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•i ,;
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'SS
I)
ALGEBRA FOR BEGINNERS.
*,.,
\\
ALGEBM FOR BEGINNERS
WITH NUMEROUS EXAMPLES.
BT
I TODHUNTER, M.A., F.RS.
NEW EDITION.
SottDon atili iSTamibYiOse :
MACMILLAN AND CO.
TORONTO: COPP, CLABK, AND CO.
i860. .
[All Righti r^erved.l
^#*'"'*.
**j«t.
*■
f -^mf
/(^ dOXo
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//
CatiiMifle:
1>IUNTBD BY 0. J. OLAT. M.A.
AT THS UNIYBBBITT PRB8B.
220514
\\
PREFACE.
Thi present work has been undertaken at the request
of many teachers, in order to be placed in the hands of
beginners, and to serre as an introduction to the larger
treatise published by the author; it is accordingly based
on the earlier chapters of that treatise, but is of a mord
elementary character. Great pafais have been taken to
render the woik intelligible to young students, by the use
of simple language and by copious explanations.
In determining the subjects to be included and the
space to be assigned to each, the author has been guided
by the papers given at the yaiious examinations in ele-
mentaiy Algebra which are now carried on in this countiy.
The book may be said to consist of three parts. The first
port contains the elementary operations in intend and
fractional expressions ; it occupies eighteen chapters. The
second part contains the solution of equkikns and pro-
blems; it occupies twelve chapters. The subjects contained
in these two parts constitute nearly the whole 6f every ex-
amination paper which was consulted, and accordingly they
are treated with ample detail of illustration and exercise.
The third part forms the remainder of the book; it con-
sists of. various subjects which are introduced but rarely
into the examination papers, and which are thereforo more
briefly discussed.
The subjects are arranged in what appears to be the
most natural order. But many teachers find it advan-
tageous to introduce easy equations and problems at a very
early stagey and accordhigly provision has been made for
>jf'
in accordance with iha «wJ«l
»»w been .elected flx)mth.^i'~"* Someofthei»
wth referenoe to points which iJ!Z7 •''* "* oonrtmcted
^^^^^^^ / authort expenence u a teacher and an
wwrki on the worfc ««!i ^'J^'^^e fluggestions. Anv ^
«»»»«llr receir^ *' ** "^^^ *«! be m«t
S» JOHH 'S COILUI,
I. TODHUITTBR
«on. and abo?3:^^;? 5^"Jf <*«'» *» ^ P««ent edi-
«• ««nged in ae^JStll^*"'!' ^^P'« »W<*
Th«e addition, have £ ^te^^rth"^*? *«« «»»»Plea
eminent teachew, in onler tTf *' '*9»«'t o' "owe
woA. ^ "" ""*** to "w*ea«e the utility of the
^^ly 1867.
# l|
.'^.■'
\ \
CONTENTS.
rAOB
I. The Frinoipal Signs i
II. Fftotor. Coeffioi«nt Power. Tenue 5
III. BemainiDg Signe. Brackets 9
IV. Ohmge of ibe order of Terms. Like Terms 19
V. Addition 16
VI. Subtraction 19
, VII. Brackets :,: 22
Vni. Multiplication «5
IX. Division 33
X. General Results in Multiplication 43
XI. Factors 49
XII. Greatest Common Measure 55
XIII. Least Common Multiple r 63
XrV. Fractions , 68
XV. Beduction of Fractious 73
XVI. Addition or Subtraction of Fractions 76
XVIL Multiplication of Fractions 84
XVIII. Division of Fractions 88
XIX. Simple Equations 94
XX. Simple Equations, continued 103
XXI. Problems 113
XXII. Problems, continued ill
XXni. Simultaneous Equations of the first degree
with two unknown quantities 136
^XrV. Simultaneous Equations of the first degree
with more than two unknown quantities... 145
XXV. Problems which lead to simultaneous equa-
tions of the first degree with more than
one unknown quantity ^ . 150
Tiu
CONTENTS,
XXVL Quadraiio Equationf 160
XXYII. EquAtiont whieh maj be lolYed like Qatd-
ratios ; 171
XXVin. Problemi wbiob lead to Quadratio EquAtioni 176
XXIX. Simultaaeoua Equatioiii involviiig Quad*
ratioi i8a
XXX. Problema wbicb lead to Qoadratie EquaUoni
witb more than one unknown quantity .<.... 1 90
XXXI. Inyolution 195
XXXII. Erolution 300
XXXIII. Indicei 318
XXXIY. Surdi ;... 125
XXXV. Batio ^30
XXXVI. Proportion 134
XXXVII. Variation , 540
XXXVIII. Arithmetioai Progression «.... 345
XXXIX. G^eometrioal Progression 349
XL. Hannonioal Progression 954
XLI. Permutations and Combinations 356
XLIL Btoomial Theorem 360
XLIII. Scales of Notation ,.... 968
XLIV. Interest 371
MiseellaneouB Examples .........1.... 375
ANSWEBS... :... 305
\\
160
lad-
nt
IfMM
176
iftd*
• • ••
189
oni
• •••
190
»95
• •••■
900
• •••
118
t • • •
««5
n^
it««
«34
• ••
940
• *•
245
• ••
149
• • •
«54
• • •
356
• ••
a6o
• •
968
• •
«;«
• •
275
305
ALGEBRA FOR BEGINNERS.
L The Principal Signi.
1. AioiBAJk is the scienoe in which we reason about
nnmben, with the aid of letteito to denote the nnmbenu
an4 of certain signs to denote the operations performed
on the numbers, and the rohttions of the numbers to each
other.
2. Numbers may be either known numbers, or num-
bers which have to be found, and which are therefore
called unknown numbers. It is usual to represent known
numbers by the first letters of the alphaMt^ a, 6, e» &c.,
and unknown numbers by the last letters at, y, ir; this is
however not a necessary rule, and so need not be strictly
obeyed. . Numbers may be either whole or fractional The
word quantity is often used with the same meaning as
number. The word integer is often used instead oftMole
number. «,
3. The beginner has to accustom himself to the use of
letters for representiiur numbers, and to learn the meaning
of the siffns; we shaU begin by explainiiur the most im-
"oortant signs and illustrating tneir use. We shall assume
ibat the student has a knowledge of the elements of Arith-
metic, and thiat he admits the truth of the common notions
required in all p^rts of mathematics, such as, if eqtude b^
added to equate the wholes are equal^ and the like.
4. The svn + placed before a number denotes that the
number is to be added. Thus a+b denotes that the num-
ber r6pre8ented4>y d is to be added to the number repre*
T. A. 1
f I
'I ■)
'Is.
..^
■J*
2
!ra^ PRINCIPAL SIGNS.
sented by a. If a represent 9 and b represent 3, then a + &
represents 12. The sign 4- is called the plu» sign, and
a+& is read thus ''ap/uf b.''
6. The sign -placed before a number denotes that the
number is to be iubttacted. Thus a -& denotes .that the
number represented by & is to be subtracted from th6
number represented by a. If a represent 9 and b repre-
sent 3, then a—b represents 6. The sign ~ is called the
minus $ign, and a-& is read thus "a minut b."
6. Similarly a+(+<; denotes that we are to add b to
a, and then add e to the result; a-\-b-c denotes that we
are to add b to a, and then subtract c from the result;
a'-b-^e denotes that we are to subtract & from a, and tiifin
add x^ to the result; a^b—e d^otes that we are to wib-
tiact 6 from a/ and'then subtract c frt>m the result
7. ^The sign «= denotes that i^e numbers between
which it is placed are equal. Thus a ~ & denotes that the
number represented by a is equal to the number repre-
sented by 6. And aA-b^c denotes that the smir of "the
munbers represented by a and b is equal to the liundber
represented by c; so that if a rewesent 9, and & represent
3j then e must represent 12. ^he sigu = is catfed" the
9ign i!(if eqwfiity, and a=^b is read thus '^a equaU b** or
*^%iiequ(dtoh,*\
8. The sign x denoted tiiiai the' nmnbers between
which it stands are to be muUiplied together. Thus
ax6 denotes that the number represent^ by, a is to J)e
multiplied bv the number represented by b. If a remre-
sent 9j and o represent 3, then axb represents 27. SSie
sign X IS called the sign qf rnuUiplicaiton, and^ ax( i»
read thus <*a into b.'' Similarly axbxe denotes the pro-
duct of the numbers represented by a, b, and c,
9. The sign of Aiultiplioation is howerer often ondtted
for the sake of brevity; thus a& is used inst^ of axb,
and has the same meaning; so also abe isiused mst^ of
axbx Cf -and has the same meanmg. ^
> The sign of multiplication ,must not be omitt^ when
limnbers are expressed in the ordmary way by %ip9A
Th|iu 4li cannot be used to represent the product of 4 api
■ M
■Sas.'StSp'^-. <mim'»h
THE PRINCIPAL SIGNS.
3
tbe
!nie
pro-
6» bedHue 4 dUlereiit meaning has already beefl tkpfigiOh
priated to 46, namely, >M^«9* ^ We moat therefore lef
present the product of 4 and 6 m another way, and 4x0
IS the way which is adopted Sometimes, noweyer, a
point is used instead of the sign x ; thus 4:5 is vaM. in-
stead of 4x5. To prevent any confusion between the
point thus used as a sign of mmtiplioation, and thenpcdnt
used in the notation for decimal finctaons. it is adyisaUe
to place the point in the latter case higher up; thus
4*5 may be kept ^ denote 4 + rr • But in &ct the point is
not uM instead pf the sign x except in cases where there
can be no ambiguity. Forexample, 1.2.3.4 may be pot for
1x2x3x4 because the ^points here will not be taken for
decimal p(^ts.
The point is sometimes placed instead of the sign x
between two letters; so that a. &. is used instead of a x &.
But the point is here superfluous, because, as we have
8aid,^isused inst^idof ax&. Nor is the point, nor the
si^ X , necesspkxy between a number expressed in the or-
dmary way b;^ a figure and a mimber represented by a
letter; so th^t, for example* da is used instead of 3^(0,
and his the saSe meaning. " ' '
. ^ , ' ■ ',-/-':^ - . ' ■ >-
The aaa -f- denotes that the number which pre-
it i^ to be divided by the number which foUows il.
lOtes that the number represented by a is to
ti^e number represeUted by K If a rem-
. represent 4, then a-i-b represefits 2. The
ted the sign qf divition, and a-^b m read
10.
cedes
Thud a-i-b
be divid
sent 8,
sign
thus
There |»- also another way of denoting that one num-
ber is to be divided hf another; the dividend is placed
over the divisor with a line between them. Thus j- is
used instead ofa-i-by and has the same meaning.
11. The letters of the alphabet^ and the siras which
we have already explained, tc^ther with those which may
occur hereafter, are c^led mgebraical symbols, because
they are Aised to represent the numbers about which W9
may be reasoning, the operations performed on them, and
1~2
f (
4 JSXAMPLS8. i.
ihefar retatibiui to eadi oUier. Any oollectioii of Akobralcal
«yinbob is called an al(f^raktU eapreirion, or htieSij an
18. We shall now giro some examples at an ezmsise
in the nse of the symbols which have been eilplalned;
these examples consist in finding the nnmerical Ttthies of
oeirtain algebraical expressions.
Suppose asl,d'=8yea3,<^==5,«=6,/-0. !nien
Ta+aft-skf+z-T-j-e-io+o-ia-io^a.
8^^^^<8»c~atf+4r«4+48-64L0=52-6=^
I T «l 00 2 * 15 8 *** *" * ^
4g-t'8g 12+80 42 - .
EzAirPLISL I,
If o=l,J&f 2, 0=8, «r=4, «=5,/=o, find the nmnerit
eal wofis of the foUowu^ expressions:
2. 4o-3a-3$4'5e;
4. B^-'bcd-h9€de''dqf,
^ aM+<iM+aft«ls+«^+fc*i «. $ + §5 + ^^?^.
o c a ^
j, 406.8(0 Sod
od oo Jo
iL
« 12a . 66 . 200
10. 7^+6orf-g^
2a4»6d . 364-20 o+J+o-l-ti?
' — ^. — - +
4
2o
12.
e
■• 04-6 . 6-i-rf . 0+0
V
1^ o+6+tf+<f+«
tf
" ..4j^-
.i
FACTOR. OOEFFICIBNT. POWER. TBRMS. 6
leal
fan
led;
m of
^^
iieri^
Ikl
8<
II. JPoctor. Oo^fieUnt. Power. Temw.
IB. When one number oonibtB of tiie product of two
or more nmnbers, each of the latter is calied a foutor of
the product Thus, for ezampleu 2x3x5=80; and eadi
of the numbers 2, 3, and 5 is a /actor of the product 30.
Or we may regard 30. as the product of the two Uxbom,
2 and 16, or aa the product of the two foctors 6 and 0^
or as the product of ih^ two fiMstors 3 aind 10. And so^ also^
we may consider Aab as the product of the two fiustors
4 and a&, or as the product of the two fiustors 4a and ft,
or as the product of the two fiustcnrs Ah and a; or we mut
reff9^ it as the product of the three fiic^ 4 and a and S.
14. When a number consists ot the product of two
fiictors, «Adi fiictor is called the ao^iini of the other
fiftctor; so that confident is equiTaleut to ed^iieior. Thua
considering Aab as the product of 4 and ab, we call 4
the coeflBdent of ab, and a& the coeflBdent of 4; and
conddering Aab as the product of 4a and ft, we call 4a
the co^fficwnt of b^ and h the codicient of 4a. Thera will
be litUe occasion to use the word coefficient in pra^lp in
any of these cases eicept the first that is the oasa ii^wbidi
4 is regarded as the coefficient oi aft; but for tjEe sake of
distuuSness wo spealL of 4 as the num&rieai ht^jficient of
oft in 4aft, or bnefly as the numerical eo^eient. Thus
when a product consists of one fiustor whidi is t^preemkiod
arUkmetteattVf ikBkt is by a figure or figures^ and of an-
other foctor whidi is represented tUgebraicaHy, that is hj
a letter or lettei!, the foimer factor is called &e itunMn-
tai eo^ficienU
16. When an the fiM^tors of a product are equal, the
product is otlled a power of that &ctor. Thus 7x7 i&
caUed the eeeond power of 7; 7x7x7is called the third
pofferoi 7; 7x7x7x7 is called the fourth power of 7;
and so on. In like manner ana is called the teeond power
of OI axaxaiscaUodthelAtnl jN>t00rof a; axaxana
iaoijQaithe/ottr<AjK>i0irofa;andsomi, And a itself i§
mmmmMmM^JkrH power ot a.
«f
f «
6 FACTOR. eOEPriaiSNT POWBB. TSBM8.
16. A power is . more briefly denoted thius : fustead of
exprefsing iul the equal &ctors, we express the fiiotor onoe,
and place orei* it^-^ linmber whieh indicateii how oftoi it
is to be repeated. Thus €? is used to denote axa; o^ is
used to denote a xax «; a* is nsed to denote ama x ^u. a;
mi. lio on. And a^ mar 1>e nsed to denote t&e first power
bif a, iliat is A iti^lf ^ so that <^ has (^ tami meaning at "d
* It. A number placed over another bo ideate- how
ma^ijr ^mes the latter occurs as a factoir in ajK>wer, 1^
billed an indtf^ qf the po^er yO/raxi eaponentqftAepifwer;
ei*, briefly, an »nde^, or tfdriMm^n^.
Thus, fpr example^ p <^^9. exponent is 3; in at* the
ffl^ponentis^ -
18. The student mudt distinjraish yeijr carefU% between
il confident md m exponent, Thus 3^ means three timet c;
hei» 3 is a eoeffkient Bat c* means c ^Htthee t ii^eeaf
hm 3 is an e^i^pondnf. That is
3e=e+o+c,
W 'i^e second po^r of a» that is d(\ is oftei^ taSM the
equarij^ a, or a squared ; and the third pow^ of Oy l^at is
^, ii ^Mif called ma cube ei a, or a cubed. There are no
fuch words in use for the higher powers; a* il read thus
**a 10 the fourth power,* or briefly ** a <o ihe fourth!*
. 8(K If an exprea&ion contain no parts connected by the
signs + and — , it is called a eimple expression, if an
expression contain parts connected oy the signs + and -*.
it It called a compound expres^on, and we parts con-
nected by the signs + and -r are called terme of ^e exr
presrion.
^ Thus ax,^, and 5aM are sim|^e expressions; <b^-i-5^~^
h a compound expression, and a*,^, and e^ are its tennis
' 21. When an expre«Hon consists of ^fimis it ii
called a binomial expression: when it ccinslstii i»i^ three
iems it is called a trincmiat expressto; any eim|li^et|
eonristipg of seyeral t^rms may m ci^ed a mfdhn^itA
©3^re8j&«B, or ajw/yiicfiwiW ex^
'
cs
lo
a
il
tb
se
til
te
di
fa
ei
hi
te
is
nv
'k
1l
FACTOR. COEFFICIBlfT, POWER. TERiiS. 1
Thus 2a+3& is a binomial ojpression; a-^b-hSe is a
trinomial expression; and a-6+<j-d-« may be called a
multinomial expfession or a polynomial expression.
22. Each of the letters wliich occur in a term is
called tk dimetmon of the term, and the number of the
letters is caHod the d^ree of the term. Thin c^t^e or
axaxbxhxbxe is said to bci of six dimensions tft of
the sixtli degree. A numerical ooefBcient is not counted;
thus 9a*M and cflb^ are of the 'flame dimensions, namely
seren dhnensions. Thus the word dimennons refers to
the number of algebndcal multiplications inyolyed in the
tern; that is, the d0gree of a tejrm, or the number qf it$
dtmennone, is the ium qf the eaponente qf itt aigebrai^
faetorif provided we remember that if no expon^it bo
es^ressed the exponent 1 must bo underlstopd, as indicated
m Art 16.
23. An expression is said iiohehomogeneoue when all its
tenns are of the same, dimensions. Thm 7a^+^t^b+4abo
18 homogeneous, for each term is of three <Hmensions.
Wo shall now siTe s<»|ie more ^camples of findhig the
numencal values of algebraical expressiomk
Then
a*l, 6=2, <r=:3, d=^i, ^=5,/«a
6»=4, ft»=8, 6*= 16, 6^=32.
35»=3x4=12, 56»=6x3=40, 9^=9x32=288.
tf"=6»=5, «»=6*=26, >=6»=12«.
<i^«.lx8=8, 3S^«*=3x4x9 = ia8.
«P+c«-7a6+/»=e4+9~14+0=69.
3c»-4c~10 27-12-10 5
fl^-2ic»+5<?-23''27-18 + 16-23'"i"'\
126 + 64 27-1
6 + 4 3-1
189 26
^-^=21~15=
>a
«'•
8
EXAMPLES. IL
1!XAMPL1& It
<<-i
vmnB^io^
, M^v <fo 32
. . - ;■■■ V-"
■€ a
M ^ . 12 ^ '':o:. 4 r '.:
12,
«• + 4<rt^ + 6aV+ 4«d^ t ft*
<^+^fr+8«M.fJ^ •
»«-|.
14
14.
16,
s.
J : -
■^1 i
'^
MEMAININQ SIQNS. BBACJCETS.
^
ioM
<^
Z'-^^'i,'
>J
III, Remaining Signt, BraekttU.
24. ^e diflerenoe of two nnmbeni is sometimes de-
noted hj the sign ~; thus a'^h denotes the cUflforenee of
the i^umben represented by a and ft; and is equal to a-^(.
or &-a^ aooorduig as a is greater than (, or less than h: hut
this symbol -^ is very rareqr required.
2^ ^The ngn >, denotes it ^Mrf^r <Aaii,.and the
i%n < denotes i9 leii than; thus a >& denotes that the
nnmber represented by a is greater than the number
representea by b, and o<a denotes that the nnmber re-
gresented by p is less than the nnmber represented by a,
^nS in both eases the^tening of the angle M tamed
towards the grater nnmb^.
, 2^ JChesign.*. denotes then or thertfore; the sign V
d<^otes M*ite0.or &0(^t(M.
^. The 99uare root of anj asdened munb«r k that
mimliar Wlili^. has the assig^ea nnmber fot its 9quat9 or
M!M|i J9^i0»r. The cube root of any assigned number is
l^lMube^WhM^ number for its eti&s or
i^mr^po^09f,\ The JburthrooT of any assigned number is
thsA number urbich has the assigned number for its fourth
poiir% Andsqon.
Thitt sinioe 48»«7', the square root of 49 is 7; and so if
a^t^f the square root of a is &. In ID&e mannw, sboe
' 125ii^) the cube roo:t of 125 is 5; and so if a^(^, the eube
rootof«ia<^
root of a may be denoted thus l/at^
4s denoted simply thus tja. The cube root
of ' fi^if "JlppTiit i^lis ff€k The lourl^ root of a is denoted
thus '^1^ so on.
mBi(f8m%
(^■ji--^ r
i^im mJ is M^d to be a ccNTUpticp of the imtial
^'^aiift
(
10 RBMAimim SWNh. BRACJCSTS,
m
^ 29. When two or more numbers are to be treated as
forming one number they are endosed within bracktU,
Thus, suppoM we have to denote that the sum of a and 6
is to be multiplied by <;; we denote it thus (a+&)xc or
{a+5} x^ or simply (a+5)c or {a-\-h)e; here weimean^hat
the fOfuik of a+5 is to be multiplied by e. Now tf ^e dndt
the bradceti we have a-^he, And this denotes tUkl 1^ onlir
is to bemuUdpllBd by and the result added too. Siwi-
hMir, {a'\-h-e)d donotes that t9io result eibfteed 1^
a+^-c is to bo multiplied by d. or that tiie l»A^ of
a+^-c is to be multiplied, by d\ but if we ^mit the
bmokets we hare a^h-ed, and this denotes that e mbi
is to bo multiplied by d and the result subtracted fl^
«+& .. • ■■.-•^"^
i ^^also (a-ft+c)x(rf+tf) denotes tiiat the leiiuHi W
priMsed ^a-d+tfis to be multi{^|ed by the resoil.^-
pressed hjd^e. This may also be denoted sitndhr thus
(«-&+<;)(<l+0); just as a X 6 is shortened into od. .
• So also ^/(a +&+<;) denotes that we are to obtaSli lihe
rwdtexpressed by a+6+<?, and then take the squaro rdol
oftiiiaresuli
80 also {aft)*denotes«&^x ofrj^and («ftj» denotes dx0n^
So aJbo (a+d-<j>-i-{rf+«) denotes that the result! ex**
pressed by a+5~<; is to be mvided %theleBu^«^^
Dyrf+«.
30. Sometimes instead of usinff l>radieti9 a Hne li
drawn oyer the numb ers whic h are t o be trotted as Ibnsd^ i
one number. Thus a-b-^cxd+e k used iiii the siWk i
meaning M (a-6+<j)x(rf+<0. A line used^ibr tWs^^^ i
pose IS caued a vinculum, . So also (a+& j|^i*^p#+«} ^ia;|^ v
be denoted thus ^^^i^; and here th<^tt» bet^ I
a+d-cand ^^e^j^^aUyAtirieuk^
(:SFlti..U
31. We hare now ex|^ned all the rigns wh^t ig
2^pi a^^jBfepa.^ We mav observe thai In. some au^li
ijmNi^ anil -ftj
tlHii^ the ft^ i^Sttbtraet^onwe i^yisp^of Mafi^i||l
lis
l?^.
sxAMnm in.
n
ihs «H^> VMning the ligiif •¥• and - ; and hi nraltipUah
tion and diviiioii we shall mak of the RuU qfStgm, mean-
i^ a role relating to the^a^ins 4- and -«
88. We shall now giro some more examples of finding
Ihe numerical values of expressions.
Suppose a»ly (■«2, c-i^ <l— 5, «»a Then
is/(»+4<?>- V(*+ 12)- >/(16)-4.
y(26+4c)-(2rf-5)4^(4<?-26>-8x4-8x2-32-16-ie.
-y(27+54+36+8)+ <y(i+4-4)-iy(m)^i-«.
ii
1
V •
■A
EzAiis^uML III.
t .-■*>■.•■ i , ■ . ■- ■ • >
If a»»l, (">2, €mZ^ if— 6, «— 8, find the nnmerical
Values of the litAov^ egressions:
% die-^d).
KH
12 CHANGE OF THB OBPMB OF TERMS.
IV. Change qf th$ order qf Termt, Like Temu,
8Si WlienallthetennsofanexpreMdonareooimected
by the sign •f it is indifferent in what order thej are
Ded; thus 0+7 and 7-f 6 giye the same retnlt, namely,
and 80 also a+b and h-ha give the same result^ namelj;
thd som of the nnmben which are represented by a and o.
We may express this fiMst algebraicaUy thtu^
a+b'»b-h<K
84. When an ezpresdon consists, of some terms pre-
ceded by the sign •(- and some terms preceded bj the
s^ — , we may write the former terms nrst in any order
we please, and the hitter terms after them in any oraer we
plefwe. This is obvious from the common notions of arith-
metio. Thus, for example,
7 + 8-2-3-8+7-2-3-7+8-3-2-8+7-3^2,
4i+6-c-«— 5+a-c— «— a+6-«-c— 6+a-«-«,
3S. In some cases we may change the order of tte
terms fiirther, by miidng np the terms which are preceded
by the sign — with those which are preceded by the sign +.
Thnsa for example, suppose that a represents 10, and b re-
presents 6, and e represents 5^ th«i
» . , , , ,
a+6~c— a-e+&— &-«+a;
for we anive without any difficult ate ^^ ab the residt in
all the cases.
Suppose however thai a represents 2, b represents 6,
and e.represlBiits 5^ then the expression a-e-hb presents a
difficulty, because we aire thus apparently required to take
a greater number from a lees, namely, 5 from 2. |t will
be conyenient to agree that such an expressiott as If<V^+^
mhea e is greater than a, shall be understood to mean the
sanpe tl^ng as a+b-e. At present we shall not nsesudi
an exprMnQH as a+d-o except when c is less than afpi
\
\
i
LIKB TBBMS.
18
\ in
I
■0 that a-^h-e will not cKDie any dUBonltT. Similar^, we
■hall consider -&+a to mean the lame thing as a-A*.
^ Thns the nnmeiloal yaloe of an enrassion remains '
the Mme^ wliaAefer maj be tlie order of the teims wMcdi
compose it This, as we hay^ seep, folloirp partly from our
notions of addition and subtraction, and partly from an '
agreem«nit as to the meaning which we asoibe to an ez-
piression when our ordinary arithmetical notions are not
strictly appticablei Snch an agl«ement is called in algebra
a eont0i^t¥mt ^nd conpin^Hori^^ |s the conesponding ad-
jectiyek
87. We Shan often, as in Art 84» haye to distinguish
the terms of an expression which are nreoeded by the sign
-«- from the terms which are preceded by the sini — , and
thiB foUowing de^nition is aocOTdingly adopted. The tiprms ..
in an exprMsion which are preceded by thie sign t(- are -
called potihW terms, and the terms whfeh are preceded
by the i%n — are cslled fM^aiive terms. This definition is
faitrodiiced merely for the m^e of breySftfi and no mea|ih|g
is to be giyen to the words potittp^ and ne^foHve beyopa
what is expressed in the defimtipn.
38. It win be seen that a term may occnr in an ex-
pression ppdceded (y no tiign^ namely the first term. ^ Snch
a term is eoimUd foOA ike wmiHve <0niit, that is it ip
treated as if the sign -f preceaedit It will be fomid that
if such a ohange be made in tiie order of the t^nns, as to
bring u tenn which originally stood first and was preceded
hf no sign, inio any other place, then it will be |^reocde4 by
the sign -f , l^or example^
heretheterm ahailK) sign befbre it m the first expre^
sion, but hi the other eqniyalent expressions it is preceded
by the sign -f. Hence we haye the following important
addition to the definition fn Art 37 ; if a term U^ecfided
l^noi^lhfi$ign'¥UtoUunder9Uiod^
'■" '
tk Terms are said to be like when th<^ do not differ
Ula^ or ^Slbr m^in thefar numerical coefficients; others
irisQ lliiQr M^ Add to be trnUi^ Thud a, ia, and 7a ^
Xi
4
f I
u
VEMi TMJUMA
like tenm; <i^ 5a^ tad 9a* are Uke teniit; d^,a5,Mid )*,
ere unlike termi.
4a An eipretdon whieh oonteini like temw may be
irfmpMflecL For example^ coneider the ezptetiilm
6a-a-fjld-f 60-5-1- 3c~2a;
bj Art 8S this expreaaion it eqniyelent to
6a-a^Sa-»-35-ft-i-5e4-8A ,
How 6a^a-8a*«3a; for whateyw number a mej re*
preaenty if we aabbract a from 6a we haye 6a left) and then
n we anbtract 2a from 6a we have 3a lefL Similariy
8J^-*M9&; and 6^4- 3e«-8a. Thua the propoied ezpraaaion
ply be piat In. the aimpler form
8a.f 2ft+8«,
Agafai; eoniider the expression ar-85«-40i TUa is
•qial to a-t& For if we have first tp subtract 35 from
a number a, and then to subtract 45 frtmi the remainderi
we ahall obtain the required result, in one ofieratieil by
Miitfacting 75 from a; this foUows from the common no-
tions c^Amhmetia Thus
^ a-35-45— a-75t
41. There will be no difficulty now in giving a
liig to auoh a statement/as i^e following,
/,: ;">.^'85-45— 75., ;.: ' ''■'''■■
Wi» oansiot aubtnUst 35 frt>m nothing and then
45 fitm therenulnder, so that the statement lust ^
not here intelligible in itself, separated fr^tnr#eit(^%^.
algebraical sentepoe in whidi it mayji6ciur) but it can |^
easily explamed thus: if in the ca|^ of an a]|e)«iilesl
operation we have to aubtract 35 frellHk number aip ^pi
to subtract 45 from the reataindw, we may subtraefr 75«t
once instead.'
\ . ■ .•* ' ' ■■''
As the student advances in the sul^Ject he may
to coi\jecture that it is possible to gfive some m<
the jNfitmosed statement by iikelt, t&t ^ a^iart
other ab^bralcal operaUon, and this jemectiire
* * ' when a laig^ treatise on JMgelNa
1 1 »■
0*
CO
ha
tei
Bu
01
eqi
*':
I
I*
}
val
\
\-
4V
■'•;,,'»
ii
f re-
ihen
lariy I
'I
■ h
n-
) '
A
-I ^
St
IK
»
u"^"^ ^ it^^^^t^^^ ^ eipknatloii which we
A ^^•i.S^ rimpBfvinf oT M^rwwioiis by oonectbff like
Gh^l^ ^^'^^ •• we •EOI see in the next two
2Si^-T#>.^t?^^ the JoDowing exprogiioM are aO
+ a\ +1x0, +^aj<l, +2j
^^]tf «*a, 5--a, c-3, ^-4, «-«, iind the nimierkel
valtiMofthei(Dllow%ezpiemtts: •»'' ««www
4u iSl?? 4<?+8rf . 5d+U \.
7.
■1<.<.
Id
ADDITION.
y. Addition..
43. It 18 convenient to make three casoB in Addition,
namely. I. When the terms are all like terms and have the
same sign; II. When the terms are all like terms but
have not all the same sign; III. When the terms are not
all like tenns. We shall take these three cases m order..
44. I. To add Hke terms which have the same sign.
Add the numerical co4ifficiente, prefix the common eign,
and annex the common letters.
For example^
6a+3a+7a^iea,
- 2&C - 7ftc— 96c« - 186<?.
In the first example 6a is equivalent to +6tf, and I6a
to +16a. See Art. 38.
45i.< II. To add like tenns which have not all the
tame sign. Add all the pontive numerical co^ffieiente
into one sum, and all the n^ative numerical coejj^icienti
into another; take the difference qf these two sums^
prefix i the sign qf the greater, and annex the common
letters.
For example,
7a-3a+ll«+«-6/^-2a=l9a-10a«9<^
2fto-t6c-36c + 46c+65c-6&c=llftc-16Jc=-6&f.
j»J^\ ^H' '^^ ^^ *®™*'' which are not all like terms.
Aoa together the terms which are like terms by the rule
tn the se<^nd ease, and put down the other terms each
preceded by its proper sign.
For example; add together
4a+5b-7c+Sd, Sa-b+2c+5d, ga-Zb-c-d,
BSid r-a+3b+4c-Zd+e. ^v^
' I* J* convenient to arranjge t!ie terms in columns, so
that like> terms shall stand in the same column: thus wo
hav6
)
th
wi
efl
W€
ho
5b
am
I
'if
flJ»4
i
is u
1
can
.
i
volv
'i
invo
1
y
■
.
:^-
ADDITIOJBT.
n
4a+Sb-'1e+2d
3a— l>+2c+5d
9a-26- c- d
-a+Zb+4c-3d+0
I5a + 6b-2c + 4d+e
Here the ietfaa 4a, 3a, 9a, and —a are all like terms;
the sum of the pesitiye coefficients is 16; there is one term
with a negative coefficient, namely — a, of which the co-
) efficient is 1. The difference of 16 and 1 is 15; so that
we obtain + 16a' from these like terms; the sign + may
howeyer be omitted by Art. 38. Similarly we haVe
5&--&— 2& + 3&r»5&. And so on.
so
W«
> i47. In the followmg examples the terms are arranged
soitably in columns : .
4a^+1a^+ a?-9
-2«3+ ^« 9*+8
-34J»- «2+10«-l
9aj"- a-l
3a"-3a&-76»
4a*+6a6+96*
a«-3a6-3J*
9a*
— <?
In the first example we have in the first column
d^+4«3^2«^— Sv*, that is 5^— &p', that is, nothing;, tins
is usually expressed by saying the terms which involve ai*
cancel each other.
Similarly, in the second example, the terms which in-
volve ab cancel each other; and so also do the terms which
involve 6*.
7a^—Sxy + a?
3a^ - y*+ 3a?-- y .
-2a;2 + 4:ry+6y"— ar-2y
-lay— t^+ da—i
4dP* +4y2- 28?
12«*- 6ii!y + 7y" + ia»— 8|^
T.A.
2
5<fc.'
18
EXAMPLES^ r.
w
r^
1.
2.
8.
5.
Examples. V.
Add together •
3a~26, 4a-&>, 7a-ll&, a4-9&.
4«"-3y», 2««-6y», -Jf«+y8, -2««+4y«.
5a+3&+c, '3a+3d+3<;, a+3& + 5c.
3«r+2y-;ir, 24r-2K + 2;ir, ^a + 2y + 3z,
7a-4ft + (?, 6a+3&-6<?, -12a+4ft
«-4af&, 3«+2fr, a-aT-5ft.
7. a+b-Cf 6+c-a> c+a-5, a+5-«,
^. #-f^+3<J^ 2a-6-2c, 6-a-c, c-a-ft,
9. a-26+3c-4<^ 36-4c+6rf-2a^ lk;-6<^+Sa-45,
7rf-4a+56-4ft
iB»-4flj"+6«~3, 2«'-7«*-14a?+6, -«»+a««+af+8.
«*-2«'+3aj*, «'+«*+d?, 4«*+6«*, 2^1^+3^-4,
-3«*-2a?-6.
a»-3a«6 + 3a6^-6^, 2a'+6<rt-6aJ»~76',
^— 2a«*+a^«+a^ «'+3aa^, 2a'-aa^-2«*, ,
2a&-3aa:*+2a>dT, 12a& + l<kMJ*-6flftp,
3a^-4ajy+y*+2«+3y-7, 2a^-4y*+3a?-^+8,^
10d;y+8^+9y, &»'-6«y+3y*+7«-7y+ll.
17. ;ir*-4«V+6«V-*»'y'+y*>4^-12«V+12dJ^-4y*,
e«V-12ajy»+6yS 4«jf»-4yS yl
18. «'+«y"+j?^-««y-«y^-^4f,
la
11.
li
la
14.
15.
16.
^
v^
"-1
j-46,
a? +8.
-4,' i
f-4irS
subtbaCtion.
YI. SubiracHon.
If
48k Suppose we have to take 7+3 from 12; the repnlt
u the same as if we first take 7 from 12, and Uien take 3
from the remamder; that is, the result is denoted by
12-7-3.
Thus 12-(7 + 3)-12-7-3.
Here we enclose 7 + 3 in brackets in the first expression,
because we are. to tsdce the tehole of 7+3 from 12; se^
Art 29.
Sunilarly 20-(5+4+2)-20-5-4-2.
In tike manner, suppose we have to take h+e from a;
the result u the same as if we first take b from a, and
then take c frx>m the remainder; that is, the result is
denoted by a— d-<;.
Thus «-(6+c)=a-6-«,
Here we endose b+cia brackets in the first expresrioni
because we are to take the whde otb+t from (k
Similarly «-(ft+c+rf)-»a-6-c-dL
49. Nelt suppose we have to take 7-3 firom 12. If
we lake 7 from 12 we obtain 12-7; but we have thus
taken too much from 12, for we had to take, not 7, but 7
diminished by 3. Hence we must increase the result by $|
and thus we obtain 12-(7-3)»12-7+3.
Sunilariy 12-(7+3-2)-12-7-3+2.
In like manner, suppose we have to take 5— « frx>m a.
If we take b from a we obtain a-b; but we have thus
taken too much frtnn a, for we had to take, not 5, but b
diminished by e. Hence we must increase the result by e;
and thus we obtain a-(&-c)»a—&+c;
Similarly a-^+c-rf)«a-6-c+dl
60. C<msider the example
a-(&+c-rf)=a-&-c+€f;
that is, if d+0-cf be subtracted from a the result !«
2—2
J«SH
i I
30
I^UMTnAaTWNL
<\. :
a—h^c-^-d. Here we see that, in the expnression to be
Bttbtracted there is a term — <f, and in the result there is
the corresponding tdrm -k-d', also i& the expression to be
jBubtricted there is a term +e, and in the result theii^e is a
term — c; also ^n the expression to be subtracted there is a
term 5, and in the result there is a term —h.
From considering this example, and the others in tii^
two preceding Articles we obtain the following rule for
Subtraction: chanpe the signs qf all the terms in the ex-
pression to be stibtracted, and then collect the terms as in
Addition,
For example; from Ax—Zy-¥2z subtract 3a? — y +4?.
Change the signs of all the terms to be subtracted ; thus
(We obtam ^^^y—z; then collect as in addition; thus
4«-3y+2«— 3a?+y-;!r=fl?— 2y+4f.
From aaj* + SiB^ - 6«* - 74? + 6 take 20^ - 2aj* + 6«* - 6a? ~ 7.
t
Change the signs of all the terms to be subtracted
fMid proceed as in addition; thus we have
3a?*+6a!»- 6a!*-7a?+ 6
.-2a?*+2a?»- 6a?2+6a?+ 7
*.
'i I
' / ««+7aj'-lla?«- a?+12
The be|;inner will find it prudent at first to go through
the operation as fullv as we have done here; but he nuur
gradtmlly accustom himself to putting down the result
without actually changing all the signs, but merely sup-
posing it done^ ,
61. We have seen that
a— (6-c)«=a— 6+c. /
^ Thus corresponding to the term — ^ in the expression
to be subtracted we nave +(? in the result. Hence it is
not uncommon to find such an example as the following
proposed for exercise: from a subtract —(?; and the result
required is a+c. The beginner may explain tMs in the
manner of Art. 41, by considering it as having a^eknlng,
not in; itself, but in connexion with some other parts of an
algelNiycal operation.
EXAMPZES. \ ri
it
It is UBiial howevw to offer some remarks which will
serve to impress results on the attention of the begimier,
imd perhaps at the same time to suggest reasons for them.
Tims we may say that a^a+c—c, so that if we subtracC
— e from a there remams a 4- <;.
Or we may say that + and — denote operations the re-*
yerse of each otiber ; thus — c denotes the reverse of -f <;, and
so - ( — c) will denote the reverse of the reverse of + e^ that
is, -(- c) is equivalent to + c.
^ut, as we have implied in Art 41, the beginner must
be content to defer untu a later period the complete expla-
nation of the meaning of operations performed on negative
quantities, that is, on quantities denoted by letters with
the sign — prefixed.
It should be observed that the words addition and
mibtraction are not used in quite the same sense in Alsebra
as in Arithmetic. In Arithmetic addition alwavs produces
increase and subtraction decrease; but in Algebra we may
speak of adding —3 to 5, and obtainmg the Algebra/ical
sum 2; or we may speak of subtracting ^3 firom 5, and
ohtsAmng the Alffeoraicai remainder S,
Examples. VI.
1. From 7a +145 subtract 4a +10&. - '
2. From 6a- 2&-C subtract 2a- 2&-3<?.
3. From 3a -26+ 3c subtract 2a- 76 -c-dL •
4. From 7«*— 8a?- 1 subtract 6«*- 6a? +3.
6. From 4a?* - 3^?* - 2a:* - 7a? + 9
subtract a?* — 2a?' - 2aj* + 7a? - 9.
6. From 2a^— 2aa?+ 3a* subtract o^—ax-^ a*.
7. Jfrom a^-Zxy-y^+yz—^z*
subtract a?* + acy + 6a?2f - 3^ — 2;8*.
a From5a?2+6a;y-12a?;y-4y2-7y^-64f' V
subtract 2a^ - 7a^ + Axz — '3y^ + 6yz — 5^^.
9. From a^-3a^6+3a6*-63 subtract -a?+3a26-Sa&'+6»
10. From 7a?'-^2aj*+2a?+2 subtract 4aj8-2a?*-2a?-14,
and from the remainder subtract 2ai^-^Ba^-¥4$fykl6:
22
BXAOKBTS.
I
t
VII. Sraekeii.
62. On aooonnt of the extensiye use which is made of
brackets in Algebra, it is necessary that the student should
obsenre yerv careftilly the roles respecting them, aiid we
shall state them here distinctly.
When an exprestion within a pair qf bracket* is pre-
ceded by the sign + the brackets map be removed,
^ When ah expression idthin a pair qf brackets is pre-
ceded by the sign — the brackets may be removed if tfte
sign qf every term within the brackets be changed,
Thufl|y for example,
The second rule has already been illustrated in Art. 50 ;
it is in fact the rule for Subtraction, The first rule might
be illustrated in a similar manner.
53. In particular the student must notice such state-
ments as the following:
+(-rf)=-<f, -{-d)^+d^ +(+^)=i+«, -(+«)= -^.
These must be assumed as rules by the student^ which
he may to some extent explain, as in Art 41.
54. Expressions may occur with more than one pair of
brackets : these brackets may be removed in succession by
the preceding rules beginning with the inside pair. Thus,
for example,
«+{&+(<?-<£)}««+{&+ c-<f}«a+6+c-<?,
a+ {&--(c-^}««+ {&-<?+ e?}«a+&—c+rf,
a''\b-¥{fi'-d)}=a-{b-\-c—d\=a-b—C'\-df
a-^-{fi'-d)}^a-{b'^c-\-d^^a'-b-\'C-{L
Similarly,
a-[5-{c-((^-^)}]=a-p-{<j-rf+«}]
UwiU be,|»en in these examples that, to prevent con*
faullinilut w eon^^arions pdrs of brackets, we use brackets
■■^w.
T^
BRAOKETA
28
of different $hape»; we might distiiigaish by wdng bradcets
of the same shape but of duTerent nze$»
A yincalam is eqniyalent to a bracket; see Art 80.
Thus, for example,
a-.p-{<j-(rf-77)}]-a^[6-{c-(rf^«+/)}]
55. The beginner is recommende^l always to remove
brackets in the order shewn in the preceding Article;
namely, by removing first the innermost pain next the in-
nermost pair of all which remain, and so on. we may how-
ever vary the order; but if we remove a pair of brackets
including another bracketed expression within it, we most
make no change in the eigne <f the included expreerion.
In &ct such an included expression counts as a single term.
Thus, for example,
a+{&+(c-<?)}«a+6+(<j-d)=»a+6+tf-rf,
a-{5-(c-rf)}«a-6+(c-rf)«a-6+c-dl
Also, a-^\h-{c-'(,d-e))]-*a-h^-{c-{d'-e)}
=:a-b+c-{d-e)^a^b+c^d+e».
And in like manner, a-jj— {c— (rf-« -/)}]
56. It is often ccmvenient to put two or more tenns
within brackets; the rules for introducing brackets follow
immediately from those for removing brackets.
Any number of terms in an expression map blB put
within a pair qf brackets and the sign + placed before
the whole.
Any number qf terms in an expression may be put
within a pair qf brackets and the sign — placed before
the whole, provided the sign of every term within the
brackets be changed, * ^^
24
EXAMPLES. VIL
Thus, for example, a-& 4-6-^4' «
— a-&+(c-</+«), or — a-6+<?+(-rf+«X
or taia~(6-c+d-^), or — a-6-(-c+rf-«).
In like maimer more than one pair of brackets may
be introduced. Thus, for example, / '
\v
EZAUPLES. VII.
Simplify the following expressions by remoYing the
brackets and collecting like terms:
1. 3a-ft--(2a-5). 2. a-5+o-(a-6-c). i
3f^l-(l-.a)+(l-a+a«)~(l-a+a«-a»). \
4. a+&+(7<i"-&)-(2a-3ft)~(6a+66).
5. a-&+c-(6-a+c) + (c-a+6)-(a-tf+&).
7. a-{6-<?-(rf->r)}.
a 2a-(25-rf)-{a-6-(2(j-2^}.
9. d^{2&-(3kJ+25-a)}. 10. 2a-{&-(a-25)}.
11. 3a-{&+(2a-6)-(a-J)}.
12. 7a-[3a-{4a-(6a-.2a)}].
13. 3a-[6-.{a+(&-3a)}].
14. 6a-[4&-{4a-(6a-4&)}].
16. 2a~(3& + 2c)-[6&-(6<J-6&) + 6<j-{2a-(<J+2ft)}], .
16. a-[2ft+{3<j-3a-(a%J)}+{2a-(6+c)}].
17. 16-{5-2a?-[l-(3-i?)]}.
la l&»-{4-[3-6ii?-(3a?-7)]}. I
19. 2a-[2a-{2<i-(2a-2a^)}].
20. 16-a?-[7a?-{ai?-(9a?-3:if-6a?)}].
2L a»-[3y-{4ii?^(6y-6ii^r7y)}]. ^^
22. 2a-[3&+(2&-c)-4c+{2a-(36-c-2&)}];
U
23. a-X5*-{a-(6c-2c-6-4&) + 2a-(a-26+<;)}J. > ■
24, f*-[>«'-{^-(4i»-l)}]-(a?*4-4a*+6a!*+4«^l)^
•
mULTIPUQATIOK Vi
is may
^ the
«—
0}.
m
i^
I
Vllh MultipKeation.
67* The student is snppoaed to know that the product
of any number of factors is the same in whateyer oraer the
factors may be taken; thus 2x3x5=2x5x3=3x5x2;
and so on. In like manner a&c»ac&««(c<i, and so on.
Thuis also dd-^b) and {a+h)e are equal, for each de-
notes the proauct of the same two meters; one &etor
being e, and the other finctor a+b.
It is conyenient to make three cases in Multiplication,
namely. I. The multiplication of simple expressions; II. The
multiplication of a compound expression by a simple ex-
piiession; III. The multiplication of compound expres-
sions. We shaJl take these three cases in order.
58. I. Suppose we haye to multiply 3a by 4b. The
product may be written at full thus 3xax4.x&,or thus
3 X 4 X a X 6; and it is therefore equal to 12a&. Hence we
haye the following nde for the multiplication of simple ex-
pressions; muUiply together the numerical co^jfidente
and put tlte lettere d^er this products
• . . *
Thus for example,
*!axZbe^2lab€,
4ax5bx3e^60ahe,
59. 77ie powers qf the fame number are mtdtipUed
together by adding the expomnte.
For example, suppose we haye to multiply o^ by a^. -
By Art Id, ii^^axaxo,
and - a^^axa;
ther^ore fl?xa'=«axaxaxaxa=«'=sa''*"V
:: Si|nilarly, c*xc'=cxcxcx<jxcxcxc=(j'=sc**'.
In like manner the rule may be seen to be true in any
other case. - -
26
MULtlPLICA TION.
60. II. Suppoflo we have to mnttiiily a4-6 bj S> We
have
Similarly, %a-¥h)»1a-¥lh.
In the aaine manner suppose we haye to multiply a4-&
bye We have
c(a+5)-«<Ja+cft.
In the same manner we have
3(a-6)— 3a-36, 7(a-J)=7a-7ft, c(«-5)««i-cft.
Thus we haye the following rule for the mnltiplioation
of a eompomicl expression by a simple expression; fntUtipiy
each term qf the compound expreeeion by the simple ex-
preseionf and put the eign qf the term before thj> rettdt;.
and eollect theee resulte to form the complete prodwst, \
61. III. Suppose we haye to multiply a + & by c + d ^
As in the second case we haye
{a-¥h)(c-{-d)-a{C'¥d)-¥h{c-¥d)',
also a(c+<l)=ac+ad^, h{C'¥d)=^l)C'^hdi
therefore (a + 5)(« + <Q = oc + a<^+ (c + M.
Again ; multiply a ~& by 6 -I- d
also a(c+d)=<w+<w^> h(fi-^d^-bc-¥hd\
therefore
(a-&)(<j+d)=a<?+adf-(5<j+M)=ac+a<?-J(j-ML
Similarly ; multiply a-^hhj c-d.
Lastly; multiply a-& by 6-^.
(a-b)(e-d)={e-d)a-(e-d)b;
also (p-d)a=ac-ad, (c-dfi^bc-^hd;
therefore
(fl'-b)(e-d)=ae-ad-(be-bd)=ae''ad-'be-^bd,
L6t us now consider the last result. By ArtJ^38 we
may write it thus,
(+a-6X+c-d)= +ac-a(^- 5c+ML .
MULTIPLWA TTOy.
n
We we that corrMpondiiig to the -fa which oocart
in the multipUoend and the +6 which ocean in the multi-
plier there is a term + oc in the product ; corresponding to
the terms 4-a and ^d there is a term —ad in the product;
corresponding to the terms -b and +6 Uiere is a term
— (ein the product; and correspondinff to the terms -6
and —d there is a term +bd in the product.
Similar obseryations may be made respecting the other
three results; and these obsenrations are briei^ collected
in the followiiog important rule in multiplication: like iign$
produce + ana untike tigne -. This nde is called the
Mule qf Signtf and we shall often refer to it by this name.
62. We can now give the general rule for multiplyhi|r
algebraical expressions; multiply each term qf the mtiAi-
pneand by each term qfthe multiplier; if the terme have
the eame tign prefix the eifm + to the product, if they
have difereni Hone prefix the^eign—; then eoilect theee
reeulte tojbrm the complete produce
For example ; multiply 2a + 3& - 46 by 3a - 4&. Here
(2a+3&-4c) (3a-46)=3a (2a+36-4<j)-4ft (2a+86r-4c)
^=6a«+9a6-12ac-(8aft+12&«-16lc)
=6fl?+9a6-12ac-8a&-12J^+l^.
This is the result which the rule will give; we may
simplify the result and reduce it to
6a^+a6-12a<j-126«+165ft
We might illustrate the rule by using it to multiply
6—3+2 by 7+3— 4; it will be found that on working oy
the rule, and collecting the terms, the result is 30, that is
5 X 6, as it should be.
63. The student will sometimes find such examples as
the following proposed: multiply 2a by -4&, or multiply
—4c by 3a, or multiply — 4(j by — 4ft.
The results which are reqmred are the following^
2ax-4&=— 8aft,
— 4cx 3a=:— 12ac^
■'"'^ -4CX-46* Ubc.
28
MULTIPLICA TTOir.
The stadent may attach a meaning to these oporatioiii
in the manner we have ahready explained; see Article 41.
Thus the statement —Acx -Ah^l^he may be under-
stood to mean, that if —Ac occur among the terms of a
multiplicand and -Ah occur among the terms of a multi-
plier, there will be a term 16&6 in the product correspond-
ing to them.
Particular cases of these examples are
2ax-4=~8ay 2x-4«=-8, 2x-l=-2.
64. Since then such examples may be given as those
in the preceding Article, it becomes necessary to take ac*
count of them m our rules ; and accordingly the rules for
multiplication may be conveniently presented thus:
^h
1 To multiply simple terms; mtdtiply together the nu-
merical co^ficientSfPut the letters <xfter thu product and
determine the sign oy the Rule qf Signs,
To multiply expressions; multiply each term in one
expression ^ each term in the other by the rule for mul-
tiplyina simple terms, and collect these partial products to
form the complete product,
65. We shall now give some examples of multiplication
arranged in a convenient form.
<r + 6 a+b a^+3x
a +b a —b « — 1
a^-hdb
+db+b*
a^+ab i^+.3af^ .
a'+2a6+ft*
<^-db+l^
a+b
a« -6* «8+2««-3«
8a«- 4a& + 65«
V a*- 2ab ■¥ 2X^
3a«- 4a»&+ 6a«62
- 6a»6+ 8aV-10a&»
+ 9a^-12a»»+166«
3a*- lOo^d + 22a*6»-22a6»+166*
m
N
MULTIPUCA TION.
Oonsider the last example. We take the first term in
the multiplier, namely a\ and multiply all the terms in the
multiplicand by it, paying attention to the Rule qf Sign*;
thus we obtain M - ia^h + 5a'5*. We take next the second
term of the multiplier, namely -2a&, and multiply all the
terms in the multiplicand by it, paying ati»ntion to the
Rule qf Signs; thus we obtain -6a*&+8a^-10a5s.
Then we take the last term of the multiplier, namely 3^,
and multiply all the terms in the multiplicand by it,
paying attention to the RtUe qf Signs; tnus we obtain
+ 9a*- 12058+ 166*.
«
We arrange the tefms which we thus obtain, so that
like terms may stand in ths same column; this is a yery
useful arrangement, because it enables us to collect the
terms easily and saielv, in order to obtain the final result.
In the present example the final result is
8«* - 10a»6 + 22a*6«- 22a&8 + 166*.
66. The student should observe that with the view of
bringing like terms of the product into the same column
the terms of the multiplicand and multiplier are arranged
in a oerUin order. We fix on some letter which occurs in
many of the terms and arrange the terms according to the
powers <iftkat letter. Thus, taking the last example^ we
fix on the letter a ; we put first in the multiplicand the'
term 3a', which contains^ the highest power of a, namely
the second power; next we put the term —4a6 which con-
tains the next power of a, namely the first power; and last
we put the term 66', which does not contain a ait all. The
multiplicand is then said to be arranged according to
descending powers of a. We arrange the multiplier in
the same way.
We might also have arranged both multiplicand and
multiplier m reverse order, in which case they would be
arranged according to ascending powers qf a. It is of
no consequence which order we adopt, but we must take
the same order for the multiplicand and the multiplier.
67. We shall now give.some more examples. .
Multiply l+2«-3iii'+a* by *'-2ir-2. Arnutge ac-
cording tadiewm^ng powers of « ''
<f
k
MULTIPLICA TION.
flj'— 2aT— 2
l^^i— — M^^MIP— — — — ^ IMIl I II ■■■,■ Mlll^
aF-6a^ +7«"+2«'-e»-2
Multiply a'+J>+c*~a6-&c-«i by a+&+tf.
Amuige acoording to descending powers of a.
a + 6 + c
if
• H
• I
-3a6<; +6^
+c»
This example might also be worked with the aid of
brackets, thus,
a + (6+c)
Then we have 0(6^- Jc + c^ - a(6 + c) (6 + c)
=a{6»-J«?+tf»-J*-26(j-c»}= -3a6c;
and (&+c)(a»-6<!+c^-M+c'. ^^
Thufl^ as before, the result is o^ + ft* + e^- 3a&(?.
MULTIBLICA TION.
Multiply together xr-a^ at-b, x-e,
X —a
X -6 .
«
a^-ax
m —e
tfi-{m+b)a^+abx
-'ea^'*'{a+b)ex—abe
«• - (a + 6 + c)«* + (aft + ac + ftc)a? - oftc
The student should notice that he can make two ezer-
dses in multiplication from every example in whidi the
multiplicand and multiplier are different compound ex-
presnons, by chaiufing the origmal multiplier into the
multipHcand, and tne ordinal multiplicand mto muUlpHer.
The result obtained should be the same^ which will pe a
test of the correctness of his work.
t «
EzAMPUS. TIIL
Multiply
1. 2«»by^. % 3a«by4a^. 8. S^ftbySoft*.
4. 3aV« by 6aV^. «• 7aVby7yV.
6. 4a*-3ft;by8aft. 7. 8a«-»oftby3a«
a &i^-4^+8;»»by2«V.
». i^-t^i^+x*a^ by «V^.
1ft. 2ayV+34Vj»-r6«*y«*by2a?^4r.
11. 2«-yby2y+A
12. SjB*+4«'+8jP4-16byav-.6.
la «^+^+«-lby»-l.
\
<»;'#
w
14.
10.
16.
17.
la
19.
20.
21.
22.^
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
EXAMPLES, VIIL
1 + 4a? - 10«*« by 1 - 6a? + 3«".
a?* - 4«" + 1 la? - 24 by a?* + 4d? + 6.
«"+4aj*+6a?-24 by «*-4aj + ll.
«*- 7«" + 6a? + 1 by 2^?*- 4d? + 1 .
«■ + 6a^+ 24a?+ 60 by a;'- 6a?*+ 12a?+12,
OJ* - 2a?* + 3a? - 4 by 4a;5 + 3aj" + 2a; + 1.
a?*-2«'+3a?*-2a?+l bya:* + 2a?*+3«»+2a?+l.
0^— 3aa?by a?+3a.
a' + 2flu? — oj" by a' + 2aa? + 0?*.
2M+ 3a6 -a« by 7a-6&.
a«- a5 + 5" by o«+ a& - 6».
a*- a& + 26* by a* + «& + 26*.
4a!» — 3ajy - y* by 3a? - 2y.
0?" — a?V + ^— y" by 0? + y.
2a!» + 3a;y + 4y« by 3a?* + 4a?y + y\
a?*+y*-«y+a?+y-l by .c+y— 1.
0?* + 2«V + 4a?V + 8^ + 16y* by Of - 2y.
81a?*+27a;V+9ajV+3ay+y* by 3a?-y.
0? + 2y — 3;y by 0? — 2y + 3<8r.
a*— flw?+ 6a? 4 &* by a + & + a?.
<;?+6'+i?*—6c~ca—a6 by «+&+<?.
0^ + 46a? + 46*0;* by a' - 46a: + 46*a;*.
a«-2a6 + 6'+c*bya*+2a6+6*-c*.
I
\
'.<!
II
1
V
Jt^
Hultiply the following expressioiis together
87. a—af a?+o, a^'+a*,
88. a+a, a?+6, a+c, ■
88. x^—ax-¥a\ a^'\-cuo'¥^\
40. «-2a, ;v~a» A?+ay
d?*l-2a.
K.: ^,
m
M VISION.
33
IX I>ivisi<m,
ea Division, as in Arithmetic^ is the inverse of Multi-
plication. In Mnltiplication we determine the product
arising from two given factors; in Division we have given
the product and one of the fiictors, and we haveito deter-
mme the other factor. The &ctor to be determined is
called the qitotient, »
The present section therefore is dosety connected with
the preceding section, as we have now in fauat to undo the
operations mere periormed. It is convenient to make
three cases in Division, namely, I. The diidsion of one
shnple expression by another; II. The division of a com*
\ pound expression by a simplo expression ; III. The division
of one compound expression by another
■^
69. I. We hare already shewn in Art. 10 how to
[denote that one expression is to be divided by another.
I For example, if 6a is to be divided by 26 the quotient is
jfadicated thos: Sa+SSc, or mow mmaUy |.
It may happen that some of the factors of the diVisor
3ur in the cuvidend; in tiiis cuse the expression for the
|uotient can be simplmed by a principle already used in.
ithmeUc, Suppose, for example, that I5cflb is to be
livided bj &fc; then the quotient is denoted by ^ .
[ere the dividend 15a*6=5a^x3&; and the divisor
^hc='2ex9b*, thus the &ctor 3& occurs in both dividend
id divisor. Then, as in Arithmetic, we may remove
common &ctor, and denote the quotient by ^;
lUS 8-^ Mr- — .'
T.A.
3
I I
'(»
DIVISION.
u
It may happen that all the fiieton which oocnr in the
diyisor may be removed in this manner. Thus sappose, for
example^ that 24a&:v is to be divided by Soar;
"So.
8047
Soar
■ -i
70. The rule with respect to the tign of the quotient
may be obtained from an examination of the cases wUch,
pccur in Multiplication.
