f • 3|yrs. Anfwer.
Tart I. Simple Interest.
83
300
3l
900
1. 11I25
20
(34.)
J. s. d.
350 12 6 amount
300 o o principal
50 12 6 intereft
5. 5I00
As III. 5s. : lyr. :: 50I. 12s, 6J..
: 4iyrs. Anfvvcr.
(35-)
As 160I. : icol. : : 264.16
100 X 12 :=: 1200
12
60
ICO
1. 160
27616
ICO
160 ) 2761600
1. 17260 Anfwer.
(56.)
1.
2ift Oa. 1782, lent 20
2,2d May, 17J54, borrowed 150
Ea;. 130
30th Ju!y, i7Si^> borrowed 15c
s.
400
2600
Bal. 280 rz 5600
21ft JuJ^', 1785, pa-d 15 18
Bal. 264 2 rz 5282
21II Aug. 17S5, paid 40
B.il. 224. 2 iz: 4482 ...
ftifl O^. 17S5, jra'd 50
Bal, J74 2 ^ 34S2 ...
Days.
579
69
356
31
61
115
Produft-s.
2316CO
179400
1993600
163742
25^.3402
400430
Carried oyer*
8+
Simple Interest.
1. S, S.
Bal. brought over 174 2 n 3482
13th Feb. 1786, paid 9 12
Bal. 164 io ~ 3290
13th June, 1786, paid iii
1070
Bal.
13th Jan. 1787, paid
53 10 =: ic
So
Bal. due to me
Intereft due to my friend
26 10
23 6 i^
Bal. in my favour
3 3 lof .
Days.
214
Produfts.
394800
Sum 3634354
Subt. 231600
7300)3402754 s.
3 lof Anf. Intereft due 23 6 i^d,
500 gui
{37.)
1. s.
700 15 Amount
525 o Principal
[75 15 Intereft
The intereft of 52 5I. for one year, at 4^ per cent, is 23!. 12s. 6d,
Then
23I. I2S. 6d. : I yr. : : 175 15s. : 7 yrs. 16036yd.
Now by counting 7 yrs. 160 days back, from the 25th September,
3788, the bond will be found dated April iStii, 1781.
(38.)
1. s.
.11)38 10
310 rate per cent, or intereft of lool. for one year.
Now to anfwer the conditions of the queftion, it is evident that th(
intereft muft be four times the principal, or 400I. Hence,
As 3|1. : 1 yr. t: 4CCI. : ii4yyrs. Anfwer.
(39-)
The intereft of 50ol.,fjr 4} years, at 5 per cent, is 11815 c
The intereft of 2 50I. fu" 9I years, at 2^ per cent, is 59 7 ^
Difference 59 7 ^
Part I.
Brokerage,
(40.)
!. s. d.
To a bin dated ift Auguft, 1786, 7 „
payable lit Oaobcr, 1786, J ^^^
5tli Odlobcr, 17S6, received 94 17 o
Bal. 863 I o
27th November, 1786, received 47 19 6
Bal. 815 I 6
i^h December, 1786, received 105 o o
Bil. 710 I 6
tft January, 1787, received 55 I ' A-
Bal. 654 10 2
J5th March) 17S7, received 101 14 o
Bal. 552 16 2
[4th May, 1787, received 110 50
BjI. 44Z II 2
T9:h Auguft, 17S7, received 140 2 6
Bal. 302 8 8
1 1 ;h September, 1787, received 50 6 6
Bal. 252 8 2
ijth March, 178S, received 2!;2 8 2
Days.
4
53
18
17
73
58
99
23
185
Sum
Prod'jas.
3^31 la o
45741 13 o
14671 7 o
12071 5 6
47779 3 ?'
32064 17 8
4^813 5 6
695s 19 4
46695 10 IQ
253<322 13 o
I. S.
253622 13 X 4 _ ,
^^^'" ^6^0 = '71. 15^. lold. Anr.vcr
'BROKERAGE.
This Example rs iMorked,
Num. A^'swERs,
(2.) 20I. los. 3-025(1.
(5.) 2I. 8s. 5i-27d.
UO 37^» 4s» ii*4ud.
Num. Answers.
(9.) 4* MS. icil.
(6.) Ill IS. oj • J9'?
86 Commission, I n s u r a n c e, &c.
COMMISSION,
Num. Answers*
(2.) 28I. 8s. 3|d. -2
(3.) 61. 17s. 8|d. -22
(4.) 51L 8s. 8id. •+
(I.)
This Example is nxorhed.
Num. Answers.
(5.) 74I. IIS. 3ld. -28
(6.) 965I. 6s. 8id, "9
INSURANCE.
Num. Answers.
(i.) i^. ics. 4fd.
(2.) il. 6s. lod. of '35
(3.) 8504I. 3s. lod. of '44
(4.) 557I. 8s. lid. -84
Num. Answers.
(^. ) 1 157I. I2S. 6d.
(6.) 7s. 6|d. -08
(7.) 6;1. i8s. iiid. -45
TURCHASING STOCKS.
- ^Ti/V Exam-pie is <v:orked^
(2.)
Tl^/V Example is nvorhed,
(3-)
If zool. : I25|l. :: 7575I. i^s- : 9J'7>- ii^--i-
(4-)
If loci. : Sg;i. :: gcel. : S03I, 5s.
(;.)
If lool. : >96^l. :: 1759!. i8s. gd. ; 3+53!. 175. 6fd.-;.
?art I, Purchasing S t o c k ai 87
5000
85I
8 150 loo
6 5
4268 15
brokerage
4275:
the purchafe
or)
7159 10
20
'
1 1190
12
10180
4
3I20
1. s.
8|7 11
d.
10| *2
18
487 19
1 1 1 '4 brokerage
6|-6
488 18
6| the purchafc
425-000
: - ^8 75
4268I75:
20
1 5 loo
I. s.
759 ^o
8
6076 o
8
48608 o
189 17 6
487197 17 6
20
19I57
12
6.90
4
3160
61 — ^61 = 4I: per cent, the difference, or lofs,
* 700C0 8 )7oo|oo
4i
87 10 brokeriTge for buying
280000 87 10 ditto for felling
350CO
3150I00
J7J
175I. the ^o»^ broker gained
3325I. the gentleman loft
I 2
£S
3) I s c o u
DISCOUNT.
5 15
2
(I.)
TI:is "Example is iV2riiJ»
' it-)
;)»i 10
3 16
ICO
8
As 103 16 8 : 100 ; : 594I. 14s. 9d, : 572]. i^^>
6 ill.
im.
ij 8
u
i 13
4
4 13
4.
100
As '104 13 4 : 4J. 13s. 4d. ;: 915I. 178. ; 40I.
6s. S-iVyd-
6 m.
5
2
10
im.
I
8
4
7
18
4
I
CO
As 107 18 4: 100 :: 75I. :691.9s. ii;d.-,yg.
(>■•)
From 22 Sept. to Chrillmas, arc 94 days.
As 365d. : 61. :: 94d. : il. ics. lojd. intereftof iccl.
for 94 days.
As loil. los. ic|d. : lool. :: 900I. : 8861.6s. 2-^y^\%d.
Anrvver.
Part I.
I> I S C O U N T«
H
(64
If36;d.: 7-1-1. :: J7d. : d. 3U gld.
If loil. 3s. 9^d. : 15000I. : : il, 3s. gfd. : 176I. 9s.
in 5 6Da ^
(7.)
From the 5th of July, 1788, to the 24th of June, 17893
are 354 days.
If 365 d. : 711. :: 554d. : 7I. 3s. ofd.
If 107I. 3s. old. : looh : : 1797I. lof/ : 1640!, 33^.
10 s^Vji^* Anfwer.
(^0
3 } 747
18
R.
d.
249
6 due
immedi
r. s.
ately, <Src,-
I.
d.
6 m. i
8
12
6 .
= 8^1.
6m.
4.m.
3
8 12
6
1 m. 6
4
6
3
4. 6
a
H
4i
z 14
2
S
7l-
7
5
100
J
100
IQS
71-
7
5
I. S; d.
ff 105!. OS. 7^d. : lool. :: 249I. 6s. : 237 7 i-|
If 107I. OS. 5d. : lool. :: 249I. 5s. : 234 10 ol
249 6 o
The prefent worth
1 3
72
D
- ^
(9-)
^O O
^ 0\0C VO
O «^ ■* O
O c^ " O
H >-< I-. c-n
^
00
t^
CO
t
•
Add
^'i-O
CO
VC
O r-.oc O
I
8
N M O O
-1"-+*
go
VC
00
1 o ^ o
H
c
•^i-
p.1 M - o
oo
H)C> »-.icl M
r^
l
- vO ro
:r I H* SIS
oo
oc
ri-
VO
VO
,
IH
-*
o
o
o
o
-o
1- O
2
J
5 rt
1. s. d. U s.- d.
Firft, If ICO i6 8 : O i6 8 :
Secoad, If loi 5 0:1 5 o :
Third, If 103 15 o : 3 15 o :
Fourth, If 104 ij 8:411 8 :
Fifth, It 10- o 0:5 O O :
1.
s.
d.
1.
s.
d.
200-
4
:
I
^3
I
133
9
4 :
1
12
"J
114
8
:
4
2
S|
300
6
:
13
3^
^i
5^
8
8 :
2
9
I If
The Difcount
.3 I ic|
Parti. Discount. 91
(10.)
I. s. d.
576 = 5|I.
3
|=:Jof3Ji6 2 6 365:d. : 5I. 7s. 6d. :: 41 d.
o 12 oj
100 o o
IX I2S. o|d.
120 15 2-5: : 100 ;: 1789I, 19s. lod. : 1482I. 5s,
7d* Anfwer.
(II.)
The difference is 500!. The intereft of loooL for 2oyrs,
is loool. but the difcount of the fame fum is only 500I.
(12.)
For 5 per cent, in this queftion> read 6.
1.
I.
1.
I. s.
d.
106 :
100 :
: 100
94 6
9i '20
112 :
100 :
: 100
89 5
8t-28
118 :
TOO :
: 100
84 14
lol -92
124 :
100 :
: 100
80 12
^^l'3S
130 :
: 100 :
: 100
76 18
5t*i4
The prefent vakie of the debt, al- ")
lowing each perfon a difcount i 425 18 9^
of 6 per cent. 3
Now,
100 + 1064- 112 4- 1184- 124 = 560!. the money which
may be made by receiving the payments as they become
due, &c.
Again,
The intereft of 425I. i8s. 9^d. for 5 years, at 6 per cent,
is J27I. I js. 7-id. which added to 425I. i8s. g^d. gives
553I. 14s. 4|d. this fubtrafted from 560I. leaves 61. 5s. 7id.
advantage, by receiving the debts as they become due.
92 Eq^uation of Payments*
EQUATION OF PAYMENTS.
This Example is ^worked.
(z.) 4I months. | (5.) 6 months*
(4.) 6^ months^
(5-)
1. s. s. d.
J 50 o X 2 m. = 3000 X 60 = 1 80000
147 17 X 74d. = 2957 X 74 =z 2 1 881 8
137 18 X 95 d. =r 2758 X 95 = 262010
65 OX 5 m. z= 1300 X 150 =: 195C00
10015 ) 855828
85^. tWiV"
or 2 m. 25 days. Anfwer*
(6.) (7.}
J 2 90
= 6 X
4 = 24
i =
18
=: 3 X
^ = 15
4 =
15 X 25 = 375
z: 2 X
7 = H
X
9
10 X 90 = 900
I X
10 rz 10
47 X 137 = 6431
12
)63
90 7706
*^
— ■
Ji
months.
85-11 days.
(8.)
I. s. d. f. d.
50018 oxfyr. =: 480864x1X21= S7757680
9C0 17 6 X I yr. i'4d. zr 864.840 x 479 :=: 41^258360
1700 18 4|X 2|yr. =: 1632883 x 91^1= »450<^05737i
»978527 ) 199202^7771
668 d. 2325661! rem*
Cr I yr. 303 days.
Part I. Compound iNTtREST^ 93
COMPOUND INTEREST,
(I.)
This Example is ivorked,
m
(2.)
I. s. d.
5I. i'-o ) 700 1 8 o principal
35 o loj '2 intereft forthe iftyear
a'<? ) 735 ^8 io| -2 amount for ditto
^6 15 / If -36 intereft ftr the zd year
"s'o ) 772 14 10 -56 amount for ditto
38 12 8J '628 intereft for the 3d year
^'^ ) 8ii 7 7 '188 amount for ditto
40 1 1 4f '2094 intereft for the 4th yeaf
8p 18 1 if '3974 amount for ditto
700 18 o principal
, 151 o II J "3974 whole intereft
(5.)
1. s. d.
41.^*5- ) 1057 17 6 principal
42 6 3 1 '4 intereft for the i ft year
25 ) I ICO 3 9i: '4 amount for ditto
44 o 1 1 '296 intereft for the 2d year
25 )ii44 3 11^ '696 amount for ditto
45 15 4+ '187 intereft for the 3d year
♦
-25 }ii89 19 3|- -883 amount for ditii
47 II Hi •67^- intereft for the 4th year
25 )J237 'I 3i '559 amount for ditto
Carried over.
54 Compound In teres t»
Brought over
1. . s. d.
25 ) 1237 1 1 3i '559 ^i^o^^t for the 4th year
49 10 Oi '462 intereft for the 5th year
25 )i2S'] I 4 '021 amount for ditto
51 9 7|J|B intereft for the 6th year
1338 10 ii|*4oi amount for ditto
1057 17 6 principal
280 13 5I '401 whole intereft
(4.)
In this and the following Examples, the amount oreach
payment, (or refult of each operation) is only put down^
the whole work would have taken too much room*
Anfwer
Anfwer
I.
s»
d.
522
2
S-i amount for the firft year
544-
6
61 ditto for the 2d
567
9
2J: ditto for the 3d
591
II
6i ditto for the 4th
616
H
4| ditto for the 5th
T.
s.
d.
733
5
amount for the ift year
768
7 ditto for the 2d
804
II
3 ditto for the 3d
842
^5
7 ditto for the 4th
882
16
2|- ditto for the 5th
924
14
io| ditto for the 6th
968
13
5 ditto. for the 7th
The three Elkmples following are folvcd by the Note
to the Rule of Compound Intereft..
Part h Compound Interest.
(6.)
Teaf/y Faymettts*
1. s. d.
52^ o o amount for the ill year
5:51 5 o tlitto for the 2d
578 16 3 ditto for the 3d
Anfwer 607 15 q\ ditto for the 4th
Half-yearly Paymefits,
1. s. d.
512 10 o amount for the ift payment
525 6 3 ditto for the 2d
538 8 10 J ditto for the 3d
551 18 I i ditto for the 4th
^6^ 14 o| ditto for the 5th
579 16 1 1 ditto for the 6th
594 6 loj ditto for the 7th
Anfvver 609 4 o^ ditto for the 8th
^larterly Payments,
I. s. d.
506 5 o amount for the ift payment
512 II 61 ditto for the 2d
518 19 8:1: ditto for the 3d
525: 9 5i ditto for the 4th
532 o 9I ditto for the 5th
538 13 9| ditto for the 6th
54 J 8 6 ditto for the 7 th
552 4 loi ditto for the 8th
599 211 ditto for the gth
556 2 85 ditto for the loth
573 4 ^1 ^^^^o ^o'' ^^^^ ^^h
980 7 6^- ditto for the 1 2th
587 12 7^ ditto for the 13th
594 19 6~ ditto for the 14th
602 8 3i ditto for the 15th
Anfwer 609 18 10^ ditto for the 1 6th
(7.)
1. s. d.
730 3 I of: amount for the ift payment
74S '+ 24 ditto for the 2d
761 II I A ditto for the 3d
777 »4 9i <iitto for^thc 4th
95
96 Compound Interest.
1. 5. a.
794 5 4 amount for the 5th payment
811 2 io| ditto for the 6th
S28 7 7I ditto for the 7th
845 19 SI ditto for the 8th
863 19 2-| ditto for the gch
882 6 5 ditto for the I oth
901 I 4^ ditto for the nth
Anfwer 920 4 4 j- ditto for the 1 2th
(8.)
this E
t
xara
iple, _/i?r 4I per cent, r^-/?^ 4 per cent,
748
s.
6
.u.
2 amount for the ift payment
7S5
15
10 ditto for the zd
^^3
7
ditto for the 3d
770
^9
8 ditto for the 4th
778
13
loj ditto for the 5;th
786
'jj ditto for the 6th
794
6
loj ditto for the 7th
802
5
9 ditto for the 8rh
810
6
27 ditto for the 9th
818
8
3i ditto for the loth
826
1 1
lit ^itto for the nth
834
17
3^ ditto for the. 1 2th
8+3
4
3 ditto for the 13th
891
12
)o|- ditto for the 14th
860
3
2 ditto for the 15 th
868
IS
3 ditto for the 1 6th
877
9
0^ ditto for the 17th
886
4
6 ditto for the i8th
895
I
9 ditto for the 19th
904
9^ ditto for the 20th
9^3
I
7 ditto for the 21ft
922
4
2| ditto for the 2 2d
931
8
7A ditto for the 23d
940
14
1 1 ditto for the 24th
950
3
0} ditto for the 25th
999
13
1^ ditto for the 26th
969
5
0^ ditto for the 27th
978
18
lol- ditto for the 28th
giS
H
8 ditto for the 29th
998
12
5 ditto for the 30th
Part I. Single Fellowship. 97
1. s. d.
1008 12 1 1 amount for the 31ft payment
1018 13 10^ ditto for the 3 2d
1028 17 7^ ditto for the 33d
'039 3 3 J ditto for the 3 4.th
1049 II 2\ ditto for the 35th
1060 I li ditto for the 36th
1070 13 i^ ditto for the 37th
740 18 o principal
Anfwer 329 15; it intereft
SINGLE FELLOWSHIP.
This Example is 'worked*
(2.)
J. s. d.
500 17 10 A's (Took
735 o o B's rtock
rem.
1235 17 10 fum
i. s. d. I. s. I. s. d. !. s. d.
1135 17 10 : 300 15 : : 500 17 lo : 121 17 9I— 19<;484 A's iTiare
1235 17 10 : 300 15 : : 735 o o : 178 17 21—101130 B's iharc
300 15 o proof
(3-)
1. s. d.
5075 18 o whol^ ft ock
574. i6 o A's ftock
947 18 6 B's — —
3044 17 o C's
4567
§08 6 6 D's Hock
K
98
S I N G L
ELL O W S H I
]. s. I. s.
io-5 18 : 1358 18
5075 18 : 1358 18
5075 18 : 1358 18
5C75 18 : 1338 18
1. s. d. 1. s. d. rem.
574 16 o : 153 J7 7I — 8825S A'sflia
947 18 6 : 253 15 5|— 77710 B^s
3044 17 o : 815 3 il — 6586 C's
508 6 6: 136 I 81— 30502 D's —
[358-18 o
Or thus, by Note I.
I. s. d. 1.
574 160— 574-8 A's flock
947 18 6 = 947.925 B's
3044 L7 o i:: 3044*85 C's
508 6 6 zr . 5c8'525 D's
5075 18 o — 5075'o whole ftock
5075-9) i35S'9coooo ( -267716, &c. con.mon multiplier.
5^4-8 x -267716, &c.
r47v 25 X •267716
3044-85 X -267716
508-325 X -267716
1.
1.
s.
d.
I53-SS3I568
—
153
17
7^ A's
fhare
253-7746893
—
253
M
5iB's
815-1550626
—
SI5
3
Il C's
I36.0S67557
±1
136
I
S|D's
—
(4-)
This Exar^i
/, ;^
-'.'"-'•■■(/ /^' Note !_/?.
1. s. d.
540 14
—
540-7 A's debt
770 18
ziz
770-9 B's
4005 14
—
40057 C's
975 18 9
'JZ.
975-9375 ^'^
3150
zn
3150. E's
9443 4 9 ~ 9443*2375 fum of the debts
whole debt. elrefls.
A:9443-2375l. : 7I74-71-
per pound.
il. : '75977 12 J 2I. zi 15s. 2|d. 'I nearly
p 410-808294
X I 585-707627
75977i2i2< 3043-415543
— I 741 489217
1^393-27931'*
1.
s.
d.
=: 410
16
H
= 585
14
li
— 3043
8
3^
— 741
9
9i
= 2393
5
7
Part I.
Single Fellowship,
99
Otherwife,
After you have found whst the
Bankrupt's efleds- will pay in the
pound, (as above, or by the Rule
cf Three) then each particular part
may be found bj^ the Rules of Practice,
fufHcieiitly exaft for common pur-
pofes. — The above cftcdls will pay
15s. 2[d. 'I per pound.-
540 14
270 7
^5 3
4 10
6
1I.6
II
c 4
3 -7
2i-8
410 16 iiA'5 fiia.
. I. s. d.
7754 17
15749 14
3497 16
5754 i« 10
3775 19
37497 19 S
—
(5.)
By Not& I.
7754-85 = A's flock in the Cargo
-j/lOV'S — C'2 "■ ■ - ■
37497*9S3' = F's
74031 4 6 zi: 74031-225 fum of their ftocka
■Loft 7975^3
Grtifl 7347
Dlft*. Whole loft 7x403
74031 '22 5 ) 72403'00, &c. ( '97800624 com. multiplier
7754-85
15749*7
3497-8
5754-9416'
3775*95
37497^.983
X
» •97800624
1. s. d.
7584-2916 := 7584 5 9I
15403-3048 = 15403 6 I
3420-8702 n 3420 17
5628-3692 zz 5628 7
3692*9026 rz 3692 18
*• 36673-2616 — 36673 5
(6.)
tons
500 A put on board
340 B
94 C
934
934
0-4
150
150
5QO
340
94
K
t. hd.
80 I
54 2
15 o
2
4l
4i
o
2i
A's lofs
B's —
C's —
D's —
E's —
F's —
g. rem.
12 510 A'a lof-:
26— 160 B's —
24— -264 G'; —
ICO
Single FsLLou-sHif.
1 4
(r)
By note 2d.
4- 3-I-4 + 54-6ZZ21 fum of the numbers
21
1680
21
1680
21
. 1680
21
• 16S0
27
16S0
21
1680
80 the id number
160 tlie 2d
240 the ^d
320 the 4th
400 the 5th
480 the 6th
. (8.)
7 -f 8 4- 9 = 24 fum of the proportional flocks
24 : 357S4 1:7:
24 : 35784 '•' ^ '
24 : 35784 : : 9 :
^C437
11928
13419
s. I.
s.
3S7?4 ' S^o : I
357F4 : 500 : :
.15784 : 500 : :
10437
11928
U419
(gO
3 + 4.
+ 7
J4 : 292 :
1 . : 292 :
14 ; 292 :
• 'i- •
s. 1. s. d.
521 17 o ] Their
596 8 o / refpeclivc
67c 19 o ) Itocks
]/ s. d.
145 16 8 I Their
166 13 4 > feparatc
187 10 oj gain»
r4 fum
62 y gallori
83 -]^ ditto
,D ditto
4- 3 + 5 = ^o1.
10
19090
10
19090
10 ;
19C90
I.
3818 A's part
5727 5's
9545 ^^
Part I.
Single F e l l o w s h i Pi
10 1
(II.)
689, 5 ■ 50, ,, 3
129I. ns. Ad. ir — ^I. 5— =: — '• A'spart. 4 -i
^ ^ ^ 3 9 9 7
= 111.
n' . ^ _ 37, „ 2
B 3 part. 4 — n — 1. 3 —
9 9 3
AslZ
IC23
"257
3
C's part.
50 31 1023 90650 72261 64449 227360
5 7 259 16317 16317 16317 16317
By reje<Stlng the common denominator 16317 (vide Note 4th, Prop.
8tli, Vulgar Fradlions) we have the following proportions :
1. s. d.
227360
227360
227360
6S9
3
689
3
6S9
3
rem.
1. : : 90650 : 91 II 4I 203840 A's (hare
I. : : 72261 : 72 19 iq| 80640 B's ■ • «
1. : : 64449 -65 2 o|— — . 170240 C's — —
To
20
(12.)
> I
> 4
15
6^'
12
To*
+ — +— 4-~ — — fum.
60 60 60 60 . 60
By rejedling the common denominator, &:c.
1. s. d. rem.
30 : 194 16 I 76 A's fliarc
20 : 129 17 4I 25 B's — —
15 : 97 8 o\ 38 C's
77 18 5I i5D's— r-
77
^ool.
77
: 500I.
77
: 500I.
77
: 500J.
12
(13.)
1. s.
57 18 A's gain
29 14 B's ~—
28 4 dlfterence of their g.ilns.
28I. 4s. •. 51!. IIS. 6d. : ! 57I. i8s. : 105I. 17s. lo^d. '^y A's flock,
from which fubtraft 51I, lis. 6d. the remainder is 54I. 6s. 4{<J. '-^y
B's ftock,
K3
r02 S r N G t E F F n t o w s K r p".
Firil,
it* I cwr. : 2I. 16s. :: 2^ocwt. i qr. 22 lb. : 701I. ^s,
A's adventure, excljfive of charges.
Ifiool. : 91I. :: 701I. ^s. : 651I. 2s. 6d. value of A's
remaining part, which dcduded from 701 1, ^s; leaves 70I. 2s.
6d. A"s lofs, exclufive of charges.
Again, If 4I. 4s. : i cwt. : : 631I. 2s. 6d. : 150CWN
1 qr. 2 lb. the quantity A faved, which deduded from
2C0Cwt. I qr. 22 lb. the quantity he bought, leaves Joocwt»
20 lb. the quantity of fugar A loft, or caft overboard.
Hence, icocwt. 2olb. x 100 zz i 0017 cwt. 3 qr. 12 lb.
their whole cargo.
Secondly,
If z^ocwt. I qr. 22lb. : loocwt. 20 lb. :: iooi7Cvvtr
3qr. 1 2 lb. : 4007 cwt. 16 lb. the whole quantity of fugar
caft overboard.
1^33 ' ^i '- ' 4007 cwt. 16 lb. : 4508 cwt. 41b. the
w hole freight of A and B.
Then 4508 cwt. 41b. — 250cwt. i qr. 22lb. = 4257 cwt.
2qr. 10 lb. B's cargo.
Ifiooi7Cwt. 3qr. i2lb. : 4007cwt. i61b. :: 4257cwt.
2 qr. 10 lb. : 1703 cwt. 41b. quantity of fugar B loft, or taft:
overboard.
Then 4257 cwt. 2qr. lolb. — 1703 cwt. 41b. =:2554cwt.
2 qr. 6 lb. the quantity of fugar B faved.
If I cwt. : 2I. 6s. 8d. :: 42^7 cwt. 2qr. lolb. : 9934I.
-;. 6d. value of B's adventure, exclufive of charges.
If I cwt. : 2I. 6s. 8d. : : 2554 cwt. 2 qr. 61b. : 5960L
12s. 6d. prime coft of the fugar B faved.
If 100 : 120 : : 59601. 12s. 6d. : 7152I. 15s. the ad-
vanced value of the fugar which B faved.
Then 9934I. 7s. 6d. — 7i52l' 15s. = 2781L 12s. 6d,
B's lofs, exclufive of charges.
Thirdly,
iooi7Cwt. 3qr. 12 lb. — 45o8cwt. 41b. = 5509cwt.
^qr. 81b. C's cargo : loccwt. 2olb. -f i703Cwt. 41b,
-= 1 803 cwt. 241b. A's + B's lofs of fugar.
40C7Cwt. i61b. — 1803 cwt. 241b. = 2203cwt. 3qr.
rolb, quantity of fugar C loft, or call overboard.
Part I. Double Fellowship. loj
Then 5509 cut. 3qr. 8 lb. — 2203 cwt. 5 qr. 2olb. nr
330^ cwt. 3qr. i61b. the quantity of fugar C faved.
701!. 5s. -f 9934!. 7s. 6d. ~ 1063^1. 12s. 6d. A's + B's
adventure, exclufive of ehargcs.
15778L 2.S. 6d. — 1063 5I. i2S. 6d. = 5142I. los. C's
adventure, exclufive of charges.
If 5509Gwt. 3qr. 81b. : 5:142!. los. :: 33OJ cwt. 3 qr.
1 6 lb. : 308 5J. I OS. prime cort of the fugar C faved.
If I cwt. : 4s. 8d. :: 330^cwt. ^qr. i61b. : 771I. 7s.
6d. C's gain by the quantity he faved.
Then 3085I. lOs. -f 77 il. 7s. 6d. = 3856K 17s. 6d. ad-
vanced value of the fugar C faved.
5142I. ros. — 3856I. 175, 6d. ~ 1285I. I2S. 6d. C's lofs^
exclufive of charges.
Laftly,
cwt. qr. lb. 1. cwt. qr. lb. 1. s. d.
If 100 1 7 3 iz : 525 r: 250 i 22 : 13 26 A's part of the charges
If 10017 3 12 : 525 : : 4257 2 10 : 223 2 6 B's
If 10017 3 12 : 525 : : 5509 3 8 : a.%Z 15 o C's —
1. s. d. I. s. d. 1. s. d.
70 2 6-fi3 2 6rr 8350 A's who'e lofs
2781 12 6 4- 223 2 6 rr 30^4 15 o B's
1285 12 6 + 288 15 o zr 1574 7 6 C's————. Anfwer
DOUBLE FELLOWSHIP.
(I.)
T'/jis Example is *worked,
L s. d. 1. s. d.
547 19 6X7 = 3835 16 6 prod, of A's flock and time
475 18 0x9= 4-S3 2 o B's
1747 14 0X41:: 6990 16 o C's —
Sum of the prod. 15109 14 6
!• s. d. 1. 1. s. d. 1. s. d. rem.
15109 14 6 : 225 : : 3S35 16 6 : 57 2 4I 2769444
15109 14 6 : 225 ; : 4283 z o : 63 15 7 2725S48
15109 14 6 : 225 :; 6990 j6 o ; 104 2 o — 1757376
[04- Double Fellowship,
Or thus, by Note 2d,
547*975 X 7 ::: 3*^35*^-5 fi"°^* °^ -^'^ ^^'^'^ ^"^^ ^'■'^s
475*9 X 9 rr 4283'i B'&
1747-7 X 4 = ^;-9^'"S C's
Sum of the prod. i5io9'72 5
35io9'725 ) 225'ooooc, &c. ( •0T4891 common multiplier,
1. s. d.
3835-825
4:
69c
iS35-S25') X r 57-11927 =r 57 2 4I A's ftare
,283-1 ^-014891^ 63*77964 ir 63 15 7 B's — .
>9cc-8 3 ^^ L 104-1 rr 104 2 o Cs — —
(3-)
7 X 13 = 91 prod, of A's oxen and time
Q X 14 = 126 B's ^
II X 25 n: 275 C's
»5 X 57 = 55^' I^'s
1047 fum of their produds
1C47 : 21 :: 91 : i 16 6 216 A's (hare
1047 : 21 :: 126 : 2 10 6t 158 B's
1047 : 21 :: 275 : 5 10 3! 135 ^'s
IC47 : 21 :: SS5 '■ ^^ ^ 7t 558 D's
(4.) .
weeks
26 time the family lodged
26 — 14 n 12 the firft 4
12 — 3 m 9 ■ ■ " thefccond 4
9 — 3 m 6 — the third 4
6 — 3—3 — the laft 4
Sum of the produfts 380
380 w. : 261. 2s. 6(?t : : I w. : is. 4^3» the fum paid per week by
each ledger.
Then,
s. d. L s. d.
260 X i 41 n 17 17 6 for the family to pay
48 X I 4i = 3 6 o for the li^ four
^6x1 It = 2 9 6 for the 2d four
24 X I 4f = 1 13 o for the 3d four
12 X I 4t =; o 16 6 for the lad four
;6 X
10 zz 260
12 X
4= 48
9 X
4= 36
6 X
4 = 24
3 X
A-— rz
Part I. Double Fellowship.
lo;
(5-)
150 X 7 = 1050
~.o
■ ICO X 5 = 500
170
270 X 6 = iCio
Prod, of A*s flock 1
31^ X 6 = 1890
KO
165 X 9 = 14S5
500
665 X 3 = 1995;
Prod. ofC'sft-ock ?
and lime j ^ ^ and time j -^'^'
ioj X 5 == 1C25
no
315 X 4 = 1260
150
165 X 9 = 1485*
I377Q
Prod, of B's {lock
and time
3170
3770
5370
1 23 10 fumoftheproc
I2JI0 ! 450 :
12JI0 : 450 :
12310 ; 450 :
I.
I. s. d.
115 17 7I-
137 1^ 3k"
196 6 oi-
By, Note 4th, Piop. I.
3170
3770
5370
fern.
• 1740 A's n»ar«
.2380 B'l «-—
J. s. d.
29 1^. 7 "9
I,
500 X 3
400 X 2 : 29 12 79 '. ' 350 X 5
1.
s.
d.
^s
II
^^
A' 3
ga
n
64.
16
3^
B*s
—
29
12
75
C'3
—
[50 o o whole gain.
(7-)
By Note 5th, Prop. II.
J- s- d. 1. s. d.
4+40 X 6x12=3182 8 o A's gain X by B's and C's time
42 16 9f X S X 5i ^n 4112 12 5I- B'sgain X by A's andC's time
79 II Zy X S X 6=13818 17 75- C'sgain X byA'saiiJB's time
11113 18 4f- furn of the produds
io6
Double Fellowshi?.
I. s. d. 1.
11113 18 4t : 227
111 13 18 4f : 227
iiij $ iS 4| : 227
J. s.
d.
1.
3182. 8
65 A'aftock
41 12 12
9l
: 84 B's
3818 17
7i
: 78 G's
(8.)
By Note 2d, Single Fellowfhlp.
A. B. C.
2 + 3 + 4 = 9 ^""^ °^ ^^^^^ proportional gains.
1.
2 ; 52 A's gain ) So far Meff. Hili and Bidt
3 ' 78 B's — ;.
234
234
254
are right ; the reft is to-
4 : 104 C's j tally falle in principle
Again, by Note 5th, Prop. II.
52 X 5 X 7 =: 1820 A's gain X by B's and C's time
78 X 3 X 7 =z 1638 B's gain X by A's and C's time
104 X $ X ^ =z 1560 C's gain X by A's and B's time'
5018 fum of .tiic produfts.
50 Id
fol8
1.
1.
s.
3822 : :
1820 : 1386
4
JS21 : :
i6$% i 1*47
11
$Hi I :
1560 ; nS8
3
844 C'i
T?7
(9.)
By Note 2d, Single Fellowfiiip,
X. Y. Z.
2 + 3+4 = 9 ^"^ of their proportional gains,
420
420
420
1. s.
93 6
T40 o
iS6 13
1. s.
93 6
140 o
iS6 13
8 X's gain ^ So far Mr. ^'j/ts' fo-
o Y's > lution is right; the
4 Z's J reft is totally falfe
in principle.
Again, by Note 5th, Prop. II.
d.
8 X 6 X 9 ZZ 5040 X's gain X by Y's and Z's time
o X 4 X 9 iz: 5040 Y's gain x by X's and Z's time
4 X 4 X 6 — 4480 Z's gain X by X's and Y's time
14560 fum of the produfts»
it I.
D y B
LE Fellow
SHIP, 10
I.
I. s.
d.
14560
14560-
14560
: 4262 :
: 4262 :
: 4262 ;
: 5040 : 1475 ^
: 5040 : 1475 6
: 4480 : ijri 7
11-.'/.°^ XV (lock
il-iW. Y's
S^iW^ Z's — ,
(10.)
By Note 6th, Prop. III.
1. s. 1. 1. s. I. s.
7815:210X9:: 76 4 : 1828 16 pr. of A'sftockandtime
7815:210x9:: 5710:1380 B's
78 15 : 210X9 :: 105 0:2520 C's _
1. s. I. s.
Then 1828 16 -^ 6 = 304 16 A's ftock
1380 -r 5 = 276 B's ■
2520 O-r- 12 ~ 210 C's: .
210 D's »
in.)
By Note 7th, Prop. IV.
1. s.
89 S
9^ 15
38 10
s. 1. s. s.
— 17S5 A's ftock. 7.!^ 10 IT 510 A's gain.
— 1855 B's 37 2 — 742 B's
-1 770 C's 24 4 — 484 C's
510 X 1855
742 X 1785
Produa of
X 770 = 728458500 A's gain X by B's aad C's ftock
X 770 — 1019841900 B's gain x by A's and C's -—
484 X 1785 X 1855 — 1602608700 C's gain X by A's and B'
3350909100 fum of the produdls.
33509P9JC0
^350909100
3350909100
23
23
23
728458500
1019841900
1602608700
5 m. A's time
7 m. B's -. — r-
ii m. C's — -
(12.)
By Note 8th, Prop. V. "
1 12 5 X 1400 X 1800 zr 28350C0000 B's gAiii X by C's and D's ftock
210 X 5000 X 1800 zi 1S90000000 C's gain x "y B's ind D's
ao2'5 X 5000 X I4<^0— 1417500000 D's gain x by B's daid C's
2835000000 t
2835000000 :
If 5000!. : 1125I
per annum.
: : 189000C000
: : 1^17500000
lool. : 22I, los. each Merchants' gain per cen
8 C's money was In trade
6 D's .
loS Loss A :: D G a i x.
Here 315I. is to be divided into two parts, as 12 to 8.
12 -J- 8 zz 20 fum of the proportional nun)bers.
20 : 315 ; : 12 : i8q B's ftock, 1 For the gains are
20 : 315 : : 8 : 126 A's j equal, therefore the
flocks are reciprocally as the times.
(I4-)
a. s. d. a, r. p. 1. s.
If I : 18 o : : 394 3 34 •* 35^ 9*32^ val.of A'sprop.
If I : 19 6 : : 417 i 14 : 406 18-08125 B's
If I 121 o : : 714 3 o : 750 9*75 C's
Whole ^uant. 1527 o 8 1512 17*15625 whole value.
If il. 5;s. 6d. : : 1512I. 17W5625S. : 416I. 14s. 4jd. '25
value of the land allotted for the tithes.
