John owett .\/: " i ■ "■'■'■; ' ..' vt' '-,<-' - •■ -t * ' ■-■''^- i.M m t^tM m m'^^i^ tZ^ or <, the smaller quantity always standing at the vertex ; thus, 8 > 6 or 6 < 8 signifies that 8 is greater than 6. 11. Three dots .'. are sometimes used, meaning hence, therefore. 12. A Parenthesis ( ), or a Vinculum •, indicates that all the quantities included, or connected, are to be considered as a single quantity, or to be subjected to the same operation ; thus, (8 + 4) X 3 = 12 X 3, or = 24 -f 12 = 36 ; 21 — 6 -r- 3 = 15 -^ 3, or = T — 2 = 5.. Without the parenthesis, these examples would stand thus : 8 + 4 X 3 = 8 + 12 = 20 ; 21 — 6 ^ 3 = 21 — 2 = 19; the sign X. in the former, not affecting 8 ; nor the sign -r-, in the latter, 21. Examples. 1. 9 + 7 — 3 + 4 = how many? 2. (9 + 15) -4- 3 == how many? 3. — - — X 14 = how many ? 4. (14 + 13) X (6 — 2j = how many? 5. 10 + (7 — 4) -4- 3 X 4 = how many? 6. 26 — (6+ 7) = how many? 7. 150 — (18 — 11) = how many? DEFINITIONS. 9 8. Prove that n5 + 8 — 49 — 14 + 190 — 54 — 16. 9. Prove that 216 — 44 + 14 > 144 + 13 — 15. 10. Place the proper sign (=, >, or <) between these two expressions, (247 -f- 104) and (546 — 195). 11. Place the proper sign (=, >, or <) between these two expressions, (119 — 41 + 16) and (311 — 104). 12. Place the proper sign (=, > , or <) between these two expressions, (417 + 31) — (187 — 72) and (127 + 179). AXIOMS. 13* All operations in Algebra are based upon certain self-evident truths called Axioms, of which the following are the most common : — 1. If equals are added to equals the sums are equal. 2. If equals are subtracted from equals the remainders are equal. 3. If equals are multiplied by equals the products are equal. 4. If equals are divided by equals the quotients are equal. 5. Like powers and like roots of equals are equal. 6. The whole of a quantity is greater than any of its parts. 7. The whole of a quantity is equal to the sum of all its parts. 8. Quantities respectively equal to the same quantity are equal to each other. 10 ELEMENTARY ALGEBRA. SECTION II. ALGEBRAIC OPERATIONS. II, A Theorem is something to be proved. 15. A Problem is something- to be done. 16, The Solution oif a Problem in Algebra consists, — 1st. In reducing the statement to the form of an equa- tion ; 2d. In reducing the equation so as to find 'the value of the unknown quantities. Examples for Practice. 1. The sum of the ages of a father and his son is 60 years, and the age of the father is double that of the son ; what is the age of each ? It is evident that if we knew the age of the son, by doubling it we should know the age of the father. Sup- pose we let X equal the age of the son; then 2x equals the age of the father; and then, by the conditions of the problem, a:, the son's age, plus 2j7, the father's age, equals 60 years; or 3a: equals 60, and (Axiom 4) x, the son's age, is ^ of 60, or 20, and 2 a;, the father's age, is 40 Expressed algebraically, the. process is as follows: — Let X = son's age, then 2x = father's ago. X -\- 2x = 60, Sx = 60, X = 20, tlie son's age. 2 J' = 10, thp fiilhfM-'rt age. DEFINITIONS. 11 2. A horse and carriage are together worth $450 ; but the horse is worth twice as much as the carriage ; what is each worth? Ans. Carriage, $160; horse, $300. All problems should be verified to see if the answers obtained fulfil the given conditions. In each of the pre- ceding problems there are two conditions, or statements. For example, in Prob. 2 it is stated (1st) that the horse and carriage are together worth $450, and (2d) that the horse is worth twice as much as the carriage ; both these statements are fulfilled by the numbers 150 and 300. 3. The sum of two numbers is 72, and the greater is seven times the less ; what are the numbers ? 4. A drover being asked how many sheep he had, said that if he had ten times as many more, he should have 440 ; how many had he ? 5. A father and son have property of the value of $8015, and the father's share is four times the son's; what is the share of each? Ans. Father's, $6412; son's, $1603. 6. A farmer has a horse, a cow, and a sheep ; the horse is worth twice as much as the cow, and the cow twice as much as the sheep, and all together are worth $490 ; how much is each worth ? OPERATION. Let X ■= the price of the sheep, then 2j; = *' " " " cow, and 4a? t= " " " " horse; and their sum 7 x :^ 490, • j7 = 70, the price of the sheep, and 2ip = 140, " " " " cow, and 4^ = 280, *' " " " horse. 12 ELEMENTARY ALGEBRA. 7. A man has three horses which are together worth $540, and their values are as the numbers 1, 2, and 3; what are the respective values ? Let X, 2 X, and 3 x represent the respective values. Ans. $90, $180, and $270. 8. A man has three pastures, containing 360 sheep, and the numbers in each are as the numbers 1, 3, and 5 ; liow many are there in each ? 9. Divide 63 into* three parts, in the proportion of 2, 3, and 4. Let 2x, 3x, and 4x represent the parts. 10. A man sold an equal number of oxen, cows, and sheep for $1500; for an ox he received twice as much as for a cow, and for a cow eight times as much as for a sheep, and for each sheep $ 6 ; how many of each did he sell, and what did he receive for all the oxen ? Ans. 10 of each, and for the oxen, $960. 11. Three orchards bore 872 bushels of apples; the first bore three times as many as the second, and the third bore as many as the other two ; how man}' bushels did each bear ? 12. A boy spent $4 in oranges, pears, and apples; he bought twice as many pears and five times as many apples as oranges ; he paid 4 cents for each pear, 3 for each orange, and 1 for each apple ; how many of each did he buy, and how much did he spend for oranges ? how much for pears, and how much for apples ? . (25 oranges, 50 pears, and 125 apples. ( Spent for oranges, $0.75 ; pears, $2 ; apples, $1.25. 13. A farmer hired a man and two boys to do a piece of work ; to the man he paid $12, to one^boy $6, and to the other $ 4 per week ; they all worked the same time, and received $264; how many weeks did they work ? Ans. 12 weeks. DEFINITIONS. 10 14. Three men, A, B, and C, agreed to build a piece of wall for $99; A could build 1 rods, and B 6, while C could build 5 ; how much should each receive ? 15. Four boj^s, A, B, C, and D, in counting their money, found they had together $1.98, and that B had twice as much as A, C as much as A and B, and D as much as B and C ; how much had each ? Ans. A 18 cents, B 36, C 54, and D 90. 16. It is required to divide a quantity, represented by a, into two parts, one of which is double the other. OPERATION. Let X ==: one part, then 2x = the other part. Sx = a, X = -, one part, o 2x = -— , the other part. o 17. If in the preceding example a = 24, what are the required parts ? A « 24 . .2a 48 ^"^- 3 == T = ^' ^^^ T = Y=^^- 18. It is required to divide c into three parts so that the first shall be one half of the second and one fifth of the third. , c 2c .5c Ans. -, -, and -. 19. Divide n into three parts, so that the first part shall be one third the second and one seventh of the third. 20. A is one half as old as B, and B is one third as old as C, and the sum of their ages is p ; what is the age of each ? ^^^ ^,^ p g,^ ^_ ^^^ ^'s ^^ y 9 y 14 ELEMENTARY ALGEBRA. SECTION III. DEFINITIONS AND NOTATION. [Continued from Section I ] 17. The last letters of the alphabet, ar, y, z, &c., are used in algebraic processes to represent unknown quanti- ties, and the first letters, a, b, c, &c., are often used to represent known quantities. Numerical Quantities are those expressed by figures, as 4, 6, 9. Literal Quantities are those expressed by letters, as a, X, y. Mixed Quantities are those expressed by both figures and letters, as 3 a, 4 a:. 18. The sign plus, -\-, is called the positive or affirm- ative sign, and the quantity before which it stands a pos- itive or affirmative quantity. If no sign stands before a quantity, -\- is always understood. 19. The sign minus, — , is called the negative sign, and the quantity before which it stands, a negative quantity. 20. Sometimes both + and — are prefixed to a quan- tity, and the sign and quantity are both said to be am- biyuoxis; thus, 8 zb 3 =: II or 6, and a zh 6 = a -j- 6, or a — h, according to circumstances. 21. The words plus and minus, positive and negative, and the signs -|- and — , have a merely relative signifi- cation ; thus, the navigator and the surveyor always rep- resent their northward and eastward progress by the sign -|-, and their southward and westward progress by the sign — , though, in the nature of things, there is nothing to prevent representing northings and eastings by — , and southings and westings by -("• So if a man's prop- DEFINITIONS. 15 erfy is considered positive, his gains should also be con- sidered positive, while his debts arid his losses should be considered negative ; thus, suppose that I have a farm worth $5000 and other property worth $3000 and that I owe $1000, then the net value of my estate is $5000 + $3000 — $1000 — $7000. Again, suppose my farm is worth $5000 and my other property $3000, while I owe $12000, then my net estate is worth $5000 _|_ $3000 — $12000 = — $4000, i. e. I am worth — $4000, or, in other words, I owe $4000 more than I can pay. From this last illustration we see that the sign — may be placed before a quantity standing alone, and it then merely signifies that the quantity is negative, without determining what it is to be subtracted from. 22t The Terms of an algebraic expression are the quan- tities which are separated from each other by the signs + or — ; thus, in the equation 4a — b z^ Sx -\- c — 1 y, the first member consists of the two terms 4 a and — b, and the second of the three terms 3 x, c, and — 7 y. 23. A Coefficient is a number or letter prefixed to a quantity to show how many times that quantity is to be taken; thus, in the expression 4:X, which equals x -\- x -^ X c\- X, the 4 is the coefficient of x ', so in 3 a b, which equals ab -\- ab ~\- ab, 3 is the coefficient of a6; in4a&, 4 a may be considered the coefficient of b, or 4 6 the co- efficient of a, or a the coefficient of 4 6. Coefficients may be numerical or literal or mixed ; thus, in 4 a 6, 4 is the numerical coefficient of ab, a is the lit- eral coefficient of 4 6, 4 a is the mixed coefficient of 6. If no numerical coefficient is expressed, a unit is un- derstood ; thus, X is the same as Ix, be as 16c. 24. An Index or Exponent is a number or letter placed after and a little above a quantity to show how many times that quantity is to be taken as a factor; thus, in the ex- 16 ' FXEMENTARY ALGEBRA. pression W, which equals 6 X ^ X &, the 3 ia the index or exponent of the power to which b is to be raised, and it indicates that h is to be used as a factor 3 times. An exponent, like a coefficient, may be numerical, lit- eral, or mixed ; thus, ^^ a:", ar*", &c. If no exponent is written, a unit is understood ; thus 6 =r 6\ a = a}, &c. Coefficients and Exponents must be carefully distin- guished from each other. A Coefficient shows the num- ber of times a quantity is taken to make up a given sum ; an Exponent shows how many times a quantity is taken as a factor to make up a given product ; thus i:X=:^x-\-x-[-x-\-x, and x*^=xy^xy^xy^x. 25* The product obtained by taking a quantity as a factor a given number of times is called a power, and the exponent shows the number of times the quantity is taken. 26. A Root of any quantity is a quantity which, taken as a factor a given number of times, will produce the given quantity. A Root is indicated by the radical sign, v', or by a fractional exponent. When the radical sign, \/, is used, the index of the root is written at the top of the sign, though the index denoting the second or square root is generally omitted ; thus, a/ X, or xh, means the second root of x ; ^Ic, or xi, " " third " " a:, &c. Every quantit}'- is considered to be both the first power and the first root of itself 27. The Reciprocai. of a quantity is a unit divided by that quantity. Thus, the reciprocal of 6 is -, and of x, -. X DEFINITIONS. 17 2S» A Monomial is a single term ; as a, or 3 x, or bhxy. 29. A Polynomial is a number of terms connected with each other by the signs plus or minus ; slq x -\- y, or 3a -\- 4:X — 1 ahy. 30. A Binomial is a polynomial of two terms ; as ^x -\- ^y, ov X — y. 31. A Residual is a binomial in which the two terms are connected by the minus sign, as x — y. 32. Similar Terms are those which have the name powers of the same letters, as x and 3 a;, or bai? and — 2ax^. But X and x\ or 5 a and 5 b, are dissimilar. 33. The Degree of a term is denoted by the sum of the exponents of all the literal factors. Thus, 2 a is of the first degree ; 3 a^ and A: ah are of the second de- gree ; and 6 a^ x^ is of the seventh degree. 34. Homogeneous Terms are those of the same degree. Thus, A:a^x, Sabc, x^y, are homogeneous with each other. 35. To find the numerical value of an algebraic expres- ision when the literal quantities are known, we must sub- stitute the given values for the letters, and perform the operations indicated by the signs. The numerical value of t a — 5* -|- c^ when a = 4, ft = 2, and c = 5 is 7 X 4 — 2* + 5^ = 28 — 16 + 25 = 37. Examples. Find the numerical values of the following expressions, when a = 2, J = 13, c r::^ 4, cZ = 15, m = 5, and n =: 7. 1. a -\- h — c -\- 2d. Ans. 41. 2. a^ -f 36c — 2cd. Ans. 40. 3. 1^1 + ^' Ans. 219. 18 ELEMENTARY ALGEBBA. 4. 6. 6. 1. (a2 _ c + h) (m + w). ^__^ X (^ - m + .0. Ans. 86t. 8. 9. 3aV6 — c X 4nv^25m. ' Ans. 1. 5m— 6n-h3d 10. -. s- . Ans. 4. 11. v^ — ^ + \/Tn. 12. (6 — a) {d — c) — m. Ans. 116. 13. 13 (4v/10 (d — m) + 14\/c. 36* Write in algebraic form : — 1. The sum of a and 6 minus the difference of m and n. (m > w.) 2. F'our times the square root of the sum of a, b, and c. 3. Six times* the product of the sum and difference of c and rf. (c > rf.) 4. Five times the cube root of the sum of a, m, and n. 5. The sum of m and n divided by their difference. 6. The fourth power of the difference between a and m. ADDITION. 19 SECTION IV. ADDITION. 37. Addition in Algebra is the process of finding the aggregate or sum of several quantities. For convenience, the subject is presented under three cases. CASE I. . 38. When the terms are similar and have like signs. 1. Charles has 6 apples, James 4 apples, and William 6 apples ; how many apples have they all ? OPERATION. 6 apples, 4 apples, or, letting a 5 apples, ^ represent one apple, 15 apples. 6 a 4 a 5 a 15 a It is evident that just as 6 apples and 4 apples and 5 apples added together make 15 apples, so 6 a and 4 a and 5 a added togeth- er make 15 a. In the same way — 6 a and — 4 a and — 5 a are equal together to — 15 a. Therefore, when the terms are similar and have like signs : RULE. Add the coefficients, and to their sum annex the common letter or letters, and prefix the common sign. (2.) (3.) (4.) (5.) (6.) (7.) b ax Sa' 4:X 6y — ^7? -bby %ax ^a\ X lOy — 2x^ -2hy 4:ax la' hx y — 7:r^ - hy 2ax Sa' Zx 13 a; ^y — 4^:^ - hy 19 ax — 16x« 20 ELEMENTARY ALGEBRA. 8. What is the sum oi ax^, Sax^, 2a2p, and 4 a a:*? Ans. 10 a x^. 9. What is the sum of ^bx, 4tbx, 6bx, and bx? 10. What is the sum of 2x2/; Qxy, 10 a: 5/, and Sxy? 11. What is the sum of — Ixz, — xz, — 4zxz, and — xz? Ans. — 13x2. 12. What is the sum of —2 b, —3 b, —6 b, and — Sb? 13. What is the sum of — a be, — Sabcy — 4: a be, and — a be? CASE II. 39. When the terms are similar and have unlike signs. 1. A man earns 7 dollars one week, and the next week earns nothing and spends 4 dollars, and the next week earns 6 dollars, and the fourth week earns nothing and spends 3 dollars ; how much money has he left at the end of the fourth week ? If what he earns is indicated by -\-, then what he spends will be indicated by — , and the example will appear as follows : — + 7 dollars, — 4 dollars, + 6 dollars, — 3 dollars, + 6 dollars, OPERATION. or, letting d represent . one dollar, [+1 d — 4rf + 6d — Sd + Qd Earning 7 dollars and then spending 4 dollars, the man would have 3 dollars left; then earn- ing 6 dollai-s, he would have 9 dollars; then spending 3 dollars, he would have left 6 dol- lars ; or he earns in all 7 dollars -j- 6 dollars =13 dollars ; and spends 4 dollars -|- 3 dollars = 7 dollars; and therefore has left the differ- ence between 13 dollars and 7 dollars = 6 dollars; hence the sum of -f 7 ^ + f — z') = x' — ^ — 2^—f + And conversely, A polynomial, or any number of the terms of a polyno- mial, can be enclosed in a parenthesis and the minus sign placed before the parenthesis without changing the value of the expression, providing the signs of all the terms are changed from plus to minus or from minus to plus. Thus, a' — b' + c' -\- d — X = a^ — (b' ~ c' - d + x). Note. — When the sign of the first term in the parenthesis is plus, the sign need not be written. (Art. 18.) According to this principle a polynomial can be writ- ten in a variety of ways. Thus, a^—3x^y+ ^^y'—i = o(^ — {Zx'^y — ^ xy"" + f) r=^-^x''y-{-^xy^ + f) = a? + ^xf—{^x^y+f) =zx^—f—{3x''y — Sxy'')&c. Remove the parenthesis, and reduce each of the follow- ing examples to its simplest form. 1. a^ — (2ab-\- c^). Ans. a'^—2ab — c\ 2. x'^ — 6ax-^a^ — 6x^y — {x'^-{- 6 ax -\- x'^ — 6 x'^ ij). Ans. — 12 ax. 3. rn^ — n''-\-2x—(4:m'^ + Sn^ — 4.c). 4. IQxy + Uc — ISy — {^Uc -]-21 y —16xy). Ans. S2xy -\-2Sc — 4:5y. 6. 4:X^y—(Sxy^~1x''y'^-\-Sx^y). b. — (-0:^+7 -25xy + /). ZO ELEMENTARY AL(]KBRA. Place in parenthesis, with the nigii — prefixed, without changing" the value of the expression, 1. The last three terms of 1 x^ — 14 xy — 3 2? -[- 4y. Ans. *ra:2— (14a:y + 32 — 4y). 2. The last three terms of ar^ -|- 3/^ — Sxy -\- 4:C. Ans. x'^ — (3 ary — y^ — 4 ^). 3. The last four terms of 4: a — 1 b — 6 c — S d -\- x\ 4. The last four terms of a^ -{- b^ -\- c'' — d^ + ct\ 5. Write in as many forms as possible by enclosing two or more of the terms in parenthesis, a^ — b^ -\- c^ 45* In subtraction, when two quantities have a com- mon factor their diflference is the difference of the coef- ficients of the common factor multiplied by this factor. Thus, ax — bx = (a — b) x. 1. From a x^ take c x'^ — dx^. Ans. (a — c -\- d) x^. 2. From 4 \/ a: take ats/ x •\- b \f x. Ans. (4 — a — &) \/ x. 3. From a a^ take bx^ — b x^. Ans. {a — b)x^-\-b x\ 4. From 4 a:^ — 6x take ax^ -\- bx. Ans. (4 — a)x^—{6Jf- b) x. 6. From 6 a* + 4 a^ — a take a^ x — a^y -\- az. Ans. (6 — x) a« + (4 +y) a^ — (1 + t) n. 6. From ab — be take ^b -\- ex. *l. From a^ — bx -{- c >^ x take bx'^ -\- ex — rf\/x. 8. From xy"^ + a:^ — x'^y'^ take f + x' y — x'^y\ MULTIPLICATION. 31 SECTION VI. MULTIPLICATION. 46. MuLTTPLicATiox is a short method of finding the sum of the repetitions of a quantit}?-. 47. The multiplier must always be an abstract num- ber, and the product is always of the same nature as the multiplicand. The cost of 4 pounds of sugar at 17 cents a pound is 17 cents taken, not 4 pounds times, but 4 times; and the product is of the same denomination as the multi- plicand 17, viz. cents. In Algebra the sign of the multiplier shows whether the repetitions are to be added or subtracted. L (+a)X(+4) = + 4a; i. e. -j" ^ added 4 times is -\-a-\-a-\-a-{-a^=-{-4ia. 2. (+a)X(~4):::.: — 4a; i. e. + a subtracted 4 times is — a — a — a — a = — 4 a. 3. (— «) X (4-*) = — 4a; i. e. — a added 4 times is — a — a — a — a = — 4 a. 4. (— «)X(— 4)== + 4a; i. e. — a subtracted 4 times is-f-<^ + <^+« + « = + 4a. In the first and second examples the nature of the product is -f- ; in the first, the -f- sign of 4 shows that the product is to be added, and + 4a added is -|- 4a; in the second, the — sign of 4 shows that the product is to be subtracted, and -|- 4 a subtracted is — 4 a. In the third and fourth examples the nature of the product is — ; in the third, the -\- sign of 4 shows that the prod- uct is to be added, and — '4 a added is — 4 a ; in the 82 ELEMENTARY ALGEBRA. fourth, the — sig-n of 4 shows Ihat the product is to be subtracted, and — 4 a subtracted is -j- 4 a. 48t Ilence in multiplication we have for the sign of the product the following RULE. Lake signs give + ; unlike, — . Hence the products of an even number of negative fac- tors is positive, of an odd number, negative. 49. Multiplication in Algebra can be presented best under three cases. CASE I. 50* When both factors are monomials. 1. Multiply 3 a by 2 b. OPERATION. 3aX25 = 3X^X2x*=3X2XaXJ = 6c6. As the product is the same in whatever order the factors Are arranged, we have simply changed their order and united in one product the numerical coefficients. Hence, when both factors are monomials, RULE. Annex the product of the literal factors to the prodwf of their coefficients, remembering that like signs give -(' and unlike, — . 2. Multiply n^ by a\ OPERATION. a^ X "' = (« X « X a) X (a X «) = a X « X n X « X a = a* As the exponent of a quantity shows how many times it is taken as a^factor, a" = a X « X a ; and c^ = a X + 3 by 4 a i — 3. 7. Find the square of 14 a^ ^^ + lO.ar^^. 8. Find the square of 4 a — b. 9. Multiply lOx + 2 by 10a: — 2. 10. Find the square of 3 a a: — 9>axy. Ans. ^a^s^ — ^^a-cry-^-Ua^T^^. 11. Find the square oi 2 a -{-h. 12. Find the value of (6 a + 4) (6 a — 4) (36 a"- + 16). Ans. 1296 a* — 256. 13. Find the square of 10 a^— 5 6^ 14. Expand (3 a^a: + 4 S/)2. Ans. 9 a^a:'^+ 24 aH a:/ -t- 16 6^V«. 15. Find the product of a^° + 1, a« + 1, a* + 1, a^ + 1, a -\- \, and a — 1. Ana. a'^ — 1. 16. Find the product of a -|- ^» ^ — ^» and a^ — i^. FACTORING. 47 SECTION IX. FACTORING. 62. Factoring is the resolving a quantity into its fac- tors. 63. The factors of a quantity are those integral quanti- ties whose continued product is the quantity. Note. — In using the word factor we shall exclude unity. 64. A Prime Quantity is one that is divisible vt^ithout remainder by no integral quantity except itself and unity. Two quantities are mutually prime when they have no common factor. 65* The Prime Factors of a quantity are those prime quantities whose continued product is the quantity. 66. The factors of a purely algebraic monomial quan- tity are apparent. Thus, the factors of d^bxyz are aXaXhXxXyXz. 67. Polynomials are factored by inspection, in accnrd- ance with the principles of division and the theorems of the preceding section. CASE I. 68. When all the terms have a common factor. 1. Find the factors of ax — ah -\- ac. OPERATION. As a is a factor of (ax — ah -\- ac) = a (x — h -\- c) each term it must be a factor of the poly- nomial ; and if we divide the polynomial by a, we obtain the other factor. Hence, 48 ELKMENTARY ALGEBRA. RULE. Write (he quotient of the polynomial divided by the com- vion factor in a parenthesis, with the common factor lyre- fixed as a coefficient. 2. Find the factors of6x9/—12xf-{-lSax^i/\ Ans. 6x^^(1 — 12^/ + 3ax/). Note. — Any factor common to all the terms can be taken as well as 6 a;y ; 2, 3, ar, y, or the product of any two or more of these quan- tities, according to the result which is desired. In the examples given, let the greatest monomial factor be taken. 3. Find the factors of a; + x^. Ans. x {I -\- x). 4. Find the factors of S a^ x"^ -\- 12 a^ x* — ^axy. Ans. 4 a x (2 a x + Zo^ 3? — y). 6. Find the factors oi h x" y'' -\- "Ih a x'' — 15 ar^/. Ans. 5a:^(x/+ 5aa:2 — 3/). 6. Find the factors of ^ ax — 8 ^^ -|- 14 a;^ T. Find the factors of 4 x^y^ — 28 x^y — 44 x^ y^ 8. Find the factors of 55 a^ c — 11 a c + 33 a* c x. 9. Find the factors of 98 a^a;^ — 294 a«x2y2 10. Find the factors o{ Ihd'l? cd — ^ aWd'' -{-X'^a^^d^. CASE II. 69. When two terms of a trinomial are perfect squares and positive, and the third term is equal to twice the product of their square roots. 1. Find the factors of a^ + 2 a 6 + ^2. OPERATION. WTe rcsolve this into c^ ■\- 1 ah -^ h^ z=z {a -\- h) {a -\- h) its factors at once by the converse of the principle in Theorem II. Art. 58. FACTORING. 49 2. Find the factors of a^ _ 2 a 6 + 61 OPERATION. We ^esol^e tl^ig j^to rt^ — 2 a 6 -|- 6^ = (a — b) (a — b) its factoi-s at once by the converse of the principle in Theorem III. Art. 59. Hence, RULE. Omilting the term that is equal to twice the product of the square roots of the other two, take for each factor the square root of each of the other two connected by the sign of the term omitted. 3. Find the factors of ^r^ — 2xy -{- 1/^. Ans. {x—y) {x — y), 4. Find the factors of 4 a^ c2 -f- 12 a c cZ + 9 d\ Ans. (2 a c + 3 rf) (2 a c + 3 rf). 6. Find the factors of 1 — A:xz -\- ^3? z^. Ans. {\ —2xz) {l—2xz). 6. Find the factors of 9 a;^ — 6 x- + 1. Ans. (3 a; — 1) (3ar— I). T. Find the factors of 25 x^ + 60 x- + 36. 8. Find the factors of 49 a^ — 14 ax -[- ^^^ Ans. (la — x) {la — x). 9. Find the factors of 16 y^ — iQa'y + 4 a*. 10. Find the factors of 12 ax + 4:X^ + 9 a"^, 11. Find the factors of 6 x + 1 + 9 a^. CASE III. 70. When a binomial is the difference between two squares. 1. Find the factors of a^ — 5^ OPERATION. We resolve this into its fac- 2 i^ / I ^\ / ^\ tors at once by the converse of the principle in Theorem IV. Art. 60. Hence, 50 KLEMENTARY ALGEBRA. RULE. Take for one of the factors the sum, and for the other the difference, of the square roots of the terms of the bi- nomial. 2. Find the factors of x^ — f. Ans. (x + 1/) {x-^y). 3. Find the factors of 4 a^ — 9 i*. Ans. (2a + 3 62) (2a — 3^). 4. Find the factors of 16 x^ — c^. 5. Find the factors of o^ 6*6'^ — x^y'^. 6. Find the factors of 81 x"^ — 49 3/^. T. Find the factors of 25 0^ — 4 t*. 8. Find the factors of m^ — n^^. Note. — When the exponents of each term of the residual factor obtained by this rule are even, this factor can be resolved again by the same rule. Thus, x* — 3/* = (^ -h V^) (^ — y^) \ but a:* — y* = (a; -f y) {x — y)\ and therefore the factors of x* — / are x' -|- y*, X -\- y^ and x — y. 9. Find the factors of a^ — h\ Ans. (a^-^lf) {a-\-h) (a — b). 10. Find the factors of a:^ — y^. Ans. {x' + y') (^' + /) (^ + y){^- y)' 11. Find the factors of a* — 1. 12. Find the factors of 1 — x\ Ans. (1 + x') (1 + x") (1 + x) (1 — ar). 13. Find the factors of a' — a^. Ans. a^(a-\- 1) {ii — 1). 14. Find three factors of x^ — a?. 71. Any binomial consisting of the difference of the same powers of two quantities, or the sum of the same odd powers, can be factored. For FACTORING. 51 I. The difference of the same powers of two quantities is divisible by the difference of the quantities. Let a and b represent two quantities and a^b, and by actual division we find (T — b"' — a-\-b, — a'j^ab-^y^, a — b ' ' ' a — b a —b a* — b' and so on. II. The difference of the same even powers of two quan- tities is divisible by the sum of the quantities. a^ — b'' a +6 a' — b* a +6 z=. a — b, — a^ — an + ab^ — W, = a'' — a'b-\-aH^ — a' h^ + a &* — b\ and so on. It follows from the two preceding statements that The difference of the same even powers of two quantities is divisible by either the sum or the difference of the quan- tities. III. The sum of the same odd powers of two quantities is divisible by the sum of the quantities. a -\-b ' "^-i^ =:a' — an + aH' — ab'+ b\ a ■\-o ' and so on. 52 KLEMENTARY ALGEBRA. 1. Find the factors of ar' — y*. OPERATION. {^ -y') -^ {^^ -y) =x' + x'y + x^f + xf + y^ By I. of this article, the difference of the same powers of two quantities is divisible by the dilTerence of the quantities; therefore X — y must be a factor ofV — if-^ and dividing or* — y* by z — y gives the other factor x* -\- x^ y -\- x"' y"- -\- x if -\- y. 2. Find two factors of c^ — d\ OPERATION. {c^ — d^) ^ {c-^d) =c^-^c^d + c'd^ — c''d^-\-cd^ — d'' By II. the difference of the same even powers of two quantities is divisible by the sum of the quantities ; therefore c -j- d must be a fiictor of c* — J^; and dividing c' — d^ by c -\- d gives the other factor c'^ — c* (/ + c' d^ — c' d^ -\. c d' — d\ 3. Find the factors of m^ -\- n^. OPERATION. (m« + n^) -r- (m + w) = m* — w^ n -[- m^ n* — 7n w^ _|_ n* By III. the sum of the same odd powers of two quantities is divisible by the sum of the quantities ; therefore m -\- n must be a factor of m' -|- n^ ; and dividing m' -\- n^ hy m -\- n gives the other factor m* — n^ n -\- rr^ r? — mr? -\- n*. 4. Find the factors of c^ — di?. Ans. (a — x) {c? -^ a X -\- x^) , 5. Find the factors of a^ + ^'^• Note. — In Example 2, the factors of c" — ^^^ the quotient thus arising will be the same as after the division of the first quantity by c — d, and can be divided hy (^ ± c d -\- d^; the third can be divided by c -f- ^j and the fourth by c — d. Performing these divisions, by each method we shall find the prime factors of c® — d® to be c-\-d, c — d, c- + c (/ + rP, and c^ — c J5 ; and let the process of dividing be as appeai-s in B) A (q the margin. Then, as the dividend is equal to qB the product of the divisor by the quotient plus ~ the remainder, A=r+qB. (1) • See Prefiwe. GREATEST COMMON DIVISOR. 