UC-NRtF ^B 53S HbT Digitized by the Internet Archive in 2008 with funding from IVIicrosoft Corporation http://www.archive.org/details/elementsofalgebrOOcolerich 'I' e (f. "^^'^, THE ELEMENTS OF ALGEBRA; DESIGNED FOR THE USE OF SCHOOLS. EEV. J. W. COLENSO, D.D. BISHOP OF NATAL. PART I. FBOM THE THIRTEENTn LONDON EDITION. NEW YORK: JOHN F. TROW, PRINTER, 50 GREENE STREET. 1867. CAJORI V ADVERTISEMENT. In this Edition (wliicli is stcreotyjped^ and so will be secured from further change) the Simpler Parts, those, namely, suited for general School purposes and required for the attainment of an ordinary B.A. degree in the University Of Cambridge, are printed separately as Part I ; to which is appended a large collection of easy Miscellaneous Examples, specially adapted to the contents of this Part, and supplying means of complete Examination in them. It will be seen tliat the easiest kinds of Simple Equations and Equation Problems are in this Edition introduced much earlier than is usual in Treatijses , on Algebra: but there can be no reason why this ^^fcanch of the subject, which is so interesting to most ^^Rudents, and gives them some idea of the practical il^)plications of the Science, should not be brought forward as soon as possible. Part II is also published, and contains the higher parts of the Subject, with such additional remarks on IV ADVKRTISEMENT. the earlier portions as M'ill suit the wants of more advanced and promising Students, -and with a similar Appendix of more difficult Miscellaneous Examples and Equation Papers. This Part may be begun as soon as the Student, having thoroughly mastered Part I, has entered upon the Miscellaneous Examples at the end of it. Fomcett St. Mart/y Nov. 1, 1849. TABLE OF CONTENTS. [^nxr. Page I. Definitions 1 II. Addition 7 Subtraction 8 Use of Brackets 9 Literal Coefficients U Multiplication 13 Division 17 Jiesolution into Factors - .... 22 III. Simple Equations 24 Problems 29 IV. Involution 33 Square of Multiiiomial 34 Evolution 36 Of square root 38 Of ditto in numbers 40 Of cube root 43 Of ditto in numbers 46 Greatest Common Measure 48 Least Common Multiple 57 VI. Algebraical Fractions. 59 Certain Properties of Fractions G 8 t Remark on the meaning of sign = 69 Ditto . . . sig7i oc 70 VII. Simple Equations continued 71 Problems ; 73 Simultaneous Equations of two u iknowns 81 Ditto of three unknoions 83 Problems 84 Vl TABLE OF CONTENTS. Chap. Pago VIII. Theory OP Indices 87 Surds 92 IX. Quadratic Equations 99 Simultaneous Quadratic Equations 105 Problems 107 Indeterminate Equations Ill X. Arithmetical Progression 115 Geometrical Progression 116 IIarmonical Progression 121 Arithmetic^ Geometric^ and Harmonic Means 123 XI. Ratio 124 Proportion 126 Variation 131 XII. Permutations 134 Combinations 137 Xlli. Binomial Theorem 140 XIV. Notation 148 Decimals 151 Interest, &c 154 Miscellaneous Examples. Answers to the Examples. ALGEBRA. PART I. CHAPTER I. DEFINITIONS. 1. Algebra is the science which reasons about quantities by means of letters of the Alphabet, and certain signs and symbols, which are employed to rep- resent both the quantities themselves, and the man- ner in which they are connected with others. Thus we might put a to represent 7, and then twice a would represent 14 ; or we might put a to represent 3, and then twice a would represent 6, tliree times a^ 9, &c. 2. The sign == {eqical) denotes that the quantities between which it stands are equal to one another. Thus, if (X = 17, then txoice a = 34. 3. The sign .*. stands for then or therefore^ and •.• for siJice or because, 4:. The sign + (plus) denotes that the quantity be- fore which it stands is added, and the sign - (minus) that the quantity before which it stands is subtracted. Thus 5 + 3 = 8, 5 - 3 = 2 ; and if cj = 3 and 5 = 4, then a + 5 = 3 + 4 = 7, « + J + 2 = 3 + 4 + 2 = 9, 10 -a = 10 -3 = 7, 10-^-5 = 10-3-4=7-4 = 3. The sign ^ is used to denote that the less of two quantities is taken from the greater, when it is not known which is the greater. Thus a^h denotes the difference between a and h. 1 5v : . I>EFINITIONS. 5. All quantities before which 4- stands are called positive^ and all before which— stands are called nega- tive quantities. If neither + nor - stand before a quantity, + is un- derstood, and the quantity is positive ; thus a means + a. 6. The sign x {into) denotes that the quantities be- tween which it stands are to be multiplied together; but very often a full-point is used instead of x , or, still more commonly, one quantity is placed close after the other without any sign between them. Thus a X 5j (X . 5j and db mean all the same thing, viz. a multi- plied by h ; and, therefore, if a = 3 and 5 = 4, we shall have ab - 12, 6a = 15, 6ah = 60 ; and if also c = 5, tZ = 0, then Aah + 3ac + 4:d-2b + 2alc-Zabcd = 48 + 45 + 0-8 + 120-0 = 213 - 8 = 205. • 7. The number, whether positive or negative, pre- fixed to any algebraical quantity is called its coeffi- cient ; thus 3 is the coefficient of 3a, - 7 of - 7aa?, &c. If no number is expressed^ the coefficient is under- stood, being 1, since a means once a. Ex. 1. If a = 6, 5 = 5, c = 4j tZ = 3, e = 2j /= 1, and g = 0^ find the numerical values of the following expressions ; \, a + 25+3c+4cZ+3e+2/+ ^. 2. 2a + 5 - 3c + 4cZ - 5/+ 6^. 3. 35 - 4a - 6c + 7^ + 2e - Ag. 4. - 3a + 25 + 3c - 2e +/. 5, db + 55c - Me + bfg. 6. Aag - 35/*+ 4cc - ad. 7. - 3a5 - 2ac + 45c - a5c. 8. bob - 8ac + \6cde - \Aaef, 9. 33a5-19ccZ + 22a5^-13c^4/: \0. abcd-2bcde ^Zcdef-^efg, 8. The sign ~ (by) denotes that the quantity which stands lefore it is to be divided by that V!\\iQh. follows it; but, most frequently, to express division, the quan- tity to be divided is placed over the other with a line between them, in the form of a fraction. DEFINITIONS. 3 a Thus a-^h and j dcnotCj either of thenij a divided by b ; and if ac Zabc + 6ac + 6ab - 35c 6, ;^^ ^^ ^ + . 2^ + 35 Zab-2ad 6c -2b 4. a« + 25^ + 3c^ + M\ 5. Za'b + 2h''c-2a''c + Wd. 6. a' - Za'^c + Zac'' - c\ 7. «* - 461^5 + 6a''¥ - 4a5^ + 5*. 8. Aalc^-Za^c . ^~. 9. ^^'^^ . -?^ _ «-^i!l£!. 2a + 5 + c Za^ a + b' 55' 10 ^^!^1iJl 1 + ^°c^ 4a + 5° + 5'c'^ a= + 2a5 + 5' * 'a^ + 5^ "" a^ + c' "^ 5*-' + c' "" "5^- 25c + c^ ' 4 DEFmiTIONS. 10. The square root of a quantity is that quantity whose square power is equal to the given quantity. Thus the square root of 9 is '3, since 3' = 9 ; the square root of a" is a^ of 64 is 8. So also the cube^ fourth^ &c. root of a quantity is that quantity whose cube, fourth, &c. power is equal to the given one. The symbol used to denote a root is V (a corruption of 7*5 the first letter of the word radix\ which, with the proper index on the left side of it, is set before the quantity whose root is expressed. Thus, V^' = «3 VG4 = 4, V3125 = 5, \Jl = 1, VI = 1, &c: The index, however, is generally omitted in denoting the square root ; thus \/x is written instead of y x. Find the values of Ex. 3. 1. V4 + 2V25 + 3V49-VC4. 2. 3vl6-4v36 + 2V0-V^1- 3. V8 + 2 V125 - 4 VI + VG4. 4. VI + 3 V16 - 2 V32 + 3 VI. If a = 25, 5 = 9, c = 4, fZ = 1, find the values of 5. ^a + 2^1 + 3Vc + 4V^. 6. ViS + ^1% + VISc- V253. 7. Z^a + 2V46'-4V9^+ ^/l6d. 8. V5^+2 V3&- V2c +4 V^- 9. v«' - 2 yh' + 3 yc' - 4 V^. 10. V^+ 3 V^-4V^^""+ Vc^». 11. Algebraical quantities are said to be liJce or unlike, according as they contain the sa77ie or different combinations of letters. Thus a and 5a, ^^a^h and la^l^ Za'^lc and -a'5(?, are pairs of like quantities ; a^ and a"^^ Zah and - 7a, Za^h and 3a5^, of unlike quantities. 12. Brackets^{\ H? []? ^^^ employed to show that all the quantities within them are to be treated as though forming but one quantity. It is of great im- portance to notice carefully the effect of using them. Thus rt-(&-c) is not the same as «-5-c; for, in this last, both h and c arc subtracted, whereas in the former it is the quan- tity, J - c, which is subtracted. DEFINITIONS. Hence, if a = 4, 5 = 3, c = 1, we have a-.&-o = 4-3-l = 0, a-(b-c) = 4:'-2=^2; 2a-35 + 2c.-8-9 + 2=I, 2a-{^b + 2c) = S-U = ^Z; 2a+ 5-c = 8 + 3-1 = 10,2(a+5)-c = 14-1 - 13, 2 (a + 5-c) =12. Sometimes, instead of brackets, a line is used, called a vinculum^ and drawn above the quantities tliat are connected; thus a—h—c is the same as a—ip—c). The line, which sei^arates the num"^ and den'^ of a fraction, is also a species of vinculum, corresponding, in fact, in Division to the bracket in Multiplication. Thus Y — implies that the wJiole quantity a + 5- c is to be divided by 4, and might have been written | (<^ + 5 - c). Ex. 4. Ifa = 0, ?^ = 2, c = 4, ^ = G, find the values of 1. 3a + (25-c)'^+ |c^-(2a + 3&)} + {?>c-(2a-\-Zl)}\ 2. Zl)^-{2c-dy + {35-(2c-cZ)}^-{3&-(2c-^'^}. 3. 2 ^d-h + 3 v3^ + 2c-l + 4^^ + h + 2c + d. 4. 3 V25=^- « + 2 V&' + c'^ + 7-V2 (5 + cf--{h + cZ)^ 5. {a^(J) + cy-d] {(a + 'by + {d--cy] |(^ + 5 + c)'»-cZ}. If a=l, 5=2, c=3, tZ=4, shew that the numerical values are equal 6. Of(?> + c + ^)(5 + c-6Z) (h-^d-c) (c + d^h) and of45V- ■[^=*-(52 + c')p. 7. Of{^-(c-5 + «)} {(^ + c)-(<^ + «)K and of d^ - (c^ + ¥) + a^ + 2 (dc - ad). 8. 0£{(h + c)-(d-a)}^+{(c + d)^(l>-ayf'+\{h + d)-(c-d)\'' + (b + c + d-ay, and of 4 (c^'* + 5^+0^ + tZ^). 9. Of K« + ^-M)f {(«+c+^-5} \c-(d-a-iy (h^c + d-a), and of 4 (a^ + dcy-{(a' + d'')-h^ + c'')\\ 10. OfcZ^-(2^-c)c+ |2(J-c) + &} l)-{2(d-c + h)-a\ a, andof {((Z-a)-(c-5)}^ 13. Those parts of an expression, which are con- nected by the signs + or — , that is, which are connect- ed by J.6Z(^^i^^(9^^/ or /S'?^5^rac^i6>^i, are called its terms., and the expression itself is said to be simple or com/pound^ according as it contains one or more terms. 6 DEFINITIONS. Thus a', 2abj and -35'*, are each simple quantities, and a^ + 2db - ^h^ is a compound quantity, whose terms are a^, + 2a&, and-3&^ Those parts of an expression which are connected by Multiplication are called li'^ factors. Thus the factors of a^ are a and a, those of 2ab are 2, a, and 6, those of - 35* are - 3, 5, and 5, or, as we should rather say, - 3 and 5*, it not being usual (except where specially required for any purpose) to break up a power into its elementary factors. Of course we might include 1 as a factor in each case ; thus, since a'' = 1 X a^^ the factors of a"^ are 1 and a*, and so of the rest : and this will be sometimes required, as will be seen hereafter, but for the present need not be attended to. It is very necessary that the student should learn at once to distinguish well between terms and factors. Thus 2a + J)-c is a compound quantity of three terms, 2a, 5, and -c; 2 (a + 'b)-c is one of two terms only, 2(a + b) and - c, of which the former, 2 (a + 5), consists o^ two factors, 2 and « + 5, the factor, a + d, being itself a compound quantity of two terms ; and 60 also 2 (a + h - c) \s a simple quantity or single term, of two factors, 2 and a + h-c,of which the latter is itself a com- pound quantity of three terms. Let it be observed then that terms are the quantities which make np an expression by way of Addition or Suhtraction, factors, by way o{ Multiplication. It may be also noticed, that it is immaterial in what order either the terms or the factors of a quantity are arranged. It is usual, however, to arrange quantities, as much as possible, in the order of the alphabet. Thus a~2b + Zc is the same quantity as - 25 + a + 3c, or 3c - 25 + a, &c., and ahc is the same as hac or hca ; but we should prefer to write a - 25 + 3c, and a5c, unless there were some reason, in any case, for arranging otherwise. A quantity of 07ie term is called a monomial, of two terms, a hinomial, of three, a tri7iomial, &c., and, gen- erally, of more than tivo terms, a midtinomial. CHAPTEE II. ADDITION, SUBTRACTION, MULTIPLICATION, DIVISION. 14. To add like algebraical quantities, add sepa- rately the positive and negative coefficients ; take the difference of these two sums, prefix the sign of the greater, and annex the common letters. Ex.1. Za Ex. 2. -125c Ex. 3. 2c« Ex.4. Za''-^2l^ -2a Zlc -7c^ -Sa' + 4l^ 5a -85c 10c' 5a* -65* 6a 55c 4c* 7a* + 35* I 7a -85c 4c* 11a* * In the last example the star is used to indicate that the terms in- volving 5* destroy one another. If the quantities are itnliJce^ we must add any that are like by the preceding rule, and write down the others with their proper signs. Ex. 5. 2a+35-4c Ex. 6. x-2y-^'^z Ex. 7. 2a+c-^cl -3a+45- c -2x+oy-4z -h+a+e 4a +75+ 7c Sx-57/-5z + c-d a- 5- 4c ^ -^ y -3a-e -/ -5a + 25- 6c 2y + 2z - 2c + 2d -2e -a +155- 8c Zx- y-4z -5 +2d-2e-f Find the sum of Ex. 5, 1. 7a-35+4c-2(Z+7. -8a + 45-6c+2^Z-ll, 13a + 35-5c+4^-4, 2a-5+c+ll, a+2d-Z. 2. 2x-Sy + 4:Z-4, x + 2y-Sz^ -Sx + 2y -5z + 7, ix-y +2^-3, 9a;-10?/ + ll2-12, x^y^z, 3. 2a*+a5 + 35*, 3a*-4a5 + 25*, 3a* + 3a5-5*, 12a* - 14a5 - 75', 3a*-12a5 + 175*. 4. ax-4J)y + 3cz, lZax-%y + 7cZj -5ax+7hy-14:cz^2ax-'by + czy "llax+lZdy -4cz, 5. 20a;^ + 20x^y - 3xy^ + Uy^, - 17a;^ + Ux'^y - Uxif - Zy\ 14a^ f +17a;*j/ + 15a'y*-5?/', - 12.c'- 13aj*y - 14a:2/*- 5^', I2x^y + Zy\ 8 SUBTRACTION. 6. 2iB' - Zx7j - 4i/^, Zxz + 22/^- s^ ic'- 2^^ + 62*, Sxj/ - 6iC2 - Zx\ Zxz-2z''+6yz, 4y''-Zyz+2x\ 7. cc'- Zo^x'^ + Za^'x-a^, 4x^- 5ax^ + 6a''x - 15a\ ZxUAax-^ 2a''x + 6a^ - 17aj' + 19aa;^ - 15a=a; + 8a^ - ISaa;'^ - 27a'a; + lSa\ 8. «'- 2a6'- ac^^^- a=Z> + 2a'(j + 2ahc, - a'^^ + i^- 26c=+2aZ;=+2aZ)c + h\ -2a^c - h^c+c" + 2abc + ac^ + 25c^ 9. Zx^ + 2^/^ + 2!^ + 8^2^ 2/' + 3a;V + 2icy^ + s' - SSaj'^g, jc» + 2xyz -\-4x^y + l^ic'^g - Oy'^z + 6yz^, 2x^- Zy^+ 4xyz-(jxy^,^ 4y^ - z^+ 6x'z - 16xyz + oy'^z - 142/2^, ^x'^z - 15xyz + Axy^ - Ix^y + G?/^2. 10. aj'*+ Sxy^- xz^+ xhj+x^z^ Zx^y"^ ^Zx'^z'^ + Zxy^z — Zxyz^ - Ox^yz^ -x^y+ y^- yz^- Zx'^y'^-^- Zx^yz, - Zxy^— Zxyz^— Zy^z + Zy'^z^- Gxy'z, - x^z + Zy^z + z* + Zx^yz - Zx^z^, Zxy^z + xz^ - Zy'^z^ + yz^ + Gxyz^. 15. To subtract algebraical quantities, change their signs and proceed as in Addition. Thus, if we take h from ^, the result will be a - 5 ; but, if we take h-o from a, the result will be greater by G than the former, since the quantity now to be subtracted is less by c than in the former case ; hence the result required will be a-h+Cj which is therefore the value of ^ - (5 - c), so that the quantities 5, - c^ when subtracted, become - 5, + pounds, he will possess the whole positive property, + a pounds, the same as if we give him + h pounds, to jmy his debt with. There will often, however, be no need formall}^ to apply the above Tiiloof changing signs, since the difference may be obtained at once, by taking that of the coefficients and annexing the common letters. I USE OF BRACKETS. 9 Thus, in Ex. 1, we may say, at once, 3a; from hx leatcb ^x y from ly leaves 6y, -Azfrom -8s leaves -4s ; though of course, if we chose to apply the Rule {change the sign of the quantity to he subtracted and proceed as in Addition) it would equally be true that -Zx added to +5a;, -y to + ly^ + 4s to- 8s, would pro- duce respectively, + 2^, + 6y, -4s, as before. Ex. 1. Ex. 2. Ex. 3. From 5a; + 7y - 8s 5a;^ - 2xy + Zy'^ - 3a^ + Adb - 55^ take 3 a; + y - 4s - 4a;^ - 2a;y + 7y^ - 7^^ -f 35' - 2c^ Ans. 2a; + Gy-4s 9a;=* ^^'^ 4a' + 4«2> - 8f'~72c' Ex. 6. 1. From 2<^- 25 + c take a + 5 - 2c;. 2. From 2a;' - 3a;y + y- take 4a;' + 4a;y - 2y^. 3. From bax - 75y + cs take ax + 25?/ - cz. 4. From 7a;' - 2a; + 4 take 2a;' + 3a; - 1. 5. From 8ii'-2a + 65' -5^5 + 5c' - 35c + 2 take a' + a + 25' + 2a5 + 3c' + 35c + 2. G. From 2a;^ - ^x'^y - 3?/' + 6 - 2a;' - 3a;2/' - 14i/» take 3a;^ + 2x'^y -y"^- 3a;?/' + a;' - 10?/^ 7. From 5a;' + (jxy- 4i/' - 12a;s - lyz -5s' take 2a;' - 3?/' + 4a;s - 5s'* + Q>yz - 7xy. 8. From 3a;' + ^xy-y"" take -a;' - 3a;?/ + 3?/', and 3a;' + Axy '-by\ 9. From a*-2ci=5+3a'5'-4a5' + 55* take 2a5^-.3a'5'-^4a^5-5a*, and 3a*-2a^5 + 6a'5'-2a5' + 35*. 10. From a'' - 4a'5' - 8a'5^- 17a5* - 125' take a^ - 2a*5 - 3a»5', 2a*5-4a'5'-6a'5^ 3«^5' - Cc^'5'-9a5^ and 4a'5^-8t*5*-125*. 16. Since the sign + or-, preceding a bracket, will imply (12) that the whole included quantity is to be added or subtracted, if we wish to remove the bracket, we must actually perform the operation indicated by means of it, i. e, we must add or subtract the quantity in question. Of course, in the case of + preceding it, this amounts to no more than merely setting down the included terms with their proper signs, because, when a quantity is added, the signs of its terms are not altered ; but in the case of - preceding a bracket, 1^ 10 USE OF BRACKETS. we shall have to change the signs of all the included terms, since they are all to be subtracted. Thus + (a-¥l-c)= a + h-c, (a^ - 2a5 - &') = a^ - 2ab - J' ; but - (« + 5 - o) = - a - 5 + c, - (a' - 2ab - J«) = - a' + 2a& + &' : BO also, in the case of a double bracket, we have Za- {a- 3c) + (25 - c) = 3a - a + Zc + 2b-c =2a + 21? + 2c. The same remark applies also to the case of a frac- tion with a num'' of more than one term, whenever the line separating its num' and den', and which (12) is a species of vinculum, is removed by any process. Thus ^ [or-i(a+5-c)] = ---- + - lov-la-ll + |c] ; and - |(a - 5)j when multiplied by 2, becomes -(a-l)^ or - a + I. Ex. 7. Reduce to their simplest forms : 1. {a-x)-(2x-a)-i2-2a) + (3-2^) - (1-aj). 2. {a'-2a''c + Zac'')-(a'c-2a^ + 2ac') + (a'-ac^-a^c). 3. (2^'' -. 2y^ - z') - (3if + 2x^ - z") - (3s^ - 2^/' - a;'). 4. (a;' + ax^^a'x)-'{y^-ly' + 5^) + («' + <^^'' + c'2) - (a;' - 2/' + e') + {ax^ + 5?/^ + cz') - (a'x - Vy + c'e). 5. a' - (&' -r) - {5^ - (c^ - a^)f + jc' - (&^ -«')}. 6. |2«^-(3a&-5^)}-{6j^-(4a& + 5')} + \2l''-{ci?-ab)\. r. ja;' + 2/' - (Sa^V + Sa;^/') } - K^' - Sx^) - (Sa-y' - j/') }. 8. {2aj~(32/-2)} - \y + (2a;-0)} + {33-(aj-22/)} - |2r-(7y-2)}. 9. l-{l-(l-4aj)f + {2a;-(3-5a;)}-j2-(-4 + 5a;)f. 10. |2a-(35 + c-2^}-|(2a-3&) + (c-2(Q} + -;2a-(3J+c)-2if -|(2<^-35 + c)-2^f. 17. It is often necessary not only to break up, or resolve^ quantities contained in brackets, but also to form such quantities, that is, to take up in a bracket any given terms of an expression. Now, in doing this, it should be noticed tliat, whatever terra we choose to set ^^ first term within the bracket, the sign of that term will have to be placed heforeX\\Q bracket, and this USE OF BRACKETS. 11 Sign will of course affect all the terms we may place within the bracket. If, then, this sign should be (+), the other terms may be set down at once within the bracket with their proper signs ; but if it should be (— ), we shall have to change the signs of all these other terms, and then set them w^ithin the bracket : for the sign (— ), which precedes the bracket, will influence all these signs, and have really the eff'ect of correcting^ as it were, the changes we have made, and will, in fact, cause the original signs to reappear, whenever we choose to resolve the bracket again. Thus + a-h-c, collected in a bracket with + « as first term, will be + {a -h-c); but, with - 5 as first term, - (5 - « + c), and with - c as first term, -(c-a + b) ; and now, if we resolve again these last two brackets, the sign (-), preceding each of them, will correct the changes Ave have made, and the quantities will be re- produced, as at first, -h + a-Cy-c+a-K So also we might use an inner bracket, and write the quantity + \(a-h)-c}, or -i-ja-(J+c)}, or -\(h-a)+c}^ or -{b-ia-c)}, &c Ex. 8. Express, by brackets, taking the terms (i) Uco, (ii) three, together, 1. 2a-b-Sc+4d-2e+Sf, 2. -&-3c+4^-26+3/+a. 3. -3e+4^-2e+3/+2«-5. 4. +4iZ-2^+8/+2a-5-3(j. 5. -2e+2f+2a-b-dc+U, 6. 3/+2a~5-3c+4(Z-2e. 7 — 12. Express the second answer in each of the above by using also an itmer bracket, including in it the latter two of the three terms within each of the outer brackets. • - — _— . 18. We have spoken hitherto only of nwnerical coefficients ; but, in fact, when a quantity is composed of two or more factors, any one of them is a coeffi- cient of the rest taken together, that is, (as the word coefficient implies) makes Ujp with them, as a factor, the quantity in question. Thus in Zabcx^ 3 is, as before, the coefficient of abcx ; but 8a is also the coefficient of bcx, dab of ex, ax of 'dbc, &c. 12 USE OF BRACKETS. Such coefiicients are called literal coeflScients, as involving algebraical letters ; and, when any terms of a quantity contain some common factor, a bracket is often employed to collect the other factors, consid- ered as its literal coefficients, into one quantity, which is set before or after the common factor. Thus we have seen already that 3a; + 2x-x = 4x, that is, = (3 + 2 - 1) a; ; and in hke manner, ax + hx - x = (a + h - l)Xj 2a - Aax + Gay = 2a (1 - 2i& + Zy), (a + 2h) x"- (25 - c) x"-- (2c - a) ic* = \{a^2l) - (25 - c) - (2c-a)} x"" = {2a- c) x\ Add {a - 2p) x^ - 2x' + {2c - 3?-) x (2p -^r 0L)x^ ^ { by + c is - Ic, and that of - 5 by -dis + id. If several simple quantities are to be multiplied together, instead of multiplying them together successively by the above rule, (thus 2a x - 35 x -4c = - 6a5 x - 4c =^ 24a5c), it will be shorter to multiply them at once together, and then prefix to this product the sign + or -, according as the number of negative factors is even or odd. 14 MULTIPLICATION. 20. The powers of a quantity are multiplied toge- ther by adding the indices. Thus «* X a' = cb^+^ = a^ ; for a^ = a.a ,a^ a'^ = a. a; .'. a^ y a"^ == a, a . a ,a,a = a^ ] and so in other cases. Hence ^ 3a"5 X Aa'h^ x - 2a'^5»=24a^ J«, 2ahc x Za^h'^c^ x - aJ'c = - Ga*h'e\ 21. If the multiplier or multiplicand consist of seve- ral terms, each term of the latter must be multiplied by each term of th,e former, and the sum of all the products taken for the complete product of the two quantities. This process is generally conducted as in the following Ex- amples. Ex.1. Sx^ - 2x7j + 42/' Ex, 2. -2a^&V 5a5» - 71* 2a^x ~4a5 Ex. 3. a + 5 Ex. 4. a + J a + 5 a-h a* + ah a^ + db + db + h* - ab-V Sa'h' -20a'^J* + 28a&» Ex. 5. a- -I a- -I a'- -db • -db + J» Ex. ^. x-^- a Ex. 7. a;' + (-5', a;' + 5a;-a'j and a;-(a + 5). 22. The student should notice some results in Mult", so as to be able to apply them when similar cases occur, and write down at once the corresponding products. Thus, (21 Ex. 3. 5) the product of a + 5 by a + 5, or the square of a + 5, is a' + 2a5 + 5', and the square of a - 5 is a' — 2a5 + 5' : by remembering these results, we may write down at once the square of any other binomial ; thus, (x + yy = a;' + 2a;y + 2/', (x - 2)'= a;'- 4a; + 4, (2a; + 2/)'= 4a;' + 4a;y + y\ (2aa;-35y)' = 4a'a;' - 12a5a;y + 95'^/'. Again. (Ex. 4) the product of a + 5 by a- 5 is a' - 5' : hence we have (a; + 2^) x (x-y) = a;' - 2/', (a; + 2) (a;- 2) = a;' - 4, (2aa; + Zhy) (2ax - 3hy) = 4a'a;' - 95'2/'. So also, (Ex. 6) the product of a; + a by a; + 5 is a;'+ (a + 5) a; + a5, where the coeff. of x is the sum of the two latter terras of the 16 MULTIPLICATION. factors, X + a, x+h, and the last term, + a6, is their product : in like manner, we shall have (x + 5) (x + 2) = x^ + (5 + 2)x + 10 = x^ + 7x + 10, ■ (x-b) (a; + 2) = a;^ + (2-5)a;-10 = ic'-3a;-10, (x + 2) (x- 2) (x + 3) (x-S) = (ic*-4) (a;^-9) = aj*-(9 + 4)aj' + 36 = a;* - Ux^ + 36, (a; + 2) (a; - 3) (a; - 4) (a; + 5) = (a;^ - a; - 6) (x^ + a; - 20) =(by common Mult") ;?;* - 27aj^ + 14a; + 120. 23. Let then these three results, ovformulcB^ be noted: (i)(^± IJ^a' ±^ah-VV', or, the square of any hinomial = the sum of the squares of its two terms together with twice their prodicct : {\i) {a -\-l){a --1))^ a" -V\ or, the product of the sum and difference of any tioo quantities = the difference of their squares : (lii) {x -{- a) {x + h) =^ x^ + {a '\-h) X -\- ah. 24. By a little ingenuity, however, the above formulae may be still more extensively applied to lighten the labor of Mulf* : thus Ex.1. {a-h + cy=: {{a-h) -^-cY^hy (\) (a -hy + 2{a-h)c-^c^ =a? ~ 2ab + Z)'* + 2ac - 2bc + c* ; or we might have written it {a -X (h-c)\^j or {(p + c) - h}^, &c., and then have expanded either of these by (i), obtaining, of course, the same result as before : but we shall give a better method hereafter for squaring a trino- mial ; it will be sufficient to have noticed this. Ex. 2. (a^ - ax + a;') (a^ - ax - x^) = by (ii) (a"^ - axY - x* = a* - 2a^a; + a'x^ - x*. Ex. 3. (a^+ax - x"") («'- ax -x^)={ (a""- x^)+ax} { (a""- x^) -ax\ « (a" - x^y - a^x^ = a* - 2a''x' + x' - a^x^ = a^ - Sa'a;' + x\ Note that the formula here employed, (a + h) x (a - h) = a"^- t', may be always applied, whenever it is seen that the two quantities to be multiplied consist of terms, which differ only (some of them) in sign, by taking for a those terms which are found iclth their signs unaltered in each of the given quantities, and the others f<.ir b: thus, in Ex. 3, a^ and ~ a;^ appear in both the given quantities, whereas in one we have + ax, in the other - ax ', hence the pro- duct required is (a^-x'^y-a^x\ as above. DIVISION. 17 Ex. 4. (a^+ ax + x^) {a^-ax+x"*) = (a^+ x^y-a^x^ = a* + aV + ic'. Ex. 5. (aVaa;-a;'.) (^2-aa;+a;'')=c*'-(aawc')^=a*-aV+2aa;'-ic*. Ex. 6. (a'^-^aj+a;') (aa;+a;''-a')=a;*-(«^-aa;)'=a5*-a*+2a=a;-aV. Ex. 7. (a + 5 + c + ^ (<^ + &-C - ^) = (a + &)* - (c + cQ' = a- + 2a5 + P-c^-2cd-d\ Ex. 8. (a + 25-Sc-^ (a-2I> + 3c-^ = (a-6r)^-(25-3(j)« = a^ -2a^ + d''-4:P + 12Z^c-0c^ Ex. 11. 1. TV'rite down the squares of a-x, 1 + 2aj^, 2(x'^ + 3, 3aj - 4y, 2. Write down the squares of 3 + 2x, 2x - 3y, a^ - 3aaj, Jo;^ - cxy. 3. Write down the product of (2a + 1) x (2a - 1), (3aa; + 1) x(Sax-d), (x-1) (x+1) (x^ + 1). 4. Write down the product of (x + Z) (x + 1), (ajV 4) (ic'-l), (a&-3) (^h + 2), (2^aj-3&) (2«aj-5). 5. Find the continued product ofx + a,x-a,x + 2a, and x - 2a, 6. Find the continued product of mx^2ny^ mx-2ny, mx-Zny^ and mx + 3?i2/. 7. Simplify 2> (a-2xy ■v2(a-2x) (a^2x) + (^x-a) (Zx+ay(2a-'Zxy, 8. Multiply ic^ + 2ajy + 2y'' by a;'-2iC2/ + %'? and 2a=-3a& + l^ by 2a'' + 3aZ> + 5^ 9. Multiply a^h ■¥ c by a + h-c, by a-h-r c, and by a - 5 - c. 10. Multiply « - 5 + c by a - 2> - c, by Z> + c - o^, and by c - Z> - a. 11. Multiply 2a+h-3chj 2a -I + 3o, and by 5 + 3c -2a. 12. Multiply 2a- 5- 3c by 2a + & + 3c, and by 5 -3c -2a. 13. Multipl}'- a -^-l) + c -^^ d by a-l + c-d^ by a-d-c + d, and by 5 + c - c? - a. 14. Multiply a - 2Z> + 3c + (Z by a + 2Z> - 3c + ^, by 25 - a + 3c + d?, and by a -f 25 + 3c - ^. 25. To divide one simple algebraical quantity by another, divide respectively the coefBcient and letters of the dividend by those of the divisor; and then, if the two quantities have the same sign, prefix to the quo- tient thus obtained the sign +^ if different, the sign -. I Thus 14a5-f-2a=-r— =75,-12a-f-10c=: — =--=-— -a-f—2c= -f -j^. 2a ' 10c 5c' 2c 18 DIVISION. The rule for the sign of the quotient is the same as that given in (19), viz. tliat like signs produce + and unlike — ; and is clearly derived from it, for if 4- ^ Qnultijplied by — h produces — a5, of course — db di- vided hj + a produces — 5 ; and so in the other cases. 26. One power of a quantity is divided by another by subtracting the index of the latter from that of the former. Thus — = c&^-»= a' ; for — = — r--= a^ : so— ^^ — = x-yz^, a^ a* a* xyz 27. If the dividend contain several terms, while the divisor still consists of only one, each term of the former must be separately divided by the latter. 2>xy oxy Zxy Zxy 3 x' p - a^c^ - 2al)c^ + 3ac' _ a^c^ ^ 2abc^ __ oac^ a 1 2e 28. But if the divisor be also a compound quantity, we must proceed as in common Arithmetic : viz. (i) Place the quantities, as in Division of Arithmetic, arranging the terms of each of them, so that the dif- ferent powers of some one letter, common to both of them, may follow in order of the magnitude of their indices, (it matters not whether in ascending or de- scending order, only the same order in each of them) ; (ii) Divide the first term of the dividend by that of the divisor, and set the result in the quotient ; (iii) Multiply the whole divisor by thefirst term of the quotient, and subtract the product from the dividend ; (iv) Bring down fresh terms (as may be required) from the dividend, and repeat the whole operation. DIVISION. 19 Ex. 1. Ex. 2. 1- X ^x^- Wy -12a;y'^ + 16y» Ex. 3. a-x) a^-x^ {a^ + ax + x^ Ex. 4. a + ic) a' + x^ (a^-ab + W. 7. Divide aUW by «- + 2a6+22*', and 4a?V + l by 2j; V-2a;?/ + 1. 8. Divide a;''-2a;V+2xV'-4a;V' + 8a;'^2/' + lG2jy'-32y* by a;^-2y^ 0. Divide l + Q^x^ + hx^ by l + 2a;+a;^ and ^^"-6^ + 5 by a'-2«+l. 1 0. Divide a;'*-4a;2/^ + Zy^ by x^-2xy + y'. and m* + 4/71 + 3 by m^ + 2m + 1. 11. Divide aJ'-Aa^l'' -Wl''-\lab^-\2l'' by a^-2a&-32;^ 12. Divide a;«-2a;Vl by a;--2a;+l, and «° + 2a'6^ + 5» by aV2a5 + Z''. 29. In some of the following Examples, the div' and div* are not properly arranged according to powers : the student must attend to this before and i)i the course o/ division. In Ex. 1, for instance, where a is taken as the letter of reference, and its powers arranged in descending order, there is found in the first rem' the terms - a"^!)^ - crc. These terms must be set Jirst, but since both involve a^, there is nothing as far as a is concerned to shew which is to be set first of the two. In such cases we take another letter, as 5, to be, as it were, next in authority to a, and so, (arranging in de- scending powers of &,) we prefer - a^l) to - a'c. Ex. 1. Divide a^ + l^ + c^ -Zahc by a + h + c' a + h + c) a^ - Zabc + 5^ + c^ {ci^ - ah - ac + 1^-1)0 + c" «' + (c^b + a^c - cc'b - a^c - oabc - a^b - ab'^ - cibc - o^c + a¥ - 2alc - a'c - abc - ac^ + al)^ - abc + ac^ + 5* + a?>' + &' + b'^c - abc + ac^ - b'^c - abc - b^c - bc^ + ac"^ + bc^ + c' + flrc^ + be' + c' The above is the most easy method in such a case ; but thv following, in which the cocff • of the difterent powers of a are col Iccted in brackets, is the most neat and compendious. DIVISION. 21 Ex. 2. a + (h + c)] a* - Sahc + (h* + c') [a' - (6 + c) a^ + (6^ - Jc + c') «' + (& + c) fl^' - (6 + c) flt'* - 3&C6^ -(b + c) a^ - (6" + 2bc + c^) <^ + (6^ - 5c + c^) a + (6' + c*) Ex. 13. 1. Divide a;' -(a +p)x^ + (q + ap) x - aq hy x- a, 2. Divide maz^ + (mb - no) z^ - (mc + nb) z + nchy mz- n, 3. Divide y^ - my*' + ny^ - ny^ + wzy -Ihj y -1. 4. Divide a^ - 6^ - c' + ^' - 2acZ + 2bc hy a-b + c-d. 5. Divide a^ + ab + 2ac- 26' + 75c -Zc^hy a-b + 3c. 6. Divide a^-6^+c' + 3a6c by a-b+c. and a'-^'-c^-SaSc by Or-b-e, 7. Divide i + x^Sy^+ 6xy by 1 + ic-2y, and 1-x^ + 8y ' + 6xy byl-a; + 2y. 8. Divide a;' - By' - 27^^ - ISxyz hyx-2y- 3z. 9. Divide x* + y^-z* + 2xY -2z^-l by x^ + y''-z''- 1. 10. Divide « by 1 + a;, and 1 + 2ic by 1 - 3a?, each to 4 terms in the quotient. 11. Divide 1 by 1 - 2aj + a;', and 1 - aa; by 1 + bx, each to 4 terms. 12. Find the rem', when x^ -px^ + qx-r is divided by aj - a. 30. We have seen above (28 Ex. 3 and 4) that a^-x^ is exactly divisible by a-x, and a^' + x^ hj a + x, and that, in the quotient in each case, the powers of a decrease continually, while those of x increase. The following general facts should be well noticed, as they will enable us to write down at once the quotient, when similar cases occur, as they often do, in practice. It will be seen that the index n is here used to denote generally any index, as the case may be : the quantity a^ is called the n^^ power of a, and read a to the n^\ If the index be odd^ a^+cc"* (like a-]-x)is div. by a+x^ gn __ ^n ^]i]^(3 a-x) . . . . by a - aj ; iftheindexbe^'y^/i, (2^+a?'*(likea''+a?'') . , . .hj neither^ a'»-a;~(like^'~a?') by loth. The student will best remember these, by thinking, in each case, of the simplest form of the same kind. ft 22 DIVISION. Thus, for a^ + x^ (index even., sign +) let him think of a* + x'; this, he knows, is div. by neither ; then a* + a;* is div. by neither : again, for a' - x^ (index odd^ sign -) let him think of ) + 3 («'- 5') - 4 (a-hy. 9. 5(a!'-t/=)H 3(a;+t/)^ 3 (a;^ - y^ - 5 (a; - ^)^ (x + yy + 2 (x^ + xy) - 3 (a;' - y^), 10. 2 («' + a'h + a&^) - (a'-¥), a' -h'- Zal (a - l\ a'-V + (a'' - ly - 2a' + 2a''l\ 32. So too we may often apply (23 iii) to resolve a trinomial into factors. Ex, 1. a;H7a;+12=(a;+3) (a;+4). Ex. 2. a;^-9a;+14=(a;-2) (a;-7). Ex. 3. flj^_5a;-14=(a;-7) (a;+2). Ex. 4. 6a;^+a>-12=(3a;-4) (2a;^3). The student may notice that, if the last term of the given trinomial be positive^ (Ex. 1, 2), then the last terms of the two fac- tors will have the same sign as the middle term of the trinomial ; but if negative^ (Ex. 3, 4), they will have one the sign +, the other -. In Ex. 4, it is clear that the first terms of the two factors might be (Sx and a;, or 3a; and 2x, since the product of either of these pairs is Qtx"^ ; and so the last two terms might be 12 and 1, 6 and 2, or 4 and 3 : it is easily seen on trial which are to be taken, that is. which serve also to produce the middle term of the trinomial. Ex. 16. Resolve into elementary factors 1. a;H6a;+5, a;H9a;+20, a;''-5a;+6, a;^-8aj+15, a;^ + 8a^+7, a;'-10;r+9. 2. a;^ + X-Q, x^'^x^^, a;^-2a;-3, a;''+2a;-15, a;V7a;-8, a;^-8a;-9. 3. 4a;'» + 8a;+3,4a;'* + 13a;+3,4a;^ + lla;-3,4a;'~4a'-3, 3a;^+4a!-4 6a!V5a;-4. 4. 12a;^-5a;-2, 12a;' - 14a; + 2, 12a;'-a;-l, a;' + a;-12, 2r'-2a;-5. 5. aV-3a'a;+2a*, a'-«'^+6aa;^3a'5+a'5'-2«5^ 12a* + rzV-a;*. 6. 2a;V+5a;'y'+2a;y',9a;V'-3a;y'-6y*, 6aV+a5a;-a', 65V-75a;'^a;*. I OHAPTEE III. I' SIMPLE EQUATIONS. 33. When two algebraical quantities are connected by the sign =, the whole expression is called, accord- ing to circumstances, an identity or an equation. An identity is merely the statement of the equiva- lence of two diflferent forms of the same quantity, and is true for any values of any of the letters involved in it. Thus it is always true, whatever be the values of x and y, that (x + y) (x- y)=x'^- 2/^5 or that (x ± yY = aj^ ± 2xy + y"^ : and so also it is always true that \ (x + y) +i(:x-y) = ^ + ^y + ^x-^y = Xj and, in like manner, that ^ ix+y)-^{x- y) =^x+^y-^x+iy=y : each of these expressions is therefore an identity. And in this way we may see one of the principal ad- vantages of Algebra, viz. that it enables us to prove once for all, and express by means of letters as general statements, results which by mere Arithmetic we could only shew to be true upon actual trial in each instance. Of this we have seen examples in the three formulas of (23) ; and so also the two last above given express the general facts, that the greater (x) of any two quantities is equal to the surn^ and the lesser (y) is equal to the difference, of their semi-sum and semi-difference. 34. An equation^ however, is the statement of the equality of two differe^it algebraical quantities ; in which case the equality does not exist for any^ but only for some particular values of one or more of the letters contained in it. Thus the equation, a; - 3 = 4, will be found true only when we give X the value 7, and x^ = Sx - 2 only when we give x the value 1 or 2. SIMPLE EQUATIONS. 25 We arc about to explain the method of finding these values which satisfy the simpler kinds of equations. 