If 19s. 9id. : 1 a. : : 416I. 14s. 4^d. '25 : 421 a. or.
16^1 p. quantity of land allotted for the tithes.
LOSS AND GAIN.
This Example is ivorked,
(2.)
I. s. d.
I5cwt, at il. us. 6d. percvvt. amounts to 23 12 6
i5twt. at 4-^d. per lb. — — 31 10 o
Diff. Gain 7 17 6
(3-)
s. d.
3 4 value of 1 20 eggs at three a penny
^ o — 1 20 ■ two a penny.
8 4 prime coft of 240 eggs
8 o felling price of 240 eggs at five for 2d.
o 4 lofs.
Part I. Loss and Gain, lOf
(4-)
12X2 — I =23 pipes faleable,
I. s.
1014 6 value of 23 pipes at 7s. per gallon.
907 4 = 75I, I2S. X 1 2 prime coll of the wine.
107 2 gain.
7s. 6d, — 5s. 4d. =: 2s. 2d. gain per yard.
Then 340 yds. at 2s. 2d. per yard, amount to 36I. i5s,
8d, whole gain.
(6.)
This Example is n/jsrked,
(7.)
If i7^d. : lool. :: i3fd. : 78I. ^s. 2|:d.-ff- which de-
duced trom lool. leaves 2 1 1. 14s. 9|d.-|;| lofs per cent.
(8.)
If 27yds. : 17I. i;3. :; i yd. : 13s, 2|d.-f prime coH;
per yard.
If 13s. 2td.-4 : lool. :: 9s. lod. : 74I. 7s. 4id.-J-I4-
which dedu<5led from lool. leaves 25!. 12s. 7;id.-T.T9 lofs per
cent. Or thus,
27 yds. at 9s. lod. peryard, amount to 13I. 5s. 6d.
If 17I. 17s. : lool. : ; 13I. 5s. 6d. : -4I. 7s. 4id.-H^
&-C, as above.
{90
If6ol. : lool. : ; 75I. : 125I.
25 gain per cent, Mr. MaU
col/ns anfwcr by mi/lake (probably in i\v^ prefi) is only 5I.
(10.)
If 7s.6d. : lool. : : 6s. zH, : 82I. los. whichdevlufted
from lOol. leavca 17I. los, Icfs per cent,
L
110 Loss AND Gain.
(II.)
^hii Example is nMorked.
If lool. : 120I. :: iis.6d. : 13s. 9*d.-| pcryard.
(13.)
If lOol. : 94I. i: 6s. : 5s. 7-|d.-i| perbundle.
(I4-)
Ifiool. : 115I. : : 12I. 12s. : 14L 9s. g^^d.-lpercwt.
Cv)
If lOol. : 82I. los. : : 7s. 6d. : 6s. 2 ^d. per yard*
(16.)
This Example is ^^juorled,
(17.)
IfSs. 9d. : 112I. :: ics. 6d. : 13+I. 8§.
34 8 gain per cent.
(18.)
If 15s. : §61. : : 21s. : 120I. 'b%,
100
20 8 gain per cent.
(I9-)
If 20I. 93. 6d. : 112I. los. : : 17I. is. 3d. : 93I. 15s.
which deduced from lool. leaves 61. 5s. lofs percent.
(20.)
This Example is nvorked,
J. s. d.
If I lb. : 7fd. : : i5cwt. 3qr. i81b. : 55 U 9 ^^Id for
125I. ; icol. : : 55I. 13s. gd. : 44 ' » opnmecoft.
II 29 whole gain.
Part L L o s 3 and Gain. i m
(22.)
1. s. d.-
coodeals,ati5d.perpiece,amountto 31 5 o felling pric<?.
If9il. : lool. :: 3il- 5^. : 3+ ^ 9HV pr- coil.
3 I 9i-vV whole
.. « lofs.
{23-)
I. 8. 1. I. s. d. L s. d.
If io6 10 : 100 : : 500 16 8 : 470 5 3IM C's pr. cofi,
38 II 6 B'sgain.
431 13 9^11 B's pr. cod.
If 104I. 6s. 8d. : lOol. :: 431I. 13s. 9i|td. : 413!. 15s.
'^\^Az\\% A's prime coil.
If i5p. : 4131. ijs* 2id..|i|^§ : : ig. : 4»»4i^' P"
gallon, Anfwer.
(24.)
^kit Example is *Wdrked*
7 y 3?| y 15 = |7S748^ ^ jS^J. 7s. 54, prime eoft of
the cloth.
If locl. : 9cl. :: 1S6I. 7s. 6d. : 167I. 14s. 9d, felling;
price.
Then 186I. 7s. 6d. — 167!. 14s, 9d. = 18I. j2s. gd,
whole lofs. Or thus.
If lool. : 90I. : : 15s. ; 13s. 6d, felling price per yard.
Then 15s. — 13 s. 6d. = is. 6d. lofs per yard, and 7 pieces
each 35I: yards, at is. 6d. per yard, amount to 18I. 12s, gd.
lofs, as before.
* MefT. Birks ftate the queftion thus :
lool. : 61. los. : : 500I. 16s. Sd. t 32!. ice.
locl. : 4-|I. : : 429I. &c. making A's prime coft 41 il. Szc. — Mr.
Vyfe has folved the queftion in a fimilar manner, — here the gain (or lofs)
of lool. is the fccond term in the ftating, inftead of its amount j which
is tlic very remark Mr. Fyft makes \>pon Mr.. Dih':orth,
L 2
JiJ Loss AND Gain.
(26.)
475 yards, at 10s. 6d. per yard, amount to 249I. 7s. 6d,
prime cod.
If lool. : 130I. : : 249I. 7s. 6d. : 324I. 3s. gd. felling
price.
Then 324I, 3s. 9d, — 249I. 7s. 6d. zz 74I. i6s, 3d. gain
in the whole.
(^70
If icwt. : 3I.cs.8d. :: 127X470^1. : 1733I. irs. prime
coft of the fugar,
£733!. IIS. 4- 52I. los. =: 1786I. IS. felling price of the
fiigar.
If 571I cwr. ; 1786I. js. :: ilb. : 6id.-4tf per lb.
Anfvver,
(28.)
I.
1400 cafks, at 2I. 5s. per calk, amount to — 3150
7C0 diito, at 2I. 15s. " ■■ ■ 1925
Value of the 700 remaining cafks 1225
Then 122^ 4- 70G =s iK ip. per cafk.
Or thu%
Since he gained 1 o% per caik by one half, he of courfe
muft lofe I OS. per eafk W the other, to make his purchafc
money ofi/j. Therefore zL 5s, -- iqs, =; il, 1554 as above,
(29.)
In order to gain as much by the v/hole quantity as four
yards are Md for, it is evident the merchant mull fell 96 yds.
at the prime coft of 100 yds. (for 100 — 4 = 96.)
Then 96yds. ; jizl. :: i yd, ; il. 3s. 4d. Anfwer.
(30.) ^
If 80I. : looI. : : 20I. : 25I. prime coft of the fnake-root.
If looI. : 125K : : 25I. : 31I. 5s. the money he ou^i>t
to have made of it, which being fold for only zoITtislofy,
in point of trade, is iil. 5s.
Part I, L O S 3 A N D G A I K, I J -
*ofi2olb. = 481b. of good tea, and | of 120 lb. =:
721b. of damaged tea. 48 lb. at los. 6d. per lb. amount to
25I. 4s. — Now in order to make a balance, the gain upon
481b. muft evidently be equal to the lofs upon 721b. viz.
3I. 12s. Therefore, 25I. 4.S. — 3I. t2s. zz 21I. 12s. the
prime coft of 48 lb. hence the tea coil 9s. per lb.
And 25I. 4s. + 3I. I2S. =1 28I. 1 6s. the damaged tez
fold for.
(32.)
800 X H^z 1 1 200 lb. at i2i;perlb. amount to i4ooood.
= 583I. 6s. 8d. the value of the anchovies, or what I re-
ceived for my 7490 lb. of tobacco.
If 117I. * : lool. : : 583I. 6s. 8d. : 4.98I. lis. 6j\\i,
the real value of my tobacco, which divided by 7490, givc^
i^|d.-||-5|} the real value per lb.
(35.)
If lool. : 140!. :: 347^1. 15s. : 4866I. i s. flerling, the
Englifh goods fold for in France.
If lOol. : 85I. : : 4866I. is. : 4136I. 2s, io|d. the
French goods fold for in England.
Then 4136I. 2s. lo^d. — 3-l75l» 15s. =z 660I. 7s. lofd,
theEnglifti Merchants' gain,
(34.)
FIrfl 50C0 ells Flem. = 3000 ells Eng. = 3750 yards,
and 42 50 guineas ~ 4462I. 10s. Then,
ells. I. s. ell. 1. s. d.
I^ -^— + 3000 : 4462 10 : : 1 : 1 5 ^.^J the
prime coft of an Englifh ell.
el!. I. s. d, ells. I. s. d
whole piece Uood me in.
• Thrs (jueftlon is in Meflirs. Blrki and Vyf^i' Arithmetic, ,(cjtnd!cd
from Clare's Introduftion to Trade and BuUnefs) with a folution upon
falje principles, fimilar to the 23d qucftion ; their error may be fccn by
comparing their fclutions with the ftating marked wi;h ;hc alkrlfm.
B X » r E (t.
BARTER.
CO
This Example is ^corked,
(a.)
e^ual
equal
93 X 400 :e 37200. Then 37200 -r 9 = 4^333 gallons,
AnlVver,
(3-)
5cvvt. I qr. 10 lb. at 2s. ^.Jd. per lb. amount to 70I. 7s. 9|d.
If 5s. 9d. : I b. : : 70I. 7s. 94d. : i^i^\\^ bufhels. Anf.
(4-)
equal
If 5000 yds. at i3l:d. k^ i value.
• at 1 8s, 6d. rS I value.
equal
500 y. 13! = 57500, this divided by 222 ( = i8s. 6d.)
gives 304 -jy yards. Anfvver.
(5-)
6 hhds. at 6s. 8d. per gallon^ amount to 12 61.
If 252 yds. : 126I. :: 1 yd. : los. Anfwer.
(6.)
288 ells, at [OS. 3d. per ell, amount to 18I.
If 19s. : I cwt. :: 18I. : i8cwt. 3^1, 2t/ylb.
(7-)
14 cwt. 3 qr. at il. 17s. per cwt. amount to 27I. 5s. 9d.
If 3s. 9d. : I gal. : : 27I. 1%, 9d. : 145 ^*j gallons. Ahf,
(8.)
3 cwt. 2qr. i61b. at il. 173. 4d. amount to 61. j6s.
If 5s. jd, : 1 2 lb, :: 61. i6s,- ; 315^-1^. Anfwer.
Parti. Barter. tij
(9;)
This Example is ivorked,
fio.)
This Example is nxjorked,
(II.)
I. s.
700 gall, at 4s. 6d. amount to 157 ^o
Paid down — — 287
To account for 129 3
Ifiifd. : I lb. : : 129L 3s. : 2695^^3 lbs. Anfwer.
(12O
1. s. d.
57 qr. 61b. at il. lis. 6d. amount to 90 19 if-
i4cwt. 3qr. i81b. at 4I. 14s. 70 i 7^-^
Balance 20 17 6 f
If 7d. : I lb. : : 20I. 178. 6d.4 : 7155J lb. Anfwer.
(13.)
1. s. d.
27 cwt. at il. IIS. 6d. amount to 41 6 o
,25 pieces at il. 19s. lofd. 49 16 lof
Balance due to B. 7 i© lof-
If3lyds. : 15s. gd. : : 120yds. : 27I. valueof A'skerfey.
. If 7s. -f* 6s. 6d. : I ; : 27I. : 40 hats and 40 pairs of
(lockings. Anfwer,
1. s. d.
73 J yards, at 8s. 6d. amount to 31276
<r32 canes, at 3s. 79 16 o
1 6 pieces, at 4I, . 64 o o
Value of A *s goods 456 3 6
ii6 Barter,
I. s. d.
B's cloth per yard — — i o o
Glafs manufadure, per lb. o i 8
Ditto finer — — 024
Value of I of each fort of goods i 4 o
If il. 4s. : I : : 456I. 3s. 6d. : 380/^ of each.
(16.)
1. s.
13784 8 value of 87 9 2 yards, at il. iis. 6d.
3446
2—
i
4-
ditto value (
Df the fugar
1723
8615
I—
5-
JL
-1
ditto value of the j")epper
ditto value of the rum
1. s.
d-
C'.Vt.
I.
s. CWt. (
IT. lb.
If
» ^S
6 :
r
• • 3446
2 : 1941
I 247*
If
7 3
9 :
; I
: : 1723
I : 239
2 ^sh
'^i
If
5s. 6d.
: I
gal.
: : 861 5I. 5
(17.)
s. : s^szSl,
gal.
I.
s.
24
B
. puncheons,
paid in cafli
at 4s. gd. per gall, amount to
Balance
478
16
10
321
6
If 714 yds. : 52 lU 6s. : : i yd. : 9s. per yard, Anfwer.
(18.)
If lool. : 117I. 69. 8d. : : 6s. 3d. : 7s. 4d. A fold the
tea for per lb.
Then 7s. 4d. — 65, 3d. 1= is. id. his gain per lb. and
3s. 9d. — 3s. 4d. zr ^d. the gain upon i lb. of tobacco. —
The number of lbs. of tobacco is to the number of lbs. of
tea it2<verfely as 3s. 9d. is to 6s. 3d. or direSllj as 5 to 3.
Therefore the gain on the tobacco is to the gain on the tea
in a compound \2X\Ci of 5 to 3 and of 5 to 13, ox Jtmply as 2 j
to 39. But the whole gain is 81. ics. 8d. which divided in
the above ratio (per Note 2d, Single Fellowship) gives the
gain upon the tea 5I. 4s. and upon the tobacco 3I. 6s. 8d.
confequently the number of pounds of tobacco bartered were
160, and of tea 96.
Parti, Exchange. 117
EXCHANGE.
This Example is wuarked,
[^■)
This Example is ivorked,
(3-)
As lojt : 100 : : 577 g, 14ft. : 546 g- 9tM P- Bank.
(4.)
As 100 : 105I : : 765g. 9ft. : 8o8g. loft. 2yV ?•
current,
(5.)
As i04|. : 100 :: 7570 g. 15 ft. : 2887 rix d. 26J||(^.
Bank.
(6.)
As i2;K : looU : : 797I. : 637L 12s. Bank.
(7.)
If 100 I u© i ! 790 J 9+8 ducati current.
(8.)
This Example is nvQi'^d,
(9O
If il. : 2450 fen. :: 4751I. 15^. : 11641787$^ fea. :=
60634 marcs, 4 fols lub. j i|- fen.
(/o.)
If il. : 36s. 6d. : : 475I. i8s. : 208444 5- f^. Flemifli,
which divided by 2 zz 104222 -j-'o fols lub. = 6513 marcs,
I4jg fols lub.
If lU : 35 folsgr. id,:: 749I, 145, : 1893742! fen. =:
9863 marcs, 3 foh lub. lof fen.
u3 Exchange.
(12.)
If il. : 3j|.foIsgr. glA, : : 754I. i8s. 9d.K.i889986j\; fen.
rr 3281 ri'x d. lofols lub. lo-j^ fen. Bank.
If 100 : io8|: : : 3281 rix d. 10 fols lub. io~ fen. :
2045909 fil^l fen. = 3551 rix d. 44 fols lub. 5*4 4 JH fen.
current, Anfwer.
('3.)
If il. : il. 15s. 9d. ;: 757I. i8s. 7d. : 1354I. 15s.
1 J id.-^V Anfwer.
(14.)
If il. : 34 fols, 4fgr. : : 754I. us. 9d. : 7781 g.
13ft. io|pennings. Anfwer.
Us-)
If il. : 54s. 7-*:d. : : 479I. 148. : 199315/5 Flemifti pence
Bank, which divided by ico, gives 19933:^0 "J^- ^« ==
1993 rix d. 7 ft. I o| pen. Bank.
As 100 : 104I : : 1993 rix d. 7 ft. ic-Jp. : 2085 rix d.
1 6 11. 1^ -iVcPo pen. current.
(16.)
If il. : 47icop. d. : : 547I. 191. lod. ! »69Sq|! cop. di
S a6o?9 copper dollars, * copper marcs, 3I ruoKks,
('7')
If il. : 48 cop. d. :: 3749I. 14s. loid. : 1 799 87 -f^^ cop. d.
= 179987 copper dollars, 2 copper marcs, 6jrunftics
(18.)
77'/> Example is <woried,
{19')
If34i^ol»gr. : il. • "• 1788 rix d. 21 fols lub. : 414I.
14s. 2id.-if
(20.)
32s. 6d. = 2340 fen. ( X 12 and 6) and 747 rix. d. 2 m.
14 fols lub. ==: 430824 (X 3, 16 and 12). Then 2340 fen,
; il, ;; 43o824fen. : 1S4I. 25. 3-rj. Anfwer.
Parti. Exchange, 119
• (21.)
Firft, I rlx dol. ( = 3 X 16 ) = 48 fols lub. hence J rix
dol. = 30 fols lub. and 120 fols lub. 1= 20 fols gros, zz il.
Fl. ••• 6 fols lub. = I fol gros, and 8 fols gros, ~ a rix dol.
Then 104 rix dol, 30 fols lub. : 100 ; : 743 rix doL 4 fols
gros, : 7 10 rix dol. i m. 14 fols lub. 4 fen. Bank money. :
33s. gd. = 2430 fen. ( X 12 and 6) and 7 10 rix dol. i m.
14 fols lub. 4 fen. = 409324 fen. Then
If 2430 fen. : il. : : 409324 fen. : 168I. 83. ii-j\ller,
Anfwer.
(22.)
ioqI : 100 : : 1749 marcs, 13 fols lub. : 1^99 marcs,
j^ fols lub. 3 fen. Bank money, — 474^7.^/ dol. 2 fols gros, r=
^48 marcs, 12 fols lub. (for 2 marcs zz: 1 J/et dol. and 2 fols
gros r= 12 fols lub.)
104I- : 100 : : 948 marcs, 12 fols lub. : 908 marcs, 15 foh
lub. 8 fen. Bank money.
mar. folsl. fen.
159.9 13 3
908 I J 8
2508 12 111= 481691 £en.
Now 120 fols lub. = il. Flemifh, and 12 fen. = i fol lub,
••• 481691 fen. =: 334I. los. id. Flemifh.
If 35s. 8d. : il. : : 334I. los. id. : 187I. iis. 5^d. fter-
ling. Anfwer.
(23.) ■
34s. 7d. : il. : : 1747I. 14s. 'jd. : loiol. 14s. 8^d.-JJ
ileriing.
^ (24.)
105:4 • ^-o • • 7495 guild. 14ft. : 7 113 guild. 7 fc. ogr.
2 p. Bank money.
If 34fc. 4gr. : rl. ;: 7113 guild, 7 fc. ogr. 2 p. : 690I,
1 2s. 4d, fterling. Anfwer.
(25.)
If 34s, 3d. : il. : : 774I. iSs. ; 4;2l. gs. ii-iY^d. Anf.
20 E X C H
A N G £,
(26.)
If 48! cop. d. : il. : ; 7i23Cop.d. i4run. : 146I. 17s.
6d. Anfwer,
(27.)
If 49 cop. d. : il. : : 5749 f. d. i cop. d, 2 c. m. 3 run.
: 352I. OS. 2|d,-||. Anfwer.
(28.)
TTjis Example is fworked,
(29-)
If 49 J d. ! irixd. :: 749I. i6s. : 3635 rixd. 2 m.
5 -i^j- fK, Anfwer.
(30-)
If 4s. 7d. : I rub. : : 7574I. 19s. : 33054 rub. 32 cop.
2 -i^ alt. Anfwer.
If4s. 9^. : 1 rub. :: 574I. i8s. : 2399 rub. 58 /j cop.
Anfwer
This Example is ivcrked.
If I rix d. : 48^ d. :: 9751 rix d. 4 m. 3 fk. : 1980I. i6s.
J4^'"tV Anfwer.
(34.)
If I rub. : 4s. 9d. : : 7454 rub. 4gr. 6 cop. : 1770I. 8s.
8 jl. Anfwer.
, (3v)
If 1 rub. : 4s. 7td. : : 7479 r. : 1729I. los. 4^. Anfw.
(36-)
If 1 24 cop. : 25^ guild. : : 4759 rub. 44 cop. : 9595"guild»
12ft. 14 pen. current.
If 103I : 100 :: 9595 guild. 12 ft. i4p. : 9271, guild.
3 ft. I p. Bank.
If 34s. 6d. : iL : : 9271 guild. 3 ft. j p. : 895I. 15s.
3^d. ftcriing.
Parti. Exchange, 121
(37-)
If il. : 34s. 8d. : : 495I. 17s. 6d. : 206284 pence, FI,
If 52 ft; : I rub. :: 206284 pence : 7934 rubles. Anf.
{38.)
If 121 cop. : 2j: guild. : : 7437 rub. 5 griv. 24 cop. :
15367 guild. 4 ft. 10 pen. current.
If 103^ : 100 :: 15367 guild. 4 ft, 10 pen, : 14901 guild.
n ft. 2 pen. Bank.
If 34s. 7d. : il. : : 14901 guild. lift. 2 pen. : 1436I.
js. io|d. fterling. Anfuer.
(39O
If270gr. : il. : : 7947 Fl. : 883I. Flemifli.
If 33s. 5d. : il. : : 8S3I. : 528I. 9s. 6ld.-|J^ Anfwer.
(40.)
If il. : 34s. 8d. : : 749I. 17s. 6d. : 1299I. 15s. 8d. FI.
If il. : 2 74grofti. :: 1299I. 15s. 8d. : 356140]- ^ grolh.
d 3957 rix dol. lOy^ groHien. Anfwer.
■f^ 273 groAi. : 6 guild. : : 4795 Fl. 4 ecu. 4 gr. :
3162 ^\ guilders current.
If io3|g- '• loog. :: 3162 /^ guild. : 3058 guild. i6ft,
I grot = 509I. 1 6s. id. Flemilh Bank.
. If 3i^- 7^* ' '^' • • S^9^' 16?. id, : 303I. 12s. i-^d. fter-
ling. Anfwer.
(42-)
If 3livr. : 3iid. ;; 636 li v. 3 fol. 9fd. : 27I. iCs.
Tl^'-H* Anfwer.
(+3.)
If I ecu. : ;5d. : : 57475 liv. 6fol. : 4230!. i6s. 4{d.^\,
Anfwer,
(44.)
If 54d. : I ecu, : : 759I. i8s. 9d. : 3377-^- crowns.
US')
If 3;49cr. 2 lir. 10 ol. : 343I. 14s. 3d. ; ; i cr, : ud.
p. ecu,
M
122 EXCHANGP^
(46.)
If4;^d. : ipla. :: 749I. 18s. : 396675^*^- piaflres.
(47.)
If I pia. : 47i-J. ; : 1347 pia. 2 r. 24m. : 266I* 13s,
2 id.- 3V
(48.)
If I pia. : 43ii. : : 9749 rials ; 21 81. 19s. 5i|d. Anfwcr.
' (490
If I pia. : 41^.1. : : 7549r. v. : 87I. os. 5id.-|. Anf,
If I m. : 65(1. :: 7434cr. 347162. : 805I. 8s. lold.--^^.
■ (,-■•)
If im. : S4td. :: 4756 m. 2 tes. 4 vin. lorez. : 1278I.
5s. old.-^h'
(52.)
If 66i:a. : i m. : : 17 881. 175. : 6456 milr
(53-)
'li'zcoom. : 6661, 135. 4d. ;; i m. ; Sod. :=: 6s. 8d.
p. mil re. Anfwer.
(54-)
ir I dol. : 53|d. : : 947 dol. : 2nl. 2s. o^d. Anfwer.
If47|d. : ipcz. : : 1749I. 17s. 6d. ; 8795 pez. 3 foL
S: den. 7ie(trly»
(56.)
if46Ad. : I pez. :: 747I. i6s.4d. : 387opez. 2 3-V\fol.
(57-)
If 1 pe/.. : i^y. : : 7439 P^^:. 9 fol. 3d. : 1499!. los.
3jd-Til-
ir,-4d. ; I pia, ;; -.,^^ ^T- '• 2^54 pia. 2 Uie. J3foK
4 den.
Parti, E X c H A N G e.' 123
This Example is i<:odcd,
(60.)
Ifiool. : 105I. 15s. :; 694l.1Ss.6d. : 734^- ^Ts- 74'-io-
(6u)
If lool. : 176I. : : 917I. i8s. ; 46151. xos. o^dol^J'.
(62.)
T/jh Example is wori^d^
(63.)
If 111]. los. : lOoU : : 6747 1. 14s. i 605 il. 14s.
(64.)
If 155I. : looL :: 475!, 14s. lod. : 352L 8-a\s.
Firft the commifli©n of gcol. at 5 per cent, is ^^]. Then
900 — 45 = ^55^' currency the Factor niuft remit according
to the courfe of Exchange.
If 125I. : lOoI. II 855I. : 684I. fterling the merchant
received.
Then 734I. 14s. gd.. — 684I. = ^oL 14s. gd. lofs.
Observatiok. The above feems to me to be the moft rational fo-
lu i m that can be given to this queftion. — A fwnilar queftion to this may
be lecn in fcveral books of Arithmetic, folved upon the following prin-
ciples. Firft 125 4- 5 ~ 130 percent.
If 130I. : looi, :: 9C0I. : 6921.64. iid.--,'j fterling.
Then 734I. 14s. pd. — 692!. 6s. l.|d.-j^j — 4il. 8s. 7d.-j\ lofs,
dift#ring from the above Anfwer by 7I. 6s. i^d.-jy. 1 fubmit to the
judgment of my reader vi^hich iolution i^ the proper one.
(66.)
T'^is Exatnple iifimiiar to the preceding.
The commifGon Gn7i5ol, 148^ at 27 per cent, is 178I,
1 5s. 4f d. Then7i;5ol. 14s. — 178I. 15s. 4|d. n 6971L
18.S. -^d. currency,, the Fadop niuft remit according to the
courlc of exchange
M2
-'4 Exchange.
IfT35l. : lool. :: 6971L i8s. 7fd. : 5164I. 73. loid.-J
fterling.
(67.)
If 1^7!. : lool. :: 7549L iSs. 4d. : 4808I. 17s. 3ld,'y^^j.
An Aver.
(68.)
Ths Example it of the fame kind as the 65th and 66th.
The commifTion of 1757I. at 25- per cent, is 43I. i8s. 6d.
Then 17^71.-— 43I. 1 8s. 6d. = 171 3I. is. 6d. fterling,
to remit to Jamaica according to the courfe of exchange.
If lool. ; 157L :: 17131. is. 6d. : 2689I. los. 6td.-||.
Anfvver..
(69.)
^his Example is nvorked.
In the Anfwer,. for gain per cent, read lofs per cent.
(700
7his Example is ixJoAed,
In the Anfvver, for kfs per cent, read gain per cent.
(71.)
1^37-9-5. : lool. :: 33s. 4d. : 89I. 1 6s. 4jd.-.Vy
'I hen iQo — 89I. 1 6s. 4jd.-TV7- = lol. 3s. 7d.-7^\ lofs
per cent.
If 33s. 4d. : looI. :: 35s. 6d. : 106I. los.
Then 106I. los. — icol. = 61. los. gain per cent.
(73.]
Jr34s. i^. ; iQol. :: 33s. 6d. : 97I. x6s. ^\^*-Y^T*
Theiiiool. — 97I. i6s. zid.-fjy = 2I. 3 s. 9id.-jY7 lofs
per cent.
(74-)
If ;s. 7id. : rool. :: 5s. ^\^. : 92I. tis. io|d.
Then lool. — 92!. us. io|-d. =: 7I. 8s. i {d. lofs per
cent.
Part Tv S'lNi^t^LB Asbitra'Tion OF Exchange. 125^
■ ' ' (750
If 5s..6c!r,:' lool. : : j's. lid. : 9:?^. ^s. 7-^d.-/y.
Then fcol. — 93!^ J3. 7'd.-/-r =± 61. i6s. 4^d.^V lofs
per cent.
(76.)
If4i|d. :.. jpoi. r: ^jfd. ; 127I. i^. 2;^d.-|f
Thea 127L r^s. 2fd.-^| — ^^^ — ^1^* '5^' 2itl'-t? 2'^^"
|>er cent.
SIMPLE ARBITRATION ok
EXCHANGE.
{77-)
Tihis Example is ivorked.
. (78.)
If 3-35. 7d. : 24od. : : 51 ^d. : 3oJ|5d. for 20 vintins^
Of 400 rez.
Then 3o|-7^ x 2-1- = 76 J^fd. fterling, p. milre..
(79.)
(This Example will be more properly exprcfTed, if' in the thiid line,
inltead of that Corrcj'pcndent, you read Rotterdam ; and in the feventh
line, inllead oi Parisy tczd Rot terdam. •^•'Thc '^z pence Flcm. Jhouid be
^z piUings.)
Then inuerfely ^^A, : 32s. : : 54d. : 31^73. Flem. per
pound fterling.
(go.)-
In-ver/elj ^od, : 4.zd. :: 53id. : 39 -j'/^^d. per dollar.
(81.)
First, By remitting to Amfieidam and drawing tipon
If 52d. Flem. : 3i^d. fter. : : 34s, 5 d. Flem. : 9924if.
r: il. OS. 8fid. the merchant made of iL fterling..
If il. ,: II..OS. 8|id. : : looI. : 103I. 8s. 3ld.-f[.
Then io3J..8s..3id.-A^ — icol. := 3I. 8s. 3;d.-f{ gain
per cent.
M3,
126 Simple Arbitration of Exchange*
Secondly, By remitting to Paris and drawing upon
Amjierdam.
If3ijd. fter. : ^id. Flem. :: 24od. fler. : 399 i^ pence
Flem.
If34s. 5d. : il. :: 39Q-j^d. : 19s. 4~'^d. the merchant
would have made of il. fterling.
Ifil. : 195. 4^VJ^3d. :: lool. : 96I. 13s. nid.-^.
Then looI. — 96I. 13s. u Jd.-i|l = 3I. 6s. o Jd..|lf lofs,
per cent.
(82.)
If I due. : ^o^d. : : 9C0 due. : 4^225 pence.
If I due. : p d. : : 900 due. : 45900 pence.
d. pia. d.
If 41 : I :: 45225 : iioj^t piaflres.
If 42J : I :: 45900 : 1086/^*9 piaft res.
Lofs i6|4f^piaftres.
(83.)
d. gu. 1.
]f 18 : I : : 500 : dSd^^ guilders.
ifiOQ : 3 :: 6666 j ; 22^ commiflion.
Diff. 6644I to remit.
If3ig. : icr. :: 6644|g. : 1 898 1| crowns.
If jcr. : 55d. :: 1898?,^ cr. : 43 5I. is. o|:d.-|f.
Then 500 — 43 5I. is. Oad.-|f = 64I. 18s, iijd.-|| the
ir.erchants' real lofs.
I. s. d.
I ool. 1 -i 1 64 18 II i-ll whole lofs.
12 19 9 J-|J the lofs percent. Anfwer.
OflstRVATicK. If the Faftor could have remitted to Bourdeaux
accordiiTg to the merchant's ord»r, he would have gained 7I. lis. i^d.-^y,
Jo that the variat'on in exchange nude 7x1. 103, ^^^i■g^^ ditVcrencf, or
comparative iof-.
Parti. Compound Arbitration of Exchange. 127
(84.)
If I04gu. : 100 gu. :: 3000 gu. : 2884/3- gwilcl. Bank.
Ifgo-id. : 1 ecu. :: 28841^3 gu. : 127401. 58s. i-Y_5Lden.
Ifi05:gu. : 100 gu. :: 30oogu. : 2 857-} guild. Bank.
If 89?d. : I ecu. : : 2857! gu. : i276cr. 56s. r-^Y-den.
Then 1276 or. 56s. id. — 127401. 58s. id. =: i crown,
jSfols, the Paris merchant gained.
COMPOUND ARBITRATION
OF EXCHANGE.
(85-)
7his Example is ivorked.
(86.)
An^cedents.
I FrecK^h crown
65 Eng. pence
100 Stampt crowns
105 Ducats Banco
I Piaftre
* Flemilh pence
Confequents^
rz 30 Engli/h pence,
rz I Stanapt crown,
rz 142 Ducats Banco,
rr ICO Piaftres.
rr 87 Fie mi Hi pence j
— 7547 cr. 15 fols.
4oyS4o|||
6 29
^g^Xi42X/';^;;^X$^X7?4.7i _ 1 864774^3 _
iX0^X/^0X/0^ ■" 455
^3 35
Flem. pence, zz 10246 guild, o ft. 4*||: pen. at Anv.
fterdam.
Then,
If I ecu. : jid. :: 'jU'J cr. i^fol. : 9622 guild. 14ft.
1 4 pen. by the dired exchange.
Hence io246g. oft. 4||^p. — 9622g. 14ft. 14 pen-. =2
623 guild, 5 ft. 6 511 pen. gained by the circular ex-
change.
128 Compound Arbitration of Exchange;
(87.)
Antecedents.
56 pence fteding
100 French crowns
T40 Ducats Banco
IJ5 Stampt crowns
* Pezzos
25 990
Confequcnts.
I French crowTit
60 Ducats Banco^
100 Stampt crowns.
ia<; Pezzos.
i32i6od. iz. 7591* fterllng.
^$:5X/g^^XX/^XX$//^p' _ 5 X^;X 990 ,^ 74250 _ ^ rs
X/pfSX^40X^/^
7 7 ^3
pezzos by the firft method
Anteeedents.
ll. fterling
165. Flemifh
J I P.ix dol.
J40 Ducats Banco
J 1 5 Stampt crowns
* Pezzos
Again,
Confequents.
~ 33s. Flemifh.
zn t Rix dol.
— 12 Ducats.
— 100 Stampt crowns.-
— 125 Pezzos.
— 759I. fterliog.
5
25 Z^
/^a//x^^P^x//^ 14 1-4
4 1 iz
2
13 2:65 r y-J: by the fecond method.
Then 2651 \\ — 1515^5 = io86|f-| pezzos,. advantage
by tlie fecond method.
(88.)
AnteccdentSi
95^ Piaftres :^
I Ducat rr
272 Maravedies, or i Piaftre rr
i^oo Rez, or i CrutaJc zz
56 Flemifh pence rr
I Ecu l^
* pence, flcrling ^
Confequents.
ICO Ducats Banco.
521 Maravedies.
631 Rcz.
50 Fiemifli pcnc^.-
I Ecu.
3i|d. fterling.
I 'Piaftre.
/dp' X 32 1 X 63 1 X ^.d X jij _ 321x631 y:^xzr\
^^xifiixi^<^xi6
19 136 4
6380396 5 __ 133805
I9XI3-X4XS6
d. fterling, p. piaftrCr
Parti. Compound Arbitration OF Exchange. 129
K s. d.
7J47pia{!res, at ^j/.^VWi^^- P- ?^^^^^> = ^733 3 J^l"
754.7 ditto, at 52d. p. piaftre, z= 1635 3 8
Whole gain 97 » 9 si
If 1635I. 3s. 8d. : 1733I. 3s. lid. : : lool. : 105I. 19s.
9?d. from which take lool. there remains 5I. 19s. gjd. the
gain per cent.
(39.)
Bj the Note. — For the Circular Exchange ^
s 00 : t ; : i : '005 general multiplier for the commiffion.
d. cr. 1.
If 54 : I :: I'JSl'l'y • 7812*2' crowns.
For commiffion |: per cent. X '005
— 39*o6i' fub.
Cro^^ns to remit to Venice 7773-161'
cr. d. or.
If 100, : 56 : : 7773*161' : 4352*9702' ducats.
For commiflion \ per cent. X '005
— 2 1 •764851' fub.
Ducats to remit to Hamburgh 4331 '205371'
du. d. du.
If I : 100 :: 4331*205371' : 433i20*537i'd. Flem.
For commiflion j per cent. X '005
— 2 165*60265' fub.
Flemifli pence to remit to Portugal 430954*934 f 5'
d. rez. d.
If 4C : 400 '/ 4309^4-93445' • 3S307'0'534»23
For commiffion \ per cent. X *oo5
— 19153*5526:0
Rcz. to remit to London 3S 11 556*98 rjjj,--^'
ijo Compound Aubitsation of Exchange.
If locorez. : 63d. :: 38115^6-981453 : 240128-089831539
= I cool. I OS. Sd, received in London,
Or thus.
Antecedents.
540. FIcmiih
100 French cr.
• I Ducat
45d. Flemi/h
loco Rez.
Cdn^cjuents.
i«Ecu zz
: 56;Ducats —
io4>d. Flemifti zr
406 Re z. iz
6;jd. llerling-cr
The product of the fccond confequcnts is 24
dividend ; and tlie produft of the antecedence Ji
quotient is 1000-5 3 3795, &c^ I. — icooi. lo-
fey the circular exchange, as above. The
found by dedufting the commijjion from the firft
ing them hi the ratio of zco : J99.
Sec. Confequenti.
•995 Crowns.
55'72 Ducats.
99*5d. Ficmi/h
398 Rez.
63d.
'757-75
I
.3i2969058o'i93;5, the
i 243000000 divifor J the
s. Sd. received in London
: fee nd confequenfs are
confcquenls> or diminilh-
Bj the DifeSl Excha?ige,
If 34.S. 7d. : il. : : 1757I. 15s. : 1016I. 10s. 7d.
Then 1016I. los. 7d. — loool. los, 8d. =: 15I. 19s. nd.
advantage by the dired method.
Antecedents.
lb.
100 Engh'fh
78 Rouen
69 Lyons
72 Geneva
121 Marfeilles
103 Hamburgh
♦ Paris
(90.)
Confcqucnts,
lb.
88 Rouen.
9^ Lyons.
