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L/i>
AD
y
ALGEBRA FOR BEGINNERS.
BY
JAMES LOUDON, M.A.
PKOFESSOR OF MATHKMATICS AND NATURAL PIULOSOPHV,
l'NIVF.R.SITY COI.LKUF,, TOUONTO.
TOROXTO :
COPP, CLARK & CO., FRONT STREET.
1 8 7 (5 .
4^CQ
PEE FACE
V
Be,.nners, for who„. this book is intended, are recon.-
mended to work according to the rules printed in itahcs, and
to om.t, on a first reading, the articles in sn.all tvpe. Those
^sho desire only a practical acquaintance with the method.
of working, may also omit Chapters X and XXI.
The author will feel obliged to tearhnrc f,..- o /
^'tot-ti lo leaciieis for any suggestion.
they may have lo communicate
SKPlKAIlJiiK. 1S75.
CONTENTS.
PACiK.
I. Introduction r
II. Addition 8
III. Subtraction i6
IV. The use of Double Signs and Brackets 20
V. Multiplicatio' 23
VI. Division 30
VII. Examples involving the application of the first
four Rules 39
VI II. Simple Equations 42
IX. Problems 49
X. Particular results in Multiplication and Division 55
XI. Involution and Evolution 61
XII. The Highest Common Measure 72
XIII. The Lowest Common Multiple 84
XIV. Fractions 88
XV. Simple Equations, continued 105
XVI. Problems, continued 108
XVI I. Quadratic Equations 113
XVI 1 1. Problems , 119
XIX. Simultaneous Equations 123
XX. Problems 129
XXI. Exponential Notation 1 33
Answers 141
I
■I
1
I
I
8
ALGEBRA.
■•o*-
CHAPTER I.
f
I
INTRODUCTION.
1. The operations of addition and subtraction, which in
Arithmetic are stated in words, are denoted in Algebra by
the signs + and — , respectively.
Thus, add together 12, 6, and 6, is expressed 12 + 5 + 6 ;
from 15 tahe 9, is expressed 15 — 9 ; add together 3, h, 5^ and
from the sum lake i and 3^, is expressed 3 + 2 + 5^—^—31;
and so on.
2. The sign + is called the plus sign, and the sign - the
minus sign.
Thus, 24 + 2 - 15, is read 24 plus 2 minus 15.
Exercise I.
Employ plus and minus signs to express the following
ope ;ations in Algebraical language : —
(1.) Add together 56, 10 and 15; 10, 12 and |; 2, land f ;
6> 10* 12> «*"^ 3«
(2.) From 29 take 15 ; from 2^ take 1^ ; from 2*5 take 1*6.
(3.) From the sum of 11, 35 and 6, take 17; from the sum
of 81 and 75 take 69 and 42.
9
INTRODUCTION.
(4.) To the difference between 15 and 7 add the sum of
8 and 9.
(5.) To the difToi'cnce between 28 and 16 add the difference
between 10 and 4.
3. The signs + and — are also used to denote that the
quantities l)efore which they are written are, respectively,
io he added to and subtracted from some quantity not neces-
sarily expressed.
Thus, +4 denotes a number 4 to he added to some number;
—5 denotes a number 5 to be subtracted from some number.
4. Quantities to be added are called positive quantities,
and quantities to be subtracted negative quantities.
Thus, +2, +7, +^, are positive quantities; and —3, — 1,
— I, are negative quantities.
5. As an illustration of positive and negative quantities,
we may take the following examples : —
(i.) A man has a certain amount of cash in hand ; he owes
$150 to one man, and $280 to another ; and there are owing
to him the several sums of $100, $210, and $120. Now, in
order to determine what the man is worth, these several
sums are to be considered in connection with the cash in
hand ; $150 and $280 are evidently to he subtracted, and $100,
$210, and $120, to be added. The former, therefore, may be
de?ioted by -150, -280, and the latter by +100, +210,
+ 120. In other words, in the process of finding out how a
man's business stands, the sign + may be used to denote his
assets, and the sign — his liabilities.
(ii.) The mercury in a thermometer rises and falls in con-
sequence of changes in the temperature, and the amounts of
these variations are expressed in degrees. In order to deter-
mine the reading of the thermometer, some of these variations
must be added to, and others subtracted from, the reading
before the variations took place. Suppose, for example, that
during the day there take place a rise of 5°, a fall of 7°, a
INTRODUCTION.
I
rise of 12°, and a fall of 8° ; 5° and 12° are to lie added to,
and 7° and 8° subtracted from, the reading of the morning.
The forjner, therefore, may be denoted by +5°, + 12°, and
the latter by —7°, —8°. If the degrees are measured from a
zero point, distances above may therefore be denoted by + ,
and distances below by — . Thus +20° means 20° above
zero, and —5° means 6° below zero.
6. In like manner the signs + and — may generally bo
employed to denote the two relations of contrariety which
magnitudes of the same kind may bear to one another in
some defined respect. Thus, if money received be denoted
by + , money paid away will be denoted by -- ; if distances
walked in one direction be denoted by + , distances walked
in the contrary direction will be denoted by — ; if + denote
games won, — will denote games lost ; and so on.
Exercise II.
(1.) A owes B $60, B owes C $30, and C owes A $20;
employ the signs + and — to denote the assets and liabilities
of A, B, C.
(2.) B and C owe A $20 each, C and A owe B $30 each,
and A and B owe C $40 each ; express in Algebraical lan-
guage the assets and liabilities of A, B, C.
(3.) A pays B $10, B pays C $7, and C pays A $4 ; express
Algebraically the amounts paid and received by A, B, C.
(4.) Denote the following variations in the thermometer ;
a fall of 2°, a rise of 5°, a fall of 3°.
(5.) Denote that the thermometer rises and falls alternately
1° per hour for 5 hours.
(6.) Denote that the thermometer £Eillsand rises alternately
2° i)er hour for 4 hours.
(7.) Denote the following readings: 25° above zero, T
below zero.
7. In Algebra the numerical values of quantities are denoted
B
INTRODUCTION.
by (1) figures, as in Arithmetic; (2) tho letters of the
alphabet, either alone or iu combination.
Thus, if there lie three points. A, B, C, situated in that
order on a right line, the distances between A and B, and
between B and C, if unknown or variable, may be denoted by
o and h feet, respectively ; and, consequently, the distance
between A and C will be denoted by a + 6 feet. If lie
between A and B, and a and h denote, as before, the lengths
AB, BC, the length AC will be denoted by a—h feet.
Again, if x denote tho length, and y the breadth, in feet, of
a room, the dimensions of a room 8 feet longer and 5 feet
narrower will be a? +3 and y— 5 feet, respectively.
8. In Arithmetical operations which are performed with
figure symbols alone, all mention of the unit is suppressed,
the results of these operations being true, whatever be the
unit in view. Thus, 10 and 5 added together make 15,
whether the suppressed unit be a pound, a gallon, or an inch.
Algebraical symbols have a still greater generality. Not only
is the unit suppressed, but the number of units is not assigned,
as in Arithmetic. Thus a may represent any number referred
to any unit. Operations performed in Algebraical symbols
will, therefore, give results which are true for any numerical
values which may be assigned to the symbols.
9. Tho product of symbols which denote numbers, is repre-
sented by writing them down in a horizontal line one after
another, in any order, with or without the multiplication
sign X , or dot . , between them.
Thus a6, 6a, «•&, b'aaxb,bx a, all denote the product of a
and b; ubc, a-b-c, axbxc, the product of a, h, c; and
so on.
Figure symbols are written first in order ; thus, da, 6ab,
6abc, f«. When there are two figure symbols, the sign x
only must be used between them.
Thus, the product of 5 and 6a is denoted by 6 x 6a, and
not by 5-6a, or 56a, whose values avejive decimal six times o,
waA. fifty-six times a, respectively.
.«f.
INTRODUCTION.
\
When thero arc throe or more figure symbols, either the
sign X or • must bo used.
Thus the product of 2, 5 and 7 is denoted by 2-5 -7, or
2x5x7.
10. The symbol a^ stands for im ; a^ for naa ; a* for aaaa ;
and so on. «^ is read a s(/Hared, or a to the power of 2 ; u^ is
read a cnbnl^ or a to the power of ^\ a* is read « to the power oj
4 ; and so on.
11. Tlio power to wliich a letter is raised is called the
iiiikx or exponent of that power.
Thus, 1 is the index or exponent of a ; 2 of tt^ j 3 of a*,
12. The quotient of one quantity, n, divided by another, &,
is denoted by either of the forms, « -f- i or ^ .
Thus the quotient of 2a divided by 36c is denoted by
2a-r-36cor by|;i.
13. Any object or result of an Algebraical operation is
called an Algebraical quantity or eapression.
14. It is to be observed that the numerical value of an ex-
pression which is not fractional in form may be a fraction,
and the numerical value of an expression which is fractional
in form may be an integer.
Thus, if to a we assign the value 2, and to 6 the value ^,
tho value of =- will be 2 -f- 2, or 4 ; whilst if the values of a
and 6 be ^ and i, respectively, the value of ah will be i x i,
or i, and the value of ^ will be J -r i, or 2.
15. The symbol = stands for is or are equal to, and is
written between the quantities whose equality it is desired to
express.
Thus a = 3 denotes that the value of a is 3.
INTRODUCTJON.
Exercise III.
If rt = 1, ft = 2, c = 3, ^ = 4, e = 5, a; = 0, find the values of the
following expressions : —
(1.) a + 2i + 8c + 4'i. (2.) 13a-4/> + 5c. (3.) ah\de.
(4.) ah-^'k—cx. (5.) 4a&c, SictZ, Gfifca;.
(6.) a2 + 6-+c2. (7.) ^.d'^-W-cK (8.) 5rt(^+4/>3-cU
a t7
(9.)
h''~e
(^^•) ^f + f-
X, I V ah ad bx
^ de be ce'
a'
3 2b Sd'e
(^^•) :x^-Q,.2w +
yZ;^""3c2(^'^8a^6**
(13.) If h = c=d=a, find what each of the following be-
comes in terms of a :
2bca,3a%2a^ + dahc-\-4:b^cd.
16. When two or more quantities are multiplied together,
each is said to be a factor of the product.
Thus, a and 6 are factors of ah ; 3 and a^ are factors of 3a'^;
and 6, h, ar d c are factors of 5&c.
17. One factor of a quantity is said to be a coefficient of the
remaining factor, and is said to be a litend or a cmtrical
coefficient according as it involves letters or not.
Thus, in 3^ and fa''' the numerical coefficients of x and a^
are 3 and *, respectively ; in ax^ and 3cd the literal coefficients
of x'^ and d are a and 3c, respectively.
Also, since a; = 1 x x, the coefficient of x in the quantity x
is 1.
The sign + or — when it precedes a quantity is also a sign
of the coefficient.
Thus, the coefficient of a; in +803 is +3, of x^ in — 5aas* is
—5a, and of dz^ in —^cdz^ is — fc.
i
INTRODUCTION.
I
f
I
In the case of fractional numerical coefficients the letter
symbols are sometimes written with the numerator ; thus,
18. Quantities are said to be like or unlike according as they
involve the same or different combiuatiors of letters.
Thus, +5a, —7a are like quantities ; and so also are +Qx-7/,
—bx^y; —2a, — a^ are unlike quantities.
Exercise IV.
(1.) Name the coefficient of x in 2aj, ^ax, fx, lex, 4ta%x,
-^(XX,
(2.) Name the coefficient of a in —a, +3a, —fa, +2ab,
— 5ax2.
(3.) Name the coefficient of x^ in -\-x^, -x^, -3ax\ -\-^dx\
(4) Name the coefficient of xym ^xy, H-Sa'^a;?/, — faxyz.
(5.) Name the numerical coefficients in ^, ^, -|, +?^'
5.15
(6.) Name the niunerical coefficients in a:, — «/, —2x2, -.3^^^
(7.) Name the like quantities among 2x, ax, x, 3ca;l
(8.) Name the like quantities among — a^, +2a2a;, +|a-',
— a'flj.
(9.) Name the like quantities among —Za^x, ax\ ahx^.
I
C 8 )
CHAPTER II.
ADDITION.
I. Figure Symbols.
19. In order to explain the meaning of Algebraical addition,
we shall, in the first instance, suppose the numerical values
of the quantities to be represented by figure symbols, as in
Arithmetic. We shall consider in order the addition of
(i.) Positive Quantities.
(ii.) Negative Quantities.
(iii.) Positive and Negative Quantities.
20. (i.) Positive Quantities.
We have seen that a positive quantity, as + 5, represents a
quantity 5 to be added to some number which may or may not
be known or expressed. The sum of any numher of positive
quantities is denoted hy writing them in a row with their signs
between them, or by a positive quantity ivhose numerical value is
their Arithmetical sum.
Thus the sum of +4, +2, and +10 is +4+2 + 10,or +16;
that is to say, the addition of 4 and 2 and 10 to a number
is equivalent to the addition of 16 to that number.
In like manner, the sum of +2, +f, +-*- and +5 is
+ 2+1 + 1 + 5 =+8/^.
The Algebraical statement
+5 + 6 + H3^ = +14f
ADDITION.
may therefore be read the addition of^^^,\ and 3^ to a nurnber
is equivalent to the addition of 14| to that number,
21. (ii.) Negative Quantities.
Since —4 denotes a quantity 4 to he subtracted from some
number, when there are several negative quantities, as —4,
—7, —8, denoting that they are all to be subtracted from
some number, the operation may be denoted by writing them
in a row, thus —4—7—8, or by a negative quantity, —19,
whose numerical value is liheir Arithmetical sum ; that is to
say,
-4_7«8 = -19.
In Algebra —4— 7— 8, or —19, is called the sum of —4:, —7,
and —8. Thus the sum of —1, —10, — |, and — f is
-1-10-t-f = -12ii.
It follows therefore that the Algebraical sum of any number
of negative quantities is a negative quantity whose numerical
value is their Arithmetical sum.
It will be observed that, instead of saying to subtract 12, we
may say in Algebra to add —12. The Algebraical statement
-2-7-10 =-19
may therefore be read, in Arithmetical language, the subtrac'
tion of 2, 7, and 10 from a number is equivalent to the subtrac-
tion of Id ; or, in Algebraical language, the addition of —2,-7,
and —10 ^0 a number is equivalent to the addition of —19.
As an illustration of the foregoing phraseology we may
take the following example : Suppose a man's gains to be
denoted by +, and his losses by — ; then the statement
-200-60-500 = -760
may be read, if a dollar is the unit understood, the sum of a
loss of 200 dollars^ a loss of 60 dollars, and a loss of 500 dollars
is equivalent to a loss of 760 dollars. It may also be read, the
subtraction of a gain of 200 dollars, a gain of 60 dollars, and a
gain of 600 dollars is equivalent to the subtraction of a gain
of 7QQ dcUars.
J, ,1
lO
ADDITION.
22. (iii.) Positive and Negative Quantities.
Since +5 denotes a number 5 to he added, and —2a number
2 to be subtracted, the performance of both these operations
may be denoted by +5—2; and since the performance of
these two operations is equivalent to addir.g 3, we may write
+ 5-2 = +3.
In Algebra +5—2, or +3, is called the sum of +5 and
-2.
Thus, +7-5, or +2, is the sum of +7 and — 5 ; +f — i,
or +i, is the sum of +? and — i.
Again, since to add 2 to a number and then subtract 5
is equivalent to subtracting 3, we may write
+ 2-5= -3;
that is, in Algebraical language, to add +2 and —5 to a
number is equivalent to adding —3.
The statement
+7-5+2= +4
may therefore be read, in Arithmetical language, to add 7 to,
then subtract 5 frort\ and finally add 2 to a number is equi-
valent to adding 4 ; or, in Algebraical language, the sum of
+ 7, —5, and +2 is +4.
So also the statement
-3+10-15= -8
may be read, in Arithmetical language, to subtract 3 from,
then add 10 to, and finally subtract 15 from a number is equi-
valent to subtracting 8 ; or, in Algebraical language, the sum
of —3, +10, and —15 is equal to —8.
As an illustration of the foregoing phraseology, we may
again take the case of a man's gains and losses. Thus the
statement
+ 10-8= +2
may be read, if a dollar is the unit understood, a gain of
10 dollars and a loss of 8 dollars are equivalent to a gain of
2 dollars. So also the statement
ADDITION.
II
-25 + 20= -5
may be read a loss of 25 dollars and a gain of 20 dollars are
equivalent to a loss of 5 dollars.
23. From the preceding cases we can deduce the follow-
ing rule for finding the sum of any positive and negative
numbers.
(i.) When the signs are all alike— i^'mc^ their Arithmetical
sum, and 'prefix the common sign,
(ii.) When the signs are different— i^«i(Z the numericcd
difference between the Arithmetical sum of the positives and
the Arithmetical sum of the negatives, and prefix the sign of the
numerically greater sum.
Examples.
(1.) The sum of +4, +3, +h, and +7 is
+4+3 + 1 + 7= +14^.
(2.) The sum of -5, -12, -f, and -3 is
-5-12-^-3= -20|.
(3.) The sum of +4, -2, -3, +5, and +7 is
+4-2-3 + 5 + 7= +4+5+7-2-3
= +16-5
= +11.
Here +16 is the sum of the positives, and -5 of the
negatives ; 11 is the numerical difference between these sums,
and has the sign of the numerically greater + 16.
(4.) The sum of +i, -2, -^, +1, and -f is
+ i-2-^ + l-|= +i + l-.2-|-|
— 4- 5 _ l_3
= -2.
Here +f is the sum of the positives, and —^-^ of the
negatives ; 2 is the numerical difference between these sums,
and has the sign of the numerically greater — ^^.
12
ADDITION.
Exercise V.
Find the sum of
(1.) + 2, + 5, +18. (2.) +i +3, +f.
(3.) _ 8, -13, - 7. (4) -i -1, -|.
(5.) -rl8, -13. (6.) -26, +20.
(7.) +i -1. (8.) -t, +1. (9.) +2-5, -3-2.
(10.) +2, -3, +12. (11.) -3, +4, -6, +7.
(12.) +5,-8, -12, +3. (13.) +^, -1, +1, -f
(14.) ~|, -1, +2, -1 +i-.
(15.) +2-58, -3-26, +1-089, -0-067.
II. Letter Symbols.
24. (i.) Like Quantities.
We shall now show how to find the sum of quantities
whose numerical values are represented by letters. When
these quantities are like quantities, their sum is obtained by
the rule —
The sum of any numher of like quantities is a like quantity
vjhose coefficient is the sum of the several coefficients.
Examples.
(1.) The sum of + 2r<, + 5a, and + 10a is
+ 2a + 5a + 10a = + 17a.
Hero + 17 is the sum of +2, +5, and + 10.
(2.) The sum of -3c, -10c, and -12c ie
_3c;_l0c-12c=-25c.
Here —25 is the sum of —3, —10, and —12.
(3.) The sum of + 8a;, — 12a5, and + Ix is
+ 8a;-12x + 7.x=+3.x.
Here +3 is the sum of +8, —12, and +7.
(4.) The sum of —10a?, +12a5, and —6a; is
— 10a; + 12a;— 5a= — 3a;.
Here —3 fs the sum of —10, +12, and —5.
Fine
(1.)
(4.)
(6.)
(9.)
(11-:
(13.
(15.:
25. (
The
writing
I between
I Thus
— 5y is
+3&, ai
I —2a; a:
-3c.
26. I
+ or -
Thufi
ADDITION.
13
Exercise VI.
Find the sum of
(1.) +a, +2a. (2.) +3a, +6a, +7a. (3.) -a, -4a
(4) -2a, -6a, -5a. (5.) +5x2, ^3^2^
(6.) -a^ +4a^ (7.) +4a, -7a. (8.) -2c, +5c, +7c.
(9.) -10c, +8c, -3c. (10.) +a;^ -7a;2, +3a;2, -a;'^.
(11.) -2a&, +lla&, +a6, -3a&. (12.) -ia, +|<i.
(13.) +|a2, -fa2. (14) _2a, +^a, -a.
(15.) -^-a,—^, +|<i, —2a.
25. (ii.) Unlike Quantities.
Tlie sum of any number of unlike quantities is denoted hy
writing them in a row in any order, with their proper signs
between them ; and each quantity is called a term of the sum.
Thus the sum of +2a and +36 is +2a+36; of +Sx and
-5y is +3x-52/; of -W md -26^ is -ia2_2&2; of +2a,
+ 36, and — 5c is + 2-* + 36 — 5c. The terms of — 2ic + 6y are
— 2a3 and +5y ,• and of — 6a+76— 3c are —6a, +76, and
—3c.
26. If a quantity contains no parts connected by the sign
+ or — it is called a mononomial.
Thus +2x, — 3a6, +6x^2/, are mononomials.
27. When a quantity consists of two terms it is called a
binomial expression; when it consists of three terms it is
called a trinomial expression ; and generally when it consists
of several terms it is called a polynomial, or multinomial
expression.
Thus +2a— 36 is a binomial, and +a— 26 + 3c a trinomial
expression.
28. The sign of a mononomial, or of the first term of a
I polynomial, if it is positive, is generally omitted.
Thus 2x^ stands for +2a;2, and a— 6 + c for +a— 6 + c.
29. Like terms when they occur must be added together.
I The operation may be conducted by arranging the several
14
ADDITION.
quantities in rows under eiich other, so that like terms shall
stand in the same column.
Exam'phs.
(1.) Find the sum of 2a+3i and 5a— 26.
2a + 3&
5r^26
Here la is the sum of %x and 5a, and +& of +36 and —26.
(2.) I'ind the sum of xy—^x and x—^xy.
—^£-\-xy
x—'^xy
^bx^lxy.
Here — 5rc is the sum of —6^- and x, and —Ixy of -^-xy
and —^xy.
(3.) Find the sum of 3 + a; + a7?/-8.T^ -3a;?/ + 2- 6a;, and
4!r?/+fl32-|.l.
3+ a; + cry -8x2
2— 6.r— 3a;y
1 + 4^xy + re''
(d-Zx^^xy-lx^.
Here the sum of 1, 2, 3 in the iirst column is 6; of +ir,
-6x in the second is -5a:; of ■\-xy, -3.r?/, +4a?/ in the
third is ^2xy ; and of -Sa;^, +a;2 in the fourth is -7x^
(4.) Add together ha-^b-ic, ^J + ic + ^a, Ac-ia-i6.
— ^-T^ + k
Here -j-^a is the sum of ^a, ia, and -ia ; - JL6 and + ^^c
are the sums of the quantities in the second and third
columns, respectively.
ADDITION.
15
Exercise VII.
Find the sum of
(1.) 2a, -3&. (2.) -oj, +3y. (3.) -2aj, -3y, -2.
(4.) 3a, 2.x, -57/. (5.) 4, -.r, %j. (6.) ««, -.52, i.
(7.) a, -' \ 3c, -c?. (8.) -a:, 2?/, -2, 1.
(9.) a-3&,8a + &. (10.) -2a'''+&c, 3a2-56c.
(11.) 3a-5?^ -4a + 2&, 5a-6&.
(12.) a-.3&, 26-5c-, 4c-3a.
(13.) 4x— 3?/+22, — 3;c + ?/— 42, a;— 4y + 2.
(14.) a-a5 + 3, 5a + 2a5-5, -2a + 7.
(15.) 2«-7, 5a+4, -6:c + 3.
(16.) a-2&+3c, &-2c+3a, c-2a+3?).
(17.) a;-2y+33-l,2ir + 3-42, 52/-2 + 7^.
(18.) a;2 + 2aa; + a^2a;2-2a^£c2-2aa; + a2
(19.) 3aS+a''^&-2a&2+&3^ 3a&2-2a26+a3, a%^al^^W.
(20.)a-?> + |, a + &-|, &-a + |.
'^"■'•^2 3^4' 2 3^1' 2 2^4'
(22.) a+56-c, 2a-4&-c, |-a+|.
( i6 )
CHAPTER III.
SUBTRACTION.
fM ■■
h-}>
30. The Alyehraical difference between one quantity and
anotlier is the quantity wliicli added Algebraically to the
latter will produce the former.
Thus the difference between 2 and —4 is the quantity
whicli added to —4 will produce 2; the difference between
—5a and 2a is the quantity which added to 2a will produce
—6a ; and the difference between 3.r and 2a;— 5 is the quan-
tity which added to 2x—6 will produce '6x.
31. The quantity to bo diminished is called the minuend,
and the quantity to be subtracted the subtrahend.
82. The difference between two quantities is found by the
rule —
Add the first quantity to the second with its sign or signs
changed.
The reason for tliis rule will appear from the following
Examples.
(1.) From 5 take 3.
Here the difference is the sum of 5 and —3=5—3=2,
because 2 added to 3 makes 5.
(2.) From 7 take -4.
Here the difference is 7 -r 4=11, because the sum of 11 and
-4=11-4=7.
SUBTRACTION,
17
i
(3.) From -6 take -4
Here the diflference = —6+4=— 2, because the sum of
-2 and -4= -2-4= -6.
(4.) From -8 take 5.
The difference =— 8— 5=— 13, because the sum of —13
and5=-13 + 5=-8.
(5.) From 2a take —3a.
The difference =2a+3a=5a, because the sum of 5a and
— 3rt = 5a — 3a =2a.
(6.) From —hx take 4.
The difference =— 5a;— 4, because the sum of —5a;— 4 and
4 is —5a;.
(7.) From 2a take -3a + 2&.
The difference = the sum of
2a and 3a— 2&=2a+3a— 2&=5a— 2&,
Iv^cause the sum of 5a— 2& and —3a + 26 is 2a.
The operation of changing signs and adding may be per-
formed mentally, and the difference exhibited as in the fol-
lowing examples, in which the minuend and subtrahend are
written in rows with the difference underneath. In this
arrangement the first row is equal to the sum of the second
and third rows.
(8.) From Zx^-^y take a;^— 5?/.
3x2+2/
2x2 +6y
Here Sx^ is the sum of 3x2 ^nd ^^-^ and +6y of +2/ and
(9.) From 5a+3&— c take a— 6+ 3c.
5a + 3?>— c
a
- &+3c
4a+46— 4c
]8
SUBTRACTION.
Hero 4a is tho sum of 5a and —a; +4Jof +36 and +&;
and — 4c of — c and —3c.
33. From tho foregoing examples it appears that, in Alge-
braical language, to subtract a positive ijuitntity is eciuivalcnt
to adding a negdiive ; and to sidUract a negative (juantity is
equivalent to adding a positive.
This phraseology may bo illustrated by taking the case of
a man's gains and lasses to he denoted by + and — , resi)ee-
tively. Thus, to subtract a gain of 10 dollars is equivalent to
adding a loss of 10 dollars ; and to subtract a loss of 25 dollars
is equivalent to adding a gain of 25 dollars.
Moreover, if a man gains a dollars and loses h dollars, we
say, in Arithmetical language, either that he gains a—h dollare,
if a is greater than b, or loses b—a dollars, if b is greater than
n. Either of these phrases may be employed indifferently if
we agree that a gain of —c dollars means a loss of c dollars,
and that a loss of —c dollars means a gain ofc dollars.
Thus if a man gains 10 dollars and loses 5 dollars, we may
cither say that ho gained 10—6, or 5 dollars, or that he lost
5— 10, or —5 dollars. Again, if ho gains 8 dollars and loses
12 dollars, we may either say that he gained 8—12, or —4
dollars, or that he lost 12—8, or 4 dollars.
Exercise YIII.
(1.) From 1 take -3.
(2.) From 1 take -1.
(3.) From —5 take 4.
