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ALGEBliA.
PKOr, 01 M • :..n.-. A.M NAT. -'UILOSf.Pl.v IN TUB RAhT LSIUA OH.!.,., ilBH"/?'
I.KViv.u, IMPKOVf .i.VU SlMVLII-IKl.,
Dv THOMAS ATKINSON. M. A,,
^*^*""'>I.iB OF COr.f. OU. COLL., CAJtBHlDSR.
.front t^; /gi,u$ ll.«:!j>jn Eoittnn, iuitt) abliitfar.i fcg ({)( tSrottrrj of :j(.
D. A J. SADLI
NEW
^A.^'JL.V\' S'livKl.T.
'(V .' : k K
No.
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AN
.3KLEMKNTARY Tin^ATISI^
' ^ ^v
OH
1
.iLG^EBRA.
• • a
• ••!>>
< « a • «
f
Bytiie L^tV;
/, ^ RET. B. BRIDajj^kD, P.R.8.,
.. ^J«J*y/J/«^Ao;;...-.'.-c««M^ j-.'lz^. IUlosoi,h'jimhEa,tLxdla Coll., Ua'tf:>ra
BEVISfeD, IH7H0-/3D, AND filMPLIFIED BY
THOMAS ATKI&X, M.A,,
LaU ScUlsr of Corp. Ch. foil. 'ccvibridQe.
i'nm t!,c £au,t imHn eMtian, «.if, .^bMti.n^ l>i, ii;e I3r.ti;cr3 of r ^
'>..j|«
*«5
7/
\-
*
Kew York:
n. & J. SABLIER & CO., 31 BARCLAY ST.
Montreal: 275 Notre Dame Street.
1S7C,
accomp]
the man
^ period
professei
fjropriat
cvjnced
in comp
These p
for tlieni
ion.
This s
*Drised ir
ADVERTISEMENT.
%'
%
The excellence of "Bridge's Algebra," as an do-
,icntary treatise, has long been ^ well known uid
extensively recognised. In the. Preface to the Second
^Edition the author expressly stAt^;, that "great pains
|werc taken to give to it all the perspicuitv and sinipli-
|City which the subject would a\im of, and to present
It m a form likely to engage th'd- 'attention ofyoun<.
f persons just entering on their jiWhematical studies''
ihe design, which he thus proposed to himself, was
accomplished with singular fem^^,~-foT not one of
the many publications on Algebra, which have durinc.
m period of forty years issued from the press, with the
professed object of producing a more simple and ap-
propriate introduction to the study of the science has
evinced such merits as justly ent J . it to be placed
in comparison with the performance of Mr I3rido-(.
These publications have accordingly failed to secm-e
flH' themselves the same measure of public approba-
This small compendium embraces all which is com-
^'^ ~ " '""--•' ''■^'i/(^ ana cu-jjcnsive editions, that
^-
Iy
ADVEimSEMEXT.
ether practicallv useful or theoreticn;iy valuable
By .ntroducng Eyuation,, and Problems at e e ""
. .est stage po.ible, a nove and in.s.r X V fure
»n.ch the eCtor is ,,ersuaded eannot fail to ex to tu
ounos,tyand stimulate the ardour of the yom?,W
l-raist, so as to induee him to pursue his studio t f. t
more than usual alacrity, intelligence, and tc ol-
nas been g,ven to the work. A great ;ariety of"!
tSdt r' ""';=T P™!':^'"^. which are'^not Z
amed m former edit^ors, have thus been interspersed
the several chapters.;; Besides these addition'm^
t. m.ty of arrangement, or of rcnderin<. ,he subiee.
boShatlis';Ir'^''''-''*">""' ""•^■'' -' -""-'
bope hat h,s ilc,n,„M-y iVeatise on Algebra' would
h"d Its wa.y ,nto <^;P>,Uk Schooh; where it vvas
.very well known, tWs branch of eduction was hi
but little attended ts-i' and if ,V „ "" ^™H'hen]
honprl tl,„t.i,- ,.'. " "°**' confidently
Goa»oBD, December, 3347. ^' A.
y valuable,
at the ear-
e feature —
' excite the
oiing alge-
■udies with
success —
ty of new,
not con-
terspersed
ons many
ic of uni.
ic subject
^ds. Mr.
vithout a
'«' would
e, it \va3
as [then]
nfidently
ice of its
to many
^ntlj at-
ing tills
n.
T. A.
CONTENTS
t*aAPTKR r DEPrN-mONS. P*®*
•^HAP. J I,
HnAi,
On the Addition, Subtraction, Muitiph^tior; '/uij 'l,-;'. "
sion of Algebraic Quantities.... ,
Addition *'
Simple Equations '\\ ^
Ou the Solution of Simple' Equation" ' .1
iVoblems "-'
Subtraction .. . .!!.'!'.".'*'.'.* ^*"^'^
On the Solution of Sirn*ple*EquatioM .*.' 11
Multiplication ^^
On the Solution of SimplVEqnVuons; *.'.'. ]!
Iroblorns. -°
Division. _ _ -9-35
HI. On Algebraic Fractions . ^ "^ ^
\^LtT7^ "^"^ 'M^iipiicati;,;: ;;.;,■ d^: ^ '
sion ot fractions
On the Solution of Sim^e Equaiions'. ." ^
Problems f55
On tlie Solution 'oV Siniple 'E^uat.c
more unknown Quantities..
61-Ct}
.ons, containing two or
Probl
ems.
60
'^a-vr. \\. On Involution and Evolution ^^'"^^
'^^.^:^,^ ^-be. anj-sim;^;':^,;^; ''
On the Involution of Comnnnn^ * \ I. !„V_'- L' V.' "H '^^
oa the Evoiuto of Aig.srQu^s;::.^!!!!!':?:;; ,'j
Tl CONTENTS.
PA«t
Oil the Investigation of the Rule for the Extraction of the
Square Root of Numbers 81
Chap, V Oii Quadratic Equations. . . , 83
On the Solution of pui-e Quadratic Equations. . , , 81
On the Solution of adfected Quadi'atic Equations 85
Problems
On the Solution of Problems producing Quadi-atic Equa-
tions .*. ...92-93
On the Solution of Quadratic Equations containing two
unknown Quantities 98
Ciup VI. O41 Arithmct'c, Geomotric, and Harmonic Progression... 10"2
Problems 105-109
On Geometric Progression 10*»
On tlie Summation of an infinite Series of Fractions m
Geoniotiic Pi'ogression ; and on the method of finding
the value of Cu'cuhiting Decimals 113
Problem , 116
On Harmonic I'rogression 117
Chat. VII. On Permutations and Combinations. II9
AJ'i'KNDix. On the Different Kinds of Numbers 125
On the Four Rules of Arithmetic 126
On the Two Terms of a Fraction 127
On Ratios rdxd Projx)i'tion3 128
On the Squares of Numbers and their Roots. 128
On the Factors and Submultiples of a Number 13G
On Odd and Even Numbers 131
On Progi'essions .......131
On Divisible Numbers without a Remainder. iaS
Properties and various Explanations. 181
Mi*cellaueous Problems. , 187
4
BRIDGE'S ALGEBRA
CHAp-pjilR I.
.* * • *
DEFSNKTIONS.
pressed by melns of Written S o.- Zboh Th^ "'", ""
u^d^todono. „u.nbo. o. ^UUies^a^'te I^.t:«
semed by the/., lette. ofl^^^^X^^r'
3. Unknown or undctermlhH quantities' aro „«, oil *
pressed b, the to. letters ortlii>,%abet "as":, ^X:''
4. 1 no m?//W^5 of quantities thnt ;« +k^ z .
quantities are ti be i^l^S^&'^X^'Z^TfJr'
34- sl: "Th""" f^^ P'-i"S. timbers Tefore'thi/f Is "C
rtear„ tfr ^ ^"-"''' » '^ ^^a^l^^Se^^ .Vu^
.« = G«. This ,vn>bo. ist„ed the' tV4«%.'"*'' '"
rore-&^+pt»:t''ir taT'i'^^ ^
siiinc th ntr as 5 • and a ^ h a. 1 , -"-""^ »^ + 2 is tiie
., whatever be the "fl^e^of at^'aXr " "^ °' "' '' ""'
titios^r^^i^neltoSEl^Xhai:;^^ ^-?^<' --^ or^
or ././.,./«.«,,/ quantities ?_3. llmv i« LiL, f''*"^"''.*^ rei>rcscnt Aown
•-6. WhatKs the sign of equality ^-o. W In \TJtror urt"'+?'^ '
^^LGEBRA.
7. Tlie sign — (read minus) signifies that the quantitv tc
which it is prefixed is to be subtracted. Thus 3 — 2 is the
same thing as 1 ; a — b means the difference of a and b, or b
tal.-en from a ; and a + 6 — .r, signifies that x is to be sub
iracted from the sum of a and b.
8. Quantities which have the sign + prefixed to them an*
called positive, and those "which have the sign — set before
them are termed tiegative quantities. When there is no sign
befoi e a quantity -\- is understood : thus a stands fbr + a. '^
9. The symbol X (read m^oj' is the sigii of multiplication.
and signifies that the quan^itje^ between Avhich it 'is placed
are to be multiplied togethcji-; - Thus, 6x2 means that (J
IS to be multiplied by 24,:aud a X b X c, signifies that
a, 6, c, are to be multiplied together. In the place of this
symbol a dot or full-point h min used. Thus, a.b.c, means
the same as (a X 6 X c. T'h©. product of quantities repre-
sented by letters is usually ^fekpressed by placing the letters
in close contact, one after another, according to the position
in which they stand in the alphabet. Thus, the product of a
into b is denoted by ab ; of a,, ^ "and .r, by abx ; and of 3. a, .r,
and y, by '6axy. """<>
10. In algebraical computations the word therefore often
occurs. To express this \vovd the symbol .*. is generally
made use of. Thus the sentence " therefore a -{• b is equal
to c + G?," is expressed by " .•! a + i = c -f c/."
•EXAMPLES.
Ex. 1 . In the algebraical expression, a 4- b — cAet a — Q
i = 7, c = 3 ; then
a + b - c= 94-7--3
= 10-3
= 13
Ex. 2. In the expression ax ■}- ay ~ xy, let i — b x — %
y n= 7 J then, to fnid its value, we have
ax-\-ay -xy— 5x2 + 5x7-2x7
=. 10 + 35 — 14
^45-14
= 31
M :, ii
':):*'''^^' rc:ia f — o. Wiuuirt iiioaiiL by wwrni-e nnd whal
oy mgative qunnlitiea i—<i. Write down tlio don of muUipUcation f T« m\
other m«r^ used to denote rnnltiplioation ? Wlien ia no Bvinbol usod?.-
1 0. W 1ml symbol is a»cd to denote the word therefore f
e quant itv tc
3-2 is the
a and b, or b
is to be sub
I to them aiv
— set bclurc
pro is no siijij
s tor -f- (I-
\ultipIication,
it is pliioed
ncans that (5
signifies that
place of this
.b.c, means
itities repre-
ig the letters
the position
product of a
ad of 3, a, .r,
erefore often
is generally
+ ^ is equal
f, let a — 9,
— 5, .r n= %
X7
DEFi.Mi OXS. 3
tlio numerical values of tl. .^s^^^l^^^^J^ ^ ' '"'
13.
3.
C5.
(1.) a + b + c + x. ^,,,
(2.) a-i + c-^ + y. ^4^^^
(3.) Ob + 3ac - ic + 4c^ ~ xy. Ans.
(4.) aic - abd + bed - acx. Ans.
(5.) 3a4c + Aacx - SAcfo; + a^ry. Ans. 176.
11. ITie symbol -r (read divided by) is the sinn of .Jh,:
^^hlch It IS placed, is to be-VJiVided by the latter. llT"
exp;esseYr"'''^f-''\ ^^^ ^^^^«'«" ^« "^o'e simp t'
expressed by makmg the ^-brn^er quantity the numerator^
^nd the latter the denominator pf a fraction : thus^mc.rs
a d^ided by ^ and is usualJ)^^r the sake of brevity, read
exaa«*i:es.
Kx. 1. If a =. 2, i = 3 ; fReh, we find the values of
/I \ ?« _ 3 X 2
_0.
15
i>^ 5X3
(2) ?l±i.= ?i<_?_!^''^'
8a -3/; 8X2 -^Vs
IG — "~ 7 ~ ^•
Ex. 2. If « = 3, i = 2, c = 1, fn,d the numerical values of
(1.)
8r/ 4- c
-^W.9.
10
(2.),-+"'-"
(3.).
3« + b~bc'
oJt -f- ac—bc
ir
Ans. 1.
. 7
Ans.--.
8
2ab—2ac + ic '
of tnnes, the product is called a now.r nf fh. ..,„..tZ.
__^ J "- - — • jtutiititV.
11. By wli.it symbol is d......^.,
division over expressed in any other
;«'">?/• of u quantity? ^
ivision denoted? W]
ninnncr?— 12. Wi
int is its nnmc
!i«t id menut i
7 tli«
ALGEBRA.
13. Pnnmrs are usually denoted by placing above the quan.
tity to tiie right a small figure, which indicates how often th€
quantity is multiplied into itself. Thus,
a - - - - X\iQ first power of a is denoted by a (a').
«Xa- - - the 2d power or s5'?/are of a " a*.
«XaX« - the 3d power or CM^^- of a " a\
« X a X a X a the 4th power of a « a\
The small figures \ \ \ &c., set over a, are respectively
called the index or exjponent of the corresponding power of a.
14. The roots of quantities. are the quantities from which
the powers are by successivf5.t«vltiplication produced. Thus
the root of the square niim|?<;r46 is 4, because 4x4= Ig'
and the root of the cube num-b^o: 27 is 3, since 3 X 3 X 3 J
15. To express the roods' }^ quantities tne symbol ^/ fa
corruption of r, the first le#f.>in the word radix,) witti the
proper index, IS employed. Thus, . ^
W or y/a, expresses f he square root of a.
/« " '• " •'•".thec^/ierootofa.
y« " « "■"■fhe/or«-^/.rootofa.
■«l
EXTAMPLES.
Ex. 1. If a = 3, 6 = 2 ; then a« =3x3 = 9 a^-^^
3X3 = 27,4^ = 2X2X2X2 = 16. •'^ « - '^ X
, ^^x. 2. If g = 64 ; then -^/a = ^'64 = 8, \/a = V04 =
^"4 X 4 X 4 = 4, Va = V64 = 2.
Ex. 3. In the expression
ax'' + /;»
6:^^rrc'^^ta=3,i = 5,c=:2
What is the numerical value ?
Here ««' + 4' = 3 X 6 x G + 5 x 5 = 108 + 25 = 133
and />.r - a« - c = 5 X 6 - 3 X 3 - 2 = 30 - -- 2 = 1 ii
ax'^ 4- h
bx -—a^ — c
133_
19
IK^wJ^rf^^TT*''"'",''*'''^^-^*- Whntaro tl.o roots of qunntities ?-
"^lilot^^l^SllT ''''''''' '''''''''''''' ^^P--^^ ^Virat'irtbe
ovc the qiian.
how often the
respectively
5 power of a.
s from which
luced. Thus,
4X4 = 16,
X3X3^
ymbol -/, (a
ix,) with the
fa.
9, a' = 3 X
a = V04 =
= 5, c = 2
- 25 =^ 133,
D - 2 = 1<)
DEFTXITXOXS.
£x. 4. If a = 1, i = 3, , ,^ 5^ ^^ ^^ ^^^ ^j^^^ .^^^j^^^^ ^^
(1.) «= + 25^.c:. .
Ans. 2.
(2.) a' + 3b' _ c>. , „
(3.) «• + 24' + 3c' + 4d'. ^„, 9j
(4.)3a'i-25'c + 4o'_4a'A ^„,' ij,;
■^ 3 ^ 3 ^ 3 • -4'«. 51.
Ex 5. Let « = 04, 4 = 81. = 1 : M the values of ' '
'■ ^''+V^"- An. n.
(2.) v/a+^i+y,. ^,_^_jg
(3.) V^
10 When several quantities are to bo taken as one onan
tj^!,, they are enclosed in bracket,, as ( ) M f 1
semed tt +1 '^ ■ ^'' -. \ ^'S"'«- *••" *e qua. tity ri J.
= 12. '^^' *"'' •■• ('' + ''-<^)-(<'-0 = 4x3
.1.-fltcnVel;ts2nt';rr\;«t%:,+iV;i-'>,t- -O-
(1.) (a+4) . (.+rf) = (3+2) . (3+5)=5i8=4"o
(2.) ("+4) <:+rf=(3+2) 3+5=5x3+5=20.
(3.) a+6c+,fc3+2x3+5=3+0+5=14
iectuely. Thus, a-i~c is the same as a-(b-c)
ofi^fr^^io^tlT-r:,:.?/. "--- ™^.de"on,iuat„r
___ " '•-' ' "' ^ ^''^'■'^ oi viiicuium, cor.
llnl^vSr"orn!;rtt r ^'"•"•'^^ ^-^^- ^hnt ,« a t
^ardcda.a:rue;;r„;;'° '^"■"•^•••^^-r "nd denominator
i.H a vinculum f Mny tlio
f>t'u liucUoii be re-
r,
ALGEBRA.
responding, in fact, in Division to the braokct in MuUlpVca,
Hon. Thus, implies that the whole quantity a-f- b-^ i
5
is to be divided by 5.
18. Z^•A•e quantities are such as consist of the same, letter or
the same combination of letters ; thus, 5«, and 7«, 4«A ^nd
9a/;, 26.c^ and 66.r^ &c., are called like quantities ; and 7//*-
like quantities are such as consist oi different letters, or oi' clif
jerent combinations of letters ; thus, 4a, Si, lax, 5bx' &o
are unlike quantities. ' ' »
10. Algebraic quantities have also different denominations
•according to the number of terms (connected by the sic^n +
or — ) of which they consist : thus,
_ a, 2b, Sax, &c., quantities consisting of one term, are called
mpiple quantities.
a4-.r, a (jtuantity consisting of two terms, is called a
fjinomial.
bx-hy-z, a quantity consisting of three terms, is call-d a
trinomial.
CHAPTER II.
ON THE ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVI-
SION, OF ALGEBRAIC QUANTITIES.
ADDITION.
20. Addition consists ii. collecting quantities that are like
bUo one sum, and connecting by means of their proper signs
those that are unlike. From the division of algebraic numv
titles into positive and nef/ative, like and unlike, there arise
Uiree cases of Addition.
Case I.
To add like quantities with like sif/ns.
21. In this case, the rule is "To add the coefficienls of the
Bevcral quantities together, and to the result annex the com-
13. Wl.at are lay nnrt what aro iinUh qnntititles ?_19. Wliat is a oimnL
quant. ty? \\ Unt hnhho,»lal and wl.at u //v;«.>,;»;«rAo. fwh^ Si
tudaion ot al-ebra consist ? Iiuo how many casos is ic divided i
ADDITION.
in MuWiplkcu
ntity a-f-i'v— <
mme letter., or
7«, \ah and
ic3 ; and vn-
ers, or ofdif
'X, 5b£\ &e.»
enominations
Y the sign +
M, are called
Js called a
I, is calkd 8
AND DIVI-
;hat are H^-e
proper signs
braic quan-
there arise
icnts of the
X the CO lu-
II at is fi simpU
III what dc««
led^
«ion sp, and the common letter or l-.t'ors •" f . -f • • i
from the comrnon principles of AH Im ic if +o / '! ^"^-"I
be -153^ ' -^^ ' ^' "^^^^ ^^g^ther, their .mn must
Ext ].
S^-f 2a— 5i
4^-j- 8a—
9^+ 4a-
5x+ 7a —
Ex. 2,
<w.
7h
Gb
9b
23x+24^^:::su
Ex. 4.
3r'+4.r'- ar
7^'+ 3x1/- 5bc
}lx'+ 5.ry- 4/;c
:»'+ 4^y- l>c
__Ji+__^/-. 2bc
^^£+23xi^m^
Ex. 5.
'7a'-3a'b^2'>b'-~3b'
4a8- aV>+ oii_ /,3
«'-2a''6+3a62__5//
5a»-3a7>4.4a/;2_oi«
Ex. 3.
4a'- 3a-4- I
2a'*— a-'-f]7
5«'-- 2a24- 4
3a«- 7a*-|- 3
loa^^— I4a-.f35
Ex. 6.,
2^V--3^+ 2
4^V-2^+ )
3.ry- 5^-1- 10
xy— ar-j_]5
,u,ttrter4S^^,ob--d tha. s„,.e of .ho
always undentood STZ.A r "u""' "«''•" «■• 1 "
Ex. 2, wo say, 1 + 1 + ,1^9+7 fo^P- "if >»' '^"''™» "f
^'+5-19; andsoofthtri "' ">"'»""«', 2+1+.,
Case II.
To add like qnanfUics with unlike si<;ns.
mfrlvf''^^ ^''^ compound quantity a4.b-c4-d . « •
positive or negative, according as the turn oT f f ' '•.' ''
torms is greater or less than thS sum of <?' ^'"^ ^''''^'^'^'^
nggregate or sum of th. q nn ^tipT o ^^ f%^'^^'^'" ^'"«% the
+2_«,^.dof;.,eciuantitieT7^^^^^^^^^ -iii '-
tor yn <^"" ^"~ ■ "^ «j(y -\~^o — j.j,'>* Will ii.i Ai-2 .
92. Sento tl,c rnio in ,!,„ ,,, „^
I
8
ALGEBRA.
Serins above the negative ones is 2a, and k the latter 4/.'
Hence this general rule for the addition of like quantities with
unlike signs, "Collect the coefficients oHhe positive terms iuti^
one sum, and also of the negative; subtract the Zm^r of these
sums from the ffreater ; to this difference, annex the sign of
the greater together with the common letter or letters, and the
result will be the sum required."
If the aggregate of the positive terms be equal to (hat of
the negative ones, then this difference is equal to 0; and com.
scquently the sum of the quantities will be equal to 0, as in
the second column of Ex. 2, following.
Ex. 1.
4ar- — 3x--|- 4
^x^—hx-\- 1
7.i:-^+2x— 4
~_^4.r+]«
1 Ix'^— 9.r4- y
Ex. 2.
— 7ai+3^;c— xy
— ab-\-2bc-\-4xi/
3a6— ic+2jry
—2ab+4:bc^Sxi/
5ab — Sbc-^ xy
-~2ab +3x//
Ex. 3.
-fxr'-f 13.r'
— 2.r'— 4r'^
7.r»+
.r'
9.c^- t4.i-^
-133:^-2.y'-
— 4.r*— (),r^
E\. 4.
4/''- 2.r+3//
— x^-{- 4x— y
7a;'— x+^y
^x^+2\x~ 2y
Ex. 5.
5a»— 2ai4- /^«
~a'+ ai--2i'
4a='— 3a<!>-f- i"
2a='+4ai-4/'>»
Ex.6.
4.iV+2a7/— 3
■ xhf— xy—\
Sxy-}-4xi/-^5
■9.ry-2./-y4-9
Case III.
^ 23. Tiiere now only remains the case where imlike quan-
titles are to be added togethei, which must be done by col.
lecting them together into one line, and annexing their proper
signs ; thus, the sum of 3jr,-2a,+56,-4?/, is Sx-2a-\-5b-.
~-' ' r ■ ■■^'■-■- «ii-^ z.!:tiftiv 4Ljaiini,ius are rnixeu to.
23. State the rule in tlio 2cl nnd 8d casos.
1 f%
'•"mm
the latter 4^)'
quantities with
itive terms into
3 lesser of these
!X the sign of
letters, and the
fual to fhat of
to 0; and f.'(>;i.
Lial to 0, as in
Ex. 3.
— 4'Z-().r^
Ex. 6.
U'y4-2A7/-3
XY— X1J—I
•2.;'jy+9
unlike quan-
! done by col.
J their proper
re mixou to-
SIMPLE EQUATIOXS. q
gether, as in the foJlowin<r eximnles whor-n ^u
maybe simplified by colfeothrC '., '"^"^^ ''^ expressions
will coalesce intolesum^^^^^^^ ^"'^''' '"^^^^ 'Quantities aa
Ex. 1.
^ab -\. X — y
4c - o^ 4. ^
r)a6 — 3c -f. t/
jy + a;^ - 2y
Collecting together ///l-e^ quan.
titles, and beginmng with 'Sub
\vehave3aH-5a6 = 8a6; -f- «'
i 2,7-"^ ^•'' 7"^-^^+ Ay
f.y y; 4c - 3c =4- p.
besides which there are the two
quantities + c/ and + x'^ whlrh X . , ^^''" ''^^^ ^^e two
Ex.2.
4.r-2xy+l --%+4a:'
4y 4-3:i:^-y2_^a:y-- ««
3£--^_ 144.2//+ 12^'2:;.
Here 4.r'~a;==3jr
^2xij-\-xy=i^xi/
+ l--15=:-i4
-3y+4y+y=^_Oy
+4ar'+3.2:H5a:-'=-flbpS
-2.r=-2x.
SIMPLE EQUATIONS.
.?'m//on. Equations ntlT/'^^r ^^.P^^««^«n ^^ called a„
problems, con\"st o "quantir S^:fr^^V^' ^«^"^^-" ^^
others nnknozvn. iZtxTilTl'yf ""^''^ ^'" '^'^^^^^^ ^^^
:^ is an ^//iX^not..^,. ^z^an//^,, anrlTi^. tni " ^" «^q"ation in which
^vill make 2^+3 L^r+T eoulun "t '\'"'^ a number as
which here saiisJi!^letlSt '^t "'^f* ^'^^ """^^^^^
ijl^stl, 4, since ix^+'^lTCanr '^T^^^'
the unknown ouamitv \eW,„h\„^ ■ J- ■'™ ™luo o(
by inspection^ ," 3^n.'"':'l,!'J!'. "I '.'"^ »«'nple been found
;*S
10
ALGEBRA.
'-]"). l!. efTccting tlie solution, the several stops of the pro-
ooss nuist be conducted by means of the following axioms,
and in strict accordance with them : —
(1.) Things which are equal to the same thing are equal to
one another.
(2.) If equals be added to the same or to equals, the sums
will be equal.
(3.) If equals be subtracted from the same or from equals
the remainders will be equal.
(4.) If equals be multiplied by the same or by equals, the
products will be equal.
(5.) If equals be divided by the same or by equals, th*'
Quotients will be equal.
(6.) If equals be raised to the same power, the powers will
be equal.
(7.) If the same roots of equals be extracted, the roots
will be equal.
These axioms, exclusive of the first, may be f/eneralized,
and all included in one very important prmci pie ^ w'hich should'
in every investigation in which equations are concerned be
carefully borne in mind ; viz., that whatever is done to one side
of an equation the same thing must be done to the other side,
in order to keep up the equality.
_ 26. If an equation contain no power of the unknown quan-
tities higher than the,^;-*-^, or those quantities in their simplest
form, it is called a Simple Equation.
ON THE SOLUTION OF SIMPLE EQUATIONS CONTAINING ONLY ONS
UNKNOWN QUANTITY.
^ 27. The rules which are absolutely necessary for the solu-
tion of simple aquations, containing only one unknown quan-
tity, may be reduced to four, each of which will in its proper
place be formally enunciated raid exemplitied.
25. What arc the axioms omployed in tlio solution of conations nnd
Btave the ireneral principio wliicli is based upon tlicni ?— 2(j. What id a'bini-
pie equation i
I
SIMPLE EQUATIOXa
eps of tho pro-
lowing axioms,
ig are equal to
jiials, the Slims
)r from equals,
by equals, the
by equals, tht»
;he powers will
;tcd, the roots
3e f/eneralised,
', which should
concerned be
'one to one side
' the other side,
mknown quan-
. their simplest
HNG ONLY ONE
y for the solu.
mknown quan-
1 in its propcr
11
Rule I.
-^qual." ^ ^ '''"^^' ^^^. q»«tients arising will be
T^x. I. Let 2..= 14; then d;vldin<; both sides of the
equation by 2, we have -=y. hnt ^^ i 1^* ^
• • -t- — 4 , y
Ex.2. Let az=b + c; then ^=*-±f. w ''"^ .
Ex.3. Let x-\-2x-\-4x-j-Qx=52.
Collecting the terms, 18a-=52.
Dividing both sides of the equation by KS,
x=4.
Ex. 4. Let Gx~~4z-{-Sx-z=SQ.
The terms being added as in Case 2d of Addition,
4.r=36.
Dividing each side of the equation by 4,
Ex. 5 10.^=150.
Ex. 6. Sz-^4x-h7.v=:84.
Ex, 7. 8jr-5.r-|-4x--2x=25.
Ex.8. ^2x-3x^4x~-x==24.
Ans. ar=15.
Ans. z~ ().
Ans. x= r>
Ans. x=i 0.
f eqiifirions, nncj
. \\ hat is 11 ttiui-
the suLjoincd cvaranles h, 1 1, ?^ , •'' "'"^ " '"■" ^'« ^<'™ I"
'" " -""ci : as jor Histi
i^
ALGEBRA.
(.V,) Dut the ost of 5 lbs. by the .juestion=30*.
M.) TFicrefore, the price of 1 ]b. in shiHings X5=305.
r/nilt. '^'''^'''' "•' ^'"^'^'"^ ^^ ^' ^'" ^^*^'" tJ^« F'*^
Thy ...v-ral steps of this soluti. a expressed a].rebr-,If...!iu
Hould take the following more compendious fonn-"^
(] .) Let a;=the price of 1 lb. in shillings.
(2.) ITien 5.r=the price of 1 lb. in shillingsx 5
(3.) But the cost of 5 lbs. is by the questbn^GO,?.
(4.) .-.bxzzzSOs.
Jvl"'"^ ''■"^'^'' ^^''^ ^^ '^' P™^ «^ l^b., as was re-
It will be seen by steps (2) and (3) of this example that
here are two dist.nct expressions for the same tl^S^' and
hat m step (4) these expressions are made equal fe' each
other. In Iraming equations from problems, this wil in all
.ases take place. As a second exaSiple let this p'blem be
A house and an orchard are let for£2S a year, but the rent
oHhe house is 6 tmies that of the orchard, -"l^'d the rent "f
The rent of the house is equal to that of G orchards- we
maj^ therefore change the house into G orchards, and w^shlu
Tnkn.u the sum of the rents of the orchard, we get
7 times the rent of the orchard=£2S
and the 7th part of each side of the equation being t^kvi^
The rent of the orcnard=£4 •
ui:i /. the rent of the house=6 times the rent'of the orciiard
=6 times £4
INow to i-ve to these operations an alffebraioal shani>
i-;et -z^i'ie rent oj the orchard in £ * '
then 8.. . '' « j^ouse «
hiMPLE EQL'ATIOXS
as was re-
But by the condition of the
Hent of the orchard
question
times ri
o/^he orchard— £28.
or, 7x=£2b.
«iHl, dividing each side of the equation by 7,
x=£4, the rent of the orchard
and ... Gx=Gx£4=£-24, the rent of the house
- ..h^.r., .hat 'onr-pa.^^^; tells r ^^ "^^
h> 9. A person unacquainted with alcreb.a mllu ^Z
.;.^^rcat_dimcu,„. s„,vi this quosLfi.'-L'"'!!;.?..;,;;^
(2.) The greater part .nust exceed the less by 9
(ii.) But It IS evident that the greater and less n^-ts ,,i i i
iogother must be equal to the whole number 35 ^ ^"''
Jii^^f^ s^i^:^ ""''^ "-^ '<=- P-' with the
(T.) Hence, twice the less part is equal to 26
2,^'iZX af ?;:!olf' ^'^' """ *^ ^'''^x"-' - «9ual to
oJi^i^:^^^, '"» ^™'- P-t -eeds the ....