< For example, we hare
A€iby.Zc^l2abci
12ahc „ I2ahc ^ ,
u
therefore
therefore
Aab ""' 3c
4a& X - 3c» — 12a5<jj
-12a6<j - -I2a^;
-4a&x3<J— --12a5c;
11
*iab.
therefore
21:: V
therefore
-lldbc
^oCf
— 12a&c
-4a6 ""' 3c
— 4a&x --3cs-12a6c;
' 12a& c
-4a&
-4a&.
—3c,
12al>c . -
,Thus it will be seen that the EiUe qf Sigm holds in
Division as well as in Multiplication.
71. Hence we have the following rule for dividing one
simple expression by another: Write the divujkiia oter
the divisor with a line^hetween them; if the exprettione
have common fcbctorif remove the common faetore ; prefix
the tign + if the expreesione have the same sign and the
sign - if they have diferent eigne. ? ,
72. One power qf any number is divided M another
power of the same numiberfhy eubtracting theind^qf
the latter power from the index qf theformar, o ^
, s
¥oT example^ 8a|ypdae we have to divide a" by-^i^
By Art 16, af=axaxiixaxa,
V^
therefore
a»
axaxaxaxa
axaxa
:Ax-a»a'=a""^,
ai n 1 ^ -excxexexexexc ^. -^-
Similarly -^ = ^ .. ^ .. ^ ,. ^ =c x <j x c— c'sc'^*.
cxcx<;xc
In like maimer the rule may be shewn to be tme in taxy-J^ >^
other case.
'^,
! \
Qr we may shew the truth of the rule thus :
by. Art 59, .-^^i^ c^xc^=c',
therefore
=c»,
73. If any power of a number occurs in tho dividend
and a high^ power of the same number in the divisor, tho
quotient can be simplified bv Arts. 71, and 72. SuppoUe^'
for example, that 4a^ is to oe divided by 3<^; then, i^e
quotient is denoted by ^^ . The fiictor 5^ occurs in botli
dividend and divisor; tUs may be removed, and the quo-'
tient denoMby g^; thus ^^ = g^.
74. IL The rule for dividing a compound expression
by a simple expression will be obtainOd from an exaddna-
tion of the corresponding case in Multiplication.
For exfunple, we have
^v
therefore
therefore
o
(a-6)x -c=— oo+Jc;
— c
=a-Ji
■■/'
■vj
86
division:
^v;
'n
V
Hence we hf^ye the foUowing rale for dividing a oom-
ponnd expreflsion by a simple expreBsion: divide each term
qf the dividend by the divieor, by the ruU in the firH
catef and coUeet the reeulte to form the comjjMe quotient.
' J^or example, — i?^=4a'-36c+aft
. 75. III. To divide One compound expresrion by
another we must proceed as in the operation called Long
.Division in Arithmetic The following rule may be given.
Arrange both dividend and divieor according to aacend'
ing powers cf eome common letter, or both according to
descending pomere qf some common tetter. Divide the
Jirst term of the dividend by the Jlrst term qf the divisor^
and put the residtfor the first term of the quotient; muh
tiply the whole divisor by this term, and subtract t%s
product from the dividend, T9 the remainder join di
many terms qf the dividend, taken t^ order, as may be
required, and repeai the whole operation^ Continue the
process until all the terms qf the diHdend have been
taken down* ■ : =
The reascm for this rule is the same as that for the
rule of hon^ Division in Arithmetic, namely, that We may
hfesk the dividend Up into parts and find how often the
divisor is contained in each part, and then the aggregate
of these results is the complete quotient
76. We shall now give some examples of Division
arranged in a convenient form.
a+bja^+2ab+l^(a+b
a^+ah
€^+db
db+I^
-db-^b^
^db^b»
a'^b)a*''l^(a+b «*+3afJ«*+2a!*-3af^«-'l
ab-b*
ab-V
DIVISION.
K
-4a^6+13a*6«-22a6»'
-4a^ft+ 8a«6«-12aft»
5a«6«-10a6»+16&*
Consider the last example. The dividend and divisor
are both arranged according to deacendinff powers of ol
Hie first term m the dividend is So* and the first term in
the divisor is V; dividing the former by the latter we
obtain 3a* for the first term of the quotient. We then
multiply the whole divisor by 3a*, and place the result so
that each term comes b^low the term of the dividend which
contains the same power of a; we subtract^ and obtain
— 4a'&+13a^&'; and we bring down the next term ot the
dividend, namely, — 22aC We divide the first term,
— 4a'&, by the first term in the divisor, a*: thus we obtain
— 4a& for the next term in tiie quotient. We then multiply
the whole divisor by — 4a& and place the result in order
under those terms of the dividend with which we are now
occupied; we subtract^ and obtain 5a'd*-"l(ki6^; and we
brinff down the next term of the dividend, namely, ISft*.
We divide fk?l^ by a*, and thus we obtain 5^* for the next
term in the quotient We then multiply the whole divisor
by 55^ and place the terms as before; we sujiytarac^ and
there IS no remainder. As all the terms in the dividend
have been brought down, the operation is .completed; and
the quotient is 3a* - 4a& + 5d\
It it qf great importance to arrange "both dividend
and diviior according to the sama order of some common
letter; and to ixttend to this order in every part qf the
operation,
77. Ijb may happen, as in Arithmetic, that the division
cannot he eaactly performed. Thus, for example* if we
divide a*+2db+2¥ by a+b. we shall obtain, as in the fbcat
examine oC the preceding Article, a -1-6 in the qaotient,
and there nfHihen he a remainder h\ This result is ex-
m
DlP^tSIOlf.
pressed in ways idniikr to those used in Arithmetio; thm
we may say that
=a+6+
t^
a+5'
6^
that is, there Is a quotient a + 5, and a fhuitional part — r •
In general, let A and B d0note two expressions, and
suppose that when A is divided by B the quotient is q, and
the remaindldr R\ then this result is eipressed algebrai-
laUly in the following ways,
A^qS^Bi or A--qB=Bf
I- I
A R A B
^^B^^-^W ^B'^^^'S'
\
Hie student will observe that eacli letter hero may re^
g^ resent an expression, simple or compound; it is oftoa
onveoient for distinctness and brevity thus to represent
an expression by a smgle letter.
We shall however consider ak^braical fractions in snb-
secmont Chapters, and at present shall confine ourselves to
e»unt»les of IXvlnon in wmch the operation can be exactly
performed.
78. We give some more examples:
IHyide 4^-5«»+7dJ»+2«*-6ay-2 by l+lto-8aj»+«*.
Arrange both dividend and divisor according to de^
Bcending powers of ;r.
- 2aj* -2«* + 6^ + 2aj»- 6«
-2ar« 4-6a^-44^-2a?
-2# +6«"-44?-2
- \\
-•— r-
DIVISION.
30
Arrange the diyidend according to. descending powers
ofo.
-J -sH
It win be seen that wo arrange these terms according
to descending powers of a; then when there are two
terms, such as crh and cfc^ whoich inyolve the same power of
Oy we select a new letter, as &, and pat the term which
contains h before the term which does not; and again, of
the terms a^ and qbc^ we put the former firet as involving
the h%;her power of 6. ' ^ ' '
This example might also be worked, with the aid of
brackets, thus:
-a^+c)-a(d*+26c+c*)
- — ' '•
a(y- ^j+^j^+y+c^
M EXAMPLES. IX.
Diyide «*-(a+6+«)«'+(aft+ac+Jc)«-aftc by m-e.
— (a + 6)«^ + (aft + flw + 6c)«- ofto
abx
Erery example of Miilti]^oati<m, in which the miiHi-
jdier and the mnltiplioand are different expresaionay wi&
ramiah two ezeroiaea in Diyiaion; beoanae u the prodnet
be ^divided by either &ctor the quotient ahonld be the other
ihctor. Thiia from the ezamplea given in the aeotion ^
Midtiplication the atodent can denye exerdsea hi DiviaioBi
and teat the accuracy of hia work. And from any example
of.Diyiaion, in which the quotient and the diyiaor aM
different expreai^na, a aecond exerdae may be obtiiMw
S' mal^ the quotient a diyisor of the dividend, ao thil^
e new quotient ought to be the original diviaor. ^f
EXAMPLBS. IX,
\.
4
6.
a
».
10.
11.
13.
14.
15.
IS*" by 8«». 2. 240^ by - 8fl^. 3. 18aV 1^ i>il>
24<iVd« by -3aV , 5. 2(to«&*«V by ^9^
4a^-8a^+Uahj 4af. 1, 3^-12d?+15<j^br%p|
fl^-3a!V+4«y'byajy. "?S:\
-15a>6S-^8a^M+12a&by-da&. %/m X
«•* 7^+12 by «-3. 1^ ajS+^-tSi byiilff;
2«»-«*+8«— 9 by 2a?-8.
SdJ* + 14;]^ - 49 + 24 by 20^ 4- 6. \^^\
9«*,+ 3«» +4?- 1 by 3i»- 1.
7«*-24e*+5av-21 by 7«-3.
V.
■'--S-vT:-"-.
v-, - • i ■ r-
/If
EXAMPLJBS.
41
>y »-«.
10 miitti^
ioiii» will
prodiMl
ifaeoUitr
Kstion ^
Dhriiio%
example
risor
obliiiu
.so
' -Mt}
*^,ik:;
■' -U:.:^
■ ■ ^^tt-:^
■ '•%%^' ■
< ■ \'«i •■ -.1- '■'.
'' ■ , ft •* ■■/-
' A'^f
'."■•^ 'vl* '
: :.v -^' ■
17. 4^-lby«-l. 18. aP-SaS^+i*bya-ei
19. «*-81y«by«-8y,
SO. «*-8«V+*^-«y"by«-y.
fil. oF-y^hya-y. 82. a"+88&'b7a+2&
53. Stf«4-S7<i('-81&«b7a+3&.
54. i^-»'«*|f+«V+«V+«y*+y"by4^4-^. .
Sft. «»4'8«V+»«V-«V-2av*-ay«by«>-y".
S«. «*-tt«»+ll««-12aT+6by«'-8«+8.
57, V+«»-9a*-l««-4bya»+4«y-4,
58. V-18d^+36by4p'+0Jr+6.
89. «*-f64by4P*+4dr+a
Sa «^+lOi^4-85dP*+0Our+24by«*+S#+4.
81. «*+4P*-24«*-3<kr4-67by«'+2<vi-8.
88. l-#-8«*r-jp»by l+2»+«*. ,
88. 4i^^Sj^4'lby«»-8<r+l.
Si «*4.S<W+.9ft*bya«-8<ift+86«.
85. <i^-ybyii^-2fl»-i-aa6»-J».
88. ••+8«»-44r*-8jj»+18i«-2*-lby«"+2jr-l. ,
87. <^+a««+S4r*+2d^^+lbydr*-a«*-f8^-S<r+l.
8a #«+«^-aby4r*+«»+l.
89. #-(a+6 +<?)«■+ (a5+aj+Jc)#-a6c
by4^-(a+5)#+a6.
4<h a^#(8eiO'-*M)«*4>^byaa^-&v+c.
41. «*-?'i^-«y»+y*by4»*+«y+y". j
48. ji»-8«y~y"-lby#-y-l.
48r 00^+ 2l4jy + 12y4f - 16;*' by 1a+3y-^,
^,- #+2a6+ftf— ^ by a+6-c.
.•♦«^*-6aft<? by <i^+4ft"+<^-<ic-8a6-26ff.
f 8<ft*t J^+c* by a+ &+«.
5i^+ft'(a--«)+«^(a-W+«^ by a+6+&
+(<i^+a&-i^-fl?6+a6* by «-a+6.
8(«+y)4f4-«" by «+y-4f.
tJ»%3(«+y)»«+8(«+|)««+«*
# V byC«+y)«+8(aT+80«+«^
QSNBRAL BBSULTS
X. QtMrql BeiuUi in MuUiptication,
7d. There are tome examples in Multiplication lAAxh
ooour flo often in alg'^braical operationB that they deserve
especial notice.
The following three examples are of great importance.
a +6
a +&
a -6
a "h
a +b
The first example gives the value of (a+5)(a+&}» that
is, of (a + 6}'; thus FC have
Thus th€ iquare qf the turn qf two numbers t$ equal to
theeum qf the squaree qf the two numbers increased by
twice their product,
^^ Again, the second example gives
(a-&)a=a»-2a&+&».
Thus the square qf the difference t^ two nttnibers is
equal to the sum qf the squares qf the two numbers
diminished by twice their products
The last example gives
(a+&)(a-&)=a»-ft«.
Thus the product cf the sum and difference qf two
numbers is equal to the difference qf their squares.
SO. The results of the preceding Article fur^di a
simple example of one of the uses of AJgcft>ra;. wo may
say that Algebra enables us, to prove general ^ihe^et^
.respecting numbers^ and also to eapress those theorems
br^y. '^ '
ip^;
5l <*
*:»■■
IN MULTIPLICATION.
48
' For exaiii|)le, the result (a+()(a-&)^<i^-^ is pirored
to be true, amd is expressed thus by symbols more coift-
paoUy than by words.
A general result thus expressed by symbols is often
called a/ormti/db
81. We may here indicate the meaning of the sign ife
whidi is made by combining the signs + and — , and which
is called the domHe Hgn,
Shice (a+&)«-a«+2fl&+&«, and (a-6)«-<i^-2a&+J«,
we may express these results in one formub thus :
(a±5)>=a«Jb2a6+6»,
where ^ indicates that we may take either the sign 4- or
the sign — , keeping throughout tJhe upper tign or the
lower eign, a ±0 is read thus^ *^ a plueor minui b/*
82. We shall doTote some Articles to explainhig the
use that can be made of the formiUse of Art 79. We shall
repeat these formulae, vad number them /or the take <\f
eaey and dietinct r^erence to them,
(a+bf =a«+2a&+6» (1) .
(a-6)« =a«-2a5+6? (2)
. (a+6)(a-&)=a^-6« (3)
83. The formulae will sometimes be of use hoi Arith-
metical calculations. . For example; required the difference
of the squares of 127 and 123. By the formula (3)
(127)'^(123)"=(127 + 123)(127-123)=250x4=1000.
Thus the required number is obtained more easily than
it would be by squaring 127 and 123, and subtracting the
second result from the first
Agam, by the formula (2) .
(29)»=(30-l)»=900-60 + l = 841 ;
and thus the square of 29 is found more easily than by
multi|^yii|g 29 % 29 directly.
Or suppose we have to multiply 53 by 47.
By the formula (3) ^
53x^=<W+3)(50-3)=«((J())«-3»=2500-9=2491.
u
GENERAL RESULTS
I
i: \
i
'I
84 Suppose that we require the sqiuure of 84?+ j^.
We can of course obtain it in the ordinary wa7» that k by
multiplying 3af+2yhjSaf+ 2y, But we can auo obtain it
in anotiier wav, namely, by employing the formula (1).
(The formula is vtae whatever numoer a may be, a^d what-
eyer number h may be; so we may put Siv for 0^ and ^
fyt b. Thus we obtain
The beginner will probably think that in such a case he
does not sain any thing by the use of the fonnula, for
he will beueve that he could have obtained the required
result at least as easily and as safely by common work
as by the use <^ the formula. This notion may be correct
in tnis case, but it will be found that in more complex
cases the formula will be of great service. i.
\
86. Suppose we require the square of w-\-p+z. J>e-
noteoT-fy by a.
Then a-hy+z=a'\-z; and by the use of (1) we have
=:a^+2a!y+p*+2xz+2yz+z'f,
/ Thus(«+y+4f)*=«»+^+;»*+2ajy+2y«+2a?;8.
Suppose we require the square of p—q+r-t. Denote
p—qij aaadr—ihyb; thenp— g+r-«=a+6.
By the use of (1) we have
(a+&)"=a»+2a6+6»=(|?-g)*+2(|>-g)(r-#)+(r~t)«
Then by the use of (2) we express (p-gf and {r-^if,
ThusCp-g'+r-*)*
=!>' ~ 2|?g + g* + 2 (pr -p# - ^r + g*) + r* - 2r# -I- «■
=i^ + ^ +**+«* + 2l?r + 2g# - 2pj' - 2p* - 2^^ - 2r#.
Suppose we require the product of p^q^-r-i and
p-q-r+i. ^
Letp—<2^=a and r~«=&; then
p-g-^r-$'^a-\'b,«Ddp'-q-^rA-»=»a-hi,
IN MVLTIFLtCATION. 4ft
Then by the we of (3) we have
(a+?>)(a-&)=<i^-J»-(p-(r)'-(r-#)»;
and by the use of (2) we have
86. The method eihibited hi the precedhig Article
is safe, and should therefore be adopted by the Eeginner;
as he becomes more familiar with the sulueci he may
dispense with some of the woric Thus in the last eiamplcL
he will be able to omit that part relating to a and ft, and
sunply put down the following process;
(p-(r+r-#)(p-(?-r+t)={p-(?+(r-«)}{p*-^-(r-#)}
= tp - g)*- (r-«)*=p* - 2pfif + g* - (r* - 2r# + «»)
=!>• - 2pg + g* - r* + 2r# - J*;
or more briefly stilli
(p-fl'+r-t)(i?^g-r+#)-(i>-g)»-(r-#)"
=*P* - 2pg + g* - f* + 2r^ - #'.
But at first the stadent will probably find it pnident to
go through the work folly as in the preceding Article.
87. The following example will employ all the three
fbrmnlflo.
Find the product of the four &ctors a+6+tf, a+6-e,
0-6+c, 6+c-tt»
Take the first two &6tord j by (8) and (1) We obtain
(a+6+c)(a+6-c)i"(«+^)"-c'««*+2a&+6*~c*.
Take the last two fiicton; by (3) and (2) we obtain
(a~6+c)(6+c--a)={c+(a-6)}{c--(a-6)}
=c«-(a-6)"=c*-a"+2a6-5».
We have now to multiply togethei^ <i*+ 2a& + d'-c* and
6*-aHia&-M "Weobftam
=2««&*+2W+2aV-tf*-&»-.c*. -
88. There aro other results in MidtipUcatioii ^yoha^
of less importance than the three fbrmn& given in Art 8S^
but which are deserving of attention. We place them here
m order that the stiicfent may be able to refer to th^
wh^n they are wanted; they can be easily verified nafs
actnal multiplication. ^
(a+&)(a^-aft+J*)=a»+5»,
(a-&)(a«+«&+68)=a^-6»,
(a+&7==(a+J)(a^+2a&+&50==a»+3a^+3a6^4-&»,«;^>
(a-&)s=:(a-. J) (a2-2a6 + d«) =*a3-3fl«& + 3a6«- 68, V
,, =a?+3a«(5+c)+3a(6»+26c+c2)+6»+36»c + 3&<^+jC»
=a8+&»+c»+3a«(6+c)+3d«(a+c)+3c2(a+&)+6a&«.
89» Useful exercises in Multiplication are formed by^
requiring the student to shew that two expressions agree^fi
givinff the same result. For example, shew that
If we multiply a-6 by &~c we obtain
db—l^—ac+bei
then by multiplying this result by c- a we obtak ,
ca6 1- cJ* - ac* + 5c" - a«6 + a6« + a^ - oftc,
thatis a*(c-6)+6«(a-c)+c»(ft-fl^). ^
Agafai; jhewthat(a-6)«+(6-<j)^+(<?^a)« ^^ ; J
/■
EXAMPLES, X A%
Bf^H^ feirnrala (2) of Art 82 we obtain
And (c-6)(c-a)=c^-ca-c&+a&,
(ar-5)(a-c)=a^-a5-ac+ftc;
therefore (e-b) (c- a) + (d-^^i) (&-<?) + {«'-^) (« - c)
therefore («-&)■+ (6-c)"^(c-af
=2(c-6)(c-a)+2(6-a)^-c)+2(a-d)(a-c)i
«
Apply the fomralsB of Art. 82 to the following rixteen
examples in multiplication:
1. (16a?+14y)a. 2. (7«*-5y^«.
a (««+2a?-2)". 4. (««-fi»+7)".
9. («"+4i>y+^)(«*-«y-y').
11. ]^^^+3«+l)(a»-2aj*+3a?-l).
• llj&i|i)«^^ 13. (a+ft)«((^-2a6-J^
14. @^ i •
I,
F I
.1
48
EXAMPLES. X
¥5
Shew that the following resnltfl are true;
la (a+d+c)"+a^+6«+c"==(a+&)«+(6+tff4.(<?+<^
19. (a-&)(&-<j)(<?-a)=6c(c-5)+ca(a^0)+iid{^w«^^^
f20. (a-ft)»+fi^-.a»«3a6(6-a).
2L (a+6+c)?-a(6+c-a)-&(a+d-^)~e(a+J-c)
22. (<^+a&+M)"-(<i^-a&+6«)««4a5(<^+J^»
A 23, (a+&+c)»-.a»-J'-.c»-3(a+6)(6+c)(c+a).
^ 24 (a+5+c)(a&+ftc+a»)=(a+6)(6+<?)(c+a)+a&R
25. (a+ft)(&+c-a)(c+a-ft) ",
=a(6»+(5^-a«)+6(c«+^4^.
Ji* 26. (a+»+c)'-(ft+c-a)»-(a-&+c)3-(a+6-c)»
=24a50.
27. (a+6+<?)P+(a+6-<j)«+(a-&+cj)«+(&+c-a)«
-2a (a+ft)«+2(a'-65+(a-6)»=(2rt)«. . V
, 29. Xa-ft)»+(6-cy+(c-a)»:^3(a-5)(6-<?)(c-a).
"^ 80. (a-5)'+(a+6)»+3(a-J)«(a+6)+3(a+&)«{a-^
: =(2<l)».
••t.
82. a(6+c)(6'+c»-<j^+5(c+a)(c«+a^-.j^
+c(a+i)(a^+5*-c^==2ttdc(a+6*<|^
33. (a-6)(«~a)(i»-6)+0>-c)(«-ft)(4&-e)
34. (a+«*+(a+c)«+(a+rf)«+(6+p)>4^4^+^+||(l
36. {(o^+W+C^y-^^WK^^+W-C^y+W}
\
FACTORS.
4»
XT, Faetor$.
90. In the precedinff Chapter we have noticed some
general resnlts in Multipfication; these results may also be
regarded hi connexion with Division, because every ex«
I ample in Multiplication furnishes an example or examples
in JMyision. ne shall now api>ly some of these results
to find what expressions will divide a given expression, or
[in other words to resolve easpretsions into theirfactors.
91. For example, by the use of formula (3) of Art 82
I we have
Hence we see that a^~5^ is the product of the four
factors 0^+6*, a* +6*, a+b, and a— 6. Thus «•-&* is
livisible by any of these £Eustors, or by the product of any
two of them, or by the product of any three of thenu
Again,
Thus o'+a^-f 6^ is the ivoduct of the two factors
f+ab-hb^ and a*-ab+b% and is therefore divisible by
bitherofthem.
Besides the results which we have already given, wo
" now place a few more before the student.
92. The following examples in ^vision may be easily
Ified.
1,
sobnu
T.A.
50
FACTORS.
Also
V
o^-rf^
'«*-«*y+«y'-y*,
x-vy
_ZIl = 4^8 - a^ + a%» - a^ + any* - y«,
cn-^-y
and so on.
Also
and so on.
a?-¥y^
a-k-y
w+y
=iiA-a!y+f/^f
<^a^-iifly+a^i^^ay^+y^.
The student can carry on these operations as far aa
lie pleases, and he will thus gain confidence in the truth of
^e statements which we awXL now make, and which are
strictly demonstrated in the higher parts of lai^r works
onAlgebitt. The following are the stotements:
iif — y" is diyisible by a?— y if n be any whole number ;
af^—y is diyisible by ar+y if n be any even whole number;
a^+y*vi divisible by a; + y if n be any odd whole number.
We might also put into words a statement of the forms
of the quotient in the three cases; but the student will
most readily learn these, forms by looking at Ihe above
examples and, if necessary, carrying the operations still
&rther.
"We may add that x*+y^ is never divisible by a+y or
;r~y, when ra is an 0r«9» whole number.
93. The student' will be asldsted in vemembering the
results of the preceding Article by noticing the simplest
FACTORS.
51
case in eadi of the four resiiltB, and referrluff other eases
to it ?or ezanvple^ suppoie we wish to eonnder whether
\t^"-y' 18 diyisible oy x—y or by x-\-y\ the hidex 7 is an
\odd whde number, and the simplest case of this Und is
\x-y, which is divisible by x-y^ but not by x-^y; so we
infer that sff-y' Sa di?isible hj x-y and not by x-¥-ff.
lAg^. take sfi—f^; the index 8 is an even whole nnmbcn*,
land the simplest case of this kind is «*— ^. whiVh is
^visible both by x-y and x+y; so we infer that u, -|^
diyisible both by x-y and x+y^
94. The following are additional examples of resohiny
expressions into &ctors.
afi-j^={a^-hy')(a^-ffy
={x+y)(a}*-xy-\-y^(x"y)(x*+xy+y'^;
86'-27c>=(26)S-(3c)8=t(2&-3c){(2&)«+-26x3<?+(3<?)«}
= (2&-3c)(46>+6&c+9c«);
:{2a&+2crf+a«+&«-<^-fiK}{2a&+2crf--a«-6«+<?+d«}
:{(a+5)»-(c-<f)«}{(c+d)»-(a-&)«}
95. Snppose that (oj' - 5jfy + 6y^ (a; - 4v) is to be diyid-
by «•— 7«y+12y*. We might multiply aj*--5a!y+6y*
«— 4^, and tiiien divide the result by afl-'*Ixy+l2y*,
ft the form of the question suggests to us to try if
'4y is not a fihctor oia^—*Jxy+l2y*; and we shall nnd
It«'-7«y+l2y»=(4?-3y)(a?-4y). Then
(a^~5ay+6^ (a?-4y) _ a^-5xy+6y^
{x-Zy){x-4y) x-dy *
by diyision we find that
afl^6xy'h6y'^
=d?-2y*
4—3
as
EXAMFLSS. XI.
m: The 8l»deiit with a little ixnictioe will Iw «U|« lO
resolve certaiii trinomials into twot moniial fiMttoffii ^^ -
For we have generally
Wppose then we wish to know if it be poadUe to Miolii.
^ +7^+12 into two binomial factors; we must itt4i/(tf
possible, two numbers, such that their smn is 7 and tWr
prodnct is 12; and we spe that 3 and 4t are sudi vmnb^n^
Thus
«"+7«+12 = («+3)(«+4).
Similarly, by the aid of the /ormtUa
(«-a)(aT-6)=«*-(^+ft)«+a&, .[
we can resolve ^— 7aT+ 12 into the fstcton («~3) (^-^ij^
And, by the aid of the formula
• (a?+a)(«-&)=d?*+(a.-&)aJ-aft,
we can resolve flj*+^- 12 into the &ctors («+4)(«-3).
We shaU noW give for exercise some misd^aalwaft
examples in the preceding Chapters. |^
' . . . .M
Examples. XI.
■If-:-
-:^t
Add together the following expresatona^ . f
1. a(a+b-'C% &(6+<j-a), c(a+6-()i
2. a(a-6+<j), ft(6-c+a), <?(<?-«+%
3. a{a^b+6+d), h{a+h^e+d), c(tf4-S4V'-<Qb
dl'-a+b+c+d), \'
4. 3a-(46-7<j), 8ft-(4<J-7a), 3<J-(4a-7ft).
5. 9a-(66+2c), 96-(6<J+2a), 9d-(Sa4^ft).
6. (a+5);»+(a+c)y, (^-c)*+(6r^o)y.
.,._,
EXAMPLES. XI
7. («-«)(a+6)+(«-y)(a-5), (a?+y)at^(jr+;if)6,
(y-4?)(i+(«-^y)6.
8. (a-ft)«+(5-c)y+((?-a);if,
9. •2(a+ft-c)«+(a+6)y+2flUr, ;
10. <i^-:(a-&+c)(a+6-c), 5» -(6- a +c) (6 :♦•<»-<?),
c^-(<;-a+6)(c+a-5).
Simplify the Mowing expressiont: ■■ ' ■
11. a-2(&+8«)-3{&+2(a-&)}.
12. (a+6)(&+c).-(<?+d)(rf+a)-(a+c)(6-rf)- J
13t 4a-[2a-{2&(«+y)-26(«-y)}]. j
14. («+6)(dT+c)-(a+6+cj(«+&)+«'+«&+&'+3aa^'
16. a-t56-{a-8(c-&)+2c-(a-2&-c)}].
16. 6a-7(ft-c)-[6a-(3&+2<j)+4c-{2a-*(6+c-a)}].
17. (i?+3)»-3(«+2)»+8(«+l)»-««.
18. («+y)»+(«+y)»y+(a?+y)^-{3*V+C|/»a?+2^;
19. »<l+«)»+(l+«)V+(l+«)y«+J^ • -^
- {3ar(a? + 1) + y (y + 1) + 2«y + 1}.
20. a(6-k<J)«+5(a+c)»+<?(a+&)«+(a-6)(a+c)(6-<r)
-(a+*)(a-.c)(6-<?)-(a-6)(a-c)(6+c). ,
{a+ft)(a+c)-(6+d)(rf+<j)
21.
22.
23.
2i
g«-B<i&+^y a«~7a&-H2y
«l--i2& "" a-36 •
If^im-^M^^m . 6a»-26a*&+40aJ'-20&»
* i");jui. T u
•l-ca*4'<ii^~12(yc+c«g+g«»)-19a5(;
2a-3& "^ ~^'
M , EXAMPLES. XL
Divide
25. ««+y«-2aVby(«-y)'.
26. sfi-^^-^-^a^hyipo^yf.
27. (<^-3fl%+5a6^-3ft^(a-2&)by<^-3a6f25».
28. (^-»«V+23a^-liV)(«-7y)by««-aiy+7^.
29. a^+a*6*+ftPby(a«-a6+ft»)(a«+<i6+ft»).
SO. a?-J^+aV(a*-M)by(«'-<*+ft')(a«+a6+5^
31. 4aV+2(3a*-2&*)-a&(5a«-llJ*)by(8a-&)(a+ftX
32. («»-3«+2)(ar-3)by««-lto+6L
33. («»-3«+2)(aT+4)by4>»+«-2i V
34. (a*+<i«+«*)(<;?+«^bya*+aV+««. *
35. (a*+a«6»+6*)(a+6)bya*+a&+6".
36. &(*'+a*)+<M?(«*-a') + a'(a?+a)by(a+6)(«+aX
Resolye the following expresdions into £EM$ton:
37. «'+da?+20.
39. aj"-16iP+50.
41. ^+ar~132.
43. «*-81.
45. fl>8-256.
47. a*+9a6+206« .
49. (a+6)"-llc(a+&)+30c".
50. 2(«+y)«-7(«+y)(a+ft)+3(a+ft)«
Shew that the following results are true;
51. (a+26)a»-(6+2a)ft»=(a-ft)(a+6)».
62.- a(a--26)»-&(6-2a)»=(a-6)(a+6)».
38. aj'+ll^+SOi
40. ^-20^+100t»
42. a!«-7a?-44.
44. 0^+125.
46. ^•-64.
48. «*-13a?y+4?y*.
\i
GBBATSST COMMON MEASURE. 55
XIL Chreatat Common Meature*
97. In Artthmetio a whole nmnber which dtvldefl
another whole number exnoUy is said to be a meaaurB of
it, or to iMOiure it; a whole number which divif^es two
or more whole numben enetly is Mid to be a eommm
fiMOffUfv of thenu
In Algebra an ennreBaion which diyides another ex-
prenion ezacUj is saia to be a meoiura of it, or to me<uure.
ft; an expression which divides two or more expresBions
exactly is said to be a comTnon meature of them.
9A. In Arithmetic the greatstt common meoiurs of
two or more whole numbers is the greatest whole number
which will measure them aU. The term greatest common
measure is also used in Algebra, but here it is not yery
appropriate, because the terms greater and ieu are sel-
dom applicable to those algebraical expressions in which
definite numerical values lukve not been assigned to the
various letters which occur. It would be better to speak
of the highest common meaeurej or of the highest common
Xdimsor; but in conformity with established usage we
Ishall retain the term greatest common measure.
The letters O.O.M. will often be used for shortness
stead of this term.
We have now to explain m what sense the term is used
Algebra.
99. It is usual to say, that by the greatest common
leasure of two or more simple expressions is meant the
jreatest expression which will measure them all; but
(his definition will not be fully understood until we have
^ven and exemplified the rule for finding the greatest
>mmon measure of simple expressions.
The following is the Rule for finding the a.aic. of
gmple expressions. Find by Arithmetw the acic. qf
numerieal coefficients; qfter this number put every
tter which is ^pmmon to all the expressions, and give
each letter respectively the lea^t tndex which it has
the expressions^ .
66 OREA TEST COMMON MEASURE.
100. For example; reauired the 0.0.11. of 16a^ and
20cfi9^. Here the nnmencal coeffidentiare 16 and 20,
And their g.o.m. is 4. The letters common to both the
expressions are a and hi the least index of a is 8, and
the least index of 5 is 2. Thus we obtain 4a^ as the re-
quired O.O.M.
Again; required the o.o.m. of 8aV<Ar"yi('. 12a*(««V,
and 16a^^a;V^ Here the nnmerical coeffiaents are 8»
12, and 16; and their o.aM. is 4. The. letters common to
an the expiessions are a, «, «, and y ; and their least indices
are respectively 2, 1, 2, and 1. Thus wo obtain 4a*eafiy as
tiie required o.cm.
101. The following statement gires the best practfeal
notion of what is meant by the term greatest common
measure, in Algebra, as it shews the sense of the word
qreate»t hera When two or more expreseions are divided
by their greatest common measure, the quotients have no
common measure.
Take the first example of Art 100, and divide the ex-
pressions by their o.o.m.; tb.e quotients are 4ae and 6bd,
and these quotients have no common measure.
Again, take the second example of Art 100, and
divide the expressions bv their a.c.M.; the quotients are
St^cjfiji^, Sc^b]^, and 4acV, and these quotients have no
common measure
- 102. . The notion which is supplied by the preceding
Article, with the aid of the Ohi4>ter on Factors, will enable
the student to determine in many cases the o.o.m. of com-
pound expressions. For example; required the, O.C.M. of
4a"(a+6)« and 606 (a«- 6"), Here 2a Is the o.o.if. of the
factors 4cfi and 6a&; and a+b is a factor of (a+t^f and
of a* -6', and is the only common factor. Tne product
2a(a + b) is then the o.o.m. of the given expressions.
But this method cannot be applied to complex ex-
amples, because the general theorv of the resolution of
expressions into factors is beyond the present stage of
the iitudenVs knowledge; it is therefore necessary te adopt
GREATEST COMMON MEASURE. 07
toother method, and we shall now glye the ufoal definition
and role.
IDS. The following may be giren as the definition of
the greatest common measure of compound exprasfions.
LH two or more compound e»pre$Hont contain power$
qf iome common letter; then the /actor <f higKeet du
meneiom in that letter which dividee all the expreaioni
ii called their greateet common meaeure.
104. The following is the Rule for fi:itdin^ the greatest
common measure of two compound expressions.
Let A and B denote the two expnseions let t^m
he arranged a4Scording to deecending powere qf f :^ie
common letter^ and euppoee the index of the h\i/,'uitt
power qf tfiat letter in A not Ices than the hidvo qf the
highest nower qf that tetter in B, Divide A ojf B:
then make the remainder a divisor and B ttte dividena.
Again make the new remainder a divisor and the pre-
ceding divisor the dividend. Proceed in this ^eay until
there is no remainder; then the last divisor is the
greatest cmimon measure required*
105. For example; required the 0.0.1c of ^~4«-f 8
md4a^-da^-\6x+l8,
4a?»-16a:*+12a;
7aj*-27af+18
7a?«-'2ar + 21
a?- 3
«-8j a^-4x-^Z ^x-l
x^-Zx
— a?+3
% —
kus sr-3 is tlie o.q.it. required.
/■ .
S8 GREATEST COMMON MEASURE.
106. The rule which is given hi Art 104 depends on
the following two principles.
(1) If P measure A, it will measure mA, For let
a denote the qnotient when A is diyided bv P; tiien
A^aPi therefore mA^maP; therefore P measures
mA.
(2) If P measure A and B, it will measore mA^nB.
For, since P measures A and By we may suppose A = aP^
and B^bP; therefore mA±nB^(pM:i^nD)Pi therefore
P measures 991^ :tn^.
B) A (r
pB
'cjBKg
qC
107. We can now demonstrate the rule which is given
in Art 104. >
Let A and B denote the two ex- B) A ( jr^' ' V
Sressions. Divide A hj B\ let /i
enote the quotient, and C the re-
mainder. Divide B by C\ let q de-
note the quotient, and 2> the remam-
dcr. Divide G by i>, and suppose
that Uiere is no remainder, and let r JO) C [r
denote the quotient rD
Thus we have the following results:
-4=i>-B+(7, B=qC+D, C=rD.
We shall first shew that /> is a common measure of
A and B. Because C=^rDy therefore D measures C\
therefore, by Art. 106, D measures ^C7, and also 0^(7+ J9;
that is, D measures B, Again, since D measures B and (7,
it measures pB+C; that is, D measures A, Thus D
measures A and B.
We have thus shewn that J) ia a common measure of
A and B; we shaU now shew that it is their grecUett
common measure.
By Art 106 every common measure of A and B mea*
surest ~p^, that is C\ thus every common measure of
A and ^ is i^ common measure of B and (7. Similtely,
every common measure of B and (7 is a conmioii In^^Miff
GREATEST COMMON MEASURE.
of C and D. Therefore eyery oommon measure of A and
^ is a measure of 2>. But no exfiression of higher dimen-
sions than 2> can divide D. Therefore Z> is the greakii
common measure of ^ and ,0.
108. It If obyiouB that, evefjf measure qf a common
meoiureqftwo or more ej^etsiont is a common measure
qf those expressions,
109. It is shewn in Art 107 that every common
measure of ^ and B measures D; that is, every common
measure of two eaipressions measures their greatest com*
mon measure.
110. We shall now state and exemplify a rule whick
is adopted in order to avoid fractioni) in the quotient; by
I the use of the rule the woik is ^mpMed. We refer to the
Chapter on the Greatest Common Measure in the laiiper
I Algebra, for the demonstration of the rule.
. Before placing a firesh term in any quotient, we mait
[divide the divisor, or the dividend, by, any expression
''ifhieh has no factor which is common to the expressions
Dhose greatest common measure is required; or, we
my multiply the dividend at^such a stage by any eX"
session which has no factor VuU occurs in the divisor,
111. For ezam^e; required the ao.H. of 2^-r74T+6.
id 3a^-7«+4. Here we take 2aj*-7«+6 as divisor;
rat if we divide Za^ by 2a^ the quotient is a fraction; ta
ivoid this we miQtiply the dividend by 2, and then divide.
2««-7«+6j6aj*-14«+ 8^3
6«*-21a?+16
7«- 7
If we now make *lx-l a divisor and 2d!*- 7^+ 5 the
ividend, the first term of the quotient will be fractional;
ut the fiictor 7 occurs in every term of the pr(^K>Bed
Ivisori and wo remove this, and ihen divide.
I
60 GREATEST COMMON MEASURK
Thus WQ obti&i «-l as the O.O.M. required.
Here it will be seen that we used the second pait of
the rule of Art. HO, at the beginning of the process, and
the first part of the rule later. The first part of the rule
should be used if ]{K>ssible; and if not, the second part We
have used the word easpression in stating the rule, but in
the examples which the student will have to solye, the
factors introduced or removed will be almost always rk^
mertMl /acton, as they are in the preceding example. ^
We will now give another example; required the q,cm,
of 2«*-7«'-4«'+«-4 and 3a?*-ll«*~2^-4«-16.
Multiply the latter expression by 2 and then take ii for
dividend.
2jjP*--7«'-4«*+»-4^ 6ar*~22a;»- 4aj"-8ar-32 (,3
6a?*-21«»-12ir8+3d?-12
- «»+ aiJ*-ll«-20
We may multiply every term of this remainder by -^-l
before using it as a new diyisor; that is, we may chauge
the sign of every term.
«»-8«*+lla?+20j2«*-^ TdJ*- 4«*+a?-4 ^24r+9
2«*-16a!8 + 22aj«+40a^
9aj8-264J*- 39«- 4
9a?'-72aj»+ 994?+ 180
46a»--138i»-18^
^ere 46 is a &ctor of every term of the remainder; we
remove it before usmg the remainder a^ a new div&or.
EXAMPLES. XIL
61
-6a"+l&»+20
Thus 0^-30^-4 is the O.O.M. required.
112. Suppose the origmal expressions to contain a
common fieictor F, which is obvious on inspection ; let
A=aF and B=^bP. Then, by Art. 109, -PwiU be a factor
of the O.O.H. Find the g.oIm. of a and 5, and multiply it
by F'f tibie product will be the o.o.m. of A and B,
113. We now proceed to the o.o.u. of more tlian two
compound expressions. Suppose we requi^ the O.O.M. of
three expressions Ay B, C. Find the q.om. of any two of
them, say of ^ and B; let 2> denote this O.O.H.; then the
O.O.M. of 2> and C will be the required o.o.ic. of J, B, and (7.
For, by Art 108, every common measure of 2> and Oh
a common measure of A, By and C; and by Art 109 every
common measure of A, B, and C7 is a common measure of
D and (7. Therefore the a.o.M. of /> and (7 is the o.ojc.
of Ay By tixid, G,
114. In a similar manner we may find the o.o.m. of
\four expressions. Or we may find the o.o.m. of two of
the given expressions, and also the o.o.if. of the other two;
then the o.o.m. of the two results thus obtained will bo
the aax. of the four given expressions.
Examples. XIL .
Fmd the greatest common measure in the following
Examples:
1. 15«*, 18«>.
5. 4(«+l)^ 6(4»"-l).
2. 16aV, 20a^bK
4. ZSan^aVy 49a«6*«V.
6. 6(«4-l)», 9(«»-l).
62
13.
14
15.
16.
17.
18.
19.
m
21.
22.
23.
,1/24.
25.
26.
27.
.^
29.
30.
31.
32.
33.
34.
35.;
EXAMPLES, XIL
12(fl?+JV, 8(a^-&*). a aJ»-y», a*-y*.
aj'+Sar+lS, iii>+9a;+20.
«»-9«+14, «»-lldr+2a
a^+2»~120, aj»-2ii?-80, *
a^-15d? + 36, «"--9j?-36.
oj* + 6** + 13a? + 12, 4;» + 7aj» + 16d? + 16.
«»-9;B*+23a?-12, «'-l(to»+2ai?-16.
a^-29a?+42, «3+aj»-36a?+49.
;88-41«-30, «*-ll«»+26a?+26.
«*+7«*+l7« + 16, ;c»+8«'+19«+12.
;ij'-10a?*+26«-8, «'-9«' + 23a?-12. i
4(aj*-«+l), 3(a?*+^+l). ^^ ^
6(««-a?+l), 4(aj«-l). ^
6«'+ii[;-2, 9«'+48«*+62d?+16. *— ^
«^-4aj»+2«+3, 2aj*-9«' + 12^-7*
«*+«'-6, a?*-3a?'+2. M
«^- 2a?* +30? -6, a?*-aj8-«*-2a?.
«*--l, 3a?*+2a?*+4aj'+2aj'+ar.
aJ*-9aj*-30a?-25, aj"+a?*-7aj*+6a?.
36a?»+47a;«+13a?+l, 42ii?*+4l«3_9;B2-.9;,,«i.
«• - Sa?* + 6a?* - 7*8 + 6«* - 3a? + 1,
a?*-aj*+2a?*-a!»+2a?»-;t+l,
2a?*-6a;'+3a?«-3aj+l, a?-3a;*+a?'-4a;»+12a?-4.
a?*-l, a?^®+af4^+.2a!^ + 2a;* + 2j;*+aj*+4?-l^l,
ir»-3a?-70, a;S_39;,.+7o, ^^s. 43,^+7^
a?*— ajy-12y«, a?*+5ajy+6y'. ^ •
?a?«+3aa?+a«, 3af'+2aa?-^al ; ' "
aj*-3a'a?— 2a', aj*-aaj*— 4^.
3dj'-3a?V+^'-"^> 4a?V-"5^+y''.
/\^.
LEAST COMMON MULTIPLE.
63
Xllt Least Common MtUtipU,
116. In Aiithmetic a whole number which is measored
by another whole nnmber is said to be a multiple of it ; a
whole number Which is measored by two or mor^ whole
nmnbem is said to be a common multiple of them. . ^^
116. In Arithmetic the least common multiple of two
or more whole numbers is the least whole number which
is measured by than all The term least common multipk
is also used in Algebra, but here it is not very apinropriate;
see Art 98. The letters L.ojf. will often be used for
shortness instead of this term.
We have now to explain in what sense the term is used
in Algebra.
117. It is usual to say, that by the least common mul-
tiple of two or more simple expressions, is meant the leaet
eo^esnon which i> measured op them all; but this defi-
nition will not be fully understood until we have giyen and
exemphfied the nile for finding the least common multiple
of smiple expressions.
The following is the Rule for finding the L.o.if. of
simple expresswns. Find by Arithmetic the L.o.if. qf the
numerical eoeffidente; after this nun^ber put every letter
which occurs in the expressions^ and give to each letter
respectively the greatest index which it has in the ex-
pressions,
118. For example; re<|uired the I1.0.M. of 16a^ and
ZOd^lfld, Here the numencal coefficients are 16 and 20,
[and their L.O.M. is 80. The letters which occur in the ex-
Ipressions are a, h, c, and d; and their greatest indices are
~ ipecUyely 4, 3, 1, and 1. Thus We obtain SOa*¥cd as the
luiredLbCM.
Ag^ ; required the L.O.M. of Sa^c^a^yj^, 12a*6cj^^,
id Wmfiifiy^. Here the l.o.m. of the numerical coefficients
48. 1%e letters wluch occur in the expressions are
s, hf Cf^fty^ and z; and theur greatest indices are respec-
Ively 4^ 3, 3, fi, 4, and 3. Thus we obtain 48a*WajV^ «
requhed X1.0.M. „
64
tEAST COMMON MULTIPLE.
119. The following statement gives the best prnetical
notion of what is meant by the tenn least common multiple
in Algebia,- as it shews the sense of the word koit here^
When the least common multiple qf two or more ewpree^
ifione ii divided by thote expreetione the quotients have no
dommon measure, »
Take the first example of Art 118, and divide the L.0.1C.
by the expressions; the quotients are SUM and 4ac^ and
meae quotients haye no common measure.
Again; take the second example of Art. 118, and ditide
the L.aM. by the expressions; the quotients are Qa^Ci^f
4i^!i?ysi?^ and ^at^a^afi^ and these quotients haye no com*
mon measure.
iT ;
120. The notion which is supplied by the i>receding«
Article, willi the aid of the Chapter on Factors, will enable
the student to determine in many cases the l.o.m. of com-
pound eamressions. For example, reqmred the l.o.m. of
4€fi(a+bf and edbia^-h'). The l.o.m. of Atfi and edb is
I2€^b, Also (a -f by And a* ~'&* have the common factor
a+b, so that (a+b){a+b)(a-b) is a multiple of {a+bf
and of a^—b^; and on diyiding this by (a+ 5)* and a'-d* we
obbiin Uie quotientB a—b and a+ (, which haye no common
pleasure. Thus we obtain 12a%(a+d)'(a~&][ as the re*
quired ii.aM.
121. The following may be giyen as the definition qf the
L.O.M. qf two or more compound expressions. Let two
or more oompopkd expressions contain powers of some
common letter; then the expression of lowest dimeni|ions
in that letter which is measured by each of th^ expres-
sions is called their least common multiple.
122. We shall now shew how to find the itSijL i^f two
compound, expressions. The demonstration howeyer will
not be fiilly understood at the present stage of the 8tm!9.^s
knowledge. ;.;
L^ A and B denote the two expressions,
ffreatest common measure. Suppose A^al>^ .
iXh^ from the nature of tiie greatest common
\
LEAST COMMON MULTIPLE^
^
and h have no common fiietor, and therefore their least
common multiple is ah* Hence the expression of lowest
dimensions ^hich is measured by oD ana hD is ad/>. And
dbD=Ah^Ba='^ . ,
Hence we have the following Rule fbr finding the L.o.if.
of two compound expressions. Divide the product qf the
expreuUma hy their o.o.m. Or we may give the rule thus :^
Divide one qf the expressions by their G.O.M., and mid'
tiply the quotient by the other expression,
123. For example; required the L.C.V. of a^— 4^+3
and 4a^ - 9a^-l6x + 18.
The G.O.M. is a?— 3; see Art. 105. Divide o^— 4^7+3
by xS; the quotient is x—l. Therefore the l.o.m. is
(a?-l)(4i»-9«*-16a?+18); and this gives, by multiplying
out, 4d?*-13a:»-6aj2+33d?-18.
It is however often convenient to havo^the L.o.ic..
expressed in factors, rather than multiplied out. We
kuow that the O.O.H., which is a;— 3, will measure the ex-
pression 4a^-da^-l5x+lS; by division we obtain the
quotient Hence the L.O.H. is
(a?-3)(a?-l)(4aj*+3«-6).
For another example, suppose we require the L.O.M. of
2«* - 7« + 6 and 3dJ» - 7a? + 4.
The O.O.M. is a?- 1 : see Art. 111.
Also (2a!«-7a?+5)-J-(«~l)=2a?-6,
and (3aJ*--7«+4)-r(a:-l)=3;i?-4.
Hence the luOM. is
(a?-l)(2a?-6)(3«-4). >
Again; required the L.O.M. of 2a?*-7«'— 4«*+d?^4,
id 3«*-.ll«»-2a:*-4a?-16.
The o.o.]f. is a;*— da;-4: see Art. 111.
Also
i2a^-1afi- 4a^+ii?-4)4-<«'- 3«-4) = 2«"- x+l.
Id
^-ll«»-2«■-4«-K)■^(«'-3a?-4)=3«■-2i^?+4. !
T.A. 5
06
LEAST COMMON MVlTIPtlB.
Henoe thd L.O.X. is
(«»- 8«-4) (2««-ar+ 1) (8««-2i+4).
124. It is obvious that, werjf rnidtipU ^f a
multiple qftwo or more ewpressiam ie a eommof^
^thoeeexpreeeU/M,
125. Every common mtdtiple of two Mpr9$iion94ia
multiple qf their least common multiple*
Xet A and B denote the two expressions M their
Zi.o.if.; and let N denote any other common multipla Sop-
pose, if possible, that when N is divided by M there is a
remainder iS ; let g denote the quotient Thus B^N—qM.
Now A and B measure JIf and N, and therefore they mea-
■iire B (Art 106). Bilt bj the nature of divbioii K is of
Uwer dimensions than Jf ; and thus there is a oommon
multiple of A and B which is of lower dimensions than
tiieir L.O.H. This is absurd. Therefore there can be no
remainder iS; that is, iVis a multiple of M.
126. Suppose now that we require the L.ojf. of three
compound expressions, A^ B%0, Find the l.o«m. of any
two of them, say of A and J7; let itf denote this ii.o.kc.:
then the l.o.x. of M and C will be the required L,p Ji. of
^^^^anda
For every common multiple of M and is a common
multiple of A, B, and C> by Art 124. And every oonunon
multiple of A and J9 is a multiple of Jf, by Art 125 ; hence
evenr common mul^le of M and C is a common multiple
of Ay Bj and O, l%erefore the l.o.v» of M and C7 is the
I..O.X. of ^, J9, and C '
127. In a similar manner we may ^d the tu^M, ct lour
Expressions.
128. The theories of the greatest oOnunonmeasaHie and
of the least common multiple are not neeessaiy for the
subsequent Qhapters of the present work, and any diffi-
culties which the student may find in them tttiiQr bo post-
poned until he has read the Theory of Equations* /Hie
examples however attached to the preceding (3hai^ and
to the present Ohapter should be carefollv woHceo^ on a«Q-
count qf the exercise which they afford In all tbe ln|^
mental'processes of Algebra.
1
v.-
EXAMPLBS. XUl
«r
BZAMPUBB. Jtlll.
Bind Ad leiit common multiple in the following ex-
ampltii; .
- 1. 4afh, edf^. ' 2. 12a«6«c, 18a5V.
-5. 4<i(a+6), 6d(a»+6»). -6. a«-&«, a^-ftl
10. »'-«Lr*+ll*-e, «»-9«»+2e»-2i,
11. «^-7d^-6, «'+8«»+l74r+10.
12. «*+«■+ 24^+flf4-l, ^**1.
13. «*-2«»-V+8dr-4> «*-6«»+2a»-ia
- 14. /p*+a?«*+a*, «*-a«*-a'«+a*.
'16. 4a'Wc, 6a6»c«, I8a«6c».
la B(€?"V), 12(«+&)", 20(a-&)".
17. 4(a+d)/ 6(a^~(>), 8(a''+&'). SHHl
la 16(a*6-afi^), 21(0^-06*), 35(aJ»+6^*
la «^^1, ii^+1, «»-l,
2a fl^^l, ^+1, aj*+l, aj8-l.
21. ««-l, «»+l, i»»-l, «»+l,
'22, #+t»+2, «>+44?+3, flj^+^p+e.
2a 4^+2ir-3, «»+3«"-»-3, a^-^4afl+a-e,
24. #+6«+I0, «»-19iJ?-30, «»-16;»-60.
6—2
ea
FRACTIONS.
XIY. FraeHoM,
129* In this Chapter and the foHowing four Oha^len
we shall treat of Fractions; and the student wfll finathat
the roles and demonstrations closely resemble thoie with
which he is already familiar in Arithmetic
130. ' By the expression 7 we indicate that a nnit is
to be cUvided into b equal parts, imd that a of snch parts
i^« to be taken. Here r is called a /^tustion; a is called
the numerator, and h is called the denominator. Thus
the denominator indicates into how many equal parts the
unit is to be divided, and the numerator mdicates how
many of those parts are to be taken.
Every integer or integral expression may be considered
as a fraction with unity for its denominator; that is, for
eza^lde,
a
I
«=-,
131. In Alffebra. as in Arithmetic, it is usual to ffive
the following Rule lor expressing a fraction as a m&ed
quantity: Divide the numerator by the denomiwttor, at
far at poetible, and annex to the quotient a fraction
hating the remainderfor numerator, and the divieor for
denominator^
Ezamples.
24a ^ , 9a
7 7
a'+3a& 2db
=/? +
«»-6ay+14
a+y
=«+3+
-df+2
«"'-3«+4
or s«+8—
«*-3«4-4
«"-3«t4*
\-
'J-'^^^
m
FRACTIONS.
6»
The rtodent is reoommended to pay parHetdar aitsn-
Hon to the last step; it is reallT an example of the use of
brackets, namely, +(-*+25=-(a7-2)L
132. Rule for multiplying a fraction by an integer.
Either multiply the numerator by that integer, or divide
the denominator by that integer.
Let r denote any fraction, and o any integer; then
will 7 X e=s -r- . For in each of *hi fractions ^ and -g- the
nnit is divided into b equal parts» and e times as many
parts are taken in -T- as in ^; hence -r* is c times r .
! This demonstrates the first form of the Rule.
Again; It ^ denote an, f^<». «.d « «^ mteg.;
then will ^^^=5* ^^ hi each of the i^?actions ^
and ? the same number of parts is taken, but each part
.
fai ? is tunes as laige as each part in r- , because in
r the unit is divided into e times as noany parts as in
g ; hence £ is 6 times T-«
This demonstrates the second form of the Bule.
133. Rule for dividing A fraction by an integer. Either
multiply the denominator by that integer, or divide the
numerator by that integer.
a
Let r denote any fraction, and cany integer; then
will y^e-Ti* For r is c tunes ^, by Art 132; and
therefore^ is -th of f .
oe e.
This demonstrates the first form of the Rule.
FSdOnONS.
ae
Again; let y denote any fraotion, and any Inti0|(fr;
ae
ae
then wiU y -{-<?= ^. For y is c times |, by Art 1S2;
and therefore r is -th of *|r- .
This demonstrates the second form of the B,nle>
134. J[f the numerator and denominator ^fanyfroi^
Hon be mvltipUed by th^ eame integer, the value qf ih$
fratOionie not altered.
'Vtft if the nnmerator of a fraction be mnltiptied by any
integer^ the firaction win be mtdtiplied by that integer;
and the result will be divided by that integer if its de-
nominator be multiplied by that faiteger. But u we multiply
any number by an integer, and then diyide the resultl>y
the same integer, the number is not altered.
The result may also be stated thus: if the numerator
and denominator of any fraction be divided by the same
integer, the yalue of the fraction is not altered.
Both these verbal statements are included in the alge-
This lesult is of very great importance ; many of the
operations in Fractions dej^nd on it, as we shall see in the
next 4^0 Chapters.
135. The demonstrations given hi this Chapter are
satisfactory only when every letter denotes some poeitive
whole number; but the results are ateumed ioloe true
whatever the letters denote. For the grounds of ^s
assumption the student may hereafter consult tiie lai]ger
Algebra. The result contained in Art. 134 is the most
important; the student will therefore observe that heiice-
forth we assume that it is ahcaye^^ana!^ in. Algebra that
r = T-, whatever a, &, and e may denote. ' ■ \
I Foi^ezampley if we put - 1 for c we have 7 = — ^ ^ :
.. P
SXAMPLE& XIV.
71
Soalfo
£1 like maimer, by amainfag that t >< ^ ii afwayt equal
00
to y we obtain ...h re«ilt. M tb. following :
a - -2a
EZAMPUfl. XIV.
Express the Mowing fractions as mixed quantities :
25* ^ 3eae+4e 8a*-f36 lag'-Sy
6.
«* + <M>* - 3a'#— 3a'
2^-6df-l
*-3
a— 2a
a
9.
Multiply
a(a-&)
10.
12.
8(a«+ft^
9i?::p)^y^(*-^>-
la
«^
by 4(a«-a&+y). 14, . >,_^ bydy+I
IMvide
15. gby2ar.
ld(a^-y)
3(a+&)
16.
9a'~4y
a+&
(«*-!)•
by3a-2&.
17.
18.
by6(a'+a6+J^.
bya^-«+l.
.72
REDUCTION OF FRACTIONS.
XV. RedudHon of FractioM.
136. The result contained in Art. 134 will now be
applied to two important operations, the reduction of a
mtctiop to its lowest terms, and the reduction of fractions
to a common denominator.
137. Rule for reducing a fhtction to its lowest terms.
Divide the numerator and denominator of the fraction
hy their greatest common measure.
For example; reduce , ,^ , to its lowest terms.
The o.o.M. of the numerator and the denominator is
ia^*; dividing both numerator and denominator by 4a'&',
we obtain for the required result -r^. That is, r^ is
equal to ^^ . , but it is expressed in a more simple
form; and it is said to be in the lowest terms, because it
cannot be further simplified by the aid of Art. 134.
■ij^"~"4^H"3 '
Again ; reduce 4^,9^^15^4.18 *^ ^^ ^^^^^ *®™*-
' The G.aif. of the numerator and the denominator is
a—S; dividing both numerator and denominator by «— 3
a? — 1
we obtain for the required result j^ — _ _ ,
\ In some examples we may perceive that the numerator
and denominator nave a common factor, without using the
rule for finding the g.o.m. Thus, for example^ ' \
(a~&)*— c* _ {a-J>-rc)(a-h—c) _ a—b+o
BEDUCTION OF FB ACTIONS.
78
ttow be
)n of a
ractiona
t terms.
s.
[nator is
by 4a»6«,
simple
)caase it
t terms.
1
138. Rule for reducing fractions to a common denomi-
nator. Multiply th€ numerator qf each fraction hy
<M the denovntnatore except ite oicn, for the numerator
correepondinff to that fraction; and multiply all the
denominatore together for the common denominator,
a c' e
For example ; reduce r> j i &nd ^ to a common de-
nominator.
a_a^ c _^ e _ebd
> b^biif* d^dlff* f'fbd'
Thus
a^f clff
and
ehd
are fractions of the same
e
AC
value respectively ^ Xf 2* *^^ ?' ^^ ^^^ ^^® ^^
common denominator b^f.
The Rule given in this Article will always reduce frac-
tions to a common denominator, but not always to the
lowest common denominator; it is therefore often con-
venient tb employ another Rule which we shall now give.