(4.) From 12 take 15.
(5.) From -3 take -8.
(6.) From -8 take -5.
(7.) From 4-56 take -6*04.
(8.) From -I'Oi take 235.
(9.) From -432 take -2-16.
(10.) From -1089 take -0*123.
SUBTRACTION,
19
(11.) From 2a take 8a.
(12.) From —5a; take 2aj.
(13.) From Sa^ take -2a^
(14.) From —3c take —5c.
(15.) From 2a take h
(16.) From -a' take 2a».
(17.) From bx^ take -Gail
(18.) From la take 4a-&.
(19.) From a + a; take a— a;.
(20.) From 5a-2a;+3 take 2a-a;-l.
(21.) From 3a2-4a& + 62 ^ake SaHaJ-i*.
(22.) From aa;— 4&y + 3c2 take 2ax-^hy—cz,
(23.) From -4a+6-l take 2a-a; + 3.
(24.) From 12a;2-5a; + l take 7a;3-16a;2+l.
(26.) From^+3/+| take |+|-;3.
(26.) From|o+6-^ take a+|6-^.
o2
( 20 )
CHAPTER IV.
TEE USE OF DOUBLE SIGNS AND OF BRACKETS.
34. The operations of addition and subtraction of positive
and negative quantities may also be denoted by the use of +
for the former operation and — for the latter.
Thus instead of saying add together 2a and —Sh, we may
employ the notation 2a -\ — 'Sh, the equivalent of which is
of course 2a— db. So also tJw sum of —5a^, +Sb, and —2c
may be written — 5cr+ +3iH 2c, which is equivalent to
— 5a'-^ + 3&— 2c; and the difference between 6a; and —7y may
be expressed hx ly, the equivalent of which is hx-\-ly.
Accordmg to this notation, therefore, 2a'^H — 3&+ +2c
means the sum of2a^, — 3&, and +2c; —60;+ +83/ H 1 the
sum of —5a?, +Sy, and —1; 7a \-2b the difference between
la and +2'>; —2x -5 the difference between —2x and —5;
4a'^ 7 the difference between 4a- and —7.
35. When any of the quantities before which the double
signs are to be used contains more terms than one, it must
be enclosed in a bracket; thus + (2a— 3&), + (—a; + 4),
- (.r-5), -(-2 + aHa;).
Thus a+(3i— c) denotes the sum of a and 36— c ; 2x
+ (4y-2) + (-2^ + 1) the sum of 2x, 4y-2, and -22 + 1;
2a2_j_ (3a2+4) the difference between 2a^-b and Sa-+4:;
and 2a+ {b—1) — (c+4) the sum of 2ct and &— 1 IckSs c+4.
Exercise IX.
Retaining the given quantities, denote by using the double
signs and brackets (when necessary) in the following opera-
tions : —
i
USE OF DOUBLE SIGNS AND OF BRACKETS. 21
(1.) The sum of 2x2, _i. 3^^^ __^y^ __^.^ 2(t, -3&,+4c.
(2.) From 2c. take —6a.
(3.) From -Stake +5^.
(4.) From the siim of 2a and — 3& take +7.
(5.) From the sum of 5 and +a; take —3a.
(6.) The sum of 5a and &— 4.
(7.) The sum of —a and —6+5.
(8.) The sum of a— 4 and 2&— c.
(9.) The sum of x^ and 2?/ + 5 less 2?.
(10.) The sum of a— 1 and 3& + 5 less —3c.
(11.) The sum of x, 2*2-1, and-3a:2-8.
(12.) The diflference between da^ and V'—c.
(13.) The diflference between a^^^ and — 2&+3.
(14.) The diflference between 2a— 5 and a2>_2a+3.
(15.) From the sum of a+&+c and a—h—c take — ct
+26-3c.
86. Double signs may be equivalently replaced by single
ones by the rule : —
Like signs produce + , and unlike signs — ; that is
+ + = -- = +,
Thus a++5=a + 5; 2x a=2x+a; 3+— 4g=3— 4c;
c— +2a=c— 2a.
37. Expressions may be cleared of brackets by the rule : —
The sign + he/ore a bracket does not change the signs ivithin^
whilst the sign — changes every sign within.
Thus 4 + (&-c)=4 + 6-c,
2a+(— x + c— 2(f)=2a— cc+c— 26^,
4a2_i_(26' + c)=4a2-l-262-c,
3a;-(-4?/ + 5)=3«;+42/-5,
03— (?/— 2)-f4— (— 3?/ + a;)=£c— y + «+4 + 3?/— .r.
22 USE OF DOUBLE SIGNS AND OF BRACKETS.
EXEBGISE X.
Eeplace the double signs \y single ones in the expres-
sions:—
(1.) 2a++3&+-c.
(3.) a;2+-4a;+-l.
Clear of brackets : —
(5.) 8a^.-(J+c).
(7.) 8a-(-2&+30.
(9.) (»+5-(2-42/)+8.
(11.) 3a;2-l-(-aj+4)+2iB-(sc2-.6).
(2.) ah—+hc c.
(4.) 5x'-\--Sx' ?a;-+8.
(6.) 8a-(5-c).
(8.) 2a-l + (&-6)+c.
(10.) a + (&-c)-(a-c).
( 23 )
CHAPTER V.
MULTIPLICATION,
38. When it is desired to denote the operation of multiply-
ing several expressions together so as to exhibit the various
factors, we enclose each in a bracket and write them together
in a row in any order.
Thus ( + 2a) (— 3&) denotes the product of -I- 2a and — 3& ;
(2a -1) l-h) the product of 2a--l and -&; (x'-d) (2a; + o)
the product of 05^—3 and 2x + 5; and («— 1) (a:; + 2) (2x—b)
the product of a;— 1, a; + 2, and 2ic— 5.
39. Each of the quantities so enclosed in brackets is called
a, factor of the product.
Thus —2a, a^— 1, and 2a— 3 are the factors of (—2a) («"— 1)
(2a-3).
40. When a factor is mononomial, it is called a simple
factor ; otherwise a compound factor.
Thus in the preceding example —2a is a simple factor, and
a^—1 and 2a— 3 compound factors.
41. In the case of a simple factor the bracket may be
omitted (i.) if the simple factor is written in the first place ;
(ii.) if the sign of the simple factor is not expressed.
Thus we generally write —2a (a3— 1), a:(a;''— 2), (a— &)a;,
(a— Z>) (c^d)x.
If the sign of a simple factor is expressed, the bracket
may be replaced by a multiplication sign, x .
24
MUL TIPLICA TION.
Thus
-2^ X +3A = -2r(( + 36),
3ic X — 5?/ = 8.t(— 5y),
(a-1) X -2tr = (rt-l) (-2^2).
JC?/^^.
Exercise XI.
Express in Algebraic<al language, retaining the given factors
and using brackets wlion necessary : —
(1.) The products of a-1, 2a2-3; -2+a, -3-a2; x-h,
-2x + 7.
(2.) The products of -%.i^, 6^-1; a^-l, -3a; 5x,
—a;- + 3.
(3.) The products of \, jk-I ; \, 2i»-3; -f, x^-S.
(4.) The products of — oo?, a:— 1, ^7 + 2; a;-— 4, +5x, 2a + 3.
(5.) The products of +8a5, —hyjXy—\\ —la, ab—'d, +86.
42. The mode of performing the operation of multiplication
whereby products are expressed as mononomials or poly-
nomials will now be explained. It is convenient to make
three cases :
I. The Multiplication of Sijiple Factors.
II. The Multiplication of a Simple and a Compound
Factor.
III. The Multiplication of Compound Factors.
43. I. The product of two simple factors is obtained by
the following rule : —
(i.) The siffn of the 'product is obtained by the rule of signs :
like signs produce -\- , and unlike signs —.
Thus the signs of +2a ( + 36), -2a (-3&), +2a (-3&),
—2a ( + 36) are, respectively, +, +, — , — .
(ii.) The numerical coefficient of the product is the product of
the numerical coefficients of the factors.
Thus the numerical value of the coefficient of the product
of ~2a and +36c is 2 x 3 - 6.
(iii.) The literal part of the product is the prodmt of the
literal parts of the factors (9).
MUL riPLICA TION,
25
Thus tho literal part of the product of — 3aj and ^ifz is
%yh.
(iv.) The 'p oditct of two powers of the same letter is a power
whose index is the sum of the ii. dices of the factors.
Thus
1 1
Examples,
(1.) The product of +2a and -36= + -2x3a6=-6a&.
(2.) The product of -5a and 2d=-5x2ad=-10ad.
(3.) The product of -^a and +3&'-=- + ^ x3a62=-.|a6='.
(4.) -a?<-3c) = 3a&c= + 3a&c.
(5.) -ia^( + ^Z>)=- + ^ • ia2&=— ia^ft.
(6.) -2x(-f3a;-)=-+2x3x. a;2=:-6a;3.
(7.) - 6ftX- Sit «) = 6x3 a3. «*= + 18al
(8.) +2a2&c3(-5«6V)= + -2 x 5 a • a^ . j . j2 . ^ . c*-
Exercise XII.
Find the product of
(1.) +3^, -26; -«, +5c; -2a2, -36; 5x, -6y.
(2.) 2a6, -7c2; -4a^ +56c; -2iB, S?/^; -6, -8a.
(3.) -ix, 3^; |a, -26; -hx, -\y; 2a\ -^.
(4.) 2^2/^ —3a;-?/ ; —ax^y, —Zxy^ ; |-a6^c, — ^a^6c^
^-Ks a __2a^ ah a-W _%c,ij^ 3x^2/' . _jxx^ _J)Oiy^
^^ 2' "3 * 5' IT ' "5"' "8" ' T' 3 •
44. II. The product of a simple and a compound factor is the
Algebraical sum of the products of the simple factor and the
several terms of the compound factor.
26
MULTIPLICA TION.
Thus the product of 2a and 3ft— 5c+<^ is the sum of the
products of 2a, 3&; 2a, —5c; and 2a, +c?; and is therefore
equal to 6a6— 10ac+2a(^.
The work may be arranged as in the following
Examples^
(1.) Find the product of —803 a;ttd 2?/'*— 403^—5,
-3a;
— 6rt2/^ + 12a3'^y + 16a;.
(2.) Find the product of 2x'^—^-\-\ and ^^^xy,
2x2-ia; + i
— fa;y
Fs'^y*
(3.) Clear of brackets the expression —2a (Sa"— 5a+l).
As this expression denotes the product of —2a and
Sa^— 5a + l, it is equivalent to — GaHlOa^— 2a.
Exercise XIII.
Find the product of
(1.) 4a-36 + c, -2x; 3a;2-2x + l, 4?/; 2a&-3c, -d
(2.) a;2«2x--5, 3^; 2rt2-3a + 7, -a»; a;'''-aa; + 2a2, -4aa;.
(3.) 2x-\-y--^z,%iyz; la^-ah^ + ah, -4ub.
(4.) a + ^ft-fc, 2a5; 2a2-3a+4, -|a; ^x^''ax-{-ia^,^ax.
^rs 4a 2ah ^ ,^ 2.k ^ , 3aa; ,.
W-) g — -g- +1,-15; _- 1+__,A|
/X'»
Clear of brackets : —
(G.) 5(^2-a^-f-4); -2(a-a& + 3); -a(a2- 2aa; + l).
(7.) (a-a;)a;; (a^h + c)c; (-!-{■ ab-Sa^)bab.
(8.) |'(6a''-9a + 12);- |'(-10 + 2x-15x«).
45. in. The product of two compound factors is the Alge-
hraiml sum of the products of one factor and the several terms
of the other.
MUL TIP Lie A TION,
2^
I
Thus the product of 2a— 36 and 4c +5^/ is the sum of the
products 2a— 3?', 4c and 2a— 36, +5<?, and is, therefore, equal
to 8ac-126c + 10a(^-156d
The work may be conveniently arranged as in the following
Examphs.
(1.) Multiply 2a;2-3a;+5 by 4a;-7.
2ic2-3a!+6
4a; -7
8a;»-12a;H20a;
-14a;H21a;-35
8a;»-26xH4ia;-35.
Here ^-llx^^-'ildx is the product of 4(r and Ix^—^x-^-^-^
*-14icH21a5— 35 is the product of —7 and 2ar^— 3a;+5; and
the sum of these two partial products, which are arranged so
that like terms stand in the same column, is the product
required.
It will be observed that in the foregoing process we work
from left to right, and not from right to left, as in the cor-
responding Arithmetical operation.
(2.) Multiply 2a-6 by c-M,
2a-6
2ac-^hc
"Qad-^-Shd
2ac •^hc— Gad + dbd.
In this case, as there are no like terms in the partial pro-
ducts, the second is placed entirely to the right of the first.
(3.) Multiply l-2x+3x- by 4(r-5a;2+2.
It will be found most convenient in this and similar
examples to arrange the given factors according to ascending
or descending powers of x ; that is, so that the exponents of
tlie successive terms shall continually increase or decrease,
in the former arrangement the numeral stands first, in the
latter last.
I..
28
MULTJPLICA TION.
iii
m
■i
In the present case let them be arranged in the order
l-2£c + 3x=^
2+4x-5»2
2-4r+6x2
+ 4:03-8** +12^3
-5a;- + lQx-"-15a;*
2 -7ar^ + 22x3- 15a;*.
(4.) Multiply 2a2-a& + Z>2 ^^y aHa&-36^
2a2-a& + 62
2a4-a3i+^r2
-6a'6H3a63-354
2aHa'6-6a''^6H4aft«-36^
Here the factors are arranged according to descending
powers of a and ascending powers of h.
Exercise XIV.
Multiply together
(1.) 2*-3, .a-{-4; 4aH-5, -.r + l; 2-3x, l+cc.
(2.) a;^-2,2.r-l; 1-a-, 2 + 3^2; \^x\^-x^,
(3.) 2a--ti + 4, 3a~2; !-« + «=, 1 + a; l + a + a^^ 1-a.
(4.) « + i, 1)1— n ; a-{-l)'-^c,7n-\-2n ; 2))i—ii, 2m -{-n,
(5.) fi;?/ + u.^a;y/-a'2; x + o:y-y\ ar-'ly.
(6.) 2«-^-5« + l,aH3^i-4; a + 26-3c, a-26 + 3c.
(7.) 3a + 2i-c,o^~2/^ + 8c,- 3xH2x2/ + 2/', «'-2a'y + 8?/2.
(8.) a;2-i,.r2-.>; « + ! ^-i ; 2a-^, a + i.
(9.) «;2_^x + l,2^— i; 3*--|:« + i, 3x— ^
aO.) ^+|-2,|-3y; ^'-2x + i3^-|.
a^ 2a
a-
2a
Ui.^g— 3- + ^' 2 ^'3~'"-^'
MULTIPLICA TION,
29
(12.) c»H2/^-a;y+a'+2/-l»«'+2/-l-
(13.) aH&Hc''— &c-ca— a&, a+&+c.
46. The <product of three expressions is found hy muUijplying
thejproduct of two of them by the third.
Examples.
(1)
Here
(2.)
Multiply together 2a;, -Sx% ^ixy^z.
2a;(-3x22/)=-6x'
Multiply together
c^y ; and —Qx^y{'-ixyh)= +|a;y2;.
X
-L aj-2, a;-3.
X
X
-I
-2
ay^—x
~2ig+2
a^-dx+2
{B8-3x2+2a;
-3a;'^+9a;~6
ar'-6x2 + lla;-6.
EXEBCISE XV.
Multiply together
(1.) -3a2, +2a% -6ab^; Ix, -^x', +|(b*;
-S:x^y,
—\y\ ^yz.
(2.) -'2x,Zxy,^x'-hy; 2a&, -^^a^, 2a2-3a& + l.
(3.) 2a;-3,4a; + l, a;-2.
(4.) a;2_a; + l, a;-l, x + 1.
(5.) a;2+2a£C+a2, a;2-2aa;+a2, a;H2a2a;2+a*.
" >
( 30 )
CHArTEB VI.
DIVISION.
MJi
47. Division being the inverse of multiplication, it follows
that, when two factors are nmltiplied together, either factor
will be the quotient of the product divided by the other.
Thus, since +a(— 26) =—2ab, +a is the quotient of — 2a6
divided by —28, and — 2& is the quotient of — 2a6 divided by
Again, since •~2x^ (x^'-4a}-\-d)=^2x^-\-Sx^—6x^, it follows
that 05^—403 + 3 is the quotient of —2x^-\-Sx^—Qx'^ divided
by —203^, and — 2a;^ is the quotient of — 223^+80^—6052 divided
by oj^— 403 + 3.
48. "When it is desired to denote that one quantity is to
be divided by another, they are enclosed in brackets and
written in a row with the sign ~ between them ; or the
second quantity is written below the first with a line between
them.
Thus (-2a;2)-f-(+3!r), or
-2ar^
+Sx'
denotes that — 203^* is to
be divided by +dx; (3a:2-2a5+6)-f-(aJ-4), or ?^-^-±_5.
03—4
that Sas^— 2a3+5 is to be divided by as— 4.
49. The first or upper quantity is called the dividend, and
the second or lower the divisor.
50. The bracket is generally omitted in the case of a
mononomial.
Thus (-203) -r (-32^) is written -2a;-7--32/; (2a;2-l)
-r(4x) is written (2x2-1) ~ 4a; ; (-3x3)-;- (2a- 1) |g
written -3a!3-r(2!»-l).
DIVISION.
3«
EZBBOISE XVI.
Betaining the given quantities and employing brackets only
when necessary, express in Algebraical language —
(1.) Divide 2a-5 by -3a; da^-Sa + l by 3a-4.
(2.) Divide 2a by -36; -a;^ by +2a;; 3a; by 2a.
(3.) Divide4a;'^by 2a5— 5; —aa;'* by oc- a.
51. The mode of performing the operation of division
whereby quotients are expressed as mononomials or polyno-
mials will now be explained. We shall consider in order
three cases :
I. When the Dividend and Divisor are Mononomials.
II. When the Divisor only is a Mononomial.
III. When the Dividend and Divisor are Polynomials.
52. I. The quotient of one mononomial divided by another
is obtained by the following rule :
(i.) The sign of the quotient is obtained by the rule of signs :
like signs produce +, and unlike signs — .
Thus the signs of -^2ah-^Scd, Sx^-7'-2x, 4a2-i--2a,
—bx^y-r- ■\-xyy are, respectively, +, +, — , — .
This rule follows from the rule of signs in multiplication;
thus, since +a (+i) = +a&, it follows that
+ a6.
+ a'
So also from the equivalent forms — a ( + &) = — o&, — a (— &)
= + a6, +a (— &) =— a6, we deduce
— a& , 7 +a6 , —at- ,
—a —a +a
(ii.) The numerical coefficient^ without regard to sign^ of the
quotient is obtained by dividing the numerical coefficient of the
dividend by the numerical coefficient of the divisor.
Thus the numerical coefficient of 12a^-^Sa is 12-7-3=4, of
2a^-T-Sx is f, and of la^^fa is I -~|=|.
(iii.) The literal part of the quotient is obtained by dividing
the literal part of the dividend by the literal part of the divisor
(12).
•= + b.
If
I'll
1:
32
DIVISION.
Thus the literal part of the quotient of 2a2 -f-Jc is ~. In
be
applying this rule the fractional form of expressing the
quotient should always bo used.
(iv.) The (jHothnt of two poirem of the same letter ia a power
inJaise index in the di'jference 0/ the indices of dividend and
diiHHor.
The reason for this rule will he evident from the following
examples : —
From (10)
a^ aaa
a a
a" aaaaa
■ aa =a2— 0^3-1 .
=iaa=a^=a'~^;
a" aaa
of aaaaaaa
=aaa = a^ssa^"* :
a
aaaa
ir
.i
So likewise
a"
±=a^-^=a\
a^
53. Since the quotient of any quantity divided by itself ia
1 , this rule can be applied when the indices of the dividend
and divisor are equal, if the zero power of a letter is considered
' a^ " d~
eiual to unity. Thus -n=l ; find, by the rule, --zra^-s—fjO.
a^ a^
therefore a°=l. Whenever, therefore, by applying the pre-
ceding rule, we get such symbols as x^, y^, c", we must
replace each of them by 1.
Examples.
(1.) -.i2-~+6=-V=-2;
-15~-10=+i^=+f;
1
4.IJ: #— _=* — -_2
f
(2.) -6aS-3/. = -«. ?=-^*.
u
(1-
(2.
(3
DIVISION.
n
^=:rt"^
(3.) -2a6-r--5c = +f^^=+|^.
c 6c
(5.) -4a6-f--7a2= + |a°-!^=+-fa3.
(6.) 2tt263c» -r 3a6c2 = ^a'-'-i ft^-i cO-2 ^ 1^52^5^
Exercise XVII.
Divide
(1.) -16 by +4 ; 20 by -4 ; -f by -|; +5 by -f,
(2.) -a by +2a;; Sa^ by -26; -Gajy by -3a.
(3.) ^ by -I; -'
^ oy g, ^ oy -^.
'A " 3
(4.) 2a» by a^\ 3a* by 3a; 8iB« by 2a;*.
(5.) -4a26 by 2a& ; lOa^JSc* by 2a6c.
(6.) a^y^ by — 3aa;2/; —ZQ^xy^ by -"Baxy.
54. II. T/je quotient of a polynomial divided hy a mono-
tiomial is the Algebraical sum of the quotients of the several
terms of the former divided by the latter.
Thus the quotient of Sx^—Qx^-\-Sx divided by —2x is the
sum of the quotients 3x'-^ — 2x, —Gx'^-i — 2x, +8x-z — 2a5 ;
and is therefore equal to — fa;"+3a3— 4.
The work may be arranged as in the following
Examples.
(1.) Divide Hx^-^x^+^x by -2x.
-•2x)S x^-4:X^+2 x
-4xH2iB-l.
(2.) Divide 6a»6-10a26='+2a& by 5ab.
5a6)5aV-10a26H2a6
a2- 2db^ + 1
(3.) Collect coefficients of x in 2aa;— S*.
"it
II
34
DIVISION,
As this means that 2aa;— Sec is to be expressed as the pro-
duct of two factors one of which shall be x, we have merely
to divide the given quantity by x to get the other factor.
Thus 2aa5-3x=(2a-3)a5.
(4.) Collect coefficients of cc^ in ^ax^—bx^+x^.
By dividing the given quantity by x'^ we get
ba3?—hx^ + flj2=(5a— 6 + 1) x^.
«t
Exercise XVIII.
Divide
(1.) 10a-15&+20by -5; -4aa! + 12--8a2by -4.
(2.) 4a2a;-3aHa'by-a2; 12x3-6!B2+9a; by 3x.
(3.) 3a''-12aH15a2by -S^^. 2a;32/-6xV+8iC2/* by 2a;«^.
(4.) 2a^6c— 3a6^c+a&c2 by a&c.
(5.) 20a26c2-15a&V+5afeby -5a6.
(6.) Collect coefficients of x in ax—hx, 2ax—cx+x.
(7.) Collect coefficients of as?/ in 4ixy—axy, Sx^y—xy^.
55. Since by (54) ^^
x-1
X
X
it follows that
likewise
g .=^-i and by (44) Ka'-l)-^-^,
and K^~l) ^J*® equivalent forms. So
2a;-3
i(2aj-3),
56. III. When the dividend and divisor are polynomials,
the quotient is obtained by the following rule : —
(i.) Arrange both dividend and divisor according to ascending
or descending powers of some letter.
(ii.) The first term of the quotient is found by dividing the
leading term of the dividend by the leading term of the divisor.
The product of the divisor and the first term of the quotient is
subtracted from the dividend, giving the first difference.
ro-
ely
DIVISION.
35
(iii.) Tlie second term of the quotient is found hy dividing the
leading term of the first difference hy the leading term of the
divisor. The product of the divisor and the second term of the
quotient is subtracted from the first difference, giving the second
difference in the process.
And so on until the last difference is zero.
y-
Examples.
(1.) Divide Qx^-hx'-'^x^-^ by 2f3!r.
3ic + 2) 6a;3-5.x2-3x + 2(2x2-3a;+l
605^ + 4tx^
-9jr;^-"-3x + 2
— 9£c^— 6a3
" 3;rT2
3x+2
— 1.
a*
So
iais,
iing
J the
isor.
nt i&
Here the divisor is first arranged, like the dividend, ac-
cording to descending powers of x. The first term of the
quotient 2x^ is obtained by dividing 605^, the leading term of
the dividend, by 'dx, the leading term of the divisor. The
product of the divisor 3«+2 and 2x'^, which is 6a5^+4a5^, is
then subtracted from the dividend, giving — Qx'^— 3£c+2, the
first difference. The second term of the quotient —^x is
obtained by dividing —9*^, the leading term of the first
difference, by 3a5, the leading term of the divisor. The pro-
duct of the divisor .ind —^x, which is — 9a5^— 605, is then
subtracted from the first difference, giving 3x+2, the second
difference. The third term of the quotient +1 is obtained
by dividing 3x, the leading term of the second difference, by
3cc, the leading term of the divisor. The product of the
divisor and +1, which is 3a3 + 2, is then subtracted from the
second difference, giving the last difference zero ; and thus
the process ends.
The latter terms of the differences need not be expressed
until the corresponding like terms in the par^i il products
are to be subtracted from them; thus in the following ex-
D 2
36
DIVISION.
I %
!ir
If
It i
ample — ITx + G is not expressed in the first difference, nor
+ 6 in the second.
(2.) Divide 6x4^5a;3 + 6x--17^ + 6 by 2.c-l.
2x-l)6xH5x3 + 6.z;'^-17u; + 6C3x3 + 4x2 + 5x-6
-12aj + 6
(3.) Divide l-2x3+&;'' by l-2ic+a;2.
l-2x + x2)l-2ie3-ha:«(l + 2a; + 3x2 + 2x3 + a;*
l-2x+u;^
2a;-x2-2:^3
2ic— 4a;^ + 2x*
3x2— 4a;^
3.c2_5a;=' + 3x*
2x3-3x4
2x3-4.r* + 2xS
a;'* — ■ia'-'' 4- x*
Here +x' is not expressed until we reach the last dif-
ference.
(4.) Divide 2a5-6a3& + 13a262-6a63-3 1<& by 2a-3&.
Here wo shall arrange the dividend and divisor according
to descending powers of a.
2a-36)2a5-3a*6-6a36 + 18a262-6a&3(a4-3a26+2a6''^
2a°-3a*6
-6a36 + 13a2&2
-6a36+ 9a262
• ^
4a=*62-.6a6'
4a262-6a63
DIVISION.
37
Exercise XIX.
Divide
(1.) a;2-7a! + 12 by a;-3; and 3x2 + 7a; + 2 by a; + 2.
(2.) a;2-4x-5 by a;-5; and 4x2-9 by 2x + 3.
(3.) 6x2-5x-6 by 2x-3 ; and 9x'- 18x2 + 26^-24 by
3x-4.
(4.) x3-4x2 + 6x-2 by x2-3x+2; and cc^ + x^+l by
x'^ + X + 1.
(5.) 4x4-15x3 + 14x2-6x + l by 4x2-3x + l.
(6.) X*— 1 by X— 1; and x^ + 1 by x + 1.
(7.) x2— 2x?/ + 2/'^ by x—y ; and x^-^-if by x + y.
(8.) aHa'&H&^bya2-a& + &2^
(9.) 20x2 + 9x?/-12x- 182/2 + 92/ by 4x-32/.
(10.) x6-2ax* + 3a2x3-3a3x2 + 2a*x-a5 ijy x3-ax2+a2x'-a8,
(11.) x^-^x^y—xy^-\-y^ by x2 + x2/ + ?/2.
