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Csimbian Stmn ct Sclgaal goohs.
ADVANCED ARITHMETIC
TOR
CANADIAN SCHOOLS.
BY
BARNARD SMITH, M.A.,
St, Peter's Collie, Cambridge,
AND
ARCHIBALD McMURCHY, M.A..
Uaiversity Collie. Toronto.
5lttt|iarue> bp t^t Counril of flubUc Jwftrttftbn of mnUxto,
TQRONTO:
COPP, CLARK & CO., 67 & 69 CpLBORNE STREET.
I 87 I.
Entered, according to, Act of the Parliament of Canada, Ixi the year
One Thousand Eight Hundred and Seventy-one, by the Rev
Egerton Ryerson, LL.D., Chief Superintendent of Education
for Ontario, in the Office of the Minister of Agriculture.
The favouraUe reception given to tke " Elementary
Arithmetic'' lias induced the Authors to prepare this
Treatise for the tise of the more advanced pvpils in the
Schools of the Dominion, with a view to completing the
course of instruction in the subject,
Toronto, July, 1870.
I
CONTENTS,
I
SEOtlON I.
Definffciong, ITotitfoii, and NtunMAtiott ^^*i
fifanpfo Addition •••-... 9
" Subtraction ,, ./....[,.,,,,, "^*
Boman Notation. !..'..*.*!..'* ^'*
Simple Multiplication. ......!.. ^
*' Division !......!.!... **
M»»»lI«neolis Qrt^stioM, &o....,.,.. ...'*.' 5
SEOTIOl^ IT.
Conctete Numbers (Tables). Money ^
Measure of Weight ^
" Lengtii,....*.. *.**.'.' ••• ^
•' Surfaoe... * •• ^
" Solidity .*.'*.'!!.'.*;.*.;; ^
" Capadly • ^
" Time ••• ^
Beduction. *'* **^
Oompoond Addition. ....'.'!.'!.*.**...*.*.*' ' * ^
" Subtracticm ............. ^
** Multiplication.... ••••.... 60
" Division ' ^
New Dedmal Coinage ..'...'.* V* ^'^
Miacellaneotta Questions *^ ** ^
• 77
^ SECTION III.
Oxeatect Common Measure
I«ast Common Multiple .".** ^
^^ ' SECTION ly.
Vulgar Fractions..!!... ' ' * '• *•
^taaitioB of VidgarFr^iM. **"*"" '*' -• ^
••........ i . i , 4 loa
ooifncNTa.
Subtraotfon of Vnlgar Praotdona . , '*°"
MaltipUoation of VulgarFractiona. **'*** ' ??1
DivUrion of Vulgar Fractions. . . ^^
Reductionof Vulgar Fractions..!!.'! ^^
Miscellaneous Questions * ^^^
Bedmal Fractions ....;../ ^^^
Addi'tion of Decimal Fractions ^^'^
Subtraction *' u '" 120,
MultipUcation " «c ""' "••••• 123
Division ♦* ti * " • • • • • 123
Circulating Decimals .....! " " ^^
Reduction of Decimals ...!!.!!.. 129
Miscellaneous Questions, &o. ....!..!! " * " * *' 1^
Practice ^^1.
Miscellaneous Questions. «5kc ^"^
^ ••••;•• • 157
'... SECTION V.
Ratio and Proportion
Rule of Three .!.....!!.. • ► . . 166
Double Rule of Three. . . !. . *! ...!.. * " ^^^
Simple Interest ' ^^^
Compound Interest ^^^
Present Worth and Discount * ' ^^^
Stocks "!;' 199'
AppUcations of the Term Per Cent. f^^
Division into Proportional Parts ..!!!!.! '
^mple Fellowship. ! ! - ... 221
Compound Fellowship ! ! ! ' ^^
Equation of Payments !!..!! ^^
Exchange 225
Value- of Foreign Coins. ^^'^
"* * ' J • 232
SECTION- VI.
Square Root
Cube Root !!..!!!!!! * ^^
Sdales of Notation. ..!... !.*.*.*!,'.*,*!"* ' ' ^^
AppUoation of Arithmetic to Geometry ' * ' ?S
ISxamination Questions . p..... xK55
• 268
ARITHMETIC.
SECTION I
« ^rZ\L^J. "^ " •""■" ' ""«'* »«-' -'■"-ft <»-w-a
2. KuMBKB is the Mme by which we signify how many objects or
h ngs .^e oons.d.red, whether ^ or more. When, for i^.tonrwl
beof Ir "o™' '"» W'»' 'h™^ yrfB, or four hours. the^\,r
8. NcMBnas are considered either as Abstbaot or Conoekti.
ti..wt^,' T" -f " f " *^°" ''''°'' ''»™ "O "f^onc* to any par-
ticular kind of unit; thus, five, as an abstract number, signifiM fi o
umts only, without any regard to particular objects.
Concrete numbers are those which hare reference to some n-r
rrh^iibrniti ;rr sr r^/n ^^^^
number, having reference to 27^2 Z:l hi ^ry^
one horse, respectively. ' ^^"»
4. Arithmetic is the science of Nombers.
M.« fi ^" "^"^^^''« ^^ «°™™on Arithmetic are expressed by means of
itselt, and nine significant figures, 1, 2 8 4 B « T « o „.,■ J j
respectively the numbers one, two thre!: fon'r fll ' . ' T"
nine. These ten figures are .Jmetrie^:^,^"^ ' '"' "™"' '^•"'
..^.m I
10
ARITHMETIC.
^ Jh. namber oae, which i. „p^„,ea by .h, fl^,o 1. i. ^^
or imri^;:"vZ°'thur»'e^"' '"""" '^ """- " «^p«-» "» «!-pi,
pre«o. ton t meTiU 11 Jr .1 ""^ """"''" "«""■ " "'*»«-
»r:;rr„orbra:;ir''^ f • -"'-"-^^^^^^
foar anils and L 2.1 "" """'• '"S'""" ""h ««» timei
for oacb xxr tb.r;r: r " ~ ""-^ • '^'^'"^ '—
. tbeL"jrrtst"tf„:Vf\rj"°^^?"*""'«^
of anils, or hundred, of 1,. ^' T *?' **' """« ^ *«■« "f »«■»
hundred, of *nT'*,t;'2''V ;«'™,''»"/™-'' of "nito ,o ten, of.
ten.ofthouZdr'of niii. 7 f ""'^ '^'°"' *ho'"«>'«ls of unit, to
ten, of tbo„,rnt It ^ ' tI'"o\l'''°r:!' ?' -'" ^ *»» of
tmweoo.etrbX°tllT^urrf;il-^^^^^^
one unit; or, t^7t fa briefly Z^ »»« ton of unite, together ;ith
17 i« ,: •""".""Ofly^HeleveK. Similarly 12, la U IK m
re«, Vwei A: XTu^f ''^,;- -"» ; '"^^ - re.peetivei;
nineteen. ' '°*"' '"''«"' eeventeen, eighteen,
twenty.fo„;7i;Tv^' tiTn": -r\::„r-''''' *"*"''-'"-•
twenty-nine. iwenty-,ix, twenty^Ten, twenty^ight,
re 1, la otllad
DEPI.WI0N3, NOTATION, AKD NUMKBATroK, U
87, 88, 89, which are respeotivei, ro«<l thirf,. .i i . ' ' • '•
eight, thirty-nine; V. thJ^ilt \.Tjj ' "''"''J'-'*™'. 'Wrty-
70 („v.„ty)^ 80 (eigM^ 90 (oT„" W ^ '^' '" ^'"'"^" "'' <""')'
ne,tnnmb6rtoThi.riOO wh?-i '^*"'*' ''* ■"»• """•i *>■•
hundred of n^i^oi h!r wift r/Z"'',''" •!•■•' °' """' "' »-
»ni^; <"...>tl.brle%;j:lr„nd;:^''^ »n.^ together with no
of hund^ of nn.t, or thon^n^f nn'it ^^i^'^Tn'titZ' f""
will reprewnt »o many ten, of thonsands of nnitra^d «, „„ ^
-Kr:i:h*«;rjtl ;t h'^^Ff - - -
and five. ' ' ^* '* '* ^"^^^J^ >'««d. t«o hundred
hundred, oTonit, Cther^.TT''' "^ """^ '»««"'»' "!«> "ven
--.its; or, .. U b Srj ™ '*"' "' ""'*»• '•Sether with no
hnnd;ed"'nd\Mr!;. ""'^"^ ''"™ million^ forty thousand, ,»v.a
.eti whi:r°:.TreX';v.riS"%r -r ^^««"- -
Notation: 1» Am^«o -^sTnonJ ~ "* two method, of
p«i^™y-^xiror:rhr:.^:::S.^^^^^^^^^^^^^
the^yJTorottttr^^'r "8T''r''t"™''"' "^ "-• »'
brought into Ekr^; L the 111 'A^^""'?'''"'''''"' "^ **"' "»
Ababio Notation It w.."^! ^^J^^ "." ^^"'^""^ "I"" ««"«« the
This meth J:rn:;atio J," ow^:l '^ '''' ^™'" '^^ "- Hindoo.
Empire, but throughout Europ" "'"°"' ""• """ """^ '» «» ^^^'^^
12
ArvITHMETIC.
Ey. I.
Exercises in Notation and Numeration,
v
Express the following numbers in figures :
(1) Sixty-tbree; eighty-one; ninefcy-nine; forty; thirteen. ^
(2) Two hundred ; three hundred and three ; seven hundred and
Bixty-four ; eight hundred and eigbty-eight.
^ (3) i'our thousand; one thousand, four hundred and seventy. one;
I ix thousand, nine hundred and thirty ; nine thousand and nine.
(4) Twenty-seven thousand, five hundred and four; thirty -three
tliousan'J ; nine thDusapd and sixteen.*
(5) One hundred thousand ; six hundred and seventy-six thousand
r.nd fifty; two hundred and two thousand, ave hundred and ninety*
three.
(6) Seven milliors, three thousand ; eleven millions, one hundred
nnd eight thousand, pne hundred and six ; fifty-four millions, fifty-
four thousand and eigi^ty-eight ; six hundred and thirteen millions,
t;7enty thousand, three hundred and three.
(7) Two billions; nine billions, thrse hundred thousand and
twenty-one; ninety-four billions, ninety millions, ninety-four thou-
sand, nine hundred and four.
Write down in words at'fuU length the following numbers :
(1) 43; 60; 88; 97; 69; 12; SI; 19.
(2) 256 i 401 ; SOO ; 999 ; 865 ; 578 ; 837.
(3) 2000; 1724; 8003; 7584; 1075; 4541.
(4) 87008; 47049; 63090; 80008; 841323.
(5) 6850406; 8080808; 7849630; 41825^,
(6) 10000001 ; 20220022 ; 92568937 ; 80180070.
(7) 2560680200; 800809560; 9738413208.
(i) 7070000423 ; 987654321 , 5707068080;
(9) 100198700010090 ; 43720370t)34108264.
ADDITION.
11. Ajjduiois la tiie method of finding a number, which is equal to
two or more numbers taken together.
ADDITION.
13
The numbers to bo added together are called Addends. •
The number found by adding two or more numbers together is
called the bum or amount of the several numbers so added. ,
12. There are two kinds of Addition, Simplb and Compouito
It^ Simple Addition, when the numbers to be taken together are
all abstract numbers; or when they are all concrete numbers of the
same denomination, as all pence, all days, aU pints.
It is Compound Addition, when the numbers to be taken to'rether
are concwte numbers of the same kind, but of different denominations
of that kmd ; as pounds, shillings, and pence ; or years, months, and
days; or gallons, quarts, and pints. . .•
U }li ^^^ l^^" "^ ' ^'''^^' ^^*'''^ ^^*=''®®° ^^^ or °»ore numbers, signifies
that the numbers are to be added together : thus 2+6+7 signifies that
2, 5 and 7 are to be added together, and denotes their sura.
The sign =, equal, placed between two numbers, signifies that the
numbers are equal to one another.
The sign , Vinculum, placed over numbers, and the sign < ) or
U, called a BEAOKET, enclosing numbers within it, are used to denote
that all numbers under the vinculum, or within the bracket, are equally
affected_by all numbers not under the vinculum or within the bracket •
thus 2+3 or (2+3) or {2+8}, each signify, that whatsoever is outside
tne vinculum or bracket which affects 2 in any way, must also affect 3
m the same way, and conversely.
The sign .*. signifies 'therefore.'
SIMPLE ADDITION.
U. jRuxE. Write down the given numbers under each other so
that units may come under units, tens under tens, hundreds under
hundreds, and so on ; then draw a straight line under the lowest line!
Find the sum of the column of units ; if it be under ten, write ii
down under the column of unit«, below the line just draw^ IT^
ceed ten, then write down the last figure of the sum under the column
of units, and carry to the next column the remaining figure or fi« r^s^
treat eanh «np.n,.«/i;n» ««i.,^^ .•_ .l. s "guio or ngures,
f 11 " « ~. ® ----^«:.:.i ;u liiu same way, and write down fii*
fan eum of the extreme left-h^nd column. The'entire ^Zt^Z^i^
down wU U the ,«m or «r.o.™t of the separate n»mbe« ,
14
ARiTBifisna
k
Ex. Add together 5469, 748, and 27.
I ^P«>««®ding by the Eule given above, wf dbt«in
5469 X
748
27
6289 '
n. «««>. oe JtuU W« app^J^ fh,fiTi>^^ eonM.raU.»^
When we toke Oe .u» of r noito wd 8 anit, ud » anita w« «t
12 hundreds; we therefore place the 2 hundreds under tLcoWo?
"zttr^ tr^^ds"^ ^ '''-' -"^ - theUoS.-:^,:;
The above example might have been worked that, vottiiiir dA*« .*
full length the local value of all the figures. ^ * ^ **
Thus 6460=5000+400+60+9
+748= +700+40+8
+27= +20+7
Now adding the columns, we get the Bom
=5000+1100+120+19
=50OO+i000+lO0+lO0+2O+io+9
(smoe 110()=1000+100, 120=100+20, and 19=10+91
, =6000+200 r 80+9. ^
(collecting the th^uaands together, the hundreds together, and «> on)
NoTB, The truth of all results in Addition mar be proved bv .^^
log them downwards; if the resolte h« *h* -.^« *v/.A_.. !r -
^ Will m aU probability h^^.r^'v^i^'^::^^'''' *^ ^"
^oit% we g«t
I of nnitg, and
iniQ of tew,
1 13 tens ; we
carry on th«
nndreds.
linndrediB, is
le column of
column, viz.,
>asan&; we
id the eotire
ing down at
od by a<?d.
I then edd*
ion In eaoli
SIMPLE ADDITION.
Ex. U.
J^mplM in Simple Addition.
15
0) 266788
21008
40036
21
looobi
(2) 627432
648201
678641
648200
968769
845678
(8) 892764
98687
9482
100
16284#
11
(6) 792086679
909672916
496987660
868268926
160470279
918066988
672409308
642984946
709866197
994888772
186988976
847090439
706050896
686676
842878
(7) 92837466821
89766689079
42900
69678009678
87064671498
46902768637
94666672990
63749668
78694698846
66789987664
49696878897
84309724606
29681371989
11234662
23678521
(4) 1807868
298748
6987
760008
247
60706
(6) 117064
92978
827669
861
777777
66666
(8) 186467902
920749866
467468729
, 814382267
876914936
743790061
210706090
634721987
698765482
119647896
962081688
749827966
67498167a
824680867
891076818
(9) 17649+86 + 8910400708 + 8666000+9687+7160030090074+498
'*'®®''^^^®®'*-^^®^+^<^»+»O8S8094O0094.
(10 ^9800071+605040+907+80056049+60005+00087078
+ 9200299200+1002008.
+8076089+882 + r485998+6789«8+l»+((«»or4.^
1(5
ARITHMETIC.
A«i ^ f«? *^^ *""'" of- 4738685, 237869513, 148794343978 «««
.eventy-d,; forty tbousand\kd thirty r 1 .l.n""'*?
hundred and one. 'uireysii, ten thousand, nine
three »aii„o. .e™nt;:i^ro:l„randXt„'rn"^^^^
eleven millions, six hundred and fifty thousand ^.i fifty ^^eh?
dred and twonty-six n^iliion., seven thonsA'ine hundred Z
ninety-one ; one thousand seven hundred Ji\ tT,! -n-
thousand seren hundred and ten ■ ^T /"n- ""''"""• »'"'
thousand an,' five. ' qnadnlhon, three hundred
«>a,Be,ardins,»aaOO0O. .i„d the total '^IfrLe'^adt:
St. Laurence, tea^rcri^ a^ ^^It^ti^^tL'^^^
lington Bar Canal ^14qi7 • q* a » "• '-'"™^ock, 1^28324; Bur-
Bideau Cai,s,,^aa^!tlktatnl^:±„r^i,,«r« ^^
»ii»7»T. find the whole sum oolleoted. " " """^ ""
vesseid.
"^(k^^i^H^u^
0, 9746, 5769,
6408, 60646,
also 936473,
t343978, 866,
, 7055591234,
7978462, 333,
1112868678,
sighty-three ;
usand, seven
I sixty; fifty
»usabd, seven
hundred and
lousand and
usand, nine
millions, six
I and sixty-
lundred and
three hun-
mdred and
illions, one
3© hundred
17000000 ;
be Ottawa,
Burlington
e Oanadiau
1866, from
, $174603 ;
324 ; Bur-
ttawa and
on vessels.
eUBTEAOlION.
1*
(20) The Exports of Canada for the years 1866 Ifififl iftK'r
the^^ a.oant of the e,p„H» of the ooa„V for th„t lyeaf
(21) Iho value of Exports of Canada for the years imTm
$10451609, $18846986 iiaglfifiift A °'^°''7 ""^ *^* ^»^e«*.
SUBTRACTIOlir.
15. Subtraction is the method of finding what number remains
when a smaller number is taken from a greater number.
The number to be subtracted is called the subtrahend- the nnm
her subtracted from, the minuend. bi^btbahend , the num-
The number found by subtracting the smaUer of two numbers from^
the greater is called the Remainder.
whf!.h" vt'"'/"' two kinds of Subtraction, Simple and Compound
which differ from each other in precisely the same way, in S
Simple and Compound Addition differ from each other. '
17 The sign ~, minus, placed between two numbers, signifies that
the second number is to be subtracted from the first number
SIMPLE SUBTRACTION.
18. RiTLE. Place the less number under the greater number b6
huirr;:? T' r "' ^^-^ ^^^^^ ^^'^^ hnndrs':;^:
iiundreds, and so on; then draw a straight line under the lower
Take, if possible the number of units r. ach figure of the lower
line from the number of units in each figure of th« «>!; i?!! ilTl'
driwn '"^T'^^'fy "^«^- it, and put the remainderVelowthe^iinr W
drawn, units under unite, tens under tens, and so on ; but f the Vi^te
18
ARITHMETIC.
in $!(ty fignre m the lower line exceed the namber of units in the figure
ftbove it, add ten to the upper Aty^ *e, and then take the number of
nnitB in the lower fignre from the iiumber in the upper figure tbui
Increased ; put the remainder down as before, and then carry one to
the next figure of the lower line. The entire difference or remainder,
ao marked down, will be the difference or remainder of the gfyen
numbers.
Ex. Subtract 4988 from 6128.
Proceeding by the Rule given above, we obtain ^
6128
4988
185
flo that the remainder iis one hundred and eighty-fire (180).
7%s reatonfor the Rule mil appear J¥om the folUming eomiderationi.
We cannot take 8 units from 8 units ; we therefore add 10 units to
the 8 units, which are thus increased to 13 units ; and taking 8 nnits
ft'om 18 units we have 6 units left ; we therefore place 6 under the
column of nnits ; but having added 1 ten units to the upper number,
-we must add the same number of units (1 ten units) to the lower
number, so that the difference between the two numbers may not be
altered ; and adding 1 ten units to the 8 ten units in the lower number,
we obtain 4 tens or 40 instead of 8 tens or 80. *
Again, we cannot take 4 tens from 2 tens ; we therefore add 10
tens or 1 hundred to the 2 tens, which thus become 12 tens or 120 ;
and then talcing 4 tens or 40 from 12 tens or 120, we have 8 tens or 80
remaining ; we therefore pkce 8 under the column of tens ; hxA having
ad^ed 1 hundred to the upper number, we must add 1 hundred to the
lower number for the reason given above ; and adding 1 hundred to
the 9 hundreds in the lower number, we obtain 10 hundreds or 1000
instead of 9Q0.
Again, we cannot take 10 hundreds from 1 hnn^red, and we there-
lore acid 10 hundreds or 1 thousand to the 1 hundred, which thus
b99ome9 11 hundreds or 1100 } and taking 10 hundreds or 1000 &om
iff eormderatiotu.
STTBIBAOnOir.
11 hundrecTs or liOO. we have 1 hundred or 100 left; we therefore
or 1 theuwnd to the upper number, we must add 1 thooaand tn thl
the 4 thousanda m the lower number, we obtain 5 th^da or ffOOOi
50OO taken from 6000 leaves ;
therefore the whole difference or remainder is 18^.
ft^lll^^Jw? ^'"'?^^^ "*^^' ^''^^ ^««" worked thus, pnttinjr down at
tan length the local values of the figures ; ^
6128= 6000 +100+ 20 +8
= 4000+1000 +100+ 20+8
=4000+ 1000+100+io+lO+S
=4000+1000+110+18
(ooUecUng the first 10 with the 100, and the second 10 with the 8), ^
4938=4000+900+80+8.
wt'^f^H '^^''^'f ^ '^' °'^°°^°^ '^^^^^ fr«°» thousands. Ac.
we get the remainder or difference ""usanas, «o.,
=100+80+5 '
=186
Ex. in.
Meamples in Simple Subtraotim,
(1) 1000000
100101
(2) 400367261
99988877
(8) 89487183
16790298
Iroitl ^"^^ *^* difference between 6548766 and 412848- 7rt6afl«* i-^
C26967; 803288384 and 192001222. ' ^ '^''^
, w-
2a
ARITHMETIC.
(^) How much greater ia 164826289 than 48476T98 ?
10000001000 than 7077070077 ?
7669030640021 than 6990040006679 f
. (6) Take two thousand aud nine, from ten thousand and ninety.
«x ; three thousand and eight, from seven thousand, nine hundred and
forty, four.
(7) Required the difference between four and four millions- also
between one hundred millions and three hundred thousand.
(8) Subtract five hundred and eighty-four thousand and seventy-'
SIX, from fifteen millions, one hundred thousand and three.
(9) The Revenue of Newfoundland for the year 1866 was
$716287.97 ; the Expenditure, $662783.15. How much did the
Revenue exceed the Expenditure ?
^^nS!?/""* *^'^ ^^^"^ ^^^^ ^^^ ^°^?°^<^^ ^^*o ^^^ Brunswick were
$10000794; the Exports, $8186185. How much more was imported
than exported ?
(11) The Imports into N'ova Scolia for the years 1865 1866 were
respectively; $14381662, $14381095. How much less wL imported'
during the latter than tl^e former year ?
19. The following method of expressing numbers was used by the
Romans, and it is still in occassional, though not in common use among
ourselves. They represented the number one by the character I • five
by V; ten by X; fifty by L; one hundred by 0; five hundred by D
or Iq ; one thousand by M or CIo.
All other numbers were formed hy a combination of the above"
characters, subject to the following Rules :
1st, When a character was followed by one of equal or less value,
the whole expression denoted the sum of the values of the sin^rle
characters; for instance, II stood for 2; III for 3; Yl for 6 • VIH
for 8 ; LY for 55 ; LXXVII for 77; OOXI for 211.
2d. When a character was preceded by one of less value, the
whole expression denoted the dtj^^erence of the values of the sinrfo
characters; for instance, IV stood for 5-1, or 4; IX for lO-l or 9-
XIX for 10+10-1, or 19; XL for 60-10, or 40: XO for lOO-lo'
or SO.
i I
?
0077?
040005679?
ind and ninety-
He haudred and
r millions; also
and.
id and seventy-
ree.
ear 1866 was
much did the
Brunswick "were
e was imported
865, 1866 were
i was imported
MULTIFLICATION.
«t
ras used by the
ion use among
laracter I ; five
hundred by D.
of the above'
I or less value,
of the single
I for 6 ; Vin
?ss value, the
of the single
r 10—1, or 9;
for 100—10.
iJl r^' 9 *"°?*^ ^ ^^ '"^'^^'^ ^^ ^«^»« of the latter
f^rth 'a / ^"«^^°°^' loo Stood for 5000 ;looo fur 60000; and^
vl; .ft 77 ^ n*^^"'^ ""^ ^"°^^«^ ^^ Ola increased the
CCrr.LT nnnL*''^^^'^' ^'' ^^^'^°^' ^^^00 3tood for lOOOoI
OOOIOOO for 100000 ; and so forth. '
vJl\ t lino drawn over a character or characters increased the
value of the latje,- a tJiousandfold ; for instance, V stood for 6000*
lor 100000 ; IX for 9000 ; and so forth. '
It follows then that either XXXXVI or XLVI will represent 4«.
Ex. IV.
222 feOOri^S.^'' ^'""'^ '^*''°''"' *^'''^' forty^ight; fifky-nine;
LAIX; COXVIU; VI; OLDCIII; MMO.
MULTIPLICATION.
20. Multiplication is a short method of finding the sum of any
given number repeated as often as there are units in another giten
number; thu« when 8 is multiplied by 4, the number produced by
r.n^f!,"''™^? .'"^"''^ '^'''^' ^''^ ^^" *^« multiplicand is to be
repeated, is called the Multipltke. '
The number found by multiplication is called the Product.
The multiphcand and multiplier are sometimes called " Factoes "
because they are factors or makers of the product. '
21 Multiplication is of two kinds, Cimplb and Compound. It is
termed Simple Multiplication, when the muUiplicaodTsTther an
abstract number or a concrete number of one denomination.
'! Itw'f;!^^!"^.? Multiplication, when the multiplicand co..
"" '""' " ^^ao one uenumination, but aU of the 8&m«
kind.
n
I i
itfimofsnc.
W. The sign X, pIuMd b«tveen tTTO nnmbert, tignifiM tbtt tb«
nombers are to be luiltiplied together.
as. The foUowing table ought to be It j-ned correctly :
1
2
8
4
5
6
7
8
9 10
11
12
2
4
6
8
10
12
14
16
18
20
22
24
6
9
12
16
18
21
.24
27
80
88
86
8
12
16
20
20
26
24
80
28
86
82
40
86
46
40
60
44
66
48
10
16
60
12
18
24
80
86
42
48
64
60
—
66
72
— -.
14
21
24
28
82
85
40
42
48
49
66
68
72
70
80
77
88
84
8
16
66
64
96
9
18
27
80
86
40
45
60
64
60
68
70
72
80
81 90
99
110
108
10
20
90
100
120
11
22
83
44
56
66
77
88
99
110
121
182
12
24
86
48
60
72
84
96
108 120
132
144
the third hne show, the sever.! pr^doots, when the figures in tt>r«^
line tt« respectively multiplied by 8 : and so on.
he ^<^-0»« of th* ftctors, namely, the maWplier. most Mcess^Ur
^.„" '"" ™'. '"'""»'".• "nee it woold be sbsLrd to^pei^I
4 "Twrc^^ ^ * """"T-.,, ^^ •"" """'P'^ « .hui" by
•iin- vT ? *'"' '""'^ «^"""8» «>ere «ra in fonr fime. rix
^; b«t there i, „o me«.i„g in .U ddmng. m„l«p«rby 1
lignifiM tbst tb«
tly:
11
12
22
24
88
86
44
48
66
60
66
72
V7
84
88
96
99
108
110
120
121
182
132
144
W8 the product
Srst line, whei
the Pesp<M$tiye
lioh theyariae:
ires in the first
ast r.ecessaril7
to «peak of 6
^ shilliiigs by
fbnr^mes six
Qltiplled bj 4
SIMPLE MULTIPUOATION.
SIMPLE MULTIPUOATIOir.
2d
! i.niS^'teS^.!!'^ ^^' T^^P"*" °"^*^ ^* mnltiplicand, «nit. under
moltiphoand, beginning with the nnita. bv the Lnr^Tn ♦k * , ,
I^tl?;,!^. *• '*""• '^" »>«-•* »<» ««• "- wm *S U^
8<S. A niMnb^r which cannot be Mparated into foAf/^r. «i.i^v
SJ. A nunbw which ou> be MpartUd into Aetdnr n»i>aiti»l.
\^^ -mity. orwhioh, in ,^., words. TprSbTSD^
.,/?\ £ '•"^** oosimuB, icnLT.n.,oATio», „a iinoe 2 xY^i
te. I^wf • f?K •■•'''*''''=^''' "• -haU of conr«'pbL„^^4
SnT* «I« w '!. °'^"P>J' -"y "-""^r by 72, or by to
Kr' ' ••'"~"'^°«» mnltipUctionj «.d «o of «,y oth^
««r«* KthanwltiDlieirdoeanotftTfliuwt io ♦!. t^_t_. ..
•nectea easUy in one line, by means of the TaWa giv#» ^abova.
— 4
24
ABiniMETIO.
Ex. Multiply 7864 by 807.
Proceeding by the Bale given above, wo obUla
7664
68678
68880
22962
8088688
ne reoion /or th« Sul$ will appear from the /olUwing eoneidmtUon^
When 7664 is to be multiplied by 7, we first take 4 seven times,
which by the Table gives 28, .'. e., 8 units and 2 tens; we therefore
place down 8 in the units* place and carry on the 2 tens: again, 6 tens
taken 7 times ^ive 35 tens, to which add 2 tew, and we obtain 87 tens
or 7 tens and 8 hundreds ; we put down 7 in the tens' place, and carry
on 8 hundreds : again, 6 hundreds taken 7 times give 42 hundreds to
Which add 8 hundreds, and we obtain 46 hundreds, or 4 thousands
and 6 hundreds ; we put down 5 in the hundreds' place, and carry on
the 4 thousands : again, 7 thousands taken 7 times give 49 thousands,
to which we add the 4 thousands, thus obtaining 68 thousands, which
we write down.
Next, when we multiply 7664 by the 9, we infact multiply it by 90 •
and 4 units taken 90 times give 860 units, or 8 hundreds, 6 tens, andO
units : therefore, omitting the cypher, we place the 6 under the tens'
place, and carry on the 8 to the next figure, and proceed wif;. m
operation as in the line above.
^ When we multiply 7664 by the 8, we in fact multiply it by 8Go • and i
4 multipUed by 800 gives 1200, or 1 thousand, 2 hundreds, tens, and
units ; therefore, omitting the cyphers, we place the first figure 2
nnder the hundreds' pbce, and proceed as before. Then adding np the
three lines of figures ^h^Jl we have just obtained, we obtain the
product of 7664 by 89j".
ce 4 seven timeg,
«; we therefore
is: again, 6 tens
e obtain 87 tens,
place, and carry
42 hnndreds, td
or 4 thousands
3e, and carry on
^e 49 thousands,
lousands, which
ultiplyitby90;
Is, 6 tens, and
under the tens*
oceed wi' . ♦>«»
itby8(n/j and
ids, tens, and
le first figure 2
1 adding up the
we obtain the
SIMPLE MULTIPLICATION.
.trS!i:Jrr;.:ftir«xr<^ '''"^^"'^^^
7654=
397=
7x1000+ 6x100+ 5x10+ 4
8x100+ 9x10+ 7
which:
.„ . *» >* 1000^^2^000785 X 10+fig
68x10000+ Wxl000+45xl00+a6xin
!?l^????^!i!:^0000+ 15x1000+1^
J<Ok 10000+ 1x100000
+ 8x100000+1x10000
+ 1x100000+1x10000+8x1000
+9x1000+ 9x100
+ 7+100+1x10
"aooooooTiuTioU^^^
..oooooo.:oo.oo..oooo.Io;iM;^r£SSSirao
=8000000+80000+8000+600+80+8
=8088688
•nd ero be maltiplied iySaw we^a"e "^ ""^"P"'" "' <««'' '
263
6200
526
1678
1680600
' •iSi !'
1824000
nil ' 1
Aii# reason is* clear • for in *u^ o j.
2 Clear, for in the first case, when we multiply bt
2$
ABIXBMBZia
Ili«^«, ^ feet irfiin«lt!|dy by 5QD; mdS ^nltipUod byfiOOglm«00-
in the second case, the 7 mnk^lied by tiio ,2 is the same au 70 mall
tmlied by 200 ; and 70 innltiplied by 200 give^ 14000.
80. Jf the MuLjjipuR contain any cypher in any other p!ace' then,
m moliiplying by tlie different figures of the multiplier we niay pl^
o^er th^ cypher i^taWng care, however, when we multiply by th; next
figure to place ttie first figure arising from that Lmltiplication mider
the third %ure of the line above instead of the second figure. Tho
mS!Snl K f. /""'^ ^^^" ""''' multiplying by 20»%to,.e
II «|^«Pl7 by the 6 we take the multiplicand 6 times, when we mul-
tow ^^ ^^ **^^ ^^ multiplicand, not jJQ times, but 200
nf i!^;J^*° ^^"^ ?°'^' "* *^ ^^ midtiplied together, it is a matter
of mdifferenoe, so for as the product is concerned, which of them be
7"!^^* f °^"P^'^*°^ ^' multiplier; in other words, the product
om^^r^'?"v J^r*'! ^^""*^ wm be the samells the product
oft^e second maltiplied by the first.
Thus, 2 x4«2+2+2 f 2=8,
4x2=4+4 =8;
therefore the results are the same, that fs, 2 :;< ^=4 x.g.
•n„r'fl!t'' P'^"°* «^/°« ^"'"^r muitipUediy ai,<^tier, will be
equal to the product of the latter multiplied hy the W.er, may
perhaps appear more clearly from the foUowing mode of showi^ thto
equahty m the case of the numbers 8 and 6.
8=1+1+1;
•%«X5=:(l + l + l)+(l+l + l) + (l + i + ij^.(l^l^^^^^^j.j
=1+1+11 '
+1+1+1
■fl+1 + 1
+1+1+1
+1+1+1J
=15.
♦.v^T:*^ we regard, the <m« fi-om left to ri^t, ^re are 8 one,
laKen o times; if we regard them taken from ton ta bntfnm
.> 1 tt
♦JWiJ^ated 8 times; and the number pf onw in eapli'^eiittio
SIMPLE MUI4TIPLIOATION.
2T
re are 8 om9
82. The accnnicy of results in Mnltiplicatioii is often tested by the
following method, which is termed "oASTma our ths rais''- add
together all the figares in tlie multiplicand, divide their sum by 9
aad set down the remainder; then divide the snm of the fignres ii!
the ranltiplier by 9, and set down the remainder; mnltiplj these i
remainders together, and divider their product by 9, aod set down the ^
remainder: if this remainder be the same as the remafaider which
results after dividing the product, or the sum of the digits in the
product, of the multiplicand and multiplier by 9, the operation is Terr
probably right ; but if different, it is sure to be wrong.
^. ?'!. *f'' ^^^""^ "P^° *^® ^^^^ ^^ "i^wy number and l&e sum
of Its digits be each divided by 9, the renudndera wiUbe the same " •
the proof of which may be shown thus : *
100=99+1,
where the remainder must be one, whether 100, or the sum of th«
digits in 100, VIZ., 1, be divided by 9, since 99 is divisible by 9 without
a remainder. * w*i«uub
Simihirly, 200=2x99+9,
800=3x99+8,
400=4x99+4»
600=6x99+8,
&o., Ac.
Hence « .ppear. th,t if 100, 200, 800, 400, 600, *«., be ewh djylded
.180 divided by 9, the two rematodere in cmIi w iriU be the tame
Also the number 632=500+80+2 "'^ii, ,, •
=6x100+8x10+3
=6x99+6+8x9+8+3:
m^e'nA^^T ^^ 5 *^" ^^^ ^^^<*^' ^^^^^ «nd^, whil
wilt Jk^'o ?°°'-''' ^' "^"^ ^^^^^ ^y »» «»« remainder
will DC 5, 8, 2 respectively; and therefore the remsinder »kft" ko? «-
umded by 9, will dearly be the same, as whpn"6+0;2 iadi^ded
2
.%^>«
2&
ASHHICBXIO.
638,
^ 67
Kon^
8781i
2665
80381,
68^5sfl x69+2=:681 +a,
67=^9)^ 6^,8^ 54+W
ft is clear, since S^i oontdna. ft without a remainder that Kfti ..kt
^ diyidmg^tlie p«4act of 633 md. 67 by 9, must be theCe
« ttyemauuier wbicb i« lef% ate divMlng thl product of 2^4
of 6^^^ ^^«« «^e Pfoduct of 67 ^d 2.=(64+3>x2, and the product
of 64 and 2 when divided by 9 leaves no remainder, therefore the
ZT^^"^ is/left alter dividing ii,e product of 683"^^^^^^
rr^^K^o '^^T '^«'«'°^«d«'- left after dividing the product of
^ejeft after the division of iiie multiplicand and multiplier respectively
i.i ^Z !? ^'"^^'"^ ^^*^^^ ^^®^' ^' ^« «^«» of its digits, which is
pr:^n~8tand'?^^^^^^^^ "^^'^^"'^ *"^^* ''''' ^ *^« — t
if cS ^^\^f ^' ^' ^^ ^^ ^'^ '""^^^P^^^' ^« committed, or
If ^j^ be mtrodnced or, omitted, the results will nevertheless a«ree
and so the error in these cases remains und^ected. ^ '
CO 87298
(E^ 840607
80
Ex. V.
(2) 16097 (3) 296897
69 88
(4) 69284
90
(6) 176
180.
CO 6^8
(8) $4@3
wVO"
I example 588
SIMPLE MULTIPLICATION.
29
(9) 2660r
5004
(10) 78847
8803f
(U) 672084664
, and^^o bjr W, 8*,»eai b;r 217a. by 7009, by wLraL'^rb^
149670 and 16790; of 664768 and 89314; of 816086 «r^ oXa .
123^6789 and 987664821 ; and of 57298492692 So^^608m'
(14) Multiply 9487862 by 4781246; 4842760 bT6M997i?m8«.
' nTv IT'.''^ "^ *''"^"««'«' ■' fi^«8661«L by 2WlS
(15) Mnlfply .« hundred and fifty thousand and niX b72«
aef r^d^ %^ T^ "''"'"""' "''"' "'•"^^ and nine.
Ja?Kd^\6tr6sr ''' "• ■""• ^' ' °"^«^' o™^.
The following alhreoiaUom in MultipUoation may be noticed,
83. To multiply a number 5y 6.
KuLE. Multiply the number ^j 10, and divide by 3.
Ex. Multiply 8768 by 6.
6=^; .-. 8768x5=8763 x:^=?^=48815
J 2
84. To multiply (t number (I) by 26 ; (2) by 126*
Rule. Multiply the number in case (1) by 100 and diVMn K^ ^
in case (2), by 1000, and divide by 8. ^ ' '^^ ^^ * '
Ex. Multiply (1) 839 by 25, (2) 7568 by 125.
(1) 25 =1^-?,. .-.889 by 25=839 xl??=?????=20975
(2) 125=152?
ft
.-.7568 by 125=7568:x '^P^I5??22£
4
'>00
8 8
85. r(^ multiply a number (1) 5y 16 ; (2) by 86 ; (8) by
..tiAKQtrtt
'V-=W i if
46; (4;&y66.
30
ARITHMETIO.
III'.?'
Ruts. Multiply the number in case (1) by 80 ; in case (2) by 70
in case (3) by 90 ; in case (4) by 110, and divide the product in each
case by 2.
Ex. Italtiply (1) 728 by ir ; (2) 887 by 86; (8) 678 by «,
(1) is=f
TO
(2) 86=^
(8) 45=^
/. 728 X 15=728 x -^^=?l|!?=io920
.-. 887 X 36=837 x ?=5?5??=29295
A 678x46=678 X
2 2
90 61020
2
=80510
86, To fiMltiply a number (1) hy 75; (2) ly 175: (8) Iv 226-
(4)Jy275. ' > \J y ^^o.
Rule. Maltiply the number in case (1) by 800 ; in'case (2) by 700 •
in case (3) by 900 ; in case (4) by 1100, and divide in each case by 4. *
Ex. Multiply (1) 973 by 75 ; (2) 687 by 176 ; (3) 978 by 226 •
(4) 1314 by 275. ^ »
/1^ tTK 800
(1) 75=— ;
700
973x75 = 973 x???=??l?2?=r2976
4 4
(2) 175=lp';./. 687x176= 687x-
* 4
900
700 480900
4
=120225
(8) 226:=
4 ' •
110
978x226= mx'-^J^^m,,0
4 4
(4) 275=^; .M314x275=l814xl^=ll^=86186
87. To multiply a number hy any number of nines,
RtJLB. Multiply the number by the same power of 10, as is fai-
dicated by the number of nines ; subtract the multiplicand from the
product, and the remainder is the required result.
Ex. 1. Multiply 789786 by 999.
999=10'' -1 ; .-. 789786 X 999=789786 (lO^* -l)
=789786000-789786=788996214
>ase (2) by 70
rodnct in each
^46.
120
95
i -
}
(8) Jy 226;
3 (2) by 700;
case by 4.
>78 by 226 ;
r2975
120225
520050
86185
0, as is in-
id from the
SIMPLE DIVISION. Si
Ex. 2. Maltiply 2686784 by 99999.
09999=lo'-l ; .-. 2686734 x 99999=268073 (lo'—l) '
=268673400000—2686734=268670718266
Similarly in any other case. •
DIVISION.
88. Dinsioir is the method of findmg how often one nnmber, called
the DivisoB, is contained in another number, called the Bividbnd.
The resalt is called the Quotibnt.
89. Division is of two kmds, Simple and Compound. It is called
Simple Division, when the dividend and divisor are, both of them,'
either abstract numbers, or concrete numbers of one and the same
denominatipn.
It is called Compound Division, when the dividend, or when both
divisor and dividend contain numbers of different denominatiors, but
of one and the same kind.
40. The sign ^, placed between two numbers, signifies that the
first is to be divided by the second.
41. In Division, if the dividend be a concrete number, the divisor
may be either a concrete number or an abstract number, arid the
quotient will be an abstract number or a concrete number, according as
the divisor is concrete %r abstract. For instance, 5 shillings taken 9
times give 30 shillings, therefore 30 shillings divided by 5 shillmgs give
the abstract number 6 as quotient ; and 30 shillings divided by 6 give
the concrete number 6 shillings as quotieiit.
SIMPLE DrVlSIOlT.
42. HuLB. Place the divisor and dividend thus :
divisor) dividend (quotient.
Take off from the left-hand of the dividend the least number of figures
which make a number not less than the divisor ; then find by the Mul-
tiplication Table, how often the first figure on the left-hand side of the
". r .„v-^ .- -u^/iivaiucw ill Liic iiioii ugure, or ine nrsc two iigures, on the
left-hand side of the dividend, and place the figure which denotes this
82
ABTEHMETIO.
11
number of time8 in the quotient: multiply the divisor by this figure,
and bring down the product, and subtract it from the number which
W^^mfiil* '.^'^f '^' ^^"^^^"^•- *^^ ^^'S down the next
proceed as betbre; if th* divisor be greater than any of thes^ re-
Z^lr V^ ^'^^^f *^ *^' ^°^'^'"*' "°^ ^""-^e ^«^° **»« ^ext figure
from the dmdend to the right of the remainder, and proceed as before.
Carry on this operation till all the figures of the dividend have been
^us brought down, and the quotient, if there be no remainder, will be
thus determined, or if there be a remainder, the quotient and the
remainder wiU be thus determined.
Nora 1 If any product be greater than the number which stands
above it, the last figure in the quotient must be changed for one of
smaUer vdue : but if any remainder be greater than the divisor, or
equ,a to It, the last figure of the quotient must be ohtoged for a
greater.
^ ^J^f a\ ^ *^® ^''^*''* ^""^^ """^ ^^°^^^ 12, the division can easily
be effected by means of the Multiplication Table*
Ex. Divide 2838268 by 6768.
Proceeding by the Eule given above, we obtain
6768) 2338268 (846
20274
81086
2708a
40648
40S48
Therefore tlio quotient is 846.
m re(uon for the BuU uill m>earfrtm the foOouii^ eomiderationi.
^J^^^?^''''J^l^^^''^^^ousmd, seven hundred and fifty-eight:
Tl tar!r .r^ ?^ *^ ^^*°^ ^^^^ ^^ ^^ ^^^^^ '^vrLit
T^^^ "^^ "^ thirty-dght thousand, and two
^ZJ^. f^f^L^^^*^^^ ^? «^- '^ «--; ^d 6768x800
.^^ -v»M««i«g ui^ iWQ ojpne«s astaeendfopoottTenienoeia
SIMPLE DIVISION.
88
working, we properly place the 4 under the 2 in the line above ; wo
subtract the product thus found, and we obtain a remainder of 8108
which represents three hundred and ten thousand, and eight hundred!
Bring down the 6 by the Rule ; this 6 denotes 6 tens or 60, but the
cypher is omitted for the reason above stated : the number now re-
presents three hundred and ten thousand, eight hundred and sixty:
6758 is contained 40 times in this, and 6758 x 40=270320 ; we omit the
cypher at the end as before, and subtract the 27082 from the 81086-
and after subtraction the remainder is 4054, which represents forty
thousand, five hundred and forty. Bring down the 8 by the Rule, and
the number now represents forty thousand, five hundred and forty-
eight: 6758 is contained 6 times exactly in this number.
Therefore §46 is the quotient of 2388268 by 6758.
The above example worked without omitting tht cyphers wooU
hiiVe stood thns :
6758) 2838268 (800-1-404-6
2027400
810868
270820
40648
40548
»
hence it appears that the divisor is subtracted from the dividehd fiOO
times, and then 40 times from what remains, and then 6 times from
what then re'-'ains, and there being now no remainder, 6768 is con-
tained exactry 846 times in 2888268.
The truth of the above methcJd might have been Aoim as fdBffwB i
2338268=2027400-f 270820+40648
6758) 2027400+270820+40648 (800+40+6
2027400
+270320
+270320
+40648
+40648
34
ABTTHiarno.
Ex. 2. DiTide 56488971 by 40W.
4064)66488971(18887
4064
15798
12192
86069
82512
86677
82612
80651
28448 .
2208
RtLE. Cat Off the cyphers from the divisor, and as many figures
from «^e Wand of the dividend, as .here are cyphers so cut oTa^
ti^e nght-hand end of the diviso.; iLen proceed with the remain
mamder annex the figures cut off from the dividend for tiie totd
Ex. Divide 687623 by 8400.
i*rooeeding by the Bule,
84,00)6876,28(168
84 ' '
197
170
276
272 .
8
fterefore 8400 is contained in 687628, 168 times with remainder
BIMPLB I»yiSI<>N.
8&
The reatM/or theBuU wUl appear fr<m the/oUowtng eM^derathnt. *
687628 is 6876 hundreds and 28, of which 687600 oontsina 8400,
168 times with a remainder 800 oyer ; and as 28 does not oontdn 8400
at all, the quotient will evidently be 168, with remainder 800-<-28i
or 828.
NoTB. The same rale applies when the divisor and dividend both
terminate with cyphers.
44. When the divisor is a composite number, and made np of two
factors, neither of which exceeds 12, the dividend may be divided
by one of the factors in the way of Short Division, and then the re-
sult by the other factor : if there be a remainder after each of these
divisions, the true remamder will be found by multiplying the second
remainder by the first divisor, and adding to the product the fint
remainder.
Ex. Divide 66782 by 46.
46
{
9
66782
6808-5
1260-8
the total remaindAT is 0x8+6, or 27+6=82.
Therefore the quotient arising from the division of 66782 by 46 Is
1260, w^*^ Ji remainder 82.'
th4 aheve Rule ia manifest from lihe following eon-
The rei.
eiderationa,
6808 is 6 times 1260 together with 8,
and 66782 is 9 times 6308 together with 6,
or is 9 times (6 times 1260+8), together with B,
or is 45 thn^ 1260+27+6,
or is 46 times 1260+32.
45. The truth of all results in Division may be proved by multiply-
ing the DIVISOR and quotient together, adding to the product the
remainder, if there be any ; the result (if the work ia correct) will bo
the DIVIDBND.
!M
8«
4fi. ^jnoe tlie poduot of th© divisor and c^notient equals iho
»mD«NDles8 the remainder; therefore, the accnracy of questions in
division may U tasted by tba proofs of cas^ out tha nines, pointed
oiU in Art. («B> -ttttp -,-*.-, ^usm»u
Ex. VI.
Examj^la in Simple Division,
(1) 14688069-*-27. (2) 8172Wa28-*-44,
iQ} 64906784^69.
(6) 70866482-4-87!
(7) 288?4646-j-12a.
(?) 1674918-«-189.
(11) 686819741-^007.
g8) ^286466800-^1440.
(16) 863008972662-4-6406.
(I'O 26799684687-^7890000.
(4) 6848734752-f.a.
(6) 649306746-1-66.
/8) 433418176^616.
(10) 81884740-S-779.
(12) lllllllllllll-i.60160.
(14) 67380625-^7676.
(16) 69996 1667212-^2468.
(18) 67illl04051-^8861.
(19) 10000000000000000-^llll, and also by Hill
(20) 684894567-M64600. (21) 671 57148372 .^ 90009
(22) 1220225292^200568. (28) 7428927416293-^8496427
(24) 60486674536845-J.79094461. (26) 65358647828-4-5678
(26) 8968901631620-S-687687948. * >
(27) Divide 162181256 by 8864, and explain the process.
(28) Divide 143266 by 4098. Explain the operation, and show that
it la oorrect.
(29) Divide 208684191 by 72.
(80) The remainder is 618, quotient 78936, divisor 878.' Fmd the
dividend.
(81) The dividend is 865866651, the quotient 86783, the remainder
2705. Find the divisor. «u«maer
(82) The distance between Liverpool and Quebec is 3060 miles .
the usual ^?^gth of a voyage by a Montreal Oc^ an steamsliip is 11
days. Slnd the number of miles which tha vessel goe& per hour.
(88) The length of the Rideau Canal is 126 miles; cost of build-
Ing, |4,880,000; length of Welland Canal, 51 miles; cost $7,000,000.
Rod, 1st, cost of each per mile ; 2d, difference of cost per mile. '
(34) The number of miles open for traffic on the Grand Trunk
Baiiway is 1877; the cost for building and equipping the ro&.%
$94^406,914: number of mUes open on. the Great Western ia 863; cosl
SniPLR DHTBION.
37
for building and eqnlppinft $24,777,480. Find. Irt, oort of each per
mile ; 2d, diflferenop of cost per mUe.
Note In tl)e above exercise, whenerer the Diviaor is a composite
number, divide, 1st, by Long Diyision, and then by its factorTand
show that the results in both oases coincide.
The following abbrevtatioru in Division may be noticed. '
47. To divide a numbir Ijf 0.
Rule. Multiply the number by 2, and divide the product by 10.
Ex. Divide «87 by 6.
1274
10. 687_687 687x3
**-2 ' ••T-lo'=-io-=
2
10
=127tV
48. To divide a number (1) hy 26 ; (2) by 125.
Rule. Multiply the number in case (1) by 4, and divide the pro-
duct by 100 ; in (2) by 8, and divide the product by 1000.
Ex. Divide (1) 541 by 25, and (2) 5600741 by 126.
541 641 641x4 2164
(1) 26=if ;
SIVA
(2) 126:
1000
' 8 '
_ 44806928 _
1000 ~*^^^T^
49. To divide a number (1) by 15 ; (2) by 86 ; (3) by 46 ; (4) bj 65.
Rule. Multiply the number in each case by 2, and divide the pro^
duct in case (l) by 80, in (2) by 70, in (8) by 90, in (4) by 110.
Ex. Divide (1) 688 by 46 ; (2) 5608 by 85.
(1) 46=?? • . 683_688 683x2 1366
70
V-/ -unr-r g- |
45 80
2
6603 _ 6603
35 '-^"j^-
90 ""80 — ^^«V
5603x2 11206 ..
TO
70
=aeo^
2
88
ABirHMXTIO.
50. To divide a number (1) hy 75 ; (8) by 176 ; (8) bif 225 •
(4) &y 275. ^ jr , V / .^r Mu ,
EuLK. Multiply the number in each case by 4, and divide the nro-
dnct in (1) by 800, in (2) by 700, ia (8) by 900, ia (4) by 1100.
Ex. Divide (1) 2097 by 76 ; (2) 28647 by 275.
10^88_.^-^
800 800 800 ""^^
4
1100
0) 75 =— • • — =222I=???I_^*
4 * " 75 800 800~^
(2) 275=:
g8647__28647_ 28647x4 94588
276 "iTO nor~~"iioo~®^*^*
51. To divide a number by any number of nines.
Rule. Divide the given number by the same power of 10 as is in.
dicated by the number of nines; repeat the same operation as often
as necessary with each successive quotient obtained; add all these
quotients together ; tJieir sum is the quotient required.
Ex. Divide 2897687 by 9999. '
289-7687
•02897687
•000002897687 .
289-792679267687
NoTB 1. If the sum of the partial remainders should be the same
as the divisor in any example (i. e. a number of nines), it is plain that
there is no remainder, but that one should be added to ihe integral
part.
Note 2, By carrying on the operation, as in the given example
the digits which recur very soon appear; for instance, as in theexampu\
9267, so that the answer above might be written 289-79267.
Ex. vn.
MiMcellaneoua Questions and Examples on the foregoing Articles,
(T) Explain the principle of the common system of numerica
*— auitiply 60S oj 48, and give the reasona for the several steps
MISCELLANEOUS QUESTIONS.
89
(2) Write at length the meaning of 9090909, and of 90909.
Find their sam and difference, and explain fully the processes
employed.
(3) A person, whose age is 78, Was 87 years old at the hlrth of his
oldest son ; what is the son's age ?
(4) Explain the meaning of the terms ♦' vinculum ", "bracket'*;
and of the signs +, -, =, .-., x.
Fmd the value of the following expression :
16 X 87153-78474- 67152^-4+40784 x 2.
«
(5) By the census of 1861, the population of Ontario was found
to be 1896091 ; of Quebec, 1111666 ; of New Brunswiolc, 262047;
of Nova Scotia, 880867; of Prince Edward Island, 80867; of New-
foundland (1867), 124288 ; British Columbia and Vancouver's Island,
84816 ; Rupert's Land, 101000. Find the whole popoktion of the
above named provinces,
IL
(1) Define "a Unit", "Number", "Arithmetic". What is the
difference between Abstract and Concrete numbers ?
(2) The annual deaths m a town being 1 in 45, and in the country
1 in 60, m how many years will the number of deaths out of 18676
persons living in the town, and 79260 persons living m the country,
amount together to 10000 ?
(B) Define *' Notation ", " Numeration " ; express in numbers seven
hundred quadrillions four hundred and nine trillions.
(4) Find the the value of
494871-94868+(46079-8177)-(64812-8987)-(1768+281)+879x879.
^(5) What number divided by 528 will give 86 for the quotient,
and leave 44 as a remainder ?
III.
(1) Define Multiplication, and Division. Shew that the product
of two numbers is the same in whatever order the operation is
performed.
(2) The Iliad contains 16683 lines, and the ^neid contains 9893
Unas; how many days will it take a boy to read through both of them,
at the f uto of oighty*nve lines a day ?
40
ABITHMKna
(4) Explain the meaning of the sign +, and find the valae at
(7854-«ia).8-(2O874-1268O)-.58-«*(8064S6-3864)+56«
fSi^ 'ogfter «core 90 runs, and ^ and (7 together score BlZ^.
find the number of mns scored by each of them. '
«ni^ »«fln» Addltton, .ad Sabtraotlon. That is meant by « prime
G™ tlp^" "• """'"" -^^ ^ "' '•^- "^ -" «^™
it in'rad^r^fs^.t^,^^r- <--''-> -.P«*
,.M ^W^^^^ \^ 21 years old when his eldest son was born • how-
old wm h.s son be when he is 60 years old, and what w^l be al
father's age when iL son is 60 years old 1 '
.Jx ^"'f '"f S""» «"» '"""'■^a "nllHons, one hundred thonsand
(4) Explain the short method of mnltiplying and dividinir >
8795678 bemg separately mnltiplied and divided by 9999
(6) The estimated population of the British American Provmee.
for the year 1870, is as follows : Ontario, 3047884 • QaeZ mrS^T
rlf/'ni^nl ""d""""'' "^'^' »""* Columbia, eoOOoT Burt's
land 116000. Find the total estimated popuWion of the atoti
provmces for the year 1870. ^*
V.
(1) Mnlt^ly478 by 148, «,a test the result by oastbiir out th.
nme^ In what cases does this method of proof f^? D^ 4^
by 99 and prove the correctness of the operatLby .nytestyr«pl^?
,a,?ll!'?' •"'-^' »"1«''»«0 "y*'?"' «ivethe'«meZaiT«
MISCELLANEOUS QUESTIONS.
41
the remaioder
»«i ©xemplij^
(8) In the city of Montreal, for every two persons who speak
English only, three speak French only, and seven both English and
I French ; and the whole population is 120000. How many speak
El glish only, French only, and both English and French ?
(4) A gentleman dies, and leaves his property thus : 10000
dolhirs to his widow ; 15000 dollars to his eldest son, on the con-
dition of his giving to a sc'aool-library 850 doUars ; 6600 dollars to
each pi his foqr younger sons; 8760 dollars to each pf his three
di^ughte|»i 4663 dollars to different societies; ^nd 699 dolUra ift
legacies to his servants. What mQiait of property di^ h^ dia
possessed of? •
(5) Pe quotient arising from the division of 9281 by a certain
numher is 17, and the remainder is 878. Find the divisor,
VI.
(1) Explain briefly tlie Ronaan method of Notation. Espress 1668.
and 9000 in lioman characters.
(2) Explain the terms ^'factor", "product", "quotient"; show by
an example how the process of Division can be abridged, if the divisor
terminate with cyphers.
(8) The remainder of a division is 97, the quotient 665, and the
divisor 91 more than the sum of both. What is the dividend f
(4) Express in words the numbers 270180 and 26784; also' write
down in figures the number ten thousand two hundred and thirly-
four; and find the least number which added to the last number wiil
make it divisible by 8.
(6) A gentleman, whose age is 60, has two sons and a daughter ;
his age equals the sum of the ages of his children ; two years since his
age was double that of his eldest son ^ the sum of the ages of the father
and the eldest son is seven times as gioaj; as that of the youngest son ;
find the ages of the children.
42
ARITHMETIC.
SECTIOJS- II.
CONCRETE NUMBERS.
• TABLES.
62. Oar operations hitherto have been oarriafl «n »■•«. , .
to abstract nmnbers, or concrete nnmbm oTo^ T • ^^ ""''
evident that if conc^te number, we°e alTof 0=^1 "'? '°"- ^* ^
instance, shiliings were the onl/n^I^ol'l:; ^'^'^.tZi ""• ""
different ^enon-inationt »drs:dl?^nX:V;e:riLr 1
Other bylOormultiDleflnf in fu«« n ""^'"^ "^"^^d from «ach
nnmbers conld be « on\? het^'v?'/'"' '"«" ~"<«'»
whole numbers BZ»eraIlr I fh „ ^^''''' '"'™ ''««' gi'"" for
does not hold U.t^ZZdL^^r'^'!'^^'''"^" "* " "'««»■>
necessary to con^lrto itoTt^birSr "'. ^^^^^^^
nnits of money together Z ^ff!^* "?/<"» "o"""!* tke different
different „„i. ItZ^lZ, '^Zlr^' "' "»«* '°<^'' ^
MONEY TABLES.
CANADIAN OUBEKNOT.
63. The Silver Coins are : a 5 cent-piece.
a 10 ** *«
a20 « "
100 cents makes one dollar, or $1
tender for more thl ^0 :^LuZT^ the\i? ''"""' '""• " "*»■
-^»:0; the^oldcoina^ofO -rBr^,- SrrCS
CONCRETE NUMB RS.
43
64.
gAT . TTAX OB OLD CANADIAN OUBBENOT.
2 Farthings make 1 Half-penny ... .^d. *
2 Half-pence 1 Penny !«}.
12 Pence 1 Shilling. Is.
6 Shillings 1 Dollar |1.
4 Dollars 1 Pound £1.
Note 1. The farthing is written thus, id ; and three ferthings
thus, |d.
BNOLZBB OB 8TEBLIN0 OTTBBBNOT.
2 Farthinijs make 1 Half-penny, or ^
2H<df-pence 1 Penny Id.
12Pence 1 Shilling Is.
20Shilling8 1 Pound. £1.
NoTB 2. The sterling pound or 80vereign=|4.86f Canadian Cur-
rency.
Pounds, shilling^, pence, and farthings were formerly denoted by
£, «, dj q respectively, these letters bemg the first letters of the Latin
words Uha^ soUdus^ denarius^ and quadrant^ the Latin names of
certain Roman coins or sums of money. £.»,<? are still the abbre-*
viated forms for pounds, shillings, and pence respectively.
The following coins are in com-
mon use in England :
COPPER COINS.
A Farthing the coin of least value.
A Half-peimy = 2 Farthings.
A Penny = 4 Farthings.
LJLVER COINS.
Three penny-piece = 3 Pence.
Four penny-piece = 4 Pence.
A Six pence
A Shilling
A Florin
A Half-Ci-owh
A Crown
= 6 Pence.
=12 Pence.
= 2 Shfllings.
^ < 2 shillings
(and 6 Pence.
= 6 Shillings.
GOLD COINS.
A Kaif- sovereign =lo"Shi]llngs.
A Sov^eign «20 ShUlings.
The following coins have been
in use at various periods in Eng-
land,- but with the exception of
the first two, which are used un-
der different names, they are now
obsolete.
SILVER COINS
A Groat =4 Pence.
A Tester=6 Pence.
GOLD COINS
£• B. d,
A Noble . =0.6.8
An Angle =0 . 10 .
A Half-guinea =0 . 10 . 6
A Mark or Merk=0 . 13 . 4
A Guinea =1 . 1 . o
A Car lus =1.3.0
A Jii obus =1.6.0
A Moidore =1.7.0 ""
44
ABITHMETIO.
pJl?^ec;2'u^r^^S*Jnt "^ •^t'"'' 'tamped, .o„to
The ^ndari of gold cotoln qZ' I f"^ *^ '^»'-
otpur, gold .„d 2 partsTC^ ti^**;" ""' ^"''""' ''^a ?««»
Tro, of standard goU^here LSd a 1 r?f ^'•<"" " P"""*
^6. 14.. M. : therefore the MintT f.^"' **'« sovereigns, or
The ,te„A«-<i of silver coin fa 87°! f ^''^=^ P<"""' Troy).
««P«-. From a po„„d Troy "f stod^'^'' ?' ^" "*«• »»<i » P«rt. of
Therefore ,he Mint price ofsflverls tlj '' ""* """'''' «« »''"«"'8^
In tlie copper coinage XZ. P"" ''™''« 8*«nd"<I.
d»pois of copper. SrefflT! "" """"^ ^"^ ^ P""""! Avoir-
Ayoirdupois. "* ^ P""^ *<»>''' ^eigh ^th of a poond
'WITED STAras CUERENCy
10 Dimes... :: ■•- ?^T ■•?•
*"'^"- •■•:•■.■.■■• iSr.vi
coined. ' ^^ °*'"® ^^ *^e towa where it. was first
Cent most likelj from the Oeltin a,«/
^ J m me uemo Cant, meamng a hundred.
I ^
MEASURES OF WEmHT*
TABLE OF TBOY WEIGHT.
brought thiAer frott &TDt ft To "P'^f • " ^""s »» have been
»ilver, diamonds, «,d other .Lcl^rf"/..'''^''*''* «""'
termining meoiflo gravitim • «T^ of a costly nature j also in do-
^ ^ omogravihes, and generally in philosophical invesUga.
TABLBSr—WEIQHT. 45
The differeiit units are grains (written grs.), penny weights (dwt8.)|
oonoes (oz.), and pounds (lbs. or flta.), and they are connected thus :
24 Grains make 1 I*enny weight. . 1 dwt.
20 Pennyweights 1 Ounce 1 oz.
12 Ounces. .... . . . . . . . i Pound 1 lb. or lb.
Note 1^ As the origin of weights, a grain of wheat was taken from
the middle of the ear, and being well dried, was used as a weight, and
called * a grainJ*
Note 2. Diamonds and other precious stones are weighed by
^GamU^^ each carat weighing about 8j grains. The term * carat'
applied to gold has a relative meaning only ; any quantity of pure gold,
or of gold alloyed with some other metal, being supposed to be divided
into 24 equal parts (carats) ; if the gold be pure, it is said to be 24
carats fine ; if 22 parts be pure gold and 2 parts alloy, it is said to be
22 carats fine.
Standard gold is 22 carats fine ; jewellers' gold is 18 carats fine.
TABLE OF APOTHECARIES' WEIGHT.
67. Apothecaries' weight only diflfers from Troy weight in the
subdivisions of the pound, which is the same in both. Th'ts t&ble is
used in mixing medicines. The diflferent units are grains (grs.),
scruples (3), drams ( 3 \ ounces ( I ), pounds (lbs. or fts), and they are
connected thus :
20 Grains. . .make 1 Scruple. . . 1 sc. or 1 3.
8 Scruples 1 Dram 1 dr. or 1 3 .
SDrams 1 Ounce lo«.orl§,
12 Ounces ,.1 Pound lib. orfi).
TABLE OF AVOmDUPOIS WEIGHT.
58. Avoirdupois weight derives its name from Avoirs (goods o^
chattels) and Poid^ (weight). It is used in weighmg all heavy aiticles,
which are coarse and drossv. or snbiAr>f. tjn wa«fa aa Knffa» vn^o^. ^^a
the like, and all objects of commerce, with the exception of medidnea,
^
46
AEITHMEno.
tinits are drams
Iiandredweigbts
gold, ailrer, and some precious stones. The diff«r«nf
(owte.), tons (tons), and they are connected thns ?
JJ^^"'^ makelOunce ..i^^
^^^^<^^' IPound.. i?r
\"-^^«- iQnarter;:::::::.-! •
20H^!:^-':V 1 Hundredweight... 1 cwt
20 Hundredweights i Ton 1 Ton
I^=19Jcwt.; 1 Great Poind^^^^ ^'»«'' ^^^<i^rof
=240 pounds. ^'^^^ ^^'^^'^s I 1 Pack of Wool
1 lb. Avoirdupois weighs 7000 grains Troy ;
lib. Troy weighs 6760 grains Troy.*
tolASURES OP LENGTH.
TABLE OP LINEAL MEASUBE.
69. Li this measure, whin^ ia no«;i *
8 Feet... ::::;::: !v"''--
^„ • 1 Yard,
6Feet - -, ,/* •
„, _. , 1 jathom 1 Av
63 Yards,,. - ^^ * ^ "^
^po.e.(.^;a.):::::::::::::;j:;;^;;.-r'«*--;p^
ift.
lyd.
8 Miles
692 MUes
^^«««e> Ilea.
1 Begree.
• •• Ideg.orr.
TABIi^iB — JJSSQTB. 47
The foUowing measurements may be added, as uaeM in certain
C&868 2
4 inches make 1 Hand (used in measnring horses),
22 Yards make 1 Chain)
100 Links make 1 Chain \ "*^^ '"^ measuring land,
a Palm=8 inches, a Span=9 inches, a Cnbit=18 inches,
a Pace =5 feet, 1 Geographical Mile=,V*' of a degree,
a Line =Y»5* of an inch.
TABLE OF CLOTH MEASURE.
60. In this measure, which is nsed by linen and woollen drapers :
2i Inches make 1 Nail.
4 Nails 1 Quarter. .. 1 qr.
4 Quarters ... 1 Yard ..... 1 yd.
6 Quarters ... 1 English Ell.
6 Quarters . . . 1 French Ell.
8 Quarters ... 1 Flemish Ell.
MEASURES OF SURFACE.
TABLE OF SQUARE MEASURE.
61. This measure is used to measure all kinds of surface or super-
ficies snch as land paving, flooring, in fact everything in which Wh
[and breadth are to be taken into account.
A Squaee is a four-sided figure, whose sides are equal, each side
bemg perpendicular to the adjacent sides. See figure below.
J A square inch is a square,.each of whose sides is an inch in length •
la square yard is a square, < ach of whose sides is a yard in length. *
344 Square Inches make 1 Square Foot. . . 1 sq. ft. or 1 ft.
9 Square Feet. ....... 1 Square Yard.. . 1 sq. yd. or 1 yd.
30i Square Yards 1 Square Pole. . . 1 sq. po. or 1 po.
40 Square Poles 1 Square Rood.. . 1 ro.
*^^oods.. 1 Acre •. lac.
48
ABITEDiffSOCIC.
1
4
7
2
8
8
3
6
36000 Sqnftre Links » 1 Rood.
100000 =lAore.
10 Ohaing » 1 Acre»
4840 Yards = 1 Acre.
040 Acres = 1 Square MUe.
Ifota. ThiA table is formed from the table for lineal measure, by
multiplying each lineal dimension by itself.
I%e truth of the above table mil appewfrom thefoHowinf^cantidera-
tians.
Suppose AB and 40 tp be lineal, yards placed perpendicular to
each other.
Then bydeflnition^^CD is a square yard. If AM; a e r n
EF, FB, AG, GH, HG=1 lineal foot each, it appears
from the figure that there are 9 squares in the square ^
yard, and that each square is 1 square foot. g
The same explanation holds gpod of the other
dimensions. _
The following measurement may be added :
A Rod of Brickwork .... =272J Square Feet.
{The worh is eujtpoted to be 14 «»., or rather more than a brici-and-a-
half, thieh)
TABLE OF SOLID OR CUBIC MEASURE.
62. This measore is used to measure all kinds of solids, or figures
which consist of three dimensions, length, breadth, and depth or
thickness.
A ouBB is a solid figure contained by m. equal squares; for in-
stance, a die is a cube. A oabio inch is a cube whose side is a square
inch. A cubic yard is a cube whose side is a square yard.
1728 Gubio Inches make 1 Cubic Foof,, or 1 o. ifc.
27 Cubic Feet l Cubic Yard, or 1 c. yd.
40 Cubic Feet of Rough Timber or
60 Cubic Feet of Hewn Timber.. 1 Load.
^ 42 Cubic Feet ^ 1 Ton of Shipping. .
128 Cubic Feet of Fire-wood 1 Cord.
i.y vUUiO JC«i3l; vi JD irS-WOOQ 1 UOfa-JOOU
ile.