Lut hy adopting the method o/ algebraic wo/^/Zn,, ft r.r
(1.) Let the /f-ss part - - . __
(•^.) Then the greater part =a-4-9
(3.) But greater part+1
part
=3;
"" ■' lum-mt^j^y^^i^ ^T-^.- ' I ll II i i ln i K wii J
U
ALGEBRA.
iliii
l!ll
(4.) ,.-. a;+9-4-.r
(5.) or2.r-j-9 ....
(6.) .'.2x
(7.) or2a; . . .
(8.) .'.X {less part)
(9.) and x-{- 9 {r/reafer part)
=35,
- =35.
. =35-9.
=26.
20
=^=13.
- =13+9=22.
29. Hjivmg tlnis explained the manner in which the several
5|-eps ni the solution of an arithmetical question may be ex-
pressed HI the language of Algebra, we now Droceed to its ex-
emplification.
PROBLEMS.
PROB. 1. A dessert basket contains 30 apples and pears
but 4 timps as many pears as apples. How many are there
01 each sort ? "^
Lot a;=the number of apples ;
then, as there are 4 times as many pears as apples,
4.r= the number of pears.
But by the question the apples and pears to(rether=30
.-. a,-+4^=30.
.■idding the terms containing .r,
5.r=30.
Dividing each side of this equation by 5,
x=G, the number of apples,
.'. the number of pears =4.r =4x0 =24.
TRon. 2 In a mixture of 10 lbs. of black and green tea,
there was 3 times as much black as green. Find the quaiititv
Ot each sort. ^ -^
Let .t=the number of lbs. of green tea
thcn3x= " « « |,]j^ck^ '
But the black tea -f the green tea = 10 lbs.
^„ . .'.^r 4-3,^ = 10 lbs.
Collecting the terms which contain x,
„. .^. , 4.r = 101bs.
.r=r4 lbs. of green tea,
.*. the black tea -3i'=3x4= 12 lbs.
s
I
,1-
t
SIMPLE EQUATIONS. 15
Prob. 3. An cqunl mixture of black tea at 5 ihUWucr^ ■, iK
«nd ot green at 7 shilllnffs a lb costs 4 .rnhnn! jj^"" '^ "^•
lbs. were there of each sort ? ^ ''''' ^^"'^' "'""J'
Let _a:=the number of lbs. of each sort •
then o.r= the cost of the black in shillings
and 7x= « « j^ ,, » '
But cost of biack+the cost of green=4 guineas=:84.
.-. 5a:+7.r=:84
12jr=84
.'.a; =7 lbs. of each sort.
Prob. 4. The area of the rectangular floor of a school-rpom
|s^^^hO square yards, and its breadth yards. ^r^H
a; X 9= the area of the floor
.'. 9.r=180 '
and .-. ar=:20 yards, the length required.
Prob. 5. Divide a rod 15 feet long into two n-irts .<. fl. ,t
the one part may be 4 times the length of the other.' '
Let ar = the less part, 1 , 15t>
then 4a:=the greater part, '~r' "~ir 1
i>ow <hese two parts make together 15 ft
.•.a:+4x—15ft.
5.r:=:I5ft.
, ^, .'.x=: 3 ft. the less part
and the greater part=4 times 3 ft. ==12 ft.
Prob. G. A horse and a saddle weie boucrht for £.10 bnt
Ajis. £30 and £4.
^'In^. Itichard's share =.-18,
-■l;*.?. Andrew's share = (}'
8aiM7f^hnf ^^ ^''"° ""'^^^ ^"^^^ "^'^"3^ '"^^'W^'s he had
saiJ, It 1 had twice as manv mm-o T .u^Ja u..,._ o^ "V^''i^J.
many had he 'I * ' ' ■"-'"'■'^ "^*^'^ '•^"' iiow
J/w. 12,
Prob. 9. A bookseller sold 10 books at a certain price, and
16
ALGEBRA.
aftcnvards 15 more at the s^xme rate, and at the latter time
Prlf"- """ ^'"^ '' ^^^ ^™-= -hat.a.th:rS
Let a:=the price of a book in shillings ;
then 10u:=the price of the 1st lot in shiHings
and 15^= » » 2d " « ^ '
Now, if the price of the 1st lot be taken from that of th<3
2d, there remains a difference in price of 35^
.:l5x-l0x=z35s.
Subtracting the 10^ from the 15^, we have
5x=St)s.
•^='^«. the price of a book.
Frob. 10. Divide £300 amongst A, B and C so ihm A
Sr^ twice as much as B, Snd C as'muchVl tl B
Let ar=B's share in £
thch 2.«;=A's share in £
and x+2x or 3x=zC's share in £.
I5ut amongst them they receive £300 •
.•.a?-f2.r-f3A—£300,
0a;=£300 ;
• •• .^=£50 B's share;
.-.As sharc=£100, and C's share=:£]50
Let .T= the number;
then 9^=nine times the number,
3.i'=threo times the number,
and 4.r=four times the number:
.-. i^x-}-Sx~4x=48,
8.i-=48 ;
.-. x=z6^ the number required
Let each boy's share =.r ;
uivu eacn woman's share=r3r
w>d each man's share =2 time's 3x=Qx;
)m that of tli«
1 as A and B
SIMPLE EQUATIONS.
Hence we have,
the share of the 4 boys =4.
<-''^' «lure of the 3 wo,n.n=37inios 'U o
,^ and the share of the 2 inen -o , ' ^'^^■^
^iit the sum of nil tL 7 ^"^".=^ times Gj:=12x :
sum of all these shares is to amount to £100 •
..4.r-f9.c+12^=£100, '
25^ =£100':
.*. each boy's share — :tU. nn.h ,
=/]'> wl":fl' t ^^^oman's=z3 times £-1
-^^- ' ^^^d each man's = tiiues £4=i:o4
«hm 'S' altf t^nr^rtTe''? ' f r p™^ ^^-- «-
third thrice, and to thV four h f ? ^' ^''"" ''^^''^'^' ^^ the
'i'-st. Whai did he give to each J'"' "'' '' "^"^''^ ''^^ *« ^^'^^
PKon 14 T.- .. ^"'' '^'" '^^'" ^^'-^ "^''^ respectively.'
thatrmfddl^re\,:,l;Tel^bleT T^^ ^^^^'^ ^-^^-"'^
tn-p]^'. the least part. ^^ *^^ ^''^'^^ '^^"^ the greatest
^'^^^^^ 5 tinSih: ^oUr^z H ?^ p«'ti l.
-ce between the first ^^^t^::;^!'^^ '"'' li;^ ^> ^^ f ^^^"-
-ixture contain^ t ame quant^'iV "' ^ ''^^ "^- ^i.^
"ow man,, lbs. arc there of eS S °' "^^ T*^ T, *''
Pkob. 17. A bill nf ^nn -'''«• 8 ibs.
sovereigns, and cmwnsTaS .Z, Ijf , '" ^^^''f'S""' ''"If
l-iiid tlio numl)cr. ^"'" ""'"•'« of each was nscd.
Paon IS T . „ ^■'"''- 400.
^^.ndfb;d^^;^;™^- ^^ the same tin. n-on,
Hks 4 miles an hour and the n!h r^ '"^l'"' ''i'^'^' the on.
^ours will they meet?' "^^^'^ ^ '"•^«^- ^i how manv
PiiOT! 10 A , ^'**- ^ hours.
luv£lV\u P^''''" ^'^"^^^t a horse, chaise nn I f
I'" AUO; the price of the hor^o u^no . • ,' ""'* harncs-.
'•■n-ncss, and tho'priee of the 'Sse 11^''?, ''^' ^r'' "^' '^"■
h-- and harness; what ^^:^Zl^:^J!:ir'^ ^^ ''<'^''
17
.1
nswe)\
•I'l'ice of harness = .£'1.3
<(
8
2*
chaise =
= 80
18
ALGEBRA.
• ■}
phi!
i
I
iiij!
SUBTRACTION.
30. Subtraction is the finding the difference hctwoen twc
filgebraic quantities, and the connecting them by proper signs,
so as to form one expression: thus, if it were I'equiied to
su})tract 5—2 (/. e., 3) from 9, it is evident that the remainder
would be greater by 2 than if 5 only were subtracted. For
the same reason, if i— c were subtracted from a, the remain-
der would bo greater by c, than if b only were subtracted.
X<nv, if b is subtracted from or, the remainder is a—b ; and
consequently, if b~c be subtracted from a^ the remainder
will be a — j-j-c. Hence this general Rule for the subtraction
of algebraic quantities, " Change the signs of the quantities to
ha subtracted^ and then place them one after another, as in
Addition."
Ex. 1. From r)rt-f-3.r— 25 take 2c— 4?/. The quantity to
•:e subtracted ?f<7A its signs changed, is— 2c-\-4i/ ', therefore
the remainder is 5a + 3.c— 26— 2c-j-4//.
Ex. 2. From 'rx'—2x -\-5 take 3.r+5x-l ;
The remainder is 7.t'— 2.ir +5 — 3.c^ — S.r+l ;
ovlx'—Sx'~2x — 5.r+5 4-l=:4x'^—7x+6.
But when like quantities are to be subtracted from each
('ther, as in Ex. 2, the better way is to set one row under the
other, and apply the following Rule; ^'■Conceive the signs of
the quantities to be subtracted to be changed, and then .proceed
»i in Addition."
Ex.3.
From 7a;"— 2.r-f 5
Subtract S.i-'^+S.r— 1
Ex. 4.
12a'^-3a4- b-\
Oa«+ a-.2b + S
Ex. 5.
5/y*— 4y-f 3f/
0//— 4//— <'
Remainder 4.c2—7.r 4-0 6a^— 4a-|-3/>~4 — .?/ * +4^
Ex. 6.
From 7xr/-\-2x—Sg
Subtract S.ri/~ x-^ y
Ex. 7.
\4:X-\-y—s-
x-^y-\-z-
5
11
Ex. 8.
13.i;8-2.c'^4-'7
iimkL i^ ' iM^i2i Lh a^r^n
s^i'. ^V!)at is mihtraction ? Stntc the rule for tlio snbtniotion of algcVniic
mMi.ti'io!^, and c.\pl:un the pnnoiplc on which it nsts.
MVrLE EQUATIONS.
19-
cc Letwocn two
by proper signs,
Bre I'equiied to
t the remainder
ibtracted. For
I a, the rernain-
ere subtracted,
sr is a— i ; and
the remainder
• the subtraction
;he quantities to
' another, as in
rhe quantity to
h4?/ ; therefore
cted fi'om each
} row under the
ve the signs of
id then .proceed
Ex. 5.
6// — 4y — a
-r
-f-4o
Ex. 8.
13i;8_Oc'J4-7
.MutUi
EM.
ruction of algcVraic j
OF THE SOLUTION OF SIMPLE EQUATIONS, C0XTA1N'IX(, CKLV OMV
UNKNOWN QUANTITY-.
Rule II.
31. "Any quantity may be transferred from or.e side of '
;.e equauon to the other oy changing its sign;" and it il
toundea upon the axiom, that "if equals b? added to ^r -
equal" '^"'^'' '^'' '""'' ^^ remainders Zm be •
Ex. 1. Let ;r+8^15; subtraci 8 from each side of the
equation, and it becomes :.+8-8 = 15-8j but 8-8^0
0,,f;iV^^'l-^=3»; f<'7 to eachside of the equation,,,
mciii 7+7_,i0+, ; but -7+7=0; .-.^=20+7=27.
tnenrf.r-2.r=o^_o^.4.9^_5 but 2.r-2j; = • 3ar-'>r-Qv^
+ 5. Now 3^-2^=;., and 9+5 = 14; hence ;:! 14.
On reviewing the steps of these examples, it appears
(1.) That a; 4-8=15 is 'identical with a;=15_8
(2.) « ;r-7=20 « with a:=204-7*
(3.) « 3.r-5=2x+9 " with 3.r-2.r=9-f 5.
Or, that "the equality of the quantities on each side of thp -
equation, is not affected by removing a quantity from one s ^
of the equation to the o-:her and chauffinr^ its s^nP
he Idl out ff \ 1- ""'' ^'f' '''^''' ^^ ^» ^q^'-ition, it n)ay .
I'e. lUt out of the equation; thus, ii'x+a=c+n then x-cl
'^~a; huta~a = 0,,\x=c. ^tu, men x^c-^-
h further appears, that the simis of all th. *««.„. ^c>
>n may be changed from -j- to -or from " "in T ""."^'i"'*;
«, .hen, by the l.ule, .r^c-a+i; change the siirns of
■)
lion
e sijijns of alt
• mm
20
ALGEBRA.
Ex.5.
Ex. 0.
Ex. 7.
Ex. 8.
Ex. 9.
Ex. 10.
-a \-C = Z, or T7:
Ans. xz=l4.
Ans. rrz=29.
Ans. x~G.
Ans. a;=9i~-2a.
Ans. x=2.
Ans. ar=10.
A71S. ar=5.
-.4;t5. a: =3.
a
' Ihe terms, then b -x-,-,, -r, in wliich ca^c b
t—a-\-b, as before.
Ex.4. 2i;+3=a:4-17.
5.K— 4=4ar+25.
7ar-9=e».r— 3.
4x+2a=3a;+9i.
15^+4=34.
8a;+7=6.r4-27.
9,r-3=4.r4-22.
Ex. 11. 17x-4jr+0=3i,'+39.
Ex. 12. ax—c=b-{-2c.
Ex. 13. 5^-(4.r-G) = 12.
The sign »— before a bracket being the sign of the wholf
quantity enclosed, indicates that the quantity is to be sub.
traded; and therefore, according to the Rule, Avhen the brack-
ets are removed the sign of each term must be chantrcd
rLiis, the signs of 4.r and of C are respectively 4- and — ,''but
when the brackets are removed they must be chann-ed to —
:tind -f respectively. The equation then becomes
5^-4.r+G = 12.
By transposition, 5x—4x= 12— 6;
.-. .r=6.
Ex. 14. 6.r--(8+.r)=4.r-(x-10).
By remo^'ing the brackets, and changing the signs cf the
. terms which they enclose, the equation becomes
Ox— 8— a:=4=:4,r— .r + 10.
Transposing, Gx~z—4x-\-z=l0-^8 ;
.-. 2x=18. ..
Dividing botli sides of the equation by 2,
Ex. 15. 4x—{Sx-\-<)=S.
Ex. la Si;-(Cu'-8)=9-(3-T).
Ex 17. 4x~(Sx-Cy)~(4r-m = l9
Ex. 18. 5a:--(8-l-3r) = 8^(-ir— 1).
Ans. .r=::12.
Ans. x=-^2.
- /-.,_
If..
Ans.
x~
:2.
Ans.
x =
12
SIMPLE EQUATICXS.
21
! sijnis f f the
^9.
X —
12.
IS.
x=.
-%
IS.
! (it
}■
2.
IS.
X —
12
PR0JJLKM8. ■
Let therefore x=.m^ Jess number;
then will a:+15 = the greater:
iJut their sum =59;
.-. a;+a;-rl5— 59,'
or2.r+15=,59. ^
And transposing 15, 2.r=59-15,
or2.r=44;
.-. ar=:22 the less number
-rxo — ^^-i-io~j7 the greater.
J3ut together they receive 27 ; '^''^ *
.*. .r+a:+5=:27,
.,, . or2.c+5=27.
lrans]-/,Kwig, 2.r=:27-5,
or2.r=:22;
••. .r=ll, the No. James receiver]
.6 -r o _ 1 o Kichard received
Let a-rrthe number;
then 4.r=4 times the number,
. 2.c= double the number,
.h.refoie, b;y the equality stated in the question
_ 4a;=2ar+12. ^
»,y trjpsposition, 4.ir— 2;r=:]2.
Oj. TO .
.'. arr= 6.
•aon. 4. At a, election 420 persons voted, and th
le succcs.*
i
ill
*|!1
1' :
!
iilli
22
ALGEBRA.
ful candidate had a majority of 4(5.
each candidate ?
How many v<itcd for
Ans. 187 and 233.
Prob. 5, One of two rods id 8 feet longer than the other
but the longer rod is three times the length of the sh(.)rter.
\\ hat arc their lengths ? Am. 4 H. and 12 ft.
Prob. 0. Five times a number diminished by 16 is equal
to three times the number. AVhat is the number 1 Ans. 8.
^ Prob. 7. A horse, a cow, and a sheep, were bought fur
£24 ; the cow cost £4 more than the sheep, and the horse £10
more than the cow. What was the price of the sheep ?
Let a:=:thc price of the sheep in ^;
then .r4-4=: " " cow •'
and a;+4+10= " " horse "
But these thi-ee prices taken together amount to £24 ;
^ .-. a:-f(ar+4)4-(a:-f-4+10)=24.
Adding together lilic terms,
3.^4-18=24.
By transposition, 3a? = 24 — 1 8,
Sx=Q ;
.-. «=£2, the price of the sheep.
Prob. 8. A draper has three pieces of cloth, which togethci
measured 159 yards; the second piece was 15 yards iongei
than the first, and the third 24 yards longer than the second.
What is the length of each piece ]
Ans. 35, 50, and 74 yds.
Prob. 9. Divide £3G among three persons. A, B, and C, in
such a manner that B shall have £4 more than A, and £7
more than B. Arts. £7, £11, and £18.
Prob. 10. A gentleman buys 4 horses ; for the second he
gives £12 more than for the first; for the third £5 more thun
for the second ; and for the fourth £2 more than for the third.
The sum paid for all the horses was £240. Find the price oi
'^Jach. ^^s. £48, £G0, £65, and£67.
Prob. 11. What nimibcr is that whose double is as much
above 21 as it is itself less than 21 ?
Let a;=:the number ;
then 2.r= double the iiiunber,
2.r— 2i=what double the number is abo/e 21,
and 11 — a:=what the number is lesi than 21 :
nany vcted fo?
187 and 233.
than the other
of the shorter.
11. and 12 ft.
by 16 Is equal
)er'? Ans.8.
ere bought for
I the horse £10
le sheep ?
t to £24 ;
ecp.
which togethci
5 yards Jongei
lan the second.
and 74 yds.
A, B, and C, in
I A, and £7
11, and £18.
the second he
I £5 more tliun
in for the third,
ind the price oJ
55, and £67.
j\e is as much
! al)o/e 21,
an 21 ;
SIMPLE EQUATIONS. . 23
By transposition, 2j:+a:=21 + 21 ^'
3u:=42';
rpi .'. .r=:14.
ine answer may easilv be nrnvn.7 f^ r
-28-21 =.7, and 21^-2^^141^7 T'-^'^'"'^
a^ n.uch above 21, as 21 is above HJainei;^;'' '''''' ^'^ ''
»5 to spare, and bv ^wZ \fl n% ^ '"t f ^'' ^'^^^ ^^V I Had
How Lny boysV?;; I'e? "'' ""'' ' '''''' ^'^ ^---^g-
^hpnJfT ^^*/=the number of bovs;
^hen. If I gave to each boy 4 oranges"' I \hr..,u^ •
times X oranfres- "^^I'lfets, i should give away 4
.;. ro/«/ «nwifr of orangcs=4^4-fi .
Agani, if each boy receivorl r^-*'*'^^-
o-angcs left ; ^ rtceucd 3 oranges, there were 12
terms of. must necessarily C::^:afi::Zll^''^'''''' '"
o , . •'•4:i^ + C=3,r+12. ^ '^
I5y transposition, 4x-~Sx=l2~0-
.: x~G^ the number of boys.
I>ut hrcmlstitLTK T ?"' '" ''''''^ "40 miles in 4 dav.
He murg?rmnes he sl"Tf '^'?''^''" ^^""^^ ^^at'
fourth da.^ less t ai the S" it^' ' '" '^^'■'' ''^"^^ ^^ ^^e
travel each day? ' ^^""^ "^^"3^ ^"'^es must b-
then^rllZ^'^^ number of miles on the 1st day
, ^-9= « « « ?^
and .r- 14=: u „ ^^ ^^
4th
.^T the number of miles whioh hn I'l
a
13 lU
aajg
^+x~.5+x~9+x~U~2.
Collecting the terms, 4x -28- 2
40.
240.
u
ALGEBRA.
Jjy transposition, 4a;=240-f28,
4.r=2G8;
.'. ar=07, the number of miles he goes on 1st Oav,
ar-5 = C2 " " " " " 2d "
g. 9—58 " " « " « 3(] "
and a; -14=53 " " « " * 4th '•
Prob. 14. It is required to divide the number 99 into five
such parts that the first may exceed the second by 3, be less
than the third by 10, greater than the fourth by 9. and Itss
than the fifth by 10.
Ans. The parts are 17, 14, 27, 8, S3
Prob. 15. Two merchants entered into a speculation, by
which A gained £54 more than B. Tlie whole gain was £49
less than three times the gain of B. What were the gains ?
J ^;is. A'sgain=£157; B's=£103.
Prob. 10. In dividing a lot of apples among a certain num.
ber of boys, I found that by giving to each I should have too
few by 8, but by giving 4 to each boy I should have 1 2 re-
maining. How many boys were there 1 Ans. lOi
MULTIPLICATION.
32. Multiplication is the finding the product of two ot
more algebraic quantities ; and in performing the process, the
four following rules must be observed.
(I.) When quantities having like signs are multiplied to.
gether, the sign of the product will be + ; and if their signs
are unlike, the sign of the product Will be — .*
* Tlii.s rule for tlie multiplication of the Signs may be thus ex-
plained : —
I. If -fa is to be multiplied by +ft, it means, that -\-a is to be added
to itself as often as there are units in 6, and consequently the product
will be -f ab.
IT. ]f — a is to be multiplied by -^b, it means, that — a ih to ho
added to itself as often as there arc units in b, and therefore the product
}b — ab.
f5<>. "What is multiplication, and what are the Rules to be observed In
multipl cation i
I
MULTIPLICATIOX.
25
Mt
(2.) Tlic coefTidcnts of the facfor^ mu^t Im r,,.,ii- i- .
g«hcr, ,o.brm the coefficient o?Zp'o2cl "'"^'^""' '"
forming the operation, the kule i, " Tn n ,u- ? J P°
..gns, ./,« the coefficic'its, J^^^ar^: ^I^H^J-'' '^
r-
the
ay be thus ex-
be observed In
Case I.
^^'■il'lTJtX^,:::. «'"^" ^™--"- ^ '■o- wi.ich the
U - »J. ""' """* "■ '. ""tl ~n»equently tl,o proiluct
Or, those four R.le, ,„i„.,,t b„ ,„ e„™pr„,,„„jej ,■„ „„, . „,^,
I J?.s"i;r'/i- '„t;w;L'3r;' :^4'ef."- »^" - *-
But a — . /j, arfc?<'(/ c ^•»^^ft«- . , = a^ — ^r
and a — 6, su btracte d d times = __ «^ |: f,j
i. e., -f- a X + c =-- 4- «/•
+ « X — (/ = —
ad
— bX~d = i.bd.
Ex. 1.
i)a
Ex. 5.
Aabc
3ac
ALG
EBRA.
Ex. 2.
Ex.3.
— Oax//*
-15aV/'c
Ex. 6.
Ex. 7.
dxhf
— 4c*f^
—2?/
2c
Ex 4.
— 5a*4c
-~_2iV
4-lOaV/cl'
>:x. 8.
— 7c/.r*?/
— 2f/A
/J^-^/b
^^^
^cW^
Case II.
34. When one factor is compound and the other simple^
''Tlien mc/i^erm of the oompound factor must be multiplied
by the simple factor as in the last Case, and the result wil'
be the product required."
Ex. 1.
Multiply Sab—2ac-^d
by 4rt
Product V2a^b—Sa^c+4a(f
Ex. 2.
Sx^ — 2x'-!-4
— 14aa?
— 42ax~-\- 28a^?^5«^-
Ex. 3.
Multiply Ix^ — 2.C +4a
by — 3a
FVoduct
Ex. 4.
12a^-2a' + 4a-l
Sx
-21a.r'-f-(k,';c— 1-.V
r
Ex, 5.
Multiply dah-^3a—x-^]
by — x^
Product ZJ^,^^ ^ y^^t-
[ Ex. 6.
! 4a;V + 3.r— 2i>
— 3.r?/
Case HI.
35. When both factors are compound quantities, each term
of the multiplicand must be mu'tiplied by each t erm of the
£x 4.
— ha*bc
>:x. 8.
MULTIPLICATION.
27
multiplier; mid then placing; Ike qnantitu- nmhr ea,\ ot\t»
Uie sum of all the terms ^viJl be the produo. required.
Ex. 1.
Multij ]y a -I- h
by a -f h
1st, by a qS^- ah
2d, by i /7A-i -//
Product aH^oHhO""
Ex.2.
a"
* -/^«
Ex. a
fl.
3 i/c
D Other simple,
: be multiplied
the result wil'
J A* /s^*
-2a;»-h4
+28a.r^-50c/-
Ex. 4.
Ex. G.
4a;V+3.r— 2^
ities, each term
ach ' erm of the
Ex. 4.
3.r*+ 2ar
4.r + 7_
i2?+~a?
±?lfM-14T
12.cH29x^-fT4j
Ex.5.
3.C'- 2x +5
•-21.g^-f-14.r-35
18.r''~33.r2+44,r"-^^
Ex. C.
14a c — 3a * -f 2
a_c — ah -}- I
-Ua'bc +3a%'^2ab
•hiiac — Sab + 2
— !tiZ!f!^±l?^+3a-^A«'^(^:f2"
Iv -i-2
Ex. 7.
jr
+ 2^.*^-- ar+4
+
IT
~4
Ex. 8. Multiply a'^+3a«34-3a/y^+63 ^y a-^b.
28
AliGEBllA.
Ex. 9. Multiply 4x'?/-f-3xy— 1 - - Ly 2.c'— a-.
Ex. 10. « a^—x^^x-b - - - by 2x'-\-x-\-l.
Ans. 2x'—x*-^ 2x''-l0x^—4x^b,
Ex. 11. « 3a«-f-2a6-i2 by Sa'^-2ab-i-lj\
Ans. 9a'-4a^lj*i-4aP-bK
Ex.12. « ar'+xV+^y'+y' by a;-?/.
-.4ns. x*—y*.
Ex.13. " ar'-Jar+l
by x'^—lx.
A /IS. a;'— |-.c^+y.r— ^.r.
ON THE SOLUTION OF SIMPLE EQUATIONS CONTAINING ONLY 0NJ5
< UNKNOWN QUANTITY.
Ex.1. 3.c-}-4(jr4-2)=3G.
The term 4(a;+2) means, l ,.it x-\-2 is to be multiplied by
4, mid the product by Case 2d is 4j;+8;
.-. 3x-^4x-\-8=S6.
Adding together the terms containing x, and transposing 8,
•7x^m-S,
7a;r=28;
• • •C' ■^— Htm
Ex.2. 8(a;+5)+4(T-f 1)=80.
Perfurming the multiplication,
8^+404-4.c+4=:80.
Collecting the terms, 12x4- 44 =80.
Transposing, 12.f=80~44,
12x=:30;
Ex.3. C(.r4-3)+4.ir=58. Ans. xt^4.
Ex. 4. 30 (x_3)4-6=:0 (.t-f 2). Ans. x^i.
Ex.5. 5(.i;4-4)-3(«-5)=49. Ans. x=:l
Ex.6. 4(3+2.r)-2(0-2.i-) = 60. Ans. x-^5
Ex.7. 3(x-2)4-4=-.4(3-.r). ^«*. .r^^o;
Ex. a 0{4-a-) -4(G-2,r)-12=0. Ans. a-^O.
SIMPLE EQUATIONS.
29
riNO ONLY ONfi
PR0DLEM8.
suin shall be 35 ? '''''^ ^^ ^ ^""^« the less.. tJie
Let ;r=:tho less numLer •
\n^ q/- ,,^^^"-^+9=thc greater. '
^^.t by the problem, 3 times the greater + 5 times the le..
.-. 3.r4-274-5.r=35,
. 8.r+27.-=35.
Iransposmg, 8.r=35-27=8;
.-. ,-r=: 1 , the less number,
and .r+9=10, the greater.
.'•"d in order to do thif o'"i. ol w''"* ^^ ^^^'^^'^*^ ^^i"^-
^-r: |uho.man;t,i:j::^K,:-^
then^tS"" """'" ^'^?"" ^^ ^^^-^>
land 7(.;+5)'^'!^' """"^^'^ "^"^''^^ the "M "
|num^.'^i,:'fP^^'^'^'" ^'^ ---,;; b!^LrLl the same
.'. bi-=7(^+5),
^ 12.c=:7,c+3o.
iransposmg, 12.r-7.r=r35,
^on-ier is in overtaking tL'f^'sf ""n.b.r of hours the .ocu,rl
kJlr^t^etLtS^;;^^^
Mow- nnmy passengers were ty/c oTleh dllf "^'"'' ^^^
Let «= the numl)er of nas«nnr,<>..,
6a' =
4>1C* ;-vi* f 1-.- "i ,,i. _»
kn.J
. p . , " 2d "
Mi
3*
tiutmitmi
SiSkd
pl!
1'
hi
1
1
(■■
1
5<'
ALGEBRA.
But these two'sums amount to £3 12s., or to 72«.
.-. 6.r+4(15-rr)-72,
6x-\ (30— 4.r=72.
Dy transposition, Gu;-- 4a; =72— GO,
2.r = 12;
.'. x = G No. of 1st class passengers ;
/. the number of 2d class pas8engers=15— a;=9.
Prod. 4. What number is that to which if G be added tvvic€
the sum wi'l be 24? Ans. C.
Piion. 5. What two numbers are those whose difference is
0. and if 12 be added to 4 times their sum, the whole will be
GO? ' Jns. 3 and 9.
Prod. 0. Tea at 6s. per lb. is mixed with tea at 4.9. per lb.,
and 161bs.jof the mixture is sold for £3 18s. How many
lbs. were there of each sort 1 -.-Ins. 7 lbs. and 9 lbs.
Prod. 7. The speed of a railway train is 24 miles an hour,
and 3 hours after its departure an express train is started tc
iHUi 32 miles an hour. In how many hours does the cxpres;-'
overtake the train first started 1 Ans. 9 hours.
Prod. 8. A mercer having cut 19 yards from each of three
equal pieces of silk, and 17 from another of the same length,
0)nnd that the remnants taken together measured' 142 yard^.
W hat was the length of each piece ?
Let a:=the length of each piece in yards ;
.*. X — 19 = the length of each of the 3 remnar.ts,
and .r— 17= the length of the other remnant;
then 3 {x-l9)-^x-]1=U2,
o^3a;-57^•a:-17=:142,
4.r-74 = 142.
Transposing, 4.r = 1 42 + 74,
4.c=216;
.', .'rr=54.
Prod. 9. Divide the number G8 into two such parts, thntl
the ditference between the greater and 84 may equal 3 timesj
the diubretit* Dotween tno less an.u 40.
Let ir=the less part,
then G8— ar=thc grsfiter;
SIMPLE EQUATIO: ^,
^ . |4oi:lLi't:ri^!,v^!i«^ ?■■ ^ «.o ...at..
31
n.n b, '-."X^^e^^f^ - e,ua. . each .W .
By transposition, ^+3^=120+68-84
4a;=I04; '
and.-.thegroater=42:""''''P"'''
to two'upon etl^Tal "CL"' <=^.■•'^^ Netted three shillm«,
•tags. How ma„; deais mV^^^f "'"'^ "« ^™» Ave sli
..•:0-:r=the number he lost;
iosfwafl,""^^"-^^ ''^'-» *e -no, won and the money
.•.2^-3. (20-.r)= 5
2^--60+3ar= 5,'
5x-.Q0= 5,
.'. «=13.
to i^iff Lt 'tfan^t^^^^^^^^^ P^^y^^l packs of cards so as
off twice as many as B ill and R .T ^ happened that A cut
'^'"^|;^=the number he'lefl,
. 4 ^^=^,*Jo number B left;
3"t the number Bci^n^ """'^^^ ^« «"* <^ff^ '
^ ,,ft . "^^^r B cut off was equal to 7 times the number
.••52-a.=7.(52-2z)
52— ar=364--14ar
transposing, |4;r-ar=364~-5o
la^=312; '
a:=24
i
Aou off48, and B cut off 28 cards.
