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Full text of "Elementary Algebra / with appendix by Alfred Baker"

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DOMATED BY 



m. ^. mqe & m'» ^§\mmmai ^tvu^. 



Elementary Algebra, 



J. HAMBLIN SMITH, M.A., 

OF GONYILLE AND CAIUS COLLEGE, AND LATE LECTUBEB 
AT ST. PETEE's college. CAMBEIDQB. 

WITH APPENDIX BY 

ALFRED BAKER, B.A.. 

MATH. TUTOR CNIV. COL. TORONTO. 

Sth CANADIAN COFYKIGHT EDITION. 
NEW BEVISED EDITION. 

Authorized by the Education Department, Ontario. 
Authorized by tlie Council of Public Ittstruction. Quebec 
Recotnmended by the Senate of the Univ. of Halifax. 



PRICE, 90 CJLIMTS. 



TOKONTO: 

^. J. GAGE & CO. 



Entered according to thf Act o/ the Parliament of the Dominion of Canada, 
in the year one thousand eight hundred and seventy-seven, by Ada* 
MiLXER & Co. , in the Office of the Minister of Agriculture. 



PREFACE 

The design of this Treatise is to explain all that is 
commonly included in a First Part of Algebra. In the 
arrangement of the Chapters I have followed the advice 
of experienced Teachers. I have carefully abstained from 
making extracts from books in common use. The only 
work to which I am indebted for any material assistance 
is the Algebra of the late Dean Peacock, which I took as 
the model for the commencement of my Treatise. The 
Examples, progressive and easy, have been selected from 
University and College Examination Papers and from 
old English, French, and German works. Much care has 
been taken to secure accuracy in the Answers, but in a 
collection of more than 2.300 Examples it is to be feared 
that some errors have yet to be detected. I shall be 
grateful for having my attention called to them. 

I have published a book of Miscellaneous Exercises 
adapted to this work and arranged in a progressive order 
so as to supply constant practice for the student. 

I have to express my thanks for the encouragement 

and advice received by me from many correspondents ; 

and a special acknowledgment is due from me to Mr. E. 

J. Gross of Gonville and Caius College, to whom I am 

ndebted for assistance in many parts of this work. 

Tlie Treatise on Algebra by .^Ir. E. J. Gross is a 
continuation of this work, and is in some important 
points supplementary to it. 

J. HAMBLIXSJ^IITH. 

Cambridge, 1871. 



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in 2009 witli funding from 

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CONTENTS. 



CHAP. PAGE 

I. Addition and Subtraction i 

II. jMultiplication 17 

III. Involution 29 

IV. Division 33 

V. On the Resolution of Expressions into Factors . 43 

VI. On Simple Equations 57 

VII. Problems leading to Simple Equations . . 6i 
VIII. On the Method of finding the Highest Common 

Factor 67 

IX. Fractions . 76 

X. The Lowest Common Multiple .... 88 

XI. On Addition and Subtraction of Fractions . 94 

XII. On Fractional Equations 105 

XIII. Problems in Fractional Equations . . .114 

XIV. On Miscellaneous Fractions 126 

XV. Simultaneous Equations of the First Degree . 142 

XVI. Problems resulting in Simultaneous Equations 154 

XVII. On Square Root • . . .163 

XVIII. On Cube Root 169 

XIX. QUADR.A.HC Equations 174 

XX. On Simultaneous Equations INVOLVING Q'"*nKATics 186 

XXI. On Problems resulting in Quadratic Equations . 192 

XXII. Indeterminate Equations 196 

XXIIi. The Theory of Indices 201 

XXIV. On Surds 213 

XXV. Oy Equations involving Surds . . . .229 



CONTENTS. 



CHAP. PAGE 

XXVI. On the Roots of Equations .... 234 

\XVII. On Ratio 243 

XXVIII. On Proportion 248 

XXIX. On Variation 258 

XXX. On Arithmetical Progression .... 264 

XXXI. Om Geometrical Progression ' . . . . 273 

XXXII. On Harmonical Progression .... 282 

XXXIII. Permutations 28" 

XXXIV. Combinations 291 

XXXV. The Binomial Theorem. Positive Integrai 

Index 296 

XXXVI. The Binomial Theorem. Fractional and 

Negative Indices 307 

XXXVII. Scales of Notation 316 

XXXVIII. On Logarithms 328 

Appendix 344 

\nswers . • . . . 345 



1 



ELEMENTARY ALGEBRA. 



I. ADDITION AND SUBTRACTION. 

1. Algebra is the science whicli teaches the use of sym- 
bols to denote numbers and the operations to which numbers 
may be subjected. 

2. The symbols employed in Algebra to denote numbers 
are, in addition to those of Arithmetic, the letters of some 
alphabet. 

Thus a, b, c x, y, z : a, )3, y : a', b', c' read 

a dash, b dash, c dash : a-^, b^, c^ read a o«e, 

b one, c one are used as symbols to denote numbers. 

3. The number o?ie, or unity, is taken as the foundation of 
all numbers, and all other numbers are derived from it by the 
process of addition. 

Thus two is defined to be the number that results from 
adding one to one ; 
three is defined to be the number that results from 

adding one to two ; 
four is defined to be the number that results from 
adding one to three ; 
and so on. 

4. The symbol +, read plus, is used to denote the opera- 
tion of Addition. 

Thus 1 + 1 symbolizes that which is denoted by 2, 

2 + 1 3, 

and a + b stands for the result obtained by adding b to a. 

5. The symbol = stands for the words " is e(iual to," or 
•' the result is." 

[S-A.] . ^ 



ADDITION AND SUBTRACTION. 



Thus the definitions given in Art. 3 may be presented in an 
algebraical form thus : 

1 + 1=2, 

2 + 1 = 3, 

3 + 1=4. 

6. Since 

2 = 1 + 1, M'here unity is written twice^ 

3 = 2 + 1 = 1 + 1 + 1, where unity is written thrte times, 

4 = 3 + 1 = 1 + 1 + 1 + 1 pur times, 

it follows that 

a = l + l + l +1 + 1 with iinity written a times, 

6 = 1 + 1 + 1 +1 + 1 with unity written h times. 

7. The process of addition in Arithmetic can be presented 
in a shorter form by the use of the sign + . Tlius if we have 
to add 14, 17, and 23 together we can represent tlie process 
thus : 

14 + 17 + 23 = 54. 

8. When several numbers are added together, it is indiffe- 
rent in what order the numbers are taken. Thus if 14, 17, and 
23 be added together, tlieir sum will be the same in w-hatever 
order they be set down in the common arithmetical process : 

14 14 17 17 23 23 

17 23 14 23 14 17 

23 17 23 14 17 14 

54 54 54 54 54 54 

So also in Algebra, when any number of symbols are added 
together, the result will be tlie same in whatever order the 
symbols succeed each other. Thus if we have to add together 
the numbers symbolized by a and b, the result is represented 
by a + 6, and this result is the same number as that which is 
represented by b + a. 

Similarly the result obtained by adding together a, b, c 
might be expressed algebraically by 

a + b + c, or o + c + 6, or b + a + c, or b + c + a, or c + o + 6, 
or c + b + a. 

9. When a number denoted by a is added to itself tlie 
result is represented algebraically by a + a, This result is for 



ADDITIO.y AXD SCBTRACTTO.V. 



the sake of brevity represented by 2a, the figure prefixed to 
the symbol expressing the number of times the number 
denoted by a is repeated. 

Similarly a + a + a is represented by 3a. 
Hence it follows that 

2a + a will be represented by 3a, 
3a + a by 4a. 

10. The symbol — , read minus, is used to denote the ope- 
ration of Subtraction. 

Thus the operation of subtracting 15 from 26 and its con- 
nection with the result may be briefly expressed thus ; 
26-15 = 11. 

11. The result of subtracting the number h from the num- 
ber a is represented by 

a-h. 

Again a — h — c stands for the number obtained by taking c 
from a — b. 

Also a — b — c — d stands for the number obtained by taking 
d from a — b — c. 

Since we cannot take away a greater number from a smaller, 
the expression a — b, where a and b represent numbers, can 
denote a possible result only when a is not less than b. 

So also the expression a-b — c can d'enote a possible result 
only when the number obtained by taking b from a is not 
less than c. 

12. A combination of symliols is termed an algebraical 
expression. 

The parts of an expression which are connected by the 
symbols of operation + and — are called Terms. 

Compound expressions are those which have more than one 
term. 

Thus a-b + c — d is a compound expression ■\nade up of four 
terms. 

When a compound expression contains 

hvo terms it is called a Binomial, 

three Trinomial, 

four or more Multinomial. 



ADDITION AND SUBTRACTION. 



Terms which are ju'eceded by the symbol + are called posi- 
tive terms. Terms which are preceded by the symb.>l — are 
called negative terms. When no symbol precede.s a t<'rui the 
symbol + is understood. 

Thus in the expression n -h + c-d + e -f 

a, c, e are called positive term.s, 

b, d,f negative 

The symbols of operation + and — are usually called posi 
tive and negative Signs. 

13. If the number 6 be added to the number 13, and if 
be taken from the result, the final result will plainly be 13. 

So also if a number b be added to a number a, and if b he 
taken from the result, the final result will be a : that is, 
a + b-b = a. 

Since the operations of addition and subtraction when per- 
formed by the same number neutralize each other, we conclude 
that we may obliterate the same symbol when it presents itself 
as a positive term and also as a negative term in the s;ime ex- 
pression. 

Thus a-a = 0, 

and a-a + b = b. 

14. If we have to add the numbers 54, 17, and 2? we may 
first add 17 and 23, and* add their sum 40 to the number 54, 
thus obtaining the final result 94. This process may be repre- 
sented algebraically by enclosing 17 and 23 in a Bracket 
( ), thus : 

54 + (l7 + 23) = 54 + 40 = 94. 

15. If we have to subtract from 54 the .sum of 17 and 23. 
the process may be represented algebraically thus : 

54 - (17 -H 23) = 54 - 40 = 14. 

16. If we have to add to 54 the difference betweon -23 ar'A 
17, the process may be represented algebraically thu!<: 

54 4- (23 -17) =54-}- 6 = 60. 

17. If we have to subtract from 54 the difference between 
23 and 17, the process may be represented algebraically ihus : 

54-(23-17) = 54-6 = 48. 



ADDITION AND SUBTRACTION. 



18. The use of brackt . is so frequent in Algebra, that 
the rales for their removal and introduction must be carefully 
considered. 

We shall first tree^t of the removal of brackets in cases 
where symbols supply the places of numbers corresponding to 
the arithmetical examples considered iVts. 14, 15, 16, 17. 

Cd«e I. To add to a the sum of b and c. 
3 is expressed thus : a + {b + c). 
a. irst add b to a, tlie result will be 
a + b. 
This result is too small, for we have to add to a a numV-ir 
/•eater than b, and greater by c. Hence our final result wili 
oe obtained by adding c to a + 6, and it will be 
a + b + c. 

Case II. To take from a the sum of b and»c. 
This is expressed thus : a — {b + c). 
First take b from a, the result will be 
a — b. 
'i his result is too large, for we have to take from a a number 
greater than b, and greater by c. Hence our final result will 
be obtained by taking c from a — b, and it will be 
a — b — c. 

Case III. To add to a the difference between b and c. 
This is expressed thus : a + {b — c). 
First add b to a, the result will be 
a + b. 
This result is too large, for we have to add to a a number 
less than b, and less by c. Hence our final result will be ob- 
tJiined by taking c from a + b, and it will be 
a + b — c. 

Case IV. To take from a the difference between b and c. 
This is expressed thus : a — {b~ c). 
First take b from a, the result will be 
a — b. 
This result is too small, for we have to take from a a num- 
ber less than b, and less by c. Hence our final result will be 
obtained by adding c to a — b, and it will be 
a~b + c. 



ADDITION AND SUBTRACTION. 



Note. We assume that a, b, c represent such numbers that 
in Case II. a is not less than the sum of b and c, in Case III. 
b is not less than c, and in Case IV. b is not less than c, and a 
is not less than b. 

19. Colk-cting the results obtained in Art. 18, we have 

a + {b + c) = a + b + c, 
a — (b + c) — a — b — c, 
a+ {b — c) = a + b-c, 
a — {b — c) = a — b + c. 

From which we obtain the following rules for the removal of 

a bracket. 

Rule I. "Wlien a bracket is preceded by the sign +, 
remove the bracket and leave the signs of the terms in it 
imchmiged. 

Rule II. When a bracket is preceded by the sign — , 

remove the bracket and change the sign of each term in it. 

These rules apply to cases in which any number of terms 
are included in the bracket. 

Thus 

a + b + {c-d + e -/) = a + b + c-d + e -f, 
and 

a + b- {c-d\-e—f) = a + b-c + d — e+f. 

20. The rules given in the preceding Article for the rt- 
moval of brackets turuish corresponding rules for the intro- 
uuction of l)rackets. 

Thus if we enclose two or more terms of an expression in a 
bracket, 

T. The sign of each term remains the same if + pre- 
cedes the bracket : 

II. The sign of each term is changed if — precedes the 
bracket. 

Ex. a-b + c-d + e-f=a-b + {c — d) + {e-f\ 
a-h-tc-d + e-/=a-{b-c)-{d -e+f). 



ADDITION AND SUB TRA C TTON. 



21. We may now proceed to give rules for the Addition 
and Subtraction of algebraical expressions. 

Suppose we have to aM. to the expression a + b — c the ex- 
pression d — e +f. 

The Sum =a + h-c + {d-e+f) 

= a + b-c + d-e+f (by Art. 19, Rule I.). 

Also, if we have to subtract from the expression a + b — c the 
expression d~e +f. 

The Difference =a + b-c-{d-e +/) 

= a + b-c-d + e-f{hj Art. 19, Rule II.). 

We might arrange tlie expressions in each case under each 
other as in Arithmetic : thus 

To a + b — c From a + b — c 

Add d-e+f Take d-e+f 

Sum a + b — c + d — e +f Difference a + b — c — d + e —f 
and then the rules may be thus stated. 

I. In Addition attach the lower line to the uyper with the 
signs of both lines unchanged. 

II. In Subtraction attach the lower line to the upper with 
the signs of the lower line changed, the signs of the upper line 
bein" unchanged. 



The following are examples. 

(1) Toa + 6 + 9 

Add a-b-Q 



Sum a + b + Q + a-b-Q 

and this sum =a + a + 6-6 + 9 — 6 
= 2& + 3. 

For it has been shown, Art. 9, that a + a = 2a, 
and, Art. 13, that 6-6 = 0. 

(2) From a + 6 + 9 

Takea-6-6 



Remainder a + h + 9 — a + h + Q 
and this remainder = 26 + 15. 



8 ADDITION AND SUBTRACTION. 

22. We have worked out the examples in Art. 21 at full 
leni.fth, hut ill practice they may he ahhreviated, by combining 
the symbols or digits by a mental process, thus 

Toc + (Z + 10 rromc + (^ + 10 

Addc-d-7 Takec-rf-7 



Sum 2c +3 ilemainder 2d + 17 

23. We have said that 

instead of a + a we write 2a, 
a + a + a 3a, 

and so on. 

The digit thus prefixed to a symbol is called the coefficient 
of the term in which it appears. 

24. Since 3a = a + a + a, 

and 5a = a + a + a + a + a, 

Sa + 5a = a + a + a + a + a + a + a + a 

= 8a. 

Terms which have the same symbol, whatever their coefli- 
cients may be, are called like terms : those which have diffe- 
rent symbols are called unlike terms. 

Like terms, when positive, may be combined into one by 
adding' their coefficients together and subjoining the common 
symbol : thus 

2a; + 5ic = 7x, 

Zy + by + 8y = 16y. 

25. If a term appears without a coefficient, unity is to be 

taken as its coefficient. 

Thus .*■ + 5x = 6a;. 

26. Negative terms, when like, may be combined into one 
t(;rm with a negative sign prefixed to it by adding the coeffi- 
cients and subjoining to the result the common symbol. 

Thus 2x-3i/-5i/ = 2x-8i/, 

lor 2x~'3y-by = 2x — (Sy + 5y) 
= 2x-8y. 

So again 3x-y-iy-Gij = 3.f -lly. 



ADDITIOPT AMD SUBTRACTION: 



27. If an expression contain two or more like terms, some 
being positive and others negative, we mual first collect all the 
positive terms into one positive term, then all the negative 
terms into one negative term, and finally combine the two 
remaining terms into one by the following process. Subtract 
the smaller coefficient from the greater, and set down the 
remainder with the sign of the greater prefixed and the com- 
mon symbol attached to it. 

Ex. 8a;-3a; = 5x, 

7x — 4x + 5a; — 3a; = 1 2aj — 7a; = 5x, 
a-26 + 56-4& = a + 56-66 = a-6. 

28. The rules for the combination of any number of like 
terms into one single term enable us to extend the application 
of the rules for Addition and Subtraction in A]g>3bra, and we 
proceed to give some Examples. 

ADDITION. 

(1) a -26 + 3c (2) 5a + 76-3c-4(£ 

3a -46 -5c 6a-76 + 9c + 4ci 



4a-66-2c 11a +6c 

The terms containing 6 and d in Ex. (2} destroying one another. 

(3) 7x-5)/+ 43 (4) 6m--13?i + 5p 

x + 'iy-Wz 8m+ n — ^ 

Zx~ y-{- bz m— n— p 

bx~'iy— z m+ 2n + bp 

IGx — ly- 3z 16m- 1 In 

SUBTRACTION. 

(1) 5a -36+ 6c (2; 3a + 76- So 

2a + 56- 4c 3a -76+ 4c 



(3) 


3a -86 + 10c 
5a -66 + 2c 
2a-66 + 2c 


(5) 


3a 

3a; + 7y + 122! 
by- 2z 





146-13-5 


(4) 


x — y + z 
x-y-z 




2z 


(6) 


7x-l9y-14z 
6x-24j/+ 9a 



Zx + 2y + 14z x+ by-2Zz 



ib ADDITION AND SUBTRACTION. 



29. We have placed the expressions in the examples given 
in the preceding Article under each other, as in Arithmetic, 
for the sake of clearness, but the same o]ierations might be ex- 
hibited by means of signs and brackets, thus Examples (2) of 
each rule might have been worked thus, in Addition, 
5a + 76 - 3c - 4rf + (6a - 76 + 9c + 4fO 
= 5a + 76-3c-4(i+ 6a - 76 + 9c + 4i 
= lla + 6c; 
and, in Subtraction, 

3a + 76-8c-(3a-76 + 4c) 
= 3a + 76-8c-3a + 76-4c 
= 146 -12c. 

Examples.— i. 

Simplify the following expressions, by combining like sym- 
bols in each. 

I. 3a + 46 + 5c + 2a + 36 + 7c. 2. 4a + 56 + 6c -3a -26 -4c. 

3. 6a -36 -4c -4a + 56 + 6c. 

4. 8a - 56 + 3c - 7a - 26 + 6c - 3a + 96 - 7c + 10a. 

5. 5a;-3a + fe + 7 + 26-3x-4a-9. 
, 6. a — 6 — c + 6 + c-d + f7-a. 

7. 5a + 106-3c + 26-3a + 2c-2a + 4c. 

EXAMPLES.— ii. ADDITION. 
Add together 

I. a + xanda-a;. 2. a + 2x and a + 3a;. 

3. a - 2x and 2a - x. 4. 3x + 1y and 5x - 2i/. 
5. a + 36 + 5c and 3a - 26 - 3c. 

a-26 + 3c and a + 26-3c. 7. 1 +a;-^ and 3-x + i/. 

2x - 3i/ + 4a, hx-^y- 2;:, and 6,x + 9)/ - 82. 

2a + 6 - 3x, 3a - 26 + x, a + 6 - 5x, and 4a - 76 + 6x. 

Examples.— iii. SUBTRACTION. 

I . From a + 6 take a - 6. 

2 3x + i/ 2x — 1/. 

3 2a + 3c + 4(i a-2c + 3i. 

4. X + 2/ + 3 x-y-z. 



ADDITION A ND SUB TRA CTION'. 1 1 

5 . From m — n + r take m — n — r. 

6 a + b + c a — b — c. 

7 3a + 46 + 5c 2a + 7b + 6c. 

8 3x + 5ij-4z 3x + 2y-5z. 

30. AVe have given examples ol' the use of a bracket. The 
methods of denoting a bracket are various ; thv.s, besides the 
marks ( ), the marks [ ], or j j, are often employed. Some- 
times a mark called "The Vinculum" is drawn over the symbols 
which are to be connected, thus « - 6 + c is used to rejiresent the 
same expression as that represented by a — (b + c). 

Often the brackets are made to enclose one another, thus 

a-[b+\c~{d-e-f)\]. 
In removing the brackets from an expression of this kind it 
is best to commence with the innermost, and to iemove the 
brackets one by one, the outermost last of all. 

Thu8 

a-[b+\c-(d-e-f)\] 
= a-[b+\c-{d-e+f)\] 
= a-[b+ \c~d + e-f\] 
= a-[b + c-d + e-f] 

= a — b — c + d — e +/. 
Again 

5x-(3x-7)- l4-2.c-(6x-3)f 

= 5x-3x + 7- |4-2ic-6x + 3j 

= 5x-3.o + 7-4 + 2x + 6a;-3 

= 10x. 

Examples.— iv. beackets. 

Simplify the following expressions, combining all like quan- 
tities in each. 

1. a + 6 + (3a-26). 

2. a + b-{a~3b). 

3. 3a + 56 -6c -(2a + 46 -2c). 

4. a + 6 — c — (a — 6 — c). 

5. 14x-(5x-9)- j4-3,r-(2x-3}/. 

6. 4x- {3x-(2x-x-a){. 

7. 15x- j7x + (3x + a^){. 



12 ADDITION AND SUBTRACTlOh. 

8. a.-[6+ja-(6 + a)[]. 

9. 6« + [4a- j86-(2a + 46)-22&}-76]-[76 h{8a 

-(36 + 4a) + 86}+ 6a]. 

10. 6-[6-(a + 6)-j6-(6-^6)j]. 

11. 2(^-(6a-6)- }c-(5a + 26)-(a-36)}. 

12. 2^- ja-(2a-[3a-(4a-[5a-(6a-x)J)])}. 

13. 25a- 196 -[36 -1 4a -(56 -6c)}]. 

31. We liave liitherto supposed tlie syml'ols in every ex- 
pression iised for illustration to represent syich numbers that 
1^'A expressions symbolize results whicli wou Id be arithmetic- 
j,lly possible. 

Thus a — 6 symbolizes a possible result, so long as a is not 
less than 6. 

If, for instance, a stands for 10 and 6 for 6, 

a — 6 will stand for 4. 
But if a stands for 6 and 6 for 10, 

a — 6 denotes no possible result, because we cannot 
take the number 10 from the number 6. 

But though there can be no such a thini,- as a negative 
number, we can conceive the real existence of a negative 
quantity. 

To explain this we must consider 

I. What we mean by Quantity. 
II. How Quantities are measured. 

32. A Quantity is anything which may be regarded as 
being made up of parts like the whole. 

Thus a distance is a quantity, because we may regard it as 
made up of parts each of themselves a distance. 

Again a sum of money is a quardity, because we mav regard 
it as made uji of parts like the whole. 

33. To measure any quantity we fix upon some known 
quantity of the same kind for our standard, or unit, and then 
any quantity of that kind is measured by saying how many 
times it contains this unit, and this number of times is called 
the measure of the quantity. 



ADDITION AND SUBTRACTION. 



For example, to measure any distaiice a]ohg a road we fix 
lip. in a known distance, such as a mile, and express all distances 
by saying how many times they contain this unit. Thus 16 is 
the measure of a distance containing 16 miles. 

Again, to measure a man's income we take one pound as our 
unit, and thus if we said (as we often do say) that a i .an's in- 
con le is 50(T a year, we should mean 500 times the unit, that is, 
£5<'0. Unless we knew what the unit was, to say that a man's 
inc. -me was 500 would convey no definite meaning : all we 
shoald know would be that, whatever our unit was, a pound, a 
dollar, or a franc, the man's income would be 500 times that 
unit, that is, £500, 500 dollars, or 500 francs. 

IN.B. Since the unit contains itself once, its measure is 
unity, and hence its name. 

;'4. Now we can conceive a quantity to be such that wheji 
piu to another quantity of the same kind it will entirely or in 
p:ii't neutralize its eff"ect. 

Thus, if I walk 4 miles towards a certain object and then 
rt turn along the same road 2 miles, I may say that the latter 
distance is such a quantity that it neutralizes part of my first 
j.iurney, so far as regards my position with respect to the point 
from which I started. 

Again, if I gain £500 in trade and then lose £400, I may 
say that the latter sum is such a quantity that it neutralizes 
liart of my first gain. 

If I gain £500 and then lose £700, 1 may say that the latter 
sum is such a quantity that it neutralizes all my first gain, and 
not only that, but also a quantity of which the absolute value 
Is £200 remains in readiness to neutralize some future gain. 
llegardii:g this £200 by itself we call it a quantity which will 
have a subtradive effect on subsequent profits. 

Now, since Algebra is intended to deal with such questions 
in a general way, and to teach us how to put quantities, alike 
• iT opposite in their effect, together, a convention is adopted, 
I ounded on the additive or subtractive effect of the quantities 
in question, and stated thus : 

"To the quantities to be added prefix the sign +, and to 
the quantities to be subtracted prefix the sign — , and then 
>vrite down all the quantities involved in such a question con- 
nected with these siKUS," 



14 A DDITION A ND SUB TRA C TTON.^ 

Thus, suppose a man to trade ibr 4 years, and to gain a 
pounds the tirst year, to k)se 6 pounds the secoii<l year, tn ga!ii 
c pounds the third year, and to lose ti pounds the iuurth year. 

The additive quantities are here a and c, which we are to 
write +a and +c. 

The suhtractive (juantities are here h ;:nd d, whidi we are to 
write — h and — d, 

:. Eesult of trading — ■\-a — h-\-c — d. 

35. Let us next take the case in which the gain for tlie 
first year is a pounds, and the loss lor eacli ot three subsequent 
years is a pounds. 

Eesult of trading = +a-a-a-a 

= - 2a. 

■ Thus we arrive at an isolated quantity of a subtractivi 
nature. 

Arithmetically we interpret this result as a loss of £2a. 

Algebraically we call the result a negative quantity. 

When once we have admitted the possibility of the inde- 
pendent existence of such quantities as this Ave may extend the 
application of the rules for Addition and Subtraction, for 

I. A negative quantity may stand by itself, and we may 
then add it to or take it from some other quantity or expres- 
sion. 

II. A negative quantity may stand first in an expression 
which we may have to add to or subtract from any other 
expression. 

The Rules for Addition and Subtraction given in Art. 21 
will be applicable to these expressions, as in the following 
Examples. 

ADDITION. 

(1) 5rt - 7ft = — 'la. 

(2) 4a-36-6a-f-76=-2rt-i-46. 

(3) To 4a To 5n-3S 
Add -3a Add -2a -26 

Sum a Sum 3a -56 



ADD/T/OX AND SUBTRACTION: 15 

(4) 6ffl-56- 4c + 6 (5) Ix-by + Qz 

-5a + 76-12c-17 -ISx + 'jy-bz 

- a- 86 + 19c + 4 - Zx-S]j+ z 



-66+ 3c- 7 -l4x-42/ + 5» 

SUBTRACTION. 

(1) Fiom X 
Take ^^y 

Remainder x + y 

or we might represent the operation thus, 

(2) a + 6-(-o + 6) = a + 6 + a-6 = 2a. 

(3) —a -b-{a-b)= -a-b-a + h= -2a. 

(4) -3a+ 46- 7c + 10 

5a- 9^+ 8c +19 



-8a + 136-15c- 9 



(5) x-y-{;3x-\-5x-(-4:y + 7x)i] 
= x — y~ [3x — \ —5x + 4y — Ix {■ ] 
=x — y — [3a; + ox - 4?/ + lx\ 

=x — y — Zx — bx + 4y~7x 
= -14x + 3i/. 

(6) 7a + 56+ 9c-12i 
-36- 12c- 8d+ 6e 



7a+ 86 + 21C- 4d-6e 



In this example we have deviated from our previous prac- 
tice of placing like terms under each other. This arrange- 
ment is useful to facilitate the calciiiation. but is not absolutelv 
necessary ; for the terms which are alike can be combined 
independently of it. 

* NoTE. — The meanhig of Subtraction is liere extm !eil so that 
the result in Art. 18, Case iv. may he true when b is less than C. 



I6 ADDITION AND SUBTRACTION. 



Examples. — v. 

(I.) ADDITION. 
Add togethe'" 
1 . 6a + 76, - 2a 45, and 3a - 5&. 

- 5a + 66 - 7c, - 2a + 136 + 9c, and 7a - 296 + 4r. 
2;e — 3?/ + 42, - 5x + 4?/ — 72, and - 8x — 9y — 32. 

4. — rt + 6 — c + (/, a — 26 - 3c + d, — 56 + 4c, and — 5c + d. 

5. a + 6 - c + 7, - 2a - 36 - 4c + 9, and 3a + 26 + 5c - '-** 

6. 5x - 3a - 46, 6?/ - 2«, 3a — 2i/, and 00 - 7a;. 

7. a + 6 — c, c — a + 6, 26 — c + 3a, and 4a — 3c. 
7a - 36 - 5c + 9*/, 26 - 3c - 5(7, and - \il + 15c. 

— 12a; - 5j/ + 42, 3a; + 2?/ - 32, and 9a; - 3i/ + 2. 



9 



(2.) SUBTRACTION. 

From a + 6 take —a — b. 
From a — 6 take - 6 + c. 
From a — 6 + c take — a + 6 — c. 
From 6x - 81/ + 3 take - 2x + 9?/ - 2. 
From 5a - 126 + 17c take - 2a + 46 - 3c. 
From 2a + 6 - 3x take 46 — 3a + 5a;. 
From a + 6 - c take 3c - 26 + 4a. 
From a + 6 + c — 7 take 8 - c — 6 + a. 
From l2x — 'Mj-z take 4y - 52 + x. 
From 8a - 56 + 7c take 2c - 46 + 2a. 
From 9p-4q + Zr take bq-'6p+r 



II. MULTIPLICATION. 

36. The operation of findins .'lie sum of a numbers each 
equal to h is called Multiplication. 

The number a is called the Multiplier. 
h Multiijlicand. 

This Sum is called tbe Product of the multiplication of h 
by a. 

This Product is represented in Algebra by three distinct 
symbols : 

I. By writing the sjTubols side by side, with no sign 
between them, thus, ab ; 

II. By placing a small dot between the symbols, thus, a.h; 

III. By placing the sign x between tlie symbols, thus, 
axh ; and all these are read thus, " a into h" or " a times 6." 

In Arithmetic we chiefly use the third way of expressing a 
Product, for we cannot symbolize the product of 5 into 7 by 
57, which means the sum of fifty and seven, nor can we well 
represent it by 5.7, because it might be confounded with the 
notation used for decimal fractions, as 5 -7. 

37. In Arithmetic 

2x7 stands for the same as 7 + 7. 
3x4 4 + 4 + 4. 

In Algebra 

ab stands for ine same as + 6 + 6+ ... with 6 written 

a times. 
{a + Vjc stands for the same as c + c + 1 ... with c written 
a + 6 times. 
[s.A.] B 



<8 MULTIPLICATION. 

38. To shew that 3 times 4 = 4 tivies 3. 

3 times 4= 4 + 4-^4 

= 1-rl-l + 1 ) 

'. I. 



+1+1+1^1 



C 



1^1 ) 



4 times ii= 3 + 3 + 3 + 3 

=1^1+1 \ 

^'^'^' II. 

+1+1+1 I 

+1+1+1 ) 

Now the results obtained from I. and II. must be the same, 
for the horizontal colunin^: of one are identical with the verti- 
cal columns of the other. 

39. To prove that ah = ha. 

ah means that the sum of a numbers each equal to h is to 
be taken. 

.'. db= 5+6+ with h written a times 

= h 
+ h 
+ 

to or lines 

= 1 + 1 + 1 + \oh terms \ 

+ 1 + 1 + 1 + to 6 temis f J 

+ \ 

to a lines. ) 

Again, 

ha= a + a + with a written 6 times 

= a 
+ a 
+ 

to h lines 

— 1 f 1 + 1 + to rt terms •\ 

+ 1 + 1 + 1 + to « terms r y, 

to h lines - 



MULTIPLICATION. 19 

Now the results obtained from I. and II. must be tlie same, 
for the horizontal columns of one are clearly the same as the 
vertical columns of the other. 

40. Since the expressions ah and ha are the same in mean- 
ing, we may regard either a or h as the multiplier in forming 

«#the product of a and 6, and so we may read ah in two ways : 

(1) a into 6, 

(2) a multiplied by h. 

41. The expressions ahc, ach, bac, hca, cah, cba are all the 
same in meaning, denoting that the three numbers symbolized 
by a, h, and c are to be multiplied together. It is, however, 
generally desirable that the alphabetical order of .the letters 
representing a product shoula be observed. 

42. Each of the numbers a, h, c is called a Factoji of the 
product abc. 

43. When a number expressed in figures is one of the 
factors of a product it always stands first in the product. 

Thus the product of the factors x, y, z and 9 i^ represented 
by 9xyz. 

44. Any one or more of the factors that make up a product 
is called the Coefficient of the other factors. 

Thus in the expression 2ax, 2a is called the coefficient of x. 

45. When a factor a is repeated twice the product would 
be represented, in accordance with Art. 36, by aa ; wlien tJiree 
times, by aaa. In such cases these products are, for the sake 
of brevity, expressed by writing the symbol with a number 
placed above it on the right, expressing the number of times the 
symbol is repeated ; thus 

instead of aa we write a^ 

aaa a^ 

aaaa a* 

These expressions a-, a^, a* are called the second, third, 

fourth Powers of a. 

The number placed over a symbol to express the power of 
the symbol is called the Index or Exponent. 

a^ is generally called the square of a. 
a? the cube of a. 



20 MULTIPLICATION. 

46. The product of a^ and o? = a^x o? 

= aax aaa = aaaaa = a^. 

Thus the index of the resulting power is the sum of the 
indices of the two factors. 

Similarly a* xa^ = aaaa x aaaaaa 

= aaaaaaaaaa = o.^** = a*+*. « 

If one of the factors be a symbol without an index, we may 
assume it to have an index \ that is 

Examples in multiplying powers of the same symbol are 

(1) axa'^ — a^~^' = a^. 

(2) 7a3 X 5a^ = 7 X 5 X a3 X a' = 35aW = 35aio. 

(3) a3 X a« X ^9 = a^+^^ = a^^ 

(4) x^y X xy^ = x-.y.x.y- = x~.x.y.y^ = x^"*"^. 1/^+2 _ ^SyS^ 

(5) a% X a¥ x a^V = a^+i+s. ji+s+r = ^8. jjU 

Examples.— vi. 

Multiply 

I. X into 3y. 2. 3x into 4y. 3, 3xy into 4xy. 

4, 3a6c into ac. 5. a^ into a*. 6. a" into a. 

7. 3a"5 into 4a.^62_ g, y^-^c into Sa^tc^. 9. 15a6''c^ by 12a%. 

10. 7aV by 4a-6c^. 11. a^.by 3rt^ 12. 4a36a; by 5a6''?/. 

13. 19x^2/3 by 4x2/ V. ^4- 17a6^2 by 3k-?/. 15. 6^y^z^ hy 8x^y-z\ 

16. 3a6cby4axi/. 17. a^i'c by 8a''6''c. 18. 9m^7ip by m%-^2. 

19. ay-z by 6x-z^. 20. lla%x by 3ft^"6^*m-. 

47. The rules for the addition and subtraction of powers 
are similar to those laid down in Chap. I. for simple quantities. 

Thus the sum of the second and third powers of x is repre- 
sented by 

x^ + x^, 

and the remainder after taking the fourth power of y from the 
fifth power of y is represented by 

and these expressions cannot be abridged. 



MULTIPLICATION. it 

But when we have to add or subtract the same powers of 
the same quantities the terms may be combined into one : 
thus 

Sy^ + 5y^ + 7y^ = 15?/', 
8x*-bx* = 3x\ 
9y^-3y^-2if = 4i/. 

Again, whenever two or more terms are entirely the same 
with respect to the symbols they contain, their sum may be - 
abridged. 

Thus ad + ad = 2ad, 

3a-b — 2a-b = a-b, 

5a%^ + 6a%^ - ga^fes = 2^363, 

1a?x— \Oa?x— 12a-x= — 15a-x. 

48. From the multiplication of simple expressions we pass 
on to the case in which one of the quantities whose product is 
to be found is a compound expression. 

To shew that (a + b) c = ac + be. 

{a + b) c = c + c + c+ ... with c written a + b times, 

= {c + c + c+ ... with c written a times) 
+ {c + c + c ... with c ^\Titten b times), 

= ac + be. 

49. To sheiv that (a — b) c = ae — be. 

(a — b)c = c + c + e+ ... with c written a — b times, 
= {c + c + c+ ... with c written a times) 

— (c + e + c... with c written b times), 
= ac — be. 
Note. We assume that a is greater than b. 

50. Similarly it may be shewn that 

(a + b + c) d = ad + bd + cd, 

(a-b — c) d = ad — bd — ed, 
and hence we obtain the following general rule for finding the 
product of a single symbol and an expression consisting of two 
or more terms. 

" Multiply each of the terms by the single symbol, and con- 
nect the terms of the result by the signs of the several terms 
of the eompound expression." 



i± MUL T I PLICA TtON. 

Examples. — vii. 

Multiply 

1. a + 6-chya. 7. 8//!.- + 9m?i + lOn^ by mn. 

2. a + 36 - 4c by 2a. 8. ^a? + 4a't6 - ZaW + 40^63 by 2a 6. 

3. a^ + 3a^ + 4a by a. 9. y?\j^ — a;-?/^ + xy — 7 by a;^/. 

4. Ba^^ 5a2 - 6a + 7 by 3a2. i o. m^ - 3 )»% ■\-'imv?-v? by w. 

5. a2 - 2a6 + V- by ah. 11. ISa^?, _ 6a262 + oah^ by 12a263 

6. o-' — 3a-62 + J3 by 3a-6. 1 2. 13.c^ - 17a;"'?/ + 5xj/2 — y^ by 8x3. 

51. We next proceed to the case in which both multiplier 
and nuiltiplicand are comjwmid expressions. 

First to nnilti]ily a + b into c + d. 

Eepresent c + d by x. 

Then (a + b){c + d) = (a + b)x 

= ax + bx, by Art. 48, 

= a(c + d) + b{c + d) 

= ac + ad + hc + hd, by Art. 48. 

The same result is obtained by the following process : 

c + d 
a + b 



ac + ad 
+ bc + bd 



ac + ad + bc + bd 

which may be thus described : 

Write a + b considered as the multiplier under c + d con- 
sidered as the multiplicand, as in common Arithmetic. Then 
•multiply each term of tlie multiplicand by a, and set down the 
result. Next multiply each term of the multiplicand by b, and 
set down the result under the result obtained before. The 
sum of the two results will be the product required. 

Note. The second result is shifted one place to the right. 
The object of this will be seen in Art. 56. 



Mi 'L TIPL ICA TIOM. 23 



52. Next, to multiply a + 6 into c — i. 

Represent c — d by x. 

Then (a + 6)(c-d) = (rt + ?))x 
= ax + hx 

= a((: - (Z) + ?)(c - rZ) 
= ac — a*-^ + 6c — M, by Art. 49. 

From a comparison of this result with the factors from 
which it is produced it appears that if we regard the terms of 
the multiplicand c — (Z as independent quantities, and call them 
+ cand —d, tlie effect of multiplying the positive terms +a 
and +b into the positive term +c is to produce two positive 
terms + ac and + he, whereas the effect of multiplying the 
positive terms +a and +b into the negative term —d is to 
produce tivo negative terms —ad and —hd. 

The same result is obtained hy the following process : 
c — d 
a + b 
ac — ad 
+ bc-bd 



ac — ad + bc — bd 



This process may be described in a similar manner to that 
in Art. 51, it being assumed that a positive term multiplied 
into a negative term gives a negative result. 

Similarly we may shew that a — b into c + d gives 
ac + ad — be — bd. 

53. Next to multiply a — b into c — d. 

Represent c — d by x. 

Then (a-b){c-d) = {a-b)x 
= ax — bx 

= a{c — d) — b{c — d) 
= {ac - ad) - {be - hd), by Art. 49, 
— ac — ad — bc + bd. 

When we compare this result with the factors from which 
it is produced, we see that 

The product of the positive term a into the positive 
term c is the positive term ac. 



±4 MULTIPLICATION. 

The product of the positive term a into the negative 

term — d is the negative term — ad. 
The product of the negative term — h into the positive 

term c is the negative term — he. 
The product of the negative term — h into the negative 

term - rf is the positive term 6(7. 

The multiplication of c - d by a — h may be written thus : 
c-d. 
a~b 



ac — ad 

- be + bd 



ac — ad-bc + bd 



54. The results obtained in the preceding Article enable us 
to state what is called the Rule of Signs in Multiplication, 
which is 

"The product of tivo positive terms or of two negative tervm 
is positive : the product of tivo terms, one of which is positive aiul 
the other negative, is negative." 

55. The following more concise proof may now be given of 
the Rule of Signs. 

To shew that (a — b){c — d) = ac — ad — be + bd. 

First, {a - h)M= M +M^M+ ... with M written a-b times, 

= {M + M + M -{■ ... with M -written a times) 
-(M+i¥ + M + ... with M written ft times), 

= aM-bM. 

Next, let M= c-d. 
Then aM= a (c-d) 

= {c-d)a Art. 39. 

= ca — da. Art. 49. 

Similarly, bM=cb-db. 

.". (a — b)(c — d) = {ca — da) — (cb — db). 
Now to subtract (cb — db) from (ca — da), if we take away cb 
we take away db too mucli, and we must therefore add dh u> 
the result, 

.". we get ca - da — cb + db, 
which is the same as ac-ad-bc + bd. Art. 33. 



MUL TI PLICA T/ ON. 2$ 



So it appears that in multiplying {a -h) {c- d) we must 
multiply each term in one factor by each term in the other 
and prefix the sign according to this law : — 

When the factors viultiplied have like signs prefix +, when 
unlike — to theprodkct. 

This is the Rule of Signs 

56. We shall now give some examples in ill'istration of the 
principles laid down in tlie last five Articles. 

Examples in Multiplication wwked out. 

(1) Multiply ic + 5 by a; + 7. (2) Multiply x - 5 by x + 7. 

x+ 5 x-b 

x+ 7 35 + 7 



x^ + 5x x^ — ox 

+ 7x + 35 +7x-35 



a;2 + 12x + 35 x- + 2a;-35 

The reason for shifting the second result one place to the 
right is that it enables us generally to place like terms under 
each other. 

(3) Multiply X + 5 by X - 7. (4) Multiply x - 5 by x - 7. 

x + 5 X- 5 

x-7 X- 7 



x^ + 5x x2_ 53. 

-7x-35 - 7x + 35 



x2_2x-35 x'''-12x + 35 

(5) Multiply x2 + ?/2 by x'-i - y'^. (6) Multiply 3ax - 5by by 7ax - 2by. 
7? + if 3ax - ■ hhy 

x^-if lax- 2hy 

X* + xh/ 210^x2 - Soabxy 

-xV-2/* - dabxy + \Ob-y^ 

X* - y* 2la-x:- - Alahxi^ + lOlj^'^ 



26 MUL TI PLICA TION. 

57. The process in the multiplication of factors, one or 
both of which contains more than two terms, is similar to the 
processes which we have been describing, as may be seen from 
the following examples : 



Multiply 


' 


(1) x^ + .Tt/ + 1/2 by a; — 2/. 


(2) a^ + 6a + 9hj a^~6a + 9. 


x^ + a;?/ + 2/2 


a^ + 6a +9 


x-y 


a^-6a +9 


01? + X^T/ + XI/2 


a* + 6a^ + 9«2 


— x^i/ - xy^ — y^ ' 


-6a3-36(i2-54a 


x^-y^ 


+ 9a2 + 54a + 81 



a*-18a2 + 81 
(3) Multiply 3x2 + ^^y _ ^2 by Zx^-4xy + y\ 

3X2+ 43;^ _ y2 
3X2- ^y.y ^ y2 

9x« + 12x3y- 3xy 

- 12x^1/ - \Qx-y- + 4x?/^ 

+ 3x-!/2 + 4x1/3 _ yi 

(4) To find the continued product of x + 3, x + 4, and 

x + 6. 

To effect this we must muUi]ily x f 3 by x + 4, and then 

inltiplv the result by ,(; + 6. 
x+ 3 
x+ 4 

x2+ 3x 

+ 4x 4- I fj 

x2+ 7x -I- 12 
x+ 6 

x-''+ 7x2 + 12x 
+ 6x2 + 42x + 72 



x3 + 13x2 + 54x + 72 



Note. Tlie numliers 13 and 54 are called the coefficients of 
x2 and X in the expression x^-" ISx^-f 54x+72, in accordance 
with Art. 44. 



MUL riPL re A T!0\ : 



27 



(5) Find the continued product of x + a, a + 6, and z + c. 




x^ + ax + bx + ab 
x + c 

7? + aa;2 + Ix^ + ahx 
+ ex- + acx + bcx + abc 



a? + {a + b + c)x^ + (ab + ac + bc)x + abc 



Note. The coefficients of x^ and x in the expression just 
obtained are a + b + c and ab + ac + be respectively. 

When a coefficient is expressed in letters, as in this example, 
it is called a literal coefficient. 



Examples. — viii. 



Multiply 



I. X + 3 by X -.'.). 2. a; + 15 by X — 7. 3. x - 12 by x + 10 

4. X — 8byx — 7. 5. a — 3 by a — 5. 6. y — 6hyy + lS. 

7. x2-4byx2 + 5. 8. x2-6x + 9 by x2-6x + 5. 

9. X- + 5x - 3 by x^ - 5x - 3. 10. a^ - 3a + 2 by a^ - 3a^ + 2. 
II. x^ — x + 1 by X- + X— 1. 12. 3:^ + xy + y'^ hy X- — xy + y\ 

13. x^ + xy + y^hyx — y. 14. a- - x^ by a* + a%^ + x*. 



i^ 

16 

17 
18 

19 
20, 
21 

22 

23 
24. 

25 
26, 



x^ — 3x- + 3x - 1 by x- + 3x + 1. 
x^ + 3x^y + 9jy'- + 21y^ by x — 3?/. 
a^ + 2a26 + 4«l- + 86^ by a - 2b. 
SftS + 4a^b + ■lab'^ + ¥ by 2a - b. 
cr - 2a26 + Za¥ + A¥ by a- - 2ab - Zb\ 
a^ + Za"b - 2a6'2 + 36^ by a-^ + 2a6 - 3&2. 
a- — 2ax + 4x2 ]jy ^2 _l 2rta; + Ax\ 
9rt- + 3ax + X- 1 ly 9a"- - 3ax + x^. 
X* — 2ax^ + 4a- !)}■ r^ -f 2ax2 + 4a-. 
a- -f 6^ -I- c^ - 06 - rf c - lie 1 ly a -i- 6 + c. 
x^ + 4xt/ -I- 5 J/- by x" - Zj?d - 2xy^ + 3y\ 
ab + cd + ac + bd bv ab • cd - ac — bd. 



Find the continued product ot the following expression 



27. X - a, X -i- a, x^ -t- a"*, X* + a* 



28. x-a, x-i-&, » — c, 



28 ML Y, TIPL ICA TION. 



29. 1 - a;, 1 + a;, 1 + a;2, 1 + x*. 

30. % — y,x + y, x^ — xy + y^, x^ + ocy + y^. 

31. a — a;, a + x, a^ + x^, a^ + sc*, a^ + x*. 

Find the coefficient of x in the following expansions : 
32. (x-5)(x-6)(x+7). 33. (x + 8)(x + 3)(x-2). 

34. (x - 2) (x - 3) (:c 4 4). 35. (x-a") (x-6)(x-c). 

36. (x2 + 3x-2)(x2-3x + 2)(x^-5). 

37. (X2-X + 1)(X2 + X-1)(X*-X2+1). 

38. (x- - mx + 1) (x^ — mx — 1) (x* — m'x — 1). 

58. Our proof of the Rule of Signs in Art? 55 is founded 
oil the supposition that a is greater than b and c is greater 
tlian d. 

To include cases in which the multiplier is an isolated nega- 
tive quantity we must extend our definition of Multiplication. 
For the definition given in Art. 36 does not cover this case, 
since we cannot say that c shall be taken — d times. 

We give then the following definition. " The operation of 
]\[ultiplication is such that the product of the factors a — b and 
cv-f? tfill be equivalent to ac — ad — bc + bd, whatever may be the 
values of a, b, c, rf." . 

Now since 

(a — b)(c — d) = ac — ad — bc + bd, 
make a = and d = Q. 

Then (0-6) (c-0) = x c-0 x 0-6c + 6 xO. 

or —bxc= —be. 
Similarly it may be shewn that 

— bx —d= +bd. 

Examples. — ix. 

Multiply 
I . a- by — b. 2. a"^ by — a^. 3. a% by - ab-. 

4. ~4a-6by — Safe^. 5. 5x^by— 6x?/2. 6. a'^ — ab + H-hy —a. 
7. 3a3 + 4^2 — 5a by — 2a2. 8. —a^ — a- — ahy—a—\. 

9. 3x"^/ — bxx/ + Ay'^ by — 2x — Zy. 



— iSiri^ — 6mn + In"^ by — m + n. 
13r--17r-45 by -r-3. 

Tx^ - 8x%! — 92- by - x - s. 

— X* + x^y — x?y- by —y — x. 

— y^ — xy- — x-y — j^ by —x — y. 



III. THVOI.TJTION. 

59. To this part of Algebra belongs the process called 
Involution. This is the operation of multiplying a quan- 
tity by itself any number of times. 

The power to which the quantity is raised is expressed by 
the number of times the quantity has been employed as a 
factor in the operation. 

Tlius, as has been already stated in Art. 45, 
a^ is called the second power of a, 
a? is called the third power of a. 

60. When we have to raise negative quantities to certain 
powers we symbolize the operation liy putting the quantity in 
a bracket with the number denoting the incfex (Art. 45) jdaced 
over the bracket on the ri-ht hand. 

Thus ( — of denotes tlie third power of — a, 
( — 2.c)* denotes the fourth jjower of — 2x. 

61. The signs of all even powers of a negative quantity 
will be ])ositive, and the signs of the odd powers will be 
negative. 

Thus (-a)2 = (_a)x(-a) = a2, 

(^-af = {-a).{-a) {-a) = a-.{- a)= -a^. 

62. To raise a simple quantity to any power we multiplv 
the index of the quantity by the number denoting the power 
to which it is to be raised, and prefix the proper sign. 

Thus the square of a^ is a^, 

the cube of a^ is a", 
the cube of - x^yz^ is - x^yh^. 



36 INVOLUTIOX. 



63. AVe form the second, third and fourth powers of a + 6 
in the following manner : 

a + 6 
a + 6 

a^ + ab 

+ «?) +6^ 



(a + 5)^ = a? + 3n-6 + 3a6'' + W 
a +6 



a4 + 3a36 + 3a26'^ + rt63 

(a + by = ft* + 4a36 + 6a%-^ + 4a¥+"b\ 
Here observe tlie following laws : 

I. The indices of (i decrease \>y unity in each term. 
TI. The indices of b increase by unit}' in each term. 
III. The numerical coefficient of the second term is always 
the same as the index of the power to which tiie 
binomial is raised. 

64. We form the second, third and fourth powers of a - 6 
in the following manner ; 

a-b 
a-h 

a^ -ab 
-ab +62 

(a-6)2 = ^2T2a6 + 62 
<7 - *: 

a^ - 2a~b + ab'' 
- a^b + -2ah--b^ 

(a-by = a^-:)a-l>^ Sali^-P 
a -b 



a* - 'Sa% + Ca-b- - ab^ 
- a^b + :Ui-b- - Sab^ + 6* 

(a - bf = a* - 4a-'6 + 6(t-6- - -iab^ + b*. 



INVOLUTION. %\ 



Now observe that the powers of a - 6 do not differ from the 
powers of a + 6 except that the terms, in which the odd, powers 
of 6, as 6', }?, occur have the sign - prefixed. 

Hence if any power of a + h be given we can write the 
corresponding power of a, - 6 : thus 
since (« + hf = a* + Sa'i?) + 1 (daW + 1 Oa-i^ + 5a6* + 6^, 
(a - If = a^ - 5a*6 + X^aW - lOa'^i^ + 5^54 _ y,^ 

65. Since (a + 6)2 = a2 + 62 + 2a6 and (a - 6)2 = a2 + 52 _ 2a&, 
it appears that the square of a binomial is formed by the 
following process : 

" To the sum of the squares of each term add twice the 
product of the terms." 

Thus (a; + yf = x- + if- + 2xi/, 

(x-5)2 = :c2 + 25-10x, 

(2a; - 7i/)2 = U- + 49i/- - 28xi/. 

66. To form the square of a trinomial : 

a + 6 + c 
a + 6 + c 



a? + 2a& + ¥ + 2ac + 26c + c-. 

Arranging this result thus a' + b'^ + c' + 2ab + 2ac + 2bc, we set 
that it is composed of two sets of quantities : 

I. The squares of the quantities a, b, c. 
II. The double products of a, b, c taken two and two. 
Now, if we form the square oi a-b-c, we get 
a-b-c 
a-b-c 



a^-ab- ac 

-ab + ¥ + bc 

-ac + bc + c^ 

a'^-2ab + ¥- 2ac + 2bc + c\ 
The law of formation is the same as before, for we have 



32 INVOLVTION. 



I. The sqTiares of the quantities. 

II. The doul)le products of a, - 6, - c taken two by two : 
the sign of each result being + or - , according as 
the signs of the algebraical quantities composing it 
are like or unlike. 

67. The same law holds good for expressions containing 
more than three terms, thus 

(a + 6 + c + (0^ = a2 + 52 + c2 + (i2 

+ lah + 2ac + 2arf + 26c + 26(Z + 2cd, 
(a-& + c-(^)2 = a2 + 62 + c2 + ,;2 

- '2ah + 2ac - 2a(i - 26c + 2M - led. 

And generally, the square of an expression containing 2, 3, 
4 or more terms will be formed l^y the following proci-.-s : 

" To the sum of the squares of each term add twice the 
product of each term into each of the terms that follow it." 

Examples. — x. 

Form the square of each of the following expressions : 

I. x-va. 2. v-a. 3. a^ + 2. 4. a; -3. 5. x^ + i/'. 

6. x^-y\ 7. n-' + R 8. a^-W. 9. X + 1/ + 2. 10. x-y + z. 

II. m + w-p-?-. 12. a'" + 2x-3. 13. 3?-Qx + l. 

1 4. 2x2 _ 7 J. + ()_ 1 5 _ yi + if- zK 16. X-* - -ix^y- + y*. 

17. a^ + P + c^. 1 8. x-'^-y^-z^. 19. x + 2y-3z. 

20. X- - '2y'^ + 5z^. 

Expand the following expressions : 
21. (x + cf)^. 22. {x-af. 23. (x + 1)^. 24. (x-1)^ 

25. (x + 2)l 26. (rt2-62)^. 27. {a + b + c)\ 28. (a-6-c)3. 
29. (»i + ?i)-.(7n - ?i)-. 30. {in+ny-.{m'- — n^). 

68. An algebraical product is said to be of 2, 3 dimen- 
sions, when tiie sum of the indices of the quantities composing 
the product is 2, 3 

Thus ab is an expression of 2 dimensions, 

aWc is an exjiression of 5 dimensions. 



DIVISTOISr. 33 



69. An algebraical expression is called homogeneous when 
each of its terms is of the same dimensions. 

Thus x'^ + xy + y- is homogeneous, for each term is of 2 dimen- 
sions. 

Also 3x^ + 4x-i/ + 5i/^ is homo<:eneous, for each term is of 3 
dimensions, the numerical coefficients not affecting the dimen- 
sions of each term. 

70. An expression is said to be arranged according to 
powers of some letter, when the indices of that letter occur in 
the order of their magnitudes, either increasing or decreasing. 

Thus the expression a^ + a^x + ax- + y? is arranged according 
to descending powers of a, and ascending powers of x. 

71. One expression is said to l)P of a higher order than 
another wlien tlie former contains a higher power of some dis- 
tinguishing letter than the other. 

Thus a^ + a-x + rtc- + x^ is said to be of a higher order than 
a^ + ax + x^, with reference to the index of a. 



rr. DIVISION. 

72. Division is the ]i!oclss liy which, when a product is 
given and we know one ot the factors, ihe other factor is deter- 
mined. 

The product is, vith reference to this process, called the 
Dividend. 

The given factor is called the Divisor. 

The factor which has to be found is called the Quotient. 

73. The operation of Division is denoted by the sign -=-. 
Thus ab-^a signifies that ab is to be divided by a. 

The same operation is denoted by writing the dividend 

owr the divisor with a line drawn between them, thus — . 

a 

In this chapter we shall treat only of cases in which the 

dividend contains the divisor an exact number of times. 

[S.A.] Q 



34 DIVISIOJV. 

Case I. 

74. When the dividend and divisor are each included in 
a single term, we can usually tell by inspection the factors of 
which each is composed. The quotient will in this case be 
represented by the factors which remain in the dividend, wlien 
those factors which are common to the dividend and the di- 
visor have been removed from the dividend. 

Thus X"^*' 

Sa^ Zaa . 
— = = 6a, 

a a 

a^ aaaaa „ 

a-' aaa 

Thus, when one power of a number is divided by a smaller 
power of the same number, the quotient is that power of the 
number whose index is the difference between the indices of the 
dividend and the divisor. 

Thus —=a}'^~-' = a\ 

15a362 _ „, 
'3ao 

75. The quotient is ^initT/ when the dividend and the 
divisor are equal. 

Thus ^ = 1; "■^'^^l; 

and this will liold true wuen the dividend and the di\-isor are 
compound quantities. 

Thus ■ — r=l; -^r— S=l. 



Examples.— xi. 

Divide 

1. .x" by x'. 2. x^^ by x-. 3. xhj"^ by xy. 

4. x?'y^^ "hy xyh. 5. 24«6-c by 4a?). 6. 72o-6-'c^ by 9a-6-c. 

7. 256«3ir(;9by USahc^. 8. 1331»i'"»'V''- b' llm-n^p\ 

^. QOa^x-if by bxy. ip. 9Ga-'6-'c-3 by 126c, 



DIVlStOM. 35 



Case II. 

76. If the divisor be a single term, while the diviJeml 
contains two or more terms, the quotient will be found by 
dividing each term of the dividend separately by the divisor 
and connecting the results with their proper signs. 

m, ax + ix - 

Thus = a + 6, 

aV + a^x^ + ftx „ „ 



ax 
12xy+16a;y-8xy2 
4x1/' 



2 — 3x^1/2 4- 4a;y _ 2. 



Examples.— xii. 

Divide 

1 . x^ + 2x2 + a; by X. 4. m/)x* + m^p-x^ 4- m^-p^ by m'p. 

2. if -y* + y^-y^ by i/^. 5.1 6a^xy - 28a^x^ + 4a''x^ by 4a'^x. 

3. 8a3 + 16a26 + 24a62by8a. 6. 72xY-36xy- 18xY by 9x2^/. 

7. 81m*ft'' - 54m^?i^ + 277/!,^/i2p by 3m2?i2. 

8. 12xY-8xy-4xy by 4x3. 

9. 169a*6 - 1 1 7a^62 + 91^25 |-,y 13^2^ 

• 10. 36l65c3 + 2286M-13363c5by 1962c. 

77. Admitting the possibility of the independent existence 
of a term affected with tlie sil^mi - , we can extend the Exam- 
ples in Arts. 74 — 76, by taking the first term of the dividend 
or the divisor, or both, negative. In such cases Ave apply the 
Rule of Signs in Multiplication to form a Eule of Signs in 
Division. 

Thus since —axb=-ah,\ve conclude that ^[— = -a, 



7 7 —ab 
ax -b=-ab, -.^ ^a, 

—ax-b = ab, — =-a; 

and hence the rules 

I. When the dividend and the divisor have the shme 

sign the quotient is positive. 
II. When the dividend and the divisor have different 
signs the quotient is negative. 



36 D/VIS/OI\r. 



78. The followinj; Examples illustrate the conclusions just 
obtained : 

(I) '^^-^.. (3) ^''=9.,. 

(5) _ . = -lP + ah--a-h + a^. 

(6) _4^^2 - - •' =3a;V-4xy + 2. 



Examples.— xiii. 

Divide 

1. 72rt6by-9a6. 6. - a V — a2x2 _ ^a; by — ax. 

2. - 60«8 by - 4a3. 7. - 34ft3 + 51 ^2 _ i7„_^.2 i-,y j -„ 

3 -84a;Vby4ry. 8. -d,a?h^'-2A(eh'^ + 2,2a~¥hy -AaW. 

4. - ISm^^i^ by Zvm. 9. - 144i';3+ 108.7-21/ -96.r?/2 ^^^ ^^x. 

5. - 128ft36-'c by - 86c. 10. ¥xh- - Wx'z^ - hhfz^ by - ¥z\ 

Case III, 

79. The third case of the operation of Division is that in 
which the divisor and the dividend contain more terms than 
one. The operation is conducted in the Ibllowing way : 

Arrange the divisor and dividend according to the 
powers of some one symbol, and ])l>ice them in the 
same line as in the process of Long Division in 
Arithmetic. 

Divide the first term of the dividend by the first term 
of the divisor. 

Set down the result as the first term of the quotient. 
Multiply all the terms of the divisor by the first term 
• of the quotient. 

Subtract the resulting product from the dividend. If 
there be a remainder, consider it as a new dividend, 
and proceed as before. 



DIVISION. 37 



The process will best be understood by a careful study of 
the following Examples : 

(1) Divide a^ + 2«6 + 6"- by a + 6. (2) Divide a? - 2ab + b- by a-b. 
a + b)a^ + 2ab + b''{a + h a-b) a^ - 2ab + 6^ (a - 6 

a" + ab a^ — ab 

ab + b'- -ab + ¥ 

ab + b"- -ab + b'^ 
(3) Divide a;^ - y^ by x^ — y^. 

x^-y^Jx'^-y'^^x^-hx^y^ + y* ■ , 

a;6 _ a;4y2 





a V - T}y'^ 




xY - y^ 
x-y* - y^ 


(4) 


Divide x^ - Aa-x^ + Aa^j-^ - o" by x"^ - a^. 


CC2- 


- 0?) x^ - 4a-x* + 4a*x2 - a^ (a;* - 3a^x^ + a* 




x" - a V 




-'3ah^ + 4a*x^~a^ 




-3a2x* + 3a%2 



aV - a° 
(5) Divide 3xt/ + x3 + 1/^-1 by i/ + x-l. 
Arranging the divisor and dividend by descending powers 
of x, 

x + y-l)x^ + 3xy + >/' - i. i^x- -xy + x + y^ + y+1 
a^ + x-y - X- 

-x-y + x^ + Smj + y^ -1 
-x^y-xy'^ + xy 

x^ + xy"^ + 2xy + y^-l 
x^ + xy-x 

xy^ + xy + x + y^-l • 
xy^ + y^-y^ 

xy + x + y^-1 

Xy + y2-y » 

x + y- 1 
x + y-1 



38 DIVISION. 



80. We must now direct the attention of the student to 
two points of great importance in Division. 

I. The dividend and divisor must be arranged accord- 
ing to the order of the powers of one of the symbols 
involved in them. This order may be ascending or 
descending. In the Examples given above we have 
taken the descending order, and in the Examples 
worked out in the next Article we shall take an 
ascending order of arrangement. 
II. In each remainder the terms must be arranged in. 
the same order, ascending or descending, as that in 

which the dividend is arranged at first. 
\ ° 

81. To divide (1) 1 -x* by a;3 + x2 + a; + l, 

arrange the dividend and divisor by ascending powers of x, 
thus : 

1+x + x^ + x^ 

-x-x^-x^-x^ 
-x-x^-a^-x^ 
(2) 48x2 + 6 - 35x5 + 58x* - 70x3 _ 333; by Gx^ - 5x + 2 - Tx^, 
arrange the dividend uud divisor by ascending powers of x, 
thus : 

2-5x + 6x2 - 7^3^ 6 - 23x + 48x2 _ 70^3 + sgx* - 35x5 (^3 _ 4^ + 5^2 
6-15x+18x2-21x3 



- Sx-I- 30x2- 49x3 + 58x* 

- 8x + 20x2- 24x3 + 28x* 

10x2 _ 25x3 + 30x* - 35x5 
10x2 _ 25^3 + 30x* - 35x* 

Examples. — xiv. 

Divide 

1. x2+15x + 50by x+10. 5. x3+ 13x2 4-54x + 72 by x-i-6. 

2. x2 - 17x + 70 by x - 7. 6. x3 + x^ - x - 1 by x + 1. 

3. x2 + X - 12 by X - 3. 7. x3 + 2x- + 2x 4- 1 by x + 1. 

4. x2 + 13x-l-12byx + l. 8. x^ - 5x3 + 7x2 + 6x + 1 by x2 + 3x + 1. 

9. X* - 4x3 ^ 2x- + 4x + 1 by x2 - 2x - 1. 
10. x*-4x3 + 6x2-4x+l by x2-2x+l. 



^1 



Dins 10 y. 39 

n. a;*-x2 + 2x-l by a;2 + x-l. 12. a;*-4x2 + 8x+ 16 l>y x + 2. 

13. x^-\- -ix-y + 3xi/ + 12)/^ by x + 4y. 

14. a* + 4a36 + 6a262 + 4^53 + 54 ^y ^ + 6. 

15. <{5 - 5«*6 + 10a362 - I0a263 + 5^54 _ ^s ijy ^ _ 5, 

16. x* - 12x' + 50x2 - 84x + 45 by x2 - 6x + 9. 

17. «■• - 4a*b + 4a362 + 4^(2^,3 _ 17^54 _ i265 by a^- - 2ah - 3//. 

1 8. 4a2xi - Ua^x^ + lSa*x- - Gu^x + a^ by 2(tx2 - 3a2x + a\ 

1 9. X* - a;^ + 2x - 1 by x2 + X - 1. 

20. X* + rt-x2 - 2a^ by x2 + 2a2. 23. x^ - y^ by x - 1/. 

21. x2 - rSxy - '30y'' by x - 15j/. 24. a- - b- + ihc - c2bya - 6 + c. 

22. x'' + (/' by X + y. 25. h - 'ilr + 36^ - i* by & - 1. 

26. tr - 62 _ c2 + ^2 _ 2(afi - 6c) ])y a + 6 - c - rf. 
27. x^ + ?/■' + ^3 - 2xyz by x + ?/ + z. 28. x^^ + ;/^" by x^ + yK 

29. ^2 4.^2 4. 223}- - 2^2 + 72?- _ 3}-2 by y- q + 3r, 

30. a^ + a«62 + a«6* + u-¥' + 6« by a* + a% + «-62 + a¥ + 6^ 

31. x^ + x^y~ + x^y^ + x-y^' + y^ by x^ - x^y + x'-y^ - xy^ + y*. 
32. 4x5 - x3 + 4,^. by 2x2 + 3^. ^ 2. 33. a'' - 243 by a -3. 

34. Z;io - k by P - 1. 35- a;^ - 5x2 _ 46.5 - 40 by x + 4. 

36. 48x3 - 76ax2 - 6ia-x + I05a^ by 2x - 3a. 

37. ISx* - 45x' + 82x2 _ 673- + 40 by 3x2 _ 4^ ^ ^ 

38. 16x* - 72a2x2 + 81a* by 2x - 3a. 

39. Six-* - 256a* by 3x + 4a. 41. x^ + 2ax2 - a2x - 2a^ by x2 - a'K 

40. 2«-^ + 3a26-2a62_363bya2-62. 42. a*- a262_ i264bya2 + 352 

43. X* - 9x2 _ Q.j.y _ y2 by x2 + 3x + y. 

44. X* - 6x^y + 9xhf - Ay^ by x2 - 3xy + 2y\ 

45. X* - 8ly* by X - Sy. 47. 81a* - 166* by 3a + 26. 

46. a* - 166* by a - 26. 48. 16x* - 81y* by 2x + Sy. 

49. 3a2 + 8a6 + 46^ + lOac + 86c + 3c2 by a + 26 4- 3c. 

50. a* + 4a2x2 + 16x* by a- + 2ax + 4x^. 

51. X* + x2j/2 + y* hy X' - xy + y^. , 

52. 256x* + 16x2!/2 + y* by 16x2 + 4xy + y\ 

53. x^ + x*7/ - x^2/2 + x^ - 2x2/2 + y^'by xi^ + x-y. 



40 DIVISION. 



54. ax^ + Sa^x^ - <id?x - 2a* by a; - a. 55. a^ - ^ byx + a. 

56. 2x2 + a;?/ - 3!/^ - 4i/z - X3 - 2;^ by 2x + 3y+ z. 

57. 9a; + Sa;* + 14x'' + 2 by 1 + 5x + x^. 

58. 12 - 38x + 82x2 - 1 12x3 + 106x* - TOx^ by Tx^ - 5x + 3. 

59. x^ + 1/^ by X* - x^y + x^y^ - xy^ + y*. 

60. (a-x^ + h\j'^ - (a?h- + x'^y'^) by ax + by + ab + xy. 

61. a& (x- + 1/2) + xj/(a2 + 62) by ax + by. 

62. X* + (262 - a^)x2 + ¥ by x2 + ax + b\ 

82. The process may in smne cases be shortened by the use 
of brackets, as in the following Exaui2:)le. 
x + 6^x^ + (a + 6 + c) x2 + (a6 + ac + 6c) x + a6c(x2 + (a + c) x + ac 
x^ + 6x2 

(a + c) x2 + (a6 + ac + bc) x 
(a + c) x2 + (a6 + 6t) x 

acx + abc 

acx + abc 

x — l)x^- mx* + na^ — ?ix2 + mx — 1 (:'•■* - (m - 1) x^ 

x^-x* -(//i.-7i-l)x2-(?n,-l)x+l. 



- (m - 1) X* 4- nx^ 
-(m-l)x* + (m-l) x3 



— (rn — n—l)x^ — nx^ 
-{m-n-1) a^+{vi-n-l) X- 



-(m- 1) x2 + mx 

- (m - 1) x2 + (in -1) X 

x-1 
x-1 

Examples.— XV. 

Divide 

1. X* - (a2 _ 6 _ c) x2 - (6 - c) ax + be by x2 - ax + c. 

2. y^-(l + m + n) y^ + {hn + In + mn) y - Imn by y-n. 

3. x^ - {m - c) x'* + {n - cm + d)oi^ + 

{r + en - dm) x^ + {cr + dn) x + dr by x^ - mx- + ')ix + r. 

4. X* 4- (5 + a) x3 - (4 - 5a + 6) x2 - (4a 4- 56) x 4- 46 by x2 4- 5x - 4. 

5. x*-(a4-64-c4-rf) x3 4-(a64-ac4-a<i4-6c4-6(i4-cd) x^ 

- (a6c 4- a6(/ 4- acd + bed) x 4- abed by x* - (a 4- c) x 4- ac. 



division: 41 

83. Tlie following Exainples in Division are of great 
importance. 



Divisor. 


Dividend. 


Quotient. 


x + y 


x2-2/2 


x-y 


x-y 


x^-y^ 


x + y 


x + y 


x^ + y^ 


x^ - xy + y^ 


x-y 


oc^-y^ 


x^ + xy + y^ 



84. Again, if vre an'ange two series of binomials consisting 
respectively of the sum and the ditference of ascending powers 
of x and y, thus 

x + y, X" + y-, y? + y^, xf^ + ?/■*, x'' + y^, a;" + 7/'', and so on, 
x-y, X-- y-, sr - y^, x* - y^, af" - y^, x'^ - 1/", and so on, 

x + y will divide the odd terms in the upper line, 
and the even in the lower 

x-y will divide all the terms in the lower, 
but none in the upper. 

Or we may put it thus : 

If n stand for any whole number, 

x^ + jj" is divisible by x + y Avhen n is odd, 
by x-y never ; 

x^-y" is divisible hy x + y when n is even, 
hy x-y alv.jys. 

Also, it is to be observed that when the divisor is a;-y all 
the terms of the quotient are positive, and when the divisor is 
x + y, the terms of the quotient are alternately positive and 
negative. 

x^ — v^ 
Thus^-^ — -^ = a^ + x'^y + xy^ + y^, 
x-y 

xJ + 'lf~ 

^ = x^- afy + xhf - oi?y^ + x^y^ - xy^ + y^, 

J- =x^- xhf + x^y^ - x^y^ + xy* - y^. 



45 Tilvisroy: 

85. These properties may bf easily remembered by taking 
the four simplest cases, thus, x + j/, x-y, x- + y-, x^-yp, of 
which 

the first is divisible by x + y, 

second x-y, 

third neither, 

fourtli both. 

Again, since these properties are true for all values of x and 

y, suppose y = \, then we shall have 

x^-\ , x^-l 

a;+l X - 1 

3? + l , , x3-l , 

- = x--x+l, '=x- + x+l. 

X+l X- I 

Also 

x^ + l , , ., , 

;- = X* - X/ + .X- - X + 1, 

x-i- 1 

x^- 1 

= x^ + X-* + x^ + X- + X + 1. 

X- 1 



Examples. — xvi. 

"Without going through the process of Division write down 
the quotients in the following cases : 

1. When the divisor is m + n, and the dividends are 
respectively 

m^ - n^, m^ + n^,'ni^ + n^, m^ - n^, m^ + ?i^. 

2. When the divisor is m - n, and the dividends are 
respectively 

vi^ - n~, m^ - n^, m* - 7i*, m^ - n^, vi' — iiJ. 

3. AVhen the divisor is a+1, and the dividends are 
respectively 

(r-1, r»^+ 1, a^ + l, a" + l, a*-l. 

4. When the divisor is y-\, and the dividends are 
respectively 

2/2-1, !/5- 1,2/5-1,7/7 _ 1^2/9 _1. 



V. ON THE RESOLUTION OF EXPRES- 
SIONS INTO FACTORS. 

86. We shall discuss in this Chapter an operation which 
is the opposite of that which we call Multiplication. In Mul- 
tiplication we determine the product of two given factors : in 
the operation of which we have now to treat the product u 
given and the factors have to be found. 

87. For the resolution, as it is called, of a product into its 
component factors no rule can be given which shall be applic- 
able to all cases, but it is not difficult to explain the process 
in certain simple cases. We shall take these cases separately. 

88. Case I. The simplest case tor resolution is that in 
which all the terms of an expression have one common factor. 
This factor can be seen by inspection in most cases, and there- 
fore the other factor may be at once determined. 

Thus a^ + ah = a(a + b), 

2a3 + 4a2 + Sa = 2a {a? + 2a + 4), 
23?y - 1 %xhf + bAxij = 9xy (x- - 2xy + 6). 



EXAMPLPS.— xvii. 

# 
Resolve into factors : 

1. 5a;2-15x. 5. a^-ax^ + hx'^ + cx. 

2. 3rc" + 18x2-6a;. 6. 3afy^-2lxY + ^'^x^y*- 

3. 49y--Uy + 7. 7. 54a%'i + 108a%'* - 24Sa^b\ 

4. 4x^y-\2x-y2 + Sxy\ 8. 45a;"(/i^ - 90^5^7 - 360xV^. 



44 RESOL UTTOJSr INTO FA CTORS. 



89. Case 41. The next case in point of simplicity is that 
in which four terms can be so arranged, that the first two have 
a common factor and the last two have a common factor. 

Thus 

a;^ + ax + 6a; + a6 = {^ + ax) + (ix + a6) 
= .r (.r 4- a) 4- & (x + a) 
= (x + 6) (x + a). 
Again 

ac - ad - be + bd = (ac - ad) - (he - hd) 
= a{c-d)-h{c-d) 
= (a-h) (c-d). 

Examples. — xvi ii. 

Resolve into factors : 

1 . x^ -ax-bx + ab. 5. ahx^ - axy + hry - y\ 

2. ab + ax — hx — x"^. 6. abx — ahy + cdx — cdy. 

3. bc + hy -cy - y^. 7. cdx^ + dmxy - cnxy - m n y'. 

4. hm + mn + ab + an. 8. abcx-b^dx-acdy + bd'y. 

90. Before reading the Articles that follow the student is 
advised to turn back to Art. 56, and to observe the manner in 
which the operation of multiplying a binomial by a binomial 
produces a trinomial in the Examples there given. He will 
then be prepared to expect that in certain cases a trinomial 
can be resolved into two binomial factors, examples of which we 
shall now give. 

91. Case III. To find the factors of 

x- + 7x+12. 
Our object is to find two numbers whose product is 12, 

and whose sum is 7. 
These will evidently be 4 ai^d 3, 

* .-. x^ + 7x + 12 = (x 4- 4) (x + 3). 

Again, to find the factors of 

x2 + 56x + 662. 
Our object is t4 find two numbers whose product is 65*, 

and whose sum is 56. 
These mil clearly be 36 and 26, 

.•. X- + 56x + 66- = (x + 36) (x + 26). 



ffF.SOLUTION- INTO FACTORS. 



Examples.— xix. 

Resolve into factors : 



a:2+llx + 30. 
X-+ 17x + 60. 
2/2+13^ + 12. 
i/ + 2l!/ + 110. 
?7i2 + zbvi + 300. 
m? + 23»i- + 102. 
a2 + 9«6 + 862, 
a;- + 13ma; + 36??;-^. 



9. j/2 + 19?!?/ + 48n2. 

10. ^- + 2^ + 1002)2. 

1 1 . a-* + 5x- + 6. 

12. u:'' + 4x3 + 3. 

13. a:2i/2 + 18a:i/ + 32. 

14. xh/-\-1x^y'+l2. 
i;. m'o + 10?rt=+16. 



93. 



16. ?i' + 27?i2+ 14032. 

Case IV. To find the factors of 
a;— 9x + 20. 



Our object is to find two negative terms -whose product is 20, 

aud whose sum is — 9. 
These? will clearly he - 5 and - 4, 

.-. x2 - 9.C + 20 = (,.; - 5) (x - 4). 



Exam ples. — xx. 

Resolve into factors : 



x--7x+ 10. 
X- - 29x + 190. 
2/2 - 237/ + 132. 
y- - 30y +- 200. 
7r-43?i+.460. 



6. 'n^-57n + 56. 

7. 3:^-7x^ + 12. 

8. a262-27a6 + 26. 

9. Mc''-ll62c3 + 30. 
10. x-yh--l3xijz + 22. 



92. 



Case V. To find the factors of 
x2 + 5x - 84. 
Our object is to find two terms, one positive and one negative, 
w liose product is - 84, and whose sum is 5. 

These are clearly 12 and - 7, 

.-. x2 + 5x - 8^= (x + 12) (x - 7). 



46 



RESOLUTION INTO FACTORS. 



Examples. 


— xxi. 


Resolve into factors : 






I. x^^'lx-m. 


6. ■ 


62 + 256-150. 


2. a:2 + 12a;-45. 


7- 


a;8 + 3ar*-4. 


3. rt2+n(j_i2. 


8. 


a;V + 3xi/-154. 


4. a2+13a-140. 


9- 


7/i'0+15rr65- 100. 


5. 6- +13?^ -300. 


10. 


7(2 +17,1 -390. 



94. Case VI. To find the factors of 
;«- - 3.C - 28. 

Our object is to find two terms, one positive and one negative, 
whose product is - 28, and whose sum is - 3. 

These will clearly be 4 and - 7, 

.-. a;2-3a;-28 = (2; + 4)(j;-7). 



Examples.— xxii. 

Resolve into factors : 



I. 


■j?-hx- 66. 


2. 


x^ - Ix - 18. 


3- 


ni^ - 9??i - 36. 


4- 


?i2_ 11,^-60. 


5- 


1/-13J/-14. 



2- -152 -100. 
.a-io _ 9.,.5 _ 10. 

c-d-'-24ctZ-180. 
in^'n- - mhi - 2. 



95. The results of the four jnvcvding articles may be tims 
stated in general terms : a trinomial of one of the forms 

X" + ax + h, x^ - ax + h, x- + ax - b, x--ax- h, 

nuiy be resolved into two simple factore, when b can be re- 
solved into two factors, such that their sum, in the fii-st two 
iorms, or tlieir difference, in the last two forms, is equal to a. 

96. We shall now give a set of Miscellaneous Examples oh 
the U'solution into factors of expressions which come under 
one or other of the cases already explained. 



RESOLUTION INTO FACTORS. 47 



Examples. — xxiii. 

Kesolve into factor.s : 

1. a;--15x + 36. 8. a;" + TOx + ?ia; + mu.. 

2. a;^ + 4.o-45. ^ 9. 1/^- 41/^ + 3. 

3. a^W' - 1 Ga6 - 36. i o. a;^ - «&x - cxy + a6c. 

4. a:^- 3?;.'x*- IOjh^. II. a;2 4. (^fj _ j^ ^. _ g5_ 

5. T/^ + 1/^-90. 12. a;- - (c - rf) X - c(i. 

6. x*-a:--110. 13. ab^ - bd + cd - abc. 

7. ar^ + 3«x- + 4ff2x. 14. 4^^ - 28^!/ + 48i/2. . 

97. We have said, Art. 45, that when a number is mnlti- 
plied by itself the result is called the Square of the number, 
and that the figure 2 placed over a number on the right hand 
indicates that the number is multiplied by itsell'. 

Thus a^ is called the square of a, 
(x - yf is called the square oi x-y. 

Tlie Square Root of a given number is that number 
whose square is equal to the given number. 

Thus the square root of 49 is 7, because the square of 7 
is 49. 

So also the square root of a^ is a, because the square of a is 
a^ : and the square root of {x - y)- is x-y, because the square 
of x-y is (x- yy. 

The symbol ^1 placed before a number denotes that the 
square root of that number is to be taken : thus ,j2b is read 
" the square root of 25." 