(12.) |a*X-\fa3x2 +|a2^3 +_3_o^a;4_^5 l^y |.f^3__A,j2^^1a;3^
57. When the division is not exact, the last difiference is
called the remainder. In this case the product of the quotient
and divisor added to the remainder will be equal to the
dividend.
Example.
Find the quotient and remainder in dividing 10x3+ 7x'*
-8x-2by 2x + 3.
2x + 3)10x3+7x2~8x-2(5x2-4x+2
10x3 + 15x2
-8x2-8x
-8x2- 12x
4x-2
4x+6
-8
Quotient =5x2— 4x + 2.
Remainder = — 8.
I i
38
DIVISION.
Exercise XX.
Find the quotient and remainder in dividing
(1.) 4x3-4a;H8a;+2by2a5+l.
(4.) 2a5«-2x* + 9x3 by 2a;' + a? + 1.
(6.) 2a;*x2x*+5a;3 by a;»+a^+»+l.
I
H
( 39 )
CHAPTEE VII.
EXAMPLES INVOLVING THE APPLICATION OF THE
FIEST FOUB BULE8.
58. In the following examples some of the given quantities
are expressed by letter symbols, and the object of the exercises
is to express in like manner other quantities which by the
conditions of the question are related to the former. When
a doubt exists as to the manner of solving a question, it will
be well to substitute numbers for letters in order to see what
operations ought to be performed in the given symbols.
59. The sign .*. will be used to mean hence, or therefore, and
the sign '.* siuce^ or because.
Exumples,
(1.) I buy goods for 2a+Sb—c dollars, and sell themfo^r
4a— 6 + 2c dollars ; what do I gain ?
4a— 6+ 2c
2a-H36-c
2a-4&+3c
.*. the gain, which is the selling price less the cost price, is
2a— 4&+ 3c dollars.
(2.) A man has 3*^^ + Tas + 2 dollars and spends aj +2 of them
per day ; how long will his money last ?
a;+2)3xH7a;+2(3a;+l
»+2
»+2
40
EXAMPLES OF THE FIRST FOUR RULES.
.*. the required number of days, which is equal to the num-
ber of times the amount of his daily expenses is contained in
the amount he possesses, is Sx- + 1.
(3.) A man walks x miles in y hours : at what rate per hour
does he walk ; how far will he walk in 5 hours ; and how long
"will he be in walking 12 miles ?
*.* he walks x miles in y hours,
„ ^ „ „ 1 hour,
and 1 mile in ^ hours.
• • »
X
Again,
X
he walks - miles in 1 hour,
y
»
>j
5x
y
,, 5 hours;
and
he takes ^ hours to walk 1 mile,
X
if i>
12y
X
» » i>
12 miles.
or OuC
The answers are, therefore, - miles, — miles, and
y y »
12^/
hours.
Exercise XXI.
(1.) A man walks x, +a, and x—2a miles in tb'^ same
direction ; how far does he walk altogether?
(2.) A man has 100 dollars, and owes 50— x dollars; what
is he worth ?
(3.) A man walks a-\-h miles and returns a—h miles; how
far is he from the starting point ?
(4.) What is the area of a room a? +2/ feet long and x—y feet
broad ?
(5.) A man walks x miles at a miles an hour ; how long is
he on the road ?
(6.) A has X dollars, B 50?/ cents, and C 76z cents ; how
many dollars have A, B, and C together ?
EXAMPLES OF THE FIRST FOUR RULES. 41
(7.) A has X pounds, B has y shillings, and C z pence ; how
many pounds have A, B, and C together ?
(8.) A spends a dollars in x days ; in how many days will
he spend 10 dollars?
(9.) How many square yards in a floor which is a feet by
X feet ?
(10.) What is the cost in dollars of painting a floor aj-hy feet
by 05— y ^66* at x'^-k-y^ cents per square foot?
(11.) A owns a acres, B 6 acres, and C 5 acres less than one-
half of A's and one-third of C's together; what is the whole
amount possessed by A, B, and C ?
(12.) A owns a+ 6 acres, B a— & acres, C half as much as A,
and D half as much as B ; how much more do A and C own
than B and D?
(13.) A walks a miles in t hours, and B half as far again in
the same time ; how far will B walk in 10 hours?
(14.) A walks 10 miles in x houis ; how long will he be in
walking a miles?
(15.) A spends at the rate of x dollars a day for a days, a
dollar a day less for twice that time, and a dollar a day
more for three times that time; how much does he spend
altogether?
c 42 )
CHAPTER VIII.
SIMPLE EQUATIONS.
I
I i;
V
60. An equation is the statement of the equality of diiferent
quantities, and these quantities are called the equation's
members or sides.
Thus 205+3=7 is an equation whose sides are 2a? + 3 and
7, and a;*— 5a; +6=0 is an equation whose sides are a;'''— 5a+6
andO.
61. An identity is the statement of the equality of two like
or different forms of the same quantity.
Thus 2a + &=2a + 6, 2a; + 3a;=5a;, x^-bx-\-6=(x-'2X«:-S),
are identities.
62. In the case of an identity the equality holds for all
values of the quantities involved, whereas in an equation the
equality does not exist except for a limited number of values
of the quantities involved.
Thus the statement a;^+2a;+l= (a; + l)^ holds no matter
what X is; but 5a;— 3=7 holds only when a;=2,and a;^+6=5a;
only when a;=2, or a;=3.
63. A symbol to which a particular value or values must
be assigned in order that the statement contained in an equa-
tion may be true is called an unknown quantity.
Thus the unknown quantity in 6a;— 6=9 is x, and in
22/2-2/=8 is y.
The letters a, &, c, Z, m, ?^, ^, q, r are generally used to
denote quantities which are supposed to be known, and x,y, z
those which are for the time unknown.
SIMPLE EQUATIONS.
43
-2/,»
&i:. Quantities which on being substituted for the unknown
reduce the equation to an identity are said to satisfy the
equation, and are called its roots.
Thus 5 is a root of 2ac— 3=7, because 5 when substituted
for X reduces the equation to the identity 10—3=7. So 2
and 3 are the roots of a;'' + 6=5a;, because when either is sub-
stituted for X the equation is satisfied.
65. The determination of the root or roots is called the
solution of the equation.
66. An equation is said to be reduced to its simplest form
when its members consist of a series of mononomials involving
positive integral powers only of the unknown.
Thus5a;-8=0, a;2-5a; + 8=0, 2a;H6a;=7, a;^- 6*2= 7a; -8
are in their simplest forms.
67. Equations when reduced to their simplest forms are
classed according to their order or degree.
68. Simple equations, or those < f the first degree, are those
in which the highest power of the unknown quantity is the
first; as, for example, 2£c=5, 5a5— 8=0, 305—7=0.
69. Quadratic equations, or those of the second degree, are
those in which the highest power of the unknown quantity
is the second; as, for example, a;^— 2xH-3=0, 2a;2=9,
4a;2_3=10a;.
70. Equations of the third and fourth degrees arc called
cubic and biquadratic equations, respectively ; thus r' + 2a;=10
is a cubic, and a;^— 2a;^=10a;— 5 a biquadratic.
71. It is proved in works on the Theory of Equations that
the number of the roots of an equation is equal to its degree ;
so that a simple equation has one root, a quadratic two roots,
a cubic three roots, and so on.
72. In order to solve an equation it is generally necessary
to reduce it by one or both of the following processes : —
I. Transposition of Terms.
II. Clearing op Fractions.
44
SIMPLE EQUATIONS.
%\
These operations will be illustrated by applying them in
order to the solution of simple equations.
I. Transposition of Terms.
73. If an equation contains no fractions it may be solved by
transposition of termSf which consists in taking the unknown
quantities to one side of the equation and the known to the
otheVy the signs of the quantities which are so transposed being
changed.
Thus, if the equation is 4a; +5=10, by subtracting 6 from
each side we get
4a; +5-5=10-5,
or 4a;=5;
and so any quantity may be transposed from one side to the
other by changing its sign.
Examples.
(1.) Solve 5a;+15=25.
Transposing + 15 we get
5a;=25-15=10.
The value of x is then found by dividing both sides by its
coefficient 5.
/. x=2.
(2.) Solve 8a;-4=2a; + 20.
Transposing —4, 8a!;=2a5 + 20 + 4.
Transposing 2x, 8a; — 205 =20 + 4 ;
6a;=24.
:.x =4.
(3.) Solve 10 + 2(6a;-l) =32-3(a;-4).
Clearing of brackets, ^;
10 + 12X— 2=32-3a; + 12.
Transposing 10, —2, —3a;, i
12a; + 3a;=32+12-10+2; [
15x=36.
• «._ 3.6 —i)2.
"^
SIMPLE EQUATIONS.
45
(4.) Solve 3(a;*+2x) + 13 = 3a;2-7+4(3a;-l).
Clearing of brackets,
3a;H6a; + 13 = 3a;2-7+12aj-4.
Transposing +13, 3a;'», +12a?,
3a!2-3a;2+6a;-12a; = -13-7-4;
Dividing by— 6,
- 6x = -24.
03 = 4.
When, as in this case, the same quantity is common to
t)oth sides, it may be struck out without actually transposing ;
thus
5a;2-6a; + 7=8a; + 5!»2_10
becomes — 6a3 + 7=8a;— 10.
(5.) Solve ax-\-'b=c.
Transposing +&, aa5=c— 6.
Dividing by a, a;=£rL.
a
(6.) Solve ax-\-'b—cx-\-d.
Transposing +&, ex,
ax—cx = d'-'b.
Collecting coeflBcients of oj,
(a— c).r = c?— 5.
Dividing by a— c,
a—c
(7.) Solve a(a;— &) = &(a; + a)— c.
Clearing of brackets,
ax—ab = hx-\-ab—c.
Transposing — a&, hx,
ax—bx = ab-\-ab—c.
Collecting coefficients of a;,
(«— &)a; = 2a&— c.
Dividing by a— &,
a;=
2a6— c
4«
SIMPLE EQUATIONS.
Exercise XXII.
(1.) 3-|-a;=6. (2.) «-6 = 4. (3.) a; + 5 = 12.
(4.) a; + 9=4. (5.) 2x-l=3. (6.) 5a;+4 = 29.
(7.) 4-3a;-5. (8.) l-aj=6. (9.) 3=6-2».
(10.) 2a; + 3 = .'r + 5. (11.) 5i»-2-2x + 7.
(12.) .'c+4 = 18-4x-4. (13.) 2r+3 = 3x-4.
(14.) 16-2«=46-5a;. (15.) 3(a:-l)+4 = 4(4-aj),
(16.) 5-3(4-2a;)+4(3-4a!)=0.
(17.) a!-l-2(a;-2) + 3(£c-3) = 6.
(18.) 6(c»-.5)+2(a;-3)-(»>-l)=9.
(19.) 2(x-.2)-3(«-3) + 4(c«-4)~5(a;~5) = 0.
(20.) 4(a;-ll)-7(x-12) = 6-(a;-8).
(21.) «=2a-». (22.) 2a-3a;=8a-5a;.
(23.) a;-2&=2a~a;. (24.) a + o:-& = a + 6.
(25.) aas— a&— ac=0. '26.) ax—a = b-—bx.
(27.) ax-a^=hx^b\ (28.) «(«-&) = c(a3-a).
I
II. Clearing of Fractions.
74. If an equation contains fractions, it may be reduced to
a form capable of solution by transposition, by multiplying
both sides of the equatio7i by the L. C M. of all the denominators
of the fractions.
In the following examples numerical denominators only
will be considered.
(1.) Solve
XXX
2 3 5
Multiplying by 30, the l.c.m. of 2, 3, 5,
15x — lOx = 6x — 30.
Transposing, 15a; -- lO.r — 6x = — 30 ;
-a; =-30.
.*. x = 30.
(2.) Solve ^^^^&« + l?^
A o o
Multiplying by 24, the l.c.m. of 2, 3, 8,
12(a!-l) + 8(2a; + 3)=:3(6.K+19);
SIMPLE EQUATIONS,
47
whence on clearing of brackets and transposing we get
It must be observed that ^=i(a;-l), ?^= i(2a; + 3),
3
6x4-19
fl-'^d y =J(6a? + 19); and therefore the brackets must be
supplied in the first step since the numerators become bino-
mial factors.
(3.) Solve
8-^:=l + ^+2^(j
2 • 3
Multiplying by 6, the l.o.m. of 2 and 3,
■18~3(a;-l) + 2(aj + 2) = 0;
whence on clearing of brackets and transposing we get
a5=25.
m
X X
Exercise XXIII.
X , X
(l.)i-i=3. (2.) ^+^=7. (3.) 1-1+1=10.
(4.) ^+^=10.
2 3 '4'
(5.) ^+1=20-^9
(6.) ^+^=4-^. (7.) ^ + 2=^-K«^ + l).
(8.) 2(a)-l)-K2a;-9) = K17-2x).
(9.) §^=4-^^-12 (10.) |(x-4) + K«'-6)=2x-15.
(11.) 3-|cc=l-J^(7a;-18).
(12.) 6(x-l)-l=f(5-2x) + .3(x + l)+4a;.
(13.) x+^^—^^-9h=0.
(14.) 4.+2,V+^-^^-^-^=^+M.
/iK\ * + 4 3a;--2 , 1 x—1
\^'^') — o -ITT- +4 =
12
3 •
..n. S-2x 6x 5_3(2x + 6) 2x
^,10.; -^-+7—7 14- -y
48
SIMPLE EQUATIONS.
,.^.?>x-\ 13-;r 7.T_^ll.x-|-33 n
(1 i .) —^ >.- —-rr+ ^. =^-
5
6
^^'•^2 3'T 5- + " 9"-^*
(20.) |(x-.8)-'^--*-g=0.
ih
•I
( 49 )
11
CHAPTER IX.
PROBLEMS,
75. When a question is assigned for solution the unknown
quantity, or number, is generally involved in the various
conditions which are proposed for its determination. The
expression of these conditions in Algebraical language leads
to an equation, the solution of which will be the solution of
the question.
76. In some cases, although there are more unknowns than
one, they are related to each other in such a manner that
when one is determined the others become immediately
known. In such cases the unknowns can be expressed in
terms of one unknown.
Thus, if the sum of two unknowns is equal to 8, we may
denote one of them by x and the other by 8— jc, or one of
them by 4: + a3 and the other by 4— a?,- if the greater of two
unknowns exceeds the less by 3, the former may be denoted
by X, and the latter by jc— 3 ; if there be two numbers of which
one exceeds 4 times the other by 7, the former may be denoted
by 4a3 + 7, and the latter by x.
In like manner, if there be three unknowns, of which the
first exceeds the second by 3, and the second exceeds the
third by 5, the first may be denoted by a?, the second by 03—3,
and the third by as— 8.
77. The following examples will illustrate the method of
solving problems by means of simple equations : —
(1.) What number exceeds its fifth part by 20 ?
Let X be the required number.
E
50
PROBLEMS.
1^'
m
Then its fifth part =. | ; and by the condition of the
question
5
/. x=25.
(2.) The Slim of two numbers is 71, and their difference 43 :
find them.
Let x be the greater number.
Then .a— 43 is the less : and since their sum is 71, we have
a; + a;-43=71.
/. a; =57, the greater;
and a; -43 = 14, the less.
This question may also be solved as follows:
Let X be one number, the greater suppose.
Then 71 -a; is the less i and since their difference is 43, we
have „^ ^ Ao
a;-(71-x)=43.
.*. a; =57,
and71-aj=14.
(3 ) A boy is one-third the age of his father, and has a
brother one-sixth of his own age; the ages of all three
amount to 50 years. Find the oge of each.
Let the boy's age z=x years.
Then the father's age —'6x years,
And the brother's age =| years.
%
\
And by the condition of the question
o
/. .x=12,
3a;=36,
^-2.
Fractions may be avoided by supposing the ages of boy,
father, and brother to be 6a?, 18a7, x years, respectively.
PROBLEMS.
5'
5:
re
we
IS a
iree
(4.) A and B start from two places, 90 miles apart, at the
same moment, A walking i miles per hour, and B 5 ; when
will they meet, and how far will each have walked ?
Let the time of meeting be x hours after starting.
Then A will have walked 4a5 miles, and B 5x miles ; and
since the sum of these two distances is 90 miles,
4a; + 5x=90.
/. aj=10.
/. 405=40, and 5x=50, are the distances in miles walked by
A and B, respectively.
(5.) How much tea at 90 cents per lb. must be mixed with
50 lbs. at $1"20, that the mixture may be sold at $1*10 ?
Let X = the number of lbs. at 90 cents, the value of which
will be '900; dollars.
Then, since there will be a? + 50 lbs. in the mixture, its
value will be 1*10 (cc + 50) dollars ; and since the value of the
50 lbs. at $1'20 is 60 dollars, we have
•90x + 60=l-10(£c + 50).
Multiplying by 100,
90a; + 6000=110(x- + 50).
.*. 3J=25.
boy,
Exercise XXIV.
(1.) Divide 25 into two such parts that 6 times the greater
exceeds twice the less by 70.
(2.) Divide 135 into two parts such that one shall be |- the
other.
(3.) The sum of two numbers is 37 and their difference 3 :
find them.
(4.) A fish weighed 71bs. and half its weight : how much
did it weigh ?
(5.) At a meeting 43 members were present, and the motion
was carried by 9 : how many voted on each side ?
E 2
5«
PROBLEMS.
Xh
(6.) Divide 326 into two parts, such that f of the one shall
be equal to the other diminished by 7.
(7.) What is the number whose 4:th and 5th parts added
together make 2i ?
(8.) Forty-two years hence a boy will be 7 times as old as he
was 6 years ago : how old is he ?
(9.) A father is 57 years old, his son 13 : when will the
father be 3 times as old as his son ?
(10.) I have made 164 runs at cricket this season in 12
innings: how many must I make in my next innings to
average 14?
(11.) My grandfather told me 10 years ago that he was 7
times as old as myself; I am now 18: how old is my
grandfather ?
(12.) If in a theatre f of the seats are in the pit, ^ in the
lower gallery, \ in the upper, and there are 50 reserved seats,
how many are there altogether ?
(13.) After losing \ of our men by sickness, and 210 killed
and wounded, the regiment was reduced by \ : how many
men did the regiment originally contain ?
(14.) In a certain examination f of a boy's marks were gained
by translation, ^ by mathematics, and -^ by Latin prose : he
also obtained 1 mark for French. How many marks did he
obtain for each subject ?
(15.) Two men receive the same sum ; but if one were to
receive 15 shillings more, and the other 9 shillings less, the
one would receive 3 times as much as the other. What sum
did they receive ?
(16.) A and B begin trade, A with 3 times as much stock
as B. They each gain £50, and then 3 times A's stock is
exactly equal to 7 times B's. What were their original stocks?
(17.) One-tenth of a rod is coloured red, one-twentieth
orange, one-thirtieth yellow, one-fortieth green, one-fiftieth
PROBLEMS,
53
•e to
the
sum
blue, oue-sixtieth indigo, and the remainder, which is 302
inches long, white : what is its length ?
(18.) Find three numbers whose sum is 37, such that the
greater exceeds the second by 7, and the second exceeds the
third by 8
(19.) Find a number such that if 5, 11, jmd 17 be suc-
cessively subtracted from it, the sum of the third, fourth,
and sixth parts of the respective results shall be equal to 19.
(20.) How much wheat at 44s. a quarter must be mixed
with 120 quarters at 60s. that the mixture may be sold for
50s. a quarter ?
(21.) How many lbs. of tea at 2s. 6d. per lb. must be mixed
with 18 lbs. at 5s. per lb. that the mixture may be sold for
4s. per lb. '?
(22.) How much sugar at 4id. per lb. must be mixed with
50 lbs. at 6^d. per lb., that the mixture may be worth 5d.
per lb. ?
(23.) A bag contains a certain number of sovereigns, twice
as many shillings, and three times as many pence ; and the
whole sum is £267 ; find the number of sovereigns, shillings,
and pence.
(24.) I wish to divide £5 4s. into the same number of
crowns, florins, and shillings ; how many coins must I have
of each sort ?
(25.) A person gets an income of £550 a year from a
capital of £13,000, part of which produces 5 per cent, and
part 4 per cent. : what are the amounts producing 5 and 4 per
cent., respectively ?
(26.) I invest £800, partly at 4| per cent., and partly at 5^
per cent. ; my income is £39 10s. : what are the sums invested
at 4 J and 62 per cent., respectively ?
(27.) A garrison consists of 2600 men, of whom there are
9 times as many infantry and 3 times as many artillery as
there are cavalry : how many men are there of each ?
(28.) My grand&ther's age is 5 times my own ; if I had
.1
54
PROBLEMS.
U\
been born 100 years ago, I should have been born 15 years
before my grandfather : how old am I ?
(29.) There is a number consisting of two figures of which
the figure in the unit's place is 3 times that in the ten's;
if 36 be added, the sum is expressed by the digits reversed :
what is the number ?
(30.) A miner works for 6 weeks (exclusive of Sundays),
his wages being it the rate of 24s. per week, but he is to
forfeit Is. besides his pay for each day that he is absent ; at
the end of the time he receives 4 guineas : how many days
was he absent ?
(31.) A contractor finds that if he pays his workmen
2s. 6d. per day, he will gain 10s. per day on the job ; if he
pays them 3s. a day, he will lose 18s. : how many workmen
are there, and what does the contractor receive per day?
(32.) An officer on drawing up his men in a solid square
finds he has 34 men to spare, but increasing the side by 1
man he wants 39 to make up the square : how many men
had he ?
(33.) If the mean velocity of a cannon-ball at effective
ranges is 1430 feet per second, and that of sound 1100 feet,
how far is a soldier from a fort who hears the report of a gun
1% of a second after he is hit ?
(34.) An army in a defeat loses one-sixth of its number in
killed and wounded, and 4O0O prisoners. It is reinforced by
3000 men ; but retreats, losing a fourth of its number in
doing so. There remain 18,000 men. What was the
original force ?
(35.) Su])pose the distance between London and Edinburgh
is 360 miles, and that one traveller starts from Edinburgh
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 they will meet.
(36.) There are two places 154 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 2 hours, and the other at
the rate of 5 miles in 4 hours : when will they meet ?
( 55 )
CHAPTER X.
PARTICULAR RESULTS IN MULTIPLICATION AND
DIVISION.
78 There are several results in multiplication and division
which should be committed to memory, as they enable us to
dispense with the labour of performing the operations. The
following cases occur most frequently.
I. Since by actual multiplication
(a + by=a^ + b^ + 2ah,
&c. = &c.
we can hence write down the square of a polynomial bv the
rule: ^
The square of a polynomial is equal to the sum of the squares
of the several terms and twice the sum of the products of every
tivo terms. ^
Thus in the last example a\ +&2, a,c\ are, respectively, the
squares of a, +&, -c; +2a& is twice the product of a and
+ &, -2ac IS twice the product of a and -c, and -2ic is twice
the product of +& and — c.
In taking the products of the terms, two and two, it will
be found most convenient to take in order the products of
the first term and every term that follows it, then the pro-
ducts of the second term and every term that follows it, and
«o on, if there be more terms than three.
m^
56
PARTICULAR RESULTS IN
Examples.
(1.) (a + 2ie)2=a2+4a!2+4aa;.
Here +4aa3 is twice the product of a and +2a;.
(2.) (2a-5£c)*=4a2 + 25a;2-20aaJ.
Here +250?^ is the square of —6x, and — 20aaj is twice the
product of 2a and —6x.
(3.) (2ar2-3a;+4)2=4a;* + 9a;2 + 16-12a;3 + 16»2_24a;
=4x^-12a;H25!»'^-24x + 16.
Here — 12a5^ is twice the product of 203^ and —3a!, +16a!r' of
2x^ and 4, and —24a; of —Sx and +4. Like terms are added
together and the terms are arranged according to descending
powers of x.
(4.) 992 =(100-1)2=10000+1-200=9801.
Exercise XXV.
Write down the squares of
(1.) 03—1, x+a, x—5, x-\-d.
(2.) 2a; + l, Sx-1, 2x+3, 3a;-2.
(3.) x"-a, 2xy + l, Sx^-'2a, aa;2-46.
(4.) 05— 2/ + Z, 2oj + 32/— s, x—2y^^z, 2a5— 42/ + 1.
(6.) 2a2 + a + 3, 3a2-4a + l, a2-2a-4.
(6.) Find the squares of 49, 98 and 995.
79. II. Since (a +6) (a— 6) =a^^b^, it follows that the pro-
duct of the sum and difference of two quantities is equal to the
difference of their squares.
Thus (2xVSy)(2x-Sy) =^x^^Qy^;
(a2+l)(a2_l) =a4_l;
(50)2+42/) (5a;2-42/) ='26x*-16y^;
(2ar»+a*)(2a;3-a^) =4:x^^a^;
501 X 499 =(500 + 1) (500-1)
=50(^-1
=249999.
MULTIPLICA T20N AND DIVISION.
57
Exercise XXVI.
Write down the products of
(1.) oj-l, x + 1; a+3, a-3; 2+03, 2-a;.
(2.) 2j5 + 1, 2x-l; 5a+2, 5a-2; 4a3+a, 4a;-a.
(3.) aHa;,a^--a5; aHl,a'-l; a'* + a;2, aO^^jZ^
(4.) 3aH26, 3a2-2&; 4a»+2a;2, 4a3-2x2; 7a*-5a3,
7a*+5a8.
(5.) Find the products of 48, 52; 95, 105; 695, 705.
80. III. Since by actual multiplication
Id'^W-k-ab) (a-&) =a3-63,
it follows that the sum of the squares less the product of two
quantities multiplied by their sum is equal to the sum of their
9ubes.
In the latter identity the two quantities are a and —h ;
the sum of their squares less their product is, therefore,
^2+62 ah=a'^-\-b^-\-ab; and, since the cube of —6 is —6',
the sum of their cubes is a^—b^.
Examples.
(I.) (x^-x + l) (x + 1) =x^-\-l.
Here the two quantities are x and 1.
(2.) (x^+x+1) (x-l)=x^-l.
Here the two quantities are x and —1.
(3.) (4a;2-2x + l)(2«+l)=8x» + l.
Here the two quantities are 2x and 1, the cubes of which
are 8x^ and 1.
(4.) {x'--a^x' + a')(x^-\-a^)=x''-\-a\
Here tlie two quantities are x^ and a^, the cubes of which
are .x^' and a*^.
(5.) (4:x^-\-6x^ij-\-di/) (2x^-Si/) =8x^-27 f.
Here the two quantities are 2x^ and —dy, the cubes of
which are Sx*' and —27y\
58
PARTICULAR RESULTS IN
\ '
EXEUCISE XXVII.
Write down the products of
(1.) m^"mn-\-ii^,in-^u; 2^"-^Pl-^Q^>P-'Q'
(2.) m^-m + l, ?/i + l; l + q-{-r,l-fj.
(3.) x^-Bx + 9,x + 3', a2+4a-l-lG,«-4.
(4.) 4a2-2a + 1, 2a + 1 ; leas^ + 4ax + a-, ^x—a.
(5.) ^a'-Qab + W, 2a + 36; 9xH 15x7 + 25^/2, 3a;- S^/.
(6.) a;< + a;2 + l, a;2-l; (c^-aV+a*, ccHa^
81. IV. By actual division it can be shown that the sum of
any the same odd powers of two quantities is exactly divisible by
the sum of the quantities.