A^-ms
m
S2
ALGEBRA.
pROB. 12. Some persons agreed to give sixpence each to a
waterman for carrying them from London to Greenwich; but
with this condition, that for every other person taken m by
the way, threepence should be abated in their joint fare.
Now the waterman took in three more than a fourth part of
the number of the first passengers, in consideration c.4* whicli
lie took of them but fivepence each. How many persons were
there at first 1
Let 4x represent the number of passengers at first ;
then 3 more than a fourth part of this number = a: -f 3, and
they paid 3 (a? +3) pence.
.-. the original passengers paid 6x4a;— 3(a;+3) pence.
But the original passengers paid 5x4a: pence ;
\ by equalizing these two values, we get
6x4a?-3 (a:4-3)=5x4ar,
i 24a; -3a; -9= 20a;.
Transposing, 24a; — 3a; — 20a; = 9 ;
. • Xzz^tj ;
and .'. the No. of passengers were =4x9 =30.
Prob. 13. There are two numbers whose difference is 14,
and if 9 times the less be subtracted from 6' times the greater,
the remainder will be 33. What are the numbers '?
Ans. 17 and 31.
Prob. 14. Two persons, A and B, lay out equal sums of
money in trade; A gains £120, and B loses £80; and now
(\.'s mo^iev is treble of B's. What sum had each at first 1
^ Ans. £180.
Prob. 15. A rectangle is 8 feet long, and if it were two feet
broader, its area would be 48 feet. Find its breadth.
Ans. 4 feet.
Prob. 16. William has 4 times as many marbles as ITiomas,
but if 12 be given to each, William will then have only twice
lis many as Thomas. How many has each %
Ans. 24 and 6.
Prob. 17. Two rectangular slates are each 8 inches wide,
but the length of one is 4 inches greater than that of the othw.
Find their lengths, the longer slate being twice the area of the
J each to a
nivich ; but
taken iii by
joint fare,
th part oi'
)n (.f whicli
jrsons were
first ;
=x-\-S, and
3) pence.
rence is 14,
the greater,
rand 31.
al sums of
) ; and now
at first 1
ns. £180.
ere two feet
adth.
IS. 4 feet.
3 as ITionias,
e only twice
24 and 6.
inches wide,
of the othcj'.
e area of the
SIMPLE EQUATIONS.
83
hZZf"' ^'''' ""^ ^ '''*^^^' '' '^ ^'"S^^ multiplied by its
Rnf :i: ^f ^"^ ^ ^'^'t'*) ''''^ ^^^ ^^^as of the slates.
But the larger slate is twice the area of the less
.•.8a;X2=8(a;+4),
16x=8x-{-S2;
.'. 8ar=32 ;
.••ar= 4, the length of the less slate,
and 0^+4= 8, « " « greater slate.
h 71' ^f'.x^"'"' rectangular boards are equal in area • the
breadth of the one is 18 inches, and that of the other 16
mches, and the difference of their lengths 4 inchef F^d tJe
length of each and the common area.
p ,^ , . , ^^*- 32, 36, and 576.
Frob 19. A straight lever (without weight) support^ in
equilibrium on a fulcrum 24 lbs. at the end of'the shorter arm
and 8 bs. at the end of the longer, but the length ofthTS
r .1. %r^^' ""^'^ *^^ ^^^t «^ the shorter. Find thi
lengths of the arms. " ^"®
Let ^=length in inches of the shorter arm •
thenar+6= - .« » j » '
Now the lever will be in equilibrium, when the weiVht at
one end multipl ed by the length of the correspond i^Sm ?s
Xg InnT"' '' '''" "'' "^^'^^^^^ b^ it--reL
.-. 24a; =8 (x-\-e>),
24a; =8^+48,
16a;=48;
V f =3 -inches, the length of the shorter arm •
anda;+6=9 " " » « longer '^ *
Pkob 20. A weight of6 lbs. balances a weight of 24 lbs on
H lever (supposed to be without weight), whose lenSh is 20
.r^hes ; if 3 lbs. be added to each weight;, what addittn must
be made o each arm of the lever, so%hLt the fulcrum mav
Ed? '^'''"^ ^'"'""' ^"^ equilibrium Btm be ;i:
This problem resolves itself into two other problems :~.
:■..■ --.-.uru uic; i.:m,r,r,.=. ..r rr.. „„ j^ ^-^^ Original posi.
<ion
iCiii-tna OI iiitj an
Let a;=the length in inches of the shorter
JO— ar= " " " « 1
lonffer
long(
arm:
X.
u
r%
I Mi
!i
\l
B4:
ALGEBRA.
Now, in order that there may be equilibrium. 24ar and 6 (20
—a;) must be equal to each other ;
.•.24a; =120— 6a:,
30a;=120;
.*. x= 4, the length of the shortei arm;
and20— ar= 16, " " " longer "
(2.) To find the addition to be made to each arm, so that
there may again be equilibrium on the fulcrum in its original
position, after 3 lbs. have been added to each weight :
Let X — number of inches to be added to each ai m ;
then the lengths of the arms become 4+ a; and 16 -fa; inches
respectively : and the weights at the arms have been respect-
ively incre"=ed to 27 lbs, and 9 lbs.
But by i..e principle of the equilibrium of the lever, 27 (4
-f-a;) and 9 (16+a:.),must be equal to each- other;
.•.27(4+a;)=9(16-fa;).
Divide each side of the equation by 9, and
3(4-|-a:) = 16+a;,
12+3^"= 16 +a;,
3a;-a; = 16-12
2a;= 4; *
Prob. 21. The condition being the same as ih the lass
problem, how many inches must be added to the shorter arm
in order that the lever may in its original position retain its
equilibrium'? Ans. 1^ inch.
Prob. 22. A garrison of 1000 men were victualled fdr 30
days ; after 10 days it was reinforced, and then the provisions
were exhausted in 5 days ; find the number of men in the re-
inforcement. Ans. 3000.
Prob. 23. Two triangles have each a base of 20 feet, but
the altitude of one of them is 6 feet less than that of the other,
and the area of the greater triangle is twice that of the less.
Find their altitudes. Ans, 6 and 12.
N. B. The area of j^ triangle = ^ base X altitude.
D - -
A
_ _ 1 Ti
iinU JD
uc^aii. i,o I'iiij yviLii equaj auras ; n
won VZs. ; then 6 times A's money was equal to 9 times B's,
What had each at first? Ans. £S.
X and 6 (20
31 arm;
•m, so that
ts original
It:
3ach ai m ;
i-x in(;hea
m respect-
aver, 27 (4
I the lasr,
lorter arm
: retain its
[^ inch.
ed for 30
provisions
in the re-
. 3000.
) feet, but
the other,
f the less,
and 13.
de.
aUiria j A
times B's.
ns, £S^
SIMPLE EQUATIONS. g^
rnvT^hut^t ''^f ^ 'l"^^"g ^^^^^ reckoning at a tavern
' ., , wcUKs Ai miles an hour morp than \ rr^
many m.le. does A walk i„ an h .r?" 'its milS
DIVISION. -^
:atl„%*f ''""'= «>»• f°"»-tom.draWyfr„„ tl.at in Mul.ipli
If +«X+4=+„S. then +1*=+J, „„d ±±=+^
4 a X-b^-ab, - - - =1^ =_6. and =11* ^^a
— aX~6=+a5, -
— a
»■ *., /j/lrff eigna
• produce-f , and
unlike signs — ^.
■f -L
iltj
ButtaXLT"" '''""' "'"'""" o'lS"""^" 9»"ti,i«,( s,... th.
SQ
ALGEBRA.
In the divisor must be subtracted from its index in the dividend
to obtam its index in the quotient. Thus,
(1.) +abc divided by +ac - . or ±^ =4-5
4-ac
(2.) +Qiibc - . - - . -2a - - or
(3.) —lOxi/z . +5y - or
6abc
~2a
— lOxyz
+5y
= —36c.
=z—2xz.
(4.) ~20a'A3 4a^y or =|?.^' =+5aa:/.
Of Division, also, there are three Cases : the same as ia
Multiplication,
' Case 1.
37. When dividend and divisor are both simple terms.
Ex. 1.,
Divide 18aa;« by Sax,
IQax'-
=6a?.
Ex.2.
Divide 15a'b^ by -5a.
+ 15a''i«
3aa;
Ex.3.
Divide —28a^y^ by — 4zy.
Ex.5.
Divide — 14a^6'c by 7ac
-14a«6'c
— 5a
= -3a6«.
-h7ac
Ex.4.
Divide 25aV by —5a»c.
+25aV
— 5aV "~
Ex. G.
Divide — 20a;y2' by — 4y«.
~20a;'yV
— 4y2.
Case II.
38. When the dividend is a compound quantity, and the
divisor a simple one; then each term of the dividend must be
divided senaratelv. and iha -refinlfina- rmonfiViVo «,:n v^ i.i-_
quotient required.
88. State the rule for Case 2d.
DIVISION.
Ex 1.
Divide 4iia»+3a5-f-12a« by Sa
42a«4-3«A+12a»
87
3a
- = 14a + i+4a
Ex. 2.
Divide 90aV-18a^»+4a'^~2ax !jy 2a*.
2<.'a;
=45aa;'--9.r-i*i*«-|.
Ex 3.
Divide 4x*-^2x'-^2x by 2^:.
Ex. 4.
Divide — 24a«a:V— Sflrary-f-Gzy by ■
—24a^x^y—Saxy -f- Gary
— 3a:y ~~"
-aty.
Ex. 5.
Divide 14ai»+7a'6»-21a'i»-f.35a«5 by 7ab
Hob "—
I
11 \
Cask III.
titits' Tn^T• ^'"^'^^"i and divisor are io^A co^j.o.mc? quan.
titles In this case, the Rule is, «To arrange both dividend
and divisor according to the powers of the sime letter Wn
mng with the highest; then find how often the fim tirS
he divisor IS contained in the first term of the dividend and
place the result in the quotient; multiply each Irm of the
«h;; n^ andplacJe the p?oict unlTth'e'cor!
it fi-om the^'. 'to tt^ terms in the dividend, and then subtract
^c .^'^.r'fV' K *« the remainder bring down as manv f^rm»
V. .uu uiviaenu as wiii make its number of terms equal'to
AnVr ^''^''^""^ "°^ '^^^«°'- '^^^ both compound quantities, what u.
: 1 1
lAej
O }
oO
ALGEBRA.
that of the divisor; and then proceed as before, till all the
Ex. 1.
Divide a^-Sa'b+Sab'^b^ by a—b.
* *
In this Example, the dividend is arranged accordincr to the
Powersoft the first term of the divisor." HaX do'ne this
we proceed by the following steps :— ° '
(1.) a is contained in a\ a' times; put this in the
tient
quo-
(2.) Multiply a-i by a\ and it gives a'^a^'b.
JSJ^ Subtract a^^a^b from a«~3a'6, and the remainder is
(4.) Bring down the next term +3a5'.
(«.) JVM7/?>/y and sM^^rocjf as before, and the remaind
ab\
er IS
(7.) Bring down the last term —b\ '
JS.) a is contained in ab\ +5« times ; put tliis hi the qua
iicnt.
(9.) Multiplif and subtract as before, and nothing remains
PIVISION.
89
•■•f2tfH-«
la'
"(n^-Sa',
Ex.2.
}-6a<xf 1 00^-7*+ 1 0aV-|-5a««-fz
"* 3a V+ 7aV4- 5aa;*
3a V4- 6aV+3aa:*
* aV+2a?>a;»
aV+2aa;*+a;»
i43<i*'f<
Ex.3.
12x*-13x*-34x'4-39a;W+2x'-5x+--f —
12.i:»-21a;* \ 4c*-7a
--20^+35^
■i .. ,fl
Ex.4.
32;~6\6a:*— 90 /2x« + 4a;'+8a:+16
5\6a:*-9C /S
*4-12a;3-9G
4-12a;^-24x-^
* 4.24a;«-96
4.24a-''-48.t;
+48a;-9G
^ I
r >l
li^
" Wlion there ia a r«mam«fcr, it must be made the numerator of a
Fraction whose denominator is the divisor ; "this Fraction must then li6
placed in the quotient (with its proper sign), the same as in common
Arithmetic.
iO
ALGEBRA.
Ex.6.
''■''~')::T::i::-^+^^-'f--^+'' -.+1
* + ar'+ar-l
Ex.6.
'"^'felffs+^^^-^^^'-l^+l
4- x'^-ix
* *
Ex. 7. Dh ide a*+4a»5+6a=6«+4ad»+i* by a+b
Am, a'+3a»6+3a6'«+fi».
£;c. 8. Divide «''-5a^^+10aV~10aV+5aa:^-;i^
Am. o*— 2<w.4.;b".
Ex. 9. Divide 2bx*^x^^2^^^^ by 5^-4^.
^ws. 5ar»+4/+3^+2,
^ 10. i^>videa^+8a3;r+24aV+32a.:3^.ie^.j^
-4ws. a«+6a«ar+12aar«+S^.
DIVISION. d\
Ex. 11. Divide al^—a^ by a — x.
Ans. a*+a*a:+aV+aar*•^x*.
Ex. 12. Divide Gx*+9j^-20x by Sj^-Sx.
Ans, 2aJ'+2x+5-
5a?
3a;'-3jr.
Ex. 13. Divide 9x«-46.r»+95a^+150x by x'-4x-5.
Ans. 9.J:*— 10ar^+5.»''-30^.
Ex. 14. Divide «*-^a^+««+|x-2 by ^x-2.
Ans. |z='— ^:r»+l. ,,
PROBLEMS PRODUCING SIMPLE EQUATIONS, CONTAINING ONL"?
ONB UNKNOWN QUANTITY.
Prob. 1. A fish was caught, the tail of which weighed 9
lbs. ; his head weighed as much as his tail and half his body,
and his body weighed as much as his head and tail. What
did the fish weigh 1
Let 2ar= weight of the body in lbs. ;
.-. 9+ar=: weight of tail+| body = weight of head
But the body weighs as much as the head and tail j
.•.2a:=(9+ar)+9,
2a:=a:+18;
.•.a;=18,
and .-. 2a;=3r), the weight of body in lbs.,
9+a:=27, the weight of head in lbs.,
and the weight of fish=36-f 27+9=72 lbs.
l*ROB. 2. A servant agrssed to serve for £8 a year an<f a
livery, but left his serv ee at the end of 7 months, and receiver
only £2 13s. 4d and his livery ; what was its value?
Let 12j;=the value of the livery in d.
But £8= 1920c?., and £2 13s. 4rf.=040d ;
then, the wages for 12 months= 120?+ 1920 ;
,*. the wages for 1 month =
12
?-f-l()0.
md .'. the wages for 7 months = (a; +160) 7,
I
i
4*
42
ALGEBRA.
Cut the waces actually received for 7 months=12x.f640'
.M2a:+640=7a;4.1120;
.*. 5a?r=480,
«=96;
and .-. l2x=U52d.=M 16.., value of li^oiy.
From the solutions of the two preceding problems it will
1^0 seen, that by assuming/or the Lknown%lantTx will
A"mption' hTri^ ''' ?"^^ ^^"^'^"^^'^^ *« "^^ke .uch
m assumption, but a more elegant solution is ei^noraU^r
thereby obtained Tl,e coefficient of . must b^a^L'S
proWem T'"'''' '' '" '^' '^^^^^^^^ ^"^^^^-^ ^nX
Prod 3. A cistern is filled. in 20 minutes by 3 pipes one
thJ Jh TT' ^^ ?"""^^« "'^^^' ^"d -»«ther 5 tnons less
than the third ;,.r minute. The cistern holds 820 rilons
How much flows through each pipe in a minute 1 ^
Ans. 22, 7, and 12 gallons, respectively.
>^. finds hLseif I160 inSr^^rdti iT:^^^
Ans. £100.
Prod 5. A met two beggars,'B and C, and having a certain
sum m his pocket, gave B A of it and C i nf Vhl ^ cemm
A now hflrl 9ft./ iLa k Ju J , ' ^^ » ^* ^"® remainder :
A now had 20(1. left ; what had he at first ? Ans. 5s.
£i^^7f\t \^rT ^^ ^V *'^^'^'' *>^^ '^ saddle worth
£0^: If the saddle be put on the first horse, his value wiJl b«v
come double that of the second ; but if it be put on thelecond:
lZuf7 ^;\^^^T^ ^"P^'^ *^^^ «f th« first, wtat is tl^
value of each horse? ^„,. ^g^ and m
PRon. 7. A gamester at one sitting lost J of his mm.Av
and then won 18.. ; at a second he lost i of the remS'
and then won 8.., after which he had 3 guineas left How
much money had he at first ? ^ ^^
Let 15-1! = t}lO rillTy»>»ow /->r «U!li: 1_ _ I 1 _ . -
having lost j of hismoneyihe had Tof iro"l2r,^m':i'iS-'
he then won 18... and therefore had 12x+18 in handTS^
i
ALGEBRAIC FRACTIONS. /l.t^
.•.(8a;+12)+3=C3,
8a;+13=:60;
/. 8a; =48,
ar=: 6;
Hence I5x=90s.=£i 10s,
CHAPTER III.
ON ALQEBRAIO FRACTIONS.
o,i?i; ^""^ K"les for the management of Algebraic Fractions
Tt^ TT ^' '^'? ^" ^'^^^^^^ arithmetic.^ Th^priSes
?oIIowtg:!!lf '"^" " '^^' "^^"^^^ -^ established aTtS
(1.) If the numerator of a fraction be multiplied or th^ ^^
nominator divided by any quantity, the frac fon t Tend^^^^^^
so many times greater in value. rendered
(2.) If the numerator of a fraction be divided or the d«
hJtlu- ^? t^Q^nV?^?^^o»' and ^denominator of a fraction
arc the foundation of tlTScit K. Hoicar'''"'' ''*' ^''""'P'"* ^^^*^''
i
44
ALGEBBA.
ON THE REDUCTION OF FRACTIONS.
41. To reduce a mixed Quantity to an improper Fraction,
Rtle. "Multiply the integral part by the denominator -f
Uie fraction, and to i\iQ product annex the numerator with iti
proper sign; under this sum place the former denominator,
luia Uie result is the improper fraction required."
Ex.1.
2x
Reduce 3a+— , to an improper fraction.
The integral part X the denominator of the fraction + the
»wwrator=3aX5a»+2a:=15a'-f-2ar-
„ 15a»4-2a;. ,
Hence, — ~- is the fraction requu-ed. .
Ex.2.
Ax
Reduce hx— ^^ to an improper fraction.
Here 5j:X6a'=30a«^; to this add the numerator with its
proper sign, viz., -4x; then ?2f!^±' ig the fraction re-
quired.
Ex.3.
Reduce 5a; —-1. to an improper fraction.
Here 5a;X7=35ar. In adding the numerator 2x-3
with Its proper sign, it is to be recollected, that the sign ~
affixed to the fraction —^ means that the whole of that
fraction is to be subtracted, and consequently that the signs
of each term of the numerator must be changed when it is
combined with 35ar; hence the improper fraction required is
ooa; — 2.g-{-3 33.P -f- 3
41. How U A mixed quantity reduced to an improper fraction I
ALGEBRAIC FRACTIONS.
45
Ex. 4. Reduce 4a6+-- to an improper fraction.
12a'6-f-2c
Ans,
8a
Ex. 5. Reduce 36'— — to an improper fraction.
156»ar-4a
^7W.
Ex. G. Reduce a~a;H -— to an improper fraction.
X
Ana.
5z
!.ion
Am n
Ex. 7. Reduce 3a;' rrr- to an improper fraction.
X
10
.^n^.
30a;'— 4ar4-9
10
42. To rerfwcc an *ot -r Fraction to a mixed Quantity.
Rule. « Observe MHu^^^h terms of the numerator are divisi-
ble by the denominator without a remainder, the quotient
will give the integral part; to this annex (with their proper
signs) the remaining terms of the numerator with the denom-
inator under them, and the result will be the mixed quantity
required." ^ -^
Ex. 1.
^ a'-^ab-^b*
i\eauce to a mixed quantity.
TT a^-^ab
^^^ — a — """J"* ^^ ^^® integral part,
and — is the fractional part;
.*. a-f-5-f J. is the mixed quantity required.
48. W hat is tho rule for reducing nn improper fhwtion to a miied (^ua^
m
td
ALGEBRA.
■ El. 2.
Keduce ^ to a mixed quantity.
15a»
'6a ^ ^® '^® ^^^^S'^^'^ part,
, 2a:-3c . ,
ana — = — is thejraciional part;
5a
2«-3c .
.-. ^H -^ jg the mixed quantity required.
, Ex. 3. Reduce — ^_ to a mixed quantity.
Ans. 2x—
Ex. 4. Reduce 'Jf±^^ ^ a mixed quantity.
5a
2ir*
4a
Ans. 3a+l-_?i.
4a
^«s. 10y+3j;-~.
ar
43, To reduce Fractions to a common Denominator,
Rule. " Multiply each numerator into every denominator
tlZ^rt '^' "'^ """aerators, and all JZSfr
fof/ether for the common denominator."
Ex. 1.
p ■, 2.r 5a; 4a
neauce — , -, and — , to a common denominator.
f^rv-^iv;— TK- f rtence the free
5^X^X5- /5a; f, new numerators ; I tions required are
r Hence the free
4^X3X^1^* \ "''' """^^''^^^^^ ; J tions required are
4aXrfX6-12a&) S jp^^ ^5^ j^^^^
5 X 0X5=156 common denominator; [ isT' \Kh> YbF
48. How aro fractions reduced to a common denominator f
ALGEBRAIC FRACTIO^^S.
Ex.2.
tveauce —g— , and — , to a common denominator.
47
Hore(2.p.f 1)X4= 8.c+4 ) new nume.
^^^^~}2:^___S rators;
5X4=20 common denomi-
nator ;
Ex.3.
'Hence the fiac-
tions required
are
— ^TTT, and -— — .
20 ' 20
Reduce — j— , ~^ and — , to a common denominator.
Here 5a;X Sx2a:=30j:» > .*. the new frac 30ar*
• '.1
{a^x) (a+x)x2x=2a'x-2x^ tions are 6^*6^
1 X(a+:r)x 3 =Sa +Sx ha'x-2.^ , 3a4 3.r
(a-i-x) X3 X2x ==Qax +Gx^j QaTTO?^ ^"^ CllJ+e?'
Ex. 4. Reduce — , g^,and -^, to a common denominator.
"":'' & l5^' ^"^ 157-
Ex. 5. Reduce --^, and — g~ to a common denommator.
Ar.j. —- — , and
3a:
3a;
Ex. 0.
4a;'-|~2a; 3a;' 2a;
Reduce — - — , — , and -, to a common denominator.
J 4Sabx^-{-24abx 45ia;» , 40aa;
C0a6 ' mab ' ^'"^ QQab'
Ex.7,
o , 7.r'~l , 4i:*— ar+2
Ze
aa"
An, '±'.f=K .ni?^Zl-:+*^
4a*x
, and
4a'at
I :
18
ALOEBKA.
44. To reduce a fraction to its lowest terms.
Rule. " Observe what quantity will divide all the terms
both of the numerator and denominator without a remainder;
Divide them by this quantity, and the fraction is reduced to
^ts lowest terms."
Ex. 1.
Reduce
14a;»+7 q ar-f21a ;»
to its lowest terms.
The coefficient of every term of the numerator and denorai-
nator of this fraction is divisible by 7, and the letter x also
enters into every term ; therefore 7x will divide both numa-
rator and denommator without a remainder.
Now
14a:»4-7aar+21a;«
iX
=2a;«+a+ai?,
and ---=5a;;
7x *
Hence, the fraction in its lowest terms is ^^'"^-JJf.
5x
Reduce
Ex.2.
20a5c~5a-+10 ac
5a^c
to its lowest term.
Here the quantity which divides both numerator and de-
nominator without a remainder is 5a ; the fraction therefore
in its lowest terms is — ^~"+ f
ac
Ex. 3.
Reduce — — -. to its lov/est terms.
(r—o*
Here a—b will divide both numerator and dcnominatoi,
for by Ex. 2 Case III. page 27. a^—b''=(a-{b){a—b)i
hence — -— • is the fraction in its lowest terms.
tt"f~0
Ex. 4. Reduco -— to its lowest terms.
2jr
Ans. — ,
44. Show how fractions are reduced to their lowest tenon.
\
ALOEBKAIO FKACTIONS.
49
Ex. 5. Reduce — -^ to its lovrest terms.
Ans. ~.
(it
Fx 6. Reduce ^ -^ ^ . to its lowest terms.
2y--3jry
Ans.
X
Ex 7. Reduce r^-^— to its lowest terms;
Vtx^
Ans.
3j:«--ar4-2
a^
ON THE ADDITION, SUBTRACTION. MULTIPLICATION, AND
DIVISION, OF FRACTIONS.
45. To add Fractions together.
Rule. "Reduce the fractions to p. common denominator
and then add their numerators together; brin<T the resulo
mg fraction to its lowest terms, and it wiU be the sum re-
quired."
Ex. 1.
"5' T' 3' ^^g^t^er.
2a;X5x3=30a; I 63a:4-30a:-|-35aj 128^
icX5x7=35a; ( •*• jQg = -y^ is the fraction
5X7X3=105J "^ required.
Jix. ^. Add -, — , ana — , together.
«X34x4a=12a''61
2aX6X4a= 8a«6
56X6X36=156'
.-. lSa^^+8«'6-fl568 __ 20a«6+15^>'
12a6» ~ 12«6^-"
6x36x4a=i2a6'J = (dividing by 6) ^^^^f is the
sum required.
46. State the rule for addiiiji fraction«i.
i
lis
■
i;
'I
50
ALGEBRA.
Ex.3.Add?^,?^i,
5 ' 2x '
i2x-\-3)x2xx7=2Sx' +42z
(3a;^l)x5 X7=105a;~35
4arx5x2ar=r40a;^
5x2a;x7=70a:
4x
and — , together.
^ 28a;»-f42g-fl0 5ar— 85-i»40r»
70a-
68a:''+147ar-35 ,
70a?
s the
Ex. 4. Add --, ~, and ~, together.
Ans.
ir-x. o. Add 2^' "5' ^^*^ 7"» together.
Ans.
sum required.
934a;
693*
105a'-f28a'6+3C6«
Ex. C. Add ?^i lf+?, and |, together.
70a6
Ans,
Ex. 7. Add
5a«+6
169a;+ 77
105 •
4««4-2^
Ans.
36 ' *^"^ "sT"' ^^g^^h®''-
SToM-m
156" '
Ex. 8 Add -1^, and — , together.
. 4«»— 7a?— 3
Atis, -
Etj, i>. Add --— , and — — together
6«
Ans.
Ex. 10. Add — -.. and — r- . v»o-PtliAr
2a?«
tt—
a + o
^n*.
^M-26«
a»-6»
ALGEBRAIC FK ACTIONS.
5J
40. To Subtract Fractional Quan.ities.
RiTLE. "Reduce the fractions to a common denominator •
and then subtract the numerators from each other, and uudw
the difference write the common denominator."
3xXl5=45a;
UxX 5=70a r
5 Xi5~75
Ex. 1.
Subtract ^ from — .
5 15
•*• 75 — if^ ~ 3 ^^ ^^® difference
required.
Ex.2.
Subtract ?f±3 from ^"^^^
3 7
(2.r+nx7=14ar+7) 15ar+6--14.r-7 ar-^1
(5a?+2)x3z=15ar+6 \ •*• ^l ~" = "sf ^^ ^^
3x7=21 ) fraction required.
Ex. 3.
From i5|r2 subtract ^1,
(10u:-9)x7=70a;-63 ) 70x-63-24a:+40 4Gr-^3
(.3^5[x8=24^-40[ .-. -^-±-=:15l^
" ' "^ '" is the fraction required.
Ex. 4.
8 X7=56
From r subtract .
a—b a+b
(a-i) (a+6)=a«~"65 J ^^ j^ ^j^e fraction required.
(
Ex. .5. Subtranft If frnm ??
5 10'
"'^•lo-
4«. Give the rule for subtracting fractious.
>n
vt:
il
62
ALGEBRA.
' I
Ex. 6. Subtract 5^ from ^i^
fix. 7. Subtract --±1 from If
x+l 5*
Ex." 8. Subtract ~-=:3 from 1^
Ex. 9. Subtract • — - from ^
a-\-b a~b'
Ex.10. Subtract --Hl from -
8 7*
^,«. lE^iZ
Ans.
Ans.
28
5ar-{-5
4x'+3
l_l£+49
66 •
8
* 47. To Multiply Fractional Quantities.
Rule "Multiply their numerators together for a nrw
numerator, and tW denominators together for a new d^
L'rmsT' '"' '^'"^' *'^ ^""^^^"S ^-<^^-- to its Wt"
Ex. 1.
Multiply ^ by 1^.
7 x9 =63 f •*• the fraction requireu is --
' ^63
Ex.2.
Multiply 1^+1 b/'
3
Here
C4«-f-l)x6«=24z»-|-6
and
3x7 =21
r 24ar«-f Gar ,
la; I * * 21 — ~ ^dividing the nu
1 JTierator and denominator bv 3''
arM-2ar * '
ij; — IS the fraction required.
(«
47. State the rule for the multiplication of fractio:,
M,
ALQEbRAIC FRACTIONS. 53
Ex. 3.
00 " a-\-b
By Ex. 2. Case III. page 27, (a^-i»)x3a'=(a-t 5)
(a-i)x3a«; hence the product is 3a'X (a+^)(a -j)^_
56x(a+6)
(dividing the numeratorand denominator by a+6) ^^'^C^ -^)
3a»-3a»6 ^^
56
Ex.4.
Multiply 54^%y '^«
14
Here
(3a:«-5a:) X 7a=21aa;'-35aa;
and
(2««-3a:) X 14=28a:»-42.r
2a;''-3ar*
^ 21a.r«-35aar ,
the numerator and denomi-
nator by 7x) ^=| is th.
J fraction required.
2x
Sx
Ex 5 Multiply -5. by ~
Ex. 6. Multiply ?^i:f by ^^
2a:*-4aj*
Ex. 7. Multiply ~ by —1'.
Ex. 8. Multiply --5fL by ^^^-30
2a:
Ans.
Ans.
Ans,
Ans.
Ix-^'
3ar-~l
ar-2"
4
2'
I-'
^
48. 0/j the Dlvmon of Fractions.
tion "^'''* "'^'*^^^^ ^"® divisor, and proceed as in Multiplira
48. Enunciate the rule for division of iVact^ona.
»jira )ll>fca wiLMWtoi i*»w» M ai nwn«M 'i i i ;/.;:*'TT^
■--'—y jwjtwi ■-
M
ALGEBRA.
II
Ex. 1.
Divide — b/ -.
Inter t the divisjr, and it becomes ~ ; hence — x —
- JqJ ~ "s ^* ^'^'"^ ^"® numerator and denominator by Qx)
s tlie fraction required.
Ex.2.
Divide ^^^ by ^^^-^
14ar-3 25'
X
6
5 -^ 25 •
(143: -3)x5 _ 703?- 15
lOr -4 10^-4 ~ 10x^=^4*
Ex. 3.
Divide ^^' by lli±.
3a^-5&«_ 5 X (a+&)(«--^) f. 5x(a-f-^^)(a~&) ^.
2a ~ 2a
4a-4-46_ 4x(a + ^)
66 " 66 '
66
JS
Ex. 4. Divide ^ by ^.
2a ^4x(a+6)
i _3 06x(g-6) 15a6--15^" .
~" 8a "^ 4a
the fraction required.
\
^'^^- 6l-
Ex. 5. Divide ~±2 by ?^i.
o 5a;
Ex. 0. Divide ~ by ^.
.4n5.
^n«.