Note. The square root of a positive quantity may be either 
positive or negative. For 

since a multiplied by a gives as a result a', 
and - a multiplied by - a gives as a result a^, 
it follows, from our definition of a Square Root, that either a 
or - a may be regarded as the square root of a^. 

But throughout this chapter we shall take only the positive 
value of the square root. 



'48 RESOLUTION INTO FACTORS. 



. 98. We may now take the case of Trinomials which are 
verfect squares, which are really included in the cases dis- 
cussed in Arts. 91, 92, but which, from the importance they 
nssume in a later part of our suliject, demand a s^eparate con- 
.-ii deration. 

99. Case VII. To find the factors of 

Seeking for the factors according to the hints given in Art 
9i, we find them to be a; + 6 and x+ 6. 

That is a;2 + 12x + 36 = (x + 6)2. 

Examples. — xxiv. 

Resolve into factors : 

1. a;2+18a; + 81. 6. a;* + 14a;2 + 49. 

2. a;2 + 26a; + 169. 7. a;2+ 10xi/ + 25t/2. 

3. a;2 + 34x4-289. 8. tn!^ + \(5mhi- + QAn*. 

4. y' + 'iy+l. 9. x« + 24.1-3 + 144. 

5. 22 + 2002+10000. 10. x-?/2 + 162x«/ + 6561. 

100. Case VIII. To find the factors of 

.r-^12x + 36. 

Seeking for the factors according to the hints given in Art. 
92, we find them to be x - 6 and x - 6. 

That is, x2 - 12x + 36 = (x - 6)2. 

Examples.— XXV. 

Resolve into factors : 
I. x2-8x + 16. 2. x2-28x + 196. 3. x2-36x + 324. 

4. 2/2 _ 40j/ + 400. 5 . c2 - lOUs + 2500. 6. .i"* - 22x2 + 1 2 1 . 

7. x2 - 30x1/ + 225?/2 8. 7H^-32?/i.2„2 + 256«*. 

n. x«- 38x3 + 361. 



RESOLUTION INTO FACTORS. 49 

101. Case IX. We now proceed to the most important 
case of Resolution into Factors, namely, that in -which the ex- 
pression to be resolved can be put in the form of two squares 
with a negative sign between them. 

Since m^ - n^ = (m + n) (m - n), 

we can express the difference between the squares of tw<i 
quantities by the product of two factors, determined by tlie 
following method : 

Take the square root of the first quantity, and thasquaie 

root of the second quantity. 
The sum of the results will form the first factor. 
The difference of the results will form the second factoi'. 

For example, let a^ - Ir be the given expression. 
Tlie square root of a- is a. 
The square root of h- is h. 
The sum of the results is a + 6. 
The difference of the results is a — b. 

The factors will therefore be a + 6 and a - 6, 
that is, a^-¥ = {a + b) (a - b). 

102. The same method holds good with resjDect to com- 
pound quantities. 

Thus, let a^ - (6 - c)- be the given expression. 
The square root of the first term is a. 
The square root of the second term is 6 - c. 
The sum of the results is a + b-c. 
The difference of the results is a — b + c. 

.. a^- {b-cy = {a + b-c){a-b + c). 

Again, let (a - by - (c - d)- be the given expression. 
The square root of the first term is a - 6. 
The square root of the second term is c - d. 
The sum of tlie results is a-b + c-d. 
The difference of the results is a-b-c + d. 
:. {a-b/- {c-df=(a-b + c-d^ (a-b-c + d). 

[6.A.] • D 



50 kESOLUTION INTO FACTORS. 



103. The terms of an expression may often be arranged 
30 as to form two squares with the negative sign between 
them, and then tlie expression can be resolved into factors. 

Thus a2 + 52_c2_f^2 + 2a6 + 2cd 

= a2 4.2a6 + 62_c2 + 2cd-d2 

= {o? + 2a6 + 62) - (c2 - 2cd + d?) 
= (a + 6)2-(c-d)2 

= (a + 6 + c - c?) (a + 6 - c + (Z). 

Examples. — xxvi. 

Resolve into two or more factors : 
I. x^-y^. 2. x2-9. 3. 4a;2-25. 

4. a* -a;*. 5. x^-,!. 6. a;''_i. 

7. x^-l. 8. m'^-lQ. 9. 36!/2-4922. 

10. Slxhf - 121a262. J I. (a _ 5)2 _ ^2, ■ 12. x^ - (m - nf. 

13. (a + 6)- - (c + (Z)2. 24. 2a;2/-a;2-i/2^1_ 

14. {x + yf-{x-yf. 25. x^-2yz-y--z^. 

15. x2-2xt/ + i/-2!2. 26. a2-462-9c2 4-12ic. 

16. (a-^6)2-(m + Ji,)2. 27. a*- 1661 

17. a2_2ac + c2-62_26ji_(^2_ ^g. l-49c2. 

18. 2bc-b^-c^ + d\ 29. a2 + 62_c2_rf2_2a5_2crf 

19. 2xy + x'^-\-y'^-z^. 30. a- - 6- + ^2 - ^2 _ 2ac + 266?. 

20. 2mn - m2 - %2 ^ ^2 + 52 _ 2^6. 3 1 . Sa'^.r'^ - 27ax, 

2 1 . (ax + byy - 1. 32. a''6'' - c*. 

22. (ax + %/ - (ax - 6?/)2. 33. (5x - 2)- - (x - 4)2. 

23. l-a2-62 + 2a6. 34. {7x + 4yy--{2x + 3y)l 

35. (753)2 -(247)2. 

104. Case X. Since 

=x2-«a: + a2, and -^ = x2 + ax + a2 (Art. 83), 

X ~r d- X — CV 

we know the following important fj^^jts ,• 



RESOLUTION INTO FACTORS. 51 

(1) The suTfi of the cuhes of two numbers is divisible by 
the swm of the numbers : 

(2) The difference, between the^ cvhes of two numbers is 
divisible by the difference between the numbers. 

Hence we may resolve into i'actors expressions in the form 
of the sum or difierence of the cubes of two numbers. 

Thus ic3 + 2 7 = :<? + 33= (./ + 3) (x- - 3x + 9) 

3/3_64 = 2/3-43=(2/-4)0/ + 42/+16). 

Examples.— xxvii. 

Express in factors the following expressions : 
I. a3 + 63. 1. a? ^W. 3. a3_8. 4. a;3 + 343. 

5. 63 _ 125. 6. x3 + 64;/3. 7. a3-216. 8. '6y? + Tif. 

9. 64a3- 100063. 10. 729x3 + 512t/3. 

Express vcijour factors each of the following expressions : 
II. x^-i/^ 12. x^-1. 13. 0*^-64. 14. 729 -j/S. 



105. Before we proceed to describe other processes in 
Algebra, we shall give a series of examples in illustration of 
the principles already laid down. 

The student will find it of advantage to work every example 
in the following series, and to accustom himself to read and to 
explain with facility those examples, in which illustrations are 
given of what may be called tlu short-hand method of expressing 
Arithmetical calculations by the symbols of Algebra. 

Examples. — xxviii. 

1 . Express the sum of a and b. 

2. Interpret the expression a-b + c. 

3. How do you express the double of x ? 

4. By how much is a greater than 5 ? 

5. If a; be a whole number, what is the number next 
above it ? 

6. Write five numbers in order of magnitude, so that x 
phall be the tliird of the five, 



RESOLUTION INTO FACTORS. 



7. If a be multiplied into zero, what is the result ? 

8. If zero be divided by x, what is the result ? 

9. What is the sum of a + a + a . . . written d times ? 

10. if the product be ac and the multiplier «, what is th« 

iiiulli[ilicand? 

1 1 . What number taken from % gives 1/ as a remainder ? 

12. J. is a; years old, and B is y years old ; how old was A 
when B was born ? 

13. A man works every day on week-days for x weeks in 
llie year, and during the remaining weeks in the year he does 
not work at all. During how many days does he rest ? 

14. There are x boats in a race. Five are bumped. How 
many row over the course ? 

15. A merchant begins trading with a capital of x pounds. 
He gains a pounds each year. How much capital has he at 

the end of 5 years 1 

•' _ « 

16. A and B sit down to play at cards. A has x shillings 
and B )j shillings at first. A wins 5 shillings. How much has 
each wiieu they cease to play ? 

17. There are 5 brothers in a family. The age of the eldest 
is X years. Each brother is 2 years younger than the one next 
above him in age. How old is the youngest ? 

18. I travel x hours at the rate of y miles an hour. How- 
many miles do I travel ? 

19. From a rod 12 inches long I cut off x inches, and then 
I cut off y inches of the remainder. How many inches are 
left ? 

20. If n men can dig a piece of ground in q hours, how 
many hours will one man take to dig it ? 

21. By how much does 25 exceed xl 

22. By how much does y exceed 25 ? 

23. If a ]iroduct has 2m repeated 8 times as'a factor, how 
do you express the product ? 

24. By how much does a + 2h exceed a-2bl 

25. A girl is X years of age, how old was she 5 ye.irs since ? 



RESOL UTTON INTO FA C TO RS. 53 

26. A boy is y years of age, how old will he he 7 years 
hence 1 

27. Express the difference between the squares of two 
numbers. 

28. Express the product arising from the multiplication of 
the snui of two numbers into the difference Ijetween the same 
numbers. 



What value of x will make 8a; equal to 16 ? 
What value of x will make 28a; equal to 56 ? 

X 

What value of x will make ^ equal to 4 ? 

What value of x will make a; + 2 equal to 9 ? 
What value of x will make a; - 7 ecpial to 16 ? 
What value of x will make a;- + 9 equal to 34 ? 
What value of x will make a.-- 8 equal to 92 ] 



Examples. — xxix. 

Explain the operations symbolized in the following expres- 
sions : 

1. + 6. 2. (I- -IP-. 3. 4(t2 + fc3_ 4_ 4(f(2^^2)_ 

5. a2_26 + 3c. 6. a + m-Kh-c. 7. ((( + ?h)(6- c). 8. s]^. 
9. ^x- + t/. 10. a + 2(3-c). II. (a + 2)(3-c), 

12. . -,-. 13. -^^^ -^ . 14. — -■- — ^- . 

4ab ■" x-y ^ ^f^ + y 

Examples. — xxx. 

If a stands for 6, b for 5, x for 4, and y for 3, find the val ue 
of the following expressions : 

I. a + x-b~y. 2. a + y-b-x. 3. 3a + 4y-b-2x. 

4. 3(a + 6) - 2(x - y). 5 . (a + x){b- y). 6. 2a + 3 r y). 

7. (2a + 3)(x + y). 8. 2a + Zx + y. 9. ^-^ti'. 

10. abx. II. ab{x + y). 12. ay{b-\-xf. 



54 * kEsdLUTION INTO FACTOkS. 



13- 


ah{x- y)^. 


14. 


v/56. 


1 6. 


isfW- 


17. 


{J^+b)\ 


19. 


^2axy. 


20. 


a^ + ¥ + y 



15- Jy^- 

18. J5bx. 

21. 3a + (2x-i/)2. 

22. |«-(&-i/)f !a-(a;-i/)f. 24. 3(« + 5 -?/)H4(/i4-x)*. 

23. (a-6-2/)2 + (a-a; + 2/)2. 25. 3 (tt - 6)^ + (4x - 1/2)2. 

EXAMPLES. — XXXi. 

1. Find the value of 

Sabc -a^ + P + c^, when a = 3, /> = 2, c=l. 

2. Find the value of 

3? + y^ -z^ + 3xyz, when ic = 3, y = 2, g = 5. 

3. Subtract «'- + c^ from (a + c)-. 

4. Subtract (x - 1/)- from x- + y'^. 

5. Find the coefficient of x in the expression 

{a + byx-{a + bxf. 

6. Find the continued product of 

2x - VI, 2x + n, x + 2m, x - 2n. 

7. Divide 

acr^ + {be + ad) r^ + {bd + ae)r + be by ar + h; 
and test your result by putting 

a = b = c = d = e=l, and r = 10. 

8. Obtain the product of the four factors 

(a + h + c), (b + c-a), {c + a- b), {o. + b-c). 
What does this become when c is zero ; when 6 + c = a; 
when a = b = c'{ 

9. Find the value of 

(a + 6)(6 + c) -{c + d)(d + a)-{a + c) (b-d), ' 

where b is equal to d. 

10. Find the value of 

3a + (26 - c2) + ! (;2 - (2a + 5b) ! + { 3c - (2a + 36)|2, 
wh(;i a = 0, 6 = 2, c = 4. 



RESOLUTION INTO FACTORS. 



11. If a = l. // = 2, c = 3, d=A, shew that the numerical 
values are ec^ual of 

j(Z-(c-& + a)f{(fZ + c)-(6 + o)i, 
and of d^ - (c^ + 6-) + a- + 2 (6c - arf). 

12. Bracket together "the different powers of x in the follow- 
in g expressions : 

(a) ax^ + bx'^ + ex + dx. . 

(/?) ax^ - h:i? - ex"- - dx} + 2x2. 
(7) A'j? - ax^ - 3x- - hi? - 5x — ex. 
(S) {n + x;'-(h-xy. 
(e) {mx- + qx. + 1 )■' - (?!x2 + qx+l )2. 

13. Multiply the three factors x-a, x-b, x-c together, 
and arrange the product according to descending powers of x. 

14. Find the continued product of (x + a) {x + b){x + c). 

15. Find the cube of a + h + c; thence without further 
multiplication the cubes of a + 'j-c; b + c -a; c + a-b; and 
subtract the sum of these three c\\ >e< from the first. 

16. Find the product of (3a + 2b) (3a + 2c - 3b). and test the 
result by making a = 1, 6=:c = 3. 

17. Find the continued product of 

a-x, a + Xj a- + x'-, a'' + a;'*, a* + x^. 

1 8. Subtract (b - a) (c - d) from {a - 6) (c - d). 

What is the value of the -result when a = 26 and tZ = 2c ? 

19. Add together J) + y) (a + x), x-y, ax - by, and a(x + y). 

20. What vaiue of x will make the difference between 
(x + 1) (x + 2) and (x - 1) (x - 2) equal to 54 ? 

21. Add together ax -by, x- y, x(x - ?/), and (a - x) {h - y). 

22. ^T:iat value of x will make the difference between 
(2x + 4) (3x + 4) and (3x - 2) (2x - 8) equal to 96 ? 

23. Add together 

2mx - 3ny, x + y, 4(m + n)(x- y), and mx + vy. 

24. Prove that 

{x + y + z)^ + x'^ + y^ + z- = {x + ij)'^ + {y + z)^ + {x + zy. 



56 RESOUmON INTO FACTORS. 

25. Find the product of (2a + 36) (2a + 3c - 20;, and test the 
result by making a = \, 6 = 4, c = 2. 

26. If a, b, c, d, e ... denote 9, 7, 5. 3, I, find the values of 
ab - cd ., - . 62 _ ^2 

___; (6c-«f?)(W-c.); ^^-; andrt«-c^ 

::/. Find the value of 

3a6c - a^ + Ir + c^ when a = 0, 6 = 2, c = l. 

;3. Find the value of 

.^ , , 2«62 • c^ , , I, 1 n 

3a- H ^:, Avhen a = 4, 6 = 1, c = 2. 

29. Find the value of 

(a-6-c)2 + (6-«-r)-' + (c-a-6)2'when a=l, 6 = 2, c = 3. 

30. Find the value ol' 

(rt + 6 - c)2 + (a - 6 + c)2 + (6+ c-fl)2 when a=;l, 6 = 2, c=4. 

31. Find the value of 

(a + 6)2 + (6 + c-)'- + (e + a)2 when a= - 1, 6 = 2, c= -3. 

32. Shew that if the sum of any two nnmhers divide the 
ditt'erence of their squares, the quotient is equal to the differ- 
ence of the two numbers. 

33. Shew that the product of the sum and difference of anv 
t\Mi numbers is equal to the difference of their squares. 

34. Shew that the square of the sum of any two consecu- 
ti\e integers is always greater by one than four times their 

jjriMluct. 

35. Shew that the square of the sum of any two consecutive 
even whole numbers is four times the square of the odd number 
between them. . 

36. If the number 2 be divided into any two parts, the 
ditlerence of their squares will always be equal to twice liie 
difference of the parts. 

37. If the number 50 be divided into any two parts, tli- 
difference of their squares will always be equal to 50 timdi tli-.- 
difference of the parts. 

38. If a number n be di^dded into any two parts, the 
difference of their squares will always be equal to n times the 
difference of the parts. 



ON SIMPLE EQUA TIONS. 57 

39. If tw'o numbers differ by a imit, their product, together 
with the sum of their squares, is equal to the difference of the 
cubes of the numbers. 

40. Shew tliat the sum of the cubes of any three consecu- 
tive whole numbers is divisible by three times the middle 
number. 



VI. ON SIMPLE EQUATIONS. 

106. An Equation is a statement that two expressions 
are equal. 

107. An Identical Equation is a statement that two ex- 
pressions are equal for all numerical values that can be given 
to the letters involved in them, provided that the same value 
be given to the same letter in every jiart of the eciuation. 

Thus, 0-<; + a)2=a;2-!-2ax + a2 

is an Identical Equation. 

108. An Equation of Condition is a statement that two 
expressions are equal for some particular numerical value or 
values that can be given to the letters involved. 

Thus, a;+l = 6 

is an Equation of Condition, the only number which x can 
represent consistently with this equation being 5. 

It is of such equations tliat we have to treat. 

109. The Root of an Equation is that number which, wlien 
])at in the place of the unknown quantity, makes both sides of 
the equation identical. 

110. The Solution of an Equation is the process of find- 
ing what number an unknown letter must stand for that the 
eipiation may be true : in other words, it is the method of 
fuuling the Eoot. 

The letters that stand for imknown numbers are usually 
X, y, z, but the student must observe tliat any letter may 
stand for an unknown number. 

111. A Simple Equation is one which contains the 
first jiower only of an unknown quantity. This is also called 
^n Equation of the First Decree. 



58 ON SIMPLE EQUATIONS. 

112. The following Axioms form the grounANVork of the 
solution of all equations. 

Ax. I. If equal quantities be added to equal quantities, 
the sums will be equal. 

Thus, if a = &, 

Ax. II. If equal quantities be taken from eaual quantities, 
the remainders will be equal. 

Thus, if x = y, ^ 

x-z = y -z. 

Ax. III. If equal quantities be multiplied b^ equal quan- 
tities, the products will be equal. 

Thus, it a = h. 

Ax. IV. If equal quantities be divided uy equal quantities, 
the quotients will be equal. 

Thus, if xy = xz, 

y=z. 

113. On Axioms I. and II. is founded a process of great 
ntilitv in the solution of equations, called The Traksposition 
OF Terms from one side of the equation "c the other, which 
may be tlius stated : 

" Any term of an equation may be transferred from one side 
of the equation to the other if its sign be changed." 

For let x-a = h. 

Then, bv Ax. I., if we add a to both udes, the sides remain 
equal : 

therefore x-a + a = b + a, 

that is, x = b + a. 

Again, let x + c = (l. • 

Then, by Ax. II., if we subtract c liom ^u,•ih. side^ the sides 
remain equal : 

therefore iC + c~c = d~c, 

l.uat is, x=d-c, 



ON SIMPLE EQUATIONS. 59 



114. We may change all the signs of each side of an equa- 
tion without altering the equalit}'. 

Thus, if a-x — h-c, 

x-a = c-b. 

115. We may change the position of the two sides of the 
e(|nation, leaving the signs unchanged. 

Thus the equation a - b = x - c, may be written thus, 
X- c = a -b. 

116. We may now proceed to our first rule tor the solution 
of a Simple Equation. 

Rule I. Transpose the known terms to the right hand side 
(>f the equation and the unknown terms to the other, and com- 
I'ine all the terms on each side as far as possible. 

Then divide both sides of the equation by the coefficient of 
the unknown quantity. 

This rule we shall now illustrate by examples, in which x 
stands for the unknown quantity. 

Ex. 1. To solve the equation, 

5x - 6 = 3x + 2. 
Transposing the terms, we get 

5x - 3x = 2 + 6. 
Combining like terms, we get 

2x = 8. 
Dividing both sides of this equation by 2. we get 
x = 4, 
and the value of x is determined. 

Kx. 2. To solve the equation, 

7x + 4 = 25x - 32. 
Transposing the terms, we get 

7x~25x= -32-4. 
Combining like terms, we get 

-18x=-36. 
Changing the signs on each side, we get 

18x = 3t). 
Dividing both sides V)y 18, we get 
x = 2, 
and the value of x is determined. 



6o 



OX SIMPLE EQUATIONS. 



Ex. 3. To solve the equation, 

2a; - 3a: + 120 = 4j; - 6x+ 132. 



that is, 






2x - 3x - 4x + 6a; = 13z - liv/, 


or, 






• 8x-7x=I2, 


therefore, 




x=12. 


U.X. 


4. 


To solve the equation, 








3a; + 5-8(13-a;) = 0, 


that is, 






3x + 5-104 + 8x==0. 


or. 






3a; + 8x=104-5, ' 


or. 






llx=99, 


therefore, 




a;=9. 


Ex. 


5. 


To 


f.olve the equation, 
()a;-2(4-3x) = 7-3(17 -■<, 


that is, 






6a;- 8 + 6x= 7 -51 + :).£, 


or, 
or, 






6x4-6:c-3a;=7-51+8, 
12x-3a;=15-51, 


or, 






9x= -36, 


therefore. 




x= -4. 



EXAMPLES.— :?cxxii. 



9. 26-8x = 80-1435. 

10. 133-3x = x-83. 

11. 13-3x = 5x- o. 

12. 127 + 9x= 12x4-100. 

13. 15-5x=6-4x. 

14. 3./ -22 = 7x-J-6. 

15. 8 + 4x=12.c-16. 

1 6. 5.r - (3x - 7) = 4x - ^6x - 3o). 

17. 6x - 2(9 - 4x) + 3 (5x - 7'i = lOx - (4 + 16x) + 35. 

18. 9x-3(5x-6) + 30 = O 

19. 12x - 5 (9x + 3) + 6(7 - 8x) + 783 = 0. 

20. X - 7(4x - 11) = 14(x - 5) - 19(8 - x^ - € :. 

21. ^T + 7)(x-3) = (x-5)(x-15). 



1. 7x + 5 = 5x+ll. 

2. ]2x + 7 = 8x + 15. , 

3. 236.c+425 = 97x + 564 

4. 5x - 7 = 3x + 7. 

5. 12x-9 = 8x-l, 

6. 124x+19 = 112x + 43. 

7. 18- 2^=27 -5x. 

8. 125-7x=145-12.''. 



PROBLEMS LEADING TO SIMPLE EQUATIONS. 6i 



22. (x-8)(x + 12)=(ic+l)(a;-6). 

23. {x - 2)(7 - x) + (x - 5) (x + 3) - 2(x - 1) + 12 =^ 0. 

24. (2a! - 7) (x + 5) = (9 - 2.r) (4 - x) + 229. 

25. (7 - 6x) (3 - 2x) = (4.C - 3) (3j; - 2). 

26. 14 - a; - 5 (x - 3) (x + 2) + (5 - x) (4 - Sx'i = 45.r - 76. 

27. (x + 5)2-(4-x)-=21x. 

28. 5(x - 2)2 + 7(x - 3)2 = (3x - 7)(4.t - 19, + 42. 

29. (3x - 17)2 + (4x - 25)2 _ (5_^ _ 09)2 = ] . 

30. (x + 5) (x - 9) + (x + 10)(x - 8) = (2x + 3) (x - 7) - 1 13. 



VII. PROBLEMS LEADING TO SIMPLE 
EQUATIONS. 

117. When we have a question to resolve by means 01 
Algebra, we represent the number sought by an unknown 
symbol, and then consider in what manner the conditions of 
the question enable us to assert thot tv:o exjjressiotis are equal. 
Thus we obtain an equation, and by resolving it we determine 
the value of the number sought. 

The wliole difficulty connected with the solution of Alge- 
braical Problems lies in the determination from the conditions 
of the question of tiro different expressions having the same 
numerical value. 

To explain this let us take the following Problem : 

Find a number sucli that if 15 be added to it, twice the sum 
will be equal to 44. 

Let X represent the number. 

Then x + 15 will represent the number increased by 15, 
fu I 2(x + 15) will represent twice the sum. 

But 44 will represent twice the sum, 
therefore 2 (x + 15) = 44. 

Hence 2x4-30 = 44, 

tliatis, *2x=14, 

or, x=7, 

and therefore the number sought is 7. 



62 PROBLEMS LEADING TO SIMPLE EQUATIONS. 

118. We shall now give a series of Easy Problems, in 
which the conditions by which an equality between two expres- 
sions can be asserted may be readily seen. The student should 
be thorouffhly familiar with the Exanijdes in set xxviii, the use 
of which he will now find. 

We shall insert some notes to explain the method of repre- 
senting quantities by algebraic symbols in cases where some 
difficulty may arise. 

Examples. — xxxiii. 

1. To the double of a certain number I add 14 and obtain 
as a result 154. What is the number ? 

2. To four times a certain number I add 16 and obtain as 
a result 188. What is the number ? 

3. By adding 46 to a certain number I obtain as a result a 
number three times as large as the original number. Find the 
original number. 

4. One number is three times as large as another. If I 
take tlie smaller from 16 and the greater i'rom 30, the remaiii- 
deis are equal. What are the numbers % 

;. Divide the number 92 into four parts, such that the first 
is greater than the second by 10. greater than the third by 18, 
and greater than the fourth by 24. 

6. Tlie sum of two numbers is 20, and if three times the' 
smaller number be added to five times the greater, the sum is 
84. What are the numbers ? 

7. Tlie joint ages of a father and his son are 80 years. If 
the nge of the son were douliled he would be 10 years older 
than his father. What is the age of each? 

8. A man has six sons, each 4 years older than the one 
next to Jiim. The eldest is three times as old as the youngest. 
Wiiat is the age of each ? 

9. Add .£24 to a certain sum, and the amount ^dll be as 
much above ^80 as the sum is below ^80. What is the sum \ 

10. Thirty yards of cloth and lorty yards of silk together 
cost £66, and the silk is twice as valuable as the cloth. Find 
the cost of a vard of each. 



PROBLEMS LEADING TO SIMPLE EQUATIONS. 63 



11. Find the number, the double of which being added to 
24 the result is as much above 80 as the number itself is below 
100. 

12. The sum of ^500 is divided between A, B, C and D. 
A and B have together ^280, A and G X260, A and D ^'220. 
How much does each receive ? 

13. In a company of 266 persons, composed of men, women, 
and children, there are twice as many men as there are women, 
and twice as many women as there are children. How many 
are there of each ? 

14. Divide i'1520 between A, B and C, so that A has ^100 
less than B, and B i;'270 less than C. 

15. Find two numbers, differing by 8, such that four time? 
the less may exceed twice the greater by 10. 

16. A and B began to play with equal sums. A won £0, 
and then three times ^-I's money was equal to eleven times B'a 
money. What had each at first ? 

17. A is 58 years older than B, and ^'s age is as much 
above 60 as B's age is below 50. Find the age of each. 

18. yl is 34 years older than B, and A is as much above 50 
as B is below 40. Find the age of each. 

19. A man leaves his property, amounting to J7500, to be 
divided between his wife, his two sons and his three daughters, 
as follows : a son is to have twice as much as a daughter, and 
the wife .£500 more than all the five children together. How 
much did each get ? 

20. A vessel containing some water was filled up by pour- 
ing in 42 gallons, and there was then in the vessel 7 times as 
much as at first. How many gallons did the vessel hold 1 

21. Three persons. A, B, C, have .£76. B has .£10 more 
ihan A, and C has as much as A and B together. How much 
lias each ? 

22. Wliat two numbers are those whose difference is 14, 
and their sum 48 ? 

23. A and B play at cards. A has £72 and B has £52 
when they begin. When they cease playing, A has three times 
as much as B. How much did A win ? 



64 PROBLEMS LEADING TO SIMPLE EQUATIONS. 



Note I. If we have to express algebraically two parts into 
which a yiven number, suppose 50, is divided, and we repre- 
sent one of the parts by x, the other will be represented by 
:.() - X. 

Ex. Divide 50 into two such parts that the double of one 
\rAxt may be three times as great us the other part. 

Let X represent one of the parts. 

Then 50 - x will represent the other part. 

Now the double of the first part will be represented bv 
2x, and three times the second part will be represented by 
3 (50 - x). 

Hence 2a; = 3 (50 -x), 

or, 2x=150-3x, 

or, 5a; = 150; 

.-. x = 30. 

Hence the parts are 30 and 20. 

24. Divide 84 into two such parts tliat three times one part 
may be equal to four times the other. 

25. Divide 90 into two such parts that four times one part 

may lie equal to five times the other. 

26. Divide CO into two such parts that one part is greater 
tlian llie other by 24. 

27. Divide 84 into two such parts that one part is less than 
t'.ie (idler by 36. 

28. Diviile 20 intn two sucli parts that if three times one 
1 art be added to five times the other part the sum may be S4. 

Note 1 1. When we have to compare the ages of two per- 
sons at one time and also some years alter or before, we must 
lie caretul to remember that hoih will be so many years older 
or younger. 

Thus if X be the age of .4 at the present time, and 2* be 
the age of B at the present time, 

The age of .4 5 years hence will be a; + 5, 
an<l the age of B 5 years hence \\ iil be 2j + 5. 



PROBLEMS LEADING TO SIMPLE EQUATIONS. 65 



Ex. ^ is 5 times as old as B. and 5 years hence A will 
only be three times as old as B. What are the ages of A and 
B at the present time ? 

Let X represent the age of B. 

Then bx will represent the age of A. 

Now a; + 5 will represent £'s age 5 years hence, 
and 6x + 5 will represent ^'s age 5 years hence. 

Hence 5x + 5 = 3 (x + 5), 

or 5x + 5 = 3x+15, 

or 2x=10; 

.'. x = 5. 

Hence A is 25 and 5 is 5 years old. 

29. A is twice as old as B, and 22 years ago he was tliree 
times as old as B. What is yl's age ? 

30. A father is 30 ; his son is 6 years old. In how many 
years will the age of the father be just twice that of the son \ 

31. A\% twice as old as B, and 20 years since he was three 
times as old. What is £'s age ? 

32. A is three times as old as B, and 19 years hence he will 
be only twice as old as B. What is the age of each ? 

33. A man has three nephews. His age is 50, and the 
joint ages of the nephews are 42. How long will it be before 
tlie joint ages of the nephews will be ec^ual to the age of the 
uncle \ 

Note III. In problems involving weights and measures, 
after assuming a symbol to represent one of the unknown 
quantities, ^ve must be careful to express the other quantities 
in the same terms. Thus, if x represent a number of pence, all 
the sums involved in the problem 7nust be reduced to pence. 

Ex. A sum of money consists of fourpenny pieces and si.x- 
pences, and it amounts to £1. IBs. 8d. The iiuniber of coins 
is 78. How many are there of each sort ? 

[s.A.] S 



66 PROBLEMS LEADING TO SIMPLE EQUATIONS. 

Let X be the number of l'ouT2)enny pieces. 

Then Aj-y is their value in fence. 

Also 78 — X is the number of sixpences. 

And 6 (78 — X) is their value in 'pence. 

Also £\.. 16s. 8(Z. is eqtiivalent to 440 pence. 

Hence 4a; + 6 (78 - a) = 440, 

or 4a; + 468- Gx = 440, 
from which we find x= 14. 

Hence there are 14 fourpenny pieces, 
and 64 sixpences. 

34. A bill of ^100 was paid with guineas and half-crowns, 
and 48 more hulf-ciowus than guineas were used. How many 
of eacli were paid ? 

35. A person paid a bill of £3. 14s. w'ith shillings and 
hall-crowns, and gave 41 pieces of money altogether. How 
many of each were paid ? 

36. A man has a sum of money amounting to £11. 13s. 4d., 
consisting only of shillings and fourpenny pieces. He has in 
all 300 pieces of money. How many has he of each sort ? 

37. A bill of .£50 is paid with sovereigns and moidores of 
27 shillings each, and 3 more sovereigns than moidores are 
given. How many of each are used ? 

38. A sum of money amounting to £42. 8s. is made up of 
shillings and half-crowns, and there are six times as many 
half-crowns as there are shilling-. How many are there of 
each sort ? 

39. I have £5. 1 1*-. 3(/. in sovereigns, shillings and pence. 
I have twice as many shillings and three times as many pence 
as I have sovereigns. How manv have I of each sort J 



YIII. ON THE METHOD OF FINDING 
THE HIGHEST COMMON FACTOR. 

119. An expression is said to be a Factor of another 
expression wiieii the latter is divisible by the former. 

Thus 3a is a factor of 12a, 
5xy of lox^y\ 

120. An expression is said to be a Common Factor of two 
or more other expressions, when each of the latter is divisil)le 
by the former. 

Thus 3a is a common factor of 12a and 15a, 

3xy of Ibx^y^ and 2\x^y^, 

4z of 82, 12^2 and I6z^ 

121. The Highest Common Factor of two or more expres- 
sions is the expression of highest dimensions by which each of 
the former is divisible. 

Thus 6a2 is the Highest Common Factor of 12a2 and 18a^, 

bx^y of 10x^1/, 15x^2/2 

and 25x*2/^ 

Note. That which we call the Highest Common Factor is 
named by others the Greatest Common Measure or the Highest 
Common Divisor. Our reasons lor rejecting these names will 
be given at the end of the chapter. 

122. The words Highest Common Factor are abbreviated 
thus, H.C.F. 

123. To take a simple example in Arithmetic, it will 
readily be admitted that the highest number which will 
divide 12, 18, and 30 is 6. 

Now, 12 = 2x3x2, 

18 = 2x3x3, 
30=-2x3x5. 



68 MET HOP Oh h IN DING THE 

Having thus reduced the numbers to their sirajylht factor?, 
it appears that we may determine the Highest Common Factoi- 
in the i'ollowing way. 

Set down tlie factors of one of the numbers in any order. 

Place beneath theiii tlie factors of the second number, in 
!-uch order tliat fact(jrs like, any of those of the first number shall 
stand under those factors. 

Do the same for the third number. 

Then the number of vertical columns in which the numbers 
are alike Avill be the number of factors in the h.c.f., and if 
we multiply the figures at the head of those columns together 
the result will be the h.c.f. required. 

Thus in the example given above two vertical columns are 
alike, and therefore there are two factors in the h.c.f. 

And the numbers 2 and 3 which stand at the heads of 
those columns being multiplied together will give the H.C.F. 
of 12, 18, ana 30. 

124. Ex. 1. To find the h.c.f. of aW-x and a%^X'. 
aWx = aaa .bb .x, 
aWx^ = aa . bbb . xx ; 
:. 'E.c.F.=aabbx 
= a'b^x. 

Ex. 2. To find the h.c.f. of 34a26»c* and 51a%*c*. 

Ma%^c* = 2 X 17 xaa . bhbbbb . cccc, 
5la%*c- = 3 X 17 X aaa . bbbb . cc ; 
.*. B..c.F. = 17 aabbbbcc 
= 17a--'6V. 



Examples.— xxxiv. 



Find the Highest Common Factor of 

\. a*h iuu\ a'b^. 3. 14x-_ir' and 24.r^. 

3. a^yh and x'-y-z-. 4. -ibm-ny and mmhip*. 



HIGHEST COMAfOA' FACTOR. 69 

5. 18rt7)'-c'd and Z^a^hcd?. 8. 17 pq~, 'i4p^q and 5\p^q^. 

6. «3?)-, arb^ and a^6*. 9. Sx^j/Sg*, Ux^z^ and 20x*3/V. 

7. 4a7), lOac and 305c. 10. 3(teY, QOx^ and 120x3t/4 

125. The student must he urged to commit to memory the 
following Table of furms which can or cannot be resolved into 
factors. Where a blank occurs after the sign = it signifies 
tliat the form on the left hand cannot be resolved into simpler 
factors. 

x^-y^ = {x + y){x-tj) x^-l=(x+l)(x-l) 

x^ + y^= a:- + 1 = 

x^ — y^ = {x — y){x'^ + xy + y^) x?-l = {x—l)(x^ + x + l) 

3^ + y^ = (x + y){x--xy + y^ a^+ l = (x+ 1) (a;^ — x + l) 

a^-y* = {x'^ + y^-){x'^-y^) x*-l = {x^+l: {x^-1) 

x* + y^= a;* + 1 = 

x^+2xy + y' = {x + yy x'^-h2x+l={x+iy 

x^-2xy + y^ = {x-yf x--2x+l = {x-iy 

05^ + 'Ax^y + Sxy^ + y^ = {x + yY x? +.3x2 + 3x + 1 = (x + 1)3 

X? - 3x2?/ + 33.^^2 _yZ^(^^_ yy x^ - 3x2 + 3x - 1 = (x - 1)3 

The left-hand side of the table gives the general forms, the 
right-hand side the particular cases in which 1/= 1. 

126. Ex. To find the h.c.f. of x^-l, x2-2x-H, and 
a;2 + 2x-3. 

X2-I=(x-I)(x+1), 

x2-2x + l = (x-l)(x-l), 
x2-i-2a;-3 = (x-l)(x-l-3), 

.". H.C.F. =X-1. 

Examples. — xxxv. 

1. a2 - 52 and a^ - b\ 4. a^ + a^ and (a -I- x)'. 

2. a- — h- and a* — b*. 5. 9x2 _ i ^j^^j ^^x + 1)2. 

3. a2_x2 and (a — x)2. 6. 1 -25a^ and (1 — bay. 

7. x2 - 1/2, (x + yY and x2 4- Zxy + 2y'. 

8. x2 — y^, x^ — y^ and x2 — Ixy + 6?/2. 

9. x2 — 1, x3 — 1 and x2 + x - 2. 
10. 1 — a"^, \ + a^ and a2 + 5^ -1- 4. 



METHOD OF FINDING THE 



127. In large numbers the factors cannot often be deter- 
mined by inspection, and if we have to find the h.c.f. of two 
such numbers we have recourse to the following Arithmetical 
Rule : 

" Divide the greater of the two numbers by the less, and the 
divisor by the remainder, repeating the process until no rr 
mainder is left : the last divisor is the h.c.f. required." 

Thus, to find the h.c.f. of 689 and 1573. 
689; 1573(2 
1378 

T95;689(3 
585 

Tb4; 195(1 
104 



9i; 104(1 
91 

13; 91 (7 
91 

.-. 13 is the H.i F. of 689 and 1573. 

Examples.— xxxvi. 

Find the h.c.f. of 

1. 6906 and 10359. 4. 126025 and 40115. 

2. 1908 and 2736. 5. 1581227 and 16758766. 

3. 49608 and 169416. 6. 35175 and 236845. 

128. The Arithmetical Rule is founded on the following 
)peration in Algebra, which is called the Proof of the Rule foi 
finding the Highest Common Factor of two expressions. 

Let a and h be two expressions, arranged according to de- 
scending powers of some common letter, of which a is not of 
lower dimensions than h. 

Let h divide a with -p as quotient and remainder c, 

c h g A. 

d c r with no remainder. 



HIGHEST COMMOX J-ACTOR. 71 

The form of the operMtiun may be shewn thus : 
fb 

d) c (r 
rd 

Then we can shew 

!. That rf is a common factor of a and 6. 

II. That any other common factor of a and 6 is a factor of 
rf, and that therefore d is the Higliest Common Factor 
of a and b. 

For (I.) to shew that rf is a factor of a and b : 
b = qc + d 
= qrd + d 
= {qr + I) d, and .'. d is a factor ot b ; 

and a=|j6 + c 

—P (3<^ + d) + c 

= pqc+pd + c 

=pqrd+pd + rd 

= {pqr+p + r) d, and .•. d is a factor of a. 

And (11.) to shew that any common factor of a and b is a 
factor of d. 

Let 8 be any common Factor of a and b, such that 
a = viS and b = n8. 

Then we can shew that 8 is a factor of d. 

For d = b-qc 

= b~q(a-pb) 

= b - qa + pqb 

=n8 - qm8 + pqnS 

= {n-qm +pqn) 8, and .". 8 is a factor of d. 

Now no expression higher than d can be a factor of d ; 
:. d is the Highest Common Factor of a and b. 



72 METHOD OF FINDING THE 



1 29. Ex. To find tlie h.c.f. of x'- + 2a; + 1 and 

x-"' + 2x- + 2a- -^ I. 

a;2 + 2x+l_;a;3 + 2x2 + 2x4-l(x 
x3 + 2x2 + X 



x+l^x2 + 2x+ 1 (,x+ 1 
x^ + x 



x+1 
x+ 1 

Hence x+1 being the last divisor is the h.c.f. required. 

130. In tlie algelnaical process four devices are frecjuently 
useful. Tliese we shall now state, and exemplify each iu the 
next Article. 

I. If the sign of the first term of a remainder be negative, 
we may change the signs of all the terms. 

II. If a remainder contain a factor which is clearly not a 
common factor of the given expressions it may l)e 
removed. 

III. We may'nmliiply or divide either of the given expres- 
sions by any number which does not introduce or 
remove a common factor. 

IV. If the given expressions have a common factor which 
can be seen by inspection, we may remove it from 
both, and find the Highest Common Factor of the 
parts which remain. If we inultiply this result by 
the ejected factor, we shall obtain the Highest Com- 
mon Factor of the given expressions. 

131. Ex. I. To find the h.c.f. of 2x2 - x - 1 and 

6x2 -4x- 2. 

2x«-x-i;6x2-4x-2(3 
6x2 -3x- 3 

- x + 1 



HIGHEST COMMON FACTOk. ^% 



Change tlie signs of the remainder, and it becomes x— 1. 

• a!-i;2!fc2-x_ 1(2x4-1 

2x2 _ 2x 



x-1 
x-1 

The H.c.F. required is x — 1. 

Ex. 1 1 . To find the h.c.f. of x^ + 3x + 2 and x^ + 5x + 6. 

x2 + 3x4-2;x2 + 5x-l-6(l 
x2 + 3x + 2 

2x + 4 

Divide the remainder by 2, and it becomes x + 2. 
x + 2;x2 + 3x + 2(x+l 
x2 + 2x 



x + 2 
x + 2 



The H.c.F. required is x + 2. 

Ex. III. TofindtheH.c.F.of 12x2 4.x-land 15x"+8x + l. 

Multiply 15x2 + 8x + l 

by 4 



12x2 + a; - 1> 60x2 + 32a; + 4 (5 
60x2+ 5a;_5 

27x + 9 

Divide the remainder by 9, and the result is Zx-k-l. 
3x+i;i2x2 + x-l(4x-l 

12x2+ 4x 

^x^ 
-3x-l 

The H.c.F. is therefore 3x + l. 

Ex. IV. To find the h.c.f, of x^ — 5x2 + 6x and 

x--Kte2 + 21x. 

Remove and reserve the factor x, which is cotuiiKJii tu both 
expressions. 



U METHOD OF FINDING TfTK 

Then we have remaining x^ — 5x + 6 and x- — lOx + 21. 

The H.c.F. of these expressions is x — 3. 

The H.c.F. of the original expressions is therefoie x^ — 3x 

Examples.— xxxvii. 

Find the h.c.f. of the following expressions : 

1. x2 + 7x + 12 and x2 + 9x + 20. 

2. x2 + 12x + 20 and x^ + 14x + 40. 

3. x2 - 17x + 70 and x2- 13x4-42. 

4. x2 + 5x-84andx2 + 21x+108. 

5. X- + X— 12 and x^ — 2x-3. 

6. x^ + 5x2/ + ^y^ ^^^ ^^ + ^^V + 9j/^- 

7. x^ - 6x?/ + 8j/- and x^ — Sxy + 16?/^. 

8. x2 - 13x1/ - 30?/2 and x^ - 18xt/ + 45i/2. 

9. x^ — y^ and x- — Ixy + 1/-. 

10. x^ + \f and x^ + 3x-i/ + 3x?/2 + 1/^. 

11. X* — 1/* and X- — 2x1/ + ^2. 

12. x^ + 1/^ and x^ + y^. 

13. X* — 2/* and 3? + 2x1/ 4 y"^. 

14. a^ _ 52 ^ 26c - c- and a- 4- 2a?> 4- 6- - 2ac - 26c 4- c-. 

1 5. 12x2 4- Ixy 4- 2/" and 28.C- 4- 3x!/ — y-. 

1 6. 6x- 4- xxj - 1/2 and 39x2 _ 22x1/ 4- 3 j/*. 

17. 1 5x2 — 8x1/ 4- 1/2 and 40x2 — 3x1/ — 1/2. 

18. x*-5x3 4-5x2-l and x*4-x3-4x2 4-x4-l. 

19. X* + 4x2 + 1 6 an,| x5 + X* - 2x3 4- 1 7^.2 _ 1 Ox 4- 20. 

20. X* 4- x2(/2 4- 1/* and x* 4- 2x''i/ 4- 3x2)/2 4- 2xi/' 4- y*. 

21. x" - Gx^ 4- 9x2 - 4 ^mi x^4-x^-2x*4-3x2-x-2. 



ftlGHEST COMMON FACTOR. 75 

22. 1 5a* + lOa^ft + ^aW + Gai^ - 36* and Ga^ + IQa^i + 8a62 _ 563. 

23. ISa:^ - Hcc^?/ + 24a;y2 - Ti/^ and 27x3 + SSx^j/ - 20x1/2 + gi/^. 

24. 21x2 - 83xy - 27x + 22!/2 + 99;,' and 12x2 _ 353.^ _ ^y. 

-33t/2 + 227/ 

25. 3a3-12a2-a26 + 10rt6-262and %a^ -\~ia?-h^M}P--W. 

26. 1 8a3 _ I8a2x + 6ax'' - Gx^ and 60a2 - 75ax + 15x2. 

27. 21x3-26x2 + 8xand 6x2-x-2. 

28. 6x* + 29n2x2 + 9rt* and Sx^ - 15ax2 + a^x - ou\ 

29. x^ + x^i/2 + x^y + T/3 and x* — ?/*. 

30. 2x3 + 103-2 + 14a; + 6 and x^ + x2 + 7x + 39. 

3 1 . 45a3a; + 3a2x2 - 9ax3 + 6x* and 1 8a2x - Sx\ 

132. It is sometimes easier to find the h.c.f. hj reversing 
the order in which the expressions are given. 

Thus to find the h.c.f. of 21x2 + 38x + 5 and 129x2 + 221x+ 10 
the easier course is to reverse the expressions, so that thev 
stand thus, 5 + 38x + 21x2 and 10 + 22 lx+ 129x2, and jjjgj^ j-^ 
proceed by the ordinary process. The h.c.f. is 3x + 5. Other 
examples are 

(1) 187x3 - 84x2 + 31x - 6 and 253x3 - 14x2 ^ 29x - 12, 

(2) 371^/3 + 262/2-50?/ + 3 and 469i/ + 7oi/ - 103?/ - 21, 
of which the h.c.f. are respectively llx — 3 and 7?/ + 3. 

133. If the Highest Common Factor of three expressions 
a, b, c be reciuired, find first the h.c.f. of a and b. If d be the 
h.c.f. of a and b, then the h.c.f. of d and c will be the h.c.f. 
of a, b, e. 

i«%. Ex. To find the h.c.f. of 

x'3 + 7x2 - X - 7^ ^.3 ^. 5^.2 _ 2; _ 5^ and x2 - 2x + 1. 

The H.C.F. of x3 + 7x2 - X - 7 ^nd x3 + 5x2 - x - 5 will be found 
to be x2 — 1. 

The H.C.F. of x2-l and x2-2x+l will be found to be 
x-1. 

Pence x— 1 is the h.C-f. of the three expressiona. 



76 FRACTIONS. 



Examples. — xxxviii. 

Find the Highest Common Factor of 

1. a;2 + 5x + 6, x2 + 7x+10, and a;2 + i^^20- 

2. x3 + 4x2-5, a;3-3x + 2, andx3 + 4x2-8x + 3. 

3. 2x2 + x-l, x2 + 5x + 4, and x^-i-l. 

4- V^-'f-V^^-, 3l/2-2i/-l, and 1/3 -2/2 + 2/ -1. 

5. 3?- 4x- + 9x - 10, x^ + 2x2 _ 3x + 20, ami 

x^* + Sx'" - v»x -»- S."). 

6. x3 _ 7a,2 + i6x - 12, 3x3 - 143.2 + igj.^ ^^^,1 

5x3-10x- + 7x-14, 

7. ■j/3 — 5?/2 + ii^_ 15j y3_y2^3y^.5 aj^Q 

2i/3-72/-+ieT/- 15. 

XoTE. We use the name Highest Common acrm n..«Tead 
of Greatest Common Measure or Highest Common Divisor lor the 
following reasons : 

(1) We liave used the word " Measure '• in An. a 
different .*ense, that is, to denote the number of tiW^~ any 
quantity contains the ^lnit of measurement 

(2) Divisor does not necessarily imply a quanntv wuich 
is contained in another an exact number of times. '1 nus in 
performing the operation of dividing 333 tiy 13, we can 13 
divisor, but we do not mean that 333 coutains 13 au eicact 
number of times. 



IX. FRACTIONt 

135. A QDANTITY a is called an Exact i)ivisoK 01 m civwn- 
tity b, when h contains a an exact number :)i mnes. 

A quantity a is called a Multiple of a (juanniy 0, M^acu a 
contains b an exact number of times. 



■FJHACTIOXS. 77 

136. Hithori-o we have treated of quantities wliicli coutniii 
the unit of »»^«^«iiremeut in each case an exact ninube:- of 
times. 

We have p'^w to treat of quantities which contain some exact 
divisor of a pnmary unit an exact number of times. 

137. We must first explain what we mean by a primary 
unit. 

We said in Art. 33 that to measure any quantity we take a 
known standard or unit of the same kind. Our choice as to 
the quantity to be taken as the unit is at first unrestricted, but 
when once made we must adliere to it, or at least we must 
give distinct notice of any change which we make with re.spect 
to it. To such a unit we give the name of Primary Unit. 

138. Next, to explain what we mean by an exact divisor of 
a primary unit. 

Keeping our Primary Unit as our main standard of mea- 
surement, we may conceive it to be divided into a number of 
parts of equal magnitude, any one of which we may take as a 
Subordinate Unit. 

Thus we may take a pound as the unit by which we mea- 
sure sums of money, and retaining this steadily as the primary 
unit, we may still conceive it to be subdivided into 20 equal 
parts. We call each of the subordinate units in this case a 
shilling, and we say that one of these equal subordinate units is 
one-twentieth part of the primary unit, that is, of a pound. 

These subordinate units, then, are exact divisors of the 
primary unit. 

139. Keei^ing the primary unit still clearly in view, we 
represent one of the subordinate units by the followinf' nota- 
tion. 

We agree to represent the words one-third, one-fifth, and 

one-twentieth by the symbols ^, -, — , and we say that if 

the Primary Unit be divided into three equal parts, - -will 
represent one of these parts. 



78 FRACTIONS. 



If we have to represent two of these subordinate units, we 

2 3 

do so by the symbol - ; if three, by the symbol - ; if four, by 

o o " 

4 
the symbol -, and so on. And, generally, if the Primary Unit 

be divided into h equal parts, we represent a of those parts by 

the symbol ■ . 

140. The symbol t we call the Fraction Symbol, or, more 

briefly, a Fraction. The number helow the line is called the 
Denominator, because it denominates the number of equal 
parts into which the Primary Unit is divided. The numbi-r 
above the line is called the Numerator, because it enumerates 
how many of these equal parts, or Subordinate Units, are 
taken. 

141. The term number may be correctly applied to Frac- 
tions, since they are measured by units, but w'e must be 
careful to observe the following distinction : 

An Integer or Whole Number is a multiple of the Primary 

Unit. 
A Fractional Number is a multiple of the Subordinate 

Unit. 

142. The Denominator of a Fraction shews what multiple 
the Primary Unit is of the Subordinate Unit. 

The Numerator of a Fraction shews what multiple the 
Fraction is of the Subordinate Unit. 

143. The Numerator and Denominator of a fraction are 
called the Terms of the Fraction. 

144. Having thus explained the nature of Fractions, we 
next proceed to treat of the operations to which they are sub- 
jected in Algebra. 

145. Def. If the quantity x be divided into b equal parts, 
and a of those parts be taken, the result is said to be the 

fraction ,- of x. 


Jfxhe the unit, this is called tlie fraction j-. 



FRACTIONS. 79 

146. If the unit be divided into b equal parts, 
y will represent one of the parts. 

r two 



T three 



And generally, 

T will represent a of the parts. 

147. Next let us suppose that each of the b parts is sub- 
divided into c equal parts : then the unit has been divided 
into be equal parts, and 

T- will represent one of the subdivisions. 
-=— two 

DC 

And generally, 

a 

— a 

be 

148. To shew that r = t- 

be b 

Let the unit be divided into b equal parts. 

Then j- will represent a of these parts (1). 

Next let each of the b parts be subdivided into c equal 
parts. 

Then the primary unit has been divided into be equal parts, 

and -J— will represent ae of these subdivisions (2). 

Now one of the parts in (1) is equal to c of the subdivisions 
in (2), 

.'. a parts are equal to ac subdivisions ; 
, a ac 
"'b^'k' 



8o FRACTIONS. 



Cor. We draw Irom this proof two inferences : 

I. If tlie numerator and denominator of a fraction be 
multiplied by llie same number, the vahie of the frac- 
tion is not altered. 

II. If the numerator and denominator of a fraction be 
divided by the same number, the value of the fraction 
is not altered. 

149. To make the important Theorem established in the 
preceding Article more clear, we shall give the following proof 

that K = o7x, ^y taking a straight line as the unit of length. 

I I I I I I I I I I M I I I I I I I I I 

A E D F B C 

Let the line AG be divided into 5 equal parts. 

Then, if B be the point of division nearest to C, 

AB is I of AC. (1). 



Next, let each of the parts be subdivided into 4 equal parts 

Then AG contains 20 of these subdivisions, 
and AB 16 

:. ABi^^^oiAJ. (2). 

Comparing (1) and (2), we conclude that 

4^1^ 

5~20" 

150. From the Theorem established in Art. 148 we derive 
the following rule for reducing a fraction to its lowest terms : 

Find the Highest Covimon Factor of the numerator and denomi- 
nator and divide both by it. The res^ilting fraction vnll be 
one equivalent to the original fraction expressed tJi the simplest 
terms. 



FRACTIONS. 8i 



151. When the numerator and denominator each consist of 
a single term the h.c.f. may be determined by inspection, or 
we may proceed as in the following Example : , 

To reduce the fraction , ^ „,, ., to its lowest terms, 

10a^6-c* _ 2 X 5 X aaabbcccc 
12a-6V^ 2 X 6 X aabbbcc ' 

We may then remove factors common to the numerator and 

denominator, and we shall have remaining -— — j- : 

" 6x0 

.'. the required result will be -^^ 

152. Two cases are especially to be noticed. 

(1) If every one of the factors of the numerator be removed, 
the number 1 (being always a factor of every algebraical 
expression) will still remain to form a numerator. 

3a'C Zaac 1 



Thus 



I2ah'^ 3 X 4 X aaacc 4ac' 



(2) If every one of the factors of the denominator be removed, 
the result will be a whole number. 

„, I2ah- 3 X 4 X oMacc 

Thus .^ , = — ^ = 4ac. 

3a-c o X aac 

This is, in fact, a case of exact division, such as we have 
explained in Art. 74. 



Examples. — xxxix. 

Reduce to equivalent fractions in their simplest terms the 
following fractions : 



4a2 
12a3' 


8x3 
^' 36x2- 




IQx^yh^ 
45x^2*' 


7o567c« 

5' 21a36V 


6. tT- 

3abc 


blay-z 




8xYz^ 


•iia-yz^' 


9- 6a^y»Z^' 


[8.A.] 




F 



§2 FRACTIONS. 



2\0mVp a? 14m*x 

lO. -. II. . 12. ■• 

42m%2jp2" • d--\-ab' 21m^p -7mx 

xy Aax + 2x^ mi + w- 

3x2/2 — 5x2^z" 8ax^ — 2x2' 3- abc + bcy' 

4a^x + 6ah j 12 aF- - 6ah „ c^-4a^ 

' ■ 8x2-18?/' ^-^^ 86-C-2C ■ ^ ' c2 + 4ac + 4a2 

3x* + 3x2|/2 ^ labhfi - 7abY 

^9' 5x* + 5x27/' ^'^ 14a%x» - 14a%2/2' 

lOx-lOy 5x9 4, 45t^2 

"°' 4x2 -8x1/ + 4?/^ ^5' lOcx" + 90crfx2' 

ax + by , 10a2 + 20a6+106* 

^ 26. 



7a2x2 - 7b-y^' ' 5a^ + 5a% 

6ab + Scd 4x2 _ g^-y ^ 4^2 

27a262x - 48c2cZV ^" 48(x-i/)2 ' 

xy-xyz - 3mx + 5?ix2 

2«3 — 2rt32' ■ 3to?/ + 5)!X1/' 

153. We shall now give a set of Examples, some of which 
may be worked by Resolution into Factors. In others the 
H.C.F. of the numerator and denominator must be found by 
the usual process. As an example of the latter sort let us 
take the following : 

To reduce the fraction „-, — ^-„ — .so ^, to its lowest terms. 

•' 2x3-9x2-38x + 21 

Proceeding by the usual rule for finding the H.C.F. of the 
numerator and denominator we find it to be x - 7. 

Now if we divide x^ — 4x2 — 19x— 14 by x — 7, the result is 
x2 + 3a; + 2, and if we divide 2x^-9x2-38x4-21 byx-7, the 
result is 2x2 + 5x — 3. 

x2 + 3x + 2 
Hence the fraction „ (. . "q i^ equivalent to the proposed 

fraction and is in its lowest terms. 



Examples. -xl. 

a2+7a + 10 ^ x2-9x + 20 x« -2x-3 

a2 + 5a + 6' '' x2-7x + 12" ^' x^-\0x^2l' 



FRACTIONS. 83 



x2-18xj/ + 452/^ x^ + x^ + l x^ + 2x3?/^ + 2/' 

x^-8x?/-105i/2' 5- a;- + x + l* ' a;^-i/6 

''■ x3 + 2x2"-3x + 20' ^"^^ m3-7m + 6 ' 

„ x^-5x-+ llx — 15_ a^ + 1 

a^_x2 + 3x+5 ^' a3 + 2a- + 2a+r 

a:3-8x2 + 21x-18 , 3ax2-13ax4-14a 

^" 3x3-16x'-^ + 21x ■ ^ ■ 7x3- 17x2 + 6x • 

x3-7x2 + 1 6x-12 14x2-34x + 12 

'°" 3x3- 14x2 +T6x • 17- 9ax2-3'9ax + 42a' 

x* + x3j/ + X1/3 - 1/^ 10a -24a2 + 14^3 



x* — x^?/- x]/'* — ?/'' * 15 — 24a + 3a2 + 6a3" 

a3 4 4a- - 5 2a63 + a6- - 8a6 + 5a 

^^' a3-3a + 2* ^^' 763-1262 + 56 * 

63 + 462-56 a3_3^2 + 3^_2 

3x2 + 2x-l a^-a-%) x^-3x^ + 4x-2 

x3 + x2-x-l "' a- + a-12' "^ x3-x2-2x + 2 
(x + y + g)2 + ( g - j/)2 + (x - g)2 + (y - a;)2 

X2 + ?/2 + ^2 

2x* - x3 - 9x2 + 1 3x - 5_ 1 5fj3 ^ab-2b'^ 

^5" 7x3-19x2 + 17x-5 ' 3^" 9a2 + 3a6 -262' 

16x*-53x- + 45x + 6 x2-7x + 10 

8x^-30x3 + 31x2^1 2" ^■^" 2x2 - X - 6 " 

4x2 - 1 2_ax + 9a2 x3 + 3x2 + 4,^ + 1 2 

^'^' 8x3-27a3 • 35- x3T4x2T4x + 3* 

6.< r^-23x2 + 16x -3 x*-x2_-2x + 2 

6x3-17x2+llx-2' ^ 2x3 -x^l"' 

x3-6x 2 + nx- 6 x3 - 2x2 - 1 5x + 36 

^9" x3-2x2-x + 2 ■ ^^' 3x2^x^- 15 ' 

7n3 + m2 + m — 3 3x3 + x2-5x + 21 

■^ ' '»i3 _• Q^-v.2 _i_ P^«v» 1^ Q* -3 ' 



3)762 + 5m + 3' :)"• 6x3 + 29x2 + 26x-2r 
x^ + 5x* — x2 — 5x X* — x3 - 4x2 — X + 1 

^'' x4 + 3x3-x-"3 ' ^^: 4x3-3x2-8x-l " 
a2 - 62 - 26c - c2 a3-7a2+I6(x-12 

?^* a2 + 2a6 + 62-?" ^O- 3^3 ::T4a2 + i6a ' 



FRACTIONS. 



154. The fraction t is said to be a proper fraction, ■when a 
is less than h. 

The fraction t is said to be an improper fraction, when a is 
greater tlian h. 

155. A whole number x may be written as a fractional 
number by writing 1 beneath it as a denominator, thus -. 

156. To prove that 5- of j = r3- 

a ocL 

\ 

Divide the unit into bd parts. 

^^'^n'^^d = 6«^^ (Art. 148) 

= r of be of these parts (Art. 147) 

= T- of 6c of these parts (Art. 148) 

= ac of these parts (Art. 147). 

But yj = ac of these parts; 

a ^ c _ac 
•'■bd~bd' 

This is an important Theorem, for from it is derived the 
Rule for what is called Multiplication of Fractions. We 

extend the meaning of the sign x and define , x t (which 

according to our definition in Art. 36 has no meaning) to mean 

r of -„ and we conclude that y >< -i = t^. which in words trives 
b d b d bd ^ 

us this rule — " Take the product of the numerators to form 

the numerator of the resulting fraction, and the product of the 

denominators to form the denominator." 

The same rule holds good for the multiplication of three or 
more fractions, 



FRACTIONS. 85 



157. To shew that r^-7= t- • 
a be 

The quotient, x, of r divided by -5 is such a number that x 

multiplied by the divisor 3 will give as a result the dividend t- 

. arc _ a 
•• li~b'' 

d p xc d . a 

.-. - of -r = - of |- ; 

c a c 

xcd _ ad 

" cd be ' 

ad 

■' ^=k- 

Hence we obtain a rule for what is called Division of 
Fractions. 

_,. a c ad 

Since r-r- j = T~) 

d be 
a c _a d 
b^d'h^'c' 

Hence we reduce the process of division to that of multiph' 
cation by inverting the divisor. 



158. The following are examples of the Multiplication an 1 
Division of Fractions. 

2x o _ ^■'^ 3a _ 6ax 2x 
I. 3„2>^'^"-3„2>^-l- = 3^ = -- 

3x_^ _3x^3a_3x 1 _ .3x _ x 
^' 26* *~26 • T~26 ^3a~6a6~2a6* 

4a^ 3c _ 3 X 4 X a-c _ 2a 
^' 9c^ ^ 2a~ 2 X 9x ac- ~ 3c' 

14x2^ '7x_ 14x2 9.v_ 9xl4xa:2? / 2x 
^ 27y^'' 9y''27y^^7x~7x27xxy^~^' 

2a 96 5c _ 2a x 96 x 5c _3 
5' 36 '^ TOc ^ 4a ~ 36 x~10or4a " 4* 



86 PR ACTIONS. 



x^ — 4x x^ + 7.x_ x(x — 4) x(x + 7) 
x^ + 7x^ x-4 x-(x + l) x — 4 
_x(x-4)x{x + 7)_ 
~x2(x + 7)(x-4)~ 
a'i _ 52 ^ 4(a2_ ab) _ «2- ?)2 a2 ^j, 

'' a^ + 2ab + b- ' a- + ab ~ a^ + 2ab + b"^ 4{a'^ - ab) 
_{a + b){a — b) a{a + b) 
~{a + b){a + b) 4a{a-b) 
_(a + b){a-b)a{a + b) _1 
~{a + b)\a + b)4a(a-b)'~4' 



Examples.— xli. 

Simplify the following expressions : 

3x 7x 3a 26 4x^ 3x 

4y^9y' ^' 4b^3a' ■ ^' df^^V' 

80253 I5xy2 Q^y2^ 20a%k 2a 46 5f 

45x2?/ ^ 24a42- 5- ioa^^c'' mxij-z' 56 "" 3c ^ 6a" 

Sx^y 5yh I2xz Ici^b* 20cM- 4ac 

4x^ ^ 6x2/ ^ 20x^" 5"c2d3 ^ 42^463 "" shd' 

9vihi^ hifiq 24x2w2 25A;3m2 "On^q 3pm 

^ -^ ^ X ^ TO X — X — ^ 

82?3g3 2x2/ 90mn" ' 14712^2 'jop'm 4k^n 



Examples. — xlii. 

Reduce to simple fractions in their lowest terms : 

a- 6 a2-62 x2 + x-2 ■t 2 - 13x + 42 

a2 + a6 a2 - a 6' ^' x^ - 7x x2 ^ gx 

x^ + 4x 4x2j^l2x x2-llx + 30 x2-3x 

X2"^X ^ 3x2Tl 2X 5 • 2-2 _ y^ + y ^ a;2 _ 53;" 

x2 + 3x + 2 x2-7x + 12 , x2-4 x2_25 

O. -r, i X ./-^ 



5x + 6 x2 + x ■ ■ x^ + S^u x^ + 2x' 

g-i _ 4 a + 3 a2-9a + 20 a2 - 7 a 

'' a^ — ba + 4 a- - 1 Oa + 21 «-' - 5a' 

62-76 + 6 62+106 + 24 6^ - 862 

8- 62:^36-4 "" 62 - 1 46 + 48 "" 62 + 66 • 



t^R ACTIONS. 87 



■y^ xy- 27/2 ^x^-xy 



II. 



13- 



x^ - 3xy + 2y^ '" x^ + xy {x - yy^ 

{a + by~c^ c2 - (g - b)- 
d' - {h - cY"" c^ - {a + hf 

{x - m)- — n- X- - (n - nt)^ 
(x - n)- - m^ X- - {m - n)'^' 

( a + hy-(c + dY {a-hf-{d- cf 
(a + cy -{b + df ""[a- cf -{d- bf 

X? — 2xy + y- -z^ x + y-z 
x^ + 2xy + y^-z- x-y + z' 



Examples.— xliii. 

Simplify tlie following expressions : 



2a 


36 


X 


~5c" 


Aa , 
rix ' 


rSab. 


bx . 


-2. 



2 l£M^^^ . 8x*^_^2x3 

14s ■ 7z' ■^' loab^ ' SOoi^ 

^ 2^; - 2 > - 1 • 5x 

11 11 



X' - a.c + 2 ■ X - r ^' x- - 17x + 30 ■ X - 15' 

158. We are now able to justify the use of the Fraction 
Symbol as one of the Division Symbols in Art. 73, that is, 

we can shew that j is a proj^er representation of the quotient 

resulting from the division of a by b. 

For let X be this quotient. 

Then, by the definition of a quotient. Art. 72, 
b xx = a. 

But, from the nature of fractions, 

, a 

X y = a; 


a 
:.-r=x. 



THE LOWEST CO^TmO^ :■ i'.^. 



159. Hi-re we may state an important : neorciu, whicn -«>' 
shall require in the next chapter. 

If ad = be' to shew that , = -,. 
b a 

Since ad = bc, 



ad 


be 


bd~ 


'bd 


a 


c 


'b~ 


'cC 



X. THE LOWEST COMMOri iviui-TIPLE. 

160. An expression is a Common Multh uk of two or 
more other expressions when the former is esLnunv divisible by 
each of the hitter. 

Thus 24x^ is a common multiple of 6, 8x^ and 12a^. 

161. The Lowest Common Multiple of two or more 

expressions is the expression of loivest dimenbi^ns which is 
exactly divisible by each of then, , 

Thus ISx* is the Lowest Couimou iuun-ipie of 6j;*, ^x^, 
and 3x. 

The words Lowest Common Multiple are abbreviated 
into L.c.M. 

162. Two numbers are said to be prime to each other 
which have no common factor but unity. 

Thus 2 and 3 are prime to each other. 

163. If a and b be prime to each other the fraction 
is in its lowest terms. 

Hence if a and b be prime to each other, uud i=^, «J>tl 

if m be the h.c.f. of c and d, 

^ 1 1. ^ 
o = — and = —. 
7/1 m 



THE L O IVES T COMMOX MUL TIPL E. Sg 

164. In finding the Lowest Common Multiple of two or 
more 'expressions, each consisting of a single term, we may 
proceed as in Arithmetic, thus : 

<1) To hnd the l.c.m. of ^a?x and 18ax'^, 



2 


4a%, 


18ax3 


a 


2a\ 


9acc3 


X 


2a\ 


9x3 




2a\ 


9x2 



L.C.M. = 2 X a X .7- X 2a2 X 9x2 = 36^83*^ 
(2) To find the l.c.m. of ab, ac, be, 



a 


ab, 


ac, 


6c 


b 


b, 


c, 


be 


c 


1, 


c, 


c 




1, 


1, 


1 



L.C.M. = a X 6 X c = a6c. 

(3) To find the l.c.m. of 12«2c, 146c2 and SGoJ*, 
2 12a2c, 14&c2, 36a¥ 



6 
a 


6a\ 

a-c, 


7bc% 

' nc\ 


18a62 
~3a62 


h 


ac, 


lbc\ 


362 


c 


ac. 


7c2, 


36 




a. 


7c, 


36 



L.C.M. = 2 X 6 X a X 6 X c X « X 7c X 36 = ^biaWc^. 



Examples 

Find the L.C.M. of 
I. 4a^x and 6<(-x-. 
Zxhj and 12.i:y-. 



-xliv. 



4a36 and 8*262. 
ax, a"x and rt2x2. 
2ax, 4ax2 and x^. 



6. ab, a-c and 6-'c^ 

7. a'^x, a^y and x-y-. 

8. blaH^, 34ax3 and ax*. 

9. 52)'q, lOq^r and 20pqr. 
I p. 18ax2, 72ai/2 and 12x?/. 



90 



THE LOWEST COMMON MULTIPLE. 



165. The method of finding tlie l.c.m., given in tlie pre- 
ceding article, may be extended to the case of compound 
expressions, when one or more of their factors can be readily 
determinea. Thus we may take the following Examples : 



(1) To find the l.c.m. of a-x, a^ — x^, and a^ + ax, 
a — x, a^ — x^, a^ + ax 
1, a + x, a^ + ax 



a — x 
a +x 



1, 1, a 

L.C.M. = (« — x){a + x)a = (a^ — .r^) ar=a^ — ax^. 

(2) To find the l.c.m. of .t^- i, x*-l, and 4x''-4.r:», 
a;2-l I x^~l,x*-l,4x^-4x* 
I 1, x^+l, 4x* 
L.C.M. = (a;2 - 1) (x2 + 1) 43;^ = (x* - 1) 4x* = 4x8 _ 43.4, 

166. The student who is familiar with the methods of 
resolving simple expressions into factors, especially those given 
la Art. 125, may obtain the L.C.M. of such expressions by a 
process which may be best explained by the following Ex 
amples : 

Ex. 1. To find the l.c.m. of a--x^ and a^-x\ 
a^ - x^ = (« - x) (a + x), 
o3 - a;3 = (rt — x) (a^ + ax + x'^ 

Now the l.c.m. must contain in itself each of the factors in 
each of these products, and no others. 
.•. L.C.M. is (a - x) {a + x) (a- + ax + x"^, 

the factor a-x occurring once in each product, and therefore 
once onlv in the L.c.Jr. 



Ex. 2. To find the l.c.m. of 

a~ — b-, a^ — 2ab + b'', and a^ 2ab + 1 
a2-52 = (rt + 6)(a-6), 
a^-2ab + b^-=(a-b){a-b), 
a^ + 2ab + b^ = la + b)(a + b); 
t.C.M. is (a + b){a- h) (a - b) {a + b). 



THE LOWEST COMMOX MULTIPLE. 91 

the factor a — h occurrinrj txoice in one of the products, and a + 6 
occurring twice, in another of the products, and therefore each 
of these factors must occur lunce in the l.c.m. 



Examples. — xlv. 

Find the L.C.M. of the following expressions : 

1. x^ and ax + x^. 10. x^ - 1, a;^ + 1 and x^ - 1. 

2. a.-2 — 1 and a;2 — X. ii. x^-x, x^— 1 and x^ + 1. 

3. a-^ — 52 and a^ + aJ, 12. x^- 1, x^-x and x^- 1. 

4. 2x-l and 4x--l. 13. 2a + 1, 4a^- 1 and 8a^ + l, 

5. a + 6 and a^^-W". 14. x + ?/ and 2x2 + 2x?/. 

6. x+ 1, X- 1 and x^— 1. 15. (a + 6)- and a2_52_ 

7. x+ l,x^— 1 andx2 + x+ 1. 16. a + 6, a- 6 and a^ — 62_ 

8. x+1, x2+l andx^+l. 17. 4(1 +x), 4(1 -x)and 2(1 -x^). 

9. X— 1, x^- 1 and x^— 1. 18. x— 1, x- + x + 1 and x^— 1. 

19. (a — 6) (a — c) and (a — c) (6 - c). 

20. (^x + l)(x + 2), (x + 2)(x + 3) and (x+l)(x + 3). 

21. x^ - ?/^, (x + 1/) 2 and (x -ijf- 

22. (a + 3) (a + 1), (a + 3) (a - 1) and a^ - 1. 

23. '3?{x — 'ijy, x{x^ — y-) and x + y. 

24. (x+l)(x+3), (x + 2)(x + 3)(x + 4) and (x + l)(x + 2). 

25. x^ — y"^, 2{x — yY and 12 (x^ + i/^). 

26. 6(x2 + x?/), ^(xy-y-) and 10(x2-i/2). 

167. The chief use of the rule for iinding the l.c.m. is for 
the reduction of fractions to common denominators, and in the 
simple examples, which we shall have to put before the student 
in a subsequent chapter, the rules which we have already given 
will be found generally sufficient. But as we may have to find 
the L.C.M. of two or more expressions in which the elementary 
factors cannot be determined by inspection, we must now pro- 
ceed to discuss a Rule for finding the l.c.m. of tv, o expressions 
which is applicable to every case-. 



92 THE LOWEST COMMON MULTIPLE. 



168. The rule for finding the l.c.m. of two expressions o 
and h is this. 

Find d the higliest common factor of a and 6. 

Then the l.c.m. of a and i = , x h, 
a 

b 
or. = 3 X o. ♦ 

a 

In words, the l.c.m. of two expressions is found by the fol- 
lowing process : 

Divide one of the expressions by (he h.c.f. and multiply the 
quotient by the other expression. The result is the L.C.M. 

The proof of this rule we shall now give. 

169. To find the l.c.m. of two algebraical expressions. 
Let a and h be the two algebraical expressions. 

Let d be their h.c.f., 

X the required L.C.M. 

Now since x is a multiple of a and 6, we may say that 

X = ma, X = 7i6 ; 

.". ma = nb ; 

fii b / J . , --v 
.-.- = - (Art. 159). 
n a 

Now since x is the Loxcest Common Multiple of a and h. 
m and n can have no common factor ; 

;. the fraction ~ must be in its lowest terms ; 
n 

:. m = T and n = -, lArt. 163). 

d d 

- Hence, since x = ma, 

b 

x = -,xa. 

d 

Also, since x—nb, 

" J, 
x = -,x 0. 
a 



THE LOWEST COMMON MULTIPLE. 93 



170. Ex. Find the l.c.m. of x2 - 13x + 42 and x^ - 19x-+ 84. 

First we find the h.c.f. of the two expressions to be x — 7. 

„, (x2-13x + 42)x(x2-19x + 84) 

Then l.c.m. = ^ ' -\ '. 

x-1 

Now each of the factors composing the numerator is divisible 

by X — 7. 

Divide x- — 13x + 42 by x — 7,'and the quotient is x - 6. 

Hence l.c.m. = (x - 6) (x^ - 19x + 84) = x^ - 25x2 _,. iggx - 504. 

Examples. — xlvi. 

Find the l.c.m. of the following expressions : 

1 . X- + 5x + 6 and .x- + 6x + 8. 

2. ft' -a-20 and n^ + a- 12. 

3. x^ + 3x + 2 and x- + 4x + 3. 

4. x2+llx + 30 andx2+12x + 35. 

5. x2-9x-22 andx2-13x4-22. 

6. 2x2 + 3x + 1 and x^ - x - 2. 

7. x^ + x^y + xy + y^ and x* - y*. 

8. x^ - 8x + 15 and x- + 2x- 15. 

9. 21x2 _ 2Gx + 8 and 7x'' - 4x''* - 21x + 12. 

10. x^ + x^y + x?/2 4- y^ and x^ - x-ij + xy^ - y^. 

11. a^ + 2a-b - ab^ - 2P and a^ - 2a'^b - ab- + 2¥. 

171. To find the l.c.m. of three expressions, denoted by 
a, b, c, we find m the l.c.m. of a and b, and then find M the 
L.C.M. of m and c. M is the l.c.m. of a, b and c. 

The proof of this rule may be thus stated : 

Every common multiple of a and 6 is a multiple of m, 
and every multiple of m is a multiple of a and b, 
therefore every common multiple of m and c is a common 

multiple of a, b and c, 
and every common multiple of a, b and c is a common 

multiple of m and c, 
and therefore the L.C.M. of m and c is the l.c.m. of a, b 

and c. 



94 OM ADDITION AND SUBTRACTION 

Examples. — xlvii. 

Find the l.c.m. of the following expressions : 
x- - 3,/; + 2, x- - 4x + 3 and a;^ - 5x + 4. 
x- + 5x + 4, a- + 4x + 3 and x- + 7x+ 12. 
X- - 9x + 20, x^ - 1 2x + 35 and x^ - 1 Ix + 28. 

4. 6x2 - X - 2, 21x2 - 17x4- 2 and 14x2 + 5^ - i. 

5. x^ - 1, x- + 2x - 3 and 6x2 _ 3; _ 2. 
x3 - 27, x2 - 15x + 36 and x^ - 3x2 _ ^x + 6. 



XL ON ADDITION AND SUBTRACTION 
OF FRACTIONS. 

172. Having established the Rules for finding the Lowest 
Common Mi;ltiple of given expressions, we may now proceed 
to treat of the method by which Fractions are combined by 
the processes of Addition and Subtraction. 

173. Two Fractions may be replaced by two equivalent 
fractions with a Common Denominator by the following 
rule : 

Find the l.c.m. of the denominators of the given fractions. 

Divide the'L.c.M. by the Denominator of each fraction. 

Multiply the first Numerator by the first Quotient. 

Multiply the second Numerator by the second Quotient. 

The two Products Avill be the Numerators of the equivalent 
fractions whose common denominator is the L.C.M. of the 
original denominators. 

The same rule holds for three, four, or more fractions. 

174. Ex. 1. Reduce to equivalent fractions with the 
lowest common denominator, 

2x + 5 , 4x-7 
-3- and -^. 



of FRACTIONS. 9$ 



Denominators 3, 4. 

Lowest Common Multiple 12. 

Quotients 4, 3. 

New Numerators 8a; + 20, 12x-21. 

8a; + 20 12x-21 



Equivalent Fractions 



12 ' 12 



Ex. 2. Reduce to equivalent fractions with the loweet 

common denominator, 

56 + 4c 6a -2c 3a -56 

a6 ' ac '' be ' 

Denominators a6, ac, be. 

Lowest Common ^Multiple abc. 

Quotients c, b, a. 

New Numerators 56c + 4c', 6ab - 26c, 3a- - 5a6. 

„ , ^ „ . 56c + 4c'- 6a6 - 26c 3a- - 5a6 

lljqiuvaient 1" Tactions — -, , = , ; . 

a6c abc abc 



Examples. — xlviii. 

% 

Reduce to equivalent fractions with the lowest common 
denominator : 

3x , 4x ^ a -b , d^-ab 

1. —-and -• 6. -— - and r^-. 

4 5 a'^6 a6'' 

3x-7 , 4x-9 „ 3 ,3 

2. — TT- and ^ —. 7. ^ and . 

6 18 1-rX 1-z 

2x-4y , Sx-Sy „ 2 ,2 

3. - ■■ ^ and ^rp-^. 8. , — and , -. 

•^ Sx'' lOx 1-2/- 1+2/'' 

4a + 56 , 3a — 46 5 1 6 

4. — r-v- and -- — . 9. and ^ „ 

-la- oa ^ 1 — X 1 - x'' 

4a — 5c 1 3a — 2c a , 6 

and ^ ^ ., -. 10. - and 



5ac 12a-c ' ' c c(6 + a!)' 

II. , ,.—,, .and 



(a-6)(6-c) (a-6)(a-c)' 

12. -r^ TT-^ , and 



a6(a — 6)(a — c) ac (a - c) (6 — c)' 



o6 ON ADD/TW.V AND SUBTRACTION 



,.Tr m 1 *.!, 4. * c ad + bc 

175. To sliew that v + ?= — I'l" ■ 

Suppose the unit to be divided into bd equal parts. 

7 

Then j-j will represent ad of these parts, 

6c 
Id 

Now ^ = g, by Art. 148, 

, c 6c 

and J = r^. 

d bd 

Hence t + -. will represent ad + be of the parts. 

But — i," will represent aa + be of the pares. 
bd 

_, „ a c ad + be 
.nerefore^ + -^=-^^. 

By a similar process it may be shewn that 
a e _ ad — be 
h~d^~~bdr' 

,„„ _,. a c ad + bo 

176. Since J 4-^=-^, 

our Rule for Addition of Fractions will run thus : 

"Reduce the fractions to equivalent fractions having the 
Lowest Common Denominator. Then add the Numerators of 
the equivalent fractions and place the result as the Numerator 
of a fraction, whose Denominator is the Common Denominator 
of the equivalent fractions. 

The fraction will be equal to the sum of the original frac- 
tions." 

The beginner should, however, generally take two fractions 
at a time, and then combine a third with the resulting fraction, 
as will be shewn in subsequent Examples. 