55 And, as the remainder is equal to the dividend minus the product of the divisor by the quotient, r = A—qB. (2) Therefore, according to the preceding theorem, from (1) any divisor of r and B must be a divisor of .4 ; and from (2) any divisor of A and B, a divisor of r ; i.e. the divisors of A and B and B and r are identical, and therefore the greatest common divisor of A and B must also be the greatest common divisor of B and r. In the same way the greatest common divisor of B and r is the greatest common divisor of r and the remainder after dividing B by r. Hence, to find the greatest common divisor of any two quantities, RULE. Divide the greater by the less, and the less by the remain- der, and so continue till the remainder is zero; the last di- visor is the divisor sought. Note 1. — The division by each divisor should be continued until the remainder will contain it no longer. Note 2. — If the greatest common divisor of more than two quan- tities is required, find the greatest common divisor of two of them, then of this divisor and a third, and so on ; the last divisor will be the divisor sought. Note 3. — The common divisor of xy and xzia x; x is also the common divisor of x and x z, or of « a: y and xz; i. e. the common divisor of two quantities is not changed by rejecting or introducing into either any factor which contains no factor of the other. Note 4. — It is evident that the greatest common divisor of two quantities contains all the factors common to the quantities. CASE I. 75. To find the greatest common divisor of monomials. 1. Find the greatest common divisor of S a^ b^ c d, lQaH'c\ and 2S aH^ c. The greatest common divisor of the coefficients found by the gen- eral rule is 4 ; it is evident that no higher power of a than a^, of 56 ELEMENTARY ALGEBRA. b than I^, of c than itself, will divide the quantities ; and that d will not divide them ; therefore, the divisor sought is 4 a' t^ c. Hence, RULE. Annex to the greatest common divisor of the coefficients fJwse letters which are comm,on to all the quantities, giving to each letter the least exponent it has in any of the quantities. 2. Find the greatest common divisor of 63 a* U^ c* d^, 2T a^ h^ c^ and 45 a^ h^ c^ d. Ans. 9 a^ IP c\ 3. Find the greatest common divisor of Ibx'y^s^ and 125 a bx^fz\ 4. Find the greatest common divisor of 99 a b^ c* d^ x^ i/^ and 22 a' b^c^d'^x^. Ans. 11 a ly^c^d^x^. 6. Find the greatest common divisor of 11 x^y^, \9x^y^, and 2\2bx'y^z^. CASE II. 76» To find the greatest common divisor of polynomials. 1 . Find the greatest common divisor of x"^ — y'^ and x^ — 2xy -\- rf-. ■x'-f)x^-r.2xy-\- /(I x^ — .r ^2xy + 2y' Eejectmg the factor 2y X- y)x^- -xy + y xy- xy- Ans. X - -V' RULE. Arrange the terms of both quantities in the order of the poivers of some letter, and then proceed according to the general rule in Art. 74. Note 1. — If the leading term of the dividend is not divisible by the leading term of tlio divisor, it can be made so by introducing GREATEST COMMON DIVISOR. 57 in the dividend a factor which contains no factor of the divisor ; or either quantity may be simplified by rejecting any factor which contains no factor of the other. (Art. 74, Note 3.) Note 2. — Since any quantity which will divide a will divide — a, and vice versa, and any quantity divisible by a is divisible by — a, and vice versa, therefore all the signs of either divisor or dividend, or of both, may be changed from -|- to — , or — to -J-, without changing the common divisor. Note 3. — When one of the quantities is a monomial, and the other a polynomial, either of the given rules can be applied, although gen- erally the greatest common divisor will be at once apparent. 2. Find the greatest common divisor of ax' — a'^x'* — 8 a^x^ and 2 c x^ — 2 a c x^ -\- 4. a^ c x^ — 6 a^ c X — 20 a'^ c. ax' — a^x^ — 8 a'a;^] 2 ex*— 2a ca:^ + 4 a- ex"- — 6 a^ca:— 20 a^c Dividing by 2 C 8a* )X^ — a3(?-\-2a''x''' — Za'x—l0a^ {I x^ — ax? — 8 a* 2a2x2 — Sa'a; — 2a* Dividing by O 2x2 — 3 ax— 2 a^ 2 a'^ X- — 3 a^ X — 2 a* ist Rem. x^— ax*— 8 a* Multiplying by 2 2x*— 2rtx^— 16a* (x2 2x* — 3ax^ — 2 0^x2 ax^+2a2x-— 16a* Multiplying by 2 2ax' + 4a2x'^ — 32a* (ax 2ax^ — 3a'x^— 2a^x la^x^-\- 2a^x — 32a* Multiplying by 2 Ua^x^-f- 4a'x — 64a* (Za'* 14a'^x''— 21a^x— 14 a* 25 a' X — 50 a* 25a^x — 50 a* 2d Rem. Dividing by 25 a' a;_2a) 2x' — 3 ax— 2 0^(2 X -fa 2 x^ — 4 a X ax — 2 a^ ax — 2 a^ Ans. x — 2 a. 3* 58 ELEMENTARY ALGEBRA. 3. Find the greatest common divisor of a* — x* and 4. Find the greatest common divisor of a* — x* and a^ _ a^ .x2. Ans. a^ — x^. 5. Find the greatest common divisor ot'2ax^ — a^x — a' and 2 x'^ -\- Z a X -\- d\ 6. Find the greatest common divisor of Q a x — 8 a and 6 a ic^ + a or-— 12 a a:. Ans. 3 a a: — 4 a. T. Find the greatest common divisor of x* — y* and x' + f. 8. Find the greatest common divisor of 3 a:* — 2i:X — 9 and 2a:^— 16x — 6. 9. Find the greatest common divisor of x^ — y^ and x'^ — y"^. Ans. x — y. 10. Find the greatest common divisor of \Qx^ — 20x^y + 30 / and x^ -{- 2x^y + 2 xy'^ + y\ Ans. x + y. . 11. Find the greatest common divisor of a* -\- a^ -\- a^ + a — 4 and a^ + 2 a» + 3 a^ + 4 a — 10. Ans. a — 1. 12. Find the greatest common divisor of 1 ax* -\- 2lax^ -f 14 a and x^ -\- x* -\- x^ — x. Ans. x -\- I. 13. Find the greatest common divisor of 27 a^ y* — Sa^y and 3 y — 2 a / + 3 a^/ — 2 a«y^ Ans. 3y^ — 2 ay. 14. Find the greatest common divisor of n^ -\- a — 10 and a* — 16. Ans. a — 2. Note 5. — The greatest common divisor of polynomials can also be found by factoring the polynomials, and finding the product of the factors common to the polynomials, taking each factor the least num- ber of times it occurs in any of the quantities. (Art. 74, Note 4.) 15. Find the greatest common divisor of 3 a j:^ — 4oj: -|- Saxy — 4: ay and a^ x — x -\- a^ y — y. 3 ax^ — 4rax-\-3axy — 4a?/ = a{x + y) (3 j; — 4) a^r — X -\- a'^ y ~ y= (x-\-y) {a — 1) (rt-^ + a + Ans. X -f- //. LEAST COMMON MULTIPLE. 59 SECTION XI. LEAST COMMON MULTIPLE. 77. A Multiple of any quantity is a quantity that can be divided by it without remainder. 78. A Common Multiple of two or more quantities is any quantity that can be divided by each of them with- out remainder. 79. The Least Common Multiple of two or more quan- tities is the least quantity that can be divided by each of them without remainder. 80. It is evident that a multiple of an}'' quantity must contain the factors of that quantity ; and, vice versa, any quantity that contains the factors of another quantity is a multiple of it : and a common multiple of two or more quantities must contain the factors of these quantities ; and the least common multiple of two or more quantities must contain only the factors of these quantities. CASE I. To find the least common multiple of monomials. 1. Find the least common multiple of 6 a^h^c, 8 a^¥ c^df and \2a^bcx. The least common multiple of the coefficients, found by inspection or the rule in Arithmetic, is 24 ; it is evident that no quantity which contains a power of a less than a*, of b less than i*, of c less than C-, and which does not contain d and a;, can be divided by each of these quantities ; therefore the multiple sought is 24 a* h^ c^ d x. Hence, in the case of monomials, 60 ELEMENTARY ALGEBRA. RULE. Annex to the least common multiple of the coefficients all the letters which appear in the several quantities, giving to each letter the greatest exponent it has in any of the quan- tities. 2. Find the least common multiple of 3 a* IP- c^ ^ a! h'' c d"^, and l^ahcx^. Ans. Z^ a! h^ c^ d"^ i^ . 3. Find the least common multiple of X^ahx, 80ai*ar^ and Zha^hxK Ans. hm a^ I/' x\ 4. Find the least common multiple of 9a'6^ Xha^hx^, and X'^axf. Ans. ^^a^h^x^y^. 6. Find the least common multiple of ISa'^c^a:, l^aVcx'y, and Z^d'y'xz. 6. Find the least common multiple of X^^xyz, 4:5 a be, and 25 m n. *l. Find the least common multiple of 10 a'^hy'^, 13 a* IP c, and ITa^&s'c^ 8. Find the least common multiple of 14 a^ IP c*, 20 a^ h c^, 2baHc^, and 2S abed. CASE II. 81. To find the least common multiple of any two quantities. Since the greatest common divisor of two quantities contains all the factors common to these quantities (Art. 74, Note 4) ; and since the least common multiple of two quantities must contain only the factors of these quantities (Art. 80) ; if the product of two quanti- ties is divided by their greatest common divisor, the quotient will be their least common multiple. Hence, to find the least common multiple of any two quantities, LEAST COMMON MULTIPLE. 61 RULE. ^ Divide one of the quantities by their greatest common di- msor, and multiply this quotient by the other quantity, and the product will be their least common multiple. Note 1. — If the least common multiple of more than two quanti- ties is required, find the least common multiple of two of them, then of this common multiple and a third, and so on ; the last com- mon multiple will be the multiple sought. Note 2. — In case the least common multiple of several monomials and polynomials is required, it may be better to find the least com- mon multiple of the monomials by the Rule in Case I., and of the polynomials by the Rule in Case 11., and then the least common multiple of these two multiples by the latter Rule. 1. Find the least common multiple of x^ — y^ and OPERATION. Their greatest common X — y) x^ — 2x1/ -\-y^ divisor is x — y, with : which we divide one of •^ y 'the quantities ; and mul- {x^ —y"^) (x^y), Ans. tiplying the other quan- tity by this quotient, we have the least common multiple (j^ — y^) (x — y). 2. Find the least common multiple of 2a^x'^, 4cX^yt a^ — X*, and a^ — a^ x^. The least common multiple of the monomials is ^c? x^y\ and the least common multiple of the polynomials is c? (a* — x*). The greatest common divisor of these two multiples is c? ; and dividing one of these multiples by a'^, and multiplying the quotient by the other, we have 4 a^ a:^ y («* — x*) as the least common mul- tiple. 3. Find the least common multiple of 3 a^ h^, 6 a^by, a» — 8, and a^ — 4:a-{- 4. Ans. 6 a^ b^y {a^ — 8) (a — 2). 62 ELKMENTAKY ALGEBRA. 4. Find the leust common multiple of 3 x^ — 24 ar — 9 and 2x^—l6x — Q. (See 8th Example, Art. 76.) 5. Find the least common multiple of a* — x* and a^ — x^. 6. Find the least common multiple of a:* — l,x^-\-2x -{- I, and {x — ly. Ans. x"" — x^ — x^ + I. 7. Find the least common multiple of x* — y* and x^ -(- y. 8. Find the least common multiple of a^ -|- a — 10 and a' — 16. Note 3. — The least common multiple of any quantities can also be found by factoring the quantities, and finding the product of all the factors of the quantities, taking each factor the greatest number of times it occurs in any of the quantities. (Art 80.) 9. Find the least common multiple of x^ — 2xy-\-y^, X* — y*, and {x -\- yy. x^— 2xy + y^={x ~y) (x — y) x^ — y^= (j?2 -|- if) {x J^2j) {x~ y) (x + yy ={x-\-y) {x + y) Hence L. C. M = {x — y) {x~y) {x'^ + y'') {x-\-y) (ar + y) = x'' — x^y'' — x'^y'-\-y\ 10 Find the least common multiple of 3a a;* — 4aar-|- S axy — 4:ay and a^ x — x -\- a^y — y. (See 15th Example, Art. 76.) Ans. o(a? + ?/) (3^7 — 4) (a^+a + l) (a — 1) =:3a*a:2_ 4:a*x -\- Sa*xy — 4a^i/ — Sax^ -[" ^^^ — Saxy + ^ay. FKACTIONS. 68 SECTION XII. FRACTIONS. 82. When division is expressed by writing the dividend over the divisor with a line between, the expression is called a Fraction. As a fraction, the dividend is called the numerator, and the divisor the denominator. Hence, the value of a fraction is the quotient arising from dividing the numerator by the denominator. XV Thus, is a fraction whose numerator is x y and denominator y^ and whose value is x. 83. The principles upon which the operations in frac- tions are carried on are included in the following THEOREM. Any multiplication or division of the numerator causes a like change in the value of the fraction, and any multiplica- tion or division of the denominator causes an opposite change in the value of the fraction. X v^ Let — ^ be any fraction : its value = xy. y ' ' 1st. Changing the numerator. Multiplying the numerator by y, which is y times the value of the given fraction. Dividing the numerator by y, y which is — of the value of the given fraction. 64 ELEMENTARY ALGEBRA. 2d. Changing the denominator. Multiplying the denominator by y, f which is - of the value of the given fraction. Dividing the denominator by y, which is y times the value of the given fraction. Corollary. — Multiplying or dividing both numerator and denominator by the same quantity does not change the value of the fraction. For if any quantity is both multiplied and divided by the same quantity its value is not changed. rr., a: y c xy x Thus, — ^ = — - = - = r. y cy 1 84. Every fraction has three signs: one for the numer- ator, one for the denominator, and one for the fraction as a whole. Thus, +=^^ — If an even number of these signs is changed from -\- to — , or — to +> ih^ value of the fraction is not changed ; but if an odd number is changed, the value of the fraction is changed from -\- to — , or — to -\-. Thus, changing an even number, -f-a:y _ __ — xy _ _ -f-ary _ , —xy _ . ^ . but, taking FRACTIONS. 65 and changing an odd number, _ + ^y _ I —JLU — _!_ + ^y _. _ — ^y ^= — x -\-y "^ -^ y —y —y The various operations in fractions are presented under the following cases. CASE I. 85. To reduce a fraction to its lowest terms. Note. — A fraction is in its lowest terms when its terms are mu- tually prime. 1. Reduce ^,—o—o to its lowest terras. OPERATION. Since dividing both terms Ua'xy _Axy __ 2 ^^ ^ ^'^^^^°" ^^ ^^'^ '^"'« 2T^"p — 6^ — Vy quantity does not change its value (Art. 83, Cor.), we divide both terms by any factor common to them, as 4 a^ ; and both terms of the resulting fraction by any factor common to them, as 2xy\ or we can divide both terms of the given fraction by their 2 greatest common divisor ^cPxy\ the resulting fraction — is the fraction sought. Hence, RULE. Divide both terms of the fraction by any factor common to them ; then divide these quotients by any factor common to them ; and so proceed tilt the terms are mutually prime. Or, Divide both terms by their greatest common divisor. 2. Reduce — ,-'^ to its lowest terms. Ans. — x-y^ ' X 7f 6. Keduce -r,r^i — - , to its lowest terms. Ans. -—„ — 408 or X y^ 3 a' y 24 X ^ z 2 '^ 4, Reduce ~ — '— to its lowest terms. Ans. — ^ Vlaxy a 6. Reduce — — ^ ,— to its lowest terms. ol a-b X y 60 ELEMENTARY ALGEBRA. c -n J 108 3^ y- 2^ 0. Keduce 120 ab~*l? ^^^ lowest terms. jpf «« 1. Reduce ^ . », ■ a to its lowest terms. Ans. , o-Dj a» — 6* ., ^ + y o. Keduce -^- _ — i-T-T» to its lowest terms. or — 2 ao -f- tr 9. Reduce — -^ — _^ -^ — ^ to its lowest terms. . a a:* Ans X — .y c (2 x« + 4 a-) 10. Reduce — -^ — a*—"** to its lowest terms. CASE II. 86. To reduce fractions to equivalent fractions having a common denominator. 1. Reduce r— and r- to equivalent fractions having a common denominator. OPERATION. We multiply the numerator and Q a^bjc denominator of each fraction by the hy V^xy denominator of the other (Art. 83, I Cor.). This must reduce them to c c y / ^ - — 53 ^ equivalent fractions having a common denominator, as the new denominator of each fraction is the pro(iuct of the same factors. 0^> In the second operation we find the a ax least common multiple, bxy^ of the dy bxy denominators by and frar; as each de- c cy nominator is contained in this'multi- J^ j^^ pie, each fraction can be reduced to a fraction with this multiple as a de- nominator, by multiplying its numerator and denominator by the quotient arising from dividing this multiple by its denominator. Hence, FRACTIONS. 67 RULE. Multiply all the denominators together for a common de- nominator, and multiply each numerator into the continued product of all the denominators, except its own, for yiew numerators. Or, . Find the least common multiple of the denominators for the least common denominator. For new numerators, mul- tiply each numerator by the quotient arising from dividing this multiple by its denominator. 2. Reduce ; — r» and —7— to equivalent fractions xy ab aoy ^ having the least common denominator. ahm n : abxy abxy' """^ abxy . abm n X y , 7? Ans. -. — f -- ---, and — , 3. Reduce -— ,» --,-» and —^ -. to equivalent frac- 15 10 ?> c 25 ac a ^ tlons having the least common denominator. , 80 a'^ erf Aoadxy , \2hx Ans. -^ — ^,^,» _„ , • . and IbQabcd 150 abed 150 abed 4. Reduce — > > and --; to equivalent fractions m n x y 5 a ^ having the least common denominator. 5. Reduce -j-- and to equivalent fractions hav- ing the least common denominator. . a^ — 2 ab -{-b"^ ^ a^m -\- abm Ans. 7, — ^ and :r^ — 7^-^- a^ — b^ a- — 6^ 6. .Reduce ^ and — ~t_~ to equivalent fractions X — 4 X — 1 ^ having the least common denominator. Y. Reduce , ;;> — ; — . and to equivalent frac- ar — y X -\- y x — y ^ tions having the least common denominator. 68 KLEAIENTARY ALGKBKA. CASE III. 87. To add fractions. h c 1. Find the sura of - and -• X X OPERATION. If anything is divided into equal 5 c h -\- c parts, a number of these parts rep- ~x ~^ 1: X resented by h, added to a number represented by c, gives h -\- c of these parts. In the example given, a unit is divided into x equal parts, and it is required to find the sum of h and c of these parts ; i. e. h ^. c h -\- c X ~^ X x It is evident, therefore, that fractions that have a common denom- inator can be added by adding their numerators. But fractions that do not have a common denominator can be reduced to equivalent fractions having a common denominator. Hence, RULE. Reduce the fractions, if necessary, to equivalent fractions having a common denominator; then icrite the sum of the numerators over the common denominator. o Ajj^ ^ J ^ A bmy-\-bnx-\-any 2. Add -; -, and ^- Ans. ' . — — ' -• n y h bny 3. Add -— > ,^> and — -• 1 Ah 2d A Ajj3a& 2x5/ J 7 m 4. Add . - , - — •', and Axy b ab S abx y g' b' - 40 abx y 30a'i-4- 16r»y2_i_35m Ans. — ' ■ 5. Add - {, - — J, and ^_ — 3 a* dcd 27 a c 6. Add - — I — and :; — — Ans. ; y 1 -|- a 1 — a 1 — a« 7. Add , *" ^ and , "7° - Ans. , . - 1 — a l-f-a 1 — o^ FRACTIONS. 69 8. Add ~"t" and — ^. 7 1 -j- a:^ 9. Add ^ ,,\ and 10. Add > > and ^^ Ans 1. xy 7JZ xz 11. Add ^r-i-. and -P^,' sr — y x^ — 7/ Ans. ^ + y 7 X 12. Add m x and ^- • 18 a Note. — Consider w x = — , and then proceed as before. . 18 a TTZ a: -I- 7 X Ans. ' 7 /v. 13. Add X -\- y and 18 a 7 a: a-j-^; 14. Add x'' + 2xy + if2indi --^- . Ans ^'' -\-^y — ^f — f-\- 1 , a: — y CASE IV. 88. To subtract one fraction from another. c h 1. Subtract - from -• X X OPERATION. If anything is divided into x I) c h — c equal parts, a ' number of these X X X parts represented by c*, subtracted from a number represented by &, leaves b — c of these parts ; i. e. = Hence, XX X B U L E . Reduce the fractions, if necessary, to equivalent fractions having a common denominator; then subtract the numerator of the subtrahend from that of the minuend, and write the result over the common denominator. 70 ELEMExNTARY ALGEBRA. o a 1 . . 7a: „ ab . ah — \Acx 2. Subtract -7- from — • Ans. x 4 8 c 8 C 3. Subtract =— from _ 7 X 6 ax 4. Subtract ,„ „ from -7— • 19x2 19a r d 1.. , 29 ac ^ 39 a: . 273 a:" — 116 acw 5. Subtract -77-, from Ans. ^77-5 ^ 11 2 a 6. Subtract , from -7- • Aus. .- — -. 1— a l-|-a a^ — 1 7. Subtract -'^—^ from ,- • X — 1 X -\- 1 o o u^ , ab-\-bc „ ab — be 8. Subtract , - -,c— from „ „ — „-»• a^x — b^x a^a^ — i^x^ 9. Subtract r from — r—r- Aus. a—b a -f 6 6= — a* 10. Subtract ^7 from ^ *" • Ans. , , , - X* — 1 a;- — 1 ar-f-1 x^ 7 11. Subtract 16 from , -^ I -\- X Note. — Consider IG = — -, and then proceed as before. . ar^— 16 a;— 23 Ans. ^ 3 12. Subtract , from xy. a — b ^ 13. Subtract x -\- b from ^ , , - Ans. , , . ■ ' 4- 4 0-4-4 CASE V. 89. To reduce a mixed quantity to an improper fraction. 1. Reduce x -\- ^ to an improper fraction. OPERATION. As eight eighths make + a Bx , a 8ar-|-a a unit, there will be in 8 8 ^^ 8 8 X units eight times x ... . 8x ,8a:, a 8a:-l-a -_ eighths ; 1. e. x = — ; and j -f - = — ^ — . Hence, FRACTIO^^S. 71 RULE. Multiply the integral part by the denominator of the frac- tion ; to the product add the numerator if the sign of the fraction is plus, and subtract it f the sign is minus, and under the I'esuU write the denominator. Note. — By a change of the language, Examples 12-14 in Art. 87, and 11-13 in Art. 88, become examples under this case. Thus, Example 12, Art. 87, might be expressed as follows: Reduce ma:-f-,cj to an improper fraction. 7 2. Reduce x^ + 4 to an improper fraction. Ans. ^ ~ — y O, I X 3. Reduce 25 a — 2b x -\- — T— to an improper frac- tion. 4. Reduce a — 1 -J- to an improper fraction. . a' — a Ans. a + l ^ X 11 I 2,*^ 5. Reduce y -\ "^ ' to an improper fraction. 6. Reduce — ^ {a -\- h) to an improper fraction. . a^ — ah Ans. 7 x^ J- \ T. Reduce x — 1 — ~- to an improper fraction. Note. — It must be remembered that the sign before the dividing line belongs to the fraction as a whole. - x'-\-l x'—i—x' — l —2 2 X — 1 ^^ r— = — j— T'or j— -' Ans. X-\-l X -\- I x-\-l x-{-l x^ -\- I 8. Reduce a: + 1 — /* to an improper fraction. 9 Ans. 1 —X 72 ELEMENTARY ALGEBRA. 2 2? a' 9. Reduce x^ — 2ax -\- a^ — — to an improper fraction. Note. — According to the same principle an integral quantity can be reduced to a fraction having any given denominator, by multiplying the quantity by the proponed denominator, and under the product writing the denominator. 10. Reduce a; -j- 1 to a fraction whose denominator is X — 1. t, ^ — 1 Ans. - — - • X — 1 11. Reduce x — 1 to a fraction whose denominator is a — h. 12. Reduce 4: ax to a fraction whose denominator is a' — z. CASE VI. 90. To reduce an improper fraction to an integral or mixed quantity. ^ 4 fl X I 5 c^ 1. Reduce _ ' to an integral or mixed quan- tity. OPERATION. a' X — 2a)x'^ — i: a X -\- b a^ {x — 2 a -\- 2a x^ — 2 ax — 2ax + 5a^ — 2ax + 4a^ As the value of a fraction is the quotient arising from dividing the numerator by the denominator (Art. 82), we perform the indicated division. Hence, RULE. Divide the numerator by the denominator ; if there is any remainder, place it over the divisor, and annex the fraction so formed with its proper sign to the quotient. FRACTIONS. 73 2. Reduce to an integral or mixed quantity. Ans. a — 4 6. 3. Reduce ^. -^ to an integral or mixed quantity. 4. Reduce — ^^ to an integral or mixed quantity. 5. Reduce — „ — ' — to an integral or mixed 0*2 j /Tj nr I. /J* 6. Reduce —^ — — ^- — to an integral or mixed quan- tity. ^ -, , 8 a a; — 106a; — 5cx ^ • . i • j Y. Reduce to an integral or mixed £i X quantity. 8. R ^i^^^^^y- ' Ans. 2a -26--^ 8. Reduce ^-—r to an integral or mixed 2a — 26 ° a — 6 x' if 9. Reduce — - to an integral or mixed quantity. X y x^ ?/' 10. Reduce - to an integral or mixed quantity X y CASE VII. 91. To multiply a fraction by an integral quantity. 1. Multiply ^±^ by c. OPERATION. According to the theorem x^y cx + cy ^" ^^^' ^^' "multiplying the q^ X g — ~ab^ numerator by c multiplies the value of the fraction c times. 74 ELEMENTARY ALGEBRA. 2. Multiply ~+ ^ by a. OPERATION. According to the theorem ah ^ ^ — — h — nominator by a multiplies the value of the fraction a in Art. 83, dividing the de- nominator by the value of th times. Hence, RULE. Divide the denominator by tlie integral quantity when it can be done without remainder; otherwise, multiply the nu- merator by the integral quantity. 3. Multiply ,1 ,-^ by m + n. Ans. -^- ^• m — n 4. Multiply ^^i^^by ai. 5- Multiply 3^^ by %y. Note. — Any factor common to the denominator and multiplier may be cancelled from both before multiplying. -7. Multiply ,lJ[^y.^ by l4(^*-y^). Ans. 2(x2 — ^2) (a + ar). 8. Multiply ^-t^ by x — y. Note. — When a fraction is multiplied by a quantity equal to its denominator, the product is the numerator. — X(.r-i,) = -+-^ = x + ^, Ans. FRACTIONS. T5 9. Multiply "^Stlll+Jl by (x - ay. 10. Multiply "^^ hj x' — 2xi/ + f. X y Ans. {a -\- b) {x — y). CASE VIII. 92. To multiply an integral quantity by a fraction. 1. Multiply x2 + 2xy + 2/' by ^-p^- OPERATION. 4(a:2 + 2:ry + 2/') - (^ + y) = ^ (x + y) We first multiply the multiplicand by the numerator 4 ; but the multiplier is 4 -^ (a; -[- ?/) ; and therefore this product is x -f- ^ times too great, and this product divided hy x -\- y must be the product sought. It is evident that the result would be the same if the division were performed first, and the multiplication afterward. Hence, RULE. Divide the integral quantity hy the denominator when it can he done without remainder, and multiply the quotient hy the numerator. Otherwise, multiply the integral quantity hy the numerator, and divide the product hy the denom- inator. 2. Multiply a« - 3 a^ & + 3 « 6^ - ^>^ by - ^, _ '^^^_^ ^, Ans. 7 X {a — h). 3. Multiply a' — x' ^y^^If^- 3 4. Multiply 7 a^ — 4 a?y by ^ o_^ • 5. Multiply \n{x^ — f) hy^-^l- 76 ELEMENTARY ALGEBRA. Note. — Since the product is the same, whichever quantity is con- sidered as tiie multiplier, by considering the inte<^ral quantity as the multiplier, Case VIII. becomes the same as Case VII. CASE IX. 93i To divide a fraction by an integral quantity. Divide -, by a. OPERATION a 1 6^« = 6- According to the theorem in Art. 83, dividing the numerator by a de- creases the value of the fraction a times. Divide r- by c. OPERATION. a re According to the theorem in Art 83, multiplying the denominator by c decreases the value of the fraction c times. Hence, RULE. Divide the numerator by the integral quantity when it will divide it without rernainder ; otherwise, multiply the denom- inator by the integral quantity. 3. Divide ^ -y- by a. 7 X* 4. Divide ,, by 14^^^- 6. Divide -^ — by Q abc. 6. Divide ^^ by 9 a 6y*. 62 y 1. Divide jf§^5 by 2 (« + x) (x + y). Ans. Ans. Ans. Ans. 46c 2/ 3 J 32 6»y» 13(x-|-y)« FRACTIONS. 77 CASE X. 94. To divide an integral quantity by a fraction. 1. Divide x by - X . , ,. . . a: -7- a = - ; but the divisor is not a, OPERATION. X but a -^b. Dividing hs a, therefore, iC * 05 - .... . a is dividing by a divisor & times too X hx great, and the quotient will be h times a^ ~a *oo small; therefore the quotient sought . X - hx ^^ IS - X o = — • Hence, a a RULE. Divide the integral quantity by the numerator, and mul- tiply the quotient by the denominator, 2. Divide 4: ax by — -• Ans. — -• 3. Divide 7 x'^ by —f— • Ans. — r— • 4. Divide a + 5 by - . Ans. ""J^tAl, . 5. Divide a'-\-2ax -|. x^ by "^i^- 6. Divide 2:'' — hx^ by -• 7. Divide 2a:' + 3/ by ^^±1. 8. Divide 1 by -. Ans. ^. Note. — Hence, the reciprocal of a fraction is the fraction inverted, CASE XI. 95. To multiply a fraction by a fraction. a , X 1. Multiply ^ by -. 78 ELEMENTARY ALGEBRA. OPERATION. We first multiply 7 by ar ; but the a ax multiplier is not x, but x ~ y^ there- b b fore the product is y times too great ; ^ _^ „ ^^ and -~ — y = -— (Art. 93) must be b ' ^ ~ by the product sought. Hence, RULE. Multiply the numerators together for a new numerator, and l}ie denominators for a new denominator. 'Note 1. — Common factors in the numerators and denominators may be cancelled before multiplication. Note 2. — Cases VII. and VIII. can be included in this by writ- ing the integral quantity as the numerator of a fraction, with a unit as the denominator. Note 3. — Mixed quantities may be reduced to improper fractions before multiplying. 2. Multiply , by ^ • Ans. be '^ dy ' bcdy 3. Multiply ^/, by ^^y A HT ix' 1 rri^ X , ac 4. Multiply —r- by ^ "^ abc '' mx 6. Multiply J^ by — - — Ans. — ^ — 6. Multiply ^^ by ^^^. ^ -^ Axyz "^ X — y 1. Multiply -j-^-+*.-j,; by ^ J-j. 8. Multiply ^fby ^^-. 9. Multiply ^^^ by ,,,^. A... ^^ FRACTIONS. 79 10. Multiply ;,,^_4 by j^^.^,^,^^ - 11. Multiply -7^^4- by jl,. Ans. ^,-^-^. 12. Multiply ^'by^\ 13. Multiply 3, 4- ^4^ by 2, - ^. Ans. -Y^ 3^. 14. Multiply together ^ ^ ^^ip^' aad ^+^. Ans. 1, 15. Multiply together ''J^^ a^4z7^' *"^ ^' 16. Multiply together a + - » 6 + - , and y — ^ • CASE XII. 96t To divide a fraction by a fraction. 1. Divide - by -r • y -^ b OPERATION. r r --4-a=— - (Art. 93); but the XX y c-y -T- a •=■ — ,. . . , a t y ay divisor is not a, but -; we nave used b _^ sy 1 ^ ^ divisor b times too great, and there- "^ ^ fore the quotient — is 6 times too ay X b X Bmall, and the quotient sought is — X & == ^ (Art. 91). It will ' ^ ° ay ay ^ ■' be noticed that the denominator of the dividend is multiplied by the numerator of the divisor, and the numerator of the dividend by the denominator of the divisor. Hence, RULE. Invert the divisor, and then proceed as in multiplication of a fraction by a fraction. 80 ELEMENTARY ALGEBRA. Note 1. — All cases in division of fractions can be brought under this rule, by writing integral quantities as fractions with a unit for the denominator. Note 2. — After the divisor is inverted, common factors can be cancelled, as in multiplication of fractions. Note 3. — Mixed quantities should be reduced to improper frac- tions before division. 2. Divide - by - • Ans. c "^ n cm ar* a^ 4 3. Divide - by t* -A-ds. -^~ - 4. Divide -—=— by 17 1^ -^ 3Ai/ 5. Divide ^^ '. — by _ -• Ads. — — . . ' _a • 6. Divide ^^^3^ by -^ X — ■y a- — ft» 4 X mr — n" n. Divide ^ by ^+^. Ads. ^^^ 8. Divide ^-±^ by 9. Divide -rr—x — X oy — i Ads. ^ j — j- 10. Divide ^^^ by ^^ • X -{- 1 '' 4 11. Divide -^ by ^7^, 12. Divide -^ r 5-^ — 5 by — ^ -^^ Ads. 3(m« + n«). 