Thus, ^±i' =1,
x+y
t±y'=x'-xy + y\
x-\ry
J-Ji-J- = 03* — xhj + x^y^ — xy^ + y^,
&c.=&c.
It will be observed that the signs of the quotient are
alternately + and — , and that the successive powers of x
are in descending whilst those of y are in ascending order.
82. V. The difference of any the same odd powers of two
quantities is exactly divisible by the difference between the
quantities.
Thus,
— X,
x—y
_ — ^=zx^-\-xy-\-y'^,
x-y
x—y
ic* + x^y + x^^y"^ + xy^ + y*
&c.=&c.
h
MULTIPLJCATJON AND DIVISION,
59
Hero tlie signs of the quotient are all + .
It may be noted that this case is included in the preceding
(81) by supj)osing the two quantities to be x and — y. Thus
the sum of the cubes of x and — ?/ is a;^— t/^ which is exactly
divisible by their sum x—y. In fact the formulas of (82)
are deducible from those of (81) by substituting —yioxy in
the latter.
83. YI. The dljference hetvjeen any the same even powers of
tioo quantities is exactly divisible by the sum of the quantities
and also by their difference.
Thus (i.)
y,
x + y
— ^ =jf-j'y-\rxy^^y^,
x-{-y
03'
P — lfi
yi. =x'^—'ji*y + x^y^—a^y^-\-xy*--y^f
(ii.)
x-i-y
&c. =&c.
x-y
-^^ =x^-{-x^y + xy^-\-y^f
x-y
/yjC _^ yd
iZ. = x** + 05*2/ + x^y^ + xy^ + xy* + y'^,
x-y
&c.=&c.
It will be observed that when the divisor isx—y, the signs
of the quotient are all + ; and when the divisor i&x + y the
signs of the quotient are alternately + and — . It should
also be noted that the formulas (ii.) are deducible from (i.) by
substituting —y for y in the latter.
Exercise XXVIII.
Write down the quotients of
(1.) x^+1 and 03^ + 1 divided by a; + 1.
6o
PARTICULAR RESULTS.
II !
i:
(2.) a;3_i and ;r'-l divided by x^l.
(3.) ^2-1 and «<-! divided by « + !.
(4.) x^-l and a;*-l divided by a;-l.
(5.) 4a2-9i2 divided by 2a + 36.
(6.) 9j^«-4a2 divided by 3.x3-2a. *
(7.) ia'-a;« divided by i«Ha;3.
(8.) Find what the quotient of x^ + if divided by aj+y
becomes when (i.) a;=2a, y=36 ; (ii.) x=a\ y=2.
(9.) Find what the quotient of x^^y^ divided by x-^y
becomes when (i.) a;=3a, y=6 ; (ii.) aj=2a2, y =36.
84.
expr(
of pc
expm^
85.
the p
the p
with
of the
Thi
i; I
Ii i: t
( 6l )
CHAPTER XL
3y a;+y
jy x—y
INVOLUTION AND EVOLUTION,
84. The process by which the powers of quantities are
expressed as mononomials is called Involution, The powei-fl
of polynomials when so expressed are said to be developed, or
expanded.
85. We have already explained the notation for denoting
the powers of a single symbol, as a, as, y. In all other cases
the power of a quantity is denoted by enclosing it in brackets
with the number indicating the power above and to the right
of the bracket.
Thus (—2a)'' denotes the third power of —2a; {a^lif the
square of d^h\ {a%c^y the fourth power of a%c^\ (a— by the
cube of a— 5; (x"— 2x+3/ the fifth power of a;'^— 2x+3.
86. The same notation is used for denoting powers of
powers of a quantity, brackets of different shapes being em-
ployed when necessary.
Thus (a^y denotes the square of a'; {(— 2icy)'^p the
cube of (-2«2/)2 ; {(x^-S)'}^ the fourth power of (x^-bf.
Exercise XXIX.
Retaining the given quantities, denote
(1.) The cubes of -a, 2x, Zxy\ 'la^Wc.
(2.) The squares of 2a-l, a-& + l, x^—l.
(3.) The squares of (xO^ (-2a)3, {4mx)\ (3a%c*y.
(4.) The cubes of (a-6)3, (jb^-I)^, {9^^3x+2y,
62
INVOLUTION AND EVOLUTION.
(5.) The squares of the cubes of x, — 2vC, aP-h^ a;— a,
(6.) The cubes of the squares of —a, x^, 4x— 1, as^— a'*.
87. A power of a power of a quantity is expressed as a
power of that quantity according to the rule
Thus,
a ) —CI .
88. So also
Thus,
(a"'¥c'^y=ia""''hP"-ci\ &G,
(abhy=a^b'V-'.
89. A rule has already been given in Art. 78 for expanding
the square of any polynomial. The expansion may also be
effected as in the following examples, in which the various
parts are arranged in rows. Jn the first row occurs the square
of the first term of the given qucDittfy ; in the second row the
^>roduct of twice the first term added to the second and the
second; in the third row the product of twice the first term
added to twice the second term added to the third term and the
third ; and so on.
Mf
Examples.
(1.) (a + 6)2=a2
+ (2a + 6)&=&c.
(2.) (&-c)^=62
+ (26-f)(-c')=&c.
INVOLUTION AND EVOLUTION.
^^h, jB— a,
(3.) (af&+c)2=a2
+ (2a + &)6
f^a\
+ (2a + 26 + c)c=:&c.
ised as a 1
(4.) (a-5-.c)2=a2
i
+ (2a-.&X~5)
1
+ (2a-2&-c)(-c)=&c.
1
(5.) (a2~& + c2-(^)2=((^'2)2
1
+ (2«'-&)(-&)
1
+ (2a2-.2& + c2)c2
J
+ (2a2-2& + 2c2-t^)(-c?v.==&C.
63
panding
also be
Various
(3 square
row the
and the
^st term
and the
Exercise XXX.
Express as powers or products of powers
(1.) (x')\ (2xy, (x^)3, (2x3)8, (3x2)*.
(2.) {ax'f, («V)2, (ah;h/)\
(3.) {ahcj, {aV)\ (2a^6V)3.
(4.) {x-ifzy, (a6V)«.
Expand
(5.) (x + l)2, (2^-3)2, {x^^bf, {x^-^a'f.
(6.) (a;2 + 2.c + 3)2, (a;2_3x + 4)^ (2x«-x«+5)2.
90. Pligher powers of polynomials are developed by tlio
Bmoraial and Multinomial Theorems, the explanation of which
may be found in more advanced works.
91. The process by which the roots of quantities are deter
mmed is called Evolution.,
92. The nih root of a quantity is denoted by writino- the
quantity under the sign V" , the line above being
sometimes replaced by brackets enclosing the given quantity.
Thus, a/2«. denotes the square root of 2 / ;
K,.2
J>
5r«2j
^«^ „ cube
^^^^'+3 „ fourth „ xH3;
^(«2-2a + 3) denotes the lifth root of «'^-2r< + 3.
!!
li 1!
64
INVOLUTION AND EVOLUTION.
93. The mth root of the wth root of a quantity is denoted
by writing the quantity under the sign "v ^ .
Thus, V V2a denotes the cube root of V2a ;
V^4^6^ „ fourth „ >^5^;
V Vo;*— ir'+l „ cube
»
94. The mth root of the wth root of a quantity is expressei
as a root of that quantity by the rule
Thus,
3
VVa = v^(i; V V23D = '>i/2«;
95. The reason of this rule will appear from the following case : —
Let s/ i>Ja-s.x, Then on cubing both sides ot this equation we have
Squaring a=a?«.
Extracting the sixth root,
Exercise XXXI.
Retaining the given quantities, denote
(1.) The square roots of 2aj, ax^^ 03^—1, a;^— 3a;+4.
(2.) The cube roots of — a;^ 3a^ a - &, (a^— 3a+4).
(3.) The square roots of the cube roots of 3aa;, a;— 1,
(4.) The fourth roots of the cube roots of 2, 3a;— 1,
2a*-aH3.
Express as roots of the quantities under the double sign
(5.) ^/^!a, y/I^, \/V3a, \/,yi^.
IN VOL VTION AND E VOL UTION.
65
(6.) \/-^x^-l, V^'-^'Jar^-O, V^^x«-6;k* + 7.
96. Since the square of a quantity is equal to the square of
the same quantity with its sign or signs changed, it follows
that there will be two square roots (if there be any), one
being derived from the other by a change of signs.
Thus, since ( + fr)^=(— a)-= + rt'-^, it follows that the square
root of +«- is +a or — tt. These two roots may be repre-
sented by the symbol ±« (read jilas or minus a) ; so that wo
have ^a^=±a; ^x*^±x^; /^dx^=±3x.
Again, since by Art. 78
&c.=&c.,
it follows that
/^a'—Saft + &''=«—&, or — a + 6,
= ±(a-b);
v'CaH&^ + c^— 2t«6 + 2ac— 2&c)=:a— /) + c, or — (i + &— c
= ±(a-& + c).
In the following examples we shall only determine thtict
square root of a polynomial whose leading term is +, the
other being derivable by a mere change of signs.
97. Since \^a;'^"'-=a;'», it follows that
j^ X ^^X" f \X ^^X , j^ X =: X \
where it will be observed the index of the root is one-half the
index of the given power.
98. The square root of a polynomial can generally be found
by the following rule.
(i.) Ari'd'iuje ih*' f/iiX'n ijiiantitu uccordiHg to afictniding or
tiescettdhvj jiowers 0/ f^oiiie letter.
(ii.) The first ternt of the root is the square root of the leading
tor lit of the given (jnuhtit^,froui wJtich <7.s sqiatre is suhtractcdy
leaving the first difference.
(iii,; The frst divisor is fn'ire the first term <f the root added
W
66
INVOLUTION AND EVOLUTION.
to the second term. The second term is the quotient of the
leading term of tlie first difference divided hy the leading term
of the first divisor. The product of the first divisor and the
second term of the root is subtracted from the first difference,
leaving the second difference.
(iv.) The second divisor is twice the sum of the first and
second terms of the root added to the third term. The third,
term is the quotient of the leading term of the second difference
divided hy the leading term of the second divisor. The product
of the second divisor and the third term of the root is then sub-
tracted fro7n the second difference, leaving the third difference.
The process is thus continued until the difference is zero.
Examples.
(1.) Find the square root of 9fl5^— 12x+4:.
9a;2-12x+4(3x-2
9x^
6a;-2)-12x+4
-12a;+4
Here the first term 3a; is the square root of the leading
term of the given quantity, from which its square 9x^ is sub-
tracted, leaving — 12ic+4. The leading term of the first
divisor is 2 x Sx=6x. This is divided into — 12j;, the leading
term of the first difference, giving —2, the second term of
the root, which is also the second term of the first divisor.
The product of 6x—2 and —2 is subtracted from the first
difference, leaving remainder zero. The root is therefore
3a;-2.
(2.) Find the square root of 4«^— 12*H5a?2 + 6x + l.
4^ *__
4x2 -.3^)« 12x3 + 5^ij + 6x + 1
''12>:^-\-9x'
4a;2-(k-l)-4xH 6a; + 1
-4^H6,r + l
INVOLUTION AND EVOLUTION.
67
of the
ig term
md the
ference,
rst and
\e third
•fftrence
product
hen siih'
fence.
s zero.
leading
is sub-
lie first
lleading
term of
livisor.
lie first
lerefore
The first two terms, 2x^ and — 3x, are found as in Ex. 1.
The first two terms of the second divisor = 2 (2/j^ — Sx)
= ^x^—6x, the leading term of which is divided into —4:x^,
the leading term of the second difference, giving— 1, the third
term of the root, which is also the third term of the second
divisor. The product of 405^ — 6aj — 1 and — 1 is then sub-
tracted from the second diff'erence, leaving remainder zero.
The root is therefore 2x^-^3x—l.
The latter terms of the differences need not be expressed
until the corresponding terms in the partial products are to
be subtracted ; thus in the foregoing example 602 + 1 might
have been omitted from the first difference.
(3.) Find the square root of x^^ix^y -\-10x^y^—12xy^ -\-9y\
x*-4.<€hj + lOxV- 12a.y + 92/* {x^-^^xy + 3y^
^Ix^ — 2x2/) — 4x^2/ + lOx'^2/'^
—4x^2/+ ^^y^
2a;2-4^^H^^^2^~6«V-12a;2/H 92/*
6a;y-12x2/^4-92/*
In this example the given quantity is arranged according
to descending powers of x, and the first two terms only of
the first difference are expressed.
99. The reason for the rule given in the preceding Article will
appear from the following method of considering the last example.
The given quantity is thei*e seen to be equal to
a?»
that is, to
^i2x"-^2xy)(i-2xy)
-ti2x^-Axy+'5y')Sy^
which by Art. 89 is equal to {x^ — 2xy-\-oy^y.
Now, since a?' = (a--)-, the first term, x^, of the root is the square root
of X*, the leading term of the given quantity. Also since —4tx^y —
2x\—2xy), it follows that the second term —2xy is the quotient of
F 2
•:H
68
INVOLUTION AND EVOLUTION.
— 4a;'y, the leading term of the first difference, divided by 2j;-, the
leading term of the first divisor. Again, since 6a;-//- = 2x^(3,7^), it
follows that the thii'd term ^xj^ is the quotient of 6,/;-^-, the leading
term of the second difference, divided by 1x^^ the leading term of the
second divisor.
100. When the process for extracting tho square root is
applied to a quantity which is not an exact square, a result
is obtained the square of which added to the last difference is
equal to the given quantity.
Example,
Find three terms of the square root of 1— 2a;.
x^
1— 2a5(l—a;— o"
_1_^
2-!c)-2a;
-2x+a;2
|-2a;-|]-
X'
-ajHccH?-
— 05^ —
X*
: r,ti' I
The
dd
03"— ^
square root is, therefore, l^x—% and remainder
Hence
(l-.-fj-.?--^ = l-%c.
EXERCISIS XXXII.
Find the square roots of
(1.) ^ti*h\ 25x2//6, ^UY^
(3.) a6x2-36.»-H9.
X
(5.) a^H^-f,^.
(2.) 16^H40a; + 25.
(4.) 1 + 6a; + 9x2.
(6.) x2~7x + ?.
INVOLUTION AND EVOLUTION.
69
2.C-, the
3/y2), it
leading
I of the
root is
- result
•ence is
linder
(7.) 4a;2-Aa: + yi^. (8.) 4a;'-'-12x2/ + V.
(9.) a;H4a;3 + 6a;2 + 4x + l. (10.) «* + 2xH3a;2+2.x + l.
(11.) .x*-4a;3?/ + 6.xV-4a:2/H?/.
(12.) 4,x«-4a;^-ll** + 14.«3 + 5ir-^-12.« + 4.
(13.) Extract to three terms the square root of 1 + ic.
101. The method of extracting the square root of a numerical
quantity is founded on the Algebraical process, as will appear by-
comparing the examples giv^n below. We shall first show how the
number of figures in the root is determined by dividing the given
quantity into periods.
Since ^\-\, ^lOO^lO, ^10000=100, ^1000000=: 1000, &c.,
it follows that the square root of a number between 1 and 100, that is,
containing 1 or 2 figures, lies between 1 and 10, and therefore contains
1 figure ; the square root of a number between 100 and 10000, that is
containing 3 or 4 figures, lies between 10 and 100, and therefore con-
tains 2 figures ; so likewise the squai'e root of a number containing
5 or 6 figures contains 3 figures ; and so on. If, therefore, we divide
a number into periods of 2 figures each, beginning at the units, the
number of such periods, whether complete or not, will be the number
of figures in the root.
In Arithmetic if the root is a+64-c4- &C'>
2a is called the p'st trial-dmsorj
2a +26 „ second „
2a + 26+ 2c „ thinl
find so on.
j>
»>
Instead of obtaining the divisors as in the previous examples, we may
form them as below, where it will be observed that the sum of a
divisor and its last term or digit gives the next trial-divisor.
Examples.
(1.) a a2+2a6+62+2ac+26c+c2(a+6+c
a a^
2a+b ) 2a6+62+2ac+26c+c2
6 2a6+62
2a+26+c ) 2flrc+26c+c2
2ac+26c+c2
)--t i
70
INVOLUTION AND EVOLUTION.
300
300
10'69'29 (300 + 20+7
9 00 00
600 + 20
20
) 169 29
124 00
600+40+7
) 45 29
45 29
In the numericiil example, since the given number contains 3 periods,
the root will contain 3 figures. The leading figure of the root,
which is also the number of hundreds, will be 3, since the given
number lies between 90000 = 300^ and 160000 = 400=. The second
figure of the root, which is also the number of tens, is obtained by-
dividing the first remainder 16929 by the first trial-divisor 600. The
third figure of the root, which is the number of units, is obtained
by dividing the second remainder 4529 by the second trial-divisor
640.
Omitting all unnecessary figures, we may arrange and describe the
work as follows: —
3
8
2
647
10'69'29 (327
9
)T69
124
)4529
4529
li
The leading figure of the root is 3, the squai-e of which is the
greatest square number under 10, tiie first period ; the square of 3 is
subtracted from the first period, and to the remainder is annexed
the second period 69 to form the first dividend 169. The first
figure of the root is doubled to give the first trial-divisor 6, the
division of which into 16 indicates the second figure of the root-
The second figure of the root is annexed to the trial-divisor to form
the first divisor 62, which is multiplied by the second figure of
the root, and the product is subtracted from 109. To the remainder
is annexed the third period to form the second dividend 4529.
Under the divisor 62 is written its last <ligit, and the sum forms
the second trial-divisor 64. The third and last figure of the root
is 7, because when annexed to the trial-divisor to form the
INVOLUTION AND EVOLUTION,
n
7TLT'""''' '^' ^''^'''' '^ '^' '-^ 7 is equal
dividend
(2.)
2
2
44
4
481
1
4825
5
48306
to 4529, the last
5'83'51'23'36 (24156
4
T83
176
751
481
27023
24125
28^836
289836
ir
( 72 )
CHAPTER XII.
TEE HIGHEST COMMON MEASURE,
^.|
102. A QUANTITY is Said to be of so many di'mensions, in any
letter, as are indicated by the index of the liighest power of
that Ujtter involved in it.
Tlids 'dx^—2x+4tis of 2 dimensions in x; 3?/ + 2?/— 5isof
4 dimensions in y ; and ax^—hx^ + cis of 3 dimensions in x.
103. A whole exvression, or quantity ^ is one which involves
no fractional forms.
Thus 3x^, —^xy, 2a;*— 3x4-4, are whole expressions, as are
also all positive and negative integers.
104. When two or more whole expressions are multiplied
together, each is said to be a measure of the product, and the
product is said to be a multiple of each factor.
Thus 1, 4, a, and h are measures of 4a6; 1, 3, x and cc+1
are measures of 3cc" + 3ji;; 5, a;-, x—\, and y'^-^l are measures
of 5x2 (^_i) (r-1).
105. It must be carefully observed that the terms measure
and multiple are to be used only in connection witli whole
expressions. In order, therefore, to obtain a multiple of a
quantity it must be multiplied by another whole quantity ; and
to obtain a measure of a quantity it must be divided by a
whole quantity, the quotient also being a whole quantity.
Thus the terms measure and multiple cannot bo us- d in
connection with facc^, a:;^— _ + 3, because they involve tractions;
whilst 1, 3, a, a;, x^, and « — 1 are measures of 3c<x^(a;— 1),
THE HIGHEST COMMON MEASURE.
73
because the quotient of the latter divided by oach of tho
former is a whole quantity.
lOG. The hlyhei't mean/i.re of a quantity is tho quantity
divided by + 1 or —1, that divisor being taken which will
make tho first term of tho quotient positive.
Thus the highest measure of — 4x- is 4:x^, oti>x—7 is 5;c—7,
and of —2x~+ x—S is 2.c^—x-\-3.
107. The Joivest mnUqjIe of a quantity is the quantity mul-
tiplied by +1 or —1, that multiplier being taken which will
make the first term of the product positive.
Thus the lowest multiple of i2x— 3 is 2x— 3, and of
—flc^ + aj— 5 is x'^—x + 6.
108. When one quantity is a measure of two or more others,
it is said to be a common measure of those quantities.
Thus 2x is a common measure of 4x^ and 2.^'-— 4x, and x—1
is a common measure of '2x—2, x^—^.*: + 1, and x^—1.
109. The highest common measure of two or more quantities
is the common measure of highest dimensions and greatest
numerical coefficient or coefficients.
Thus the common measures of li;^ and Qxhj are
1, 2, X, 2x, 2x%
of which the last is called the highest common measure
(h.o.m.) ; the common measures of 4:{x'^—T) and 6(£c— 1)^
are
1, 2, x-1, 2(x-l),
of which the last is the h.c.m.
110. We shall consider the process for finding the h.c.m.
in the three following cases, namely — I. When one of the
quantities is a mononomial. II. When the two quantities are
liolynomials, neither of which has a mononomial measure.
III. When the two quantities are polynomials, one or both
of which have nioiionomiai measures.
111. I. (i.) When the given quantities consist of two or
more mononomials, their h.c.m. is the product of the G.C.M.
74
THE HIGHEST COMMON MEASURE.
of the numerical coefficients and the highest power or powers
common to the several (jiven quantities.
Examples.
(1.) Find the h.cm. of ISa&^x^ and Iha^'W.
Here the g.cm. of 18 and 15 is 3;
the highest power of a common to both is a ;
ft it ^ f> M " »
and there is no power of x common to both ;
.'. H.O.M. is 3aP.
(2.) Find the H.O.M. of 12xyz\ IQxYz^ and 28x*yz\
The G.CM. of 12, 16, and 28 is 4;
the highest power of x common to the three quantities is x"^ ;
It
i»
y
a
it
»
»
z
it
M
y ;
i
.*. H.O.M. is ^a?yz^.
112. (ii.) The h.cm. of a mononomial and a polyn^ni •'
is the H.O.M. of the mononomial and the h.cm. of the several
terms of the polynomial ; and may be found by the following
rule : —
Express the polynomial as a product one factor of which is the
H.CM. of its several terms: the H.CM. of this simple factor and
the given mononomial will he the h.cm. required.
Examples.
(1.) Find the h.cm. of &aW and 8o:'W-VlaW.
Here ^a^h^-VlaW^^a^W (2&-3a), where 4^262 is the h.cm.
of U%^ and \2a%^; and the h.cm. of ^^h' and &aW is 2ai^
the H.CM. required.
(2.) Find the h.cm. of 15x7/V and lQx'y'^z--l^xhfz^
+20a;Vs^
Here lOxhfz^ - Ibxhfz^ + IQxhfz^ = 6xYzX'2x - 3?/ + 4^),
where 5a;2?/V jg the h.cm. of lOxYz^, I5:>y^y^z^, and 20.c:^y^z^ ;
THE HIGHEST COMMON MEASURE.
7$
and the h.c.m. of bxhfz^ and li^xifr} is 6xyh'\ the h.o.m.
required.
Exercise XXXIII.
Find the h.c.m. of
(1.) 12rt&2 and IGa^fts. (2.) 15a^6 and 20ah\
(3.) da.7i/ and SQxyz. (4.) 'Jax\y^ and 15a^.r2;.
(5.) 4:2a^x^y and 35a^ic'^2/'*' (^O «&^cw'y and Sa'^twy'^i^.
(7.) 8a*&, 12a»6^ and IGa^i^.
(8.) 30a«<2/^ 426a;32/3, and IQcxY-
(9.) 4a&2 and Ua^hx-^SahY
(10.) 10tt&2cand30a»6H45a2i*.
(11.) 10a&2a;y and 4:2a¥cx~70h*ci/.
(12.) Swv^t^; and 12uhv^-24:u^viv^+S6uHo*.
113. II. When two polynomials, neither of which contains
a mononomial measure (other than unity), involve powers of
a single letter, their h.c.m. can be obtained by the following
rule : —
(i.) Having arranged the given quantities according to descend^
ing powers^ choose that one which is not of lower dimensions than
the other as divisor.
(ii.) Divide this into the other multiplied hy the least number
ivhich will make its leading term a mtdtiple of the leading term
of the divisor. When this numher is unity, actual multiplication
may he dispensed with.
(iii.) Divide the first difference hy the highest mononomial
measure contained in it. When this measure is +1, actual
division may he dispensed with.
(iv.) Bepeat the steps (i.), (ii.), (iii.), ivith respect to this last
quotient (or diff'erence) and the first divisor; and so on, until there
is no diff'erence.
The last divisor ivill he the H.C.M. required.
i
76
THE HIGHEST COMMON MEASURE.
,.*
It will be observed that no fractions occur in the process,
and that the leading signs of all divisors are made positive.
Examples.
(1.) Find the h.o.m. of 2x2-7a; + 5 and Sx^-Tac+i.
2
2x2-7x+5)6x'^-14a;+ 8(3
6a;2-21x + 15
7)7^£T
X- 1
Here since the dimensions of the two given quantities are
equal, either one may be made the divisor. 205^—70; + 5 being
taken as divisor, Sx^— 7^ + 4 is multiplied by 2 in order to
make the leading term Gx^ a multiple of the leading term 1x^
of the divisor. The first difference 7i»--7 is divided by its
highest mononomial measure 7.
In the next step x—1 and 2x^—lx-\-b are to be treated in
the same manner as the given quantities.
a;-l)2a;2_7a; + 5(2a;
2x2-2x
—5) -5a? + 5
03 — 1
The leading term of ^x^—lx-\r^ is a multiple of the leading
term of u:— 1, and therefore the multiplication of the former
by 1 is omitted. The difference — 5» + 5 is divided by its
highest mononomial measure —5.
In the next step the quotient x—1 and divisor x—\ are to
be treated as the previous quotient and divisor were.
a;~l)a;-l(l
OJ-l
Th(
X —
oft
the
THE HIGHEST COMMON MEASURE.
77
process,
sitive.
;ities are
- 5 being
order to
^erm 2a5^
d by its
eated in
I leading
former
by its
1 are to
The process thus terminates and the h.c.m. is the last divisor
a;— 1.
Whenever as in the second step the difference is a multiple
of the divisor, the division may be continued and the work of
the last step avoided. Thus
a;-l)2a;2-7a; + 5(2a;-5
2x2 -2x-
— 5x+5
-5x + 5
le whole work
may
be arranged j
JB follows :
3x2
-7x+ 4
2
2ic2.
-7aj+5)
Gx'^-
6x2.
-14x+ 8(3
-21x + 15
7)7x- 7
:=a;~l«
X- 1)2x2
2x2
-7x+5(2x-
-2x
-5
H.O.M.
-5x+5
— 5x + 5
(2.) Find the h.c.m. of x2+2x— 3 and xH5x + 6.
a;2+2x-3)x2 + 5x + 6(l
X- + 2X-8
~3)3x + 9
^'+3)x2 + 2x~3(x-l
X" + 3x
H.C.M.=:X + 3. — X — 3
Here the multiplication of x2 + 5.r + 6 by unity is unneces-
isary. The other steps are similar to Ex. 1.
(3.) Find the h.c.m. or2x''-7x-2 and 2':^-x-G.
78
"//£ HIGHEST COMMON MEASURE,
2a;2-a;-6)2x3--7a'-2(a;, 1
2i«;^— x"— 6.
X
X
^--x-^
2x2-2x-4:
2x2-
X-
■6
-l)-a: + 2
X
--2)2x2-cc-.6(2x+3
2x^—4:0;
fi.o.M.=:a;— 2.
3x--6
Sic— 6
Here 2x'^— cc— 6 is used as divisor in the second step, the
dimensions of the first difference being 2. The partial
quotients x, 1 of 2x^— 7x— 2 and 2x^—'2x—4: divided by
2x2— cc— 6 ^j,Q separated by a comma to distinguish them
from parts of an ordinary quotisnt.