53 Geneva.
100 Marfeilles.
100 Hamburgh,
1 01 Paris.
1 London*
47X53x100x10 1 ^
39X69X9X11 X103 ^
// 47
$$X^-^X53X/jE^^Xiooxioi
/^5T7fx69X^x7?>loo3
39 9 '«
L^li?-o j^^ ^^ p^^.^ ^ ^ j^^ ^^ ^^^^^^^ ^^^^ ^ 25159100
27440127 27440127
2281027 7700245 , .1. r> • iu
n '- = 1 oz. c -!-!■ ^ drs. the Pans lb. ex-
27440127 •'27440127
ceeds the Englilh avoirdupois lb. Or the Englifh lb, is
to the Paris lb. as lOO : 109 nearly.
Part I. Involution.
'3t
INVOLUTION.
This Example is 'worked^
(2.)
'754 X 1754= 307^5:i'5-
(3.)
549 X 549 = 3^1401-
(4.)
3^1416 X 3*1416 X 3-1416 = 3 1 •006494199296.
(5.)
•7854 X -7854 X '7854 = -484476471864.
(6.)
I + I = 2.
57'*5 X 57*5 = 3306-23%
2 + 2 =1 4.
3306-25 X 3306-25 = io93i289-o62y.
(7-)
I + I = 2.
1*732 X 1-732 =: 2-999824.
24-2+1= 5.
2*999824 X 2-999824 X 1-732 zi 15*586171061650432,
(8.)
I + I + I = 3.
735 X 735 X 735 = 39706537?-
Z -\- Z + 3 =
397065375 X 397065375 X 397065375 =
9-
6260 1 689 1 5 5608 1 39974609375.
(9.)
^l + I + I r= 3. .
365^ X 365 X i^^s — 48627125.
3 + 3 = 6.
-48627125 X 48627125 = 236459728.
I J2 E V O L U T I O N. — S QJJ A R L R O O
E V O L U T I O N.— S Q^u A R E Root.
STit/V Example is muorked^
7'>6/> Exa'mple h ^worked.
(3.)
. . . root
393129(627
(4v)
• . ... root
3272869681 ( 57209
122)33
107)772
1247)8729
1142)2386
1 14409 ) 1 02968 J
(5-1
, root
5241578750190521 (12345-6789
22)52
243)*4»
2464)11257
24685)140187
246906)1676250
2469127 )i948i4i9
24691348)219753005
216915569)2222222121
(6.)
root
57i32*oocooo(239*e23,&c,
43)'/'
469)4232
47802 )iioooo
478043 ) 1 439600
o
I + 8
o
proof
5471 rem.
part I. EvoLUTio N.— S a,u a s e Ro o T. 1 3 j
(7-)
, root
75* :547 0000000 (8- 6 802 6497 29, &c.
166)113+ (8.)
1728)13870 ... ... root
i788-57'777777{42-29i5»&c-
173602)460000
82)188 6
1736046)11279600 7 + 8
8449 )77377 6 ^
168906 • proof
84581)133677
863324
12662
See contracted r lo
7
845825)4909677
Divifion of 680552 rem.
Decimals.
(9O
•4325000000
root
( -65764
L, &c.
125)725
1307 ) 1 0000
5
1+4
•X-
13146)85-100
131524)622400
proof
96304
rem.
N
/
ji34 Evolution. — S <^u a r e Root,
root
5'3'3353333 ( 2-3094.. &<:.
4609)43335 ^ ^^
43)133 2
o -f 2
X
46184)^85233 2
' proof
497 rem.
(II.)
This Example is ^worked.
This Example is ivorkei,
567 81 V 8j 9
(14.)
or
, \/i4822C 38c f
27; X 539 = 14822: and 1 i = ^-^ =-^root.
ill =.15 ,,d /ir^-iroot.
539 49 V 49 7
(^50
45 X 94 = 4230 and 2LL^ = - ^ ^^ =
^^ ^^ "^ 94 94
•6918984, &c. root.
('6.)
-V/15I = >/i5'625 = 3*9 > 2^4 root"
(17-)
V29>5 = 'v/29-i6 = 5*4 root.
Parti. Evo L u Ti o N.— Sq^u A RE Root. 135^
(18.)
(2oL IS. = 2401s. and v' 240 1 = 49s. = 2I. 9s,
3oJ;d. = 121 farth. and ^^ 121 = 11 men, they fpent
each 1 1 farthings, confequently each man drank a pint and
a pennyworth, or it pint.
(20.)
£4 X 24 =: 576 =: A B x A B
18 X 18 = 3^4 = B C X B C
v'goo nz 30 ~ A C the.
length of the ladder*
(zu)
50 X 50 r: 2500 zr E C x EC
30 X JO iz 900 = B C X B C
^1600 iz 40 — BE
50 X 50 = 2500 zr E C X EC
40 X 40 ZZ. 1 600 zz A C X AC
V500 zz 30 zz A E
Then ABzzAE + EBzz
4o + 3oz= 70 ft. zz23|yd5. the
breadth of the tlreet.
Firft,
^•449 zz B G
64- —AD
B E
z6-249 iz E G
97 X 97 = 9409 zz D G X D G
26-249 X 26-249 — 689-01 — E G X E G
N 2
156 E V O L U T I O N. S QJU ARE R O O T,
^8719-98 — 93-3808 ri D E = A B the iiitsnce of the httom of
the tower A f-om ttat of B i 76 -f 93*3803 rr 169-3808 r: A C the
ci.lance of the bcrtcm of the tower A Irom tbat of C zr K K.
Secondly,
64—50—14 — DK=:AD — CH
^ 369-3808 X 169-3808 — 28689-8554:1: KH X KH
14 X 14 — ^S^"
z:DK X DK
^^28885-8554— 169-958 — DH the
dlftance of the t':b of the tower A from tLat of C.
Thirdly,
90*249 -^ 5c = 40*249 — LG — BG — CH
And HL —EC
76 X 76 — 5776 = HL X HL
40*249 X 40*2-49 =^ 1619*982 ::: LG x LG
v^7395-9S2 r:"85-999, &c. = 86 «jr/jfj
the diA»ace of the /op of the tower B from ibat of C.
'3->
For that of C
50teet, -ead thit,
of C 28 feet.
D
Then,
The fide of aa
3 cq«l lateral trian-
gle divided by the-
fqaare root of 3>
gives the radius
of .t? circumicri-
bing circle.
50
v:
i*73Z,£L-c.
— aS-SSy feet ir AG the diftance of each tower
from the centre of the garden, and
30 + 54 + 28 __
— 3o| the height
of a mean tower.
30^ X 3 of = 940*444444
a8-S67 X 28*867 — 833*;c5689
•f the ladder nearly
Vj773-748i33 =: 4Ji»ii5 = PD the length
Part I. E V o L u T I n. — C ubeRoot, 137
1773-745133 — PD X PD
30 X 30 m 9<^'0« ~ AD X AD
n/S73-748i33 = 19-559 = AP
177^742133 = PD X PD
34 X 34 zr'iisO^ = BD X B D
v'617.748133 = i4'S54 = B P
i77r74Si33 rr PD X PD
28 X 22 iz: 784* = CD X CD
v/989-748i33 lt 31-46 = C P.
Observation. Had the height of A been 38, B 42, and C 45,
and the diftance from A to B z= 50, B to C zz 40, and from C to A rr
47 feet, the operation wouVi have been more difficult.— The length of
a ladder in this caft would have been 49*552j and hence, the didances
would have been found as above.
CUBE ROOT.
T^is Exa?n^le is <WQrked,
(2-)
122615327232 ( 4968 root
4 X 4 X 4 = 64
4 X 4 X 300 — 4800 ) 58615 relbl.
9x9X9= 729
4 X 30 X 81 — 9720
4800 X 9 zn 43200
53649 fub.
49 X 49 X 300 = 7203CO ) 4966327 refol.
6x6x6 zi: 216
49 X 30 X 36 = 52920
720300 X 6 zz 4321800
4374936 fub.
496 X 496 X 300 — 73804^00) 591391232- refol,
8x8x8 zz 512
496 X 30 X 64 rz 95232c
73804800 X 8 zz: 5904384C0
■59139123Z
N3
J38 Evolution. — Cube Root,
(3.)
41421736 ( 346 root.
3 X 3 X 3 = ^7
3 X 3 X 30^ = 51700 ) 14421 refol.
4x4x4 ==
64
3 X 30 X 16 —
1440
2700 X 4 —
10800
12304 fub.
34 X 34 X 3C0 = 346800 ) 21 17736 refol'
6x6x6 = 216
34 X 30 X 36 =r 3672a
346800 X 6 zz 2080800
2117736 fufe.
{4-)
705'9i9947284{S'9jQ4 root«,
8 X 8 X 8 = 512
2 X 8 X 300 — 19200 ) 193919 refol.
9X9x9 = 729
8 X 30 X 81 r= 19440
J9200 X 9 — 172800
192969 fub.
S90 X 890 X 300 — 237630000 ) 950947284 refo].r
4x4X4 =: 64
890 X 30 X 16 zz 427200
237630000 ^4 — 950520000
950947264 fub.
Rem. 20, &■;,
Parti. Bto L u Ti ON.— C u B E Root. 139
is-)
. . ; root.
i7*5:4ooooooo(2'598. Sec.
2x2x2 = 8
2 X 2 X 3C0 = 1200)9540 refol.
5X5X5 = 125
2 X 30 X 25 ^::r I ;oo
1200 X > -■= 6coo
7625 fub.
25 X 25 X 3C0 = 187500 ) 1 91 5000 refol,
9X9X9 = 729
25 X 30 X 81 =: 60750
187500 X 9 zz 1687500
1748979 fub.
259 X 259 X 300 = 20124300) 1 6602 1 000 refoL
8 X 8 X 8 = 512
259 X 30 X 64 = 497^80
20124300 X 8 = 1609944.00
1 61 492 1 92 fub.
Rem, V4528808
1^0 Evolution. — Cube Root.
(6.)
254358061056000 ( 63360 rooti
6 X 6 X 6 =z aj6
6 X 6 X SCO n loSoo ) 3S35S rex"'!.
3X3X3 = ^7
6 X 30 X 9 = ^^^^
icSoo X 3 == 3MQ3
34C47 fub.
63 X 63 X 3C0 => 1190700)4311061 refoU
3X3X3 = 27
63 X 30 X 9 = 17^10
11907C0 X 3 = 3572TOO
35S9137 fub.
633 X 633 X 300 m 120206700)721924056 refoU
6x6x6 rr 216
633 X 30 X 36 =. 683640
120Z06700 X 6 zz. 72124020a
721924056 fub,
••• • ceo
(7-)
. . . root.
•573450000000 ( '830^, &c»
8X8x8 = 512
8 X 8 X 300 — 19200 ) 61450 reW,
3X3X3= 27
8 X 30 X 9 = 2j6o
19200 X 3 = 57600
59787 fub;
83c X 830 X 3C0 :^ 206670CC0 ) 1663C00000 refol,
8 X 8 X 8 rr 512
83c X 30 X 64 r= 15936C0
»o66700oo X 8 = 1653360000
1 6549 541 12 Alb.
8045888 rcmr
Parti. Evolution. — Cube P. o o t, 141
(3.)
.... . root.
75:'3857ooooo ( 4*224, Sec,
4X4X4 = 64
4 X 4 X 300 = 4800 ) 1 138 J refol.
10088 fub»
42 X 42 X 300 = 529200 ) 1297700 refol,
1:6344^ Tub.
422 X 422 X jco zii 55425x00) 2342520C0 refol.
213903424 fub.
20348576 rem.
«^ (9)
root,
•785400000000 ( -9226, Sec,
9x9X9= 729
9 X 9 X 3C0 zz 24300) 564.00 refol.
49688 fub.
92 X 92 X 300 = 2539200 ) 6712000 refol.
5089448 fub.
922X922X300 z= 255025200 ) 1622552C00 refol.
1531147176 fub.
91404S24 rem.'
14*'. E V o t u T I N, — C e B E Root,
(10.)
root,
517-375475000000 (8*0278, &c,
8x8x8 = 512
80 X 80 X 300 =z 1920000)5375475 refoL
3849608 Tub.
802X802X300= 192961200)1529867000 refoL
1351907683 fub.
fo27xSo*7X 300 = 19329818700)173959317000 rcfolr
I546<;396i952 fub.
19305355048 renit-
(n.J
roof.
20874107909304)27534
2x2X2 = 8
2 X 2 X 300 = 1200 ) 12874 refoU
1 1 683 fub.
27 X 27 X 300 = 218700 ) 1 191 107 refol,
1 1 1 3 87 5 fub.
275 X 275 X 300 = 22687500} 77232909 refol.
68136777 fub.
i7 S3 X 2753 X 300 = 2273702700 ) 9096 1 32304 refol.
9C961 32304 fub»
Part I. EroLUTio n. — C u b s R o o t. 1 43
(,2.)
root.
1551328-2159785156^5 (115.7625
X X I X I = I
X X I X 3^0= 300)551 refol.
331 fub.
ji X u X 300 :::^ 36300 ) 2Z0328 refol.
189875 fub.
115 X 115 X 300 — 3967500) 30453215 refol.
27941893 fub.
3157 X 1157 X 300 — 401594700)2511322978 refol.
2410817976 fub.
I1576 X 11576 X 300 — 40201132800) 100505002515 refol.
20403654728 fub.
115762X115762X300—4020252193200)20101347787625 refol.
20101347787625 fub.
(13.)
*Tbts Example is ivorked
(14.)
7i54'i09i6753o ( 19*26921, &c,
1 X I X I = 1
jy, ZOO zz 300)6154109
19 "\20513 NoTi. The 3d figure In the
•\- 9 JiS root by the rule would be 5, but
■ .■ by Involution I find that too big.
285 ) 1 5 1 3 —-See the Note to the Rule.
1425
144 E V O L U T I O N.— C U B E R O O T.
7I54MC9I675300COOCO
IC2 X 19a X 192 ~ 7Q778SS
192. X 3CCC0 zz 5760000 ) 7682116753C000000
1926 \ 13337008251
4- 6 / 11556
J9329 \ I78IOO
+ 9 / 173961
193382 \ 413982
-|- 2 / 336764
1933841 ) 272I85I
I933S4I
7SS010, &c.
(•5-)
Flrft, root.
8302348CCCC00, &c. ( 20248'8475, See*
2X2X2 = 8
2 X 300 = ^^° ) 3^^348
2C2 ) 503
99
Second,
S3C234S0COCOOOOOOOCO00000
2C2 X 202 X i02 — S24240S
SO^XSCCOCCO— 6060COOCO ) 59C4QOCOOCOOOOOOOCCOOGO
2024 ) 98910891089108
40288 ) I7950S
402968 ) I72049I
2029764 ) 965870S
20297687 ) 1 53965291
202976945 ) IIS8I4820S
J73263483, fcc.
Fart I. Evolution. — Cube Root. 14J
(16.)
Flrft, . . • root.
I'oooooo ( i'25992io49S94, &c.
1 X 1 X I = I
I X 300 zi 3c>o ) 1000000
^^ ) 3333 '^^''^ 3^ figure would be a 6,
• but that, by Involution, is to*
14^ ) 933 ^'§> ^ therefore take 5.
57
Second,
2 '000000000000000000
125 X 125 X 125 — 1953125
S25 X 30C00 n 375^000 ) 4637500COC0000000
1259 ) 1250000000Q
12689 ) 116900
;269S2 ) 269900
1269841 ) 1593600
323759, &c.
Tiiird,
•^ 2 '000000000000000000000000000000000000
[259921] =r 1999999762590486961
i2.:;oc2iX 3000C00 \ c
_Y775;630ococo ) ^37609513039000000000000000000
12599I104 ) 6-863600982125069, &c,
1259921089 ) 12466759332
12599210988 ) II274695SII2
125992109969 ) ii9532702o35o
X25992I0997S4 ) 6I39803II2969
1100118713833, dtc.
This laft operation would give the root true to n;ar twenty placet of
ilLcimals, if neceflary.
o
146 Evolution. — Cube Root.
(.7-)
Firft, .... root.
•0C0135700CO0 ( •05i387799i2> &c.
5X5X5 = 125
5 X 300 = 1500 ) 10700C00
51 ) 6666
523 ) 1566
Second,
I 357oooooooocooooocoooooooe
513 X 513 X 513 = 1350^5697
513 X 3CCCC00 — 15350CCC00 ) 6943040000000C00CC000000
5135)451139670110461, &c.
51467 ) 400996
5H747 ) 4072770
5147549 ) 469 541 "
51475581 ) 63617004
514755222 ) 1214142361
JS4630717, Sec.
(18.)
Firft, . . . root.
J3'666666 ( 2'39cS6o3o, &c.
a X 2 X a = S
a X 300 zz 600)5666666
23 ) 9444
^69 ) 2544
123, &c.
Part I. EvoLUTio n. — C u b e Root. 147
Second, . ........
13-666666666666666666666666
239 X 239 X 239 = 13651919
239 X 3C00000 — 717000000 ) 14747666666666666666666
23908 ) 2056E572756857, &c.
239166 ) 1442172
23917203 ) 71767568
: 505957, Sec.
(r9.)
Firft) . . . • . root.
92398647506217 ( 45208*6846, &c,
4 X 4 X 4 = 64
4 X 300 zz 1200 ) 28398647
45 ) 23665
5C2 ) 1165
161
Second, • ....?. . .
9239?. 647 5062 1 70C00000000CO
452 X 452 X 452 = 9^345408
45^ X 3000000 =. 13560C0000 ) 5323950621700C0000COCOC
45208 ) 3926217272640]
452166 ) 30 ,5772
4521728 ) 3827767a
45217364 ) 210384804
452173686 ) 2951540801
2384- 8685, .^'C.
Observation. When the reader has cnce m ide this n-.eth- d of ex-
trading the Cube Root fam;i'ar to hitn^ i am peiluad-d he will ftnd it
not only th-.- <<i/T/, but the /^eft method hitherto made uie of. — I know
of none that converges iaftcr, or is Icls troublefome in tlic proceft,
O 2
J48 E T O L U T I O N. C U B E R O O T.
(lO.)
This Example is ix:orkcd»
(21.)
This Example is ^worked*
J^ -^68 - -71638, &c.
3 /iL. -- 3 / ^7 _ J
V 375 V 125 5*
^ /-^ = '82207, &c.
(26.)
y |i = -9859^ &c.
_^ (27.)
Asi,^ : •5236 : rTsl^ : 1767.15 ttet. Anfwcr.
(28.)
^^31185 — 3 1*476 1, the fide required.
(29-)
ai50'42 X 60 X 8 zr 1032201 '6 cubic inches content of the fcia.
Thcav/io32ici'6 z: lOi'C? inches. Arfwer.
Part I. EvoLUTio n. — C u b e Root. 149
(30.) -
lb.
— >., 3 /';coo X Sooo
ai7 : 2c)* : : 9000 : / r: 69*2294. length.
V ^'7
217 rTfl' : : 9000 : / — . '. r: 5i'922 breadth.
— PAI ^ /90OO X 5 '2
Z17 •. 8p : : 9000 : / — r: 27*69x7 depth.
FIrft,
V 3 X i^5;^ rr i-SV^S =: 180*28 the keel.
v/3 X ~T|'^ = 25\/3 = 36-05 miJA'p beam.
v^3 X T5I' n 15 \/ 3 ;= 21*6 depth of the hold.
Secondly,
V^:- X 12^^ 1= iz$\/^ rr 99-21 the kee!.
V/ I X 25I rr 25 V^l zn i9"?4 midfnip beam,
v/i X 15)^ — 15 V^'t^ r: 11-905 depth of the hoU.
(3^.)
Admit the folidity of the coine t? be i, then
I '."^ : : 1 : / -— = 20 ^T - 13.^672 the heigh
cf the (op part.
3 ibcoo
I7'47i6 -- i3*S672 — 3-6044 }-.e"ght of the middle par:.
; ^°l - i = J ~~r = ^^ l/^ ~ 17-47I and
Alfo 20 — 13-8672 + 3.6044 — 2.5284 height of the bottom part,
ia tnchei, Anfwer,
This Example is nxorJic£i.
o 3
150
E V O L U T I O N. A NY R O O T.
(34-)
. . . . root.
a«oooooo ( I "414, Sec,
"14,* z: 196 fab.
14 X 1 — -8 ) 40 fecond div.
14?!* — 198S1 fub.
141 X 2 — 2S2 ) 1190 third div.
1414I* rr 1999396 fub.
Rem. 604, &c.
i3S')
.... root.
5'ocooocooo ( I '709, &c.
X X I X I = I
I* X 3 1:1 3)4° fif^ <^*^*
TtI^ z: 49^3 ^^^•
T70]* X 3 ~ 867CO ) 870C00 fecond div.
"1705]^ = 4-991443S29 l"'-b'
Rem. S556171, &c.
{36.)
root.
i728*oooococooooo ( 6'447, &c.
6x6x6x6r= 1296
""^^ X 4 m 864 ) 4320 firft div»
64*4 zr 16677216 fiib.
64^3 X 4 ~ X048576 ) 6017S40 fecond div, _
"'v>44^* ZZ 172005949696 fub.
644!^ X 4= 1063359936 ) 7940503040 thrrd^iv.
^447)4 r: 17275502185S8481 fub.
Rem. 449731411519, &c.
Fart I. Evoi. UTI9N. — Any Root. 154
(37-)
. . . . rout.
5y54ccooo6cooooco ( i*2<-9, ic:r
2X2X2xaxz r: 3*
T|4 X 5 = So) 255 firftdiv.
22 1> -zi 515363a fub.
12^+ X 5 ~ 1 171280 ) 6003680 fccond div.
224^ S ~ 563949338624 fub.
"22^'^ X 5 — 12588154880 ) 1 1450661 3760 third dlv.
Z249V rz 57537008386886249 fub.
Rem. 2991613:13751, &c.
(38.)
. • root.
3 '141600000000000000000000 ( 1*2102, &C,
TXIXIXIXIXI = 1
1^X6 ir 6)21 firft div.
T2,6 — 2985984 fub.
7?|5 X 6 IT 1492992 ) 1556160 fecond div.
I2i\^ ZZ 3138428376721 fub.
.55l^476l6oc7oo}3'"«^3»790ocoooo tlnrd div.
12102^5 — 31415421541C9890900723264 fub.
Rem. 57845890109099276736, &c,
In order to make room for more ufeful matter, I rtiall only fct down
pirt of the following folutlons.
(39-)
root.
547'5oooooooocococoococoo ( 1*461, &c.
'5^
Duodecimals.
(40.)
root.
547»30ooooooocoooococcoocooo ( 2' 199, &c.
(4'.)
i-53i328ii557^5>5625 ( 1-05 r3«t.
DUODECIMALS.
(I.)
'hlj 'Example h ivorked.
(2.) 33 ft. II in. iTpts.
(3.) 81 ft. 6 in. 6pt5.
(-4.) 38 ti. 6 in. 2 pts.
(y) 8^6 ft. I in. 6 pts.
(6.) 92 ft, 2 in. 5 pts.
(-J 2 2:;2 ft. 4 in. 6 pts.
(S.) 4203 ft. 3 in. 3 pts. o" Ci"\
(9.) 253 ft. II in. opts, ii'^
1 ]'" 2IV IV,
{10.) 80 ft. I in. 7 pts. 7" 9'".
(II.)
f. in. pt5.
25 II 6 S" 7''
25 II 6 8 7
649 OHIO 7
23 9 7 I ^o 5^^-
1 o II 9 4 3 6'''
1538 5 8 8yi.
13 T 8 II O ivii.
674 I i' 4" 7'" 11^^ I'' 8^^ 1^".
(12.)
24 ft. 9 in. 5 pts.
Part I. Duodecimals,
. (13.)
S5
m,
^ 'M Heights
^ ^ f add
4 3J
16
115 6 heights together.
Mult, by 3 6 breadth.
Prod. 404 3 = 404*25 feet area.
Then 404-25 X 14*5 = 5861*625 pence, zz 24I, Ss.
5 [d. Anfwer.
ft. in.
24 5 length.
12 7 breadth.
9)307 2 II area.
yds. 34 I 2 II
(14.)
ft. in.
2
8"
3 4
5. d. yds,
3 4 lil 34
4i '777
i-962
•987
'370
d, 5f '096
5 '3 4
si add
(15.)
Firft,
Find the value of the whole court at half-a-crown a foot,
and the foot-path at fixpencej the fum of thefe values will
be th« whole colt.
'54-
D U O D E C I
ft. in.
62 7 length
44 5 breadth
Prod. 2779 8 II area of the whole yard.
s. d. ft.
2 6 III 2779
6
347 7 6
o 1 loj
347I. 9s. 4:Jd. val.
c'
I
^
6'
4'
1'
6
4
s. d.
2 6
I 3
5
i:
of the court-yard at 2s. 6d,
And
I '333
•833
IS. ioJ.d. -166
ft. in.
62 7 length
Si
344 2 6 area of the foot-path,
in.
d.
d. ft.
6 l,VI 344
8 12
id.
812 I J value of the footpath at 6d. per foot.
347 9 4J; ditto of the court-yard at 2s. 6d.
356I. IS. cfd. whole value. Anfwer.
(16.)
230219796I. -s. gd. 1= 227828377*5:ii'qo476' guineas,
or fquare inches; theie divided by 1296 fquare inches, or
gu neas in a fquare yard, ^ive 175793*501166593, Sec. yds.
rr 99 miles, 7*0633, Sec. furlongs ; fo that the national debt
would pave a foot-^ath of a yard wide r/earh from London to
Baih»
Part I. Duodecimals. ijj
(>7-)
ft. in.
22 7 = "-583; 7 Mult.
13 II = i3'9io j
9 ) 3I4.-28472' feet «
34.'920524, &:c. yards
Mult. Ill-
Prod. 401-586053 pence, r= il. 13s. ^fd. Anf,
(18.)
#fc. in. ft. in. ft. ft.
74 loxii 7 ==: 74*83'xii-5S3' — 866'Sic4' area ofthe whflle room
7 6x 3 9 Z= 7*5 X 3-75 — 2,2' 1250' area of the door once
6 8x 3 4X5rz6'6' X 3'3'X5 — lUMiii'dittoofthe ihuttersoncc
8 + 8 + 3*3' + 3-3' X I* 16' X 5 n 131-222^' ditto of the breaks once
Sum of tlie areas ii38'2777'
Dedud 6 ft. 9 in. X 5 ft. zr 33'75 ditto of ttie chimney
Area of the whole work iio4'5277' feet
122 7253 yds. X 81i. —
io43'j65 pence — 4I. 6s. iid. Anfwer.
(19O
ft. in.
31 5 o breadth of the building
15 8 6 half the breadth
47 I 6 breadth of the roof
57 7 o length of ditto
100)2713 7 4 6 area
27 fq. 13 ft. 7 in. 4pts. 6" at half-a-guinea per fquare,
amounts to 14I. 4s. ii'ii6d. Anfwer.
(20.)
ft. in,
14 II height
•947
2
6
5
3)
4736
6
272)
1578
8
2
5 rods.
21
8 ft.
(2r.)
ft.
in.
ft.
in.
ft.
in.
28
10
28
10
^5
8
20
20
28
10
8
4f42
576
8
576
8
45^
8
10
5
4
*r
3
28 10
1883
2306
8
^3SS
2
302 9
ft. in.
2SS3 4 content of the firft 20 feet X by i bricks
2306 8 ditto of the fecond 20 feet x by t ditto
1355 2 ditto of the i j ft. 8 in. x by -j- ditto
302 9 ditto of the gable X by t ditto.
3)6847 II fum.
2282 7 B content of the wall, reduced to theHandard
thicknefs. Then
If 272 ft. : 5I. i6s. : : 2282ft, 7 in. 8 pts, ; 48I. 135
5-2d. Anfwer,
Part I. Bills of Parcels, &c. lyy
BILLS OF PARCELS, exercising COM-
POUND MULTIPLICATION, the
RULE OF THREE, PRACTICE,
TARE AND TRET, &c. '
♦
(!•)
(2.)
(3.)
rames Lamb,
Efq.
Sir John Guchim.
Andrew Wines, Ef^,
1. s.
d.
I. s.
d.
,. 1. s. d.
3 17
6
2 II
^1
109 8 2i
lo 9
8 15
151 13 2
8 i6
Si
8 10
I
157 14 io|
91 10 9i
11 8
i
8 I
li
13 2
257
5
59 19 iii
* 15
90 5 3
284
1170 17 ol
50 9 I — ^— 660
(4.) (5.) (6.)
rge VereS;
,Efq.
Hugh Abbot.
MifsEvltt. *
I. s.
d.
1. B. d.
I. s. d.
6
5i
651 18 8
9 6 J
9
°i
92 9 7.1
21 7 oi
14
8|
374 7 ?e-j
12 13
7
3S 5 ^i
13 15 9?
80 9 io|
7
^t
35 10 5
5
5i
10 4 o|
169 10 5|-
(7.) (8.) (9.)
Mr. Crowl
J. s.
^ 14
3 19
378 12
7 12
30 17
J22 S
:her,
d.
i
6i
Mrs.
J.
13
11
79
189
^3
Mertown.
s. d.
8 8
19 6^
7 II
5 6:-
19 7
Mr. Ochterlony
I. s. d.
405 3 4
369 IS
10 15 6
267 .4 4^
So 10 5
318
I 3
3j.-; ^3 4
5^^ 5
Jl
1476 :i II J
* In the firft line of this bill, /or per ell, read per yard,
P
158
Bills of Parcels, &c.
' (10.)
(11.)
(12.)
lir George Lovell.
Mr. Meafure
;well.
Mr.
Cudworth.
1. s. ' d.
1. s.
d.
h
s. d.
3 12 o
14 14 8
9 10
7 9 10
6 5
7 8
6 12
6 15
3i
6
70
135
66
13
6 5i
15 I
19 6i
40 14
ai 9
5 '1
29 16
?!
2
89
8 1;
12 4
96
62 I-l
(13.)
Anthony How, Efq.
1. s. d.
122 8 o
a82 19 6
133 17 6
(14.)
(15.)
Dr. 539
5
*59
13
10
I
9
3
Cr. 539
5
Geo. Germaine, Efq. William Weft, Efq.
3
7i
7i
1.
s.
d.
71
4
9i
66
II
4i
15
2
4i
41
13
74:
88 13 5
56 II s\
339
1.
s.
7
17
4
9
9
19
36
z
12
9
5
I
76
(16.)
Theodore King, Efq.
I.
s.
d.
12
18
3l
4
8
2
99
14
2
rl3X
17
3
7
5
126
13
4
{17.)
(18.)
Mr.
Torin.
Valentine Fawkes, 1
1.
s. d.
J. s. d.
29
15 10
55 17 7f
19S 3 H
37
4 H
17
3 iot
31 7 H
II
9 »
39 19 4
9
6 4|
'6 3 4i
8
I 6
34 5 31:
382 16 ai
J13
375 16 5I
•art I.
BiLLi OF Parcels, &c,
iS9
(19.)
Mr.
Carpenter.
1.
s.
d.
i8z
10
7^
66
s
2
63
8
3
14
9
10
98
424
13
lOl
(21.)
Mr. Cole.
Neat
we
ght.
cwt.
qr.
lb.
15
I
7
20
I
20
18
Z
9
XI
n
4
3
26
7
r
Value.
1. s.
d
53 "
85 16
i(
43 13
27 19
26 3
37 6
(
20.)
Mr.
WiUet.
1.
s.
d.
158
14
7|
39
5
55
12
i°i
9?
II
4
148
10
4
214
9
711
14
i^T
(21
•)
Mr. George Lane.
cwt. qr.
lb.
29 1
27 grofs
2
9 tare
27 I
18 neat
and the value is
I
351- US.
4id.
274
(230
Meflrs. Langton and Co
Neat weight.
cwt. qr. lb.
1520
*3
18
8
8
I
2 oX*
I 25
3 7
I iX
1 3I
Value.
1. s. d.
75 u 3
123 7 111
28 o 4|:
27 I 2|
23 16 6?
8 17 4^
286 14 71
Mr. Henry Chapman.
Neat weight. Value,
cwt. qr. lb- 1. s. d.
9 3 i3i'
II o 13
II 22 ^672 i6 o
8 3 21
9 i ^S{
* In working this (and every one of the following Examples which do not
divide even in lbs.) I have taken the neareft ^ or I lb. and rejefted the reft,
frQlumingtbat fufficiently accurate for Bufinefs.— Thus 16 lb. \-j\ 27 cwt.
I qr. 191b. now the true tare would be 3 cwt. 3 qr. 18-7 lb. but the 5
that is over I multiply by 4, and then divide that produft (20) by 7, and
the quotient is 2 or i, lo that I make it 3 cwt. 3 qr. 18^ lb. Had the
quotient been 1, I (hould have called it i, or 3 iz !> Sec.
P Z
tSy Bills of Parcel*^ 3tc»
(J5-) {2.6,)
FrancU Clarice, Efij^
Neat weight,
ewt. qr. lb.
3 1 i^Jl Value
7 a 4t
7 2 26i > 70!.
6 X 19^
8 I iSL
Mr. Amuitie.
Neat weight. Value
cwt. qr. lb. k s.
d.
II 2 14 58 19
X
1204 11
H
33
(27.) (28.)
J Granville -King, Efq. Mr. Jolm Grant.
Neat weight. Value.
cwt. qr. lb. L s. i,
o 3 iifl
o 3 "f I
o 2 2i|: S 23 5 iij
Neat weight.
Value.
cwt. qr. lb.
1. s. d.
18 18 —
91 2 li
4 2 7i
73 9 S
35 5t
15 4
71 13 4i
19 3 12
41 S 7i
45 7 7{
a 3 2i|
357 iS 3i
o 3 22i
2 2l|: S
1 o 23
3 2:iJ
22^
(29.) (30.) (31._)
Wiimer WiUct, Efq.
Mr. Fenton.
Sir James Lawfon
Neat weight. Value
.
1. s.
d.
I. s. d.
cwt. qr. lb. ]. s.
d.
2 10
3^
10 19 jI
9 2 i3n
2 7
II 1 2^
2 7|
9 ' 5i|
3 3
8 oj
8 2231J
54 I 17
i|
5 I
15 17
I 16 6i
I J3 5;:
5 5: ■
24 13
53 n
'1
27 16 4|.
Part L
Bills of Parcels, Sec,
i6i
(3^0
(33.).
Mr. Adams.
Mr. L. ThirKvall
1. s. d.
1. s. d.
62 3 z}
61 14 31*
60 8i
9 9;;
016 2 '
5 9l
4 16 9i
r83 18
(34.)
Lumley Tutt, Efq.
{
I.
s.
d.
19
4i
3
14
8
4
3l
19
7i
17
19
10
23 17
* In this ftating, for 9d. per dozen, rfj</ 9s.
(35-
(36.)
Sir Leonard Hurt.
Mr.
:fham.
Contents.
7 fqrs. 40 ft. I In.
214 yds. 3 ft. 5 in.
Values.
1. s. d.
53 9 5
72 15 ii|
10 4 lOi
136 10 3^;
Contents. Values.
1. s. d.
10 r. 96 f. 2 in. 10 p. — 60 J I 4^
15 rods, 227I ft. — 103 14 Sl
3367 ft. 1 in. 6' — 49 2 oi
862 fqr. 31 ft. 3 in. — 1810 17 li
2024 5 o
(37.)
John Carttar, Efq.
Contents.
204ft. Sin. 3pts.
399 ft-
Nofe, 6 fcore
deals make a
hundred
Values
1. s. d.
— 1 18
— 29
20 14
27 19
64 17
(38.)
Meflrs. Mount and Sc
Contents.
Value3»
1. s. d.
4I 2062 ft. 5 in. 1 pt. —31 o 7^
[J7 li
20 yds. 6 ft. _ 4 6 6|
15 r. 255 f. I in. 8 p. — 86 13 8i*
1 1 yd. loin. 11' 2" 8"'— 2 o 5I-
8 fqrs. 91 ft. 9 in. — 57 10 4I
181 II 8i-
* For 5s. 9d. in this bill, read 5I. 9^
i6x
Invoices,
&c.
(39.)
Mr. Conftant.
Contents. Value.
- 1. s. d.
7 fqrs. — 47 19
68 yds. 4 ft. 8 in 63 7 7
120 make a hun- C 11 1 l|
dred. / ^ ^4 ^
10 I jI
(40-)
Mr. Craven.
Contents. Value.
1. s. d.
6 fqrs. 79 ft. 6 in. — 43 5 2|
2C yds. 8 ft. 5 in. — 17 0'-
395yds. 3f. 4in. 6' — 4 2 4!
48 14 71
141 3 'Sl
INVOICES, &c.
(4'-)
Invoice from Dubliv.
1. s.
Value of the butter - ^3^ 3
Ditto of the pork - 6713
Sum of the neceffary expenccs, as 7
fpecified in the Invoice J 34 ^ '
CommiiTion (on 43 81. 8s. i|d.) \
at 2i per cent. ] '^ ^9
d.
4
3
449 7 3t ^un^-
If 112I. : lool. : : 449I. 7s. 3|d. : 401I. 4s. ^{A, fter-
ling. Anfwer,
(42.)
Invoice from Jargiaica.
The neat weight of the tobacco is 7 « j
82791b. and its value is j^^ ' -^
Sum of the neceffary expaaces, as 7 ^
fpecified in the Invoice j ^
Commiffion (rn324l. 2S. 2|d.) at / /q j. '
r I I O lO;^
3i per cent, '
Sum
S3S 9 i;
If 125I. : lOoI. : : 335I. 93, ijd. : 268I. 7s. jjd. Her*
ling. Anfwer,
Part T.
I N T O I
E 8, &C.
F65
(43.)
Invoice from Amjierdam,
g. ft. p.
Value of the Holland - 587 3 10
Ditto of the Cambric - 412 13 2
Ditto of the Ghentifli - 108 13 12
Sum of the necelfary expences, as '
fpecified in the Invoice
Commiffion (on 1178 g. 15ft. 7 2^ g 5
i-
5 o
29 9
8 p.) at at per cent.