. , . a e ad — be 
Also, since ^-5=--^^-, 

the Rule for Subtracting one fraction from another will be, 



OF FRACTIONS. gy 



•' Reduce the fractions to eijuivalent fractions having the 
Lowest Common Denominator. Then suhtract the Numerator 
of the second of the equivalent fractions from the Numerator 
of the first of the equivalent fractions, and place the result as 
the Numerator of a fractijjn, Avhose Denominator is the Common 
Denominator of the equivalent fractions. This fraction will be 
equal to the difference of the original fractions." 

These rules we shall illustrate by examples of various degrees 
of difficulty. 

Note. When a negative sign precedes a fraction, it is best 
to place the numerator of that fraction in a bracket, before 
combin'ing it with the numerators of other fractions. 

177. Ex. 1. To simplify 

4x - 3|/ 3x + 7|/ _ 5a; - 2)/ 9x + 2y 
7 ^ ^ ~I4 2l " "*" ~42~' 

Lowest Common Multiple of denominators is 42. 

Multiplying the numerators by 6, 3, 2, 1 respectively, 

24x-a % ^ii^l y_ _ lOx-4?/ 9x + i'y 
~~ 42 "*" 42 42"~^'''~42~ 

24x - 1 8;/ + 9x + 2 1 7/ - ( 1 (\c - 4y) + 9x + 2y 
~ 42 

_2 4x-18?/ + 9x + 21?/-10x + 47/ + 9x + 2;/ 

42 
_ 32x 4- 9 ;/ 

42~'' » 

Tv o 'p • ^•r 2x+l 4x + 2 1 
Ex. 2. To simplify — h =. 

^ 3x ox 7 

Lowest Common Multiple of denominators is 105x. 
Multiplying the numerators by 35, 21, 15x, respectively. 
70.C + 35 _ 84x + 42 1 5x 
lU5x l()5x 105x 

_ 70x 4- 35 - (84x + 42) + 15a; 
~ lOSx 

70x4-35-84x-42-H5x x-7 
^ 105x ~T05x' 



98 ON ADDITION AND SUBTRACTION 



Examples.— xlix. 

4x + 7 3a; -4 3a -46 la-h + c 13a -4c 

I. ^— +— ,-v— • 2. ^ + .,— . 

5 15 / .J 12 

4x - 3i/ 3a; + 7i/ bx — 2?/ 9x + ly 
3- ~y~"'^~l4 21^'^~42~^" 

3a;-2i/ 5x-7y 8x + 2y 
^' ~5x '*' 10a; "^ ~ 25"* cv 

4a;2 - 7i/2 3a; - 81/ 5-2 j/ 
5' "3x2" + ^6x~ "'' ~~[2~' 

. 4a2 + 56_2 3a^26 7^2a 
W^^ 56 "^ 9 • 

4x + 5 3x - 7 9 
^' ""3 5x~''"l2x2" 

5a + 26 _ 4c - 36 6a6 - 76c 
3c 2a 14ac 



2a + 5c 4oc - Zc^ bac - 2c^ 
3X1/-4 51/2 + 7 6x2-11 

10. 5—5 ' 6 T - \ 

jc^-' xy* x-^y 

a - 6 4a - 56 3a - 76 
a?h a^bc b^c^ 

178. Ex. To simplify 

a-6 a+b 

a + b a-b' 
L.C.M . of denominators is a2 - 62. 

Multiplying the numerators by a-6 anri a +6 respectively, 
we get 

a'^ - 2ab + b- a- + -lab + 6^ 
a2-62 "^""^a— ftl 
_a2- 2a6 + 62 + a^ + 2ab + b- 

a2 - 62 
_2a2^+^62 
~ fi2_j2-- 



or FRACTIONS. 99 





EXAMPLES.- 


-1. 






1 1 

a; - 6 a; + 5' 


1 1 
^' x-7 x-3' 




3- 


1 1 

1 +X l-X 


?±1 _ ?J"_1 
x-y x + y' 


1 2 

5' l-X 1-X2* 




6. 


a (ad - be) x 
c c{c + dx) 


XX o 

7. + . 8. 

X+y x-y 


1 

X- 


— + 


X 

(x-yy 


2 3a j^ 
"* x + a {x + ay 


2a 


1 + 1 . 

(a + x) 2a (a - a) 


179. Ex. 1. 


To simplify 

3 5 6 

1 + y 1-1/ 1 + 1/2" 








Taking the first two fractions 










3 5 

1+y l~y 




1 






3-3?/ b + by 
~1-/ ' l-t/2 










8 + 2y ^ 
-1-7/' 








we can now combine with this result the third of the original 
fractions, and we have 




3 5 

1 + 7/^1-7/ 1 


6 


- 






+ 7/2 






8 + 27/ 6 










1-7/2 1+7/2 





^ 8 + 27/ + 87/2 ^27/3 _ 6 - 6?/2 

~ ' 1-jr* "l^p 

_ 8 + 27/ + 87 /2 + 27/ 3 - 6 + 6y2 
1^7/* 

_ 27/3+147/2-|.27/ + 2 

1-y* 



too ON ADDITION AND SUBTRACTION 



Ex. 2. To simplify 

2 2 2 

(a-6)(6-c)"'"(a-6)(c-a)^(6-c)(c-a)' 

I..C.M. of first two (lenominatoi-s being (a - 6) (6 - c) (c - a) 

_ 2c -2a '26 - 2c 2 

~ (a - 6y (6 - cH^ a) ^ (a-b) (6-"^H7-T) ■*■ (6 - c) (c - a) 

26-2CT _ 2 

~ (^Ujlh~=^cj~{^^ ^ (6^ r-) (c - «)• 

L.C.M. of the two denominators being (a -h) {b- c) (c - a) 

26-2ffl + 2a-26 



{a -b)(b- c) (c -a) (a- b) (6 - c) (c - a) 



=0. 



Examples.— li. 

1 Jl_ 2a _J^ 1 26 4¥ 

'• H-ft''"l-a''"l-a2" "^^ «-6 a + b a'^ + b^ a*-^b*' 

1 1 2a; X y x^ 

1-x 1 + x l+X' y x + y x' + xy 

X x^ X ^ x + Z x-4 x + 5 

^ x-l x-1 x-3 
a;-2 x-3 x-4 

8 -^ _^_ _ 5a^ 
X - a (x - a)"-^ (x - a)3* 

1 1 3 

^' x-l x + 2 (x+l)(x + 2)' 

1 3 

lo. 



(r + 1 ) (x + 2) (x + 1) (x + 2) (x + 3)* 



X X 

- 1 "*'x^'T'^x+r 
1 1 

£2. 



(d + c) (a + d) (tt + c) (a +• e)' 

a-6 6-c c-a 

'^" (ftTcncTo) (c + a) (aT6) (a + 6) (fTc)* 



OF FRACTIONS. 



X — a x — b {a - b)' 

x-b x-a (x - a){x-b)' 

x + y 2:c x-y-x^ 

'■ y x + y y{-''--y')' 



1 6. 



17- 



a + 6 6 + c c + ii 

(6 -c){c- a) (c -a) (a- b) (a — b){b- c)' 

x 2xy 

x^ + xy + y^ x^ — y^' 



i8 2,2^2 ^( a-br- + {b-cy + {c-af ^ 
a-b b-c c — a {a — b){b — c){c-a) 

a + b 2a a-b - a^ 
'9- 'b^~^+b^a^b^¥' 

1 1 1 

^°' {n+l){n + 2) ()i+l)(?H-2)(?i + 3) (?i + l)(7i + 3)* 

a^ — be b- - ac c^ — ab 

2 1 1 + 

(a + 6)(a + c) {b + a){b-\-c) {c + b){c + a) 

„ .-,• O'b -, — ab . ►,« 

180. bmce X~"' ^ ~T'~^' ^'^' ' 

ab _—ab 
'b~~^T' 

Fioiu this we learn that we may change the sign of the 
(lenoiiiinat(ir of a traction it we also change the sign of the 
numerator. 

Hence if the numerator or denominator, or both, be expres- 
sions Avith more than one term, we may change ibe sign of 
every term in the denominator ii we also change the sign of 
every term in the numerator 

„ a — b —(a-b) 

c-d -{c-a) 

~ - r+i ' 

or, writing the terms of the new i'raction so that the positive 
terms may stand first, 

_b — a 
d-c' 



I02 ON ADDITION- AND SCB TRACTION 

181. t-X. To simplify — ^-- . 

Changing the signs of the numerator and tleiiominator of tlu^ 
second fraction, 

X (a + x) - bax + x- 
a-x a—x 

_a!}c + x^-( — 5ax + x-) ax + x^ + bax -x^ _ 6ax 
a — x a — x a-x' 

182. Again, since —ab= the product of -a and h, 

and «6= tlie proiUict of +a and h, 

the sign of a product will he cliani,'ed hy changing the sic;ns of 
one of the factors composing the product. 

Hence (a — b)(b- c) will giv. a set of terms, . 

and (6 -a) {b- c) will give the same set of terms icith dif- 
ferent signs ' 

This may be seen hy actual multiplication : 

(a - h) {b -c) — (ib-ac-b^ + he, 
(b- a) (b - c)= — ah + ac + Ir — he. 

Consequently if we have a fraction 

1^ 

(a - h) {h - c)' 

and we change the factor n-h into h-a, Ave shall in effect 
change the sign of every term of the expression which Avould 
result from the multiplication of (a - b) into {b - c). 

Now we may change the signs of the denominator if we also 
change the signs of the numerator (Art. 180) ; 
1 -J 

" (a - 6) (6 - c) ~ (b-a) [b- c)' 

If we change the signs of two factors in a ilenominator. the 
sign of the numerator will remain unaltered, thus 
1 1 

ia-b)(b-c)~(h-n)(c'-by 



I 
I 



OF FRA C TICNS. io3 



183. £x. Simplify 

1 1 



(a-6)(6-c) {h-a){a-c) (c-a){c-b)' 

First change the signs of the factor (b-a) in the second 
fraction, changing also the sign of the numerator ; and change 
the sigTis of the factor (c - a) in the third fraction, changing 
also the sign of the numerator, 

, . 1 -1 -1 

the result is , rr-r, r + ; Tx / \ ~ 7 w i\- 

(a -b) {b- c) (a - o) (a - c) [a -c) [c- b) 

Next, change the signs of the factor (c - b) in the third, 
changing also the sign of the numerator, 

.11.- 1 - 1 1 

the result is ^ tttt ^ + -. jv-? ; - 7 ttt — ;. 

(a -b) [0- c) [a -o) {a- c) {a - c) (6 - c) 

L.c.M. of the three denominators is (a -b) {b- c) {a - c), 
_ a-c -b+c a-b 

~{a-b){b- c) (a - c) (a- b) {a -c) (b- c) (a -b)(a- c) (6 - c) 

a-c-b->rc- {a -b) 



(a -b)(b- c) {a - c) (a - h) (6 -c){fl- c) " 



.0. 



Examples.— lii. 

X x-y 3 + 2x2 — 3a; 16a! — a;^ 

' x-y y-x' ' 2 — x 2-f-x x^ — 4' 

jc x x^ 114 

^- x+l~l-x'''x2-l* ^' 61/ + 6 ~ 2?/ - 2 "•■ 3^7?" 



5- 



1 2 1 



(m-2)(m-3) (m-l)(3-m) (m - 1) (to - 2)" 



^- (a-6)(x + 6)"*"(6-a)(x + a)* ''" a2-62 a^ - fe3 "•" a^ + F 
1 1 1 



°' 4(l+x) 4(x-l) 2(l + x2)' 

1 ^ 1 • , 1 ' 

1 _-.__„ 1 

a(a-b)(a — c) b{b-a)ib-c) cic-a)(c-b)' 



I04 ADDITION AND SUBTRACTION OF FRACTIONS. 



184. Ex. To simplify 



1 1 



a;2-llx + 30 a;--12x + 35' 

Here the denominators may Le exprcssi.il in lacinrs, and "Wo 

have 

1 1 



(x — 5)(x — G) (re — 5) (x — 7)" 

The L.C.M. of the denominators is (x — 5) (x — 6) (x — 7), and 
we have 

X— 7 X— 6 

+ - 



(a;-5)(x-6)(x-7) (x-5) (x-G) (x-7) 

_ 2x-13 

"~(x-5)(x-6)(x-7)' 



Examples. — liii. 
1 



: + -^ 



x''' + 9x + 20 x2 + 12x + 35* 
1 1 



x2-13x + 42 x2^15x + 54' 

, __i + i 

^' x2 + 7x-44 x2-2x-143" 

1 2x 1 

^' x2 + 3x + 2'*'x- + 4x + 3"^x^ + 5x + 6* 

m 2ni 2?7),?i 



15. — h — 

^ n m + n {m + nf 

1+x 1-x 2 

l+x + x^ 1— x + x- 1+x^H-x** 

5 2 7x 7x 

^* 3^( 1^^) ~ T+x "^ 3x2T3 " 3x- - 3' 

1 1 J j 

8(x-l)^4(3-x) 8(x-5) (l-x)(x-3)(x-5) 

X* 
Q. 1 - X + X- - X^ + - — — . 

^ 1+a; 



XII. ON FRACTIONAL EQUATIONS. 

185. We shall explain in this Chapter the method of 
solving, first, Equations in which fractional terms occur, and 
secondly, Problems leading to such Equations. 

186. An Equation involving fractional terms may be 
reduced to an equivalent Equation without fractions by mul- 
tiplying every term of the equation by the Lowest Common 
Multiple of the denominators of the fractional terw^. 

This process is in accordance with the principle laid down 
in Ax. III. page 58 ; for if both sides of an equation be multi- 
plied by the same expression, the resulting products will, by 
that Axiom, be equal to each other. 

187. The following examples will illustrate the process of 
clearing an Equation of Fractions. 

EX. 1. 1 + ^.8. 

The L.c.M. of the denominators is 6. 
Multiplying both sides by 6, we get 





6.5 6x ,„ 
T+6=^^' 


or, 


3x + x = 48, 


or. 


4x = 48; 




.-. cc = ]2. 


Ex. 2. 


X x + \ _ 
2 + -7-"-'- 



The L.c.M. of tie denominators is 14. 
Multiplying both sides by 14, we get 

14x 14X + 14 ,^ 

-— + 14x-2a^ 



(o6 ON FRACTIONAL EQUATrON"^. 

or, • .7a; + 2a; + 2 = 14x-28, 

or, 7a; + 2x-14x= -28-2, 

or, -5x=-30. 

Changing the signs of both sides, we get 
5x = 30; 
.-. a; = 6. 

188. The process may be shortened from tlie foUowin.i,' 
considerations. If we have to multiply a fraclion by a multi])le 
of its denominator, we may first divide the multiplier by the 
denominator, and then multiply the numerator by the quotient. 
The result will be a whole number. 

Thus, - X 12 = a;x 4 = 4x, 

^^x56=(x-l)x8 = 8x-8. 

EX. 1. M + 1-39. 

The L.c.M. of the denominators being 12, if we multiply the 
numerators of the fractions by 6, 4, and 3 respectively, and the 
other side of the equation by 12, we get 

6x + 4x + 3x = 468, 

or, 13x = 468; 

,-. x = 36. 

Ex, 2. §-^ + ^ = ^. 
" X 2x 3x 12' 

The L.c.M. of the denominators is 12x. Hence, if Ave mul- 
tiply the numerators by 12, 6, 4, and x respectively, we get 

96 -90 + 28= 17x, 

or, 34=17x, 

or, 17x = 34; 

/. x=2. 



O.y FRA C no A' A L EQ UA TIONS. 107 



EXAMPLES.— liv. 

,. 1 = 8. . f = ». , ,M = 8. 

X X „ .-,„ 4x ^ , 2a; 176 -4a; 

4. --- = 3. 5- 36--g- = 8. 6. -=-^— 

2a: , 7x ^ a; + 2 X - 1 a- - 2 

o 2x 4.7; a; x_ 3 X 

^- "3" + ^^- 5+^- '^- 2 + 3-^4-4- 

3x , 5x ^ x + 9 2x 3x-6 „ 

9. — + 5 = — + 2. 19. -—4-^ = —^ — + 3. 

'^4 6 ^4(0 

7x , 9x „ 17 -3x 29-llx 28x+14 

,0. _-5 = ---8. 20. _^- - = _-^_+_^_ 

5x _ ^. 7a; 2x-10 . 

II. -r--8 = 74-T7i. • 21. — = — = 0. 



9 12* " 7 

X 

6 



X , ,, X 3x + 4 4x-51 

12. ,- — 4 = 24--. 



22. 


7 + 47 


- = u 


23- 


^-3 = -l-L 

X X 




24. 


12+x ^ 6 

X X 




25. 


1 1 1 

4"+10^ + 20^ = 


= 40. 



13. 56-| = 48-^|. 

3x 180 -5x _- 
^4- -4- + — 6— = '^- 

3x , , X - 8 
^5- T-^^ = -2- 

, X X X 13 , -1 3-x _5 ._,! 

'6- 2 + 3 + 4=12- ^6- 24^ + -2-=V-% 

.-,3 31 325 

^'^^ ""4 x~x 100' 

^ „1 18 -X ,1 1 3-2x 2 
^^- 22 + -3-=¥ + 3 + -10- + 5- 

X X 5x _- ,2 ^„ 

^9- 3+4-6-- 12= ¥-^^- 

7x+2 ,- 3x 3x + 13 17x 
3°- -10—12-^=-^-^. 



108 ON FRA CTION.A L EQUA T!Oy<!. 



189. it must next be observed tbat in clearing an equation 
!)[■ fractions, whenever a fraction is precedeii by a negative sign, 
u e must place the result obtained by multiplying that nume- 
: ilor in a hraclcet, alter the removal of the denominator. 

For example, we ought to proceed thus : — 

Ex. 1. ^±! = ^^Z^_^ 
5 2 7 

Multiply by 70, the l.c.m. of the denonii:)ators, and we get 

I4x + 28 = 35.7; - 70 - (10.c - 10), 

or 14.c + 28 = 35a;-70-10x+10, 

I'.om which we shall find a; = 8. 

Ex.2. 12-2^_.4J? + 2^1. 
bx 'ix 

Multiplying by 15a;, the L.C.M. of the denominators, we get 

51-6x-(20a; + 10) = 15x, 

or 51-6jj-20x-10 = 15x, 

from which we shall find a;= 1. 

Note. It is from want of attention to this way of treating 
fractions preceded by a negative sign that beginners make so 
many mistakes in the solution of equations. 

Examples.— Iv. 

x + 2 „, 5x 5a; 9 3-x 

1. 5x— 2- = 71. 4- T-T = 4— 2-- 

3-x ,2 „ 5x-4 „ l-2x 

^- ^—3- = '3- 5- 2..- -—=.—— . 

5-2.7; „ 6x-8 , x + 2 14 3 4 5x 
3. -— - + 2 = x —. 6. -2- = -9- ^ . 

5x + 3 3 - 4x x_31 9-5x 
7' ~8~~ 3 ^l~^ "~6~" 

„x + 5x-2.(; + 9 „x + 2x 

^- -l—~^-^U- '°- ^-='— 8- = 3 

x+1 x-4_x + 4 x + 5_x + -2 X — 2 



GN FRACTIOXAL EQUATIONS. tog 



x + 2 x-2 'a;-! ^ 2x x + 3 ., _. 
ID. -i; ^— =3x-21. 



5 2 7' 7 

+-7_a 

11 



x + 9 3x-6 „ 2x 2i; + 7 9x - 8 x-11 

H- — , r- = 3--. 17. ^ 



7x-31 8 + 15x_7x-8 

■ 4 26~~~22~" 

8x-15 llx-1 7x + 2 
19. 



3 7 13 ■ 

7x + 9 3x+l_9x-13 249-9X- 
20. ~^- 7— -4 T4- • 

X -^ XX X 10 -X -„3 

190. Literal e(|nations are those in wliicii known quantilifs 
are represented by letters, usually the first in the alphabet. 
The following are examples : — 

Elx. 1. To solve the equation 

ax + bc = bx + ac. 



that is, 


ox -bx — ac- be, 


or, 


{a-b)x = {a — b)c. 


therefore, 


x = c. 


Ex. 2. 


To solve the equation 




a-x + bx- c = h-x + cx- d, 


that is. 


(i^x + bx- b'-x -cx = c-d, 


or. 


(a^ + b- b- - c)x = c — d, 


therefoi-e, 


c-d 
" a^ + b-b'-c 



Examples.— Ivi. 

1. ax+bx = c. 4. dm - ox = bc- ox. 

2. 2a — ex = 3c — 56x. 5. abc- a-x = ax — a-b. 

3. bc + ax — d = a^b-fx. 6. 3acx — 6bcd=l2cdx + abc. 



ox FRACTIOXAL EQUATIONS. 



7. A;- + ?><xckx 4- 3A; = ^x + Zahh — li^ - ackx. 

8. — ac^ + b'^c + obex = abc + cmx — ac-x + h-c — mc. 

9. {a + X + b) (a + b - x) = {a + x) {b — x) — ab.* 

10. (a — x){a + x) = 2a^ + 2ax — x'. 

11. (a2 + a;)2 = x2 + 4a2 + a*. 

1 2. (o" - a;) (a- + x) = a* + 2ax — x-. 

ax-b x + ac m (p-x + x^) mx^ 

13. l-a = . 17. — ^ ' = mqx-\ . 

•^ c c px p 

3a-bx 1 „ X , c 

TA. ax 7; — = ^. lo. — o = -, — x. 

^ 2 z ad 

Aax - 26 x'^ — aa-x2xa 

15. 6a 3— = x. 19. -fc^— 2,- = y--- 

, 6x4-1 a{x^-\) 3 ab-x- 4x-ac 

16. ax = ~^ -. 20. , = . 

XX c ox ex 

ab + x b' -x x — b ab-x 



22. 



¥ a'^b a^ 6- 

3ax — 2b ax — a ax 2 



36 26 



, ax 05 - 

2"?. am — 6 — -^H = 0. 

■^ 6 m 

^a263^ 62^ 3a2c- _ 3acx _ ^-^^ab^x 

(a + 6) a(a4-6) a + 6~ 6 (a + 6) " 

ax^ ax ^ a6 , , 1 

25. r + a + — = 0. 27. — = 6c + rf+ . 

o~cx C X X 

, a(d- + x-) ax _ m(a — x) 

26. ^— J ^ = ac + -j-. 28. c = a+~' -^. 

ax a 3a + x 



29. (a + x) (6 + x) - ffl (6 + c) = ^ + x'i 



ace (a + by.x , „, 

30. — T-— ^^ ox = ae-36x. 

- d a 

191. In the examples already given the L.C.M. of the 
denominators can 'generally In- deterniined by inspection. 
When compound expressions appear in the denominators, it 
is sometimes desirable to collect the fractions into two, one 



ON FRA CTTONA L EQUA TIONS 1 1 1 



on each side of the eqiiution. When tliis has been done, we 
can clear the equation of fractions liy multiplying the nu- 
merator on the hft by the denominator on the right, and the 
numerator on the right by the denominator on the left, and 
making the produ ts equal. 

For, if ^ = -j, it is evident that ad = bc. 
' a 

F 4x + 5_13x-6_2.'c-3_ 

10 7x-l-4 ~ 5 ' 

4a: + 5 2x-3_13a:-6^ 

■■ To 5~~ 7.C + 4 ' 

4x + 5 - (4a; - 6) _ 1 3x - 6 ^ 
' 10 7x + 4 ' 

ll_13x-6^ 
•■ 10~ 7x + 4 ' 
.-. ll(7x + 4) = 10(l;ix-6); 

whence we find a; = -— r-. 

06 

Examples.— Ivii. 

3x + 7 3X-1-5 ,2 5 ^ 



4X-I-5 


4x + 


3" 


x-l-6 


X 




2x + 5 


2x- 


5' 


2x + 7 


4x- 


1 


x + 2 


2x- 


1' 


5x-l 


5x- 


3 


2x + 3 


2x- 


3* 


1 


2 
4-^ — 





1 - 5x 1 - 2x 
1 1 3 



7- —- + 
8 



x-1 x+1 X--1' 
4x + 3 8x+19 7x-29 



9 18 5x-12" 

X .'■- - ox _ 2 
3 ox — 7 3* 
3x + 2 2x - 4 ^ 

'' 3^^-^^-"3="' '"• ^:rr+^T2-=^- 

II. l(x + 3)-^(ll-x) = |(x-4)-l(x-3). 

(x+_lM2x + 2)_.^_ J, x+_l_^l_ 

(x-3)(x + 6) " ■ ^ x + 1 x-l~l-x2- 
(2x + 3)x 1 , 2 8 4.5 

•^ 2.(- + i 3x ^ 1 - X 1 + X 1 - x^ 



ON FRA C TIONA L EQUA TIONS. 



, 4_ _3^ iA = _.? 

■ x-8''2j;-16 24 3a;-24' 
:c*-(4a;2-20a; + 24) , „ 
a;2 - 2.C + 4 

„ 2rc* + 2x3 -23x2 + 31a; 

18. ., —r ; = 2X'' - 4x - 3. 

X- + 3x - 4 

(A 2\ 1 3x-(4-5x) 

192. Equations into which Decimal Fractions enter do not 
present any serious difficulty, as may be seen from the follow- 
ing Examples : — 

Elx. 1. To solve, the equation 

•5x = -03x + l-41. 

Turning the decimals into the form of Vulgar Fractious, 
we get 

5x_2x_ 141 

10 ~ loo "^ loo* 

Then multiplying both sides by 100, we get 
50x = 3x+141; 
therefore 47x=141; 

therefore x = 3. 

Ex. 2. 1 •2x - i^^^ = -ix + 8-9. 
■5 

First clear the fraction of decimals by multiplying its 
numerator and denominator by 100, and we get 

i-2x-— |=^ = -4x + 8-9; 
DO 

^, , 12x 18x-5 4x 89 
therefore -^-^y ___ = _ + _; 

therefore 60x - 1 8x + 5 = 20x + 445 ; 

therefore 22x = 440; 

therefon- x = 20. 



ON' FRA CTIONAL EQ UA TIOMS. 1 1 3 



Examples. — Iviii. 

1. •5.-c-2 = -25x + -2x-l. 

2. 3-25X-5-1 H-x — ■75x = 3-9 4-"5x. 

3. •125x + -01x=13--2x-+-4. 

4. -S.c + 1 -SOS.t + -Sx = 22 -95 - • 1 95x. 

5. •2x--01x + -005x=ll-7. 

6. 2-4x-:^^^;:-^^ = -8x + 8-9. 

7. 2-4x- 10-75 = -25x. 8. •5x + 2- •75a;=-4.r.- 11. 
9. ^^ + 3-875 = 4-025. 



10. 2-5x ^=^{ i~^)~"^ 



2 + .'.•/! ^\ . 5x4-3 
8 "' 



8-5 -2 J 1--1X -48x 3-4x .„__ 

"• Y-x=^4 -.^' ■ ''■ ^—^-=^^^3. 

2-3x 5x _2x-3_x-2 ^^7 

14. ?i^ + --. ■04(x + -9) = 24I-2. 

•45X--75 1-2 -Sx-'e 
15...5X + ___ = _-___. 

, , 3-5x 24 -3x .^, 

16. -5 ^ „ =-3/5x. 

X — 2 8 

•135X--225 -36 -09x--18 

17. •15X + = - ^^. 

193. To shew that a simple equation can only have one root. 

Let x = a be the equation, a form to which all equations of 
the first degree may be reduced. 

Now suppose a and /3 to be two roots of the equation. 
Then, by Art. 109, 

a = a, 
and /? = «, , 

therefore a = P\ 

in other Avords, the two supposed roots are identicaL 

XS.A.1 H 



XIII. PROBLEMS IN FRACTIONAL 
EQUATIONS. 

194. We shall now give a series of Easy Problems resulting 
for the most part in Fractional Equations. 

Take the following as an example of the form in which such 
Protjlems should be set out by a beginner. 

"Find a number such tliat the sum of its third and fourth 
parts shall be equal to 7." 

Suppose X to represent the number. 

Then - will represent the third part of the number, 

o 

and - will represent the fourth part of the number. 

X X 

Hence ^ + t "^^'i^^ represent the sum of the two parts. 

But 7 will represent the sum of the two parts. 

Therefore ^ "^ 4 ^ "^^ 

Hence 4a; + 3x = 84, 

that is, 7x = 84, 

that is, a; =12, 

and therefore the number sought is 12. 

Examples. — lix. 

1. What is the number of which the half, the fourth, and 
the tilth jiaits added together give as a result 95 ? 

2. ^^"llat is the number of which the twelfth, twentieth, 
and fortieth parts Ridded together give as a result 38 ? 

3. What is the number of which the fourth part exceeds 
the Ulth part by 4 1 



PR OR f. EMS IX FRACIIOXAL EQUATIONS. 115 



4. Wiiat is the iniiuber of wliicli the twenty-fifth part 
exceeds the thirty-tifth jjart by 8 \ 

5. Divide GO into two such parts that a seventh part of one 
may be ec^ual to an eighth part of the other. 

6. Divide 50 into two such parts that one-fourth of one 
parr being added to five-sixths of the other part the sum may 
be 40. 

7. Divide KK) into two such parts that if a tliird part of tlie 
one be subtracted from a fourth part of the otlier the remainder 
may be 11. 

8. \Yhat is the number which is greater than the sum of its 
third, tenth, and twelfth parts by 58 ? 

9. "When I have taken away from 33 the fourth, fifth, and 
tenth pai'ts of a certain number, the remainder is zero. Wliat 
is the number ? 

10. What is the number of which the fourth, fifth, and 
sixth parts added together exceed the lialf of the number 
by 112? 

11. If to the sum of the half, the third, the fourth, and the 
twelfth parts of a certain number I add 30, the sum is twice as 
large as the original number. Find the number. 

12. The difference between two numbers is 8, and the 
quotient resulting from the division of the greater by the less 
is 3. What are the numbers ? 

1 3. The seventh part of a man's property is equal to his 
whole property diminished by £1626. What is his property ? 

14. The difference between two numbers is 504, and the 
quotient resulting from the division of the greater by the less 
is 15. What are the numbers ? 

15. The sum of two numbers is 5760, and their difference 
is equal to one-third of the greater. What are the numbers ? 

16. To a certain number I add its half, and the result is as 
much above 60 as llie number itself is below 65. Find the 
number. 



ii6 PROBLEMS IN FRACTIONAL EQUATIONS. 

17. The difference between two numbers is 20, and one- 
seventb of the one is equal to oue-third of the other. What 
are the numbers ? 

18. The sum of two muubers is 31207. On dividing one 
by the other the (^uolient is fouud to be 15 and the remainder 
1335. What are tlie numbers ? 

19. Tlie ages of two brothers amount to 27 yeai-s. On 
dividing the age of the elder by that of ihe younger the quo- 
tient is 3i. What is the age of each ? 

20. Divide 237 into two sucb parts that one is four-fifths of 
the other. 

21. Divide £1800 between A and B, so that 5's share may 
be two-sevenths of ^'s share. 

22. Divide 46 into two such parts that the sum of the 
quotients obtained by dividing one part by 7 and the other by 

3 may be equal to 10. 

23. Divide the number a into two such ]iarts that the sum 
of the quotients obtained bv dividing one part by 7?i and the 
other by n may 1*6 equal to h. 

24. The sum of two numbers is a, and their difference is h. 
Find the numbers. 

25. On multiplying a certain number by 4 and dividing 

the product by 3, I obtain 24. Wiiat is the number ? 

5 

26. Divide £864 between A, B, and G, so that A gets — 

of what B gets, and C"s share is equal to the sum of the shares 
of A and B. 

27. A man leaves the half of his property to his wife, a 
sixth part to each of his two children, a twelfth part to his 
brotlier, and the rest, amounting to £600, to charitable uses. 
What was the amount of his property ? 

28. Find two numbers, of which the sum is 70, such that 
the first divided by the second gives 2 as a quotient and 1 as 
a remainder. 

29. Find two niimbers of Avliich the difference is 25, such 
that the second divided by tlie tiist ^jives 4 as a quotient and 

4 as a renjainde:. 



PROBLEMS IN FA' ACTIONAL EQUATIONS. W; 

30. Divide tlie number ^08 into two parts snch that the 
sum of tlie fourtli of the tjreater and the tliird of the less is 
less bv 4 tliau four times the difference between the two parts. 

31. There are thirteen days between division of term and 
the end of the first two-thirds of the term. How many days 
are there in the term ? 

32. Out of a cask of wine of which a fifth part had leaked 
away 10 i^allons were drawn, and then the cask was two-thirds 
full. How much did it hold 1 

'^3. The sum of the ages of a f;ither and son is half what it 
will be in 25 years : the difference is one-third what the sum 
will be in 20 years. Find the respective a.ges. 

34. A mother is 70 years old, her daughter is e.xactly half 
that age. How many years have passed since the mother was 
3J times the age of the daughter ? 

35. A is 72. and B is two-thirds of that age. How long is 
it since A was 5 times as old as B ? 

Note I. If a man can do a ]iiece of work in x hours, the 
part of the work which he can do in one hour will be repre- 
sented by -. 
•' X 

Thus if A can reap a field in 12 hours, he will reap in one 
hour — of the field. 



Ex. A can do a piece of work in 5 days, and B can do it 
12 days. I 
do the work ] 



in 12 days. How long will A ami B working together take to 



Let X represent the number of days A and B will take. 
Then - will represent the part of the work they do daily 

Now - represents the part A does daily, 
and Yg represents the pai-t B does daily. 



irS J'ROBLEMS IN FRACTIONAL EQUATIONS. 



Hence - + -- will represent the part A and B do daily. 

.1 1 1 1 

Consequently ^4-^^ = -. 

Hence 12x + 5x = 60, 

or 17x = 60; 

60 

•■• ^ = 17- 

9 

That is, they will do the work in 3r— days. 

36. A can do a piece of work in 2 days. B can do it in 3 
days. In what time will they do it if they work together ? 

37. A can do a piece of work in 50 days, B in 60 days, 
and G in 75 days. In what time will they do it all working 
together ] 

38. A and B together finish a work in 12 days ; A and G 
in 15 days ; B and G in 20 days. In what time will they 

finish it all working together ? 

39. A and B can do a piece of work in 4 hours ; A and G 
in 3- hours ; B and C in 5= hours. In what time can A do 
it alone ? 

, 40. A can do a piece of work in 2;^ days, B in 3.^ days, 

and G ill ?> days. In what time will they do it all working 

together ? 

41. A does - of a piece of work in 10 days. He then calls 

in B, and they finish the work in 3 days. How long would B 
take to do one-third of the work l>y liimself ? 

Note II. If a tiip can fill a vessel in x hours, the part of 
the vessi'l llllcd 1>\ it in om- Innir will be represented by . 

Ex, Three taps running separately will fill a vessel in 20, 
30, and 40 minutes respectively. In what time will they fill it 
when thev all run at the same time \ 



PROBLEMS IN FRACTIONAL EQUATIONS. 119 



Let X represent the number of minutes they will take. 

Then - will represent the part of the vessel filled in > 
minute. 

Now - represents the part filled by the first tap in 1 minute, 



1 

30 

J_ 

40 



second . 
third.. 



1111 

Hence 20 + 30 + 40 = ? 

or, multiplying both sides by 120ic, 

6a; + 4x + 3.x = 120, 
that is, 13a; = 120; 

120 

••• ^=-iy 

3 
Hence they will take 9 ^^ minutes to fill the vessel. 

42. A vessel can be filled by two pipes, runnincr separately, 
in 3 hours and 4 hours respectively. In what time will it be 
filled when both run at the same time ? 

43. A vessel may be filled by three different pipes : by the 

first in I5 hours, by the second in 3- hours, and by the third 

iu 5 hours. In what time will the vessel be filled when all 
three pipes are opened at once ? 

4i|. A bath is filled by a pipe in 40 minutes. It is emptied 
by a waste-pipe in an hour. In what time will the bath be 
full if both pipes are opened at once ? 

45. If three pipes fill a vessel in a, 6, c minutes running 
separately, in wliat time will the vessel be filled when all three 
are opened at once ? 



I20 PROBLEMS IN FRACTIONAL EQUATIONS. 



46. A vessel containing 755 "allons can be filled bv three 
pipes. The first let<? in 12 gallons in Z- minutes, the second 

15- gallons in 2r minutes, tlie third 17 gallons in 3 minutes : 

in what time will the vessel })e filled by the three pipes all 
running together? 

47. A vessel can be filled in 15 minutes by three pipes, 
one of which lets in 10 gallons more and the other 4 gallons 
less than the third each iiiiniite. The cistern holds 2400 gallons. 
How much comes throug'n each pipe in a minute ? 

Note III. In questions involving distance travelled over in 
a certain time at a certain rate, it is to be observed that 

Distance ^t^. 

— .is = linie. 

Rate 

That is, if I travel 20 miles at the rate of 5 miles an hour, 

number of hours I take = -^. 
5 

Ex. A and B set out, one from Newmarket and the other 
from Cambridge, at the same time. The distance between the 
towns is 13 miles. A walks 4 miles an hour, and B 3 miles an 
hour. Where will they meet ? 

Let X represent their distance from Cambridge when they 
nu^et. 

Then 13 -a: will represent their distance from Newmarket. 

X 

Then - = time in hours that B has been walking. 



13- 
4 


X 








A 








And 


since 


both have been walking 


the 


same 


time, 








X 


13- 


- X 














3" 


4 


» 












or 


4.x = 


= 39- 


-3x, 












or 


7x = 
.'. x = 


= 39; 

39 
" 7' 











PROBLEMS IN FRACTIONAL EQUATIONS. 121 



4 
That is, they meet at a distance of 5- miles from Cam- 

bridge. 

48. A person starts from Ely to walk to Cambridge (wliich 

4 
is distant 16 miles) at the rate of 4- miles an hour, at the 

y 

same time that another person leaves Cambridge for Ely 

walking at the rate of a mile in 18 minutes. Where will they 

meet ? 

49. A person walked to the top of a mountain at the rate 
of 2- miles an hour, and down the same way at the rate of 

o 

3^ miles an hour, and was out 5 hours. How far did he walk 
altogether ? 

50. A man walks a miles in 6 hours. "Write down 

(1) The number of miles he will walk in c hours. 

(2) The number of hours he will be walking d, miles. 

51. A steamer which started from a certain place is fol- 
lowed after 2 days by another steamer on the same line. The 
first goes 244 miles a day, and the second 286 miles a day. In 
how many days will the second overtake the first ? 

52. A messenger who goes 31 ^ miles in 5 hours is followed 

after 8 hours by another who goes 22- miles in 3 hours. When 
will the second overtake the first ? 

53. Two men set out to walk, one from Cambridge to 
London, the other from London to Cambridge, a distance of 
60 miles. The Ibrmer walks at the rate of 4 miles, the latter 

3 

at the rate of 3- miles an hour. At what distance from Cam- 
4 

bridge will they meet ? 

54. A sets out and travels at the rate of 7 miles in 5 hours. 
Eight hours afterwards B sets out frrnu the same place, and 
travels along the same road at the rate of 5 miles in 3 hours 
After what time will B overtake A. ? 



122 PROBLEMS TN FRACTIONAL EQUATIONS. 





Note IV. In problems relatincj to clocks the chief point to 
be noticed is that the minute-hand moves 12 times as i'ast as 
the hour-hand. 

The following examples should be carefully studied. 

Find the time between 3 and 4 o'clock when the hands of a 
clock are 

(1) Opposite to each other. 

(2) At right angles to each other. 
(3^ Coincident. 



«g3 



(1) Let ON represent the position of the rainiite-hand in 
Fig. I. 

OD represents the position of the hoiu-luind in Fig. I. 
M marks the 12 o'clock point. 
T 3 o'clock 

The lines OM, OT represent the position of the hands at 
3 o'clock. 

Now suppose the time to be x minutes past 3, 

Then the minute-hand has since 3 o'clock moved over the 
urc MDN. 

And the hour-hand has since 3 o'clock moved over the 
arc TD. 

Hence arc MDN= tvelve times arc TJX 

If then we represent MDN by x, 

we shall represent TD by . 

Also we shall represent MT by 15, 
and DX in- 30. 



PROBLEMS IN FRACTIOiVAL EQUATIONS. T?3 



Now MDN = MT ^TD-\- UN, 

that is, x=15 + — +30, 

or 12a; = 180 + x + 360 

or llx = 540; 

540 
.•.x=— . 

Hence the time is 49-- niimites past 3. 

(2) In Fig. II. the description given of the state of the 
clock in Fig. I. applies, except that DN will he represented hy 
15 instead of 30. 

Now suppose the time to he x minutes past 3. 

Then since 

MDN= MT+TD + DN, 

x=15 + ^ + 15. 

from which we get 

360 

8 ^ 

that is, the time is 32— minutes past 3. 

(3) In Fig. III. the hands are both in the position ON. 

Now suppose the time to be x minutes past 3. 

Then since 

MN=MT+TN, 

IK ^ 

^=15 + ^2, 

or 12x=180 + x, 

180 
or x = --, 

4 
that is, the time is 16 — minutes past 3. 

55. At what time are the hands of a watch opposite to 
each other, 

(1) Between 1 and 2, 

(2) Between 4 and 5, 

(3) Between 8 and 9 ] 



124 PROBLEMS IX FRACTIONAL EQUATIO.VS. 

56. At what time are the hands of a vatch at light angles 
to each other, 

(1) Between 2 and 3. 

(2) Between 4 and 5, 

(3) Between 7 and 8 \ 

57. At what time are the liands of a watch together, 

(1) Between 3 and 4, 

(2) Between 6 and 7, 

(3) Between 9 and 10 ? 

58. A person buys a certain number of apples at the rate 
of five for twopence. He sells half of them at two a j)enny, 
and the remaining half at three a penny, and clears a penny 
by the transaction. How many does he buy ? 

59. A man gives away half a sovereign more than half as 
many sovereigns as he has : and again half a sovereign more 
than half the sovereigns then remaining to him, and now has 
notliing left. How much hud he at first ? 

60. ^Miat must be the value of 71 in order that 
may be equal to -— wlien a is - ? 



3u + 69a 



61. A body of troops retreating before the enemy, from 
which it is at a certain time 25 miles distant, marches 18 miles 
a day. The enemy jairsues it at the rate of 23 miles a day, 
but is fiist a day later in starting, then after 2 days is forced 
to halt for one day to repair a bridge, and this they have to do 
again after two days' more marching. After how many days 
from the beginning of the retreat will the retreating force be 
overtaken ? 

62. A person, after ]iaying an income-tax of sixpence in the 
pound, gave away one-tliirteentli of his remaining income, and 
had .£540 left. What was his original income ? 

63. From a sum of money I take away £bO more than the 
half, then from the remainder £.10 more than the filth, then 
fiom the seconil remainder ;£20 more than the fourth part : 
and it last onlv i;iO remains. W!;at was the original sum ' 



PROBLEMS IN FRACTIONAL EQUATIONS. 125 



64. I bou;4lit a certain number of eggs at 2 a penisy, and 
the same nuuiher at 3 a penny. T sold tlieni ut 5 for twopence. 
and lost a petiny. How man}' eg;,'S aid I Luy ? 

65. A cistern, liolding 1200 gallons, is tilled by 3 pipes 
A, B, C in 24 minutes. The pipe A re'Uiires 30 minutes more 
than C to fill tlie ci-stern, and U) gallons le.~s run tl.rough C per 
minute than through .4 and B togellier. What time would 
each pipe take to till the cistern by itstdf ? 

66. A, B, and (-' drink a barrel of beer in 24 days. A and 

4 
B drink „rds of what C does, and B drinks twice as much as A. 
o 

In what time would each separately drink the cask ] 

67. A and B shoot by turns at a tari;et. A puts 7 bullets 
out of 12 into the centre, and B puts in 9 out of 1-. Between 
them they put in 32 bullets. How many shots did each fire? 

68. A farmer sold at market 100 head of stock, horses, 
oxen, and sheep, selling two o.xen for every horse. He obtained 
on the sale £2, 7s. a head, li he sold the horsgs, oxen, and 
sheep at the respective prices .£22, £12, lOs., and £1, 10s., how 
many horsesi^oxen, and sheep respectivir-ly did he sell ? 

69. In a Euclid paper A gets 160 marks, and i> just passes. 
A gets full marks for book-work, and twice as many marks 
for riders as B gets altogether. Also B, sending answers 
to all the questions, gets no marks for riders and half marks 

for book-work. Supposing it necessary to get - of full marks 

in order to pass, find the number of marks which the paper 
carries. 

70. It is between 2 and 3 o'clock, but a person looking at 
the clock and mistaking the hour-hand lor the minute-hand, 
fancies that the time of day is 55 minutes earlier than the 
reality. What is the true time ? 

71. An army in a defeat loses one-sixth of its number in 
killed and wounded, and 4(X)0 prisoners. It is reintbrced by 
3000 men, but retreats, losing a fourth of its nundjer in doing 
so. There remain 18000 men. What was the original force / 

72. The national debt of a country was increased by one- 
fourth in a time of war. During t\\ enty years of peace widen 



ii6 Oy MISCELLANEOUS FRACTrON^. 

followed £25,000,000 was paid off, and at the end of that time 
the interest! was reduced from 4J to 4 per cent. It was then 
found that the interest was the same in amount as before the 
war. What was the amount of the debt before the war ? 

73. An artesian well supplies a brewery. The consump- 
tion of water goes on each week-day from 3 a.m. to 6 p.m. at 
double the rate at which the water flows into the well. If 
the well contained 2250 gallons when the consumption began 
on Monday morning, and it was just emptied when the con- 
sumption ceased in the evening of the next Thursday but one, 
what is the rate of the influx of water into the well in gallons 
per hour ? 



XIV. ON MISCELLANEOUS FRACTIONS. 

195. In this Chapter we shall treat of various matters con- 
nected with Fractions, so as to exhibit the mode of applying 
the elementary rules to the simplification of expressions of a 
more complicated kind than those which have hitherto been 
discussed. 

196. Tlie attention of the student must first be directed 
to a point in which the notation of Algebra difiers from that of 
Arithmetic, namely wktn a whole number and a fraction stand 
side by side vdth no sign between them. • 

3 3 

Thus in Arithmetic 2'- stands for the sum of 2 and -. 
/ 7 

But in Algebra x- stands for the product of x and ". 

So in Algebra 3— — stands for the product of 3 and ; 

° c c 

. „a + b 2a + Zb 

that 18, 3 = — - — 

c c 



ON MISCELLANEOUS FRACTIONS. 127 



Examples. — Ix. 

Simplify the following fractions : 

1, a + x + 3-. 3. ^ + 2—*^. 

X ^ X x-y 

a- + ax jx-a .a + b ^a'^ — b^ 

2. • s- -2 — -, 4. 4 ,~2--. — jT,. 

x^ X a-h a^-\-¥ 

197. A fraction of which the Numerator or Denominator 

is itself a fraction, is called a Complex Fraction. 

y X 

Thus -, ■% and — are complex fractions. 
a a m 

b n 

A Fraction whose terms are whole numbers is called a 
Simple Fraction. 

All Complex Fractions may be reduced to Simple Fractions 
by the ]nocesses already described. We may take the follow- 
ing Examples : 

a 

b_am_a n _an 
^ m~b ' n b m bm 
n 



b___d_/a c\ /m _p\_ad-bc , mq-np 
^~' m p \b dJ \n q/ bd ' nq 
n q 

_ad — bc nq _ nq (ad - be) 

bd mq - np bd {mq — np)' 



,„, 1+x ,, . /, 1\ ,, , x-t-1 
(3) _ = (i+x)^(^l + ^j = (l+x)--— — 

_l+a; X _x(l +x' 



1 + - 

X 



1 "x+1 1+x * 



t28 OM MISCELLA.VF.OUS FRACTTOyS. 



1 1 



^^ X . 1 Vl-a; 1 + x/ ■ Vl-x 1 + x' 
1+x-l+x cc + x^+l-. 



1-x 1+x 



(5) 



1_3;2 • l_a;2 

_ 2x l-x^_ 2j 
~ 1 - a;2 1 + x=^ ~ 1+ x"-^" 

3 3 3 3 



3,3, 3(1 -.c) , 3-,ix 
3 l-x+3 l-x+3 4-x 

1 — X 1 - X 

3 3 (4 - x) _ 12 -3x 

'^ 4-x + 3-3x ~4^x + 3-3x~ 7-4x' 
4 — X 

Examples.— IxL 

Simplify the following expressions : 

4 « « 

5 X 7/ X • 1 -g* 

^' 7 ~ 1' ^' x-u" ^' r 

3— ^ 1+- 

23 ^^a 



0-D 



2-x + -„ 1+i 



1 XX 



a^ -x + rtx-a 2x 
7. — ^- 8. 2I • 9- r 



a x^-a'' 1-rx^ 

x-u x + V 

.X , 1 x+y X -y 

7+1 pj. II. s. 

1 x + 1 x-y x + y 



1 + 



X+y x-v 



ON MISCELLANEOUS FRACTIONS. 129 



., 2771 - 3 + — 

1 + a: + X- m 

■ '^- -T-T- ^4. 2m- 1 

1 + - -I .,- 

X x' m 

a + b _b_ J_ L i. 

b a + h ah ac be 

a 6 aft 

198. Any fraction may Tie split np into a number of trac- 
tions equal to the number ol tenns iu its numerator. Tiius 

a^ + x^ + x + l x^ I!? X 1 
X* X* x'* :f^ X* 

1111 

X X- X^ X* 



Examples.— Ixii. 

Split up into four fractions, cacli in its lowest terms, the 
following fractions : 

a* + 3a3 + 2a- + 5a 9«3 - 1 2^2 + 6a - 3 

'• 2a* ■ ^ 108 ■ 

a^bc + alri + abc^ + feed' 18;?-+ 12y^-36r2 + 72sg 

abed, ' 'Spqrts 

x^-3x2y + 3x?/--?/ 10x3 - 25x2 + 75x- 125 -^ 

5' x'-y ' ■ 1000 ■ 

199. The quotient obtained by dividing the unit by any 
fraction of that unit is called The Reciprocal of that fraction. 

Thus -, that is, -, is the Reciprocal of ?-. 
a a ^ 6 

b 

200. "VVe have shewn in Art. 158, that the fraction symbol 
r is a proper representative of the Division of a by b. In 

r.s.A.] 1 



no ox MISCELLANEOUS LR ACTIONS. 



Chapter IV. we treated of C3=es of division in which the divisor 
is contained an exact number of times in the dividend. We 
now proceed to treat of cases in which the divisor is not con- 
tained exactly in the dividend, and to shew the proper method 
of representing the Quotient in such cases. 

Suppose we have to divide 1 by \-a. We may at once 

represent the result by the fraction . But we may 

actually perform the operation of division in the following 
■way. 

\-a) 1 (1 +a + a2 + a3-t-... 
\-a 

a 



i3-a4 



The (^lotient in this case is interminable. We may carry 
on the operation to any extent, but an exact and terminable 
Quotient we sliall never find. It is clear, liowever, that the 
terms of the Quotient are formed by a certain law, and such 
a succession of terms is called a Series. If, as in the case 
before us, the .scries may be indefinitely extended, it is called 
an Infinite Serie.s. 

If we wish to express in a concise i^tiu the result of the 
operation, we may sto|) at any term of the quotient and write 
the result in the following way. 

_!__ _a_ 
l-a~^'l-a' 

1 - a 1 - a' 

1 , ., «^ 

;; = 1 + a + a- + :; , 

\-a \-a 

= 1 + a T ((- + a^ + :; , 

I -a 1-a' 



ON MrsCELLANEOUS FRACTIONS. 



13' 



always bein^ careful to attach to that term of tlie quotient, at 
which we intend to stop, the remainder at that point of the 
division, placed as the numerator of a fraction of which the 
divisor is the denominator. 



Examples. — Ixiii. 

Carry on each of the following divisions to 5 terms in the 
quotient. 

1. 2 by \+a. 7. 

2. m by m + 2. 8. 

3. a - 6 by a + 6. 9. 

4. a^ + X- by a^ - x^. 10. 

5. ax by a- X. 11, 



1 by 1 + 2x- - 2x2. 
1 + X by 1 - X + x^ 
1 + h by 1 - 2&. 
x^ — 6^ by X + 6. 
a^ by x-h. 



b bv a + x. 



1 2. a^ by (a + x)^ 



13. If the divisor be x-a, the quotient x--2ax. and the 
remainder 4a^, what is the dividend ? 

14. If the divisor be m - 5, the quotient m^ + 5m^ + Ibm + 34, 
and the remainder 75, what is the dividend ? 



201. If we are required to multiply such an expression as 
x^ X 1 , X 1 
¥ + 3 + 4^^^2-3' 
we may multiply each term of the former by each term of the 
latter, and combine the results by the ordinary methods of 
addition and subtraction of fractious, thus 



a;2 X 1 




X 1 




2 3 




X^ X^ X 




4+-6- + 8 




X^ X 


1 


6 9 


12 


«* X 


1 


t ' 72 


LX 



t32 Or^ MISCELLAiWEOUS ER ACTIONS. 

Or we may first reduce tlie mulliplicaiid and the multiplier 
to single I'ractioiis und proceed in the loUowing way : 



(-2+3 + 4)42-3) 



_ 6.x2 + 4x + 3 3a:-2_ 18ar^4-x- 6 
12 ^ 6 ~ 72 

"72+72 72~ 4 +72 12 
This latter process will be louud the simpler ty a beginner. 



Examples.— ixiv. 

Multiply 

a- a \ , a \ 11,11 

'• y-6 + 3^'-^'4-5- 5- ^ + 6^by^-p-. 

, 11, 1 ,111,111 

xa;-^ X a c ' a c 

7. 1 + - + -r by 1 - - + -^. 

8. l+-a:f-.r-byl--x + -x--x3. 

5^ 37 2 1_1 

9' 2x2 + x'3 ^x-^"x 2" 

10. pr + -5- + 2 by j:r - -T - 2. 
2b2. If we have to divide such an expression as 



^ o 3 1 

X x^ 



by X + -, we may proceed as in the division of whole numbers, 

carefully observing that the order of descending powers of x 
is 

*^' ^' *' t' X2 ' X3 



ON MISCELLANEOUS ER ACTIO. \S. 133 



Any isolated digits, uo 1, 2, .j ... will stand between x 

, 1' 
and -. 

X 

Tims the expression 

■! 1 r. O y ^ 5 

arranged according to descendinfj powers of x, will stand thus, 

5 3 1 

a^ + 3x2 + 5x + 4 + - + _ + 

The reason for this arrangement will be given in the Chapter 
on the Theorv of Indices. 



Ex. x + l ]x3 + 3x + ^ + -,l x2 + 2 + 4 

x/ a. x^ ^ X'* 

x^+ X 



2x 


3 

X 




2x 


2 

+ - 

X 






1 

- + 

X 


1 




1 

- + 

X 


1 

X3 



Or we may proceed in the follov/ing way, which will be 
found simpler by the beginner. 



(x3 + 3x + ^ + l3)-(x+^) 



3 1 

)-^lx+ , 

x/ 

x^ -f 3x^ + 3x- + 1 , x2 + 1 

x^^ ■ X 

x« + 3x* + 3x2 + 1 3. 



x^ x- + 1 

X* + 2x' +1 X* 2x^ 1 , ^ i 
= = —, + -^ + - = x- + 2 + -.. 

X- X^ X' X- X' 



134 ON MISCELLANEOUS FRACTIONS. 



Examples.— Ixv. 

Divide : 

2 1 V, 1 fill,' 

1. %'■ — nDya; + -. 4. c° — t- bv c — -5. 

1 1 X V^ X 1/ 

2. a--j-„hy a--.. 5. -5 + 2 + % by - + ^. 

b^ •' b y^ x^ -^ y X 

3. m-' + -3bym + -. 6. -4 + -wg + ri bv — ,--r 4 rs- 



a-' w'^ X y . X y 

7. -.-^,-3-4-3'- by --4 
1/-* x-* 1/ X y X 

_ 3x5 , , 77 , 43 „ 33 „- , a;^ 

8. -r - 4x* 4- — x3 - —X- - ^x 4- 27 by — - X 4- 3. 

a^ ¥, a b 1113,111 

9. i:i + ^byT + - 10. -, + M + ~i — jrDy- + rH — • 

^ ¥ a? ■' a a^ ¥ c^ abc ■' a b c 

203. In dealing with expressions involving Decimal Frac- 
tions two methods may be adopted, as will be seen from the 
following example. 

Multiply -Ix - -21/ by •03x 4- -4?/. 

We may proceeil thus, applying the Eules for Multiplication. 
Addition, and Subtraction ot Decimals. 

•\x—-2y 
•03x 4- -Ay 



•003x2 -Ouexy 

4-04 xy--08y» 

•003x"2T-03l--,v--0%2' 

Or thus, 

_ x-2y 3x4^40y 
~ 10 ^ lOOT" 
^ 3x2 4-3 4x?/-80y3 
~ 1060 

= -003x2 4. .034x2/ - -083/2. 

The latter method will be found the simpler for a be^nner. 



ON MISCELLANE O US ERA C TIONS. 1 35 



Examples.— ixvi. 

Multiply : 
(. -Ix- -3 by -53; +07, 2. •05x + 7by-2 -3, 

3. -Sx - -2!/ by •4x + -ly, 4. 4-3x + b-2y by •()4x - -06?/. 

5. Find the value of 

a^ - 6^ + c^ + Zabc when a = -03, h=-\, and c = -07. 

6. Find the value of 

01? - 3ax2 + 3a^x - n^ when x = '7 and (i = 'OS. , 

204. When any expression E is put in a form of which /is 

E 
a factor, then -^ is the other factor. 

Thus a + h = a{ \ 

c 7 T , ab-\-ac-¥hc 
So (U) + ac-^oc = aoc. -, 

and a?+2an/ + 2/2 = x2.(?l±^^±^') 

EXAMPLES.— Ixvii. 

1. Write in factors, one of which is a^x, the series 

a^x + a^ocr + a^x^ + a4X'* + . . . 

2. Write in factors, one of which is xyz, the expression 

xij -XZ + yz. 

3. Write in factors, one of which is x^, the expression 

X- + x!/ + y'. 

4. Write in factors, one of which is a + 6, the expression 

(a + 6)3-c(a + 6)2-d(a + 6) + e. 



136 O.V MISCELLANEOUS FRACTIONS. 

205. We s!i;ill now give two examples of a process by 
which, when certain Iractiuns are known to be equal, otlier 
relations between the q^uautities involved in them may be. 

deteruiineJ. 

This i)rocess will be found of great use in a later part of 
the suliject. and the student is advised to pay particular 
attention to it. 

(1) If ^= J, shew that 

0, 

a — b c — ^' 

T ^ a 

Let r = X- 



Then 3 = X ; 

a 

.■. a = \b, 
and c = \d. 
Now a + 6^X6+^&^6(X + l)_X + l 

a'-b \b-b~bl\-l)~\-l' 
c + d_\d + d d{X+l) X + 1 



and 



c-d Xd-d (Z(X-l) X-T 



TT C'-^b , c + fZ - . , , ;^+i 

tience — _-^ and ~-_-j being each equal to — — are equal to 

one another. 

(2) If =- = = .shew that m + u + r = 0. 

a — 0- c c - a 



Let 



a-b~^' 

o-c 

r 

= X, 

c-a 

then m = Xa-X6. 

n = \b- Xc, 

r='Kc - Xa ; 

.•. m + n + r = X(T -X^-L \6-Xc + Xc-Xff = 0. 



ON MISCELLANEOUS FRACTIONS. 137 



Examples. — Ixviii. 



O. C 

1. If r = -7 prove the following relations : 

. 'lZ^ — —A ^r\ 8a + & _ 8c + d 

, . rt _ _c , c2-6-_a6 

, N 3ft _ 3c . ll« + 6_13a + 6 

^^^ 4a - 56 "" 4c - 5rZ" ^^^ TlcTrf ~ 13c+"5" 

/ ^ "' + ^^_ C' + tf - «2-rt6 + 6-_c-'-ff(' + d* 

W a2_p-^rr"ci2- ^^) a2 + a6 + 62-c^ + cf/ + f;2- 

Tr ^ m 71 , , 

2. 11 r = T = , then i + 7/1-}- ?i = 0. 

a — n — c c — a 

^ jfO c e ^, ^ « la + vic + ne 

3. ii r = -, = 7, T>rove that y = .-7 1 > 

a J' ' h Ib + md + nf 



a+h b+c_c+t 
c a 



4- -11 — 7;— = = . prove that a = 6 = c. 

'■ r, ■ i- 



3- 11 1- =r = r> sliew that fJ= ^/ ^- . •*. 

6. II T, -J. J he in descending order of magnitude, shew 

xi-fi + c-l-e., , a , , e 

i_j7T— ->is less than ^- and greater tlian ■^. 

7. If -'=^ shew that ^^A^4x, + 5y,^ 

2/1 2/1. '^i + y^/i '^2 + 92/a 

T4-^ <^ 1 ^1 , rt--t-a6 ab-b- 

8. Il5 = ^,shewthat-^^ = ^^_^, 

9. If^ = .%hewthat7«-+.^, = I^,. 

o a 6a + ob 3c-f5(i 



138 Oy MTSCELLANEOUS FRACTIOf^S. 



lo. If r be a f roper fraction, shew that -r is greater 



than r, c being a positive quantity 

6 + 



II. If r bt- an improper fraction, shew that t — : is less 



than r, c being a positive quantity. 

206. We shall now give a series of examples in the svorking 
of which most of the processes connected with fractions will 
be introduced. 



Examples.— ixix. 

I. Find the value of Sa^ H y^ when 

a = 4, b = ^, c=l. 

^- Sin^pMy 7x^-12x + 5 -'^"'^ a^'-H4a-45 - 

3. Simplifv(^t^_^--^)^(«_tP + ^-n 
-^ ^ ' \a-j9 a+p/ \a-^ a+p/ 

4. Add together 

a;2 t/2 2^ ^2 j,2 3.2 j;2 j-i y2 

4" e'^S' 4""6"*" 8 ^""^ 4"^ 6"^8' 

and subtract 2- - x- + ^ from the result. 

5. Find the value of -5 — .,~ „ — :r^ wheii 

a=4, 6 = -, c = l. 

6. Multiply |x2 + 3ax - \a^ by 2x2 -ax-%, 

01 xi ^ a^-ft- „, 36- 

7. Shew that -. tw = « + 26 -t- r- 

' (a - 6)-^ a-h 



OM MTSCELLANEOUS FRACTIONS. 130 



8. Simplify ?^ + ^^ + 4-^!. 
^ X x-y x^ — xy- 

01 ^, , 60x3 -17x2- 4x + l ,- „, ^ 49 

Q. Shew tnat , ., . ^ = 12x-25h ^. 

^ 5x- + 9x - 2 .j; + 2 

„. ,.„ x*-9.r'' + 7x2 + 9x-8 

10. fei«^Pl'fy^4 + 7^3Z9^2_7^V8' 

11. Simplify ^^+ j— . 

1-.-- J 

1 2. Simplify a+'ah + 6^ f a + «6 + ^V^tt- )• 

13. Multiply together U + 1)\}' + j>)y ~ \ 

14. Add to!::'ether -, -, — -, -, and shew that if their 

^ ° a+V h+V c+l 

sum be equal to 1, then ahc = a + 6 + + 2. 

^. ., X b h^ h b^ , 

15. Divide--! 5 + - + --, by x-a. 

a b c . 
r-^cH '-a-\ =-0 

16. Simplify r , and shew that it is equal 

--rC + i-^aH ^6 

a c 

to -^^ ' — T^ — if 2s = a + 6 + c. 

be 

17. Shew that -^ + p+ _- = --— ^. 

a-^x a~x a^-T-x^ 

1 8. Simpllly r + r - 2-:^ — r-,. 

^ •'a-b a + b a^ + b- 



_. ,.. 6 a + 6 a2 + 62 
19. Simplify--^ --2 - + 2^^^ZTy 

„. ,.,. a--ab + ¥ (i--ly^ 



I40 ON MISCELLANEOUS FRACTIONS. 



2 
2 1 . Simplify r-j — ,-Tr, — 



(x2_i)-' 2x--4x + 2 l-x2 

... (^^-V'-vlah-c^ a->rh->rc 

22. Sunpuly-n T. — ,., — ^TT-^i— -• 

'^ '' c^ - a^ — b' + 2ab b + c — a 

23. Simplify /-^■.l--j-\-^-^^-x-^^y 

^ X X 

. /x-rt\3 x-2« + 6 , a + 6 

24. rind the value of I r I ^j, when x = — s"~' 

\x-6/ x + a-26 2 



a" - (6 - c)2 Zy- - (fi - c)- c- - (a - 6)^ 



, „. ,.„ (x2-4x)(x2-4)a 

26. Simplify^— ^,-^3-i-. 

27. Simplify ^^^^,^^,---^ 

28. Simplify ^ + — -5 r-r, + -s r---rr-^ rr^ 

•^ - ar X- X (x^ + l)^ x2+l 2--(x- + l)' 

T,,. . , x^ X a a^ , X a 

29. Divide -, - - + b by . 

a'^axx^-'ax 

30. Simplify |2—_-_^^-^^4-^-^.}^^-. 

31. Simplify ^^ " ^ + ^^'^ ^ (^ : '^l:^ ( V ")'^ ^ ^" - ^^\ 

„ , l-x-3x2 - l+3a2 + 2x3 

3^- ^^^^ (3-2.c-7x^)3 ^^""^ (-332^^x-^v 

33. Sim^lih(i±f,-i--t)^(^J-'^^). 
*'-' -^ • \X'-y- X' + y/ \x-y x + y/ 



34. si.p,ir,Q-:->)(-!^-i).(^-OC-.|^e-0 

35. Simplify 

a--(<6 g- 4- a6 + 6- /_ 2a3_ _ , \ / , _ ^gft \ 



OM MISCELLANEOUS FRACTLOXS. 141 



36. Simplify 



1 _ 1^+ 



2(a;-l)- 4(x-l) 4(a;+l) (x - 1)-' (x + 1)" 



37. Prove that 

1 1 

— + 



s—as-h&-c /111 \ 

4- + ... =s(- +r+ -+ ... )-". 

\a be I 



ab.c a {a - h) {x - a) h{b — a) {x - h) x {x - u) {x - 1)' 

38. Tf s = a4 ?) + c+ ... to 71 term?, slie-\v that 

-b s- 

T- -, — + 

a c 

39. MuUith-(^,-^^.>y -^$^/?-_. 
jy i- •■ \ X' - y- X- + y-/ X- — y-y + {x- + ij-)- 

, a-x , a- — x^ 

1 + 1 + -, r. 

,.r « + •'<; rt-4X' 

4.0. Simpluv — •.. 

^ , ^ •' a-x U- — X- 

a + x a- + X' 

41 . Divide x^ + 3 - s( ^ - x- ) + 4f x + - ) liy x+ -. 

42. If s = rt + 6 + c + ...tow terms, shew th;it 

s-a s — b .« - r , 

+ + +...=71-1. 

5 S S 

43. Divide (-"- — ^- - I l.y ( ..^^-. + X0- 
^•^ \x-|/ x + 2// \x- + y X'-y-y 

1--^. /I 

44. bimphfy ^—^ -T 

45- If r_-a6 =^;j:ri' P^«^'^ that --p^-^ = a6.i 

a c a 

46. Simplify 

p* + 4p^q + 6p-q' + 4p(f + 2^ ^ 1^ + ^P'1 -*- •^P^ '^ * 'f 
pi - 4p^q + Gjy-q' - 4pq^ + q* ' p^ - 3p-q + 3pg- - ^ 




142 SIMULTAXEOUS F.QUATIOXS 



48. Simplify 



1 1 y(xi/3 + x + a) 

y+~ 
^ z 





1 1 1 a: y 




^ iniBlifr * " ■'' " ~ '^ ^"^ ~ ^ • ' ^"^ " ^^' 


49. 






(a - 1/) (a - x)2 (a - 1/;2 (« _ x) 




3 


50. 


c.:..-„i:f.. f'^'^ 3-a-6-c 


i'"--' 1 1 1 a^ 0-c ' 




T)c ca ah 




Simpliiy -l{a'^-b'^. 

a — - — 


51- 



♦ b 



XV. SIMULTANEOUS EQUATIONS OF 
THE FIRST DEGREE. 

207. To determine several unknown quantities we must 
have as many independent equations as there are unknown 
quantities. 

Thus if we had this equation given, 
x + y = 6, 
we could determine no definite values of x and y, for 



a; = 2) x = 4) x = 3) 



or other values miglit he given to x and y, consistently with 
the equation. In fact we can find as many pairs of values of 
X and 1/ as we please, which will satisfy the equation. 



OF THE FIRST DEGREE. 143 



"We must have a second equation independent of tlie first, 
and then we may tind a pair of values of x and y wliicii will 
satis/ u both equations. 

Thus, if besides the equation x + y = Q, we had anotlier 
equation x-y = 2, it is evident that the values of x and y 
which will satisfy both equations are 

x = 4 

y = 2 
since 4 + 2 = 6, and 4-2 = 2. 

Also, of all the pairs of values of x and y which will satisfy 
one of the equations, there is but one pair which will satisfy 
the other equation. 

We proceed to shew how this pair of values may be found. 
208. Let the proposed equations be 

2x + 7i/ = 34 

5x + 9i/ = 51. 

Multiply the first equation by 5 and the second equation by 
2, we then get 

10x + 35j/ = 170 

10.c+18?/ = 102. 
The coefficients of x are thus made alike in both equations. 

If we now subtract each member of the second equation 
from the corresponding member of the first equation, we shaU. 
get (Ax. II. page 58 j 

35-j/-18i/=170-102, 
or 17(/ = 68; 

-■• 2/ = 4. 
We have thus obtained the value of one of the unknown 
symbols. The value 01 the other may be found thus : 

Take one of the original equations, thus 
2x + 7j/ = a4. 

Now, since y — 4,7y = 28; 

.: 2x + 28 = 34; 
.-. x = 3. 

Hence the pair of values of x and y which satisfy the 
ecjuations is 3 and 4. 



144 SIMULTANEOUS EQUATIONS 



Note. The process of thus obtaining from two or more 
equalioiis an equation, Ironi wliicli one ol the unknown quanti- 
ties has disappeared, is called Elimination. 

209. ^Ye worked out tlie steps fully in the example given 
in the last article. We shall now work au example in the lorm 
in which the process is usually given. 

Ex. To solve the equations 

5a; + 4)/ = 58. 
Multiplying the first equation hy 5 and the second by 3, 
15x4-35^ = 335 
15x4- 12// = 174. 

Subtracting, 23^=161, 

and therefore 2/ = 7. 

Now, since 3x4- 7;/ = 67, 

3x4-49 = 67, 
.-. 3x = 18, 
.-. x = 6. 

Hence x = Q and y=*7 are the values required. 

210. In the examples given in the two preceding articles 
we made the coetticients of x alike. Sometimes it is more con-" 
venient to make the coefficients ol" y alike. Thus if we have 
to solve the equations 

29x4-27/ = 64 

13x4- 2/ = 29, 
we leave the first equation as it stands, and multiply the 
second equation by 2, thus 

29x4- 2?/ = 64 

2Gx + 2!/ = 58. 

Subtracting, 3x = 6, 

and therefore x = 2. 

Now, since 13x4-y = 29, 

264-?/ = 29, 
.-. 2/ = 3. 
Hence x = 2 and ?/ = 3 are the values required. 



OF THE FIRST DEGREE. 145 



Examples — Ixx. 

I. 2a; + 7i/ = 41 2. 5.<; + 8i/=101 3. 13.r+ l??/^ 189 

3a; + 4i/ = 42. 9x + 2y = 95. 2a;+ i/ = 21. 

4. 14x + 9?/ = 15G 5. a; + 152/ = 49 6. 15;(;- 19?/= 132 

7x + 2)/ = 58. 3x+ 7)/ = 71. Sax + 17?/ = 226. 

7. 6a; + 4?/ = 236 8. 39.j; + 277/ = 10o 9. 72a; + 14?/ = 330 

3a; +15?/ = 573. 52a; + 297/ =133. G3x+ 7?/ = 273. 

211. "We .shall now give some examples in which negative 
signs occur attached to tlie coelUcieut ol y in one or both of 
the equations. 

Ex. To solve the equations: 

6a; + .^o?/ = l77 
8a;- 217/ = 33. 

Multiply the first equation by 4 and the second by 3. 

24x+140?/ = 7()3 
24x- 63?/ = 99. 

Subtracting, 2037/ = G09, 

and therefore y = ^- 

The value of x may then be found. 



Examples.— Ixxi. 

I. 2x + 7y = 52 2. 7x- 47/ = 55 3. x + 7/ = 9G 
3x-5// = 16. 15x-13// = 109. x-7/ = 2. 

4. 4x+ 9?/ = 79 5. a; + 197/ = 97 6. 29x-14?/ = 175 
7x-17?/ = 40. 7x-53?/ = 121. S7.:;-5G?,'-4y7. 

7. 171x-213?/ = 642 8. 43x+ 2?/ = 26G 9. 5x + 9?/ = lfi8 
114x-326?/ = 244. 12x-17?/ = 4. 13x-27/ = 57. 

fs.A.l - 



146 . STMUL TANEOUS EQUA TIONS 



212. We have hitherto taken examples in ■which the 
coetiicients of x are both positive. Let us now take the lolljv . 
ing equations : 

5x -7y = 6 

9y-2x = 10. 

Change all the signs of tJie second equation, so that we get 

5a; — 7i/ == 6 
2x-9y= -10. 

Multiplying by 2 and 5, 

I0x-Uy = l2 
10x-45j/=-50. 

Subtracting, 

-141, f 45?/ = 12 + 50, 
or, Sly = 62, 
or, y = 2. 

The value of x may then be found. 

Examples.— ixxii. 

I. 4x~7y = 22 2. 9.c-5?/ = 52 3. 17x + 3i/ = 57 

7y-'Sx=l. 8y-'3x = 8. 16i/-3x = 23. 

4. 7y + 3x = 7S 5. 5.c-3i/ = 4 6. 3x + 2i/ = 39 

19i/-7x = 136. 12!/-7x=10. 32/-2a;=13. 

7. 5y-2x = 21 8. 9?/-7x=13 9. 12a;+ 7y=176 

13x-4y=l-20. 15x-7t/ = 9. 32/-19a; = 3. 

213. In the preceding examples the values of a; and y have 
been jiositive. We shall now give some equations in which x 
or y or both have negative values. 

Ex. To solve the equations: 

2x-9y = U 
3x-4y = 7. 

Multiplying the equations by 3 and 2 respectiveiy, we get 
6x-27y = :yi 
6x- 8}f=14. 



OF THE FIRST DEGREE. i^'j 



Subtracting, 

-19*/ = 19, 

or, 192,'=-19, 

or, y=-l. 

Now since 9y= - 9, 

2x - 9y will be et[uivalent to 2x - ( - 9) or, 2x + 9. 

Hence, from the first equation, 

2x + 9 = ll, 
.-. x = l. 



Examples. — Ixxiii 

I. 2x + 3i/ = 8 2. 5x~2y = bi 3. 3x-5y = ol 

3x + 7y = 7. 19a; -3?/ = 180. 2x + 7i/ = 3. 

4. 7i/-3x=139 5. 4x+ 9?/= 106 6. 2x-7i/ = S 

2x + by = i)l. 8a; + 172/=198. 4!/-9x=19. 

7. 17x+122/ = 59 8. 8x + 3y = :i 9. 69i/-17x=103 

19x- 4?/ = 153. 12x + 9(/ = 3. 14.c- 13^/= -41. 

214. We shall now take the case of Fractional Equations 
involving two unknown quantities. 



Ex. To solve the equations, 

2x-^:-^=4 
5 

3y = 9— 3-. 

First, clearing the equations of fractions, we get 
10x-y + 3 = 2O 
9y = 27-x + 2, 
from which we obtain, 

10x-y = 17 
x + 9y = 29, 
and hence we may find x = 2, i/ = 3. 



148 SIMULTANEOUS EQUATIONS 



Examples.— Ixxiv. 



I. 1 + 1 = 7 2. iOx + | = 210 3. ^ + 7j/ = 251 

1 + 1 = 8. 10!/-? = 29(). | + 7x = 299. 

4. —,-^ + 5 = 10 5. 7a; + Tr = '413 6. — --^ = 10-^ 
3 2 5 3 

^--1 + 7 = 9^ 39x = 142/-l609 ^^=| + 1. 

7. x-'^-^ = 5 10. ^;^ + 8, = 31 

4u :^— = 3. =i-— +10x = 192. 

3 4 

8. | + 8 = |-12 II. ?^J^ + 3x = 2y-6 
x + ?/ w 2x-'V „^ 1/ + 3 1/-X „ ^^ 

3x-5;/ „ 2x + y x-2 10-x 1/-IO 

2 o 3 4 

x-2y/_x 7/ 2>/ + 4_4 x + y+13 

S ____ + _ _- ^ . 

13. — ---^ + 3x = 47/-2 
■^ 13 

5x + 6i/ 3x-2v_5, _ .^ 

5x-3 3.7;-19 , 37/-x 
.4. -2 -o" = ^-^3- 

2x + y _ 9x - 7 _ % t? _ f^ + ^ 
~2 8 "4 16 ■ 

4x + 5?/ 
^5 -40-" = ^-^ 



2x - 1/ „ 1 

-3-^"^' = 2- 



OF THE FIRST DEGREE. 149 

215. We have now to explain the nietliod of solving Literal 
Equations involving two xniknowu quantities. 

Ex. To solve the equations, 

^ ax-\-by = c 

px + q>j = r. 
\. 

Multiplying the first equation Ly p and the second by a, we 

get 

apx + bpy = cp 

apx + aqxj = ar. 

Subtracting, bjnj — aqij = cp — ar, 

or, {bp -aq)y = cp - ar ; 

_ cp - ar 
bp — aq' 

We might then find x by substituting this value of y in one 
of the original equations, but usually the safest cour.se is to 
begin afresh ami make the coefficients of y alike in the original 
equations, multiplying the first by q and tlie second by b, 
which gives 

aqx + bqy = cq 

bpx + bqy = br. 

Subtracting, aqx — bpx = cq- br, 

or, {aq — bp)x = cq-br; 
_ cq- br 
aq-bp' 

Examples.— ixxv. 

I. 'mx + ny = e, 2. ax + by = c 3. ax-by = m 

px + qy =f. dx - cy =/. ex + ey = n. 

4. ex =dy 5. mx-ny = r 6. x + y = a 

x + y = e. m'x + n'y = r x-y = b. 

7. ax + by = c 8. abx + cdy = 2 o. , = ^; 

' -^ •' ^ h + y 3a + x 

dx + fy = c-. ax-cy= -rj-. ax + 2by = d. 



156 ^TMtJLTAYF.OTJS F.QUATIOI^S 



\o. bcxi-2b -aj = II. {b + c)(;x + c — b) + a(7j + a) = 2a^ 

^, a(c^-b^) 2h^ , ay (b + cy 

^ be c (b-c)x a' 

(8b-2m)bm 

12. 3x + 5i/ = — , ij „ — 

0- - r/r , 

6-a; - , t-(b + c + vi) my = ■ni^x + (b + 2m) &m. 



216. We now proceed to the solution of a particular class 
of Simultaneous Equations in which the unknown symbols 
appear as the denominators of fractions, of which the following 
are examples. 

Ex. 1. To solve the equations, 

a b 
- + -=c 
X y 

m TO J 
=(L 

X y 

Multiplying the first by m and the second by a, we get 



am bm 

+ ^ =cm 
X y 




am an , 
X y ~ ' 




bm an 

— H = cm- 

y y 


-ad. 



Subtracting, 

bm + an . 

or, =cm-ad, 

y 

or, bm + an = (cm - ad) y, 

_^bm + an 
^ cm -ad' 

Then the value of x may be found by substituting this value 
of y in one of the original etjuations, or by making the terms 
containing y alike, as iu the example given ia Art. 215. 



6P THE FIRST DEGREE. t^t 



ElX. 2. To solve the equations: 
X 3y~27 

Ax'^y 72" 

Multiplying the second equation 1->y 8, we cret 
2__5 _-i_ 
a~3i/~27 





2 8_11 
X y~ 9' 


Subtracting, 


5 8_ 4 11 

•3y y~27 9" 


Changing signs, 


5 8 11 4 
Sy'^y~9 27' 


or. 


5 + 24 33-4 
dy 27 ' 


whence we find 


y=9, 



and then tlie vahie of x may be found by substituting 9 lor y 
in one of the original equations. * 



Examples. — Ixxvi. 



X y 




2. 


1 2 

- + - = a 
a; y 




3- 


a b 
- + - = c 
X y 


i + =^ = 20. 
X y 






3 4 . 

- + - = ?). 
X y 






b a 

- + - = d. 
X y 


a b 
- + - = m 
X y 




5- 


X y 




6. 


5 2 „ 
Sx by 


a b 






X y 






7 1 ^ 


X y~ 










6x 102/ ""■ 


2 
ax 


3 


:5 




8. 


m n 

1 = m + n 

nx my 


5 
ax 


2 

by" 


3. 






n 

- + 

X 


TO ., ., 
— =TO- + ?l-. 



I.<2 SIMULTANEOUS EQUATIOh'S 



217. There are two other methotls of solvin.n Simultaneoua 
El [nations of which we have hitherto made no mention, because 
they are not generally so convenient and simple as the method 
which we have explained. They are 

I. The metliod of Substitution. 

If we have to solve the equations 

a; + 3?/= 7 
2x + 4?/ = 12 

we may find the value of x in terms of y from the first equa- 
tion, thus 

a; = 7-3(/, 

and substitute this value for x in the second equation, thus 

2 (7 -37/) + 47/= 12, 

from which we find 

i/ = l. 

We may then find the value of x from one of the original 
equations. 

II. The method of Comparison. 
If we have to solve the equations 

5x + 2j/ = 16 

7x-3!/= 5 
we may find the values of x in terras of y from each equation, 
thus 

x = — - — -, from the first equation. 
x = — — -, from the second equation. 

Hence, equatini; these values of x, we get 
16^27/ _5+_3y 
5 '~ 7" ' 
an equation involving only one unknown symbol, from which 
we obtain 

!/ = 3, 

and tlun the value of x may be found fr<ini one of the ori-inal 
e(iuati'>'>s. 