10.g
3 •
4ar--12
5 "'
_ _ _ 9r«— 3rr rs*
fix. /. jnvide — :; — by — .
Ans,
n« o
SIMPLE EQUATIONS.
6u-
ON TIIK SOLUTIOif OF SIMPLE EQUATIONS, CONTAININO ONLf
ONE UNKNOWN QUANTITV.
Rule III.
Inc?!;.^"-!*^"^..^'?" "'^>' ^^ ^^^^^^^ «^ fractions by mullinlv..
SsTnVutlt^ ^^"^'^" '- ^'^ denominators^of thS
eac^'Llde Tth«''" ""? ^e cloared of fractions by multiplying^
s^iititf trSiot^^^ '^" ^^^'^-^'^ -/.>/.y thf :
This Rule U derived from the axiom rd\ fi,of ,•<• i
2"rjf^^ "^"^^!p"^<i by thetr^j^;.* ^i^ :s
quantities), the products arising wUl be e laal ^ ^ ^ ^
Ex. 1. Let 1=6.
Multipli^ each side of the equation by J>, then (since tha-
multiplication of the fraction | by 3 just takes away the de.
nominator and leaves x for the product) we have
ar=6x3=18.
Ex. 2. Let |4-|=7.
Multiplj^ each side of the equation by 2, and we have
^Ag,m,«„/.>/yeach^,HerfJhi, equation by 5.^ „.
7a:=:70';
Var=10.
Ex.3. Let^ + -=i3__f
Multiply each side by 2, then a:4.- =2G-??
3 4
Mullinlv oanh oJ/la !»». O j c!._ . ^ __
6^
4'
Multiply each side by 4, and 12x-\-8x=zm^^(kc.
S8
ALGEBRA.
By traiispt iition, 12j;-f 8a;+-G.r=::312,
26a:=312;
.-. ar= 12.
Tliis example might have been solved more simply, Ly mul
rL'])l3ing each side of the equation by the least common multi
pie of the numbers 2, 3, 4, which is 12.
Multiply each side by 12, -7; — |-
2
:156-1?A
or, Qx-^4x=zl56—Sx.
By transposition, 0a;-f-4a:4-3a?=156,
* 13x=156;
,\x= 12.
Ex.
4.
Let
2x X
3 "^4"
:22.
Ex.
5.
Let
7x
4"
5x
a'
55
-6'
Ex.0. Let ^.+5=31-1.
<« «> o
Ex. 7. Let ^-~+f =44.
5 () 2
Ans. a: =24.
Ans. ar=:10.
Ans. x=SO.
Ans. ar=GO.
60. In the application of the Rules to the solution of simple
i.juations in geocral containing only one unknown quantity, il
will be proper to observe the following method.
(!) To clear thj equation of fractions by Rule III.
(2.) To . ol'ect the unknown quantities on one side of the
-Cfjuullon, fjid ike known on tl e other, by Uule II.
(3.) j'o find the value of the unknown quantity by di-
viding TAcU g/iie of the equation by its coefficient, lis in
Rule I.
60 F.) I A -' ■: tie thrco Btups Vr which B'mpie equation containing
only oivj •.'•-'/> -vn iiaantity rany be BolVoU.
SIMPLE EQUATIONS. 57
Ex. 1.
l^uvi 'Jie value of a: in the equation — - l=f-|-l?.
7 5 5'
Multiply by 7, then 3^-fV=-+-.
Multiply by 5, then 15j;4-35=7ar-j-91.
'X)llect the unknown quantities on )
one side, and the known on. the [ 15.r— 7;i:=9I— 35
other; 1 '^'
or 8z:i=5G.
Divide by the coefficient of x, ar3=— — 7
' 8 ~ *
Ex. 2.
Find the value of a: in the equation ^i?- 1=:2— -
5 7*
Multiply by 5, then ar+ 3- 5 = 10-—-
7 *
Multiply by 7, then 7x4-21 ~25=70-5ar.
Lollect the unknown, quantities )
on ont side, and the known > 7ar-f-5ar=70— 21-f-35.
'.•n the other ; \
orl2ar=84;
••''■-l2~^-
Ex. 3.
Find the value of x in the equation
x—\
2:e~2
~5
-24.
Multiply by the least) .^ r , ^ ,^
(vmmon multiple (10), f 40«-5ar+5=10.r4-4.r-4+240.
Uy transposition, AQx-^bx—lOx— 4a'=240— 4— 6.
or40a;-19.r=231,
i. e. 21a:=231 ;
231 ,,
IWA
As the Jirst step in this Example involves the case " where
the sign - stands before a fraction," when the numerator of
./ -M^^mmm^
^
ALGEBRA.
!
that fraction is brought down into the same line with 40ar the
signs of both its terms must be changed, for the reasons as.
Signed m Ex 3, page 44; and we therefore mai«e it --5.r-f 5
and not 5af— 5. '
Ex. 4.
Find the value of a: in the equation 2a?— - -f 1 =5ar~2.
Multiply by 2, then 4ar--a;+2= 10a?— 4.
By transposition, 4+2=10a:--4a;-f ar,
or 6= 7x;
6
6
t or x=-.
Ex. 5.
What is the value of a; in the equation 3az-{-2bx=Sc+a'i
Here Sax-{-2bx={Sa+2b)xx',
.•.{Sa+2b)xx=:Sc+a.
Divide each side of the equation by Sa+2b, which is iha
coefficient of a:; then a;==?^il^
'Sa-i-2b'
Ex. 6.
Find the value of a? in the equation Sbx+a=2ax+4f,
Bring the unknown quantities to one side of the equati(A. ui/^
the kno%on to the other ; then,
36a;-— 2aa;=4c--a; '
but 36a;— 2aar=(36--2a) Xar ;
.'. (36-2a)a;=4c-a.
Divide by 36-2a, and «=|^.
Ex.7.
Find the value of a; in the equation 6a;+.r=2a;+8a
Transjose 2a;, then bxA^x 2a?=i3a,
or bx— a;=:3a ;
but 6a?— a;=(6— l);r;
. (6—1^ a;=3a.
and x-rz
m*
w^tmmmmi^m-'r'nk
ssasfi
Ex. 8 a:4-4.-~ii
SIMPLE EQUATIOXS. ^j^
Ans. xz=.Q.
" 5^4^3 2^ •
^ 2^3 4~2'
-E:x.\12. 3^+i *+5
9~"~F'*
— Ex. ^3. -f«5=:,2^_2^
- Ex.|l4. C^-^-.9=.5:r.
"Ex.15. 2x~^^+i5^1?i±20
3 6
-^ Ex.lG. ^Z:?+^-2C-^-^
2 ^S"-"^^ ~F"-
7
-4?w. ar=rl4.
-4««, ar=3G.
Ans. «=12
-4n*. a?=:18.
2ar--l
Ex. 17. 5^~ffll^4.1-3^.fi:2.^ .
3 ^^--«>*i ^ — f-7. Am. z=8.
Hoc. 18. 2az-{-b='6cxi-ia,
Am. z=^.
Ex. 19.
Multiply by 15, 45x-.60-2S^4'm^7oj_A^ a
45ar-28ar^4r=72l4+ 60-30
13i;=:92; '
";
t^
""^B^w^!*".
eo
ALGEBRA.
Ex. 20.
4«-y 7a;-29_8ar+10
18
; fmdx.
126ar~522
Multiply by 5;r-12, 126:^- 522=65.^-150
126a?-65;r=522-150!
61«=366;
.-. ar=6.
* Ex. 21.
Given ~? L - 1
«~1 a:-|-7"~7"(«-l)' *<> fi»«* *
Mult, by 7(x-l), 7-.lifei:l)^i^
»* "f* 7
6=:iiferJ)
a;-H7 •
Divide by 2, sJltnl}
3ar4.21=7:c-7,
7«— 3ar=2l4-7,
4aJ=28 ;
.-. a?=7.
Ex.22.
Let ?^i4-2ir?-.I«^+J5 . 2J • ^
14 ^6«+2~' 28 r-^;find«
Mult by 83, ,6.+ 10+l^r|f=i«,+ ,5^„]
196;r— 84
-==14
196ar--84=84ar4.28. *
~^-\
V.
V.
0,
the
iJgfflSWft*-
SIMPLE EQUATIONS.
63
V.
4 3 -^i 9~--
Ex. 24. ?f±?2_l^-12, ar
36 ~5^34^-4'
Ex. 25. ?2^±^ . 5f+20__4.r 80 '
25 ■^9jr-16~"5'^25*
V
Ant. 0.
^n*. 8.
An9, 4.
Ex, 26. ?^S_i2fr5 , f _7« ^+16
9 17a;-32"*"3~12 36~* ^"*- ^•
Ex.27. 4(5.-3)-G4(3~.)->3(12.~4)=96. Ans. 0.
Ex.28.10(.-f|)^0.(l^j).23. ^,,^
Ex. 29. -?±?f4.5£±?f_,.^ 48
^««. 3.
t:H-3
PROBLEMS.
Let a;=:the number required •
then ;.+ loathe number, Wth 10 added to it.
Now Jths of (ar+10)==?(a:^-10)=l(f±i£)_3£^-30,
But, by the question, Jths of (ar+ 10) =66 • ^
• Hence, 5f+22^ee.'
Multiply by 5, then 3af+30=330 ;
.•.3^=330-30=300; or.r=?^=ioo.
the quotient shall be 20? ^ ' ^ ""^^ ''''' ^'^'^^'^ V »
Let ar=the number required ;
tjien 6^=the number multiplied hv fi.
Ox-t-io=ihe product increased by'is/
6.r-}-18 , •'^ °»
ana — = — -ih^it sum divided bv 9
^
9
6
:
1!
J
62
ALGEBRA.
Hence, by the qu^ ion, ^i+l^^g^^
Multip]j^by9,then6:r+18=:l80
1 ROD. 3. A no«!<- io i*k • 1 6 ~" *
^et out of tho C fe tSf ' ^^^j^ -ate,^ and 13
I^et .^.length ofThe1>o:t": j.^^^ ^^^^« P-t ?
then |=the part ofit in the earth.
3ar
y=^the part ofit in the water,
Butparur'ii^nr'^'""'''^^^^^^*^^-
whole post ; ' ^ ^ ■ ' " ^al'^ + part out of water =
I -^
(t) '•
^■2 s= X.
Multiply b.y5, then. + '•-_, 65^5^.
Multiply by7, then 7^+ ,5^1:455^35;
Prob 4 Afv . 13 -"^^ ^^"gtli of post in ft.
had^«- f V ^fte^ paying away kh and ith nf
ftad i^5 left ,n my purse. WhJ rnnnlv^v-^^ r "'^ ^«««>^ »
Let ^^.>oney in purse a? fim ; ^ '' ^'^'^ ^
then j+-=money paid away.
(4+7J = 85,
Multiply by 4, then 4z~.x-^^S40'
Multiply by 7, then 28.-7.^41=2380.
.-. 17.=2380:
X
■=27.
, and 13
t?
^aterss
iinft.
ney, \
?
nirg;
SIMPLE EQUATIONS.
98
^rol^ll^fZsl^^^^ «<Jd 20, and
suDtract 1^, the reraaaider shall be 10 ?
Ith'^W ^"^^ """'"'^ " "'»' ^h"^ id part cxceeda U,
^«*. 540,
thi« difference divided bvTttn.-^''"^!^^^ ^^^
arc the numbers? ^ ' ' ^""'^'^"^ 7^^^ ^« «. What
4»ff. 21 and 16.
mainder will be 5. What a,! thenJllt ^'''""' ""^ ^"
p ^w*. 35 and 41
the sto„4Jl!;Ztll\S ^ '^' """""■■ ^""^""^ "'
P„„ > , » ^"''- •*"• ^'*= "'"J 13-
mcome. t. l oi it ^^m« ij^s. p,e(|uired his
^/<*. £150.
Prob. 12. A gamester at one aittina lost Ifh nf k.-c
guineas left. wStr„e^ fa'^R fit,' ""'^'^t: '^^ '
lo the same quantity 7 ' "''•'' "" ''« «q«»l
^ •'^' -^'"- 18, 23, 10, 40.
»/«TpeHb^bT^t"^^^*" Jy■."f.^«^ «~...<'- worth
-u™ he take ,» fi.m a oheVtV To4 ^:.^^CZS,
Ana, 33 at IS*. M.
1
''1 at 9*. 6ci
i#
i I
!i:i
64
ALGEBKA.
<;
((
((
1 uoB. 15. Three persons, A, B, and C, can separately reao
ft i.eJd of com m 4,8, and 12 days respectively, lii how
luanj days can they conjointly reap the field ?
I [.et ;r = No. of days required by them to reap the field;
then It 1 represent the work, or the reaping of the field,
■J^=the part reaped by A in 1 day.
j_ « ;; « •' g ^»
T^= " " " C
.'.i+l-\-j'^=z « « « all three
But the part reaped by all three in 1 day multiplied by the
number of days they took to reap the field, is equal to the
whole work, or 1 ;
•••(i+i+A)^=l:
Clearing of fractions by multiplying by 24,
(6+3+3) a;=24,
llar=24;
i .'. x=2^^ days.
Prob. 16. a man and his wife usually drank a cask of
*»eer in 10 days, but when the man was absent it lasted the
wife 30 days ; how long would the man alone take to drink
'^' Ans. 15 days.
Prob. 17. A cistern has 3 pipes, two of which will fill it in
3 and 4 hours respectively, and the third will empty it in C
hours ; in what time will the cistern be full, if they be all set
a-running at once? jins. 2h. 24m.
Prob. 18. A person bought oranges at 20c?. per dozen ; if
he had bought 6 more for the same money, they would have
cost 4c/. a dozen less. How many did he buy ?
Let ar=rthe number of oranges ;
thena;+6= « " " " at 4d less per dozen.
Price of each orange in lstcase=fft=4d'.
and « " *' « « 2d " =11 =k
,*, the cost of the oranges =-
5x
3*
■r
But we have also
inv cifSi ui tne orauaes=* ia;-f-0}.
Two independent vrlues have therefore been obtained foi
9
f\\
SIMPLE EQUATIONS.
65
the cost of the oranges ; these values must necessarily be coual
to each other ; •' ^
to each other ;
5^
.••3=1(^+6).
Multiplying each side of the equation by 3,
p ... •'• «=24, the No. of oranges.
rROB. 19. A market-woman bought a certain number of
apples at two a penny, and as many at three a penny, and
sold them at the rate of five for twopence ; after which she
found that instead of making her money again as she expected,
she lost fourpence by the whole business. How much money
had she laid out] ^ Ans.Ss.4d''
Prob. 20. A person rows from Cambridge to Ely a di«^
tance of 20 miles, and back again, in 10 hours, the stream
Howmg uniformly in the same direction all the time : and he
mds that he can row 2 miles against the stream in the same
time that he rows 3 with it. Find the time of his ffoin<r and
returning. *' '°
Let 3a;=:No. of miles rowed per hour with the stream •
,'.2x=== " « « « « « against " '
^ow the distance divided by the rate per hour gives the time ;
**' ¥x~ of hours in going down the river,
3x
J 20
and — - =
2x
(C
((
It
u
coming up
((
But the whole time in going and returning is 10 hours •
20 20
"Fx'^2:v^'
Dividingby 10, .-^+1=1.
r>.5 X
Multiplying each term of the equation by 3x,
2+3=3^;
.'.x=-=ll
and .-. 32- =5, miles per hour down
- .t .. ._ . . 2^
,-. ine :ime m gomg down rhe river=:— =4 ^ours
5
ne of returnin
on
wquciitly the
0*
: 10-4 =6 hours.
0li
4
' . ■ 'Wl^^ «?E»«^
m
ALGEBRA.
Prod. 21. A lady bought a hive of bees, and found that the
pr,ce came to 2.. more than |-ths and Jth W the price fI^^
Ans. £2.
Prob. 22. A hare, 50 leaps U^c<. .. .oyhound takes 4
eaps for the greyhound's 3 ; but W.., of tUe grVho^id's leaps
are equal to three of the k. .'.. How mafy iapTwIl ?he
greyhound take to catch the hare ? ^
Let X be the No. of ieapn taken by the greyhound ;
then - will be the corresponding number taJ, a by the hare.
Let 1 represent the space covered by the hare hi 1 leap;
then - « 4i «4 u u , ,
2 " . " greyhound «
.4a: 4x ,
. . g X 1 or y wil' be the whole space passed over by the
hare before she is take. ; and :. x | or | .vill be the space
passed over in the corresponding time by the erevhound
Now, by tho problem, the difference betwo'n ?l7spaces
Srps^;^"'' ^^" '^ the greyhound and harl LTx"
.3a: 4x _
• ' 2 - 3 =^^^'
Oar— 8a;=:300;
.'. ar=300 leaps.
ON THE SOLUTION OF SIMPLE EQ. ^ONb, C0NTAI^.VO TWO
OR MORE UNKNOWN QUANTITIES.
51. For the solution of equr 'om. coutaining tw , or more
u known quantities, as many independent equations arrrl
quired as there are unknown quanUties. The two It;.m
necessary for the solution of the case when t^vn „nl n
«es are concerned, may be ^sI^V ^ ^ ^"^
ax-{-by:szc
a'x-\-b'y=:c\
Where a, 6, c, a\ b\ c>, represent known quantities, and z, y,
SIMPLE EQUATIONS.
67
thQ unknown quantities whose values are to be found in terms
ot these known quantities.
There are three different methods by which the value of
fMic of the unknown quantities may be deterniin d.
J^
ml
FIRST METHOD.
^i*^'^ t^e value of one of the unknown quantities in tenns
ol the other, and the known quantities by the ruJ. s already
given. Find the value of the same unknown quantity from
the sofond equation.
Put these two values equal to each other ; and we shall
then have a simple equation, con^Mning only one unknown
quantity, an hich may be solved as before
•gx [■ to find X and y.
a)
Ex. 1. Given x-\-y=S - ... (1)
x—yz=4: - -
From (1) y=S~x - -
" (2) y=.-4
Putt'ng these two values equal to each other, we get
ar~4=.8--ar,
2^=12;
X=z:6.
By ( y=8-a:='8--6=2.
Ex.2. Let 4y=.16 (1)
4. , y~M (2)
From equation (1), v have a;=16— 4y.
(2) « «« 34— y
16
in
((
u
« « XT-'
Hence by the rulo^ ^ =10 -'4y,
34-y=6 -16y,
15y=30j
. . y=2.
It has already bee shown that ar=:I«-_4y=/'since ?/ -2'
and .-. 4y=8) 16-8 8. ^ ^ '
/d
'/
51. For tlio soiution of equations containing
titios, how uany independent equations arc
niHliod ol'auiution.
•- more unkno 1 qaan
S^^asawr**-
1
Q8
Ex. 3I
Ex.4.!
ALGEDRA.
\
5z+3y=38 )
4x— y=10J"
2^-~3v = — 1 )
3x-2y=6 f
Ans. 1^=4
SECOND METHOD.
1^1
j From either of the equations find liie value of one of (h,.
unknown quantities in terms of the other and the known quau.
Jties, and for the same unknt wn quantity substitute this value
(n the other equatior, and there will arise an equation which
Jontams only one unknowi quantity. This equation can be
solved by the rules already laid down,
I Ex. 1. i/-'X=2 . -
' From (1) y=2+x. (a)
This value of ?/ being substituted in (2), eivet
2a!r=6;
And by (a) y=2+a;=24-3=5-
Ex. 2. x+2 , ^
-3-+8y-31 (1)
~ + 10a:=192 (2)
Gearing equation (1) of fractions,
a;+2+24y=93, or a;-f-24y=91
Uearing equation (2) effractions,
y+54-40a;=768, or i/-\-40x=z'7QZ
From (a) ar=91--24y.
Substitute this value of a:, according to the rule in equation
^3)1 and ^
y+40(91-.24y)=763,
or, y+3640-960y=:763;
.-. 959y=:3640~763=2877,
and y=3.
By referring to equation (a) we have «= 91 -24y=f since
'=3; and .-.24^=72) 91 -72=19. !/ K ^
Enunciate the second method of solution.
■mtm^m^x^
SIMPLE EQUATIONS.
Ex.8. 4:j;4.3y-_3i
Sar+Sy
=3!)
=22 f
6»
Ans,
=4
5
THIRD METHOD.
Multiply the first equation by the coefficient of x in tho
Bccond equation, and tfien multiply the second equation by the
coofticient of x m the first equation; subtract the second of
these resulting equations from the Jlrst, and there will arise an
cquution which contains only y and known quantities, frhm
which the value of y can be determined
It must bo observed, however, that if the terms, which in the
remlitnff equations are the same, have unlike signs, the re-
suiting equations must be added, instead of being subtracted,
e'uatbns) "" ""^^ ^^ ^Uminaied {i e., expelled from tho
Ex. 1. Given 5z-f 4^=55 . . . (l)
3a:+2y=31 - - - (2)
To find the values of a: and y
Mult. (1) by 3, then 15ar+12y=165
" (2)^7 5, " _15;»+10y=155
.'. by subtraction, we have 2y= 10
.*. y= 5.
Now from equation (1) we have
55~4y
x=z
5
55-20
Ex.2.
_35
~5
=7.
Let the proposed equations be
ax-^byz=c - - - (1)
a'x-\-b'x-c' (2)
Mult. (I) by a', and aa'x-{-a'by=za'e
" (2) by a, " aa'x-\-ah'y=ac' ',
How are equations solved bj the third method f
16
in
/5
i-.f-
■j^-mm":
'SI '
i
I '
I I
6 70
ALGEBRA.
a'c-^ac'
\
t
c
Mult. (8) by 4, and a'bx+hb'v=bc'
B7 subtraction, (ai'Z'b)lJtl_M;
JtL ■■ ab'-a'b'
m'^'"'-*- Let3.t+4y=29 (n
>«!». rn by 3, tl,e„ 9?+l|z^? (2)
, *l'-(2)by4,thenC8:.-lg:-fl4
4-rfroi\^i:rtr,r:tr -^ --'t'- •••"
together; and then ^ equations must be added
77ar=231 ;
rom (]) ve have 4y=29~3i?,
=29--<), (smce ar=3 ; and .-. 3^=9)
.•.y=:5. '
ilx. 1
'x. 5.
^x. 6.
=31)
=22 f
u^)
!-x. 7.
q :.v, 8.
n
k. 9.
Let 6.r+3y=:33)
13^~4y=19f
4ar+3y=:31
3ar+2j/
3ic+2y=40 r
2x+3y=35 f
5a?~4y==iai
4.r+2y=36 f
ai?+7y=:79 )
2y--|a:= 9f
^l?^+3
r
• ■
If ^
'J
|y=4.
\ y ~7.
(a;=ll
/; J-
y
)V
-f o
5
/i^
^!?.
^i>..
li^
SnrPLE EQUATIONS.
Ex. 10.
x-\-y
3
2x—^y
23
Ex. 11. H^^+y=7
^^7-^^^;= ^
h'l^W^ti
9
5a;~-13y=
67
^r''/§./-
-<4«s.
Ex. 12. 2iziZy=2^+H-l
3
8 ~^=6
5
mn^.Lr^^f /'""'^ unknown quantities are concerned, t'ne
most gor-oral form under which equations of this kind can be
expressed, is ax-\-by-{-cz^d (I)
a'a:+6V+c'«=c/' (2)
a"x+b"y\-c"z^d" (3),
and the solution of these equations may be conducted as in
the follow^ig example :
Ex. l.# Let 2a:4-3y4-4s=29 (1) ) , ^ ^ ^ ,
3ar+2y+5^=32 (2) ' to fmd the values
4a;4-3y+2^=25 (3)
of X, y, r.
r. Mi^^tiply /I) by 3, then 0.r+%+ 12^=87 (4)
Multiply- X2) by 2, then 0.r+4y 4-1 0-^=04 (5).
Subtract (5) from (4) then 5^+ 2^=23 (a).
Multiply (2) by 4, then 12ar4-8y +20^=128
Multiply (3) by 3, then 12a;4-9//+ 6^=75
'ijj Subtract - - - . -y4-14z=53 (/3).
^■^ JI. Ilenrjo the given oquations are rednnfirl tn
5//-f 2i'=23 (a)
/5
J ^
<> ''
9
L
1ft
7
>
,[**;""" <»
'^ 5nip ;»i->i,-/r
^^^
^
ALGEBRA.
72 •
Mult (S^f\ I ' ^^+ 2.=23
By addition . . . 70^—900 ^ «,
I'rom equation (/3) . . „_,. ^„ "
\^) y= 142—53=56-53=3.
29~3.y~4s
III. From equation (1)
Ex.2.
29-2:
Ex.3.
ir+y+2=90
2^+40=3y+20 ^
2ar+4O=40 4-lO )
^-h y+ 0= 53
ar+2y+32=105
a:+3y 4-4^=134
-o
J*R0BLEM8.
greater kdd'ed toldThe t's is7^u7l T> '^^ ' *'"^^« '^-
greater be suhtraL* from 6 timl. L f^' ^"^ ^^ ^^^'^^ ^^^
der divided by 8, The q^otien? ^ ^V""!' ^"*^ ^^^ ^e"»^'»-
numbers ? -^ ' '"® quotient will be 4. What are the
Let ar= the (/reaier number
y = the /e«5 number; '
Then 3«+|=:36
^jj. 9:c+ y=108
' 6y-.2a;= 32;
Or, y 4-90.= 108 (1)
W then6y- 2a:= 32;
then 56,r=616-
610 '
56 —
From equation (1) j,= 108Jo.=,08-09.,9
... ru"""' ^- ''^eie is a certain fraction. ,,..1. ,i,„. .-,• , . ., -
". "- r.um.™to,-, ,« ™i™ will be jd ■ and if'r;ubtmro,«
4.
-".='WMC(*BrtWMHi«*H', , ,
SIMPLE EQUATIONS.
73
!^
from the denominator, its value will be |th. What is the
traction ?
Let a;=its numerator, ) , .,/.,. . x
y= denominator; \ *^^" ^^^ ^^^«*^«" '^ -
rr+3_l ^
~3
Add 3 to the numerator, then
y
3^+0= y
^ .r 1
Subtract one from denom'., and — ^ — =-
y—l 5 J
By transposition, v—^x—9 (1)
y-5x^l (2).
Subtract equation (2) from (1), and we have
2x==8 ;
.*. 2r=-=4, the numerator.
From equation (1) y=9+3j;=9+12=2l, the denominator.
4
Hence the fraction required is — .
21
Prob. 3. A and B have certain sums of money ; says A
to B, Give me £15 of your money, and I shall have 5 times
as much as you will have left ; says B to A, Give me £5 of
your money, and I shall have exactly as much as you will
have left. W hat sum of money had each %
Let a;=A'a money ) , ^ • i k_ j what A would have after
y=B's money f ^"^^ a;-|- lo- ^ receiving £15 from B.
y~ 15= what B would have left.
Again, y+ 5= \ ^^^^ ^ would have aftei
° ' ^ ( reccivmg £5 from A.
x—- 5=:what A would have left.
ITencc, by the question, a;+15=5x (2^— 15)=5y— 75, )
and ?/-}- 5~iK-— 5. j
By transposition, 5y— a; =00 (1), )
andy— a;~ — 10 (2). f
8ct down equation (1) 5y— x=90.
Multiolv eo". ^2A by 5, 5?/— !>i
__ n{\
Subtract (2) from (1) 4ar™H0;
n
'I
74
|i '
ALGEBKA.
• . ,7* "-": 0(r 1 f
A — 'J», A s money.
4 •-"? ^1. s money,
■^rom equation en ^//-ort .
. .. 125
J^KOB. 4. Whaf ^ ^ -•>. Rsn.oney.
,,„o ? '^'--"- ' -btract o«., ,h, reUld:?^i;I„';7''
PROB 5 THaj-o • "^"*'- ^» '^''^J 3-
to its mmevator itV '''''^'''", ^^^^^^'^n, such th.,t if i 7 ,
■"^ggarsdid he relieve ? '" '" '"^ P^^ket; and howf,„V
Tl-en 2ixy, o,f = J No of .,;«,. ,,1,,, ,„„,^
"d 2Xy, or 4= ' '"'".^ S'*'™ »' 2»- 6i each
Hence, by the q„ostio„,|'=,^^3 (1) "' ^4, each.
Sub'. (2) fr„„ (,) „, y_ ^ >
^??.?. 56, and 33.
It)
Proc. 8. Thoro k ^^''- ^^» ^
<^'git«. Tho...,of thnJS;"."'i'"^^«'S consisting
of ?.«i.-rt
^0 added to
t
'****'««*'™i»»««**««««i,i
^ngmmmt^-^sm,,;:
SIMPLE EQUATIOXS.
75
io:
A.
What is the
St Zh^""" '"""■' *' *8its will be inverted.
right.ii.d diguTl's: ^JiTxi'A!''^''"" '^«" i^'"^ "-
Let xz=le/t.hand digit,
y-riffhi./iand digit.
? ,^^-^+2'=the number itself!
Hence bvfht !."";' ""^"^'^ ^^'*^ '^'g't^ ^>^''^^e<^-
on.i in f '^-^ *"® question, .r+y-5 nr
Subtract (2) from (1), tl.,n 2y=G, and ;~|"-~' ^^^^
.rzrS— y = 5— 3—2 •
Ajji^ ,•'• t-^c number is (10^+vW23 '
Add 9 to this number, and t betomTs 30 whJ.^ •
number with the diffits inverted ''''''"'^^ *^^' ^^'^^ is '.ne
^the, ..eater, the remaind^'^ajT^J^r;^^
the digits will be '■»..« wLTtttLefuSr?""'''"' "'•
-t be ta^e to r.™ » 2^ ^'JZ. SSttS
^«5. 33 at 13.V. (id.
Prob. 12. A vessel CO imininr ion n ^.^ ^' ^■'''- ^*''-
mimitesbytv.ospol^rng^^^^^^^^^ '^ ^0
gallons in a minu c, the other O^lull' ' ^"^ ''''''' ^'^
what time has eac-A'spout run ? ^ "' '" * "^'""^^- -^'"•
Ans. 14 gallon spout runs minutes,
Ptton. 13. To find thr^^ • ■ ' "^"i"^ f""^ ^ minute.,
I the sum of tiL . If^^^r ';::?'is \ ^'lU'^ &'' -^^'
with Jfh fhn,liff!.....,„„ /"^ M " ®"«^1 ^« 120; the secoml
I thoi.unr;^7he\h;;;— --^^ -d
^l«5. 50, 06, 75.
i
t
76 ALGEBRA.
CHAPTER IV.
ON INVOLUTION AND EVOLUTION.
ON THE INVOLUTION OF NUMBERS AND SIMPLE ALGEBRAIC
QUANTITIES.
53. Involution, or " the raising of a quantity to a given
power," is performed by the continued multiplication of that
quantity into itself till the nu|ttUer of factors amounts to the
number of units in the index-%fthat given power. Thus, the
square oC a=aXa—ri^ -, the cube of b=bxf>Xb=b^; the
fourth poioer of 2 =2x2x2x2-- 16; the Jifth power of 3
=3 X 3 X 3 X 3 X 3p=243 ; &c., &c.
^ 54. Tlie operation is performed in the same manner for
simple algebraic quantities, except that in this case it must be
observed, that the powers of negative quantities are alter-
nately + and — ; the even powers being positive, and the odd
powers negative. Thus the square of +2a is 4-2a x +2a or
4-40"; the square of — 2u. is — 2aX — 2a or +4a^; but thp.
cube of — 2a=— 2aX— 2aX— 2a=+4a*X — 2a=-8a^
The several powers of - | And the several powers of——,
are,
a a a*
a a a a*
cube =^X^X^==^a-'
2c'
Sq>i.
>
= -2cX
|4l»lil>— -— A7-A7A7
»Si:c.=&;c.
a
'6
a
V
2c
b
-+4c*'
b
Cub. t)^^~''()v^ 2c~ °-"
4th
power
2c'" 2c'
___b__ _J_
~ 2c^ 2c^
=+___, &e.=.&c.
b_
"2c
*1
8c»'
b
ON THE INVOLUTION OF COMPOUND ALGEBRAIC QUANTITIES.