(4.) Find the h.o.m. of
4x2 + 3x-10 and dx^+Tx^-Sx-lS.
^^■{■3x-^10)ix'-\-7x'- 3x-15(x + l
4a;H3x-2-10x
4a;2 + 7a;— 15
4lx^-\-3x-10
Sic^)4a;H3x-10(a; + 2
H.C.M,=:4x — 5.
8.-10
8r-10
In this example there is no necessity to introduce or
suppress any mononomial factors.
114. The process of the foregoing examples will frequently
enable us to find the h.c.m. of polynomials involving powers
of several letters, as in the following
bep, the
partial
cled by
li them
2
ice or
lently
kowers
THE HIGHEST COMMON MEASURE.
79
Exam.'ple.
Find the h.c.m. of %^-^xy~6if and 3x2—40:^+2/'^.
2
_6x2 + 3.i7/-%2
-lly)-ll.rv/ + 112/2
H.C.M. =£13 — ?/•
cc — 7/) 2«;2 -^xij— 2>if- (2x + By
'dxy-^'dif
Here the mononomial factor — lly is suppressed.
115. The reason for the rule in Art. 113 will appear from the follow-
ing proposition and its application in the next Art.
V/licn one quantdij is a mea'^ure of two others, it will measure the sum
and difference of any multiples of them.
Let the quantities be A, L\ C : and let A measure B and C, so that
JB = inA, Cz:=7iA, where m and 7i are wliole quantities.
Take any multiples pB, qC of B,C, where p and q are any whole
quantities whatsoever. Then, since ^9^ =/wi^4, qC=qnA,
pBdzqC = p}nA±qnA = (pm±iqn)A.
. pB±.qC ,
A
that is, A is a measure of j)BdzqC, the sum or difference of any
multiples of B and C\ because the ([uotiont pia-±.qn is a whole
quantity.
Thus, *Ja'-, which is a measure of (j.r-' ;ind 8,r-y, will measure
(U"'( - -la) - ^jrii{ - ox), (Jx^-f y.r-//, ^ix^ - 8x'-//(4-.i7/), &ic.
11(3. Sui)pose, now, that A and J! denote two ]iolynomials (as in
Art. ll'i), neither of which contains a mononomial measure other than
unity ; and let the dimensions ot' A be not greater than the dimensions
of B. Divide A into // n)ultiplied by a mouonomiid whole quantity a,
which makes its lirst term a multiple of the lirst term of ^4 ; and
8o
THE HIGHEST COMMON MEASURE.
divide the difference G by the highest mononomial meatiure which it
contains, and lot the quotient be D.
B
A)aBib
64
c)C
D
Now, C being equal to aB—bA, or the diffbrence of two multiples
of A and B, is a multiple of all the common measures of A and B, and
therefore of their H.c.M.
Again, every common measure of C and A is a measure of C+bA, or
ctB, and therefore of B, because A has no mononomial measure.
Hence the H.C.M. of A and B is the same as the H.c.M* of ^1 and C,
which is the same as the n.C.M. of A and D, because A has no mono-
nomial measure.
The problem is thus reduced to finding the H.c.M. of A and 1),
These two quantities, A and I), are then treated in precisely the
same manner as A and B ; and the process is continued until it
tciminates as /> " ws, when the last divisor, F (suppose), is a measure
of the last divide. 1 Q.
P)Q(r
rP
The problem is thus finally reduced to finding the n.c.M. of P and
Q. This is evidently P.
Hence the last divisor in the above procef^ will be the H.C.M.
required.
PJXERCISE XXXIV.
luiid the H.c.M. of
(1.) 3.r2 + 2.t;-'il ?vA .^r>;2-fl3.r-6.
(t2.) 2.tH;r-auivi8:<;-.-l^i 1.
(3.) a;2-5.r + 6 and .'.'^-f;:^:-! 9.
(4.) i>jHi0a; + 21 and x2-2.r-15.
THE HIGimST CPMMOJV MEASURE.
8i
v'hich it
lUltiples
i i)\ and
'+6^, or
«
A and C,
lO mono-
sely the
until it
measure
f P and
H.C.M.
(5.) 2xHa;--15and2a;^-19a; + 35.
(6.) x2-4a; + 3 and 4«3-9a;2- 15^ + 18.
(7.) ct" + 10.rj + 25 and x^ + 15^;2 + 75 ,. ^ i25.
(8.) ;/;3-6a-- + ll*-6 and a;3-it;^-Ux- + 24.
(9.) ;t3-3x2-9x + 27 and 3,c3-a;a-27.« + 9.
(10.) 3.>;2-22x + 32 and x^ -\\x" ^■Z'lx-'l^,
(11.) 7a)2 - 12x + 5 and Ix? + .*- - Sa? + 5.
(12.) 5.xH 2x^-15*;-. 6 and 7x^— 4cb2_21xH-12.
(13.) 2x3 + 9^-2 ^ 4,^, __ 15 j^nd 4a;3 + 8^2 + 3x + 20.
(14.) a;3-6x2 + lla5-6 and a;4-2x3-13a;2 + 14a; + 24.
(15.) a;*-2x2 + l and a;*-4x3 + 6x2-4x + l.
(16.) x^-^xy— Ykf and x^ — bxy + 6?/-.
(17.) 2x2+3x2/ + ?/2and3x2+2x2/-2/'^.
(18.) £c^ + a;22/ + ^2/ + 2/'^ ^Ji<i *''■~■2/^•
(19.) 5*2 + 26jry + 33?/2 and 7x^ + 19i»2/ - Qtf.
(20.) 3a;*-a;V-2i/^ and2x-* + 3x3?/-2xV-3a;2/».
117. III. The H.o.M. of two polynomials involving mono-
nomial measures is found as follows :
Express each polynomial as the product of a moiioiioimal and
a polynomial luhich contains no mononomial measure. Then
the H.C.M. required is the product of the H.CM. of the rnono-
Qiomial factors and the H.C.M. of the polynomial factors.
Example.
Fin( '. the h.c.m. of
'6x'^y + l%9y + ^xhj and Qx^y^—Qx^y^ — l^xy"^.
Here the given quantities arc equal to
4^2^(2x2 + 3,/j + l) and ijxif(^-x^'^.
The H.C.M. of 4^7/ and (Sxy^ is %ry ; and the h.c.m. of
2*^ + 3x- + 1 and x- - x - 2 is u; + 1.
.*. the H.C.M. re(iuired is %icy{x + 1).
118. The H.C.M. of three polynomials is the H.o.jr. of any
one and of the H.c.31. of the other two.
o
82
THE h 'CHEST COMMON MEASURE.
Example.
Find the h.c.m. of x^-l, ccH 2^2-3, and 2x^ + 2x3 + 3a; + 3.
The H.C.M. of x^—1 and a;^ + 2x^—3 is cc^— 1; and the h.c.m.
of x^— 1 and 2x'* + 2;*3 + 3x + 3 is as + l, the h.c.m. required.
Exercise XXXV.
Find the h.c.m. of
(1.) 12aa;^-27aa;2and2a2ic3+aV-3a2a;.
(2.) 10x2+40^ + 30 and 4a;3-16a;2-84a;.
(3.) 2x«-6x3-4x2and3x*-3x3-12x.
(4) 2x2+x-3, x2-l, and x'-^^.x-h.
{L) 6x2-a;-2, 21x2-17x + 2, and 15x2+5x-10.
119. When all the component factors of the given quan-
tities are known or can be determined, the h.cm. may be
found by the rule of Art. 111.
Examples.
(1.) Find the h.c.m. of
4(x-l)2(a; + 2)3 and 6(x-l)3(a;+2).
The g.o.m. of 4 and 6 is 2 ;
the highest power of x— 1 common to both is (x— 1)*
)}
X
+ 2
})
03 + 2.
the h.c.m. required is 2(x--l)2(ic + 2).
(2.) Find the h.c.m. of
By Art. 80, Sn^x(x'--l) = H(fx(x-tXuHx-hi);
Pj Art. 79, 12r/,TV-l) = 12ax'-^(.t;2-l)(a;^ + l)
-I2r/x2(x-l)(x + l)(x2 + l);
^;^fJ^fffffSrC0MM0J^ MEASURE.
Now the O.O.M. of 8, 12, and 20 is 4
the highest power of
and the other factors a. + l, ^2^'i^ ^2
IS a;
a common to the three quantities
X
x—\
a:— 1;
. +, + »'• + 1 arc not common
. . tlie H.C.M. required is 4ax(x--l).
Exercise XXXVI.
Find the h.o.m. of
(2.) 6a2(^+2Xa.-3)and8a^(«:-3)0
(3.) ax2_2aa; + a and 2a^x'^^2a'^
X
+3).
(4.) a;2-l,a;3+l
and a?*— 1.
C5.)a;+2,a;2_4,andx3+8
(60 3^3_8i,«,2^6^^9^^^^2^3__
18a7.
I
1 11
g2
li I'
( 84 )
i.
1-
\
CHAPTER XIII.
THE LOWEST COMMON MULTIPLE.
120. When one quantity is a mnltiplc of two or more
others, it is said to be a common multiple of those quantities.
Thus l%t^ is a common multiple of 2.c and 3*^; and
Xhx(x—V) of 3, 5, 15.^- and a; — 1.
121. The lowest common midtiple (l.c.m.) of two or more
quantities is the common multiple of lowest dimensions and
least numerical coefficient or coefficients.
Thus of the follo\*ing common multiples of 1x and Sx^,
namely,
&x\ 12^2, 18x2,
6x^ 12xS 18xS &c.,
the first 6x2 ^g q^\\q^ the lowest.
122. The L.C.M. of two quantities is found by the following
rule : —
(i.) If they contain no common measure except unity, their
L.C.M. is their product.
Thus, tho L.C.M. of 4x and Tab is 28a/;x.
(ii.) If they contain a common measure, their L.c.m. is c^jual
to one of the given qtiUiitifies multijilied by tjie quotient (f the
other dlnlded liy their H.c.M,
It will be generally found most convenient to expresK the
L.C.M. as the product of several fuciors.
THE LOWEST COMMON MULTIPLE.
85
(1.) Find the l.c.m. of ^.rhj and Sixif,
The H.o.M. of these quantities is '6xy,
.-. by the rule, l.c.m. J^f^- x Sixy'^X^xhf.
'6xy
(2.) Find the l.c.m. of 2x2-7a: + 5 and Zx^'-lx^^,
The H.c.M. is found to be x—\\ and since
= 2x-5, the L.C.M. will be (2a;— 6) (^x'-lx-Y^).
2a;2-7a; + 5
X'
123. The following is the proof of the rule given in the preceding
Article : —
Let the two quantities be denoted by A and B^ and their ii.c.M. by
C; and let A — aC^ B — bC, where a and h are whole expressions which
have no common measure except unity.
Then, since the L.C.M. of a and b is ab, the L.C.M. of aC( = Ji) and
bC( = B) IS abC= —— =-^= -^.B=-^.A.
124. The L.C.M. of three quantities is the L.C.M. of any one
and the l.c.m. of the remaining two.
The L.C.M. (if four quantities is the L.C.M. of any one and the
L.C.M. (f the remaining three.
And so on.
J/Jxample.
Find the l.c.m. olUx^ifz, Gxyz^, and lOx^yz^.
The L.C.M. of 3x'i/'z and 6.///^!- js Gx'VV; and the l.cm. of
QxYz^ and lOx'vjz^ is SOx'-y'z^
125. When alx the component factors of the given quan-
tities are known or can be found, their l.c.m. may also be
obtained hy multiplyimj the l.c.m. of the numerical factors by
the Jiiyhest j)oiuer or 'powers of the several factors that occur in
the fjiven quantities.
86
THE LOWEST COMMON MULTIPLE.
I-
I f'
!
.^
d
Examples.
(1.) Find the l.c.m. of 6x^^22^ i.x?u\ and 8xYz.
The L.C.M. of 6, 4, and 8 is 24 ; the highest power of x
which occurs amongst the factors of the given quantities is
£c* ; and the highest powers of y and z are, respectively, y^
and z\
Therefore the l.c.m. required is 2ix* ijh^.
(2.) Find the l.c.m. of 15«&(a— ft), 21a(a + bXa—h), and
The l.c.m. of 15, 21, and 35 is 105; the highest powers of
a,h,a—h, a + b, wliich occur amongst the given quantities,
are, respectively, a, U^, a—b,a-\-b.
Therefore the l.c.m. required is 105ahXa—b){a-\-b).
(3.) Find the l.c.m. of x^-^l, a;^— 1, and cc^+l.
Here x^-l = (x + l)(x^l);
a;3-l = (a;-l)(,;:2 + aj + l);
x'^+l = (x-\-lXx'-x-\-l).
/. the L.O.M. = (cc + l)(x - l)(a:;2 + a; + l)(a;2-a; + 1)
= lx'-lXx'+x^ + l).
Exercise XXXVU.
Find the l.c.m. of
(1.) Sahx, 2bxy. (2.) Sa^xy, Uax% (3.) a¥, bc\ ca\
(4.) Sa%c, 12ah% 24Mhc\
(5.) Uhcu^, lQcav\ 20ab7v'-, 40a2'>V.
(6.) x^-7x-\-12,x^-^x-G. (7.) 2x'-5cr-3,4a;2+4« + l.
(8.) 3a;^-ll« + 6,2a;2-7a; + 3.
(9.) cc^— 4aa;^ + 5a2£c— 2a^, o;'-?-^^*— 4a'.
(10.) 8(a;2-l), 12(x-l)2.
.0^
TJIEJ^ESrcOMj^O^ MULTIPLE.
(13.) a^-{.a%, ab"b\ a^-~b'i
(14.) 2x(:c^ + a; + i)^3^,3_3^^^a_^
«7
(15
/^ +r,i>'-?^i>H«3^
(16.)
r -I,i>*-1,7>«-.1.
(17.) (a-&) (a-,)^ (j.,.>> ^j_^^^ ^^^^^
(18.) 8a«6(a«6), 12a6(6-a), 3(a^-J2), W(j, J^,^^
IMAGE EVALUATION
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( 88 )
CHAPTER XIV.
FBACTI0N8.
hi
126. When one quantity is not exactly divisible by another,
the quotient is represented by writing them in the form of a
fraction.
Thus, — ^ denotes the quotient of —2 divided by +3;
n?- the quotient of -2a divided by -^Sx; .'^"■^ ^ the
quotient of a;— 1 divided by a;'*— 3a; +4.
127. A fraction is not altered in value by multiplying
or dividing the numerator and denominator by the same
quantity.
-2_-2x3_--6 ±6 _ +^6 X -4 _ -24
Thus, ^3 - ^3^3 - ^9; _8 " -8x -4 "" +82^
-20 _ -20-j — 5 _ +4, -4 _ -4xf __ -^f .
-25- -25-r— 5~ +5' +5
+ 5x|
-^'■r
a ac —ax ^
h ~~ he ~ ~-hx '
x-l __ (x-l)(.r + l)
2:^-3 ~ (2x-3)(x + l)
a;2-l
2x2-a;-3'
128. The statement in the preceding Article depends on the two
following propositions : —
I. 77ie numerical value of a fraction is unaltered by multiplying or
dividing its numerator and denominator by Vie sam^ quantity,
(i.) Let a, 6, m, be integers. Then, since ^ denotes a of the 6 parts
into which a unit is divided, it follows that
FRACTIONS.
89
b ' mb
a
b
ma
. (1)
. (2)
. (3)
Before considering the case where a, 6, m are fractional, we shall
show how the operations of multiplication and division of numerical
fractions must be performed.
mc
Let Of, 6, c, c?, m be integers. Then, since — =m, it follows that
But by (3),
a mo a , ma
-.— =-xm=--
h c b
ma _mao
1 'be
, a mc_mac
"^ c bo
. (4)
Hence, if - be multiplied by an integer in a fractional form, the
b
product L a fraction whose numerator is the product of the numerators
and denominator the product of the denominators. If, now, we wish
ft o o
to find the product of - and -, where - is not equal to an integer, the
*^ b d a
operation of multiplication must be in accordance with this rule ; for
any application of the term multiplication to cases where its primaiy
meaning (which is repetition) does not apply, must not be inconsistent
with the cases where it does apply.
, a c _ac
"VlTbd
(5)
Again, since diyision is the inverse of multiplication, both in its
primary and extended applications, it follows from (5) that
ac
a
bd d b
__acd
'~bcd
__ac d
~bd' c
by (3)
by (5)
^i;
i \
I '9'
M'l
90
FRACTIONS.
Also
ac ,a
hdrV
_c
'd
_abc
abd
bd a
by (3)
by (5)
* /I
Hence it follows that the quotient of the fraction - divided by
b
-. is equal to the product of - and — : that is,
d b
a.c a d_ad
6
d
_a _
b c he
. (6)
(ii.) Now, let a, 6, m be fractional, which will include the case
where some of them may be integers.
Let
Then
y u d
X
z yz
z
u
And
mh'
c
c
X ex
cz
by (5)
d u du
__cxdu
czdy
by (6)
=^ by (3)
yz
••T
.Tna
mb'
II. If, in the fraction -, a and 6 denote positive and negative
quantities, the sign of the fraction depends on the signs of the nu-
merator and denominator. These signs will be either like or unlike,
and on multiplying or dividing by a positive or negative quantity they
will still be like or unlike, and the sign of the fraction will therefore
remain unchanged.
FRACTIONS.
91
ivided by
the case
negative
the uu-
r unlike,
tity they
therefore
For example, multiplying by — 1,
I 2 — 2 2
— „ = — - = +- , a positive fraction ;
+ — o o
_i_3 _3 q
— - = — _ = - a negative fraction.
— 4 +4 4
129. A fraction which involves fractional coefficients in the
terms of the numerator and denominator can always be
reduced to one whose numerator and denominator are whole
expressions by multiplying the numerator and denominator
by the l.o.m. of the several denominators.
Thus, n^ii_3j ■j ^ is reduced by multiplying nxmierator
and denominator by 12, the l.c.m. of 2, 4, and 6, to the
. . _ 12x-6
equivalent from 24^2^:y^^2-
130. A fraction is said to be in lowest terms when its
numerator and denominator contain no common measure,
except unity. Hence a fraction is reduced to lowest terms hy
dividing its num>erator and denominator hy their H.C.M.
Examples.
(1.) Eeduce " lOa^S'-^y/ *^ lo^^st terms.
The H.O.M. of SaWx and Ua^b^y is 4a^h\
Sa^b^x _%x
•'• UaWy "" 3ay*
■x^—1
(2.) Reduce -3-r-i to lowest terms.
The H.O.M. = 03 + 1.
x^—\ x—\
^2^~^Tl*
ceHl ~
as'
2^2 I g^ 2
(3.) Eeduce 03,2^53; 7 9 to lowest terms.
The H.O.M. = 2x— 1.
2^3^-2 _ a^-f^
•*• 2a;2-5;r + 2 "■ x-2'
!
i, '
92
FRACTIONS.
m
!-M
I,
I
I.
'I
Exercise XXXVIII.
Reduce to lowest terms —
(5.)
(8.)
(11.)
(13.)
6a2Z>2
(3.)
Qx^yT?
Sxy'^^
O"
a»-l
^-7«J-10
3 + l%+3a;2
S-\-Sx-3x^'
(6.)
OJ + l
C7) -"^
(9.)
x^—y^
a^—y^'
(10.)
jc"— .4
a;-3
(12.)
a;2-4x + 3'
a;H7a; + 12
a;2-ic-20'
n4N g;'^-3ag-70
'Ti^ cc^— 6a;— 9
^ '^ a;* + 3x3-9a;-9'
(16.)
12x^-15x+3
6*3_6x2+2x-2*
(17.)
a;^— 4a3V
x3-6a;'^^ + 12ay-82/3
nS) «'^— 3a;'^y + 3a;;/''— y ^
x^—x^y-'xy^-^y^
131. Fractions ate said to be like or unlike, according as thej
have the same or diflferent denominators.
2a
Thus -, - are hke fractions, as are also
SiC
X X
a;2--r a;=^-l' '3o
2a
jr- are unlike fractions.
Sx
132. Unlike fractions are reduced to like fractions by mul-
tiplying the numerator a:>d denominator of eadi fraction by the
quotient of the L.O.M. of the several denominators divided by
lis deno7ninator.
The common denominator will accordingly be the l.c.m.
of the several denominators.
Examples.
(1.) Reduce p^, -^ to like fractions.
00 4»
The L.C.M. of 3& and U k i%d.
1
6ah
4x+3*
3
-2*
ig as thej
SiC
2a^
^-1' 3»/
s 6y mul-
lon by the
ivided by
;he L.c.M.
FRACTIONS.
93
The multiplier for the first fraction is -w. -^l\
•* Wh~md'
The multiplier for thti second fraction is -j-^- = 36 ;
. 3c__ %c
" 4a5 126c^'
(2.) Eeduce
a—b ' 6— a
to like fractions.
The L.o.M. of a—b and b—a is a--6, the quotient cf which
divided by 6— a is — 1 ;
6— a a— 6*
(3.) Eeduce
1 x-l xj-2^
x-r x^+x+v x^-i
to like fractions.
The L.c.M. of the denominators is x^—l, the quotients of
which divided by x-1, jrHas + l, jk»-1 are, respectively,
tr/^ + cc + l, x—1, 1.
a;-l __ a;'^-2a; + l
aHaJ + l a^'-l '
as— 2»'^_£c— 2ar
£C^— 1 os^— 1'
EXEECISB XXXIX.
Eeduce to like fractions : —
1 2
(1.)
a
^^•^ i^ ' ~ab'
(3.) '^ , -5^ ,
cc xy xyz
(5.) i , J- . A.
a —a ax- — xy ««;«/
(4.)". ^
a
4
la
(6.) - - , -^-y .
(7.) -
x-\-l u;+3
W --r,
3— a;
x
-r i-x-
(9.)
a
•a
a—b^ b—a
m
94
FRACTIONS.
1.1
(10.)
(12.)
(13.)
(14.)
4 3^ 1_
2
x^\
3aJ
(^^•) a^^-x + l'a^ + l'x^+l*
3 1
(a: + 2) (x-lj ' (1-u^) (2-x) ' (a— 2) (x + 2)
1 _ __ 1. _ 1
(6_a) (c-ii) ' («-/>) (c-&) ' (a-c) (6-c)'
1^ 1 1
a((0—b) (.c— a) ' h(h—aj (x—h) ' a&x'*
i ^t
Addition and Subtraction.
133. The operations of addition and subtraction of frac
tions may be denoted by the signs + and — respectively.
(c-2
a
-2&
;2-3
-2b
Thus, the sum of , -^ , ajid may be denoted by
_a , — 'J<> , a?— 2.
x-l x^-3 dx-^'
and ^Ac diference between -_,^ , a?^c? v^— * by
Sx"^—! oa;--4
4a5
a;-l
3a;''^— i 5a;— 4*
Thus also +? denotes the fraction ? ^o be added to, and
— - the fraction - to be subtracted from some quantity not
expressed.
184. Sums and differences of fractions when expressed as
single fractions are said to be simplified, the operation being
IDerformed according to the three following rules : —
(i.) The sum of any number of UJce fractions is a like fraction
ivhose numerator is the sum of their numerators.
Thus — i- + ~^ + ^'^— ^ = 4--a; + 2a;— 3_a? + 1
i»— 1 x—1 OS— 1 a;— 1 cc— 1'
(ii.) The difference between two like fractions is a like fraction
whose numerator is the difference between their numerators.
FRACTIONS,
3a;
af frac-
vely.
loted by
to, and
ity not
ssed as
11 being
''raction
raction
95
(1.)
(2.)
Examples,
5a;~4 _ 6 _ 5a;-4- 6_ 5a;~10
2x-3 2a;-3 2a;-3' 2aj-3*
4ag --2|K_4a: + 2x_ 6aj
03'
It will be observed that the — before the second fraction
changes the sign of — 2ic.
^ '^ " X+l X^ + 1 X^ + 1
_Sx^-x-]-6-x^-{-Qx + S
_.2x'^ + 5x + 13
a^ + 1 "
In this case the — before the second fraction changes the
signs of all the terms of a;^— 6a3— 8.
It appears from the preceding rules that
_ a—b _ — a + 6
c c
and conversely ; in other words, the suhtraction of a fraction
is equivalent to the addition (fa like fraction, whose numerator
is the numerator of the former icith its sif/n or signs changed.
Thus,
a;2-2a: + 3 , -a-2 + 2x-3
+
x^-l
-2^5
X^—1
_ 5
3a;2-7*
2^-b
(iii.) Addition and subtraction of unlike fractions are per-
formed by reducing the unlike to like fractions and proceeding
as above (i.), (ii.).
Examples.
(1.) Simplify .- + - +
be ca ab
>h
III
||.'
)|i*'"
|t|H
i;.
■#'
96
FRACTIONS.
einco the l.o.m. of he, ca, ah is ahc, this sum is equal to
a
abc ubc ^ abc
"^ nh,.—
a^\■h■\^c
ubc
2r*
+ i-^. - l_a2
(2.) Simplify 1^^ . i_^^
Tlie L.c.M. of tho denominators is l--a\
1
2«
•'• 1 + a "^ 1-a "l-a^ - l-a2 + i_^2 -JZ:^
1— « 1+a 2a
1— a + l + a— 2a
- l-a"
_2--2a
~ l-a2
2
~ 1+a
(3.) Simplify ^^—y - ^^::;^ - (^3^-
The L.O.M. of the denominators is (x-\-yy (x—yY
. 1 1 1_
" (x+yf y'-x^ (x-yy
__ (j^—yf y'^-^x^ (?i-\-yf
- (a;2-2/2)2 - (^2_2/2)2 - (052-2^2)2
_ x'^—^xy^y'^—y^-\-x^ ~'X^'-^lxy'^i^
x^—Axy—y'^^
~~ (x^—y^y
Simplify —
EXEBCISE XL.
(^•) 2a"'"(/
X
/ox- 1 2
FR^CT/ONS,
97
1 2
2 8a 1
Wx~3^' (S-^^+^^-i^'* <6-)
a;— 1 ^—8
2a5 ~ a" '
2a-3fe 3a--26 a-36
<.y-; a~6 a + 6
a+6 a-&
(8-) ^3^6+^6-
(10.) i:i^.+ii:^-
(^^•-^ a'*"a + 6 + a*+a6 ' ^^^'J a"2x-l""4x2-l *
^^^•''a;2 + a; + l+a;2-a;+I ^^*'^ 2"(»-l)"2(a; + l)"a;«'
(■"•^0 ^c^iy'-^^'iy ■*■ x^^^p ■
.... 2 1_ a;+3
. 2y' .
^ -^ a:;=' + a;?/+2/^^a;'*— as*/ + «/'*'»* +33 V+!/
/iQ\__^""l JL_ g ^ + 1 1
2 2 1
(^^•) (¥:^"XiT2)~5^)(a;-2)''"(x-2)(a;+2)'
ah ac
(^0-) (a-6)(6-c),'''(a-cXc-6)'
V^J-O (a;-6XaJ-c)'*"(a;-cXiC-a)+(a;-a)(x-6)
(ic— a) (as— ft) (as— c) *
OS— a
cc— 6
as- c
(22.) (fc-aXc-«)"*"(a~&Xc-«»)'''(«^0(^-^c)'
(^^•) ("^:^(^:::^+(6-«x^-c)+((;-6Xc-a) '
135. A mixed quantity is the sum of a whole expression and
a fraction; as, for example,
6 „ 1 3a; _ as— 1
FRACTIONS.