Sum 1208 4 14
If34s. 6d. : il. :: 1 208 g. 4ft. 14 p. : 11 61. 14s. 9 Jd.
fterlinor, Anfwer
(4+.)
Invoice from Bourdeaux,
Value of the claret
lb.
liv. s. d.
75 o o
Grofs weight of 13 cafks of prunes 9768
Ditto of 1 2 calks 1061 1
Grofs weight of the whole
Tare zz 79ilb. x 25 z=
Neat weight
20379
I9b7
839 ^^
1^ J> 529 JO
If loolb. : 2I. 17s. 7d. : : i839i^Ib. :
^29 liv. ics. 5d. the value of the prunes.
1. s. d.
Cuflom, &c. of the wine 10 o o
Ditto of the prunes ^ 118 15- o
, Sledage, &c. - o 15; o )»I5'8
Ditto for the prunes - 1 1 ^ o
To the Ship Broker - 500*
Poor's box, &c. - 12 5 7*
Commiflion (on 7 62 liv. us.) at 2| per cent. 20 19 4
Sum 7^3 10 4
^ If 3 liv. : (;4td. : : 783 liv. ics. 4d. ; 59I. 6s. i ^d. fter-
ling. Anfwer.
* The reader will obferve that the tonnage is taken on the whole grofs
weight, and that I have made ufc of the Englijh lb.
i64 Invoices, Sec,
No. 45, 46, 47, 48, and 49, being drafts, &rc. require no
anfwer.
(50.)
If il. : 34s. 4d. : : 571I. 18s. : 981I. 15s. 2|d. Flemifh.
Anfwer.
(5>-)
If 35s. 4d. : il. : : 7494 g. 14ft. : 707I. os. iijd. fler-
ling. Anfwer.
" (52.)
If I cr. : 4s. 3d. : : 5000 cr. : 1062I, los. fterling. Anf,
(53.)
If I p. : 54id. : : 1576 : 357I. 17s. Sd.
(54-)
If54^d. : r due. :: 174.9I. iSs. : 7741 i|-J due. Anf.
Th^ reji require ?i$/olution.
K
E
TO t. H B
COMPLETE PRACTICAL
A R I T H M E T I CI- A N.
- <
PART
II.
ALLIGATION.
(
I.)
rf>:s
Example is nuorked.
(2.)
(3.)
qrs.- s.
lb. d.
16 X 5S
= 928
50 X iri- = n5
30 X S3
=: 15^90
4-0 X 14 =r 560
21 X 39
= 819
27 X 30 1= 810
I J X 34.
=2 510
87 X ^6 1=3132
82
)3847{4-6|i«.
204. }5:o77(24i,?^d.
"—
= 2I. 6s.
ioJd..|-^
Anf.
= 2S. oJd.-|*. Aafwer.
66 Alligation.
(4.)
qr. 1.
2-5X2 = 5-
4*5 X 1-2 = 5-4
5* X -8 = 4-
i2 ) 14*4 ( I'zl. = ll. 4s.
Sd.
Car.
19.
Ts-T]
Ti&/V Example is nvorhdm
(6.)
r= 6 ib. at lod. "J
rr 2 — at 6 i
— a— at 4 3
(7.)
5 I sofasT s r
2+i|3of2ol3l
I iofi8>.5^^
I lof 17 I ^1
4 4ofi4J"L
24-4= 6 ib.
2
2
Anfwer*
Anfwert
Other anfvvers may be found by Unking the fimples drfferently,
(8.)
Firft,
4S —
36—
24— I
Anfwer.
8
8 or 2
4
4—1
4
16+4
28
4— I
20—5
28 — 7
Second,
4
28+4
16
Anfwer.
4 or I
8 — 2
4— I
32-8
16 — 4
Third,
Anfwer.
48
8
8 or 2 1
36—1
4
4—1
ao.
24-T-
8
8 — 2
16—!
16
16-4
12
28+4
32-8
or wheat
Of rye
Of barley
Of peas
Of oats
Ofvrheat
Of rye
Of barley
Of peas
Ofoate
Of wheat
Of rye
Of barley
Of peas
Of oats
Part II.
ao.
A L
A T I
Fourth,
48
8
8
4
4
28 + 16
8 or 2
8— 2
44—11
24—,
16— 1
12
Fifth,
!1— I
28
i I 16+4
Anfwer.
4 or I
8 — 2
8—2
28 — 7
2Q 5
Sixth,
Anfwer.
8
8 or 2
8+ 4
12— 3
4
4— I
164. 4
20— 5
28 + 16
44—11
N.
Of wheat
Of rye
Of barley
Of peas
Of oats
Of wheat
Of rye
Of barley
Of peas
Of oats
Of wheat
Of rye
Of barley
Of peas
Of oats
167
The following Anfwers, in addition to the above, may be found by
linking the prices difterentiy.
7th,
4 or 1
4— 1
8— 2
44—11
4— I
8 th,
12 or 3
4— I
8— 2
44 — 11
32- 8
9th,
8 or 2
4— 1
12—3
20— 5
32—8
loth,
12 or 3
4— I
4— I
48 — 12
28- 7
Of wheat
Of rye
Of barley
Of peas
Of oats
nth,
4 or I
12— 3
4— I
48 — 12
16 — 4
15th,
12 or 3
8— 2
8— 2
28—7
48 — 12
r2th,
4 or I
4— I
12—3
48 — 12
4— I
1 6th,
12 or 3
8— 2
12— 3
32—8
48 — 12
8 or 2
8— 2
12— 3
4— I
48—12
17th,
12 or 3
12— 3
8— 2
44—11
48 — 12
[4th,
8 or 2
12— 3
8^ 2
16—4
48 — 12
1 8 th,
8 or 2
12— 3
12— 3
20—5
i 48 — 12
Of wheat
Of rye
Of barley
Of peas
Of oats
Of wheat
Ot rye
Of barley
Of peas
Or eats
More anfwers may be found by nnklng, and thelc may Ic infri'ttdj
increafed by obfervirg the Note given to this Rule.
x6S
A L i I
^ T I • M.
(9-)
^his Example is icorked.
8.
-J
4 —
diff.
4 + 1 =: 5-
(lO.)
difF. gal. dlff.
rr 4- As 5 : So
n 4- 5 •• 80
: 4 : 64 ga!
: 4 : 64 gal
•I
Anfwer,
12
8-,
6—1
s.
12
diff.
I.
difF.
I.
2|.
5-"
I.
(II.)
diff. lb. diff.
As 5 : 28 : : 2I :
5 : 28 : : r :
< : 28 : : 1 :
14 lb.
5|Jb.
5|lb.
Anfwer.
diff.
As 1
I : aS
I : 28
Or thus,
lb. diff.
28
5
2S)b."->
70 lb. >An
140 lb. 3
fww.
Orher Ajifwers may be obtained by linking differently.
(12.)
40.
d.
48 T
36-i
24 —
18
diff. diff. b.
224-44-16^=42 As 8 : 24
S rz 8. 8 : 24
8 — 8. 8 : 24
8 — 8.
(13.)
7his Example is nvorkej.
diff.
42 : 126 b.
8
8 :
: 126 b. 7
: 24b4
: 24b.i
Anf.
(14.)
j6.
d.
24"
ao-
12-
diff. s. diff. lb.
8 As 24
4 i4
4 24
8 24
diff.
: 72 : t 8 : 24 lb. *>
: 72 : : 4 : 12 lb. /
: 72 : ■• 4 : 12 lb. J
: 72 J : 8 : 24 lb. J
Anfwer.
Sum of the 'if. 24
Other Anfwert «iiy be oltained.
Part II.
Single Position.
16'}
$•
(15-)
s. dlf?.
s. diif. gal.
diff.
8 . 5
As 14 ; 16
•• 5,-
5 f g^l
7— 1
4
14 : 16
• 4 ••
4yg^^
5-
1-1
2
14 : 16
: 2 :
27 gal
10 i 3
14 : 16
: 3 :
sH'^i
Sum of the dift'. 14-
"*■
Or thus.
s.
diff. s. diff.
1:19 As28 :
16 :
iiff.
8 i
5+4
. 9 • 5
7 1
5+4
= 9 28 :
16 :
9 • 5
1 1
3+2
= 5 =^S :
16 :•
5 •• 2-
n
_
34-2
= S 2S :
16 :.
5 = 2-
Sum of the diff. a 8
Other Anfwers may be found.
H
(i5.) .
d;ff.
s. diff.
6+5 = 11
As 31 : 64-
: II
22|t carats.
10 n 10
31 : 64:
: 10
20jx carats.
10 zz 10
Anf.
^Anf.
Sum of the diff". 31 31 : 64 : : 10 : aof? carats.
SINGLE POSITION.
T^is Example is 'worhd*
(2.) _
Sappofe 12 to be divided according (o thefe parts, asbe'iig
the leaft whole number divifible by 2, 3 and 4, without a
remainder.
i = jcsi !-"•
cUm 13 fhoald be 20.
I70
Sing
L E
Position.
I. s. d.
13 : 20 : : 6 : 9 4 -j'^.J^ A'sfliare)
13 : 20 : : 4 : 6 3 ol--.^ B's i^Anfwi
13 ^ 20 :: 3 : 4 12 3^4? C's ]
(3-)
Suppofe they could finidi it in 210 days.
days. work. days.
7 •
I ;
: 210
30 by A
5 •
I :
: 210
: 42 by B
6 :
I :
: 210
35byC
Sum
07
Hence it appears that A, B, and C working together for
210 days (the time fuppofed) can perform 107 times the
work. Confequently
work. days. work.
If 107 : 210 :: I : i4|ydays. Anfwer.
(4.)
Suppofe the army confided of 30 men.
Then j of 30 = 6
\ =5
TO — 3
Sum 14 And 30 — 14 = 16, but
— fhould be 4000. ' Hence,
If i6m. ; 4000 m. :.: 30 m. : 7500 men. Anfwer.
(50
Suppofe the cidern would be emptied in 1287 hours,
h. cif. h.
9 ■
I :
: 1287 :
143 by D
II :
I :
: 1287 :
117 by E
>3 :
I :
: 1287 :
'99 by F
Sum 35:9
Part ir.
Double Position.
17:
Hence it appears that D, E, and F fet open for 1 287 hours,
would run out 3^9 times the contents of the ciftern.
Ifjjgc. : 1287 h. :: ic. : 3||| hours. Anfwer.
See the 9th of die Promifcuous Examples, at the end of Vulgar
Fradions.
(6.)
Suppofc the fum delivered was 5'cr,
Int€reft thereon for 3 yrs. at 4 per cent, 6
The amount is 56I. but ihould be
17 61. 8s. Hence,
If ^6U : 116U 8s. : : 50I. : 157I. los. Anfsvcr.'
DOUBLE POSITION.
(I
•)
T^is Example ii tiiarlcd^
(2.)
FIfft, Second,
1. I
Suppofe. the firft horfe worth 30
Chaife, furninare, &c. 150
I.
Suppofe the firft horfe worth 1 50
Chaife, &c. J 50
3)180
3 )3oo
Vali\? of the fccond horfe 60
Chaife, <«c» 150
Value of the fecond horfe 100
Chaife, &c. 150
2 ) 210
2)250
105
Should be 30
125
Should be 1 50
Error. — 75
Error. +25
CL2
172 Double Position.
By Rule I.
itfertor,
ift fuppofit'on 30 Kvyl — 75
2d fup£of)tion 150 1^^ H-25
75 i.'S frrcr. 30
lJ^$o 750
^5 + 75 m 100 ) J2CCO
izoi. value of the firft horfe, and 50I.
:^ value of the fecoci.
(5-)
First, Suppofe it was 8 o'clock in the morning, then it
v.as 4 hours after fun-rife, and 12 hours before fuu-fet;
••; I + 4 of 12 = II, but fnould be 8, hence the error
is — 3.
Second, Suppofe it was 12 o'clock, then it was 8 hours
after fun- rife, and 8 hours before fun-fet ; •.* | -f i of S =? 1 -
but ihould be 1 2, hence the error is -f 2.
By Ruje I.
its error,
ift fuppofition 8 k>< — 3
2d fuppoiition 12 rS 4-2
3 //J- <f?Tjr. 8
36 16
3 -r 2 = 5 ) )»
iOj= loh. 24'
fo tliat it was 24 minutes pad 10 o'clock in the morning.
(4-)
First, Suppofe o* the vahie of a French crown, then
will 7 dol. be worth 4I. ics. lod. =: logod. and i dal. =
* It is often of advantage to make o and i the fuppofitions. l may be
made a conilant fuppcfition in ail queflionsj and in moit cales it i? pre-
ferable to any other nuaiber.
Part 11. Double P o s i t i o k. 175
i55-5d. alfo 4 crowns -j- 3 dol. = 46-]-}d, Then ^S-jjd, —
42od. ( = il. I 5s.) zi — 47f the firft error.
Second, Suppofe id. the value of a French crown, then
will 7 del. be worth (1090 — 11) =: i079d. and i dol. =
it^4^d. alfo 4 crowns -f 3 dol. = 4d. -f 462^ 1= ^66},
Then 466! —420 = —464 thefecond error.
By Rule II.
^ ^ ^^^" = iii = 65d. thecorreftion. Then 65 -f 1
.477 --46f 5
r: 66d. = 5s. 6d. the value of a French crown, confequently
a dollar is worth 4s. 4d,
Otherwife,
1. s. d. d. cr. dol,
4 10 10 = 1090 =11 + 7
1 15- o = 420 = 4 + 3
3270 = 33 +21
2940 = 28 H- 21
530 = 5 +
••• I or. = -^^ z= 66d, z= 5s. 6d. confeqoently i doU
zr 4s. 4d. as before.
ExpLAK ATioK. I multiply logod, and its equivalent nirmber of
crowns and dol. by 3, alfo the 4Zod. and its equivalent number of crowns
and dol. by 7, in order to cancel the dol. and fubt.a^ one produft from
tie other, fo that there remains 5 crowns rr 330J.— For if equal quan-
tit'es be multiplied by equal quantities, the prod^ifts will be equal 5 and
if equal quantities be fubtradled from equal quantities, the ren;lainder wiU
be equal.
••
' Remark, By multiplying the firft line by 4, and the fecond by 11,
and then tdking their "difference, the crowns would have been cancelled,
and we fliould have had 5 dol. ZZ- 26od, confequently i dol. i::! 3id, zz
First, Suppofe there were 6 children.
Now 6 X 2 z: rzd. hence he had i6d.
Alfo 6x3= iSci. and 18 — 16 = 2d. but fhould
be 10, hence the error is — 8.
Q.3
174 Double Position*
Second, Suppofe there were 12 children.
Now 12 X 2 rz 24d. hence he had zSd.
Alfo 12x3=: 36d. and $6 — 28 = 8d. l)ut fhould
be I o, hence the error is — 2,
By Rule II.
Z2 — znz corredion. Then 12 + 2 ::= 14
8 — 26
crhildren. Anfwer.
(6.)
First, Suppofe^ ~ 3000, irs Jog. ::r 3*477i2i3,
Then 3000 x'137^'^ = ^05. 3000 + 6 x log.i37ar: zz'-^oiz^^^
and 4''X 430^1^ — ^^S' 4 + ^ x log. 436S rz 22-4437556
Diff. •1425097
ihould be o. ••• The error is + •1425097.
Second, Suppofe^ ir 2900, its log. rr 3«46239So.
Then 2900 X 147^1^ ~ 1"S- -5^0 + 6 x log. 1472 = 22.469S448
' Should be 22-4437556
Error — '0260392
3JtI7Il!lJri:£^i2!l4^2l££i! - .C0227S3 the correaion
•1425097 + •02006^2
By the Rule,
•0260S92
7^ ~
Then 3*4623980 4- '0022783 :r: 3*4646763 the log. of the value of
», the neareft number anfwering to which is 2915-3, 1 therefore take g
— 2915*3 for a fecond operation.
Then 291 5-3 x'T456^|^— 1^6' ^9i5'3 + ^ X log. 1456-7
— 22-4449038
Should be 22'4437556
Error — 'Ooii/.Si
By the Rule,
^•AeASS-ll — 3462'Jq8o X '0011482 „.
? ^^ — ^ ^ — ^ — — '0001052 correflion.
•0260892 — •001x482
Then 3-4646832 + '0001052 zz 3^4647884 the log. of the value
%Fgy the neareft number anfwe.Ug to which is 2916, the true value of
g, Anfwer.
Part II. Arithmetical Progression. 17^
Note. I have inferted this foigt'ion at full length in orJer to eluci-
date the note to the fecond rule 5 I would have inferted the following
likewife, but for want of room.
(8.)
Anfwer i -o^*
(10.)
Anfwer 6'098.
ARITHMETICAL PROGRESSION,
^his Example is 'worked.
(7-)
Anfwer 3.
(9-)
Anfwer i '05.
3 + io3 X V = 777 fum,
(3-)
I + 12 X '5 = 78 Anfwer.
(4-)
The perfon mufl travel the ground twice over to fetch the
100th egg. •••
2 + 200 X '1° = loioo yards =z 5nules 1300 yards,
M
This Example is ^worked,
(6.)
108 — 5
3= 8A common diff, Anfwer,
14 — I
>S Arithmetical Progression,
2^3 ___ 2
-: = '} common difF.
8 — 1 ^
2 lea. firil day's journey ,-
2 -h 3 =r 5 2 —
5-^3 = 8 3
8 -f 3 =z II 4
11+3 =14 5 — .
14 -4- 3 = 17 6
f] + 3 =20 7
20 + 3 = 23 8
^"his Example is n^orhed,-
(9-)
i2 — mi zz 21, and 21 + i = 22 the number of terms,
5
(10.)
?i_ir — ri 7, and 7 4- i = 8 the number of terms.
3
9^/V Example is ^worked,
(12.)
22 1 X 5 r= 105^, and loS — 10 j = 3 theleail term,
(T3.)
(^ I X 4 — 20, and 40 — 20 = 20 the lead term,
(14.)
This Example is ^werhed*
(15.)
,22 — I X ^ ,
1221 -f- 22 = 555:, ^^ = 52 a-
And ^5i — 52i = 3 ^^^ leaft term.
Fart II. Geometrical Progression. 177
(16.)
12 1X4 , T
300 -T- 1 2 = 25, ^ = 22, and 25 — 22 z= 31.
Aufwer.
(^70
This Example is lAJorked,
(18.)
22 X 5 +3 =113, and 113 — 5 5s 108 the greateft term,
(19O
JOG X I 4- 2 =s 102, and 10a — i =: lois. sa 5I, is.
Anlwer,
GEOMETRICAL PROGRESSION.
('■)
This Example is ^worked,
1.2.3.4.5.6.7, &c. indices
2 . 4 . 8 . 16 . 32 . 64 . 128, Sec, leading terms
-y -f. -7 4. ^ 2= 19 index to the igth term
12S X 128 X 3^ = 524288 the 19th term, Anfvvcr.
(3.)
1.2.3.4, 5;. 6. 7, &c. indices
3 . 9 . 27 . 81 , 243 . 729 . 2187, &c. leading terra*
7 + 7 -f 6 = 20 index to the 20th term
2187X2187X729 = 3486784401 the 20th terra. Anfwer.
(4-)
This Example is <workecl.
178 Geometrical Progression.
0.1.2.3.4,5.6, &c. indices
7 , 14 . 28 . 56 . JIT . 224 . 448, &c. leading terms
6 -f 6^ zr 12 index to the 1 3th terra
448 y 448
= 28672 the 13th term.
7 ' /
12 + 6 = 18 index to the 19th term
28672 X 448 „ „ ., ,
-^-!-- =z 1835C0S the 19th terra.
7
(6.)
0.1.2. 3.4. 5^ 6. 7, &c, md'ces
4 . 12 . 36 . ic8 . 324 . 97s . 2916 . 874S, &c. Uading terms
7+7 =: 14 index to the I5rh term
SL 111- n 19131876 the 15th term.
4
14 4- 5 zr 19 index to the aoth terra
'^'^'^^^ >< ^'' = 4649045868 the to* .e™.
4
7680 X 64 r: 491520 barley-corns in a bu(hel,
barley-corns d. barley-corns
451520 : 30 : : 4649045S68 t 1182I. 6s. 3d. AnfMrw^
M
This Example ;i Vfcrked.
(8.)
0.1.2.3.4, &c. indices
4 . 12 . 36 . loS . 324, &c. leading terms
44-2— 6 index to the 7th term
324 X ^6 , , ,
^—^ zr 2916 the 7th term.
4
Then — ^^— — ^ zr i456,and J456 4--JniG— 4--? fumof the term:*
6.1.2.-;. 4 . 5 . 6 . 7
J . 2 . 4 . 8 . 16 . 32 . 64 . I2S . ^, ); .C
8 -f- 7 n 15 index to the J 6th tt.
256 X 128 — 32768 the 16th term.
J5 4. 8 -|- 8 :=: i^ jrdex to the 320 r.
3=768x256 X 256 n 2147433,6.3 -- ^- --'-^
2147483648 — I -f- 2147483648 rr4-9V o^ ^^
4473!;24l. 55. s^d. value of ths fir"
Part 11. Geometrical Progression. 179
Again,
, 3x768 — I -f 32763 rn 65535 ftrthlngs, — CSl. 5s. ^IJ. value -of
t^e fecond iiorfe— T he dillcrcncc ot their values is 47738561. Anlwer.
(10.)
I . z . 3 . 4 . 5 . 6, &c. Jnuircs
10 . 100 . 1000 . icoco . loocoo . lOOOOOO, Sec. leading terrns
6 + 51= II index to the nth term
1000000 X icoooo zz looooooocooo the nth term.
loooooooocoo — lo
— iiiiiiliiio and luiiliiiio +
10 — I
loooooooocoo rZ IIIIIIIIIIIO wh-a. . nv jr. Vi' '.vh.;i.- =
7680 X 64 — 491520 wheal-C'-ra: ' : .1 bu'hci.
1^491520 : 4s. :: iiiiniiiiio : 452111. 4s. 6|:J. Anfwer.
(II.)
0.1.2.3.4.5, &c. iid'czs
1024 . 1536 . 2304 . 3456 . 5184 . 7776, ^c. leauing terms
5 -H 5 n: 10 index to the nth term.
7776x7776
. zz 59041;}. the eldeft fon's fortune.
1024
59049 — • 1024
f rr 29C12-4!. which incieafcd by 59049I. gives
SS061I. los. the nobleman died worth. ®
(.2.)
This Example is 'worked.
in-)
T I
, — X — =
2 2
ium required.
(14.)
1 i_* ' ' '. '.* ^ \, c
"3 9*" "9* 3 I" 9' 9*9""'» ""^
required.
JO «— 9 n I and 10 X 10 — 100, then 100 -r i n iccinilcs. Anf.
I So Variations and Combinations.
VARIATIONS.
(I.)
This Example is nvcrked^
(^•)
1X2X3X4x5X6x7x8X9= 362880. Anf\V-r»
(3-)
1X2x3X4X5X6x7 = 5<^40 variations or days, r: 13 years,
-95 <l'»ys« Aniwer. '
(4-)
JX-2X3X4X':X6X7X8 X 9 )k 10 X 11X12 =479^^0^600 changes,
4- 10 m 47900160 rr.inuicsj 365 d. 6 lirs. zz 525960 minutes.
Then 4-900160 -f- 525960 n 91 years, 26 days, 6 hrs. ringing with-
out intcrnailfion. Annvcr.
(5-;
This Example is ivorked^
__^)
12 X li-"—! X 12 — 2 X 12— 3 X 12—4 X 12 — 5 X 12 — '-
IZ i2XiiXic;X9X8x7X6 zz 39916S0 changes. Aniwer.
(7-)
26 X 26 — I X 26 — 2 X 26 — 3 X 26 — 4 zr 26 X 25 X 24 X
3 X 22 n 7893600 words. Anfwer.
COMBINATIONS.
(I.)
This Example is nuorhd,
(2.)
1X2X3X4X5x6x7X8x9X10 = 3628800 dlvifor.
100 X 100 — I X ICO — :■ X 100 — 3 X 100 ~ 4 X 100 — 5
X ICO — 6 X 100 — 7 X ICO — 8 X ICO — 9 = 100 X 99 X 98
X 97 X 96 X 95 X 94 X 93 X .2 X 9i = 6281 56 50955529472000
ijjvidend, which divided by 36:>.88oo, the divilbr gives fbr the quotient
17 3 10309456440 farthings zz. 18031572350I. 9s. zd. Anfw-er.
i^art IL SiMFLE iNThSEST BY DECIMALS. iSi
SIMPLE INTEREST by DECIMALS.
T.his Example is ivorked,
(2.)
'^35 X yiS X •<^5 + 23^- =1 279-06251. = 279I. IS. 3d.
.maunt.
(3-)
550 X 5 X '035 = 96-251. =z 96I. 5s. iritereft.
■ (4.)
700*5 X 5-25 X 'oi -\- 700-5 =:: 810*828751. = 810I.
1 6s. 63d.-}.
^?-)
715-75 X 7-5 X -04.25 -f 715-75 ~ 943*S953i25l. =
943I. 17 s. lo^d.-^. the amount.
(6.)
•002739726 X 240 = -65753424.
71 5-75 X-05X -65753424 1=23-5315066141. =231. 1C3,
7 Id. -246. AniVer.
(7-)
•002739726 X.6!^ =''17808219.
357-5 X •I78082I9 X -05 1= 5-183219146251. = 3I. -y.
7]d. -89.
(8.)
'002739726 X 120 -f 5 = 5»328767i2.
510 X 5-32876712 X ^05 -f 510 = 645-883561561. =:
645I. 17s. S*055d. .
(9-)
This Example is ivoih\^*
R
i82 Simple Interest by Decimals.
(lO.)
This Ex auntie is ivorked*
(II.)
279I. IS. 3d. = 279 0625I.
j'75 X '05 + I = 1*1875' divifo:.
Then 279-0625 H- 1*1875 = 235I. Anfwer.
(12.)
96I. 5s. = 96*251. 5 X -035 rr '17^ divifor*
Then 96-25 -r- '175 ^ 550I. Anfwer.
(13-) ^
Siol. 163. 6-|J.-| = 810-S2875I. 5-25 X -03 -f I ==
i-i;;75 divifor.
Then 810-82875-7- 1*1575 = 700-51. =: 700I. los. Anf,
(M-)
943I. 17s. io3d.-i=: 943-8953x251. 7-5 x-o+25 + I ==
1-31875 divifor.
Then 94-3'S953i25 "^ 1*31875 = I^^'IS^- = 7^5^- ^S^^
Anfvver.
(I5-)
23I. 10s.7id.-fl n 23*5315067 dividend.
•002739726 X 240 X -o; = -032876712 divifor.
Then 23-5315067 -r -032876712 =: 7i5-75l- = 7'$^*
15s. Anfwer.
(16.) ,
3I. 3s. 7|d.-|i- = 3'i832r96I. dividend.
•002739726 x6^ X -05 = -0089041095.
Then 3-1 b32 196 -^ •0089041095 ~ 357'5l. = 357l. losc
Anfwer.
(17.)
679I. 8:J. 4y-7V = 679*417808221.
•002739726 X 120 + 5 = 5-32876712 the time.
5*328-6712 X -05 + I =z 1*266438356.
Then 679-41780822 -7- 1*266438356 =: 536*4791781. 3
536I. 9s. 7-oo27d.
Part II. Simple Interest by Decimals. i?3
(i8.)
This Example is ivorked,
279]. IS. 3d. = 279'o625l. and 279*0625 — 235 :«
44."o625l. dividend.
235 X 3'75 = 881*29 divifor.
Then 44-0625 -^ 881*25 = '05 the ratio, hence the rata
is 5 per cent.
(20.)
96*25 -r 5 5o"r^ 5 := '035 the lotlo, hence the rnte is jl-
per cent.
(21.)
810I. i6s. 6|d.-| = 810*828751.
Then 810*82875 — 700*5 ~ 1 10*32875 dividend.
110*32875 -^ 700*5 X 5-25 z=. '03, hence the rate is 3
per cent.
(22.)
943I. 17s. I o|d.-f = 943*89^3125!. from which fubtraft
7i5'75, there remains 228*1453125 dividend,
715*75 X 7*5 = 5368*125 divifor.
Then 228*1453125 -i- 5368*125 = '0425 the ratio, hence
the rate is 4^.
(23-)
•002739726 X 240 = '65753424, 23I. 10s. 7 Id. z=
23-53125-
•65753424 X 715*75 = 470-63013228 divifor.
23*93125 -^- 470-6301-3228 HI *o49999, <S:c. zz. ^05, hence
the j^ate is 5 per cent.
(24-)
•002739726 X 61; z=. •17808219, 3I. 3s. 7-4d,-|-|- rz
3-1832196!.
•178082196 X 357*5 = 63^664385C7 divifor.
Then 3*1832196-7- 63^664385c-; = •05, lie nee the rate
is 5 per cent,
R z
(no
In this Example y for 5I. 10s. read 510I.
•002739726 X 120 + 5 =1: 5-32876712 yrs. 679k 8s.
4-|d.-y3- = 679-417808221.
679-41780822 ■ — 510 = 169-41780822 dividend.
5-32876712 X 510 = z-jiyG-jiz^iz divifor.
Then 169-41780822 -r- 27i7'67i23i2 = -062339332 the
r.ulo, hence the rate is 6*2339332!. = 61. 4s. 8*i4396d,
[cr cent.
(26.)
T/:/'s Example is 'worked*
(2;.)
279I. IS. 3d. =: 279-0625, and 279*0625 — 235 =:
44-0625 dividend.
^3S ^ '^35 ~ 8*225 '^i^'i^or.
Then 44-0625 — 8-225 — 5'3S1H» ^^' = SUsY^^-
Note. The five Examples followlng^are the fame with the 20th, zift,
22d, 23d, and 24th piecedii^g, except in the divifors, which are found
by muluplying the principal by t'ue ratio inllead of the time 5 I daou^ht
h therefore unacceffary to infert their folu'.ions at large.
(28.)
(29.)
5 years.
5 i years.
(30.)
(31-)
(32.)
7 k years.
240 days.
6^ days.
i33'}
The dividend is 169-41780822. Sec Example i^th.
And 510 X -05 = 25-5 divifor.
Then 169-41780822 -7- 25-5 1= 6-64383, Sec, yrs. Anf.
PartlT. Discount atSimpleIntereit, Decimals. iS^
DISCOUNT AT SIMPLE INTEREST,
BY DECIMALS,
('•)
Ti'ls Example :'s 'VJorkcd,
(2-)
49j'9 X -083' X ^ X -0375 = 7'74^437S^- i^tereft or
the debt for 5; months.
•083' X 5 X '0575 + ^ ^ 1*015625 amount of il. for
ditto.
As 1-015625 : il. : : 7'74-375 • 7-629231. r:: 7I. 12s,
7 "o 1 5d. the difcount-. Anfwer.
(3-)
1507I. 145. gd. = 1507*73751. dividend,-
•083' X 7 X -05 + I = 1*02916' divifor.
Then 1507*7375 -4- 1*02916' = 1465*00811. r: 1465],
OS. i|d, '773 the prefent value.
(4.)
7147-7 X '002739726 X 175 X -0375 =i 128-511729 -f
intereft of the debt for 175 days.
•002739726 X 175 X *0375 -f r n: 1*017979452 amount
of 1 1, for 175 days.
As 1*017979452 : il. :: 128*511729 : i26-24i9676I. ~
126I. 4s. 1 00 7 2d, the difcount.
(5-)
1789I. 13s. 4d. ~ 1789*66' dividend,
•002739726 X 29 -f 3*75 ■=. 3*8294^2 years.
3-829452 X '043' + I := ^''■65943 c^ivifor.
1 hen 1789-666' -i- r 76594-3 ~ 1554-9692621, r: 1534?,
195. 4td. -492 the prefent worth,
R3
86 Ec^ATioN OF Payments, &c.
EQUATION OF PAYMENTS at SIMPLE
INTEREST, BY DECIMALS.
Ofi MALCOLM' s PRINCIPLES,
(I.)
This Exa?nple is ^workej,
(2.)
Firft,
100 + 105 zr 205 furrv of rhe debts.
so 5
ICO X 2 X '05 — 10. Then Y i 11:21 '5 the firft number found.
Secondly,
105 X 2 -r 100 X '05 zi 42 the fecond number found.
Thirdly,
VzT^i — 42 rr V'42o-25 rr 20-5, and 21-5 — 20*5 — i year,
reckoning from the firll payment, •,• the equated time is 2 years.
(3-)
Firil,
From Michaelmas 17S8, to Michaelmas 1793, are 5 years. The fum
of the debts is 522I.
32c X 2 X '05 — 32, and {- 2*5 n iS*3i25 the tail number
3^
fuund.
Secondly,
:02 X 5 -r 32-- X 05 ~ 63*125 the fecond number found.
Thirdly,
\/ i8-8iz7^' — 63M25 — V'290*785i5625 rz: i7'0524, and
^•8125 — 17*0524 zTl I'76ci year, r= i year, 277 days. Hence tht
jta fhould be received the 3d of July, 1790.
COMPOUND INTEREST by DECIMALS,
This Example is ixorked,
1*05 X 1*05 X 1*05 ~ i'i57625 araoufit of il. for 3* years.
And I-J57625X275 — 318-3468751. — 318J. ^%% ir^-:l, the amount
Part II. Compound Interest by Decimals. 187
(3-)
1 -04 X I '04 X I -04 X I -04 X I -04 X I '04 X I -04 =
1 '3 1 5-93 1 8, &c. this mukiplieJ by 'joo-]^, produces
922-139208851. =:_922l. 2S. gid. -64 the amount; from
which dcduifl the principal 70075, there remains 22 il. 7s.
9^d. •64intereft.
(4.)
! '05 X I 'O^ X I '05 X I -05 X r '05 X I •05' X I '05 X I '05 X
1-05 rr 1-5513282, &c. See Ex. ift. Involution.
1-5513282 X 8co •— 800 = 441I. IS. 3-oi4d. intercd,
(5.)
T/j/s Example is njjorked,
(6.)
318I. 6s. iijd. = 318-346875, and 1-05 ' = ri;7625.
••• As 1-157625: il. :: 318-346875 : 275I. Anfwer.
(7-)
819I. 15s. 6|d. '2504832 zr 819-77838592, and 1*04 X
1-04 X 1*04 X 1*04 = 1*16985856.
As 1-16985856 : il. : : 819-77838592 : 700-75!. z=
700I. 15s.
(8.)
1241]. IS. 3'0i7467875d. — i24i»o625727S2oi25\
r^9 — 1.551328215978515625.
. As i'55i32S2i597S5i5625 : il. : : i24i«o625727S23i25 : 8oo].
Aiifwcr.
(9-)
This Example is 'worked,
(10.)
31SI. 6s. ii-;d. = 318-346875.
318-346875 ^275 = 1-157625, the cube root of which
is I -05, hence the rate is 5 per cent.
(II.)
819!. 15s. 6|d. -2504832 =819-77838592, this divided
by 700-75, gives 1-16985856, the 4th root of which is 1-04;
or the fquare root is i -08 16, and the fquare root of i 'oS 1 6 =:
1-04, hence the rate is 4 per cent.
i83 DiscouKT AT Compound Interest,
(12.)
1241!. IS. 3»oi7467S-'cd. iz: I24i«c625727828iz5, this 4- 800
gives I'55i3282i59785i56z5, the cth root of which is 1*05, or tVe
uberootis i'i57625, and \/ i«i57625 — i^oc, hence the rate is 5
;.;r cent.
7'his Example is ^worked,
(H-)
The amount -f- principal gives i'iz^^6it^ {'vide Ex. 10.)
this divided by 1*05, and that quotient by i'C5, Sec. till
nothing reir.ains, the number of divificns will (hew the time,
3 years.
. ^'^'^
The amount -h- principal z=. J'i6c)^y9>ii6 (fide Ex, 11.)
this divided continually by 1-04, will fliew the time,
4 years.
{16.)
Theamonnt -r principal = 1 '5 91 3282 15978^1 5625 ("j/dt
Ex. T 2.) this divided continually by I'O^, will ihew the time^
9 years.
discounTat cOxMpound interest.
^his YjXawfh is ^jjorked.
1*05 X vo^ X 1*05 X roj =: 1-21550625 the amount of
il. for 4 years.
Then 4C0 -H r'2r 550^25 = 329*081 -f - 3291. is.
7 id. '76 the prefent w'orti^ which deduced from 400L leaves
70I. 1 8s. 4£d. '2 difcount.
(3-)
i-C5\fi in I '340095640625, and 643I. 4s, lid. — 643*24583'.
Then 643'24583' -r- 1*340095640525 — 4S0I. (nearly) t'le prefcir*
worth.
i'art II. Eqjjation of Payments at. Sec, iSg
M
Ratio — : i«o6 ) ioo.( 94- 3 3962.2 prefent worth for the ift
year
I o^'^n i'i236 ) 100. ( 88*999644 ditto for the 2d year
I'obj ^zz i'i9ioi6 ) loO'C 83*961928 ditto for the 3d year
J -06 4ri: 1*26247656 ) 100. ( 79*209366 ditto for the 4th year
i-o6^5:^ 1*3382255776) 100.(74*725817 ditto fertile 5th year
The fqm is the prefent wortli 4^1*236378 — 421I. 4s. 8*73Td,
Anfwer.
EQUATION OF PAYMENTS at COM^,
POUND INTEREST.
Th's Example is •worked,
(2.)
^•051^ X 3^*^ "^^ 408*4101 amount of the ift payment
96* laft payment
504-4101 fumofthe amounts
320 + 96 rr 416I. fum of the debts
_. /3P-. 504-4101 — /opr. 416 „ ^
Tnen '^ ^ ^ ^ ^ . — -0836905 -r '021 1893 n
kg. 1-05
3*90496, hence 5 — 3*90496 zzL 1*09504 years, the true equated time.