Ineal measnre, bj
^XUnoing coruidera-
perpendicular to
A E F n
a
B
1
4
7
2
6
8
3
6
reet.
an a hrick'and-a-
E.
solids, or figures
, and depth or
squares ; for in-
e side is a square
ard.
, orlcffc
I, or 1 c. yd.
pping. .
TABLES — ^WBIQHTa XSjy MEASURES ^g
JiAB, AC, and Al} be perpendicular
to each other, and each of them a lineal
jard m length, then the figure DE is a
I cubic yard.
Suppose BE 2, lineal foot, and HELM
a plane drawn parallel to side I>G,
Bj the table Art. 61, there are 9 square
feet in side DC. There will therefore be
9 cubic feet in the solid figure BL,
Similarly if another lineal foot EN
were taken, and a plane NO were drawn
^CrXo,^"" "^"^ "^^^'' '^ ' ^^^^° ^-* --*-«^ ^ the salid
Shnilarly,.there would be 9 cubic feet in the solid, figure KB.
Icubb y::r^ *""' ^^ ^^^^^^^ '^^ - *^^ solidfigSe^^,or
haWcotdf"''' "°^' ' '''' %b, 4 feet wide, and 8 feet long,
MEASURES OF CAPACITY
TABLE OP WINE MEASURE.
oeptionof^^^jf i^'"""'"' ^^^^'"'^^^^"^ and all liquids, with the ex-
peption of malt liquors and water, are measured
t^^^l" •••...makelPint....\..lpt.
>Q°^«- 1 Gallon..... iLi
ITT"."- IHogshead.. Ihhd.
^^T^^*'^^ IRpe Ipipe.
64.
leasured:
TABLE OF ALE AND BEER MEASURE.
In this measure^ by which aU malt liqnors and
2^"»ts make 1 Quart..... i .
^^°arts IGallon.....!
\-^ water are
.if.
gal.
!•
66.
■
!
ABTTHMETIO.
OGalloM. IFirlrin 1 fir. ,
18 Gallona 1 Kilderkin. . 1 kil.
86 Gallons 1 Barrel 1 bar.
1^ Barrels, or 64 Gallons. . 1 Hogshead. . 1 hhd.
2 Hogsheads 1 Bait 1 bait
2 Butts 1 Tan 1 tun.
TABLE OF BRT MEASURE.
2 Pints.... J... malce 1 Quart 1 qt ^
4Qaarts . 1 Gallon 1 gaL
2 Gallons 1 Peck 1 pk.
4Pecks 1 Bushel 1 bu.
86 Bushels. 1 Ohaldron . . 1 oh.
Note 1. Grains are generally sold by weight, as under.
»
66. J84 Pounds make 1 Bushel of Oats.
48 Pounds 1 Bushel of Buckwheat.
48 Pounds 1 Bushel of Barley.
60 Pounds 1 Bnshel of Beans.
66 Pounds 1 Bushel of Rye or Indian Corn.
60 Pounds 1 Bushel of Wheat, Peas, or Clover Se«d.
MISCELLANEOUS TABLE.
67. laXTnita... .make 1 Dozen.
12 Dozen 1 Gross.
12 Gross. 1 Great Gross.
20 Units ..1 Score.
24 Sheets of Paper. 1 Quire.
20 Quires 1 Ream.
100 Pounds 1 Quintal.
196 Ponnds 1 Barrel of Flour.
200 Pounds 1 Barrel of Pork or Bee£
Note. A sheet folded into two leaves is called a folio, into 4 leave
8 quarto, into 8 leaves an octavo, into 16 leaves a 16mo, into 18 leave
an 18mO| dco.
, Iflr. ,
IkU.
. 1 bar.
Ihhd.
. 1 butt
1 tun..
. Iqt
IgaL
. Ipk.
. Ibu.
. 1 oh.
TABLsa-^miAsuMs OF time:
51
under.
I Corn.
or Olover SmcU
BS.
Flour.
Pork or B6e£
a folio, into 4 leave
L6mo, into 18 leave
68.
MEASURES OP TIMB.
TABLE OF TIMBL '
1 Second is written thus 1".
60 Seconds make 1 Minute' r
eOMinntes IHour... Tv
24Hours tDay.... "'l^'
^^'^^^ IWeek.... : l^^'
^ "'^"<>r common year... lyr.
of the/ollov^J^re': ^''' ""^^ ^« ^^^^ remembered by meanJ
Thirty days hath September,
April, June and November;
February has twenty-ei^ht alone.
And all the rest have thirty-one :
But leap-year coming once in four
February then has one day more, j'
NOTB.-A civil or common y«ar= 52 wks., 1 day.
A leap-year =366 days. ^
point;id"r.:r:^^^^^ ^^ ^^^^^^
ment of time, and it is thrml' ^^ """^^ ^^^ <^« ««««re.
tween two su cessive ^7^7^"''' *^« ^^^^'^ ^^P^ be-
place. *"''*" ""^ *^^ S"^ across the meridian of any
■r
ii
52
AIUTHMETIO.
would, of coarae, in time b6 very considerable, and oauBe great con-
fasion.
Jnllus Offiuar, in order to corteot this error, enacted that every 4th
year should consi^t of 866 days; this was called Leap or BmextiU
year. In that year February bad 29 days, the extra day being called
** the Intercalary " day.
But the solar year contains 866-242218 days, and the Julian year
contains 866'25 Ot 866^ days.
Now 866-26— 866-242218=-007782.
Therefore in one year, taken according to the Julian calculation,
the Sun would have returned to the same place in the Ecliptic 007782
of a day before the end of the Julian year.
Therefore in 400 years the Sun would have come to the same
place in the Ecliptic -007782x400 or 8-1128 days before the end of the
Julian year ; and in 1257 years would have come to the same place,
•007782 X 1267 or 9;7819, or about 10 days before the end of the Julian
year. Accordingly, the vernal equinox which, in the year 825 at
the council of Nice, fell on the 21st of March, in the year 1582
(that is, 1267 years later), happened on the 11th of March ; there-
fore Pope Gregory caused 10 days to be omitted in that year,
making the 15th of October immediately succeed the 4th, so that
in the next year the vernal equinox again fell on the 2l8t of March;
and to prevent tha recurrence of the error, ordered that, for the
future, in every 400 years 8 of the leap years should be omitted,
viz. those which complete a century, the numbers expressing which
century, are not divisible by 4; thus 1600 and 2000 are leap years,
because 16 and 20 are exactly divisible by 4; but 1700, 1800, and
1900 are not leap years, because 17, 18, and 19 are not exactly divi-
sible by 4.
This Gregorian style, which is called the new etyle, was adopted in
England on the 2nd of September 1752, when the error amounted
to 11 days.
The Julian calculation is called tho old etyle : thus old Michaelmas
and Old Ohristmaa take place 12 days after New Michaelmas and
Ketr Christmas.
In Russia, they still calculate according to the oU ttyle, but in the
6tM tfOtmtriw Gi £firot>e tho new style is nsed. Sir Harm Nioolao
Qse great ooa-
that every 4th
IP or BunextiU
Kj being called
^he Julian year
ian calonlation.
Ecliptic -007782
16 to the same
the end of the
the same place,
id of the Julian
le year 825 at
the year 1582
March ; there-
in that year,
le 4th, BO that
2l6t of March ;
>d that, for the
lid be omitted,
(pressing which
are leap years,
1700, 1800, and
lot exactly divi-
was adopted in
error amounted
old Michaelmas
Kfichaelmas and
ityU^ bnt in the
: Harris Niooiao
TABLES-IMPERIAL STAitoARD MEA8CRE. 63
TABLE OP ANGULAR MEAStTRS.
60 Beoond. make 1 Minnte. . . i „.„ „ „
90 Degree. , S a " " J " " " • ' ^«ff- " 1°.
^ ^ ^'K"" Angle.. . . 1 rt. ang. or 90'.
Tlie clrcnmferenf ■> of ever.- ni~.i. i
860eq«.I part., e.el«f wS •^l!!'"'"?' ^«"^ t" 'o^vided into
«n angU of r .t the cent™ of the cS^e! " '^*^' " " »"*•■«"
70. An Act of Parliament "fob A«^.,^
It 18 thereby enacted, ^' ^^-
l-ahrenheif. therraometerr andLf ^^r^""*"'"" »' «2° by
shall be the unit or oSy atldl!, Jf" ^"P*™' Stan-iani Yard,
from or whet^by ril other meLtl^f T"^ "' ^^'«"«'™. '"'ere-
the same be lineal, a»perfd7or ' n/ r^ "'""o^ver, whether
«nd ascertained, Tnd thai tie <«I 5,t ' ^' *"''*<'• «<»°P°ted,
a» Inch. "'" • "* '*«'-«2'-»»«A part of thi. yard shall bo
w^'n'r vZt?i„d^:'rir„^™f « "^"^ ^^ *• '•«"""« <>?
B«eh inches, i. .. 39 ;„„, ^ht and'lf tT^" '""'" '» "^ ""-^^'S
snob inch. ■'"' '^'' •»" thonsandths of another
'^om^ttttTZlTr'^l' *' r-Perial standard TaM
S-' -red :is trr:r^-- --t
Secondly. Tha^-^'A^^— -• ' - -
-n in the <^;^r^^^^^^^;^^^.S,
M
54
AUITHHETIO.
Standrf *" Meeuure of WeigU^ from which all other weights shall be
derived, compated and aecertaiued ; that 6760 grains shall be contained
in the Imperial F^andard Troy Pound, and YOOO such grains in the
Avoirdupois Pound.
Now the weight of a euUo inch of distilled water is 252-458 grains
Troy the barometer being at 80 inches and the thermometer at 62".
This aflFords the means o| recovering the Imperial Standard Pound
should it be lost. In fact, the brass weight of 1758 was destroyed or
lost at the above-mentioned fire.
8d. That the Standard Measure of Capacity for Liquids arid
Dry Goods shall be " the Imperial Standard Gallon,''^ containing 10
Pounds Avo'rdupois weight of distilled water, weighed in air at a
temperature of 62° Fahrenheit's thermometer, aq^ the barometer being
at 30 inches.
Now this weight fills 277*274 cubic inches, therefore the Imperial
Standard Gallun cuntnins 277*274 cubic inches.
The Imperial Bushel, consisting of eight gallons, will consequently
be 2218*192 cubic inches.
REDUCTION.
71. Wheu a number is expressed in one or more denomuiations,
the method of finding its value in one or more other denominations is
Reduction. Thus, £1 is of the same value of 240<?., and 7». l^d. is of
the same value as 342 farthings, and conversely : the method or process
by which we find this to be so, is Rbdxjotion.
72. First. To express a number of a higher denomination or of
high&r denominationa in units of a lower denomination.
Rule. Multiply the number of the highest denomination in the
proposed quantity by the number of units of the next lower denomina-
tion contained in one nnit of the highest, and to the product add the
number of that lower denomination, if there be any m the proposed
quantity. > • -n
Repeat this process for each succeedmg denomination, tiU the
required one ib arrived at. ,
■c<—
<« TT^^.^ .»««» ^A-n^a in AIO.'T 1K9
unation or of
aation, till the
REDITOTION.
R-ooeeding by the Rnle given abore,
$127.15 ■^^^^'^^^n for the process.
100 Since 100 cents make one dollar •
or }|;i27.16=12n6 cents. =12716 cents
or 1127.15=12715 cents.
Ex. 2. Reduce 27 acres, 1 rood, 82 poles, to poles.
«»<a-e». rood. poles.
2^ . 1 . ^2 *
_^ (add the 1 rood) '
109 ro.
_fO (add th^- 32 poles),
4392 poles. •
Ex. 8. How ^^, ^^^ i„ ,„, „y^ ^ ^_^^ _ ^ ^^^^ ^ ^^^
55
jl<
yds.
2}
n»"e8. ftir. poL
10« . 6 . 25 .
_8 (add the 6 fur.)
8o4 fur.
_ "^O- (add the 25 po.)
84185 poles
^ (add the 2| yds.)
170927^
j^OQ^g (product of the I)
188020 yards
8 ,»
•664060 ffeet
6768720 incites
56
•Aiy^lS^^lCRpQb
^ Rule. Divide fhe given nnsiLl^v by tiie iMmibeF of units which
connect th&t denoonination with the next higher, imd the remainder,
if any, will be the number of sorplas units of the lower denomina-
tion.
Oarrj on thk process, till yon arrive at the denomination re-
quired.
< Ex. 1. In 17895$ eents, how many dollars and cents f
BytheBnie,
17892 ' Reason for the Rule, v
iY39_2 ^®^ cents=|l.
.'. 17392 cent8-i-100=$173+ 92 cents.
♦178-92 cts. .*. 17392 cents=$178.92 cents.
flO
100 ^
1 10
NoTS, — From the above example, we see that by cutting off the last
2 figures on the right of any number of cents, gives the dollars, and
the figures so cut off will be the cents.
Ex. 2. Beduoe %^ acres, 28 poles, 10 yards, 8 feet, 112 inches, to
inches. Prove the result.
ao.
49
4
po. yds. ft. hi.
28 . 10 . 8 . 112
196 ro.
: 40 (add the 28 po.)
7868 . poles
80i (add^the 10 yds.)
286050
1967 (product of i)
288017 yards
9 (add the 8 ft)
2142161 feet
144 (add the 112 i|L)
8568756
8568644
^142161
80847129^ ^iphil
ominfttion re-
112 inches, to
VJSBvcao^,
IVoof
144 / ^^ I SOsIVlW-
m
_Jll2aq.|n.
i la 25706941
»l 2142161-8 sq.ft.
238017
l^ow, since 80J or ifi gn vila-i -«
redaoe. the «,. yd. into q Je^ rfl^^' T.."""'^' "' *• »""<*
121 f " J!?!?!??-r 1 40 q„.rt«« of «,. ^
lu[_86661-8/opio,q.yag.
Ex. vnt
[PJeces. «»ruu«igs, aoa 670 <aro^nis to fompeimy
B«l.oe. varifyiag the «»„lt to ^h ^ ^, ^„^^__
- — — - -T Maa- 4 _
\m
I I
58
ABITBMEIIO.
(8) 69 lbs., 7 02., 14 dwt3., 19 gre., to grains ; and 87400167 gre.
to lbs.
(9) 66832006 Bors. to lbs. Troy; and 636 lbs. to drams and
scmples.
' (10) 7 tons, 16 cwt., 2 qrs., 16 lbs. to oance? and 7568241 drs.
to tons.
(11) 6838297 oz. to tons ; and 83 tons, 17 cwt., 3 qrs., 27 lbs.,
16 drs. to drains (cwt.=l"'2 lbs.).
(12) 17 lbs., 2 1 , 2 3 to grains ; and 84678 grs. Apoth. to oz. Troy
(18) 8 m., 7 fur., 8 po. to yards ; and 573 miles to inches.
(14) 1364428 in. to leagues; and 74 m., 3 fur., 4 yds. to inches.
(16) 4 lea., 2 m., 2 in. to barleycorns ; and 50 m., 3 po. to yards.
(16) 7 far., 200 yds. to chains ; and 6 cubits, 1 span to feet.
(17) 84 yds., 1 q^. to nails ; and 56 Eng. ells, 1 qr. to iiails.
(18) 88 Fr. ells, 3 qrs. to nails : and 73 Fl. ells, 1 qr. to nails. .
(19) 85 ao., 2 ro. to poles ; and 56 ac, 2 ro. to yards.
(20) 8 ro., 87 po., 26 yds. to inches ; and 3 ac, 30 po. to feet.
(21) 16 ac., 3 t6. to links ; and 50000 po. to acres.
(22) 29 cub. yds. to feet ;. and 158279 cub. in. to yds. .
(23) 17 ca'o. yds., 1001 cub. in. to inches ; and 26 cnb. yds., 19
onb. ft. to inches.
(24) 663 gals, to pints ; and 3^5843 gills to gallons.
(25) 760 bus., 8 pks. to quarts; and 2875646 quarts to bus.
(26) 250 chaldrons to bushels ; and 186043 pks. to chaldrons.
(27) 56 reams, 19 quires to sheets ; and 52073 sheets of paper
to reams.
(28) 86 wks., 6 d., 17 hw. to seconds ; and 1 mo. of 30 days, 23 hrs., .
69 sec. to seconds.
(29) How many barrels, gallons, quarts, and pints are there in
1386881 half-pints?
(80) On^ year being equivalent to 866 days, 6 hours, find How
many seconds there are in 27 years, 246 days.
(81) From 9 o'clock p. m., Aug. 6, 1852, to 6 o'clock a. m.j March 3,
1868, how many honrs are there, and how many seconds ?
(32) In Great Britain and Ireland there are 121838 square miles;
In British North America, 8389345 square miles ; in the United States
bf America, 8306000 square miles : how many acres in each of those
countries?
00MPOUOT> ADDmON.
59
COMPOUND ADDmOK.
iimi, into one eum^ ^^"^ ^^'^^ deDominations of that
_ ^*^- Grange the numbers, so that thow of <J»« «»«« ;i •
fined in this 12 °*^ ''«'"" dono'-iMtioa ar, oon- ,
Proceed thus with aU the columns. '
Ex. 1. Add together $87.96, $86.87, $97.48.
By tlie Rule,
*36 87 Th« sum of the right-hand column is 20 • writi
36.87 „nder that column, and carry 2 to the next : \he^m
of the next column together with the 2 carried is 28™
write 8 under that column and carry 2 to the nelrt!
Rule, and for tC^ZZl^n, '''''' "'" " "" "^'^^ ^" *^^ «-Pl«
£stl2!'llit *'^'*'"^'- ^- ^^-^ ^8- «•• IW., m l&.,and
Proceeding by the Rule given above,
2
8
9.
4
6
16 . 15
d.
n
S3
12 . 11|
^64 . 18 . 6i
Reamifor the above proem. :^
The ,um Of 2 farthing., 1 mrthing, and 3 farthing^ , B fi^j^
*
fllMi_
mBSPCBMMa!90*
-1 penny, and 1 farthing ; we therefore pnt down i^ that is, one
ftfthiiag, and carry 1 pminy to the celumn of pence. Then
ok* d ihSOingg, jtulid 6 i>enb6 ; w6 therefore pn^ (fothi 6(2., toA o&ftj on
the 2 to the colnixtn of ahillings.
Then(2 + 12+16+5+4>.=88».=(20xl + 18)«.=£l., andl^. ; Urietherfe^
fore pat down 18t., and carry on the 1 ponnd to the cfdmnn of pounds.
Then (l-t-B8-tl6+8+2) poands=£64
Therefore the result is £54. 18«. Bid,
T^&rt, The method of proof is the sattie as ^ait in Siniple Addliion.
Ex. 8. Add together 84 tons, 15 cwt., 1 qr., 14 Ihs. ; 42 tons, 8 cwt.,
18 lbs. ; 18 tons, 19 owt, 8 qrs. ; 7 cwt., 6lbs. ; 2 qrs., 19 lbs. ; and 8
tons, 7 lbs.
a
torn. «wt.
<pB.
Hm.
84 . 10
. 1 .
14
49 . 8
. .
18
18 . 19
. 8 .
. r
. .
6
.
. 2 .
19
8 .
. .
r
Ans. 99 . 6
14
Ex. IX.
J.
(1) 8
86
17 . 10|
. 11
8 . 4i
73 . 19 . 8f
80 . 1^* 5^
iOBB. «wt. qn. Ibg.
(4) 16 . 17 . 2 . 24
18 . 10 . . 20
17 . 15 . 2 . 19
84 . . 8 . 23
11 . 11 . 1 . 11
£.
(2) 63
88
41
6
76
c
15
8
7
17
9f
101
If
£.
(3) 628
854
678
607
859
«.
14
19
18
14
d.
Uf
4
in
03C dra. BO.
(6) 22 . 8 . 2
66
8
15
79
2
6
4
1
2
1
1
gn.
19
.10
11
9
10
•0.
(6) 82
18
20
6^
45
to.
2
8
1
3
po.
24
14
27
80
iy that is, one
Chen
a,y taA oaifj on
Ifti. ; UriB therfei
lamn of pounds.
limple Addliion.
42 tons,^ 8 cwt,
, Id lbs. ; and i
coMfd^m ^btnoN.
£.
«.
d.
J28 .
14 .
nf
J54 .
19 .
4
)78 .
18 .
H
m .
.
Of
J59 .
14 .
in
10.
to.
po.
J2 .
2 .
24
.8 .
8 .
14
50 .
1 .
27
i€ .
.
:5 .
3 .
80
61
prove the re,«It fa e»ohW^ "^^ "^ .•*^»«»- «•• "K t ,mi
(8) Add tqgttber a lbs 9 o» i j_. «.
^d 46 lbs., U 0.., 16 dwt. 9 gT' ^ ni) „■ i ^'•' '" «^ !
4179 oz., 11 dwt., 14 g«; 8497 oMs'dwT Sil. M^'^' "'«*^ ••
ir grs. ; and 1088 oz., 4 dwt., 14 M- 7«^ *t" ' "*^* "^^ " ^'^.
(9) Add together 10 lbs 8 otT^ ^ *"'' *^"-
j9ib., 9 o. s-^d^ 3";!T'mS:^t;.^.::a' w^' ir- «*-'
1 «cr. : also 18 lbs., « oz., r drs 3 mr Vff !? , ' " <»'•> * **,
18 grs. ; 86 Iba, 8 ^z., 3 «r.. M ^ Voz fiL 'J "*' " *»- * "■<"•
96 grs. : explain the p«0M, fa oS ol ' ' '^' ^«^ i »»d 17«lb<4
9 oz. , and 19 cwt, 2 W W Ibl u'o, ! ' V"^- ^ «■••' «> »>«.
a ITS., 15 lbs. J 781 tinXwt 8 It?; ol** "" *<"^ " o^'-
tons, 12 owt, 16 lb,, 12 oz. ; 'es ton^ 17 o'w^T^r ^ "^ «l ^
8 qrs., 15 oz. ; 19 oTt. 27 Ibi • ...,1 ntt T?.. ''• ^* "*• ' '^ toM.
results (owt=112 lbs.).' • '"'' ' '""^ ^« «*. " oz. ! proTo ^
-d.5..,i^ ,9poU,i!^;,««,-;«*^.««po,8y*.;
? B., U in. ; 12 m., 6 fiir 1 tA <> » o • ' 1 ""^•> *« PO., 2J yds-
8* yds., 1 ft., 10 ,„. 86 fa., «'ft' \fl "ii ^W ^"•. " PO.:
W »., 1 ftar., 21 po., 8 vd^l ft ? 1 '^V ^/^' ^ft-. 8 in. ; Ld
108 yds., 67 lea, 8 ^.,^7^^' n'^^l f""? '«*■• ^ ">., 8 fe.,
-piai^^iT4fyS,t;^;^'».ilj^
^
ABTUBnuEiia
4 qi^, 2 na. ; 87 Eng. ells, 4 qrs., 8 na. ; and 79 Eng. ells, 8 na. : and
pi;ov$ each resalt.
(18) Find the sam of 25 ac., 2 ro., 16 po. ; 80 ao., 2 ro., 26 po. ;
26 ao., 2 ro., 85 po. ; 68 ac., 1 ro., 81 po. ; and 84 ac, 2 ro., 29 po. :
also of 5 ac., 2 ro., 15 po., 25^ sq. jds., 101 sq. in. ; 9 ac., 1 ro., 86 po.,
12i sq. yds., 87 sq. in. ; 42 ac, 3 ro., 24 po., 28f sq. yds., 67 sq. in. ;
12 ac, 2 ro., 5 po., 18f sq. yds., 23 sq. in. ; and 17 ac, 24 po., 80 sq. yds.,
118 sq. in. : explain each proems.
(14) Find the sum of 8 c. yds., 23 c ft., 171 c in. ; 17 c yds.,
17 c ft.. 81 c. in. ; 28 c yds., 26 c ft., 1000 c in ; and 84 c yds.,
28 0. ft., 1101 c in.
(15) Add together 89 gals., 8|qts., 1 pt. ; 48 gals., 2 qts., 1 pt. ;
56 gals., 1 pt. ; 74 gals., 3 qts. ; and 84 gals., 8 qts., 1 pt..: also 2 pipes,
42 gals., 8 qts. ; 86 gals., 1 qt. ; 5 pipes, .48 gals. ; 12 pipes, 58 gals.,
8 qts. ; and 27 pipes, 2 qts., of wine : also 19 hhds., 10 gals,, 8 pts. ;
29 hhds., 50 gals., f pts. ; 116 hhds., 46 gals., 5 pts. ; 2 hhds., 2 pts. ;
and 285 hhds., 1 bar., 3 qts., of beer.
(16) Add together $19.28, $27.35, $37.89, $216.16, $152.98,
$225.17, and $g3.19 ; also $2796.28, $8878.15, $737.85, $6797.2*7,
$9689.21, $5298.78, $6925e.a6, $52678.38,- $27812.15.
(IT) Add together 4 mo., 3 w., 5 d., 23 h., 46 m. ; 5 mo., 1 d., 17 h.,
57 m. ; 6 mo., 2 w., 1 h. ; 1 w., 6 d., 23 h., 59 m. ; and 11 mo., 1^.,
58 m. : also 7 yrs., 28 w., 3 s. ; 26 yrs., 5 w., 6 d. ; 68 yrb., 6 d.,
23 h., 69 s. ; 43 w., 23 h., 50 m., 12 s. ; and 124 yrs., 14 w., 19 h.,
87 s.
(18) When B was born, A's nge was ,2 yrs., 9 mo., 8 w., 4 d. ;
when (7 was born, J5'3 age was 13 yrs., and 3d.; when J) was born,
(Pb age was 9 mp., 2 w., 3 d., '23 h. ; when £1 was bom, J^a age was
6 yrs., 11 mo., 23 h. ; when F was born, IPb age was 7 yrs., 8 w., 5 d.,
15 h. What was JL's age on ^'''s 5th birth-day ?
COMPOUND SUBTRACTION.
75. Compound Subtbaotiox is the method of ^ding the differ-
ence between two numbers of the same kind, but opntaming different
denominations of that kind.
Is, 8 na. : and
2 ro., 26 po. ;
3 ro., 29 po. :
1 to^BH po.,
s., 57 sq. in. ;
o., 80 sq. yds.,
. ; 17 c yds.,
nd 84 c. yds.,
»
2 qts., 1 pt. ;
also 2 pipes,
ipes, 53 gals.,
' gals., 8 pts. ;
lihds., 2 pts. ;
o., 1 d., 17 h.,
11 mo., 1^.,
)8 yrb., 6 d.,
14 w., 19 h.,
ing the differ-
ining different
OOMPOUOT) SUBTRACTION.
bers'l^.e^lre dLV^r tv^t'" *f ^ ^^''^^^ "^ "^^^ ^^ -^'
column, and ArawlTZZ Zn "'" "^' ^^' ^ «^« «™
underneath. '^ Btend^abore it, and set the remainder
ber!l"t%rtri!:" 1*'^ '^'^^ "°^ ^ ^-^^^ «^- «»e nnm-
mination as make one u^VrtleTe^V ^ °/**^ ^' *^^ ««»« ^«°-
as before, and carr;:n:tl"LUwt^^^^^^ -^^-*
tion in the lower line. * ^^^^^ denomina-
Prooeed thns throughout the columns.
Ex. 1. From $278^26 take $1783.29.
$2782.26
$1783.29
$998.96
This example is worked in the same way aa
Simple Subtraction. ^ "
Ex. 2. Subtract £88. 18*. 8id. from £146. 19,. m
Proceedmg by the Rule given above,
^ ■ •• d,
M» . 19 . 6i
88 . 18 . aj
£S8
8»
Eeaum/or the above pneeu.
Since ^d. ia greater than id., we ad<l f n X7 ^ r »i-
thus raising it to 6 fartlungsfald 'henVfJ^- * *""""«' <»• 1 P'-ny.
6 ferthlngs, we have 3 ferthin^ T » . ^' *™ '"'"'^'ed from
<«.d in pS;, to tolUeTeifwif • ''!,*''*"'''"« write down JA:
»«mber;we add 1 pt^'t,''.: Jr^'J^ ""' '^"'"' '''^ «"« ""P-
pen^ » LToTpZr; '"""it""; ' "•""•• "« '"'"f"" '-'d 12
I M
64
mitiTHMXfrrOi
It f§ msoffM thUt !n this process, vh«!»«¥«r wi add to tlte npp«t line,
tre also add a omnber of the same value to the lowet line, so that the
final difference is not altered.
Ex. 8. 6L')traot 106 lbs., 11 oz., 16 dwt., from 144*lbs., 8 oz.,
Udwts.
Jh. 01. dwt.
144 . 8 . 14
106 . 11 . 16 <
87
8
tt. ^
18
£.
§.
d.
£.
t.
d.
(1)
843 .
18
. 6t
(2)
66S .
6
. 11|
-
11 .
18
. »*
849 .
19
• H
owt.
qr.
lbs.
OS.
ftir.
po.
ydB.
(8)
63 .
. 18 .
1
(4)
14 .
84
. 6
68 .
1
. 12 .
10
1 .
88
. 4
M.
TO.
pa
'
qn.
bua.
pk. gal*
(B)
63 .
1
. 29
(•)
64 .
8
.1.0
67 .
2
. 88
8 .
6
.8.1
(7) Subtract £466. 15«. llfd!. from £534. 13«. 10-ld. ; and prove the
result.
Find the difference between the following numbers, and verify the
results: ' .
(8) 6836 lbs., anl 49T6 lbs., 7 oz., 13 dwt., 19 grs.
(9) 26 tons, 2 qrs., 23 lbs., and 19 tons, 8 cwt., 8 qrs., 18 lbs.
(10) 144 lbs., 9 oz., 4 drs., 1 so., and 129 lbs., 7 drs., 8 se*
(11) 418 yds., 1 qr 1 na., and 387 yds, 8 qrs., 8 na.
(12) 16 yds., 1 ft., 5 in., and 18 yds., 2 ft., 7 in.
(18) 18 m., 6 fur., 85 po., 3^ yds, and 12 m., 88 po., 4 yds.
(14^ a ro.. 28 po., 27 sq, yds., 7 sq^ ft, and 1 ro., 89 po., 28^ sq. yds.,
8 sq. ft.
he npptt line,
te, so that the
14 lbs., 8 oz.,
oowocTO KWamjOA-noH.
d.
; . IH
I . 9f
yds.
) . 4
18. pk. gal«
I . 1 .
► . 8 . 1
and ppoTe the
and verify the
.,18 lbs.
3 se*
ryds.
).. 88^ sa. Tdfl..
1280?„M ""•• '"*■ >' '""• '^- <«^™''- •-•. "4 M c,b. ,a... 24 c«b. ft.
(20) What ,„™ .aa^ to £M7 trr^w n " 'f ">
(81) A ftrnis!,ed Louse taloth Im^i"""'* ^""^ '
worth ♦6978.50. By how 11 Z, ,1,* ^^-^^ ' mftrniehed, it b
tbe valae of the hou^ f ""* ''*'"* "^ "» f»""">« «ceed
COMPOUND MULTIPLIOATION.
^p-^Top^x^-^iutr-^ 1 ""^'"^ '^*
composed of different denominat^T. w n' ^"' "^ "'^ •"' """nhef
it is repeated a given number of ttaeV "" "' ""* """' ''"^ ^^^^
MalaX^d!"" *"' ■""'"""" """''' '"* '—' denomination of the
»dtd ttnSf °aLr„^r '7r"''*'<'° '•^ *-"«P«-.
this first product ; if ft™ be . '*, '^/'"°"''''»'''''' '"»'*•'-'«» '"
-oond produot, m^I^J^'^Jt *": f' "^^ " "''""' *» '"<>
ttultiplicand by the multiDlier .„!^ <- T' ^«''»""''»«on in tl,e
mentioned "amter of nSJ ^' f'tb'^'''''^ *° '' '"« '""'^-
produot ^ P ''***^ '""' ''f* "^"It as with the first
Carry this operation thronith witji .n a a-^
of the mnltiplioand. *■•* **'^'«°' denominations
. «
Ex. 1. Multiply $212.13 by 12.
$212.13 '
-^ Mu,ii^rrrr°j^v-'^*-'''«'"p'e
»25«.6» . arating the dollars-.nd".S„;;inTp;;:;;;i"''-
66
ARrraMEno.
Ex. «. Multiply JBB6. 4#. 6|<Z. by 5.
Frooeediug by the Bale given above,
£.
66
4
6
2
8i
, ., £281
Heaton for th6 above proeeti,
\d. multiplied by 6 is the same M(\*\+l4:^+{)d.^6 balf-penoo
e2^. ; we therefore write down }d.y and carry 2d, to the denomina-
tion of pence : »
Qd. multiplied by 6=80i. ; therefore (2+6x6)d!.=82i.=(2xl2+8)<f.
=28.+8d. ; we therefore write down SdL, and carry 2». to the denomina-
tion of sliillings : •
4». multiplle,d by 6 = 20«. ; therefore (2 + 4 x 6>. =» 22«. = (20 + 2)*. =
= £1 + 2«. ; we therefore write down 2«., and carry £1 to the deno-
mination of ponnd^ :
Now by Simple Multiplication £56x6=£280 ; therefore £(1+66x5)
=£(1+280)=£281.
Therefore the total amount is £281. 2». 6ld.
77. When tljo multiplier exceeds 12, a convenient method is to split
the multiplier into factors, or into factors and parts: thus 16=3 ii6;
17=8x5+2; 28=4x5+8; 240=4x6x10: and at) on.
Ex. Multiply £56. 12*. Hd, by 28.
£. ». d.
66 . 12 . 9i
4
222 . 11 . 1 =ralue of £5r. 12s. W. multiplied by 4.
1112 . 15 . 6 =value of £222. lU Id. multiplied^y 6, or of
£56. 12*. 9^(1. multiplied by (4x6), or 20.
166 . 18 . 3?= value of £55. 12*. 9^ multiplied by 8.
£1270 . 13 . 8|fe: value of £56. 12«. 9K' multiplied by (20+8), or 33.
Note For an example, when the multiplier is & large number, see
&2XV J^i'Sm'OAV&iJ
iji* l^T7 wiV
, J..
«v.
ooMPouin> DivMioir. 57
Norn When the multiplicand contains ferthinga. if one of tha
Multiply "^* '^^•
(1) $217.86 separately by 8 and U.
(2) £7. 19». 7ld. separately by 10 and 13.
(8) £721. Os. bid. separately by 81 and 96
S «V 'o.lf ' " ^"'- " «"• "^P'wtoly by 8 and 88.
S «» ^'t ""■' * '■'"'•' ^ '"• "P'^'^'y by 12 and est
S »! .' ^'■•' ^ ""• "»P»™te'y by 9 and 68.
/,n^ ,, ^ ' ^ "■' ^^ '"• '^P^ot'ly by 7 and 29.
(10) 16 80., 8 ro., 88 po., 27 yds., 2 ft. by U. ■ '
na^ T T' '."■' '^ P"- "'P'^te'y by 12 and 106.
(ia> 67 gaK, 8 qts. separately by 10 and 267
n2 r T'' f ]""•' ^ P""- ««P"-»«e'y by 18 and 840.
as LTm' .^^ '', "- " "■• "'P*™'*'^ "y " "nd 889.
n « f ' '^*''-' ' ''•• ^ P'- of !>«« separately by 39 and 7M.
fcm'i* T,V.'' ""'" "' *''' "^""S" of ^3^- <>•• 2K for overy
8 of them and 17 horses at 87 guineas t^aoh . th^ « - ^
them aU home amount to in Sas m , ^ '^^'"^^^
from his iv^nlcers to pay f^ Hi wTolHutlaT' ""' ""' '^ '^^" '
^r.r®^ ^^T. "^ ? ''^^''' ^^ ^'^^«"- ^ ^^^^ Chest there are 18
rlZmt T' '"r ' '^^^^^^"^ ' -^ - -^ division tTe^ei!
placed $25.26. How much money is deposited in the chests ?
- . ^^■
OOMPOUKD DIVISION
tained in another of the same kind. ' ' ^'^'^'^ ""-^^ ^ <^"-
ea
juqscBusxia
Carry on this procees through the whole dmdenfi,
Ex.1. Divide ;ei99. 6«. 8<?. by 180.
Proceeding by the Role given aboye,
j6. «. d.
180) 199.6.8 (1£.
180
20 (add the to.)
180) laea (lOfc
1800
86
13 (add the 8<?.)
laO) 1040 (8«?.
1040
Therefor© the answer is ^1. lOa. ^
The work is usually written thns :
180) 199 . 6*. 8*(U. lOi, 8dL
180
69
20
1886
1300
84
^13
1040
^^onfltr the above proem.
We first subtract £1 t^iken 180 times, from £199 6« m o„^ *t.
remains £69. 6*. 8^ ^ ®*' ^^'> ^^ *^«'«
. Therefore £1. in« fi>7 :- — x-i__, - .. . _
.^. ^^, ^o wwai«ia^a im titnes in JB199. Ai; M,
^ or not larger
then find bow
ination of the
multiply -ia in
>e next iDferior
ination in the
written thns:
f.f and there
mbtpact 10«.
t 6d. taken
COMPOHBTp pHfl^lON.
Ex. 9. Uym ^,1076. 4». 8H by 627,
627)1076 . 4. 3i(2£.0,.l0ef.||l.
1054
• 22
20 (add the 4<.)
444 (0#.
12 (add the 8<?0
627) 6381 (lOtf.
627_
«1
f (add the 8y.)
^- 627) 247 (Oy.
Therefore the result is £2 fta in/7 o«^ *i,
♦ u j« .J ^ , • ^"^•» *"^ *^ere remain 247 farfWTt.To
Therefore the quotient is £2. 0«. 10<?. Ofl?^.
Ex. XII.
m S^t^': '^''' 00^478.14.6^^.^12.
(3) 459 lbs., 4 oz., 5 dwt., 22 grs. h-29.
S 'f "^,^"^^:-'«/^«-»2««.^68. (5) £1288. 1,. 8^.^764.
(6) 2fur.,10po.,lyd.,lft.,10in-^85. (7) £165. 15* gJ • U
rS^ £2'79ft -t.TA^ V«>'*ioo. 10*. »!<?.— 180.
H9) 1738 0. yds., 1286 c.in.+798.
(11) 266 tuna, 88 gaLB.^10».
(13) $61411+217.
(16) £180264. 9*. 6^.^9416.
(17) 178 cwt., 8 qrs., 14 lbs. -^68.
(19) 684 d., 8 h., 9 m.-.47.
(21) 76 cwt. .^968,
(23) 18 ac, 1 ro.-i-147.
(25) 97 qrs., 8 bus., 8 pk8.-^107
(8) £2728.^744.
(10) £87. 8«. l<?.+74.
(12) £492710. U 8^-5-6362
(14) £1746-5-2787.
(16) 1288 cwt., 4 lbg.-j.7$,
(18) 206 mo. of 28 days, 4 d.-i-26.
(20) 15cwt.,271b.,lloz.~F.466.'
(22) 75 ac, 8 ro., 89 po.-i-26.
(24) 9t yds., 2 qrs., 1 na.-i-903.
(26) $455466-^687.
79, It-may soTnAtimpa Vka 'A-wt^,,^ . ...
into fector,: ihus. " """' "»"'«'"«'" '» "^k "P th» divi«,r
4
■ ■
i. i-
70
ABTEHMETio.
IP
Now, dividing by the factors 6 and 8 we get
^- « d.
21
48
{
I —
ir
ly rem. 4
^1 ^t rem. 5
48) 181 2 . 8J (2£. 14.. 71^. II.
85
JO
48) 702 (14*.
* J8
223
192
80
12
48) 868 (7d,
J?2£
b2
_4
48) 130 (2^.
96
34 • •
Therefore the qnotient is £2. 14*. ^U. ^g.
tlJLJL^''^^''''''^^ '^^^-re U compound numler. of
2 . 14
Therefore the true remainder
=(5'<6+4)q.=34;7., '
and since our divisor > 48, we write the
remainder; thus J |.
9.
J12
68
4
265
d.
16 .
20
818
12
8825j
4
15300
265) 15300 (60
1680
Therefore 60 is the answer.
18
9.
3 factors 6 ana 8.
ra 6 and 8 we get
rem. 4
\ rem. 5
ler
?..
^ we write the
nd numhert of
ion: divide as
uired.
.?
MKCELLAOTDOUS EXAMPLE WORKED OUT. 71
Reason for the alme proem.
6». 8H=255 farthings,
^ A^r . * ^^^' ^^- 9^-15300 farthiniw •
rifot ''"*"'"^ "■'•'"'"^ «» «- ^'^'" l^so'c farthings !«., no
Ex. xm.
(1) £2. 12«. S^.-j-U 4^rf.
S^ ff • ^®*- ^<'H-£2. 8*. lid.
(3) £160. 4.. 8i,.^£i. 10,. 6K
(4) £401.4..8rf.^£2.]U6idr.
=$60
Ex. 2. A spring of water, which vie?^<. 'tk „„n ,
eOO families : how much wat;r .r^'aetL^'lf^^^^^^^^ ^"^P^^
The daily supply of water=(76 x 24) gallonsf '
therefore each family may use daily ^5^
600
pals., or 8 gal&
8miles=(3.M60)rard8=5280Taras
ana smoe the wheel passes over 4 yards in one revolutfon •
6280 '
-^- or 1320=nomber of revolutions required.
Ex.4. How many guineas, soveroi^no i.„ir-_-__ . .
and of each an equal number, are therein £^246?'"''''''' ^^'^ ^^"""^
i
72
ABEEHOtEno.
Now, 1 guine»+l sorereign+i half-crown+l Omng
=(43+40+6+2) siip^noea t
-80 Bixpeiices ;
aiid iJl246=(1246x20.2) si:rpen<^=4084^,j^^^^^^
the qnestion therefore is reduced to thia • tt««, ^
contained in 49840 sixpences , ^** *^ * ^^^ <>«=«« are 89 .ixpenoes
Number reqnired=l??l?=660
Cost of oask=($3) K 60=1180=18000 cents
$1.92=192 cents " '
therefore 1|?»2, „ 98t=the nnmber of gallon, which the ^ ^^
contain, in order that its contents may be sold at *1 09 . „
Therefore (9Si-«01 or sa» ♦!,. Z Z , * ^^ * K*"""-
tave to be added ^' *="" ''""'^' "^ gallons of water which
Ex. 6. A tiaveller walk- 22 niiJeq ft «fa«- ««^ a , ,
miles, another foHows him at tte ^te of sl'J^f "^ ^'. ^ Sone 84
wm the second traveller o-erlTeCflLtt ^ ^' " ^'"'* ""^
The second traveller has to walk ovai- m -,:i ,
before he cu overtake him. • °''*' ""'" '^"' «>« ««'
Each day he walks (34-22) or 12 miles more than the first •
therefore ?t or 7 is the nnmher of days required
price per gallon !«««; ih. L- . ^^^"''''' ^* ^*- ^^' « ga»on ; at what
neither gai^ nor Z bt ) 'T ^ '' "^'' '^^' ^^* *^^ «^"- ^-7
8 gallons at 12*. lOd. cost 5' . 2
^ gallons at lOs. 6d. cost - 8 . 18
10 gallons at 9«. 1(?. cost 4 . 10
therefore 26 gallons cost £{^
82
inferic
1
1 are 89 sixpences
isk containing 60
$1.92 a gallon ?
ih the cask must
3 a gallon,
of water which
' he has gone 84
r ; in what time
re than the first
fie first;
ired.
!».10<?. a gallon,
gallon; at what
the seller n^ay
7 gain £l, 13*.
"JSCEtLAMpCB EXAMPUB WOBKED 0T3T 78
Snd. If he is to gain £1. 18,.
26 gaUons must be sold for £18! r...£l. 18,., or £16 ■
therefore, 1 gallon mnst be sold for ^ ■ whi,), i. .' '
«. he price reonired. '' ' ^^-^-^ed out, gi.es 13,.
each wotfian. ^ '~^' '"^ •»«■» >»«° thrice as much a.
8in<» each woman's sh.re=twioe ea<A boy's sh.r.
therefore 6 women's shares=ia boye'Ths™ ' *^
Again, since each man's 8h«« tL- ^^^
oys «^ares+12boy8»share8.7 boys' shares =: W i
oi- 49 boys' 8hares=49 "^
- ^^«refore, each boy's share|i?=|i.
Therefore, each woman!8 share=|2.
• • • • each uan's share=$6. •
» ' _
NEW DECIMAL COINAGE
aaoS-a^i^rbw^rrdii^:^;]:'^ - »^,,««-^. '-„ds to-
of which the nnit shall be th7 T ^ ' ^"' "" P™'«°' «>. hot
coinage as is proposedtbe^s^r^n'^S' t'c"""* ''*^ «""<•
r;'^"-"-!--'>^''»r^-S. ^hetbu^rrtatd
10 mils (m.) make 1 cent, Ic
10cents.........ifl,,i^-fl^ V
^^^"'•'n«- 1 pound, £1.
^ 82. In such a system, muoh nf fi,. i„v__ .- _ , .
uu^rior aenouonatlo^^ and the co^— --;-.^™^^^^
74
^^JMTHMEno.
^nld at once say, £24. 8 fl. 7 o. 2 ni.-24372 m Rin^ k . .
the operation ofreducing at length, we obSn ^^ P^'^on^iDg
- ^
0. m.
24 . 8 . r . 2
10
240+8, or 248 fl.
10
2480+r, or 24270.
10
or
vemight8ay£24.3fl. 7o.2 ni.=£^72
24870+2, or 24872
m.
for, pro
Simila
(
'ceeding
trly,£24.8fl.7c.2m.=
Conversely 24372 mil8=£24,
in.=248-72 fl., or=2487-2
by Rnle (Art. 73), we get
8fl.7c2m.
10
10
24372
2437-2 m.
243-7 (
24-8 fl.
hence 24372 m,=£24. 8 fl. 7
Again, £264. 6^ o.=£264. 6-5
c. 2 m.
100
26400 0.+5-5C. *
=25406-6 0.
=264066 m.
Also, £264. 6i fl. =£264. 6-25 fl.
10
2540 fl. +6-25. fl.
=2646-26 fl.
' 5=26462-6 c. -
=254626 m. -
Ex. xrv.
y^°^^:f '"" ^^ ™<"=-'- inferior denomiutlon «„,
(1) £16. 6 fl. to mii8,.and 6 fl. a n a r« *^ --.-I-
( f
loe b7 performing
xn.
J.,
EQiii&tion and
NEW BBOHfAI. OOIKAGE.
75
(2) £80. 9M. to mil«. and :eflft 1 fl « « .
8» J^ ;:.""• '"•^«"''^*«- 2*0. ton,!,
examples. J^I«™rmed, as will be eridentfrom the following
Ex. 1. Fin<Ithesninof£18 Sfl »« k ..
*18. 6 fl. 8 ©. 5 ni.sl8685,
9£. 9in.s 909,
£24. 1 m.=r24001,
- 8 0. 2 m.= 82,
6ifl.« 625,
£.
ora 18-686,
ors '909,
or=s 24-001,
<»= -082,
or= -626,
or=je4410a,
44102 m.,
each of whicli results=£44. 1 fl. 2 m
3SX.2. I'rom£l6.8c.2m.,8ubtract£l4.4fl.9m.
m,
*16. 8 c. 2 m.=:16082 .f*
£l4.4fl.9.m.=14409 0^-1^-09%
i440», or:= 14-409 ,
162am., or= £1.628. '
each of which results=:£l. 6 fl. 2 o. 8 m. )
Ex.8. Maltiplj£i6.8c.2m.b728. ./
4 £16. 8 0. 2 m.=16082 m., or»£i6.032. ^
4^.
16-082
V. A
n.
16082
28
«???1 J2064
8(r^86m, ^gg-^
each of the above rfl;^ult8--«£368. r fl. 8 c. 6ni.
BiTide £3aS. Tfi. 8 c. 6 m. by 28. 1 .,-.»i^'
--*;.
V o
7G
-^^MlHMKno.
In other words, diyide 86878« n». b^ 28, or £868-786 by 28.
28) 868786 (16082
28
188
79
69
46
46
28) 868-786 (16-082.
?L
183
188
78
69
46
46
^ ot the above retaits=£I6, fl. 8 c. 2 m.
2>tOT«. ^piUar advantages would result fm«. *^, . .
■jitem in weights and meXe^ ^ ""^ *"' ' ^^""^^
■ , ►
Addtogeth^ . ^ ^^-
ilnd the difference between
(8) ^19. 6 fl,, and £16. 8 fl. 9 c.
(4) £20, and £19.9 fl. 9 0.9 m.
(B) ^6.6ifl.,ahd£4.4io.
Multiply :
(fi ^^«- 8 fl. 8 m: separately by 5 and 68.
. (7) 9 fl. 2] c. separately by 18 and 1008. ^
(8) £160. 6 m. separately by 2006 and IBB^^
Divide
(0) ^194. 6 fl. ire. 6 m. by 6.
^ 00) £10764. 2 iim. by 11.
,(11) Je842186.8fl.by7380.
e of a deolmal
»™«™^»l»Ora OramoW AND EXAMPLB9. 77
(8) What 18 meant bj * OomDonn#i ir«i*i .'*"***""«'«»
»amb«r, of the «me or dIff«^Slt *'^S'' °" '""^
th« reuon. Wh,t fa the eort of ^h^ ""-"'plW togwherr Gl,e
ohUdren at $9.12. each? ' «ooommodrtio» for W760
(*) Aperaon boaght 1768 thk!. «* -i-.v » »,
l««»^ how mud. win f.^ rlLve.*^ "** '*^' ««»« «
i.^^Vuzr^^'oh^^rrcr'^r^-™' ■'-♦'^^
(8) D«««e • . „w .1 Iw J "•■«"« ttehonwworthf
"""•y cubic feet tt^J™ V ?'1^TJ'«"'? "^ • »««•«' how
«nb. io. to enb. yd,. • L tod^-.t ^"^ ^'"* mamm
(«) A seiTiiiit'. wagM «» £10. ar , ^. . i .i™ .
~o.ive for r week. («pporf„g . j^t, 'o^'^Z^^f'' *• «•
78'
ABITHMETrO.
m.
(1) What afe the different uses to which Troy weiirhf •« i a« •
lJ^L!\T^V^ ^"""^''^ ** ^^ P^''^^' ^^ the cost of cirriaire ia
IK per lb. ; they are sold at £4. 10s. per owt. • what is th«^oT
loss per cwt. ? ^"^ " "ie gain or
W, b,. oaoptea to co^/r„ral:Sr:"^^^^
W A gentleman laid up in ^ne year 1851 tun ka v •
<i.Uy $7.80: what w«, hia income inTstZr* " '"' "*■"
~oo„<,, ana the fourth 4 tilt l^I It'l ViT " """"" " ""^
IV.
(2) Two persons bay postage-stamps at 12 a shillini?- c-.a i.«fo?i-
them at 11 for a shilling, and the other at 18d %T!7 '
the gains on selling the tae number o/^lmpl* ' '^"'^ ^ ""^'^^^
(3) A hnndred sovertigns ah oqaally UebL are wnrfh «!„ . /s
""T Rid" " *^' ^ "^ '° '^"''"^' ™
1. ne sum of £27. 3 o. B m. ; £660. 2ifl. ; £80. 8 o. 7in.
8. The quotient of £406.6 fl. 8 0.6 m. by 16
«f !?• ^fT? '*^' "'" *^*-*' '■» »P''it' «t tl-28 a gallon- some
to^m^lt^ "■": in th a„i.ge; however, ie sold ^er el'aTe
»m20 at the rate of 11.80 . gallon : how many gallons leaked out ?
Snde" '' ****" °"' "' '*■ '""""""^ "-«"- "rtterein thi
«ar^?8K?T ""°^ '"'"1 '"'™ ^"P"^ "■"=» «■« Wrth of Christ to the
rear 1863, ,nppo.,ng each year to consist of 365 days, 6 hours.
MISGELLANEOUS QUBBHOire Am> EXABfPLIS. 79
V.
(1) Explain how the statnte defines * a yard', with referenoo to A
natural standard of length. Find the corresponding linear unit, when
an acre is one hundred thousand squnr^ units.
(2) How many barley-corns will roach round the earth, flupposintf
the cn-curaferf loo of it to be 25000 miles ? » i-r "*»
». ^l\ Y \!ll^^^ ^'^'^^^ ^'^'^ ®^ ^^°^' ^^^ "'^"y ^o^ens can be
bought for $415.20. •.
(4) How many times will a pendulum vibrate hi 24 hourg. which
vibrates 6 times in 2 seconds?
(6) If the sum paid for 247 gal'ons of spirit amount, together with
the duty, to $859.66; and the duty on each gallon be Jth part of its
original cost; what is the duty per gallon? *
(6) 12 persons on a journey each spend £28. 4 c. 6 m. in board and
lodging ; 6 ot them agree to pay the travelling expenses, the share of
each amounting to £18. 1 m. Find the amount of expenditure during
the journey. *
VI.
^(1) What is the meaning of the word ♦ carat' as applied to gold,
and as Wl^^f to diamonds ? How many 'carats ' fine is standard gold ?
If from 2793461 lbs. Troy of gold there be coined £130524466. 4^ 6<?
find the value of each lb. *
(2) A wheel makes 51 4 revolutions in passing over 1 mile, 467 yards.
1 foot: what 13 Its clrcumfereiice?
*o/^ i"^ TT"^ ^"^' * liogshead of sugar, containing half a ton, for
|90^ana retails It at 11 cents per lb. ; how much mo ey does he make ?
at it ^/]'''^''''^''y\]l«'^^^^^ ISgallons
at 14*. 6d. a gallon ; and 18 gallons at 15*. 9d, a gallon : what will be
hisouTyf '''''*"''' '""'^^^ he may gain £2.6*. 6c?. on
.A <^>/ e^°f ^"*^ ^distributed $198 among 12 men, 16 women, and
30 children ; to every man Le gave twice as much as to a woman
and to every woman three times as much as to a child: what did each
receive ?
(6) A merchant emend. £1686. 5*. on equal quantities of wheat at
£2. 2*. ,^ quarter bari., at £1. 1,. a quarter, and oats at 14*. a quarter:
what onantitv nf ^anh n,?n i, v o i » ^c* .
89
ASnBKUld
vu.
b. so trillion, of mile, d bUntl *''**"'• '"fP"^ *"
W, given np Zl^^n „* wtX x Vol"" T ™'^? '" '»
of the IS rooeire I ' " ""^ '""^ O"*
„„^? A?"'*' '*" '"'' *""' '»" **«»<X' more than he left hi, .econd
fether leave tThl Z^t '"'*"' ' *"' *■"' ""» <>** *•
vni.
by il. iH/ mT.'l "J. "T '"f W *» «^« ""ItipBction of 6* M.
(2) A carriage-load is found to weiirh 1 tnn ft ««f -i
con^Bt, of ai« eqaa, package.; wh.tTalt 1X^1;^.' "- "^ *'
what ia hia inoomef ^ ' ^ '^* '"^ "^ "■« ?«"••
ao «. «.d . pavi.«.„t,<rfru ta ft, £ . tlLtt, fte'ro,"' "*• '^
cost of the house? "*•««*. wiwt u the whole umoal
MMGELLANEOIJI QUWHOIW AND EXAMPLM. - 81
IX.
n iP^ r^w n!° ^* ?'^'"^*'' " °'*'^ *° "■•• ^n "^^n® 21 o' 1851, the
hi° wrtJ. ^ ^"^'^^r^^'' ^^ *^*» ^y *°<i y^*^ of
th.?i7wt^M ^^'t^ ""! ^""""^^ «• 10 fe«* i« circumference, and
the hbd-wheel Is 16 feet : how many revolatlons will one make more
than the other in 100 miles f •Comoro
(8) Sound travels at the rate of 1142 feet a second : if ajfun be
t'fZ^ir "^'.f'"^^ ^'^^ ^^^ J^ow long will it be, after^ein^
tlie flash, before I hear the report?
JcM^^TaTZ""" ""^ ' ""'• '"■* """^ »"• •'■'"*«™- •"*•
. iiuL^''Z\y n"' '^.""° •" '" "'^'nefron' Toronto to Brampton.'
. dfatMoeof 18 mile., when he tJcesUO step, of 2ifeet every minWt
™i lif •»"«''!>«">"■• employ. 60 men «.d « boy., who re.peoU7elT
work 10 «.d U hoar, per d.y daring 6 day, of the week, Ld hM
ll.^' "L T'"""' ?"' ' •** "»" «'°«'^" «<^- P«' hour, uid
ewh boy SA per hoo.-: what i. the «noant of wage, paid 1.^ the
year! (ay aarsSa week.). ■ •" "uo
X
inn^^ A gentleman Mnt a tankard to hi. rtveBmith, which weighed
100 ox, 16 dwta, and ordered him to ma)™ it into spoon., iaoh
wei^mg a oz, 16 dwt.. : how many qKwn, did he mjeive?
fj^i it «*""*"»"'•«"«*«. fi"- ""» » y«ar. ending with 18«, yielded
£1227. 16.. i how maoh coald he .pend one day with anoli|.er, „ a. to
lay by ISSgninewt ■»• i"</«oi«
beln^! M?!.'"'^'' °' * ^^", '"'°« '""^ ^"^ "^ ^^ »' » '»■'« month
L LT* . ^""^ "^ ' *^'"' "' '""«• i^ «""»' !« "ortl. ♦1.20 per
V« A™f*\ oo-'i'te of 1000 .hekel., each weighing 218 grain.?
and «ld B^r^' ► ^"of ^ '"'~' "'"'o""' <"«* 87 y.^., for »2««.88 ;
and »Id 60 yard, at »1.27 per yard ; at what mnat he seD 3ie remainder
per yard m order to grin |17.04 on the whole f "m<«M»r
tim«..t."°''t"' '"""'^^ "^ ""*"' *""«« " m«ny women, and three
time. a. many boy. earned in 6 day. £7. 1». ; each man earaed UM.
^cWeman lOd, and .«h boy 84 . day. ' How Z^^Zf^'^
i
62
AKXHUEno.
SECTION III,
GREATEST OOJOTON MEAStTRE.
common measure of 18, 27, and 86 " ' » » »
««J.b.r, ex„t>,.. tho, 9 i, ,,e greyest JLtf^t^.^ 0^18^7^
^* If a number mewure each of two tifkA** >• --.vr ?
Thus, 8 being a common measure of 9 An/i ik ^mi ,
»™ their diff^eucej ^ .«. », ZjL ^eK'o ~' '^'^ '
The sum of S ami 16=9+16=24=3x8 ;
. therefore 8 measures their sum 24
The difference of 16 and 9=15-9=6=2x8 ;
^ ^ ^^^'•^^^re 8 measures their difference 6
Agam, 86 is a multiple of 9, and 86=8 x 12 •
therefore 8 measures this multiple of 9- L,! Ar^x\ i
multiple of 9. ^ 01 », and similarly any othor
Again, 75 is a multiple of 16 ; and 75=8 x 25 •
therefore 8 measures this multiple of 15 • ««!? «• m i
multiple of 16. "^u^ipie ot 15 , and similarly any other
B8. "^0 find m greatest <^monmmmre of two number,.
Bulk. Divide the greater number by the less- if ff.«r. K.
mma«., divide the first divisor by it; if' th^ret '.^^aTe^'^d^
©BEATBST OOMHOK ICEASUBS.
63
i will divide
a measure
r is said to
contains it
I a nnmber
iQs, 3 is a
lore giyen
the giv^en
of 18, 2r,
9 measuro
4m.
mre their '
07 other
ly other
he a re^
nainden
^vid« the second divisor hy this remainder, and so 6n ; dwajs dividing
the last preceding divisor by the last remainder, tiU nothing remain^
The hist divisor will be the greatest, common measure required.
Ex. Required the greatest common measure of .i76 «nd 589.
Proceeding by the Rule given above,
475) 689 (1
475
114) 475 (4
466
19) 114 (6
114
therefore 19 is the greatest common measure of 475 and 589.
.Beoionjbr the above process.
' Any number which measures 689 and 473
also measures their difference, or 589-476, or 114, Art. (37)
akomeasuresanymultipleofll4, andtherefore4xll4,or466,Art.(87);
and any number which measures 466 and 476,
. also measures their difference, or 476-456, or 19 ;
and no number greater than 19 can measure the original numbers 589
and 475 ; for it has just been shown that any number which measures
them must also measure 19.
Again, 19 itself will measure 589 and 476.
For 19 measures U4 (since 114=6x19);
therefore 19 measures 4 x 114, or 456, Art. (87) ;
therefore 19 measures 456+19, or 476, Art. (87);}
therefore 19 measures 475+114, or 689;
therefore since 19 measures them both, and no number greater than x»
can measure them both,
19 is their greatest common measure. '\
89. To find the greatest common memare of three or more nun^me.
RtTLE. Find the greatest common measure of the first two numbers ;
then the greatest common measure of the common measure so found
and the third number : t^an ihaf. nf fba «"»«,«^« ~.- ^- ^ -^ - _ -
1
If
i
84
JoarmiBxm,
and th#.?ftnirt3i jinmber. find «n «n tu i _^
8) 18 (a
u uuv greawst OQmmpn measure of 8 and 18.
8) 18 (3 ^
16
2)8(4
tterefb,^2«th.^«^^
«;ce .jfrwteBi common measure of 16, 24, and 18.
Find tte greatest
(i) 16 and V:2,
(4) 65 and 121.
(0 272 and 426.
(10) 826 and 960*
(18) ire and 1000.
(16) 689 and 1678.
(19) 2028 and 7681.
(32) 8444 and 2268.
Ex. XVH.
common, measure of
(2) 80 and 75.
(6) 128 and 824.
(8) 894 and 672.
(11) 775 and 1800.
(14) 1286 and 1682.
(17) 1729 and 6P50.
(20) 468 and 1266.
(28) 6644 and 6562.
(8) 63 and 99.
(6) 120 and 820.
0) 720and8So.
(12) 856 and 986.
(16) 6409. and 7896.
(18) 5210 and 6718.
(21) 2484 and 262^.
(24) 4067 and 2678.
measures 8
18, is tho
118.
199.
Jd820.
»d 8fl(o.
d 986.
nd r895.
Qd6Tl8.
Dd!262^.
ad 2673.
(2ff) 10895 and 16819.
(27) 1242aud232S.
(29) 42^87 and 75683.
(81) 10858 and 14877
(8«) 14, 18, and 24.
(«ff) 18, 62, 416, and 78.
(87) 805, 1811, and 1978.
(89) 504, 6292 and 1620.
MlAflT OOmrOK VOLTXPLB.
88
(26) 80934 and 110881.
(28) 18536 and 28148*
(80) 2867U and 999999.
(32) 271469 and 80599.
(8*) 16, 34, 48, and 74.
(36)887 1184, and 1847.
(88)28, t ,164, and 848.
m 896i 6184, and 6914.
tEAST COmiON miLtiPLE
of times Without a reLTntr Thus^TJl °°"'"*' ^^ exact number
8, 9, 18, and 24. ^"^'^ ^^ "^ * ^^^^^i multiple of
^e Least OoMkoirf MtrLxiPtE YL If^ n/ f '
nmrtbers is the least number wKch wm \ • """^ ^'' '«o»'o gtven
nnmbe« an exact numberTtim^^^^^^^ .^^^^ **^ «»« «^-««
the least common multiple of ^9 ir-nd 24 ""'""'''^''' ^"^^ ^
18) so (1 '
18 .
12) 18 (1
6) 12 (2 •
a.e«fb™ 6 1, the great,.* common »el„ of 18 «.a 80
41. ^ 18x30-f-6=90
therefore 90 is the least common multiple of 18 and 30 '
^-^a^on/or the above proem.
«. „ ^ 18=8x6, and 30=6x6.
Since 3 and 6 are prime f^tn^ ;. z. .,. ..
^ ,, „ „,^„ .^^. g |3 jj^^ greater
ae
ABITHMJBI'IO.
^.i'
I
common measure of 18 and 80 ; therefore their least oomm*^^ mnltiple
must contaia 8, 6, and 6, as factors.
Now every multiple of 18 must contain 8 and 6 as factors • and
every multiple of 80 must contain 5 and 6 as factors; therefore every
number, which is a multiple of 18 and 80, must contain 8, 5 and 6 as
fectors; and the least number which so contains them is 8x6x6 or 90
; Now> 90=(8 X 6) X (6 X 6), divided by 6, '
=18x30, divided by 6,
s=18x30, divided by the greatest common measure of 18
and 80.
•92. Hence it apifears that the least common mnltiple of two
numbers, which are prime to each other, or have no common measure
but unity, is their producv.
' 98. Tojind the least common multiple of three or more numbera.
Exile Find the least common multiple of the first two numbers:
then the least comracm multiple of that multiple and the third number
and 3o on. Tbe last common multiple so found will be the loast
common multiple required.
Ex. Find the least common multiple of 9, 18, and 24,
Proceeding by the Rule given above,
Since 9 is the greatest common measure of 18 and 9, their least
common multiple is clearly 18.
Now, to find the least comnion multiple of 18 and 24^
18) 24 (1
18 , •
6) 18 (3
18 .
therefore 6 is the greatest common measure of 18 and 24 ;
therefore the least common multiple of 18 and 24 is equal to (18x24)
divided by 6, ^
24x18-^6=73
therefore 72 is the least common multiple required.
[ JReasonfor the above procfiss.
Every multiple of 9 and 18 is a multiple of their least common mul-
tiple 18; therefore every multiple of 9, 18, and 24 is a multiple of
'•■ ;■???'.
LEAST COUHOir ItCLTTFIJE
89
■'A mnltiple
Jtors; and
fore every
5 and 6 as
► '♦C, or90.
rare of 18
e of two
u measure
mbers,
numbers ;
I number,
the loast
iieir least
(18x24)
on mul-
tiDle of
18 and 24; and therefore the least common mriltiplir of.0^, 18, and 24 ia
the least common multiple of 18 and 24: but ^2 is the least common
multiple of 18 and 24; therefore 72 is the least ooi^mon multiple of
9, 18, and 24. *^
94. When the least eomnummumpleqfiewralm^mhersUriquhwL
the most emumientpraetieal method U that given ly Ue/ollowing RuU,
Rule. Arrange the numbers in a line from left to right, with a
comma placed between every two. Divide those nuwoers which bav^
a common measure by that common measure and place the quotients so
obtained and the nndivided numbers in a line beneath, separated aa
before. Proceed in the same way with the second line, and so on with
those which follow, until a row of numbers is obtained in which there
are no two numbers which have any common measure greater than
nnity. Then the continued product of aU the divisors and the numbera
in the last line will be the least common multiple required.
Note. It will in general be found advantageous to begin with th«
lowest prime number 2 as a divisor, and to repeat this as often as can
be done ; and then to proceed with the prime numbers 8, 6. duo. in tho
same way.
Ex. Find the least common mnltiple of 18, 28, 80, and 42,
Proceeding by the Rule given above,
2
2
a
7
E.
18, 28, 80, 42
9,
14, 16,
21
9,
7,15,
21
8,
% 6,
7
8, 1, 6, 1
therefore the least common multiple required
=2x2x8x7x8x6=1260.
Measonfor the above process^
Since 38--:2x8x8; 28=2x2x7; 30=2x3::6; 42=2x3x7; it is clear
that the least common m-c >. of 18 and 28 must contain as a factor
2x2x3x3x7; and this act ,- itself is evidently a common multiple
of 2xSx3, or IS, andof 9<2x7, or 28; now the least number which
contains 2x2x3x3x7 as a factor, is the prodact of these numbers;
therefore 2x2x3x8 -7 is the least coramos multiple of 18 and 28:
also it is clear that the least -3Q!«mn« r»»i4^£>.i^ _* ^o ««__■. «^
88
ABxxiijaBnc.
of 2>«9K8«8>ct and 80, or of 3.2x8.8x7 and 2x8x« «i„^
I Therefore thU pr<^„" oT^«^ f fr^?** "' '^ '"^"^
nmnber in the smne line- fottl^XT ^^ '" ""J" "*«'
leart common multiple ofV 4 8 18^ IH ^'^""'^ «» «»« ti>.
ridenstioD, and 240 tha W' I^™" m 48, ni.y be left oat of con-
Ex. XVUL
Find the least common multiple of
(1) 16 and 24. (2) 86 and 75.
(4) 28 and 85. (g) 819 and 407.
(7) 2961 and 799. (g) 7568 and 9604.
(10) 6327 and 23997. (H) 6m and Jom
(12) 16868 and 21489.
(8) 7 and 15.
(6) 833 and 604.
(9) 4662 and 6476.
(18) 12, 8 and 9.
(16) 6, 10, and 15.
(17) 27, 24, and 15.
(19) 19, 29, and 88.
(21) 68, 12, 84, and 14.
(28) 6, 16, 24, and 26.
(26) 15, 85, 68, and 72.
m 64, 81, 68, and 14.
(29) 1, 2, 8, 4, 6, e, r, 8, and 9.
(80) 7, 8, 9, 18, 24, 72, and 144.
(31) 12, 20, 24, 64, 81, 68, and 14.
(82) 225, 266, 289, 1028, and 4006
(14) 8, 12, and 16.
(16) 8, 12, and 20.
(18) 12, 51, and 68.
(20) 24, 4:, 64, and 193.
(22) 5, 7, 9, 11, and 15.
(24) 12, 18, 80, 48, and 60.1
(26) 9, 12, 14, and 210. f
(28) 24, io^ 82, 45, and 26.
most oofD-
evidently a
Jx6 or 80;
Bt Gommoa
mltiple of
let oontaili
mtlyiteelf
iber which
Lombera.
1 mnltipla
i^jeotin^
^7 other
find the
3,4* 8,16,
at of con-
1 48, wm
5.
1604.
id 6476.
'}
nukmiojn, 39
sEcTioisrir.
^^iui'Z^C^^ brtl>.:K=,^ Which W.wffl,„.
oneofsuoh parts .ID ia« foot or one-thiri j n L -
pwt, of tbo yard, «nd it !• denoted thn i ' ' f f " *
frerf <«*6M«Q 3 t»o of a,e« ^^ or two foel^ thu. } (r«id (», tt*^) .
thm of then. ^ or th,« f-t, or the whote jsri &,^ | or L '
If Mother eqiul portion JBFot a Moond yard WJvided in ih.