139. Rule for reducing fi«ctions to their lowest com-
mon denominator. Find the least common multiple qf
the denominator»f and take this for ,the common denomi-
nator; then for the new numerator corresponding to any
qf the proposed fractions, multij^^ the numerator qfthat
fraction hy the quotient which is obtained by dividing
the least common multiple by the denominator qf that
fraction,
l^fst example; reduce — , — , — to the lowest com-
*^ yz^ zx^ say
mon denominator. The least common multiple of the de-
nominatoFB is xyz\ and
an
ayz*
zx
xyz
e
scy
ez
ocyz'
74
MXAIBPLBS. Xr.
N.,.,
EZAMPLBS. XV,
Rodace iho following fractions to their lowest terms:
1.
4.
7.
9.
II.
13.
15.
17.
19.
21.
23.
25.
2.
5.
12a*W«
18a«6«y'
10a*a?
5a'«— 16ay2*
;i^+3a?+2
.'c2+6«+5*
2a^+a?-15
2«2-i9aj + 35'
dg'-(a+&)j?-t-a6
a?'»-10a?+21
;»»-46a?-2l'
«3~10«=*+21a? + ia'
20a^+a?-12
12«»-6«« + 6a?-6'
2iB»-6«*-8a?-16
2a6 •
4(a+&)>
3.
6.
aj"+10a?4-21
8.
10.
12.
)4,
18.
Y.20.
af2-2a?-16*
/g*-i-(a+5)d?+<i&
«*+(a+c)a?+a<5'
3a^ -1-233? -36
4a>« + 33a?-27'
d?^+5;i? + 6
;«.•* + a? +10'
a?* + 9a?+20
aj3 + 7a!*+14;r+8'
e^j-11^+5
3d?»-2i5*-r
d;^-2g ;g-t -<i^
«»-2a«'+2a2ar_a»"
2a;'+lla?« + 16af + 16* ^ 2a;3+aa»+a^«-4a»
fl^"-8a?~3
«*-7«"+l*
je»~a^-7a?+3
«*+2«» + 2«if-l*
^ Oft 3:g *"14af-9a?+2
* ^^' 2«*-9a?»-14a?+3'
vS
27 3^— 7ydy
'• 2**+13aV+15a*'
EXAMPLES. XV.
75
29.
31.
#* + «» + 4P* + *+l
'-1
28.
30.
32.
Reduce the following fhustions to their lowest common
denominator :
33 ^ ^ -^
'*^* 4^' 64^' 12^'»-
34.
8
L 35.
36.
3/.
38.
39.
40.
a X
a"
0?— a' a-j?' o^-c^* c^-a^'
c
«-6' a+6' o«-62' a;'+6«'
1 ar 3 4 5
^-1' (^-1)«' *+l' (Of+l^' W^'
it a+a ax
1 1 a^
a^-ax+a** x^+ax+a^^ ^*+aV+a*'
«2-(a+6)a;+a&' x^-{a+e)x+ac*
1
2
'8*
76 ADDITION OB SUBTRACTION
XVI. Addition or Subtraction qf Fractiom,
140. Rule for the addition or flubtraction of frac-
tions. Reduce the fractions to a common denominator^
then add or subtract the numerators and retain the, com-
mon €lenomin(Uor,
Examples. Add -r~"t®^x"«
Here the fractions have already a common denominate^
and therefore do not require reducing;
' ^" a+e ^ a''e _ a+c+a—c 2a \
From take -•
e e
4a-3h Za-4b _ 4a-Zb'-{3a-'4h)
~ c e
II ■ 4a-3&~3g+4&_a+6 '
The student is recommended to put down the work at
fully ap we have done in thb example, in order to ensure
accuracy. •
Add — 7T to
a+6 ^ a-V
Here the common denominator will be the product of
a-f & and a-d, that is a^-&^.
e _ <?(a~&) c _ c(a-l-&)
a+6" a«r.6«' a-^" a^-d* *
Then.fon) _^ + _£. ^ ^_(«r^±^)
_ ca—cb'\' C a-\-db _ 9xa
OF FRACTIONS.
F^^ «±_? take «-^
a-6
a+6*
The common denommator is a*--(^
a-f-6 (a-f-&)» 0-6 _ (g--6) «
Therefore
0+& a-5 (a+6)'-(a-&)"
a— 6 a+6 o^— 6*
q*+2g&+y - (a» - 2a6 + 6*) 4a6
From
a*-6* "" a^-ja*
a?+l , , 4a^-3a?+2
take
aj2~4«+3 -ia^-Soj'-lSay+lS*
By Art. 123 the l.o.m. of the denominators is
(a?- 1) (a?- 3) (4ij2+ 3iP-6);
a?+l
(;l?+l)(4^* + 3^-6)
«*-4a?+3 (a?-l)(;i?-"3)(4iB«+3a?-6)*^
4^V3a?+2 ^ (4;ir^-3a?+2)(a'-l)
4aj»-9aj2-16«+18 (a?-l)(a?-3)(4««+3a?-6)'
Therefore
«+l
4«'--3a:+2
«"~4a?+3 4^-9ic2-15a?+18
^ (a?+l)(4a:«+3a?-6)-(4a^--3j?-i-2)(;g~l)
(a?-l)(aj-3)(4«» + 3;»-6)
4a^+7a!'-3a?-6-(4j?»-7a?*+ga?-2)
(a?-l)(aT-3)(4«»+3i»-6)
14aj*~8av 4
(4f- 1) (4?-3)(4af»+3a?-6) •
77
78
ADDITION OR SUBTRACTION
141. We have sometimes to reduce a mixed qtumtity
to a fraction; this is a simple case of addition or sul^
traction of fractions.
Examples. «+-=-•+- = — + - = .
^ c I c c e c
2ah _ a 2a& _ a(a+b) 2db a^+Sab
a+b~ I a-i-b" a+b a+b" a+b
fl?+3-
«-2
af-2
=£±2
a?*~3a?+4 1 «J*-3af + 4
l^■,
:f,.^^:' :':'■■
'iy.,
V
, «2-3a?+4 ""^-3d?+4
jg'--5jy+12-(a?-2) _fl ^-'5a?-fl2~a?4-2 a;*--6a?+l4
~ «"-3a;+4 " ;u2-3«+4 *" ^-3a?+4 *
142. Expressions may occur involving both addition
and subtraction. Thus, for example, simplify
a db
+
a'
// '
a+b'^a^-b* a2+6»-
The L;aM. of the denoramators is (a2-&2)(a«+52),
that is «*-&*.
a __a{a-b){a^-¥V) a^-cfb^'an^-'dl^
a+b
d'-b*
a*-b*
db _ db(a^+b*) c^b + dt^
a^+ft*" a*-&* " a*-6* '
a
a&
o'
Therefore — tti- -r" Ta — am
c^-d^b+am-dJ^r^cPb+dlt^-a^+d!^ _ 2a»/^
OF FRA0TI0N8>
79
\uantity
or Bub-
ih
•60? + 14
-3a?+4 *
addition
^(aH6^,
The bednner should pay particular attention to this
example. He is very liable to take the product of the
denominators for the common denominator, and thus to
render the operations extremely laborious.
The second fraction contains the factor h—a in its de-
nominator, and this factor differs from the &ctor a-h,
which occurs in the denominator of the first fraction, only
in the sign of each term ; and by Art. 135,
& h
{h-'C){h-ar (b-e){a-by
Also the denominator of the third fiuction can be put
in a form which is more convenient for our object ; for by
the Mule of Signs we have
(c-a)(c-&)=(a-<j)(6-c).
Hence the proposed expression may be put in the form
a h c
(a-b){a-c) {h-c){a-b)'^(a-c){b-'cy
and in this form we see at once that the L.O.M. of the de-
nominators is (a— &) (a — c) {b — c).
By reducing the fractions to the lowest common deno-
minator the proposed expression becomes
a(b—c)—b{a'-c)+c(a—b)
{fl-b){a-c)(p-'C) '
ab—ac—db+bc+ac—bc . , . . ^
— , that IS 0.
that is
(a— &)(a-c)(6— c)
143. In this Chapter we have shewn how to combine
two or more fractions into a single fraction; on the other
hand we may, if we please, break up a single fractioD into
two or more fractions. For example,
Zbe—4^ac-\-bdh _2hcAac 6o&_3 4 5
nibc " abc abc abc" a be'
EXAMPLES, XFf.
^xAumjuk XYL
Find the yalue of
3a- 5ft gg-ft-c a+h^e
4^ "^ 3
12
a-ft ^+6'
«■»
a-6 a+6*
ftc oc ^
6.
7.
a
9.
2y
l+3d? l-3dy
l-3a? l+3a:*
a
a (a— a) a(a''W)*
a b
2a-2&"26-2a*
2aaf
10. + — ^- „ — i.
11 <»-2& 6-3c 4a6'f3ftg
3c
^^ «-ft
12. -~ +
2a 6ac
2a a'+
ft_6 a^h-¥'
., 2ft-a . &-2a , 3a?(a-:&)
/ ■
EXAMPLES. XVI,
81
1&
3 2tfT
<r-« «+2^(ar4-2)«'
l«.-i^4 ^
5-:^"^5T? S^::]^-
17. — — + — -• — , . * »
1&
19.
»+l a;+2 «+3*
"T*
a;-l «+l d?-2*
20. i^-^Zy + ^±£,
y a?+y »-y
21. «
«— 1 a+V
22. ^— ^+ ^
23.
1 ;^ 1 2
x—a x+a X
25. ^ + -^+ *
26.
27.
28.
29,
«*-l a?-l «+l'
a~aj a+x^€^+a^'
3 1 a? -HO
2«-4 «+2"'2aj»+8*
d?-3
«■
«+4"*«*--4i+ 16 "*■«*+ 64*
T.A.
e
82 EXAMPLES. XVL
«
^•fy* ^ y*
+
80.
31.
88.
88.
8i.
85.
86.
37.
39.
40.
41.
42.
43.
44^
«■+! '^•P^-ar+l «+l*
1 '2 1
(«-3) («-4) ■" (fl;-2) (d7-4) "** (»-2)(a^-3)'
1 2ay~3 1
fl7(j; + 1) "" « (d? + 1) (jp + 2) "** 47 (» + 2) •
l-2jy ay+1 1
3(dJ*-«+l)'**2(««+l) 6(a:+l)'
^-y . 1 , ^
«^— «y+y2 a;+y a^+y^'
+
«— y /py— 2aj*
iP-y «*+a;y+y* ^-y* *
tff+1
^-1
2
^*+^+l 4;'-a?+l »*+dr*+l'
g4-& a--& 2(a»ay+yy)
aa+by ax—hy a*ai^+l^ '
2ay 1 1
2
3
^■-7a?+12 a;"-4d;+3 <r»-6^+4*
d?+a
1 4a
2a
a-a a^-A* a^+a*'
2&
4&»
a-6 a+6""a*+9". a*+6*'
_J__ _ 1 _3 3_
i-Sft^^+Sa 47+a *c -a'
V
I ;.
BXAMPLBS. XVL
^
45.
4e.
47.
6
-26""a^"^a""a+6'*'a-»-26*
(#-a)(a-.6)'*'(«-6)(&-a)'
a h
(«-a)(a-6) **■ («-d)(6-a) *
48. .^4.^^ ^
49.
(»-a)(a-6)"*"(«-d)(6-
i 1
(a-6)(a-c)^(6-a)(6-c)-
^' (a-6)(a-c)'*"(6-a)^-c);
"• (a-6)(a-c)'^(6-a)(6-c)'*" (c-a)(c-d)*
52.
53.
54.
<i(a-ft)(a-c)'^&(6-a)(ft-<j)""aft<>*
6»
<^
(a-6)(a-c) "^ (6-a)(6-c) "^ (c-a)(c-^
_ 1 , 1
«*— (&+c)ii?+fcc'
55.
'«+«
«+&
flj'-(a+ft)«+a6 «*-(a+c)«+a<?
m-^a
^•-(6+c)«+6c
66.
^
6—2
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WIlSTiR.N.Y. I4SM
(716)t73-4S03
'^
84 MULTIPLIOATION OF FRACTIONS,
*.'
m
XYII. MuUiplkation qf Fractions.
144. Rule for the multiplicatioii of fractions. Multi-
ply together the numeratore /or a new num$ratorf and
tMdmominatori for a new denominator, *
145. The following is the usual demonstration of the
Bule. Let ? and ^ be two fractions which are to be
a
xnuliiplied together; put |==«, and ^=:y; therefore
a=baB, and e=
ae=bdaif;
ac
therefore
diVideby^thn. ^=*y.
But
«y=jX3
therefore
a c _ac
b^d'bd'
And ae is the product of the numerators, and hd the
product of the denominators; this demonstrates the Rule.
Similarly the Rule may be demonstrated when more
than two fractions are multiplied together.
146. We shall now give some examples. Before multi-
plying together the factors of the new numerator and the
factors of the new denominator, it is adyisable to examine
if any &ctor occurs in both the numerator and denomi-
nator, as it may be struck out of both, and the result will
thus DC simplified; see Art 137.
Multiply a by - .
^=1'1
ah
a^l__ah
— X — — ~"" f
e e
Hence a- and — are equiyalent; so, for ex^ple,
c c
,w 4a • 1,« «» 2a?-8
4j: = TT^ ; and i(2«-3)=-x— .
thu
(^
MULTIPU04TI0N OF FRACTIONS. 95
Muiti'
Tf and
'ft
of the
to be
hd the
Rule.
\ more
multi-
ndthe
cami&e
enomi-
iltwiU
<^ple,
Multiply J by 2.
thus
©■■?•
8a 8<?_ 8ax8<? 2g x 12g 2g
46^9a'"46x9a'*86xl2a"35*
Tir«i««i- 3g' , 4(a«-6«)
Multiply -nrrm ^y
(a +6)5'
3a6
30^ 4 (<iy~y) 4a(a-&) x 3a(a4»6) _ 4a(a-5)
(a+&)« Soft " 6(a+6)x3a(a+6) " 6(<i+6) '
Multiply 1 + ^+1 by f + J-1.
6"*'a a6"*'a6"**a&"
a
a'
;« 6« a& a«+6«-a6
db cib db
db
a^-^^^-^db g'+y-gft _ (««+V+a6)(ft«+y--rt6)
a6
a&
a«6«
_(a«+y)«-g«y _ a*+5*+a«6»
a^
Or we may proceed thus :
a»6»
(f*l")(M-')-(M)'-
(M)"-®"*'SI-(D"-
.+2+5
%*
86 MULTIPUCATION QF FRAOnONS.
therefore
(M*')S*l-')-?*'*5--S*S*'-
The two nralta agree^ f« n+g-H= *^^i *
Hdtiplytogetlier J=J, 1^, «»* 6+j^. '
We might multiply together the first t?ro iSMto% and
then midtiply the product sepiarately by h and by ^ZTg * *^
add the results; but it is more conyenient to reduce the
nib
miaed quanHty b + r— - to a single fraction. Thus
Then
R I, ^ _ ^(l"-<*)+a& _ ft
• 1-a 1-a "1-a'
I
<ls^„-^
lif^ i-^ v ft _ (i"<^(i-y)ft i-ft
ft+ft«^a+a* l-a""ft(l+ft)a(l+a)(l-a)'" a '
147. As we have already done in former Chapters, we
must here give some results which the student must a#-
nms to be capable of explanation, and which he must use
as rules in working examples which may be proposed. See
Arts. 63 and 135.
Multiply jby-j,
ft^ d^b^T"" bd ''
ae
bd'
Multiply -I by
e ^ae
b'^d
X -.=
d bd
ae
bd'
a
Multiply -T by -
\
r-3=T
—a — «_«?
^ d "bd
fS.
BXAMPLES. XVIL
87
itor% and
l-a'
dducethe
18
1-J
■" a •
ipters, we
must OM-
must use
lecL Be«
Etakpltm. XVIL
Hud tbeTtloa of the following:
«v y*^ ^^ *
^ a« 6* <^
2, r-x — X -r.
&c oc a&
4. r X
«-l
6.
xa
a+a
e-D-
'• K4.)('-^''
8.
9.
a^a-a)
a{a+x)
fl^+2awT+«^ a*-2aaf+a^'
af^^^
«*+y" \«-^y *+y/
98
DIVISION OF FRACTI0NI3.
XYIII. Division qf FracHom,
148. Rule for dividing one fraction by anoiher. Invert
the divisor and proceed as in MuiHplieation. •
149. The Mowing is the usual demon&tration of the
Rule. Suppose we have to divide r ^7 3) poi g^dr,
and j=Vf therefore
a=:haf, 9Me=dy;
ad^hdx, and&c=^y;
ad _ Mx _ m
"be "Idy y'
therefore
ilkerefore
Bat
a a e
therefore
a
b
e
d
ad
a d
ISd. We shall now give srane examples.
Divide
Divide
a a h _a e _ae
""i/ l'*'c""l'^5""6'
3a, 9a
4b^^Sc'
3a^9a_3a 8c_ 2cxl2a _2c
46'**8c""46^9a""36xl2a 36*
Divide
a6-J»
a&-y ^ y
(a+b)* ^^ aa-6«*
ft*
X
(a4-6)« ^-F""(a+ft)«'' ft»
_ &(a-ft)(a+&)(a->6) _ (a--6)»
ft«(a+ft)« "6(a+6)'
i
I
8
C
C
DIVISION OF FRACTIONS.
89
Invert
of the
a
151. Complex fractional ezpreaaions may be simplified
by the aid of some or all of the roles respeoting fractions
wnich hare now been given. The following are examples.
Simplify /^+^U(^-^|.
'^ ^ \a-h a-^h) {a-h a+b)
a+b ' a~&_ (a4-ft)«+(a-6y _ 2a'4-2y
a-b^a+b" {a-b)(a+b) ~ 4fi-V '
a+b a-b (g+&)«-(g~6)' 4a&
«-6 a+b ia^b)(fl+b) "a«--6»*
2a»4-2y . 4db 2a«+26« a«-5» _ a»4-y
a2_/^ • a«-j»~ a»-6» ^ 4ab " 2ab '
In this example the factors a—b and a+& are mtdU-
lied together, and the result a*-&> is used instead of
a+6)(a— &); m general however the student will &ftd it
advisable not to mtdtiply the factors together in the
course of the operation, because an opportunity may occur
of striking out a common factor from the numerator and
denominator of his result
f.
^-v.
Simplify
a+
14
a+l
3-a
- a4-l _3-<8 g+l _ 3~q+g+l
3-a""3--a'*"3-a"' 3-a
1^ 4 _1 3-:a_3--a
3-a 14 4 *
3-a 4a . 3-a 3 + 3a
, 3+3a 14 4
1-r — :; — = r X
4
3-a*
1 3+3a 3+3a*
90
DIVISION OF FRAOTIONa.
. 2ab a 2a&~a(a+5) a&— a'
2^— a=s — _— _s i 1 ss r-:
2ar-6«
"■ a+ft •
a+6 1
a+b
Therefore
2x-b a+b a+b ^ a+b ^ ab-l^
tlierefore
ab-a* ajp—a)
'^db-l^^bia-by
/ 2g~a Y_ / a\*_ ^
\2af'-bJ''\b)'''b**
a
b'
. , a db a(a+b)-ab a^
Again, a-4r«j-— j=-^^^j3 ^^5
. _b db b(a+b)-iib ^ V
l^a+b" a+b a+h'
a— «
a* a+b
Therefore ^— ^= — x-«--^x= -tt
* / b'-a a+b a+b a+b
Therefore ^^ _ __ = t _ _ =o.
6«'
agam
152. The results given in Art. 147 most be given
'- ^ re in connexion with Division of Fractious.
ae
ae
Since ?x *-5=-£j, and -iX^=-?^
bd
ae
b d bd
ae
we have -rj-^ — j=rj a»d — rj-T- j=— i-
»(]{*<;&
bd' d
ae
Also dnce -¥ x - js= £3, we have
^ M'
ae
id'
\
BXAMPZSa. XVttL
91
V
-6^
1
jfiyen
BxiMFLu. xyni.
TA,
' 1 1
8. 3 — 3 by .
*• a(a+ft)«'^?a(a^-t^'
6.
8«*
by
4«'
9.
10.
11.
12.
.1
^+»ff±y^
by
«^-!^
jy*— 8j?+2 - «"-5»+6
('*3('-J)
by
y
13. ««»-^Jby«+^
14. oF-^bya--
15.
** <i*
-:3^y::--
5*v .4?*-^
BXAMPUBS.
16. T-**-*-
a
120^
,^ j^ 1 - # . 1 ■ 1
17. -3-- by -»+- + -.
18. 5+1+ J by 2-1+?.
Simplify the foUawing expressioiui:
21.
3x a-l
2 "^ 3
»-l +
?(*+»)-f-2i
22.
6
«-6
«^-24
8
23.
// 3 2jy-l .
2 2
24.
«— 6
a— a
X-
{x^mx-cY
25. 1-
X
26. 1 +
AT
l+« +
l-«
.4.
27.
1—
1+i
or
2a
1 +
X
l+«+
2d^
l-«
V
a^
ft^+y^ v^
-A
,x+y x-y sfi
*-W* Wy^^^A
BXAUPLBS. XVtn.
^
L.
y(«y«-f «•!-«)*
I^ tiie Talnoi of the Mowing expresgioiui:
83* V when ars= . r«
«# 1 — wnen aT= — =-.
o a a-6
^ i + 5:::5-5T5''^*=ft^•
3g. ^_£_2!y ^iiena=5and6=-.
37.
38.
39.
40.
Sx
a-htf «-y «*-^
«+2tf . «-2a
I
4a&
26-d? 26 + « 4&*-««
whenoTs
a;+a-26
when ;v=
a+&
when HB^
a»+l
andyss
ab+a
^
1
M
SIMPLE BQUATIONS.
XIX. Simple EqmUcm,
168. When two algebraical ezpreBdons are oonneoCed
bj the sign of equality the whole ia called an equation.
The ezpreMions thus connected are called Hdn of the
equation or fn«nd>er9 of the eqnation. The ezprawion to
the left of the sign of eanality la called the JUrH aide^ laid
the ezpreasion to the rignt ia called the 9eGond aide.
1C4. An idenHeal equation ia one in which the two
aides are equal whateyer numbers the letters represent;
for ezamplei the followhig are identical equations,
t (ir+a)(«-a)=:«*-a^,
(«+a)(«'-«a+a«)=fl^+a^ ;
that is, these algebraical statements are true whatever
numbers a and a may represent The student will see
that up to the present point he has been almost ezchisiTely
occupied with results of this Idnd, tbskt is, with identical
equations.
An identical equation is called briefly an identity.
155. An equation qf condition is one which is not true
whateyer nunibers the letters represent, but only when
the letters represent some particular number or numbers.
For example, a +1 — 7 cannot be true unless 47=6. An
equation of condition is called briefly an equation,
156. A leifter to which a particular yalue or values
must be giyen in order that the statement contained in an
equation may be true, is called an unknown quantity, Buch
particular yalue of the unlpiown quantity is said to eoil^
the equation, and is called a root qf the equation. To
eitlve an equation is to find the root or roots.
157. An equation inyolying one unknown quantify la
said to be of as many dimensions as the index (^ tlie
faighest power of the unknown quantity. Thus, if m dehote
i
\
lation.
if the
Ion to
Belaid
e two
Mont;
\
liever
ill see
flively
ntiou
true
when
iben.
An
flues
in an
Bach
fthe
SIMPLE BQUATION&
M
the unknown quantity, the equation is said to be of oim
dimension when a oocnrs only in the ^rH power; snob an
equation is also called a $impU equaitonf or an equation of
the /irtt degree. If «* occurs, and no higher power of «,
the equation is said to be of two dimensions; such an
eouation is also called a guadroHo equationy or an equation
of the $eeond degree. If «* occurs, and no higher power
of Wf the equation is said to be of three dimensions ; sui^
an equation is also called a eubie equation^ or an equation
of the third degree. And so on.
It must be obsenred that these definitions suppose both
members of the equation to be integral expreenone eojar
a$ r^atee to x.
158. In the present Ohapter we shaU shew how to soke
simple equations. We have first to indicate some opera-
tions which may be performed on an equation without
destroying the equality which it expresses.
159. If every term on each side qf an equoHon he
multiplied ly the tame number the reeuUe are equal.
The truth of this statement follows from the obyious
principle, that if equals be multiplied by the same number
the results are equal ; and the uee of this statement will be
seen immediately.
Likewise if wery term on each tide qf an equation^
be divided by the tame number the retultt are equal,
160. The principal use of Art 159 is to dear an eqwh
turn qf fractumt; this is effected hj multiplyinff every
term by the product of all the denominators of t£e frac-
tions, or, if we please, by the least common multiple of those
denominators. Suppose, for example, that
^846
Multiply every term by 3 x 4 x 6 ; thus
4 X 6 X 07 + 3 X 6 X iT + 3 X 4 X « = 3 X 4 X 6 )$ 9,
that is, 24«+1&i;+12«b648;
divide evcv^ term by 6 ; thus
^ .4;p+3a;+2j9=108.
96 SIMPLE EQUATIONS,
Instead of multiplying every term by 3 x 4 x 6, we may
multiply every term oy 12, which is the l.o.m. of the deno-
mimitors 3, 4, and 6 ; we shoiild then obtain at once
4^+3^+2^=108; , \
that is, 9^=108;
divide both sides by 9 ; therefore ^
108 ,o
^=— = 12.
Thus 12 is the root of the proposed equation. We may
renfy this bv putting 12 for x in the original equation.
The first side oecomes
i IQ IQ |0
V + ^ + ^, thati84+3+2, thatis9;
o 4 o
which agrees with the second side.
161. Any term may he transposed from one side qf
an equation to the other side by changing its sign.
Suppose, for example, that x—a^h-y.
Add a to each side ; then
x—a+a^h-y+a^
that is x^h—y-ha.
Subtract h from each side ; thus
Here we see that —a has been removed from one , side
of the equation, and appears as + a on the other side ; and
+& has been removed from one side-and appears as — & on
the other side.
162. If the sign of every term qf an equation he
changed the equality still holds, \
This follows from Art. 161, by transposing every t^nxi.
Thus suppose, for example, that ;r—a~o—2f«
SIMPLE EQUATIONS.
97
9 may
deno-
emay
latioii.
ide qf
,side
and
n he
By tra]U^K)Bitioxi
that 18,
a-~w=y-'b;
and this result is what we shall obtain if we change the
sign of eyery term in the original equation.
163. We can now give a Rule for the solution of any
simple equation with one unknown quantity, dear the
equation of jf'ractionSj if necessary; transpose all the
terms which involve the unknown quantity to one side cf
the eqwUion^ and the knovon quantities to the other side;
divide both sides by the coefficient^ or the sum qf the co-
ejfficients, qfthe unknown quantity^ and the root required
u obtained,
t 164. We shall now give some examples. '
Solye 7«+26=»35 + 5a?.
> Here there are no fractions ; by transposing we have
7a?-6a?=35-265
that is, 2a?=10;
divide by 2; therefore w=-^=5.
We may verify this result by putting 6 for « in the
original equation; then each side is equal to 60.
166. Solve 4(3^-2)-2(4a?-3)-3(4-a?)=0.
Perform the multiplications indicated; thus
12d? - a- (Sa? - 6)- (12 - 3a?) = 0.
Bemove the brackets; thus
12a?-8-8ii?-4-6-12 + 3i»=0;
collect the terms, 7«?-14=0;
transpose, *Jaf=l4;
14
divide by 7,
a-
= 2.
- The student will find it a useful exercise to verify the
correctness of bis solutions. Thus in the above example^
98
SIMPLE EQUATIONS.
/
if we put 2 for x in the original equation we shall obtun
16-10-6, that is 0, as it should be.
166. Solye «-2-(2a?-3)=^-^.
Remove the brackets; thus
fl?-2-2a?+3=— ~,
3a? + l
2-2a?==3a?+l,
2-l = 2a?+3a?;
l=5a?, or6ii?=l; t
1
o
tliatisy
multiply by 2,
transpose,
lihatis,
«
therefore
167. Solve
6a?+4
= 6}--
10
28
5}= — ; the L.O.M. of the denominators is 10; multiply
by 10;
thus
that is,
transpose^
that is,
therefore
6(6a?+4)-(7a?+6)=28x2-6(aj-l);
26ar+20-7d?-6 = 66-6a? + 6;
25a?-7^+6^=66 + 5-20 + 6;
23^=46;
46 «
X— — =2.
23
The begmner is recommended to put down all the work
at fuU, as in this example, in order to ensure ar^curacy.
Mistakes with respect to the si^ns are often made u\ clear-
ing an equation of fractions. In the above equation the
faction -1^ ha. to be multipHed by 10. «d i^ I. ad-
visable to put the result first in the form —(7^+5), and
afterwaids in the form — 7«-5, in order to secoie Iktten-
tioa to the signs. y>
tl
ti
tl
J
th
tr
cl
tl
tl
EXAMPLES. XfX
99
obtain
lultiply
168. Solve |(&»+3)-^(16-5«)=37-4«.
By Art 146 this is the same as
Multiply by 21; thus 7(6iP+3)-3(l6-64?)=21(37-4p),
that is, 364?+21-48 + 16««777-84«;
>
transpose, 35;i;+ 15a; +84^=777 -21 +48;
that is, 134a;=804;
therefore
169. Solve
804 ^
^=134=^- ,
607+15 a»-10 Ax-l
11 7 5 •
Multiply by the product of 11, 7, and 5 ; thus
36(6a?+15)-55(a»-10)=77(4a?-7),
that is, 210a7+525-440a;+550=30&v-539;
transpose, 210a? - 440a? -30a»=- 639 -525 -560;
change the signs, 440a?+308a?-210a;=539 +525+ 550,
that is, 5380?= 1614;
1614
therefore
0?=
638
=3,
1^
> work
iuracy.
dear-
\xi the
is ad-
), and
Itten-
KXAMPJJES. XIX.
1. 5a;+50=4a?+56. 2. 16a?-ll=7a7+70.
3. 24a?-49=19a?-14. 4. 3a? +23 =78- 2a?.
{fc y{a?-18)=3(a?-14). 6. 16a? =38 -3 (4 -a?).
1 7(a?-3)=9(a?+l)-38. 8. 5 (a? -7) +63= 9a?.
»t 59(a?-7)=61(9-a?)-2. 10. 72(a?- 6)=^ 63(5-0?).
X . X
11. '28(o?+9)=27(46-a?). 12. «+o+3-ll
7-2
100
EXAMPLES. XIX.
Of X
5 3
21. ~ + 12=^+6.
3 o
83. '|-6=^*-a
4 O
14. ~+24=2rf?+.6.
4«
16. 36-~=a
3^ 5dT ^
18. ^+fi=T + 2.
.4 6
2a? 176 --4a ?
"3" fi~'
24. f-8=74-g.
26. f-.^=*-2.
27. 4(«-3)-7(«-4)=6-a?.
3""3 4'*"4'"6~6"6^6'
' X 2a? So?
30. 2a?-
19-2a? 2a?-ll
2
2
„ a?+l 3a?-l'
31. — = = — =a?-2.
3
92. 0?+
3a?~9
=4-
ffa?-12
S3.
34.
10^+3 6ar-7
3
2
3 • .
= 100?- 10,
5a?-7 2a?+7
3
=30?- 14.
36. 4?-l
o?-2 . a?-3
'.A>'-.-;.'^
2
3
=a
>,-'v
6.
2.
*»
w*'
n -;=■-
»"«* J
>i i '*.-
V .
I
EXAMPLES. XIX.
»+3 . «+4 . «+5
101
37.
88.
89.
40.
,«•
48.
44.
45.
46.
47.
7«+9
8
=7+«-
4
r=l«.
8a?— 4 ftp--5 _ 8a?-l
2 """i 16 •
«-8_«--5 «— 1
4 """T""*'T"*
<p~l «— 8 . «-5
8
4 •^"T-'='^
7^+5 6^+6 8-&9
6 4
«+4 »— 4
=2+
12 •
84^-1
5 "■ 15 •
m-X 2af+7 ;«+2
2
a
•;=9.
«— 1 «-2 . «f-8 2.
2 3
2iP-5 . 6d?+8
6
8
=&»-l7J.
48. T-
« 5^+8 2ir~9
6
8
8aT+6 2dT+7
8«
.„ « ar . «-2 «
*^ 8-4 +--5-=*
8
+io-^=a
49.
J^, i(3«-4)+5(5i»+3)=:43-&l?.
^ « , fl» « « w. MO *
d? dr~2 «4-3 2
2 8
3'
m
iXAMPLSS.
53.
fi-Zx (kp 8 3-&P
4 **■ 8 ""2"
54. |(27-2^)=|-l(7^-5f).
55. 5«-[a»-3{16-6«-r(4-5«)aa:e.
fSA ^-^ 4-5JJ ^ 13 ^ •
3 6 42 '^^
-^ »+l « -1 , 2«— 1
51 -J _4.4^=,i2+f^.
8
59. «£zl^9^-5 »4?-7
24'
11
^•
60. ^±i-.2z? = ?£z£4.1
2 8 12 *4'
61v/|(8-ir)+a?-l§=|(af+C).|.
62.
34?
-1 13-4T 74? 11, .■
5 "~2~"'T"^i"^*'*'^
2aN-l 6^-4 7^+12
^ 5 "^^ 7 "ir^v
3
2^6
vr
'.■^
M-
SIMPLE EQUATIONS.
10»
we r^
JtJu Simple Epiaticnif continued,
170. We shall now give some examples of the soluiion
of simple equations, which are a little more difficult than
those m the preceding Chapter. Tlw i^dpmfc will gee tibat
it is sometimes adyantageous to dear of ihictions
tiaiijf , and then to effect some redactions, b efoEP
IBoye the remaming fractions. *" """^
17L Solye-jj- __ + __=.5j+-_ .
Here we may conyenientiy multiply by. ]i2; thiu^
12(;P4-6)
11
16
4(a»-18)+3(2«+8)« J X 124'34;+4,
that is, i?^ji5)~8f+72+6«-f 9=^+3«+4.
By transposition and reduction we obtain
Multi^y by 11; thus 12 («+ 6) =11 (6a?- 13),
thatis^ 12^+72=5&i;~143;
by transpofidtion, 72+143=6&i;-12dr, <^
that is, 43^=215;
therefore
216 ^
172. Solye^^.2..^^=6A-?5i^^
JBere we may conyeniently multiply by 24; thus
M
sM?
11^
w
1
M8d;4-16d;-16=24x
H-C"-")
m
sm^Ls mu4m>m
tliatb,
By t»n«poda<m and rednction
1444y~820
iiWltlply!^15-2fl7. thug
thepeftw 144»+ar=320+60,
<^*V 162»=880j
«-7 «+9*
«*teict ««fix)in each sid^ of the equation^ thus
4«-45s-4;ir^21;
4»+4«=45-21,
847=24;
transpose^
Ihatis,
ti&erefbre
24
\
\
>th
i <
SIMPLE MQUATION&
174 Sohe-j^-jj^ + gj^j.
105
Here it ie oonyenient to multiply by ix-¥i, that ii by
4(«+l);
therefore 8»+12-4»-5=:i^^l
.V
thatiis
M;^tiply by 8«+ 1 ; thus (SdT-l- l)(4«-f 7)== I2(«+ 1)*;
thatis, ' 12Ai^+2&v+7»12d^+24p+12.
Subtnust 12«* from each glde^ and transpose; thus
2&9-24cv=12-7,
that is, 47=5.^
175. Bdye
Wehaye
And
d^-2 "" «-3 ~ «— 6 ~ «— 6*
J?-l 4?--2 ^ (jp-l)(jp~3)-(a?-2y
»-r2~«-3 («-2)(i-8)
(«-2)(«-3) "■(aT-2)(«-3)*
«-5 ar^e
fl^~lftg-f24-(jg>-l(to4.25) ^
(;»-e)(«-6)
(4T-6)(«-6)
1
■(«-5)(«-6)*
Thus tibe proposed equation becomes
" • • - 1 L 1
"'(«-2)(«-3) («-5)(«-6)'
106
BIMFUt SqUAnONS.
OhMge the «ign.,.tlia. ^j_^j_„^_jj^.
Clear of fraotioiiB; thus («-5)(«-6)=:(tf-S)(«-8}f
■» f
ttiatifl,
therefore
that is,
therefore
therefore
a^-lld;+S0=«'-&»+6 ;
6a7=:24;
n^ Soke *Q«4
•4&r-'75 1-2 -a^-'B
I
•«
•2
•9
To ensure aoenraey it is advisable to express all the
dedmahi as oommon finictions ; thus
-I 10 6 \100 100/ " 2 ^ 10 9 \10 10/'
««p«^. M(i-i)=«-(M)' -
thatis, ^^??-5=6_£ + ?,
^ 2 4 4 3 3
Multiply by 12, 6^-f 9a;-15-72-4a7+a;
1907=72+8+16=95; ^
96'
^=i9=«-
therefore
177. Equations may be proposed in which letUir$ are
used to represent known quantities ; we shall oontmiie to
represent the unknown quantity by x, and any other tetter
wul^ be supposed to represent a known quantity. We ^ will
sdye three such equations. i
.Vv«,
SIMPLE MQlTATTOim
107
\
17a CMto f+f«a
lAiH^by a&; thus (dr+AVaoBt
diyidebya+d; thiu
179. Sohro (a+«)(d+«)-a(6+c)+^+^.
Here a5+aar+ddr+«««a6+ac+^+««;
1 !
therefore
diyidebya+ft;thn8 «=2.
18a. Solve ^ = (f ;) ; .
Clear^ of fractions : thus
(ar-a)(ai?-&)«=(af-ft)'(ai?-a)«j
thatia, («-a)(4i?«-4^+ft2)=(a.-ft)(4aj»-4^+<^
Multiplying out we obtain
4aj»-4««(a+6)+«(4a&+5*)-ai!?«
s=4iJ»-4aj«(a+&)+«(4a&+«^-a«&;
therefore af^-dt^z^axfi^aHf;
therefore a?(a«-d^=a«d-a6«=a&(a-&);
.a&(a~ft) db
f s
X-
a«-^ ~a+d"
v»
MXAMFLBSk XX.
181. AUhoiigh the fbOowiM MpMilta
bekms toJOie proNiit Ohnter w^ ghrt it
no duBoaltj in ibllowing tbe ileiM aT «tM
win www M a model for ifanikr
Kfomblee those ab«edy solred, hi the
we obtehi onlj a Wf^i;^ nhie of the
Mve iy«+V(«-l«)«a
Sy tnmspodtioii, ^(a-lB)^S'-^a;
■qnarebothiideB; thiie «-16=(8-V«?=e4-l6,/«4-#;
therefore -16~64-16V«;
*«n»poii^ 16^a?r=64+16»80| i
therafora ijxsilii
Qi»ni<ore
4rs2ff.
EzAMPtnB. XX
//
1 1?4.-L ?E
« 12ar~24'
a
128 216
ZsB"^ 6x^-9'
2.
4
42 85
«-2 «-3*
45 57
) -..
20^+3 iaf-a*
5. -2 — i-^-r-r^-^-V
A 2 . 4 . 4-jg 10-«
'• «H)n'(f-l)-*
ft Uj.5*r^-<! ag-^8
o o
9. r^4.j«#-5C|p
t 'J
Im4-mi
\
b»-l
<^^R^n>vK*'v <• ^mh^RWf J * ^PijivlM
11.
109
#-1 7#-21
IB. #-3-(8-#)(«+l)»#(«-8)+a
le. JI-«-«(«-l)(«-l-2)-(«-8)(5-2<p). ,
17. ^^-l+?~=7ar. la («+7)(4r+l)=(#f 8)«.
'i». |(to--10)--i(8flT--40)=15-i(57-afX •
^ to 4- 8 i9-i-88 -
lu. ^ . . , =-z =1,
2L
2ar4>l «4-12
4y~l «--5 15-2» 9-» 7
4 "* 82 ■*"^'^"' 2~"8'
/.J
28.. ^+*(«-2)«(«-l)«.
24. ^+(«-l)(«-2)^«»-2i»-4U
^ 8g'-2jp~8 (7ir~2)(ag-6)
■1
EXAMPLES. XX.
29.
«-5 «-6 a?-8 *«— 9'
30.
i» «-9 a?+l a?-8
31.
3-2a? 2ii?-6 4aj>-l
l-2a? ^-7~ 7-16a?+4«8*
32.
3+a? 2+a? 1+a? -
3-iP 2-a? l-o?"" •
33.
d?-6 as^+6 «2-2 aj'-ar+l
7 3^2 6
34.
Oc+l)(x+2)(a!+Z)
f3.
35.
36.
//
37.
38.
39.
40.
41.
«
43.
44.
45.
= (a;-l)(a?-a)(a?-3)+3(4«-2)(a?+l).
(«-9)(a?-7)(«-6)(4?-l)
=(a?-2)(a?-4)(aj-6)(a?-10).
(8a?-3)2(a?-l)=(4a?-f')2(4a?-6).
as^-a+l . aj2+a?+l „
— ^ _ — =2«'.
•6a? - 2 = •25a? + '20? - 1.
•54? + '60? - '8 = '75^ + •25.
•135a?-^225 '36 •09;»-^18
•15a? +
•6
•2 '9 •
.- a?-a ,a?45
42. a— r^ + 6 — ^ = 0?.
a
a— a ,6+0?
a —J- o =0?.
a
x{pD''a)-{-x{x-'b)=2{x-a){ps-h), \ '
(a?-a) (« - J) (« + 2« + 26)
«(a?+2a) (a?+26) (4T-<i-d).
(a?+l).
»-10).
i#«^ EXAMPLES, XX.
48. («-a)(«-5)»(a;-a-5)>,
a ft a— 6
111
48.
49.
50.
51.
tB-^a x-^b W'^c*
a h _ a-\-h
I 1 a-b
x—b «*— oft'
1 1
x—a
1
fnx—a—b mx—a-^c
n nx—c—d" nx—b^d*
52. {a-b){x-c)-(p-c)(x-a)-'{C'-<i^{x^b)^^
64, {a—x){b-x)^{p-{-x){q-{-x),
x—a x—a-l x—b
55.
x-b-l
x—a—X x—a— 2" x—b— I x-b—2'
56. (x+a)(2x+b+ef={x+b)(2x+a+cf.
67. (a? + 2a) («- a)* = (a? + 26) (^ - &)". , !
58. (a?-a)*(a?4-a-26)=(a?-6)»(«-2a+&).
6d. ^/(4d?)+^/(4i»-7) = 7.
60. V(«+14)+/s/(«-14)=14
61. ^/(a?+ll)+V(a?-9)=10.
62. ,J(9x+4)+sJ(Bx-l)=S.
63. V(«+4a&)=2a-Va?.
64. V(«-a)+V(«-.6)=V(«-&)f
h^X
112
PROBLEMS.
XXI. Problems.
183. We shall now apply 'the methods explained in the
precedmg two Chapters to the solution of some problems,
and thus exhibit to the student specimens of the use of
Algebra. In these problems certam quantities are given,
and another, which has some assigned relations to these,
has to be found; the quantity which has to be found is
called the unknown quanlity. The relations are lusually
expressed in ordinary language in the enunciation of the
problem, and ihe method of solving the problem may be
thus described in general terms: denote the unknown
quantity by the letter x, and express in algebraical
language the relations which hold between the unknown
quantity and the given quantities; an equation will thui^
be detained from, which the value of the unknown quantity
may be found.
183. The sum of two 'numbers is 85, and their differ-
ence is 27 : find the numbers.
Let X denote the less number ; then, since the differ-
ence of the numbers is 27, the greater number will be
denoted by «+27 ; and since the sum of the numbers is 86
we have
] fl?+a?+27=86;
that is, ai?+27 = 86;
therefore 2«=86-27«68 5
therefore
58 ««
Thus the less number is 29 ; and the greater number is
29+27>thatis56.
184. I^ivide £2, 10s. among A, B, and C, so that B
mav have 5s. more than A, and C may have aa much as A
and B together.
Let X denote. the number of shillings In A*b share,
then a+5 will denote the number of shiUings in B*m share,
and 2^+5 will denote the number of shillings in 0*b share.
PROBLEMS. ilZ
The whole number of shillings is 50 ; therefore
«+« + 6+2aT+6=605
that is, 4d;-l- 10=50;
therefore 4a;=50-10=40;
therefore a; =10.
Thus A*% share is 10 shillings, ZTs share is 15 shillings^
and (7's share is 25 shillings.
185. A certain sum of money was divided between
Ay B, and C; A and B together receiyed ;£17. 15«. ; A
an<i together received £15. 15^. ; B and C together
received .^12. 10«. : find tiie sum received by each.
Let a: denote the number of pounds which .^1 received,
then B received 171—^ pounds, because A and B
together received 17| pounds; and C received 15f— 4f
pounds, because A and C together received 15} pounds^
Also B and C together receiv^ 12^ pounds; therefore
12j=l7i-af+15i-i»;
that is,
12j=334-2d?;
therefore
2a?=33j-12j=21:
therefore
21 ,^1
«=2=10J.
( «
Thus A received ;£10. 10^., B received £*!. 5s.f and O
received £5, 58,
186. . A grocer has some tea worth 2«. a lb., and some
worth 3». 6a. a lb, : how many lbs. must he take of each
sort to produce 100 lbs. of a mixture worth 2«. 6d, a lb. ?
Let ic denote the number of lbs. of the first sort ; then
UK)-* or will denote the number of Iba of the second sort
The value of t^e x lbs. is 2x shillings ; and the value of the
T.A.
8
114>
:PROBLBMS.
100- a; lbs. is ^(100-^) shillings.. And the whole value
is to b^3 ^ X 100 sinkings; therefore
5x100= 2a? + 5(100-0?);
multiply by 2, thus 600 = 4a; + 700 - 7^ ;
therefore 7d?-4iP= 700-600;
that is, 3a?=200;
200
therefore
«=•
a
\
Thus there must be 66|lbs. of the first sort, and
33ilbs. of the second sort.
< ■ • ■ , "
187. A line is 2 feet 4 inches long; it is required to
divide it into two parts, such that one part may be three-
fourths of the other part
Let X denote the number of inches in the larger part ;
then -7- will denote the number of inches in the other part.
4
The number of inches in the whole line is 28 ; therefore
3^ «o
«?+-j=2S;
therefore
4a?+3a?=112;
that is,
7a?=;=112;
therefore
a?=16.
Thus one part is 16 inches long, and the other part 12
inches long.
188, A person had £1000, part of wlHch he Jent at
4 per cent., and the rest at 6 per cent.; the whole annual
interest received was j£44 : how much was l^t at 4 per
cetot. I
Mm.
PROBLEMS.
lis
per
Let X denote the number of pounds lent at 4 per cent ;
then 1000— d; will denote the number of pounas lent at
5 per cent The annual interest obtained from the former
is 7^ , and from the latter mn^ *
100
therefore
therefore
that isy
therefore
100
4^ 5(1000~d?) ,
100"^ 100 '
4400 = 4a? + 6(1000-a?);
4400 = 4^ + 5000 - 5aT ;
a?=6000-"4400=600.
Thus ;£600 was lent at 4 per cent
1 I
189. The student will find that the'only difficulty in
solving a problem consists in translating statements ex-
pressMl in ordinary language into Algebraical lan^^uage;
and he should not be cuscouraged, if he is sometmies a
little perplexed, since nothing but practice can give him
readiness and certainty in this process. One remark may
be made, which is very importont for beginners; what is
called the unknown piantity is really an unknown number^
and this should be distincUy noticed in forming the equa-
tion. Thus, for example, in the second problem which we
have solved, we begin by sayii^, let x denote the number
of shillings in A^% share; beginners often say, let a?=^'s
money, which is not definite, because A*% money may be
expressed in various ways, in pounds, or in shillings, or as
a nuction of the whole sum. Again, in the fifth problem
which we have solved,^ we begin oy saying, let x denote
the number of inches in the longer part; oeginners often
say, let a;= the longer part, or, let a; = a part, and to these
phrases the same objection applies as to that already
noticed.
190. Beginners often find a difficulty in translating a
problem from ordinary language into Algebraical language,
because they do not understand what is meant by uie
ordinary language. If- no consistent meaning can be as-
signed lo the words, it is of course impossible to translate
them; but it often happens that the words are not ab-
8—2
EXAMPLES. XXL
solutoly nnintelligiblejVut appear to be sosoeptible of more
than one meaning. The student should then select one
meaning, express that meaning in Algebraical symbok. and
deduce from it the result to which it will lead, jlf the
result bo inadmissible, or absurd, the student should try
another meaning of the words. But if the .result is satis-
factory he may mfer that he has probably understood the
words corrector ; though it may still be interesting to try
the other possible meanings, in order to see if the enun-
ciation really is susceptible of more than one meaning.
191. A student in solving the problems which are
eiven for exercise, may find some which he can readily solve
by Arithmetic, or by a process of g^ess and trial ; and he
may be thus inclined to undervalue the power of Algebra^
and look on its aid as unnecessary. But we may remark,
that by Algebra the student is enabled to solve all these
problems, without any uncertaintv ; and moreover, he will
find as he proceeds, that by Algebra he can solve pro-
blems which would be extremely- difficult or altogether
impracticable, if he relied on Arithmetic alone.
/ Examples. XXI.
1. Find the number which exceeds its fifth part by 24.
% A father is 30 years old, and his son is 2 yeak old :
In how many years will the fiEither be eight times as old as
the son?
3. Tho differonco of two numbers is 7; and their sum
is 33 : find the numbers.
4. The sum of .£165 was raised by -4, B, and tog^
ther ; B contributed ^15 more than A, and C ^20 more
than B : how much did each contribute i
5. The difference of two numbers is 14, and thefa* sum
is 48 : find the numbers.
\
, S. AiA twice as old as B^ and seven years ago thoir
united, ages amounted to as many years as now t epCM ii it
the age of .4: find the ages of ^ and A . ^liv >
. •
.■VS',!^;^;"^.-;,;
EXAMPLES. XXL
117
moxo
P8IIIII
thoir
7. If 5d be added to a certain number, the result is
treble &at number: find the number.
8. A child is bom in November, and on the tenth day
of December he is as many days old as the month was on
the day of his birth : when was he bom ?
9. Find that number the double of which increased by
24 exceeds 80 as much as the number itself is below 100.
10. There is a certain fish, the head of which is 9
inches long; the tail is as long as the head and half the
back ; and the back is as long as the head and tail toge-
ther : what is the length of the back and of the tail ?
11. Divide the number 84 into two ^arts such that
three times one part may be equal to four times the other.
. 12. The sum of £76 was raised by Ay By and C toge-
ther; B contributed as much as A and j£10 more, and (7
as much as A and B together : how much did each con-
tribute?
13. Divide the number 60 into two parts such that a
seventh of one part may be equal to an eighth of the other
part.
14. After 34 gallons had been drawn out of one of
two equal casks, and 80 gallons out of the other, there
remained just three times as much in one cask as in the
other: what did each cask contain when full ?
15. Divide the number 75 into two parts such that
3 times the greater may exceed 7 times the less by 15.
16. A person distributes 20 shilling among 20 per-
sons, giving sixpence each to some, and sixteen pence each
to the rest : how many persons received sixpence each ?
17. Divide the number 20 into two parts such that
the sum of three times one part, and five times the other
pari^ may be 84.
I8«^> The price of a work which comes out in parts is
£% ld». 8^. ; out if the price of each part were 13 pence
more than it is, the price of the work would be £Z, 78. 6d, :
how niany parts were there 1
19. Divide 45 into two parts such that the first divided
by It fdiaU be equal to the second multiplied by 2.
118
EXAMPLES. XXL
20. A father is three times as old as his son; four
years affo the father was four times as old as his son then*
was : what is the age of each ?
21. Divide 188 into two parts such that the fourth of
one part may exceed the eighth of the other by 14.
22. A person meeting a company of beggars gave four
Eence to each, and had sixteen pence left ; ne found that
e should have required a shilling more to enable him to
give the beggars sixpence each : how many beggars were
there?
23. Divide 100 into two parts such that if a third of
one part be subtracted from a fourth of the other the re-
mainder may be 11.
24. Two persons, A and B. engage at play; A haS;
£*J2 and B has j£d2 when they begin, and after a cei*t»in •
number of games have been won and lost between them,
A has three times as much money as B : how much did A
wini
25. Divide 60 into two parts such that the difference
between the greater and 64 may be equal to twice the
difference between the less and 38.
26. The sum of ;£276 was raised by A, B, and C^ toge-
ther; B contributed twice as much as A and £,\2 more;
and C three times as much as B and £12 more: how much
did each contribute ?
27. Find a number such that the sum of its fifth and
its seventh shall exceed the sum of its eighth and its
twelfth by 113.
28. An anny in a defeat loses one-sixth of its number
in killed and wounded, and 4000 prisoners ; it is reinforced
by 3000 men, but retreats, losing one-fourth of its number
in doing so ; there remain 18000 men : what was the ori-
ginid force 1
29. Find a number such that the sum of its fifth and
its seyenth shall exceed the difference of its foiu*th and its
seventh by 99.
30. One-half of a certain number of persons receiye^
eighteen-pence each, one-third received two shilUngs each^'
and the rest received half a crown each ; the whole sum
distributed was £2. 4#. : how many persons were ^ere t .
EXAMPLES, XXL
119
SI. A peraon had ^^00 ; part of it he lent u Ihe rate
of 4 per cent^ and part at the rate of per cent, and he
received equal sums as interest from the two parts : how
much did he lend at 4 per cent. ?
32. A father has six sons, each of whom is four years
older than his next younger brother; and the eldest i»
three times as old as the youngest: find their respectiTO
ages.
33. Divide the number 92 into four such parts thai
the first may exceed the second by 10, the third by 18, and
the fourth by 24
34. A gentleman left ;£560 to be divided among four
servants A,BfC,D; of whom B was to have twice as
much OS A, V M much as A and B together, and D as
much as C and B together : how much had each ?
35. Find two consecutive numbers such that the half
and the fifth of the first ieiken together shall be equal to
the third and the fourth of the second taken together.
36. A sum of money is to be distributed among three
persons A, B^ and C; the shares of A and B together
amount to £60 ; those of A and C to £80 ; and those of B
and C to £92 : find the share of each person.
37. Two persons A and B are travelling together ; A
has £100, and B has £48; they are met by robbers who
take twice as much from A as from Bf and leave to A
three times as much as to ^ : how much was taken from
each ?
38. The sum of £600 was divided among four persons^
so that the first and second together received £280, the
first and third together £260, and the first and fourth
together £220 : find the share of each.
39. After A has received £10 from B he has as much
money as B and £6 more ; and between them they have
£40: what money had each at first ?
40. A wine merchant has two sorts of wines, one sort
worth 2 shillings a quart, and the other worth Ss. Ad. a
quart; from these he wants to make a mixture of 100
quarts worth 28, id, a quart: how many quarts must he
take from each sort f
120
EXAMPLES. XXI.
41. In a mixture of wine and water the wine composed
25 gallons more than half of the mixture, and the water
gallons less than a third of the mixture : how many gal-
lons were there of each ?
42. In a lottery consisting of 10000 tickets, half the
number of prizes added to one-third the number of blanks
was 8600 : how many prizes were there in the lottery ?
43. In a certain weight of gunpowder the saltpetre
composed 6 lbs. more than a half of the weight, the sulphur
fflbs. less than a third, and the charcoal 3 lbs. less than a
fourth : how many lbs. were there of each of the three
ingredients ?
44. A general, after having lost a battle, found that
he had left fit for action 3600 men more than half of bis ;
army ; 600 men more than one-eighth of his army were
wounded ; and the remainder, forming one-fifth of the
army, were slain, taken prisoners, or missing : what was
the number of the army ?
46. How many sheep must a person buy at £7 each
that after paying one shilling a score for folding them at
mght he may gain £19, I6s, by selling the^::^ at £S each ?
46. A certain sum of money was shared among five
persons A, B^ C, D, and E; B received £l(i less than A ;
t7 received £\^ more than B ; 2> received £6 less than C\
and E received ;£16 more than D ; and it was found that
E received as much as A and B together : how much did
each receive ?
47. A tradesman starts with a certain sum of money ;
at the end of the first year he had doubled his original
stodc. all but ;£100 ; also at the end of the second year he
had doubled the stock at the beginning of the second year,
all but j^lOO; also in like manner at we end of the third
year ; and at the end of the third year he was three times
as rich as at first : find his original stock.
48. A person went to a taveni with a certain sum of
money ; there he borrowed as much as he had about him,
and spent a shilling out of the whole ; with the remainder
he went to a second tavern, where he borrowed as nluch as
he had left, and also spent a shilling ; and he then went to
a third tavem,- borrowing and spending as before, after
which he had nothing left : how much had he at first !
c
o
81
n
ii
tl
m
m
a-
th
is
th
th
mi
th
th
PROBLEMS.
121
XXII. Prciblemif continued.
192. 'We shall now giye some examples In which the
process of translation from ordinary language to algebrai-
cal language is rather more difficult than in the examples
of the preceding Chapter.
193. It is required to diyide the number 80 into four
such parts, that the first increased by 3, the second dimi-
nishea by 3, the third multiplied by 3, and the fourth
divided by 3 may all be equaL
Let the number x denote the first part; then if it be
increased by 3 we obtain •;»+ 3, and this is to be equal to
the second part diminished by 3, so that the second part
must heof+S; again, d7+3 is to be equal to the third part
multiplied by 3, so that the third part must be — ^— ; and
«+ 3 is to be equal to the fourth part dirided by 3, so that
the fourth foxt must be 3(a: + 3). And the sum of the parts
is to be equal to 80.
Therefore
that Is,
that Is,
fl?+a?+6-l-~^+3(j?+3)=80,
2a?+6+^|^+3ii?+9=80,
5«+^=80-16=66;
multiply by 3; thus 16a?+a5+3=196,
that Is, 16a?=192;
a— — =12.
16
therefore
Thus the parts are 12, 18, 5, 45.
I2i
PROBLEMS.
I94k A alone can perform a pieoe of work in 9 dam
and B alone can perform it in 12 days : In wh«t time wUi
they perform it if they work together 1
Let w denote the reqnhred number of dayi. In one.day
A can perform ^ th of the work ; therefore in w daya he can
perform ^ths of the work. In one day B can perform
l^th of the work; therefore in x days he can perform
gether perform the whole work, the ram of the fradioM
of the work must be equal to «m<y; thfttia.
Multiply by 36 ; thus
thatis,
ther^i^re
9 + 12 *•
7«=86;
36 ,.
\
195. A dstem could be filled with water by means of
one pipe alone in 6 hours, and by means of another pipe
alone in 8 hours ; and it could be emptied by a tap in 12
hours if the two pipes were dosed : hi what time will the
dstem be filled if tne pipes and the tap are all open %
Let m denote the requured number of hours. In, one
hour the first pipe fills ^thof i^e dstem; therefore in o)
hours it fills - ths of the dstem. In one hour the second
pipe fills - th of the dstem ; therefore in a hours it fills
X 1
r ths of the dstem. In one hour the tap empties -^ th
PRaBLBMA
12a
of the oiiterr: ; iherefore In # hours it emptiei ^tfai of
the dsteni. And ihioe in « honn the whoU oiitoni is
iUled, we haye
6-'8-i2='^-
MnUipl7b7 24; thus 4«*i-8d7-2«s24,
thati% 6«=24;
24
therefore
«=
M-
196. It is sometimes oonyenient to denote by ^ not
the unknown quantity which is explicitly required, but
some other quantity froni which that can be easify deduced;
this will be illustrated in the next two problems.
197. A colonel on attempting to draw up his regiment
in the form of a solid square finds that he has 31 men
over, and that he would require 24 men more in his regi-
ment in order to increase the side of the square by oue
•man : how many men were there in the regiment 1
Let w denote the number of men in the side of the fint
square ; then the number of men in the square is afi and
the number of men in the regiment is 0^+31. If there
were x+\ men in a side of the square, the number of men
in the square would be (^+ 1)* ; tnus the number of men
in the regiment is (« + 1)* - 24.
Therefore (a?+l)«-24=«*+31,
thatisy fl^+2aj+l-24=«'+31.
From these two eqr I expressions we can remOTO o^ which
occurs in both ; thuti
24^+1-24=31;
therefore 2fl7=31- 1+24=64:
therefore
^=y=27.
Hence the number of men in the reghnent is (27)* +31,
that is, 729 + 3r, that is, 760.
124
PROBLEMS.
198. A starts from a certain place, and trayels at the
rate of 7 miles in 5 hours ; B starts from the same place
8 hours after A^ and travels in the same direction at the
rate of 5 miles in 3 hours: how far will A travel before he
is overtaken by JB ? '
Let X represent the number of hours which A travels
before he is overtaken; therefore B travehi xS hours.