(3-)
1-05]- X 100 -zz. 110*25 amount of the firft payment
105 laft payment
215-25 fum of the amounts
[CO 4. 105 zr. 205I. fum of the debts
henca 3 — 1 — 2 years, the true equated time.
/o.T. 215-25 — /o^. 205 - -
Then .^ i — l i^ 2 — .0211893 -r •0211S93 =: i,
/.^. 1*05 ^^
1^0 Annuities in Arrears
(+■)
J. 05^4 X ICO r: 121*550625 amount of the ift payment
ditto of the 2d
■*"<-'5i* X 3^° ^^ 35^*"5 '^'"o of the 3d
fee laA paynieRt
iiS3'825625 fum of the amounts
ICC -f- 2CO -f 3CC + 5C0 zi iiccl. fum ef the debts.
„, /»r. ii83'82!:62C — /rff-. iioo
Trcn -i i 1 - t — •031S950 -7- -0211893 r::
h^g. 1-05
2-50554, hence 5— i*Soi*4 — 3*49475 years, the true equated time.
ANNUITIES IN ARREARS at SIMPLE
INTEREST.
77'/; Example is nxarled*
142 + 3+4+5 + 6+7 + 8 = 36
One year's intereft of Sol. =4
Whole intereft due - - 1 44.
80 X 9 =^ '20
The amount /. ^64
i + 2 + 3 + 4 + r+^ = 2*
One year's intereft ot 560I. zr 22*4
Whole Intereft - - 470'4
560 X 7 = 3920
£' 4390'4 = 4390'
8s. the a mo u at..
Part II. AT Simple Interest. 191
(4.)
1+2 + 34-^ -i- 5+6 + 7 = 28
One year's intereil of 173I. at 2^ percent. = 3*937^
Whole intereft 1 10-25
17^ X 8 = 1400*
15101. 5s. the amount,
is-)
1 + 2 + 3+4, <l-c. to 19 = 190
One year's intereft of 1 7'5h at 1 1 per cent. = '2 1 87 ^
Whole intereft 41-^625
17-5 X 20 = 3 JO
/ , £• 391 -5625 ~
391I. us. 3d* the amount.
(6.)
T^j's Example is <v:orkcch
(7-)
S64X 2 zr 1728 — 2a thedividcnd} 9x9 X '05 + 9x2 r= 22-05^::
•;- + zr, and -05x9 zi '45 ^^ tr. Then 172S -7- zz'Of — -45 :::: 80I.
he annuity.
(8.)
4390*4X2 :z:27So-8 ~ za the dividend ; 7 X 7 X -04. -}- 7 X ^ r= 1 5-96
— ttr + ZA, and •04X 7 — '28 — tr. Then 3780-8 ~ i5-<^o 28 z:
560I. the annuity.
(9-)
1510-25X2 = 3020.5 = 2 J the dividend J 8x8x -0225 + b' X2 r±
17-4+ = ttr^\'^ty and '0225x8 — .18 — t>; Then 302C'5 -^
17*44 — '18 IZ 175I. the half-yearly payment, hence the annuity
i. 350I.
(/o.)
39^*5625 X 2 = 733-125 ~ xa the dividend j 20 x 20 x '0125 -f-
20 x2 = 45 :r ffr -f. zr, and -0125x20 — -25 nrr. Then 783-125
~5" 45 •— • **5 -^ I7*5l« the quarterly payment, hence the annuity is 70!^
192
Annuities in Arrears
(II.)
This Exn^rple is ivorhed,
•864x2 2/j 2 — -05 2 — r
-43-=— J _- ^ I - 19-5- — — i
So X '05 r.r -05 X
Then v/43- -{- 19-5'.* — i9'5 "TL \/ 812-25 — i9'5 =9 years.
(13.)
4390*4 X 2 _ ^ , _ f;f . - — '^4 _ , _^ 2 — r
560 X -04 ~ ^^^ "" ^r' -04 X 2 •+ 5 — ^^ »
Then v/ 39^ + -4-5r "" --'■*-5 = "' >'"^'**
JC1C-25X2 61136 la 2 — '0225 395'5 - — '
175 X 02-5 — Si ~" 7r' '0225 X 2 ~ 9 ~ ^^
Then /^^-^36 ^ 595->" 395" 5 _ 467-S_ ^9^5 7^ _,
/'s/Sx 9. 9 9 9—9
8 payments, hence the time is 4 years.
(i;.)
39,-56.5 X 2^33S,^^. i^:£!is ^ L,-^^.
17.5 X 0125 i.r -0125 X 2 ZT
Then VX3580 + 79-5' ^ — 79'5 rz 20 payments, hence the time is
5 years.
(16.)
This Example is ^worked,
('70
S64— bo X 9 = 144 = ^— «r j 9 — I X 00x9= 576c = f —
X tr. j Then 144X2 -r 720 r= -os, hence the rate is 5 per cent.
(18.)
439c«4— 500 X l — L-iO'Xzza^nt^ 7 — 1 X 5^° X 7 = 23520
— ; — I X f« 5 470*4x2 -r 23520 :z: '04, hence the rate is 4 percent.
Part lU AT Simple Inter Es«r. tg$
(I9-)
ISI0'25--»I7$X« = no-25 rStf — nf ; 8^i K 175x8 = 9800
*^ f — I X ffi'i iiO'2 5X^ -f 9*00 it 'oaajh theintercftof tool, for
I year, hence the rate is 4^ per cent.
(26.)
391*5625 — 17-5X20 tfr 41*5625 :=ta-^itti 20— t X iJ'SX
tn 6650 -- f— I X f«i 4i-562SKi -r 6650 ;2: •oii^l. the iatsreft
of looi. for ^ year^ hence the rate is 5 per centt
The present vtoKtU ot ANNUlTIiES m
ARREARS AT SIMPLE INTEREST*
This Exam/)/e is ivorkedi
M
*Thc amount of this annuity (vide Ex. id of the precedift|[
Art*) is 864!* and 9X*o5 + i d i*45 ^^ amount of lU
for 9 years.
Then 864 -r- 1*45 = 5:95;*862o6897l. &c, = 595I. 175.
i|d, •58621 the preicnt value.
1+2 + 3 + 4+5 + 6 = 21
One year's intereft of 560I. = 28
Whole intereft 588
560 X 7 ^ 3920
The amount 4508
7 X'05 + I = 1*35 the amount of il. for 7 year*
Then 4508 •+ r35 sz 333r^'i9'^' = 3339'* 5^. 2$d.
♦he prefent value*
S
194- The PRESENT WORTH OF AMNUltllS IN ' -
(4-)
, ..rXhe amount^of .this annuity (vide Ex. 4th cfthe^ preceding
^ty) is)i5io'25L,;and4X*045 + i = ri8, the aniSunt of
il. for 4 years. .. • .' i; : -.1 ,..
Ihen I 910-25 -i- i-i8 rz 1279-8728813, &c. n 1279].
17s. 5'492d. the prefent value.
The amoi'int of this annuity (vide Ex. 5th of the preceding
Art.) is 39i*5625l. and 5 X '05 + i = 1*25, the amount
of 1 1, for 9 years.
Then^ 391 -^625 —1*25 zz 5 13*251. rr 313I. 5s. tlie
prcfent'^'Qrthfc
■•■ ■ ' (6.)
This Example, is <vJorkcd.
iv)
Flrft,
595I. 17s. 4d. 3-58&2I ^ 595'Sd2o6897l. tliis divided by 9, gives
9>C.ax-o5 + 2 =: 2-9 z= 2^r + 2; 9 + -o5 + 2— -05— 2*4:^
(8.)
rr -f 2 — r. Then — ~ X 66'2o6896552' zi Sol. the annuity,
2-4
Firft,
3339^* 5^' i-f'^' ^^ 3339'*'59'^' '^'^ divided by 7 gives 477'o'37'l.
= -^i 7X2X-05 -f2zz2-7=:2/r4 2j 7 X'os+i — 05=1 » 3
n fr + 2 — r; Then — - X 477'0'37' ~ 560I. the annuity.
2-3
(9-)
1279I. 17s. 5'492d. — I279»37288i3l. this divided by S zr
i59«.984iioi625 rr -y 8 x^ X •^^225 + 2 — 2«36.rz a/r-f 2 ;
2.36 .
— X
2-1575
^59*9^41101625-— 175I. the half-yearly payment, hence the annuity
i^ 350^-
SX'0225 + 2 — '0215 — 2*1575 n rr+ 2 — r ; Then ■■
Fart II. Arrears, AT Simple Int»erb8'T. 19J
(io.)
■p '
3T3'25 -r 20 = 15-6625 — : — "j 20X 2 X -0125 +2 =z a'5 n
2;r 4. 2 ; aox*ci25 + a.— •<?i25=: 2*2375 ~''' + a — '•J
"a-k *-'",,■■"
Then ^ X 15*6625 — IT'S', the quarterly payment, hence
2-2375
th* Annuity is 70I; . • -
^hii Example it nvorked,
(12.)
3339I. 5s. 2fd. = 3339-i'59'i'
3339-i'59'Xa . , ., 'ay 56o~i66'o62'96^ , __
■ : • — 230*5 10 ~ - - J ' 2 1 " ■"* 2 —
50oX'C5 rn 56ox*o5.
X3-537 03' =: =^ — f 5
Then V 23S*5'i8' -f iTsTFos^^— I3-537'C3'= 20-537'o3'
*— i3*537'o3' iz 7 years {exa^ly.)
{is-)
595I. 17s. 2d. 3-8621 — 595'862o6897l,
595-86206897x2 5, 2/.
80 — 29'793ic34485 ^ w — r/>
^— r-^-^^i-^^-il-?— -I = 12-05172414=: i — li
i>o X -05 ^ • rn
Then V 297*931034485 -f 12-05172414V — 12*05172414 —
9' yeai-s.
(H-)
, I2791' ^7^* 5"492d« r: 1279*8728813!.
1179-8728813X2 £ /: ^Z*
— : zz 650-09416193 iz -^ ;
I75X-0225 ^ ^^ ^^ r« '
175—28-70713982025 . r o " — '/' T
^ — J zz 36-63o8i)5i2 — ^~';
i75X'0225 . J J :) ^^ .>
Then V 650-09416193 + 3^^^88512] ^ — 36-63088512 — 8
payments, -hence the tinae is 4 yearB.
1^ The FKBiKK'i^ WORTH or Anwuities
i7'5X*oiz5 "" -iiBys "" 49 "" ra *
^7-5 — 3-9'5625 _ ^ _ ^3-475 _ 43 ^'it _ n-^r^ ^ ^^ ^
lTSXO}zs ^""•21875"" 7 ~~ rn *'
Then / ^OS^^ . 43^-2.Y 43'-g _ S?!'^ ^^^ 43'-^ _
N/49 yj 7""7 7""
~ 20 payments, hence the time is 5 years.
(.6.)
This Exavt^le is ivorked*
(I7-)
595!. 17^. 2d. 3'5S6i ~ 595'862o6897l.
80x9 — 595*86206897 — i24»i3793io3 m fit-^fi
595-86206897x2 + 20 — 80X9 = 551-72413754= */ + « — »r/ J
Then 124/13793103 ~ 551-72413794 X "I = '05, hence the rate is
5 per cent.
(18.)
33391. 5s. 2|d. r= 3339-2'59'l-
560x7— 3339-2'59' = 58o.7'4o' =«?—/>}
Ts39*2'59'X2+56o — 560x7 = 33iS'5'iS' zz zp^»^ rt j
58o'7'4o' H- 33i8.5'i8' x y zz ii6i'4'8i' -J- 23229*6'a9' n '05,
hence the rate is 5 per cent.
1279I. 17s. 5«492d. n: 1279.87288131.
175 x 8 — 1279-8728813 = 120-1271187 =r vt — />j
1279-8728813 X 2 -h 175 — 175x8 ir: 1334*74576-6 zn zf-^t n
— w^ ; 120-1271187 -^ 1334.7457626 X |- n -0225, the intereft of il.
for I year, hence tlie rate is 4^ per cent.
(20.)
i7'5 X20 — 3i3'25 r= 36-75 =: rr— / ;
313-25 X 2+ i7'5 — 17-5x20 — 254 — 2;) + «--wf J
36-75 + 294 X a\ = '0125, the intereft of il. for^yeaj, hence the
rate is 5 per cent.
Part ir. Rev£Rsion, AT Simple Interest. 197
The' prf.sent worth of ANNUITIES int
REVERSION AT SIMPLE INTEREST.
^lis Example is ^worked,
(2.)
Fir ft,
i-^2-f-3-f-4+ 5 + 6-I-7 — 2^j fum of the Teries.
20X '04x28 + 20 X^ n 182-41. amount of the reverfion.
Ani i82'4 -=8^ '04 + 1 rr; \i%*ii%' the piefent worth of the
legacy were it to commerce immediately.
Secondly,
i3S«i'8' -7- 5X'04 + I — i^S'i's'^* — 115^' 3'o'3's« the value
of the legacy,
(5.)-
i + 2 + 3^4'ir:*lo, fum of the feries.
50 X "05 X 10 4- 50 X 5 rz 275l» amount of the reverfion.
And 275 -7- 5X'05 4- I r= 220I. the prefent worth of the leafe,
fuppofmg. it to commence immediately*
Secondly,
220 -~ 3X'05 4- I ~ i9i'3043478i6 zn 191I. 6s. i«C4 3478d. &c»
the value of the leafe.
(4.).
Firff,
T+2 + 3+4, &c. to iQ rr 190, fum of the feries.
^icooX'OSX 190 -f- 1000x201:129500, amount of the reverfion.
29500 -7- 2CX'05 + I rz: 14'' 50 the prefent wor-th of the revcifioo,
fappofing it to oomnren«€ inunediately. ,
Secondly,
T^IS^ -r 5X'05 + I — I iSool. the value of the leafe.
. (,'.)
^his Example is 'VJorked^
S.3 ...
198 Annuities in Arreaas
(6.)
115I. 3-o'3's. =: iis-i's'i- = /> ^ = 8,7=: 5.
SX2X-04 + 2 X 5X-04+ I n 3'*68 — 2rr + 2 X Tr + i>
8x8x-o4 + 2x8 — 8x-o4=: 18-24 zi rrr + 2^— rr;
Then 3- 168 -r 18-24. X iiS'i's' = 20I. the legacy.
(7.)
191I. 6s. i'0435d. r: i9i'3043479 + = /> /=5>7'=:3.
5X2X'05 + 2 X 3X'05 + I zz 2-875 = a/r + 2 >^ Tr + i j
5X 5 X -05 + 2x5 — 5X'05=: II =ur + zt — rrj
2*87C
Then x 19»*S043479 + = 50I. the annuity.
(8.)
11800 = /, f~20, andTn5.
20X2X-05 X a X 5X05 + 1 — s — 2rr + 2 X Tr 4-15
aoX2oX'05 + 20x2 — aox*o5z= 59 r:/rr+ 2r— rrj
Then -^ X 11800 iz loool. the yearly rent.
59
ANNUITIES IN ARREARS at COM-
POUND INTEREST.
('.)
This Example is fworked,
(2.)
l + i*05+'i-o5)*4. 1.05P + I05J4+ 1^5+ i:^t +1^7 +
t'^Srz '^^ "" ■4- ^05]' = ii»0265634, this X 80 produwa
-i 1.05^—1
t8zM25]. — 882I. 2S. 6d. amount.
(3.)
!• + 1*05 +1-05)* +■^05]'+ I^4+ 7^ J +ro5i6 =
V^^^ ^ +"i^o5J* = 8*1420076, this X 560 produces
4559-5242561, zz 4559l« xos* 5i^«
Part II. AT CoMPoVND Interest. ig^
(4-)
^his Example is <worked,
M
Here «= 175, y=:i'0225, and /=3.
Log,r :=. log. 1*0225x8 = '0773064, the number an-
fwering to which is 1 '19483 := r ;
Log.r — I = log. '19483 = — i'2896j5:8
Log, « = %. 175 = + 2-2430380
+ 1-5326938
Log, r — I =/c/. '0225 =z — 2-3521825
X<?g-. of the amount + 3*1805113 thenuro-
bcr anfwerirrg to which is 151 5'344l. = i $; 1 5I. 6s, 1 •o5d.
(6.)
Here «n 17*5, r= 1*0125, ^'^^ /=20.
Z'Tg'. r z= log, 1*0125x20 = '1079000, the number an-
fwcring to which is 1*282035 =: »• ;
Log, r — I = log, '282035 -~ ■"■ ''45^03030
Log, n^ log, 17-5 = -f 1-2430380
+ 0-69^3410
Log,r — I zzLog,*oii^ = — 2-0969100
Log, of the amount -f 2-59643 1 o, the nun:«
ber anfw'ering to which is 394*849 = 394I, i6s, n|d.
(7-)
This Example is nnorhd,
(8.)
882I. 28. 6*04d. =882'i25:i6I. =^7;
1*05|9 — I = '551328216 = / — I ;
'iSijzTzla ^ ^82*12516 = Sol. the annuity.
I '05 — I
•4-07 1 CC4
2C0 Annuities in Arrears
(9.)
45591. ics. 5d. 3*7444 = 4559*52473 + = ^;
I'^i^ — I zz •4071C04 = r — I ;
X 4559*52473 == 560!. the annuity.
(10.)
97v> ExamJ>/e is ^worked,
(II.)
1515I. 6s. I'id. -zz 1515-3451. = a-.
By the 5th Example, r — i = 1-02251^ — i rz '19483
.' Log. r— J =: log. -012^ —: — 2-3921825
log.a-Iog.isiS'HS =+ 3"i8o5iij
H- 1-5326940
ir?^. '19483 ZZ — 1-2896558
Sum -f 0-2430382, the num-
ber anfwering to which is 175I. the half-yearly payment,
hence the annuity is 350I.
(,2.)
394I. i6s. icid. — 394'8428I. =.a',
By the 6th Example, r — i = ioi25i'° — i=:'282035
Zcg..r — I ^zlog. -0125 =: — 2-0969100
Log. fl =:- /og. 354-8428 n + 2-5^)64242
4- 0-6933342
Log. '282035 = — 1-4503030
Sum -4 r24303i2,thenuni^
ber anfuering to which is 17-5 {»earlj) the quarterly pay-
ment, heucc the annuity is 70I.
(13.)
^his Example is worked.
Part II. AT COMFOVND InTIREST, 201
(h)
882I. 2S. 6*04^. = 882'J2^l6l. =r <jr,
882'! 25 16 X*05 =44*106258 =5 r — 1 y.a\
Then ^''°^'^^ + , = i-5f 132815 = 7^' hence
/ may be found by repeated divifions (or rather by logarithms)
;;= 9 years.
(15.)
4^591. losi 5d. 37444 = 4559'5H 73 + /=^'
4559-52475 X •05' = 227*9762^65 =r — I X a\
227-9762365 _^ ^ _ 1.4071004=^051' hence may be
found as above, = 7 years.
(16.)
?">&« Example is ^worked,
ill')
1515I. 6s. i-id. = 1515*3451. =5^;
i5i5*34-5 X 1-0225 — 1515-345 + 175 = 209'0952 =
ar — a -{- n;
Log. 209-0952 = 2-3203441
Log, nz=. log, i""!^ = 2-2430380
L'yg, 1-0225 == -0096633 ) 0-0773061 ( 8 pay-
ments, hence the time is 4 years.
(iS.)
394I. 165. lojd. ir: 394*8428 =: ^;
394- ^4^^ X 1-0125 — . 394-8428 + 17-5 = 22*43554 =
ar — a -\- n i
Log. 22-43554 = 1-3509366
Log, 17-5 ;= 1-2430380
Log, 1*0125 = -0053950 ) 0-1078986 (20 payments,
hence the time is 5 years.
lot ThEPRTSEN-T WORTH OF AkNUITIES I"
^h:i Examplt is ivoried.
. 882I. 2s. 6d. -0+ =: 882-12516 = c;
882'i2fi6xr Q 88'2-i2ci6 — So
: — — r^ Zi : now ov tnai
^O .... ;•..! 8.Q. . '. • -'
and frror, as in Example 6th of Double Fofition, I find r ss
1 05, hence the rate h 5 per i:ent.
The present worth of ANNUITIES in AR-
REARS, AT COMPOUND INTEREST.
(^•).
7^/j Example is nvcrked^
(2.)
The amount of 80I. for 9 years, at 5 per cent, is 882*1 25I.
(vide Example 2d of the preceding Article, and 1*051 9 =
1*551528216 + the amount of 1 1. for 9 years.
- Then 882-125 : il. •• i'55i3282i6 4- : 568*62564l. =
568I. I2S. 6-i54d, the prcfent worth.
(3.)
The amount of 560!. for 7 years, at 5 per cent, is
4J59-?242~561. (vide 3d Ex. of the foregoing Art.) and 1-05V
— ■ I -407 1004 the amount, of il. for 7 years.
Then 1-4071004 : il. :: 4559-524256 : 3240*368821. =
3240I. 7s. 44d. the prefent worth.
This Example is ivorked,
(5-).
Here « m 175, r n I '0225, and /rr S.
L:g. i*c2Z5x8«rr •0773064, this fubtradlcd from the%. of i rr o,
iji%es — I*9226q3'6 for the kg. cT — , the number anAvering to which
r
■V36938G, hence, i — -— ::z: i — •8569386 — •163061-:.
r
Part lI/'Ai^REA**, AT Compound -Interest. 20 j
Log. -1630613 ' =:: — i'2i23509
Log, n-z^ og. 175 - + 2-24303550
,\ ,.\ i
+ 1-4553289
Log. r — I •=: log. -022.$ ~ — 2-35-1^25
Leg. of the prcfcnt wortli -f" 3*1032064, the number
anfwcring to which is i268«254l. zz 1268I. f-oSs. Anfwer.
Here ^«?^ J7'»5, r— i-oizj, and f n 20.
L^l^. I'0I2 5X20 — •10,79000, this fubtratted from the Ys^. of t — o,
leaves — i •8921000 for the 7?^. of --, the number anrweriiig to which
r
h •7800097, hence i — i* — -7800097 — •2199903.
- r
r. . • . . ,
lo^. •2199903 =—1-3424035
Log. n zzlog. IJ'S ZZ + 1-2430380
+ o 5854415
. Lo^, r — I zz log. ^0125 zz — 2-09.09100 ,
Lijj;. of the prefent wortli. 4-2-4835315, the number
anfwering to which is 307-986 3I. zz 307I. 19s. 8^-d. Anfwer.
(7-)
'This "Example is ivorked,
568!. 12s. 6-i77d. IT 568-62571. zi p\
5681-6257x-t)5 =r28-43i2»5 zz r—i xp-y
I -05] 9 11: 1-551328216 = r J I — 1-7- i'55i3282ib ZZ
1 — -6446089 ir •3553911 ilr 1 } Then 28-431285-7- *
' - r^
'j'553911 — So^' the annuity.
2842!. 7s. g-oiSifd. =: 2842-38757 rr /> ;
*_a842«387'57X'05 = i42'ii93785 zz r — i x /> j
1 -05) 7^— ' i' '407 icro4 rr r j I — 1 -r 1-4071004 r: i — '7106813
= -2893187 = i'—_L} Then i42«ii937'S5 ^ •2893187 —
r
491 '2208561. 13 49 il. 4s. 5d. the annuity.
to4 The TxisitfT wotra of AuMfftrtti tn
(ro.)
Thit Example it wjorked*
(...)
Utxtt nS, rz: I'oaaj, and 1268I. 5'o8s. 2^ I468»254J. ~/> j
By the 5th Example, l — — — — •1630613,
t
Log. r-^ t 1^ log. '02%$ r:— i*?S*>S25
Leg. f n log. 1168 '154. ir + 3 '1032064
+ 1-455388-9
Log, »x63o6i3 rr — i*2i235o9
Diff* 4- 2-2430380, the number
•Tifvverlng to which is 175I. the half-yearly payment, hence the annuity
k 3501.
(12.)
Hercr^ao, rr:i*oi25, 307I. 19s. 9»6d. i= 3o7*99l»=^}
By the 6th Example, i — «-— -^z •2199903,
r
Leg. r— I :zz leg. '0125 zz — 2*0969100
Log. p zn log, 307-99 rr + 2-4885366
4- o 48 5446 6
Log, •S199903 :i — 1 •3424035
Sum + J '2430431, the number
•ofwcring to which is 17-5 {nearty) hence the annuity is 70).
(>3)
Ty?"// E-xample is nvorkeJ,
(14O
5fr81. 12s. 6.i77d. -jz 568«6257l. zr /* }
80+568-6257 — 568-6257 X I-C5 = 5i'5687i5 ~ w+j>— /r j
£0 -7- 5i'56S7i5 ir: i'55i3282 — zz. i-05\'» hence the vaVieof r may
be found by repeated divifions, or rather by logarithms, r: $ yeari.
Part IL Arrears at Compound Interect. to^
('$■)
$842!. 7s. 9'oi83d. r: 2842'38757U -r /» j
$6o+2S42-38757— 2«42-38757Xi'05r: 4-i7-88o62i5i::b+/>— ^rj
560-^417-8806215 — 1*34000956 zz i-ojy hence the value off nKyr
be fouad as above, zH 6 year*.
(16.)
Ti^/j Example is nvorked,
(■7.)
1*681. 5-o8s. n 1268-254!. zz^i
175 + 1268-254— I268-254X i«oa25 — i46'4643— fi-f^ '— ^^ j
Lig* n ~ log. 175 — ?.-243038o
ii^. 146-4643 ~ zibsT^ii
Log. r ir 1.0225 rS •0096633 ) '0773064 ( ^paymcnu,
hence the time Is 4 years.
(»8.)
307I. 195. 9-6d. =^ 307-991- = /;
17.5 ^ 307-99 — 307-99 X 1-0125= 13-65 nr+^—^rr
Lo^. n -n log* 17 •$ zz Ji 4430380
Log. 13-65 — 1.13513x7
Log, Tzz log. 1-0125 — •005395 ) 0-1079053 (20 payments,
hence tlie time is 5 years.
Tiis Example ij n,jorifd,
(20.)
568I. I2S. 6-i77d. = 568-62571. =: /> ;
568-625 7 -i- bo 9 9+1 80 „ , «.
563.6257 ■' +'■-•'■ = j^TelTi' ^^ ''^"^""
648-6a57 x '-9 — 568-6257 X r^"" :=. ''>o, by trial and error r miy be
found zz 1-05, hence the jate is 5 per cent*
zo6 The irREsrNT worth of Akkwities in, 5tc.
The present worth of ANNUITIES in RE-
VERSION AT COMPOUND INTEREST.
Th's Exajfiple is fworked*
(2.)
5.515631*5, this X 50 producesi76*»Si562 5l. amount of die reverfion,
i'05^,5 rr i'i7628i6, amoount of il. for 5 years.
As 1-2760816 : 11. ! : 276-2815625 ! 2i6«47383, tlic prcfcnt
worth of the lekfe were it to Commence immediately.
As i-o^]^ ,= I»157625 : il. •• : 2i6'473S3 : 187I. (nearly) the
p.efent wortli of the reverflon.
(5-)
— V -. I -Of; ^9 — I . — ^.
i«4-i'05 + i-05i*-&c. to 19 — — -^ — + i«o5V9 =:
33'^)954'*» *^*^ X l^^co produces 33o65«9542l. the amount of the
reverfion.
As i"^05"*<==i 4*6531977 •• iJ. • •• 33°6S*9542 ' i2462'iio3l. the
^refent ^v^rth of the leafe, fup^ofirtg it to commence immediately.
As 1^05^5 =: 1-2762816 : ij. : : i2462'Zio3 : 9764*46781. —
9764I. 9s. 4id. "088, the prefent worth «f the leafe,
(4.)
7 his Example is *workfJ,
M
187I. 0».©|<i.--4i76 = 127-003561.=:/} 1-05— I = -05 = r — i;
r WT^"* -7^8 —,.4774554, this X •05X187-00356
~ 13.814471 =r— I xr X/i r — I =r I'osl — I=:'a76i8l6i
The» I3-SI447X -r '2761816 = 50I. the annuity.
Part II. The rt'RCHASE OF Freehold IvjTATEt. ao?
(6.)
5764I. 9s. 4|cJ. .cSS ~ 9764-46781. —/ 5 1-05 — i=-o5— r — 1 ;
5764'4678 rr 1653-297669 ~r — ixr X/jr — in 105^^0
-* I rr i-653z.;77 ; Thea i653-z97669 ~- 1-6532977 rr loool.
(ricurly) the annuity.
The purchase of FREEHOIJD ESTATES,
OR. PERPETUAL ANNUIIIES.
(')
J. 06 — T = '06, the ratio lefs i.
Tfcen 250 -7- '06 =: 4.166-6' :::4i66l. 13s. 4d. theprefent
worth.
(3-)
voz — 1 =: '02, the ratio lefs r.
125 -7- '02 =z 6250I. theprefent worth.
Or thus. See the Note to this Example,
1*019804 — I ■=:: '019804 -f , t! e ratio lefs i.
Then 125 -^ -019804 = 6311*8^61 = 631 il. 17s. ijd.
the prefent worth ; exceeding the above anfwer by 61I. 17s.
i-|d. and of courre a more correct anfwer, for it is certainly
more advantageous to receive the rents half-yearly than yearly.
(4-)
1*01125' — I =:*oii25, the ratio lefs r.
Then 500 -^- -01125 = 44444-4'l. — 44444I. 8s. ic^d.-^
the prefent worth; but the anfwer would be the fame if the
rents were received annually inftead of quarterly. Hence by
the fecond table 500 -^ •011065 —4.5187-52824- 45187L
los, 6|d. exceeding the above anfwer by 743I. is. 8d.
(5-)
''Jhis Example is tvcrked,
T z
»08 fivYINe IKD S.ELLINC Fl-EKHeLD EsTATft.
(6.)
4t661. 13s. 4d. rr 4i66«6'l. i*q6 — 1 r= '06, tlie ralio
Jefs J. Hence 4i66'6'x*o6 z:^ 250I. the annuity.
(7-)
1*045 — ' — '^iS> ^^^^ ^^*^o ^**s I.
1 760 X '045 =: 79'il. = 79I. 4s.
T/t// Example is loorked,
2^0^4166*6'— '06, and '06 + 1 z: i*o6 the ratio. Hence
the rate is 6 per cent.
(10.)
60 -f- 1200 = *05, and *C5 -f 1 z= 1*05 the ratio. Hence
the rate is 5 per cent.
The buyikg and selling FREEHOLD
estates, according to a number of
year's rent for the purchase-money.
0*)
Ihh Example is n/jorked,
I2C0 ~- 20 = 60I.
(3-)
7hiS Example is ^worked,
M
6iy -i- 2j = 2j years.
(5-)
This Exa?nple h iforked,
(6.)
45 X 2j =: 625I. Anfwer.
Partll. The PUHCRASE Of Freehold EsTATts. 20Q
(7-)
This. Example is ivorked,
(8.)
22|-f- 1 4- 22| = 209 4- 200 = I '04^ the ratio. Hence
the rate is 4^ per cent.
(9-)
This Example is ivorked,
(10.)
I -f- *04 = 25 years, Anfwer.
The purchase of FREEHOLD ESTATES, or
PERPETUAL ANNUITIES, m reversion.
This Example is njoorked,
6o*5 -4- 'O^ = 1 210, value of the eftate if entered on im-
mediately.
As iToT''^ = 1*6288946 : il. : : 1210I. : 742 83504!.
r= 742I. 16s. 8;^d. the prefent worth of the reverfion,
(3-)
290 -r '04 = 72501. value of the eftate if entered on
immediately.
'As I -041* = IM698586 : il. : : 7250 : 6i97*33l. =
6x97!, 6s. 7d. the prefent worth of the reverfion.
(4.)
This Example is fworkedv
210 The Purchase of Freehold Estates,
M
742I. 1 6s. 8:Jd. '8 = 742-8351.
i-o5Vo X 742*835; = 1*6288946 X 742*835 =: 1210
(nearly) the amount of the purchafe-money to the time when
the reverlion begins.
1 210 X '05 = 60*51. = 60I. 10s. the yearly rent.
(6.)
6197I. 6s. 5|d. = 6197*32292 -f £.
"i'04'|4 X 6197*32292 = n698586 X 6197*32292 =
7249*991 5 1 5> the amount of the purchafe-money to the time
when the reverfion begins.
7249*99/515 X '04 = 289*9996, &c, = 290L (nearly)
the yearly renl.
rhe End of th SOLUTIONS.
A N
APPENDIX
TO THE
COMPLETE PRACTICAL
ARITHMETICIAN.
CONTAINING A
SYNOPSIS OF LOGARITHMICAL
ARITHMETIC,
Shewing their Nature, Conftrudlion, and Ufe, in the
plaineft Manner poflible.
TABLES OF COMPOUND INTEREST anb
ANNUITIES,
Extending from One to Forty Years,
A L S 0>
GZNZRAL AND UNIVERSAL DEMONSTRATIONS OF TKK PRINCIPAS,
RULES
IN THE
-COMPLETE PR ACTICALARITHiMETICI AN.
By THOMAS KEITH,
Teacher of the Mathematics, &c.
Mathejis mentis expurgatio, H I E R o C L .
LONDON:
miNTED FOR B, LAW, AVE-MARIA-LilNE*
MOCCXCi
A N
APPENDIX, Sc.
$. I. The nature aad formation of Logarithms,
Definitioft,T OGARITHMS are a feries of numbers fo
J J contrived, that by thetn the work, of malti-
plication is performed by addition, and diviUon by fub-
traftlon.
If a feries of numbers in arithmetical progreffion be placed
jis indices, or exponents, over a feries of numbers in geome-
trical progreffion, the fum or diiference of any two of the
former willanfwer to, or Handover, theprodud, or quotient,
of any two of the latter.
Thus 0.1.2.3. 4* 5* ^» ^^' ^"'^5^' feries, or indices.
1.2.4.8. 16. 32. 64, &c. geom. feries.
a 4- 3 = 5» or 3 — I =: z.
4 X 8 — 32, or 8 -r 2 rz 4'
Now the arithmetical feries, or indices, have the fame
properties* as logarithms, and therefore fucb a feries may be
taken for their foundation. Conlidering logarithms as indices
to a feries of numbers in geometrical progreffion, it isevident
that there may be as many kinds of logarithms as there can be
taken geometrical feries; and it is equally evident, that if
any one fy. em of logarithms be once obtained, an infinite
number of others may eafily be derived from it. But, though
ail thefe different fyflems will be equally perfeft, if calculated
with equal accuracy, yet the moft convenient rank of con-
tinual proportionals is a decuple progreifion.
» Thus o, I, 2, 3, 4, 5, &c. indices, or logs,
I, 10, 100, 1000, 10000, 100000, nat. numbers.
But if the natural number be lefs than an unit, its index,
or log. will be negative,
^c.— 4, --3, —2, —I, o, I, 2, 3,
&c. 'ooooi, •ooor, 'ooi, 'oi, i, 10, 100, 1000,
4, &c. indices, or lo^s.
10000, &c, natural numbers.
* Thefe properties of numbers, which arc the foundation of l.garltlrtst
are declared by Archimedci in his Arcnarius i but the invention ot'toga-
richms belongs to Lord Njpier.
4 ASyncpsisof
From this feries-it appears, that the index to any logarithm
will always be an unit lefs than the number of figures in the
natural whole number to which it belongs; hence fiat index
will always determine how far the tirft figure of the ahfolute
number is diflant from the unit's place, of which 1 fhall
have occaiion to fpeak more particularly farther on: I now
proceed to the confrrufticn of logarithms. ♦
^. II. A Jhcrt and eafy RitJe for the Conjiritff ion cf Logarithmic
either common or loyferholiCy illujir cited by E.^amplei,
The Rule*.
1. Let the fum of the propofed number, and its next kfs,
be called n^ by which divide '8685889638, &-c. and referve
the quotient.
2. Divide the referved quotient by the fquare of zr, re-
ferring this quotient, which divide by the fquare of «r, as be-
fore. Proceed thus as long as divifion can he ma-^e, and write
the referved quotients under each other.
3. Divide thefe quotients refpeftively by the odd numbers'^
I, 3, 5-, 7, 9, II, \^y 15-, 17, &c. viz. divide the firflf
referved quotient by i.thefecond by 5, the third by 5:, ^lC,
Let the quotients found by thefe divifions be written regu-
larly under each other, and added together; to this fum add
the logarithm of the whole number, next lefs than the pro-
pofed one, and the fum will be the logarithm required.
Note. If, inftead of the nuir.ocr •8685^^^9638, &c. we irake uii of
the number 2, ihe. rule will fervc fj/ the conftni^ion of hyperbolic loga-
rithms.
Ex. I, Required the common logarithm of 2.
Here the next lefs number to the propoft-d one is i ; hence
2 + 1 z= 3 ~ w, and ■i,y.'3^ IT 9 rr the fqu.ire of //.
•S635889638 ~3zz*2895296546-i- i =-28952965:46
•2899296946-r 9 — •032i6996r6T- 3 •0107233205
•O32i6996i6-7-9rz:*C035'74440i-i- 5 = *ooo7i4888o
•003 574440 1 -T- 9 == '000397 ■' 600-f. 7 = '0000567 3 7 1
•0003971600-^ 9ir-occo44i2 88-r- 9 = -cooc 049032
•0000441 288-1. 913-0000049032^ 1 1 = -0000004457
•0000049032-^ 9::^ •0000005448— 1 3 rz '00000004 1 9
•0000005448-^-911: '000000060^-7- 1 5 — '0000000040
Sum '3010299950
To which add the log. of i =: •coocoooooo
Common logarithm of 2 rr '30102999^0
♦ An inreftigation of this rule was given in the original eflay, but is
here omitted as b.-ing of little or no ufe to tho yoiir.g Arithmeticiaa, who
cianoc
Log ARi THMi c AL Arithmetic. 5
Ex. 2. Reaj/ired the €om?n:n logarithm of ^,
Here the next lefs nuinber to the prc-vjfed one is 2 ; hence
3 -f 2 — 5 nz^, and 5 X 5 := 25 zz ihc fquareof »,
•8685889638 -r $ — '\l^l 111^,^1 -^ I := -1737177927
•173 ; 177927 -^ 25 rr '0069487 116 *j- 3n ooi3 162377.