^^Z^' "^ "- •^'^ «'»^. « fo/i^SllSd-
Snch eipreeeiona, reprewnting any ntmiber of th« eqn,l „,«. of »
K^JofCrr ''""'' " •'^"°'^* "' ^' - »«e^ BBol'
»6. A Fbaotion denotes one or more of ti>6 equal parte of a nnit •
U „ expressed by two numbers placed one above tt,e ofw wL a^^e
^dTw^*"/ ?"'"'*' nmnberisoalled the D™o«mA^^i!\
ri,S.T t° "^ '"'"''■ *''''' P"*' t*" ""' » divided i the npwr
« oalled the KrMB^ioB (Nam'.), and shews how n,any of snch S
•re taken to form the fraction. ^^
iheLtiS"" '^"^'"~'"^* •'"''^•^* "' ^ «"»-*« bX
t Thns j=2+8 ; for we obtain the same resnlti whether we divide
one nnit, ^IB or 1 yard, Into 3 eqnal parts AJ>, JDF. SB each = 1 »
Z * '--r!,"^ 2 oesnch part, ^(represent^ by |r= ntWt
to^M^^t ...or . nnm or = . . S. ^ence |^f d .^t haX'
fl.JL^Zt""*'™'^'* ^*"°"^ '» ♦''o ""»»»«• above expUiaed,
they are called Vuisab FBAomo»e. F"^™,
Fractious, whose denommatow are ooiunft^d o' rn ™ ,» ..._,.
3
90
AKCFBMS^O,
m. (
..| ,
bj itself; anj number of timeB, are often denoted in a different manner ;
and when so denoted, they are called Dsoimal FnAorxoars.
VULGAR FRACTIONS.
90. In treating of the snbjeot of Vulgar FraotionB, it ia nanal to
make the following distinctions :
(1) A PfiOPEffi Fbaottok is one whose numerator is less than the
denominator ; thus, |, $, ?, are proper fractions.
' (2) An Impbopbb Fit action is one whose numerator is equal to or
greater than the denominator ; thus, |, |, ][, are improper fractions.
(8) A SnfPLB FBAcriON Is one whose numerator and denominator
are simple integer numbers ; thus, j, | are ample fractions.
(4) A Mixed Nitmbeb is composed of a whole number and a
fraction; thus, 5^, 7J are mixed numbers, representing respectively
6 units, together with |tU of a unit ; and 7 units, together with fths
of a unit.
(5) A Compound Fkaotion is a fraction of a fraction ; thus, i off,
I of I of ^% are compound fractions.
(6) A Complex Fbaotion ia one which has either a fraction or a
'^ fl 21. 8 2 *
mixed number in one or both terms of the fraction ; thus, ~i — -» -rrj =7*
' f 8 4| of
100. It is clear from what has been said, that every integer may be
considered as a fraction whose denominator is 1 ; thus, 6s|, for the
unit is divided into 1 part, comprising the whole unit, and 6 of such
parts, that is 5 units, are taken.
101. To multiply a fraction hy a whole nuTfi^r. 1
Rule. Multiply the numerator of the fraction by the whole nmnber.
Thus, |x8=?.
are complex fractions.
Season for the above process.
In ^ the unit is divided into 7 equal parts, and 2 of those parts arc
taken : whereas in | the unit is divided into 7 equal parts, and C of those
parts are taken ; i. e. 8 times as many ^arts are taken in f as are taken
in f , the value of each part being the same in each case.
it maimer;
Ignsiial to
I than the
Bqnaltaor
ictions.
inominatdr
^er and a
3spectlv6ly
"witk fths
lotion or a
1\ i. ^T
8' ^' 61^
fer may be
=f, for the
6 of such
le nnmber.
parts ara
1 e of those
I are taken
VULGAB VBACnONU.
Et. XIX
(1) Multiply 1^ separately by 8, 9, 12, 8«.
(2) Multiply Jj separately by 7, 15, 21, 45.
102. To divide a fraction hy a wTioU number.
Rule. Multiply the denominator of the fraction by the whole
number.
3
»« 2 2
Thus, ~^Z=z-f-^ .
^7 7x8 21
IteoMonfoT the above proeea.
In the fraction ?, the unit is divided into 7 equal parts, and 2 of
those parts are taken ; in the fraction /^, the unit is divided into
21 equal parts, and 2 of such parts are taken : but since each part in
the latter case is equal to one-third of each part in the former case,
and the same number of parts are taken in each case, it is clear that A
represents one-third part of ^, or f-i-8.
Ex. XX.
(1) Divide | separately by 2, 8, 4, 5, 10.
(2) Divide jVt separately by 11, 20, 25, 45.
108. If the numerator and denominator of a fraction be both mul-
tiplied or both divided by the same number, the value of the fraction
toill not be altered.
Thus, if the numerator and denominator of the fraction ^ be mul-
tiplied by 8, the fraction resultin|t will be 5^, which is of the same
value as f . ,
JReasonjor the above process.
In the fraction f the unit is divided into 7 equal parts, and 2 of
those parts are taken ; in the fraction i, the unit is divided into 21
equal parts, and 6 of such parts are taken. Now there are 8 times as
many parts taken in the second fraction as there are in the first
fraction ; but 3 parts in the second fraction are only equal to 1 part in
the first fraction ; therefore the 6 parts taken in the second fraction
equal the 2 parts taken in the first fraction ; therefore %=:{j.
104. Hence it follows that a whole number may be converted into
a vulgar fraction with any denominator, by multiplying the nnmW
92
ABITHMETIO.
^by tiie required denominator for the numerator of tlw^ fraction, and
plaoiog the reqairqd denominator nnderneath ;
and to convert it into a fraction with a denominator 6 or 14, we hare
g_6_6x5 80
1 lx5~6»
g_^_6xU_84
l""4xl4~U*
Ex. XXI.
Eeduce (1) 7, 0, and 11, to fractions with denominators 8, 7, and 23
respectively; and (2) 20, lOO, 117, and 125, to fractions with denom-
inators 2, 6, 18, 28, ahd 35 respectively.
106. Multiplying the numerator of a fraction hy any number, it tho
tame in effect a$ dividing the denominator ly it, and conversely.
For if the numerator of the fraction f be multiplied by 4, the re-
Bultmg fraction is V ; and if the denominator be divided by 4 the
resulting fraction is |. '
Now the fraction V- signifies that unity is divided into 8 equal parts,
and that 24 such parts are taken; these are equivalent ';^'&^ unitk:
also f signifies that unity is divided into 2 equal parts, and that 6 snfch
parts are taken; these are equivalent to 8 units: hence ^/ and fire
equal. The proof of their equality may also be pat in this form : that
aince the unit, in the case of the second fraction, is only divided intb 2
equal parts, each part in that case is 4 times as great as each part in
the case of the first fraction, where the unit is divided into 8 equal
parts; and therefore 4 parts in the case of the first fraction are equal
to 1 part in the case of the second ; or the 24 parts denoted by the first
ar^ equal to the 6 denoted by the second ; or, in other words, the
fractions ^ and | are equal.
Again, if we divide the numerator of the fraction | by 2, the re-
sulting fraction is f ; and if we multiply the denommator by 2, the
resulting fraction is ^.
Now, I signifies that the unit is divided into 8 equal parts, and that 8
of such parts are taken ; and ^ signifies that the unit is divided into 16
equal parts, and that 6 of such parts are taken : but each part in | is equal
to_2 parta in^ ; and therefore J is of the same valno as ?^> or ^
' ■' 1« 16 %
VULGaB FBACmONS.
9d
ttion, and
r, and 23
I denom-
•«r, it tho
I, the re-
7 4, the
lal parts,
fe UDltk :
It Gsobh
nd f Are
m: that
)d into 2
t part in
8 equal
^e eqnal
the first
rds, the
the re-
7 2, the
d that 8
into 16
id equal
6
^16- .
106. To TgpreHnt an improper J¥aetion aa a whole jt mixed nwnler,
EuLB. Di to tlio DDmerator by the denominator : if there be no
remainder fhe quotient wiU be a whole number; if there be a re-
mainder, p. down the quotient as t e integral part, and the remainder
aa the nume Mot of the fractional part and the giv* a denominator as
the den ominator of the fractions' art.
Ex Reduce V and V to whole or mixed numbera.
Bjr the Eule given ahove,
V^^i A "v^iiole number;
Reason for the above proeeu,
fl. 25 6x6 5
Since -^=—=-_x6, (Art. 1
and since | signifies that the unit is divided into 1 ^qnal parts, and
that 6 of those parts are taken, which 6 parts are equal to the whole
unit or 1 ; therefore V=i x 6=1 x 6, or 6.
6x6+5
' 6
Again, ?=?«+«
6- 6
A V K
Which equals -y- together with |, that is, =6 together with f by what
has been said above ; or, as it is written, 6|.
Ex. XXTL
Express the foUowing improper fractions as mixed or whole numbers:
(1) ¥.
(6) ^'-.
(») w.
(18) l^f^.
(2) -V-.
(6) W.
(10) H^K
(14) ^VtV-^.
(18) ia^ffL.
(8) -Y.
(T) W.
(11) ni
(16) -4fS^.
(i») -Wi^v
(4) H^.
(8) ^^/.
(12) W.
(16) A^fl.
(20) -V^^.
107. To reduce a mixed number to an improper /ration.
Rule. Multiply the integer by the denominator of the fraction,
and to the product add the numerator of the fractional part; the re-
sult will be the required numerator, and the denominator of the
fraotional part the required denomhiator.
IMAGE EVALUATION
TEST TARGET (MT-3)
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Photographic
Sciences
Corporation
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:'^^iJ^
'^'^^'"^v ^^
33 WEST MAIN STREET
WEBSTER, NY. M580
(716) 872-4503
94
AEITHMEllO,
Ex. Convert 2^ into an improper fraction.
Proceeding by the Eule given above,
2x7+4 18
2*— -
HeoMnfor the cibovie proeess.
^ n is meant to represent the integer 2 with the fraoUon 4 added
Bnt 3 i, the Mn,e as ^ or ^; ,md therefore 2} »„,t b„ fto
t^V^' '^'"'■^"^'^ "P'«^-=» "-oh parts together with
Ex. XXItt
Eednoe the foUowing mixed number, to improper fractions •
(1) 2^.
(6) 26|i.
(9) 2008|.
(18) 3/4.
t^(2) 6?.
(6) 43,\.
(10) m\i
(14) 26|§^.
(18) 17fS^|.
(8) 4|.
('J') 25tV.
(11) 57|^
(15) 16411 f.
(19) 427V|^.
(4) 71.
(3) 14||.
(22) 13j\.
(16) 106]}?-.
(20) lOOji^'.
(17) 167f||.
RuLB. Multiply the several numerators toffether fnr f),- «.,«,« *
itiS: """""- ""^ "■*•--"' ^-s:.^ c^T— :
- Ex. Convert f of J into a simple fraction. '
Proceeding by the Rule given above,
-iof~=?ii?-?!
6 8 6x8~4(|*
Reason /or the above process.
of part L^^^^^ .r" 'r'^' ^"'° ' ^^"^^ P*^*«'^"<i « -^ eaclTet
of parte be tdcen, then each of the parte win be one-fortieth p J^of
VULGAR FRACXIONS.
95
a^ A<l<l^d
St be the
nnity is
her Tfith
is: ■
4.13
00*1"'
simple
merafcor
for its
IbyJ:
m, and
wb-set
wt of
tlie original unit, and the nnmber of parts taken wiU be 8x7. or 21.
*v -._-». - - 21 8 x7 '
the result therefore is — or
40 " 6 X 8 •
8
that is,
— of-l-^JL?
6 8 ""fix 8*
^fr'^^'^'A'' '■^^"'''"^ compound fractions to simple ones, we may
stoke out factors common to one of the numerators and one of^e
denommators: for this is in fact simply dividing the nuJerltld
denc^mator of the fraction by the same number. Art. (108)
ThuBio{2j\ of 1,->,=| of f^ of H. ^'
_ 8x2gxl 6_gx jx^x^x4 _4
5x12x15 ^x^x^xaxa^T"
(Striking out the factcs 8, 6, 6, 4 from the numerator and denominator).
Ex. XXIV.
Reduce the foUowing compound fractions to simple ones .
(4)|ofH. (5)2of|of7. (6)|of?oflofAof28
(7) TVof2iof|of 10-5. (3) s\fl\iof|o7,1f TofO.
(9)fVofJof?|of|ofV»,of2ofA. ' V* *
(10) f of gof^ofrojof y3^ofij7, of ur.
d«ni!?-*n f ^"^"^^^ ^« ^'^ ^*^ ^o^^ST T«BM8, whcn its numci^tor and
denommator are phime to each other. '"^"M'or ana
Note. When the numerator and denominator of a fraction are not
uZ k" r-r ''? 'r ^^'^- ^'^ '^ — - factor^aTtht
nm y If we dmde each of them by this, there resu,ite a friction eW
to the former, but of which the terms, that i,, the numerl/Td
mavT^nlir r; T "T" ''^" *'^" ^^ ^^^ ->«-^ fraction and ^
may be considered to be the same fraction in louder terms. When thl
that IS, have no con.mon factor greater than unity, it is clear tiiat it^
terms cannot be made lower by division of this kind, aLd oTt^k
account the fraction is said to be in its lowest te^s ^ ^^
110. To reduce a fraction to itt lowest terms.
Rum. Divide the numerator and denominator by their greatest
common measure. ^ •'/ wwr greacesi
^6
JLBIXHMlCna
Er. 1. Beduce f ||| to its loveafc teems.
The G. 0. M. of 6466 and 7385 is 16 : 481, 489 are the qnotienta ^
the numerator and denominator, respectively divided by theaO.M. i6 •
therefore the fraction in its lowest termsar^^ '
JRwuon for the above proeem,
K the numerator and denominator of a fraction be diyided by the
same number, the value of the fraction is not altered (Art. W) ; and
the greatest number which wUl divide the numerator and denominator
is their greatest common measure.
Note. Sometimes it is unnecessary to find the greatest common
measure, as it is easier to bring the fi-action to its lowest terms by
wicceteive dividons of the numerator and denominator by commoa
actors, which are easily determined by inspection,
Ex. 2. Reduce |J§ to its lowest terms,
4II=II» divid&g numerator and denominator by 10;
=»H» dividing numerator and denominator by 8.
Beduce each
a; 1.
(S) If.
(9) IH.
(21) /^.
m iim.
Ex.
of the folldwioif
(2) f?.
<«) If.
<10) HJ.
(W) m.
(22) tHt.
(26) mth
XXV.
fractions to its
(8) If.
ai) Hf.
(IS) m-
(28) |f?|.
(27) ff^f.
(81) hmn
loWMt terms t
(4) H.
02) ^1^
(i«) WiA.
(20) mi'
(24)
(28)
. (32)
afianr*
111. To reiuee fraeUom to equivalent onet with a common denom-
inator.
Run. Rnd the least common multiple of the denominators : this
wiU be the common denominator. Then divide the common multiple
BO found by the denominator <tf each fi-aetion, and multiply each
quotient so found into the numerator of the fraction which belongs to
it for the new numerator of that fraction. •
mi»^^^
notleqti ^
led bjthe
^ M) ; and
mominator
t common
t temis by
r oommoa
It
nil.
*xtk\ tills
multiple
ply each
eloogs to
88xl6»
or
ilit ^^^^ "''22 17x16
12x44' iSm* 24^'
«« HI, m,
^^M»fm for ihle al&te jpfoeen,
inaction, » *x::?'^^r^«r^*r^fhl' " "" ""• ''^«"
nominator of that fr«^f inn \ u^« . ^^^ *^® nnmerator and de-
the fraction l^nott altt:^^^^^^^ ^.r^ **^^ ^°^ ^'
then be eoaal to th« w ^ ^' "*** *^® denominator wiU
» this betn wi^ oU ITT °^°?^P^« ^^ •« the denominatoi^
manoer, red^^fo^l:? ^/r L^^^^^^ ^^^^
^omm^multipleof an thedenomi.^^^
then^::i«plA:!:it^^^ wmn.,
own, for a new numerator fo" frf ti™ ^^rf °"' *^^P* *«»
together for the common d^omtlt ' ""^ "" ""^ denominato«
^Be<^ucei,U,toe,aivale-tfh.o^^
The least common multiple of the denominators
therefor© the fractions become
=s5x7x9;
illiS 2x5x9 1x5x7
6x7x9' 7x6x9' 9^:6l7»
«^ ift^ M.
or
D8
AsirHMEna
Ex. XXVL
Reduce the fractions in each of the foHowing sets to equivalent
fractions, having the least common denominator :
(1) ilrand}.
(8) I, I and i.
(6) I y^, and 11
(7) I j^, and U.
(9) I, Vu, and j|. .
(11) f, I, f and |.
(18) i, II, and jVV
(IS) I, t\» is, I'l, and V(r.
(1^) I, f A» iV, iWt and ,Vv.
(19) lit *J, iJ, iJy, and f.
(2) f,and^
(4) f,and^,.
(6) i, I, and f
(8) -A, A, and II.
(10) I, if and ti-^
(12) I, T*r, t\, and f.
(14) T^, I ih and if.
(16) i,l,l,T\,5\,andH..
(1®) W» TTTr» TuVffi and xrJ^
(20) II, H, *f , and rV
Note. Whenever a comparison hag to be made between fraotionfl,
in respect of theip magnitudes, they mnst be reduced to equivalent
ones with a common denominator ; because then we shall have the
nnit divided, in the case of each fraction so obtained, into the same
number of eqnal parts; and the respective numerators will show us
Jiow many of such parts are taken in each case; or which is tho
greatest fraction, which the next, and so on.
Ex, Compare the values of ,\, |J, |, ^, and |. -
The least common multiple of the denominator8=1080 ;
therefore the fractions become
Cx4Q 11x45 CxlSO 4x72 8x216
27x40' 24x46' 6x180' 16T72' x"216» ^^
90«
^8«
«4 8
therefore | is the greatest, f the next, |} the next, r*y the next, and JL
the least. ' ■
Ex. XXVII.
Compare the values of
(1) i,^,and^ (2) i, ?, I, and |.
(8) i of I ,V, and i off (4) ^, ^ ^4, and 2|.
(«) I A, 1%^ A, and §i. (6) I fi, ^, V'^, and ||.
(7) f of I of 4, T?r of I of 6, J of I of 4f, and l\.
equiralent
9
3
ri-
I fractions,
equivalent
I have the
the same
il show us
ilcli is the
T
ADDinOIf OF VULGAR FEACnOWS.
(B) V-, 8^, ondfofai.
99
(10) I, A, i\, ,%, and
s
(») ?, i2, IJ, «. and If.
,,„, , (") 'i» Hi, I5S, fH, and ?f§. ,
(12) V, 81, ? of 9|, and ^ of | of |.
Find the greatest and least of the fractions
- (i3) I ,\, g, i, and i. (U) ij, |g, J 7, ^^^ ^^^ ^ j^
ADDITION OF VULGAR FRACTIONS.
.112. Rule Reduce the fractions to equivalent ones with the
least common denominato'' • n^A nil fT^« ««^ ^ 7
««^^, 4.^. ' """\'"«atO' , aaa all the new numerators together, and
under their sum write the common denominator.
Ex. Find the sum of 1, 1?, and ||.
Proceeding by the Rule given above,
The least common multiple of the denominators=105 •
therefore the sum of the frnctions is '
49 . 60 . 48^49+50+48 _ 147
105 ~inK = lf'
105 105^105
106
JSeaaonfor the Rule.
In each of the equivalent fractions, we have unitv divid.«? infn iak
equn parts, and those fractions repUsent respe tf^^^^^^^^^ t 3
48 of such parts; therefore the sum of the fra'cti^Lt Vep^^^^^^
49+50+48 or 147 parts, that is, must be H
105*
it« wl/; ^^ *^' T"^ ""^ *^' ^'""°*^"°^ ^« ^ fraction which-is not in
Its lowest terms, reduce it to its lowest terms; and if Te result b«
b;:?ri:L'"f .1 ^^^^ -^"- ^^ ^^ ^ whole o t^ ;:i
Fractions'"" ^^- ^'^ '""^ ''"^''^ applies to all results in Vulgai
100
ABlTHMJffriO.
and
Ex, Find the sum of -?-, 8f }, 10|, and ~.
8 16 6 **'22 1820 1820 1820"*" 1820
w
therefore the whole snm
_ 495 + 1232+ 528 +540 _^ .
1320 —^Tft ?
=18+2jVf=15^.
Add together,
0)fand4.
(4) If and sV. ,
(7) i^andyV
(10) ^ and j\,
(18) f, I, and tV
(16) V^, i, and /j.
(19) ^, I, and 3JV.
(22) I 2\, and ISf^.
(24) iof|off,6|,and^
(26) If, ,S, and ^.
Ex. XXVIIT.
(2) fand^
(5) j\ and ^j.
(8) i^andf^.
(11) 3|and7|.
(14) I I and J.
(17) f, i, and ,-\.
(20) i, h h and i.
(8)
(6)
(9)
I and|.
f\ and if.
i and 2^.
(12) 4|and9f^
(15) i §, and ^
(18) f, 1J, and f
(21) 4, tI, and ^.
(23) i$of^,and9^^
(25) I,f,i,andiJ.
(27) I, tV ^, and i^.
(29) i, 6|, and I of J.
(81) 261J, I74f, andfoflOl,
(28) i, fa, ^, and ^,
(30) lOOf, 64|, J of 701.
(32) 387i, 285i, 894J, and f'of 8704.
Find the value of
(88) H + lV6+THTr+TT,*ff*T.TJ. (34) H + U+I^H+i|.
(36) ^ + ^+T\4.f|-ffJ.
(86) 2H+6ii+Jf+ion?+ff+|of2^
(87) 2|+3i+4|+6^+6f (88) H+SJ^+ff +7^+/,+! of J.
(39) 6i4|off of3i+9T\+^off of4.
(40) I of 12+f of ^3^ of 1| of ,-V5 + ff Of 8f of ^ of 1/y.
(41) 270|+650tf'«5+6000i+68J+l7V.
(42) iof 1+^ of (1+H)+Sf +IJ of |l+f f.
t
SUBTEACnON OF VUIOAB FBAOTIOira. 101
Vr;
aid if.
id2i.
nd 9f ».
and^
andi.
, and ^.
>flO|.
I
SUBTRACTION.
118. Rms. Reduce the fractions to equivalent ones Tvith tbo
leiwt common denominator, take the difference of the new numerators,
and place the common denominator underneath.
Ex.
1 7
Subtract — from --.
2 8
Proceeding by the Rule given above, since 8 is clearly the least
common multiple of the denominators, the equivalent fractions will bo
% and I,
and their difference^=— ^= ^
/
JUasonfor tJie JRule,
8 8*
The unit in each of the equivalent fractions is divided into 8 equal
parts, and there are 7 and 4 parts respectively taken, and therefore the
difference must be 8 of such parta, or, in other words, the difference
of the two fractions is |.
KoTB 1. Remember always, before applying the above Rule, to
reduce fractions to their lowest terms, improper fractions to whole or
mixed numbers, and compound fractions to simple ones.
Note 2. If either of the given fractions be a whole or mixed
number, it is most convenient to take separately the difference of the
integral T>arts and that of the fractional parts, and then add the two
results together, as in the following examples.
Ex. 1. From 4| subtract 2 J.
Here 4-2=2, and i-i=|-|=i;
therefore the difference of 4^ and 2i=2|.
Ex. 2. Take 2| from 4^.
Now f cannot be taken from i, since it is the greater of the two ;
Z. Ir rl ^ *' *' "°^ ^^^ ' ^'""^ 1 +i or| ; and then, in order
that the diffference may not be altered, we add 1 to the 2c
Now 1—2— J — 3 _7
4^3=1;
therefore the difference of 4i and 2^=1 J. ■ ^
102
AErniMETIO.
For the process expressed at length is
whicL=4+l+i-(2 + i + 2) adding and sabtraotingj).
Ex. XXIX.
/
(2) 5andi.
(8) Jand^V.
(6) A and ^.
Find the difference between
(l),inndf
(4) jVandrV (g) f ^ anj ^^^ .
.»„^ . , ^"^ i "f S Of * «""] * Of J.
(21) By howm„ch does J of ,V-} of ,«, exceed 7of ^.,-1 of Af
(28) From the su,n of llj and 8J Bubtract DM.
MIJLTIPLIOATION-.
m meaning « ^«^« ,, ,-,rf„^, ^^.^ ^^^ ^ J-ARr/o/Ttim
Si. Multiply ^ by -i.
1
:J),
^tV.
Dd SOtV-
3
h
It.
cceed tho
-°j exceed
used to
20.
■
MULTIPLICATION OP FRAOnoJIB.
103
Proceeding bj tlio r.ulo Given above,
Sx5 15
7 X 8""6ft
Iiecuon/or the Eule,
If ? be multiplied by 6, tho result is VrArt. (97)
than 5 or in o.her Avon^s, is one-eighth part of 6. Consequently the
proa.,ct above, viz, V- must be divided by 8, and V-^8=if!Art. (98)!
.rr^T ^' i^-'" ?'"' ''''''''''''« ^'" '^I'P^^' ^^'«^«^«'' ^« tl^« number
effractions wliicli iiave to be multiplied together.
r..i.^T. ^* ^'^''' ."^^^^'"^ ^^^^ "^°^" ^^"^^' '"^•^^d ^'^"^bers must be
reduced to improper fractions.
fnviT ^;- M-^"' .^°''' '^'°'''" ^^"* ^ ^'•^'*'°" '^ ^^^"^«d *« it" lowest
teims by dividing its numerator and denominator by their greatest
common measure, or, in others, by the product of those factors which
are common to both : hence, in nil cases of multiplication of fraction.,
It will bo well to spilt up the numerators and denominators as much
as possible into t!ie factors which compose them ; and then, after
putting llio several fractions under the form of one fraction, the sign
of X being placed between each of the factors in the numerator and
denominator, to cancel those factors which are common to both before
carrying into effect the final multiplication. Thus, in the foUowing
Examples : ®
Ex. 1,
Multiply — and -^ together.
•jv ,,___3x4 8 .
— 4x^~"6' ^i^'i^ing num'. and den', by 4.
Ex. 2. Multiply—, Vl ?t, and — too-PfhAi.
9 *24' SO ^ 60 ''^Setner,
Prod'.=
8x16x27x45
9 X 24 X 80 X GO
J? ><lx2)x(2x^<^x ?) X (3 X 3 X J?) X (? X ?[ y ^
(>ixiJ;x(^x;ix;ix3M^xii<x^)x(3l^^ir^^
j^=y dividing by 2x2x2x2x2x2x3x8x3x3x3x5.
104
ABirmama
Ex. 8. Muldpljr 21 8J.40J. 20|, and 5,V, together.
2x(2x;«)x>{xV>rx^
__ gx8x9x9 x81 37666
21^2 =-4-=W16i.
Ex. 4. SImpllfr (« of U ofi|+8f of2K-2})x8f
Valae=(|-oflofl*+7 52^M 27
\7 4 15^3 21 9;^T
2x5x2x7. 7x2x26
. f8x
Yj 26 8\ 27
2x2x8x6
26
8
2x8x
8x26
8
26 ^\
7"~8y
27
-8 27 21 27
~^y=T'<7-=27.
II
Ex. XXX.
jytaltiplj
(1) i by f . (:j) I by ^.
(5) * J by f ^. (6) 7^ by f
(9) 12 by f of 5.
(8)fbyf (4)Wby.W.
(7)8Jby2f. (8)7jbyioff
(10) ioff bySfofS.
(n)|fof3|byl,VofKof|.
(12) H of Ij,^ of ^ by r\ of 87^ of 84- of A
(18) I of 2^ of 1^ by 8^ of yv of H.
^ <^^^ ^^ «^ 8i of ,1, of 84 by T^, of /, of 1 J of 19.
JJind tbe continued product of
nn *d' « *;?f ^ ^"^ *^' **• «• *■ ""^ «•
/-ox . . **• ** "^ ''^' **' 'ASr. 6A of 49, and A.
(20> W, 1^, Mf, iS^, and li|4. ^
DIVISION-.
So^ ^ ^^"^"^"^**^^ " * --°»*"tor, and proceed as in uZ
pa
to
a
TT
do(
as:
I
DIVISIOIT OP VRAonojsm.
106
»7ioff
at
foflf.
denom-
1 Haiti-
Ex, Divide ibj-i.
Proceeding by the Rule given above,
XI 5 ""11 8 ""88'
lUaionfor the Rule^
2
If jY be divided by 8, the result is
TuTq ^' 88 ^^^- ^^•
This result is 6 times too small, or, iu other words, is only one-flfth
part of the required quotient, since, instead of dividing by 8, we have
t» divide by I, whioli is only one-fifth part of 8 ; and the quotient of
TT divided by g must therefore be 5 times greater than if the divisor
were 8. Henoo the above result ^ must be multipUed by 6 in order
to give the true quotient.
10
88*
Therefore, the quotient = ^ x 6 = ^ " '^
88
83
Note 1. Before applying this Rule, mixed nmnbers must ht re-
duced to improper fractions, and compound fractions to simple ones,
as in the following Examples ; -^
Ex. 1. Divide ij by 2f .
11
62 _
* 88 " **•
Ex. 2. Divide|.of-|.byl|ofr.
^8x7 16x1
"4 X 8** 15x7
8x7
4x8
16x7
Idxl
. 8x7x16
'4-. 8x16x7
""^x2x^x3>r6xr~10'
NoTB 2. OoHF^itr Fbaomons may by this Rule bo reduced to
umple ones.
106
ABrrHMEXIO.
TJlBS
Again,
Again,
(Art. 96) = -^ X i
4 5
^-i = i-» 80 9 1
80 ao ■Y-""2"'^T=2-''3o
30 80
a
ft
1
2
80 2
1 9 3 ^^
quantity «hich, wl.e „ t-Cjed bvTo d"" ^^t' *■"" """'"^ "
dend ; multi^lU being n„ emood i, T, '" ""^ ""'"^ ""« D'^'"
andi.artor^ar^.ofatim^ io^rmany Un.es or ho«r m,ny ,i^„
SliX^a, ^St?SV Sfn;t ■^-'-^•
(H) a^ofSJof 3J bys^of ,«, .,f V ft/,- ^ ,
13)5onof8CJofO,y?j;jof ofSJ. ^^-^^''">^3-,
05) Compare tho product Apd quotient of2J by ST .
Eedaco to rimple fractions the following complez fraoL •
(10)
n
(19) 4
iff
(17) g.
(20)
13^
(18) ?|i.
- (21) f i
(22) •'"'"^'^
18^
"lU*
KEDUOTION OF FRACTIOm
IHi
*« -wwiiipyuaa i/iyision. ' — — — --«4iuawi-, as
WDtroTtoir'' ap 'M^kcrnxxsm.
107
Ex. 1. Hud the value of—, of £1.
8
Proceeding by the Role given abov«,
-of ;fil-'^^2^* '^'^^
85
andiof U=l^^.==6<i:./
therefore the value required=17». «<?.
ReoMtrnfoT the ahoee proeea.
of £1 is the same as 7 times -— of £1,,
8
and i of £l=20f:
8 8
8
5«.
= 2-;
6s. 85«.
therefore 7 times ~ of £1 =7 tunes ~ =^^=17|t.
=17*. 6<f.
Ex.2. Find the value ofA of 8 tons.
-J of 3 tons=-^ of a ton=l| tons,
'^ ^p ^4. 7x20 7x5
-^ of a ton =-^ cwt. =I|£ owt.=17i cwt.,
8
r.-,* 1X4'
=-2— qfs.
8
1
therefore required vajue =1 ton, 17 cwt. 2 qrs.
Ex. 8. Find the value of f of a bushel- 1 of a peck.
-TT of a bush.=-^ pics. =^ pks.=:2| pks.,
9
2 , 2x8 16
- pk.=-^ qts.=- qfs; =17 qtg. ;
9
therefore -^ of a bush. =2 pks., 1 J qts.,
5x8
y of a pk.=-^ qts. =*? qts==^5l nfg. j
therefore required value =2^pk9., 1^ qtg.-.6| qti.
=lpt.74g*jqts.
108
A^OTBumic
Ex. XXXII.
Find tiie respective yalucs of
(8) 7f ofalb.Avoird.; Uofalb Trov 9^ nf a „.i a bI'' ^'
(Q\ flJL r^f^ ».kj ir V *"* "^"' ^roy , J, or agal. ; 4A of an acre.
(»> 8^ of* hhd. of beer; 2^ of a tun of wine; 6^ (rf^ bue.
ao)|of|oflO|br»,£l^l.n,,|l,,^
. ^^^^ 'S^ll*'^^^^-®*-^^-'- ^o^l!ofl2^of^of$2x^.
^- (12) *of4lx5f; |offof$l^j.
,^!!^ ^ ""^^^ of $2.52+1,.+^ Of 60 cts.
(17) ^3|+7|*.+4H
(18) -:]p:of6tons+^4cwts.+|ofaqr.
(19) iofaton+|ofaowt.+|Ib.
il?^ W^'^y+i 1^- Troy-~5 oz. Troy.
(21) TVofamiIe~|ofafur.+^po.
(22) TiTCub.yds.+2|cub.ffc.
r24^ a nfrl^^^ 1*"^* ^'••^^ of a ba«.-^ of a qr.
(24) i<>f^;"r.,29po,8iyds. + |of6mi.,8fur,37po,4|yds.
^ 726) ^ of«l ^ ^,^^^f^i<J- + 3,»,of|wU+^;f5/hr/'^
(26)V.of91ac.,8ro.,86po.,2Jyds.-.|of6ac.,2ro.:irpo..26|yds.
f Jl'l!lf ril'.l«^:?^^^^^^^^^ ^^r^'-^c^on, and aisothennmberor ^
____ ^ _ „,,,,„„ „, ^^.^^^ ^. ^ ^^ ^^ ^^^^^^^^ ^ ^^^ respective
eqdvalent valaes ia terms of some one und the siiiw denottHnitlon •
then the fraction of which the former b made the mimerator, and the
hitter the denominator, will be the friiction required. J -« ««»
Ex. 1. Reduce | of £1 to the fraction of 27t,
Proceeding by the above Bnle,
I Of £1 :=b20 times | of U,
6x20
6x6
■^27~6i'
6x5 27
25
8
2T#.=2r*.
6x5
therefore fraction required^— -
27
_6x5
~" 2
inJlT ^^"^^^^ f*^ 27 equal parts; and f of £1 is divided
hito ^ of such par^; therefore the part of unity, or 27., whieh Z
Utter represents, is ^=?~.
' 27 64
Ex. 2. What part of J of a ton is 2f of 1^ of 4 of a cwt t
2| of H of 4 of a cwt.=f of I of 4 of a cwt.
=|x^x4cwt
l^of aton=iyicwt.
Therefore fhujtion pequired=iiitiii
Ex. xxxm.
Seduce
(1) $2.26 to the fraction of $4 ; and $8.15 to the fraction of $6
/!^ tl I *"'*''*'''° ''^ ^*- ' ^°^ ^'- ^- ^ *^« fraction of £l'.
of 7! 9! ^'^' ''' *^^ ^'*'"'''' "^^ "^^ ' *^^ ^'- ^^- ^<^ *^« fr*c«on
tion tf 4 or*' ^^^ "'^' ^ ^^* ^*^*'''" ^"^ * **''' ' *°^ ^^* ^^*- ^ *^* ^''^
lio'
orr-AinTmanKj.
I
(5) 8 qrs^ 4 lbs. to the fraction of 2 cwt. ; and 5 oz., 2| dra. to the
fit uon of a grain.
(6) 8 ro., 27i po. to the fraction of an aore ; and 26f so. yds. to the
fraction of 2 acres. i j"o. fu mo
. J^\\^^ ^^^'' ^ ^*-' ^ '"• **" *^® ^""^^"^^ o^ a mile; and 6 cub. ft
100 cub. in. to the fraction of a cubic yard. '
(8) 2 qrs. 2| na. to the fraction of an Eng. ell ; and 8 h., 8 m. to the
iraction of a day. » -r-
(9) 2ac., Iro. to the fraction of 9ao., 2ro.; and 154Cyds., 2 ft.,
9 m. to the fraction of 2 miles. ' *'
(10) 1 ft., ^ in. to the fraction of a sq. yd. ; and 2 qts., U pt. to the
fraction of a barrel. . ^ * y^/i/. vwmw
on^"\lu^^;' ^^'*^'' ''^•' ^''°'- *^ *^^ ^'««*^»«« ^f a day; and Iro.
20 po. to the fraction of an acre.
(12) 4 bush., 2f qts. to the fraction of a load ; and 8 quires, 7 sheets
to the fraction of a ream. i , . «^w.
JctLoni'l'''* '' *^' ^''*'*^'"'^ ''^'' '^''^"•' ^dHyds.tothe
fractionlflf *^^ *"" *^' ^''^''*''''' ""^ ^^'^^^ "°^ f of a ferthing to the
fracttLlf if9 '^' ^^** ^' *'' *^^ ^''^''"'''' ""^ "^^^ ' ^^ * ""^ ^^^ *^ *^^
(16) I of a dwt. to the fraction of 1 lb. ; and g of 2 lbs. to the frac
tion of 2^ tons.
(17) I of a lb. to the fraction of a cwt. ; and | of a yd. to the frac-
tion of a mile.
Ji tVrfri2:vA^t:d*^ ^' ' ^^- ^^^^^^-^^^^^ -^ ^^^ ^^ *
fractltfl/iSSir '"' '"'*"' '' ' '^^^^ ^ ^"^ '^ ^"^^«°^ ^<>'*^^
(20) § of 7i of 16^ yards to the fraction of a furlong; and a, of
alb. Troy to the fraction of a pennyweight.
(21) ^ of a lb. Avoird. to the fractLn of 2 lbs. Troy; and i of a
Trench ell to the frnction of a yd. ***'*.
(22) ^ of a sq. in. to the fraction of a sq. yd. ; and 4 of a yd to the
fraction of an English ell. * ^ ^^ ^"®
(23) What Dart of £ft ia 1 nf _9 «f o« cj o
(24) What part of a second is ^ Wt^j i. of a day ?
\ drs. to the
. yds. to the
I 6 cub. ft.,
8 ip. to the
? yds., 2 ft.,
^ pt. to the
and Iro.
es, 7 sheets
yds. to the
liDg to the
^6 to the
the frac-
' the frac-
Dgs to'the
^nd ^5 of
ad I of a
rd. to the
MJSOLLi ^OUS EXAMmiW Bf TOL^AR FBACEKXira. 1 1 J
(25) What part off, of a leagu6 is 5 of a mile?
(26) What part of 8 weeks, 4 days, is ^ of 6^ see. ?
(27) What part of \ of an acre is 25^ po. ?
(28) What part of i^ of a min. is ^-^ of a month of 28 days ?
IX. —^ (29) Whatpartof Jcf4tun9ofwineisJ»Jhhd8 ?
(30) What part of 8 fathoms is V^ of ^ of a pole?
/ •- ' (31) What fraction of ^ cwt. together with 8 qrs., 14 lbs. wUl give
a ton and a half? (cwt.=112 lbs.) ^ » . e «
118. MvMllaneou»Emmpl^inFTacti(miiw<yrTceAouU\
Ex. 1. What number added to ^+/y will give 2J ?
This quesdon in other words is the following: "What number
will remain after ^ + j-"^ has been suhtracted from 2 J ? »
Now '^\-(X^h)-^\-l-h-^ir'-l-h
Therefore the number required = |.
Note It will be remembered, that all quantities within a vinculum
are equally affected by any sign placed before tlie vinculum.
Thus m the above expression, -(|^,y means that the sum of 1
th«t^. r. K 'f'^'''^^ fr^"^ 2i; whereas -2.,*, would mean
the rLulf ^'^^^'^^^tedfrom 2^, and then ^^ hud to be added to
Ex. 2. What number subtracted from \\\ wiU !eave n for a re-
mamder?
Number required
i
= 14i~lJ = (U + l + |)_(i + i+^) ,.;
= (14 + V)-(2 + |) = 14-2 + (Y-f)=:12|.
Orthus, 14i-lf=18g-f = ioJ-o = i«i = 12|.
Ex. 8. What number multiplied by 1 \ wiU produce 14f ?
Dy Ig, what will Che quotient be ? "
XrUb
i4? 4". .
Therefore the number required « lOjJ^.
\
in
I Ex. 4. WhAt nnmber dirided by 1| will produce lOf'V t
^ Thi* question ir other words is the following : "What is the dkk
duct of 1| and U ? *^
I The prodool of If and lO^J^
i Ez. 6. Beduce the expression
to its simplest form.
fa. 3 -iofiUi3-/^^,f 5.4V ,
Ex. 6. Bunplify the expression yl^l^L^.
6 + 4 + 8
2^8^ 4^
±1±\
13
Si 8i^4| I^T"? —816^
Ilx. 7. Sbnplif J the expression
N'ow,
therefore
8 + i.
1
1 ^ 89 89
1 + i "Iti ! + /« 89 + 4''48»
8 + i V-
13°'l + i
18 ^Us 48"
8 + i
7 13 1
is the prcK
f = |.
The expression
^ ( 11 . 176 2
IIS
I
228
88 8 ) 806
(tie least common mutiple of 4, 88, and 8, « 88 x 2 x 8)
^ j 11 X 19 X 8 + 1 75 X 2 X 8 - 2 X 88 K a
-1
88 X 2 X 8
) 228
) 806
627 + 1060 - 162 ) 228
228 f "806
.1677-152 228 1625
288
806 806
«6.
Ex. XXXIV.
Miscellaneous QueeHom and ExampUe on ArU, (94-118).
I
^ 1. Define a fraction ; what is the di8tinct«'>n hetween a VtfiMr
ana Decimal fraction ? How many diflferent kinds of Vulgar fractiono
are there? Give an example of each kind.
2. Find the sum and difference of ^ of 71, and If divided by 2^ ;
and the sum of 6^, | of 8^, and i-^f
8. . Simplify
(1) {i+Iof5i[x{|+|+8f}. (2)8riTrof8^-f-iVVof9.
^^4f 4i"2^- <*) 4ix4i-i • (^>«*rrs-
4. Show that the fraction ~~\m between the greatest and
least of the fractions, f, I, and f.
oAiT' --"=—-' vx .Tvv iiuuiuors 13 lofsi UM greater nmnbep is
20 J|: find the smaller number.
114
'ABniuQnne.
n.
Q ,„, / "" "/ -^iff ot ^*y produces 3i of * ?
*. Simplify "^""m'Jr'o 'heir lowest temu.
en S y-l
^^«i*iiT|-JA. (2)«}of6jofj-jofA.
(8) (A * A)■^(8-i),(J^J). (4) ^ ofMrflii
to red., the fracti,. ^r^tralr^''"''''""-'''^^''"-
. IIL
that /nmb"!Xote^7;. "^ ^'ff'"" » ^^^^-f 1.0 denominator by
^ 2. Simplify
4
T
(4)i*^ + 8-2J
4i + 3^
the ^Iir^Le ;^:^i^::/,'':^i^-r'^^^^ *»
1 A » '^ ^'"^'^ •'>' 2i, the »um will be
IV.
nJ;ratt;dI^S:5-!^!r; -™P~'«^^-ti:„, a„d ,„ete™.
QUESTIONS AND EXAMPLES IN FBACTIONS. 116
Pl-ove by menns of an oxamplo the rule for the mnKiplication of
fractions; and multiply the sum of f of ^ and IJ by the difference of
2. Reduce to their most simple forms the following expressions •
(1) i X ^ l%i -h gths of m . I). (2) J - ,v^ a - ^0. (8) Fri^.
(4) iVof(l + 5^) + |ofAof(r-2f)-f (6) V-^^-f? .
8 What number added to J of (J + J -^4^ + j) ^^^3 gj j ^^ ,^j^^
nnmber divided by i of J of ^ will give ^*j ?
1 ^ y} ^nf '''^^^ ^ °^ ""^ "'°"^y' *^^" ^ of what remainai and then
i of what still remains ; what fraction of the whole will be loft ?
6. Explain the method of « comparing ' fractions.
Compare the product and quotient of the sum and difference of 64-
and 5^. . *
V.
1. State the rules for multiplying and dividing one fraction by
another ; and prove them by means of an example.
2 + 8 4 + 85^ ,
^ 6T5I ' ""^"^ "multiply the sum of f , 1 1 and |, by the
Divide
4 + 5 "'•'C + 6^
difference of >, and ^''0, and divide the product by ^ of 1||.;
\ 2. Reduce to their simplest forms
0) (|-!)-a-J). (2) M-T^-'-T
If
(^-T
8
(8)|ofii.^ofi§ + ^of^.
1
T*
114 -Vt
(6)
iof^L+lof^
'^ B»rr. (5) 2 J
3.'
(r;
8^ + 5/y •
lofA-lofT^/
8. What is meant by the symbol | ?
Find the hast fraction which added to the sum of ?, I and 25. shall
make the result an integer. '**
4. Find the sum of the /-reatest and least of the fractions 2 J' *
and ,;^; the sum of the other two; and the difference of these anS
0. A man has ^ of an estate, he gives his son ^ of his share; what
portion of the estate has he then left?
116
ABrrmcBiio.
VT.
1. state the rules for addition and robtraotlon of Tnkar fraotioM j
and prove them by means of an example w*ottons ;
S. Bimpliiy
0) |ofJ-i|of^^4jofl||. (2) ^llli^. JL
(4) ^+?i
(8) H''l'<13H + Ux3 + 40f.
m^thJ^f ° V .^'^'^' *^''^' *"^ '^^"'^^ fr*°*io°^' Explain the
method of reduomg a compound fraction to a simple one.
Ex fof^of jVroflJ.
aiimt'n!^'',! ^L'^'T". ""^ ^'^ ^""""^^^^ ^^°^ ^ ^'•^««o° i« ^ected if the
same number be added to its numerator an.l denominator.
5. Multiply8iby8^„anddivlde^*byaandfindthediirerence
Letween the sum and difference of these results.
6 TOat number added to | J + j| will produce 8?|2 ? and what
number divided by 2,^ will produce * » *^'
r^?
VU.
f^f-^th^r: r=Tr' ^'•^^^^-*^-^^^=^?; «^-t i of
. 2. Simplify
(3) f(8iof4l)}^.^2i^-.|)of(3i-i)}.
(4){(^of8i)*(|^||)}_JJ^.Mj^(2_^)^
^* ^'""^^'^ i^of|ofi| > ^°^ *^^® *^« ^«s^fc from the sum of lOi
4. Add together J, f , J, and J, subtract the sum from 2, multiply
the result by | of f J of 8, and find what fraction this is of 99
6. In a match of cricket, a side of 11 men made a certain number of
runs, one obtained ith of the number, each of two others ^th. a^ eLh-
of three others ^,th, the rest made up between them 126 ; whi^h was^
remainder of the score, and 4 of these liu.t «.nr^ n ♦;«... L f. „_ ^' T
other. What wasthewholennmberofrunVrandU^^c^r:;^^^^
DEOntAlJS.
117
DECIMALS.
hichwasthe
119. Figures In the units* place of any nnmber express their timpU
values, while those to the Itft of the units' place increase in value tetifold
At each step froni the units' place ; therefore, according to the same
noUtlon, as we proceed from the units' place to the right every suc-
cessive figure would decrease in value ten/old. We can thus represent
whole numbers or integers and certain fractions under a uniform
notation by means of figures in the units' place and on each side of it ;
for instance, in the number 5678-241, the figures on the left of the dot*'
represent integen, while those on the right of the dot.denote ^ace«m#.
The number written at length would stand thus :
6 X 1000 + 6x100 + 7x10 + 8 + — + — + -L.
10 100 1000*
The dot is termed the decimal point, and all ^gures to the right
of it are called Duoimals, or Decimal Fbaotions, because they are
fractions with either 10, 100 or 10x10, 1000 or 10x10x10, &c., as their
respective denominators.
The extended Numeration Table will be represented thus :
7664821-23466r
*^ S W »;3
|i
3
00
5*
I
00
.a
a
00'
s
I
c
o
H
9
§
a
w ;^ 4
120. 10, called t^Q first Power of 10, is written thus, 10».
10x10, or 100, called the second Power of 10, is written thus, 10».
10 X 10 X 10, or 1000, called the third Power of 10, is written thus, 10^
Ann art tvn • e{mi1a*.1*T t\.P ^tio. ~ i ^i .. a...
. . „,^,^^^j ^ viiici liuwiuurs : inua me Biiu power ot 4 is
4x4x4x4x4) and is written thus, 4*.
118
ARimATETIO.
The small Hgures 1, 2, 8, Aei, at tlio right of the number, a little
above the liue, are culled Indices.
121. From the preceding it appears that
I
Fint^
.2346 = 1. ±.--1-. -A.
10 100 lOUO lUUUJ*
Now the least common multiple of the denotninators of the fractions
is 10000 : therefore, reducing the sovcr.il fractions to equivalent ones
with their least common (ienominator, wo got
8 100 4
.„„.« 2 1000 8 100
Zo40 = r-x- X :: + x +
10 lUOO 100 100 lOUO
JO 5
** 10 ^ 10000
* 20&0_+300 + 40 + 5,
lOuOO
•00324 = — + ™
10 100
8
2345
loubo'
2
Secondly.
10 100 1000 ' lOUOO JOUOuO
, (the least common multiple of the denoniinatori is 100000)
8 100 2 10 4
. ^ 10000 1000
" 10 ** 10000 "^ 100 " 1000 ^ 1000
300 + 20 + 4 324
^ 100
"^ 100 "*■ lOOOJ
JO
" ** i'o ^ loojuo
100000 100000*
Thirdly, 66-816 = 5 x 10 + 6+ ^V + tJtj + rAij
(the least common multiple of the denominators is 1000)
6
5 X 10 JOOO ^^lOOO's 100 1 10
1 ''looo'" 1 '*iuoo''io''ioo"'ro"o*'io"'iooo
_ fiOOOO + 6000 + 800 + 10 + 6 66816
1000
lOUO*
^ Hence, we infer that every decimal, au. rvery aumber composed of
integers and decimals, can be put down in the fonu of a vulgar fr.iction
with the figures comprising the decimal oc those composing the inte-er
and decmial part (the dot being in either case omitted) as a numeraror,
and with 1 followed bv as many zeros as there are decimal places in
the given number for the denomina:or.
ti2. Conversely, any fraction having 10 or any power of 10 fi^r its
<^.e caiuator, as 4fy«^% m
lumber, a lUtlo
For
DEOVdAIBi -
66816 5 X 10000 + Q x lOOQ + Q xJ^OO + 1 x 10 + 6
lOOO'" ~
119
1000
6 X ICOOO 6 X 1000 8 X 100 1
+ ^^— ■»• ; +
10
1000 1000
louo iooo ■ lopo
= 5xlO-f6 + -,«ff+TU-^T»''oi
= 56-816 (by the notation wo have assumed).
128. Again, by what has been saiU above, it appears that
•327
827
•0327 -
827
•8270 =
8270 827
1000' lOUOU' loouo'iw'
We see that •327, •0(i27, and -8270 are respectively equivalent to
fractions which have the snme numerator, and the first und third of
which have also the same denominator, while the denominator of the
second is greater.
Consequently, -827 is equal to -8270, but ^0327 is less than either.
The value of a decimal is therefore not affected by affixing cyphers
to the right of it; but its value is decreased by prizing cyphers:
which effect is exactly opposite to t^at which is produced by affixing
and prefixing cyphers to integers.
124. Hence it appears that a decimal is muUipUed by 10, if the
decimal point be removed one place towards the right hand ; by 100
if two places; by 1000, if three places: and so on: and conversely, a
decimal is divided by 10, if the point be removed one place to the left
hand ; by 100, if two places ; by 1000, if threi places ; and so on.
Thus 6^6x10 =4«xlO =66.
5-6 X 1000 i= f » X 1000 = 5600.
6-6-.10 =44x^, =^5^ =.66.
5-6 + 1000 = f « X ^^^, .-. ^^^6^, = .ooo6.
125. The advantage arising from the use of decimals consists in
this; viz. that the addition, subtraction, multiplication, and division
of decimal fractions are much more- easily performed than those of
^Igar fractions ; and although all vulgar fractions cannot be reduced
to finite decimals, yet we can find decimals so near their true valae:
.. ...,,. ^, aiicxiig liuiii uamgiiae aecwaai iiialead of the vulgar
fraction is not perceptible.
!
!
f
120
AKITHMEno.
Ex. XXXV.
1. Express as vulgar fractions in their lowest terms :
•075 ; -848; 3-02; 8-484 ; 848-4 r03484; -060005; 230-409; 2-80409-
2137-2; 91300-0008; 24-000625; 8213-7169125 ; -00083276- 1-0000009*
•000000001. , v»,
' 2. Express as deciuals,
8. Multiply
•7 separately by 10, 100, 1000, and by 100000 ;
•006 separately by 100, 10000, and by 10000000;
•0481 separately by 100, and by 1000000 ;
16-201 separately by 10, 1000, and by a million ;
9-0016 by ten hundred thousand, and by 100
4. Divide
•61 separately by 10, 1000, and by 100000 ;
•008 separately by 100, and by a million ;
. 6-016 separately by 1000, and by 100000 ;
8780186 separately by 1000, and by a million.
6. Express according to the decimal notation, five-tenths ; seven,
tenths ; nmeteen hundredths ; twenty-eight hundredths ; five thou,
sandths; ninety-seven tenths; one millionth ; fourteen and four-tenths-
two hundred and eighty, and four ten-thousandths ; seven and seven-
thousandths ; one hundred and one hundred-thousandths ; one one-
thousandth and one ten-millionth ; five billionths.
6. Express the following decimals in words ;
•4; -25; -75; -745; -1; '001 ; -00001 ; 23-75 ; 2-375; -2376; •00002375-
1-000001; -1000001; -00000001, '
ADDITIOlSr OF DECIMALS.
126. Rule. Place the numbers under each other, units under units,
tens under tens, ifec, tenths under tenths, &c. ; so that the decimals be mm
«i_^ -,„ ^,.^^, ^ „„^ „_ .^ vviiyi© numoers, ana place the decimal §■"
pomt m the sura under the decimal point above.
» tbUIi ; tAWi
ADDinO ir OF DEOIMAIS.
121
Ex. Add together 27-603r, -042, 842, and 2-1.
Proceeding bj the Rule given above
27-6037
•042
842-
. 2-1
871-6457
WoTE The same method of explanation holds for the fhndamental
rales of decimals, which has been given at length in explaining the
Eules for Simple Addition, Simple Subtraction, and the other fLa-
mental rules m whole numbers.
i ^^<*9on /or the above process.
If we convert the decimals into fractions, and add them together
as such, we obtain tt'«-*w
27-5037 + -042 + 842 + 2-1,
270037 42
10000
■^ 1000 "^
342 21
1 "^IC
(or reducing the fractions to a common denominator),
_ 275037 420 3420000 21000
10000 ■" 10000 ■" 10000 "^ ioooo
8716457 ^
" 10000" = '^^1*6467, (Art. 122).
2376; -00002876}
Ex. XXXVI. ^'
Add together :
(1) -284, 14-3812, -01, 32-47, and -00075.
(2) 23216, 3-225, 21, -0001, 34005, and -001304.
(3) 14-94, -00857, 1-5, 6607-25, 630, and -0067.
Express in one sum :
(4) -08 + 166 + 1-327 + -0003 + 2760-1 + 9.
(5) 346 H- -0027 -f -25 + -186 + 72-505 + -0014 + -00004.
(6) 6-3084 + -006 + 36-207 + -OOOl + 864 + -008022.
(7) 725-1201 + 84-00076 + -04 + 60-9 + 143-713 '
(8) 67-8125 + 27-105 + 17-5 + -000376 + 255 + 3-0126.
Add together :
Tu^l 20068, -Oim, -mm, l-OOOOOOS, Sr, «.d in -, and proT. the
122
I 1
ARITHMETIC.
(10) -0008025, 29-99987, 143-2, 5-000025, 9000, and 8-4073; and
verify tlie result.
result ^ ^^*^*' *^^^' ^^^*^°^^^» '^^^^^^^^ ^°d 4957-5 ; and yerifjr the
(12) Five hundred, and nine-Iiundredtlis ; three hundred and
3. J; r ' ;^ '^"'""^ '"^ eighty-four, and seventy-eight
hundred-thousaudths ; eleven irnllions, two thousand, and two hnndred
and nine niilhonths ; eleven millionths : one billion, and one billionth.
SUBTRACTION OP DECIMALS.
127. Rule Place the less number under the greater, units under
nmt,, tens under tens, &e., tenths under tenths, &c. ; suppose cypher:
to be supphed ,f necessary in the upper line t. the right of the decimals
then proceed as in Simple Subtraction of whole nurr.bers, and place the
Uecimal point in the rem lindei- under the decimal point above.
Ex. Subtract 6-473 from 6-23.
Proceeding by the Rule given above,
6-23
5-473
-757
■Reason for tlie alone process.
If vve convert the decimals into fractions, and subtract the one from
the other as such, we obtain
623 _5473^ 6230 5473
100 1000" 1000 ~ 1000
6-23 - 6-473 =
= 1000 = ■^^'^' (^^*- 122).
Ex. XXXVII.
(1) Find the diflference between 2-1354 and 1-0436 ; 7-835 and
2-0005; 15-67and 156-7; -001 and -0009; -305 and -000683.
Find the value of
(2)213-5-1-8125. (8) -0516 --0094187.
(4) 603 - -6584008. /'5^ tt'K _-io.a(\Aa
(6) -582- -09647.
(7) 9-283 - -OSSfi.
nd 8-4073 ; and
; and Terify the
ct the one from
MULTIPLIOATION OF DEOIMAIS 123
\.All l^?.?l ^''°''^' ^^'^^ ^^"^ 718-00688; 85-009876 from
66078 ; 27148 from 9816 ; and prove the trath of eich result.
(9) Required the diflference between seven and seven tenths • also
between seven tienths and seven miUionthsj also between seventy,
four + three hundred and four thousandths aud one hundred and
I seventy-four + one hundredths ; and verify each result
MULTIPLIOATIOK OF DECIMALS. *
128. Rule. Multiply the numbers together as if they were whole
numbers, and point off in the product as many decimal places as there
are decimal places in both the multiplicand and flie multiplier; if thero
are not figures enongh, supply the deficiency by prefixing cyphen.
Ex. Multiply 5'34 by '0021.
Proceedmg by the Rule given above,
6-84
•0021
■i^
634
1068
11214
The number of decimal places in the multiplicand + the number of
those in the multiplier = 2 . 4 ^ 6 ; but there are only 5 figure" in the
Reason/or the above prouas.
6-34x.0021=5?*x 21
_ 11214_
100 10000 1000000
= -011214.
Ex. XXXVIII.
Multiply together :
(1) 3-8 and 42 ; -38 and -42 ; 3-8 and 4-3 ; -088 and -0043
m 417 and -417; -417 and -417; 71956 and -000026.
(3) 2-052 and 0031 ; 4-07 and -916 ; 476 and -00026.
Multiply fprorinff the trnfh nf flio ,.an„u ; %. ^
(4) 81-4682 by -0878. (6) 27-86 by 7-70071. (6) •04876by'0764^
I
}
i
I
--
124
ABITHMKHa
' CO '0046 by T'8^ (8) -00846 by -00824 (9) -SM by -0021
l-a ^n?o V^i«n ^^^,'*"^^ prodnctof 1, -01, -001, and 100 ; also of 12
V . • f "^ ^^® ' *"^ P""^^^ ^^ ^^^^ o^tl^e results. '
JBind tbe yalne of
^ (14) 7-6 X -071 X 2-1 X 29.
<^^) '007 X 700 X 760-8 X -00416 x 100000.
DIVISION OF DECIMALS.
129. Jif.^.^^ tke number of decimal places in the dkidendl
«ftwfc the nmib^ <if deeimal places in the ditiJr. »«'«w«<»
nm^ll^f f '?^^'"1''' ''^''^^ '''''^^''^ '""^ "^'^^ «ff i° th« quotient a
nmnber of decimal places equal to the excess of the number of decimal
places m the dividend over the number of decimal place7i„ the diZr
if there are not figures sufficient, prefix cyphers as in Multiplicatir '
Ex. 1. Divide 1-1214 by 5-34.
Proceeding by the Rule given above,
6-84) 1-1214 (21
1068
084
584
«f fr *J^V«^?«r of decimal places in the dividend - the number
of decimal places in the divisor = 4-2 = 2; namoer
tberefore the quotient = '21.
; Ex. 2. Divide -011214 by 53-4.
63-4) -011214 (21
1068
634
634 •
= 6-1 = 5:
thfwfore we prefix three cyphers, and the quotient is -00021.
'-'lL'iLRa»l-JBt!«VSaB
DIVISION OF DECIMALS.
125
«« in the dkid&ndX
ttd - the namber j
- the number of !
jReasonfor the aibove process.
n214 534
1 •1214-^5-84 =
lOpOO ' 100
11214
21- 1 / .
^-r- ^ rrr : ( Bince
1 100 ' \ 634
11214 100
10000 ** 634
100
10000
11214 100
534 10000
= ?.l, and
looy' •
- 21 „ .
"roo=*^^-
Again, •011214-^53•4 =: ^l?l^-:-5?! =J1^1L.12.
11214 10
7— X
634 1000000
"lOOOOOO"* 10
1 '* lOGOOO
1000000 ** 634
21
= •00021.
100000
ISO. Secondly, When the number of decimal places in the dividend
is less than the number of decimal places in the divisor.
Etjle. Affix cyphers to the dividend until the number of decimal
places in the dividend equals the number of decimal places in the
divisor ; the quotient up to this point of the division will be a whole
number ; if there be a remainder, and the division be carried on
further, the figures in the quotient after this point will be decimals.
Ex. Divide 1121-4 by -534. -
Proceeding by the Rule given above,
-634) 1121-400 (2100
1068
6^4
634
JReasonfor the above process.
11214
1 121 -4-5- -534-=
534
10
11214
1000
!ooo
11214 1000
X --
10
634
--= 21 X 100 = 2100.
534 10
Note. In order to prevent mistakes in tl)e proof of examples in
Division of Decimals, always c -ntrive in the process to separate 10,
100, &c. in the two fractions from the otlier figures, as in the above
examples ; and be sure never to AfFp^i. flio T.i,iU;r,K^«4-!^., ,*^i.i. v_ x
Jett m the denominator ; nor, if there be tens left in the numerator,
to effect it until the last step of the operation.
•!i ; !
I i
It !
126
AEITHMEno.
Ex. Divide 172-9 by -142 to three places of decimal*.
•142) 172-900000 (1217-605
809
284
250
142
1080
994
860
852
800
710
* 90
f^ ' Here wd'must affix 5 cyphers to 172-9 • for if w« nffl^ *«,«
^^the rule the division up'to that point^m ^I^ f ^^^^^^^^^
the quotient only, and therefore as the quotient is fn Z u! ?^ !\
three places of decimals, we must affiVthr e "^^^^^^^ H
, Tre must affix five altogether. cypner* mpre, that is,|
^ , Reason for tJie above process.
172-9^-142 = — ^^
1^2 _ 1729 1000
10 * 1000 ~ I^
10
Now 172900000
1729 100000 _ 172900000 i
142 1000 lia— " looJ
142
therefore the result
= 1217605... from abore;
12176^5...
1000
= 1217-605.
Ex. XXXIX.
Divide, (proving the truth of each result by Fractions):
(1) 10-836 by 5-16, and 34-96818 by -881.
(2) -026075 by 1-003. anrl .naoifl t>„ .aa-.«
(3) -00081 by 27, and 1-77089 by 4-735.
li]
_ ;- I
DIVISION OP DECIMALS.
127
(4) 1 by -1, bj -01, and by -0001.
(6) 81-5 by -126. and 5-2 by -82.
(6) 82ir by -0626, and -08217 by 6260
• (7) 4-68638 by 81-34, and 16-4646 by -019
(8) -429408 by 59-64, and 2147-04 by -086
(9) 12-6 by -0012, and -065341 by -000476.
(10) 3-012 by -0006, and 298916-669 by 641-283
(11) 180-4 by -0004 and by 4, and 46-634205 by 4807-65
(12) 1-69 by 1-3, by -18, by 13, and also by -013.
(18) -00281 by 1-405, by 1405, and by -001405.
(14) 72-36 by 86 by -0036, and -008 by 1-6
(15) 6725402-3544 by 7089, and by •70§9. *
(16) 10363284-75 :>y 396-25, and -09844 by -0046
(17) 816 by -0004, and •00196106£r2875 by 2-38645
(18) 18368830-5 by 2815, by 231-5, and by ^2315
(19) -00005 by 2-5, by 25, anS by -0000025.
(20) 684-H97 b/l200-21, and also by -0120021.
.« w? */i'''' ^\r! ^l^'^^^^^ each of the following, and prove
le truth of the results by Fractions :
(21) 82-5 by 8-7; -02 by 1-7; 1 by -013.
(22) -009384 by -0063 ; 51846-734 by 1-02.
(23) 7380-964 by 028 ; 6-5 by 8-42 ; 25 by 19
(24) 176432-76 by -01257: 7457-1345 by 6535496-2. *
(25) 37-24 by 2-9; -0719 by 27-53.
Find the quotient (verifying each remit) of
(26) -0029202 by 157, and by 1-67
(28) (7iofi + H)byW5; of 81-008 by ||| of IJ of ^»^; -7576
Certain Vulgar M-actions can he expressed accurately
as
181.
decimals.
W. Eednce the fraction to its lowest tenns; then place « dot
CL*! ™°'*™'°',.''"^ "^'^ .'=yPl'»™ for ^«»imals; divide by the
.!i !
128
ABITHMBTia
Ex. 1. Oonrert ^ into a decimal.
6 |_8;0
•6 ^
There is one decimal place in the dividend mid none in the divisor-
therefore there is one decimal place in the qacient.
Note. In reducing any such fraction as /, or .^^ to a decimal, we
niay proceed m the same way as if .e were reducing §; taking caTe
however in the result to move the decimal point one place further to
the left for each cypher cut off.
Thus
T = ^'
8
= •06,
8
= •006.
50 ""' 600
. Ex, 2. Reduce :r^ to a decimal.
,10
■ 16) 6-0000 (-8125 - '
48
20 or thus,
40
82
80
80
^ 16 (^ 11
U 1
00
/. T*ir = *8125
2600
■8125
Ex. 8. Convert ^ and ^ into decunals.
K"ow 612=8 X 64=8 X 8 X 8
8
8
8
8-000
•375000
•046876000
-005869876
^' ^ ^® equivalent to •006859876,
61200 ^^ ^*i''^^^°^^ent to •00006869876.
u
)ne in the divisor ;
according as the
VULGAR FRACTIONS BXPBB9BBD AS DiBOIMAIfl. 129
Ex, 4. Convert j + SJ + 2^„ + e^ into a dedmal.
8 I 1-000
•125
5
6
6
11
2-20
•440
4 I 9-00
225
.-. A = 225
•088
ierefore J = '6, ^ = -1 25, f% = -225, ^, =, -088 ;
!ierefore the whole expression
= 11 + •e + -125 + '226 + '088
=» 12-088.
Ex. XL.
Rednoe to decimals :
(3) 6HT 5^; rt; Trh\ l^iUh.
(4) 8|of;Hhr.
(7) |+^06]
7-75
(10)
9
(6)
(8)
nf ^^ .^ 20
n
i+i+tSr+?V- (6) ^x-0064.
*H-i. (»)fof|a.
(11) 6Tj|Tr+*75of|of7i.
(13) f ^w^^;.2oo^.|,.
(12) 8j^+^+81t^+^.
132. We have seen that, in order to convert a vulgar fraction into
[decimal, we have in fact, after reducing the fraction to its lowest
Irms and affixing ciphers to the numerator, to divide 10, or some
lultiple of 10 or of its powers, hy the denominator : now 10=2x6 and
lese are the only factors into which 10 can he hroken np; ther^re
pen the fraction is in its lowest terms, if the denominator be Hot
mposed solely of the factors 2 and 5, or one of them, or of powers
r2 and 6, or one of them, then the division of the numerator by the
I " """ "^^"^xuauc. sjvinuioisoi bnis &ma are called
Idetermmate decimals, and they are also called daouLATmo, Rbpbat-
I I! If
i
^^^ ABiTfliamo.
»o, orRiouBBiNo Dboimals, from the fact that, when a decimal does
not terminate, the same figures must come round again, or recur, or be
repeated : for since we always affix a cipher to the dividend, whenever
any former remainder recurs, the quotient wili also recnr. Now, when
we divide by any number, the remainder must always be less thin that
numW, and therefore some remainder must recur before we have oh-
dh^Bw * '''^^^'' **r'*^"*^°^^''' ^^"^ ^ *^« »"°>ter of units in the
188. Pure OiBOULATmu Decimals are those which recur from the
begbniDg: thus, -SSS.., -2727.., are pure circulating decimals.
Mixed Oiboulatino Decimals are those which do not begin to recur
till a^r a certam number of figures. Thus, -128888.., -0118686... are
mixed circulating decimals.
Pure and mixed circulating decimals are generally written down
only to the end of the first period, a dot being placed over the first and
last figures of that period.
Thus -8 represents the pure circulating decimal -888..
.*^? -3636..
*®®? • • • f. •63968S..
•188 •-• mixeif* 1388
'^^'^^^ •0113636..
184. Pure CircuUUng DecimaU may he converted into their eouiva-
lent Vulgar Fractions ly the following Rule.
RuuB. Make the period or repetend the numerator of the fraction
and for the denominator put down as many nines as there are figures
m the period or repetend.
This fraction, reduced to its lowest terms, will be the fraction
reqmred m its simplest form.
Ex. Reduce the following pure circulating decunals, .8, 2-7 -857142
to then- respective equivalent vulgar fractions. *
Proceeding by the Rule given above,
•3=1=1. 27-??-^
•857142=—— _ 142857x6 6
9S9990 142667 X 7"" 7~*
1 be the fraction
omatJuurmG deodcais.
m
^^J^^^^^ these r^Uvnllapprn'/^ths/olM^
y=-mill 4o., hence -i ==-4444 &a, ^=-7777 4a
Agaio,
therefore
9-8*
1 1
g0=y+n=-llllll &0.-M1
•01010140. 1
I hence l=.0707O7 4o., i^= -171717 4o. ;
99
therefore
I In like manner,
J— .1 111
999~ 9 "*■ "
99'
^99 11*
•mill &0.+111 =-001001 &a,T
land
1 _1 „,,
gggg~y4 1111= -111111 &o.-f.llll=00010001 4c.;
•82U82U4a;
f'""'' §r9=-206206 &c., and |||
Itherefore •857142=?i^^-^^2®^'^><<^ «
■ 999999 142867 x7~y*
186 Jf^ OireulatmgJ)eeimals may ieeofwerted into their eqttwa^
yent Vulgar M-actiom by the following EuU,
RuxE Snbtract the figures which do not circolate from the figures
tak^ o the end of the first period, as if both were whole number.