*j
Now since A travels 7 miles in 5 hours, he travels - of a
o
*lx
mile in one hour : and therefore in x hours he travels ~
6
■»
miles. Similarly B travels r of a mile in one hour, and
3
t
therefore in ^—8 hours he travels - («-8) miles. And
when^S overtakes A they have travelled the same num-
ber of miles. Therefore
mul^iplyby 15; thus 25 («~ 8)= 21a?,
that is, 25ar~200=21a;;
therefore 25a; -210; =200,
that is, 4^=200;
therefore
200 _^
4
*lx 7
Therefore y = gX60=70; so that A travelled 70 miles
before he was overtaken.
199. Problems are sometimes s^ven which suppose the
student to have obtained from Arithmetic a knowledge of
PROBLEMS.
125
of a
the
of
the meaning of proportion; this will be illustrated in the
next two problems. After them we shall conclude the
CSiapter with three i>roblems of a more difficult character
than those hitherto given.
200. It is required to divide the number 56 into two
parts such that one may be to the other as 3 to 4.
Let the number a denote the first part; then the other
part n^ust be 56 -x; and since a;i8tobcto56-«iras3to4
we have
a 8
66-^"" 4'
Clear of fractions; thus
4^=3 (56 -a?);
that is, 4a?=168-3a?;
therefore 7^=168;
therefore
«=-^=24.
Thus the first part is 24 and the other part is 66-24,
that is 32.
The i>recedinff method of solution is the most natural
for a beghmer; w.e following however is much shorter.
Lot the number 3x denote the first part; then the
second put must be 4d;, because the first part is to the
second as 3 to 4. Then the sum of the two parts is equal
to 56: thus
that is,
therefore
3a?+4d;=56,
7a?=56j
Thus the &nt part is 3 x 8, that is 24; and the second
partis4x8,thatis32.
126
I'ROBLEMS.
201. A cask, A, oontainB 12 gallons of wine and 18
gallons of water; and another cask, B^ contains 9 gallons
of wine and 3 gallons of water: how many gp&Ilons must be
(kawn from each cask so as to produce oy their nqisture
7 gallons of wine and 7 gallons of water ?
'^ Let X denote the number of gallons to be drawn from
A\ then. since the mixture is to consist of 14 gallons,
14-^ will denote the number of gallons to be drawn from
J?. Now the nimiber of gallons in ^ is 30, of which 12 are
12
wine; that is, the wine i<9 35 of the whole. Therefore the
X gallons drawn from A contain -r— gallons of wine.
vO
Similarly the 14 - a? gallons drawn from B contain ^ \Z \
gallons of wine. And the mixture is to contain 7 gallons
of wine; therefore
m 9(14-0?)
30 ^12 :'
thatis,
2x Z(U-x)
6^ 4. "-'*
therefore
8a?+16(14-a?)=xl40,
thatis,
8a:+210-15d?=140;
therefore
7^=70;
therefore
«?=10.
Thus 10 gallons must be drawn from Ay and 4 from B,
202. At what time between 2 o'clock and 3 o'clock is
one hand of a watch exactly over the other 9
Let X denote the required number of minutes. after
2 o'clock. In X mimttes the long hand will move over
X divisirTis of the watch 'feuse; and as the long han4 o^oyes
twelve times as &st as the short hand, the short hwd wUl
move over r^ divisions in « nunutes. At 2 o'clock the
PROBLEMS.
127
Bhort hand is 10 divisions in advance of the long hand; so
that in the x minntes the long hand must pass over 10
more divisions than the short hand; therefore
therefore
therefore
therefore
^=^ + 10;
12d;=«+120;
lla?=120;
120 ,^,o
the
203. A hare takes four leaps to a g^honnd's three,
but two of the sreyhound's leaps are equivalent to three of
the hare's; the hare has a start of fifty leaps: Yam many *
^leaps must the greyhound take to catch the hare?
Suppose that 307 denote the number of leaps taken by
the greyhound; then Ax ^U denote the number of leaps
taken by the hare in the same time. Let a d^iote the num-
ber of inches in one leap 4^ Jlie hare; then 3a denotes the .
number of inches in three leaps of the hare, and therefore
also the number of inches in two lei^ of the greyhound;
therefore ~ denotes the number of inches in one leap of
the greyhound. Then Zx leapa of the greyhoiypd will con-
3(1 ^ •
tain 307 x~ inches. And Hd-^Ax leaps of the .hare will
contain (50+ 4or}a inches; therefore
Divide by a; thus -^a60+4o7;
.therefore 9oj=100+8o?; ...
therefore o;=100.
Thus the greyhound must take 300 leaps.
The student will see tbat we have introduced an auzl-
liaiy symbol a, to enable us t6 form the equaUon e^ily;
and tmit we eaa- remove it by division when the equation is
formed.
128
EX4MPLE8. XXII.
204. Four gamestera, A, B, C, D, each with a ^^Ifiimt
stock of money, sit down to plav; A wiiis half of I^wt
stock, B wins a third part of C\ C wins a fourth part of
jD's, and D wins a fifth part of ^'s ; and then each of thio
gamesters has £2^. Fmd the stock of each at first '
Let X denote the number of pounds which D won from
A\ then 6x will denote the number in A'% first stodk.
Thus 4ar, together with what A won from By mi^e up 23;
therefore 23— 4a; denotes the number of pounds which' ^
won from B, And, since A won half of JErs stock, 23—4^0
also denotes what was left with B after his loss to A,
Again, 23-4ar, together with what B won from C,
make up 23 ; therefore Ax denotes the number of pounds
which B won from C, And, since B won a third of C*f^
first stock, \2x denotes (7's first stock; and therefore 8«
denotes what was left with C after his loss to B,
Again, ^x, together with what C won from D, make up
23; therefore 23— 8a; denotes the number of pounds which
O won from 2>. And, since C won a fourth of D's first
stodi, 4(23—80;) denotes i>*s first stock; and therefore
3(23— 8a;) denotes what was left with D after his loss to (7.
Finally, 3 (23 -8a;), together with or, which D won from
^1 make up 23; thus
23=3 (23 -8a?) + a?;
therefore
therefore
23a;=46;
a;=2.
Thus the stocks at first were 10, 30, 24, 28*
Examples. XXII. >
1. A privateer runidng at the rate of 10 miles an hour
discovers a ship 18 miles off, running at the rate of 8 miles
an hour: how many miles can the ship run before jt is
overtaken 1 .
2. Divide the number 60 into two parts such that if
three-fourths Qf one part be added to five^tbs of the
other part the sum may be 40.
EXAMPLES. XXIL
129
from
hotv
[miles
It is
3« SuppoM the distance between London uid'Bdin-
liiirghvis 360 miles, and that one traveller starts from
Edinboigh and travels at the rate of 10 miles an hour,
while another starts at the same time from London and
travels at the rate of 8 miles an hour : it is required to
know where tiiey will meet.
4. Find two numbers whose difference is 4^ and the
difference of their squares 112.
5. A sum of 24 shillings is received from 24 people ;
some contribute M. each, and some IS^dL each : how many
contributors were there of each kind ?
6. Divide the number 48 into two parts such that the
excess of one part over 20 may be three times the excess
of 20 over the other part
7. A person has ;£98 ; part of it he lent at the rate of
6 per cent simple interest, and the rest at the rate of
6 per cent simple interest; and the interest of the whole
in 15 years amounted to j£81 : how much was lent at ^
per cent?
8. A person lent a certain sum of money at 6 per cent
simple interest ; in 10 years the interest amountea to £Vl
less than the sum lent : what was the sum lent ?
9. A person rents 25 acres of land for ^^7. 12«. ; the
land consists of two sorts, the better sort he rents at 8t.
per acre, and the worse at 5«. per acre : how mapy acres are
there of each sort %
10. A cistern could be filled in 12 minutes bj two
pipes which run into it ; and it would be filled in 20 minutes
bv one alone : in what time could it be filled by the other
alone ?
11. Divide the number 90 into four parts such that
the first increased by % the second diminished by 2, the
third multiplied by 2, and the fourth divided by 2 may all
be equfd.
. 12. A person bought 30 lbs. of sugar of two different
sorts, and paid for the whole 19f. \ the better sort cost
10<l. per 1d„ and the worse *ldn per lb. : how many lbs.
were there of each sort ?
T.A.
9
N
ISO
EKAMPLE8. XXII.
18. Diyide the number 88 into four parte rach that
the first increased by 2, the second diminished by 3, the
third multiplied by 4, and the fourth divided by 5, may all
beequaL
14. If 20 men, 40 women, aad 60 children receive ^50
among them for a week's work, and 2 men receive as much
as 3 women or 5 children, what does each woman receive
for a week's work?
15. Divide 100 into two parts siich that the difference
of theur squares may be 1000.
16. There are two places 164 miles apart^ from which
two persons start at the same time with a design to meet ;
one travels at the rate of 3 miles in two hours, and the
other at the rate of 5 nules in four hours : when will th^y
meet? ^
17. Divide 44 into two parts such that the greater in-
creased by 5 may be to the less increased by 7, as 4 is
to 3.
18. A can do half as much work MBf B can do half
as much as O, and together they can complete a piece of
work in 24 days: in what time could each alone complete
the work ?
19. Divide the number 90 into four parts such that if
the first be increased by 6, the second diminished by 4, the
third multiplied by 3, and the fourth divided by 2, the
results shall all be equal.
20. Three persons can together complete a piece of
work in 60 days ; and it is found that the first does three-
fourths of what the second does, and the second four-fifths
of what the third does : in what time could each one alone
complete the work ?
21. Divide the number 36 into two parts such that one
part may be five-sevenths of the other.
22. A general on attempting to draw up his army in
the form of a solid square finds that he has 60 men over,
and that he would require 41 men more in his^army in
order to increase the side of the square by one man : how
many men were there in the army ?
i..jKaV\
EXAMPLES. XXIL
131
li that
3, the
nay all
re £60
I much
receive
ference
[Which
meet;
iod the
iUth^y
Biter in-
as 4 is
do half
dece of
mplete
that if
4»the
2, the
lece of
three-
Ir-fifths
alone
it one
ly m
over,
rmv in
\\ how
23. Divide the number 90 into two parts such that one
part may be two-thirds of the other.
24. A person bought a certain number of eggs, half of
. them at 2 a penny, and half of them at 3 a penny ; he sold
them again at the rate of 6 for two pence, and lost a penny
by the bai^n : what was the numl&r of eggs ?
K 25. ^ and j9 are at present of the same age; if ^'s
age be increased by 36 years, and B\ by 52 years, their
ages will be as 3 to 4 : wnat is the present age of each ?
26. For 1 lb. of tea and 9 lbs. of sugar the chai^ is
8«. 6df. ; for 1 lb. of tea and 15 lbs. of sugar the charge is
12«. 6(f. : what is the price of 1 lb. of sugar ?
27. A prize of ^£2000 was divided between A and J?,
so that their shares were in the proportion of 7 to 9 : what
was the share of each ?
28. A workman was hired for 40 dajrs at 3«. Ad, per
day, for every day he worked ; but with this condition that
for every day he did not work he was to forfeit 1«. Ad, ; and
on the whole he had £Z, 3«. Ad, to receive: how many days
out of the 40 did he work 1
29. A at pl^ first won £5 from B^ and had then as
much money as 6\ but By on winning back his own money
and £5 more, had five times as much money as A : what
money had each at first ?
30. Divide 100 into two parts, such that the square of
their difference may exceed the square of twice the less
part by 2000.
31. A cistern has two supply pipes, which will singly
fill it in 4^ hours and 6 hours respectively; and it has also
a leak by which it would be emptied in 5 hours : in how
many hours will it be filled when all are working together ?
32. A farmer would mix wheat at 4f . a bushel with
r^e at 2«. 6«?. a bushel, so that the whole mixture may con-
sist of 90 bushels, and be worth 3#. 2ti?. a bushel : how
many bushels must be taken of each ?
9—2
132
t!XAMPLE8. XXIL
33. A bill of £^, Is, Bd. was pud in lialf^croiviuB; and
florins, and the whole number of coins was 28 : how many
coins were there of each kind ?
34 A grocer with 56 lbs. of fine tea at 5t. a lU Would
mix a coarser sort at Ss, 6d. a lb., so as to sell the whole
together at As, ed, a lb. : what quantity of the latter sort
must he take?
35. A person hired a labourer to do a certain work
on the agreement that for every day he worked he should
receiye 2s., but that for every day he was absent he should
lose 9d. ; he worked twice <\s many days as he was absent,
and on the whole received ^£1. Ids, : find how many days
he worked.
36. A regiment was drawn up in a solid square ; wh^i
some time after it was again drawn up in a solid square
it was found that there were 5 men fewer in a side; in tiie
interval 295 men had been removed from the field: what
was the original number of men in the regiment ?
37. A sum of money was divided between A and j9,
BO that the share of ^ was to that of S as 5 to 3 ; also the
share of A exceeded five-ninths of the whole sum by £50:
what was the share of each person ?
38. A gentleman left his whole estate among his four
sons. The share of the eldest was ^£800 less than half of
the estate; the share of the second was ^120 more than
one-fourth of the estate; the third had naif as much as
the eldest; and the youngest had two-thirds of what the
second had. How much cud each son receive ?
39. A and B began to play together with equal sums
of money; A firat won ;£20, but afterwards lost half of all
he then bad, and then his money was half as much as that
of B : what money had each at nrst ?
40. A lad]r gave a guinea in charity among a number
of poor, consisting of men, women, and chUdren ; each man ,
had 12a^ each woman Qd., and each child Sd, The number
of women was two less than twice the number of men; and
the number of children four less than three times the
number of women. How many persons were there re-
Keved?
(
EXAMPLES. XXIL
133
^ and
TXMSf
Would
whole
)r sort
I work
should
should
ibseilty
y days
;wi
square
in the
: what
and ^,
Jso the
y;£60:
lis four
half of
than
iuch as
lat the
sums
of all
that
dumber
m man,
lumber
[n; and
les the
^re re-
41. A draper bought a piece of cloth at 3«. 2<f. per
yard. He sold one-third of it at 4«. per yard, one-fomrtn of
it at 3«. 8^. per yard, and the remainder at 3». Ad, per
yard; and his ^ain on the whole was \A». 2d. How many
yards did the piece contain?
42. A grazier spent ;£33. 7s. 6d. in buying sheep of
different sorts. I*or the first sort which formed one-third
of the whole, he paid 9«. 6(1?. each. For the second sort^
which formed one-fourth of the whole, he paid 11#. each.
For the rest he paid I2s, 6d. each. What number of sheep
did he buy]
43. A market woman bought a certain number of oggiu
at the rate of 5 for twopence; she sold half of them at
2 a penny, and half of them at 3 a penny, and gained 4d.
by so doing: what was the number of eggs 1
44. A pudding consists of 2 parts of flour, 3 parts of
raisins, and 4 parts of suet ; flour costs 3d. a lb., raisms, Bd.^
and suet Sd, Find the cost of the several ingredients of
the pudding, when the whole cost is 2s, Ad,
45. Two persons, A and B, were employed together
for 60 days, at 6s. per day each. During this time A, by
spending 6d. per day less than B, saved twice as mucn as
B, besides the expenses of two days over. How much did
^ spend per day?
46. Two persons, A and B, have the same income. A
lays by one-fifth of his ; but B by spending £60 per annum
more tlmn A^ at the end of three years finds himself £100
in debt What is the income of each ?
47. A and B shoot by turns at a target A puts 7
bullets out of 12 into the bull's eye, and B puts in 9 out of
12; between them they put in 32 bullets. How many
shots did each fire ?
48. Two casks, A and i?, contain mixtures of wine
and water; in A the quantity of wine is to the quantity of
water as 4 to 3 ; in ^ the like proportion is that of 2 to 3.
If A contain 84 gallons, what must B contain, so that when
the two are put together, the new mixture may be half
wine and half water?
134
EXAMPLES. XXIL
49. The squire of a parish bequeaths a sum equal to
one-hundredth part of his estate towards the restoration
of the church; j£200 less than this towards the endow-
ment of the school ; and j£200 less than this latter sum
towards the County Hospital After deductiog thepe lega-
39
cies, jg of the estate remain to the hehr. What was the
value of the estate t
60. How many minutes does it want to 4 o'clock, if
three-quarters of an hour ago it was twice as many minutes
past two o'clock?
61. Two casks, A and B, are filled with two kinds of
sherry, mixed in the cask A in the proportion of 2 to 7,
and in the cask B in the proportion of 2 to 6 : what qiian-
tity must be taken from each to form a mixture which
shall consist of 2 gallons of the first kind and 6 of the
second kmd ?
62. An officer can form the men of his regiment into
a hollow square 12 deep. The number of men in the
regiment is 1296. Find the number of men in the front of
the holKm square.
63. A person buys a piece of land at ;£30 an acre, and
by selling it in allotments finds the valae increased three-
fold, so that he clears j£l60, and retains 26 acres for him-
self: haw many acres were there?
64. The national debt of a country was increased bT
one-fourth in a time of war. During a long peace which
followed j?26000000 was p^^d ofi; and at the end of that
time the rate of interest «/as reduced from 4^ to 4 per
cent. It was then found that the amount of annual In-
terest was the same as before the war. What Was the
amoimt of the debt before the war 1
66. A and B play at a game, agreeing that the loser
shall always pay to the winner one shilling less than half
the money the loser has ; th^ commence with equal quan-
tities of money, and after Ja has lost the first g^me and
won the second, he has two shillings more than A : hoW
much had each at the commencement?
EXAMPLES, XXIL
135
laal to
Dration
Bndow-
sr smn
lelega-
ras the
look, if
idnutes
indfl of
2 to 7,
b quan-
» whicti
of the
mt into
in the
front of
Ire, and
three-
>r him-
Bed bT
which
of that
4 per
ual in-
as the
loser
half
quan-
le and
how
(
66. A clodc has two hands turning on the same centre ;
the swifter makes a revolution every twelve hours, and the
slower every sixteen hours: in what time will the swifter
£^ just one complete revolution on the slower?
67. At what time between 3 o'clock and 4 o'clock is
one hand of a watch exactly in the direction of the other
hand produced?
68. The hands of a watch are at right angles to each
other at 3 o'clock: when are they next at right angles?
69. A certain sum of money lent at simple interest
amounted to ;£297. 12«. in eight months; and in seven more
months it amounted to £306 : what was the sum ?
60. A watch sains as much as a clock loses; and 1799
hours by the dock are equivalent to 1801 hours by the
watch: find how much the watch gains and the clock loses
per hour.
61. It is between 11 and 12 o'clock, and it is observed
that the number of minute spaces between the hands is
two-thirds of what it was ten minutes previously: find the
tune.
62. A and B made a joint stock of j£600 by which
they ^ined £160, of which A had for his share £32 more
than n\ what did each contribute to the stock?
63. A distiller has 61 gallons of French brandy, which
cost ;iim 8 shillings a gallon; he wishes to buy some £n-
gli * brandy at 3 shillings a gallon to mix with the French,
and sell the whole at 9 shillings a gallon. How many gal-
lons of the English must he take, so that he mav cain
30 per cent on what he gave for the brandy of both
kinds?
64. An officer can form his men into a hollow square
4 deep, and also into a hollow square 8 deep; the front in
the latter formation contains 16 men fewer than in the
former formation: find the mmiber of men, '
186 SIMULTANEOUS SIMPLE EQUATIONS
XXIII. Simultaneout equatiom df thejint degree with
two unknown quantitiet,
206. Suppose we have an equation containing two un-
known quantities a aiid y^ for example 30^-7^=8. For
every value which we please to assign to one of the
unknown quantities we can determine the corresponding
value of the other ; and thus we can find as many pairs
of values as we please which satisfy the given equation.
Thus, fbr example, if y=l yre find 3^=15, and therefore
«=5; if y=2 we find 8^=22, and therefore a=H; and
soon.
Also, suppose that there is another equation of the
same kind, as for example 2a; +6^ =44; then we can alsb
find as many pairs of values as we please which satisfy this
equation.
But sui)pose wo ask for values of a and y which satisQr
both equations; we shall find that there is only one value
of a and one value of y. For multiply the first equation
by 5; thus
16a?-36y=40;
and multiply the second equation by 7 ; thus
1407+35^=308.
Thereforei by addition,
150^-352^+1407+35^=40+308;
that is, 2907=348;
therefore
348 ,„
^=29=^2.
Thus if both equations are to be satisfied a mutt equal 12.
Put this value of w in either of the two given equations,
for example in the second; thus we obtain
\
. . -
24 + 5^=44;
therefore
5y=20;
theretbre
y=4.
ONS.
SIMULTANEOUS SIMPLE EQUATIONS. 187
9e foith
two un-
B. For
of the
}ondiDg
ly pain
luation.
lerefore
rj; and
of tlie
ian al86
sfythis
i satisfy
e yalue
^uation
>?■■
iiall2.
ations,
806. Two or more equations which are to be satisfied
bj the same values of the unknown quantities are called
itmultaneotu equatiofui. In the present Chapter we treat
of simultaneous equations inyolving two unknown quanti-
ties, where each unknown Quantity occurs only in the first
degree, and the product of the unknown quantities does
not occur.
207* There are three methods which are usually giyen
for solvinff these equations. There is one principle com-
mon to allthe methods; namely, from two giren equations
containing two unknown quantities a single equation is de-
duced containing only one of the unknown quantities. By
this process we are said to eliminate the unknown quan-
tity which does not appear in the single equation. The
single equation contaimng only one unknown quantity can be
solved hj the method of Chapter XIX ; and when the value
of one of the unknown quantities has thus been determined,
we can substitute this value in either of the given equations^
and then determine the value of the other unknown quantity.
208. First method. Multiply the equatione hy $ueh
numbers as will make the confident qf one qf the unr
known quantities the tame in the resulting equations;
then hjf addition or subtraction we can form an equation
containing only the other unknown quantity.
This method we used in Art 205 ; for another example^
suppose
8«+7y=100, M
124;-6y=88.
If we wish to eliminate v we multiply the first equation
by 5, which is the coefficient of y in the second equation,
and we multiply the second eq|uation by 7, which is the
coefficient of y in the first equation. Thus we obtain
40.r+35^=500,
'84ar-35^=6I6;
therefore, by addition,
40a?+84a?=500 + 616;
that is, 124^=1116;
therefore «=9.
138 SIMULTANEOUS SIMPLE EQUATIONS.
Then put this yaloe of iP in either of the given equatiom^
for example In the second ; thus
108-6ys=88;
therefore 20= 5y; ^
therefore y-4.
Supposo, however, that in solving these equations we wish
to begin by eliminatmg x. If we multiply the first equa-
tion by 12, and the second by 8, we obtam
964^+84^=1200,
96a?-40y=704.
Therefore^ ly itibtraeticn^
84y+40y=1200-704;
that is, 124y=496;
therefore y=4.
Or we may render the process more simple ; for we may
multiply the first equation by 3, and the second by 2;
thus
24a;+21y=300,
' 24a?-10y=176.
Therefore, by subtraction,
21y+l()y=300-176;
that is, 31j(=124;
therefore y=4.
209. Second method. Expreti one qf the unknown
guantitiei in termt (^ the other from either equation, and
eubetitute this value in the other equation.
Thus, taking the example given in the preceding Arti-
cle^ we have firom the first equation
therefore
a»=100-7y;
,_100-7y
\
X-
8
.J^^i^
BaB^aa.
SIMULTANEOUS SIMPLE EQUATIONS. 139
irti-
Substitate this yaliie of ^in the second equation, and we
obtain
12(100-7y)
g wy-oo,
that is,
8Ji00z!LL5y=88;
therefore
8(100-7y)-10y=l76;
that is,
300-21y-10y=176;
therefore
300-l76=21y + lQy;
that is,
31^=124;
therefore
y=4.
Then substitute this value of y in either of the given equa-
tions, and we shall obtain or «: 9.
Or thus : from the first equation we have
7y=100-8a?;
1OO-&0
therefore y = — = — .
Substitute this value of y in the second equation, and
we obtoin
therefore
that is,
therefore
therefore
,2^_«000^)^885
84a?-5(100-8ar)=616;
84^-500+40^=616;
124^=500 + 616=1116;
fl?=9.
210. Third method. Eapren the tame unknown
quantity in termt qf the other from each equation, and
equate the expremom thut obtained.
Thus, taking again the same example, fix)m the first
100—72/
equation «= — o~^} <uid from the second equation
88+5y
12 •
«=
140 SlMULTANEOUa SIMPLE BQUATtONS,
Therefon, mfv^^,
Olear of fractions, by multiplying by 24; thus
3(100-7y)=2(88 + 6y);
that is, 300-21y=l76+10y;
therefore 800-176=21y+10y;
that is, 3l2(=124;
therefore y=4.
Then, as before^ we can deduce «= 9.
Or thus: frt)m the first equation y= — = — , and
from the second equation y= — j^^^ — ; therefora
O ^
100 -8« 12a? -88
■
From this eouation we shall obtain x=d\ and then, as
before, we can aeduce 2(=4.
211. Solve 19a?-21y=100, 21a?-l9y=140.
These equations may be solved by the methods already
explained; we shall use them however to shew that these
methods may be sometimes abbreviated.
Here, by addition, we obtain
19a;- 21^+210?- 19^= 100 +140;
that is, 40a? -40^ =240;
therefore «-y=0.
Again, from the original equations, by subtraction, we
obtain
21«-19y-19a?+21y=140-100;
that is, 2a? +'2^=40;
therefcm a'¥y=%0.
I '
SIMULTANEOUS SIMPLE EQUATIONS 141
Then since dr--y=6 and «+y=20, we obtain by addi-
tion 2x = 26| and by subtraction 2^ = 14 ;
therefore
0^=13, and y-*J.
212. The student will find as he proceeds that in all
parts of Alffebra, particular examples may be treated bv
methods wmch are shorter than the general rules; butsucn
abbreyiations can only be suggested by experience and
practice, and the beginner should nOt waste his time in
seeking for them.
218. Solve
^2 + ?-.8
X y
If we cleared thesfe equations of fractions they would
involve the product 4^ of the unknown quantities; and
T.ri strictly they do not belong to the present Chapter.
. It they may be solved by the methods already given, as
«<«9 shall now shew. For multiply the first equation by 3
and the second by 2, and add ; thus
a y a y
that is,
?? + 5*=80;
?2=30r
that is,
therefore 90=30^;
therefore a —3.
Substitute the value of x in the first equation ; thus
therefore
therefore
therefore
5=8.
4=4;
8=4y;
y=^2.
142
EXAMPLES. XXIN.
214. Solre a'^+d'ysc', tm-^ly^e*
Here w and y are sapposed to denote tmkmwn mttati-
tiei, while the other lettuB are rapposed to denote mown
quantities. . , •> • , .,
Multiply the second equation by h^ and subtract it from
the first; thus
that is,
therefore
Substitute this yalue of or in the second equation; thus
■ . i
a— 6 a— a—
" 6(a— o) 6(6-a)
Or the yalue of y might be found in the samd way as
that of » was found.
EZAMPLBS. XXIII.
1. Za—4y-2f
2. 7«-6y=24,
3. 847+2y=:32,
4. ll«-7y=37,
5. 7«+5y=60,
6. 6«-7y=42,
7. 10»+9y=290,
8. 3«-4y=:18,
7a>-9y=7.
4a;-3ysll.
20ay-3y=l.
av+9y=41.
13ar-lly=10.
7a?-6y=76.
12«~ny=130.
3i»+2y=0.
9. 4.v-|=ll, 2«-3y=0.
\
S^AMPLBS. XXIIL
m
10.
11.
it
18.
14
15.
16.
17.
18.
20.
21.
22.
28.
24.
8
+ay-7,
4dH-2
-4.
e«-6yal, 7«-4y=8j.
2<r4-
3«
«-2__
=21.
4^-1-
6
=::29.
r^-f5y»13, 2«+
4-7y_
= 33.
f+^=^lOj, 2^-y=7.
fly-ky . v-x _
3
2
= 9.
» «+y__
2 "^
=5.
??-^=l ?f4.?^=
3 ' 6
6.
^±y^«^
3
+«=16,
^*' 8 **■ 4 8
+ 12.
7« . By__
T"*" 3""
8
2a Zy
6
3
10.
3ar 2y
^ + -f-i6j, "^-^^lej
3 2
fl?-l , y-2
2 3
8
=2.
2a:+
2y-5
3
1 '
«21.
^+«jf=2o. : ?-i.?;y-.2^-7.
8
2;P4-3y
= 10-
y
5
"3'
l-3i»
7 "*■
3y-
5
i-
4y-3ay_8^
2.
6
3«+y
11
= ^+1.
+y=9.
25. 2(2»+3y)=3(2;p-3y) + 10,
447~3y s 4 (6y- 2x) + 3.
*/
144
EXAMPLES. XXllL
26. 3d?+9y = 2-4, '21«-'06ys-03.
27. •3j7+'125y=a?-6, 3«-'5y=28--26y.
28. •08i»--21y=-33, •12«+-7y=3-64.
29. -»-i=I
18 20
= 16.
30. «-4y
-,4y=7. -?4.11 ^rSf
3y 10 6^
31.
a?+l a?-l 6
y-1
y
32. 4a?+y=ll,
«-y=l.
y _ 7a?-y 23
(^ 307
16
33.
^3
«-6
i
•+7=0,
3y-10(iP-l) , x-y
- 4
+1=0.
i:
34. 5+1=2
ha-ay=0.
36. a?+y-a+&, ba+ay=2ab.
36.
£+?=!. f+y=i.
a
a
37. (a+c)x-'by=bc, a?+y=a+&.
38.
"— -I- — =:i
a
?-«=0.
39. «+y=tf, <M?-&y=c(a-&).
40. a(a?+y)+&(a?-y)=l, a(«-j()+6(i»+y)=l.
41.
X
-av-b^
a
=0.
w+y-b ^ x-y-a
a
=0.
42. (a+6)a?-(a-6)y=4a5,
(a-&)ii?+(a+&)y=2a'-26'
43.
or
y
a+b a-b
2a,
X
-y _ of+y
2ab a^d*'
\
44. (a+A)a?+(6-A)y=<?, (&+*)«+ (a-Aj)if=c.
SIMULTANEOUS SIMPLE EQUATIONS 145
XXiy. ^muUaiMmu equattoni qf ths Jlnt degree with
fnore than two unknown quantitiet,
215. If there be three simple equations containing
three unknown quantities, we can deduce from two of tiie
equations an equation wmoh contains only two of he un-
known quantities, by the methods of the preceding Ohap*
ter; then from the third given equation, and either of the
former two, we can deduce another equation which con-
tains the same two unknown quantities. We have thus
two equations containing two unknown quantities, and
therefore the values of these unknown quantities may be
found by the methods of the preceding Chapter. By sub-
stitutinff these values in one of the given equations, die
value of the remaining unknown quantity may oe foimd.
216. Solve 7«+3y-24f=16 {l\
2«+^+3;8f=39 (2X
6«- y+64f=31 (3).
For convenience of reference the equations are num-
bered (1), (2), (3) ; and this numbering is continued as we
proceed with uie solution.
Multiply (1) by 3, and multiply (2) by 2 ; thus
21a?+ 9y-64f=48,
4aJ+10y + 6;8r=78;
therefore, by addition,
25.r+192^=126 (4).
Multiply (1) by 5, and multiply (3) by 2; thus
35a?+15y-10;y=80,
lOiP- 2y+104f=62;
therefore, by addition,
4&v+13y=>142 (6).
T.A. 10
Cj
I*
f
Ik-
146 SIMULTANEOUS SIMPLE EQUATIONS.
We haye now to find the Talnes of w and y from (4)
and (5).
Miiltipiy (4) by 9, and multiply (6) by ; thus
226d?+l71y=1184,
2250?+ 66y= 710;
therefore, by snbtractiony
106y=4245
therefore y=4.
Substitute the Talne of y in (4); thus
26a?+76 = 126; i
therefore 26«= 126-76=60; ^
therefore a=2.
Substitute the values of47 and y in (1); thus
14+12-2af=16;
therefore 10=24?;
tiierefora «=6.
217. Sobo i + |-|=l ...(1),
Multiply (1) by 2, and add the result to (2); thus
2 4 6 6 4 . 6 „.„;
-+--:: + - +- + -=2+24:
X yz X y z ' ^
that is, - + -=26 .........(4).
^ X y ^ ^
. I.
{SIMULTANEOUS SIMPLE EQUATIONS 147
Multiply (1) by 8, and add the result to (3) ; thus
a y z a y »
that is,
?-i-' <»
Multiply (5) by 4| and add the resnlt to
•
f.!H.U?.M+«,
X y X y
that is,
X
therefore
47=94ar;
therefore
Substitnte the Talne of x in (6) ; thus
20-?=l7;
therefore
?=20-l7=3;
therefore
2
y=3-
Substitute the values of x and ^ in (1) ;
-
2+3-1=1;
z
therefore
1-'^
therefore
; •-!•
10—2
: ;
148 SIMULTANEOUS SIMPLE EQUATIONS.
218. Solve
f-H ('^
'I «
M=» (*^
5 + 1=4 (8).
Subtract (1) from (2) ; thus
h e a *
that is, J -^=2 (4).
By subtracting (4) from (3) we obtain
—=2;
iiherefore -=1; therefore or = a.
By adding (4) to (3) we obtain
therefore -=3: therefore z=^e.
c
♦
By substituting the value of a; in (1) we find thai y-^.
S19. In a similar manner we may proceed if the nun^-
ber of equations and unknown quantities should exceed
three. ^
EXAMPLES, XXIV.
149
Examples. XXIV.
1 fi«-6y+4jf=15, 7«+4y-air=:19, 2;r+y 4- 6:2^=46.
8. 4i»-5y+4f=6, 7A'-lly4-2«=9, dr+y+3^=:12.
4. 7«-3y=30, 9y-6;»=34, «+y+;2r=33.
6. 3«-y+;»=17, C«+3y~2;»=10, 7«+4y-0;r=d.
6. x+y+z=5, 3«-5y+7jif=75, 9;v-ll;8r+lO=0.
7. «+2y+34f=6, air +4^+2;? =8, 3ay+2y+air=101.
63f— a?
8. S^=l,
= 1,
9.
3;»-7 *' 2y-3z
a-h2y _ 3j/+4z 6a}+Qi6
7 ■" 8 " 9
y z *'
y~2^ _i
3y-2« •
t
a+y—z=l26»
aye*
X y z
11. y+;»=a, z-ha^b, a?+y=c.
12. af+y+z=a'hh+e, flj+a=y+6=«+<J.
13. y+;y-a?=a, ;8r+d?-y=6, a?+y-;8f=<?.
*
,- a h e ^
16. - + -+-=3,
a? y ;2f '
a c b '
a? y ;» '
& a <;
2a b c_
a y z
16. i>+«+y+;y=14,
2«+a7=:2y-f ;7~2,
8»-«+2y+24f=19,
8*4^6 + 2 *'
*
.■t«.4«r^; '.■ ■>\''.**;Jt--
150
PROBLEMS.
XXV. PrdbUmt which lead to iimultaneom equaiiont
f^thejlrtt degree with more than one unknown quantity,
220. We shall now solye some problems which lead to
simultaneous equations of the first degree with more than
one unknown quantity.
Find the fraction which becomes equal to = when the
3
4
numerator is increased by 2, and equal to = when the de-
nominator is increased by 4. i
Let a denote the numerator, and y the denommator of
the required fraction ; then, by supposition,
y ""3' y+4"7'
Clear the equations of fractions ; thus we obtain
av-2^= -6 (1),
7a?-4y= IG (2).
Multiply (l) by 2, and subtract it from (2) ; thus
7a?-4y-6a?+4y=16 + 12 ;
that is, ^=28.
Substitute the value of x in (1) ; thus
84-2y=s-6;
therefore 2^=90 ; therefore ^=45. •
28 '
Hence the required fraction is 7^.
221. A sum of money was divided eaually among ia
certain number of persons ; if there had oeen six more,
each would have received two shillings less than he did ;
and if there had been three fewer, each jv^ould have re-
ceived two shillings more than he did : finclnthe number of
persons, and what each received.
(f,V:'', , --i^ya*.
PROBLEMS, 151
Let w denote the number of personi, and y the number
of shillings which each received. Then xy is the number of
shillings In the sum of money which is divided ; an(^ by
supposition,
(a?+6)(y-2)=ajy (1),
(4?-3)(y+2)=«y (2).
From (1) we obtain
flfy + 6y - 2ii? - 12 = «y ;
therefore 6y-2a7=:12 (3).
From (2) we obtain
«y + 2aT-3y-6 -ary ;
therefore 2x~Zy=^ (4).
From (3) and (4), by addition, 3y = 18 ; therefort y^^
Substitute the value oty in (4) ; thus
2^-18=6;
therefore 2x^2^ ; therefore ;i;=12.
Thus there were 12 persons, and each recdved 6
shillings.
222. A certain number of two' digits is equal to five
tunes the sum of its digits ; and if nine be added to the
number the digits are reversed : find the niunber.
Let X denote the digit in the tevi :' riace, and y the digit
In the units' place. Then the numuci is 104? +tf; and,Dy
supposition, the number is equal to five times the sum of
its digits; therefore
10a?+y=6(aj '-:/).., (1).
If nine bo added to the number its digits are reversed,
that is, we obtain the number \(^ + x ; therefore
10a?+y+9=10y+i» (2).
From (1) we obtain
6x=Ay (3).
From (2) we Obtain 9^+9 = 9y ; therefore « + 1 =y.
152 PROBLEMS.
Babstitate for ^ in (3) ; thns
therefore ^=4.
Then from (3) we obtain y- 5.
Hence the required number is 45.
223. A railway train after travelling an hour is detained
24 minutes, after which it proceeds at six-fifths of its
former rate, and arrives 16 mmutes late. If the detention
had talcen ^lace 5 miles further on, the train would have
arrived 2 mmutes later than it did. Find tiie original iftte
of the train, and the distance travelled. \
Let 6^ denote the number of miles per hour at which
the train originally travelled, and let y denote the number
of miles in the whole distance travelled. Then y—bx will
denote the number of miles which remain to be travelled
after the detention. At the original rate of the train this
distance would be travelled in ^ — hours; at the in->
/ bx '
creased rate it will be travelled in
\.^ y-6x
eof
hours. Since
the train is detained 24 minutes, and yet is only 16 minutes
late at its arrival, it follows that the remainder of the
joumev is performed in 9 minutes less than it would have
been if the rate had not been increased. And 9 minutes
9
is --of an hour; therefore
y—Sx _ y—5x 9^
ex " bx "60
a).
If the detention had taken place 6 miles further on,
there would hbve been y-bx—b miles left to be travelled.
Thus we shall find that
\
y—bx—by—bx—b 2_
ex ■" bx ~60
^(2).
PROBLEMS. 153
Sabtaract (2) from (1) ; thus
6«""6a? 60'
therefore 50 =80 -2a;;
Uierefore 2a; = 10 ; therefore a; = 5.
Snbstitate this yalue of a; in (i), and it will be found by
Bolying tiie equation that ^=47^.
224. Af Bf and O can together perform a piece of
work in 30 days; A and B can together perform it in 32
days; and B and O can together perform it in 120 days:
find the tune in which each alone could perform the work.
Let a denote the number of days in which A alone
could perform it, y the number of days in which B alone
could perform it, z the number of days in which alone
could perform it. Then we haye
a^y z 30 ^*^'
111 . .
5'*'y""32 ^^^*
y"^i"l20 (^^•
Subtract (2) from (1); thus
z 30 32 ""480*
Subtract (3) from (1) ; thus
1^2 1^1
5 30"l20 40'
Therefore a; =40, and ;v=480; and by substitution in
any of the giyen equations we shall find that y= 160.
225. We may obsenre that a problem may often be
Bolyed in yarious ways, and with the aid of more or fewer
letters to represent the unknown quantities. Thus, to
take a yery simple example, suppose we haye to find two
154
EXAMPLES. XXV,
numbers snch that one is two^hirds of the other, and ^ehr
sum is 100.
Wo may proceed thus. Let x denote the greater
number, and y the less number; then we have
2a?
y=g-, «+y=100.
Or we may proceed thus. Let x denote the greater
number, then 100— a? will denote the less number; there-
fore
100-a?=j.
Or we may proceed thus. Let Zx denote the greater
number, then 2x will denote the less number; therefore >
2.1?+ 3a? =100.
By completing any of these processes we shall find that
the required numbers are 60 and 40.
The student may accordingly find that he can soke
some of the examples at the end of the present Chapter,
with the aid of only one letter to denote an unknown quan-
tity; and, on the other hand, some of the examples at the
end of Chapter xxn. may appear to him most naturally
solyed with the aid of two letters. As a general rule it
may be stated that the employment of a larger number of
unknown quantities renders the work longer, but at the
same time allows the successive steps to m more readily
followed; and thus is more suitable for beginners.
The beginner will find it a good exercise to solve the
example given in Art. 204 with the aid of four letters to
represent the four unknown quantities Which are required.
Examples. XXY,
1. If A*% money were increased by 36 shillings he would
have ^hree times as much as ^; and if ^s money were
diminished by 5 shillings he would have half as much as
A : find the sum possessed by each.
2. Find two numbers such that the first with half the
second may make 20, and also that the second with a third
of the first may make 20. ^
EXAMPLES. XXV.
155
3. If If were to gite ;£25 to A they would have equal
toma of 9(ioney; if A were to give £22 to ^ the monev
of B womd be double that of A : find the money whicn
each actoally has.
4 Find two numbers such that half the first with a
third of the second may make 32, and that a fourth of the
first with a fifth of the second may make 18.
5. A person buys 8 lbs. of tea and 3 lbs. of susptr for
£1, 29, ; and at another time he buys 5 lbs. of tea and 4 lbs.
of sugar for 15«. 2d, : find the price of tea and sugar per lb.
6. Seyen years ago A was three times as old as ^
was; and seven years hence A will be twice as old as J?
will be : find their present ages.
7. Find the fraction which becomos equal to i when
the numerator is increased by 1, and equal to i when the
denominator is increased by 1.
8. A certain fishing rod consists of two parts; the
length of the upper part is to the length of the lower as
6 to 7 ; and 9 times the upper part together with 13 times
the lower part exceed 11 times the whole rod by 36 inches:
find the lengths of the two parts.
9. A person spends half-a-crown in apples and pears,
buying the apples at 4 a penny, and the pears at 5. a
penny; he sells half his apples and one-third of his pears
for 13 pence, which was the price at which he bought them:
find how many apples and how many pears he bought.
10. A wine merchant has two sorts of wine, a better
and a worse; if he mixes them in the proportion of two
quarts of the better sort with three of the worse, the
mixture will be worth U. 9d, a quart ; but if he mixes them
in the proportion of seven quarts of the better sort with
eight of the worse, the mixture will be worth Is. lOd. a
quart : find the price of a quart of each sort
11. A farmer sold to one person 30 bushels of wheat,
and 40 bushels of barlev for ;£13. 10«. ; to another person
he sold 60 bushels of wheat and 30 bushels of Imrley.
for ;£17 : find the price of wheat and barley per bushel.
.rf,?'-fla^
..*
156
EXAMPLES. XX K
12. A fEurmer has 28 bushels of barley afc Sir. 4di i
bushel: with these he wishes to mix ire at 8t. a bui^V
and wheat at 49. a budiel, so that the muture may consist
of 100 bushels, and be worth Ss, 4d. a bushel: find hxm
many bushels of rye and wheat he must take. "
13. A and B lay a wager of 10 shillmp; if ^ loses
he will have as much as B will then haye; if B loses he
will have half of what A will then have: find the money
of each.
14. If the numerator of a certain fraction be increased
b;^ 1, and the denominator be diminished by 1, the Talue
will be 1 ; if the numerator be increased by the denomi-
nator, and the denominator diminished by toe numerator,
the yalue will be 4: find the fraction.
15. A number of posts are placed at equal distanora
in a straight line. If to twice the number of them we add
the distance between two consecutiye posts, expressed in
fedt, thQ sum is 68. If from four times the distance be-
tween two consecutiye posts, expressed in feet, we subtract
half the number of posts, the remamder is 68. Find the
distance between the extreme posts.
16. A gentleman distributing money among some poor
men found that he wanted 10 shillings, in order to be
able to giye 5 shillings to each man ; therefore he giyes
to each man 4 shillings only, and finds that he has 5
shillings left: find the number of poor men and of
shillings.
17. A certain company in a tayem found, when they
came to pay their bill, that if there had been three more
persons to pay the same biU, they would haye paid one
shilling each less than they did ; and if there had been
two fewer persons thOj would haye paid one shilling each
more than they did : find the number of persons and the
number of shillings each paid.
18. There is a certain rectangnilar floor, such thai
if it hadi^een two feet broader, and three feet longer, it
would tS^ been sixty-four square feet larger; but if it
had]>e^J||Me feet broader, and two feet longer), it would
haye been slxtj-eight square feet larger : find tfie length
and breadth of tlie floor. 1 1 v
i llJl ^^iertain number of two digits is equal %^
EXAMPLES. XXV.
167
timeB the sum of its digits ; and if 18 be added to the
mnnber tiie digits are reversed : find the number.
20. Two digits which form a number change places *
on the addition of 9; and the siim of the two numMrs is
33 : find the digits.
*-- 21. When a certain number of two digits is doubled,
and increased by 36, the result is the same as if the numberl
had been reyersed, and doubled, and then diminished by^-
36 ; also the number itself exceeds four times the sum of
its d^ts by 3 : find the number.
22. Two passengers have together 5 cwi of liuw;age,
and are chained for the excess above the weight imowed
ffff. 2<f. and 9«. lOdf. respectively ; if the luegage had all
belonged to one of them he would have been chained
19«. 2a. : find how much luggage each passenger is allowed
without chai^ge.
23. A and J9 ran a race which lasted 6 minutes; B
had a start ef 20 yards ; but A ran 3 yards while B was
running 2, and won by 30 yards: find the length of the
course and the speed of each.
24. A and B have each a certain number of counters ;
A gives to J9 as many as B has already, and ^ returns
back again to ^ as many as A has left ; A gives to J? as
many as B has left, and B returns to ^ as many as A has
left; each of them has now sixteen counters: find how
many each had at first
> 25. A and B can together perform a certain work in
30 days; at the end of 18 days however B is called off
and A finishes it alone in 20 more days : find the time
in which each could perform the work alone.
^ 26. A, B, and Ccan drink a cask of beer in 15 days;
A and B together drink four-thirds of what C does ; and
C drinks twice as much as A : find the time in which each
alone could drink the cask of beer.
f. 27. A cistern holding 1200 gallons is filled by three
pipes A, By O together in 24 minutes. The pipe A requires
30 minutes more than (7 to fill the cistern; and 10 gallons
less run througii C per minute than through A and B
tcffiether. Find the time in which each pipe alone would
iilTthe ci !«m.
mmmm
mni^^mmm
mmmmmmm
158
EXAMPLES. XXV.
28. A and B nm a mile. At the first heat A gives B
a stu^ of 20 yards, and beats him by 30 seconds. At the
second heat A gives B a start of 32 seconds, and beats him
by 9^ yards. Find the rate per hour at which A rnna
29. A and B are two towns situated 24 miles apart,
oiw the same bank of a river. A man goes from 4 to ^
in 7 hours, by rowing the first half of the distance, and
walking the second half. In returning he walks the first
half at three-fourths of his former rate, but the stream
being with him he rows at double his rate in going ; and
he accomplishes the whole distance in 6 hours. Find his
rates of walking and rowing.
30. A railway train after travelling an hour is detftined
16 minutes, after which it proceeds at three-fomrths of itp
former rate, and arrives 24 minutes late. If the detentio^
had taken place 5 miles further on, the train would have
arrived 3 minutes sooner than it did. Find the original
rate of the train and the distance travelled.
31. The time which an express train takes to travel
a journey of 120 miles is to that taken by an ordinarv train
as 9 is to 14. The ordinary train loses as much time in
stoppages as it would take to travel 20 miles without stop-
ping. The express train oulj loses half as much time m
stoppages as the ordinary train, and it also travels 16 miles
an nour quicker. Find the rate of each train.
32. Two trains, 92 feet long and 84 feet long respec-
tively, are moving with uniform velocities on parallel rails;
when they move in opposite directions they are observed
te pass each other in one second and a half; but when they
move in the same direction the faster train is observed to
pass the other in six seconds: find the rate at which each
train moves.
33. A railroad runs from A \^ C. A goods' train
starts from 4 at 12 o'clock, and a passenger train at 1
o'clock. After going two-thirds of the distance the goods* ,
train breaks down, and can only travel at three-fourths of
its former rate. At 40 minutes past 2 o'clock a collision
occurs, 10 miles from C, The rate of the passenger train
is double the diminished rate of the goods' frain. Find the
distance from A to C7, and tiie rates of the trains.
EXAMPLES. XXV,
159
34. A certain sum of money was divided between A,
E, and C, so that A^a share exceeded four-sevenths of the
shiures of B and C by ^£30 ; also ^'s share exceeded three-
eighths of the shares of A and C by £30; and C'b share
exceeded two-ninths of the shares of A and E by ;£30.
Find the share of each person.
35. A and E working together can earn 40 shillings
in 6 days; A and C together can earn 54 shillings in 9
da3rs; and E and C together can earn 80 shillings in 15
days: find what each man can earn alone per day.
36. A certain number of sovereigns, shillings, and six-
pences amount to £8. 6s. 6d. The amount of vthe shillings
IS a guinea less than that of the sovereigns, and a guinea
and a half more than that of the sixpences. Find the
number of each coin.
<^ 37. A and E can perform a piece of work together in
48 days ; A and C in 30 days ; and E and C in 26j days:
find the time in which each could perform the work alone.
38. Thiere is a certain number of three digits which is
equal to 48 times the sum of its digits, and if 198 be sub-
tracted from the number the digits will be reversed; also
the sum of the extreme digits is equal to twice the middle
digit: find the number.
39. A man bought 10 bullocks, 120 sheep, and 46
lambs. The price of 3 sheep is equal to that of 5 lambs.
A bullock, a sheep, and a lamb together cost a number of
shillings greater by 300 than the whole number of animals
bought; and the whole sum spent was £468. 6«. Find the
price of a bullock, a sheep, and a lamb respectively.
40. A farmer sold at a market 100 head of stock con-
sisting of horses, oxen; and sheep, so that the whole realised
£2, Is. per head ; while a horse, an ox, and a sheep were
sold for £22, £12. 10«., and £1. 10«. respectively. Had he
sold one-fourth the number of oxen, and 25 more sheep
than he did, the amount received would have been still the
same. Find the number of horses^ oxen, and sheep, respec-
tifely which were sold.
ti
160 QUADRATIC EQUATIONS.
XXVL Quadratic Equationt.
226. A quadratic equation is an equation whicll con-
tains the tquare of the unknown quantity, but no hi^er
power.
227. A pun quadratic equation is one which contains
only the square ot the unknown quantity. An affected
quadratic equation is one which contains the first power
of the unknown quantity as well as its square. Thus, for
example, 2^*= 50 is a pure quadratic equation; and
8^- 7^+3=0 is an a^eeted quadratic equation. .
228. The following lis the Rule for solving a pure
quadratic equation. Find the value o/* the equare qf the
unknown quantity by the Mule /or sotving a simple equO'
Hon; Mm, by extracting the square root, the values (/the
quantity are found.
=6.
For example, solve — r— + -^jr- •
1 1 ' 3 10
Clear of fractions by multiplying by 30; thus
10(ii^-13)+3(««-6)=180;
therefore 13:1^=180+130+15=325;
therefore
^=—=25;
extract the square root, thus 4?= :k5.
In this example, we find by tiie Rule for solving a
ample equation, that a^ is equal to 25; therefore x must
be such It number, that if multiplied into itself the pro-
duct is 25. That is to say, x must be a square root of
25. In Arithmetic 5 is the square root of 25; in A^ebra
we may consider either 5 or -5 as a squareXroot of 25,
since, by the Rule cf Signs — 5x — 5a5x5. Hence x
majr nave either of the values 5 or —5, and the equation
will be satisfied. This we denote thus, d;« dbfi.
QUADRATIC EQUATIONS.
161
ties.
289. We proceed to the solution of adfected quadnir
If we multiply ^+ r by itself we obtain
SB
(«+|)(;r+|)
ax
r««+2 2- + ~=««+a»+j.;
thus d^+av+^ is a perfect square, for it is the square
of « + f. Hence a^+a« is rendered a perfect square
by the addition of 7-, that is, hy the addition qfthe square
(fha\f the co^cient qf ao. This fact is the essential part
of the solution of an adfected quadbratic equation, and we
shall now give some examples of it.
oi^-¥Qx\ here half the coefficient of or is 3; add 3', and
we obtain ^ + 6;v + 3^ that is {jxs-\-2if.
a^-Sa; here half the coefficient of 4; is ~o; &dd
(-rj, that is (5), and we obtain x^^Sx+y^, that
to (*-g'.
4x 2 ' /2\s
«a+y ; here half the coefficient of a? is g; add m ,
and we obtain ^+ x + (5) » *^** ^ (^'*'ft J •
a^—-j-; here half the coefficient of a is —3; add
( - |Y, that is f lY, and we obtain x^-^ + (l)\ that
A
A
is (a.~iy.
The process here exemplified is called completing the
square,
T.A. 11
162
QUADRATIC EQUATIONS.
i
230. The following is the Rule for solving ftn adfeoted
quadratic equation. By tran$po$ition and reduction
arrange the equation eo that the terme which involve the
unknown quantity are alone on one tide, and the coifficieni
qf a^i$ + 1 ; €uid to each tide cf the equation the equare
of hd^f the coefficient qf a, and then extract the equare
root cfeach tide.
It will be aeen from the examples which we shall now
solye that the above rule leads us to a point from which
we can immediately obtain the values of the unknown
quantity.
231. Solve ^~10ar+24=0.
By transposition, ii^-^l0x=-24', i
add(^Y, «2«iQ^+5a=:«24+25=lj
extract the square root^ a-'5= Jbi •
transpose^ ^=5akl=5 + l or5-l;
hence a-B or 4.
It is easv to verify that either of these values satisfies
the proposea equation; and it will be useful for the stu-
dent thus to verify his results.
232. Solve ai!*-4»-65=0.
By transposition, 3^-4^=55;
divide by 3,
. 4a? 66
3 3
•**(IX
3*\3/ 399'
2 13
extract the square root, «- ; = ±-5- ;
.WW \
trmupose,
2 13 . 11
k.
QUADRATIC EQUATIONS
233. Scire 2^4-3«-35»0.
By transpoBitioii, 2j*+ 3drs 35;
divide by 2,
2 2 '
-"©■
^ 2 W 2 16 16 *
3" )'7
extract the square root» ^"^1^*^*
transpose^
8^17 7
4 4 2
163
234. Solre «»-4«-l = 0.
By transpositioii, ai^-4a=l;
add 2*, «'-4^+2'=l + 4=5;
extract the square root, a-2=^ ^ J5;
transpose, a^2^ J5.
Here the sanare root of 5 cannot be found exactljr;
but we can find dy Arithmetic an approximate value of it
to any assigned degree of accuracy, and thus obtain the
values of ;i; to any assigned degree of accuracy.
235. In the examples hitherto solved we have found
two different roots of a quadratic equation; in some cases
however we shall find really only one root. Take, for ex-
ample, the equation a^-> 14^+49=0; by extracting the
square root we have a? - 7 = 0, therefore a? = 7. It is how-
ever found convenient in such a case to say that the quad-
ratic equation has two equal rooti,
11—2
164 QUADRATIC EQUATR r:!^
236. SolTe «>-r6df-l-13-6.
By tramposiUon, d^-6^*-13;
adds*, «»-6ar+3'--13+9--4. *
If we try t6 extract the square root we hare
But —4 can have no square root, exact or approximate,
because any number, whetner positiye or negatire, if mul-
tiplied by itself, giyes a positiye result In this case the
quadratic equation has no real root; and this is sometimes
expressed by saying that the roots are imoffinary or
imposiible.
237. Solve
2(i»-l) ««-l
1
i'
\
Here we first clear of fractions by multiplying by
4(^'-l), which is the least common multiple of the de-
nominators.
Thus 2(a?+l)+ 12=^-1.
By transposition, al* —Zof ^15;
addl« «2-.2a?+l = 16+l = lC;
extract the square root, «—l= :b 4;
therefore . ^asldB4=5or -3.
238. Solve ??+3^-«<>
12a?+70
190
16 3(10+a?)
Multiply by 670, which is the least common multiple of
16 a^d 190; thus
therefore
therefore
190(3a?-60)
=210-400?;
10+w
190(.^»-60)-(210-40«)(10+i)J
QUADRA TIC EQ UA TIONS.
that if, 070^-9500 =2100- 190^-40^;
therefore AOafl + 760a; » 1 1600 ;
therefore «* + 19« = 290 ;
19 S9
extract the square root> a+^^^^^i
2 2
165
therefore
^=:-^-.f=10or-29.
239. Solye
/g+3 fl?-3 _ 2^-3
a+2 «-2"" «-l '
Clear of fractions; thus
, (aT+3)(i»-2)(«-l)+(i»-3)(«+2)(«-l)
«=(2«-3)(a;+2)(aT-2);
that is, a^-7a?+6+«'--2«*-5«+6=2«'-3a'-a»+12;
that is, 2aj»-2iij3-12i»+12=2«"-3aj«-&»+12;
therefore «*— 4iir=0j
add2«, «8-4a?+2«=:4;
extract the square root, A;-2=:sfc2,
therefore a; = 2 it 2 = 4 or 0.
We hare given the last three lines in order to com-
plete the solution of the equation in the same manner as
m the former examples; but the results may be obtained
more simply. For tne e€[uation d^— 4a;= may be written
(a;— 4)ar=0; and in this form it is sufficiently obvious
that we must have either or— 4=0, or «=0, that is,
«=4 or 0..
The student will observe that in this example 2a^ is
found on both sides of the equation, after we have cleared
of fractions; acoordingly it can be removed by subtraction^
and so the equation remains a quadratic equation.
"■■■^ ■' '';.dBti>v.-.^."tf;.,-.j',;,/
166
Q UAD RA TIC EQ UA TIONS.
240. Everp quadratic equation can be put in the
form x^+px+ q«0, lehere p and <| represent tome known
numbers, whoie or fraetional, positive or negative.
For a quadratic equation, by definition, coiitains no
Eswer of the unknown quantity higher than the second,
et all the terms be brought to one side, and, if necessary,
change the signs of all the terms so that the coefficient of
the square of the unknown quantity may be a positive
number; then divide every term by this coefficient^ and
the equation takes the assigned form.
For example, suppose 7x—4a^=5, Here we have
7a?-4a?*— 6=0;
therefore 4a^-1x+5=0i
i
therefore
^ 7a? 6 ^
^ Ik
Thus in this example we have p= -^ ^'^^ ^=i*
J. 241. Solve
' By transposition,
a^-^-px+q^O,
a^-¥px=-q\
add ri^'
^.;>-(fy=-..?=^;
extract the square root, « + § = * v^^"* ^f .
2
2
thei^fore ^^^t^tMz^^ZP^^!^^:!^ ,
2 2 2
242. We have thus obtained a general formula for
the roots of the quadratic equation ;r+j7;r+g=0, namely,
that X must be equal to
-y-f-N/(j>«-4g) _^^ -j>-^(p«-4g )
. orto ^ .
\
We shall UvW deduce from this general formula some
▼er^ important inferences, which wiu hold for any quad-
ratic equation, by Art. 240.
qUADBATIC EQUATIONS.
167
243. A quadratic equation cannot have more than
tteo roots.
For we have seen that the root must be one or tho
other of two assigned expressions.
244. In a quadratic equation where the terms are
all on one side, and the coefficient of the square of the
unknown quantity is unity, the sum of the roots is equal
to the coefficient qf the second term with its sign changed^
and the product qfthe roots is equal to the last term.
For let the equation be «* -{-px + g = ;
the sum of the roots is
-P*^f-*^) ^ -p-^f-*^) , thati. -p;
SI J»
the product of the roots is
that is
p"-(p*-4</)
that is q.