•000)4871:6 -r 25 ii •000277,4^4 -~ $ zn 'ooo ,555896
•0002771,484-;- 25 — •ooonijji79 ^ 7 rr 6000015882,
•C000111179 -^ 25 — •000000-. j.43 -^ 91::; 000D000493
•0000004443 -r 25 rr '0000000177 -^ n tz: •0000000016
Sum •1760912586
To which add the log. 01 2 rr -3010299956
Common log. of 3 — : '4771212541
Ex.5. Required the common logarithm of /^,
Here 2 X 2 =: 4. *.• 2 X /<5i. 2 n %. 4."
Hence '301029995X2 = '602059990, corr-mon log. of 4.
Ex. 4. Required the common logarithm of ^.
Here 10 -r 2 := 5. •.* Log. id — hg. 2 ri log, ^,
I — '30102999; zz -698970004, common log. of j,
Ex. C. Required the common logarithm of 6.
Here 3 X 2 =z 6. •.' %. 2 -j- leg. 3 z= log. 6.
Hence '301029995 + '4771212542 = '7781512492, com-
mon log. of 6.
Ex. 6. Required the common logarithm of '^,
Here the next lefs number to the propofed one is 6 ; hence
•7 + 6= 13 z= », and 13 X 13 = 169 zz the fqiiare of;/.
•8685880638 -r 13 — '0668145356 -r I — '0668145356
•0668145356 -r 169 z=. -0003053522 -r 3 r: '0001317840
•0003953522 -r- 169 rr '0000023393 -f- 5 n: •ooccoo4r.78
•0000023393 -f- 169 ~ •0000000138 -r 7 n: '0000000019
Sum •0669467893
To which add the log. of 6 — •7781 5 12492
Common log. of 7 zz '8450980385
Ex. 7. Required the common logarithm of ^.
Here 4x2 — 8. •.' log. 2 + log. 4 = log..^.
Hence '301029995 + '602059991 zz '903089956, com-
mon log. of 8.
canrrot reafonably be fuppofcd to be acquainted cither with the method of
JiuxiarSi or the ftature and properties of the iyftrinlaf whence this ralr
is dedaccd.
6 A Synopsis or
Ex. 8. Refuired the common logarithm of(^.
Here 3x^ = 9. *.• /o^. 3 -f log. ^zzlog.^; or 2 X tog, j
= V- 9-
Hence '4771 2 1 2 J47 X 2= '9 5424.25094., common log* of 9.
Ex. 9. Required the common logarithm of \o.
Here 2x5 = 10. '.• %. 2 + Ir.g. 5 = /o^. 10 ; but the
common logarithm of 10 is known to be i»
Ex. 10. Required the common logarithm of \\,
Here the next lefs number to the propofed one is ic ;
hence 11 -f 10— 21 = », and 21 x 21 n 441^ the Iquacc
of/-/.
•868588963R -^ 21 tr •0413C13792 -^ I iz •0413613792
•C41 361 3792 -T- 441 zr •0C00937S9.. ■4- 3 ir: '0000312633
•0000937899 -i- 441 ^ •ocOoot)2»26 -r 5 ±r •O«oc0ooo425
Sum •e4i39268 5X
To which add the log. of 10 :fi i-ooooooooo»
Common log. of 1 1 -r 1*041 3926851
I
In this manner may be found the logarithms of any other
rime numbers; it may be obferved that the operation wiH
e fhorter the larger the prime numbers arc; for any number
exceeding 350, tne firft quotient, added to the logarithm of
its next lefs number, will give the logarithm fought, true to
7 or 8 places of decimals. The logarithms of the compofite
numbers are found, by addition or fubtra6lion, as above;
hence it will be eafy to examine any logarithm in the tables
which we fufpecl to be falfe ; or, if any logarithms be torn
out or obliterated, we can readily calculate them, and fupply
the defea.
Observation. If JV~ any number, as 1, 2, 3, 4, 5, &c. then
will A' X ^ — 1» ^"'l A^ X 6 4- 1, coniVitute a feries which will contain
ail the prime numbers above tlie prime number 3.
Ex. 1 1 . Suppo/e a per/on ^working a mathematical problem ^
nvhereiti he has occnjion to make u/e of the logarithm of ^-ji ; but^
up'jn locking in the tables, finds that logarithm obliterated , or
fufpe^s it to be falfe, hoiu mufi he find jt ?
Here the next lefs number to the propofed one is 570 ;
hence 571 -|- 570 =: 1 141 = n,
•8685S89638 -i- 1T41 zr •0C07611520 -7- 1 rn •0007612520
By Sherwin's tables, the log. of 570 — »'7 558749
Su.-n 1$ the tr« log. of 571 :r 2*7 566361
L O G A R I THMICAL ARITHMETIC. *J
§. III. The ufe of a Table of Logarithmic
The methods of difpofing the logarithms (when made) into
tables are various, and therefore no general direftions can be
given here for taking a logarithm out of thefn ; nor, indeed,
is it neceflary, fmce in every book which contains tables
there is, or fhould be, a rule given for that purpofe. — Before
I proceed farther, it may be proper to obferve that every
logarithm confifts of two parts, an integer, or whole number,
and a decimal fradion. The decimal part is always ajfr^
mati'ue, but the ;>//^^^r, which is generally called the index,
or ckarafleriftic, may be either affirmati've or negati've ; for, if
the natural number be greater than r, the index of its lo^^a-
rithm will be affirmative, (or -f ;) but if the natural number
be lefs than i, the index of its logarithm will be negative,
(or — ,) as I have already obferved. This index is not put
down in fe.eral books of logarithms, as Sherwin's, &c. be-
caufe all natural numibers, (confifting of the fame fignif.cant
figures,) whether they be whole numbers, or pure or mixed
decimals, have the decimal part of their logarithms the fame,
the only difference being in the indtx, or whole number, as
in the following examples.
Nat. numb. Logarithms.
•94621 = — 1-0743320
•094261 - — 2-97433^(5
•0-9426! = — j.974.3320
•0-09^261 = — 4-9743 ;;20
•000094261 = 9*9743320
•0000094261= — ^'9743320
From the preceding Examples it appears, that, if the /irft
figrihcant figure of the natural number (r ckoning fr -rn fhc
lefi hand towards the right) be units ^ te\s, hundreds, ihjujands,
&:c. the index to its logarithm will be c, i, 2, 3, 6cc. re-
fpedively ; or, if the index to any logarithm be o, 1,2, ^,-
&c.'the firft figurt* of the natural nuaijcr will be Lnir, tens,
hundreds, thoufan-'.s, Sec. refpecflively. Likewife, if the
finl fignificant figure of the natural nuuiber be te,'it,b., J:un^
dndths, tUufandths, Sec, the ind x to i ; Ic'garithm v. ill be
— i, — 2, — 3, &c. and, confe-]uentiy, \i ihc index to any
logarithm be — 4, — 2, — 3, -—4, — 5, &c. tho irrft i.gni-
ficant figure of the aumber anfwering is \Y.z jirJi,fscoiid, thirds
fourth, fyih, Sec, place of decimals.
u
Nat. numb.
uogariihms.
9-4261
=:=
0-9743320
94-261
~
1*9743320
942-61
~~
2*97433^0
9426-1
•~-
3-9743320
9f26ir
•*-*
4-9743320
942610*
rz
5-9/-43320
9426100
-z:z
6-9743320
S ASynopsisof
Proposition i.
To Jin d the logarithm of a 'vulgar fraHion*
RcLi. Subtraft the logarithm of the denominator from the logarithm
of the numerator: if you carry i, add to it the index of the logarithm 1
of the denominator; then take the aitfercnces of the indxes for the ^
index of the bgari:hi:n of the fraftion; prefix the fign of— or -f before
k, according as the fraftion is proper or improper. Obferve to reJucc
compound, mixed, &c. fraftions to limple ones.
In this and the following propofitions, whenever an index
is written without a fign, it is always affirmative or +.
Ex, I. To find the logarithm 5/" -Jl.
Log. of 93 = J '9684829
Log. of 94 = 1-9731279
Log, of II = —1-9953550
Ex. 2. To find the logarithm of •^-^*
48
^6^ 36x3 + 2 no . , r ». » •
Here -^ = ^ ^^ = , the log. of which is
48 48x3 144
— rS83«302.
PROPOSIT40N 2.
To find the logarithm of a ciicuUting decimaU
RpLE. Reduce the decimal to its equivalent vulgar fra^ioa, and thc»
ikid it3 logantkm by prop. i.
1. Required the logarithm of *i^6^\
Here •c6,/ = I^lHi^ = i£2, the log. of which is
^ -* 900 900
^17507655.
2. Required the logarithm ^5"3'54'»
Here ^3^5+' = ^^^^=^^ = -^^» ^^« ^^S- o^'^^'^^ ^
^^^+ 999 999
^ 0*7287071.
3. Required the logarithm of 'C 6^2 S 803 .
Here -o^'ziio^' = '^iH£3 = 1^1^ ,he H- cf
9999990 3/0370
which is —i' 7975 1 50.
LoC A R I TH M I C AL ARITHMETIC. ^
4. Required the logarithm of •oo84'9 7133'.
Here -oog+'gy . 33' = il?IIl^ = -ll2I^ = 4, .
999999000 999999COO 9768
the log. of which is — 3-9292724.
Proposition 3.
T^ot':? cr worr numbers being given ^ to Ji?id their prcduf! by
logarithms.
Rule. Add the logarichnQS of t'le nurTibcrs together ; when you come
to the indices, add the affiimatives and wha: you carry into one i'am, and
the negatives (if any) into ani-ther, and take t!ie difference of the fums,
to which put the fign of the grer.tcr fum, and this will be the index for
the fum of the bgirithms, the correfponding number to which witl be
the produfti
E X . I . Required the produB c/ 84X^6X37X8.
log. 84 = 1-9242793
log, j6 = 1*7481880
% 37 == 1*5682017
log, 8 =: 0*9030900
Prod. 1392384 log, = 6*1437590
Ex. 2.
Reqm
'red the
? produa (7/ '84 X ' 05 6 X
•37'
log.
•84
=
— 1-9242793
l;g>
•056
=
— 2'748i88o
log.
•37
=
— 1*5682017
Sum of the decimals = •\- 2*2406690
Sum of the neg. indices = — 4*
Log. of the produfl '=■ — 2*2^06690, the number
anfvvering to which is 'oi 74048.
Ex. 3. Required the produd of '37 X 426 X '5 X '004 X
•275x336.
hg. "37 n — i'56820i7
^ kg. 426 — 4- 2-6294096
log. '5 — — i'698o70o
Lg, '004 n — 3 '6020600
/er.-27 5 = — i-43933a7
leg. 336 tz + 2-5263391
Sum of the decimals and affirm, indices n + 7'4643i33
Sum of the negative indices r: — 6-
Logarithm of the produdl: — 4- i*4643i3J> the numbef
anfwering to which is 29«i28i76.
U z
to ASynopsisof
Ex. 4. Required the produS 0/ 56]- X /© X 'J X ^ X
47
Log, ^e\ = hg,^i^ - + 1-75076^:4
leg. 7^ =: — 1-8450980
log. 4d = — 1-9777236
L.g. f^ =z log. 11^ = -1-9778857
S'^raofthedecimais and affirm, ind. zr +5'3473528
Sum of the negative indices =1 — 4*
-Logarithm of the prod 11(51 rr -f 1-3473528, the num-
ber anAvcri.ig to which is 22*25 n 6.
Kx. 4. Renuir^d the produfi of •0594405' X •583'x
•OJ229I6' X -423571' X -I'S'.
Log. -0/94405' n: %. ;^^ = %. ~ =-2-774o829
995-^9^^ 2SO
Z.?-.-c8y=:%.li-^-^=^=%^^ =%.-! = — 1-7659168
* ^ "^ "* 9OQ "^900 ^12 * ^^
, 322016— 32291
"* -^ ^ "^ 9COCOCO
1 8
Loy.'i'^'-hg. — = —1-2596373
99
Sum of the decimals = -42-9407508
Sum of the negative indices = — 7*
-Logarithm of the produci = — 5'94^7SO^«
the number anfwcring to which is •000087247.
LOGARITHMICAL ARITHMETIC.
II
Proposition 4.
To find the quotient of one mimher by another, by logarithms.
Rule. From the logarithm of the dividend fubtraft the
logarithm of the divifor ; but take care when you come to
the index of the divifor to change it from affirmative to nega-
tive, or from negative to affirmative; then, if the indices
have unlike figns, take their difference*, prefixing the figa
of the greater index to it; but, if they have like figns, take
their fum, prefixing the common fign thereto, for the index
of the logarithm of the quotient.
Obferve, when there is an unit to carry from the decimal
part of the logarithm of the divifor, to add it to the index of
that logarithm, if affirmative, or fubtrad it from it, if nega-
tive, before you change the indices as above.
Examples 'wherein the indices of the logarithms of the divifon
and dividends are affirmati<ve, or -f •
Ex. I. Divide 785*925 by 25.
Log, 785-925 = 2-8g538ri
Log. 25 = 1*3979400
Quotient 51*437 /o^. =: 1 •497441 1
Ex. 2. Divide 250 by 78*5.
Log, 250 = 2*3979400
Log, 78*5 =: 1-8948697
Quotient 3*18471 log, n 0*5030705
Here the indices of the dividends are greater than thole of
the divifors; hence thefe and fuch like examples are ma*
naged like common fubtraftion.
Ex. 5. Divide 51*2^ by 125.
Log, 31*25 = ] -4948500
Log, 125 = 2-0969100
Quotient '25 log, = — 1-3979400
* The word d'lfferenct In this and the preceding rule is not to be un-
derftood in an aigebra\c fenfe, but fimply the difference of the 'wdic»i ^
thus tbe di/Jerence between 4-5 and — 2, is not +7 or ^—7, but +3,
Us
12 A Synopsis of
Ex, 4.. Divide 125 by 312^.
Log, 125 = 2 "0969 1 00
Log. 3 12 J = 3-49+8500
Quotient '04 %. = — 2-6020600
Here the indices of the divifors are greater than thofe of
the dividends, and when they are changed are negative,
••• the indices of the quotients are negative.
Ex. 5. Divide 962*5 by 770.
Log. 962*5 = 2*9834007
Log, 'J 'JO = 2*8864907
Quotient 1*25 log, z= 0*0969 100
Ex. 6. Divide 317*55 by 870.
Zc^. 317*55 = 2*5018121
Leg. 870 = 2*9395^93
Quotient '$6^ log, r= — 1*5622928
Here the indices of the divifors and dividends are equal,
and the difference in the 5th example is o ; but, in the 6th,
there is an unit to carry from the decimal part of the loga-
rithm of the divifor, which mud be added to its index, be-
caufe it is affirmative ; then it becomes +3; changed it is
— 3 ; then the indices are different, and their difference is
— I.
Examples ^wherein the indices of the logarithms of the di'vifari.
end dividends are negative , or — .
Ex, 7. Divide '0565 by •25.
Log. '0565 = — 2*7520484
Leg, -25 = —1-3979400
Quot. '226 log. = — 1*3541084
Ex. 8. Divide '00379 ^V -0678.
Log. -00375 = —5 '57403 13
Log, 'o6-jS r: — 2*8312297
Quot. 1*05530973 log, = —2*7428016
Logarithmic A L Arithmetic. ij
Here the indices of the logarithms of the dividends (not
regarding their figas) are greater than thofe of the divifors.
In the 7th example, the index of the divifor is — t, changed
it becomes -{- 1 ; then the figns are different, and their dif-
ference is — 1. In the 6th example, there is one to carry
from the decimal part of the logarithm of the divifor, which
mud be fiibtrafted from its index, becaufc it is negati\'e ;
then it becomes — i ; changed it is -|- 1 j then the figos are
different, and their difference is — 2.
Ex. 9. Divide -9072 by •0084.
Lo^. -9072 = —1-9577030
Log. -0084 = --3-9242793
Quotient 108 log, z=. -f 2 •0354237
Ex. 10. Divide •4606 by '0094.
Log. -4606 zr — 1*6633239
Log. -0094 == —3*937^279
Quotient 49 log. = +1*6901960
Ex. II. Divide '001404 by 'ooriy.
Log. -001404 = —3*^73671
Log, •00117 = — 3*0681859
Quotient 1*2 log. = 0*0791812
Ex, 12. Divide '3515 by •37.
Log, '37 = —1*5682017
Quotient '95 log, = — 1*9777236
Examples fwherein the indices of the dividends are qfirmathV4,
and thofe of the divifors negative,
Ex. 13. Divide 6543*9 by '27863.
log. 6543*9 = +3*8158366
Log, -27863 = —1*4450279
Quotient 23485*97 kg, r: +4*3708087
14 A Sywopsis Of
Ex. 14. Divide 54498 by '09 5.
Log' 54498 = +4*7363806
Log, -093 = — 2-9684829
Quotient 586000 log, == +5*7678977
Ex. 15. Divide 83-1*4 by -00112.
Log. 834*4 =z +>. •9213743
Log. •001 12 zz — 3-04.92180
Quotient 745000 Jog. z=z +5*8 7 21565
Ex. 16. Divide 19-206 by '0194.
Log, 19*206 = 41*2834369
Log. '0194 = —2*2878017
Quotient 990 log. = +2-9956352
Ex. 17. Divide 9?5*68 by -oSSS.
Log. 985-68 = +2*9937359
Log, -0888 := — 2*94.84130
Quotient 11100 /(3g-. =z +4-0493229
Ex. 18. Divide 9890*1 by •00999.
Log, 9890*1 ~ +3-9952007
Log, -00999 = — 3*9995<^55
Quotient 9900CO log. zz +5*9956352
From the 6 preceding examples we may infer, that, when-
ever a logarithm with a negative index is to be fubtrafted
from one with an affirmative index, if there is nothing to
carry from the decimal part, the index to the logarithm of
the difference will always be equal to the fum of the indices ;
and, if there is i to carry, it will be equal to the fum lefs i,
^d in both cafes affirmative*
LOGARITHMICAL ARITHMETIC. I5
Zxamples 'wherein the indices of the di-uidtnds are negat've,
and thofe of the divifrs affirinativt.
Ex. 19. Divide 11^ by ^\\^.
Leg. \\^ = —1-6930450
Log. -\\- =: +2'0-oiSco
Quotient '00419628 log. r= — 3-6228650
Ex. 20. Divide 1- by 7^°-
Leg. 4 = — 1-744727^
Log. 75^00 _ ^^.Q„,g.,3
Quotient •0005^92' log. = — 4*7727562
Hence, by a comparifon of thefe examples with fome of
the preceding, we may inter, that, if a:W logarith:n with an
atfirmative index is to be fubtracted i'rom one with a nega-
tive index, the index to the iogarithm of the diftereiice ^yill
always be equal to the fum of the indices, if there is nothing
to carry from the decimal ; but, if there is i to carry, it will
be equal to the Turn more i, and in both cafes negative*
Proposition ^.
71? involve a gi'ven nwnher to any poiver.
Rule.
Cafe I. Wheu the index to the logarithm of the frfi pi^vji^^
or ront, is afirmnti--ve, {or -f-.) Multiply the l:)^arithm of
the root (rejedino- its Index) bv the index of the po\ver re-
quired, and referve the produ<!t ; then mmtiply the md:x ot
the logirithm by the index of the p'.)w^'r ; this prod uilit, added
to the former, will give the logarithm of the power required,
witia its index always affirmative.
CaHj 2. IVhen the index to the logarithm ^f the fir ft p<nver,
or root, is 7icgative [sr — .) Multiply th^; logarithm of the
root (rejeOi'ling its index) by the index of the power required,
and referve the product, which will always be affirmative;
the'i multiply the index of the logarithm by the index of
the power; this produ(^; which will always be negative,
muft he fubtraa-d from the former to obtain the logaritlun
of the power required.
i5 ASynopsisof
Note. In fubtrafting the above produifls, when you
come to the indices, or whole numbers, their difference mull
be taken, and the fign of the greater (which will generally
be negative) prefixed, for the index to the logarithm of the
power. If there be an unit to carry from the decimal part
of the lower produ«5t, it muft be added to the index of that
produdl before you take their difference, as above.
I . Examples ivhcreia a 'whole number or mixed decimal is in-*
'vol'ved to any ponx'er.
Ex, I. Innjolve \\"]Z ta the third pQ^joer,
log, 14*72 = + 1 •1679078
Index of the power 3
Produfl of the decimal -f 0*503 7 2 34
+ 1x3 = -f 3 prod, of the index.
Power 3189*506 %= +3*5037254-
VIZ. 147721^=3189*506
Ex. 2. Required the '^(i^th poiver of VOO^S"]*
Leg, 1*00567 zr -|- 0*0024555
Index of iLe power ^6^
Pfodudl of the decimal +0*8962575
+ X 365= -fc prod, of the index.
Power 7*875125 log. =1 -|-o*8962575
viz. 1*00567]^*' =7*875125
Ex. 3. IfTJohe 1*05 to the ^oth po-ivert
Log, 1*05 =: 0-0211893
Index of the power 40
Produft of the decimal +0*8475720
+ X40 =. -fo* prod, of the index.
Power 7-0399887 log. zz +-0-8475720
Viz. ro5l+° =: 7-0399887, the amount of iL for 40
years at 5 per cent, compound intereft.
Log AR iTH MIC A L Arithmetic, 17
Ex. 4. Required the $"] ^ pwer of i^"ig.
Log. 14-79 = +1-1699682
Index of the power 3*75
Produft of the decimal +0*6373807
+ I X3-75 = +3*75 prod, of the index.
Power 24399-49%. =: +4*3^73807
Viz. 14-79] 3. 7 5 =24399-49 = 1479I**
Ex. 5. Required the 1*26' ponver of ']Z^,
^°S' l^S ^^ +2-8948697
Index of the power i '26'
Produft of the decimal + 1 'i 33 5:0 1 6
+ 2 X 1-26' = -^2'S333333 prod, of the index.
Power 4643*387 %. == +3-6668349
Viz.l87l«-*<^' =4643-387 =7'8?|H
Ex. 6. Required the '075 ponver <?/' 147*5.
Log, 147*5 = +2*1687920
Index of the power -075
Produft of the decimal +0*0126594
+ 2 X *075 z= +0-150 prod, of the index.
Power 1*4543 ^og' = +0*1626594
Viz. 14^51 -^7 5-- ,.4^42 =i47*5i4:V
Ex. 7. Required the *3^'s^' p^^'^t^ of 94*75'*
^^'g' 94*75' = U' ^W = +1-9766047
Index of the power '34' 5^'
Produft of the decimal +0*3373636
+ 1 X-34'54' = +o-34'34V54
Power 4«8i736 lag, = +0-6828090
Viz. 9f77i'^*'**' =4-81736 =^4*771^5inK
iS A Synopsis o *
2. Examplei luherein a pure decimal is invol'ved to any ftr.Ki<.
Ex. 8. iKvoh'e '0725/0 the third piywer^
Log, '0725 = — 2*8603380
Produft of the decimal -|-2'58ioi40
— 2x3=: — 6* prod, of the index,
Power •000381078 log, •=. — 4-5810140
Viz. •0725] = -000381078.
Ex. 9. Required the 6'2!^ /cnuer of 'co^t.
Log, -0032 = — 3*5051500
^
Prodttfl of the decimal. +3*1571875
— 3 X 6*25 = — 1875 pr.ofth<f
(index.
Power '00000000000000025538 /^^.=: — 1 6-407 1 S75
Viz- -0032/-*^ = -0000000CO000CCC25 ^38= •C032V*
Ex. 10. Required the -625 ponver cf 'OO^l,
Log, -0032 zz — 3-5051500
•625
Produft of the decimal +0-3 157 1875
— 3 X -625 = — I '875 prod, of the index.
Power -0275879 /<?^. = —2-44071875
Viz. -0032 -^*' = '0275879, Arc. =: •0032,*"
Ex. l.r. Ini.-ol've -09475' to *34'54* ponver.
Log. -09475' = %. jViT = —2-9766047
*34 H
Prod u ft of the decimal +0-33736364
— 2 X •34'54' = — o-69'o-'(;o89o8, Scq,
Power '4^.0; log. n: — 1-6464727
Viz.^^^^-^-"^' = -++307 = ■5VA,"»'=
LoGARITHMlCAL ARITHMETIC. I9
Proposition 6.
I'c cxtra8 anj root of any given po'wer.
Rule. Di\ ide the logarithm of the power to be extrafted
by the index e::prefling the root, the quotient is the loga-
rithm of the root. — If the index to the logarithm be negative,
and does not exactly contain the divifor, increafe it by fucb
a number as will make it exadly divifible, and increafe
the logarithm alfo by the fame number, before you begin to
divide »
Note. The method of increafing the -negative index is bjr
adding equal numbers to the negative and affirmative parts;
thus, the logarithm of '0711 in — 2*8 5 18696 is equivalent
to — 3 + i'S5i8696, or = — 4-f-2'85iS696, or zz — ^
-f-3'8^i8696, or it is = — 365-f 563'S5i8696, 6cz. where
the latter part is entirely affirmative : hence the fquare, cube,
fourth, fifth, 365th, or any other root of 'O'jii., or any
number, may be readily extract ji^
I. Ex<imples njjherein the poucer is a <vjho!e number or mixed
dccimah
Ex. I. What is the/quare-root of ^?
Log. of 3 rz -1-0*4771213 ; this, divided by 2, the index,
gives 0*2385606 for the log. of the root, the number an-
fwcring to which is 1*73205.
Ex. 2. Extras the cube-root o/* 3189*505.
Log. of 3189*506 z= +3'5037234; this, divided by 5,
the index, gives 1*1679078 for the root, the number an-
fwering to which is 14*72, the root required.
Ex. 3. Extra<flthe^6^throotofyS']p2^,
Log. 7*875125 =: -|-o*8962574; this, divided by 365,
the index, gives 0*0024595 for the logarithm of the root,
the number anfwering to which is 1*00567 ;
Viz. 7*875125']tJt = i •^00567.
X
20 A b Y N' O r S I S OF
Ex. 4. ExtraH the yj ^ root cf 24399'4g.
Log. 24.399'49 = +4*3873807, which, divided hy 3-7$'.
gives 1-1699681, the logarithm of the root, the number an-
iVering to which is 14*79 ;
Viz. 24399-49~lrTT = 14-79 = 2^J99^i*T
Ex. J. Extrad the 1*26' root 0/ j\.6j{.y^S-j.
Log. 4643*387 = +3*6668348, then 3-6658348 + 1*26'
rz: 3*66683480 — -36668348 + 1*26 — -12 r: 3-3001 9 132
-^ I '14 = 2-894869 J, log. of the root, the niimber an-
fwering to which is 785, extremely iiear^ the fmall error ari-
fing only from the imperfedlion of the logariihms;
Viz. ^Hy^%-l\-^'\-^ — 78; zz 4643*3^^'^
Ex. 6. Extracl the 'O"] ^ root of 1*4^4318.
'Log. 1*45^4318 zi +0*1626^93, which, divided by •075',
gives 2*1687910, log. of the* root, the number anfwcring to
which is 147*5 "(^^ffj'i
Viz. i-4543i'8l-^Ti = i47'5 = 1 '4543^^1 V
Ex. 7. ExtraB the "i^4^ !^Ji^ root of /^'%l'] -^6,
Log. 4-81736 r= +0-6828091 ; then 0*6828091 -f-
•34'54' rz '6821262909 +■ '3451 = 1*9766047, log. of the
root, the number anfwering to which is 94*75'. f^ide Ex, 7,
in lu'volutmu
Viz. 4-8i736i*3-4/T4^=:94-75'=: 4^81736] "J^tt
2. Examples ^wherein the panver is a pure dccirnaL
Ex. 8. Extract the fquare -root of *02^.
Log. '025 = — 2-3979400, which, divided by 2, the
index, gives — riQ89700 for the log. of the root, the num-
ber anfwering to which is -1581 1.
Ex. 9. ExtraB the cuhe-root of '000381078.
Log. .000381078 =: —4*5810139 = —6 +2*5810139;
this, divided by 3, the index, gives — 2 + '8603379 =
— 2 "8603 3 79, log. of the root, the number anfwering to
which is '0725 uearlj ;
Viz. •000381078)1 = '0725.
L G A R 1 T H M I C A L A R I 1 H M E T J C 21
Ex. 10. Required the 6*25 root of '0000000000000002553804.
Log. '0000000000000002553804 = — i6'J^07i875 =r
— 31-25 -f 15-25 + •407JS75: = T^^'^'i + rr657i«75 J
this, divided by 6-25, the index, gives — 5 -("''S^S • 5^<^ ==^
■ — 3-50,' 1 500, log. of the root, the number anfwerin^ to
whieh is 'oo'^ 2, the root required.
Ex. 1 1. Extras the '62^ root of ^02758791.
Log. -02758791 = — 2*4407 187 = —18-75 + 1.^*7 J +
•4407187 s= — 18-75 + n* 1907 1 87, which, divided by
•625, the index^ gives — 30 -f 27*5051500 ~ — 3-5051^00,
log, of the root, the number anfwering to which is -0032,
the root required.
Ex. 12. Required the •l,\<^)J r.ot cf'xi^lo-^Of^,
Log. -4430705 "=■ — 1-6464727 zz -~3-4'5+' + 2 4^5+' -f-
•6464727 = — 3-4'54'-|-3-ioo92 7i5'44', which^ divided by
•34'54^ the index,( = 3-45] +3-0978262273 -f- -3451), gives
— 10+8-9766045 = — 2-9766045, log. of the root, the
mimber aniwering to which is '09475' [extretnely near), the
loot required.
Having now given examples of all the varieties that can
pofiibly happen m the practice of involution and evolution,
where the index of the power is a whole number, or a mixed
or pure decimal, I (ball add but one obfervation more to-
wards rendering the practice complete, viz. that, whenever
we meet with an expreffion with a fradional index, as 475!^,
the numerator denotes the power to which the root is to be
raifed, and the denominator (hews what root of that powtr
is to be extraded.
In fome of the preceding examples in involution the powers
are lefs than the roots, and in evolution the roots are greater
than the powers ; fhould the critic be difpofed to cenfure the
naifiC 1 have given them by aflcrting that '* powers are properly
dilHnguillied by the elevation of roots, or involution ; and
roots by the depreffion of powers, or evolution ;" 1 have-
only to reply, that I may as properly put fuch examples undtr
the above titles, as the product, or quotient, of a pure de-
cimal can be placed under the title of multiplication or di-
viiicn.
X 2
22 The construction of Decimal Ta-bles
The application of LOGAPvITHMS to
COMPOUND INTEREST,. crV.
A .principal ufe of Logarithms, as the celebrated Dr.
Ilariey obferves, is to folve all the cafes of Compound In-
tereft, which are not v.ithout great difficulty attainable by
the rules of common Arithmetic. — In the Arithmetician I
have given Logarithmical theorems for all tlie different cafes
of Compound intereft and Annuities; I now proceed to the
conftrudlion of Decimal Tables at 3, 'i,\, 4^ 4^, and 5 per
cent. Compound Intereft, by means of which the ajifwer to
mo'l: queftions in Compound Intereft and Annuities may be
readily obtained.
The cojrJiruSIion of Table I.
/-:j. i'03z:o*or2S372 /o^. of the amount of il. for i year.
0-0128372
I -0609 - 0-0256744 log. of the amount of il. for 2 years.
•092727 r::o-0385i 16 /^^. ofthe amountof il. for 3 yrs. kc.
The CO ijlruilion of Table II.
The amount of il. for a Atcj at 3 per cent, will be the
36cth root of 1*03, by the univcrfal theorem, Pfage rzth, of
I. •. .irilhmeticiaJi.
L'jg. I 03 .— 0*0128372, this -^ -x^d^y giv^slog. of the
root, = 0*0000351, the number anfwering to which is
1-0000:09, the amount of il. for 1 day.
Then 0*0000351
0-000035 [
o'ocoo702, the number anfwering to which is
1-0001619, the amount of il. for 2 days, &c. &c.
The confiru^iion of Table III.
The logs, of the fereral amounts in Table I. fubtraf^ed
from an unit, leave th* lo^>, of the prefent worths in this
table.
FOR Compound Interest. 23
fable. Thus i — •0256744.= — 1-974325:6, the number
anfwering to which is '9425959, the prefent worth of il,
for 2 years, at 5 per cent, &c.
T^e cor^Jiruakn of Table IV.
This table fhews the amount of il. a-nnulty, and is cor-
^ruded from the firft table thus,
I '0000000
1-0300000 amount of il. for r year at 3 per cent.
2*0300000 amount of il. annuity for 2 years.
1 '0609000 amount of the feeond year.
3*0909000 amount of il. annuity for 3 years, 5cc,
"^he conjiruahn of Talle V.
Thi? tabic (hews the prefent worth of i-l. annuity, and^
is conitru^d thus,
Firft year, table III., at y per cent, is '9708738
Second year, ib. — — '9425959
Sum.. Second ycar^ table V. i-9i34/)97"
Third year, table III.. -(^i^i^iy
Sum- Third year in tabic V- &c.. 2*82861 14,
Th€ conflruaimi of Table VI.
This table is eonftruCled from the 5th table by fubtracling
the log. of the prefent worth of il. from an unit;, thus the
log. of 2*82861 14 is 0*4515732, this fubt rafted from i leaves
-=- 1*5484268, the number anfwering to v.hich is •3535304y
the annuity which il, will purchafe for 3 years at 3 per
cent,
X 3
TABLE I.
The amount of il. for years.
•r.^i ■; per cenc. ] 3^ uer Cent.
I|l .0300000
2jI.o6o900C^
I 0927270
[255088
1.194052;.
229873^
1. 266770c
^•304773^
^•3439^63
1-384233^
1. 425760c
1.4685337
.5125S97
1.557967.1
i.6o47o6.:j
I 6528476
1.702433:
1.753506c
1. 8061 1 1:
1.0350000
1. 0712250
1.0871780
1.147523c
I 1876865
'29-555
1.2722791
1. 3 1 6809:
I 362897;
1.410598-
^•4599^:
1.51 1068^
1.563956c
I 61S694;
^•67534^^
1.733986.
1.794^755
I 857489:
I 92250J '
1.989708;-
I 8602945
1.9 16 1034
1.9735865
2.0327941
2-093777^
2.156591
2.221 2 89'j
2.287927
2-3565655
2 427262
2.5000803
2.575082-^
2-652335
2.7319055
2.8138624
2.8982783
2.9852266
3.0747834
3.1670269
3.262037;
4 per ^,^{\i
2.059131.
2 . I 3 I 5 I I C
2.2061I4J
2 283328.
2-363244'
2.44595b
2.531567]
2.62CI7IC
2.71 1877c
2.8067937
2.90503 u;
3.0067075
3.1 II 9425
3.2208603
V3335904
3.4502661
3.5710254
j. 6960 II
3-82537^7
? 9592597
1 .040000c
1 .0816000
1.124864c
1. 1698586
1.2166529
1.265319c
315931*^
3685691
4233 1 1 c^
1.4802445
1.0450000
1.092025c
1. 141 1661 I
1. 1925186 I
I 2461819
1539454^
1.601032:
1 6650735
1.7316764
1.8009435
4|perCenr.
I 3022601
360861:8
I 4221006
I 4860951
ij_i29^9.
1.6228530
1. 6958814
1.7721961
I 8519449
1.9352824
1.8729812
1 9479005
2.0258165
2.106849:
2.1911231
2.3699188
24647155
2 5633042
2.665 8363
2.7724697
2.8833685
29987033
3.1 1 865 14
3-243397 5
3-3731334
3 50B0587
3.6483811
3-7943^63
3.9460889
2.022370
2.1133768
2.2084787
2.3078603
2 41 1714c
2.520241 1
2.633652c
7521665
.876013^
3.0054344
3.14C679C
3.2820095
3.4296999
3.5840364
3-7453iSi
4-1039325
4 2680898
4.4388134
46163659
4.8010206!
5 p'. rCei.t.
050OOOC
.IO250OC
.157625^
.2155063
.2762816
.3400956
.4071004.
•4774554
•55132
.6288946
•7103393
7953563
.8856491
.9799316
0780282
3-9138574
4.089981c
4.2740301
4.4663615
4^7147^
. 37737"84
5.0968604
5.3262192
5.5658990,0
.1828746
.29^018^
.4066192
.5269502
•6532977
7.^59626
.9252607
.071523!
2251000
3^63549
•7334563
9201291
1161356
3219424
•5380395
7649415
0031885
.2533480
.5160154
7918161
0814069
3854773
704751
039988-
TABLE II. The amount of i1. for davs.
3 \ er Cent.
1 .OOO0B09
1 .00016 I 9
I 0002429
c.ooo324'j
I 0004050
1.000486c
1.000567c
1 .000648c
1 .0C07291
1 .0008101
3-^ v-crCcnC.
1.0016209
1.0024324
1.0032445
1.0040573
I.OO4870S
7
80
90
160
no
120
3
14c
150
1 6c
1-0000942
1 .0001 885
1.0002827
I 0003770
I 0004713
I 0005656
1 .0006600
1.0007542
I 0008486
1.0C09420
i.ooi'8867
1.00283
1.0037771
1.0047236
1.005671C
4 p;r Lent
I.OOCIO74
1. 0002 I 49
1.0003224
I 0004299
1.00053
1 .0006449
1.0007524
1 .0008600
1.0009675
1. 00 1 07 5 1
4.-^ [:er Cent.
l-OOOI2C6
I.COO24I2
1.0003618
I.COO4824
1.000603
1.0007238
1.000844
1.0009652
I .0010859
1.0012066
5 per C.:u
1.0056849
1.0064996
1.0073151
1.008131 1
1.0089479
1.0097653
1.010583^
1 .01 1402'!