35. The last letters of the alphabet a?, y, 2^ (fee, are usually employed to denote those quantities, to which particular values are to be given in order to satisfy the equation, and are said to be the unJcnown quantities. An equation is said to be satisfied by any value of the unknown quantity which makes the values of the two sides of the equation the same. This includes the case when all terms of an equation lie on one side and on the other, as in a;^ - 3ic + 2 = 0, which is satisfied by 1 or 2j cither of which, being put for x^ makes the first side = 0. Those values of the unknown quantities, by which the equation is satisfied, are called its 'roots. Thus 1 and 2 are the roots of the equation aj^ - 3a; + 2 = 0, 7 is the root of a;-3=4, 1, 2, and 3 are the roots of a;'-6=6aj^-lla;j &c. 36. An equation of one unknown quantity is said to be of as many dimensions as is denoted by the index of the highest power of the unknown involved in it. Thus aj - 3 = 4 is an equation of one dimension, or a simph equation ; a;' = 3a; - 2 is of two dimensions, or a quadratic equa- tion ] x^~6 = 6x^ is of three dimensions, or a cubic equation ; x*- 4aj =13 is of four dimensions, or a Mquadratic equation; &c. &c. It may be noticed, in passing, that it can be proved tliat every equation of one unknown quantity has as many roots as it has dimensions, and no more. 37. Every term of each side of an equation may he multiiMed or divided hy the same quantity^ without destroying the equality expressed hy it. Thus, if 3a; + |a; = 34, multiplying every term by 4, we have 12a; + 5a; = 13G, or 17a; = 13G ; therefore also, dividing each term by 17, x = ^-^ =8. I 26 SIMPLE EQUATIONS. Again, if 12ir + 6a; = 144, dividing every term by 6, 2x^ aj = 24, or 3a; = 24; hence also, dividing each term by 3, x = 8. We find, therefore, that 8 is the root of each of these two equations. 38. Hence an equation may be cleared of fractions^ by multiplying every term by any common multiple of all the den". If the l. c. m. be employed, the equa- tion will be expressed in most simple terms. Thus, \^ \x + \x + |a; = 13, multiplying every term by 12, which is the L. c. M. of 2, 3, 4, we have ¥^ + V^^ + V^^ = 1^^> ^^ (jx + Ax -^Zx- 150 ; hence 13a; = 15G, and x = V¥ = 12. 39. A qumitity onay he transferred from one side of an equation to the other hy changing its sign^ without destroying the egiiality exjyressed hy it. Thus ii^x — a=^y-{-l)^ adding a to each side of the equa- tion (which, of course, will not destroy the equality) we have x—y+l)+a^ and, subtracting J from each side, we have x—h^-y+a ; where w^e see that the — a has been transferred to the other side with its sign changed to +, and so also the +5, w^tli its sign changed to — . Hence if the signs of all the terms of both sides of an equation be changed, the equality expressed by it will not be destroyed. Simple Eguations of one unhnown Quantity. 40. To solve a simple equation of one unknown. (1) Clear it, when necessary, of fractions (38) ; (2) Collect all the terms involving the unknown quantity on one side of the equation, and the known quantities on tlie other, transposing them, when ne- cessary, wirti change of sign (39) ; SIMPLE EQUATIONS. 37 (3) Add together the terms of each side, and divide the sum of the known quantities by the sum of the coefScients of the unknown quantity; and thus the root required will be found. Ex. 1. 4x + 2 = Zx + 4. There being no fractions here, we have only to collect the terms ; .*. 4x - 3ic = 4 - 2j or a; = 2, the root of the equation. Ex. 2. 4jj + 5 = 10.C - 16. Here lOaj - 4.^ = 5 + 16 ; .-. 6x = 21, and aj = ^'- = 3f = 31 Ex. 3. 5 (a? + 1) - 2 = 3 (a; - 5). Here, removing the brackets, 5aj + 5 - 2 = 3a; - 15 ; .-. collecting terms, 5a;-3a;=- 15-5 + 2, or 2a;=-18, and .*. x=-9. Ex. 4. hx-r2x-a = Sx + 2c. Here 2>a; + 2a; - 3a; = (5 - 1) a; = a + 2c ; ,\ x= -^ — ::-. Ex. 17. 1. 4a;-2 = 3a; + 3. 2. 3a; -f r = 9a;-5. 3. 4a; + 9 = 8a;-3. 4. 3 + 2a; = 7 - 5a;. 5. a; = 7 + 15a;. 6. mx + a = nx + d. 7. 3(a;-2) + 4 = 4(3-a;). 8. 5-3(4-a;) + 4(3-2a;) = 0. 9. 13a; -21 (a'-3) =10-21 (3 - a;). 10. 5 (a + x)-2x= ^ (a- 5a;). 11. 3(a;-3)-2(a;-2) + a;-l = a; + 3 + 2 (a; + 2) + 3(a; + 1). 12. 2a;-l-2(3x-2) + 3(4a; - 3) - 4(5a; - 4) = 0. 13. (2 + a;) (a - 3) = - 4 - 2aa;. 14. (m + n) (m -x) = m (n - x). I Ex. 5. ia;-fa; + fa; = 11 + |a;. Here we first clear the equation of fractions, by multiplying every term by 24, the l. c. m. of the den", and (observing that in the first fraction-^/- = 12, in the second, -^^ = 8, and so in] the others) thus we get 12a; -8x2a;+6x8a; = 264 + 3a;, or 12a;-16a5 + 18a; = 264 + 3a; collecting terms, 12a;- 16a; + 18a; - 3a; = 264 ; .-. 11a; = 264, and x==-^^ = 24. Ex. 6. l(a; + 1) + » (a; + 2) = 16-^ (.r + 3). Multiplying by 12, we have 6 (a; + 1) + 4 (a; + 2) = 192 - 3(a;+3), or Ox + 6+4a;+8=192-3x-9; collecting, 6a; -f 4a; + 3a; = 192-9- 6 - 8 ; .-. 13a; = 169, and x ^ Vr = 1^. 28 SIMPLE EQUATIONS. Ex.7. 1 (Zx'+xy^ i2xUx) + l (x'^xy2^\=x'^-f^+i ix'-^x)-^(x'^5x). Expressing the mixed number 2/^ as an improper fraction f J, we then multiply by GO, the l. c. m. of the den"; and, olserting the remarh at the end of (l(j)^ we thus obtain 90ic' + 30iB-40a;''-20iC + 15a;^ + 15x-129=G0a;' + 8 + 10a;' + 10aj-5a;'-25aj ; collecting, we find that the terms involving x^ destroy one another (otherwise the equation would be a quadratic), and we have the result 30a; - 20a; + 15a; - 10a; + 25a; = 129 + 8 ; /. 40a; = 137, and x = 3^ J. Ex. 18. 1. la;+^a;=a!-7. 2. •|a;-^a;={aj~l. 3. Aa;— Ja;+|a;=2-Ja;+/^ic• 4. |a; + ^(a;-2) = 2a;-7. 5. |a; + i(a;-l) = a;-4. ^ 6. i(9-2a;) = ^-yV(7^-18). 7. a; + i (14 - a;) = ^ (21 - a;). 8. 2a;~^=f(3-2a;)+|a;. 9. | (2a; + 7)-yV (9«^-8)=H«-ll)• 10. \{x^a)-\C2x'-Zl)-\{a-x) = Q. 11. i(3a;-l)-f(a;-l) = H^-3)-^(a;-5) + 5i 12.ia;-lf=8? + 2(?a;-l)-^(a? + 8). In some of the following examples the common multiple of all the den" is too large to be conveniently employed. In such a case, we may see whether two or three of the den" have a simple common multiple, and get rid of their fractions first, observing to collect terms, and simplify as much as possible, after each step. Ex. 8. tV (2a; + 3)-^(a;-12) + J (3a; + 1) = 5J- + yV (4^ + 3). Here the l. c. m. of all the den" would be 132 : but as 12 will include three of them, multiplying by it, (having first changed H to ^{^, we get If (2a; + 3) -4 (a;-12) + 3 (3a; + 1) = 64 + 4a; + 3; /. If (2a; + 3)-4a; + 48 + 9a; + 3 = 64 + 4a; + 3; hence, collecting terms and simplifying, we have ff (2a; + 3) -4a; + 9a; -4a; = 64 + 3-48-3, or jf (2a; + 3) + a;=16,- .-. 12 (2a; + 3) + 11a; = 176, or 24a; + 11a; = 176 - 36 ; .-. 35a; - 140, and x = W = 4. I SIMPLE EQUATIONS. 29 Ex. 19. 1. A(2a5-3)-J(3.T-2) = |(4aj-3)-3/^. 2. 5(^_9)^7(^.5) =1(05-7) + If. 3. tV(2^-1)-iV(3^-2) = tV(^-12)-3V(^ + 12). 4. I (7x + 20) -j\ (3aj + 4) = yV i^^ + 1) - ^V (20 - 8a;). 5. f (^-2x)-f (2«-a') + ^(a;-a) = J-|(«+ir). 6. ,V(9^-10)-TV(2^-7) = f^-3V(5 + aO. 7. A(4^-l)-A(2^+ l) = 51-/2^- 8. |{a-(5-a;)}-J|oj-(5-a)}-f{5-(^+aj)}=||a; + a-5}. 9. |(4aj-21) + 7| + |(a;-4) = a; + 3f-i(9-7a;) + 7V- 10. -J (a; - a) - -5^^ {m - (a - aj) } = ^ (m + ic) - ^V (^^^ + lOiii), 11. J^ (2a; + 7)-yV (S.'c-T) = 1|- 2V (3aJ + 4). 12. i \x-'(2a-Zc)\ -i^-^{7a-5 (x-2c)\ = ^\{S (a+lOc) '-(2c-x)\. Problems jproducing Simjple Eqitations. 41. We shall now see the practical application of the above in the solution of many entertaining Arithmeti- cal questions. In treating these, however, ^ter having observed the methods used in the following examples, the student must be left very much to his own inge- nuity, as no general rule can be stated for their solu- tion. The only advice that can be given is to read over carefully and consider well the meaning of the ques- tion proposed ; then it will always appear that som.e quantity, at present unknown, is required to be found from the data furnished by it : put x to represent this quantity, and now set down in algebraical language the statements made in the question, using x whenever this unknown quantity is wanted in it. AVe shall thus (in the problems w^e are now considering) arrive at a simple equation, by means of which the value of X may be found. Ex. 1. What number is that to which if 8 be added, one-fourth of the sum is equal to 29 ? Let X represent the number required ; 30 SIMPLE EQUATIONS. adding 8 to it, we have a; + 8, one-fourth of this is J (aj + 8), and this is equal to 29 ; we have, therefore, the equation ^ (x + S) = 29, whence x = 108. Ex. 2. What number is that, the double of which exceeds its half by 6? Let X = the number ; then the double of x is 2x, the half of a; is ^a* ; hence 2x-\x = 6, whence x = A. Ex. 3. A cask, which held 270 gallons, was filled with a mix- ture of brandy, wine, and water. There were 30 gallons of wine in it more than of brandy, and 30 of water more than there were of wine and brandy together. How many were there of each 1 Let X - no. of gals, of brandy ; .*. a; + 30 = • . . . wine, and 2a; + 30 = .... wine and brandy together ; .*. 2a; + 30 + 30 or 2a; + GO = gals, of water 5 but the whole number of gallons was 270 ; .-. a; + (a; + 30) + (2a; + 60) = 270, wlience x = 45, the no. of gals, of brandy, a? + 30 = 75, wine, 2a; + CO = 150, water. Ex. 4. A sum of £50 is to be divided among A^ B, and C, so that A may have 13 guineas more than i?, and C £5 more than A : determine their shares. Let X = jS's share in sMllings : .-. a; + 273 = ^'s, and {x + 273) + 100 or a; + 373 = C's ; .-. , since £50 = IOOO5, {x + 273) + x + {x-^ 373) = 3a; + 646 = 1000 ; .-. 3a; = 354, and x = 118, a; + 273 = 391, x + 373 = 491, and the shares are 391«, 118^, 491^, or £19 II5, £5 18s, £24 11«, respectively. Ex. 5. A^ B, C divide among themselves 620 cartridges, A taking 4 to j5's 3, and 6 to 6"s 5 : how many did each take ? ^ Let X - A^s share ; then fa; = B^s, |a; = 6"s ; .-. x + ^x + ix=^ 620, whence x = 240, J;r = 180, ix = 200. AYe might have avoided fractions by assuming 12a; for A's share, when we should have had 9a; = i?'s, and 10a; - C^s; .-. 12a; + 9a; + 10a; = 620, whence a; =^ 20 ; and the shares are 240, 180, 200, as before. SIMPLE EQUATIONS. 31 Ex. 20. 1. What number is that which exceeds its sixth part by 10 ? 2. What number is that, to which if 7 be added, twice the sum will be equal to 32? 3. Find a number such that its half, third, and fourth parts shall be together greater than its fifth part by 106. 4. A bookseller sold 10 books at a certain price, and afterwards 15 more at the same rate, and at the latter time received 35s. more than at the former : what was the price per book ? 5. What two n^s are those, whose sum is 48 and difference 22 ? 6. At an election where 979 votes were given, the successful candidate had a majority of 47 ; what were the numbers for each ? 7. A spent 2$ 6d in oranges, and says, that 3 of them cost as much under Is, as 9 of them cost over 1^: how many did he buy ? 8. The sum of the ages of two brothers is 49, and one of them is 13 years older than the other : find their ages. 9. Find a number such that if increased by 10, it will become five times as great as the third part of the original number. 10. Divide 150 into two parts, so that one of them shall be two-thirds of the other. 11. A post is a fourth of its length in the mud, a third of its length in the water, and 10 feet above the water ; what is its length ? 12. There is a number such that, if 8 be added to its double, the sum will be five times its half. Find it. 13. Divide 87 into three parts, such that the first may exceed the second by 7, and the third by 17. 14. Find a number such that, if 10 be taken from its double, and 20 from the double of the remainder, there may be 40 left. 15. A market-woman being asked how many eggs she had, re- plied, If I had as many more, half as many more, and one egg and a half, I should have 104 eggs : how many had she? IG. A and B began to play with equal sums ; A won 30*, and then 7 times A^s money was equal to 13 times J5's : what had each at first ? 17. A is twice as old as B; twenty-two years ago he was three times as old. Required J.'s present age. 18. A and B play together for a stake of 5^ ; if A win, he will $2 SIMf»LE EQUATIOKS. have thrice as much as B ; but if he lose, he will have only twice as much. What has each at first ? 19. Divide £G4 among three persons, so that the first may have three times as much as the second, and the third, one-third as much as the first and second together. 20. A workman is engaged for 28 days at 2s Qd a day, bub instead of receiving anything, is to pay Is a day, on all days upon which he is idle : he receives altogether £2 12s (jd ; for how many idle days did he pay ? 21. A person buys 4 horses, for the second of which he gives £12 more than for the first, for the third £G more than for the second, and for the fourth £2 more than for the third. The sura paid for all was £230. How much did each cost ? 22. A person bought 20 yards of cloth for 10 guineas, for part of which he gave lis (xZ a yard, and for the rest 7* Qd a yard. IIow many yards of each did he buy ? 23. Two coaches start at the same time from York and London, a distance of 200 miles, travelling one at 9| miles an hour, the other at 9| : where will they meet, and in what time from starting ? 24. A cistern is filled in 20 min. by 3 pipes, one of which con- veys 10 gallons more, and another 5 gallons less than the third 2)er minute. The cistern holds 820 gallons. How much flows through each pipe in a minute ? 25. A starts upon a walk at the rate of 4 miles an hour, and after 15' B starts at the rate of 4J miles an hour ; when and where will he overtake A 1 26. A garrison of 1000 men was victualled for 30 days ; after 10 days it was reinforced, and then the provisions were exhausted in 5 days: find the number of men in the reinforcement. 27. A and B have together 8s, A and G have 10s, B and C have 12s. What have they each ? 28. What was the total amount of a person's debts, who when he had paid a half, and then a third, and then a twelftli of them, had still 15 guineas to pay? 29. A father's age is 40 and his son's 8 : in how many years will the father's age be triple of the son's ? 30. IIow much tea at 4s (Sd must be mixed with 50 lbs. at C^ that the mixture may be sold at 5s CZ ? CHAPTEE IV. INVOLUTION AND EVOLUTION. 42. Involution is the name given to the operation by which we find the powers of quantities. We have already (22) had occasion to notice the square of a binomial: but all cases of Involution are merely ex- amples of Mult", where the factors are all the same. It should be noticed, that sitij powe?' of a power of a quantity is obtained by multij)lyiiig together the indices of the two powers. Thus the cube of a;', that is (x^y = a?® ; for it = x^ ^ x^ ^ x* ^ ^2 + 2 + 2 (20) = a;« : and, similarly, (x^ = a;« = (x^y, that is, the square of the cube is the same as the cube of the square of any quantity, &c. So also (a»)^= a'^= {a% (2xYy='^^Y, (-2xy'z'y= "SxY^', Hence, we may shorten the operation of finding the 4th power of a quantity by squaring its square ; and, similarly, to find the 6th, 8th, &c. powers, we may square the 3d, 4th, &c. So also to find the cube, or 3rd power, we may take the product of the 1st and 2nd, that is, of the quantity itself and its square ; to find the 5th, we may take that of the square and cube ; to find the 7th, of the cube and 4th ; and so on. Thus we shall have (a + by :=(a+h) (a' + 2ab + 5^) = a' -^ 3a«6 + 3«J» + h\ by Mulf, (a - by = (a-b) (a^ - 2ab + b') = a' - 3a^6 + 3a6» - b\ (a ± by = («^ ± 2ab + b^) (a^± 2ab + b^) =a*± 4a=&+6rt'^6^± 4a6^+ b\ (a ± by = (a^ ± 2ab + b') (a'± Sa^b + 3a6^± Z>=) ^a'±5a*b-^lQa'h''±l0a^'^5ah^±b', 2* k 34: INTOLUTION. The above results should be remembered and ap- plied in the following Examples. The expansions of higher powers are generally best obtained by the Bi- nomial Theorem, which will be given hereafter. Ex. 1. (a + h+cy= {a+ (b+c)y= a'+ Za^b+c) -^Za (b + cy+ (& + c)' =a'+ 3a^b + Za^c + Sab^+ Oabc + '6ac^+ 6'+ Zb^c + Zbc^+ c\ Ex.2. (a^b-cy^{a-(b+c)]'=a'-Za^(b + c) + Za(b + cy-(b-^cy =a' - Za^b - Za^c + Zab^ + Gabc + Zac^ -¥- Z¥c - Zbc'' - c\ Or thus : {a-b-cy= {{a-b)-cY ^ {a--by-Zia-by c ^ Z{a-by-c\ which, of course, when expanded, would give the same result as before. Ex. 3. {2x - 3)*= {2xy- 4.3. (2xy + G . 3\ (2;r)'- 4 . Z\ (2x) + 3* x= ICj;^ - 96.r' + 21Ca;' - 21Ca; + 81. Ex. 21. 1. Find the values of(2^6'')», (-3a'5V)», [—^\: (-^-^)*- Write down the expansions of 2. (.c+2)'. Z, {x-2y, 4. (ic+3)». 5. (l + 2a:)*. C. (2m-l)'. 7. (3a; +1)*. 8. (2a;-«)\ 0. (3a;+2a)». 10. (4a-3&)^ 11. (ax-yy, 12. (ax^x^, 13. {2am''my, 14. (a-5+c)'. 15. (l-a;+ic')^ 10. (a + 6aj+caj'^)». 17. (l+aj+x^)*. 18. (l + ic-ic^)*. 10. (l-2x+a;-)». 20. (a-2b^cy, 21. (l + 2a5-3x^)*. 43. The following result is worthy of notice, as it exhibits the form of the square of any Multinomial, (a + 5 + c + (? + &€.)'= a'+ 2a (& + c + ^ + &c.) + (5 + c + + c, &c. INVOLUTION. 35 We see from (i) that the square of any multinomial may bo formed by setting down the square of each term and then the 2>'^oduct of the double of each term hy the sum of all the terms thatfollovj it. Another form of this result is given in (ii), to which reference will be made hereafter. Ex. 1. (1 + 2a; + Zxy = 1 + 2 (2a; + Sx^) + ^x^ + 4a; (3a;'0 + Ox' = 1 + 4a; + 10a;' + 12a;* + 9a;*. Ex.2. {a + hx + cx'^-i-dx* + ex* + &.c.y=a^ + 2alx+2acx^ + 2adx*-t-2aex* + &c. + h^ x^ + 2hex^i^2bdx* + &c. + c'a;* + &c. ^ aU2ah x+ (2ac+¥) x^-y2 (ad+hc) xU \2 (ae-¥l>d) + c^\ x*-i-&c. Ex. ?>. (1 - 2a;)« = (1 - Ca; + 12a;« - Sa;')' = l-12a; + 24a!'-16a;* + 36a;' - 144a;' + 9Gx' + 144a;* - 192a;» + C4a;« == 1 - 12a; + COa;' - 160a;' + 240a;* - 192a;» + 64a;*. Ex. 22. Find the expansions of 1. (1 + a; + a;')«. 2. (1 -a; + 2a;'0'. 3. (3 -2a; + a;")'. 4. (a'-2a& + ZP)\ 5. (2x-Zy + Azy, 6. (3aa; + 2hy + cz)\ 7. (1 - 2aa; - a'x'')\ 8. (2a' - a - 2)\ 9. (1 - a; + a;' - x')». 10. (l + a;)^ 11. (a;''-2a;' + 3a;+4)'. 12. (l-f2a;-3a;'+4a;T. 13. (a' - 2a'b + 2ah' - h')\ 14. (a - x)\ 15. (1 -2a; + 3a;' - 2a;» + a;*)'. 16. (a*-2a'a;+a'a;'-2aa;'+a;*)'. 44. Let tlie student notice the following remarks : (i) Since any even number of like signs, whether all 4- or all -, will give + in mult", it follows that any eve7i power of a quantity is the same, whether that quantity be taken positively or negatively ; thus, (+a)' and {-ay are each = + a^y and (1 -x+xy is the same as \-{i-x + irr)]\ or (-1 +x-xy] §(ii) No even power of any quantity can be negative ; (iii) Any odd power will have the same sign as the I 36 EVOLUTION. 45. Evolution is the name given to the operation by which we find the 7'oots of quantities. Since the G\\\)Q;power of a' = a\ therefore the cube o'oot of ^' is a? ; so Vet" — a", \/16a''b* = 2a% &c. ; and so we may often extract a required root of a simple quantity, by dividing it^ index by that of the root. If, however, the index of the quantity cannot be exactly divided by that of the root (as e, cj, in the 5th root of a?^ where the 2 cannot be divided by 5,) then we cannot find the root of it ; but can only indicate that the root is to he extracted, by writing down the quantity, and the sign 4/ before it, w^ith the index of the root in question ; as y^'', \d. Such quantities are called surds^ or irrational quantities, 46. It follows from (44), that (i) Any even root oi Vi ^positive quantity will have a double sign ± ; (ii) There can be wo evenrooi of a negative quantity; (iii) Any odd root of a quantity will have the same sign as the quantity itself. Thus y V= -3j. v-8.^v« = -2.y^ /-8r/= ^--sf ^^- Hence, when we have a surd expressing an odd root of a nega- tive quantity, we may write the quantity positive under the sign of evolution, and set the negative sign outside ; thus \/ -x^ -- \/x^^ ^6 _y _ l^ = a- + \jh'. But this cannot be done with an eten root of a negative quantity, such as -J -x^, which must be left as it is, and is called an impossible or imaginary quantity; the diifer- encc between surd and impossible quantities being that the former have real values, though v/e cannot exactly find them, while there cannot be a quantity, positive or negative, an even power of which would produce a negative quantity. Imaginary quantities, liowcver, are employed in some of the higher applications of Algebra ; but for the present we shall leave the consideration both of these and of surd quantities. EVOLUTION. 87 Ex. 23. 1. Find the square roots of 4a^h*c\ A9x*yh^, 100a"5"c*». 9a^xY 49ajV 25a; V" 2. Find the square roots of - 25s^ ' 64a^ ' Ua'^b* 3. Fmdy/-^-, 4/-2^r, |/i25^r., |/ 343— 46. To find the square root of a compound quantity. "We ^7i6>i^ that the square of « + 5 is a' + 2a5 + V ; let us see then how from a^ + ^ah + 5'', we might deduce its square root a-j-h. a^+2aJ> + P(a + h Let us write down then the quantity a^ a^ + 2ab + Z>^. Now a, the first term of 2a + h) 2ah + h^ the root, may be found immediately by 2ab + &'■* taking the square root of its first term : set a then on the right, and then subtract a^ ; we have now remaiu- ing 2a'b + h^j and if we divide 2ah by 2a, we get + ^, the other term in the root : lastly, if we add this h to the 2a, multiply the 2a + h, thus formed, by h, and subtract the product, there is no remainder. Now we may follow this plan in any other case, and if we find no remainder, the root will be exactly obtained. Ex. 1. Ex. 2. 9a;- + 6x1/ -f y^ (3.^' -f y Ua^ - 66ah + 49&' (4a ~ 7b 9a;* IGa^ 6a; + 1/) Qxy + 2/* 8a - 7h) - 56a5 + 49&* 6xy -f y '' - 56ab ^ 493>* Ex. 3. 4a^ - 4ab - 5* (2a - b 4a^ 4a -h)- 4a.b - b^ - 4ab -f l^ _ -2b^ Here we find a remainder - 2b^ ; we conclude, therefore, that 2a- J is 710 1 the exact root of 4a^ ~ 4ab - b^^ which is a surd, and can only be written Vl«*^- 4a?; - b^. I 8S EVOLUTION. Ex. 24. Find the square roots of 1. Ax^ + 4xy + y\ 25a'' - SOah + Ob\ 25a;* + ZOx^y + 9x*y\ 2.4:9a^h'-Ua^b+a\ 16a;y +40aJ2/'s+25i/V, 25a*5V + 10»'5c'+cl 47. If the root consist of more than two terms, a similar process will enable us to find it, as in the fol- lowing Example ; where it will be seen that the divi- sor at any step is obtained by doubling the quantity already found in the root, or (which amounts to the same thing and is more convenient in practice) by doubling the last term of the preceding divisor^ and then annexing the new term of the root. Ex. 16a;«-24a;' + 25aj*-.20aj' + IQx'-^x + 1 (4x«- Zx" + 2a;-l ^x^-Zx')-2ix'' ^25x' ^x^-.Qx' + 2x) 16a;* -20a;* + 10a;* iea;*-12a;'+ 4a;* 8a;^-G.7;* + 4a;-l)-8a;*+ 6a;*-4a;+l - 8a;' + 6a;' - ^'aj + i 48. The reason of the above method may be thus exhibited by considering the square oi a + h + c. + c + d,we may find a + & + c as before, and put it = a" : then (a + 1 +c + dy=(a"+d)* = a"^ + 2a"d+d'^, and, as we shall have already subtracted (a +J^+c)' or a"^, the third remainder will be 2a"d+d^ ; and, therefore, tak- ing 2a" as trial-divisor (obtained as before by doubling the last term of the preceding divisor 2a + 2b + c), we may get ^, &c. It will be seen that the successive subtrahends in the above operation are a'^^ (2a + &) 5, (2a' + c) c, (2a" + d)d, &c., and of course, the sum of them all, that is, the whole quantity sub- tracted, is (43 ii) (« + 5 + c + ^ + &c.)^. 49. As the 4:th jpoive7\of a quantity is the square of its square (4^), so the 4th root of a quantity is the square root of its square root, and may therefore be found by the preceding rule. Thus, if it be required to find the 4th root of a* + 4a^x + (ja-x* + Aax^ + a;*, the square root will be found to be a' + 2ax + x^, and the square root of this to be a + ic, which is therefore the 4th root of the given quantity. 50. It should be noticed as in (45) that all eve^i roots have double signs^ Thus the square root of a^ + 2a'b + 5^ may be -(a + J), that is, -a -5, as well as ^ + &: and, in fact, the first term in the root, which we found by taking the square root of a*, might have been - + 8a;*i/^ - lOa;'^/' + ^^'2/' - 4a;y' + y". 10. Of 1 - 6a; + 15a;^ - 20a;' + 15a;* - Gx' + x\ 11. Of 4 - 12a + 5a^ + 14a' - 11a* - 4a' + 4a«. 12. Of j9^ + 2pqx + (2j9r + q"") a;^ + 2 (p« + qr) a;' + (2q8 + r") x* + 2rsx^ + s^a;®. Extract the 4th root 13. Of 1 - 4a; + Ga;^ - 4a;' + x', and of a* - 8a' + 24a^ - 32 a + IG. 14. Of IGa* - 9Ga'6 + 21Ga^6^ - 21Ga6' + 816*. Extract the 8th root 15. Ofa;«-lGa;' + 112a;''-448a;' + 1120a;*-1792a;' + 1792a;M024a;+25G. IG. Of a^-8a^6 + 28a^6=-5Ga'6' + 70a*6*-5Ga'6' + 28a''6»-8a6' + h\ 51. The inetliod of finding the square root of a numerical quantity is derived from the foregoing. Since l=r, 100=10^ 10000 = 100^ &c., it follows that the square root of any number between 1 and 100 lies between 1 and 10, that is, the square root of any number having one or tioo figures is a number of one figure; so also the square root of any number between 100 and 10000, that is, having thr^ee or four figures, lies between 10 and 100, that is, is a number of two figures, &c. Hence, if we set a dot over every other figure of any given square number, beginning with the units figure, the number of dots will exactly indi- cate the number of figures in its f=^quare root. a be Ex. 186624 (400 + 30 + 2 100000 a^ (2a + 6) 800 + 30 = 830) 26624 24900 (2rt + b)b (2a'+ c) 800 + 60 + 2 = 862) 1724 1724 (2a' + c)f EVOLUTION. 4i Here the number of dots is three, and therefore ihe number of figures in the root will be three. Now the greatest square- number, contained in 18, the first period (as it is called), is 16, and the number evidently lies between 160000 and 250000, that is, between the squares of 400 and 500. "We take therefore 400 for the first term in the root, and proceeding just as before, we obtain the whole root, 400 + 30 + 2 = 432. The letters annexed will indi- cate how the difierent steps of the above correspond with those of the algebraical process in (48), from which it is derived. Ex. 1. The cyphers are usually omitted in practice, 186624 (432 and it will be seen that we need only, at any step, take down the next period, instead of K^^ the whole remainder. gQ2)i724 Ii^ Ex. 2, notice (i) that the second re- 1724 mainder 49 is greater than the divisor 47 ; Ex. 2. this may sometimes happen, but no difficulty 7784 i (279 can arise from it, as it would be found that if 4 instead of 7 we took 8 for the second figure, 47)378 the subtrahend would be 384, which is too 329 : large : And (ii), that the last figure 7 of the 4941 ^^^^ divisor, being doubled in order to make - — ~ the second divisor, and thus becoming 14, 10291^64 (3208 ^^"^^^ ^ *^ ^^ 2lMq^ to the preceding figure, 4, 9" which now becomes 5. In fact the first di- 62)129 visor is 400 +70, which, when its second term 124^ is doubled, becomes 400 + 140 or 540. 6408)51264 In Ex. 3, we have an instance of a cypher r 1 OCA. *iJi_ occurring in the root. 52. If the root have any number of decimal places, it is i^lain (by the rule for the mult" of decimals) that the square will have twice as^many, and therefore the number of decimal places in every square decimal will be necessarily even^ and the number of decimal places in the root will be half that number. Hence, if the given square number be a decimal, and therefore one of an even number of places, if we set, as before, the dot upon the units-figure^ and then over every other 42 irv^oLUTiox. figure on loth sides of it, the number of dots to the left will still indicate the number of integral figures in the root, and the number of dots to the right the number of decimal places. Thus 10.291264 would be dotted 10.291264, the dot being first placed on the units-figure ; and the root will have one integral and three decimal places, that is, would be (Ex. 3 above) 3.208. If, however, the given number be a decimal of an odd number of places, or if there be a rem'' in any case, then there is no exact square root, but we may ap- proximate to it as far as we please by dotting as before, {^remembering to set the dot first u/pon the units figure^) and then annexing cyphers (which by the nature of decimals will not alter the value of the number itself) and taking them down as they are wanted, until we have got as many decimal places in the root as we desire. Ex. Find the square roots of 2 and 259.351, to three decimal places. Ex. 1. 2 (1.4U abUV Sa-h + Zah"" + P: by dividing the first term of this remainder by 3a^, we shall get h, the other term in the root, and then, if we subtract the quantity 3a^5 + Zah^ + l^y there will be no remainder. Pursuing the same course in any other case, if there be no remainder, w^e conclude that we have obtained the exact cube root. Here the quantity corresponding Sx^+12x'^i/ + 6xy^+y^ (2x+y to the trial-divisor 3a^ is 3 (2xy 8ic' = 12.i?', that to Za'^b is 12x''y, that to 12a;^) 12x^y + Gxy^ + y^ ^aV is Ga-y', and that to h^ is y^ ; so 1 2x'^y + (Sxy^ + ?y' that the whole subtrahend is \2x'^y-^(jxy'^ + y*. By attending however to the following hint, the subtrahend may be more easily constructed. a'+ Zo}h 4- Zal^-^- b^ (a-^h a' Za + h 3a* (3a+J)J Za^ + aah + P Za^h + Zah'-^ h' Za^h + Zah''+ h* Set down first 3a, some little way to the left of the first re- mainder, and then, multiplying this by a, obtain 3a' as before ; by means of this trial-divisor find i, and annex it to the 3a, so making Za+b] multiply this by J, and set the product (3a + 2») b or Zab + b* under the 3a'j and add them up, miking3a' + 3a5 + 6'; then, mul- tiplying this by 5, we have Za^b +3aZ»'-f P, the quantity required. Tho value of the above method, in saving labour, will be more fully seen when the root has more than two terms, or, if numeri- ?\ more than two figures. ^^S iJ^iore thai u Ex. 6x + y 12x' + Gxy KyOLU-nON. 8a;'' + I2x''y + Qxy^ + y^ (2x + y 12x^y + 6.T^^ + 2/' 12aj^ + Oxy + y' 12ic^y + 6xy^ + y* Ex. 27. Find the cube roots 1. Of x^+6x''y + 12xy'' + SyK 2. Of a^-9aU27a-27. 3. Of a;*+12jj^+48ic + G4. 4. Of Sa' -ZOa'h^ 6 iah'' -27 h\ 5. Of a'+24a^Z>+192a6^ + 5126^ G. Of 8^'-84a;V+294iC2/'-343i/^ 7. Of m^ - 12m''nx + 48/?i;.V - G4/iV. 8. Of oV - I5a''hx' + 75ab''x^ - I25b'x\ 54. If the root consist of more than two terms, as a + i> + c, we may (just as in the case of the square root) first find a+6 as above, and put this = a' : then, at this point of the operation, we shall have subtracted first a^ and then 3a'^b + Zab^ + b\ that is, altogether . (a + by or a'^ ; and therefore, since the whole quantity (a + & + c)' =(a'+c)'=a''+3a'^c + 3a'c^ + c', the remainder will be no other than Sa'^c + 3a'c' + c^ [Tn fact, as was done in the case of the square root, it may be easily shewn to be identical with this.] If, there- fore, we take now as trial-divisor 3a'*, just as before we took 3rt'*, we shall get c the third term in the root, and subtracting the quantity Sa'^c + ^a'c^ + c^, we shall have no more remainder. Now the process of finding 3a ^ is much simplified by observing that it =3 {a+by=Za^ + Oab + W \ but, if we add b\ the square of the last term in the root, to the two lines „ , o i L ? the sum ' da^ + oab + 0^ will be exactly 3a^ + GaZ) + 3Z/', the quantity required. By this means then we get 3a"-', and then have only to set to the left of it 3a' or 3a + 36, (which may be found by tripling the last term of the preceding divisor 3a + b) and proceed just as we did befoi*e when we had set down 3a and 3a* — that is, first finding c, and then forming, as before, 3a'-c + 3a'c* + c', which we subtract, making, with a'' already subtracted, (a + cf or (a + h + c)^ sub- tracted altogether. And so on, if the root were a + b + c + d^ &c The student should study carefully the first of the two follow- ing Examples, as it is the type to which all others .are reft-rred. EVOLUTION. 45 I CO + . . W CO ^ t^ ^* ^ K^ •* CO _, I O "-J to CO ^ sr » >-• r^ v.^ CO ^ CO ^ 5* H. I I CO ttj CO c* »> o hQ CO ^, H. p -• - ii 1 ? 1 H 1 to 1 1 CO 1 CO 1 t c t 1 C 1 1 + + + c c C s^ »» + + ? ? j + + ^ ^ 1 CO + 1 CO "j~ 1 1 f CO CO + + 1 + >— 1 h- ' «u t— ' + + 1 1 1 1 1 CO 1 00 I CO I CO f 00 I I to CO CO Si CO CO Si CO CO Ci II + CO CO Si + CO CO CO + CO CO Si CO + CO Si + CO + CO + CO 40 EVOLUTION. Ex. 28. Extract the cube roots 1. Of a'' + 6a^ + 15a* + 20a' + 15a^ + Ga + 1. 2. Of x' - 12a;^ + 6ix* - 112x' + 108x' - ASx + 8. 3. Of a' - Sa'b + Oa*b'' - 7a'b^ + Ca^5* - Zah" + 5«. 4. Of aj° - 12ax'^ + 60aV - IGOa^aj' + 240aV - I02a'x + 64a«. 5. Of 8x^ + ASx'y + 60xY - ^^^Y - ^OxY + lOSxif - 27y\ 6. Of x' - Zx^ + ^x' - 10a;"' + 12a;* - 12a;* + 103;=* - ^x" + 3a; - 1. 7. Of a^ - &' + c' - 3 (a-5 - a\ - ay - ac" - Vc + W) - ^abc. 8. Of l-6a;+21a;^-56.t;= + llla;*-174a;*+219a;«-204a;U144a;«-64a;». 55. Since 1 == 1^ 1000 = 10^ 1000000 = 100', &c., it follows that the cube root of any number between 1 and 1000, that is, having one^ two^ or three figures, is a number of 07ie figure ; so also tlie cube root of any number between 1000 and 1000000, that is, having fouT^five^ or six figures, is a number of two figures, &c. Hence, if we set a dot over every tliird figure of any given cube number, beginning with the units- figure, the number of dots will exactly indicate the number of figures in its cube root. If the root have any number of decimal places, the cube will have thrice as many ; and therefore the num- ber of decimal places in every cube decimal w^ill be necessarily a multiple of three^ and the number of decimal places in the root will be a tliird of that num- ber. Hence, if the given cube number be a decimal, and consequently have its number of decimal places a multiple of three, by setting as before the dot upon the units-figure^ and then over every third figure on l)oth sides of it, the number of dots to the left will still indicate the number oi integral figures in the root, and the number to the right the number of decimal places. If the given number be not a perfect cube, we may dot as before, (always setting the dot first upon the units- EVOLUTION. 47 figicre)y and annex cyphers as in the case of the square root, so as to ai)proximate to the cube root required, to as many decimal places as we please. It will be seen, by the following example, how the numerical process corresponds to the algebraical. The cyphers are omitted, except that in the numbers corresponding to 3tt*, 3«j'', &c., it is necessary to express two at the end : thus a is really 4000, and therefore Za^ is 48000000 ; but as in the first remainder we only need the figures of the first and second periods, corresponding to 43 in the root, we may treat the a as 40, and thus 3a' will be 4800 and Za will be 120, so that Za + l will become 123. Ex. 80677568161 (4321 64 4800 369 5169 16677 15507 554700 2584 1170568 557284 1114568 55987200 12961 56000161 56000161 56000161 123 1292 12961 Ex. 29. Find the cube roots of 1. 9261, 12167, 15625, 32768, 103.823, 110592, 262144, 884.736. 2. 1481544, 1601.613, 1953125, 1259712, 2.803221, 7077888. 3. 12.812904, 8741816, 56.623104, 33076.161, 22425768. 4. 102503.232, 820025856, 264.609288, 1076890625, 2.116874304. 