13. Divide 1 + ^-±^ by* — ^^ 1. A„8. ^J^4:y) + (^ + y)' c(c — x — y) 14. Divide x + y — -^ by ;^ + a: -4- y. FRACTIONS. 81 15. Dmde ^-^ + ^-^ by j-p^. Ads ^^^+^y Note. — The division of fractions is sometimes expressed by writ> a ing the divisor under the dividend. Thus, — . Such an expression y is called a Complex Fraction. A Complex Fraction can be reduced to a simple one by performing the division indicated. 16. Reduce -^ to a simple fraction. Ans. =— • 1 ^ 7 X 5 i+? It. Reduce r- to a simple fraction. 7 . cx-\- ac c Ans. ' — r- . i ex — ox 18. Reduce ^_ ^ to a simple fraction. x-\-l 1 19. Reduce — . . to a simple fraction. 1 —X a^ — V" 20. Reduce ^ , '^^ to a simple fraction. Ads. — i xJry 21. Reduce — ^^ to a simple fraction. x — y ^ a + 6 Ans. {x -\- y) {a -\- b) . ]S5"0TE. — A Complex Fraction can also be reduced by multiplying its numerator and denominator by the least common multiple of the denominators of the fractional parts. Thus, if both terms of the fraction in Ex. 16 be multiplied by 5 a:, or both in Ex. 17 by car, the result will be the same as above. 4* F 82 ELEMENTARY ALGEBRA- SECTION XIII. EQUATIONS OF THE FIRST DEGREE CONTAINING BUT ONE UNKNOWN QUANTITY. 97. An Equation is an expression of equality between two quantities (Art. 9). That portion of the equation which precedes the sign = is called the first member, and that which follows, the second member. 98. The Degree of an equation containing but one un- known quantity is denoted by the exponent of the highest power of the unknown quantity in the equation. An equation of the first degree, or a simple equation, is one that contains only the first power of the unknown quantity. For example, An equation of the second degree, or a quadratic equation, is one in which the highest power of the unknown quantity is the second power. For example, x^ — aa; = i-|-c, or ax^ — Jr=17. An equation of the third degree, or a cubic equation, is one in which the highest power of the unknown quantity is the third power, and so on. 99. The Reduction op an Equation consists in finding the value of the unknown quantity, and the processes involved depend upon the Axioms given in Art. 13. The processes can be best understood by considering an equation as a pair of scales which balance as long as an equal weight remains in both sides : whenever on one side any additional weight is put in or taken out, an equal weight must be put in or EQUATIONS OF THE FIRST DEGREE. 83 taken out on the other side, in order that the equilibrium may remain. So, in an equation, whatever is done to one side must be done to the other, in order that the equality may remain. 1. If anythinp^ is added to one member, an equal quantity must be added to the other. 2. If anj^thing is subtracted from one member, an equal quantity must be subtracted from the other. 3. If one member is multiplied by any quantity, the other member must be multiplied by an equal quantity. 4. If one member is divided by any quantity, the other member must be divided by an equal quantity. 5. If one member is involved or evolved, the other must be involved or evolved to the same degree. TRANSPOSITION. lOOo Transposition is the changing of terms from one member of an equation to the other, without destroying the equality. The object of transposition is to bring all the unknow^n terms into one member and all the known into the other, so that the unknown may become known. 1, Find the value of x in the equation x -\- 16 = 24, Subtracting 16 from the first OPERATION. member leaves x ; but if 1 6 is sub- a: -f- 16 = 24 tracted from the first member, it ar = 24 — 16 = 8 must also be subtracted from the second. 2. Find the value of x in the equation x — h^=a. OPERATION. Adding h to the first member ^ J = a gives x ; but if h is added to the ^ 1 T first member it must also be added X = a -\- to the second. 84 ELEMENTARY ALGEBRA. 3. Find the value of x in the equation 2 x == x -\- 16. OPERATION. 22; z= X 4- 16 Subtracting x from both mem- 2x xzz^ia bers, we have 2ar — x= 16, or i« a: =16. cc = 16 It appears from these exan)ples that any term which dis- appears from one member of an equation reappears in the other with the opposite sign. Hence, RULE. Amj term may be transposed from one member of an equation to the other, provided its sign is changed. 4. Find the value of x in the equation 8a: — 15 = 4a; + o. OPERATION. 8a; — 15 = 4a:-f 5 Transposing, 8a: — 4a:=r 5 +15 Uniting terms, 4x=20 Dividing both members by 4, a:= 5 5. Find the value of a: in 4 a: -f- 46 = 5 a: -|- 23. Note. — Reducing, we liave — x = — 23. If each member of this equation is transposed, we shall have 23 = a: ; i. e. 23 equals X, or X equals 23. Dividing both members by — 1 will give the same result. Hence, the sic/ns of all the terms of an equation may be changed without destroying the equality. 6. Find the value of x in 17 a; + IT = 19 or + 13. Alls. x=z2. 1. Find the value of a; in 8 a: — 14 = 13 a: — 29. 8. Find the value of a: in 5 a: + 25 = 10 ar — 25. Ans. X = 10. 9, Find the value of .r in 24a; — IT = 11 a: + 74. 10. Find the value of x in 3T a: — (4 + T) = 41 a: — 23. EQUATIONS OF THE FIRST DEGREE. 85 CLEARING OF FRACTIONS. 101 • To clear an equation of fractions. 1. Find the value of x in the equation - — 2 = --J- 1. o o OPERATION. If the given equation is mul- ^ 2 = - 4- 1 tiplied by 6, the least common ^ ^ multiple of 6 and 3, it will give 2x — 12 =zx -{- 6 2x — 12 = a:-|-6, an equation a; 1= 18 without a fractional term. Hence, RULE. Multiply each term of the equation by the least common multiple of the denominators. Note 1. — In multiplying a fractional term, divide the multiplier by the denominator of the fraction and multiply the numerator by the quotient. Note 2 — An equation may be cleared of fractions by multiplying it first by one denominator, and the resulting equation by another, and so on, till all the denominators disappear ; but multiplying by the least common multiple is generally the more expeditious method. Note 3. — Before clearing of fractions it is better to unite terms which can readily be united; for instance, the equation in Ex. 1, by transposing — 2, can be written ^ = ^ -|- 3. o o Note 4. — When the sign — is before a fraction and the de- nominator is removed, the sign of each term that was in the nu- merator must be changed. 2. Given ^ — ^ + 25 = 33 — ^^ • # OPERATION. Transposing 25, X 4 X 5 ~ = 8 — - — 6 2 Multiplying by 20, bx- -4:r = = 160- -10a; + 60 Transposing and uniting, 11.2r = = 220 Dividing by 11, aj = = 20 86 ELEMENTARY ALGEBRA. X Note. — The sign of the numerator of — r is -f"* ^"^ '""st be o changed to — when the denominator is removed ; for — (-j- 4 x) = — A x\ and so the sign of each term of the numerator of the fraction X — 6 — — - — must be changed when the denominator 2 is removed; for — (-j- 10 a; — 60) = — 10 a: -[- 60. 102i To reduce an equation of the first degree contain- ing but one unknown quantity, we deduce from the preced- ing examples the following RULE. Clear the equation of fractions, if necessary. Transpose the known terms to one member and the un- known to the other, and reduce each member to its simplest form. Divide both members by the coefficient of tJie unknown quantity. Note 1. — To verify an equation, we have only to substitute in the equation the value of the unknown quantity found by reducing the equation. For instance, in Ex. 2, Art. 101, by substituting 20 for X, . X m-- 1 + 25 = 33 X — 6 — — , we have 20 4 ■f + 25 = 33 20 — 6 2 ' 5 - -4 + 25 26 = 33 = 26. — 7, Note 2.- ed. - When answers are ffbt | given, the work shoul 103. Since the relations between quantities in Algebra are often expressed in the form of a proportion, we intro- duce here the necessary definitions. ' EQUATIONS OF THE FIRST DEGREE. 87 104. Ratio is the relation of one quantity to another of the same kind ; or, it is the quotient which arises from di- viding one quantity by another of the same kind. Ratio is indicated by writing the two quantities after one another with two dots between, or by expressing the division in the form of a fraction. Thus, the ratio of a to h is written, a : b, or - ; read, a is to b, or a divided by b. 105. Proportion is an equality of ratios. Four quan- tities are proportional when the ratio of the first to the second is equal to the ratio of the third to the fourth. The equality of two ratios is indicated by the sign of equality (=) or by four dots (::). (1 c Thus, a : b ^= c : d, ov a : b: : c : c?, or y = -^ ; read, a to 6 equals c to d, ,or a is to 6 as c is to d, ova divided by b equals c divided by d. The first and fourth terms of a proportion are called the extremes, and the second and third the means. 106. In a proportion the product of the means is equal to the product of the extremes. Let a : b z= c : d i. e. Clearing of fractions, A proportion is an equation ; and making the product of the means equal to the product of the extremes is merely clearing the equation of fractions. Examples. . 1. Reduce ^ + 10 = I-^ + 13. Ans. rr = 30, 2. Reduce 17 a; — 14 = 12 cc — 4. Ans. x = 2. a b c d ad be 88 ELEMENTARY ALGEBRA. 3. Reduce 6 a: — 25 + x = 135 — 3 a: — 10. Ans. X =^ 15. 4. Reduce 3a: + 5 — a: = 38 — 2a:. Ans. x — 8f 5. Reduce ~~- + | = 30 — "^-i^- Ans. x = 12. 31 3 X 6. Reduce x — Tt]- = • Ans. x = IlyV- 7. Reduce f + ^ + '^ + ^= 154. Ans. x = 120. 2 3 4 5 8. Reduce ^ + ^ = 16 + ^ . Ans. x = 24. 9. Reduce --l-«=r- — ' -4- d. OPERATTON. Multiplying by5c^, cAx-|-«ic^==:icx — b hx-\-bc dh Transposing, c kx — b c x-\-hhx=ibc d h — abch Factoring 1st mem., (c A — bc-\-bh)x=zbc d h — abch b c dli — abch Dividing by coeflScient of x, x = c h — b c -\-b h 10. Reduce x-\-mx = c. Ans. x = —-, — . ' 1 -f- ^» 11. Reduce 3 = 7. Ans. x = ~ » X 10 12. Reduce — \--z=x. Ans. x= - •" . a c a c 13. Reduce 4- - r= c. Ans. a: = ^ *" • X ^ X c 14. Reduce 8 = -^—- + 6. Ans. x = 9. x— 2 "^ ieT.j 23a . «4-l 15. Reduce = c. Ans. x = — ' — • XXX c EQUATIONS OF THE FIRST DEGREE. 89 16. Reduce | -f | + | = 39. 17. Reduce — --l-c = d. a ' X 1 18. Reduce {a — &) x -| — = - • 19. Reduce ar — /| — |) = 6. Ans. a: = 6. 20. Reduce 6 — ^^^ = x — 4. 5 oi T? J o Ox — 29 _Q 6a: -1-11 21. Reduce 2 x ^ == 18 i 4 . 5 Ans. a; = 9. ■ ^ _, - • 117— X „ a:— 95 22. Reduce = 3 a? -[- — 23. Reduce 2 a: ^^ — 14 ^"p— ^"s. x=zb. 24. Reduce 6x-hn-| = 9^-T+^- Note. — Before clearing of fractions, transpose 7^ and unite it .,,, X .....llx with 9^; also transpose — -, and unite it with — -. 25. Reduce 4 a: -j ' — = 5 -| ^- 26. Reduce ^^ — ^^ = 21 — ^i?. Ans. x = 39. 2 6 6 27. Reduce — [- -: -\- - = d. Ans. a: = r — ; ; — v a * o ^ c be -^ ac-\- ab 28. Reduce — ~- — 6 = ^^ 4- Y. 3 4 ' onT^j a;— 1 _ 22 — a: 34-a: . _ 29. Reduce — - — = 6 ^ i — Ans. a: = 7. 6 5 5 90 ELEMENTARY ALGEBRA. on r> ^ in I 2a;— 22 3x—75 , 284 — 4 a: 30. Reduce 19 -| — = 1 ~ . oi -D J 4a: 4-5 5a: — 5 x 4- 1 31. Reduce — s^ — = — ~ 1. 5 4 o Ans. X = 5. «« T^ 1 18 — 5a: 3x4-3 * ,^ , 5x4-3 32. Reduce ^ ^ = 4a: — 17 -\ ^• «« T. J . a:— 12 , ^ 20x4-21 1 33. Reduce 4 a; h ^ == ?- t • o ' 4 4 34. Reduce = - • Ans. x = X c m ' bm -\- cd X CL X 35. Reduce v ^^ := 1 — 3 a c. h c oa ^ A 5x'-|-3 , _ 3X-I-15 6x4-10 36. Reduce — ^^— -f- 6 ^ = 4 + — ^— • OK r» 1 o 3x— 19 „ 23— X , 5x — 38 , ,^ 37. Reduce 3 a; ^r 8 = ^- 10. J 4 3 Ans. X = 19. «o T. J 13 — 3x 3x4-2 ^ ^ , 8x— 13 38. Reduce ,„ J— = 7 — 6 x 4 «^ T, J 4 (x — 7) , 3 (x 4- 1) 7 X — 1 7 , X 39. Reduce -^-^ + -^j"^^ = — ^y- + 21 * ^^ _, , 4x — 6 , _ 19— 4x 5x — 6 , 7x-|-8 40. Reduce a: \-^ = — ^3 j 1 ^ — 41. Reduce -+tH [--> = '». a * b * c ' d .0 r» ^ 7X-I-5 , 6x— 30 . , 42. Reduce - ^' 1- ^ ^ __^ = a: + 1. Note. — Multiply by 7, transpose, and unite. 43. Reduce 2 (3 -f a:) : 6 a: — 9 = 2 : 3. Ans. x= 6. 44. Reduce I + I : ,-^+4^ = 11 : I- 45. Reduce b : c 4- d = in. EQUATIONS OF THE FIRST DEGREE. 91 PROBLEMS PRODUCING EQUATIONS OF THE FIRST DEGREE CON- TAINING BUT ONE UNKNOWN QUANTITY. 107i The problems given in this Section must either con- tain but one unknown quantity, or the unknown quanti- ties must be so related to one another that if one be- comes known the others also become known. 108i With beginners the chief difficulty in solving a problem is in translating the statements or conditions of the problem from common to algebraic language ; i. e. in preparing the data, and forming an equation in accord- ance with the given conditions. 1. If three times a certain number is added to one half and one third of itself, the sum is 115. What is the number ? SOLUTION. Let X z= the number ; then by the conditions of the problem, 3^+1 + 1 = 115 Clearing of fractions, lSx-\-3x-\-2x=. 690 Uniting terms, 23 x = 690 Dividing by 23, a: = 30 VERIFICATI(5N. 3X 30 + ^ + ^^=115 115 = 115 In this problem there is but one unknown quantity, which we rep- resent by X. 2. There are three numbers of which the first is 6 more than the second, and 11 less than the third ; and their sum is 101. What are the numbers? 92 ELEMENTARY ALGEBRA. SOLUTION. Let X ■^= the first, In this problem then X — 6 = the second, there are three un- and y + 11 = the third. known quantities ; mi • ^i i c if^V but they are so re- Their sum, 3 X -4- 6 = 101 , , -^ - _ lated to one an- 3 a; = 96 *u *u * r other that, if any x= 32. the first, one becomes known, X— 6 = 26, the second, the other two will a: + 1 1 = 43, the third. be known. VERIFICATION. 32 + 26 + 43 = 101 101 = 101 From these examples we deduce the following GENERAL RULE. Let X {or some one of the latter letters of the alphabet) represent the unknown quantity ; or, if there is more than one unknown quantity, let x represent one, and find the others by expressing in algebraic form their given relations to the one represented by x. With the data thus prepared form an equation in accord- ance with the conditions given in the problem. Solve the equation. The three steps may be briefly expressed thus : — Ist. Preparing the Data ; 2d. Forming the Equation ; 3d. Solving the Equation. 3. The sum of three numbers is 960 ; the first is one half of the second and one third of the third. What are the numbers ? Ans. 160, 320, and 480. 4. Find two numbers whose difference is 18 and whose Bum 112. Ans. 47 and 65. EQUATIONS OF THE FIRST DEGREE. 93 6. A man being asked how much he gave for his horse said, that if he had given $ 70 more than three times as much as it cost, he would have given $445. How much did his horse cost him ? 6. A man being asked how many sheep he had, replied that if he had as many more, and two thirds as many, and three fifths as many, he should have 8 more than three times as many as he had. How many sheep had he ? 7. Divide $575 between A and B in such a manner that B may have two thirds as much as A. Ans, A's share, S 345 ; B's " $230. 8. A father divided his estate among his three children so that the eldest had $ 1440 less than one half of the whole, the second $500 more than one third of the whole, and the youngest $250 more than one fourth of the whole. What was the value of the estate ? SOLUTION. Let X = whole estate. Then X 2 "~ 1440 = share of the eldest, 1 + 500= " " " second, 2 + 250 = " " " youngest, Their sum 13x 12 690 = X, whole estate. X = 8280, whole estate. 9. A gentleman meeting five poor persons, distributed $7.60, giving to the second twice, to the third three times, to the fourth four times, and to the fifth five times as much as to the first. How much did he give to each ? 94 ELEMENTARY ALGEBRA. 10. Divide 795 into two such parts that the greater di- vided by 3 shall be equal to the less divided by 2. Note. — To avoid fractions, let 3 a: = the greater and 2x=the less. Ans. 477 and 318. 11. Divide a into two such parts that the greater di- vided by b shall be equal to the less divided by c, SOLUTION. Let X := the greater, then a — x = the less. . - X a — X And T = b c Clearing of fractions, c x ■= ab — bx Transposing, bx -{- c x :=z ab Dividing by b -\- c, x = ■, the greater, ab ac ^ , , a — x^=. a — , ,- = ,— r^-, the less. b-\-c b-\-c 12. What number is that which, if multiplied by 7, and the product increased by eleven times the number, and this sum divided by 9, will give the quotient 6 ? 13. If to a certain number 55 is added, and the sum divided by 9, the quotient will be 5 less than one fifth of the number. What is the number? Ans. 125. 14. As A and B are talking of their ages, A says to B, "If one third, one fourth, and seven twelfths of my age are added to my age, the sura will be 8 more than twice my age/' What was A's age ? 15. A farmer having bought a horse kept him six weeks at an expense of $20, and then sold him for four fifths of the original cost, losing thereby % 50. IIow much did he pay for the horse? Ans. S150. 16. A man left $ 18204, to be divided among his widow, three sons, and two daughters, in such a manner that the widow should have twice as mtich as a son, and each son as much as both daughters. What was the share of each ? EQUATIONS OF THE FIRST DEGREE. 95 n. If a certain number is divided by 9, tiie sum of the divisor, dividend, and quotient will be 89. What is the number? Ans. 72. 18. If a certain quantity is divided by a, the sum of the divisor, dividend, and quotient will be h. What is the quantity? ^^^ a h - a\ • a-fl 19. Verify the answer to the preceding problem. 20. A farmer mixed together corn, barley, and oats. In all there were 80 bushels, and the mixture contained two thirds as much corn as barley and one fifth as much bar- ley as oats. How many bushels of each were there ? 21. Three men, A, B, and C, built 572 rods of fence. A built 8 rods per day, B 7, and C 5. A worked one half as many days as B, and B one third as many as C. How many days did each work ? 22. What number is as much greater than 340 as its third part is greater than 34 ? Ans. 459. 23. A man meeting some beggars gave 3 cents to each, and had 4 cents left. If he had undertaken to give 5 cents to each, he would have needed 6 cents to complete the dis- tribution. How many beggars were there, and how much money did he have ? SOLUTION. Let X =z the number of beggars ; then, according to the first statement, 3 a: -|- 4 = the number of cents he had, and, according to the second statement, ^b X — 6 = the number of cents he had. Therefore, 5a? — 6 = 3a; + 4 2^=10 X =z b, the number of beggars, and 3ar -|- 4 = 19, the number of cents he had. 96 ELEMENTARY ALGEBRA. 24. A boy wishing to distHbute all his money among his companions gave to each 2 cents, and had 3 cents left ; therefore, collecting it again, he began to give 3 cents to each, but found that in this case there was one who had received none, and another who had only 2 cents. How many companions, and how much money had he ? Ans. 7 companions, and 17 cents. 25. What two numbers whose difference is 35 are to each other as 4 : 5 ? 26. A man being asked the hour, answered that three times the number of hours before noon was equal to three fifths of the number since midnight. What was the time of day ? SOLUTION Let X = the number of hours since midnight, i. e. the time ; then 12 — x =z the number of hours before noon. Then 36— Sx=^-^ o Clearing of fractions, 180 — 15 a: = 3 a: Whence 18 a: =180 X = 10. Ans, 10 o'clock. 27. A gains in trade $300; B gains one half as much as A, plus one third as much as C ; and C gains as much as A and B. What is the gain of B and C ? Ans. B's, $375; C's, $675. 28. What number is to 28 increased by one third of the number as 2 : 3 ? Ans. 24. 29. What number is that whose fifth part exceeds its sixth by 15? 30. Divide $3740 into two parts which shall be in the ratio of 10:7. 31. Divide a into two parts which shall be in the ratio of b : a . ab ., ac Ans. ,— . — and bJ^c b-irc EQUATIONS OF THE FIRST DEGREE. 97 32. What number is that the sum of whose fourth part, fifth part, and sixth part is 37 ? 33. What quantity is that the sum of whose third part, fifth part, and seventh part is « ? ^^^ 105 a 34. A farmer sold IT bushels of oats at a certain price, and afterward 12 bushels at the same rate ; the second time he received 55 shillings less than the first. What was the price per bushel ? 35. A certain number consists of two figures whose sum is 9 ; and if 2t is added to the number, the order of the figures will be inverted. What is the number? SOLUTION. Let X = the left-hand figure ; then 9 — x = the right-hand figure. As figures increase from right to left in a tenfold ratio, 10 a: + (9 — x) rz=z 9 X -\- 9 = the number ; and when the order of the figures is inverted, 10 (9 — x) -\- X =z 90 — 9 X = the resulting number. Therefore 9x + 9 + 2t = 90 — 9a; Or 18 a; = 64 Whence . a: = 3, the left-hand figure, and 9 — x= 6, the right-hand figure. Ans. 36. 36. A certain number consists of three figures whose sum is 6, and the middle figure is double the left-hand figure ; and if 198 is added to the number, the order of the figures will be inverted. What is the number ? Ans. 123. 37. Two men 90 miles apart travel towards each other till they meet. The first travels 5 miles an hour and the second 4. How many miles does each travel before they meet ? 98 ELEMENTARY ALGEBRA. 38. A man hired six laborers, to the first of whom he paid 75 cents a week more than to the second ; to the second, 80 cents more than to the third ; to the third, 60 cents more than to the fourth ; to the fourth, 50 cents more than to the fifth ; to the fifth, 40 cents more than to the sixth; and to all he paid $68.15 a week. What did he pay to each a week ? 39. What number is that to which if 20 is added two thirds of the sum will be 80 ? 40. What number is that to which if a is added - of c the sum will be rf? . cd Ans. ~ a. b 41. A man spent one fourth of his life in Ireland, one fifth in England, and the rest, which was 33 years, in the United States. To what age did he live? 42. A post is one fifth in the mud, two sevenths in the water, and 18 feet above the water. How long is the post ? 43. What number is that whose half is as much less than 40 as three times the number is greater than 156? Ans. 56. 44. Two workmen received the same sum for their la- bor ; but if one had received $ 15 less and the other $ 15 more, one would have received just four times as much as the other. What did each receive ? 45. Of the trees on a certain lot of land five sevenths are oak, one fifth are chestnut, and there are 32 less wal- nut trees than chestnut. How many trees are there ? 46. Divide 474 into two parts such that, if the greater part is divided by 7 and the less by 3, the first quo- tient shall be greater than the second hy 12. Ans. 357 and 117. EQUATIONS OF THE FIRST DEGREE. 99 47. Two persons, A and B, have each an annual income of S 1500. A spends every year $400 more than B, and at the end of five years the amount of their savings is $6000. What does each spend annually? Ans. A $1100, and B $T00. 48. In a skirmish the number of men captured was 41 more, and the number killed 26 less than the number wounded ; 45 men ran away ; and the whole number en- gaged was four times the number wounded. How many men belonged to the skirmishing party ? Ans. 240. 49. A and B have the same salary. A runs into debt every year a sum equal to one sixth of his salary, while B spends only three fourths of his ; at the end of five years B has saved $ 1000 more than enough to pay A's debt. What is the salary of each ? Ans. $ 2400. 50. A man lived single one third of his life: after hav- ing been married two years more than one eighth of his life, he had a daughter who died ten years after him, and whose age at her death was one year less than two thirds the age of her father at his death. AVhat was the father's age at his death ? SOLUTION. Let X = his age j then - =r his age at marriage, o - -f- - -|- 2 = his age at daughter's birth, and X — f ^ -)~ u + 2 ) = her age at his death. Then ._f_^_2+10=.L--l Transposing and uniting, — - = — 9 o X = 72, the father's age. 100 ELEMENTARY ALGEBRA. 51. Divide S 864 among three persons so that A shall have as much as B and C together, and B $5 as often as C $11. 52. A father and son are aged respectively 32 and 8. How long will it be before the son will be just one half the age of the father ? 53. A man's age was to that of his wife at the time of their marriage as 4:3, and seven years after, their ages were as 5 : 4. What was the age of each at the time of their marriage ? 54. One fifth of a certain number minus one fourth of a number 20 less is 2. What is the number? Ans. 60. 65. There are two numbers which are to each other as J : J ; but if 9 is added to each, they will be as | : ^. What are the numbers ? Ans. 9 and 6. 56. A person having spent $ 150 more than one third of his income had $ 50 more than one half of it left. What was his income ? 61. A merchant sold from a piece of cloth a number of yards, such that the number sold was to the number left as 4 : 5 ; then he cut off for his own use 15 yards, and found that the number of yards left in the piece was to the number sold as 1:2. How many yards did the piece originally contain ? Ans. 45. 58. Four places, A, B, C, and D, are in a straight line, and the distance from A to D is 126 miles. The distance from A to B is to the distance from B to C as 3 : 4, and one third the distance from A to B added to three fourths the distance from B to C is twice the distance from C to D. What is the distance from A to B, from B to C, and from C to D ? 59. A laborer was hired for 40 days ; for each day he wrought he was to receive $2.50, and for each day he was idle he was to forfeit $1.25. At the end of the time ho received $58.75. How many days did he work? Ana 20. EQUATIONS OF THE FIRST DEGREE.',,' ;'•.; ; ''"POJ 60. A cask which held 44 gallons wa^ , fi'Ued xHth-' j^ mixture of brandy, wine, and water. There were 10 gal- lons more than one half as much wine as brandy, and as much water as brandy and wine. How many gallons were there of each ? 61. Two persons, A and B, travelling each with $80, meet with robbers who take from A $5 more than twice what they take from B; then B finds he has $26 more than twice what A has. How much is taken from each ? Ans. From A, $69 ; from B, $32. 62. Four persons, A, B, C, and D, entered into part- nership with a capital of $84816; of which B put in twice as much as A, C as much as A and B, and I) as much as A, B, and C. How much did each put in ? 63. In three cities. A, B, and C, 1188 soldiers are to be raised. The number of enrolled men in A is to that in B as 3 : 5 ; and the number in B to that in C as 8 ; 7. How many soldiers ought each city to furnish ? Ans. A, 288 ; B, 480 ; C, 420. 64. Divide $65 among five boys, so that the fourth may have $2 more than the fifth and $3 less than the third, and the second $4 more than the third and $5 less than the first. 65. A merchant bought two pieces of cloth, one at tho ^rate of $ 5 for 7 yards, and the other $ 2 for 3 yards ; the second piece contained as many times 3 yards as the first times 4 yards. He sold each piece at the rate of $6 for 7 yards, and gained $24 by the bargain. Hov many yards were there in each piece ? Ans. First, 84 ; second, 63. 66. A drover had the same number of cows and sheep. Having sold 17 cows and one third of his sheep, he finds he has three and a half times as many sheep as cows left. How many of each did he have at first? 162 ELKMENTARY ALGEBRA. Ct. A' fip.ur ^dealer sold one fourth of all the flour he had and one fourth of a barrel ; afterward he sold one third of what he had left and one third of a barrel ; and then one half of the remainder and one half of a barrel; and had 15 barrels left. How many had he at first ? SOLUTION. Let X = number at first; 3 X 1 then — = number after first sale, 4 4 I— -) — s^^^o — 2^^^ number after second sale, d 2(2 — 2) — 2^^4 — 4^^ number after third sale. 3 an ' Then ^ _ ! =, 15 4 4 Clearing of fractions, x — 3 = 60 Whence x = 63, number at first. 68. A merchant bought a barrel of oil for $50; at the same rate per gallon as he paid, he sold to one man 15 gallons ; then to another at the same rate two fifths of the remainder for $ 14. How many gallons did he buy in the barrel ? 69. Two pieces of cloth of the same length but dif- ferent prices per yard were sold, one for S5 and the^ other for $1M. If there had been 5 more yards in each, at the same rate per yard as before, they would have come to $15.47^f-. How many yards were there iu each? Ans. 21. 7i(l A and B began trade with equal sums of money. The first year A lost one third of his money, and B gained $750. The second year A doubled what he had at the end of the first year, and B lost S150, when the two had again an equal sum. What did each have at first? EQUATIONS OF THp: FIRST DEGREE., , ; , , ,^-08 Tl. A man distributed among his laborers ^-2.60 ^p^eQ^, and had $25 left. If he had given each $'^ as long as his money lasted, three would have received nothing. How many laborers were there, and how much money did he have? Ans. 68 laborers, and $195. T2. A man who owned two horses bought a saddle for $35. When the saddle was put on one horse, their value together was double the value of the other horse ; but when the saddle was put on the other horse, their value together was four fifths of the value of the first horse. What was the value of each horse ? T3. From a cask two thirds full 18 gallons were taken, when it was found to be five ninths full. How many gallons will the cask hold ? 