1 .012221 c;
1.0130415
17CI1.6138623
1.0146.837
1.0155057
I. 0163284
1.017151S
18c
19c
20c
210
22c
23c
240
25e
26c
28c
29c
SO
31c
32c
34'^
36
1.0179759
1.0188006
1.0196260
1.0204520
1. 0212788
1.OC66193
1.007^685
1. 008 j 1 86
i .0094696
1.C104214
I 0021513
1 .003228b
1.0043074
1.0053871
1.006468c
1.0001336
1.0002673
1 .000401 1
1.0005348
1 .000668;
1.0008023
1.00C9361
1. 00 1 0699
1 .001203}
1. 001 3376
1 .0075501
I 0086333
1 .0097 1 77
I oio8®33
1. 01 1890C
1 .0024148
1.0036243
1.0048354
1.0060479
I 0072618
rToc84773
1.0096942
1 .0109125
1.0121324
I-OI33537
1.CI1374
1.0123279
1.0132825
1.0142379
1.0151945
1.0129779
1.014067c
1 .01 5 1 572
1.0162487
1.0173412
1.0221062
1.0229342
1.0237630
1 0245924
1:0254225
1.0262532
1.0270847
1. 02791^68
1.028749
i.o29q83c
1.0161516
1.C17109S
1.01806S9
1.0190288
I. 0199897
I. 0209515
1. 02 1 9 142
1.022877S
1.0238424
1.024807S
1.0257741
I 0267414
1.0277096
1.0286786
1.0296486
1. 0306195
1.0315914
1.0325641
1-033537^'
lo^^i^iz
1.002677c
I.OC40I82
1 .005 36 1 1
1.0067059
1 0080525
1.0094009
1.010751 1
1.0121031
1. 0144565
1.0 148*25
1.0145765
1.0158007
1.0170265
1.0182537
1-0194824
1.018435c
1. 019.5299
1.0206261
I.02I7233
1.0228218
1.0239215
1.0250223
1. 026 1 243
1.0272275
^ 02833^9
1.0294375
IC305443
1 0316522
I.03276I4
I 0338717
1 0349832
1.036596
1.0372099
I 038325
1.0207126
1.0219442
1.0231774
1.0244120
I.02564S1
1 .026885S
1.0281249
1.0293655
4.0306076
1.03I85I2
1.0330963
I 0343429
I.03559IO
1.0368406
1.038001
i 0393444
1.040598.5
i 0418542
1.0431114
1.03944131-1 C44370C
1 .0161699
1.0175291
1 .01 88902
1.0202531
1.0216178
1.0229843
1.0243527
1.0257228
1.0270949
1.0284687
1.0312219
1.03^6013
1. 0339^25
I •0353 -^56
1.0367^05
1-0381373
i -^^395^59
1.0409164
1.0422087
1.04^629
1.0450900
1 .0464969
1.0478967
1 0402084.
TABLE IIT,
The prefcnt worth of il. for years.
I
3 pciCenc.
.9708738
3ipeiCe.nc.
4 per Cent
41 per Cent.
5 per Cent.
.9661836
.9615385
.9569378
.9523809
2
.9425959
.9335107
9245562
.9157299
.9070295
3
•9i5H'7
.9019427
.8889964
.8762966
Mi^i-je
4
.8884870
.8714422
.8548042
.8385613
.8227025
5
6
.8626088
.8419732
.8219271
.8024511
.7835262
.7462154
•8374843
.8135006
.7903145
.7678957
7
.8130915
.7S5992O
.7599178
•7348285
.7106813
8
.7894092
.7594116
.7306902
.7031851
.6768594
9
.7664167
•73373IC
.7025867
.6729044
.6446089
lO
II
•744^939
.7224213
7089188
.6755642
•0439277
.6139133
.6849457
.6495809
.6161988
.5846793
I 2
.7313799
.6617833
.6245971
.5896639
.5568374
»3
6809513
.6394041
.6005741
.5642716
.5303214
H
.6611178
,6177818
•5774751
•5399729
5050679
^5
i6
.6418619
.6231669
5968906
.5552645
.5339082
.5167204
.4810171
.5767059
.4944693
.4581115
.171.6050164!
.5572038
•5133733
.4731764
•4362967
18
.5873946
.5383611
.4936281
.4528004
.4155207
19
.5702860
.5201557
.4746424
4433018
•3957340
20
21
•5536758
.5025659
.5855709
.4563870
.4146429
3768895
•5375493
•4388336
.3967874
.3589424
2;i
.5218925
.4691506
.4219554
.3797009
.3418499
23
.5066917
.4532856
.4057263
•3633501
•3255713
H
•4919337
'^119'^!'^
.3901215
•3477035
.3100679
25
26
.4770056
.4231470
.3751168
.3606892
.3327306
.2953028
.4636947
.4088378
.3184025
2812407
27
.4501891
.3950123
.3468166
.3046914
.ze-jz^'^oi
28
.4370768
.3816543
^^^^h^lls
.2915707
.2550936
2r>
.4243464
.3687482
.3206514
.2790150
.2429463
1°
31
.4119868
.3562784
.3083187
.2964603
.2670000
•2313775
.3999871
.3442304
.2555024
2203595
J2
.3883370!
.3325897
.2850579
.2444999
.2098662
33
•3770263.
.3213427
.2740942
.2339712
.1998726
34
.3660449
.3104761
.2635521
.2238959
.1903548
35
36
•3553834
•34503^4!
.2999765
.2898327
•2534155
,2436687
.2142544
.2050282
.1812903
.1726574
37
•3349829I
.2800316
.2342969
.1961992
.1644356
38
.3252262!
.2705619
.2252854
.1877504
.1566054
39
•3157536I
.2614125
.2166206
.1796655
.1491479
40
.3065568-
.2525725
.2082890
.1719287
.1420457
TABLE IV.
The amount of il. per annum^ or annuities for years.
3 per Cent.
n ?«-•-'■"
I ooooooo
2.0300000]
3 0909000J
4 1836270]
J-J2211SS
6.4684090
7.6624622
8.8923360
^10.1591061
lOji 1 .4638793
12 8077"^
-_ 14.1920296
13^5-6177904
I4I17. 0863242
IJ
16
17
18
»9
^^•5989139
20 1568813
21.7615877
23 4H4354
25.116868^
1 .ooooooo
2 O35OOOC
3. 1062250
42149429]
5 362+659
4 per Cent
1 OOOOOOO
2 0400000
3.12 16000
4.24.^464
5.4163226
6 5501522 6.6329755
7-7794075 7-8982945
9.0516866 9.2142:63
10.36S4958 10582795.
II. 7313931 12.0061071
13.1419979 13.4863514
14. 6019616 15.0258055
16.1130303 16.6268377
17.6769864 iS. 29191 12
1 9 29 56809 20.0235 ^
20. 9710297J21. 8245311
22.7050i58j23-6975i24
24.4996913 24.6454129
26.3571805 27.67 12294
20-26 870.3745 28 2796818129.7780786
21128.6764857 30 269470713 1.969201
22j3o»5367.So3l3 2.3289022!34-24r^698
23132.4528837134.4604137136.6178886
2 r_, __■ ._' _'_'j ._^^
■ i. 04. 5:0003
4^ per Cent.
5 pir Csot.
1 .OOOQOOOl
2.0450000!
3.1370250'
4 278191 1
5.4707097
I. ooooooo
2.050COO;
3.1525000
4.310125c
5.5256312
6.7i689!7l
8.0191 58! I
9.3800156;
10.8021 142'
12.2S82094
4I34. 4264702 36 6665282139 082604
2 5 1 3M5 9i^3 j 3^-949^567! 4i-6|5 9o"
26|3a.553o.;..22;4i 3131017144.3117446
27'40.7096335|43.759o6o2i47-o^42i44
2.^142.9309225
29:45.2188502
30 47_5754i57
3 IJ50. 0026782
^027585
34s
46.29o6273;49-96'^85<
48.9107993 52.9662863
51. 6 226773: 56.0 849-^77
54.4294710 59-32833'i2
57-334502562 7oi4C)87
3155.0778413:60 3412101 66 2095274
34157-7301765163.4531 524169.8579085
35!6o. 4620818 66.6740127173 6522248
3^|63-27T9H3
37|66. 1742226
"^'169.1594493
39!72. 2342327
4075.4012597
70.007603 2! 77 -59^3 '3'^
73 4578693IS1. 7022464
77. 028894-J85.9703362
80.724906c 90.409 1 497
84.5502778I95 0255[57l
13.8411788
15.4640318
17 1599133
18.9321094
.20 784 0543
22.7193367
24.7417069
26.8550837
29.0635625
31. 3714228 I
337^3'368
363033779
38.9370299
41.6891963'
44.5652101'
6.8019128
8.142C084
9.5451089
1 1.0265643
12.5778925
14.2067871
15.9171265
17.7129828
19.5986320
21.5785636
47.5706446
50.7113236
53-9;33332
57-4^30332
61.0070697
64 7523878
68.6662452
72 7562263
77.0302565
81.49661 Ko
80.1639658
91.0413443
96.13S2048I
01.4644240:
07.030323 I'l
23.6574918
25.840366^
28.1323847
30.5390039
33-065954
35.7192518
38.5052144
4>-430475»
44.5019989
47.7270988
51.1134538
54.6691265
58 402582^
62 3227 J ig
_66__4388475
70.7607899
75.2^88294
80 o63:'7oS
85 0669594
90 3203073
95.8363227
01.6281388
07.7095458
14.095023;
20.799774-
TABLE V.
TI\e prefcnt worth of il. per annum, or annuity fpr years.
yr.- 3 per Cent.
3i per Cent.
4 per Cent.
4^ per Cent.
5 per Lent. 1
0.9523800!
I.8594JC '
2. 7232. >
3-54593^
4-329470/
1 0.9708738
2 I.9I34697
3 2.82861 14
4 3.7170984
J 4-5797072
6 5.4171914
7 6.25C2S29
8 7.0196922
9 7.7861089
10 8.5302028
11 9.2526241
12 9.9540040
13 10.6349553
14 1 1. 2960731
Li '^-9379351
0.9661 836
1.8996943
2.8016370
3-6730792
4.5150524
0.9615385
1.8860947
2-7750910
3.6298952
4.4518223
O.956937I
1.8726678
2.7489644
3-5«75257
4.389976;
5-3285530
6.1145439
6-8739555
7.6076865
8.3166053
9.0015^10
9-6633343
10.3027385
10.9205203
IJ.5174IC9
I 2. 0941x68
12.6513206
I3.1896817
13.7098374
14.2124033
14.6979742
I 5.167I248
15.620410;
16.0583676
16.4815146
5.2421369
6.0020547
6-7327448
7-4:533'4
8.1 1089;; 5
8.7604763
9-5850733
9-9^56473
[0.5631223
11.1183868
5.157872s
5.8927C09
6.595SS61
7.2687905
7.9127182
8.5289:69
9. 1 1 85808
9.6820524
10.2228253
^0.7395457
1 1.234c I 51
11.7071914
12.1599918
12.5932936
13.0079365
5.07; 69 21
5-7^63734
6.4632128
7.1078217
7-7 2 17349
8.3064142
8.8632516
9-3935750
9.8986409
10.3796580
10.8377695
11.2740622
1 1.6895869
12.0853208
12.4622105
16 12.561 1020
17 13.1661185
'8 ^3'7S3Sn^
'9 '4-3237991
20 14.877474F
11.6522949
12.1656680
12.6592961
13-13393^5
13-5903253
14.029158c
14.451 1142
1 4.8 568405
15.2469619
15.6220787
21 15.41 50241
22 15.9369166
23 16.4436084
2416.9555421
25 17-4131477
13.4047239
13.7844248
'4-1477749
14.4954784
14.8282089
15.14661 15
15.45,3028
15.7428735
16.0218885
16.2888885
12.8211 527
13.163002'
13.488573. '
13.7986418
14-0939445
26 17.8768424
27 18.3270315
28 18.7641082
29 19.1884546
so 19.6004413
31 20.00042 8 ^
3220.3S8765S
33 20.765791S
34 21.1318367
3r 21.4S722CO
36 21.8322525
37 22.1672354
38 22.4924616
39 22.80S215]
4c 23.1147719
16.8903523
17.2853645
17.6670188
18.0357670
18.3920434
18.7362758
19.0688656
19.3902082
19.7006842
20.ccc66i 2
15.9827678
16.3295844
16.663061 b
16.9837132
17.292031 S
'4-375^853
14.6430336
14.8981272
15.1410735
15.3724510
17.5884921
17. 873550c
18.1476441
18.41 1 1962
I 8.66461 I 6
16.5443909
16.7888909
17.0228621
17.2467580
17.4610124
15.5928104
15.8026766
16.0025491
16.1929039
16.3741942
2o.290493h
20.5705254
20.8410874
21.1024990
21-3550723
.8.9082S03
19.1425771
19.3678625
19.5844831
19.7027721
17.6660406
17.8622398
I 8.0499902
18.2296557
18.4015844
16.5468516
16.7112872
16.8678926
17.0170406
17. 1 590867
TABLE VT.
The annuity which il. will purchafe for any number of years.
yrs
3 pc. C Bt.
I .0300000
3i per Cent.
1.0350000
4 per Cent.
1 .04000.00
4.{- per Cent.
5 perfent.
1.0450000
1.0500000
2
.5221)108
.5264005
.5301901
•5339976
•5378049
3
-3535304
•3569342
•3603485
•3637734
.3672086
4
.2690271: .2722511
.27549^.1
•'2-l'^l\ll
.2820118
5
6
.2I83S46; .2214814
2.' 46271
.1907619
2277916
.1938784
.2309748
.1970175
.IS45975] .1876682
7
.1605064' .1635445
.1666096
.1697015
.1728198
8
.1424504! .1454767
1485279
.1516097
.1547218
9
.12^4339
.1314460
.1344930
•1375745
.1406901
IC
.1172305
.1202414
.1232909
.263788
.1295046
u
.1080775
.1110920
.1141490
.1 172482
.1203889
I2
.1004621
.1034840
.1065522
.1096662
.1128254
13
.0940295
.0970616
.1001437
.1032754
.1064558
H
.0885263
.0915707
.0946690
.0978203
.1010240
IS
16
.0837666
.0796109
.0868251
.0899411
.0858200
.0931138
.0890154
.0963423
.0922699
.0826848
17
•075^525
.0790431
.0821985
.0854176
.0S86991
18
.0727087
.0758168
•0789933
.0822369
.0855462
'9
.0698139
.0729403
.0761386
.0794073
.0827450
20
21
.0672157
.070361 I
.0735818
.0712801
.0768761
.0802426
.0648718
.0680366
.0746006
.0779961
22
.0627474
.0659321
.0691988
.0725457
•0759705
23
.0608139
.0640188
.0673091
.0706825
.0741368
24
.0590474
.0622728
.0655868
.0689870
.0724709
^5
.0574279
06c 674c
.0640120
•0674390
.0709525
26
.0559383
.0592054
.0625674
.0660214
.0695643
27
.0545642
.0578524
.0612385
.0647195
.0682919
28
•0532932
.0566027
0600130
.0635208
.0671225
29
.6521147
.0554454
0588799
.0624146
.0660455
3?
.0510193
/05437^3
.0578301
.G613915
.0650514
31
.0499989
•0533724
.0568554
,0604435
.0641321
32
70490466
.0524415
.0559.^86
.0595632
.0632804
33
.0481561
.0515724
.0551036
.0587445
.0624900
34
.0473220
.0507597
.054314*^
.0579819
.0617554
35
36
•0465393
.0499984
•0535773
.0572705
.0610717
.0458038
.0492842
.0528869
.0566058
.0604345
37
.0451116
.0486133
.0522396
.0559840
.0558398
3^
.0444593
.0479821
.0516319
.0554017
.0592842
39
0438439
.0473878
.0510608
.0548557
.0587646
40
.0432624
.0468273
.0505235
•O54343J
.0582782
30 The use of the Decimal Tables.
The ufe of the preceding Talks,
TABLE I. AND II.
Proposition i,
GI-VC72 the pHncipaly rate, and timey to find the amount.
Rule. Multiply the amount of il. (in table I or II.) at
tlie rale and for the time given, by the principal, and the
produd will give the amount.
Note. If the amount be required for any number of year:-
or days exceeding thofe in the tables, divide the given num-
ber of years or days into two or more fuch numbers as are in
the tables, and multiply the amounts anfwering thereto into
one another continually, and the laft produft by the prin-
cipal, which will give the amount required.
Example. Vvhat will 2Col. amount to in 6 years, at j
per cent per annum. Compound Intereft. Vide Ex\ I.. Co?n-
pound Intereji.
Agaiaft 6 y. under 5 per cent, ftands 1.3400956 amount of il
200 principal.
a681' OS. 4{d. ~ Prod. ^T. 268'Oi9i2oo amountrequired.
In a fimilar manner the amount of 70I. for 40 days, at 5
per cent, will be found to be 7o*37(j277l. per table II. And
the amount of 82-51. for 75 years, at 5 per cent, will be
found, by the note, to be 3203*69661.
Observation. Anfwcrs to qnellions in any of the other
three propofitions in Compound Intereft, may be eafily ob-
tainted by "a little attention to the preceding Rule and Ex-
amples.
TABLE III.
Proposition.
Any fum of money , due fame time hence , being given, to find
its prejent Kjalne to the creditor y difcounting at any rate per tent.
Compound Interefi,
Rule. Find the prefent worth of il. at the rate and for
the time given (by tabic III.) which multiply by the debt,
aad the piodud will be the prefent worth required.
The use of the Decimal Tables, 31
Example. What ready money will difcharge a debt of
243I. 2^*55. due 4 years hence, difcounting at 5 per cent.
Compound Intereit ? f^'ide Ex. i. Difcuunt at Com^mni
hitereji,
Agaiuft 4y. under 5 percent, (lands •82270^5 prefent worth Oi' il.
243!. 2^V' = 243 '1012 5 the debt.
Produft £,- 2CO the prefent worth.
TABLE IV.
Proposition i.
Cii'eu the annuity ^ the rale per cent, and time, to find the
amount.
Rule. Find the amount of il. per annum at the rate and
for the time given (by table IV.) which multiply by the
annuity, and the produdt will be the amount required.
Example. What is the amount of an annuity of 80I. to
continue 9 years, at 5 per cent, per annum, Compouatl Ivk'*
tereft ? Fide Ex, 2. Annuities in An cars,
Againft 9 y. under 5 per cent, ftands ii'Qz6^(]^7, amour.t oi' il.
So annuity,
S82I. 2,s. 6d. — ProJud /. S82T251440 amount.
Observation-. Anfwcrs to queftions in any of the other
3 Vropofitiom in Annuities in Arrears may eaiily be obtained,
by paying a little attention to the above Rule and Example,
TABLE V.
Proposition i.
To^nd the frrfi:ni luorth of an antmity at Compound Interejf,
tee lime of its con'tinuance and the rate per ant, beijig
given.
RuLK. Find the prcfent worth of il. at the
far the time given (by table V.), which multiply by
rats and
^ the an-
p:i
Y
52 The use of ihe Decimal Tables.
Example. Required the prerent worth of an annuity of
Sol. to continue 9 years, at 5 per cent, per annum. Com-
pound Intered, Vide Ex. z. Pre/ent Worth 0/ Annuities.
Againft 9y. under § per cent, ftands 7 '10782 17 prcfent worth of il.
8.0 annuity.
568I. J2S, 6d. n Produft ;^. 568-6257360 the prefent worth.
Proposition 1,
^ofind the frefent tvorth of mi Aumiity in Reaver/ion at Com"
Poiind Initreji,
Rule. Find the prefent worth of il. per annum, at the
given rate, for the time being, and alfo for that time and
the time in reverfion added together (by table V.). Sub-
tra<rt the lefs value from the greater, and multiply the re-
mainder by the annuity, the produd will anfwer the queftion.
Example. What ought a perfon to pay in ready money
for the re'verfion of lOOcT. a year, to continue 20 years on a
leafc which cannot commence till the expiration of ^ years,
allowing the purchafer Compound Intereft at 5 per cent, ?
t^ide Ex. 3. Annuities in Re-cerfo72.
The prefent worth of il. for 20+5 i:; 25 yrs. 7 ^
at 5 per cent, is — — \ '4'0959443
Againft 5 yrs. under 5 per cent, ftands 4-3294767
Diff. 9-7644678, which X
jooo, produces 9764'4678I. zr 9764I. 9s. a^z\. Anfwer.
TABLE VI.
Proposition.
^o find tuhat annuity y ta continue a ginjen number of years ,
4i^y fum of money nvill furchafey at a certain rate per cent.
Rule. Find the annuity which il. will purchafe at the
rate and for the time given (by table VI.}, which mul-
tiply by the purchafe -money, the product will be the an-
nuity required.
The use of the Decimal Table*. ^^
ExAMPLF. What annuity to continue 9 years will
568*62 5736I. purchafe, allowing the purchafer 5 per cent,
per annum. Compound Interei^, for his money ? See Ex. 8.
Vrefent Worth 0/ Annuities,
Againft 9 yrs. under 5 per cent, ftand?; •1406901, the annuity which
i!, will purchafcj this x 56^'6a£736, giveb Sol. the annuity required.
Observation. The above Rules and Examples arefut*-
ficient to elucidate the ufe of the tables.
r t
GENERAL AND UNIVERSAL
DEMONSTRATIONS
or THE
PRINCIPAL RULES
I N T H E
COMPLETE PRACTICAL
ARITHMETICIAN.
I. Demonji ration cf the method of proving Addition y hj cajiing
out the 7iines,
THIS method of proof depends upon this theorem,
** any number divided by 9 will i^r.-.e the fame re-
mainder as the fam of its digirs divided by 9." Let N be
the number, a, by c, dy the digits which compofe it, then by
the nature of notation loooa -f loob ■\- 10c + d -zz N. But
I Geo y^azz. 999 ■{- 1 Xa-=za-\- a X 999 ; 1 00 X ^ = 99 -|- 1
X^ = ^+^X99; ioX<:zi9+i Xt=r -\- cy.g\ \' icoo«
-f- ioob-^ioc-i-d= gg^a -f-^ 4- 99^ + <^ + 9^+ ^ + dizN,
N icooa-\-ioob-\-ioc^d ggga ~{-ggb -i- gc
and — zz n: +
9 9 9
'■ ; but 999^ -f 99 5 4- 9 f is evidently divifible
9
1 ^^' -t, . 1 r -4 a-^b-^c-^d
by q. •.•—will leave the fame remainder as
- ^ 9 9
Now the excefs of nines in two or more numbers being
feparately taken, and the excefs of nines alfo taken out of the
fum of the former exceffes, it is evident, that the lait excel,
muft be equal to the excefs of nines contained in tlis tot.i'.
fam cf all thefe, the fum of the parts being equal to the
whole. ^. K, D,
Demonstration of the Rule or Three. 3^
II, Bemonjiration of the method of pryving Multiplication ^ by
cafting out the nines,
** If any two numbers are feparately divided by 9, and
the two remainders multiplied together, and that produ(fl
divided by 9, the laft remainder will be the fame as if you
divide the produft of the two firll numbers by 9."
l^tti^A-^-a and <^B-\-b be the two numbers, a and b Hr^t
remainders. Their produft is ^y.(^A-\- <^Ba-\-<^Ab -^ ab,
but the three firft terms are evidently divifible by 9. •.• there
is no remainder but what is had by dividing; a X b by q.
Wha-t has been demonftrated refpefting the number 9,
holds good with refpe*^ to the number 3. A little attention
to the preceding demonftrations will render the method of
proving Divifion, Square Root, &c. obvious.
Note. The rule of Addition, v/hich is founded on this
fimple axiom, " That the whole is equal to the fum of its
parts ;" and the rule of Subtradion,. which may be deduced
from the fame axiom, feerti to require no demonltration. —
The rules of Multiplication and Divifion are likewife bell
explained by examples, the former by Addition and the latter:
by Subtraction ; the Compound Rules and Redudion carry
their rationale along with them,.
II I. Demotifi} ation of the Rule of Three,
Deftnition. Four numbers are faid to be proportional,,
when the firil contains the fecond,, as often as the third con-
tains the fourth..
Viz. If^ = ?-, then J'.B;: C:D,orB:J:: D:C,,
which is the definition I have given of proportion.
Theorem. If fbur numbers are proportional ; A i B : :
C : D, the produd of the extremes is equal to the product
of the means, viz. A x D zz B x C.
A C
Demonstration. Let — = -^ = r, then A zz Br,
B JJ
and C=zDr, htncc AD=zBrD, and^C = BrD. •.• AD
— BC, i^ E. D, Hence we deduce a demonftration of the
Rule of Three,
Y3
36 DEMO^'STRATION OF THE RuLE OF FiVE, ScC.
If J : B : : C : D.
Then .^ X D = B X C. No;v ./ .....1 i..n.';...
be divided by equal numbers, the quotients will be equal.
B X C
',' D =z 7—, Nvhich is the rule, i^ E. D.
CoT.D= ~xC = ^xB=zC-^^--B^^. See
the notes.
Note. The Rule of Three Inverfe may be made a Rule
of Three Dired, by making the third term the firil, and by
proceeding forv.ard to the other two terms ; therefore the
above demcnftration will ferve for both rules. A fmall
attention to the notes annexed to the Inverfe Rule will render
it very plain.
IV. T>emoTijiratiott cf the Rule of Five, ^c.
All queftions that come under this rule may be fohed by
two ftatings in the fmgle Rule of Three, either both dired,
or one diretft, and the other inverfe. The reafon of this
rule will therefore appear obvious from the firll, fecond,
and third example.
I. Hen- the blank falls under the third term.
If 7m. :izd.^i.6a. ^ _,6X3XIZ6
7 X la
Ey fwo <Iire£^ ftatings,
.r * ^ c 126X16
If 7 m. * : 126 a. : : i5 m. : a.
7
126x16 16X3X12G
If 12 d. ♦ : a. : : 3d.: . •
7 7X12
By one dired, the other Inverfe.
If 7 m. : 12 d. : : i5 m. * : d.
16
,r 7X12 4 , , 16X126X3
If -— d. ♦ ; iz6 a. : : 3d.: -»
j6 ^ 7X12
II. Here the blank falls under the frji term.
If 7 m. : 12 d. : 126 3.
3
: 72 I 3X"6i
General principles and theory of Fractions. 37
By two dired^ flatings.
If 12 0.* : 126a. :: 3d.: a.
12
, i26x^* 7X12X72
Ir —a. : 7 m. : : 72 a. : -— .
12 3X126
By one diie<ft, the other inverfe*
1-2 y 7
If 12 d. : 7 m. : : 3d. * : m.
3
T- C * 7XT2 , 7X12X72
Ir 126 a. * : m. '. : 72 a. : m»
3 3X126
III. Here the blank falls wider the fecottd term*
7XI2X7S
If 7 m. : 12 d. : 126a
XI
16 : * : 72
16x126
By t\vo dire ft ftat'mgs.
Tr * /: ^ 16x126
If 7 m. * : 126 a. : : 16 m. *. a.
7
7X12X72-
12 m. : : 72 a. : — -r-in*
16x126
By one direct, the ot'uer inverfe.
Tf /: * 7X72
If 126 a.* : 7 m. : : 72 a. : r- m,
126
.. lY.^'i- , . ^ 7X12X72
If — -m. : 12 d. : : 16 m. * : — -• m»
126 16X126
Agreeable to the ruk, and whatever the qucftion may be,
it is evident there will be the fame reafon, for the blank muft
fall under either the firll, fecond, or third term : therefore
the rule is true, ^ E, D. The univerfal rule of proportion
is demonftrated from a fimilar manner of reafoning ; for the
firft example, which ccnfifts of 13 terms to find a I4.th, may
b(f eafily reduced to 5 terms to find a 6th.
V. General principles and theory of FraSIionf,
Proposition i.
The reafon of the rule to this propofition , may be clearly
fhewn from the firft example: thus, 24. meafures 192 fince
o remains i it alfo meafures 216, which is feme multiple of
38 General principles and theory of Fractions.
192 with 24 over. For 192 X r -f 24 rr 216 ; for the fame
leafon it meafures 408, which is fome multiple of 216 with
192 over. For 216 X i + 192 =408, &c. v» 24.meafures
both 216 and 408.
It is the greateft common meafure ; for, if there were a
greater, then fince the greater is fuppofcd to meafure 216
and 408, it will alfo meafure 192 and 216; and fince it
meafures 192 and 216 by fuppofition, it ought alfo to m.ea-
fure 54 and 192, which would be abfurd, for a greater num-
ber cannot be contained in a lefs. Whatever numbers are
propofed, the manner of reafoning will be the fame, •.• the
rule is true,. ^ E.. D.
Cor-. Fknce if there are more numbers than two, we can
find the greateft common meafure ot two of the given num-
bers, then of that common meafure and a third number, &c.
Thus the greateft common meafure of 72, 60, and 28, is 4.
Proposition 2.
The rule to this propofition is founded on this axiom,
•* That dividing both the terms of any fraction by any num-
ber whatever, will give another fraftion of equal value with
the former."
Proposition 3, 4, and j, are felf-cvident.
Proposition 6»
r.
Let -^ reprefent a complex fra^ion in its moft fimple
'd
florm, to which form every complex fraftion muft be reduced
before it can be folved by this rule. Now every fraftion
denotes a divifion of the numerator by the denominator, and
its value is equal to the quotient obtained by fuch a divifion 3
n
'~^ n , N »D ^ ^ ^
hence ^=;j -^F = Nd'^'-'^'
D
General fp^inciples and theoryof Fractions. 59
Proposition 7,
That a compound fradion may be reprefented by a fimplc
one is evkfcnt, fince a part of a part muil be equal to fome
part of the whole. The truth of the rule may be iliewn thus.
a f. b , \ . b h
Let the compound fraclion be— of — , tnen — ot — _ — -
^ n m n m in
b . r , ^ r ^ ^ ^ ^
'^nzn — , andconfequently— ot — — — x «= S
if the compound fraftion confifls of more numbers than two \
wc may firll reduce two of them to a fimple fradion, then that
one and a third, &c. ^. £. D,
Cor. 1. \iN~A-\-B-\-Cy$iZ, then— of A^ZZ— oi A ■\- '
iof£+ iofC, &c. Ex.20— ii+8» s* J; ofac = iof li
+ i ot i> zi 5.
Cor. 2. —Qi A — A X — of u Ex. | of 2=1 of iriaXiot i-
n n "
Cor. 3. — of » n -5 of I. Ex. \ of il. — | of il.
A A
Cor. 4. — of/?: zi — of ^. Ex. 4- of 2 zr ^ of 3.
n n ^
CoR. 5. — of b zn — of axb. Ex. I of 5 zz |of 10.
n n
Proposition 8.
The numerator and denominator of each fraftion are equally
multiplied, viz. by the denominators of all the other frac-
tions, confequenily the fradions produced are equivalent.
Thus in the firll example,
^7X11—303' 7 4Xii'~3oS'ii 7x4 3^^'
The fecond rule under this propofition is equally obvious,
for -^of 5o8r2-of3oSx3 = -X ^t!' ^'- ^^' ^'
4 4 - 4 7 >vii
Note. The following rules in vulgar I'raaions will appear
fufficiciilly plain, from an attentive obferNancc of the nature
40 The THEORY of Circulating Decimals.
of the examples. — Tfee truth of the rules for working finite-
decimals will llkewile appear more clearly from the defini-
tions, and the nature of the operations, than they can be
rendered by a multiplicity of words.
\I. Demonjirathns of the rules of Circulating Decimals »
D E F I N I T I o K I.
** It is afferted that if the denominator of a vulgar
fradion, in its loweft terms, is not compounded of 2 or 5^
the decimal produced from fuch. a vulgar £ra<ftion will be
infinite*"
K
Demonstration, Let— be the given vulgar fraclion,.
tken fince D has not in its compofiticn r or ^, nor any mul-
tiple thereof, there may be found a certain number of 9'$
which D win mdafure ; for by dividing 1000, &c. by any
prime number whatever, except 2 or 5, or fome multiple
thereof, the figures in the quotient will begin to repeat
©?er again as foon as the remainder is iv And fmce 999,-
&c. is lefs than icoo by r. •.• 99Q,<i-c. divided by any prime
number whatever excepting thofe above, will leave no re-
mainder when the repeating figures are at their period..
Suppofe the number of 9*8 which D will meafure to be
reprefented by 99, &c. take D : 99, Sec, : '. N : R, then
fince D meafures 99, Sec, N muft meafure R.
T, QQ, Sec. R „ . . . . TX IT
For^— — = — , •.• i^is ao integer. Again, D : N ::
CO, Sec. : R and — =1 , but — - — is evidentljr
D QC),^c. 99, &c.
a pure circulate, the repetend of which begins immediately
N
after the decimal point, •.* yr is a repetend, ^ E. D^
Proposition i^
Every pure circulate is, from the nature of a decimal
iiradion, a feries of decreafmg fra<f\ions with equal numera-
tors, and their denominators a geometrical feries increafing
in a tenfold proportion. Let R reprefent the repetend or
aiimerator of the fraftion, 1 o" = r the ratio,.
The theory of Circulating Decimals. 41
R R R R
Then will — •{- —; A — T H 7i ^c. ad infinitum, reprefent the
decimal, and the fum of this feries by the laws of progreiTion ^
^ . ^ ^ ^ „ fi,^ which is the rule.
R^ ^R R L. ^ __ i? _ -R
— ^_j — 10"— I ~9'^'<=
^ £. D.
Proposition 2.
Let /^ reprefent the numerator of the finite part 10^, its
o
denominator, then will :^ — be the vahie of the pure
g9,cvc. ^
fcpetend by the laft propofition, •••
lo" 99,<!s;c.xio« 9;,&c.xio" 99, &;c. x io»
Fx ^00, Sec. + R~"F ^ ^ _. ......
rr the traction reprefenting the value ot any
99, &c. X lo' "^ " -^
mixed rcpetend, which is the rule exa(5lly. ^ E, D,
The notes which follow are corollaries, deducible from the
preceding dcmonftrations.
Proposition 5.
N
Let "Tj l^e a circulate, D having fome other prime in it
than 2 or ^, then D may be reprefented by Ay.i", or J X
5'", otJxz" X 5'", &c. where ^ is a number which has no
power of JO, 2, or 5 in it.
„ iV 7Y iV I ^ r
Hence — = zr -r X ; but is
D .:^X2"X5'" ^ 2"X5'» 2"X5'"
a fraftion, haying no other prime than 2 or 5 in its denomi-
nator, and \- is reducible into a finite decimal which has
2" X 5'", or fonie power of 10 for its denominator; fuppofe
10 ; then -;; :;; may be expreffed by •
42 Demonstration of the common rule of
Hence^=jX — =-^-,o ; Now —
may be an improper fradion, but it cannot be an integer ;
Na r ,N
for if it were, then — p -^ i o = — mufl either be an
integer or a finite decimal, contrary to the fuppofition :
Na
••• if —r- be a proper fradion, the decimal equivalent will
be a pure circulate (prop. i.)» and if it be an improper
fraftion, it will refolve itfelf into a mixed decimal, the finite
part of which will be a whole number. Now if this decimal
quotient be divided by lo'', it is plain the figures will not
be changed, the decimal point will only be removed as
many places to the left hand as there are units in lo^, there-
fore the repetend begins after fo many places as are exprelTed
by ?•, and hence the whole is evident. ^ E, D,
Whoever underftands the preceding demonftrations, which
are the foundations on which the other rules for managing
circulating decimals arc built, will readily fee the reafon
of the feveral operations. — The rules of Pradi^ce, Tare and
'i'ret, Intercft, Brokerage, cSrc. are felf- evident.
VII. Demonftration cf lie common Rule cf Equalkn of
Pnjmtnts,
I. 0]i Cocker's principle^ 'vide note id to the rule,
J (■ .^ 1= a debt due a years hence, 1 and .v for the
^ I -5 zz a debt due h years hence, ^ic. J equated time,
Kovv .V muft neceffarily fall between a and h, hence .v —
a zz time A is at inierelt, and h — x ■=■ time B is at in-
tereft : alfo x — a X Ar =. intcreft oi A fcr its time, and
X yi B r =^ intereil of ^ for its time. •.* x — a x A .
=z b — X X Brhy the fuppofition, oi Arx '\- B r x = Ara
r. , , Aa^ Bb ^. . . ,
•\- Brb^ whence .v = -— which is the rule, and the
A -^ B
faTiC may be fliewn fcr any number of pajinenrs. j^. E, D,
E c^u A T I o N OF Payments. ^j
II. 0/i Hatton's principle y <vide note ^d to the rule,
j^ J ^y4 = a. debt due a years hence, 7 and x = the
I B zz: a debt due 6 years hence, J equated time.
Then J -^-B x fx = interefl: of the fum of the debts from
tlie time of thcquellion to the equated time ; Jra:=: interefl:
of ^ for the time<7, and Brb =■ interefl of/? for the time b.
\' A-^-BXrx-zzAra-^rBrh by the fuppofition, or
Arx •{• Brx :z Ara -{-Brb as above, hence x :z: — — -
A -{- B
which is the rule, and the Tarns may be fhevvn for any number
of payments. ^ E. D,
III. G// Moreland's or Burrows' principle, 'vide note \th.
Let
A 1:1 2i debt due a years hence T
B zz a debt due b years hence ( ^ — ^^^
C zz 2i debt due c years hence f equated
D rr a debt due ^ years hence, &c. J ^^™^'
d — ^ rz time ^3^ is at interefl:,
d — b 2Z time B is at interefl.
d — c zz time C is at interefl:.
^ — d zz time D is at inierell.
A -\- A X d — a X r = amount of^ for its time,
B -{- B X d—b X r = amount of 5 for its time.
C -{- C X d — c X r zz amount of C for its time.