OF TFJE FIRST DEGREE. 153 

218. If tliere be ihrRe, unknown symbols, their values may 
be found from tliree independent eijuations. 

For from two of the equations a third, which involves only 
tioo of the unknown symbols, may be found. 

And from the remaining equation and one of the others 
a fourth, containing only the same two uukuown symbols, may 
be found. 

So from these two equations, which involve only two un- 
known symbols, tlie value of these symliols may be found, and 
by substituting these values in one of the original equations 
the value of the third unknown symbol may be found. 

Ex. 5x-6y + 4z=15 

7.'c + 47/-3.-;=19 
2x+ 7/ + 6.v = 46. 
Multiplying the first by 7 and the .second by 5, we get 
35.<;-42i/ + 283=105 
3Jx + 20?/-15s = 95. 
Subtracting, 

-62)/ + 433 = 10 (1). 

Again, multiplying tlie first of the original equations by 2 
and the third by 5, we get 

10.c-12r/ + 83 = 30, 
lOx + by + 30z = 2Z0. 

Subtracting, - 17?/ -222= -200 (2). 

Then, from (1) and (2) we have 

62>j-43z= -10 
17(/ + 222 = 200, 
from which we can find ij = 4 and s = 6. 

Then substituting these values for tj and z in the first equa- 
tion we find the value of x to be 3. 

Examples. — Ixxvil. 

1. 5x + 7y- 22 = 13 3. bx-3y + 2z = 21 
8x + 3!/+ 2 = 17 8x- y-3z= 3 

x-4?/ + 103 = 23. 2x + 3i/ + 2z = 39. 

2. 5x + 3)/-6.i; = 4 4. 4x-5y + 2z= 6 
3x- y + 2z = S 2x + 3?/- 2 = 20 

x-2y + 2z = ± 7x-4.v + 32 = 35, 



154 PROBLEMS RESULTING TiV 

5 x+ y+ z= 6 8. 4x-3j/4- 2= 9 

5x + 47/ + 3;2 = 22 9x + 1/ - 62 = 1 6 

15a; + 10)/ + 62 = 53. x-4i/ + 3z= 2. 

6. 8x + 47/-3.v = 6 9. 12a; + 5?/ -42 = 29 

a; + 32/— z = 7 13x- 2^4-52 = 5b 

4x-52/ + 4z = 8. 17a;- ?/- z = 15. 

7. x+ y+ 2 = 30 10. y-x + z=- 5 
8a; + 4?/ + 22 = 51) z - ?/ - x = - 25 

27a; + 9;/ + 32 = 64. x + ?/ + 2 = 35. 



XVI. PROBLEMS RESULTING IN SIMUL- 
TANEOUS EQUATIONS. 

219. In the Solution of Problems in which we represent 
two of the numbers sought by unknown symbols, usually x and 
y, we must obtain two independent equations from the condi- 
tions of the question, and then we may obtain the values of 
the two unknown symbols by one of the processes described in 
Chapter XV. 

Ex. If one of two numbers be multiplied by 3 and the 
other by 4, the sum of the products is 43 ; and if the former be 
multiplied by 7 and the latter by 3, the difference between the 
results is 14. Find the numbers. 

Let X and y represent the numbers. 

Then 3a; + 4*/ =43, 

and 7x — 'iy = 14. 

From these equations we have 

21x + 28!/ = 301, 
21a;- 9?/ = 42. 
Subtracting, 37j/ = 259. 

Therefore J/ = 7. 

and tlu'U tlie value of a; may be found to be o. 
Hence the numbers are 5 and 7. 



A. 



SIMULTANEOUS EQUATIONS. 155 



Examples.— Ixxviii. 



The snm of two numbers is 28, and tlieir difference is 4, 

find the numbers. 
ill 2. The sum of two numbers is 256, and their difference is 

10, find the numbers. 

3. Tlie sum of two numbers is 13'5, and tlieir difference is 

1, find the numbers. 
■6 ^4. Find two numbers such that the sum of 7 times the 
"^greater and 5 times the less may be 332, and the product of 

their difference into 51 may be 408. 
.jjl 5. Seven years ago the age of a father was four times that 
•^of his son, and seven years hence the age of the father will be 

double that of the son. Find their ages. 
^^6. Find three numbers such that the sum of the first and 

•second shall be 70, of the first and third 80, and of the secoud 

and third 90. 

7. Three persons A, B, and G make a joint contribution 
which in the whole amounts to ^400. Of this sum B contri- 
butes twice as much as A and £20 more ; and G as much as A 
and B together. What sum did each contribute? 

8. If A gives B ten shillings, B will have three times as 
much money as A. If B gives A ten shillings, A will have 
twice as much money as B. What lias each ? 

9. Tlie sum of £760 is divided between A, B, G. The 
shares of A and B together exceed the share of G by £240, 
and the shares of B and C together exceed the share of A by 
£360. Wliut is the share of each ? 

^(^ 10. The sum of two numbers divided by 2, gives as a quo- 
"^tient 24, and the difference between them divided by 2, gives 

as a quotient 17. What are the numbers? 
^^ II. Fiml two numbers such that when the greater is 
divided by the less the quotient is 4 and the remainder 3, and 
when the sum of the two numbers is increased by 38 and the 
result divided by the greater of the two numbers, the quotient 
is 2 and the remainder 2. 
/A 12. Divide tlie number 144 into three such parts, that 
(/'when the first is diviiled by the second the quotient is 3 and 
the remainder 2, and when the third is divided by the sum 
of the other two parts, the quotient is 2 and the remainder 6. 



IS6 PROBLEMS RESULTING IN 



H 



13. A and B buy a horse for £120. A can pay for it if B 
will advance half the money he has in his pocket. B can pay 
for it if A will advance two-thirds of the money he lias in his 
pocket. How much has each ? 

.^14. "How old are you?" said a son to his father. The 
father replied, "Twelve years hence you will be as old as 1 was 
twelve years ap;o, and I shall be three times as old as you were 
twelve years ago." Find the age of each. 

1,^5. Eequired two numbers such that three times the 
greater exceeds twice the less by 10, and twice the greater 
together with three times the less is 24. 



^7' 



li6. The sum of the ages of a father and son is half what it 
"^vill be in 25 years. The difference is one-third what the sum 
will be in 20 years. Find their ages. 

, / ' 17. If I divide the smaller of two numbers by the greater, • 
/ the quotient is '21 and the remainder "OIjT. H" I divide the 

greater lunuber by the smaller, the quotient is 4 and the 

remainder '742. Find the numbers. 

18. The cost of 6 barrels of beer and 10 of porter is £51 ; 
the cost of 3 barrels of beer and 7 of porter is £32, 2s. How 
much beer can be bought for £30? 

19J The cost of 7 lbs. of tea and 5 lbs. of coffee is £1, 9s. 4il. : 
the cost of 4 lbs. of tea and 9 lbs. of coffee is £1, 7s. : what is 
the cost of 1 lb. of each ? 

20. The cost of 12 horses and 14 cows is £3S0 : the cost of 
5 horses and 3 cows is £130 : what is the cost of a horse and a 
cow respectivel}' 1 

21. The cost of 8 yards of silk and 19 yards of cloth is 
£18, 4s. 2d.: the cost of 20 yards of silk and IG yards of doth, 
each of the same quality as the former, is £25, 16s. Sti. How 
much does a yard of each cost ? 

22. Ten men and six women earn £18, 18s. in 6 days, and 
four men and eight women earn £(>, Cs. in 3 days. What are 
the earnings of a man and a woman daily ? 

1 1)23. A farmer bought 100 acres of laud for £4220, part at 
^^£37 an acre and part at £45 an acre. How many acres had 
lie of each kind? 



SIMULTAi^EOUS EQUATIONS. 157 

Note I. A number consisting of two digits may be repre- 
sented algebraically by lOx + y, where x and y represent the 
significant digits. 

For consider such a number as 76. Here the significant 
digits are 7 and 6, of which the former has in consequence of 
ils position a local value ten times as gre'it as its natural 
value, and the number represented by 76 is equivalent to ten 
times 7, increased by 6. 

So also a number of which x and y are the significant digits 
will be represented Ijy ten times- x, increased by y. 

If the digits composing a number lOx + y be inverted, the 
resulting number will be lOy + x. Thus if we invert the digits 
composing the number 76, we get 67, that is, ten times 6, in- 
creased by 7. 

If a number be represented by lOx + y, the sum of the 
digits will be represented by x + y. 

A number consisting of three digits may be represented 
algebraically by 

100x+ lOy + z. 

Ex, The sum of the digits composing a certain number is 
5, and if 9 be added to the number the digits will be inverted. 
Find the number. 

Let lOx + y represent the number. 
Then x + y will represent the sum of the digits, 
and lOy + x will represent the number with the digits inverted. 
Then our equations will be 

x + y = 5, 
10x + y + 9 = l0y + x, 
from which we may find x = 2 and ?/ = 3 ; 

.". 23 is the number required. 

^'^ 24. The sum of two digits composing a number is 8, and if 
36 be added to the number the digits will be inverted. Find 
the number. 

jvi^25. The sum of the two digits composing a number is 10, 
and if 54 be added to the number the digits will be inverted. 
What is the number ? 



1S8 PROBLEMS RESUL TING IN 

26. The sum of the digits of a munber less than 100 is 9, 
and if 9 be added to the number the digits will be inverted. 
What is the nundjer? 

27. The sum of the two digits composing a number is 6, 
and if the number be divided by the sum of the digits the 
quotient is 4. '\Vliat is the number ? 

28. The sum of the two digits composing a number is 9, 
and if the number be divided bv the sum of the digits the 
quotient is 5. What is the number ? 

29. If I divide a certain number by the sum of the two 
digits of wliich it is composed the quotient is 7. If I invert 
the order of the digits and then divide the resulting nund)er 
dinnnished by 12 by tlie difference of the digits of the original 
number the quotient is 9. What is the number ? 

A 30. If I divide a certain number by the sum of its two 
digits the quotient is 6 and the remainder 3. If I invert the 
digits and divide the resulting number by the sum of the digits 
the quotient is 4 and the remainder 9. Find the number. 

31. If I divide a certain number by the sum of its two 
digits diminished by 2 tlie quotient is 5 and the remainder 1. 
If I invert the digits and divide the resulting number by the 
sum of the digits increased by 2 the quotient is 5 and the re- 
mainder 8. Find the number. 



^i 



32. Two digits which form a number change places on the 
addition of 9, and the sum of these two numbers is 33. Find 
the numbers. 

33. A number consisting of three digits, the absolute value 
of eacli digit being the same, is" 37 times the square of any 
digit. Find the number. 

34. Of the three digits composing a number the second is 
double of the third : the sum of the first and third is 9 : the 
sum of all the digits is 17. Find the number. 

.1 (35. A number is composed of three digits. The sum of the 
digits is 21 : the sum of tlie fust and second is greater than the 
third by 3; and if 198 be added to the number the digits will 
be inverted. Find the number. 



SIMUL TANEOUS EQ UA TIONS. I J9 



Note II. A fraction of which the terms are unkno\vn may 
be represented by -. 

Elx. A certain fraction becomes ^ when 7 is added to its 

denominator, and 2 when 13 is added to its numerator. Find 
the fraction. 

Let - represent the fraction 

a; + 13_ 

are the equations ; from which we may find a; = 9 and i/=ll. 

9 
That is, the fraction is yy. 

36. A certain fraction becomes 2 when 7 is added to its 
numerator, and 1 when 1 is subtracted from its denominator. 
What is the fraction ? 



37. Find such a fraction that when 1 is added to its 
1 
3' 



numerator its value becomes -, and when 1 is added to the 



denominator the value is -. 
4 



38. What fraction is that to the numerator of which if 1 be 
1 

^2 



added the value will be ^ : but if 1 be added to the denominator. 



the value will be ;^ ? 

39. The numerator of a fraction is made equal to its 
denominator by the addition of 1, and is half of the deno- 
minator increased by 1. Find the fraction. 

40. A certain fraction becomes - when 3 is taken from the 
numerator and the denominator, and it becomes - when 5 



i6o PROBLEMS RESUL TING IN 

is added to the numerator and the denominator. "What is the 
fraction ? 

7 

41. A certain fraction hecomes ^ when the denominator is 

20 
increased hv 4, and — ^ when the numerator is diminished by 

15 : determine the fraction. 

42. What fraction i.< that to the numerator of which if 1 be 
added it becomes , and to the denominator of which if 17 be 

added it becomes - ? 

o 

Note III. In questions relating to money put out at 

simple interest we are to observe that 

T Principal x Rate x Time 
Interest = , 

where Eate means the number of pounds paid for the use of 
£100 for one year, and Time means the number of years for 
which the money is lent. 

43. A man puts out £2000 in two investments. For the first 
he gets 5 per cent., for the second 4 per cent, on the sum 
in\ested, and by the first investment he has an income of 
£10 more than on the second. Find how much he invests in 
each case. 

44. A sum of money, put out at simple interest, amounted 
in 10 months to £5250, and in 18 months to £5450. "What 
was the sum and the rate of interest ? 

45. A sum of money, put out. at simpie interest, amounted 
in 6 years to £52(10, and in 10 years to £6000. Find the sum 
and the rate of interest. 

Note IV. "When tea, spirits, wine, beer, and such com- 
modities are mixed, it must be observed that 

quantity of ingredients = quantity of mixture, 
cost of ingredients = cost of mi.\ture. 

Ex. I mix wine which cost 10 shillings a gallon with 
another sort which cost 6 shillings a gallon, to make 100 



SIMULTANEOUS EQUATIONS. i&i 

gallons, -which I may sell at 7 shillings a gallon vithout profit 
or loss. How much of each do I take ? 

Let X represent the number of gallons at 10 shillings a gallon, 
and \j 6 

Then a; + 2/=100, 

and 10x + 6t/ = 700, 

are the two equations from which we may find the values of 
X and y to be 25 and 75 respectively. 

46. A wine-merchant has two kinds of wine, the one costs 
36 pence a quart, the other 20 pence. How niucli of eacli must 
he put in a mixture of 50 quarts, so that the cost price of it 
may be 30 pence a quart ? 

47. A grocer mixes tea which cost him Is. 2fZ. per lb. with 
tea that cost him Is. %d. per lb. He lias 30 lbs. of the mi.vture, 
and by selling it at the rate of Is. 8(Z. per lb. he gained as 
much as 10 lbs. of the cheaper tea cost him. How many lbs. 
of each did he put in the mixture? 

Note V. If a man can row at the rate of x miles an hour 
in still water, and if he be rowing on a stream that runs at the 
rate of 1/ miles an hour, then 

X + 1/ will represent his rate down the stream, 
X — ?/ wp 

48. A crew which can pull at the rate of twelve miles an 
hour down the stream, finds that it takes twice as long to come 
up a river as to go down. At what rate does the stream How ? 

49. A man sculls down a stream, which runs at the rate of 
4 miles an' hour, for a certain distance in 1 hour and 40 minutes. 
In returning it takes him 4 hours and 15 minutes to arrive at 
a point 3 miles short of his starting- place. Find the distance 
he pulled down the stream, and the rate of his pulling. 

50. A dog pursues a hare. The hare gets a start of 50 of 
her own leaps. The hare makes six leaps while the dog makes 
5, and 7 of the dog's leaps are equal to 9 of the hare's. How 
many leajJS will the hare take before she is caught ? 



l62 PROBLEMS RESUL TIXG IN 

51. A grevhoimd starts in pursuit of a hare, at the distance 
of 50 of liis own leaps Irom ber. He makes 3 leaps while the 
bare makes 4, and he covers as much ground in two leaps as 
the hare does in three. How many leaps does each make 
before the hare is caught ? 

i;2. I lay out half-a-crown in apples and pears, buying the 
apples at 4 a penny and the pears at 5 a jtenny. I then sell 
half the apples and a third of the pears for thirteen pence, 
•which was the price at which I bought them. How many of 
each did I buy ? 

53. A company at a tavern found, when they came to pay 
their reckoning, that if there had been 3 more persons, each 
would have paid a shilling less, but had there been 2 less, 
each would have paid a shilling more. Find the number of 
the company, and each man's share of the reckoning. 

54. At a contested election there are two members to be 
returned and three candidates. A, B, and C. A obtains 1056 
votes, B, 9S7, C, 933. Now 85 voted for B and C, 744 for 
B only, 98 ibr C only. How many voted for A and C, for 
A and B, and for A only ? 

55. A man walks a certain distance : had his rate been 
half a mile an hour faster, he would have been H hours less 
on the road; and had it been half a mile an hour slower, he 
■would have been 2h hours more on the road- Find the distance 
and rate. 

56. A certain crew pull 9 strokes to 8 of a certain other 
crew, but 79 of the latter are equal to 90 of the former. Which 
is the faster crew ? 

Also, if the faster crew start at a distance equivalent to 
four of their own strokes behind the other, how many strokes 
will they take before they bump them ? 

57. A person, sculling in a thick fog, meets one barge and 
overtakes another which is going at the same rate as the 
former ; shew that if a be the greatest distance to which he 
can see, and b, b' the distances that he sculls between the 
times of his first seeing and passing the barges, 

2^1 I 
a h h'' 



STMUL TA NEOUS EQUA TTONS. 1 63 

58. Two trains, 92 feet long and 84 feet long respectively, 
are moving with uniform velocities on parallel rails in opposite 
directions, and are observed to pass each other in one second 
and a half ; but when they are moving in the same direction, 
their velocities being the same as before, the faster train is 
observed to pass the other in six seconds; find the rate in 
miles pei» hour at which each train moves. 

59. The fore-wheel of a carriage makes six revolutions 
more than the hind-wheel in 120 yards ; but only four revolu- 
tions more when the circumference of the fore-wheel is increased 
one-fourth, and that of the hind-wheel one-fiith. Find the 
circumference of each wheel. 

60. A person rows from Cambridge to Ely (a distance of 
20 miles) and back again in 10 hours, and fihds he can row 

2 miles against the stream in the same time that he rows 

3 miles with it. Find the rate of the stream, and the time of 
his going and returning. 

61. A number consists of 6 digits, of which the last to the 
left hand is 1. If this numl>er is altered by removing the 1 
and putting it in the unit's place, the new number is three 
times as great as the original one. Find the number. 



XVII. ON SQUARE ROOT. 

220. In Art. 97 we defined the Square Root, and explained 
the method of taking the square root of expressions consisting 
of a single term. 

The square root of a positive quantity may be, as we 
explained in Art. 97, either positive or negative. 

Thus the square root of 4a'- is 2a or - 2a, and this ambiguity 
is expressed thus, 

J4a^=±2a. 

In our examples in tnis chapter we shall in all cases regard 
the square root of a single term as a positive quantity. 



l64 OKT SQUARE ROOT. 

221. The sfjuare root of a product may be found by taking 
the square root of each factor, and multiplying the roots, so 
taken, together. 

Thus y/^' = ab, 

222. The square root of a fraction may be found by taking 
the square root of the numerator and the square root of the 
denominator, and making them the numerator and denominator 
of a new fraction, thus 



V4a^_2a 
8lP" 9b 

4 



96' 

2bx-y^ _ 5ory^ 
492'^ ^~72^' 



Examples. — Ixxix. 

Find the Square Root of each of the following expressions ; 
2. Slants. 3. 121mio?ii2,.u. 

5. 11289a*b^z^. 6. lG9a^%^c^^. 

1 25a^6<' 



I. 


4x-y'^. 


4- 


Ma'^b^^cl 




9a2 


y- 


1G62- 




256x^2 




289/" 



4a2c** ^* 121x«i/">' 

625«2 
"• 3246'-i* 

223. We may now proceed to investigate a Rule for the 
extraction of the square root of a compound algebraical 
expression. 

"We know that the square of a + 6 is a'^ + '2nb + b'^, and there- 
fore a + 6 is the square root of a- + 2ab + b'. 

If we can devise an operation by which we can derive a + b 
from a^ + 2ab + b', we shall be able to give a rule for tlie 
extraction of the square root. 

Now the first term of tlie root is the square root of the first 
term of the square, i.e. a is the square root of a^. 

Hence our rule begins : 

"Arrange tlie terms in the order of magnitude of the indices 
of one of the quantities involved, then take the square root of the 



ON SQUARE ROOT. idg 



jirst term and net down tlie result as the first term of the root: 
subtract its square from the given expression, and bring down the 
remainder :'' thus 

d^ + 2ab + b- (a 

a- 



2ab + b'^ 

Now this remainder may he represented thus &(2a + 6^: 
hence if we divide iab + b"^ by 2a + b we shall obtain rh the 
second term of the I'oot. 

Hence our rule proceeds : 

'" Double the first term of the root and set 'fowr the result as the 
first term of a divisor:'' thus our process up to this point will 
stand thus : 

a^ + 2ab + b^ [a 

a? 



2a , 2a6 + &2 

Now if we divide 2ab by 2a the --eRult is b, and hence we 
obtain the second term of the root, and if we add this to 2a 
we obtain the full divisor 2a + b. 

Hence our rule proceeds thus : 

'• Divide the first term of the remainder by this first term of the 
divisor, and add the result to the first term of the root and also to 
the first term of the divisor:" thus our process up to this point 
will stand thus : 

a^ + 2ab + b-{^a + b 
a2 



2a+b 



2ab + 62 



If now we midtiply 2a + 6 by 6 we obtain 2ab + b^, which we 
subtract from the first remainder. 

Hence our rule proceeds thus : 

^'Multiply the divisor by the second term of the root and sub- 
tract the result from the first remainder :' tiius our process will 
stand thus : 



i66 



ON' SQUARE KOOT. 



a2 + 2a6+6%a + fe 

o2 



2tH-6 



2a6 + 62 
2a6 + 62 



If there is now no remainder, the root has been found. 

If there he a remainder, consider the two terms of the root 
already found as one, and proceed as before. 

224. The following examples worked out will make the 
process more clear. 

(1) o2-2a6 + 62(^a-6 

ft2 

2a- 6 I -2a6 + 62 
■ -2a6 + 62 



Here the second term of the root, and consequently the 
second term of the divisor, will have a negative sign prefixed, 

because ->, — = -o. 
2a 



(2) 



(3) 



6p + 42 



101-6 



9p^ + 24pq+l6q-(^3p + 4q 
9p2 

24pq + 16^2 
24pq + IQq- 



25x2-60x + 36(5x-6 
25x'- 



- 60z + 36 
-60X + 36 



Next take a case in which the root contains three terma. 

a- + 2ab + b- — 2ac - 2bc + c-{^a + b — e 
a2 

2a + 6 



2ab + b--2ac-2bc + c^ 
2ab + 6"- 



2a + 26 - c 



- 2ac - 26c + c^ 

- 2ac - 26c + c* 



ON SQUARE ROOT. 167 

When we obtained the second remainder, we took the double 
of + 6, consiflereJ as a single term, and set down the result as 
the first part of the second divisor. We tlien divided the first 
term of the remainder, — 2ac, by the first term of' the new- 
divisor, 2a, and set down the result, - c, attached to the part 
of the root already found and also to the new divisor, and then 
multiplied the completed divisor by -c. 

Similarly we may proceed when the root contains 4, 6 or 
more terms. 

Examples.— Ixxx. 

Extract the Square Eoot of the following expressions : 

1. 4a- + V2ab + 9b\ 6. x^ - 6x^ + I9x' - 30x + 2b. 

2. lG¥^-24kH^ + 9l^ 7. 9x^+12x3+ 10x2 + 4x+l. 

3. a-b-+l62ab + 65Gl. 8. 4r*- 12)-3+ 13?---Gr+ 1. 

4. /-38?/3 + 361. 9. 4)i* + 4)i3-7n2_47i + 4. 

5. 9a26V - 102a6c + 289. 10. l-6x+ 13x2-12x3 + 4x* 

11. x8- 4x5 + 1 Ox* -12x3 + 9x2. 

1 2. 4y* - 12yh + 2oyh^ - 24yz^ + 16a*. 

13. a^ + 4ah + 4¥ + 9c' + 6ac + '[2bc. 

1 4. a^ + 2a'6 + 3a^b- + 4a"¥ + 3a-b* + 2ab^ + W. 

15. x8-4x5 + 6x3 + 8x-' + 4x+l. 

1 6. 4x* + 8ax3 + 4a2x2 + 1 662x2 + \<oab-x + \ 66*. 

[7. 9 - 24x + 58x2 - 116x3 + 129x* - 14.0x5 + ioOx«. 

f 8. 1 6a* - 4Qa?b + 2ba?b' - 80a62x + 646'x2 + 64a26a;. 

1 9. 9a* - 24a^p^ - •^OaH + 1 da-f + 40apH + 25^2, 

20. 4?/*x2 - 1 2 y^x^ + 1 7i/2x* -\2yx^ + 4x^. 

2 1 . 25x*2/2 _ 30x37/3 + 29x2?/* - 1 2xif + 4y^. 

22. 16x* - 24x3?/ + 25x2y2 _ 12x»/3 + 4y\ 

23. 9a2-12a6 + 24ac-166c + 452 + i6c2. 

24. x* + 9x2 + 25-6x3+10x2-30x. 

25. 25x2 _ 20x2/ + 4 )/2 + 9^2 _ 1 2i/3 + SCtea. 

26. 4x2 (a;2 _ ^) + ^^3 (j, _ 2) + y2 (43.2 4. 1). 



i68 



ON SQUARE ROOT. 



225. "When .any fractional terms are in tlie expression of 
which we have to liiid the Scjuare Root, we may pmceed as in 
the Examples just given, taking care to treat the fractional 
terms in accordance witli the rules relatiun to fractionsi 



8 16 

Thus to find the square root of '•''^ — K^ + o^• 

9 81 

.,8 16/ 4 

X" — -.c + ■ 



9 81 



{^-\ 



2x- 



8 16 

9^ + 81 

8 16 

■9^-*-8l 



Since 



8 c_8_2_8 1_4 
9^ 9 • 1~9''2~9- 



8 16 

Or we niis^ht reduce x^--^z + -^ to a single fraction, whicli 

9 oi 



would he 



81.T--72X4-16 
8l ' 



and then take the square root of each of the terms of the 
fraction, with the I'oUowin result : 

9x-4 , . , . ^, 4 ' 

— - — , which IS the same as x - -. 



Examples.— ixxxi. 



I. 4a'' + -~ - a^b-. 
lb 

9 ^ a-' 
a- 9 



4- TT + 2 + -T. 



5. x*-2x^^ + 2x--x + - 



3. «*-2 + -7. 



6. X* + 2x3-.T+ ,. 
4 



353 54 

7. 4a- - 12a& + ah"- + 9)h- - t + tt-- 
' 2 lo 



ON CUBE ROOT. '^9 



16 32 

8. x^ + 8x- + 24 + ^ + --,-. 
X* x^ 

9 ^ . 16 p ., .^ , „ , ,16 , 

lb y i 

1 4 9 4 6 12 

a;^ y'' z- xy xz yz 

71 5 n^ 25 .)». 

12. a^J^ - Gffkrf + " ■'- + 9c'ch + -Jq- ~. 

4x2 j,2 9!/2 • ^ 6?/ 12xw 

^-^ X- 3- X 2^ 

4??i2 9,i,2 16^ 247i 

14. -—+-^ + 4 + — . 

n- nf n iii 

cfi ^ h' c- d? ah 2ac ad be bd cd^ 

^ 5 ■ ¥ "^ iTi "^ 2^ '*' T ~ 6 "*" Ts" ~ y ~ To "^ T " y * 

1 6. 49x4 - 28x3 - 1 7x2 + 6x + ?. 

4 

1 7. 9x* - 3ax^ + 66x3 + abx'^ + b^x^. 

4 

1 8. 9x* - 2x3 - l^lx'' + 2x + 9. 



XVIII. ON CUBE ROOT. 

226. The Cube Root of any expression is that expression 
whose cube or third power gives the proposed expression. 

Thus a is the cube root of a^, 
3b is the cube root of 276^. 

Tlie cube root of a negative expression will be negative, for 
since 

(-a)3=— ax —ax -a=-a^, 
the cube root of - a^ is —a. 



170 ON CUBE ROOT. 



So also 

- 3a; is the cube root of - 27:c-'', 
and — 40^6 is the cube root of - %\a^}?. 

The sA^mliol I] is used i-o denote tJae operation of extracting 
the cube root. 



Examples. — Ixxxii. 

Find the Cube Roots of the following expressions : 
I, Sal 2. 27xY- 3- -125m3n3. 

4. -216ai263. 5. 34361^8. 6. - lOOOa^iVa. 

7. -1728m2in24. 3. 133100/^13/ 

227. We now proceed to investigate a Rule for finding the 
cube root of a compound algebraical expression. 

We know that the cube of a + & is a^ + 3a-5 + 3a6- + 5^, 
and thereiore a + 6 is the cube root of a'' + 3a-6 + 3a62 + ft'. 

We observe that the first term of the root is the cube root of 
the first term ol the cube. 

Hence our rule begins: 

"Arrange the terms in the orrler of magnitude of the indices of 
one of flic quantities involved, then take the cube root of the first 
term and set down the result as the first term of the root: subtract 
its cube from the given expression, and bring down the remainder;" 
thus 

a^ + 3a-b + 3ab^ + ¥{^a 
a3 



3a-b + -3ab' + P 



Now this remainder may be rejiresented thus, 
b {3a- + 3ab + b-) ; 
hence if we divi<1e 3a-b + 3ab- + P by 3a- + 3ab + b^, v:e shall 
obtain +b, the second term of the root. 

Hence onr rule proceeds : 

" Multiply the sqiiare of the first term, of the root by 3, and xet 
doivn the result as the firU term of a divisor:" thus our process 
up to this point will stand thus : 



OiV CUBE ROOT. 171 



a3 + 7,orh + 3a&- + h^ (a 



a 



3a2 I 3a26+3a62 + 63 

Now if we divide 3a-6 by 3a2 the result is 6, and so we 
obtain the second term oi the root, and if wc add to 3a2 the 
expression 3a6 + 6'- we obtain the full divisor 3ti- + 3a6 + 6^. 

Hence our rule proceeds tlius : 

" Divide the first term of the remainder by the first term of the 
divisor, and add the result to the first term of the mot. Then take 
three times the product of the first and second terms of the root, 
and also the square of the second term, and add these results to 
the first term of the divisor." Thus our jsrocess up to this point 
will stand thus : 



a^ + 3a^6 + ZalP' + 6^ (^a + 6 

«3 



3a2 + 3a6 + 62 



Za'-h + ZaV' + l^ 



If we now multiply the divisor by h, we obtain 
3a26 + 3a6- + 6^ 
which we subtract from the first remainder. 
Hence our rule proceeds thus : 

"Multiply the divisor by the second term of the root, and sub- 
tract the result from the first remainder :" thus our process will 



stand thus : 



a3 + 3a26 + 3aZ)2 + &3(,a + 6 



a 



3a2 + 3a6 + 62 



3a- 6 + 3a 6- + 1^ 
Za-b + Zalfi + h^ 



K tliere is now no remainder, the root has been found. 

If there be a remainder, consider the two terms of the root 
already found as one, and proceed as before. 

228. The following Examples may render the process more 
clear: 



172 



ON CUBE ROOT. 



Ex. 1. 

3a2-12o + 16 



rt3_i2a2 + 48ft-64(a-4 

«3 



12a'- + 48a -64 
12a2 + 48a-64 



Here observe that the second term of the divisor is formed 
thus : r- 

3 times the product of a and — 4 = 3xax -4= — 12a. 

Ex. 2. a;« - 6r^ + 15a;* -20x3 + 15x2 -6x+l(x2-2x+l 



3x-* - 6x3 + 4_j2 _ (jj.-, + X5^.4 _ 20x3 + 15x2 - 6x + 1 
-6x-'+ 12x^-8x3 



3x'*-12x3 
+ 15x2-6x+l 



3x*- 12x3 + 15x2 -6x + l 
3x* - 12x3 -f 15x2- 6x + l 



Here the formation of the Urst divisor is similar to that in 
the preceding Examples. 

The formation of tlie second divisor may be explained thus: 

Regarding x2 — 2x as one terra 
3 (:c2 - 2x)2 = 3 (x* - 4x3 + 4^2) = 3_^4 _ I2x3 + 12x2 
3x(x2-2x^xl = 3x2-6x 

12 = i 



and adding these results we obtain as the second divisor 
3x* -12x3 + 15x2 -6x + l. 



Examples. — Ixxxiii. 

Find the Cube Root of each of the following expressions: 

1. a^-2>d^h-vZa\r-h^. 2. 8a3 + 12a2 + 6a + 1. 

3. o3 + 24a26 + 192a62 + 51263. 

4. a3 ^ 3(^2^ + 3rt^2 + 53 + 3„2(. 4. 6a5g ^ 352(5 + 3^^^ + "ihc- + cl 

5. JC3 - 3x2)/ + 3j;j^2 _ y^ + 3j.2~ _ g_j.y.. ^ 3j^2~ + 3jv2!i _ 3y~2 + j^^ 

6. 27x" - 54x° + 63x* - 44.t3 + 2 1x2 - 6x + 1. 



ON CUBE ROOT. 173 



7. 1 - 3a 4- 6a- - 7a3 + 6a* - 3a'' + a". 

8. x^ - 3x-i/ + 3x1/2 _ ^3 + 833 + 6x^2 - Vixxjz + G?/-2 + 12x32 _ i2y;:2_ 

9. a" - 12a-^ + 54a* - 1 12a3 + 108a2 - 48a-+ 8. 

10. 8»^'' - 367/t= + 66»i* - %Zm^ + 33??i2 - 9m + 1. 

11. x^ + 6x-.v + 1 2x.i/2 + 8?/3 _ 3x-2 - 1 2xw2 - 1 2 v^s + 3a;~2 + ^y^i _^z_ 

12. %m? - 36?7i-?i + 547JiJi,2 — 2Tu' — Mm-r + 36m?(?- — 27'7i-r 

4- 67?ir2 — 9?i7"2 — r'. 



1 3. ?7i3 + 3??i2 - 5 H 5 =. 

"^ ?/(." 7?l'* 



229. The fourth root of an expression is found by taking 
tlie sejuare root of the s(|iiare root of the expression. 

Thus 4/16aS6* = ^l^a^h" = 2a26. 

The sixih root of an expression is found by taking the cube 
root of the scj^uare root of the expression. 

Thus 4/64ai266 = .^8a663 = 2a26. 

Examples.— Ixxxiv. 

Find the fourth roots of 

1. 16u*-96a3x + 216a-x2-2T6ax3 + 81cc*. 

2. l + 24a2 + 16a<-8a-32al 

3. 625 + 2000X + 2400x2 + 1280x3 + 256a;^. 

Find the sixth roots of 

4. a« - ^w'h + 15a*62 - ma?\? + ISa^t* - Gai^ + ^6. 

5. x6 + 6x5 + 15x'- + 20x3+ 15x2 + 6x + l. 

6. m" - 12771^ + 60771* - 160?jr + 2407?i2 - 192m + 64 



XIX. QUADRATIC EQUATIONS. 



230. A Quadratic Equation, or an equation of two dimen- 
sions, is one into which the square of an unknown symbol 
enters, without or with the tirst power ol' the symboL 

Tims a;2=16 

and x--i-6x = 27 

are Quadratic Equations. 

231. A Pure Quadratic Equation is one into which the 
square of an unknown symbol enters, the fii-st power of the 
symbol not appearing. 

Thus, x-=16 is a. pure Quadratic Equation. 

232. An Adfected Quadratic Equation is one into which 
the square of an unknown symbol enters, and also the lii-st 
power of the symbol. 

Thus, x^ + 6x = 21 is an adfected Quadratic Equation. 



Pure Quadratic Equations. 

233. When the terms of an equation involve the square 
of the unknown symbol oiibj, the value of this square is either 
given or can be found by tlie pi-ocesses described in Ciiapter 
XVII. If we then e.xtract the square root of each side of the 
equation, the value of the unknown symbol will be determined 

234. The following are examples of the solution of Pure 
Quadratic Equations. 



QUA DRA TIC EQUA TIONS. I7S 

Ex. 1. x^=\%. 

Taking the square root of each side 
x=±4. 

We prefix the sign ± to the number on the right-hand side 
of the etjuation, for the reason given in Ait. 220. 

Every pure quadratic equation will therefore have two roots, 
equal in magnitude, but with different signs 

Ex.2. 4a;2 + 6 = 22. 

Here 4x'- = 22-6, 

or 4x^=16, ' 

or x- = 4 ; 
.-. x=±2. 

That is, the values of x which satisfy the equation are 2 
and - 2. 

Ex. 3. '^^ -^^ 



Here 128 (5x2-6) = 216 (3x2-4). 

or 640x2-768 = 648x2-864, 
or x2= 12 ; 
.-. X=±V12. 

Examples.— ixxxv. 

I. x'^=Qi. 2. x2 = a252_ 3. x2- 10000 = 0. 

4, x2-3 = 46. 5. 5x2-9 = 2x2 + 24. 6. 3ax2=192a5c6. 

x2 — 12 x2-4 o 

7. - — - — = — -. — . II. mx- + n=q. 

3 4 ^ 

8. (500 +x) (500- x) = 233359. 12. x2-ax + 6 = ax(x- 1) 

8112 45 57 

9- — =3x. 13. 2^J-3 = 4^:r5- 

rl ■> ,„ r.. /^ OW 42 35 

10. 5-x--18x + 6o = (3x-3)2. 14. ^^32 = ^733- 



176 QUADRA TIC EQUA TTO.VS. 



Adfeded Quadratic Equations. 

235. Adfectecl Qnailratic Equations are solved by adding 
a certain term to both sides of tiie e(.|uatiou so as to make the 
left-hand side a perl'ect sq^uare. 

Having arranged the equation so that the first term on tlie 
left-hand side is tlie square of tlie uui<no\vn symbol, and the 
second term the one containing the lirst power of tiie unknown, 
quantity (the known symbols being on the right of the equa- 
tion), we add to both ddes of the equation the square of half th-e 
coefficient of the second term. The left-hand side of the equa- 
tion then becomes a perfect square. If we then take the square 
root of both sides of tiie equation, we shall obtain two simple 
equations, fiom which the values of the unknown symbol may 
be determined. 

236. The process in the solution of Adfected Quadratic 
Equations will be learnt by tlie examples which we shall give 
in this chapter, but before we proceed to them, it is desirable 
that the student should be satisfied as to the way in wliich an 
expression of the form 

x^ + ax 

is made a perfect square. 

Our rule, as given in the preceding Article, is this : add the 
square of half the coefficient of the second term, that is, the 

square of 5, that is, -^. We have to shew then that 

4 
is a perfect square, whatever a may be. 

This we may do by actually performing the operation of 

extracting the square root of x'^ + ax + —, and obtaining the 

result X + A with no remainder. 



QUADRATIC EQirATIOiVS. 1 77 

237. Let us examine this process by the aid of numerical 
coefficients. 

Take one or two examples from the perfect squares given 
in page 48. 

We there have 

x^+ 18x+ 81 which is the square of x+ 9, 

a;2 + 34x + 289 x+ll. 

ic- — 8x + 16 X— 4, 

a;2-36u; + 324 cc-18. 

In all these cases the thinl term is the square of half the 
coefficient of x. 

For 81= (9)^ = (\^)', 

289 = (17)^ = (=^y, 

324 = (18)2 = (''|y. 

238. Now put the question in this shape. What must we 
add to X' + ax to make it a perfect square i 

Suppose b to represent the quantity to be added. 

Then x'^ + ax + 6 is a perfect square. 

Now if we perform the operation of extracting the square 
root of x- + ax + b, our process is 

x^ + ax + hi x + - 



X 



2x + H ax + 6 



2 



a" 

ax + --r- 

4 



'-T 



faA.] M 



178 Q VADRA TIC EQUA TlONS. 

Hence in order that x^ + ax + 6 may ba a perfect square we 
must have 

t 
4 



i-?-o, 



or t=-, 



(ly 



That is, 6 is equivalent to the square of half the coefficient 
ofx. 

239. Before completing the square we must be careful 

(1) That the square of the unknown symbol has no coeffi- 
cient but unity, 

(2) That the square of the unkno^vn symbol has a positive 



These points will be more fully considered in Arts. 245 and 
246. 

240. We shall first take the case in Avhich the coefficient of 
the second term is an even number and its sign positive. 

Ex. a;--l 6x = 40. 

Here we make the left-hand side of the equation a perfect 

square by the following i)rocess. 

Take the coetficient of the second term, that is, 6. 

Take the half of this coefficient, that is, 3. 

Square the result, which gives 9. 

Add 9 to both sides of the equation, and we get 
x2-|-6x + 9 = 49. 

Now taking the square root of both sides, we get 
x + 3=±7. 



QUADRATIC EQUATIONS. 179 



Hence we liave two simple equations, 

a; + 3=+7 (1), 

and 35 + 3= -7 .'. (2). 

From these we find the values of x, thus: 
froui (1) x = 7-3, that is, x = 4, 

from (2) x= — 7 - 3, that is, a;= - 10 

Thus the roots of tlie equation are 4 and - 10. 

EXAMPLES. — IXXXVi. 

I. x- + 6x = 72. 2. a;-+12x = 64. 3. a;2 + 14x = 15. 

4. x2 + 46x = 96. 5. x-+128x = 393. 6. x- + 8x-65 = 
7. x2+18x-243 = 0. 8. x'-^ + 16x- 420 = 0. 

241. We next take the case in which tlie coethcient of the 
second term is an even number and its sign negative. 

Ex. x^-8x = 9. 

The term to be added to both sides is (8-7-2)^, that is, (4)-, 
that is, 16. 

Completing the square 

x2-8x+ 16 = 25. 

Taking the square root of both sides 
z-4=±5. 

This gives two simple equations, 

a;-4=+5 (1), 

a;-4=-5 (2), 

From (1) x=5+4, .-. x = 9; 

from (2) x=-5 4-4, .-. x=-l. 

Thus the roots of the e'j;iation are 9 and - 1. 
I 



1 80 Q UADRA TIC EQ UA TIONS. 

EXAMPLES. — IXXXVii. 

I. a;2-6a: = 7. 2. x--Ax = ^. 3. a;2-20x = 21. 

4. a;2-2x = 63. 5. a;2- 12x+ 32 = 0. 6. x2-14x + 45 = () 

7. x'' - 234x + 13688 = 0. 8. (x - 3) (x - 2) = 3 (5x + 14). 

9. x(3x-17)-x(2x + 5) + 120 = 0. 

10. (x - 5)- 4- (x - "7)- = X (x - 8) + 46. 

242. We now take the case in wliich the coefficient of the 
second term is an oiii number. 

Ex. 1. x2-7x = 8. 

The term to be added to both sides is 

Completing the square 

, ,. 49 ^ 49 

o ^ 49 81 
or, x'^ - 7x + -r = -r- 
4 4 

Taking the square root of both sides 

7 .9 

•"-2=±2- 

This gives two simple equations, 



7 9 
^-2=+2 <!>• 

7_ 9 ^^ 

^~2~~2 ^''^• 

From (1) ^"^9+2' or!^=9-) •■•x = 8; 

9 7 —2 

from (2) x= - - + -, or, x = — -, .-. x= -1. 

Thus the roots of the equation are 8 and - 1. 



QUA DRA TIC EQUA TIONS. l8l 



Ex. 2, a;2-x = 42. 

The coefficient of the second term is 1 
The term to be added to both sides is 

/. a;' - x + :: = 42 + - 
4 4 

1 169 

or, x--x + ^= — ; 

1 ^13 

2 - 2 

Hence the roots of the equation are 7 and —6. 



Examples. — Ixxxviii. 

I. a;2+7a; = 30. 2. x2-llx=12. 3. x2 + 9x = 43-. 

4. x2-13x=140. 5. x2 + x = — . 6. x2-x = 72. 

7. x2 + 37x = 3690. 8. x2 = 56 + x. 

9. x(5-x)(-2x(x-7)-10(x-6) = 0. 
10. (5x-21)(7x-33)-(17x+15)(2x-3) = 448. 

243. Our next case is that in which the coefficient of the 
second term is a fraction 0/ which the numerator is an even 
number. 

Ex. i?-jx = 2\. 

5 

The term to be added to both sides is 

4 4 4 

5 2o 2o 

„ 4 4 529 



r82 Q UADRA TIC EQUA TIONS. 



2 ^23 
5 - 5 

21 

Hence the values of x are 5 and - -=-. 





Examples. — Ixxxix. 

„ 2 35 ,4 3 „ 28x 1 ^ 

I. x2--x = -g-. 2. ^^ + 5^= -25- 3. ^''-9- + 3 = f^- 

,83_ ,43 -„16 16 

4- ^ -n^-ll = ^- 5- ^^ + 35^ = 7- 6. x==-y-'; = y. 

7. x2-|^x + ^| = 0. 8. a;2_4 ^45_ 

244. "We now take the case in which the coefficient of the 
second term is a fraction u7i-ose numerator is an odd number. 

Ex. ^-W^- 

The term to be added to both sides is 

2 7 49_13G 49 
''•'' ~3'' + 36~ 3 "^36' 
„ 7 49 1681 
°' ^-3-^-^36 = -36-' 

6-6 

17 
Hence the values of x are 8 and — ^. 



Examples.— xc. 



I. X2-2T=8. 


2. x2-j2; = 98. 
5 




3. .x2 + _a. = 39. 


4. x^ + ^x=76. 


g 

5. x2--x=16. 




6. x--^x + 6 = 0. 


7. x2-— X- 


-34 = 0. 


8. 


^-3 3 
/ 4 



QUADRATIC EQUATIONS. 183 



245. The square of the unknown symbol tnust not be pre- 
ceded by a negative sign. 

Hence, if we have to solve the equation • 

6x — X- = 9, 
we change the sign of every temi, and we get 
x2-6x= -9. 

Completing the square 

a;2-6x + 9 = 9-9, 
or x^ - 6x + 9 = 0. 
Hence a; - 3 = 0, 

or x = 3. 

Note. We are not to be surprised at finding only one 
vaJue for x. The iuterpretatiuu to be j>laced on such a result 
is, that the two roots of the equation are equal in value and' 
alike in sign. 

24(5. The square of the unknown symbol must have no 
coefficient but wiity. 

Hence, if we have to solve the equation 
5x2-3x = 2, 
we must divide all the terms by 5, and we ^.et 

, 3x 2 

X" — — = -. 
o o 

2 
From which we get x = l and x= — -. 

247. In solving Quadratic Equations involving literal co- 
efficic-nts of the unkiiown symbol, the same rules will apply as 
in tlie cases of numerical coefficients. 

Thus,' to solve the equation 

?^-?-2 = 0. 
X a 

Clearing the equation of fractions, we get 
2a2-x2-2ax = 0; 
therefore -x^-2ax= -2a\ 

or x^ + 2ax = 2a^. 



r84 Q UADRA Tld EQUA TIONS. 



Completing the sqiiare 

X- + 2ax + a^ = Sa^, 
whence* x + a=^±. ^J'i • a ; 

therefore a; = - o + ^3 . a, or x = — a - ^^3 . a. 

The following are Examples of Literal Quadratic Equations. 

EXAMPLES.— XCi. 

7m- 
I. x2 + 2ax=a2. 2. x^ — 4ax = 7a2. 3. x^ + Zmx = -^ . 

, 5n ._372^ _rt2 fe2 

4. ^--Y^-~2~- ''■ (x + a)2 (x-«)2-*- 

5. x^ + (a-l)x = a. 8. adx-acx- = bcx-bd. 

6. x'^+ {a-h)x = ab. 9. cx-{ — -y = (a + b)3:?^. 

a"x^ 2ax b- ^ 

10. -To +-2 = 0- 

^ , Sa^x 6a2 + a6-262 JZx 

11. abx--{ = „ . . 

c c^ c 

12. (4a2 - 9cd-) X- + (4a2c2 + 4abd'-) x + (ac^ + bd^^ = 0. 

248. If both sides of an equation can be divided by the 
nnknowTi symbol, di\ide by it, and observe that is in that 
case one root of the equation. 

Thus in solving the equation 

x^-2x2 = 3x, 
we may divide by x, and reduce the equation to the form 

x2-2x = 3, 
from which we get 

x = 3 or .r= - 1. 

Then the three roots of the original equation are 0, 3 and - 1. 

We shall now give some Miscellaneous Examples of Quad- 
ratic Equations. 



Q UADRA TIC EQUA TIONS. 185 



Examples.-— xcii. 

I. x2-7a; + 2 = 10. 2. x--5x + 3 = 9. 3. a;2-llx-7 = 5. 

4. x2-13x-(j = 8. 5. x'- + 7x-18 = 0. 6. 4x - -"---^ = 22. 

x-3 

7. x--9x + 20 = 0. 8. 5x-3— ~4 = — J,— ■ 

x-3 2 

9. x--Gx-14 = 2. 10. —^ -—:!?- = 2. 

^ x-^3 2x + 5 

4x X- 7 
x + 7~2xT3' 

14- 2^^""~3^+~8 = ^- ^5- '^•^ - = 26. 16. 2x- = 18x-40. 

4 + 3x 15 — X 7x — 14 „ , o 

3x-5 ^f__l 7_2x-5_3x-7 

'^' 9x 3x-25~3' ^°' 4~^+y""~27~' 

4X-10 7-3x 7 , ^,., , ,, 

21. ; =-. 22. (x-3,- + 4x = 44. 

x + o X 2 ^ -^ 

x+11 „ 9 + 4x ^ , „ ,11 

21. —^='-—^—' 24. 6x- + x = 2. 25. x--^x = ^. 

26. x2-x = 210. 27. — % + - = 3. 28. ^-11=5. 

' X 4- 1 X 3 3 

X 3 x-1 

= 15. 



1 


2 

x + 2' 


3 

"5' 


10 
3- ¥" 


14 -2x 

X- 


22 
" 9" 


12 


8 


32 



30. 


15x'- — 7x = 46. 


32. 


4x 20 - 4x 
5-x X 


34- 


X 7 
x + 60 3x-5' 



— + -- = 2-- 
' 5-x'4-x x-f2' •'"' 7-x X 10' 



JV ^- + 7-^ = -,-o- 36. ;^— ; + — — = 2 



37. x-+(a + 6)x + rt6 = 0. 38. x2-(6-a)x-a6 = 0. 

39. x^ - 2ax 4- rt' — t- = 0. 40. X- - (rt- -a^)x — a^ = 0. 

, a 2a- „ a- + 62 

41. x2 + ^x--^ = <-. 42. x---^x + l=0. 



XX. ON SIMULTANEOUS EQUATIONS 
INVOLVING QUADRATICS. 

249. For the solution of Simultaneovis Equations of a de- 
gree hi^'Iier than the first no lixed rules can be laid down. We 
shall ])oiiit out the methods of solution which may be adopted 
with advantage in particular cases. 

25('. If the simple power of one of the unknown symbols 
can be expressed in terms of the other symbol by means of one 
of the given equations, the Method of Substitution, explained 
in Art. 217, may be employed, thus: 

Ex. To solve the equations 

x + y = 50 
xy = 600. . 

From the first equation 

x = 50-y. 

Substitute this value for x in the second equation, and we 
get (50 - y) . y = 600. 

This gives 50-/- i/^ = 600. 

From which we find the values of y to be 30 and 20. 

And we may then find the corresponding values of x to be 
20 and 30. 

251. But it is better that the student should accustom 
himself to work such equations symmetrically, thus : 

To solve the equations 

x + y = 50 (1), 

x!/ = 600 (2), 

From ( 1 ) x'^ + 2xy + y- = 2500. 

From (2) 4x7/ = 2400. 



O^ SIMULTANEOUS EOUArrONS. &^c-. 187 



Subtracting, x^ — 2xy + y- = 100, 
:. x-y=± 10. 

Then from this equation and (1) we find 

X = 30 or 20 and y = 20 or 30. 

Examples. — xciii. 

I. x + y — 40 2. x+v = 13 3. a; + i/ = 29 

xj/ = 300. xy = m. xy=\ 00. 

4. .c — 1/=19 5. x-y = 45 6. x-?/ = 99 

•c?/ = 66. xy = 250. i^l/ = 1 00. 

252. To solve the equations 

x-y=\2 (1), 

a;2+!/2 = 74 (2). 

From (1) '*x2-2.r;/ + i/2 = 144 V^)- 

Subtract this t'roui (2), then 

2x1/= -70, 
.• 4x1/= -140. 

Add this to (3), then 

x^ + 2xy + y- = A, 

:. X + 1/ = ± 2. 

Then from this equation and (1) we get 

X = 7 or 5 and y= — 5 or — 7. 

Examples. — xciv. 

I. x-y = A 2. x-y=l() 3. x-«=14 

x2 + 2/2 = 4o. x2 + 2/2 = 178. x2 + i/2 = 436. 

4. x-t-i/ = 8 5. x + y = l-2 * 6. x + i/ = 49 

x2 + i/2 = 32. x2 + i/2=104. ' x2^y2=,ie81. 



i88 ON SIMUL TANEOUS EQUA TJONS 

253. To solve the equations 

a^ + 2/3 = 35 (1), 

a; + 2/ = 5 (2). 

Divide (1) by (2), then we get 

x^-xy + f^l (3), 

From (2) a;2 + 2xi/ f i/^=25 (4), 

Subtracting (3) from (4), 

3a;2/ = 18, 
.". 4^2/ = 24. 

Then I'rum this equation and (4) we get 
x''^-2x?/ + i/2= 1, 
:.x-y=±\; 
andl'rom this etiuation and (2) we find 

x = 3 or 2 and ^ = 2 or 3. 



Examples.— xcv 

I. ftr' + i/3 = 91 2. a;^ + i/ = 341 ' 3. x3 + i/3 = 1008 

x + y = l. x^y = \\. x + i/=12. 

4. ic3_y3 = 5(J 5, 3.3_^3 = 98 6. x3-i/3 = 2T9 

x-y = % x-y = 2. x-y = S. 

'254, To solve the equations 

s+r6 ^'^' 

1 1 13 ,-, 

F + ?=3«-' ^')- 

From (1), by squaring it, we get 

1 2 1 25 ,.,, 

-^ + — + -2=^ (3). 

x^ xy y' 3d 

From this subtract (2), and we liave 
^_'12 
xi/ 36 ' . 

^_24 
xn 36' 



IXVOLVING QrADA\4 7/CS. 189 



Now subtract this from (3), and avo gi't 

x^ xy y'^ 36' 

X y - o • 
and from this equation anil (1) we find 

x = 2 or 3 and i/ = 3 or 2. 

Examples. — xcvi. 



I. 


1 1 9 

x'^y~20' 


2. 


1 1_3 
X y~A' 


3- 


1 1 r. 

X 1/ 




1 1 41 




1 1 5 

x2"^/~16' 




-^ + -4 = 13. 
X- 1/- 


4. 


1 1_ 1 

X y~\-2 


5- 


1-1=2^ 
X y 2 


6. 


1-1 = 3. 

X ?/ 




I 1 7 
x^ 1/- 144' 




x^ ?/^ 4" 




X^ i/^ 



255. To solve the equations 

x2+3xj/=7 a), 

^ + 4y'-=18 (2} 

If we add the equations we get 

x- + 4xi/ + 4!/- = 25. 
Taking the square root of each side, and taking only the 
positive root of the riglit-hand side into account, 

X + 2?/ = 5 ; 
.•. x = 5- ly. 

Substituting this value for x in (2) we get 
(5-22/)2/ + 4!/2=18, 
an equation by which y may be determined. 

Note. In some examples we mu-t subtract the second 
equation from the first in order to get a perfect square. 



190 ON SIMULTANEOUS EQUATIONS 

256. To solve the equations 

x^-f = i^ (1), 

a;2 + xi/ + i/2=p (2). 

Dividing (1) by (2) we get x-i/ = 2 '3), 

squaring, x'--'±xy-\-y- = \ (4}. 

Subtract this from (2), and we have 
3^2/ = 9; 
.'. 4x?/ = i2. 

Adding this to (4), we get a;2 + 2.rj/ + ?/2= 16 ; 
.-. X + ?/ = ± 4. 

Then from this equation and (3) we lind 

a; = 3 or — 1 , and ;/ = 1 or — 3. 

257. To solve the equations 

a;2 + y2 = (;5 ^Y\ 

01/ = 28 (2,. 

Multiplying (2) by 2, we have 



! + i/2 = 65) 
2a:?/ = 56) ' 



.-. X2 + 2X!/ + J/2=121^_ 

a;2-2x2/ + i/= 9) ' 

.-. x + i/=±ll (A), 

x-xj=± 3 (B). 

The equations A and B furnTsh four pairs of siiii])lr 
equations, 

x + i/=ll, .r + ?/=ll, a- + !/=-ll, a: + )/=-ll, 

x-i/ = 3, x-2/=-3, x-2/ = 3, x-?/=-3. 

from whicli we find the values of x to be 7, 4, -7 aixl -4. 
and the corresponding values of ?/ to be 4, 7. —4 and - 7. 

258. The aititiee, l\v wliich the solution of the equation.^ 
eiven in this article is eti'ected, is a])plicable to cases in wliich 
the equations are homogeneonx mid <>/ the savie orilcr. 



INVOLVING QUADT?ATICS. 19^ 

To solve the equations 

x2 + xy = 15, 

Suppose y = mx. 

Then x'^ + mx^=l^. from the first eonation. 
and mx'^ -m'^x^ = z, from the secona equation. 

Dividing one of these equations by the other, 
x^ + mx^ _15 
mx^ - m^x~ 2 ' 

x^n+m) 15 

or ^^ — = — 

x^ {m - m^) 2 ' 

1 + m .15 

o^ 2= o- 

m — m^ 2 

From tliis equation we can determine the values of m. 

2 
One of these values is ^, and putting this for m in the 
o 

2 

equation x- + 7nx-= 15, we get x'^ + -x''=l5. 

From which we find a;= ±3, 
and then we can find y from one of the original equations. 

259. The examples which we shall now give are intended 
as an exercise on the methods of solution explained in the 
four preceding articles. 

Examples.— xcvii. 

I. a;^ — y^ = 37 2. x- + 6x?/ = 144 3. x'^ + xy = 2l0 

x- + xy + y- = 37. 6xy + 36?/ = 432. y- + xy = 23 1. 

4. a;2 + 7/2 = 68 5. x^ + y^=l52 6. 4x- + 9xy=l90. 

xy=l(J. x^-xy + y-=l9. 4x-5y = l0. 

7. x^ + xy + y- = 39 8. x^ + xy = 6Q 9. 3x- + 4x!/ = 20. 

3y'--5xy = 2o. xy — y' = b. 5xy + 2y- = l2. 

\o. x^-xy + y- = 7 11. x'^ — xy = 35 12. 3x^ + 4xy + 5ij- = 7l. 

:ix^ + l3xy + 8y- = liy2. xy + y^ = 18. ox + 7y = 29. 

i2,.'X^ + y-- = 212S 14. u;2 + 9xy = 340 15. x^ + y-==225 

x^-xy + y-=l24:. 7xy -y'^=\7l. xy=]()8. 



XXI. ON PROBLEMS RESULTING IN 
QUADRATIC EQUATIONS. 

260. The method of stating problems resiiltiug in Quad- 
ratic Equations does not require any general explanation. 

Some of the Examples which we shall give involve ane 
unknown symbol, others involve tivo. 

Ex. 1. What number is that whose square exceeds the 
number by 42 ? 

Let X represent the number. 

Then x^ = x + 42, ^ 

or, a;'''-x = 42; 

thereiore x- — x + - = —j- ; 

4 4 

whence x - ^ = ± -^. 

And we find the values of .'• to be 7 or — 6. 

Ex. 2. The sum of two numbers is 14 and the sum of 
their squares is 100. Find the numbers. 

Let X and y represent the numbers. 
Then x-f-!/ = l'i, 

and .x2 4-r = i00. 

Proceeding as in Art. 252, we find 

x = 8 ur 6, y = 6 or 8. 
Hence the numbers are 8 and 6. 



ON PROBLEMS RESULTING, dj'c. 193 



Examples. — xcviii. 

*<j \ I. What number is that whose half multiplied by its third 
part gives 864? 

2. What is the number of which the seventh and eiglith 
parts being multiplied together and the product divided bj' 

2 
3 the quotient is 298^ ? 

^ 3. I take a certain number from 94. I then add the 
number to 94. 

I multiply the two results together, and the result is 8512. 
What is the number ? 

{.4. What are the numbers whose product is 750 and the 
quotient of one by the other 3- ? 

5. The sum of the squares of two numbers is 13001, and 
the difference of the same squares is 1449. Find the numbers. 

* 6. The product of two numbers, one of which is as much 
•^ above 21 as the other is below 21, is 377. Find the numbers. 

A \ 7. The half, the third, the fourth and the fifth parts of a 
^certain number being multiplied together the product is 6750. 
Find the number. 

8. By what number must 11500 be divided, so that 
the quotient may be the same as the divisor, and the re- 
mainder 51 .' 

g. Find a number to which 20 being added, and from 
which 10 being subtracted, the square of the first result added 
to twice the square of the second result gives 17475. 

10. The sum of two numbers is 2G, and the siim of their 
squares is 436. Find the numbers. 

11. Tlie difference between two numbers is 17, and the 
sum of their squares is 325. What are the numbers ? 

1 2. What two numbers are they whose product is 255 and 
the sum of whose squares is 514 ] 

a-/ '3- Divide 16 into two parts such that their product 
' added to the sum of their squares may be 208. 

[S.A.] N 



194 ON PROBLEMS RESULTING 



\ 



14. What number added to its square root gives 'as a 
result 1:332 ] 

3 

15. What number exceeds its square root by 48^? 

16. What number exceeds its square root by 2550 ? 
^ 17. The product of two numbers is 24, and their sum 

niultiiilied by their difference is 20. Find the number.*. 

18. What two numbers are those whose sum multiplied 
(; by the greater is 204, and whose difference multiplied by the 
less is 35 ? 
f\ 19. What two numbers are those whose I'.ifference is 5 
'and their sum multiplied by the greater 228 ? 
; 20. Find three consecutive numbers whose product is 
V equal to 3 times the middle number. 

^ 21. The difference between the .squares of two consecutive 
niimbers is 15. Find the numbers. 

3 2. The sum of the squares of two consecutive numbers is 
481. Find the numbers. 

23. The sum of the squares of three consecutive numbers 
is 365. Find the numbers. ' 

Note. If 1 buy x apples for y pence, 

- will represent the cost of an apple in pence. 
If I buy X sheep for z pounds, 

- will represent the cost of a sheep in pounds. 

Ex. A boy bought a number of oranges lor 16(f. Had he 
bought 4 more for the same money, he would have paid 
one-third of a penny less for each orange. How many di<l 
he buy ? 

Let as represent the number of oranges. 

Then — will represent the cost of an orange in pence. 

„ 16 16 1 

Hence — = - — , + ^, 

X a; + 4 3 

or 16(3x + 12) = 48x + x2 + 4x, 

or x2 + 4.i- = 192, 

from which we find the values of x to be 12 or — 16. 

Therefore he bouglit 12 oranges. 



IN Q UA DRA TIC EQCA TIOXS. 19$ 

24. T buy a number of handkerchiefs for £\l. Had I 
bought 3 more for tlie ^^anle money, they would have cost one 
shilling each less. How many did I buy '{ 

25. A dealer bought a number of calves for £80. Had he 
bought 4 more for the same money, each calf would liave cost 
£\ less. How many did he buy ? 

26. A man lx)ught some pieces of cloth for £33. 15s., 
which he sold again for £1. 8.?. the piece, and gaiueil as much 
as one piece cost him. What did he give for each piece ? 

27. A merchant bought some pieces of silk for £180. 
Had he bought 3 pieces more, he would have paid £3 less for 
each piece. How many did he buy ? 



28. For a journey of 108 miles 6 hours less would have 
sufficed had one gone 3 miles an hour faster. How many 
miles an hour did one go ? 

29. A grazier bought as many sheep* as cost him £60. 
Out of these he kept 15, and selling the remainder for £54, 
gained 2 shillings a head by them. How many sheep did 
be buy ? 

30. A cistern can be filled by two pipes running together 
in 2 hours, 55 minutes. The larger pipe by itself will fill it 
sooner than the smaller by 2 hours. What time will each 
pipe take separately to fill it ? 

31. The length of a rectangular field exceeds its breadtli 
by one yard, and the area contains ten thousand and one 
hundred square yards. Find the length of the sides. 

32. A certain number consists of two digits. The left- 
hand digit is double of the right-hand digit, and if the digits 
be inverted the product of the number thus formed and the 
original number is 2268. Find the number. 

33. A ladder, whose foot rests in a given position, just 
reaches a window on one side of a street, and when turned 
about its foot, just reaches a window on the other side. If tlie 
two positions of the ladder be at right angles to each other, 
and the heights of the windows be 36 and 27 feet respectively, 
find the width of the street and the length of the ladder. 



tg6 ON PROBLEMS RESULTING, dr-r. 



34. ('lot]), bein<,r wetted, shrinks up - in its length and 

o 

~- in its width. If the surface of a piece of cloth is di- 

3 

minished by 5- square jards, and the length of the 4 sides 

by 4- yards, what was the length and width of the cloth % 

35. A certain number, less than 50, consists of two digits 
whose difference is 4. If the digits be inverted, the difference 
between the squares of the number thus formed and of the 
original number is 3960. Find the number. 

36. A plantation in rows consists of 10000 trees. If there 
had been 20 less rovvs, there would have heen 25 more trees in 
a row. How many rows are there ? 

37. A colonel wished to form a solid square of his men. 
The first time he had 39 men over: the second time he in- 
creased the side of the square by one man, and then he found 
that he wanted 50 men to complete it. How many men were 
there in the regiment ? 



XXII. INDETERMINATE EQUATIONS. 

261. WHEisr tlie number of unknown symbols exceeds that 
of the independent equations, the number of simultaneous 
values of the symbols will be indefinite. We propose to ex- 
plain in this Chapter how a certain number of these values 
may be found in the case of Simultaneous Equations involving 
two unknown quantities. 

Ex. To find the integral values of x and y which will 
satisfy the equation 

3x + 7y=lO. 
Here 3a;=10-7|/; 

.-. x=3-2j/ + ^^. 
Now if X and y are integers, -— must also be an integer. 



INDE TERMINA TE EQUA TIOXS. 197 

1 —1/ 
Let — ;— = in, then 1 — ^ = 3?7i ; 

.". i/ = 1 — 3m, 
and a; = 3 — 2i/ + m = 3 — 2 + 6m + ?^^ = 1 + 77w ; 

or the general solution of the equation in whole numbers is 
x = l + 1 m and y = \ — 3?7i, 

where rii may be 0, 1, 2 or any integer, positive or 

negative. 

If m = 0, x= 1, 1/= 1 ; 

if m=l, x= 8, 3/= -2; 

if m = 2, x= 15, 1/= -5; 

and so on , from which it appears that the only positive inte- 
gral values of x and y which satisfy the equation are 1 and 1 . 

262. It is next to be observed that it is desirable to divide 
both sides of the equation by the smaller of the two coefficients 
of the unknown symbols. 

Ex. To find integral solutions of the equation 

lx->rby = Z\. 
Here by = ^\-lx: 

1 - 2x 

1 — 2e 
Let — --^ = m, an integer. 

Then 1 -2x = 5m, whence 2x=l -5fli; 
1 — m 



2 -2"^- 



T . 1 — m- . , 

Let =n, an integer. 



Then 1 -m = 2n, whence m = l -2n. 

Hence x = n-27?i = ?i — 2 + 4n = 5n — 2 ; 

y = 6-x + )n = 6-5?i-l-2 + l-2«, = 9-7». 

Now if n = (K x= -2, 2/=: 9; 

if n = l.x= 3,2/= 2; 

if n=2, X— 8,ys:~-5. 
and 80 on. 



198 INDE TERMLVA TE EQUA TIO.VS. 



263. In how many ways can a person pay a bill of £13 
with crowns and guineas? 

Let X and y denote the number of crowns and guineas. 

Then 5a; + 21?/ --=260; 

.-. 5a; = 260-211/; 

x = 52-4v-|. 
^ 5 

Let ^ = m, an integer. 

Then y = 5m, 

and x = 52-4y — m = 52 -21m. 

If 771 = 0, 2 = 52, y= 0; 

m=l, x = 31, y= 5; 
m = 2, a;=10, y = 10; 
and higher values of m will give negative values of x. 

Thus the number of ways is three. 

264. To find a number which when divided by 7 and 5 
will give remainders 2 and 3 respectively. 

Let X be the number. 

x-'2 
Then — „— =an integer, suppose m; 
I 

and =an integer, suppose n. 

Then x — 7-ni + 2 and x = 5?i + 3; 

.-. 7?rt + 2 = 5?i + 3; 

2m-l 



:. 5n = 7m - 1, whence n = m + - 



5 



Let — ' — =p, an integer. 

Then 2m = 5p+l, whence m=2p -»-— g— v 

Let ~n— — 9.} *^ integer. 

Then j9 = 2g-l, 

m = 2|) + (7 = 4(7-2 + 7 = 5g-2, 
x=7m + 2 = 357- 12. 



INDE TERM IS' A TE EQ L 'A T/ONS. I99 



Henct if 


» q = 0,x=-l'2; 


if 


Q = \,x= 23; 


if 


q=2,x= 58; 


and so on. 





Examples. — xcix. 

Find positive integral solutions of 

I. 5x + 72/ = 29. 2. 7x+192/ = 92. 

3. 13x+19!/=ll70. 4. 3x + 5i/ = 26. 

5. \Ax-by = l. 6. llx+15?/ = 1031. 

7. llx + 7i/ = 308. 8. 4x-19?/ = 23. 

9. 20x-9!/ = 683. 10. 3x + 77/ = 383. 

II. 27x + 4i/ = 54. 12. 7x + 9^ = 653. 

13. Find two fractions with denoiuinators 7 and 9 and 
their sum -^~. 

DO 

14. Find two proper fractions with denominators 11 and 

82 
13 and their difference -77:. 
14.3 

15. In how many ways can a debt of £\. 9s. be paid in 
florins and half-crowns ? 

16. In how many ways can £20 be paid in half-guineas 
an(f half-crowns ? 

17. What number divided by 5 gives a remainder 2 and 
by 9 a remainder 3 ? 

18. In how many different ways may £11. 15a-. be paid in. 
guineais and crowns ? 

19. In how many different ways may £4. lis. Qd. be paid 
with half-guineas and lialf-crowns .' 

20. Shew that 323x- 527?/ =1000 cannot be satisfied by 
integral values of x and y. 



INDETERMINATE EQUATIONS. 



21. A farmer buys oxen, sheep, and hens. The whole 
number bought was 100, and the whole price .£100. If the 
oxen cost .£5, the sheep ;£1, and the hens Is. each, how many 
of each had he? Of how many solutions does this Problem 
admit ? 

22. A owes B 4s. lOd.; if A has only sixpences in his 
pocket and B only fourpenny pieces, how can they best settle 
the matter ? 

23. A person has £12. 4s. in half-crowns, florins, and shil- 
lings ; the number of half-crowns and florins together is four 
times the number of shillings, and the number of coins is the 
greatest possible. Find the number of coins of each kind. 

24. In how many ways can the sum of £h be paid in 
exactly 50 coins, consisting of half-crowns, florins, and four- 
penny pieces \ 

25. A owes B a shilling. A has onlj' sovereigns, and B has 
only dollars worth 4s. 3d. each. How can A most easily pay Bl 

26. Divide 25 into two parts such that one of them is 
/ divisible by 2 and the other by 3. 

27. In how many ways can I pay a debt of £-1. 9s. with 
crowns and florins ? 



:b 



28. Divide 100 into two parts such that one is a multiple 
of 7 and tlie other of 11. 



29. The sum of two numbers is 100. The first divided by 
3 5 gives 2 as a remainder, and if we divide the second by 7 the 
remainder is 4. Find the numbers * 



^ 30. Find a number less than 400 which is a multii^li^ <•» 7^ 
'f and which when divided by 'z, o, *, 5. 6. gives as a iciiiiuuder 
in each case 1. 



XXIII. THE THEORY OF INDICES. 

265. The number placed over a symbol to express tlie 
power of the symbol is called the Index. 

Up to this point our indices have in all cases been Positive 
Whole Numbers. 

We have now to treat of Fractional and Negative indices ; 
and to put this part of the subject in a clearer light, we shall 
commence from the elementary principles laid down in Arts. 
45, 46. 

266. First, we must carefully observe the following results : 

(a3)2=a6. 

For a^ X a^ = a . a . a . a . a = a^, 

and (a^y = a^.a^ = a.a.a.a.a .a=a^. 

These are examples of the Two Rules which govern all 
combinations of Indices. The general proof of these Rules we 
shall now proceed to give. 

267. Def. "When m is a positive integer, 

a" means a, a. a with a written m times as a factor. 

268. There are two rules for the combination o^ndices. 
Rule I. a'"xa" = a''^. 

Rule II. {'(*")••=«-. 

269. To prove RvLE 1. 

a"^ = a.a .a to m factors, 

a'-'a^a.a to /i factors. 



402 THE THEORY OF INDICES. 



Therefore 

^" X a" = (rt . o . a to m factors) x (a . a . a to ?i. factors) 

= a .a. a io -^m + n) factors. 

= a'"+", by the Definition. 

To prove Rule 11. 
(a'")" = a'' .oT' .a" tc n lactors, 

= (a.a. a m m taei-ors) (a . a . a ... to m factors) . . . 

repeated n times, 

= a.a .a to mn factors, 

= 0"", by the Definition. 

270. We have deduced immediatehj from the Definition 
that when m and n are positive integers a" x a'' = a'*+ . When 
m and n are not positive integers, tlie Definition has no mean- 
ing. We therefore extend the Definition by saying that a" and 
a", whatever m and n may lie, shall be such Uiat a" x a' = a"*+", 
and we shall now proceed to shew what meanings we assign to 
a" in consequence of this definition, in the following cases. 

p 

271. Case I. To find the meaiiing of a', p and q being 
positive integers. 

? p p,p 
a''xa'> = a'' ', 

P P T fj.? ? ?+?+? 

«» X «' X o» = a» 1 xa'' = a'' » »; 
and by continuii^g this process, 

xa'x to (/ lactors = a» « « 

But by the nature of the symbol 4/ 

i^a^ >^ ^a'' X to q factor8 = a'; 

p p 

:. a^xa'' X to q factors = ^/a' x ^o' x . . . to ^ factora ; 

p 



THE THEORY OF INDICES. 203 



272. Case II. To find the meaning of a~\ s beirig a po.si.- 
tive number, ivhole or fractional. 

We must first find the meaning of a". 

We have 



Now 



a 


"xa" 






.•. </," 


— i. 


a' 


X a~' 


= 1; 




:. a~' 


_ 1 

~ a'' 



273. Thus the interpretation of a*" has been deduced Irom 
Rule I. It remains to be proved that this interpretation 
agrees with Rule II. This we shall do by shewing that Rule 
II. follows from Rule I., whatever m and n ma\' be. 

274. To shew that {a"')" = a'"" for all values of m and ?i. 

(1) Let n be a positive integer : then, whatever m may be, 
(ft"*)" = a" . rt"* . o"" to n factors 

™m-f-"i-i-"»+ ... to n t«rm3 



(2) Let n be a positive fraction, and equal to ~,p andV 
being positive integers ; then, whatever be the value of m, 

^ - ^ + ^+...109 terms 

(a"*)' X («")« X to 5' factors = (a")' ' 

= «"•", by (1). 
But a' xa' x to 5' factors = a ' ' 



that is, Ca''y = a'^. 



204 THE THEOR V OF INDICES. 

(3) Let n— —s., s being a positive number, whole or frac- 
tional : then, whatever m may be, 

(«")- = - — _ bv Art. 272, 
(a*")" ^ 

= -;j, by (1) and (2) of this Article ; 
that is, (0°*)"= ^— - 



275. We shall now orive some examples of the mode in 
which the Theorems established in the preceding articles are 
applied to particular cases. We shall commence with exam- 
ples of the combination of the indices of two .single terms. 

276. Since x'" x x" = x'''+", 

(1) x" X af— = yf^-' = X*. 

(2) X' X X = x'+i. 

(4) ft^—'.fc" ''xa"-'".6''-".c 
= a'"-"+"-"'.6"-''+^-".C 

= 1.1. c 

= c. 



Since (x"')" = x"", 

(1) (.c6)3 = x6^3 = a.ij. 

(2) (x^)^ = x*'^=.o'. 

(3) {a^^ = a"'^=^(iK 



278. Since x' = 4/x-% 

(1) x^= Vr*. 

(2) x^=^i2; 



THE THEORY OF INDICES. 205 

Note. When Examples are given of actual numbers raised 
to fractional powers, they may often be put in a form more fit 
for easy solution, thus : 

(1) 144-^ = (144)3 = (V141)'=12-'=1728. 

(2) 125^ = (125^)''i = ( 4/125)''^ = 5- = 25. 

279. Since (ic"*)" = a;™", 

(1) j(a;"')"j'' = (x""')'' = x""'P. 

(2) {(«-"*)-"}'■ = («'""/ = «"•"'. 

(3) I (x-")" i*" = (x-'"") " = X-"""'. 

280. Since x-" = — , 

X" 

we may replace an expression raised to a negative power by 
the reciprocal (Art. 199) of the expression raised to tlic. same 
positive power : thus 

(1) a-i = -. (2) a-^= \. (3) a~^= -\. 

Examples. — c. 

(1) Express with fractional indices : 

1 . ^x5 4- 4/x2 + ( Jxy. 3. 4/^^ A ( ^'af + a J^. 

(2) Express with negative indices so as to remove all p<n\ er.s 
trom the denominators : 

1 a 6^ 3 o(? 5x^ X 

X x^ x'^ X* '^ 42/-3'' Tt/s-* yz 

x^ 3x 4 ocy I z 

y^ y^ y*' ^^^ bx^y'^ x?y^' 

(3) Express with negative indices s<- as to remove all powers 
from the numerators ; 



206 THE THEORY OF INDICES. 



1 x X? x* 4o-6- 3f( l]x 

t t f M^ ih^^ ^/(T'ofi 

(4) Express with root-symbols and positive indices : 

-i o -» -I 

2 12 X^SX-X* 

y-^ y-i 3y-i • 
-2 -i -f 

I o X X X 

2. x"3 + (/~S + 2-3. 4- "X "^ ir^ "*" ~^' 

281. Since x"-^x" = — = x'" .x~" = x"-", 

X" 

(1) X«-=-X^ = X«-3 = X*. 

(2) x3H-xS = x3-8 = X-5 = -1 

(3) x"-^x"-" = x"-''"-"'^x"'-'^" = x". 

(4) a'-ra'^ = a'-<^' = a'-»^=a--=— . 

(5) x*-^x^ = x5 ^ = x* 

(6) X^^X^ = X^"6=X^"^ = X~^ = X~^=^- 

x^ 

282. Ex. Multiply a^ -a-' + a'-\ by a' + 1, 

a^ — a^ + a'-l 
o'+l 



a*' - a*' + a*'-a' 
ay-1 



EXAMPLES.— Ci. 
Multiply 

1. x'' + x'y' + T/*' by .r*' - x'j/' + y*. 

2. a'" + 3a^y' + 9a-t/*" + 27y'" by a" - Sy". 

3. x**- 2ax*' + 4a« by x*^ + 2ax='' + 4a\ 



THE THEORY OF INDICES 207 



a"* + 5" + c' by a"' - 6" + c". 
a" + 6" - 2c' by Sa"* - 6 + c''. 

x*" — x"i/" + 2/-" by x-^" + ar''^/" + 1/**. 
ap*+p _ 5?° 4. cp by a''"-' + 6*-'' + c^"*. 
Form the square of ar''' + x' + 1. 
Form the square of x^ — x^ + 1. 



283. Ex. Divide, x*" - 1 by x" - 1. 

x" - 1) x*" - 1 {x'" + x'*" + X' f 1 



x^p . 


-1 






a?p. 


-x^ 








7?" 


-1 






T?"- 


-xJf 








xf- 


-1 






af- 


-1 



Examples.— cii. 



Divide 
I . X*" — y*"* by X"' - y". 3, x*' - y'' by x' - y. 



10 



x"* + 1/'" by x" + y\ 4. o"'' + 6'°' by a^*" + 6' 

x'" - 243 by x" - 3. 

a*" + 4a-"'x^" + 16x*" by a*" + 2a"'x" + 4x^" 

Qx" + 3x*'' + Ux'" + 2 by 1 + Sx' + x^''. 

14&*"'c'" - ISi'^c'"* - 56''" + 4b-"'c'"' by 6»~ + 6"c^'" - 26^'"c" 

Find the square root of 

a*" + Ga'"" + IQa*"" + 20a^ + 15a-'» + Ga" + 1. 

Find the square root of 



2o8 THE THEOR V OF INDrCES 

I 

Fractional Indices. 

284. Ex. Multiply J - ah^ + b^ by J + 6*. 
J-ah^ + b^ 
a^ + b^ 

a - a%* + 0*6* 

+ ah^-ah^ + b 
a +b 

Examples. — ciii. ' 

Multiply 

1. x^-2x^+lhj x^-l. 

2. 2/* + 2/^ + ?/^+ 1 by J/*- 1. 
3* a* - x^ by a^ -t a^x^ + x-^. 

4. a^ + b^ + c^ - a*b^ - a^c^ - b^c^ by a* +b^ + c^. 

5. 5x^ + 2x^y^ + 3x^1/2 + 7y^ by 2xi - ??/*. 

4 31 sa 12 4, a 1 

6. to"' + TO-'7i' + 7*i,-^?i-^ + 7?i"?i'^ + "" by TO" - n", 

7. m^ - 2dhn^ + 4d- by m^ + 2dhn-> + -ld~. 

8. 8 J + 4ah^ + 5ah^ + 96^ by 2«^ - 36*. 
Foi-m the square of each of tlie following expressions : 

9. x^ + a^. 10. x^-a^. II. x'^ + y^'. 

12. a + ti 13. x2-2x* + 3. 14. 2x' + 3x'+4. 