5. Extract to 4 figures the cube roots of 2.5, .2, .01, 4. CIIAPTEE V. GREATEST COMMON MEASURE I LEAST COMMON MULTIPLE. 56. When one quantity divides another without remainder, it is said to measure it, and is called a rtieasure of it. ThuSj 3, «, J, 3g^j ab, a^^ &c. are all measures of Za^h. A common measure of two quantities is one which divides each of them without remainder. Thus, a, &j 3^5 35, ab^ 3a5, are all common measures of Za/h and V^abc ; and their greatest common measure, that is, the largest common factor they contain, is 3a5. 57. It is commonly easy to detect Tjy inspection^ i, e. by looking at the two quantities, their largest common measure, if it is a simple factor, that is, if it consists of only one term ; because then it will be found as a factor in every term of each of them. Thus, Zxy will divide every term of Zx^y - Gxy^ and also of Zxy-9x^y' ; it is therefore a common measure of them : and since when these are divided by Zxy^ the quotients cc^ - 2i/' and 1 - Zxy have no common factor, Zxy is their greatest common measure (g. c. m.). So 2a^Z> is the greatest divisor of (ja'h^ - 8a% and a-c of 2a'c' — 5a^hc ; and a^, which is the g. c. m. of 2a^b and a^c, is plainly therefore the g. c. m. of Oa^b^ - Sa*b and 2a'c' - 5a'6c. Ex. 30. Find the g. c. m. of 1. 3a;' and I2x^y ; 4:a^b^ and - 6ab^ ; - I2x'yh* and 8i/zK 2. Zax^ - ^a'x and a V - Zabx ; 3a» + la'b - 5a5' and 2a'6 + 2aV ; Caj'y ~ 12x^y^ + Zxy^ and 4aa;' + 4:axy + 4a'a;. 58. In like manner we may sometimes find by inspection the g. c. m. of two quantities, when not a simple factor, if it happens to be easy to separate them into their component factors. GREATEST COMMON MEASURE. 49 Ex. 1. The G. c. M. of GaV (a* - x^) and Aa^x (a + xf is 20^ X (a + x). Ex. 2. The g. c. m. of a" (ci'x''- Zax^ + 2x') and a;' (a* - 4«V), that is, of a^x"^ (a* - Zax + 2x'^) or a^x^ (a - 2x) (a - x) and a'x^ (a'' - 4x'), is ^^o;^ (^ - 2x). Ex. 31. Find by inspection the g. c. m. of 1. 4x^ (a+xy and 10 (a^'x-xy. 2. o)^ (a^-x^y and (a'*a;+aic^)». 3. (a'b-aPy and «& (a^-5^)^ 4. 6 (x^-1) and 8 (iC^-3iC+2). 5. (;i'^ + 0.-)' and ic^ (x''-x-2). C. 4 (o;^ + a^) and 6 (a;'-2aa;-3a»). 7. a' (a;^ -f 12^ + 11) and aV - Ua"x - 12a\ 8. 9 (aV - 4) and 12 (a'^x'' + 4aaj + 4). 59. But if the greatest common measure of two quantities be a compound quantity^ it cannot generally be thus easily found by inspection, but may always be obtained by a method we are now about to ex- plain, the proof of which will be given hereafter. Def. An algebraical quantity is said to be of so many diw^nsions^ as is indicated by the highest index of its letter of reference. Thus aj^ - 7ic + 10 is of tico dimensions, aj^ + 1 of three. If it also involve other letters, it is said to be of so many dimensions in each of them, according to the highest indices of each. Thus x*y+Sx^y^+x^y^ is of four dimensions in x, and three in y. If the dimensions of each term are the same, the quantity is said to be homogeneous, and of so many dimensions as is indicated by the sum of the indices in each terra. Thus the last quantity is homogeneous^ and of five dimensions. The word dimensions has been adopted from the language of Geometry ; — such quantities as «, 6, &c being compared to lines (which have only one dimension, viz. length), and called linear 3 60 GREATEST COM^ION MEASURE. quantities ; such quantities as a?^ ab^ &c. to areas (which have two dimensions, length and breadth) ; and such as a^ a"^!)^ ahc, &c. to solids (which have three dimensions, lengtli, breadth, and thick- ness) : beyond this we have no corresponding quantities in Geom- etry ; but the term dimensions, having been once employed in Algebra, has been retained in all other cases. 60. Let there be given then two algebraical quan- tities, of which it is required to find the g. c. m. Ar- range them according to powers of some common letter, and divide the one of higher dimensions by the other ; or if the highest index happen to be the same in each, take either of them for dividend. Take now, as in Arithmetic, the remainder after this division for divisor, and the preceding divisor for dividend, and so on until there is no remainder : then the last divisor will be the g. c. m. of the two given quantities. Ex. Find the g. c. m. ofx" - 7ic+10 and Ax"" - 25x^ + 20a; + 25. x'-7x+10)4cx'-25x'' + 20x + 25(Ax + 3 4x^ - 28a;^ + 40x ^x^ - 20aj + 25 3iu' - 21a; + 30 5)a;'»-7a; + 10Cr-2 x^-5x -2a; + 10 Ans, a; - 5. - 2a; + 10 We may as well observe, that the expression Greatest c. m., which has beeji adopted from Arithmetic, must be understood in Algebra as applying not to the numerical magnitude, positive or negative, of the quantity, but to its dimensions only, on which account it is sometimes called the Highest c. m. Thus it would be quite immaterial whether, in the above example, we consider the G. c. M. to be a; - 5 or 5 - a; : and either of these, in fact, might be made numerically greater than the other, by giving different values to x. GREATEST COMMON MEASURE. 61 Ex. 32. Find the g. c. k, 1. Of 3.^' + ic -2 and 3a)' + 4ic - 4. 2. Of 6a;' +7x -S and 12a;' + 16a; - 3. 3. Of 9a;' - 25 and Ox'' + Sx ~ 20. 4. Of 8a;' + 14a; - 15 and 8a;' + 30a;' + 13a; - 30. 5. Of 4a;' + 3.^ - 10 and 4a;' + 7a;' - 3a; - 15. 6. Of 2a;* + a;' - 20a;' - 7a; + 24 and 2a;* + 3a;' - 13a;' - 7a; + 15. 61. If the given quantities have both or either of them, in any case, simple factors, as in (57), these ninst be struck out, and the Kule applied to the re- sulting quantities. Then the g. c. m. of these, being found as above, will be the same as that of the given ones ; ^except it should happen that we have to strike factors out of hoth of them, and that these factors them- selves have a common factor. In this case the g. c. m. found, as above, of the resulting quantities, must be multiplied by this common factor, in order to produce that of the given ones. So also, whenever we convert a remainder, accord- ing to the Eule, into a divisor, we may strike out of it any simple factor it may contain. Here, however, there is no restriction, as in the former case ; because no part of such a simple factor can be common also to the new dividend, which, being the same as the former divisor, will be already clear of simple factors. It is only with \\\q first pair, or gi'ven quantities^ that we shall have to attend to this. And if, moreover, the first term of any such re- mainder is negative, we may, for the sake of neatness, before taking it as a new divisor, change the signs of all its terms, which is equivalent to dividing it by - 1* Tliis can only affect the signs of the g. c. m. 52 GREATEST COMMON MEASURE. Ex. Find the g. c. m. of 2a;^ - 8:c^ + 12.i;' - Sx^ + 2x and Zx^ - 6:c« + 3^. Here, striking out of the first the factor 2x (which is common to all its terms) and of the second the factor Sx, we reduce the quan- tities to X* - 4x^ + Qx"^ -4x + 1 and x*-2x^ + 1; but as 2x and Zx have themselves a common factor, x, it is plain that the original quantities have a common factor x, which these latter quantities have not ; hence the g. c. m of these, when found, must be multi- plied by X to produce that of the given quantities. x'-2x^ + l) x'-4.x^+6x''-ix+l (1 x'-2x+l) ic*-2aj' + l (aj' + 2a;+l x*-2.t^ + l x^-2x^+x ^ - 4:x \-4x^+Sx^-4x 2x^-Sx^ + 1 x''-2x +1 2x^-4x^-^2x a;^-2j;+l ic'-2j+l In this Example, the first remainder is reduced by dividing it by ^4x ; and, the g. c. m. of these two quantities being x"^ - 2x + 1. that of the two given quantities will be x (x"^ - 2a; + 1) or x^ -2x^ + x, Ex. 33. Find the c. c. m. 1. Of a^+.^•'anda'+2«aJ + ic^ 2. Of ic'+ a;-2 and a;'- 3a; + 2. 3. Of 2x^ +ex^ + Qx + 2 and Ox^ + 6x^ - ex - 6, 4. Of 2if - lOy"" + 12y and 3^/* - IS^/' + 2hf - 24. 5. Of x^ - eax"" + 12^=0? - ^a^ and a;* - 4aV. 6. Of 2x^ + 10a;' + 14a; + 6 and a;' + a;' + 7aj + 39. 7. Of 3a;' + 3a;'' - 15a; + 9 and 3a;* + 3a;' - 21a;' - 9a;. 8. Of a;' + xhj + xy"^ + y^ and a;* + x^y + xy^ - y*, 9. Of 2a' + a'b - A.arh'' - Zah' and 4a' + a'b - 2a^^ + al\ 10. Of Za'+15a'b-Za'I>''-l5a'h'' and lOa'-ZOa'b-lOa'h^-i-ZOaP. 11. Of X* - 2x^y + 2xy^ - y' and x' - 2x^y + 2a;'7/' - 2xy^ + y\ 12. Of x' + Ca;' + 11a;' + 4a; - 4 and x' + 2a;' - 5a;' - 12a; - 4. 62. If now, having first attended to the directions of (61), we findj at any step of our process, that the first term of the dividend is not exactly divisible by the first of the divisor, then, in order to avoid fractions in the quotient, we may multiply the whole dividend by GREATEST COMMON MEASURE. 53 Gucli a simple factor, as will make its first term so divisible. Ex. Find the g. c. m. oi(jx'^y + ^xy'^-2y^ and ^x'^ + Ax'^y-^iXy^, Stripping them of their simple factorSj 2y and 4a', (and noting that these contain the common factor, 2), we have Zx^ + 2xy — y^ and 2x^ + xy - y^, and proceed with these quantities as follows : " + xy' Sx' 2 + 2xy' -y' '(3 2x' -y')6x' ex'' + 4:xy-2y' + Zxy-Zy' y\xy + 2/' x + y) 2x'' + 2x'' + xy- 2xy -y'i2x- -y - xy- -y" -^ xy- ■y' The G. c. M. then will be 2{x + y\ it being plain that the G. c. M. of 2 (Zx^ + 2xy-y^) and 2x'^ + xy-y'^ will be the same as that of Zx' + 2xy - y'^ and 2x'^ + xy - 2/^ because the 2 introduced into the first is no factor of the second quantity. Ex. 34. Find the g. c. m. 1. Of Gx'^ + 13^ + 6 and 8ic^ + Caj-9. 2. Ofl5x^-.aj-Gand9iC^-3aj-2. 3. Of 6aj= -aj- 2 and 21aj3- 26 aj^ + 8a7. 4. Of 6^3 - (jx'' + 2^-2 and 12.r^ - 15aj + 3. 5. Of 3aj^ - 22aj- 15 and 5a;* + a;«-54a;2 + 18aj. G. Of 3aj^ - Zx'^y + ir^/^ - 2/^ and ix^—x'^y - Zxy'^, 7. Of ic^-So; + 3 and x' + Sx^ + x + 3. 8. Of 5x' + 2x'' - 15a; - G and - 7x^ + 4a;' + 21a; - 12. 9. Of 20.T* + x^-l and 25a;* + 5a;^-a;-l. 10. Of (jx^-xhj-Zxhf + Zxy^-%/ and ^x^-?>xhj-2x''yUZxy^-y\ 11. Ofl2a;M2a;'y- + 12a;V-3a;7/*& 12a;' + 8a;V-18.i'y-6a;"^»+4a;2/*. 12. Of x' - 2^ h x"" - 8a; + 8 and 4a;3 - 12a; '^ + 9a;- 1. 63. In order to prove the Eule above given, it will be necessary to shew first the truth of the following statement. 54 GREATEST COMlklON MEASURE. If a quantity c he a common measure of a and b, it will also oneasure the sum or difference of any mul- tiples of a and b, as ma ± nb. For let c be contained jp times in a^ and q times in h ; then a = ^(?, 5 = qc^ and m(2 ± ^^^ = '?^^<^ ± n^, and therefore c measures ma ± 7i5. Thus, since G will divide 12 and 18 without remainder, it will also divide any number such as 7 x 12 + 5 x 18, 11 x 12-3 x 18, 12 (or 1 X 12) + 7 X 18, 5x12-18, &c.,^. e. any number found by adding or subtracting any multiples of 12 and 18. 64. To prove the liule for finding the greatest Com- mon Measure of tioo quantities. First, let the two given quantities, denoted by a and J, have neither of them any simple factor. Let a be that which is not of lower dimensions than the other; and suppose a divided by 5, with qnotient j9 and remainder , h and ) ; and of 6 (x^'y + xy^\ 9 {x^ - xy'^), 4 (y^ + icy^). YO. Every common rriulti/ple of a an^^ b ^6* a multir fie of their l. c. m. For let J/be any common multij)le of ^ and J, and m their l. c. m. ; and let M contain m (if possible) T times with remainder 5, which will of course be less than the divisor m ; hence we should have M=- rm + 5, and, therefore, s = M- rm : but since a and 5 measure both J/" and ?;?, they would also (63) measure If-rm^ or ,5; i. e. 5, which is less than m, would be a common multiple of a and J, contrary to our supposition that m was their least common multiple. Hence Jf will contain m with no remainder, and will therefore be a niidtvple of m. CHAPTER VI. FEACTIONS. Algebeaical Fractions are for the most part pre- cisely similar both in their nature and treatment to common Arithmetical Fractions. We shall have, therefore, to repeat much of what has been said in Arithmetic ; but the Kules which were there shewn to be true only in the j^articular examples given, will here, by the use of letters, which stand for any quan- tities, be proved to be true in all cases, 71. A Fraction is a quantity which represents a part or parts of an unit or whole. It consists of two members, the numerator and de- norninator^ the former placed over the latter with a line between them. Now we have already agreed (8) that such an expression shall denote that the upper quan- tity is divided by the lower; and, in accordance with this, it will be seen presently that a fraction does also express the quotientof the num'' divided by the den^ The den"" shews into how many equal parts the unit is divided, and the num' the number taken of such parts. Thus y- means that the unit is divided into 1 equal parts, a of which are taken. Every integral quantity may be considered as a fraction whose den "■ is 1 ; thus a is - . Y2. To multi;plij a fraction by an integer, we may either multiply the num'or divide the den' by it; and, conversely, to divide a fraction by any integer, we may either divide the luiiu'' or multiply the den' by it. 60 FRACTIONS. Thus ■=■ X X = -J-: for in each of the fractions ^ , -7- > 00 00 the unit is divided into h equal parts, and x times as many of them are taken in the latter as in the for- mer ; hence the latter fraction is x times tlie former, ^. ^. ax a 1 v • -1 . ax ^ a that IS -=-= - X X : and,by smiilar reasoning, - — -x=j , A^cain - ^ rx^' = — - ; for in each of the fractions -7, — - ° & Ix' Vox the same number of parts is taken, but each of the parts in the latter is ~th of each in the former, since the unit X in the latter case is divided into x times as many parts as in the former ; hence the latter fraction is -tli of the X former, that is, ^~— ~j-^x\ and, similarly, ^— x a? = ^. ' 'hx b ' '^^ Ix b 73. If any quantity be both multiplied and divided by the same quantity, its value will, of course, remain unaltered. Hence if the num'' and den^ of a fraction be both multiplied or divided by the same quantity, its value will remain unaltered. a ax a^ „ , a^h a ac „ Ihus -=--=—-- &c. and ---- =-=-—- ~ &c, ox ab a^oG c c* 74. Since c^ = - (71), and, therefore, a divided by b ^---^b= J (72), it follows, as stated in (71), that a fraction represents the quotient of the num^ by the den'. In factj we may get -- tli of a units, (or a -*- h.) by taking - th part of each of the a units, and this is the same as a such parts of one unit, which (71) is expressed by y. Hence it is that, in Arithmetic. I of £3 is tlic suiiic as ^' of £1. ^.c. FRACTIONS. 61 75. To reduce an integer to a fraction with a given denominator, multiply it by the given denominator, and the product will be the numerator of the required fraction. Thus a, expressed as a fraction with den' oj, is — : or, with den' - c, is —^ . 6-c The truth of this is evident from (73). 76. The signs of all the terms in both thenum"^ and den'^of a fraction may be changed without altering its value : thus — ;r- is identical with — - — •. dax-x' X -"dax This follows also from (73), as the process is equiva- lent to that of multiplying both num"" and den"" by-1. 77. To reduce a fraction to its lowest terms, divide the numerator and denominator by their g. cm. ^x^y"^ o?x^if axy Ex.1. a^xy + axy"^ axy {a + y) a + y ^ ^ (y a^ + x^ _(a + x) (a^ -ax + x-) _ «' - ax + aj* a^ -X' (a + x) (a-x) ~ a-x ' ^ ^ x^ + 4x + o (aj 4- 3) (a? + 1) x + 1 Ex. 3 V / V / _ Ex.4. x'^ + 5x + 6 (x + Z) (x + 2) X + 2' x^ + x^ + Zx-6 (x - 1) (x^ + 2ic + 5) x"^ + 2x + 5 x^-4x+S (.<;-l)0c-3) aj-3 * Of course, the student should consider for a moment whether he cannot obtain the g. c. m. as in (58) hy mere inspection. Ex. 36. Reduce to their lowest terms axy+xy^ cx+x' llm'^ + 22mx I4x^-7xy 5«'6-15a^i* axy ' a^c + a\v' S'6 (rrr-ix^) ' lOax-^iay' 20ah^+l0a-i'^' 6x'^y-12xy^ 2m^n-^2mu^ dHic^alPc^abc^ Vlx^y--2\xy''' al)c-\- %c-5c^ ae + hy + ay + he acx'^ + (ad-hc) x-hd 2ahdf-\-\mif~i{)cdj of^2bx + 2'>x-^hj a^x'-b- 62 FRACTIONS. . aj'-l a;*--a* «•-&• x^-h'^x a^ - ah + ax- Ix ax+x x^—a^x^ a^-V x'^ + 2bx+b^ a^ + ab + ax + Ix x'-ix + Z x"" + 2x-S g'- al>-2b^ '^^2x - a' x^ + bx + fj 'a"- Zab + 21)'' Qa'^-lZax+Q^x'' (ja'^ + lax-Zx' ic^+iC-12 ^•*-« T « -I o o ^1 -I - •} /» a* J.-J» 5a;* - ISoj^y + ll^y^ - Gy'*' "" a^oj + 2a V + 2aa;* + x^' a;* + 3a;'^ - 4 a;^ - 3a; + 2 a;* + a^a;* + a* - 3a^a;* - 2ax'^ - 1 a:* + ax^ - a^x - a*' ' 4a^x^ - 2a'^x^ - '6ax^ + 1' 78. If the niim'' be of lower dimensions than the den'', the fraction may be considered in the light of a proj)er fraction in Arithmetic ; if greater, in that of an iinproper fraction, which may be reduced to a mixed fraction^ by dividing the num*" by the den^, as far as the division is j^ossible, and annexing to the quotient the remainder and divisor in the form of a fraction. Conversely, a onixed fraction may be reduced to an improper fraction, by a process similar to that em- ployed in Arithmetic. _^ , 3a;« + 2a; + l ^ ,. 41 Ex. 1. = 3a; - 10 + j. a;+ 4 a; + 4 Ex. 2. a;^ + a; + 1 + x-\ x-V Ex. 37. 1. Reduce to mixed fractions 3a;V6a;4-5 a'^-a x^x'' 2a;'' + 5 10^17«^+10a;' 16(3a ;' + l) a; + 4' a + a;' a;-3' 5a -a; 4a; -1* 2. Reduce to improper fractions , ^ 3a;(3-a;) , . . , Ca;» a^-ay^y'' x^-^x ^-TT^, a^-2aa;+4a;^ tt-, a;-a + 2/+ • ^-^. a; - 2 ' a + 2r a; + a FRACTIONS. 63 Shew that 3. 1 + ^r—. = i^— , , and 2ah 2ab ' a^ + h^ - c^ _ (a - h + c) (h - a + c) 2ab 2ab a .w .h^-c'X^ (a + h + c) (a + h — c) (a + c-h) (b+c-a\ [ ^ 79. To reduce fractions to a common den"", multiply the num"^ of each fraction by all the den" except its own, for the new num' corresponding to that fraction, and all the den'^ together for the common den^ The truth ofthis rule is evident; since, the numerator and denominator of each fraction being both multiplied by the same quantities, viz. the denominators of the other fractions, its value will not be altered, though all the fractions will now appear wuth the same de- nominator. Ex. Reduce - , -, -,, to a common denominator. Z>' c d For the nunV a x c x d = acd I X b X d = b^d and the required fractions are c X b X c =bc'^ acd ¥d be" bed) bed) bed' For the den' b x c ^ d = bed ; 80. If, however, the original den''^ of the fractions have, any of them, common factors, this process will not give them with their least common den"", which, as in Arithmetic, will be found by forming the l. c. m. of the given den" : and the num' corresponding to any one of the given fractions will be obtained, by multiply- ing its numerator by that factor, which is obtained by dividing the l. c. m. by its denominator. Ex. Eeduce ^7—, -— — , to a common denominator. Ibx Gabxy oacx Here the l. c. m. of the denominators being Gabcxy, the fractions . , ^a^cy c^ 2b'y Gabcxy^ Gabcxy' Gabcxy' 64 FRACTIONS. Ex. 38. Reduce to common denominators, X y z iB^ 1/ s^ , 2^^f/ 33;* 4y^ 5xy* ^' a' h' c' 2ab' Tac' 4J^' "3a^' 4a^' 6a6^' W ' x^ y^ a + X a-x Ax^ xy ^'a' + b"-' a'-b'' a-x' a + x' 3(a + &' 6(a»-&^)' 3. 1 ^ 1 81. To add or subtract fractions, reduce them to common den", and add or subtract the num'"' for a , new num'', retaining a common den'. ^ ^ x y z hex + acy + ahz Ex. 1. - + ^^ + - = ^ . a c aoc Ex. 2. Add :; 1 + Ans, 1 + X + x"^ 1 -X + x" (l+x)(l-x + x"") + (1 - x) (1 + x + x') (1 + a; + x'') (1 - ic + oj') 1 + if' Ex. 3. From ^ -„ take Ans, 1 + X + x"^ 1 - X + x"^' (1 + ic) (1 - ic + a;^) - (1 - ic) (1 + cc + x^") 2x* (L + X + x'^) (I - X + x'^) 1+x^+x* Ex. 4. Find the value of 2 + ~ — ,-„ r. a^ - b^ a + b 2 (g^ - b'') + (g' + b^) -{a- I) {a - b) ^a" -\-2a b -21^ Arts. ^—^-^i ________ Ex. 39. Find the vahie of 1 ^ _ (^ ~ ^) iL (^ "^ ^^^ ^^ ~ ^^ ^''^ ~l-e \ba - Ac ' 2b 2 (a + bf 2b "^ Z~(cr^y 2~" ^ ^ 12 p ci* ^ a b a b a-b ab a -b ■ a + b a-b' a-b a + b a+ b a^ - b"^' a (ad - be) x a^ + b* a-b 2.c^ - 2xy + y"" ' x • - - —7 tV, -,-;. ± 4. c c(e + dx)^ a^ -b^ a + V x"^ - xy x-y' 1 1 a a~be-ah-c 2(a-jc) 2Qi + x) a" + a-^' ab ae be ' FRACTIONl*. 65 6. rr = + 2(a;-l) 2(a;+l) cc^ ' ' 2a+b 2a-b 4a^-h^' 7 ^ (a''-'b'' )x a(a^-&V J^ 1__ ^-1 5 5'^ " ■*■ b''{b + ax) ' ' x" (a;^ + iy "*" ic'+l* 9 ^yl _ __^._ + _5_ 10. 1 - ^^ + ^' . -?^y a? -2/ ic-y ii'+T/ ic+2/ or-y^ x^ + y^ 11.4, ^Zfi^. 12. 2-?;:^!.5:i<. 13. 4- -f^^. a^ a(a+x) x^ + y x'-y^ a^ a(a-x) 14. ^'^y ^ x^y-x^ 1- ^ ^^ ^* 82. To multiply one fraction by another, multiply the numerators together for a new numerator, and the denominators for a new denominator. Suppose that we have to multiply - by - : let - =a?, j=y; .', a=: hx, c = chj, and ac = hdx?/; CiC hence, (dividing each of these equals by hd)^ :~—'xy\ bd , , a c ^ac axe product of num" but w = Y X - , and ^, = -. — - ~ , ^ , o d bd bxd product 01 den" whence the truth of the rule is manifest. Similarly we may proceed for any number of fractions. a+b a-b 3 _ Z{a+b) {a- b) 3 {0" - &^) Tld"" c-d'' 2~ 2{c + d) {c -d) ^ 2 (c^ - d"") ' 83. To divide one fraction by another, invert the divisor and proceed as in Multiplication. Suppose that we have to divide -7 hy , : let ~z=x^ ~ = y] .*. a = bx, c= dy ; , 7777 77 -3 ctd hdx X hence ad = bdx. be = bdi/, and :=— = 7-- = - ; ' •^' be bdy y' 66 FRACllONS. , ^ a? a c . ad a d but -=zx -^7/ =: y-f--^, and -— = - X -, y -^ h d^ ho h c' whence the truth of the rule is manifest. 2a + h ^ c-cl _ {2a+U) (2a - Zh) _ 4a" - 96" c + d ' 2a-W~ {c-^d)(c-d) " c'-d^ • In mult" and div" of fractions, it is always advisable, before multiplying out the factors of the new num ' and den', to see if some of them do not exist in loth the num' and den', in which case they may be struck out, and the result will be more simple. ^..acx_ac_ac ' bx d id Id' p 5ax xy + y"^ _ 5a (x + y) _ 5ax + 5ay '6cy x"^ - xy 3c (x — y) Zc (x — y) ' ^ o 4aa; a"^ - x^ he + hx Ax (a + x) Aax + 4x^ Zby c^ - X' a^ - ax Zy {c - x) Zy (c-x)' p . X' + xy X* — y* ic" + xy (x — yy _ x JilX. 4. ■ -i- - r^ = ■ X - -— = — - - , x — y {X- yy x — y x^-y^ x^ + 2/ The student should leave the denominators of fractions with their factors unmxdtijolied^ as in Ex. 2 and 3 ; unless they happen to combine very simply, as {a-^x) (a+xy into (a+xy, or (a + x) (a-x) into a^ - a;^. The convenience of this will be found in practice. Ex. 40. Find the value of _ 2x Zah Zac ax a'-x^ a /, hx\ /_ a \ 1. _ X . X -—-, X ■ —- -- X [0+ — X 1 . a c 2b (a-xy ah ' bx \ a] \ a + xj 9, i -^l\(^ ^\ ^^'~^' ^ ja-^xy 2a {x'^-yy x^ ' \ a j \x ay a^ + x^ (a-x)'^ ex "" i^-y) (x+yy' a" + 2a& ^ «&_-2&^ ^'^y ^ {x-yy I 4_ ^\ ^ aV+q Z>a;' ^ ax * a' + W "" a"'--4>' ~x^y "" l^y^' \^ ~ ic= / ax+l~ "^ a^^' 4. "-^?-:^^-|, tiyl^-'--y^y\ (u^\^H,A] . fi j: a^-V a + h x'-y^ x-y \ x) \ x] \ x^ ^ I. V\ (a h\ a^-Za'^b^ZaW-b' 2ab-2h^ a^+ab V '^y"*"U^^J' ^"^^ '^~~J~"'^b' x*-b* x'' + bx ic'-JV ic*-2Ja;'+6V x^-2hx + b'^ ' x-h x^ + h^ x'^-bx + h'^ FRACTIONS. 67 84. A complex fraction, ^. e, one in which the niim^, or den^j or both, are fractions, may be simplified as follows. ^ X 2-x 2 _ ^J^ _ 2-x l^ _ 2-x Ex. 1. 4x ~ 4x ~ 2 "" 4x~ ' 8x '' T Hence observe that, when a complex fraction is put into the form of a t; ;— , the simple expression for it will be found by taking the product of the upper and lower quantities, or extremes, for the num', and that of the two middle ones, or means, for the den' ; and that any factor may be struck out from either of the extremes,' if it be struck out also from one or other of the means. Ex.2. 2x 2x I 1 ~ 3a; - I ~ ex ^x - r Ex.3. 20 -aj 5-lx 4 60 - 3a; . ic + 11 3a! + 4 4 (3a; + 4) 3 x+2 x+2 Ex. 4. (a!+2) (2a;-a;= + 3) 6 + 7a;-a;' (l-x) (2a;-a!V3)+a; 3-3a;''*+a;^* 3-a; 3a; + 2A 2| - j a; x-^ x' X + 2i' 31 ' fx - Ip 2 J - Ix 2 ^- i(^^-^) 6a; - f (3 4- bx) ' 2^ -1 (x~2 ) l|-|(a;+2) 3 ' 2^ ' |(^4-l)-4l' tVC-c + 1) o} -vV^ 25^ « + a; r?- - a; 2o thus r = i, - = -— - „ V = ^J &c. a ± c ± d c c ± d c + d c -d So that, If any two fractions are equals we may com- bine hy Addition or Suitr action^ in any way^ the nur)i^ and den^ of the one ^ provided that we do the same with the other, 86. The above results may be yet further generalized. _ .^a c ,. m a m c ma mc For, \1y=~t, then — x - = — x -, or — r- = — .\ ^ d n n d 710 nd and, therefore, by what has been above she«rn, ma±n'b mc±nd . ma±n'b mc±nd , ma±nh me±nd . whence • = , and = ; ma nc ^ a c pa j>^ ma±nh mc±nd ma±nc mh±nd ma±nc mh±nd so also Y— = —3 — , = — , = -z — , pb pd pa P^ P^ P^ ma + nh me + nd ma + nb ma - nb „ ma - nb 7nc + nd) mc + nd mc — nd) . . . ma ± nh mc ± nd .pa±ob pc ± qd Again, since = , and^^ ^ = --, a c ' a c ma ± nb mc± nd . 2ja ± qb pc ± qd) Hence we see that the statement of (85) is true of any mnltlpUa whatever of the numerators and denominators of the fractions. 87. Further, rf ^ = ^-^ then^,- = ^„ -^, = ^, &c. ^- = ^-. Hence the previous results hold with a", b'\c''^d'*^ instead of «, K c, d. FKACTIONS. 69 For let T = ^ = 3 = ^ > t^i^Ji a = dx, c = (Zaj, 1, it becomes greater and greater. Hence, when a; < 1, the fraction = expresses approximately, and with more and more of accuracy, according as we take more terms, the value of the series \ + x + x^ + &c. ; whereas, when x > 1, it does not at all express the value of the series, unless we take account also of the remainder = ■ . 1 -X Thus, if a; = I, we have —-j or2 = l + | + |^ + J+ &c., the sum of which series approaches more and more nearly to 2 as its Limitj without ever actually reaching it. But if a; = 2, we have =— - OP -1 = 1 + 2 + 4 + 8 + &c., the sum of which series departs 1 — ii more and more from -1 : the error, however, will be corrected, if we introduce the remainder at any step ; thus 1+2 + 4+ ~2 = 7-8 = -l. 70 FKACTIONS. In all such cases we may consider the sign = as expressing, not the actual equality of the two quantities, but merely that the fraction can be made to assume the form of the serieSj and there- fore may be used as an abridgment for it. 90. If a; = 1 in the above, 'then = — = = 1 + 1 + &c., that is, X = an infinite number of units, which is, of course, an infinitely great quantity, and is denoted by oo (read infinity). The meaning of this result may be thus explained. If ic = 1 'cery nearly^ so that \-x is 'cery small, then j — ^ will be, of course, very great, and may be made as great as we please by still farther diminishing 1-x, that is, by taking x still more nearly = 1. T/^hen, therefore, we write ^ = oo , we are not to suppose the denominator actually zero (in which case the division by which we obtained the scries would be absurd), but only a very small quantity ; and by using the sign . . n . rV . l"l^^ 2f-f^V(a;-l) Id. 1^ {oa; - f (1 + a;;[ + ~p- = ^: -, X a + X 2a -h ^^ x + 4 _, 3a; + 8 14. = ^ — . 15. ^ ^ + If = o — ^ • « + a; X 2x 6x + b "^ 2a; + 3 16. ^V (11a; - 13) + i (19a; + 3) - 1 (^^ " 25^ = 281 -^ (17^ + 4). 10a; + 1 7 12.r+2 _ 5a;-4 a; +11 lO-a; _ 4-|a; 1 " 18 i3a;-16~ 9 * 3 " 3|~~"Tr"TI* 19. |(a;-lt|-)-TV(2-6a;)=a;-^V|5a;- 1(10-3^)}. .. 6a; + 13 3a; + 5 2a; oi ^-'' 2a;- 15 1 15 5a!-25 5' ' a; + 7 2x-6 2(a! + 7)* „„ 132x H- 1 8a! + 5 „, 7a! + l 35 a'+4 24. ^tl 1^=_J_.25 _1L_.-A___J_ 6« + 17 3x-10 l-2« 12a; + 11 C« + 5 42! + 7 26. ias - H2^- 3) -H3^-l) _ 3 x- - jx ^ 2 i(a!-l) 2" 3x-2 • 27. A (7a;+Ci) + Jj |ll«-i (a;-li)} = J (3a;+l) +^ j43a>-i (S-Sa')}. '^'*-13-2x'^^'^ 2r^-*T5 3 • 29. 4a!-i(*-2)-[2x-aa!-J,jl6-i(« + 4)()] = |(x+2). g, 6-Sa! 7 -2a!'. _ 1 + 3 a; _ 2a; -2j ^ _1 15 14(x-l)~ 21 C * 105" SIMPLE EQUATIONS. 73 93. Tlie toUowing are additional Problems in Sim- ple Equations, presenting somewhat more of difficulty than those given under (41). Ex. 1. A fish was caught whose tail weighed 9 lbs; his head weighed as much as his tail and half his body, and his body weighed as much as his head and tail. What did the fish weigh? It is sometimes convenient to take x to represent, not the quan- tity actually demanded in the question, but some other unknown quantity on which this one depends. It is only experience, how- ever, and practice which can suggest these cases ; but this ex- ample is one of them. Let X = weight of body ; .'. 9 + |.r = weight of tail + \ body = weight of head; but the body weighs as much as head and tail ; .*. a; = (9 + ^ x) + 9, whence a; = 36, weight of bod}^ ; .'. 9 + ^ic == 27, weight of head ; and the whole fish weighed 27 + 3G + 9 = 72 lbs. Ex. 2. A gamester at one sitting lost J of his money, and then won IO5 ; at a second he lost^ of the remainder, and then won Zs ; and now he has 3 guineas left. How much money had he at first? Let X = number of shillings he had at first ; having lost 4- of it, he had | of it, or | a; remaining; he then won 10s, and had, therefore, | a; + 10 in hand ; i)sing \ of this, he had f of it remaining, that is, f (| a; + 10) ; and he then wins 3s, and so has f (| ^ + 10) + 3 shillings, which, by the question, is equal to 3 guineas, or 63s; hence f (| a; + 10) + 3 = 63, w^hence x = 100s = £5. Ex. 3. Find a number such that if % of it be subtracted from 20, and y\ of the remainder from ^ of the original number, 12 times the second remainder shall be half the original number. I ' Let X = the number ; \\ 20-| X =lst remainder, and 5- ^ - fi (20 - i a-) = 2nd remainder ; .*. 12 {\x- y\ (20 -^x)} =1 re, by the question ; whence x = 24. Ex. 4. A certain number consists of two digits whose difference is 3 ; and, if the digits be inverted, the number so formed will be f of the former : find the original number. 4 k 74 SIMPLE EQUATIONS. Let x = lesser digit, and .'. oj + 3 = the greater: then, since the value of a n° of two digits = ten-times the first digit + the second digit (thus 67 = 10 X G + 7), the n" in question = 10 (a; + 3) + cc ; similarly, the n" formed by the same digits inverted = 10j;+ (x+Z) ', hence, by question. 10a; + (a; + 3) = -f { 10 (x + 3) +a; J-, whence aj = 3, a; + 3 = 6, and the n" required is 63. Ex. 5. A can do a piece of work in 10 days ; but after he has been upon it 4 days, B is sent to help him, and they finish it together in 2 days. In what time would B have done the whole ? Let a; = n" of days B would have taken, and TT denote the work : W W ,\ — r, — , are the portions of the work, which A, B would do in 10 a;' 4Tr one day ; hence in 4 days, A does -y^-, and in 2 days, A and B ,^ ^ 2W 2W 4Tr 2W 2W ^ ^ tofiretherdo -^t,- + : .\ -ttt + -^-n + = f^'y whence x = o. ^ 10 a; 10 10 a; It is plain that in the above, we might have omitted W al- together, or taken U7iiti/ to represent the work, as follows: A, i? do y^, - of the work respectively in one day, and therefore, 4 2 2 reasoning just as before, tk + th + ~ = ^^c whole work = 1. [In all such questions the student should notice that, if a person does - ths of any work in 1 day, he will do - th of it in — th of a n n m day, and therefore the wJiole work in — days. Thus if he does ^ in one day, he will do | in ^ of a day, and .•. ^ or the wJiole in 5 = 2^ days]. Ex. 6. A cistern can be filled in half-an-hour by a pipe A^ and emptied in 20' by another pipe B : after A has been opened 20', B is also opened for 12', when A is closed, and B remains open for 5' more, and now there are 13 gallons in the cistern : how much would it contain when full 1 Let X = number of gallons that would fill the cistern : then, in 1', A brings in 75^5^3; gals., and B carries out -^x gals. ; but A h opened altogether for 32', and B for 17' ; .*. f|aj - \lx = 1^ whence a; = 60 gals. SIMPLE EQUATIONS. 75 Ex. 7. Find the time between two and three o'clock, at which the hour and minute-hand of a watch are exactly opposite each other. Let X = number of minutes advanced by the Jiour-hand since two o'clock : then 12ic = number of minutes advanced by the minute- hand, since it ti-avels GO' while the other travels 5' ; but. by ques- tion, the minute-hand will have advanced (10+a;)+30=a;+40 min. ; .-. 12ic = ic + 40, whence x = Z^j^ and the time is 2h 43yy'. Ex. 8. There are two bars of metal, the first containing 14 oz. of silver and 6 of tin, the second containing 8 of silver and 12 of tin ; how much must be taken from each to form a^bar of 20 oz. containing equal weights of silver and tin ? Let ic = n" of oz. to be taken from first bar, 20-0? from second ; now ^J of the first bar, and therefore of evevi/ oz. of it, is silver ; and, similarly. 2^ of every oz. of the second bar is silver ; and there are to be altogether 10 oz. of silver in the compound ; ... |4^ + _Cg. (20 -ir) = 10, whence x = 6|, and 20 - a; = 13 J-. Ex. 44. 1. The stones which pave a square court would just cover a rectangular area, whose length is six yards longer, and breadth four yards shorter, than the side of the square : find the area of the court. 2. Out of a cask of wine, of which a fifth part had leaked away, 10 gallons were drawn, and then it was two-thirds full : how much did it hold ? 3. A person bought a chaise, horse, and harness for £G0 ; the horse cost twice as much as the harness, and the chaise half as much again as the horse and harness : what did he give for each ? 4. The value of 50 coins, consisting of half-guineas and half-, crowns, is £16 55 : how many are there of each? 5. A^ after spending £10 less than a third of his ycarlv income, found that he had £45 more than half of it remaining : what was his income 1 6. A boy, selling oranges, sells half his stock and one more to -4, half of what remains and two more to i?, and three that still remain to C: how many had he at first ? k 76 SIMPLE EQUATIONS. 7. In a garrison of 2744 men, there arc two cavalry soldiers to twenty-five infantry, and half as many artillery as cavalry : find the numbers of each. 8. A person dies worth £13^000: some of this he leaves to a Charity, and twelve times as much to his eldest son, whose share is half as much again as that of each of his two brothers, and two-thirds as^uch again as that of each of his five sisters: find the amount of the bequest to the Charity. 9. A farm of 270 acres is divided among A, B, C: A has 7 acres to 11 of i>, and G has half as much again as A and B together : find the shares. 10. Divide 150 into two parts, such that if one be divided by 23 and the other by 27, the sum of the two quotients may be G. 11. ^ had I85 in his purse, and B, when he had paid A two- thirds of his money, found that he had now remaining two-fifths of the sum which A now had : what had ^ at first ? 12. The first digit of a certain number exceeds the second by 4, and when the number is divided by the sum of the digits, tho quotient is 7 : find it. 13. The length of a floor exceeds the breadth b}^ 4 ft. ; if each had been increased by a foot, the area of the room would have been increased by 27 sq. ft. : find its original dimensions. 14. A met two beggars, B and (?,, and having a certain sum in his pocket, gave 2^^ of it to^, and f of the remainder to C: A had now 20^ left ; what had he at first ? 15. In a mixture of copper, lead, and tin, the copper was 5 lb less than half the whole quantity, and the lead and tin each 5 lb more than a third of the remainder : find the respective quantities. 16. A sum of money was left for the poor widows of a parish, and it was found that, if each received 4« 6d, there would be Is over ; whereas, if each received 55, there would be 10s short ; how many widows were there ? and what was the sum left ? 