D -{- D X d — d y. r zz. amount of Z> for its time.
Now the fum of the amounts according to the fuppofitiou
muft be J 4- J X d — x X r, putting s {or A -\- B -{- C -{- D,
A X d—a X r -Y B X. d-^b x r + C X d—c X r -f
Z) X d — d X r = s X d — X X r, by di\'iding by ;• and re-
ducing the equation, we obtain A d — Aa-\-B d — B b -\- C d
^Cc-{-Dd—Ddzzsd-^sx, then A+Bi-C-^-D^d
^Aa-^Bb-^Cc^Ddzz sd-^s,\\
Z
44 Demonstration of the
Hence x = =: - — - — - — -- — -- —
■ s - A'+B + C + D
tv-hich is the rule. ^ E. D.
Hence the rule is univerfally true according to any of the
fuppofiiions, anc^asthefe fcveial pi-inciples produce the fame
final fimple equation : it there- ore follows that they are ma-
nifeft confequences of each other, as I have aiferted at note
5th to the rule,
VIII. Demon/? rat ion of the Rules of FelloivJJjip,
BY MR. V/. ADAMSON, TFACHLR OF THE MATHEMATICS,
AT BURLINGTON, YORKSHIRE.
I. Shigle FellQnx>Jhip,
Merchants. Stocks. Gains
A o
B h y > Notation.
C c z. )
Ey the general definition of Fellowlhip it muft be
As i7 : A- : : ^ : _>• : : t' .: z, hence by the dod\rine -of
proportion we have a-\-b\c : a 4-j-f 2 : : a \ x, ^E, D.
II. Double Fellonjc/bip.
chants "
I. n
> Notation.
Admit u4
. B .
c.
Sec.
merchants 1
Put in a
. b .
ey
&c.
ftoclis 1
For i
• P '
9^
&c.
months j
And gain x
• y
Zy
Sec.
pounds J
Now ;f ~ J's gain in the time /; and by Single Fel-
lowfl:iip,
bx _, ....
zz B s gain in the time /.
a
ex
a
•zz C's gain in the time /.
— '.: p : -^ = J — -^'s gain or lofs in his time/.
a ta
— '.'. q : -^-~ :zi z zz C's gain or lofs in his time y.
ia
Rules of Fellowship.
45
Hence we (hall have their refpedive gain expreficd three
ways.
I. II. III.
A. x—x
B, y-x X —
•^ at
C K-=zx X ~
tp
x—y X
c q
x—z X
^—2; X
qc
a t
Let G = the whole gain, then wtf' fhall have the three
following equations.
bp
I. X ■\- x X— +XX -^ zzG-, or at + bp -^c-^ XX ::z:aiG.
at at
II. yx'r -fv+J x\^ — G, or at -i-op-^ca Xy=.bpG.
hp ' op
III. XX-- ■\-7^-k~-\-z — G)OVat-\-vp-^cqX^—cqG,
cq cq
Now let. the fum of thefe produds be put = s, and wc
*^ sx = atG '.' s : G :: ai : X )
sy = bpG -,' s ', G -.'. bp ', y >^ E. D.
jz :ii c gG •.' f ; G :: C7 : z}
Proposition i. Note 4.
By the rule <)f Double Fellowlhip we have the following
proportions {x being given).
at + [>/> : X -^y :: at : -v 7 j_j^^^g
a t -^ c q : X -{- z, : : a t : X I
bpx •=! aty '.' at '. X : : bp :y 1 ^ j^^ ^^
cqx zz atz. '.• at '. x :: cq : ~J
Proposition 7. Note ^.
^ut .v-f-j-f 2 rz G the whole gain, and at^b p -^-c q-= m
the fum of the pr jdufts of each man's itock and time, per
Double Fellowlhip.
m : G
m : G
m : G
: at : x'i
: bp '. y^
: c a \ ^\
Hence ^, = - — ^ * ^"^^
G X y ^
from thefe equations we deduce the following.
Z a
46
Demonstration of the
I.
A' a :n a
C. i — aX —
II.
a — by,
b- b
c-b X
9y
III.
qx
a-=.c X —
t s
Now putting s for the fum of the ftocks, we have
I. o-^uX — -^<iX — —
p^
II. b X^- -^-b+bX^- - s.
1^
'iy
Hj
III. cx'—\cx^^^rc~
:3 px.
P i .V + ^ ?>+'f ~X:i IT / J .V i.
fiX^--tqy\tpz.Xo — t<iy 5.
••• pqx Ar tqy Jr tpz, : t
' ■ pq X -^ rqy + tp^ : s
••• p3--< + r^y -i- 'f^ •• i
p q>:-\-tqy-\-lp^X C ZZ t p % t.
P
p qx '. al
t qy : b L'^ £. D.
Proposition 3. Note 6. '^
By prop. I. f/e have hpx zz c/r, and cjx zzatz* Now
let a l'€ given. TliCn
A- : ^ / ; : y ; hp, and — = ^
;«■ ;<?/:: s : f<7, and — =z c
^^./>.
Proposition 4. Note 7.
By prop. 2. we have — = -^^ = - , and from thefe e-
;c J a
quations we deduce the following.
II.
I.
A. t — t
^ ^y
C. q — t X —
bx
t—p X —
ay
P -P
b^
qZZp X —
cy
III.
q X
'y
? = ?
k tf LE S O F L OS S A N D G A I N. 4^
Xow by putting j for the fum of the times, we fhall have
-ay , z-
I. /+ X 7^ + r X - =15
II. p X — 4-/>4-/X — = 5.
ay cy
III y X — H ?x 7^3 + ? — i.
i^TA-rf 'it-y-j-ii ^ <i.X<
6 cx-\-a cy-\ub z XpZHa c y .
cy :
Proposition ^. Note 8.
•. • hex •\- a c y + «''''« : ^ •• : ^ f x : t')
'.' I c X ■\- a c y -f- a b z. : j : : rt c >• '. p >^ E. D.
••• Zt^ + <2f_y + <^bx, : i :: abz, : qy
By prop. I. we have bpx rr city, and c^x = /7/^;
multiply the firft of thefe equations by c, ami the fecond by
b, and we (hail have cb^x rr c atjy and c bqx ::=. batj,^
let / be given.
Then b c x : t : : c cy : p} ^^ p ^
And ^ r .V : / : : ^ rt 2 : 5^ f "^*
The method of demonftration made ufe of above is very
general, and will be the fame let the number of partners be
ever fo many.
IX, Demoyiftration of the Rules of Lof and Gaiiu
Proposition i.
This rule is felf-evident.
Proposition 2.
p.' r Prrprime coft ) of an f to find G tliegain,
\ S — felhng price ) integer, [ L the lo's per c^n
or
nt.
r : d : : loo : — - — or f : loo :
I GO S
S : -— l,y
the
rule, and agreeable to reafon. — Mow if 6" be greater than F,
ioo5 .„.,,, , „ 100.9
— —- will evidently be greater than lOO. Hence — —
100 n G, for — - — is the amount of loolr
Z 3
4$ Demonstration of the
Again, if 3" be lefs than P, then will — -- be lefs tha»
100. ••• ICO — — p— = ^ t"^ ^0" per cent.
Proposition 3.
^. J P=primecoft of an integer 7 to find 5 the
I G=thegain,orZ = the lofs per cent. ) fellingprice.
It has been already (hewn that — p- — 100 1= G, and
100 — — ^ zi L, Hence 100 X S = P x G ■■\- 1 00,
nd 100 XS zz P X 100 — L.
••• I CO : 1 00 -f G : : P : .S if gain, and 1 00 : i co — Z.
; P : ^ if lofs. ^E,D.
Proposition 4.
or /the
given fei-
^. V 5 rrfeUmg price of an integer 7 f"/"'^ ^ '^'.^'»'"!
Given -^ ^ — , • T ►u I r ^ * ?■ 'ofs at any other gi
|G:=thegaiii,Lir thelofspercent. ^,5^^ p^. J^^ &
From the equations in the preceding propofition, we have 100 4- G
looS ...
! 100 : : 5 : P ZI if gain, or 100 — L : JOO : : 5 : P
lOO+G
"=°^ if loft.
100 — 1/
Nov;, by propofition 2. we have
^ lOQp , ioo/>
P : 100 \ '. p '. — — i i then — — i- — 100 zr ^, and loo —
— •' n /, in thefe equations for P, fubftitute its values ■ — -,
ioo5 ico-i-Gx/'
, -. Then w.U — 100 zz r, and 100 —
and 100 — L S ^
zz 1} or joop 4- Gp — 100 5 4- Sgi and ico* — i/>
o
ICO S — SI.
•.• 5 : ICO 4. G : : /> : 100 + ? if gain, and 5 : 100 — • L : : /> :
JOO— /if lofs. i^£. £>.
Rules of Lossand Gain. 49
Proposition ^,
^. 5" S =the feUing price of an integer 7 ^^ ^"^ ^}'' wjiole grn
^■'''"j G zzche gain, or L = thelofs per cent. \°' ^^^' by the fale of
(, 3 ^"y qi^antity ^
By the firft dating in the l?.ft: propofitioa we have
lOO ^- G : lOO : : 5 : P the prime coil of an integer, if gnin,
loo — L t loo : : S : P , if iofs.
Now if S and P be equally Increafed hj any quantity P,
the proportion will evidently continue the Cun.c,
. . J TOO + G : loo : : 5 X ^: Z' X ^if gain ) ^ _
• J loo — Z, : loo : : 6" X i^: /- X i^if Iofs | ^^"^ '^
^ ^is the whole value at which the goods were fold, and
P X i^is the whole prime cod.— ^Hence the difference l)e-
rveen S x i^and P X ^ mull be the whole eain or Iofs.
9. E. D,
Proposition 6.
?^= che quantity of goods bought 1 to find the
P — the prime coil of an integer > whole gain
^ G n the gain, or Z-zz the Iofs per cent. J or Iofs.
Here P X i^ =: the whole prime coft of the goods.
From the lail propofition we have
loo : lOO + G :: Px^: 5 X ^ if gain.
ICO : loo — I :: P X^: 5 X ^ if Iofs.
Hence the difference between S % ^ and P x ^ = the
whole gain or Iofs. i^ E. D.
The rules of Barter and Exchange will appear plain by
obferving the nature of the feveral Examples. — The rule of
Compound Arbitration of {'Exchange may feem to want de-
monltration; but whoever takes the trouble to folve any
one of the examples by two, or more flatings, in the fingle
Rule of Three, in the fame manner as I have proved the
Rule of Five bv ingle flaiings, will immediately fee the
truth and nature of the rule.
Any perfon who can extra*?l the fquare and cube-root in
Algebra, will not be at a Iofs to demonftrate the rules of
fquare and cube- root J and to thofe who cannot, a demon-
ilratioQ
50
Demonstration or the
flration would be of little or no ufe. It may not be amifs
to remind the reader, that the 2d rule for extrafting the
cube-root is deduced from Mr. Ward's Mathematician's
Guide, page 151. Sch edition.
X. Demonjiration of the Rules of Alligation.
PROPOSITION I,
This rule needs no demoi\ftration.
Proposition 2.
Suppofe four fimples A, B, C, D are to be mixed ; let the
mean price be m, the price oi A =^ m -\- a, of /? = m -\- b,
of C = m — Cy and oi D z=l m — d. And let the quantities
to be taken of A, B, C, D, be x, j, z, nj refpedively.
Then
prices, quantities.
m -f n
mean
m -\- b
m
m — c
m — d
Now if each quantity be multi-
^ plied by its price, the fum of the
produfts will evidently be equal
to the fum of a.l the quantities
multiplied by the mean price.
Viz. m -{- a X X -\- m -j- b X y -{■ m — c X. z. -{- m
zn X -\-/v -\- y -\- z X m.
dX
Let m -^ a X X -\- m — dX'vrzLX-^-HJXm,
And rn -\- b Xy + m — c X z z= y -\- z X m.
Then mx -f ax -f- m^v — d -z! =■ mx + m'v.
Alfo fny -J- by -f fnz — c z '^. my -\- mz.
\ A 1^— ^!*but here are four unknown quan-
tities and but two equations, •.* x and y may be taken at
pleafure. Take xzza and y—c, then will ^vzza and z=b,
and confequently the work will ftand thus.
m + a"
m + b-
m — c-
^1
c /Which is the rule.
Rules of Position. 51
Cor. Since ax = dn; and by = cz^ take x :=s m d and
y :=z uc \ then will <v z=: ma and z ■=. nb. Then by putting
md, nc, 71 b, ma^ for ;f , j', 2;, f refpeftively, iniiead of the
differences dy r, b, a, we lliall have m dy ncy nby ma, which
is a dired demonftration of the latter part of the note.
♦ Pr ©position 3.
Here one quantity is given, and confequently all the
quantities «, b, c, d mull be increafed or decreafed in pro-
portion. Hence the rule is evident.
Proposition 4.
Here the lum of the quantities is given, confequently tlfe
other quantities muft be taken in proportion, fo that a -^ b
-\-c-\'d may be to the whole quantity as any of the differences
<7, by 8cc. to the refpedUve quantity required.
XI. Demo^i/I ration of the Rules of Fofitlon,
I. SINGLE POSITION.
Such queflions properly belong to this rule as require the
muhiplieation or dlvifion of the number fought by any pro-
pofed number, or when it is increafed or diminiihed by ii-
felf. or any part of itfelf a certain propofed numl^er of times.
^-For in thefe cafes the refults will be proportional to the
fuppolitions agreeable to the definition. Thus
X s
— I X : '. — '. s
n n
XX s s
^ + —y Sec. : X : : ~~ + — , Sec : s. Hence the
n -^^ m n — ?n
whole is evident. 9^ E. D.
II. DOUBLE POSITION.
This rule is founded on a fuppofition, that the firft error is
to the fecond, as the difference between the true and firil
fuppofed number is to the difference between the true and
fecond fuppofed number.
52 Demonstration of the
'Let A and B be produced from a and b by any fimilar
operation, to find the number frorn which N is produced by
the like operation. Put % for the number fought,
^' — A z= r, and A' — B z=.s, then by the fuppofition
r : s : : z — a : z — b •.' rz. — rb =■ sz. — j£, and by
tranrpcfi.tion rx — sz zn rb — sa,
Hence z = ; or if j be negative, or B greater
r — s
than A', then will z = I-iif which is the fiift rule.
r-i-s
Secondly,
Let s be the lefs error, being the error of b, and r the
correction ; then if B be lefs than A^, b + c ^ z, and c zz,
— A — ^^"~-^^ /.^ y3 — sa — rb^ ib _ b — gXjf
~" r— / "" r — i r— i
But if 5 be greater than N, then b — r = s, and r =: ^ — «
, rb-\-sa rb-\'sh — rb — sa b — tfX/ ,.« •
=£> — = ; -zz , which IS
r-\-s r + s r+s
the fecond rule, ••• the rules are univerfally agreeable to
their principles. ^ E. D,
Scholium. It has been fhewn that the nuoiber fought
will come out exadly by this rule, when the errors are
cxadly proportional to the differences of the fuppofed num-
bers from the true one. It therefore follows, that when the
errors are nearly proportional to thefe diferences, that the
anfwer will come out nearly true. And thefe proportions
will be the nearer to an equality the nearer thefe fuppoled
numbers are taken to the true one. Hence we fee the reafon
of the fecond note to this rule.
Rules in Arithmetical Progression,
XII. Demonjiration of the Rules in Arithinetical
Progrejfion,
^Z
f/ = the leaft term
g =■ the greateft
. n = the number of ten
I J = the fum of the ten
[ ^/ = the common dirlei
Then will
/+/+^+/+2^+/+3^-f/+4^, &c. be In Arithmetical
Progreffion.
Now it is evident that the fum of the extremes will always
be equal to the fum of any two means that are equidiftant
from the extremes; and confcquently if the number of terms
be odd, the fum of the extremes will be equal to double the
mean ;
That is /+/-f 3^=/-f^+/+2./= 2l-\-2d
I-}-l+4.d=zI-i-2d+/-i-2d=2/+^d
l+l-^-zd—l-^d^l-^d— 2l+zd, Sec,
Hence Propofition i. s = i-\-gx~. For
2
r /4-/-I- d 4- 1 A- id -^rTTd ■+
Add
V /+/+^ + i+zd +l-\jd -hI-\.^d,Scc.zz{um
I l^^d-\rTTTd +/+2^ -{- /+7 + / zrfum
2/+4^+2/ + 4^/-f 2/+-y-{-2/++^+2/+4^, &C.
*.♦ twice the fum is equd to as many times / -f / 4- ^d, or
I -\- g as there are terms in », confequently s rz l-{-g x
^. E. D.
n
2
Inferences deduced from this equation
\ c T 7 2/ — 712
Interence I. / iz ^,
n
II. n = ^.
Tir i!—"^
m. i = -;;-.
IV. s = "J±:i.
^4- Demonstration of th
Propofition 2. g — I zz n — i X d,
1.2.3 .4 • 5> &^c. to n,
l^TTd-^T+n-\rJTTd-\-JTTdl &c. to^.
Here the common difference d is evidently as often re-
peated as there are terms in the feries wanting one ; that is,
evexy terra, except the firft, is equally increafed by dy *,•
J-— / = «— r X^. ^E.D,
Inferences deduced from this equation.
Inference J. I =z g — « — i x d,
n. . = -^±^.
HI. d- ^^^.
IV. g=: I + n—l X d.
Kow by the afliftance of the above deraonllrations, if any
three of the iive terms /, g^ », s, d he given, the other tw©
are readily found.
Proposition i«
Given /, g, and n, to find s and d.
(Prop. 2, Inf. 3.) d= ^^^^; and (prop. i. inf. 4.) s =:
7 '^
/+; X -.
Pro position 2.
Given /, g, and /, to find n and </.
{Prop. I. Inf. 2.) » =z -^, =. ^"^ ."" (prop. 2. Inf. 2.)
Hence we deduce ITieo. III. and IV,
Proposition 5.
Given /, g, and </, to find n and /.
(Prop. 2. Inf. 2; « = ^-~—, = > (prop. i. Inf. 2«
Hence we deduce Theo. V. and VI,
Rules in Arithmetical Progression. $$
Profosition 4.
Given /, «, and /, to find g and d.
Prop. I. Inf.3,)^ = ^~-^ = ^+^— ix^ (Prop. 2.
Inf. 4.)
plence we deduce Theo. VII. and VIII.
Proposition 5.
Given /, », and d, to find g and s.
{Prop. 2. Inf. 4.) ^ = / + «— I X^= (Prop, i.
Inf. 3.)
Hence Theo. IX. and X. are deduced.
Proposition 6.
Given /, s, and J, to find g and ^,
(Prop. I. Inf. 3.J ^ =z =/-f //— I Xd (Prop. 2 .
Inf. 4.)
(Prop. I. Inf. 2.) «r=-ii=rC±4ll^ (Prop. 2. Inf. 2.)
Hg d
From tbefe equations we deduce Theo. XI. and XII.
Proposition 7.
Given gy ;/, and s, to find / and d,
fProp. I. Inf. I.) / ~ 1-11^=: ^— ^117 Xi (Prop. 2,
h.i'. i.)
Plence v/c deduce Theo. XIII. and XI'v' .
Proposition 8.
Given gy 7/, and d, to find /and s»
(Prop. 2. Inf. I.) l—g-^n-^i Xdzz-- ''^ (Prop. i»
Inf. I.)
Hence we deduce Theo. XV. and XVI,
A a
5-6 Demonstration of the
Proposition 9.
Given j-, j, and d, to find / and 11,
(Prop. 1. Inf. 2.) '^=;A^ = '^^^~^ (Prop. 2. Inf. 2.)
(Prop. 2. Inf. I.) / =:^— a— i X ^ z= J^^'^ (Prop, u
Inf. I.)
From thefe equations we deduce Theo- XVIL and XVIII.
Proposition 10.
Given ;/, /, and dy to find / and g.
Prop. 2. Inf. 4.) g =:i-i- n — i Xdzz (P^op* '•
Inf. 3.)
(Prop. 2. Inf. I.) 7=:^— ;J^ X^=: ^i^— ^ (Prop. i.
iQf. I.)
From thefe equations are deduced Theo, XIX. and XX.
XIII. DemonJIratlon of the Rules in G^oinetrkal
ProgreJJion,
Before I proceed to invefligate the Theorems, it may not
he improper to explain the rules to the firft and fecond pro-
pofition, page 95, — In the firil propofition, where the firft
term is equal to the ratio, the reafon of the rule is evident ;
for as every term is fome power of the ratio, and the indices
point out the number of multipiiers, it is plain, from the
nature of multiplication, that the produd of any two terms,
will be another term correfponding with that index, which
is the fum of the indices ftanding over tliefe refpedtive
terms.
And, in the fecond propofition, where the feries dees
not begin with the ratio, it appears that every term, except
the firlt and fecond, contains fome power of the ratio mul-
tiplied into the firft term, -.• this rule is equally evident
%\ith the former.
Holes in Geometrical Progression-. ^y
La /r:the leaft teniT
^r=:the greatell
vnthe number of" terms
j~rhe fum of the terms
r— :the ratio
log. iiilogarithm of any let-
[tcr, or term.
Then /, /r, /V% /r% /;4,
Geom. Progreflion.
[hen /, /r, /V% /r% /;4, &^'. -J
Or, /, — , ~, — , --, &:c. (
r r^ i^ r^ \.
Kow fincc the above progreflion is compofed of two ranks.
The equals /, /, /, /, /, 1 .... ,
Geometrical proportionals i, ^, rS r^ ^4^ J &«^ »t 'oUows th.t
the moft natural Geometrical Progrefiioii is that which bcg'ns wlr.h i,.
both incieafing and dccreatinjr.
Proposition r.
Demon. For / : /r : : s — g : s — /) By Geometrical
But / I Ir : : I : r 3 Proportionals.
"Whence 1 : r : : s — g • ^ — ^«
'.• s — I^ s — g X r. i^ E. D.
Inferences deduced from this equation.
Inference I. s =: ^^i^^ z= ^-^ + g.
r — I r — I
II. r zz ..
HI. / = s -h ^g — rs = rg — r — I X /»
,-- sXr — 1+/ * s-i-/
IV. g = =z s .
^ r r
Pr OFOSITION 2.
r X / = g.
For /, ir, Ir^, lr\ /r^ &r.
And I^ = /r^-' ... //-^ =^. ^ £. Z>.
j;8 DlMONSTRATlON OF THE
Inferences deduced from this equatidn.
n — I
liiference I. ^ = /r
T
II. / = -'^.
/; — I
r
, III. „ = !£I:f=i£I:> + ,. for/-=-f.
Jog, r i
IV. rzn-^p'
Hence by the affiflance of the preceding demonftrations, if
any three of the terms /, g, », ;, r be given, the other two
are readily found.
Proposition i.
Gi^en /, g, and », to find / and r,
1
(P/op. 2. Inf. 4.) r = 4i'*""' and (Prop, i. Inf. i.) ; =
~ — . Hence we deduce Theo. I. and IT,
r — I
Proposition 2.
Given /, g, and r, to find » and r,
(Prop. J. Inf. 2.) r=: ^-^^, and (Prop. 2. Inf. 5.) » =r
^'^ ^^ — {-/. Hence we have Theo. III. and IV,
l;>g. r
Proposition 3.
Given /, ^, and r, to find « and s,
(Prop. I. Inf. 1.) / = ^— z= -^ + g-
r — I r — I
/Prop. 2. Inf. 3.) « =: -^f ~-^?:- + i, the fame wirh
Theo. V, end VI,
Rules in Geometrical Procressioit, 59
Proposition 4.
Given /, », and /, to find g and r,
(Prop. 1. Inf»4.) g n ^ = Ir (Prop. 2r
Inf. 1.) Hence we deduce Theo.. VIIT,
ft I
(Prop. 2. Inf. I.) ^ = ir (Prop. i. Inf. 2.) r =
J /
. Hence by fubftitution and redudion we deduce
Theo. VIL
Proposition 5*.
Given /, /?, and r, to find g and /•
(Prop. 2. Inf. 1.) g — Ir which is Theo. IX»
Hence r^ = /?•.
But (Prop. I. Inf. I.) ;=: ^^^llll/ =: ^llH^ bv fubai-
r — I r — I
r" — I
tution, V • y, 1 =z s, which is Theo. X.
r — 1
Proposition 6.
Given /, s, and r, to find'^ and //.
(Prop. I. Inf. 4.) g z=. . (which is Theoi
XI.) =: /r (.Ptop. 2. Inf. i.) Hence we deduce
Theo. Xir»
Proposition 7.
Given g, k, and j, to find / and r.
(Prop. 2. Inf. 2.) / = — ^ (Prop, i. Inf. i.) s =
« — I
r
-^ , by fubftitution and redudion we get j r r— x ;■
!= gr — g, and hence we deduce Theo. XIV.
Theo, XIII. is the fame as Theo. VIII.
Aa 3
6o Demonstration: '. 7 run
Proposition 8.
Given g, », and /•, to find / and r*
From 'the preceding propofitlon we have Theo. XV. and
from the equation sr — sr =^^ — g* I" the
fame prop. \^e get Theo. XVI, by divilion.
Proposition 9,
Given g, s, and r, to find / and «.
(Prop. I. Inf. 3.) Izzrg + s — rs, which is Theo. XVIL
(Prop. 1. Inf. 3 and 4.}4 = ^ = ""^
(Prop. 2. Inf. I.) which is Theo. XVIII.
Proposition 10.
Given n^ s, and r, to find / and g^
(Prop. I. Inf. 4.) g = i^irjuH:/ -- // ^ (P,op. 2.
Inf. 1.) Hence we deduce Theo. XIX.
Eyprop. S, sr — sr =z gr — g. Hence by
divifion we eet Theo. XX.
Cor. In the original equation s — /= rX s — g, when.
the !eaft term /:= o, then s = -l^, which is Theo. XXI*
r — I
and hence the reft are deduced. In the demonftration to
prop. I. in tlie theory of circulating decimals, it is afferted
, n R R R , . ^ . R
liiat — -i — - -{ — : -] , &c. ad infinitum, rz .
r ?•* r' r+ r — i
Demon, x ~ r x s — Rj from above, hence by rcduc-
n
ti«nj=: . ^£. Z),
Rules of Variations, Sec, 6i
XIV. DemonJI ration of the Rules of Variations,
Pro position i.
The reafon of the rule to this propofition is evident. For
one thing, as a^ is capable of bat one ppfition a. And if
there are two things, a and h^ they are only capable of thefe
two variations aby bay and thefe may beexpreffcd by i X2.
Suppofe there are three things, abc\ abc | be a
then any two of them leaving out the acb cab
tiiird will have 1X2 variations; and bac \ cba
confequently when the three are taken in,
there will be 1X2x3 variations. Sec, Sec,
PROroaiTION 2.
" The nunnber of changes of ^ things, taken » at a time,
is equal to m changes of «i — i things, taken k — i at a
time."
•* For fuppofe ^ things, a b c d e be given, firfl leave out
<7, then we (hall have the four b c d e, out of v/hich let there
be taken all the 2's bcy b dy &c. put 'v = the number of
variations of every two out of the four quantities bcde.
Now if <7 be put in the finl place of each of them, it wilt
znake abcy a bd, See. then will each confift of three letters,
viz. t; = number of variations of every 3 out of 5, abcde,
when a is firft,
"In like manner, if b, c, d, e be fucceffivcly left out, the
number of the variations of all the 2's will alfo be ru ; and
putting be de in the firft place to make 3 quantities out of 5-,
there will ftill be 'v variations as before. But thefe are all
the Tariations that can happen of 3 things out of 5, when
a, by e, dye are fucceffivcly put firft ; and therefore the fum of
aTl thefe is the fura of all the changes of 3 things out of 5-. But
the fum of thefe is fo many times ~o as is the number (A
things; that is, jo; or m'v zz all the changes of 3 things
out of 5. And it is evident the fame way of reafoning may
be applied to any numbers ;;;, ?i." This being premifed, we
deduce a dired demonftration of the rule, for
" Suppofe any number of things a b c d e fg\ here fn=z-i,
and let // = 3, Subtra(5l i from 3, and there remains 2;
fubtraiii 2 froii; 7, and there rcTi^iLns 5. Hence it is evi-
dent.
6z Demonstration of the
dent, that the number of changes that can be made by taking
I by I out of 5 things, will be 5=-^.
** Then when »i = 6, w =: 2, the number of changes = r/n/
^ 6 X 5 zz 1; a fecond time*
" Again,, when m = 'j^ n z::z ^, the number of changes
= m'v IT 7 X 6 y 5, that is rzTwX m — i x m — 2, con-
tinued to 3 or « terms. And the like may be ihewn for any
other numbers." ^. E. D. Vide Mr. E?nerfon^ permu^
tations, prop. 2, and prop. 3.
XV. Demonjlratlon of the Rule of Cjmhi nations.
The number of combinations of 2 things out of 5, ah cde
are 10, as follow.
For with Ul V^'V;uV:
the different < c >are combined J •• v ^f, ^^, ''^^
things 1^1 \— ''''/;
Hence the combination of 2 tilings in 5p ~ la
The number of combinations of 2 things in 5 is fhewa
to be 1 + 2 + 3 + 4, and by the fame manner of reafoning
the combination of 2 in 6 will be 1+2 + 3+4+5, hence
univerfally the number of combinations of m things taken
2 by 2 will be 1+2 + 3+4+5, &c. \q m — 1 terms, an-
fwering to a figurate number of tlie third order, the fum-
, . . m m — I
whereof is — X .
1 2
Again, the combination of 3 things out of 5 are 10, as
follow.
With a. With h. With r.
a b c, a b dy a b e \ ^
• .., a c dy a c e \ 2
• . ., . . ., a d e\ I
bcdy ^f ^ I 2 I cde I I I
, . ,. bde\i
Rules of Simple Interest, Sec, 6j
The number of combiRations of 3 things out of 5, are
fr.evvn to be i -f 3 -f 6 zz 10, and the number of 3 things
at a time out of 6, by the fame manner of reaioning, will
be i-f3 + 6-fiozz2o; this ieries anuvers to a figurate
number of the 4th order> the fum whereof is — X
I 2
— — , which is the rule, P. £, />•
3 ^
XVI. Demoriftration of the Rules of Shnple Interefly
by Decimals.
Ufing the notation prefixed to the rules, we have, by
proportion, i : r : : p : rp, the inferell of / for one year ;
and I '. rp :; t I prt, the intereH ©f p for the tiir.e /;.
alfo p -\- prt HZ the arrear at the end of the time /.
Hence prtzzi, and prt -\- p = a, from which equations
the rules are deduced. The rule of Difcount is the fame-
with the rule to prop. 2, putting D in the place of a.
XVII. Dsmonf} ration of Malcolm'j Rule of Equation
of Paymerits.
The principles on which this rule is founded are given
in the note to the rule.
Let / n the firft payment, P = the fecond, / = the
time between the two payments ; r zz the ratio, or rate per
cent, divided by 100, and ,v zr the equated time from t!fte
firlt payment.
Then prx = the intereft of/ for the time x.
And zz difcount of P for the time / — .v.
rt — rx -\- I
Hence, according to the fuppofition on which the rule is
Pr* Prx
founded, /r A- = 1. Now by reducing th&
rt — r.v 4- I
equation, Scq. we get xzzt^ 4- — + / ^^ + -H — I"
but it will be found, upon examination of the problem, that
only one of thefc values will aufwer the conditions of the
queftion; and by a fynthetical demonftration that value is
< + /^-5-. + — — l-l, which is the rule. ^E.D^
64 , Demonstration of the
XVIII. Demonfiration of the P^ulcs of Compound
Inter £ji^ by Dtcimals.
Here we fliall ufe the fame notation as is prefixed to the
rules.
r zz the amount of il. for i year,, and by proportion,.
1 : r :: r : r^ amount of il. for 2 yeari;.
I : r :: r^ : r^ ditto • for 3 years,
I : r : : r^ : r"^ ditto for 4 years.
And if the number of years, or payments, be denoted by
/, the amount of il. for / years will be r , hence it appears
that the amount of any other principal fum / for'/ years,.
will be /;- for I : r : : / : pr ,\- ^r z= a, and hencc
all the rules are deduced*
XIX, Demonfiration of the Rule of Equation of Pay-
mentis at Compound Interej},
Let Ay By Cy D debts be payable at the end of a, hy Cy d
years, x =: the equated time from the firft payment. Then
d — X zz. the time from the equated time to the laft term.
j4-\-B-\- C-^- D = s. Then by Moreland's or Burro'vj's prin-
ciple (vide demon. 3 of Equation of Payments at Simple
Intereft,) we get ^/~''-f b/~~^^ c/'^' + D =z P =
sr ; by dividing by s and extrafting the root by loga-
rithms, we get x = d Qg- — og.s ^ which is the rule
log. r
exadly. ^ E. D^
Cor. I. If we divide the equation by r we get yr ZZ Ar -f-
—b ——c D i
•Br •\- Cr + — :, hence by the nature of negative indices — zr
A , B C B V
' + — 7 + + — ; ^P, ••• Pr^ = 5, and x —
ra / ^c ^d
log. 5 — lo g. P , . , . ^ ^ ,. ,
, which IS Kerfy% rule.
Cor. z. By Malcolm'^ prindple j fuppoling x tKe equated time
to fall between the fecond and third payment, we have At —
X — ^ r D
A+Br — B fum of the intcreft zn C — H -^ "— '.
Sum of the diCcov«t$. By tranfpofition ^r'^"^^+ ^r*"" +■
II
Rules for Annuities in Arrears at, &-c. 65;
CD X
h -— — J-\-B4-C-^D zz ^J divid:; by r and we havs
rr— X d — .V
y r
^a ^b C D S S yj B
Ar + 5 r ^ + = > or — ^ (r
j,c ^a ^x J.X ^.a ^o
— - A , the very fame exprefllon as we deduced from Ktrfty'^
principle.
Cor. 3. Hence we may obferve that the rule is univeifally tin?,
whether we argue from Burrc'zv''s, ATer/Vy's, or Malcolm^ s principle.
But if we argue from Cockex^^ or Hattons principle, we deduce a different
theorem. For in the former cafe we get
^/""'— >^+Br'"~*— J^zrc/"""^— C+Z?/""""— D, and In the
By proportion <
latter y/+5+C+^ X^ — Ar ^Br +Cr +Dr -
Scholium. From what has been demonftrated, Art. 7, 17, and 19,
it plainly appears, that all the rules are true according to the fuppnfitioni
on which their refpeftive authors have founded their demonitrations ;
and the reafon why they differ, arifes only from the injuftice of Simple
Intereft. I fhall only remark, that what ^'fr. Todd has written on this
fubjeft, in his folution to his own problem, at page 39 of the Ladiei'
Diary for 1789, is a compofitioa of error arid inconfifliincy.
XX. Dejnonjlration of the Rids s for Annuities in Arrears
at Simple Intercfi,
Here we ihall ufe the notation prefixed to the rules.
J- I : r : : n '. rn the fecond year's intereft,
I : r : : 2« : trn the third year's intereft.
I : r : : 3« *• 3r« tlie fou.th year's inteteii.
I : r : : 4" : \rn the fifth year's intereft.
1 : r :: t — JX« : r—iX»'' the rth year's intereft.
J 4. 2 ^ 3 4.4, &c. to /—I years x '«> gives thcwho'e intereft due
orv the annuity agreeable to the rule to prop, i, or ttj -p 2 rn + 3 r« -t-
4r.«, &c. to /— 1 years zz — ^"j the rules of progrefiion.
Hence -f" '" — ^> ^"*^ hence all the theorems are deduced.
XXI. Demonji ration of the Rules for the prcfcnt worth
of Annuities at Simple Interejt,
By Art. i 5, ^r r +/' = <»> and by the preceding article '—
4- t rznay ••• prt ■{■ p zz + tity and hence al the rules
arc<4c>luced,— The truth of the rule to prop, i, is evident.
Demonstration of, ice.
r> r- L . t:r-\-Zt — tr , 1
Cor. From above we get p zr X «» and h — ■
putting T in the place of /, and a zz l, then '-^ " ■ x
r+3 Tr + 1
ZZ p, the prefent worth of an annuity in reverfion at' Simple Interell,
and hence the tules are deduced,
XXIL DemonJiraUon of the Rules for Annuities in
ylrrears at Compound Inter eft.
Here we fliall ufe the notation prefixed to the rules.
« rr the money due at i years end.
» + r» rr arrear at 2 years end.
« + rfi + r*«rr arrenr due at 3 ye^rs end.
n -^ r >i •\' r^ n ■]- r' n ZZ. arrears at 4 years end.
K+rw-fr^w -}- r's + &c. to «/•' ' rz:. arrears due at the end of t
years. By the rules of geometrical progreflion the Turn of i -|- / + r^
4- r^, Sec, ... to r'""'ziLlZi ^^^ .unount of il. for / years.
r — I
rf I
*.* X «Zl^j and hence air tlie other theorems arc deduced.
XXIII. Demonjlration of the Rules for the prefent worth
cfAtinuitiei at Compound interejl^ ^c.
By Art i?. ^/ zz^j and by the preceding Art. X « — 'J*
/ »■' — I
'*• t r zz. X «> from this equation all the theorems are deduced.
1— V
Cor. I. From the above equation we get X nznf, by
■dividing the former part of this equation by r we get
,_ix.-^+^
X rZZ py which" is Theo. I. of the prefent worth of Annuities in Rever-
fion at Compound Jnteieft j and hence we deduce Theo. II.
Cor. 2. From Cor. i. we have X « — ^5 but when t is
infinite, r^ is infinitely greater than i, or i is infinitely fmall in com-
parifonwithr ••• ?— i ZZ r , and hence X n ZZ py from
which equation all the rules are deduced for buying and felling Freehold
FINIS.
.m
I IP lamiv ""^^-^O^-
-Vl^-'
.h:'
'%■
1^*
^
^■\
^' ■:-•.■ ^■■:.^: ■'/:'''■' ^■^'
":','•. ; . / "