55. The powers of compound algebraic quantities are
mj^ TiTi
Involution performed for simMe al)?8braic quautitioa ?— 55. How are tht
powofii of comp mnd (luivntitJ m raised I
GKBRAIC
to a given
tion of that
3unt8 to the
Thus, the
<6=6'; the
potoer of 3
manner for
} it must be
s are alter-
and the odd
iX +2a or
a^; but tho.
s of—--,
2c~ 8c»'
b b
2c
%c
UNTITIES.
mtities are
iiui iimiiner la
Howr are tht
INVOLUTION. ^
it^lK: (Anr^^f "?^3" o' '■- I^"'« <•- expound
E-v, I. What is the square
ofa-f-26?
a +23
. a'+2a6
^+2ai+4i»
Square =a^+4a,^q:4^5
Ex. 2. What is the cube of
a^—x
a^—x
- a.'x+ x»
Square r=a*~2a«a;-fp;,.
a^—x
«'~-2a*^+^
-- «*ar+2aV— a^
Cube=a''~-s„-..j.:j:3;;3-^rr7i
-Ex. 3.
What is the 5th power of a+5?
a -i~b
«'+ afi
+ ab +5«
«*+ 2oi+i^=:Square
g + b
a»+ 2a'5+""^
a«+3a^6+ 3^F^b'=:Ci^-
g + 5
a*+3««5+ 3a'^^~^3
__+_5!'^+ 3««6''+ 3a//Vi*
«^+4aV/Tl?^4a53+^/=.4th Power
t±J^il:i2S!±iO«V.«+5a6^+i^=:5th Po«.
T8
ALGEBRA.
Ex. 4. The 4"^ power of a+Sb is a*+12a«6+54a*A'^+108ai»
Ex. 5. Tlie S!7Mare of 3a;'4-2r+5 is 9x*+12a;'»+34x'+20*
+25.
Ex. 6. The cwJe of Sx-6 is 27a:«-135a:''+ 225a: -125.
Ex.7. ThecMieof a;'-2jr+l is z'-Qx'+15z*-20x''+l5x*
— Cx+1.
Ex. 8. The square of a+i+c is a'+2ab+b^+2ac-{-2bc+c\
ON THE EVOLUTION OF ALGEBRAIC QUANTITIES.
56. Evolution, " or the rule for extracting the root of any
quantity," is just the reverse of Involution ; and to perform
the operation, we niust inquire what quantity multiplied into
itself, till the number of factors amount to the n\imber of
units in the index of the given root, will generate the quantity
whose root is to be extracted. Thus,
(1.) 49=7 x7 ; .-. the sq. roo^ of 49 (or by Def" 15,y'49)=7.
{2.)—b^=—bX—bX—b\ .'.cube root ot—b^{^:z:j;z) = —b.
. 16a*__2a 2a 2a 2a 4 /16a*_2a
^ ^^8r6'^~3^^36^36^3^;**' V816"*~36'
(4.) 32=2X2X2X2X2;.-. ^32=2,
(5.) a^=za?Xa?Xd?', ,'.^a^=a\
Hence it may be inferred, that any root of a simple quan-
tity can be extracted, by dividing its index, if possible, by the
index of the root,
57. If the quantity under the radical sign does not admit
of resolution into the number of factors indicated by that
sign, or, in other words, if it be not a complete power, then its
exact root cannot be extracted, and the quantity itself, with
the ladical sign annexed, is called a Surd. Thus -y/37, }/a^,
V//, J/47, &;c., &c., are Surd quantities.
.«. Wha" {mlMluMon? How is it performed ?— 57. Wlint ie a Surd qiumtity f
EVOLUTION.
79
derived from thos" of ZZlT PoTin" "" T' f ■''^ "^^^
Ex. 3 , the square o{a+!ua'+<laf,In T''''' i^^ ^'- ^S,
arranged according 10^0 nowtflf''" ' 7^"'^ ""' '«'™s.-"-e
with a'+2a4+4', we observe fh,. ?K « ."" eomparing „+/,
tllf Tr-f'''tsrtet';!?t-^' '-"> "^"-^ ?--
root (a). Put a therefore for the fir^f «s_lo a . w /
term of the root, square it, and ubtrac J+^"^+*' (-+'
that square from the first term of the
2^aT;.» 1"? ^^J" f^^ °'^^^ t^^« terms 2^4T|2«i+i«
-2«6+Z> , and t/oM6/e the first term of the l2ab+b'
root; set down 2a, and having divided ~^^
he first term of the remainder (2a6) by •==— •
It, It gives b, the other term of the root-
and smce 2ai4-A«— r9«a.;.\;. -c 1 « ',
being subtracted fr&Vo^tL'"';l:ht'^'+'" \*'"''
remains. ™^ oi ought down, nothing
60. Again, the square of a+b4-r ( Av^ f^K v c. x .
2ab-^b'4-2ac4-2bc4-r^- in 7^- (Art. 55, Ex. 8.) is a"-+
deriTed from t ' ' '^''' "^^" *^^ ^^ot may bo
the power, by '','+2«*+*'+2ac-f 25c-f c'/a-f b-i-c
continuing the « "■ 7.^ \ '
process in the *^" + X"* "^ *'
iast Article. goi+i'
Thus, baving 2a-^2bi-d2ac+2bc+c
found the two }ftac+2bc~ -'■
first terms (a+i)
of the root as
befor
4'i\t%ry^t
f»_-
ri '
'iOC f-HOf
»i?SVttSl«t»:; jr* '''''""'^^^*^
'\
80
ALGEBRA.
i;,creased, and it ^f ^X+st + J ^hiXSg subtrac.od
iklhU manner the following Examples are solved.
Ex. 1.
4**
4:c'4-
|>'+T
89 ,
6x»+|a:'
\ 20.tM-15a;+25
Ex. 2.
^^ .
2a^+2a;')4a;'+2a;*
Ex. 3. Tne sqaare
rootof4x'+4a:.y+2/'is2a:+y.
„. . rn,_.n„.«rnotof25a' + 30a6+9/ris5a+36.
^" " ^"^' " " ■" ' root of 9.*+12.«+22.'+12.+a
Ex. 5. Eind the square
^Ins. 3x'+2a;+3.
__ *^^^^^^^ ■-■ai.
•»--^
►.-wfci-j
EVOLUTION.
81
», it gives f, the
term (6) of the
the divisor thus
iltiply this n*vr
being subtrac.od
res no remainder,
solved.
-|x+5.
Ex. 6. Extract tlio square root of 4a;*~-16x»4-2 ic'—Wx-^
Ans. 2x^—4x-T 2.
Ex. 7. Find the square n jt of SCir^— 3Gi-^+17^«--4jr4l
y
o
Ex. 8. Extract the square root of «*-f 82;'+ 24-f — -f-i?.
a;- «*
4
ylns. ar'+4-j — .
2a;*-a:-f2
■4
4
is 2;c4-y-
-9P is 5a +35.
Ins. 3x^4-2x+3. .
ON THE INVKSTIOATION OP THE RULE FOR TJIE EXTK N
OF THE SQUARE BO^ T OF NUMBERS.
Before we proceed to the investigation of this Rule, it wili
be no'cssary to explain the nature of the common arithmeti-
cal notation.
01. It is very well known that the value of the ficrures in
the common arithmetical scale increases in a tenfold°propor.
tion from the right to the left ; a number, therefore, may be
t xpressed by the addition of the units, tens, hundreds, &c , of
which It consists. Thus the number 4371 may be eipressed
in the followmg manner, viz., 4000+300+70+1, or bv 4^
1000+3x100+7x10+1 ; hence, if the digits* of a nuU^r
hanHhen ^ "' ' '' ^' '' '^°'' ^^S^nning from the left
A No. of 2 figures may be expressed by lOa+i
« 3 figures ' by lOOa+lOJ+c.
4 figures « by 1000c +1006+ lOc+cf.
<^c. &c. &c.
62. Let a number of three figures (viz., lOOa+105+c) be
* \^^^^/!!^'J^ of ^ number are meant the figures which composa
n, cons,(?ered independentlv of the value winch they possess in thHSh^
metical scale TJius the cTi^it. of the number 537 aK simply the num-
Zn'nf' ^1^^' ^\^''^^' **^" ^' ^on^i'lered with respect tJits place n
the numeration scale, means 500, and the 3 means 30 ^
HiS;? S Gi ^\?'"™?"."'''*""®*'°''^ s^"'^ of notation. What ig a
digit ?-b2 ShoNV the relation between the algebraical hfid ml nerioJ
methoi of ox^raot.ng tbo square root, and tliut tfey are Jden icJ
/I:
f r
m
ill
MICROCOPY RESOLUTION TEST CHART
(ANSI and ISO TEST CHART No. 2)
I.I
1^ 13.2
1^
Vi
m
i4.0
•uuu
2.5
""Z2
zo
1.8
J /APPLIED IIVHGE
inc
1653 East Main Street
Rochester, New York 14609 USA
(716) 482 - 0300 - Phone
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./ 'Wsmnmmmixmm'-.'
h:\
82
ALGEERA.
squared and its root extracted accordiiig to the Rule in Art
00, and the operation will stand thus ;
I^00a'+2000a5 + 100i'+O00ac+o05e+cXl00a^
20(k+l0Z;) 2000ab + l00b'
2000ab_j-im^
200a+206+c) 200acH-206c+c«
200«c+20ic4-c2
*
* *
II. Letff=2] , ,
6=3 [ ^''? ^'\^ operation is transformed into the fol.
c_lj lowing one;
^^^^^+12000+900+400+00+1^200+30+1
400+30)l2000+900+400
/ 12000+ 900
400+G0+l\400+60 + l
7400+00+ 1
* *
5330i/231
II. But It IS evident that this operation would not be
affected by collecting the several nJimbcrs which star d m
the same luie into one sum, and
leaymg out the ciphers which are
to be subtracted in the several parts
of the operation. Lot this be done •
and let two figures be brought down
at a time, after the square of the
first figure in the root has been sub-
tracted; then the operation may be
exhibited in the manner annexed-
from whwili If ni-»i->/->r.-..„ «^i,„i. iL _ '
;. trj-j."^«i3 tiiui, uic square
loct of 53,361 is 231.
4m urn
|l29_
4011401
1401
]
. ^»— «?<#|«l|!l,'««!i-;
'''^»«^<^>^^^^-
QUADRATIC EQUATIOXS.
83
square root of 100 is 10- nf lAonn • inn ' r ' ^'"^® ^''^
1000 &c &.ro . v. f 11 ' ?, ^T^^ '^ 1^^' of 1,000,000 is
^""^t" 100 ;, , t^^^^^^^ T'^<^ root of'a n'un.bor
listing of rr%ures '^^' ^^"^''^ ^«°' ^^ ^ '--^er ccn
£x. 1. Find the square root of 105,625. . Ans. 325.
Ex. 2. Find the square root of 173,050. A71S. 41Q.
Ex. 3. Find the square root of 5,934,090. Ans. 243G.
CHAPTER Y.
ON QUADRATIC EQUATIONS.
<U. Quadratic Equations arc divided into pure and adfec/Pd
v/i^r,7^ of the unknown quantity, such as x'^SG- ^4 5-
« ,U i^'i *^'' . ^'^'^^^^'^^ ^"^^^r^tic equations' a, etlio'e
UHch involve Ijoth the square ind .v.-.«7>/. ^o^m- oHhc un
I It
i
ill
■^'.:>:?mf^ mw rjm m.. :
84
ALGEBKA.
r
ON THE SOLUTION OF PUUE QUADRATIC EQUATIONS
necessary by the coefficient of;.'; then extract' ttf ^
root of each side of the equation, and ifwrnTvl the tC
Ex. 1.
Leta;'+5=54.
By transposition, a;^=54— 5=49.
-Extract the square root )
of both sides of the f then xz= + W49--^7
equation, j —V^^-—'.
Ex. 2.
< L'et 3j;'— 4=71.
By transposition. 3;c'=71 4.4=^75^
Divideby3,a:''=^=25.
o
Extract the square root, x=j^^25=+5,
Ex. 3. "~
Let ax^—bz=ci
then ax^=zc-{-bf
and :r*=f±i
a
V a
=244 - - Ans. x= +7,
=3a;'+63 Ans. x=z-{-^^
Ans. ar= + 10.
Ex. 4.
Ex. 5.
Oar' 4. 9
4a;'+5
9
=45
Ex. 7. i;j
65. State tha rn\a ^nr. o«I..: . ~~ —
._ .-.. ^..-iTiuj,' juiro qxiauruiio oquatloug.
of the
QUADRATIC EQUATIONS. flg
ON THE SOUTIOK OF ADPHCrED QUADRATIC B0UAT.O.,-S
y or / actional. Divide each side of this equation l.y
a: + -a; =-. Let --=;9, - =q ; then this equation is
KULE I
Let x^^px~q.
Add |! to each -<3e ) ,^ ^^^^^. ^,^^^
of the equation, then j 4 4"^'^ 4
Extract the square root
of each side of the
equation, then
andar=il:5!+47+7>
2
will arise, on the lejjnd sld/'of tt "Sn' f ^' "^•''•^*
which IS a complete square; and by pvI^?;"! Z^' '' '^"•'"'^'^>'
binco the square of +« fg ..„? „„j »,
of P=+4, ™ay be expressed by ± /,?:^ ' ''"" "" ^^"-^^^ '-*
reM^irr^fry:;- Pjj'^JlJ^n^ Can U b.
8
86
./,
!i
-ALGEBRA.
one corresponding to .he sic^n 1 i f, ^^ill have two values ;
of the radical quan% ^" "^' '"^ '^^ ^^^^^^ *« the sign-,'
Ex. 1.
. . , , Let x'+8x=Gr\
^^ Add ft. y^<'Jl^J^^O)^a^ ,Me of the equation,
E.U.-aeUHos,„a;etor„JtKrrl e,u.io. .1
ar-f4=+ V'81 = +a
and a; = 9 — 4 — "s". '
or a; =—9-4- -13.
Ex. 2.
Let a;'-4ar=:45.
Add the square iof )
2 (e. e. 4), then f ^ -4a;+4=45+4=49.
Extract the square root, and x-2= + y'49= +7
anda;=7+2=9; ~~ *
or, ar=2-7=:-'5
- ■ ^«*- ^= 6 or -18.
X — 14e — '1I J
. — 01 . . . , ji^g^ ar=17or— q
a:'- Gx= 40 . ^ ' or J.
^^ - - - - Ans. a;--=10or — 4.
Ex. 0. a;« 5a;=::(J
In .his exa„,plo ,ho oo.^ri«, „f , ;, 5, „„ ^,^ _^^^^^^^^
ts hains -; a„d ... adding to each .ide of the eq„a«„„
(2) "" T' "'" e«t
I
x'-~-5x+(^Y=G4-^-^+^^ 40
\2/ ^ 4 -~4 -^ J.
5 _l7
Ex. 3.
Ex. 4.
Ex. 5.
extracting the sjuar* root, ar-5 - +7 •
2 "~ — o '
2 ~2
_54.7_^
2-2
r— O, or —I.
■pwnciisiaWkiS
■■., .«?-i^i^S^«ft^^
QUADRATIC EQtJATIOXS. 3)
Extracting the square root, x~l=-h^ .
^ — 2'
1-4-5
• • ^=^±2=3 or-S.
«r-^-/ar_7» Ans. xz:zQ or ^IS.
Ex. 9. a;24-3^=28 . . j .
Ex. 11. a;2j- aj—on ,
Ex.12. Let7^«-.20^z=32; find or.
Dividing by 7, x^-^^x=^-^.
^:pJ;; \ .^-^.+ (^y.L!52.^^4 100,324
I J 7 V7/ 7^41) 40^49 -4ir*
Hence, :.-:L?= + ,/??4_ + 18,
anda:=— ±l?=4or-i|.
Ex. 13. • 6x«+4ar=273.
Dividing hy 5, a;«+-^=?Z2.
5 5
.idfadd' ^ I (%v± and .'+ t+l-?I? . 4 __ 1369
^ W 25 5 ^25- 5 ^25-~25^
"• Extracting the square root, «+?- +?Z
' ^5~— 5*
Ex. 14. 3a''+2^— ini A
ii:.
S8
ALGEBRA.
Ex.
Ex.
15.
10.
-5z=
-2x=
=117 . . .
=280 . . .
Am.
Ans.
■• Oor
:10or
- 0*
Ex.
17
4a;'-
-7x=
:49? , . .
Ans.
X=z
■12,or
3
-101,
A qt.adratic equation seldom aonpnr^ in o ^ . .
as those of the preceding exaTples^ ft I t)f 7"" '^ ''""^'^
foundnecessaiTtoemnrorinT 'i V ^^^^^^^^'^ generallv
following reductiol.^ ^ '^'' '°^"^^^" ^^ ^ ^^^'^^''^tic the
(1.) Clear the equation of fractions.
(2.) Transpose the terms involving ^« nnrl ^ f^ .u , .v
ha«d and th„ numbers to the ri«hth^„/.4"V.he%;t:^
^^,V3.) B:v,do all the tornas of the equation by the ..V.C
(4.) Complete the square.
Ex. 1.
4a;«
1
3 ll=r-
Multiply by 3, and 4a;'-33=.r.
By transposition, 4.r— a; =33
Divide by 4, and x'-.\x~^
4 4'
Complete the ) 1 1 33 1 ^oa ,
square, \ ^"~7^+;rr=^+i-=2l24.1_
^ 4 04 4^64 64^64- 64
Extracting the sq. root, x—-=: +??
■.8—8"
529
I4-23
Ey. 2.
9 4
9+-
4rJ-J.
a;
•=5ar-f-5.
•«*«^:«i»«SR%g|^:-?^,
Ex. 3.
Ex. 4.
Ex. 5.
Ex. G.
Ex. 7.
Ex. 8.
Ex.9.
Ex. 10.
Ex. 11.
Ex. 12.
both
—4
of
QUADRATIC EQUATIONS. gft
5 25 5^2o~25'
5 —5'
andar=-+-=2or-l
6 ^--*+lI Ans. x=:zl2or-6.
2^ 1 7
3 ~*"^'^3* • - • - - Ans. x=3 or J.
3 2~" ^ns. x~G or —1.
2
a;+l'^a:"~ -^w*« a?=2 or — ^.
a:»-34=|^. Ans. x =6 or -^l
5+- =5] Ans. ar=25 or 1.
, 24 ^
'^'^^^ZiT^^''''^- - - - ^«^. a:=5or-2.
a;+l a? ~"6" ' * ■ ^"*' ^=2 or -3.
x-\-2 — ^"-^•- - - Ans. ar=10or— i
Given a;'4-8a:= -31; find ar.
^+8a:4-lG=lG-31 = ~15,
x-\-4= + V—l6;
/ — 15, & ar= —4 — V — 15^
lich r-re impassible or imaginary values of x,
- 8*
90
ALGEBRA.
Ans. ar=1+^V — 1^
Ans. x=l5 or 1.
Ex 13. a;'- 2:,= - 2. . .
Ex. 14. £'-lQ£=z-lo. . -
Ex. 15. Let 132«+2t=G0.
Divide by 13, 3;'' + ?:?=^.
^ ' ^13 13
square of ^ j - +T3^1oi)=l3+nu>=I(i9+loi)=10
Extract the )
781
109
1
.4- -V/T81
square root r^"^i3~ — 73"
=,+
• • aC ■ I-
±27.94-1 20.94
27.94
• 13
13 ^ Is"^^-^^ ^^ -2.226.
Ex. 16. ar'-6.r+19=rl3. - Ans. a:=4.732 or 1.268.
Ex.17. 5.c=-f4.r=25. . . Ans. ;r=1.871.
_ Any equation, in which the unlcnown quantity is found onlv
ni two terms, with the index of the higher power double
that of the lower, may be solved as a quadratic by the pre
cedmg rules. J i
Ex. 18. Let a;'— 2^^ =48.
Complete the square, .r*— 2.c'4- 1 =49 •
Extract the square root, .r^— 1 = + 7-
.-. x^=8 or —6 ;
and .-. X =2 or yZTT
Fi. 19. 2x~7^x=m. ^
7 , 99
^-2V-=2-'
•7 . . /7\« 99 49_841
2 "^lO^Te*
2» ^U/ '>- i«~^^
4 4 2
•. by squaring both sides, a- =81 or ~.
:-^mLmm
QUADRATIC EQUATIONS.
9).
; 781^
>r -2,226.
or 1.2G8.
'oiind only
er double
y the pre
Ex. 20.
Ex. 21.
^+4,r'^=12.
- Aus. z=±^2 or ±y/Z:?,
• Ans. ar=3or^i:TJr
Rule IJ.
Multiply each side of ) . ^^^ ^^'±^^-^=<^, '
tlie equation by 4a, \ *"^" 'i«'-i'l:4a4.r=4ac.
Add b- to each side, ) . , ,
we have f 4a V+^ 4adz-i b^=4ac-i-l,\
Extract the square root as before, 2ax±6 = ± ^4^;^^
and X =i V^^^TZ"'^:^
rom which we infor thni- wp , .
U multiplied by /„",•;« 1 "^ T^ ''^^ °^ "'O "q^t^n .
side there be Idl{ rJl^^T' "^f: ""<> *° '»'* '
quantity on the left.h„„d sWe of 1 ""■^""" "^ "' "'«
square of 2a.+4. Ex net tt f ''™"°;' "'" ^^ ""-'
tho equation, a"Sd there Irifes! PI ''''"' "'^ *^''«''' ^'''^ °<"
the value of'. „ay be determLd "^'^ ''"'""'"• '■™"' "'''»'> '
i..'tL:7a!;*heSre°?he'p1"'="* '".^^ '■°™ -•+P-^=?;
Pl>ing each sTde ofThe equatt Z\ ''' ^^'^^''^ ^T^-nuk
of the coefficient of " " ' ^^ *' ""•* '"'''"'g ">e square
spondingto theSL + Id'lV^l "^ ™'"'^; <""> corre-
radical luantityT ' ""^ ""'"• '» *« ^'g» -. of the •
Ex. I.
Multiply each side of the )
equation by (4„) ,3 ; j 36.-'+ 60^=504.
«ISt£S;I° ,"!!:'*,'?.„«•■> -M.bo found In .!,„»«, ^„„. .
I
^H2
ALGEBRA.
-^^cM (/>«) 25 to each side )
of the equation, we have f ^Cx'-f-GOjr + o-.^g^j^ ^25=520
.-. 6jr= +23-5=18 or --28•
28
(J
or a-=-r-= _42
Ex.2.
14
3
Leta;'— 15jr=— 54.
Multiply by 4, then 4^^-C0^-_oig
■ Add (i«) 225 ) ,,,
to each side [ ^"^ 4x»-60x+225=225~216=9.
Extract the square root, 2ar— 15=+ ^9—4-3 .
18 12 •'•2^=lS±3=18orl2.
and;r=_or~==9orG.
\
ON inE SOLUTION OF PROBLEMS PRODUCING QUADRATIC
EQUATIONS.
values of the unknown l;„t ™'r ""^ "f ""^
' lonniVa,! m, . . "™' quMtity, Will answcr the conditions
. 3r.et:^l:.\7rnr: :?;t -h,t i^i>r -
Problem 1.
: ,.-Ltsi;lVb: cV '" '"*° '™ ^-•' p''^'. "■»' 'h^--
Let x—one part,
then 56— a? = the o//<^r part,
and X (56-a:)= ;;rocf«c; of the t%Yo parts.
Hence, by the question, x (56-.r)=640,
or 56x— a:':=640.
gUADRATIC EQUATION-:
H f 25 = 520.
equation, whick
i:23;
8 or —28;
-n-
1=9.
[13;
8 or 12,
ADRATIC
ve quadratfc
■ one of the
e conditions
Mys be verv
L itself.
s, that theii
3.
9S^
,j , . % transposition, a;»-5Cj:=: -040
ii>- completing the square, ) „ .
(UuleI.) j-^-^^-^-f784=784-.040z=M4
anda:=^8 + 12=40or 10.
Ill th,^ case It appears that the two values of the unknown
I If".?"- ~' ^"^ "^ '"■" ""mbcrs whoso dilTerenoo is 7 ai.fl
nun^oei. VV hat are the numbers?
Let a;=the less number,
then «+7=the r/reaier number,
and - +30=3 half their product plus 30.
Hence, by the question, ^±Z)+3o=^. (,j,,,, ^f ^^^^^^
or
+30=ar».
Multiply by 2 . z'-{.7x--].Q0 = 2x\
nf u?!' transposition . x"--7x=Q0,
Multiply by 4, and add ) , ,, ^„
49 (Rule 11.) f 4^-28^;+ 49=240+49=289,
/. 2a;-7=y'289=17
2^=17+7=24, or ar=12 less numL3r;
hence a:+7=12+7=19 preaier number.
Prob. 3. To divide the number 30 into two such parts.
Let a'=the /ess part,
then 30--ar=the greater part,
and 30-ar-a; or 30-2a:=their c/^/e,-e«ce.
1 Tf^nna ^\\T +Vi
. V > /r>/\
AV'^v— a;j=«x(oU— 2.ir),
or30x-a;°=240-lCj:.
I
'i
m
I i
ALGEBRA.
Bj transposition, a;«-w46^-- _24Q^
'bmplete the square.) , ,^
(UvLEl) [^-46^+529=529-240=280;
.-. ar-23= + ^289= + 17,
and ar=23±17=40 or 6=/m part ,
30-ar=30- 6=24 =<7rea/*r part,
(n thi8 case, the solution of the equation ijives 40 and
.<0, we take 6 for the less part, which gives 24 for the ZlaZ
i^da/red ' ''' '"^ """^'^'*^' ^ ^"^ «' ~ ^h^ tSS:
sofen'at&'T n''"^^^ '^^'l^^' ^^'^ ^^^^ ^h'^h he
.oi V i'^ 85., per piece, and ga ned by the bar<raln
as much as one piece cost hin,. Req'uired thi number^of
LetW=the number of pieces.
Ihen — =the number of shillings each piece cost,
. ^^ f r'l" """'^'' ^^ '^^"^^^'^^^ ^^ *^^^^ the whole for;
. . 4Sa:-675=what he gained by the bargain.
■675=5Z5.
X
Ilonce, by the problem, 48j?
By transposition ) 225 225
and division, J * ""'le'^^'io'*
= -7^+
1024 ~~ 16 • 1^
. ^_225_ /(
32 ~V 1024
255+225
1024*
225 /05025 255
32*
=15.
and X:
32
Prob. 5. A and B set off at the sam^ time to a place at
^JT^^'l '^ ?^^ "^"^^' ^ t^^^^« 3 miles anC>Z faster
than B, and arrives at his journey's end 8 hours aiid or
minutes be/ore him. At what rate did o Jh nToL"";!,!^
per iiuur '^ ~ * '" •''"■=-'■
.>vmm.r^i«m:
240=289;
!89=i:l7,
G=less part ,
realer part.
fives 40 and
'' be a pari of
for the (/reader
the conditions
15s., which he
)y the bargain
le number of
Be cosif
the whole for ;
QUADRATIC EQUATIONS.
95
5_65025
'""1024*
255
32'
15.
a place at
1 hour fasier
)urs and 2C
Then 'fUT"" r. ^"'"^ "'„*"-'' B < -vels.
And — =number of hours for whi^h B travels.
150
x-\~S~~ " ** " A «
ney\" e^d't^L'er "' "'"'" ^'' '""^^ '''"''' '' ^'^ J^^
i50 150
150 25 150
By reduction, x^+Sx=54.
Complete the square, ;i:«-f 3^-1- -=544.^. _ 225
, . ox-t ^__o4i-- ----(Rule I.):
Hence -^l^Clj_fii_ ^^"
or
2 V 4 2 '
4 - ■ 4
225 15
f
, 15--3
and x^ — — =G miles an hour f)r B,
2
a:-|-3=9 « ^
flight'?h;L^e^'oo^tTfh^.t? f.f '^ "P^^ * *^^^5 at 'one
Iths of ther two bees ttn 5 *^^ ^^"i^^vay ; at another
on the tree ?* " remamed. Honv many alightc^d
Let 24;»=the No. of bees
then xi- ~-+2=2x\
or9ar+16.r»+18=18.c»
.-. 18u:'-.16a;«— 9x=18,
/r. or 2.^''— 9a;=18
(UuLE II.) Multiply by 8
16i:^--72.r=144.
^^^./S'lb Jl!^^- .'^•- M^ Strachey's tran«lation of t,..
pudson. be foumi^t.;acr:mrwmw£%/;-'7'" 7^7"?" r"'' "P*'" ««'°'
8ta..d. h, that tnu.slution. p 02 '^''"'^ °^ '^^"^•°"' «» i<
96
i I
<■?
II
I ! i
ALGEBRA.
Add 81; then lG^»-72ar+81=225,
or4x~9=15;'
.•.4^=15+9 =24,
* „ ^ 24
and ir= — —a.
4 '
••• 2^'=72, No. of bees.
tnlTrolIt''^^^^^^^ ^6^ "'" '' "^^^ ^^^. -^^ P-*^ '^-^
P o WL ^iw«. 27and6.
JlZLY^I '"" """"'"" «^^ ""'- -h"^ -m is 29,
p Q rn. ^''*- 25 and 4.
Of tS AS f '^rwtTo rn\^x^.V"^ *'" ^-^
p .- _, ^««. 13 and 8.
.?xa„ „/«/« ^.,„,,,. Vhat fro th:"'u™beri ?' "^'"" '° "'"
P 11 m. ^«s. 17 and 11.
p ,„ . ^ ^««. 15 and 8.
B gives £5 aV„e to each '^I ^'IP^hT;" H " ''' "'"^
persons were relieved by A and B rj^ecttve^-j " """-^
^«». 120 by A, 80 by B.
We cost him TshmLsTss H^' '""^ '^^^ """^
there? sumings Jess. How many sheep weru
for^S'7.- HavinStsTs o?"f^ " ^r'" "'"»''^' "^ ^''eep
8 shillings a.ird%'„fit\:^-:''.rC- ."''.''•f ""■"'""'^ «'
many sheep did he buy?
igjiin. How
Am. 38.
4
' bees.
such parts thai
Ans. 27 and 6.
hose sura is 20,
4aw. 25 and 4.
s 5, and jth part
ers?
1/is. 13 and 8.
is 6 ; and if 47
be equal to the
i?
!«. 17andll.
sum is 30; and
uare of the ksj-
ns. 21 and 9.
product is 120.
>m the greater,
1 also be 120.
''«. 15 and 8.
mong a certain
>re than B, and
- How many
A, 80 by B.
ibcr cf sheep
sheep u-ould
Y sheep were
Ans. 40.
iber of sheep
remainder: at
argttin. How
Am. 38.
QUADRATIC EQUATIONS. 97
rhe'^di^tlcLfso^^ to a place at
an hour faster than T) . d i vesttl' ' '''' '' "" ^"^"
hours before him. A'" ,'\^^fllZ%A^ ^u J^"^"^^ « end 10
Let ar=6ji^ part,
t^enl6-rc=t't^' 'other part.
Hence x (16-a:) or 16x-^^?rt70.
Transpose, and ar'—l'diii— 70.
Complete the square, ' *.'.'.'
a:«~ 16^+64= ~r0.4^iij4== _6^
.-. a;--8=: + -v/irO, or a:=8+>/IlG»
Let ar= one factor;f .'*
then -=the other factor.