245. The preceding Article deserves special attention,
for it furnishes a very good example both of the nature of
the general results of Algebra, and of the methods by
which these general results are obtained. The student
should verier these results in the case of the quadratic
equations already solved. Take, for example, that in
Art 232; the equation may be put in the form
- 4« 56 ^
and the roots are 5 and — -^ ; thus the sum of the roots Is
o
4 55
r , and the product of the roots is - -x-*
>^
168 QUADRATIC EQUATIONS.
246. Solve aa^-{l'X^C'mO,
By transposition, aa^+hx^^-ci
divide by a,
a a
extract the square root, a; + r- = db ^^ — '- ;
therefore
2a
247. The general formulsu given in Arts. 241 and 246
may be employed in solving anv quadratic equation. Take
for example the equation S^c'*— 4a?— 55=0; divide by 3,
thus we have
■^ 4a? 55 ^
Take the formula in Art. 24 i, which gives the roots of
A. f\f\
a^+px+q=0; and put p=-^) and q=—-^; we shall
thus obtain the roots of the prciy>«ed equation.
But it is more convenient to use the formula in Art. 246,
as we thus avoid fractions. The proposed equation being
3aj2-4a?-55=0, we must put a=3, d=— 4, and<J=— 5r,
in the formula which gives the roots of aa^ + hx H j = 0,
that is, in
-^=fax/(&g-4ac)
2a
Th^ ^e have i±^M±«««), that U, *±fm,
6
thatiSy
4J^26
6 •
that iff, 5 or —
U
1
EXAMPLES. XXVL
169
Examples.
1. 2(d^-7)+3(a!»~ll)=33.
5.
5 • 4
k 4 1
6.
a?-3 fl?+3 3'
7; «8-3d?+2=0. 8.
9. «8+10aj=24. 10.
11. 3a?»-4»=39. 12.
13. (4?+l)(2aJ+3)=4^-22.
16. 4(aj«-l) = 4a?-^l.
17. 8aJ*- 17a? +10=0.
XXVL
2. («-16)(a?+16)=40a
3(^-11) 2(a?'-60) _
5 y— -36.
^ 4_£ 9
4 af~9 «?•
aj'~6a? + 6=0.
2^2_i = 6;j. + 2.
;»^ + 10a? + 3 = 20^2 _ 5^ ^. 53^
14. (il?-l)(a?-2) = a
16. (2;»-3)2=8a
18 5.-^-2
19. fl?=2 +
4a?*
2+^ x—afl
£1. ^^-^^«l-a?+a;8.
20. ic2-3=
22. a?+
^-3
a?-3
6
«5.
/
<«« .• 12-a? «^
23. 4a?- -^^=22.
a?— 1
25. — s + 2a?=12.
27. 8a? + ll + - = ^.
X 7
00 2 ,a?+3_10
"^^^ ^+3"^~2~'" 3'
31. J^ + £+?=2.
33.
fl?+2
ar
2^1?
so+l a?+4
-. = 1.
„. 2d?+ll - a?-6
SJ6. f + -^=6f
7 a?+5 ^
28 ^+2 a?-2_13
a?-2 a?+2 6'
30..^M-5M„5.
«+l a?-l
32 .-iL_.^+l_13
•JA ^ + 2 ^+1 _ 13
'**• a?+l"*'a?+2" 6'
170
EXAMPLES, XXVI.
35.
37.
39.
41.
43.
45.
47.
49.
5l!
53.
55.
57.
59.
61.
63.
47+1
ay~2 9
"5*
^~2 a?— 4
«-3 a?— 1
«-l 0?— 3
36. ^ + ^-7.
14
16'
11
ar— 4
3
«-2 12*
1
2(««-l) 4(;»+l) 8'
2x+l 3a?-2 _ll
«-l 3a:+2 2 '
3a? -fl 2a?~7 5_
3(«-5)"2«-8"'2'"
3jp--2 2af-5 10
2«-5'*' 347-2" 3 •
(4?-3)*=2(««-9).
5 3 14
x+2 at
«+4*
OH- 1 a— I 2a?- 1
a?+2 «-2 «— 1
2
38.
40.
47-4 47-2
47-3 _ a7-l
47-2 47-4'
1 2
6
'5*
47-2'"47 + 2 5*
47 16-74?
42.
4J«-1 8(1-47)'
^^ 247-1 247-3 1 ^
44. r 5-+r = 0.
47-1 47-2 6
AR 2^-3 347-5 5^'
347-5 247-3'" 2*
47+2 4-47 7
48.
47-1 247 3*
50. (47+ 10)2=» 144(100-47*).
4.5 12
47-1 5 ^
4f+l" 6 *" 7(47-1)*
47-1 47-2 _ 247+13
47+1 47 + 2"" 47 + 16'
247-1 347-1 5^-11
47+1 47 + 2 47-1
a*47»- 2a"47 + a* - 1 « 0.
47 a 47 ft
a 47 & 47
52.
54.
56.
47 + 1 47+2
47 + 3*
47-2 47 + 2 47+3
47 + 2 47-2"" 47-3'
4.5 12
47 + 2 47 + 4 47 + 6'
^. 47 + 1 . 47 + 2 247+13
Oo , r + = -^ r— .
47-1 47-2 47 + 1
60. ;.-ii^r»„^z3.
847-3 47+1'
62. 4a«47=(a^-6»+4?)3.
a 47+&
a «+&'
EQUATIONS LIKE qUADBATICH 171
XXVII. Equativni which may he r'^ved like
Quadratics,
248. There are many equations which are not ttrictly
quadratics, but which may be solved by the method of eoni^
pleting the square; we will give two examples.
249. Solve ^•-7ic»=8.
7 9
extract the square root, «•- 5= *« »
«'=|*|=8or-l;
therefore
extract the cube root, thus «=2 or ~ 1.
260. Solve ««+3«+3/v^(«'+3a?-2)=6.
Subtract 2 from both sides, thus
«2+3^_2 + 3^/(«*^"3a?-2)=4.
Thus on the left-hand side wo have two expressions,
namely, ij(a^ + S^v — 2) and ai*+3x- 2, and the latter is the
square of the former; we can now complete the equare.
Add l-j , thus
«» + 3«-2 + 3 ^/(««+3a?-2) + Q^ =4+^ = ^;
extract the square root, thus
therefore V(^+3a?-2)=-2±2»l or -4.
1 1
172 EQUATIONS LIKE QUADRATICS,
First suppose /^(dj2 + 3ii?-2)=l.
Square both sides, thus o;^ + 3^ - 2 = 1.
This is an ordinary quadratic equation; by solving it
we shall obtam x=
-3db s/21
2
Next suppose ^/(«2+3a?-2)=-4.
Square both sides, thus ^+ 3a;- 2 =16.
This is an ordinary quadratic equation; by solving it
we shall obtain ^ = 3 or — 6.
Thus on the whole we have four values for ^f namely,
-3db J21 I
2 • ^
3 or -6 or
An important observation must be made with respect
to these values. Suppose we proceed to verify tnem.
If we put a? =3 we find that a;2+3a?— 2=16, and thus
J {a? + 3a? ~ 2) = tfc 4. If we take the value + 4 the original
e(}uation will not be satisfied; if we take the value —4 it
will be satisfied. If we put a?= — 6 we arrive at the same
result And the result might have been anticipated,
because the values x = Z or — 6 were obtained from
is/(a?2+3ii?- 2)=r-4, which was deduced from the original
— 3 ± /21
equation. If we put ;»= —■ — we find that
oc^-\-Zx—2—\, and the original equation will be satisfied
if v/e take is/(a?2+3^— 2)= +1; and, as before, the result
might have been aD.ticipated.
In fact we shall find that we arrive at the same four
values of ^, by solving either of the following equations,
aj2+3;i?-3V(a;2+3a.-2) = 6,
ic24. 3-1.4. 3 ^(^+ 3^_2) = 6;
but the values 3 or -6 belong strictly only to the first
_3± ^21
equation, and the values ~ — belong strictly only to
the second equation.
?^:£«hw<'. .-i^- ••'^ii
mVATIONS LIKE QUADRATICS. 173
V
ing it
dng it
lamely,
respect
them,
d thus
)ridnal
I -4 it
e same
ipated,
1 from
)riginal
that
itisfied
result
le four
)ns.
le first
mly to
251. Equations may be proposed which will requiro
the. operations of transposing and squaring to be per-
formed, once or oftener, before they are reduced to quad-
ratics ; we will give two examples.
252. Solve
Transpose,
square,
transposo, .
divide by 3,
2j?-;y(aj*-3a?-3)=9,
2aj-9=V(aj3-3a?-3);
4aj2 - 36a? + 81 « a^ - 3a?-« 3 ;
30)2- 33a? +84=0;
a;2«ii^+28=0.
By solving this quadratic we shall obtain a? =7 or 4.
The valjae 7 satisfies the original equation; the value 4
belongs strictly to the equation 2a? + v (aj^ - 3a? - 3) = d.
253. Solve ^/(a?+4)+^/(2a?+6)=^/(8a?+9).-
Square, a?+4+2a?+6 + 2V(a?+4}^(2a?+6)=8a?+9;
transpose, 2.y(a?+4),y(2a?+6)=5a?-l;
square, 4(a? + 4)(2a? + 6) = 26aj2 ~ 10a? + 1 ;
that is, 8a?'+66a?+96=25a!2-l0a?+l;
transpose, 17a?' -66a? -95=0.
By solving this quadratic we shall obtain a?=!:5 or —
19
17*
The value 5 satisfies the original equation; the value
19
-r^ belongs strictly to the equation
V(2a? + 6) - ^/(a? + 4) = V(8af + 9).
254. The student will see from the preceding examples
that in cases in which we have to square in order to re-
duce an equation to the ordinary form, we cannot be
certain without trial that the values finally obtained for
the unknown quantity belong strictly to the original
equation.
174 EQUATIONS LIKE QUADRATICS.
265. Equations are lometimes propoBod whieh ara
intended to be lolyed, partly bj inapeouon, and partly by
ordinary methods ; we will give two examples.
256. Bolye
x^^ «+4""9-«"9+«*
Bring the fractions on each side of the equation to a
common denominator ; thus
1?^ ^ 81-«» ^
that is,
\^
36«
««-16 81-«**
t
Here it is obyions that ^=0 is a root To find the
other roots we begin by diyiding both sides of the equsr
tion by 4v; thus
4 9
«»-16""81-«*'
the]:efore
4(8l-««)=9(««-16)j
therefore
130^=324+ 144::=468;
therefore
«*=36;
therefore
«=tfc6.
Thus there are three roots of the proposed equationi
namely, 0,6, -6.
*
257. SoWe fl^-7a?a'+6a*=0.
Here it is obvious that 4;= a is a root. We may
write the equation a^-a^=s7a*{w-a); and to find the
other roots we begin by diyiding hj a-a. Thus
By solving this quadratic we shall obtain 4;= 2a or -3a.
Thus there are tlffee roots of the proposed equation,
namely, 0,20, -3a.
. ^^
h tre
rtlyby
m toa
nd the
luatioiii
EXAMPLES, XXVIL
175
EXAMPL1& XXVIL
1. 4?*-13aj'+36=0. 2. «-5>/«-W=0.
5. 2V(^-2»+l)+«*=23 + ap.
6. »*-2aj»+«^=86. 7. V(^-8«+16)+(«-3)"=ia.
8. 9y(«^-94^+28)+9«=«»+36.
9. 2aj»+6iP=226-V(«'+3aT-8).
10, fl?*-4aj8-2>/(«*-4aj«+4)=31.
11. «+2Ay(^+&»+2)=ia
12.' 3«+Ay(«»+7i»+6)=19. 13. «=7V(2-«a).
14. ^/(«+9)=2^/a?-3. 15. ^/(^+8)- V(«+3)=Vj?.
16. 5>/(l-««)+6a?=7.
17. ^/(3«-3)+^/(6a?-19)=V(2«+8).
la V(2ar+l)+^(7i»-27)=N/(3«+4).
19. ^/(ft"+aa?)-^/(aH6«)=a+6.
20.
21.
23.
24.
25.
26.
2a?^/(a + «*) + 2a?» = a' - A
£+_5/(12fl^-£) __ a +1
ii?-^/(12a^--ip)""a-l'
a?+7 fl?-l «+l a;-7
22.
= 0.
34?
1-4? 1+4? 1+4J*'
= 4?.
4?-V(«8-l) 4? + V(««-l)
^ =8^/(45*-l).
fl?+a 4?-a &+4? &-
4?
4?-a 4?+a d-4? t+4?"
27. «»+3a4J*=4a^.
28. 5««(a-4f)=(a«-4;^(4?+3«).
176
PROBLEMS.
XXYIII. Problems which lead to Quadratic
EquatioiM,
258. Find two numbers such that their sum 'is 15,
and their product is 54.
Let X denote one of the numbers, then 15i— a; will
denote the oUier number; and by supposition
a; (16— a?) =54.
By transposition, . «2-.l5;i?=-.54j
/15V -. 225 9
therefore
«»-15a?+|
54+^^ =
4'
therefore
therefore
15 3
a?=^*-=9or6.
If we take ^=9 we have l^-x^Q, and if we take
d;= 6 wo have 1 5 —47= 9. Thus the two numbers are 6 and 9.
Here although the quadratic equation gives two values of
OB^ yet there is really only one solution of the problem.
259. A person laid out a certain sum of money in
goods, which he sold again for ^£24, and lost as much per
cent, as he laid out : find how much he laid out.
Let X denote the number of pounds which he laid out ;
then x^24. will denote the number of pounds which he
lost. Now by supposition he lost at the rate of x per cent.,
X
that is the loss was the fraction y^ of the cost ; therefore
X
therefore
^^100=^^24;
a;2-i00a?=-2400.
From this quadratic equation we shall obtain ^=40
or 60. Thus all we can infer is that the sum of money laid
out was either MO or ;£60; for each of these numbers
satisfies all the oonditions of the problem.
PROBLEMS.
177
is 15,
'X will
. \
76 take
|6 and 9.
ktlues of
m.
mey in
uchper
d out ;
hich he
)r cent.,
^erei^ore
260. The tnm of £*J. 4m, web divided equally among^
a certain number of persons ; if there had been two fewer
persons, each would have received one shilling more : find
the number of persons.
Let X denote the number of persons; then each person
144
receired — shillings. If there had been x-2 persons
each would have receiyed
supposition,
144
x-2
144
x-2
144
shillings Therefore, by
X
Therefore 1444>=144(j?-2)+«(4f-2) ;
therefore «'-247=288.
From this quadratic equation we shall obtain xts\^
or — 16. Thus the number of persons must be 18, for that
is the only number which satisfies the conditions of the
problem. The student will naturally ask whether any
meaning can be given to the other result, namely —16,
and in order to answer this question we shall take another
problem dosejy connected with that which we have here
solved.
261. The sum of £*J. it, was divided equally among a
certain number of persons ; if there had been two more
persons, each would have received one shilling let9 : find
the number of persons.
Let X denote the number of persons. Then proceeding
as before we shall obtain the equation
144
144 ^
«2+2d?=288;
«=16 or -18.
therefore
therefore
Thus in the former problem we obtained an applicable
result, namely 18, and an inapplicable result, namely — 16 ;
and in the present problem we obtain an applicable result,
namely 16, and an inapplicable result, namely —18.
T.A.
la
IMAGE EVALUATION
TEST TARGET (MT-3)
1.0
1.1
J25
m
ttt Uii 122
11.25 IHU
I
1.6
^
^
7
'/
Fhotogra{^
Sdaioes
Corporation
33 WIST MAIN STMIST
WnSTM.N.Y. )4SM
(71«)t72.4S03
Is
;\
178
PROBLEMS.
262. In Bolvmg problems it is often fomid, as in Aii 260,
that remilti are olraamed which do not api^y to the poUent
actaally proposed. The reason appears to be^ that the
algebraical mode of expression is more general than oidir*
nary language, and thus the equation whidi is a oroper
represen&tion of the conditions of the prooiem wul juso
apply to oUier conditions. . Experience will cdnvmce the
suident that he will always be able to select the result
which belongs to the problem he is solving. Jind it will be
often possil^, by suitable changes in the enunciation of the
original problem, to form a new problem corresponding to
any result which was inapplicable to the orighial problem ;
this is illustrated m Article 261, and we will now give ano-
ther example.
263. Find the price of eggs per score, when ten moiie
in half a crown's worth lowers the price threepence p^
score. ■ ^
Let X denote the number of pence in the price of a
score of eggs, then each egg costs ^ pence ; and therefore
the number of eggs which can be bought for half a crown
It AAA
is 30 -7-^, that is -xr* ^ ^® P^^^^ '^^^ threepence
20'
X
x-Z
per score less, eadi egg would cost -^ pence^ and the
number of eggs which could be t>ought for half a crown
would be -^-:l . Therefore, by supposition,
«-3*
therefore
therefore
X-Z X +^"'
60;»=60(«-3)+a?(«-3) ;
ai*-3a?=180.
From this quadratic equation we shall obtain «=15
or ~ 12. Hence the price required is \M, per score. It
will be found that l^d. is the result of the following pro-
blem; find tiie price of eg^ per score whra tp fewer
in half a crown's worth roMti the price threepence per
score. . _
/\
<»»-
EXAMPLES. XXrilL
179
proUMtt
Juit the
AH ordir<
k proper
ince the
le result
twill be
m of the
ndingto
>roblem;
^veano-
»n more
ence pd^
rice of a
tJierefore
a crown
reepence
and the
a crown
■$">
n «=15
core. It
inngpro-
n fewer
ence per
/\
Ktamplbr XXYIIL
1. IMyide the nnmber 60 into two parti Buch thi^
their prodnot may be 864.
2. The Bom of two nnmbers is 60, and the sum of
their squares is 1872 : find the numbers.
3^ Tbib difference of two numbers is 6, and theur pro-
duct is 720 : find the numbers.
4 Find three numbers such that the second shall be
two-thirds of the first, and the third half of the first ; and
that the sum of the squares of the numbers shall be M9.
5. The difference of two numbers is 2, and the sum of
thdr squares is 244 : find the numbers.
. 6. IHvide the number 10 into two parts such that
their product added to the sum of their squares may maJie
76.
7. Find the number which added to its square root
willmidLe2ia.
8. One number is 16 times another; and the product
of the numbers is 144: find the numbers.
9. One hundred and ten bushels of coals were divided
ainong a Certain number of poor personp; if each jperson
had received one bushel more he would have received as
many bushels as there were persons: find the number
of persons.
10. A company dining together at an inn find their
bill amounts to £8. 15f. ; two of them were not allowed to
pay, and the rest found that their shares amounted to 10
shillings a man more than if all had paid : find the number
of men in the company.
11. A dstem can be supplied with water by two
I^pes; by one of them it would be filled 6 hours sooner
than by the other, and bj both together in 4 hours: find
the time in whidi each pipe alone would fill it
12—2
180
BXAMPLBS. xxrm.
'I '
12. A jMnoD booffhi * 6«rft«lii nombtr of pfiOii of
c^loth for £SB, I6i,, which ho fold afiln st d% $§, por ploooi
iiid ho galnod Miniioh in tho wholo ii ft liiii^ piooo ooii}
And tho numbor of piocoi of oloth.
18. A ftnd 3 iogoihor cm porform * ploof of work In
14} dayt : and A alono Cftn porform » In 18 dftjo loot
than JSr ilono: And tho tlmo m which A alono can par*
fomiit*
. 14. A man bought a certain quantify of moat to
IB fhlUingf. If moat woro to rlfo In jnto ono penny
por lb.| ho would got 8 Ibi. loM fior tho Muno fum. find
now much moat ho bjought
10. Tho pri«90 of ono kind of fUffar nor atono of 141bf.
If 1#. 9d* moro than that of another kina ; and 6 Ibi. leu Of
tho flret kii»d can bo got for £1 than of tho aecond: find
the price of each kinaper itono.
16. A penon apent a certain fum of money In goodi,
which he wud again for £^4, and gained aa much por cent.
aa the goods coit him: find what the goode coat.
17. The iide of a sauare la 110 inches long; find the
lengtii and breadth of a rectangle which shall hare its
penmeter 4 inches longer than thM of tho square, and its
area 4 square inches less than that of tho square.
18. Find the price of eggs per dozen, when two less In
i^ shilling's worth raises tho price one penny per dozen.
19. Two messengers A and B were despatched at the
same time to a pU^ at the distance of 90 milea: the
former by riding one mile per hour more than the latter
arri?ed at the end of his Journey one hour before him: find
at what rate per hour each trareUed.
fiO. A person rents a certain number of acres of pas-
ture land for £10i he keeps 8 acres In his own possession,
and aublets the remainder at 6 shillings per acre more than
he gare^ and thus ho corers his rent and has £p orer :
find the number of acres.
^
roik in
yi Urn
la ptr-
Ulbf.
d: find
I goodie
nr cent
nd the
iftTd iti
and it*
leu In
at the
I! tllO
ktt«r
i: find
EXAMPLES. XXrill
151
Sl« Vmn two placM «i » diftanoe of 890 nfloi, two
poraona A and JS^ aot oat In ordor to moot oaob otbor,
A tfiraOod 6 nllaa a day more tban Bi and tho mmibM' of
di^a In whkli tlufjr mot waa otjoal to half tiio nnmber of
mdoi J9 want In a day. Find how te oach travalM bafofo
th^oiot
8S. A poiiondrowaaiiaatltjofwInoilromnftdlToaaal
wUch hold 81 gallona, ana thon flllod np tho yoaial with
water. He then drew from the mlitore aa mneh aa he
before drew of pure wine: and It waa found that 64 aidlona
of pure whie remained. Bind how mnch he drew ead time,
28. A certain company of a(ddleri can he formed Into
a iolld iqnare; a battalion contliUnff of aeren each eqnal
companlea can be formed Into a hoUow agoarOi tho men
bemg fonr deep. The hollow aqoaire formed bj the ba^
tallon la alzteen timea aa hu|;e aa the aoUd aouare formed
bjoneeompanj* Find the nnmber of men In the company.
24. There are three e^ reaaela A^ P, and {7; the
firat contalna water, the aeoond brandy, and the thfad
brandy and water. V the contenta al B md O be jmt
together, It la found that the fraction obtained by difldfaig
the quantity of brandy by the quantity ct water la nfaie
thnea aa gfetii aa If the contenta of A and O had been
treated in like manner. Find tlie proportion of brandy to
water in the reiad 0,
25. AperaonlendaifMOOatacertafairateoflntereat;
at the end of a year he reoelTea hla intercat^ ipenda £25 of
it and adda the remainder to hla ci^tal; be then lenda
hia capital at the aame rate of intereat aa before, and at
the end of another year finda that he haa altofether
^6882: determine the rate of intereat
ion,
than
over:
iSi SIMULTANEOUS EQUATIONS
XXDL Sitntdtaneoiu Eguaiion» inwivinff. QuadraHa,
264. We shall now solre Bome examplM of limnltme-
0118 equations involying quadratics. There are two oases
of frequent occurrence for which roles can be giyen; in
both iiese cases there are two unknown quantities and two
equations. The unknown quantities will always be den6ted
by the letters AT and y.
* ' - ■ • ,
265. First Can, Suppose that one of the equations
is of the first degree, and the other of the second degree.
Rule. From the equation qfthejlrat degree find ^
value qf either qf the unknown quantitiee in terme kf
the other, and eiAetitute thie value in the equoHon qf
the eeeond degree.
Example. Sdve 3^+4^=18, 5a^-3^=2.
From the first equation y- — ^ — ; substitute this
Tsilue in the second equation; therefore
4
therefore 204j»-54^+9«*=:8;
therefore 29«*-64aT=8.
4
From this quadratic equation we find x^2<ift -rr;
• 267
then by substituting in the value of y we find ^ =3 <^ -gg •
266. Solye 3«2+6^-8y=36, 2aj*-3«-4y«3.
Here although neither of the giyen equatiodb is of the
first degree, yet we can immediately deduce from them an
equlltion of tne first degree.
INVOLVING QUADRATICS;
18S
tdroHci,
Biiilteiie-
wo
[yen; in
and two
den6ted
qmUoBB
Legree.
find me
terms ^
ate
this
^ 29'
,267
8 of the
;hem an
For multiply the first equation by % and the second
by 8; thus
6a^+l(to-16y=72, 6«'-PaT-12y«:9;
therefore^ by subtraction, 104;-16y+9«+l^si72-9 ;
that is, 194^-4^=63.
From this equation we obtain y=s — -7 — ; substitute
this value in the first of the given eq]aations ; thus
3«*+&P-2(19a>-63)=36;
therefore 3a)'- 33d; +90=0;
therefore d^-lldr+30=0.
From this quadratic equation we shaH find that«=5
or 6; and then by substituting in the Value of y werfind
that y=8 or 12|,
267. Second Case, When the terms involving the un-
known quantities in each equation constitute an expressioa
which IS homogeneous and of the second degreis; see
Art. 23.
Rule. .Assume y=vz, and siibstUute in both e^[wi»
lions; then by division the value qfyean be found.
Example. Solve «'+a!y+2^=44, 2«'-ii?y+y»=16.
Assume y=«;p, and substitute for y; thus
y(l+»+2»^=44, d?8(2-t>+»>)=16.
Therefore^ by division,
l+ g-t-2g' _44_ll
therefore
therefore
therefore
2-»+»* 16
4(l+i>+2r!0=ll(2-»+i?^;
Sc'-lSc+lS^O;
r2-e»+6=0.
184 81MULTANE0(TS EQUATIONS
"Ftook this qnadraiio eqiuitioii we ihall obtain 9s8 or ai
In the equation a^(l+e+2«*)s44 put 2 for e; tiins
wts rft2; uid ilnoe y=v«) we haye y^ ik4. Again, in tiie
same equation put 3 for « ; thns «» db j^ ; and atnod
ys94r, we have ysASi '
\
Or we might proceed thne: mnltiply the forat of the
giyen equations by 2; thna
2«*+2jEy-l-4y*s88;
the aecond equation ia 2^-4^ 4- y'== 16.
Byfubtraotion 8a)Sf4-8^=:72y therefore ^=24-«y.
Again, multiply the aeoond equation by 2 and lubtract
the firat equation; thus
3^-30^8-12; therefore «>=«y— 4.
Hence, by multiplication
«V = (24 - «y ) («y - 4),
or 2«y-28«y=-9G.
By solying this quadratic we obtain d^=8 or 6. Sub-
stitute the former in the given equations ; thus
// «'+2y*=36, 2«»+y*=24.
Hence we can find a^ and ^. Similariy we may take the
other Talue of ^, and then find «^ and ^.
26a Solye 2«»+3ajy+y*«70, 6a*+«y-^=60.
Assume yse«, and substitute for y; thus
«»(2+3»+fJ«)=70, ii^(6+e-e«)=«60.
Therefore by diyision f
2+3g+e* ^70_7.
6+e-e« *60 6'
therefore 6(i4-3»+e*)= 7(6 +«-«*);
therefore 12ij*+ 89-32=0; \
therefore 39*4-2e~8=0.
2oraL
; tiiiii
in the
i linQe
of tho
ibtract
\
Biib-
kethe
INVOLVINO QUADRATICS. 189
Fiom tbift q^iadimtic equation we shall find car <Kr --8.
In the equation d^(2+8«+«')»70 pot ^ for v; that
«Bik8; and tinee ys«a? we have yasdki. The talne
«s «2 we shall find to be inapplioable ; for it kads to the
inadmissible resnlt «* x s 70. In fiust the equations from
which the yalue of « was obtained may be written thm^
««(2 +«)(!+«) =70, ii^(2+»)(8-e)=50;
and henoe we see that the Tslue of v found from 2+e=o
IB inapplicable, and that we can only haye
, 269. Equations may be proposed #hich do not fall
under either of the two cases iniich we haye diseussed,
but which may be solyed by artifices whidi can only be
suggested by trial and experience. We will giye some
examples.
270. Solve dr+y=6| «*+y'=65.
Bydiyision, ^^^
that is, «'-«y+y*=sl3;
then from this equation combined with x+p ~t we can
find X and y by the first case. Or we may com^^lete the
solution thus,
«+y=6;
square - «*+2«y+y'=25 (1).
Also «*-«y+y*=13 (2),
Therefore, by subtraction, 9xy = 12 ;
therefore «y»4;
therefore 4rysl6.... (3).
Subtract (3) from (1); thus
4^-2«y+y*=9;
extract the square root, x^y=*tZ,
n :
\\
\
180 SIMULTANEOUS EQUATIONS.
We hftye now to find a and y from the ilmple eqnatknui
these lead to «=)l or 4, y=4 or 1.
271. Solve «*+y*=41, d^=20. •
. These equatioiis can be Bolved by the second case; or
they, mav be sd?ed hi the manner Just exempUfled.. r'or
we can deduce from them
«*+y«+2«y =41 +40=81,
^+y«-a»y=41-40=si.
then by extracting the square roots,
w+y=rdk9f «-y=Al.
And thus finally we shall obtam
w=:k6br'^4, 2^r3d,4 or ifeff.
272. Solve a^+«y+y*=19, a^+aV+y*al88.
^y^^^"' ^-f^+y« =19"'
that is, ««-«y+y*=7.
We have now to solve the equations
' / «*+«y+^=19, «*-«y+y»=7.
By addition and subtraction we obtain successively
11^+^=13, ay =6.
Then proceeding as in Art. 271, we shall find
47=^3 or :£:2, y^±2 or db3.
273. Solve w-y=2f «"-*-y'=242.
a^-yg _ 242
•«-y " 2 ' *
a?* + «V + «V + «y* + y* = 121,
i»*+y*+«y(as*+y*)+«V=121 (l)i
«-y=2; .
square a5»-2a^+y»=4 ;
therefore «'+y«=2iy+4 .,«....,....*...,.*,. (2).
By division,
that is,
that is.
Now
EXAMPZB8. Xr/X
liT
Square t^'¥%^'¥y^™ia^-\-\^y-¥l%t
therefore «'+y*^2d^-f 16dV-l-16 (8).
Substitote from (2) and (8) in (1); thui
S^V + 16^ 4- 16 + «y (2d^ + 4} + «V - 121 ;
thatifl, &BV+2(keys=lO0;
therefore 0^+4^^=21.
From this qoftdratio equation we ahall obtain «if=8
or - 7. Take aey^^t and from this combined with dr-y s 2,
we shall obtain 4^=3 or —1, y=l or -8. If we take
«y=: ~7, we shall find that the values of » and y are im«
possible; see Art 236.
Examples. XXIX.
1. «-y=l, «*-«y+y*=21.
2. 2i»-5y=3, «2+«y=20.
a x-k-y^K.x-yX fl)»+y*=100.
4. 5(aj»-y»)=4(««+j^, «+y=a
5. «--^=3, «*+y«=66.
6. 4/»-5y=l, 2«2-ajy+3y*+8ii?-4y=5 47.
7. 4«+9y=12, 2a^+ay=SyK
' S. (at-6)a+(y-6)»+2«y=60, 6y-4a?=l.
6
9. 4a^+2a!y+j^+^{4x+y)=4l, 4a-y^4.
a y
*v. 12 10 ""* *
7a?y ^
15 "" 3
11. ^^2y^6xyf l\Sxrr4y=4ay,
12. «y+2=9y, scy+^-x,
13. 8(«y + 1) =58y, 4(«y + 1) = 33a?.
14. xy-x+y, ax=by.
188
MXAMPLSS. XXIX.
10L
1«.
5 + *.
a^b.
' >
W, ? + ?-% «^+y«-a»+ay.
la ?+?«!
19.
SO.
81.
29.
83.
84.
85.
86.
87'.
8&
89.
80.
81.
82.
83.
34.
8^-«ys56, 80^-^*^48.
a*-8«yBl5, «y-8^«i7.
«*+3Jlfss88, «y+4/s8.
«'+«y-$y*=21, «ry-2y'»4.
d^-f8«ys04^ 4V+4y*a:118.
i-y «+y"2'
««+y««9a
4^=48.
g'+y* _25
4y-fy ig-y ^lO
«-y «+y"3'
a^-t^^Z.
{
a^(«+y)-fy(^-y)sl58, 7dr(«+y)s7^(4*-y).
fljVC^+y)*^, A:>y(2«-'3y)=80.
2aj'-«y+y*=2y, 2a^+4^=5y.
«*+y*=(*.
ay-fy fly~y __ a*-f 1
<p-y *+y a '
4^+«y=a(a+^), fl^+y»=a'+6*.
fl^+2av-y*=d'+8a-l,
(a-l)»(«+y)=a(a+l)y(«-y).
8fi. 4F~y=2^ a^-y*=sl52.
li^.*-
I I
MXAMPLM& XXIX. W
44. 4fl^.-fy>+2(2«+y)»6, 4«y(^+l)s3.
45. «*-l-«yB&p+3, y'+^sSy+C.
46. d^-«y-i2«+ff, «y-y's2y+2.
47. 8«+y-l-6>/(2«+y+4}»23, 4i^-ejrsyi4-9y.
48. 18+9(«+y)«2(«+y)«, 6-(«-y)=(«-y)*.
49. «^-fly-a(«+l)+6+l,
a»
«y
62. «*=a«+^, y'aay+&;v.
ffS. a^xisa, ai/^z^b, »yz^=e,
60. 8y4rH-2«r«->4a^sl6/ 2y;i?~3;ra;+^s5y
4yz-za-Zxy=16,
68. 6(«^+if*+«^«l3(«+y+«^)=:-^, «y««*.
«y-y*=ay+6.
6a ;^ + g=18,
a* f/*
61. ^-£=12,
«• P'
= 1.
=2.
vv^i 4ite^ »t^^i
190
PROBLEMS.
XXX Prdtlemi^hich lead to QuadraUe Equatton*
with more than one unknown qttantity,
274. There is a certain number x>f two jdigits; the Bum
of the squares of i^ cUsitB is e^nal to the nnmbw in-
creased by the product of its digits; and if thirty-six be
added to the number the digits are retersed: find the
number.
Let a denote the digit in the tens' place, and if the
digit in Uie units' place. Then the number is lOa+y; and
if tike digits be reversed we obtain lOy +>. Therefore^ by
supposition, we have
«"+y»=4;y+lOa?+y ,..(1). 1
10a:-fy+36=10y+a?.... (2).
From (2) we obtain 9y=9a?+36; therefore ^=^+4.
Substitute in (IX thus
thei^fore «*-7«+12=0.
From this quadratic equation we obtain a; =3 or 4;
and therefore y=7 or 8. Hence the required number
must be either 37 or 48 ; each of these numbers satisfies
all the conditions of the problem.
275. A man starts from the foot of a mountain to
walk to its summit His rate of walking during the
second half of the distance is half a mile per hour less than
his rate during the first hal^ and he reaches the summit in
6^ hours. He descends in 3| hours by walking at a uni-
form rate, which is one mile per hour more than hia rate
during the first half of the ascent. Find the distance to
the sununit, and his rates of walking.
\
Let 2af denote the number of miles to the summit, and
suppose that during the first half of the ascent the man
fi
£t
th
th
Tl
th
„w-^'
PROBLEMS.
m
X
Item
ieBam
Ksr in-
•six be
id tbe
r
tf the
y; and
ore^by
t-4
or 4;
amber
.Usfies
i& to
g the
B than
nut in
a nni-
rate
ice to
walked y mfles per hour. Then he took - hours for the
w
first half of the ascent^ and — ^ hours for the second.
y-
2
Therefore - +-£^=6J
(1).
V"
2
SimUarljr,
From (2),
iherefore
2a;
y+1
T— «fj <
(2).
Therrfore^ I^ Babstitolioii, '
therefore 15(y+l)(4y-l)=44y(i2y-l),'
therefore 28y'-89y+16=0.
From this quadratic equation we obtain y=3 or ^ .
The value ^ is inapplicable, because by supposition y is
1 15
greater than ^. Therefore y=3; and then ^=-^9 so
that the whole distance to the summit is 15 miles.
t>and
) man
192
EXAMPLES. XXX.
Examples. ZXX.
1. The lam of the squares of two numbers is 170, and
th% difference of their squares is 72 : find the numbers.
2. The product of two numbers is 108^ and their sum
is t^ce their difference: find the numbers.
3. The product of two numbers is 192, and the sum of
their squares is 640 : find the numbers.
4. The product of two numbers is 128, and the differ-
i enoe of their squares is 192 : find the numbers. \
6. The product of two numbers is 6 times their sum,
and the sum of their squares is 325 : find the numbers.
6. The product of two numbers is 60 times their differ-
ence, and the sum of their squares is 244 : find the numbers.
7. The sum of two numbers is 6 times their difference,
and their product exceeds their sum by 23 : find the num-
ber&
8. Find two numbers such that twice the first with
three times the second may make 60, and twice the square
of the first with three times the square of the second may
make 840.
9. Find two numbers such that their difference multi-
plied into the difference of their squares shall make 32,
and their sum multiplied into the sum of their squares
shall make 272.
10. Find two numbers such that their difference r.dded
to the difference of their squares may make 14, and iheir
sum added to the sum of their squares may make 26.
11. Find two numbers such that their product is equal
to their sum, and their sum added to the sum oi their
squares equal to 12.
EXAMPLES. XXX.
193
0, and
r sum
tarn of
differ-
> sum,
8.
differ-
ibers.
renoe,
nam-
with
|uare
, may
nulti-
e 32,
iiares
ided
heir
qual
;neir
12. Find two numbers sttch that their sum increased
by their product is equal to 34, and the sum of their
squares diminished by tiieir sum equal to 42. .
13. The difference of two numbers is 3, and the dif-
ference of theuf cubes is 279 : find the numbers.
14. The sum of two numbers is 20, end the sum of
their cubes is 2240 : find the numbers.
16. A certain rectangle contains 300 square feet; a
second rectangle is 8 feet shorter, and 10 feet broader,
and also contains 300 square feet: find the length and
breadth of the first rectangle.
16. A person bought two pieces of cloth of different
sorts; the finer cost 4 shillings a yard more than the
cotoer, and he boiufht 10 yards more of the coarseir than
of the finer. For we finer piece he paid £18, and for the
coarser piece £16. Find the number of yards m each piece.
17. A man has to travel a certain distance ; and when
he has travelled 40 miles he increases his speed 2 miles
Sor hour. If he had travelled Mth his increased speed
uring the whole of his Journey he would have arriv^ 40
minutes earlier; but if he Imd continued at his original
speed he would have arrived 20 minutes later. Find the
whole distance he had to travel, and his original speed.
18. A number consisting of two digits has one decimal
place ; the difference of the squares of the digiiis is 20, and
if the digits be reversed, the sum of tiie two numbers is II :
find the number.
19. A person buys a quantity of wheat which he sells
so as to gain 6 per cent on his outlay, and thus clears £16,
If he hf^i sold it at a gain of 6 shillings per quarter, he
would have cleared as many pounds as each quarter cost
him shillings: find how many quarters he bought, and
what each quarter cost.
20. Two workmen, A and ^, were emplo^red by the
day at different rates ; A at the end of a certain number
of days received £4, 16«., but Bj who was absent six of
T. A. 13
194
EXAMPLES, XXX.
those days, received onlv £2, 14«. li B had worked the
whole time, and A had been absent six days, they would
have rei^ived exactly alike. Find the number of days,
and what each was paid per day.
21. Two trains start at the same time from two towns,
and each proceeds at a miiform rate towards the other
town. When they meet it is found that one train has nm
108 miles more than the other, and that if they continue
to run at the same rate thev will finish the journey in 9 and
l6 hours respectively. Find the distance between the
towns and the rates of the trains.
22. A and B are two tdwns situated 18 miles apLtii on
the same bank of a river. A man goes from .^ to i? in
4 hours, by rowing the first half of the distance and walkiiffi
the second half. In returning he walks the first half at
the same rate as before, but the stream being with him, he
rows 1^ miles per hour more than in ffoing, and accom-
plishes the whole distance in 3^ hours. jPind his rates of
walking and rowing.
23. A and B run a race round a two mile course. In
the first heat B reaches the winning post 2 minutes before
A* I In the second heat A increases nis speed 2 miles per
liour, and B diminishes his as much ; and A then aiTives
at tne winning post two minutes before B, Find at what
rate each man ran in the first heat.
24. Two travellers, A and B^ set out from two places,
P and ^, at the same time; A starts from P with the
design to pass through Q, and B starts from Q and travels
in the same direction as A, When A overtook B it was
found that thev had together travelled thirty mil^, that
A had passed through Q four hours before, and that By at
his rate of travelling, was nine hours' journey distant from
P. Find the distance between P and Q,
INVOLUTION.
195
2XXL Involutions
276. We have already defined a power to be fhe pro-
dact of two or more equal factoraf and we hare explained
the notation for denoting powers; see Arts. 15, 16, 17. The
frocess of obtaining powers is called Involution ; so that
nvolution is only a particular case of Multiplication, but
it is a particular case which occurs so often that it is
convenient to devote a Chapter to it The student will find
that he IS already familiar with some of the results which
we ^hall have to notice, and that the whole of the present
Chapter follows immediately from the elementary laws of
A^ebra.
277. Any even power qf a negative quantity is poii-
tive^ and any odd power is negative.
This is a simple consequence of the Rttle of Signs. Thus,
for example, ~-a x -a—a^^ —a x -a x ^a—a^ x — a= -a';
— ax — ax — ax — a=— <rx — a=a*; and so on. In the^'*^
following Articles, when we use the words give the proper
sign, we mean that the sign is to be determined by the
rule of the present Article. (See Art 38.)
278. Rule for obtaining a power of a power. Multiply
the numbers denoting the powers for the neyj exponent,
and give the proper sign to the result.
Thus, for example, (a^^=a^; (~a8)»=-a»; (a«)5=a";
(— a*)3= —a'". This is a simple consequence of the law of
powers which is demonstrated in Art 59. For example,
The Rule of the present Article leads igimediately to
that which we shall now give.
279. Rule for obtaining any power of a simple integral
. expression. Multiply the index of every factor in the ex-
pression by the number denoting the power, and give the
proper 8ign to the result,
13-2
196
ifk^ZUTIONi
Thus, for example, ^
(-a«y'c*)»=-a">>»c^; (2a6»(j»)«=2«a*^>"<j"=64a^6«c»».
280. Rule for obtaining any ^wer of a fraction, i^auf
^<A the numerator and denominator to ihat power, and
give the proper eign to the result.
This follows from Art 145» For example,
d«'
/ a«\» _fl^ /2aY
3^6*
I6a«
816**
28h Some examples Of Involution in the ease <^f
binomial eapressions have already been given. 809
Arts. 82 and 88. Thus
(a+5)«=a»+2a&+&»,
(a+lif=a?+Za^b+Zdb^+J^.
The student may fbr exercise Obtain the fourth, fifth
and sixth powers of a+ b. It will be found th&t
(a+e>)»=a?+5a*6+10a»^*+lOa26>+6a&*+«>^.
(a 4- &)• =^ «• + 6a*6 4- 15a<6* + 20a?ft» + 16a«6* + 6a&« + e^.
In like manner the following results may be obtained : :
(a-.6)a=««--2a&+&«,
(a-ft)*=a*-4a^6+6a26a-4a&8+t*,
(a-5)»=a«-6a»Hl5a*6*-20a^6»+15a«6*-6afc*+e*.
Thus in the results obtained for the powers of 'a— 5,
where any odd power of b occurs, the negative dgp is pre-
fixed; and thus any power of a-'b can be immdiaiely
deduced from the same power of a+b, by changing tiie
8:|;ns of the terms which involve the odd powers ofW
iirrozuTioN.
191
■•-. f
JRaiss
8r, and
I6a»
816**
Hise of
. ScSp
h, fifth
2^.
ned:
•a-6,
Bpre-
the
282. The stadent will see hereafter that, by the aid
of a theorem called the Binomial Theorem, any power
of a binomial expression can be obtained without the
labour of actual multiplication.
283. The formuloB c^ven in Article 281 may be used
in the way we have already explained in Art 84. Sup-
pose, for example, we require the fourth power of 2sB—Zy,
In the formula for {a-hf put 2x for a, and 3y for h \ thus,
(2a;-3jf)*=(2a?)*-4(2a?)«(3y)+6(2a?)«(3y)«-4(2aO(3y)'+(3y)*
= 16ar«-96aj»y+216aiy-216a^+81y*.
284. It will be easily seen that we can obtain required
results in Inyolution b^ different processes. Suppose, for
example, that we require the sixth power of a +6. We
may obtain this by repeated multiplication by a+&. Or
we may first find the cube of a +6, and then the square of
this result; since* the square of {a-^Vf is (a+&)'. Or we
may first find the sauare of a +6, and then the cube of this
result ; since the cube of (a + 5)^ is (a + h)\ In like manner
the eighth power ot a-¥h may be found by taking the
square of (o •(-&)*, or by taking the fourth power of (a+&)^.
285. Some examples of Involution in the case of
trinomial expresnons have already been given. See
Arts. 85 and 88. Thus
(a + 6 + c)5 =5 a' + B» + c? + 2aH 26c + 2flk?,
(a+6+c)»= •
a'+6^+c^+8a«(6+c)+86a(a+c)+3c*(a+6)+6a6&
These formulsB may be used in the manner explained in
Art 84. Suppose, for example, we require (1— 2d;+3a^'.
In the formula for {a+h-\-cf put 1 for a, -207 for 6, and
3«* for c; thus we obtain
(l-2a?+3«*)«=
(l)»+(-2a>)?+(3ipV+2(lX-2a?) + 2(-2a:)(3a;^+2(l)(3««)
= l+4«"+9a;*-4a?-12aj»+6aj»
==l-r4iP+10««-12«»+9;i?*.
198
EXAMPLES. XZXL
■ f
Similariy, we hare
+ 3(I)V-2d?+3a5»)+3(--2a?)«(l+3^+8{«a5«(;i-jtey
+6(l)(-2ar)(8ifj^ /
= 1- &» + 21dJ*- 44** + 63«* - W«" + 27«^.
286. It is found by obsenratioii that the aonare of any
multinomial expression may be obtained by either of two
rules. Take, for example, {a-hb+c-k- d^» It inU be found
that this
- a* + 6* 4- c* + <f * + 2a& + 2a<; + 2tf(f -f S&i? 4- 22»df -f- 2ctf :
\
and this may be obtained by the foUowingrndeTlA^igiMrtf
<^ any muainomial eshfrtmitm eonritt* qfithe ^iiar0 <if
each term, together with twice the ptroduot est wtry pair
oftetiM*
Again, we may put the result in this Imii .
=a'+2a(6+c+<Q+d*+26|«f4^«^4*^^
and this may be obtained by the folloi^og. rpjejj^jf
qf any muftinomiai expressum ctmtisft't^'wiegiua^ ^
each term, together with twice the froduet <2^%i^i0nii
by the 9um qfaU the terms iohich/cmfoM
BXAMPLESL XXXI.
Find
1. (2ajy^'
3. (-3aJ«c»)*.
2. ^(^2«V^
X.'^
N^
)-86a«»
of any
of two
» found
ted; V
y pair
EXAMPLES, XXXL
138
11. (iHM)^
13. (l+«)«.
15. (2fly+8)l -
6. U
(•
«»V
Y
io. (!-»)•.
12. (3-a»)».
14. («-2)*.
16. (oiT + "by^ + (flfdj - 6y /.
17. .(ei«+8y)*+(««-?y)*. 18. (l+«)»-(l-«)».
20. (l+4?+«»)*.
22. (l+-4?-«^.
24 (l-&»+3«*)*.
19. (l+if)*(l-a?)*.
21. (1-4^+^.
23. (l+8«+2««)».
26. (2+3»+4a?«)a+(2-3a?+4»»)«.
26. i(l+4r4«*y. 27. (l-iP+aj«)«.
28. (r+4^-«»)*/ 29. (I+3;r+2«»)^.
80. (i^a»+aif)»..
81. (2+a»+4;r»)»-(2-3a?+4aj«)».
82.^^-7fl>4^«»+«»jl 83. (l+24y+3;iJ»+4a^\
36. Xt+8i»+a<»*+«^V 37. (l-6ar+12a?«-8«^».
3a (l+4f«r+6«*+4ij'+aj*)V
8SI- .<l.r-jt)^(l +iP+<j»)*. 40. (1 -ar+«»)»(l + «?+«»)».
200
EVOLUTION.
XXXIl. Evolution,
287. Eyolntioii is the inyene of Involaiioii; so that
Eyolution is the method of finding any proposed root of
a given number or expression. It is usual to emploj the
word extract and its deHvatives in connexion with the
word root; thus, for example, to extract the tqmre root
means the same thing as to find the tquare root.
In the present Chapter we shall ^^^'fj^ ^7 >tatinff three
simple consequences of the Ride qf Signe^ we shful then
consider in suocessioi^ the extraction of Qie roots of simple
expressions, the extraction of the square root of compound
expressions ai)d numbers, and the extn^on of the cube
root of compound expressions and numbers. '
288. Any even root qf a positive quantity may he
either positive or negative*
Thus, for example, axa=aK and —ax —d^a*] there-
fore the square root of a' i9 either a or -a, tiiat is, eiUier
+ aor —a.
'289. An^ odd root of a quantity has the same sign
as the quantity.
Thus, for example^ the cube root of a' Is a, and the cube
root of -a» is —a.
290. The^e can he no even root qfa negative quantity.
Thus, for example, there can be no square root of -a*;
for if any quantity be multiplied by itself the r^ult is
a positive quantity.
The fact that there can be no eyen root of a negatiye
^uantit^ is sometimes expressed by calling su^ a root an
impossible quantity or an imaginary quantity,
291. Eule for obtaining any root of a simple integral
expression. Divide the index qf every factor in the
expression hy^ the number denoting the root, and give
the proper sign to the result.
evolution:
201
9ign
cube
ThuB, for example, J(\Qa^b^) = ^(4^t^ » ^ ial^,
y ( - WVc>») = y ( " 2 V W) = - 2aV<?*.
t'
292. Rule for obtaining any root of a firaction. IHnd
ihe root <^the numerator and denominator^ and give the
proper eign to the retuli.
For example, ^(^ = ^(p^')- J-g.
293. Suppose we require the cube root of a*. In this
case the index 2 is not divirfble by the number 3 which
denotes the required root{ and we haye, at present^ no
other mode of expressing uie result than l/aK Similarly,
JOf ^(^, ^o^ cannot^ at present, be otherwise expressed.
Such quantities are called turcXf or irrationcU quantities ;
and we shall consider them in the next two Chapters.
294. We now proceed to the method of extracting the
square root of a compoimd expression.
Thesquarerootofa'+2a&+&*isa-K&: and we shall be
led to a general rule for the extraction of the square root
of any compound expression by observing the manner in
which a+ & may be derived from 0*4- 2a& + S^.
Arrange the terms accord- a'+2a&+^^a+&
ing to the dimensions of one
letter a; then the first term is
aK and its square root is a,
wnich is the first term of the
required root. Subtract its
square, that is a\ from the whole expression, and bring
down the remainder 2ab+hK Divide 2ab by 2a, and the
quotient is b, which is the other term of the required root.
Take twice the first term and add the second term, that is,
take 2a 4- 5; multiply tins by the second term, that is by d,
and subtract the product, that is 2ab+b\from the remain-
der. This fimshes Uie operation in the present case.
a*
2a+&j2a^+6*
2a(+&>
IF^T'f
202
EVOLUTION.
If there were more terms we •honld proceed with a +5
08 we did formerly with a; its square, that is, a*-¥2ah-^h\
hM already been subtracted from the proposed expression,
so we should divide, the remainder by 2(a-i-()for a new
term in the root Then for a new subtrahend we multiply
the sum of 2 (a +5) and the new term, by the new tcirm.
The process must be continued until the required root
is found.
\
. 2S95. Examples.
40^ + 120):^ -f 9^ ^2d? -f 3y
12«y+9y"
4i»*-2ae»+ 37^-3(to+ 9 (^2a«-6aT+ 3
4^
4«*-5«; -20««+37aj«-3(to+9
-20;i5'+25«*
12^-3(to+9
2a«-4ajy+3y»;6«V'- 12«y»+9y*
, 6djV'-12«y'+9y*
\
SVOLUTION. 203
4ar»+4ir*
1 1
2;c» + 4«"-4d?-l^ - 2«»-4«»+4«+l
- 2«*-4«*+4«+l
296. It has been already observed that all even roots
admit of a double sign; see Art. 288. Thus the square
root of a'+2a6+^ is either a+6 or —a-h. In fact, in
the process of extracting the square root of a*4-2a&-f&*,
we begin by extracting the square root of a'; and this
may be either a or —a. If we take the latter, and con«
tinue the operation as before, we shall arriye at the residt
— a-&. A similar remark holds in eyery other case.
Take, for example, the last of those worked out in Art 295.
Here we begin by extracting the square root of afi\ this
may be either a^or —a^. If we take the latter, and con-
tinue the operation as before^ we shall arriye ^t the result
~«»-2«*+2«+l.
297. 'She fourth root of an expression may be found
by extracting the square root of the square root ; similarly
the eighth root may be found, by extracting the square
root of the fourth root; and so on.
298. in Arithmetic we Imow that we cannot find the
square root of eyery number exactly; for example, we
cannot find the square root of 2 exactly. In Algebra we
cannot findl^^ square root of eyery^ proposed expression
204
EVOLUTION,
eJeactlp, We Bometimes find such an example as the follow^
ing proposed; find foor terms of the square root of 1 -2ar.
l-2*(^»-*-2-2
2-a»-^ !-«»
4
a?
«*
s-s^-^-f;-**-!
•-flj'+aj^H 1 —
6*U* sfi iXr
'T""2"" 4
U /pi /n8 /m8
Thus we have a remainder — 4"'" 2*""T' *^'
findmg four terms of the squpre root of 1 - 2^; and so we
know that (l-*-5-0=l-ar+^ + ^ + ?.
» '
299. The precedm^ investigation of the squar«i root of
An Algebraical expression will enable us to demonstrate
the nue which is jp^ven in Arithmetic for the extraction o^f
the square root of a number.
The square root of 100 is 10, the square root of 10000
is 100, the square root of 1000000 is 1000, and so 6a ; hence
it follows that, the square root of a number less than 100
must consist of only one figure, the square root of a
MrOLUTJO^,
205
iitimber beiwden 100 and 10000 of two plaoes of figdreii, of
a number between 10000 and 1000000 of three plaoea of
fissures, and so on. If then a point be placed over eyerj
second fiffure in any number, be|;inning with the figure in
the units' place, the number of points will shew the number
of figures in the square root. Thus, for example^ the
square root of 4$5ft consists of two figures, and the square
root of 611^24 consists of three figures.
30b. Stippose the square root of 324^ required.
2500
100+7J749
749
Point the number according to the
rule ; thtis it appears tiiat the root
must consist of two places of figuf es.
Let a+5 denote the root^ where a is
the value of the figure m the tens'
place, and h of that in the units' place.
Then a must be the greatest multiple
of ten, which has its square less than ^200; this is found
to be 50. Subtract a\ that is, the square of 50. from the
giyen number, and the remainder is 749. Diyiae this re-
mainder by 2a, that is, 1^ 100, and the quotient is 7,
which ?3 the value of h. Then (2a+&)5, that is, 107 x 7 or
749, is the number to be subtracted ; and as there is now
no remainder, we conclude that 50 + 7 or 57 is the required
square root.
It is stated above that a is the greatest multiple of ten
which has its square less than 3200. For a evidently can-
not be a greater multiple of ten. If possible, suppose it
to be some multiple of ten less than this, say x\ then since
X is in the tens' place^ and h in the units' place, «4- 6 is less
than a ; therefore the square of a;+& is less thaii a\ and
consequently a; + & is less ilum the true square rooi
' If tiie root consisi of three places of figures, let a re^
present the hundreds, and h the tens; then having ob*
tained a and h as before, let the hundreds and tens
together be considered as a new value of a, and find anew
valuo of h for the units*
.viiSim**
206
woLunoN.
301. The <^herB may be omitted for the sake^ of
brevity, and the following rule may be obtained from the
process.
Point every second figure^ beginning <^ 3fi4d ^67
voith that in thA unit^ place, and thus 25
divide the whole number into periods,
Find the greatest number whose square 107 J 749
is contained in the first period; this y^g
is the first figure in the root; subtract its
square from tlie first period, and to the
remainder bring dozen the next period. Divide this
quantity, omitting the last figure, by tvnce the part of the
root already found, and annex the result to the root and
also to the divisor; then mtdtiply the divisor <is it noy)
stands by the part qfthe root last obtained for the suhtrc^
hend. If there be more periods to be brought down, the
operation must be repeated..
302. Examples.
Extract the square root of 132496, and of 5322249.
lS249d{364 53^2^4^(^2307
9 4 '
'f
4
66^424
396
43^ 132
129
724 J 2896
2896
4607 J 32249
32249
In the first example, after the* first figure of the, root is
found and we have brought down the remainder, we have
424; according to the rule we divide 42 by 6 to give the
next figure in the root: thus apparently 7 is the next
figure. But on multiplying 67 by 7 we obtain the product
469, which is greater than 424. This shews that 7 is too
large for the second figure of the root, and we ac^rdingiy
try 6, which succeeds. We are liable occasionally in this
manner to try too luge a figure, especially at the early
stages of the extraction of a square root.
EVOLUTION.
207
Id the second example, the student should notice the
occurrence of the cypher in the root.
, -riiJipi
t'
303. The rule for extractinp^ the square root of a
decimal follows from the preceding rule. We must ob-
serve, however, that if any decimal be squared tiiere wUl
be an even number of decimal places in the result, and
therefore there cannot be an exact square root of any
decimal which in its simplest state has an odd number of
decimal places.
The square root of 32'49 is one-tenth of the square
root of 100x32*49; that is of 3249. So also the square
root of '003249, is one-thousimdth of the square root of
1000000 X '003249, that is of 3249. Thus we may deduce
this rule for extracting the square root of a decimal. PtU
a point over every second figure^ beginning with that in
the units^ place and continuing both to the right and to
the l^ qf it; then proceed as in the extraction qf the
square root of integers^ and mark off as many decimal
places in the result as the number qf periods in the deci-
mal part qf the proposed number. In this rule the stu-
dent should pay particular attention to the words beginning
with that in the unitf^ place*
^T
304. In the extraction of the square root of an integer,
if there is still a remainder after we have arrived at the
figure in the units' place of the root, it indicates that the
proposed number has not an exact square root. We may
if we please proceed with the approximation to any desired
extent, by supposing a decimal point at the end of the
proposed number, and annexing any even number of cy-
Shers, and continuing the operation. We thus obtain a
ecimal part to be added to the integral part already
found.
Similarly, if a decimal number has no exact square
rootj we may annex cyphers, and proceed with the approxi-
.mation to any desired extent.
208
EVOLUTION,
305. The Mowing is thd extraction of the square root
of '4 to Beyen decimalplaces :
0'4606... (^•6324565
3d
123J400
369
//
1262^3100
2524
12644^57600
50576
126485^702400
632425
1264905^6997500
6324525
12649105; 67297500
63245525
4051975
306. We now proceed to the method of extracting the
cube root of a compound expreission.
The cube root of a>+3a'&i-3a^4-&' \& a-¥h\ and we
shftU be led to a general nile for the extraction of the cube
root of any compound exi>re8sipn by observing the manner
in which a + & may be derived from c? + 3a^ii> 3aft^ + Jfi^
Arrange the temid ac^ rt*+3a2j+3ai'+ft'(rt+6
cording to the dimensions ^t
of one letter a; then the
tint term is a', and its cube
root is a^ which is the first
term o^ the required root.
Subtract its cube, that is
(j^y from the whole expression, and bring down' the re-^
3aV3a'6+3rt62+2,^
3a«6 + 3<i^«+6»
EVOLUTION.
20ft
mainder Sa*ft -I- 305^+ (*. . Divide 3a% by 3a', and the quo-
tient is hi which is the other teim of the required root;
then subtract 3a% + Sad' +&* from the remainaer, and the
whole cube of a+ ^ has been subtracted. This finishes the
operation in the present case.
If thete were more terms we should proceed with a + & as
we did formerly with a; its cube, that is (^+ 3a'& + 3a^ + ^,
has already been subtracted from the proposed expression,
80 we should divide the remainder by 3(a+&}^ for a new
term in the root ; and so on.