Make the result the numerator, and write down as many ninee
as there are figures in the circulating part, followed by as many eeroe
las there are figures in the non-circulating part, for the denominator.
loii^^ 1^^.°'^ ^^^ following mixed circulating decimals, -14, -0188.
r24l8, to their respective equivalent vulgar fractions.
Proceeding by the Rule given above,
•14=
14-1 18
z=~\ -0188=
90 ""90'
•2418
138--18 125 1
9000 '"9000""72'
9990 "'9990^4996*
r-il
irii
i I
'I V
*"iiiiuiiit"i9tiiiM^^vnmiMM#
188
ABITHKimOL
l%tf€atM itf iht ruls will appnar/rom tks^lhwinff ^mtiimvH&na.
Let |27886 be the mixed circulating decimal,
we have 87*880 by mnltipIyiDg, in this case, the given deoimal by 100
=27fH Art. (184).
But this valne is 100 times too great ; ^
o/T 886
therefore, =T7:7i+ ^^^^A true valae
100^»9»00
27x999 + 886
99900 "~ 99900
27000-27 + 836 27886—27 '27809
27 X (1000-1) +886
99900
99900 ""99900*
Note 1. Always multiply by such a number as will make the non-
circulating part a whole number.
Note 2. Sometimes a dechmal of very long period may be carried
out easily to many places, as in the following example :
Bednce t^ to a decimal.
17)1-00 (-OSSStV.
85
160
186
•140
186
4
hence -^ = •0688tV ; .*• ^ = '2852^,
hence ^ = •068828624^ (by substitution) ;
16
/. jf^ = -94117632^,
hence j^ = -05882352941 17632i^ (by substif.).
•0588285294117647
TT'
By the above process, we double at every step the number of figures
yrevioBsIy obtained.
Ex. XLI.
1. Reduce the following vulgar fractions and mixed numbers to
oironlating decimals :
a) « ; 1^; W; h (2) H ; H! ; i4; 15^^^
(8) fftt ; 'r^^ ; ji^^W (4) 24,1^1, ; 17^^, ; 2UU§
(6) tV; A; j*»; jV
OraOTTLATINO DEOnrALS.
183
lU inakt tha non-
may be carried
^ (by Bubstif.).
umber of figures
sed numbers to
Find the yulgar fractions equivalent to the recurring decimals:
(6) 7; 'Or. -227. (7) '688 ; .iSB ; -268.
(8) -00186 ; .8-6al ; -01236. (9) -142867 ; -897916 ; 88214286?;
(10) -807692; -6307692; 2-7857142. (11) -842768; -03182182; 8-02088.
(12) 86-60806; 8-6428671; 127-00022096.
136. The value of the circulating decimal -999... is found by Art.'
(184) to be ^ or 1 ; but since the diflerence between 1 and -gzzi-l be-
tween 1 and -99=01, between 1 and •999=-001, &c., it appears 'that
however far we continue the recurring decimals, it can never at any
stage be actually=l. But the recurring decimal is considered =1,
because the difference between 1 and -99... becomes less and less, the
more figures we take in the docimal, which thus, in fact, approaches
nearer to 1 than by^asy difference that can be assigned. ,
In like manner, it is in this sense that any vulgar fraction can be
said to be the value of a circulating decimal ; because there is no
assignable difference between their values.
137. In arithmetical operations, where circulating decimals are con-
cerned, and the result is only required to be true to a ceri^n number
of decimal places, it will be sufficient to carry on the circulating part
to two or three decimal places more than the number required :
taking care that the last figure retained be increased by 1, if the suc-
ceeding figure be 5, or greater than 6 ; because, for instance, if we
have the mixed decimal -6288, and stop at -628, it is clear that -628
18 less, and -629 is greater than the true value of the decimal: but
•628 is less than the true value by -000888.... and -629 is greater th^
the true value by -000111...
Now •000111...is less than -000888...
Therefore -629 is nearer the true value than -628. \
Ex. 1. Add together -88, -0482, 2-345, so as to bo correct to 6
places of decimals.
•8333383
i
•0432432
2-7220811
Ant. 2*72208.
134
ARITHMETro.
; Ex. 2. Subtract -2916 from -989583, so as to be correct to 6 places
of decimals. .
•9895833
•2916667
•6979166 Am. -69791.
Note. This method may be advantageously applied in the Addition
and Subtraction of circulating decimals. In the Multiplication and
Division, however, of circulating decimals, it is always preferable to
reduce the circulating decimals to Vulgar Fractions, and having found
the product or quotient as a Vulgar Fraction, then, if necessary to
reduce the resujt to a decimaL
Ex. XLII.
Und the va^ue (correct to 6 places of decimals)* of
(1) 2^418+l-16+3-d09+^7354+24^042.
(2) 234^6+9^928+-6l23456789 + -6044+466.
(3) 6^45-^3 ; and 7^72-6^045 ; and 309-^94f24.
^ (4) Express the sum of ff, |^g, and ^\, and the difference of 18A
and 4t^, as recurring decimals.
Multiply
. (5) 2^3 by 5-6; •7675 by •366.
(7) 7^62 by 48^8 ; 368 by 4.
Divide
(9) 195^02 by 4 ; -37692 by •05. (10) 54* by •it; 13^2 by 6-6.
(11) 411-8519 by 68-7646; 2-16595 by -04; -6559903 by 48-76.
REDUCTION- OF DECIMALS.
188. To reduce a decimal of any denomination to its proper value,
Rule. Multiply the decimal by the number of units connecting the
next lower denomination with the given one, and point off for decimala
as many figures in the product, beginning from the right hand as
there are figures in the given decimal. The figures on the left of 'the
(6) -406 by 62 ; 825 by -36.
(8) 3-146 by -4297 ; 20^ by -84.
lorrect to 6 places
d in the Addition
[ultiplication and
lys preferable to
md having found
, if necessary, to
KEDTJCaaON OP DECIMALS.
135
decimal point will represent the whole numbers in the next denomina-
tion. Proceed in the same way with the decimal part for that denom-
ination, and so on.
Ex. 1. Find the value of -0484 of £1.
Proceedmg by the Rule given above,
£.
•0484
20
•9680».
12
ll-6160rf.
4
For, £-0484 of £1 = ^^^ of £1.
9680'
10000
:».
116160
10000
■d.
11^ 2464
^ 1000^*
2-*<^^^- . = lU, . 2,V\^.
= 11*1!^.
therefore the value of -0484 of £1 = llf^y^
Ex. 2. Find the value of IS'SS^^ acres.
Acres.
18-3375
4
1-8500 ro.
40
14-0000 po.
therefore the value is 13 ac, 1 ro., 14 po.
Ex. 3. Find the value of -972516 of £1.
Ist method.
je.
•972917
20
19-458340«.
12
5-500080<2.
4
2-000320^.
therefore
Note, The 2"* method
2d method.
•97291 « nf PI 972916 - 97291 ^ „, .
«7-i916 Ot £1 = r,,,:,^r:r-. of £1 Art. (186),
900000
875625
900000
of £1
/467 \ .
467 ,^
the value is 19«. h\d. nearly,
is generally the better one to adopt.
i.L
i ! I
136
ABITHMETIO
1 1
Ex. 4. Find the value of i?? of 8| tons-sios of If qrs. +:?i?5*2
*o26
of 1 cwt., 63 lbs.
J33
400 °^ 3* *0°s
tODS
183 x»'
80x4
399
_/133 15\
Uo0^4J
/133x3 \
=24 cwt., 3 qrs., 18| lbs.
•3405ofl|qrs.=f?155=?ofAUrs
^ ^ \ 9990 s) ^^^•»
= (9990 ""y^ 26 jibs.,
/21x25\„
= ^-87— jibs. =14^ lbs.
tons,
cwt.
■'''''' of 1 cwt., 68 l,s.=('-^?^,Z^,^m ^^^^3j ^
•826
'[' 900000
96007
826
lbs.
900
=106-^ lbs.
therefore the value of the expression
=24 cwt., 3 qrs., 18f lbB.~14^1bs.+106^1bs.
=24 cwt., 3 qrs., 4^^^^- lbs. + 1 cwt., 6^ lbs.
=1 ton, 5 cwt., 3 qrs., lly^ lbs.
xun.
Find the respective values of
(1) -45 of $1 ; -16875 of $4 ; -87708 of $6.
(2) -28125 of £1; -7962 of £1; -359375 of £2.
(3) -086 of $5 ; -5783 of $10 ; -075 of $16.
(4) -875 of a lea. ; 2-5884875 of a day : -6 of 1 Ih. Tro^
(5> -85076 of a cwt.; -07326 of a cwt. ; •045ofamUr
f ^ m'K ;4 i' ^"t ii Uf ir*>*-
.-.i%.^:iKS,Si«3SS«i^£i
EEDUCTION OF DECIMALS.
137
(6) 4-16525 of a ton ; 3-625 of a owt. ; -06 of an acre.
(7) 3-8843 of a lb. Troy ; 2-46875 of a qr. ; 4-106 of 3 owt., 1 qr.,
21 lbs.
(8) 8-8376 of an acre ; 8-5 of 18 gallons.
(9) -925 of a furlong ; -34375 of a lunar^onth. v
(10) 5-06325 of $100 ; 3-8 of an Eng. ell.
(11) 2-25 of 8^ acres ; 2-0396 of 1 m., 580 yds.
(12) 4-751 of 2 sq. yds., 7 sq. ft. ; 2-009943 of 2 miles.
(13) -383 of $1 ; -47083 of $4 ; -4694 of 1 lb. Troy.
(14) -5740 of 27». ; -138 of 10«. 6d. ; 2-6 of 5«.
(15) 4-05 of 1^ sq, yds. ; -163 of 2^ mUes ; 4-90 op 4d., 8 hrs.
(16) 3-242 of 2^ acres;
-09318
•568i
of 2tV of 2-5 days.
(17) Find the difference between -77777 of a pound and Ss. 6*6648<? }
and between -70323 of $4.80 and 3*5646 of 24 cents.
(18) -268 cwt. + -0562 ton— -5786 qr.
(19) £-684875 + -025 of 25«. + -316 of 30«.
(20) 2-81 of 365i days+5-75 of a week-f of 5| hours.
(21) I of ^jof 3 acres— 2-00875 square yards + -0227 of 3^ square feet,
139. To reduce a number or fraction of one or more denominations,
to the decimal of another denomination of the same hind.
EuLB. Reduce the given number or fraction to a fraction of the
proposed denomination ; and then reduce this fraction to its equivalent
decimal.
Ex. 1. Reduce 13«. G^d. to the decimal of £1.
13«. 6id, = 162J<?. = JSl|^.
£l = 240d;
649
therefore the fraction =
649
240 960
= •67.
1-00
6-25
18-52083
138
or tbas, 4
12
2,0
•6760416
by 20, which is '6760416
Ex. 2 Reduce 8 bns., 1 pk. to the decimal of a load: and Terif7
the result. »wiAjr
1-00 ^
ABETHMBTIO.
We first reduce ^. to the fraction of
a penny, which is -25 ; next 6-26<?. to
the decimal of a shilling by dividing
by 12, which is '52088«; then 18-62088
to the decimal of a £1 by dividing
40
i
4
8
5
3-25
•40625
i !
•08125
therefore -08125 is the decimal required.
ii I -08125 Id.
6
' -40625 qrs.
8
8-25000 bush.
4
1-00000 ^)k.
therefore 0-8125 of a load = 8 bus., 1 pk.
Ex. 8. Express the sum of -428571 of $72, i of i or* f of $7-68
and j- of 12 cts., as the decimal of $48.
' • •^28671 of $72 = ^Ul^ of $72.
= fof$72 = $^a
= $30.85f
i of- of I of $7.68 = i of ^^ of ^ of $7.68
= 64f cts.
f of 12 cts. = 6^ cts.;
therefore the sum = $80.85f + 54f cts. + 6f eta.
= $8].47f
»l>^.^r ^ Av_ j__f 1 . , 81.47-2.
sitivi^i.vsjs jiiio uucuuai required =
48
^ = •665714
BEDXJOnON OP DECIMALS.
139
Ex. 4. Oonyert £17. 9*. 6d. into pounds, florins, &a ; and verify
the resalt.
First reduce 9«. 6d. to ths decimal of £1.
12
2,0
6-0
J9;5_
•475
.% £17. 9«. U. = £17-475
= £l7.4fl. 7c.5m.
Again, £17. 4 fl. 7 c. 5 m.
= £17-475
20
9-500*.
12
6'OQOd.
,\ £17. 4 fl. 7 0. 5 m. = £17. 9«. 6d,
Ex. 6. Beduce the difference between a cent (New Coinage) and a
penny to the decimal of 3«. 4d,
ld. = £j^; Ic. =£^1^^;
.-. difference = £(y^ - ^) = £j|4|^ = £^^
= (t7o . X 20 X 12)f?. =^.
Zs. 4:d. = 'iOd,
.-. fraction = X = ^ = ^1^ .
.*. decimal = 'OSS.
XLIY.
Beduce
(1) $1.25 to the decimal of |2 ; and $8.75 to the decimal of $4.
(2) 4s. lyi. to the dec', of £1 ; and 16«. 11^ to the dec", of £1.
(8) 10». OJ. to the dec', of £1 ; and 6«. 8f<f. to the dec', of £5.
(4) 2 oz., 13 dwts. to the dec', of 1 lb. ; and 4 lbs., 2 sc. to the dec*, of 1 02.
(5) 2 qrs., 21 lbs. to the dec', of 1 ton ; and 8 owt., 8 oz. to the dec*, of
10 cwt.
140
AEITHMETIO.
"" dJof'alt^f *°''''"^-«^»»^-^<'-.- -Oa-.^Opo. .0 the
f (8) 4 days, 18 hrs. to the dec', of a wk • anrl 1 1 c^« * ^i. ^ C'
; W llb.Troytothedec.. oflIb.Avo,>d andLtt .^^ f '"^"•
I ("^?t^^'-'othede..of.*L';Orttr2^t^^^^^^^ ,
1 cwt., 9,i qrs. ' * ** "^^^ *^® ^eo'. of
(12) 3 wks., 5ld. to the dec , n ; ira • and 1 min oi .
irV of a lunar month. " "'•' ^'^^^^^"•^^isec.totnedec' of
(18) 8 reams to the dec", of 19 sheets • anfl qi o« * *t. :. .
(14) 83 yds. to the dec', of a muf bTb y fV k' ^^ "''* ^^^ ^^•
(a dollar being 4*. 3d) ; a" d 7. 8 -*\f tl^t. . ^?' / ^ ^^"^^
(16) A of $7 to the dec', of $1^ and 67ctt tn II ^ r' 1 '^*- '^•
(16) H of $8 to the dec', of $7 •' and 'nk to ll ^" ""; ^^ * *'^-
(ir) f of a guinea to the dec' of £2 ^nd ^' /' ' '' ' ""'''
of a day. ^. oi a^ , and ^j^ of a year to the dec'.
(18) I of ^ of 40 yds. to the dec', of 4 of 2 mis • «n^ i .. «.
to the dec', of 2 ac, l.ro. ^' * ''^ ^ '"^«- » a^* i of 8| sq. yds.
(19) I of 4| hrs. to the dec'! of 8654^ davs • nnd fl_fi_ ^^ 1 1 i \
of 8^ of 8 bush. ^ ^ ' ^^ ""^ ^ P^«- *o the dec'.
(20) 8 lbs., 6 02. Troy to the deo> of in iKc a -^
to the dec', of /oz. Troy ""''• ' '"'J * "^ Ayoird.
(21) Add together | of a day, | of an hoar and « of « i.
m-, r"^ "'u '""" ■" *^ ' ""^ «f « "eek '^ ' """"^ • """
(23) Add 5i owt. to 3-125 ,rs.; and reduce the sn. to a.e deouna.
"^ oX^no^an:r.la-^^^^^^^^^ --
I' y- 2. lOd 8. ^d. 4 5.
6. 10«. 6d. 6. 16,9. . *'
S.£54.7..<W. 9. £20.19..7i<e. 10. l^Ud
12. £2. 16.. n-088A 13. f8.0,."lW8-04y.
11
1 A ~ a.l n t
MI80ELLAKE0US QUESTIONS.
141
0. to the dee*.
20 po. to the
lec'. of 6 days.
Iec'.of2|iulB.
ofliqts.
) the dec', of
the dec', of
ofSJsq.yda.
of a dollar
'f 10«. Qd.
'f a ton.
iqrs.
to the dec'.
f 3|sq.yds.
. to the dec',
oz. Avoird.
ours; and
3f $1-20 as
le decimal
^ Decimal
Id. Z-Qiq,
XLV.
MUcellaneoua Questions and Examples on Arts. (119-189).
I.
(1) Define a Decimal ; and show how its value is affected by affijc-
ling and prefixing ciphers. Reduce '0625, 3'14169 to fractions; and
I express th difference between 20^^ and Vj^r^ as a decimal.
(2) Find the value of lO^Jrl^+ t\-\-\1 both by vulgar fractions
and by decimals : and show that the results coincide.
(8) Find the sum, difference, product, and quotient of 578*005 and
•000754:; and of 1-015 and '01015, and prove the truth of each result.
(4) If a vulgar fraction, being converted into a decimal, do not
terminate, prove that it must recur. What must be the limit to the
number of figures in the recurring part ? Is ^^\^ convertible into a
terminating decimal ?
(5) Simplify 1. 2^72 1 + 316^2-875. ' 2. •026649-f-2||.
^ ' .{•18 + •009}-^•016.
8.
l_-05 3-'8
5 + -5 " 8-8 • 10'
4.
(6) Divide
1-0714285.
48f
hy
U
TT
1085xV 174tV'
Divide 91-863 by 87*56.
II.
reduce the quotient to the form
(1) "Write down in a decimal form seven hundred thousand four
hundred and nine billionths. Express 12*1345 as a fraction, and
Ti-ii^hsis as a decimal.
' (2) State the effect as regards the decimal point of multiplying and
dividing a decimal by any given power of 10. Write down in words
th3 meaning of 397008*405009 ; multiply it by 1000, and also divide it
by lOOO ; and write down the meaning of each result in words.
(3) What decimal multiplied by 125 will give the sum of |, ^, f,
•09375 and 2*46 ?
(4) Multiply 1.05 by 10*5 ; and reduce the result to a fraction in its
lowest terms. Divide •8727588 by 1620 ; Had the value of
reduce iV + tStt - ^ *<> a decimal.
- -0003 X -004
^
•006
! I
142
ARITHMETIO.
(8) SUnm, oxpr,^siog each resale in a decimal form " ' "
rt^ t P^+*^' *■ ^=^ + 1^^ --6,^+2.000876.
W *ina a number whiVh TnnU.vi; i - ^ ^
duct which differs only Tn ^ 7^, d^ f f ^^ «^«2.458 will gi.e a pro-
y m tne 7th decimal place from 7823-6572.
iir. •
# (1) Divide 684'1 197 bvlQftft.oi «^i 1 ,
■* 2 It:?!
6 -•0626*
^- (i-i)x(f + l^).
•035 •
8. l + -14+f of 1-0784. ,. ,^-.^,^,..,,,
- (8; What IS meant bv a « p^«««-- -r^ .
vnlgar fractions produce lhdJri7'slT''', """" «"•» »'
any recnrring decimal to a v„i ^ . "** "'^ "■"'«» for reducing
and divide VU ^^Tooui L £1 'r ^'^^ ,*'""'P'^ ^'^i b, -U;
(4) SI.OW that if lA 1a 3 wf r ^ 7 ''™"'"« ''«''™«1»
«o», and (2) a, deci-nl^ thrrtSt^oi'^cidf '"'^"'"' <'^ "» *-
. -a-; fi/''t>'eamo„'„t„fhUda"^"2;!'"'*'"'^'"' """'^■^ >«•»«
w^ii?rpa':.-;:i:~r:iiS
IV.
fra^on corresponding the rllt' "r .''fj '"<' ""O '■'o -"tear
■".ai.^Tori^rr^rri^^;-''"'^^^^^^^^^
^^^ When the last is ZmoZ^^J^LT""''"'^ "' «"> «^o
W oon>p„. the valuea of 6 x -05, l-ex'^T^'a^,, 2-626^6
MBOELLANEOUS QUESTIOKIB.
143
r+2-000875.
11 give a pro-
J-6672.
; and 694*27
the position
ecimal form,
hat kind of
or reducing
i by -4583,
f deoimal ?
(1) as frac-
three first
Iked 13-95
Jiis share ;
decimals,
he vulgar
equivalent
thousand
the saitid
(4) Find the product of •0147i4f by -833; and the quotients of
•12693 by 19-89 ; of 132790 by -245 ; of -014904 by 8yV; of 61061 by
8-05 ; and of 6106-1 by 805000.
(5) Shew that the decimal -90487582 is more nearly represented by
'90488 than by -90487; and find the value of
Ji. L_4._J_ 1 ^^ ? 4 "
( 5 8x58"*"6x6»~'7xF''*'®°' J~
16x
289
8x58 ^ 6x6*
accurately to 5 places of decimals.
(6) A person sold -15 of an estate to one person, and then -j^ of the
remainder to another person. What part of the estate did he still
retain ?
V.
(1) Express |(6^ + 2f - 8), f?|f , and also the product of 8| and
iH ~- 1) of T as decimals.
(2) Simplify
. 4-255 X 032 « ,, , ,
^' .00016 • 2- <Hj+i+i+iV)-a+^ + i+^.)
8. (A of 35^ - 8i) + (2-5625 + 7i). 4. •593-4-l-78x-86-j-072.
(8) State at length the advantages which decimals possess over
vulgar fractions ; what disadvantages have they ?
Shew whether ^ or |^ is nearer to the number 3-14159.
(4) Find the value of 1 + -1 + ^ + ^^^ + &c., to 7 places
of decimals ; and also of
1 ; / ^ 8x4
103 ''V^ 102 ""172
•+ -
8x4x5
''lOV
10* ' 1 X 6T3
' expressing it (1) ao a decimal, and (2) as a fraction.
(5) Find the Earth's equatorial diameter in miles, supposing the
Sun's dia^ieter, which is 111-454 times as great as the eiuatorial dia-
meter of the Earth, to be 883345 miles, v
(6) In what- sense is a vulgar fraction said to be the value of a
recurring decimal ? Explain how a sufficient degree of accuracy may
be obtained in the addition and subtraction of circulating decimals to
any given number of decimal places, without converting the decimals
into fractions.
Ix. Find the sum of -125, 4-163, and 9-457, correct to 5 places of
decimals.
144
ARrraOkCBTIO.
(1) Prove the Rale for Mnltiplicadon of dccimalB by means of
the example 404-04 multiplied by -080808. Multiply .346 by IH . ^.
divide -04818489963 by -6693, and -006698 ^'^ '
(8) Reduce to their lowest terms ?^?i? and ^^'^^^
1033'2'
5-7980*
(4) Shew that .^?^ii:?!^zi:??6ro25 2 ,
•875 — 025 = 5"' ^^ ^afc
8 + ~+ J
2Q-= 8-14159 nearly.
rt)' Whit?''' f ^^ 'r""^^°^ ^'^'^ fr^^^-'^-
W Whatdecimal added tothe sum of 1^ -^ nnri is ^-ii i xt^
sum total equal to 8 ? ^' ^' ^ ^^^ '"^^^ *^®
(6) The quotient being 2^ and the divisor -15, find the dividend.
VII.
What aro a da/, wages of eli. Z^ZT """ '" ""^ '""^ "'""^'
(8) The wages of 25 men amount to •P'7fi lOn ^ 7 • -,« ,
many boys r:ast work 24 days to c Xf li™ itf- V.T' ""^
of the latter being one-half those of the formed ' * '""^'^'^^
the work Ineln'Z B i'nVd""" "' T'^ '""' ^'■''- ^ "-'O ^o
Pleeed in .days;"po^- X^"::i<^^r""^"''--
. ..yl ..^ ?*'•=»■' bnys 3Ibs. of tea at 74 c™c, ,«, ,k ..^ _.,_ ..
-en 6 ibs. at 66 cents per lb.. What will 2.brof hU IT;;;: hi^r
^ f^""" ' m I' ll
PBAOnOE.
145
PRAOTIOE.
140. An aliquot part of a number is such a part as, wheu
taken a certain number of times, will exactly make up tbat number.
Thus, 8 is an aliquot part of 9 ; $6 of $18.
TABLE OF ALIQUOT PABTS.
Pa/rt» of a ewt. (100 lU.)
50 lbs. or 2 qrs. = ^ cwt.
25 lbs. or 1 qr. = ^
20 lbs. ' = ^
10 lbs.
_ 1
- Tir
(;
u
((
5 lbs. = ^y
Note. The parts of a $ the same
as of the cwt. (100 lbs).
Parts of a £1.
10«.
6a. M.
5a.
4«.
8a. 4^.
2a. 6^.
2«.
la. M,
la. 4:d.
Is. Sd.
U
- 1
- ^
= *
= i
_ 1
- Tir
_ 1
- T5
_ 1
- rj
_ 1
- TJ
It
u
((
((
u
u
n
u
<(
Parts of a ewt. (112 lbs.)
56 lbs. or 2 qrs. = ^ cwt.
28 lbs. or 1 qr. = ^
16 lbs.
14 lbs.
r lbs. = ,v
4 lbs.
2 lbs.
- T
_ 1
- ¥
_ 1
- 5«
_ 1
- IF
(t
((
(t
t(
u
Parts of a shilling.
6d.
4d. .
8d.
2d.
IK
Id.
id.
id,
Id.
= iofla.
= *
- 1
- ja
i_ 1
- TT
_ 1
- at
_ 1
((
((
u
Note. In working examples in Practice, the above tables will often
have to be varied ; the knowledge which the scholar now has, will
render him expert in taking such aliquot parts as he may require in
in any particular example.
141. Practice is a short method of finding the value of any
number of articles by means of aliquot parts, when the value of a
unit of any denomination is given. Practice m£^y be divided into two
cases, Simple and Compound. -
146
ABITHMETIO.
BIMJf»LB PRACTIOm
I. In this case the given number is expressed in the same denomi-
nation as the uuit whose value is given ; as, for instance, 27 bushels of
wheat at $1.10 per bushel.
The Rule for Simple PraoUoe will be easily shewn by the foUowine
examples. ^
Ex. 1. Find the value of 1296 things at 16». lOK each.
The method of working such an example is the following:
If the cost of the things be £1 each ;
then the total cost=£l296.
eaoh=^ of the above sum. . .
each=i the cost at 10*. each.
eaoh=i the cost at 6*. each,
.*. cost at
10«. Od.
58. Od.
Is. Sd.
0*- W . eaoh=i the cost at la. Sd. each,
/. by adding up the vertical columns,
cost at 16«. lO^d. each
The operation is usually written thus :
£.
8.
d.
= 648
.
.
= 824
.
.
= 81
.
.
= 40
.10
.
=£1098
.10
.
10«.=iof£l.
5«.=f of 10a.
Is. Bd.=zi of 6s.
7^.=^ofl«. 8d.
£.
1296
s.
648
324
81
40
10
£1093 . 10
d.
= cost at £1 eacB.
= cost at 10». each.
= cost at 5«. each.
= cost at U. Sd. each.
= cost at 7^d. each.
= cost at 16«. lO^d. each.
Note. The student must use his own judgment in selecting the
most convenient 'aliquot' parts; taking care that the sum of those
taken make up the given price of the unit.
Ex. 2. Find the value of 826 bushels of wheat at $1.80 per bus.
^ If 1 bus. cost $1, cost of 825 bus. =$826 at $1 each.
$825.00=value at $1 each.
165.00= value at 20 cts. each.
82.60 =yalue at 10 cts, each.
20 cts. = J of $1.
10cts.=^of20ots.
$lU72.60a:vaIue at $1.80 each.
PRACmCK.
147
«.
d.
.
.
.
.
.
.
.10
.
COMPOUND PR ACTIOS,
II. In this case tho given number is not wholly expressed in the same
(lenoraination as the nnlt whoso vuluo is given ; as for instance, 1 cwt.
2 qrn,, 14 lbs. at $10.24 per owt.
The Rule for Compound Practice will bo easily shown from the fol-
lowing examples.
Ex. 1. Find the value of 60 cwt., 8 qrs., 6 lbs. of sugar at $8,60 per,
cwt.
The method of working such an example is the following:
The value of 1 owt. of sugar being $8.50 ; . ^
.•. value of 60 cwt. =$(8.50 x 60)
2 qr8.=^ (value of 1 cwt.)
=^- ($8.50)
1 qr.=^ (value of 2 qrs.)
=\ ($4.25.)
5 lbs.=} (value of 1 (qr.)
=1 ($2,120
Therefore adding up the vertical columns,
value of 60 cwt., 8 qrs., 5 lbs.
The operation is usually written thus .
2 qr8.=^ cwt. I $8.50= value of 1 cwt.
10
=$610.00
= $4.25
= $2.12^
= $0.42^
=$516.80.
1 qr. =\ of 2 qrs.
61b8.=loflqr.
86.00=value of 10 cwt.
6
TH_ rt "Pt^J *V*v
510.00=value of 60 cwt.
4.25
2.12i
.42^
$516.80 =value of 60 cwt., 8 qrs., 6 lbs.
i»/^ /^roiQ /.nrf ft nisi 1ft Ihfl fth £9 \^. 6d. oai*
XT A tr XT TT V»T
cwt.
148
-ieithmeho.
1 1
2 qrs.=:|^o-vrt.
£.
2
12
d.
6
10
value of 1 owt.
subtracting
1 qr. = I of 2 qrs.
14 lbs. = 1 of 1 qr.
2 lbs. = I of 14 lbs.
26 . 5 .
4
105 . .
8
840 . .
2 . 12 .
6
887
1
r
6
13
6
= value of 10 owt.
= value of 40 cwt.
= value of 820 cwt.'
= value of 1 cwt.
6 = value of 319 cwt.
3 = value of 2 qrs.
1^ = value of 1 qr.
6f = value of 14 lbs.
Hi = value of 2 lbs.
^£839 . 14 . 41 = value of 319 cwt., 3 qrs., leibs.
XLVI. '
Find the value oi
(1) 275 articles at 25 cents each ; 125 articles at 30 cents each.
(2) 92 articles at 45 cents each ; 80 articles at 50 cents each.
(3) 120 articles at 75 cents each ; 215 articles at 85 cents each
(4) 225 articles at $1.10 each; 350 articles at $1.25 each.
(5) 128 bus. oats at 53 cts. each ; 75 bus. wheat at $1.10 each.
(6) 318 yds. cloth at 72 cts. a yd. ; 48 bus. pease at 63 cts. a bus.
(7) 7 tons, 2 cwt., 3 qrs., 10 lbs. of sugar at $10 per ton.
(8) 87 ac, 2 ro., 22 per., at $8 an acre.
(9) 210 lbs. tea at 42 cts. a lb ; 812 lbs. sugar at 10 cts. a pound.
(10) 6261bs.,7oz.,19dwts. at$1.27perdwt. 78 things at $2.36 each.
(11) Find the value of 282 ac, 17 per. at $0.60 per perch.
(12) Find rent of 100 acres at 87^ cents a rood.
In the following examples the cwt. =112 lbs.
(13) Find the value of 5 cwt., 2 qrs., 14 lbs. at £2. 5*. U. per owt
(14) Find the value of 33 cwt., 3 qrs., 7 lbs. at £6. 7«. 8d per cwt.
(io) Find the laiue of 72 cwt., 3 qrs., 17 lbs. of sugar at £1. 4». U,
per owt.
1
s
e
l
t
ij
c
PBAOnOB.
149
.
(16) Find the value of 60 cwt., 8 qrs., 12 lbs. at£r. 18«. Qd. per cwt.
(17) Find the value of 8 cwt., 2 qrs., 16 lbs. at £3, 7s. 6d. per cwt.
(18) Find the value of 9 yds., 2 ft., 10 in. at 6«. 7id. per yd.
(19) Find the value of 39 cwt., 10 lbs. at £Z. 15s.''7^ per cwt.
(20) Find the cost of 30 cwt., 2 qrs., 14 lbs. at £1. 17«. 8id. perqn
(21) Find the value of 15oz., 6 dwt, 17 grs. at 5«. lOd. per oz.
(22) What will 2789 lbs. of pork, cost at $8.50 per 100 lbs.
(23) Find value of 28800 ft. fire-wood, at $6 per cord.
Find the amount of the following account :
(24) 24 lbs. crushed sugar at 12 cts. a lb. ; 7f lbs. tea at 75 ots. a lb.,
4f lbs. coffee at 32 cts. a lb. ; 5 lbs. rice 7 cts. a lb. ; 20^ lbs. cheese at
11 1 cts. per lb. ; 17^ lbs. ham at 19 cts. a lb.
142. Examples which are usually classed under particular Rules,
such as the Rule of Three, &o., can nevertheless be readily solved in-
dependently by means of the foregoing principles.
The following examples, which are worked out, are intended to ex-
emplify various methods of reasoning. In the examples for practice
which follow them, questions will be found the solution of which may
be easily arrived at in a similar way : the number of such questions in
this place must necessarily be very limited, and therefore the fctudent
is strongly recommended to apply to all questions which are hereafter
classed under particular Rules, an independent method of solution, as
well as the one denoted by the Rule to which they are respectively aflBxed.
Ex. 1. Express a degree (69|^ m.) in metres, 32 metres being =85 yds.
85 yards =32 metres.
32
.*. 1 yard=^ metres ;
.*. 1 degree=(69^ x 1760) yards=(139 x 880) yards.
^139 X 880 X 82\
— gg ] metres = 111885^ metres.
-f
Ex. 2. If §ds of a lottery ticket be worth $880, what is the value
of Y^ths of the same ?
.*. §rds of the ticket = $880.
.♦. ^rd of the ticket = $440.
.% whole ticket = |(440 x S) = $1320.
.-. Aths of the ticket = tV of $1820 = $l?^ili= $860.
150
AEITHMETIO.
1 1
Ex. 8. A person has ?ths of an estate of 4000 acres left him ; he
sells fr Is of his share: how many acres has he remaining, and what
. fraction of the whole estate will they be ?
Q Q Q
He sellsy of y of 4000 acres, or y of 4000 acres.
.% he has remaining (y of 4000 -y of 4000 J acres •
=y of 4000 acres=57lf acres.
^ Ex. 4. The sum of $1000 is to be raised in a school section, the
assessment of which is $100000; what is the rate in the dollar? i
$100000 produce $1000,
' 100000 ,
"^'iooooo^^'-^^^"""*-
Ex. 6. After taking from ray purse J of my money, I find thai f of
what is then left amounts to 7«. 6d. ; what money had I in my purse
at first? J f
Let unity, or 1, denote the sum in the purse at first. After taking
away i, f remains. Now by the question
2 8 2 3
yof-^ of unity, or y of— of the sum in the purse at first=7«. 6d,
or y of the sum in the purse at first=7s. 6^7.
.*. sum in the purse at first=15*.
Ex. 6. A met two beggars, B and 0; and having ?i of— ^ of—
® 4f 7i 540
of a moidore in his pocket, gave i? y of y of that sum, and C' — of iho
remainder ; what did each receive ? •
40 ^
A had at first ii of 1 of ;2^ of 278.. or li *.
' . 7 2 - -
q
h
n
w
a
ti
al
so
MISOELLAITEOUS EXAMPLES "WOEKED OFT. 151
rr
"
B received y of -^ of — « ., or -^ »., or U,
A hadleft afterwai;d3 {— - ^ ».=—».,
.-. G received y of y «-, or —-«., or 2«. Qd.
Ex. 7. A farmer pays a corn-rent of 6 quarters of wheat and 8
quarters of barley, ^iVinchester measure: what is the money value of
his rent, wben wheat is at 60«., and barley at 54s. per quarter, imperial
measure; 32 imperial gallons being=33 Wincbester gallons?
Rent is 5 qts. of Wheat "Win. raea. + 3 qrs. of barley "Win. mea.
32
But 1 Win. Gal.=--imp. gal.
oo
.*. 1 Win. qr.=— imp. qr.
oo
.-. rent is 5 x ~ imp. qrs. of wheat + 3 x ^^ imp. qrs. of barley, -
32 32
.-. money value of rent=(5 x 33 x 60 + 3 x ^^ x 54) «.=£22. 8».
Ex. 8. If £1. sterling be worth 25 francs, 60 centimes ; and also
worth 6 thalers, 20 silbergrosohen ; how many francs and centimes is
a thaler worth? (One tbaler=30 silbergrosohen, 1 ftano=100 cen-
times.)
6 thalers, 20 silbergroschen=25 francs, 60 centimes,
or 6§S- thal§rs=25/5V ^anos,
1 thaler=(25J-j-6|) francs
384
~ioo ^^^°°^~^ francs, 84 centimes.
Ex. 9. Standard gold contains 11 parts of pare gold to one part of
alloy, and 201b. Troy are coined into 934 sovereigns and a half,
sovereign ; find the weight of pure gold in a sovereign.
Number of part»=ll + l=12, of which j^ is pure gold.
152
AEITHMETia
B7 the' question
9341 sovereigns weigh 20 lbs. Troy,
20 X 2
.*. 1 80V. weighs -j^ lbs. Troy
(11 20 X 2\
jgX^sgg jib. Troy
■ = ll^ffij grs.
Ex. 10. If a person, travelling 13f hours a day, perfonn a journey
in 27| days, in what length of time will he perforih the same if he
travel 10^ hours a day ?
If he travel 13f hrs. a day, he does the journey in 27| days,
l^r , (271 xlSg) days,
102 hrfl 27p X 1 3y
lu^nrs , , — ~ — days,
which, worked out, gives 36 H^l days.
Ex. 11. If 85S men in 6 months consume 234 quarters of wheat,
how many quarters will be required for the consumption of 979 men
for three months and a half?
858 men in 6 months consume 234 quarters,
234
.'. 1 man in 1 month consumes^^^ — ;; qrs..
oOo X O ^ '
.*. 979 men m 1 month consume -^yr, — tt- ors..
858 X 6 ^ '
(979 X 284 7 \
ggQ ^ g X — j qrs., or 156| qrs.
Ex. 12. If 5 men or 7 women can do a piece of work in 37 days •
in what time will 7 men and 5 women do a piece of work twice as
great ?
N
5 men =7 women,
7
.'. 1 man=^ woman,
49
.'. 7 men =-v- women.
• 8 •Lwxi c&m;
/49_\ 74^
vii— i -^ T y i vvuiiieii=-v- women.
an
$1
pai
16
Of
'"
MISOELLANEOTJS EXAMPLES WOBKEB OUT 163
Now by the question,
7 women in 37 days do the piece of work,
.*. 1 woman in (37 x 7) days does
87 X 7
.\ 74 women in —f—- days do
74 . . 87 X 7 X 5 ,
.*. -g- women m =g — days do
74 . 87 X 7 X 5 X 2
•*• ~5 ^<^°i^^ i» 74 or in 35 days do twice as much.
Ex. 13. A bankrupt owes three creditors, A, B, and C, $250, $880,
and $420 respectively, and his property is worth $125 ; how much will
fcach creditor receive, and how many cents in the dollar ?
Debts amount to $(250 + 330 + 420), or $1000.
If the bankrupt has $1, he pays — • part of debt,
125
$125 j^^ part of debt,
y part of debt.
.-. A gets $31.25, B gets $41.26, and G gets $52.50. He pays \ of
$1, or 12^ cents in the dollar.
Ex. 14. Gunpowder being composed of nitre 15 parts, charcoal 8
parts, and sulphur 2 parts ; find how much of each is required for
16 cwt. of powder.
The whole number of parts = (16 + 3 + 2) = 20
Of every 20 parts,
^^« ^ • -,. 3 . - ,2 1 .
2^ or — is nitre, — is charcoal, go ^^ io ^^ sulphur.
•*• -^ of 16 cwt., or 12 cwt. = quantity of nitre required.
o
^ of 16 cwt., or 2f cwt. = charcoal
1 •
j^ of 16 cwt., or 1| cwt. = sulphur
Ex. 15. The price of a work which comes out in parts is £2. 16». 8i.
154
ARITHMETIO.
price of the
But!f theprice of each part were 18^. more than ifc i^ t
work would be £3. 7*. 6d. How many parts are there ?
£2. 16». 8d. + (number of parts xl8)d.=£S. 7«. 6d.
.-. (number of parts x 13) i. = io«. lOd, '
= 130^.
.'. number of parts = Ys" = 10.
as it'et^ *« ^i'f 1,'T f '""'' ^''''''° ^' ^' *^^ ^' «^ *^** «« often
as ^ gets $5^ ^ shall get $4, and as often as B gets |3, C shall get $1.
It IS clear that ^'s share = 3 times (7's share,
4 times A's share = 5 times £'s share,'
f'
or, ^'s share = - times -B's sbare.
= f -^ X 8 j times C's share,
but ^'a share + B'b share + C7's share = 1860 dollars •
15 '
Cb share + 8 CT's share + (7's share = 1860 ^lollars,
• •
4
**^ (t ^ 7 ^'s share = 1860 dollars,
81 ^ ,
or -^ CT's share = 1860 dollars ;
(1 ARO A. \
-J- X —j dollars = 2 iO dollars.
-B's share = 720 dollars, and ^'s share = (24O . ^) dollars = 900 dollar.
i J""' l!' ^^ V""!.''''' ^^""^'^^^ * ^^*^^ ^^°S« ^'^ «^ *J^e same name,
i of another, i of a third, and ^ of a fourth, and there are 5 besides •
how many are there of each name ?
Representing the whole dynasty by unity, or 1.
1 ^
Y = number of kings of one name.
4
J_
8
1^
12
of a second...,
of a third...,
• • > ■ < s > . . vjL a, ivuitii,,,.
.
Be of the
as often
get $1.
lollars.
name,
asides :
MISCaULLAirEOUS ETAMPT^ WORKED OUT. 165
Nowia.i.i.y
8 4 8*12 24'
.-. whole dynasty - ^, or 1 - 1?, or A == ^o. of remidning kings in it
But by the question,
2j of unity, or ^ of the whole dynasty « 5 ;
r
.'. 1, OP the whole dynasty, = 6 x — = 24 •
.-. there are 8 kings of the 1st name, 6 of the 2nd, 8 of the 8rd, and
2 of the 4th. ,
Ex. 18. A can do a piece of work in 5 days, B can do it in 6 daya,
and C can do it in 7 days ; in what time will A, B, and O, all working
at It, finish the work ? Find also in what time A and B working
together, ^ and G together, and B and C7 together, could respectively
Representing the work by unity, or 1.
In one day A does y part of the work,
In one day B does — - part of the work,
o '
'S
. C does-=- ;
^+i;+(7do(|4+l),orHp^.
A tune in which ^+jB+ C would finish the work
1 J 210,
= j^ days = j^days= 1|«^ days.
210
Again in one day^l+^do (y+y], orii,of the work; therefore
time in which they would finish it=-- or 2^ days.
11
80
In like manner, it may be shown that ^ and (7 would finish the
work in 24-4 clays j and if and Cin 3i\ days.
I
1667
ABITHMBTIO.
Ex. ,;&. It being given that A and B can do a piece of work in 2JL
days; and that A and Ccan do the same in 2U days L that VJa
^can do it in 8^ days : find the time in whichT^ 'a^^wlld "o
the work: working, first, aU together, secondly, sepamtely.
In one day A and -5 do li of the work,
^andCdoi?
85 »
.^andCdo—.
42
/. by addition,
Ihoneday2^+2^+2(7woulddo/'~+— +!5^ 214
/. in one day u4+^+ ^do ~
210
.% time reqdred=--=j-^ days=lio3 days.
' 107 107
■ . fl 210
f Again,
work done by ^+^+ C in one day - work done by B+ in one day,
or, work done by A in one day =— - 12~JL .
., ^ 210 42~"6 '
therefore time required, in which A could do the work, =6 days
, In hke manner it may be shown that B would do the work in fl
days, and that G would do it in 7 davg. ^"^^ ^^ ®
Ex. 20. A cistern is fed by a spout which can fill it in 2 hours how
it?f iTw? *' ^^ "'^'^" "''''''' ^'' * leak which wouWempty
In one hour spout fills -^ of the cistern.
leak empties—
Therefore in one hour, when the spout and leak are both owfin +>.«
part of the cistern filled by what runs in - what runs out ^ '
V2 loy 6'
; .-.time required for filling the cistern = -ihrs. = |-hr3. = 2^ hrs.
M
a
t
'>
ork in 2^j
hat B and
would do
J work,
>ne day,
ork in 6
ars, how
d empty
*j
MBOELLANEOUS QUESTlOlffB AND EXAMPLIB. " 167
Ex.21. ^ and ^ can do a piece ofwork in 15 and 18 days respect-
ively ; they work together at it for 8 days, when B leaves, but A con-
tinues, and after 8 days is joined by C, and they finish it together in 4,
days ; m what time woald G do the piece of work by himself ? ^
Representing the work by unity, or 1.
In one day ^ + ^ do ^1 + Ij of the work,
in 8 days they do ^1 + iU a
or
19
11
80'
•'• gQ <>f the work remains to be done.
In 8 days more A does — or 4- of the work •
ID o *
/. when A is joined by C,
19 1 13 ^ ^
80 "S"' °^ 80 ^ ^^® ^^^^ remains to be done.
In 4 days more A does — of the work •
15 ♦
.*. work which has to be done by ^ in 4 days
~80 16~30"'ir'
.-. part of work to be done by C in one dav - i
24?
.-. time in which O would do the whole work = 24 days.
)en, the
rs.
Ex. XLVn.
Miscellaneous Questions and Samples on preceding Arts.
I.
(1) State the rules for the multiplication and division of decimals,
and divide 34-17 by 3J. ^
(2) What is the value in English monernf IfiKfl.ftK f. _v—
the exchange IS at 24-25 francs per J? =, «*
'*
158
ABITHMEno.
of /!l^f ?"" ViK^ ;iV -^ A to a decimal fraction. What decimal
or a cwt. 18 1 qr. 7 lbs. ?
of it?^ IM of an estate b. rorth $4818.60, what ig the value of f
on ilu:ftmt ''" '' '"" " *'^ '^""' "'^* ^"^ "- ---^
f (6) A person possessing ^, of an estate, sold | of i of his share for
£120| ; what would i of VV of the estate sell for at the same rate ?
» (7) A man, his wife, and 8 children earn $24.75 a week ; the wife
earns twice as much as each child, and the man three times as much as
ills wife ; required the man's weekly earnings.
(8) If £1. sterling be worth 12 florins, and also worth 25 francs, 58
centimes ; how many francs and centimes is one florin worth ? ?100
centimes = 1 franc.) ^
(9) The wages of 6 men for six weeks being $406, how many weeks
-will 4 men work for $540.
11.
(1) What is meant by saying that one sum is a certain fraction
(for example f ) of another ? If 26 francs are equivalent to a pound
what fraction of a shilling is a franc ? Give the reasons for the process
which you ado )t in answering the question.
(2) Express f of 1^ of a mile in terms of a metre, supposing 82
metres = 35 yards. » i r e «"»
(8) A, B and C rent a pasture for $192. A puts in 8 cattle, B 9
and 6, 11 : how much should each pay for his share ?
(4) Reduce 3H to the decimal of 10«., and divide the result by 12-5
Explain the process employed.
(5) If the property in a town be assessed at $288000, what must be
the rate m the dollar in order that $12000 may be raised?
(6) If the circumference of a circle = Diameter x 3-14159 • find the
number of revolutions passed over by a carriage-wheel 5 ft. in diameter
in 10 miles.
^7) A farmer has to pay yearly to his landlord the price of 7^ bushels
.Tf} ^^^ ^^' P^' ^"'^'^' '°*^ ^^ °^ "^^^^ «* S*- 3^-» and 61 of oats
at IS. 4a. What is the whole amount of his ren* ?
\
I
t deoimal
line of f
) receiyed
share for
rate?
the wife
much as
rancs, 56
h? (100
ly weeks
fraction
t pound,
I process
>sing 82
le, B, 9,
by 12-5.
must be
find the
iameter
bushels
of oats
\
I
MISCELLANEOUS QUESTIONS AND EXAMPLEsf ,159
If there were a decimal coinage of pounds, florins, Ac how manr'
of them would he have to pay ? , «c. , now many
12 hoL"^ fi'r.,"*'!.'^'' ''. ^''''' ""^ """'^ ^ ^^ ^°""' «°d ^ <'«° <Jo it in
ml^l the time n which both working together con do it.
W Ten excavators dig 12 loads ofearth'ji 16 hours wliilat 12 oth«r-
<»^% oo„ 9 load, in ^, l,o„„; in what ..o ^2;IZ^SZ
in.
ty.Ji\ ^'?^ ^^^T' ^^''*-' ^^'•'•' ^"*° 86 equal portions: and find'
«h.lf ^? r. f '^'•' ^ ^^- '^ ^°- '' ^^« ^^^i'""! of ^ chain. If one
oha n = 10 chamlets = 100 llr,ks= 100. linklets; express the above in
chains, chainlets, links, linklets. •
I? o!f, f "^^'""^ ^® "^^'^^ ^^ ^«°^«' 9 Baiocchi (Roman) and be'
must be paid by an estate whose rental is £115. 12s L
in what time w.U he gain $97.20 with a capital of $1512. '
(6) In the civil year 97 days are mtercalated in 400 years : what Ib
the average length of the year ? '
I (7) If 15 horses and 148 sheep can be kept 9 days for £76 IK* ^
what sum will keep 10 horses and 182 sheep for sTayssufp^^^^^
horses to eat as much as 84 sheep ? ^ ' Bupposmg 6
I (8) A, B and Care three workmen : A can do half a piece of work
ge her do the whole in 2^ hours. Shew that C7can do in 5 hours L
much as B can do in 9 hours.
L^^?K ^Tk ^Tr' ^^'"°'^ ^^ *'^^" ^^^^^5 «"« having put in $2400
receive? ^ '' "''* '"' '' ''^ ^^^^^ ^^^^^ -<^' P--- t^
IV.
(1) Explain how whole numbers are represented in the decimal or
amon system of notatir)n. ^u■nu^,^l„ nac. i-_ «*, „^ ,.,_.,
"• — « "«ixiucia are representea in ti
common system of notation. Multinlv '7910 i^ o'r «„^ ._ ..
procesfl ''''' '"' "'' '""' "*^" ^^i'*^^^ ^^9
process.
160
AIOTHMETIO.
(2) Add together the fifth of a $0.24, two-sevenths of a $1.20, and
four-ninths of a $6.04; and reduce the result to the decimal of $123.
(8) Taking the circumference of a circle at 3| times its diameter,
find the cost of a marble column of two feet breadth, and 5 yards
height, marble being at 16». 6d. per cub. ft. (Area of circle =1 circum-
ference X semi-diameter.)
.(4) If a certain number of men can throw up an intrenchment in
12 days, when the day is 6 hours long, in what time will they do it
when the day is 8 hours long? ^
(6) Find the entire cost of 10 lbs. of tea at 4«. M. per lb., 18 lbs. of
coffee at Is. S^d. per lb., 28 lbs. of sugar at 4^^. per lb., and 16 lbs. of
oandles at ltd. per lb., and divide the araonnt equally among 14 persons.
(6) Reduce 2375| Spanish dollars to English lucney, the exchange
being at 8«. 4:d. per dollar And find the value of 1,000,000 rupees at
2«. Sid. each.
(7) The roller used for rolling a bowling-green, being 6 ft. 6 in. in
circumference, by 2 ft. 8 in. wide, is observed to make 12 revolutions as
it rolls from one extremity of the green to the other ; find the area
rolled when the roller has passed 10 times the whole length of it.
(8) Divide $1400 among A^ jB, and C, in such a manner that as
often as A gets $5, B shall get $4, and as often as B gets $3, shall
get $2.
(9) A fraudulent wine-merchant sells, as brandy, a mixture of brandy
and rum at $5.40 a gallon, which is the proper price of his brandy, that
of his rum being $2.52 a gallon. Supposing one-third of the whole
mixture to be rum, ascertain how much a gallon he gains by his dis-
honesty.
V.
Ex-
(1) Divide 550974 by 1472 ; find the quotient and remainder,
plain the oj eration, and prove the result.
(2) Shew that the value of a fraction is not altered by multiplying
the numerator and denominator by the same number.
Express the fractions ^%, 4> ^°^ tj ^Y corresponding fractions hav-
ing the same denominator, and find the sum.
(3) If 1 lb. Avoirdupois be equivalent to 7000 grains Troy, and 1869
sovereigns weigh 40 lbs. Troy, how many sovereigns will weigh 1 Avoir-
L.20, and
iiamoter,
6 yards
oircum-
iment in
ley do it
8 lbs. of
6 lbs. of
persons,
ixobange
upees at
. 6 in. in
ations as
the area
'it.
that as
, CshaU
f brandy
idy, that
e whole
r Lis dis-
er. Ex-
Itiplying
)ns hav-
md 1869
1 Avoir-
.
MKOELLAITEOUS QXTESTTONB Am EXAMPLES. 161
(4) A qnarter of wheat is consumed Anmaii*, t. v
EncIaDd- if whPflfr h« of AK ^"''^'^"^^^ annually by each person in
^1 of .1 r °^ ^^*- ^ *1"^'*^«''' «°d the population 27 BOO 000
wbat ,s the value of a quarter of a year's consumption ? ' ' '
W A certain number of men mow 4 acres in « hn««. a
number of others mo. 8 acre, in 6 h„„;. i„i'7."J "mT'^'"
mowing n acres, if all work together? * **" *^*' ^
(6) If a man can do a piece of work in fti /i«„„ v
. da^ 1.W .an, Ws a I, .irw^rk^t^i^Vi" X' »"""
(7) If 7 men or 11 women can finish a piece of work in 17 L„
(8) A bankrupt owes A $2476, 5 »1963.60, and «1406 62 • hi.
:rtVii. rta^fc^ - r -- "^ -'^ '^ - -- -
(9) How many francs must be transmitted from Paris to Berlin f.
r2:s^r rrour :ti:r ^""« ^^^^^^^^^^
VI.
n^ltXX "' """""^ '* """"P"^'-' *»«<"-«' "■"•«P'^
(2) A bankrupt's assets amounted to *9finQ o«^ v j.x
eeiv^ 56 cents in .ho aoiiar : find thf alt!^\rde\r^'°"
twi hlrds'ofltcttoo";'"'; """'"'"' '''"' " ^"* ""«"-" "hich is
iUtrue lengfhT "'' "''""' '" "^ ^^i ^^^^ '""K. ^""t i»
P»r pound When the cost of the whoL h'as heen'rldnced'C » 1 7 ""
. ^iot/f "*■' ''°y" ^"''""- 8^9. 21 lbs. of sugm' for I64 80 «n^
s^^hif^s^^' "' '"«'"'« "- •■» -» " pe-rpo^^rt^er.'
how many hours will he be in getting to the top of a nl ««° ,!lY'
(8; !,<, profits of a tradesman average m. 6,. M.' p^" weeMut
162
ARITHMETIC.
of whicli be pays 8 foremen, 10 shopmen, and 5 assistants, at the rate of
2 guineas, 1 guinea, and 17«. 6^. per week repectively : his yearly out-
goings for rent, &c., amount to £723. lis. 8d. Find his net annual profit.
(9) In an orchard of fruit trees, I of them bear apples, i pears,:
^ plums, and 50 cherries ; how many trees are there in all?
vn.
(1) "What is meant by a fraction ? Find the value of f of i of $6 f
and then express the result as the fraction and decimal of $237.50
(2) By what number must £5. 6s. 3^^. be multiplied, in order to
give as product £85. 0«, 4d. ? Divide £34. 13s. into 3 parts, one of
which shall be twice and the other 4 times as great as the third.
(3) If a year consist of 365-242264 days, in how many years will
its defect from the civil year of of 365J days amount to 1 day ?
(4) If 15 men take 17 days to mow 300 acres of grass, how long
will 27 men take to ^mow 167 acres ?
(5) If 20 men can perfoim a piece of work in 12 days, how many
men will accomplish another piece of work, which is six times as great,
in a tenth part of the time ?
(6) I am owner off of | of ^ of a ship worth |30,000, and sell ^th
of the ship ; what part of her will then belong to me, and what will it
be worth ?
(7) A bankrupt owes $900 to his three creditors, and his whole
property amounts to $675 ; the claims of two of his creditors are $125
and $375 respectively ; .yhat sum will the remaining creditor receive
for his dividend?
(8) Tiaere are in a manufactory a certain number of workmen who
receive $13 a week, twice as many who receive $10 a week, and
eleven times as many who receive $8 a week, and the total amount of
the workmen's wages for one week is $847 ; find the number of work-
men.
(9) Beduoe £405. 6s. 8^. to francs and centimes, at the rate of 25^
francs to £1, and 100 centimes to a franc.
vni.
(1) Find the value at $15.60 per oz. of 18 lbs. 9 oz. 8 dwt. of gold
dust.
he rate of
early out-
inal profit.
9, i peats,;
»fiof|6;
L order to
ts, one of
rd.
years will
r?
how long
low many
3 as great,
id sell jth
iiat will it
his whole
I are $125
or receive
:men who
7eeky and
imount of
r of work-
ate of 25^
t. of gold
MISCELLANEOUS QUESHONS AND EXAMPLES. 163
resent ItuZ, '^ "^'^ *'^ ^°^* ^' ^^^^^ ^^^^ --ber will rep.
f^acL off:;X • f.rr -' - ^-^ « ^— > -b-
in f h« ,^,f ^'^^^t'"'^'' ''^ ^ ^'^* °^ ^^'^20.80 a dividend of 61| cents
m the do lar, and he receives a further dividend, upon the deficiency
of 18i cents in the dollar ; what does the creditor eceive in trewhole ?
IhheL^Tt^!^ '° ^"^^"^^/^ ^200 a year; an mcome-ta^ is estab-
hshedof rdm the pound, whUe a duty of 1^ per lb. is taken off sugar •
a) If A can do as m-ch work in 5 hours as B can do in 6 hoars
or as Ccan do in 9 hours, how long will it take O to compira pTece
the^n^^rr;^^-:^- -^^^^ w^- are e,nivalent to
expfnlJ$mT4Vr''"'?^ " undertaking average $14400, and the
X and tl In ^r ""'''' r^t''"'"^ "^^^^ remainder is putasido for
wear and tear, and the annual charges amount to $115880.08. WhaF
IS the net annual profit? (1 year = 52 weeks.)
IX.
(1) Explain the process of Long Division.
Eeduce ^«- ^ x.^4^ to its equivalent whole number.
(2) Shew how to convert any proper fraction into a decimal
Reduce f and ^Ify to the decimal form
How masy cente sliould bo given in exciange for il* of a dollar?
(4) If two-tl,irds of an aondemio term e«eod onl-half of it bv 181
days, how many days are there in the whole term » »"'••? l^i
tuueu lor xi». 17«. O^d. ? How much is lost by the exchange ?
164
ARITHMETIC.
■il
!
(6) A butler concocts a bowl of punch, of which the following are
the ingredients : milk 2^ quarts, the rind of one lemon, 2 egg3, 1 pint
of rum, and half-a-pint of brandy. Compute the value of the punch
reckoning milk at Zd. a quart, lemons at 2s. a dozen, eggs at 16 a
shilling, rum at ]3». per gallon, and brandy at £1. 4«. 8^. per gallon.
(7) A Cochin China hen eats a pint of barley and lays a dozen eggs,
while an English hen eats half-a-pint of barley and lays five eggs.
Supposing the eggs of the English hen to be half as large again as those
of the Cochin China, which is the more economical layer? ^
(8) If 72 men dig a trench 20 yds. long, 1 ft. 6 in. broad, 4 feet deep,
in 3 days of 10 hours eacli, how many men would be required to dig a
trench 30 yards long, 2 ft. 3 in. broad, and 5 feet deep, in 15 days of
9 hours each ?
(9) A crew consists of 420 men, and a certain number of boys ; the
men receive each $14.40 per month; and the amount of wages of the
whole crew is $720p per month ; find the number of boys supposing
each to receive $7.20 per month.
X.
(1) Explain the rule for the addition of decimals ; add together f
and -061 ; subtract '003 from -02 ; and divide '0672 by -006.
(2) Subtract |- of | from | of 3^, and multiply the resiilt by ^ of |,
(3) If £1 sterling = 10 florins = 100 cents = 1000 mils, shew that
JE25. 10«. 7id. = 255 florins, 3 cents, li mills.
(4) If 6 men earn $90 in 7^ days, how much will 10 men earn in
Hi days?
(5) A person expends $345.60 iri the purchase of cloth, how much
can he buy at the rate of 52 cents a yard?
(6) "What is the cost per hour of lighting a room with ten burners,
each consuming 4 cub. in. of gas per second ; the price of gas being 6s.
for a thousand cubic feet ?
(7) "What is the value of 8 qrs., 5 bushels, 3 pecks of wheat at $1.20
a bushel ?
If 8 qrs., 6 bushels, 2 pecks of malt cost £21. 3«., what is. the price
per bushel?
(8) If 36 men, working 8 hours a day for 16 days, can dig a trench
72 yards long, 18 wide, and 12 deep, in how many days will 82 mea
I
MISCELLANEOUS QUESTIOira AND EXAMPLES. 165
working 12 hoars a day, dig a trench 64 yards long, 27 wide, and 18
deep ?
(9) If a sheet of paper 51 feet long by 21 feet broad be cut into
strips an inch broad; how many sheets would be required to form a
strip that would reach round the earth (25,000 miles) ?
XI.
(1) Express Jg j as a decimal ; and thence find its value when unit
represents $300.
(2) A per-son has city property yielding a rental of $3070 ; a rate
of 2 ct9. m the dollar being levied, what will he have to pay ?
(3) Find the price of 2 tons, 16 cwt., 17 lbs. of sugar at 20 cts. for
2 2 lbs.
(4) If 1 cwt. of an article cost $33.60, at what price per lb. must it
be sold to gain -j^^ of the outlay ?
(5) Find in inches and fractions of an inch the value of -00003551136
of a mile. Explain the process employed.
(6) £xpre?3 each silver coin now current in England by a decimal
of 2ld. If^Vth of 2ld. be the unit of money, what decimal will express
a halfpenny ? ^
(7) A Canadian dollar is 4s. ^d., and is 5-42 francs ; find the num-
her of francs in £1 sterling, and express both a dollar and a franc in
terms of the unit of money mentioned in the last question.
(8) A and B can do a piece of work in 6 days, B and C in 7 days
and A, B, and G can do it in 4 days ; how long would A and C tie
to do it ?
(9) A bag contains a certain number of sovereigns, three times as
many shillings, and four times as many pence and the whole sum in the
bag IS £280 ; find how many sovereigns, shillings, and pence it con-
tains respectively.
166
AETTHMETIO
SEOTIOK Y.
RATIO AND PROPOETIOIT.
143. "Wb may ascertain the relation which one abstract number
bears to another abstract number, or one concrete number to another
concrete number of the same kind, in respect of magnitude, in two
different ways; either by considering how much one is greater or less
than the other; or by considering what multiple, part, or parts, one is
of the other, that is, how many times or parts of a time, or both, one
number is contained in the other. Thus if we compare the number 12
with the number 3, we observe, adopting the first mode of comparison,
that 12 is greater than 3 by the number 9 ; or, adopting the second mode
of comparison, that 12 contains 3 four times, and is thus '/- or four times
as great as 3. Again if we compare the number 7 with tho number 13,
,re observe, according to the first mode of comparison, that 7 is less
than 13 by the number 6 ; and, according to the secoml, that as 1 is one
ttirteent'i part of 13, so 7 is seven thirteenth parts of 13, or j'^gths of 13.
144. The relation of one number to another in respect of magni-
tude ife called Ratio ; and when the relation is considered in the first
of the above methods, that is, when it is estimated by the difference
between tho two numbers, it is called Akithmetioal Ratio ; but when
it is considered according to the second method, that is, when it is
estimated by considering what multiple, part, or parts, one number is
of the other, or, which is seen from above to be the same tiling, by the
fraction which the first number is of the second, it is caVed Geome-
trical Ratio. Thus for instance, the arithmetical ratio of the numbers
12 and 3 is 9 ; while their geometrical ratio is y. or 4. In like manner
the arithmetical ratio of 7 and 13 is 6, while their geometrical ratio is ^\.
145. It is more common, however, in comparing one number with
another to estimate their relation to one another in respect of magni-
tude according to the second method, and to call that relation so esti-
mated by the name of Ratio. According to this mode of treatment,
which we shall adopt in what follows, " Ratio is the relation which
one number has to another in respect of magnitude, the comparison
being made by considering what multiple, part, or parts, the first num-
RATIO AND PEOPORTION.
167
press the mnldpTe or p^t or bT . ''r'"."'^""'- ""' ""^"^ «"
second, orthen'uLberof Le „;l:o"at ^e'^lr'r " "' *^«
second is contained in the trTt CsuJlZZ^ ' "{' ""*' *"•
8, the fraction V, which is eauiv„l7„f LZt t , """''*'" ^^ «°'J
the multiple which l"i,of%'^ . * ' '^'"''* ""'"•'" *. »I>ew,
12. Ana^a ■:?^: rail ti^Vu'irr'ri^^^^^^^^^^^^ r^r- '^
express the part or parts which the nZberr is rf 13 oTT,1 '' "'"
the part or parts of a time that 18 is co^Uinld in 7 L, ^ "^"v-"
teenth part of 13. so that t m,„t h. °°"™°™ "» 7: for 1 13 one thir-
is, ,lths of it : and 1 i tlTea , „r in rT^t'^^^^^ "' ^'' ''»'
tained only ,' ths of a time in r W T J . ' '^ """'' ^^ <=»"-
of one number to another maV Z 7»'f V'''""'''"'^ """ ">« '»«<>
fraction in which the fomor number is ^r """ '"'"*"«'' "^ '"^
the denominator. *' " ""* numerator and the latter
a co"^^rnt:L!"7hrt tr°'^;;'^^ f™°'^""'--^
As we have shewn tt at the ril!.f ^ " ^! " '''"°'^'' "^ ^ = !»•
expressed by the fracfon in wh ^h thtl""" " !." -"'hermay bo
thelatterthedenominator.lletlfr: ;ri" *'^b:rr°T'
which form a ratio are p«n»xi ,.= < Ii „ '^'^' "^ '■''° numbers
theANTEOEDLraudtheet'a;™^^^^ ""> ^I,'' "-"ber being called
148 Tf t. * "'«/^''™'J ""mber the Consequent, of the ratio.
n>iles in respect Tmaltur. bT "°'"^"'"'' "•^'**^ ^ "^"^^ """J ^^
days; and iUs clear" rrdt^ v^, L™1L"'"'"^ ' '''^^ "'* ^^
-- number^tXIb^s^Z^llr^Vr IX';^^^^
by the fraction ,V Since Si. reduced to the fraction o' 12. = J. it
rufr/.tll'/iri!"'- '-concrete numbers Of the .an.^^^
reduce them'7;;;e"rrthT' "\"™' '° "''" '" ^"^ *'-'' ^t";
them as ablLt 1™::!^' '°^' denomination, and ma, then treat
m
■f
168
AEITHMEno.
149. When two Ratios are eqnal, in other words, when they can
be expressed by the same fraction, they are said to form a Peopoktion,
and the four numbers are caUed Pbopoetionals. Thus the ratio of
8 to 9 is equal to that of 24 to 27, for 8 : 9 = — , and 24; 27 = — = —.
The Ratios being equal, Proportion exists among the numbers 8,^9,
-<S4, 27; and those numbers are Proportionals.
150. The existence of Proportion between the numbers 8, 9 24 27
s denoted thus, 8 : 9 = 24 : 27, or 8 : 9 :: 24 : 27, which is usually read
thus 8 13 to 9 as 24 is to 27. j *«««
161. It has been stated that proportion is the equality of two ratios,
and we have explained that the two numbers constituting a ratio must
either be both abstract, or (if concrete) both of the same kind. In a
proportion if one of the ratios be formed by two abstract numbers, the
other may arisp from two concrete numbers. For it has been explained
(ArL 148) that if a ratio consist of two concrete numbers, we may
reduce them both to the same denomination, and then treat the result-
ing numbers as abstract, the ratio of those abstract numbers being the
same as that of the two concrete numbers from which they have arisen
ior the same reason, one of the two ratios constituting a proportion
may be formed from concrete numbers of one kind, while the other
IS formed from concrete numbers of a different kind; for 7 days • 13
days :; 7 miles : 13 miles, each ratio being in fact that of 7 to 13. Indeed
1 appears by (Art. 148) that the ratio of two concrete numbers may
always be expressed by a ratio of two abstract numbers. If both or
either of the ratios in a proportion be formed from concrete numbers
we may thus replace each such ratio by one arising from abstract
numbers, and in this way every term of the proportion will become
an abstract number ; so that, notwithstanding the remark in note
(Art. 23), any one of the terms may then be multiplied or divided by
any other. "^
152. In any Proportion, as 8 : 9 :: 24 : 27, the product of the 1st and
4th, t. e. the extreme terms = the praduH of the 2nd and 3rd, i. e. the
mean terms;
24
27
9
X y X
27=
2-1
27
X 9 X 27, or 8 X 27 = 24 X 9.
EUIJ! OF THEFI.
169
instance, 8, 9, ai^arTo pTo;!!^!'''^" " ''' """■'"•^ '^-- ^"^
.-.27: 24:: 9; 8.
For since in anj Proportion
1st term x 4th term = 2nd term x 3rd term •
•••^«**--^ = ~^,2ndterm.i!ili^'
f^^' 3rd »
3rd term = l!^, 4th term = ^J}^^^
Ex 1. Find the 4th term in the proportion 2, 3^18. *,
2 : 3 :: 18 : 4th term,; ...4th term = l^ = 2r!
Ex. 2. Find the 2nd term in the proportion 8^ 82, and 24.
8 : 2nd term :: 32 : 24 ; ... 2nd term = 1^-8
82 ~"*
Ex. XLVIir.
Eind the 4th term in each nf i^hc ^^n •
fl) 18 : 4« . 1 « *•" f!"°""S proportion, :
(2)
t- *
•• JO .
0) 18: 48:: 16:
Kn, /I ';*•■=*= W) 1-2: 8-6:: 1.3:
I-md the 2od to™ in eacB of the fonowi„gp.„po,„„„,,
W.10::4J:15. (8) : If, :: 294 : -072.
EtJLE or THREE.
the sa.e ratio to the thM r rrndt ^JT!^'-'!^--' "l"
170
ABITHMETIO.