Hence x'^+^—h
x" *
or ar'^-f a'^ar*.
fro;|tV;Sp;SttTt?e^*J;;^^^^^^^ -'-^^ can „n.
may be divided is when these two lZ\l iT 'f ""''J ^'^'^^ ""'"l^^*'
jluct therefore/winch couIdTrir?romn,rr''?""^' I''" ^'•^»*^''t P'"»
•nto two part., is when each of Is 8 'Tn/^- *''' ""•"?'"'• ''
(livide the number 16 into two si c , Trt- ♦! * *i "''''• '" •"^q"'^"^ "to
70 ' the solution of the querontr/r*):' ''''' P^^^^^ should be
Plitd tL<tw!S'!!:!_"!^""A *»>« two numbers which bdn. n,.,,,:.
I
! ; 9 1
■ II
II
98
ALGEBRA.
By KULE 17.
2 ^
^.... -^^5. 6 and 3.
ON T,« acuTIO^ OP QnADRAT,0>SQCAT.O»S COKTAi™ ,we
UNKNOWN QUANTITIES,
ver/ well kiown these cli.es, the two following are
* V
Case t,'
equation;" i ^h ch case X P„^- ''t'!™'"''''' '^ ^ "''"^'^
then submitute Lit Z v„i ")■ ""*.' .'™P''' '^1»^'»». "d
-Mvcd fey th^e :?S; rI;;? ^ ■J^^^^^'-- -W^h may be
*)lwtion of these equatiotcrntnf^t/'^^^ ^''^ ^^"^'•al
of higher dimensio2rtharqt!;,Vralfcs '*'^ ^^ '"'^'^"^ °^ '^'^''^'»
equatio,.. to one quadr^i^^f 'ti;^ IZ^ZmT'^'"'^ ^' '"^"^"'^ ^'«' *^
) i
"^^■^^^'^^^mmk.:'^^0mm.^
o two such fee
^ns. 6 and 3.
CONTAINING TWC
n quantities, in
in a quadratic
I by means of
) following are
the values of
3, is a simple
a value of one
equation, and
ther equation ;
^hich may be
equations cm
his:
t the general
na of eqnationa
Jclucing the two
QUADRATIC EQUATIONS.
T * . « ^X. 1.
Let a:+2y=7 )
and x^-\.Zxy-y =23 [ ^^ ^"^ ^^^^ values of ^ and y.
From 1st equation, .=7~2y, ... a:«=49-28y+4r ;
Substitute these values for x and x^ i„ the 2d emmtion
then 49~28z,+V+21y-.63,«-y=2S, ^ '
R I? ir °r 3/-+7y=49~23=2G.
By Rule II. 3G/+a%+49=312+49=361
:"./0y+7=19
6y==19-7=12, ory-2
ar=7-2y=7--4=:3.
99
J * • 1 •
Let|±-^= 9
Ki, 2.
to find the values of x and y.
and 3a'y=210
From 1st equation, 2a;+y=27 •
.-. 2^=27— y.
apd xJitry
' ' 2
Hence, 3^y=3x^^Xy=210,
or3x(27-y)xy=420
8ly-3y«=420
27y- y»=140;
R T> ,T ory'~27y=_l40.
By Rule H., 4y«--108y+729=729~500=109 ;
.•.2y-27=13,ory.?Z|:!3_.^3,>^
andx^^Lz^^^^
ill!
! If!
j 1 1
III I'
! i !
l\\
I !'
nil I
100
ALGEBRA.
.to'nuXl '° "" """" °' "-^ '"^•'-•^ ^!g!'- What I.
Hence, a:=3y ) ,
and lOjr+y— 12= a;'' | ">^ "^^ question ;
.•• by sub.) „. . .' .
stitution r^^+y-12=:9/,ffi,i.Mo^=,30y,and ;r»=0/)
3
31 -12
J:
.-. 2,'--y=
ByRuLE I.,y'-?-v4-— -2^' ^^ 061-432 529
9^ 324~3^^^=-3oT-=324-
, Hence, y-?^=??. or v--^'*~q
« '•'is 18'?^ ^-18=^'
a^=3y=9^;
and consequently the number is 9^..
E<4. Let2ar-3y= 1) *'*•'
2a;*4-ary-5/=20 j ^^-^"^ ^^e values of x and y.
•°' ^ws. ar=5, y=3.
35; and if four times theelJ^rhl'^--/J"^""'r *"' l^»
tl.o less plu, one, the quotifr^n vf w \ ^^''^ "'mo^
ber. w'hat are 'the nSo™ /'" "' '''™'i'l *?,'^" """"
-d«5. 13 and 4.
^wj?. 78.
Case II
71 When x\ y\ or a?y, is found in every term nf fJ,. .
equations, they assume the form of ^ ^^^ ^'^'^
n.ay ho effected :-as in the following E^ampiesr" ^"^""""
^'^^^^fS^**''^'
digit. What ia
^rt. 61, lOx+y
liber.
question ;
rj, and x^—^xf)
-432529
24 -324'
=3,
es of X and y,
ar=5, y=3.
if the less be
ainder will be
by three times
the less num-
B, 13 and 4.
i^hose digits is
digits will be
Ans. 78.
w of the two
fk
QUADRATIC EQUATIOXS.
Ex. 1.
Let2a''-f3ary+y''~20
5^+4^'=41 ;
AMumex=.y, then2.y+3y/+y^==oo or ,/»- ??
101
and 5i;V+4/=41, or v'- -i]
2i;-'+3y-f i
41
20
41
Hence
2y*+3?;+l ~~^^p^^
which reduced is, Qv'-^^\v^~.\i.
41v__^I3
" 6 ~ "6"*
5z''''+4'
.•.i>'
% KuLE I., z/'-iJf .lggl_13C0
6 144 144 >
.•.t.-i^=:±2!.or.-41±37 13 .
12 lo>or._-__^^_,,..
Let v=5, then v«=-^L__ ^1 _369
v=2;y=^X3=l.
, Ex. 2.
theg.tlH:77Ta'„TXse'dT^ ^"^ -"^^'P^-d by
i« equal to 12 ? ' ^""'^ ^'^^^^«^« multiplied b^' the less
Let ar=greater number. '
2/= less
Then (^•+y)X:r=;^r»4:a-v=.77
Assume x-=.vy'
Then j;y+^y.^77 ) or /-^ j
andt,y«-y^=12'[ if"
<ii 1
w
M
M !
i I
1 *l '
ihl
102
ALGEBRA.
or 12v«4-12v=77i;— 77;
65 77
which gives v* v=i-- —
19. 1
12
12 ^576~57G'
. „_65±23 88 or 42 11 7
24 24 ~ 3 "^4*
£'////«• value of v will answer the conditions of the question ,
''' - 12 12 48 48
but take v=-.; then i/
v-l 1-1 7-4~~3"" '
and y=4,
7
a;=vy=:-x4=7.
4
Hence, the numbers are 4 and 7.
i
Ex. 3. Find two numbers, such, that the square of the
greater minus the square of the less may be 56 ; and the
square of the Iessj»/ws ^ their product may be 40.
Ans. 9 and 5.
"^r.J^^^^ ^''^ *^° numbers, such, that 3 times the
square of the greater plus twice the square of the less is 1 10 •
and half their product j)/«« the square of the less is 4. What
are tne numbers?* ^«,. 6 and 1.
i
i
CHAPTER VI.
ON ARItHMKTICAL, GEOMETRICAL, AND IIARMONJCAL
PROGRESSIONS.
1?' ^f ^A^^^^ o^ quantities increase or decrease by the
contmual addition or swi^rae^/o/t of the same quantity, then
those quantities are said to be in Arithmetical Progression.
llh^7n^ ^^""^ ''*"i°*^ °^ questions relating to quadratic equations
hicn contain twn nnV-nnwn r,„^^Ht.i t»i ..^ > . • . ^. _ '
Which contain two unknown nnantiMV-
tern*. "^ "'"'"*
•^-■jiutaicai j-ror.
I or 42 11 7
24 -=-3°' 4-
ns of the question ,
48 48
■==3-=16,
<4=7.
the square of the
iy be 56 ; and the
ly be 40.
Arts. 9 and 5.
, that 3 times the
of the less is 110;
fie less is 4. What
Ans. 6 and 1.
IIARMONJCAL
)r decrease by the
Tie quantity, then
'Ml Progression.
quadratic eqiiationa
mmrnm^'.^^l^i^
AHITIIMETXCAL PHOGKEobiO.,.
J nils the numbers \ o. n a r n
the addition of 1 ^o ,\nh f' ' ''.' ^' '^^'- (" ^icli increa,^' U
^-- V^2 from :^ach%u Lt ;i J^^"^^ '"''^r V che .;./.,..;
H'o series itself bo exprelerf h """"'? ''i''"""^- ""m r ,v
"eco.-di„g «s rf i. po.mlltZ^ZT'"' " " *-™"«/one;
«"'e:, 1. e., the coefficient of j/^r' ;"""=>'"■'« it is ttr«,
««.Jy than the number Vhieh ITJ '!,™ 'f """^'^ '« '"
'" '^•^ »«■«• Henee, if the numhel /t'^'"" "y'""' '"■»'
be denoted bv (,A tV ' ,u """iibei- of tenns in the serie,
w"i be aH'L^'^i't:%iJzt 'r r ^^^ ^^^
^- then ^ "''' " "le /ith term be represcnte.t by
I? , T.. ^=«+(«-iK
i-x. 1. Fjiid the 50th term of H,n • ,
TT Luxixi or the series 1 3 ^^ -y ^
Ex. 2. Find the 12th term offk
'" '^^^"^ or the series 50 47 aa a,^
Here a= 50 1 • / rt\ , /,« ' ' ^' ^^-
Ex.3.Fi„dthe25th~.e™„f;;,„,3_,_3_^j_^_^
'^"- " ^'^'" ' IM,,f "
'>»f4n-l!;S'4| ""'"'"""•'' '"»- ior intennedftTte™!)-
. ^^ere the number of terms i^ fi ,.0 1 ,
!!!:!!::::^^;:^fs|~m^ «■ -™%. ;h„ c^te„ns to b.
V
fi
104
ALGEBRA.
! It
m
m
I II I
')i;
I : I J
I !
I i
0= 1 / But a-f (n — 1) d~l
l=^4S[ .-.1+7^=43; .
n= 8 y .-. c/=0.
And the means required are 7, 13, 19, 25, 31, 37.
Ex. 6. Find 7 arithmetic means between 3 and 59.
Ans. 10, 17, 24, 31, 38, 45, 52.
Ex. 7. Find 8 arithmetic means between 4 and 67.
Ex. 8. Insert 9 arithmetic means between 9 and 1 09,
74. Let a be the Jirst term of a series of quantities in
arithmetic progression, d the common difference, n the num-
ber of terms, I the last term, and S the sum of the series :
Tiien
^=a+(a+(/) + (a4-2(/)+ +/
and,, writing this series in a reverse order,
Sz=l+{l-~d)-{-{l-2d)-\- ... -fa.
These two equations being added together, there results
2 >Sr=(«+Z) + (a+/)4-(a+/)+ --- +(a+0
=(a+Z)Xw, since there are n terms j
.n
.-. fc(a+/)^ -
(1).
Hence it appears that the sum of the series is equal to th«
sum of the first and last terms multiplied by half the number
of terms :
And since l—a-^{r: — 1) d;
.'.S=.i2a-^{n-l)dl^ (2).
From this equation, any three of the four quantities a, d^
n, s, being given, the fourth can be found.
Ex. 1. Find the sum of the series 1, 3, 5, 7, 9, 11, &c.
continued to 120 terms.
Ilore a= n .., g^ | 2a4.(n_l) ,/ j- x|
d=z 2
«=120J =|2X1+(120-1)2[X-^.
:=:(2+119x2)x60=240x60=14400
f=43; .
J5, 31, 37.
u 3 and 59.
7, 24, 31, 38, 45, 52.
sn 4 and 67.
en 9 and 109.
3ries of quantities in
iifference, n the num,'
be sum of the series:
. - - 4-a.
er, there results
... - +(a+0
^ terms;
(1).
e series is equal to th«
cd by half the number
■'-- (2).
e four quantities cr, d^
d.
1,3,5,7,9, 11, &c.
44
-1)2[X-^.
0=240X60=14400
ARITHMEIICAL PP.OGRESSIOX. I05
^c^^tolo SLV'^ «"- ^^ ^^^ -ies 15, 11, 7, 3,^1, ^5
.= -4l^^^ ^■'^"-')<'l^-
=|2xI5+(20-l)x-4[x«
flf=- 4
«= 20J =|2X^5+(20-l)x~4[x??
= (30-.19x4)xl0 ^
= (30-76) X 10
= -40X10= -460
14, &/ ^"' ^^^«"-«^25 terms ofthe series 2, 5, S H
34,V ^''' ''' -- o^^« -- of the series^- ^30'
li^%f"^^^-""^«^^^^--o^theserie^r^t|!
Herea=|. ''■ ^= ^^+{n—l) dl!!,
^=i i =|2XH(150-l)xj[i^
n=150 =/?4.149\ 151 ^ ^
^ I3+T"/ '^=-3-X75=377.5.
2it &c. ^"' *'^ ^""^ «^ 3^ terms of the series 1, 1^, 2.
^««. 280.
p PROBLEMS.
term? ""J ^^^^^ ^0. What is the /rii
2&
Here A8'-„i 240 ) . c f« , ^ »
•^^" .. >S'=|2a+(n-l)rfl^
'^"■" ^( ^240= J2a+(20-l)> ./ J ?(
-= 20 J =(2«-19x4)10 ^^
l24=2a-76;
•■• 2«= 124+76=200.
Hence the series is 100, 96, 92, <feo.
Ill
I I
IH
' fl
f i t
IN
108
ALGEBRA.
Ptt' 2. Tlic «wm of an arithmetic series 13 H55, the ^r«J
r -W , . eaid the nwnber of terms 30. What is the common
nijferei "^
IIer« ^=14551 j 2.4 (n--l)rf 1 1*=5
«= ^r--. |2x5[-(30-l)cfl^=1455
n= 30 J »t0+29ff) 15=1455;
Dividing both sides by 15, 10+29(f=97,
29£/=87;
.*. c/=3.
Hence the series is 5, 8, 11, 14, &c.
Pbob. 3. The sum of an arithmetic series is 567, the^/'«/
Urm 7, the common difference 2. Find the number of terms.
Here 5=507 1 .-. since J 2a+ (n-l)t/ j. ^=5
a= 7
rf= 2J
i2x7+(n-l)2l^=5G7
7i'+Gn=567.
Completing the square, n'4-6n4-9=576.
Extracting the square root, n4-3=,+24;
.-, n=21 or -27.
Prob. 4. The sum of an arithmetic series is 950, the com-
mon difference 3, and number of terms 25. What is the,/??-*/
term? Ans. 2.
Prob. 5. The sum of an arithmetic series is 105, theirs/
term 3, and the number of terms 10. What is the common
difference? Ans. 3.
Prob. C. The sum of an arithmetic se: ' ' 4t'0, first
term 3. and common diffe.^:dce 2. What >- Hit }.t ."}>er of -
terms P • « -a.^.. 20. |
Prob. 7. The sum of an arithmetic series is 54, the ^rst
ifer»n 14. and common difference —2. What is the number m
tei ' « i
Ans. 9 or 6.
jeries I's 1455, thajirsl
What is the common
30
1455
5=1455;
ic/=87;
cf=3.
JC.
series is 567, the Jirat
the number of terms.
= S
=507
567.
►=570,
1= +24 ;
(=21 or -27.
series is 950, the com'
f 25. What is the first
Ans. 2.
series is 105, theirs/
What is the common
A71S. 3.
;ic seiii'- ' ■ 4*0, Jirst
''hat '*•■■ •■; , J>er of
} series is 54, the first
What is the number ni
Ans. 9 or 6.
AHITILMETICAL I'ROGKESSIOX. j^?
tfnrd, and so on. Jn how man t ^ ^-^H *''^'*^» 20 the
journey's end? ""^"^ ^*.^=* ^'^ he arrive at his
Uere is giver. «= 30 i
SZ ~'iQly''^'''^'^^'^-^^^r of terms.
.-. |2x30+(n»l)x-2|^^=198,
(31— «)n=i98^
w'-31«+
©'=?
■198=
109
n-
4 - 4
31 ,13
••"=2-±2-=22or9.
pos1tL"?lir of rS it"'^ r '"^^ ^-- 'he two
the traveller's arrival at his ifJrn/' '''^ '^i^^-^^^ ;^^r^o^. of
that if the proposed serieiHs^r^'' \^ "^"^^ ^^^-^-'
terms, the 16th term will htL;; ' ' *^?'' ^^ ^""^ed to 22
terms will be «^^«^eW L wh nh "'^' !"^ '^^ ^^"^^'"'"g ^^
traveller on the 16th 'day anfh' ^
^^.c^^OM during the six days foIIowin^"'^ ''u/^'' 'P^''^^'
hini again, at the end ofX 9oTa^' ^""^ ^'^'^ ^^^^ brin«
^vhieh he was at the end of the l^h '^' '? '^^ ^^"^^ P«i»t at
place whence he set out '^' '''"•' ^^^ "^"^^ from the
pthering up 200 sSiS pdntl"^ f!?'" ^^'^ ^^^^' h>
-.^ 2 feet from each other js^tl?^'!?. k"'' ^' '""''''^^^
^l^Oly to a basket stand nffaf-f^iw ^^^™g^^^«h stone
[he first stone, and C^X^Zt^^T^ ^"^ ^"^'^^ ^^^^
basket stands? ^^"^^ '^^^^ the spot where fh^
3 jj
ft:
t
/*
%
108
ALGEBRA.
i-ii
M III
'//Sb."- "'^"'')' "«-»''(#•-- 2 feet, and «„„J„
Here a= 60 ) « f n ».
«=200) =(120+398)X100.
=518x100=51800 feet,
ffenee the distance required=103"600= I'q .""'i""' .'"'•
What i, ««-a^:;'eSsf' ' '"' '"^f 5?ro--
the .a4, aifd so o^' "wfft' ml'^ H"? '*™>-«4"
ohaHt,at.he,endofihe^\™"^^t..''lcrir6Vi'''
before B; B follows him Tl !^ I '"""''^ ""'• 20 mhiutos
« the second, 7 the Mi™ and so? ' f 1"""^^ ""'>"' hour
B overtake A» ' '° ""• '" '"'»' many hours «-i|l
^M. in 8 hours.
arithr«e' proIL'slo^' LT"^ """"'f "^ 1™«ities u,
whose «»» is ela to ll?!- T'.' ''^^'•^«« ^^ 2. and
if 13 be added to heZ„!;r''"'°'';T""'"-' """-"'ver;
by the numiJofV^, S Lto« "; ""?,? ?'^ ^-^ ''<' divided
«™>. What aro^Smberrr """ ''^ "l"*' '» *«>«
Let the/r«< term=x, }
and iVb. 0/ tenns^y; ] ^nen the wcowrf term will be a'+2.
'^ ''' ^^^''"^^'^'^ ^^^+(^^^X|, substitute .for .,2 for ft,
and.for., and it becomes 2l+(^riy3^|(^,y..__ ' "
li>r the swm of the scries.
By the problem, a^y+y^^y^^^^ ^^ ^^^^^^
unci — : — -:. '-!-.,
GEOMETRIC PROGRESSIOX.
et, and numitef
109
ftirbngi,
feet.
G40
?ave 1 shilling
ird, and so on.
w. £110 95.
giving away a
hree farthings
disposed of in
^ lis. 6fd
of 6 miles an
1 20 minufes
hQ^first hour,
17 hours \vill
n 8 hours.
[uantities in
« is 2, and
; moreover,
1 be divided
1 to thojirsi
2.
•r a, 2 for b,
Hence, ^^?=;r, or ;t«-8x=-15;
.*. «'-8^+lC = 16-15=l,
anda:--4= + l;...;r='5or,1,
Fmm which it appears that there are ^^^J'sets of numt,.
...•^"'^'':- ^"^^ ^''''''' '^ ^ ^^^^a^n number of quantities in
arithmetic progression, whose >.^ term is 2 and wW Lm
Am. 2, 5, 8, 11, 14.
ON OEOMETHIO PROORESSION.
ra. If a series of quantities increase or decrease hv .!,„
contnmal ,«,,//,>&„&„ or division by tl,e same qum.Utv then
continual ^^.-Wiif ' ^AfanlVet^^^^^^^ Y '•>;
m,dl,pl,cal,on by J), are in Geometrical Progression
76. In general, if a represents the first term of sncl, .
series and r the common mulliple or ratio then mw h!
series itself be represented by a or or' n^' „Ta. ""y '''«
will evidently be aL-„«..«„/o"r' ZZ^i^; 'ser es „e;,„:S
eri;s"u,e1l;':f '••"^ ''^™^^'--^'™''°'' '" "» °'^°g" g
""'" " "' geoirietncul progression found f
~5
tiie common
"' -mmiikm ei-ii),,
no
ALGEBRA.
iJere the common ratio -?-o
"~1 — ^•
^ 2, rind the common ratio of the series ^ ^ ^ .
3' <>■ 27' ^^
I" this example the common ratio = ^^2_2
9'3-3'
•^x. 3. imd the commo,, ratio in the series 5 1 ^ ^ ^
TO. Lot S be the sum of the aeries a, „' a^ J „
Mu]t,p„ .he eWtion b, ., a„a itbeeot; " ' = ^^
S"i'-ttho.,,.e,„..„,„„.:,:-j-=^
•••'~a=.^^^,orO.-.l)^^„,..
-a;
and therefore, S=^^^~:Zf^
_« the convenienee of ea,cu,ation, .he-efore, it is better
'" *" '"'"' '^ "-"-pose the equation into S - «-'"■" t
'"i'ltiplj-ing the numerator and dnn„ • . ~ ^^^' ^
— - hv -1 denominator of the fraction
r—1 y ^'
™- "■ ' bo the last term of „ series of this Icind „
W"tion.lheref„re,if„„y,h_^/f7'^, '-l' ^""^ *^'»
l, ke given, the fourth maVbefoun *"/"'"' '»"'""■«- ^' «. ^
78. Whnt i« fi,« „ ^ 7 ! ~~ ■ — — ___
^ «'^«'"«trieal K^gr^^i^t" '"^ "'« «»'» a^" « terms of u series of „„n,bo«
cal
progression
4
8
2
3^
2
3
5'
25' *^^-
A
3
ns. r.
o
^,
^c, then
= S.
ar"
=rS.
we
have,
^rs are Jess
it is better
he fraction
kind, then
>om this
?s ;S; a, r,
>nnimb«n
GEOMETRIC PKOGRESSIOX.
HI
Ex.1.
Find the sum of the series 1, 3, 9, 27, &c. to 12 terms.
Here a = 1
r= 3
«=12J
.^_^"^2^_ 1X3*^-1
r-1 3-1
_8P-1
"" 2 •
531441-1 531440
■ 2 =~"2 — ='*^5'^20.
Ex. 2. ^-
Find the sum often terms of the series l4-?4.^-. ® ^
3^9"^27' ^
a..- 1
2
r_= -
3
f?=rlO
3
Now L2)'"=2L»^ ^4
■■-sr-
3'« "59049 '
1024^58025
59049 ~59049'
3x58025 174075
and S=z
59049 59049 *
81^^^- ^"'^ '^'' ^""^ "^ ^ '''"^^ ^f ^h- series, 1, 3, 9, 07
Ex. 4. Imd the sum of 1, 2, 4, 8, 16, &c. to 14 terms.
, , , -^ns. 1C3S3.
Ex. 6. Find the sum of 1, -, - -1 &« m ft f
3 9' 'n' • '^ ^ terms.
'3' 9' 27'
. 3280
Ana. .
Ex. 6. Find three geometric moans between 2 and^f
Herea= 2) And ar-^'^/
/=32
I
• 0..4
2r* -32,
:JG.
And the means required are 4 8 10,
rii
112
ALGEBRA.
iiii !
:ii:|iiF
11
^. 7. .Find two geometric meana between 4 and 256.
P « -P- J , * ^^^' ^^ ^^^ 64.
J^x. a Find three geometric means between | and 9.
p^ -p. , "^^^- i» 1, 3.
T «? tSf,"^^t"« "lean between a and /.
Ihen a, x, /, are three terms in geometric progression.
and -= _
a X
or x^=zal
.'.xz=y^
Ex. 10. What is the geometric mean between 16 and 64 1
Ex. 11. Jr^ert four geometric means between -^ and'sT
Atis. 1, 3, 9, 27.
PROBLEMS.
squares to 21. ^ *^ ^ » ^"^ ^^^^ «wm o/M«V
I'-fc p ^, :^y, be the numbers. Then by the problem,
X
X*
'~,+x'+xy=.21 . . (2)
Trom equation (1), x{l+l^.y\ ^^
'• Vsquanng, -'g+?+3+2y+y]:=49
.% by subtra«tir.ii, a:'^^^^2+2y\ ~I^
or 14z=28
*. X=: 2.
4 and 256.
'S. 16 and 64.
>n I and 9.
Am. ^, 1, 3.
ndl.
' progression,
en 16 and 64?
Ans. 32
1 } and 81.
1, 3, 9, 27.
5 progression,
3 sum of their
problem,
GEOMETRIC PROGRESSrOJf.
Tliis value of a; being inserted in (1),
us
5±3 ^ 1
Hence, the numbers a: e 1, 2, 4; or 4 2 1.
.•.y=
Ans. 2, 4, 8 ; or 8, 4, 2.
nhosT";^- islr^nll'/^'' """J^f ' '" S^^"^^^'« profession
Ans. 3, 6, 12.
^>,o . °°' ^;P°If ''''^ **'''^^ numbers in geometric procrression
's 100. What are the numbers? Ans. 2, 10, 50.
Prod. 5. There are three numbers in geometric proffressio,.
atrr„'^er 'r^ *" ^^ o' *^>-™^ 'j«°i«S
-4/js. 1, 5, 25.
I
OS THE SUMMATION OF AN INFINITE SERIES OF FRACTIONS ,H
ThTvaZ oTr """"'' ^•"' "^ ™--™0O0F Z,N„
THE VALUE OF CIRCULATINa DECIMALS.
«nJ?' ^"^ ^^"''''^^ expression for the sum of a ffcometi-ln
sct^es^hose common ratio (r) is a fraction, is flTl^)
'^'=T=7"' ^"PPOse now n to be indefinitely great, then
'•• ('• being a proper fraction) will be ii defin tely ma//,» »j
* When r is a proper fraction, it is evident that r" dccrea^n as »
increase: let r=- for instance, then r^=l- ..--J= ^-__I .
114
;l I
* 'i:
■ I fl
ALGELRA.
tt^CraTor'::^^^^^^^^^^^ ^-^^ -Peot to a in
v^hen the number of terms is infmite is-^ ^
' 1-/
Ex. 1.
Find thesum of the series l+I + U^ a,, , • ^ .
Here a=:l ) _ ,
2
Ex. 2.
Findtho value.ofl-L-ij _L,. _, . ,
Ex. 3. Pind the value of H-i. -,-1 . ± . c , . ^ .
Ex. 4. Find the valueof l+l+i. . 27 ^ . ^
Ex. 5. Find the value of 1+1+ A . ... . • t""'' ^'
bers composing whiIhL<,''"^f^"^ ^'"■"^«^^' ^^e num.
_____^_»J^^^^ progressions, whose
80. W»iD» to *u . \ ~ ' ■ ~ — ■ ■
uun.bcr ortenua^; alter" '"''"' '""' of ^geometric series, when th.
■ha4i
respect to a in
[ing the valua
'>S' approaches,
infinitum.
' infinitum.
Ans,^,
infinitum,
dna. 4,
'■nitum,
Ins. ^.
tJous me.
the 11 um-
»s, whose
'i when thi *
GEOMETRIC PROGEESSIOX.
llo
1 1
1
common ratios are — ~ * » •,.
10' 100' 1000' ' *c^<^rding to the num.
ber of factors contained in the repeating decimal.
^- 1
rind the value of the circulating decimal .33333 &c llils
d^ecmal is represented bj the geometric series '
TO'^'IOO'^IOOO"^*^''-' ^^^'« >•«' '^^»* is A and com»ao«
1 ^"
10
Hence a= — .
10'
-2
**~10'
.-. aS'=
3
10
1-r
1— L ^^-^
10
9~3
Ex.2. Find the value of .32323232, (fee. at/ m> //«;;,.
32
Here a= .
100'
r=
1
100'
1— r
32 32
1
1 100-1 ~99*
100
Ex. a. Find th« value of .713333, &c. ad infinitum.
The series ^of fractions representing the value of this
decimal are — + (geomet-io series) --1-+-J_ . &.
100 ^ lOQO^lOOOO ^ ^^
100"^ •
Here
,.__
.-. S=
3
1000
3
1
J 1 1000-100~900~306'
10
Iknce the value of the decimal=/~+^)Zi4. .1.^214
\tvy X'^OO /iOO 3oo"^a0(j
107
150'
116
ALGEBRA.
^^' 4. Find the vaJue of .8134^4^4 a
2re a=r-?f 1 -^^^4^434, &c. ad injinittm.
Here a=Jl_ 1
10000
\
1
34
.^=--l_^_iO00O^ 34
'"'=100 I ^~''' 1— i- ^^^o^o-ioo'^y^
^ 100
And value of the decimal =ii . e 81 , ^
^x.5..Findthevalueof.77777,<S.c.a.f/.>,V.n.
100 100^9900-i)900-
Ex.6.
Ex.7
EjT.a
Ex. 9.
« «
9
.232323 &c. ac/ m//»7«m.
M
Ans.
23
99'
M
.83333, &c. arf ^V/^^Vw;;^.
Ans. --.
•'^l4l4I4,&c.a«?e;y?;,,V..m.
. 707
Ans.
u
•956806. &c.<,rf ,■„>,•,„„,
Ans.
990*
287
Let S=: .813434
.-. 10000 *^=8134.3434
a^dj!00^__8i.3434 .' .* * .*
.-. 9900 >S=8053 "
. <._8053
' * 9900' ^^ ^^^"^-e.
-"V "-ves over ^ a mile the ^mw" '"-■^"^^Vg cause, it
*m;,i«r second, | the M,Vd;
fnjiniium. \
f 34
-lOQ-yyoo
8053
A 23
99
im.
ns.
Q'
itum,
707
990'
ns.
im.
ns.
287
300*
found 1
-ause, it
HAKMOXIO PROGRESSION.
117
and so on Show that, according to this law of motion, the
body, though it move on to all eternity, will never pass over
a space greater than 2 miles.
ON HARMONIC PROGRESSION.
81. A series of quantities, whose reciprocals are in arith.
luetic progression, are said to be in Harmonic Pronression.
Ihus the numbers 2, 3, 6, are in harmonic progression, since
heir reciprocals ^, ^, J, are in arithmetic progression I -4
being the common difference). ^ '
Ex. 1. Find a harmonic mean between 1 and — -
3*
Let X be the mean required :
^^" ^' T' "2"' ^^^ ^" arithmetic progression,
Andi-l=i_i
X 2 X
X ^2
5
.*. af=
2
4
5*
Ex. 2. Find a third number to be in harmonic pro^re«8loi
mth 6 and 4. i o ^
Let X be the number required :
Tnen — , ~, —, are in arithmetic progression.
All 1
And - — --:
4 6
1
*
X
1
1_
X 4
-i-i.
2 6
1— i
1
111
1'-'
r.l
I: if
ar=3
118
V i
ALGEBRA.
rhe re.,rocak of 9 and 3 arc I and ^ ., • k T
and last term of an arithmer ' ^' ''' '^'•^^'■^'
amr^Uc means a^tf r^inSeTt' V^^^ ^^^^ 3
according to Art. 73— "^^erted. We have therefore.
3
^=5j
And a+(n— Ijc/:--/^
3 9
.*. d=z
3^
'9
9
18'
9
Hence i 1 ^ l^'
'^6' 9 '18'-^ the arithmetic ..«., to be inserted
between ^ and ^, and therefore tkeir reciprocals 6 ^ ^^
are .three harm^^^ -- -.uired. ' ^' ^'
• '• ^"' ^ ^^^--- — between 12 and C.