74. A farmer had two flocks of sheep, and sold one flock for $60. Now a sheep of the flock sold was worth 4 of those left, and the whole value of those left was $8 more than the price of 8 sheep of those sold, and the flock left contained 40 sheep. How many sheep did the farmer sell, and what was the value of a sheep of each flock ? Ans. Number sold, 15; value, $4 and $1. 75. A man has seven sons with 2 years between the ages of any two successive ones, and the sum of all their ages is ten times the age of the youngest. What is the age of each ? 76. Divide 75 into two parts such that the greater in- creased by 9 shall be to the less diminished by 4 as 3 : 1. 77. Divide a into two parts such that the greater in- creased by b shall be to the less diminished by ■ '2j;+ 3^ = 2 ) (3a:-y=zl6) '' — '^^ G. Given - ( 2x — i/ = ui 8. Given - -4-^ = 4^5 5 ^-4 108 ELEMENTARY ALGEBRA. CASE III. 114. Elimination by combination. 1. Given -< "^ f- , to find x and y. X'lx—Zy — Z) ^ 2x — Si/ = S OPEKATION Zx — 2y= 1 (1) 6x — 4y=14 (3) 6a: — 9j^= 9 (4) by— b (5) y= 1 (6) 2x (2) (8) If we multiply (1) by 2, and (2) by 3, we have (3) and (4), in which the coefficients of x are equal; subtracting (4) from (3), we have (5), which contains but one unknown quantity. Redu- cing (5), we have (6), or ?/ = 1 ; substituting this value of y in (2), we obtain (7), which reduced gives (8), or z = 3. 2. Given 2-4- ^ 2 + ^=12 L 3 — 2 , to find X aud y. OPERATION. !-!=« (1) - + ?-12 3 — 2 (2) (6) =c-| = 12 (3) 9-1= 6 ^-^^ (4) y=12 (7) x=18 (5) If we multiply (1) by 2, we have (3), an equation in which y has the same coefficient as in (2) ; since the signs of y are different in (2) and (3), if we add these two equations together, we have (4), which contains but one unknown quantity. Reducing (4), we have (5), or a: = 18. Substituting this value of x in (1), we have (6), which reduced gives (7), ory— 12. Hence, EQUATIONS OF THE FIRST DEGREE. RULE. 109 Multiply or divide the equations so that the coefficients of the quantity to be eliminated shall become equal ; then, if the signs of this quantity are alike in both, subtract one equa- tion from the other ; if unlike, add the two equations to- gether. Note. — The least multiplier for each equation will be that which will make the coefficient of the quantity to be eliminated the least common multiple of the two coefficients of this quantity in the given equations. It is always best to eliminate that quantity whose coefficients can most easily be made equal. B}'- this method of elimination find the values of x and y in the following equations : — 3. Given |^- + 3^-33) 4. Given j 8. + 6, = 6) (lOa: — 3y = 4) Ans. Ans. • . \*--r!,_^ ^ 19. Given - 5 PROBLE MS PRODUCING EQUATIONS OF THE FIRST DEGREE CON- TAINING TWO UNKNOWN QUANTITIES. 116. Many of the problems given in Section XIII. con- tain two or more unknown quantities; but in every case these are so related to each other that, if one becomes known, the others become known also ; and therefore the problems can be solved by the use of a single let- ter. But many problems, on account of the complicated conditions, cannot be performed by the use of a single letter. No problem can be solved unless the conditions given are sufficient to form as many independent equa- tions as there are unknown quantities. 1. A grocer sold to one man T apples and 5 pears for 41 cents; to another at the same rate 11 apples and 3 pears for 45 cents. What was the price of each ? EQUATIONS OF THE FIRST DEGREE. 113 SOLUTION. Let X =z the price of an apple, and y= '' " " a pear. Then, by the conditions. 7a; + 5y = 41 (1) and 11 a; + Zy=. 45 (2) 55x+15^ = 225 (3) 21x4- 15^=123 (4) 21+5y = 41 a) 34 a: =102 (5) y= 4 (8) x= ' 3 (6) We multiply (2) by 5 and (1) by 3, and obtain (3) and (4) ; subtracting (4) from (3) we have (5), which reduced gives (6), or a; = 3. Substituting this value of a: in (1), we have (7), which re- duced gives (8), or 2/ = 4. 2. There is a fraction such that if 2 is added to the numerator the fraction will be equal to ^ ; but if 3 is added to the denominator the fraction will be equal to ^. What is the fraction ? SOLUTION. Let - = the fraction. y Then, by the conditions, ^4-21 /ix J ^ 1 /n\ 3x = y+3 (3) 2x + 4 = y (4) x — 4 = 3 (5) x=1 (6) 144-4=18=y a) ^ = ^3 (8) Clearing (1) and (2) of fractions, we obtain (3) and (4) ; sub- tracting (4) from (3), we obtain (5), which reduced gives (6), or a; = 7. Substituting this value of x in (4), we have (7), or y= 18. 114 ELEMENTARY ALGEBRA. 3. There are two numbers whose sura is 28, and one fourth of the first is 3 less than one fourth of the second. What are the numbers ? Ans. 8 and 20, 4. The ages of two persons, A and B, are such that 5 years ago B's age was three times A's ; but 15 years hence B's age will be double A's. What is the age of each ? Ans. A's, 25 ; B's, 65. 6. There are two numbers such that one third of the first added to one eighth of the second gives 39 ; and four times the first minus five times the second is zero. What are the numbers ? 6. Find a fraction such that if 6 is added to the nu- merator its value will be ^, but if 3 be added to the de- nominator its value will be ^ ? * Ans. ^j. 7. What are the two numbers whose difierence is to their sum as 1:2, and whose sum is to their product as 4 : 3 ? SOLUTION. Let X = the greater and y = the less. Thenx — ij:x + i/=l:2 (1) x + ?/:xy = 4:3 (2) 2x — 2i/ = x + i/ (3) Sx + 3f/ = 4:xy (4) x = 37/ (5) 9y + 33^=12/ (6) x = S 0) \=.y (8) Having written (1) and (2) in accordance with the statement in the problem, we form from them (3) and (4) by Art. 106. Re- ducing (3), we obtain (5) ; substituting this value of x in (4), we have (G), which, though an equation of the second degree, can be at once reduced to an equation of the first degree by dividing each term by y ; performing this division and reducing, we obtain (8) or y = 1 ; substituting this value of y in (5) we obtain (7), or a* = 3. EQUATIONS OF THE FIRST DEGREE. ' 115 8. What are the two numbers whose difference is to their sum as 3 : 20, and three times the greater minus twice the less is 35 ? 9. There is a number consisting of two figures, which is seven times the sum of its figures ; and if 36 is sub- tracted from it, the order of the figures will be inverted. What is the number ? Ans. 84. 10. There is a number consisting of two figures, the first of which is the greater ; and if it is divided by the sum of its figures, the quotient is 6 ; and if the order of the figures is inverted, and the resulting number divided by the difference of its figures plus 4, the quotient will be 9. What is the number ? Ans. 54. 11. As John and James were talking of their money, John said to James, " Give me 15 cents, and I shall have four times as much as you will have left." James said to John, " Give me 7^ cents, and I shall have as much as you will have left." How many cents did each have ? Ans. John, 45 cents ; James, 30 cents. 12. The height of two trees is such that one third of the height of the shorter added to three times that of the taller is 360 feet ; and if three times the height of the shorter is subtracted from four times that of the taller, and the remainder divided by 10, the quotient is 17. Re- quired the height of each tree. Ans. 90 and 110 feet. 13. A farmer who had $41 in his purse gave to each man among his laborers $2.50, to each boy $1, and had $15 left. If he had given each man $4 and then each boy $3 as long as his money lasted, 3 boys would have received nothing. How many men and how many boys did he hire ? 116 • ELEMENTARY ALGEBRA. 14. A man worked 10 days and his son 6, and they received $31; at another time he worked 9 days and his son 1, and they received $29.50. What were the wages of each ? 16. A said to B, "Lend me one fourth of your money, and I can pay my debts." B replied, "Lend me $100 less than one half of yours, and I can pay mine.'' Now A owed $1200 and B $1900. How much money did each have in his possession ? Ans. A, $800 ; B, $1600. 16. If a is added to the difference of two quantities, the sum is b ; and if the greater is divided by the less, the quotient will be c. What are the quantities ? . be — ac ,6 — a Ans. , and -• c — 1 c — 1 17. A man owns two pieces of land. Three fourths of the area of the first piece minus two fifths of the area of the second is 12 acres ; and five eighths of the area of the first is equal to four ninths of the area of the second. How many acres are there in each ? Ans. Ist, 64 acres ; 2d, 90 acres. 18. A and B begin business with different sums of money; A gains the first year $350, and B loses $500, and then A's stock is to B's as 9 : 10. If A had lost $500 and B gained $350, A's stock would have been to B's as 1:3. With what sum did each begin ? Ans. A, $1450; B, $2500. 19. If a certain rectangular field were 4 feet longer and 6 feet broader, it would contain 168 square feet more; but if it were 6 feet longer and 4 feet broader, it would contain 160 square feet more. Required its length and breadth. EQUATIONS OF THE FIRST DEGREE. 117 20. A market-man bought eggs, some at 3 for 1 cents and some at 2 for 5 cents, and paid for the whole $2.62 ; he afterward sold them at 36 cents a dozen, clearing $0.62. How many of each kind did he buy? 21. A and B can perform a piece of work together in 12 days. They work together 1 days, and then A fin- ishes the work alone in 15 days. How long would it take each to do the work? Ans. A 36 and B 18 days. 22. " I was ten times as old as you 12 years ago/' said a father to his son; "but 3 years hence I shall be only two and one half times as old as you." What was the age of each ? 23. If 3 is added to the numerator of a certain frac- tion, its value will be § ; and if 4 is subtracted from the denominator, its value will be J^, What is the fraction? 24. A farmer sold to one man T bushels of oats and 5 bushels of corn for $12.76, and to another, at the same rate, 5 bushels of oats and 7 bushels of corn for $13.40. What was the price of each ? 25. Find two quantities such that one third of the first minus one half the second shall equal one sixth of a ; and one fourth of the first plus one fifth of the second shall equal one half of a. . 34 a , 15 a ^ Ans. ^ and — • 26. A person had a certain quantity of wine in two casks. In order to obtain an equal quantity in each, he poured from the first into the second as much as the second already contained; then he poured from the sec- ond into the first as much as the first then contained ; and, lastly, he poured from the first into the second as much as the second still contained ; and then he had 16 gallons in each cask. How many gallons did each origi- nally contain? Ans. 1st, 22 ; 2d, 10 gallons. 118 ELEMENTARY ALGEBRA. SECTION XV. EQUATIONS OF THE FIRST DEGREE CONTAINING MORE THAN TWO UNKNOWN QUANTITIES. 117. The methods of elimination given for solving equa- tions containing two unknown quantities apply equally well to those containing more than two unknown quantities. ( ^+ y— ^= ^) 1. Given • J2x + 3y+4^r=17k (Sx — 2y+ bz= b) OPERATION. to find X, y, and z. x+y— 2=4 (1) 2a: + 3.y4-42=l7 (2) 3ar — 2?/+ 52= 5 (3) 2x-{-2y — 2z= 8 (4) 3ar + 3?/— 32= 12 (5) y + 62= 9 (G) 5y- 82= 7 (7) 5.y + 302 = 45 (8> x+3— 1=4 (13) ^ + 6=9 (11) 382 = 38 (9) ar = 2 (14) y = 3 (12) 2= 1(10) Multiplying equation (1) by 2 gives equation (4), which we sub- tract from (2), and obtain (6) ; multiplying (1) by 3 gives (5), and subtracting (5) from (3) gives (7). We have now obtained two equations, (6) and (7), containing but two unknown quantities. Mul- tiplying (6) by 5, we obtain (8), and subtracting (7) from (8), we obtain (9), which reduced gives 2=1. Substituting this value of z in (6), and reducing, we obtain y = 3. Substituting these values of y and z in (1), and reducing, we obtain a; = 2. 2. Given X -\-y =2Q y +z=z2^ z + w; = 66 w) + M = 81 n + X = 46 - , to find u, w, X, y, and z. EQUATIONS OF THE FIRST DEGREE. 119 OPERATION. x + y = 26 (1) y4-z = 29 (2) z + xv=5Q (3) u' + j/ = 81 (4) u + x = iG (5) y + x = 2Q z—x= 3 w + x=53 u—x = 2S z — x= 3 (6) w + x=m (7) M — .T=28 (8) 2x = 18 (9) y = n (11) Z--12 (12) IV ^U (13) « = 37 (14) a;= 9(10) Here we subtract (1) from (2), and obtain (G) ; then (6) from (3), and obtain (7); then (7) from (4), and obtain (8); then (8) from (5), and obtain (9), which reduced gives (10), or a: = 9. Sub- stituting this value of x in (1), (6), (7), and (8), and reducing, we obtain (11), (12), (13), and (14), or ?/ = 17, 2 = 12, w = 44, and M= 37. Hence, for solving equations containing any number of unknown quantities, RULE. From the given equations deduce equations one less in number, containing one less unknown quantity; and con- tinue thus to eliminate one unknown quantity after an- other, until one equation is obtained containing but one unknown quantity. Reduce this last equation so as to find the value of this unknown quantity ; then substitute this value in an equation containing this and but one other unknown quantity, and reducing the resulting equation, find the value of this second unknown quantity ; substitute again these values in an equation containing no more thayi these two and one other unknown quantity, and reduce as before ; and so con- tinue, till the value of each unknown quantity is found. Note. — The process can often be very much abridged by the exercise of judgment in selecting the quantity to be eliminated, the equations from which the other equations are to be deduced, the method of elimination which shall be used, and the simplest equa- tions in which to substitute the values of the quantities which have been found. 120 ELEMENTARY ALGEBRA. Find the values of the unknown quantities in the fol- lowing equations : — 3. Given y -\- z -\-w-\- u=z 18 X -{- z -\-w-\- uz= 17 X -\- y -\-w-\- u= 14 x-\-y-\-z-\-uz= 15 Note. — If these equations are added t(^ether and the sum di- vided by 4, we shall have x-\-y-\-z-\-w-\- m = 20; and if from this the given equations are successively subtracted, the values of the unknown quantities become known. 4. Given + ^y + ^2=10 Ans. xz=2. y = 3. z = Q. U=4:. w= 5. Ans rx=z2. .]y = ^. (z=6. 6. Given 2x+ Sy-{-^z = Q1 2y + 2=:25 r.= 1. An8.-<^y= 1. izz= 11. 6. Given X x + 2 2x 2y — 102= 1 V' 4y+ 3z= I) rx=z b. Ans. -Jy = 3. (z = l. 7. Given -< lx + ly + l. = 22 1.1.1 4^ + 4^+, =^ = 24 1 . 1 . I x^ + «y + «^=io Ans. E a: = 20. 12. z =32. EQUATIONS OF THE FIEST DEGREE. 121 8. Given < -+- = - + ' = - y^ X 12 - + '=' Note. — The best method for this example ia that used in Ex- ample 3, without clearing of fractions. 9. Given ( ^+ y+ ^= 6j fx = 42x + By + 4:2 = 20y, Ans. -lt/ = idx4-1vA-5z=zd2) (z = x=l. 2. 3. x + iy = S1 10. Given -iy -\-i 11. Given 12. Given 13. Given x + y = a rx-f-y = ax rx = bl . of each ? Ans. 1st, $12; 2d, $14; 3d, $15. 2. Three boys. A, B, and C, divided a sum of money among themselves in such a manner that A and B re- ceived 18 cents, B and C 14 cents, and A and C 16. How much did each receive? Ans. A, 10 ; B, 8 ; C, 6 cents. 3. As three persons, A, B, and C, were talking of their ages, it was found that the sum of one half of A's age, one third of B's, and one fourth of C's was 33 ; that the sum of A's and B's was 13 more than C's age ; while the sum of B's and C's was 3 less than twice A's age. What was the age of each? Ans. A's, 32; B's, 21 ; C's, 40. 4. As three drovers were talking of their sheep, says A to B, "If you will give me 10 of yours, and C one fourth of his, I shall have 6 more than C now has." Says B to C, "If you will give me 25 of yours, and A one fifth of his, 'I shall have 8 more than both of you will have left." Says C to A and B, "If one of you will give me 10, and the other 9, I shall have just as many as both of you will have left." How many did each have? 5. Divide 32 into four such parts that if the first part is increased by 3, the second diminished by 3, the third mnltipliod by 3, and the fourth divided by 3, the sum, difference, product, and quotient shall all be equal. Ans. 3, 9, 2, and 18. EQUATIONS OF THE FIRST DEGREE. 123 6. If A and B can perform a piece of work together in 8^2- days, B and C in 9^^ days, and A and C in 8^ days, in how many days can each do it alone? Ans. A in 15, B in 18, and C in 21 days. 7. Find three numbers such that one half of the first, one third of the second, and one fourth of the third shall together be 56 ; one third of the first, one fourth of the second, and one fifth of the third, 43; one fourth of the first, one fifth of the second, and one sixth of the third, 35. 8. The sum of the three figures of a certain number is 12 ; the sum of the last two figures is double the first ; and if 297 is added to the number, the order of its fig- ures will be inverted. What is the number? Ans. 417. 9. A man sold his horse, carriage, and harness for $150. For the horse he received $25 less than five times what he received for the harness ; and one third of what he received for the horse was equal to what he received for the harness plus one seventh of what he received for the carriage. What did he receive for each? Ans. Horse, $225; carriage, $175; harness, $50. 10. A man -owned three horses, and a saddle which was worth $45. If the saddle is put on the first horse, the value of both will be $ 30 less than the value of the second ; if the saddle is put on the second horse, the value of both will be $55 less than the value of the third ; and if the saddle is put on the third horse, the value of both will be equal to twice the value of the second minus $10 more than one fifth of the value of the first. What is the value of each horse ? Ans. 1st, $100; 2d, $175; 3d, $275. 11. The sum of the numerators of two fractions is 7, and the sum of their denominators 16 ; moreover the sum of the numerator and denominator of the first is equal 124 ELEMENTARY ALGEBRA. to the denominator of the second ; and the denominator of the second, minus twice the numerator of the first, is equal to the numerator of the second. What are the fractions ? Ans. f and ^. 12. A man bought a horse, a wagon, and a harness, for $180. The horse and harness cost three times as much as the wagon, and the wagon and harness one half as much as the horse. What was the cost of each ? 13. A gentleman gives $600 to be divided among three classes in such a way that each one of the best class is to receive $10, and the remainder to be divided equally among those of the other two classes. If the first class proves to be the best, each one of the other two classes will receive $5 ; if the second. class proves to be the best, each one of the other two classes will receive $4f ; but if the third class proves to be the best, each one of the other two classes will receive $2. What is the number in each class? 14. A cistern has 3 pipes opening into it. If the first should be closed, the cistern would be filled in 20 min- utes ; if the second, in 25 minutes ; and if the third, in 30 minutes. TTow long would it take each pipe alone to fill the cistern, and how long would it take the three together ? Ans. 1st, 85f minutes ; 2d, 46y\ minutes ; 3d, 35y\ minutes. The three together, 16rfy minutes. 15. Three men, A, B, and C, had together $24. Now if A gives to B and C as much as they already have, and then B gives to A and C as much as the}^ have after the first distribution, and again C gives to A and B as much as they have after the second distribution, they will all have the same sum. How much did each have at .first? Ans. A, $13 ; B, $1, and C, $4. EQUATIONS OF THE FIKST DEGREE. 125 SECTION XYI. POWERS AND ROOTS. 119. A Power of any quantity is the product obtained by taking that quantity any number of times as a factor; and the exponent shows how many times the quantity is taken (Art. 24). Thus, a =z a} is the first power of a ; a a^=zcP' " second power, or square, of a ; « « rt = a^ " third power, or cube, of a ; a a a a:=.a^ " fourth power of a ; and so on, 120. In order to explain the use of negative indices, we form, by the rules of division, the following series: — a\ a\ a\ a\ a, 1, 1 a' 1 1 a'' 1 a*' 1 «^ a\ a\ a\ a\ a\ a-\ a-\ «-^ a-^ a-\ We form the first series as follows: a^ divided by a gives a*,* a* by a, gives a^ ; a? by a, gives a" ; a^ by a, gives a ; a by a, gives 1 ; 1 by a, gives - ; - by a, gives - j ; — ^ by a, gives — ^ , and so on. The second series is formed in the same way from a^ to a; but if we follow the same rule of division from a toward the right as from a^ to a, viz. subtracting the index of the divisor from that of the div- idend^ a divided by a, gives a" ; cP by a, gives a-^ ; read a, with the negative index one ; a~^ by a, gives a~^ ; a~^ by a, gives a~^ \ and so on. From this we learn, Ist. TJmt the power of every quantity is 1 ; 2d. That a~\ a~^, a~^, &c., are only different ways of writmq -, „, -,, dec. 126 ELEMENTARY ALGEBRA. Any two quantities at equal distances on opposite sides of a°, or 1, are reciprocals of each other. 121 1 The rules given for the multiplication and divis- ion of powers of the same quantity (Arts. 50 and 54) apply equally well whether the exponents are positive or negative. For a« -?- a-2 = a^ -h \ = a« X «' = a' a"' -T- a"^ z= -. -i- a* = -n* or a~" a' a" The following examples m multiplication are to be done according to the rules for the multiplication of powers of the same quantity by each other, given in Art. 50 ; and those in division, by the rule for the division of powers of the same quantity by each other, given in Art. 54. 1. Multiply x' by x~^. Ans. x^. 2. Multiply a* by a~^. 3. Multiply a?* by x'^. Ans. ar^ or 1. 4. Multiply y'' by y*. 5. Multiply a-*x^f by a'^x-^f/'^. Ans. a-^aPip, or ar^y"'. 6. Multiply 4:X~^y-^z by ^x^^z^. 7. Multiply llx'^y^z-'^ by 4 ar-^y"-* z"*. 8. Multiply ""iLtSll by 5 a* b-^ c\ o 9. Divide x^ by a;"". Ans. x**. 10. Divide x^ by x~''. 11. Divide a:"® by x~\ Ans. x'*. EQUATIONS OF THE FIRST DEGREE. 127 12. Divide y-^ by y^. 13. Divide y''^ by y'"^. Ans. y'^. 14. Divide a-^hc^ by d^}r^c~'^. Ans. ar^Wc^. 15. Divide l^x'y-^z by 4a:2y-'^23 16. Divide 4 o^"^ y-^ ;^ by 2 « x-'^ y-^ z^. 17. Divide laHx-^y^ by 1 a J-^ x* ^"^ ;22 18. Divide Ui: a"^ b c-'^ x"- y'^ z by 16 a^J-^c-^a^^/. 122. It follows from the preceding article that a factor may he transferred from the numerator of a fraction to its denominator, or vice versa, provided the sign of the expo- nent of the factor is changed from -\- to — , or — to -{-. For I r=a«X^. = «'X^-^ X gf' 1 5_1 , 1 1_^ _5 1 - — - ^^~y~^~y ' ^ ~ar^y — = ' X i = — X dJ X dJ X 1 . Transfer the denominator of -^—^ — r to the numerator. o c^ y-^ 2. Transfer the numerator of -i-„ — , to the denominator. Ans. a! h~^ c~'^ x^ y . ,6 ^4 ^2 2-1 Ans. 2 .V— I ^2 J-7 ^-6 ^ 2/' 3. Transfer the denominator of — ~^ to the numerator. a x^ M — * 2 4. Transfer the numerator of — r^-j— to the denominator. oca 128 ELEMENTABY ALGEBRA. 6. Free from negative exponents _^ . _4- Ans. , — :, 7 acrxy 6. Free from negative exponents — 3— j— j- T. Free from negative exponents — ^ i^^^ — . Ans. 8. Free from negative exponents j ' :^_, . ' (x — y)-* (x -{- y) us. ^;~^> (a^ + y) 9. Free from negative exponents -^rK^~m ' INVOLUTION. 123. Involution is the process of raising a quantity to a power. 124. A quantity is involved by taking it as a factor as many times as there are units in the index of the re- quired power. 125. According to Art. 48, (+«)X(+o) = + <»'. (+«) X (+a) X (+a) = (+a^) X (+«) = + a'. and so on ; and (—0 ) X (—a) = -|- a^, (—a) X (—a) X i-a) = (+«') X {-a) = — a\ (— a) X (— a) X (-a) X (— «) = (—a') X (-a) = + aS and so on. Hence, for the signs we have the following RULE. 0/ a positive quantity all the powers are positive. Of a negative quantity the even powers are positive, and the odd powers negative. INVOLUTION. 129 INVOLUTION OF MONOMIALS. 126. To raise a monomial to any required power. 1. Find the third power of 2 a^ b. OPERATION. (2 aHy = 2anx2anx2aH (1) = 2. 2. 2. a^ a^ a^hhh (2) r=: 8 aH^ (3) According to Art. 1 24, to raise 2 a^ 6 to the third power we take it as a factor three times (1) ; and as it makes no diflference in the product in what order the factors are taken, we arrange them as in (2); performing the multiplication (Art. 50) expressed in (2), we have (3). Hence, RULE. Multiply the exponent of each letter by the index of the required power, and prefix the required power of the nu- merical coefficient, remembering that the odd powers of a negative quantity are negative, while all other powers are positive. Note. — It follows that the power of the product is equal to the product of the powers. 2. Find the square of 2 x. Ans. 4 x^. 3.- Find the cube of 3a?2. Ans. 21 x\ 4. Find the fourth power of a^ W. Ans. a^^ h^, 5. Find the third power of 4a^a;. Ans. 64 a® a:^. 6. Find the square of 2 x~^. Ans. 4 x~'^. 7. Find the cube of '^x'^y^. Ans. 27 x"^/. 8. Find the with power oi ah. Ans. a'"5"'. 9. Find the third power of — 3 a^ h. Ans. — 27 a^ i^ 130 ELEMENTARY ALGEBRA. 10. Expand {—2a^xy. Ans. 16 a^^a:\ 11. Expand (Sa^ir)"', Ans. S'^a^'"^^". 12. Expand (2x^i/)\ 13. Expand (—4:a^x'')\ 14. Expand {-^Sx^y)\ 15. Expand (— a'^. Ans. — ar^. 16. Expand (x-^y^y. 17. Expand ( — 4x"^^)^ Ans. — |-. 18. Expand (Sa^'x^y. 19. Expand (— 2a:-8^-")». 20. Expand (— 3 a^-^" y'"/. 21. Expand (— 9a-^6-''^a;2^'*)». INVOLUTION OF FRACTIONS. 127. To involve a fraction. 1. Find the cube of V2 6/ ~ 2 i 26 OPERATION. According to Art. 124, 6 '^ 2^ ^ 26 8^ quantity we must take it three times as a factor; taking —7 three times as a factor, and performing the multiplica- tion by Art. 95, we have — r:. Hence, RULE. Involve both numerator and denominator to the required power. INVOLUTION. 131 2 a 4 a* 2. Find the square of —5. Ans. ^• 3. Fmd the cube of — zr-o-,- Ans. — 21 &d? 4. Find the fourth power of " _^ 6. Find the fifth power of — ^^'. — 2 a a;-^ 6. Find the third power of t. Find the mth power of 3 x-'" y^ 2 c? x~^ 8. Find the fourth power of — ,—^ — ^,-. 3 a^ 6-* C-* 9. Find the third power of — ^ m^ dr^\' X 'IT Z 10. Find the fourth power of — _ o" , " 2 a;~^ w^ z 11. Find the fifth power of — o^-Tpas* INVOLUTION OF BINOMIALS. 128. A BINOMIAL can be raised to any power by suc- cessive multiplications. But when a high power is re- quired, the operation is long and tedious. The Binomial Theorem, first developed by Sir Isaac Newton, enables us to expand a binomial to any power by a short and speedy process. 129. In order to investigate the law which governs the expansion of a binomial we will expand a -\- b and a — ^ to the fifth power by multiplication. 132 ELEMENTAEY ALGEBBA. a +6 a +b a^ + a6 a6 +, &» a2 4-2 a6 + 6^ . • • • a +h , a« + 2 an + «6« a'b-^- 2aJ'' + 6^ a« + 3 an-\- 3a62 +, . « +6 a* + 3 a«6 + 3 a^ ^2 _|. aW an-\- 3 a2 //- + 3 aV" + i' a^ + 4 aH + Qan^-\- 4 ah^ + i* . « +i a^ + 4^ a^5 + 6a«^ + 4 aH^ + « i* a*6 + 4a«62 4. 6 dn^ + 4a i« + ¥ 2d power. 3d power. 4th power. a5 -^ 5 an + 10 aH2 + 10 a2 63 4- 5 a ^.* + b^ 5th power. a — h a — 6 o^ — aft a^ — 2ah -\-h'^ 2d power a — h df:^2 a'h-\- ^ a u -|— a u — g'^ 6 + 2 g 6-^ — 6» a8 — 3g26+ 3 a 6'^ — 6« .... 3d power. a — h a* _ 4 ^8 ^ _|_ 6 g2 62 — 4 a 63^> . 4th power. a —b ati^^a*b-{' QaH'^— 4 a'^" + ab* — a*b+ 4taH^— 6aH''-^^a b* — 5» o8 _ 6 a* 6 + 10 a8 6'^ — 10 o^ b^~^5^b*~^^^^^ 6th power INVOLUTION. 183 By examining the different powers of a -\- h and a — 6 in these Examples, we shall find the following invariable laws governing the expansion : — 1st. The leading quantity (i. e. the first quantity of the binomial) begins in the first term of the power with an ex- ponent equal to the index of the power, and its exponent decreases regularly by one in each successive term till it dis- appears ; the following quantity {i. e. the second quantity of the binomial) begins in the second term of the power with the exponent one, and its exponent increases regularly by one till in the last term it becomes the same as the index of the power. Thus, in the fifth power the Exponents of a are 5, 4, 3, 2, 1. Exponents of b are 1, 2, 3, 4, 6. It will be noticed that the sum of the exponents of the letters in any term is equal to the index of the power. 2d. The coefficient of the first term is one ; of the second, the same as the index of the power ; and universally, the co- efficient of any term, multiplied by the exponent of the lead- ing quantity, and this product, divided by the exponent of the following quantity increased by one, will give the co- efficient of the succeeding term. Thus, in the fifth power, 5, the coeflScient of the second term, multiplied by 4, a's exponent, and divided by 1 plus 1, 6's exponent plus 1, = —— — = 10, the coeflfi- cient of the third term. The coefficients are repeated in the inverse order after passing the middle term or terms, so that more than half of the coefficients can be written without calculation. The number of terms is always one more than the index of 134 ELEMENTARY ALGEBRA. the power ; i. e. the second power has three terms ; the third power, four terras ; and so on. When the number of terms is even, i. e. when the index of the power is odd, the two central terms have the same coefiBcient. 