156. BuLE. Find out of the three quantities whioh are given, that
whiph is of the same kind as the fourth or required quantity ; or that
which is distinguished from the other terms by the nature of the ques-
tion : place this quantity as the third term of the proportion.
Now consider whether, from the nature of the question, the fourth
term will be greater or less than the third ; if greater, then put the
larger of the other two quantities in the second term, and the smaller
in the first term ; but if less, put the larger iu the first term and the
smaller in the second term.
Then multiply the second and third terms together, and divide
by the first, treating all three as abstract numbers. The quotient will
be the answer to the question, in the denomination to which the third
term was reduced.
Note 1. The first and second terms must be brought to one and
the same denominations.
Note 2. Although we have said in the Rule, multiply the second
and third terms together, and then divide their product by the first; it
will be found in most cases advisable not to perform the actual multi-
plication until we have discovered, by putting the expression in the
form of a fraction, whether there be any factor or factors common to
the numerator and denominator, and if so, have rejected such factor or
factors.
157. It may be proper to observe that the Rule of Hiree isap-
plioable in two different kinds of cases, according to which it is called
the Kule of Three Direct or the Rule of Three Inverse. The method
just stuted (Art. 156) is applicable to both kinds of cases.
The Rule of Three Direct is that in which more requires more, or
less requires less ; or. in other words, in which a greater number re-
quires a greater answer, or a less number a less answer. Thus in the
question, " If 4 acres of land cost $250, find the cost of 15 acres, at
the same rate." The 15 acres being more than the 4 acres, will require
a larger sum than $250 for their purchase, and so, in this case, more
requires more. Again in the question, " If 15 acres of land cost $937.50.
find the cost of 4 acres, at the same rate," the 4 acres being less
than the 15 acres, will require a less sum than $937.50 for their pur-
chase, and therefore, in this case- less renuires less. Such
to the Rule of Three Direct.
I i i
BULB OF THBEB.
171
er, and dividd
gbt to one and
The Rule of Three Inverse is that fai which more requires less, or
less requires more: or, in other words, in which a greater number
requires a less answer, or a less number a greater answer. Thus in the
question, " If 4 men can mow a certain meadow in 3 days, find the time
in wliich 6 men ought to mow it," the six men being more than the
four, sliould perform the work in less time, ^nd so, in this case, more
requires less. Again, in the question, "K 6 men can mow a oertaift
meadow in 2 days, find the time in which 4 men ought to mow it," the
4 men, being fewer tl.an the 6, will require a longer time for perform-
ing the work, and therefore, in this case, less requires more. Such
cases belong to the Rule of Three Inverse.
Ex 1. Find the value of 87 yards of silk, when 25 yards cost $«0.
There are here three given quantities, 25 yards, 37 yards, and $50,
and we have to find a fourth which will be the price of 37 yards. It is
manifest that the three given quantit 25 yards, 37 yards, $50, and
the required sum, must form a proportiuQ, because the 25 yards must
have the same relation in respect of magnitude to the 87 yards, which
the $50 (cost of 25 yards) has to' the required sum (cost of 37 yards).
Proceeding tlien by Rule (Art 156) we observe that $50 is of the same
kind as the required term, viz. money ; we make that the third term of
the proportion ; and since the required sum (cost of 87 yards) must
necessarily be greater than $50 (cost of 25 yards), we make 87 the
second term, and 26 the first. We have thus the first three terma
arranged as follows :
25 yds. : 37 yds. :: $50.
And the entire proportion will be as follows:
25 yds. :: 37 yds. :: $50 : required cost.'
The first and second terms are in one and the same denomination, and
require no reduction. And by previous reasoning we must now treat
the numbers as abstract, therefore
cost required=$^l^=$74.
Meaaonfor the above process.
We have-the cost of 25 yards given, viz. $50, in order to ea&hlA ni
find the cost " "
to
9
37 yards.
172
ABireMBTio.
tV^TJi. """'''' """ '" ""' ""'' '* "'^'^ '». ^"""^ ir /aril
do t ; ^ • , ''* '*''"'' number which indioatee bow man;
loU.v.U« required sum contains to the abstract number 60 al m^^
(rfthe fonner nu W ^ ^^^ ^^ ^^^__ .^^_^ »««4 be e'pCed b^
the fraction "^"'""^ ""'"'«■ ■ ^
JO
nianrar, bo expressed by the fraction ~.
. reqnired number 87
I ' ' 60 ~26 '
^ reqnired number 87"
or
reqnired nnmber x 50 37 x 50
60
25
or required number =?!^^, ^Art. 103),
60x37
**' =-^r--
J m result shews that if we arrange the three given terms 25 v«r^o
87 yards, and $50 in the following mamier, ' ^*'^''
yds. yds.
25 : 37 :: $60,
~.a then consider the numbers to be abstract, as if they had been writ-
25 : 37 :: 50,
ten we shall obtain the abstract number which will «}..«, « i.
> flame relation in
26 yards ; that ia,
» that of 87 yards
lired sum to $50,
ioates how many
aber 60, an< may
be expressed by
19 that of the ab-
therefore, in like
RULE OF THREL.
>3),
erms, 25 yards,
had been writ-
' ns how many
he second nnd
rai-. • n-rtA 4.1 ■
-^'■j UXIU itii\iJS
173
by treating this nnmber as concrete, that is, as so many dollars we
have the required answer in dollars. '
thus?^ '^'"'^'''^ ^' '''''''^^^ '''^^''^^ ^"^ "^""'^ ^^'^^ ^^^"^P^^ ^^^<^^^ntiy,
26 yds. cost $50. • .
.'. 1 yd. costs $2g or $2 ;
.-. 87 yds. cost $(87 x 2) = $74.
Ex. 2. Tf a workman earn £17. 6«. in 102^ days, Low long will he
be m earning 50 guineas ? » ^ > g wiu ne
Hero the required quantity is time, and as the given quantity of that
kind IS 1021- days, we must place that as the third term in the propor-
ern'ln Jnf TJ"."^ '' '' ^"^''''' ^^" '^^^'^^ ^ ^^"«^^ *5"^« th«« the
earnmg of £17.6..: we must therefore place the 50 guineas as the
second term, and the £17. 6«. as the first.
Therefore the proportion is
£17. 6<<. : 50^. :: 102^ days : required number of days,
UU : 1050a ;: 205 half-days : required number of half-days;
.-. required number of half-davs = ^^''^ ' ^^^ - 622 ^ «
.-. required number of days = 622^\ -^ 2 = 811 m .
Indepen nt method. *
206
A man earns £17. 65 or 846*. in 102^ days, . .• ?55 days ;
•* , 1«. in
1050*. in
205
2 X 846
205
<3ay8;
xl060
* 2 X 846
= 811gi^days.
$4lS'20 ? ^^ *^' ^''^ *''' ^^^^ ^' ^^^' ^^' ""^^^ ^^" "" *^^ *^^ ^^
^ The $69. 12 being of the same nature with the sum reauired. must
.-. placed as tue third terra in the proportion;, and as the required tax
174
■i!
ARITHMETIC.
«»86:, 4195.20:: ,69.12: the re,„irc,Uar,
.-. the required tax = |^1 95-20 v 69-12
936
7^, _ ^ = $809.80 nearly
independent method, ^
The tax on $930 is $69.12
$1....$
69«13 -ge
930- ""' ^l8
•96
I ; I
•'• ^196.20.. $^^,4195.2
= $309.80 nearly.
6« "ufeuc i to be carried for $38 61 ?
111.88: $38 01:: 198 n,.:requi,.e.idisranco.
.*. Required di8t"nce=??51iii^^;, _3801
1188 "''^- — Q- n»it'8 (cancelling by 198)
643^ raiies = 648^ miles.
than th.; ter::\;;:, :rot";rtt 1""^' *" '^^' •'■«"' -»-
distance? ^ '^'^''- ^^^' 13 lbs. carried the same
Required cost = $7l.3o.
contains as much cloth as t,,lCr, " '' ^"'"^ '""«- ^"^ ^"ch
BULE OF THREE.
$4195. 20 as t^«
175
tax.
.88, how far at
?
0.
illiug by 198)
3t sight more
'83 in certain
: rae |16.90,
ed the same
on, since the
)st; whence
I J-nrds in
and which
As the length of the sooond piece is less than that of the first its
breadth .nust necessarily be greater, in order that the content ma; bo
breadth and so the example belongs to Rule of Three Inverse
Wo have the breadth of the second piece to find. That of 'the first
-i.eco s ,1 yard : place this therefore as the third term. Now the re
.mred breadth is to be greater than this. Therefore pface tLloyar^^^^^
%s the second term, and 12 yards as the first.
12 yds. : 20 yds. :: f yd. : required breadth in yds.
• .-. Required breadth =?^yd8. = « yds. = IJ yds.
K. J""' "^V f ^ ^^ T"" '^° '^^^ ^ ^'^^ ^° * ^^^«' ^^ ^hat time can the •
Bame work be performed by 82 men ?
,n«n ^ " f '''.^^^"* ^^ "'''' '^° P''^"^"^ ^^« ^^'•k ^° a less time than 12
taen, and so the t.me required will be less than 4 days, the third term
X'rZfirsT^ "-''' "- ^^- ^'^ '^ - ^^« -nd t:;^
82 : 12 ;: 4 days : required time in days.
Required time =^^days = ^ days = f days = 11 days.
Independen t meth od.
12 men can r^ap the field in 4 days ;
•'• ^™an in48daysS
•••S2men i^ | days,
=1^ days.
In^f^""* \ Jo^^ ^""^ *^^ P"'^ ""^ ^^'^* P^^ l'"*!^^! ^hen the penny
Here are two numbers, viz. 1 bushel and 1 penny, which can erl-
dontly have no effect on the answer, for if any other measure hrb^n
named .n place of the bushel, and any other loaf in place of the penny
loaf, the answer would be the same.
.. _.„,.,„„ i„ ucaici", or as ine price is more, the weiffht of anv
given lorf ia le.,, and conversely, as the weight of a given I^iJie^^
I
III,
I!
i I
f
i ■ !
If
;!
176
ARITHMETIO.
( i
t«™ of the proportion. TkeZZtl °" '''"' """'* "^ '■>" '"-^
and the 8 oz. the first. ^" '" "'■ """' >>« '!»« second term,
8oz. : 10 02. :; I2j».
Required price=^^, _^0 x 3
8 ^«. = 15*.
b.th:Ru,e!rrT,t' """ '" •"" ^""""■''■«- -■• -% worted o„t
• For every lasSrf;,"'?!^*^''''*' ''"""''' "'<=<>''ie?
income-tax r"^ ^^ "'"* ^^ """^ '^^». '- '-ad £l. before he p.id ,,3
I ••• 19s. 5^. : £248. 10s. 8d. :: £1 . „„,„•„ 1 •
-''-e, rcuired i„eomo-:S:'""^°"^'
at starting, J^TZ!:! 5 STh^d ^'^ ^^"/^^"^ "o^^ ^™
had the hare gone when she was o»lt hvt t'* '-"n 7 yards : how far
For every 5 yards the harT„Ttl I f^^''°™<l?
he Has gained 130 yards he will ha™ ca^gtu.eT"' ' ^''"*' "'"' '''""'
•*. 2 yds. ; ISO yds •• f^ v^o ^. , '
mneshyrailwayriSifa,":,'^^^^- f;2' "■""' .^""^ »*" ^^^
eaves £ for ^ ; the train, meetat the efl / A ■"' ""' "'»' " "•«!■>
to 5 having travelled 16 mile 1 L ""'''' "'^ ""'" '™>» ^
mary miles did each travel™ ho^r? '"°'' """^ *^ °"'^'- How
by one train, and therefore 54 - 19 V^V T"'' *'^^^"^^^ P^^' ^""^
t^e other. * ^9, or 35 = ,„iies travelled per hour by
Ex. 13. A clock, which is 4 min ft " „ ,
- o. X„.day, loses . mia. ^ ^:^^ ^ - '.a.^^ne
EULE OF THBEE
ired must clearly
lust be the third
the second term,
177
5ily worked out
>f Id. in the £,
ne?
fore he paid his
come
ii'ds before him
ai-ds: how far
id?
rds, and when
yards ;
■ch other 324
e that a train
train from A
other. Howr
en the trains
hem.
aios in 1 hr.,
= 54;
'd per hour
pcT hour by
If-past nine
>i/» 4.x •
uo ciujo m-
dicated l«r the clock at a quartei-past five p.m. on the following
[Friday? 6
From ^ A.M. on Tuesday, till 5i p.m. on Friday, there are m
hours. '' *
.-. 24 hrs. : 79^ hours ;: 2'.45" : time lost by clocJ:.
whence, timeiost -by c^ ?,k = 9'.8^" ;
.-. time by the clock at S^^Np.m. on Friday
= 4'.8^"+5 hrs. 15'- 9'.8^ ' = 5 hrs. lOnm.
Ex. XLIX.
(1) If 4 yards of cloth cost $2.88, what will 96 yards of the same'
Cloth cost?
for ^^ ^^ ^ ^^^^^ ""^ ''^''^^ ''''^* ^^^'^^* ^^"^ ""^^ ^^^^^ °^ ^® "tio-^^t
(3) If 7 bushels of wheat be worth $8.82 what wUl be the value
of 3 bushels of the same quality ?
(4) The rent of 42 acres of land is $68, how many acres of the'
same quality of land ought to be rented for $273.
(6) If the cost of 72 tons of coals be $432. what will be the cost
of 64 tons ?
(6) How long will a person be saving $14.4D. if he put by 30 cents
per week ?
(7) Find a number.which shall bear the same ratio to 9 which 20
does to 15.
(8) If 2 cwt., 3 qrs., 14 lbs. of sugar cost $28.90, what quantity
of the same quality of sugar can be bought for $142.80 ?
(9) If 3 cwt., 3 qrs, cost $33.75. what will be the price of 2 cwt
2 qrs. ? *»
(10) Find the value of 23 yds., 1 ft. of cloth, supposing 4 yds., 31 in.
of the same quality to cost $18.
(11) What will be the income-tax at 1^ cents in the dollar, on
$267.50? '
(12) What is the tax upon $1450.46, when $2061.18 is rated at $3 24?
(13) If one bushel of malt cost $1.40, how much can I buy for
$129.60? ^
(14) Find the price of 2 tons, 3 cwt., 14 lbs. at $2.11 per quarter, i
178
AKlTiniETIC.
M I
! I
ii f
li i!i;
(15) ^ pays half yearly an income-tax of £10 Is Sd • £n/i y.i.
income, the tax being U. in the £. '. ' ^""^ ^'^
% ^^9 ^»d th^ ^'"ount of a servant's wages for 215 4s at 55 cents
' t860^^fiO ^""^"T;^ t^'' ^^""^* *" ^^^3-^^' «°<1 his assets to
f860.16 ; how much in the dollar can ho pay?
to ^dsl^flT^r' '''' ''! r^'^ '■" '''' '^"^^'^^^^ hi« -««^ts amount
to 5P4560 ; find tho^amount of his debts.
fn. ir^ ^^ "" ^r' containing 400 ac, 2 ro, 20 po. be let at $1201 87^
for the year, what is the rent per acre ? *i-wi.»7j
(20) Find a fourth proportional t., the numbers 3, 3-75, and 40.
Ji5Un i:z "^ ""^ ' '-'' ^^ '' ''^'^ ''^ ^- --^ ^^^«
waS milLr ^"^' '' ^'^^ ^^ ^ ^^^^^' ^^^ ^-- -°^ ^^^« -i" he
ch.fiV^^'oT.r'' V^' "'''''^ ^^ "^"^^ ^ y^^^ ^^«t bo given in ex
change for 936^ yards worth $4.35 per yard ?
for sixf : fmr; ' '^''" '' -''■'•• " "'^- "^ -«- »* ^» -*»
S2 2 ^^ ''''• ^'''''''- "'^'^ ^^ •^^' ^hat will 30^ lbs. cost ?
(28) How many men must be employed to finish a piece of work in
15 days, which 5 men can do in 24 days ? .
renfo'f ai'rr* ' "' ''* ^"^ "' """"" ^' ^«»^- '''■ '<>'■' -'>'" -«>« '
mtl" '^''™-' '^'''- ""'' ^"'- ^'' 'i'-' ^'"" '» t"^ price of
(32) ^ borrowed of S 400 dollars for 6i months afterwards .<
w„„,d reguite B's kindness by ien.ing him ^0: how Mng Zm hi
RULE OF THREE
179
■ Bd. ; £nd his
lys at 55 centa
his assets to
assets amount
at $1201.871-
5, and 40.
w many days
' days will he
i given in ex-
hat sum will
' at 20 cents
a journey in
journey if he
?
3 of work in
, what is the
he price of
40 yards of
ierwards A
f should he
will be its
(34) If the price of 1 lb. of sugar be $0.0626, what is the value of
•75 of a owt. ?
p, (35) If 3^ shares in a mine cost $54, what wUl 28f shares coat?
' (36) If S^ yards of cloth cost £12. 7«. Hid., how many yards can
be bought for £3. 19s. Of ^. ?
(37) Find the rent at $7.20 an acre of a rectangular field whoso
Bides are respectively 50 chains 40 links, and 56 chains 25 links.
(38) In what time will 25 men do a piece of work which 12 men
can do in 3 days ?
I (39) If -3 of 4-5 cwt. cost $11.55, what is the price per lb. ?
' (40) A piece of gold at £3. 175. lO^d. per oz. is worth £150 ; what
will be the worth of a piece of silver of equal weight at 64«. 6d
per lb. ?
(41) If a piece of building land 375 ft. 6 in. by 75 ft. 6 in. cost
$566.40, what wiU be the price of a piece of similar land 278 ft 9 in
by 151 feet.
(42) A servant enters on a situation at 12 o'clock at noon on Jan. 1,
1870, at a yearly salary of $224, he leaves at noon on the 27th of May
following ; what ought he to receive for his services ?
(48) A was owner of a j\ of a vessel, and sold ^ off of his share for
$1600 ; what was the value of ^r of f of the vessel ?
(44) A exchanged with B 60 yards of silk worth $1.68 a yard for
48 yards of velvet ; what was the price of the velvet a yard ?
(45) A person, after paying 3 cents in the $ for income-tax on his
income, has $7838.12 remaining ; what had he at first?
(46) A watch is 10 minutes too fast at 12 o'clock (noon) on Mon-
day, and it gains 3'. 10" a d .y ; what v ill be the time by the watch at
i^ quarter past 10 o'clock a.m. on the following Saturday?
(47) The circumference of a circle is to its diameter as 3-1416 : 1 ;
find (in feet and inches) the circumference of a circle whose diameter is
22^ feet.
(48) If the carriflge of 3 cwt. cost $2.40 for 40 miles, how much
ought tt> bo carried for the same price for 25| miles?
(49) If I spend 20 dollars in a fortnight, whnt must my income b«
that I may lay by $200 dollnrs in the year 1855 ?
(50) TIte hoUSe-tftX nnrm n >iAnqa ••o+n-l
Hd. ; what will be the tax upon one rated at £120 ?
]m
n4- ttrtl • ._ ttJt ^h.
m
III
Hi';
I i V
i i
Mir.
180
ARITHMETia
•on^fP A Silver tankard, -hich weighs lib., 10 oz., lOdwt. cogt
. f 29.70 ; what is the value of the silver per ounce ?
(52) A man, working ^ hours a day does a piece of work in 9 daVg •
how many hours a day must he work to finish it in 4| days ?
(63) If a pound of silver costs $1C.84, what is the price of a salvei-
which weighs 7 lbs., 7 oz., 10 dwt, subject to a duty of 36 cts. per
ounce, and an additional charge of U cts. per ounce for the workman-
^ (54) How much did a person spend in 63 days, who with an annual
income of $3925 is 90 dollars in debt at the end of a year ?
• if^i ^^^^^^^^'^^^^^^^'^^^l 9 boys, can complete a piece of work
ID 60 days, what time would 9 men, 15 women, and 18 boys take to do
lour times as much, the parts done by each in the same time being a>
the numbers, 3, 2, and 1 ? ^
(56) A person possesses $800 a year ; how much may he spend per
day in order to save $48.25 after paying a tax of $5 on every $100 of
income?
^ (67) If 3 cows or 7 horses can eat the produce of a field in 29 days
in how many days will 7 cows and 3 horses eat it up ?
(68) How many yards of carpet f yard wide will cover a room
whose width is 16 feet, and length 27^ feet ?
(59) A person buys 100 eggs at the rate of 2 a penny, and 100 more
at the rate of 3 a penny : what does he gain or lose by selling them at
the rate of 6 for 2d J j & ^>
(60) A church-clock is set at 12 o'clock on Saturday night- at
noon on Tuesday it is 3 minutes too fast: supposing its rate regular,
what will be the true time whfen the clock strikes four on Thursdav
afternoon ? "^
(61) A person after paying a poor's rate of 4 cents in the dollar has
$7200 remaining ; what had he at first ?
(62) If a piece of work can be done in 50 days by 35 men working
af; it together, and if, after working together for 12 davs, 16 of the men
were to leave the work; find the number of days in which the remain-
mg men couM finish the work.
(63) A regiment of 1000 men are to have new coats; each coat is
to contam 2^ yards of cloth U yards wide ; md it is to be lined with
shalloon of f yard wide; how many yards of shalloon will be re-
quired ?
RULE OF THREE.
isr
, lOdwt. cost
rorkia 9 days;
ays?
ice of a salvei"
of 36 cts. per
the workman-.
with an annual
?
piece of work
3ys take to do
time being as
lie spend per
Bvery $100 of
Id in 29 days,
5over a room
and 100 more
lling them at
ty night; at
rate regular,
on Thursday
he dollar has
•
nen working
6 of the men
tho remain-
each coat is
e lined with
will be re-
(64) IfSonnces of silk can be spun into a thread two furlongs"
and a half long, what weight of silk would supply a thread sufficient to
reach to the Moon, a distance of 240,000 miles ?
(65) How man: revolutions will i carriage-wheel, whose diameter
IS 3 feet, make in 4 miles? (See Ex. 47.)
(QG) If 8 oz. of sugar 1^^ worth $0.0525, whatis the value of -75 ofaton?
(67) The price of -0625 Ib.s. of tea is -4583^. ; what quantity can be
bought fcr £61. 12».?
(68) Two watches, one of which gains as much as the other loses,'
viz. 2'. 5" daily, are set right at 9 o'clock a.m. on Monday; when will
there be a difference of one hour in the times denoted by them ?
(63) How many, yards of matting, 2-5 feet broad, will cover a room
9 yards long, and 20 feet broad ?
(70) A person bought 1008 gallons of spirits for $3072 ; 48 gallons
leaked out: at what rate must he sell the remainder per gaUon so as
not to lose by his bargain ?
^ (71) If a soldier be allowed 12 lbs. of bread in 8 days, how much
will serve a regiment of 850 men for the year 1856 ?
(7*^) If 2000 men have provisions for 95 days, and if after 15 days
400 men go away ; find how long the remaining provisions will serve
the number left.
(73) A gentleman has 10000 acres ; what is his yearly rental, if his
weekly rental for 20 square poles be 3 cents? (1 year = 52 weeks!)
(74) If an ounce of gold be worth £4-189583, what is the value of
•86822916 lbs. ?
(75) If 1000 men have provisions for 85 days, and if after 17 days
150 of the men go away ; find how long the remaining provisions will
serve the number left
(76) What is the quarter's rent of 182-3 acres of land, at £4-65 per
acre for a year ?
(77) A grocer bought 2 tons, 3 cwt., 3 qrs. of gouds for .$576, and
paid m for expenses ; what must he sell the goods at per cwt. in order
* J clear $294 on the outlay? (cwt. = 112 lbs.)
(78) What must be the breadth of a piece of ground whose length
is 40 ,> yards, in order that it may be twice as great as another piece of
ground whose length is 14f yards, and whose breadth is 13^^ yards?
(79) If 3-75 yards of cloth cost ^3-825. x^h&t will .q« ^a.
3 nails cost 3
a/-
¥ H
'i»'ii
I,
fi'.-l!;
1182
ARITHMETIO.
tain field; andrcowseatas much .s 9 horses ; what must be the ize
9 cows 1 ' '' ''' '''""' "'^^' "^^^ «"Pi>-* 1« ^-- -d
(81) ^ alone can rea:> a fleM in 5 days, and B in 6 days, working
DOUBLE RULE OF THREE.
n I li
f '
n''ii
158 Thero are many quostiom, which are of Ihe same nature with
those belonging to the Rule of Three, but which if worke, 1.1
means of tl^at Rule as before gi™„, would requ tw^ Ir mo 'e d" tin'^
aw, , cations of it. Every such question, in fact, may be Zsiier^tl
and when each of those questions has been worked out by means of
the x^.le the answer obtained for the last of them will be'^heanswer
to the origmal question, 'luswer
159 The following example may serve to Ulustrate the nrecedin,,
observations. " If the carrii™ rf m ,.,.,* e -,t '""'^^ v^^oamg
„i * , , ., carnage ot 15 cwt. for 17 miles cost me *20 40
what would the carriage of 21 cwt. for 16 mUes cost me? " '
The above question may be resolved into the following two
The first question may be this : " If the carriage of 15 cwt for 17
miles cost me $20,40, what would the carriage of Si cwt f!r '/° 11
cost me r- In this question the 17 miles would hav ol^ffect ton
he answer because the distance is the same in uoth parta ofthe oues
Uon and tl,e answer would clearly remain unaltered if at ote^
number of miles, or if the words " a certain distance " L^ T I
instead of the 17 miles. This number maythl?!: b ne Wen
rtrrrrh'rtot"otd''s:i"™r''~'^"-— ^^
Three, we find t Jthran^er wiflt^Lt: '""'"' "' "'^ ^"'^ »•'
The second question may be this: " If the carriage of 21 c«t for
osrm:;"%::f ^-^^^ ^'^'r '- "^""'=- "^ ^^ "" ^"-"-
th 27cw. wi, b7""°"'^ '■^'"°"' ^'""'"^ '0 'to'^ before given,
^ , „^ „^„ j,^^ uuiswer 10 be $26.88.
DOUBLE RtTLE OF THREE
183
?i'as8 on a oer-
List be the size
18 horses and
lays, working
P it together,
e nature with
►rked out by
more distinct
considered to
ule of Three,
by means of
the answer
le preceding
t me $20.40,
two.
> cwt. for 17
for 17 miles
effect upon
of the ques-
f flny other
1 been used
leglec'-ed as
remaining,
the Rule of
21 c^vt. for
fc^r 16 miles
ifore giveDj
le Rule of
From thenjonnection of the two questions with that originally pro-
posed, we observe that $26.88, thus obtained through two distinct
applications of the Rule of Three, must be the auswer to the origmal
question.
160. The DoTTBLB Rule of Theek is a shorter method of working
out such questions as would require two or more applications of the
Rule of Three ; and it is sometimes called th6 Rule of Five, from the
circumstance, that in the practical questions to which it is applied
there are commonly five quantities given to find a sixth.
161. For the sake of convenience, we may divide each question
mto two parts, the supposition, and the demand; the former being the
part which expresses the conditions of the question, and the latter the
part which mentions the thing demanded or sought. In the question
"If the carriage of 15 cwt. for 17 miles cost me $20.40, ^hat would
the carriage of 21 cwt. for 16 miles cost me ? " the words " if the
carriage of 15 cwt, for 17 miles cost $20.40," form the supposition;
and the words, "what would the carriage of 21 cwt. for 16 miles
cost me ? " form the demand. Adopting this distinction we may give the
following rule for working out examples m the Double Rule of Three.
162. Rule. Take from the supposition that quantity which corre-
•eponds to the quantity sought in the demand ; and write it down as a
third term. Then take one of the other quantities in the supposition'
and the corresponding quantity in the demand, and consider them with
reference to the third term only (regardmg each other quantity in the
supposition and ita corresponding quantity in the demand as being
equal to each other) ; when the two quantities are so considered, if
from tlie nature of the case, the fourth term would be greater than the
third, then, as in the Rule of Three, put the larger of the two quantities
in the second term, and the smaller in the first term ; but if less, put
the smaller in the second term, and the larger in the first term.
Again, take another of the quantities given in the supposition, and
the corresponding quantity in the demand ; and retaining the sj»ne third
teru), proceed in the same way to make one of those quantities a first
term and the other a second term.
If there be other quantities in the supposition and demand, proceed
m like manner with them. *
In each of these statings reduce the first and the second terms to
Wm
n\r.
f**
w
l;
fi<!
184
ARITHMETIC.
may then bo treated as abstract numbers ^'"' ""•°"
.ocona'tet:"t:!;:thr tr: ':T' '-;■ -'="""'"' 'o™' -^ -" '"«
third term In a « , '""""' ""■'"' '"<' '•'-''»in «'« f"™er
note".::^?r:Sterit;„td":i^!~ »"-- ^--r-^.
ho J'fong wm UtatTim "^ " """.'"" "' *'""» sain $50 in 3 months,
w long will It take l„m with a capital of $3000 to gain $:r5 ?
«8000 as a tir t tZ Tl toon™' "'°,*'*'^ ' "^^^'"'"^ ^™ P'»««
- Of $50 from thf;; oiui L" t^.tTf":! m tT;,'"?^ ": n
considering them in li'ke manner tit^ ef L^L toTe t "'"^'^r^
term, we see that if the amount of gn n be no eied tb T '" "" '"?
a given capital would produce it mu,t .'""''''=''''' *''« '™° '" ^i'^h
the fourth termwoul be g o 'e^ tan thTrf ^f' f" ""* ""«
place the $60 as a first term and the *7w ' T^ "''''"'"™ *«
have the following stateVne^is *"' "' " ""™" '"™ = *'>' ^«
$3000 : $3000 )
$50: $173 \ ■■•■Bm.
Proceeding according to our E„,o wehave the following statement:
^ 3000 X 50 : 2000 x 173 :: 3,
and the required number of months r ^^^^ x 1T5 x 3
T>,^ r.^ ' ^ 2000 X 50
Ihe required answer is therefore 7 months.
Reason for the above process.
The tradesman, with a capital of $2000 ..:.. c^n. :,
= r.
3 months.
DOUBLE RULE OF THREE. ' lg5
msllTfT'' '^ *'' ^1° °t ■"'™^' ^°" '""^ '"' ^°«l'l "e in gaining
fl/6 With the «aw0 capital. Thus , b« "mg
$50 : $175 :: 3 m. : required time.
Required time = (~^] months. „ '
frnttr *^' *'^^''"^^° '"^^^ ^ ^^Pital of $2000 would gain $176
'° \~W~) ""^"^^^ ^^^ "^ ^e^t find, bj the Rule of Three, how long it
must have the answer to the original question. Thus ' ^ ^®
$3000 : $2000 ::^ll months : required time.
. Required time in months =.- (~— x 200o)-f. 3000
_175 X 3 X 2000
50~x 3000
_2000 X 3 X 175
3000 X 50 '
whence it appears that if we arrange the quantities given by the
question as follows: ^ ^ ®
$3000: $2000) ^
$50 : $175 f •= ^^»
and treat the numbers as abstract ; and then multiply the two first
terms together for a sinple first term, and the two second terms
together for a single second term ; and thei- divide the product of the
second and third terms by the first, we shall obtain the answer in that
denommation to which the third term was reduced.
Independent method of worUng above example.
A capital of $2000 gains $50 in 3 months,
$1 .... $50 in (3 X 2000) months,
*1 *. . /3 X 2000\
*1 • $1 in f ^ — ~ — j months,
(fcOAAA *■< . / 3 X 2000\
>.......... .povuv? -fi mi—- 1 months
V50 X 3000/ "'^°^"S»
* *f til
1 S 1
if
n\
. il
^°^ AEITHMEI'IO.
A capital of I80UO gains 1176 in (^-^^^-9 '^ l^^Vn, .ft,
^ »*'"in ^ 50x8u00 ) ^"^^^H
/2000 X 175 X 8\
that is, if we arrange the given quantities .as follows,
$3000: $2000)
$50: $176 j " ' ■
we obtain the required time in months by multiplying the two fir«f
terms together for a final first term, the two second CL tetTer fo
a final second term; and then dividing the product of The secon^^^^^^^
thud terms by the first term. ^
r da^s'fo; ifsl'""^ '^ '^'* '' '''' '^^ ^''^ ^- -^ -" ^^ ^ept
r days • 20 davs ) "^7 ^'''''' ''^ ^^ ^^P* ^^^ * ^*^^
il4 .'jftpo [ -^ 7 horses ; ^^'^ ^/^^^^y for 7 days than for 20 days,
* ^'^^ ^ «^d "^«re horses can be kept for a given
number of days for $28 than for $14.
the required ni ^ber of horses = ^^ '^^ " ..'^
7 X 14
= 40.
Ex. 3 If 20 men can perform a piece of work in 12 davs find fl,.
The first piece of work beinfj reckonArl n« i ^t.«
reckoned as 3. ^ reckoned as 1, the second must be
1 ■ 8 V
V- days:' 12 days]" ^^°^^°- ,
.-. req- number of men = !i^il^ ^ ^^^^
Independent method.
In 12 days work is done by 20 men
. -'-J" J^^y (20xl2)mGn,
• . in 1 .... 3 times work.. (20 x 12 x3) men,
.•.rnJjf-days 20x12x3
or "800 men.
DOUBLE iJULE OF THREE.
187
montbs,
nonthd:
the two first
s together for
le second and
' will be kept
t for a given
I] for 20 days,
>t for a given
a for $14.
ays, find the
c 3 times as
nd must be
Ex. 4.#If 252 ...on can di- b trench 210 ynrd^ long, 8 wide, and 3
deep, in 5 days oi 1 hoars each ; i/i how many days of 9 hours each
will 22 ine-i dig a trench of 420 y \^. long, 5 ido, and 8 c'oep?
The first trench oimtains (210 x 8 x 2) cubi ; yds.
= 1200 cubic yds.
Thy ticoond (420 x 5 x 3) cubic yds.
= G300 cubic } ds.
«
On the supposition, therefore, that 252 men can remove 12G0 oubic
yds. of earth in 55 hours, we have to find in how many hours 22 men
can remove C300 cubic yds.
Then we have the following statements .
22 men : 252 men |
1260 cub. yds. : 0300 cub. yds. J
55 hrs.
,, .. 252 X 6300 X 55
. . req . time = — — ^ ^^^-^ — working hours
= 850 days of 9 kours each.
Ex. 6. If 660 flaj-stones, each 1 J feet square will pave a court-yard,
how many will be required for a yard twice the size, each flag-stone
being 14 in. by 9 in. ?
Superficial content of each of former flag-stones ' ,
=(1| X U) sq. ft. = (r] x'-3) sq. ft. = I sq. ft.
Superficial content of each of the latter flag-stones
=(tI '< A) sq. ft. = (I X f ) sq. ft. = I sq. ft.
Considering the first court-yard as 1, and therefore the escond as 2
our statements will be
^sq. ft.:^sq. ft. ) ^.^ „ ,
1-2 j *' ^^^-fl^g-stones,
which by our Eule, will give us the following single statement :
I : I X 2 :: 660,
.-. req"' number of flag-stones = (f x 2 x 560)^-J
= (i x560x 8) =
9 X 560 X 8
2 X 7
= 2880.
Ex. 6. A town which is defended by 1200 men, with provisions
enough to sustain them 42 days, supposing each man to receive 18 oz. a
day, obtains an increase of 200 men to its garrison ; what must now be
IMAGE EVALUATION
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Photographic
Sciences
Coiporation
23 WEST MAIN STREET
WEBSTER, N.Y. 14380
(716) 872-4503
<V
iV
^^
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6^
1^^ • AEITHMEnO.
the aUowance to each man, in order that the provisions may mtVQ the
whole garrison fbr 64 days? '
1400 men : 1200 men )
64 days ; 42 days j'-^^^**
.-. number of oz. req- = l?2?_!li2jcW _ 13
^ 1400x64 ~^^'
Ind^endent method,
1200 men for 42 days have each daily 18 oz. ;
•*• ^"*° (18x1200)02.,
•': ^ Idayhasdaily (18 .^ 1200 x 42) oz.,
••• 1 54 days has daily 18 x 1200 x 42 ^^
64 '*
..1400 men.. .. have each daily ?iJlJMj?:f? o^
^ 1400 X 64 »
' or 12 ozr
• Ex. 1.
\k ^' ^^'^,fr''^°o''*P^*^'^'^'''^^^°^'^'^°^^any men will reap
16 acres ift. 14 hours? *^
^ 2. It- 3 men earn $75 in 20 days, how many men will earn $78.76
m days, at the same rate? v •«
3. If 16 horses eat 96 bushels of corn in 42 days, in how many
dayL will 7 horses eat 66 bushels ?
4. If 800 soldiers consume 6 sacks of flour in 6 days, how many
will consume 15 sacks in 2 days ? ^
will «'>, ^^ ^^ ^»«^«1« be consumed by 6 horses in 13 days, what quantity
will 8 horses eat m 11 days, at the same rate ? -i J
6 16 horses can plough 1280 acres in 8 days, how many acres
will 12 horses plough in 6 days ? j «"» w
9Q J\^^^^ cwt can be carried 12 miles for $1.50, how far can 36 cwt.
23 lbs. be carried for $5.25 ? ♦
8 If the carriage of 8 cwt. of goods for 124 miles be $30.24, what
we>ght ought to be carried 53 miles for half the money ?
9. If 6 men on a tour of 11 months., spend $1540, how much at
the same rate would it cost a party of 7 men for 4 months?
in It, r ""''^n u^^^} ""^ ^^^^^ ^ tradesman gain $100 in 6 months,
in what time will he gain $49.50 with a capital of m5
DOUBLE PwULE OF TIIBEE.
189
Ls maj ngTTQ the
2) oz.,
-oz..
oz..
|r men will reap
ill earn $78.76
, in how many
lys, how many
I what quantity
)w many acres
far can 86 cwt.
Q $80.24, what
how mnch at
) in 6 months,
11. If it cost $84 to keep 8 horses for 7 months, what will it cost
to keep 2 horses for 11 months? '
12. The carriage of 4 cwt., 8 qra, for 160 miles costs $3.85 ; what
weight ought to be carried 100 miles for $30 ?
13. If 1 man can reap 345 f sq. yds. in an hour, how long will 7
I such men take to reap 6 acres ?
14. If 20 men in 3 weeks earn $900, in what time will 12 men
I earn $1500 ?
15. If the carriage of 1 cwt., 3 qrs., 21 lbs. for 52i miles come to
j 17*^. 5d., what will be charged for 2] tons for 46^ miles ? (cwt. = 112 lbs.)
16. If 10 men can reap a field of 7i acres in 3 days of 12 hours
leach, how long will it take 8 men to reap 9 acres, working 16 hours
I a day j '
17. If 25 men can do a piece of work in 24 days, working 8 hours
I a day, how many hours a day would 30 men havo to work in order to
j do the same piece of work in 16 days?
J 18. If the rent of a farm of 17 ac, 8 ro., 2po., be £39. 4«. 7<Z.,
what would be the rent of another farm, containing 26 ac, 2 ro., 23 po.,
it 6 acres of the former be worth 7 acres of the latter ?
1 9. If 1500 copies of a book of 1 1 sheets require 66 reams of paper,
! how much paper will be required for 5000 copies of a book of 25 sheets,
! of the same size as the former ? •
20. If 5 men cm reap a rectangular field whose length is 800 ft.
[ and breadth 700 ft. in 8i days of 14 hours each ; in how many days cf
12 hours each can 7 men reap a field whose length is 180aft. and
breadth 960 ft. ?
21. If a thousand men besieged in a town with provisions for 5
weeks, allowing each man 16 oz. a day, be reinforced with 500 men
more, and have their daily allowance reduced to 6§ oz. ; how long will
the provisions last them ?
22. If 20 masons build a wall 50 feet long, 2 feet thick, and 14
feet high, in 12 days of 7hrs. each, in how many days of lOhrs. each
will 60 masons build a wall 500 feet long, 4 thick, and 16 high?
23. If 10 men can perform a piece of work in 24 days, how many
men will perform another piece of work 7* times as great, in one-fifth
of the time ?
24. If 125 men cuii make an embankment 100 yards long, 20 feet
wide, and 4 feet lush, in 4" davs. wnriinw iq hf^^^r^a n /•«., v^^
190
ARITHMEno. ' .
W 8 ft °1 ' ^ ' •"• '''^^ "''- " W<«k of til" S S ft
%itZ "' • «f f •'• «-• -^^ep, weigh* 7600 lbs. J
26. If 100 men drink $96 wortb of wine At *i oa „. v .., ,
many men will drink $345.60 worth at iiao !!;k 17 ^".''' '""'
time, at the same rate of drinkir? * "*' '° """ »"">
When oL is 55 cent' TLZl l""' "^^ '' """^ "^ '^^' ">' *^»"«
ble of holdingT ' ''°" """^ ^«^' 0^ ™'«' « it c^Pa-
. J^^ i^i.""' * "*"'* '""^ "^"'^^ ^'SBIbs. when wheat is At ti ia
I '6^:1:^5 r"' "- '- '-'' '- *^^ ■^^- °^ <>-"« Xtt-it
aig'oLt Aow deep wm"; r^lYh"^"^' ,"""'"' ""'' *"•« '»
which cists $21 60 J ^ '"'"^'"""' '*"'«'"' ^ft- by «ft.
material, whose breadth is 3+ Ztviff "*'"' "^ ""' """«
2028 lbs.? * ^*'' *'""'^'«'« T* inches, and weight
86. If 12 oxen and 85 sheep eat 12 tons 12 ^nrt „fi,» • ^^
how much will it cost per month (ofmT' l, . f ^ '" ^ ^°'">
.beep, the price of hJZnT^^ a^, ^"''l *° ^""^ " «^«" """l 12
to eat as m„ch as r sheep? ' "'"' ' '"'"' '•""^ '"PP"^
86. If 1 man and 2 women do a piece oYwork in 10 days, find in
SIMPLE INTEREST.
> yards long, 16
a day?
6 in. long, 6 ft.
same stone 6 ft.
bs.?
er bottle, how
Lie, in the same
nd 16 quarters
3pt for 1205.15
lis. 2d., what
t?
and 4: ft. dscp,
If the number
fcer is it capa-
tt is at $1.14
3n wheat is at
cost 114.40 to
Vft. by 9ft.
a room, sup-
^ould it come
I 1820 yards
' a field 1320
hick, weighs
>f the same
and weight
ly in 8 days,
oxen and 12
Qg supposed
lays, find in
191
how long a time 2 men and 1 woman will do a piece of work 4 times
as great, the rates of working of a man and woman being as 3 to 3.
37. A person is able to perform a journey of 142-2 miles in 44-
days when the day is 10-164 hours long; how many days will he be in
travelling 505-6 miles when the days are 8-4 hours long?
38. If the sixpenny loaf weighs 4-35 lbs. when wheat is at 6-r5»
per bushel, what weight of bread, when wheat is at 18-4». per bushel
ought to be purchased for 18-13«. ? '
39. If a family of 9 people can live comfortably in England for
$7862.40 a year, what will it cost a family of 8 to live in Canada in
the same style for seven months, prices being supposed to be S of what
they would be in England?
INTEREST. *
163. Intekest is the sum of money paid for the loan or use of
some other sura of money, lent for a certain time at a fixed r'>U ; gen-
erally at so much for each $100 for one year.
The money lent is called the Pkinoipal.
The interest of $100 for a year is called the Rate peb Cent.
The principal + the interest is called the Amoitnt.
Interest is divided into Simple and Compound. When interest is
reckoned only on the original principal, it is caUed Simpo! Intebest.
When the interest at the end of the first period^ instead of being
paid by the borrower, is retained by him and added on as principal to
the former principal, interest being calculated on the new principal for
the next period, and this interest again, instead of being paid, is re-
tained and added on to the last principal for a new principal, and so on ;
it is called Compound Intebest.
SIMPLE INTEREST.
164. Tojind the Interest of a given sum of money at a giaen rate
per cent, for a year.
Rule. Multiply the principal by the rate per cent., and divide the
product by 100.
Note 1. * The interest for any given number of years will of course
be found by multiplying the interest for one year by the number of
192
AEirmfEno.
years ; and the interest for anv parts of a year may be fonnd from the
interest for one year, by Practice, or by the Rule of Three.'
NoTS 1. If the interest has to be calculated from one given day to
another, as for instance from the 30th of January to the 7th of Feb-
ruary, the 80th of January must be left out in the calculation, and the
7th of February must be taken into account, for the borrower will not
have had the use of the money for one day till the 81st of January.
Note 2. If the amount be required, the interest has first to be
found for the given time, and the principal ha^ then to be added to it.
. Ex. Find the simple interest of $250 for one year at 8 per cent, per
annum.
Proceeding according to the Rule given above,
^250
I
8
$20.00
therefore the interes^t is $20.
Measonfor the proceBs. /
The sum of $100 must have the same relation ia respect of magni-
tude to $250 as the simple interest of $100 for a year has to the simple
interest of $250 for a year; and thus the $100, $250, $8, and the re-
quired interest must form a proportion. (Art. 148.)
We have then •
$100 : $250 : : $8 : required interest,
whence, required interest =$— — -1_ (Art. 156.),
which agrees with the Rule given above.
Independent method,
$100 for 1 year gives $8 int.
.•. $1.........
'W^'-
/. $260 $
(260. A)i,,,
or $20.
it 8 per cent, per
SIMPLE INTEREST.
193
Examples worhed out »
1 year, lO months, at 4^ per ct^nt.
417.
7
9
4|
417
1669 . XX
156 . 10
.
•4J
£18-26 . 1 .
20
41
6-2U
13
2-56d
«. d.
7. 9
8
3 1252 .8.8
156 . 10 . 4J
£.
.*. Int. for 1 year -is
= 18
Int. for 6 mo., or i^ of 1 year = 9
Int. for 4 mo., or | of 1 year= 6
d.
5 . 2
56|
100
6
2
1
O 01
17 01
A227
"3 3 IT
.'. Int. for 1 yr., 10 mo =83 . 9 o--^-
.-. amount = £4ir. 7«. 9<?.+£83. 9«. Q^Jll
= £450. 17«. 8ff J^.
FoTB. In examples like the above we may reckon 19 mn«tT.« * *v.
yoar, but tf calendar montts are given, thrLertr,^urfbe be't
onnd by the Rule of Three ; a. for iastance in the foUowilg Ltlpk :
Ex. 2. Find tl.e simple interest and tho amount nf ikinA ^
June 15, 1843, to Sept. 18. 1843, at 7^ per cent. ^ ' ''""^
$106^ The number of days from June 16 to Sept. 18
742
53 =16 + 81+31 + 18
^7^95 =95.
.*. $7.95 is the interest for 1 year
Hence, 365 days : 95 days :: $7.95 : interest required
whence, it wiU be found, that interest required = $2 OftM-
^ .-. amount =» $106 + $2.06fl = $108.06fl.
194
ABITHMiaia
^ Ex. LL
Find the simple Interest.
(1) On |85 for 1 year at 8 per cent. "•
(2) On $310 for 1 year at 7 per cent.
" (8) On flOOO for 1 year at 6^ per cent.
(4) On $475 for 8 years at 7^ per cent.
(5) On $936.50 for 2 years at 6 per cent.
(6) On $656.76 for 6 year. 8 per cent. . '
(7) .On $946.40 for 2 years at 7 per cent.
(8) On £198. 6s. 8d. fcr 1 year at 3^ per cent.
(9) On £236. 68. 8d. for 2^ years at 3 per cent.
(10) On £98. 16«. lOd. fur | year at 2^ per cent.
Find the amount
(11) Of $1000 for 2 years at 7 per cent.
(12) Of $2^38.25 for 4^ years at 6 per cent.
(13) Of £1050. 6 fl. 2 c. 5 m. for 6 years at 8 per cent
(14) Of $139.80 for 3^ years at 7^ per cent.
(15) Of $1895 for 4f years at 6| per cent.
(16) Of £1634. 6s. 2d. for 1| years at 8 per cent.
Find the Simple Interest and Amount
(17) Of $375 for 8 years, 8 months, at 7 per cent.
(18) Of $446.50 for 3 years, 3 months, at 8 per cent.
(19) Of $220 for 7 montlss at 7^ per cent.
(20) Of $243.80 for 2 years, 5 months, at 8 per cent.
(21) Of 40 dollars from March 16, 1850, to Jan. 23, 1851, at 8 per
cent.
(22) Of $320.75 for 2 years, 35 days, at 7 per cent.
(23) Of £34. 10«. from August 10 to October 21, at 6 per cent.
165. In all questions of Interest, if any three of the four, (principal,
rate per cent., time, amount) le given, the fourth may le found: a^yfor
instance, in thefollowiny examples.
Ex. 1. Find the amount of $225 for 4 years, at 8 per cent, per
to
tei
anc
of]
annmn.
$100 for 1 year gives. . . .$8 int.,
^^ • *ioo^**»
Inte
i^IMFLE DTEEBEST.
195
1851, at 8 per
per cent.
mnd; as^for
>er cent, per
1228 for 1 year gives.. . . .$
(j5|5 X 226 K ijiat.
.•.$225 for 4 yearn $
or $72 int.;
.*. Amoont r= |226 + $72 s |297.
Ex. 2. In what time will $225 amount to $207 at 8 ner oant.
simple interest ? *^
$297 - $226 = $72, which is the intereri; to'be obtained on $226 in
order that It may amoont to $297. " .^^
But Int. of $226 fori year = $18; whi6h mnst have the Mme rela-
tion m respect of magnitude to the $72 as the 1 year has to the re-
quured tune ; if •«« «-
/. $18 : $72 :: 1 year : required number of years,
whence, required number of years = 4.
Ex 8. At what rate per cent., simple interest, wiU $226 amount
to $297 m 4 years?
In other words, at what rate per cent. wiU $226 give $72 for -in-
terest in 4 years, or ?^-^ or $18 in one year?
Then $226 : $100 :: $18 : required rats per cent,
whence, required rate per cent =8.
Ex. 4. What snm of money will amount to $297 in 4 years, at 8
per cent, simple interest ? / ■» •
$100 in 4 yrs. at 8 per cent, amounts to $100 +(8x4)$, or $182 •
and this $182 must be to the $297 as the $100 is to the wqnired sun!
of money;
.♦. $182 : $297 :: $100 : required number of dollars,
whence, required number of doDars- $226.
Ex. IJI.
Inte Lr** "^"^^^ *"'''"* ^ ^^^^^ * years, «t 8 per cent simple
8
196
ABITHMXmo.
« 2. At what rate rate per cent, will $640 amount to $928.80 in
9 yeari, at simple interest?
1 8. In what time will $850 amonnt to $448, at 7 per cent, simple
interest!
4. At what rate per cent, will $825.25 amonnt to $398 '6525 in 8 J
years, at simple interest ?
6. In what time will $142.50 amount to $242.26 at 7 per cent.
dmple ioterest? •
6. At what rate will $157 amonnt to $892.50 in 35 years at sim-
ple interest ?
7. What sum will priodnce for interest $87.75 in 2^ years at 6^
per cent, simple interest ?
8. What sum will amount to $1014.67^^ in ^ years at 7 per cent,
■imple interest? \
0. What sum will amount to £387. 7«. *f\d. in 8 years at 4 per
cent, simple interest?
10. In what time will £1276 amount to £1549. lU at 8| per
cent, simple interest if
11. At what rate per ceut., simple interest, will £986. 18«. 4d,
amount to £1157. 7«. ^d.^ in 4} years ?
^ 12. In what time will $125 double itself at 5 per cent, simple in-
terest f
18. What sum will amount to £425. 19». 4|<f. in 10 years at*8|
per cent, simple interest, and in how many more years will it amount
to £458. 11«. 7d.
14. What sum of principal money, lent out at 10 per cent, per
annum, simple interest, will produce in 4 years the same amount of
interest as $250, lent out at 6 per cent, per annum, will produce in
8 years ?
Kon. Though questions are given in Simple Interest, in which the
time is for some years, or several payments are made ; yet, in all such
cases Oompound Interest is the only fair method to both lender and
borrower, and is the method employed by Building Sucieiies, Insurance
Oompanies, dso* .
•/» V V.
COMPOUND INTEBEST.
197
COMPOUND INTEREST.
1C6. To find, the Compound Intereit of a given sum qf money at a
given rate per cent. /or any number 0/ years.
Rule. At the end of each year add the interest of that year,
found by Art. (164), to the principal at the beginning of it; this will
be the principal for tlie next year ; proceed \A the same way as fir as
may be required by the question. Add together the interests so arising
in the pevetal years, and the result will be the cumpouud interest for
the given period.
The reason for the chove Rule is clear from what has been stated
in Arts. (163 and 164).
Ex. 1. Find the Compound Interest and Amount of $2000 for
2 years at 6 per cent, per annum.
By the Rule,
$2000
5
$100.00
.'. $lOO=Int. forl*»y'.;
.'. $2100=Prin'. for 2''* y'.;
5
^ of $2000=
y»o of $2100=
or thus, since -^—^^
2000=1" yVsPrin'.
100= Int
2100=2"'yVsPrin«.
106= Int.
$2205=8"* y'.'sPrin'., or
Amount ;
Comp* int. =$2205 —$2000 =$205.
$105.00
.-. $105=Int. for 2"' y'. ;
.-. $100+$105 or $206=Compound Int. for 2 years,
and $2100+$105 or $2205=Amount.
Decimals may advantageously be employed in working questions in
Interest.
Ex. 2. Find the Compound Interest of $2000, for 2 years at 6^ per
cent, per annum. • • »,
%^ int. on $100 is ^, or $'065 int. on $1.
.♦. Int. on $2000 =(2000 x -065)$ =$130 ;
. .•.2»*y«ar'sPrin'^=$2000+$130 =$^180.
198
ABrrHMEno.
lat oa $2180 =(2180 x -066)$ =$188.46 ;
/. 8"* year'i Prin'.=$2180 + $188.46=12268.46.
Hence, Oompoand Int.a=$180 + $188.46 r^$268.46.
Kon 1. It is customary, if the oompoand interest be reqalred for
ftnj number of entire years and a part of a year (for instance for
6) years), to find the compoand interest for the 6th year, and then
talce f ths of the last interest for the fths of tiie 6th year.
KoTB 2. If the intercut be payable half-yearly, or quarterly, it is
dear that, the compound interest of a given sum for a given time will
be greater as the length of each given period is less ; the simple interest
will not be affected by the length of each period.
Ex. LIII.
(1) Find tl^e compoiind interest of $2000 in 2 years at 6 per cent,
per annnra.
(2) Find the amount of $800 in 8 years at 7 per cent., allowing
compound interest.
(8) Find the compound interest of $270 in 2 years at 8 per cent.
(4) Find the amount of $690 fur 8 years at 7 per cent., compoand
interest.
(6) Find the amount of $280.76 for 8 years at 6 per cent., com-
pound interest.
(6) Find the difference in the amount of $416.60, put out for
4 years at 7 ycr cent., Ist at simple, 2nd at compound interest.
(7) Find the compound interest of $130 in 8 years at 8 per c6nt.
(interest being payable half-yearly).
(8) What will $1750.60 amount to in 2|^ years, allowing 8 per cent,
compound interest ?
(9) A person lays by $280 at the end of eacTi year, and employs
the money at 7 per cent, componnd interest ; what will he be worth at
the end of 8 years ?
(10) Find the difference between the simple and componnd interest
of $416 for 2 years at 6 per cent.
(11) What is the difference between the simple and the componnd
?ntorest of $13,883 for 6 years at 6 per cent.
(12) Find the amount of $180 in 8 years at 8 per cent, compoand
interest (interest being payable quarterly).
FBxsmrr wobth.
199
(15) What ram of money put oat to coii-^yoand interest for 2 jetrk
at 7 per oent. will amount to $100?
(14) What sum at 8 per cent, compound interest will amount in
2 years to $2(14?
(16) A and £ eaoh lend £266 for 8 years at 4| per oent. per annum,
one at simple interest, the other at compound interest: find the differ-
ence in the amount of interest they respectively receive.
PRESENT WORTH AND DISCOUNT.
167. A owes B $600, which is to be paid at the end of 9 months
from the present time : it is clear that, if the debt be discharged at once
(interest being reckoned, we will suppose, at 7 per cent, per annum),
B ought to receive a less sum of money than $600 ; in fact, suoh a sum
of money as will, being now put out at 7 per cent, interest, amount to
$600 at the end of 9 months. The sum which B ought to receive now
is called the Present Worth of the $600 due 9 months hence, and the sum
to be deducted from the $500, in consequence of immediate pnyment,
which is in fact the interest of the Present Worth, is called the Discount
of the $600 discharged 9 months before it is due.
We may tiierefore define Pbesent Wobth to be the actual worth at
the present time of a sum of money due some time hence, at a given
rate of interest ; and we may define the Discount of u sura of money
to be the interest of the Present Worth of that sura, calculated from the
present time to the time wb .a the sum would be properly payable.
PRESENT WORTH.
168. Bulb. Find the interest of $100 for the giyen time at the
given rate per cent., and state vhus :
$100+its interest for the giyon Ume at the givoi rate per cent. : given
sum :: $100 : present worth required.
Ex. 1. Find the present worth of $500, dne 9 months hence, at 8
per cent, per annum.
Proceeding according to the above Rule,
Interest oi $100 for 9 months at 8 per cent, is $6,
.*, $106 : $500 :: $100 : required present worth,
-whence, required present worth =$471.6911.
200
ABITHMEnC.
T^tf reaafm for the ahovf, process is clear from the consideration,
that $100 in 9 m nths at 8 per cent, interest would amount to $108,
and therefore $100 is the present ralue of $106 due 9 months hence:
and GOQsequentljr we haye
1" debt : 2"" debt :: l" present worth : 2°"' present worth.
Independent method.
Since Interest on $100 for 9 mo. at 8 per cent.= $6
.*. P. W. of $106 due 9 mo. hence at 8 per cent. =$100
_ 100
~*106
$1
r 1 t .
.100
....$600...... =$^xaoo
=$471.6911.
Ex. 2. Fmd thb present worth of $838, due 19 motflhs hence, at
6 per rent, simple inteiist.
Sinc^ the interest of $100 for 19 months, at 6 per cent.
=$afx6)=$V— $9^,
.-. $109|^ : $838 :t $100 : required present worth,
whence, required present worth=^765.29ff|.
I! '
I
DISCOUNT.
«
^ 169. Rule. Find the interest of $100 for the given time at the
given rate per cent., itnd state thus :
$100+ its interest for the given time at the given ra*c p^r cent. :
gi-en sum :: mterest of $100 lor tho given time at the given rnte per
cent. ; 'discount required.
Ex. 1. Find tJie discount of $5C0, due 9 months hence, at 8 per
cent, per annum.
Proceeding according to the above Rule,
The interast of $100 for 9 months at 8 per cent. =$6 ;
therefore, $106 : $600 :: $6 : required discount,
whence, required discount =$28.80^^.
Th$ reason for the above pro<^ is clear from the oonsideration that
Disoouirr.
201
leratioB,
to $106,
9 hence :
)0
lence, at
) at thd
r cent :
rrtte per
it 8 per
$6 is the interest for 9 months, at 8 per oeut., of $10G, the present
wortib of $106 daeat the end of that time ; and oonseqnentlj we haW
1" debt : 2°*' debt :: discount on 1*' debt :*di8Goant oi;l 2** debt
Ex. 2. Find t^e discount on $1000, due 16 months hence, at 6 per
cent, per annum.
Interest of $100 for 15 months at 6 per oent.=4| of $6=:$7|v
.*. $107^ ; $1000 :: $7^ : Discount required;
.*. Discount required=$
1 000 X 7i
107^
=f69.76H.
on, that
Ex. 8. Find the diiscouut on £127. 2«. for half a year at 6 per cent
jeiOOf : £127^5 :: £| : required discount ;
Whence, required discount=£3. 2«.
KoTB 1. Discount = given sum leu Present Worth ; Present
Worth =:given sum less Discount
NoTB 2. lo the discharge of a tradesman's bill it is usual to deduct
interest iostead of disconoL ; thus, if B contracts with A a debt of $100,
A giving 12 months' credit, it is usual io business, if the interest of
money be reckoned at 5 per cent, per annum, and the bill be discljarged
at once, for^ to throw off $5, or for J to. receive $95 nstead of $100;
but if A were to put out the $95 at 5 per cent interest it will not
amount to $100 in 12 months; therefore such a proceeding is to the ad-
vantage of B : the sum of money wliich in strictness ought to have been
deducted, was not $5, tl-d interest on the whole debt, but $4. Td^^,, the
interest of ^he present worth of the debt, i. 0., the discount
NoTB 3. Bankers and Merchants in discounting bills calculate in-
terest, instf^ad of discount on the sum drawn for io the bill, from the
time of their discounting it to the time when it becomes due, adding
THBEK DAIS OF ORAOB, wbich days are allowed usually after the time
a bill is KOMiNALLT due, before it is leoallt due ; which is, of course,
an additional advantage. When a bill is payable on demand, the days
of grace are not allowed.
NoTB 4. If a bill, without the days of grace, should appear to be
due on tlie 81st of any month which contains only 80 days, the last
day of that month, and not the first day of the next, is considered as
thst dav on which the bill is dae^ Thus a bill drawn on the 81st of
October, at 4 months, would be really due, adding in the days of grace,
202
ASTTHMEIia
on the 8rf of Maroh. Also bills which fell doe on a Snnday, are paid
on the previons Satnrdaj.
Ex. A bill of £1000 is drairn on Feb. 16th, 1861, at 7 months' date •
it is disconnted on the 8th of July at 5 per cent. What does the banW
gain by the transaction ?
The biU is legally due on Sept. 19 ; and from July 8 to Sept. 19 are
78 days.
»» . je. #.
The interest of £1000 for that time=10 .
The true discount = 9 , ig,:^^
.*. the banker's gmn IVtS,
Ex. LIV.
Jind the Present Worth of
(1) $821 !
t2) $251.66
(8) $688.28
(4) $944.92
(6) $468.50.
(6) £890
(7) $856.96
(8) $1252.40
(9) $1250
(10) £2110
(11) £275.6s.6d,
(12) £918
(18) $600
(14) £2197
due 1 y. hence, at 7 per cent, per ann"., simp*. int.
6
6 months
6
.. 7.......
.. 8 .......
.. 1
.. 8
.11
.15
, . 4 years 5
.19 months . . . 5J
. . 8 years 4
. 8
.7
. 6
. 6
12
. 7
. 6
. 4
compound interest
Wmd the Discount on
(16) $64 due 4 months hence, at f per cent, per annum, simp*, int.
(16) $1880
(17) $107.26
(18) $125.46
(10) $487
(30) $340
(21) £S640
(22) £818. 9«.
9
.. 6
.. 8
.. 6
.. 6
..10
8
5
8
7
6
. 1} year 4f
X I iisuiiiiUS 9
(»4) $102 ...146 days .6
are pai4
IB* date ;
»banlcer
t.19 are
mp'.int.
8T00X». 203
(26) A bill of £649 is dated on Jime §8, 1858, at 6 montlia, and k
discounted on July 8, at 8i p6r cent ; what does the hanyr
gain thereby?
(26) Find the true discount on a bill drawn Ifarch lY, 1958 at 8
months, and discoonted Kay 2, at 8 per cent (days of grace
allowed).
(27) Rnd the simple interest on $545 in 2 years, at 7 per oeqt per
annum; and the disoonnt on $621.80 due 2 years hence at
the same rate of interest. Explain dearly why these two
sums are identical.
(28) Ezplaiu the difference between Discount and Interest.
Five volumes of a work can be bought for a certain sum
payable at the end of a year; and six volumes of the same
work can be bought for the same sum in ready money: what
is the rate of discount ? * '
(29) A tradesman marks his goods with two prices, one for ready-
money, and the other for one year's credit allowing discount
at 5 per cent. ; if the credit price be marked at JB2. 9«., what
ought to be the cash price?
interest
np*. int.
■••••••
STOCKS.
170. If the 6 per cent. " Dominion of Oanada " stock be quoted in
the money market at 105|, the meaning is, that for $105^ of money a
person can purchase $100 of such stock, for which he will receive a
document which will entitle him to half-yearly payments of Interest
or Dividends, as they are called, from the Government of the country,
at the rate of 6 per cent/ per annura, on the stock held by hin^ until
the (Government choose to pay off the debt.
Similarly, if shares in any trading company, which were originallj
fixed at any given amount, say $100 each, be advertised in the share-
market at 86, the meaning is, that for $86 of monoy one share can be
obtained, and the holder of such share will receive a dividend at the
end of each half-year upon the $100 share according to the state of the"
finances of the company.
Sxoc^ may therefore be deHued tu be the oapitiU of trading com-
204
ABITHMBXIO.
pAiiies; or to be„tlio mqnej borrowed bjour or anjothar Goyemmeot,
at so nmoh per ceut., to defray the expenses of the nation.
The amoantof 4«bt owing by the Government is celled the Na-
TiovAL Debt, or the Funds. The Government reserves to itself the
option of paying off the principal or debt at any future time, pledging
itself, however, to pay the interest on it regulai'ly at fixed periods in the
mean time.
From a variety of caases the price of stock is continnally varyiiig.
A fnndholder can at any time sell his stock, and so convert it into
money, and it will depeud upcm the price at which he disposes of it, as
compared with the price at which he bought it, whether he will gain
or lose by the transaction.
KoTB 1. Purchases or sales of stock are made through Brokers,
who generally charge $|, or 12 J cts. per cent., upon the stock bought
or sold : so that, when stock is bought by any party, every $100 stock
costs that party $} more than the market-price of the stock : and when
stock is sold, the seller gets $^ less for every $100 stock sold than the
market-price.
Thus, the actual cost of $100 stock in the 8 per cents, at 94^, ia
t{9^ -f- i), or $94f . The actual sum received for $100 stock in the 3 per
cents, at 94^, is $(94f-|), or $94.
Unless the brokerage is mentioned, it need not be noticed in work-
ing examples in stocks.
NoTB 9. When $100 stock costs $100 in money, the stock is said
tobeatjpor.
When $100 stock costs more than $100 in morey, the stock is said
to he At & premium.
When $100 stock costs less than $100 in money, the stock is siud
to be at a diaeaunt.
All Examples in Stoeht depend on the principlee of JPfoportiony and
mcey therrfore be worJeedly the Mule of Three.
Th(Me of most frequent occurrence will now be given.
Ex. 1. What sum of money will purchase $3600 6 per cent, stock at 94 ?
$100 stock (st.) costs $94 in money ;
.*. $100 St. : $8600 st. :: $94 : req\ sum ;
* th<
be:
at
br<
— . t J
86 00 X 94
100
STOOKB.
205
Brnmest)
the Na-
tself the
pledging
da in the
varyiiig.
t it into
I of it, as
nrill gain
Brokers^
: bought
00 stock
nd when
Lhanthe
t 94^, is
the 3 per
n work-
k is said
k is said
c is sajd
ioTiy and
»kat94?
Independent inethod.
$100 stock is bobght for $94
$1
•ioo»
94
.-.$8600 f^x8600
, =$3884.
Note. The student is strongly advised to work the questions by
the independent method. ,
Ex. 2. Find the cost of £2858 8 per cent. Consols at 90^, brokerage
being ^ per cent.
£100 St. coFts £(90^ +1), or £90^ ; '
.-. £100 St : £2858 st. :: £90J : req". cost ;
.-. req-. cost=£??5|^i?5J=£2129. 9». Z^. J^.
Independent method.
£100 stock costs £(90 J + 1), or £90 ^ ;
.'. £100 stock costs £90 5,
^90-5
£1
lOO"*
90-5
.•.Je2853. £^x2353,
or £2129. 9«. 3J<f. 1^.
Ex. 3. A person who has $10000 Bank-stock, sells ont when it is
at 40 per cent, preminm ; what amount of money does he reoeive,
brokerage being ^ per cent. ? '
$100 St. sells for $f 140— ^J, or $139| money ;
.'. $100 St. : $10000 St. :: $139] : req^ am*, of money;
... req'. a,n'.=ti520»iiM=|i8987.BO.
Independent method.
$100 stock sells for $(140-i), or ^189| ;
.'. $100 stock sells for $139-876, .
$1
100
2M
AEOTsaanic.
.-.Iloooo |i^><ioooo,
or $18987.60.
F '. 4. What incotnM will $8500 of 7 per cent stock, and $8600
inveeted in fhe 7 per cent- stock at 102|, resnectivelj produce?
in 1st, since every $100 stock gives $7 int. ;
.•. income from $8600 of 7 per cent. 8tock=$ — ^ ^ =$696. ^
2d, ^ince $100 stock, which gives $7 int, costs $102f ;
/. every $102| gives $7 int. ;
.'. $102f : $8600 » $7 : req'. income:
-.8600 X 7 ^ ^
/. Mq^ moome=$— j^j^|~=$679'54|^.
Ex. 6. One per^n bays £800 Consols at 90^ and sells out at 98 ;
another invests £800 in OonsoU at 90 J and sells oot at 98 ; what sum
of money does each gain ?
Ist man gains £(93— 90|^), or £2|, on every £100 stock ;
.-. his whole gain=£(2f x 8)=£21. 6a. Sd,
2d man gdns £2| on every £100 stock, i. e, on every £90|^ of his
money whidi he invests ;
.•. £90J : £800 :: £2| : whole gain ;
800 X 24
/. whole gain=£— 5^— =£28. 12«. 4d. nearly.
Ex. 6. A person invested some money in the 8 per cent. Consols
when they were at 90, and some money when they were at 80 ; find
the rate of interest he obtained in each case, and the advantage per
cent, of the second purchase over the first.
£90 : £100 :: £3 : rate per cent, in 1st case,
£80 : £100 :: £3 : rate per cent, in 2nd case,
/. rate per cent, in 1st case=^£ — g^— =£8. ^.6d.;
/100x8\
• •••••••..••••a. 2(1 . • ■ • ssJdI 5J» I —xo. LOS, J
.% advantage— £3. 16«.--£3.8#. 8<?.=»8«. 4(2.
Ex. 7. A person invests £1087. 10«. in the 8 per centi. at 88 ; the
ftu
at
f
frc
hit
at
thi)
the
8TOCI0B&
207
f
ftmds rise 1 per cent ; he then tnuwfen hie capital to ti^ft 4 per o^nfa.
at 96 : find tiie alteration m his income.