17. A horse was sold at a loss for 40 guineas; but, if it had been sold for 50 guineas, the gain would have been three-fourths of the former loss : find its real value. 18. A can do a piece of work in 10 days, which B can do in 8 : after A has been at w^ork upon it 3 days, B comes to help him ; in what time will they finish it ? 10. There is a number of two digits, whose difference is 2, and, SIMPLE EQUATIONS. 77 if it be diminished by half as much again as the sum of the digits, the digits will be inverted : find it. 20. A and B have the same income : A lays by a fifth of his : but i?j by spending annually £80 more than A, at the end of 4 years finds himself £220 in debt. What was their income V 21. A number of troops being formed into a solid square, it was found there were GO over; but, when formed into a column with 6 men more in front than before and 3 less in depth, there was Just one man wanting to complete it. Find the number. 22. A person has travelled altogether 3036 miles, of which he has gone seven miles by water to four on foot, and five by water to two on horseback : how many did he travel each way ? 23. A and B can reap a field together in 7 days, which A alone could reap in 10 days : in what time could JB alone reap it ? 24. A cistern can be filled in 15' by two pipes, A and J3, nin- ning together : after A has been running by itself for 5', B is also turned on, and the cistern is filled in 13' more : in what time would it be filled by each pipe separately ? 25. What is the first hour after 6 o'clock, at which the two hands of a watch are (i) directly opposite, and (ii) at right angles, to each other ? 20. A person played twenty games at chess for a wager of 3^ to 2^, and upon the whole he gained 5s : how many games did he win ? 27. I wish to enclose a piece of ground with palisades; and find that, if I set them a foot asunder, I shall have too few by 150, whereas, if I set them a yard asunder, I shall have too many by 70 : what is the circuit of the piece of ground ? 28. A and B began to pay their debts : ^'s money was at first f of i?'s; but after A had paid £1 less than f of his money, and B had paid £1 more than J of his, it was found that B had only half as much as A had left. What sum had each at first ? 20. A can build a wall in 8 days, which A and B can do together in 5 days : how long would B take to do it alone ? and how long after B has begun should A begin, so that, finishing it together, they may each have built half the wall ? 30. A person wishing to sell a watch by lottery, charges Cs each foi: the tickets, by w^hich he gains £4 ; whereas, if he had made a third as many tickets again and charged 5^ each, he would have gained as many shillings as he had sold tickets : what was the value of the watch ? 78 SIMPLE EQUATIONS. 31. A mass of copper and tin weighs 80 lbs, and for every 7 lbs of copper there arc 3 lbs of tin : how much copper must be added to the mass, that for every 11 lbs of copper there may be 4 lbs of tin ? 32. A does f of a piece of work in 10 days, when B comes to lielp him, and they take three days more to finish it : in what time would they have done the whole, each separately, or both together ? 33. A cistern can be filled by two pipes, A and B, in 24' and 30' respectively, and emptied by a third C in 20' : in what time would it be filled, if all three were running together ? 34. A and B were employed together for 50 days, each at a day, during which time A^ by spending 6d a day less than B, had saved three times as much as B, and 2| days' pay besides ; what did each spend per day ? 35. Divide £149 among A^ B, C, i>, so that A may have half as much again as B, and a third as much again as B and G to- gether ; and I) a fourth as much again as A and C together. 3G. There are two silver cups and one cover for both. The first weighs 12 oz, andj with the cover, weighs twice as much as the other cup without it ; but the second with the cover weighs a third as much again as the first without it. Find the weight of the cover. 37. A man could reap a field by himself in 20 hrs, but, with his son's help for 6 hrs, he could do it in IG hrs: how long would the son be in reaping the field by himself? 38. A horsckeeper, not having room in his stables for 8 of his horses, built so as to increase his accommodation by one half, and now has room for 8 more than his whole number : how many horses had he ? 39. A grocer bought tea at 6s Gd per lb, and a third as many lbs again of coflee at 2s Gd per lb ; he sold the tea at 8.9, and the coffee at 2s Zd, and so gained five guineas by the bargain ; how many lbs of each did he buy ? 40. Find a number of three digits, each greater by unity than that which follows it, so that its excess above one-fourth of the number formed by inverting the digits shall be 36 times the sum of the digits. 41. A man and his wife could drink a cask of beer in 20 days, the man drinking half as much again as his wife; but, ^| of a gallon having leaked away, they found that it only lasted them SIMPLE EQUATIONS. 79 together for 18 days, and the wife herself for two days longer: how much did it contain when full ? 42. A and B have each a sum of money given them, which will support their families for 10 and 12 days respectively ; but J.'s money would support JB^s family for 15 days, and i>'s money would support ^'s family for 7 days, -with 2s Gd over: what were the sums ? 43. A person being asked how many ducks and geese he had in his yard said, If I had 8 more of each, I should have 8 ducks for 7 geese, and if I had 8 less of each, I should have 7 ducks for 6 geese : how many had he of each ? 44. A man, woman, and child could reap a field in 30 hrs, the man doing half as much again as the woman, and the woman two- thirds as much again as the child : how many hours would they each take to do it separately ? 45. If 19 lbs of gold weigh 18 lbs in water, and 10 lbs of silver weigh 9 lbs in water, find the quantity of gold and silver in a mass of gold and silver, weighing 106 lbs in air and 99 lbs in water. 4G. From each of a number of foreign gold coins a person filed a fifth part, and had passed two-thirds of them, gaining thereby 355, w^hen the rest were seized as light coin, except one with which the man decamped, having lost upon the whole IfStf as much, 'fiS*^ as he had gained before : how many coins were there '^first ? 47. A and B start to run a race : at the end of 5', when A has run 900 yards and has outstripped B by 75 yards, he falls ; but, though he loses ground by the accident, and for the rest of the course makes 20 yards a minute less than before, he comes in only half-a-minute behind B, Ilovr long did the race last ? 48. A and B can reap a field together in 12 hrs, A and C in IG hrs, and A by himself in 20 hrs : in what time could (i) B and together, (ii) A, By and 0, together, reap it ? 49. Fifteen guineas should w^eigh 4 oz : but a parcel of light gofd, having been weighed and counted, was found to contain 9 more guineas than was supposed from the weight, and it appeared that 21 of these coins weighed the same as 20 true guineas : how many were there altogether '? 50. A, B, travel from the same place at the rate of 4, 5, and 6 miles an hour respectively, and B starts two hours after A : how long after B must start, in order that they may both overtake J. at the same moment ? 80 SIMPLE EQUATIONS. Simultaneous Equations of one Dimension. 94. If one equation contain t%m unknown qnanti* ties, there are an infinite number of pairs of values of these by which it may be satisfied. Thus in ic = 10 - 27/, if we give amj xalue to t/, we shall get a corresponding value for x, by which pair of values the equation will of course be satisfied ; if, for example, we take 7/ = 1, we shall getaj = 10-2 = 8; ify = 2, aj=G; if2/ = 3, aj = 4; &c. One equation then between two unknown quantities admits of an infinite number of solutions ; but if we have as many diflferent equations, as there are quan- tities, the number of solutions wnll be limited. Thus, while each of the equations x = 10-2y^Ax ■¥ 4 = Sy, separately considered, is satisfied by an infinite number of pairs of values of x and t/, there will only be found one pair common to both, viz. a; = 2, y = 4, which are therefore the roots of the pair of equations," a; = 10-2?/, and Ax + 4:= Zy. Equations of this kind, which are to be satisfied by the same pair or pairs of values of x and y, are called shmdtaneous equation s. If there be three unknowns, there must be three equa- tions, and so on: and moreover, these equations must all be different from one another ; i. e, must all express different relations between the unknown quantities. Thus, if we had the equation a? -- 10 - 2?/, it would be of no use to join with it the equation 2x = 20-4?/ (which is obtained by merely doubling it), or any other, derived, like this, immediately from the former ; since this expresses no new relation between X and 7/, but repeats in anotlier form the same as before. • It may be observed, that if any two or more equa- tions be given, any equations formed by adding or subtracting any multiples of these equations, will be also true^ though expressing, in reality, no new rela- tions between the quantities. Thus if ic + 3?/ + 42 = 9, and 2>x-2y ^ Viz = 25 j then, subtract- ing the second from three times the first, we have \\y - 62 = 2. SIMPLE EQUATIONS. 81 95. There are generally given three methods for solving smiultaneous equations of two unknowns; but the object aimed at is the same in each, viz. to com- bine the two equations in such a manner as to expel, or, as the phrase is, eliminate from the result one quantity, and so get an equation oione unknown only. dQ, First inethocl. — Multiply, when possible, one equation by some number, that may make the coeff. of X or y in it the same as in the other; then, adding or subtracting the two equations, according as these equal quantities have different or same signs, these terms will destroy each other, and the elimination w^ill be effected. Ex.1. Here mult. . (ii) by 4, but 4aj + 7/ = 34 ) (i) Ay + X = 1^) (ii) lC>y^Ax=U, 2/ + 42/ = 34; (i) ,*, subtracting, and (ii) 152/ = 30, and . aj=16-42/=16-8 = 8. Ex. 2. Here and, mult, (i) by 4, 4x- 2/= 7) (i) 3ic + 42/ = 29f (ii) 3aj + 42/ = 29, IQx -42/ = 28; .-. adding, 19a; = 57, and .*. ^ = 3; and (i) y = Ax- 7 = 12-7 = 5. Sometimes we cannot make the coefKcients equal by multiplying only one of the equations ; but shall have to multiply both by some numbers, which it will be easy to perceiv^e in any case. Ex. 3. Mult, (i) by 3, - (ii) by 2, 2x+2>y= A) Zx-2y = -l\ 6a; + 92/ = 12 Qx _ 42/ = - 14 CO (ii) subtracting, and (i) 2x = 4* 132/= 26, = 4-3y = 4-6 = and /. 2/ = 2; -2; .•.x = - 82 6BIPLE EQUATIONS. 97. Second riuthod, — Express one of the unknown quantities in terms of the other by means of one of the equations, and put this value for it in the other equation. Ex. 4. 1x + \ (2y +4) = IG ^ or reducing, 35a; + 2?/ = 76 ^ (i) Zy-lix +2)= 8^ 127/- a; = 34^ (ii) Here from (ii) x=l2y- 34, and from (i) 35 (12?/ -34)+ 2y = 76, whence y = 3, and .*. ic = 2. 9 Third metliod, — Express the same quantity in terms of the other in both equations, and put these vahies equal. Ex. 5. hx-\ (5?/ + 2) = 32 j) or reducing, 20a;-5?/=130 ) (i) Zy + -5 (.c + 2) = 9 ^ 97/+ x= 25 \ (ii) Here in (i), y = J (20a; - 130), in (ii) y = l (25 - x) ; .-. ] (20a; - 130) = -J- (25 - x), whence x = 7, y = 2. The first of these methods is generally to be preferred ; but the second may be used with advantage, whenever either x or y has a coefficient unity in one of the equations. Ex. 45. 1. 2x + 9y = n) 4x + y = 5 ) 2. X + y ^ a ? 3. 2.r - 7/ = 8 ) 27/ + a; = 9 ) ax + hy - Ir ^ 4. ax + y = h) X + hy = a) 5. 2x- 97/ = 11) 3a; - 127/ = 15 \ 6. Ix + ay = h ) ax -ly = a) 7. 2a; + 3?/ - 8 = ; 7x- 2/- 5 = 0^ 8. ax = ly) X + y = c ) 9. 5a; + 47/ = 58 J 3a; + 7y = 67 i 10. x(y + 7) = y(x + l) 2a; + 20 = 3?/ + l ? ^1- i ix^ly^Ul lx^{y= 5^ 12. ia; + ]7/ = 43) {x^iy=42\ 13. X y a h X y c d 14. .-...■^=.1 b c «± . ^ = c a 16. ax + ly = c*" 17. 18.^4-1-? =0i h + y a + X J -_^= 1 h a a c 19. i {2x + 3y) + Ix = .?! 20. J(2a;- -y) + 1 = 1(7 + x) ^(7y-3a«)- y = i (3- 4j) + 3 = ^(5^^-7) SIMPLE EQUATIONS. 83 21. a;-|(y-2) = 5^ 22. .yyiy + | (a;-Cy + 1) = J(x-3)| J(x-5y + 8) = |(3»-13y) + ef 23. j\(3x + 4y + 3)- Jj(3x-y ) = 5 + Ky-8| 42/-^(« + 10) = 3i 24. 2« - 4y - -'- (9y + 5a;-8)-:[(a! + jr) = JyC^a; + C) 2/ + 3 „ Si/ — 2x 4^5 8-05 241- 2y+l 25. ic- 2/ + 2/- 05-18 '=zoJ± 3y 99. Simultaneous equations of three unknown quan- tities are solved by eliminating one of tliem by means of any pair of the equations, and then the sa7ne one by means of another pair : we shall thus have two equa- tions involving the same two unknown quantities, which may now be solved by the preceding rules. Similarly for those of more than three unknowns. Ex. 1. 05 - 2y + 3^ = 21 2.r - Sy + z = l\ 3o5- y + 2z = 9] From (i) 2o5 - 4y + 62 = 4 (ii) 2o5 - 3y + z = l - y + 52=3 hence (a) y = 5^ - 3 = 2, (ii) Again (i) 3.?5 - 6y + 9^=6 (iii) (iii) 3o5 — y + 2z=0 Ex.2. X y~ r 1 a c - + - 05 Z I c - + - y z and 05 = - z p 2pqra (i) (ii) (iii) -52/+ 72=-3(/3) but - 5?/ + 252= 15 (a) (a) .-. - 182 = -T8, and 2 = 1 : and (i) 05 = 2 + 2y - 32 = 3 : 5_ 1_1 P From (ii) and (iii) - - ^ ' X y q ;, ... a h 1 and (1) - + - = - X y r 2a 1 1 1 _ ((/ + r)jo- X q r p pqr or soy: 2pqrh Ipqrc (q + 7')p-q7*' ^^ ^ {p + r) q-pr^ " ip + q) r-pq^ which latter values may be written dovrn at once from the Sym- metry of the equations, since it is obvious that the values of y and z will be of the s:ame/ x-i=y + z J ?>x + 2y=z-l J c=x+y J 2^2=3(2/ + 2)J 8.2(x-y)^Zz-2 ] 9. \x^yy=\2-\z 1 10. 2/ + l2=Ja; + 5 1 0^+1=3(^+2) ^ \y^lz= S-^ix i(-^-lH(2/-2)=A(2 + 3) 2^--h3.=4(l-2/) J 1^+|2=10 J a.'-J(27/-5)=lJ-^V J Ex. 47. ]. What fraction is that, to the numerator of which if 7 be added, its value is f ; but if 7 be taken from the denominator its value is J ? 3. A bill of 25 guineas was paid with crowns and half guineas ; and twice the number of half guineas exceeded three times that of the crowns by 17 : how many were there of each ? 3. A and B received £5 17s for their wages, A having been em- ployed 15. and B 14 days ; and A received for working four days II5 more than B did for three days : w^hat were their dail}^ wages ? 4. A farmer parting with his stock sells to one person 9 horses and 7 cows for £300 ; and to another, at the same prices, G horses and 13 cows for the same sura : what was the price of each? 5. A draper bought two pieces of cloth for £12 13^, one being Ss and the other 9^ per yard. He sold them each at an advanced price of 2s per yard, and gained by the whole £3. "What were the lengths of the pieces ? 6. There is a number of two digits, which, when divided by their sum, gives the quotient 4 ; but if the digits be inverted, and the number thus formed be increased by 12, and then divided by their sum, the quotient is 8. Find the number. 7. A rectangular bowling-green having been measured, it was observed that, if it were 5 feet broader and 4 feet longer, it would contain IIG feet more ; but, if it were 4 feet broader and 5 feet longer, it would contain 113 feet more. Find its present area. 8. Find three numbers A, B, C, such that .4 with half of i?, B with a third of G, and C with a fourth of J, may each be 1000. 8IMPT,E EQUATIONS. 85 0. A train leij Cambridge for London with a certain number of passengers, 40 more second-class than first-class ; and 7 of the former would pay together 2s less than 4 of the latter. The fare of the whole was £55. But they took up, half-way, 35 more second-class and 5 first-class passengers, and the whole fare now received was J as much again as before. What was the first-class fare, and the whole number of passengers at first ? 10. A person rows from Cambridge to Ely, a distance of 20 miles, and back again, in 10 hours, the stream iiowing uniformly in the same direction all the time ; and he finds that he can row 2 miles against the stream in the same time that he rows 3 miles with it. Find the time of his going and returning. 11. The sum of the two digits of a certain number is six limes their difference, and the number itself exceeds six times their sum by 3 : find it. 12. A grocer bought tea at 10s per lb, and coffee at 2s 6J per lb, to the amount altogether of £31 5s: he sold the tea at 8s, and the coffee at 4s (jd^ and gained £5 by the bargain : how many lbs of each did he buy ? 13. A and B can do a piece of work together in 12 days, which B working for 15 days and Cfor 30 would together complete ; in 10 days they would finish it, working all three together ; in what time could they separately do it ? 14. A sum of £12 18s might be distributed to the poor of a parish by giving | a crown to each man and Is to each woman and each child, or ^ a crown to each woman and Is to each man and each child, or ^ a crown to each child and Is to each man and each woman : how many were there in all ? 15. Divide the numbers 80 and 90 each into two parts, so that the sum of one out of each pair may be 100, and the diflerence of the others 30. 16. Some smugglers found a cave, which would just exactly hold the cargo of their boat, viz. 13 bales of silk and 33 casks of rum. While unloading, a revenue cutter came in sight, and they were obliged to sail away, having landed only 9 casks and 5 bales, and filled one-third of the cave. IIow many bales separately, or how many casks, would it hold ? 17. A person spends 2s Gd in apples and pears, buying the apples at four, and the pears at five a penny ; and afterwards 86 SIMPLE EQUATIONS. accommodates a neighbour with half his apples and a third of his pears for IM. How many of each did he buy ? 18. A party was composed of a certain number of men and women, and, when four of the women were gone, it was observed that there were left just half as many men again as women : they came back, however, with their husbands, and now there were only a third as many men again as women. What were the original numbers of each ? 19. A and B play at bowls, and A bets B Zs to 28 on every game : after a certain number of games, it appears that A has won 3s ; but had he ventured to bet 5s to 2s, and lost one game more out of the same number, he would have lost 30^. How many games did they play ? 20. A person, being asked how many oranges he had bought, said ^ These cost me Is 6d a dozen ; but if I had got the five into the bargain which I asked for, they would have cost me 2^d a dozen less.' How many had he ? 21. Having 45s to give away among a certain number of per- sons, I find that if I give 3s to each man and Is to each woman, I shall have Is too little, but that, by giving 2s 6d to each man and Is Od to each woman, I may distribute the sum exactly. How many were there of men and women ? 22. Find a number of three digits, the last two alike, such that the number formed by the digits inverted may exceed twice the original number by 42, and also the number formed hy putting the single figure in the midst by 27. 23. A party at a tavern, having to pay their reckoning, and being a third as many men again as women, agree that each man shall pay half as much again as each woman ; but, a man and his wife having gone off without paying their share, lOfZ, the rest had each to pay 2d more. AVhat was the reckoning ? 24. A, B, 0. sit down to play : in the first game, A loses to each of ^and 6'as much as each of them has, in the second B loses similarly to each of A and C, and in the third C loses similarly to each of^ and B; and now they have each 245. What had they each at first ? CHAPTEE VIII. INDICES, AND SURDS. 100. It was stated in (45), that, wlien any root of a quantity cannot be exactly obtained, it is expressed by the use of the sign of Evolution, as V3, V2ac, V d: + c^^ and called an Irrational or Surd quantity. It was also stated in (46) that there cannot be any even root of ^negative quantity ; but that such roots may be expressed in the form of surds, as V-3, V-<^^ ^-{a'^+h'')^ and are then called im-possiUe or wiagi- nary quantities. These we shall considermoreatlength in thischapter. It was seen in (20), that powers of the same quantity were multiplied by adding their indices ; we shall now prove this rule^ to be generally true, which was there only shewn to be true in particular instances. 101. To prove that a^^ x a^=ra"^+*^, iche7i m a7id n are any positive integers. Since by (9) a''^^ = a x a x &c. {m factors) and a^ = a X a X &c. (71 factors), it follows that a'^xa'^=^axax &c. (m factors) xaxax &c. {n factors) =:a xax &c. {ni+n factors) =:a*^+% by (9). 102. Hence (c^'^)^^=a^'^=(^^^)"^; for(<^'^)^=a'^.^'^.a^.&c. n factors=a'"+^^+*^+*^°- ^^^^'^'z^a""*, and(a^)^=a^a".a^&c. m factors=a"+"+«+^°-'"^^^°^^=a^''* ; - .-. since a^^^r^"""', we have {c(/^Y^a''^^={a'y^ \ that is, the n^^ power of the m}^ power of a = the m*^ power of the n^^^ power of a, and either of them is found by multiplying the two indices. 103. Hence also Vc^^^CV^')"": for let V a'''=x''\ then a^= {x"')''=: (i^«)"* by (102) ; hence a — a?^, and .*. \/a = x, and (V ci)"^ = ^''^; 88 INDICES, but also, by our first assumption, \l a^ = a;*"; hence we have yoJ"" = (^ ciY' ; that is, the ii^^' root of the m^^ power of 2^.^ the m^^ power of the n^^ root of a. lOi. These results refer as yet only to positive in- tegral indices, which (9) were first used to express briefly the repetition of the same factor in any product. But now, suppose w^e WTite down a quantity, with a jp i^'O^iiiWQ fraction for an index, such as a^, and agree that such a symbol shall be treated by the same law of Multiplication as if the index were an integer^ viz. cC\cf' = t^^'+": — what would such a symbol, so treat- ed, denote? Since it follows from this law, in the case of positive integers^ that ici'^^y^ = a"^"', we should have here also {a^yz=:i'i=a'P\ and hence it appears, that cif^ would denote such a quantity as, when raised to the c^^power^ becomes equal to a^. But that quantity, whose q**^ power=:a^, is (10) the q^^ root of oP ; and, therefore, a^ z=z ya'P, or = (V ay by (103). Hence, when a fractional index is employed with any quantity, the nicmerator denotes ^ power ^ and the denominator a root to be taken of it. Thus a^ = 2°"* root of 1*' power of a = V «j «* = V<^j ^^=V «) ^^ €? - cube root of square of a = V^'^^ or = square of cuhe root of a = (^ ay ; so a^=\la'^ or (V«)^ a^=a^=a^=&c., or ^a=*/a'=ya''=&c. 105. Again, if we write down a quantity with »? negative index, as a~^ (where p may now be integral or fractional), and agree that this symbol shall be treated by the same law of Mult" as if the index were positive, what would such a symbol, so treated, denote? AND SURDS. 89 Ey this law we should have a^'-^^ x a~P=a"»+p-^=a^ ; but ^VQ have also a''^-^^' -?- a^^ = = — -— =a"' : so that, to multijjly by a"^, is the same as to divide by ccP: and, therefore, 1 x cr-^ = 1 -^ a^, or ,-. VI Hence also any power in the numerator of a quan- tity may be removed into the denominator, and vice versd^ by merely changing the sign of its index. Thus «-iv- = ?:5: = -fC = 5!£: = &c. c o~'c a** 106. Lastly, if we write down a quantity with zero for an index, as a\ and agree that this symbol shall be treated as if the index were an actual number, — what then would it denote ? Since, by this law, cd" x <^^=a°+'''=^^, it follows that a° is only equivalent to 1, whatever be the value oi a. In actual practice, such a quantity as a° "would only occur in cer- tain cases, where we wish to keep in mind from what a certain num- ber may have arisen: thus (a^ + 2fl^^ + 3a+&c.)-f-a^=a+2+3a"'^ + &c., vrhere the 2 has lost all sign of its having been originall}^ a coeff. of some power of a \ if, however, we write the quotient a+2a° + 3a~^+ ower of ajpower of a quantity, we must multiply together the two indices. 1 3^x 11-3 -1.+ 3 _l Thus a'xa^=a'~'- fl. a*-^a -= a ^^- a"^^ a^-^ a'^=a^ ^=ai^ Hab-Wab j ' = I ah' . ah^ V = (a^5'^") ' = a'b*. AND SURDS. 91 Ex. 1. Multiplication, a^ + a^l^ + o^lfi + ah + a^b'^ + h^ A 1. S. 1 4 15 - ahi - a'b^ - ah ~ al^ - a^b^ ~ ^' a' * * * * 5^ _ 52 Ex. 2. Division. a;2 - 4f'^ + 3 + 2a~^b + a'^b'^, 17. Find the square root of at- _ 3a + 3^-a^ - 21a^ + 45 - C3a"^ + 90a'^ - 108a-^ + 81a"i 18. Find the cube root of a ^a;2 - Zar'^x + Ga -o;^ - 7 + ^a^x ^ - 3aaj"^ + a-x ^. 19. Find the fourth root of x^y'^-ix^y"^ ■\-^xy^-Ax'^y's -^xY" 9 3 3 3 9 20. Find the fourth root of lGaj"-96a;22/T+21Ga;V2-21Ga:2y* +8I2/'. 108. SincG every fractional index indicates by its denominator a root to be extracted, all quantities hav- ing sucli indices are expressed as surds. When a oiegative quantity has the denominator of its index (reduced to its lowest terms) even (46), the expression will be imaginary. Thus ^-3 or (-3)-^, ^-9 or (-9)^, are imaginary quantities; but ( -4)^ is not so, since it is the same as (-4)^, where the root to be taken is odd. 109. In the case of tx numerical surd, expressed with a fractional index, should the numerator be any other than imiti/y we may take at once the required power, and so have unity only for the numerator, and a simple root to be extracted. Thus 2^ = (2^)'^ = 4^ or V4, 3"? = (3-')^ = (^V)* or VoV- AND SURDS. 93 110. Quantities are often expressed in the form of surds, wliicli are not really go, i, e, when we ccm^ if we please, extract the roots indicated. Thus ^Ja^ V7. (a'+a5 + &^)* are actually surds, whose roots we cannot obtain ; but -.y<^^, y27, (4fi^ + Aal) + P)- are only appa- rently so, and are respectively equivalent to «, 3, 2a + h. Conversely, any rational quaiitity may be expressed in the form of a surd, by raising it to the power indi- cated by the denominator of the surd-index. Thus 2=4^=VS='S:c., a=\la\ fc= (|c^)^, a+x= (a? + 2ax + X')^ 111. In like manner a mixed surd, ^^ e. a product partly rational and partly surd, may be expressed as an entire surd, by raising the rational factor to the power indicated by the denominator of the surd-index, and placing beneath the sign of Evolution the prod- uct of this power and the surd-factor. Thus 2V3 = V4 X v3 = V12: 3.2^ = 3 yi = V^^ x V^ = V108, 2a^h ^ V4«^6, Aa |/~^ = y''-^ = V32c.^c. Conversely, a surd may often be reduced to a mixed form, by separating the quantity beneath the sign of Evolution into factors, of one of which the root re- quired may be obtained, and set outside the sign. Thus V20 =^/I1^5 = 2^5, y24 = VsVs = 2r/3, Vfl^ = ^a-^^, V|K^ = tadV2a^, 112. A surd is reduced to its simplest form, when the quantity beneath the root, or surd-factor, is made as small as possible, but so as* still to remain integral. Hence, if the surd-factor be Vi fraction, its nwmJ and den"" should both be multiplied by such a number, as will allow us to take the latter from under the root. /2 72.3 1 ^^ 5 s /24 ^3/3-3 /3.5^ , __ 94 INDICES, These latter forms allow of our calculating more easily tho numerical values of the surd quantities. Thus to find that of ^f, we should have had to extract both ^2 and y'o, and then to divide the one by the other, a tedious process, since each would be ex- pressed by decimals that do not terminate ; whereas in ^^6, we have only to find ^6, and divide this by the integer, 3. Similar surds are tliose which have, or may be made to have, the same surd-factors. Thus, Z^a and *Ja^ 2a \/c and 3& y^, are pairs of similar surds ; and ^8, -^50, ^18 are also similar, because they may be written 2V2, 5V2, 3V2. Ex. 50. 3 2 3 3 1 3 1. Express 4*, 9^, 3'^, 2"^, (|)"^, (i)" ^ with indices, whose numerator is unit}^ 2. Express 5, 2i, f «, |a", l(a + ?>), as surds, with indices ^ and J. 3. Express 3"^, (3J)-^, «-^ ab'^c-^^ with indices J and - {, Reduce to entire surds 4. 5V5, yi f.S* yill{\)-\ 25 (l|)-i 5. 3V2, 8.2- J, 4,2», S.S"?, f (§)-^, i (J)"?. 6. 2Va, 7aV2i, » («^)-') (» + *) («' - ^°)' ^ (« - ^) ('^' - ^')-'- „ /25 „ /2« 2a' /3J 2n!3 / 9 , , /«-« 7. a y^-, 3ax ^/g-, -^-^^^ -, -^^, (a.x) ^/— . Reduce to their simplest form 8. V45, V125, 3V432, V135„3V432, Vi, 2^% 3Vi, 4^3?. 9. 8S 32'^, 72^, (li)-2, (201)-V(30S)-UVV,5V-^.,|^9J. 10. Shew that V12, 3v75, ^Vl47, f VA, Vrc. and (144)"^^ are similar surds. 113. To co.mpare surds with one another in magni- tude, express them as entire surds, and then reduce their indices, if necessary, to a common denominator, simplifying as in (109) : their relative vahies will be Ti0%v apparent. AND SURDS. 95 Thus 3 >^2 and 2 -^3, expressed as entire surds, are -^18 and Vl2, and it is at once plain which is greatest : but 3 ^1 and 2 ^3^ or their equivalents >^18 and ^24, in which different roots are to be taken, cannot be at once compared; here then 182=18^= ^5832, 1 2 and 24^=24^=^576, and now their comparative values are evident. 114. To add or subtract surds, reduce them, when similar, to the same surd-factor, and add or subtract their rational factors. Thus V^ + V^O - Vl8 = 2 V2 + 5 V2 - 3 V2 = 4 V2, 4aV'^^&V8^-\/i25^*=4a=5 V^+2a^5 ^&-5a^6 \/h=a'h\/h. Dissimilar surds can only be connected by their signs. 115. To multiply surds, reduce them (113) to the same surd-index, and multiply separately the rational and surd factors, retaining the same surd-index for the jproduct of the latter. _Thus V8 X 3 V2=3 VlG=12, 2 V3 x 3 VlO x 4 ^0=24 Vl80=144 V5 2V3 X 3V2=2V27x3V4 = GV108. Compound surd quantities are multiplied according to the method of rational quantities. Ex. 1. (2 ± V3)' = 4 ± 4^3 + 3 = 7 ± 4 V3. Ex. 2. (2 + V^) (2 - V3) = 4- 3 = 1. Ex. 3. (2 + V3) (3 - V2) = 6 + 3 V3 - 2 V2 - V^. Ex. 4. (1 + V2)* = 1 + 4 V2 + 12 + 8 V2 + 4 = 17 + 12 V2. 116. Division of surds is performed, when the divi- sor is a simple quantity, by a process similar to that for multiplication. _Thus(8V2-12v3 +3v6-4)-5-2v6 = 4Vf-6vf + i- 4 = jV3-3v2+i-W6> (2 V3 - 6 V2) -*- V^ = 2 Vf - eVatr = V2 - V864. 96 INDICES, 117. But, if the divisor be compound^ the division is not so easily performed. The form, however, in which compound surds usually occur, is that of a iinomial quadratic surd, i, e, a binomial, one or both of whose terms are surds, in which the square root is to be taken, such as 3 + 2 |/o, 2 |/3-3 4/^5 or, generally, Vet :k |/J, w^here one or both terms may be irrational ; and it will be easy, in such a case, to convert the operation of division into one of multiplication, by putting the dividend and divisor in the form of a fraction, and multiplying both num"^ and den'^ by that quantity, which is obtained by changing the sign be- tween the two terms of the den^ By this means the den' will be ycl^Aq rational : thus, if it be originally of the form \Ul i 4/5, it wall become a rational quantity, a-b^ when both num'and den' are multiplied by V<^±V&' 2+V3 ^ (2-fV3 ) (3-V3 ) _ 6 + 3v3^2v3- 3 _ S + yS 3 + V3 (3 + V3) (3-V3)~ 9-3 " 6 * P, o __J ^ 2V2-^ V3^ 2V2^ V3 "• 2 V2 - V3 « - 3 5 Fractions thus modified are considered to be reduced to their simplest form, for the reason mentioned in (112). Ex. 51. 1. Compare GyS and 4v7; 3^3 and 2 \/lO\ 2 \/\^,4:\/2, and 3 Vo ; V^ and ^11 ; -JV^ and ^ \/21 ; V5, 2 Vf , and 3 {A\)'K 2. Simplify Vl28 - 2 V^O + ^12 - ylS, ^40 - i V320 + »/135. 3. Simplify 8 V|-i V12 + 4 V27-2 VA, V72-3 »/|4-6 V2H. 4. jNIultiply 3 V8 by 2 V6, 3 ylS by 4 V20, and 2 V4 by 3 V54. 5. Find the continued product of 3^8, 2^6, and 3^54; and cf2v24, 3V18, and4V24. G. Multiply 3 V3 + 2 V2 by ^2>-^2, and 2^1^-^(S by V^ + V^- 7. Find the continued product of 4 + 2 V2, 1 - yS, 4 -2 V2, V2 + V3, 1 + V^, andV2-V3. 8. Div. 2V3 + 3V2+V30 by SyG, and 2V3 + 3V2+V30 by 3v2. an£) surds. 97 9. Rationalize the denominators of 1___ 4 3_ 8-5V 2 3-f V 5 4 v7-f 3v2 2V2-V3' V^-^' V5 + V2' 3-2V2' Z-^V b^2^2^1 10. Divide 2 + 4 V" by 2 ^7 - 1, 3 + 2 V^ by 2 ^5 - 1, and 5 - 2 VG by 6 - 2 V^. Simplify Va+ii'+Va-a; 1 1 ic+Vic^-1 x-^/x^-\ 11. -/-=^ -s/a+x-^a-x a-^a^-x^ a^-vaF-x"^ x-'s/x'^-l x-^^x^~\' Vi^+V.T^ VivT-V^^ ,1 1 1 12 , -^=^ +---r.rr= ;r=:=r, & - 'V.ij'^ + l-Va;^-! V^^'+l + V^^-l' 4(1 + V^) 4(1-V.?^) 2(l+ic)- 18. The following facts should be noticed. (i) The product of two dissimilar surds caniwt he rationed. Let ^x X Vy = m.^ a rational quantity ; .*. xij ~- m^ ; , ^;^^ m'* - VI hence ?/ = — = -^^a?, and Vy = — Vti', or Vy may be made to have the same surd-factor as \lx ; that is, ^Ix and \ly must be similar surds (112). (ii) A surd cannot equal the sum or difference of a rational quantity and a surd^ or of two dissimilar surds. For let ^/a = x ± Vy, .*. (^ = a^' ± 2x -^y + y\ whence i '^x\ly — a -x^-y^ and ± ^Jyz= — ^^ or a surd = a rational quantity, Avhich is absurd. , Again, let \la = \/x ± Vy, .\a = x ±2 \lxy + y, whence ifc 2 -Jxy :=z a-x-y^ and ± ^xy = ^ ((^-x-y)y or the product of two dissimilar surds = a rational quantity, which is impossible. (iii) 7/^ a + Vb = X + Vy, the'?i a = x, a7id Vb = Vy. For since a + ^/b =x-{-^y, we have V5= {x-a)+Vy ; so that, if ccbe not equal to a, we shall have V5= sum 5 I 98 INDICES, AND SUKDS. of a rational quantity and a surd, which is. impossible*, hence a? = <2, and .*. V5 = Vy. Hence also, \i a + -Ji =a?+V2/j then a-\/l=x-^y ; and, if <^ + ^h—0^ we must have separately (3^=0, and J = ; otherwise we should have V& = - ^j or a surd = a rational quantity. (i v) If Va 4- Vb — X + Vy, then Va-V b = x - V y. For since V^ -j- V5 = a? + Vy, we have, squaring, a + -Jh=d'-\-2x-Jy-\-y\ ,\a—x^-\-y^ and V5=2a?Vy; wdience a-^'b = x'-2x Vy + Vy, and ^a-s/h =^ x- Vy So also, if V(^ -{- ^5=V^+ Vy, then Vc^ - Vi = V^-Vy 119. :Z(> extract the square root of a hinomial surd^ one of whose terms is rational^ the other a quadratic surd. Let a + y5 represent the given surd ; assume ^a ■\- 4b =^ \lx-\- Vy, .'. '^a-^b =^ \fx —Jy ; hence, multiplying these equations, ^d'-b=x-y \ but, since a+^b=x + y + '^^xy^ ,\ also(118, iii)a=cc+y ; .-., adding and subtracting, a+'^a^-b^^^^x, a-^a'^-b=^2y ' .\x = i{a + ^d' - b\ y = * {a-^a'-b), &V(^±V^')=Va?±Vy=VB(^+^^^±VB(^-^^l- Ex. Find the square root of 7 ± 2 -^10. Let V7 + 2vio = v^ + Vy) .-. V7-2vio = va;-vy; and V49 - 40 = ic - y, whence 3 = a; - y ; but, since 7 + 2 -^10 = a; + 2/ + 2 Va^y, .•. also 7 = a; + y ; .-. 10 = 2a;, 4 = 2y, or a; = 5, y = 2; and V7 ± 2 ^10 = V^ ± V^- Ex. 52. Find the square roots of 1, 4 + 2 V3. 2. 11 + 6 V2. 3. 8 - 2 Vl5. 4. 38 - 12 VlO. 5. 41 - 24 V2. 6. 2} - V^. 7. 4 J - J V^- 8. j J J - \ ^2, Find the fourth roots of 9. 17 + 12 V2. 10. 5G-24V5. 11. fV^ + 31. 12. 48^V+W15. CHAPTEE IX. QUADRATIC EQUATIONS. 120. Some equations involving surds are reducible to simple equations, as in the following examples. Ex. 1. Vl2T^ = 2 + V-^- Squaring, we have 12 + x=4:+iyfx+x .\ 4V^=8, and V^=2j or ic=4. Ex. 2. S +x- Vif^ + y' = 2. Here Va;^ + 9 = 1 + a* : [observe in other similar cases to take this step, when possible, by which we get the surd h^ itself on one side, and so it will disappear upon squaring:] hence x"^ + 9 = 1 + 2x + x"", and x = 4. Ex. 53. 1. V5 (a; + 2) = V5a; + 2. 1 2. 4. G. 8. 10. ^/xh + V^ (« + ic) = x'^' o. V&a; + a;^ = 1 + a*. 