11;. x^-y^ + zK 16. x* + 2i/J-a* 



THE THEoky OF INDICES. 



2c 9 



285. Ex. Bividt a-bby ija- i/b. 

1 1 

Putting a^ for ^'a, and b^ for i/b, we jjroceed thus ; 

J -b^)a-b{J + ah^ + aihi + },i 

3 1 

a-a^b'^ 



ah^-b 
ah^-a^b^ 

ah^-b 



a^b^-ah^ 

ah^ - h 
ah^-b 



EXAMPLES.— Civ. 



Divide 

1. x-yhy x~--y^ 

2. a — bhy o^ + 6* 

3. x--y \)y x-^ -y' 

4. a + b by a^ + b^ 

1 I 

5. x + yhyx^+y'' 

1 1 

6. m — n by ?)i^ — 7i''. 



.s _ a,/i. 



7.x- Sly by x^ - 3y^. 

8. 81a-166by 3«5^-2Ai. 

9. a-x by a;-^ +a~. 

1 

10. 1*1 — 243 by m" — 3. 

1 1 

11. a;+17x-- + 70 by a:2 + 7. 

12. x^ + x^ - 12 bv X* — 3. 



13. 63 _ 3t 5 + 36 _ 5I by b^ - 1. 

14. x + y+z- 3.'?:3y3;.'3 ^y x* + 1/* + 2^. 

g 1 1 

15. X - 5x3 - 46x3 - 40 by x^ + 4. 

1 .!_ 1 1 i 1 

16. m + m^n^ + n by m2 -m*ri* + ?i2, 

17. ^ - 4^)* + 6p2 •_ 4pi + 1 by ^- - 2pi + 1. 

18. 2x + x^y^ ~3y- 4i/^3^ - xh^ - 2 by 2x^ + 3^/2 + z^. 

4 31 2g la 4. 

19. x + 1/ by x-"^ -x^y^ +x^y" -x-'^y^ +y". 



Sio THE THEORY*OF INDICES. 



Negative Indices. 

28(5. Ex. Multiply x~^ + x~-y~'^ + x~'^y~^ + y~^ by x"' — y~^. 
x^^ + x~~y~^ + a:~^T/~^ + y~^ 



- x~^i/~^ — x~^y~^ — x~'i/~^ - y" 
x"*-i/~* 



Examples. — cv. 

Multiply 

I. «-i + 6-1 by (i-i - 6-1. 2. x-3 + 6-2 by xr^ - 6-2. 

3. x^ + x + x-i + x-^ by x-x-i. 4. X-- I4-X-2 by x2+ l+x~2. 
5 . a-2 + 6-2 by a-2 - 6-2. 6. a"! - 6-1 + f-i by a-i + ft-i + c-i. 
7.1 + «6-i + a-6-2 by 1 - a6-i + a'^b-^. 

8. a26-2 + 2 + a-262 by a26-2 - 2 - a'-b-. 

9. 4x-3 + 3x-2 + 2x-i + 1 by x-2 - x-i + 1. 

r o. ^x-2 + 3x-i - 1 by 2x-2 - x~i - J. 
2 3 -^ 2 

287. Ex. Divide x^ + l+x'- by x-l+ x~\ 

X-l+X-lJ x2+l +X-2 (^X+1+X~l 
X2 - X + 1 



X + X-2 

X - 1 + x-^ 

l-x-i + x-3 

l-X-l + X-2 

Note. The order of the powers of a is 

a', a^, (1(1, a", a~^, a~'-, a'^'.. 

u serii^s which may be written thus 

3 2 1 1 1 1 



a a-" a** 



THE THEOR Y OF INDICES. 



EXAMPLES.— CVi. 

Divide 
I. a;2 - X"- by *; t i ' j. > 6~^bya — 6~^ 

3. 771^ + ?i~^ by 7?i + 7^~*. 4. c^ - tZ~^ by c - d~^. 

5. x^~^ + 2 + x~-y'^ by x?/~i + x~^y. 

6. a-* + a-'-^t-s + h-* by a-^ _ ^-i^-i + 5-2. 

7. x^y~^ - x~^y^ — Sxy~^ + '3x~^y by xy~^ - x~hj. 

^ 3x-5 . . 77x-3 43X-2 33x-i „, 
g. — -4x-4--g J-+27 

a;-2 
by 7:;^ — x~'^ + 3. 

^ 2 

g. a^6~^ + «~^6^ by «6~^ + rt"'6. 
10. a~^ + 6"^ + c~^ - 3a~^6~^c'i by a~^ + ft-^ ^-c~^ 

288. To shew that (rt5)'' = a". J", 
(at)" = a6.a6. a?)... to 71 factors 

= (a . a . a . . . to n iactors) x (6 . 6 . 6 . . . to ti factors} 
= a" . 6". 

We shall now give a series of Examples to introduce the 
various forms of combination of indices explained in this 
Chapter. 

Examples. — cvii. 

1 . Divi de x^ - 4x!/ + Ax^y + Ay^ by x* + 2x~y^ + 2y. 

_i_ _i_ 

2. Simplify )(x»"*)3.(x6)-'j3-». 3. Simplify (.r^o* . xi«^)^-l 

j _i i_U 

,.- ) 1 1 x + a x-a 

4. Simpliiv < -TT — 5 — ^ 2 r: — 3- 

^ ^ - jx''-a^ x^ + a'' x- + a- 

\ a 



THE THEORY OF INDICES. 



5. Multiply |x-2 + 4x-i - 1 by -isr'^ - 2xr^ - \. 

01 it 

af^' i""* x*~^ 

6. Simplify ' — ^ . 7. Divide x^ - 2/"" by x" + ?/". 

8. Multijjly (a^ + 6^)-' by a^ - 1^. 

9. Divide a — 6 by 4^rt - 4/^. 10. Prove that (a^)" = (a")-. 

11. If a"'" = (a'")", find 7?i iu terms of n. 

1 2. Simplify cc''+*+' . a;*+'~' . x'*~'+' . x*^^. 

13. Simplify(^--j-^(^-,^) . 14. Divide 4^' by —. 

15. Simplify [j (a-")- }^]-[ j (a"')" I"']- 

1 6. Multiply iC + 1"- 2c" by 2a"' - 36. 

17. Multiply a'"-"})"-" by a"^6'>-"c. 

18. Shew that --+<^^")^:("^^^^ = ^^^. 

19. Multiply x^ + x^ + 1 by x^ - x^ + 1 

and their product by x^ - x^ + 1. 

20. Multiply a" - 6a"— ^ x + ca"— ^ a;2 by a" + 6ti"-' x - ca'-'h^. 

2 1 . Divide x^*"*-^' - i/2«<^-ii by x*^*"" + !/«<'-". 

22. Simplify j (a")" "•i'»+i. 

23. Multiply x^"" + x'^yf" + x'y^ + y"'' hy af—y'. 

24. Write down the values of 625^ and 12~^. 

25. Multiply .•-•'"•-'"• - 2/'— 1)" by x" - y". 

26. xM u] tiply x^ + 3.C- - 1 by x^ - :>x"i. 



XXIV. ON SURDS. 



289. All numbers which we cannot exactly determine, 
because they are not multiples of a Primary or Subordinate 
Unit, are called SurdS. 

290. We shall confine our attention to those Surds which 
originate in the Extraction of roots where the results cannot 
be exhibited as whole or fractional numbers. 

For example, if we perform the operation of extracting the 
square root of 2. we obtain 1-4142..., and though we may 
carry on the process to any required extent, we shall never be 
able to stop at any particular point and to say that we have 
found the exact number which is equivalent to the Square 
Root of 2. 

291. We can approximate to the real value of a surd by 
finding two numbers between which it lies, differing from each 
other by a fraction as small as we please. 

Thus, since V2 = 1-4 142 



14 15 1 

a/2 lies between :— and -—, which differ by :r- ; 
10 10 ■' 10 

also between -—- and -— -, which differ by tt^k, 
100 100' ■^ 100 

also between ^ and ^^^, ,vhich differ.by -^-. 

And, generally, if we find the square root of 2 to n places 
of decimals, we shall find two numbers Ix'twec^; wliich ^2 lies. 

differinLT ironi eucli other by the fraction ,^- . 



214 ON^ SURDS. 



292. Next, we can alwaj's find a fraction differing from the 
real value of a surd by less than any assigned quantity. 

For example, suppose it required to find a fraction differ- 
ing from ^'2 by less than ^o- 

Now 2(12)''^, that is 288, lies between (16)- and (17)2, 
.'. 2 lies between ( t;^) and (rs) ; 

.•. ^2 lies between -^ and j^ ; 

.-. J2 differs from r— by less than r-^. 
12 -^ 12 

293. Surds, though they cannot be expressed by whole or 
fractional numbers, are nevertlieless nuinlx-rsof which we mav 
form an approximate idea, and we may make three assertions 
respecting them. 

(1) Surds may be compared so far as asserting that one is 
greater or less than another. Thus ^^^3 is clearly greater than 

^'2, and 4^9 is greater than ^fS. 

(2) Surds may be multiples of other surds : thus 2 ^^2 is 
the double of J2. 

(3) Surds, when multiplied together, may produce as a 
result a whole or fractional number: thus 

V2x ^2 = 2, 

294. The symbols ^^a, ^a, ^/a, i^a, in cases where the 
second, third, fcurth, and n*^ roots respectively of a ainnot be 
exhibited as wliole or fractional numbers, will represent surds 
of the second, third, fourth, and Jt"" order. 

These symbols we may, in accordance with tlie principles 
laid down in Chapter XXIII., replace by a*, a^, a*, a". 



ON SURDS. 215 



295. Surds of the same order are those for which the root- 

symljol or surd-iudex is the same. 

1 
Tliiis ^a, 3 >Ji^h), 4 i^l(inn), r^ are surds of the same order. 

Like surds are those in which the same root-symbol or surd- 
index appears over the same quantity. 

Thus 2 sja, 3 Ja, 4a^ are like surds. 

296. A whole or fractional number may be expressed in 
the form of a surd, by raising the number to the power denoted 
by the order of the surd, and placing the result under the 
symbol of evolution that corresponds to the surd-index. 

Thus 0= Mja\ 

b ' Ib^ 



297. Surds of different orders may be transformed into surds 
of the same order by reducing the surd-indices to fractions 
with the same denominator. 

Thus we may transform ^fx and ^y into surds of the same 
order, for 

and. ^y = y^^y^ = ^l/y\ 

and thus both surds are transformed into surds of the twelfth 
order. 

Examples.— cviii. 

TransforTn into Surds of the same order : 
I. Va;and ^y. 2. 4/4 and ^2. 3. ^(18) and 4/(50). 
4. 'J^2 und ;i/2. 5. ^/rt and ;;/6. 6. 4^(a + 6) and i^{a-b). 

298. If a whole or fractional number be multiplied into a 
surd, the product will be represented by plaqing.the multiplier 
and the multiplicand side by side with no sign, or with a dot 
(.) between them. 

Thus the product of 3 and ^f2 is represented by 3 ^^2, 

of 4 and 5 v'2 by 20^2, 

of rr and Jc by a ^/c. 



2i6 ON SURDS. 



299. Like surds may he combined by the ordinary pro- 
cesses of addition and subtraction, that is, by adding the 
coefficients of the surd and placing the result as a coefficient 
of the surd. 

Thus ,v/« + «/« = 2 V**? 

X Jc- ^/c = (.C - 1) i^C 

300. We now proceed to prove a Theorem of great ini- 
]>ortance, which may be thus stated. 

The root of any expression is the saw,e as the product of the 
roots of the separate factors of the expression, that is 

sj(ah) = ^la . ^h, 
^{xyz)=^x.^y. »/z, 
;:/(pqr)= ;'2). s^q.^fr. 

We have in fact to shew from the Theory of Indices that 

1 11 

(aby =0" . h". 

Now \(ahy>r = (abf = ab, 

11 11 '^ 1 

and Irt". ?)" j" = (a")". (6")" = rt". 6" = rt.6; 

^ 111 

.". (ab)" =rt". b". 



301. We can eometimes reduce an expression in the form 
of a surd to an equivalent expression with a whole or frac- 
tional niimber as one factor. 

Thus v'("2) = V(-fi X 2) = ^/CM) . ^/2 = 6 ^/2, 

4/(128) = ^(64 X 2) = ^(64) . ^72 = 4 ^f'2, 
!j{a'x) = a^a" . Zfr = a . ^J/x. 



O.V SURDS. 



Examples.— cix. 

Reduce to equivalent expressions with a whole or fractional 
number as one factor : 

I. V(24). 2. ^/(50). 3. V(4a3). 

4. s'{l2fiaH^). 5. v/(32?/s3). 6. ^/(lOOOa). 

7. V(720c2). 8. 7.V(396x) 9. 18.J(^x3). 

-T-. II' \^('t^ + 2a-x + ax-). 

12. V(a^-2x2|/ + XJ/2). 13. v'(5Oa2_iO0a64-5O&-). 

14. V(63c*?/-42cy + 7y3). 15. 4/(54rt662), 

16. 4/(1 60xV). 17. 4/(108m9ni»y. 

18. 4/(1372ai65i6). ig. 4/(3;* + 3x31/ + 3xV- -^ xr)- 

2a 4/(rt*-3a36 + 3a262-a&3). 

302. An expression containing two factors, one a surd, the 
other a whole or fractional number, as 3 »J2, a ^x, may be 
transformed into a complete surd. 

Thus 3 v'2 = (32)i V2 = V9 . V2 = ^/(18), 

a^fx = {a^)K ^x= 4/a3. ^x= ^{a^x). 



Examples.— ex. 

Reduce to complete Surds : 
I. 4V3. 2. 3^/7 

4- 24/6. 5. 3/^''^ 

7. 4«V(3x). 

9. (»^+^)-^G~-3- 



3. 54/9. 


6. 3 V«- 


^■•W(£> 


'«+«(,7y- 



\x + v/ ' \x- - 2xv + v'-' 



2i8 ON SURDS. 



303. Surds may be compared by transforming them into 
surds of the same order. Tlius if it be required to determine 
whether s/^ be greater or less than 4^3, we proceed thus : 

V2 = 2^ = 26= 4/23= ^8, 

4/3 = 3^ = 3^=4/32=4/9. 
And since 4^9 is greater than ,^8, 
^3 is greater than ^'2. 

Examples. — cxi. 

Arrange in order of magnitude the lullowing Surds : 

1. J3 and 4/4. 6. 2 ^87 and 3 ^33. 

2. VlO and 4/15. 7. 2 4/22, 3 4'7 and 4 V2. 

3. 2 V3 and 3 ^'2. 8. 3 ^/19, 5 4/I8 and 3 4/82. 

4. ^J'l^rld^{~). 9- 2 4^14, 5 4/2 and 3 4'3. 

5. 3^7 and 4^3. 10. ^ ^72, | ^3 and ^ v'-i. 

304. The following are examples in the application of the 
rules of Addition, Subtraction, Multiplication, and Division to 
Surds of the same order. 

1. Find the sum of ^'18, ^a28, ami ^'32. 

v/(18)4- v/(128)+ V(32)= ^'(9x2)+ ^/(64x2)+ v'(16 x 2) 
= 3V2 + 8s'2^4^'2 
= 15 ^'2. 

2. From 3 ^/(75) take 4 ^/(12). 

3 ^/(75) - 4 x/(12) = 3 x/(25 x 3) - 4 ^'f4 x 3) 
= 3.5.^'3-4.::. x'3 
= 15 ^3 - 8 J3 
= 7^3. 



ON SURDS. 



3. Multiply v/R ^y V(12). 

^/8x v/(12)= v/(8xl2) 
= V(96) 
= V(16 X 6) 
= 4^6. 

4. Divide ^/32 by ^18. 

x /(32) _ >v/(16x 2) ^ 4^2 ^ 4 
V(18) V(9x2) 3V2 3" 

Examples. — cxii. 

Simplify 

1. V(27)+ 2^(48) + 3^(108). 11. ^6 x */8, 

2. 3^(1000) +4^(50) + 12^(288). 12. ^(14) x ^(20). 

3. a VC^^a;) + & sj(t»^x) + c sj{c^x). 1 3. ^/(50) x V(200). 

4. ^(128) + 4/(686) + 4/(16). 14. 4/(3rt26) X 4/(9a6''!). 

5. 7 4/(54) +3 4/(16) +4/(432). 15. 4/(12a6) x 4/(8a^6S). 

6. V(96)- V(54). 16. ^/(12)- V3. 

7. V(243)-V(48). 17. x/(18)-vio'V 

8. 12 ^(72) -3^(128). 18. 4/(rt-^)-^ 4/(«?'-). 

9. 5 4/(16) -2 4/(54). 19. 4/(a36)-^ 4/(a63). 

10. 7 4/(81) - 3 4/(1029). 20. V(^2 + ^,3y) ^ ^/^^ + 2a;2y + a;^)/^). 

305. We now proceed to treat of the Multiplication of 
Compound Surds, an operation which will be frequently ?e- 
quired in a later part of the subject. 

The Student must bear in mind the two following Rules ; 

Rule I. sjax Jb= ^/(ab), 

Rule II. ^ax ^a = a, 

which will be true for all values of a and b. 



ON SURDS. 



EXAMPLES.— cxiii. 



Multiply 

s]x by ^y. 

V(3;-2/)i>y Vy- 

6 ,^x by 3 sjx. 

7V(* + l)by 8V('X+1) 

lO^a^by 9V(a;-l). 



9- \'35 ^y - 'J^- 

10. V(^-l) by - sl{^.-l). 

11. 3 ^/.c by - 4 ^x. 

12. - 2 ^a by - 3 ,^a. 
13- \/(a;-7)by - ^x. 

14. -2 V(a; + 7) by -3 ^a:- 

15. -4Vra2-l)by -2^(a2-l). 
V(3x) by ^/(4x-) . 16. 2 V(a^ - 2a + 3) by - 3 ^{a^ - 2a + 0). 



306. The following Examples will illustnite tlie wny of 
proceeding in forming the products of Compound Surds. 

Ex. 1 . 1 o multiply ^x + 3 by ^a: + 2. 

^/x + 3 

Vx + 2 

« + 3^x 
+ 2v'a; + 6 

X + 5 v'a; + 6 

Ex- ?. To multiply A^x + Zjy by 4 ^r - 3 ^hj. 

AJx + ZsJy 
4 Vx - 3 y/j/ 

16x + 12v'(x?/) 

-12V(xj/)-9?/ 

16x - 9?/ 



Ex. 3. To form tlu' scpiare ofV(a;-7)- ^^x. 
V(x-7)- Vx 
^(x-7)-Va; 

x-7 - ^/(x2-7x) 

- ^/(x2-7x)+a; 

2x-7-2\/(x''2'-7x) 



ON SURDS. at 



Ex AMPLES.— CXiv. 

Multiply 

I. ^x + 7 hj ,^'o: + 2. 2. v'-^-5 by Vx+3. 

3. J(a + 9) + 3 by ^f{a + 9) - 3. 

4. V(a-4)-7byV(a-4) + 7. 

5. S^x-1 hj i^x + 4. 

6. 2^/{x-i>) + 4hy3J{x-5)-6. 

7. ^(6 + x) + ^fx by ^/(6 + x) - v'x. 

8. V(3-;;+l)+ v/(2x-l)byV3x- V(2a;-1). 

9. s^a + J{a - x) by .Jx - J(a - x). 
I o. V (3 + x) + Jx by ^/(3 + x). 

11. sjx+ ijy+ ijz hy Jx- Jy+ Jz. 

12. Ja+ J(a — x)+ Jx hy Ja- J(a-x)+ Jx. 

Form the squares of the following expressions : 

13. 21+ ^/(x2-9). 17. 2V^--3. 

14. J{x + Z)+ J(:x + 8). 18. J{x + y)- J{x-y). 

15. JX+ J{x-A). 19. Jx.J{x+l)-J{x-l). 

1 6. J{x - 6) + v'a;. 20. ^/(.c + 1) + V^ . V(-^' - 1 • • 

307. We may now extend the Theorem explaineil in 
Art. 101. We there shewed how to resolve expression^ df 
the form 

a2-6« 

into factors, restricting our observations to the case of perfect 
squares. 

The Theorem extends to the difference between any tivo 
quantities. 

Thus 

a-b={Ja+ Jh){Ja- Jb). 

» 
x^-y = {x+ Jy) {x- Jy). 

l-x==(l+ Jx) (1- Jx\ 



222 ON SURDS. 



308. Hence we can always find a multiplier which will 
fVie tVoiii surds an expression of any of the /oitr forms 

I. a+ s/b or 2. Ja+ Jb, 
3. a- s,fh or 4. Ju- Jb. 

j.'Oi- since the first laid third of these expressions give 
as a product a'^~b, which is free from surds, and since the 
second and fourth give as a product a-b, which is free from 
surds, it follows that the required multiplier may be in all 
cases found. 

Ex. 1. To find the multiplier which will free from surds 
each of the following expressions: 

I. 5+V3. 2. ^6+^5. 3. 2- ^o. 4. x/7- ^'2. 

The multipliers will be ^ 

I. 5-^3. 2. V6-V5. 3. 2+^5. 4. V7+V2. 

The products will be 
I. 25-3. 2. 6-5. 3. 4-5. 4. 7-2. 

That is, 22, 1, -1, and 5. 

ct 
Ex. 2. To reduce the fraction ^_ ^^ to an equivalent. 

fraction with a denominator free from surds. 

Multiply both terms of the fraction by 6+ ,^c, and it be- 
comes 

ab + atjc 
b^-c ' 
which is in the required form. 

Examples.— cxv. 



Express in factors : 








I. c-d. 


2. c2-d. 


3- 


c-d^. 


4. 1-1/. 


5. 1-Sx\ 


6. 


5m- -\. ■ 


7. 4a2-3x. 


8. 9-8?i. 


9- 


11«--16. 


10. p'^ - 4r. 


1 1 . jj - 83^ 


12. 


rt*" - b\ 



ON SVRDS. 223 



Reduce the following fractions to equivalent fractinus Avitli 
denominators free from surds, 

,, 1 „ N^L ,- 4 + 3V2 

16 ^ 17 V3 2-V2 

V«+Vx V(™'+*1)- V('h2-1) 

^" Va - V*' ■ \/(»i2 + 1) + ^{m^ - i) 

* 1- Vrc' ~^' a~ s/(a2-l)- 

V(a + x) + v'(a - a;) ^ a+ sj{a^-x-) 

^{a + x)— sjia — x)' ~ ' a- sj{a? - x^)' 

309. The squares of all numbers, negative as well as posi- 
tive, are positive. 

Since there is no assignable niamber the square of which 
would l)e a negative quantity, we conclude that an expression 
which appears under the form sfi - 'i^) represents an impossible 
quantity. 

310. All impossilile square roots may be reduced to one 
common form, thus 

V(-«2)=Vla-x(-l)f=>2.N/(-l) = a.V(-l) 
^(-a;)=VI^ x(-l)\=Jx .^'■-1). 

Where, since a and sjx are possible numbers, the whole 
impossibility of the expressions is reduced to the appearance of 
^( - 1) as a factor. 

311. Def. By ,^/(-l) we understand an expression which 
ivhen multiplied l»y itself produces - 1. 

Therefore 

}n/(-i)P=U'(-i)!--v^(-i)=(-i)-v^(-i)=- V(-i), 
*U/(-i)l*=U'(-i)!MV(-i)P=(-i)-(-i)=i, 

«V-d so on. 



:524 ON SURDS. 

Examples.— cxvi. 

^lultiply, oLservini,' tluit 

^ - ax ^1 -h= - ^fab. 

1. 4+ ^/(-:3)l_.y4- V(-3). 

2. V3-2V(-2)l,y ^'3 + 2 ^/( - 2). 

3. 4V(-2)-2V2l.y^^/(-2)-3V2. 

4. V(-2)+ V(-3j+ x/(-4)by V(-2)- V(-3)- ^/(-4). 

5. 3 V( - «) + x^( - b) by 4 V( - «) - 2 v/( - 6). 

6. a + s,f( - a) by a - ^f( - a). 

7. a^{-a) + b^'{-h) by a .yA; - a) - 6 V( - &)■ 

8. a+/5v'(-l) bya-/ix/(-l). " 

9. 1- V(l-e') by 1+^/(1-62). 

I o. t''^'-" + e"''^ '-" by e""^ '-'' - e"^^'-". 

312. "We sliall now gi\ e a few Miscellaneous Examples to 

illustrate the principles explained in this Chapter. 



Examples. — cxvii. 

1. bnnphly ^^^^^J-^^-^A 

2. Prove that |1+ ^(-1)^+11- v/(-l)j2 = 0. 

4. Prove that 11+ ^f{-\)'r- \1- ^{-l)\^= ^'(-16). 

; . D i \- i 1 1 e .r^ + (I M ly x- + ^/2ax + a^. 

6. Divide 7a'' 4-?c' by m-— ^^f2mn + n-. 

7. Siniplil'y ,^f {x^ + 2x-y + xy-) + ^' {x^ - 2x-y + xy^). 

8. Simplify- , „ , yr, ami verify by puttniL: 

„ -. ;) and i = 4. 



ox sunDS. 225 



9. l^iiul the square of « >» /r - sf{cd). 

10. Find the square of aV'^ — -j^r 

11. Siniplit'y 

12. Smipuiy — — — ^ i. 



x/(l-^'-') 

„. ,.,. ic-l ( a;-l \-x ) 

13. Simphtv { — , r- + T- > , 

14. Form the square of . /( \v + •' ) - /i / ( ! - 3 ). 

15. Form the square of i^(^x + a) - sj{x — a). 

16. :\Iiiltiply J/(a^'»-"6''"'+V-"') by xy(a''&"^*c— ^0- 

17. Raise to the 5"" power —\ — a^l{- 1). 

1 8. Simplify 4/(81 ) - ^l{ -512)+ 4/(192). 

19. Simplify ^-^y( 3-3 ). 

20. Simplify ~~„ j '4/(32:'-a;^ - GSjjV + 441^^^: ._ 1029^52) j . 



X— I 



21. Simplify 2('h -\)^ ( - _-, -1 5— ,- \ 

•' - ^^V 2/i*-6/i3 + 6?t^-2?i/ 

22. Simplify 2(?i - 1) ^(63) + \ v/(112) - ^'(^j!^ 

2 



-../!l75(n-l)2c^!xA_2 /(^;j 



23. Wliat is the difference between 

s/jl7- v/OW)!x VI17+ v/(33)J 
and 4/ ! ( i") + ^/( 1 :!! )) I X 4/ ) f;5 - V( 129)1 

[S.A.] 



226 ON SURDS. 

313. We have now to treat of the method of finding the 
Square Root of a Binomial Surd, that is, of an expression of 
one of the following forms : 

m+ s]n^ m— /y/n, 

where m stands for a whole or fractional number, and tjn for 
a surd of the second order. 

314. We have first to prove two Theorems. 
Theorem I. If Ja = m+ ^n, m must fee zero. 

Squaring both sides, 

a='mP'-\- 2m ^n + n ; 
.". 2ni ^11 = a — m- — n ; 
, a — Tn? — n 

that is, V*i, a surd, is c(iual to a -whole or fractional number, 
which is impossible. 

Hence the assumed equality can never hold unless in =0, in 
which case ijn= s,hi. 

Theorem II. 7/"fe+ ^'a=^m^ Jn, then must fe=7?i, and 

For, if not, let b — m, + x. 

Then m + x+ ^a=m+ ^/n, 

or x+ ^fa= x^i ; 

which, by Theorem I., is inipossiblii unless a; = 0, in which case 

h = vi and ^'(f= i>^^n. 

315. To find the Square Boot of : + ^fb. 

Assume V(*+ V^)= >/•*+ \'v- 

Then a+ ,Jb = x + 2 VC-r.V) + y ; 

••• x + y = a (1). 

2n'(-'-."V v7, ^i), 

froni which we have to tiud x auu ij. 



ON SURDS. 227 



Now from (1) »2 ^ 2x1/ + 1/^ = a-, 

and from (2) 4xy = h ; 

.•. x"-2x2/+ (/- = a2 — 6; 

Also, x + y = a. 

From these equations we find 

and y-- 



2 " i2 ' 

Similarly we may show that 

^'(» - ^'») = ^l " "4""- *' } - ^^^^f^l . 

316. The practical use of this method will be more clearly 
seen from the following example. 

Find the Square Root of 18 + 2 VC^T). 

Assume V{ 18 + 2 ^(77) | = V« + Vy. 

Then 18 + 2 V(77) = a; + 2 ^(xy) + y ■ 

.-. a; + 2/ = 18 ) 

2V(a^) = 2V(77)r 

Hence x^ + 2xy + 1/- = 324 ) 

4a-j/ = 308J"' 

.'. x^ - 2xy + y"^ = \^ ; 

:.x-y=±A; 

also, x + y=l8. 

Hence a = ll or 7, and y = l or 11. 

That is, the square root required is ^^(11)+ ^^7. 



228 ON SURDS. 

Examples. — cxviii. 

Find the square roots of tlie following Binomial Surds: 

I. 10 + 2^/(2^. 2. 16^2^(55). 3- 9-2^(14). 

4. 94-42V5- 3- 1-3-2^/(30). 6. 38-12^(10). 

7. 14-4V6. 8. 103-12^/(11). 9. 7.^> - 12 ^/(21). 

10. 87-12v'(42). IT. 3_^-v/(10). 12. .57-12^/(1.5). 

317. It is often easy to determine the square roots of 
expressions such as those given iu the preceding set ot 
Examples hxj insjiedion. 

Take for instance the expression 18 + 2 \/(77). 

What we want is to find two numbers wliose sum is 18 and 
whose product is 77 : these are evidently 11 and 7. 

Then 18 + 2 V(7V) = 11 + 7 + 2 ^(11 x 7) 

= U/(ll)+^/7p. 
That is v/(ll)+ \'' is the .s(iuare root of 18 + 2 ^/(77). 

To effect this resolution by inspection it is necessary that the 
coefficient of the surd should be 2, and this we can always ensure. 

For example, if the proposed expression be 4+ /v/(15), we 
proceed thus : 

8 + 2V(15) 5 + 3 + 2^(5x3) 



4+ V(15) = 



2 

V2 



~\ J2 J ' 



:. — 75^ is the square root of 4+ \/(15). 

Again, to find the Square Root of 28 - 10 is/3. 
28-10^/3 = 28-2^/(75) 

= 25 + 3-2v/(2.-)'x.3) 
= :5- V3)2; 
:. 5 - ^3 's f '16 sipuire root required. 



XXV. ON EQUATIONS INVOLVING SURDS. 

318. Any equation may be cleared of a single surd, by 
transposing all the other terms to the contrary side of the 
equation, and then raising each side to the power correspond- 
ing to the order of the surd. 

The process will be explained by the following Examples. 

Ex. 1. ^'.r = 4. 

Raising both sides to the second power, 
a; =16. 

Ex. 2. 4/x = 3. 

Raising both sides to the third jjower, 
a; = 27. 

Ex. 3. Via;2 + 7)-x=l. 
Transposing the second term, 

J(a;2 + 7) = r+a;. 

Raising both sides to the second power, 
X- + 7 = 1 + 2x + a;2, 
.-. x = 3. 

Examples.— cxix. 

I. Jx = 1. 2. v/-c = 9. 3. x^ = b. 

4. 4/a; = 2. 5. x- = Z. 6. 4/x = 4. 

7. v/(x + 9) = 6. 8. ,./(x-7)-7, 9. V(.r-15) = 8. 

10. (x-9)^=12. II. ^(4x-16) = 2, 12. 2()-3Va; = 9. 



230 ON EQUATIONS INVOLVING SURDS. 

13. 4/(2a; + 3) + 4 = 7. 17. ^/(4x2 + 5x-2) = 2x + l. 

14. h-\-CsJx = a. 18. x/(9x2-12a;-51) + 3 = 3x. 

15. V0^"-9) + x = 9. 19. v^(^''-"^ + '^)-«=-K- 

16. ^(x^- 11) = x- 1. 20. ^'iLoy? — '^inx-Vii)-hx = m. 

319. When ^iro surds are involved in an equation, one at 
least may be made to disappear Ly disposing the tenns in 
such a way, tliat one of the surds stands by itself on one side 
of the equation, and then raising each side to the power cor- 
responding to the order of the surd. If a surd be still left, il 
can be made to stand by itself, and removed by raising each 
side to a certain power. 

Ex. 1. ^(x-16)+ v'-c = 8. 

Transposing the second term, we get ' 

^/(x-16) = 8- ^Ix. 

Then, squaring both sides (Art. 306), 

3;-16 = 64-16V« + a;; 
therefore 1 6 ^/.c = 6 i + 1 6, 

or 16Va: = 80, 
or /y/x = 5 ; 
x = 25. 

Ex. 2. V(^ - 5) + sK^ + T) = 6. 

Transposing the second term, 

V(-c-5) = 6- ^'Crr';). 

Squaring both sides, x - 5 = 36 - 1 2 sj{x + 7) + x + 7 , 
therefore 12 ^'(x + 7) = 36 + x + 7-x + 5. 

or 12V(x + 7) = 48, 

or V(x + 7) — 4. 

Squaring both sides, x + 7 = 1 6 ; 

therefore x = 9- 



r V EQUA TIONS LWOLVIXG SURDS. 23 1 



Examples. — cxx. 

1. v''(16 + x)+ Jx = 8. 6. 1+ v/(3a; + l)= x/(4x + 4). 

2. ^f{.C-\(3) = 6- s,tx. 7. l~ ^J{l-■6x) = •ls/(^■-'•c)■ 
2,. s/{x + 15) + ^.0= 15. 8. a - ^/(x - a) = ^x. 

4. ^'{x -21)= ^'x - 1. 9. V^ + v/(x - 7?l) = y. 

5. v'(-c-l) = 3- v/(u; + 4). 10. V(x-1)+ ^'(:c-4)-3 = 0. 

320. When surds appear in the denominiitors of fractions 
in equations, tlie equations may be cleared of fractional terms 
by the process described in Art. 186, care being taken to 
follow the Laws of Combiualiou of Surd Factors given in 
Art. 305. 



Examples.— cxxi. 

36 28 

2. Vx+,/(.-21) = ^^. 4. V(x-15)+V^ = -^i*^-^. 

9a 

^I{ax) + h^ b-a jjx+l6 _ s'-'- + S2 

'' x + 6 ~h- ^{ax)' 9- ^/x + :r~^Zr+12' 

o /I , / N /I / \ 4+ ,^/x v/:c-8 ./;/;- 4 

8. (1 + Vx) (2 - Va;) = — ^- . 10. \_- ,. = >, — -. 

321. The following are examples of Surd Equations result- 
ing in quadratics. 

Ex.1. 2^x^^^-'5. 

r'learing the equation of fractions, 2a; + 2 = 5 ^jz. 



232 ON EQUA TIONS INVOL VING SURDS. 



Squaring both sides, we get 4x2 4-8x + 4=25x; 

whence we find re = 4 or -. 
4 

Ex.2. V(-'' + 9) = 2V^-3. 

Squaring both sides, a; + 9 = 4x - 1 2 ^/x + 9 ; 
therefore \1 sjx = 3a;, 

or A: ,Jx=x. 

Squaring both sides, 16x = x2. 

Divide by x, and we get 16 = a;. 

Hence tlie values of x which satisfy the equation are 16 
and (Art. 248). 

Ex.3. v/(2x+l)+2^x = ^^-^-j^. 

Clearing the equation of fractions, 

2a; + l + 2v'(2x2 + x) = 21; 
therefore 2 ^(2x2 + x) = 20 - 2x, 

or V(2x2 + ;i;) = 10-x. 

Squaring both sides, 2x"^ + x = 1 00 - 20x + x*, 
whence x = 4 or -25 

322. We sliall now give a set of examples of Surd Equa- 
tions some of which are reducible to Simple and others to 
Quadratic Equations. 



Examples.— cxxii. 

I. 4x - 12 ^/x = 16. 4. V(6x -11)= V(249 - fix^). 

2." 45-14Vx=-x. 5. >/(6-x) = 2- ^/(2x-l). 

3. 3V(7 + 2.c2) = 5^/(4x-3). 6. x-2 ^',4-3x) + 12 = 0. 

7. v/(2x + 7) + V(3x -18)= v\7x + 1). 

8. 2 V(204 - 5x) = 20 - ^'(3x - 68). 



ON EQUATIONS INVOLVING SURDS. 233 

9. Vx-4 = -^^. 14. V(x + 4)+ V(2x-1) = 6. 

10. V:c+ll=^^?^. 15. V(13x-1)- v/(2x-l) = 5. 

>y X — 11 

11. V(.c + 5). V(a; + 12) = 12. 16. V(7x+1)- V(3x+1) = 2. 

1 2. V(a; + 3) + V(a; + 8) = 5 ^x. 17. VC-l + x) + V^ = 3. 

525 
13- v'(25 + x)4- V(25-x) = 8. 18. v/x+ V(a! + 9975)=-7=. 

20. V(x2-l) + 6 = -^^^ 



. 21. V(('^-«)" + 2«/) + 6-S=a;-a+i. 

22. Vl(^ + «)' + 2aft + 6-J=6-a-a;. 

23. V(x + 4)- V^=J(,r + |). 

a; — 1 5 

24- ^;/^I^=^ + 4- 26. V(a; + 4)+ V(a; + 5) = 9. 

V(a;-4)- 



25 . V(4 + .r) - v'3 = ^x. 27. ^fx + ^{x - 4) = -j^ 



28. x2 = 21+ ^(^2-9). 

29. V(50 + a;)- V(50-x)=2, 

30. V(2xr4)- J(|+6) = l. 

31. V^3 + .r)+^/x= ^ 



V(3 + x)' 



1 _J ^1^ ^ 

3^- V(a; + 1) ■*" \./{x ~i)~ ^/{x' - !)• 

3x -r ■^f(4x — x^ 



XXYI. ON THE ROOTS OF EQUATIONS. 

323. We have already proved that a Simple E(iuation can 
have only one root (Art. 193) : Ave have now to prove that a 
Quadratic Equation can have only two roots. 

324. We must first call attention to the following fact : 

If m7i = 0, either m = 0, or n = 0. 

Thus there is an ambiguitv : but if we know that m cannot 
be equal to 0, then we know for certain that n = 0, and if we 
know that w cannot be equal to 0, then we know for certiiin 
that m = 0. 

Further, if lmn = 0, then either 1 = 0, or 7?i = 0, or n = 0, and 
so on for any number of factors. 

Ex. 1 . Solve the equation (x - 3) (x + 4) = 0. 
Here we must have 

x-3 = 0, or x + 4 = 0, 
that is, X = 3, or X — — 4. 

Ex. 2. (x - 3a) (5x - 26) = 0. 

Here \m must have 

x-3a = 0, or 5x — 26=0, 

, . 26 

that IS, «=3a, or x = — . 

o 



OM THE ROOTS OF EQUATIONS. 23$ 



Examples. — cxxiii. 

I. (a;-2)(a;-5)=0. 2. (x-3) (x + 7) = 0. 3. (a; + 9)(x + 2)=0, 

4. (x-5a)(a;-6?*) = 0. 6. (19x-227) (14a; + 83)=0. 

5. (2a; + 7)(3.c-5) = 7. (5x-4m)(6x- lln) = 0. 

8. (a;2 + hax + Sa^) (x^ - Tax + 1 2a2) = 0. 

9. (x^ - 4) (x- - 2«x + cfi) = 0. 

10. X (x^ - 5x) = 0. 

11. (acre - 2ffi + 6) {bcx + 3a - 6) = 0. 

12. (ex - (f ) (ex - e) = 0. 

325. The general form of a quadratic equation is 

ax^ + bx + c = 0. 

Hence aix^ + -x + -) = 0. 

\ a a/ 

Now a cannot =0, 

.-. x^ + -x + - = 0. 
a a 

... b e 

Wnting x> for - and q for -, we may take the following 

as the type of a quadratic equation of which the coeflBcient of 
the first term is unity, 

x'^-irfx + q — O. 

326. To show that a quadratic equation has only two roots. 

Let x^ +px + 5' = he the equation. 

Suppose it to have three different roots, a, b, c. 

Then a'^ + ap + q = (1), 

¥+bp + q = i..-(2), 

c2 + cp + q = (3). 

Subtracting (2) from (1), 

a^-b^+(a-b)p = 0, 
or, {a-b){a + b-\-p) = 0. 



236 ON THE ROOTS OF EQUATIONS. 

Now a-b does not equal 0, since a and 6 are not alike, 

:. a + h+p = (4). 

Again, subtracting (3) from (1), 

a^ — c^ + (a — c) p = 0, 
or, {a — c){a + c+p)=0. 

Now a — c does not equal 0, since a and c are not alike, 

.-. a + c+p = (5). 

Then subtracting (5) from (4), we get 

6-c = 0, and therefore h = c. 

Hence tliere are not more than hvo distinct roots. 

327. We now procet-d to show the relations existing be- 
tween the Roots of a quadratic equation and the Coefficients 
of the terms of tlie equation. 

328. x'^^-px + q=0 

is tlie general form of a qtiadratic equation, in which the co- 
efficient of the first term is unity. 

Heni'e x'^+px= —q 

x'^ + 'px+^-^=^-q, 






Now if a and /? be the roots of the equation, 

«=-i-V('i-') '"• 

^--i-V(t-') <''• 

Adding (1) and (2), we '.y\ 

a + j3= -p (.3). 

v 



ON THE ROOTS OF EQUATIONS. 



Multiplying (1) and (2), we get 

or a/3=^--^-+2, 

or ay8 = (2 (4^. 

From (3) we learn that tiiM, sum of the roots is equal to the 
coefficient of the second term with its sign changed. 

From (4) we leam that the product of the roots is equal to 
the last term. 

329. The equation x'^ + px + q = has its roots real and 
different, real and equal, or impossible and different, according 
as 'p- is > = or < Aq. 

For the roots are 

2 



-i-V(?-')'"" 



and _2 _/(»?. A „,r-ti4£!zil). 



i-V(?-'> 



First, let p~ be greater than Aq, then >J{p^ - Aq) is a possible 
quantity, and the roots are different in value and Ijoth real. 

Next, let2'^ = 4g', then each of the roots is equal to the real 

quantity -^. 

Lastly, let ^^ be less than Aq, then \f{p- — Aq) is an impos- 
sible quantity and the roots are different and both impossible . 



Examples.— cxxiv. 

I. If the equations 

ax- + bx + c = 0, and a'x^ + h'x + c' = 0, 

have respectively two roots, one of which is the reciprocal of 
the other, prove that 

(aa' - cc')^ = {aV - he') {a'b - b'c). 



238 ON THE ROOTS OF EQCA 



2. If a, /? be the roots of the equation ax- + 6a; + c = (), prove 
that 

.) no &^ — 2ac 
' a- 

3. If a, ^ be the roots of tlie equation ay? + 6x + c = 0, prove 
that 

ac'j? -i- {2ac ~ b'^) x + ac = ac \-^~ r,)\^— )- 

4. Prove that, if tlie roots of the equation ax- + bx + c = i) be 
equal, nx- + bx + c is a perfect square witli respect to x. 

5. If a, y8 represent the two roots of the equation 

x^-{l + a) a; + ^(l + a + «") =0, 
show that a- + /3'- = a. 



33O. If a and /3 be the roots of the equation x^+px + q=Oy 

th en x'^ + jjx + 3 = (a; - a) (x - yS). 

For since ^= - (a + /?) and q = afB, 

3? + px + q = x- ~ {a + (B) X + a/B 

= {x~a){x-(3). 

Hence we may form a quadratic equation of which the roots 
are given. 

Ex. 1. Form the equation whose roots are 4 and 5. 
Here x-a = x — 4andx-/3 = x-5; 

.•. the equation is (x - 4) (x — 5) = ; 
or, x--9x + 20 = 0. 

Ex. 2. Form the equation whose roots are ^ and - 3. 



2 

.111(1 r— /?=)•-!- 3 ■ 



Here x - a = x -- and x - ^ = x + 3 ; 



th« 



equation is f x - ^ j (x + 3) = ; 



er, (2x-l)(x + 3)=0; 

or, £3:- + 5x-3 = 0. 



UN THE ROOTS OF EQUATIONS. 239 

Examples.— cxxv. 

Form the equations whose roots are 
I. 5 and 6. 2. 4 and -5. 3. -2 and -7. 

12 5 

4. 2 and-. 5. 7aud-- 6. v/3 and - ^3. 

7. m + 7i and. m — n. 8. - and . 9. -7^ and-. 

a jd pa 



331. Any expression containing x is said to be a Function 
of X. An expression containing any symbol x is said to Ll; a 
positive integral function of a; when all the powers of x con- 
tained in it liave positive integral indices. 

3 1 

For example, bx^ + 2r^ + ^x* + j~x^ + 3 is a positive integral 

■ 1 • 

function of :r, but Qx^ + Scc^ + 1 and 5a-" - 2x~^ + 3x- + 1 are 

1 
not, because the first contains x^, of which the index is not 

integral, and the second contains a;"^, of which the index is not 

positive. 

332. The expression 5x^ + 40;' + 2 is said to be the expres- 
sion corresponding to the equation 5x^ + 4x^ + 2 = 0, and the 
latter is the equation corresponding to the former. 

333. If a be a root of an equation, then x-a is a factor 
of tlie corresponding expression, provided the equation and 
expression contain only positive integral powers of x. This 
principle is useful in resolving such an expression into factors. 
We have already proved it to be true in the case of a quadratic 
equation. The general proof of it is not suitable for the stage 
at which tlie learner is now supposed to be arrived, but we 
■will illustrate it by some Examples. 



240 ox TUF. ROOTS OF EQUATTOA^S. 



Ex. 1 . Rrsul ve 2oc2 - 5x + 3 into factors. 

If we solve the equation 2x^-5.15 + 3 = 0. we shall find thai 
its roots five 1 and -. 

Now divide 2/--5.r + 3 by x-1 ; the quotient is 2j6-3 
that is o(.,:- I); 

.'. the L;i\'eii eA])ression = 2 (a; - 1) ( x - ^ I. 

Ex. 2. Eesolve 2x^ + a;-— ll.f- 10 jnto factor.?. 

By trial we find that this expre.ssion vanishes if we put 
x= - 1 ; tliat is, — 1 is a root of the e(jiiation 

Sx^ + x^-ll.t- 10 = 0. 
Divide the expression l>y x4- 1 : the quotient is Sx^-x- 10 ; 
.'. the expre.ssion = (2x- - x - 10) (x + 1) 

= 2(x^-|-5)(. + l). 

We must now resol\-e x- - -b into factor.s, by solving the 
corresponding equation x^ ~'- — b=0. 

The roots of this equation are - 2 and g; 

.-. 2x3 + x2 - 1 Ix -10 = 2 (.»• + 2) (x - ^) (x + 1) 
= (x + 2)(2./;-5)(x + l). 

Examples.— cxxvi. 

Resolve into simijle factors the following expressions : 

I. .t3-11x2 + 36x-36. 2. x^-7.c2 + l4x-8. 

3. x-"* - 5.1-2 - 4(i.-- - 40. 4_ 4x3 + 6.1-2 + X-1. 

5. 6.r3+ll.»;2-9x-14. 6. 3?^y^ -^^-Zxyz. 

7. a^-P~c^-2ahc. 8. 3x3-x2-23x + 21. 

9. 2.f3 - 5x2 _ i7.r + 20. ■ 10. 15.1-3 + 41.r2 + 5.r - 21. 



ON THE ROOTS OF RQClATlONS. 241 

334. Tf we can find one root of such an equation as 

2a;3 + a;2-llx-10 = 0, 
we can find all the roots. 

One root of the equation is - 1 ; 

.-. (x + l)(2x2-x-10) = 0; 
.-. x+l = 0, or2a;2-a;-10 = 0; 

.. x= - 1, or —2, or -. 

Similarly, if we can find one root of an equation involving 
the 4"" power of x, we can derive from it an equation involving 
the 3'* and lower powers of x, from which we may find the other 
roots. And if again we can find one root of this, the other 
two roots can be found from a (quadratic equation. 

335. Any equation into which an unknown symbol or ex- 
pression enters in two terms onl3', having its index in one of 
the terms double of its index in the other, may be solved as a 
([uadratic equation. 

Ex. Solve the equation x^ — Qx^ = l. 

Regarding x^ as the quantity to be obtained by the solution 
III the equation, we get 

therefore x^-3=±4; • 

therefore x^=7, or x^= — 1. 

Hence x= ^'7 or x= ^ -1^ 

and one value of ^^ - 1 is - 1. 

336. In some cases by adding a certain quantity lo both 
sides of an equation we can bring it into a form capable of 
solution, thus, to solve the equation 

x2 + 5.>; + 4 = 5 ^'(x^ + 5,/; + 28), 
add 24 to each side. 

Then x^ + 5x + 28 = 5 s'{x'^ + 5x + 28) + 24 ; 

or, a;2 + 5a; + 28-5 V(a'-"^ + 5x + 28) = 24. 

This is now in the form of a quadratic e([uation, the un- 
known quantity being ^f{x- + 5x + 28), and completing the 
square we have 

fs-A-l Q 



242 0\ THE kOOTS OF EQUATIONS. 

95 191 

.-. V(a;2 + 5.r + 28)-|=±^; 

whence »J{x^ + 5x + 28) = 8 or - 3 ; 

.-. .r'' + 5x + 28 = 64 or 9; 
from which we may find four values of x, viz. 4, - 9, ani 
5+ V(-51) 



Examples.— cxxvii. 

Find roots of the following equations : 

I. x-*- 12x2=13. 2. x6+14x3 + 24=0. 

3. x»-t- 22x^ + 21=0. 4. x-'" + 3x"' = 4. 

5. x--3X-» = ^. 6. ^=-20;-=-. 

7. x-2 + 3x-i = ^. 8. x-'"'-x-'' = 20. 



9. x2-2x + 6(x2-2x + 5)2 = ll. 

10. x^-x + S V(2x2-5x + 6) = — ^ — . 

11. x2-2V(3x2-2ax + 4) + 4 = |*(x + ^ + l). 

12. ax + 'i >J{x--ax + a'^=x^ + '2,a. 

337. Every equation has as many roots as it has dimen- 
sions, and no more. This we have proved in the case of 
simple and quadratic equations (Arts. 193, 323). The general 
proof is not suited to this work, but Ave may illustrate it by 
the following Examples. 

Ex. 1. To solve the equation x-^- 1=0. 

One root is clearly 1. 

Dividing b}- x — 1, we obtain x- + x + 1 = 0, of which the roots 

-1+ V-3^^.. -l-x/-3 

are ^r^'— and ^ . 

2 « 



ON RA no. 243 



Hence the three roots are 1, ~ and ^^ . 

Ex. 2. To solve tht equation x^-\=0. 
Two of the roots are evidently + 1 and - 1. 
Hence, dividing by (x- l)(x + 1), that is by a;^- 1, we obtain 
a;2 + 1 = 0, of which the roots are v^— 1 and - v''— I- 

Hence the /our roots are 1, - 1, ^^- 1, and — \'— 1. 

The equation x^ — 6x^ = 7 will in lii<;e luauner have six 
roots, for it may be reduced, as in Art. 335, to two cuT)ic 
equations, x^ - 7 = and x^ + 1 = 0, 

each of which has three roots, which may be found as in 
Ex. 1. 



XXVII. ON RATIO. 

338. If a and B stand for two unequal quantities of the 
same kind, we may consider their inequality in two ways. We 
may ask. 

(1) By ichat quantity one is greater than the other ? 

-The answer to this is made by stating the difference be- 
tween the two quantities. Now since quantities are represented 
in Algebra by their measures (Art. 33), if a and b be the 
measures of A and B, the difference between A and B is 
represented algeljraically by a-b. 

(2) By how many times one is greater than tlie other? 

The answer to this question is made by stating the number 
of times the one contains the other. 

Note. The quantities must be of the same kind. We can- 
not compare inches with hours, nor lines with surfaces. 

339. The second method of comparing ^4 and B is called 
finding the Eatio of A to B, and we give the following ileti- 
nition. 

Def. Eatio is the relation which one quantity bears to 
another of the same kind with respect to the number of t,ime: 
the one contains the other. 



244 ON RATIO. 



340. The ratio of A io B is expressed thus, A : P>. 
A and B are called the Terms of the ratio. 

A is called the Antecedent and B the Conskquent. 

341. Now since quantities are represented in Algebra hy 
their measures, we must represent the ratio between two 
(quantities by tlie ratio Ijetween their measures. Our next 
step then must be to sliovv how to estimate tlie ratio between 
two numbers. This ratio is determined by finding how many 
times one contains the other, that is, by obtaining the quotient 
resulting from the division of one by the other. If a and 6, 

then, be any two numbers, the fraction j- will express the ratio 

of a to b. (Art 136.) 

342. Thus if a and b be the measures of A and B respec- 
tively, the ratio of A to JB is represented algebraically by the 

fraction r. 



343. If a or b or both are surd numbers, the fraction ^ 



may also be a surd, and its approximate value can be found In' 
Art. 291. Suppose this value to be ' , where m and n are 
whole numbers : then we sliould say that the ratio A : B is 
aj (proximately re])resented by — . 

344. Ratius may be compared witli each other, by com- 
paring ihe fractions by wliich they arc denoted. 

Thus the ratios 3 : 4 ami 4 : 5 may be compared by com- 

3 4 

paring the fractions - and -. 

These are equivalent to — and ^ resi)ectlvely ; and since 

gx is greater than 7,—, the ratio 4 : 5 is greater than the 
latio 3:4 



ON RATIO. Ms 



Examples. — cxxviii. 

1. Place in order of magnitude the ratios 2 : 3, 6 : 7. 7 : 9. 

2. Compare the ratios x + 3y : x + 2y and x + 2ij : x + //. 

3. Compare the ratios x-5y : x — 4y and x-Zy : x - -ly. 

4. What number must be added to each of the terms of th« 
latio a : h. that it may become the ratio c : d? 

5. The sum of the squares of the Antecedent and Conse- 
quent of a Eatio is 181, and tlie product of the Antecedent 
and Consequent is 90. What is the ratio? 

345. A ratio of greater inequality is one whose antecedent 
is greater than its consequent. 

A ratio of less inequality is one whose antecedent is less than 
its consequent. 

This is the same as saying a ratio of greater inequality is 
represented by an Improper Fraction, and a ratio of less in- 
equality by a Proper Fraction. 

346. A Ratio of greater inequality is diminished by adding 
the same nuwher to both its terms. 

Thus if 1 be added to both terms of the ratio 5 : 2 it becomes 

6 : 3, which is less than the former ratio, since g, that is, 2, is 

less than -. 

And, in general, if x be added to both terms of the ratio 
a : h, where a is greater than 6, we may compare the twu 
ratios thus, 

ratio a + x : 6 + a- is less than ratio a : b, 

if -i be less than -y, 

b + x V 

.„ ' ab+bx . ■, ,, ab + ax 

it ,-i5 — r- be less than -=- — ;— , 

62 + bx V + bx^ 

if ab + bx be less than ab + ax, 

if 6x be less than ax, 

if b be less than a. 

Now b is less than o ; 

:. a +x -.b + x \?, less than a : h. 



246 CyV l^A no. 

347. We may observe that Art. 346 is iiieiely a repetition 
of that which we proposed as an Example at the end of the 
chapter on Miscellaneous Fractions. There is not indeed any 
necessity for us to -vvearj' the reader with examples on Ratio: 
for since we exjiress a ratio by a fraction, nearly all that we 
mi.t,'ht have had to say about Ratios has been anticipated in 
our remarks on Fractions. 

348. The student may, however, work tlie following Theo- 
rems as Examples. 

(1) If fl : 6 be a ratio of greater inequality, and x a positive 
quantity, the ratio a — o:: b — x is greater than the ratio a : b. 

(2) If (/ : h he a ratio of less inequality, and x a positive 
quantity, llie ratio a + x : b +x is greater than the ratio a : b. 

(3) If a : i be a ratio of less inequality, and x a positive 
quantity, the ratio a — x: b — x is less than the ratio a : b. 

349. In some cases we may from a single equation involv- 
ing two unknown symbols determine the ratio between the 
two symbols. In other words we may be ahle to determine the 
relative values of the two symbols, though we cannot determine 
their absolute values. 

Thus from the equation 4x = 3?/, 

X 3 

we get - = -. 

y 4 

A^ain, from the equation 3x2 = 2?/-, 

■we"et'., = ^; and therefore =-\t. 
" 2/- 3 y x/.3 

Examples. — cxxix. 

Find the ratio of x to y from the following equations : 
1. 9.1 = 6)/. 2. ax = by. 3. ax-by = cx + dy. 

4. x-4-2a-?/ = 5?/-. 5. a;2- 12,r)/= 13;r. 6. x' + mxy = n^y-. 

7. Find two numbers in the ratio of 3 : 4. of \\ hich the 
sum is to the sum of their scjuares :: 7 : 50. 

8. Two numbers are in the ratio of 6 : 7, and when 12 is 
addid to each ihe resulting numbers are in the ratio 1 I 12 : 13. 
Find the nimibers. 



OK RATIO. 247 

9. The sum ol' two iiujuLers is 100, and the nunriieis are 
in the ratio of 7 : 13. Find them. 

10. The ditt'erence of the squares of two numbers is 48, 
and the sum of the nvimber^ is to the difierence of the num- 
bers in the ratio 12:1. Find the numbers. 

11. If 5 gold coins and 4 silver ones are worth as much as 
3 gold coins and 12 silver ones, find the ratio of the value of a 
gold coin to that of a sih'er one. 

12. If 8 gold coins and 9 silver ones are Avorth as much as 
6 gold coins and 19 silver ones, find the ratio of the Aaliie of a 
silver coin to that of a srold one. 



350. Ratios are compounded by multipljing together the 
fractions by wliich they are denoted. 

Thus the ratio compounded of a : 6 and c : fZ is ac : hd. 

Examples. — cxxx. 

Write the ratios compounded of the ratios 

1. 2:3 and 4:5. 

2. 3 : 7, 14 : 9 and 4 : 3. 

3. a;- — y- : x^ + y^ and x- - xy + y- : x + y. 

4. a^ — b^ + 2bc - c^ : a^ - 6- - 2hc - c^ and a + b-rC : a + h - r. 

5. m^ + n^ : vi^ - n^ and m — n : m + n. 

6. x^ + 5x + 6 : y'-' — ly + 12, and y^ - 3)/ : x- + 3x. 



351. The ratio a^ : b'^ is called the Duplicate Ratio of a ; 6. 
Thus 100 : 64 is the duplicate ratio of 10 : 8, 
and 36a;2 : 2oy^ is the duplicate ratio of 6x : by. 

The ratio a^ : ¥ is called the Triplicate Ratio of a : &. 
Thus 64 : 27 is the triplicate ratio of 4 : 3, 
.and 343x^ : 1331)/^ is the triplicnte ratio of fx : lly. 



248 ox PROPORTION. 



352. The definition of Ratio given in Euclid is the sanu! jf^ 
in Algebra, and so also is the expression for the ratio that one 
quantity bears to another, that is, A : B. But Euclid cannot 
employ fractions, and hence he cannot represent the value of a 

ratio as we do in Altjebra. 



XXVIII ON PROPORTION. 

353. Proportion consists in the equality of two ratios. 

The algebraic test of Proportion is tlud the two fractions 
representing the ratios must he equal. 

Thus the ratio a : b will be equal to the ratio c : d, 

and the/o?(?- numbers a, b, c, d are in such a case said to be in 
proportion. ^ 

354. If the ratios a ■ b and c ; d form a proportion, we 
express the fact thus : 

a : b = c : d. 

This is the clearest manner of expressing the equality of the 
ratios a : b and c : d, but there is another way of expressing 
the same fact, thus 

a : b :: c : d, 
which is read thus, 

a is to 6 as c is to d. 

The two terms a and d are called the Extremes. 
, b and c the Means. 

355. When four numbers are in proportion, 

product of extremes = product of means. 
Let a, b, c, d lie in jiroportion. 



ON PROPORTION. 249 



Multiplying both sides of the equation by M, we get 
ad = he. 

Conversely, if ad = bc we can show that a : b=c ', d. 

For since ad = be, 

dividing both sides by bd, we get 

ad_hc 
hrVd' 

that is, h^ d' ^'^' " '•^ = '^ '• ^' 

356. liad = bc, 

Dividing by cd, we get - = j, i-fe- a : c = b : d; 

d c 
Dividing by ab, we get r = -, i-t^- '^ : 6 = c : a ; 

Dividing by ac, we get - = -, i.e. d : c = b -.a. 

357. From this it follows that if any 4 numbers be so 
related that the product of two is equal to the product of the 
other two, we can express the 4 numbers in the form of a pro- 
portion. 

The factors of one of the j^roducts must form the extremes. 

The factors of the other product must form the means. 

358. Three, quantities are said to be in Continued Pro- 
portion when the ratio of the first to the second is equal to 
the ratio of the second to the third. 

Thus a, b, c are in continued projjortion if 
a : b = b : c. 

% The quantitj' b is called a Mean Proportional lietween 
a and c. 



25© ON proportion: 

Four quantities are said to be in Continued Proportion 
when the ratios of the first to the second, of the second to 
tlie third, and of the tliird to the fourth are all equal. 

Tlius a, b, c, d are in continued proportion when 
a : b = b : c = c : d. 

359. We showed in Art. 20.5 the process by wliicli when 
two or more fractions are known to be equal, otlier relation? 
between the numbers involved in them may be determined 
That process is of course applicable to Examples in Ratio and 
Proportion, as we shall now show by particular instances. 

Ex. 1. li a : b = c : d, prove that 

a^ + b'^ : a^- -¥ = <:'- + d^:c^- d^. 

Smce a : o = c : d, t= j. 

a 

Let r=X. ThenT = \; 
d 

:. a = \b, and c = \d. 



Now 
and 



ft^ + 6' _ X^b- + ¥ _ />-(\^+l) _ X2+J 
c'^ + d^- _ \\l- + d'^ _ d- (X^+l)_X^ + l 



a^ + b'^_c- + d- 
^t^b^~^~d^'' 



Hence 
ti^at is, a2 + 6« : a2 _ 52 = c2 + ^^2 . ^2 _ ^2^ 



Ex. 2. If « ; fe :: c : d, prove that 

a:c:: ^{a* + ¥): i/{c* + d^), 

LetJ = X. Then^ = X; 

d ' 

,: a = \b, and c = \d. 



ON PROPORTION. 251 



a _yb _h 
c y^d d' 



i/ic'+d^) ~ ^(Md* +'d') ■" ;^d^. */^\* 4-1) ^¥~'d- 



Hence 






that is, a:c:: ^{a^ + b*) : ij{c* + d^^ . 

Ex. 3. ] f a : 6 = c : f? = e : /, prove that each of these ratios 
is equal to the ratio a + c + e: b + d +f. 

Let - | = \, | = X, ^ = X. 

Then a = \b, c = Xrf, e = X/. 

^ a + c + e 'Kb + \d + \f_H b + d+f) _^ 

°^'^ b + 'd+f~~b + d+f " 'b + d+f 

TT a + c-re a c e 

^"^^^ b^drrb=d=f' 

that is, a + c + e : b + d +f— a : b = c : d = e :/. 

Ex. 4. If a, b, c are in continued jiroportion, show that 
a"^ + b'^ : b'^ + c^ ~ a : c. 

Let ~ = X.- Then- = X. 
c 

Hence a='\b and 6 = Xc:. 

a^ + b'^ _X-b- + b-_b\ \^+l )_b-(\^' + l)_b'^ _ac_a 
b'^ + c'^ ~ T- + C-' ~ T-c^^Tc- ~ f^X'-4-l)~ c''^ ~ c^ ~ c" 

Ex. 5. If Uxi + b : 15t- + d=l2a + b: 12c + d, ])rove that 

a :b = c : d. 
Since 15a + 6 : 15c + c?=12a + ?* : 12c + ci, 

and since product of extremes = product of means. 



252 ON PROPORTION. 



(15CH-6) (12c + i) = (15c + d) (12a +6), 

or, 180ac+ 126c + 15ad + 6d = 180ac + 12a<Z+ 156c + 6rZ, 

or, 126c + Ybad = 12ad + 156c, 

or, 3ad = 36c, 

or, atZ = 6c. 

Whence, by Art. 355, a : b = c : d. 

Additional Examples will be found in page 137, to which 
we may add the following. 

EXAMPLES. — CXXXi. 

1 . li a : b = c : d, show that a + b: a = c + d :e. 

2. U a : b = c : d. show tliat a^ - 1- : b^ = c- - d"^ : d^. 

"?. It a, : Oi = a2 : 6,, show that — ^ ,- ==-. 

4. If a : 6 :: c : c?, show that 

3a- + ab + 26'' : 3a- - 26^ : : 3o- + cd + 2d^ : 3c2 - 2d-. 

5. If ffl : 6 = c : rf, show that 

ft2 + 3ab + ¥ : c^ + 3tY/ + d- = 2«6 + 362 . 2cd + 3d-. 

6. Ifa:6 = c:rf = e :/ then a : b — mc — ne : md-nf. 

7. If — a, —6, any parts of a, b, be taken from a and 6 

n n 

respectively, show that a, b, anil the remainders form a propor- 
tion. 

8. If a : 6 = c : d = e :/, show that 

ac : bd = la^ + mc- + ne- : lb- + ind- + nf-. 

9. If (/, : 6i = aj : 63 = 03 : 63, show that 

(/,- 4-^,2 + ^^2 . 5^2 ^5^2 + 5^2 .. „^-.- . i,i^ 



av PROPORTION. 253 

10. If ai : 6i = rt2 : 62 = a3 : h-i, show that 

a^a.2^ + a^Og + 03(11 : h^.^ + 6063 + 6361 = a-^ : 6,2. 

-rpa2-a6 + 62 c--cd + c?2 ,,,.,, a c a d 

11. It „- — , --, „ = -.,— — — „, snow that either r = 3 or t = -• 

12. If a2 + 6- : rt^ - /*- = c2 + c?2 . cii - rf2^ phow that 

a: b — c : (^. 

13. If rt : 6 = c : (Z, show that 

(rt - c) (a^ - r-') (6 - d) (¥- d^y 

14. If rtj : h^ — a., : /*.,, show that 



On the Geometrical Treatment of Proportion. 

360. The definition of Proportion (viz. the equality of 
ratios) is the same in Euclid as in Algebra. (Eucl. Book v. 
Def. 6 and 8.) 

But the ways of testing whether two ratios are equal are 
quite different in Euclid and in Algebra. 

The algebraic test is, as we have said, that the two fractions 
representing the ratios must be equal. 

Euclid's test is given in Book v. Def. 5, where it stands 
thus : 

" The first of four magnitudes is said to have the same ratio 
to the second which the third has to the fourth, when any 
equimultiples whatsoever of the first and third being taKen 
and any equimultiples whatsoever of the second and fourth : 

" If the multiple of the first be less than that of the second, 
the multiple of the third is also less than that of the fourth : 



" If the multiple of the first be equal to that of the second, 
the multiple of the third is also equal to that of the fourth : 
or. 



254 ON PROPORTION. 



" If the multiple of the first be greater than that of the 
second, the multiple of the third is also greater than that of 
the fourth." 

We shall now show, first, how to deduce Euclid's test of the 
equality of ratios from the algebraic test, and secondly, how to. 
deduce the algebraic test from that employed by Euclid. 

361. I. To show that if quantities be proportional accord- 
ing to the algebraical test they will also be proportionai 
according to the geometrical test. 

If a, 6, c, d be proportional according to the algebraical 
test, 

a _c 

Multiply each side by — , and we get 

ma _mc 
nb ruT 

Now, from the nature of fractions, 
if ma be less than nb, mc will also be less than nd, and 
if ma be equal to nb, mc will also be equal to nd, and 
if ma be greater than nb, mc will also be greater than nd. 

Since then of the four quantities a, b, c, d equimultiples have 
been taken of the first and third, and equimultiples of the 
second and fourth, and it appears that when the multiple of 
the first is greater than, equal to, or less than the multiple of 
the second, the multiple of the third is also greater than, 
equal to, or less than tlie multiple of the fourth, it follows that 
a, b, c, d are proportionals according to the geometrical test. 

362. II. To dediioe the algebraic test of proportionality 
from that given by Euclid. 

Let a, h, c, d be proportional according to Euclid. 

Then if s- is not equal to -3, 

let , be equal to , (1). 



EXAMPLES ON RA TIO. ±%^ 

Take to and n such that 

via, is greater than nh, 

but less than n (i + x) (2). 

Then, by Euclid's definition, 

TOC is greater than nd (3). 

But since, by il), -77-,— ^ = —7? 

and, by (2), wa is less than i!(6 + x), 

it follows that 7/i.c is less than nd (4). 

The results (3) and (4) therefore contradict each other. 

Hence (1) cannot be true. 

Therefore -7 is equal to -^. 

We shall conclude this chapter with a mixed collection of 
Examples on Ratio and Proportion. 



EXAMPLES. — CXXXii. 

1. VL a-h -.h-c •.-.h : c, show that i is a mean proportional 
between a and c. 

2. If a : 6 : : c : rf, show that 

a^^W : ^\ = c'' + d^:-^-. 
a+b c+d 

and a : b :: ^/{ma* + nc*) : i/(^mb^+ iid*). 

3. li a : b :: c : d, prove that 

ma — nb _ mc - nd 
ma + nb mc + nd' 

4. If ba + 2b': 7a + 36 : : 56 + 3c : 76 + 3c, 
6 is a mean proportional between a and c. 

5. If 4 quantities be proportional, and the first be the 
greatest, the fourth is the least. 

If a + 6, TO 4- n, m-n,a — b be four such quantiti««j show that 
h is greater than n. 



25^ k^AMPLES ON RA TlO. 

6. Solve the equation 

x-\ : a;-2 = 2x + l : x + 2. 

7. If — , — = — 5—, show that the ratios a : b and c : d are 

b a 

also equal. 

8. In a mile race hetween a bicycle and a tricycle, their 
rates were proportional to 5 and 4. The tricycle had half-a- 
niinute start, but was beaten by 176 yards. Find the rates of 
each. 

9. li a : b :: c : d and a is the greatest of the four quanti- 
ties, show that a- + d- is greater than b~ + c^. 

01 .1, .-plOa + fe 12a + 6 , , , 

10. bhow that it vt; i=rEi 3) i^hen a : :: c : a. 

lOc + d 12c + d' 

11. U X : y :: Z : 2 and x : 25 : : 24 : 1/, find x and y. 

12. If a, b, c be in continued proportion, then 

(1) a : a + b :: a-b : a-c; 

(2) (a2 + f^) (62 + C-) = {ab + bcf. 

13. If a : : : c : a, show that — j— = — t— ; 

and hence solve the equation 

ah — bc — dx_a — h — c 

bc + dx b + c ' 

14. If a, b, c are in continued proportion, show that 

a y- nib : a - mb :: b + vie : b - mc. 

15. li a : h :: ."> : 4, find the value ot the ratio 

1 3 

16. The sides of a triangle are as 2- : 3- : 4, and the peri- 

2 4 '^ 

meter is 205 yards: tiiid the sides. 

17. The sides of a triangle are as 3 : 4 : 5, and the peri- 
meter is 480 y^rds : find tlie sides. 



AND PROPORTION. 257 

1 8. Assuming a + 6 :^ + 9 '■'-p — i • a-b, prove that the sum 
of the greatest and least terras of any proportion is greater than 
the sum of the other two. 

'^'1 19. A waterman rows 30 miles and back in 12 hours, ■md 
he finds that he can row 5 miles with the stream in the same 
time as 3 against it. Find the rate of the stream. 

A,^ 20. There are three equal vessels A, B, C ; the first con- 
tains water, the second brandy, the third brandy and water. 
If the contents of B and G be put together, it is found that the 
mixture is nine times as strong as if the contents of A and G 
had been put together. Find the ratio of the brandy to the 
water in the vessel G. 

21. A factor buys a certain quantity of wheat which he 
sells again so as to gain 5 per cent, on his outlay, and thus 
clears £16. Had he sold it at a gain of 5s. a quarter lie would 
have cleared as many pounds as each quarter cost shillings. 
How many quarters did he buy, and what did each quarter 
cost him ? 

22. A man buys a horse and sells it for £144, gaining as 
much per cent, as the horse cost him. What was the price of 
the horse 1 

23. I buy goods and sell them again for £96, gaining as 
much per cent, as the goods cost. "What is the cost price ? 

24. A man bought some sheep and sold them again for £24, 
gaining as much per cent, as the sheup cost him. What did he 
give for them ? 

^^ 25. A certain crew, who row 40 strokes per minute, start 

'at a distance equivalent to four of their own strokes behind 

another crew, who row 45 strokes to the minute. In 8 minutes 

the former succeed in bumping the latter. Find the ratio 

between the lengths of the strokes of the two boats. 

26. The time which an express train takes to travel a 
journey of 180 miles is to that taken by an ordinary train a»s 
9 : 14. The ordinary train loses as much time from stoppnges 
as it would take to travel 30 miles without stoppini;. The 
express train only loses half as mucli time as the o:hfi- in this 



25S OA' VARIATION. 



manner, and it also travels 15 miles an hour quicker. Sup- 
posing the rates of travelling uniform, what are they in miles 
per hour ] 

, 27. An article is sold at a loss of as much per cent, a? it 
/lis, worth in pounds. Show that it cannot be sold for more 
than ^25. 



XXIX. ON VARIATION. 

363. If a sum of money is put out at interest at 5 per cent, 
the principal is 20 times as great as the annual interest, what- 
ever the sum may be. 

Hence if x be the principal, and y the interest, 
x = 2()i/. 

Now if we change x we must change w in ike same propor- 
tion, for so long as tlie rate of interest remains the same, x 
will always be 20 times . as great as y, and hence if a: be 
doubled or trebled, y will also be doubled or trebled. 

This is an instance of what is called Direct Vari.\tion, 
of which we may give the I'ullowing definition. 

Def. One quantity y is said to vary directly as another 
quantity x, wlien y depends on x in such a manner tiiat any 
increase or decrease made in the value of x produces a propor- 
tional increase or decrease in the value of 1/. 

364. If x = my, where m is a constant quantity, that is. a 
quantity which is not altL-rcd by any change in the values of j, 

and y, 

y will vary directly as x. 

For any increase made in the value of x must produce u 
proportional increase in the value of y. Thus if x be doubled, 
y must also be doubled, to prt-serve the e(|uality of x and my, 
since m cannot be changed. 



ON VAI^IATION. 259 



365. Suppose a man can reap an acre of corn in a day. 
Then 10 men can reap 60 acres in 6 days, 

and 20 men can reap 60 acres in 3 dayss 

So that to do the same amount of work if we double the 
number of men we must halve the number of days. 

This is an instance of what is called Inverse Variation, 
of which we may give the following definition. 

Def. One quantity y is said to vary inversely as another 
quantity x, when y depends on x in such a manner that any 
increase or decrease made iu the value of x produces a propor- 
tional decrease or increase in the value of y. 

366. If a; = — , where m is constant, 

y 

y will vary inversely as x. 

For any increase made in the value of x must produce a pro- 
portional DECREASE in the value of y. Thus if x be doul)led, 

y must be halved, to preserve the equality of x and — . 
■ For 2x = = — . 

y y 
2 

367. If 1 man can reap 1 acre in 1 day, 
5 men can reap 20 acres in 4 days, 

and 10 men can reap 80 acres in 8 days. 

That is, the number of acres reaped will depend on the 
product of the number of men into the numl^er of days. 

This is an example oi joint variation, of which we may give 
the following definition. 

Def. One quantity x is said to vary jointly as tw^o others 
;/ and j-, when any change made in x produces a proportional 
change in the product of y and z. 

368. One quantity x is said to vary directly as y and 

inverseh' as z when x varies as -. 

z 



2bo O.y \ AklAT.'OS'. 



369. Theorem, ll x varies as y when % is constant, and 
«s z when '/ is eonstiiiit, then wlien y and z are both variable, 

% varies as yz. 

Let x — tth. yz. 

Then we have to show thnt in is constant. 

Now when z is constant, 

X varies as y ; 
.". mz is constant. 

Now z cannot involve y, since z is constant when y changes, 
and therefore m cannot involve y. 

Similarly it may be shown that m cannot involve z ; 

:. m is constant, 

and X varies as yz. 

370. The symbol oc is used to express variation; thus xocy 
stands for the words x varies as y. 

371. Variation is only an abbreviated form of expressing 
proportion. 

Thus when we say that x varies as y, we mean that x bears 
to y the same ratio that any given value of x beiirs to the 
corresponding value of y, or 

x : J/ = a given value of x : the corresponding value of y. 

And similarly for the other kinds of variation, as will be 
.seen from our examples. 

Ex. 1. If xoc y and i/oc,t, to show that xocj. 
Let x=my, and y = iiz. 

Then substituting this value of y in the first equation. 
x = m}U ; 
•Old therefore, since mn is constant, 

• OCS. 



ON VARIATION. 2C1 



Ex. 2. If a;cci/ and xocz, then will xcc ^/(i/js). 
Let x = mi/, and x = ns. 

Then x^ = mnyz\ 

;. x= v/(mTO) . V(l/«)- 
Now J(mn) is constant ; 

.-. re ex: VM- 

Ex. 3. If y vary as x, and when x=l, i/ = 2, what wiU be 

the value of y when x = 2 ? 

Here 1/ : x= a given value of y : corresponding value ot x; 
:.y:x = 2:l: 
.-. y = 2x. 
Hence, when x = 2, y = 4. 

Ex. 4. If A vary inversely as B, and when A = 2, B=12, 
what will B become Avhen A =91 

Here yl : -j, = a given value of A 









corresponding 


value 


ofS' 


A 


1 

12" 

9 
12" 


- ^ . 

2 
*^ 

2 


1 

\2' 

1 






B 


24 
9 


8 
~3' 


-1 







Hence, when ^ = 9, 



whence 

Ex. 5. If A vary jointly as B and G, and when yl = 6, J5 = 6 
and (7= 15. find the value of A when 5= 10 and C=3. 

Here 
A : BG= a given value of A : corresponding value of BC\ 
:. A :BC='6 : 6x15; 
.-. 90.4 =6B0. 



262 Uy iARIAlJOW 

Henee, when J5= 10 and C=3, 

90^ = 6 X 10 X 3 ; 

••"*-90-^- 

Ex. 6. If z vary as x directly and y inversely, and if when 
2 = 2, x = 3 and i/ = 4, what is the value of z when x=15 and 

TT a; -. ■ 1 r corresponding value of x 

Here % : - =*'a civen value of z : ^ — ^.-^ = jr- ; 

y " corresponding value oi y 



X 

\ z :- 

y 


-:!. 


. 32 

■■ 4 


_2j; 


nd j/ = 


8, 


32 

4 


30 

8' 


.'. «-= 


120 ^ 

^T4- = ^- 



Examples. — cxxxlii, 



II 

1. If ^oc-^ and Boc — then will ^ocC. 

2. If .-loc^then Avill^oc^. 

3. U Acr.B and Coc D then will ^ C<x 5D. 

4. If xccj/, and when x = 7, J/ = 5, find the value of z whew 
y = 12. 

5. If xec- , and when x» lO, y = t, find the value of y when 
:c = 4. 



ON VARIATION. 263 

6. \i xazyz, and when x = l, i/ = 2, a = 3, find the value of y 
when X = 4 and ^ = 2. 

7. If xoc^, and when x = 6, ]/ = 4, and 2 = 3, find the value 
of X when 1/ = 5 and a = 7. 

8. If 3x + 5?/ oc 5a; + 3y, and when a; = 2, 1/ = 5, find the value 

. X 
of -. 

y 

9. If ^cci> and B^ozC^, express how J. varies in respect 

of a 

10. If z vaiy conjointly as x and y, and 2=4 when x=l 
and 2/ = 2, what will be the value of x when s = 30 and y = Si 

11. If ^ocB, and when A is 8, 5 is 12; express A in 
terms of B. 

12. If tlie square of x vary as the cube of y, and x = 3 when 
7/ = 4, find the equation between x and y. 

13. If the square of x vary inversely as the cube of y, and 
« = 2 when i/ = 3, find the equation between x and y. 

14. If the cube of x vary as the square of y and x = 3 when 
i/ = 2, find the equation between x and 3/. 

I?. If XOC5J and i/oc-, show that xoz-. 
^ z' y 

16. Show that in triangles of equal area the altitudes vary 
iiurersely as the bases. 

17. Show that in parallelograms of equal area the altitudes 
vary inversely as the bases. 

18. H y=p + q + r, where p is invariable, q varies as x, and 
r varies as x^, find the relation between y and x, supposing 
that when a;=l, y = 6; when x = 2, y=ll ; and when x = 3, 
2/ = 18. 

19. The volume of a pyramid varies jointly as the area of 
its base and its altitude. A pyramid* the base of which is 9 



264 ON ARITHMETICAL PROGRESSION. 

feet square and the height of which is 10 feet, is found to con- 
tain 10 cubic yards. What must be the height of a pyramid 
upon a base 3 feet square in order that it may contain 2 cubic 
yards ? 

20. The amount of glass in a window, the panes of which 
are in every respect equal, varies as the number, length, and 
breadth of the panes jointly. Show that if their number varies 
as the square of tlieir breadth inversely, and their length varies 
as their breadth inversely, the whole area of glass varies as the 
square of the length of the panes. 



XXX. ON ARITHMETICAL PROGRESSION. 

372. An Arithmetical Progression is a series of 
numbers wliich increase or decrease hxj a constant difference. 

Thus, tlie following series are Arithmetical Progressions: 

2, 4, 6, 8, 10; 
9, 7, 5, 3," 1. 

The Constant Difference being 2 in the first series and - 2 
in the second. 

373. In Algebra we express an Arithmetical Progression 
thus : taking a to represent the first term and d to represent 
the constant ditt'erence, we shall have as a series of numbers in 
Arithmetical Progression 

o, a + d, a + -Id, a + ;j(/, 
and so on. 