£88 : £1087. 10». ;: £100 : quantity of 8 per cent, at ;
.-. quantity of 8 per cent. at. bought=£i5?I|jli??_.£i250.
bo
The ftinds have riseu 1 per cent, therefore to trantfer £1260 atook
from the flinds at 84 to the foods at 96,
£96 : £84 :: £1250 stock : quantity of 4 per cent stock, (sinoe the
higher the price of the stool? the less will be the amount purchased) ;
.-. quantity of 4 per cent stook=£— ~?=jB1098. 16#.
Ist Inoome=£-^^^ =£87. 10*. *
2nd Income=£ — j.il_=£48. 16,. ;
.*. alteration in income=£48. 16«.— £87. 10».=:£6. «•.
Ex. 8. TThich is the best stock to invest £1000 in, the 8 per cents,
at 89^, or the 8 J per cents, at 98J?
In the first case,
every £89J of money gives £8 interest ;
,\ et^ery £1 of money gives £—-, or £~ uitereat.
In the second case,
every £98i^ of money gives £8 J interest ;
.% every £1 of money gives £-g|, or £—, interest ;
6 7
' ^ and comparing the fractions-r^ and ~f
since 7 X 179 is>6 X 197,
th4) 2»* fraction is greater than the 1", and therefore the 2"* investmenr
the best
Br. LV.
Und the quantity of stock purchased by investing :
\i.} ^aouv ia soe o per censs. &i (&.
(3> $712 hi the 7 per cents, at 89.
208
ABTEHHBTIO.
(8) $604 in the 8 per cents, at 99.
(4) $8741 in the 7 per cents, at 8r.
If (6) $D00 in the 6 per cents, at 88f .
(6) $800 in the 8 per cents, at 15^.
[T) £4311. 8a. 9af. in the S| i)er cents, at 85|.
(8) $2858 in the ft per cents, at 90|, brokerage \ per cent
(9) £3277 in the 4 per cents, at 105|, brokerage \ per. cent.
Find the money value of '
(10) $2600 in the 7 per cents, at 98. •
(11) $1920 in the 6 per cents, at 77j.
(12) $3000 in the 7^ per cents, at 92J-.
41") $2240 in tlie %l per cents, at 81|.
(14) £1000 4 per cent, stock at 97f per cent., brokerage \ per cent
(15) £2153. 10«. bank stock at 188J per cent., brokerage \ per cent.
Find the yearlj^ income arising from the investment of
'(16) $1008 in the 6 per cents, at 84, ^
IVT) $5580 in the 8 per cents, at 93.
(18) $1638 in the 7 per cents, at 93f.
(19) $2000 in the 6 per cents, at 88|.
(20) £8425. 15«. 2d. in the 8 per cents, at 91|.
(21) £4788 in the 8^ per cents, at 106.
(22) £3500 m the 3 per cent, consols at 94J, brokerage \ per cent.
"Wliat snms of money must be invested in the undermentioned
stocks in order to produce tlie following incomes?
(23) $120 in the 6 per cents, at 85.
(24) $288 in the 6 per cents, at 67. •
(25) $170 in the 7 per cents, at 90.
(26) £37 in the 3 per cents, at 74|, brokerage \ per cent.
(27) £37. 10«. in the 4 per cents, at 93i, brokerage \ per cent.
At what rate per cent, will a person receive interest who invests
his capital?
(28) In the 6 p r cents, at 91.
(29) In the 7 per centn. at 94.
(30) In the 8 per cents, at 96^, brokerage \ per cent.
(31) In the 7 per cents, at 102 J, bri>kerage \ per cent.
(82) If $7927.60 be laid out in purchasing Canadian Bank of Com-
STocas.
200
dot.
per cent
per oent.
per cent,
entioned
lent.
> invests
of Oom-
meroe Stock at 105, yielding annnal dividends of 8 per cent, per anntim •
what jearlj inci>me w.ll be derived from this investment after dcdact^
ing an income tax of 1 J cents, in the dollar?
, (88) A person invested money in JRoy.l Canadian Bank Stock at'oo
and some more at 80 ; find the rate of interest he obtained in each case'
and the advantage per cent, of the second purchase over the first. The
bank's yearly dividends being r per cent.
B Jk fiJi" ^^r ?''"''' ® ^'' ''^*- '" ^'^ ^*P^^^ ^y investing in
Bank Stock yielding 7 per cent, per annum, what is the price of the
stock, anil h..w much stock can be purchased for $1200 ?
Sf Jf i fr^ ""^^ ™""'^ """"^ ^ ^'"^'^ ^^^^"^^ ^^ ^°^t of MontVeal
ft^Ko/i ir* "J^'^'"'" ^'*° '*°^^ income as if he had invested
$660^ when the stock was at 163 ? "*««u
«.ii^^^l'^«r"\^''^' ^^^^^ ^^^*^ Canadian Bank Stock at 66. and
sells «mt at 63 ; what doe. he gain by the transaction ?
(87) A person invests $9000 in Bank Stock at 168, which pays
yearly d.v^dends of 12 per cent., and sells out when it has sunk to IsT
how much does he losa by the transaction ? *
at what rate may the same quantity of stock be bought in the 81 per
cents, with equal advantage? fueo^per
in T^^^ t ^TT '"'^''^' ^'' '^^^^ ^** * ^^^^^^ of $1200, which is a third
InJr"; ,.":'T ^* ''' P«^^"^ ^ P^^ «^°*- P«^ ^^^um interest
find his half-yearly dividends. '"i-eresi ,
160^^/ f 7" ^'•^"^^^^^ ^5000 from the Bank or Montreal stock at
tll'h^lf 1 . °rr'^*''^'*''^' ^"^ "Iteration in his income,-
XcS '' said stocks being 6 and 4 per cent,
«n^^tLir-^**'°T'''''"*^^^^'*^"^ P«y^°^ lialf-yearly4 per cent.,
See ? '"^ ^ ^'^ '""''• ^'''"^''''''' ®'°'^ *' ^^^' ^^«P«o*iv«Jy
in. K^i ^'""^ *T'? P'^^"'^^ ^^ ^^^^^ Merchants' Bank stock, pay.
L !t m ^^^ ^ °^ "" P'' '"'''•' '"^ ^'^ ^*^°^ ^^«° *^« «^oct
fift3^tn .)^ P^^^t'^^^f^'^ ^3000 stock from the 3 per cent, consols at
89,, to the reduced 8f per cents, at 98^ ; find what quantirv of the lat,
s«r ne wiii hold, and the alteration Jn his income. ' " -
(44) The stocks of the Canadian Bank of Comm. and tJie Quebec
"■'•«*..
nfm"
210
ASnBMXSLO,
'"\rVLl^i:io» o/'S. e plrtnu. «bic. mto,. In 1878
v./t^t ioT- how much money mu.t b« inverted in them to produo.
i:'"rlTn;.m. of .SSo, after dedocting an income t„ of 2 o«.U. in
^fT'l ner«.n inye.to £1087. 10.. in the 8 per cent.. «t 88, .nd
.h "L^^rL. risen 1 per cent he tr«.sfe» his cpit- U. th,
; -««♦- afftfl- find the alteration in h»8 income.
* '^(ItThow ru^h in tt S per cenU. .t 98 must be «.ld out to pa, .
bill of f 16H » >»on«» "'*" " "««»""'' ""'• '*^ *"~ "
*"%"KvSlro;T::::l «nount of » percent. .to* ^-
m„,^2 fn 18';ear. to £8081 How much stock was there, .nd wh^
^fi i\z :^;- p- :-;- ^ -:is;5 r^ r:t;
toresB^eonthcwboleagamofflSO, afiernaTiogpu tf
oommissibn on each f^^'}^} ^„t. g„„tk g^a Annnitle^
(60) A person had £10,000 » Jf« » P« "* ^^^^ ^ the rate of
„a the Government "^^"^ '» ^^^tinuH^To' Ux the £10,000 in
a^per cent, for eve^ ^^^^ °f j^^.^'^^L ^as prrferred, and on the
*** ""rj.":^ a was -iiSn consols at »S. Howmuch would
rhtSrtntrb^ be accepted the first proiK,sal, and what
-'';r^^:.^.^:rJ:7l^U ^^ the interest on a pnbi^
aebf:f^£T,SSo,OOOwererednc^.^8,^^^^^^^^^^^^
:::rwSbetbrpUerofXUolders be dimini^^
APPLIOATIOKS OF THE TEEM "PEE CENT."
171. There are many <"»>" -- 'V,'^^ *;rcfr^ia':^
occurs beside those already mentioned ; we w,U mentio
buying or selling goods for another.
At charges for
AFPLIOATOONB OP THE TZSBH FEB OKNT. 211
Bbokebaob is of the same nature as Commission, but has relation
to money transactions, rather thao dealings in goods or mercbandise.
lusDBANOE is a oontract, by which one party, on behig paid a oer-
tain sum or iVm««w by another party on properly, which is subject
to rislc, undertalces, in case of loss, to malce good to the owner the
value of that property. The document which expresses the contract is
called the Policy of Inmrance,
Life Assurancb is a contract for the payment of a certain snm of
money on the death of a person, in consideration of an annual premium
to be contmued during the life of the Auured, or for a certain number
of yearp.
Questions on Commission. Brokerage, and Insurance, these charges
being usually made at so much per cJent., amount to the some thing as
findmg the interest on a given sura of mopey at a given rate for one
year, and may therefore be worked by the Rule for Simple Interest or
by the Rule of Three.
Ex. 1. A Commission merchant sold 80270 bushels of wheat at
11.16 per bushel; the Commission being 2 per cent. : how much will
he receive?
Amount obtained from the sale of the wheat is $84810.50 ;
Therefore, $100 : $84810.60 :: $2 : Commission required ;*
.-. commission rftquired=$^^^^^^;f ^ '*-=$696.21.
Independent method.
Commission on
100
$100 is $2;
••• 184810.60.. $^x 34810.60,
or $696.21
Ex. 2. What isHhe brokerage on the purchase of $7260 6 per
cent. Toronto debentures at \ per cent. ?
Bt. 8t
$100 : $7260 :: $| : brokerage required;
.-. brokerage required™$?^^li— $86.26.
212
ABITHMBno.
s»
$m : %2W :: $} : premium required ;
.'. premium required = $27 xf
=120.26.
In every 12 parts 1 part is dross;
.-. 12 : 100 :: 1 : percentage of dross ;
.'. percentage of dro88=-i5?^^__oi
12 —''«•
Ex. 6. Archimedes discovered that the crown made for TTm^ wt
consisted of goli and silver in the ratio of 2 . , '^^^^^^^^'f^SfHIero
was gold, and how much per cent, stverl ' ^'"^ '""'^ ^^^ ^•"'•
Out of every 8 parts, 2 were gold and 1 silver;
.*. 8 ; 100 :: 2: percentage of gold;
.-. percentage of gold = J^^— = qq^\
and percentage of 8ilver=88 J.
172. All questions Which relate to gain and loss in mA,.n««*'t x '
actions fall under the head of Profit and LoZ """""'^'^^ <*«^
or J;^^*^;^f"^ "«•'•«"»•« ^b^ir Profit or Loss by the actual amount g.ined
^r lost,^or by the amount gained or lost on every |100 of the' ^^^
J^i, sis^;t:;xr::;;::f «^ ^^^- -- ^^^o, what is
The gain =selling price ?tfM first cost ;
the selling price. 4- ^S ^ x 84) =$294 ;
therefore the g.^i . .. ^ g/ $21 0=$84.
, Ex. r. A ream of paper cost m< $5.20. what ,n,i«f. T «.n «. .. -. „
so realize w per cent l " " "'^ ''"**®
APPIIOATIONS OF ran; TERM FEB CENT. UlS
wm'J;~:4^ ^^»^--i,vif |100gain|20. or produce $120. wUt
.-. 1100 : $5.20 :: $120 : required an. .nut in dollars,
whence, required umount=|6.24.
15 f^T^inU ^ ^"^ ^**^ "' ^^' ^^** " *"°' ""^""^ '""'^ ^ ''" ^* "* ^ ^*>««
In this case every £100 would re .lizo £(100-15), or £86 ;
.-. £100 : *"4. 103. :: £'.5 : required amount in pounda,
whtncr-^ required amount=r£4. U. "J^d.
Ex. 9. I person buys shares in a railway when they are at £191.
£.6 having been pai.l, and sell, th.m at £32. 9*. when £23 has been
paid : how much per cent, does lio gain ?
£m 1.7' "It '^r "' ^^^^' ""^^ ^' after..ards pays upon it
shl""]?o' Tn ;, 'V'' "''^' ^™^ ^'^ «^'"^' ''^ ^'^ P«i^ ol each
£29. 10«. which he h;is paid (£82. 9«.-£29. 10a.)=^2. 19». ;
.-. £29 J : £100 :: £2^2 : gain per cent, in pounds ;
whence, gain per cont.=£lO, or gain is 10 per cent.
for ^QfiR^ Wh.t was the prime cost of au article, which when sold
for $2.88 realized a profit of 20 per cent. ?
Here what cost $100 would bo sold for $120 ;
.-. $120 : $2.88 :: $100 : prime cost in dollars,
whence, prime cost=$2.40.
If the above example had been, - What wa. the prime cost of an
article which when sold for $2.88, entails a loss of 20 p.r cent ? »
then $80 : $2.88 :: $100 : prime cost in pounds,
whence, prime cost=:$3.60.
be S.fedr'"'^ ""*"'' °' '°*"°" ""* "' "' """^ """P'" ""V
Since 20 is the I of 100,
therefore, 1 + h orf =selling price,
I of selling price=prime cost,
(t 5 ^f An aa ihe\ tn
$2.40 is therefore cost price.
2U
ARITHMEDC
f
^^H|
■
f
i
j
1
■
!
1
I
H
.. i
I Agaia, since 20 is i of 100 :
j tJiereforel-J, or *;=selling price,
* " i selling price =prime cost,
I or f of $2.88=$8.60.
f $8.60 is therefore cost price. »
^ T v^^ ^]' J^^^ ^^"'"'^ * hcr^ofor £40 I lose 20 per cent., what mmt
I Jiave sold him for so as to gain 10 per cent. ?
Here what wcuUl co.t mo £100 must be sold in one case for £80,
and m Uie other for £110 ; and tlierefore we get this statement ; selling
pnoe of £100 in Isi case: selling price of horse in 1st case ;: seUing
price of £100 in 2od case : selling price of horse in 2nd case ;
or £80 : £40 :; £110 : selliDg price in pounds;
Avhence, selling price=£55.
* ■.nf;^^. ^^''''''^ ^"^' ^ '''''*• ""^ ^"S^^ ^* ^^' a lb-» 2 cwt. of sngar
at lOK a lb., and 2^ qrs. of sugar at 1*. a lb. ; aud mixes tiiem : he
^ sells 4 cwt. of the mixture at 9d. a lb. What must he sell the remamder
at, m order to gain 25 per cent, on his outlay ?
Scwt., or336lb8., at6t?. alb.,cost 8*. 8.0
2 cwt., or 2241b8., at lO^d. a lb., cost 9 . 16 .
21 qrs., or 70 lbs., at Is. a lb., cost 3 . 40 .
.-.680 lbs. cost.. 21 . 14 .
In order to gain 25 per cent, on £21. 14«., it must realize £27. 2». 6d. ;
£. a, d.
.-. he must sell 630 lbs. for 27 . 2 . 6
but he sells 448 lbs for.... 16 . 16 .
.-. by Subf he must sell 182 lbs. for ... . lo . 6.6
.-. he must sell 1 lb. for ^i5i|i^, ^r IS.-^e?.
178. Tables respecting the increase or decrease of Population &o
are constructed with reference to the increase or decrease on every 100
of such population ; Education returns are constructed in the same
way ; and so are other Statistical Tables.
ww'. '1 ^" " ^"^ ^^^^"^ ^^ ^^® chudren, 126 learn to write,
wnat u the percentage?
;., what mnit
ase for £80,
aent ; selling
ase :: selling
e;
wt of sugar
s Ijiem : he
e remamder
d.
.
,
I2t28.6d.;
lation, &i).,
i every 100
. the same
to write.
«
APPUOATIONB OF THjB TEBM PEB CENT. 315
In other words, what nnmher bears the same ratio to 100. wliioh
126 bears to 160? , «*
.'. 160 : 100 :: 126 : percentage;
.-. percentage=i^=:83J
Ex. 14. Between the years 1861 and 1861 the population of the
city of Toronto increased about 78 per cent., and in the latter year it
was 44821. What was it in 1861 ?
For every 178 persons in 1861 there were 100 persons in 1881 ;
.-. 178 : 44821 :: 100 : number required ;
.-. nmnber requh-ed=li?^|I52=26180 nearly.
Ex. 16. In 1842 the number of the members of the University of
Cambridge was 5852, and in 1852 the number was 6897: find the in-
crease j)er cent.
Subtracting 5853 from 6397 we obtain 644, the increase on 6868
members ; the question then is this ; if 5853 members give an increase
of 644, what increase do 100 members give ? >•
.*. 6858 : 100 :: 644 : increase per cent. ;
.-.increase per cent.=^^=9y??.
^ 6853 6853
Ex. 16. The numbers of male and female crimmals are 1286 and
988 respectively ; while the decrease in the former is 4*6 per cent.» the
increase in the latter is 9-8 per cent. ;' find the increase or decrease per
cent, in the whole number of criminals.
1st 100 : 1235 :: 4*6 : whole decrease of male criminals;
.-. whole decrease of male criminals= — tt^ — =66*81.
2nd. 100 : 988 :: 9*8 : whole increase of female criminals ;
.'. whole increase of female criminals= — r— — =96*824 •
.-. m (1286+988) or 2223 persons there is an increase of
(96-824-56-81) or 40-014 persons.
.*. 2223 : 100 :: 40014 : percentage required ;
- 4001*4
.-. percentage required ="222^ =1-8.
216
AEnatMEno.
• Ex. LVl.
(1) What is the percentage on 66894 at 4 : « ; 4 . v« • --n . i Km o
(2) How much per cent, is 15 of 96- IQ nf si o^'/ ' ^^^*'
of 782176; 63 of 11080-6? ' 19 of 81 ; 23 of 266; 186^
(8) Write in a decimal form i- 9a.. ai . k^ ««.
600-0138 per cent. * ' ^ ' ^' ^^5 26J ; 230-06 ;
(4) Bought 200 cords of wood af <k4. 9K rx«« i , ,
fo. ,C pe. eord. W, at .. .Cl^lt ..7^^^;^ "'" " "^'"
(6) It 5 owt. 8 qra., Ulbs. be bought for £9 s. ^r.A ,a ,
^"V^.^"'':- 7'-' ^"- ™-f gain Per'o^U St im J ' '
the r^te of Sfor $29 rsT '' ^"'""^ " '»'' "^ ««"^S '"o"' •'
(7) A dask, which contained 2005 gallons, leaked 87 ,>., *
how much remained in ihe cask? S-uions, jeated 27 par cent.,
.ha itc^p-r^^^^
how much per cent. l,e gaiL or loses ' ' " ^"^ ' ^"^
.. /"i ;i.f Xa^r a:7i irp^d .t -^rf "" * -^
iacraii''jr^:i:;;-t::nt*"-" - -'-^ -^^ ^-
pe/owi f r isCg •: :r^ z ^::;:t -' "^ '^- »' *--
whole number ofrfek people ""° "'"■ <"'"'• "' ""'^"'^ '" '"e
(16) The pODUlation nf Trolor./! Hhr^^.^^ . ^.
1841, 6516794' in 1861" FinHihT-"" "°'*"^ '" ^*^-^^' ^^^^^24 in
, io/«4 1861. Fmd the increage per cent, in the first ten
APPPIOATIOHB OF THE TERM PER CENT. 217
years, die decrease per cent, in the second ten years, and the decrease
per cent, m tbe 20 years from 1881 to 1851.
(16) The population of a city is a million; it rises IJ per cent for
3 years successively ; find the population at the end of 3 years.
(17) A school contains 383 scholars, 3 are of the age of 18 yoars^
6 per c.nt.^of the remainder are between the ages of 15 years and
lOand 12, and the remamder under that age ; find the number of each
(18) An article which cost 84 cents is sold for 93 cents: find the
gain per cent. ' ^°^
♦o.^», Tvn.it IS his gam per cent. ?
wh«t Iho l,orse oust him ; wimt was the original co^
(21) Sugar being composed of 49-856 per cent of oxyem 4S-2B15
per cent of carbon, and the remainder hydrogen; find Tow tanr
poa, ds of each of these materials there are in one toi of sng,lr
(32) In 1853 the number of the graduates of the University of
(28) A mer(|iant buys 13600 bushels of whent at «1 05 a bushel
l\ rrf, 1" " ""'^" ' "» '^"^ ^^ P«^ «-'• of the remainder :^
tl a bushel, 20 per cent, at »1.05 a bushel, and the rest ar*! 26 ,
bushel; what does he gain or lose by the transaction?
(24) If the increase in the number of male and female criminals
t » per cent., and the increase m the number of femnles is 9« Tn™
pare the number of male and female criminals respectlrely. """
(25) By selling an article for 5<,. a person loses 6 per cent • what
was the pnme cost, and what must he sell it at to gain 4i per cenT !
cen uion ti: ™ f' "' " T '' *''"' = '"" «'-- "Hale 5 per
ruflv?! ''t!°f„L^r-' =*'"^*''.'' I"'"' '' "- -'• "Pon ae "hole
. ,,.,^ ociiixig pi-iuu or ine uook.
per^'ii' gi:^'ri:: ift;: ii^r:^^,- « - -• - '»'^ --
218
ABITECBfSna
M i (28) I bonght 600 sheep at |6a-head; their food cost me $1.25
B-head: I then sold them at $10 a-head. Find my whole gain, and
also my gain per cent.
(29) A pef son having bonght goods for £40 sells half of them at a
gain of 6 per cent. ; tor how much mast he sell the remainder so as to
gain 20 per cent, on the whole ?
(30) A vintner buys a cask of wine containing 86 gallons at $2.40
per gallon ; he keeps it for four years, and then finds that he has lost
6 gallons hy leakage ; at what price per gallon must he sell the re-
mainder in order that he may realize 20 per cent, upon his outlay ?
(31) A person rents a piece of land for £120 a year. He lays ont
£625 in buying 50 bullocks. At the end of the year he sells them
having expended £12. 10a. in labour. How much per head must he gain
by them in order to realize his rent and expenses, and 10 per cent,
upon his original outlay ?
(32) A grocer mixes two kinds of tea which cost him 88 cents and
44 cents per lb. respectively; what most be the selling price of the
mixture in order that he may gain 15 per cent, on his outlay?
(33) A stationer sold quills at 1 1«. a thousand, by which he cleared |
of the money ; he raises the price to 18«. 6d. What does he dear
■ per cent, by the latter price?
(34) A smuggler buys 6 cwt. of tobacco at 1*. Bd. per lb. ; he meets
with a revenue-officer, who seizes ^d of it: at what rate per lb. must
he sell the remainder, so as, 1st, neither to gain nor lose; 2nd, to gain
5 guineas ; and 3rd, to gain cen'.;. per cent. ? •
(35) A farm is let for £96 and the value of a certain number^ of
quarters of wheat. When wheat is 38«. a quarter, the whole rent is
16 per cent, lower than when it is 56«. a quarter. Find the number
of quarters of wheat which are paid as part of the rent.
(86) A person bought an American watch, bearing a duty of 26 per
cent., and sold it at a loss of 6 per cent. ; had he sold it for $8 more, he
would have cleared 1 per cent, on his bargain. What had the first
party for it? ♦
174. Questions are often given, in which the term " Average "
occurs ; a few examples of such a kind will now be worked by way of
illustration, and others subjoined for practice.
Ex. 1. A gentleman in each of the following years expended the
j g jii BJgrwM-jtf.j'i' !*'
3 cost me $1.25
whole gain, and
lalf of tbem at a
mainder so as to
gallons at $2.40
that he has lost
'< he sell the re>
his outlay ?
IT, He lays out
r he sells them,
ad must he gain
md 10 per cent.
im 88 cents and
ng price of the
utlay?
ich he cleared |
) does he dear
r lb. ; lie meets
te per lb. must
e ; 2Qd, to gain
•tain number^ of
whole rent is
id the number
duty of 25 per
'or $8 more, he
at had the first
n "Average"
ked by way of
expended the
APPLICATIONS OF THE TEEM PER CEim ^ 21^
following^sumS: in 1858 ^500, in 1859 $G0O, in 18G0 $600, in 1861 $600
in 18G2 $700, iu 18C3 $700, in 18C-i $700. Fiiid Ins yearly avoia^
expenditure. '^
The obj'ct is to find that fixed ?nm wl.icli ho might have spent ia
each of iho seven years, so that his total expenditure in that ease
miglit bo the snme as liis total expenditure was in the above question. ;
Adding the various sums together we obtain the total expenditure
which equals $4400 ; this sum divided by 7 gives $628.59| as the aver- '
age yearly expenditure.
Ex. 2. In a school of 27 boys, 1 of the boys is of the age of 17 years,
2 others of 10, 4 others of 15J-, 1 of 14f, 2 of 14^, 5 of 13f, 10 of 12^,'
and 2 of 10 ; find the average ago of the boys.
The object is to find, what must be the age of each boy supposing '
all to bo of the same age, that the sura of their ages may = the sum of
the ages in the question.
sum of ages in question = 17+33 + 62+14|+29+68f +122^+20 = 366 ; ^ '
366
.•. average age = — = 13^ years.
Ex. 3. In a class of 25 children, 19 have attended during the week.
Bays attended by children: 5 for 5 days, 6 for 4^, 3 for 4, 2 for 3^,*
1 for 3, 1 for 2, 1 for ,> day. Find the average number of days attended
by each child.
The whole number of days attended by class
= (5x5 + 6x4|- + 3x4 + 2x3^- + lx3 + lx2 + lx^)
=25 + 27 + 12 + 7 + 3 + 2 + ^ = 76| days;
/. average attendance = 1^ ='^^^ ^^^
25 50 100
= 3-06 days.
Ex. 4. In a school the numbers for the week were :— Monday
moVning 67, Tuesday morn. 60, Wednesday morn. 65, Thursday morn. 68,
Friday morn. 62, Monday afternoon '5 more than the average of Monday
-..,-1 ^«^:,vicij iiivnnjij^r, iuoouuy uib. yy, » euDesaay ait. 'u iess tban the
average of Tuesday, Thursday the average of Monday morn, and Tuesday
aft., Friday ^ft, GO, Find the average attendance for the week.
220
ARirmiETia
Number of children who attended on
Monday = C7 + 64
V ' Tuesday = GO + 50
"Wednesday = 05+59
Thursday = C8 + 63
Friday = C2 + GO ,
.-. the total number of children who attended on the 10 occasions = 027 ;
.-. average attendance = ~= 62-7
10 • *
f w n ^; ^ ^T ""^ ^^^ "'"'' '' ^'^ '^ ^ ^''^""^^^t eqnnUy appor-
tioned between wheat and barley ; it is valued at £930 a year when the
average pnce of wheat is 6. a bushel, and that of barley' Ja buThe
find 1,0 ren when wheat rises to the average price of Ts. Id. per bushd
and barley to that of 5«. 3df. per.busheh «• per Dusiicl,
First we must find the number of bushels of wheat and barley at
the given rent of £930. ^ ^'
£930
~2- = ^465 the sum to be raised by each kind of grain;
4G5 X 20
> /. - -^ = 155 X 10 = 1650 bushels of wheat ;
465 X 20
' •'• "~~4 465 X 6 = 2325 bushels of barley ;
. .-. rent in latter case = (1550 x 7^ + 2325 x 5i>.
= £1191. lis. Bd.
_ Ex. LVir.
county wal'/eso' Tl''\'\' '^^^'''' ''^''^ ^''' '' ^ *^^«^^^- ^^ «
JJ.-f . ' *^^ lowest, $84: the highest salary paid in n city
1 fTA ''f ''""'' ^''''- '''' ^'^Shest in a town |1 000 thi
owe3t,$140: the highest in a village, $6C0 ; t],e lowest $2T Tnd
I.e average of the highest salaries, (2) the average of the lowe^
(3) the average salary of a teacher for the year 1865, in Onta-io
2. The number of quarters of trvmn Jmnnnf-,! ,-„*.. ^ . ' . -.
woocisave years woro 2679438, 2958272, 8030293, 8474302, 2248161,
amm
cjcaslons = 027' ;
eqmUy appor-
year when the
7 4:3. a bushel ;
Qd. per bushel,
and barlej at
gram ;
I teacher in a
aid in a city
llOOO; the
$270. rind
f the lowesf,
atai'io.
onnirj in 11
02, 2248161,
DIVISION INTO PEOPOETION-^ PAEIS. 221
2827782, 2855525, 2588234, 820C482, 2801204, 8251901 ; find the
average importatiun dunng that i-eriod.
8. If 60 quarters <,f wheat are sold for $8.40 per quarter and 100
quarters for $8.80 pev quarter ; what is the average price per bushel ?
4. In a class of 23 children, 8 are boys, 15 girls. The age of the
boys-4 of 8, 2 of 11, 2 of 12. Of the girls->5 the average age of the
bojs, 4 of 9, 2 of 10, 4 of 13. Find the average age of (1) the bovs
(2) the girls, (3) the whole class. ^ ^^ ^ '
5. There are 25 children on the register of one class in a school.
19 have been present at one time or other during the week. The sum
of days on which t!ie children have attended is S^. W.,at is the
average number of days per week attended by each child ever present
dunng the week, there being no school on Saturd:iy or Sunday ? Give
the answer in deciranls.
6. In a school of 7 classes, the average number of days attended bv
each child in Class I. is 4 5 ; Class II., 4 ; Class III, 3-9 : Cla^s IV 4-1 •
Jass v., 3-6; Cla.s VI., 4-2; Class VII., 3-3. Find the avera4
number of days attended by each child in the school. "^
7. A Farm is valued at the yearly rental of $1812 ; one-third of
the rent is payable in money, one-fourth in wheat, and the rest in
barley, the average prices being as follows: wheat $1.61 a bushel and
barley 75 J cents a bushel. What will the rent amount to when the
average prices of wh- H and barley are $1.75 and 85 cents per bushel
respectively ?
8. A tithe-rent of £310 per annum is commuted in equal parts into
a corn-rent consisting of wheat at 565. per qr., barley at 32.9 pe- qr
and oats at 223. per qr. ; find its value when wheat is at 04* per qr" '
barley at 445. per qr., and oats at 245. per qr. *- ■ ^ •»
DIVISION INTO PROPORTIONAL PARTS.
^ 175. To divide a given number into parts which ihall le propor-
tional to certain other given numbers.
A w?rf ''"' ""^ "''' ^'^""^ "'^^ ^^ "^^^'^ ^y *^^ "method employed in
Art (156), or by the following.
Rm-E. As the sum of the given parts : any one of them : : the entire
quantity to be divided : the corresponding part of it.
222
ARITHMEnO.
This statement mnst be repeated for each of the parts, or at all
events for all but t!:e last part, which of course mny either be f.,und
by the Rule, or by subtractiBg the su.n of the values of the other paita
from the entire quantity to be divided.
Ex 1. Divide 1128 among A, B, and 0, so that their portions may
be as 7, 11, and 14 resptctively.
Proceeding according to the Rule given above,
82: 7::$128; J^'sshare;
82: 11:: $128: i?'d share;
whence A'b share = $28, and ^'s share = $44.
Cajh&TQ may be found from the proportion
82: 14:: $128: C's share;
whence C's share = $56 ;
or by subtracting $28 + $44, or $72 from $128 which leaves $56
as above. ' '^ '
The reason for the above process is clear from the consideration, that
$128 ,s to be divided into 32 equal parts, of which A is to have 7 parts.
x» 11, and Cl^. ^ * '
Independent method.
$128 is to be didived into 82 equal parts ;
therefore — of $128 = $4,
n i
((
i(
((
32 of $128 = $28,
11
of $128 = $44,
32
g2 of $128 = $56.
f'A ^\^: ?'J''^^ f^^^^^ """^"^ ^ P®''°°'' ^' ^' ^. A in the propor.
ti<5ns of 1^, I, J, and |. ^ *
77
iSnm of shares =
60'
•.77 1
' 60 ' 2" *' ^^^^^^ - -^'s share in pounds ;
whence. A'a share = £4285. 14*. B^d,
Wi
FELLOWSim* OR PARTNERSHIP.
223
Similarly,
^'3 Bliaro = £2:57. 2s. lO^J., C'b share = £2142. iTs. 1«J.
jD's share = £1714. 5s. S^tZ.
Ex. 8. Divide $45000 among A, B, C, and D, so that ^'a share : ^s
share :: 1 : 2, I?'s : C's :: 8 : 4, and C's : i>'8 :: 4 : 5.
la this case, ■ '
B'b share = 2 ^'s share, 8 C'a share = 4 J5's share,
4 i>'8 share = 5 C's share ;
.-. we have ^s share = ^ 5's share = | ^'s share,
and D'b share = | C'd share = ^ ^'3 share ;
,'. ^'s share + ^'s sharo + C's share + D's share
= A'b share (1 + 2 + f + Y),
= 9 w4's share ;
.-. -4's share = $5000, iJ's = $10000, C's = $13383. 83 J,
i>'s = $16666. 66|
FELLOWSHIP OR PARTNERSHIP.
176. FELT-o-wrsniP or Partxeeship is a metliod by which the re-
spective gains or losses of partners in any mercantile transactions are
determined.
Fellowship is divided into Simple and Compound Fellowship : in
the former, the sums of money put in by the several partners continue
in the business ^'^^' the same time ; in the latter, for different periods
of time.
iiPLE FELLOWSHIP.
177. Examples in this Rule are merely particular applications of
the Rule In Art. (175), and that Rule tlierefore applies.
Ex. 1. Two merchants, A nnd B, form a joint capital ; A puts in
$1200, and i? $1800 : they gain $400. How ought the gain to bo
divided between them ?
$(1200 + 1800) : $1200 :: $400 : A\ share in dollars,
whence, ^'s share = $160 and .'. B'a share = $240.
creditors by the same method.
224
ARITH3IEnO.
Ex. 2. A bankrnpt owes three rrerlitors. 4 7? atiI n /'i*-- ^m,.
^050 ; £175 ;: £422J : vi'a share,
£650 ; £210 :: £422i : i?'s share, >
Whence ^'s share = £118. 15.., i?'« share = £180. 10*. •
.*. C's share = £172. 5«. *
COMPOUi^D FELLOWSHIP.
1 y^}; ^'^'^' ^^"^"^^ «" the times into the .ame denomination an.!
"^^ '"'^"'^ '''"^ '' ''' ''- ^' --^ conanuanciraT'th:'
As the sum of |ill the products : eadi particular product •• th« wi .7
quantity to be divided : the corresponding share " '^'
■^f* ^' ,-^"^^^^"*^'-^°f« partnership; -4 contributes $15000 for 9
Proceeding by the Rule given above,
$(15000 X 9 . 12000 . G) : $(15000 . 9) :: $5750 : ^'s share of gain
or $207000 : $135000 :: $5750 ; ^'s share of gain '
and e207000 : $V2000 :: $5750 : ^'s share of gain ; '
whence, J's share := $3750, and ^'s share = $2000.
The reason for the alove process is evident from the consideration
y tiirjes $15000 f .r 1 month ; and one of $1^000 for 6 month +..
of e times 112000 for I ™.nth : Lence, ul i„l!, a 'C^'tC Z
Kdered, Iho qaestioa then booomos ono uf Simple FellcveWp
fr every 10,,. t a a n,a„ paid, a wo»a„ paid C... and a servant I I
Ivantp^r *" ^"= ""^ ""^•'^ •=" <>-'' ->. — . -a
,. !• "'° "^ l"'- 1^."'' " -'"O °t 1'- 80 -women at 6,. = ISO at 1,.. and
iu ourvauia aii as. =. ao at 1». ; and 200 + 180 + SO = 410.
EQUATION OF PAYMENTS.
225
Hcnco we have
410 : 200 :: £41 : 20 men's slmro (in pounds) ;
410 : 180 :: £41 : 30 women's sliaro (in pounJB) ;
( 410 ; 80 :: £41 : 15 servants' sliaro (in pounds);
.*. 20 men's shares - £20, 80 women's bhares = £18,
and 15 servants' shares = £8 ;
( .*. each man paid £1, each woman 12s., and each servant 4*.
EQUATION OF PAYMENTS.
1T9. "When a person owes nnother several sums of money, duo at
different times, the Rule by which wo determine the just timo when
the whole debt may be discharged at one payment, is called Iho
Equation of Payments.
Note. It is assumed in this Rule that the sum of the interests of
the several debts for their respective times equals the iutcrest of tho
sum of tho debts for tho cqunted time.
Rule. Multiply each debt into tlio time which will elapse before
it becomes due, and then divide the sum of the products by the sum
of the debts ; the quotient will bo the equated time required. ^^
Ex. 1. A owes B $50D, whereof $200 is to be paid in 3 monthr,
and $303 in 5 months : find the equated time.
Proceeding according to the Rule given above,
then (200 x 3 + 300 x 5) = (200 + 300) x equated tune in months,
whence, equated time = 4|- months.
The reason for the above process^ in accordance with our assumption,
is clear from the consideration that tho sum of the interesfs of $200
for 3 mofiths, and $300 for 5 months, :s the same as the interest
of $(G0O + 1500), or $2100 for 1 month; if therefore A has to pay
$jGO in one sum, the question is, how long ought ho to hold ib so that
tlio intcro^t on it may be t':o same as tho interest on $2100 for 1 month.
Tho statement therefore will be this:
whence, reqilired number of months = ^\ months;
226
ARrntMETIC,
In this caso, <
(200 X 8 . 800 X 8 . COO X nun.ber of montl.s required) = 1000 x 9
or 500 X number of montlis required = GOOO ; * '
whence number of months required = 12. '
ill:
Ex. LVIir.
8 fown/ '-7'"^ '^'^"^^^" consisting of 72 men is fo be raised from
1^2 ^;'-'--^-" respectively 1500, 7000, and 9500 men Ho^
many must onch town provide ? .
as 5 f 16."'"''° *'*•'' '""' '"° porta which shall bo to cch other
^^i:il:,^: "'''""- «=«° -'« p«'-t. which .h^H ho >„ the
n, i*; '^.'""'';''"P'. °^«' ^ £230. 6..8d.,S£20S. IC, and (7 £141
13».4J.i h,s estate is worth £431. Uj how .nuch wil ASaniO
roceive respectively ? ^ "»" ^a, x., ana c/
5 A mass of counterfeit metal is composed of fine gold 15 na^ts
dver 4 parts,, and copper 3 parts: find how n.uch of each is requir^
in makin^j 18 cwt. of the composition, (cwt = 113 ) ^
and t'ho 'llT^Z7 ''':'f':''''^l- ^^-^^ ^^^^^ ; the one put in $10560
and the o>hor $oGiO ; wh.t is each person's share of the profits?
lMn/;i ""' ff*"'".^^b^tance there are 11 parts tin to 100 of copper
Xmd the weight c.f tin in a piece weighing 24 cwt. ? ^
8. A man leaves his property amounting to £13,000 to be dividpd
amongst h,s children, consisting of 4 sons and 3 dangrrs the t" r e
find i: It: :f ::ci:"' '- - ''''''-' -"^ ^^^ - '-^'^^ ^^^^^ i
9. Tvvo person., A and i?, are partners in a mprcantile concern
and contnlute $6760 and $9000 capital res^ectiv.lv • i f^- --^
uxoHAiraE.
227
eo3 witli I ho
10 per cent, of the profits for managing the basinees, and tlio remain-
ing profits to bo divided in proputtiou to the capital contributed by-
each ; the entire profiL at the year's end ia $8840 ; how much of it
mast each receive ?
10. Divide $480 among Aj B, C, and D, bo thfit B may receive as
mucli as -4 ; (7 as much as A and B together ; and I) as much as A, j?,
and C together.
11. Divide £11,876 among A, B, and G, so that as often as A
gets £4, B sliall get £8, and as of.tn as B gets £0, C fhall get £5.
12. A commences business with a capital of $1000, two years
nfterwards ho tnkes B into partnership vvitli a capital of $15,000, and
in 8 years more thoy divide a profit of $1500 ; required i?'a share.
18. $700 is due in 8 months, $800 in 5 months, and $500 in 10
months ; find tlio equated time of payment.
14. Find the equated time of payment of £750, one hu'f of which
is due in 4 months, {^ in 5 montlis, and tlio rest in 6 months.
16. A owes B a debt payable in 73V months, but ho pays J in 4
months, i in 6 months, ^ in 8 months ; when ought the remainder to
be paid?
16. Ay B, and Crent a field for £11. 6«. ; A puts in 70 cattle for 6
months ; B 40 for 9 months ; and C 50 for 7 months ; what ought C
to pay ?
17. A, B, and G invest capital to the amount of $7000, $5000,
and $8000 resjiectively ; A was to have 26 per cent, of the profits,
which amount to $4500 ; wliai share of the profits ought C to have ?
18. A and i? enter into a speculation; A puts in £'.0 and J5putg
in £45 ; at the enl of 4 months A withdraws i- his capital, and at the
end of 6 months i? withdraws J of his; C then enters with a capital
of £70 ; at tlio end of 12 months their profits aro £254 ; how ought
this to be divided amongst them ?
EXCHANGE. /
180. ExonAXGE is the Rnlo by which wo find how much money
of one country is equivalent to a given sum of another country, accord-
iiAr to a iriven course of Exchani'Q^ _
228
ABITHMEno.
,1'81. By the OoTTBSE of Exohangb is meant the variable sum of tha
money of any place whiclj is given in exchange for ajized sum of money
of another place. The Couese of Exchange between any two
countries will be affected by whatever causes may increase or diminish
the demand for Bills of Exchange between them : tlius, for instance
in London, one pound sterling, a fixed sum, is given for a variable
number of French francs, more or less, according to circumstances.
182. By the Par of Exchange is meant the intrinsic value of the
coin of one country as compared with a given fixed sum of money of
another.
The par of exchange depends on the weight and fineness of the
coins, wliich are known either from the Mint regulations of the differ-
ent countries, or by direct as3:iy. If the metal from which the par is
calculated be not a standard of value in both countries, its market value
in that country in which it is not a standard must be taken into account.
Thus, in the United Kingdom gold is Jie only standard of value.
183. In orde^ to facilitate mercantile transactions between persons
residing at a dlstanco from eacli other, payments are usually made by
Promimry Notes, Drafts or Bills of Exchange. The holder of either
of these BUls being entitled to obtain its value in gold from the party
on whom it is drawn.
A Promissory Note is a written engagement to pay a sum of money
after the expiration of a certain time.
FORM OF PROMISSORY NOTE.
iamcd t^o/te.
Wu^^^4 ^, /c^/dK
^
EXCHAKGE.
229
Note. Tlio noU *ntf*^be stamped^ and on the stamp some important
element of the notb must be written, such as the amount of the note or
the signiiture of the maker. After being signed, no alteration whatever
should lo made.
FORM OF DRAFT.
^/a c/<^cuu4, ^tiauUe, /Ae 4«jn cf Kywc Aune^ea ana
^itit/pe ^cuaU^ fauee ieeeipeaj ana c^aiae ^e dam^
/c account c/^
0^/^ VW. ^imeiea/.
^C^i€ ^
?A
FORM OF BILL OF EXCHANGE.
tyta^ty im^d a^ei aian^ cf mM f ts^Ha^J vf
Wtzcnanae /'^ cfecc^m ana fyAuacf ^Ae tsame ^(enci €ina
<mfe €m/zafaj , Aat^ ^ /Ae cic^i c4^ ^a?ned Cfaneuuincu
WiaA/ Ati^aiea Aoan<Ai C/^eiUft^^ tta/ae iecetuea, anee
/aace /Ae dafne 'wcm oi it^'mccU Acci/Aci amj^ce /a »n'U aceoan/,
S^ de ^a^c'cna/ mind c/ ^<^nJ,
230
ABITHMETIO.
mr
Note 1. Three Bills of Exchange, constituting a set, called First,
Second and Third of Exchange are usually made for the same amount
and sent off by different conveyances, to provide against delay, if the
first, should miscarry: only one of course is paid, the others being
cancelled.
Note 2. The party who signs either of the above forms is called the
maher or drawer of the bill ; the party on-whom drawn, is called the
drawee, and after accepting it, the acceptor.
The usual way of accepting a bill is for the drawee to write his name
across the face or back of the bill ; the meaning of which is, that he
undertakes to pay the bill When it becomes due. The party who buys
a bill, is called the luyer or remitter; the party who has possession of
it the holder, and the party to whom the money is to be paid, the payee.
If the holder or payee of a bill wishes to dispose of it either by
selling or transfering it to another party, ho writes his name on the
back of it, i. e. he^endorses it, and is then called the endorser.
If a bill is not paid, or refused to be paid, it is protested; protesting
a hill consists in a party called a Notary Public notifying the maker,
endorser, «feo., of the non-payment, or non-acceptnnce of the bill.
Note 8. By an Act of the Parliament of the United Canadas, passed
in 186-, the dollar was declared to be in value, one-fourth of the
pound currency. The pound currency to be of the weight of 101-821
grnins Troy and of the standard fineness of the gold coinage of the
United Kingdom.
From the above it follows, that the legal value of the sovereign or
pound sterling = $4.86^, which also is its intrinsic value.
But by an Aot passed mnny years ago, tlie sovereign was declared
to be only equal in value to $4-444, or £9 (sterling) = $40; ppd tiiis is
the value which almost invariably is quoted in mercantile trmsnc-
tions: the premium on this depreciated value of the sovereign which
will make it equal to its intrinsic value is ^ per cent.
Ex. 1. A merchant in Toronto has to remit to one in London £785
sterling; how many dollars will he have to give for the bill of ex-
change ; exchange at 109 per cent., commission \ per cent. ?
By old statute £9 = $40
A.C\
.•.£1 = $~;
iP
EXCHANGE.
rat6 of exchange to the buyer is 109+.J=109J ;
109J
100
231
.-.£1 = $^/
73
/. £735 = $11 X 786
= $3577.
o7S" '°^°^^° "^'^^^^^ -^" ^- *o P*7 13577 for the Bill
whet 594 Irt'lso Te"" '' "''"^.' '^*"^^^ ^^"^^" ^^ ^^^o-
then 694 mils., 480 rees : 1 mil •• £-1^0 ir^ q^
'• '*'^'^*'- 16s. 9f?. : course of exch*
or 694-48 mils. ; 1 mil. :: 3812H. • *
whence course of exch. = 041 24 d
S^ ;:; S :; pX^z «•- ^- -• ^^ -e.. ,o«.a .,
tlKse place's i. Wn It [fcalVsT^ "n"''°°' "°'' """'• *»• "^
To.o„^"or:r::f^rxt::^'"^^^^^^^^^^^
remit directly through the Bank o Sa^tK^J't " ^T '°
on Paris lelng a. 8 per cent, diacoun o^ J wal ofNew ?T'
change on Paris bei.ig at 2 per cent np.mv i ^'"*' "'
rate of e.chnnge between Paris and J 17 T'- " ^^ '"^ "^ ^"""o"-
tn-een London'and Canada I iga^°"ot.r"' "• "''' "^ "'"' ^«-
bebg J per cent. '^ ^*' oouimission either way
^ Value offiano =10-r cents.
8i per cent, discount = -7 ..
.-. eiohange value of franc = 19 cents •
232
.ABITHMEnO.
.'. bill of exchange costs $5100 x — ^
= $969.
U. S. value of franc = 18-6 cent?,
2^ per cent, premium = 'SllS .. ;
;•. exch. value of franc = 19'1115 cents ;
.-. bill of excli. costs $?^ x 191115
= $974.69.
British value of Irauo = 9.7od. ;
.-. 5100 franc3 =£207*1875,
exchange between London and Canada, together with commission, is
at 108 per cent. ; therefore value of £207*1875 stg. is $994.50 Canadian
currency.
Hence it is cheaper to remit by the Bank of British North America.
Ex. 4. £1 iEnglish being = 25*1 francs, 3*75 francs being = 105
kreutzers, 60 k'rentzers being = 1 florin ; find in English money the
value of 1143 florins.
1143 florins = (1143 x 60) kreutzers,
/ 3*75\ ,
= ( 1143 X 60 X -^TTn: 1 franca,
105.
/ 8-75
= £fll43 '<60x-^ X
= £96. 88. 6^.
25'4y'
Value oi Foreign Coins.
U.S.
BovereTgn $4.84
Guinea 5.00
Crown. English 1.06
Shilling piece 23
Franc, France and Belgium 186
*' (franc = 100 ceniimes)
Florin, or Gulden, Holland 40
Florin, Austria and Augsburg. . .486
Florin of S. Germany States. . . .40
Ducat, Austria (gold)
Intrinsic.
. $4.86
. 5.11
. 1.216
. 243
. -197
.418
.486
.417
2.286
U.S.
Klx-dollar or Thaler, Prussia . , $0.69
(thaler = 80 silber groschen = 860 pfennings)'
Marc Banco, Hamburgh 35
Specie-dollar, Denmark l.05 ,[,][
Pwigsbank dollar " 52 '
Specie-dollar, Norway 1.06
• Milree (1000 rees), Portugal.... hl2 ........[[
" Brazil 54
Real de Vellon, Spain . .' 05
Peal de Plata, " iq
Pillar-dollar.
u
1.00
Ruble, Russia (silver) 75
Imperial, " (gold).,.. .' .*
Dou oon, Mexico (gold) 15. 60
Dollar, « (silver) 1.00
Dollar, « (gold) i^qo
Lira, Sardinia jsG
Eaole, U. S. America (gold) ... 10 00
i>^Jiar, " u :.__ 1^00
233
. $0,727
Note. In the above table the column marked U. S
. »30
. 1.106
. .552
. 1.105
. 1.086
. .53
. .05
. .133
, i.or
.786
7.974
15.74
1.07
.986
.197
10.00
1.00
Mnitm ar^ irubner was made use c. for obtaining the weight and
nneness each coin, and British standard silver'taken atl . M
steilmg, Its average market price, per ounce.
486. New Brunswick has the same currencj as Ontario and Quebec.
IS en^rrr 1^ *'^^^-^^^^.^-S<io-> - the currency of Nova Scotia,
1. equal to $5 ; the silver coins in proportion to their value of the gold
T«l/n"i ^''"'^ ^'^"^''^^ ^^^'°'^' *^'^ ^"'^=«^ sovereign equals 80 shillings
Island cnrronoy, silver coins in proportion. °'
Tho American Engle ($10) is legal tend.r for £3. Wand Carrency.
In Newtoundl.md the British sovereign is legal tender for $4.80,
Sliver corns m proportion to their v.due of the sovereign.
The American Eno-lA iq 1^1,^1 +^«-i x-— ^^ «- '" , ..
in proportion. ~~°" " '^"""^ '"""" "''" *'*''""' ""^ "^*^"«* ^'"''^
234
ABTEHHETIO.
MONEY.
(English.) i
1869 sovereigns are coined from 40 pounds Troy standard gold,
which is l^ fine ; therefore it follows that
Weight of a sovereign =- 123-27447 grains,
Weight of pure gold = 11300159 "
A Pound Troy of standard silver, which is f J fine, is coined into
66 shillings ; therefore '
"Weight of a shilling = 87-27273 grains
Weight of pure silver = 80-72727 "
NoTB. ^Mint value of an ounce standard silver is 5s. 6d,. but usual
market price is 5«. 2d,
(Feenoh.)
The fineness of gold and silver coins in France is the same, viz. yV
The mode of exiire?sing the fineness of the coinage adopted by French
assay ers, is to state the number of pans of the pure metal which are con-
tained in 1000 parts, and to say tiiat the metal is so many millUmes fine.
One kilogramme of standard gfild is coined into 3100 francs
silver 200
u
i(
t\
u
u
(United States.)
By the Act of 1852 the weight of the-iilagle was ordered to be 253
grains -^jf fine ;
.-. Weight of the Eagle = 258 grains
Weight of pure gold = 232.2 "
The fineness of the silver coins is the same as that of the gDld.
The silver dollar coined 1657 is 412.5 grains in weight.
(Canadian.)
By an Act of Parliament of the United Canada?, the poi:nd currency
was ordered to be 101-321 grains in weight, of gold of the standard
fineness prescribed by law for the gold coins of the United Kingdom
on the first day of Aug'is*-, 1854.
By law the dollar U defined to be one-fourth of the pound. Tho
gold raglo of tho United States coined since 1852, is legal tender for
ten 4oiiars.
iSXCHANGE.
235
Ex. LIX.
«.o ^}\ ^ '^^^^'^^''^ ^'^ Toronto has to remit to ono in Berlin (Prussia)
612 thalers ; how many dollars will lio have to give in order to par
the amount, commission i per cent., exchange at par?
(2) Convert 4750 milrees, 280 rees into English money, at 64K a
milree, and bring the amount into Canadian currency, exchange
at 108 per cent. •' » &
(3) Convert £246. 15.. Gd. into piastres and rials, exdhange being
at 47'}a. a piastre. (1 piastre = 8 rials.)
(4) By an Act of the U. S. Congress in 1834, it was enacted that
the weight of the eagle should be 258 grains, and its fineness 899-2
miih^mes. Frojn this calculate the par of exchange between G B.
and U. S. of America.
(5) By an Act of the U. S. Congress, 1837, it was ordered that
the dollar shon]d weigh 412t. grains of silver, ^ fine. Calculate the
silver par, British standard silver being 5.9. l^d. per oz.
(6) By an Act of the U. S. Congress, 1853, it was enacted that in
thecomageof half-dcllar., quarter-dollars, &c., the lialf-dollar should
weigh 1 92 grains of silver, j% fine. Calculate the par at Bs. l^d. per oz.
(7) A merch.iKt in London is indebted to one at St. Petersburg
15,000 rublei: the exchange between St. Petersburg and En-land
13 60^. per ruble, between St. Peters\)urg and Amsterdam 91d. per
ruble, and between Amsterdam and London " 36«. 3^. per £. sterling-
which will bo the m6st advantageous way for the London merchant to
be drawn upon ?
(8) What snm in English money must be given for 500 francs,
when 25-G francs is exchanged f(,r £1 ? What is the arbitrated price-
between London and Paris, when 3 francs = 480 rees, 400 rees = BU
Flemish, and 355. Flemish = £1 ? ^'
(9) A person in London o.ees another in St. Petersburg a debt
of 460 rubles, which must be remitted through Paris. He pays the
roquisita snm to his broker, at a time when the exchange between
London and Paris is 23 francs for £1, and between Pa. is and St Pe-
tersburg 2 frnncs for one ruble. The remittance is delayed until the
rates of exchange are 24 francs for £1, and 3 francs for 2 rubles.
What docs tlio broker crain or loao hv th^ fmnoo^fS.
(10) A gentleman has £3000 in the 3 per cents at 97^: ho wishes
236 H
ABITHMETK.
to sell nnd invest tho proceeds in Canada Dominion stock at 106,
yielding 7 per cent, dividends annually. Find tho iilterntiun inhia in-
come, exchange betwe^in tho United Kingdom and Cannda being at 8|
per cenr. premium, commission of ^] per cent, being allowed on each
transaction.
SECTION YI.
SQUARE ROOT.
186. The Sqitaee of a given number is the product of that number
multiplied by iiself. Thus 36 is the square of 6.
The square of a number is ft-equently donotoci by placing the figure 2
above the number, a little to the riglit. Thuc 6- denotes the square
of6, so that6-'=^3G,
187. The Squaiie Root of a given number is a number, which
when multiplied by itself, will pro(hico the given number.
The square root of a number is sometimes denoted by placing the
sign V before the number, or by placing the fraction ^ above tho
number, a little .to the right. Thus V3G or (36)i denotes the square
root of 36 ; £0 that V36 or (36) J = 6. - •
188. The number of figures in the Square Root of any number
may readily be known from the following considerations:
The square root of 1 is 1
100 is 10
10000 is 100
1000000 is 1000
&c.
IS
&c.
Hence it follows tliat tho square root of any number between 1
and 100 must lie between 1 and 10, that is, will have one figure in its
inteirral part; of any number between lUO and 10000, must lid between
10 and 100, thfit i«, will have two figures in its integral pnrt ; of any
number between 10000 and 1000000, must lie between 100 and 1000,
that is, mui^t have three figures in its integral part; and so on.
"whoreri.'io, if u point ho piuced over tlie units' plaee of the irairiber,
and thence over every second figure to the left of that place, the points
SQUARE EOOT.
237
will she-w the number of figures n the integral part of the root. Thns
tlio square root of 91) conjist?, so far ns it is integral, of o?ic figure;
tjjat _ot 193 of two fignrco; that of 17G432 of three figures; lliat of
1764321 of four figures; and so on.
Again the square root of -Ol ij 1
•0001 is -01
•000001 h -001
•00000001 is -0001
&a &c. .
it appears, that in extracting the square root of decimal:!, the decimal
places must first of all bo made even in number, by affixing a cypher to
the right, if this be necessary ; and then if points be placed over every
second figure to tlie right, beginning as before from the uuits' jjlace of
whole numbers, the number of such points will show the number of
decimal iilaoes in the root. '
189. Bute for extracting the Square Boot of a number,
' Place a point or dot over the units' pLice of the given number ; and
thence over every socoml figure to the left of that place ; and thence also
over every second figure to the right, when the number contains de-
cimals, annexing a cypher when the number of decimal figures is odd;
thus dividing the given number into periods. The number of points
over the wholo numbers and decimals respectively will shew the num-
ber of whole numbers and decimals respectively in the square root.
Find tha greatest number Avhose square is contained in the first
period at the left ; this is, the first figure iu the root, which placa in
the form of a quotient to the right of the given number. Subtract its
Dquare from the first period, and to the remainder bring down, on the
right, the second period.
Divide the number thus formed, omitting the last figure, by twice
the part of the root already obtained, and annex the result to the root
and also to the divisor.
Then multiply the divisor, as it now stands, by the part of the root
last obtained, and subtract the product from the number formed, as
above mentioned, by the first remainder and second period.
If there be more x>eriod3 to bo brought down, the operation must bo
repeated.
238
AEITHMETIO.
Ex. 1. Find tlio square root of 13G9.
incb (37
9
C7 ■
4G9
4G9
After pointing, according to the Rule, we take the first period, or 13,
and find the greatest number whose pquare is contained in it. Since
the square of 3 is 9, and that of 4 is 16, it is cleiir that 8 U the greatest
number whoso square is contained in 13 ; therefore place 3 in the form
of a quotient to the right of the given number. Squire this nuniber,
and put down the squnro under the 13 ; subtract it from the 13, aud to
the remainder 4 afl3x tl>e next period 09, thus formini; the number 4G9.
Take 2x8, or 6, for a divisor ; divide the 409, omitting tlie last figure,
that is, divide the 46 by the 6, and we obtain 7. Annex tl»o 7 to the
8 before obtninqd and to the divisor G ; t!ien muUiplvin,; the 07 by
the 7 wo obtain 469, which being subtracted f oin the 409 before
formed, leaves no remainder ; therefore 37 is tlia square root of 13G9.
Reason for the above process
Since (37)^=1369, and therefore 37 is the square root of 1309; wo
have to investigate the proper Rule by wliich the 37, or 30 + 7, may be
obtained from the 1309.
Now 1369=900+d69=90C +49+420
=(30)2+7=+2x30x7
=(30)3+2x30x7+7^
where wo see that the 1369 is separated into parts in which the 80 and
the 7, together constituting the square root, or 87, are made distinctly
apparent. Treating then the number 1369 in the following form, viz.
(30)2+2x30x7+7^
we observe that the square root of the first part, or of (30)^ is 80 ; which
is one part of the required root. Subtract the square of the 30 from the
whole quantity (30)^ +2 X 30 X 7+ 7^, and v/o have 2x30x7+7= nmainlng.
Multiply the 80 before obtained by 2, and we see that the product is
' contained 7 times in the first part of the remainder, or in 2 x 30 x 7 ; find
nn . tr
XT- * rt Ct/\ Xl~.»— .^— ..1—!— — t. Mta\ . rr ->•-. *rrr «.)k«*.v
tihr i.X.i
--I, -■■■
1-*.*,.
quantity is contained 7 times exactly in the remaining 2x80x7+7 or
rioil, or 13,
1 it. Siiico
he greatest
in the form
lis number,
) 13, and to
urabcr 409.
last figure,
1)0 7 to the
-: tlieCTby
4C0 before
)t of 13G9.
fl360; wo
+ 7, may bo
the 80 and
distinctly
form, viz.
3 30 ; which
30 from the
nmnining.
5 product is
< 30 X 7 ; nnd
mis iuuLci
< 30x7+7 or
SQUARE BOOT,
239
469 ; 80 that by this division we shall gain the 7, the remaining part
of the root. If we had found that the 2 x 30+7 or 67, when multiplied
by the 7, had produced a larger number than the 469, the 7 would
have been too large, and we should have had to try a smaller number,
as 6, in its place.
The process will be shewn as follows ;
(30)3+2x30x7+7^(30+7
(80)3
2x80+7
2x80x7+73
2x30x7+7=*
This operation is clearly equivalent to the following :
900 + 420 + 49 (30 + 7
900
60+7
420 + 49
420 + 49
This again is equivalent to the following:
1369 (37
9
67 '
469
469
which is the mode of operation pointed out in the Rule.
Note 1. The reasoning will be better understood when the
student has made some progress in Algebra.
Note 2. The divisor obtained by doubling the part of the root
already obtained, is often called a trial divisor, because the quotient
first obtained from it by the Rule in (Art. 189), will sometimes be too
large. It will be readily found, in the process, whether this is the
case or not, for when, according to our Rule, we have annexed the
quotient to the trial divisor, and multiplied the divisor as it then stands
by that quotient, the resulting number should not be greater than the
sumbef from whlcli it ought to be «ubtracted. If it be, the quotient
is too large, and the number next BmaUer should be tried in its place. .
240
AErrnMEiic.
Ex. 2. Find tho square root of 71C90612350625.
71*660512850625 (8467025 ^
i2x8=16f
{2x84=168 J
{2x846=1692;
164
1686
16927
04
709
f.56
11806
10116
118912
118489
5 (2 X 8467=1 6934) ) 1 693402
1(2x84670 = 169340) f
4233506
3386804
16934045
84670225
84670225
' .'. 8467025 is the reauired Bqnaro root. .
190. As fho decimal notation is only an extension or continuance oJT
the ordinary integral notation, and quite in agreement with It, the reason
given for the process in -whole nnmlers will apply also to decimals.
191. To extract the sqnare root of a vulgar fraction, if the nume-
rator and denominator of the fraction bo perfect squares, we may find
the square root of each separately, and the answer will thus be ob-
tained as a vulgar fraction ; if not, we can first reduce tho fraction to
a decimal, or to a whole number and decimal, and then find the root
of the resulting number. The answer will thus be obtained either as
a decimal, or as a whole number and decimal, according to the onse.
Also a mixed number may be reduced to an improper fraction, and its
root extracted in the same way.
Ex. 8. Extract the square root of 53111*8116.
53ili-8il6 (230-46
4 . -
43
181
129
460'
i
21181
18416
'
4tJU
an
2/6510
276516
itinnance of
t, the reason
lecimals.
f the nome-
we may find
thus be ob-
fraction to
find the root
ned either as
f to the onse.
3tIoD, and its
SQUABE BOOT.
241
Ex. 4. Find tho sqnaro root of 4.
This mny bo done by first reducing ^ to a decimal, and then ly
tractjng the square root of the decimal, thus 4=-7l428G...
ez-
•714285 (-845...
.^ 04
104
1685
743
G50
8085
G425
200
or
«>-V| = i/(?^)=;
V86
,7-
1 100
, 1181
85000000 (5-910
1000
981'
1900
1181
11826
71900
70966
therefore
944
5 6-916
/t =
= -845...
Ex. LX.
Find the sqnare roots of
(1) 289; 670; 1444; 4096. (2) 6661; 21025; 173056.
(3) 08596; 37249; 11G64. (4) 998001; 978121; 824464
(5) 29506624; 14356521; 5345344.
(6) 236144689 ; 282429536481 ; 282475249.
(7) 295066240000 ; 4160580082500.
(8) 167-9616; 23-8369; 57648-01. (9) -3486784401; :9-15380329
vviiijo/o; '00203401. (11) 6774409; 6*774409.
(12) 120888-68879026; 240898-012416.
242
AEITHMEHa
(IS) IG; 1-6; 'io; -016.
(15) -0004; -00031; 879-861.
(14) 235-0 ; -1 ; '01 ; 5 ; '5.
(16) 20i; 153}; i; ff^.
1f^TT,-^,4j. (18) 1^; 1,^%; 23-1; 42;
io four places of decimals in each case where the root does not ter-
miaate.
(17)
. 21- rl
> ^2^5 77-
CUBE ROOT. ^
192. The Cube of a given number is the product which arises
from multiplying that number by itself, and then multiplying the re-
sult again by the same number. Thus 6x6x0 or 216 is the cube of 6.
I The cube of a number is frequently denoted by plicing the figure 3
above the number, a little to the right. Thus 0^ denotes the cube of 6,
60 that 63 = 6 X 6 X 6 or 216.
193. The Cube Root of a given number is a number, which, when
multiplied into itself, and the result again multiplied by it, will produce
the given number. Thus 6 is the cube root of 216 ; for 6 x x 6 is = 216.
Tiie cube root of a number is sometimes denoted by placing the sign
y ^before the number, or placing the fraction ^ above the number, a
little to the right. Thus V^TO or (216)3 denotes the cube root of 216 j
so that V216 or (216)i = 6.
194. The number of figures in the Cube Root of any number may
readily be known from the following considerations :
, The cube root of 1 is 1
1000 is 10
lOOOOOO is 100
1000000000 is 1000
&c. is &o.
Hence it follows that the cube root of any number between 1 and
1000 must lie between 1 and 10, that is, will have one figure in its
integral part; of any nnmber between 1000 and 1000000, must lie
between 10 and 100, that is, will have two figures in its integral part ;
of any number between lOOOOOO and 1000000000, must lie between 100
and 1000, that is, must have three figures in its integral part; and
nf til A
5'^ \Sii.
tXTT ^f^— .
:e _
^d-ks.-v4> 1-kr^ V\ I ^ A /^ # 1
T/\.. + l-fc A
i,v,t:vi \'rci iiiv
sill t t,tj
r\lfina
]■■■"■—
number, and thence over every third figure to the left of that place, the
points will shew the number of figures in the integral part <rf the root.
1; 42;
!s not ter-
licb arises
ng the re-
:ube of 6.
16 figure 8
cube of 6,
lich, when
ill produce
6 18 = 216.
ig the sign
number, a
30tof216j
mber may
ireen 1 and
igure in its
0, must lie
sgral part;
etween 100
part; and
ittna of the
t place, the
^ the root.
CUBE EOOT.
243
Thus the cube root of 67t consists, so far as it is integral, of one figure ;
that of 198999 of two figmes ; that of 134198999 of thret figures; and
so on.
Again, since the cube root of -QOl is %
the cube root of -000001 is '01,
the cube root of 'OOOOOODOl, ia -001,
&c. is&c.
it appears, that in extracting the cube root of decimals, t!iO decima:i
places must first of all be made three, or some multiple of three in
number, by affixing cyphers to the right, if this be necessary; awl
then if points be placed over every third figure to the right, beginning
as before from the units' place of whole numbers, the number of such
points will shew the number of decimal places in the cube root
195. HuU for extracting the Cube Root of a number.
Place a point or dot over the units' place of .the given number,'
and thence over every third figure to the left of that place; and
thence also over every third figure to the right, when the number con-
tarns decimals, affixing one or two cyphers, when necessary, to make
the number of decimal places a multiple of 8 ; thus dividing the given
number into periods. The number of points over the whole numbers
and decimals respectively will shew the number of whole numbers
and decimals respectively in the cube root. •
Find the greatest number whose cube is contained in the firpt
period at the left ; this is the first figure in the root, which place in
the form of a quotient to the right of the given number.
Subtract its cube from the first period, and to tho remainder bring
down, on the right, the second period.
Divide the number thus formed, omitting the two last figures, by 3
times the square of the part of the root already obtained, and affix tiie
result to the root ,
Now calculate the value of 3 times the square of the first figure in
the root (which of course has the value of so many tens) +8 times the
product of the two figures in the root+the square of the last figure in
the root Multiply the: value thus found by the second figure in the
roof- nnd Hnhfrnr^f. fTia Y'r^c-.M- A.^ — A-u- i /•
'•'--• ^vntiiu iiuiii Liio iiuiuuer lormea, as above men-
tioned, by the first remainder and the second period. If there
more perio^ to be brought down the operation must be repeated.
be
244
ABITHMEnO,
Ex. 1. Find the cub© root of 15625.
. 15625 ^26
, 2 =8
8x22==12
8 X (20)3 = 8x400 =1200
x20x5 = 800
53= 25
1526
multiply b y 5
7625
7625
7625
'After pointing according to the Rule we take the first period, or 16,
and find the greatest number whose cube is contained in it. Since the
cube of 2 is 8, and that of 3 is 27, it is clear that 2 is the greatest number
whose cube is contained in 15 ; therefore place 2 in the form of a
quotient to the right of the given number.
Cube 2, and put down its cube, viz. 8, under the 15 ; subtract it
from the 15, and to the remainder 7 aflax the next period 625, tlms
forming the number 7625. Take 3 x23, or 12, for a divisor ; divide 76
by 12, 12 is contained 6 times in 76 ; but when the other terms of the
divisor are brought down, 6 would be found too great, therefore take 5.
Annex the 5 to the 2 before obtained; and calculate the value of
8x(20)2+3x20x5+5«, which is 1525 ; multiplying 1525 by 5 we obtain
7625, which being subtracted from 7025 before formed leaves no re-
mainder,
therefore 25 is the cube root required.
Eeaaon for the above process.
Since (25)3=15625, and therefore 25 is the cube root of 15625 ; we
have to investigate the proper Eule by which t^o 25, or 20+5, may bo
obtained from 15625. , - n -4
Now 15625=8000+7500+126
-=8000+6000+1500+125
.t(20)8+3x(20)2x5+8*x20x52+5^
the 6, together constituting the cube root or 26, ara pftde distinctly
otl, or 16,'
Since the
}t number
form of a
attract it
625, tlms
divide 76
ms of the
►re take 5.
I value of
we obtain
res no re-
5625; we
5, may be
distinctly
OUBB SOOT.
245
apparent Treating then the nwnber 16626 in the following form viz.
I (20)'+8x(20)»,<6+8x20x5»+6», *
we observe that the cube root of the first part or of (20)» is 20 • which
is one part of the required root. Subtract the cube of the 20 from thai
whole quantity, and we have 3x(20) x5+3x20x6a+53 remaining.