5. I'^llx - 26 + ^ = l^Jj-. a -\- X — Va^ + a;^ = h. 7. Va? - 0^ = ^aj + V& + a;. - -^aj + V aj + 2V«a; + a^= V^- 9. a + X- \l2ax ■¥ x"^ -l. a + X + V»^ + Jx + a;' = 5. 121. QiiadraiiG Equations are those in which the square of the unknown quantity is found. Of these there are two species : (i) Pure Quadratics, in wliich the square only is found, w^ithout tlie first power, as a?^ — 9 = 0, &c. ; (ii) Adfected Quadratics, where the first power en- ters as well as the square, as aj"" — 3i» + 2 = 0, &c. 122. Pure Quadratics are solved, as in simple equa- ,tions, by collecting the unknown quantities on one side, and the known quantities on the other. We shall thus find the value of cc', and thence the value of cc, to which we must prefix the double sign (±). 100 QUADRATIC EQUATIONS. Such equations therefore will have two equal roots, with contrary signs. Ex. 1. ic' - 9 = 0. Here x" = 9, and a; = ± 3. If wc had put ±x = ± 3, we should still have had only these two different values of a;, viz. a; = + 3, a; = -3; since - jc = + 3 gives aj = - 3, and - a; = - 3 gives ic = + 3. Ex. 2. \ {Zx" + 5) - -J {x" + 21) = 39 - 5x\ Reducing, 121a;' = 1089 ; .-. a;' = 9, and a; = ± 3. ^ _ ^/a"^ + x"^ + X h T^ ,„- .^ -/a^ + x^ h + c Ex. 3. = - . Here (85. vO = r ; Va= + aj^-a; ^ ^ ^~^ a^ + x"" fb + cV ^ x^ (7) - cy a (b - c) .*. ^— = , and — = -^-r^^ — 5 or a; = ± ,, ._ ^ . x^ \b - cj a^ 4bc ' 2^^^ ' The above method of reduction from (85. vi) may alwaj^s be ap- plied with advantage to an equation of the above form, wVien the unknown quantity does not enter in both sides of it. Ex. 54. 1. ia;'=14-3a;^ 2. aj' + 5=-V-aj'-16. 3. (a; + 2)^=4a; + 5. 4. , + = = 8. 5. J- - — - = ^ 6, 8x + -= -_— . 1 + a; 1 - a; 4a;'' 6a;' 3 x 7 3a;' 15.2;' + 8 _ , ^ ^ a;' a;' - 10 _ 50 + a;' <. — i ^ = ^X — O. O. -z 7"= = 7 — -r- , 4 6 5 15 25 ^^^ 90 4- 4.7 ;' _ 4 a;'+5 _ 2a;'- 5 _ 7a;'- 25 * a;' + 3"^ a;' + 9 ~ 10 15~~ 20 * 10.z;' + 17 12a;' + 2 _ 5a;' - 4 14a;'+1 6 2a;' + 8 2a;^ 18 11a:' -8 ~ 9 ' 21 8a;'-ll~T- 2 2 _ 1 1 13. z + -X 14 _ a;+V2-i' a;-V2-a.» * ' «_Va'-x' a+^a'-x^ a;'"* 15 V^-g-V^^"^^" _c ^. , Tig' Va'-x' + V6'"+;r' " d * ■^^- ^+Va' + a;'= :;/^r^/ 123. An adfecied quadratic may always be reduced to the form, cc'+paJ+^'^O, where the coeflF. of cc' is +1, and^, g', represent numbers or known quantities. Now, in this equation, we have ^ -\-jpx = -<7, and, adding {^jpf to each side, we get a;'+pa?+J^/=ij9' -jt QUADRATIC EQ.JJAJTONS, ' TCI by this step, tlie first side becomes a complete square ; and taking the square root of each side, prefixing, as before, the double sign to that of the latter, we liave x + ijy= ± '^\f-(i^ and x = -\p ± ^\xf-q^ ; which expression gives us, according as we take the upper or lower sign, two roots of the quadratic. 124. From the preceding we derive the following Hule for the solution of an adfected quadratic : Keduce it to its simplest form ; set the terms involv- ing iz?^ and X on one side, (the coeff. oix^ being +1?) and the known quantity on the other ; then, if we add the square of half the coeff. of xto each side^ the first will become a complete square ; and taking the square root of each, prefixing the double sign to the second, we shall obtain, as above, the two roots of the equation. Ex. 1. x" --Cix = 7. Here ic^ - Ga; + 9 = 7 + 9 = 16 ; whence ic - 3 = ± 4, and 0^ = 3 + 4 = 7, oraj = 3-4 = -l so that 7 and - 1 are the two roots of the equation. Ex. 2. x" + 14^ = 95. Here x" +n4a; + 49 = 95 + 49 = 144 ; whence a; + 7 = ± 12, and a; = - 7 + 12 = 5, or a; = - 7 - 12 = - 19. Ex. 55. 1. x'- -2^=8. 2. x'' + lOjj = - 9. 3. x^-Ux^i20. 4. x-"- - 12^ = - 35. 5. a:^ + 32a; = 320. G. cc^ + lOOaj = 1100. 125. If the coefiicient of x be odd^ its half will be a fraction. In adding its square to the first side, we may express the squaring, without efi'ecting it, by means of a bracket. Ex. 1. x'-^x^ -G. Here x''-hx^ (|)2_:_6+-y-=i (-24+25)=|- j whence a; - 1 = ±\^ and ;r = | + ^ = | = 3, oric=|-| = | = 2. Ex. 2. x^- X ^l. Here x'' -x + ^(4)' = f + ]- = 1 T whence x '-\- ±\ and .i' = -^ + 1 = 1^, or x- \-\ = -\. lOS 'QtJADItA ^IC EQUATIONS. Ex. 56, 1. aj^ + 7a; = 8. 2. x" - Ux = G8. 3. x' + 25x = - 100. 4. x'' + 13a; = - 12. 5. x^ + 19a; = 20. G. a;^ + Ilia; = 3400. 126. If the coefficient of a? be a fraction, its half will, of course, be found by halving the numerator, if pos- sible — if not, by doubling the denominator. Ex. 1. X' + \'-x = 19. Here x'' + -'^x + (^y = 19 + -V- = ^^ ; whence a; + | = ± Y", ^^^ x = -^ + -^^-=^,or x = - ^- Y: = - ^j- Ex. 2. x'+ y^x = 74. Here x''^ ^^-x + {\iy = 74 + i«| = i^ ; whence x + U= ±fj, and x~ - T|+fJ=7f , or a;= - fj - 14 = - l^* Ex. 67. 1. a;''-^a;=34. 2. a;^-fa; = 27. 3. x' + Ja; = 8G. 4. a;^ - -2y«a; = 144. 5. a;^ + yV^ = 145. G. a;'^ - f f a; = 147. 127. In the following Examples the equations M-ill first require reduction ; and since theKule requires that the coeff. of a?' shall be + 1, if it have any other coefF., we must first divide each term of the equation by it. Ex. 3a;^ - 20a; = 5. Here a;'- -%^-x = |, and x^- -^/x +l^ = i^; whence x= J (10 ± VH-^)? th* roots being here surd quantities. Ex. 58. 1. a; = 5 + ^x\ 2. 2a; = 4 + - . 3. -jl-a;^ - |a;= ^V (H^ + 18)- 4. lla;'-9a;=llj. 5. J (a;^-3) =i (a;-3). C. 2a;'' + l=ll (a; + 2). 7. a; - -y— - =2. 8. ^ + ^ + ^— ^ = 0. a;^ + 5 o o + a; o + 2a; _ a; + 22 4_9.r-6 a; + 2 4-a; ^•~3~~~^-~2~' , ^^' ^i- ~2r-^^' 12_ _4_ _ _32_ jr x_+_l _ 13 *5-a; 4: -x~ X + 2' 'a; + l x ~6* 128. An equation of the form ax^ + hx+c = 0, or ax^ + hx = -c (where a, &, c, are any quantities whatever), may, however, be Bolved as follows, without dividing by the cocflBcient of a*'. QUADRATIC EQUATIONS. 103 Multiply every term hjAa, and add h^ to each side ; then Aa^x^ + 4ahx + 1)^ = h^- Aac. whence x = ■ zz . 2a Ex. 1. 2a;'-7aj+3=0, or 2x^-7x = - 3. Here, mult, by 4x2=8, and add 7' = 49 to each side ; then IGoj^ - 6(jx + 40 = 49 -24 =25 ; .-. 4;r - 7 = ± 5, and a; = J (7 ± 5) = 3 or |. The advanced student will find it well to accustom himself to apply at once (by memory) the formula above obtained for x, Ex. 2. (3.r - 2) (1 - x) = 4, or ^x^-5x + (j = 0. Here x=^ (6 ± V25^72)=^ {5± ^-41}, the roots being impossible. Ex. 59. ^ _1 1__J^ 2 48 ^ 165 g 'x-1 a; + 3~35' 'a; + 3~a; + 10 ^ x + 4 7-x 4.r + 7 , . Sa;-7 4a; -10 ^, 3. —5 = — -r 1. 4. + ■ ^ = 31 3a;-3 9 x x + 5 ^ ^ 2x 2x-5 ^, ^ 2x + 9 4x-Z ^ 3a;-16 5. 5 + ?.- =-8^ O. Pi + -z = 6 + r-^r • a;-4ic-3 ^ 9 4a; + 3 18 ^ 6x 3a; - 2 4a; + 7 5 - a; _ 4a; "• ^74" 2:^33 = ^* "l9~ ■" 3T5 " T' 129. The Yootsofx'+px+q=:0aYe{12S)-^2^±^ip'-q : hence, (i) ifip^>q, we shall have j^j/ -q positive^ and .'. '^^i^' -q Vi possible quantity : and since, in one root, it is' taken with +, and in the other with -, the two roots will be 7'eal and different in value ; (ii) if i^/ = 2', we shall have ii/-y=0, and, there- fore, the two roots will be Q'eal and equal in value. (iii) if ^p^ < q^ w^e shall have \p^ -q negative^ and V\p'-q impossible^ and so the two roots Avill be im- possible. Hence, if any equation be expressed in the form x'+px-^-q^O^ its roots w^ill be real and different^ real and equals ov impossible^ according as^^>, =, or<4(^. So also in the more general equation, ax^+hx+c^O^ the roots w^ll be real and different^ real and equals or impossible^ according as J" >, ==, or <4(X- '^jy-^; .-. a+^ = -p, and a^ = \p^- (ip"-q) = q^ Hence, when any quadratic is reduced to the form x^ +px -\- q =z 0, vre have coeff. of 2"^^ term, with sign changed, =5w^;z of roots. and S""*^ term =product of roots. ThuSj in (124), the equation, when expressed in this form, is x"^ - Gx-7=0, and the roots are there found, 7 and -1 ; and here + 0=7+ (-1) - sum of roots, and -7=7 x (-1) -.product of roots. So also ax^-\-hx+c=^0^ expressed in this form, becomes "b G 1) C ' x^+ ~x-\- - = 0' .•.--= Slim of roots, - = vroducL a a a a 131. If a^ phe the roots of x^ +px + q — 0^ then x" +px ■\- q^ix-a) {x-^). For, (130) x^+px + qz=x'-{a + ^)x-\- a/S = x''-ax ^'I3x -f a^={x-a) {x-/3). So also if a, /She the roots oi*ax^-rhx-\-c=Oj we have ax^ + hx + c = a ix'' + - x+ -j = a{x-a) {x- ^). 132. Hence Ave may form an equation with any given roots. Thus with roots 2 and 3, we have (;r-2) (rr-3) =a;'-5j + G=0 ; with roots - 2 and ^, we have {x + 2) (aj - J) = a;* + Ja; - ^ = 0, or, clearing it of fractions, Ax^ + 7a; - 2 = 0. This law is not confined to quadratics, but may be shewn to be true for equations of all dimensions. Thus the biquadratic whose roots are - 1, 2, - 2, 3, is (a; + 1) (a; - 2) {x + 2) (a; - 3) = x' - 2x^ - Ix" + 8ar + 12 :- 0. QUADRATIC EQUATIONS. 105 133. If one of tlie roots be 0, the corresponding factor will be ^ - or x. Thus, with roots 0, 1, 3, wc have x{x-V) {x-Z)=x^-\x^ ^■Zx-^, In such a case then x will occur in every term of the equation, and may therefore be struck out of each ; but let it be noticed that, whenever we thus strike an x out of every term of an equation, it must not be neglected, since such an equation, as it originally stood, would be satisfied by fi?=0, which is therefore one of its roots. ThuSj in the ahove equation, we may strike an x out of every terra, and thus reduce it to a?* - 4ic + 3 = 0, which gives us the two roots, 1 and 3 ] but, besides these, we have the root ic = 0. Ex. 60. Form the equation with roots 1. 7 and - 3, 2. f and - #. 3. 3, - 3, f, - \, 4. 0, 1, 2, 3. 5. 0, - i, li, - 1. G. 0, - 1, 2, - 2, \. We shall now give a few examples of quadratic equations of two unknowns. The solution of these is generally more difhcult : but there are three cases of frequent occurrence, for which the following observa- tions will be useful. 134. (i) Express, when possible, by means of one of the equations, either of the unknowns in terms of the' other, and put this value for it in the other equation. PI 1 2^H-y^ Ex. 1. ic + - = — o— X ^ y _\x-y __ _ ______ (i) (ii) Prom (i) we get y-x^\ ; and, putting this value for y in (ii), wo have = -~^, whence x-1 or - 1 and .-. 2/=a;+l=3 or f. The given equations have, therefore, two pairs of roots, a? = 2 and y = 3, or a* = - ^ and y -l» 106 QUADRATIC EQUATIONS. 135. (ii) When either of the two equations is homo- geneous with respect to x and y, in all those terms of it w^iicli involve x and ?/, put y = vx, by which means we may generally without difficulty obtain an equa- tion involvii^g V only, which being determined, x and y may then be found. Es. 2. x'' + xy + y^ = 7) (i) 2x+ Zy = s\ (ii) Here putting vx for y^ x' (1 + v + v^) = 7, («) x(2^ 3r) = 8 ; (/3) •. dividing (a) by the square of (/3\ the x"^ disappears, and we have l+i) + ?)^ 7 ^ ^ ,^ -(27T.)" = 64'''^"°''^^^^"^'1^5 and from (3), ic (2 + G) = 8, or aj = 1, and y = ra; = 2, or X (2 + 54) = 8, or x --= -J, and y = vx= 2*, (iii) When each of the two equations is symmetrical with respect to x and ?/, put ?^+?; for x and '^/-t' for y. Def. An expression is said to be symmetrical with respect to x and y, when these quantities are similarly involved in it : thus x^ + x-y^ + y\ 4xy + 5^ + 5y - 1, 2x^ - Zx^'y - Zxi/ + 2y\ are symmetrical with respect to x and y. Ex.3. ai" + y^=lSxy-) (i) OJ + 7/ = 12 \ (ii) Put '?i + V for 0^, and ?/ — v for ?/ ; then (!) becomes (a + vy + (u - i^y = 18 (u + v) (u - -c), or u'' + 3?^y^ = 9 (w^ -?)-); (a) and (ii) becomes (tc + ^c) + (u -v) ~ 12, whence u = ] putting this for u in (a), 210 + 18y' = 9 (36 - v'), whence v = ± 2 ; ,'._x = u + V = C) ± 2= 8 or 4, and y = 'w-'y=G±2 = 4or8. 13C. The preceding are general methods for the solution of equations of the kinds here referred to, and will sometimes succeed also in other equations ; yet in many of these cases a little inge- nuity will often suggest some step or artifice, by which the roots may be found more simply, but for which no rules can be given. The methods pursued in the two following examples are worthy .of notice in this respoct. QUADRATIC EQUATIONS. 107 Ex. 4. Zx" - 2xy = 15 ) (i) 2a; + 3y = 12 ^ (ii) Mult, (i) by 3, 9a;- - Q>xy = 45, .... (ii) by 2a;, 4a;' + (Sxy = 24a; ; .'. adding, 13a;*=45 + 24a;, or 13a;'-24a;=45, whence a;=3 or -1^. and from (ii) y = \ (12-2a!)=2 or 4||.* Ex. 5. a;' + y'' = 25 ; (i) 2xy=2A\ (ii) Here adding, x^ + 2xy + y^ = 49, whence x + y = ±7 1 subtracting, x^ - 2xy + y^ = 1, whence a; - y = ± 1 : and X -y = + 7) x+ y = + 7 ) = + IS x-y = -1 j .*. 2a; = 8, and a; = 4, 2x = 0. and a; = 3, 2y = 6, and y = 3; 2y = S, and y = 4 : similarly, by combining the equation x + y = —7 with each of the two x-y=±l, we should get the other two pairs of roots a; = - 4, 2/ = - 3, and a; = - 3, y = - 4. Ex. 61. 1. i^{^x-^5y)-vl(4.x-Zy)=:^n 2. a;^ + 2/'=25) 3. x'^.y''=2^l 3a;= + 22/^=179^ x+y= 1) 4^ + 3a;=:24^ 4. 2 (a;-?/) =11 ) 5. a;' + a'2/=66) 6. a;-2/=2 ) a'2/=20 \ X'-y''=ll ) 15 (a;^-2/-)=lGa'2/ S ^ aj*-* _ 85 4a; 1 8. a;?/=(a;-^)(y + f) ) 9. a'+y=G) ' V " ¥ " y I 2;-2/'= (a;' + 3) {y-^4) \ ^3 + 2/'=72 \ ~ x-y^2 ] 10. 3a;y + 2a; + y = 485 ) 11. a;-y= 1) 12. a;^ + «/* = 189 ) 3a; = 2y ) a;^- 2/'= 19 \ xhj + xy"" = 180 S 13. x-^y=a ) 14. a;?/=aM 15. V'^+V^=^? 1^- ic' + ^'2/=«'; aj*+2/^=5^^ x-y-h f a^+2/=9) y' + a;y=&^^ 137. In the solution of Problems, depending on quadratic and higher equations, there may be two or more values of the root, and these may be real quantities, or impossible. In the former case, we must consider if any of the roots are excluded by the nature of the question, which may altogether reject fractional^ or nega- tive, or surd answers: in the latter case, we conclude that the solution of the proposed question is arithmetically impossible. 108 QUADRATIC EQUATIONS. Ex. 1. What number^ when added to 30, will le less than its square hy 12 1 Let X be the number ; then 30 -^ x = x'^ ~\2. whence a; = 7. or - 6 : and here the latter root would be excluded, if we require only positive numbers. Ex. 2. A pefson bought a number of oxen for £120; if he had bought 3 more for the same money ^ he would have paid £2 less for each. Hoio many did he buy f Let X be the number he bought : then the price actually given 120 120 120 for each was , and .*. ^ = 2, whence x - 12, or - 15, X X ^Z X which latter root is rejected by the nature of the Problem. Ex. 3. The sum of the squares of the digits of a number of two places is 25, and the product of the digits is 12. Mnd thejiumber. Let X. y be the digits, so that the number will be lOa; + y ; then x-+y^= 25, and xy = 12, from which equations we get x = o. y - 4^ or a? = 4, 2/ = 3, and the number will be 34 or 43. In this case both the roots give solutions. Ex. 4. Mnd two numbers sucJi, that their sum, 2^roductj and difference of their squares may be all equal. Hero assume x + y and x-yfor the two numbers : [this step should be noticed, as it simplifies much the solution of problems of this kind :] then their sum = 2x, their product = ic' - y^^ and the difference of their squares = 4r?/ ; .*. (i) 2x = 4.ry, (ii) 2x = x^- y'^ ; from (i) y = ^, from (ii) 2x =: x' - \^ whence aj = |(2 ± ^5) ; and .-. a; + 2/ = 1 (3 ± ^5), x-y = \(\± ^5), the numbers required. Ex. 5. Find two numbers ichose difference is 10, and 2>roduct one-third of the square of their sum. Let X =r. the bast, and .t + 10 = the greater ; then x(x+ 10) = J (2x+10y, wlionce a;=-5±5V-3, which are impossible. The question in fiict amounts to asking for two numbers x and y, such that xy = i(x + yy^ or Sxy = x' + 2xy + t/^, or xy = a;'^ + t/', which may be easily shewn to be impossible : for (x - yy, or x^- 2xy + ?/^ is necessarily positive (being a square quantity) whatever x and y may be, and .-. ar f y" must be greater than 2xy. QUADRATIC EQUATIONS. 109 Ex. 62. 1. There are two numbers, one of which l<3 f of the other, and the difference of their squares is 81 : find them. 2. The difference of two numbers is | of the greater, and the sum of their squares is 35G : find them. 3. There are two numbers, one of which is triple of the other, and the difference of their squares is 128 : find them. 4. In a certain court there are two square grass-plots, a side of one of which is 10 yards longer than a side of the other, and the area of the latter is -^j of that of the former. "What are tho lengths of the sides ? 5. What two numbers make up 14, so that the quotient of the less divided by the greater is ^V of the quotient of the greater divided by the less ? 6. A draper bought a piece of silk for £16 4s, and the number of shillings which he paid per yard was ^ the number of yards. How much did he buy ? 7. A detachment from an army was marching in regular column, with 5 men more in depth than in front ; but on the enemy com- ing in sight, the front was increased by 845 men, and the whole was thus drawn up in 5 lines : find the number of men. 8. What number is that, the sum of whose third and fourth parts is less by 2 than the square of its sixth part ? 9. There is a number such that the product of the numbers obtained by adding 3 and 5 to it respectively is less by 1 than the square of its double : find it. 10. There is a rectangular field, whose length exceeds its breadth by 16 yards, and it contains 960 square yards : find its dimensions. 11. The difference between the hypothenuse and two sides of a right-angled triangle is 3 and 6 respectively : find the sides. 12. What two numbers are those whose difibrence is 5, and their sum multiplied by the greater 228 ? 13. A labourer dug two trenches, one 6 yards longer than the other, for £17 I65, and the digging of each cost as many shillings per yard, as there were yards in its length : find the length of each. 14. The plate of a looking-glass is 18 inches hy 12, and it is to be framed with a frame of uniform width, whose area is to be equal to that of the glass : find the width of the frame ? 110 QUADRATIC EQUATIONS. 15. There are two square buildings, paved with stones, each a foot square. The side of one building exceeds that of the other by 12 feet, and the two pavements together contain 2120 stones: find tlie sides of the buildings. IG. A person bought a certain number of oxen for £240, and, after losing 3, sold the rest for £8 a head more than they cost him, thus gaining £59 by the bargain : what number did he buy ? 17. A tailor bought a piece of cloth for £147, from which he cut off 12 yards for his own use, and sold the remainder for £120 5«, charging 5 shillings 'per yard more than he gave for it. Find how many yards there were, and what it cost him per yard. 18. The fore-wheel of a carriage makes 6 revolutions more than the hind-wheel in going 120 yards ; but if the circumference of each were increased by 3 feet, the fore-wheel would make only 4 revolutions more than the hind one in the same space. What is the circumference of each ? 19. By selling a horse for £24, I lose as much per cent, as it copt me. What was the prime cost of it ? 20. Bought two flocks of sheep for £15, in one of which there were 5 more than in the other ; each sheep in each flock cost as many shillings as there were sheep in the other flock. How many were there in each ? 21. A and B take shares in a concern to the amount altogether of £500 : they sell out at p^r, A at the end of 2 years, B of 8, and each receives in capital and profit £297. How much did each embark ? 22. A and B distribute £5 each in charity : A relieves 5 persons more than i>, and B gives to each \8 more than A, How many did they each relieve ? 23. There is a number of three digits, of which the last is double of the first : when the number is divided by the sum of the digits, the quotient is 22; and, when by the product of the last two, 11. Find the number. 24. Find three numbers, such that if the first be multiplied by the sum of the second and third, the second by the sum of the first and third, and the third by the sum of the first and second, the products shall be 2G, 50, and 5G. INDETERMINATE EQUATIONS. Ill We have seen that when we have only one equa- tion between two unknowns, the number of solutions is unlimited^ and the equation is indeterminate. We shall here make a few remarks upon the simpler kinds of such equations. 138. If one solution be given of the equation ax ±,hy =^ c^ all the others may be easily found. For let aj==a, 2/=/3, be one solution of the equation ax + ly—G ; then ax +hy =^ c =^ aa-^- h^^ or a (x—a) + ^ (2/~yS)=0, which equation is satisfied by x-a= - ht, y-^=iat, where t may be any quantity whatever, pos- itive or negative. Hence the general values of x and y are given by the expressions x = a-U, y =z ^ -\- at If the given equation be of the form ax - h/=c, we should obtain in the same way, x=a + It^ y=:^-\- at, the same as w^e get by writing ~ h for i in the above. If we require only integral values of a? and y, the n"* of solutions will be limited ; i\\Q above results will still apply, only we must now have a, y8, t all integers. 139. It may be shewn however that there can be no integral solution oi ax± 'by=c, if a and Shave any conimon factor, not common also to c. For let a=^mdy h^nd, while g does not contain d ; thenmdx±7idy=c,ovmx±ny=:z -3=a fraction, whichis, of course, impossible for any integral values of x and y. We shall suppose then in future that a \s> prime to 5. 140. To solve the equation ax^hj ^^ c m integers. If w^e can discern one solution, we may apply (138). Thus 13a; - 9^ = 17 is satisfied by a; = 2, y = 1 ; whence ISaj-Oy = 17 = 13 x 2-9 x 1, or 13 (aj-2) = 9 (y-l), which is satisfied by a; - 2 = 9^, y - 1 ^ 13^, so that the solution is ic = 2 + 9#j y = 1 + 13^, where t may have any integral value. 112 INDETEEMINATE EQUATIONS. But the following examples will shew the simplest general method of solving such an equation. Ex. 1. Find the integral solutions of ^x + 5y = 73. Divide by the lowest coefficient, and express the improper frac- tions wliich niay arise as mixed numbers ; then a; + y + f 2/ = 24 + J, or a; + y - 24 = J - |y = — ^ — . l-2v Now. since a; + y - 24 is integral, so also is — —- , and any o multiple of it ; multiply it then by such a number as will make the coeff. of y din, hy the derC with rem'' 1, i. e. in this case, mult, it by 2 ; 2 -All 2 - V . 2 - V . . then — :j-- or — ^ - y is int., /. — ~- is mt. = t suppose ; hence 2 - 2/ = 3«, or 7/ =2 -3^, and a; = J (73 -5?/) = 21 + 5f. Thus, if we take t = 0, then aj = 21, y = 2 j if ^ = 1, aj = 26, 1/ = - 1 ; if « = - 1, ic = 16, y = 5 ; &c. If we require only positive integral values of x and y, then we caanot take t positively > f, nor therefore >0, or negatively > -^/, nor therefore > 4 ; hence the values for t range from - 4 to in- clusively, and thus there will be only 5 positive'miQgv^X solutions. N. B. It may be shewn that it is always possible to find such a number for multiplier as we have employed above, which shall be less than the denominator : and this is the reason why we divide by the least of the two coefficients, in order to have the multiplier as low as possible. But when the denominators are both large, a little ingenuity will save the trouble of searching for such a number, by some such reasoning as that in the next Ex., it being noticed, that the point to be aimed at is, to get the coefficient of y (or of a;, as the cas^ may be) in the numerator to be unity, Ex. 2. Solve in positive integers 39a; — 5Gy = 11. 1, ,, i7y + 11 . . ^ , 342/ + 22 Here x-y-l'^y = l^] .-. — ^ — is mt, and .-. —^ — , 342/ + 22 by-22 ^ 40^-176 , y - 20 and .-. y - -^ or-^-, and .-. -- — or 2/ - 4 + ^-3^- ; 7/-20 let ^^^^ = ^; .-. 2/ = 39i + 20, and x = 3V (U + 56?/) = 56^ + 29. If we take ^ = 0, then a; = 29, 2/ = 20, which arc the least positive integral values they admit of: but the number of such values is here unlimited^ since we may take a?iy positive value for t. INDETERMINATE EQUATIONS. 113 Ex. 3. Find the least number which when divided by 14 and 5 will leave remainders 1 and 3 respectively. Let the number required I^=l^x+l=6i/ + ^ ; then 14a;-52/=2, and here 2x+^x-i/=^j or 2x-y= — ^ — ; hence — ^— is integral, and .'. also — - — -. and —z — , which put =^ t: o . whence x = Z - 5t, fxnd y = ^ (Ux - 2) = S - UL If we take t - 0, we have a; = 3, y = 8, which arc the least positive integral values they admit of, and therefore the least value of iVis 14.3 + 1 = 5.8 + 3 = 43 ; but the n° oi positive values is unlimited, since we may take any negative value for t. N.B. It appears from Ex. 1, 2, 3, that when only positive integral solutions are required, the n* of them will be limited or not, ac- cording as the equation is of the form ax + hy =c^ or ax -hy = c, Ex. 4. Find the least integer which is divisible by 2, 3, 4, with remainders 1, 2, 3. Let ]^=2x + l = 32/+2=42J + 3: then (i) 2a; - 3y = 1, whence, as before, a;=3^-l, 7/r=2^-l ; and (ii) 2a;-4^=2, or 3^-2^=2, whence t=2t\ 2=3^'-l : .-. aj=6i5'-l, y=W-\, s=3^ -1, whence, putting t' = 1, we get a? - 5, and JSf- 2a; + 1 = 11. Ex. 5. In how many ways may £80 be paid in £s and guineas ? Let a; = n'* of £s, y = n** of guineas ; then 20a; + 2\y = n" of shillings in £80=1000, and x-\-y+^^y=%0: put 2^2/=^ 5 .\y=20t^ and X = 2V (1000 - 2ly) = 80 - 21^, which gives /bi^r solutions, or rather three^ if we omit the solution t -- 0, which gives y = 0. [In the Answers we shall omit all zerO'Valiies for x or y.] Ex.63. 1. Find the positive integral solutions of 2a; + 3y = 9, ^x + 29?/ = 150, 3a; + 29y = 151, Ix + 15?/ = 225. 2. Find the least positive integral solution of 19a; - Uy = 11, 17a; =ly + 1, 23a; - 9y = 929, 8a; = 23y+l9, 3. Find the number of positive integral solutions of 3a; + 4y = 39, 8a; + Uy = 500, 7x + Uy = 405, 2x + 7y=: 125. 4. Given x-2y + z = o and 2a; + y - s = 7, find the least values of a;, y, «, in positive integers. 5. A person distributed 4s 2d among some beggars, giving 7d each to some, and Is each to the rest : how many were there in all ? 114 INDETERMINATE EQUATIONS. 6. In how many ways could 12 guineas be made up of lialf- guineas and half-crowns ? In how many ways, of guineas and crowns ? 7. How man}^ fractions are there with denominators 12 and 18j whose sum is f | ? 8. A wishes to pay B a debt of £1 12s, but has only half- crowns in his pocket, while J3 has only fourpenny-pieces ; how may the)'- settle the matter most simply between them ? 9. "What is the least number, whichj divided by 3 and 5, leaves remainders 2 and 3 respectively? What is the least, which divided by 3 and 7, leaves remainders 1 and 2 ? 10. A person buys two pieces of cloth for £15, the one at Ss, the other at llsjo^r yard, and each containing more than 10 yards : how many yards did he buy altogether ? 11. In how many ways can £1 be paid in half-crowns, shillings, and sixpences, the number of coins used at each payment being 18 ? 12. A person counting a basket of eggs, which he knows are between 50 and GO, finds that when he counts them 3 at a time there are 2 over, but when he counts them 5 at a time, there are 4 over : how many were there in all ? 13. If I have 9 half-guineas and G half-crowns in my purse, how may I pay a debt of £4 Ils6d7 14. A person in exchange for a certain number of pieces of foreign gold, valued at 29s each, received a certain number of sover- eigns under fifty, and Is over : what was the sum he received ? 15. A French loicis contains 20 francs, of which 25 make £1 : how can I pay at a shop a bill of 45/?* most simply, by paying Eng. and receiving Fr. gold only ? Shew that I cannot pay a debt of 455. 16. A person bought 40 animals, consisting of calves, pigs, and geese, for £40; the calves cost him £5 a piece, the pigs £1, and the geese a crown : how many did he buy of each ? 17. Find the least integer whi«h when divided by 7, 8, 9, respectively, shall leave remainders G, 7, 8. 18. Three chickens and one duck sold for as much as two geese ; and one chicken, two ducks, and three geese were sold together for 25s : what was the price of each ? 19. Find the least odd number which when divided by 3, 5, 7, shall leave remainders 2, 4, G. 20. Find the least multiple of 7, which divided by 2, 3, 4, 5, 6, leaves always u?iit7/ for remainder. CHAPTER X. ARITHMETICAL, GEOMETKICAL, AND HARMONICAI. PROGRESSION. 141. Quantities are said to be in Arithmetical Pro- gression^ when tliey proceed by a common difference. Thus, 1, 3, 5, 7j &c., 8, 4, 0, -4, &€., a^ a + d^ a + 2d^ a + M, &c., are in a. p., the common differences being 2, -4, d. respectively, which are found by suhtj^acting any term from the term following, 142. Given a the first term, and d the common dif- ference of an AR. series^ to find 1 the n"^ term^ and S the sum of n terms. Here the series will be a, a+cZ, «+2cZ, a+Sd, &c.y where the coeft". of d in any term is just less hij one than the No. of the term : thus in the 2"*^ term we have d^ i. e, Id^ in the o'^^ ^d^ in the 4*^, ScZ, &c., and so in the n^^ term we shall . have {71 -l)d\ hence ^=^ + (^1-1) d, Ag2i\xiS=a+{a+d)+{a+'ld)+&Q,+{l-2d)+{l-d)+l, and also S=l + {l-d) + {I -2d) + &c. + {a + 2d)-\- {a + d)-\- a\ .-. 2S=:{a-tl)'\-(^a-\-l)-^{a+l)+&Q.=:{a+l) n] n n .'. S= {a-\-l)-=\2a+{n-l)d\-, since? = a-\- 2 . ^ {n-l)d, Ex. 1. Find the 10*^ term and the sum of 10 terms of 1, 5, 9, &c. Here a-1, d = i, 7i = 10 -, ... ^ == 1 + (10 - 1) 4 = 1 + 9 X 4= 37 ; ^= (1 + 37) x y> = 190. Ex. 2. Find the 9*'' term and the sum of 9 terms of 7, 5^ 4, &c. Here a = 7, d= - ^, 71 = 9 ; ,'.l = 7 + (9-1) x''-f = 7-8 xf = -5;>Sf=(7-5)xf = 9. Ex. 3. Find the 13'^ term of the series - 48, - 44, - 40, &,c. Here a = - 48, cZ = 4, ti = 13 ; ... Z = _ 48 + (13 - 1) 4= - 48 + 12 X 4 = 0. 116 ARITHMETICAL, GEOMETRICAL, AND Ex. 4. Find the sum of 7 terms of ^ + ^ + ^ + &c. Here «=|, d= -|, n=l ; and here we are not required to find I : .'.J using the second formula, ^5^= (1 + G x —J) l = (1 - 1) J = 0. In this case the series, continued, is |, -J-, j, 0, - J, - -J-, - |, itc. where the first 7 terms together amount to zero. Ex. 64, Find the last term and the sum of 1. 2+4+G+ &c. to IG terms. 2. 1 + 3 + 5+ &c. to 20 terms. 3. 3 + 9 + 15 + &c. to 11 terms. 4. 1 + 8 + 15 + S'=30,« = 10, ^ = - 2 ; .-. by (iii) 30.= 1 20 - 2 (w-l)}^; and the roots of this quadratic are 5 and G, either of which satis- fies the question, since the sixth term of the series is zero : (2) S'-= 28, a = 10, d = - 2 ; and the values of n are 4 and 7, either of which also satisfies the question, since the 5^^ G"*, and 7"* terms of the series, viz. 2. 0. - 2, together = zero. Ex. 3. How many terms of the series, 3, 5, 7,*&c. make up 24? Here /S'= 24, a = 3, d = 2; whence 7i = 4 or -G, of which the first only is admissible by the conditions of tlie Question. HARMONIC AL PROGRESSION. 117 Ex. 4. Insert 3 ar. means between 6 and 2G. Here we have to find three numbers between 6 and 26, so that ihajive may be in a. p. This case then reduces itself to finding d^ when 0^ = G, l^ 26, and n = b ', we have then by (i) 26 = 6 + 4^7, whence d= b^ and the means required are 11, 16, 21. Ex. 5. The sum of three numbers in a. p. is 21, and the sum of their squares, 170 ; find them. Let a-d^ a^ a + d, represent the three numbers (which is often a convenient assumption in problems of this kind) ; then {a-d) + a -v {a + d) = 21, and {a - dy + a^^ {a + dy = 179, from which equations a = 7, d= ±4, and the Nos. are 3, 7, 11. Ex. 65. 1 . The first term of an ar. series^s 2. the common difference 7, and the last term 79 ; find the number of terms 2. The sum of 15 terms of an arithmetic series is 600, and the common dificrence is 5 ; find the first term. 3. The first term is 13 y*^, the common difference - f , and the last term § ; find the number of terms. 4. The sum of 11 terms is 14^, and the common difference is ? ; find the first term. 5. Insert 4 ar. means between 2 and 17, and 4 between 2 and -18. 6. Insert 9 a. m. between 3 and 9, and 7 between - 13 and 3. 7. Insert 10 a. m. between - 7 and 114, and 8 between -3 and -J. 8. Insert 9 a.m. between -2| and 4 J, and 9 between -3f and 2 J. 9. Find the 3 Nos. in a. p., whose sum shall be 21, and the sum of the first and second = f that of the second and third. 10. There are 3 Nos. in a. p., whose sum is 10, and the product of the second and third 33 1 ; find them. 11. Find 3 Nos. whose common difference is 1, such that the pro- duct of the second and third exceeds that of the first and second by J. 12. The first term is n'- n + 1, the common difference 2 ; find the sum of n terms. 13. How many strokes a-day do the clocks of Venice make, which strike from one to twenty -four ? 14. How many strokes does a common clock make in 12 hdVirs ? and how many, if it strikes also the half-hours ? 15. A debt can be discharged in a year by paying one shilling the first week, three the second, five the third, &c. : required the last payment and the amount of the debt. 118 ARITHMETICAL, GEOMETRICAL, AND 16. One hundred stones being placed on the ground at the dis- tance of a yard from one another, how far will a person travel, who shall bring them, one by one, to a basket, placed at the dis- tance of a yard from the first stone ? 144. Quantities are said to be in Geometrical Pro- gression^ when they proceed by a common y^^^^r. Thus 1, 3, 9, &c. 4, 1, i, &c. -^, *5, -\^, &c. «, ar, ar\ &c. are in G. p., the common factors or ratios (as they are called) being 3, J, - 1, r, respectively, which may be found by dividing any term hy the term preceding, 145. Given a the first term and r tlie common ratio of a GEOM. series^ to find' 1 the n^^ terin and S the sum of n ter7ns. Here the series will be a^ ar^ af^ ar"^^ &c., where the index of r in any term is just less hj one than the number of the term : thus, in the 2"^^ term we have 7*, i. e. r\ in the S""*^, r'', in the 4^^, /•', &c., and so in the n^^ term we shall have r^'^; hence I = ar"~\ Again S = a + ar + a?'^ + &c. + ar^'\ and .-. rS = ar + a?'^ + a?'^ -f for greater than, and < for less than.] Smce A = -^, and J7= — j, .\ ^5^= -^r- x = ab = G'' i 2 ' a+b 2 a + b ' .'. C'^ = ^/AII, or 6^ is the geom. mean between A and H, 41 J ,-r •<» ^ + ^ 2a& .- „ ^ , •,„ . , Also A> U.if — r- > -, or if a^ + 2ab + b"" > 4ab, 2 a + b ' or if a^ + b"^ > 2ab ; and, this being the case (137), /. A> H, and, of course, > (r, whose value (being the geo.m, mean between them) lies between those of A and H. HARMONICAL PROGRESSION. 123 151. Three quantities a, b, c, are in ar., geom., or haem. prog. according as a—t a a a -. = - , or = ^ , or = - . - c a c (\) 4- = - = 1 ; .'. a—h = h - c, and a. h, c, arc in a. p. : ^ b - C a ' ; ^ ; h c (ii) ah -h^ = aJ) — ac, orb^ - ac ',.'.-- y , and a, J, o, arc in o.p. : (iii) ac-hc-ab—ac^ or, (dividing each by flSc,) v = -, whence - , -^ , - are in a. p., and therefore a, b, c are in h. p. a^ b c ' Ex. 70. 1. Find the ar., geom., and harm, means between 2 and 4-J. 2. Find the ar., geom., and harm, means between 3J and IJ. 3. The sum and difference of the ar. and geom. means between two numbers are 9 and 1 respectively ; find them. 4. The HARM, mean between two numbers is |f of the ar., and one of the numbers is 4 ; find the other. 