3d. When both terms of the binomial are positive, all the terms of the power are positive; but when the second term is negative, tlwse terms which contain odd poivers of the following quantity are negative, and all the others positive; or eveiy alternate term, beginning with the second, is negor tive, and the others positive. 1. Expand {x -\- yY. OPERATION. According to the law, the first term will be a:", and the second term -|" ^ ^ y* 4 The coefficient of the third term will be — ^— , and the third term -\-2^j?i^. 2 28 \x ■«. The coefficient of the fourth term will be — - — , and the fourth term -\- b^3?t^, 14 The coefficient of the fifth term will be — P — i and the fifth term 70a^y*. Having found the preceding coefficients and the coefficient of the middle term, we can write the others at once. Hence, (X + y)8 = xs + 8a:7y + 28a:«y2 + 56a:5y3 4. TOx^y* + 56x^y5 .j. 28i2y« + %xy^ -f y8. 2. Expand {a — h)\ Ans. a«— 6a^H-15«**"— 20o«6«+16a2i*— 6a^+6«. 3. Expand (m + ny. 4. Expand {b -r- yY. 6. Expand {a — xf^. INVOLUTION. 135 6. Expand {b -+- cY^. 1. Expand (x + 1)^ Note. — Since all the powers of 1 are 1, 1 is not written when it appears as a factor; but its exponent must be used in obtaining the coefficients. Ans. x^ -{- b x^ + 10 x'' -\- 10 x'' + 5 X + I. 8. Expand (1 — y)^ Ans. 1 — 6y + 15 f ~ 20/ + 15y* — 6/ + /. 9. Expand (a — 1)^ 130. When the terms of the binomial have coejfficients or exponents other than 1, the theorem can be made to apply by treating each term as a single literal quantity. In the expansion, each factor should be enclosed in a parenthesis, and after the expansion of the binomial by the binomial theorem, the work should be completed by the expansion of the enclosed factors, according to the rule for the expansion of monomials. 1. Expand (2 x — y^)*. OPERATIOX. (2 xy - 4 (2 xy (f) + 6 (2 xy (fy _ 4 (2 a.) (/)« + (fy Expanding each factor as indicated, we have * 16 x^ — 32 x^f + 24 x'^y^ _ 8 x/ + / 2. Expand (Sx^— 2yy. (3a:2)5 _ 5 (3 a:2)4 (2y) + 10 (3x2)3 (2^)2 _ IQ (3x2)2 (2y)3 -|- 5 (3ar2) (2y)4 — (2^)5. Ans. 243x^'' — 8103^y-{-1080x^f—720xUf-\-24:03^y* — 32y^. Note. — Any letters, as a and 6, might be substituted for 3 a;' and 2 y, and the expansion of (a — by Written out, and then the values of a and b substituted. 3. Expand (a^ _ 3 by. Ans. a«— 12a«i + 54a^62_ lOS a^ b^ -\- SI bK 4. Expand (x^ — fy. 136 ELEMENTARY ALGEBEA. 5. Expand (2 a + T)«. Ans. 8 a« + 84^2 + 294 a + 343. 6. Expand {2 a c — a:)*. Ans. 16aV* — 32aVa: + 24a2c2;r2 — 8acx» + a:*. 7. Expand (a^a; — 2^f. ^-•1^ + 1 + ^ + 2-^ + -*. 8. Expand (s + ^) • 9. Expand (I- ly. 32 48~ "1 36~ 54 ^ 162 243 10. Expand (| — lY. U. Expand (V^-L^)'. , 8i» . . 9:i:j/« 27.v» Ans. — -r'y-l- -^ ^. 12. Expand (i + iy. Ans. ^ + ^^ + ^ ^i?+F 13. Expand (ac — -J. « 14. Expand (x-{--\. 15. Expand (l — -V. 16. Expand ^2 a^ _ IV. 17. Expand + iV. 18. Expand (i--^. EVOLUTION. 137 131. The Binomial Theorem can be applied to the ex- pansion of a polynomial. Thus, in a -]- b — c, a -{- b can be treated as a single term, and the quantity can be written (a -\- h) — c. In like manner, a -\- b -\- x — y can be written (a -\- b) -\- (x — y). In such cases it is easier to substitute a single letter for the enclosed terms, and after the expansion to substitute the proper values. 1 . Expand (a -\- b — c)^. OPERATION. Put a -\- b = X {x — cy = x^ — Sx^c-[-Sxc^ — c^ Substituting for x, its value, a -\- b', (a + 6 — cP = a3 + 3o26 + 3a62 + 63_3a2c — 6a6c — 362 c +3ac3 + 36c2 — c3 2. Expand (2 a — b — c — dy. Note. — For 2 a — b — c — d write (2 a — b) — (c -}- d). Ans. Aa^—Aab-\-W—Aac—Aad-\-2bc +2 W-[-c2+2cc/-f d^. 3. Expand {^x — ^y — a-^ bf. 4. Expand {lx — a-{-bf, EVOLUTION. 132. Evolution is the process of extracting a root of a quantity. It is the reverse of Involution. 133. A ROOT of any quantity is a quantity'' which taken as a factor a given number of times will produce the given quantity. The number of times the root is to be taken as a factor depends upon the name of the root. Thus, the second or square root of a quantity is a quantity which taken twice as a factor will produce the given quantity ; the third or cube root is a quantity which taken three times as a factor will produce the given quantity ; and so on. 138 ELEMENTARY ALGEBRA. A Root is indicated by the radical sign \/, or by a fractional exponent. Thus, t^ X, or a:^ indicates the square root of x. y/lc, or a:^ " " cube " " " V^x, or J^ " " mth " " " 134, A root and a power may be indicated at the same time. Thus, v^x*, or x^, indicates the cube root of the fourth power of x, or the fourth power of the cube root of X ; for a power of a root of a quantity is equal to tlw i same root of the same power of the quantity. -^8=^ or 8^ is the square of the cube root of 8, or the cube root of the square of 8, i. e. 4. 135» A perfect power is a quantity whose root can be found. A perfect square is one whose square root can be found ; a perfect cube is one whose cube root can be found ; and so on. 136* Since Evolution is the reverse of Involution, the rules for Evolution are derived at once from those of Involution. And therefore, as according to Art. 125 an odd power of any quantity has the same sign as the quantity itself, and an even power is always positive, we have for the signs in evolution the following RULE. An odd root of a quantity has the same sign as the quan- tity itself An even root of a positive quantity is either positive or negative. An even root of a negative quantity is impossible, or im- aginary. EVOLUTION. 139 SQUARE ROOT OF NUMBERS. 137i The Square Root of a number is a number which, taken twice as a factor, will produce the given number. 138. The square of a number has twice as many figures as the root, or one less than twice as many. Thus, Roots, 1, 10, 100, 1000. Squares, 1, 100, 10000, 1000000. The square of any number less than 10 must be less than 100; but any number less than 10 is expressed by one figure, and any number less than 100 by less than three figures; i. e. the square of a number consisting of one figure is a number of either one or two figures. The square of any number between 10 and 100 must be between 100 and 10000; i. e. must contain more than two figures and less than five. And the square of any number between 100 and 1000 must contain more than four figures and less than seven. Hence, to ascertain the number of figures in the square root of a given number, Beginning at units, point off the number into periods of two figures each ; there will be as many figures in the root as there are periods, and for the incomplete period at the left, if any, one more. 139. To extract the square root of a number. 1. Find the square root of 5329. From the preceding explanation, it is evident that the square root of 5329 is a number of two figures, and that the tens figure of the root is the square root of the greatest perfect square in 53 ; i. e. v/49, or 7. Now, if we represent the tens of the root by a and the units by b, a -\- b will represent the root ; and the given number will be (a -\- by == a^ -{- 2 a b -\- b\ Now a" == 70^ = 4900 ; therefore, 2 a 6 -[- 6^ = 5329 — 4900 = 429. But 2ab-\- W ={2a-\-b)b; 140 ELEMENTARY ALGEBRA. If therefore 429 is divided by 2 a -|- h, it will give h the units of the root. But b is unknown, and is small compared with 2 a ; we can therefore use 2 a = 140 as a trial divisor. 429 -^ 140, or 42-J-14 = 3, a number that cannot be too small but may be too great, because we have divided by 2 a instead of 2 a -f- &. Then 6=3, and 2 a -f 6 = 140 -j- 3 = 143, the true divisor ; and (2 a -f 6) 5 = 143 X 3 = 429 ; and therefore 3 is the unit figure of the root, and 73 is the required root. The work will appear as follows ; — OPERATION. 5 3 2 9 (7 3 a^TO 49 b= 3 2a + 6 = 14 3)429 (2a-\-h)h= 429 Hence, to extract the square root of a number, RULE. Separate the given number into periods of two figures each, by placing a dot over units, hundreds, &c. Find the greatest square in the left-hand period, and place its root at the right. Subtract the square of this root figure from the left-hand period, and to the remainder annex the next peHod for a dividend. Double the root already found for a trial divisor, and, omitting the right-hand figure of the dividend, divide, and place the quotient as the next figure of the root, and also at the rigid of the trial divisor for the true divisor. Multiply the true divisor by this new root figure, subtract the product from the dividend, and to the remainder annex the next period, for a new dividend. Double the part of tlie root already found for a trial di- visor, and proceed as before, until all tlie penods have been employed. EVOLUTION. 141 Note 1. — When a root figure is 0, annex also to^the trial di- visor, and bring down the next period to complete the new dividend. Note 2. — If there is a remainder, after using all the periods in the given example, the operation may be continued at pleasure by annexing successive periods of ciphers as decimals. Note 3. — In extracting the root of any number, integral or deci- mal, place the first point over unit's place ; and in extracting the square root, over every second figure from this. If the last period in the decimal periods is not full, annex 0. 2. Find the square root of 46225. OPERATION. We suppose at first that a rep- resents the hundreds of the root, 46225 (2 15 and b the tens ; proceeding as in 4 Ex. 1, we have 21 in the root. A \ A o Then letting a represent the ' hundreds and tens together, i. e. A 1 o ' 21 tens, and b the units, we have 4 2 5) 2 1 2 5 2 a, the 2d trial divisor, = 42 2 12 5 tens ; and therefore 6^5; and 2 a -f 6 = 425 ; and 215 is the required root. 3. Find the square root of 5013.4. operation. 50 1 3.4 6(7 0.80 5-1- 49 140.8)113.40 112.64 141.605) .760000 .Y08025 4. Find the square root of 288369. Ans. 537. 5. Find the square root of 42849. Ans. 207. 6. Find the square root of 173.261. Ans. 13.16-f. 7. Find the square root of .9. Ans. .948-t-. 142 ELEMENTARY ALGEBRA. 8. Fini^ the square root of 2. Ans. 1.4l42-f-. 9. Find the square root of 484. 10. Find the square root of 48.4. 11. Find the square root of .064. 12. Find the square root of .00016. Note. — As a fraction is involved by involving both numerator and denominator (Art. 127), the square root of a fraction i.s the square root of the numerator divided by the square root of the denominator. 13. What is the square root of | ? Ans. §. 14. What is the square root of ^| ? 15. What is the square root of ^^j ? -^^:=^^. Ans. f. Note. — If both terms of the fraction are not perfect squares, and cannot be made so, reduce the fraction to a decimal, and then find the square root of the decimal. A mixed number must be re- duced to an improper fraction, or the fractional part to a decimal, before its root can be found. 16. What is the square root of | ? Ans. .53-f-. 17. What is the square root of ^fy ? 18. What is the square root of y\ ? 19. What is the square root of 7|? CUBE ROOT OF NUMBERS. 140. The Cube Root of a number is a number which, taken three times as a factor, will produce the given number. 141* TJie cube of a number consists of three times as many figures as the root, or of one or two less than three times as many. Roots, 1, 10, 100, 1000. Cubes, 1, 1000, 1000000, 1000000000. The cube of any number less than 10 must be less than 1000; but any number less than 10 is expressed by one figure, and any EVOLUTION. 143 number less than 1000 by less than four figures; i. e. the cube of a number consisting of one figure is a number of less than four figures. The cube of any number between 10 and 100 must be be- tween 1000 and 1000000; i. e. must contain more than three figures and less than seven. And in the same way we see that the cube of any number between 100 and 1000 must contain more than six figures and less than ten. Hence, to ascertain the number of figures in the cube root of a given number, Beginning at units, point off the number into periods of three figures each ; there will he as many figures in the root as there are periods, and for the incomplete period at the left, if any, one moi^e. 142. To extract the cube root of a number. 1. Find the cube root of 428T5. From the preceding explanation, it is evident that the cube root of 42875 is a number of two figures, and that the tens figure of the root is the cube root of the greatest perfect cube in 42 ; i. e. ^I~21, or 3. Now, if we represent the tens of the root by a and the units by 6, a -\-h will represent the root, and the given number will be (a -I- &)» = rt« -f 3 0== & + 3 a 6^ _|_ ^,3, Now 0^= 30^ = 27000; therefore, Z a^h -\- Z aV -\- h^ = A2%lb — 27000 = 15875. But 3a=&-f 3ai2_^6^= (3a2+ 3rt6 + 62)6. If therefore 15875 is divided by 3 a^ _[_ 3 « & _|_ ft2 \^ ^'^\\ gj^^ j^ the units of the root. But &, and therefore 3 a 6 -j- W, a part of the divisor, is unknown, and we must use 3a^=^2700 as a trial divisor. 15875 -|- 2700, or 158-^27 = 5, a number that cannot be too small but may be too great, because we have divided by 3 a^ instead of ihe true divisor, Z a^ -\- Z ah -\- V^. Then & = 5, and 3a2_j_ 3a^_|_j2^ 2700 -f- 450 -}- 25 = 3175, the true divisor; and (3 a'* -f- 3 a 6 -|- 6^) 6 = 3175 X 5 = 15875, and therefore 5 is the unit's figure of the root, and 35 is the required root. The work will appear as follows: — 144 ELEMENTARY ALGEBRA. OPERATION. 4 2 8 T 5 (35 Root. 27 Trial divisor, 3 a^ = 2 7 Sab— 45 ^== 25 True divisor, S a^ -\- Sab + b^ — 3 11 b 15 8 7 5 Dividend. 15875 Hence, to extract the cube root of a number, RULE. Separate the number into periods of three figures each, by placing a dot over units, thousands, &c. Find the greatest cube in the left-hand period, and place its root at the right. Subtract this cube from the left-hand period, and to the re- mainder annex the next period for a dividend. Square the root figure, annex two ciphers, and multiply this result by three for a trial divisor ; divide the dividend by the trial divisor, and place the quotient as the next figure of the root. Multiply this root figure by the part of the root previously obtained, annex one cipher and multiply this result by three ; add the last product and the square of the last root figure to the trial divisor, and the sum will be the true divisor. Multiply the true divisor by the last root figure, subtract the product from the dividend, and to the remainder annex the next period for a dividend. Find a new trial divisor, and proceed as before, until all the periods have been employed. Note 1. — The notes under the rule in square root (Art 139) apply also to the extraction of the cube root, except that 00 must be annexed to the trial divisor when the root figure is 0, and after placing the first point over units the point must be placed over every third figure from this. Note 2. — As the trial divisor may be much less than the true EVOLUTION. 145 divisor, the quotient is frequently too great, and a less number must be placed in the root. 2. Find the cube root of 18191447. 1st Trial Divisor, OPERATION. 3 a'' =1200 3a6 = 360 6*^ = 3 6 18191447(263 8 10191 1st Dividend. 1st True Divisor, 3 a^ -{- 3 ah -\- b^ =15 9 6] 9 576 2d Trial Divisor, Ba^ =20 2 80 01 Sab == 2340 6^= 9 6 1544 7 2d Div. 615447 2d True Divisor, Sa^-fSaft + ft^ =205149 We suppose at first that a represents the hundreds of the root and 6 the tens: proceeding as in Ex. 1, we have 26 in the root Then letting a represent the hundreds and tens together, i. e. 26 tens, and b the units, we have 3 a^ the 2d trial divisor, = 202800 ; and therefore 6 = 3; and 3 a^ -{- 3 a b -\- b\ the 2d true divisor, = 205149; and 263 is the required root. Note. — Though the 1st trial divisor is contained more than 8 times in the dividend, yet the root figure is only 6. 3. Find the cube root of 687I6.4T. OPERATION. 6 8 7 1 6.4 T (4 0.9 5+ 64 4 8 0.0 1 8.0 .8 1 4 7 1 6.4 T 49 8.8 1 60 18.43 6.13 50 .0 25 6 24.56 7 5 4417.929 2 9 8.541000 25^L2 2 8_37^ 4 7.312625 146 ELEMENTARY ALGEBRA. 4. Find the cube root of 2924207. Ans. 143. 5. Find the cube root of 8120601. Ans. 201. 6. Find the cube root of 36926037. 7. Find the cube root of 67917.312. 8. Find the cube root of 46417.8. 9. Find the cube root of .8. Ans. .928+. 10. Find the cube root of .17164. 11. Find the cube root of .0064. 12. Find the cube root of 25.00017. 13. Find the cube root of 2.7. Note. — As a fraction is involved by involving both numerator and denominator (Art. 127), the cube root of a fraction is the cube root of the numerator divided by the cube root of the denominator. 14. What is the cube root jV ? ^^s- f • 15. What is the cube root of j^^ ? 16. What is the cube root of f^f ? ^ = ^f|. Ans. ^. Note. — If both terms of the fraction are not perfect cubes, and cannot be made so, reduce the fraction to a decimal, and then find the cube root of the decimal. A mixed number must be reduced to an improper fraction, or the fractional part to a decimal, be- fore its root can be found. 17. What is the cube root of ^^ ? Ans. .899-f-. 18. What is the cube root of ^\ ? 19. What is the cube root of 3f ? 20. What is the cube root of 117^1' EVOLUTION. 147 EVOLUTION OF MONOMIALS. 143. As Evolution is the reverse of Involution, and since to involve a monomial (Art. 126) we multiply the exponent of each letter by the index of the required power, and prefix the required power of the numerical coefficient, Hence, to find the root of a monomial, RULE. Divide the exponent of each letter by the index of the re- quired root, and prefix the required root of the 7iumerical coefficient. Note 1. — The rule for the signs is given in Art. 136. As an even root of a positive quantity may be either positive or negative^ we prefix to such a root the sign ± ; read, plus or minus. Note 2. — It follows from this rule that the root of the product of several factors is equal to the product of the roots. Thus, y/"36 = v^l y/'O = 6. 1. Find the cube root of 8a^y^ Ans. 2xy^. 2. Find the square root of 4 x^. Ans. ±2x. 3. Find the third root of — 125 a^x. Ans. — 5 a^ x^. 4. Find the fourth root of 81 a"^ b. Ans. ± 3 a-i h^. 6. Find the fifth root of S2 a^H^ Ans. 2a^b^. 6. Find the cube root of — 129 x^ if. Ans. — 9xy*. 7. Find the fourth root of 256 a;'*/. 8. Find the cube root of — 512 a-^^ 9. Find the fifth root of 243 ar^/^ 148 ELEMENTARY ALGEBRA. Note. — As a fraction is involved by involving both numerator and denominator (Art. 127), a fraction must be evolved by evolving both numerator and denominator. 4 ^2 2 a 10. Find the square root of — j- Ans. ± r--^- Perform the operations indicated in the following ex- pressions : — 11. v^—T29 aH«c». 12. (49a2a;4/)i 13. y 36iV 14. ^a'^af*". 15. (266a*a:i<^3^i<5)^. 16. ^81aH^ IT. \^a'^b^"'(f\ SQUARE ROOT OF POLYNOMIALS. 144* In order to discover a method for extracting the square root of a polynomial, we will consider the rela- tion of a + ^ to its square, a^ -{- 2 a h -\- b^. The first term of the square contains the square of the first term of the root ; therefore the square root of the first term of the square will be the first term of the root. The second term of the square contains twice the product of the two terms of the root ; therefore, if the second term of the square, 2 a b, is divided by twice the first term of the root, 2 a, we shall have the second term of the root b. Now, 2ab-{- b'= (2 a-\-b) b; therefore, if to the trial divisor 2 a wc add b, when it has been found, and then EVOLUTION. 149 multiply the corrected divisor by h, the product will be equal to the remaining terms of the power after a^ has been subtracted. The process will appear as follows : — OPERATION. Having written a, the square a^ -\- 2 a h -\- h^ {a -\- b root of aS in the root, we sub- a^ tract its square («^) from the 2a-\-b)2ab-\-h'^ g^^^" polynomial, and have 2 ab A- b^ 2 a 6 -|- ^'^ left. Dividing the first term of this remainder, 2 a &, by 2 a, which is double the term of the root already found, we obtain 6, the second term of the root, which we add both to the root and to the divisor. If the product of this corrected divisor and the last term^ of the root is subtracted from 2ah -\-h^, nothing remains. 145. Since a polynomial can always be written and involved like a binomial, as shown in Art. 131, we can apply the process explained in the preceding Article to finding the root, when this root consists of any number of terms. 1. Find the square root of 0^+ 2ab-\- b'^ — 2ac—2bc + c\ OPERATION. ^.2 2a + b)2ab + b'^ 2ah + h'^ 2a-\-2b~c)—2ac — 2bc-\-c^ — 2ac — 2bc-{-c'^ Proceeding as before, we find the first two terms of the root a-\~b. Considering a -\~ b as a single quantity, we divide the remainder — 2 ac — 2bc -\- c^ by twice this root, and obtain — c, which we write both in the root and in the divisor. If this corrected divisor is multiphed by — c, and the product subtracted from the dividend, nothing remains. 150 ELEMENTARY ALGEBRA. Hence, to extract the square root of a polynomial, RULE. Arrange the terms according to the powers of some letter. Find the square root of the first term, and write it a^ the first term of the root, and subtract its square from the given polynomial. Divide the remainder by double the root already found, and annex the result both to the root and to live divisor. Multiply the corrected divisor by this last term of the root, and subtract the product from the last remainder. Proceed as before with the remainder, if there is any. 2. Find the square root of 4 x^ — 4:Xi/^ -\- ^*. Ans. 2x — y^. 3. Find the square root ofa^ -\-2ab-\-b'^-\-4:ac + 4 5 c + 4 c*. Ans. a-\- b-\-2c. 4. Find the square root of 9x* — I2a^ -{- 4:X^ -\- 6ax^ — 4:ax-\-a^. Ans. Sx'^ — 2x-\-a. 5. Find the square root of4a^-|~ Sab — 4:a -\- 4:b'^ — 4i+l Ans. 2a + 2b—l. 6. Find the square root of 25 a:* — lOx* -j- 6x^ — x H-i. Ans. 5x^ — x+^. 1. Find the square root of x^ -{-2x^ — x* — 2x* + ^*- 8. Find the square root of 4 0^ — 4a6 + b'^ — iac Ans. 2 a — b — c — d. 9. Find the square root of x^ — 4a:^ + 6 a:* — 6x* -\-5x^ — 2x+ 1. 10. Find the square root of 4 a* + 8«'6 — Sa'H^ — 12a6» + 9i*. EVOLUTION. 151 Note l. — According to the principles of Art. 136, the signs of the answers given above may all be changed, and still be correct. Note 2. — No binomial can be a perfect square. For the square of a monomial is a monomial, and the square of the polynomial with the least number of terms, that is, of a binomial, is a trinomial. Note 3. — A trinomial is a perfect square when two of its terras are perfect squares and the remaining term is equal to twice the product of their square roots. For, (a-\-by==a^-\-2ab-{-W (a — hf = a^ — 2ab -\- I? Therefore the square root of a^ ± 2 a 6 -]- 6^ is a ± 6. Hence, to obtain the square root of a trinomial which is a perfect square, Omitting the term that is equal to twice the product of the square roots of the other tivo, connect the square roots of the other two by the sign of the term omitted. 5; y 2 2 11. Find the square root of ~ — ^ + ~- Ans 12. Find the square root oi x^ -\-2x-\- 1. Ans. X -\-\. 13. Find the square root of 43:^ — ^xy -\-^y^. . 14. Find the square root of - — 2 a i + 9 5^ 15. Find the square root of 16/ _|_ 40^2^2 _|_ 25^4 Note. — By the rule for extracting the square root, any root whose index is any power of 2 can be obtained by successive extractions of the square root. Thus, the fourth root is the square root of the square root ; the eighth root is the square root of the square root of the square root; and so on. 16. Find the fourth root of a^ — 12a«& + 540^^^ — 108 a^i^ 4-81 5^ Ans. a2-^3i. 152 ELEMENTARY ALGEBRA. 11. Find the fourth root of -, + ~ + ::^ + — , 4- 1 X ' y 18. Find the fourth root of a:^ — 4x'^ + ^Ox^ — IQx^ 4-19ar*— 16a:»+ lOar^ — 4ar+ 1. Ans. ar^ — x+1. 146. To find any root of a polynomial. Since, according to the Binomial Theorem, when the terms of a power are arranged according to the power of some letter begin- ning with its highest power, the first term contains the first term of the root raised to the given power, therefore, if we take the re- quired root of the first terra, we shall have the first term of the root. And since the second term of the power contains the second terra of the root raidtiplied by the next inferior power of the first term of the root with a coefficient equal to the index of the root, therefore if we divide the second term of the power by the first term of the root raised to the next inferior power with a coefficient equal to the index of the root, we shall have the second term of the root. In accordance with these principles, to find any root of a polynomial we have the foUowing RULE. Arrange the terms according to the powers of some letter. Find the required root of the first term, and vjrite it as (he first term of the root. Divide the second term of the polynomial by the first term of the root raised to the next inferior power and multiplied by the index of the root. Involve the whole of the root thus found to the given power, and subtract it from the polynomial. If there is any remainder, divide its first term by the di- visor first found, and (lie quotient will he the third term of the root. Proceed in this manner till the power obtained by involv- ing the root is equal to the given polynomial. EVOLUTION. 153 Note 1. — This rule verifies itself. For the root, whenever a new term is added to it, is involved to the given power, and whenever the root thus involved is equal to the given polynomial, it is evident that the required root is found. Note 2. — As powers and roots are correlative words, we have used the phrase given power, meaning the power whose index is equal to the index of the required root, and the phrase next inferior power meaning that power whose index is one less than the index of the required root. 1. Find the cube root of a^ — 3 a5 + 5«^ — 3a— 1. OPERATION. Constant divisor, 3 a*) a^ — 3 a^ + 5 a^ — 3 a — 1 (a^ — a — 1 — 3 a*, 1st term of remainder. a« __ 3 a^ + 5 a^ — 3 a — 1 The first term of the root is a^, the cube root of a^. a^ raised to the next inferior power, i. e. to the second power, with the co- efficient 3, the index of the root, gives 3 a*, which is the constant divisor. — 3 a^, the second term of the polynomial, divided by 3 a*, gives — a, the second term of the root, {d? — ay = a^ — 3 a* -j- 3 a* — a' ; and subtracting this from the polynomial, we have — 3 a* as the first term of the remainder. — 3 a* divided by 3 a* gives — 1, the third term of the root, (a^ — a — 1)^= the given poly- nomial, and therefore the correct root has been found. 2. Find the fourth root of 16 a;^ — 32 x^ / + 24 a;^ ^/^ OPERATION. 4 X (2a;)' = 32r^) 16 a;* — 32 a:^ y^ -[- 24 r^ ?/* — 8 ar / -f ?/ (2 x — ?f 16x* — 32arV-f 24: sc" y^ — S x y" -f- f 3. Find the cube root of a^ + S aH + Z ab^ + b^ — Sa'^ c — 6abc — Sb^c + Sac''-\-Sbc'' — c\ 4. Find the fourth root of 16 a' c' —32a^c^x + 24 a^ c^ x" 7* 154 ELEMENTARY ALGEBRA. SECTION XVII. RADICALS. 147. A Radical is the indicated root of any quantity, as \/x, a^, \/"2, 3^, &c. , 148. In distinction from radicals, other quantities are called rational quantities. 149. The factor standing before the radical is the co- efficient of the radical. Thus, 2 is the coeflScient of \/2 in the expression 2\/2. 150. Similar Radicals are those which have the same quantity under the same radical sign. Thus, \/a, 2 \/a, and X a/ a are similar radicals ; but 2 \/« and 2 \/6, or 2x^ and 2x" are dissimilar radicals. 151. A Surd is a quantity whose indicated root cannot be found. Thus, \/2 is a surd. The various operations in radicals are presented under the following cases. CASE I. 152. To reduce a radical to its simplest form. NoTB. — A radical is in its simplest form when it contains no factor whose indicated root can be found. 1. Reduce ^x/TSa^'J to its simplest form. OPERATION. We first resolve 75 a' h into two factors, one of which, 25 a", is the greatest perfect square which it contains ; then, as the root of RADICALS. 155 the product is equal to the product of the roots (Art. 143, Note 2), we extract the square root of the perfect square 25 a^, and annex to this root the factor remaining under the radical. Hence, RULE. Resolve the quantity under the radical sign into two fac- tors, one of which is the greatest perfect power of the same name as the root. Extract the root of the perfect power, multiply it by the coefficient of the radical, if it has any, and annex to the result the other factor, with the radical sign between them. Reduce the following expressions to their simplest form : — 2. -v/12a:. Ans. 2 ^'6x. 3. s^4:9x\ Ans. 1 x^ sTx. 4. ^12an\ Ans. 2a4^9b^. 5. 5-^64a6*. Ans. I0b^4=a. 6. S\^U1aH\ Ans. 2lab^s/S. 1. 25 ^~56x. Ans. 50 ^Tx. 8. 4.\/l2Sx^y. 9. -^343x». Y i28V 10-/ -W / 27 a" c __ / 9 a' /?_£__ ?^ / Y 128^ ~y 64^ Y 2y ~8a:^^ Y 3c ■ — I Ans. 2y 11. ^ 12. \/16x2y2_32^4^6 ^ 16 xV _ 32 x\y^ =: V' 16 a:^/ V" 1 — 2 ipV = 4xy Vl — 2x^^, Ans. 15G ELEMENTARY ALGEBRA. 13. 4:\/Sla^c + 21a^ Ans. 12a^3c + l. 14. (a + i) V3a2_6a6-[-362. Ans. (a^ _ ^) >^ 3. 