Multiply the square of the 20 before obtained by 8, and we see that
i^/^ol?*f '^''^'^f ^ ^^^' ^° *^" ^'^* P«'* *>f ^^ remainder, or.
m 8 x(20)^ x5 ; and adding 8 times the product of the two terms of the
root+the square of the last term of the root, thus making 3x(20>«*8
.f'^iu*'^^ see that this latter quantity is contained 6 times ex-
actlym the remamder 3x(20)-x6+3x20x5»+63, so that by this diW-
won we shall obtain the 6, the remaining part of the root
The process will be shown as follows :
(20)»+8x(20)3x5+3x(20)x68+63 (20+6
(20)8
divisor=:f;xC20)2, 1
and
3 X (20)3x6
=6;
^x(?0)2x5+8x20x63+6»
'3x(20)^x5+3x20x53^.g8
8 x (20)3
.-. i3x(20)2+8x20x5+53}x5=
This operation is clearly equivalent to thb following :
8000+6000+1600+126 (20+6
8000
8 X (20)3=1200, and f|»«=6'
(1200+300+25) X 6=
6000+1600+126
6000+1500+125
This again is equivalent to the following :
16626(25
8
8x28=8x4=12, and ff =6
8 X (20)3 ^1200
8x20x6 = 800
+63 = 16
1626
7625
rantt
traaw
which it the mode of operation pointed out ip the Rnk
m^
jjEixaiisfto.
has mado some prdgif^s iti Algebi^
i((ytk i, The divisor wtiich is obtained nooording to the Rn^ '^Iren
M (Ait. 195) Is sometiuiies called a trial divisor, because the number
iirOtii ^6 divi^oQ may be too large, as was the case in the above Itx-
lAmpl«r td Which oftse #e must try a smaller number. We shafl readily
ilfldcittiith Whether the number obtained from the division is too large
i>t ii6tj hecautte if it be too large, the quantity which we ought to
jRibtnidt frOtti the number formed by a remainder and a period will
ttuii out in that Cas^ to b6 largeir than that number^ which of course
it Idug^i hot to 1)6, &ad so we must try a smaller number.
NoTK 8. If at any point of the operation, the number to be divided
by the trial divisor be less than it; we affix a cypher to the root, two
cyphers to the trial divisor, bring down the next period, and proceed
according to the Rule. ,
Ex. 8. ilnd the cube root of 223648548.
223«48543 (607
i5=»=216
tJfialdivisor=Sxd'^ =slOS
trial divisor=i8xti6b)'»il0800
8 X (600)2 =-1080000
8x600x7»: 12100
1092649
7
7648548
t648 76 is not divisible by 108;
7648548 bring down the next period
and affix to the root ;
iSlfv 8*^®^ ^ tim«9, and 7
Beeths likely to be the figure
required; since 7? =848, and
8 is the final figure in the
7648548 remainder.
Therefore 607 is the eube root required^
196. As the decimal notation is only an Extension oi* continuance
of the ordinary integral notation, and quite in agreeMent with it, the
reason given for the process in whole numbers, will apply also to
decimals.
197. To extract the cube root of a vulgar fraction, if the numerator
and denominator of the fraction be perfect cubes we may find the cube
f 9ot of ^ch separately ; and the answer will thus be obtained as a
V
O
81
O
n
a:
8^
8:
8)
8>
OITBB B0OT.
24^
rnlgar fraetion; if not, we Qftn first reduce tbe fraction to a decioiptl,
or to a.^hote nuniber and decimal, ai^d the^ find the roofe ol: the x»-
bvlMix^ nnmber. The answer, will tbns be obl«iiied either w a. d»«i«)i^
or as a whole number and deeimal, according, to the eaoei Mt^, t^.
mixed nnmber maj bo reduced to an improper fraction, and its root
Axtraoted in the same way.
"Rx. 4. Find the cube root of *000007 to three places of deoinialp,
•000*007000 (019
1
8x(10)» =800
8x10x0 .fro
9 W 81
m
9
8x12
=8
6869
6000
6869
141
Ex. 6. Find the cube root of 4 to three places of decisaaln.!
|=-665656556..,
•66i«5b856 ('822
83=611
8x83=192
8x(80)» =19200
8x80>(a » 480
2«^ 4,
- » 19684
5
89868
8x(82)«.2C172
8x(820)* =2017300
8x820x2 <- 4990
2^*= 4
20^124
2.
404424a.
48665^
89868
41fT«55
4044248 :
143807;
248
ABITHMEnO.
198. Higber roots than the square and cube ean somotimes be
extracted by meang of the Rules for square and cube root ; thus the 4th
root is found bj taking the square root of the square root ; the '"^^ root
by taking the square root of the cube root, and so on. '
Ex. LXI.
Find the cube roots of ,
(1) 1723; bSrS; 29791.' (2) 64872; 110S92; 800768.
(8) 681472; 804867; 941192. (4) 2406104; 69426531; 8866427.
(6) 251289591 ; 28372625 ; 48228544.
(6) 17173512; 259694072; 926859376.
(7) 27054036008; 219365327791.
(8) '889017; 82-461759; 95443-993; -000912678;
•001906624; -000024889.
(9) 8, -3, -03. --^ ® 250
(11) 405^; 7|; 3-00415.
^^^^27 5 iii 5 ^•«-
(12) -0001; 1?^^,
16384
to three places of decimals, in those oases vhere the root does not
terminate.
(13) Find the cube root of 238-744896, and also the cube root of the
last-mentioned number multiplied by -008.
04) The cost of a cubic mass of metal is iJl0481. 1». 4<l. at 10». Sd,
a cubic inch. What are the dimensions of the mass ?
(15) A cubical block of stone contains 50653 solid feet ; what is the
•ea of its side ? ♦
(16) A cube contains 56 solid feet, 568 solid inches ; find its edge.
SCALES OF NOTATION.
r 199. From what has been said (Art. 6), it appears that numbers
In the common or decimal system of notation increase in a uniform
[manner from right to left by a ten-fold ratio. The number 10, there-
[Tore is called the radia or base of the scale. It is plain that in any
Ibere are miits in the radix of the scale.
IF-
es be
ie4th
^ root
5427.
3 not
fthe
s. hd.
s the
*
Ige.
ibers
form
lere-
any
5 as
■
SCALES OF NOTATIOIT. 249
Thns, in the Common Scale, we have ^m different characters. If i
the radix were 2, then we would only have two; viz., 1, 0. ,
If 8, then we would have three^ 1, 2, ; and so on. If we select a
radix larger than 10, in that case, we would have to immt a twio
character for each additional unit. Thus, if the radix were 12, for
10 and 11, wo might use the characters r,€.
200. The scales are called, according as the number 2, 8, 4, 6, 6, 7,'
8, 9, 10, 11, or 12, is the radix, the Binary^ Ternary^ Quaternary^
Quinary, Senary, Septenary, Octenary, Nonary, Denary^\
Undenary, and Duodenary Sq^qb,
Note 1. The different scales are sometimes denoted thus; 231 in!
the quarternary scale is written (281)^; 9r*45 in the duodenary is'
written (9r645)ig; and similarly of other numbers.
Note 2. The operations of Addition, Subtraction, Multiplication, ;
Division, Square and Cube Boots, are performed on numbers in any
given Scale as in the common Scale of Arithmetic, care being'
taken that instead of using 10 and its powers, as in the common or
denary scale, we use that number and its powers, which denotes,
the particular scale in which we are working.
201. To express any given nnmber in any assigned scale.
Rule. Divide the given number by the radix ; divide agam the
quotient by the radix ; so continue the division, as long as possible :
finally write the remainders in reversed order ; the result thus obtained
is the number expressed in the proposed scale.
Ex." 1. Reduce the number 48046 ordinary scale to the duodecimal
scale, and prove the truth of the result.
■ 1
t
Proceeding laj the Rule given above. ♦
12
12
12
48046
8587-2
298-11 ore'
24-10 or r
2-0
'■%»i
Therefore the number is 20rt2,
250 ,,
Proof 10
2Prc2
10
25r8-a
10
afT-4
' 10
37-0
4-8
AEITHMETIO.
^ 20r(2 is in the soalo whose radix
is 12; .•.20=2xl2+0=r24,andl0iii^24
goes 2 and 4 over j then 4r=4xl2+10
=58, and 10 in 58 goes 5 times and
8 over ; then 8e = 8xl2+ll= 107,
.*. (20 rf2)i2 = (43046) ^ oj and 10 in 107 goes r and 7 over ; then
72 =» 7x12+2 = 80, and 10 in 86 goes 8 times and 6 over, and so on.
NoTB. The operation of transforming a number from any scale to
the denary, or common scale can be niore readily done as follows.
Take the number in the last example, 20rf2. .
20re2
- • ■\J2 . ■ . .
24 =2x12+0
12
\
298 =
12
\
2x122+0x12+10
8587 =2xl23+0xl2-+10xl2+ll
12
43046 =2 xl2*+0x 123+10x122+10x12+2;
which is the same result as was obtained above by a different method.
p Ex. 2. Transform the nnmber .5056 from the septenary to the
quaternary soale
The division must be performed in the septenary scale.
4
4
4
4
4
4
5056
1165-0
214-3
36-1
6-3
1-2
J^
0-1
.•. number requirea = 123130 ;
and since 128130
= L45+2.4*+3.43 +1,43 +3.4+0 = 1024+512+192+16+12 = (1756),
.'. ftom Ex. 1, we see that the result is correct.
'^
radix
)in^24,
12+10
8 and
= ior,
; thou
A.
sale to
Hows.
ethod.
]o the
ll A •
SCALES OP TiCflATlOTX.
KoTx. 6056 might first have been expressed in the denary soalift
thus^ 5x73+0x7»+6xr+6 = m6+86+6 = l756, and 17^6 then t^s-
iformed from the denary to the quaternary thtis,
4
4
4
4
4
4
1766
489-0
109-8
27-1
'••il
6-8
1-2
0-1
.*. number in quaternary s<Jale as before «r 128180.
Ex. 8. Ilxtraot the square root of 25400544 in the seoAry
25400544 (4112
- 24
121
1221
140
121
1505
1221
12222
24444
24444
E±. LXtl.
r (1) Add together 1445, 22601, 56482, 87, 677, and 6 in the octe-
nary, and also in the nonary scale; the first three numbei-s in the
septenary scale; and the last three numbers in the midenary, and
duodenary scales.
(2) Subtract lr864 from 7r348 in the undenary, and also in the
duodenary scale ; and 50543 from 61210 in the septenary scale.
(3) Multiply together 35 and 61 ; 2064 and 812 ; 67264 and 675 ia-
the octenary scale ; 468 and 701 ; 86t and 734 in the undenary : 9294
and 844; 6rl2 and 814; €7t8 and 2rr9 in the duodenary: 1466 wid
6541 in the septenary ; and 80122 and 822 in the quaternary.
(4) Divide 14332216 by 6541 in the septenary scale ; 29x96580 by
aitir ^ auM. iruvoa;7^a sjj uj rr=i lu Luu uuoueiiary J SUivUS DJ io iU tJUi
quaternary ; and 24510502 by 4381 in the senary.
252
▲BITHMETIO.
(5) Extract the squar6 root of 12044424 in the senary wale ;
82*Y5721 in the duodenary ; aud 47610870 in the nonary. ,^
(6) Expresi 1828 in the septenary Bcale ; 7681 in the hinary;
29 and 49 in the octenary ; 1000 in the undonary ; 1000000 in the
■enary ; 80198 in the duodfenarv.
(7) E4>res8 (62tO,2 and (11^34) e in the common system; (84528)8
In the duodenary ; (654321), a in the septenary.
(8) Transform 28784 and 587 from the nonary to the duodenary
Boale ; 4321 from the quinary to the septenary ; and 2304 from the
quinary to the undenary scale ; and prove the truth of each result.
(9) Transform (8978)^1 and (3256)., to the duodenary scale, and
find their product.
202. A proper fraction is converted from one scale into anofher "by
the following rule :
Bulb. Multiply the numerator of the fraction hy the radix of the
given scale, and divide hy the denominator ; repeat the same operation
as often as necessary ; the result is the given fraction transformed into
the required scale.
17
Ex. 1. Express 8 j^ in the septenary scale, and prove the truth of
the result.
8io=(ll)7;
BytheRnle,?^i^=2-?-., ?^=8; ...g)^„=C28)t;
,*. rnmber required=ll'28.
Pmf. U,=(lxr+l),(,=8jo> ♦
W7=(T+|f)..=(^)..=©i.'
Ex. 2. Convert 26*5 into the quaternary scale, and prove the result.
4x1
(26)io=(122)4; '5=1; proceeding by rule, -^^^i
or thus, '5x4=21.
.•.(26-5)io =(122-2)4.
Proof. (122-2)4«lx4a+2x4+2 + f=16+8+2+-5=(26-5)io.
BOAIiid OF DOTATION.
Ex. 8. Convert 828010'22n2 from the quaternarj to the oot
Male.
Work in quaternary soale.
828010 .22112
8
m
8
8
8
8
18120—4
823-0
18-.8
0-r
5-02800
3
1*13000
8
8-00000
.•. required number is 7804-618.
NoTB The number might have been trausformed into the dei
scale, and thence into the octenary.
Ex. 4. Oonvert(466-16),, to the ternary scale, and prove the trut
w tne result.
"Work in duodenary eoalo.
•16
JB
0-46
3
1-16
8
8
8
8
8
8
8
456
16r ~0
5€--l
U-2
_7 -3
2 -1
12
12
12
0-46
8
-2 i-ig
,\ number required is 212210010i.
Work in ternary scale.
^i??10 The value of .bi
1222-6
11-5
= 2^, (Art 184)
0-4
and
\22J8-\8Ji
proceeding by Bale l?iil-i| :
12x1
B
=A
.'. number above found in ternary scale=456-16 in duodenary seal©,
Ui
AJkBSrBMBFaO,
lSr«ri. Sinoe 1 ft.Bia in. or 12' j 1 lii«^9'V l'*'*!^'", Ac», the
Jcodedmal Scale is often applied tu examples in?olving the oalooliflDii
of areas of lonlaoos and contents of solids.
Ex. 6. B^qnired the area of a room 17 ft. 8 in. long, and 18 ft. 10 in.
broad.
n. B.\
irft. 8in.=(lfi*8)i«i and 18fl.lOk.«=:(ll-T)i.
ltJ-8
111-
158
168
lYr-re
■q. ^ iq. ft.
(Wt-7«>,.»(«W)..+
(^+i5)'«-«-
=288 sq.ft. 90sq. in.
Ex. 6. Find the product of fl yds. 2 ft. 2 in. 8 pts. and 6 yds. 11 in.
7pts.
B yds. 2il. 2 in. 8 pt8.=17 ft. 2 in. 8 pt8.t=(16'28), , ft.
6 yds. 11 in. 7 pts.«16 ft. 11 in. 7 pt8.«=(18-c7)i a ft
16^28
18c7
T089
18909
48^9
1528
ltT^4r09*(274), . sq. ft. + ^±+^ +-|-^ sq. ft.
=i274r sq. ft. +4i superficial primesH- 10 superficial second8+ 9 super'
I ^eial'tftwirtlis.
Si. 7. a
265
278 8q.ft.68iiq.In.*(i^9.fl8)^;^ •
12-7) lrfl-58 (lff.9
127
8^
7B9
"^
V#8
* 'ft. * " — '
- 'w Ex. ixm.
valae of (38'(^6),. ^ "" ^^^^^t^^^wyso^e; Wifedtiie
11
APPriOATIOK OF AEITHMETIO TO
GEOMErRT.
208. A itcometrical Poi^t is that *],««i. i.
ma^tude. '" *^ ^^«^ has no parts orao
^ ^(?«>m.«rfcaZZ*n. has lengtti only. '
256
ARITHMETIC.
. 206. Otlier lines than straiglit lines are called ouBVBD or OBOOMSD
XINEB. ^
if the points A and B; 0, E,
andD be joined, ae in the fig*. ABy y" ^ C'
CJg'i)/ the lines^^andCJS'Dare / \ Bi
OTTRTSD or OSOOKBD LIKES.
207. A line, or linea/r content^ is mearj/red Arithmetieally by the
nnmber of times, or parts of a time, it contains a certain fixed Ime,
which has been fixed upon as the unit of length or measurement.
Thus, if we take one foot as the unit of measurement, and call it 1,
a line of 3 yds., or 9 ft. in length will be denoted by 9, a line of 2 yds.
18 in, by T^ ; a line of 1 in. by iV »^^ ^^ ^^'
' 208. A FiGTjEE or Body is a portion of space enclosed by one or
more boundaries.
f 209. The Supeefioibs, Subfaob, or aeea of a Body has only length
and breadth, and not thickness, and may be defined to be the outward
coat or face of the body. It is called a plake supbbfioieb, stjepaob,
or ABBA, or simply a plane, when it is such, that whatever two points
are taken in it, the straight line between them lies wholly in the I
superficies.
210. A superficies, surface, or area, or is measured Arithnustir
cally by the number of times, or parts of a time, it contains a certain
fixed area, which has been fixed upon as the unit of measurement.
Thus, if 1 sq. ft. be called 1, 1 sq. yd=9 sq. ft. will be denoted by 9,
and 1 sq. in. by yfy, and so on.
211. A PLANE BBOTILINEAL ANGLE is the
inclination of two straight lines, which meet
together in a point, but are not in the same
straight line.
The straight lines AB^ ^(7 meeting to- b
getter at the point B, but not both of thorn
in the same straight line BA or BO, form at the vertex B, the angle {l)
ABG, or GBA,
zABGia said to be greater than zDBO, and less than zEBO
The student will hence observe that the magnitude of a plane reo-
,
or OBdOXED
v
,
Ei
eally by the
in fixed line,
leasarement.
nd call it 1,
ine of 2 yds.
ed by one or
J only length
the outward
IE8, 8TIEPA0B,
it two points
holly in the
I Arithmeti-
ins a certain
neasnrement.
denoted by 9,
the angle {/)
yS a plane reo
SCALES OF NOTATlOlf
SSj -Ctrl? .Tcr :;?"« r -• -«"..„«
line to the other. • ^^^^^"^ ^^ ^®«« inclination of the one
212. When one straight line .i^ stand
TaZZ "T^ T""'"'' '^' ^^ -^ -"he
^ABC equal to the adjacent zAKj) *t!:
each of the .'s ^5^, Al> is cdled f '. ' ' ^
r^' and the line AB is said to be perpen
dionlar, or at right angles to t7i). ^'"^^^V ^
An OBTUSE ANGLE ViV / TPnT\ •
the oxK<.trro.i"atirrtti srr,''^»"« «"« '»"«<^
certain point within it, called ™ltLtl T "^'"^ '''"° »
eqaal to one another. ountbis, to the ciromnferenoe are
The plane superfloies, surface n, »
ofthit.*;t::r:fetnt^rsi^^^^^^
[215. THEDiAMETEBofacircleisafltri»VKf V \
centre of the circle, and ter.inlte A^ f "^^ ^^^7" *'T^^'*^^
. n^'. ^^^^' ^S"^^' *^« straight line^ir« J- ,"^°'^'''''""'
ABODE. ^ "°® ^^ 's a duiTneteT of the oirole
S 2' 5i! "'^^-^ "^ -7 ciroIe=it, diameter K^.
IfoTBS. Tl.earea„fanyoirole=theeqaareofltaradia3x5:
216. Reo'""' TT^"" • »
258
/ ',
A^O^nBtBiCBSSO.
' ill. 'A *rBiAifOiJB is a plane snperficibs, Btirfaoe, or area #hich
i8 botin'dfed by «Ar«« straight lines. .The plane super- .
ficies, surface, or area contained by the straight
lines AB^AG, and ^C, is called the triangle ABC,
or bSA, or BAG, whose sides are AB, AG, BG,
and ^080 ^8 are ABGoi GBA, BGA or ^C7^, and
BAGotdAB.
2ia lions of the 2 s of a triangle, -as A GB in the trian^ld AM^, "be
a fight angle, the triangle ABO is sa;id lo be t^ 'right-
angled triangle, and the side ul5 opposite to the angle
ACB is called the hypothenuse, BC the heue, and AC
the vertical height or altitude of the triangle.
'Kbli 1. In Hhe right-angled triangle ABCy fke
^iSght angle ' beihg ^(!7P, square described on AB=
■"^u»e di^iioribed'on ^Cr+sqiiare described ou jBC,
"^If one of the angles of a triangle be obtuse, the triangle is said to be
Stuse angled.
If all the angles of a triangle be acute, the triangle is said to be
<teute angled.
If all the angles of a triangle be egtial, the triangle is said to be
eguidngular. . ., . ,
' If oZr the Bidfes of a iiidXigL& he equal, the triangle is said to be
' ^uildierat.
If two of the sid68 of a triitigle be «2"MaZ, the triwigle is said to be
isoaeeles.
NotK 3. At(»a of BiijA^i' iiiy side of it x perpendicular drawn
■upon that side, or that side produced, firom the opposite angle.
If ABC, A'BC be triangles, and AD, AD' be drawn from ^ aUd A^
perpendiculars on BC, and on B C produced ;
idular 3rawn
Qgle. ^
om ui aUdjdL',
I
V
«re« of A X^C7'=j of i-C'x^'iy.
« called ,to la^ ,„ ^ ^^ ^ _
Wendionlar trora A or i) on BC oZo *~ '^ ^-<r~i.
ogram. ' """*""'' *^« <^«»»"««». or<fiay<««j^ of theparXu
-/^b ff i^r^L" ^St^' ""* "- "^ »^ ''^- ^-.'-^
of Z.at ^riju "irrnj: « «U.r.m .Mch ha, each
opposite eidea equal to ^^'o^^ °°' "^ '" ''"^ «9ual, bat only tho
IZ i' ^""' '"•'"' '"^'•"«"<'S^»"'="« base X altitude,
.id^ • "*' ''"'"'^'^' "' ""^ -tUinoal flgnre=tl.e sam of it.
m A S„^ . that Which hath length, breadth and thicUs.'
%of-spa:^„™;^xjL'^';'::;ra:drr ''*«''»-
contain, or tal^ea np ; and it\ ^It.'rf^vL^i^';,''''?'''^'' -"«" «
or times, or parts of a tinifi i^ ^««f„- -'cx«t,x*j,cy uj- une xiumber
has been fi.^ on as S.^i;l;~ l:^ ^ "^^^ ^^^"^^^ ^^
'I
Jill
•!•«*■
^■IMIfRMn^M*^
m i ig i mmimi]j| ii
260
ABITHMETIO.
226. A Rbotaxoulab Paeallblopipbd is a solid, contained .by six
right-angled quadrilateral figures, whereof every -q ■
opposite two are parallel.
The figure ABCD is a rectangular parallele-
piped. Q
■ 227. A CUBE is a solid figure qontained by six
equal square?.
228. A BIGHT cTLiNDEE is a Bolid described by the revolution of an
oblong round one of its sides which remains fixed.
The side of the oblong' which remains fixed is
called the axis of the cylinder. The side of the
oblong opposite to the fixed side or axis traces out
...^ cylindrical afrface. The circles traced out by
the other sides of the oblong are called the lases
of the cylinder.
The figure AJBGD is a right cylinder, traced
out by the revolution of the oblong JiJi^^ about
the fixed side or axis BF.
^ 229 In the Tables of Square and Oubio Measure we have seen
that length multiplied by length prt)duces area, and area mult^phed by
length produces capacity ; the units in the products in these cases
differing in kind from the units in the factors ; thus, a rectangular area,
whose adjacent sides are 4 and 3 feet respectively, is divisible into
(4x3) or 12 equal squares, as shewn by the accom-
panying figure, the length of a side of each square
being one linear foot. The rectangular area in
this case is said to be the product of the two ad-
jacent sides, represented respectively by numbers,
the units in the numerical product being no longer
linear feet, but square feet. Similariy, if the adjacent edges ot a rec-
tangular parallelepiped bo 3, 4, and 2 feet, respectively, the capaci y
of the solid is e<>uivalent to 24 cubes, each containing one cubic foot;
and thus the capacity of the parallelepiped is correctly expressed by
the product of the three acijaceut eages ieprcsoutc^ i^ri.~-----v -.-
numbers, the units in the numerical product being no longer hnear feet,
as in the factors, but cubic feet.
1
5
9
2
6
S
7
4
8
12
10
11
Ex.
and 18 f
areas
Ex. i
and brea
Area
.*. cost
Ex. 8.
and 2 ft.
cub. ft
Cent*.:
Its vail
KOTE.
hat sq. f(
inearffc. >
Qd 12 sq. i
Agaip, I
»at cub. f
[. ft. give
lb. ft. -4-1 2
Ex. 4.
ft. 5 in. I
Area of
ad ihj six
B
p
tion of an
n
^c
have seen
iltiplied by
these cases
igular area,
?^i8ible into
2
6
S
7
4
8
12
10
11
AEPLIOATIOir OP ABmoiEno TO GEOMETET. .261
-Smmpfe, in Sgmn and GuM, lr«um>
and 18 ft 4S"br:ar"'' "' ' "'"•°«''" ~"'-^"'» "^'^ « »• >ong,
-1 sq. ft. ^
«««=(17ft. 6in.)x(18ft. 4in.)=17|ft.xl8J.a-/??x^
roo \2 8,
—3- »q. ft.=288i ,q. ft. ^25 ,^ ^^ , ^^ ^ ^ ^^_ -_^
«.at,.it8 ft!':* rpei^':^r " "'°'' ^'-^ """^^ '^ «^^ ^'"•
Area of floor=(88 ft. a In.) X 18 ft.=83i ft. X 18 ft.
/1»9 18\
/"wo 18\
■•■••""' "^ P''™^ "»-=(« '<Txf)'=893*.=£19. 18..
Oonf.=(rft. 8in.)x(lft. rin.)x2ft.=rift.xlAfl.x2ft.
_::y29 19 \
""VT "^ 12 "^ ^) °"^- ^<=*=22 cub. ft. 1656 cub. in.
;es of a rec-
he capacity
I cubic foot ;
xpressed by
rWV a4- 1 TTZil XT n V
r linear feet,
«d 12 sq. ft..:-4 linear ft'la iiL'r ft '' •"^' """' "•==* "-« «•'
I. ft. gi.e Hne» ft'^lirY^lr ft';-.' :1 <'t«^\'""'*' "'
■b. fl..M2 linear ft.=4 so ft J^Ta^ \\t ': •=** ""''• **•• ••■ ^8
ar n._4 sq. It., and 48 onb. ft.-i-4 sq. ft.=i2 linear ft.
Ex. 4. Find the expense of carpeting a room IB ft <n„ i
Sft. 5 in. broad, with carpet | yd. Le, 'a 1 11 a "a'd "' ''°^' "^^
Area of floor =(15^ x 12vU so. ft. =. ^?? . li^^ .. ..
763 149
=iT^i2-^
|)sq.yds.=(^X^jsq.yds.
II!
263
ABITHIICBTIO*
8
Now required length of carpet in linear yds. x -— linear yd.
=no. 01 «!• yds. in area of floor= ( — x ^^1 sq. yds. ;
/. length of carpet in ]m%ar yds.
J
T U9\ -
-T-X^jsq.yds.
-— linear yd.
4
7 149 4
=T^
12 ^ 3 '
Ctr V 149\
lx^^^^^j$=28.97f.
Ex. 5. Find the expense of painting the walk and ceiling of a
room whose height, length, and hreadth are 17ft. (Jin., 8P ft. 4in., and
20 ft. re^peotiyely, at 15 cts. per sq. yd.
Area of the 2 length walls ==2(heiglit x length), ©r 3(Zrx L).
of the 2 breadth wall3=2(lieightx breadth), or i{IIx B),
.... of the ceiling =length x breadth, ov LxB,
.', area to be painted
=z2CB:xL)+2(ffxB)+I,xB-%J3:x{L+B}+LxB
=(2 X ir^) ft. X (35^+20) ft +(35J X 20) sq. ft.
7930 ^, 7980
=(2 X I7f X 55|.+^ X 30) aq. ft. ="j- sq. ft.=g^ aq. yds. ;
(7930\
15 X g^-g j cts.=$44;05|-
280. Another method of working examples in square and onbio
measare is styled Okoss Mtjltiplioation or Dtjodkoimals, and it is
generally employed by painters, bricklayers, <feo., in meaouring work.
They take the dimensions of their work in feet; inches, part?, &c.,^
decreasing from the left to. the right in a twelve-fold proportion;
thus,. 12 inohe8=l foot, 12 part8=l inch, &c. : the inches, parts, &c.
are termed primes, seconds, thirds, &o., and are distinguished by thoj
I If itt
accents , , , 6co., placed a litwo tu the ight above the^ nuuxw .;; s
which they belong.
ar jd.
caillng of A
ft. 4 in., and
TxZ).
ITx B),
I- yds.;
ire aqd onbio
iLB, and it is
oaring work.
)s, part?, &c.,'
. proportion;
B8, parts, &c .
ished by tho j
SQUAEB ATO OtTBIO MASnER 4^3
-»«& /or Crm MuUiplioatim.
tipli«md. boginnmg «t them; b^^.'^T'^*'""'"' «" ""■--
'."otofthedenominaUono/feet bv^ .f, f '""'' P'"^""** '''»1<"'
•Prodnol, «ad pl«, a,, ^emirinder ande' 'f ''«"J"°"«"' to the next
j".t used, whea the denominTon ofl , '"" "^ *^'' ""'aplio^id
amoved to the right ^CT^X^l^Ty^^ » ^-^ o"" Pl.ee
«^ when it fe third,, &„ j? 2 ' ? "''^^ ^^^ " i" «e<>on«e,
-e^ 13, and the aon^Vmbelhe ^s^T""^ '"'^"•"> '^'^'^ ^ ^
Ei.l. Knltiply4ft7fe.tygjj_jj/_
By the Role,
ft.
The prodnot=43 so. ft + . .v. „, ,
«'.-*:ia.^/««asthe'.ilC I-VV ^°"
flcial prime, i. , , 1 / " ' °'" "I^-
I .dgfu lit r^' "' '"^ "'»•«''»-«- tie capacity of , eabe .hose
The product
r
6'
=18 cub.
US'*" 144'*' 1728
onb. ft.
10
J//;
6
=18ctib.ft.+
=18oub.ft.+?g,„b.ft
««18 cub. ft. + 1664 cub. itt.
cub. ft.
?P4
AlUTHMETIO.
Ex. LXIY.
L will stand for length, B for breadth, 5" for height. ^
(1) Find the circumference of a wheel whose diameter is 4 ft. 8 in. :
how many times will it turn round in lOi miles?
(2) IIow much space does a circular pond occupy, whose diameter
is 15 ft.? . ■
(8) 1. Find the diameter of a wheel which turns 4290 times in
15i miles?
2. A circular pond contains ^ acres ; find its diameter.
1^ (4) 1. A horse in turning a mUI moves round at a distance from
its center of 6 ft. 5 in., and makes on the average 85 circuits every
8i min. ; how much is his pace less than Smiles an hour?
2. A circular flower-bed, 16 ft. in diameter, has a grass border
round it 4 ft. wide : find the number of sq. yds. in the border.
(5) 1. 'yriiat will it cost to fence a circular bowling-green, whose
radius is 52 ft. 6 in., at 84 cts. a yard ?
2 A cow, tethered by a rope 7 yards long fastened to a stake in
the middle of a pasture, has its rope doubled in length ; how much
greater space is it allowed than at first ?
^ (6) 1. Find the hypothenuse of a right-angled triangle, whoso
other sides are 24 ft., and 27 ft. 6 in. „ v i, •
2. The hypothenuse of a triangular plot is 4ch8. 25 Iks., the base is
2 chs. 65 Iks. ; find the other side.
(T) 1. The circumference of a circular spot is such that it encloses
1386 sq. yds. ; how much is its radius less than the side of a square of
the same area as the circle?
2. How long will it take a person, who walks 8 miles an hour, to
walk twice round a square field containing 32 ac. 64 po. ?
(8) 1. -If from the extremity of a path 24 ft. widt . i ladder reachej
1 ft. Y in. over the top of a house 45 feet high on the other side of thi
PAth : find the length of the ladder.
2 If the end of the ladder be shifted 2 feet further from the hous^
and then just reach to the top of a house 40 feet high on the otha
Bide of the street j find the width of the street.
(9) Two engines start from the same station, the one due JNort
St the rate of 35 miles an hour, and the other due East at the rate o
11\ miles an hour ; how far will they be apart at the end of 4 hom-si
«QUABB AND CUBIC MBAflFKB.
i8 4fb. Sin.:
LOse diameter
290 times ia
er.
listanoe from
ircuits every
grAss border
der.
green, whose
to a stake in
I ; how much
iangle, whose
cs., the base ia
lat it encloses
f a square of
3S an hour, to
ladder reacho
ler side of th(
rom the housei
1 on the othei
one due Nort
at the rate o
I of 4 hours!
Wffet 7"' '"' "~o^«*H.ngle Whoso b.« i. « r.et, .„d ..«t„d.
,8ft.9in.,.B=8ft.8ux. 6. Z-fltu ,^ t""'' ^=* ^ " '"• «• ^=
(13) ^o/'cb.C^o^Muuia i ^'^.I'Jr.f^^t*'"- '
by 1yd. ift. ir. 3. ryd,.6^io''bvn ^ ^ »; ^- "^^"-^
.6. 2 yd«. 1 ft. B'.8" by » it 59". 8 n ft r ;, k*' 1^ «' ""y « *• 10".
(14) Find the solid oonteit of lh«\„ ^ '*"•*'•
i?=7ft.6in.,.ff=3ft.l0in. 7 l-20ft r?!"' '' -^"lOft-^ln..
«• Of 8 enbe whose edge is 6ft rtw',"^^^ ^*- * '"- -ff=lft. 21n.
■ (15) Kndthelongthofthefnilll "'(^^''^ ""•<'» '»"'l«PlioatioD.
=6 sq. ft.. 5x9 in. 4 Irea-iri ''•,"'•' ^=^ ^- «>"■ » Ire^
(16) ^indtbeareaof [h ;w^Kt.'^-''-;^=^«^^-2«.
1. Z=82ft.,5=i8ft »-_Tii^***^"»"''°8''<>'>n'a:
fxl2ft.4i, 8 Z=29ft'^="3tft V=f'*-*'°-^="«-8i"-.
^=16ft. 9in., JJ-=i3ft. 8i'n 5 i-^,^=".*^- *• ^=82fl.6in.
,18 ft. 6 in.. deductiDgin feT for » fl~ i "•' -^=""- » '-•. ^'^
.tioixr:: n:"2fr8rr ^- "? -"-^^-ftUin.
63 cts. a square foot "• '""S' ""^ . 16 ft. 9 in. wide, at
by nl ^•.^^'"'"-^'-'^P^I-- i.v4 wrj^ win co.er awalll5ft.8in.'
>bfi«Ltsrw";auL'a:a''ofrrLn'' """'- ■■'--»-«
2. Find the nnmber oT,™ * • ^=^ '"• ^ "• ''^"Iv.
4 chs. 60 lies. °^ "='^ ■" » »V<«-« field whose side is
8. Arectangolarfleldisyoha quit. 1 • ,.,
is to be out off from it bv I lS^'Zu\T^.' "^ ^ '^^ ^""^ ! H ~.
this Im. h^ A ^ , « '>y a Jme parallel to its braadtl, . „i,».. _._ .
im
AlOTHMEfllO.
' (20) 1. What length of carpet, 1 yd. 4 in. wide, wfll be reqirjad
for a room whoae length is 16 ft. 6 in., and width 10 ft. 8 in. t ^
2. A Aemidronlar plot of ground, whose radius is 12 yards, haa
inside the circumference a path 2 yards wide ; ♦^'> ^est of the space is a
flower-bed ; find the size of the bed.
(^1) Find the cost of carpeting the folio w^ing rooms :
1. Z=20 ft. 8 in., i?=20 ft. 8 in., with carpet f yd. wide at 4i, Hd.
a yd. 2. Z=20 ft. Bin., -B=17 ft. 4 in., with carpet | yd. wide,
at 4a. 2d, a yd.
(22) How many yds. of paper, 1ft. 4 in. wide, wil^ be itj»iUired for
a square room, whose side is 18 ft. 9 in., and height 18 ft. 4 in. ?
' (28) What fs the cost o^papering a room, Z=24 ft. 4 in., -B=26 ft.
6 in., if=18 ft., with paper 28 in, broad, 6 cents per yard ?
(24) Tmd the cost of papermg a room, 19 ft. 8 in. wide, 24 ft. 4 in.
long, and iSift. high, with paper 21 it. wide, which costs 11«. per piece
of 12 yds.; the windows and parts not requiring paper making np a
silth of the whole surface.
(26) 1. Hnd the weight of water in a bath, 6 ft. long, 8 ft. wide,
and i ft. Sin. deep, the weight of 1 cub. ft. of water being 1000 ounces,
i 2. The bottom of a cistern contains 16 sq. ft. 128 sq. in. ; hpw
dee^ must it be to contain 1216 gallons ? 1 gallon contains 277i cub. in.
nearly.
(26) 1. A cylindrical pail is 14 in. in diameter, and 14 in. in height,
how often can it be iiUed from a cubical cistern each of whose inside
edges is 7 ft. 8f in.?
2. How many bushels of malt are thero on the floor of a cylindrical
kibi, the diameter of the floor being 6^ yds., and the depth of the malt
being 14 in.? Note. 1 bus. =2218- 192 cub. in,
8. The diameter of the base of the standard bushel being 18 J in.
nearly; find its height.
(27) 1. How many flag-stones each 5-76 ft. long and 4-15 ft. wide
are required for paving a cloister which encloses a rectangular court
45-77 yds. long and 41-98 yds. wide: the cloister being 12-45 ft. wide ?
j 2. A moat of the uniform width of 15 yds., and depth of 7i ft.,
gurronnding a square plot of ground containing H acres is quite full of
water: how many gallons will it contain? Notb. 1 gallon contains
277*274 cub. in.
5^ cub.
SQUAEE AITD OITBIO MEASUBB.
267
ystdBy has
space 18 a
) at 4*, Hd.
yd. wide,
»«iUiredfor
,5=26 ft.
24 ft. 4 m.
h per piece
aking up a
8 ft. wide,
)00 ounces,
.in. ; hpw
r7i cub. in.
, in height,
hose inside
cylindrical
of the malt
eing 18|iD.
•15 ft. wide
igalar conrt
t5 ft. wide ?
th of 7i ft.,
juite full of
Ion cont&ins
(28) 1. If 12000 copy-books be used yearly, and each book contain
20 leaves euch leaf being 7i in. broad and 9 in. long, find laow many
Ihelroult ^''^''''''^ ^^ ^^"^ ^^^^'' ""^^^^"^ copy-books spread out on
I 2 The area of a rectangular field whose length is four Umes its
breadth is Sacies 1280yards; find its perimeter.
Tf /^^'^ k ^ .'''°*^P^^^ ^^"'•^^ i« 80 yards long and CO yards broad.
It has paths jummg the middle poifit« of the opposite sides 6 feet wide
and It has also paths of the same breadth running oil round it on the
inside. The remainder is covered with grass. If the paths cost i. 8d
per square foot, and the grass 8.. per square yard, find the whole cost
of laymg out the court.
^h.I'A^ ^'"''^ ^^^ ^''''^' ^'^ P^'^' ^^'""^ ^ ^^"* *° P«t i°to two boxes.
who e dimensions are-the larger one, 4^ ft., 2 ft. 8 in., and 2 ft.; the
smaller, 4 ft., 2^ ft, and ^ ft. I can get 50 books into the smaller how
many wi 1 remain unpacked when I have filled both the boxes, the
books being aU of the same size ? '
1 rTJl'-^ *'!f '"''''^'''' °^ *^' ^"^'''' ^^^^^ ««lid contents are
X. 5 It. 621 m. 2. 14706 ft. 216 in.
n,J^? ^\^T "J'Y °"^'' ""''^'^ '^«^' *'« ^^«^ 2Jin. can be cut
out of a cube of which each edge is 22 in. ?
th. .!V^^'.* T^ \^ '^' l^eig^'t of a cylindrical column of marble,
S cub. ft '? '*'' '" ^ '"'^^'"^ '" ^'^'' *''"^ '^ "^^^ ^^°<^^
K ^!^^no^ "^"""f *^ f ''^ ^'""'^' ^'^ *^" ^«^^ °f ^«" » said to be
about 108 feet m length, and to have an average transverse se- tion of
113 8q.fr If shaped for an obelisk it would probably lose on -third
of Its bulk, and then weigh about 600 tons. Determine the number
of .ubic yds. m such an obelisk, and the weight in pounds of a cub ft.
of granite.
(38) 1. If the diameter of a cylindrical well br 5ft. 2 in audits
depth 27 ft. 6 in. ; how many cubic yds. c: earth were removed in order
to form it ?
is 2 ft. 8 in. % half full of water ; how .nany gallons does it contain ? "
8. How many gallons must be drawn /^ff f^«»„i,*AV. ^_
smk one foot? ,„
268
▲BiTHiacTia
EXAMINATION QUESTIONS.
The fbllowing qnestions have 1)een selected' from the KatricttS-
tion Exftmination papers ^et for seTernl years at the Universitiea of
McOiLi^ QuEBN^ Trinity^ and Toronto.
I.
(1) What conditions must he satii^ed in order that one vnlgar'
fraotibn may he capable of being added to or subtracted from another ?
If these conditions be fulfilled, explain why it is necessary to cliange
the foroos of the fractions before performing the operations. How are
these changes efEecte4 in the case of decimals ? ^
Add together %^\ and 1.85 of a £ currency, and subtract ^ of a
£ sterling. (The £ sterling to be taken as equal to £1. 4«. 4d. currency.)
(2) Three students, Jl, B^ C, are to (fivide between them at the
end of a term of 9 weeks a sum of $125^^, the sthare f each being pro-
portional to the work d<me by him. B can do half as much again as G
in the same time, A twice as much. G works steadily 8 hours a day,
B works 7 hours a day for the first 2 weeks, 6 for the next 2, and 3 for
the remainder, except the last week, when he works II. *. During the
first 7 weeks, A works only 2 hours a day for 4 days in the week ; but
faring the last fortnight he v/orks 14 hours a day ; but he finds that
v^ the last 4 hours of each day he gets through no more than G could.
Find how much each should receive.
(3) The freight by a steamer to a certain port is $19.40 per ton
measurement ; by a sailing vessel it is $6, Insurance is at IJ^ per cent.
by steamer, and 4f by sailing vessel. Find whether it will be more
advantageous to send (and insure) by steamer or by sailing vessel a
package whose dJouensions are 5 ft. 6 in., 4 ft. 4 in., and 3 ft. 3 in.,
and value $780. A ton measurement is 40 cubic ft.
(4) The pound sterling being £1. 4«. 4^. currency, find the exact
T^lue of the sterling sliilliog in cents.
(6) "Water flows into a tank from two taps which running sepa-
EXAMIITATION QUESTIONB.
269
IfatricnS-
ersities of
me vulgar*
I another ?
to Ciiango
How are
•act ^ of a
Durrency.)
em at the
jfeing pro-
igaia a3 G
ura a day,
and 3 for
)iiring the
veok; but
finds that
1 C could.
to per ton
per cent.
be more
I vessel a
J ft. 3 in.,
the exact
ling sepa-
am leyel \
WTv^fr^t'^^'^^^- ''P'"" • ^°*^^ *^P* ^"°«'°«' ^°^ t^^ tank being
kept (illoa to this level, in what time will the waste-pipe dlschame a
quantity of water equal to that in the tank ?
(6) A buildicg-lot 80 feet in front by 120 in depth is sold at $100
per foot frontage ; liow much is that per acre ?
(7) DcBcribe the respective adv'anto^es of the use of tulgar and
<wom«^ fractions.
Divide 8654 by 2-03, explaining each step in the process.
(8) State what diflferent units of weight occur in the Engtish
system. ^
Fifteen guineas weigh 4 oz. Troy, the metal consisting of 11 parts
gold and 1 alloy, and the value of the alloy being ^^ths that of an
equa weight . 2 gold. Find the price per lb. Avoirdupois of the alloy.
(9) A quantity of pulp, filling a trough 8 ft. deep, 10 ft. 7 in. long
and 11 inches wide, is made into paper of the same width and of such
a thickness that 12 sheets would measure a quarter-inch; find the
length of paper made, the pulp losing |ths of its bulk in the manu-
facture.
(10) A grocer buys 150 lbs. of coffee at Mots, per lb., and 89 lbs
of chicory at 6 cts. per lb. ; he pays an import duty of 12^ per cent.
ad valorem, and mixes and sells them at 25 cts. a lb., but by the use of
a false balance gains i oz. on every apparent lb. sold. Find the profit
per cent, made on his outlay.
(U) Give a short description of the ordinary numerical notation,
and prove that every vulgar fraction can be expressed by a decimal
either terminated or repeating. If the base of the system had been
two.hy how many figures would 1000 have been represented, and how
would the equality, twice two are four have been written in figures?
(12) The mint price of pure gold is £4. 4s. ll^^^d. per oz. Troy, and
tbe gold of the coinage has one part out of 12 alloy (the value of which
may be neglected); find Uie Avoirdupois weight of a sovereign.
II.
(1) Explain the common system of notation. Multiply 857 by 284'
explaining the different steps of the process. '
(2) "What is the first step von take, wbnn vnn wSclT +« ^aa 4.^^
vulgar fractions together ? Explain clearly why this step is neoessli^y"
270
AEITHMETIO.
6 14
Simplify _ ^ _ X --
s^y^y-^
(3) The population is 1842265 souls, and the revenue from cus-
toms is $3595754 by an average duty of 12^ per cent., and the con-
sumption falls off one-tenth ; how much is the average taxation per
head altered ?
(4) State the rule for finding the quotient in division of decimal
fractions.
Divide -034695 by -000241.
Find the number of yards, feet, and inches in -084 of a mile.
(5) "Wliat is the interest at 7 per cent, per annum of £133. 6«. 6d.
from the Ist of Janntry, 1862, to the 15th April, 1863 ?
(6) Extract the square root of
(a) 74684164.
(/3) -03275 to 4 places.
(7) State and explain
to a vulgar fraction.
Exampl
Express as a decimal
the rule for
B, -075386.
1, 3
-^.
(8) A note for $400 at 3 months is cashed by a bi'olcer at a dis-
count of 2 per cent, per month. At maturity it is protested for non-
payment, but is renewed for a farther term of 3 months, upon the
maker paying tiie protesting charges, which amount to $1, and interest
at the rate of 2| per cent, per month in advance. Find the rate per
cent per annum which the maker on retiring the note will have paid
for the money originally advanced.
(9) Out of a mass of metal consisting of 3 oz. of gold, 18 oz. of
.silver, and 10 oz. of nickel, a jeweler makes 40 snoona. whfin irnlrl ia
from cus-
d the con-
ixation per
of decimal
uile.
133. 68. Sd.
Dg decimal
3r at a dis-
I for con-
upon the
id interest
16 rate per
have paid
, 18 oz. of
an ffold is
EXAMINATION QUESTIONS.
271
-worth $15.60, silver 95 cents, and nickel 25 cents. He can then sell
each fpoon for 02.25, but afterwards tlif' value of gold rises 15 peif
cent., and that of silver 12 per cent., while nickel falls to 20'ceuts an
ounce. Find the price at which he must sell to make the same profit.
(10) Explain the reason of the rules for poiutin^j in the multiplica-
tion and division of decimals.
Divide 0-000279 by 300000 and multiply 23'4i59 by 0-083*9.
(11) "What fraction of a lb. Troy is an oz. Avoirdupois?
(12) What fraction of 19«. S^d. is lis. 11^ ?
III.
(1) "Whence does it appear that a vulgar fraction may always be re-
duced either to a terminated oi' circulating decimal ? Explain how to de-
termine which kind of decimal any given fraction will produce. Reducd
33 3
to decimals y— , — , and express as vulgar fractions in their lowest
terms 3-0561, 15-G013789.
(2) "What is an aliquot part ?
Find by " Practice " the value of
(a) 1589 bushels at $3.75 per bushel.
(/3) 1 ton, 6 cwt., 2 qrs., 6 lbs., 4 oz., at $17.13 per ton.
(3) Explain what is meant by Interest and Discount.
Find the time for which the discount on a certain sum of money
will be equal to the interest on the same sum for a year ; the rate of
interest in both cases being 5 per cent.
(4) A piece of timber is 8 ft. G in. wide, 5 ft. 9. in. deep, and
20ft. 8f in. long; at what distance from one end must a piece be cut
off so that the remainder may contain 23 cubic yards ?
(5) A railroad runs through an estate for 50 miles, taking up a
space 22 yds. wide, and the ground is valued at £55 an acre. The
owner receives in exchange a square field worth £10 a rood, and pays
as balance £360.0. "What is the length of the field's side ?
I (6) £356*3^yY«. is paid as the present value of a bill due seven
months from that date ; at the same rate per cent, interest £126 is
paid as the discount of £1726 for one year anjd nine months. Find
the amount of the first bill.
^
*&
'^i
til
in'' M
272
ABITHMEnC.
(7) How can £11. 7«. Id. sterling be divided into crowns, half-
crowns, sliillings, sixpences, and pence, so that there will be an equal
number of. each,
(8) What is meant by sajing that gold is at a premium in the
United States of America ?
If the premium on gold be 105, find the discount on American
treasury notes.
(9) I purchase in Toronto American silver on which there is a
discount of 4 per cent., and taking it to New York, where gold and
silver are both at a premium of 80, 1 there buy American paper money
with the silver ; gold falling to 150, 1 buy gold with my paper money,
and upon my return to Toronto find that I have made just enough to
pay my expenses, which Avere $120 in Canadian currency. What was
the sum originally invested ?
(10) What is meant by " the Funds? " Explain why the English
Funds rose on the birth of the Prince Imperial of France.
(11) A person holds stock, in the English 3 per cents, which are
at 98, to the amount of £1500 sterling. This he transfers to Canadian
Government 6 per cents, which are at 105 : find the alteration in his
income in dollars if the £1 sterling is worth $4.87.
(12) When gold is at 250 in Wall street, New York, what further
rise will make a reduction of one cent in. the dollar? .
TV.
(1) The value of the old Spanish dollar (which was the unit of
exchange between America and England) was 4^. 6d. sterling, but gold
became the dtnndard of the currency of the United States of America,
by the acts of 1834-7, which made the gold eagle weigh 258 grains,
being nine-tenths fine. The English coinage is of metal 22 carats fine,
40 lbs. being coined into 1869 sovereigns. With these data explain
why the bank par of exchange between New York and London is said
to be 109^.
(2) Add three-fifths of 4s. 7^. to seven-twentieths of la. 5k7., and
subtract from the result thirteen-forty-eighths of 5s.
(3) Shew that every vulgar proper fraction can be reduced to a
terminating or circulating decimal; and examine the form of a frac-
EXAMTNTATIOK^ QUB8TI0NS.
273
i^ns, half-
an equal
im in the
imerican
here is a
gold and
Br money
r money,
nough to
Ihixt was
English
rhicli are
Canadian
on in his
t further
nnit of
but gold
America,
3 grains,
•ats fine,
explain
n is said
Id. J and
3ed to a
a frac-
tion which gives Hse to a decimal consisting of p digits which do not
recur and of q 'digits -vrhich are repeating. t
Reduce jV to a decimal; multiply the decimal 1-4, and divide the
product by -j^.
(4) The old standard bushel was defined by statute to contain
2150 cubic inches, but on examination was found to contain only 2124.
By the Act of 1824 the bushel was declared to contain 2218 cubic
. inches. Examine the real loss on the rental (£1075) of a farm (which
was calculated on a certain percentage of the selling price of the corn
grown), supposing the price per bushel to remain the same.
(5) Having 8 separate parcels of powders weighing respectively
84 lbs., 3 oz., 15 dwts., Troy; 45 lbs., 10 02,, 4 drs., 12 grs., Apothe-
caries; and 32 lbs., 7 oz., 3-712 drs.. Avoirdupois ; how can I subdivide
them into parcels weighing the same integral number of grains?
(6) The link of Gunter's chain being 7U inches, -prove that ten
square chains make an acre. .
^ The Scotch ell being 87*069 inches, and 24 ells making the Scotch
chain, what diflPerence (in square feet) is there between 65 English
and 42 Scotch acres?
(7) A grocer buys a stock of tea and sells |ths of its nominal
amount at 82 cts. per lb., thus-clearing $190 ; he now calculates that if he
sells the remainder at 85 cts. per lb., he will on the whole make 80 per
cent, on his outlay ; but he has forgotten to take into account a loss in
weight of 2 per cent, by waste in handling. How mnch less cash will
he receive than he expected?
(8) Explain the distinction between Simple and Compound in-
terest.
What rate per cent, per annum interest is discount on s note for
one year at 7 per cent. ?
What rate per cent, per annum iaterest (compound) is discount
on a note for half a year at S^ per cent. ?
(9) Prove^the following rule for computing interest at 6 per cent.'
for a period of months and days, the substance of which was g^ven in
the Leader of March IKh, 1865.
Multiply the number of months by 5, and odd one-si^th the num-
ber of days ; multiply this sum by the principal expressed in dollars ;
the result will be the interest expressed in mills.
274
ABITHMEnC.
(10) A tradesman who givea e\x months' credit abates 6 per cent,
for cash ; find the rate of interest in order that this may be the t. ae
discount ,
(11) Shew that changing the position of the decimal point is
equivalent to multiplying or dividing by a power of 10 whose index is
the number of places by which the position of the point is changed.
Multiply 23-58 by '0005, and divide-the result by -36 ; in each case
shewing the correctness of the rule by which the position of the point
is determined.
(12) A person has an income of £100 from Bank of England stock,
which pays 6| per cent, dividend, and sells at 198|-. Find his income
in dollars if he sells out and invests in the Canadian 6 per cents,
at 1121, the £ sterling being equal to $4.86
(1) Reduce, tne numbers 3954 and 6872 from the denary scale to
the nonary; obtain their product when thus transformed, and reduce
the result to the septerary,
(2) A person purchases a quantity of goods in Liverpool for
£37. 10«. sterling, and sells in Montreal for £65 Canadian currency;
he pays in Montreal an ad valorem duty of 4| per cent. Neglecting
other incidental charges, what is tlie gain per cent., supposing a pound
sterling to be $4.87?
(3) A vessel contains 120 gallons of wine ; 20 gallons are drawn
therefrom and the vessel filled with water; 15 gallons of this mixture
are then drawn and the vessel again filled with water. If this opera-
tion be performed six times, 20 and 15 gallons being drawn alter-
nately, how much wine will the mixture contain ?
(4) Reduce the decimals -21816, -31249, and -8934 to their equiva-
lent vulgar fractions.
(5) A, B, and engage m trade ; A contributes £150, B £200,
and C £250. At the end of two years A draws £100, and one year
after B £150. When the partnership is wound up, at tbs end of four
years, it is found that there is to the credit of tlie firm £1000. Are
the data sufficient to enable us to make an equitable distribution ?
Give reasons for your answer. If sufifioient, determine the amount to
which each is entitled.
BXAMHTATION QUESTIONB.
"276
(6) If the decimal point in any urnnber be moved one place to the
left, and then again, and so on, and the numbers thus formed be added
together, the sum is the result of dividing the original number by 9.
(7) "Which of the following statements is more nearly correct ?
10 ^ ,. 10
(8) Prove that the square of 99-9899995 differs from 9993 by
little more than a unit in the eighth decimal place.
(9) The distance between the earth and moon being expressed
by 59-9643 with reference to the earth's radius as unit, and this radius
being 8962*8 miles, each of these numbers being exact to the nearest;
decimal ; what can be known of the moon's distance from the earth in
miles ?
1 (10) The imperial gallon is defined as containing 27V.2 cubic inches,
and as holding 10 lbs. weight of water. What would be tlie error in
saying that a cubic foot of water weighs 1000 oz. ?
(11) In the French system, a cubic centimetre of water is said to
weigh one gramme. The metre is 39-371 inches, and the gramme
15-434 grains. How does this compare with the English statementj
and what is the reason of the difference?
(12) FroE^he fact that ten square chains make an acre, deduce
the length of a link in inches.
(1) Sterling gold is 22 carats fine, and from 40 lbs. of it are coined
1882 sovereigns. Jewelers' gold is 18 carats fine. An ornament made
of tJie latter, and weighing 22 oz. was sold at an advance of two-thirds
on its value by weight, and the jeweler's profit was equivalent to
^^^ per oz, on the pure gold contained in it. What was the charge
foi workmanship disregarding the value of the alloys ?
(2) Two persons, A and B^ bor^-ow $300 on joint mortgage from a
building society, A taking $200 « id / ;^100, the amount being repay-
able principal and interest by oau:»i monthly instalments to which
A and JB contribute proportionally. After a few payments have been
made, they desire each to borrow $200 additional, and propose to
-^1
,;|
merge tljjp old ^ebt and the iiew 4p|» a single mortgage for f^QO,
[rli^reopoa the account stands thus :
Amount required to pay off old mortgage $2^
" of new loan 4Q0
Surplus 55
Total $700
Discuss the interest each has in this surplus of $55, and the propor-
tion in whioh thej should contribute to the iustalments payable in
future.
,(8) From "IQ square chains make one fcre." deduce the length
jQifk link in inches.
(4) Find the a 0. M. of 1859 and 8042.
(5) Find the prime factors of 6300.
(6) Find the value of 1^ + 2| - 3f.
(7) Find the yal^e of ^ yard + | foot - 1 in.
(8) Ueducei f"™*^ to its simplest form.
(9) Find the square root of 213*536 an ! ^.
(10) Find the cube root of 47045-881.
(IJ) Divide -558 by -024.
(1) Defire a numerical fraction. Distinguish between Vulgar and
Decimal Fractions. Express 5 fur., 3 per., 3^ yds. as a fraction of a
mile in each system.
(2) Divide -01 by '00001, and multiply the quotient by -3.
(3) Light travels at the rate of 192000 miles per second. How
Jon^ will it take to come from the Sun to the Earth, the distance
being 95000000 of miles ?
(4) Extract the square root of 373*45.
(5) Shew that if any number expressed in the decimal scale be
divisible by 9, the sum of its digits will be divisible by 9.
(6) Find ^ of £2. Ss. 9dl., and express the result as a fraction of
£1. 2«. 6df.
(7) Reduce 0-85278 to a vulgar fraction. Prove the rule.
(8) If the gas consumed by one burner cost 17». %d. for 40 days,
what will be the charge for another burner for 56 days ; 220 cubic feet
EXAMINATION QUESTIONS.
277
or I7Q0,
K)
)5
10
> propor-
tyable in
e length
Igar and
on of a
I. How-
distance
scale be
5tion of
iO days,
ibic feet
of gas being consumed by the latter, whUe 115 are consumed by the
former? ■
(9) ^Extract the square root of 0*000008.
(10) Reduce 828 to the binary scale.
(11) Find the amount of £5 in 2i years at 3 per cent. 'compound
interest ; the interest payable yearly.
(12) The national debt of the United Kingdom amounted, in the
year 1860, to £801477741 ; the interest paid on it was £26833470;
calculate the average rate per cent, paid as interest.
The total revenue for the year ended June, 1861, was £71863095 ;
how much per cent, was the total interest of the total revenue ?
VIII.
(1) A book consists of 21f sheets of 16 pages, each page contain-
ing 38 lines; how many sheets wiU it run to, if printed in sheets of
24 pages, each page containing 32 lines ; the length of the line in the
latter case being || that of the former ?
(2) A bankrupt pays his creditors £1915. 10«. 6^.; calculate the
whole amount of his debts, the composition being 9s. 5d. in the £1.
(3) Divide 358-3 by 1-27, and from the quotient subtract lof 4
of 12. T T
(4) Reduce 3 furlongs, 5 yards, 2 feet, 1 inch, to the decimal of a
mile.
(5) Add f + f + 1|, and from the rem't subtract ^ of 2.
(6) Reduce the circulating decimal -634 to the equivalent vulgar
fraction.
C^) li the yearly rent of 325 acres 2 roods of land bo $450, what
would bo the rent at the same rate of a square mile ?
(8) Find the interest on £485. 7«. 6^. sterling for 3 years and
8 months at 6 per cent., and reduce the result to dollars and cents :
£1 sterling being worth $4.86.
(9) Extract the square root of -075, to 4 figures.
(10) Find the value of f of -^ of 25 cwt. 3 qrs. 1 lb., and reduqe
the result to a decimal of 100 cwt.
(11) Add together the fractions ^ + 2i + 5^ + f; multiply the Bum
by f, and divide the product by 4 times the third of 7.
Mk
278
ABITHMEnO
(12) Find the interest on $667.40 for three months and 10 days, at
8 per .cent. Convert the result into sterling money, a pound being
worth $4.86. * * •
IX.
(1) Add together 2J + f + 1^ ; subtract from the sura the half of |,
and divide the reiiiaioder by 6.
(2) The total value of the Imports of Canada for the year 1861
was $48054836, and the total duty on them was $4768192.89. What
was the average rate per cent, levied ? •
(3) Find the interest on $19876.54 for 3 years and 3 months &t^
per cent.
Gynverf the result into Halifax currency.
(4) Extract the square root of 2 to 4 decimal places.
(5) Express 305 yds. 2 ft. 5* in. as a decinwl of a mile, and verify
tiie result by reducing the decimal to a vulgar fraction, and finding the
value of tliat fraction of a mile>
(6) Calculate the ratio of the English mile to the French kilo-
metre ; the kilometre being equal to 1000 metres, the m^tre = 89"371
inches.
(7) Find the value of f of 55. 6d., bring it to the decimal of
£1 currency, and convert the result into dollars and cents.
(8) If $100 in Canadian bank-notes be worth $103.50 in United
States silver, what is the value of 367 United States silver dollars in
Canadian currency ?
(9) Find the interest on $650 at 6 per cent., for 8 years and
8 months.
(10) Add together the sum, the difference, the product, and quo-
tient (the greater being divided by the less) of ^ and |.
I Give the reasons fur the rule in each proces^mentioned. .
(11) The weight of a cubic inch of water is 252*458 grains, a gallon
of water weighs 10 lbs. Avoirdupois ; find the number of cubic inches
in a gallon.
(12) Reduce tlie fractions in question (10) to decimals ; solve the
question then, and shew that the two results coincide.
EXAMINATION QUESHONB.
279
T»
X
(1) Give the rule for division of decimals and the reason for \t
(2) If gold be at a premium of 49 per cent, when purchased with
ITnited States notes, what is the gold value of $357 in notes?
(3) To what sum will |600 amount in 6 years, 6 months, and
20 days at 6 per cent, per annum, simple interest ?
(4) Extract the square root of 82-56.
(5) Acid together | and | : multiply the sum by 1^, and divide the
result by 4|.
Beduce the above vulgar fractions^ to decimals, perform the same
operations, and shew that the results obtained by the two methods
coincide.
(6) If 6 men will dig a trench 15 yards long and 4 broad in three
days of 12 hours each, in how many days of 8 hours each will 8 men
dig a trench 20 yards long and 8 broad ? I
(7) Divide the sum of 10 and yV l>y tli« diflference, and also the
difference by tlie sum, and find the difference of the two quotients.
(8) Find the value of •439£. +''l-256». + 3-7l8<?.
(9) If 21 men mow 72 acres of grass in 5 days, how many must
be employed to mow 460 acre?, 3 roods, 8 perches in days ?
(10) What sum must be put out on interest at 4| per cent, to
amount to £4027. 19«. 4^. in 5^ years ? i
(11). Reduce £557. 19«. 5^^. sterling to dollars and cents (the value
of £1 sterli^ being $4,867), and then convert the dollars and cents to
Halifax currency.
(12) Reduce the cumulating decimal '8325 to a vulgar fraction.
'•:;i!
XL
(1) What is the present worth of $3560 payable in 8 months, dis-
count being at the rate of 6 per cent, per annum.
(2) A bar of gold is 4-17 inches long, 0*64 wide, 0-31 inches deep ;
a bar of silver is 13*22 inches long, 1*14 inches wide, 0*65 inches deep;
find the ratio of the first bar to that of the second, if the weights of
any equal bulks of gold and silver be in the ratio of 1935 to 10*51.
(3) Add i + 3| + 6] ; reduce the result to a decimal form, and
divide it by the half of f of 4. •
280
AlUTlIMi^ilO.
(4) Find a number such that the square of it shall be one-and-a-
half timesi 86.
(6) Find the interest on $3450.35, for 185 days, at 6j\ per cent.
per annum.
(6) Find how mucli per cent, is 53 of 65 ?
(7) Find the greatest common meusure of 1281 and 7259.
(8) From the sum of ^ + 8|+ 2^ ; take t'le difference of ^ and ^,
and divide the remainder by the half of fj.
(9) Convert ff into a decimal, and divide the square of the result
by '0012.
(10) The volume of a sphere whose radius is r is fwr^' (where
»r = 8'14159); find hence in lbs. Avoirdupois tlie weight of a hollow
globe i of an inch thick, the diameter of whose internal surface is
8 inches, if the weight of one cubic inch of the material be 500 g< tins.
(11) Calculate the ratio of the En;:'lish mile to the French kilo-
metre ; the kilometre being 1000 metres (the m^tre = 3D-371)
(12) What U the difference between the income arising frjrf £2500
invested in 5 per cent, stock, when the price of the stock is 114 and
the same sum invested in 3 per cent, stock at 92?
xn.
(1) The greatest amount of sea-salt which 10 lbs. of pure water
can dissolve is 87 lbs. How much salt will be required to saturate to
an equal degree of saltness, 2 gallons and 8 quarts ?
(2) The area of a circle (radius = r) is n-r^ and the volume of a
cylinder with circular base is equal to area of the base multiplied by
the height. Hence, find the lieight of a cylindrical jar which exactly
contains a gallon (10 lbs.) of water, if the diameter of the base of the
jar be 8 inches, and the weight of one cubic inch of water be 252*5
grains.
(3) British standard silver contains 37 parts in 40 of fine suver,
and 1 lb. Troy of standard silver is coined into 66 shillings. Calculate
the value of the money which can be coined from 100 lbs. Avoirdupois
of fine. silver.
(4) The moon revolves in her orbit round the earth in 27 days,
7hrs., 43 rain., Usee. Through how many degrees of her orbit does
she move in 7 days ?
i
EXAM UOTON QTTESTIOI^a
le-and-a-
per cent.
281
^ and I,
he result
(where
i hollow
irface is
) grains,
icli kilo-
A £2600
114 and
re water
.urate to
mo of a
)lied by
exactly
3 of the
36 252-5
e Buver,
ulculate
irdupois
J7 days,
bit does
I
f ifrT!^!"!! '^!! r',°^'^ ^" ^* <1ays,how for ^ui another
b^ in 6 days, .f she can sail 8 miles for the form
(6) The popu
er'g 7 ?
, . ,„, - - 'oa of the oitj of London in 1801 was 86484'?
.nd^ m IWl inooos*. Find the rate per cent, of the iZJT^;
(7) At prese t tlie value of fhe British wereign is ,' 1 883 • it )»
fte value of .he sovereign shall be »6.Mi. Calculate what sui in the
cuTn" " "°' "'"'" '' '''"'™"^"' '" *2««» of ">« P"-°t
Ti-nrfn^ .^l'" ""T ^'"""^ "' ^™"y '■""' ">« Sun is 0-38 times the
Earth s distance from the Sun. Assuming the earth to move in a ciro e
^r,?u, "^ «0 g'^S^opl'ieal mUes «qual 695 statute miles; find
the disl.ice of Mercury from the Sun in nte miles.
nf wi? *^k"' ?f * """'*' ™'"^'' ■"'"'"* "<'<' ■'» 2 feet and which is full
of waler, 5 gallons arc drawn. Find by h„.v many incho., tl« d.pth
weigns 1011)8. and a cubic foot 1000 ounces.
(10) Add together f + | + 21 , divide the result by half the differ-
en.e between | and ^, and reduce the quotient to a decimal
for r mon^ITs^ '""^ ''''"""'' "" ^^^'' ''*' '^^ '' '^ ^'' ^«^^- P^^ «"°^°^
XIII.
weii?oJTTnv'^''ff ^"'I '".^ Apothecaries' weight. Comparative
weight of Troy and Avoirdupois pounds; advantages of requiring the
use of only one kind of lb. *«quirinfe uie
W ^^^/^"^ ^''"^ ^!''^' '"^ ^ '^^^' ^^"* ^^ y^^^' i" a Voh. Square
cut '?/•'""' r'" "^ ^" '^^^- ^"^^° i'^^^'- - acubicfoorand
cubic feet m a cubic yard.
of 13| by I of ^.
<^'^^^^|-'^'lJ--^ivide|-ofA
(4) Divide '025 by -12 ; 594-27 by -047.
(5) Square root of -00089; cube root of 140- value of |. of acre.
IMAGE EVALUATION
TEST TARGET (MT-S)
t
1.0
I.I
1.25 II
1.8
U ill 1.6
iJ
w
V2
^'/W
'' m
-^ A^'^/- ;^,
&3^.Jh
^^?&'
%
/
O^^^'y,
ll 4.^
liL
Sciences
Corporation
23 WEST MAIN STREET
WEBSTER, N.Y. 14580
(716) 872-4503
,V '^g
sr AT 4
^
^M
HM
282
ASITHMEnO,
(6) 75 yds. at 8H
(7) Interest of £60 at 10 per cent, for ono year. Of £27. 10#. at'
6 per cent, for one year. ^ t
(8) If 5 men can build a wall in 6 days, how many can bnUd it
in one?
(9) A gentleman pays in all 50«. to Ms work-people, to eacli
^man !«., each woman Si., and each boy 4d., hhe number of eftch being
eqnal ; what was the numhor of each ?
(10) If a family of 3 persons expend £200 in 9 mos., how much
will serve a family of 18 per.^ons 12 mos. ?
(11) What is the price of 60 lbs. at 2«. Gd. a lb. ? at 3«. 4d, a lb. !
(12) Give the cost of 1875 lbs. at $3 a ton.
XIY.
(1) Chang* •327 into a vulgar fraction.
(2) Find the least fraction which added to the sum of 1.2, '12,
•012, and 210, will make the result a whole number.
(3) Give the square root of 1*3 to four places of decimals.
Give the cube root of -=- to two places of decimals.
7
(4-) Divide 8 days 8 hours by 2 hours and 40 minutes.
■ (5) If 15 pumps working 8 hours a day can raise 1260 tons of
water in 7 days, how many pumps working 12 hours a day will raise
7650 tons in 14 days ?