5. The dilFerence of the ar. and harm, means between two num- bers is 1|; find the numbers, one being four times the other. 6. Find two numbers whose diflference is 8, and the harm. mean between them 1|, CHAPTER XI. KATIO, PROPORTION, AND VARIATIOIT. 152. The liatio of one quantity to another is that relation which the former bears to the latter in respect of magnitude, when the comparison is made by con- sidering, not hy Jiow much the one is greater or less than the other, hut what number of times it contains it, or is contained in it, i, e. what multiple^ ;pm% ovparts^ or, in other words, ^NlidXfr action i\\Q first is of the second. This is, in fact, the way in which we naturally, and, as it were, unconsciously, compare the magnitude of quantities. Thus the mere numerical difference between 999 and 1000 is the same as between 1 and 2 ; but no one would hesitate to say that 999 is much greater^ compared with 1000, than 1 is, compared with 2. The reason is, that the mind considers intuitively that 999 is i\ much gi-eater fraction of 1000 than 1 is of 2 ; and this is what we should express by saying that the ratio of 999 to 1000 is greater than that of 1 to 2. On the other hand, we should say at once that 1001 is much less^ compared with 1000, than 2 is, compared with 1, the fraction in the former case being less than in the latter. The ratio, then, of one quantity to another is repre- sented by the fraction obtained by dividing the for- mer by the latter. Thus, the ratio of 6 to 3 is 5 or 2, that of 15 to 40 is || or J, that of 4a to 6Z) is ttt or -kt Co oh Of course the two quantities compared (if they are not mere numbers, or algebraical quantities express- ing numbers) must be of the same kind, or one could not be a fraction of the other. RATIO, PROPORTION, AND VARIATION. 125 Thus, the ratio of £9 to £12 is the same as that of 9 cwt. to 12 cwt., or of 9 to 12, or of 3 to 4, or of | to 1 ; since, in each of these pairs of quantities, the first is J of the second, and hence J is the value of each of these ratios ; in saying which we may- suppose, if we please, a tacit reference to 1, i. e. in saying that the ratio of £9 to £12 is f , we may either imply that £9 is -J of £12, or that the ratio of £9 to £12 is the same as that of f to 1. 153. Tlie ratio of one quantity to another is ex- pressed by two points placed between tliem, as <^ : J ; and tlie former is called the antecedent term of the ratio, the latter the conseqiient, A ratio is said to be a ratio of greater or less in- equality, according as the antecedent is greater or less than the consequent. The ratio of a^ : JMs called the dujplicate{i,e, squared) ratio of ct : h, a^ : 1/ the triplicate ratio of a : 5, &c. 151:. Problems upon ratios are solved by represent- ing them by their corresponding fractions, which may now be treated by the ordinary rules. Thus ratios are comjyared with one another, by re- ducing the corresponding fractions to common den", and comparing the num" ; and, if these fractions be multiplied together, the resulting fraction is said to bo the ratio com/po^indedofthe ratios represented by them, Ex. 1. Compare the ratios 5 : 7 and 4 : 9. ■^^^^- Ifj 11 ; whence 5 : 7 > 4 : 9. Ex. 2. Find the ratio of 4 : f. Ans. -? -*- J = -f x J = ||. Ex. 3. What is the ratio compounded of 2 : 3, G : 7, 14 : 15 ? Ans. f X 4 X II = yy or 8 : 15. 155. A ratio of greater inequality is diminished, and of less inequality increased, by adding the same quantity to both its terms. For - ^ - — ^-. as ah\ax ^ cib-rhx. as ax hx. as dJ ^ 5. h < h-{-x < ' < ' < 126 RATIO, PROPORTION, AND VARIATION. In like manner it may be shewn that a ratio of greater inequalit}^ is increased, and of less diminished, by subtracting the same quantity from both its terms. Ex. 71. 1. Compare the ratios 3:4 and 4 : 5 ; 13 : 14 and 23 : 24 ; 3:7, 7 : 11, and 11 : 15. 2. Ofa+i : a-l) and a^ + l"^ : a^-V^^ which is >, supposing a > hi 3. Which is loss of x + y :y and 4x:x + yl ofx^+y^-.x+y and x^+y^ :x'^ + y'^7 oix^ + y'^ and x'^ + y^ : x^-x^y+x^y^-xy^+y*7 4. Find the ratio compounded of 3:5, 10 : 21. and 14 : 15 ; of 7 : 0, 102 : 105, and 15 : 17. e -n- J XT. X- 11/. a'^ + ax+x'^ , cr-cix+x^ 5. Fmd the ratio compounded of ^ — — r and . a'^-a^x+ax-x^ a+x ^ G. Compound x""- 9a; + 20 : x''-(jx and a;''-13a; + 42 : x''- bx. 7. Compound the ratios a+l) : a-b^ a' + h^ : (a+hy, (a^-hy : a*-h\ 8. What is the ratio compounded of the duplicate ratio of a+h-^a -h, and the difference of the duplicate ratios of fl^ : a and a : &, supposing a>h1 9. What quantity must be added to each term of the ratio a : J, that it may be equal to the ratio c:dl 10. Shew that a-h-.a + 1) a* - 5' : a^ + 6^, according as a : & is a ratio of less or greater inequality. 15G. When two ratios are equal^ the four quantities composing them are said to be proportio7ial to one another: thus, iia : h=c : d, i,e. if ^ — - , then a, 5, c?, c?, are proportionals. This is expressed by saying that a is to h as c is to d^ and denoted thus, a : i : : c : d. The first and last quantities in a proportion are called tlie Extremes^ the other two the Means, Problems on proportions, like those on ratios, are solved by the use of fractions. 157. When four quantities are projyortionals^ the pro- duct of the extremes is equal to the product of the means. For if _ = -^ , then ad = le. d RATIO, PROPORTION, AND VARIATION. 127 Hence, if three terms of a proportion are given, we can find the other ; thus 1)G 7 ad ad -, Ig d G h a Cor. li a:h::l): g^ then ac = V, 158. Jf the product of two q^iantities he equal to that of two others^ the four are proportionals^ those of one product ieing the extremes^ and of the other the means. For if ad = he. then -v = -^? or - = - ; d G d and .\a\l)\\G\d^ or a\ g::1): d^ in which propor- tions <2, d are the extremes, and 5, g the means. ^0 \i ac =^ V^ a :!) : :!) : c. 159. If 3 quantities are prop^% the first has to the third the duplicate ratio of that which it has to the secon d. ■r. .p^ 5 ^1 a a h a a c^ i^ or It ^ = -, then - ^ ~ y^ -■=.-- y^ ~ — -~\ G G G .\ a\c is the duplicate ratio of a : 5 (153). 160. When four magnitudes are proportionals^ if any equijmdtiples whatever he taken of tJiefrst andthird^ and any whatever of the second and fourth^ then^ if the midtiple of the first 5^ >, =r, < that of the second^ the multiple of the third shall he >, =, < that of the fourth. y^ ...a G , ma mc , . hov it - = ~, we have — j- = —-.where manan may d no nd be any quantities whatever ; and hence it follows that, if m^ >, =, < n5, so also is mc >, =, < nd, 161. Conversely, If there he four magnitudes such^ that^ when any equimultiples whatever of the first and third are talcen, and any lohatever of the second and fo^irih^ it isfound^ that if the multiple of the first he >j r=j < that of the second^ that of the third is always 128 RATIO, PEOPORTION, AND VARIATION. >5 =5 < that of the fourth^ then these four quantities are projyortionals. For, let «, 5, c, d be such that, any equimultiples, ma^ mc^ being taken of tlie first and third, and any nl)^ nd^ of the second and fourtl), it is found that ac- cording as 7na>^ =, ^ =, •*• — r = — ^^^ f^H values of m b e nb ne and n ; suppose m and n to be taken such that ma=^nl)^ then also 772(? = -yif? : but when ma = nb^ by our hyp., a c mc =: nd ; hence 72cZ = ne. or cZ = ^ ; and /. ^"=15 <^^' 6 a. , 5, c, cZ are proportionals. 162. If^ : b : : c : d, and b : e : : d : f, then a : e : : c : f. ■r^ a G ^b d, abed a c For - = -, and - =:-. ; .-. ^ x - = - x -., or - = 2- b d e f b e d f e f This is the proposition ex ceqtiali, referred to in Euc. v. 163. 7)^a : b: :c : d, a^id e : f : :g : h, then ae : bf : : eg : dh. ^ a c 1 e q ae eg This is called coinpounding the two proportions, and so we may compound any number of such proportions. 164. If 4 quantities form a proportion, we may derive from them many other proportions, all equally true. Thus, if ^ = - , then —^ = -^, or 772a :mb::c:d: b d mb d similarly nia :b\: mc : c7, a\ mb : : :: C : cl, and 171 :n::p '• q, then ma + nh : ^wa — nZ» -pc + qd -.pc — qd, 7. If a : 5 :: Z) : C, thcil a^ - h^ : a::h^ - c"^ : C. 8. If a : 5 :: C : iZ :: =& : c, then a + Z> + c : « - Z> + c :: Ta + & + c)^ : a* + Z<^ + y -Z:(jy-\2x~\\' (iv) a;:27::j^:9::2:a;-2/. 14. What number is that to whicli if 1, 5, and 13 be severally added, the first sum shall be to the second as the second to the third ? 15. Find two numbers in the ratio of 2-J- : 2, such that, when di- minished each by 5, they shall bo in that of 1 J : I. IC. A railway passenger observes that a train passes liim, mov- ing in' the opposite direction, in 2", whereas, if it had been moving in the same direction with him, it would have passed him in 30" : compare the rates of the two trains. 17. A and B trade with different sums : A gains £200, B loses £50, and now yi's stock : i>'s :: 2 : ^ ; but, if A had gained £100 and B lost £85, their stocks would have been as 15 : 3^ ; find the original stock of each. 18 A hare is 50 leaps before a greyhound, and takes four leaps to his three ; but two of the greyhound's leaps are as much as three of the hare's : how many leaps must the greyhound take to catch the hare ? 19. Divide £500 among A^ B, G in the proportion of 3, 4, 5, and also in the proportion of J, ^, | ; and if ^4's portion be to J5's::9 :8, and to (7's::G: 5, shew that the shares of J[, B, are in the proportion of H. IJ, 1|. RATIO, PROPORTION, AND VARIATION. 131 20. A quantity of milk is increased by watering in the ratio of 4 : 5, and then three gallons are sold ; the rest, being mixed with three quarts of water, is increased in the ratio of 6 : 7 ; how many gallons of milk were there at first ? 16S. The value of any Alg. quantity will, of course, depend on the values we give to the letters it contains. Def. "When two quantities are such, that their ra- tio is constant^ that is, remains the same, wliatever values we give to the letters they contain, one of them is said to vary as the other. The sign used to denote variation is Gc(read varies as). Thus, x^ + 3a; a ^x"" + (Sx^ since -■- — — =-. - , whatever be the value of X. 167. Hence if -4 oc ^, (where A and B are used to denote, not numerical or constant^ but algebraical or variable quantities, such as admit of diflferent values by giving different values to the letters they contain) then, according to the above definition, the value of the ratio A : B will remain constant, whatever may be the values of the quantities A and B themselves. If then weputm A todenote this constant value, we have— =m, or A=mB ; SO that, whe7i one quantity varies as anotlier^ they are connected hy a constant Qnvltiplier, Thus ic'+ 3a; = ^(2aj^ + Ga;), from which it follows necessarily that .VT-TT- = vi • for ali values of x, or, as above stated, x^ + Zx^z 2x'^ + 6x, 2aj* + 6a; 2' ^ ' ' 168. Hence also if J. gc ^, and a, 5, be any pair of values of A and B, tlien for any other values of ^ and B, we have A : B = 77i = a : h, that is, when mie qiicmtity varies as another^ if<^^y two pairs of values he taken of them^ the four willhe 2yroportionals : pr since A\a\:B:h^ we may state tins by saying that if one of them be 182 RATIO, PROPORTION, AND VARIATION. changed from any one value {A) to any other value («), the other will be changed in the same proportion {vom the value (B) corresponding to the first to the value (J) corresponding to the second. 169. The following are terms used in Variation : 1. If A=mB^ then A is said to vary directly as B; 2. If J. = ~, A is said to vary inversely as B; 3. If ^ =:mBC, then A is said to vary jointly as J? and (7; 4. l{A=7n --- , then A is said to vary directly as B^ and inversely as C, 170. ThefollowingresultsinVariationarenoticeable. (i) If ^ oc ^ and B a: C, then Ack C For letal=m^, B=nC\ then A=rnnC\ and .•. ^ oc {?, since, 77i, ?i, being constant, so also is mn. So also, if J. a J? and B qc~^ then J. oc — ^ . (ii) If^oc CandBcK C\A±Bcc C,and^{AB)cx: C. For let J. = 7?i{7, B=:nC\ then Adc:B~mC±nC={m±7iyC, and .\^1±j5oc C and V (^^) = V (m6' x nC) = V (m7i(7^) = V (m/i) 6; and therefore V (^^) c/: (7. (iii) If Act, BC\ thenBa:-, and 6^oc 4- For Iet^=mi?6^, then B= 1.:^, or ^oc 4.; so (7a: 4 (iv) If ^ cr. ^, and Co: D, then .4(7oc BD, For let^=mj5, 6^=?ii>; then7l6'=w?ji?i>, or^C^cc BD. (v) If^oci?, then JL^»oc^». (vi) If J. cc jB, and 7^ be any other quantity, then ^7^ X ^7^, and 4^-^r RATIO, PROPORTION, AND VARIATION. 133 171. -5^ A, B, C, he variable quantities^ depending on 07ie another^ and it is observed that^ when C is Ttept constant^ KccH^ and lohen B is hejpt constant^ A cc ; tken^ generally^ that is, when all three are allowed to change their values together, A (X BO. For since Ao: B, when C is kept constant, A must be of the form mB, where m is some constant, and 7nay, therefore, contain the constant C\ but not B. [From this we see that A must contain ^ as a factor, but not B"^, B^, &c., and may contain C] Again, since A cc (7,.when B is kept constant, A must be also of the form nC, where n is some coii- slant, and ma.y, therefore, contain the constants?, but not C. [From this we see that A must contain C, as a factor, but not C, C% &c., and 7nay contain B, as, in fact, w^e have ah^eady shewn it does,] Upon the whole^ then, it appears that A must con- tain both B and C as factors, but no other powers of B or C, and therefore must be of the form ^^^6', where ^ is a constant, containing neither ^ nor C; hence, since A =pBC, we have A qc BC, when all three are allowed to change their values together. The above result may similarly be proved for any number of quantities, B, C, D, &c. ; so that, if any quantity vary separately as each ofseveral others, when tiie rest are kept constant, it varies as their product, when all are allowed to change their values together. Ex. 1. If a oc y^c. and 1, 2, 3, be contemporaneous Tallies of a, &j c, express a in terms of & and c. Since a oc V^c, .'.a- mVc, where we have to find m ; now, when ^=2 and c=3. a becomes 1; .*. l = 12w. or w^yj, and .*. a^^¥c. 134 RATIO, PROPORTION, AND VARIATIO'N. Ex. 2. If y = the sum of two quantities, one of which oc x and the other ,oc x^^ and when ic = 1, y = 6, when a; = 2, y = 20 ; express y in terms of x. Here y = mx + ??x', where we have to find m and /i . noW; by the Question, when a; ^ 1, y = 6, .*. (i) 6 = «»i + n, and when ic = 2, y = 20, .*. (ii) 20 = 2/?! + 4/1 ; from which equations w = 2, n = 4, and .•. y = 2a; + 4a;'. Ex. 73. 1. If xy cc X' + 2/') ^^d ^j 4, be contemporaneous values of x and y^ express xy in terms of x"^ + y-, 2. If y= the sum of two quantities, whereof one is constant and the other oc x inversely^ and when a; = 2, ?/ = 0, when a; = 3, y = 1, find the value of y^ when x = (j. 3. If 2/ = the sum of two quantities, whereof one is constant, and the other xy^ and when a;=2, y = - 2J. when a; = - 2. i/=l, express y in terms of x. 4. If y = the sum of three quantities, which vary as a;, a;', «* respectively, and when aj = 1, 2, 3, y = 6, 22, 54 respectively, ex- press y in terms of x, 5. If 2/ = the sum of three quantities, of which the first oc a;*^, the second oc a;, and the third is constant; and when a; = I, 2, 3, y - G, 11, 18, respectively, express y in terms of a;. 6. Given that zee x -^ y^ and y cc a;-, and that when x-\^ the values of y and s are * and J, express z in terms of a;. z y 11 7. If a; cc - and z"^ oc ~, shew that a; different Perm"^ Tlie same would be true for each of the iV^Perm"' ; hence, if tlie^ a'§ were changed to differentlQitQr^^ we should have altogether 1.2.3 p x iV different Perm"^ of 7i letters, whereof still q are 5's, r are c's, &c. So if in these the q Vs were changed to different letters, we should have 1.2.3 qx 1.2.3 . , , ,2?xN different Perm"^ of ti things, whereof still r would be o's, and so we may go on until all the n letters are differ- ent ; but when this is the case we know (174. Cor.) that their whole number of permutations=1.2.3 n ; hence 1.2.3 . . . .^ x 1.2.3. . . . ^ x &c. x iV^=1.2.3 .... n, . -j^_ 1.2.3... . 71 anc i\ - ^^^ _ _^ X 1.2.3 ....(/ x i^c' ^ Ex. 1. How many changes can be rung with 5 bells out of 8 ? How many with the whole peal ? Here V, - 8. 7. 6. 5. 4 = 6720, P = 8. 7. G. 5. 4. 3. 2. 1 = 40320. Ex. 2. How many different words may be made with all the letters of the expression a^l'c ? 1 2-3 4 5 6 Of these 6 letters, 3 aro a'^, and 2 ¥s j .-. IT= i~9 3 Vl ^^ " ^^' Ex. 3. What No. of things is that, whereof the No. of Yar"*, taken 3 together, is 20 times as great as the No. of Yar"" cf half the same No. of things taken 2 together? , Here, if n denote the No. of things required, we have n (71 - 1) (71 - 2) = 20 {\n) (i?i -r 1), whence n = 6. 138 VARIATIONS, PERMUTATIONS, Ex. 74. 1. How many changes may be rung with 5 bells out of G, and how many with the whole peal? 2. In how many diflferent ways may 7 persons seat themselves at table ? 3. How many different words may be made of all the letters of the words division^ insincere^ commencement, haccalaureus ? 4. How many different words may be made of the letters of the expression a^h^c'dl 5. The No. of Yai*-, 3 together : the No., 4 together : : 1 : 6 ; find the No. of things. G. How many diffei'ent words may be made of all the letters of the words mammalia^ carCtvansera^ Oroonolo^ Mississippi ? 7. The No. of things : the No. of Var", 3 together : : 1 : 20 ; find the No. of things. 8. The No. of Yar'" of n things, 3 together : the No. of Var" of 71 + 2 things, 3 together : : 5 : 12 ; find n, 9. The No. of Var*" of n things, 4 together : the No. of Var"" of f 71 things, 4 together : : 13 : 2 ; find n» 10. If the No. of Var**' of n things, 3 together, be 12 times as great as the No. of Yar^ of \n things, 3 together, what is the No. of Perm"^ of the same n things ? 11. Of what No. of things are the Perm^' 720 ? 12. There are 7 letters, of which a certain No. area's ; and 210 different words can be made of them ; how many a's are there ? 176. The Corahinations of any No. of quantities are the different sets that can be made of them, taking a certain No. together, without regard to the order in which they are placed. Thus, the Comb*" of c^, &, c, (Z, 3 together, are dbc^ abd, acd, led. It is readily seen that each Coiiib^ will supply as many corresponding Far"', as the No. of quantities it contains admits of Perm"'. Thus, the ComV ale supplies the 1.2.3 or G Var" dhc^ acl, laCj hca, cab, cla. AND CO^IBINATIONS. 139 177. The No, of ComTf^ ofn different things^ taken r together^ is n(n-lUn-2) •_!_• - (n-r + 1) i.:4.5 . . 7 . r For (176) each Coriib^ of r things will supply 1.2.3 „.r Var'^' of r things; hence, if C^ denote the No. of Comb"'' oin things, v together, we have 1.2.3 ry^Cr — No. of Var"' of n things, r together z:^ Y^:=n{ii-\) (7^-2)....(n.--r + l); •*• ' r:2.3 ....?• rx TT n 't'^ n n{n-Y) ^ n{n-l){7i-2) J. CoR. Hence C = -, C,= --^^\ ^*"~^2 3 ' Now it will be seen hereafter that these are the same as the coefficients of the binomial (1+a?)", so that (1 + j^)^ = 1 + c,x + cy+ &c. + c\x\ Hence, putting^=l, we have2'*=l+C;+C;+&c.+C;; or the sum of all the Comb"' that can be made of n things, taken 1, 2, 3, &c. n together = 2^-1. 178. The expression for (7;., (by multiplying both num'^ and den*^ by 1.2.3 ... (?i-r)) may be put into the form n{7i-l){n-^) (71-r+l) X (yi-r) 3.2.1 1.2.3 r X 1.2.3 {n-r) 1.2.3 n Y- ~ 1.2.3 .... 7" X 1.2.3 .... {n-7')~' |^r \7i~r if we use \n to denote the continued product 1.2.3 . . . -^i. Hence, writing n- r for ?', Ave have ^nr — =z ^r J \9l-r \7' (7^1^1-r or the No. of Comb"" of n things taken n-7' together = the No. of them taken r together. The Comb"^ of one of these sets are said to be Buppleineniary to those of the other. 140 VARIATIONS, PERMUTATIONS, AND COMBINATIONS. Ex. 1. Find the No. of Comb" of 10 things, 3 and G together ? „ ■ 10.9.8 ,^^ , ., ^ 10.9.8.7 ^.„ Here d = ^tt^ " ^^^' ^'^^^ ^" = 6'4 = yr^-g-^ ^ 210. Ex. 2. How many words of G letters might be made out of the lirst 10 letters of the alphabf^t, with two vowels in each word ? In these 10 letters, there are 7 consonants and 3 vowels ; and in each of the required words, there are to be 4 consonants and 2 vowels : now the 7 consonants can be combined four together in 35 ways, and the 3 vowels, two together, in 3 ways ; hence there can be formed 35 x 3= 105 different sets of G letters, of which 4 are consonants and 2 vowels : but each of these sets of G letters may ha 2^ermuted 6.5.4.3.2.1 = 720 ways, each of these forming a different ^cord^ though the whole 720 are composed of the same G letters : hence the No. required = 105 x 720 = 75600. Ex. 75. 1. How many Comb'"' can be made of 9 things, 4 together? how many, G together ? how many, 7 together ? 2. How many ComV' can be made of 11 things, 4 together? how man}^, 7 together ? how many, 10 together ? 3. A person having 15 friends, on how many days might he invite a different party of 10 ? or of 12 ? 4. How often might a common die be thrown, so as to expose five different faces ? 5. Find the whole No. of Comb" of G things, 1, 2, &c., 6 together. G. Four persons are chosen by lot out of 10 ; in how many ways can this be done ? and how often would any one person be chosen 1 7. How often may a different guard be posted of G men out of GO '] on how many of these occasions would any given man be taken / 8. The No. of Comb" of \n things, 2 together, is 15 ; fmd n. 9. The No. of Comb" of n things, 3 together, is y\ of the No., 5 together ; find n. 10. The No. of Comb" of t? + 1 things, 4 together, is 9 times the No. of Comb" of n things, 2 together ; . find n. 11. The No. of Comb" of hi things. 4 together, is 3J of the No. of Comb"* of ^n things, 3 together ; find 7i. 12. How many words of 6 letters may be made out of the 26 letters of the alphabet, with 2 out of the 5 vowels in every word ? CHAPTER XIII. THE BINOMIAL THEOREM. 179. The Binomial Theorem is a formula, discovered by Sir Isaac Newton^ by means of wliicli any binomial may be raised to any given power, without going through the ordinary process of Involution. It may be stated as follows : AVhatever be the value of n^ positive or negative, fractional or integral, j-^ a .^ + Otc. , where the coefficient of + C) x^ + {ah + ac + he) x -r abc. Assume this Law of Formation to hold for ?i - 1 factors, so that (x + «i) (x + ai)...{x + ti„i) = a;**"^+jpia;""^ + p^x"" " + &c. +^n-i) where ^i=«i+^a+«3 + &c.j iH=aiai+aiaz-^a-iaz + &c.. &c. = &c. 2>n\= aia^az ... a„ i , then, multiplying by another factor, x + (X„, we have (;c+ai) (aj+a2)...(a;+fl^„)=a;''+^iic""^+ jp2a;"^ + &c.+ i)^^x + ct^a;**'* +^ia„a;"''' + &c. +7?„ ^a^x -^PmCtn =a;"+2'xa;"'*+ $'2ic"^ + &c.+ $'n.ia;+ 5-, where qi=^pi + «»= «i + «a + «3 + &c. +.^„, q,2. -pi + ^iCt„ = ai^a + <^^i«3 + en as n is e^eii or odd^ it follows that, if n be even, there will be one middle term, but if odd, two middle terms, which, by (182), will have equal coeff', and on each side of which the same coeff* will occur in order. When, therefore, in expanding a binomial with a 'positive integral index, we have passed the middle term or terms, we shall find all the coeff* repeating themselves ; and, instead of calculating those of the remaining terms, we may write down, in inverted order, the coeff" already found, as in the following examples. Ex. 1. (1 +if)* = l + ja; + |^ic^-{-&c. = 1 + 4a; + Go;^ + 4^' + a;*. We shall not, however, give any more examples of the 3"*, 4* and 5*^ powers of a binomial, which the Student should be able to write down as in (42). 144 THE BINOMIAL THEOREM -^ ^ .1 ^T 1 '^ 7.G , 7.6.5 3 ^ Ex. 2. (i-xy = 1 - i ^+ J72 ^ - 1:2:3 ^ ■" ^^• = l-7aj + 2lx^-Z5x^ + 35a;^-21aj* + 7a;«-aj^ =729x«-G X 243aj'^ x ly + 15 x 81x* x i2/'-20 x 27x» x {y* ■i-15x9x^x^*-6xZxx^y^+-^y^ =729x'-729x'y-i-^-^^xY-H-^Y+^^Y Ex. 76. 1. (1 + xy. 2. (a + ic)^ 3. (1 - xy. 4. (a - xy. 5. (1 + xy\ 6. (1 - 2xy\ 7. (« - 3ic)«. 8. (2aj+a)». 9. (2c^-3a;)'. 10. (l-ixy\ 11. (1-iic)^^ 12. aa;-^y)^^ 183. To 2^^'ove the Binomial Theorem^ lohen the index is fractional or negative. It will be sufBcient if we can prove the Theorem for the expansion of (1+ ct')"^, that is, if we can shew that iov all values of n, (l+a:^y'= l+C^x-\-C^d'-\- &e. For then, since 0^ + x = a\l+ -] , we shall have — a^+ C^a^'^x-r C,a"V + (fee, as required. Let then the series 1+ —a?+ '\ — ^a?''+&c., what- 1 1.2 ever be the value of 7?i, be denoted by the symbol^/* (771). Now, when ra is a positive integer, we know that this series represents the expansion of (1 + x)^^ that is, /(m) = (l+ajy% when in is a positive integer. We shall now shew that this is the case for all values of m. THE BINOMIAL THEOREM. 145 mce f {rri)—\'\--x-\ \—- — - x^ + &c. 1 1.2 and /(m) x/(;0=l+p'«+ '^-^- x' + &c. + '^ 1 mn a?' + &C. 1 n {n — 1.2 1) a;' + &C. VI l-i-ccx+ h^ + &c. where we use a, 5, &c. to denote the coeff% found by addition, of a?, a?', tfec, so that -f n^ I = -1-2 "^ ^^^^ "^ 12' Now 5, 0, &c. might be reduced to much simpler forms than these, but the process would be tedious : we may find them however, immediately, by the following consideration. Since the above multiplication does not at all depend upon the actual values of m and n, we should still have, by the addition, the same values as above for a^ 5, &c., whether 77i and tI' stand for positive or negative, integral or fractional, quantities. But when m and n 2iXQ positive integers^ we know that /(m) = (l+irr, f{n) = {l+xr, and .\f{m) x /(»)=(! +a;)'» x (l+a;)"=(l +«)'"■'"; and since m + n is here a positive integer, we know also that (l+a.r-=l + "TLJ^x^ {m + nUm+n-D ^, ^ ^^_ 1 l.J Here, therefore, we have the values of ^, 5, &c. when m and n are positive integers : hence also they will 7 146 THE BINOMIAL THEOEEM. be the same, whatever be the values of m and n, and we have, therefore, in all cases, or, since this series would be denoted by/ (m+n), we have/* (771) x/*(7i)=y(m+7i), for all values of m and n. The student may easil}'' satisfj^ himself that the values just obtained for a, Z>, c, &c. are identical with the former, though simplified in form ; thus m (m - 1) n in-X) m (m - 1) + 2m n + n (n - 1) h = -~j- ^nm^ "j;2~ " TS _m(m-l + n) + 7i {m^n-\) __ {m + n) (m -i- n - 1) Hence /(m) x /(?i) x f{p)=^ f{m + n) x f(p) = f{m-{'7i +i>)? and similarly for any No. of such fac- tors ; i. e, the product of any two, or more, such series, as that denoted byy(m), produces another 'series of precisely the same form, Now, (i), let there be n factors, each=/[—V where m and n are positive integers; then since m is a positive integer; , in "hW /. taking the n^^^ root on both sides, (!+«?)« =/(— Hence/ (m) is the series for (1 + x)''\ so long as the index \^ positive^ whether it be integral ox fractional. Again, (ii), let n^=^-7n^ where in \^ positive^ but may be integral or fractional; then / W x/ (- m) =/ {vi - m) =/(0) = 1, THE BINOMIAL THEOEEM. 147 (since the series becomes = 1, if we put for m in it); 1 1 . . . ,-. fi-iii) = — , — r = Tz r- i Since m is positive, fip) (i+^-r z=z il+x)'"^^ bj the Theory of Indices. Hencey(-7?^) is the series for {\+xy"\ where the in- dex is neffative,(iiid maybe QitliQv integral or fractional. It follows then that for all values of the index, we have {l^xf =f{n) = 1 + % + ^'l^izl) of + &c. J. 1. J 184. We have seen (181) that, Avhen the index is a positive integer, this series will stop after n+1 terms; when fractional or negative, it will never terminate, but consist of an infinite number of terms, since we cannot then find any value of r, which Vvill make 7i-r-\-l=0. Ex. 1. (l+oj) ^^=1+ -—-a; + — ^ 2 (-2-1) , -2 (-2-1) (-2-2) 1.2.3 &c. 2 2.3 2.3.4 =1- IT x+ Yo^^~ tVq ^^ "^ ^^' = I - 2x + 2x' - ix^ + &c. In this Ex. there is some trouble in simplifying coefT', and getting rid of superfluous signs : to save this, it will be useful to remember the result of the following general example. Ex. 2. a.xr-i-^x.:^^^t^ &c. Ex. 3. (l-^x)-' = l-~x + jl x'- ~~^ x' + &c. = l-3x + 6x''- 10j;= + &c. Ex. 4. (1 . ..)i= 1 . f ... iiiri> .' . ^^^ ^ &c. 148 THE BINOMIAL THEOREM. Here also it will be well to notice the following general results, Ex. 5. (1 ± a;)7 = 1 ± j^a; + — j;^— x'' ^ I.2.3 * So also (\±x) ?=l±^-a;+ -T4r4 « ^ — ^".r^— 3-^ ic' + &c (m). Ex. G. (1^^^ = 1.30.. j-^:^ 2.5.8 1.2.3.3« 1 + fa; + |aj^ + f ja;'' + &c. iC* + &c. Ex. 77. 1. (l+ar)-\ 2. (l-3a;)-\ 3. (l + 3ic)-'. 4. (l-2icr». 5. (l-ia;)-«. 6. (l + Ja;)-». 7. (l + 2;7j)t 8. (1 - 3jr)* 9. (l-xyk 10. (1 - a;^)^. 11. ^ . 12. Vl - a; vr Ex, Ex.8 :.,.<..^,-...-.(.4)-.,t-|(5).g(iy-.o.( - 8a-^a; + 40a-''aj'' - KjOar'x^ + ifcc. ,.-.,i..^(.-|)-L.-iju|(5).a(-?)-..c.i -f^. .cc 15 aj^ 35a;» . ( a 2 a' 2 a' =a*2 + Sa"2^ + -ya"2ic- + -^-a'^v^ + &c. Ex. 78. 1. i2-x)-\ 2. (3-2a;)-*. 3. (a+&a')-». 4. (a-5»a!)-«. 6. (a"^-J^-«. C. (a'-x^)^, 7. (a'K&"V'. 8- (a-x)K 9. (a*-x')'y. 10. (a'-a;')^. 11. (a'-aj»)"* 12. (aa;-2«)"*. I CHAPTER XIY. NOTATION, DECEVIALS, INTEREST, &C, 185. Notation is the method of expressing numbers by means of a series of powers of some one fixed num- ber, which is said to be the radix or hase of the scale^ in which the different numbers arc expressed. Thus in common Arithmetic, all Nos. are expressed in a scale whose base is 10 ; for 3578 denotes 3000 + 500 + 70 + 8, i. e, 3.10^ + 5.10^ + 7.10 + 8 ; so also 370, when expressed in a scale \vhos3 radix is 12, is 274, since 2.12^^ + 7.12+4=288 + 84+4-376. 186. Ifx he any integer^ any No, "^may he expressed in the form ]Sr=:p"r"+pn.ir"^+&c.+p2r+PiT+Po5'^^^^^'^ the coefficients p^, Pn.i, &c. are integers all less than r. For divide iV" by the greatest power of r it contains, suppose r**; and letthe quotient be/>,j (which will, of course,be r'^+iV^p Similarly N,=^]y,,.{^-'''''+N,,N,=:p,,.,r'''''VN,,&^^^ and thus continuing the i>rocess until the rem"" becomes 3>''* Some of the coefficients j^o, ^„ j^i'i ^^' ^^"^^7 vanish, but none can be > r. Their values then may range from to r-1, and these different values are called the digits of the corresponding scale. Ilence^ includ- ing zero^ there will be r digits in the scale of/*. Thus in the scale of 12, the digits will be 0, 1, 2, 3, 4, 5, C, 7, 8, 0, i and„, p^.^^ &c. may be found. Divide iTby r ; then we shall have j^^p^r-' + &(t.+p,r+p,+^, L e. we shall have an integral quotient, ^„r'***4-&c.+pj (=:iV^jj suppose,) withremainder^j)^; hence the remain- der, upon dividing iT by ?', is j^^, the last of the digits. Again, divide iV^i by r ; then we shall have V r r hence the rem^, upon dividing N^ by r is^^i, the last hut one of the digits ; and so dividing N^ by r, we get jpg,&c. Ex. Express the common number 3700 in the quinary^ and convert 37704 from the nonary to the octenary scale. Ex.2. 8)37704 Ex. 1. 5) 3700 5) 740 ... 5) 148 ... 5) 29 ... 3 5) 5.. .4 Ans. 104300. 1...0 8) 4311... 5 8) 480 ... 1 8) 54 ...4 6...1 Ans, C1415. Notice that in Ex. 2, the radix is 9, and therefore, when, in beginning the division, we arc obliged to take the two figures 37, these do not mean tldrty-seccn^ but Zx9 + 7=thirty'fonr : hence 8 in 37 will go 4 times with 2 oyer 3 3 in 27 (not ticenty-seve/i, but 2^^ ^1 ^ticenty-Jlve) will go 3 times with 1 overj and so on. NOTATION, DECIMALS, INTEREST, &C, 151 Ex. 79. 1. Express 1828, 34705 in the septenary scale. 2. Express 300 in the scales of 2, 3, 4, 5, 6. 3. Express 10000 in the scales of 7, 8, 9, 11, 12. 4. Transform 444 and 4321 from the quinary to the septenary. 5. Transform 27^ and 7007 from the undenary to the octenary. 6. Transform 123 and 10000 from the nonary to the quaternary. 189. The common processes of Arithmetic are car- ried on with these, as with ordinary Nos., observing that when we have to find what K^os. we are to carry in Addition, (fee, we must not now divide by 10, bnt by the radix of the scale in question. Ex.1. Addition, r=:4 r = 7 32123 65432 21003 54321 33012 43210 22033 1444 31102 65001 332011 226041 201210 102221 21212 Suhtrnction. 3_ r=12 7^8 5^6^4 1^864 Ex. 2. Multiply together 68 and 71 in the undenary scale ; express also and multiply these Nos. in the nonary scale, and compare the results, by reducing each to the other scale. Here 68 and 71 in the undenary = 82 and 86 in the nonary : 68 82 9) 4378 11) 7823 71 86 9)633...3 11) 642.. .8 f 543 9) 65...2 11) 52...7 1?L 1^ "7...8 X.S 4378 7823 It will be seen that in the last two operations we have shewn that 4378 in the undenary = 7823 in the nonary, and mce versdy as it should be. Ex. 3. Divide 234431 by 414 (quinary), and extract the square root of 122112 (senary). 414) 234431 2302 (310 122112(252 4 423 • 414 45) 421 401 ' 41 542) 2012 1524 44 There is a rem' here in each case. 152 NOTATION, DECIMALS, INTEREST, &0. Ex. 80. 1. Take six terms of the series 1, 10, 10^, &c. ; express and add them in the senary scale, and reduce the result to the denary. 2. ]Multiply the common Nos, G4 and 33 in the binary and quaternary, and transform each result to the other scale. 3. Transform 175G and 345 from the octenary scale to the nonary ; multiply them in both scales, and divide the result in each case by the first of the two numbers. 4. Divide 51117344 by G75 (octenary), 37542027 by 42t (ud- denary), and 29^96580 by 2tt9 (duodenary), 5. Extract the square roots of 25400544 (senary), 47610370 (nonary), and 32^75721 (duodenaiy). 6. Express in common Nos. the greatest and least that can be ' formed with four fiprures in the scales of G, 7, and 8. 190. A decimal fraction may be considered as a vul- gar fraction, whose den"^ is some power of 10, the ISiO, of decimal places pointed off from the right being the same as the index of the den^ Hence, if P represent the digits, or, as they are called, i\\Q sigyiificant party of a decimal of ^ places, its equivalent vulgar fraction It is obvious that decimals, having the same sig- nificant part, P, may difi'er much in value, in conse- quence of the difference in the value of j(>, i. e, in the position of their decimal points. Thus 1.23 = J-?f, -^123 == 1|? 12.3 = Ig. 191. To prove the rule for j^ointmg in MuJP of Decimals, Let Jfand iV^be two fractions, which, expressed as decimals, give the significant parts Pand (>, with jt> and q places of decimals respectively ; then M^^~, N^ J-, and Jfxi\^= — x ^^E^, l()p^ . IC? 10^ 10^ 10^ + ^' KOTATION, DECIMALS, mTEREST, «feC. 153 PO Now — -^ represents a decimal, whose significant part is PQ (tlie product of the two decimals as whole Nos.) and liaving j9+2' decimal places ; hence the rule: Multiply as in v)hole Nos, / and in the product point ojf as many decimal places as there are in tJie Midtiplier and M%dtiplicand together, 192. To prove the rule for pointing in Div^ of Decimals, Let My JVy jP, Q,Pj q be the same as before; '^®^iy"~10^ • 10^~10P^ Q ^'10^' M P \ P p > 'n- Q-iQ^"'''^- q'""' = ^.10",as_p = <7. p Now yr is the quotient obtained by dividing P by Qy as in whole Nos. ; hence the rule : Pivide as in xvhole Ifos, ; then (i) If the No» of places in the Dividend exceed that in the divisor ^ point off in the quotient a iTo. of deci- vial places equal to that excess y (ii) If the No, in the dividend he the same as that in the divisor ^ the q^iotient will have no decimal places; (iii) If the No, in the dividend fall short of that in the divisor^ annex to the quotient a No, of cyphers equal to that defect. Notice that any cyphers, annexed to the dividend in the process of Division, must be reckoned as so many decimal places : thus 1 ^ 12.5 = -^—-^ = .08. 