15. •? ^250a:«/— 125arV. 16. (x — 1/) {a-'x — a'^y)^. IT. (a«+ a«Z;2^i 18. V- -16. 16 19. ^- 1250. Vl6 V— 1=4\/-^, Ans. 20. Vl9a' — 4^. 153. When a fraction is under the radical sign, it can be transformed so as to have only an integral quantity under the radical sign, by multiplying both terms of the fraction by (hat quantity which will make its denominator a perfect power of the same name as the root, and then re- moving a factor according to the Bute in Art. 152. 1. Reduce yy - to its simplest form. OPERATION. (/5 = ^/^ = V^^^^=5^ Transform each of the following expressions so as to have only an integral quantity under the radical sign. 2. l^l- Ads. ^ >/ 6. ^|. Ans. VV9. l\/l- ■ Ans. 1^343. - a I 17 ^ 1 , 3. 4 4. RADICALS. 157 6. ^^^ . A„S.1V^. 10 833- Ans. g-5^/30. -•'^/^ ^■2^8-.- Ans. -VUx. ^ — _ 11. (« + ^) y/^. Ads. V^^^=^. CASE II. 154* To reduce a rational quantity to the form of a radical. 1. Reduce Sx^ to the form of the cube root. OPERATION. Since 3 a;'' is to be placed 3 <^2 ^ 3^27 x^ under the form of the cube root without changing its value, we cube it and then place the radical sign, ^, over it. It is evident that (/"27^ =3 3^. Hence, RULE. Involve the quantity to the power denoted by the index of the root required, and place the corresponding radical sign over the power thus produced. 2. Reduce 4a^5 to the form of the square root. Ans. \^l6aH\ 3. Reduce 2aPc~'^ to the form of the fifth root. 1 4 Ans. 4/S2a'b^'c-^. 4. Reduce - a^ c^ to the form of the cube root. 158 ELEMENTARY ALGEBRA. 2 cPh 6. Reduce -^ — : to the form of the fourth root. Zxyk 6. Reduce x — 2y to the form of the square root. Ads. \/ x^ — 4a:y-|-4y*. 155t On the same principle the rational coeflBcient of a radical can be placed under the radical sign, by involv- ing the coefficient to a power of the same name as the root indicated by the i^adical sign, multiplying it by the radical quantity, and placing the given radical sign over the product. 1. Place the coefficient of ht^ly under the radical sign. OPERATION. 6 >^ 2^ = /^ 125 ><^ 2y = -^ 250y In the following examples, place the coefficient under the radical sign. 2. Z4/^x^y. Ans. -^F = ^ 2Y ) Ans. 3. Reduce \/ ^ and v^ ^ to equivalent radicals having a common index. Ans. ^^ib ^"^ ^ s^- 4. Reduce i / - and tV - to equivalent radicals having a common index. 5. Reduce \/ a, a^ a — b, and \/ a -\- b to equivalent radicals having a common index. Ans. i^^^ ^ (a^^y, and ^ (a+~by. 6. Reduce \/ 2, -C^ 4, and -^ 3 to equivalent radicals having a common index. 7. Reduce \/ x and \^ y to equivalent radicals having a common index. Ans. v'a;"* and v^y*. 160 ELEMENTARY ALGEBRA. CASE IV. 157. To add radical quantities. 1. Add tsf X and ^/ y. Ans. ^fx -^ ^~y. It is evident that the addition can only be expressed. 2. Add Zi^x and bsfx. Ans. 8 /v/x . It is evident that 3 times the y/ x and 5 times the ^~x make 8 times the y/ x. 3. Add'V'S and \/5U together. OPERATION. In this case we make the radi- ^8 -_ 2 ^~^ cal parts similar by reducing them -V-- -^ to their simplest form (Art. 152), ^ HI — — — and then add their coefficients as Sum =1^/2 in Example 2. Hence, RULE. Make the radical parts similar when they are not, and prefix the sum of the coefficients to the common radical. If the radical parts are not and cannot he made similar^ corir nect the quantities with their proper signs. 4. Add 2^^0ax and S\/9Sax. Ans. 31\/2aa:. 5. Add 4-^24^8 and a:/^81. Ans. llx-^3. 6. Add \/27 and \/363. Ans. 14 \/ 3. 7. Add -^512a:< and Ayi62y*. Ans. (4:X + Si/)^2. 8. Add A^5 and \/"^. Vi = \/2»F\^5 = iV5; V5 + iV5 = fV5, Ans 9. Add t/^ and ^1|^. Ans. -g^ ^T2. KADICALS. 361 10. Add \/|, 10\/^V^ and 6a/~20. Ans. 13 V^- 11. Add V'lO and \/^. CASE V. 158. To subtract one radical from another. 1. From A/lb take ^/~21. OPERATION. We make the radical parts sim- ,-hT r /~o il^r hy reducing them to their sim- plest form (Art. 152). And 3 y/^ taken from 5 y/ 3 evidently leaves 2 v/"3. Hence, V27 i=3v/3 2^/3 EULE. Make (he radical parts similar when they are not, sub- tract the coefficient of the subtrahend from that of the min- uend, and prefix the difference to the common radical. If the radical parts are not and cannot be made similar, indi- cate the subtraction by connecting them with the proper sign. 2. From /^"sT take /^ 3. Ans. 2 a^J. 3. From 9\^a^xy^ take Za^/xy^. Ans. ^ays/x. 4. From T \/ 20 ar take 4 \/ 45 a:. Ans. 2 >/ ^ ^. 6. From -^500 take -^l08. Ans. 2 ^4. 6. From 2 \/7^ take ^/^. Ans. t^^ V^. T. From \/ f take \/^. Ans. -i^ \/Ta 8. From 2/^n6a:« take >^891a;S. 9. From a /^a:^ take 1 ^ a^x". 10. From -^1174 take -^1892. 162 ELEMENTARY ALGEBRA. CASE VI. 159. To multiply radicals. 1. Multiply 3/v/« by h m/I). OPERATION. 3 \/ a X 5 s/1) = 3 X 5 X \/ « X \/"^ =15 's/ah As it makes no difference in what order the factors are taken, we unite in one product the numerical coefficients ; and ^ a y^ ^ h = ^ab (Art. 143, Note 2). 2. Multiply 4\/2a6 by 5\/Say. OPERATION. We reduce the radical parts to equivalent radicals 4vza6r= 4\/ 8a 6 having a common index 5^Sax= 5^ 9a^x^ (Art. 156), and then multi- Product = 20 ^l2aW^^ P^>^ ^ ^" *^^ preceding ex- ample. S. Multiply >^ a hy \f a. OPERATION. , a^ X a^ = ^cFX~c^—\f~i^\ or J Multiplying as in the preceding examples, we have ^ a*, or a* ; but t = ^ + ^ ; i- e. the Index of the product is the sum of the indices of the factors. From these examples we deduce the following RULE. I. Reduce the radical parts, if necessary, to equivalent radicals having a common index, and to the product of the radical parts placed under the common radical sign prefix the product of their coefficients. II. Boots of the same quantity are multiplied together by adding their fractional indices. RADICALS. 163 4. Multiply 3 \/ 10 by 4 V 5. Ans. 60 V 2. 6. Multiply ^f^ax^ by a\/a:. Ans. ^axt^d^x. 6. Multiply av'c by hf^c. Ans. a 6 c. T. Multiply isf X by i^ x. Multiply ^7 by V^. Ans. Q^. 5 8. Ans. 76, or -^16807. 9. Multiply s/ ^ bj^ s/ X. Ans. V^'" + ". 10. Multiply 2 ^/a + 6 by 6 a: ^a + 6. Ans. \2x ^ {a + hy. 11. Multiply \/a?, Va:, and ^a: together. Ans. x^^, 12. Multiply 3 V^ by 2 VS. Ans. 6 ^2. 13. Multiply a^x~^y by b\/xy. Ans. a^y. 14. Multiply (a + 6)* by (a — 5)^ Ans. {a^ — h^)^. 15. Multiply ^Vf by 3\/7. 16. Multiply 2 V'S by 4 V^. CASE VII. 160. To divide radicals. 1. Divide 60\/15x by 4V5ar. OPERATION. As division is finding 60 s^lbx H- 4 ^/bx=^ 15/^/3" a quotient which, multi- plied by the divisor, will produce the dividend, the coefficient of the quotient must be a number which, multiplied by 4, will give 60, the coefficient of the dividend, i. e. 15; and the radical part of the quotient must be a quantity which, multiplied by y'Sa;, will give v/15a:, i.e. ^ Z\ the quotient required, therefore, is 15 y' 3. 164 ELEMENTARY ALGEBRA. 2. Divide 6 \/Ty by 2 >^ 2y. OPERATION. 6 V42/ -^ 2 /^2y = 6 .^64^ -j- 2 >^4y2 = 3 >^ 16y We reduce the radical parts to equivalent radicals having a com- mon index (Art. 155), and then divide as in the preceding example. 3. Divide /sj a by /^ a. OPERATION. a* -f- a^ r= ^ a»"^r^2 __ ^- ^^ „i Dividing as in the preceding examples, we have ^ a, or a«. But \ = \ — \\ i- e. the index of the quotient is the index of the divi- dend minus the index of the divisor. From these examples we deduce the following RULE. I. Reduce the radical parts, if necessary, to equivalent radicals having a common index, and to the quotient of the radical parts placed under the common radical sign prefix the quotient of their coefficients. II. Boots of the same quantity are divided by subtracting the fractional index of the divisor from that of the dividend. 4. Divide 16 a/ ax by 8\/a^. Ans. 2a/~^. 6. Divide 4:\/a^ — l^ hj 2A/a — b. _ Ans. 2 \/ « + *.- 6. Divide 6^2*7 by 3\/3. Ans. 6 7. Divide \^x by v^a;. Ans. "v^x*"" 8. Divide ^/"^ by ^1. Ans. r/^ 9. Divide 3 by ^"3. Ans. y/S RADICALS. 165 10. Divide x by ^~x. Ans. f^'^, 11. Divide 4,ar\^x by 2a-^/^y, Ans. 2a^U-' 12. Divide a/T by a^T. 13. Divide ^7 by -^T. 14. Divide /^a by -^a! 15. Divide?^? by I ^f. CASE VIII. 161 1 To involve radicals. 1. Find the cube of Sa^x, OPERATION. (3\/^y= 3\/^X 3\/^ X 3\/"ac In accordance with the definition of involution, we take the quan- tity three times as a factor. By Art. 159 the product is 27y'r*. 2. Find the square of 2^ a. OPERATION. ^^ *^^^ ^^^ ^® ^^^® "^^^ _ X Z ^^® fractional exponent, and (2 ^ a)2 = (2 a^y = 4 a^ found the square of the given quantity by multiplying its exponent by the index of the required power, according to Art. 126. Hence, EULE. I. Involve the radical as if it were rational, and placing it under its proper radical sigh, prefix the required power of its coefficient. II. A radical can be involved by multiplying its fractional exponent by tlve index of the required power. 166 ELEMENTARY ALGEBRA. Note. — Dividing the index of the root is the same as multiply- ing the fractional exponent. Thus the square o( ^ a is if a. For (a«)2 = a^, or ^a. 3. Find the cube of 3 a? \/ a. Ans. 27 a x^ V^i or 27 J x^. 4. Find the square of 4 a^. Ans. 16 a^, or 16 ^a^. 5. Find the fourth power of B\/x. Ans. 81 x'^. 6. Find the nth power o^uAyx. Ans. d^ /^Ixf', 7. Find the fourth power of 5\/i- Ans. 25. 8. Find the cube of 3 \/T. Ans. 189 VT. 9. Find the fourth power of 10. Find the cube of 2V4x. CASE IX. 162. To evolve radicals. 1. Find the cube root of 8 a^ t^ a^x^. OPERATION. As the root of the product is ' ^ ^ (Art. 143,. Note 2), Ave prefix to the cube root of the radical part the cube root of the rational part. The cube root of the radical part must be a quantity which, taken three times as a factor, will produce ^a^z'; i. e. ^ax, 2. Find the fourth root of f^ x. OPERATION. In this case we have used' — — / i\i jL the fractional exponent, and y/ ^a: =: \x^) = x^, or '(/ a; found the fourth root by di- viding the exponent of the given quantity by the index of the required root, according to Art. 143. Hence, RADICALS. 167 RULE. I. Evolve the radical as if it were rational, and, placing it under its proper radical sign, prefix the required root of its coefficient. II. A radical can be evolved by dividing its fractional exponent by the index of the required root. Note. — Multiplying the index of the root is the same as divid- ing the fractional exponent. Thus, the square root of ^a is ^a. For (a3)2 = a6^ or ^ a. 3. Find the square root of ba^^x. (5 a A^Tx)^ z= (^500^'^)^ = ^ 500 a' X, Ans. 4. Find the cube root of a:"^^a^6. Ans. i/^r-* 5. Find the fifth root o^x^/s/x. Ans. ^/ x. 6. Find the fourth root of { ^~J. Ans. /^^. 7. Find the cube root of 1 \/3. Ans. >^l47. 8. Find the square root of 12\/5. POLYNOMIALS HAVING RADICAL TERMS. 163. It appears from the principles already established, that the laws which apply to calculations with quantities which have exponents, apply equally well whether the exponents are positive or negative, integral or fractional. The following examples, therefore, can be done by rules already given. 1. Add 4 a — 3 \/y and 3 a + 2 \/y. Ans. la — A^i/. 2. Add 3 a: + ^135 and 7:zr — -^1080. _ Ans. 10 a: — 3/^5. 168 ELEMENTARY ALGEBRA. 3. Add 2 V 28 — V 27 and 2 V 63 + \/48. 4. Subtract 15 a: — \/ bO a from 13 a: — \/^a. Ans. 3\/2a — 2ar. 5. Subtract /s/ aoc^ — a^ 4:h from \/ ax — \/i6 3. Ans. /s/ ax — x\/ a — 2\/A. 6. Subtract /i^ 32 — ^"242 from — 3 -^T — 7 V 3. •?. Multiply \/« — \/6 by \/« — V^» OPERATION. \/ a — \/ x a — /s/ ah — j>J ax -\- s/hx 8. Multiply a;y + \/a6 by 4 — /^~ah. Ans. 4a?y + (^ — ^y) \/«^ — «^' 9. Multiply T +/\/rO by 6 — ^10. Ans. 32 — V 10. 10. Multiply \/a-(T-\/6 by \/a — \/6. Ans. a — h. 11. Multiply VS — 4/^3 by V^S + \/9. 12. Multiply i VI + T \/ 3 by ^ \/l — 7 \/ 3. 13. Divide t>J ax -\- f^ ay -\- x •\- 1>/ xy by \/Qf + \^a: OPEKATION. ^ ax 4" ^ \/ay +V^ RADICALS. 169 14. Divide /\/ac — j^^ad — ^/hc -\- s/^dhj s/ c — \/rf. Ans. \/ a — \/6. 15. Divide (^ x-\-^ x-^cf-y^ -^}^y^- hj x^y^. 16. Divide x — ^ by ^Ic — ^ y. Ans. \/ x -\- \/y. 11. Divide 4:xy-\-4: \/a h — xy s/ a h — a 6 by 4 — s/ a b 18. Expand (\/^ + V7)^- -^^s. x + 2 \/^ + ^• 19. Expand {a^ — r^)^. Ans. a _ 2 */^ + i. 20. Expand (V a — V^)*. Ans. a2 — 4a*6i + 6a6 — 4aU^ + 62. 21. Expand (4 — v^f . Ans. 100 — 51^/3. 22. Expand (a-4 — a;-^)*. Ans. a-t — 3 a'^ a:-^ + 3 arix'^ — a?"*. 23. Expand (1-y/iy Ans l-JU-4--^ ^4-i. • 16 2v^a 2a ay^a "^ a' 24. Expand (^l-y'ly. Ans. ^-^^* + xy-i4^ + f 4 v^6 ' ^ 3v^6 ' 9 25. Find the square root of a — 2a^6^ -f- 6^. Ans. Va — a7'6^. 26. Find the cube root of ic^ — 3 x^^/^ -f- 3 x^/^ — y. Ans. ic — y^. 2Y. Find the fourth root of 16 a — 32a^/ + 24a*y^ — 8 a^i/2 _j_ ^1 Ans. 2 a^ — t/1 170 ELEMENTARY ALGEBRA. SECTION XVIII. PURE EQUATIONS WHICH REQUIRE IN THEIR REDUCTION EITHER INVO- LUTION OR EVOLUTION. 164* A Pure Equation is one that contains but one power of the unknown quantity ; as, \/x -|- a c = i, 4 ar^ -j- 3 1= 7, or I4.3f = ab. 165. A Pure Quadratic Equation is one that contains only the second power of the unknown quantity ; as, 6x^—Ua = 5lb, af=lScd, or ac z' = 14. 166( Radical Equations, i. e. equations containing the unknown quantity under the radical sign, require Invo- lution in their reduction. 167. To reduce radical equations. 1. Reduce \^x — 3 = 8. OPERATION. V^ — 3 = 8 Transposing, \/ x =^ 11 Squaring, x = 121 2. Reduce ^x — 4 + T = 10. operation. ^a: — 4 + 7 = 10 Transposing, a^ x — 4 = 3 Cubing, a: — 4 = 27 Transposing, ar = 31 RADICAL EQUATIONS. 171 3. Keduce ^^±^ = sTa. OPERATION. Clearing of fractions, ^ d"^ -\- ^/ x ^= a Squaring", c?^ + V^ = ^^ Transposing, ^ x =: a^ — d'^ Squaring, xz=z{(j? — ^2^2 Hence, to reduce radical equations, we deduce from these examples the following general RULE. Transpose the terms so that a radical part shall stand by itself; then involve each member of the equation to a power of the same name as the root ; if the unknown quantity is still under the radical sign, transpose and involve as before ; finally reduce as usual. m 4. Reduce 4 + 1 + 3 f^lc = ^{-. Ans. x = 16. 5. Reduce -4/- = -. Ans. x = 2. 4 y X 2 6. Reduce (a/Ic + 4)^" — 2. Ans. x = 144. 7. Reduce V 11 + a; = \/a? + 1. Ans. a? = 25. 8 Reduce V^ — 7 — /v/x + 18 — >v/5- Ans. x = 27. 9. Reduce ■ ^^ = ~ . Ans. x = X — ex ^ X 1 — <^ 10. Reduce X_^:=ll = >^x—X ^^^ x^^, V^x+lO v^a:+23 172 ELEMENTARY ALGEBRA. 11. Reduce s/ x 4- \/a: — a = . yx — a 12. Reduce s/ x — 30 + \/a: + 21 = ^/ x — 19. 13. Reduce /s/9a:4- 13 = 3 Va^+ 1. Ans. x = 4. 14. Reduce V ' — = V ' Ans. a: = 5. V/5a; -|- 1 ^5x-{- 2 15. Reduce V^' — 32 = x — ^V 32. 168. Equations containing the unknown quantity in- volved to any power require Evolution in their reduction. 160. To reduce pure equations containing the unknown quantity involved to any power. 1. Reduce -^ 3 _ ■ 7 ~~ 97 35 OPERATION. At^ 3_ 97 Clearing the given equation of 5 7 ~ 35 fractions, transposing, and divid- 282^2 — 15 = 97 ing, we have x' = 4; extract- 28x2 = 112 • ing the square root of each mem- x^ = 4 ber of this equation, we have X=z ±2 ar=±2. (Art. 136.) 2. Reduce Tx* — -89z = 100. OPERATION. f a:« — 89 = 7x« = 100 189 Transposing and dividing, we have a:* =s 27; extracting the cube root of each member of x» = 27 this equation, we have x = 3. X = 3 Hence, RULE. Reduce the equation so as to have as one member ike un- known quantity involved to any degree, and then extract that root of each member which is of the same name as the power of the unknown quantity. PURE EQUATIONS ABOVE THE FIRST DEGREE. 173 Note. — It appears from the solution of Example 1 that every pure quadratic equation has two roots numerically the same, but with op- posite signs. 3. Keduce ^ a;^ + T = ? a;^ 4- 3. Ans. x^ ±6. 4. Reduce a =5 ^ • c a ^ Ihcd — . . , ., ^ ^ acd Ans. x=z ± i/ ^— . 4 j;2 24 5. Reduce r = 10. Ans. x = ± 4:. 14 1 6. Reduce 3a:^+ 3 = 9* Ans, x=-. 7. Reduce --+50 = 1. Ans. x = — 1. 8. Reduce^! — r— = — ^P—- Ans. a:=±\/ — 5. 2ar-|-l X -\-4 ^ 9. Reduce 4a:' — 4ar« = 0. 10. Reduce 5a:2— 3a: = 8x2 — 3a; + 50. 1 7 11. Reduce x -{- - = -^ 1. 12. Reduce 2a; + 2 = (a:+ 1)2. 13. Reduce 1 + 14 a?"^ = 2 — 2 a--^. 14. Reduce 3 a:-^ — 5 x'^ = 2 a;-^ _|_ 3 x-^ — |. 15. Reduce (c + a:)« — 6 c^ar = (c — x)^ + 16 cl 16. Reduce^2-3^-p3 = 3^. n. Reduce ^_r f- + -J^tt" =2. 174 ELEMENTARY ALGEBRA. 170. Equations containing radical quar^ities may re- quire in their reduction both Involution and Evolution ; and in this case the rule in Art. 167, as well as that in Art. 169, nnust be applied. Which rule is first to be ap- plied depends upon whether the expression containing the unknown quantity is evolved or involved. 1. Reduce 17 — V^""-^^ = 12. OPERATIOX. 17 — \/;r3-^2 = 12 Transposing, &c., \/a:^ — 2 = 5 Squaring, a:» — 2 = 25 Transposing and uniting, a:^ = 27 Extracting the cube root, x = S 2. Reduce (Va:« — 4 + 3)« = 125. Ans. x = 2. 3. Reduce i/^— ^^ — = Vx. Ans. x= ± - 4. Reduce \/x -{- a = \J X — a Ans. x—± /v/2a-2-f2a6 + 6». 5. Reduce ^-^3 (.^ + 11).^:^. 6. Reduce ^'2 x*-\-%x^ + 24^^+"32~i^ = x + 2. 7. Reduce ^9 (x* -f 19) + 100 — ^ == ^^^ 171. Equations which contain two or more unknown quantities may require for their reduction involution, oi evolution, or both. In these equations the elimination is effected by the same principles as in simple equations. (Arts. 112-114.) (2x2 4-v = 64) 1. Given •<( 5 4 ^, to find x and y. PUKE EQUATIONS ABOVE THE FIRST DEGREE. 175 OPERATION. 'f l^u (1) 2x' + y= 54 (2) ^-.^ 56 (3) -/ = no (t) 15-?= 14 0) x2= 25 (5) !/= 4 (8) a; =±5 (6) Adding four times (1) to (2), we obtain (4), which reduced gives (6), or X = -j- 5; substituting this value of x in (1), we obtain (7), which reduced gives (8), or y = 4. Find the value of the unknown quantities in the follow- ing equations: — 2. Given = ±5. 3. Given |3.-4y=2y) ^^^ ^xr=±^. (^x^z=z2Q \ {x=i ±\. 4. Given -j 2x 2r = 10 >- - Ans. ^ 2^ = ± 3. (32^2: = 45) (z=±b. 5. Given |i:1-^I=n. Ac«. j^^^^T. 6. Given 1-^ + 2/^ = 97 >. (a: — y =:y — 2x} 7. Given (x=* — 2y2=14 ). 176 ELEMENTABY ALGEBRA. PKOBLEMS PRODUCING PURE EQUATIONS ABOVE THE FIRST DEGREE. 172. Though the numerical negative values obtained in solving the following Problems satisfy the equations ibrmed in accordance with the given conditions, they are practically inadmissible, and are therefore not given in the answers. 1. A gentleman being asked how many dollars he had in his purse, replied, " If you add 21 to the number and subtract 4 from the square root of the sum, the remainder will be 6/' How many had he? SOLUTION. Let X = number of dollars. Then, V« + 21— 4= 6 Transposing, v'^ + 2i= 10 Squaring, a: + 21 = 100 Transposing, x = T9, number of dollai^. 2. Divide 20 into two parts whose cubes shall be in the proportion of 27 to 8. Ans. 12 and 8. 3. "What two numbers are those whose sum is to the less as 8 : 3, and the sum of whose squares is 136 ? Ans. 10 and 6. 4. What number is that whose half multiplied by its third gives 54 ? 5. What number is that whose fourth and seventh multiplied together gives 46f ? Ans. 36. 6. There is a rectangular field containing 4 acres whose length is to its breadth as 8:6. What is its length and breadth ? PURE EQUATIONS ABOVE THE FIRST DEGREE. 177 *J. There are two numbers whose sum is It, and the less divided by the greater is to the greater divided by the less as 64 : 81, What are the numbers? Ans. 8 and 9. 8. The sum of the squares of two numbers is 65, and the difference of their squares 33. What are the numbers ? 9. The sum of the squares of two quantities is a, and the difference of their squares b. What are the quantities ? Ans. ± \/i {a + b) and ± \/ ^ {a — b). 10. A gentleman sold two fields which together con- tained 240 acres. For each he received as many dollars an acre as there were acres in the field, and what he received for the larger was to what he received for the smaller as 49:25. What are the contents of each? Ans. Larger, 140; smaller, 100 acres. 11. What are the two quantities whose product is a and quotient 6? — /"^ Ans. ± V « 6 and ± i / t • 12. What two numbers are as m : n, the sum of whose squares is a ? msia , , ni/a Ans. ± -. — v__ and ± -, — -^ 13. What two numbers are as m:n, the difference of whose squares is a ? ,— ,-- Ans. db ~, and ± ^ 14. Several gentlemen made an excursion, each taking $484. Each had as many servants as there were gentle- men, and the number of dollars which each had was four times the number of all the servants. How many gen- tlemen were there? Ans, 11. 15. Find three numbers such that the product of the first and second is 12 ; of the second and third, 20 ; and the sum of the squares of the first and third, 34. 8* L 178 ELEMENTARY ALGEBRA. SECTION XIX. AFFECTED QUADRATIC EQUATIONS. 173. An Affected Quadratic Equation is one that con- tains both the first and second powers of the unknown quantity ; as, 3 x^ — 4 ar =: 16 ; or a a; — bx^ ■=€. 174. Every affected quadratic equation can be reduced to the form x^ -\-bx=zc, in which b and c represent any quantities whatever, posi- tive or negative, integral or fractional. For all the terms containing oc^ can be collected into one term ■whose coefficient we will represent by a ; all the terms containing x can be collected into one terra whose coefficient we will represent by d\ and all the other terms can be united, whose aggregate we will represent by e. Therefore every affected quadratic equation can be reduced to the form a3^-\-(lx = e (1) Dividing (1) by a, a:^ -f- ^x = ^ (2) d e Letting - = h, and - = c, we have x^ -\- bx ^ c (3) 175. The first member of the ecjuation x^ -{- bx z= c cannot be a perfect square. (Art. 145^ Note 2.) But we know that the square of a binomial is the square of the first term plus or minus twice the product of the tuv terms plus the square of the last term; and if we can find the third term which will make x^ -\- bx a perfect EQUATIONS OF THE SECOND DEGREE. 179 square of a binomial, we can then reduce the equation Since h x has in it as a factor the square root of a:^, x^ can be the first term of the square of a binomial, and h x the second term of the same square; and since the second term of the square is twice the product of the two terms of the binomial, the last term of the binomial must be the quotient arising from dividing the second term of the square by twice the square root of the first term of the * square of the bino- mial; i. e. the last OPERATION. . /• ^1, u- term of the bmo- x^-\-hx = c (1) , ^ . hx h vaidl is -- = - ; and therefore the third term of the + *^ + 4-=4+<' (2) ^ + l = ±v/r + « (3) i / T" ~r ^ \^) square must be (2)^4- ^"^'^^"S = -2^ V4+^ (4) — to each 4 ber, we have (2), an equation whose first member is a perfect square. Extracting the square root of each member of (2), and transposing, we obtain (4), or x = — 9 -"- v/ aT "^ ^' ^^'^^^ ^® ^ general expression for the value of x in any equation in the form of a:^ -f" ^ ^ = *^' Hence, as every affected quadratic equation can be re- duced to the form x"^ -]- bx = c, in which b and c repre- sent any quantities whatever, positive or negative, integral or fractional, every affected quadratic equation can be re- duced by the following RULE. Reduce the equation to the form x^ -\- bx = c, and add to each member the square of half the coefficient of x. Extract the square root of each member, and then reduce as in simple equations. 180 ELEMENTARY ALGEBRA. 1. Reduce Tx2 — 28 ar+ 14 = 238. OPERATION. Tar*— 28 ar 4- 14 = 238 Transposing", 7 a:^ — 28 x = 224 Dividing by 7, x^ — 4x= 32 Completing the square, x^ — 4ar -|- 4 = 36 Evolving, X — 2 =±6 Transposing, a:=:2±6 = 8, or — 4 Note. — Since in reducing the general equation a:^ -|- 6 x = c we find x = — oil/j ~l~<^» every affected quadratic equation must have two roots; one obtained by considering the expression - -|- c positive, the other by considering this expression nega- tive. Whenever 4 / - -[- c = these two roots will be equal. 2. Reduce -— --U-^'^-l-f. 5 10 ' 20 2 "^ 4 OPERATION. ^ ^ 1^ 13 Tr I X 5 10'" 2"0 2"'~4 Clearing of fractious, 4x'^ — 2 a? -|- 13 = 10 a:^ -{- bx Transposing, — 6a:'^ — 7ar = — 13 Dividing by — 6, x' + Ix _ 6 _ 13 Completing the square, ^' + () + 40 _ 144" _ 49 " 144 : + is _ 6 ~ Evolving, ■^ + 7 12 = ± 19 12 Transposing, x —k^ 19 _ ll ~ = 1, or- -H 861 144 EQUATIONS OF THE SECOND DEGREE. 181 ]*foTE. — In completing the square, as the second term disappears when the root is extracted, we have written ( ) in place, of it. 3. Reduce Sx^ — 25 -\- 6x = SO. Ans. X =^ 5, or — T. 4. Reduce x = 3. Ans. x = 6, or — 4. X 5. Reduce 2x + — ^ = 1. Ans. x = 2, ' X — 1 Note. — In this example both roots are 2. x" -I- 4 6. Reduce 1 x -^ = 5 x — 1. X — 4 Ans. X = 8, or — 1. H r> J iH U—x IS — X . 1. Reduce n ^ — — g _ ^ + ^^- Ans. a: = t, or — 21. 8. Reduce ^ -{- -= ^. Ans. a; = 10, or — 1§. X — j— O o 10 9. deduce '-' + '-'^^ = i. ,^ _, - 16 100 — 9a: 10. Reduce —^ — = 3. X 4ar ' 176. Whenever an equation has been reduced to the form x"^ -{- bx = c, its roots can be written at once; for this equation reduced (Art. 175) gives x = — - ± » /- -\-c. Hence, ITw roots of an equation reduced to the form x'^ -\- bx = c are equal to one half the coefficient of x with the opposite sign, plus or minus the square root of the sum of the square of one half this coefficient and the second member of the equation. 182 ELEMENTARY ALGEBRA. In accordance with this, find the roots of x in the fol- lowing equations : — 1. Reduce x^^ 8x==65. a: = — 4 ± V 16 + 65 = 5, or — 13, Ans. 2. Reduce a:^ — lOa; = — 24. ar z= 5 ± yv/25 — 24 = 6, or 4, Ans. 3. Reduce a;^ _ 6 x == — 5. Ans. or = 5, or 1. 4. Reduce a:^ -|- 7 a: = 170. 7 / 49 5. Reduce a;'* -f- 9^ = 9- =-5 ±\/r6 + ^ = i °" "-^' Ans. 1 9 6. Reduce x'^ + -a: =-• Ans. a: =1, or — IX. 00 •7 -D J 2 ^ ^ A 3 1 7. Keauce x^ — 7^= — ;;^- Ans. a: = -> or -• 5 2d 5 5 8. Reduce - = - -f- 6^. Ans. a: = 7, or — 5^. SECOND METHOD OF COMPLETING THE SQUARE. 177i The method already given for completing the square can be used in all cases ; but it often leads to in- convenient fractions. The more difficult fractions are in- troduced by dividing the equation by the coefficient of a:^ to reduce it to the form x'^-\-hx=zc. To present a method of completing the square without introducing these fractions, we will reduce equation (1) in Art. 174. EQUATIONS OF THE SECOND DEGREE. 183 1. Reduce ax^ -\- dx •= e. OPERATION. ax^ -\- dx = e (1) a^x^-\-adx=zae (2) a'x'^ + adx + '^ = '^-^ + ae (3) ax At 2 = ± y 7 + «« W Multiplying (1) by a, the coefficient of x^, we obtain (2), in which the first term must be a perfect square. Since ad x^ the second term, has in it as a factor the square root of a^ x^^ a? x^ can be the first term of the square of a binomial, and ad x the second term ; and since the second term of the square is twice the product of the two terms of the binomial, the last term of the binomial must be the second term of the square divided by twice the square root of the first d (1 jc d term of the square of the binomial, or ^ — = - ; and therefore the term required to complete the square is — , which is the square of one half of the coefficient of x in (1). Adding — to both members of (2), we obtain (3), whose first member is the square of a binomial. Extracting the square root of (3) and reducing, we obtain (5), or 1/ '^o. /^'j \ Hence, to reduce an affected quadratic equation, we have this second RULE. Reduce the equation to the form ax^ ~\- dx = e ; then mul- tiply the equation by the coefficient of x^, and add to each member the square of half the coefficient of x. Extract the square root of each member, and then reduce as in simple equations. 184 ELEMENTARY ALGEBRA. Note 1. — This method does not introduce fractions into the equa- tion when the numerical part of the coefficient of x is even. When the coefficient of 7? is unity, this method becomes the same as the first method. Note 2. — If the coefficient of j? is already a perfect square the square can be completed without multiplying the equation, by add- ing to both members the square of the quotient arising from dividing the second term by twice the square root of the first. This method also becomes the same as the first method when the coefficient of a^ is unity. Note 3. — As an even root of a negative quantity is impossible or imaginary, the sign of the first term, if it is not positive, must be made so by changing the signs of all the terms of the equation. 2. Reduce 3 x^ + 8 a: = 28. OPERATIOIf. 3a:2+8a; = 28 Completing the square, 9 x^ + ( ) + 16 = 16 + 84 = 100 Extracting square root, Sx-\-4:= ± 10 Whence, 3x=: — 4 ±10 = 6, or— 14 And x = 2, or — 4§ 3. Reduce 252?^ — 10 a: = 195. OPERATION. 25x^— 10x=:195 Completing sq. by Note 2, 25x^— ( ) + 1 = 1 + 195 = 196 Extracting square root, bx — 1 = ± 14 Whence, 6a:=l ± 14=15, or— 13 And x= 3, or — 2a 4. Reduce 5 a:'^ — 20 a: = —15. Ans. a? = 3, or 1. 6. Reduce 7a:2 — 8a:= 12f. An8. x = ^ ± ^V^'^- 6. Reduce 7 x^ — 4 a a: = -, • Ans. x = ~ , or — - • / 7 7 7. Reducer- ^^I— =a; — 3. Ans. a;= 10, or 3J. 14 — X o EQUATIONS OF THE SECOND DEGREE. 185 8. Reduce x^ -j — = 4. 4 4 r' — 4 X— 3 3a:— 7 9. Reduce THIRD METHOD OF COMPLETING THE SQUARE. 