Ex. 5. The numbers 4 «n,i p . ^»»- §•
progression; find a third te™ ^^ '"° '"™= »'''' ha™onic
E^-O.Eindt.o harmonic n.e.„s Ween S4 .:!":;"
E..7.InseruhreeharmoniemeansheJ::r5:;fdf
S2. Let a. i c t/ ^ ^ T. 2 ' ''' 4 •
*«-».p4re:sio;:;hli'i%"r7°'"^™"«"-"
« ' b' c ' (/' 7' ^C" are in arith
and 3.
are thejlrst
een M-hlch 3
} therefore.
inserted
' 2' 5^'
ns. 8.
trmoiiic
'. 12.
56.
163.
is.
15
I*
ties in
arith,
HARMONIC PROGRESSION.
119
a^
From (1)
-i-i-i- 1
b a c b
..(I)
• ■ - (2)
^ c e d
(3)
&c. = &c.
a--6 h—c
ab c
a—h b—c
a c
a a—b
c b—c'
o converting this equation into a proportion,
a:c::a — b:b~c
Similarly from (2) b:d::b^c:c-d
(3) cie-.-.c^did—s
and so on for any number of quantities. .
These proportions are frequently assumed a/fh*. /.w •.•
to quantities in Aamo^^zc progression nnf,!. 'Y'''^!!'''
expressed in words-— if anv//,;.^T!l.. "^^>' ^^ *^"«
gression be taken M. f^, / quantities in Aa,v«on/c pro-
a«=jy; .'.a—b»
c*=b^', .'.c=bi
(1).
r2\.
By multiplying (1) by (2) ac=b>
! +
■J, • ; I
■I
ii
i-i;!i
-s?.«^^M«e.t*-»n*f^»^t«.v;'J^
I I I
$
=1
V Vl
120
ALOEBKA.
l^ut by geometric progression ac=A»
and .-. 2=il+il
y X z
or i-i=i_l
y X Z y'
, CHAPTER VII.
ON PERMUTATIONS AND COMBINATIONS.
83. By Permutations are meanf t]io «., i,
«-lHch any quantities, n i c Te V ""^^^ ^^ ^'''«'^^^'
spect to theirorder, when takon'/ ' T ""^'^""^ ^'^h m
f , «?., are the difre;ent pe^ml^e^^^^^^ ^«' ^^' < ^«.
^ ,<•, c/, when taken two aTdtZfLf^^ ^T ^"^"tities a,
7^, ci., of the ^Am quantifies Vr/.' "*'', "^•^' *^^' *-«
Mre. together, &c., &j"^"^'^*'^ «' *' <^. when taken ^Are, „ J
'^'^m:^z:z:t^ th., hy
(«-!) be substituted for ;. in tho l^^Vo^-T''?''''''^'- *^'^'' '^
of permutations of n-^l Z!t ft ^'^f ^^' ^^^" ^he number
will be (.-1) (,^,) ^ jJj^^X '±Vr.f!!' ^- together
-> c, a, ^ O.C., taken t^o and two\og^^ a^e'^^J^;:!!^;
fsiWi, ^r.^Agmm^^,*^^me'—- ■.
PERMUTATIONS.
121
f changet
» with m
her, three
•*, cf/, </«,
m titles a,
<'>'0'c, Acer,
?Aree and
then, by
ations in
B(n--l)
0^, ^, &o.
■he form
e. ; i. e.
d two
iS
)gGt her,
For if
lumbei'
3get her
/ions of
quantities a, 6, c, d e^, &c, taken //.r.. fl;.rf three together ir
which a may stand first for the same reason there a?e (n-U
il;i ^' ""'"*'' ; ''' '^^ permutations of his kind viU
therefore amount to . (/i— 1) (w— 2).
r ^fetHer. ^'''^ '^'' """'*'' '>/;>^m./a^/o„, of n things taken
By Arts. 85 and 8(5—
llie No. taken /m;o together — ;i («- 1) --"
" " '/'^^^ " =n (n-1) (,,«o)
Sm.dar]y " four « =n (n-1) (n>2) (.-3)
supposed 'tX'ttLZ'^^t is ff ^;h^ P""f ^^ ^'^^^•^' '^
taZs of. things rcf !('^t tXi; t ^rgetw r--
n (,.-1) (,,^2) (/.-r+2)
will be by putting .-I ^ :):: ^X^^;:-^^^^^^-'
{n-l){n-2) («~r+l)
Now,^if a be placed before each of these permutations, t^'^ore
{n-l){n-2) («-r+l)
ft1s"ciraftLl'e>"-ir''"ry *^^^^^"^' ^" ^-^'^h - ^^-^d^/'--^^.
ic IS Clear tnat there will m like manner be the same nnm)u.n
here arl IS *' ^'h? *!•' ^'»"d mr.o&./y /„, ,. »,„, „,
.vel_y_sta„d first,, that is, „ times^ri)'' [t^^g] "^.'•f';';
'or«(„_l)(„_2) („_,.+ ,)
nJ! • *"\''<*™ P>-oved, that if the law by whioh the c^
^TI^JH^l ".'"""-.of permutations of ^, thln's t^kt!;
- "-„„_.:ci 13 iouna, oe true, it is also true for tho nov*
uperjor number, or when „ tiling, ,,« tak „ r,i::, !
but the law of .he expression ha? been found to l,f,lj fi','
5 XI
1:1
p»»«(f^.^«,»««^^jf^,,,,.V.^^^^^
122
ALGEBRA.
m m
taken four togetfer k U t.^ T '°«f ''^'•' '"'^ '^ "■"« «to
and si on fo?any number no? „ "/""T ''^™-^^'^ '"S'^'her,
taken together ^ ' «'''''"*'■ "">" »> »Weh may be
mathematical sconces ® ^''"" "''l>°'-t»»'» in the
titicI'al^;=%Ln (Itn'-^'tr "^"f "" *^ •)"""-
mutation, which might be' fo;m;dV,.^^,'^,*,^""™''''• "'' P"
the word •.«,.„... aVflxS/xTxiLlar '°'"''°''""
thefit^t\^trbi::tutSrrTr"'''-°^'''"-.
mutations by the number Tf 1? .' "^^hole number of per.
arisen if .^/j;j:ZsXScTi;,st::d"Si"°^^
of the same letter. Thus if the samrilft 1 Z,''^ repetition
then we must divide by 2x 1 ?/,•!"■ m" °°™'' ""'«'
.nust divide by 3X2X I ; if p t ,ncs bt f of ''"" "T' ^
any other letter which mnO „„i ^ ,• • •■?'! »"'' ^o for
the^eneral e.xpre2"n forV , IC'o^f ?" Tt^ ^'^™»
thmgs. of Which there are , l^'T^lX IZmZ' "' o'
another, &e., &e is ^ii!iriijfcl2H!ii:3) • . 2.1
the permutations which may be formPfl Z.r ,i \ ..
posing the word "m.m..."iince .TccmVl; ' ^'""'' T^^"
8.7.6.5.4.3.2.1 '''*'^' ^ ^"■'''') '^^'^
1.2.3. ><T2~~^^^'^-
can\^; U'":;^VlTat^Sr,r^^^^^^ «"'«'■
The number=l X2x3x4x5xO=T20.
-pS'biHr'"' ""' """"''" °f '''""S- -»"<=•■ can be rm.
The No. o. <">a„ges=lx2x3x4x5xcx7x8=40320.
^^^^mm.
w together
taken three
smonstratcd
* true when
i-'e together,
ich may be
nonstmiive
mce in the
the quan-
' them will
>er of per
composing
I* of times,
^erofper-
ould liave
repetition
cur twice^
'hrice, we
nd so for
. Hence
ions of n
her, q of
. Thus
Bi's com-
oice) are
ts which
be rung
0320,
COMBINATIOXS. ^23
Jl\I:t " ""S^ °f ''"'■^'-' «<"»"-■ how ™a,.y u.jnaU
The number of signals, when the/aj, „e taken-
Singly, are
Two together =5.4 . .
Three . . . =5.4.3 - .
Four . . . =5.4.3.2 -
Five . . . =5.4.3.2.1
= 5
= 20
= 60
= 120
= 120
.'. the total number of signals = 325
ietfeS,tuti ariiiT""""'"' ■=» ''''''7:^:^^''
^I'f ^Ai ^^^ ^"f"^ permutations can be formed out of th..
Xn af S ""' ^''"^^'^ '-^-'™'^. ^. thetSf j!^
Aris. 2o20 and 1680.
ON COMBINATIONS.
w|^!h ?j„ttr:7onrofr;t,rtr/f t"'
are the ««6»„&„ which ean hef^tl/lfJl.'/'
quantities a, i, c, d, tal<en too and Jo toUther at { J
icd, bed, the combinations which may bo formed'„nr;,f- ,j '
same quantities, when talcen three and iree toXll &c'
i.n..;e.iiately deduce the theo^^t ' & ui ' th TmtvTf
r»«4»,a .o,« of,, things tal<cn in the same manner H'oMhi
IT'TIT "' : '^''T "'''™ """ «'"' '-together Ling ,
t« - 1 ), and as each eomb,m,lion admits of as many »ot«,,2„
us mnjr bo made by two things (which is aTlf tlTnun r
li vXd by rreTnumb^""' r'" *?-""""'^^ 'Ut^r,;':
uiv.aca By ^, I. e. the number of comSwarions of » things takes
.'"'0 and two toccther is " ^" ^) v^^ .1
i.uj,i.im,i IS ^ ^ J, or the same reason, th«
79 I
ALGEBRA.
co^nbinations of n things, taken three and three together, n.usi
be equal to !Lferi)_(!i-2) ^ .
1.2.3 ; anci m general, the combhia.
'S^lU^S^^^ together »„. bo e,u„, to
The number - 8X7x6x5 x4
P o ..r. . 1X2X3X4X5-^^-
be t;nL^Sfof 6 oT''' ?r ^'.' "^ combmations which can
lurinca out ol 6 colours taken in every possible wav ?
No. of combinations when the colours are taken-^ ^
1 at a time
4
5
G
_6.5
~1.2 - - -
6.5.4
1.2.3 ' " "
_ 6^.4. 3
"1.2.3.4 ■ '
.6.5.4^3^
"1.2.3.4.5 "
.C-5^.jk2J.
"1.2.3.4,5.6
= 6
= 15
= 20
= 15
= 6
= 1
Hence the total number = 63
_Ex._3. Find how many different combinations of 8 loftn,,
i^ken m every possible way, can be made. ^^'"'
%♦ Several other useful and interesting subiects of ^Z' ^^^'
character yet remain to bo treate-l^ofTh-E^litn^! '""*"'"'"
for puMacafon a Second Part, wh.da will elnbrltlULa^"?*
gether, musrt
he combina-
be equal to
hich can b«
is which can
! way?
APPENDIX.
MinCBAL PRINCIPLES, PROPERTIES OF NUMBERS AVn otu-t,
8 letters,
s. 255.
ilementary
preparing
e subjecU
ON THE DIFFERENT KINDS OF NUMBERS,
1. A number expresses single unif« nr na^fa .e • .
m-.^, or at the ,,a,„/timo, unit? and pXfSs °' " """''
By unit we understand a whole 1 ai-d hv Tzt, e
c- ^fraclion, all that is below the vkue o/l" ^"'^ "' " ™"'
2. The number which expresses units only is called a whnh
uiL.^ hucn as 1,2,3,4, is called an abstract mimlwr
J; :»r."''CTh";r:^d^wf.hi «« '5 "7 '^vV-''^/";
an odd number. ' ' ^' ^' ^^ ^' '« called
V^c, is called a comj./.a: numberr ^~ '' "' "'^ '^'' '"' ^•■*' ^^' ^•'^
1]*
126
APPENDIX.
The exact divisors of a complex number (or multiple) ar«
called submultiples. Thus, foi example, 1, 2, 3, 4, 6, 8 and
12, are submultiples of 24.
¥ I
ON THB FOUR RULES OF ARITHMETIC.
6. By Addition we add two or more numbers so as to
make but one. The result is called sum or total.
By Subtraction we cut off a number from another num.
o ??^ ^ ^^"^*^ ^^ ^'^^^^^ ^^'^ ^^"^^^^(^^r, surplus, or deference.
By Multiplication we take a number called multlvlicand
as many times as there are units in another number called the
multiplier. The result is named the jororfe^c^. The multipli-
cand and the multiplier are called the two factors of the
product. *
By Division we ascertain how many times a number called
the divisor is contained in another called the dividend. The
result IS named the quotient. The dividend and the divisor
are na:ned the two terms.
7. The unit neither multiplies nor divides.
8. To multiply by a number less than the unit is to di
nuimh the number multiplied ; hence it follows that to mulli-
pi'/ IS not always to increase.
To divide by a number less than the unit, is to increase the
number divided; hence it is that to divide is not alwavs to
air.uiisk. "^
9. Of the two factors of a product, the one being multi-
pbcd and the other divided by a like number, the result re-
mams the -lame.
10 The two terms of a divisor being multiplied or divided
by a like number, the quotient remains the same.
11. The general product of several numbers is always the
same in whatever order they are multiplied.
12. A quantity multiplied or divided by a number give"
the same product or the same quotient mul'tiplied or divided
successively by the factors of that number.
13. If the quotient be greater than the unit thp. a\rr\Ai^i'4
IS greater than the divisor, and vice versa. If it be the unit
the divisor is eoual to the dividend.
Ml
idMh
APPENDIX.
dtiple) are
t, 6, 8, and
127
1 so as tc
ther num*
difference.
iltlplicand
called the
3 multipli-
rs of the
ber called
^.end. The
le divisor
is to di
; to multi-
3rease the
ilways to
ng multi-
result re-
r divided
ways the
)er, gives
r divided
the unit,
14 If the product of two factors be less than their sum
It IS bpcausc that one of them is necessarily the unit.
15. To d<mhle treble, quadruple, centriple, &,c., ii nmuhcr,
IS to multiply It by 2, 3, 4, 100, &c.
10. A quantity multiplied and divided in turn by a Hke^^
^nmber, becomes again what it M^as at first. In this case '
Ind dividint" '""' """^^ '' '" ^'^^^"^^ ^'^h ^"^^h multiplying [
17. The product of two nmnbers divided bv one ol ^
them, gives the other. ^
18. The quotient multiplied by the divisor gives the-
Jvidend; the dividend divided by the quotient «iv?s the-
divisor.
OK THE TWO TERMS OF A FRACTION.
19. ^very fraction is composed of two terms: the fr<t
mentioned is called the numerator, the second denominator
ihe numerator indicates how many equal parts of the unit
are contained in the fraction : the denominatk- gives the name
(»i those parts. ^^
These two terms of a fraction are assimilated to the two
crms of a division. The numerator represents the dividend •
the denominator the divisor. >'«t.iiu,
20. If the numerator be equal to the denominator the frac-
tion IS equal to h Jf the numerator is less than the denoin-
n.ator, the fraction ,s less than 1. If the numerator is
gi cater than the denominator, the fraction is greater than 1.
21 Of two fractions having the same denominator, the
greatest is that which has the greatest numerator. Of two •
fractions having the same numerator, the greatest is that
which has the smallest denominator.
22. To render a fraction greater, the numerator is multi
pi.e. without touching the denominator, or the denominator is
divided without touching the numerator.
lo ronder a fraction smaller, the numerator I^ divided mth.
out torching the denominator, or the denominator is m„lt;.
pnca witijuul louciiing the numerator.
23. Tlie two terms of a fraction being multiplied or dlri
d.vl by a like number, its value remains the same.
irt
I f
S f
lii
it
1123
APPEN DIX.
24. Any whole numoer may always be put indiferently
vundei- the form of a fraction : ^he only thhig is to give it the
unit for denominator.
25. To take any part or fraction of a number, is to mnlti-
ply it by that fraction.
Thus to take the | of 12=12x| = ^^=8.
Then if the fraction to take has the unit for. numerator, \vt
have only to divide by the denominator. Thus to take the
h- h 01" i, &c., of a number, is to divide that number bv 2
3, or 4, ' *^ '
20. To incrcfise a number by any fraction of itself, is to
multiply it by a new fraction whose denominator equals the
sum of the two given terms, and whose denominator remains
the same,
Ttus to increase^GO by ^^j=60Xi2 = * j§"=:85.
ON RATIOS AND PROPORTIONS.
27. The ratio of two numbers is the quotient of the first
number divided by the second. Thus the ratio of 15 to 5 is 8.
The connection of two equal ratios is called the rjeometrlai,
. proporUoa. Thus 15 : 5 : : G : 2 is one proportion, as the ratio
ot 15 to 5=3, and the ratio of 6 to 2=also 3.
The first term of a ratio is named antecedent; and the
second consequent.
Ihe first and the fourth term of a proportion are called
ihQ extremes ; the second and third are called \kii. mediums.
The mediums liiay always exchange places without dis-
turbing the proportion.
, 28. The product of the extremes alwavs equals that of the
mediums.
29. We determine the fourth term of a proportion whnn
' unknown, by diuding the product of the mediums by th fir:^•t
; \erm.
m 1
ON TflE SQUARES OF NUMBERS AND THEIR ROOTS.
The sqtiare number or second power of a number is that
number multiplied once by itself.
Thu ,sqi,<ne m-H. or second root of a number is the very
number which has been raised to the sauare.
• iSiiilfe^-jii'saai^ JiA
APPENDIX.
differently
;ive it the
5 to multi-
orator, we
take the
iber by ",1^
ts:clf', is to
:quals the
»r remauis
129-^
* the first
► to 5 is JJ.
eometricui.
3 the ratio
• and the
ire called
'iiiums.
hout di*
lat of the
ion whnn
y-th first
)TS.
r is that
the ver^
Tins IS the. natural series of squares up to 100
Squares. 1 4 9 16 25 80 49 64 81 100
Joots. 12 3 4 5 G 7 8 9 10
(->bserve that the successive difference befwpnn f l.o
*iways exceeds itself by two un ^^ "lT.n= f '^''*''?'
30. A number which is i ot a nerfprf cmm^^ • ^^ ^
<^oot, add to 25 the doubll of 5lV^~i i* ^? ^'''■' ^ "' *^^
25 + 11=36, with tlu^l^lfe'^"'^ ^'"^^ '"''^ ^^^^^
off^trlt-^I^ZrSot'r ^ --^"^-'-t i.
Le the number be 53, of which the v/=7+thc rfm.ind..
-15 cut'offfh'' '^' ^'.°'i' '"'"^ '''' do^.bIe of the "ot 7 +
64-15=49, of which theXT ^~"^ = ^^'-^«" ^^''^Ihave
given number, ^'ampl . ' '''^' '^' '^^'^^ ^^'^"^ ^he
Sujiposc tiio number is 86, of which tho /-o_Lfi
>iov 5. Ju order to have tl^ roo -ft f,! '^~ i"^ uf ''''^'","-
rf,ot9_i_]7 nr]>^ ihl ^'^.^oot=:8, to tho doublc of tie
^2. Then l£lSf_nI ^^f ''^•"/J^r 5, you will have 17+5=.
34. Knowing the differonco of tw
-xtractei
M
iu:nb.3rs and that <jf
m
|:J^H
s^^H
;..■
H
180
APPENDIX.
B* 11
their squares, the quotient of the second difference divided by
the first, gives the sum of the two numbers, which enables m
to determine each of them immediately.
ON THE FACTORS AND SUBMULTIPLES OF A NCMDER.
35. Every number has at least two factors; it may I.it»
more If it have but two factors, it is necessarily a r ;•/;««
number, (5 ) and those two factors, consequently, are that
number Itself and the unit. ,. ^
36. To reproduce by multiplication a number which has
niore than two factors, the greater factor must be muliiplied
by the lesser ; ^
Or the factor immediately inferior to the greater by the
factor immediately superior to the lesser; so on, all through,
always following the same order. Example •
Q f 'f Pq "fo ^^ 5'i™^^'' ^^ ^' ^'^'^^ ^^^ ^he cifht factors, 1. 2,
^, % 0, », 1^, and 24 ; to reproduce that number we shall have •
24X1=24, or 12x2=24, or 8x3=24, or 3X4 ^-24.
In this example the quantity of the factors is even: if it be
ooTrf then It is the medium factor which, multiplied by itself
produces the number in question. Example • ^ »
o !"EP?^^ *i!f "7"^"^^ ^^ ^^' ^^'^^ has the seven factors, 1,
*, % o, 10, 6Z, 04 ; to reproduce that number we shail have-
64X1=64, or 34x2=64, or 16x4=64, or 8x8=64.
^ From this article we shall deduce tiie three following prin-
37 The greatest factor of a number is that number itself,
and Its least factor is the unit.
38. The greatest factor of an even number (that number
Itself excepted) is always the half of that number, and its
least factor (the unit excepted) is always 2.
39. Every number which has an uneven quantity of fio-
tors IS a square number whose root is the mean factor.
40 Observe that every number has always a submtdthU
less than it has factors. Indeed a number, whatsoever it be.
hgures Itself amongst its factors, but never amongst its sub!
multiples. °
piication of two of its sub multiples, the unit will never be
diviclo4 ^y
enables ua
MDER.
may I.^t?
y a /r/?««
r, are that
which has
multiplied
T by tlie
1 through,
Jtors, 1, 2,
hall have :
iur24.
; if it he
by itself,
'actors, 1,
Kill have:
S=64.
r'ing prin-
3er itself!
number
. and its
y of fio-
r.
or it be,
its sub-
le mui ti-
le ver be
appe:ndix
131
;l
^1^^ tS:^^^;f ^' ^^^^"^ ^^-^ ^t -not in any way
without exception ti i^fuUi;!^^^^ ^'^ """^^^' ^Ij
• ON ODD AND EVEN NUMBERS.
Thl'of^tto'X'XrTl" ir'"^ ^^'^" even number.^
an ... n.nbe. andt^^j^^irbrisr.r n.^^^^^^^ ^^
^^^^oV'^'t^^^^^^^^ J- --numbers is an e.en
number. That between «n. ""x!"^""' '' ^^«^^ ^^^^ -^'^
i.s an odd number ''^ ""'^^'^ ^"^ ^" «^^^ "umber
Thlf JMrLlb!^^ r— n"-ber.
-- number and an ^ddZ^Ci.t r ntnbJf " ^^ ^"
^,U,^ Every oc^.„un.ber has only o./cf factors and submuL
n^^^^'ZZ:^, "-'''' ''^ -^^^^ --Ption of the
ON PROGRESSIONS.
4G. Tne arithmeticnl vronre^^inn 1= «
cessively increased or dimSed bJ . ri.'''' ''^ *"""^ «"^
former case the pro JsZl is o.lL^ • ' ^"^"''*^- ^» ^^^
latter decreasing. ^ The Sel b f ^^^^rm^^ng, and in the
proportion, "^ clitteience between the terms is called
The geometrical prooreisinn \a n cr. ' ly
ly multiplied or divided bv. l if ' ""^ ^^ *^''"' successive.
«i«etlmVogression ssaid^toL."^"^^^^ ^" the former
^^:!^:::?^;L:^;ir :::i ^J: -? -i^- p-.-io. ;
ii'^ terms, mid the pyoporlion or ratio' ''" '"""' '"" """ ''■''
47. n,e last te™ of an ar.thme,:;., p..„g,,^i,„ ,.3 ^,„
ii
1S2
Al PEXDIX.
lili
posed of the first tern, more us many times the ratio a» Ihert
are terma before it.
^ Bxamplc-Su^pose -2, 5, 8, 11, 14, 17, of which the ratio
19 S. Kemark that the last term 17, which has Jive terms bo
Kire^i^n 19 composed of the first term 2+ (the proportion 3x
^ 48. The sum of the terms of an arithmetical proffrcRsi( n
js composed of the sum of the first and last tefms. muKipliod
by halt the number of terms. '
^a«i^/..--Suppose -5, 7, 9, 11, 13, 15, of which the half
of the number of terms=:3. Then the first terms 5-ftho
last 15=20 whichx3=60, the sum of the six terms.
49. In all arithmetical progressions the sum of the fir«»t
and last terms equals that of the second and last but one or
that of the third an(| the antepenultima, and soon all through,
:5till observnig the same order.
Example -Suppose^4, 7, 10, 13, l^^ 19,22, 25. Remark-
that the 1st and the 8th term =4 +25 =29
that the 2d and the 7th " =7+22=29
that the 3d and the 6th " =10+19=29
that the 4th and the 5th " =13+10=29.
^ In the example just given the number of terms is even U
It be odd, It IS then the double of the middle term, which equals
the sum of the other taken two by two as above.
^oram^^^.—Suppose ^ 1 1 , 1 6, 2 1 , 36, 3 1 . Remark
that the 1st and the 5th tcrm = ll+3l— 42-
that the 2ci and the 4lh " =16+26=42 •
that the 3d doubled =21+21=42.'
50. If the number of the terms of an arithmetical pit.,
gicssion be odd, the middle term always equals the sum divi.
ded by the number of the terms, that is to say, that if tl at
number be 3, or 5, or 7, &c., the middle term =4 or ^ or 1
Kc, of the sum of the terms. * ?'
^^a;w;>/^._Suppose -^10, 20, 30, 40, 50, of which the sum
= 150.
Then that sum divided by 5, the number of the terms
— 5 =30.
Hence, as we see, the middle term =30.
If the number of the terms be even, the two mea^is taken
€
r
J^|^|-f^^jrams '■-
tio a* Iherft v
h the ratio
' term? be
ortion 3x
rogros'ilc n
Tiultipliod
h the half
IS 5-f-thu
IS.
' the first
ut one, or
1 througli,
. Remark
Al PENDIX
133
even. \{
eh equals
rk
ical pit),
ium divi.
it if tl:al
r \ or f
the suiw
le terina
\s takes
i
tS"' '^'"^ ''^ ^"" ^'^'^^^ ^^ J-'^ the number of the
.umlTof lfo?lT5 d'-.'^A'^'^' 2^' 3^' «f -h>ch the
terms=:io5-35 lln^Af ^ ^^' '^^ ^'^^^ ^he number ol
first'mdtifiS w'h^^'ati'riSrr P^^^""^^'"' ^^"'^^^ -^^
equal to th^ num/er of the termsll " ^^'' '^ "^ ^^^^^
The,,, oWve tlrl^riS— fel^Tx1.^=i^'-
term is takon from the product 3lH ^ ■ ""f " ' ""^ '''^'
by the ratio, lessened by a uS '''""""'^" '' ^'"'^"''
rJ'>t:t';fsrz:-o!o «/'« = ^, ■■ '«2, of which ,,,0
=4b. From 480 „".;« the fiTt ™ o" '^™ '"^^^
Now tl,is remainder 484 : (thera.ro t-xtZT^ll^!; "^^
"r the terms. " ^^— j — -;4^, sum
p.y'^^r'i/V^o" tf:if -' P""f'-P'- o„ progression „p
applicable t:^„wrr;pfoS„?iri,o"', '" """'" "■■™
change the words/,.,!^ Crdlf, ilL^r"""^^ '^'
ox WV,SIBt„ NUMBERS w,T„OUT A REMAINDER
titule o'^ t™nrS,:^S:ferthr '''r '" " """
bcr exactly divisible bv another Th '^"'^^'' ^"^ """'
n-de without any remainder ''"'' '"" "^'^-^ '-^'^'^^^ «''
oonsKiered as simple unit ™!w it ^ll^^^rt"^ '-^"■^^'
rigl^atV^diSelfr ^^'^^^^^^ ^^^^ ^^^ ^^ %-s on th.
n; ?" nil ""'"^'''' germinated by a 5 or an n
i>> 0, all even numbers a
lH
12
idy divisible by 3.
i I
iU
APPEXpiX.
> ■ill
I' Jili
.if'
•lit
, I
000 Vc^' '^^^' ^^^^' <^Cm all numbers ending with 0, 00.
^J%}^\fSTJ''' ''''''"" """^bers whereof the sum of :hi
4rh n h ft.. !. «'''f ' ^'^•' ^' ^^"^^ ^'^ the sum of the 5>.]
ik 19 II ^^-'OrwhosedilFerenceis 11 or a multiple of IJ.'
Ti o- 1 ''^^'' numbers already divisible bv 3 and 4
liy ^o, all <;i;en or «/jei;g;i numbers whereof the two last
hgures on the right are divisible by 25.
PROPERTIES AND VARIOUS EXPLANATIONS.
4-fhi' SI ^^"^ TT^^l 'iT^'"" ^^^ S^-eatest equal (the sum
-f the d flerence) (fiyided by 2, and the smallest equal (tlS
^wm-the dfence) divided also by 2; whence it follow
that the sum mcreased by the difTerenee equals .'e./crthe
greater number, and that, diminished by the difference il
equals twice the smaller. ^ -aicicace, ii
,.t^'Jf ''*^?^ two unequal numbers, wiihoui altering thetr
sum the greater ,s dmanished by half the difference, mid the
lesser increased by the other half.
50. The difference between two numbers cannot be superior
LpiTse" ''' '''-''' """^^-' ^- ^^ -^ ^^-» or
67. The least difference possible between two whole num-
bers, even, or between two whole numbers, od I, is 2
58. Knowing how many times A is greater 'than B, wo at
o^ice perceive how many times B is smaller than A. pre e7wno
the same numerator to the given fraction, and forming « new
denomniator from the sum of the two terms. Examples-
A being > B by ^, B is <A by 4-.
A being > B by I, B is < A by /j^.
stantly determme how many times B is greater than A pre-
serving the same numerator to th. given fraction, and flVrmTno
a new denomniator of the iriven ,hmnu>u.., ^„ a, . _.u. , ",\"8
numeratcn- is subtracted. Axamples : ' •— u uio
figures on
hose ligurc:!
vith 0, 00.
sim» of :1m
of ilie JM,
tipleof li.
and 4.
e two last
APPEXDIX.
(the sum
equal (the
it follows
twice the
fcrence, it
nng tketr
e, and the
e superior
equal or
lole
II urn-
B, wo at
reserving
ng & new
npks :
3, we in-
• A, pre-
formino
A benig < 13 by |-, 13 is > A bv t'
i&c
of the givol fractiot iw«> '°™"^"* "'« '»•" '«■•-
Aboing|ofB, (i=i„,^A.'
Abcingjof iJ,U=|of A
co,fmi„f;h:ii.iit:„rw'„ ir '•■"' r"^ "^ •- ""-"'-«.
number to the o.^.^^Z^'Ch.Z^''. P"''""'"" "^ --
«"« its den„n,i„ator. S seco, d .w'™'''";, "'^ ""= ''■•«''
proportion of the lesser m,nbert„ .h *'" ""^'^^^ ">'■•
it will expres.s the nroporZn of , f S'™"'"""' """^ "»"^''i
i'-^am^k.-T.^vrtT^'^Zl 13 T'"f 'V"? 'r"^-
12 and the difference 2- the ™!„ „ . ' °^*l'"''' thesu,n =
the other. Then 6 in 'a ,Ltbn rC'''^ ""T'^"' « "'"'^
second fraction such as it i, we «1> S, 1='' "'"' '^'"«"'g «"
t'.at the lesser nn,„b:r=V:fThV",,t-'' *• "»' - to' sav
the s„„,=95 ™f Ihe d Le,T-'°" ' " ^'"'^ ^=- "^ -^^
«,„. o.:.__ . . ^ aincrei.ce 2. -ie ,„e contains, there
ore 3 tunes J the other. Then
t differ bv 9 fh*. ^.-fli; "> by or a multiple of 9 W
itditrerby^^fotb;?"^^^^^^ isV' ,f
the two figures is 2 3 W r^ \^^'' ^^^<^'fferencebetM.> o
fta Tk« II , "'"erence of the figures H nnA i t
tW. The smaller the difference -^ hr.u. . ^~^-
No. 1.
Sum a
3x5=15 Sum 9.
,4x4=l(>
^X / = 14
3X6=18
4X5=20
«
' fu
I'if
186
APPENDIX.