307. It will be convenient in extracting the cube root
of more complex expressions, and of numbers, to arrange
the process of the preceding Article in three columns,
as follows:
3a +& 3a«
(3a+J)&
3aV3a&+6«
a' + 3a*& + 3a6* + 1* (,a + 6
a^
3a26+3a&«+6»
3a26+3a2>2+&»
Find the first term of the root, that is a; put a' under
the given expression in the third column and subti'act it
Put 3a in the first column, and 3a^ in the second column;
divide 3a% by ^\ and thus obtain the quotient &. Add
h to the expression in the first column ; multiply the ex-
pression now in the first column by 6, and place the pro-
duct in the second column, and add it to the expression
already there; thus we obtain 3a2+3ai&+&2. Multiply
this by by and we obtain 3a%+3a2>2+&^, which is to be
placed in the third column and subtracted. We have thus
completed the process of subtracting (a+d)' from the
original expression. If there were more terms the opera*
tion would have to be continued.
T. A.
14
210
BVOLUTION.
3a+3&
}
308. In oontintdiig the operation we mnst add locfa a
tenu to the first column, as to obtain there three timee the
part qf the root already found. This is oonyeniently
effected thus; we have already in the first
column ,3a+&; place 2b below b and itdd;
thus we obtain Za+Sby which is three limes
a+b, that is, three times the part of the root
already found. Moreover, we must add such a
term to the second column, as to obtain there
three times the eqtiare qf the part of the root already
found. This is conyenientlY effected thus; we have already
in the second column (Sa + b)b, and below
that 3a^+Zdb+b^; place V^ below, and
add the expressions in the three lines ;
Uius we obtain d(ir+6aib+db^j which is
three times (a+bf, that is three times
the square of the part of the root already
found.
(3a+5)& ')
da*+3a&+ (3>
W
1-
3aS+6a&+35*
309. Example. Extract the cube root of
au"-36«' + 102a?*-l7l«»+204««-144a?+64.
6««-3a?) 12aj*
Sof) ~&»(6««-3«)
6««-9a?+4 12aj*-18«>+9««
12a;*-36«»+27aj«
4(6««-9i?4-4)
12a;**-36«»+61«»-36«+ 16
8aj»-36a;"+102a;*-l7l«»+204;««-144»+64(,2i*-3d?+4
- SO;!?" + 102a?« - 171«* + 204aj» - 144a? + 64
-36«*+ 64aj*- 27«"
48a?*- 144a!'+204a^- 144a?+ 64
48.i?*-l44a!»+204iJ«-l444?+64
BVOLVTWN.
211
The ettbe root of d^ is 2a^, which will be the first term
of the requhred root; put 8d^ under the given expression
in the thfrd column aikd subtract it» Put three times 2a^
in the first column, and three times the square of 2a^ in
the second column $ that is, put ^3^ in the firat column,
and 12d^ ill the second column^ Divide ^36a^ by 12^.
and thus obtain the quotient —Str, which will be the secoUa
term of the root; place this term in the first column, and
multiply the expression now in the first column, that is
60^—307, by — 3«; place the product under the expression
in the second column, and add it to that expression \ thus
we obtain 12^ - 18;i^ 4- 9a;' ; multiply this by — ' 3ar, and place
the product in the third column and subtract. Thus we
have a remainder in the third column, and the purt of
the root alroEuly found is 2ix^—^, We must now acMust
the first and second columns in the manner explained in
Art 308. We put twice - 3ar, that is - 6a;, in the first column,
and add the two lines; thus We obtain 6a^->-9ar, which is
three times the part of the root already fbund. We put
the square of -3a;, that is 9a;', in the second column, and
add the last three lines in this column ; thus we obtain
12a;*-36a^+27a;^, which is three times the square of the
part of the root already found.
Now divide the remainder in the third column by the
expression just obtained, and we arrive at 4 for the last
term of the rooi^ and with this we proceed as before.
Place this term m the first column, and multiply the
expression now in the first column, that is 6a:'-^9a;+4,
by 4; place the product under the expression in the
second column, and add it to that expression; thus we
obtain 12a;*~36a;'+51a;'--36a;+16; multiply this by 4
and place the pix)duct in the third column and subtract
As there is now no remainder we conclude that 2a^-3a;+ 4
is the required cube root
310. The preceding investigation of the cube root of
an Algebraical expression will suggest a method for the
extraction of the cube root of any number.
The cube root of 1000 is 10, the cube root of 1000000 is
100, and so on; hence it follows that, the cube root of
14—2
212
EVOLUTION.
'
a number less than 1000 most condst of onW one figure^
the cube root of a number between 1000 and 1000000 of
two places of figures, and so on. If then a point be placed
oyer every third figure in any number, begmning with the
figure in the units' place, the number of points "will shew
the number of figures in the cube root Thus, for example,
the cube root of 40&224 consists of two figures, and the
cube root of 1^81^^904 consists of throe figures.
Suppose the cube root of 274G25 required.
180+5 10800 274625 (,60 + 5
926
1172S
216000
68625
68625
Point the number according to the rule ; thus it appeto
that the root must consist of two places of figures. Let
a-\-h denote the root, where a is the value of the figure in
the tens* place, and h of that in the units' place. Then a
must be the greatest multiple of ten which has its cube
less than 274000 ; this is found to be 60. Place the cube
of 60, that is 216000, in the third column under the given
number and subtract Place three times 60, that is 180,
ih the first column, and three times the square of 60, that
is 10800, in the second colunm. Divide the remainder in
the third column by the number in the second column,
that is, divide 68625 by 10800; we thus obtain 6, which
is the value of 6. Add 6 to the first column, and multiply
the sum thus formed by 6, that is, multiply 185 by 6; we
thus obtain 925, which we place in the second column and
add to the number already there. Thus we obtain 11726;
multiply this by 6, place the product in the third column,
and subtract The remainder is zero, and therefore 66 is
the required cube root. -
' The cyphers may be omitted for brevity, and the pro-
cess will stond thus :
185
108
925
11726
27462^(65
21^6
58626
5S625
EVOLUTIOir.
213
811. Bzample. Extract the cube root of 1092. 352.
127
271
14 f
1418
10^21536^^473
64
45215
39823
5392352
5392352
48
889
5689^
49]
6627
11344
674044
After obtaining the first two figures of the root, namely
47) we adjust the first and second columns in the manner
explained in Art 308. We place twice 7 under the first
column, and add the two lines, giving 141 ; and we place
the square of 7 under the secona column, and add tlie last
three lines, giving 6627. Then the operation is continued
as before. The cube root is 478.
In the course of working this example we might have
imagined that the second figure of the root would be 8 or
even 9 ; but on trial it will be found that these numbers
are too large. As in the <^e of the square root, we are
liable occasionally to try too laige a figure, especially at tho
early stages of the operation.
. 312. Example. Extract the cube root of 8653002877.
605) 1200 g65d00S87t(,2053
10
}
6153
3025
123025
. 25
}
8
653002
615125
37877877
37877877
126075
18459
12625959
In this example the student should notice the occur-
rence of the cypher in the root.
814
EVOLUTION.
313. If the root have any nrnnber of decimal itoei^
the cube will have thripe as many ; and therefore the nnm-
ber of decimal places in a decimal number, which is a
perfect cube, ana in its simplest state, will neoessaHlT be a
multiple of ihrte^ and the number of decimal placte m the
cube root will necessarily be a third of that number. Hence
if the giyen cube number be a dedmal, we pUce a point
imr the Jigure in the unitt* place, and o>?er every third
fiffure to the right and to the left of it, and proceed as in
the extraction of the cube root of an integer; then the
number of points in the decimal part of the proposed
number will mdicate the number of decimal places In the
cube root
814. Example. Extract the cube root of 14102*31)7296.
64
8,'
721
2/
12
256 n
1456 •
16,
Ill02'32}29d(24*l6
8
6108
6824
7236
1728
72r
278327
173521
173521
1.
104806296
104806296
174243
4341
6
17467716
316. If any number, integral or decimal, has no exact
cube root, we may annex cyphers, and proceed with the
approximation to the cube root to any desired extent.
- The following is the extraction of th^ cube i>oo| of *4 to
four decimal pUu^s:
913
}
2196
12
}
22088
n
EXAMPLES. XXXII.
215
15987
13176|
1611876>
36)
1625088
176704
162685504
•400... (•7368
343
57000
46017
10983000
9671256
1311744000
1301484032
10259968
EzAMPLXS. XXXIt
Find the valae of
1. ,J{9€^.
4. ^(16a«6Pc»).
//25a^6»\
6.
8
2. 4^(8a»6»). 3. ^(-64a»Z»«).
3// 216a^6»\
9.
VV326"j*
10. y(^^
Find the sqnare roots of the following expressions:
11. 16a*+40a6+256«. 12. 49a*-84a^+36&«
13. 36a^+124!'+l. 14 64a*+48aJc+96*c'.
15.
25<^-f20a6+4y
250^ rf 2006+40**
16.
9^^-24^Mhl6
44J»-l2»+9 •
m
216
EXAMPLES. XXXIl
■
17. «*+2aj'+34^+24T4.1. 18. l-2»+«««-4«»+4<ir*.
19. «*+6;ij'+26a;*+4ar+64. 20. «*-4«»+8a?-»-4.
21. l-4a? + 10««-12d:» + 9««.
22. 4d:»-4dJ*-7^ + 4a?'+4.
23. «*-2aaj*+5AV-4a'4?+4rt*.
24. a?*-2a««+(a«+268)««-2a&»«+ft*.
25. ««-12i8*+60«*-160a:' + 240«*-192a? + 64.
26. «• + 40^ - 1 Oa V + 4a*4? + a*.
27. l-2ar+3«*-4aj*+6ar*-4«'' + 3aj*-2a?'+iC*.
4fl^ a? 16aj* , 9y" . 6ajy . 160?*
\
^®' 9y« ;» 15y^ ^ 16^* "^ 6xf« ^ 264^ •
Find the fourth roots of the following expressions :
29. l+4;»+6«'+4dj"+a?*.
30. iear*-96;»3y+216ic22/»-216a!2/»+81y*.
81. l-4a?+10«*-16«'+19ar*-16^+10a:«-4«'+««.
32. {a?*-2(a+&>c»+(a»+4<»6 + 62);ij2-2a6(a+6>+a^}*
Find the eighth roots of the following expressions :
33. ««+8a^4-28aj«+66a?» + 70ii?*+66a?»+284^+ap+l.
84. {a^-2afip+3a^^-'2a!^-hp*}*, ^
Find the square roots of the following numbers :
35. 1156. 36. 2025. 37. 3721. 3a 5184.
39. 7569. 40. 9801. 41. 15129. 42. 103041.
43. 165649. 44. 3080*25. 45. 41^1^.
46. *835396. 47. 1522756. 48. 29376400.
EXAMPLES, XXXIL
217
49. 88492401. 50. 4981*5364. 51. 64128064.
52. -24373969. 53. 144168049. 54. 254076*4836.
55. 3-25513764. 56. 4*54499761.
57. '5687573056. 58. 196540602241.
Eztnot the square root of each of the following nam-
bers to five places of decimals :
59. -9. 60. 6*21. 61. *43. 62. -00352.
63. 17. 64. 129. 65. 347*259. 66. 14295*387.
Find the cube roots of the following expressions:
67. aB»+36a?!^ + 54ary«+27y'.
6a i72aB»+i72apV+fi76ajV+^V.
69. «»-3a:«(a+&) + 3a<a+6)2-(a + 6)».
70. ««+5aj* + 6«*+7aj3 + 6«»+3a?+l.
71. ^•-3aa»+5a'«»-3a*'d?-a«.
72. 8a«+48c«»+6065«*--80c»aj»-90c*aj*+108c»ar-27A
73. l-94?+39^-99;i?» + 156dr»-144a:«+64ar».
74.1-3aj+6«»-10a;»+12ar*-12a?» + 10«*-6«'+3«'-«».
Find the sixth roots of the following expressions :
75. 1 + 12ir+60aj*+160aJ»+ 2400;* + 192^ + 64a:«.
76. 729aj«-145aB»+1215«*-540«* + 135«*-18a?+l.
Find the cube roots of the following numbers:
77. 19683.
80. 226981.
83. 2628072.
86. 60236'28a
89. 1371330631.
7a 42875. 79. 167464.
81. 681472. 82. 77868a
84. 3241792. 85. 54010152.
87. 191*102976. 88. *220348864.
90. 20910518875.
91. 91398648463125. 92. 5340104393239.
218
INDICES,
XXXIII. Indieef.
316. We haye defined an index or smxment in Art. 16,
and, according to that definition, an index has hitherto
always been a positive whole number. We are now about
to extend the definition of an index, by explaining the
meaning of fractional indices and of negative indices.
317. If m and n are any poeitive whoie numbers
a'"xa"=a'"'*'".
The truth of this statement has already been sheVn
in Art 59, but it is convenient to repeat the demonstra-
tion here.
«r=axax(ix to m factors, by Art. 16,
a^-a'i^axa'K to n fiu^tors, by Art 16 ;
therefore
a«*xa"=ax<ixax...xaxaxax ...to m+n factors
.'/ =a*"^", by Art 16*
In like manner, \ip is also a positive whole number,
and so on. ^
■ *
^ 318. If m and n are positive whole numbers, and tn
greater thui n, we have by Art 317
-ii=a'
m^^
therefore
This also has been ahready shewn; see Art 7S^
> 319. As ihustional indices and negative i|i|d|^!^
not yet been defined, we are at liberty to give wha| MM*
tions we please to them ; and it is found oonveiliJIt lo
INDICES.
21»
give sudi clefiQiiioDs to them as will mi^e the hnportant
relation a* X <»"-'»"•+" -' " *
may he.
relation a**xa"=a"*'^" tdways true, whatever m and n
« For example; required the meaning of aK
By supposition we are to have a* x a" = a^ = a. Thus a ^
must be such a number that if it be multipli^ by itself
Uie result is a; and the eqiiare root of a is by definition
such a number; therefore or must be equivalent to the
squarerootof (z, thatiSya^=>/a. .
Again; required the meaning of a^,
. By supposition we are to have
a xa >ia =a
a^=a.
Hencei as before^ a^ must be equivalent to the cube
root of a/that is a^» /J^a.
Again; required the meaning of a V
faff
By supposition, a xVxa xa =a';
therefore
a*= ^a\
These examples would enable the student to under-
stand what is moant by any fractional exponent; but we
will give the definition in general symbols in jttie next two
Artides.
X 320. Required the meaning qf a" where n U any
positive whole number.
By supposition,
111
111 ^^
a'xa'xofx ...ton factors = a" «•**•*
=a^=ai
therefore a" must be equivalent to the n^ root of a,
that is,
a*^ff^
220
INDICES.
. 321. Required the meaning qf a" where m andn are
^ any positive iohols numbere.
By supposition,
a"x a"xa" X ... to » factors=a" " • "* =a"»;
therefore a* must be equivalent to the n*** root of a*",
that is, a^^lja"^.
M
Hence a" means the n*^ root of the w*^ power of «;
that is, in a fractional index the numerator denotes a power
and the denominator a root.
\
322. We have thus assigned a meaning to any positfve
index, whether whole or fractional; it remains to assign a
meaning to negative indices.
For example, required the meaning of a"'.
By supposition, a' x a"" =«"-'=«*= a,
therefore
a^ or
We will now give the definition in general symbols.
< 323. Required the meaning qf a~'/ wh^ren ie any
positive number whole orfraetionaL
By supposition, whatever m may be, we are to have
a"*xa~"=a"'~".
Now we may suppose m positive and greater than n^
and then, by what has gone before, we have
m >
fit
rt""" X «• = a"» ; and therefore a""* = -» •
Therefore
therefore
a
,m
a*
\
INDICES,
221
In orddr to express this in words we will define the
Yford reciprocal. One quantity is said to be the recipro'
col of another when the product of the two is equal to
unity ; thus, for example, a is the reciprocal of - .
Hence a~" is the reciprocal of a" ; or we may put this
result symbolically in any of the following ways,
«--•-„
a "'
a xa
—II—
1.
y 324. It will follow from the meaning which has been
given to a negative index that a'"-T-a"=a'"~" when m is less
than Uf as well as when m is greater than n. For suppose
m loss than n ; we have
o
a
—- = ^-(»-«") = a"*"".
n—m
Suppose fn=n; then a'^-r-a' is obviously =1; and
cr-*=ia\ llie last symbol has not hitherto received a
meaning, so tliat we are at liberty to give it the meaning
which naturally presents itself; hence we may say that
325. In order to form a complete theory of Indices it
would be necessary to give demonstrations of several pro-
positions which will be found in the larger Algebra. JBut
those propositions follow so naturally from the definitions
and the properties of fractions, that the student will not
find any difficulty in the simple cases which will come be-
fore him. We shall therefore refer for the complete theory
to the larger Algebra, and only give hero some examples as
specimens.
326. If m and n are positive whole numbers we know
that (a'")"=a"^; see Art. 279. Now this result will also
hold when m and n are not positive whole numbers. For
example,
(a*)^=a^.
For let (a»)*=d?; then by raising both sides to the
fourth power we have a^=^; then by raising both sides
f
i2i
INDICES,
to the third t>ower we have a=a;^'; thel«fore x^a^, which
was to he diewn.
i(
327. If n is a positive whole number' we know that
a^xlf^iflbf. This result will also hold when n is not
a positive whole number. For example, a^x&^=(a&A
For if we raise each side to the third power, we obtain in
each case ah\ so that each side is the cube root of ah
In like manner we have
111 1
a" X &" X (^ X . . . =i (a&c. . .)".
Suppose now that there are m of these quantities
a, by c,..., and that all the rest are equal to a\ thus we
obtain
1 1^
(a")* = (a*)" j that is, ( ya)- = yo*.
Hius the m^ power of the n^ root of is equal to the
n* root of the «i* power of a»
' 328. Since a fraction may take different forms without
any change in its value, we may expect to be able to give
Afferent forms to a quantity with a fractional index, with-
out altering the value of the quantity. Thus, for example,
4 9 4
g we may expeet that a^=a^ ; and this is the
smoe
3
3
case. Fof if we raise each side to the sixth power, we
obtain a*; that is, each side is the sixUi root of a*.
\t
r
329. We will now give some examples of Algebraical
operations involving fractional and negative exponents.
Multiply a^^M by ah^fi.
2 17
3"*"2"C'
3 1 13
4 3"'12»
thet^fore
3-^5
?=^L
:>-•
INDICES.
223
Divide a^jfi by «^y*.
4 2""4»
?«i_l.
therefore
Multiply
8 6 %*
x-^-ar+ar* by a^-i-x'^-aTK
Of +«■ +»"'
;»^+2aj^-M
-^-t
Here in the first line «* x a?=a?^**=i»^, a?^ x a^ »x\
^'x^'^soj^ssl; andsoon. ;
Divide
— .
ww«w«»i>*»«i < "I 'l
224
EXAMPLES. XtklTL
6. (81)-^.
9. ya-».
Examples. XXXIII «
Find the value of
I. 9"*. 2. 4"*. 3. (100)"*. 4. (lOOOili
Simplify
6. {dF)-\ 7. ' (a-")-». 8. Va"*.
10. o^a*xa"i
Multiply
II. i^+y^ by a?^-y^. 12. <fi ■{■ n^h^ ^h^ by a*-6*.
13. a?+ar^+2 by a?+a?*-2.
14. «?*+«* + 1 by x-^-x'^+l, \
16. a~*-J-a~* + l by a"^-l.
16. a*-2+a"^ by a*-a~^.
17. (a+aMj-a?*y* by(a+aM^a?V.
18. a^-ay^+a^y-y^ by a?+a?^y*+y.
Divide
19. a!^-y^ hy a^-^, 20. a-& by a*-6*.
21. 64a?-^+27y-* by 4aj"^+3y"i .
22. x^-a^y^+a^y-y^ by a?'-y*.
23. o^ + ah^ + h^ by a* + a^&* + &*.
. 24. a^ + &* - c^ + 2aM by a* + 6* + A
25. ii?^-2aM+a' by ir*-2aM + a. f
26. ii?*--4a?^y* + 6a?^2^-4ar*y^+y* by «*-2a:*^+y*.
Find the square roots of the following expressions : j
27. iP^-4+4a?~*. 28. (af+«'"^)*-4(a?-«-*).,,
I. a?»-4a?^+2a?^+4a?-4a?^+iA
29
30.
4^t_12;»t
12;i?' + 25 - 24a?~* + 16*
rl
.-f
SUBDS.
225
XXXIV. Surds,
330. When a root of a number cannot be exactly
obtained it is called an irrational quantity, or a surd.
Thus, for example, the following are Burds;
And if a root of an algebraical expression cannot be
denoted without the use of a fractional index, it is also
called an irrational quantity or a surd. Thus, for ex-
ample, the following are surds ;
. Ilie rules for operations with surds follow from the
propositions of the preceding Chapter; and the present
Chapter consists almost entirely of the application of those
propositions to arithmetical examples.
331. Numbers or expressions may occur in the form
of surds, which are not really surds. Thus, for example,
J9 is in the form of a surd, but it is not really a surd, for
1/9=3; and ^(a*+2ab+b^ is in the form of a surd, but
it is not really a surd, for fj{a^ + 2ab + J^ = a + &.
332. It is often convenient to put a rational quantity
into the form of an assigned surd ; to do this we raise the
quantity to the power corresponding to the root indicated
by the surd, and prefix the radical sign. For example,
3=>/32=V9; 4=4/43=4^64; a=i/a*; a+b= i/{a+h)\
333. The product of a rational ouantity and a surd
may be expressed as an entire surd, by reducing the
rational quantity to the form of the surd, and then multi-
plymg ; see Art 327. For example, 3 ^/2 = ^9 x ^2 = ^ 18 ;
24/4= ^(8x^4= J/32; a^/&=^/a«x ^b=jj{a%
334. Conversely, an entire surd may be expressed as
the prodnct of a rational quantity and a surd, if the root of
one mdor can.be extracted.
T.A.
15
226
SURDS.
For example, ^/32 = ^/(16 x 2) = ^16 x ^2 = 4 ,^2 j
4^48= j|/(8x 6)= ysx 4/6 = 24/65
i/ia*b*)=i/a*xi/b*=ai/b\
335. A surd fraction can be transformed into an
equivalent expression with the surd part integral.
™ 1/3 /3x2 /6 Je
jFor example, iJ ~ — U ■■a/ — = 5l_ • ^
*wi cA»uip*o, V g ^8x2 ^16 4 '
336. Surds which haye not the same index can be
transformed into equivalent surds which have ; see Art 327.
For example, take ^/5 and 4/11: V6=6*» /yil = (ll)*j
6^=6*= 4/6»= 4/125, (11)*=11^=: 4/(1 1)«= 4/121.
337. We may notice an application of the preceding
Article. Suppose we wish to know which is the greater,
1^5 or ^11. When we have reduced them to the same
index we see that the former is the greater, because 125 is
greater than 121.
338. Surds are said to be Bimilar when they have, or
can be reduced to have, the same irrational factors.
Thus 4 ^7 and 5 Jl are similar surds ; 5 4/2 and 44/16
are also similar surds, for 4 4/ 16 = 8 4/2.
339. To add or subtract similar surds, add or subtract
their coefficients, and affix to the result tbe common
irrational factor. 1
For example, ^/12+^/75~^/48=2^/3+6^/3-4^3
= (2 + 5-4)^/3=3^/3.
V3^.1 V256 2 Vl2 1 V64xl
V2'*"4^T=3'^8"^4^'^'"W
_2^12 1 44/1224/12
3 2 4 3 3 •
SURDS,
227
340. To multiply simple sards which have the same
index, multiply separately the rational factors and the
irrational factors.
1/ For example, 3^/2x^3=3^6; 4^5x7^6-28^30;
24^4x3^^2 = 64^8=6x2 = 12.
341. To multiply simple surds which have not the same
index, reduce them to equivalent surds which have the same
index, and then proceed as before.
For example, multiply 4 >/5 by 2 4/11.
By Art. 336 ^/6=4^125, 4/11=4^121.
Hence the product is 84^(125 x 121), that is, 8^15125.
342. The multiplication of compound surds is per-
formed like the multiplication of compound algebraical
expressions.
For example, (6/^/3-5 ^2) x (2 ^3 + 3 J2)
= 36 + 18/s/6-10^/6-30 = 6 + 8^/6.
343. Division by a simple surd is performed by a rule
like that for multiplication by a simple surd; the result
may bo simplified by Art. 335.
w 1 « /« ^ i« 3n/2 3 /2 3 /6 >/6
For example, 3 V2-i-4 ^/3=^ = - V ^ = ^ V ^ = ^- ;
4^5^-^11-2;^- ^121 -^'^121-^'^121x(ll)*
24^1830125
~ n ^'
The student will observe that by the aid of Art. 335 the
results are put in forms which are more convenient for nu-
merical application; ti^us, if we have to find the approxi-
mate numerical value of 3 ,^,/2-r4 ^/3. the easiest method is
to extract the square root of 6, and divide the result by 4.
16—2
228
SURDS.
•
844. The only case of division by a eomponnd inrd
which is of any importance is that in which the divisor is
the sum or difference of two quadrcUie surds, that is, surds
inyolving square roots. The division is practically effected
by an important process which is called rdtionmiting the
denominator cf a fraction. For example, take the fraction
4
5 724-2 /a * ^ ^^ multiply both numerator and denomi-
nator of this fraction by 6 J2 -2JZ, the value of the frac-
tion is not altered, while its denominator i» made rational;
4 4(6^2-2 ^/3)
thus
5^2 + 2^/3 (5^/2 + 2^/3){5<>/2-2,ys)
^ 4(5^/2^2^/3) 10^/2-4^/3
60-12 * 19 ' •
fiimikrlv N/3 + V2 _ (^/3^^^^2)(2^/3•^-^/2)
^'2 3-^/2"(2^/3-^/2)(2^/3+^/2)
^ 84-3i^6 _8-f3j6
12-2 10 •
\
845. We shall now shew how to find the square root of
a binomial expression, one of whose terms is a quadratic
nurd. Suppose, for example, that we require the square
root of 7+4/^3. Since Ua!'h^v^=^w+y+2j(ap\ it is
obvious that if we find values of of and y fi*om x+y=*I,
and 2 tJiay) -- 4 ly/S, then the square root of 7 + 4 ^3 will be
tjx + Vy. We may arrange the whole process thus :
Suppose
square,
7+4,/3=A;+y+2^/(a?y).
Assume ^+^=7, then 2^(a^)=4^/3;
square, and subtract, (w + y)^~'4ay =: 49 - 48 = 1,
that is, (;»-y)*=l, therefore a?-y=l.
Since a; -I- ^=;: 7 and a; - 2^ = 1, we have or = 4, j^ = 3 ;
therefore /s/(7 + 4V3)=>/4+ .^3=2+1/3.
Similarly, s/(7-4 J3)=2- ^3.
EXAMPLES. XXXIV.
229
EXAHPLES. XXXIY.
Simpliiy .
1. 3^/2+4^/8-^/32. % 2^4 + 6^32-4/108.
3. 2^/3 + 3V(li)-^/(6l). 4. ^"i^- •
Multiply
6. V5+V(li)-;^by^/3.
*
^- 'y^-yr6-^^2^yy^
7. l+is/3-V2by ^/6-^/2.
8. V3+N/2by-^+^.
Rationalise the denominators of the following fractions:
9.
11.
3+V2
2-V2*
2^6+^3
10.
n/3+ .^2
>/3-n/2-
12.
2/s/3+^2
3;/6T2;73' '*• 3^/3-2^5•
EitrsMst the square ropt of
13. .14 + 6^6. 14. 16-6*77. 15. 8 + 4^/3.
16. 4-^16.
Simplify
17.
^/(6+^/24)*
18.
n/(7-4V3)-
19: ^^f^^ll 20. V(3+>/5)+V(3-V5).
230
RATIO,
XXXV. Ratio.
846. Ratio is the relation which one quantity bears
to another with respect to magnitude, the comparison
beinff made by considi 4ng what multiple, part^ or parts,
the first is of the second.
Thus, for example, in comparing 6 with 3, we observe
that 6 has a certam magnitude with respect to 3, which
it contains twice ; again, in comparing 6 with 2, we see that
6 has now a different relative magnitude, for it contains
2 three times; or 6 is ^eater when compared with 2 than
it is when compared with 3.
347. The ratio of a to 5 is usually expressed by two
points placed between them, thus, a:b\ and the former is
called the antecedent of the ratio, and the latter the conte-
qmnt of the ratio.
, 348. A ratio is measured! by the fraction which has for
its numerator the antecedent of the ratio, and for its
denominator the consequent of the ratio. Thus the ratio
of a to ( is measured by t ; then for shortness we may
say that the ratio of a to & is equal to ^ or is r .
349. Hence we may say that the ratio of a to 5 is equal
a
to the ratio of c to d, when r = 3
. h d
350. ffthe terms of a ratio he multiplied or divided
by the 9am£ quantity the ratio i» not altered.
a ma
For g = ^ (Art. 136).
361. We compare two or more ratios by reducing
the fractions which measure these ratios to a common
denominator. Thus, suppose one ratio to be that of a to d,
and
a
6'
RATIO.
231
and another ratio to be that of <; to </; then the first ratio
? = «-*» and the second ratio :> = !-;,.
h ha a ha
Hence the first ratio is g^reater than, equal to, or less
than the second ratio, according as ad? is greater than,
equal to, or less than be,
352. A ratio is called a ratio of greater inequality ^ of
leti inequatity^ or of eqtudity^ according as the antecedent
is greater than, less than, or equal to the consequent
353. A ratio qf greater inequality it diminiihed^
and a ratio qf lest inequality it increated, by adding
any number to both tennt qfthe ratio.
Let the ratio be ^ , and let a new ratio be formed by
adding x to both terms of the original ratio: then .-— -
o-\-x
is greater or less than y according as & (a + or) is greater or
less than aifi-^-x); that is, according as bx is greater or less
than ax, that is, according as 6 is greater or less than a,
354. A ratio qf greater inequality it increased, and
a ratio qf lett inequality it diminished, by taking from
both termt qf the ratio any number which it lett than
each qf thote termt.
Let the ratio be v , and let a new ratio be formed by
taking x from both terms of the original ratio; then ^^
is greater or less than ^, according as b{a-x) is greater
or less than a(b-'X); that is, according as &a; is less or
greater than ax, that is^ according as & is less or greater
than a.
355. If the antecedents of any ratios be multiplied
together, and also the consequents, a new ratio is obtfuned
which is said to be compounded of the former ratios. Thus
232
RATIO.
the ratio ocxMSa said to be compounded of the two ratios
a :&andc : dL
When the ratio a : & is compounded with itself the
resulting ratio is a' : ^ ; this ratio is som^imes .called the
duplicate ratio of a : &. And the ratio a? : 2^ is sometimes
called the triplicate ratio of a : &.
356. The following is a very important theorem con-
cerning equal ratios.
ace
Suppose that ^ = ^ = ^, then each of these ratios
where jp, q, r, n tre any numbers whatever.
\
For lot A;= r = ^ = -^j then
Kb=af kd=c, ^=e;
J9(*6)"+g(A^"+r(^)"=i?a"+2'6"+r«";
"ph^-^qd'+r/**
therefore
therefore
therefore
1
The same mode of demonstration may be applied, and
a similar result obtained when there are more ttum three
ratios given equal
As a particular example we may suppose n = 1^ then we
fE c e
see that ^^ i:-^-fy ®<^ ^^ ^^®<3^ ratios is equal to
^f ^^ — >; and then as a special case we may suppose
po+qa + rj
p=zq=ryEo that each of the given equal ratios m equal to
a+c+e
b+d+f'
EXAMPLES. XXXV.
2da
Examples. XXXV.
1. Find the ratio of fourteen shillingB to three guineas.
2. Arrange the following ratios in the order of magni-
tude; 3 : 4, 7 : 12, 8 : 9y 2 : 3, 5 : 8.
3. Find the ratio compounded of 4 : 15 and 25 : 36.
4. Two numbers are in the ratio of 2 to 3, and if 7 be
added to each the ratio is that of 3 to 4 : find the numbers.
5. Two numbers are in the ratio of 4 to 5, and if 6 be
taken from each the ratio is that of 3 to 4 : find the numbers.
6. Two numbers are in the ratio of 5 to 8; if 8 be
added to the less number, and 5 taken from the greater
number, the ratio is that of 28 to 27 : fii^d the numMrs.
7. Find the number which added to each term of the
ratio 5 : 3 makes it three-foiuihs of what it would have be-
come if the same number had been taken from each term.
8. Find two numbers in the ratio of 2 to 3, such that
their diiSerence has to the dif erence of their squares the
ratio of 1 to 25.
9. Find two numbers in the ratio of 3 to 4, such that
theu* sum has to the sum of their squares the ratio of
7 to 50.
10. Find two numbers in the ratio of 5 to 6, such that
theur sum has to the difference of their squares the ratio of
1 to 7.
11. Find X so that the ratio x : 1 may be the duplicate
of the ratio 8 : x,
12. Find x so that the ratio a—xxh-x may be the
duplicate of the ratio a : h,
13. A person has 200 coins consisting of guineas, half-
sovereigns, and half-crowns; the sums of money in g^mneas,
half-sovereigns, and half-crowns are as 14 : 8 : 3; find
the numbers of the different coins.
14. If &-a :&+a=4a-& : 6a-&, find a :&.
15. If — £ = i: — = ,then/+m+n=Q.
2S4
PROPORTION.
XXXVI. Proportion.
357. Foar numbers are said to be proportional when
the first is the same multiple, part, or parts of the second
as the third is of the fourth ; that is when 7 = ^ the four
o a
numbers a^hfC^d are called i>roportional8. This is usually
expressed by saying that a is to & as c is to </; and it is
represented thus a\h iieidjOv thus a : &=c : d.
The terms a and d are called the extremes^ and h and e
the means.
358. Thus when two ratios are equal, the four numbers
which form the ratios are called proportionals ; and the pre-
sent Chapter is devoted to the subject of two equal ratidis.
359. When four numbers are proportional the pro^
duct of the extremes is equal to the product qfthe means.
Let a, hfCydhQ proportionals ;
a_c
multiply by hd\ thus ad=h€.
If any three terms in a proportion are given, the fourth
may be determined from the relation ad— be.
U b-e we have ad^b^; that is, if the first be to the
second as the 'second is to the third, the product qf the
extremes is equal to the square qfthe mean.
When a:b::b:d then a, b, d are said to be in con-
tinued proportion; and b is called the mean proportional
between a and ^. ,
360. ff the product <^ two numbers be equal to the
product qf two others, the four are proportionals^ the
terms qf either product being taken for the means, and
the terms qf the other product for the extremes,
X b
For let ^ = db ; divide by ay, thus - = - ; \
then
or xiaiibiff
(Art 357).
PROPORTION.
233
361. lta\hv,eid, and e:d ::€ :f, then aibiie :/,
e
a e
For ^ ^ ^, and ^ = ^ ; therefore ^ = ^;
or a : b :: e :/•
362. fffour numbers be proportionahf they are pro-
portionals when taken inversely; that ia, it a :b ::e :d,
then b : a :: d : c.
a
For 3 = 5J divide nnity by each of these equals;
thus - = -', or b : a :: d : c.
a e *
363. If four numbers be proportionals, they are pro-
p3.^'onaIs when taken alternately; that iafHaibiieid,
tii'^ iciibid
For| = |; multiply by-; thus2 = _.
or a :c lib :d.
364. jj/* four numbers are proportionals, the first
together with the second is to the second as the third
together with the fourth is to the fourth; that is
if a :b::c:d, then a + b:b::c + d:d,
a e
For T = ^; add unity to these equals; thus
a
r + 1 = J + 1, that is -X
b d ' b
._ a + 6 c-^-d
" d
; or a •¥ b \b \: e •{- d \d.
365. Also the excess of the first above the second is to
the second as the excess of the third above the fourth is to
the fourth,
a c
For 7 = ^; subtract unity from these equals; thus
<* • c , . « J. . a—b c—d , , « J J
jT - 1 = ^ - 1, that IS -r- = -^ or a- : 6 :: c-a : a.
238
PROPORTION.
T
366. Also the first is to the excess qf the first above the
second as the third is to the excess qf the third above the
fourth.
By the last Article ^- = ^; also f = -^ ; .
^ b d * b d' '
therefore
a— 6 b c—d d a-b e-^d
X - =
a
d
or
a
or a-b : a :: c-d : c; therefore a : a-b :: c : e-d, v
367. Whenfournimbers are proportionals^ the sum
qf the first ana second is to their difference as the sum
qf the third and fourth is to their difference; that is, if
a'.bwcid^ then a-¥b : a'-b :: c-{-d : c—d
•o A^ n/i.i ji»/.-^ + & c-^d J a-b c—d
By Arts. 364 and 365 — r- = --r-» and -=— = —-=-":
' b d * b d -
a-^b a-^b e-hd c-d ., . . a+b c+d
^ -^ , that IS
therefore
b d ' d * """""" a-b
or a+6 : a-6 :: c+d : c-d.
c-d*
368. It is obvious from the preceding Articles that if
four numbers are propoHionals we can derive from them
many other proportions; see also Art 356.
369. In the definition of Proportion it is supposed that
we can determine what multiple or what part one quantity
is of another quantity of the same kind. But we cannot
always do this exactly. For example, if the side of a
square is one inch long the length of the diagonal is de-
noted by J 2 inches t but ,J2 cannot be exactly found, so
that the ratio of the length of the diagonal of a square
to the length of a side cannot be exactly expressed by
numbers. Two quantities are called incommensurdbie
when the ratio of one to the other cannot be exactly ex-
pressed by numbers.
The student's acquaintance with Arithmetic will sug-
gest to him that if two quantities are really incommei^-
surable still we may be able to express the ratio of one to
the other by numbers as nearly as we please. F9r example,
we can find two mixed numbers, one less than iJ2, and the
other greater than V2, and one differing from the other by
as small a fraction as we please.
PROPORTION.
237
870. Wo will ^ve one proposition with respect to tbo
comparison of two mconimensorable quantities*
Let X and y denote two quantities; and suppose it
known that however great an integer q may be we can find
another integer p such that both x ana y lie between
^ and ^- — : then x and y are equal.
q q
For the difiference between x and y cannot be so great
as - ; and by takmg q large enough - can be made less
than any assigned quantity whatever. But if x and y wero
unequal their difference could not be made less than any
assigned quantity whatever. Therefore x and y must bo
equal
371. It will be useful to compare the definition of pro-
portion which has been used in this Chapter with that
which is given in the fifth book of Euclid. Euclid's defini-
tion may DO stated thus: four quantities are proportionals
when if any equimultiples be taken of the first and the
third, and also any equimultiples of the second and the
fourtn, the multiple of the third is greater than, equal to,
or less than, the multiple of the fourth, according as the
multiple of the first is greater than, equal to, or less than
the multiple of the second.
372. We will first shew that if four quantities satisfy
the al^braical definition of proportion, they will also
satis^ Euclid's.
For suppose that a \ h i\ c \ d\ then^^^ ^ ; therefore
^ = ^, whatever numbers p and q may bo. Hence pc is
greater than, equal to, or less than qd^ according as j9a is
greater than, equfd to, or less than qh. That is, the four
quantities a, &, c, d satisfy Euclid's definition of proportion.
373. We shall next shew that if four quantities satisfy
Euclid's definition of proportion they will also satisfy the
algebraical definition.
For suppose that a, &, Cy d are four quantities such that
whatever numbers p and q may be, pc is greater than,
238
PROPORTION.
equal to, or less than qdy according 2A pa\& greater than,
equal to, or less than qb,
Furst BuppoBe that c and d are eommensurable ; take
p and q such that pc=qd; then by hypothesis pa=^qb: thus
■^ = 1 = ^; therefore 1=3. Therefore a : 6 :: c : ^.
^0 ga a
Next suppose that e and cf are tn(;omm^^ura&^(9.
Then we cannoi find whole numbers p and q, such that
pc=qd. But we may take any multiple whatever of d, as
qd, and this will lie between two oonsecutire multiples of c,
say between pc <u>d {p+l)c. Th™ g w less than unity,
and ^^ ' is greater than unity. Hence, by hypothesis,
^ is less than unity, and ^^r — is greater than unity.
Thus ^ and r are both greater than - , and both less than
~ — , And since this is true however great/? and q may
be, we infer that v and -3 cannot be unequal; that is, they
must be equal: see Art. 370. Therefore a i b :: c : d.
That is, the four quantities a, b, c, d satisfy the alge-
braical definition of proportion.
374 It is usually stated that the Algebraical definition
of proportion cannot be used in Geometry because there is
no method of representing geometrically the result of the
operation of division. Straight lines can be represented
geometrically, but not the abstract number which expresses
ow oftein one straight line is contained in another. But it
should be observed that Euclid's definition is rigorous and
applicable to incommensurable as well as to commensur-
abU quantities ; while the Algebraical definiti<|n is, strictly
speaking, confined to the latter. Hence this consideration
alone would furnish a sufficient reason for the definition
adopted by Euclid,
EXAMPLES. XXXVL
239
Examples. XXXVL
Find ihd yalue of x in each of the following propor-
tiona
].
3.
6.
6.
7.
7 :: a: :
9 :: 16 ;
42.
X,
4 : 7 :: 8 : or. 2. 3
H I X V, X \ 45. 4. X
«+4 : a?+2 :: «+8 : «+6.
«+4 : 2a?+8 :: 2d?-l : 3a?+2,
Sor-f 2 : a;+7 :: 9^-2 : 50^+8.
8. «»+a?+l:62(a?+l)::«*-a?+l :63(ii?-l).
9. 007+^ :&a7+a:: »M;+n :na7+m.
10. If pq=ri, and qt=8Uf then p :r ::t :u,
11. If a : & :: c : ei?, and a' : &' :? ^ : d', then
aa^ :W :i ctf : <W and ab' : «'6 :: cd' \(fd,
12. If « : 6 :: 6 : c, then (a* +68)(&*+<^ =(«& + &<?)«.
13. There are three numbers in continued proportion;
the middle number is 60, and the sum of the others is 125:
find tiie numbers.
14. Find three numbers in continued pronortion, such
that their sum may be 19, and the sum of their squares
133.
If a : 5 :: c : ef, shew that the following relations are
true,
15. a(<j+dt) =<?(«+&). 16. «>/(<?*'' +<^=cV(a'+ 5^
'• (a-c)(aa-c2)~(6-d)(62-cf^)'
pa^-\-qah'¥rh'^ pc^+qcd+rd^
18.
Id^+mab+nb^ Ic^+medi-nd*'
b e
10 ^ 1_ 1 J__ I ia b <? ,)
^^' a''2b^Zi'^4d'~^U'l''2'^^i'
20. a : b :: ^'(ma'+nc') : tfinibf-^ndl^).
240
VARIATION.
XXXYII. Variation.
L i:
375. The present Chapter consists of a series of pro-
pos'tiona connected with the definitions of ratio and pro-
porLon stated in a new phraseology which is conyenient
for some purposes.
• ■ . ■
876. One quantity is said to vary directly as another
when the two quantities depend on each Other, and in such
a manner that if one be changed the other is changed in
the same proportion. •
Sometimes for shortness we omit the word dirm
and say simply that one quantity varies as another.
377. Thus, for example, if the altitude of a triangle be
invariable, the area varies as the base; for if the base be
increased or diminished, we know from Euclid that the
area is increased or diminished in the same proportion.
;We may express this result with Algebraical symbob thus ;
let A and a be numbers which represent the areas of two
triangles having a common altitude, and let B and h bo
numbevB which represent the bases of these triangles re-
A B
spectively; then ~- = t'* And from this we deduce
A a
■g = ^ , by Art 363. If there be a thurd triangle having the
same altitude as the two already considered, then the ratio
of tiie number which represents its area to the number which
represents its base will also be equal to ?. Put r=^)
A
then Q=9ny 2Xl^ A=mB. Here A may represent the
area of any one of a series of triangles which have a com-
mon tdtitude, and ^ the corresponding bas4 and m re-
mains constant Hence the statement that the area varies
as the base may also be expressed thus, the area has a
VARIATION.
241
constant ratio to the base ; by which we mean that the
number which represents the area bears a constant ratio
to the number which represents the base.
These remarks are intended to explain the notation and
phraseology which are used in the present Chapter. When
we say that A varies as B^ we mean that A represents the
numerical yalue of any one of a certain series of quanUties,
and B the numerical value of the corresponding quantity
in a certain other series, and that A=mBf where m i"
some number which remains constant for evei7 correspond-
ing pair of quantities.
It will be convenient to give a formal demonstration
of the ralution A=mB, deduced from the definition in
Art 376.
378. XfA vary as B, then A is equal to B multiplied
by some constant number.
Let a and b denote one pair of corresponding values of
the two quantities, and let A and B denote any other pair;
then — = T , by definition. Hence ^ = t 5 = mB, where
mis equal to the constant I .
379. The symbol x is used to express variation ; thus
A Qc B stands for A varies as B,
380. One quantity is said to vary inversely as another,
when the first varies as the reciprocal of the second. See
Art. 323.
Or if -4=^, where m is constant, A is said to vary
inversely as B,
381. One quantity is said to vary as two others jbtn^^y,
when, if the foi-mer is changed in any manner, the product
of the other two is changed in the same proportion.
Or ifA=mBC, where m is constant, A is said to vary
jointly as B and C,
T. A. 16
ipii
242
VARIATION,
882. One qiumtity is said to vary directly as a second
and invenely as a third, when it varies jointly as the
second and the reciprocal of the third.
Or \tA = -jy-y where m is constant^ A is said to vary
dhrectly as B and inyersely as C,
383. Ifk oc BiaitdTQ oc 0, then A oc 0. x
For let^=m^, and B=nCf where m and n are con-
stants; VienA=mnC; and, as mn is constant^ ^ oc (7.
384. I/* A oc 0, an(i? B ocO, ^A«n AifaB oc C, and
V(AB) oc 0. 5
For let A =mC, and B=nOf where m and n are con-
stants ; then At^^B = (m dt n)C ; therefore A^B <x: C»
Also ^/(^-B)=^/(m«C72)=C^/(mn); therefore ^/(-4-B) xC.
385. yA« BO, /AwBqc ^,an(^0oe4.
For let ^=»*5C; then -B = -i ^ j therefore J3 « ^ .
Similarlyi C7 X -g •
386. ^A X B, and x B, then AO x BD.
For let A=^mBf and (7«n2>; then AC=fnnBI>;
therefore AC ac BD,
Similarly, if AccB, and CxZ>, and JSccF, then
ACE^BDF; and so on.
387. JE/*Ax B, #AenA"x B*.
/ For let A=^mBy then A^=m*B*\ tl^erefore -4" x J5".
VARIATION.
248
38a if A<xi B, then AP oc BP, where P i$ any
quantity variable or invariable,
"Eoft let A = mB^ then AP=^ mBP; therefore ^ P oc BP,
389. j^^ A oc B tehen is invariable, and A qc when
B if t9t9ana&/(0y ^A«n A « BO when both B an«f are
variable.
The yariation of A depends on the variations of the
two <]puitities B and C; let the variations of the latter
qnantities take place separately. When B is changed to b
A B
let A be changed to a' ; then, by supposition, T/ =» -r •
Now let C be changed to c, and in consequence let a* be
a' O
changed to a; then, by supposition, - =-. Therefore
a c
B O
-7 X — = ^ X — J that is, — =
A BC
a
be
therefore^ oc BO,
A very good example of this proposition is fiimished in
Oeometry. It can be shewn that the area of a triangle
varies as the base when the height is invariable, and that
the area varies as the height when the base is invariable.
Hence when both the base and the height vary, the area
varies as the product of the numbers wmch represent the
base and the height.
Other examples of this proposition are supplied by the
2uestion8 which occur in Arithmetic under the head of the
double Rule of Three. For instance suppose that the
Quantity of a work which can be accomplished varies as
the number of workmen when the time is given, and varies
as the time when the number of workmen is eiven ; then
the quantity of the work will vary as the product of the
number of workmen and the time when both vary.
390. Tn the same manner, if there be any number of
qnantities B, O, 2), ...each of which varies as another
quantity A when the rest are constant^ when they all vary
A varies as their product.
16—2
244
EXAMPLES. XXX VIL
Examples. XXXYII.
1. A varies as B^ and A=2 when ^=1; find the
value of A when B=2,
2. If A^-^-E^ varies as A^-E^^ shew ihat A-\-B
varies as u4-j?.
3. ZA + HB varies as 5^ + 3^, and wi =5 when B=2\
find the ratio A \ B,
4. ^ varies as nB + (7; and ^ =4 when ^ a l, and
C-2; and ^ = 7 when B=2, and 6'=3: find n.
5. ^ varies as B and (7 jointly; and ^ = 1 when
B= 1, and C^ 1 ; find the value of A when B=^2 and C7=2.
6. ^ varies as B and (7 jointly; and ^ = 8; when
i?= 2, and C= 2 : find the value of BC when ^ = 10. •
7. ^ varies as B and (7 jointly; and ^ = 12 when
J9=2, and (7=3: find the value of ^ : .8 when C=4.
8. ul varies as B and C7 jointly ; and A = a when
^=6, and C^c\ find the value of A when ^=&^ and
C7=ca.
9. ^ varies as B directly and as C inversely ; and A -a
when B=K and C=e\ find the value of A when J?=(; and
(7=6.
10. The expenses of a Charitable Institution are partly
constant, and partly vJFvry as the number of inmates.
When the inmates are 960 and 3000 the expenses are re-
spectively j£ll2 and ^180. Find the expenses for 1000
inmates.
11. The wages of 5 men for 7 weeks being £\1\ 10*.
find how many men can be hired to work 4 weeks for £ZQ,
12. If the cost of making an embankment vary as the
length if the area of the transverse section and height be
constant, as the height if the area of the transverse section
and length be constant, and as the area of the transverse
section if the length and height be constant, and an em-
bankment 1 mile long, 10 feet nigh, and 12 feet broad cost
£9600 find the cost of an embankment half a mile long,
16 feet high, and 15 feet broad.
ARITHMETICAL PROGRESSION. 245
XXXVIIL Arithmeticai Progression,
391. Quantities are said to be in Arithmetical Pro-
ffression when they increase or decrease by a common dif-
ference.
Thus the following series are in Arithmetical Fro-
gression,
2,6,8,11,14,
20, 18, 16, 14, 12,
a,a+&, a+2&, a+3&, a+4&
The common difference is found by j;ubtrar *^ing anr
term from that which immediateljr follows it In the fir -^
series the common difference is 3 ; in the second series n is
-2; in the third series it is 6.
392. Let a denote the first term of an Arithmetical
Progression, h the common difference; then the second
term is a+&, the third term is a+2&, the fourth term is
a+ 3&, and so on. Thus the n^ term is a+ (ra - 1) &.
393. To find the sum qf a given number of terms of
an Arithmetical Progression, the first term and the com-
mon difference being supposed known.
Let a denote the first term, h the common difference, n
the number of terms, / the last teri^i;^ 9 the sum of the
terms. Then
*=a+(a+5)+(a+2&)+ + ?.
And, by writing the series in the reverse order, we have
also
f=/+(/-&) + (/-2e>)+ +a.
Therefore, by addition,
2«=(^+a)+(^+a) + tow terms
=w(/+a);
n
therefore »=*o(^'*'*) (^^*
V /■'' '■■■
246 ARtTHMEtlCAL PBOQBESSION.
Also /=a+(n-l)6 (2),
thus «=^{2a+(n-l)6} (3).
The equation (3) gives the value of # in terms of the
quantities which were supposed known. Equation (1) also
gives a convenient expression for «, and furnishes the
foUowmg rule: the Bum qf any number qf termt in
Arithmetical Progretsion is equal to the product of the
number qf the terms into ha{f the sum <^ the first and
last terms.
We shall now apply the equations in the present Article
to solve some examples relating to Arithmetical Pro-
gression.
\
394. Find the sum of 20 terms of the series 1, 2, 3, 4,...
Here a=l, &=1, n=20; therefore
«=^(2 + 19)=10x21=210.
395. Find the sum of 20 terms of the series, 1, 8, 5, 7,...
Here a=l, &=2, n=20; therefore,
*=|'(2+I9x2)=|^x40=(20)"=400.
396. Find the sum of 12 terms of the series 20, 18, 16,. . .
Here o=20, 6=s-2, n=12; therefore
«=^(40-2xll)t=6(40-'22)=6xl8=ipa.
397. Find the sum of 8 terms of the series A t ::. 7f L-
12'6 4 3
11
Herea= — , 6=Y2,n=8j therefore
8/2 . 7\ . 9 ^
of the
[1) also
^es the
in
of the
Yst and
■Article
Pro-
I ■
3, 4,.„
»5,7,...
EXAMPLES. XXXVJIL 247
398. How many terms mnst be taken of the Beiies
15, 12, 9,... that the sum may be 42 ?
Here #=42, a=15, 5= -3; therefore
42=||30-3(n-l)} =^(33-3n).
We have to find n from this quadratic ej[nation ; by
solving it we shall obtain n=4 or 7. The series is 15, 1^
9, 6, C 0,--3, ; and thus it will be found that we ob-
tain 42 as the sum of the first 4 terms^ or as the sum of the
first 7 terms.
399. Insert fiye Arithmetical means between 11 and
23.
Here we have to obtain an Arithmetical Progression
consisting of seeen terms, beginning with 11 and ending
with 23. Thus a=ll, /=23, n=7 ; therefore by equation
(2) of Art 393,
23=11+65,
therefore &a2.
Thus the whole series is 11, 13, 15, 17, 19, 21, 2a
EXAXPLB9. XXXVIII.
Sum the following series :
1. 100, 101, 102, to 9 terms.
2. 1, 2|, 4, to 10 terms.
3. 1, 2}, 4^, to 9 terms.
4. 2, 3f, 5^,..., to 12 terms.
6. X, -, 1, tolSterms.
6. 2' "3' ""?'••• tol5terms,
7. Insert 3 Arithmetical means between 12 and 20.
8* Insert 5 Arithmetical means between 14 and 16.
us
EXAMPLES. XXXVIIL
9. Insert 7 Arithmetical means between 8 and -*4.
10. Insert 8 Arithmetical means between — 1 and 5.
11. The first term of an Arithmetical Pro|^ession is
13, the second term is 11, the sum is 40: find we number
of terms.
12. The first term of an Arithmetical Progression is
5, and the fifth term is 11 : find the sum of 8 terms.
13. The sum of four terms in Arithmetical Progression
is 44, and the last term is 17 : find the terms.
14. The sum of three numbers in Arithmetical Pro-
gression is 21, and the sum of theur squares is 155 : fii^ the
numbers. ^
15. The sum of five numbers in Arithmetical Prcwpres-
sion is 15, and the sum of their squares is 55: find the
numbers.
16. The seventh term of an Arithmetical Progression
is 12, and the twelfth term is 7; the sum of the series is
171 : find the number of terms.
17. A traveller has a journey of 140 miles to perform.
He goes 26 miles the first day, 24 the second, 22 the
third, and so on. In how many days does he perform the
journey?
18. A sets out from a place and travels 2^ miles an
hour. B sets out 3 hours after A^ and travels in the
same direction^ 3 miles the first hour, 3^ miles the second,
4 miles the third, and so on. In how many hours wUl B
overtake A 1
19. The sum of three numbers in Arithmetical Pro-
gression is 12 ; and the sum of their squares is 66 : find
the numbers.
20. If the sum of n terms of an Arithmetical Pro-
gression is always equal to n\ find the first term and the
common difference.
OEOMETRICAL PROGRESSION. 249
XXXIX. Geometrical Progression,
400. Quantities are said to be in Geometrical Pro-
gression when each is equal to the product of the i>recediDff
and some constant factor. The constant factor is called
the common ratio of the series, or more shortly, the ratio.
Thus the following series are in Geometrical Progres-
sion.
Xf Of 9f 27) ol)..».>t
1 1 1 J_
^'2*4'8V16'
a, ar, ar^, ar^, ar*,.
The commcn ratio is found by dividing any term by
that which immediately precedes it. In the first example
the common ratio is 3, in the second it is - , in the third
itisr. ^
401. Let a denote the first term of a Geometrical Pro-
gression, r the common ratio; then the second term is ar,
the third term is at^, the fourth term is ar*, and so on.
Thus the n^ term is ar^'K
402. To find the sum qfa given number tf terms qfa
Geometrical Progression, the first term and the common
ratio being supposed knoton.
Let a denote the first term, r the common ratio, n the
number of terms, s the sum of the terms. Then
f=a+ar+ar' + ar' + ...+ar"''*; .
therefore sr-ar-k-at^+ai^-^-.^.-^-ar^'^-^-ar^.
Therefore, by subtraction,
sr—s—ar^—af
therefore a(r*-l) ^^^
25a GEOMETRICAL PROGRESSION.
1(1 denote the last term we have
l=ar^'i
rl-a
therefore
i-
r-1
.(2X
.(3).
Ednation (1) gives the value of « in terms of the
quantities which were supposed known. Equation (8) is
sometimes a convenient form*
We shall now apply these eouations to solve some ex-
amples relating to Qeometrical rrogression.
403. Find the sum of 6 terms of the series l, 3, 9, 27,. . .
Here a=l, r=3, n=6; therefore
3«-l 720-1
#=
3-1 3-1
=364.
404. Find the sum of 6 terms of the series 1, -3,
9, ^2if» ^"
Herj a=l, r= -3, n=6; therefore
(-3)»-l 729-1 ,^„ -k
405. Find the sum of 8 terms of the series 4, 2, 1, r ,. ..
Here a = 4, r = - , n = 8 J therefore
^ V^ " ^ J _ ^ V "¥) 255 2 255
I-'
'-I
64
32
406. Find the sum of 7 terms of the series, 8, -4,
2 -1 i
Here a =8, r=-=, n=7; therefore
-!-.
-^'
16 3 8
'Vf
of the
(8) fa
me ex-
9,27,...
h "3,
OEOMETRICAL PROaRESSIOK 251
407. Insert three Geometrical means between 2 and
32.
Here we have to obtain a Qeometrical Progression
consisting of Jtte terms, beginning with 2 and ending with
32. Thus a =2, /=32, n=5; therefore, by equation (2)
of Art 402,
32=2r*,
that is r*^16=2*;
therefore r=%
Thus the whole series is 2, 4, 8, 1^, 32.
408. We may write the value of », given in Art. 402,
thus
«(l-r")
*=
1-r
Now suppose that r is less than unity; then the larger
n is, the smaller will r** be, and by taking n large enough
r" can be made as small as we please. If we negle<^ r^
we obtain
a
and we may enunciate the result thus. In a Qeometrical
Progression in which the common ratio is numerically
less than unity, by taking a sufficient number of terms
the sum can be made to difer as lUtle as we please
from r-^.
i—r
409. For example, take the series 1, = , - , - , . . .
^ 4 o
1 /»
Here a=l, **=o; therefore y— =2. Thus by taking
a sufficient number of terms the sum can be made to differ
as little as we please from 2. In fact if we take four
terms the sum is 2—-, if we. take five terms the sum is
8
2- — , If we take six terms the sum is 2^ — , and so on.
The result is sometimes expressed thus for shortness,
the sum of an irfinite number qf terms qfthis series is
2; or thir, the sum to infinity is 2» *
It
252 EXAMPLES. XXXIX.
410. Recurring decimals are examples of what are
called infinite Geometrical Progression. Thus for example
3 24 24 24
•3242424... denotes j^ + j^ + ^^ + j^, + ...
Here the terms after z-z form a Geometrical Progres-
24
sion, of which the first term is j^, and the common ratio
Is rr^. Hence we may say that the sum of an infinite
24 / 1 \
number of terms of this series i^ Tp -^ ( 1 "*To2j » *^** ^^
24
^^. Therefore the value of the recurring decipaal is
3^ 2£ ^
10 9»0*
The value of the recurring decimal may be found prac-
tically thus:
Let 9= •32424...;
then 10*= 3-2424...,
and 1000«=324'2424...
Hence, by subtraction, (1000 - 10) « = 324 - 3 = 321 ;
*!. f 321
therefore '"ogo'
And any other example may be treated in a similar
manner.
Examples. XXXIX.
Sum the following series :
1. 1,4,16, to 6 terms.
2. 9,3,1, to 5 tends. %
3. 25,10,4, to 4 terms.
4. 1, V2, 2, 2V2, ... tol2terms.
leciinal is
>und prac-
EXAMPLES. XXXIX. 258
5. g> |> g, to 6 terms.
2 3
6. ^» "■1>2» to 7 terms.
7. 1, ""3» 9> to infinity.
8. 1, J, jg, to infinity.
9. 1» ""o* 4* to infinity.
2
10. 6,-2,-, to infinity.
o
Find the Talue of the following recurring decimals:
11. -ISISIS... 12. -123123123...
13. -4282828... 14. -28131313...