We observe that the terms of the series differ only in the 
coefficient of d, and that each coefficient of d is always less by 1 
than the number of the term in which that particular coefficient 
stands. Thus 

the coefficient of d in the 3rd term is 2, 

in the 4th 3, 

? in the 5th 4. 



OM ARITHMETICAL PROGRESSION. 265 

Consequently the coefficient of d in the m"" term will be 

Therefore the v!^ term of the series will be a -i- (n - 1; d.. 

374. If the series be 

a, a + d, « + 2(i, 

'anil % the last term, the term next before z will clearly be 2 - d. 
and the term next before it will be s - 2d, and so on. 

Hence, the series written backwards will be 

2, z - fZ, 3 - 2rf, a + 2d, a + rf, a. 

375. To find the s^im of a series of numbers in Arithvietical 
Progression. 

Let a denote the iirst term. 

... d the constant difference. 

... z the last term. 

... n the number of terms. 

... s the sum of the 7i terms. 

Then s = a+(a + d) + (rt + 2(f)+ +{z~2d) + (z-d) +z. 

Also s = z + {z-d)+ {z-2d)+ +(« + 2d) + (a + rf) + a, 

the series in the second case being the same as in the lirst, Vmt 
written in the reverse order. 

Therefore, by adding the two series together, we get 

•2s={a + z) + (a + z) + (a + z)+ + {a + z) + (a + z) + (a + z) ; 

and since on the right-hand side of this equation we have a 
series of n numbers each equal to a + z, we get 

2s = n{a + z)', 

This result may be put in another form, because in the 
place of z we may put a + {n—l)d, by Article 373. 

Hence s = ~\a + a + {n- i.)d\, 

thatio, =|{2a + (n-l)di. 



/ 



266 ON ARITHM£:TICAL PROGRESSION. 

376. We have now obtained the following results : 

a(-=a + (M-l)rf (A), 

«=|(« + ^') (B), 

< = |)2« + (n-l)cf( fC). 

From one or more of these equations we have in Examples 
to determine the values of a, d, n, s or z. We shall now jjro- 
(•ee<l to j^ive instances of such Examples. 

Ex. 1. Find the LAST TERM of the series 
7, 10, 13, ...... to 20 terms. 

Taking the equation z = a+ {n — \)i, 
for a put 7 and for n put 20, and we get 
« = 7 + (20-l)i, 
or, !J! = 7+19d. 

Now d is always found by taking the first term from ike second^ 
and in this case, 

r^=10-7 = 3; 
.-. 2 = 7 + 19x3 = 7 + 57 = 64. 

Ex. 2. Find the last term of th - series 
12, 8, 4, to 11 term.=^. 

In the equation z = a-^ {n—\)d, 

]iut a = 12 and n=ll. 
Then z=\<i + \Od. 

Now d = 8-12=-4. 

Hence, a = 12 -40 =-28. 

Examples. — cxxxiv. 

Find the last term of each of the following siiries • 

1. 2, 5, 8 to 17 terms. 

2. 4, 8. 12 to 50 terms. 



On arithmetical PROGRESSIOiV. Z'oJ 



3- 


^ 29 15 . -.^ ^ 

7,-r,-Tr to 16 terms. 

' 4 2 


4. 


1 5 

^,—1, -- to 23 terms. 


r.. 


5 11 . n. 

6' 2' 6 to 12 terms. 


6. 


-12, -8, -4 to 14 terms. 


7- 


-3, 5, 13 to 16 terms. 


8. 


w-ln-2n-3 

, , to ?i terms. 

n n n 


9 


(x -^yf-^x^ + y-, {x-ijf to n terms. 


lO. 


a- fe 4a- 36 7a — 56 

., ^, — -^— to 71 terms. 



a + 6' a + 6' a + 6 

377. Ex. 1. Find the sum of the series 
3, 5, 7 to 12 terms. 

In the equation s = -{2a+ (« - 1) (l\ 

put 3 for a and 12 for n, and we get 

19 

Now d = 5 - 3 = 2, and so 



s = ^{6 + 22; =6x28 = 168. 



Ex. 2. Find the sum of the series 

10, 7, 4 to 10 terms. 



s = |!2a + (»-l)d{: 



put 10 for a and 10 for n, then 



,=12j20 + 9d(. 



26S ON ARITHMETICAL PROGRESSION. 



Now rf«=7 - 10= - 3, and therefore 



a = l?)2<v-27!=5xf- 7)= -35. 



EXAMPLES. — CXXXV . 

Find the sum of the following series : 

1. 1, 2. 3 to 100 tdins. 

2. 2, 4, () to 50 t(M-m3. 

3. 3, 7, 11 to 20 terms. 

4. -, , -7 to 15 terms. 

4' 2 4 

5. -9, -7, -5 to 12 term 

6. -. -, - to 17 terms. 

6' 2' 6 

7. 1, 2, 3 TO n terms. 

8. 1, 4, 7 to ?i terms. 

g. 1, 8, 15 TO n terms. 

n - 1 ?! - 2 ?i - 3 . , 

10. , , to 7? terms. 

n n n 

378. Ex. What is tlie Constant Differekci!; when the 
first term is 24 and the tenth teiiii is - 12? 

Takintf tlie equation (A), 

z = a + (n - l)d, 
ajid re^'ardinij tlie tenth as the last term, we get 

-12 = 24 + (10-l)rf. 
or - 36 = Off, 

whence v,m obtain d= — 4. 



ON ARITHMETICAL PROGRESSION. t j 

Examples. — cxxxvi. 

What is the Constant Difference in the following cases % 
I. When the first term is lOo and the twentieth is — 14. 
2 c fifty-first is - x. 

3 —-= forty-ninth is 5-. 

3 3 

4 — ^ twenty-fifth is -21-. 

5 -10 sixth is -20. 

6 150 ninety -first is 0. 

379. Ex. What is the First Term when 

the 4()th term is 28 and the 43rd term is 32 ? 

Taking equation (A), 

2 = a + (n- l)c?, 

and regarding the last term to be the 40th, we get 

28 = a + 39(7 (1). 

Again, regarding the last term to be the 43rd, we get 

32 = a-f42tf (-:) 

From equations (1) and (2) we may find the value of a to 
be -24. 

Examples. — cxxxvii. 

I. What is the first term when 
(i) The 59th terra is 70 and the 66th term is 84; 

(2) The 20th term is 93 - 356 and the 21st is 98 - 376 ; 

(3) The second term is ^ and the 55th is 5-8 ; 

(4) The second term is 4 and the 87th ir; - SO ? 



fijro ON ARITHMETICAL PROGRESSION. 

2. The Sinn of the 3rd and 8th terms of a series is 31, and 
the sum of the 5th and 10th terms is 43. Find the sum of 
10 terms. 

3. The sum of the 1st and 3rd terms of a series is 0, and 
the sum of the 2nd and 7th terms is 40. Find the sum of 
7 terms. 

4. If 24 and 33 Ije the fourth and fifth terms of a series, 
what is the 100th term ] 

5. Of how many terms does an Arithmetical Progression 
consist, whose difference is 3, fir.st term 5 and last term 302 ? 

6. Supposing that a body falls through a space of 16^^ feet 
in the first second of its fall, and in each succeeding second 
32- feet more than in the next preceding one, how fir will a 
body fall in 20 seconds? 

7. What debt can be discharged in a year by weekly pay- 
ments in arithmetical progression ; the first pajTiient being 1 
shilling and the last ^5. 3*'. ? 

8. Find the 41st term and the sum oi 4i lernis in each of 
the following series : 

(1) -0,4,13 

(2) 4a2, 0, -40.2 

(3) 1 + a-, 5 + 3.r, 9 + 5x 

(4) -4 -1'4 

V.5.) 4> 20 

9. To how many terms do the following series extend, and 
what is the sum of all the terms ? 

(1) 1002 10,2. 

(2) -0, :: ,186. 



ON- ARITHMETICAL PROGRESSION. 271 

(3) 22X, -Sa; -72-3je. 

/ X 1 1 

(4) 2' 4 -^ 

(5) m-\ 137(1 -m), 135>a -m). 

(6) a; + 254, x + 2, x-2. 

380. To insert 3 arithmetic niMyis between 2 and j.O. 
The number of terms will be 5. 

Taking the equation z = a + {n- i) d, 

we have 10 = 2 + (b-l)d. 

Whence 8 = 'id; :. d=2. 

Hence the series will be 

. 2. 4, 6, 8, 10. 

Examples. — cxxxviii. 

1. Insert 4 arithmetic means between 3 and 18. 

2. Insert 5 arithmetic means between 2 and —2. 

2 
■*. Insert 3 arithmetic means between 3 and -. 
^ 3 

4. Insert 4 arithmetic means between - and -. 

381. To insert 3 arithmetic menus between a and b. 

The number of tenu^ in the series will be 5. since the;, 
are to be 3 terms in addition to the iirst term a and the last 
term b. 

Taking the equation 2 = a + (n — iy a, 

we have to find d, having giveii 

a, z = b and n — b. 



272 UN ARl'l HAfETICAL PROGRESSION. 



Hence h = a-\-(^-\)d. 

or, 4<i=6-a, .". d=— p . 

Hence the series will be 

6 — a h — a 3(6 — a) . 
«, a + —4—, « + -2"' '^ + ~ 4 — "' °' 

that IS, a, -^p-, — g— , —J—, 6. 

Examples. — cxxxix. 

1. Insert 3 arithmetic means between rfi and n. 

2. Insert 4 arithmetic means between m + 1 and in-\. 

3. Insert 4 arithmetic means between 11^ and ?!- + 1. 

4. Insert 3 arithmetic means between x^ + y- and a;'-^ — y'-. 

382. We shall now give the general form of the proposition 
" To iiisert m arithmetic means beticeen a aiid b." 

The number of terms in the series will be 7?i + 2 

Then taking the equation z = a + (n-'\)d, 
we have in this case b = a + {m + 2 - 1 ) f/, 
or, b=a+{m + l)d. 

Hence d= ,, 

m + V 

and the form of the series will be 

, 26 - 2a , b — a . 
m+1 m+V ' 



bm-b + ia bm + a , 

m+1 ' m + l' ' 



a 


b-a 
' m+1' 


26 -2a 

a + — , 

m+1 ' 


that 


is, 






am + b 


avi - a + 26 




'^' »r+T' 


m + 1 ' 



XXXI. ON GEOMETRICAL PROGRESSION. 

383. A Geometrical Progression is a series of mnuliers 
which increase or decrease by a constant factor. 

Thus the following series are Geometrical Progressions. 

2, 4, 8. 16, 32, 64; 

12 3 2 A ^• 

^^' "*' 4' 16' 64' 

_1 ^ __! Jl 
2' 16' 128' 1024' 

The Constant Factors being 2 in the tirst series, - in tin- 

4 

second, and — - in the third. 

8 

Note. That which we shall call the Constant Factor is 
usually called the Common Ratio. 

384. In Aljj;eV)ra we express a Geometrical Progression 
thus : taking a to represent the jfirst term and / to represent 
the Constant Factor, we shall have as a series of numbers in 
(ieonii'trical Progression 

a, of, af'^, af^, and so on. 

We observe that the terms of the series differ only in the 
index of/, and that each index of/ is always less by 1 than the 
number of the term in which that particular index stands. 

Thus the index of/ in the 3rd term is 2, 

in the 4th 3, 

in the 5th 4 

Consequently the index of/ in the nth term will be n - 1. 

Therefore the ?ith term of the series will be a/"~'. 
[s.A,] 8 



274 ON GEOMETRICAL PkOGKESSlON. 

Hence if z be the last term, 

385. If the series contain 7! terms, a being the first term 
and / tlie Constant Factor, 

the last term will be a/""', 

the last term but one will be a/"~*, 

the last term but two will be a/*~*. 

Now a/"-' x/=a/'-i x/i = o/'-i+' = a/", 

a/"-^ X /= «/--^ X /' = a/"-'+' = a/-\ 

a/^s X /= a/'-=' X /I = a/"-='+' = a/-*. 

386. We may now proceed to Jind the tmm of a teries of 
numbers in Geometrical rrogression. 

Let a denote the first term, 

/ the constant factor, 

71 the number of terms, 

s the sum of the n terms. 

Then s = a + af+ af+...+ af-^ + of-'' + af-K 

Now multiply both sides of this equation by/, then 

fs = af+ af^ + af+ ... + af"-^ + «/"-' + af". 

Hence, subtracting the first equation from the second, 

fs-s=^af"-a. 

■•• «(/-!)=« (/"-I); 

•■'- f-i • 

Note. The proposition just proved presents a difficult}' to 
a beginner, which we shall endeavour to explain. When we 
multiply the series of ?! terms 

a + af+af-i- + af'-^ + af-^ + c^f^ 



OiV GEOMETRICAL PROGRESSION. 275 

by/, we shall obtain another series 

af+af- + af + + a/'-' + «/--» + a/", 

which also contains n terms. 

Though we cannot fill up the gap in each series completely, 
we see that the terms in the two series must be the same, 
except the first term in the former series, and the last term in 
the latter. Hence, when we subtract, all the terms will dis- 
appear except these two. 

387. From the formulae : 

2 = a/"-' (A), 

.."-^' (B, 

prove the following : 

(a) sJj^. (y) a=fz-{f-\)s. 



f 



s — a 
z' 



388. Ex. Find the last term of the series 
3, 6, 12 to 9 terms. 

The Constant Factor is -, that is, 2. 

In the formula 

3 = a/— », 

putting 3 for a, 2 for/, and 9 for n, we get 
3 = 3x25 = 3x256 = 768. 

Examples.— cxl. 

Filid the last term of the following series 

1. 1, 2, 4 to 7 terms. 

2. 4, 12, 36 to 10 terms. 

3. 5, 20, 80 tu 9 terms. 



276 ON GEOMETRICAL PROGRESSION. 



4. 8, 4, 2 to 15 terms. 

5. 2, 6, 18 to 9 terms. 

6- ^' 1^' 4. to 11 term.. 

2 1 1 ^ n, 

7- -3' 3' -6 to 7 term*. 



389. Ex. Find thf sum of the series 

3 

2 



3 
6, 3, ^ to 8 terms. 



Generally, s= — 7— j — 

anti here a = 6,/=^, « = 8, 

2 2 

6__ 6_ 

256 256 _ 766 

" _1 1 ~~"6T* 

2 2 



EXAMPLES.— CXli. 

Find the .sum of the following series : 

1. 2, 4, 8 to 15 terms. 

2. 1 , 3, 9 to 6 terms. 

3. a, ax^, ax* to 13 terms. 

4.. a, -, -., to 5) terms 

,r ./- 

0^--x-, a- X, — ; — to 7 terms. 

' a + x 



osr Geometrical pkockESSioM. 277 

6. 2, 6, 18 to n terms. 

7. 7, 14, 28 to ?i terms. 

8. 5, -10, 20 to 8 terms. 

2 1 1 ^ r, ^ 

9- -3J 35 -g to 7 terms. 

390. To find the sum of an Infinite Series in Geometrical 
Progression, when the Constant Factor is a proper fraction. 

If/ be a proper fraction and n very large, 
/" is a very small number. 

Hence if the number of terms be infinite, f" is so small that 
we may neglect it in the exjiression 

,_«(/"- 1) 
/-I, ' 
and we get 

-a 



"I-/' 

391. Ex.1. Find the sum of the series 5 + 1 + 7 + to 

infinity. 

Here /=1-| = !' 
4 
_ 0^ 3 16_-1 

•'•*~i-7~7~3~T~^3- 

^-4 

3 2 8 
Ex. 2. Sum to infinity the series g ~ o + 07 ~ 

Here /=-|^|=-!; 

3 3 

a 2 V 2 27 



/ 4\ , 4 26" 
-(-9) ^+- 



9 



27^ ox GEOMETRICAL PROGRESSION, 



Examples. — cxlii. 



Find the sum of tht I'olldwiug infiniie series; 
I. 1, i \ 9- 4^ 2*. .. 



2' 4' 



2. 1. -. ~ lo. 2z^, - -SSa;^ 

-i in 

3- 3, -, - II. o, 6, 

o Z t 
2 11 11 

4- o. o> 5, 12. 



3' 3' 6' 10' 10^' 

13. X, -I/, . 



3 1 

)• 4' 4' 



11 ^ 86 

2" ~3 ''^' 100' 10000 

•7- 8, I, 15. -54444 _ 



3' 



8. l|, -5, 16. -83636, 



392. To inst'ti 3 geometric means heticeen 10 and 160. 

Taking the equation z^af"^^, 
we put 10 for a, 160 for z, an.l 5 for ?i, and we obtain 
160=10./'-': 
.-. 16=/*. 

Now 16 = 2x2x2x2 = 2*; 

• 2* =/*. 

Hence /=i;. and the serie.s will be 

10, 20, 40, 80, 160. 



ON GEOMETRICAL PROGRESSlOX. if^ 

Examples.— cxliii. 

1. Insert 3 geometric means between 3 and 243. 

2. Insert 4 geometric means between 1 and 1024. 

3. Insert 3 geometric means between 1 and 16. 

4. Insert 4 geometric means between - and — -. 

393. To insert m geometric riieans between a (ind b. 
The number of terms in the series will be m + 2. 
In the formula z = af'''''^, 

putting b for z, and to + 2 for n, we get 

or, 6 = f^"'+l; 

•••' ~a' 
or, /=~t:. 

Hence the series will be, 

1 _i_ _}_ 1 



rt, a X — p , a X 



6-^— r-, 6-^— T-, t, 



that is, 

II J 1 

a, (rr . &)-+', (ft"-i.62)m+i^ ^ (a^S—y+i, (-; . Z/"-)"^!, />. 

394. AVe shall now give some mixed Examples ou Aiitl:- 
raetical and Geometrical Progression. 

Examples. — cxliv. 

I. Sum the following series : 

(i) 8 + 15 + 22+ tol2terms. 

(2) 116 + 108 + 100+ to 10 terms. 



28o ON GEOMETRICAL PROGRE.SSTOM. 



(3) 3 + 2'^12"^ to infinity. 

'4) 2 - - + — - to infinity. 

4 oz 

1 2 11 

(5) 2~3~y ^"^"^ terms. 

112 

(6) 9~o+q— to 6 terms. 

1 5 

(7) g-1-^- to 29 terms. 

(8) s + l + l?+ toSterms. 

(9) 3 + 9 + 27"^ ••••■■• to infinity. 

, , 3 14 Ol i ^/^i . 

(10) V — T7^-r-- to 10 terms. 

5 10 lo 

('0 /v/?- v'6 + 2V(l-''')- to 8 terms. 

V 5 

, , 7 7 35 ^ - i 

(12) -^ + s — r-+ to .5 terms. 

o 2 4 

2. If the continued product of 5 terms in Geometrical 
Progression be 32, show that the middle term is 2. 

3. If a, h, c are in arithmetic jirogression, and a, ?/, (• :i;v 

.1 • 1 *i 4. ^ <* + <' 

in geometrical jirogressioii, show that 17 = 5 — 77 — r. 

4. Show that the arithmetical mean between a and h i- 
i:reater than tlie geometrical mean. 

5. The sum of the first three terms of an arithmetic series 
is 12, and the si.xth term is 12 also. Find the sum of the first 
6 terms. 

6. What is necessary that «, 6, c may be in geometric pro- 
grescsion ? 



ON GEOMETRICAL PROGRESSION. 281 



7. If 271, X and -^r- are in cfeometric protrression, what is x? 

8. If 2n, ?/ and — are in aritlimetic progression, what is 1/? 

9. The sum of a geometric progression whose firet term is 
1, const nit factor 3, and number of terras 4, is equal to the sum 
of an arithmetic progression, whose tirst term is 4 and constant 
difference 4 ; how many terms are there in the arithmetic pro- 
gression? 

10. The tirst (7 + ?i) natural numbers when added together 
make 153. Find n. 

11. Prove that the sum of any number of terms of the 
series 1, 3, 5, is the square of the number of terms. 

12. If the sum of a series of 5 terms in arithmetic progres- 
sion be 95, show that the middle term is 19. 

13. There is an arithmetical progression whose first term is 

1 4 

3„, the constant difference is 1,;. and tlie sum of the terms is 

22. Required the number of terras. 

14. The 3 digits of a certain number are in arithmetical 
progression ; if the number be divided liy the sum of the digits 
in the units' and tens' place, the quotient is 107. If 396 be 
subtracted from the number, its digits will be inverted. 
Required the number. 

15. If the {'p-Vfjf' term of a geometric progression be'm, 
and the {p — qf" term be n, show that the 2^"" term is >^f{mn). 

16. The ditt'erence between two numbers is 48, and tlie 
arithmetic mean exceeds tlie geometric by 18. Find the 
numbers. 

17. Place three aiithmctic means between 1 and 11. 

18. The first term of an increasing arithmetic series is "034, 
the constant difference •0004, and the sum 2-748. Find the 
number of terms. 

19. Place nine arithmetic jueans between 1 ami - 1, 



282 ON HARMONIC AL PROGRESSION. 

20. Prove that every term of the series 1, 2, 4, is 

greater by unity than the sum of all that precede it. 

21. Show that if a series of my) tenns forming a geometrical 
progression whose constant factor is r be divided'into sets of p 
consecutive terms, the sums of the sets will foim a geometrical 
progression whose constant factor is r'. 

22. Find five numbers in arithmetical progression, such 
that their sum is 55, and the sum of their squares 765. 

23. In a geometrical progression of 5 terms the difference 
of the extremes is to the difference of the 2nd and 4th terms 
as 10 to 3, and the sum of the 2nd and 4tli terms equals twice 
the product of the 1st and 2nd. Find the series. 

24. Show that the amounts of a sum of money put out at 
Compound Interest form a series in geometrical progression. 

25. A certain number consists of three digits in geometrical 
progression. The sum of the digits is 13, and if 792 lie added 
to the number, the digits will be inverted. Find the number. 

26. Tlie population of a county increases iu 4 years from 
10000 to 146 il ; what is the rate of increase ? 



XXXII. ON HARMONICAL PROGRESSION. 

395. A Harmonicai Progression is a series of numbeis 
of which the reciprocals form an Arithmetical Progression. 

Tims the series of numl)ers «, h, c, <l, is a Harmon ical 

• .•1 .1111. .-.,., 

Progression, 11 the series , r> -> "7? ^s an Anthmetical 
a bed 

Progression. 

If a, b, e be in Harmonicai Progression, b is called the 
Harmonicai Mean between </ and c. 

Note, There is no way of finding a general expression for 
the sum of a Harmonicai Series, but manv problems with 



OA' HARMONICAL PROGRESSTON. 283 

reference to sncb a series maybe sobbed by inverting tbe terms 
and treating the reciprocals as an Arithmetical Series. 

396. J/a,,b, c he in Harmonical Progression, to show that 

a : c :: a — b : b — c. 

Since -, -,) ' are in Arithmetical Protrression, 

c 6 5 a' 

b-c a—b 
DC ao 

ab a-b 

a a — b 

or - = T^ — . 

c b — c 

397. To insert m harmonic means between a and b. 

First to insert m arithmetic means between - and t- 

a h 

Proceeding as in Art 357, we have 

a ' 

or a = 6 + (m + l).a6(£ 

, a-b 

ab (?«.+ 1) 

Hence the aritlimetic series will be 

11^ a-b 1 , 2 (ffl - 6) 1 m{a-b ) 1 

a' rt'a6(ni+l)' a ab{m+iy a a6(m+l)' b' 

1 6?M + a 6//1 + 2a-b am + b 1 

a' a6(»i-rl)' ab{m+l)' ah m + iyb' 

Therefore the Harmonic Series is 

ab(m + \) abjm+^l) ab{m + l) 

' 6m -r a ' hm + 2a-b' am + b ' 



284 ON HARMONICAL PROGRESSION. 



398. Given a and h the first two terms of a series in Har- 
monical Progression, to find the n*'' term. 

-, T are the first two terms of an Arithmetical Series of 
o 

which the common difference is t — . 

6 a 

The w"' term of this Arithmetical Series is 

1 (n — 1) (o - 6) _ 5 + Tia - a — n6 + ft 

a ah ah 

* 

(Tta - g) - inh - 2b) _ ( n - 1) a -{n-2) h 
ah ~ ah 

.'. the n** term of the Harmonical Series is 
(rr^)a-(n-2)6" 

399. Let a and c be any two numbei-s, 

6 the Harmonical Mean between tliem. 

1111 
Then t — = --i-> 

b a c 

2 a + c 
or T= ; 

ac 

,_ 2ac 

~a + c' 

400. The following results should be remembered. 
Arithmetical Mean between a and c = —^ — . 

Geometrical Mean between a and c= ^ac. 

2ac 

Harmonical Mean between a and c = — — . 
, a + G 



ON HARMONICAL PROGRESSION. 285 

Hence if we denote the Means by the letters A, G, H 
respectively, 

A X 11=——- X 

= ac 

that is, (? is a mean proportional between A and H. 

401. To show that A, G, H are in descending order of 
magnitude. 

Since ( ^'a - aJc)- must be a positive quantity. 

( V« - */c)^ is greater than 0, 

or a — 2 ,Jac + c greater than 0, 

or a + c greater than 2 i^fac, 

a + c . , — 

or -— greater than ^ac ; 

that is, A is greater than G. 

Also, since a + c is greater than 2 ^ac, 

Jac (a + c) is greater than 2ae ; 

,— . , 2ac 

:. Jac IS greater than — ; — ; 
^ * a + c 

i.e. G is greater than H. 



Examples.— cxlv. 

I. Insert two harmonic means between 6 and 24. 

2 four 2 and 3. 

3 three - and -. 

4- foiir -and—. 



286 ON HARMONICAL PROGkESSlON. 



5. Insert five harmonic means between — 1 and 2~^. 

6 five ^and--. 

7 SIX 3 and — . 

8 n 2x and By. 

9. Tlie sum of three terms of a harmonical series is Yg> *"*! 
the first term is - : find the series, and continue it both ways. 

10. The arithmetical mean between two numbers exceeds 
the geometrical by 13, and tlie geometrical exceeds the har- 
monical by 12. What are the numbers? 

11. There are four numbers a, 6, c, d, the first three in 
arithmetical, the last three in harmonical progression ; show 
that (t : 6 = c : rf. 

12. If X is the harmonic mean between m and n, show that 

_1_ _1_ = J_ 1 

x-tii x-n m n 

13. The sum of three terms of a harmonic series is 11, and 
the sum of their squares is 49 ; find the numbers. 

14. If X, y, z be the //"", 5"*, and r* terms of a h.p., show 
that {r-q)y- + {P -r)xz + {q- p) xy = 0. 

15. If the H.M. between each pair of the numbers, a, b, c 
he in a. P., then b'-, a-, c'^ will be in H.P. : and if the h.m. be in 
H.P., b, a, c will be in H.P. 

16. Show that ^ — ~+ =4, >7, or >10, according as 

c — c — a 

c is the A., G. or H. mean between a and b. 



XXXIIi. PERMUTATIONS. 

402. The different arrangements m respect of order of suc- 
cession wliich can be made of a given number of things are 

called Permutations. 

Thus if from a box of letters I select two, P and Q, I can 
make two permutations of tliem, placing P first on the left and 
then on the right of Q, thus : 

P, Q and Q, P. 

If I now take three letters, P, Q and R, I can make six per- 
mutations of them, thus : 

P, Q, B ; P, R, Q, two in which P stands first. 

Q,P,R; Q,R,P, Q 

R,P,Q; R,Q,P, R 

403. In tlie Examples just given all the things in each case 
are taken together ; but we may be required to find how many 
permutations can be made out of a number of things, when a 
certain number only of them are taken at a tinie. 

Thus the permutations that can be formed out of the letters 
P, Q, and R taken tivo at a time are six in number, thus: 
P,Q; P,R; Q,P; Q,R; R,P; R, Q. 

404. To find the nuvdicr of jJermutations of n different things 
taken t at a time. 

Let a,h, c, d ... stand for n difi'erent things. 

First to find the number of permutations of the n things 
taken two at a time. , 

If a be placed before each of the other things 6, c, d ... of 
which the number is n— 1, we shall have n—\ permutations 
in which a stands first, thus 

ah, ac, ad, 



2^8 PERMUTA TIONS. 

If I be placed before each of the other thiiifjs, a, c, d ... we 
shall have « - 1 permutations in which b stands first, thus : 
ba, be, bd, 

Similarly there will be n- 1 permutations in which c stands 
first: and so of the rest. In this way we get every possible 
permutation of the 71 things taken two at a time. 

Hence there will be n . (n - 1) permutations of n things taken 
two at a time. 

Next to find the number of permutations of the n things 
taken three at a time. 

Leaving a out, we can form (n- 1) . (n — 2) permutations of 
the remaining (n - 1) things taken tivo at a time, and if we 
place a before each of these permutations we shall have 
(«- 1) . (?i- 2) permutations of the n things taken three at a 
time in which a stands first. 

Similarly there will be (n - 1) . (n — 2) permutations of the 
n things taken three at a time in which b stands first : and so 
for the rest. 

Hence the whole number of permutations of the n things 
taken three at a time will be n.(n-l). {n-2), the factors of 
the formula decreasing each by 1, and the figure in the last facto? 
being 1 les^s than the niuiiber taken at a tinu. 

We now assume that the formula holds good for the number 
of permutations of n things taken r—1 at a time, and we shall 
proceed to show that it will hold good for the number of per- 
mutations of n things taken r at a time. 

The number of permutations of the n things taken r—1 at 
a time w iU be 

n.{n-l).(n-2) [„- } (r- 1) - I [], 

tliat is ?i..(?i-l). («.-2) (n-r + 2). 

'Leaving a out we can form {n - 1) . (n - 2) (« - 1 — r + 2) 

permutations of the (n-l) remaining things takrn r — 1 at a 
time. 

Putting a before each of these, we shall have 

(n-l). {n-2) (n-r+l) 

periiuitatiniis of the n things taken r at a time in which a 
stands fir>l. 



PERMUTA TIONS. 289 

So again we shall have (to — l).(n — 2) (?i-r + l) per- 
mutations of the n things taken r at a time in whicli h stands 
first ; and so on. 

Hence the whole numtier of permutations of the n things 
taken r at a time will be 

n.(«-l).(?i-2) (7i-r+l). 

If then the formula holds good when the n things are taken 
r- 1 at a time, it ■will hold 'good when they are taken r at a 
time. 

But we have shown it to hold when they are taken 3 at a 
time ; hence it will hold when they are taken 4 at a time, and 
so on : therefore it is true for all integral values of r* 

405. If the 71 things be taken all together, r = n, and the 
formula gives 

n. (n— 1) . (?i-2) (n — n-l- 1) ; 

that is, n.(n-l).(7i-2) 1 

as the number of permutations that can be formed of n dif- 
ferent things taken all together. 

For brevity the formula 

TO. (71- 1). (71-2) 1, 

which is the same as 1.2.3 to, 

is written 1 77. This symbol is called /a ciorwZ n. 

Similarly \r is put for 1 . 2. 3 r ; 

[r-1 for 1.2.3 {r-\\ 

Ohs. i 7i = n . 1 71 - 1 = n . (ti — 1) . ?! — 2 = &c. 

406. To find the numbei- of jpermutations of n things taken all 
together ivhen certain of the things are alike. 

Let the n things be represented by the letters a, b, c, d 

and suppose that a recurs p times, 

b q times, 

c r times, 

and so on. 

* Another proof of this Theorem may be seen in Art. 475. 
£s.A.l ^ 



290 PERMUTA TIONS. 



Let P represent the whole number of permutations. 

Then if all the p letters a were changed into f other letters, 
different from each other and from all the rest of the n letters, 
the places of these -p letters in any om permutation could now 
be interchanged, each interchange giving rise to a new permu- 
tation, and thus from each single permutation we could form 
1.2 p permutations in all, and the whole nutnber of per- 
mutations would be (1 . 2 ...^) P, that is [p . P. 

Similarly if in addition the g letters h were changed into 5 
letters different from each other and from all the rest of the 7i 
letters, the whole number of permutations would be 

k.l^.P; 
and if the r letters c were also similarly changed, the whole 
number of permutations would be 

ind so on, if more were alike. 

But when the^, g, and r, &c., letters have thus been changed, 
we shall have n letters all different, and the number of permu- 
tations that can be formed of them is \ n (Art. 405). 

Hence P .\p . \q .\r = ?i ; 



\p.\q. [r 



Ex AMPLES. — CXivi. 

1. How many permutations can be formed out of 12 things 
taken 2 at a time ? 

2. How many permutations can be formed out of 16 things 
taken 3 at a time ? 

3. How many permutations can be formed out of 20 things 
taken 4 at a time 1 

\ 4. How many changes can be rung with 5 bells out of 8 ? 

5. How many permutations can be made of the letters in 
the word Examination taken all together \ 
y,^. In how many ways can 8 men be placed side by side ? 



CO MB IN A no MS. igt 



7. In how many ways can 10 men be placed side by side ? 

8. Three flags are required to make a signal. How many 
signals can be given by 20 flags of 5 different colours, there 
being 4 of each colour ? 

9. How many different permutations can be formed out of 
the letters in Algebra taken all together ? 

I o. The number of things : number of permutations of the 
things taken 3 at a time = 1 : 20. How many things are there? 

11. The number of permutations of in things taken 3 at a 
time : the number of permutations of j?i + 2 things taken 3 at 
a time = 1:5. Find m. 

12. In the permutations of a, b, c, d, e, f, g taken all 
together, find how many begin with cd. 

13. Find the number of permutations of the letters of the 
product a^b^c* written at full length. 

14. Find the number of permutations that can be formed 
out of the letters in each of the following words : Conceit, 
Talavera, Calcutta, Proposition, Mississippi. 



XXXIV. COMBINATIONS. 

407. The Combinations of a number of things are the 
diflerent collections that can be formed out of them by taking 
a certain number at a time, without regard to the order in 
which the things stand in each collection. 

Thus the comliinations of a, b, c, d taken tu-o at a time are 
ab, ac, ad, be, bd, cd. 

Here from each combination we could make tico permuta- 
tions : thus ab, ba ; ac, ca ; and so on : for ab, ha are the same 
combination, and so are ac, ca. < 

Similarly the combinations of a, b, c, d taken three at a time 
are abc, abd, acd, bed. 

Here from each combination we could make six permuta 
lions ; thus abc, acb, bac, bca, cab, cba : and so on. 



igi COMBIXATIONS. 



And, generally, in accordance with Art. 405, any combina- 
tion of n things niuy he made into 1 , 2 . 3 ... n permutations. 

408. To fuul tJie number of combinations of n different things 
taken x at a time. 

Let C, denote the nnmher of combinations required. 

Since each conibinatinn contains r things it can be made 
into I r permntations (Art. 405) ; 

.•. the Avliole number of permutations = : r . (7,. 

But also (from Art. 404) the wliole number of permutations 
of n tilings taken r at a time 

— n{n—\) (n-r + 1); 

.-, I r . C, = n (?i - 1) (?i - r + 1) ; 

. ^ _ n{n-\) (?t-r + l) 

409. To show that the number of combinations of n things 
taken t at a time is tlis same as the number taken n — r at a 
time. 

_, n. (n- 1) (7i-r+l) 

^'" 1.2.3 r ' 

and c n,(n-l) \n-in-Hl\ 
1.2.3 (n-r) 

_ n.(n-l) (r+1) 

~ 1.2.3 {n-r) ' 

Hence 

C, _ n.(n-l) (n-r+l) 1.2. 3 (n-r ) 

C^~ 1.2.3 r ^n.{n-l) (r+1) 

n.(?i-l) (n-r+l). (n-r) 3.2.1 

"^ 1.2.3 r. (r+1) (n-l).n 

\n 

= 1. 
That is. O.-'O,^ 



COMBINATIONS. 293 



410. Making r=], 2, 3 r- 1. r, r+ 1 in order, 



--, _ p _ " ''^ — 1 /-I _ 'I 71-1 71-2 



^ ^^71.01-1) (»-r+2) 



1.2 (r-l) 

„ n.(n-X) (7i-r + 2). Oi -r+1) 






1 .2 (r-1). '• 

71 . (n — 1) (7? — r + 1 ) . (71 — r) 



1.2 r.(r+l) 



c;.=i. 

Hence the general expression for the factor connecting Cv, 
one of the set of numbers Cj, Cj, C^i C',, with C^i, 

that which stands next before it, is , that is, 

^^^7^-r + l 
r 

With regard to this factor , we observe 

r 

(1) It is always positive, because 71 + 1 is greater than r. 

(2) Its value continually decreases, for 

7! - r + 1 71+1 



r -1' 



which decreases as r increases. 



11 J- -^ 1^ 

(3) Though continually decreases, yet for several 

•» 

successive values of r it is greater than unity, and therefore 

each of the corresponding terms is greater than the preceding. 

(4) When r is such that '^— is less than unity the cor- 
responding term is less than the preceding. 



294 . COMB IN A T/OXS. 

71 — 7* -*- 1 

(5) If 11 and r be such that '■ — = 1, C, and C^, are a 

pair of equal terms, each greater than any preceding or suLse- 
quent term. 

Hence up to a certain term (or pair of terms) tlie terms in- 
crease, and after that decrease : this term (or pair of terms) is 
the greatest of the series, and it is the object of the next Article 
to determine what value of r gives this greatest term fur )iair 
of terms). 

411. To find the value of t for which the number of combina- 
tions of n thincjs taken r together is the greatest. 

n.(n-l ) ( n -rH-2) 

^r-,- - jf_2 (r-i) 

^ _ n. (n-1) fw-7- + 2 ) (n-r + 1) 

' 1.2 [r-[) * r 

„ _n.(7i-l) (n-r+l) n-r 

^^^ 1.2 r r+l ■ 



Hence, if 0, denote the number of combinations required, 

C C 

j^- and -^ must neither of them be less than 1. 

a n-r+l 

-But Jt— = -y 

Cr r+l 
and rr~ = — • 

C'^i n-r 

vt r+l. T* + l. 

Hence is not less than 1 and is not less than 1, 

r n — r 

or, n — r+ 1 is not less than r and r+l not less than n -r, 

or, n + 1 is not less than 2r and 2r not less than n — l; 

:. 2r is not greater than ?! + 1 and not le.ss than n—l. 

Hence 2r can have only three values, 7i — 1, n, n + 1. 

Now 2r must be an even number, and therefore 

(1) If n be odd, ?! - 1 and 7i + 1 being both even numbers, 
2r may be equal to 7i - 1 or ?» + 1 ; 



COMBTNA rroNs. 295 



n— 1 w+ 1 

(2) If n be even, n-\ and n + 1 being both odd numbers, 
2r can only be equal to n ; 

n 
■■' = 2- 

Ex. 1. Of eight things how many must be taken together 
that the number of combinations may be the greatest pos- 
sible ? 

Here « = 8, an even number, therefore the number to be 

taken is 4, which will give = — - — - — - or 70 combinations. 
1x2x3x4 

KXi 2. If tlie number of things be 9, then the numlier 
9 _ 1 9 -I- 1 
to be taken is — v— or — g— , that is 4 or 5, which will givf 

respectively 

9x8x7x6 



1x2x3x4 
9x8x7x6x5 



, or 126 combinations, and 
or 126 combinations. 



1x2x3x4x5 



Examples. — cxlvii. 

/^ I. Out of 100 soldiers how many different parties of 4 can 
be chosen ] 

( 2. How many combinations can be made of 6 things taken 
' 5 at a time / 

A 3. Of the combinations of the first 10 letters of the alphabet 
/ 'taken 5 together, in how many will a occur ? 

^ /\ 4. How many words can be formed, consisting of 3 cnn- 
sonants and one vowel, in a language containing 19 consonants 
and 5 vowels ? 

5. The number of combinations of n things taken 4 at a 
time : the number taken 2 at a time =15 : 2. Find n. 

6. The number of combinations of n things, taken 5 at 



296 COMBINA TIONS. 



3 

a time, is 3_ times the number of combinations taken 3 at a 


time. Find n. 

. 7. Out of 17 consouants and 5 vowels, how many words 
\ r can be formed, each containing 2 vowels and 3 consonants ? 

. Q 8. Out of 12 consonants and 5 vowels how many words can 
be formed, each containing 6 consonants und 3 vowels ? 

9. The number of permutations of n things, 3 at a time, is 
6 times the number of combinations, 4 at a time. Find n. 

10. How many different sums may be formed with a guinea, 
a half-guinea, a crown, a half-crown, a shilling, and a sixpence ? 

; -^ II. At a game of cards, 3 being dealt to each person, any 
one can have 425 times as many hands as there are cards in 
the pack. How many cards are there ? 

I 12. There are 12 soldiers and 16 sailors. How many dif- 
/ ferent parties of 6 can be made, each party consisting of S 
soldiers and 3 sailors ? 



//, 



13. On how many nights can a different patrol of 5 men be 
drfiughte<l from a corps of 36 ? On how many of these would 
any one man be taken \ 



XXXV. THE BINOMIAL THEOREM. 
POSITIVE INTEGRAL INDEX. 

412. The Binomial Theorem, first explained by 

Newton, is a method of raising a binomial expression to any 
])ower without going through the process of actual multipli- 
cation. 

413. To investigate the Binomial Theorem for a Positive 
Integral Index. 



THE BINOMIAL THEOREM. 29? 



By actual multiplication we can show that 

(x + aj (z + a,) = x^ + («! + a^) X + Oja, 

(x + ai) (x + Oa) (x + Og) =x3 + (a^ + Oj + 03) x^ 

(x + aj) (x 4- a^) (x + 03) (x + a^) = X* + (oj +02 + 013 + 04) i? . 
+ (ttitta + ajCTj + c^a^ + a^a^ + a.ja4 + 0304) x* 
+ (cfi<*2% + ffliffl2«4 + d-fl'^di + a2a3a4) x + a^ajCtja^. 

In these results we observe the following laws : 

I. Each product is composed of a descending series of 
powers of x. The index of x in the first term is the same as 
the number of factors, and the indices of x decrease by unity 
in each succeeding tenu. 

II. The number of terms is greater by 1 than the number 
of factors. 

III. The coefficient of the _^rs< term is unity. 

of the second the sum of a^, a.^, tij ... 
of the third the sum of the products of 

%, rtj, rt3 ... taken two at a time. 
of the/o?t?-;/i the sum of the products of 
Oj, «2, ^3 ... taken three at a time. 
And the last term is the product of all the quantities 

«1, «2> «3 

Suppose now this law to hold for 7i — 1 factors, so that 

(x + tti) (x + aj) (x + «3) (x + a„_i) 

= x"-i + S'l . rc"-2 + S^ . x"-^ + ,^8 . x"^+ + S„_i, 

where .S\ = a^ + aj + 03 + . . . + a„_i, 

that is, the sum of aj, a.,, 03 ... a„_i, 

Sf=aja.2 + a^a.^ + a^Oj + . . . + aia„_i + a„a„_i + ... 

that is, the sum of the products of dj, a^, a^ ... a.»_i, 
taken two at a time. 



298 THE BT.VOMIAL THEOREM. 

S3 = a^a^a-i + aiCiM^ + . . . + a^aM^^i + aia^a„_y + ... 

that is, the sum of the products of Oj, aj...a^„ 
taken three at a time, 



that is, the product of a^, a^, 0.3 ... dn-i- 
Now multiply both sides hy x + a„. 
Then 
{x + ai)(x + a.,) ... (.T + a„_i) {x + a„) 

=x" + Si X"-' + ,S', X"-- + S3 x"-^ + ... 

+ a„ x"~^ + a„Sj x"'^ + a„S.^ x"^^ + . . . + a,S„ _i 
=x'' + {Si + a„) x"-i + (S3 + a„Si) x"'"^ 

+ {S3 + a„S.,) x"-^ +... + a„S„_i. 

Now Si + a„ = ai + a„ + a3 + ... +a„_i + a„ 

that is, the sum of Oj, a.^, 0-3... a„, 

/Sj + a„Si = S^, + rr„ (f?! + Oj + . . . + a._i), 

that is, the sum of the products of a^, aj,..a,„ 
taken two at a time, 

Sg + a„S.2 = S3 + a„ {aia.2 + a^a^ +...), 

that is, the sum of the products of a^, aj...a,, 
taken three at a time, 

that is, the product of Oi, a,, ^3 ... o,. 

If then the law holds good for n-l factors, it will hold good 
for n factors : and as we have shown that it holds good up to 4 
factors it will hold for 5 factors : and hence for 6 factors : and 
so on for any number. 



THE BINOMIAL THEOREM. ig^ 



Now let each of the n quantities a^, a^, a^... a„he equal to 

a, and let us write our result thus : 

{x + a^) {x + a.^) ...{x + a„) ^x' + Ai . x"~* + ^2 . x"-'+ ... +A^. 

The left-hand side becomes 

{x + a) {x + a)...{x + a) to n factors, that is, {x + a)'. 

And on the right-hand side 

Ai = a + a + a+ ...to n terms = ?;«, 

A^ = a^ + a^ + a^+ ...to as many terms as are equal to the 
number of combinations of n things taken two at a time, that 
. n .(n-l) 



. _ n.{n-l) 

.. A^- ^^ .a, 



A3 = a^ + a^ + a^+ ...to as many terms as are equal to the 
number of combinations of n things taken three at a time, that 
. n.{n-l) .{n-2) 



1.2.3 



_^. (n-l). (n-2 ) 
^' 17273 •"'' 



A„ = a . a . a ...to n factors = a". 
Hence we obtain as our final result 

/ N„ - - 1 n . (n - I') „ . , 
{x + a)" = x" -t- Tiaa;""' -| ^^ — ^-■' a-x*-^ 

n.(n-\) . (n-2) , .^ 

1.2.3 -r...-rt* 

414. Ex. Expand (x + a)6. 

Here the number of terms will be seven, and we have 

^6.5.4.3 ,2,6.5.4.3.2 , . 
+ 1727^74 "^^17273:476 " ^^^" 

— x^ + Qaufi + 15aV -I- 20aV -i- 15a*x*-i- 6a^x + afi. 



300 THE BINOMIAL THEOREM. 



Note. The coefficients of terms equidistant from the end 
and from the beginning are the same. The general proof of 
this will be given in Art. 420. 

Hence in the Example just given when the coefficients of 
font terms had been found those of the other three might have 
beeu written down at once. 



Examples.— cxlviii. 

Expand the following expressions : 

I. (a + x)*. 2. (6 + c)8. 3. (a + 6)^ 

4. (x + i/)8. 5. (5 + 4a)*. 6. {a^^hcf. 

415. Since 

1 n . (« - 1) , , 
(,-c + a) " = a:" + naT^"'^ + -^ — „— ' . aV'* 4- ... + a", 

if we put x= 1, we shall have 

(1 +a)" = l +na + — Y~a~~ -^ "^ ••• +<* • 

416. Every binomial may be reduced to such a form that 
the part to be expanded may have 1 for its first term. 

Thus since x + a = x(l+-Y 

(x + a)- = x"(l+^); 

and we may then expand (l + - j and multiply each term of 

the result by x". 

Ex. Expand (2.c + 3y)\ 
(2x + 32/)6=(2x)''.(l+||y 

, 5-4.3.2 /3i/y /3yy| 
"^1.2.3.4-\2x/ ■^V2x/ I 



THE BINOMIAL THEOREM. -^oi 



= 32x5 + 240x*y + 720x3i/2 + loSOxy + SlOxi/^ + 243?/. 

417. The expansion of (x — a)" will be precisely the same as 
that of {x + a)", except that the sign of terms in -which the odd 
powers of a enter, that is the second, fourth, sixth, and other 
even terms, will be negative. 



Thus 



[X - a)" = X" — 7utx"~' + — ^ — - — . ah:"'* 



TO ■ (w - 1) . (?i - 2) 
1.2.3 



for (x — a)'=\x + (-a)\' 

^x' + ni-a) x'-' + ^_lC^_rLl) ( _ afx'-' + &c. 

Ex. Expand (a - c)*. 

/ ^^ ^ r4 5-432 5.4.3,, 5.4.3.2^ ^ 
(a - cf = a» - 5a*c -- j— g ~ r~2"~3 "^ 12 3 4 ~ 

= a^ - 5a*c + 10u\'2 - lOa^c^ + 5ac* - c^ 

Examples. — cxlix. 

Expand the following expressions : 

I. (a-x)«. 2. (b-cy. 3. (2x-3j/)». 

4. (l-2x)5. 5. (l-x)io. 6. (a^-by. 

418. A trinomial, as a + b + c, may be raised to any power 
by the Binomial Theorem, if we regard two terms as one, thus : 

+^4^*-(«-»)-'-'*-^ 



302 THE BINOMIAL THEOREM. 

Ex. Expand (l+x + a;2)3. 

(l+x + x-)^ = (l+x)3 + ;3(l+a;)'''.a;2 + ^-|(l+x).x* + x« 

= (1 + 3ic + 3a;''^ + x3) + 3 (1 + 2x + a;2) o-? 

+ 3(l+u,).c-' + x« 
= 1 + 3a; + 3a;2 + ic^ + 3:c2 + 6x3 + 3x^ + 3a;'» 

+ 3XS + .7-'' 
= 1 + 3x4-6x2 + 7x3 + 6x* + 3xS + x«, 



Examples.— cl. 

Expand the following expressions : 
I. (ft + 26-c)-'. 2. (l-2x + 3x2)3. 3. (x3_a.2 + j.-^3_ 

4. (3x^ + 2x« + l)3. 5. ^x + 1--). 6. (a^ + 6^-c^); 



419. To jind the r"" or general term of the expansion of 
(z + a)". 

We have to determine three things to enable us to write 
down the r"" term of the expansion of (x + a)". 

1. The index of x in that term. 

2. The index of a in that term. 

3. The coefficient of that term. 

Now the index of x, decreasing by 1 in each term, is in the 
r* term ?i — 7-+ 1 ; and the index of a, increasing by 1 in each 
term, is in the r"" term r— 1. 

For example, in the tliird term 

the index of x is n — 3 + 1, that is, n-2 : 
the index of a is 3 - 1, that is, 2. 

m assigning its proper coefficient to the ;•"' term we have to 
determine tlie last factor in the denominator and also in the 
numerator of the fraction 

n.{n-l).{n-2).(n-Z) 

1.2.3.4 



TffE BINOMIAL THEOREM. 303 

Now the last factor of the denominator is less by 1 than the 
number of the term to which it belongs. Tlius in the 3'* term 
the last factor of the denominator is 2, and in the ?•"■ term the 
last factor of the denominator is r— 1. 

The last factor of the numerator is formed by subtracting 
from 7i«the number of the term to which it belongs and adding 
2 to the result. 

Thus in the S"* term the last factor of the numerator is 

71-3 + 2, that is ?i— 1 ; 

in the 4* 71- 4 + 2, that is 9i-2 ; 

and so in the j-"" ?i - r + 2. 

Observe also that the factors of the numerator decrease by 
unity, and the factors of tlie denominator increase by unity, so 
that the coetticient of the r"" term is 

n.(n-l). {n - 2) {n - r + 2) 

1.2.3 (r-1) ~"' 

Collecting our results, we write the r* term of the expansion 
of (x + a)" thus : 

n.(7i-l).(n-2) in-r + 2) ' 

1.2. 3 (r-1) •" ••" • 

Obs. The index of a is the same as the last factor in the 
denominator. The sum of the indices of a and x is n. 



Find 



Examples. — cli. 

The 8*'term of (1+x)". 
The 5'" term of (a^ - ft^)". 
The 4'" tf rm of (« - 6)i«>. 
The 9"" term of (2a6-cd)». 
The middle term of (a — 6)^^. 
The middle term of (a^ + h°)^. 
The two middle terms of (a - by^. 
The two middle terms of (a + x)^. 



304 THE BINOMIAL THEOREM. 

9. Show that the coefficient of the middle term of 
1.3.5 (4n-l) 



(a + as)*" is 2"" x 



1.2 .3 2n 



10. Show that the coefficient of the middle term of 
(a + xr- is 2-^ X ^-2!^) (^" + ^) (^"- 1) (^^-^ 1) 



1.2 



420. To &how that the coefficient of the r"" temi from the 
btyianing of the expansion of (x + a)" is identical with the coeffi- 
cient of the r"^ term from the end. 

Since the number of terms in the expansion is n+ 1, there 
are n+l—r terms before the r"" term from the end, and there- 
fore the r^'term from the end is the (n — r + 2)'^ term from the 
beginning. 

Thus in the expansion of (x + a)*, that is, 

X* + 5ax* + lOa^x^ + lOa^x^ + 5a*x + a°, 

the 3rd term from the end is the (5 - 3 + 2)"', that is the 4"" term 
from the beginning. 

Now if we denote the coefficient of the r*^ term by (7„ 
and the coefficient of tbe (?) -r + 2)"' term by C«_^2, 
we have 

n.{n-l) (n-r + 2) 



C,= 



C'.-H-l — 



1.2 (r-1) 

_ n.(n-l) {n-(n-r + 2) + 2{ , 

1.2 (71-7- + 2-1) 

n. (n- 1) r 



1 .2 (n-r+l) 

Hence 

C, n.(n-l) (n-r + 2) 1. 2 (n-r+l) 

CZ^,~ 1.2 (r-l) "" n.(n^) r 

n.(Ti-l) (n- r + 2 ).(n- r+ 1) 2.1 

- 1.2 (r- 1) .r (n - 1). n 

In , . , , 

= 4^:= = 1, which proves the proposition, 
n 



THE BINOMIAL THEOREM 305 



421. To find the greatest term in the expansion of (x + a)% n 
being a positive integer. 

Tlie r* term of the expansion {x + a)' is 

n.{n-l) {n-r + 2) , 

1.2 (r-1) 

The (r + 1)"" term of the expansion (x + a)" is 

n.(n-l) (n-r + 2). jn-r + l) 

1.2 (r-l).r 

Hence it follows that. we obtain the (r + l)* term by multi- 
plying the r"" term by 

n - r + I a 
r ' x' 

When this multiplier is first less than 1, the r"" term is the 
greatest in the expansion. 

Now . - is first less than 1 

r X 

when na-ra + a is first less than rx, 

or na + a first less than rx + ra, 

or r (x + a) first greater than a {n + 1), 

r first greater than — ^^ -. 



or 



x + a 



re \, 1 *. <^ ('"' + ^) i.u n-r+1 a . , ,, 

If r be equal to , then .- = 1, and the 

x+a « r X 

(r + l)"" term is er^ual to the r*, and each is greater than anv 

other term. 

Ex. Find the greatest term in the expansion of {4 + ay, 
when « = „. 

Here -±±^^£11.)^^^^^.^^. 

X + a ^3 11 11 ^' 

^ + 2 T 

The first whole number greater than 2^ is 3, therefore th« 
greatest term of the expansion is the .3rd. 

[s.A.j 17 



2o6 THE BINOMTAL THEOREM. 

422. To find the sum of all the coefficients in the expansion 
of{l+x)\ 

a- /^ \, 1 n . (71— 1) „ 

Since (1 +x)" = l +71X+ — - — ^— V + 

n.(?z — 1) , 
-<-— ^-g- V-' + ruf^ + af 

putting x = l, we get 

_. , n.(n-i) n.(n-l) 

2"=l + n + — 1 2~ '*' — 1 "2" ' 

or, 2" = the siiiu of all the coefficients. 

423. To shon: that the sum of the coefficients of the odd term 
in the expansion of (1 + a;)" is equal to the smn of the coeffi^yients 
of the even terms. 

Since 

,, , , 7i,(n-l) , n. (n- l).(n — 2) , 
(l+x)-=l + T?x + — j-^'^ + 123 ^"^ 

putting x= - 1, we get 

(1-1) _l-n + -j--2~ j-2-3 + 



or. 



,|„^ .,.(,-,.(.-., ^ } 

= sum of coefficients of odd terms - sum of co- 
efficients of even terms ; 

.". sum of coefficients of odd terms = sum of coefficients of 
even terms. 

Hence, by the preceding Article, 

2" 
sum of coefficients of odd terms = — = 2»~^; 

2" 
sum of coefficients of even terms = g- = 2"~*. 



XXXVI. THE BINOMIAL THEOREM. 
FRACTIONAL AND NEGATIVE INDICES. 

42-L We have shown that when m is a positive integer, 

N_ T m.(m-\) , 
(l+x)" = H-mz+ l^-r — ^ X-+ 



We have now to show that this equation holds good when 

. . , . 3 . . 

TO is a positive fraction, as -, a negative integer, as - 3, or a 

3 

negative fraction, as - --. 

We shall give the proof de-vised by Euler. 

425. If m be a positive integer we know that 

,, ,_ , m.(m.— 1), m . Cm-1) . lm-2) , 
(l+a;)"=i-r7nj;+ p-^ — -3^ + ^ , ^ g ^x^+ 

Let us agree to represent a series of the form 
m . (m — 1) „ 

1+77!X+ p2 ^ + •' 

by the symbol /(m\ irhatever the valw of m maii be. 

Then we know that when m is a positive integer 

(l+x)"'=/(m) ; 

and we have to show that, also, when m is fractional or 

negative 

■ ■' (l+.r;;-=/(»0. 

o- J-/ \ -, - m.(m-l) , 

Sine* f{m) = l + 'mx-\ ;-- ■ z-+ 

/(??) = 1 + nx + — ~-^ — X-+ 



3o8 THE BIAOMTAL THEOREM. 

If we multiply together the two series, we shall obtain an. 
expression of the form 

1 + aic + &x2 + cx^ + dx*+ 

that is, a series of ascending powers of x in which the coeffi- 
cients a, 6, c are formed by various combinations of 

m and n. 

To determine the mode in which a and h are formed, let ua 
commence the multiplication of the two series and continue it 
as far as terms involving a;^, thus 

,, , _ TO . (m - 1) 
/(m) = l+mx4-- — ^^—^ — --x'+ 

/(n) = l +«a; + f-^" 



f{m) xf(n) = l+mz+ — pg"" ■*" 

+ nx + mnx^+ 

n.(n-l) , 
^ 1.2 



/ s (m.(m — 1) 
l + (m + n).i + | ^-g-^ 



n . (u — 1) , , 



1.2 ( 



Comparing this product with the assumed expression 

l+ax + bx- + cz^ + dx* + 

we see that a = m + n, 

. , to.(to-I) n.(n—l) 

and b = — i"^ +mn + — ] o 

m^ -m + 2mn + 7i- — n 
"" T72 

(m + n) . (m + n— 1) 
" 172 • 



P/^ACIYOXAL AND NEGATIVE INDICES. 309 

Similarly we could show hy actual viultiplication that 

(m + n) . (m + n — l). {m + n-2) 
*" 1T273 ' 

,_ (m + 7i) . (m + n — 1) . (m + n — 2) . (m + n - 3) 

TTaTsTi • 

Thus we might determine the successive coefficients to any 
extent, but we may ascertain the law of their formation by the 
following considerations. 

Tlnj forms of the coefficients, that is, the way in which m 
and n are involved in them, do not depend in any way on the 
values of m and n, but will be precisely the same whether m 
and 71 be positive integers or any numbers whatsoever. 

If then we can determine the law of their formation when 
m and n are positive integers, we shall know the law of their 
formation for all values of m and n. 

Now when m and n are positive integern, 
/(m) = (l+x)", 
/(u) = (l+x)"; 

" /(™) x/(w) = (1 + x)" X (1 + x)- 
= (H-a;)"'+- 

, , , (m + n) .{m + n -\) , 
= l + {m + n)x + ^ -Y-a — V+ ... 

=/(m + n). 

Hence we conclude that whatever he the values of m and n 

f(m)x f{n)=f{m + n). 

Hence f{m + n+p)=f{m).f{n+p) 

=f(m).fin).f{p), 
and »o generally 

/{m + n+p + ...)=f{m).f{n)./{p)... 



3 to THE BLVOM/AL 'IHEOREM. 



Xuw let m = n=p= ... =j-, h ami k being positive integ(;rs, 



then 

^h h h 



^fh h h ^ , \ 

/i^^ + j + ^+.-. tot terms j 

=/a)./(J)./©...to. facto. 



h (h \ 

, h k-\k V, 
k 1.2 

which proves the theorem for a positive fractional index. 

Again, since f{m).f{n)=f{m + n) for all values of m and n, 
let 71= -VI, then 

/(m)./(-m) =/(«!-?») 

=/(0). 

, . , mJm-l) ., 
Now the series l + mx+ -^—h ^'+--' 

becomes 1 when vi = 0, that is,/(0) = l ; 
.-. /(m)./(-770=l; 

•■■•^(-™^=/(7^=(TT^-=^^ + ^>'^' 

.-. (l+x)-=/(-7n) 

■■ / \ - 771 ( - 771 - 1) 2 , 
= l+(-77l)x+ j—g -X2+ ... 

which proves the theorem for a negative index, integral or 
fractional. 



FRACTIONAL AND NEGATIVE INDICES. 3:1 



426. Ex. Expand (a + x)2 to four tenna. 



.a'' .J? ... 



1 



1.2.3 
3 



111 4. -T ft 5 

z z o 

= a2+ g + — ^ 

2a 2 8a2 16«- 



Or we might pmceed thus, as is explained in Art. 416. 

a-0 .= ia-')a-^) ^ 



1 /I 



= aMl + ^ ^^ 



2 a 1.2 o^ 

= a^ -'1+ „H , ... ^ 

( 2a 8a2 16a3" j 

i , X X2 X2 

2a2 8a^ IGa^ 



1.2.3 



Examples. — clii. 

Expand the following expres-sions : 
I. (1 + x)2 to five terms. 7. (1 -x-)^ to five terms. 



2. (1 +a)-^ to four terms. 

3. (a -f- x)^ to five terras. 
4 (1 + 2x)2 to five terms. 

5. ( a + —^ H to four terms 

1 i i 

6. (o^ ^x*;-' to four terms. 



8. (1 - a^)^ to four terms. 

9. (1 — 3x)^ to four terms. 

10. (x- — -^y^to four terms. 

1 1. (1 — x)* to four terms. 

12. ( o" "" g ) ^^ three terms. 



312 THE BINOMIAL THEOREM. 

427. To exjxnid (1 +x)~'. 

(1 +x)- = l + (-«). x + ^1^^"^^ X* 

-n.(-n-l).{-n-2) 

1.2.3 

= 1 na; + -j--2~x ^ ^ ^ .x-+ 

the terms being alternately positive and negative. 
Ex. Expand (1 +x)~^ to five terms. 
/, , x, , o 3.4, 3.4.5, 3.4.5.6 . 

= 1 - S.c + 6x2 _ 10x3 4- 15x* - ... 

428. To expand (1 - a;)—. 

_ -n{ -n-l)(-n--2) . 
1.2.3 

n.(n + l) „ n. (?i + l)(n + 2) , 

the tenns being all positive. 

Ex. Expand (1 -x)~^ to five terms. 

„ ,, , „ 3.4, 3.4.5, 3.4.5.6, 

(l-x)-3=l+3x + — .2+^_^_^^+____.^+... 

= l + 3x + 6x2 + 10x3+ 15x*+ ___ 

Examples.— cliii. 
Expand 

1. (1 +a)~2 to five terms. 4. ( 1 - 5) to five terms. 

2. (1 - 3x)-i to five terms. 5. (a2-2.r)~^ to five term*. 

3. ( 1 — T ) to four terms. 6. (a^ — x^)""" to lour tenus. 



THE BINOMIAL THEOREM. 313 



429. To ex'pand{\+x)-'n, 

11 \ n / 



n 
2 "*" 



_l(_l_l)(_I_2) 



1.2.3 



x'^^ ... 



n 2?i^ fan-* 



Examples. — cliv. 
Expand 

I. (1 + x2)~2 to five terms. 4. (1 + 2a;)~2 to five terms. 

3. (1 — c'^)~2 to five terma. 5. (ft^ + x*)'^ to four terms. 

2 _i 

3 (a^ + is*) ^ to four terms. 6. (o' + x^) ' to four terms. 



430. Observations on the general expression for the term involving 
X' in the exjjayisions (1 + x)" and (1 - x)". 

The general expression for the term involving x', that is the 
(r+ l)"" term, in the expansion of (1 +x)'' is 

n.(n-l)...(7i-r+l) 
1.2 r ' •'■'■ 

From this we must deduce the form in all cases. 

Thus the (r+ l)* term of the expansion of (1 -x)" ifl found 
liy changing x into ( — x), and therefore it is 

n.(n-l)...(7i-r+l) 

r72~;:~r -^""^^ 

*"' V^^ 1.2 r ' 



314 THE BINOMIAL THEOREI.I. 

If n be negative and = — m, the (r+ 1)* term of the expan- 
sion of (1 +x)" is 

( — m)(-m— 1)... -m — r + 1) , 
1.2 r *' 

(-1)'. j?n..(m+l)...(m + r-l)jx'. 

°'"' 1727::::^.:::::::^^ 

If n be negative and = -m, the (r+ 1)* term of the expan- 
sion of (1 + x)" is 

(-l)-. )m.(m+l )...(m + r-l) { _ 

1.2 r "^ ^' 

m. (m+ 1) ... Cwi + r- 1) , 
1.2 ....r -•^• 



Examples. — civ. 

Fiuil the r"' terms of the following expansions : 
1. (l+a-y. 2. (l-x)i2. 3. {a-x)\ 4. (5x + 2!/)». 

5. (1+x)-^ 6. (l-3x)-^. 7. (l-a;)"-. 

9. (l-2x)~^. 10. (a2-x2)"5. 

1 1. Find the (r + 1)"" term of (1 -x)-^. 



'* S. (a + x)^'. 



12. Find the (r+ 1)* term of (1 -4s;) -. 

13. Find the (r + 1)'*" term of (1 + x)''. 

14. Show that the coefficient of x'+* in (1 +x)''+^ is the sum 
of the coefficients of x' and^ j;"^* in (1 +x)", 

I ;. What is tlie fom th term of ( « — ) - ? 



• 16. "What is the tiftli term of (a^-i'-)^ ? 

17. Wliat is the ninth term of (a2 + 2x-)- ? 

18. What is the tenth term of (a + 6)"" ? 

ig. What IS the seventh term of ya-^b)" \ 



THE BINOMIAL THEOREM. 315 

431. The following are examples of the application of the 
Binomial Theorem to the approximation to roots of numbers. 

(1) To approximate to the square root of 104. 
V104 = ^/(lOO + 4) = 10 ( 1 + ~-f 

=io|i+i.-i-+tk:'J.(Ay 

( 2 100^ 1.2 VlOO/ 



^2V2 /V2 /.(J \% 



ia-')a- ^) 

= 10|l4-A__?__ + _J . 1 

( 100 10000 1000000 [ 

= 10-19804 nearly. 

(2) To approximate to the fifth root of 2. 

4/2 = (1 + 1)^ 

= 1 + 1 + 1. l(l-l) + l.l.Cl-l)(l-2U... 
5^2 5\5 / 6 5 \5 / \5 / 

^j^l_^ _3 21_ 

5 25 "^250 2500"^'" 

= 1 + -^ + nearly 

25 2500 

= 1-1236 nearly. 

(3) To approximate to the cube root of 25. 

4/25=4/(27-2) = 3il-.-U. 

Here we take the cube next ahove 25, so as to make the 
second term of the binomial as small as possible, and then 
proceed as before. 



Examples.— clvi. 

Approximate to the following roots : 
I. ;/31. 2. ^108. 3. 4^260. 4. 4^31. 



XXXVII. SCALED OF NOTATION. 

432. The sjinbols employed in our common system of 
Arithmetical Notation are the nine digits and zero. These 
digits when written consecutively acquire local values from 
their positions ■s\-ith respect to the place of imits, the value of 
every digit increasing ten-fold as we advance towards the left 
hand, and hence the number ten is called the Radix ot the 
Scale. 

If we agree to represent the number ten by the letter t, a 
number, expressed according to the conventions of Arithmetical 
Notation by 3245, would assume the form 

3<3 + 2?2 + 4« + 5 

if expressed according to the conventions of Algebra. 

433. Let us now suppose that some other number, as^re, 
is the radix of a scale of notation, then a number expressed in 
this scale arithmetically by 2341*will, if five be represented by 
/, assume the form 

2/3 + 3/2-r4/+l 

if expressed algebraically. 

And, generally, if r be the radix of a .scale of notation, a 
number expressed arithmetically in that scale by 6789 will, 
when expressed algebraically, since the value of each digit 
increases r-fold as we advance towards the left hand, be repre- 
sented by 

67-' + 7r* + 8r + 9. 

434. The number which denotes the radix of any scale will 
be represented in that scale by 10. 

Thus in the scale whose radix is five, the number fire will 
be represented by 10, 



Scales of xo ta tion. 3 17 

In the same scale seven, being equal to five + two, ■will 
therefore be represented by 12. 

Hence the series of natural numbers as far as tv:enty-j\,vt will 
be represented in the scale whose radix is five thus : 

1, 2, 3, 4, 10, 11, 12, 13, 14, 20, 21, 22. 23, 24, 30, 31, 
32, 33, 34, 40, 41, 42. 43, 44, 100. 

435. In the scale whose radix is eleven we shall require 
a new symbol to express the number ten, for in that scale the 
number eleven is represented by 10. If we agree to express 
ten in this scale by the symbol t, the series of natural numbers 
as far as twenty-three will be represented in this scale thus : 

1, 2, 3, 4, 5, 6, 7, 8, 9, t, 10, 11, 12, 13, 14, 15, 16, 17, 

18, 19, It, 20, 21 

436. In the scale whose radix is tioelve we shall require 
another new symbol to express the number eleven. If we 
agree to express this number by the symbol e, the natural 
numbers from nine to thirteen wdll be represented in the scale 
whose radix is twelve thus : 

9, t, e, 10, 11. 

Again, the natural numbers from twenty to twenty-five will 
be represented thus : 

IS, 19, If, le. 20, 21. 

437. The scale of notation of which the radix is two, is 
called the Binary Scale. 

The names given to the scales, up to tliat of which the 
radix is twelve, are Ternary, Quaternary, Quinarj-, Senarv, 
Septenary, Octonary, Nonary, Denary, Undenary and Duo- 
denary. 

438. To perform the operations of Addition, Subtraction, 
Multiplication, and Division in a scale of notation whose index 
is r, we proceed in the same way as we do for numbers ex- 
pressed in the common scale, with this difference onlv, that r 
must be used where ten would be used in the common scale : 
which -will be understood better by the following examples. 



3 1 8 SCALlLS OF NOTA TION. 

Ex. 1. Find the sum of 4325 and 5234 in the senary scale. 
4325 
5234 

the sum -14003 

which is ohtained by adding the numbers in vertical lines, 
carrying 1 fur every six contained in the several results, and 
set' ing do%vu tlie excesses above it. 

Thus 4 units and 5 units make nine units, that is, six units 
together with 3 units, so we set down 3 and carry 1 to the 
next column. 

Ex. 2. Find the difference between 62345 and 53466 in 
the septenarv scale. 

62345 
53466 

the difference = 5546 



which is obtained by the following process. We cannot take 
six units from five units, we therefore add seven units to the 
five units, making 12 units, and take six units from twelve 
units, and then we add 1 to the lower figure in the second 
column, and so ou. 

Ex. 3. Mul iply 2471 by 358 in the duodenary scale. 
247 1 
358 



17088 
e < e 5 
7 193 

8333 18 

Ex. 4. Divide 367286 by 8 in the nonary scale. 
8 ; 367286 
~42033 

The following is the process. We ask how often 8 is contained 
in 36, which in the nonary scale represents thirty-three units ; 
the answer is 4 and 1 over. We then ask how often 8 is con- 
tained in 17, which in the nonary scale represents sixteen units; 
the answer is 2 nnd no remainder. And so for the other digits. 



SCALES OF NOTAtiON. 



M$ 



Ex. 5. Divide 1184323 hy 589 in the duodenary scale. 

589; 1184323 ('2483 
e5fi 

22f3 
KeO 



3e32 
39i!0 



1523 
1523 



Ex. 6. Extract the square root of 10534521 in Vhe senary 
scale. 

10534521 ( 2345 
4 



43 


253 
213 


504 


4045 
3224 


5125 


42121 

42121 



I 

2 

3 

4 

5 
6 

7 
8 

9 

lo 

scale. 



Examples. — clvii. 

Add 23561, 42513, 645325 in the septenary scale. 
Add ,3074852, 4635628, 1247653 in the nonary scale. 
Subtract 267862 i'roni 358423 in the. nonary scale. 
Subtract 124321 from 211010 in the quinary scale. 
Multiply 57264 by 675 in the octonary scale. 
Multiply 1456 by 6541 in the septenary scale. 
Divide 243012 by 5 in the senary scale. 
Divide 3756025 by 6 in the octonary scale. 
Extract the square root of 25400544 in the senary scale. 
Extract the square root of 56898(1 in the duodenary 



^io SCALES 01' NorATiohr. 



439. To transform a given integral number from one scale to 
another. 

Let N be the given, integer expressed in the first scale, 

r the radix of the irw scale in which the number is to 
be expressed, 

a, b, c in,}:!, q tlie digits, n + l in number, expressing 

the number in the Jiew scale ; 
so that the number in the new scale will be expressed thus : 
ar" + br"""^ + cr"~- + + mf- + jn- + q. 

We have now from the equation 

J\r= ar" + 6r"~^ + cr""2 ^ ^ ^,ij-2 ^ pj. ^ q^ 

to determine the values of a, 6, c m, p, q. 

Divide N by r, the remainder is q. Let A be the quotient : 

Uien 

A = ar"-^ + br"-- + cr"^^ + +mr+p. 

Divide A by r, the remainder is p. Let B be the quotient ; 

then 

B = ar''-^ + br"-^ + cr"-^+ +m. 

Hence the 
iirst digit to the right of the number expressed in the 

new scale is q, the first remainder ; 

second p, the second remainder ; 

third m, the third remainder ; 

and thus all the digits may be determined. 

Ex. 1. Transform 235791 from the common scale to the 
scale whose radix is 6. 



6 


235791 


6 


39298 remainder 3 


6 


6549 remainder 4 


6 


1091 remainder 3 


6 


181 remainder 5 


6 


30 remainder 1 


6 


5 remainder 



I remainder 5 

The number required is therefore 5015343. 



SCALES OF NOTATIOiY. 



32i 



The digits by which a number can be expressed in a scale 

whose radix is r will bel, 2, 3 t- 1, because these, with 0, 

are tiie only remainders which can arise from a division in 
which the divisor is r. 

Ex. 2. Express 3598 in the scale whose radix is 12. 



12 
12 
12 
12 



3598 



299 remainder t 



24 remainder e 



2 remainder 

remainder 2 

.•. the number required is l^&i. 

440. The method of transforming a given integer from one 
scale to another is of course applicable to cases in which both 
scales are other than the common scale. We must, however, 
be careful to perform the operation of division in accordance 
with the principles explained in Art. 438, Ex. 4. 

Ex. Transform 142532 from the scale whose radix is 6 to 
llie scale whose radix is 5. 



5 


142532 




5 


20330 


remainder 2 


5 


2303 


remainder 3 


5 


300 


remainder 3 


5 


33 


remainder 3 


5 


4 


remainder 1 







remainder 4 



The required number is therefore 413332. 



Examples. — clviii. 

Express 

1. 1828 in the septenary scale. 

2. 1820 in the senary scale. 

3. 43751 in the dnodenarv scal(». 

rs.A.i 



SCALES OF NOTATIOK. 



4. 3700 in the quinary scale. 

5. 7631 in the binary scale. 

6. 215855 in the duodenaxy scale. 

7. 790158 in the septenary scale. 

Transform 

8. 34002 from the quinary to the quaternary scale. 

9. 8978 from the undenary to the duodenary scale. 

10. 3256 from the septenary to the duodenary scale. 

1 1. 37704 from the nonary to the octonary scale. 

12. 5056 from the septenary to the quaternary scale. 

13. 654321 from the duodenary to the septenary scale. 

14. 2304 from the quinary to the undenary scale. 

441. In any scale the positive integral powers of the num- 
bei which denotes the radix of the scale are expressed by 
10,100, 1000 

Thus twenty-five, which is the sqiiare of five, is e.xpressed in 
the scale whose radix is live by 100: one hundred and twenty- 
five will be expressed by 1000, and so on. 

Generally, the ?;"' power of the number denoting the radix 
in any scale is exjjressed by 1 followed by n cyphers. 

The highest number that can be expressed byj:^ digits in a 
scale whose radix is r is expressed by ;•'' - 1. 

Thus the highest nuni\)er that can be exjjressed by 4 digits 
in the scale whose radix is five is 

10^ - 1, or 10000 - 1, that is 4444. 

The least number that can be expressed by 2' digits in a 
scale whose radix is r is expressed by j-^-^ 

Thus the least numlier that can be expressed by 4 digits in 
the scale whose radix is five is 

JO*-' or 103, tiij^t is 1000, 



SCALES OF NO TA TION. 323 

442. In a scale whose radix is r, the sum of the digits of 
an integer divided by (»■- 1) will leave the same remainder as 
the integer leaA^es when divided by r — 1. 

Let iV be the number, and suppose 

Then 

JV=a(r''-l) + 5(r''-i-l) + c(r"-2-l)4- ... +m(r2- l)+p(r- 1) 
+ jrt + 6 + c+ +m+j:> + 7{. 

Now all the expressions r" - 1, r""^ — 1 r-- 1, r— 1 are 

divisible by r - 1 ; 

N . ^ a + b + c+ m + p + q 

.. ;-= an integer +- :; — - ; 

r— 1 ° r— 1 

which proves the proposition, for since the quotients differ by 
an integer, their fractional parts must be the same, that is, the 
remaindt^ra after division are the same. 

Note. From this proposition is derived the test of the 
accuracy of the result of Multiplication in Arithmetic by cast- 
ing out the nines. 

For let A = Qm + a, 

and B = ^n + h ', 

then AB=Q{Qmn + an + 6m) + ah ; 

that is, AB and ah wlien divided by 9 will leave the same 
remainder. 

Radical Fraction?. 

443. As the local value of each digit in a scale whose radix 
is r increases 7'-fold as we advance from right to left, so does 
the local value of each decrease in the same proportion as we 
advance from left to right. 

If then we affix a line of digits to the right of the units' 
place, each one of these having from its laosition a value 
one-r"" part of the value it would have if it Avere one place 
further to the left, we shall have on the right hand of the 
units' place a series of Fractions of which the denominators 



324 SCALES Of- NOTAlIOAr. 

are successively r, r''-, r^, , while the immerators may be 

any numbers between r— 1 and zero. These are called 
Eadical Fractions. 

In our common sy.^tem of notation the word Radical is 
replaced by Decimal, because ten is the radix of the scale. 

Now adopting the ordinary sj'stem of notation, and markins^ 
the place of units by putting a dot ' to the right of it, we have 
the following results : 



246-4789 = 2 X 10^4 4x 104-6 + A + _^"^^ + _|_^_9^. 



In the denary scale 
246-4789 = 2: 
in the quinary scale 

324-4213 = 3x 10^+2. 104-44-A + _L+_l_ + J_, 

remembering that in this scale 10 stands ioifive and not for teii 
(Ai-t. 434). 

444. To shoio that in any scale a radical fraction is a proper 
fraction. 

Suppose the fraction to contain n digits, a, b, c 



Then, since r - 1 is the highest value that eacli of the digits 
can have, 

- + -5 + ... is not greater than (r- 1)^-+ -t, + ... to n terms) 
r r^ \r r- / 



than(r-l) 'y~ 



not greater 

--1 
r 



( r" — \ ") 

not greater tlian (r - 1) ■; — I ; 

(?-'\r-l)j 

not greater than ; 

not greater than I - — 
r" 



SCALES OF iVOTA TTON: 325 



Hence the criveu Iraction is less than 1, and is therefore a 
proper fraction. 

445. To transform a fraction expressed in a given scale into 
a radical fraction in any other scale. 

Lut F be tlie given fraction expressed in the first scale, 

r the radix of the new scale in which the fraction is to 
be expressed, 

a, b, c.the digits expressing the fraction in the nev/ 
scale, so that 

r r- r^ 

from which eqnation the values of a, 6, c.are to be deter- 
mined. 

Multiplying both sides of the equation by r, 

TP be 

r r* 

b c 
Now - + ^+ ••• is a proper fraction by Art. 444. 

Hence the integral part of Fr will =a, the first digit of the 
new fraction, and the fractional part of Fr will 

b c 
= - + -,+ ... 
r r- 

Giving to this fractional part of Fr the symbol F-^^ we have 

Alultiplying both sides of the equation by r, 

F,r = b + -+ ... 
r 

_ Hence the integral part of F^r=^b, the second digit of the new 
Taction, and thus, by a similar process, all the digits of the 
lew fraction may be found. 



\26 SCALES OF NOTATION. 

3 

Ex. 1. Express = as a radical fraction in the quinary 

Kcale : 

7 7 7' 

1-5^5 

5 , 25 ., 4 

7 / 7 

4 . 20 ^ 6 

i?x5 = — = 2 + -, 
7 / 7' 

6 , 30 , 2 

7 7 7' 

2 , 10 , 3 

^xo = -=l+-; 

therefore fraction is •203241 recurring. 

Ex. 2. Express •84375 in the octonary scale : 
•84375 
8 









6-75000 














8 










6^00000 




The 


fraction required 


is •66. 








Ex. 


3. 


Transform ■ 


42765 from the 


nonary 


to the 


senary 


scale. 




• 


•42765 
6 

2-78133 
6 










5 •23820 










6 










155430 










6 










365800 





TTie fraction required is •2513... 



SCALES OF NOTATION. 



327 



Ex. 4. Transfonn 6l24-i275 from the duodenary to the 

quaternary scale : 

•^275 
4 

3-4«58 
4 

1-75 i8 
4 

2-5e68 
4 

l-e(28 



Number required is 10223230-3121 



4 


el24 




4 


2937- 


- remainder 


4 


834- 


-remainder 3 


4 


20e- 


- remainder 2 


4 


62- 


- remainder 3 


4 


16- 


- remainder 2 


4 


4- 


- remainder 2 


4 


1- 


- remainder 




0- 


- remainder 1 



Examples. — clix. 