136. A mixed quantity may he expressed as a fraction by
considering a whole expression as a fraction whose denominator
is unity ; and conversely, a fraction may be expressed as a
mixed quantity when part of its numerator is a multiple of its
denominator.
liii
ii i
[11
Examples,
,- . _ 1 2a; 1 2aa; + l
(1.) 2x+-=-j- + -=— - .
, 1 a;— 1 1 a^
(3.) p-+i-»"+i=,;=-TT — r=x^+i-
a;'— 1
It will be observed here that —a^ + 1 = —(a;''— 1)= — -| —
(4.) Express ^.^ as a mixed quantity.
On dividing 2a5*— 3a;+4: by ac+1 we get quotient 2a;— 6 and
remainder 9.
x-\rl x + 1
(5.) Express ^±^^^^ as a mixed quantity.
In this case the quotient is a? +2, and remainder — aj+1.
. a;8 + a;2_2a;+3_^ . o. -fc + 3^
*, 5 x-r^-t-A — i .
a;''— a; + l US'*— a; + l
= a; + 2-
a;-l
a;'^— a; + l '
Here the + before the fraction is changed to — by changinp:
at the same time the signs of all the terms in the numerator
-a;+l.
iion by
minator
ed a8 a
)h of its
—5 and
FRACTIONS.
99
Simplify —
(1.) 1 + 5.
(4) ^+a-l
X
EXEBCISE XII.
X
(2.) ^,-1.
(5.) ??-o+l.
X
(3.) 2-?.
a
(6.) 2+?-i.
(7)8-
03*
05 — 1
(12.) 4a; -1-
(10.)-^, -a.
(11.) 2-
a + 6'
2a; + l
a; + 3
Express as mixed quantities —
(13.) xHiry + t/'^-^"^'.
(14.)
(17.)
(19.)
(22.)
a
2a;2-aj + 3
X
a;' + 3a; + 4
a;«-3 a;H2
(15.) ^-i.
a;
(18.)
(20.) ^^-6
(16.)-^
4-3:^ + 6a:''
2-3a;
3x '
'Sx
(21 >i ^ + ''^''
(23.)
6a:'— 4a; + 5
2a;=^-a; + r'
(24.)
jc*— a; + 5
^\
B + 1.
langmg
nerator
Multiplication.
137. To denote that two or more fractions, or a fraction and
a mononomial whole quantity, are to be multiplied together,
they are written in a row with the multiplication sign x , or
• (dot), between them.
Thus ? X -,, or V * 4 denotes that ~ is to be multiplied by
d a b
c, x-l 20^-3 jej^oteg the product of '^"- and ?^'"?.
d' a'
x^-l
X--1
^^^ X — 3aaj denotes the product of ^. - and — 3«a:.
b
h2
100
FRACTIONS,
1^
K j
Hi ^
lit *
■S v-
Sums and differences when they are factors must bo
enclosed in brackets.
Thus, _„A_/a'^— aaj+cc^ denotes the product of -3^ and
a^— ic^
a^— as'
a^— ax+a;2; _^(a;+^) denotes the product of -^ — and
cc+aX XI flj + a
0; + ^; -^(^+?-i\ denotes the product of 4^:? and
X 05-^— 1\2 X a2V cc*— 1
138. Multiplication of fractions is performed according to
the following rule : —
The product cf any number of fractions is a fraction whose
numerator is the product of their numerators and denominator
the product of their denominators.
A whole quantity is to be considered as a fraction whose
denominator is u iiity ; and sums and differences of fractions
and mixed quantities must first be reduced to fractions.
(1)^
x^—a
Examples.
ax _(x^—a~)ax_x—a
2ax x + a 2ax(x-^a) 2
Factors common to any numerator and any denominator
may be struck out. Thus ax which is common to the first
denominator and second numerator, and x+a which is com-
mon to the first numerator and second denominator, may
be struck out before multiplying, and the result will be in
lowest terms. When possible, therefore, the component
factors of the several numerators and denominators should
be obtained.
(^\ 4a^— 4aa;^ hc-\-hx_4ia(a+x)(a-^x') ^ &(c + cc)
^ "^ Sbc^-Sbx^ ' o^^ac" 35(c + a;) (c-a;) * a(a-x)
__A(a+x)
3(c-a;)'
FRACTIONS.,
lOI
Here a,a^Xy b, and c+a; are struck out, being common to
the numerators and denominators.
(3.)
X
(a; + 6)(a; + 7)
(a;2-49) =
X
(x + 6)(a; + 7)
x(x-7)
(x + 7)(x^7)
\ x^—i)x^ + l x^—1 x^ + 1
xHl
x-l
(x-tl)(x-l) JcHl
1
x+r
ExEBCiSE XLU.
Simplify —
yo\ b^c <?a o^}>
KP'J ~~~-\ • — o • — 5«
2/2S 2i» ^32/
^2
a
c?-W
(5.) . — ^.
.ON a- <>- c-
oc ca ao
(4.) '^ y
(6.)
1—03 ' l + ai*
(7.)
(9.)
^S + p • ^2^2,2
1 ac + l a rl
^°'^ -(7-6)3 --^3 •
05 + 1 0)2 + 1 a;*+l
(11.) /I+I+IW
\x y zj
a3.)(..:.«i)(«-g.
(17.) (t,— ■^-+i)^
\x* xy y^Jx
(10.)
x^—l x^—\ a;*— 1
x^y^
(12.) -^(^-«\ .
(14.) 5;+* f(-5 1-).
ae.) ^-(&+^Ui— ^V
6x\ <* / \ a + xj
(18.) (l±'+^\4dt' .
I- • *
1
i ■
w
it :
102
FRACTIONS.
Division.
139. To denote that one fraction is to be divided by another,
they are written in a row with the sign -f- between them.
The same notation is employed when the dividend or divisor
is a mononomial whole quantity
Sums and differences when they are the object of division
must be enclosed in brackets.
Thus, --f-- denotes that - is to be divided by - ;
4a
a
|4-(2a-l)
4a
,2
a
»>
ii
X
y
it
X'
n
ii
a
Ii
it
a
ii
Ii
3c;
2a-l;
3a;
X
2a
X
.2__:
a
The same thing may also be denoted by writing the
quantities which are the objects of division in the form of
a fraction.
2^ a
Thus, i==27j-^a* 2^n=6-^-(2a-l);
a
a. — •*'
X
140. When the product of two quantities is unity, each is
said to be the reciprocal of the other.
Thus, since axl=l , ? . ^ = 1 , ^^ . ?£r§_i :. r^,
lows that
f'R ACTIONS.
103
1 ..
a
a
h
2x-3
is the reciprocal of a , and
ft
)}
tt
»i
b
a'
2x-S
x-l
»
■5 »
1.
a
a .
a " b'
2.r-3 x-1
a of
b
n
x-1 " 2a:-3'
141. Division of fractions is performed according to the
following rule : —
The quotient of one fraction divided hy another is the product
of the former and the reciprocal if the latter,
A whole quantity is to be considered as a fraction whose
denominator is unity ; and sums and differences of fractions
and mixed quantities must first be reduced to fractions.
Examples,
^ ''' bxy'' ' luxhj 5xy^ ' 2ab^~ 'ET'
(2.)
a'
^y 5xy^
a^ + ax _ a^
by
a^-}-ax + x^
a^—a^ ' a^^ax+x^ a-'—x^ ' a^ + ax
. a^
(a— a;) (a^ + a.x + a;^)
a
a^—x^'
a^-\-ax-\-x^
a(ci-\-xj
Wh) ' \a b) ab" ' ~ab
_bx-\-ay ab
ab bx—ay
_bx-\-ay
bx—ay
_y^—xy-\-x^
x^y"^
ix+y)(x^-xy-\-y^)
■M
1 1 *
I
I
aV(a;+2/)'
ini
i •■■■■'•
II :
I'
W':
104
FRACTIONS.
EXBBOISE XUn.
Simplify—
(3.) — , -^ -ir-
^ ' cp + l • as— 1
(9.)
a;
y
(11.) ^-r^+y.
05—2^ as+y
^ - ^ -
i+6 S"*'G
a a
(6.)
(4.) -J^-:--^-.
a+cc • a— as
... /I 1\ . 1 .g. ia^h a^^\a^
6
a+ac a— a;
/in\ a—x'_a±x
(12.) ij^ 1
(B 9? i9
«?
(14.)
8~10 10~li}
S+io 10+12
( I05 )
CHAPTER XV.
SIMPLE EQUATIONS (continued).
I ,•
142. We shall give in this chapter some examples of equa-
tions involving fractions with literal denominators. Such
equations may be cleared of fractions by the rule already
given in Art. 74. In some cases before applying this rule
it will be found more advantageous to simplify parts of the
equation separately.
(1.) Solve
Examples,
x—a x~-b
b ~ a
Multiplying by ah, the l.c.m. of the denominators, we get
(x—a)a = (x—h)b.
Clearing of brackets and transposing,
ax—bx^a^-^hK
Collecting coefficients of x,
(a-'b)x = a^—b\
Dividing by a—h,
(2.) Solve
x = r- = a+&.
a—b
3a;-l 4a;-2
Multiplying by the l.c.m. 6 (2x— 1) (3c5— 2),
6(3a;-lX3x-2) . 6(4a;-2X2a!-l)=(2x-l)(3a!-2).
io6
SIMPLE EQUATIONS.
m
w
n-T I
Clearing of brackets,
54a;2-54:a;H-12-48a;2+48x-12=6x2-7x+2.
Transposing,
54x2-48x2-6a;2-54a;+48a;+7a;=2-12+12,
.*. a;=2.
a;— l__a;— 2_a7— 4__aj— 5
03—2 as— 3 as— 5 as— 6*
(3.) Solve
Simplifying the sides separately,
(sc-1) (a;-3)-(a;-2y_ (a;-4)((g-6)-(a;-5)a
(a;_2)(a;-3) (!B-6)(a;-6)
Clearing the numerators of brackets,
-1_ ^ -1
(a;-.2Ki»-3) (ir-5)(a;-6)*
Multiplying by the l.o.m. (as— 2) (ai— 3) (as— 5) (as— 6), clearing
of brackets, and solving,
as =4.
Solve-
EXEBOISE XLIV.
a\ 12 1 _ij
•-^¥"^12^-^*'
(2.)
(3.)
(5.)
(7.)
16
27
3aj-4 5a;-6*
a;-l_7a;-21
a;_2 7a;-26'
_^+2 = -^.
a; + l as + 2
(4.) .-
42 _ 35
a;-2 a;-3*
45 57
^QN 5as— 3 __ 2as + 3 _j^
(ll.)^=:i+«^~^
2a; + 3 4a;-5*
C6 "i 2as— 6 _ 2 as— 5
'^ '^ 3x-8 3^^*
xo N 2as— 3 , 2as— 1_Q
^^'^ 2HHhl + 2^+3~'''
/I n\ Gas + 13 2as_ 8.r + 5
2a;-5
(12.)
g!-14 _ 2a;-29 _ 1
a? 2aj-20 2!b'
(13.)
(2aj+3)(a!-5) (3x-2) (jb-11) '
SIMPLE EQUATIONS.
107
X
-^ = 0.
/15) JP a;--l_g;— 3_ a;— 4-
''^ a-l a;-2 a;-4 a;-5'
(16.) zr"'
6— a? 6— a; 4— a;~
Uo-; — T~ + = r + -.
a, a
(20.)^-^+^=!.
(22.) ^— ^ -L^'"^ - ^"^^ I ^jj.
(17.) 2-5= c.
a b
(19.) 5+»=5+*.
(21.) A-.«=j2-a2.
( io8 )
-it--
1'
I'm'*
i ■■
I':. *^
CHAPTEE XVI.
PROBLEMS (continued).
143. We shall give in this chapter some examples whicl
are more difficult than those in Chapter IX.
Examjples,
(1.) If A can perform a given work in 60 days, and B in 4C
days, in how mauy days will A and B, working together, be
able to perform it ?
Let w denote the work to bo done, and x the required
number of days. Then
w
amount of work done by A in one day=gQ»
B
AandB
$$
»
li
W
io
M
f9
t>
X
w w w
1 1
Divide by «;,- = gQ+|^-
Multiply by 120a;, 120=2x + 3a;.
.*. a:!=2-l.
(2.) At what time between 5 and 6 is the minute hand of
a watch 5 minute divisions behind the hour hand ?
PROBLEMS,
109
les whicl
id B in 4C
;ether, be
required
Lot X =5= the number of minute divisions between the hour
hand and 5; then 5— a; = the number of minute divisions
between the minute hand and 5. But the number of minuto
divisions between 12 and 5 is 25 ; therefore the number of
minute divisions between 12 and the minute hand is 25—
(5-a;)=20 + a;.
The hour hand thus moves over x minute divisions while
the minute hand moves over 20 + a;; and since the latL.r
moves 12 times faster than the former, it follows that
20 + a:=12a;.
• T — l-fi-
• > •*' — "'■ 1 T*
Hence tho required time is 12!r=21^Y minutes past 5.
(3.) A grocer bought 200 lbs. of tea and 1000 lbs. of sugar,
the price of the sugar being \ of that of the tea. He sold the
tea at a profit of 40 per cent., and the sugar at a loss of 2i
per cent., gaining on the whole $4550. "What were his buy-
ing and selling prices ?
Let the cost price of the sugar per lb. =x dollars.
•*• a ii tea „ =033 „
Then „ „ 1000 lbs. of sugar = 1000a; dollars.
„ „ 200 „ tea =1200a! „
40
The profit on the tea-TQ^* 1200a; =480a;
2J
The loss „ sugar = jQQ- 1000x = 25a;
/. 480a;~25x=45-50
a;=-10
.". the buying price of sugar is 10 cents, and the selling
price 91 cents per lb. ; the buying price of tea is 60 cents,
and the selling price 84 centa per lb.
( ■■
;
te hand of
Exercise XLV.
(1.) I arrange 1024 men 8 deep in a hollow square : how
many men will there be in each outer face ?
no
PROBLEMS.
If
!!l-
Ik
(2.) A regiment containing 700 men is formed into a hollow
square 5 ranks deep : how many men are there in the front
rank ?
(3.) A man has a number of cents which he tries to
arrange in the form of a square ; on the first attempt he has
180 over ; when he increases the side of the square by 3 cents
he has only 31 over. How many cents has he ?
(4.) On a side of cricket consisting of 11 men, one-third
more were bowled than run out, and 3 times as many run out
as stumped ; two were caught out. How many were bowled
and run out, respectively ?
(5.) Water expands 10 per cent, when it turns to ice. How
much per cent, does ice contract when it turns to water ?
(6.) A manufacturer adds to the cost price of goods 20 per
cent, of it to give the selling price ; afterwards, to eflfect a
rapid sale, he deducts from the selling price of each article a
discount of 10 per cent., and then obtains on each article a
profit of 8 shillings. What was the cost price of each article ?
(7.) A person invests £14,970 in the purchase of 3 per
cents, at 90 and 3r per cents, at 97. His total income being
£500, how much of each stock did he buy ?
(8.) A and B join capital for a commercial enterprise, B
contributing £250 more than A. If their profits amount to
10 per cent, on their joint capital, B's share of them is 12 per
cent, on A's capital. How much does each contribute ?
*(9.) In a concert room 800 persons are seated on benches
of equal length. If there were 20 fewer benches, it would be
necessary that two persons more should sit on each bench.
Find the number of benches.
*(10.) A man travelled 105 miles, and then found that if he
had not travelled so fast by 2 miles an hour, he would have
been 6 hours longer in performing the journey. Determine
his rate of travelling.
(11.) An express train running from liondon to "Wake-
field (a distance of 180 miles) travels half as fast again as an
These questions belong to Exercise XLVIII.
PROBLEMS.
Ill
a hollow
he front
tries to
)t ho has
^ 3 cents
rac-third
J run out
e bowled
ice. How
,tcr?
ds 20 per
) efifect a
article a
article a
article ?
of 3 per
ne beincr
rprise, B
nount to
is 12 per
e?
benches
ivould be
h bench.
hat if he
uld have
etermine
o Wake-
tiin AS an
ordinary train, and performs the distance in two hours less
time ; find the rates of travelling.
(12.) A can do half as much work as B, B can do half as
much as C, and together they can complete a piece of work
in 24 days; in what time could each alone complete the
work ?
(13.) Three persons can together complete a piece of work
in GO days ; and it is found that the first does ;I of what the
second does, and the second % of what the third does : in what
time could each alone complete the work ?
(14.) What is the first time after 7 o'clock when the hour
and minute hands of a watch are exactly opposite ?
(15.) The hour is between 2 and 3 o'clock, and the minute
hand is in advance of the hour hand by 14i minute spaces of
the dial. What o'clock is it ?
(16.) At what time between 3 and 4 o'clock is one hand of
a watch exactly in the direction of the other hand produced ?
(17.) The hands of a watch are at right angles to each
other at 3 o'clock : when are they next at right angles ?
(18.) How much water must be mixed with 60 gallons of
spirit which cost £1 per gallon, that on selling the mixture at
22s. per gallon a gain of £17 may be made ?
(19.) How much water must be mixed with 80 gallons of
spirit bought at 15s. per gallon, so that on selling the mix-
ture at 12s. per gallon there may be a profit of 10 per cent,
on the outlay ?
(20.) If 16 oz. of sea-water contain 0'8 oz. of salt, how
much pure water must be added that 16 oz. of the mixture
may contain only 0*1 oz. of salt ?
(21.) I have a bar of metal containing 80 per cent, pure
gold, which weighs 30 grains : how much must I add to this
of metal containing 90 per cent, pure gold, in order that the
mixture may contain 87 per cent. ?
(22.) How much silver must I add to 2 lbs. 6 oz. of an
1 ■.
l!,
112
PROBLEMS.
iii>
oUoy of silver and gold containing 91*7 per cent, of pure
gold, in order that the mixture may contain 84 per cent, of
gold?
(23.) A person started at a certain pace to walk to a rail-
way station 3 miles off, intending to arrive at a certain time ;
but, after walking a mile, ho was detained 10 minutes, and
was in consequence obliged to walk the rest of the way a milo
an hour faster. At what pace did he start ?
(24.) A person started at the rate of 3 miles an hour to
walk to a railway station in order to catch a train, but after
he had walked \ of the distance he was detained 15 minutes,
and was obliged in consequence to walk the rest of the way
at the rate of 4 miles an hour. How far off was the station ?
(25.) A wins the 200 yard race in 28i seconds, B the con-
solation stakes (same distance) in 30 seconds: how many
yards ought A to give B in a handicap ?
(26.) A wins a mile race with B in 5' 19". B runs at a
uniform pace all the way ; A runs at \^ of B*s pace for the
greater part of the distance, and then doubles his pace, win-
ning by a second : how far did A run before changing his
pace ?
(27.) A boy swam half a mile down a stream in 10 minutes ;
without the aid of the stream it would have taken him a
quarter of an hour. What was the rate of the stream per
hour ; and how long would it take him to return against it ?
(28. A contractor undertook to build a house in 21 days,
and engaged 15 men to do the work. But after 10 days he
found it necessary to engage 10 men more, and then he accom-
plished the work one day too soon. How many days behind-
hand would he have been if he had not engaged the 10
additional men ?
(29.) Two crews row a match over a four-mile course; one
pulls 42 strokes a minute, the other 38, and the latter does
the distance in 25 minutes; supposing both crews to row
uniformly, and 40 strokes of the former to be equivalent to
36 of the latter, find the position of the losing boat at the end
of the race.
( "3 )
CHAPTEB XVn.
QUADRATIC EQUATIONS.
144. Wb have already defined Quadratic Equations in
Art. 69. They are further called adfeded or pure, according
as the term involving the first power of the unknown quan-
tity does, or does not, appear.
Thus, 4«;2-5x + 7=0, «;2+6x~3=0, a:^-3x=0,areadfected
quadratics; 3a;2-8=0, 0^24-6=0, are pure quadratics.
145. I. Pure Quadratics are solved bi/ transposition of
terms and extraction of tlie square root.
Examples.
(1.) Solve jb2-4=0.
Transposing, a?=L
Since the square root of a positive quantity is either + or
— , we have, extracting the square root,
a:=±2.
Thus the two roots are +2, —2.
(2.) Solve 0^+6= Var'-ie.
Clearing of fractions,
3a;H15=10ay»-48.
Transposing and dividing by —7,
X
^=9.
.-. X =±3.
Thus the roots are +3 and -3.
I
114
QUADRATIC EQUATIONS.
if'i:i;
!
!!'■« lil,;
*il
n Li's'
146. n. Adfected Quadratics may be solved by one of
the following three rules : —
(i.) When tlie equation is in the form of the product of two
factors, each containing the unknown, equated to zero, the solu-
tion is effected by equating to zero each factor in turn.
Examples.
(1.) Solve (x-i-l) (2a;-3)=0.
Since, when the product of two factors vanishes, one or
other must be zero, we have
either a; +1=0, and /. a;=— 1;
or 2a5— 3=0, and .*. a;=f.
Thus the two roots are —1 and f.
(2.) Solve a;2-5aj=0.
Factoring, x{x—b)=0. *
:. either a;=0,
or cc— 5=0, and /. £c=5.
Thus the roots are and 6.
Whenever, as in this case, the terms of an equation are
divisible by the unknown x, we can infer that one root is
zero.
(3.) Solve (2x-5) (ax-43}) =0.
Here either 2x— 5=0, and .*. a;=f ;
46
or aaj— 4&=0, and /. a;=
a
ib
Thus the roots are f and --
■* a
Q UADRA TIC EQUA TIONS.
"5
•y one of
\ct of two
the solu-
one or
;ion are
root is
Exercise XL VI.
(1.) a)2„36=o. (2.) 5a;2=45.
(4.) 2(a;2-7)+3(x2-ll)=33.
(5)- K«^'H4) + Ka^H3) = a;2 + l.
(3.) I =27.
,(.. _6 7_
(7.) x'=Sx. (8.) 2+6^^=0. (^') i(x-'-4x)=6x.
(11.) 4x2+1=0.
(10.) x^-'~=0.
(12.) a;2— ^=2a:2+iK. (13.) (x-S)(x-5)=0.
(14.) (a5 + 5)(a;-7)=0. (15.) (a; + l)(a; + 3)=0.
(16.) (2i»-l)(3x-4)=0. (17.) (3x-5)(2a; + 7)=0.
(18.) (5u; + 6)(6a; + 7)=0. (Id.) (ax^b){cx-{-d)=0.
(20.) a;^— aaj=a£c--a5'^.
147. When tlie quadratic is not in a lorm adapted for
applying rule (i.), it may be solved by either of the following
rules : —
(ii.) Having transposed the unknowns separately to one side,
make the coefficient ofs? unity by division (if necessary). Then
add the square of^ the coefficient of x, and the solution ?s effected
hy the extraction of the sqtcare root of both sides.
148. (iii.) Having cleared the equation of fractions (if neces-
sary), and transposed the unknoums separately to one side, mid-
tiply both sides by 4 times the coefficient of x^, and add the
square of the coefficient of x. The solution is then effected by the
extraction of the .iquare root of both sides.
Examples.
(1.) Solve a;2- 12a; + 35=0.
By rule (ii.), transposing,
a;2-12r=-35.
i2
- p. 4
kiu*
'*!'
I
I
I
[^'■1' !
ii6
QUADRATIC EQUATIONS.
Adding the square of one-lialf 12,
a;2-.12^ + 6"=:36-35=l.
Extracting the square root,
a;-6=±l;
that is, a:— 6=1, and /. a; =7,
or 05— 6= —1, and ,*. a;=5.
Thus the roots are 7 and 5.
(2.) Solve 2a;2+5;«-3=0.
By rule (ii.), transposing and dividing by 2,
Adding the square of one-half -|,
«.2 1 5^ I /5\2 — 2 5 _1_3— 4.9.
X -\r-^x-t Vi^ --^6^^2--l6•
Extracting the 6quar(3 root.
»+^=±I.
• • flj—
-5±7
4 =¥ or -3.
Thus the roots are \ and— 3.
(3.) Solve a;2+i)a;+g'=0.
By (ii.), transposing,
03^+^03=— g'.
Adding the square of one-half^,
Extracting the square root,
QUADRATIC EQUATIONS.
Thus the roots are
2 2
(4.) Solve 2a;2 + 5a;=3 by rule (iii.).
Multiplying by 4x2=8,
16x2+40x=24.
Adding the square of 5,
Vox' + 40a; + 5^=25 + 24=49.
Extracting the square root,
4ir+5=-fc7.
/. a;=2, or — 3.
(5.) Solve aic2+6ic+c=0.
By (iii.), transposing,
aa)2+5x=— c.
Multiplying by 4a and adding W,
Extracting the square root,
2aa; + &=±V62— 4ac.
j^^-6±V6''-4ac
2a
Thus the roots are
" 2a"
and
— &-V&^— 4ac
2tt
117
i
Exercise XLVn.
(1.) (B2-6ir+8 = 0.
(3.) a;2+4a;-21==0.
(5.) l-a:^=|.
(2.) x2«4a,«6=o.
(4.) 2*2_5^^2=0.
(6.) ^=x^+-^.
^ -i
ili
f
1^'
lis
QUADRATIC EQUATIONS.
(7.) 4a;=^-4a;=15.
(9.) |'-a.=12.
1-a; 2-
03
(8.) 6a;2-lla;+4=0.
(10.) a;+l="
(14.)
£C
a; + 60 3x-5*
(17.) a;=5— i-.
a? — o
(19.) ?±i_5fe±|)=8. (20.)
(21.) (a;-3)2-3(x-2)(a;-7)=21.
(22.) »2-(a+6)a;+a6=0.
y-ip s 2a7— 3 a;— 3 _t
'^ "^--^ 2^+l"3^T2-^
(18.) ^-2xz:^=f .
2a;-5 3x--2
a; + 2
4— a;
1,#
( "9 )
CHAPTER XVm.
PROBLEMS.
149. When the Algebraical statement of a problem leads to
a quadratic equation, the unknown quantity will be one of
the roots. In some cases either root may be taken, but it
will generally be found that one of the roots must be rejected
as being inconsistent with the conditions of the particular
question proposed.
Examples.
(1.) 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 out how much he laid out.
Let X = number of dollars laid out.
.'.05—24:= „ „ lost.
But the loss is also x per cent, of x =— x»= ^
100
100'
X"
^x-IL
" 100
a;2-100a;=-2400.
03=40 or 60.
The amount laid out was, therefore, $40 or $60. Thus
both roots satisfy the conditions of the problem.
(2.) A person buys a certain number of shares for as many
dollars per share as he buys shares; after they have risen as
many cents per share as he has shares, he sells and gains
$100. How many shares did he buy ?
I20
PROBLEMS,
Hi**
#;
Let X = the number of sLares bought, the price of which
at X dollars per share is x^ dollars.
The rise being x cents or ^ dollars, the price for which
he afterwards sells the x shares at a^+^Tw^ dollars per share is
(ic+=^W But the gain is $100.
a;2=10000.
/. X = + 100, or -100.
As the negative root would not answer the conditions of
the problem, it must be rejected. The answer is, therefore,
100.
Exercise XL VIII.
(1.) A rectangular room which contains 1800 square feet is
twice as long as it is broad : find its dimensions.
(2.) Divide 20 into two parts whose product shall be 91.
(3.) Find a number whose square increased by 20 is 12
times as great as the number itself.