15. Insert 3 Geometrical means between 1 and 256.
16. Insert 4 Geometrical means between 5} and 40j^.
17. Insert 4 Geometrical means between 3 and -729.
18. The sum of three terms in Geometrical Progression
is 63, and the difference of the first and third terms is 45:
find the terms.
19. The sum of the first four terms of a Geometrical
Progression is 40, and the sum of the first eight terms is
3280 : find the Progression.
20. The sum of three terms in ij^eometrical Progres-
sion is 21, and the sum of their squares is 189 : find the
ierms.
254 HARMONWAl PROGRESSION,
XL. Harmonicat Progretnon,
411. Three quantities A^ B, C are said to be in Har-
monical Progression when A : :: A- B : B — C,
Any number of quantities are said to be in Harmonica!
Progression when every three consecutive quantities are in
Harmonical Progressiott
412. The reciprocals qf quantities in Harmonical
Progression are in Arithmetical Progression,
Let Af Bf C he in Harmonical Progression; then
A: C::A'-B : B-0. \
Therefore A (B-C)=C{A-'B).
Divide by ^5C; thus i -i = -i - 2 •
This demonstrates the proposition. *
413. The property established in the preceding Article
will enable us to solve some questions relatmg to Har-
monical Progression. For example, insert five Harmonical
2 8
means between - and r-. Here we have to insert five
9 lo
Arithmetical means between ^ and -3-
3 1 •
therefore ^&=-, therefore &= — .
o 10
3 25 26
Hence the Arithmetical Progression is - ,
HencOi by equa-
tion (2) of Art 393,
27 28 29 15.
16' 16' 16' 8 '
2 16 16
16' 16'
and therefore the ^a^monical Pro-
16 16 16 8
greasionis-, ^g, -, ^, -, -, -.
EXAMPLES. XL.
255
414. Let a and e be any two quantiUes ; let ^ be
their Arithmetical mean, G their Geometrical mean, H
Uieir Uarmonical mean. Then
A-a=e-A \ therefore A = " (a+c).
a \ Q \\ G : e\ therefore G= tj{ac).
a : e :: a^H : if— c; therefore J7=
2ae
a+c"
1 1
Examples. XL.
1. Continue the Harmonical Progression 6, 3, 2 for
three terms.
2. Continue the Harmonical Progression 8, 2, 1^ for
three terms.
3. Insert 2 Harmonical means between 4 and 2.
4. Insert 3 Harmonical means between - and rr.
5. The Arithmetical mean of two numbers is 9, and
the Harmonical mean is 8 : find the numbers.
6. The Geometrical mean of two numbers is 48, and
the Harmonical mean is 46^ : find the numbers.
7. Find two numbers such that the sum of their Arith-
metical, Geometrical, and Harmonical means is 9f , and the
product of these means is 27.
8. Find two numbers such that the product of their
Arithmetical and Harmonical means is 27, and the excess
of the Arithmetical mean above the Harmonical mean
is IJ.
9. If a, &, <! are in Harmonical Progression, shew that
«+<?— 2& : a— <? :: a-c : a+<?.
10. If three numbers are in Geometrical Progression,
and each of them is increased by the middle number, shew
that the results are in HarmonicEd Progression.
■■ ■!> " ■'
256 PERMUTATIONS AND COMBINATIONS.
XLL Permutations and CombinatiQns,
415. The different orders in which a set of things can
be arranged are called i\iea permutations.
Thus the permutations of the three letters a,&, c, taken
two at a time, are a&, &a, aCy ca, be, cb,
416. The combinations of a set of things are the
different collections which can be formed out of them,
without regarding the order in which the things are placed.
Thus the combinations of the three letters a, &, o, taken
two at a time, are ab, ac, be ; ab and ba, though different
permutations, form the same combination, so also do ac
and ca, and be and cb,
417. The number qf permutations of n things taken
r at a time is n (n— l)(n— 2) (n-r+ 1).
// Let there be n letters a, b, c, d, ; we shall first find
the number of permutations of them taken two at a time.
Put a before each of the other letters; we thus obtain
n-1 permutations in which a stands first. Put b before
each of the other letters; we thus obtain n— 1 permuta<
tions in which b stands first. Similarly there are n— 1
permutations in which e stands first. And so on. Thus,
on the whole, there are n(n — l) permutations of n letters
taken two at a time. We shall next find the number of
permutations of n letters taken three at a time. It has
just been shewn that out of n letters we can form n (n— 1)
permutations, each of two letters; hence out of the n—l
letters b,c,d, we can form (n-1) (n-2) permutotions,
each of two letters : put a before each of these, and
we have (n- l)(n— 2) permutations, each of threer letters,
in which a stands first Similarly there are (n-1) (n-2)
permutations, each of three letters, in whicb b stands first.
Similarly there are as many in which e stands first. * And
so on. Thus, on the whole, there are n (n — 1) {n—2) per-
mutations of n letters taken three at a time.
PERMUTATIONS AND COMBINATIONS. 267
From considering these cases, it might be conjootmvd
that the number of permutations of n letters talcen r at a
time is n(n-l)(n~2)...(n-r+l); and we shall shew
that this is the case. For suppose it known that the num-
ber of permutations of n letters taken r— 1 at a time is
7t (n-l)(n- 2).. .{n-(r-l)+l}. we shall shew that a similar
formula will give the number or permutations of n letters,
taken r at a time. For out of the n-\ letters h^ e, d,...
we can form (n-l)(n-2) {n-l—(r— 1)4-1} permuta-
tions, each of rr>l letters: put a before each of tnese, and
we obtain as many permutations, each of r letters, in
which a stands first Similarly thr-r* are as many permu-
tations, each of r letters, in which b stands first Simi-
larly tnere are as many permutations, each of r letters,
in which e stands first. And so on. Thus on the whole
there are n(n— l)(n-2)....(»-r+l) permutations of n
letters taken r at a tima
If then the formula holds when the letters are taken r- 1
at a time it will hold when thev are taken r at a time.
But it has been shewn to hold when they are taken three
at a time, therefore it holds when they are taken four at a
time, and therefore it holds when they are taken five at a
time, and so on : thus it holds universally.
418. Hence the number of permutations of n things
taken all together is n (n — 1) (n — 2) ... 1.
419. For the sake of brevity n(7i- l)(n~2)...l is often
denoted by \n; thus \n denotes the product of the natural
numbers from 1 to n inclusive. The symbol \n may be
resAf factorial w.
420. Any combination of r things wiU produce [r
permutations.
For by Art. 418 the r things which form the given
combination can be arranged in [r different orders.
421. JTie number qf combinations qfn things taken r
r,*^4'^^ • n(n-l)(n-2)...(n-r + l)
aJt a time %s -^ — r — ^ •
T.A. 17
258 PERMUTATIONS AND COMBII^ATfOm
For the number of permutatiom of n things takesi r at
atimei8n(n-l)(n-2)...(n-r+l)byArt417; and each
combination proauces [r permutations by Art 420; hence
the number of combinations must be -^ ' „
n(n--l)(n-2)...(n-r4-l)
__.
' If we multiply both numerator and denominator of
In
this expression by |n-r it takes the form -, — -= — , the
value of course being unchanged.
422. To find the number qf permutations of n things
taken all together which are not all different.
Let there be n letters; and suppose p of them to be a,
q of them to be &, r of them to be c, and the rest of them
to be the letters dy e, ..., each occurring singly: then the
number of permutations of them taken all together will be
lPL2l£'
For suppose N to represent the required number of
permutations. If in any^ one of the permutations the p
letters a were changed into p new and dififerent letters,
then, without changing the situation of an^ of the other
letters, we could from the single permutation produce \p
different permutations: and thus if the p letters a were
changed into p new and different letters the whole number
of permutations would be iV x [£. Similarly if the q letters
h were also changed into q new and different letters the
wliole number of permutations we could now obtain would
he Nx\px\qj And if the r letters c were also changed
into r now and different letters the whole number of per-
mutations would be iV X [p X [£ X [r . But this number
must be equal to the number of permutations of n different
letters taken all together, that is to [n.
\ \n
Thus iVx[£x [^x [r = |n; therefore iv= . , . -*
And similarly any other case may be treated*
EXAMPLES. XLL
259
423. The siadent should notice the peculiar method of
demonstration which is employed in Art 417. This is called
inathemaiical induction, and may be thus described: Wo
shew that if a theorem is true in one case, whateyer that
case may be, it is also true in another case so reUted to the
former that it may be called the neat case; we also shew
in some manner that the theorem t« true in a certain case;
hence it is true in the next case, and hence in the next to
that, and so on; thus finally the theorem must be true
in eveiy case after that with which we began.
The method of mathematical induction is frequently
used in the higher parts of mathematics*
ExAUPLESv XLI.
1 . Find how many parties of 6 men each can be formed
from a company of 24 men.
2. Find how many permutations can be formed of the
loiters in the word company, taken all together.
3. Find how many combinations can be formed of the
letters in the word longitude, taken four at a time.
4. Find how many permutations can be formed of the
letters in the word consonant, taken all together.
6. The number of the permutations of a set of things
taken ybur at a time is twice as great as the number taken
three at a time : find how many things there are in the set.
6. Find how many words each containing two conso-
nants and one vowel can be formed from 20 consonants
and 6 Towdsi the rowel being the middle letter of the
wordk
7. t^te pelhM>nB are to be chosen by lot out of twenty:
find in how many ways this ctm be done. Find also how
often an assigned person would be chosen.
hi A boat^s crew consisting of eight roWei^s and a
bVeersman is to be formed out of twelve persons, nine of
whom can row but cannot steer, while the other three can
steer but cannot rowt find in how many ways the crew
can be formed ^ind ite in how many ways the crew
could be formed tf one of the three were able both to row
and to steer.
17—2
mtm
26a
BINOMIAL THEOREM.
XLII. Binomiai Theorem^
424. We liave already seen that («+aJ2=aj8+2aya+a2,
and that (ar+a)'=«*+3«*a+3a?a2+a*; the object of the
present Chapter is to find an expression for {a+a)* where
n is any positive integer.
425. By actual multiplication we obtain
(a; + a) (d7 + &) (« + c) = aj« + (a + 6 + c)dj2 + («6 + &<; + ca)^ + a&c,
(a+a)(a-{-b)(x+c)(:!B+d)=a!*+(a+h+c+d)a}^ \
+ (€ib+ac+ad+bc+bd+cd)a^ \
+ {abc+bcd+€da-i-dc^)a}+dbcd,
Now in these results we see that the following laws
holdt
I. The number of terms on the right-hand side is one
more than the number of binomial factors which are multi-
plied together.
II. The exponent of or in the first term is the same as
the number of binomial factors, and in the other terms
each exponent is less than that of the preceding term by
unity.
in. The coefficient of the first term is unity; the
coefficient of the second term is the sum of the second
letters of the binomial factors ; the coefficient of the third
term is the sum of the products of the secon^ letters of
the binomial factors taken two at a time; the coefficient of
the fourth term is the sum of the products of the second
letters of the binomial footers taken three at a time; and
so on; the last term is the product of all tiie second letters
of the binomial factors.
We shall shew that these laws always Bold, whatever
be the number of binomial factors. Suppose the laws
to hold when n-1 factors are multiplied together; that is,
BINOMIAL THEOREM.
261
suppose there are n-l factors a;+a, d7+&, d?+c,...«+A,
and that
{x f «) (a? + 6). . .(a? + /b) = af"^ + jooj""* + ga?"-' + raJ"** +... + «,
\v here ^ = the sum of the letters afb,c,...ky
^=the sum of the products of these letters taken
two at a tune,
r=the sum of the products of these letters taken
three at a time,
t<=the product of all these letters.
Multiply both sides of this identity by another factor
a+l, and arrange the product on the nght hand accoixling
to powers of a; thus
(a? + tf ) (a? + &) (a? + c) . . . (a? + A;) (a? + Q = ;»* + tp + Z)af "*
"Now p+l=a+l>-\-c+..,+k+l
' = the sum of all the letters a, h, Cy.^k, I ;
q+pl=q+l(a+h+c+,.,+k)
=the sum of the products taken two at a
time of all the letters a, b,c,.,.k,l;
r+ql=r+l{€ib+ac+bc+ ,.. )
=the sum of the products taken three at a time
of all the letters a, b,Cf., k,l;
td = the product of all the letters.
Hence, if the laws hold when n-l factors are multi-
plied together, they hold when n factors are multiplied
together; but they have been shewn to hold when /our
factors are multiplied together, therefore they hold when
/vd factors are multiplied together, and so on: thus they
hold uniyersally.
262
BINOMIAL THEOREM.
We shall write the res* Jt for the multiplication of n
factors thus for abbreyiation :
(« + a) (« + 6). . .(« 4- Aj) (a? + = af + P«""* + ^«"""
Now P is the sum of the letters a, 5, «,...^ t^ which are
91 in number; Q is the sum of the products of these
letters two and two, so that there are .^ ^ of these
products ; i2 is the sum of 7 » 7 products; and so
on. See Art. 421.
Suppose &, (;,...A!, I each equal to a. Then P becomes
na, Q becomes !t^^:^a%R becomes ^(^-^(^-f) ^.
and so on. Thus finally ^
w(n-l)(n-2)(n-3) , „.,
"*" 1.2.3.4
+a*.
426. The formula just obtained is called the Binomial
Theorem; the series on the right-hand side is called the
expansion of (x+ay, and when we put this series instead
of (w+aY we are said to expand (a+a)\ The theorem
was discovered by Newton.
It will be seen that we have demonitrated the theorem
in the case in which the exponent n is a positive integer;
and that we have used in this demonstration the method
of mathematical indtiction.
427. Take for example (a}+ a)«. Here w = ^,
w(w-l)_6»5_ n(w~l)(n~2) 6.5.4
1.2 1.2 ' 1.2.3 "l.2.3
w(»-l)(n-2)(»-3) 6.6.4.3
20,
1.2.3.4
1.2.3.4
15,
w(n-l)(n-2)(»~3)(n-4) __ 6.5.4.8.2
1.2.3.4.5
1.2.3.4.5
-6;
BINOMIAL THEOREM.
263
m of n
V.
lich are
theso
|f theso
and so
ecomcs
-2)
a
3.
a»;l?»-3
nomial
9d the
Dstead
eoreijii
9orem
ethod
thus
(;» + a)* = «• + 6<Mj' + 1 5ii?aj* + 2(ki'aj' + 1 6a*aj* + 6a«a? + a«.
Again, gnppose we require the expansion of (b^-hcp)*:
we have only to put &^ for w and c^ for a in the prece<ung
identity; thua
(J« + cy)» = (62)« + 6<^ (&«)»+ 1 5((^)2(&2)4 + 20(cy)»(62)»
+ 20cy 6* + 16cy&* + 6c«/6" + c«/.
Again, suppose we require the expansion of {x—cf] we
must put —c for a in the result of Art. 425 ; thus
1 • ^
1.2.3 "^"^^ ^••
Again, in the expansion of {x+aY put 1 for a;; thvs
(1 +«)-=l +n«+^^>a»+^^-^««+ ...
and as this is true for all values of a we may put x for a ; thua
/t . \« t. . w(w-l) , n(»-l)(«-2) «
428. We ma^ apply the Binomial Theorem to expand
expressions containing more than two terms. For example,
required to expand (1 + 2a?— a^)*. Put y for 2a?— a?^. then
we have (1 + 2a? - a;^)* = (1 + y)* = 1 + 4y + 6y* + 4^* + 2^
= 1 + 4(20?"" a?») + 6(2a?-a!«)2 + 4(2a?-aj»)8 + (ai?-".'c^)*.
Also (2«-«*)2=(2a?)2-2(2a?)ar' + (a?3)2==4iB»-4a?3+a?*,
(24? -aPf= {2xf - 3(2a?)«a?3 + 3(2a?) {jx^f- {x^^
=8a?»-12a?* + 6a?»-««,
(2a? - aj«)* = (2a?)* - 4 (2a?)»a?" + 6 (2a?)«(aj*)2 - 4 (2a?) (a?2)' + (^«)^
= 16a?* - 32a?« + 24a?« - 8a?7 + a?8.
- I l»IFt«lfti«Wi faiwto
264
BINOMIAL THEOREM.
Hence, collecting the terms, we obtain (1 + 2a? ~ a^^
429. In the expansion of (1+x)" the coefficients of
terms equally distant from the beginning and the end
are the same.
The coeflScient of the r*^ term from the beginning is
n(n-l)(n-2)...(w-r+2) , i*. i . u lu ^ a
'^ ' ^ ^; by multiplying both numerator
|r-l
In
and denominator by [ n — r + 1 this becomes , rT= r .
•' L [r-l [ w-r+1
The r* term from the end is the (w-r+2)*term from
the beginning, and its coefficient is .
n(n-I)..,{w-(w-r+2)+2} thatia ^^^""'^^'"^ '-
by multiplying both numerator and denominator by I r- 1
\n '
this also becomes . ^ . _^^^ .
430. Hitherto in speaking of the expansion of {x-\-af
we have assumed that n denotes some positive integer.
But the Binomial Theorem is also applied to expand
(d?+a)" ■vhen w is a positive fraction, or a negative quan-
tity whole or fractional. For a discussion of me Binomial
Theorem with any exponent the student is referred to the
larger Al^bra; it will however be a useful exercise to
obtein vanons particular cases from the general formula.
Thus the student will asiume for the present that whatever
be the values of Xy a, and n,
1.2 1.2.3
n(n-l)(ii~2)(n-3) .
^ 1.2.3.4 ^^ ^-T-
If n is not a positive integer the series never ends.
rt^0i^
BINOMIAL THEOREM.
265
431. As an example take (1+^)^. Here in the formula
of Art. 430 we put 1 for x, y for a, and ^ for n.
n{n-\) 2\2 ) 1
1.2 1.2 ""8*
n(n~l)(n-2) 2\2"^Jv2" / 1
1.2.3 1.2.3 16'
n(w~l)(n-2)(n-3) _ 2(2"v(2"^}(2~V A
1.2.3.4 "■ 1.2.3.4 ""128'
and 80 on. Thus
i
(.i^yf^iy^y-'ly^^t.^'^^y'^
As another example take (1 +^)~^. Here we put 1 for x^
2^ for a. and —^ for
n.
n
^_1 M(n~l) _3 n(n-l)(n~2) ^ 5
1.2
8
1.2.3
16
w(n-l)(w-2)(ffl-3) 35
1.2.3.4
{\+yy^=.l^\yAy%^
^~ , and so on. Thus
16^ 128*^
8
Again, expand (1 +y)~"». Here we put 1 for ar, y for «,
and -wforw
n=-m,
n(w — 1) w(m+l)
1.2
1.2
n(n~l)(n-2) _ #t(w+l)(m+2)
1.2.3
1.2.3
n(n-l)(n-2)(w~3) m(w-i-lXm+2)(m + 3)
1.2.3.4
1.2.3.4
and so on.
WHs&i
266
EXAMPLBS. XML
Tho. (1 ^y)-= 1 -my ^.^i^V- '"^^.^^ ^ jf
m(m+l)(m4-2)(m4-3) ^_
1.2.3.4 ^ •••
As a particular case sui^se m=l ; thus
(l+y)'"»=l-y+y«-y»+y*-...
This may be verified by diyiding 1 by 1 + ^.
Again, expand (1 +2dr-d^)^ in powers of x, Pat y for
2.«- «*; thus weJiave (l+2«-a5*)'~(l j-y)^
= 1+2^-8^+162^-128^+ •
\
:=l+|(2a;-«»)-|(2»-4jV+^(2«-«y-j~(2«-«»^^
Now expand (2a;-a3*y, (2a?-«2j8^,,, and collect tho
terms : thus we shall obtain
1 3
(l+2«-«*)'=l+a?-a;*+«»-s«*+...
Examples. XLII.
1. Write down the first three and the last three terms
of(«-a?)w
2. "Write down tho expansion of (3- 2;»y,
3. Expand (l-2y)'.
4. Wrif» down the first four terms in tho expansion
of(a^^-2y)". •
0. Expand (l+a?-**)*. ^
6. Expand (!+«+«*/.
EXAMPLES. XLir.
267
7. Expand (l-2i» +«?■)*.
8. Find the coefficient of o^ in the expanBion of
9. Find the coefficient ot sfi ixk the expansion of
(l-2ii;+3««)».
10. If the second term in the expansion of («+ y)" bo
240, the third term 720, and the fourth term 1080, find
Xi y, and n.
11. If the sixth, seventh, and eighth terms in the ex-
pansion of {x+yY be respectiyely 112, 7| and 2» find x^ y,
andn.
12. Write down the first five terms of the expanrion
of (a -2a?)*.
13. Expand to four terms
14. Expand (l-2a?)-\
(-S-)-'-
15. Write down the coefficient of ^ in the expansion
of(l-a?)-«.
16. Write down the sixth term in the expansion of
{2a-y)'l
17. Expand to five terms (a-SbyV: shew that if
a=l and &=- the fourth term is greater than either the
third or the fifth.
18. Write down the coefficient of of in the expansion
of(l-a?)-*.
19. Expand {lA-a-^a^^ to four terms in powers of a,
20. Expand (1 - « + «*)'* to four terms in powers of x»
W/' ■''->,i_>ka^
268
SCALES OF NOTATION.
XLIII. Scales of Notation,
432. The student will of course have learned from
Arithmetic that in the ordinary methdd of expressing
whole numbers by figures, the number represented by each
figure is always some multiple qf some power qf ten, Thng
in 523 the 6 represents 6 hundreds, that is 6 times 10';
the 2 represents 2 tens, that is 2 times 10^; and the 3,
which represents 3 units, may be said to represent 3 times
10^ see Art. 324.
This mode of expressing whole numbers is called the
cS^mmon scale qf notation, and ten is said to be the hose
or radix of the common scale. a
433. We shall now shew that any positive integer
greater than unity may be used instead of 10 for the radix;
and then explain how a given whole number may be
expressed in any proposed scale.
The figures by means of which a number is expressed
are called digits. When we speak in future of any radix
we shall always mean that this radix is some positive
/ integer greater than unity.
434. To shew that any whole number may he express-
ed in terms of any radix.
Let N denote the whole number, r the radix. Suppose
that r" is the highest power of r which is not greater than
N\ divide iV by r"j let the quotient be a, and the re-
mainder P: thus
N=ar*+P,
Here, b^ supposition, a is less than r, and P is less
than r*. Divide P by y"-^j let the quotient be &, and the
remainder Q : thus
Proceed in this way until the remainder is less than r:
thus we find N expressed in the manner < shewn by the
following identity,
iV=ar"+&r"~^+<rr"*2+ ■\-hr+h.
SCALES OF NOTATION.
269
d from
)res8ing
by each
I, Thng
168 10';
i the 3,
3 times
led the
)he hme
integer
e radix;
may be
[pressed
y radix
positive
express-
Suppose
er than
the re-
' is less
and the
Each of the digits a, h, e, h, k is less than r; and
any one or more of them uter tho first may happen to be
zero.
435. To express a given whole number in any pro-
posed scale.
By a given whole number we mean a whole number
expressed in words, or else expressed bv digits in some
assigned scale. If no scale is mentioned the common scale
is to be understood.
Let iVbe the given whole number, r the radix of the
scale in which it is to be expressed. Suppose ky A,. . .c, 6, a
the required digits, n+1 in number, begmning with thAt
on the right hand : then
J\r= ar" + &r""^ + cr""* + . . . + Ar + A;.
Divide iVby r, and let M be the quotient; then it is
obvious that itf=ar""^+&r"~'+ +/*, and that the
remainder is k. Hence the first digit is found by this^
rule : divide the given number by the proposed radix,
and the remainder is the first qf the required digits,
Agam, divide ilf by r ; then it is obvious that tho
remainder is h\ and thus the second of the required
digits is found.
By proceeding in this way we shall find in succession
all the required digits.
436. "We shall now solve some examples.
Transform 32884 into the scale of which the radix is
seven.
7132884
7 1 4697 ...5
7 1 671 ...0
7 95.. .6
13... 4
Thus 32884=1. 7*+6.7*+4.7'*+e.7"+0. 7^ + 6,
so that the number expressed in the scale of which the
radix is seven is 164605.
270
SCALES OF NOTATION.
Transform 74194 into the scale of which ilie radix \%
twelve.
12 1 74194
12 1 6182 ...10
12 1 515... 2
12 1 42 ...11
3 ... 6
Thus 74194=3. 12<+6.12«+11.12«+2. 12 + 10.
In order to express the number in the scale of whicU^
the radix is twelve in the usual manner, we require two
new svmbols, one for ten^ and the other for eleven: we will
use t ror the former, and e for the kttor. Thus the number
expressed in the scale of which the radix is twelve is
36^^. \
Transform 646032, which is expressed in the scale of
which the radix is nine, into the scale of which the radix is
eight
8 [ 646032
72782. ..4*
Th3 divisioii by eight is performed thust First eight is
^ot contained in 6, so we have to find how often eight is
contained in 64; hare 6 stands for six times nine^ that is
fifby-fouf, so that the question is how often is eight oon^
tainod in fifby-eight> and the answer Is seven times with
two oven Next we have to find how often eight is con-
tained in 25. that is how often ei^ht is contained in twenty^
three) and the answer is twice with seven t^Vet. Next we
have to find how often eight is contained tn 70, that is hoW
often eight is contained in sis^-three, and tne answer is
seven times with seven, over^ Next we have tq find boW
often eight is contained in 73> that is how often eight |S
contained in sixty-six, and the answer is eight times \dth
two oven Next we have to find how often eiffht is con-
tained in 22, that is how often eight is contained in twenty>
and the answer is tvdce with four oven Thus 4 is the first
of the required digits.. ' y
We win indicate the )*emiunder of the pfoeess ; the
student should carefully work it for himself, and then com>'
EXAMPLES. XLIIL
pare his result with that which is here obtained.
8 1 72782
81 8210. ..2
271
8 1 1023 .. .a
8|113...6
8|22...5
A • t < 3*
Thu8thenumber = 1.8' + 3.8*+6.8*+6.8»+S.8*+2.8+4,
80 that, expressed in the scale of which the radix is eight, it
is 1366324.
437. It is easy to form an unl' od number of self-
verifying examples. Thus, take two numbers, expressed in
the common sade, and obtain their sum, their difference,
and their product, and transform these into any proposed
scale; next transform the numbers into the proposed
scale^ and obtain their sum, their difference, and their pro-
duct in this scalo ; the results should of coiurse agree re<
spectively with those already obtained.
Examples. XLIIL
1. Express 34042 in the scale whose radix is fire.
2. Express 45792 in the scale whose radix is twelvOi
3. Express 1866 in the scale whose radix is two.
4. Express 2745 in the scale whose radix is eleven.
5. Multiply Ht by te\ these being in the scale with
radix twelye^ transform them to the common scale and
multiply them together.
6. Find in what scale the number 4161 becomes lOlOI.
7. Find in what scale the number 5261 becomes 40205.
8. Express 17161 in the scale whose radix is twelve,
and divide it by te in that scale.
9. Find the radix of the scale in which I3> 22, 33 are
in geometrical progression.
10. Extract the square root of ^^001> in tha scale
whose radix is twelve^
^.
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IMAGE EVALUATION
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PhotDgraphic
Sciences
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23 WIST MAIN ST^KT
WEkSTIR.N.Y. MStO
(716) •72-4503
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274
INTEREST.
XLIV. Interest.
438. The gnbject of Interest is disciissed |[n treatises
on Arithmetic; but by the aid of Algebraical notation
the rules can be presented in a form easy to undersl^d
and to remember.
43d. Interest is money paid for the use of money.
The money lent is called the Principal. The Amount at
the end of a given time is the sum of the Principal and the
Interest at the end of that time.
440. Intereeft is of two kinds, simple and compound.
When interest is char|^ed on the rrincipal alone it is called
simjpfe interest ; but if the interest as soon as it becomes
due is added to the principal, and interest chaiged on the
whole, it is called compound interest.
441. The raf« of interest is the money paid for the use
of a certain sum for a certain time. In pntctice the sum is
usually ;£100, and the time is one year; and when we say
that the rate is £4. 5s. per cent we mean that £4. 6s.f that
is £4if is paid for the use of ;£100 for one year. In tfieory
it is convenient, as we shall see, to use a symbol to denote
the interest of one pound for one year.^
442. To find the amount qf a given sum in any given
time at. simple interest.
Let P be the number of pounds in the principal, n the
number of years, r the interest of one pound for one year,
expressed as a fraction of a pound, M the number jof
pounds in the amount. Since r is the interest of one pound
for one year, Pr is the interest of P pounds for one year,
and nPr is the interest of P pounds for n years; therefore
3f=P+Pwr=P(l+wr).
443. From the equation M= P (I + nr)f if any three of
the four quantities M, P, n, r are given, the fourth caabe
found: thus \
J, M JIf-P M^P
Pa , ■ , na ^ , r=
1+nr*
Pr
'Fn
INTEREST.
273
treatises
notation
deratand
money.
nount at
and the
mpound.
is called
becomes
»d on the
r the use
le sum is
m we say
. 5f ., that
[n theory
k> denote
ny given
)al,nthe
me year,
miber jof
nepomid
Dne year,
therefore
three of
ti ciin.be
444. To find the amount qf a given sum in any
given time at compound intereet.
Let P be the number of pounds in the prmcipal, n tlie
number of years, r the interest of one pound for one year,
expressed as a fraction of a pound, M the number of pounds
in the amount Let M denote the amount of one pound in
one year ; so that E=l+r, Then PE is the amount of P
pounds in one year. The amount of PE pounds in one
year is PEE, wPE^; which is therefore the amount of P
pounds in ^too years. Similarly the amount of PE^ pounds
m one year is PE^E, or P/2', which is therefore the amount
of P pounds in three years.
Proceeding in this way we find that the amount of P
pounds in n years is PE^; that is
M^PET,
The interest gained in n years is /
Pi?»-PorP(ir~l).
445. The Present value of an amount due at the end
of a given time is that sum which with its interest for the
given time will be equal to the amount. That is, the Prin--
cipcd is tiie present value of the Amount; see Art 439.
446. Discount is an allowance made for the payment
of a sum of money before it is due.
From the definition of present value it follows that a
debt is fairly discharged by paying the present value at
once: hence the discount is eqim to the amount due
diminished by its present value.
447. To find the present value qf a sum qf mofiey due
at the end qfa given time, and the discount.
Let P be the number of pounds in the present value^ n
the number of years, r the interest of one pound for one
year expressed as a fraction of a i>ound, M the number of
pounds m the sum due, J> the discount.
Leti2=l + r.
T. A*
18
m
EXAMPLES. XLir.
At simple interest
Jf = P(l + »r), by Art. 442 ;
therefore P=
3f
l+nr*
J)=M"P=
Mnr
1+w*
At compound interest
M^PR", by Art. 444;
therefore P=~; 2>=iJf-P--^^^^.
448. In practice it is rery common to allow the
interest of a sum of money paid before it is due instead of
the discount as here defined. Thus at simple interest in-
stead of ,- the payer would be allowed Mnr fbr im-
l+nr
mediate payment.
.": ■• '-. .'V
I Examples. XLIY.
1. At what rate per cent, will £a produce the same
interest in one year as £b produces when the rate is £c
percent?
2. Shew that a sum of money at compound interest
becomes greater at a given rate per cent, for a g^yen number
of years than it does at twice that rate per cent for half
that number of years.
3. Find in how many years a sum of money will double
itself at a given rate of simple interest.
4. Shew, by taking the first three terms of the Bi-
nomial series for (1 +r)**, that at five per cent compound
interest a sum of money will be more than doubled in nfteen
years. \
MI8CELLANE0 US EXAMPLES. 27 o
,*
Miscellaneous Examples.
1. Find the Tahies when a=5 and & = 4 of
ii^+3a«&+3a6a+6», ofa^+lOab + ^, of (a-6)»,
andof(tf+96)(a-&).
2. Simplify 6a?-3 [2«+9y-2{3ar-4(y-4?)}].
3. Square 3-5d?+2aj*.
4. Divide 1 by l-or+a;* ^q fQ^j. terms: also divide
1— a? by 1—0? to four terms.
5. Simplify
6aj*-17ay+12*
6j7'-5;»-6, and
6. Find the L.O.M. of 4^-9,
-+ 2 -+-+2
7. Simplify —-—-- + -—----.
a—a a+a
- o , a?— 2 «+6 74?-6
a Solye-5- + -^ = -g-.
9. The first edition of a book had 600 pages and was
divided into two parts. In the second edition one quarter
of the second part was omitted, and 30 pages were added
to the first part; this change made the two parts of the
same length. Find the number of pages in each part in
the first edition.
10. In paying two bills, one of which exceeded the
other by one tinird of the less, the change out of a £5 note
was htUi the difference of the bills : find the amount of each
bill
11. Add together y+x;8f-g;r, »+2^~'l^*^'^2^''3^'
and firom t^e result subtiract g^-V^o^*
^ 18-2
,^
27e MISCELLANEOUS EXAMPLES.
12. If a= 1, & a 3, and tf">5, find the yalne of
2a?-6»+c'+a«(6-c)-6«(2a-c)+c»(2a+6)'
13. Simplify(a+6)*-(a+&X«-&)-'{«(2&-2)T(a^-2a)}.
14. Divide
15. Reduce to its lowest termB
«*+«»+l
16. Find the UkU. oi ii?«--9a?-l0, a?2-7a?-30,
(a?+l)(a?+3)(«-10), and««+4a?+3.
17. Simplify
__ 2 3 6
<IJ»-94J-10 «*-7^-30*"«2+4d?+3*
la Solve « 3~"T~*"6*
19. Solve |(a?-l)-|(dP+2)+j(a?-3)*:i4»
\
■I
20. Two persons A and j& own tog^ether 175 shares in
a railway commny. They agree to divide, and A takes 85
shares, while B takes 90 shares and pays ;£100 to A, Find
the value of a share.
21. Add together a^2^^y4-24&, 3a-4«-2y-8l6,
a?+y— 2a + 66&;
and subtract the result from 3«t 4 & + 3a? + 2^^
22. Find the Value of^+^7a6(2c''-a6)-(^*-3fc)*,
when<i7:3, 5e2)}andi^=2i '
23. Simplify {a:(ir+a)«-a(dJ-o)}{af(«-a)-a(d~«)}.
24. Divide --^l-g^^ by l-g; and verify the
result by multiplication. V
' 25. Find the o.CH. of a?^+ 3iB»- 10 and a?*-? 8«*#^i^
MISOELLANSOUS BXAMPLSS. 277
27. Find the l.o.m. of a^-^ 4ji^^7x-2, and
28. SoIto
2a
3
0?— 1 io?— 1
+
15
6
29. A man bouffht a snit of clothes for £4, *1$, ed.
The trowBers cost half as much again as the waistcoat, and
the coat half as much again as the trowsers and waistcoat
together. Find the price of each garment.
30. A. farmer sells a certain number of bushels of
wheat at 7#. 6^. per bushel, and 200 bushels of barley at
4s. 6d. per bushel, and receives altogether as much as if he
had sold both wheiskt and barley at the rate of 5f. 6d, per
bushel How much wheal did he sell ?
31. If a=l, 6=2,c=-|,<?=0,findtheV5gueof
a-h+c ad-he
«-6-c bd+ac
"VV^ "*<:»/
■^*'
32. Multiply together d;~a, x-bj x+a, and a+h;
and divide the result hy a^+a{a+b)+ab,
1
33. Divide Sa^''a^+^t^hy2a-\'tf.
34. Find the O.O.M. of 4F(aj»+10)-26a?-62 and
a^-lx+lO.
12a*— 16«y+3w*
35. Reduce to its lowest terms e^^6^4.2a?y»-2y» '
36. Simptify
1 +
*--!
37. Solve
9 4
2ar-l . 2-3d? ^
+ — r::— = 0.
14
30
,, ..;^!,,r^-.
\
278 MISCELLANEOUS EXAMPLES,
-, , Zx-l «+4 &»-l
88. Solve -^ r-^-W
^ 39. ^ can do a piece of work in one hour, B and
each in two hours: how long would 4» A |u»d C/take,
working together 1
40. ^ having three times as much money as B gave
two pounds to S, and then he had twice as much as B
had. How much had each at first?
41. Add together 2a?+3y+48r, w-Zy+Sz, and
42. Find the sum, the difference, and the product of
3«2-4ajy + 4y' and 4a^+2xy- 3y*.
« 43. Simplify
2a-3(6~c)+{a-2(6-c)}-2{a-3(6-(?)}.
44 Find the O.O.M. of
»*+67«^+66 and a^+2aj«+2aj»+2a?+l.
45. SimpUfy ^ X ^j^j-^^^^_^.
/ 46. FrndtheiuCM. ofaj2~4,ic2-6a?+6, and«2-9.
47. Beduce to its lowest terms g^^na^, 10^+7 •
48. Solve 3(d?-l)~4(a?-2)=2(3-a?).
49. Solve ^(9+4;i?) = 6-2V^.
50. How much tea at Ss, dd, per Ih. must be mixed
with 45 lbs. at 3«. 4d, per lb. that the mixture may be
worth 3«. 6dL per lb. ?
61. Multiply 3a*+a6-&2 \^j a^-2db-^i\ and divide
the product by a + &.
5i Find the g.o.m. of 2a: (a?- 3) +3 (or- 68) +15 and
2«»-6i»»-6a?+15.
53. Simplify , + , *
1-
1 + 4?
1-0?
MI80BLLANE0US EXAMPLES.
279
' - o , 1.2 a»+3 i-a** y „ ^ ^
y X xy SB a
«- fl 1 3 7 , 2 26
56. Solyea^ + ^=:^, 8«-y = -3.
57. Sol?e2(4r-3)-|(y-3)=3,
3(y-5)+|(«-2)=10.
^ 58. Sol?e
, 7yj»=10(y+;y), 3*a?=4(4f+«), 9a:ys:20(4;4-y).
59. Solve - + -=m, ----n.
X y X y
60. The denominator of a certain flraction exceeds the
numerator by 2 ; if the numerator be increased by 5 the
fraction is increased by unity : find the fraction.
61. Divide ic'-p by i»---.
62. Reduce to its lowest terms
3.V^-49ar-10
210^8 I4«*-29;»-10*
63. SimpUfy /a-^y^g + i-l).
^ x'
64 Solve 3(a?-l)+2(a?-2)=«-3.
65. Solve
a?~l _ y+1 2£-3 __ 13 -2^
3 ~ 4 * 5 7 '
66. Solve 5ar+2=3y, 6ajy-10a5*+^^^=8.
67. Sdye^-^=3, 3J^^9M=|.
il
i
.,*
■ 'Tfer^fn
280 . MISCELLANEOUS EXAMPLES
6a Solve ^/(^+40)=«+4.
69. Solve
or+l
w+2
T
70. A father's age is double that of his lOft ; 10 yean
ago the father's ase was three times that of his son: find
the present age of each.
71. Find the value when ^s 4 of
V(2..I)-(..-^)-(3-4=).
72. Reduce !^"!?^'*"??^'"? to its lowest terms;
and find its value when /v >-• 3.
I
if
5f *»IN
< /
I
*li^
II
S4
73. Resolve into simple factors o^ - So; + 2, o^ - 7^ + 10,
and«'*6a;+0.
t 74. Simplify ^__^^2 + ^-7!^+l6";c«-L+5-
75. Solve i(3;»+y)-^(4^-.2|)=|(6a;-i).
76. Solve 9«»-63aT+68=0.
4
77. A man and a boy being paid for certain days' work,
the man received 27 shillings and the boy who had been
absent 3 days out of the time received 12 shillings : had the
man instead of the boy been absent those 3 days they would
both have daimed an equal sum. Find the wages of each
per day.
78. Extract the square root of 9a;* - e^e* + 7^— 2;v + 1 ;
and shew that the result is true when x=s 10. '
79. Ifaibiie: d, shew that
80. If a, bfCfdhe in geometrical progression, shew that
a'+<^ is greater than J* +c*. \
81. If n is a whole positive number 7*"'*'* + 1 is divisible
by8.
MISCELLANEOUS EXAMPLES. 281
81 Find the least common mnltiple of «'-4^,
A'+CaV+lJto^'+Sy*, and d^-e«V+12«ir-8y*.
83. Solye | + -
14 3.
2' *'"y'=^*-
84. Solve «■+a» + 2^/(d^+2ar + l)=47.
1,
80. The sum of a certain number consisting of two
digits and of the number formed by reversing the digits is
121 ; and the product of the digits is 28 : find the number.
86. Nine ffallons are drawn from a cask Aill of wine^
and it is then filled up with water ; then nine gallons of tho
mixture are drawn, and the cask is again filled up witii
water. If the quantity of wine now in the cask be to the
qiiantity of water in it as 16 is to 9, find |iow much the cairic
holds.
87. Extract the square root of
16aj« + 25^- 304?^* - 24a?*y" + 9a?V* + 40«V'
88. In an arithmetical progression the first term is 81,
and the fourteenth is 159. In a ^ometrical progression
the second term is 81, and the sixth is 16. Fmd the
harmonic mean between the fourth terms of the two pro-
gressions.
89. If ^/5= 2*23606, find the value to five places of
decimals of -7^ — r, ,
90. If ^ be greater than 9, shew that kjx is greater
than^(a?+18).
91. Divide (a?-y)»-2y(4?«y)«+y«(a?-y) by (j7-2y)«.
92. Find the ao.H. and the l. cm. of
24(«'+«V+*^+^) and 16(«'-a?'y+^-y').
93. Simplify
» . y . _J 1
(I
#
'.*;>■,
282 MiaCMLLANBOUS EXAMPLES.
4
//
94. Solve -33- + -y-=8 ip.
96. Solve
4jy+20(«-y)=0, y«+30(y-*)=0, 84T-2Jrf>0.
96. Solve 3«*-2ar+V(av>-4ar-6)=18-f2«.
97. ^ rowf at the rate of 8) miles an hour. He leaves
Oambridge at the same time that B leaves Ely. A spends
12 minutes in Bly and is back in Oambridge 2 hotkrs and
20 minntes after B gets there. B rows at the rate of 7^
miles an hour; and there is no stream. Find the distance
firom Oambridge to Ely.
98. An apple womim finding that apples have this
year become so much cheaper that she could sell COimore
than she used to do for five shillings, lowered her price and
sold them one penny per dozen cheaper. Find the price
perdosen.
99. Sum to 8 terms and to infinity 12 + 4 + 1} 4- ...
100. Find three numbers in geometrical progression
such tiiat if 1, 3, and 9 be subtracted from them in order
they will form an arithmetical progression whose sum is 15.
101. Multiply «?*-«•+«* --«*+«^-«+«*-l by «*+l;
and divide 1-0^ by l-ar».
102. Find the L.aM. of a^-a^, ^+a', ^-l-aV+a^
a^-aa^'-a^w-^c^, and a^+oi^-a'a?- a*.
2P"^
103. Simplify
a»-6»+
1 +
a+b
a-b
104. Solve
6
5(|.?)-|(3.^)=
4ay-14 . «+10 i
— :: — +
10
105. Solve -—-+-—:
«-l «-5
18
a+l «+5
MISCELLANEOUS EXAMPLES. 288
loe. Solve
afy--y£-zx=47.
1Q7. A and B travel 120 miles together by roil. B
intending to oome back again talces a return ticltet for
which he pays half as much again as A: and they find that
B travels cheaper than A by it, 2d. for every 100 mUes.
Find the price of A'§ ticket '
108. Find a third proportional to the harmonic mean
between 3 and z, and the geometric mean between 2
and 18.
109. Extract the square root of
y\ yj «\ a yj
no. If a :&;:&: c, shew that b*=: .f "V^t'*'^-* *
111. Dhideai-a'i hyJ-a'^.
112. Reduce
«»+3aj»-20
ar*-«»»-12
find its value when «=2.
to its lowest terms, and
113. Solve
af-3
13
3
a+2
a-4 3 3(6-«)'
114. Find the values of m for which the equation
mV+(m'+m)aa;+a^=0 will have its roots equal to one
another.
Solve 8a^+;c*= 10, 5ay-2a^=2.
115.
116.
Solve i + i=6, 2 + ^^
Of y * y w
2*.
117. Fhid the fraction such that if you quadruple the
numerator and add 3 to the denominator the fraction is
doubled ; but if you add 2 to the numerator and quadruple
the denominator l^e fraction is halved.
ti
284i MISCELLANEOUS EXAMPLES.
118. SimpUfy {-(aj»)*}"*x{~(-a?)-»}*.
119. The third term of an arithmetical progression is
IS; and the seventh term is 30: find the sum of 17 terms.
120. If — g-, J, -g- be in harmonical progression,
shew that a, &, c are in geometrical progression.
1
121. Simplify a-
&+
6+
ah
a—b
122. Extract the square root of
37a?V-30ajV+9^-20aJ2^+4y*. i
123 Besolve 3a^~ 14a^ - 24^ into its simple factors.
,«^ a 1 ^+5 3(5a?+l) 4 ^,
124. Solve s r ^ — — r^ = s T -24.
207-1 6d?+4 2«-l ^
125. Solve aj»+i = ^.
126. Solve a^-y^=^9, a?+4=3(y-l).
127. Solve y+^/(dJa-l)=2, V(«+l)- V(a?-1)= Vy.
128. If a, &, c, (^ are in Qeometrical Progression^
a : b-^d :: c* : c^+cP.
129. The common difference in an arithmetical pro-
gression is equal to 2, and the number of terms is equid to
the second term : find what the first term must be that the
sum may be 35. /
130. Sum to n terms the series whose m^ term is
2x3'",
ni Simnlifv l+N/a~2^) , ^-N/a-2^ )-
131. Smiphfy J— ^^j-^+ ^ ^ .
132. Find the o.CM. of 30aj*+16iB'-50o;*-24a? and
24o?*+14aj»-48aj"-32o?.
)88ion is
I terms.
ression,
MISCELLANEOUS EXAMPLES
133. Solve «'-a?-12=0.
285
134. Form a quadratic equation whose roots shall be
Sand -2.
135. Solve «*+-=«*+ -\.
Or a*
136. Solve
a?^
= 1 +
V(aj2 + 6) "^^/(dJ2+5)•
137. Having given ,y/3= 1*73205, find the value of
to five places of decimals»
s/3-1
138. Extract the square root of 61 - 28 ^/3.
A* J. «#
139. Find the mean proportional between — - and
140» If a, 6, e be the first, second and last terms of an
^arithmetical progression, find the number of terms. Also
find the sum of the terms.
/'
141. If d, c, h, a are 2, 3, 4, 5, find the values of
a+b+e ah-cd , /a-l
142. In the product of l+4a?+7^+10;»' + l5a:* by
1 + 6a? + 9d;8 + 13^?* 4- 17a:*, find the coefficient of «*.
Divide 2ii»»-2a?*-70a;*-23a?«+33a?+27 by 7a:"+4a?*9.
«*-&* a-b
"^ ^^^^WTM^¥:^k>
and
fjx-tja'' tjx^r nja" x-\-a
286 MISCELLANEOUS EXAMPLES.
144. Solve the following equations:
(1)
(2)
60-a? 3^-6 24~3;i;
"14 7~=^ — r~"
a?+4 5d? + 12
a?+3""43a? ^ '
, . 3a?-t-5y . 5^-3y ^ ^+l_?
^^^ ~20r"^~"8~""^' y + 2~3-
145. Solve the following equations :
(3) 3«2-4a?y=7, 3a!y-4y2=6.
146. A bill of J2(} is paid in sovereigns and crowns,
and 32 pieces are used: find how many there were of each
kind.
147. A herd cost £180, but on 2 oxen being stolen, the
rest average £\ a head more than at first : find the number
of oxen.
148. Find two numbers when their sum is 40, and the
«•
sum of their reciprocals is 7t .
48
149. Find a mean proportional to 2^ and 5|; and a
third proportional to 100 and 130.
150. If 8 gold coins and 9 silver coins are worth as
inuch as 6 gold coins and 19 silver ones, find the ratio of
the value of a gold coin to that of a silver coin.'
151. Remove the brackets from
(a? - a) (a? - 6) (a? - c) - [&c (oj - a) - {(a + & + c)>a? - a (6 + c)} a?].
152. Multiplya+24^(a2&)+2^/&bya-2y(a^)+2V6.
153. Fmd the o.cm. of aJ*-16«'4-93«"-234a?+216
and 4«'- 48aj" + 186a? - 234.
MISCELLANEOUS EXAMPLES. 287
154. Solve the following equations :
13a?-l
(1)
(2)
4 •
28-6d? _^ 3a?+l
= 17-
3
2a?-8
8
3a?+9 3a?-f3'
(3) .-j,=3, 3(14)=llg-i).
155. Solve the following equations: «.
(1) ^/(a? + l)+^^(2ar)=7.
(2) 7a?- 20^0?= 3.
(3) 7djy-5«a=36, 4ajy-3y8=105.
156. A boy spends his monej in oranges ; if he had
bought 5 more for his money they would have averaged
an half-penny less, if 3 fewer an half-penny more : find how
much he spent.
157. Potatoes are sold so as to gain 25 per cent, at
6 lbs. for bd, : find the gain per cent when they are sold at
5 lbs. for 6(^.
158. A horse is sold for £2Ay and the number ex-
pressing the profit per cent, expresses also the cost price
of the horse : find the cost
159. Simplify V{4a2+V(16a2a;2+8aaj'+ a?*)}.
160. If the sum of two fractions is unitv, shew that the
first together with the square of the second is equal to the
second together with the square of the first.
161. Simplify the following expressions :
a- [&-{«+ (6 -a)}],
25a-19&-[36-{4a-(6&-6c)}]-8a,
[{(a-")-}-^-L{(«*")-"}*"].
288 MISCELLANEOUS EXAMPLES.
I
162. Find the o.CM. of 18a«-18a24?+6«a:*-6a^, and
163. Find the L.O.M. of 18(a^-y2), I2{x-yf, and
24(^ + 2/8).
164. Solye the following equations :
2af-4 . 3a?-2
(1)
(2)
(3)
=7.
9a? +20 _ 4jy-12 x
36 "■5a?-4 4'
ill
2
r
(
(4) 2(d?-y)=3(a?-4y), 14(d?+y) = ll(a?+8).
165. Solve the following equations :
(1) 32a?-5a?2=12.
(2) ^/(2a?+3)^/(;l?-2)=16.
(3) a?2+y2=290, ajy=143.
(4) 3;c2-42/8=8, 6«8-6ii?y=32.
4^ 166. ^ and B together complete a work in 3 days
which ^would have occupied A sione 4 days: how long
would it employ B alone 1
2
167. Find two numbers whose product is - of the sum
of their squares, and the difference of their squares is
96 times the quotient of the less number divided by the
greater.
168. Find a fraction which becomes ^ on increasii^g its
numerator bgr !» and ^ on similarly increasipg its denomi-
naton
MISCELLANEOUS EXAMPIES. 289
169. U a \h \\ c \ df shew that
1 + 1 .l-l..l + i . 1«1
170. Find a mean proportional between 169 and 266,
and a third proportional to 25 and 100.
171. Bemove the brackets from the expression
&-2{6~3[a-4(a-&)]}.
172. Simplify the following expressions :
y xy a^ arV '
(j'' - fl' — *w)p — (w + ^ -^) 3' + (g + w) wi + m (/? — m) + j'*,
173. Fmd the o.o.m. of ar*+<M;'-9aV+lla»d?-4a*
and 0?* - aaj* - 3a V + 6a?« - 2a*.
174. Solve the following equations :
ai?+l x+7
m
(2)
(3)
(4)
X—
3
10j?+1?
18
6 •
12a? +2
13a?-16
6a?-4
9^+1^=70, 72^-i|^=44.
6a?+7 2a?+19
3a?+l x+7
1 75. Solve the following equations :
(1) ;p+4..?:£Z§ = 3.
^ ' X
(2) 2aj«-3y2=2, xy=20,
(3) 2y2-ic2=i, 3«a-4a!y=7.
(4) x-\ y^6, i»»-f y»-126,
T. A.
ti
19
290 MISCELLANEOUS EXAMPLES.
176. "When are the dock-bands at right angles first
after 12 o'clock ?
177. A number divided by the product of its digits
gives as quotient 2, and the digits are Inverted by ad£ng
27: find the number.
178. A bill of ^£26. 16«. was paid with half-guineas and
crowns, and the number of half-guineas exceeded the num-
ber of crowns by 17 : find how many there were of each.
179. Sum to six terms and to infinity 12+8 + 5^-1-....
180. Extract the square root of 66 — 7 ^/24.
n/3 + 1
»y3— 1
.., - , and y=^^— — - , find the value of
181. Ifi» =
182. Reduce to its lowest terms
3a?«-16a?-12
««-8iB*-12a?4-144*
183. If two numbers of two digits be expressed by the
same digits in a reversed order, shew that the difference of
the numbers can be divided by 9.
184. Solve the following equations:
3a?-3 3i»-4 21-4a?
(1)
(2)
(3)
4
2a?+3y
+ f=8.
9 •
-3a?
-y=ll.
. 14-a? .,
4a? r- = 14.
«+l
185. Solve the following equations:
(1) V(«+3)x^/(3a?-3)=24.
(2) ^/(a?+2)-^..y(3a?+4) = 8. ,
(3) «*-«2(2a?-3)=2a?+a
186. Find two numbers in the proportion of 9 to 7
such that the square of their sum snail be equal to the
cube of their difierencoi
MISCELLANEOUS EXAMPLES 291
187. A traveller sets out from A for B^ going 3} miles
an hour. Forty minutes afterwards another sets out from
B for Ay goinpf 4j^ miles an hour, and he goes half a mile
beyond the middle point between A and J? before he meets
Uie first trayeller; find the distance between A and B,
188. Two persons A and B play at bowls. A bets B
four shillings to three on every game, and after playing a
certain number of games A is the winner of eight shillings.
The next day A bets two to one, and wins one game more
out of the same number, and finds that he has to receive
three shillings. Find the number of games.
189. If m=«-a?~^and»=y-y~*,
shew that iwn+V{(«»'+4)(n*+4)}=2rajy+— j.
190.
9 3 3
Sum to nineteen terms 7 + 5 + 7+ ....
4 ifi 4
191. Multiply --^ + 1 by j + 5-2'
Divide ?^-4»*+^«^-^«8-^a?+27by^-ar+a
192. Reduce to its lowest terms
4a?»-27a^+58a?-39
aj*-9«"+29«*-39iP+18*
193. Fuid the L. 0. M. of «?"+ 24?*y + ^'^■^ 8^ and
laj'-2«^+4a:y*--l
194. Solve the following equations :
(1) l(,,+6)-i(l6-3;ir)=4j.
6a?- 9 23-2i»
(2)
13
9
=3a?-20.
(3) ^(«+y)=J(2«+4), |(a?-y)-|(a^-24).
3
2
19—2
292 MISCELLANEOUS EXAMPLES,
ii
195. Solve the following equations :
(1)
|(^-3)=^(aF-3).
(2) K/(^ + 3) + /s/(3a?-3) = 10.
(3) a?+y=6, (aj2+y2)(-B3 + y8) = i44o.
196. The express train between London and Cam-
bridge, which travels at the rate of 32 miles an hour, per-
forms the journey in 2k hours less than the parliamentary
train which travels at the rate of 14 miles an hour: find
the distance. ^
197. Find the number, consisting of two digits, which
is equal to three times the product of those digits, and is
also such that if it be divided by the sum of the digits the
quotient is 4.
198. The number of resident members of a certain
college in the Michaelmas Term 1864, exceeded the num-
ber in 1863 by 9. If there. had been accommodation in
1864 for 13 more students in college rooms, the number in
college would hiEive been 18 times the number in lodgings,
and the number in lodgings would have been less by 27
than the total number of residents in 1863. Find the
number of residents in 1864.
199. Extract the square root of
a*-2«'&+3a^-2a6'-f &*, .
and of (a + 6)*-2(a2+&2)(a+5)2+2(a4+&4).
'■ 200. Find a geometrical progression of four terms
such that the third t^gpm is greater by 2 than the sum of
the first and second, and the fourth term is greater by 4
than the sum of the second and third.
201. Multiply 8-3^+
by 9 -2a? 4
38ar-6a?»-58
7~2a?
7aj*-65 + 30aj
t ■.-■.
6-3« '^
202. ^ FmdtheG.C.M.ofir*-K4«*-H6anda:*-^-Har-a
203.
Take
204.
205.
206.
at the ra
station
then 4^
a mile tr
been out
207.
units dig
is to the
208.
and is s
equal to
worth at
the sum.
209.
210.
first ten
MISCELLANEOUS EXAMPLES 293
nd Cam-
lour, per-
iamentary
our: find
fits, which
ts, and is
digits the
a certain
the num-
>dation in
lumber in
t lodnngs,
ess by 27
Find the
)ur terms
he sum of
eater by 4
203. Add together
1
Take
l+x+afl
from
1 &P~5 a^-x^-^
2 + 3a?* (2 + 3a?)«» (2 + 3d?/ *
1
204. Solve the following equations :
(1) — g— =39-6a?.
(2) (a + &)(a-a?)«a(&-a?).
(3)
16
12
2i/=3.
I'+ar-a
205. Solve the following equations:
(2) 4(a^+3a?)-2^/(4?«+3a?) = 12.
(3) aj2+icy=16, ^+ary=10.
206. A person walked out from Cambridge to a village
at the rate of 4 miles an hour, fmd on reaching the railway
station had to wait ten minutes for the train which was
then 4^ miles off. On arriving at his rooms which were
a mile trom the Cambridge station he found that he had
been out 3^ hours. Find theMistance of the village.
207. The tens digit of a number is less by 2 than the
units digit, and if the digits are inverted the new number
is to the former as 7 is to 4 : find the number.
208. A sum of money consists of shillings and r^wns,
and is such, that the square of the number of cro\. .i^ a.
equal to twice the number of shillings; also the sum * .
worth as many florins as there are pieces of money: find
the sum.
209. Extract the square root of
4il?* + 8aaJ3+4a24^^ + 16&«a;2^.lgaJ2^+1654, .
210. Find the arithmetical progression of which the
first term is 7} and the sum of twelve terms is 348.
( f
294 MISCELLANEOUS EXAMPLES
ii
211. Divide C
212. Multiply
^ ^ 'l2+41a^+36«*
8 + 60?--
25;rV+47«V-^5«y+62«y*-46y»
by6-2«+
264r-8««-14
4+7«
213. Reduce to its lowest terms
4aj»-46a^ + 162a?- 185
3-44T
«*-16«»+81d?*-186aT+150'
214. Solve the following equations:
(1)
3^-2 1-ga?
5 11
=9.
(2) a?+-y=l7, y+^«=8.
18'
215. Solve the following equations :
I 1
6*
(1) i
a?+3
(2) lO«y-7^=7,' 5y2-3a?y=20.
(3) a+y=6, ;b*+^=272.
216. Divide ;£34. 4s. into two parts such that the num-
ber of crowns in the one may be equal to the number of
shillings in the other.
217. A number, consisting of three digits whose sum is
9, is equal to 42 times the sum of the middle and left-hand
digits; also the right-hand digit is twice the sum of the
other two: find the number.
218. A person bought a number of railway shares when
they were at a certain price for ;£2626, and afterwards
when the price of each share was doubled, sold them all
but five for ^£4000: find how many shares he bought
220.
221.
mi
222.
223.
2la;8-2
1^24.
225.
226.
rate of
at the I
tanceu
he run'
227.
to bed
daught
MISCELLANEOUS EXAMPLES. 295
By*-46y»
-14
\.
the num-
iumber of
me sum is
left-hand
m of the
ires when
fterwardfl
them all
ugbt
219. Four ntimbera are in arithmetical proffrewioii;
their sum is 60, and the product of the second ana third is
156: find the numbers.
220. Extract the square root of 17 + 12 j^
221. Divide aj»- 1 by 4b*- 1 ; and
tn{qa!^-'ra!)+p(mx*—na^—n(qx~r) hjma-n,
222. Simplify
a^bx—b^afi
a»-ft2_^.26c
223. Find the L. 0. M. of 7^- 44^-2107+ 12 and
21aj8-26«+8.
^24. Solve the following equations :
a»-4 2-a»
(1)
(2)
(3)
=7.
7 6
174?-I3y=144, 23a?+19y=890.
l^l^l 1 1_1 1_1_8_
^""y""8* w z~'9^ z y 72'
225. Solve the following equations :
(1) -^_ 21^=1
^ ' 100 25d? 4*
(2) •0076a!« + '75d?=160.