25 



1. Express :^^ in the senary scale. 

3 

2. Express — in the septenary scale. 

3. Express 23' 125 in. the nonary scale. 

4. Express 1820"3375 in the senarj' scale. 

5. In what scale is 17486 written 212542 ? 

6. In what scale is 511173 written 1746305 ? 

7. Show that a number in the Common scale is divisible 

(1) by 3 if the sum of its digits is divisible by 3. 

(2) by 4 if the last two digits be divisible by 4. 

(3) by 8 if the last three digits be divisible by 8, 

(4) by 5 if the number ends with 5 or 0. 



32^ ON LOGARITHMS. 



(5) by 11 if the difference between the sum of the digit* 
in the odd places and the sum of those in the even 
places be divisible by 11. 

8. If iV be a number in the scale whose radix is r, and n 
be the number resultinsr when the digits of N are reversed, 
show that iV- 7i is divisible by r- 1. 



XXXVIII. ON LOGARITHMS. 

446. Def. The Logarithm of a number to a given base 
is the index of the power to which the base must be raised to 
give the number. 

'^ Thus if m = a', x is called the logarithiu of m to the base a. 

For instance, if the base of a system of Logarithms be 2, 
3 is the logarithm of the number 8, 
because 8 = 2^: 
and if the base be 5, then 

3 is the logarithm of the number 125, 
because 125 = 5^ 

447. The logarithm of a number in to the base a is written 
thus, logaWi ; and so, if vi = a', 

X = log^m. 
Hence it follows that m = a'"^"". 

448. Since 1 = 0", the logarithm of unity to any base is 

zero. 

Since a = a}, the logarithm of the base of any .system 
i3 unity. 

449. We now proceed to describe th;it wliish is called the 
Common System of logarithms. 

The ba,8e of the system is 10. 



ON LOGARITHMS. 



%n 



By a system of logarithms to tlie base 10, we mean a succes- 
sion of values of x whicli satisfy the equation 

771=10" 

for all positire values of m, integi-al or fractional. 

Such a system is formed by the series of logarithms of 
the natural numbers from 1 to 100000, which constitute the 
logarithms registered iu oiir ordinary tables, and which are 
therefore called tabular logarithms. 



450. Now 



1 = 100, 

10 = 101, 

100 = 102, 

1000 = lO-'', 



and so on. 

Hence the logarithm of 



1 is 0, 
of 10 is 1, 
of 100 is 2, 
of 1000 is 3. 



and so on. 



Hence for all numbers between 1 and 10 the logarithm is a 
decimal less than 1, 

between 10 and 100 the logarithm is a decimal between 1 
and 2, 

between 100 and 1000 a decimal between 2 

and 3, and so on. 

451. The logarithms of the natural numbers from 1 to 12 
stand thus in the tables : 



No. 


Log 


1 


0-0000000 


2 


0-3010300 


3 


0-4771213 


4 


0-6020600 


5 


0-6989700 


6 


0-7731513 



No. 


Log 


7 


0-8450980 


8 


0-9030900 


9 


0-9542425 


10 


1-0000000 


11 


1-0413927 


12 


1-0791812 



The logarithms are calculated to seven places of decimals 



33<5 ON LOGARITHMS. 

452. The integral parts of the logarithms of numbers 
higher than 10 are called the characteristics of those logarithms, 
and the decimal parts of the logarithms are called the mantisscB. 

Thus ■ 1 is the characteristic, 

•0791812 the mantissa, 
of the logarithm of 12. 

453. The logarithms for 100 and the numbers that succeed 
it (and in some tables those that jirecede lOOj have no charac- 
teristic prefixed, becfiuse it can be supplied by the reader, l)eing 
2 for all numbers between 100 and 1000, 3 for all between 
1000 and 10000, and so on. Thus in the Tables we shall 
tind 



No. 


Log 


100 


0000000 


101 


0043214 


102 


0086002 


103 


0128372 


104 


0170333 


105 


0211893 



which we read thus : 

the logarithm of 100 is 2, 

of 101 is 2-0043214. 

of 102 is 2-0086002; and so on. 

454. Logarithms are of great use in making arithmetical 
computations more easy, for by means of a Table of Logarithms 
the operation 

of j\Iultiplication is changed into that of Addition, 

. . . Division Subtraction, 

... Involution Multiplication, 

...Evolution Division, 

as we shall show in the next four Articles. 

455. The logarithm of a product is equal tc. the sum of the 
logarithms of its factors. 



Oy LOGARITHMS. 33» 



Let 




m = a', 


and 




n = a". 


Then 




mn = a"'^' ; 




••• log, 


mn = x + y 

= log„m + ^O'^ji. 



Hence it follows that 

log^mnp = log^TO + \og^n + log^^'j 
and similarly it may be shown that the Theorem holds good 
for any number of factors. 

Thus the operation of Multiplication is changed into that of 
Addition. 

Suppose, for instance, we want to find the product of 246 
and 357, we add the logarithms of the factors, and the sum is 
the logarithm of the product : thus 

log 246 = 2-3909351 
log 357 = 2-5526682 



their sum = 4-9436033 
whicli is the logarithm of 87822, the product required. 

Note. We do not write logio246, for so long as we are 
treating of logarithms to the particular base 10, we may omit 
the suffix. 

456. 77ie logarithm of a quotient is equal to the logarithm of 
the dividend diminished by the logarithm of the divisor. 

Let m = a", 

and n = a^. 

Then -^a'-"; 

n 

, m 
.: los„——x — y 

= log^7)i — log„n.. 

Thus the operation of Division is changed into that of Sub- 
traction. 



532 ON LOGARITHMS. 



If, for example, we are required to divide 371 "49 by 52-376, 
we proceed thus, 

log 371 -49 = 2-5699471 
lo" 52-376 = 1-7191323 



their difference = '8508148 
which is the logarithm of 7-092752, the quotient required. 

457. The, logarithm, of any jwiver of a number is equal to the 
product of the logarithm of the number and the index denoting the 
poiver. 



Let 


771 = a". 


Then 


TO' = a"; 




.•. logjn' = rx 




= r . lo" m. 



Thus the operation of Involution is changed into Multipli- 
cation. 

Suppose, for instance, we have to find the fourth power of 
13, we nauy proceed thus, 

log 13 = 1-1139434 
4 



4-4557736 
which is the logarithm of 28561, the number required. 

458. The logarithm of any root of a number is equal to the 
quotient arising from the division of the logarithm of the number 
by the nurnher denoting the root. 



Let 


m = a*. 




Then 


1 X 

m' = a" ; 
1 -' a; 






_1 

r ' 


log^TO. 



Thus the operation of Evoluiion is changed into Division. 



ON LOGARITIJMS. y^. 



If, for example, we have to find the fifth root of 16807 we 
proceed thus, ' 

5 |_4|2254902, the log of 16807 

•8450980 

which is the logarithm of 7, the root required. 

459 The common system of Logarithms has this advanta-e 
overall otliers for numerical calculations, that its base is th" 
same as tlie radix of the common scale of notation. 

Hence it is that the same mantissa serves for all numl.ers 
which have tlie same significant digits and difler only in the 
position of the place of units relatively to those digits. 

For, since log 60 = log 10 + log 6=1+ log 6, 

log 600 = log 100 + log6 = 2 + log6,' 

log 6000 = log 1000 + l(3g 6 = 3 + log 6, 

t is clear that if we know the logarithm of any number as 6 

ve also J.novv tlie logarithms of the numbers resulting' from 

nultiplymg that number by the powers of 10. 

So again, if we know that 

log 1-7692 is -247783, 
TO also know that 

log 17-692 is 1-247783, 
log 176-92 is 2-247783, 
log 1769-2 is 3-247783, 
log 17692 is 4-247783, 
log 176920 is 5-247783. 

.1^ unit^^'' ™"'^ ""'^'^ *^'^' ""^ '^' logarithm., ot numbers less 
Since \ = \Q\ 

•^=^o=l«"^ 



334 ON LOG/.RITRMS 

the logarifhm of a numl^er 

between land "1 lies between and -1, 

between -land '01 -land -2, 

between •()! and '001 —2 and -3, 

and so on. 

Hence the logarithms of all numbers less than unity are 
negative. 

"We do not require a separate table for these logarithms, for 
w^e can deduce them from the logarithms of numbers greater 
Lhan unity by the following process : 

log-6 =log j^ =log6-loglO =log6-l, 
log-06 =logi^ =log6-loglOO =log6-2, 
log -006 = log ^^- = log 6 - log 1000 = log 6 - 3. 

Now the logarithm of G is •7781513. 

Hence 

log-6 = - 1 + -7781513, which is written 1-7781513, 
log -06 = - 2 + -7781513, which is written 2-7781513, 
log -006= - 3 + -7781513, which is Avritten 3-7781513, i 

the characteristics only being negative and the mantissse 
positive. I 

461. Thus the same mantissce serve for the logarithms of 
all numbers, whether greater or less than unity, which have the 
same significant digits, and differ only in tlie position of the 
place of units relatively to those digits. 

It is best to regard the Table as a register of the logarithms 
of numbers which have one significant digit before the decimal 
point. 



ox LOGARITHMS. 335 



No. ! Log 

For instance, when we read in the tables 144 | 1583625, we 

interpret the entry thus 

log 1-44 is -1583625. 

We then obtain the following rules for the characteristic to 
be attached in each case. 

I. If the decimal point be shifted one, two, three ...n 
places to the right, prclix as a characteristic 1, 2, 3 ... n. 

II. If the decimal point be shifted one, two, tlu'ee...TO 
places to the left, prefix as a characteristic f, 2, 3 ... w. 

Thus log 1-44 is -1583625, 

.-. log 14-4 is 1-1583625, 

log 144 is 2-1583625, 

log 1440 is 3-1583625, 

log -144 is 1-1583625, 

log -0144 is 2-1583625, 

log -00144 is 3-1583625. 

462. In calculations with negative characteristics we follow 
lae rules of algebra. Thus, 

(1) If we have to add the logarithms 3-64628 and 2-42367, 
M-3 first add the mantissoe, and the result is 1-06995, and then 
add the characteristics, and this result is 1. 

The final result is 1 + 1-06995, that is, -06995. 

(2) To subtract 5-6249372 from 3-2456973, we may arrange 
the numbers thus, 

- 3 + -2456973 
-5 + -6249372 



1 + -6207601 



the 1 carried on from the last sul)traction in the decimal places 
changing — 5 into — 4, and then — 4 subtracted from - 3 giving 
1 as a result. 

Heuce the resulting logarithm is 1-6207601. 



336 ON LOGARITHMS. 



(3) To multiply 3-7482569 Ly 5. 

3-74825C9 
5 



12-7412845 

the 3 carried on from tlie last multiplication of the decimal 
l>laces being added to — 15, and thus giving — 12 as a result. 

(4) To divide 14-2456736 Ly 4. 

Increase the negative characteristic so that it may be exactly 
divisible by 4, making a proper compensation, thus, 

14-2456736 = 16 -i- 2-2456736. 

14-2456736 16 + 2-2456736 - 

Then ^ = ^ =4 + -5614184; 

and so the result is 4-5614184, 



Examples. — clx. 

1. Add 3-1651553, 4-7505855, 6-6879746, 2-6150026. 

2. Add 4-6843785, 5-6650657, 3-8905196, 3-4675284. 

3. Add 2-5324716, 3-6650657, 5-8905196, -3156215. 

4. From 2-483269 take 3-742891. 

5. From 2-352678 take 5 4286 19. 

6. From 5-349162 take 3-624329. 

7. Multiply 2-4596721 by 3. 

8. Multi])ly 7-4296S3 by 6. 

9. Multiply 9-2843617 by 7. 

10. Divide 6-3725409 by 3. 

1 1. Divide 14-432962 by 6. 

1 2. Divide 4-53627188 by 9. 

463. "We shall now explain how a system of logarithms 
calculated to a base a may be transformed into another system 
of which the base is 6. 



ON LOGARITHMS. 



Let m be a number of which the logarithm in the first 
system is x and in the second y. 



Then m = a*', 

and 771=6". 

Hence h^ = a', 





■•^=iog.^; 




■ X logj) ' 


\ 


■y-io^y^- 



Hence if we multiply the loyiirithm of any number in the 

system of which the base is a by = we shall obtain the 

logarithm of the same number in the system of which the 'base 
is h. 

This constant mulliiilier -. — , is called The Modulus of the 
log„o ■' 

system of which the base is b with reference to the system of 

which the base is a. 

464. The common system of logarithms is used in all 
numerical calculations, but there is another system, which we 
must notice, emj^loyed by the discoverer of logarithms, Napier, 
and hence called The Napierian System. 

The base of this system, denoted by the symbol e, is the 
number which is the sum of the series 

of which sum the first eight digits are 2-7182818. 

465. Our common logarithms are formed from the Loga.' 
rithms of the Napierian System by multiplying each of tha 

[s.A.j Y 



338 ON LOGARITH.\fS. 

■latter by a common multi])lier culled Tlie Modulus of the 
Couimon System 

Tliis modulus is, in accordance with the conclusion of 

Art. 463, i— ^. 
log, 10 

That is, if I and iV V)e the logarithms of the same number in 
the common and Napierian systems respectively. 



Now log, 10 is 2-30258509 ; 
1 . ] 



or -43429448, 



■ ■ log, 10 2-30258509 
and so the modulus of tlie common system is -43429448. 

466. To prove that log,6 x log(,rt = 1. 

Let ft; = log A 

Then 6 = a'; 

.-. - = log,a. 

X 

Thus loga6 X logjft = .-; x - 

= 1. 

467. The following are simple examples of the method ot 
applying the princi})les explained in this Chapter. 

Ex. 1. Given log 2 = -3()l()3()0, log 3 = -4771213 and 
log 7 = -8450980, find log 42. 

Since 42 = 2x3x7 

log 42 = log 2 + log 3 + log 7 

= -3010300 4- -4771213 + -8450980 
=r 1-6232493. 



ON LOGARITHMS. 339 



Ex. 2. Giveu log 2 = -3010300 and log 3 = -4771213, find 
the logarithms of 64, 81 and 96. 

log 64:* log 26 = 6 log 2 

log 2 = -3010300 
6 



log 64 = 1-8061800 



log 81 = log 3* = 4 log 3 

log 3 = -4771213 
4 



log 81 = 1-9084852 



log 96 = log (32 X 3) = log 32 + log 3, 
and log 32 = log 2-^ = 5 log 2; 

.-. log 96 = 5 log 2 + log 3 = 1-5051500 + -4771213 = 1-9822713. 

Ex. 3. Given log 5 = -6989700, find the logarithm of 
^(6-25). 

log (6-25)^ = i log 6-25 = ^ log ~g = J (]og625-log 100) 
= ^(Iog5''-2) = i(4log5-2) 
= i (2-7958800 -2) = -1136657. 



Examples.— clxi. 

1. Given log 2 = -3010300, find log 128, log 125 and 
log 2500. 

2. Given log 2 = -3010300 and log 7 = -8450980, find the 
logarithms of 50, -005 and 196. 

3. Given log 2 = -3010300, and log 3 = -4771213, find the 
logarithms of 6, 27, 54 and 576. 

4. Given log 2 = -3010300, log 3 = -4771213, log 7 = -8450980, 
find log 60, log -03, log 1-05, and log -0000432. 



340 ON LOGARITHMS. 



5. Given log 2 = -SOinS'OO, log 18 = 1 -2552725 and 

log 21 = 1-3222193, find log -00075 and log 31-5. 

6. Given log 5 = -6989700, find the logarithms of 2, -064, 

J, 

and (5,0; • 

7. Given log 2 = -3010300, find the logarithms of 5, -125, 






8. What are the logarithms of -01, 1 and 100 to the base 
10? What to the base -or? 

9. What is the characteristic of log 1593, (1) to base 10, 
(2) to base 12 ? 

10. Given —^ = 8, and x = 3i/, find x and y. 

11. Given log 4 = -6020600, log 1-04 = -0170333 : 

(a) Find the logarithms of 2, 25, 83-2, (-625)"~. 

(6) How many digits are there in the integral part ot 

(1-04)0000? 

12. Given log 25 = 1-3979400, log r03 = -0128372 : 

(a) Find the logaritlims of 5, 4, 5r5, (•064)"'~. 

(6) How many digits are there in the integral part of 
(1-03)000? 

13. Having given log 3 = -4771213, log 7= -8450980, 

log 11 = 1-0413927: 

find the logarithms of 7623, - -^ and ^^r^. 

14. Solve the equations : 

(i> 4096'=-^. (4) a-?)-=<;. 

(2) (:4y = 6-25. (5) a^.\^-^-^c^-\ 

(^) a^.]f = m. (6) a'lr =&~\ 



ON LOGARITHMS. 34! 

468. We have explained in Arts. 459 — 461 the advantages 
of the Common System of Logarithms, wliich may be stated in 
a more general form thus : 

Let A be any sequence of figures (such as 2-35916), having 
one, digit in the integral part. 

Then any niiniber iV having the same sequence of figures 
(such as 235-916 or -00235916) is of the form A x 10", where n 
is an integer, positive or negative. 

Therefore logjjxY= logjo( J. x 10") = log^,^ + n. 

Now A lies between lO*' and 10\ and therefore log ^ lies 
between and 1, and is therefore a proper fraction. 

But logjjiV and logjo.4 differ only by the integer 71 ; 
.". logjp^4 is the fractional part of logu,iV. 

Hence the logarithyns of all numbers having THE same 
SEQUENCE OF FIGURES have the same mantissa. 

Therefore one register serves for the m.antissa of logarithms of all 
such numbers. This renders the tables more comprehensive. 

Again, considering all numbers which have the same 
sequence of figures, the number containing t'co digits in the 
integral part =10. J., and therefore tlie characteristic of its 
logarithm is 1. 

Similarly the niimber containing m digits in the integral 
part = 10". A, and therefore the characteristic of its logarithm 
is m. 

Also numbers which have no digit in the integral part and 
one cypher after the decimal point are equal to A . 10~' and 
A . 10~^ respectively, and therefore the characteristics of their 
logarithms are - 1 and — 2 respectively. 

Similarly the number having m cyphers following the decimal 
point = ^ . 10-<™+"; 

.'. the characteristic of its logarithia is ~{m + 1). 

Hence we see that the characteristics of the logarithms of all 
nuvibers can be determined b)j inspection and therefore need not be 
itj'istered. This renders the tables less bulky. 



34:2 ON LOGARITHMS. 



469. The method of using Tables of Logarithms does 
not fall within the scope of this treatise, but an account of 
it may be found in the Author's work on Elementaky 
Trigonometry. 

470. We proceed to give a short explanation of the way 
in which Logarithms are applied to the .solution of questions 
relating to Compound Interest. 

471. Suppose r to represent the interest on .£1 for a year, 
then the interest on P pounds for a j^ear is represented by 
Fr, and the amount of P pounds for a year is represented, 
by P + Pr. 

472. To find the amount of a (jiven sum for any time at 
conifpound interest. 

Let P be the original principal, 

r the interest on £\ for a year, 
n the number of years. 

Then if P^, P„, P.^...P„ be the amounts at (he end ol 
1, 2, 3 . . . n years, 

Pi = P +Pr = P (1 + r, 

P2 = Pi + Pir-P,(l+r)=P(l+7f 

P3 = P„ + P.,?- = P., (1 + 7-) = P (1 + if, 



P,. = P(l+r)". 

473. Now suppose P„, P and r to be given : then by the aid 
of Logarithms we can find n, for 

logP„ = log !P(l + r)"| 

= log P + nlog(l+r) ; 

_ log 7'„-l()gP 
log(i+r) 



I 



ON- LOGARITHMS. . 343 

474. If the interest be payable at intervals other than a 
year, the fornmla P^ = P(1 +r)" is applicable to the solution of 
tlie question, it being observed that /• represents the interest 
on £\ for the perio'l on wliich the interest is calculated, half- 
yearly, quarterly, or for a*iy other period, and n represents the 
number of such periods. 

For example, to find the interest on P pounds for 4 years 
at compound interest, reckoned quarterly, at 5 per cezit. per 
annum. 

Here r=l of A = l^ = .0i25, 

n = 4x4 = 16; 
.-. P„ = P(1 + -0125)16. 



Examples.— clxii. 

N.B. — The Logarithms required may l)e found from the 
extracts from the Tables given in pages 329, 330. 

1. In how many years will a sum of money double itself 
at 4 per cent, compound interest ? 

2. In Iiow many years will a sum of money double itself 
at 3 per cent, compound interest \ 

3. In how many years will a sum of money double itself 
at 10 per cent, compound interest ? 

4. In how many years will a sum of money treble itself 
at 5 per cent, compound interest ? 

5. If £F at compound interest, rate ?•, double itself in n 
years, and at rate 2r in m years : show that in : n is greater 
than 1 : 2. 

6. In how many years will £1000 amount to £1800 at 
5 per cent, compound interest ? 

7. In how mnny years will £P double itself at 6 per cent, 
per ann. compound interest payable half-yearly 1 



APPENDIX. 

475. The following is another method of proving the prin- 
cipal theorem in Permutations, to which reference is made in 
the note on page 289. 

To prove that the number of pernjfiitatioHs of n things taken r at 
a time is n . (n - 1) (n — r + 1). 

Let there be n things a, h, c, d 

If n things be taken 1 at a time, the number of permutations 
is of course n. 

Now take any one of them, as a, then n - 1 are left, and 
any one of these may be put after a to form a permutation, 

2 at a time, in which a stands first: and hence since there are 
n things which may begin and each of these n may have n - 1 
put after it, there are altogether n (n — 1) permutations of n 
things taken 2 at a time. 

Take any one of these, as ab, then there are n-2 left, and 
any one of these may be put after ab, to form a permutation, 

3 at a time, in which ab stands first : and hence since there 
are n{n — 1) things which may begin, and each of these n{n - 1) 
may have n-2 put after it, there are altogether n(n — 1) (ti - 2) 
permutations of n things taken 3 at a time. 

If we take any one of these as abc, there are ?i - 3 left, and 
so the number of permutations of n things taken 4 at a time is 
n.(n-l){n-2){n-3). 

So we see that to find the number of permutations, taken 
r at a time, we must multiply the nvimber of permutations, 
taken r— 1 at a time, by the niimber formed by subtracting 
r— 1 from n, since this will be the number of endings any one 
of these permutations may have. 

Hence the number of permi;tations of n things taken 5 at a 
time is 

n(n-l)(7j-2) (n-3) x (n-4), orn(n- 1) (h -2) (n-3) (n-4); 
and since each time we multiply by an additional factor the 
number of factors is equal to the number of things taken at a 
time, it follows that the number of permutations of n thinga 
taken r at a time is the product of the factors 
n.(n-l)(n-2) (n-r+1). 



A :^ S W E R s. 







i. 


(Page 10.) 






I. 


5a+ 76-f 12c. 


"7 


a + 3b + 2c. 


3- 


2a + 26 + 2a 


4- 


6a + 2h + 2c. 


5- 


'2x-7a + 3b-2. 


6. 


0. 


7- 


126 + 3c. 











ii. (Page 10.) 
I. 2a. 2. 2a + 5A 3. 3a — 3x. 4. Sx + Sr/. 

5. 4a + 6 + 2c. 6. 2ti. 7. 4. S. 13x-y-6z. 

9. 10a— 76- X. 

iii. (Page 10.) 
I. 26. 2. a: + 2]/. 3. a + 5c + d. 4. 2y-{-2z. 

5. 2r. 6. 26 + 2c. 7. o-36-c. 8. By + z. 

iv. (Page 11.) 

I. 4a -6, 2. 46. 3. a 4-6 -4c. 4. 26. 

5. 14x + 2. 6. 2x + a. 7. 6x — a. 8. a. 

9. 2a -6. 10. 2a. 11. c. 12. x + 3o. 

13. 29a -276 + 6c. 

^r. (Page 16.) 
Addition. 

I. 7a-26. 2. -106 + 6c. 3. -llx-Sy-6z. 

4. -66-5c + 3d. 5. 2a. 6. -2x-2a + b + 4y. 

7. 7a + 46 - 4c. 8. 7a — 6 + 7c. 9. — 6?/ + £<;. 

[S.A.] ^* 



34& 



AxYSlV£RS. 



I. 

4- 

7- 

lo. 



Subtraction. 

2a + 26. 
8x-l7j/ + 5. 
- 3a + 36 — 4c. 
6rt - 6 + 5c. 



2. a - c. 3. 2a - 26 + 2c. 

5. 7a -166 + 20c. 6. 5a-36-8x. 

8. 26 + 2c -15. g. llx-7i/ + 45;. 

II. 12^-95 + 2r. 



I. 


2xy. 


5- 


a?. 


9- 


180a^65c*. 


13- 


76x*i/%3. 


16. 


12a-6can/. 


19. 


ahx^yh^. 



Vi. (Page 20.) 
2. \2xy. 3. 12a;2i/2. 

6. a*. 7. 12^561 

10. 28a"6r'". 11. Ba' 

14. 51a6*c-2/2;. 

17. 8a"6-V. 

20. 33a206i6m2x. 



4. 3a26c2 

8. 35a66c*. 

12. 20a*b^xy. 

15. 48x8t/i<'2«. 

18. ^mhi^p^. 



Vii. (Page 22.) 

I. a2 + o6-ac. 2. 2rt- + 6a6 — Sac. 3. a^ + Za^ + Aa^. 

4. 9a5-15a'*-18a^ + 21a2. 5. a'j _ 2a252 + ^js^ 

6. 3a56-9a*63 + 3a264. 7. 8/)i% + 9m2ji2+ lOmiiA 

8. 18a66 + 8a562-6a*63 + 8a36*. 9. a; Y - ary + x^ - 7xi/. 

to. m^n - 2m-n^ + Smn^ — 71'*. 11. 1 44a^6* - 72a'*6^ + 6()a^6^ 

1 2. 104a;*i/ - 136.c'!/2 + 4t)x-)/'' - ^xyK 



I. 


:c- + 12x + 27. 


4- 


x^- 15.r + 56. 


7. 


jc* + a;- - 20. 


9- 


.T*-31x2 + 9. 


II. 


.v;-* - .X- + 2x - 


14. 


a« - .r«. 


16, 


a;*-81y*, 



viii. (Page 27.) 

2. x2 + 8x - 105. 3. X' - 2x - 1 20. 

5. «2 — 8a+15. 6. i/' + 7j/ — 7S. 

8 . X* - 1 2.c3 + 50x2 _ 84x + 45. 

10. a" — 3a^ — 3a* + 1 3a' - 6a- — Ga + 4. 

12. .x* + .>;2v2 + (/■». 13. x^-y^ 

15. a-^-5.i;3 + 5x-- 1. 

17. a* -166*. 18. 16a* -6*. 



ANSWERS. 347 



19. a^ - Aa*h + 4aW + Aa^h^ - llah* - 1265, 

20. a= + ba*h + aW - lOa-1? + 12a6^ - 2}y>. 

21. a* + 4a-x- + J6j;*. 22. Sla^ + Qa^x^ + x*. 
23. x8 + 4a-x*+16a*, 24. a^ + 6^ + c3 _ 3a6c. 

25. x^ + x*y - 9x3?/2 _ 20x2^/3 + 2x7/< + 15?/*. 

26. a^fi- + c-d- - rt-c- - 6-cZ-. 27. s? - a\ 

28. x^ - ax- + 6.';- - cx^ — abx + acx - hex + abc. 

29. 1 c*. 30. x^ — y^. 31. a^S-x^^ 32. -47. 
33. 2. 34. -14. 35. ab + ac + hc. 36. -60. 
37. 2. 38. m^. 

ix. (Page 28.) 

I. -a%. 2. -a*. 3. _a363_ ^ l^aW. 

5. -30xV. 6. -a3 + a26-a62. 7, -6a5-8a*+ lOal 

8. a* + 2a3 + 2a2 + «. 9. - 6x3y + x^y^ + y^j/S _ 1 9y4_ 

lo. 5m3 + 7/i,2/i- 137H7i2 + 77i3. II. - IS/'^ - 22?-2 + 96r + 135. 

12. - 7X* + X^Z + 8x2^2 + 9x^2 + 923 

13. x« + xy. 14. x* + 2x3?/ +2x22/2 + 2X2/3 + 1/4. 

X. (Page 32.) 

I. x'^ + ^ax + a^. 2. x2-2ax + a2. 3. a:- + 4x + 4. 

4. x2_gj.^.9_ g_ x* + 2.r2y2 + ,y. 6 x4-2x2?/2 + y4. 

7. a6 + 2a363^56_ g_ a6-2rt363 + 56 

9. x^ + j'2 + ^2 + 2x?/ + 2x2 + 2yz. 

10. x2 + 1/2 + ^2 _ 2x?/ + 2x3 - 2yz. 

11. m2 + n2 + 2)- + 7-2 + 2 m n - 2mp -27nr~2iip-2nr+ -Jj r. 

12. x* + 4x3-2x2-12x + 9. 13. X* - 12x3 + 50x2 -84x + 4y. 

14. 4x* - 28x3 + 85x2 -126x + 81. 

15. x* + i/ + ^+2xY-2x-z^-2y^'^, 



34^ ANSWERS. 



1 6. cc8 - 8x«2/2 + 1 8x*i/* - 8a;2i/8 + f, 

17. a6 + 66 + c« + 2a3i3 + 2a3c3 + 26V. 

18. x^ + 2/8 + 2^ - 2x^2/3 - 2x^»^ + 2yh^. 

19. x^ + 4t/2 + Qz^ + 4x1/ - 6x2 — 1 '2,yz. 

20. X* + 4?/* + 252* - 4x^2/2 + 10x2g2 _ 2O2/V. 

21. x^ + 3ax2 + Sa^x + a^. 22. x^ - Sax^ + Sa'x - o*. 
23. x3 + 3x2 + 3x+l. 24. x3-3x2 + 3x-l. 

25. x3 + 6x2+12x + 8. 26. a6-3a<62 + 3a26*-6fi. 

27. a' + 3a26 + 2,a¥ + 6^ -|. c' + 3a«c + 6a6c + Zhh + 3ac2 + 36c2. 

28. a3 - 3a26 + ZaV^ -h^-c^- Zah + Gahc - W-c + Sat- - Zhc-. 

29. m* - 2?>i-?i2 + n*. 30. m* + 2m^n - 2mn^ - n*. 

xl. (Page 34.) 
I. a^. 2. x*. 3. x^?/. 4. x*y^. 5. 66c. 6. 8c*. 
7. 16a266c8. 8. 121m«ri«2>^ 9- 12a3xy*. 10. 8a*6c^. 

xii. (Page 35.) 

I. x' + 2x + l. 2. 2/' -1/2 + 2/-!. 3. a* + 2rt6 + 36-. 

4. X* + m2?x- + m^p*. 5. Aay -Ix + x^. 6. 8x^1/^ — 4.r-!/2 _ 2y. 

7. 27m%*-18m%'* + 97ny. 8. 3xy - 2x!/^ - y*. 

9. 13u26-9a62 + 76. 10. 196V + 12&V- 76c*. 

Xiii. (Page 36.) 
I. -8. 2. 15a^ 3. -21x't/'. 

4. -6m2rj. 5. 16a^6. 6. a-x-Jrax + l. 

7. -2a2 + 3a-x*. 8. 2 + 6a=6 - 8a*66. 

9. — 1 2x2 4. 9_^. j^ _ 8y2_ 10. - x^ + i^x V + fry*. 

Xiv. (Page 38.) 
I. x + 5. 2. x-10. 3. x + 4. 4. x+12. 

^. x2+7x+12. 6. a;--l. 7- x^ + x+l. 



ANSIVERS. 349 



8. x3-3x2 + 33;+l. 9. X--2X-1. 10. x^-'ix+l. 

II. x^-x + l. 12. x3-2x2 + 8. 13. x- + 3y'^. 

[4. a^ + ^a^ + Zah' + y^. 15. a* - 4a35 + Ga^fcs _ 4a{,3 + j4 

16. x2-6x + 5. 17. a^ — ^a~h + Zah^ + A¥. 

1 8. 2rtx^ - 3a-x + a'. 19. x^ - x + 1 . 20. x^ — a^. 

21. x + 2i/. 22. X* - x^y + x^i/^ - X1/3 + 2/*. 

23. x^ + x*2/ + x-'ff^ + x-y' + xj/* + y^. 24. « + 6 — c. 

25. -6 + 2&--61 26. a-6 + c-d. 

27. x^ — xy — x.: + y^ — i/z+2^ 28. x*' — x^2/- + x^!/* - x^j/" + y*. 

29. 2J + 29-r. 30. a* - 0^6 + a'6'^ - a6^ + 6*. 

31. X* + x% + X"2/2 + xt/' + 7/*. 32. 2x' - Sx''^ + 2x. 

33. a^ + 3a3 + 9a2 + 27a + 81. 34. ^-7 + ^* + ^. 

35. x2-9x-10. 36. 24x^-2ax-35a2. 

37. 6x2-7x + 8. 38. 8x3+12ax2-18a2x-27a». 

39. 27x3 - 36ax2 4- 48a2x - 64a3. 40. 2a + 36. 

41. x + 2a. 42. a^-Alfi. 43. x'^-3x-y. 

44. x--3xy-2y-. 45. x^ + Sx^y + 9xi/2 + 27i/3. 

.46. a^ + 2a% + 4ab^ + 8¥ 47. 27a3-18a26+ I2a62-8R 

48. 8x3-12x2i/ + 18x?/2_27i/». 49. 3« + 26 + c. 

50. a2-2ax + 4x2. 51. x^ + xy + y" 52. I6x--4xy + y'. 

53. x^ + xy-y-. 54. flx2 + 4«-x + 2flA 55. a-x. 

56. x-y-z. 57. 3X--X + 2. 58. 4-6x + 8x'-'-10x'. 

59. x + y. 60. ax + by-ab-xy. 61. bx + ay. 

62. x^ - ax + 6-. 

XV. (Page 40.) 

I. x2 + ax + 6. 2. 2/2 - (^ + to) 1/ + Zm. 3. «;'4-cx + (/. 

4. x^ + ax-b. 5. x2 - (6 + 0?) X + 6ci. 

xvi. (Page 42.) 

I. m~n, m^ — mn + v?, m* — in?n + ni^n- - mn^ + n*, 

vv" - mhi + &c., m* - m'n + <Ssc. 



556 ANSWERS. 



2. 'm-^n,w?^- mn + n-, w? + mhi + &c., m* + mhi + &c., 

m® + m*n + &c. 

3. a - I, a^ - a + I, a* - a^ + &c., a^-a^ + &c., a^ - a^ + &c. 

4. y + l,y'^ + y+l, y* + y^ + &.c., y^ + y^ + &c., y^ + y'' + &,c. 

xvii. (Page 43.) 

I. 5a; (x- 3). 2. 3x{x^ + Gx-2). 3. 7(7i/2-2i/ + 1). 

4. 4a;y (a;2 - 3x1/ + 2?/2), 5. a;(x^ — ax^ + 6x + c). 

6. 3xY (x^i/ - 7x + V). 7. 27a%^{2 + 4a%'^-9a^b3). 
8. 45xy(xV-2x-8i/). 

xviii. (Page 44.) 

I. (x-a)(x-6). 2. (a-x)(6-!-x). 3. (b-y)(c + y). 
4. (a + m) (6 + n). 5. (ax + y) (bx - y). 6. (a6 + cd) (x - j/). 

7. {ex + my) {dx - nyy 8. (ac - bd) (bx - dy). 

xix. (Page 45.) 

I. (x + 5)(x + 6). 2. (x + 5)(x + 12). 3. {y + U){y + l). 

4. (!/ + ll)(i/+10). 5. (»i- + 20)(?n + 15). 6. (m + 6)(m + 17). 

7. (a. + 86) (a + 6). 8. (x + 4?)i)(x + 9m). 9. {y + 3n)(y + l6n). 

10. (s; + 4^j) (2 + 25;?). II. (x^ + 2) (x2 + 3). 

12. (x^+l)(x3 + 3), 13. {xy + 2){xy+l6). 

14. (xY- + 3) (xy + 4), 15. (m5 + 8)(7rt5 + 2). 

16. {n + 20q){n + 7q). 





XX. 


(Page 45.) 


V (x-5)(x-2). 






2. (x-19)(x-10). 


3. {y-U)iy-l-2). 






4. iy-20)(y-10). 


5. (n- 23) (71 -20). 






6. (7i-56)(ji-l). 


7. (.x3-4)(x3-3). 






8. (ab - 26} {ab- I). 



9. (6'-c»-5)(6V-6). 10. (xi/~-ll)(xy«-2). 



Al^SlVEHS. 35» 



xxi. (Page 46.) 

I. (a; 4- 12) (a; -5). 2. (x + 15)(x-3). 3. (a+12)(a-l). 

4. (a + 20) (a -7). 5. (& + 25) (6 - 12). 6. (6 + 30) (6 -5). 

7. (x* + 4)(x*-l). 8. (x!/+14)(x2/-ll). 

9. (m5 + 20)(m5-5). 10. (7i + 30) (ji- 13). 

xxii. (Page 46.) 

I. (x-ll)(a; + 6). 2. (x-9)(.r; + 2). 3. (m- 12) (/?!, + 3). 

4. (7i-15)(n + 4). 5. (2/-14)(i/ + l). 6. (3- 20) (2 + 5). 

7. (x5_i0)(x5 + i). 8. (cd-30)(cd + 6). 

9. (m% - 2) (m% + 1). 10. (;>Y - 12) (i^V + '^)' 



xxiii. (Page 47.) 



I. 


(x-3)(x-12). 




2, (x + 9)(a;-5). 


3- 


(a6-18)(a6 + 2). 




4. (x* - 5m) (x* + 2m). 


5- 


(l/3+10)(j/3-9). 




6. (x2+10)(x2-ll). 


7- 


z (.r^ + Zax + 4a2). 




8. (x + to) (x + n). 


9- 


(2/3-3)(r/3-l). 




10. (xy — ab) (x-c). 


II. 


{x + a) (x - 6). 




12. (x - c) (x + d). 


13- 


(a6 - d) (6 - c). 




14. 4.(x-47/)(x-32/). 




xxiv. 


(Page 48.) 


I. 


(x + 9)2. 2. (x + 


13)2. 


3. (x + 17)2. 4. (2/ + 1)2. 


5- 


(2+100)2. 6. (X2 + 


■7)2. 


7. (x + 52/)2. 8. (m2 + 87*2)2, 



9. (x3 + 12)2. 10. (X7/ + 81)2. 

XXV. (Page 48.) 
I. (x-4)2. 2. (x-14)2. 3. (x-18)2 4. (7/ -20)2. 

5. (3-50)2. 6. (X2-11)2. 7. (x-157/)2. 8. (77^2 - 1 67*2) ». 

9. (it'- 19)2. 



3S2 ANSH'EH^. 

xxvi. (Page 50.) 

1. {x + y){x-y). 2. (x + 3)(x-3). 3. (2x + 5) (2x - 5). 

4. (a2 + x2)(a-'-x-^). 5. (a; + l)(a;-l). 6. (x3 + 1) (x^- 1). 

7. (:c* + 1) (x* - 1 ). 8. (m2 + 4) {m^ - 4). 

9. (61/ + Tz) (6?/ - 72). 10. (9xr/ + lla6) (9xi/-lla6). 

II. {a-h + c) {a-h-c). 12. (x + m-n) (x-m + n). 

13. (a + b + c + cO (« + ^-<^'~^)- H- 2xx2y. 

15. (x-i/ + z)(x-i/-z). 

16. {a-h + m + n) {a-h-m-n). 

17. (ffl-c + 6 + (0 (^-c-^-c^)- 18. (a + 6-c) (a-6 + c). 
19. (:c + t/ + z) (x + y-a). 20. (a-6 + m-n) (a-6-m + n). 
21. {ax + h]i+l){ax + hy-\). 22. 2axx2by. 

23. (H-a-6) (l-a+'O- 24. (l+x-i/)(l-x + 2/). 

25. (X + 2/ + 2) (X-1/-2). 26, (a + 26 -3c) (a -26 + 3c). 

27. (rt2 + 46)(rt2-46). 28. (1 + 7c) (1 - 7c). 

29. {a-b + c + d){a-b-c- d). 30. (a + 6 - c - rf) (a - 6 - c + d). 

31. 3ax(ax + 3)(ax-3). 32. (a^t^ + c*) (a-^t^ - c*). 

33. 12(x-l)(2x + l). 34. {9x + ly){5x + y). 

35. 1000x506. 

xxvii. (Page 51.) 

I . ((( + /;; (cr -nh + b"^). 2. (a - b) (a« + a6 + i^). 

3. (« - 2) (rt2 + 2a + 4). 

4. (x + 7) (.r--7x + 49). 

5. (6-5) {b- + 56 + 25). 6. (x + 4?/) (x^ - -ixy + 16?/2). 
7. (a-6)(rt2 + 6rt + 36). 8. (2x + 3^) (4x2 - 6xj/ + 9i/»). 
9. (4a - 106) (Ifia^ + 40a6 + 10062). 

' 10. (9x + Sy) (8 lx-2 - 72xy + 64 j/2). 

II. {x + y) {jc- - xy + y-) {x - y) {x- + xy + y-)- 



ANSWERS. 353 



20. n^. 


21. 25 -z. 


5. x-5. 


26. 1/ + 7. 


29. 2. 


30. 2. 


34. 5. 


35. 10. 



12. (x+l)(x2-x + l)(x-l)(a;2 + x+l). 

13. (a + 2) (a2 - 2a + 4) (a - 2) (a2 + 2a + 4). 

14. (3 + 2/)(9~3y + 2/2)(3-2/)(9 + 37/ + 7/2). 

xxviii. (Page 51.) 

I. a + 6. 2. Take 6 from a and add c to the result. 

3. 22/., 4. a -5. 5. x + l. 6. x— 2, x-1, x, x+l, x + 2. 

7. 0. 8. 0. 9. da. 10. c. II. x-i/. 12. x-y. 

13. 365 -6x. 14. x-10. 15. x + 5a. 

16. A has X + 5 shillings, B has 1/ - 5 shillings. 

17. x-8. 18. xy. 19. 12-X-2/. 
22. y — 25. 23. 256r/i*. 24. 4b. 
27. x2_^2 28^ (x + 2/)(x-?/). 
31. 28. 32. 7. 33. 23. 

XXiX. (Page 53.) 

1. To a add b. 

2. From the square of a take the square of h. 

3. To four times the square of a add the cube of b. 

4. Take four times the sum of the squares of a and b. 

5. From the square of a take twice b, and add to the result 

three times c. 

6. To a add the product of m and b, and take c from the 

result. 

7. To a add m. From b take c. Multiply the results 

together. 

8. Take the square root of the cube of x. 

9. Take the square root of the sum of the squares of x and y. 

10. Add to a twice the excess of 3 above c. 

1 1. Multiply the sum of a and 2 by the excess of 3 ab^ve c. 

[S.A.] g 



354 ANSWERS. 



I. 


2. 


2. 


0. 


3- 


17. 


4- 


31. 


7- 


105. 


8. 


27. 


9- 


14. 


10. 


120. 


•3- 


30. 


14. 


5. 


15- 


3. 


16. 


4. 



12. Divide the sum of the squares of a and h by four times 

the product of a and h. 

13. From the square of x subtract the square of y, and take 

the square root of the result. Then divide tliis result 
by the e.xcess of x above y. 

14. To the square of % add the square of ?/, and take the 

square root of the result. Then divide this result by 
the square root of the sum of x and y. 



XXX. (Page 53.) 

5. 20. 6. 33. 

II. 210. 12. 1458. 
17. 49. 18. 10. 
19. 12. 20. 4. 21. 43. 22. 20. 23. 29. 24. 41536. 25. 52. 



xxxi. (Page 64.) 

I. 0. 2. 0. 3. 2ac. 4. Ixy. 5. a^-^h"-. 

6. 4x* + (6m - 6?i) x' - (4m ^ + 9??in + 4?r) x'- 

+ (6™^?i — 6m?i-) X + 4m^n^. 

7. cr^ + dr + e. 8. - a* - 6* - c* + 2*262 + 2tt2c2 + 2i-c2. 

When c = 0, this becohies - a* - 6* + 2*262. When 
6 + c = «, the product becomes 0. When a = h = c, it 
becomes 3a*. 9. 0. 10. 34. 

12. (a) (a + 6)x2+(c + rf)x. (/S) (a-6)x3-(c + (Z-2)x2. 
(7) (4-a)x3-(3 + ?))x2-(5 + c)x. (5) a^ - 62 + (2a + 26) x. 
(e) (7)1.2 _ ^2^ a^s ^ ^271! 2 — 2?((;) x^ + (2wi — 2?i) x2. 

1 3. .x^ _ ((,, 4. 6 + c) x2 + (((6 + rtc + 6c) X - a6c. 

14. x^ + (« + 6 + c) x2 + (rt6 + ac + 6c) X + a6c. 

15. (a + 6 + c)3 = a3 + 3a-6 + 3rt62 + 63 + c3 + 3rt2c 

+ 6a6c + 36'-c + Zac^ + 36c2. 
(d + 6 _ c)3 = a^ + 3a26 + 3<»6'- + 6^ - c^ - 3rt2c 

- ^a\)c - 36-c + 3(a-2 + Zhc", 



ANSCVERS. 



(6 + c-a)3=-a3 + 3a26-3rt62 + 63 + c3 + 3a2c 

-6a&c + 362c-3ac2 + 35c2. 
(c + a - 6)3 = «3 _ 3f^25 + 2aV' -h'^ + c^ + ^ah 

- 6rt6c + 362c + 3rtc2 - 36c^. 
The sum of the hi?t three subtracted from the first gives 
24a6c. 
1 6. 9a2 + 6ac-3«6 + 46'--662. 17. a^^-x^^. 

1 8. 2ac - 26c — 2«fZ + 26c/. The value of the result is — 26c. 

19. a6 + a:i/ + (6+ l+2a)a; + (2a-6- 1)2/. 

20. 9. 21. 06 + 2:- + (a -6+1) a; -(a + 6 + 1)7/. 
22. 2. 23. (7m + 4?i + l)a;+ (1 -6>i — 477i)?/. 

25. 4a2 + 6ac + 2a6 + 96c-662. 26. 3; 128; 3; 118. 

27. 9. 28. 44. 29. 20. 30. 35. 31. 18. 









xxxii. 




(Page 60.) 








I. 


3. 


2. 2. 


3- 1- 


4- 


7. 5- 2. 


6. 2. 


7- 


3. 


8. 


4. 


9. 9. 


10. 


A 


?!s. 54. 








II. 


2. 


12. 9. 


13- 9. 




14. -7. 


15. 3. 


16. 


7 


17- 


2. 


18. 8. 


19. 10. 




20. 6. 


21. 4. 


22. 


lit. 


23- 


3. 


24. 15. 


25. 1. 




26. 2. 


27. 3. 


28. 


4. 


29. 


6. 


30. -1. 


xxxiii 




(Pa^e 62.) 









I. 70. 2. 43. 3.' 23. 4. 7,21. 5. 36,26,18.1:2. 
6. 12, 8. 7. 50, 30. 8. 10, 14, 18, 22, 26, 30. 9. .iC 
10. 12 shillings, 24 shillings. 11. 52. 

12. A has £130, B il50, C jElSO, D £90. 

13. 152 men, 76 women, 38 children. 14, £350, £450, £720. 
15. 21, 13. 16. £8. 15s. 17. 84, 26. 18. 62, 28. 

19. The wife £4000, each son, £1000, each daughter £5no. 

20. 49 gallons. 21. £14. £24, £38. 22.31,17 



356 AX.sirhRS. 

23. £21. 24. 48, 36. 25. 50, 40. 26. 42, 18. 

27. 60, 24. 2.S. 8, 12. 29. 88. 30. 18. 31. 4a 

32. 57, 19. 32,- -4. 34- SO. 128.. 35. 19, 22. 

36. 200, 100. 37. 23, 20. 38. 53. 318. 39. 5, 10, 15. 

xxxiv. (Page 68.) 
I. a%. 2. x-y-z. 3. 2x-y. 4. 15m2?ijs. 5. 18a&c(f. 
6. a2j2_ 7_ 2. 8. 172)2. 9- 4a;2j/222. 10. SOxV- 

XXXV. (Page G9.) 

I. a-h. 2. a'^-fcl 3. a — x. 4. a + x. 5. 3x + l. 
6, l-5a. J. x + y. 8. x-y. 9. x-1. 10. 1+a. 

XXXVi.' (Page 70.) 
I. 3453, 2. 36. 3. 936. 4. 355. 5. 23. 6. 2345. 

xxxvii. (Page 74.) 

I. x + 4. 2. x+10. 3. x-7. 4. x + 12. 

5. x-3. 6. x + 2y. 7. x-4!/. 8. x-l5y. 

g. x-y. 10. x + y. 11. x-y. 12. x + y. 

13. x + y. 14. a + 6- c. 15. -ix + y. 16. 3x-!/. 

17. bx-y. 18. x* + x^- 4X- + X + 1. 19. x--2x + 4. 

20. x^ + xy + y"^. 21. x^ + x"- — x-1. 22. 3a^ + 2a6-6-. 

23. Zx — y. 24. 3x-lli/. 25. 3a-6. 

26. 3(a-x). 27. 3x-2. 25. 3x2 + al 

29. x2 + 2/l 30. x + 3. 31. (3a + 2x)a-. 

xxxviii. (Page 76.) 

I. x-f2. . 2. x-1. 3. x + 1. 4. y-1. 

5. x2-2x + 5. 6. x-2. 7. J/- -2!/ + 6. 



ANSWEfiS. 357 



xxxix. (Page 81.) 

J^ 2a; _56 2x2 

aW'c^ , 4xy 3y _ 5h-c 

5" ~3~' 36c* 7* 2aa* * 4a^' 

4 5 m a 2??ix 



3x^2/^' .P ' " a + 6* ■ Sm^p — x 

1 2a + x jr^ o2 

3?/ -5x3* 4ax- — x' -*' 6c' ' 2x-3y' 

3ab . - c-2a 3 

17. sr . 18. ^. 19. -. 

' 2bc + c c + 2a ^5 

5 1 2 

20. ^ — -. 21. s srr- 22. 



2x-2y* ' 7ax-7by' ' 9abx—12cdx' 

xy 62 1 , 2a + 26 

23. .r-^. 24. ^.-. 25. TT- 26. ;;— . 

1 - X 

-7. 12- 28. -. 



Xl. (Page 82.) 

a + 5 x-5 a + l 

a + 3* * x-3' '■ 03-7* 

. ^JL^y x^-x+l 6 — +^' 

^ x + 7y ^ oi?-y^ 

x-2 X--3 x2_5a; + 6 

^* x+"4* ■ x + r ^" 3x2 -7x ■ 

x2 - 5X + 6 x2 + XT/ - y' 

3x2 _ 8x ' * x2 - xy - 2/2* 

a2 + 5a + 5 6^ + 56 m2 + 4m 

'"• a2 + a-2* ^^' 6^ + 6-5* ^^ m2 + m-6' 

a2-a + l 3ax-7a 14x-6 

"5* a2 + a+l' ^ ■ 7x2 -3x' ^7- g^x - 21a' 

g 10fl -14a2 2a62 + .3a6-5a 

' ' 15-9a-6a2' '9" 762^I"56 ^* 



358 ANSWERS. 



a^-a+l 3x-l a-b 

20. , „ ;,. 21. „ ,, 22. 5. 

a;- - 2x + 2 - 2x- + 'ix-b 

23. -^^32-- 24. 3. 25. ^^_5 • 

4x- + 9x + l 2x-3a „ x-3 

2x--3x-2" ^'^* 4x2 4.6ax + 9a2- ^^- x-^' 

m — 1 x^ + 5x 

2Q. T- 30- 5- 3'- 

6a + 26 

x2 + 4 , X3 + X--2 

^5' X- + X+1' ^ 2x^ + 2x + r ^'^' 

x^-2x + 3 x3-2x^-2x + l 

^ 2x2 + 5x-3" ^^' 4x2-7x-l " '^°* 3a2-8a 




x + 3 ■ 


x-5 
2j; + 3' 


a:2 + x-i2 


3x + 5 ■ 


a2_5a + 6 



Xli. (Page 86.) 



I. 



\2f 



5. ax. 



1 

^- 2- 


2x' 
2- 3y3- 


^•1- 


3 

7- 8- 


bkm? 
10. -j 

4pq 





4- 


by 

9ax' 


8. 


Sa^c' 


9d2' 



4- 



3mnxi/ 
423g2 



xlii. (Page 86.) 

a-h 4 (x 4- 2) (x - 4) 

~W ^' 3' ^' x(x-2) • 

(x-l)(x- 6) x-6 g (x-2)(x-5) 

x''' * ^* X — 3' * a^ ' 



1 01 n V c- a+b 

7. 1. 8. 0. 9. — ^— . 10. f. 

x—y c-a —0 

x — m + n - x — y-z* 

II. . 12. 1. 13. ~ . 

x + m-n "' x + y + z 










JJVSIV£J?S. 






359 




xliii. 


(Page 87.) 








lOae 
^- 2bx- 


3 




3. ^. 4. 


4 

36?ix" 




3 

5- 4- 


, hx 
^- 4i' 


5x 
7- I4- 




^- x-2- 




9- 


1 
x-2' 



Xliv. (Page 89.) 

I. 12a3x2. 2. 12x2?/2. 3. Sa^ftz 4. a^x*. 

5. 4ax3. 6. aW(^. 7. a3x22/2. 8. 102a2xi 

9. 20p222r. 10. l^ax^y^. 

Xlv. (Page 91.) 

I. x2^(j^a.)_ 2. x^-x. 3. a{a'^-}p). 

4. 4x2-1. 5. a3 + 63. 6. x2-l. 
7. (x5-l)(x + l), 8. (x2 + l)(x3+l). 

9. (X + I)(x3-1). 10. X*-l. 

II. x(x3-l) (xHl). 12. X (x + 1) (x^ - 1). 

13. (2a-l)(8a3 + l). 14. 2.(;2 + 2x2/. 

15. (a + 6)2 (a -6). 16. a2_ft2_ 

17. 4(l-x2). 18. x3-l. 

19. (a - 6) (a - c) (6 - c). 20. (x + 1) (x + 2) (x + 3'). 

21. (x + 2,')2(x-i/)2. 22. (a + 3)(a"''-l). 

23. x2(x2-i/). 24. (x + l)(x + 2)(x + 3)(x + 4). 

25. 12(x-?/)2(x3 + 2/3). 26. 120x1/ (x2- 1/2). 

Xlvi. (Page 93.) 

I. (x+2)(x + 3)(x + 4). 2. (a-5)(a + 4)(a-3). 

3^ (x+l)(xH-2)(x + 3). 4- (x + 5)(x + 6)(x + 7). 

5. (x-ll)(x + 2)(x-2). 6. (2-,: + 1) (x+1) (x-2). 



36o ANSWERS. 



7. (x2 + y)(x + y)(x2 + i/2)(x-2/). 8. (x-5)(x-3)(x + 5). 

9. (7z-4)(3x-2)(x2-3). 10. (a;2 + j/2)(x + i/)(x-2/). 

II. (a2- 62) (a + 26) (a -26). 

xlvii. (Page 94.) 

I. (a;-2)(a;-l)(x-3)(a;-4). 2. (a; + 4) (x + 1) (a; + 3). 

3. (a; -4) (a; -5) (a; -7). 4. (3x - 2) (2x + 1) (7x - 1). 

5. (x+l)(x-l)(a; + 3)(3x-2)(2x+l). 

6. (x-3)(x2 + 3x + 9)(x-12)(x'--2). 



xlviii. (Page 95.) 

15x 16x 9x-21 4x-9 

^' W "20"* ^' 18 ' ~T8~' 



4x-8?/ 3x2 -Bxy 20a + 256 Q>a'-%ah 

lOx- ' 10x2 • 4- iOrt-^~' 10a2 • 

48a--60ac 15a- 10c , ah-W a^-a% 
3 - 3x 3 + 3a; 



2j^2!/2 2-2y2 



1-X-" 1-X2' • l_2/4' i_y 

5 + 5x 6 a6 + ax b 

lO. 



l-x2' l-x2' • c(6 + x)' c{b + x)' 

a—c b—c 

{a'^'b)(h~c)(a-cy (a-b) (b-c) (a-c)' 

c{b-c) 6( a-6) 

o6c(a-6) (a-c) (6 — c)' abc (a - 6) (o - c) (6 - c}' 



xlix. (Page 98.) 

15X+17 71a -206 -56c 32x + 9tf 

I . 2. . 1. . 

15 84 -^ 42 

16x» + 55 x'+ 4x1/ - 55!/ , 27x2 - 2x2;/ _ ^ gj-y _ 28^2 

^ "■ 50x ■ '■ li:.-'- 



AI^SWERS. 361 

ISOffl^ + 54ffl6 + 3316^ - 20a62 S Ox^ + 64 x2 + 84a; + 4 5 

9062 • 7- gQ^2 

35rt2 + 23a6 + 2l6c-42c 2 Aa?c - Zac^ - 3ac + 7c» 

2 lac ■ ^" a-c^ 

lly2-8xY-4xy-7x' 

3a* - 7a^b + 4a%c - 5ahh + ahc^ - ¥c* 







aWc^ 






1. (Page 99.) 




2x-l 


4 ^ 2 




(x-6)(x + 5y 


- (x-7)(x-3y •"■ (i+x)(i-xy 


4- 


4x1/ 


- 1 y. a + bx 
1+x" 'c + dx' 


(x + i/)(x-3/y 


7- 


2x2 


2x-i/ 2x4- 5a 
(x-t/)2- 9- (^^^jr 


{x + y){x-yy 


10. 


1 




(a + x) (a-x)" 








U. (Page 100.) 


I. 


2 


4x 2x 86' 
1-x*" ^" 1-x*" "^^ a8~l> 


5- 


x + y 

y 


, 3x3 + 20x2 _ 32a; _ 235 


• (x + 4)(x-3)(x+7)" 


7. 


3x3- 24x2 + 60x 


-46 3x2-2ax-6a- 


(x-2)(x-3)('x 


-4)- ^- (x-a)3 • 


9- 


6 


X 


(x-l)(x + 2)(x+l)' — (x+l)(x + 2)(x + 3)" 


1 1 


3x2 

x2-r 


e-d 




■ {a + c){a + d){a + ey ^^' ' 


14- 


2. 15. 


y . 16. 0. 17. ";^-^^ 



362 ANSWERS. 



i8. 0. 19- -A^, 20. 0. 21. 0. 

a + 6 



lii. (Page 103.) 



?/ 1 3x2 y + 6 

^ ^ 3(1-^ 



^' x-1/' ^" 2 + « ^* x-'-r "^ 3(1-2/2) 



5. 0. 6. , r-^ rr. 7 

^ (x + a) (x + 6) c* -</ 

1 2 1 

1-x* ^ (x-2)(y-8;) aoc 



liii. (Page 110.) 
2x + ll 2(x-8) 



(x + 4)(x + 5)(x + 7)* ■ (x-6)(x-7)(x-9)' 

2x - 1 7 2 7/1^ + 4m2n + m?!^ 



4- rr^- 5- 



■'■ (x-4)(x + ll)(x-13)' ^ x + 3" ^" n(m + 7i)2 

, - Ilx3-x2 + 25x-l „ - 1 

6. 0. 7. ;r-7^ ,, . 8. 0. 9. :, ■. 

' 3(l-x*) ^ l + x 



liV. (Page 107.) 



I. 


16. 


2. 


12. 


3- 


15. 


4. 


28. 


5- 


63. 


6. 


24. 


7- 


60. 


8. 


45. 


9- 


36. 


10. 


120. 


II. 


72. 


12. 


96. 


13- 


64. 


14. 


12. 


15- 


28. 


16. 


1. 


17- 


8. 


18. 


9. 


19. 


7. 


20. 


4. 


21. 


5. 


22. 


1. 


23- 


1. 


24. 


3 
2' 


25. 


100. 


26. 


24. 


27. 


2 


28. 


6. 


29. 


24. 


30. 


4. 



IV. (Page 108.) 
16. 2. 5. 3. \. 4. 1- S- 8. 



AA'SU'EI^S. 



363 



^•4 

12. 12. 

18. 9. 



7. 9. 

13- 8. 
19. 9. 



8. 2. 



9. 11. 



14. 7. 15. 9. 
20. 9. 21. 10, 



10. 6. 
16. 7. 



II. 2. 

17. 7. 



14- 



25. 



c 

a + 6' 

6c — rfm 
a — 5 

3a&-2Jfe-3 



4ac-l 



a 
10. -2- 



3a +1 

„ ahd + ac 
18. — J — T. 
aa + a 

22. 1.^ 



Ivi. (Page 109.) 

3c -2a 
^" 56^T' 

6 (a + c) 
^ 1 + a 



15 



2. 

18a + 2& 



4a + 3 ■ 
19. 6-1. 



23. 6m. 



6c 
c2-6" 



o a (m - 3c + 3o) 
c- a + m 



26. 



29. 



12. 0. 
16. ^, — . 



0^6 — bc + d 

6bd + ah 
3a^l2cl 

{a + hf 
b — a' 



13- 



_6_ 

a-r 



-f- 



21. 



2a^ 

F-"i- 



3a36c + 2a%^ + «6^ 
^"^ 63 + 3a3cT3a26c + 2a2p- 



c 

ac 
T' 



27. 



30- 



a6-l 
6c + d' 

a'-e (c — d) 
Xofi'+Wjd' 



I. 


2. 


6. 


1 

7* 


I. 


9. 



16. 12. 



Ivii. ^Tage 111.) 



2. 15. 



7- 5 



2' 



3- 1- 



4- 



13" 



8. 6. 9. -7. 



12. 19. 13. 1. 14. 4. 

1 
2" *^- 8* 



17. 2. 18. \. 19. i 



7 
5- To- 



10. 6. 



15. -- 



20. 3. 



35 



364 AmiVERS. 



Iviii. (Pacre 11:1) 

4f)Q 

I. 20. 2. 3. 3. 40. 4. ~. 5. 60. 

J ^ 46 ^ 

6. 10. 7. 5. 8. 20. q. 3. 10. -^. 

II. 8. 12. 100. 13. 0. 14. -1. 15. 5. 
16. -. 17. 5. 



liX. (Page 114.) 



I. 


100. 


2. 


240. 3. 80. 


4. 700. 


5. 28,32. 


6. 


A.-\ 




7. 24, 76, 


8. 120. 


9, 60. 


10. 


960. 




II. 36. 12. 


12,4. 


13. £1897. 


14. 


540, 36. 




15. 3456, 2304. 


16. 50. 


17. 35, 15. 


18. 


29340, 1867 


19. 21, 6. 


20. 


IO5I, 13l| 



21. X has £1400, B has £400. 22. 28, 18. 

m (nb - a) n (mb -a) a + b a — b 

23. — !^ ■% -5^ . 24. -^r-, -5-. 25. 18. 

■^ n-m m-n ^2' 2 ^ 

26. £135, £297, £432. 27. £7200. 28. 47, 23. 

29. 7,32. 30. 112,96. 31. 78. 32. 75 gallons. 

33. 40, 10. 34. 20. 35. 42 years. 36. 1^ days. 

37. 20 days. 38. 10 days. 39. 6 hours. 40. I53 days. 



41. 


4- days. 




42. 1.:, hours. 


43- 48'. 


44. 


2 hours. 


45- 


abc 
, , minutes. 

ab + ac + oc 


46. 48|. 



47. 51;r, 6I.3, 47.J gallons. 48. 9_ miles from Ely. 

000 i 



ANSWERS. 365 



, , -1 ac Id ^13 

49. 14 miles. 50. -J, — . 51. 11—. 

30 

52. 42 hours. 53. 30.-- miles. 54. 50 houi's. 

55. (1) 38^ past 1. (2) 54^- past 4. (3j 10-- past'S. 

56. (1) 27-- past 2. (2) 5^ and also 38— past 4. 

9' 6' 

(3) Slj- past 7, and also 54— past 7. 

57. (1) 16^ past 3. (2) 32^ past 6. (3) 49^ past 9. 

58. 60. 59. £3. 60. ^. 61. ISidays- 
62. .£600. 63. ^£275. 64. 60. 

65. 90', 72', eC. 66. 126, 63, 56 days. 67. 24 

68. 2, 4, 94. 69. 200. 70. 2*, 5—. 

71. 30000. 72. X200000000. ' 73. 50. 

Ix. (Page 127.) 



z* + ax + 3a 
I. . 

X 


a2 + 3ax-2x' 


^" x{x-yy 


2a3 + 6a26 + 6a62 + 26» 


"^ (o-6)(a2 + 62) • 




Ixi. (Page 128.) 


8-13X 


xy V y 4 j^_^_ 


x^ + ox'^ + l 
5- 2a;2-x3 + r 


, x^-x + l a^ + a + 1 

D- - — -. • 7- • 

X a 



3«6 


ANSWERS. 




8. 


1 

X. Q. -. lO 


X 
X. II, - 


-2x]/ 




a(a2 + 2a6 + 262) 
(a + 6)2 


14. «t-L 


I- 1 


^3- 


'^' c(a-6-c)' 




Ixii. 


(Page 129.) 




I. 


13 15 


2. 


fi 6 c (i 
-1 + - + J + -. 
a c a a 


3- 


^_3 + ?_y. 
y'^ y X 3?' 


4- 


i2~¥"^18~3(j' 


5- 


6p 4} 12r 24s 
grs jjrs fqs fqr 


6. 


100 40^^40 !5 



Ixiii. (Page 131.) 

1. 2-2a + 2a2_2a3 + 2a* 

, 2 4 8 16 

2. 1 + — 3 + —4 

m m" 771"* m* 

, 26 262 265 26* 
-' a a- a"^ a* 

2x2 2x* 2x« 2x» 

x2 x^ X* X^ 
5. X + — +-2 + -3 + -4 

-' a a^ a^ a* 

^ 6 6x 6x-' hs? 6x* 
a a^ a^ a* a° 

7. 1 - 2x + 6x2- 16x3 + 44x* 

8. l + 2x + x2-x3-2x* 

9. 1+36 + 66= + 126' + 246* 

, ^ ,, 263 26* 
10. x--6.c + o^ + -^ 

X X^ 



ANSWERS. 367 



a2 a26 0^52 a^ a^ 

X x^ x^ x* x^ ' 

, 2x 3x2 4a;3 5a4 
12. 1- — + .. - , +—r.... 

n. n- n9 n* 



a a- a'" a* 
13. x^-3ax- + 2a^x + 4a\ 14. m< - lOm^ - 41to - 95. 



Ixiv. (Page 132.) 



J x^ x2 23x 1 , a^ _49a^ la 1 

9^4'''l20'''20' ^' 20 "600''" 60" 15- 



3- 


^*-^- 4. 


x*+l + - 

X* 


5 ---^ 




6. 


12 11 

a^ ac b'^ c^' 


7- l+a2 + a^- 


«■ '4'-f 


x« 
'64 


9- 


5 7 107 5 7 
X*"^2'i3-i2x2 + 6x'^6- ^°- 


¥ a* 0*^ ^• 





Ixv. (Page 134.) 

1 ,1 , m 1 

I. «--. 2. a + T. 3. m2-- + - 

X b n n^ 

, c' c2 c 1 XV 

^ a d^ d^ d^ •' y X 

6. -^ + -7 + 75. 7. -0-2 + ^2. 8. -x5-5x2+--x + 9. 

62 ""^o2- ^°- a2 ab ac"*" 62 6c "^"?- 



Ixvi. (Page 135.) 
I. -05x2 -•143x- -021, 2. •01x2+l-25x-21. 

3. -12x2 + -13x2/ --141/2. 4 -172x2- -05x2/- -3 12?/2. 

5. 0. 6. -300763. 



368 ANSWERS. 



Ixvii. (Page 135.) 

/, «o «, „ a4 , \ 

1. OiXl 1+— a: + — x2 + — x3+ ... I. 

^ aj ttj ttj / 

2. a;i/2( + -). 3. x2(l+^ + ^). 

" \z y xf \ X xV 

4- (a + 6) I (a + 6)2-c(a + 6)-d + -^l. 



Ixix. (Page 138.) 

2x2 + 3x-5 ,a2 + 5(j_i4 2«}7 

I. 46. 2. — = r — and — — r — . 3. -v- - , 

7x-5 a + 9 -^ a^+f'^ 

37x2-71/2-1922 11 

4- 24 • 5- -9- 

, 60x* + 42ax'-]07a2x2+10aSx + 14o* 
6. j2 • 

„ x' + xhi + 2w^ X - 8 x2 a 

8- tY^ Sr-- lO- —,-5- II- ^i 4- 12. :; — , 

x{T/ — y^) X + 8 1 - X* 1-6 

„ 1 ah + ac + hc + 2 a + 26 + 2c + 3 

^' ~F a6c + a6 + ac + ic + a + 6 + c + 1' 

1 6 62 52 8a252 6(a2 + 6») 

15. 2 5- 18. 4 ,4. 19. -7-5 — roi. 

■' a ttx a^x ax- a* — o* ^ a {a^ - 6^) 

a^ + ¥ 1 a + b- c 

22. 



^ (a-6)2.(a2 + 6-')" 2(x + l)2- "" a-6 + c 

A 1 ^ (x-4)(x + 2)2 
23. X. 24. 0. 25. 1. 26. ^^ -. 



27. 



X 



,2 



29- ^2 + i 30. 1. 



W X' 



-2 + 5x + 17x2-ll.r '-21x* 
3^- (3^2x'^7x?V* 



28. 


X3(x2 4 

31- 


^1 

3. 






33- 


r-' + y^ 




34. 


2. 



i 





ANSIVERS. 




369 


35- 


2a-6 . r. 
r- 36. 0. 


^9- :^,(;--i + 2/2)- 


X 
40, -, 




41. 


x2 + 3x + 3-- + -,. 

X X- 


(.^ + 2/2)2 

^^- x« + 3/* 


44- 1- 




46. 


^ + ^ A7 


1 


48. 1. 




p-q ^'' (x2 + 


l)(a;3 + l)- 




49- 


2a2 - ax - ay. 50. 


a + 6 + c 


(a3-63)2. 





Ixx. (Page lib.) 

I. x = 10 2. x=9 3. x = 8 

2/ = 3. y = 7. i/ = 5. 

4. x = 6 5. x=19 6. x = 5 

l/ = 8. i/ = 2. 2/ = 3. 

7. x = 16 8. x=2 9. x = 4 

y = 35. y=l. i/ = 3. 

Ixxi. (Page 145.) 



I. 


x=12 


2. x = 9 




3. x = 49 


4. x = 13 




y = A. 


2/ = 2. 




2/ = 47. 


t/ = 3. 


5- 


x = 40 


6. x = 7 




7. x = 5 


8. x = 6 




!/ = 3. 


l/ = 2. 




i/=l. 


y = 4. 


9- 


x = 7 
2/ = 17. 














Ixxii. 


(Page 146.) 




I. 


x = 23 


2. x = 8 




3. x = 3 


4. x = 5 




2/ = 10. 


2/ = 4. 




!/ = 2. 


t/ = 9. 


5- 


x = 2 


6. x = 7 




7. x=12 


8. x = 2 




t/ = 2. 


y = 9. 




y=9. 


2/ = 3. 


9- 


x = 3 

|/ = 20. 

rs.A.] 








sa 



370 


1 




ANSWERS. 














Ixxiii. 


(Page 


147.) 






I. 


x = 7 


2. 


x=9 


3- 


x = 12 


4- 


x= -2 




2/= -2. 




J/=-3. 




1/=-3. 




i/ = 19. 


5- 


x= -5 


6. 


x=-3 


7. 


x = 7 


8. 


1 
^ = 2 




2/ = 14. 




y=-2. 




3/= -5. 




■^ 3 


9- 


x=-2 

y=i. 




Ixxiv. 


(Page 


148.) 






I. 


x = 6 


2. 


x = 20 


3- 


x = 42 


4- 


x = 10 




2/ = 12. 




2/ = 30. 




2/ = 35. 




y = 5. 


5- 


x = 9 


6. 


x = 4 


7- 


x = 5 

• 


8. 


x = 40. 




2/= 140. 




2/ = 9. 




y = 2. 




2/ = 60. 


9- 
13- 


x=12 
2/ = 6. 

x = 6 


lO. 

14. 


x=19 
1/ = 3. 

x=19^ 


II. 

15. 


x = 6 
2/ = 12. 

1 


12. 


3201 
^~ 708 

278 
^=59- 



y= -17 



y=-^' 



5* 



Ixxv. (Page 149.) 

eg -nf ce + bf em + bn 

x^ * 2 X= '^ "^ x = 

mq — np ' bd + ae ^' ae + bc 

_ mf— ep _cd-af _ an — cm 

mq - np' ^~ bd + ae ^ ae + bc' 

de n'r + n/ , a + b 

x = — r^ c. x= — , 7- 6. x = — ^— 

c + d ■' mn +mn 2 

_ ce _ to/ — mV _ a — 6 

^~c + d' ^~mn' + m'n' ^~ 2~ " 

c(/-6c) „ 1 2b^-6a^ + d 

x= ^;; , / 8. x = -T- 9. x= 5 

a/-6(i aft ^ 3a 

_c(ac-rf) _J_ 3a'-fe^ + rf 

^~ af-bd- '-''cd- y~ 36 • 



ANSWERS. 371 



a ofi bm 

10. x = T- II. x = i — 12. x = r 

be b + e b — m 

_a + 2b _ ¥-c^ _ bm 

^ c ■ ^~~a ■ ^~b + m' 



Lxxvi. (Page 151.) 

_ 1 _ ^_ _ 6- - g'' 

'• ^~2 ^- ''~b-2a 3- -"'bd-ac 

1 2 _ ¥-a^ 

^4- '^~S^^- y~bc-a<r 

2a 61 , 1 

26 61 1 



103' ^ 5' 



x = - 8. x = - 

a n 

1 1 



Ixxvii. (Page 153.) 



I. x=\ 


2. 


x = 2 


y = 2 




y = 2 


2 = 3. 




2 = 2. 


5. x = l 


6. 


x = l 


2/ = 2 




2/ = 4 


2 = 3. 




2 = 6. 


9. x = 2 


10. 


x = 20 


t/ = 9 




2/ = 10 



3. x = 4 4. x = 5 

2/ = 5 y = 6 

2=8. 2 = 8. 

2 8. x = 5 

7- ^ = 3 , = 6 

y=-7 2!=7. 

2 = 36^. 



2=10. 



Ixxviii. (Page 15'.) 
I. 16, 12. 2. 133, 123. 3. 7-25, 6-25. 

4. 31, 23, 5. 35, 14. 6. 30, 40, 50, 



372 ANSWERS. 



7. £60, £140, £200. 8. 22s., 26s. 9. £200, £300, £260. 

10. 41, 7. II. 47, 11. 12. 35, 11, 98. 13. £90, £60 

14.60,36. 15.6,4. 16.40,10. 17.503,1072 

18. 10 barrels. 19. ^s.,\s. Sd. 20. £20, £10. 

21. 15s. \Qd., 12s. 6(f. 22. 4s. 6d., 3s. 23. 35, 65 

24. 26. 25. 28. 26. 45. 27. 24. 28. 45. 

29. 84. 30. 75. 31. 36. 32. 12. 33. 333. 

34. 584. 35. 759. 36. I 37. A 38. I 

2 7 35 19 

39.3- 40.^9. 41.41. 42.40- 

43. £1000. 44. £5000, 6 per cent. 45. £4000, 5 per cent. 

46. 31^, 18| 47. 20, U). 48. 3 miles an hour. 

49. 20 miles, 8 miles an hour. 50. 700. 51. 450,600. 

52. 72, 60. 53. 12, 5s. 54. 750, 158, 148. 

55. 15 and 2 miles. 56. The second, 320 strokes. 58. 50,30. 

5 
59. 4 yd. and 5 yd. 60. -, 6, 4 miles an hour respectively. 

61. 142857. 

IxxiX. (Page 164.) 
I. '2rt]. 2. 9af6*. 3. IIw^hV. 4, Sa?lf>c. 

5. 267a26x». 6. ISa'S^c^. 7. ~. 8. — V 

^ ' Ah 2a? 

5a-63 16x6 25a 

^- llxV '°- 171/2- ^'- "1S6- 



Ixxx. (Page 167.) 
I. 2a + 36. 2. 4fr^-3P. 3. a6 + 81. 4. 1/8-19. 

5. 3a6c-17. 6. x--Zx + b. 7. 3a;- + 2x + l. 



^N^IVERS. 373 

8. 2r2-3r+l. 9. 2u2 + 7i-2. 10. l-3x + 2x2. 

II. x3-2x2 + 3x. 12. 2^2-32/3 + 42-. 13. a + 26 + 3c. 

14. a^ + arb + aki' + h\ 15. x^-2x- — 2x-l. 

16. 2j;2 + 2ax + 46-. 17. 3 - 4x + 7x- - lOx^. 

18. 4a2_5o6 + 86x. 19. Za^-Aap^-bt. 

20. 2i/2a; - 3yx2 + 2x^. 2 1 . 5x^2/ - 3x?/2 + 2 y^. 

22. 4x2 - 3x2/ + 2i/2. 23. 3a -26 + 4c. 24. x^ — Zx + b. 

25 . 5x - 2i/ + 3^. 26. 2x2 - y + i/2. 

IXXXi. (Page 168.) 

3 a - 1 

5. x2_a;+ 6. x2 + x--. 

8. x2 + 4 + ^. 9. -i^a^x + ia---. 

x' 6 4 

II. 6m --+^. 
n 5 

2x 3i/ z 

13. — --^ + -. 

"^ z z X 

a b e _d 
^5- 3"4''"5 2" 

« o ftX , 

17. 3x2- — + OX. 



Ixxxii. (Page 170.) 



I. 


4 


4- 


a b 
b^a- 


7- 


2a-36 + ^. 
4 


10. 


1 2 3 

+ -. 

X y z 


12. 


ab - 3cd + Y- 


14. 


2m 3» 


16. 


7x2-2x-| 


18. 


3x2-1-3. 



I. 2a. 


2. 3x2t/2. • 


3- 


- binn. 


4- 


- 6o<6. 


5. 7V>c<'. 


6. -lOafc^c*. 


7- 


- I'lm'n^. 


8. 


llo'6«. 



%1A 






AJ^SWMRS. 










Ixxxiii. (Page 172.) 




I. 


a-h. 


2 


!. 2a + 1. 3. a + 86. 


4. a + 6 -r c. 


5- 


x-y + z. 




6. 3x2-2x+l. 7. 


1 - a + o2 


8. 


x-y + 2z. 




9. a2-4a + 2. 10. 


2m2-3m+l. 


II. 


x + 2y — z. 




12. 2m-3;i-r. 13. 
IXXXiv. (Page 173.) 


1 
TO 4- 1 . 

TO 


I. 


2a - 3x. 




2. 1 - 2a. 


3. 5 + 4x. 


4- 


a-h. 




5. x+1. 
IXXXV. (Page 175.) 


6. TO - 2. 


I. 


±8. 




2. ±ah. 3. ±100. 


4- ±7. 


5- 


± v'(ll). 




6. ±8a2cl 7. ±6. 


8. ±129 


9- 


±52. 




10. ±4. II. ± J(' 


7?l / 


12. 


W(.- 


I> 


13. ± x/6. 14. ±2v^2 
Ixxxvi. (Page 179.) 




I. 


6, -12. 




2. 4, -1(5. 3. 1, -15. 


4. 2, -48. 


5- 


3, -131. 




6. 5, -13. 7. 9, -27. 
Ixxxvii. (Page 180.) 


8. 14, -30. 


I. 


7,-1. 


2. 


5,-1. 3. 21, -1. 


4. 9, - 7. 


5- 


8,4. 


6. 


9, 5. 7. 118, 116. 


8. 10±2v'34. 


9- 


12, 10. 


lO. 


14,2. 
Ixxxviii. (Page 181.) 




I. 


3, -10. 




,^ ■■ 7 25 

2. 12, -1. 3- 2' -y 


4. 20, -7. 


5- 


1 5 

4' 4" 




6. 9, -8. 


7. 45, -82. 


8. 


8, -7. 




9. 4. 15. 


TO. 290, 1. 



ANSWERS. 37S 



Ixxxix. (Page 182.) 



I. 


7 
3' 


5 
3' 


2. 


1 3 
5' 5" 


3- 


3.^- 


4- 


1, 


3 
11* 


5- 


3 5 

5' f 


6. 


^.-1 


7. 


8, 


2 
3" 


8. 

xc. 


(Page 182.) 






I. 


3, 


8 
3' 


2. 


u.,-|. 




3- ^. -¥■ 


4- 


8, 


19 
2* 


5- 


^.-^- 




^- ^. 1 


7. 


8, 


17 
4" 


8. 


7 3 
2' 14" 







xci. (Page 184.) 

I. -a±V2-a- 2. 2rt±^/ll.a. 3. 2'~"2* 

, ^ , cfi + ah a^ — ah 

5. 1, -a. 6. 6, -a. 7. — • 

•^ a-b ' a + b 

c+ J{c'^ + 4ac) c- ^ (c^ + 4ac) 

^" 2 (a + 6) ' 2 (oTi) ' 



4- 


Zn, 


n 
~2' 


8 


d 


h 




c' 


a 


0. 


62 


¥ 


ac' 


ac 



2a -b 3a + 26 
II. , , 

ac be 



ac^ + bd^ ac- + bd^ 

2a + 3d Vc' ~2a^3f?v'c' 



XCii. (Page 185.) 



I. 8, -1. 2. 6, -1. 3. 12, -1. 4. 14, -1. 

9 
4' 



5. 2, - 9. 6. 6, 5. 7- 5, 4. 8. 4, - 1. 9. 8, - 2. 



376 ANSWERS. 



lo. 3, -^. II. 7,^. 12. 12, -1. 13. 14, -1. 

14. \-\ 15- 13, -y. 16. 5,4. 17. 36,12. 

o ^ c 25 5 „ 10' „ 10 

18.6,2. 19.18,-3. 2°-7,-y. 31. .,-y. 

^ r o 1 12 2 1 

22.7,-5. 23.3,-2. 24.2,-3. "5.3.-,. 