1st. The one of the two numbers which, laultiplied one bv tho
other, g.vc the smallest product, is alwkys tht un^C^hethe
thi; sum be ocW or even; ' ''"*'"'*'^
'o , f.'f, ^^'° S'-eator product is equal to half the sum;
od. if the sum he odd, No. 2, the two numbers M'hich give
the g, eater product differ between themselves by the unit.
04. There are some quantities which, according to the
lutuie ot the question, can only be fractionary. Thus fov
n.stance, when, m speaking of workmen, birds, eqqs &c ' wg
mentioix the /.«//; thirds^ Quarters, &c., it is n^cefsarit kp-
po ed that those quantities are exactly divisible by 2, 3,4, &c.
flf! nf /^ k'!^ Tl ^''"'''^'>'' ^^^ ""»^^«r whereby i quan-
tity of his kind becomes divisible, if increased by one of it.
parts add the two terms of the fraction which expresses that
part, the sum will be the answer.
\:ollT' ^''' f '""Ple, if 1 increase by \ the contents of a bas-
kct of eggs, I conclude that those contents, at? first exactly di-
visible by 7, is now divisible by^+5=A 12
If, on the contrary, the quantit/of one'of its parts be di-
« .imshed, you will determine the number by which it becomes
divisible, taking the numerator from the denominator. The
relTianider will be the answer.
I '^^^".^'/^'•/^^a'^Pje, if I diminish by f the birds of an aviarv
I conclude hat their number, at first exactlv divisible by 7 Ts-
now (divisible by 7— 5 = A. 2. " ":/*,^^
In fv/.^l 'f ?K ^P'^'f «" ^'hich often occurs in my solutions.
In Oder that the reader may perfectly understand it, J am '
flbout te give here an explanatory example.
Let us suppose that the question is to divide |'r4-4^ i„fn
tvjo parts, one of which = the f of the other. Make the su.u
o( the two terms of the given fraction, you will have S-f 5 ^
N which indicates that the lesser part ought to have the 2 -»i
the number (a:+4) and the greater the |! *
Opera* n ; (a:+4) f =:(2f+i^ the lesser number.
m^
t>+4}| =
8
the greater number.
1th thcvfol
APPENDIX.
1 PT
one by tha
I'fc, whether
1 numbers
sum;
which givo
he unit.
ng to (he
Thus, fov
S, &C., AVG
iarily sup-
i, 3, 4, &;c.
)y a quan-
one of its
esses that
I of a bas-
ixactly di-
rts be di-
; becomes
tor. The
Ml aviary,
e by
7 , IS •
solutions,
it, 1 am
f 4) into
the sum
s 3-1-5^
the I «)i
aer.
MISCELLANEOUS PJiOB] EMS.
lien"- o?S!ft"erS"?„rr "?' *« ^"'" <"■''- 'i-.
•nay be 10. ''"« """ 'V 7 and tho other by 3
07. Divide *lfrft 1 . , -^'(s. 28 and IS.
t-nally to theif ag^s :'b W t"? TT ^' ^' ^' P^P-
ofA,whoisbutL]fofC'rWh^ ;[^?'^^^^^ ^^^" ^'^'^t
J A *^ *^*^ ^^^''e of each ?-'
What' is the capital V^*' '"''"' '^ ^^^ ^'"«"»t of $8208.
69. A capitalist placed the * of V . , "^'''' ^^^^^'
;...o,-e arte,, .hid. \Z2^,^ mo'"fr'' "'" '"" '" *''»
hi-.st? "-" ''-'"• Wow many were in ii
'2- Says A to B, Give ,„c *10(. i , "'"'• *^^-
«l.at thou hast. Ill m^U'ad «lTV,.f 'i-nlf ™/'""'''«
'3. Find two „u,„he.s whose dr *''"''""'^ ®™"-
-y be to one a^^o^J' :^-J:t:--<'^;;, r ''"""''•
7r, A J r» ^^"'^'^ are the numbers? Ans ^ m^A a
'' would ha™ .;s'm™h kJ A Imd'clrr^",^--^'^"^
(ais lii
IS each ]
--■^^.mnutcr^t
■is I
-'J
133
APPENDIX.
76. What is that number whose seventh part multiplied bv
Its eighth and the product divided by 3 would give 298| for
It. What are two numbers whose product is 750 and
whose quotient is 3^ ? Aus. 50 and 15.
78. A person being questioned about his age, replied • My
mother was twenty years of age at my birth, and the nun.boV
of her years multiplied by mine exceed by 2500 years hgr
ige and mine united. What is his age ? Jus. 42.
.eJ^'i-^ iffl^T"^ ^,^"S^^ ^^"^^ furniture and sold it shortly
after for |1 44; by which he gained as much per cent, as k
CA>st. Kequired the first cost. Ans. $80.
80. Determine two numbers whose sum shall be 41 and
the sum of whose squares shall be 901. Ans. 15 and 26.
81. The difference between two numbers is 8, and the sum
ot their squares 544. What are the numbers ?
8x5. Ihe product of two numbers is 255. and the sum of
their squares 514. What are the numbers? Ans. 15 and 17.
83. Divide the number 16 into two such parts, that if to
their product the sum of their squares be added, the result
will be 208. Ans. 4 and 12.
^■^V^^* l^o^i'^ "'""^^'' '^^'''^ ^^^^^ to its square root the
sum shall be 1332? W 1296.
85. What is the number that exceeds its square root by
*' Ans.5Ql.
?^L ^i"i *^° numbers, such that their sum, their product,
and the difference of their squares, may be equal ?
Ans .^tVi> l±-/5
2 ' 2~*
87. Find two numbers, whose difference multiplied by the
difterencc of their squares gives for product 160, and 4ose
sum multiplied by the sum of their squares gives 580 for pro
fit .V, Ans.7md2.
88. VN hat is the ratio of a progression by difference of 22
terriM, the first of which is 1 and the last 15? Ans ^
89. J here is « nmviKni. ^e t.^.-^ £. _-- « .i _. .^
vide It by the sum of its figures, then inverting the numbci
rjultiplied bjj
ive 298 1 for
Ans. 224.
is 750 and
•0 and 15.
•eplied : M/
tho number
years hsr
Ans, 42.
Id it shortly
cent, as it
dns. $80.
be 41, and
3 and 26.
.nd the sun)
1 and 20. "
the sum (»f
15 and 17.
, that if to
, the result
t and 12.
ire root the
IS. 1296.
•e root by
ns. 56^.
sir product,
i + v^s
2~*
ied by the
and whose
^0 for pro
r and 3.
snce of 22
Ans. |.
ii yuu di.
i numbci
APPEXDIX.
139
lot, I answeirSMhe »Tf ?.?' t«f "-''^ P"' " » ■-"'
of «n egg exceeded the f'of the wS h^T '"''''' ^•■»- ">« J
all the eggs employed "''" ''>' "■« ^<i"'"-e roof of
n.e'fh.tergTTdf S Te ""' ■""?'"'»"• •"" '^ «'vo'L
instead of £5° that I lost fr""^'"'^'"' »/ "vy n,„„ev, an.l
many pounds have I now^' ™ ""*"' •^"' §'""• ""w
»"Mnrtr:to';?pil%^H„d.^^
dren have*!? '^'^ '">^ ^^""'y- How many ehil-
96. Two numbers are sunh fKof ♦ • 7 , , ^'^"'''' ^^•
more than the greater and if ft *"^^' ^t" ^^'^ '« 3 units
i\ and the U^s^'tsT the fori h ^''"''' ^ augmented its
What are they ? *' ™^' ^^^"'"^'^ <^ouble the latter
97. A brother said to his sister • f m^'"' ^^ ""^ ^^•
founds to make £30- eive niHl' T"^^ ''^''^ * «^ vour
t^e sister, | of yours'to^ havellt will v""'' """^- ^^^^'''^^
How many pounds had each% ' /^" gT." '"« ^^em ?
yS. By means of „ i " . ''• "^^ ""<^ -^^0.
Michael L spend $2 a dTan^^^^^^^^ ^"'"^"'^'^'^ ^'^ '— ^
legacy. Whit was it? ^ ^ '^^ "^ ^^«'-'>' ^he |^ of tho
99. A fruiterer )mn„Kf ^''*- *^00.
J n
14U
APPENi IX
jvould have been but 10 shillings less than half the first cost
Itoquu-ed the first cost and selling price.
Ans. 100 and 120 shillings.
101. Determine two numbers whose difference shall b«
equal to one ol them, and whose product shall be 18 more
than triple their sum. Ans. 12 and 6.
102. Of three numbers the mean is ^ greater than the less,
jnd the turmc-r is i less .han the greater ; now if each was re-
rtuced ^, tiieir sum would be reduced 19 units. What are
the three numbers ? Ans. 24, 18, and 15.
103. Three times a number, less 20, is as much above
fr^V wr^'-"T^^'"^f ^^' ^°"'^^ part, plus 2, is undei
Its half. What is the number ? Ans. 24.
104. Louisa buys 2^ lbs. of sugar, at 6d. per lb., andgives
in payment a piece of silver such that the square of the Siece
returned exceeds trie triple of the expense by a sum equal to
the return. Required the value of the piece given by Louisa.
Ans. "is. 8d.
105. By what number should 3 be multiplied in order that
the f J of the product may be equal to the sum of the two
^^^^^'''^ ' Ans. By 12.
106. When the granddaughter was born, the grandfather
was 3^ times the granddaughter's present age, and 10 years
af^er, the latter was but i of the grandfathers aforesaid^a«e.
Kequired their respective ages. Ans. 90 and 20?^
107. Determine three numbers, of which the greater, equal
U) the sum of the two others, is also equal to the I of their
product, and of which the less is but the I of the other two
^^''^''' aI. 24, IS, and G.
1 ^^?"..^" yncle claims the /^ of an inheritance, the nephev*
I and the niece the remainder. The product of the parts of
the two latter IS 13 millions less than the square of the m-
clespart. What was the inheritance? ^ns. $12 000.
.,\^^:.y''^'^^'^<'J'»^^^^^^^'^,yvhose sum is equal to4 time?*
their difference, and the greater of which, plus the difference
?Kceeds the less by 48 units. What are the numbers ?
Ans. ()0 and 3t>.
My wafch is very methodical in its time. Were ( to.
iiU.
ie first costii
shillings,
ce shall be
>e 18 more
12 and 6.
lan the less,
iach was re-
What are
'<, and 15.
luch above
2, is undei
Ans. 24.
»., andgives
•f the piece
m equal to
by Ix)ui8a.
f. Is. 8d.
order that
>f the two
. By 12.
grandfather
1 10 years
•esaid age.
and 20.
ater, equal
J of their
other two
, and G.
le nephew
e pai-ts of
of the lUi-
12,000.
to 4 time?*
lirteronce,
TS?
md 3tj.
Wan- f tf>
APPENDIX.
141
1 1 1 -p^,,, , ^^^' i minute fast or slow
od'r\ZZrCti7;.' Z:':*"' T ^ <'^*-^ brother- Tea,.
J of the sister's ^eatiesHL/nT -T" «•■« brother will be
ofeaoh? ^^O's less than the sister. What is the age
112 HnH I ^ u, ■ ®'"'^'"' '8; lirother, U
< -M Hve tripled ^.r^^bral' tt |. J^'J' ^^
^■^rz\r'tt^-^ttfX-:i.ei
114 I k„ . ^^"•*'- -A^t 72 and at 40
-^- .'ow^^;israri::;rz'^« "-^ '?—
net, just as you please nf ,ZT [I *"«'■■""'• the prod-
my sister's ai;d Zt^:/'^^ .^;::'JX "^"^ '''--«■"'
115. Subtract /oT.';tot3''^ '--■ 2 years each.
««e,you shall hlem^Lthit,''-''^''-T'J''''<"y^^tor',
years will have lessen^lt " Wh^^lhTf ,f„f ^ h f '"^'^
onVo^r teoiro^tf s but »^— '^^ •
.u. is but half ,„, ,J, ^Z: I^l^tstelt''
mayi4"„^airjTtl;:rat' '"1 1 ^''™- o^'" I-'
lt« i, may exceed'the lessftv I' " ®?'"'' *"""'f-ed
US Tk J ^«». 20 and 4.
.«m, and thr,ttie"„t Ts " Wh^ ^"^'."'^ '^^^^ «-- '''eir
4 i"«"r IS rf. vv hat are the numbers ?
119. Square I plus 1 of thn . ^'''' ^^ ^''"^ "^
qimrter of a n«n?„f, 1. ^.^*^^.3r««»•« tti.'it I am .hort of.
r* ? "" " "' '^"" •> "" ^"' P'-oduce my age. What is
120 A gamester lo8» at the first
Am. 16.
game the srjujire of ^ o|
142
APPENDIX.
the dollars he ha4 about him; but at the second round he
quintuples his remainder, and withdraws neither gainer nor
loser. How many dollars had he ? Ans. 80.
121. I am going to add 5 new shelves to my library, each
of which will hold 20 vol'umes more than the ten already ex-
isting and so I shall have 1000 volumes in all. How manv
^'«^^^"«^^ ^...600. ^
122 Multiply half the father's age by half the son's age and
you will have the square of the son's age; this square is
equal to double the sum of their ages. How old is each ?
Ans. 40 and 10.
123. The breadth of my room is but the f of its length —
As broad as it is lo-jg, it would contain 144 square feet more.
Kequired its dimensions. ^ns. 24 by 18
124. The difference between the | of my age, less 5, and
Its g, plus 3, IS the square root of my age. What is it?
Ans. 3(5 years.
125. The product of two numbers is 220. If from tho
greater you subtract the difference, their product will lessen
yy units, i^md the two numbers by one unknown term.
Ans. 20 and 11.
126. What is the number of your house? Tlie sum of its
digits, considered as units, is equal to ^ of the number. Find
*^ Ans. 54.
127. The square of the difference of two numbers is equal
to Its sum, and I of the former is equal to | of the latter.
What are the two numbers ? Jns. 10 and 6.
128. An officer gave the following indicatjon of the number
of his regiment : One of its factors is to the other • • 1 • 5
and their sum is to their product :: 6 : 25. What was thr
luniiber? ^ ,o-
Ans. 12a.
129. My age is composed of two figures, and read back-
wards It makes me I older. What is it? Ans. 45.
130. The product of two numbers is ^ more than their
sum, and is equal to triple their difference. What are they ?
Ans. 2 and 6.
131. When the brother's age was the snu.irf^ nf fh^ =-!=f.-.T-.»=
/he was | of the brother's present age, and 8 years hencrthe
round he
gainer nor
Ans. 80.
irary, each
il ready ex-
low many
ns. 600.
n's age and
5 square is
i each ?
and 10.
? length. —
feet more.
t by 18.
ess 5, and
is it?
> years.
' from tho
nil lessen
term,
and 11.
3um of it8
>er. Find
ns. 54.
's is equal
he latter.
' and 6.
e number
?r : : 1 : 5,
It was thr
*5. 125.
jad back
ns. 45.
h.in their
^re they ?
and 6.
.^ „:_i. »_
hence thr
APPENDIX.
143
« their ages will be augmented it. I. What is the age
and at the sam? perioj th^eTe of ^ ''''" ^' ^"' '^^ t ^^^''■^
i of the three agL united'^ f hi te the'^^S^ '""^''^^^'^
134. Divide a baskpf .f ^'''* ^^' ^^' «"<^ 1^2.
the part of the' o^t m ;2^thaT^f .t"^ ^'■^^^'•^' ^ ^^^^^
aiid that of the second to the parf f h ' '''^"'^ = = * ^ ^ 5
greatest ^ubmultS ? i d2'"''h *'"*! divided hy t.
pounds d'd he gain ?' ^^'I'shed £12. How manj
number. What is it f °"^" "' ""^"itude, i, f „f the
137 J gQ]^ , ^ -4wa-, JiH.
in anoiher. Triple tto r'mri^,dl"*^1 ''" ""' house, and 25
tlie primitive ccftents. What h^' it" ^"" ^'""' '•''P'-"i''oe
">y'?^t;&,;itS;re?ttsv''^--^^
r^i^eS etir ^■"' — "z:i mZ..i,-
I.IQ rr I K„^ -11 ^'**' 1^ and 8.
Jave bee^lettr^* CltohaTlf • '1 "'^ -"'"
a«d I pay for it? "uuuie wiiat it cost me. What
140. One of the fapfnr« „p , • , ^''*- ^^'-
■■"" " - "■■ ■"-- -att !;ttt
IS
144
APPENDIX.
pres, ut time, plu^ its root, is half greater than less its root
What o'clock was it ? 4^^ g
142. The sum of the four terms of an arithmetical pro.
gression is 44 ; tliat of the two first, 18. What are the four
^^""'^- ^1/^^.8,10,12,14.
143 There is 4 difference between two numbers, and theii
sum is less tiian their product. Required the two numbers.
Ans. 2 and 6.
144. The difference of two numbers equals 4 of the jjreater
and represents the square of the less. What are the num'
^^^^- ^ln«. 30and5.
145 There are two unequal numbers; the less is equal to
I ot their sum, and their sum equals ^j of their pr )duct.
Ans. 6 and 4.
ini^^' I^u '"??^"^ ^'^^ numbers, plus their difference, makes
100, and their difference joined to their quotient equals 45
What are the numbers? Atis. 50 and 10. *
147 I received this morning a basket of peaches: I laid
i by for myself, and the remainder, a prime number, 1 divided
amongst my children in equal parts. How many children
^^^^^^ ;in.. 11.
148. The I of the numerator equals the f of the denomi-
nator, and the sum of the two terms is 11 more than the pro.
duct of ^ of the denominator by ^ of the numerator. What
IS the fraction? j„„ s
149. 1 bought a horse yesterday and sold him at a profit
equal to the f , less £11, of my outlay, bv which 1 gained 20
per cent. Required the first cost and selling price.
Ans. i>20 and £24.
150. The four terms of a proportion make, together, 100.
Ihe first i« equal to the third and the ratio is 4 What are
the terms? Ans. 40 : 10::40: lo!
151. The dividend is equa' 10 the square of the divisor and
their sum added to their qu nent, is equal to 840. What is
the dividend ? what is the divisor ? Ans. 784 ar. d 28.
152. One square is quadruple another, and their sum added
to the sum of the t.wn rm^ta \a f^QA \,\7l.„*. .1
— - ... .,._.._., TT iK-tL arc tne twc
^"^^^^«- ^-Im'. 400 and 100.
APPENDIX.
3S its root
Ans. 5.
etical pro-
re the four
, 12, 14.
, and theii
umbers.
2 and 6.
be greater,
the nura
> and 5.
5 equal to
)duct.
> and 4.
ice, makes
squals 45,
and 10.
!s: I laid
, 1 divided
children
ns. 11.
denomi-
1 the pro-
'. What
A.ns. |.
it a profit
gained 20
di;24.
her, 100.
•Vhat are
0:10.
isor, and
What is
r.d 28.
m added
the two
i 100.
145
153. A milkmaid sells hens' eggs and ducks' oa^. tu •
mean price is 16 cents the dozen Now a a ^f '. .^^''""
are worth 10 dozen of the former pin • 1T" ^^ '^' ^^"*^^
dozen. "'^''' Required the price of each
.r. „ru '^'**- J 2 and 20 cents.
io4 What cost these sl^ pounds of sucrar ? rr .k
»^orth 3 cents per lb more mv nnM ^ ,1 . ^^^^ ^^""o
more. Whatliditcorre'rT/. '^ would have been I
l^r m, . ^ •• ^ns. iDctS.
they « ^ *" "^ '"" proportional. What are
two numbers? -^ '^^ What are the
1K7 A , ^w*. 6 and 6.
. .xth\\etSe'rXbr4:taVt roof "r^^/T
square? 'i "mes its root. Required the
• 158. An annuity placed at iqi r.. . ^"''' *^^'
duces monthly a lenteauJfn-J ^ ''^"*' P^*" ^""'""' P'"^
annuity ? ^ ^^"^^ ^^ ^^" ^^^^''^ '^ot. What is the
i^jf. Ihe product of two numbers is 120 A,l^ i . l
and their product shall be 150. wLt a '^he dumber; ?'"^
160. Two numbers are equal If o k. fj'l' ^ ^"^ ^•
product will increase 51 What are Th. ! ^^^ '? ''^"^' '^^'^
«» »'i. w nat are the two numbers ?
Iflo Tk« « 1- - '^'**' 4 and 12
ho less b, 2. m Jr th'/thtettr ""^ <i-''™' »"'
164. Th« «n„„..« „, .u. .. . '^"'- H 4, and 2l'.
nominatorrandThes'.lm'nfZ?'''''?'''''.'""™ "■»" *« dfr
their difrer'enee '"wh,™ r/the fra^tioT"" " ' "'""' '"l"'" T'-^
146
APPKiVDIX.
1«5. The two torms of a division mm are the san e as the
two te: ,ns of another, but in an inverted order, ^e sum ot'
the fimr tenns is 80, and that of their Quotients 2i What
are the terms of these division sums? ' Am. 24 and hi '
«.n^^t" i!'^'5 ^]'.^ d'^'^^nfl is equal to the square of the divi-
sor ; halt the divisor equals the square of the quotient. Re-
quired the <lividend and divisor. Am. 128 .and 8
tn lTrl'P^7u """""^r ^? V ® '' ^- ^"^^'"^^^ 5 f'-'^'" the first
to add to the -econd, and they will be : : 7 : 6. What are thp
two numbers? ^„„ ^n ^ «k
-^"«. 40 and 25.
«nn ^®Vt"''^' 'V^^/f '^«« fro"^ A to B ? said an inquisitive per
son He received for answer that their numbor had but two
factors,- whose sum was 20. What is that number?
TT, Ans. 19.
A^i , u P^'^'^'J^^^Pf the two terms of a fracti..n is 120 —
Add 1 to the numerator, subtract 1 from the denominator their
quotient will be 1. What is the fraction ? Ans! '{^
17.0. The failure of an insolvent debtor took away the ! of
the capital that .had placed in his hands. The interest of' the
remamder placed at 5 per cent, is equal to the square root of
the hrst capital. What was it? ^n.%10,000.
,..^^^-i/^ ^?" ^""^ '"^^ ^^"^ ""^'^^ ^^^^ "^y ag« backwards, and
you will make me the J younger. What was the person's
^^ • , Ans. 81.
,x,^lu'' F''f "^.^ t^'ee numbers in arithmetical progression
the third of which shall be equal to the square of the firTt'
and the second triple the ratio. Ans. 2 3 4
*. '"'^n J i?T"u^^ ,^' ,""'^^^ ^" hour, said a pedestrian : had I
travelled 7^, I should have arrived 8 hours sooner. Required
the length of the journey. Ans. 240 miles.
174. Jf you double the denominator of a fraction, the sum
l'ln'''''-irTT"Jf 22^ and if you triple its numerator,
the sum will only be 21. Determine the fraction, ^/j,. |.
175. Increase the contents of my purse £3, and it will be-
come, a perfect square. If, on the contrary, you lessen it £3
di^pin w!T ^"^^^ ^' *^® aforesaid square. What are
176. A lady questioned about her age, answered : Increase
APPENDIX.
^an e as. the
rhe sum of
II What
and 1 «.
)f the divi-
tient. Re-
8 and 8.
Ti the first
Kat are thp
and 25.
isitivo per
d but two
ins. 19.
is 120—
lator. their
Ins. f^.
i the I of
est of the
i"e root of
10,000.
ards, and
person's
ns. 81.
)gression,
the first,
i, 3, 4.
n : had I
Required
miles,
the sum
merator,
[ns. I
will be-
en it £3,
Vhat are
s, £6,
Increas*
w
it the f, and in that state lessen it the * and vn,, .h.u u
made mo 25 years younger. What wat the la'd^s age ? ''^^
resent* my^tT ' ^ "butlf'^ ^-"d a-ng thosf thaf rep
figures thJtlshe^nl be mV''"''*^^^'^^^''^^ ^^o tZ^
' ins 18
,^„ „ ■ ' ' ^ns. 10 o'clock,
ivy. How many children have vArt%1r? tu - . .
«„]®wh/""^ T """""""^ whose>iim : their difference ■ • 7 ■ <l
■ Ans. 15 and 6.
at 6 •tLTrl7ir|31'iog''^l'^'iP^"™'••''■''-"•-'^-
p.n is e,ua, . , ;Lfo^/rsee!„^ "^^^1:1^^
182 P.„. V ." ^'"- **5,000 and $20,000.
tio'fi/Zerrri™^-t:i~r;hr7:"r^ '^^ '«■
of the second by the third is Tll;:;:^' thfCteC ™'
,S, TT,„ . -<"». 4:8::8: 16.
ISA Ihe number x exceeds the number « hv fho u ,
root of X, and ^ of ,: equals the A of ./III '•y."'<> whole
numbers with o*ne unk,?own tenn'^ '^ ^A^Ts^nfT"
oflt.^^a^or'nilrgTbl'ittvl'S,"'"'!^^/^'-''-
and my expense was tie samf at h iftt;!' tT''
mea,, pr,ce was 35 shillings. Required the 'prtje rf^a' ^JS:?
^'w- 30 and 40 shillings.
>„l!!i'lr!'r..';™'=.«' '"M ? Kv, my daughter's a„,. will
to-dav ?
""'^ "' ""'*J' '■'^ «as o years ago.
186. The .sum of two numbers is 4
What is her age
Ans. 10 years,
times their difference;
I
i B!
148
APPENDIX.
and the d.fferenofe ,i, of their product. What .re the two num
Jn )^Ia , " °®*'^'" ^^'"& questioned about the number of men
wh e sumtTl Zrr' '■ '^t"^""'"^^^ ^- but 3 factors
wnose sum 18 31. What was their number? Ans 25
188 Waiter, your bill of fare amounts to so much, does it
hat^hl. K^" IT'' r'V T^^^- ^^^"' here is another sum
that lacks but i dollar of the J of the first, and let's say no
IrinnV*' ^' '"' "^^^-^iheless, hard. sir. to lose the double,
plus 1, of the sq .arew.of my bill. What was the nmountt
* '" ^w«. 144 dollars.
ih«t ft! T\ °^ '^'^ pandpapa and grandson are such,
that their nuotient is equal to | of their product, and the sum
of eaeh^ q^o^^ent.aftd. product is 320. What is the ^ge
^^ • ■ - ■ Ans. 96 and 3 years.''
lyo. 1 mixed two pipes'-Af Vine ; one cost 180 shilJinffs and
he other 140 shillings. The first contained 20 bo ties rr^ e
than the second and cost 5d. l^ess per bottle. What is the vZl
of a bottle of the mixture ? = ;,. ^,,. ^^^-^'^^
191. Ihe quotient exceeds the divisor by half phis 1 and
the sum of the divisor and quotient exceeds doubeCsoi^ire
root of the dividend plus 1. .R-equired the divid!'. id visor
and quotient.
Ans. 400, 10, and 25.
u!hfr ^7," ^*"^et«J"nni»g tog\^ttter, filled a basi,, in 3 hours
If the first had run but 2 hours, it would have taken the second
6 hours to do the remainder. What time would it take each
tunning alone? ^... 4 and 12 hours
193 T^e father's age has two factors, of which one repre-
sents his daughter's age, and the other is 18 less. Sqimie ?h^
iXtZVf^^ ''^'' the daughter's age, and tSis Lu
U> the sum of both ages, and the general result will be 100
Required the age of each. a ns. 03 and 2 1
-, lit .'T^e product of two numbers exceeds their sum ov U
and the.r difference is 2. What are the two numbers ? ^ ^
Ans. and 4,
_195. A number of three figures is a multiple of 11. and rim
".-iiio is quaarupiu the huiiuieds. What is the number?
Au,. 154.
>
he two num
10 and e.
Tiber of men
but 3 factors,
Ans. 25.
luch, does it
and receipt
mother sum
let's say no
the double,
le amount *
i dollars.
n are such,
nd the sum
t is the affe
3 years.
liliings and
ottles more
is the value
. 2s. 5d.
plus 1, and
the square
id, divisor,
and 25.
in 3 hours,
the second
t tal<c each
2 hours.
one repre-
Square this
this result
11 be 100.
and 21.
um oy 14,
rs?
) and i.
1 . and rlia
er?
(6-. 154.
APPKNUIX
ue
at !hf ricdit^oH^ft o1 T I?^"'"." ^"\^' ^^«^ '^» ^-^ r'«-<i
t.^7 TT n^^'"^ ^-^--ratrt t etnZrS
i\es, the bird-seller answerpfl • If_i_,v. a •"i^«-'«>i lugi-
■tiv.„u„,her. which Tas:t;r''4"aL''rdrY80T'';H
now he reduced J. What was the prim tive nt 1, 'r f w'llil
B their present number' ^ I" w j cV""
/!«». I.'jO mid 84.
n.nl ik""!"'"'"'"'' <l™o™ple 3r.<)ther, and their sum i, 4
r;h:'trs;r:,"'^''"'■''''''"™^^■^«-•^;"ft"-o:v^^^
^ ' ; -^'tA. ^5 and 100.
flute p,„. the square rL of^^^lZf Z %^. ''"w^j ^
the price of each instrument, if the sum of ,« tL
quadruple the pounds given to l>oot bylwrenLr "'"^ "
9nn A u ^ .. .'o ^^*«. illo and £25.
diJfs i.V""" ri?^'^'''.^^"''"^ ''^ «"«h, that the sun. <.f its
digits ,s lb; and by m verting the number, then addin./tt!
m." wtt ;:r:itrt'" ^•'" •^'^ '^''*-^ v'^--
of?L,tetroroVits^^^^
their sum is to their product • • 6 • 05 vVKa. ' ' u' ' ^
ber? F^wuuci .. o . .iD. What was the num-
202 Two sisters have unequal .ums for their purchases
;tn«rfa:^^"™^- -- ^^^- - f ?^^*^wi^^
^08. What IS ihe number, whose scmaie reduced t„ it.
■luarter, excneds by { three times the | ',f the nuXV?
tinferaslrari"iU:.!;r .\:°T f ^--<'. ^^i:: is^
squares of inarble of a^^riainXiTirsi^i^St'Z uWt/
n.e mason answered, that if the length was but double Te
l.S*
150
APPENDIX.
CJD
breadth, it would have taken 800 less. What do vou
dudo fi-oni this answer ? ' i uo ^ou
Ans. That it would have taken 4000.
.Jf' ^ T^'^^"^ g^^« 2^ "^' his profit to the poor. At thr
>ear s end his alms amounted to $390. I demand what vvas
the amount of his sales, if half was at 10, I at 15 and k
remainder at 18 per cent, profit ? ^Ans. $6(^000
fhnT' J^^ '""" ""f the'foJr terms of a proposition is 63-
the first ,s 4 more than thVk:c.ond ; the quotient of the th rd
by the second ,s>8i; and .the product of the means I 136
Required the four terms. ' .;: . . Ans. 8 : 4 • • 34 17
^ A*jf . * : o : ; 4 • 1 'i .
TTDC IWD.
do
you cjD
ikeii 4000.
oor. At the
nd what was
15, and the
. 160,000.
ber was rep
representing
at the sum
t what num.
No. 210.
ition is 63 ;
of the third
sans i,« 136.
: 34 : 17.
^ion i'i 576 ;
term k 10,
What an>
:4- 1','.
I
,i>
5'
t ■
<
\ .?«
x
'•s
<
% f-
\
h
il
»V™.
apiTroved school books
D. k J, SAP! fE'' c<' ;l qi p„ I «
f
OJii odiiion.
^rui. u
#
tun in Puor Sc/^oU
cnuls■^lA^ uw^muiiis' fikst
•in.-" PRAfTJi VI .. -"^- Lie
* ■' u ',' ''"«■ Dioct-se of Boston. ",'!'
, , ThVam..'^i^!i>rib1e^X?h """^ ^''"«*-"^^ -''^ «« cut-. „."
%• Any of thi^boTe Book* will be ^ui hTIaii u *
CO.