(3) J{a!+y) + s/ia!-y)==^c,
b{x-a)+a{b'-y)=0.
226. A person walked out a certain distance at the
rate of 3^ mues an hour, and then ran part of the way back
at the rate of 7 miles an hour, walking the remaining dis-
tance in 5 minutes. He was out 25 minutes : how Du* did
he run?
227. A man leaves his property amounting to ^£7500
to be divided between his wife, his two sons, and his three
daughters as follows: a son is to have twice as much as
II
296 MISCELLANEOUS EXAMPLES.
A tlanghter, and the widow £60^ more than all the five chil-
dren together : find how much each person obtained.
% 228. A cistern can be filled by two pipes in 1} hours.
The larger pipe by itself will fill the cistern sooner than
the smiuler oy 2 hours. Find what time each will sepa-
rately take to fill it.
229. The third term*of an arithmetical progression is
four times the first term \ and the sixth term la 17 : find
the series.
230. Sumtonterms3j+2i+l}4-...
231. Simplify the following expressions :
_6 <g + & g'+^g
a+h" 2a 2a(a-&)*
\
a«-a&+&2
a2-52
«8+lla?+30
232. Reduce to its lowest terms
//
9«»+63a!»-9«-18*
233. Solve the following equations :
1 7
(1)
(2)
(3)
1 \_
x'^2x 3a? ~ 3*
l+dT l—X
= 8.
4a+5y _ 2x
~40 ^"^' ~
3 '-^^»-l:
234. Solve the following equations :
(2) a«2+62+c»=a2+25c+2(5-c)jp^/dE.
(3) /v/(a?+y)+V(a?-y)=4, «*+y2=41.
MISCELLANEOUS EXAMPLES. 297
235. A body of troops retreating before the enemy,
firom which it is at a certain time 26 miles distant, marches
18 miles a day. The enemy parsnes it at the rate of 23
miles a day, but is first a day later in starting, then after
two days' march is forced to halt for one day to repair a
bridge, and this they hare to do again after two days'^more
marching. After how manv days from the beginning of the
retreat will the retreating lorce be overtaken 1
236. A man has a sum of money amountins to ;£23. 15#.
consistinff only of half-crowns and florins ; in all he has 200
pieces ofmoney : how many has he of each sort ?
y 237. Two nmnbers are in the ratio of 4 to 5 ; if one is
increased, and the other diminished by 10, the ratio of the
resulting numbers is inyerted : find the numbers.
•
X: 238. A colonel wished to form a solid square of his
men. The first time he had 39 men over; the second time
he increased the side of the square by one man, and then
he found he wanted 60 men to complete it. Of how many
men did the regiment consist ?
239. Extract the square root of
«? + 2a»6 + 3a*6* + 4a«6« + 3a2&* + 2a&» + Z)«,
and of a'+46*+9c^+4a&+6a<J+125<?.
240. Multiply «V^ - 2ary + 4a?M by x^ + 2yK
241. Simplify
40iry-(9a?-8y)(5d?+ 2y) --(4y-3a?)(16a?+4y),
, 1+d? . 1-a? l-a?+aj* l+a?+«2
and -z + -— s : :;— +2.
l-X 1+0?
\+a^
l-a^
242. Find the o.o.m. of o^+aix?'\-2a^ix?'¥^x-¥a^y
and a?* + 00^3 + 2aV + ^^x + aWx + a* + aV;^,
243. Two shopkeepers went to the cheese fair with the
same sum of money. The one spent all his money but &8,
in buying cheese, of which he bought 250 lbs. The other
298
MISCELLANEOUS EXAMPLES,
bought at the B&me price 850 lbs., but was obliged to
borrow 35«. to complete the payment. How much had
they at first ?
244. The two digits of a number are inverted; the
number thus formed is subtracted from the first, and
leaves a remainder e(psX to the sum of the digits; the dif-
ference of the digits is unity: find the number.
245. Find three numbers the third of which exceeds
the first by 6, such that the product of their sum multi-
plied by the first is 48, and the product of their sum mul-
tiplied by the third is 128.
246. A person lends ;£1024 at a certain rate of
interest ; at the end of two years he receives back for his
capital and compound interest on it the sum of j£115G :
find the rate of mterest. \
247. From a sum of money I take away ;£50 more
than the half, then from the remainder £30 more than the
fifth, then from the second remainder £20 more than the
fourth part; at last only £10 remains: find the origimd
sum.
248. Find such a fraction that when 2 is added to the
idumerator its value becomes ^, and when 1 is taken from
o
the denominator its value becomes 7 .
249. If I divide the smaller of two numbers by the
f eater, the quotient is *21, and the remainder is *04162 ; if
divide the greater number by the smaller the quotient is
4, and the remainder is '742 : find the numbers.
250. Shew that ^-^^-^ — \-^ = -r — 7. \
«? + y a;* + i/a »
^: \
251. Simplify- v j
- 6a+[4a-{8&-(2a+4&)-226}-7&] "^
-[7&+{8a-(36+4a) + 86}+6a].
252. Multiply a—x successively by a + a?, a* + oj*, a* + a?*,
«'+«*; also mmtiply a*""" ft""' by a*""* 6'*" c
MISCELLANEOUS EXAMPLES 299
253. Find the O.O.M. of 45a'a7+3aV-9aa^+6;i?*and
254. Solve the following equations :
a?-2 ir+23 10 + d?
(1) X-
(2)
(3)
3
y
I'-ti^^'^
2-7-46.
a-a=^{a^-a!j{4a^-7x^).
255. Divide the number 208 into two parts, such that
the sum of one quarter of the greater and one third of the
less when increased by 4, shall equal four times the diffe-
rence of the two parts.
256. Two men purchase an estate for ^£9000. A
could pay the whole if B gave him half his capital, while B
could pay the whole if A gave him one-third of his capital:
find how much money each of them had.
257. A piece of ground whose length exceeds the
breadth b^ 6 yards, has an area of 91 square yards : find
its dimensions. * * ;
258. A man buys a certain quantity of apples to divide
among his children. To the eldest he gives half of the wholes
all but 8 apples; to the second he gives half the remainder,
all but 8 apples. In the same manner also does he treat the
third and fourth child. To the fifth he gives the 20 apples
which remain. Find how many he bought
259. The sum of two numbers is 13, the difference of
their squares is 39 ; find the numbers.
260. A horse-dealer buys a horse, and sells it again for
j£l44, and gains just as many pounds per cent, as the horse
had cost him. Find what he gave for the horse.
261. Simplify , , ...
I
"<'
SOO MISCELLANEOUS EXAMPLES.
262* Multiply ^+aj«+«*+i»*+l by aj*-l; fmd
a 2x , , X 2a _
1 by r-+l.
07 a
X
263. What quantity, when multiplied by 0?—-^ will
give«._^_(^_iy,
264 Simplify the following expressions:
3a?'-13aj2+23a?-2I
6a:3+i»2__44^+21 '
-6
26 '
//
\ a^-h a-h 26^ \a
t2(a-6) 2(a+6)'**a2-.&2j j
265. Solve the following equations :
/i\ ga?+3 . 2^-3
(2) ^/(3+^)+^/^=;7^.
(3) y + 9y=91, ^ + 9a:=167.
266. Solve the following equations:
(1) i»»-a?-6=0.
Vix
a?+l a? + 2 2a?+13
.r
^ -^ x-l x-2 x+l
(3) a!^-'xp+y^==7, X'hy=5.
267. The ratio of the sum to the difference of two
numbers is that of 7 to 3. Shew that if half the less be
added to the greater, and half the greater to the less, the
ratio of the numbers so formed will be that of 4 to 3.
268. The price of barley per quarter is 1^ shillings
less than that of wheat, ana the value of 50 quarters of
barley exceeds that of 30 quarters of wheat by £7. 10#.:
find the price per quarter of each.
/-'
MISCELLANEOUS EXAMPLES,
269. Shew that
={bc-ad)(ca''hd){ab-ed),
270. Extract the square root of
301
r /
and of
12 3 9'
33-20^2.
271. If a=y+z-2Xj b=z+x—2p, 03idc=a-hp~2z,
find the value of l^+c^+2bc-aK
272. Divide «*-21a?+8 by l-3^+a;».
273. Add together ;; — -, ::-rzy and
_, , Sa+x - 27a* +3<w? +7^:2
Take — — - from
a^-a^'
3a~a
16a«+aa?-2«8 *
274. Multiply 3a?- 77 Z^ by ix- ^^ *^
4a-^Zx
»*"d«^-n^''y^+A
6a -2^7
275. Simplify
a +
and
a«
'" ax a^
6+
2
a
'« "^ oo? "*■ a;*
C + dJ
276. Solve the following equations:
^ ' X X X
(2) 6y-3«=2, 8y-6a?=l.
(3) — ^ 2—1, 3+2-*-
302 MISCELLANEOUS EXAMPLES.
277. Solve the following equations:
(1) a2(ii?-a)«=62(a?+a)«.
... X 6ay+l „
(3) ^/(13^-l)-s/(2a?-l) = 6.
278. A person walked to the top of a mountain at the
rate of 2^ miles an hour, and down the same way at the
rate of sl miles an hour, and was out 5 hours : how far
did he wiQk altogether 1
279. Shew that the difference between the square of a
number, consisting of two digits, and the square of the
number formed by changing the places of the digits is divi-
sible by 99.
280. If a : & :: c : (^ shew that
281. Find the value of ^^r/!"f» 4- n/{^<^"(^-^)? ,
ivfaena=3, 6=4.
-■,... I r ?- ' . \"
282. Subtract (6 —a)(C'- d) from {a-b)(c--d): what is
the value of the result when a = 26, and d=2c1
283. Reduce to their simplest forms :
V-2<w?-24a^ J. x-y x , y V
-1 — ;;: TT'i ftJid — : — — H . ;
«r— 7<w?— 44<r x-\-y x—y y—x
I
284. Solve the equations:
4 19
(1)
(2)
S+of X *Jx*
Sx-Zy x-y
= 1 ^ + 2^-4 \
6 ^ 2 '32
(3) ^/(2^-l)+^/(3«+10)=^/(ll«+9X
MISCELLANEOUS EXAMPLES. 303
285. Solve the equations:
(1) 10;»+j4i=9-
(3) a^-a?y+y«=7, 6a?-2y=9.
286. Id a time race one boat is rowed over the course
at an average pace of 4 yards per second ; another moves
over the first half of the course at the rate of 3^ yards per
second, and over the last half at 4^ yards per second,
reaching the winning post 15 seconds later than the first.
Find the time taken by each.
287. A rectangular picture is surrounded by a narrow
frame, which measures altogether ten linear feet, and costs,
at three shillings a foot, five times as many shillings as
there are square feet in tne area of the picture. Find the
length and breadth of the picture, .
288. li a\h v.cid, shew that
a-^h-¥c-\-dia-k-h-c-di\a—h-\-c-d :a— 6-c+dL
289. The volume of a pyramid varies jointly as the
area of its base and its altitude. A {pyramid, the base of
which is 9 feet square, and the height of which is 10
feet is found to contain 10 cubic yards. Find the height
of a pyramid on a base 3 feet square that it may contain
2 cubic yards.
290. Find the sum of n terms of the arithmetical pro-
'^\.
291. Find the value of a'-J^+c'+Satc, when a=03,
J=-l, c=07.
304 MISCELLANEOUS EXAMPLES
4\
292. SimpUfy(^^:^^^|^±^-a«, and 8hew that
-.(a+&+c){a2(&-p) + &8(c-a)+c»(a-&)}=0.
293. If a+b+c=0, shew that a^-i-b^+c^=3abc.
294. Reduce to its lowest terms ^
a?*+2aj3+6;i?-9
//
295. Solve the following equations:
n\ 10^+17 12a?+ 2 _ 5x-4
^^ 18 13^-16" 9 "
(2) 6a?-5y=l, y-x=lZ-
„ i.
(3) f+8y=66, 1+8^=129.
o o
296. Solve the following equations :
a?+l 3a?+l
' f^ ■■■■^
^■.j
»*■ 't>fi
.»,. .^#-
(1) '^4— +— TT=4'
^ ' 4 iC+4 ;, ., vj
(2) V(2^+2)^/(4^-3) = 20. ' .
(3) V(3a? + 1)-V(2a?-1) = L
297. A siphon would empty a cistern in 48 minutes,
a cock would fill it in 36 minutes ; when it is empty both
begin to act : find how soon the cistern will be ^led.
298. A waterman rows 30 miles and back in 12 hours,
and he finds that he can row 6 miles with the stream in
the same time as 3 against it. Find the times of rowing
up and down.
299. Insert three Arithmetical means between a-h
and a+b, '
300. Finda?if2«^:2*»::8:l.
»♦
that
ANSWERS.
1. 22.
2. 26.
0. 274.
10. 89.
6. 10.
11. 6.
7. 6.
12. 5.
3. 89.
8. 6.
13. 9.
4. 564.
9. 34.
14. (^.
II. 1. 65. 2. 81. 3. 94. 4. 8. 5. 27.
6. 81.
7. 12.
8. 11.
9. 21.
10. 15.
11. 10. 12. 3. 13. 2. 14. 127. 15. 6. 16. 1.
III. 1. 6. 2. 16. 3. 9. 4. 224. 5. 469.
6. 7.
7. 74.
8. 12.
9. 8.
10. 238.
11. 420. 12. 144. 13. 43. 14. 15. 15. 9. 16. 2.
IV. 1. 7. 2. 88. 3. 43. 4. 2. 5. 72.
6. 1. 7. 1. 8. 16. 9. 14. 10. 5. 11. 7.
12. 5. 13. 11. 14. 7. 15. 4. 16. 2.
V. 1. 16a--96. 2. Sx^-Zy\ 3. 9a+96 + 9<?.
4. 4a-h2y-\-4z, 6. a-b, 6. 3^-3a-2&. 7. 2a+2b.
8. a+b+e. 9. '-2a+7b + 2d. 10. ai?'-2aj*-8d?+10.
11. 5ar*+4a?'+3aj8+2«-9. 12. 4€fi + 2a^-4atf^+b^-1b^,
13. a*«+3a^. 14. 6a&-9a*a?+7aaf2+aaj». 15. 5ay^
16. 10«*+8y«+12a?+12. 17. a?*. 18. af^ + y^'+x^-Sxyz,
VL 1. 3a+4&. 2. 4a+2<?. 3. a+5b+4c+d,
4. 2«*-2a:-4. 5. 3a;*-«»-14a?+18.
6. «*-»»+ 2a'. 7. ^6a!y-5xz+2y^+y/i;,
a 3««+13ajy-16a?^-y8-13y^. 9. 2a»-6a26+6a6'-26'.
10. 3«'+4a?+16, «'+8d?l
VII. 1. a. 2. 2<?. 3. a+a". 4. a-Sb,
5. -2&+2<?. 6. 3aj + 3y-;2r. 7. a-b+c+d-e,
8. a-6+2c-<f. 9. 3c. 10. 3a-3&. 11. 2a-b.
12. 6a. 13. a. 14. 4a. 15. 4a- 166 -2c.
16. 3a-2c. 17. 9+3a?. la 7a:+6. 19. a.
20. 16-12;p. 21. 12»-15y. 22.4c. 23. 3a -2c.
24. -ai^-aft
T.A. 20
1 1
*-?^'
S06
^-!
Via 1. 8«^ 17«^ ^ % 4flflBR.
7. 24<l*-27a"&. 8. 64r*y-8«y+10a^«. '
10. 4iijV«* + 6^^-10^^. 11. 2««+3ay-2^.
12. 64r*-96. 13. ar*-2«+l.
14. l-a»-31aj*+72a:»-30«*. 16. a;»-41af-120.
16. a;" + 161a?-2S4. 17. 2«»-18a?* + 39«»-25«*+iP+l.
18. ^•+iOO8«4-720.
19. 4ii^-6a?'+8aj*-104j'-8j^-6a?-'
20. ^•+2«"+3^+2aj«+l. 21. ^-da^a.
^. a*+4a"ar+^aj«-a?*. 23. ^106'-a6«+26a*-7a^.
^ a*-flW»+2a&»-&*. 26. a*+3a*62+4J4. |
86. ;12«»-17«V+3ai^+2y»- ^7- ««-aV+«y~y«.
Sa 6«^+l7«V+26«y+19a:j^+4^.
29. «*+y'+3ajy-2a?-2y+i. 30. «"-32y«.
8Jt, 243a?"-y». 32. «»-4|^+12y<?-9;»».
33; a»+fl«+aft«+6»+2ft2;c-.(a-&)«a.
34. <i^+6»+c'-3aftc. 36. a*+85*^(a!-2)+ 166*4?*.
/ 36. a*-2aV+&*+4a&c"-c*. 37. «*-«*.
38. afi+j^(d+h+c)+a(ab+ae+l>c)+abc,
39. afi+a*a^+€fi» 40. «*-6aV+4a*.
IX. 1. Soj*. 2. -3a». 3. 3i»y.
4. -8a«W. 6. 4««6V. 6. «"-2a?+4.
7. -a'+4«~6. 8. a^~day+4^, 9. 6aV+a&-4.
10. 16<»V-12aJ'+9a5c"-6c*. 11. «-4. 12. a?~8.
13. a^+x+Z. 14. 3aj"-ap+4. 16. 3aj*+2iP+l.
16. flj"-3a:+7. 17. «*+ic*+aj*+«*+«+l,.
18. fl^+a6-6'. 19. «»+3«V+9*y'+2V»
20. a^-r-a^y+ai^, 21. a*+a^y+a^+ayi^+f^,
22. a^--2a»6+4a*&"-~8a6»+166*. '
2a 2<?-6a»6+18a6»-.276". 24. s»»+^+y.
2& aj*+2a!y+3y*. 26. «'-2a?+2, 27. ;c"^3a?-l.
fiS. V-6iP+6. 29. «^-4a?+8. ^jK^* 4j"+6a?-l:6.
r
fr*.
W^mBm
wi
8(J, i^+Sto^+goft'+ft'. 86.
87.
89.
42.
44.
46.
48.
flr*+2«»+3«*+2« + l. . 88.
m—e, 40. oaj* + &a?+ c.
«*+d?(y+l)+y*-y+l.
a+6+<?. 45.
a* +'a(26 - c) + 6» - J<j + <:l
«*-8a!»+4«+l, .
aj"-«*+2«*-2. .
41. afl-'2xy'^f^.
43. Ix-^-Az. .
47. a(5+<j)-ftc.
«»-;p(a + J)+aft. 49. 4?+y-;2r. 50. x-k-y-^z.
X. 1. 225aj«+420;il2(+196y'. 2. 49««-7ai!V+25y*.
3. a?«+4«3-8d?+4. 4. «*-10«'+394^-70»+49.
5. 4a?*-12aj»-7«*+24a?+16.
6.«»+4y2+9^2+4a?y+6ar4r+iaiC^7. ««+2«V+«y-^.
s! a?*+«?V'+y*- 9' a?*-ii!V-2i«y*-y*.
10. ««-ai?2/*+2a?y»-y. 11. aJ'+2a;*+5aj«-L
i»«-ia»»+8I. 13. a*-4a?6«-4a6»-6*
12.
14.
16.
164?* + 96«V + 144aj^« - 81y*.
tf*a?*-2a«fc2a.y+2>y.
15. a*««-5V'
y
8.
6.
8.
9.
10.
12.
16.
19.
22.
24.
26.
28.
31.
XL 1. £^+6«+c». ^
a? + ft' + c* + <?+ 2ac + 2W.
2(a + 6 + c). 6. 2&(a? + y).
«(2a + c) + y (2& + a) + ;8? (2c + 6).
2(a+ft+c)(ir+^+;y).
2(a*+6'+c'-a&-6c-(ja).
B"-i^'. 13. 2a+42>y.
2a-6&+4<j. 17. 6.
36.^ 23.
2. a'+ft'+c^.
4, 6(a+6+c).
7. 5d;+<^+(a+6)4f.
11. 6-lla.
14. (a?+a)". 16. a.
18. «»+iBV+^+^«
20. 12aftc. 21. a-Th-\-e-{^d.
9a*~30ad+25ft«.
--6c*+c(9a+4&)-^6a&. 25. (aj*+ajy+^/.,
{jx^-xy-^tff, 27. a*-2a&+36'.
«"-8ajy+152/». ^ 29. a<-aV+6*. 30. «*--M
2a*-3a&+4Z>», 32. x-l, 33. («~l)(a?-h4}.
20—2
■4i
^#r-*^'';;^.
87. (»+4)(»+6). 8i^1N^I#ir+6).
89. («-6)(ar-10). 40. («-10)F. 41. (i-ll)(«-|.12).
42. (»+4)(»-ll). 43. («-8)(#+8)frl*+9).
44. («+5)(««-e«+26), ^
45. («-2)(»+2)(a>"+4)(«*+16).
4e. (;»-2)(4T+2)(«"+2«+4)(«*-2»+4).
47. (a+4&)(a+66). 48. {«-6y)(«-7y> ^
49. (a+6-5c)(a+6-6c).
50. (2a;+2y-a-6)(a?+y~3a-35).
XII. 1.
4. 7aV«V.
7. 4(fl^+ft^. 8.
n. 0^-10.
14. d?"-6«+3.
17. «+3.
20. V-«+l.
23. «"-2.
'' 26. 09^+3^+5.
28. flr*-2«*+3«"-2a;+l.
80. 47+1. 31. «+7.
2. 4aW 3.
5. 2(^+1). 6.
«*-2/". 9. aj+5.
12. «-12. 13.
15. a^-6«+7. 16.
18. a^-4. 19.
21. 3^+2. 22.
24. «-2. 25.
27. 7«"+8«+l.
12a>y^.
3(a7+l).i
10. *-7
^■+3a?+4.
flj"-6dJ-6.
«■-»+!.
29. a;*-3*+l.
32. x+Sff, 33. A; + a.
34. a-2a.
36. a?-y.
XIIL 1. 12a»6'. 2. 36a»W, 3. 24aVajV«
4. (a+6)(a-&)". 5, 12a6(<^+6»). 6. (a+&)(iiP-n
7. (a»+l)(«+3)(«-4). 8. («+2)(4y+4)(4>«+3»+i>
9. »(2»+l)(3«-l)(4i»+3). /
10. (ii^-5a?+6)(«-l)(a?-4).
11. (4j"+3«+2)(«~3)(ay+5).
12. («i«+i»+I)(«»+l)(^+I)(*"I)« *
13. («»-^-4i»+4)(«-.l)(ar-4).
14. («"-<i«+<^(a^+«»+a")(i»-a)P. ^
.16. aOaVtf". 16. 120(a+6)^a-&)*,
16.
5.
r
1)4
-Za+4.
hi.
fl.
*M
17. t4(a-*»)(<i^-f^. ' 18. 105a('(a-|.5)(a-»X
19. 1^-1* 20. V-l. 21. «»-l.
22. («+l)(«-l'2)(«+8). 23. («+l}(aT+2)(A^-f24T-8).
24. (iB»-19«-80)(«*+&»+10).
XIV. 1. 3;p+^. 2. 4ac+^. i 2a+?5.
7 9 4a
7.. «^+3a«+3a*+
9. «»+«»+a?+l4
5. flr+
3a»
« + 3*
B. 2i»-
2
8. a?-l-
2j?-1
«*-d?+l*
12j
8(a«-i-5^
3(a+&) '
4af
—- ^. 10. *»-««+fly-l. 11.^.
*— 1 So
13.
2(a+by
14.
Al>
(«-l)«(d?+l)*
15. ^. 16. ?^^ 17. |(?rW. 18. ^^:^±1\
Sp a+b Z(a+b) «*+l
3y 26 a-6 oa?-
5(a-6) a-& aj+5 fl?-5 fl?-7
,0 2±5. 11. *^,
12.
34T-4
14.
17.
20.
23.
26.
29.
g-t-3
w—a
«*-3ar+l'
3a^+ay+2
2«*+«+3*
1
15.
18.
4a?-3'
«"4-7a?+3'
6a?- 5
13.
16.
19.
a+b-a-^e'
a+5
6se+4
^^- i»+4- ^^' 2x'+dax+4a*'
24.
a;+a
«*+<M?+a'*
25.
d?-3
30.
a?'
iu'-aV*
31.
1 «/""*
310
Am
?%.•'
83.
86.
88.
40.
I2afl
84.
4(«»-iy
a(a-fft)(a^+y)
(«-a)(«-6)(«-c)
,I«.J4.
87.
89.
« . . ■
iii'
{iiKi)(flN^iy
>•••
XVI. 1.
4. -
2eh
6a-66-c
a+J+o
2a
a+of
aa
9.
12.
19.
23.
26.
29.
31.
37.
40.
44.
abo
a+h
2a-26'
a
6.
¥::&'
8.
aa-6i '
«-y
10
4a
- a+w
11.
l-9««;
2a«+9g4
606
16
^ „ , ^ 2iP— 3
^::6- ^3. ^^z^' 14. ^(4^«i)- 1^- (^-2)(«+2)»-
17. _^ ' ' ,^ — . 18.
aa-
(a;»-l)(a?-2)
2a«
6
a*-a^
. 20.
4aj»
(at + l)(a;+2)(«+3)*
. 2a^ 2^
24.
2a*+6tf'y
25.
3«2
..27.^^^±i^. 28.
4^-16 ^
aj«-l'
2a?'-9ig+44
a;*+64 •
■^*
30.
2a ^ a?—2x
31. L 32. ^^rirr^. 33. 0.
«(« + l)(il? + 2)*
38.
35.
«»+l •
4a?8
48a^
(«»-^)(aj»-9aa)
41. 0.
(1+«2)(1+^)"
39. ,
Si6.
2j8»
2y« ^o 2a;»+2 ^^ 4(a*a^~5V )
45.
246*
a(a?-&«)W*-46»)
13.
13.
16.
20.
23.
26.
*
30.
r
/^':
^MM
811
At
33. 0.
4ft
49.
47
w(a-¥b)^ab
i0^4(m^V)' *^- («-a)(*-i)- ^' (w-a)(m^h)
(«-<?)(«-6)
50.
(c-a)(c-6)
61. 0.
52. -
55.
c(c-a)(c-ft)*
(«— a)(«— 6)(«— c)
XVII. 1. ~. 2. 1, a
53. 1. 54.
({r-a)(«-ft)(«--<j)
56.
(«-a) («-ft) («-c)
5. W'-a*
5a
6.
9j
10.
(«-l)(«+2)'
f:::P'
a
a«
11.
X
x-y
12.
13.
afi—asfiA-fj^x—cfi
a*x^
14.
«■ , a* y* 6"
+ -a-TS--s.
«»""P
XVIIL
w{a+2x)
g-f 5— g
c+a-6*
13. 5a?~l.
. Bay
hA^
leS?'
3.
(a-g)«-y
o^c *
15. 1.
3(a-6)«
5.
9.
6.
^
7.
«+y
8.
«— a
a~y x+y
10. -yU. 11. f^Y. 12. «^.
a*+a*+l ,« (j?'4-g')(^-fg*)
14.
16.
20.
23.
26.
xa
17.
x-y
y
18.
15.
«*-3«*a+3a'^+a*
aV
i»+l*
24.
21. 1.
x{a+b+c)-hc'
19.
22.
25.
24MX *
fly --4
47 — 5*
1
27. 4?+l.
30. X, 31. 1.
32.
28.
1+j?. '
29.
a
^+y*
Cr a
7\T%i
312
35. 0.
36.
9
37. 2f 88. 0. 39. 0. 4a a.
XIX. 1. 6.
2. 9.
3. 7. 4 11.
6. 2.
11. 18.
16. 63.
21. 45.
26. 6.
32. 3.
37. 6.
42. 12.
7. 4.
12. 6.
17. 60.
8. 7.
13. 2.
18. 36.
6. 21.
10. 6.
22. 24. 23. 120. 24. 72.
9. 8.
14. 2Y. 15. 16.
20. 96.
25. 12.
19. 64.
27. 6.
33. li
38. 2J.
43. 4.
28. 1.
29. 6. 30. 2. 31. 2.
47. 6i. 48. Ij.
62. 7.
s
67. 3.
62. 2.
53. 1.
58. %
63. k
34. 7.
39. 3.
44. 3.
49. 10.
54. 12.
59. 3.
64. 2.
35. 1}.
40. 7.
45. 7.
50. 6.
55. 5.
60. 28.
65. 4.
36. 11.
41. 11.
46. 3.
51. 10.
66. \.
61. 5.
66. 2.
XX.
1. 10.
2. 8.
3. 12.
4. 6.
6.
-7.
6. 16.
7. 6.
8. 3f .
9. -6.
10.
5.
11. 8.
12. 7.
4
13. 3.
14. 2.
15.
7.
16. If
17. 1.
la 1.
19. 17.
20.
2.
21. 6.
22. 2. 23.
6. 24. 7.
25. 2.
26.
•
2.
27. 2.
28 5?
^?- 29-
29. 7.
30. 4.
31.
-1.
32. |.
37. 0.
33. -23.
34. 3.
35. 5^.
36.
4
13-
38. 20.
39. 3.
40. 5.
41. «-&. 42. a+b.
45^ 2(a+6). 46.
48:
61.
a+b+e+d
m+n
a+b
49.
43. 6 -a.
47.
44.
a^+ab+l^ ,^ db
2ab
a + b'
2db
a+b'
62. e.
a+b-e'
a+b
SO.
53.
6-a'
r
ANSWERS.
313
3(a+&) •
60. 50. 61. 28.
54.
57.
55. i(a+&+3).
58. 5(a+5).
13
56.
59. 4.
62. ^. 63. (a-6)«. 64. «.
XXI. 1. 30. 2. 2. 3. 13,20. 4. 35,50,70.
5. 17,31. 6. 28,14. 7. 28. 8. Noyember 20th.
9. 52. 10. 36,27. 11. 48,36. 12. 14,24,38.
13. 28,32. 14. 103. 15. 54,21. 16. 8.
17. 8,12. 18. 10. 19. 36,9. 20. 36,12.
21. 100, 88. 22. 14. 23. 24, 76. 24. 21.
25. 36,24. 26. 24,60,192. 27. 840. 28. 30000.
2^. 420. 30. 24. 31. 500. 32. 10,14,18,22,26,30.
33. 36,26,18,12. 34. 50,100,150,250. 35. 5,6.
36. 24,36,56. 37. 88,44. 38. 130,150,130,90.
39. 13,27. 40. 75,25. 41. 85,35. 42. 1000.
43. 18,3,3. 44. 24000. 45. 80. 46. 26,16,32,27,42.
47. ^140. 48. lOi^.
XXII. 1. 72. 2. 20, 30. 3. 200 miles from
Edinbm^h. 4. 12,16. 5. 8,16. 6. 32,16.
7. 48. 8. 30. 9. 9, 16. 10. 30. 11. 18, 22, 10, 40.
12. 6, 24. 13. 10, 15, 3, 60. 14. 10 shillings. 15. 56, 45.
16. At the end of 56 hours. 17. 27, 17. 18. 168, 84, 42.
19. 16,25,7,42. 20. 240, 180, 144 da^ 21. 15,21.
22. 2560. 23. 36, 54. 24 60. 25. 12. 26. 8 pence.
27. 875,1125. 28. 25. 29. 10,20. 30. 20,80.
31. 5/7. 32. 40,50. 33. 11,17. 34. 28.
35. 24. 36. 1024. 37. 450,270. 38. 2200,1620,
1100,1080. 39. 60. 40. 7 + 12 + 32. 41. 30.
42. 60. 43. 240. 44. Zd.9d.l8Ad. 45. 60d,
46. £133}. 47. 24. 48. 60. 49. £120000.
50. 25. 51. 4i,3i. 52. 39. . 53. 40-
314
54.
55. (5s. m •^ 57. 48^ minutes
pa^ three. 58. 32^ imnntes pftsi tbim > 59. £2^8.
60.
200000000
tree,
seconds,
61. .40 miniktet past eleyen.
62. £300 and ;£200. 63. 14. 64. 640.
XXIII. 1. 10 ; 7.
4. 4 : 1.
5. 5 : 5.
2. 17 ; 19.
6. 21 ; 12.
3. 2; 13.
7. 20: 10.
20. 13; 5.
24. 5 ; 7.
28. 12; 3.
32. 2; 3.
21. 9 ; 7.
22. 10 : 4.
8. 2; -a 9. 3;2, 10. 3; 2. - 11. 3i; 4.
12. 10; 7. 13. 19; 2. 14. 38i; 70. 15. 6; 12.
16. ftf ; W 17. 10; 5. 18. 12; 12. 19. 20; 20.
23. 4 ; 9.
27. 10 ; 8.
31. 3 ; 2.
35. a I &.
25. .2i; 1. 26, -2; '2.
29. 3; 2.
/33. 4; 12.
30. 63; 14.
34. a ; b.
36.
39.
ab 4h
— •
a+b* a+b'
ae be
V 37. 5 ; a.
38.
40.
;0.
a-hb* a+b' a+b
42. a+bia-b, 43. (a+b^iia-b)\ 44.
41. a; b,
e e
a+b* a+b*
n
XXIV. 1. 2; 1; 3. 2. 3; 4; 6. 3. 2; 1; 3.
4. 9; 11; 13. 5. 4; 0; 5. 6. 5; -5; 5.
7. 45;-2Ul. 8. 10; 7; 3. 9. 51;76;1.
lA 2 3 2
■ 3' 4' 5'
2
12. a?= g (a + 6 + c) -a, &c.
abc
14. a:=y = z = .^^^^^^^,
16. «?=3,a?=4,y=5, ;2f=2.
11. a?=-(6+c-a), &C.
13. x=^(p+c\^.
15. a}=a,y=bfZ=e,
XXV. 1. 42; 26. 2. 12; 16.
4. 24; 60. 5. ZOd.;Sd. 6. 49; 21.\ 7.
3. 116 ; 166.
4_
15*
8./45; 63. . 9. 72; 6^ 10. 30<;.; 15<iK. Ih 5#.; 3».
10.
14.
18.
22.
26.
29.
33.
37.
4t
45.
49.
r
ANSWERS,
515
■ 5
e
11 20; 52. 13i 70;50. 14. |. 16. (24-1)20.
10. 15; 65. 17. 12; 5. 18. 14; 10. 19. 24.
20. 1;2. 21. 59. 22. 100 lbs. 23. 150 yards;
30, 20 yards per minute. 24. 21 ; 11. 25. 50 ; 75.
26. 70 ; 42 ; 35. 27. 90 ; 72 ; 60. 28. 12 miles.
29. 4 miles walking, 3 miles rowing, at first; 30. 33^
miles per hour ; 48^ distance. 31. 45 ; 30 miles per hour.
32. 30; 50 miles per hour. 33. 60 miles; passenger
tram 30 miles per hour. 34 150; 120; 90.- 35. 3J«.;
39.; 2i«. 36. 4; 59; 55. 37. 120; 80; 40. 38. 432.
39. 420; 35; 21 shillings. 40. 2; 4; 94.
XXVI. 1. i4. 2. ±25. 3. ±7. 4 ±9.
t, ±9. 6. ±6. 7. 1,2. 8. 2,3. 9. 2,-12.
10. 3, -|. 11. 4i,-3. 12. 10,5. 13. 5,-5..
14. 6, -3.
15.
2' 2*
16 ? ^
^^- 2V 2:
17. 5,?.
18. 3,-9. 19. 2J, -5. 20. 15, -li 21. 1,2.
22. 4. 2a 6,|. 24. 11,3.
26. 44, -2.
27. 7,-3^.
29. 3, -2i. 30. 5,-3. 31. 2.
33. et2.
16
34. 1, -4. 35. 3, -^.
37. 6, y. 38. 7,^. 39. 8, 2,^.
5
41. 3, -5. 42. 3, -y 43. 2, -1.
25. 5,3}.
28. 10, -10.
32. 2,-3.
36. 6, 2;.
40. 3, -41.
44. 4, -1.
45. 7,3H. 46. If, 1. 47. 4}, ^
49. 3,-9. 50. -10, 9|{. 61. 3, -ij. 52. 3, -1§.
48. 3, "" j: .
'^m,
t
810
ANSWERS.
53. 4,0. 54. 1},0. 55. 18,^. 56. 6,-3).
61. a±^. 62. (a±5)«. 63. *V(a&). 64. a, -^^\
XXYII. 1. lb 2, Ik 3. 2. 49. 3. 4. 4. «b4.
5. 5,-3. 6. 3,-2. 7. 6,0. 8. 12,-3.
9. 9,-12. 10. ±3. 11.2,-151. 12. 4,11).
3 4
5' 5
a-1
13. 1}. 14. 16. 15. 1. 16. ^, ^. 17. 4.
21. ft«»
18. 4. 19. *J2^lE!(pEl. 20. , .
(a— 0}" 2
22. 0, ^-^. 23. 0, ±5. 24. 0, >fc ^2. 25. 2, dbl.
26. 0, ^J(db). 27. a, -2«i) -2a. 28. a,^, -|.
XXYIII. 1. 36,24. 2. 36,24. a 30,24.
4 18, 12, 9. 5. 12, 10. 6. 4, 6. 7. 196.
8. 3,48. 9. 11. 10. 7. 11. 6,12. 12. 15.
13. 24. 14. 27 lbs. 15. S9.9d.,7s. 16, £20,
17. 126,96. 18. Sd, 19. 10, 9 miles. 20. 56.
21. 192,128. 22. 9 gallons. 23. 64. 24. Equal.
25. 4 per cent.
XXIX 1. 5, -4; 4, -6.
2 4 -?2. 1 J-1
* 7 ' * 35*
3. a8; ^6. 4. 6, 12; 2, -4. 5. 7, -4; 4, -7.
6.4,-g;3,-g. 7. -24,?; 12,^. 8.6,-i;5,|f.
9. 2, -|^;4, -y. 10. 6,0; 5,0. lUg,0;|,0.
12.3,6;|,|. 13.4,|;8,^. 14. ^,0;^,0. 15.0,5.
ANSWERS.
317
4.
,e.«,e»z«l».j,«!i;S?. ,7,
a+&
a+^
20. ik5: db4.
V2
iaa»0;Oy&. 19. 1^4,^4^; "^3,
21. di7; 1^6. 22. ikl5; 1^7. 23. ^4^ ikH; ikl, >v4^
23
24. d>9; iii4. 25. ikS, >i>36; >t5, 7 ^-^ 26. ik9; ikS.
27. •kS; di6. 28. ^2; ^\. 29. ±9, :^8^/2; si.7, ^^1%
30. db 4 : db 1.
15
9
31. 0, 1, ;;^ ; 0, 2. ik
22
32.
(g-i-l)ft . . {a-Dh
34. :fea. db
^/(2a8+2)'^^(2a>+2)•
a+1 .«-!
33. ±a,:fe
a-¥h
^/2
±&,±
22'
7^'
36. 6,-4:4,-6,
> ^>
36. 5, 4: 4, 5. 37. 4, 2 : 2, 4. 38. 4, -3; 3, -4.
39. 1|2;2,1. 40. :b4» :fe3; :fe3, <fe4. 41. 2,1
2 1
3' 3'
48. >ii5: ifa3.
43. 2,1,-1, -2; 1,2,-2, -1.
44. ^
-2*^3 -1*>/13
2'- 2
45. 3, -r ; 6, -
1, -2*^3,
-1tn/13
3
2
3
46. 5, -=; 2, -r
3
3"
47. 2: I,
48 4?.l -?.2?^ -? ?
«• *>2»4» 4' ^'2* 4'4-
49. «+6+l, -
a+5+l
a+1
a-fl
60.
3
^tb*
51.
±2&. 52. 0, a+ft, i(a-&)±5N/{(a+8d)(a-6)}i
'^>«+^s(«-^)=Fo'^{(<»+3&){a-&)}. 53.a?=a-s-/y(adc); &c.
2
2
C4. («+yXy+*X*+*)= *<*<?; Ac 55. *1; *2; ±3.
^8338..
•^ 3' 2' a' 3' *
^#?
B18
XXX. 1,
5. 10; 15.
9. 5; 3.
7; 4.
60; 10.
4NS1VEB8,
' f.
H; 7. 2. 6; 18. 3. 8; 24 4. 8; 16.
6. 10; 12. 7. 7; 5. a 18; 8: 6^, 16.
10. 4; 2. 11. 2; 2. 12. 4; 6.
la 7 ; 4. 14. 12 ; 8. 15. 20 ; 15. 16. 80 ; 40.
,17. 60; 10. 18. 6*4. 19. 160; ;£2. 20. 24^; 4».; 3».
21. 756; 36; 27. 22. 4^ walking; 4^ rowing at first
23. 10 ; 12 miles per hour. 24. 6 miles.
XXXI. 1. 8aj«y«^".
5.
2. -8aj«2^V.
64^
27y«'
3.
81a*W«.
6.
7. «'+ 7a«J+21a»6"+35a<6»+35a^+21a«6«+ 1a¥+lff.
9. .a«-3«*6*+3a'6*-6«. 10. l-3a?+3aj«-«».
II. 8 + 12a?+6«'+«'. 12. 27-64a?+36a!»~84!'.
13. l + 4a?+6a?2+4a?» + ii?*. 14. d;*-8a:»+24aj2-32a?+16.
16. I6i»*+96«»+216aj"+216a?+81. 16. 2a^a?-\-Qaxl^\
17. 2a*d?«+12aV6y+26V. 18. 2(5^+lOa^+aj»).
19. l-4a^+6aj*-4««+«8. 20. l + 2a?+3«*4-24J»+^.
22. I + 2a?-«2-2a;'+ar*. 23. l + 6a?+13ar'+12a;»+4a?*.
24. 1 -6a?+ 150^- 18a? + 9j^. 25. 2(4 + 25iB*+16aj*).
j26. l + 3^+6iB2+7aj'+6a?*+3ar'+a;'.
28. l + 34?-5a;' + 3«"-««.
29. l + 9a?+33a?'+63a?+66ar*+36A''f8a?.
30. l-9a?+36«2-81aj» + 108a?*-81aj»+27a?«.
31. 2(36a?+171«'+144aJ»). 32. l-2a:+3a:2-i»*+2a;»+^.
33. 1 + 4a? + lOa?* + 20a?* 4- 25a?* + 24a:» + 16a:».
34. ^{db^ad-^-hc+cd), 35. 2(a2+2a<j+<?2+6a4.2j(/+^).
36. l + 6a?+15a:24.2bflj»+i6a;*+)5a;*+aj«.
57. l-12a?+60a?-160a? + 240a?*-192d^+64as«./
38. l+8a?+28a?+56aj'+70aj*+56a?» + 28a!«^+8aj'+aJ*.
39, l-3aj»+3aj*-aj». 40. l + 3a:2+6..p4+7j56^.g^+3^io+^ii
I,
XXXII. 1. 3a'6*. 2. 2a5, 3. -4<i6«. 4. 2a6V,
5. -a6V.
5a6
^- 7?-
7. -
6ay
8.
3^
9 ^
ANSWERS.
819
f.11
10.
2a5S
14. %a+Zlc,
11. 4a+^&. 12. 7a«-e&. 13. e4J»+L
15.
5a+2&
16.
3^-4
17. aj*+a?+l.
fia+2c' *"• 2a?-3'
la l-«+2«*. 19. iB*+3a?+8. 20. «»-2a?-2.
21. l-2a?+3^. 22. 2ai(«-a:»-2. 23. «*-«*+2a«.
24. «• - flw? + 6*. 26. «" - 6«^ + 12iP - 8. 26. a:" + 2aa^ - 2a*a? - a».
27. l-a?+a;»-^+«*. 28. ?^-l?-?2f. 29. l:».ar.
3y 0;2f ^Z
31. l-a?+aj\ 32. a;«-(a+ft)a?+a&.
34. a^-ay-hf/^. 36. 34. 36. 46.
30. ai?-3y.
33. XP+ 1.
37. 61.
41. 123.
46. 6*42.
' 49. 620-1.
63. 12007.
67. -76416.
38. 72.
42. 321.
46. -914.
60. 70'68.
64. 604*06.
68. 443329.
62. -09233.
39. 87.
43. 407.
47. 1234.
61. 8-008.
66. 1*8042.
69. '94868.
63. 412310.
61. -66574.
66. 18'63488. 66. 119*56331.
68. I2a^+4y^. 69. a?-a-&.
71. sfi-ax^a\ 72. ^a^-^Acx-ZcK
74i l-iP+aj«-aj». 75. l + 2ar.
77. 27. 78. 36. 79. 64. 80. 61.
83.138. 84.148. 86.378. 86.39*2. 87.5*76.
88. -604. 89. 1111. 90. 2755. 91. 45045. .92. 17479.
40. 99.
• 44. 65*6.
48. 6420.
62. -4937.
66. 2*1319.
60. 2*4919a
64. 11-3678L
67. 2a?+5y.
70. «2+a?+l.
73. l-3J?+4a?2.
76. Zx-l.
81. 88. 82. 92.
2 \
^' 8*
4. 100, 5, 2=.
XXXIII. 1. ^. 2. g. ^.^^
6. «"•. 7. a'. 8i a"*. 9, a'K 10. a^'. ii. a?*-y^
12. a-&. 13. «2+2a?*+a?-4. 14. x*+l+x"*. 15. a-^-l.
16. a«-3a*+3a"*-a"^, 17. a«+2aM+a&-a?V.
18. a^ +arp - 0^^- tr, ■ 19. a?' + x^y^ + a? V + P>
20. a^+aM+6*. 21. ^164?"^-12a;~*y"^+9y~\
f
V
SSO ANSJVSBJS,
22. »+y. 28. J-ahKbK 24 aKb^n-A
25. «*+2ara^ + 3a^*a+2«*a*+a'. 26. w^-ix^y +yK
27. «*-2«"^. 28. a-Z-a'K 29. aj*-5«i+«^.
80i 2a^-8+4«"i
XXXIV. 1. 7^/2. 2.9/^4 8. |^/3. 4.^.
_ 13^/15
^- 10 •
9. 4+|^/2.
6.
6J^2
3
7. 2+2^/2-2^3. 8. 2+^V«,
10. 5 + 2^6.
11.
K
12. J ( 18+9^6+4 V15+6V10
)
6
33
K 8-^/7.
15. ^6+1^2.
16.
13. 8+^5.
V 2" ^2'
^7« fj^-fj^* 18. 2+^/3. 19. V3. 20. ^/lO.
XXXV. 1. |.
2 1
5 2 3 8
8' 3' 4' §•
3 ^
3- 27-
14,21. 6.
24, 30.
6. 20,32.
7. 1.
15, 10. 9.
6,8.
10. 35,42.
11. 4
12. ^^. la 50, 60, 9a 14. 0, 2 : 5.
XXXVI. 1. 14 2. la 3. 15. 4. 12. 5. 4.
6. 4. 7. 2, 2i. 8, 6. 9. 1^ -1. 13. 45, 60, 80.
14. 4, ^9.
•II
XXXVII. I. 4. 3. 5:2. 4. 2. 5. 4
6. 0. 7. 8. 8. a((?. 9. -p-. ao. j$113}.
lU 15.^ 12. £15360.
5.
9.
la
16.
i^ifiimMii'-^
ANiiWJERS.
821
27'
XXXYIII. 1. 936. S. 77i 8. 69. 4. 139}.
6. 37}. 6. -116. 7. H16,18. a 14}, 14J,...
9. 6J,5,... 10. -|, |,... 11. 10,4, 12. 82.
13. 5,9,13,17. 14. 5,7,9. 16. 1,2,3,4,5.
16. 18, 19. 17. 7. 18. 5. 19. 1, 4, 7. 20. 1, 2.
XXXIX. 1. 1366. 2.13}. 3.40}. 4.63(^2 + 1).
5.
666
648'
11 1'
"• 33'
6.
463
96*
12. 11.
^ 333*
7 ?
7. ^.
212
^^•496'
9 ?
14.
10. 4}.
667
1980*
15. 4,16,64. 16. 8,12,18,27. 17. -9,27,-81,248.
18. 3, 12, 48; or 36, -64, 81. 19. 1, 3, 9,... 20. 3, 6, 12.
XL. 1. g, -,1.
2 1 2
2 ^ A 1
^' 5' 13' 2*
3 a i?
3. 8, J.
4, jj, jg, ^. 5. 6, 12. 6. 36, 64. 7. 1, 9. 8. 3, 9.
XLI. 1. 134696. 2. 6040. 3. 126. 4. 30240.
5. 11. 6. 1900. 7. 16604; 3876. 8. 27; 99.
XLII. 1. a«-13a"a?+78a"aj«...-78a«a?»+13a«M-«".
2. 243-810ajVl080«*-726aJ«+240«'-32»*!
3. l-14y+84y«-280y'+560y*-672y»+448y*-128y'.
4. «f +2««->y+2n(«-l);U"-V+ — ^^-*^^«""V.
o
5. l+4a?+24:»-8a;^-6««+8a?'+2««-4«'+i»'» 6. l+6«
+ 16aJ»+ 30«» + 46dJ*+ 61«»4- 45««+ 30a?'+ 1&B»+ 5«»+ «»•.
7. l-8«+28«»-66««+70d?*-66a?»+28««-8«'+«'.
8.5922. 9.1690. 10.af=2,y=3,»=5. ll.«=4,y-2,n=a
1 a"^a? ZaT^o^ la'ha^ *J*Iar^ai^
!. «• 7i 7^ rs 77J5 — •
12.
13. 1 + ^+^ +
2 8 16 128
2^ Z^ 64 •
T.A.
14. l+2a?+4«»+8«'+.M
21
322
4NSrrBRS.
\. ■
16. r+1. l«: tIJJjJ»J!(to)-l-y.
.U _!• .1* 1A40 t* 4040 **
18.
(r+l)(r+2)(r+3)
1.2.3
1 S^ 8«*
20 1 + 5-.^-^
^"- ^^2 8 16'
XLIII. 1. 2042132. 2. 22600. 3. 11101001010.
'4. 2076. 6. H592. 6. Radix a 7. Badix 6.
8. 9e&l\U, 9. Radix 5. 10. eee.
XLIV. 1. 5£.
a
3. n = -.
.. MlSOBLLANVOns. 1. 729, 369, 1, 41. 2. 41^-51^.
3. 9-30*+37«*-20««+4aj*. 4. l+a?-4J»-«*,
l_ay+aj«-a:*. 5. ^.t^r^ . 6. (4«?-9)(9a«-4).
8. 3.
3«-4
9. 240,360.
10. £2, £21.
n. ^%|^.|,^+1|? + |, 12. 1. 13. 3J«.
fl!* + « + l*
59
14. 2«»-d??y-22/» 15.
16. :(^-10)(..+ l)(a,+3). n> (^-10)(a^-H)(ar.f.3) -
18. 5. 19. 7. 20. £40. 21. 2a-26-flf-2y,
a+36+4j;+4y. 22. 11. 23. «*-«*. 24. ^ + ^-7.
2 3 4
25. ««-2. . 26. ^^. 27. (16«»-l)(«=»-4).
28. 6.
31. i.
\
62-4a2*
29. 14«, 21*, 52J#. 30. 100.
d2. (a^-.a«)(^-&2),' (a?-aK«-ft).
ANSWERS.
8S3
a 1
83. 44?*-2«v+«V*-«y'+^.
80.
8(4r-y)
34. ^-2.
36. 1. 37. 4. 38. 2. 39. 30 minutes.
2(3««+j^'
40. £\Sym. 41. 1047+10;2r. 42. 7d^-:2d72^-t-2^*,
-«"-6«y+7y*, 12jr*-10^y-jr»y«+20«y8-12/.
;^. 46. (««-4)(4;>-9).
43. a+6-& 44. «•+!. 46.
47.
4P*+jr+2
48. 1.
49.
16
50. 30 lbs.
fil. 3a*-6a»&-12a«&»-aft» + 3&*, 3a^-8a«5-4a&«+3&'.
52. 2aJ-5. 53. 2, 64. (^+51*. 55. 1.2. 66. 3; 6.
67. 5; 8. 68. 4; 6; 2. 69. -^ ; ^^.
60. I .
61. 4^+d;a+l+l + i.
62.
iij?+2
63.
2<LP
64. 1. 66. 4; 3.
7«=*+7d?+2* ^"^ ^+1
66. 2; 4. 67. 3; -3. 68. 3. 69. 2.
3d7-l 8
70. 20; 40 years.
71. 1.
72.
2^-1' 6'
73. (a7-2)(^-l), («-2)(d;-6), (a7-l)(^-6). 74 0.
17 4
76. |.
76.
^ , Q . 77. 3 shillings, 2 shillings.
7a 3«»-ar+l. 82. («»-4y«)». 83. 3; -2.
84. 6. 86. 47 or 74. 86. 46 gallons.
87. 4a^-3j?y«+6^. 88. 52^. 89. 4-86409, 91. a?- y.
92. 8(«»+2/*);48(«*-y*). 93. ^i^. 94.1. 95.4^5,6.
■r
96. 3, -r. 97. 20 miles. 98. Present price 3 pence
per dozen. 99. 18^1-^}; 18. 100. 4, 8, 16.
824
ANSWERS.
101. «*-l, l+«*+«*. 102. (^-a")(«*-rt^ 103. a.
104. 1.
105. 13.
735
106. ikS: iii4: i^S:
<nr AS: ii>5: *ii4.
107. 20 Bhiniiigs.
108. 4a
a
y
109. - + i-^
V
a
112.
6
111. «»+l+ar"".
SIS
113. 7, J . 114. 1 or -3.
llff. iii2; rfil.
117.?
lift 1111
"®- 8* 2' 2' 8
3
118. -«-•.
119. 612.
121.
2a'y-ay+<i^-3ad-t>y
2aft«-6»+a-6
122. 3a>*-6;»y+2J^"
123. «(3«+4)(«-6). 124.
17
125. 2. ^
lAA M 13 . 5
126. 5, — T ; 4, 7
127. 1, ^; 2: ^
3
129. 3.
//
130. 3(3»-l).
131. -
X
132. 2dr(3a;+4).
133. 4, -3. 134. «»-i»-6=0. 136. «*a=a* or -;,
136. *2. 137. 819615. 138. 7-2^3. 139.^^.
ay
140.
g4-5— gg («+«)(<?+ 6— 2a)
6-a *
2(6-a)
141. 3,2,2.
4xa
142. 197, 3«»-2a?'-6«-3. 143. a(a«+68),-i^
144. (1)4. <2) 0,6. (3) 6; 7.
146. (1) 3, g. (2) 8.
(3)* 7; ±6. 146. 16; 16. 147. 20. 148. 16, 24.
149. ^,169. 160. AB6tol. 161. «". 162. a^+46.
163. «-3. 164. (1) 6. (2) 3. (3) 7; 4.
156. (1) a (2) 9. (3) rfi9; iii7. 166. 30 pence.
ANSWERS.
325
107. 80. 158. £20. 159. 4^+20.
161. a, 21<i-27&+ec, a"***. 162. 8(a-«).
163. 72(«-y)»(«»+y8). 164. (1) 9. (2) a (3) 12,
(4) 20; 2. 165. (1) 6, |. (2) 11. (3) All, 413;
A13, All. (4) :k2; w\. 166. 12 days. 167. 4, a 168. ~.
15
170. 208; 400. 171. 236- 18«. 172. 2,p«,«w.
173. «"-3a««+3a«iir-rt^. 174.(1)13. (2)4. (3) 6; 10.
(4)3. 175. (1)2,4. (2) db5; *4 (3) *1, *7;
ipl, db5. (4) 1, 5; 5, 1. 176. 16^ minutes after 12.
177. 36. 178. 40,23. 179. 86 (l-^), 36.
3:i;4-2
180. 7->/6. 181. 15.
(2) 6; 8. (3) 4, -j. 185. (1) 13, -15. (2) 7. (3) 2, -1.
186. 288,224. 187. 29 miles. 188. On the first day
A won 8 games and lost 4 games. 190. -85j^.
18aj*+12«'-43«"+36a?-18 6aj*-20aj«+d?+36
191.
192.
144
4««-15«+13
«"-6«*+lla;-6'
(2) 7. (3) 40; 16.
4,2; 196. 56 miles.
199. a«-a6+6«, a«+6».
1 + 9j?-13«»
193. aj*-16y*. 194. (1) 8.
195. (1)|, -|. (2)13. (3)2,4;
201.
197. 24. 198. 23+15.
200. 2,4,8,16.
16«»
3(7-2iif)
202. a^-2ii?+4. 203.
(2+3ar)»'
2^
n^fiJ. 204. (1)9. (2)^. (3)6;&
.»■'»■
326
ANSWSJRA
206. (1) 7, |. (2) 1, -4 (3) is J ±2. 206. lO ttdlea.
6
207. 24 , 208. 6 crownB+ 18 shiUings.
209. 2«»+2aa?+46«. 210. 7,11,15,...
211. 3aj»-2ajV+3d^-6y".
213. 4^-26^+37
212.
^
12 + 6«-28«**
^-lO^^rral^Zao- /^^^ (1)9. (2)16;4
(3) 3; 6; 9. 216. (1) 3, -6. (2) J.7; *6. (3) 2,4; 4,2.
216. 114 of each. 217. 126. 218. 21. 219. 11,12,
13,14 220. 3+2^/2. 221. «•+«*+ 1, paj»+g'aT-r.
2^- 5^' ^S^^ ^'^- (7^-4)(3^-2)(««\.3).
224 (1)9. (2) 23; 19. (3)12; -24; 36. 226. (1)28, -3.
ae he
(2) 100, -200. (3)
2a+2V(aa-6a)» 2a+2^/(a«-6«)*
226. j^ of a mile. 227. 600; 1000; 4000. 228. 2 hours;
,'/
r
4 hours.
230. f|(9-n).
<i^+5»
229. 2, 6, 8, ....
* ^ ' a(aa-62)» (a-6)3i(a«+6«)-
"^-^-^ 233. (1)|. (2)*|. (3) J; J.
5— <;ifaa
232.
9«»-«-3*
234 (1) 6, y. (2) *^^^7^. (3) 6; ±4 235. 19.
236. 160,60. 237. 40,50. ^238. 1976.
239. a^+a^b+at^+li^y a+26+3<?. 240. «V*+8«V«
2(l + aj*-«»)
241. liay,
lings.
l-«>* •
244 64 246. 3,6,8.
242. «r+a. 243. 106 shil-
246. \ 6} per cent
247. 2S^ 248. ^.24^. 6-678, 1234. 251. 2a- d.
■'W^,
?62. a"-a^«, e,
ANSWERS.
253. a(2ai'2at).
S27
264. (1) 6.
(2) 114; 77. (3) 0, |. 256. 112; 96.
256. ^has
£6400, J9 has ^200. 257. 7; 13. 258. 80. '
269. 8; 6. 260. ^080. 261. c>+2&<;.
262. fl?»»-l, -^,(ai?*+3flWJ»-4aV-3a»a?+2a*).
263./-^+l+i+i. 264.gl^3,l. 265,(1)?.
(2) 1. (3) 18; 9. 266. (1) 3, -2. (2) 5, ?. (3) 2, 3; 3, 2.
a I
268. 45 shillings, 30 shiUings. 270. a^+^-:^» 5-2^/2.
3(a»+^
271. 0.
272. «»+3a?+8.
2 3'
273.
12a«-8aa?+6a^ ^ 4a?* -.n-^r^
15aa+<M;-2a;a • ^'** (4a -3a?) (6a- 2a?) ' ''^'^ "^^^
276.
a6(<j+rf)+a+c+rf' a''+aa?+aj'*
276. (1)2.
(2)ll;7. (3)4;f . 277.(1)^\^. (2)4,7.
(3) 6. 278. 7+7maes. 281. ~. 282. 2(a-6)(c-<^,
-2Jft 283. 1^, -^. 284. (1) 4. (2) 6; 4.
(3) 6, |. 285. (1) |, ^. (2)20, -a, a, -\. (3) 1, |;
47
-2, r^. 286. Second boat 16 minutes. 287. 3 feet;
2 feet 289. 18 feet. 290. 2(A- + ^-?^l-
a;»+3
a?»+2a?+8*
291. 0.
292. (*.
294.
1* ■■•i
328
AUSWERS.
295. (1) 4 (2) 61; 73. (3) 16; 8. 296. (1) 7> -S.
(2) 7, -—. > (3) 1,5. 297. liiraiimtei.
4
298. 4^ hours with the stream, 7^ hoiirs a|;aiiist the
stream. 299. a-^h, a^ a^r-h* 300. 3,-1.
//
THE MD.
OAMBBIDOl: PBINTBD AT TBI UNIVIB8ITT niSS.
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