26. 15,-14. 27. 2, -|. 28. 3, -^. 29. 2,|. 

o 23 o 14 .5 o 21 

30. 2, --. 31. 3, -— . 32. 4, -„. 33. 3 



15' -^ ■ ' 3' -^- ' 3' •'■^" ' ir 

58 
13" 



58 
34. 14,-10. 35. 2,--. 36. 5,2. 37. -a, -6. 38. -a,h. 



, , o , a 2a. ft 6 

39. a + 6, a -6. 40. a-, -a'. 41. ^, -y. 42. p -. 



xciii. (Page 187.) 

I. a; = 30 or 10 2. a; = 9or4 3. 2; = 25 or 4 

y = 10 or 30. y = 4 or 9. j/ = 4 or 25. 

4. 2 = 22 or -3. 5. x = 50 or - 5 6. a;=100 or - ] 

j/ = 3or-22. i/ = 5or-50. i/=lor-l()0 

XCiv. (Page 187.) 

I. x = 6or-2 2. a;==13or-3 3. x = 20or-6 

y = 2 or — 6. i/ = 3or — 13. y = Gor-20. 

4. x = 4 5. a;=10or2 6. x = 40 or 9 

y = 4. 1/ = 2 or 10. j/ = 9or40. 

XCV. (Page 188.) 

I. a; = 4 or 3 2. a: = 5 or 6 3. a- = 10 or 2 

i/ = 3or4. J/ = 6 or 5. t/ = 2 or 10. 

4. a;==4or-2 5.a; = 5or-3. 6. x=7or-4 

y = 2 or — 4, 1/ = 3 or - 5. y = 4 or - 7. 



i 



AA'SIVERS. 377 



XCVi. (Page 189.) 
1 . 2 = 5 or 4 2. a; = 4 or 2 

y ==-4 or 5. y = 2 or 4. 

1 

4 a; = 3 5. x = ^ 

y = 4. y = 2. y = 

xcvii. (Page 191.) 



3- 


1 1 

x = — or — 
3 2 




1 1 
^ = 2°^ 3 


6. 


1 



I 



I. 


j:=^-i or- 


-3 


2. 


x=±6 




3- 


x=±10 




y — 'i 01- 


-4. 




2/= ±3. 






y=±n. 


4- 


x=±8 




5- 


2 = 5 or 


3 


6. 


95 

x = 5or-^ 

33 

!/ = 2or-y. 




y=±2. 






y = 3 or 


5. 




7- 


x=±2 

y=±5. 




8. 


x=6 
y=5. 




9- 


z=±2 
2/=±l. 


10. 


x=±2 




II. 


2=±7 




12. 


11 
2=3 or - 

!/ = 2or|. 




y=±3. 






i/=±2. 




















'3- 


a- = 10 or 


12 


14, 


x = 4 or 


85 
"8" 
19 
8" 


15- 


2=±9 or ±1-2 




?/ = 12 or 


10. 




t/ = 9 or 




y=±l2 or ±9 



xcviii. (Page 193.) 

I. 72. 2. 224. 3. 18. 4. 50, 15, 5. 85, 76. 

6. 29, 13. 7. 30. 8. 107. 9. 75. 10. 20, G. 

II. 18,1. 12. 17,15. 13. 12,4. 14. 1296. 15. 56^ 

16. 2601. 17. 6, 4. 18. 12, 5. 19. 12, 7. 20. 1, 2, 3. 

21. 7,8. 22. 15,16. 23. 10,11,12. 24. 12. 25. 16. 

26. £2, 5s. 27. 12. 28. 6. 29. 75. 30. 5 and 7 liours. 

31. 101 yds. and 100 yds. 32. 63. 33. 63 It., 45 It. 

34. 16 yds., 2 yds. 35. 37. 36. 100. ^j. 1975, 



378 ANSWERS. 



XCix. (Page 199.) 

1.35 = 3 2. x = b 3. 05 = 90, 71, 52. ..down to 14 
i/ = 2. 2/ = 3. |/ = 0,13,26 upto52. 

4. a: = 7, 2 5. x = 3,8,13... 6. x = 91, 76, 61 ...down to 1. 
y=\,A. i/ = 7,21,35... i/ = 2,13,24 up to 68. 

7. a; = 0, 7, 14,21,28 8. a; = 20,39... 9. a; = 40,49... 

i/ = 44,33, 22, 11,0. i/ = 3,7... i/=13,,33... 

10. a- = 4, ll...uptol23 II. x = 2 12. x = 92,83....2 

1/ = 53, 50... down to 2. y = 0. y=l, 8... 71. 

4 3 8 2 

13. i^ and-. 14. yyand— . 15. 3ways, viz. 12,7,2; 2,6,10. 

16.7. 17.12,57,102... 18.3. 19. '2. 

21. 19 oxen, 1 sheep and 80 hens. There is but one other 

solution, that is, in the case where he bought no oxen, 
and no hens, and 100 sheep. 

22. A gives 5 11 sixpences, and B gives A 2 fourpenny pieces. 

23. 2, 106, 27. 24. 3. 

25. A gives 6 sovereigns and receives 28 dollars. 

26. 22, 3 ; 16, 9; 10, 15; 4, 21. 27. 5. 28. 56, 44. 
29. 82, 18 ; 47, 53 ; 12, 88. 30. 301. 

C. (Page 205.) 

^27 2 2 3 jf 

(1) I. x"2+x^ + x2. 2. x^y + x^y^+x'y. 

4 5-fi. 121312 

3. o5 + a=i+a2. 4. x-^yz^ + a^yH + a-''yz-'- . 

(2) I. x-' + ax-^+b^x-' + 3x-*. 2. x^y-* + 3xy-* + 4y-*. 

x^y~h~^ 5xhrh~^ , , 

3. ^^ +7 — +a-r*2"'- 

4. ^^ + ^g'- + x-''y-*z. 

11 1_ ^ 1_ _1_ J^ 



ANSWERS. 379 



4 3 1 

a ^b~c^ X ^y 
1 1 1 



(4) :. 24/.^.34/(x,^).i,. 2. ^-^^ + i 



ci. (Page 206.) 

I. x*'' + x'^''y^'' + y*''. 2. a*"-81i/*". 

3. x*'' + 4aV + 16a'», 4. tr" + Sa^c' - 6=" + c^ 

5. 2a- " + 2a'"6" - 4a"'c'' - a'"b - b"+^ + 26c' + crc- + b"c^ - 2c'^"-. 

6. x^' + x""'"". 2/""'"" — ^''-j/""- 7/"'"""+"'. 7. x*'' + ar"y-'' + y*\ 

8. a='^ - a'''-'' b"^ + a^^-^ c" + a''°+'' . S^-^* - 6 + S'-'" c" + a''^+'' c'-" 

- b^^c^-" + c. 

9. x*' + 2x«'' + 3x2'' + 2x''H-l. 10. x*''-2x'"' + 3x-''-2x''+l. 

cii. (Page 207.) 

1. x^" + ar'^y'" + x'"if"' + y^"'. 

2. x^" — x'"!/" + x^'y-" - x^'y^" + y*". 

3 . r'' + x^'i/' + x^'y^ + x-'y^' + x'y*' + ]f'. 

4. a'^" - a"6^ + a*''6*» - a^''&*'' + h^. 

5. x'-* + Sx** + Qx-" + 27x'' + 8] . 

6. a-" - 2a"'x" + 4x-". 7. 2-x'' + 3x*^ 

8. 46"'c"'-552"'. 

9. a'"' + 3«="' + 3a'" + l. lo. a"' + 6'' + c', 



38o ANSWERS. 



ciii. (Page 208.) 
I. x-3a;^ + 3x^-l. 2. y-\. 3. a^-x^. 

4. a + h + c-ZaH^c^. 5, 10a;-lla;*i/* + 5xi?/*-21?/. 

4 12 
6. vw — w. 7. m-^ + 4d^m3 + 16rf. 

8. 16a + 8a"^6^ + \Qah^ + 18a^6^ - 2Aah^ - 12a^6^ - 15a^ b? 
-276. 

fi 1 2 a 1 1 A 

y. -. +2a^x^ + a^. 10. x3-2a3x3+a3. 

4 3 2 A '^ i 

II. x^ + 2x^y^ +y^. 12. a- + 2a65' + 62^ 

13. x-4x* + 10x2-12x* + 9. 

1 4. 4x* + 1 2x' + 25x'^ + 24x'^ + 16. 

15. x^ - 2x^2/^ + 2x^z* + 2/* — 22/*2* + 3*. 

16. x2 + 4x%4 - 2x*a* + 41/2 - Ay^z^ + a^^ 

Civ. (Page 209.) 
I. x^ + j/* 2. a^ — 6^, 3. yfl -{-T^y* + y'\ 

2112 ^ZlZS^XSik 

^. a^-a^h^ + o^. 5. x^ -x^y -\-x^y^ -x^y ^y. 

6. m 8 + m* w* + m2 71,3 4. m* «, 2 ^ jnByj,^ + n^. 

7. X* + 3x^1/^ + 9x*i/2 + 27t/^. 

8. 27a^ + 18a^6i + 12ai6^ + 86^. 9. a^-xi 

10. ?>i"+3m5 + 9m5 + 27m5 + 81. 

11. x2 + 10. 12. x^ + 4. 13. -h + 2h^-h^. 

a 11 1152 11 

14. x^ -x^y^ —x^z^ +y^ + z^ —y^z-^. 

15. x3-9x3-10. 16. m^ + 1111^11^ + 11^. 

11 111 11 

J7. 'p~-2p^ + \. 18. x- -(/--;:-. 19. x'^+y^. 



ANSIVERS. 381 



CV. (Page 210.) 
I. a-2-&-2 2. x-«-6-*. 3. a;4-x-4. 

4. iC' + l+oj-*. 5. a-'*_j-4_ 5_ a-2 + 2a-ic-i-6-2 + c-2. 

7. l + a26-2 + a*6-4. 8. a^J"* - a-*6* - 4a-262 _ 4. 
9. 4a;-5-x-* + 3a;~' + 2a;-2 + x-^-Vl. 

cvi. (Page 211.) 

I. x-x-^. 2. a + 6~^ 3. m2-mn-i + w,-2. 

4. c* + c3ci-i + c2i-2 + cd-3 + d-4. 5, xy-i + x-Y 

6. a-'^ + a-'^h-^ + h~\ 7. a;2?/-2 _ 2 + a;-2,^2_ 

8. |x-3-5x-2 + lx-i + 9. 9. a26-2-l + a-262 
I o. a-2 - a-ift-i - a-ic-i + 6-2 - &-ic-i + c-2. 

cvii. (Page 211.) 

2 11 y?*+I2 

I. x*-2x%2 + 2y. 2. X '"-' • 

108+18a 
3. X 3i-2 . A 



2a 



(x*-a*)^' 

^ , 22 _, 421 _, 10 . 1 
5. 7x-*+ g-x 3-^^a; «-yx-i + -. 6. x". 

7. x"-i/". 8. a2 + 2a26'5-2a^65-6^ 

9. a^ + a3 63+63. u. 771 = ™"^'. 12. x^+2H-2=, 

13. x^*. 14. I6a'^. 15. a'^'-p. 

16. 20^"" + 2a" 6'' - 4(( c"-3a"6-3/>''+i + 66c". 17. c. 



382 AJ\rslVE/?S. 



19. x^ + x^-hl. 

20. a"^ + 2a'"+»-* . bcu^ - a""^-^ b-x- - a"^-^ c V. 

21. x'^'-"-i/^'^". 22. a"~^ 23. x^'-tf". 

~^' ''' 144 ^^' ic'"''-xV~''"--c""""""i/" + r"- 

26. x + Zx^-2x^-7x^ + 2x~^. 

cviii. (Page 215.) 
I. 4'x3, 4'j/'. 2. '4^(1024), jys. 

3. 4/(5832), 4/(2500). 4. •";'2", "■;/2'-. 5. ';/a', 'V^/h". 

6. 4/(a2 + 2a6 + 6'-0, 4/(a3-3a26 + 3a52-63). 

cix. (Page 217.) 

I. 2v'6. 2. 5^'2. 3. 2a Ja. 4. oa-d ^(5d). 

5. 4zV(2y2). 6. 10x/(10a). 7. 12c ^'5. 

V5x „ ,a 

II. (a + x).,Ja. 12. {x-y)i^lx. • 13. 5(a-6).^'2. 

14- (3c2-t/).V(7l/). 15- 3u^4'(26-). 

16. 2x?/2 . 4/(20a»7\ 17. 3m3„3^/(4„). 

18. va^fe^ 4/(46). 19. (x + y).^x. 20. (o-t).4/a. 

ex. (Page 217.) 
I. V(48). 2. V(63). 3- 4^(1125). 4- v.96). 

5. I~. 6. V(9a). 7. >v'(48a^x). 8. ^'{Zah . 



AmWERS. 383 



CXi. (Page 218.) 

The numbers are here arranged in order, the highest on the 
left hand. 

I. x^3, 4/4. 2. J\0, 4/15. 3. 3^/2,2^/3. 

6. 2 ^/87, 3 s^33. 7. 3 4'7, 4 ^'2, 2;'22. 

8. 5 4/I8, 3 ^'19. 3 s"^:^. 9- 5 ^'2, 2 ^^14, 3 4/3. 

10. |x'2,^,^3,i,'4. 

cxii. (Page 219.) 

I. 29 ^'3. 2. 30 ^ao+ 164^/2. 3. {a^ + h^ + c^) ^'x. 

4. 134/2. 5. 33 4'2. 6. V6. 7. 5^'3. 

8. 48^2. 9. 44/2. 10. 0. II. 4v'3. 

12. 2^'(70). 13. 100. 14. 3a6. 15. 2a6 4/(126). 



,6. 2. ,7. I .8. 4/? ,9. J 

V. 



X 

20 



.+X1/ 

cxiii. (Page 220.) 

I. ^{xy). 2. s'{xy-y^). 3. x + !/. 4. s'{v^-y^). 

5. ISx. 6. 56(.f+l). 7. 90v'(-r--x). 8. 2x^3. 

9. -X. 10. 1-x. II. -12x. 12. 6rt. 

1 3. - s'{^ - 7x). 1 4. 6 v^(x2 + 7x). 15. 8 (a2 - 1 ). 
16. -6a2+12a-18. 

CXiv. (Page 221.) 
1. x + 9^'.r+14. 2. x-2^/x-15. 3. a. 

4. a-53. 5. 3x + 5^/x-28. 6. 6.v-54. 7. 6. 

8. V(9x^ + 3x) + ^'(6x2 - 3x) - ^'(6x- - x - 1 ) - 2x + 1 . 



3^4 AmWE}?S. 



s/iax) + sj{ax — x^) - js/{a^ -ax)-a + x. 

3 + X+ ^{Sx + x"^). 11. x-y + z + 2s,/xz. 

2x + 2j{ax). 13. 4Z2 + 42j{x^-9)+x^. 

2x+ll+2V(a;2+lla; + 24). 15. 2x - 4 + 2 ^(a;''' - 4x). 

2a;-6 + 2V(a;2-6x). 17. 4x + 9-U^x. 

2x-2s/{x--y-). 19. x- + 2x-l-2y/{x?-x). 

x2 + l + 2V(a^-x). 



cxv. (Page 222.) 

I. {^c+ ,^d)(Jc- ^fd). 2. {c+ ^d){c- ^d). 

3. ( s'c + d){Jc-d). 4. (1 + -Jy) (1 - Vy). 

5. {\+ ^•i.o-:){\~ >JZ.x). 6. (V5.m + l)(V5.m-l). 

7. |2a+ v/(3x)n2a- V(3x){. 8. |3 + 2V(2n)} j3-2v/(2n){. 

9. U'(ll).« + 4nV(ll).«-4i. 10. {p + 2^r){p-2Jr). 

II. (v'p+ V3.2)(ViJ- V3.2). 12. {a" + 6^'na"-6^}. 

«W6^ oW^. 15. 24+17^2. 

■^ a- -b a — o 

16. 2+^2. 17. 3 + 2^3. 18. 3-2^/2. 

a + x + 2 yj{ax) 1 + x + 2 ^x 
IQ. • 20. ^ . 

^ a-x 1-x 



0+ V(a2-x2) 



22. TO-- v'(w*-l)- 

2a2-x2 4-2aV(a2-x2) 



23. 2a2 - 1 + 2a V(a2 - 1). 24- 

CXVi. (Page 224.) 

I. 19. 2. 11. 3- 8-26v/(-l). 4- 5+4^/3. 

5. 2h-\-2 ^'{ah)-\2a. 6. a- + a. 7. i^-a'. 



I 



AA'SW^/^S. 385 





cxvii. (Page 224.) 


I. 


x+y ^ x+y 
SJixy)- 3. 2^{xy)' 


5. x^- v'2.ax + a-. 


6. 


vi^+ ^f2.vin + n^. 7. 2x i^x. 


2« v''^ - 26 Va 
0. - , . 
a-b 


9- 


ah J „ fd 
-^+cd-2ac^-^. 10. 


-'^'-^ + aU 


II. 




x-l 
'3- ,,.• 



14. l-^\/{^-^y ^5. 2x-2V(x2-a2). 

16. a'^¥c. 17. -l4-5a2(2-a2) + a(10a2 a4_5)^(_l) 

18. 8 + 74/3. 19. 4^/(3cx). 20. x^^CS^?'"). 

21. -4/(-47i0. 22. (9?i-10).^7. 23. 0. 



t cxviii. (Page 228.) 

"l. v/7+V3. 2.^/11+^5. 3. ^/7-v/2. 4.7-3^5. 
5. v/10- V3. 6. 2^5-3^2. 7. 2^3- v/2. 8. 3^11-2. 
9. 3 V" - 2 -v/3. 10. 3 V7 - 2 v/6. 11. h ^10 - 2). 12. 3 ^/5 - 2 ^3. 



I 



cxix. (Page 229.) 

•I. 49, 2. 81. 3. 25. 4. 8. 5. 27. 6. 256. 

7. 27. 8. 56. 9. 79. 10. 153. 11. 6. 12. 36. 
13. 12. 14- '^ — 2^' 15- 5- 16. 6. 

17. 3. 18. 10. 19. _. 20. ^3^ 

.rs,A.i 2 p 



:86 ANSWERS. 



cxx. (Page 231.) 
I. 9. 2. 25. 3. 49. 4. 121. 5. 1^. 

7.0,-8. 8. (-2-)- 9. (-^). 



6. 8,0. 
10. 5. 



cxxi. (Page 231.) 



I. 26. 2. 25. 3. 9. 4. 64, 5 ^^ 



o 



6. -^ 7. a. 8. \ or 0. 9. 64. 10. 100. 

5 4 

cxxii. (Page 232.) 
I. 16, 1. 2. 81, 25. 3. 3, 2^. 4. 10, - 13. 

5. 5.? 6. -4.-32. 7.9,-3? 8. 28.1||f 

9. 49. 10. 729. II. 4, -21. 12. 1 or—. 13. ±24. 

145 25 

14. 5 or 221, 15. 5or— r-. 16. 5 or 0. 17. i^. 18. 25. 
^ ^ 121 ' 36 

19. ±9^/2. 20. ± ^/65 or ± *y5. 21. 2a. 

22. -2a. 23. gOJ^-^e- '^- 4- "=5- 1-^. 

26. ^1^-. 27. ^. 28. ±5 or ±3 ^'2. 29. ±14. 

1,30. 6or-y-. 31. 1. 32. ^. 2>2,. 2or0. 34. or ^j^. 

cxxiii. (Page 23.'i.) 

I. 2,5. 2. 3, -7. 3. -9,-2. 4. ba,Gh. 

_7 5 A ?^ _§^ ^ Yk^l 

5- 2' 3' l9"' 14" ^' 5 ' 6 ■ 



A.VSlVERS. 387 



8. -2a, - 3a and 3a, 4a, 9. ±2, a. 10. 0,5. 



2a - & & - 3a d e 

, — r — . 12. -, -. 

ac be c c 



CXXV. (Page 239.) 

I. a;2-llx + 30 = 0. 2. .x^ + x- 20 = 0. 3. x2 + 9x + 14 = 0. 

4. 6x2-7x + 2 = 0. 5. 9x2 -58a; -35 = 0. 5 a;2-3 = 0. 

7. x^ — 2mx + m--n- = 0. 8. x^ i^-c + — ^ = 0. 

a/3 a/3 

2 a^ - /3- 

a/3 • 

cxxvi. (Page 240.) 
I. (x-2)(x-3)(x-6). 2. (x-l)(x-2)(x-4). 

3. (x-io)(x+i)(x+4). 4.4(x+i)(x+l::^)(x+^-:!^). 

5. (x + 2)(x+l)(6x-7). 

6. (x + 1/ + 2) (.(;2 + y^ + z^ — xy — xz- yz). 

7. (a-fe-c) (a2 + 6^ + c2 + a6 + ac-6c), 

8. (x-l)(x + 3)(3x-7). 9. (x-l)(x-4)(2x + 5). 
10. (x+1) (3x + 7)(5x-3). 

cxxvii. (Page 242.) 

I. Vl3orV-l. 2. 4/ -2 or 4/- 12. 3. ^V _ 1 or 4/- ?1 . 

4. lorV-4. S.^aor^--. 6 25.,,.; 

1 L 

7- ::9F^7- ^- (D'*^^- (-D- 9- 1 or 1 ±2 VI- 

„ 1 5± V1329 

10. 3 or — - or ~ . 

3 4 



A.VSWERS. 



„ a + 6 a±2J(a2-3o) 

11. a + 2, or - --— , or - ^\ \ 

o o 

1 2. 0, or a, or ^'- — '. 



cxxviii. (Page 245.) 
I. 6 : 7, 7 : 9, 2 : 3. 2. The second is the greater. 

3. The second is the greater. 

4. ^iz^, 5. I0:9or9:10. 

« 
CXXix. (Page 246.) 

I. 2:3. 2. h:a. 3. 6 + (Z:a-c. 4. ±^6-1:1. 

5. 13 : 1, or, - 1 : 1. 6. ± s,l{rn^ + 4m^ - in : 2. 7. 6, 8. 
8. 12,14. 9. 35,65. 10. 13,11. 11. 4:1. 12. 1:5. 

CXXX. (Page 247.) 

8 8 x-v a-h+c 

1. TT- 2. pr. 3. -. 4. , 

15 9 -^ x + y ^ a-o-c 

m^ - mn + n^ , (x + 2) y 

^' !«?■■¥ wn + n^' ' (y — 4)x' 

cxxxii. (Page 255.) 

6. X = 4 or 0. 8. 440 yds. and 352 yds. per minute. 



II. x = 30, 2/ = 20. 


62 

^3- T 


9 
15- 41. 


16. 50, 75 and 80 yards. 


17- 


120, 160, 200 yards. 


19. I5 miles per hour. 




20. 1 : 7. 



21. 160 quarters, ^2. 22. £80. 23. £60. 

24. £20, 25. 90:79. 26. 45 miles and 30 miles. 



ANSWEI^S. 389 



cxxxiii. (Page 262.) 
4. 16|. 5. 5. 6. 12. 7. 3^. 8. I 

9. Aoz(P. 10. 5. II. A=Ib. 12. 64x2 = 91/3. 

13. x2 = — 3-. 14. 4x3=271/2. 18. i/ = 3 + 2x + x2. 19. 18ft. 



CXXXiv. (Page 266.) 



I. 50. 2. 200. 3. lo|. 4. -32.^ 



5. -2| 6. 40. 7. 117. 8. 0. 

9,20/ o\ 3an - 26?^ - 2a + 6 
9. x^ + y^-2{n-'2)xy. 10. r . 



CXXXV. (Page 268.) 
I. 5050. 2. 2550. 3. 820. 4. 30. 

5. 24. 6. -34. 7. ^^^^l 8. ?^^^l 

b 2 2 

7n2 - 5» w - 1 



10. -2-. 



CXXXvi. (Page 269.) 



I. -6. 
5. -2. 



2. 


X 

"25" 


1 

3- 8- 


4- 


7 
8* 


6. 


-4 









cxxxvii. (Page 269.) 

1. (I) -46. (2) 36-2. (3) ?. (4) 4-4. 

2. 155. 3, 112. 4. 888. 5. 100. 



390 ANSWERS. 



6. 6433|. 7. £135. 4s. 

8. (i) 355,7175. (2) - ISGa^, -3116«- 
(3) 161 + 81a;. 3321 + 1681X. (4) 119^, 2357^. 

(5) 8^, I74I. 

9. (i) 126, 63252. (2) 25, 2250. 

(3) 45, -1570-5X. (4) 99, -1163^. 

4 

(5) 71, 4899(1 -m). (6) 65, 65x + 8190. 



cxxxviii. (Page 271.) 
I. 6, 9, 12, 15. 2. \\, |, 0, -| -1^. 

„_5^ 5 1 ^ 1^ ? 11 



CXXXiX. (Page 272.) 
3m + TO ?n. + n m + 3?i 





4 ' 


2 ' 4 ■ 






5m 4- 3 


5mH-l 5m — 1 


5to-3 


2. 


5 ' 


5 ' 5 ' 


5 • 




6n2 + l 


5n2 + 2 5?i2 + 3 


5n2 + 4 


J- 


5 ' 


5 ' 5 ' 


5 ■ 


4- 


2x2 + 1/2 
9 ' 


^2 2x2-^2 





CXl. (Page 275.) 

I. 64. 2. 78732. 3. 327680. 4. -J-. 

^ ^ 2048 

5. 13122. 6. 16384. 7. -— . 

' ' 96 



ANSWERS. 391 



CXli. (Page 276.) 



I. 


65534 






2. 364. 


3- ;^-i- 


4- 


x^ {x- 


■1) 
1)' 




(a-x)jl- 
5" (a + u;)». (1 


(^ + ^)^t 6. 3--1, 
— a-x) 


7- 


7(2»- 


■!)• 




8. -425. 
CXlii. (Page 


43 
9- -96- 

278.) 


I. 


2. 




4 
^- 3- 


27 
3- 8-- 


^ 1- 5. ll. 


6. 


•3. 


7- 


4 


3.4 


^-1 16x5 
9- ^^3- ^°- 8x2+1- 


II. 


a2 
a-V 




12 


1 
• 9- ^3. 


x2 86 
x + y '■^- 99" 


15- 


49 
90" 






. 46 
^6- 55- 





cxliii. (Page 279.) 



I. 9, 27, 81. 


2. A, 


, 16, 64; 


,256. 


3- 2,4, 


3 9 27 


81 








^ 4' 8' 16- 


32' 










cxliv. 


(Page 


279.) 




I. (i) 558. 


(2) 800. 




/ ^ 18- 

(3) T 


(4) f 


(5) -2-- 


(^) 486- 


(7) 


1189 
2 ■ 


(8) 13^. 


(9) 1- 


(10) -84. 




(II) - 


9999 V3 

(Vio + i).V5' 


, , 3157 










5. 42. 


6. ac=b'^. 


7- 


±1. 


8. n+i-. 

471 



392 












ANSWERS. 








9- 


4. 






10, 


. 10. 




13- 


4. 




14. 642. 


1 6. 


49, 1. 








17- 


3| 6, 


4 






18. 60. 


19. 


4 3 

5' 5' 


2 
5' 


1 
5' 


0, 


1 
5' 


2 
6' 


3 

~5' 


4 
"5" 






22. 


3, 7, 


11, 


15, 


19. 




23- 


5, 


15, 


45, 135, 405. 


25. 


139. 












26. 


10 


per 


cent. 



cxlv. (Page 285.) 
1111 „ „ , 2 



^ 


6' 9' 


12' 


15* 


>• 


-^, ^, ^, L, 


3* 




6. 


3 3 

4' 2' 


0°, 


3 3 

2' 4' 


7- 


6 3 6 3 

5' 4' 11' 7' 


6 
17' 


3 

10' 


8. 


6a5?/ (n + 1) 
3?i2/ + 2a; ' 


6x1/ (% + l) 

3?ii/ + Ax — 'iy 


» 


6x1/ (n + l) 
■■' 2nx + Sy ' 






9- 


1 
4' 


1 

2' 


1 1 1 

'*' 2' 4' 6' 


5 
^"^31' 


5 5 15 
24' 17' 2' 3' 




5 

4* 


10. 


104, 


234. 


cxlvi. 


13- 
(Page 


2, 3, 6. 
290.) 






I. 


132. 




2. 3360. 


3- 


116280. 


4- 


6720. 


5. 


Ill 
8' 


6. 


40320. 7. 


3628800. 8. 125. 


9- 


2520. 


10. 


6. 




II. 4. 


12. 


120. 


13- 


1260. 



14. 2520, 6720, 5040, 1663200, 34650. 

cxlvii. (Page 295.) 

I. 3921225. 2. 6. 3. 126. 4. Ii628a 

5. 12. 6. 12. 7. 816000. S. 3353011200. 

9. 7. 10. 63. II. 62. 12. 123200. 13. ;376992 ; 52360 



ANSWERS. 393 



cxlviii. (Page 300.) 

a* + 4^x + %d?x^ + Aax^ + x*. 

66 + 66=c + 156^c2 + m?(? + 1552c* + Gtc^ + c«. 

a" + 7a66 + 21a562 + 35^453 + 35^35* + 'iXaP-lP + TaS^ + h\ 



7? + 8x^y + 283fiy^ + 563^y^ + '70x*y* + 563^y^ + 28x2y« 

+ 8xy-'-{y\ 
625 + 2000a + 2400a2 + 1280a3 -i- 256a*. 



cxlix. (Page 301.) 

1 . a^ - Ga^x + 15a'*x2 - 20a V + 1 5a2x* - 6ax^ + x^. 

2. V - Wc + 216^02 - 356*c3 + 356V - 216-V + Ihc^ - c\ 

3. 32x5 _ 240x*i/ + 720x31/2 - lO^Ox-y"^ + 810xi/* - 243i/5. 

4. 1 - 1 Ox + 40x2 _ 80x3 + SOx* - 32x5. 

5. l-10x + 45x2- 120x3 + 210x*- 252x5 + 210x6 -120x7 

+ 45x8- 10x9 + xi». 

6. a24 _ 8a2i62 + 28ai86* - 56tti566 + 70ai268 - 56a96^o 

+ 28a66i2_8o36u + 5i6^ 

9 

Cl. (Page 302.) 

1 . a3 + 6a26 - 3a«c + 1 2o62 _ 1 2a6c + 3ac2 + 863 _ 1 262c + 66c2 - c^. 

2. 1 - 6x + 21x2 - 443^ + 63x* - 54x5 + 27x6. 

3. x9-3x« + 6x"-7x6 + 6x5-3x* + x3. 

4. 27x4-54x^ + 63x3 + 44x2 + 21x3+6x^ + 1. 

5. x3+3x2-5+?3--,. 
-' X- x^ 

6i o^ + 6^ - c^ + Zah^ + 3a«6^ - 3a^ci - 36^ci + 2a^c^ 

i 1 111 
+ 365c^-6a*Z)*c*. 



594 ANSWERS. 



Cli. (Page 303.) 
r. 330a;7. 2. ^^ha}%^. 3. - IGlTOOa^^^s^ 

4. 192192a666c8d8. 5. 12870a86s. 

6. TOa^ii 7. -92378ai069and92378a96i«. 

8, I7l6a7a;6 aud iheaSx^. 



Clii. (Page 311.) 



1 1 2 _L 3_ ^ 4 

I. , l+^aJ-gX +^ga; ^28** 

, la a^ 4a^ 
i X a;2 5J.3 loa;* 

3. «^ + -| — 5+ — I n- 

3aS 9a5 Sla^ 243a^ 

4. 1 + X - -X2 + -X^ - -X*. 

I 1 1 _& „ 5 -a , 

•^ 6 54 

1 4 _i 1 2 -- i 4 
6. «-^+5-«"-^*-25-*"^'-125' 

x^ X* x^ 5x^ 
7- 2~8~16~ l28" 

_ , 7 „ 14 , 14 - 
8. l-3a^ + ^«*-,-ia«. 

9x 27x2 135 
9- ■^-T~~32' 128-'^- 

10. a=^-^y + 6^+5i^- 



ANSWERS. 395 



,5 __5^ , 35 

6'^ 72"^ 1£96 



/2\2 2 /3\i _i 3/3U 4 , 



Cliii. (Page 312.) 
1 . 1 - 2a + 3a2 - 4a'' + 5rt^ 2. 1 + 3x + Qx^ + Tls? + 81:c* 



5 „ 5 , , Zr? x^ 5x^ 



5 . a''" + 10a-'' c + 60a-'*x- + 280a-'«./;3 + 1 1 20a-' V. 

, 1 , 6x^ 21.x^ 56x 
a'^ I I a^ 



Cliv. (Page 313.) 

, x2 3a;* 5a:6 35»8 
2 8 16 128 
3x2 15^ 353.6 315^ 
^- ^^ 2""^ 8 "*" 16 "^"128 • 

2 7 98 

3. X ^X Z +^^X Z -—X . . 

j_ 3x2_5x3 35^ . 1 «' 3x* 5x6 

"^^ . 2 2 ■*■ 8 • '• a 2^3--8,j5-i6^^r 

, 1 x^^ 2x6 143.9 
■ a"3a5"''9a^~81ai<>- 



Clv. (Page 3 U.) 

I. 7^;9-;) ,^1. 3. (-i)-.12^L^(M::r) ,- 

1-^ ;••(»•-!) ^ ^ 1.2...(r-l) • • 

3. (_i)^i. 8.7...( 10-r) ., 



396 



ANSJFEJiS. 



4- 


i.V;;.y_iy • (5^)"- • (22/)-^ 5- ( - 1)--^ »• • x-^ 


6. 


r.(r+l).(7- + 2) „ 1.3.5...(2r-3) /xX-^. 
6 •^^''^ • 7. i.2.3...(r-l)-W 


8. 


1.2.5...(3r-7) / x\-\\ 
1.2.3...(7--1) "V 3a/ • 


9- 


7.9.11...(2r + 3) 
1.2.3... (r-1) •* • 


lO. 


a"2 3.7.11... (4r- 5) /^Y^-^^ 
4'-^* 1.2. 3. ..(r-1) *\cJ • 


II. 


(r+l)(r+2) 1.3.5...(2r-l) 

2 •''• '- 1.2.3..:r -^^ ^• 


13- 


1.3.5...(2r-l) 5 1 
1.2.3...r •^^''^' '5. le-^i^. 


1 6. 


3 .,„ 429 xi« 
128 •«^'- »7 -128-a^- 


1 8. 


1.2 9 •'' •^• 


19. 


(1 - 5?7i) (1 - 4m) (1-m) l-« 

1.2 6m6 •" 



clvi. (Page 315.) 

I. 3-14137.... 2. 1-95204.... 

3. 3-04084.... 4. 1-98734.... 



Clvii. (Page 319.) 

I. 1045032. 2. 10070344. 3. 80451. 

4. 31134. 5. 51117344. 6. 14332216. 

7. 31450 and remainder 2, 8. 522256 and reinainder 1. 

9. 4112. 10. 2437. 



ANSWERS. 



397 



clviii. (Page 321.) 

I. 5221. 2. 12232. 3. 2139e. 4. 104300. 

5. 1110111001111. 6. atee. 7 6500145. 

8. 211021. 9. 6^12. 10. 814. 11. 61415. 

12. 123130. 13. 16430335. 14. 27^ 



I. -41. 

4. 12232-20052. 



Clix. (Page 327.) 
2. -162355043. 
5. Senary. 



3. 25-1. 

6. Octonary. 



I. 1-2187180. 

4. 4-740378. 

7. 5-3790163. 

10. 2-1241803. 



Clx. (Page 336.) 

2. 7-7074922. 

5. 2-924059. 

8. 40-578098. 

II. 3-738827. 



3. 2-4036784. 

6. 3-724833. 

9. 62-9905319. 

12. 1-61514132 



Clxi. (Page 339.) 

1. 2-1072100 ; 2-0969100 ; 3-3979400. 

2. 1-6989700; 3-6989700; 2-2922560. 

3. -7781513 ; 1-4313639 ; 1-7323939 ; 2-7604226. 

4. 1-7781513; 2-4771213; -0211893; 5-6354839. 

5. 4-8750613; 1-4983106. 

6. -3010300; 2-8061800; -2916000. 

7. -6989700; r0969100; 3-3910733. 

8. -2, 0, 2 : 1, 0, -1. 



9. (I) 3. 



(2) 2. 



9 

10. a: = 5,i/ = ; 



ANSWERS. 



11. (a) -3010300; 1-397940C; 1-9201233; 1-9979588. (6)103. 

12. (a) -6989700; -G020600; 1-7118072; 1-9880618. 
(6) 8. 

13. 3-8821260; 1-4093694; 3-7455326. 

14. (i) x=r {2)x = 2. (3)a; = 



(4)a; = 
(5)x = 
(6)a; = 



6' ^ / • V J/ Yog a + log 6" 

los c 



log a + 2 log h' 
4 lo" 6 + log c 



2 log c + log /) - 3 log a* 

log c J 



log ft + ??i lo^ 6 + 3 log c 



clxii. (Page 343.) 

I. 17-6 years. 2. 23-4 years. 

3. 7 2725 years nearly. 4. 22-5 years nearly. 

6. 12 years nearly. 7. 1 1-724 years, 






APPENDIX. 



The following papers are from those set at the llatrinulatioL 
Examinations of Toronto, Victoria, and McGiil LTniversi- 
t^es. and at the Examinations for Second Class Provincial 
Certificates for Ontario. 



UNIVERSITY OF TORONTO. 

Junior Matric., 1872 Pass. 

1. Multiply ^x'-lxy + y^hjlid' + lxy-y'. 

Divide a* - 816* by a ± 36 and {x + af - {y - by 
by X + a — y + b. 

2. What qnantity subtracted from x^ + px + q ''ril] 
make the remainder exactly divisible by a; — a .? 

Shew that 

(a + b + cf- {a+ b + c) (a' + b'' + c' -ab -be - ca) 
- 3abc = 3 (a + 6) (b + c) {c + a). 

3. Solve the following equations : 
(a)^{2x-3) + Hi^x-7) = Ux-^). 

4a; — 7 3a; — 5 

(^)i.^r=Tn-^r=r2=20- 



i<^) 



X — 1 ^x — 2 
11-11 



4 X — 5 X — 6 



2/+I y x + 2 11 

(d) « + -2-=l' 3-^-5- =18- 

4. In a certain constituency are 1,300 voters, 
ind two candidates, A and B. A is elected Vjy a 



tt APPENDIX. 



certain majority. But the election having been de- 
clared void, in the second contest {A and B being 
again the candidates), B is elected by a majority of 
10 more than A's majority in the first election ; find 
the number of votes polled for each in the second 
election ; having given that, the number of votes 
polled ior B in the first case : number polled in the 
second case : • 43 • 44. 



Junior Matric, 1872. Pass and Honor. 

1. Multiply a; + y + 2* - 2y^ zi + 2zi a^ - 2ar4 yi by 

X + y + zi + 2yi zi — 2zi x^ — 2x^ yi, and 
divide a' + 86' + 27 c^—18abc by a* + 45» + 9 c'— 
2ab — Sac — 66c. 

2. Investigivte a rule for finding the H. C. D. of 
two algebraical expressions. 

If X + c be the II. C. D. of af + px + q, and x* -f 
p' X + q', show that 

{q-q'Y-p {q-q) (p-p) + 9 (p-pY^^^- 

3. Shew how to find the square root of a binomial, 
one oi' whose terms is rational and the other a quad- 
ratic surd. What is the condition that the result may 
be more simple than the indicated square root of the 
given binomial 1 Does the reasoning apply if one of 
the terms is imaginai y 1 Show that *y/ — 4m' = ^m 

+ ^ -m. 

4. Shew how to solve the quadratic aquation aa^ 4 
6a; + c = 0, and discuss the results of giving difierent 
values to the coethcients. 

If the roots of the above equation be as p to 9 

, 6» {p + qY 

show that — = -• 

ac 



APPENDIX. iB 

6. Solve the equations 

xy +y»-10 = 0. 



(c) 



a:' + 6ic + 2 £c'+6x+6 a:*+6a + 4 



x* + 6a; + 8 

a* + 6 a; +10* 

* • 

(i) 6 x' - 5 af* - 38 ic* - 5 aj + 6 = 0. 

6. Shew how to find the sum of w terms of a geometnp 
series. What is meant by the sum of an infinite 
series ? When can such a series be said to hav? • 
sum % 

Sum to infinity the series 1 -j- 2r + 3 r* -(- Ac. 
and find the series of which the sum of n terms i& 

aF — . 

a — \ 

7. Find Che condition that the equations 

ax-\-hy — cz — ^. 
a, a; + 6, 3/ — Ci 2 = 0. 
e«, a; + 6, y — c, » = 0. 
may be satisfied by the same values of x, y, z. 

8. A number of persons were engaged to do a })Iece 
of work which would have occupied them m hours if 
they had commenced at the same time ; instead of 
doing so, they commenced at equu! intervals, and then 
continued to work till the whole was finished, tne 
payments being proportional to the work done by 
each ; the first comer received r times as much as the 
last : find the time occupied. 



APPENDIX. 



Junior Matric, 1872. Honor. 

1. There are three towns, A, B, and C ; the road 
fi'om B to A forming a right angle with that from B 
to C. A person travels a certain distance from B 
towards A, and then crosses by the nearest way to the 
road leading from C to A, and finds himself three 
miles from A and seven from C. Arriving at ^, he 
finds he has gone farther by one-fourth of the distance 
from B to C than he would have done had he not left 
the du-ect road.* Requii-ed the distance of B from A 
and C. 

2. If ay -\r h x jx + a^ _bz + cy ^ ^^^^ ^^ 

c h a 

ay* 

a h e 



3. Solve the equations x* — yz = a*, y' — zx-b*, «* — 



a:y = c\ 

4. If a, h, and c be positive quantities, shew that 

a« (6+c) + 6» (c + «) + c« (a + 6) > %abc. 

5. Find the values of x and y from the equations 

o 5?/ + 3 

2y + -^ = 1, 

x ' 

£c* + 5x + 2/ (y - 1) = 24. 

6. A steamer made the trip from St. John to Boston 
via Yarmouth in 33 hours ; on her return she made 
two miles an hour le.ss between Boston and Yarmouth, 
but resumed her former sjieed between the latter place 
and St. John, thereby making the entire return pas- 
sage in 11 of the time she would have required had 
her diminished speed lasted throughout ; had she 
made her usual time between Boston and Yarmouth, 
and two miles an hour less between Yarmouth and 



APPENDIX. T 

St. John, her return trip would have been made in 
i-J of the time she would have taken had the whole 
of her return trip been made at the diminished rate. 
Find the distance between St. John and Yarmouth 
and between the latter place and Boston. 



Junior Matric, Honor. 1 
Senior Matric, Pass. ) 

1. Solve the following equations : 

, . i x^- 2a;?/ + 2?/" ^ xj/ 

(a) .... I 



1874. 



a? + xy + tf = 63. 
4a;— ?)xy = 171. 
Zy-A:xy= 150. 
1 1 1 
a? xy y^ 
1 1 1 o 

— . + — r-„+ -.= 133. 

X* u^y^ y" 
Am i find one solution of the equations 

{d) .. 



(&) 



y* — x*^ 68. 
a^ + sf x = y. 



2. Find a number whose cube exceeds six times the 
next greater number by three. 

3. Explain the meaning of the terms Highest com- 
mon measure and Lowest common multiple as applied 
to algebraical quantities, and prove the rule for finding 
the Highest common measure of two quantities. 

4. Reduce to their lowest terms the following 
fractions : 



\a) . 
(6) 



i Sar* + i^^^^x — 10^ ' 

x' + lOx' + 35a;- + 50a; + 24 



x' + IBa;^ + irj,/2 + 342.^- + 360 



▼1 APPENDIX. 

5. Find the sum of n terms of the series — |, \, — 
\, (fee, and the ccth term of the series 

a; + 1 2 2> — x 

6. Find the relations between the roots and co 
efficients of the equation ax* +;yx + ^ = 0. 

Solve the equation 

a;* + 6a;'+10x« + 3a;=110. 

7. A cask contains 15 gallons of a mixture of wine 
and water, which is poured into a second cask con- 
taininif wine and water in the proportion of two of the 
former to one of the latter, and in the resulting mixture 
the wine and water are found to be equal. Had the 
quantity in the second cask originally been only one- 
half of what it was, the resulting mixture would have 
been in the proportion of seven of wine to eight of 
water. Find the quantity in the second cask. 

8. What rate per cent, per annum, payable half- 
yearly, is equivalent to ten per cent, per annum, pay- 
able yearly. 

9. A is engaged to do a piece of work and is tn 
receive $3 for every day he works, but is to forfeit 
one dollar for the first day he is al)sent, two for the 
second, three for the third, and so on. Sixteen davs 
elnps • l)efore he finishes the work and he receives §26. 
Find the number of days he is absent. 

Cliange the enunciation of this problem so as to 
apply to the negative solution. 



Junior .Vatric, 1876. Pass. 

1. Explain the use of negative and fractional in- 
dices in Algebra. 

Multiply, -A by i/'«'' and the product bv V" 



APPENDIX. vii 



Simplify , writing the factors all in one 

line. 

2. Multiply together a* + ax ■¥:!?, a + x, a*-ctx + 3^, 
a-x, and divide the product by a^ - ar*. 

3. Divide 1 by 1 - 2x + x^ to six terms, and give 
the remainder. Also divide 27a;''-6x^ + ^ by Sa;^ + 
2x + J. 







a, ^ n a»-« 




■> + • 


4. 


Multiply 


a +6 


by a 


+ 6 . 


6. 


Solve the 


equations : 








(!)• 


3a; + 4 7a; - 3 
5 2 


a;-16 
4 • 




(2). 


( X (y + z) = 
■ly{z + x) = 
{z{x + y)-- 


.24, 
.45, 
= 49. 






^ior Matric 


, 1876. 


Hnvn 



1. An oarsman finds that during the firet half of 
the time of rowing over any course he rows at the 
rate of five miles an hour, and during the second 
half, at the rate of four and a half miles. His course 
is up and down a stream which flows at the rate ol 
three milus an ho\ir, and he finds that by going down 
the streani first, and up afterwards, it takes him one 
hour lojiu:er to go over the course than by going first 
up and then down. Find the length of the course. 

2. Shew that if a^ 6^ tr" be in ^.P., then wilJ h - <■, 
« + rt, a + 6 be in II.P. 

Also, if a, 6, c be in A. P., then will 



he 7 ca ah 

a + , b + , cr - 

-^ c c + a a-^ It 



be in U.P. 



APPENDIX. 



3. If s = ffl + 6 + c, then 



y/{as + be) (bs + ac) (cs + ah) = (s-a) (s — b) (s - c) 

4. If a, + fta + +a„^ —, then 

(6- - «,)*+ + (s - a„)^ = a,2 + a/+ +a''«. 

5. If the fraction - — - — - , when reduced to a re- 

2n + 1 

petend, contaijis 2n figures, shew how to infer the last 
n digits after obtaining the fii-st n. 

Find the value of -Jy by dividing to 8 digits, 

6. Solve the equations 

X — y + z-S, 
xv + xz = '2 + 1/z, 



Junior Matric, 1876. Honor. 

1. Shew that the method of finding the square 
loot of a number is analagous to that of finding the 
square root of an algebraic quantity. 

Fencing of given length is placed in the form of 
a rectangle, so as to inchnle the greatest possible area, 
which is found to be 10 acies. The shape of the 
field is then altered, but still remains a rectangle, and 
it is found that with 162 yards more fencing, the 
same area as before may be enclosed. Find the sidea 
of the latter rectangle. 

2. Prove the rule for finding the Lowest C^ommon 
Multiple of two compound algebraic quantities. 

Find the L.O.M. of a» - 6=» + c' + 3a6c and d-(b-^c) 

-^{c + a)^^ (« + b)+abc. 

3. If a, p be the roots of the equation 3^+px + g = 
0, shew that the equ;ition may be thrown into the 
form (x — a) (x- f3) - 0. 



APPENDIX. Lx 

3 + v/2 is a root of the equation aj* — 5a;' + 2a^ + a; 
f 7 = : find the other roots. 

4. (1) Shew how to extract the square root of a 

binomial, one of whose terms is rational, 
and the other a quadratic surd. 

(2) Find a factor which will rationalize x^ — y^. 

5. a, b are the first two terms of an H. P., what is 
the nth term ? 

JI a,l, che in. H. P., shew that 

h^ci - cf = 2c-{b - a)^+ 2a"(c - by. 

6. A and B are to race from M to 'N and back. A 
moves at the rate of 10 miles an hour, and gets a start 
of 20 minutes. On A's returning from N, he meets 
B moving towards it, and one mile from it ; but A is 
oveiiaken by B when one mile from J\I. Find the 
distance from ]\I to N. 

7. Solve the equations 

(1). ar' + 8-2a^+lla;+U. 
X 51 



(2). 



V JC 12 a:y 



Second Class Certificates, 1873. 

1. Multiply +_+iby f+--l. 

b a •' b a 

„ ^, , a'- 3ab + 26- a^ - lab + 1 26' 

2. Shew that —. — „, 

0.-26 a- ob 

;an be reduced to the form 36, 



APPENDIX. 



3, Reduce to its lowest terms the fraction, 
. b^ 1 



"^ "*■ 12 + 9 



^ 1 



4. (a) Prove that a^ - ?/" is divisible hj x-y with- 
ut remainder, when m is any positive integer. 

ih) Is there a remainder when a;""- 100 i>i 
^ided by a; - 1 ] If so, write it down. 

0. Given ax + by = 1, 

,xy 1 
and - + , = -^' 
a ah 

Find the difference between x and y. 
6, Given 3 - n^^zM _ j'^ t! - 

^x-\) 3(x+i) "• 

Find X in terms of wi, 

^. X 2 ^ 7a; + 16 

/. Given - =-5. Find the value of ;= n*' 

y 3 73^ + 24 

-<. Given = 1, 

x-y X -¥y 

, 6 10 

and 3. Find x and v. 

X ~y x-\-y ^ 

9. There is a number of two digits. )^y inverting 
:.Iie digits we obtain a number which is less by 8 than 

hree times the original number ; but if we increase 
bach of the digits of the original number by unity, 
and invert the digits thus augmented, a number is 
obtained which exceeds the original number by 29. 
Find the number. 

10. A student takes a certain number of minutes 
to walk from his residence to the Normal School. 
Were the distance ^th of a mile greater, he would 
need to incKMs. his pace (number of miles j^er hour) 



APPENDIX. 



by ^ of a mile in the hour, in order to reach the 
school in the same time. Find how much he would 
have to diminish his pace in order still to reach the 
school in exactly the same time, if the distance were 
■^^ of a mile less than it is. 



Second Class Certificates, 1875. 

1. Find the continued product of the expressions, 
a + b + c, c + a-6, b + c-a, a + b-c. 

a' + a^b a{a-b) 2ab 

2. Simplify ^,^~ y - ^^^ - -^-^,- 

3. Find the Lowest Common Multiple of 3a^ - 2a; - I 
and ^a?-'2.x'-^x+\. 

4. Find the value of x from the equation, ax — 

a* — '6bx 6bx — 5a^ bx + 4 i 

- — ab^ -bx+ — K — — -f 

a 2a 4 

5. Solve the simultaneous equations, 



05 V 


-jrt. 


c d 




- +— = 
x y 


-n. 



6. In the immediately preceding question, if a 
['U})il should say that, when nb — md, and be ^- ad, the 
values of x and y obtained in the ordinary method, 
have the form f, and that he does not know Iio'a- io 
interpret such a result, what would you reply ? 

7. Two travellers set out on a journey, one with 
%^ 00, the other with $48 ; they meet with robbers, 
who take from the first twire as much as they take 
from the second ; and wliat remains with the first is 
5 times that which remains witii the second. How 
i;aMir'> mor^ey did each traveller lose '? 



APPENDIX. 



8. A and B labor together on a piece of work for 
two days ; and tben B finishes the work by himself 
in 8 days ; but A, with half of the assistance that B 
coukl render, would have finished the work in 6 days. 
In what time | could each of them do the whole work 
alone % 

9. P and Q are travelling along the same road in 
the same direction. At noon P, who goes at the rate 
of j?i miles an hour, is at a point A ; while Q, who 
goes at the rate of n miles in the hour, is at a point 
B, two miles in advance of A. When are they to- 
gether \ 

Has the answer a meaning when m — n is nega- 
tive 1 Has it a meaning when.m = ?i? If so, 
state what inter^jretation it must receive in these 

cases. 

10. P is a number of two digits, x being the left 
hand digit and y the right. By inverting the digits, 
the number Q is obtained. Prove that 11 (a; + y) 
(P— Q) = 9(a;— 2/)(P + Q). 



Second Class Certificates, 1876. 

1. Divide (1 + m) o^ — {m -¥n) xy {x — y) — {n — 1) y' 
by ar^— iy + 2/«. 

Shew that {a + a^U + hY—{a — aiM + 1)^ is ex- 
actly divisible by 2ai6i. 

2. Resolve into factors x* + layii (z* — t^) — y*, 

11^ {h — c) + b^{G — a) 4- c-{a — h), »ind 25a;* + 
bx^ — X — 1, 

3. If x^ -hpci? -k- q.r + r is exactly di\'isible by a:' + 
mx + n. then nq — io^ = 7in. 

4. Prov»^ that if m be a common measure of p and 



APPENDIX. xiii 

q, it will also measure the difierence of any multiples 
of p and q. 

Find the G. C. M. oi x*—^x^ ^ {(i—\)x^ + yx— 
q and x* — qoi? + {p — l)a^ + qx — p and of 1 + 

x^ + x-¥^ and 2x + 1x^ + 3a,-* + 3a;^' 

5. Prove the rule for multiplication of fractions. 

a* — (y — zY 2/* — (2 — cc)^ «^ — {x — ?/)' 



Simplify 



(2/ + zf—zc' (z + a;)2— y* (x + yf 
a? 



and -5— Ts, — -^ — r2 + 



6. Wliat is the distinction between an identity and 
au equation ? If a; — a = y + b, prove x — b = y + o. 

Solve the equations (2 -^x) (7/1 — 3) = — 4 — '2ii'x, 
16a;— 13 40.x— 43 32a^-30 20:^24^ 

7. What are simultaneous equations ? Explain why 
there m\xst be given as many independent equations 
as there aie unkno'vn quantities involved. If there 
is a gieatei- number of squations than unknown o'^.^.n- 
tities, what is the inlerence '( 

Eliminate x anci v irom xhe eor.atioiib ax ^ by 
= c, ax + bv = c , ax + U'y = <~. 

8. Solve the equations — 

( 1 ) y/n + x-^ 'Wn — X = m 

(2) 3a; + w + s=i;j 
3'y + 2 + a;= lo 
'iz -^ x + y - 17 

9. A j)erson has two kintls of foreign money ; it 
takes a pieces of the first kind to make one £, and b 
pieces of the second kind : he is ofieied one £. for c 
pieces, Low many pieces of each kirul must he take 1 



rfr APPENDIX. 

10. A person starts to walk to a railway station 
four and arhalf miles olF, intending to arrive at a 
certain time ; but afiur walking a mile and a-liaif he 
is detained twenty minutes, in consequence of which 
he is obliged to walk a mile and a-half an hour faster 
in order to reach the station at the appointed time. 
Find at what pace he stai-ted. 

11. {a) If y = ^ then will ^j^, = ^,. 

(6) Find by Homer's method of division the 
value of 
a*+ 290a;'+ 279ar'—2892j:*—586a>— 312 when 
a; = —289. 
(«) Shew without actual multiplication that 
(a + 6 + cf — {a + 6 -^ c) (a* — ah + 6'— 6c + c*— ac) 



McGILL UNIVEE.SITY. 



First Year Exhibitions, 1873. 

1. The clifFerence between the first and second oi 
four numbers in geometrical progression is 12, and 
the difference between the 3rd and 4th is 300 ; find 
them. 

2. Find two numbers whose difference is 8, and 
t])e harmonica! mean between them 1|^. 

3. Prove the general formula for finding the sum 
of an arithmetical series. 

4. The diflerences between the hypotenuse and the 
two sides of a right-angled triangle are 3 and 6 
fe.s]»ectively ; find the sides. 

6. Solve the equations 

ce^ + 2/^ = 25 , x + y=l; 

X a; + 1 13 
a; + 1 X ~ 6 ' 
x + y + 2; = 0, x + 2/ = s-t; x-7)=y \ z 
03+4 3x + 8 

+11= . 

3a3 + 5 2a3 + 3 

6. A cistern can be filled by two pipes in 24' and 
D '' respectively, and emptied by a third in 20' ; in 
what time would it be tilled, if all three were running 
together. 

7. Shew that 

aj^ ^}? - ^ (a + b + c) (a + b-c) 
^^ 2ah ~ 2ab 



tvi APPENDIX. 

8. Prove the rule for finding the gi'eatest common 
measure of two quantities. 



First Year Exhibitions. 1874. 

1. The sum of 15 terms of an arithmetic series is 
600, and the common ditierence is 5 ; find the first 
term. 

2. Find the last term and the sum to 7 terms of 
the series 

1-4+16-&C. 

3. Find the arithmetical, geometric, and harmonic 
means between 3| and 1^. 

4. The difierence between the hypotenuse and each 
of the two sides of a right-angled triangle is 3 and G 
respectively ; find the sides. 

5. The sum of the two digits of a certain number 
is six times their difierence, and the number itself 
exceeds six times their sum by 3 ; find it. 

6. Solve the equations : — 

X- y = l; «'-?/'= 19 
3.C - 7 4a;- 10 .^, 
X + x + b -^^' 

x-\{u-1) = b\ 4?/-i- (a;+10) = 3. 

232:c+l 8^5 .^ 

7. A man could reap a field by himself in 20 hours, 
but with his son's help for 6 hours, he could do it in 
16 hours ; how long would the son be in reaping tho 
lield by himself? 

8. Find the value in its simplest form of 

x + y 2a; x' y - x^ ^ 
y ~ x + y 3^y-y^ 



tvii APPENDIX. 

9. Find the greatest common measure of 
Sar* -f 3a:^ - 15a; + 9 and 3a;^ + 3a;' - 21a;* — 9^. 



First Year Exhibitions, 1876. 

1. Solve the equations 

12a 



\a + a; + /rt — X = =-,— ^, 
\ \ O V ff + a; 

■X y X y X y 

- ^ - = 1 _ -; _ f _= 1 +_. 

a b cab c 

2. Reduce to its siuipiest form the expression ; — 

7 V54 + 3 VFe + ^' 2 - 5 4/128. 

3. Find the greatest common measure of 

2x'-h^^ — Sx + 5 and 7x' — 1 2x+5. 

4. Simplifjfc 



5. A nuoaoer consists uf two digits, of which the 
lett is twice the right, and tlie sum of the digits is 
one-seventh of the number itself. Find the n;imber. 

6. Solve the following : — 

X y X z 1/ z 

_ + 1^ + 1, - + - =2, - + - =3; 

a b 'I r. be 

1 1 

X y 

7. Find the sum o\' n terms of the series 1, 3; 5, 
7, &c. 

(a.) Shew that the reinprocals of the first four 
terms, and also of any consecutive four terms, are ir» 
harmonical proportion. 



tjNIVERSITY OP VICTORIA COLLEGE. 



Matriculation, 1873. 

1. What is the " dimetision " of a term ? WLtn k 
ail expression said to be " homogeneous " ? 

2. Remove the bx-ackets from, and simplify the 
following expression : — 

{•la — Zc + id) - \M ~ {m ^ Za)\ + |5a — {— 4 
__j)|_ ^3a — {4a — 5rf — 4)|. 

3. P'.ove the " E.ule ot Signs" in ISIultiplication- 

4. INLultiply a — hjx + . 

ax 

5. Di vide ax^ + bx'' -*- cx-\- dhy x — ♦•. 

6. L^ivide 1 by 1 + a;. 

7. ^'''nd the Greatest C mmon Measure of 6a* — 
aV — \'Ik and ya'' - \'2a^a^ — 6a-.r - Sx*. 

tt, i'rom 3a — 2c — - ~ _ subtract; 2a — x — 
x^ — 1 
a — X 

(1+1^42^ 
9. Civ en ■: ' ,> to find a; and v. 

I'l l>ivide tne iiui.i )er a into four such pai-ts that 
t!if M^coud shall SAceed the first by m, the thiid shall 
exi-i.(- 1 tilt; >^oooad oy n, ana the fourth shall exceed 
the third hv p. 

) I. .V .sum o^ moi^e^ pat out at siiople intersd 



APPENDIX. xU 



amouxits in m months to a dollars, and in n monthi 
to h dollars. Required the sum and rate per cent. 

12. Given a-' + a6 - 5x•^ to find the values of x. 

13. Divide the number 49 into two sucn parts that 
the quutient ot the greater divided by the less niay 
be to the quotient of the less divided by the greater, 
as I to |. 

14. L>ivide the number 100 into two such parts that 
thcii- product ma^ be equal to the dilierence of their 
squares. 

j" ar* 4 a;?/ — 56, ] 

15. Given I >tofind\aluesofa;aud2/. 

l«y+2/-60j 

16. A farmer bought a numbei of sheep for $80, 
and if he had bought four more for the same money, 
he would have paid %\ less for eu,ch. How many did 
he buy 1 



Matriculation, 1874. 

1. Find the Gieatest Common Pleasure of 26^ — 
iO'ib'' + 9>a% and 9^4^ — 2>a¥ + Zd^U" — 9rt '/?. rti.! de- 
moiistrate the rale. 

a* + 7^ n o* — (UB 

2. Add tc'-ether a — x + , 3a — , 

=^ a + x a + x 

„ 3a* — 2af , . a- + x 

2x — , and — 4a — ;. 

a — X a — XT 

3. Divide + by — =-' 

1+a; 1 — X 1 — X i+x- 

and reduce. 

4. Given I (x — a) — lo {2x — 3b) — ^ (a — x) 
= 10a 4 116 to find x. 

5. A sum of mone/ was divided ainoii!:^ tin ee per- 
sons, A, B, aiid C, at> follows : tJie slian of 1 
©.,.rewded 4 of the sha.-es of h and C >\v §1J0: th^ 



APPENDIX. 



share of B, f of the shares of A and V)y 81-0 ; 
and the share of C, | of the shares of A and B by 
^120. What was each person's share? 

6. Given | ^3 ^ ]f^_'^^_4^ . 12 | ^^ ^^-^ * ^'^'^ 2/- 

7. Shew that a quadratic equation of one unknown 
quantity cannot hav^e more than two loots. 

8. Given ~ .— — ; to find the vahie of x. 

4 + V a; >/x 

9. The e is a stank of hay whose len-fth is to its 
breadth ab 5 to 4, and whose height is to its Vueadth 
as 7 to 8. It is wotth as mai y cents per mibic foot 
as it is feet in breflcith; and the whole is worth at 
that rate 224 times as many cents as bhere art square 
teet on the bottom. Fjnd the cdu^ensions of the stack. 

y/ocy + b \ 

10- Given ^ /" V to find x and y. 

J^ = y/xy — 4 I 
x->ry / 

11. In attempting to arrange a number of countei-s 
in the form of a square it was found there wer«t se\ en 
over, and when the side of the square was incie:ised 
oy one, there was a deficiency of 8 to complete the 
square. Find the number of countere. 

12. Reduce to its simplest form 

g' — (6 — cf ^ b^ — {c — ay _^ c» — {(i-^b)* 
(a + c)« — 6» (a + by — c» (6 + c)' — a"' 

13. A and B ciin do a piece of work in 12 days; 
in hew many days could each do it alone, if it would 
lake A 1 days longer than B 1 




y w I ^ ^^ 

1 4. Given ) x— y = A ) x, y, z, 

\ X* + 1/ -^z' + w' = 62^ 



APPENDIX. larf 



15. Find the last term, and the sum of 50 terms, 
of the series 2, 4, 6, 8, ifec. 

( M ' 

IG. Writo down the expansion of {x — - > 

17. How many u..ut»ir)nc swains may be rung on 
tea different bells, supposing all the combinations to 
produce diffe-ent notA" ' 



ANSWERS- 



Junior Matric l87'J. I'af^s. 

{x + ay + {x + a) {y-b) + {y-bY. 2. a^^ap + q 

3. (a), 1^; {b), U; (c), 4^; (J), ^ ^. 4. 640, 660. 



Junior Maine. , 1872. jPcws a/w? Honor, 

1. I s-i + (.rJ - yi) I ' I 2i _ (xJ — .vi) I ' = 

I «j _ (a^ , .,*)» I '; ,, + 2b + 3c. 2. We have 

c* — ;?ci- 5- = aiid c* — ^'c + g-' = 0, fron? 
vhicli to elim-Date c. 

4. If /3 be one root, - - _ fi ^1 + -Y '' = i8'-^ 

aad, eliminating j, — = ^ — 2Z . 
ac pg 

6. (a), 4, — 7, ^(— 3± v/277) ; (6), 3, 2, ; — 3,— 3 
— _-, -L;- -i, - ~ (c-),-3 

^_ /2. ((f), Divide tL rough bv x^ and put y for 
jc+ -, and :.y^ — 2 for a:^+ — , then y = 

_ or — - and a; = 3. i, — 4 or — 3. 
3 2 ' * 



ANSWERS. xxUi 



6. ; >a 4-a -J-a -f- ..-. V 

«• rrr- 



Junior Mdtric, 1872. Honor. 

L 8 ami 6 miles. 2. Each of the first set ol' 

fractions may be shewn equal to 

X y 

labc « or 2ahc h or 2ahc 

b' + c'-a* c' + a^'-b'' 

s 

, which are therefore equaL 

a' + b'-c^ ^ 

3. Multiplying the equations successively by y, z, a 
and 2, X, y, we obtain c^x + a^y + h^z = 0, 

b'x + c'l/ + a'z - ; thence —. — j^, = . . ^ . , = 
a —o'c —car 



and X — 



4. «* + 6^>2a6,.-.c(a» + 6^)>2rt6c, &c. 

5. 3,0;- 2,- 5; -3, 6; -8, 1. 6. 00 and 2-tO mla 



Junior Matric, Uonor. \ .„_ 
Senior Matric, Pass. 

1. (a), From first x— 'ly or .//, and then solutions are 
3, 3; -3,— -3^y2r, v/21; --V2l,— V2f: 
(6).>e(41drs/7"69^H-37±/7Gy). (c). i.i; 
-i.-i; ii;-i>-i ('0, 4,18. 2. 3. 

, , , 33x-«+61a;+ 10 ,,, a;^ + 3x + 2 
^ '' a: + 2 ^ T^ + llaj + 30 



ixiv ANSWERS. 



_ X (3— x) 

6. a; - 2 and a; + 5 are factors, and roots are, 2, — 5, 
H-3±^/35): 7. 7^ gals. 

8. 4.88 percent. 9. I days. 

He receives $3 every day the work continues ; 
he returns nothing the first day he is idle, 
$1 the second, and so on, and the number of 
days he works is IG. 



Junior Matric, 1876. Pas$. 

1. a' ; a""'' 6""' c~V 2. a« - cc» ; a'+^, 

3. 1 + 2a; + 3ar^ 4- 4a^ + 5a;* + Gar* + ; rem. 7a:^- 

Gx'. 90.-^ — 6a;+l. 

4. a^+^ab)^' '' -^ (ab) + b'^'. 

5. (1), 2. (2), 2, 5, 7; or -2,-5,-7. 



JiMiior Matric., 1876. Honor. 

1. 35 mis. 2. (2), These quantities are in H. P. if 

,&c., are in A. P., i.e., if a, 6, c 

ah + ac + be 
are in .4. P. 

5. It may be shewn that the remainder at the nth 

decimal place is 2n ; hence if the nth digit be 
increased by unity, and the whole subtracted 
from 1, the remainder is the remaining part 
nf the period. 

6. a = 4,a; = 2or-3^=3or-2;a--l,« = 2*^ro; 

y--2-.7l0. 



ANSWERS. xxf 



Junior Matric, 1876. Honor, 

1. 121 and 400 yards. 

2. (a — h + c) {ah + 6c -f ca) (a* + 6* + c* + rtb + be — ca) 

3. Iri4,tionaI roots go in pairs/. 3 — i/iT is a rooi , 

and other roots a"e ^ (-—1 zbyZIg). 

S 13 2 la. 

4. X- + le'y^ + ic-_y^ + .ry -f- xiy^ + j/^. 

•"»■ i— 7 ^T-7 A\- ^- ^ "'^^• 

0+ (?^ — 1) (a — b) 

1. (1), Plainly x + 2 divides both sides, and roots 

are— 2,24- /f. {-2), x= 3, >/ ^ i cv I ; x = 

— 3, y- — 4 or — i. 



Second Class Certificates, 1873. 

^- Kb^a)-^=b^^^-a^- 
2. {a-b)-{a-\b) = 2,b. 

5. ('/. — t*) (a; — y) = ; .*. if a be not = 6, x -y — ; 

it' a — b,x — y may have any value. 

4 3-1 4to, , , . , , , 

6. , , o- '« • n, provided x be not = - 23 ; 
1 i//i- 1 ) "* ^ ' ' 

then fraction becomes § and is indetei minate. 

o 1 1 

x-y ' a; + / ^ ' ' -^ 

9. 13 10. ^ of a mile per hour. 



«xvi ANSWERS. 







Second 


Class Certificates, 


IS 


7f 


1. 


2{a'h' 


+ hh- + c^a*) - 


(a^ + b' + c'). 




2- -\ 
a f o 


3. 


(3x + 


\){\x^- 


2^- 


3x+l). 


4. 


2a(26-^ - 5) 
4a - o6 


5. 


he 


— ad 


^6c- 


-a^ 







7?ic — na 

6. X and y are indeterminate : there Ls but one 
equation. 7. 8SS, .$44. 8. 1 4 Jays, 1 1 g Jays. 
2 

9- ^i^ hrs. m — n negative means that thev 

m — n ° 

2 

were together hi-s. before noon, m — n, 

n — n 

they are neve»^' together. 
10. Each side equals 99(x-^ — y-). 



Second Class Certificates, 187C. 

1. {\+m)x-{l-n)ij. 2. (x + y)'(.r-y); {a-h) 

{b-c) {c-a); (S.yr-l) (5ar' + x-r 1). 

3. Let the other factor be x-¥ a; multijjly and eijuato 
co-etiicients ; eliminating a, nq — 'n- - na; other 
condition in pn — mn — r. 4. x— 1; 1 ^ -t». 

g { x + y-z) {x-y + z) {>/ + z- x) . _1__ 
[x + y + zf a - b 

6. -§; 1. 

7. a'ih'c - be) + b''[^ac' — dr) + c'{a'h — ah) = 0. 

8. (1,) Cube, and 3(« + x)i (>i-x)i (n») = m* - ^jj, 

q a (c — 6) 6 (a — c) 
a — 6 a — i 

10. 3 miles an hour. 



AA.i'^l^'ER^* xxvii 

11. (a), See §359. {b), 2,000. (c), Snnstitute suc- 
cessively — b, —c, —a for a, b, c, in tho left 
hand side, and it appears that a + b, b + c, 
c + a are factors, and /. expression is of form 
N{a + b) (6 + c) {c + a); putting a-b-c- 1 , 
we get iV'= 3. 



First Year Exhibit/ions, 1873. 

1.3,15,75,375. 2. 9 and 1, or V^ and- H- 4.9,12. 
5. (a), 4, -3; -3,4. (6),2,-3. (c), 4,-5, 6. {d),-\ 

6.40'. l.J^l±^Ll^ = . 
2ab 



Firxf. Year Exhibitions, 1874. 

1. 5. 2. {—if; 3277. 3. 2^; 1\; 2^^. 
4. 9, 12. 5. 75. 

6. (a),3,2;— 2,— 3. (6), 7 or— If (c),5,3. {d),U. 

7. 30 hours. 8. JL. 9. 3(a; + 3). 

x + y 



First Year Exhibitions, 1876. 





11111 


1 


1 


4 3a 6 c a b 


c 




b ' b ' \ 1 1 ' 1 1 


,1 




a« ^^ 7 «=" 6"* 


c* 


2. 


— 12 '^/2. 3. X — 1. 4. m. 




5. 


21, 42, 63, or 81. 6. o, b, 2c; 1, 1 





ixviii ANSWERS. 



Matriculation, 1873. 

2. \\a — 3c — 5(Z + m. 4. — ax. 

5. ax' + (ar + 6) aj -|- (a/ ' + br + c) -f 
ar' + 6a^ Jfcr + d 
X — r 
C. l—x¥ x'— x^ ^ .. .. 7. 3ft' + 4a--*. 

8. (^t — .r) {x'—2)^ g 144, 216. 

x-^ — 1 

10. \ {a — 2>i)i — "In — ^j), lire. 

^^ vib~na 1200 (a — 6) 

?u — ?i mh — na 

12. drJv'^- 13. 28, 21. 

U. 50 {VI —1), 50 (3-/5). 

15. x= zfc 10, ^= =F10; x = ±W2, y ^ ^ 3»/2 

16. 16. 



Matriculation, 1874. 

1 1 o 4a* + a'x — 2rt;c' + ar* • . 

1. a— 6. 2. ;^ 3w 1. 

cc* — a^ 

4. _ 5a — 36. 5. 600, 480, 360. 

6. 2, 4 ; 4, 2. 8. 4 or 9f 

9.20,16,14ft. 10. 40, 10; 10, 40. 11.56. 

12. 1. ' 13. 30 and 20 days. 

14. 6, 2, 41, U,or-2, — 6,— 1^, — 4|. 

15. 100, 2550. 

16. x^ — 7a^ + 21af'— 35.r + 35a:-' — Sla:"* + 7«-» 

— x-\ 17. 1023. 



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(a TIIOROIGII EXAMIKA7I0S GIVEN). 

St. Thomas, Nov. 30th, 1878. 
To the Trosident and Members of tlie County of Elj^in Teacher's .Associa- 
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of junior pupils, and we would urge its authorization on the Goverimient, 
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Signed, A. F. Bitler, Co. Inspector, .1. McLean, Town Inspector. 
J. Millar, M. A., Head M.aster St. Thomas High School. 
A. Steele, M. A., " Orangeville High SchooL 

N. Campbell, " Co. of Elgin Model School. 

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EXAMINATION SERIES. 

Canadian History. 

Bv James L. Hughes, Inspector of Public Schools, Toronto. 
Price, 25 Cents. 

HISTORY TAUGHT BY TOPICAL METHOD. 

A PRIMER IN CANADIAN HISTORY, FOR SCHOOLS AND STUDENTS PREPARING FOR 
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3. Examination questions are given at the end of each chapter. 

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7. Constitutional growth is treated in a brief but comprehensive exer- 
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Epoch Primer of English History. 

By Rev M. Creighton, M. A., Late Fellow and Tutor of Merton College, 
Oxford. 

Authorized by the Education Department for use in Public Schools, 
and foi admission to the High Schools of Ontario. 

Its adaptability to Public School use over all other School Histories will 
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In a brief compass of one hundred and eighty pages it covers all the 
work required for pupils preparing for entrance to High Schools. 

The price is less than one-half that of the other authorized histories. 

In using the other Histories, pupils are compelled to read nearly thi-ce 
times as much in order to secure the same results. 

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Used in separate schoo 3. M. Stafford, PRiPiST. 

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This little book, of one hundred and fort}' pages, presents history in a 
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The epochs chosen for the division of English History are well marked 

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With a perfect freedom from all looseness of style the interest is sn 'tH 
sustained throughout the narrati\c that those who commence t^. 
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Comprehensive. Litkrart World. 

The special value of this historical outline is that it gives the reader a 
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THE BEST ELEMENTARY TEXT-BOOK OF THE YEAR. 



: Gage's Practical Speller. 

I A MANUAL OF SPELLING AND DICTATION. 

' Price, 30 Cents. ! 

: Sixty copies ordered. Mount Forest Advocate. ■ 

I After careful inspect on we unhesitatingly pronounce it the best spell- j 

I ing book ever in use in our public schools. The Practical Speller secures : 
I an easy access to its contents by the very systematic arrangements of the 
! words in topical classes ; a permanent impression on the memory by the 
( frequetit review of difficult words ; and a sa\i!ig of time and effort by the 
1 selection of only such words as are difficult and of common occurrence | 
I Mr. Reid, H. S, Master heartily recommends the work, and ordered some | 
I sixty copies. It is a book that should be on every business man's table as 1 
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we have seen no book which we can recommend more heartily than the one ' 

liefore us. ! 

o • ( 

Good print. Bow.maxville Ob.server. ] 

The " Practical Speller" is a credit to the publishers in its general get ! 
".p, classification of subjects, and clearness of treatment. The child whe 
uses this book will not have damaged eyesight through bad print. i 

What it is. Strathrot Age. 

It is a series of graded lessons, containing the words in general use, i 
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Every teacher should introduce it. Caxadian- Statesman. } 

It is an improvement on the old spelling book. Every teacher should 1 
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The best yet seen. Colchester St7<, Nova Scotia. 

It is away ahead of any"speller"that we have heretofore seen. Our public 
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8E. S- ®«9^ * ^'^- <^^to (Ebucational SRorke. 

WORKS FOR TEACHERS AND STUDENTS, BY JAS. L. HUGHES. 



Examination Primer in Canadian History. 

On the Topical Method. By Jas. L. HruiiK.s, iti^iwctor of Schools, To. 
rooto. A Primer for Students preparing for Examination. Price, 25c 

Mistakes in Teaching. 

By Jaj. Lauoiilin IIooiies. Second edition. Price, 50c. 

AIK>PTKS B7 8TATR UNIVRRaiTT Of IOWA, AS A.S BI.EMKNTART WORg FOR 081 
OF TKACH8R8. 



This work discuascf) in a terse manner over one hundred of the mistakes 
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Manual of Drill and Calisthenics for use in 
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Life Guards, Orill Instractor .\ona»l a-xl Model School*. Toronto 



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Suitalsle for Intermediate Examinations. 

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Wm. Scott, B. a., Head Master, Provincial Model School. 



PRICE, - SO CKNTS. 



Thit vohnne contains papers on ArUhmetie, Euclid, Geography, 
Algebra, Book-keevitui, History, Statics and Hydrostaiies, English 
liiUrtHure, French (July, 18S0), Chemistry, English Orammar. 



FROM THE PREFACE. 

In reapoDM to the desire of a larxt: numt>er of Teachers, we reprint 
the KxuninatioD Papers suitable foi the Annual Intermediate Ex 
fcmination, which have appeared in the number*, for 1881, ot Gage's 
"School Kxaiiiiner and Student's Assistant." 

The steadily tncreastng circulation of this monthly magazine, and 
the numerous letters received testifying to the great value of the 
questions in the vmHoos cubjeots reouired for the Examinations, 
plainly indicate that sach k periodical U a mo«t oaeful aid to both 
MAcher and student. 

The almost exhsastire nature of the questions on each subject 
bringv the student Into clot^ acquaintance with every needful point; 
and the drill experienced in thinlving and working out the answers is 
of Incalculable practical benefit to those who wish to exo«l at written 
examiDatloua. 

When we state that the editors of this department ot tiM SeJwol 
Examiner are Messrs. T. Sirkland, M.A., and W. Scott, B.A., we con- 
sider tiiat It is a suihcient guarantee for the excellenos and appro- 
priateness of the work, 04 these gentlemen have eam«d a wide reputa- 
tion as specialists in science and hterature. 

lo consequence ot numerous applications (or the PrwKsh Paper given 
at the Intermediate Elxamination, 1880, w« reprodoo* it in this book. 

Hints and Answers to the Above, 50 Cents. 

W. J. OAGE <& CO.. 

t. (OTBK, 



Vm. J. 6iigc & doQ. Ilclu €butational 



eMorks. 



The Canada Schooi Journal 

HAS RKXEIVED AN HONORABLE MENTIOM AT PARIS EXHIBITION, 1878 

Adopted by nearly every County in Canada. 
Reconniiended by the Ministe of Education, Ontario. 
Recoiiiniended by the Council of Public Instruction, Quebec. 
Recomniendfd by Chief Supt of Education, New Brunswick. 
Rcccniniendcd by Chief Supt. of Education, Nova Scotia. 
Reconinicnded liy Chief Supt. of Education, British Columbia. 
Recommended by Chief Supt. of Education, Manitoba. 

IT IS EDITED BY 

A Committee of some of the Leading Educationists in Ontario, assisted 
by able Provircial Editors in the Provinces of Quebec, Nova Scotia; New 
Brunswick, Prnce Edward Island, Manitoba, and British Columbia, thus 
having each s«H;tion of the UomiiiioD fully represented. 

CONTAINS TWENTY-FOUR PAGES OP READING MATTER. 

Live Editorials; Contributions on important Educationa! topics; Selec- 
tions- Readings for the School Room ; and Notes and News from eacb Pro- 
vince. I 

PiiACTicAi, Department will always contain useful hints on methods of 
teaching different subjects. 

MAniBMATicAL Departmbnt gives solutions to difBcult problems also on 
Examination Papers 

Official Department contains sueh regulations as may be Issued trom 
time to time 

Sulscription. 81 00 per annum, strictly in advance. 

Read TUB F:i!,i,"u ■ i i'tter frov John Oreenleaf Whittibb, thb Fa- 
ucis AMI.RH-'AN I'ul.l 

I b"^ve also rt'cei\til a .No of the •' Canada School Journal,' which seems 
to me the brightest and most readable of Educational ilagannes I am very 
truly thy friend, John Greenleaf Whittier. 

A Club of I.Ojo Subscribers from Nova Scotia. 
(Copy) Edccation Office, Halifax, N S . Nov. 17, 1878. 

Messrs. Adam Miller & Co., Toronto, Ont 

Dear Sirs,— In order to meei, the wishes of our teachers in various parts j 
of the Province, and to secure for them the adxantajje ot vcur excellent 
periodical, i hereby subscribe in their behalf for one thousaii'' (1,000) ccpits > 
at club rates mentioned in .\our recent esteemed fa\or subscription!-- -vil, 
begin with January issue, and lists will be forwarded to your office in a li» 
days. Yours truly, 

David Allison, Chief Supt. of Education. 
Address. W. J. GAGE & CO., Toronto, Canada. 



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