(4.) Divide 15 into two parts such that their product shall
be 4 times their difference.
(5.) By what number must I divide 24 in order that the
sum of the divisor and quotient may be 10 ?
(6.) Find three consecutive numbers such that the square
of the greater shall be equal to the sum of the squares of the
other two.
(7.) A ladder 34 feet long just reached a window of a house,
when placed in such a position that the height of the window
above the ground exceeded the distance of the foot of the
ladder from the wall by 14 feet. Find the height of the
window.
PROBLEMS.
121
(8.) A horse is sold for £24, and the immber expressing
the profit per cent, also expresses the cost price of the horse :
what did he cost ?
(9.) An article is sold for £9 at a loss of as much per cent,
as it is worth. Find its value.
(10.) A and B start together for a walk of 10 miles; A
walks 1-2 miles an hour faster than B, and arrives 1^ hours
sooner than he does : at wl)at rate did each walk ?
(11.) After selling a part of an estate, and the same part of
the remainder, I find I have left nine- tenths of the part first
sold : what part did I sell at first ?
(12.) An uncle leaves 14,000 dollars among his nephews
and nieces, but 3 of them having died in his lifetime, the
others received 600 dollars apiece more : how many nephews
and nieces were there originally ?
(13.) A number is composed of two digits, the first of
which exceeds the second by unity, and the number itself
falls short of the sum of the squares of its digits by 26. What
is the number ?
(14.) The sides of a rectangle are 12 and 20 feet : what is
the breadth of the border which must be added all round
that the whole area may be 384 square feet ?
(15.) One hundred and ten bushels of coals arc distributed
among a certain number of poor persons; if each had
received one bushel more, then he would have received as
many bushels as there were persons. How many persons
were there ?
(16.) A sum of £23 is divided among a certain number of
persons ; if each one had received 3 shillings more, he would
have received as mary shillings as there were persons. How
many persons were tnere ?
(17.) A company at an inn had £7 4s. to pay, but before
the bill was settled 3 of them left the room, and then those
who remained had 4s. apiece more to pay than before; of
how many did the company consist ?
■ I
'' i
122
PROBLEMS.
, ii
(18.) A person rents a certain nuu* jer of acres of pasture
land for £70; he keeps 8 acres in his own possession, and
sublets the remainder at 5s. an acre more than he gave, and
thus covers his rent and has £2 over. How many acres were
there?
(19.) An oflBcer can form the men in his battalion into a
solid square, and also into a hollow square 12 deep ; if the
front in the latter formation exceed the front in the former
by 3^ find tho number of men in the battalion.
t ,,
li'^ii
of pasture
ission, and
gave, and
acres were
( 123 )
lion into a
3p ; if the
ihe former
CHAPTER XIX.
SIMULTANEOUS EQUATIONS,
150. If two unknowns are to be determined, there must
be two independent equations. These equations are called
simultaneous equations, because the same values of the un-
knowns X and y must be substituted in both equations.
Thus if
2x^y = 9,
the only values which satisfy both these equations at the
same time are x=7,y=6.
151. It must be borne in mind that there is an infinite
number of values which will satisfy either equation sepa-
rately.
Thus, in the equation 203— 2/ =9,
ifa;= 1, 2--2/=9, and .'. 2/=- 7;
if 05= 2, 4— y=9, and /. 2/=— 5;
if a:=10, 20-2/=9, and .*. y= 11 ;
and so on.
152. If three unknowns are to be determined, there must
bo three independent equations ; and generally the number of
unknowns must be the same as tho number of independent
equations connecting them.
153. The solution of simultaneous equations is effected by
deducing from them other equations, each of which involves
one unknown. This process is called elimination, and may be
conducted according to one of the following methods :~
I
m
124
SIMULTANEOUS EQUATIONS.
I. Substitution.
II. Comparison.
III. Cross Multiplication.
I. Method of Substitution.
154. Tins method consists in finding from one equation the
value of one unknown in terms of the other, and substituting the
value so found in the second equation, which is thereby reduced to
a simple equation in one unknown.
For convenience of reference the given equations and others
which arise in the process of solution are numbered (1), (2),
(3), &c.
Example.
Solve x-\'y=d .... (1),
2aj+2/ = 4 .... (2).
From (1) we find 2/ = 3— a; .... (3).
Substituting this value of y in (2),
2a; + 3-a;=4:.
Substituting this value of x in (3),
2/=3-l=2.
Thus the solution is x~l. y=2.
II. Method of Comparison.
155. This method consists in finding from each of the pro-
posed equations the value of one and the same unknown i7i terms
of the other f and equating the values so found.
Solve
Example.
7a:~32/ = 19 .... (1),
4a;+7y=37 .... (2).
lation the
tuting the
'educed to
Qd others
^ (1), (2),
ne pro"
n terms
SIMULTANEOUS EQUATIONS.
125
From (1) we find
y
__7i»-19
• . (3);
and from (2)
y 7- .... (4).
Equating these values of ?/,
7aj-19_37-4a;
3 T~'
,\ a;=4.
Substituting this value of x in (3)
Thus the solution is a;=4, y=6.
III. Method of Cross Multiplication.
156. T/u-s ^^e^7,of^ co«,s/sf.s in Tnulfiplyhifj the qiven equations
{reduced to the form ax + by=c) Inj such quantities as will
render the coefficients of the same nuhioum mtmerically equal
By adding or suhtracting the equations so found, ive obtain a
simple equation in one unknoivn.
Examples.
(1.) Solve lx-2y= 5 . . .
13«+4?/-30 . . .
Multiplying (1) by 4 and (2) by 9,
28x-36v/= 20 . . .
117a^ + 36;y=270 . . .
Adding (3) and (4),
145ir = 290,
Again, multiplying (1) by 13 and (2) by 7,
91X-117?/- 65 .... (5),
91as-+ 28^=210 .... (6).
(1),
(2).
(3),
c-
\^
■
f!
126
SIMULTANEOUS EQUATIONS.
Subtracting (5) from (6),
145?/ = 145,
/. 2/ = l.
Thus the solution is x = %y = \.
\ (2.) Solve 8a; + 25?/ = 9 . . . .
(1).
^ |, 12a;-10iy = 4 ....
(2).
,!|"' Multiplying (1) by 2 and (2) by 5,
[ 16x + 502/ = 18 . . .
. (3),
;Li. 60a;-50?/ = 20 . . .
.(4).
Adding (3) and (4),
76a; = 38,
1, * rf — 1
1 ..»«'— a •
Again, multiplying (1) by 3 and (2) by 2,
1 i 24a; + 75?/ = 27 . . .
. (5),
j ; 24a;-20//= 8 . . .
.(6).
Subtracting (6) from (')>
! ; 95?/ = 19,
.1
V • ?y-i
! • • 2/ - 6'
-i^ Thus tho solution is x=\, ij=^.
I*'^-
m
Exercise XLIX.
(1.) 4x + 2/=ll, a; + 4?/=14.
(2.) 2x + 32,'=21, 3a; + 5,?/=34.
(3.) 3x=:23-2?/, 10 + 2.t=5?/.
^^•^ 2 3' 32
^^■^5^6~2^'^' 3 *'10 4*
(6.) 3a;-22/ = 3(6-a;), 3(4a;-3?/) = 72/.
(7.) 7(a)-l)=3(2/+8), ^+?=5^.
SIMULTANEOUS EQUATIONS.
127
^ON 2a;—;/ 3 3y «
(SO "1-^-2 = -4 -^-2, a7+y = 8.
(10.) 2a;-2^3=5fr2 g.,_^~5_7?/-7
5 2 ' *^ 3 2~'
(ll.)T^(aj + ll)+Ky-4)=:a;-7,-Ka; + 5)-.K2/-7) = 3y-a:.
(12.) a;-24 = |+16, Ka'+y)+x=3(2y-a;) + 105.
(13.) K3^-72/)=K2a;+2/ + l), 8-K«'-2/) = 6.
(14.) ^^±^=l + 2(2a;-6?/ + l) a;_,,
(15.) a; 4- 2/ = a, a; -2/ = 6.
(16.) aa; + a2/ = a2+6^ a;=a.
(17.)^ + |=l,c.+y=c.
157. When there are three simultaneous equations containing
three unhnoivns, the solution is effected hy eliminating one of the
unknoivns hetiveen the first and second equations, and also he-
twecn the first and third, or second and third. Tu-o equations
are thus obtained involving two unknoivns, ivhich may he found
hy the methods already exi>lained. The value of the third un-
knoivn may then he found hy substitution.
Example,
Soke 2.r-32/+ 2= 1 ... . (1),
305-52/ + 4^= 3 . . . . (2),
4x + 22/-32=13 .... (3).
Multiplying (1) by 3 and (2) by 2,
6a;- 92/ + 32=3 .... (4),
6a;-102/ + 82=6 .... (5).
>}'
128
SIMULTANEOUS EQUATIONS.
if :■■
Subtracting (5) fr^^m (4),
2/--5,i=— 3 .... (6).
Thus a- is eliminated from (1) and (2).
Again, multiplying (1) by 2,
4a— •62/ + 22= 2 . . . . (7),
and 4a; + 2y-32=13 .... (3).
Subtracting (7) from (3),
8?/-5^= 11 ... . (8).
And y—5z=-'3 .... (6).
From (6) and (8) we find y=2, 2=1.
Substituting these values of y and z in (1), (2), or (3)
we get a; =3.
Thus the solution is x=3, y=% 2=1.
n.
EXEKCISE L.
(1.) x + By-^2z=ll, 2x + y-i-3z=U, Sx -{-^y + z=ll.
(2.) a; + 2^ + 37;=13, 2.b + 3?/ + 2=13, 3x-{-y + 2z=10.
(3.) 2x-{-Sy-'4:Z=10, 3x-4:y + 2z=6, 4:X-9aj+Sz=21.
(4.) lOx-2?/ +42=10, 3a; + 52/+32=20, a;+32/-2.2=21.
(5.) 3^ + 2^=13, 3?/+22=8, 32+2x=9.
(6.) ^-f 1+1=3, 4:x + 5y-{-6z=77,z+x=2y.
4 5 D
(7.) 3a;-2.y=6, 3y-22=5, 32-2a;=-2."
(8.) |^(«-l)-2/=35, %-5z=43, x+y-{-z=60.
(9.) 2/— 2 + 3=0, z;--iC=-5, a3 + 2/=6.
(10.) 2,' + 2=a, z + x=h, x-\-y=c.
( 129 )
CHAPTER XX.
!), or (3)
LI.
10.
=21.
2=21.
PROBLEMS.
158. In the following problems the various unknowns are
expressed m terms of separate and distinct symbols Td th^
ot equations. If two symbols x and y be employed thA
tit and 1 *'' -"f ^^". "^^* '"^^ two'indepentn tiua!
tions, and three mdependent equations will be required to
determine three unknowns, x, y, and z, ^ *"*
Examples^
(1.) A fraction becomes equal to 1 when 1 is added fn fli«
rni^rir Lot* '''^" '^^-^^ *^^-^-
Let - be the fraction.
y
Then by the conditions of the question,
a+l_
y
X
1,
the solution of which is x=5, y=.Q,
Hence the fraction is |.
^^«Z^^? ^^""^ numbers such that the sum of the first
one-fifth the secondhand one-tenth the third, shall be equal
K
130
PROBLEMS,
li 'I
H :
to 4; the sum of one-half the first, the second, and one-
tenth the third equal to 7 ; and the sum of one-half the first,
one-fifth the second, and the third equal to I'i.
Let aj= the first, y=. the second, and 2= the third number.
Then by the conditions of the question,
2 5
Multiply these severally by 10,
10a;+ %j-\- 2= 40 ... . (1),
5x + 10y-f 2~ 70 . . . . (2),
hx\ 2?/-|-10;^=120 .... (3),
2x(2)-(l), 182/-h2=100 .... (4),
(3)-~(2), -87/ + 92= 50 ... . (5>,
9x(4)-(5), 170y/=850.
.•.?/=5.
Therefore from (4) 2=100-18?/=10 ;
and from (1) a; =2.
The required numbers are thus 2, 5, and 10.
Exercise LI.
(1.) One of the digits of a number is greater by 5 than the
other. When the digits are inverted, the number becomes
I of the original number. Find the digits.
(2.) In a division the majority was 162, which was i\ of
the whole number of votes ; how many voted on each side ?
(3.) The sum of two digits is 9. Six times one of the
numbers they form is equal to 5 times the other number,
i'ind the digits.
PROBLEMS.
131
md one-
tho first,
lumber.
than tlio
becomes
as -i\ of
side?
of the
Qumber.
(4.) If the numei^ator and denominator of a fraction be
each increased by 3, the fraction becomes 2 ; if each be in-
creased by 11, it becomes f . Find the fraction.
(5.) A number consists of two digits whose sum is 12, and
Btich that, if the digits be reversed in order, the number pro-
duced will be less by 36. Find the number.
(6.) Three towns A, B, and C are at the angles of a
triangle. From A to through B, the distance is 82 miles ;
from B to A through C, is 97 miles; and from C to B
through A, is 89 miles. Find the direct distances between
the townSk
(7.) The diameter of a five-franc piece is 37 milUmetres^
and of a two-franc piece is 27 milUmetres. Thirty pieces laid
in contact in a straight line measure one metre exactly. How
many of each kind are there ?
(8.) At a contested election there are two members to bo
returned and three candidates. A, B, C. A obtains 2112
votes, B 1974, C 1866. Now 170 voted for B and C, 1500 for
C and A, 316 for A and B. How many plumped for A, B, C,
respectively ?
(9.) A boat goes up stream 30 miles and down stream 44
miles in 10 hour >. Again, it goes up stream 40 miles and
down stream 55 .irui>k
stream and boat.
in 13 hours. Find the rates of the
(10.) At a contested election there are two members to be
returned, and three candidates, A, B, C. A obtains 1056
votes, B 987, and C 933. Now 85 voted for B and C, 744 for
B only, 98 for C only. How many voted for C and A, how
many for A and B, how many for A only ?
(11.) Seventeen gold coins, all of equal value, and as many
silver coins, all of equal value, are placed in a row at random.
A is to have one-half of the row, B the other half. A's share
is found to include seven gold coins, and the value of it is
£6. The value of B's share is £6 15s. Find the value of
each gold and silver coin.
k2
it
V
»32
PROBLEMS.
(12.
road
and
)m A to D passes through
cessively. The distance between A and B is six miles greater
than that between C and D, the distance between A and C
is ^g of a mile short of being half as great again as that
between B atd D, and the point half-way from A to D is
between B and C half a mile from B. Determine the dis-
tances between A and B, B and C, C and D.
(13.) Fifteen octavos and twelve duodecimo volumes are
arranged on a table, occupying the whole of it. After six of
the octavos and four of the duodecimos are removed, only |
of the table is occupied. How many duodecimos only, or
octavos only, might be arranged similarly on the table ?
(14.) Three thalers are worth \d. more than 11 francs.
Five francs are worth \d. more than 2 fijrins. One thaler is
worth %l. more than a franc and a florin together. Find the
value of each coin in English money.
(15.) Six Prussian poimds weigh \ oz. more than 5 Austrian
pounds. Twenty-five Austrian pounds weigh \ oz. more than
14 kilogrammes. One kilogramme weighs 1 oz, less than the
sum of the weights of a Prussian and an Austrian pound.
Find the number of ounces in each foreign measure of
weight.
(16.) A person walks from A to B, a distance of 9i^ miles,
in 2 hours and 52 minutes, and returns in 2 hours and 44
minutes, his rates of walking up hill, down hill, and on the
level being 3, 3*, and 31- miles an hour, respectively. Find
the length of level ground between A and B.
( 133 )
CHAPTER XXI *
EXPONENTIAL NOTATION.
u
159. Although the notation adopted in the preceding pages
is sufficient for the purposes of the operations herein treated
of, yet it is found expedient, before proceeding farther, to
employ another notation to express roots, powers of roots,
and their reciprocals. This notation, which consists in em-
ploying fractional exponents instead of radical signs and
integral exponents, and negative exponents instead of reci-
procal forms, possesses the great advantage of reducing to a
few uniform laws the operations of Multiplication, Division,
Involution and Evolution, with respect to powers, roots,
and powers of roots, of a quantity, and their reciprocals.
160. The exponential notation consists in writing
and
a" instead of y^a'",
1
a-P
Thus, according to this notation.
a^=^a,
a^=^a\
a^=ya»,
a
1
a
■3 1
-8
a =
1
a
■^ 1
1
a
•f_
_1
1
* This Chapter may be omitted by those who do not intend to read
more advanced works on Algebra,
%
« !
1 1
:l
jr,:i
134
EXPONENTIAL NOTATION,
-3 2
2a =^,'
161. When the exponential notation is employed, the
quantity is said to be raised to the -power indicated by tho
exponent.
Thus
a^ is read a to the 'power ^ ;
a
it
if
i»
a
X
X
it
n
a
a
f;
a
-3;
a
-|.
The term poiver in this extended Algebraical sense thus
includes the terms power, root, root of a power, reciprocal of a
power, reciprocal of a root, reciprocal of a root of a power, as
used in tho ordinary or Arithmetical sense.
U
i::
EXEBOISE LII.
Express in the Arithmetical notation—
,_ ^ .1. A 2. 3.
(1.) a^, a«, «», a2.
(2.) x-^, x-\ x-'^'^.
(3.) m 3, n '^, p *.
(4.) 2a*, 3a:- 2, Gm'^.
Express in the Exponential notation —
(5.) ^x, ^m, ^n.
X a'^ a'' a*
(7.) ^'] ^^% V^,
^^'^ ^x ^x j;/^'
EXPONENTIAL NOTATION.
135
(9.)
sd, the
I by tho
ise thus
)coX of a
oweif as
2 3 10
m rr p^
(10.) -!,. 4. '
^x ^x ^x^'
162. The utility of employing the exponential notation will
be exhibited in the statement of the three following rules,
which are usually called Index Laws.
I. The product of any two powers of the same quantity is a
power whose exponent is tlie Algebraic sum of the exponents of the
factors.
Since the product of a quantity and its reciprocals 1, this
rule cannot be applied when the exponents of the two factors
are numerically equal and of opposite signs, unless tlie zero
power of a quantity be considered=l.
Examples.
(1.) a'^ ' a3 = a2 3=^6.
^8 8. 3 + 6. XS
(2.) a* • a*^=a'^ ^=a^^,
(3.) a^ .a-i=a3-i=a2.
(4.) a-2 . a-3=a-2-3=ra-6.
-^ ^ —A 1 — .' i
(5.) a- a •^=a '■'=a'\
5. -3 5_8 JL
(6.) a^ • a * = a« * = ai2.
(7.) a- a-i=ai-i=a°=l.
(8.) a^ ' a"2=ra" = l.
This law may also be expressed briefly as follows :—
where m and n are any quantities whatsoever, positive or ne^Of
Uve^ integral or fractiotial, including zero ifsP=l.
136
EXPONENTIAL NOTATION.
I'
■'%''■
m'
%
m
163. Proofs of the preceding rule will be exhibited in the
following particular cases.
(1.) If m and n be positive integers,
a"* • a.^-:^aaa . . . . (w factors) xaaa . . . . (w. factors),
= aaa .... (w+n factors),
_ „♦»+»
(2.) aa . a8=Va ^a^^~Q^ ^a2 = ^a'^=o« = a2^3.
Here v^ = >v/^> because each of them when multiplied
by itself six times produces a*. So also ^a = y~€?-\ and
generally, if m, %, j9, are integers,
because each of these quantities when multiplied by itself «p
times produces a'"^.
(3.) If m, n^p^qhe positive integers, then
= ^a»»«+p», by Ex. 1,
m p
(4.)a».a-»~=^=a->=a'^-».
a* a
Exercise LIII.
Find the products of
(1.) 2a;, Sec"; x^, 405*"; dac", a;^™,
(2.) a;2, 2a;*; 3a; s, 2x2; 6a;*, 5a;^.
(3.) as", aj*; 2a;2, Sec"; a;2^ a;}f.
EXPONENTIAL NOTATION.
>37
(4.) 2a», a- a ; a-\ Sa^; 5a, ^a'\
(5.) Q^, a" 3; 2a*, a"«; a, a'^.
(6.) aa, a"3; a«, a~^; a^", a"^.
(7.) a», a-3; a», a-; 2a, 3a-»; wa^, Ma-«.
164. n. When one power of a quantity is divided hy another y
the quotient is a power whose exponent is the Algebraic difference
letween the exponents of the dividend and divisor.
Since the quotient is=l when the dividend and divisor
are the same, this rule cannot be applied in the case of equal
powers unless the zero power of a quantity he considered=.l.
a
(1.) -3=a«-s=a8.
2
(3.) -3= a =a •
a
(5.) ''=|=a-^t=aA.
Examples.
(2.) «^=at-t=a*.
ai 4-i A
ai
(4.) — r— a =a^
a
a"
(6.) -,=a»-fi=aO=l.
This rule may also be expressed briefly as follows :—
a
a"
TO
-Qfn^n
where m and n are any quantities ivhatsoever, positive or nega-
tive, integral or fractional^ including zero ifaP=l.
165. The proof of this rule rests on that of the preceding.
For since a"*-". «"=«"', it follows that
a"*
il
If ¥'
1*1
I3S
EXPONENTIAL NOTATION.
Also, since t^ =«"•*, the second rule may be considered to
be included under the first. Thus
a"
W y^— M __ /»W1"»II
ssa™. a "=«'
.■T
'iSf'
Divide
Exercise LIV.
(1.) a^"* by a*"; a"" by a*.
1.
3 .
2
.3
(2.) or by a^ ; a by o^
(3.) a;^ by a; ; ^ by a*.
(4.) 01? by a;-^ ; x^ by a;'^.
(6.) a;-^ by a;-2 ; a;-^ by as'".
(6.) a;^ by a;'^; a;^ bya;-^
(7.) a;"" by aj-^"; a; 2 by a?"".
166. m. The power of a power of a quantity is a power
whose exponent is the product of the numbers expressing those
powers. In other words,
where m and n are any quantities whatsoever^ positive or nega-
tive j integral or fractional, including zero ifaP=X.
Examples.
(1.) (a^y=a^; {a'y=--a'^.
(2.) (a^)2=a«; (««> = «*.
(3.) (a"^)a = a«; (a^y = a^.
(4.) (a-l)2=a-2. (^-2)3^^-6^
(5.) (a-i)-2=a2; (a-3)-4=ai2.
(60 ia^r^ = a~^; (a^y^z=a'K
EXPONENTIAL NOTATION.
139
167. Proofs of the preceding rule will be exhibited in the
following particular cases : —
(1.) («")=^=a2 . a2 . a2=««=a3'<2.
(2.) (a3)2=a'3 • a3=a^"'3=a«
(3.) If n be a positive integer,
(»"»)•»= a*" • a"
— /ym+w-f
=«•»".
. to n factors,
to n terms
(4.) If ^ and i? be positive integers,
(a")«=a«,
because each of these quantities when raised to the ^th power
produces a"^.
EXEEOISE LV.
Express the following as powers of a : —
(1.) (aO^(«^y;W^
(a-i)2; (a-2)3. (^-3)4^
(a2)-3; (a-2)-3; (a-3)-*.
(2.)
(8.)
(4.)
(5.)
(6.)
(a3)2; (0^4)3 J (a"4)3.
-tx-1
(c.-^-^ ; (a-t)-^ ; (a-t)-!.
168. The Index Laws (I., IL, III.) are thus seen to be true
on the assumption that
-1' 1
for all values of m and w, positive or negative, integral or
fractional, including zero.
Instead, however, of treating the subject of the Index Laws
and notation as in the preceding Articles, we may proceed as
follows : —
If m and n be positive integers, it may bo proved, as is done
in Arts. 163, 165, 167, that
i I
n
if
h" ■
140
EXPONENTIAL NOTATION.
L
II.
III.
a
m
^n—^m+n^
^ sa"'-", w being greater than n.
a
(a"')'»=a»'*.
These laws, which are thus proved to hold in the particular
case where m and n are positive integers, are then assumed to
be true when m and w are any quantities whatsoever, posi-
tive or negative, integral or fractional, including zero ; and
from this extension of these laws we deduce that
a" must=v a"*, a"
'^=0^, and a°=l.
Thus, by I.,
But
a/ a. /s/a =a.
/. a^=^a.
By m.,
(afy=:a\
But
(*/'^y=a\
.-. a^=Va\
ByL,
a* . a^=a\
But
a' X l=a'.
/. a«=l.
ByL,
a3 . a-3=a»=l.
But
a^
• rt-3 — -^
a*"
169. From the remarks of this chapter it will be thus seen
that, although there is no absolute necessity for using such
3. 4, —5.
symbols as a*, a" , a 3, still their introduction gives rise to a
uniformity in certain Algebraical processes ; and the number
of rules which otherwise would be required to meet the
different cases that arise in those operations thus becomes
largely reduced.
( 141 )
ANSWERS.
1.
(1.) 5r 10+15; 10+12+f ; 2+i+f ; a+^+|i+^.
(2.)i 5; 2i-li; 2-5-1-6.
(3.) 11 + 35 + 6-17; 81+76-69-4a
(4) 15-7+8 + 9.
(50 28-16 + 10-4.
U.
(1.) A's -60, +20; B's +60, -30; C's +30, -20.
(2.) A's +20, +20, -30,-40; B's -20, +30, +30,
C's -20, -30, +40, +40.
(3.) A -10, +4; B -7, +10; -4, +7.
(4.) -2°, +5°, -3°.
(6.) +1°, -P, +P, -P, +P.
(6.) -2°, +2°, -2^ +2°.
(7.) +25°, -7°.
m.
-40;
(1.) 30.
(2.) 20.
C3.) 22.
(4.) 22.
(5.) 24, 120, 0.
(6.) 14.
(7.) 7.
(8.) 16.
(9.) ^'
El
14^
ANSWERS.
(iO.) h (11.) M.
(13.) 2a^3a^2(i2 ^3^3 +4^4^
IV.
(1.) 2, 3a, I, &c, 4a*-^6, k.
(2.) -], +8, -f, +26, -5i»2.
(3.) +1, -1, -8a, +|c(£.
(4.) -1, +3a^ -faz.
(6.) 1, -1, -% -3, +i -f.
(8.) -a2, +|a^ +2a2a;, ^a^x.
(9.) -Sa^^a;, +2a*£c; aos^, ^ax\
(5.)^>|,4, +f,
(7.) 2£c, oj.
'f*
if
(1.) +25.
(4.) -21^.
(7.) +^.
(10.) +1L
(13.) -H.
(1.) V6a.
(4.) -13a.
(7.) -3a.
(10.) -4x2.
(13.) -bV*'-
(1.) 2ct-3&.
(4.) 3a + 2a;— 5!/.
(6.) d'-^li'^h
(8.) -«;+22/-2 + l.
V.
(2.) +43^.
(8.) -28.
(5.) +5.
(6.) -6.
(8.) -aV
(9.) -0-7.
(11.) +2.
(12.) -12.
(H.) +1/5.
(15.) +0*342.
VI.
(2.) +15a.
(3.) -5a.
(5.) +2x=».
(6.) +3a2.
(8.) +10c.
(9.) -5c.
(11.) -*-7a&.
(12.) +>
(14.) -fa.
(15.) -f|a.
VII.
I.) -a; + 82/.
(3.) -f:!;z;-3?/-2'.
(5.)
4-a) +2^.
(7.)
a-2ft + 3c-(?.
(9.)
4a-26.
(10.) a2-45c.
(12.) -2a-&-c.
(14.) 4a + 07 + 5.