178. The method of the preceding Article iutroduces fractions whenever the numerical coefficient of x is not even. To present a method of completing the square without introducing any fraction, we will again reduce equation (1) in Art. IH. 1. Reduce a:/? -\- dx^^e. OPERATION. ax^ -\- dx=:.e (1) „ , d e x'^-\- - X =- (2) ^ _. d j_ (P ^_L^ ^ ■T"a^'+"4a^ — 4a^"^"a (3) 4:a^a^ + 4:adx + cP = d^-\-4.ae (4) 2ax + d= ± ^d'--^^ae (5) — d± v/tP-f-4ae ^ — (6) Dividing (1) by a, the coefficient of oc^, we have (2) ; then com- pleting the square according to the Rule in Art. 1 75, we have (3) ; and if we multiply (3) by 4 a^ it will give (4), an equation free from fractions (unless a, d, or e in (1) are themselves fractions), and one whose first member is the square of a binomial. To pro- duce this equation directly from (1), we have only to multiply (1) by 4 a ; i. e. by four times the coefficient of 3^, and add to both members d^; i. e. the square of the coefficient of x. Reducing we have (6), which is a general expression for the value of x in any equation in the form of a r* -j- rf ar = c. 186 KLEMENTARY ALGEBRA. Hence, to reduce an affected quadratic equation, we have this third RULE. Reduce the equation to the form ax^ -\- dx :^ e ; then mul- tiply the equation by four times the coefficient of x^ and add to each member the square of the coefficient of x. Extract the square root of each member, and then reduce as in simple equations. Note. — The third Note under the Rule in Art. 177 is applica- ble in all cases. 2. Reduce b x^ - -1x = 24:. OPERATION. 5a:2 — 7ar = 24 Multiplying by 5 X 4 and adding 7^ to each member, lOOx^ — ( ) + 49 = 49 + 480 = 529 Extracting the square root, lOx — 7 = ±23 Transposing, 10 a: = 7 ± 23 = 30, or — 16 Whence, a: = 3, or — 1.6 Note. — The multiplication of the coefficient of s^ need only be expressed. Its coefficient after evolving is double its original coef- ficient. 3. Reduce ^^^— ^^a: _ ^^^ OPERATION. 44x2— 15a: = 293 7 Clearing of fractions, 44 x^ — 1 5 x = 2051 Completing square, 176 X 44x2 — {) -f 225 = 225 + 360976 = 361201 Evolving, 88 X — 15 = ± 601 Transposing, 88 x = 15 ± 601 = 616, or — 586 Whence, x = 7, or 44 Jr. Reduce Y a;* — 15 x = — 2. Ans. a: = 2, or ^. EQUATIONS OF THE SECOND DEGREE. 187 6. Reduce Ans. X =z 1, or — ■28, 6. Reduce 10 1 9 _ 5. Ans. a: = 3, or — ■H 1, Reduce a:+ 1 ' a; : 3. Ans. x = = 2, or - -i 8. Reduce 4 a: -f- .4 Sx — 3 __ lOar-f 10 X 2x — 1 ~ 3x 9. Reduce 1 + 7 — 2a; "^ 2x 3 _ 13 4-4 ~ 10* 10. Reduce ^^3-_ra^ = ■.x — b. Ans. X = - ± V 126 ' 11. Reduce b — 3x-^=z UOx-\ Note. — Mult iply by 3^. 12. Reduce ^-^ _|_ ^^3 = 6 V^. Note. — Divide by \/ x. 179. The rules which have been given for the solution of affected quadratic equations apply equally well to any equation containing but two powers of the unknown quan- tity whenever the index of one power is exactly twice that of the other. By the same reasoning as in Art. 174, it can be shown that all such equations can be reduced to the form ax'^'' -^ dx"" = e, or It will be seen that the first member is composed of two terms so related that they may be the first two terms of a binomial square, and we can supply the third by one of the rules already given for completing the square. 188 ELEMENTARY ALGEBRA. 1. Reduce a:« — 2^^ = 48. OPERATION. O- oince the square root x' — 2x' = 4.S (1) of a:« is a:», it is evident a;® — 2a?^ + 1 = 1 + 48 =: 49 (2) that the second terra _8 1 I >T /o\ contains as one of its ^ — 1 = ± 7 (3) - ^ lactors the square root a^ z= 8, or — Q__ (4) of the first term ; i. e. X =z 2, or ^ — 6 (5) the first member of the equation is composed of two terms so related that they may be the first two terms of the square of a binomial. Completing the square, we have (2) ; extract- ing the square root of each member of (2), we obtain (3) ; transpos- ing we have (4), and extracting the cube root of (4) we have a; = 2, or ^^^ 2. Reduce 3a;^ — 4a:^= 160. OPERATION. 3a;* — 4a;^=160 (1) 36 x* ~ ( ) + 16 = 16 + 1920 = 1986 (2) 6 a;^ — 4 = ± 44 (3) Q'x^ = 48, or — 40 (4) x^ = 8, or — 5j0- (5) J = 2, or^-^- (6) a:= 16, or (— ^3C^^. Z 4 32 4. Reduce s/ar* 4-^-^ a: =: 1. Ans. a: = |, or — 8. EQUATIONS OF THE SECOND DEGREE. 189 5. Reduce x — § \/x = 44^. Ans. x = 49, or 40^. 6. Reduce x^ — x^ = 0. Ans. x = 1, or 0. Y. Reduce 3x^ — 2x^ + 3 = 228. Ans. X = ± S, or ± 5 \/ — ^. 8. Reduce 3 ^r^" — 2 ^r'* = 8. Ans. a; = / J a:-4-4 4a:-f-7 , , 7 — x 13. Reduce -^t _r_ + i ^ ___. 14. Reduce 2 V^ — V^ — 7 = 5. Ans, X = 16, or 7^. 15. Reduce 2 \/T:^:^ + 3 ^/2lc = L liJ" ^. \ X — a Ans. a: = 9 a, or — a. 15 16. Reduce 4\/x — \/2a;+ 1 = , ' v^2a:4-l Ans. a: = 4, or — 2f . 17. Reduce 5V25 — a; = 6V25 — a; + a:— 13. Ans. X = 16, or 9. 192 ELEMENTARY ALGEBRA. 18. Reduce ^-i^ = Vi+ \/2^^^. Alls. X = 12, or 4. 19. Reduce 6 + 4a:-i — I'ioj-^ = lOOar-^. 20. Reduce i ^x' + 6 ^ x — '-^ = -• 21. Reduce 30:4 — 24^72 — 80 = 304. 22. Reduce ^' + 10 = 1 + 4a:». Ans. a: = ± 4, or ± 2 V— 2. :*. Ans. X = ^'9, or ^. 23. Reduce 5;r4 — 3a:2 4- — = 27. ' 4 Ans. X = ± J\/6, or ± Sy/ — ^, 24. Reduce 2x^ — 5a:^ + 4 = 2. 25. Reduce 6a:^+ 1184 = 5a;i 26. Reduce V^ + 3 — -^x + 3 = 2. Ans. X z= 13, or — 2. 27. Reduce x^ — Vic^ + a: — 5 = 25 — a:. Note. — By transposing — x and subtracting 5 from each mem. ber, make the expression without the radical in the first member like that under the radical; then complete the square, &c. * 28. Reduce x^— 2x-{-Hx^2x^ — 6x— ll=x-\-^3. Ans. a: = 6, or — 3^ or J ± jj^ V273. 29. Reduce 21 x^ _|- U a:^ _ 69 = 321 — (11 x* + 6ar»). Ans. x= ± ^\/iS, or ± ^A^—i5. 30. Reduce ^^^,==^ + ^^.. 31. Reduce (x2-_4a:)2z=12a: — 3x«. 32. Reduce ar + (ar» — . a:)^ = ar» -f 51 12. EQUATIONS OF THE SECOND DEGREE. 193 PROBLEMS PRODUCING AFFECTED QUADRATIC EQUATIONS WITH BUT ONE UNKNOWN QUANTITY. 182. Though the numerical negative values obtained in solving the following Problems satisfy the equations formed in accordance with the given conditions, they are prac- tically inadmissible, and are therefore not given in the answers. 1. Divide 40 into two parts such that the sum of their squares shall be 1042. SOLUTION. Let X = one part ; then 40 — x = other part Then, a:^ -|- (40 — xf = 1042 Expanding, a:« -|- 1600 — 80 x -f- a:^ ^ 1042 Transposing and uniting, a^ — 40 a; = — 279 Whence, a; = 20 ± 11 = 31, or 9 And, 40 — a; = 9, or 31 2. Divide 20 into two parts such that their product will be 99|. Ans. 9^ and 10^-. 3. The ages of two brothers are such that the. age of the elder plus the square root of the age of the younger is 22 years, and the sum of their ages is 34 years. What is the age of each ? Ans. Elder, 18 ; younger, 16. Note. — The other answers found by reducing the equation, viz. 25 and 9, satisfy the conditions of the equation only upon consid- ering y/ 9 = — 3. To make the problem correspond to these an- swers, the word "plus" must be changed to "minus." 4. A merchant had two pieces of cloth measuring to- gether 96 yards. The square of the number of yards in the 194 ELEMENTARY ALGEBRA. longer is equal to one hundred times the number of yards in the shorter. IIow many yards are there in each piece ? Ans. 60 and 36. 6, Find two numbers whose difference is 3, and the sum of whose squares is 117. Ans. 9 and 6. 6. A merchant having sold a piece of cloth that cost him $42, found that if the price for which he sold it were multiplied by his loss, the product would be equal to the cube of the loss. What was his loss ? Note. — If the word " loss " were changed to gain, the other an- swer, — 7, or as it would then become, -\- 7, would be correct. Ans. $6. t. Find two numbers whose difference is 5, and prod- uct 176. Ans. 11 and 16. 8. There is a square piece of land whose perimeter in rods is 96 less than the number of square rods in the field. What is the length of one side ? Ans. 12 rods. 9. Find two numbers whose sum is 8, and the sum of whose cubes is 152. 10. A man bought a number of sheep for $240, and sold them again for $6.75 apiece, gaining by the bargain as much as 5 sheep cost him. IIow many sheep did he buy? Ans. 40. 11. Find two numbers whose difference is 4, and the sum of whose fourth powers is 1312. Note. — Let x — 2 and x -|- 2 be the numbers. Ans. 2 and 6. 12. A man sold a horse for $312.50, and gained one tenth as much per cent as the horse cost him. How much did the horse cost him? Ans. $250. 13. The difference of two numbers is 5, and the less minus the square root of the greater is 7. What are the numbers? Ans. II and 16. EQUATIONS OF THE SECOND DEGREE. 195 14. A and B started together for a place 300 miles dis- tant. A arrived at the place 7 hours and 30 minutes be- fore B, who travelled 2 miles less per hour than A. IIow many miles did each travel per hour? Ans. A, 10 ; B, 8 miles. 15. A gentleman distributed among some boys $15; if he had commenced by giving each 10 cents more, 5 of the boys would have received nothing. How many boys were there ? Ans. 30. 16. Find two numbers whose sum is a, and product b. a ± yJ a- — Ah , a T \/ d^ — 4 6 Ans. ^~~ ■ ^^d ^ ^ -• n. A merchant bought a piece of cloth for $45, and sold it for 15 cents more per yard than he paid. Though he gave away 5 yards, he gained $4.50 on the piece. How many yards did he buy, and at what price per yard ? Ans. 60 yards, at 75 cents per yard. 18. A certain number consists of two figures whose sum is 12 ; and the product of the two figures plus 16 is equal to the number expressed by the figures in inverse order. What is the number ? Ans. 84. 19. From a cask containing 60 gallons of pure wine a man drew enough to fill a small keg, and then put into the cask the same quantity of water. Afterward he drew from the cask enough to fill the same keg, and then there were 41 1 gallons of pure wine in the cask. How much did the keg hold ? Ans. 10 gallons. 20. There is a rectangular piece of land 75 rods long and 65 rods wide, and just within the boundaries there is a ditch of uniform breadth running entirely round the land. The land within the ditch contains 29 acres and 96 square rods. What is the width of the ditch ? Ans. .5 of a rod. 196 ELEMENTARY ALGEBRA. SECTION XX. QUADRATIC EQUATIONS CONTAINING TWO UNKNOWN QUANTITIES. 183t The Degree of any equation is shown by the sum of the indices of the unknown quantities in that term in which this sum is the greatest. Thus, 4:xy — 2x = 7 is an equation of the second degree, hx^y'^-\- xy^=^a^c " '' " fourth " . Note. — Before deciding what degree an equation is, it must be cleared of fractions, if the unknown quantities appear both in the denominators and in the numerators or integral terms ; and also from negative and fractional exponents. 184. A Homogeneous Equation is one in which the sura of the exponents of the unknown quantities in each term containing unknown quantities is the same. Thus, 4:X^ — 4:xy +/ — 16 or a;«+ Sxf+Bx'y + y' = 21 or X* — 4:X^y + Gx^/ _ 43,^8 _|_ ^4 _ 256 is a homogeneous equation. 185. Two quantities enter Symmetrically into an equa- tion when, whatever their values, they can exchange places without destroying the equation. Thus, a:- — 2a:y + / r= 25 or a:» + 3x2y4-3a:.y2 4- /= 8 or x^'^2xy-\-y^+ 2x + 2y::=24: QUADRATIC EQUATIONS. 197 186. Quadratic equations containing two unknown quan- tities can generally be solved by the rules already given, if they come under one of the three following cases : — I. When one of the equations is simple and the other quadratic. II. When the unknown quantities enter symmetrically into each equation. III. When each equation is quadratic and homogeneous. CASE I. V 187. When one of the equations is simple and the other quadratic. 1. Given 1^^ + ^^,'" ^^l, to find x and 3^. OPERATION. 2^:4-2^ = 22 (1) Sx'-{-7f=lll (2) y=ll—x (3) 3ar'-fl21 — 22a; + a;2=lll (4) 4a:-— 22a: = — 10 (5) 42x2— ()-j- 112= 121—40 = 81 ^Q^ 4a:=ll ±9 = 20, or 2 (7) y = 6,orlO^(95 a:=5, or^ (8) From (1) we obtain (3), or y = 11 — x. Substituting this value of y in (2), we obtain (4), an affected quadratic equation, which reduced gives (8) ; and substituting these values of x in (3), we obtain (9). In this Case the values of the unknown quantities can generally be found by substituting in the quadratic equation the value of one unknown quantity found by reducing the siviple equation. 198 ELEMENTARY ALGEBRA. 2. Given < ^-^ f- , to find x and y. ^^ = 28 (1) OPERATION. ar — y = 3 (2) V — 2xy + 2/'= 9 (3) 4a:2/ =112 (4) x' + 2xy + f=\2l (5) x + y =±11 (6) 2x = 14, or— 8 a) 2y = 8, or— 14 (8) X = 7, or — 4 (9) y = 4, or — 7 (10) Adding four times (1) to the square of (2), we obtain (5) ; ex- tracting the square root of each member of (5), we obtain (6) ; adding (2) to (6), we obtain (7) ; subtracting (2) from (6), we ob- tain (8) ; and reducing (7) and (8), we obtain (9) and (10). Note. — Though Example 2 can be solved by the same method as Example 1, the method given is preferable. By this method find the values of x and y in the fol- lowing equations ; — 3. Given j^ "2' = H- Ans. 1^ = 6- 4. Given j*+^=in. Ans'' = ^'»^«- ix^-\-fz=Sf>) (^, = 6, or7. 5. Given I :^y = 20) < 5 X + y = 29 t 6. Given \ x3, = 24> iSx — 2ij= 10) QUADRATIC EQUATIONS. 199 CASE II. 188. When the unknown quantities enter symmetrically into each equation. 1. Given i „ ' ^ f- , to find x and y. I :c3 + / r= 152 ) • OPERATION. a:+y= 8 (1) x« + / = 152 (2) x^ + 2xy + / = 64 (3) (5) (6) a) (8) Sxy =45 xy =15 X- - -2xy+y'= 4 a: — y = ±2 2 a: = 10, or 6 (9) 2y = 6, or 10 (10) X z= 5, or 3 (11) y = 3, or 5 (12) Squaring (1), we obtain (3); dividing (2) by (1), we obtain (4); subtracting (4) from (3), we obtain (5), from which we obtain (6) ; subtracting (6) from (4), we obtain (7) ; extracting the square root of each member of (7), we obtain (8) ; adding (8) to (1), we ob- tain (9) ; subtracting (8) from (1), we obtain (10) ; and reducing (9) and (10), we obtain (11) and (12). Note 1. — It must not be inferred that x and y are equal to each other in these equations ; for when x ^ 5, y = 3 ; and when ar = 3, y == 5. In all the equations under this Case the values of the two unknown quantities are interchangeable. Note 2. — Although af^ -\- y^ = 152 is not a quadratic equation, yet as we can combine the two given equations in such a manner as to produce at once a quadratic equation, we introduce it here. 200 ELEMENTARY ALGEBRA. 2. Given \ . ^^ ^ ^ I , to find a: and y. x^ + f — 2x — 2y OPERATION. xy^e (1) x'-]-f—2x—2y= 3 (2) 2xy =12 (3) (x-^yy-2(x^y) = 15 (4) (^ + yy- ()4- 1=16 (5) x-\'y=l±4 = 5yOT — B (6) — 3±\/- x = 3,or2, or ^ -''a^ -SqpV^- -15 y=2,or3,or— -^"^ (8) Adding twice (1) to (2), we obtain (4) ; completing the square in (4), we obtain (5) ; extracting the square root of each member of (5), and transposing, we obtain (6) ; and combining (1) and (6) as the sum and product are combined in the preceding example, we obtain (7) and (8). In Case II. the process varies as the given equations vary. In general the equations are reduced by a proper combination of the sum of the squares, or the square of the sum or of the difference, with multiples of the product of the two unknown quantities; and finally, of the sum, with the difference of the two unknown quantities. Note S. — When the unknown quantities enter into each equation symmetrically in all respects except their signs, the equations can be reduced by this same method ; e. g. a: — ;/ = 7, and 3^ — ^ = 511. In such equations the values of the unknown quantities are not inters changeable. Note 4. — The signs ± q: standing before any quantity taken in- dependently are equivalent to each other ; but when one of two quan- tities is equal to ± a while the other is equal to ^ ft, the meaning is that the first is equal to -}- a, when the second is equal to — h\ and the first to — a, when the second is equal to -\-h. QUADRATIC EQUATIONS. 201 By this method find the values of x and y in the follow ing equations : — 3. Given \^\-^^yr ''I' Ans. j-^^^orS. 4. Given \^ - V ^ H ^^^ (a: = 9,or-l. U3_/=z728i \y=z 1, or— .9. 5. Given |^+^/^+y =14) Note. — Divide the second equation by the first. 6. Given 1^ -V^ + y = U. (x2+ xy^y'^lZZ) CASE III. 189. When each equation is quadratic and homogeneous. 1. Given -< ^V \ V f- > to find x and y. XZx" —xy— 10) ^ OPERATION. 2xy+y^5 (1) 3:r2 — a:y=10 (2) Let X =11 vy 2vy'^-\-y'^ = b (3) 3 ^^3/2 _ ^^2 _ ^q ^4^ 2 y -f 1 3 y2 — y '^ ' >' 15 ^2 — 5 V = 20 y + 10 (8) 3^2 _ 5^ — 2 (9) t; — 2, or — ^ (10) ^ = 4^' or^^^-- (11) 2/= ± 1, or±-v/15 (12) .T = vy = ± 2, or q= ^ \/T5 (13) 9* 202 ELEMENTARY ALGEBRA. Substituting vy for x in (1) and (2), we obtain (3) and (4) ; from (3) and (4) we obtain (5) and (6) ; putting these two values of y' equal to each other, we obtain (7), which reduced gives (10) ; sub- stituting this value of v in (5), we obtain (11), which reduced gives (12) ; and substituting m x = vy the values of v and y from (10) and (12), we obtain (13). Examples under Case III. can generally be reduced best by substituting for one of the unknown quantities the product of the other by some unknown quantity, and then finding the value of this third unknown quantity. When the value of this third quantity becomes known, the values of the given unknown quantities can be readily found by substitution. Note. — Whenever, as in the example above, the square root is taken twice, each unknown quantity has four values ; but these values must be taken in the same order, i. e. in the example above, when y = -}-l, a; = -f-2; when y = — l,a: = — 2; when y = -[- \'~ib, x^= — i y/ lo ; and when y = — \/15, a: = -j-^V/l5. By this method find the values of x and y in the follow- ing equations : — 2. Given | ^^-^y =14) 2y^ Ana. < y= ± 5, or ± 11 V — i 3. Given j-^+3-y = 2T) ( V = A„8. f^=±3,or±9V-J. (j/z= ± 2, or q= 8V— i- 4. Given |x' + 4xy = U - 2/) (.a;y — 3/= 3 — x' ) An«. fx=±2,or±24V-j^. (. x" — 3=y + 2 ) QUADRATIC EQUATIONS. 203 190. Find the values of x and y in the following Examples. Note. — Some of the examples given below belong at the same time to two Cases. Thus in Example 1 both the equations are symmetrical, and both are quadratic and homogeneous, and there- fore it belongs both to Case II. and Case III. Example 3 belongs both to Case I. and Case II. 1. Given j ^ ^y = ^n. Ans. I^ = 2. Given \ ^-^^ ^ I. 3. Given .^ ^ +^= ^ I. U2+.y = 32-(:r + y2)> = ± 5, or ± 4. ± 4, or ± 5. Ans. 5^ = 4' or 3. <^ = 3, or 4. 4. Given \ ^^ = 12 ) \x^-\-x = Z1—y — y'-S \ ^='^. Ans. j- = ^.or3. 6. Given 6. Given | x-y=. 2) r:^/^ 2^:5^ = 1295) Note. — Considering a:y a single quantity, find its value in the second equation. 7. Given fx^y-xy^=30) Note. — Subtract from the second equation three times the first, and extract the cube root of each member of the resulting equation. 8. Given i 2^^ + 2x/ = 168| . ( a;» 4- v« = 91 ) A ns. -l w = 3, or 4 204 ELEMENTARY ALGEBRA. 9. Given 5 3a-^ - 3./ = 18) 10. Given \^ ^ ~ 3 ^ • ( a:y=10 ) An8. |^=±5. 11. Given |^-^-2.y = 88> Ans f^=±4, or± 66 V^. (2^= ± 3, or q:n5Vij(^5. 12. Given f x^ - 2x + 2^/ = 30 -y^ ) ( 4ic2/=z60 > 13. Given i ^ -V^^/) Ans. 1^-1000, or 8. (y= 625, or 1. 1. n- f 3;ryiiiil8) 14. Given \ ^ v - (a:^ — y = 65i . (ar== ± 3, or ± 2>C^ — 1. Ans. < -1- ; -1- "V (y = ± 2, or ± 3/i/— 1. 15. Given |3 (x - y) === 3 (^x + ^Z^) ) ( a:yr=36 ) X = — , or 9, or 4. y = — ^^ ^ , or 4, or 9. 16. Given V^" V^ = -T""* I. 17. Given |* ^* — '|. (x — y =19) QUADRATIC EQUATIONS. 205 18. Given -l^ -r -^ if c . 19. Given | ^~' — ^"' =^ ^"^ I . 20 Given j ^' + 2:^'y + 2a:y2 _^ / = 95> 21. Given I ^^ == H . (a;4 + 3/4 = 272) 22. Given |^ +.^/ = H PROBLEMS PRODUCING QUADRATIC EQUATIONS CONTAINING TWO UNKNOWN QUANTITIES. 191. Though the numerical negative values obtained in solving the following Problems satisfy the equations formed in accordance with the given conditions, they are practically inadmissible, and, except in Example 4, are not given in the answers. 1. The sum of the squares of two numbers plus the sum of the two numbers is 98 ; and the product of the two numbers is 42. Wh^t are the numbers ? Ans. 7 and 6. 2. If a certain number is divided by the product of its figures the quotient will be 3 ; and if 18 is added to the number, the order of the figures will be inverted. What is the number ^ Ans. 24. 3. A certain number consists of two figures whose product is 21 ; and if 22 is subtracted from the number. 206 ELEMENTARY ALGEBRA. and the sum of the squares of its figures added to the remainder, the order of the figures will be inverted. What is the number? Ans. 37. 4. Find two numbers such that their sum, their prod- uct, and the difference of their squares shall be equal to one another. Ans. f ± ^ V5 and ^ ± ^>v/5. 6. There are two pieces of cloth of different lengths ; and the sum of the squares of the number of yards in each is 145 ; and one half the product of their lengths plus the square of the length of the shorter is 100. What is the length of each ? Ans. Shorter, 8 ; longer, 9 yards. 6. Find two numbers such that the greater shall be to the less as the less is to 2f, and the difference of their squares shall be 33. 7. The area of a rectangular field is 1575 square rods ; and if the length and breadth were each lessened 6 rods, its area would be 1200 square rods. What are the length and breadth ? 8. Find two numbers such that their sum shall be to 6 as 9 is to the greater, and the sum of their squares shall be 45. Ans. 9 VT and 3 \^'^, or 6 and 3. 9.. The fore wheels of a carriage make 2 revolutions more than the hind wheels in going 90 yards ; but if the circumference of each wheel is increased 3 feet, the car- riage must pass over 132 yards in order that the fore wheels may make 2 revolutions more than the hind wheels. What is the circumference of each wheel ? Ans. Fore wheels, 13J feet; hind wheels, 15 feet. 10. Find two numbers such that five times the square of the greater plus three times their product shall be 104, and three times the square of the less minus their prod- uct shall be 4. KATIO AND PEOPORTION. 207 SECTION XXT. RATIO AND PROPORTION. 192. Ratio is the relation of one quantity to another of the same kind ; or, it is the quotient which arises from dividing one quantity by another of the same kind. Ratio is indicated by writing the two quantities after one another with two dots between, or by expressing the division in the form of a fraction. Thus, the ratio of a to h is written, a : 5, or r ; read, a is to h^ or a divided by h. 193. The Terms of a ratio are the quantities compared, whether simple or compound. The first term of a ratio is called the antecedent, and the other the consequent ; and the two terms together are called a couplet. 194. An Inverse, or Reciprocal Ratio, of any two quan- tities is the ratio of their reciprocals. Thus, the direct ratio of a to b \s a : b, i. e. r; and the inverse ratio of a to i is 1 1 . 1 1 & , - : 7> 1. e. - -i- 7 = -' or 6 : a. a aba 195. Proportion is an equality of ratios. Four quan- tities are proportional when the ratio of the first to the second is equal to the ratio of the third to the fourth. The equality of two ratios is indicated by the sign of equality (==) or by four dots (: :). Thus, a : b = c : d, or a : b : : c : c?, or t = -, ; read, a to b b d equals c to d, or a is to J as c is to d, or a divided by b equals c divided by d. 208 ELEMENTARY ALGEBRA. 196. In a proportion the antecedents and consequents of the two ratios are respectively the antecedents and con- sequents of the proportion. The first and fourth terms are called the extremes, and the second and third the means. 107. When three quantities are in proportion, e. g. a : b =z b : c, the second is called a mean proportional be- tween the other two ; and the third, a third proportional to the first and second. 198. A proportion is transformed by Alternation when antecedent is compared with antecedent, and consequent with consequent. 199. A proportion is transformed by Inversion when the antecedents are made consequents, and the conse- quents antecedents. 200. A proportion is transformed by Composition when in each couplet the sum of the antecedent and consequent is compared with the antecedent or with the consequent. 201. A proportion is transformed by Division when in each couplet the difference of the antecedent and conse- quent is compared with the antecedent or with the con- sequent. THEOREM I. 202. In a proportion the product of the extremes is equal to the product of the means. Let a : bz=z c : d a c Clearing of fractions, ad = be RATIO AND PROPORTION. 209 THEOREM II. 203t If the product of tvjo quantities is equal to the prod- uct of two others, the factors of either product may be made the extremes, and the factors of the other the means of a proportion. Let ad = be Q C Dividing hj bd, f ^^ ^ i. e. a : b = c : d THEOREM III. 204. If four quantities are in proportion, they will be in proportion by alternation. Let a : b = c : d By Theorem I. ad = be By Theorem II. a : c = b : d THEOREM IV. 205. If four quantities are in proportion, tJiey will be in proportion by inversion. Let a : b = c : d By Theorem I. ad^^bc By Theorem II. b : a = d : c THEOREM V. 206. If three quantities are in proportion, the product of the extremes is equal to the square of the mean. Let a : b = b : c By Theorem I^ _ ac = b"^ THEOREM VI. 207. If four quantities are in proportion, they will be in proportion by composition. ELEMENTARY ALGEBRA. a'.b = :c:d a b — c d lember, l + ^ = 1 + ^ b ~ c-f-rf d a + b:b = c + d 210 Let i, e. Addi or i. e. THEOREM VII. 208. If four quantities are in proportion, they will be in proportion by division. Let a : b = c : d a c i = i Subtracting I from each member, v — I =^ ~ — 1 a—hc—d i. e. a — b : b=z c — did THEOREM VIII. ^09. Two ratios respectively equal to a third are equil to each other. Let a '. b =z m '. n and c : d = m : n . ' a m . c m 1. e. T = - and ^ = - on d n Hence (Art. 13, Ax. 8), i. e. a : b=zc : d a c b~d THEOREM IX. 210. If four quantifies are in proportion, the sum and difference of the terms of each couplet will be in proportion. RATIO AND PROPORTION. 211 Let a : h =z c '. d By Theorem VI. a -\- b : h = c -\- d : d {I) and by Theorem VII. a — b.b = c — did (2) From (1), by Theorem III. a-{-b: c-\-d=b: d From (2), by Theorem III. a — b:c — d=b'.d By Theorem VIII. a-\-b: c-\-d=a — bic — d Hence, by Theorem III. a -\-b : a — b = c -\- d : c — d THEOREM X. 211. Equimultiples of two quantities ham the same ratio as the quantities themselves. For by Art. 83, ? = ^ •^ ' h mb i. e. a : b =: ma : mb Cor. It follows that either couplet of a proportion may be multiplied or divided by any quantity, and the result- ing quantities will be in proportion. And since by Theo- rem III. if a : b = ma : mb, a : ma = b : mb, or ma : a =z mb : b, it follows that both consequents, or both ante- cedents, may be multiplied or divided by any quantity, and the resulting quantities will be in proportion. THEOREM XI 212. If four quantities are in proportion, like powers or like roots of these quantities will be in proportion. Let a : b = c : d i. e. Hence, i. e. a"" : b"" =^ c"" : d^ Since n may be either integral or fractional, the theorem is proved. a b = c d 6« == d^ 212 ELEMENTARY ALGEBRA. THEOREM XII. 213. If any number of quantities are proportional, any antecedent is to its consequent as the sum of all the antece- dents is to the sum of all the consequents. Let a : b = c : d=: e :f Now ab=:ab (1) and by Theorem I. ad = bc (2) and also af=:be (3) Adding(l),(2),(3), a{b + d+f)=b{a-\-c-^e) Hence, by Theorem II. a:b=za-\-c-{-e'.b-\-d -\-f THEOREM XIII. 21 4 1 If there are two sets of quantities in proportion, their products, or quotients, term by term, will be in proportion. Let a : b=z c : d and e:f=g :h By Theorem I. ad =z be (1) and eh=fg (2) Multiplying (1) by (2), ad eh ^ b efg ; (3) Dividing (1) by (2), . ad be (4) From (3), by Theorem II. a e : bf =^ e g : dh and from (4), a b c d PROBLEMS IN PROPORTION. 215. By means of the principles just demonstrated, a proportion may often be very much simplified before making the product of the means equal to the product of the extremes ; and a proportion which could not oth- erwise be reduced by the ordinary rules of Algebra may often be so simplified as to produce a simple equation. RATIO AND PROPORTION. 213 1. The cube of the smaller of two numbers multiplied by four times the greater is 96 ; and the sum of their cubes is to the difference of their cubes as 210 : 114. What are the numbers ? SOLUTION. Let X = the greater and y =. the less. Then 4a;/ = 96 (1) a^^ + ^/^a:^ — / == 210 : 114 (2) From (2), by Theo. X., Cor. x^ -\-y^ : x^ — y^ = Zb : 19 By Theorem IX. 2a;« : 2/ = 54 : 16 By Theorem X., Cor. x^ '.y^ = 21 :^ By Theorem XI. x:y = Z:2 By Theorem I. 2 ;r = 3 y (3) From (1) and (3) we find a: = 3 and y = 2. 2. The product of two numbers is V8 ; and the differ- ence of their cubes is to the cube of their difference as 283 : 49. What are the numbers ? SOLUTION. Let X = the greater and y = the less. Thena;2/=78 (1) 3^ — f : xr^ — 3 sfiy -{- 3xf — f = 283 :49 (2) From (2), by division, 3x'y — 3xy'^ : (x — yY = 234 : 49 Dividing 1st couplet by a: — y, 3xy : (x — yY = 234 : 49 Dividing antecedents by 3, xy : (x — yY = 78 : 49 Substituting the value of xy, 78 : (a: — yy = 78 : 49 •Dividing antecedents by 78, 1 : (x — yy = 1 :4d Extracting the square root, 1 : x — y = l : 7 Whence, x — y == 7 (3) From (1) and (3) we find a: = 13 and y = 6. 