(6) If 12 men can dig a ditch in 4 days, in what time can 32 men
perform the sanne work ?
(7) How many yards of carpeting 27 inches wide will cover a
room 14 X 16?
(8) Find the present value of $1 due in 8 months at 8 per cent.
(9) A person buys goods for £5. 17«. 6i. and sells thenr for
£9. 18«. Gd. H')vv much per rent, does he gain ?
(10) If 3 oz. of gold be mixed with 9 oz. of silver, what is the value
of 1 oz. of the alloy, gold being $18 and silver $1.25 per ounce?
(11) 5 lbs. of tea at $1, 9 lbs. at 90 cts., and 14^ lbs. at 80 cts.,
what is a lb. of it worth ? and how many lbs. of each i^t the above rates
most be taken to make a compound worth 85 cts. a poundi?
'EKAMm kTION QUESTIONS.
"283
I
xy.
(1) Find the value of (i) j^-^ of 5 hrs. 25 min. 40 sec.
C^i a + l)£. + (^ + f>. + a H- i)d. (Ill) £3. 18s. 6d. X 7561.
/ (2) The furewheel of a carriage is 6 ft. C in. round, and the hind-
wh*>el is 11 ft. 4 in. ; how far must the carriage travel before each
wheel shall have made u Humber of complete turns ? How often will
this happen in 10 miles ?
(3) Define a decimal fraction, and give the rules for pointing in'
the multiplication and division of decitrals.
Divide -001. by 1 x -01 x 100; -20736 by 1-2 x -012 x 120, and
98-8452864 by 76-8 x -0987.
(4) Write down the table of time. How do we determine whether
any particular year is a leap-year ? Are 1864, 1900, 1950, 2000 leap-
years ?
(5) What is the interest at 7 per cent, per annum of £138. 6«. 8d.
from January 1st, 1862, to April 15th, 1864 ?
(6) Extract the square root of 74684164 and -03275 to 4 decimal
places.
(7) Divide £23. 15«. 7J^. by 37, and 571 yds. 2 qrs. 1 nail by 47.
(8) A wall that is to be built to the height of 27 feet was raised
feet high by 12 men in 6 days ; how many men must be employed
to finish it in 4 days ? ,
(9) Eeduce to their lowest terms -— and ^^^
1536
(10) Find the value of (i) -~ of an acre.
8
364*
(u) ~oe48.10d.
(m) -009943 of a mile, (iv) -625 of a shilling.
ai) Reduce (0 ^ of a pound to the fraction of a penny.
2
(ii) ycwt. to the fraction of a lb. (in) -26<?.todecimalof a pound.
(iv) -056 of a pole , o decimal of an acre.
(12) Find (a) the interest on £517. 15«. for 3^yr8. at 6f per cent,
per annum.
(/S> The present value of £720 due in 4 yrs., at 5 per cent.
interest.
(18) A square fishpond co^icains an acre ; find the length of a side.
284
ABITHJO^O.
XVL
(1) Add together f, 2|, 13^, and reduce f of 2«. 4^. to tLe frac-
tion of 2«. 6^.
(2) Reduce 3«. 4^d. to the decimal of £1, and 8^. Qd. to the decimal
of£2. 10». • ,
(8) Prove the rule for pointing in the extraction of the square
root of the numbers. Find the square root of 534-5344, and prove that
ay s- 2V7 -..
(4) The prime cost of a cask of wrine of 38 gallons is £25, and
8 gallons are lust by leakage : at what price per gallon must the re-
mainder be sold so as to gain IC per cent, in the whole prime cost?
(6) If in Toronto there ia a discjunt of ^ per cent, on English
gold, when excbafrige in London is quoted at 112, shew that a merchant
who wishes to send money to London will save nearly 2 per cent, if
instead of buying exchange he sends the gold, having given that the
par of exchange is 109^, that when exchange is at par the pound
sterling is worth $4.87, and that the charge for freight, insurance, &c.,
on golt' from Toronto to London is f per cent
F
(6) Prove that
ys + V3 _ ys
2y2 y5 - y3 + y2
= iand(^2 + yii + y2)' bV2+V2 + ^/2
V'2 fi/2-V2
(7) Describe Gunter'a chain, and explain fully how it is nsed to
find the acreage of a field.
(8) Define Present Value and Discount If the discount on £567
be £34. 14s. 3fd, simple interest being reckoned at 4^ per cent, per
annum, when is the same due ?
(9) What is meant by "course of exchange" and by "par of
exchange ? " Explain briefly the cause of fluctuation in the price of
exchange.
EXAMINAllON QUESTIONS.
286
Exchange between Toronto and London being qnoted at im
what must I give for a Bill of Exchange for £18. 19*. stg. ?
fhf^^ ^''^^'l" *^^ "^'^^ ^^ transforming circnlating decimals into
the^ equivalent vulgar fractions, taking as examples 'krU
ad^Ual '^' ^"^^'""'"^ expressions, brieflj explaining any artifices
(1) -9286714 + •821^-2857 + '48 + 2-87.
(ii) 11-036 -3-9876.
(m) -163 X -06.
(iv) 1-015873 -^ 1-636904761.
(11) The true length of the year is 865-24224 days ; find in what
w^me the error in the common reckoning will amount to ; d^y
(13) Define a vulgar fraction, and prove the rule for multiplying
fractions together, taking as example ^ x |. Shew which is greater
Vf or VT without finding their actual values. .
xvn.
fT,.f ^^^ T.^"""^ ^""T "^^""^ "^en must £105. 8.. 4d. be divided in order
that each man may have £10. 10s. lOd. ? •
(2) Express the fractions #1 JtA- us oa ^,.„«*« i. .
1 . cv^uivyus, j-g^j TSTi T-ff ftS tractions hnvino o
common denominator, and express the differLce-of the first twTfs a
fraction of the difierence of the second two.
(3) How many ounces are there in a huudred-weight, and how
many square yards in an acre ? ^ ' ^
(4) Divide 220-8864 by 72-66 and 2-208864 by -07266
perafre.''"^ *'' '"''' ''''''*"' 2 roods, 27 poles at* £1.^8^
ponnd LTrt:;^ ^'^ ^"^^^^^^ ^^ '''' '^ ' ^^^ ^^ « P- -t. com-
fir . ^^ I ™'"; ^ '^^™^°' ^ ^°^' «' ^ ^''^^ «^ do a piece of work in
eCaays; how long will it take 1 man, 2 women, 3 boys ZiToiZ
working together? ' ^ ' " * 6*"^
(8) Find the difference in the expense of carpeting a room 17 f^
in. long and 12 ft. 6 in. broad with Brussels carpet i of a "L" ^t
me
ARITHMETIC.
at 4s. Qd. per yard, and with Kiddonninster | of a yard wide at 8*. 6^,
per yard.
(9) "Whac Bum will amount to £425. 19«. 4f d in 10 years at 8J
per cent, simple interest ?
(10) "What is the yearly interest arising from the investment of
£385. 7«. S^d. in the purchase'of 3 per cent, stock nt 94} ?
(11) "Write down tho- tables of Troy measure, and of square
measure.
(12) Divido 109339 by 35 by short division, explaining the method
of finding the remainder.
/IN o- IV /^ 23760
(1) Simplify (I) -61^.
XYIII.
(II) (f of6^) + iof(2^+Gi).
(m)
X
^4+-
8 +
4 +
__ Waof^of^VHC/ronoff). (v)| ''i jj'f .
5*
(2) Find the value of
(i) £4^+ 11 Js. +7tV^. (ii) 1416 A '.. R 16 P-5-^ of (4ac. 3ro. 27pc.).
(3) Reduce | of Is. 9d. to tlie fraction of 3s. 4d
(4) Find the value of (i) 2-7- -913. (ii) 91 78 x -381.
(in) -00044406 -f- "0112. (iv) 2-27-5-1-136.
1 1 1
(V) 1 +
+ ... to 5 places of decimah.
12 1.2.3 1.2.3.4
(5) Find by practice the value of 6 yds. 2 ft. 9 in. at 5s. Z^d. ijer
ibot.
(6) If a lb. of standard gold which is 22 carats line bo woxth
£46. 14s. 6<Z., find thevalue of the Mohur of Bengal of weight 7 dwt,
23 grs., and fineness 993 parts pure gold to 7 of alloy.
(7) What snm invested at 4 per cent, simple interest will amourt
to £111 in 5 years?
(8) If £1000 of 3 per cent, stock at 72 ba transferred to the 4 per
cents, at 90, find the change of income.
Notation.
ANSWERS TO THE EXAMPLES.
EX. I. (p. 12.)
1. 63; 81; 99; 40; 13.
8. 4000; 1471; 6930; 9009.
.6. 100000; 676060; 202693.
6. 7003000; 11108106; 64064088; 613020803.
Y. 2000000000; 9000800021; 94090094904.
2. 200; 303; 764; 888.
4. 27604; 83000; 0016.
Numeration, • ' ^
1. Forty-three; sixty; eighty^ight; ninety-seven ; fifty-ninc; twelve-
twenty-one; niueteen. ' "^'J'"'"®' ^^^w^e,
2. Two hundred and fifty^ix; four hvndred and one; five hundred, nine
hundred and nnety-nine; three hundred and sixty-five; five hundred td
seventy-eight ; eight hundred and thirty^even.
♦»,. ^' V""" ^'^T'^^''^ 5 o^e thousand, seven hundred and twenty-four three
^ousand and three ; seven thousand, five hundred and eighty-four ; one thou!
sand and seventy-five ; one thousand, five hundred and forty-one
^i^fA^^'T'^^'V^T"^ ^""^ '^'''' f«^^y-««^e" thousand and forty-nine;
sixty-three thousand and ninety; eighty thousand and eight ; three hundred
and forty-one thousand, three hundred and twenty-three
«• S« "'ilHons, eight hundred and fifty thousand, four hundred and six-
eight miU.ons, eighty thousand, eight hundred and eight ; seven maiions, eieht
hundred and forty-nine thousand, six hundred and t^; four hu2ed f d
eight ^Pn thousand, two hundred and fifky-four.
«a„dtn/r 'f '""^ "^^ """'' twenty millions, two hundred and twenty thou,
sandand wenty-two; nmety.twomillions,fivehundredandsixty-eightthousand.
Tda^^tve:^;'^'^^-^^^^^*' *'^'^^"'"^^- -^ ^^^'-' Ld%ht7th:i
thoulJT ^T""!' f ' ^""^'"'^ "^'^ '^*y "^"^°"^ fi^« t^d'-^d and thirty
thous^id, wo hundred ; eight hundred milUons, three hundred and nine thou^
sand five hundred and sixty; nine billions, seven hundred and thirty-llht
mUUons^our hundred and thirteen thousand, two hundred and eight '^
288
ANSWERS (pp. 16-^9.)
^. Odo hundred trillions, one hundred and ninety-eight billions, seven
hundred milliooa, ten thousand and ninety ; forty-eight quadrillions, seven hun<
dred and twenty-six trillions, eight hundred and seventy billions, six hundred
and thirty<four millions, one hundred and three thousand, two hundred and
sixty-four.
Ex. IL (p. 16^
2. 8611911. 8. 1148890.
t>. 8794787869.
9. 16237839200806.
12. 14547 ; 48829 ; 82891.
14. 149036967938; 16696688926; 142228910946.
16. 4304268. 17. 1000002783686293.
19. $840086. 20. $80081668.
1. 423578.
6. 1881390.
8. 8665743090.
11. 61463796.
4. 2923088.
7. 713878689U0.
10. 9691400863.
13. 779264; 2926618.
16. 98929.
18. $22640000.
21. $126246091.
« Ex. III. (p. 19.) 7
1. 899899. 2. 800368384. 8. 73646889.
4. 6130908; 7036970; 111232112.
5. 116849491 ; 2922930923 ; 668990634342.
6. 8087 ; 4986. 7. 3999996 ; 99700000. 8. 146169P.7.
9^ $68624.82. 10. $1814609. 11. $567.
Ex. IV. (p. 21.)
1. XXX ; XLVIII ; LIX ; CCXX ; DO ; M.DeCO.XLin.
2. Twenty-three, 23 ; sixty-nine, 69 ; two hundred and eighteen. 218 ; five
tihousand and one, 5001 ; one hundred and fifity thousand, six huudred and
three, 160603 ; two millions^ one hundred, 2000100.
4. 6235660.
V. 4843162 8. 3270069.
11. 614796033260.
Ex. V. (p. 28.)
1. 401?<'^08. 2. 949723. 8. 24642451.
6. 67248660. 6. 83076.
9. 128137428. 10. 694090141.
12. 4222404, 6802762, 12432634;
61964682, 87860370, 397683780;
686269802,2868835636,2681382769;
182681498641, 68943103679, 7093523Y486, 67108866880.
ANBWEE8 (pp. 8^-88.)
289
18. ie822m; 213777000; 2361710800; 21810149152-
T6340824080; 121932631 1126852C9;
40,165,302,248,806,278,764,132.
14. 44886996200692; 2606661667240; 128672881324016-
16232906283422680 ; 1,630,188,063,103,649,208,285. '
15. 1966470720; 684763647963886.
16. 8876; 64096923986; 440966790820. 17. 21084100; 1408008;
8. 930622, rem. 86.
6. 11806669.
9. 8862.
12.. 22161387, rem. 47191.
16. 6629947^
18. 14830201.
Ex. VL (p. 86.)
1. 548817. 2. 18674687.
4. 71840987. 6. 814646, rem. 17.
7. 284916. 8. 70474«.
10. 40980, rem 270. n. 691863.
13. 6719070. 14. 7676.
16 243096269. 17. 3396, rem. 6094687. _. .,„,„,
19. 9000900090009, rem. 1 ; and 900009000090, rem. 10.
20. 8854, rem. 26167. , 21. 746116, rem. 83837.
^^- ®^^*' 23. 874869. 24. 764096.
26. 11717201, rem. 645. 26. 5771, rem. 542962567
27. 89486, rem. ?211. 28. 86. 29. 2826863, rem. 66.
80. 68911741. 31. 9862. 82. ll^J.
88. (1) 134761.90^?, 1137254.90^^ (2) $102493 nearly.
84. (1) 168659.12^^, $70390.42|if. (2) |1831.30 nearly.
Ex. Vn. (p. 38.)
^ 1. 28944. 2. Nine millions, ninety thousand, nine hundred and nine •
ninety thousand, nine hundred and nme ; 9181818 ; 9000000. '
8. 86yeaw. 4. 548501. * 5. 8481622.
II.
2. 5 years. 8. 700409000000000000. 4. 638242.
III.
2. 800 days, and 76 lines remaining. 3. 13008.
6. A, B, and C score respectively 18, 57, and 33 runs.
B. 19062.
4. 9376.
290
AJTBWBBB (pp. 4(MJ7.)
IT.
1. 24570. 2. 29; 71. ' V
8. 100100101 ; one thousand and ten milUouB, one hundred and one
thousand and ten; 1840. 6. 4649206.
T.
1. 69788; 48, with remainder 91. 2. 816. 8. 20000 English only
80000 French only ; and 70000 both English and French. 4. $68412.
6. 624.
Ti.
1. M.D.LXin, IX. 8. 667842.
4. Two hundred and seventy thousand, one hundred and Ujlrty ; twenty-
six thousand, seven hundred and eighty-four; 10234; 6.
6. 81, 18, .16 years are the ages of the children.
Sx. Vin. (p. 87.)
1. 87828 cts. ; 102787 cts. 2. 18680<?. ; 8744J.
8. 201 halfpence ; 9l6g. 4, 80426^. ; 188663 halfpence.
6. 8866O7. ; 8660 fourpenny-pieces. 6. ^6909. 18«. Hd.
7. 200 half-crowns, 1000 sixpences, 1600 fourpences.
8. 843666 grs. ; 6493 lbs., 19 dwta., 21 grs.
9. 196597 lbs., 2oz., 17dwts., 12 grs. ; 61466 drs. \ 164368 so.
10. 249056 oz.; 14 tons, 16cwt., Iqr., 18 lbs., 14 oz., 9 drs.
11. 162 tons, 17 cwt., 8 qrs., 26 lbs., 9 oz. ; 19489376 drs.
12. 98920grs.; 72oz., 4dwts., 22gr8. 13. 6864 yds. ; 36306280 in.
14, 7 lea., 4 fur., 10 po., 5 yds., 2 ft., 4 in. ; 471 2644 in.
16. 2681126 barleycorns; 88016^ yds. 16. 79 chains, 2 yds. ; 9 ft., 9 in.
17. 1348 nails; 11 24 nails. 18. 2004 nails; 880 nails.
19. 6680 po. ; 273460 sq. yds. 20. 61 88724 sq. in. ; 188847^ sq. ft.
21. I5760008q.links; 312ac., 2ro.
22. 783 cub. ft. ; 8 cub. yds., 10 cub. ft., 1031 cub. in.
28. 794163 cub. in. ; 1246888 cub. in.
24. 4604 pts.; 11432 gals., 2 qts., 3 gills.
26. 24344 qts. , 89863 bus., 3 pks., 1 gal., 2 qts.
• 26. 9000bush.; 1291chald., 84bush., 3p!M.
27. 27386 sheets; 108 reams, 9 quires, 17 sheets.
28. 22266000 sec.; 2674859 sec.
■ 't^li0r
AJR8WEB8 (pp. 60-6^)
n
2«l
89. 668190ipt8.; 884096^ qts.; 83623}^ gals.; 2820^ luw
80. 878223200 Hec. 81. B026hra.; 18090000 eea
82. 77976820 bq. ac. ; 2189180800 sq. ac ; 21 15&40000 sq. ac.
Ex. IX. (p. 60.)
1. £168.16«.8K 2. £271. 10». 8. £8829.8«.1K
4. 148 tons, 16 cwt., 4 qrs., 21 lbs. 6. 14 lbs., 9oz., 2dn., 19gra. ^
6. 228 ac, 8 ro., 16 po.
7. £66851. 0«.4ii. ; £79261. 16«.0K ; £76^861. 16«. 2^
8. 49 lbs., 10 02., 18 gra ; 198 lbs., 9 oz., 19 dwta. ; 1767 lbs., 1 oz., 18 dwta.,
I4gra.
9. 282 lbs., 4 oz., 4 drs., 1 so. ; 246 lbs., 4 oz., 2 dra., 17 grs.
10. 2 tons, 18 cwt., 8 qrs., 24 lbs., 8oz. ; 2214 tons, 9 owt., 2qrs., 261bfl. ; '
168 tons, 9 cwt., 2 qrs., 2 lbs., 2 oz.
11. 199m., 2 fur.; 166m., 7 fur., 18 po., 1yd., 2ft., lin.; 126 lea., 2m.,
4 fur., 198 yds. 12. 186 yds., 1 qr., 3 na. ; 182 Eng. ells.
18. 181 ac, 16 po. ; 87 ac, 2 ro., 26 po., 14^ sq. yd., 2 sq. ft., 93 sq. io.
14. 86 c yds., 9 c. ft., 676 c. in.
15. 804 gaL, Iqt.; 47 pipes, 55 gals., Iqt.; 403hhds., 86 gals., 7 pts.
16. $701.47 ; $178937.93.
17. 28mo., 1 wk., 19 hrs., 40 m. ; 216 yrs., 89 wks., 6 d., 17 hrs., 61 m., 51 sea
18. 86 yrs., 7 mo., 2 wks., 8d., 13lirs.
1. £831.19«.lHd
8. 4 cwt., 8 qrs., 5 lbs., 7oz.
5. 6 ac, 2 ro., 31 po.
7. £77.17«.10H
9. 6 tons, 16 cwt., 3 qrs., 6 lbs.
11. 80 yds., iqr., 2na.
18. 1 m., 6 fur., 86 po., 5 yds.
Ex. X. (p. 64.)
2. £313.6«.1H
4. 12 fur., 36 po., 1 yd.
6. 56 qrs., 5 bush., 1 pk., 1 gal.
8. 859 lbs., 4 oz., 6 dwts., 5 grs,
10. 16 lbs., 8 oz., 4 drs., 1 sc.
12. 1yd., 1ft., 10 in.
14. 1 ro., 28 po., 28 sq. yds., 8 sq. ft.
16. 1 cub. yd., 20 cub. ft., 1306 cub. in. 16. 15 tuns, 2 hhds., 63 gals., 1 qt, 1 pt.
17. 5bar., 8fir., 8qta. 18. 2 mo., 1 wk., 3 d. 19. 7°, 61', 26".
JO, j562.0«.4K 21. $5978.50.
IS
293
AMtmMMB (pp. ar<«^)
Ex. XI. (p. 61.) \
1. 11738.80; $3042.90. 2. £79. I6«.8(/.; £WAI^.9d.
8. £58402. I5«.6i</.; £69218. 2«.
4. £8791 13. 9«. 2^d. ; £399746. 9«. 8\d. ; £1222447. 9«. 7i<i ;
i^floneee. 6«. 8K
£. 698 lbs., 2oz., lldwts., 16grs. ; 8119 Ibo., 6oz., 12 dirts., 12gif.
6. 88ton8,2cwt., 2qr8.,22lb8., l&oz. ; 228 tons, 16 cwt., 8 qn., 10 lbs., ISOB.
^. 647 lbs., 6oz., 4dr8. ; 8102 lbs., 8oz., Idr., Isa
8. 606 yds., 1 qr., 2na. ; 8670 yds., 8qrfr., 2 no.
9. 2 fur., 10 po., 1 yd., 1 ft., 10 in. ; 1 m., 1 fur., 14 po., 1 ft., Sin.
10. 186 ac, 8 ro., 27 po., 26 J yds., 4 ft.
11. 4671 ac, 1 ro., 24 po. ; 40880 ac, 2 ro., 83 po.
13. 677 gal., 2 qts. ; 14841 gal., 3 qts. 18. 199 Ids., 1 qr., 7 bos., 2 pks. ; 8681 Ida,
14. 1 yr., 18 wks., 4 d., 8 h., 84 m. ; 88 yrs., 46 wks., 2 d., 18 b., 6 m.
15. 072 tuns, Ipipe, 86 gals., 8 pts. ; 7706 tuna, Ihbd., 10 gals.. 2qUi.
IC. 1691 bar., 17 gal., 2 qts., 1 pt. ; 38186 bar., 30 gal, 2 qts.
17. £1068.4«. lOK 18. $26452.
, Ex. XII. (p. 69.)
1. £86.8«.8|d. 2. £39.9«.6^d 8. 15Ibs., lOoz., Idwt, Ugn.
4. 228 lbs., 1 oz., 2 drs., 1 sc 6. £1. Ua. 'Id,
6. 2 po., 8 yds., 6 in. and 28 in. rem. 7. £1. 88. \(i\d.
8. £8. 18«. 4(/. 9. 2 c yds., 4 c ft., 1391^|| c in.
10. 108. O^d. 11. 2 tuns, 1 pipe, 27 gals., 2 qts
12. £77. ll».4i<i: 13. $283. 14. 12«.9-/f^.
16. £13.16«.8ii. 16. 17cwt., lOlbs., lloz., S^idi-s.
17. 8cwt., lqr.,121bs., 8oz.,.7Hdrs. 18. 7 mo., 26 d.
19. 14 d., 18b., 27 m. 20. 81bs., Uoz., 14Hdn;
21. 7 lbs., 14oz., 4|H«'ra-
23. 2ac, 8ro., 27 po., 198q.yJs., 7 sq.ft., l^'^sq.in.
28. 14po., 12sq.yds., Osq.ft, 119-4V8q. in.
94. 1^ nails. 26. 7bus.,Ii>{^pk. 26. $710.
^^PP-W'^^^^Pm^BPIPW
1« ovtillUM,
5. S^iiS ^s*"
im^Em (pp- fM70
Ex. Xin. (p. 71.)
a. 28tiiiie«. 8. lOStimM.
fl. S6 timed. 7. ediiuies.
9. 186JU3} times. 10. IStimea,
'^fi
4. 16« timet,
t. i$8ti0XM.
Ex. XIV. (p. 74.)
1. 166 fl., 1660 c., 16600 m.; 68*2 e., 682 m.
2, 809-8fl., 8095c., 80960m.; 961-29fl., 9612-9o., 96129in.
8. 180-66 fl., 1806'io., 18066m. ; 926 fl.. 92-6 o, 926m.
4. WO-Ol fl., 10001 c, lOOOl m. ; 460 26fl., 4602 60., 46026m.
Ex. XV. (p. 76.)
1. £264. Ifl. 8-. 6ro. 2. £662. 7fl. 7c. 7m.
8. £3. Ifl. lo. 4. Im. 6.*£l. 6fl. 6 m.
6. £884. 1 c. 6 m. ; £4838. 2 fl. 8 c. 9 m. 7. £16. 6 fl. 6 c. j £932. 4 fl.
8. £8007i;0. 2 a 6 m. ; £2786492. 8 fl. 8 c.
9. £88. 9fl. 1 c. 6m. .10. £678. 6fl. 6<j. 4m. ll. £46. «fl. 6c.
Ex. XVL (p. 77.)
I.
1.844. 8. ei26400. 4. $687.67. 6. £1712. 6. 64 gaL
II.
1. £87. I2«.»fi. ». $28782.96. 8. 90090; .179«)000.
4. £2.4«. 8<l. ; £1.9«.6d: 6. 66dozeDa. 6. £1. 8<.
ni.
1. 68 lbs., ISdwts., Bgw. 2. 116 yds. 8. £1. 6». Hd.
6. $4268-66. 6. 6 dollars, 12 dollars, 36 dollara, 144 dollars.
IV.
2. Gain in fid cue x 12=gain iu 1st case x 11. 8. 19i.
6. 19~gaL 6. 1000 perches. 7. 16228806.
V.
2; 475^Q0QpOO. 8. 4HdoBeii8. 4.216000. 6. Moenti.
6.,£|84, 6fl.6o.8ra.
294
AH8WXK8 (pp. 79-08;)
1. £i6 Ut. td.
4. l6s.M.
1. 6260820 min.
j). 26|yds.
Tl.
2.47d8., 1ft.
i. $7.92; 13.96; $1.8
8. $90U
6. 425qiuut0nk
TII,
2. 16 tons, IScwt., Iqr., 14(Ib0, ; 48in., SAnr., Ipo.
4. 1205 dbys, 13hrs., 11', HH''* *• H^.Sa
6. $92040; $220660.
Tin.
2. 71iNL,««B^ iHdrs. 8. $844200. 4. $3724.Bg.
6 £1811. tfl.Bc. Iro.; £60S. 9m.; £6727. 8fl. Sm. 6. £109.
1. May 1, 176f.
4 144640..
ix:.
*. 11300.
A. ifhsi., 7 m., 12 sec.
8. 20Hf gee.
6. £6791. 10a
1. 86.
.4. $647.80.
2. lit. 1«Ht!f9^
5. $1.60.
8. 286A\niO.
8. 6 men, 12 womei, 13 boyr
1. 8.
7. 17.
18. 8.
19. 7.
26. 11.
81. 87.
2. 16.
8. 2.
14. 12.
20. 6.
26. 123.
82. i7.
87. 23.
Ex. XVn. (p. 34.)
5. 9. %. n.
9. 20. 10. 16.
16. 491. 16. 18.
21. 86. 22. 84.
It 28. 28. 85.
83. 2. 84. 2.
£8. 7. 89. 4. -
Hx. XVIII. (p. 88.)
i. 4. 8. 40.
11. 29. 12. 8. .
17. 18. 18. 2.
28. Sd4. i4. 83.
29. 2223. 80. 14.2087.
86. 13. la, 8.
40. 2.
1. 48. 2. 900. 8. 105.
ti. 11803. 6. 18648. 7. 60^^37.
9. 844S88. 10. 2663667. H. 10867906.
18. 72. 14. 48. 16. 80. 16. 120.
18. 20*. 19, 1J02. 20. 192. 21. 282.
28.600. H. ^9. 26.2620. 26.1260..
SO. 7200. 29. p^^9, 99* 1008. 81. 22680.
4. liO.
8. 408672.
12. 11764488.
17. 1080.
22. 846a.
27. 1184.
82. 2017790775.
quuienk
(Anr., Ipo.
0. iiiaa
109.
AirSWXBS {pp.M-#5.)
J 16 46 60 180
'. 12' 12' 12' 12"'
' 1«* 24* 32* 40* 80*
Ex. XIX. (p. »1.)
2 287 616 861 1846
68' 68' 68' 68~*
Ex. XX. (p. 91.)
16 16
2.
16 16
6467 11740 14676* iwl?
Ex. XXI. (p. 92.)
, 21 49 164. 27 68 198. S3 77 242
^•TTT2-'TrT2*r r-2^-
g 62 ISO 338 698 910 218 646 1417 2607 8816
* 2' 5 13* 28* 36 5 T' V IT'^T* IT'
B?l, £??, ^^^V 2691 4096 260 626 1626 2876 4876
86 » '
2
--, _ — , ____, _ — . ___^ ___j ___^ ____^
13 23
2 6 13 23 86
m
ISboyr
. 40.
.8. ^
. 2.
. &d.
. 14.2687.
. 2.
1. 81.
6. 86 1 ^.
11. 6.
16. 969VV.
- 480
•• IT-
11 ^^
"• 43 •
16.
90326
8S1 *
Ex. XXir. (p. 93.)
2. 12f 3. 6i. 4. 12J.
7. 45 ,^. 8. Sif. 9. Slff.
12. 37i«yr. 13. 90tVV. 14. 1614^1-
17. 22t}H- 18. aeSiViV 19- 26U}H.
Ex. xxni. (p. 94.)
2.
7.
12.
17.
ii8
7*
826
13"*
1100
84 •
21631
137 "
- 2?.3
18. |?l
18.
224"
6120 8
2869"'
4 H
„ 14022
6468
202 •
45689
107 *
14.
19.
6. 9.
10. 92^.
16. 69i|}.
20. 12i^.
. 811
10.
16.
20.
11162
18 •
72443
441 '
72418
720 •
ts.
)776.
1 1
^' 16*
1
Ex. XXrV. (p. 96.)
2 ^
86*
8 ^
4 M
*• 72*
9 ^
*• 16*
10.
246
48'
6399
AiriHwii (^p. fPi^
Ex. XXV. (p. 96.)
1.
6.
16.
21.
26.
1
T
8
16
26
m
296'
6
12'
647
7.
2
T
7
16*
«-|-
"• n-
8. r^
18. r4^
~. 17.
22.
27.
8
4*
27
1624'
2081
8068'
18.
11'
276
903*
128
127'
9 11
^' 67'
62
T
18
17'
6.
10.
28 ^^
28.
901
1000'
19.
24.
29.
628
6452'
6
T'
2
T'
20.
^5.
80.
108
186
288
6641'
18
20
81 ^^
""'W
1.
6.
8.
16 20 24
80* 80' 80'
12 10 11
2§* 28' 28'
20 21 26
48' 48' 48'
16
Ex. XXVI. (p. 98.)
16 86
40' 40'
8 ® ^
^' 12' 12'
10
12-
6.
9.
18 27 20
86' 86' 86*
25 27 28
30' 80' 80*
.66
27' 2?-
11.
18 20 21
429
24' 24' 24' 24*
28
12. ^^, ~A
68 66 68
72' 72' 72r
36 60 60 63
90' 90' 90' 90*
1092 1617 2002
10.
16.
17.
19.
-g 240 175
• 4W>' 400' 400*
808 180 ^ if 11
896' 396' 896' 896' 396*
486 824 189 72 48 81
729' 729' 729' 729' 729' 729*
8256 1190 8276 60 3600
6800' 6800' 6800' 6800' 6300'
18.
20.
8003' 8003' 8003' 3008'
147 216 ^ 89
262' 252' 262' 252'
224 688 560 482 72 !j67
672' 672' 672' 672' 672' 672'
90()0 900 90 9
10000' i3ooo' 10000' 10000'
434 297 636 189
756' 766' 766' 766'
Ex. XXVII. (p. 98.)
la order of value the fractions will stand thus :
1.
C 7 8
"Plo't^
' 8» 6' 4' 2
. 4 .6 7 1^.8
'•T^'Y'i2'T^^T'
Ajmnaa (pp. 10(^10%^
297.
,8tl0££ K^ 1. a 1. 1. ti ?^ 1. ?L 1. ^
uO' 21' 12* 16* 26' 11* 18' 7' 22" ««' i''-' ^"' '*'
82' 10' 40' 16' 8*
h ^«p5 .. 2 -8 .^14 1 .1 »..
7. ~af_of4,-of-of6,-,-of-of4f.
8 ' 11 6
, £ 6 29 13
^' ^^* V T' 66' 28-
id 1. 1 1 1 1 11 IJL !^ *^1 ^1* ^'^
9' 22' 18' 11' 36' ^' 748' 448' 162' 76*
8- X'^^'T^^^J-
12.^,8i.|of9|,lof-^ofi..
I* 8 .1
18. -and ^.
47 7
Ex, xxvin. (p. 100.)
The eums viU be :
1. W
- T-
#r 29 -
^•86- ®-
18. 2iV.
18. 2iW.
28. IOtj^u-.
28. mi
88. 18H.
26
82*
8. m.
9. 2|^.
14. lAV
19. Uf
24. em.
29 75*f .
84. 4^M.
89. 18AV.
,^ 149
166
16. 1.
20. m.
26. 8iV
80. 68Sf.
85. 2im.
40. 8i^V
6.
17
64'
11. iHd'
16.-1.
21. ItVjt.
26. ItVjt.
SI. 4441.
86. IItV.
41. 6976.
6. Itt.
12. 14f.
17. 1t*A.
22. 16|^^
27. 2^.
82. 2548|^.
87. 23HJ.
42. 8^.
..1.
^•Ti-
ll. «3fr.
16. ?l21.
21. By ^.
• 27
Ex. XXTX. (p. 102.)
12. 11^.
17. 81H.
99
8. 4rJT]r.
"• re-
22. 8W. 254.
*• 20*
9. 1|.
28. lOUf.
*^- 86-
10. SHI
13. 13Ht. 14. 64IHH. 18- 19Hf.
20. 4-.
6
24. 2^H.
B5. The iiam of (he fHAciioQS is 6 timea as great as their differenoek
ns
4^SWEBS (pp. 4-108.)
Ex. XXX. (p. 104.)
1 1?
*• 21*
6. 2^.
11. 1.
3
104
*• 186-
12.
86
16.
425'
1152'
17. ojjjj.
18.
10
1. 4.
, 163
9 ii
^■852'
18. 34Y|*
17. |-. 18. 6i.
29'
18. 242^.
Ex. XXXI. (p. 106.)
8. lA.
4. ~.
9. 40.
14. 2.
19. J-.
85
6. -^
5
10. 6f.
1
16.
6
«>s-
2. 2^.
8
6.
21'
7 ^
25
4. liV.
8 2^
®- 60- .
14. 1^.
19. 6|.
11. 3iV 12. 168.
-_ 4805 , 320
15. and — .
496 496
16. 4'
5
-I-
21. 86. 22. 7t%.
Ex. XXXII. (p. 108.)
1. 76ctB.; $2; $2.60; 75 lbs.; $7.60.
2. £1.2«.6rf.; £1.6».8<f.; Hid.; 8$.
8. $4.08. ; £1. lis. 2d. ; 3 tons, 8 cwt., 2 qrs., 7|lb8. ; $30.
4. 62 cents ; £1. 16a. 4|d. ^q. ; $8.66J ; 1«. ll^i. |g.
6. 7«.8K; £3.17«.3o?.; la. 4c?.
6. Iqr., 6Jlb8. ; 12 oz.; 6 fur., 88 yds.; 2ro., 20po.
7. 3 fur., 25 po., 2 yds., 1 ft., 6 in. ; 7 hrs., 12 m.; 2 ft. ; 3 qrs., 21 lbs.
8. 7 lbs., 9 oz., 9| drs. ; 1 lb., 9 oz. ; 2 gals., 1 qt., 1^ pt. : 4 ac, 1 ro., 2 po..
8 yds.; 1ft. 94Hin. *
9. 3hhds., 22 gals., 2qts.;- 2 tuns, 1 hhd., 81 gals., 2 qts. ; 6bus., Spks.,
Ijfegals- 10. 51irs., 36ra.; £2; $3.12i.
11. £2.16.8rf.; 6f|i cents. 12. $4.; 83^ cents.
18. 112 lbs., 4oz., 18dwt., 12grs.; 2 mis., Ifur., 22 po., fift.
14. $2.l7i. 15. 38. Hd.^^q. 16. 45ct8.
17. £4. 0«.4f«?. 18. 1 ton, 11 cwt, 8 qrs., 20 lbs.
19. 12cwt,2qrs., 181bs.,2o2.,10fdr8. 20. lib., loz., 12dwt8.,6igri.
21. 4 fur., 39 po., 2 yds. 22. 6 cub. ft., llOff cub. in.
on. A.h Hr. Am j| Am 1 «... 1 1 «^.. KS :_
26. 2726 days, 18 hrs., 84m. 26. 4ao., Iro., 28 po., Siydfc
16 ' 100*
Aifswmw (pp. totr-nr.)
Ex. XXXin. (p. 109.)
147, 659
16 ♦ 661*
38®
12' 160"
8.
19, 2660
200' 1 •
g £ 161
• 16' 480*
8
6 ^. " ^
64' 1100* ^•
S* ^;7nrT-r. 10.
4 JL 1£5:
' 1000' 2 •
761 2817
10660' 11664'
1169 11
11.
9269 8
480 ' T'
"• T' m-
12.
16.
88 ' 8840*
298 79
2880' 480'
-L. _i_
400' 3000'
10368' 576*
18.
8026 86
6696' 1*
,- 8 1
800 ' 2112*
14.
— • JL
16' 60*
1ft 1^2 6
19.
81
2240.' 18i'
28. -L.
160
28.
2822400
61 •
24.
20. ??. t^
80' 79-
108
21.
1225 6
2304' 6'
^. 22.
126'
29^^
^^' 64-
26.
28
81*
26.
24
6400000"
81.
1863 ' 26*
«>, 213
''•So-
288
20* ,
Ex. XXXIV. (p. 113.)
I.
9. (1) 87f. (2) 8H.
2. 4^^g and8T5F:8i.
(4)4i*. (6)8T»ft. 6. 6^.
4. (1)
_87
976*
XI.
2. 21fand3H.
(2) H.'iV.
(8)
8. — ;- and — .
113 466
9
247*
6. 16.
III.
2* (1) 6000. (2) -. (3) 2. (4) lyfj.
1474
8. i- of 4 is greater by 1. 4. ^^^^.
». liWir.
wp
unmii (pp. iMkJ^ioi;)
IT,
*• H- 2- <^) ^- (2) — . (8)
86
22'
8. 2Hand
1
946'
684'
*4-
wfj. WiWV
6. The quotient is 144 times as lai:ge as the product.
r
1 1. A
• 9 ' a*
2.(1)8^ (2)£?. i,)g i,)^.
<*>»ii- ("U- <'>a- *T
Tl.
«. (1)1. (2) 1. 8)^. (4)8. 8. i?.
4.
860'
0.
8
II
6. H. «.2A;^.
i
*• (1) 5^t (2) ISiVt. (8) a. (4)
Til.
830
8. 18Jand8|§. 4. tHit.
8726 > ' •«' ^ ' - --'468"
6. The whole score was 240 ruus, and the score of each 80, 24, 24,
12,12,12,80,80,80,80,6.
Ex. XXXV. (p. 120.)
1 ± . 1^ . 11^ . 1^ . lain . 1'71'y 10001^
* 40' 126* 60 * 600 * 6 * 60000/ 200000'
230409 230409 10686 114125001 88401
, • _ .
1000 * 100000* 6 ' 1260
667097363 20819 10000009
• 1600
1
80000 ' 2500000 ' 10000000 ' 1000000000'
2. -1; -8; 7; -63; 07; 003; 9178; 9178; '09178; -0091; '00009;
B20'8; -9; 8-0t42; 6-72819; -000672819; 6728-19.
8. 7; 70; 700; 70000. -6; 60; 60000. 4-31 ; 43100.
16201; 16201; 16201000; 9001600; 90016.
4. -061; -00051; 0000061. -00008; -000000008. -005016; "00006016
•8780186; -0003780186.
5. =5; -1; 19; 28; 005; 9^7; OOOOOi; 14-4; 280-0004; 7-007;
100H)0001 ; 1-0010001 ; '000000006.
019A-
6.
8
Ifisff*
24, 24,
I!
«. Fourwtenlbs ; tweoty-fii e hundredths ; sermtj.fiTc hnr.^b«dibf ; mt«ii
hundred and forty-five thpiwandtha ; onthttnth ; Qnethpuawdkh ; ono h^adred
' thousandth ; twent^three i^nd serenty-five hundredths ; two and thne hundred
and seTenty-fire thousandths; two thousuid three hundred and 8«ventj.fiTe
tm thousandths; two thousand three hundred and serenty-five hundred
millionths; one and one milUonth ; one milUon and one ten miUionths: one
hundred millionth.
Ex. XXXVI. (p. 121.)
1. il'QSHQS.
4. 2985«073,
1. 963-77886.
10. 91 81 •6074970.
2. 29q-381404.
.6. 418-94614.
8. 870 •480876.
11. 6082-8192996.
8. 6168 70427.
6. 4q6-629622.
9. 62-6868U9.
12. 1011022969 090788191.
Ex.XXXVn. (p. 122.)
1. 10918 ; 68846 ; 14103 ; -OpOl ; •304817.
2. 211-6876. 8. •0421813. 4. 602-8416997.
6. 4-4954. 6. -48668. 7. 91794.
a -09; 666.30283; 21-068124; 9788*862. 9. 6 •S; -699998- 99*^08
XXXTIJL (p. 123,)
I. 169-6 ; 18V6 ; 16-96 ; -0001696. 2 178-889 ; -178889 ; 1-7M9.
8. -0063612; 372812; 12376. 4. . 8D7980896. 6. 210-6144186.
6. '00329876. 7. -03611. 8. -0000274104. 9. 0006694.
10. -00007614. 11. -065767692. 12. -27492. 18. -001; -20786.
14. 82-86164. 16. 164974»6-82.
•00009;
)006016
7;
Ix. XXXIX. (p. 126.)
1. 21 ; 91 ^78. 2. -026 ; 24-3. 8. 00003 ; '874.
4. 10,100,10000. 6. 260; 16-26. 6. 61472; -0000061478,
7. -057; 818-4. ' 8. -0072; 69640. 9. 10600; 187-66.
10. 8020 ; 643. 11. 82600O ; 32-6 ; -(mi. 12. 1-8 ; 18 ; -18 ; 180.
13. 002; -000002; -2. 14. gOl ; 20100; .001876.'
16. 948-7096 ; 9487096. 16. 26168-4; 21-4.
17. 2040000; 00082176 18. 7984-7; 79347; 79847000.
19. !Oq002 ; «0q002 ; 20. ,80. •67 ; 67000.
Wf
139-
>
SI. amtf; 'OIH; 76-9280.
SS. 880912-478S; 1'9006 ; l-815t.
ti. 19-8418; -0026.
VI. HIHttSeSSe ; 208266 ; 266-266.
22. 1*4896; 60880-1818.
24. 14086019-0980; -0011.
26. •0000186; -00186.
28. 4860; 108*86; -04646.
Ex. XL. (p. 129.
1. -26 ; -76 ; -628 ; '86 ; '8126 ; -96.
2. •616625; *482; 286; 1-86; -00626.
8. 6171876 ; -2875 ; -06078126 ; -005869876 ; 16H)076264.
4. •007080078126;^ 6. -84875. 6. "OOOl. 7 -^61. 8 -676.
9. -79876. 10. ^6. 11. 11^7678126. 12. 86-497. 18. 662-926.
Ex. XU. (p. 182.)
l. •»; -is; -027; -428671.
5. -66; -743; '197680864; 16-166.
t. -91789772; 7-286714; •00017.'
4. 24-009; 1701867142; 21678482.
6. '662681578947368421 ; •6434782608695662173918 ;
*0S4482758620689655 1724 137981 ; *63-i>268064616129.
7 7 6 *
»"• 90* 22' 496' 87' 800* * 540* 888 * 1875*
o JL HI. m?l'
7 * 480 ' 184680*
,^ 4 10619 89
10- T^;
11.
114137 . 1043 886
838000* 83300* 48'
12.
13* 16836* 14'
1 284121 61 4028867
15000 ' 14' 31680 '
Ex. XLIL (p. 134.)
1. 81-871638. 2. 700-672301. t. 6*116666; 1-681818; 808-062762.
4. 2*2884616 ; 13-72619047. 5. 13-2 ; *27. 6. a6-218 ; 800.
7. 863*6746; 246-8. 8. 1-85169.. .; 17-46.
9. 48-76; 6*76. 10. 808*76; 2*8. 11. 7; 48*784; •0184.
(pp-
)
-ftt---.
\ -576.
62-926.
Ex. XLUL (p. 186.) i
1. 45oti,; 67ict«.; 14-88^4. 2. 6«.7K; lB#.11088dL; Ut.iid,
8. 43 eta.; 15-788; $1.20.
4. 2m., llOOyds. ; 2d., 12br8., 66.21"; 7o«., 4dwt.
5. 8qr8., 10 lbs., 1 -21602.; 7 lbs., 62 oz.; 14 po., 2 yds., 7*2 fa.
6. 4 tona, 8 cwt., 1 qr., 6 lbs., 8 oz. ; 3 cwt., 2 qre., 12 lbs., 8 ot. ; 8 fq. po.
7. 81b«., 10 oz., 6-668 grs.; 2 qre., 8 buah., 8 pks. ; 14 cwt., 20 lbs., 108 16 OS.
8. 3ac., Sro., 14po.; 63gal8. 9. 87 po. ; 9 d., 16 hrs.
10. 1606-328 ; 19 qra. 1 1. 7 «c., 8 ro., 20 po. ; 2 m., 1 1 60 yds., 2-062 ft.
12. 13 sq. yds., I sq. ft., 111-6 sq. in. ; 4 m., 6 po., 1 yd., 2 ft., 11-97696 fa.
18. 88ict8.; |1.88i; 6oz., 12dwt8., 16gr8.
14. 16«.6(f.; 1».6K; 13».4rf.
16. 6 sq. yds,, 108 sq. in. ; 3 fur., 10 po., 3 y^s., 2ft.; 20 d., 6 hrt.
16. 8A^ac.; 20hr8., 30 m. 17. 7».; $2.62.
18. 1 OWL, 24 lbs., 13 oz., %J drs. 19. £1 . 2«. 9}dL
20. 162wks., 6d., lOhrs., 64i^8ec.
21. 1 ro., 89 po., 28i sq. yds , ^^ sq. in. 22. -0231 of a gufaea.
7
75-
»5275S.
)00.
,84.
I. -626; '9876.
4. -22083 ; 48*083.
Ex. XUV. (p. 139.)
2. -23126; -796876.
6. -0366; -300176...
8. -603125; -05729W.
6. -27329646; -07S916.
7, •2786493827160; -876.
9. -82286714; -000016....
11. 1-916; 14-24.
18. 76-789 ; 6212-807692.
8. -67867142; -00002646296.
10. -0384821....; 82 5.
12. 114-64; -00061
14. -01876 ; -806 ; -7317.
15. 13125; -3. 16. -30612; 013671876. 17. -225; -611.
18. -00248... ; -000080. .. ' 19. -000304. ..; -066625.
20. -288; -646876. 21. 11826396. 22. 1.69.... 23. '8140625,
24. (1) 2 c. 6 m. (2) 4 c. IJ m. (3) 1 c. 8^ m. (4) 2 II. 6 c.
(5) 6fl. 2c. 6m. (6) 8fl. (7) £6. 6fl. 2c. 5m.
o; jju-x. vs it. ^y; X.SV. V U. O C. IJ HI. (lu) 7 H. 'i C. 9'7VIQ m.
(ll)7fl.8e.4m. (11)£2. 7fl. 9 c. 6tm. (13)£3.4c.9m.
IM
'^. uimm^
!• TSt fi^i^? ; 8-2788096288.
Sz. XLT. (p. UL)
I. ^
^. 1801
16* lOUOOO
8. 678'005764; 678004246; -48204677; 769968*5...; l-OfSlS ;
1'004((6 ; -01030226 ; 100.
, 4. Y«g. 6. (1) 894. (2) -009072. (8) 1. (4) U-|129.
6. 1-06 nearijT.
U.
24260
1. 000700409; ^f^; -0082646.
3. Three hundred and ninety-seven thousand and eight, and four hundred
and fire thousand and nine qullionths; 897008403 009; 897-008406009.
Three hundred and ninety-seyen millious, eight thousand four hundred and fire,
and lune thou^n4ths. Three hundred and ninety-seven, and eight miUions
four, hundred and five, thousand and nine thousand-million ths.
441
8. -08493. 4. 11026; -j--; -00068874; '0002; 0642.
6. (1) 000091804. .. . (2)2-618. (3) 626. (4)10 0046.
6. 2*4976096088.
in.
1. '67 and 67000 ; 12644042. . . .
2. (l)^andi>.
8. 2*6 ; 8685 ; no.
(8) ^ and 1 6488. (4) 1 and -2916.
6. 16*86 miles.
6.
240'
IV.
1. 124-86668. 8UH; ^iUh
2. 8006006 ; three hundred thousand, six hundred and five-tenths.
Z, In o^1cr of magnitude they stand thus 1-6 x -76 ; 2-626-f-6 ; 6 x^06.
4. -0649 ; '12698. An», 006646 ; 642000 ; 0046
20020; -02002.
m A.I J ^ trn
.6» -r-.
• '.-ft
1-01816 ; •
11$1SI9.
r hundred
8405009.
and- five,
i roiUions
1642.
100045.
-2916.
I
($f.immfn)
dot
'<r.
1. $088; 2-8'?296; 8026'7857142.
l (1) 881. (2) 1-60546876. (8) 85-4878. (4) l'6888oi.
8. ji^ IS the nearer.
6. 7925-7 niilos nearlj. 8. 18*74696.
'«. 2-7182818 ; -00097061 ;
97061
leooooooo'
TI.
1.12-24862412; -0089147....; 0780091:7-30091. 2.'^.
869
«. ~ em
4.
6401
49500'
yii.
8. -72.
« mB.
1. A ought to receive 60 cents ; 5, 86 cents ; C, 12 cents.
2. $656.67gV; $786.80J?; $1101.52^^, each person ought to pay 26}f oti.
8. 45 boys. 4. l^mlies per hour; 1 hr.
8. A should have $8.60 ; £, $2.88; and the boy, 72 cents. 6. $1.28f
Ex. XLVI. (p. 148.)
1. $68.76; $37.60. 2. $41.40; $tO.
4. $247.60; $437.50. 5. $67.84; $82.50.
7. $71.42^. 8. $701.10.
10. $191006.73; $184.08. 11. 27082.20.
18. £12. 16.». IIH 14. £216. 16». 8H
16. £467. Is. 6K fgr. 17. £12. 6«. lOK ^•
19. £147. l%8.llid.^q. 20. £280. 16«. 8^ ij-
22.$237.06i. 23. $1860.
Ex. XLVII. (p. 157.)
1.
8. $90 ; $182.78.
6. $228.96; $30.24.
0. $88.20; $81.20.
12. $850.
15. £89.6«. IK
18. £2.15«.llidL
21. £i.9$.6i(L\iq.
24. $16.12i.
IS.
Sx'OB. ^
; -0046
1. 10-51884616.
4. $2698:36.
7. $18.50.
2. £64. 4».
5. $3001.86.
8. 2 francs, 18 centimes.
8. -628; r82.
6. £236.8«.6d
9. lOwlca.
II.
2. 16681^ metres.
iM
(pp. i68^m.)
t. A'i ibaf«=|t;4.88f, JTs shar6=|61.71f, CTa 8liare=|YB.4tf.
4. 0036. 5. 4iota. 6. 8861-8. ..rer.
r£4.17«.U|J. 8. SyArhn. 9. 74^hri.
III.
V
1. ISowi, 2qri., 21Ibf.; $66126.
2. 8496; 8 chains, 4 ohaiuleti, 9 links, 6 liukletf.
4. 6d.\ £Z.l1$.ld, 6. 6 mo. 6. 8662426 dayi.
7. £60.10*. 9. $666,661; 11183.83^.
IT.
1. 26978. 2. 0219288096. 8. £86. 10«. SK f j'*
4. 9 days. 6. £4. 4«. 2$^ ; 6a. ^d.
6. £396. 19«. 2d. ; £116146. 16a. 8d. 7. 195 sq. ydi.
8. A'b ihara=$600, ^'s share=$480, Ca 8hare=$d20. 9. 96 oM.
>
T.
J
*• 816 8l6' 816' ^^•
1. 874 quotient, and 446 remainder.
$. 8}H}S07. 4. £16468760. 6. 8} lira.
6. lObrs., 12 m. 7. 71^ days.
8. ^Mlhcts.; $1866.72-^^; $1466.67AW; $1065.16Hiff
9. 1512.
Tl.
L £96385. a7t.9<;. 2. $4732.72/r- 8. 10 yds., 11 in.
4. 12 cents. 6. 4 cents. 6. 4 cents.
7. 686iHr8. 8. £1000. 9. 600 trees.
1. $1; tIt; -0041.
8. 129Hfyw.
7. $800.
1. $2576.84.
VII.
2. 16 ; £4. 109. ; £9. lf«. ; £19. 16«.
4. 6H|day>. 6. 1200 men. 6. Vir; $1000.
8.98. 9. 10284 fr.,66| cent.
•fUL
2. 16H- 8. i^r.
4. $978.28.
6. -rtW; 00284876. 6. 8cwt., 87ilb8. 7. 67f hrs.
8. 6«. 8d. 9. $388784.80.
1. 2148. 2. -4; 04.
6. £19. 8fl. 7 c. 7 m. ; -,Vm.
8. 45 tu<ia.
IX.
8. Yea; VO wits. 4. 80 days.
6. 4«. . 7 3 English hen.
9. 160 ;,o,7S.
AmWlBS
(PP* 164^181.)
9m
Weta,
liVr.
If
16«.
•1000.
laya.
len.
7. 188.70. 8. 24day^ 9. 600000 sheeti.
ft. u
zi.
1. •082706766917293288; |24.81tH. 2. $61.40.
4. 1449.86. 4. 86 96cent8. 6. 2iin.
6. 26; 12-6; 6; 2-6; 16; 1-28; 2088. 7. 25Afr.; 216: 89iH.
8. 6idays. 9. 240 bov. ; 720«. ; 960dl
EX, XLVIIl. (p. 169.)
I. 40.
2. i9. 8. i
4. 8-9.
8. i.
0. -79986. 7, 8.
EX. XLIX. (p. 177.)
8. 60.
1. $69.12.
2. 72 yds. 8. $8.78.
4. 82acs.
6. $824.
6. 48wk8. 7. 12.
8. 14cwt., Iqr., 81b*
9. $23.60
10. $86.40. 11. $8.86i.
12. $2.28.
18. 92fbn8h
14. $364.102V. 16. £690.
16. $118.26.
11- 87i cents.
18. $8047tV. 19. $8.
20. 60.
SI. 8 days.
22. 8-8709 . . . days.
28. 4676 yds.
24. £6012. 2fl., J
Jc. 7im. 26. $608.12.
26. 18 days.
21. $280.04.
28. 8 men. 29. £6. 6a. Sd.
80. £6.4«.8idL
81. 24ydH.
82. 2^mo3. 83. $172.
84. $4.68}.
86. $443. 67f.
86. 11 yds. 87. $2041.20.
88. iHdays.
89. 7-7 cents.
40. £8. 14».11K^/^^.
41. $840.92fff.
42. $89.60.
43. $13200. 44. $2.10.
46. $8080.68ftf.
46. 10br8.,40',86-/y". 47. 70 ft., 8-282 in. " 1
48. 4cwt., 2qrg.,
MrSrrlbs. 49. $721f
60. £4.10«.
61. $1.82.
62. 16hr8. 63. $193.98.
64. $698.
66, -iOGdnys.
66. $1.96. 67. lOidays.
68. 66Arydfc
8^ ^ :.. 8w:
eo. 6iV?fr' before 4 o'clock
61. $7600. i
iiiji. Vodays.
63. 4 166 J yds.
64. 240000 lbs. 1
66. 22401^1^.
66. $167.60.
67. 168j}f7ll|fc '
68. Monday fortnight, at 6h. 36m., p.m.
69. 72 yds.
^0. $3.20.
^1. 466660 lbs.
72. 106 days. 1
^3. $124800.
74. £18. 10a. Sd: nearly.
76. 80 days. J
16. £211.19«.8<jr
11. $20.16.
78, 9iyds. 1
81. 8 days. . |
iV. 4»dU«Oi20.
80. 207:82.
Mxmmaa ^p. is8&i«9i;>
1. 16 men. 2.
6. 19:^-btiu8. 6.
9. $1Si.
12. 69cwt.,22^Ibs.
16. 8d.,6hrs. 17.
20. 9 days. 21.
24. 2400 men. 25.
27. 69 J J days.
80. $1.69^1%. 81.
U. ISHft. 85.
88.
Ex. L. (p. 188.)
7 men. 8. 66 days.
600 ac. 7. 12^11 mi.
10. 11 mo.
13. 12hr8. 14. S^wkH.
lUhrs. 18. £60.8«. 9i.
Swks. 22. 64uuy8.
47 tons, 17 OWL, 66 lbs.
28. £382. 6s. 2-,V.
8 ft. 32. $10.86.
$463.69^.. 88. 36 days.
49-3 lbs. 89. $2440.08.
29.
4. 7200ioldierfl.
8. 9Bcwt.
11. $88.
16. £20.
19. 600 reams.
23. 360 men.
26. 824 men.
2268 cab. ft.
83. 1320 yds.
87. 19-36 days.
1. $6.80. ^
5. $112.38.
9. £17.14».6ct
12. $3698.22f.
Ex. LI. (p. 194.)
2. $21.70. 8. $66.
6. $267.24. 7. $132-366.
10. £1.4».8K^.
13. £1664. 9 fl. 2 c. 6 m.
4. $106.87i.
0. £6. 18«. lOd
11. $1140.
14. $176.491.,
16. $2618.83^.
I7.$9tf.26+; $471.25 +
19. $9,621; $229,621.
21. $2.74H; $42.74^.
16. £1738. 16s. 6H<i
18. $116.09; $662.69.
20. $47.1311; $290.98ti.
22. $47.06Hf ; $367.80||f.
23. 8s. imH ; £34. ISs. 1H8K
Ex. LII. (p. 195.)
1. $118-63.
6. 10 years.
9. £346.17«. 6dL
2. 8 per cent.
6. 6 per cent.
10. 5 J years.
13. £316. 10<.8(il; 21 years.
8. 4 years.
7. $600.
11. 4^ per cent.
14. $225.
4. 6 per cent
8. $816.
12. 20 years.
1. $247.20.
4. $845-2796
7. $34-49 + .
10. $1-4976.
18. $87*3466.
Ex. LIII. (p.. 198.
2. $9800344.
6. $274.83 nearly.
8. $2135.58+.
11. $360.41+.
14. $226.33 ....
3. $44-928.
6. $12.79 + .
9. $739-427.
12. $228.28+.
16. £l.lU<7dL nearly.
U0mm (pp.2<i»-sift.)
tm
oldieni
rt.
iaxoB,
len.
ten.
ft.
yds.
idays.
ler cent,
yearfl.
Ex. LIV. (p. 202.)
1. $800. 2. |!23'7.325V. 8. $657.
6. $400. 6. £375. 16s. O^d. nearly.
8. $1240. 9. $1228.60 nearly.
11. £262.4*.6Ki?- 12. £766.
14. £l968.2».6rf. 16. $1.26^.
17. $2.fiUt. 18. $2.46. 19. $18.80 //v.
21. £140. 22. £48. 9«. 23. $26.62^.
26. 2«. lid. nearly. 26. $l-06-i-
29. £2.6«.8<;.
-Ex.LV.(p. 267.)
4. $912.96Hf.
7. $824.
10. £2000.
18. $462.47 + .
16. $78.1 IH*
20. $8.29H.
24. $2.
28* 16) per cent.
1. $8800. 2. $800.
5. $697|^. 6. $1069.60i^.
9. £3091. 10«. 2iH
12. $2776. 13. $1834.
16. $72. 17. $480.
20. £112.0».8HH
23. $1700. 24. $8216.
27. £876.7«.9K
80. $8.27H!. 81
83. $7.77^ ; $8. 75 ; 97| cents.
86. $5297.64ia,V
88. £104.8a.4i. 89. $16.21iJ.
8. $625.
7. £6050.
10. $2418.
14. £972.10*.
18. $122.60;
21. £159. 12».
26. $2186.71f .
28. $6.59^.'
$6.79^1,-.
4. $4300.
8. $2600.
11. $1488.
15. £4048. 1U7K
19. $135.69VVr.
22. £111. 8s. IHK
26. £2164.2«.6dL
29. $7.44}f.
32. $694.94.
34. 87|; $187lf
36. $320. 87. $331.28-lif.
40. $\U, half-yearly.
42. $280, half-yearly ; $7360.
41. $200; $141-601^, half-yearly.
43. £2729rir; £5. 10«. S-A^rc?.
44. The Canadian Bank of Commerce Stock. 45. $10612. 24||.
46. £6.6». 47. £1666.13».4A 48. £7900; £8310. 2*. 6dL
49. £86. 60, £26; £22.1U7^. 61. £20000; £226000.
I nearly.
Ex. LVI. (p. 216.)
1. 187 98; 862-4626; 2266*76; 4286-944; 6689-4; 84872-97.
2. 16-626; 28-466...; 8-984376; 2-680...; -048...
8. -006; •0276; -048; -06626; .263; 2-8005; 6.000138.
4. $360. 6. 8s. Sd. 6. $763.60 ; $268.08 gain.
7. 1463-65 gals. 8. C437.6bu8. 9. 1|J. 10. 16|gain.
11. 9 cents. 12. $9.18i-i-. (cwt.=1121bs.) 18. 66AeaijaL
SIO
AN8W1EB8 (pp. 21&-237.)
U. lO-fa...; 8-Yl...; 10.44.... 16. 6-24...; 20*29...; IC'll....
16. 1046678'376 persons.
17. 8 of the age of 18 years ; 19 between 16 yrs. and 18 yrs. ; 88 between
12yr8. and 16 yrs, ; 183 between 10 yrs. and 16 yrs., and 190 under
that age.
18. lOf. 19. S^. ' 20. $168.76.
21. Weight of oxygen=997121b8. ; weight of carbon =866 8 lbs.; weight
of hydrogen=lS768 lbs.
22. 204f ; 163|H; for the whole time, 672^ per cent., or 42^^ per cent.
per annum.
28. He loses $91.80.
24. Ko. of male criminals : No. of female criminals :: 6 : 4.
26. 6».Z^^d.\5s.6d. 26. $2.10. 27. £37.UllK A«.
28. $1376 ; $37H. 29. £27. 3 >. |3-4fi6.
81. £8.18*. 82. 4718 cents. 83. £.6. 7.'».3A<i
84. 1». lOid. ; 2». 1^4. iq. ; 8«. 9d. 86. 80 quarters. 36. $10.
• Ex. LVn. (p. 220.)
1. (1) $896, (2) $178.60, (3) $634.26. 2. 2861607TVqi'8. 8. $1.08^.
4. Average age of boys=9i yrs. Average age of girls=10s(V yrs. Aver-
age age of whole class = 1 O^f yrs.
5. 4-447 days. 6. 8-942857i. 7. $1976. 8. £372. 18«. lirf.^^
Ex. LVin. (p. 226.)
1. 6, 28, 88.
2. $20.10; $64.82.
8. 616, 860, 1204, 1892; $717fJ; $861fj, $820Jt.
4. £l79.8«.8rf. ; £142. 9».; £99.39.4d.
6. 12cwt., SOi^rlbs. ; 3cwt., 30 -ft- lbs. ; 2 cwt., 60H1*>8-
6. $1900.80; $1666.20; - 7. 2cwt., Iqr., 12 lbs., 18^02.
8. £3280; £2166. 13».4rf.; £.1083. 6».8(/. 9. $1680, $2160.
10. $60, $60, $120, $240.
11. ^'s share=£6000, B'b share=£8780, (Pa sl)are=£3125.
12. $1350. 18. 5H months. 14. 4f months.
16. 12 months. 16. £3. 10s. 17. $1266.62^.
18. A ought to have £80., B £90, and C £84.
▲N8WEB8 (pp. 286-S6S.)
Sll
Ex. LIX. (p. 288.)
1. H^I'Ua. ... 2. £1271. 13«. 9-i%d. ; $6104.11.
8. 1246 pia«.,6^f reals. 4. $4.87. 6. $4.67. 6. $6.02.
7. The direct way. 8. £19. 10». 7K ; 25 francs. 9. £11. 5#.
10. $480.08^ gain ; the income in England ia taken at par.
1 17; 24; 38; 64.
4. 999; 989; 908.
Ex. LX. (p. 241.)
2. 81; 146; 416.
6. 16867; 681441; 16807.
8. 12-96; 6-37; 2401.
10. -207; 0374; -0461.
12. 347 6905; 490-304.
14. 16-3492...; -3162...; -1; 2.2360.
8. 814; 193; 108.
6. 6432; 3789; 2312.
7. 643200; 2039760.
9. -69049; 6-2673.
11. 2403; 2 403.
13. 4; 1-2649...; '4; '1264
• • • • « «
-7071
15. -02; -0284...; 194901 16. 4i; 12-4007.
•6778,
47
99*
Ex. LXI. (p. 248.)
t 13; 16; 81.
4. 134; 411; 208.
7. 8002; 6081.
2. 88; 48; 67. 8. 88; 98; 98.
5. 631; 806; 364. 6. 268; 6S8; 975.
8. -73; 3-19; 45-7; '097; -124; 029.
9. 1-442...; -669...; -810 10. A; 1.. 3-646
11. 7f; 1-930...; 1-442 12. -046...; -426.
18. 616; 1-232. 14. Each edge=27-2in. 16. 1869 sq. ft
16. 8 ft., 10 in.
Ex. LXII. (251.)
1. 108544, 82821; 114111; 609, 6<?8. 2. 6/694, 6«6M; 10384.
8. 2616; 660410; 61117344: 2/3568; 674097: 2704064; 476/968-
29/96680:14332216:23033210. ^ *
4. 1466; «7/8; /4/e«; 10232; 8402. 6. 2604; 62/«; 6643.
«. 6221; 1110111001111; 36,61; 82/; 33233344; S/4«2.
7. 10787; 418; 2/43; 16430385. 8. 9294; 844; 1466; ilt,
9. 476/968.
joxvKjm <pp. SSM«5)
Ex. LXIIL (IK 255.)
1. 11-41, lF-84; 101-382, 81-714; 4204-3514..., 664-7921... •023S49...,*Q<6;
18214.
2. 102-601t.., 146-408873..., 173-S5186...; 21-14,26-1; 271; mrhUtlh,
2218-4610642782, 8146-412172...; 312-«38i, 467-4765841280,
670-424365...
8. 22111-210111..., 8211-302323; 12012000-12211002..., 823010-22112i
102-120211...; 23-2112.
4. 572640; 26^A». . 6. 12 ft. 9 in. 6,£11.16«.
Ex. LXIV. (p. 264.)
. 1. 4 yds. 2ft. 8 in.; 8690 revolationa.
a. 19 sq. ^ds. 6 sq. ft. 113f sq. in.
8. (I)2yd8. Oft. Zx^in.
4. (1)738 yds. 1ft.
6. (1) £19. 8«.
6. (1)86 ft. 6 in.
7. (1)16-229 yds.
, 8. (I) 52 a 7 in. (2) 6018.., ft.
10. 42 sq. yds. 4 sq. ft. 73 sq. in.
"i2. (l)628q.yds.
(8) 8 sq. yds. 6 sq. ft. 1^8 sq. in.
(5) 8 sq. yds. 6aq. ft. 12 sq. in.
18. (1)85 sq, ft. 10".
(8)208q.ft.9'. 10"..!'". 10"".
(6)70eq.ft.3'. 8". 10'".4"".
14. (1)888^ eft. (2)S6c.ft.
16. (1)6 yds. (2) 6 yds. 9 in.
16. (1) 122 sq. yds. 2 sq.ft.
(8) 181 sq. yds. 2 sq. ft. 86 sq. in. (4)
(6)
17. (n !,060. (2)
18. (I, UC yds. 4 in. ♦ (2)
19. (2)2ac. 4po. (»)
20. (1) 17 yds. 1ft. 9i in. (2)
21. (1)£21. (2) £18.
(2)
(2)
(2)
(2)
(2)
(2)
(4)
(6)
(2)
(4)
(6)
(3)
(3)
(2)
124 096... yds.
27 sq. yds. 8 sq. ft. 61^ sq. In.
462 sq. yds.
Schs. 40 Iks.
86 min.
9. 166-52... mi.
11. 822 yds. 2 ft.
1 2 sq. y ds. 4 sq. ft. 1 8 sq. tn.
2sq.yd8. 1148q.ia.
15 sq. yds. 6 sq. ft. 112. sq. fak
76 sq.ft. 7'. 6".
68 sq.ft. 8'. 8"'.
878q.ft. 5". 4''.
286 c. ft 659 cu. in.
2 yds. 2 ft. (4) 18 yds. 1ft.
107 sq. yd8.8 sq. ft. IC sq. in.
145 sq. yds. ISsq.in.
121 sq. yds.
$279.44fV.
4 ft. 9|in.
8 chs. 50 Iks. off ihe length.
100 sq. yds. 6 sq. ft. 20^ sq. in.
22. 260 yds. 23. tl3.07f.
ANBWEES (pp. 265-267.)
SI 3
(8) 7.987.
24. £5. 10». 26. (1) 1968 lbs., 12 oa.
26. (1) 366HI times. (2) 251049... bus.
27 (1) 300. (2) 31.^6002 nearly.
28. (1) 2 ac, 1 ro., 39 po., IS yds., 6 a, 86 in. (2) 1000 yds
29. (1) £1046. 8.. (2)20. 30. (1) ISsq. ft. 548q in!
(2) 3601 sq. ft. 72 sq. in. 81. (1) 8. (2) 3 ft, liin
82. SOlicyds.; 166^5,lbs. (cwt. = 1121b3., *
83. (1) 21iWiV (2) 48H nearly. (8) 189,28.
In compUance with the request of several masters, the answers to the
fwl'^f ^"'''' (?^^^mi.^ are not given. They will appear inlLe ily
which 18 in course of preparation. «• wc ^ey,