12.0 193. To prove the ride for reducing a circulati7ig decimal to a vulgar fraction. 154 NOTATION, DECIMALS, INTEREST, SzC, We need here consider only the fractional part of a circulating decimal. If there be any figures hefore the decimal point, these may be kept separate, and connected with the vulgar fraction equivalent to the other part, so making a mixed No. Let iV^be a circulating decimal, in which jP repre- sents tlie figures not recurring, and Q the period or recurring jDart; and let P and Q contain p and q digits respectively. Then iY:= ,PQQ &c. and 10^ .N =^ P ^QQQ ^^c. and 10 P^'^.N^I'Q.QQQ ike. .'.{lOP-^^-lO^) jV=PQ-P, 10?>-f(?_10P 10^(10^-1) Hence the rule — (since 10^-1 will be expressed by q nines, and 10^ is 1 followed hj 2^ cyphers) — Por the mime'rator, set down the decimal to the end of the first period, and subtract from it the non-recwr- ring part; and for the denominator , set down as many 9'^ as there are recurring fig%iresy followed hy as many cyphers as there are Qiaii-o'ccurring figures. 194. Let ^ be a proper fraction in its loioest terms. Then if h can be but in the form 2*^ 5 , i. e. the pro- duct of any powers of 2 and 5, the fraction may be reduced to a terminating decimal, in which tlie num- ber of places will be the greater of the two, m and n. l^or it m > ??, then — -^r- = ^ ^ ■= , which, expressed as a decimal (190), has m decimal places : and it m < ?? , tlien -—— = — = , which, expressed as a decimal, has 71 decimal places. NOTATION, DECIMALS, INTEREST, . 196. To find the Amount of a giveoi sum, in any given time^ at Simple Interest, Let P be the principal in jpounds^ n the length of time in years^ r the interest of £1 for 1 year ; then the interest of P pounds for 1 year will be jPr, and for n years, will be Pm^ which is the whole interest re- quired ; and the Amount^ M=P+Prn—P {l-\-rn). If J[f=2P, or the original sum has doubled itself, we have 2P = P (1 + rn)^ and ^ = 1 -i- r, ^=1 -j- n. Thus at 4 per cent., since here we should have r - y|^, and .*. n = -^J- = 25, it appears that any given sum will double itself in 25 years ; but to have doubled itself in 15 years, it should be put to interest at 6f per cent., since then we should have n = 15. and .-. r = yV, andl00r = 6f. CoR. Hence the Simp, Int. on any sum, is propor- tional, (i) to the Principal^ when the Eate and Time are given, (ii) to the Rate^ when the Principal and Time are given, (iii) to the Time^ when the Principal and Eate are given {Arithmetic^ 96) ; but the Amount only in the first case. 197. To find the Amoimt of a given Sum, in any given time, at Compound Interest, Let P, n denote, as before, the Principal and Time ; R the amount of £1 with its interest for 1 year=:l+r; II 156 NOTATION, DECIMALS, INTEREST, AC. then PR will be the amount of £P with interest for 1 year, and this becomes \\\QPrincipal for the 2nd year: .\PE X R^PR will be the amount of £P for 2 years, and this becomes the Principal for the 3rd year: .-. PR' xR = PR' will be the amount of £P for 3 years, &c. hence J}£=PR^=P (l+r)% the amount of £P for n years : and the interest =P7?^-P«P(^'»-1). Coi^. Hence the Com/p, Int, on any sum, as also the Amount^ is proportional to the Pri7icijpal^ when the Rate and Time are given ; but the corresponding state- ment will not hold good, for the other cases of (196 Cor.). 198. To find the present Value and Discount on any Bum for a given time^ (i) at Simple (ii) at Compound Interest, Let Y represent the present value, D the discount, of a sum P due at the end of ti years ; then, since Fis the sum, which at Int. for the given time will amount to P, we have (i) P= F(l+m), (ii) P— F(l+r)^ ; hence (i) F=: -^, and Z>=P~ F=:-^, (ii) F= -^. ^ 1+m' l+m' ^ ^ (l+r)'* Ex. 1. What sum will in 9 months amount to £600, at 5 per cent, per annum. Simple Interest ? Here Jf = 600, r = y^^ = .05, n = J = .75, to find P: 7> ^ 600 600 ^^„^ ^^ .'. P= T = , ^ — ^^ = Tl^^zrz = £5^ 8 65 3^ nearly. 1 + r/i 1 + .05 X ./5 1.0375 ^ Ex. 2. In what time will £91 13« 4^^ amount to £100 at 3 per cent., Simple Interest ? Hero P= 91f, r = j?^, if = 100, to find n : .-. 100 = 91f (1 + y3^7i), whence n = -V/ = SjV years. Ex. 3. Find the Comp. Int. on £275 for 3 y6ars at 5 per cent. Hero P = 275, n = 3, P = 1.05, to find M: \ M= 275 X (1.05)» = £318 6« ll}r7, and Interest - IT- P = £43 6a 11 Jc?. MISCELLANEOUS EXAMPLES : Part L 1. Multiply a" - lax -V ^Ix by V + ax, 2. Divide '6x'^ + 4aZ?.c- - (jo^lrx - 4a^5' by 2a5 + x, 3. If a; = 1, y = - 2, 5 = 3, find the value of Zx' - 2xy + 5?/^ + 53- + 2y2 + 2xz 4x^ + 2x1/ + 32/'^ + 25'' + yz- xz ' 4 Ecducc— ^'-^^-- and ^* ^ '^V' ^ ^^ a (inr + rr) - man x* + 2x^y + Sx'y^ + 2xy^ + y* 5. Extract the square roots of 1//^ and GG.455104. r c- rr (H-^)-i(r-U) _ 1 1 aj 4-3 6. Simplify ^j^--^^--.^ and ^^ - 2^^ - ^^^. 7. Sum the a. p. 7 + 8 J + &c. to 8 and to n terms. 8. Insert an h. mean between 1^ and 1}. 9. Reduce to their simplest forms Vl^S, V98a^c, f V^lyV 10. Expand (1 - 2a!) 2 to five terms. 11. (i) H^^-^)-i(^^-^) = H (ii) ic + 7 = V5^?TT9 (iii) ix-iy=^l ) (iv) ic^-f7/ = 13) G (a; + 2/) - 3 (.2J - ?/) = 13 (a; - 1) S a;y = 6 ^ 12. A certain fraction becomes 1 when 3 is added to the num' and i when 2 is added to the den' : find it. 13. Write down the square of 1 + 2a; - aj' - |a;'. 14. Divide 51a;y + 10a;*-48ajV-152/*-f4aJ2/' by 4x7/--6x^ + Zy\ 15. Find the value of x* -2a (a -l))x^ + (a^ + P) (a-h) x- a'l\ when a=lj 5 = - 2, a; = 3. IG. Find the g. c. m. of ox^ - x""?/ - 2y^ and 10a;* + 15a;V - lOa;^' - loxif. 17. Extract the cube roots of 1953125 and 5. _ o c- rr 2 (a;^ - J) , -J cc^ + Sa'a? + Sax^ + a;' (a + xY 18. Simphfy — > ~ + i and ^ s- -^^ ^—> 19. Sum the g. p. 3 - 1 + &c. to 5 terms and ad infinitum. 20. Simplify {{aH'h^'^^Y'' 2iTi^x-^y"^z^{xyz'^. 21. Expand to five terms g — ^ — -. - V ^. .. ., — ,^ , .. — ,,.. . ., — r--r-„. on c- vr l-ill-ia-a')} , :^ + 2 2-a; x 20. Simplify ^ 1 -7-71 — fr ana j— — -- + -^^-r—T^ " -?-?• •^ 1-J {l-^(l-a;)t 2(a;+l) 2 (a;-l) a;^ + l SO. Find the square roots of 19321, 1.9321, and 19.321. 31. Obtain a fourth proportional to ?, J, |-, and a mean propor- tional to .017 and .153. 32. Sum the g. p. | - f + &c. to 71 terms and ad infinitum, 33. Expand {ax - x") ^ to five terms. 34. In how many ways may a sum of 40 guineas be paid in dollars (45 6d) and doubloons (13s) ? and how may it be paid with fewest coins ? 35.(i)^-^i?.S7i-?n^ (n)ix-12 = ly.8 ) .^.J:^^^^^^, i{x+y)+lx=l{2y-x') + Z5] ^^x + 2 x '^' 30. A can correct 70 pages for the press in 1 ^ hr, B can correct 150 pages in 2 J hrs : liow long will they be in correcting 425 pages jointly ? 37. IMultiply {a + h + c) {a + h - c) hy {a - 1) + c) {h + c - a), 38. Divide 1 - ^a; by 1 - ^a; - Ix"^ to five terms. 30. If « = - a; = i, 6=0, find the numerical value of "x* -{a~l)x' + {a - h) ¥x - l\ 2x^ - a;'* + a; + 1 40. Reduce to its lowest terms 2a;' + 3a;^ + 3a; + 1* AUSCELLANEOUS EXAMPLES. iU 41. Find the cube roots of 2C85G10 and J. 42. Simplify the fraction i^^^^^^^^-^ 43. Expand (a* - 4a'a;^)T to five terms. 2a s / ^ Sx * /80y* 44. Reduce to their simplest forms -^ ,i/ — and ~ a/ Trp^ . 45. Sum the a. p. ^ + f + &c. to 31 and to n-2 terms. 4G. Transform 1828 into the septenary scale, and square it ; re- duce the result to the nonary, and extract the square root •, and express the latter two results in the denary. 47. (i) Zx-i(x-l^)=:d-i{ox-7) (ii) X- ij -z= Gl (iii) a(:x + y) -I) (x- y) = 2a^) Z?/-x-z=::12[ (a'-h-)(x-y)=4a'bS 7z -y -x = 24: \ 48. Two men can do a piece of work in 12 days, and one of them can do half as much again in 24 days ; in what time could the other do a third as much again ? 40. Simvrify i {la -(h-a)\-ilib- la)- I {a.~-l(h-ia)]]. 50. If a = 1, 5 = 3, c = 5, fmd the numerical value of |«_(5_c)p + |5_(c-a)p+ \c-(a-h)}\ 51. Expand and simplify the quantities in the preceding question. 52. Find the g. c. m. of T.i;'-2a;'7^-G3;r2/'' + 18?/' and 6x'-Zxhj-AZxY^ + 27xy^-18y\ 53. Extract the square roots of IIIOOIG and 9 + 2 Vl4. 54. Simplify a+h+ — I -i- (a + 5+ -7- and {a-h+ — , a \ I \ a+b a-hj 55. Sum .2 + .02 + .002 to n terms and ad ivfmitum, 5G. IIow many terms of the series 17, 15, &c. will make 72 ? 57. Expand (a* - bx)"^ to five terms. 58. IIow many different throws can be made with two dice ? UO. (i) _A_ = 8-2 ^'^'^ "■ ^^ ^"^ 5^ + 7y = 43 ) ic+1 \x+Z) llaj + 9y=G9) (iii) x-y — xy- = G = 2xy, tV person bought cloth for £12: if he had bought one yard )ss for the same money, each yard would have cost him \% lore ; how many yards did he buy ? IV SUSCELLANEOUS EXAMPLES. CI. Multiply 2y + Zx^i/- x^ by 7x^ - Syi C2. Divide x' + 4x + S by a;^ - 2^ + 3. C3. If a = 1, & = 2, c = 3, find the value of Va (6' + , ^ ^'^^ j—q q {x^ - 2/ ) (.^ +2/ ^2/) cc^ - oa; + 3 7G. Add together 7vG3 + 2 V252 + 11 V28. 77. Find V3.14159, and the fourth root of a;*- ^o; + fa;' + iV - 2^*- 78. Shew by the Bin. Theor. that V^ = 1 + i - i + iV " tI? + ^c. 79. Sum the a. p. f + 2 + &c. to 9 and to n terms. 80. Form the equation whose roots arc 2, - 2, 1 + -^5, 1 - ^5. 81. What number is that which is the same multiple of 7, that its excess above 20 is of its defect from 30 ? 82. How many different arrangements can be made of the letters of the word Notogorod? IIow many with two o's at the be- ginning and two at the end ? 83. (i)i(7a; + 5)-f (.c + 4) + G = # (;i'+ 3) (ii) X + y -8 = 0= i (x-y) + Ux-\y + 2) (iii) a; + V5a; + 10 = 8. MISCELLANEOUS EXAMPLES. 84. Out of £5000, a person leaves £20 to an old servant, and the remainder among three societies, A^ i?, and 6', so that I) may have twice as much as C, and A three times as much as B: how much docs each receive ? 85. Multiply V^' -^ 1 + ^^ by V'l-' - 1 + ^- 80. Divide ^a^ + ^a'X - 2x^ hj \a + x. 87. If a = 1, 5 = I, ic = 7j 2/ = 8, find the numerical value of ^{a-l) V {a+x) yUa-h\fJa7x)i/- '^^if - {a - V3 (x+2h)\\ 88. Simplify 1^-^ |1-J (x-i)\ and \a-^(a-ih)\-^{h-^ {a + 'il)\. 89. Write down the quotient of ax~^ + 2>^ by a"x~^ + Z>^. 90. Find the square root of (x + x'^) -2(x'^ - x'^) - 1. 91. Sum the a. and g. p. | + 2 + &c., each to n terms. Can the latter series be summed ad infinitum ? 92. Expand Vl + 4.c to five terms, and square the result. 93. Find two numbers in the ratio of 1^ : 2f , such that, when increased by 15, they shall be in the ratio of 1| : 2\. 94. In how many ways ma)^£24 IGs be paid in guineas and crowns? 95. (i)-^9:c + 7)-jaj-4(a;-2)} =3G (ii) 05+ 1 : 2/ :: 5 : 3 ) |:r-i(5-2/)=3^-i(2^-l)5 9G. A messenger starts with an errand at the rate of 3| miles an hour ; another is sent half-an-hour after to overtake him, which he does in 2 hours: at what rate did he ride? Find also in what time he will do it, if he rides 12 miles an hour. 97. Simplify | {x (x^l) {x-^-2) + x{x-V) {x-2)] + § {x-l) x (aj+1). 98. Divide a' - -'/crl'' + ^al^ + ^l' by a^ + 2ah + 4^h\ 99. Find the g. c. m. of Zx^ + 4;c^ - 3u; - 4 and 2x' - Ix' + 5. 100. Reduce -^fJQS^ and ^ ^ V^^ . {x- + V- 2bx) (bx + x') x-a^y 101. Find the cubo root of C9.42G531. 5. ^2 4 .4 2 3 _2 2 102. Multiply 1 + a** -a; i^ -f a3" +2? ^ + rt^ic ^ by oj 3- ^a + 1. VI MISCELLANEOUS EXAMPLES. 103. Find the common difference of an a. p., "when the first term is 1, the last term 50, and the sum 204. 104. If a : h :: c : d^ shew that 7 a + h : oa — 51 :: 7c + d: 3c- 5cZ. 105. Divide 100 into two parts, so that ^ the greater ma}^ be greater than ^ the less by J their difference. lOG. Employ the septenary scale to find the side of a square which contains a million square feet. 107. (\)l-(x■,^)-un-x) = |(x-4)-iJ(x-^) 2x-l 2x^;^_o (iii) 3a;-y + 5 = 17 1 ^^'-^ 2a; + 1 "" 2aj - 1 ~ 5 (aj + 2/ -2) = 2 (2/ + s) [ 4(.^ + 2/ + 2) = 3(l-a; + 32) J 108. A and B engaged in trade, A with £275, B with £300 ; A lost half as much again as B^ and B had then remaining half as much again as A : how much did each lose ? 100. lia-l) - x = Z and « + & + ic = 2, find the value of {a-l)\x''-2ax'' + a''x-(a+ 1)1'']. 110. Shew that (2a + 5"^) (25 + or') = {2ah^ + ah'^y, 111. Find the l. c. m. of 6a;^ - 13a? + C, Gaj'* + 5a;- 6, and Oa;^ - 4. 112. Obtain the square root of Ja;* + \a*- \ax(2a^ + 3a;'-4aa;). 113. Obtain -^G to four places, and thence find -^J, ^f, -^1^. 114. Siraphfy and r + , + - — r-r 115. Square a - 25 - 3c and 2a - ihx - icx^ +2dx\ IIG. Sum the G. p. 5 + 2 + &c. to 7i terms and ad infinituin, 117. The trinomial ax"^ + hx + c becomes 8, 22, 42 respectively, when X becomes 2, 3, 4 : what does it become when a; = - J ? 118. Expand Vl-4a; to five terms, and obtain the same by Evol". 119. (i) ^(4a;- 21) + 3f + i (57 - Zx) = 241 - yV (5a;-9G)- 11a? (ii)lla?^ + l = 4(2-a;)' (iii)i(3.c-2y + l).l(.^_2/) = S2/l 5__3 _ 15 I ~x 2y~2xr J 120. A and B sold 130 ells of silk, of which 40 were ^'s and 90 i?-s, for 42 crowns ; and A sold for a crown ^ an cU more than B did. IIow many ells did each sell for a crown ? MISCELLANEOUS EXAMPLES. Vll 121. Write down the quotient of IG - 81fit by 2 + 3 \la, 122. Multiply «" + §(« + V)x-\3y and a?-l{a-l)x-^ ix^ 123. Reduce to its lowest terms -^-^ — —-i — . Vlx^-bx^ + 4aj-4 124. Find the l. c. m. cf a? ± ic', (a ± xY, and a'' ± a;'. 125. Obtain the square root of 1^ and of 12 + G^3. 12G. Simplify a-ih-c) -\l-{ci-c)\ -{ci- .;2& - (a - c)|l, and ^, ^ « + c 5 + c ic + c shew that - — r—. - - — — = --. — . {cL - 0) (x - a) (a - h) (x -h) {x- a) {x -I) 127. Sum the a. p. ^ + ^ + with tho use of fewest coins, if A have only guineas and B crowns ? 120. Simplify VS {a'x-vax'^yiQa'x' and(V«)*"M(«^^V^r^^*' 130. Compare the numbers of combinations of 24 different letters, when taken 7 and 11 together; and also when the letters a, Jj c occur in each of such combinations. -., ..^Gjj+18 ., ll-3:c „ ._ U-x 21-2.it 131. 0) -^-41—^^ = 5^-48-— ^_ (ii) 1 -f lOy + 5) - 1 (T^- 6) = 10-f,- (3a^ - 10 4- 72/ ) J(12-a'):5.iJ-l(14 + 2/)::l:8 J On) 5^ - _-^ = 2x . — ^ . 132. A party at a tavern had a bill of £4 to pay between them, but, two having sneaked off, those who remained had each 28 more to pay : how many were there at first ? 133. Shew that {ac ± Id)'' + {ad ± lcy= (a^ + 1-) (c- + cV\ and ex- emplify this identity when a=l=-d^l) = 2=— c. 134. Obtain the product of x+2^/xhj + 2^/yhjx-2\'x-y->r2jy. 135. Divide x"^ - (ctr - h - c) x' - (l -c) ax + he by x'^ - ax -i- c. 13G. Reduce ?f:^ ^ ^\% and ^^ " ^^'^' " ^^' 10(6=^ - Oay - 9y' Ox^ + 53.r= - 9a; - 18 * 137. Find the l. c. m. of mhi-mn^, m" + w?z-2;i-, and m--r7in-2n\ 138. Obtain the square root ofa^-2a^ + Za^-2a'^^ + 1. 139. l^ a : h : : c : d, express (h + d) (c + d) in terms of a, 5, c, 140. Find -^24, and thence deduce the values of 5 V2 2V3 + V2 1 -f V21G via MISCELLANEOUS EXAMPLES. 141. Insert two a. and two ii. means between 1 and 3. 142. Expand (1 - 4aj)"^ and (1 - 4:r)"^ to five terms ; and shew that the former sericSj when squared, coincides with the latter. 143. (i) ix-i(x-2) = i {x^l.(2i-x)\ - \(x-5) X x-d _x + 1 x-S ..... 7a; + 1 80 fx- i\ 144. A farmer bought 5 oxen and 12 sheep for £G3, and for £90 could have bought four more oxen than he could have bought sheep for £9 : what did he pay for each ? 145. Find the continued product of {x + a) (x + h) (a-2x) (b-x), 14G. AYrite down the square and fourth powers of a - \^Jax - 2x, 1A7 c- IT (x'-4x)(x'-4y . (a^-l)K-l) 147. Simplify 1__— A— and ' {x''-2xy' (a+ ly (a^-af 148. Reduce to its lowest terms r-^ — j ,o o 2 o "i — u-i — T7{ — • Sa^ + 46^ + 3c^- Hah - Sbc + lOac 149. Extract V-Ol to four places of decimals. 150. Obtain the square root of (x + ly - 4^/.r (x - ^^ x + 1, 151. Determine which is the greater ^2 -f- \/3 or ^Jo -f- 1/5, 152. Sum the g. r. -^ + ^ + &c. to 71 terms and ad injinitum, 153. Given — 1 to be a root of the equation ic* - Ix' - 6a; = 0, find the other three roots. 154. In how many different ways could a farmer lay out a sum of £G3j in buying sheep and oxen at 30s and £9 respectively ? 155. (i) a {x -J)) z= I (a - x) - (a + l)x 3 5 4 (iii) 2a;^ + 3x2/ = 26 ) ^"^l-Sa;"" l-'ox'^2x~l~ Zif^2xy^Z^\ 156. A and B can do a piece of work together in 4 days : A works alone for two days, and then they finish it in 2^ days more : in what time could they have done it separately ? 157. Fmd the value of ^\/\^l + ^^^~^+ VS^=^^ + 4J- when a = J, 5 = ]. MISCELLANEOUS EXAMPLES. IX 158. Divide a^ + 2ahi + h^ by a^ + 2a^ h^ + h. 159. Find the g. c. m. oi:x' + 7x'' + 7x'-~15x and aj'-22;'-13a;+110. IGO. Simplify 5 -: - -^ — 7 and :, , \, .-, \ . . IGl. Multiply together a; - 1 + V-) a; + 2 + V3, a; -- 1 - V2, and aj + 2 - V^- 162. Find the 7*^^ term of S + SI + CJ + Ac, and its sum to IG terms. 1G3. If a : Z> :: & : c :: c : tZ, shcw that a : I) :: ^ b , ^1 2.^+1 4a,' + 5 1 / 3. Fmd tnc sum of 7. and of 1 + pj- ^- - -pr- — ^ x-a x-b- 2 (a; - 1) 2 (ic+ 1) 174. Extract -^15, and thence obtain the square roots of |, f. 2f^ 412 ■ 1x3. 175. Sum the a. p. 13 + 11^ + &c. to 5 and to n terms, beginning in each case with the ninth. 8* X MISCELLANKOrS EXAMPLES. 17G. If a; =c ^-^, y = ^4. ^^^ the talue of x^ -^ xy ^ y\ 177. Expand (1 + -^.x) ^ to five tcrmSj and obtain from the result tlio scries for (1 + -^.c) *. 178. Find three numbers in the proportion of |, §, f, the Bum of whose squares is 724. x-\ 179. (i) 6a;-a:4a?-6::3aj+ &:2aj + « (ii) 3(a;-J) p: = 5 ic + 2 (iii) {x + 5)'^ + (?/ + C)= = 2 (a^z/ -- 24), y = cc + 1. 180. A docs § of a piece of work in G days, when B comes to help him ; they worlc at it together for ^ of a day, and then jB by Ijimself just finished it by the end of the day: in what time could they have each done it separately ? 181. Find the continued product of a + ic, a + \y, a^\z\ and deduce from the result the value of {a + lif, 182. Multiply \x + 3^"^ -:^a^ by 1x - a'M ^ \a^, .183. Simplify x-\\Q.\-x)-\{^\-x)-l (H -a;-2i)}. T o I T. 1 X -x 1 X .L ic" - oj^ + 3.7?^ - 2.C + 2 184. Kcduce to its lowest terms 7 — z-;. — -z — . x^ — b7j' + Oil' — 5 185. Find the l. c. m. of ax^ - a%, ax^ - 1, and «ic' + 1. 18G. Extract the square root of a^^^ ^ + \a ''V - a'l) + 2db \ 187. Find the sum of 77-- r-r + —- - - — - — r^. 2 (x - \y 2 (a; - 1) 2 {X- + 1) 188. Multiply together 3 ^8, 2 ^G, V^^. V^O ; and find V2+Tv7. 189. Sum the g. r. G - 2 + &c. to 7 and to n terms. 190. A watch which is 10' too fast at noon on Monday gains 3' 10" daily : what will be the time by it at 7h 12' a.m. of the fol- lowing Saturday ? 101. (i) Jr (3x + I) - ^ (4x - C§) = i (5.r - C) (ii) 5x+4y=38i + i {Zx-y), x^5^\ - J )i («+y) - {{x-y)]. ,„., 2 3 5 192. A man and his wife w^ould empty a cask of beer in IG days ; after diinking together G days, the woman alone drank for 9 days more, and then there were 4 gallons remaining, and she had drank altogether SJ gallons. Find the number of gallons in the cask at first. . MISCELLANEOUS EXAMPLES. XI 193. If Aa = 5h= 1, find the yalue of iC^" x""' 1 1 1 (J + a-V);*-vQ u + «"^ -a + «6-)'*}]. 194. Find the sum of - . ^ „ i • iC' - 1 iC" + 1 iC'* - 1 .'C" + 1 195. Simplify the surd expressions 3V2 + 2V3 3V2-V3 3_V^+_2Vi 196. Reduce 7-^ »// .. ^^ and (m'^ - a^) (m^ - am - 2a^) a' - 2a\t - «ai* + 2a;'' 19T. Find the l. c. m. of 3a;- -2:c-l and 4.i^-2a;- - 3a; + 1. 198. Sum the scries 3 -2 + IJ— &c. to ti terms and ad ivfinitum, 199. Prove that the sum of any number, ti, of consecutive odd numbers, beginning with unity, is a square number. 200. Given if ^ a^ - a;^, and when x - ^d^ -h^, ai/= h', find the value of X when y = |&. 201. A person distributed £2 Is Sd among some poor people, giving 9^d to each man and 6^d to each woman : how many men were there, it being known that the whole number was a multiple of 10 ? 202. Expand (1 + VO" ^^ ^^'^ terms, and obtain from the result by Evolution the series for (1 + yx)"^. 203. (i) i {I + t(a; + 2)f -f {IJ- (l^-o^)} = h% (ii) ahx"^ -{a + h)x+l==0 (iii) i(x+y) =x-y= '\/x + 2y-l. 204. A and B lay out equal sums in trade ; A gains £100, and B loses so much, that his money is now only f of ^'s ; but if each gave the other J of his present sum, ^s loss would be diminished by one half. What had each at first, and what would ^'s gain be now ? 205. Shew that \ (x^ + y^) + z^-^xy + xz-yz and (y ~ zy become identical when -x = y = a, 20G. Divide mpx^ + (mq - np) x^ - {mr + nq) x + nr by mx - n 207. Multiply <*^ + a'^" + 2 - a^ + ^"3 by cC^-a^ + 1. 208. Reduce to its lowest terms — — — — 3 • x^ -ax + rt-a;-a^ 1 - 2aj 1 + a; 1 209. Obtain the sum of V Zif-x^X) 2(a;- + l) 6(a;+l)' Xii MISCELLANEOUS EXAMPLES. 210. Find the square root of 40.14290404 and the cube root of 8242408. 211. The Z'^ and IS^'^ terms of an a. p. arc 3 and l: find the 14* term, and the sum of 20 terms. 212. Simph'fy the sUrd expression {ah-^ . V«^^ . Vab* . VaJ}}'^. 213. The forc-whcel of a carriage makes G revolutions more than the hind wheel in 120 yards, and the circumference of one is a yard less than that of the other : find that of each. 214. Transform 1000000 from the quinary to the septenary scale; and extract its square and cube roots in the latter. 215. (i)i(x-l)(x--2) = (x-2^(x-l^ (ii) 2x + S?/ = 5 = -(21/ + Sx) (iii) xU xy=a'', y''+x2/-^^h\ 216. Find the time in which A and B can do together a piece of work, which they can do separately in m and n days. How long must A work to do what B can in m days? 217. Find the difierence between (n + 2) (n + 3) (n + 4) and 24 ^7^- 1(7.-1)} {n~IO^-2)} {n-^(n-li)\. 218. Divide a + &^ + c' - 3 V^6V by a^ ^ h^ + c, 219. Fmd the sum of + ^ ^ x-h X -^ a (x - a) {x - b)' 220. Find the l. c. m. of x^+ xhj + xy^ + y^ and x^ - xhj + xy^ - y\ 221. Obtain ^\0^ and thence derive the values of | ^f, ^4 J, ^'24, (V5 + v2)-*-(V-5-V2), and (VS-V^) - (^ V2-2V5). " 222. Sum (10-^ + 2-» + (2f)-^ + &c. to n terms and ad injinitxm, 223. Expand {a" + 2.r-)"^ and (2^-3ir)-2 each to five terms. 224. A servant agrees with a master for 12 months, on the con- dition of receiving a farthing the first month, a penny tho second, fourpence the third, and so on : what would his wages amount to in the course of the year ? 225. Given two roots of the equation ic* + 4aj = bx^ to be 1 and -2, find the other three roots. 226. A person changed a sovereign for 25 pieces of foreign coin, some of them going 30 to the £, the others 15 : how many did he get of each ? 227. (i) 2ax^ + {a-2)x-\ - (ii) ax^\^hy^\=ay-^hx ... X aj + 2_8j;-13 MISCELLANEOUS EXAIklPLES. XlJl 228. Find the time in which A, B, and C'can together do a piece of work, which A can do in m daj^s, B in n days, and C in i (m + n) days. 229. Divide 5?/* + lay^ -^ -Y^'-aY' + -^2/ + W ^7 iv"" + oay-la\ 230. Obtain the products of ^x^ + a V^;' + a"^ (i) by ^x^-aijx^^-a^^ (ii) by ^x^ + aXjx^-a^^ (iii) by ^x^'-aXjx^-a'^. 231. Find the g. c. y.. of 3a^-a"6^-26* and lOa^ + 15a^5-10a-Z>^-15ai^ 232. FindtheL.c.M.ofii'^-3a,-^+3a;-l, a^^-rc^-ic+l, x'-2x^^2x-X anda;*-2ic« + 2a;'-2.'c + 1. 238. SimpHfvr _-i^ and ^-^2- ^. 234. Extract the fourth root of 2 4 li> 8 5_ 1 2 in Y^x'^' -^x^'y^ + -Y-x^'y^ -2!J0x^y'^' + G25?/"^". 235. Sum IG^ + 14| + 13 + &c. to 11 terms, and | + |+|;- + &c. to n terms and ad inf, ; and insert 8 h. means between 1 and 2. 236. Given y"^ '-¥(ax ^ «, and when x = h.y = a, find the value of y when x = 3a» 237. Four places lie in the order of the letters A, B^ (7, JD, A is distant from I) 34 miles, and the distance from -4 to -5 is I of that from C to I) ] also J of the distance from -4 to i? is less than thrice the distance from B to hy | of the dis- tance from C to B. Find the respective distances. 238. If (l + xy = l+ Atx + &c., and (I + x)"" =1 + Ba -^ &c., shew, by finding the actual values of J^i, i?i, &c. that A, + A^Bi + A1B2 + i?3 = 0. 239. ({) 3x^20 = 7-^]Z-^(x-l)\ (ii) - + - ^ a, - + ^=5 y X y (i) 3x* 20 = 7-i ■13- •H^- ■1)} (ii) X (iii) Gy- -4x 5z- - X y- -22! = 1. 33- -7 Zy. -2a; 240. If in (228) A work for \ {Zm - 2i\) days and B for i (37i-2w) days, in what time will G finish the work ? 241. Write down the quotient of x"^ -y^ by x'^ + y"^. and divide jc^ - 2ax' + (rr + a& - Jr) x - a"J> + aJi^ by a; - « + &. 242. If a = 16, 5 = 10, a; = 5, 2/ = 1, find the value of (x-b) {^a- h) + V(rtt -b) (x + y) and (a - a;)^ - (& - a'^) - V (<^? - .c) {h + y). 243. Find the g. c. m. of 3 00^^3^265 j-'^ + 50r + 2\ and OO.i'-53.T-f 4. XIV MISCELLANEOUS EXAMPLES/ 244. Simplify — — - and I3 . -^^j-^j x j- - ^^^^ ^. 245. Find ^-G, and obtain by means of it the values of V^t, ^/^^ (V3 - V2)', and (2 V3 + 3 ^2) + (3 V3 - 2 V2). 24G. Shew that V{«' + V^^} + Vl^' + Va^*} = (a^ + P;2. 247. Divide 48 into nine parts so that each may just exceed that which precedes it by i. 248. Given the coefficients of the 4*^ and G'^ terms of (1 + a-)*^"^^ equal to one another : find n. 249. In the permutations of the first eight letters of the alphabet how many begin with ahl "^ 250. Express 12345G54321 in the scale of 12, and extract its square root in that scale.. 251. (i) f(.r-5)-tV(-r-13J)=15--H19--Ja') (ii) ax -hj = a"" ) ( 8x - 3\2 _ 4a; - 5 hx-a7j = h^\ ^^"-^ [i^'i) ~ x-i 252. Find the time in which A, B. C can together do a piece of work, which (i) A can do in m days, and B and C together in ^ {m + n) days, or (ii) A can do in m days, A and B in n, and A and 6^ in ^ (m + n) days. 253. Find the coefficient of a; in (a; + 2) (x - 6) (a; + 10) (x - 5), and of a;-* in (1 + -j^a; + Ix^ + ix^ + &c.) x (1 - |a; + ]a;^ - -^a;^+&c.). 254. Divide a; * + 7/ by a;'5" + y^ and a;^ - ma-x^ + max^ - a^ by a;^ - ar^, 255. Find the l.c.m. of Gx'' - Ux" + 5a; - 3 and 9a;'' - 9a;' + 5a; - 2^ 25G. Simphfy :{ ^ wi o ^? and reduce -^ r^ .. l-^(l + 2.r)' a^+a^'b-a-h 257. Find the sum of a ac a h-2c ^ 2-2a;- 1 -, and b b{b + c) b b-c' ^/([^^y VE^'' 258. A walks at the rate of 3 miles an hour, ^ starts 2 hours after him at 4 miles an hour : how many miles will A have walk- ed before B overtakes him ? Find also how long B should start after A, in order that A, when overtaken, may have walked G miles. 259. Simplify bVSa^ + 4aV^* - Vl25a«^*. MISCELLANEOUS EXAMPLES. XV 260. If the first term of an a. p. be G, and the sum of 7 terms 105, find the common difference, and shew that the sum of n terms : sum of 7i - 3 terms :: ?i + 3 : 7i - 3. 261. Which is the greater of the ratios a + 2x:a + 3x and a^ + 2ax + 2x'^ : a"^ + ^ax + ox' 1 262. Of 12 white and 6 black balls how many different collections can be made, each composed of 4 white and 2 blackballs? 263. (\)(x-ll)(x-2^) = ia^ix)(x-l) 00 ^x-iij + z = 7,ix + y-iz = l,y + iz-x + 10^:0 264. A market-woman bought eggs at two a penny, and as many more at three a penny ; and, thinking to make her money again, she sold them at five for twopence. She lost, how- ever, 4.d by the business : how much did she lay out ? 265. Shew that (x + ar')^ - (y + y-^Y = (xy-xrY') (xy'' - ^'y\ and exemplify this result numerically when aj = i, y = - f . 266. Find the g. c. m. of 4qjV + 9«V + 2ax'' - 2a^x-AaudzJx^ + 6ax^ - ct^x + 2. 267. Find by Evolution Va + 6x to five terms, and square the result. 268. Simplify 3a-[Z^+ \2a-(h-x)]] +i-f 2x + r 14 9 269. Find the sum of ,^ ;; - + 2ic + 2 x+ 2 2 (ic + 3) lx + 2)(x + 3)' 270. A gamester loses J of his money, and then wins 10s ; he loses -J of this, and then wins £1, when he leaves off as he began. What had he at first ? 271. The sum of n terms of the series 21 + 19 + 17 + &c. is 120 ; find the n^^ term and n, 272. Divide 100 into two parts so that one shall be a multiple of 7 and the other of 11. 273. Into how many different triangles may a polygon of n sides be divided, by joining its angular points ? 274. Convert 85 and 257 to the quaternary scale ; multiply them in that scale, and reduce the result back to the denary. 275. (i) ix + ^x-l = i {Zx-i(x-l)} (ii) ax + y = x + by= ^(x + y) + 1. (iii) Zx^y = 144 = 4xy^ 276. A and B can reap a field of wheat in m days. B and C in ?i days, and A can do p times as much as C in the same time* in what time would the three reap it together ? XVI MISCELLANEOUS EXAMPLES. 277. Find the value o{ ax -^ly - c when mc - nh . Ic - na X = y> and y = y. . 7na ^ lb lb - ma 278. When ^ = 4, ic=-8, 2/ = l, shew that a^x'' -^y^ ^ {a'^x^^-y^){a-'x^ - a'^x^ y^ + 279. Reduce to its simplest form 3a V + 5a 'x - 12 a-V-8^rV~12a-^a; + 63* 280. Find the l. c. m. of ax'^-lj ax^ + 1, (crx-iy, (a'-'x + 1)", a-x^ - 1, a-x* + 1. 4 1 281. Obtain the square root of x^ - Ax + ^x^ + 4. 282. Simplify \jA0-^%jZ20+yn^, and 8vJ-i V12+4 V27-2VrV 283. Shew that the sum of the cubes of any three consecutive numbers is divisible by three times the middle number. 284. lia:b'.:c:d^ shew that 2a'' - W -. 2c^ - ZcP :: a"" + P:c^ + d\ 285. Two thirds of a certain number of poor persons received Is GcZ each, and the rest 2s (jd each: the whole sum spent being £2 155, how many poor persons were there ? 2SG. The No. of Comb"" of n letters taken 5 and 5 together, in all of which a^ &, and c occur, is 21 : find the No. of Comb" of them taken G and G together, in all of which a, 5, c. d, occur. 1 2 _ ' 3 ic+3 iC + G a; + 9 287. (i) VfcB + (1 - xy = 1 - ir. (ii) (iii) a;' + icy + 2/' = 37, x + y = 7, 288. A certain number of sovereigns, shillings, and sixpences amount together to £8 Os G^, and the amount of the shil- lings is a guinea less than that of the sovereigns and 1^ guinea more than that of the sixpences : how many were there of each ? 289. What is the difference of a (b + cy + h (a+cy+c(a+iy and (a+h) (a-c) (b-c) + (a-b) (a-c) (b + c)-(a-b) (a + c)(b-c)7 290. Prove the preceding result when a = - -^j 2) = 1, c = - J. 291. Multiply 1+ia'^ x-^ia'x^hjl-^ ar^ X + ^a"^ x"^-^ a V. 292. Obtain the coefficient of ;?,« in (1 - 2a; + Sa-'-' - Ax^ + &c.y, 293. Extract the square roots of 7-^, .064, and 31 - 10 V6. MISCELLANEOUS EXAMPLES. XVll 294. Simplify 295. Given two numbers such that the difFerence of their squares is double of their sum, shew that their product will be less than the square of the greater by the double of it. 20G. Sura to n terms — ^r + -^ + -^ + &c. and - + 1 + - + >/5, and of 25?- - ^^-xy' + j%x-Y - -'^x-'y + ^\ x''y-\ 317. Expand^/ and a/-^—. each to five terms. ^ y a-x y a +x' 318. Multiply together 1 + 2V2, 4- V3, V2 + V3, 4 + V3, 2V2-1, V^- V^. 319. Find the n*'^ term and the sum of ?i terms of the a. p. a — n a- 2n a - ^n ■ + ' + + &c. n n n 320. If the sum ordifTerencc of two numbers be 1, shew tUiitthe difference of their squares is the difference or sum of the numbers respectively. ^ 321. A servant agreed to live with his master for £8 a year and a livery, but was turned away at the end of 7 monUis, and received only £2 13s 4fZ and his livery : what was it worth? 322. How many different sums might be made of a sovereign, half-sovereign, crown, half-crown, shilling, and sixpence ? and what would be the value of them all ? ...Ix^a x-h ^ax + (a - 5)' OZO. (1) = = = T ^^ b a ab ,..^x + 14- a; + 12 1 ..... ^ ^ ^")^T2 2(":^TT9)==2 ("'^ ax-€y = = ayi hx-cxy E24. Two girls carried between them 25 eggs to mark«;t : they sold at different prices, but each received the sam^ amount upon the whole : the first would have sold them c.ll for 1a- the second for 13c?: how many did they each sell ? MISCELLANEOUS EXAMPLES. XIX 325. Write down the square of 1 - -|a; + ix^, and square the result. 32G. Divide -2x''ij-^+17x'ij-'-6x'''2-LxY hy-x^i/-^+7x'^y~' + SxY' 327. Find ^7, and thence V?, V^Ij V^i "^ V^ij 2 -*- (4 - V"^). 328. Find the value of ^ ^ — + 77— j — ^r — , vrhen x = i (a -i I). 2na-2nx 2nb-2nx^ 329. Simplify VJi^ %^ and |^-^^ V^" ooO. Find the sum of ^ . ^ - - - ^-^^-^^ . ^,-^ - ^^.-^y),. 331. If w=^+^ + rj where p is constant, ^ oc xy^ and 7'oc ^1*2/"^ and when a; = 2/ = 1, -w = 0, when a; = y = 2, w = 6, and when aj = 0, -i^ = 1, find u in terms of x and 2/. 332. Shew b}^ the Bin. Theorem that V3 = l+f - -J+||- if J + &c. 333. In how many wa}- s could I distribute exactly 555 among the poor of a parish, by giving Is Q>d to some and 2s (Sd to others ? 334. How many words can be formed of 4 consonants and 2 vowels, in a language of 24 letters, of v/hich 5 are vowels ? 335. (i)^(..i]=i.-^f .^(1.1 a - c\ xj (a — c)x a - c\ x^ (ii) 4a; - 5^/ + mz = 7x— Ihj + nz = x + y + pz- (^ 336. A boat's crew rowed 3^ miles down a river and up again in 100' : supposing the stream to have a current of 2 miles an hour, find at what rate they would row in still water ? ^ 2ac ^ , ,, , ^Va + dx + Va - bx 667. li x = ^r~z ~, find the value of ^G+^) ^a + bx-^a-lx QOQ T> 1 * -I- 1 -i Sax^ - 2a^x'' - ah 3oO. Reduce to its lowest terms Oco^x' — a^x — 1 330. Find the coefficient of a;° in (I + ^a; + ^x^ + |a;= + &c.)' 340. Fmd the sum of a^^l) - 1/ (a + 1)1) a-" - h^' 341. Simplify a'bc Var^c - ¥c V^f^W^ + a'l'c'' V2-iZar'h-''c-\ 342. Obtain the cube roots of 51.0G4811, and 1 -Ga; + 21a;'- 44a;' + 63a;* - 54 a;^ + 27a;^ 343. The prime cost of 38 gallons of wine is £25. and 8 gallons are lost by leakage: at what price per gallon should the re- mainder be sold, to gain 10 per cent, upon the outlay ? XX MISCELLANEOUS EXAMPLES. 344. Ua:'b'.:c:d, shew that 5 3^ _5 345. Expand {2fl-3 vax'f^ and {Sa-2va^Xf -, each to live terms. 346. From a company of 50 men, 5 are draughted off every night on guard : on how many different niglits can a different selection be made ? and on how many of these will two given soldiers be found upon guard ? 347. (i) ^^^^^^ = ax.V ('"> f 2' = ' f' ^ "/I i ^ ^ a + X Ixy - c {ax - by) ) (ii) 5.r - l\y^+\Zz^ = 22, ^x+Gy^+C^zi = 31, a; - y^' + s^ = 2. 348. A person, having to walk 10 mileSj finds that, by increasing his speed half a mile an hour, he might reach his journey's end IGf minutes sooner than he otherwise would : what time will he take, if he only begin to quicken his pace halfway ? 349. Divide (.r'» -I) a^-(x^ + x^-2)a'' + (4a;' + Zx + 2)a-Z (x+1) by (x - 1) a^ -(x-1) a + 3. 350. Multiply l/a"^ + ^(a^c)^ by ^a^-\/(a^c)^. ' 351. Ifa;= Vi-4^+ V (i^'" - 2?!?') h ^^n^ the value Grx'^rx^+^\q\ 352. Extract the square root of ^x^-^x^y^ ■h-\Y-^^y-l^'^y^ +^%xy\ 353. Addtogethor-J4Ll^^^^ and -,Kip^)^l_; ^ a;^ - -^ (1 + ^5) aj + 1 aj^ - i (1 - V^) a; + 1 354. Find the sum to n terms and ad ivf, of the g.p., whose first two terms are the a. and 11. means between 1 and 2. 355. "What is the least number which is divisible by 7 and 11 with remainders G and 10 respectively? 35G. A privateer, running at the rate of 10 miles an hour, dis- covers a ship 18 miles off, making away at the rate of 8 miles an hour : how long will the chase last ? 357. Expand -J 2^^ - 3 yax] ^ and { Za - 2 va-a?[ -, each to five terms. 358. In what scale will the common number 803 be expressed by 30203 ? What are the greatest and least common numbers that can be expressed with five digits in it? 359. (i)--i^''-.