(16.) 2(1 + 26+ 2c.
(18.) 4a;l
X
ANSWERS.
M3
(21.) |+i%2/+i^^.
(11.) 4a-9&.
(13.) 2.x-.6y-2!.
(15.) 5a— 4x.
(17.) lOx + Sy-.-,
(19.) 4a' +46^ (20.) a + &.
(22.) 2a+|5-|c.
(1.) 4.
(4.) -3.
(7.) 10-6.
(10.) -0-96G.
(13.) lOa^
(16.) -W.
(19.) 2«.
(22.) — aa;— 5&2/+4:C2.
(24.) -7ccH2az;2-5x.
(26.) ia+i&--ic.
(2.) 2.
(5.) 6.
(8.) -3-39*
(11.) -a.
(14.) 2:;.
(17.) \\x\
(20.) 3a-cc+4.
(3.) -9.
(6.) ~3.
(9.) -21G.
(12.) -7a;.
(15.) 2a-|.
(180 3a + &.
(21.) -5a& + 26=^,
(23.) -6a + &+«-4.
(25.) 1+14^.
IX.
(1.) 2*2+-l; 3a; + -42^+-5; 2a+-.36++4c.
(2.) 2a- -5a.
(4.) 2a+-8/>-+7.
(6.) 5a + (6-4).
(8.) a-4 + (26-r).
(3.) ~6-+5x.
(5.) 5++a: 3a.
(7.) -a +(-6 + 5).
(9.) a;2 + (2i/ + 5)-;^.
(10.) a-l + (36 + 5)--3c-.
(11.) ic + (2x''-l) + (-3x^-8). (12.) 4a2-(62-c).
(13.) «2 + 4-(-26 + 3). (14.) 2a-5-(a2-2a-+ 3).
(15.) a + 6 + c + (a-6-c)— (-a + 26-3c).
■I
144
ANSIVERS,
J*'
(1.) 2a+36-(;.
(3.) x'^^x-l.
(5.) 8a— ft— c.
(7.) 8a+26-3c
(9.) a;+ll+4y.
(10.) h.
XI.
(2.) o&— 6c+c.
(4.) 5a;»-3a;2+7xB-8.
(6.) 8a-6+c.
(8.) 2a-6 + 6+c.
(11.) 2aiH3a;.
<■ 1
1
(1.)
(2.)
|i
(3.)
11
(4.)
(1.) (a-l)(2a2-3); (-2+a)(-3-a2); (a5-5)(-2x+7):
(2.) -2a2(62«i); (a2-l)(_3a); 5a;(~a;H3).
(3.) K^'-l); K2^-3); -1(0)^-5).
(4.) -5<a;-l)(a; + 2); (a;«-4)( + 5iB)(2a+3).
(50 +8<-52/)(a:y-l); -7a(a6-3)(+86).
xn.
— 6a&; — 5ac; +6a*6; — SOajy.
— 14a&c''; — 20a'*6c; — 16a3yz; +48a.
-fa^?^; -fa&; +iajy; -fa^J.
— 6a5^«/^; +3aa"''«/''; — 4a'fe^c*.
w
a^b*
(5.) -g ; 15 ; ~"
"20"' ''""12^
xm.
(1.) -8aa; + 66aj-2cx; 1203=2/ -8a;y+43^; -2a6c?-|-3c(f.
(2.) 3x»-6»2-15i»; -2a'''+3a*-7a^ -4aa;H4aV-8a»a;.
(3.) 4a;V + 2^y'^^-6ic^2^ -28a362+ 4^253 _ 4^2 j2^
(4.) 2fi26 + a?>2-fci&c; -a3 + fa2-2a; iaa;^- ia*^;* +|a'a;.
(5.) -12a + 10a6-15; ^-^ax^^\^ax-\-^%\
(6.) 5x'*-5a; + 20; -2a+2a6-6; -a'+2a'»aj-a.
(7.) ax -a;*; ac -&c + c*; -Saft + Sa^i^-lSaSj,
(8.) 4a3-6a2+8a; +2x'-§a;H3a;».
ANSWERS.
»4S
XIV.
(1.) 2a;2+6a;-12; -^x^^x+h; 2-!c-.3a;2.
(2.) 2a;«-a;2_4^^2; 2-2x+3a:2-.3a)«; 3+3x2-a;3-.a;«
(3.) 6a3-7a2+14a-8; l+a^; i_^3^
(4.) am-an ^bm-hn; am +hm^cm + 2an+ 2bn^2cn'
(5.) a;y-x*; a;3+a,3y-a;2^2_2a;y-2a;2/H2y3.
(6.) 2aHa3-22a2+23a-4; a2-4&H126c-9c2.
(7.) 3a|-4^&+8ac-46H8&c-3c2; 3a.*-4a;3y+6xV+4a;y»
(8.) a)*-fa.Hi; a^^a-i; 2a='-i
(9.) 2^3-|x2+|ia.-i; 9a.«-|a;2^.|^_^^
(11.) ia^-^H|a-l. (12.) a;«+3a:y-2a:+y3-2y+l.
(13.) a»-3a6c + 63+c3.
XV.
(1.) +30a«J»; -J^a;7. ^i3^y,^',^
(2.) -24a;V+30a;V; -2a'>6+3a*62-a85.
(3.) 8x3-26x«-17a.+6. (4.) a.*.a:3+^_i,
(6.) a;8-2aV+a8.
XVI.
(1.) (2a-6)-r-.3a; (4a2..,3a + l)-f.(3a-4).
(2.) 2a-f— 3&; -a^-j- + 2x'; 3a;-r-2a.
(3.) 4a;2-j-(2c-5); -aaj2_j>(a;-a).
xvn.
(1) -4; -6; +f ; -V. (2.) ^±. -?^'
. "^ 2a;' 26
+
2«y
a
146
ANSWERS.
(3-) -|;
6«a5.
W
56'
(5.) -2a; 6a6V.
14a2-
(4) 2a'; a»; 4a^.
(6.) -ia;?/'; +fay.
xvm.
(1.) -2a+36-4; aa;--3+2a2.
(2.) -4a;+3-a; 4a;2_ar + 3.
(3.) -aH4a-5; a;2-3a;i/+4/.
(4) 2a-36+c. (5.) -4ac2+3&c'-l.
(6.) (a— 6)0?, (2a— c+l)a;. (7.) (4— a)a;s', (3a3— y)a;y.
XIX.
(1.) a!-4; 3x+l. (2.) «+!; 2a;-3.
(3.) 3a5+2; 3a:2-2(»+6.
(4.) a;-l; oj^-jb+I. (5.) x^-Zx->^\,
(6.) a;Ha;Ha;+l; (c^-ac'+cc^-aj + l.
(7.) a;-3/; x^-xy-\-y\ (8.) aHaJ+ft^.
(9.) 5a; + 6y-3. (10.) aj^-aaj+a''.
(11.) a;2-2i»2/+2/'. (12.) faa;-2a;2.
XX.
(1.) ar2-3a5+ V , - f . (2.) x-a, '2a\
(3.) !r2_aa;+aa, -2a3. (4.) a-2-a;-f4, -ac-4.
(5.) 2a;2+3, -5a;2-3a;-3.
XXI.
(1.) 3a3— amiles.
(2.) 50 +x dollars.
(3.) 26 miles.
(4.) x^—y^ square feet
(5.) - hours.
a
(6.) .+|+3^
<7-) -+^+4-
/Q \ lOoj
(8.) ^ .
ANSWERS,
1
147
(9.) |.
(!«•) 1^00-
100*
(11.) ^+f -5.
(12.) 3& acres.
(13.) ^^ miles.
V
(14.) 1^ hours.
(15.) 6aa;+ a dollars.
XXII.
L.
(1.) 2.
(2.) 10.
(3.) 7.
•y)xy.
(4.) -5.
(5.) 2.
(6.) 5.
(7.) "h
(8.) -5.
(9.) f . 1
(10.) 2.
(11.) 3.
(12.) 2. j
(13.) 7.
(14.) 10.
(15.) 2f 1
(16.) i.
(17.) 6.
(18.) 61 1
(19.) 7.
(20.) 13.
(21.) «. f
(22.) 3a.
(23.) a +6.
(24.) %. 1
I
(25.) 6+c.
(26.) 1.
(27.) a2+a& + &^ 1
1
(2«.) '^*-^'.
•1
I
a— c
i;
1
1
XXIII.
s
1
(1.) 12.
(2.) 12.
(3.) 24. 1
(6.)6A. !
1
(4.) 30.
(5.) 23i.
^^^1
(7.) 6.
(8.) 3.
(9.) 3.
" i
I
(10.) 9.
(11.) 4.
(12.) 5. '
I
(13.) 3i.
(14.) -2.
(15.) 33.
I
(16.) 2i.
(17.) 2.
(18.) 2.
1
(19.) 120.
(20.) 13.
1
XXIV.
{
1
(1.) 15 and 10.
(2.) 60 and 75.
(3.) 20 and 17.
1
(4.) 14 lbs.
(5.) 23, 17.
(6.) 181 and 145.
l2
148
ANSWERS,
(7.) 5. (8.) 14 years. (9.) In 9 years.
(10.) 18. (11.) 66 years. (12.) 400.
(13.) 700. (14.) 30 for translation,
5 for mathematics.
4 for Latin prose.
(15.) 21 shillings. (16.) A's £800, B's £100.
(17.) 400 inches. (18.) 18, 11 and 8.
(19.) 35. (20.) 200 quarters. (21.) 12 lbs.
(22.) 150 lbs. (23.) 240 sovereigns, (24.) 13.
480 shillings,
720 pence.
(25.) £3000 at 5 per cent., (26.) £450 at 4i per cent.
£10,000 at 4 per cent. £350 at 5i per cent.
(27.) 1800 infantry, (28.) 17 years.
600 artillery,
200 cavalry. (29.) 26. (30.) 12.
(31.) 56 workmen; 150 shillings. (32.) 1330.
(33.) 4290 feet. (34.) 30,000 men.
(35.) 200 miles from Edinburgh. (36.) In 56 hours.
XXV.
(1.) a;2-2i» + l, a;2 + 2aa; + a^ a;2-10a;+25, a;H6a; + 9.
(2.) 4a;H4a5 + l, 9a;2-.6a; + l, 4a;H12a; + 9, 9a;2-12a!H-4.
(3.) £c*--2aa;2 + a2^ 4xV + 4a;2/ + l, ^x^ -VHax^ ^-^a^, a'x''
-8a6x2 + 1662.
(4.) a;2 + 2/H2^—2iC2/ + 2x2-2^2, ix^ + dy^ + z^ + 12x1/ -4:xz
-62/z, x^+4ty^+2bz^'-ixy^lOxz-\-20yz, 4a;- + 16y + l-16a7/
+4a3— 8y.
(5.) 4a* + 4a3 + ISa^ + 6a + 9, 9a* - 24a» + 22a^- 8a + 1,
a*-.4a3-4aH16a + 16.
(6.) 2401, 9604, 990025.
XXVI.
(1.) ar^-l; a2-9;f4-a;2.
(2.) 4ar»-l; 25a2-4; lex^-a^
ANSWERS.
149
years.
) 12 lbs.
) 13.
)er cent.
»er cent.
.) 12.
.) 1330.
) men.
hours.
9.
H-4.
\xy—4:xz
1— 16ay
[•
(8.) a*-a^; a«-l; a»-a;*.
(4.) 9a*-462 ; 16a«-4a;< ; 49a8-25a«.
(5.) 2496 ; 9975 ; 489975.
xxvn.
(1.) mP-{-n^; p^—g^.
(2.) m^+l; 1-2'.
(3.) a;»+27; a^-QL
(4.) 8as + l; 64a;»-a3.
(5.) 8a»+276^; 27x3-1252^^
(6.) x^-l; x^^-a^
xxvni.
(1.) 03^—03 + 1; aj*— 03^+03'^— aj+1.
(2.) x^+x+1; x^+x^+x'^+x+1,
(3.) x-1; a;3-a!2+a;-l.
(4.) x + 1; x^+x^+x + l.
(5.) 2a-3&.
(6.) 3a;3+2a.
(7.) W-x\
(8.) 4a2— 6a5 + 9&2; a*-2a2+4.
(9.) 9a2 + 3ab + &2 ; 4,^4 ^ 5^2 j ^ 95a,
XXIX.
(1.) (-a)3, (2x)3, (ari/'^)^, {2a*hhy.
(2.) (2a-l)^ («-& + in (x3-l)2.
(3.) { (xy ] \ { (-2ay } \ { {^axf \ \ { {^aVf ]
(4.) { (a-hf ] ^ { (x'-iy ] ^ { {x-'-3x^2f ] \
(5.) (a^^)^ U-2x)M^ {{a'^hfW [(x-afW
{{x'-ax^Vf]K
(6.) {(-a)M^ {{x'^fW { (403.1)2 }», {(a^-a»)M
A3 I 2
XXX.
(1.) jbS 8< a;», 8ic», 81a;«.
(2.) aV, a^r^ oVy'.
\
150
ANSWERS.
^.
(3.) aWc\ a«6V, Sa'fe'c".
(4.) x^Y^z^y a'&'*c".
(5.) «H2a; + l, 4a;2-12a; + 9, a!*-10a;2+25, a;«-4aV+4a*.
(6.) a;*+4a;H10a;2 + 12a; + 9, a;*-6a!'+17a;2-24a; + 16,
4a;«~4a;»+a;*+20cc3-10xH25.
XXXI.
(1.) V^, Ja^\ V^^^. Va:2-3a;+4.
(2.) ^":::^«, ^"3^3, ^^^ir^, ^(a3-3a+4).
(3.) V^^x, V^^^-l, Vl/'^^l .
(4.) \/^2, V4^3^^1, A//y2(?^^^^iH3. ^
(5.) 4^"^, ^'^, y W, !p^ .
xxxn.
(1.) 2a'&, 5a;2/', 9a; V. (2.) 4aj + 5.
(3.) 6a;-3. (4.) l+3a^. (5.) x^h
(6.) a;-|. (7.) 2a;-J^. (8.) 2a;-32/.
(9.) a;2+2a;+l. (10.) jcHaj + l.
(11.) aj2-2a;i/+2/''. (12.) 2a;»-aj2-3a;+2.
(13.) 1+|-|'; remainder l"-!^,
(1.) ^l\
(5.) 7aV2/.
(9.) 4a6.
XXXIII.
(2.) 5a&. (3.) ^xy.
(6.) aftwv. (7.) 4a26.
(10.) baW. (11.) 26^
(4.) 3aaj.
(8.) 6xV.
(12.) 4ww;.
(1.) 05 + 3.
(4.) x+3.
XXXIV.
X-
(2.)
(5.) 2ir-5.
(3.) a;-3.
(6.) x-3.
ANSWERS.
iS«
(7.) (B2+10a;+25.
(10.) 05-2.
(13.) 2a3+5.
(16.) aj-3y.
(19.) a;+3y.
(1.) a!r(2a;+8).
(4.) a;-l.
(1.) a(aj— a).
(4.) aj+l.
(8.) a;2-5a;+6,.
(11.) a;-l
(14) a;-2.
(17.) a; +2/.
(20.) £c2-/.
XXXV.
(2.) 2(i»+3).
(5.) 3x-2.
XXXVI.
(2.) 2a(aj-8).
(5.) a; +2.
XXXVII.
(2.) I^^xhj.
(5.) 240a2&2c2«fcVM;2
(9.) cc2-9.
(12.) a;2_3.
(15.) a;2-2a; + l.
(18.) a;2+y.
(3.) x(a;-2).
(3.) a(a;-l).
(6.) cc-a
(3.) a^W.
(1.) 6a&scy.
(4) 24a26V.
(6.) (a;2-.7 + 12)(a; + 2) = (a;2-a;-6)(a;-4).
(7.) (2a;2-5a;-3) (2iK + 1) = (4*2 +4a; + 1) (a;-3).
(8.) (3a;2-lla;+6)(2x-l) = (2x'-^-7x+3)(3x-2).
(9.) (x^ - 4aa;2 + ^o?x - 2a») (x" f 2aa; + 2a2) = {x' - 2a2a;
.4a3)(aj2-2aa; + a2)- (10.) 24(a? + l)(a;-l)2.
(11.) (!r-l)n«^+l)'. (12.) xhfix'-y^),
(13.) a3&(«'-&')- (1^-) 12(a;=^-l)(a'Ha;+l).
(15.) ip^i-q'')(p'-q'Kp'-pq+f)'
(16.) (i>*-l)(i>*+pHl). (17.) (b^c)(c^a)(a~b),
(18.) 24a2&2(a2-62).
XXXVIII.
a
<^-) fc
(2.) -
cy
(4.)
2a&
a
:::&■
(^•) ^f^
362-6a«
(3.)
(6.)
[JB
3_
4^2'
x::!
152
ANSWERS,
bvw
I
(7.)
5+2*
(10.) J,^.
(13.)
(16.)
3+aj
3-a;'
3(4a;~ l)
2(3a;!*+l)
(8.)
(11.)
(14.)
(9.) _^±y,.
(12.)
(15.)
(17 ^ ^'(^+2y) ris^
g! + 3
05 — 5*
a;-3
cc+y'
a>. 6 2a
XXXIX.
(3.)?^^, 3?., 4
(5.) ::?- ,
y
a;?/^ xyz xyz
-2a 3
aajy axy axy
,«x a(a; + l ) 2a ^^^ a; + 3 2( g; + l)
^''•' a;2-l ' x^-r ^''^ (a; + l)(a; + 3) ' (x + 1) (x + 3) '
(9.)
a
(10.)
(11.)
X
JC — 1 * 03 — 1*
4(a;'^-l) 3a;(g;-l)
gi'-l g;'^-a; + l 3a;
g-l
a—b * a—b'
nS.^ 2(«^~2) 3(a;+2)
^ ^ (a;-l)(a;-2)(x+2) ' (a;-l)(£c-2)(a;+2) *
(13.)
a;-l
(a;-l)(a;-2)(cc + 2)*
a—c
(fc«c)(c-a)(a-fe) ' (b-rc)ic-d)(a-b) '
6— a
(6— c) (c— a) (a— &) *
.,.^ bx(x—b) ax(a—x)
> ^^ abx(a-b)lx-d)(x~b) ' a&a;(a-&)(a;-a)(c»-6) '
(q— &)(ag~a)(ag— &)
qhy(a-^h) (r —a)(.r~ h)
ANSWERS,
«S3
(1.)
(4.)
(6.)
(8.)
(11.)
(14.)
(17.)
(19.)
(21.)
'+6»
ah
3a;~-2
a;2~3a;+6
2x^ '
2a'' +26'
a»-5a *
a+h
a
V-1)
2a; + 2y
(2.)
XL.
aa;+2
2aa •
(3.)
4a2a; + 6a+6
12aa
(5.)
(7.)
8x«+12ag;- 1
66-17a
60
(9.)
(12.)
(15.)
4:db
~2
a;(4a;2-l)
(10.)
(13.)
1-
a
2a;»
a;* + a;2 + l"
a5-2y
(16.)
2x + 6
a;*-l'
x^-)rxy-^y^
X
-9
(aj-l)(a;-2)(aJ+2)
(18.)
(20.)
2a;(2a;Hl)
£»'
«-.l
a^
(«—•&)(«— c)
(a+6+c)aj
(a; — a) (a; — 6) (as — c)
(22.) 0,
(23.) 0.
a.)
(4.)
(6.)
(8.)
(10.)
(12.)
3+5cc
a+aic— 05
2xHg;~l
^^ •
^-«2+2a;-3
^^
g— •a'-f a&
a— 6
4a;g+9a;-4
(c + S •
(2.)
XLI.
cc— a'^
a'
(5.)
(7.)
(3.)
3a— aa;+a3
as
2x2 + 1
2a— OS
a
OS
2 •
^n\ l + 3a?
a+36
a + 6 '
x^—x y
(11.)
(13.)
«S4
ANSWERS.
*■
S;i:
(14.) S+|.
a
0.7.) 2aj-l+?.
a;
3a;+5
(19.) 14
a;2-l'
(21.) 4-j-^^.
(23.) 8- ^"^
aaj^-oj+l'
(1.) ^.
(4.) ^y
(7.)
(9.)
(a+6)2 •
1
(11.) 2/»+2aj+ajy.
(13.) ^.
(17.)
oj+y'
(4.) «:::^.
(15.) a-±.
a;
(16.) ^-1.
(18.) ^-1+2..
(20.) 5--i-^.
(22.) .-3+1=7,
(24.) a;2-
XLH.
(2.) 1.
a+b
x^+x—5
x^+l '
(5.)
6 •
(6.)
x^yz"
^±abVb^
a^-ab + b^'
(10.) ,4^Y2-
(a;'*— 05 + 1/
(12.) aj-a.
(14.) -^ .
(16.) 1.
2
(a~6)
(7.)-
a+a?
(2.)
(5.)
(8.)
(18.)
XLUI.
4ax^y
a^+ab + b^
a^^ab + b^'
2
(a^br
r
(3.)
(6.)
(9.)
x+r
a
2ap '
ANSH^EKS.
m
(10.)
(12.)
ax
(11.) ^^rf.
(1.) 10.
(4.) 6.
(7.) -^
(10.) 20.
(13.)f|.
(16.) f .
(19.) a+&.
a+6 '
(22.)
(13.) M.
XLIV.
(2.) 8.
(5.) 8.
(8.) -1.
(11.) V.
(14.) -i
(17.) i^.
(20.) a.
XLV.
(2.) 40.
(14.) 1^.
(3.) 12.
(6.) 2.
(9.) -f.
(12.) 15.
(15.) 3.
a+6 *
(18.) 2.
(^^•) 4-
(1.) 40. (2.) 40. (3.) 355.
(4.) 4 bowled, 3 run out. (5.) ^■^. (6.) £5.
(7.) £9150 of 3 per cents., £5820 of 3i per cents.
(8.) A £1250, B £1500. (9.) 100.
(10.) 7 miles an hour. (11.) 45 and 30 miles an hour.
(12.) C 42 days, B 84 days, A 168 days.
(13.) 240, 180, 144 days. (14.) 5^^ minutes past 7.
(15.) 26^ minutes past 2. (16.) 49^^ minutes past 3.
(17.) 32^^ minutes past 3.
(19.) 30 gallons.
(21.) 70 grains.
(23.) 3 miles an hour.
(25.) 10 yards in 200.
(27.) Imile; half-an-hour.
(29.) ^6 of a mile behind.
(18.) 10 gallons.
(20.) 112 oz.
(22.) 21 oz.
(24.) 4i miles.
(26.) 1430 yards.
(28.) 5f .
156
ANSWERS.
mk
a.) ^ -6.
(4.) 4, -4.
(7.) 0, 3.
(10.) 0, f.
(13.) 3, 5.
(16.) i, f.
(19.) ^, -1
(1.) 2, 4.
(4) 2, h.
(J.) h -f.
(10.) I, f .
(13.) 2, «^.
(16.) -1, -I-.
(19.) 1, 2.
(21.) 6, f .
XLVL
(2.) 3, -3.
(5.) 2, -2.
(8.) 0, -12.
(11.) 0, -^\.
(14.) -5,7.
(17.) h -h
(20.) 0, a,
XLVn.
(2.) 5, -1.
(5.) h -2.
(8.) i |.
(11.) 2, i.
(14.) 14, -10.
(17.) 4, 4.
(20.) 3, -|.
(22.) a, 6.
(3.) 9, -9.
(6.) V26, -V26.
(9.) 0, 19.
(12.) 0, -|.
(15.) -1,-3.
(18.) -f, -h
(3.) 3, -7.
(6.) h h
(9.) 6, -4.
(12.) V, -10.
(15.) 4, -1.
(18.) 1, V.
XLVIII,
(1.) 60 ft. by 30 ft. (2.) 13 and 7.
(3.) 2 or 10. (4.) 12 and 3.
(5.) 4 or 6. (6.) 5, 4, 3 ; or -1, 0, 1.
(7.) 30 ft. (8.) £20. (9.) £90 or £10.
(10.) A's rate 4 miles ; B's 2^ nail^s an hour.
(11.) |. (12.) 10. (13.) 87.
(14.) 2 ft. (15.) 11. (16.) 23.
(17.) 12. (18.) 56. (19.) 1296.
ANSWERS.
157
(1.) 2, 3.
(4) 6, 12.
(7.) 4, -1.
(10.) 2, 3.
(13.) 13, 3.
(16.) a, f .
XLIX.
(2.) 3, 5.
(5.) 30, 12.
(8.) 3, 5.
(11.) 9, 4.
(11) 4, 1.
aib--aG
(3.) 6, 4.
(6.) 4, 3.
(9.) 2, 3.
(12.) 60, 40.
nx!^\ a+& a—h
^^^'^ "2"' "2"*
(17.)
ah— he
b—a * a—h
L.
(1.) 2, 1, 3. (2.) 1, 3, 2.
(4.) 3, 4, -3. (5.) 3, 2, 1.
(7.) 4, 3, 2. (8.) 19, 7, 4.
nn^ ^+c— a c+a— & a+6--c
y.^^-) — o — * — o — » o •
(3.) 5, 4, 3.
(6.) 4, 5, 6.
(9.) 2, 4, 7.
LI.
(1.) 72. (2.) 378 and 216.
(3.) 5 and 4. (4.) V (5) 84.
(6.) A to B 37 miles, B to C 45 miles, C to A 52 miles.
(7.) 19 five-franc pieces, 11 two-franc pieces.
(8.) 296 for A, 1488 for B, 198 for C.
(9.) The stream 3 miles an hour; the boat 8 miles an
hour.
(10.) 148 for A, 750 for C and A, 158 for A and B.
(11.) The gold coins are half-sovereigns, the silver coins
are crowns.
(12.) A to B Hi miles, B to C 7 miles, C to D 5| miles.
(13.) 24 octavos or 32 duodecimos.
(14.) A thaler = 2s. lid. ; a franc = W. ; a florin =
Is. lUd.
158
ANSWERS.
ll
,«.r
m
(15.) A Prussian pound = 16^ oz. ; an Austrian pound =
191 oz. ; a kilogramme = 35i oz.
(16.) 3i miles.
(1.: '/a, «/a, ^a\ ^a^
(3.)
Ml.
(2.)
L L L.
^m V»i° tIP'
(4.) 2^ a
- 3
cc-
./wi^
1. 1. 1
(5.) x^, m^, n
(7.)
a. 3L i
a;2 ass gjt
(6.)
(8.)
x~\ a~\ a'\ a
-8
-1 -i.
X
-3
(9.) 2m-i, 3»-2, lOp
(1.) 6a;"+*; 4a;'»+2; 4a;*».
a;
-1
_a
(10.) 2«"2, 5a;"3, 7a;~»
Lin.
(2.) 2.x* ; 6x6 . 30a;
:l
2n+l
3n
fin
(3.) jc 2 ; 6a;=^ ; a;«
(5.) a6; 2a"'^; o^.
(7.) 1; 1; 6; wiw.
(4) 2a; Ba'^; 30a
-1
(6.) a^; a^; a\
(1.) a"; a2«.
(4.) a? ; (B^
Sn
(7.) as"; a;2.
12. «8
a': a
(1.) a
(3.) a . .
(5.) a; a^; a*.
LIV.
(2.) a»; ai
(6.) a?; aj^.
LV.
(!2.) a-=
(3.) x^; X.
(6.) (c; (B*.
a
-6. „-12
-«»• a«: ai2.
a=« ; a"; a
9. „io
(4.)
(6.) a3 ; as ; a^.
id =