^- = . (ii)^ i^=<, h + x -X ^ b + X b -X X y x-a y-b (m) -- + - ~l - — — + ^ "^ a b b a MISCELLANEOUS EXAMPLES. XXI 360. -4, B^ C reaped a field together in a certain time : A could have done it alone in 9 J hrs more, B in half the time that A could, and C in an hour less than B. What time did it take them? 301. Divide sJx'Y - z -Jx'y'' - f aj VV + f ^V Vjj" V' by \/xi/ - "I -Jx^y^. 3G2. The edges of three cubes are a, h, a + h; shew that tho greatest : difference between it and the sum of the others 363. Extract the square root of a; + 1 -2^0? (1 + ^x) + 3 ^x. 364. Simplify ^72 - 3 VJ and V2^^ - V2ax^-4ax-i- 2a, 365. If oj = ^ (V3 + 1), find the value of 4 (x^ - 2x^) + 2x + 3. 366. A^s money with ^ of i?'s would be ^ as much again as before ; and if 2s be taken from JL's present sum and added to i?'s, the latter amount will be I of the former. What had they each at first 1 367. Find the value of Va +6x, and square the result. 368. If the difference of two fractions be vi?i ^, shew that m times their sum = n times the difference of their squares. 369. The first term of an a. r. is 71^ -n + 1, the common differ- ence 2 ; find the sum of n terms, and thence shew that 1 - IV 3 + 5 - 2^ 7 + 9 + 11 = 3^ &c. 370. Find the area of a court 250 ft long by 200 ft broad, (i) by the senary, (ii) by the duodenary scale. 371. (i) — + = (ii) 7ix + - = na ^ha^ ^ ao - ax be - ox ac - ax x B (iii) a;^ + 7/^ = 2a'^, x + y-.x-y.-m-.n ^72. A cistern has three pipes A^ B, and C: by A and B together it can be filled in 36', and emptied by C in 45', whereas, if J. and C were opened together, it would be emptied in 1^ hr : in what time would it be filled, by A, by B, or by all together '? Ill n 373. Find + + , when s = — (171-11+ p), mn - mz np — nz mz — mp m 374. Multiply 7?iaa;^ + {m-l)a''x'^ + (m-2) a^x ^ by a'^ \Jx*Ajx, 375. Extract the square root of 1 + m^ + 2 (1 - ni^) -^m + Sw-?7i'* XXll MISCELLANEOUS EXAMPLES. 37G. Simplify -^l- ± l/.-^, . ^ . ,__. ' x-y Y {x-yy x-y x^\lxy 377. Find a number of two digits such that its quotient by their Bum exceeds the first digit by 1, and equals the other. 378. IIow many terms of the scries -7 -5-3 - &c. amount to 9200? and how many of G + 4 + 2J + &c. amount to 14|? 379. A certain number of men mowed 4 acres of grass in 3 liours, and a certain number of others mow 8 acres in 5 hours : how long would they be in mowing 11 acres, all working together ? 380. If «, J, Cj d are in c. p., shew that (r/. + 5 + c + (Tf = (6? + Vf + (c + dy + 2(Z; + cf, 381. The No. of Var°"of n things, r together: the No., r-1 to- gether -10 : 1, and the corresponding Nos. of Comb"' arc as 5:3; find n and r. 382. A person makes 20 lbs. of tea at 4^ OtZ, by mixing three kinds at 3^ 6^, 45 CcZ, and ^os : how can this be done ? 383. (i)x(^.i^5)_Ll^^^^_.^i_i5^_|(l_.3^.)i (n) x + a + h + c = (in) - + [f] =2} a+ b + c + X ^ \aj \h J \ ay + hx = J 384. A trader maintained himself for 3 years at an expense of £50 a year, and in each of these years increased tliat part of his stock which was not so expended by -J- thereof: at the end of 3 years his original stock was doubled : find it. 385. Divide (C«=-7(jJ + 260 a'' + (5a'-3a=Z>-5a2*H36V+ (fi'-i^')'^ by {2a - h) x + cC - h\ 3SG. Find the l. c. m. of X* -(p^ + l)x' + p^ and x* - (p + ly ic' + 2 (^ + 1) px -p*, 387. Obtain the values of (i) x - ^fxy + y, and (ii) of x^ ^ xy ^ %f^ when X = y^g (4J + V"t); V = iV (H - V^t)- 388. Simphfy (a-^l)\ — i-^, + --!-„ ^ + 2 ^ -^ LI l(x+ ay (x + by) (x + a x + b\ 389. Obtain the square roots of 1V2 -2v/2 a'c ^ ^ If 2 + a + rt and — -^ cj - 2ac a/ ^ 390. The n^ term of an a. p. is ^n - J : find the sum of n terms. MISCELLANEOUS EXAMPLES. XXlll 391. The diagonal of a cube is a foot longer than each of the sides : find the solid content. 392. Find the first time after noon when the hour and minute hands of a watch point exactly in opposite directions. 393. In how many ways ma)^ £10 be paid in crowns, scvenshilling pieces, and moidores (27s) thirty coins being used ? 394. Out of 5 white, 7 red, and 8 black balls, how many different Bets of G balls could be drawn, (i) two of each colour, (ii) one white, two red, three black, (iii) three red, three black ? 395. (i) X + Vaj' - 2ax + h"" = a + h (u) = (in) y 1 + ~ + /4/ 1 — - = 1| X ^ a X- c X + x-c r cr f a^ ^ S9G. Two vessels, A and B^ contain each a mixture of water and wine, A in the ratio of 2 : 3, ^ in that of 3 : 7. What quan- tity must be taken from each, to form a mixture which shall consist of 5 gallons of water and 11 of wine ? 397. Shew that {ay -Ix)' + {ex - azy + (Iz - ct/Y = {a^ + &^ + c^) {x" + y^ + s^) — {ax + hy + czf, 398. Find the g. c. m. of Zx^ + (4a -2V)x- ^al + a"" and x^ + {2a -d)x^- {2ah -a'')x- a'5. 399. From-i {x^ + Zx'^) {x^-^x"^) take i {x^ + 2x'^) {x^ - 3a;"^), and multiply the result by 6 (1 -a;^)'\ 400. Extract the square root oix'^ ■^2x'^ + ?>x^ -2x ^ ^ x ^ - 1. 401. Multiply together n+l m-\ n-1 ''yjaTbyf'^\ ^{a + hy^^ ^{a + ly^^ y{a + Z))"2 . 402. Simplify \ l + x 4:X Sx l-x} ^ jl+g;^ Ax'' l-x^l (1^ '^l + x''^ 1+x' U'x) ' il^x^'^Ui^T^^S 403. Sum {a + ic)^ + {a^ + x^) + {a -xy + &c. to 5 and to n terms. 404. Find two numbers such that their sum, product, and differ- ence of their squares may be equal. 405. Apply the Bin. Theor. to find (l.Ol)"^ to nine places. 400. Find the least integer which, when divided by 7, 8, 9, re- spectively, shall leave remainders 5, 7, 8. 407. (i) a; + 3 = V2(a;-r3) + 4 (ii) ahx' - {a + 5) ex + c^ = «(iy*(i)"='.M=«- XXlv MISCELLANEOUS EXAMPLES. 408* A person bought 38 sheep for £57 ; but, having lost a cer- tain number, n^ of them, he sold the remainder for n shil- lings a head more than they cost him, and so gained upon the whole IGs: how many sheep did he lose? 400. Shew that (a^ + 5' - 1)'' +(a" + V' -1)^+2 (aa' + Uy = (a" + a"" - ly + Q)^ + &" -1)^ + 2 {ah + a'h'y. 410. Find the g. c. m. of xy + 2x^ - Zy^ + Ayz ■¥ xz-z^ and 2^' - ^xz - ^xy + 4s' - ^yz - 127f. 411. Find tho foiirth term of (^2 + ^3)", correct to four places. 412. Obtain the square root of l + x-^\/x (1+ ^Jx) + ^Jx(2+ j%^x). 413. If the r^^ term of a series be a?^ ^ -7*, shew that the sum of .u , »v -, .1 ^ N»vi m^+mn + n'* the m*^ and n^^ terms exceeds the (?7i-f??r'' by ; r a. mn {m + n) 414. If X-' = (a-c) (h- c), y'^ (a -h)(h- c), z' = (a-h)(a- c), find the values of a; — y + ^ ^^^ ^^^^ ~ ^^Z/ + ^^^* 415. If P, §, i?, be the jp"*, q^^ and r*^ terms of any h. p., shew that (p-q) FQ + (q-r) QE + (r-^?) EP = 0. 41C. Two parcels of cotton, weighing lbs and 16 lbs, cost lis 6d and £1 Os Ad respectively^, and (he charge for carriage was proportional to the square root of the weight : how much per lb. was paid for the purchase of the cotton? 417. If a : Z> :: & : c, shcw that a + h-.l) + c::a^ (h-c) :h^ (a^h), 418. Find tho least number which being divided by 2, 3, 5, shall leave remainders 1, 2, 3. 419. (i) (x-1) + 2 (x-2) + 3 (a;-3) + &c. to six terms = 1 4 .... 2x(a-'X) , ..... X y ^ X z 420. A square court-yard has a rectangular walk around it ; tho side of the court wants 2 yds of being six times the breadth of the w^alk, and the no. of sq. yds. in the walk exceeds by 92 the no. of yds in the periphery of the court : find its area. ANSWERS TO THE EXAMPLES. 1. 1. 48. c. -1. 2. 1. 11. 6. -64. 3. 1. 25. 6. 22. 4. 1. 46. 6. 135. 2. 12. 7. -178. 3. -8. 8. 150. 2. 1. 7. 16. 3. 0. 8. 264. 2.-15. 7. 7. 3. 12. 8. 13. 2. 24. 7. 8. 3. 35. 8. 120. 4. 1. 5. 106. 9. 450. 10. 192. 4. 94. 9. 5. 4. 6. 9. 15. 4. 10. 9. 384. 5. 89. 10. 3. 5. 21. 10. 4. 5. 7200. 10. 4. 5. 1. 15a + Zh-Gc+ ed, 3. 23a^-26ab+Uh\ 5. 5aj' + 50a;V - Uxi/ + 4yK 7. - 9a;* + 2ax^ - ZWx + 16^^ I 9. Cx^ + 4f + 2" - 24a;y;?. 2. 14a; - 9y + 10« - 12. 4. Ghij - 7cz, 6. 2x' + 2if + 22'. 8. a* + b^ -\- c* +6a6(?. 10. ic* + y* + 2* I 6. 1. « - 35 + 3c. 2. - 2a;' - 7a;y + Zy\ 3. 4aa; - 95y + 2cz. 4. 5a;' - 5a; + 5. 5. 7a' - 3a + 4&' - 7ah + 2c' - CJc. 6. - a;' - 6a;'y - 2?/' + 6 - 3a5' - 4y' 7. 3a;' + 13a;y - ^f ^ 16x2 - 13y2. 8. x^ + xij + y\ 9. 3a* - 4a'b - 4a&» + 25*. 10. 0. 7. 1. 4a -4a;. 2. 4a» - 4a'c. L4. 2aa;' + 25y' + 2c2'. 5. a' - 36' + 3c'. 7. 0. 8. "Zx-y+iz. 9. 8a; -8. 3. a;'-3y'-32». 6. 2a5 + 45'. . 10. - 4c + 4d!. O) ANSWERS TO TIIE EXAMPLES. . 1. (2a-h)-(Sc-^d)-(2e-2f),(2a-h-Zc) + (U-'2e+Zf), 2. -(& + Zc)+(4cd-2e) + (3/+ a%-(b + 3c -4^ -(26 -3/- a). 3. - (3c - M) - (2e - 3/) + (2a - h\ - (3c - 4^ + 2e) + (3/+ 2a-hy 4. (U-2e) + (3/ + 2a) - (Z> + 3c), (4d-2€ + 3/) + (2a - 5- 3c) 5. _ (2c - 3/) + (2a -b)- (3c - 4fZ), - (2c - 3/- 2a) -(b + Zc-id) G. (3/ + 2a) - (6 + 3c) + (U - 2c), (3/ + 2) - (3c - 4d; + 2c). 7. {2a-(h + Sc)\ + {id-(2e-^f)\. 8. -|6 + (3c-4fZ)}-{2c-(3/+a)}. 9. -{3c-(4^-2c)} + j3/+(2a-&)}.10. { 4tZ-(2c-3/) | + { 2a-(& + 3c) 11. -j2c-(3/+2«)}-lZ>+(3c-4cr)}. 12. {Sf+(2a-b)]-{Sc-(4d'-2€)\ 1, (a^b + c) x^ - (b - c + d) x^ ^ (c + d + e) X, 2. 2(ax - by). 3. (a + b) x^- (a - 55) a??/ + (^^ - c) i/'*. 4. 2 (aaj + cy), 2b (x + y. 5. ^(a-bb)^ + (2a + Zb + c)y^ (a-4b-c) x ^- (a~Zb- 2c)y, (b-c)x -\- (3mnx^ - n^a;* ; a'' + a(m+ 2n) x- \ a (m + /i) - 2mn ] x^- (in? + 2n^) x"^ + mnx*, 18. a'x* - a*-* (5 - c + cZ) x'^y - (abc - abd + acd) xy'^ + bcdy^. 19. 4x* + 6 (w-7i) a;'- (4m^ + 9wi/i + 4ii^) %^ + Cm7i (w-;?) a;+ 4w'/i*. 20. a;*-(2a^+25» + a5)a;='+ (aH a^5+ a'b''^aV^V)%'-(a^l)a'b\ (2) ANSWERS TO THE EXAMPLES. 11. 1. 2. 4. 5. 7. 9. 10. 11. 12. 13. 14. a'''-2ax+x\ l + 4ic' + 4x*, 4a* + 12a* + 9, 9a;' ^24ji/ + 16y\ 9 + 12aj + 4x\ ix^ -12xy + Oy\ a* - Wx + 9a'a;-, V'x'' - 2hcx^y + cV^/'. 3. 4a' - 1, 9'- 2hc + c». 4a' - &' + G5c -9c', - 4a' + 12ac + &' - 9c'. 4a' - &' - G5c - 9c', - 4a' + 4a5 - 5' + 9c'. a' + 2ac + c' - Z;' - 22>cZ - ^', a' + 2acZ + eZ' - 5' - 26c - c\ h^^2lc +c'-a'-2a^-^'. a'+ 2ad + ^'-46'+ 12Z'c-9c',9c'+ 6ccZ + ^-cZ'. 12. ^. &c', 5^2/', -35Z?a;. 2. 3a;y-2a;^+32/2, -a-/>' + 7a&c'-4c*, ■ -3aa;+36y — . a 2??i 4??i' 3?^ 3/2,' 5a 5 2Z»' 2^'"^^2^"'«;^ n_ 3^' 3m' 26' 4. X ■\- 5, m'^'^m + 3. 5. 3a-26, 3a; + 2y. G. 2ab-U\ 7. a'-2a6+26', 2a;'i/'+2a;y+l. 8. 'a;*-2a;V + 4a;'y'-8a'i/V16i/*. 9. l-2a;+3a;'-4.t;«+5a;*. 10. a;'+ 2a;y + 3?/', m'-2m + 3. 11. a'+ 2a'& + 3aJ'+ 46». 12. x' + 2a;^ + 3a;' + 2a; + 1, a*->2a'6 + 3a'6'-2a6' + h\ 13. 1. a;'-^a;*'+ ^. 2. as' + 6s--c. 3. y*-(ni^ 1) y^-(m- 71^ 1) y' - (w - 1) y + 1. 4. a + & - c - xy (ay-x^) (ay + a;'), 2a¥c (a -2c) (a + 2c). 3. x^ (ox - a) (5a; + a), a* (a - ^¥) (a + 3i'), (2a;- 3) (4a;^ -^ Cx + 9), (a - 26) (a'^ + 2ah + 46^, 4. (X + 2) (a;* -2a;'+ 43;^^ -8a; + 16), x* (a + 3a;) (a^-Zax + dx") (2x' + y') (4a;« - 2a;V' + y% (ah' + c =) (a6^ - c'^) (a'6« + c*\ ahc (a + cy. 5. (3a;-l) (3a;+l)(9a;ni), (a;-2) (a;+2) (a;V2a!+4) (a;'~2a;+4). x^x-hy, x''(x-ay(x + ay, 6. (4a; -5) (2a; + 1), (a + 36) (a-&), 7 (x-y)(x + y), 7. (a;-2/)' (a; + 2/)', (c + a- 6) (c-a + 6), 8a6. 8. (x + 2/)', ^/i (m-n), 6& (a- 6). 9. 2(a; + 2/) (^x-y\ 2 (x-y) (^-x), Ay(x + y), 10. (a + I) (a'' + a 6 + 6^^), (a-hy, 0. 16. 1. (a; + l)(a; + 5), (a; + 4) (a; + 5), (a; - 2) (a; - 3), (a; - 3) (a? - 5), (a; + 1) (a; + 7), (a;-l) (a;-9). 2. (a; + 3) (a;-2), (a;-3) (x + 2), (a;-3) (x 4- 1), (a; + 5) (a;-3), (a; + 8)(a;-l), (a;"9) (a; + 1). 8. (2a; + 3) (2a; + 1), (4a; + 1) (a; + 3), (4a; -1) (x + 3), (2a; - 3) (2a; + 1), (3a; - 2) (x + 2), (3a; + 4) (2a; - 1). 4. (4a; 4- 1) (3a; -2), 2(6a;-l) (a;-l), (4a; + 1) (3a; - 1), (x + 4)(x-Z), (3a;-5)(a; + l). 5. a- (x-a) (x-2a\ a (a- 3a;) (a + 2x), ah (3a- 25) (a + h\ (4a' -a;^ (3a' + a;'). 6. xy (2x ^y)(x^ 2y\ Sy' (Zx + 2y) (x - y\ a' (Zax - 1) (2aa;+ U a;' (26 - 3a;) (3 6 -f x), (4) ANSWERS TO THE EXAMPLES, XV. 1. 5. 2. 2. 3. 3. 4. t-. 5. ^i. 6. '^-t m-n 7. 2. 8. 1. 9. 4. 10. -1^. 11. -4. 12. |. 13. -f. 14. ^l n 4. 5. 18. 1. 42. 2. 12. 3. 12. 5. 7. G. 4. 7. 5. 8. h 9. 7. 10. yV (25a - - 18^). 11. 7. 12. -8. 19. 1. 4. 2. 2. 3. 18. 4. 8, 5. -a. 6. 6. 7. 4. 8. l-a. 9. 7. 10. a-w. 11. 10. 4. 7.9. 12. 2(a + c). 20. 1. 12. 2. 9. 3. 120. 5. 35, 13. 6. 513, 46G i. 7. 15. 8. 31, : 18. 9. 15. 10. 90, 60. 11. 24 ft. 12. 16. 13. 37, : 50,20. 14. 20. 15. 41. 16. £5. 17. 88. 18. 85^, 35s. 19. £36, £12, £16. 20. 5. 21. £45, £57, £63 , £65. 22. 15, 5. 23. 9S§ miles from L, lOf h. 24. 22, 7, 12 gals. 25. 1 h 20' i from B's s tartiiig, (j\ milej 3. 26. 3000. 27. 3s, 5^, 1& 1. 28 . £189. 29. 8. 30. 25. 21-1. 4/y.V)4 _ 97/y.67i«^i3 81«V/ x''y 2. 4. 5. 6. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 32 x" + ^x" + 12a; + 8. 3. ic* - 8.c^ + 24a;^- 32a; + 16. x'' + 15x* + 90a;' + 270a;= + 405a; + 243. 1 + lOo! + 40a;' + 80a;' + 80a;* + ^2x\ 8w' - 12w^ + 6m - 1. 7. 81a!* + 108a;' + 54a;' + 12a; +1. 16a;* - 32aa;' + 24a'a;' - Sa\v + a\ 243a;^ + SlOax' + lOSOa^a;' + 720aV + 240a*a; + 32a*. 64a' - 144a=& + 108a&' - 275'. a^x^ - Zct^x^if + 3aa;?/* -y^, a'^x'^ + 4a'a;^ + ^a/x^ + 4aa;'^ + a;*. 32a^m' - 80a*/?i« + 80a'w^ - 40a'm« + lOar^i' - 'm}\ a' - 3a'?>> + Za?c + 3a5' - ^dbc + 3ac' - 5' + Zlyc - 35c' + o>. 1 - 3a; + 6a;' - 7a;' + ^x" - Zx^ + a;«. a' + 3a'6a; + 3a Q? -v ac) X' + (6ac + 5') 5a;' + 3 (ac + 7r) ca;* + 35c'a;» + c'a;^ 1 + 4a; + lOa;' + 16a;' + 19a;* + Ux^ + lOa;" + 4a;^ + x\ l+5a; + 5a;' - lOo;' - 15a;* + lla;^ + 15a;« - lOo;' - 5xU5x' - x'\ 1 - 6a; + 15a;' - 20a;' + 15x* - Gx^ + x\ (0) ANSWEES TO THE EXAMPLES. 20. a*-8a^6 + 4a*c + 24a''^'-24a^5c + (ja'c^-Z2ah' + iSal'c-2ial(^ + 4ac^ + 165'- 326V + 24&V - She* + c*, 21. 1 + lOaj + 25x^- 40a;» - 190a;* + 92^» + 570aj« - 360a;^ - 67 5x* + 810a;'' - 243a;^°. 22. 1. 1 + 2a; + Sa;'* + 2a;^ +x\ 2. 1 - 2a; + 5x* - 4a;' + Ax\ 3. 9-12a;+10a;^-4a;'4-a;\ 4. a*-4a^h + l0a'h''-12ah^+9l\ 5. 4aj' + 92/' + 162* -123-2/ + lQxz-2ij/s, 6. 9a''a;' + 4&*2/' + ^^"^^^ + 12a5a;2/ + Gacxz + Ahcyz, 7. 1 -4«a; + 2a-x'' + 4a V + a\^*. 8. 4a*-4a^-'ra' + 4a + 4. 9. 1 -2a; + Sa;'* - 4a;^ + 3a;*-2a;'' + x\ 10. 1 + 6a; + 15a;' + 20a;» + 15a;* + Ca;' + x\ 11. x'-4x^ + 10a;* -4a;' -7a;' + 24a; + 16. 12. 1 + 4a; - 2a;' - 4a;' + 25a;* - 24a;* + lOa;". 13. a' -- 4a'h + 8^*6' - lOaW + 8a'5* - 4aV + h\ 14. a^ - Sa'x + 28a V - 56a'a!' + 70a*a;*- 56a'a;' + 28a'a;«- ^az'+x^ 15. l-4a; + 10a;'-16a;' + 19a;*-16a;'^+ 10a;«-4a;^ + x\ 16. a^-4a'x + 6aV-8aV + lla*a;*-8aV + 6a'a;«-4flx' + a^. 23. 1. ± 2a5'c', ± 7a;'i/'2, ± 10a*&V. 2. 3aa;'?/' 7a;?/' Sa;^* "^ 55 ' ^ 8a ' ^ 4a6' * 3. a'a;'y 2a2/' 45'c' 6aZ)C* 2 ' 3a;' ^ 5a* ' 7 ' 4. ^2a;2/' ^ 3a5V 2a5' 2a;'2/ 5a' ' 4a;* ' c' ' "^ 3s» ' 24. 1. 2a; + 2/, 5a - 36, 5a;' + Sx^/. 2. 7a5 - a', 4a;?/ + 52/2, 5a'6c + c*. 25. 1. l + 2a;+3a;'. 2. 3a;' + 2a;+3. 3. 3a+25 + c. 4. x''^xy+4i/\ 5. 2a'-3a+4. 6. 4a;'-2a6 + 26'. 7. a;'-2a;' + 3a;-4. 8. 3a-6 + 5c + ^. 9. x^-2xhj-^2xy''-y\ 10. l-3a;+3a;'-a;3. 11. 2-3a-a' + 2a'. 12. p + qx ■¥ rx^ -i- 8X*. 13. 1 -X, a -2. 14. 2a- 36. 15. a; -2. 16. a -I. (C) ANSWERS TO THE EXAMPLES. 26. 1. 421, 347, 69.4, 737, 1046, 4321. 2. 2082, 20.92, 1011, 20.22, 129.63. 3. 3789, 75.78, 15.156. 8642, 2211. 4. 4.164, 8328, 2568.2, 11307, 31230. 5. 4.044, 8088, 5055, 6633, 15165. 6. 1.5811, 44.721, .54772, .17320, 10.535, .03331, .06324^ .07071. 27. 1. « -f 2y. 2. a - 3. 3. ic + 4. 4. 2a - 3&. (). a + 85. 6. 2x - 7y, 7. m- Anx, 8. ax - 5lx. 28. 1. a^ + 2a+ 1. 2. x"" - Ax + 2, 3. a'' - ah + l\ 4. 05- - Aax + 4a\ 5. 2a;- + Axy - 3y\ 6. x^-x'' + x-l, 7. « - & + x^y\ ax'y - axy'', ah'' - ad\ 2. 120a*2>^ lOa'^i!^^ ISOOaV. ^_ 8. 6 (a'' - ¥\ 12a (a' - 1), 120xy (x' - j/^). iH 4. 24a'5^ (a'^ - ¥), Z6xy' (a;^ - y'*). I 36.1.^,^, 2. ANSWERS TO THE EXAMPLES. m 7a? a' - Zab a ' a^ ' 3 (w- 2ar) ' 5a ' 26 (a + 2V) ' (a; - 2^^) ' w + 7i ' a + 6 +c ' Ax-ly ^ c c + y ex + d . a; - 1 a;* + «' a* + a'J' + J* a;' - 5aj a -h a ' a;* ' a^ + h"^ ' a; + 5 '\ a + & * 5. r . 6. . /. . 8. ^- . 2a -Zx a; + 4 7.^-2?/ 2a + 3a; ' ' x^^2x + 1 ' bx"^- Zxy + 2y* ' -^ ba^ {a-^x) -^ a;'+4a;+4 -^ a;'+a!-2 a; (a^ + aa;+a;^) a;'+a; + l aj^' + oaj+o - - a;^ - «aj + a^ , . 3aa;' + 1 15. — — . IG, 4a^a;*+ 2aa;^-l 37. 1. 3a;~6+~-., a-2a; + — ,2a;4-6+-^ a; + 4 a + X x-o 2a-3a; + --^ , 12a; + 3 + --^- . oa-x 4a; - 1 a;(a!'-2 a; - 3) a^+ 2x^ x^ + x y + y* a;-2 '« + 2a;' a; + a oo 1 ^^^> ^^y? ^^^ ^^^^^5 4Z>2/^, 3rt2' 2. «6c ' 12a6o ' 40Z>^r7/, 45a?>^>c^ 48a-62/^ 50a^ry^ G0a=6^ aV-&V, «7/^_+^V ^^ ^ 2r^a; + ar*. a* - 2aa; + a;* 5^!_Z ^^^iJ^L Q ^-^N a + a;, 2« ao 1 ^* + ^' 3a'--a5 + 2Z>" 25a-20Z> * •2(a + J)6' G(a-2»)Z> ' 12 * ^ «& ^IjL^ ^llJ*' a^-aJ + Z/'' 'a-6' a^-J«' a^^^i^' ~~^F^b^ ' 2 g + ^a ; 2a'^-2ah + 2h^ 2ah x-y (8) ANSWERS TO THE EXAMPLES. 4 ^^ 5 ^ • 6 ^ 7 ?^? -. «^ + a;^ -^ 2aj*+4a;y-22/* a' + a;' 11. —r— r-. U. T • lO. —- r . a^ (a + a; j x*-y* _ ^^ (a; - a) y - , a? - 3a;^ + 3a;' ^ . 1 + 2aj + 3a;' 14.—^—. lo. — -j -^— . 16. —-m ir* ' b(a-x)' ' Z> a* - a;* a^ + 2a''a; + 2fl5a;' + a;' 2ax'^ (x - y) a^x ' {a-'X) {a? - aa; + a;^) ' c ^ ab X a^x (ax - 1) . a' + 6' _ a; - 1 (a^-l'^ )b Za x^' + b^ ^' a' '2b' ^' x-b • 4- 3a; 3a; 6-2^ 18a; -fl4 27 -4a; 12a;-40 10 ' 15 -2a;' 2a;+5' 21 ' 2 (4a; -9)' 33-2a; ' 1 0- 13a; 20 - 3a; 14 -20a ; 6 ' ^ ^' 2a;-25' 9(a;+l)* 1 J^ a^ + a;'^ , _4 a; (1 + a; + a;') a^ + 5tf ace ^ abc cd-be' ' a + b' 4. ace + Jc^ - b'^e b{ae-c') ' 5.1 c «4- 7 ¥h '' a/-^2bc-bfg' Ua 25 {a + 1)' 0.4. 2. f. 10. 5^. 43. 1. 1. 3. 1. 4. - f . a'-a6 + 6'' 6.1^. ^ b'-a' |. 8. -1. 9. f • 10. -(a+c-b). 11. 1. 12. Za, 15. 2. 14. - 2^ '-^y '»•-■!■ 16. 8. 17. 4. 18. Sf. 19. 11 20. 20. 21. 8. 22. 14. 23. - 107. 24. i. 25. -J. 26. 0. 27.-^. 28. H. 29. 10, 30. 4. (9) ANSWERS TO THE EXAMPLES, 44. 1. 144sq.yds. 2. 75 gals. 3. £36,£1G, £8. 4. 25 of each. 5. £210. C. 22. 7. 2450, 19G, 98. 8. £200. 9. 42, GG, 1G2. 10. C9, 81. 11. £5 Ss. 12. 84. 13. 15 ft. by 11 ft. 14. is Sd, 15. 20 lb, 15 lb, 15 lb. 16. 22, £5. 17. £48. 18. 3J days. 19. 75. 20. £125. 21. 1504. 22. 1540, 880, GIG. 23. 23» days. 24. 37i', 25'. ' 25. 7h. 5j\\ Gh. IGyV- 2G. 13. 27. 110 yds. 28. £72, £108. 29. 13 J days, 2f days. 30. £32. 31. 10 lbs. 32. 18, lOi, Gf days. 33. 40'. 34. 45 4itZ, is lO^d, 35. £48, £32, £4, £G5. 3G. 6§ oz. 37. 30 hrs. 38. 40. 39. 90, 120. 40. G54. 41. 12 gals. 42. 25s, 20^. 43. 120, 104. 44. G2, 93, 155. 45. 76, 30. 46. 12, 21Js. 47. 36'. 48. 21 p\ hrs, lOiJ hrs. 49. 189. 50. 1 J hr. 45. 1. ic = 1, y = 1. 2. a; = - 6, 2/ = a + J. 3. a; = 5, y = 2. . a- 1)^ h - a* /. x=l, y-2. 8. jr = — - y =. — 9. ic=6, 2/=7. 10. aj=l, y=7. 11. a;=10, y=2i. 12. ic-144, y=216. 13. ic= — ^~j — -■ , y= — — — 7— . 14. ic = 2, 2/ =x 3. /ir/7-LA/. > ^ ad + be ' ^ ^^•^-~^-2^-'2^=^-26- 1 Q _ ^^^ { J<5 - 6^ (J + c) } _ «Jc { h (a + c) - ae\ 19. ic=G, 2/=8, 20. ii'=3,y=2. 21. a;=5,y=2. 22. ir=-2, y=-J. 23. a;- 7, 7/ = 9. 24. aj=5, y = 5. 25. a; = 21, y = 20. ' 46. 1. a; = 1, y = 2, 2 = 3. 2. a; = 7, y = 10, 2 = 9. 3. aj = 5, y = 6. 2 = 7. 4. a; = 4, y = - 5, 2 = 6. 5. a; = - 5, y = 6. 2 = - 2. G. x==i(l) + c- rr), y = i (a + c - Z>), 2 = ^ (a + J - c). 7. a; = 1^, y = 2f, 2 = - 12. 8. a; = 2, y = - 3, 2 * 4. 9. a: - 12. y = 12, 2 « 12. 10. ic = 5, y « 7, e «= - 3. (10) ~" ANSWERS TO THE EXAMPLES. 47. l: tV 2, 21, 40. 3. 5^, 3^. 4. £24, £12. 5. 17 yds, 13 yds. 6. 48. 7. 108 sq ft. 8. G40, 720, 840. 9. 18^, 90. 10. 4 hrs, 6 hrs. 11. 75. 12. 40, 90. 13. 20, 30, 60. 14. 222. 15. 30, 50, and 70, 20 : or CO, 20, and 40, 50. 16. 24, 72. 17. 72, 60. 18. 12, 12. 19. 34. 20. 3L 21. 12, 10. 22. 255. 23. 3^. 24. 39^, 2b, 12^. 34^2 224 1 5 a5^+ a^-va^h^ <- ab^ , ah^ + ab^ + ah^+ ah\ a"- + 2 J ' + 3c'''+ 4a&^ + ba'^l; a^l}-^^ Za'h- + 5^5^ + 4a- »5 + 2a-»6»: 12 3^ _5 13 5 42 ia^h-^'c-^ + 4:a~^b-'c^ + 2ar'bc + \a-'b-'c-\ \abc^ + ^aWc" + fa'^o'V? + 5a-^Z»"*c; and 3a-»6''c=' a'^Jc-^ ab-'c-' 'dabc ' 1 2 3 5 3 2a-'b-'c^ Za'^b-'^c-'^ Aa^h^C^ ab^c'^ V« + 2 V«' + 3 Va' + i\/a + ya\ \la y^b 2V«? V^ Vb^ \/b' "■ 2V(j "" 3 v^^' "" TvTt ■" 5Va'* 5c_ ac_ 1 ^ + 5a 2^,2 « ct 252, 16. ab-' + 1 + «-'&. 17. a^ - fa^ + 3 - 6a'K 9a"^. 18. a"*ic^- 1 + a V*. 19. iC2/"^ - «"*2/^- 20. 2x2 . Zy^. 60. 1. 64* 8r^, a)\ (#, (SA S\ 2. 25* (-#, (K)* (faO*, {i(a' + 2a& f 5')}^; 125^ (i|^)5-, (^V«)t (-V-^'A U(«'+3a'& + 3aZ)«+&«)i* 6561"*, (^-v^«-)"*, K)'*, (^'y*. 4. V125, V3, Vl'2, V?5 V^, V320. 5. V54, V25G, V2048, ^3, VI) Vt^- 6. Via, VOSa'-'o;, , J a/ -, , -^ 7 — Ti' * /4^j2 » /2fi 8. 3V5, 5V5, 3GV3, SyS, 18 «/2, iV^, V12, VH 6. 9. 4 V2, 8 V2, C V48, f V2, /y V2, t V2, ? V21, I V^SO, V375. 10. 2V3, 15 V3, iV3, AV^, >V3, W3. 51. 1. V108, V112 ; V81, V^^ 5 ^120, V128, V135 ; V125, V121 • Vi, V-}; V125::^144,V1G2. 2. V2, 3V5. 3. -2/ V3, 9 v'9. 4. 24v3, 120v3, 3G. 21G 'yCj. 288 "V72. C. 6-J(j, Gv3+3v30. (;12) ANSWERS TO THE EXAMPLES. 10. 11. 52. 1. 5. 9. 63. 1. 6. 64. 1. 7. 13. 65. 1. 4. 66. 1. 4. 67. 1. ' 4. 68. 1. 5. 9. 59. 1. 5. 60. 1. 3. 6. 16. 8. i (V2 + V3 + V5), W^ + i V32 + i VISO. J (2V2 + V3), V5 + 1, V5 - V2, 4 + V2, i (7 + 3 V5), ^V(7V14-13). ^V (58 + 8 V7), tV (8 V5 + 23), J (3 - V6). 1 (* + V«'-ic^ 2Va'-a;^ 4a;V^^-l. 12. 2x\ l-x'' V3 + 1. 2. 3 + V2. 3. V5-V3- 4V2~3. 6. iV^-l- 7. 2-^v3. V2fl. 10. V^-l. 11. i(V5 + l). 4. 2V5-3V21 8. fV2-i. 12. V^+iV3. 2. 2+./^r 3. 2^^ 2(^^)) • r^^2vs* 4. 7. «'-5. 8. T>. 9. 26 5-2' 10. 5. a ?> (Z>-2 a) '6b-2a ' ±2. 2. ±3. ±|. 8. ±5. 3. ±1. 9. ±3. 4. ±i. 10. ±5. 5. ±i. 11. ±2. 12. ±2. 6. ±2J. :.V3. U.±%i ^r.^ ,^ajc^l-nc.^^ ,, (n-l)« 2 (c^ + d') 16.'- V2?2-l 4, -2. 7, 5. 1, -8. -1, -12. 6, -5§. 14, -lOf. 2. -1. -9. 5. 8, - -40. 2. 17, -4. 5. 1, - -20. 2. 6, - -4i. 5. 12, -12tV 10, 2. If, 2, if. Ih 2. 3, - 1. 3. 2, - f G. 7, - LV. 7. 2, i. 10. 3, -|. 11. i(27±V57), 11, -13. 2. 5, 5f. 3. 5, 21. 6, 3yV 6. 5, - 41-. 7. 1, 10§. 3. 20, - 6. 6. 10, - 110. 3. -5, -20. 6. 25, - 130. 3. 8|, -10. 6. 13, - 113j. 8. i(-9±3v3). 12. 2, - 3. 4.7, -1?. 8. 3,-8^. aj^ - 4aj - 21 = 0. lGaj*-153aj^ + 81 = 0. 2. 6^^ + 5.2; - 6 = 0. 4. iC*-6^'+lla;^-Gaj = 0. 6. 4a;* + 3a;*-17ic'-12a!^+4x = 0. ANSWEES TO THE EXAMPLES. 61. 1. a;=7, 3/= ±4. 2. x=4. y=-3, ) 3. x=4, 2/=3, ] x=-S,y=4.] x=^lVj,y=m\ 4. a;=8, y=2i, ) 5. ir:=6, 2/=5, ) 6. ic=5, y=3, ) = -8.^ iC=-6, 2/=-5.^ a;=J,y=-lJ.S =,3 ) «=i A, y= - t'i^. ) x= - l-fV, 2/= -2tV 5 9. a;=4, y=2,> 10. a;=10, y-15, ) fl;=2,y=4.5 x=-lOly=-l^.\ x=-2ly^ 7. ic=5, y=,3 ) «. a;=3, ^=4, ; 11. aj=3, y=2, ) 12. a;=5, y=4, ic= - 2, y= - 3. ^ a;=4, y=5. ^ 13. a;=i{a±V26^£^},) 14. x=i {± V^^Th^ + h\,\ y=i\aT V26-^- a'}, t y= i j± VS^Tj' - 5}. ) 15. a;=8, y=l, l,y=8.i 16. x= ± y- ^=\y=^A v^v;p'^ v^^ 62. 1. ± 12, ± 15. 2. i 10, ± 16. 3. ± 4, ± 12. 4. 15 yds, 25 yds. * 5. 8 and G, or 56 and -42. 6. 27 yds. 7. 4550. 8. 24 or - 3. 9. 4 or - 1 J. 10. 40 yds by 24. 11. 9, 12, 15. 12. 12 and 7, or -9i and -14^. 13. 10 yds, 16 yds. 14. 3 in. 15. 26 ft, 38 ft. 16. 16. 17. 49, £3. 18. 4 ft, 5 ft. 19. £60 or £40. 20. 10, 15. 21. £275, £225. 22. 25, 20. 23. 264. 24. 2, 5, 8. 63. 1. aj = 3, ) ic = 23, ) a; - 31, 2, ) a; = 30, 15, | y = lj y= 2,f y= 2,5,f 2/= 1, 8.r 2.05 = 5,) x= 5,) 05=49,) 05 = 11,) 33559, 4. a; = 5, y = 3, 2 = 6. 5. 5. 6. 4, 2. 7. 4. • 8. A gives 14 pieces, J5 9. 9. 8 ; 16. 10. 21, 12. 11. 4 12. 59. 13. 8 h. g. and 3 h. c. 14. £13 Is or £42 1«. 15. By paying £5 and receiving 4 louis. 16. 3, 21, 16, or 6, 2, 32. 17. 503. 18. 2s, 45, 5s. 19. 209. 20. 301. 64. 1. 32, 272. 2. 39, 400. 3. 63, 363. 4. 694, 3475a 5. 9. 16. 6. -1,0, 7. -28. 8. -275. 9. 16i. 10. -84^. 11. 336|. 12. -84. (14) ANSWERS TO THE EXA3IPLE3, 65. 1. 12. 2. 5. 3. 20. 4. - ^^. 5. 5,8,11, 14; -2,-6,-10, -14. 6. 3§, 4i, 4f , 5f , 6, 61 7 1 7^, 8| ; -11, -9, -7,-5, -3, -1, + 1. 7. 4, 15, 26, 37, 48, 59, 70, 81, 92, 103 ; - 2i, - 2^, - 2i, -2,-11, -H,-li,-l. 8. -2, -li, -i, i, 1, If, 2i, 3i,4; -2|, -2^, -1|, -1, -la, f, lf,2. 9. 5,7,9. 10. -3^,3^,10. 11. - i, i, li. 12. n\ 13. 300. 14. 78, 90. 15. £5 35 ; £135 4^. 16. 5 miles, 1300 yards. 66. 1. 64, 85. 2. 1280, 1705, 5. 4096, 3277. 6. 9. 4AV 3. 96,189. 4. -256,-170. 10. 2|ii. 512,-341. 7.j\%, 12. 72^V 67. 1. 8, 7. 3 2. H. 8.1. 3. J. 4. A. 9. 1/,. 10. H. 6.4. 11. lOJ. 12. -2jV 0. ^. 68. 1. 4. 2. 3, 6, 12, &c. 3. 4, - 8, 16, &c. ; or — t - |, -YS &c. 4. A- 5. 3, 15, 75, 375 ; or - 2, 10, - 50, 250. 6. ±4, 8, ±16; ±2, 8, ± 32. 7. i -1,1,-1?; - 1, H, -j2J, 3f. 8. 2 + 1 + 1+ &c.; or4-$ + |--&c. f-&c. 10. l^or6J. 9. 3-1 + 69. 1. -4,0), 4, ...f, f, ^;i^,|f,lr^^,...15,-7i,-3; A, J, J, ... If, 2i, 31. 2. 2|,3;li,f,JL,^^VV,rV 3.24. 70. 1. Si, 3, 2i«. 2. 2^, 2i, 2yV 3. 8 and 2. ^ 4. 1 or 16. 5. 8 and 2. 6. 9 and 1, or f and - 7J. 71 1 JLS 1« 158 161. Jt95 735 847 4aJ 2. X + y x^ + y^ ^ x" - \\x + 28 6. r- (15) a -1) A 4 5. a' + d^x^ + .t* : 7.1. 8.^;^^. 9. 6^ (a - h) ad — be d* AKSWEES TO THE EXAMPLES. 72. 1. 10,4^,2^. ^ 2. 9,4i,iJ. 3. G, If, 1|. 13. (]) x = h (-^7^) ; (ii) x = a+hor I (a-l) 5 (iii) a;=l, y=4 (iv) a; = ± 9, y = ± 3. 14. 3. 15. 25, 20. 16. 8 : 7. 17. £200, £150. 18. 300. 19. £125, £166f, £2081; £212J|, £159^^, £127J|. 20. G. 73. 1. icy=i|(a;'+i/). 2.2. 3.y = ^-^. 4. 2/=3ir+2aj«+fl;». 5. y = a;^ + 2aj+3. Q»: z = -^^x + lx\ S. iACBC, 74. 1. 720, 720. 2. 5040. 3. 6720,45360, 3326400, 19958400. 4. 12600. 5. 9. 6. 1120, 831600, 336, 34G50. 7. 6. 8. 7. 9. 15. 10. 3628800. 11. 0. 12. 4. 75. 1. 126, 84, 36. 2. 330, 330, 11. 3. 3003, 455. 4. 6. 5. 63. 6. 210, 84. 7. 50063860, 5006386.* 8. 18. 9. 12. 10. 11. 11. 12. 12. 43092000. 76, 1. 1 + 6a; + 15a;'* + 20a;* + 15a;* + Gx"" + x\ 2. a' + 7a'x + 21aV + 35aV + 35aV + 21aV + 7ax* + x\ 3. 1 - 8a; + 28a;' - 56a;» + 70a;* - 56a;^ + 28a;° - 8a;^ + x\ 4. a'-9rt«a;+36<:iV-84aV+126aV-126aV+84aV-36aV + 9ax^ — x^, 6. 1 + 12a; + 66x^ + 220a;' + 495a;* + 729a;* + 924a;«+792a;U495a;» + 220a;' + 66a;^'' + 12a;'^ + x^\ 6. 1 - 20a;+180a;'' - 960.i;^ + 3360.?;*-8064a;*+ 13440a;' - 15360a;^ + 11520a;» - 5120a;' + 1024a;^''. 7. a' - ISa'x + 135aV - 540aV + 1215aV-1458«a;*+729a;«. 8. 256a;« + 1024ara;^ + 1792a'a;« + 1792aV+ 1120a V+ 448a V + 112aV + lOa^a; + a\ •9. 128a-^-1344a«a;+ 6048a V-15120a*a;'+ 22680a V-20412a'a;» + 10206aa;«-2187a;^ 10. l-5a;+V^'»-15a;'+i«^a;*~-V-a;* +-'^a;«-i|a;U/3V^»--^3V» 11. 1 - -y-^ + .^^x' - -Vx' + :jVV-Vt**''+i|4««-|i§x'+ jfi,*' (15) ANSWEK9 TO THE EXAMPLES. 3. 1 - (jx + 27x^ - lOSx^ + 405x' - &c. 4. 1 + Ca; + 24^;^ + 80^=» + 240aj* + &c. 7. 1 + x-\x''^\x^-\x^^k(^, 8. l-2a;~a;'-|a;'-^a!* -&c. 9. 1 + f ic + -V-i»'' + tI^' + tII^* + <^c. 10. 1 - Ix' + -V-aJ* - -||a;« - -^'j.aj^ - &c. 11. 1 + la; + fa;^ + ifa;* + -^i^x^ + &c. 12. \-\x^ -^ -Ja;* ~ -i^a;' + ^y^ ^' - &c. 78. 1. i + Ja; + f^a;' + ^^a;* + /^a;* + &c. 3. a^ — oT^x + a'Wx"^ — <^^5'a;^ + a'^JV — &c. 4. <*^ + 2« 'J'a; + 3a *&V + 4a^i V + ba'^b'x' + &c. 5. a^ + 6ah^ + 21a^&? + 5Ga«5 + 126/^"p + &c. 2 9 .13 23 r^n 6. a^ - }a'^ aj' - 2%a ^ a;* - rl^a'"^' ic* - AV^"^ •''^^ ~ ^^' 7. a- Zah'^ + 6^^*r« - lOa'^b' + 15a^&'^ - &c. 8. a^ - ^a'^x - ^a'^ x^ - -ijof^x^ - i^^a'^'x' - &c. 9. a^ + J^a-V + 2\a-"aj^° + jy^a-''x'''+ ^^^a-^'x""' + &c. 10. a^ - ^aJx"" + f a"3a;* + /ya^^'aj" + -.h^"^'^^ :- &c. 11. a' 2 + !«"%« + ^cf^'x* + -^^a'^x^ - lj||a""^'ic* + &c. _1 .1 .4 2 .15 10 8 13 11 12. a^x^ ^\a %^ + la ^x^ + \\a ^x^ + oV^a^^ic"^ + &c. 79, 1. 5221, 203116. 2. 100101100, 102010, 10230, 2200, 1220. 3. 41104, 23420, 14641, 7571, 5954. 4. 235, 1465. 5. 511, 22154. 6. 1212, 1212201. CO. 1. 1 + 14 + 244 -f 4344 + 114144 + 2050544 = 2214223 (sen.) = mill (den.). 2. 100001000000 (bin.) = 201000 (quat). 3. 1756 X 345 =701746 (oct.), 1337 x 274= 381011 (non.), 345, 274. 4. 57264, 95494, e7^8. 5. 4112, 6543, 62i^^. 6. 1295, 216; 2400, 343 ; 4095, 51? m MISCELLANEOUS EXAMPLES: Part L 1. (a^ - h^) h^ + (a» - Zah^ + P)x- {2a - I) ax\ 2. 3aj- - 2al)x - 2«^Z»^- 3. 3»-. 4. (w + n) a, ^^f^^- ^ x^ + xy + y^ 5. lp\, 8.152. G. g^, 1^^. r. 98, in{Zn + 25). 8. lyV 9. 5^5, 7aV27c, ^4. 10. l+a;+5a;^+|a;» + -V-a;* + &c. IL (i) ic = 5 ; (ii) a; = 5 or - 1^ ; (iii) oj = 4. 7/ = 3 ; (iv) a; = ± 3, y = ± 2, or ic = ± 2, 7/ == ± 3. 12. |. 13. \-'r^x^2x''-bx^-x^+x''+lx\ 14. -2ai'^ + 8iry-5/. 15. 68. 16. X' - y\ 17. 125, 1.709. 18. x, ^-^*. 19. 2v., 2}. 'x-y 20. a%^c^, aj'^A 21. a"^|l+a-^a;H-2a V+-VVa:»+-V.^-V + «5cc.[. 22. 1232, 11313, 363, 1044. 23. (i)cu = 3f^; (ii)aj=2or-f; (iii) ^ == 5, y = 41. 24. 12 days. 25. x' - a\ 26. Ix" - 5x- + Ix + 9. 27. f. 9« ^' ^^- *-l on ^Ll? 2^ • ic^ - y' 4a^ + 2a - 1* 5 - a;' ic^ - 1* 30. 139, 1.39, 4.3955. 31. 2^% .051. 32. fljl - ( - 1)"(, ||. 33. (rtic)"?jl + la-'x + i§« =aj' + t%\«-V + ff|a-*a;^ + &c.}. 34. 7 ; 22 dollars and 57 doubloons. 35. (i)ic=17; (ii) a; = 60, 7/ = 40 ; (iii) a; = 3 or - | J. 36. 3f hrs. 37. 20"!)^ + 2rrc^ + 2¥c' -a* -I*- c\ 38. 1 - ^aj + ^x' + ^f^^ -f y^a;* + &c. 39. {. 40. -^-^^'■4. 41. 139, .6933. 42. ^. 43. a' {1- 7a V + 4/-a V + icr'x' + Y'^V + &c.}. 44. i Vm, V^\ 45. 93, yV (71^ + n - 6). 46. 5221, 40255141, 6252711, 2451, 3341584, 1828. 47. (i) a; = 2J; (ii) a? = 39, y = 21, ^ = 12 ; (iii) x=a . r, y=:a . -^-. 48. 64 days. 49. a-h. a-Jy a + h ^ 50. 9 + 1 + 49 = 59. 51. 3 {a" + J» + c') -2(a& + ac + lc\ 62. a;' - 9^^^- 53. 1054, ^1 + V2. (18) ANSWERS TO THE EXAMPLES. ^^'l^ ^y ^5. f|l-(^)«M. 56.6. 57. a^ {l + ^a-^^bx+^^a^b^x^ + fi^a-'h^x^ + i^^a 'h'x' + &c.\ 58. 21 59. (i) x=-i; (ii) x=Z, y=i; (iii) a;=3, y=l ; (iv) x=-l, y=-^. GO. 16. Gl. 2(Sx^y^-x^y-lx^ - 10y\ G2. x^+2x + 1. G3. 1. G4:. a'hUb -^ c\ G5. 2.C-1. CG. --]^%- . X {4x^ - 1 ) 67. 12.747. C8. (a^'x)'^^ {l + ia'x-^la-^x^ + j^a'x'-^^%a*x' + &c,} 69. 30. 70. 250, G0300, 13874000. 71. (i) oj = 9 ; (ii) iz; = 3^ or - 4 ; (iii) x = -J-, y - ^. 72. 8 hrs; 17| hrs,24 hrs, 40 hrs. 73. G (x + 2x^ + 4/^ + Sx-"). 74. |. 75. aj, -.-^"^ . 76. 55 ^7. 77. 1.772452, x-i. 79. 50f, Jti (3/i+i). 80. x' - 2x^-Sx^+Sx+1(j = 0. 81. 28. 82. 15120, 120. 83. (i) ic = 1 ; (ii) x=2ly = ^; (iii) x=3. 84. £553 J, £11061, £3320. 85. x' + l+ x-\ ^ 8G. a' + ax-2x\ 87. 7. 88. i (1 -f ^), ?^-:!^ . 89. Jx'''-Jxhh hi ' 36 - 2a 90. cc^ -1 - x-i, 91. ]- 71 (37i + 1), j\ { (I)" - 1 } . 92. I +x-^x^ + liC* - -V^J* + &c., 1 + 2a; - 2a;* + 4a;' - 10a;* + &c. 93. 27, 48. 94. \ 95. (i) a;=9 ; (ii) a;=4, y=Z ; (iii) a;=6 or i. 96. 4j\ miles an hour ; 134 minutes. 97. x\ 98. a- - 2ab + U\ 99. a;' - 1. 100. ?-^' , ^^^. ^ ^x-'s/a'^y 101. 4.11. 102. l-«' + a;-^ + Zcfix'i, 103. 7. 105. 75, 25. 106. V11333311 s^^jf. = 2620 = 1000 den, 107. (i) a? = 9 ; (ii) a; = ± -J V^ 5 ("i) ^ = "^i V^^i ^ = ^• 108. £135, £90. 109. 48. 111. 36a;* - 97.t;* + 36. 112. ^x'-'ax + \a\ 113. 2.4494, .4082, .8164, 1.2247. 114. (ah-')Hm+i)^ 1. 115. a'-^^ah-Gac + AV + 12^c + ^c\ 4a^ - 2abx - («c - i?/) .^;^ + {^ad+ lie) a;"- - (2&^Z- ^c") x'- cdx* + 4cZV. 116. 2_5 |i _ (|)"i^ 81. 117. - 11. 118. 1 - 2a; - 2.z;* - 4.^' - 10.i;* - &c. 119. (i) a'-21 ; (ii) a;= - 3 or f ; (iii) a;=5, y=3. 120. 31 3. 121. 8-12a^+18^^-27aT. 122. aUla'hx-i(a'-h')x^+^ax^-ix\ S r- + 4x + 2 123. Y-i ^ — ^. 124. a^°-a°a;*-aV + a;^°. 4x^ + X + 2 125. i.2247, 3 + v3. 126. c. 127. 0, j\n (7-7i}. 12^ A gives 26 guineas and receives 10 crowns. (19) ANSWERS TO THE EXAMPLES. 129. 2(a-x) V2^, i \/a. 130. 33 : 238, 1 : 34. 131. (i) 2^=10 ; (ii) x=Z, y=l ; (iii) x=^ or - 1. 132. IG. 133. With upper signs, IG +9=5x5; with lo^cer^ + 25=5x5. 134. aj'+4?/. 13d. x^ + ax + h* 130. -z — , jr-r ^-. *^ ba + oy^ 9a;-ic--3 137. mn (w' - 71^ (wi- - 4/1^). 138. a^ - aj'^ + 1. 139. ^-^(a+h) (a + c), 140. 4.8089, .G803, 4.4494, 1.550C, 3.4494. 141. If, 2i ; If, If. 143. (i) a; = 7 ; (ii) a; = 4 ; (iii) a; = 2 or |i. 142. l+a;+|a;'+Y-^»-iJA;;* + &c., 1 + 2j; + C.-c' + 20.^' + 70aj* + j' + IGa^oj^ + IGa;^ 147. a;^-12-lG.r-S a^ + l + a-'. 148. -^'^4-. 1^^- -2154. ' a -26 + 3c 150. cc-2V^ + l. 151. V^-^V^. 152. f {l_(f)«l,l|. 153.0,3,-2. 154. G. 15G. 5idays,16days''. 157. 3j|. 155. (i) x = —\: (ii) a; = f ; (iii) aj = ± 2, y = ± 3. 158. a^ - 2ah^ + zJb - 2ah^ + h\ 159. x-5. 160. ^^^^, A. IGl. x'+2x'-Sx'-ex-l. 1G2. 9, 160. ic* - 16 * 163. (&+c)^ 164. 1147. 165. l-6a:H24a;^-80aj« + 240x«-&c., «^ {1 + Za-'b + #a-^&^ - la-»Z>» + ^a'*h* - &c.}. 166. 33233344, 4344 = 1000 de?u, 244 = 100 deii. 167. (\)x=li ; (ii) a? = -^-^-j , y = -r~i 5 (m) aj = ± G. 168. £40, £28, or £28, £52, according as A had more or less at first tlian K 169. ^{a^ + h^} y(a'^h')^=Zl/2S9=10.SZ4, 170. X-' - 4a;" V + 3y\ 171- frl>~^^ ^-^i" • 1-0 1 1 a o2 ^^r, • ax~h^ x + 2 1<2. 1 -i«a!- -2^^^ 1/3. ^- — jr. -^ - . (.^-a)(a;-?>)' a;*-l 174. 3.8729, 1.2909, .'7745, 1.5491, 6.4549. 175. -10, 1/1(7- 3;?). 176. 15. 177. 1 - 2x^ + Zx - 4xi + 5a!' - &c , 1 - 4a;^+ 10a;-20.T^+ 35a;' - &c. 178. 12, 16, 18. 179. (i) ^~' ; (ii) 2 or -1 J ; (iii) a'=.49, y =50. (20) ANSWERS TO THE EXAMPLES. 180. 10 days, 3 J days. 181. a» + ia'' (2x + y-z) + la {2xy^2xz-^z) - i^y^, which becomes a^ + Za^b + Sah^ + ^', by putting X = h = iy :^ - iz, or x= b, y = 2h, z = - 2b, 182. ox'' + laKi - -'^-a'h + ^a-'x^ + Ja'^ 183. a; + 1. 184. f "^ ^ -. 185. «*a; (a^ic* - 1). 18G. ah-' - ia^'b + !• X + X — o ^^^- (i^^T^^r ^^^- 720,4(lfV7). 189. 4iff,|{U(-J)-}. 190. 7h 37' 12". 191. (i) If ; (ii) x=4, y=5 ; (iii) 4}. 192. 10, 193. V'j. 194. «'» + 2. 195. 5 + 2 V6, V^, 6 (5 + 2 V6). 196.f!^^^^V, ^1^4^. 197. 12x'-2..'-ll.'.l. \m + a J ^ a* - X* 108. § {1 - (- f)"[, 1|. 199. 72,^ 200. ± 5a. 201. 15. 202. l-6^^-f21aj^-56aj+12Gaj^-&c., l-3^^+6a'l-10^+15aj^- &c. 203. (i)f; (n)ar'ovb-'; (iii) y = 3, y = 1, or aj = f , y = J. 204. £800, 0. 206. px^ + qx- r. 207. a - ar' + 4. 208.^1^-^:1^^11^. 209. --—1-—-. 210.7.0102,202. x-a (ic^ + l)(a;'+l) ' 211. yV, 20. 212. (aZ))T"2. 213. 4yds, 5 yds. 214.63361,236,34. 215. (i)4orl|; (ii) :c = -5, y=5 ; (ii^) x ==-£=, y = -j^' ^a' + b'' ^a^ + b^ 216. -^^ days, ^ days. 217. 2(n + 4). 218. a^ + j'^ + c' - a^^ - a^e - b%. 219. 2. 220. x^ - y*. 221. 3.1622, .12649, 2.1081, 1.5811, 4.4414, .31622. 222. l{l- {lY}^ 2f. 223. a'^ |l-3«- V+\-^a-V-^a-V +-^4^o^-V-&c !, i«-' (1 + 3a-»ir+ ^^-^tt-V + V«~V + -\Va"V + &c.!-. 224. £5825 85 5H 225. 0, - 1, 2. 226. 20, 5. 227. (i) . = «-or-i; (ii) . = -^^_, ^ = -_^^_ ; (iii) ^=3 or |. 228. T?"" "" ''I ^ays. 229. 2y'-ay-la\ 230. ic* + a-ic^ + .t*, a;' + 2ax^ + a'a;^ - a\ x* - a^'x^ -'2a^x^ - a\ 231. a" - b\ 232. x' - 2.?j« + a;^ _ a;V 2a; - 1. 233. yV» s~^-- 234. ioj^- 5yl 235. 88, | U - QTh % 1}. IJ, 1?. (21) ANSWERS TO THE EXAMPLES. 236. la ^ h, 237. 12, 4, 18 miles. 240. ^' "" ^' days. 4W7l 239. (i) x:= ~6i ; (ii) x= ^^- , y = , 5 (iii) a;=10, y=7, «=3. 241. ic- —xi/^+ x^y ^ -y *, x^ - (a + h) X + ah. 242. 3G, 125. 243. 5x + 4. 244. ^~, J. 245. .8164, 1.6320, 2.0412, .1010, 3.2549. 247. 3^, 3f, 4^, «S:c. 248. 7. 249. 720. 250. 248064^/60, 54373. n-i ^N -i-f r-x a* + ah + b"^ ah ,.... 2ol. (,) .: = 17 ; (n) x = -^^^ ' ^ = ^TTj ' <'"> ^ = "• 252.^^days, r^!'^"> , day.. ' 253. 140,VyVV 43T21 11 2a— 6 254. ic'J -x^y^ + 2/^, a;^ - (r/i - 1) «%* + a. 256. 1 J, -,::7j- 255. 18.r* - 45aj» + ^7x' - 10.^ + 6. 257. ~--^, J-^- , h (h^ - c') V(l-a;^)» 258. 24 miles, i hr. 250. a^ji 260.3. 261, a+2x: a+Zx. 262. 7425. 263. (i) cc = 4 or If ; (ii) ic = 10, y = - 3, 2 = 4. 264. Ss id. 265. -\'- - -^3^"- = Y = i >< t''2- 266. A + 2. 267. -V + &c.;, (3<^)"^ {1 + ^akx^ + f|a"^a;^ + fl«~'a; + |||a'M + &c.}. 346. 2118760, 17296. 347. (i) x=a ^ ; (ii) a;=l, y=4, g=27 ; 349. (x" + X + 1) a - (» + 1). S50. a'^ - a^e^. 351. 0. 352. # - |.,4 . |.V 353. 4^-14^ 354 ^til-d)-}, 13i. 355. 7G. 356. 9 hrs. (23) ANSWERS TO THE EXAMPLES. _n 3 357. (2ay^ {1 + ^%^ + 5a-'x + ^a'ix^ + ^Ya-'x* + &c.}, (3a)a { 1 - lah^ + iah^ - /^a-'aj - yf ^a"^^^ - &c. }. 358. 4; 1023, 256. 360. 2f hrs. 361. x^i/^-x^yh. 359. (i) ± y -^-— '- ; (ii) ; (iii) x=-— , y =^ j—. f c c a—0 o—a 363. «* - ic* + 1. 364. V^, V2^. 365. 1. 366. 24«, 16a 367. a^ {I + 2«-»a; - 4«-V + ^^-a" V - ^J^c^-^o;* +