3. The sum of the cubes of two numbers is to the cube of their sum as 13 : 25 ; and 4 is a mean proportional be- tween them. What are the numbers? 214 ELKMENTARY ALGEBRA. 4 The difference of two numbers is 10 ; and their prod- uct is to the sum of their squares as 6 : 37. What are the numbers ? SOLUTION. Let X = the greater and y = the less. Then x—y=10 (1) ary : r» -|- ^ = 6 : 37 (2) From (2), by Theorem X., Cor. 2zy::c2-}-y^=12:37 By Theorem IX. x" -\- 2 x y -\- f : x"" — 2 xy -{- f = 40 : 2o By Theorem XL x -J[- y : x — y= 7:5 By Theorem IX. 2 i: : 2 y = 12 : 2 By Theorem X., Cor. x : y =6:1 By Theorem I. x ^ 6 y (S) From (1) and (3) we find a: = 12 and y = 2. 5. The product of two numbers is 136 ; and the dif- ference of their squares is to the square of their differ- ence as 25 : 9. What are the numbers ? Ans. 8 and 17. 6. As two boys were talking of their ages, they dis- covered that the product of the numbers representing their ages in years was 320, and the sum of the cubes of these same numbers was to the cube of their sum as 7 : 27. What was the age of each ? Ans. Younger, 16; elder, 20 years. 7. As two companies of soldiers were returning from the war, it was found that the number in the first multi- plied by that in the second was 486, and the sum of the , squares of their numbers was to the square of the sum as 13 : 25. How many soldiers were there in each company? Ans. In 1st. 27; in 2d, 18. 8. The difference of two numbers is to the less as 100 is to the greater ; and the same difference is to the greater as 4 is to the less. What are the numbers ? Note. — Multiply the two proportions together. (Theorem XIIL) PROGRESSION. 215 SECTION XXII. PROGRESSION. 216. A Progression is a series in which the terms in- crease or decrease according to some fixed law. 217. The Terms of a series are the several quantities, whether simple or compound, that form the series. The first and last terms are called the extremes^ and the others the means. ARITHMETICAL PROGRESSION. 218. An Arithmetical Progression is a series in which each term, except the first, is derived from the preced- ing by the addition of a constant quantity called the com- mon difference. 219. When the common difference is positive, the series is called an ascending series, or an ascending progression ; when the common difference is negative, a descending se- ries. Thus, a, a -\- d, a -\- 2d, a -\- '6d, &c. is an ascending arithmetical series in which the common difierence is d ; and a, a — d, a — 2 c?, a — 3 c?, &c. is a descending arithmetical series in which the common difference is — d. 220. In Arithmetical Progression there are five elements, any three of which being given, the other two can be found : — 1. The first term. 2. The last term. 216 ELEMENTARY ALGEBRA. 3. The common difference. 4. The number of terms. 6. The sum of all the terms. 221. Twenty cases may arise in Arithmetical Progres- sion. In discussing this subject we shall let a = the first term, / = the last term, d = the common difference, n = the number of terms, aS^=: the sum of all the terms. CASE I. 222. The first term, common difference, and number of terms given, to find the last term. In this Case a, d, and n are given, and I is required. The suc- cessive terms of the series are a, a-\-d, a -f 2 J, a-\- 3d, a -\- Ad, &c. ; that is, the coefficient of d in each term is one less than the number of that term, counting from the left ; therefore the last or nth term in the series is a -}- (n — 1) d or 1 = a -j- {n — 1) d in which the series is ascending or descending according as d is posi- tive or negative. Hence, RULE. To the first term add the product formed by multiplying the common difference by the number of terms less one. 1. Given a = 4, d =:2, and w = 9, to find /. l = a-\-{n — l) rf = 4 + (9 — 1) 2 = 20, Ans. 2. Given a = T, d=3, and n = 19, to find L Ans. /=61. PROGRESSION. 217 3. Given a = 29, d = — 2, and n = 14, to find /. Ans. 1 = ^. 4. Given a = 4.0, d= 10, and n = 100, to find /. 5. Given a= I, d =: ^, and n = 17, to find I. 6. Given a z= ^, d = — J^, and w = 13, to find I. 1. Given a= .01, d = — .001, and n= 10, to find /. CASE II. 223. The extremes and the number of terms given, to find the sum of the series. In this Case a, I, and n are given, and S is required. Now S = a 4- (a + rf) + (a + 2rf) + (a + 3rf) + + / or, inverting the series, S= Z + ( Z — rf) + ( / — 2rf) + (Z — 3d) 4- +a Adding these together, 2 S= (a + + (a + Z) + (a + Z) + (a + + + (a + Z) And since (a -{- Z) is to be taken as many times as there are terms, hence 2S = n{a-{-l) or -S? = 9 (« -j- 0- Hence, RULE. Find one half the product of the sum of the extremes and the number of terms. Note. — If in place of the last term the common difference is given, the last term must first be found by the Rule in Case I. 1. Given a = S, 1= 141, and n = 26, to find S. S=l(a-irl) = ^{S+ 141) = 1872, Ans. ^. Given a = i, 1=25, and n = 63, to find S. Ans. S=19Si. 3. Given a = 4, d =z2, and n = 24, to find S. Ans. 5=648. 4. Given a == — 3, c? = 2, and w = 4, to find S. Ans. S=0. 218 ELEMENTARY ALGEBRA. 5. Given a = ^, d = — ^, and n = 3, to find S. 6. Given a = .07, /= .11; and n=n, to find S. 1. Given a= — 4^, . n (71 — 1) 11. Find the formula for the value of a, when d, n, and S are given. . 2 S — n(n--V)d Ans. ct — — 2n 12. Find the formula for the value of a, when d, I, and S are given. ^ns. a = ^ ± y/ (-^ + ?)' - 2 rf & 226. To find any one of the five elements viheo. three others are given. RULE. Substitute the given values in that formula whose first mem- ber is the required term, and whose second contains the three given terms. 1. Given d=:2, / = *21, and S= 120, to find a. OPERATION. d ^=2 ±v/G+0-'^^^ ^^) l^jn y/(^ + 2l) -2.2.120 (2) 3, or — 1 (3) In Example 12, Art. 225, we find (1), the required formula; substi- tuting the given values of • are ascending series ; _54, _i8, _ 6, — 2, &C.I ^ 16, 8, &c.) , ,. >- are descendine: — 32, —64, &c. ) ^ 64, 32, _ .' ^ Y are aescendin^ series. 8, — 16, If the ratio is negative, the terms of the progression are alternately positive and negative. Thus, if the ratio is — 2 and the first term 3, the series will be 3, — 6, + 12, — 24, -f 48, &c. ; but if the first term is — 3, — 3, +6, —12, 4-24, —48, &c. The positive terms of these two series constitute an as- cending progression whose ratio is the square of the given ratio ; and the negative terms a descending progression having the same ratio. 231. In Geometrical Progression there are five elements, any three of which being given, the other two can be found. These elements are the same as in Arithmetical Progres- sion, except that in place of the common difference we have the ratio. 10* * o ' 226 ELEMENTARY ALGEBRA. 232. Twent}'^ cases may arise in Geometrical Progres- sion. In discussing these cases we shall preserve the same notation as in Arithmetical Progression, except that instead of d ^ the common difference we shall use r = the ratio. CASE I. 233. The first term, ratio, and number of terms given, to find the last term. In this Case a, r, and n are given, and I required. The successive terms of the series are a, ar, ar^, ai^^ a 7*, &c. That is, each term is the product of the first term and that power of the ratio which is one less than the number of that term count- ing from the left ; therefore the last or nth term in the series is ar" or l = ar"-^. Hence, RULE. Multiply the first term by that power of the ratio whose index is one less than the number of terms. 1. Given a = 7, r = 3, and w = 5, to find /. l=ar-^-'^z=1 X 3^ = 667, Ans. 2. Given a = 3, r = 2, and w ::= 9, to find /. Ans. /= 768. 3. Given a = 64, r = ^, and n ^ 10, to find I. Ans. / = i- 4. Given a = — 7, r =. — 4, and n = 3, to find /. Ans. / = — 112. 6. Given a = — ^, r = ^, and w = 6, to find /. Ans. /= — j^^. 6. Given a = 5, r = — \, and w = 10, to find /. 7. Given a = — ^, r = ^, and n = 8, to find /. 8. Given a=: — 10, r = — 2, and n = 6, to find /. PROGRESSION. 227 CASE II. 234. The extremes, and the ratio given, to find the sum of the series. In this Case a, I, and r are given, and S is required. Now S = a -{- ar -\- ar" -\- ai^ -{- -^ I (l) Multiplying (1) by r, r S = ar-\- ar" -\- a?^ -}- ^IJ^lr (2) Subtracting (1) from (2), r S — S = Ir — a Whence, S= —. Hence, r — 1 RULE. Multiply the last tei^Tn by the ratio, from the product sub- tract the first term, and divide the remainder by the ratio less one. 1. Given az=2, 1= 20000, and r = 10, to find S. ^^l_r-a^ 20000 X 10 - 2 _ ^2222, Ans. r — 1 10 — 1 2. Given a = 1, Z = 45927, and r = 3, to find S. Ans. ^r= 68887. 3. Given a = — 5, 1= — 405, and r = 3, to find *S'. 4. Given a = — t7T5) ^ = i> and r =z — 7, to find S. Ans. S=r,\% CASE III. 235. The first term, ratio, and number of terms given, to find the sum of the series. In this Case a, r, and n are given, and 5 required. The last term can be found by Case I., and then the sum of the series by Case 11. Or better, since lr'= ar^ Substituting this value of / r in the formula in Case II. we have r^ — 1 S = ~ X a- Hence, 228 ELEMENTARY ALGEBRA. RULE. From the ratio raised to a power whose index is equal to the number of terms subtract one, divide the remainder by the ratio less one, and multiply the quotient by the first term. 1. Given = 4, r = 7, and n = 5, to find S. S='^^^ Xa= '^^ X 4 = 11204, Ans. 2. Given a = |, r = 5, and n = 6, to find S. Ans. *S'=558. 3. Given a = ^, r = ^, and n = 1, to find S. Ans. S=m. 4. Given a z= — 6, r = — 4, and w = 4, to find aS^. Ans. .S'=255. 5. Given a = — |, r = 6, and n = 5, to find S. 6. Given a = §, r = — 3, and n = 6, to find S. 7. Given a = — }, r = 2, and n = S, to find S. In a geometrical series whose ratio is a proper frac- tion the greater the number of terms, the less, numeri- cally, the last term. If the number of terms is infinite, the last term must be infinitesimal ; and in finding the sum of such a series the last term may be considered as noth- ing. Therefore, when the number of terms is infinite, the formula S= becomes 1 1 — r Hence, to find the sum of a geometrical series whose ra- tio is a proper fraction and number of terms infinite, RULE. Divide the first term by one minus the ratio. PROGRESSION. 229 1 Find the sum of the series 1, ^, |-, &c. to infinit3^ '5 = l^.= l^i = 2. Ans. 2. Find the sum of the series -|, |, ^5, &c. to infinity. Ans. j%. 3. Find the sum of the series -, ~^, -, &c. to infinity. . 1 Ans. c — 1 4. Find the sum of the series 6, 4, 2|, &c. to infinity. Ans. 18. 5. Find the value of the decimal .4444, &c. to infinity. Note. — This decimal can be written -^ -\- j^ -\~ y^j^^, &c. Ans. |. 6. Find the value of .324324, &c. to infinity. 1. Find the value of .32143214, &c. to infinity. CASE IV. 237. The extremes and number of terms given, to find the ratio. In this Case «, /, and n are given, and r is required. From Case I. Z = a;"-i Whence, r= »/-. Hence, RULE. Divide the last term hj the first, and extract that root of the quotient whose index is one less than the number of terms. 1. Given a == 7, / =: 667, and n = 5, to find r. "-yi 4/567 3, Ans. 2. Given a == 6|, 1=1, and n = 6, to find r. Ans. r = ^. 230 ELEMENTARY ALGEBRA. 3. Given a = — ^, / = 31^, and n = 4, to find ••. Ans. r = — 5. Note. — This rule enables us to insert any number of geometri- cal means between two numbers ; for the number of terms is two greater than the number of means. Hence, if m = the number of m+l/J means, m-|-2 = n, orw-|-l==n — 1; and r = i/-« Having found the ratio, the means are found by multiplying the first term by the ratio, by its square, its cube, &c. 4. Find three geometrical means between 2 and 512. Ans. 8, 32, 128. 5. Find four geometrical means between 3 and 3072 Ans. 12, 48, 192, 768. 6. Find three geometrical means between 1 and y'^. Ans. ^, i, i. Note. — When m = 1 , the formula becomes Multiplying by a, ar = a ^ - =^1 al But a r is the second term of a series whose first term is a and ratio r ; or the geometrical mean of the series a, ar, a r*. Hence, the geometrical mean between two quantities is the square root of their product. 7. Find the geometrical mean between 8 and 18. Ans. 12. 8. Find the geometrical mean between ^ and 343. Ans. 7. 9. Find the geometrical mean between ^ and ji^^. 10. Find the geometrical mean between — ^ and — yiVr- PROGRESSION. 231 238. From the formulas established in Arts. 233 and 234, l=ar^-^ (!)• S^^^-^ (2) r — 1 ^ ^ can be derived formulas for all the Cases in Geometrical Progression. * From (1) we can obtain the value of any one of the four terms, /, a, n, or r, when the other three are given; frona (2), the value of 5, Z, r, or a, when the other three are given. Formulas for the remaining twelve Cases which may arise are derived by combining the formulas (1) and (2) so as to eliminate that one of the two unknown terms whose value is not sought. 1. Find the formula for the value of *S', when /, n, and r are given. From (1), Substituting this value of a in (2) or (r_l)r»-i Note. — The four formulas for the value of n cannot be derived or used without a knowledge of logarithms ; and four others, when n ex- ceeds 2, cannot be reduced without a knowledge of equations that can- not be reduced by any rules given in this book. 239. To find any one of the five elements when three others are given. RULE. Substitute in that one of the formulas (1) or (2) that con- tains the four elements, viz. the three given and the one re- quired, the given values, and reduce the resulting equation. If neither formula contains the four elements, derive a for- mula that will contain them, then substitute and reduce the resulting equation ; or substitute the given values before deriv- ing the formula, then eliminate the superfluous element *and reduce the resulting equation. I V — : a ir ' 1 r —\ l(r^- 232 ELEMENTARY ALGEBRA. 1. Given r — 3, n — b, and S= 726, to find I. l-af-' (1) S=.^'-^ (2) /-81a (3) 726 = '"-" (i) g^ = a (5) a = 3 ; — 1452 (6) 8/- 1452=1 (7) 242 ;=r 1452 X 81 (8) / = 486 (9) Substituting the given values of r, n, and S in (1) and (2), we obtain (3) and (4) ; finding the value of a, the superfluous element, from (3) and (4), and putting these values equal to each other, we form (7), an equation containing but one unknown quantity. Re- ducing (7) we obtain (9), or I = 486. 2. Given a = 4, r = 5, and S= 15624, to find I. Ans. lz=z 12500. 3. Given a = 2, w = 5, and /= 512, to find S. Ans. >S'=682. 4. Find the formula for the value of a, when r, n, and S «^« gi^e"- Ans. a = ^'~V- r» — 1 5. A gentleman purchased a house, agreeing to pay one dollar if there was but one window, two dollars if there were two windows, four if there were three, and so on, doubling the price for every window. There were 14 windows. How much must he pay? Ans. $8192. 6. A man found that a grain of wheat that he had sown had produced 10 grains. Now if he sows the 10 grains the next year, and continues each year to sow all that is produced, and it increases each year in tenfold ratio, how many grains will there be in the seventh harvest, and how many in all? ^^^ f In 7th harvest, 10000000 grains. tin all, 1111 11 1 1 grains. PROGRESSION. 233 PROBLEMS TO WHICH THE FORMULAS DO NOT DIRECTLY APPLY. 240. In solving Problems in Geometrical Progression, if we let X = the first term and y = the ratio, the series will be X, xy, xy'^, xy^, &c. It will often be found more convenient to represent the series in one of the following methods : — 1st. When the number of terms is odd, or > ', ^y, f) for three terms ; X' a? if —, x^, xy, 7f,~ for five terms. y x 2d. When the number of terms is even, a? if -, X, y, — for four terms; y ■^ 3? x" y" f o ' 2?' y' ""' ^' x' x^ ^'"^ *^^™®- Which method is most convenient in any case will de- pend upon the conditions that are given in the problem. 1. There are three numbers in geometrical progression, the greatest of which exceeds the least by 32 ; and the diflerence of the squares of the greatest and least is to the sum of the squares of the three as 80 : 91. What are the numbers ? SOLUTION. Let X, xy^ and xy^ represent the series. Then a;/ — x=32(l) x*?/* — x2:x2 + x27/2-|-x2r/*==80:91 (2) y_l:l_|_y2^y4=80:91 (3) 91y4_91=:80-(-80?/2-|-80?/* (4) 11 2^4—80 2/='= 171 (5) x=4 (7) 2/ =3 (6) 234 ELEMENTARY ALGEBRA. Dividing the first couplet of (2) by x*, we obtain (3) ; from (3) We form (4), which reduced gives (6), or y = 3. Substituting the Value of y in (l), we obtain (7), or x == A. Ans. 4, 12, 36. 2. The sura of three numbers in geometrical progres- sion is 39, and the sum of their squares 819. What are the numbers ? SOLUTION. Let X, \^xy, and y represent the series. Then x + ^Vy + y = S9 (1) x^ + xy + f = ^^^ (2) x-^JoJ+j^2i (3) 2 x^-\-2y = Q0 (4) x+ i/ = SO (5) 2-v/'^=18 (6) xy = Sl (1) Dividing (2) by (1), we obtain (3); adding (3) to (1), we ob- tain (4), which reduced gives (5) ; subtracting (3) from (1), we obtain (6), which reduced gives (7). Combining (5) and (7) as the sum and product are combined in Example 1, Art. 188, we obtain a: = 27 and y = 3. Ans. 3, 9, 27. 3. Of four numbers in geometrical progression the dif- ference between the fourth and second is 60 ; and the sum of the extremes is to the sum. of the means as 13 : 4. What are the numbers ? SOLUTION. Let X, xy, xy^, and a;y* represent the series. Then xf — xy — m (1) a;/ + x:xy'' + xy=13:4 (2) y'-y+l:./=13:4 (3) 4^ — 4y + 4 = 13y (4) 64a: — 4* = 60 (7) 4y»— 17y = — 4 (5) x= 1 (8) y = 4 (6) PROGRESSION. 235 Dividing the first couplet of (2) by X7j -\- x,yve obtain (3) ; from (3) we form (4), which reduced gives (6), or y = 4. Substituting this value of y in (1) and reducing, we obtain (8), or x=\. Ans. 1, 4, 16, 64. 4. Of four numbers in geometrical progression the sum of the first two is 10 and of the last two 160. What are the numbers ? Ans. 2, 8, 32, 128. 6. A man paid a debt of $310 at three payments. The several amounts paid formed a geometrical series, and the last payment exceeded the first b}^ $240. What were the several payments? Ans. $10, $50, $250. 6. In the series x, \/ xy, and y what is the ratio? Ans. ^\ Y. In the series -, x, y, and - what is the ratio? 8. There are four numbers in geometrical progression whose continued product is 64 ; and the sum of the series is to the sum of the means as 5 : 2. What are the num- bers ? Ans. 1, 2, 4, 8. 9. There are five numbers in geometrical progression ; the sum of the first four is 156, and the sum of the last four *r80. What are the numbers? 10. There are three numbers in geometrical progression whose sum is 126 ; and the sum of the extremes is to the mean as 1*7 : 4. What are the numbers? 11. The sum of the squares of three numbers in geo- metrical progression is 2275 ; and the sum of the ex- tremes is 35 more than the mean. What are the numbers ? 12. Of four numbers in geometrical progression the sum of the first and third is 52 ; and the difference of the means is to the difference of the extremes as 5 : 31. What are the numbers ? 23(3 p:lkmentary algebra. SECTION XXIY. MISCELLANEOUS EXAMPLES. 1. From 6ac— [>ab-{-c^ take Sac— {3a6 — (c — c^) + '7c}. Ans. 3ac — 2a6 + 2c2-f6c. 2. Reduce x^f — (— xf + x^ — ^) xy — x^ (— {/ — y {^1/ — ^'0 1 ) to its simplest form. Ans. 2x^f + x^. 3. Reduce (a — b -\- cy — fa (c — a ^ b) — ^b (a -\- b -\- c) — c (a — b — c) } j to its simplest form. Ans. 2(a2 + 62 4-c2). 4. Reduce (x -{- a) a -\- i/ — | (^ -|_ ^) (^x + b) — y (a: + a — 1) — (x -\- y) (b — a)} to its simplest form. Ans. a^ — i-. 6. Reduce (a^ — b"") c — {a — b) [a {b-\- c) —b {a — c)\ to its simplest form. Ans. 0. 6. Reduce {a -{- b) x— (b — c) c — ^{b-^x) b— {b — c) (5-|-c)} —ax to its simplest form. Ans. 2bx — be. 1. Multiply a^ + 2aH — SaP by — (— 3aH -\- aH^). 8. Multiply a* -f- 6 a2 _|_ 9 by a* — 6 a^ + 9. 9. Multiply a-\- b — c by a — b -\- c. 10. Divide 28a2 _ 6a^ — Ga^ — 4a^ — 96a + 264 by 3a2_4a + 11. 11. Divide 1 — 18ar«4- 81a:* by 1 -|-6x4- 9x\ 12 Divide 9 0^+ 1 —4a* — 6 a by 1 -f 2 a^ — 3 a. 13. Divide 9 x^ — 1 x^ f -}- 2/ by S x* + 2x''i/ — t/\ MISCELLANEOUS EXAMPLES. 237 14. Divide 23 a — 30 — 7 a^ + 6 a* by 3 a — 2 «= — 6- 15. Find the prime factors of a^ — h^. 16. Find the prime factors of 4m^?^^ — 49m*w^^. 17. Find the prime factors of .t^ — ^xy-^-y"^. 18. Find the prime factors of x^ — ^^ 19. Find the greatest common divisor of 5 ar* — lOx^y -j- 15/ and 4. x^ -\- ^ x"^ y -\- ^ x y'^ -\- ^ y^ . Ans. x -\- y. 20. Find the greatest common divisor of 8 « i^ -f- 24 a i* + 16 « i and U' + U' + 7 ^* — Ul Ans. ¥ -f h. 21. Find the greatest common divisor of 6 a:;^ -[" *^ ^V — 3y2 and 120^24- 22x^ + 6/. 22. Find the greatest common divisor of 4 a; -f- 4: x'^ — 40 and ^x^y — 48 y. Ans. x — 2. 23. Reduce 7 — ,^ , ., , ^"\ , ..^^ to its lowest terms. 24. Reduce , ,, to its lowest terms. («- -6) (a-2 4-2a6 + 6^) a'- -3a2^,-|_3a6- — ^' t a^ — y' a^ — 2/* 25. Reduce -r^, — ^—ri, — ^ — -1 — ^r to its lowest terms. 26. Find the least common denominator and reduce :; :; — i — ; — 5 — :; — — ^ to a smglc iraction. 1 — a 1-j-a l-j-a' 1 — a^ ° ^ Ans. — -— i — ^• 1 -f- « 27. Find the least common denominator and reduce 1 _|_ ^2 1 _ „i2 . , . , . :; — — . — - — i — 7i to a Single fraction. 1 — rrc 1 -[- m- ° 28. Find the least common denominator and reduce 4a^-3a5~^- 16a^-9a^6^ *° ^ '"'^^^ ^'^^^'^"' 29. Reduce to one fraction with the least possible de- nominator r r- ^^- ^ V- -r-, • bed cd ' hd 238 ELEMENTARY ALGEBRA. 30. Reduce to one fraction with the least possible de- . . a-\-b b 4-c , a-f-c nominator 7^ ^. r^ — 7 vt ix + n tt r; • (b — c) (c — a) (a — c) (a — b) ' (/> — a) {c — b) Ans. yj -7 — ^ r- = 0. (0 — c) (c — a) (a — b) 31. Find the least common denominator and reduce 32. Reduce to one fraction with the least possible de- . , I -\-x Ax \ — X . 2a: -+62:' i^ominator ^j^, - j— ^ - jy^-^,. Ans. -p^:^.- 33. Reduce a — c ^^ ^ to its simplest form. I + -I1 34. Reduce ^^^ to its simplest form. 35. Reduce -— ?! 1- a; and — f— | 2y each to a X — 2y ^ X -\-2y ^ single fraction and find their product. Ans. j^-^ x» X 36. Subtract -^— from 2^ — a; a: -hy X X — ct 37. Subtract Bx4--r from a: 88. Multiply (-il)\y ^/-g^- 39. Divide by -; i, and multiply the result by a* a — X '^ c? — ar ^ '' *' 40. Divide '^^^-^^y. by -^^. 41. Divide ^' by (-^^ + -±-\ MISCELLANEOUS EXAMPLES. 239 42. Divide -7^7 — ;— t:v/ by — r^- b (a -{- by "^ rt- — b- T^. .1 a A- X , a — x^ a -\- x a — x , 43. Divide — ^^- j — by — ' i — , and give a — X ' a-{-x *' a — x a -\- x ^ a" + x^ the answer in its lowest terms. Aus. -—-^ d Cl X 44. Reduce ;— r = 1 • What is the value of a a -f- a — X, if a = — 2 and 6 = 3? 45. Reduce x — = - -j- 2 7 ' 2 fj^c hx^ 46. Reduce {a -\- x) {h — x) — a {b — c) ^ = 0. What is the value of x, if a = 2, J = — 3, and c == — 1. 47. Reduce ~'' ^=- What is the value 0/ b a x, if a = 2, 6 =r — 1, and c — Zt \ A- x 48. Reduce a — ■— ^- = 0. 1 — X 49. Find the value of x in the equation x = ^ "^ — ^— -^ in its simplest form. « — * « + * 60. A man spends $2. He then borrows as much money as he has left, and again spends $2. Then borrowing again as much money as he has left, he again spends $2, and then has nothing left. How much money did he have at first? 61. If 5 is subtracted from a certain number, two thirds of the remainder will be 40. What is the number ? 52. Having a certain sum of money in my pocket, I lost c dollars, and then spent one ath part of what re- mained and had left one 6th part of what 1 had at first.' What was the original sum ? What does the answer be- come if a = 3, b =z 9, and c = 6 ? 240 ELEMENTARY ALGEBRA. 53. If I buy a certain number of pounds of beef at SO. 25 a pound, I shall have $0.25 left; but if I buy the same number of pounds of lard at SO. 15 a pound, 1 shall have $1.25 left. How much money have I? 54. Divide 84 into three parts so that one third of the first, one fourth of the second, and one fifth of the third shall be equal. 55. In a certain orchard 25 more than one fourth of the trees are apple trees, 2 less than one fifth are pear trees, and the rest, one sixth of the whole, are peach trees. How many trees are there in the orchard ? 56. A merchant spent each year for three years one third of the stock which he had at the beginning of the year; during the first year he gained $600, the second $500, and the third $400. At the end of the three years he had but two thirds of his original stock. What was his original stock ? 57. From a cask of wine out of which a third part had leaked, 84 liters were drawn, and then the cask was half full. What is the capacity of the cask ? 58. A gentleman has two horses and a chaise. The chaise is worth a dollars more than the first horse and h dollars more than the second. Three fifths of the value of the first horse subtracted from the value of the chaise is the same as seven thirds of the value of the second horse subtracted from twice the value of the chaise. What is the value of the chaise and of each horse ? What are the answers if a =: — 50 and 5 := 50 ? 59. A had twice as much money as B. A gained $30 and B lost $40. Then A gave B three tenths as much as B had left, and had left himself 20 per cent more than he had at first. How much did each have at first ? MISCELLANEOUS EXAMPLES. 241 60. A number of mea had done one third of a piece of work in 9 days, when 18 men were added and the work completed in 12 days. What was the original number of men 1 61. A boatman can row down the middle of a river 14 miles in 2 hours and 20 minutes ; but though he keeps near the shore where the current is one half as swift as in the middle, ft takes him 4 hours and 40 minutes to row back. What is the velocity of the water in the middle of the river ? Ans. 2 miles an hour. 62. A had three fifths as much money as B. A paid away $80 more than one third of his, and B $50 less than four ninths of his, when A had left one third as much as B. What sum had each at first? 63. A farmer hired a man and his son for 20 days, agreeing to pay the man $3.50 a day and the son $1.25 for every day the son worked ; but if the son was idle, the farmer was to receive $0.50 a day for the son's board. For the 20 days' labor the farmer paid $67. How many days did the son work ? 64. I purchased a square piece of land and a lot of three- inch pickets to fence it. I found that if I placed the pick- ets 3 inches apart, I should have 50 pickets left ; but if I placed the pickets 2^ inches apart, I must purchase 60 more. How much land and how many pickets did 1 pur- chase ? Ans. 18906^ square feet and 1150 pickets. 65. A criminal having escaped from prison travelled 10 hours before his escape was discovered. He was then pursued and gained upon 3 miles an hour. When his pursuers had been on the way 8 hours, they met an ex- pressman going at the same rate as themselves, who had met the criminal 2 hours and 24 minutes before. In what time from the commencement of the pursuit will the crimi- nal be overtaken ? Ans. 20 hours. 11 p 242 ELEMENTARY ALGEBRA. 66. In February, 1868, a man being asked the time, answered that the number of hours before the close of the month was exactly one sixth of 10 less than the number that had passed in the month. What was the exact time? Ans. February 25th, 10 o'clock, p. m. 6t. A and B owned adjoining lots of land whose areas were as 3 : 4. A sold to B 100 hectares of his, and after- ward purchased of B two fifths of B's entire lot ; and then the original ratio of their quantities of land had been re- versed. How much land did each own at first ? Ans. A, 300 ; B, 400 hectares. 68. A laborer was hired for 70 days; for each day he wrought he was to receive $2.25, and for each day he was idle he was to forfeit $0.15. At the end of the time he received $118.50. How many days did he work? 69. A sum of money was divided equally among a num- ber of persons by giving to the first SI 00 and one sixth of the remainder, then to the second $200 and one sixth of the remainder, then to the third $300 and one sixth of the remainder; and so on. What was the sum divided and what the number of persons ? 70. A besieged garrison had a quantity of bread which would last 9 days if each man rec^ved two hectograms a day. At the end of the first day 800 men were lost in a sally, and it was found that each man could receive 2| hectograms a day for the remainder of the time. What was the original number of men ? 71. Find a fraction such that if 1 is added to the de- nominator its value will be J; but if the denominator is divided by 3 and the numerator diminished by 3, its value will be 5. 72. If 7 years are added to A's age, he will be twice as old as B ; but if 9 years are subtracted from B's age, he will be one third as old as A. What is the age of each? miscp:llaneous examples. 2 43 T3. A, B, and C compare their fortunes. A says to B, "Give me $700 of your money, and I shall have twice as much as you retain." B says to C, "Give me $1400, and I shall have three times as much as you retain." says to A, "Give me $420, and I shall have five times as much as you retain." How much has each? H. An artillery regiment had 39 soldiers to every 5 guns, and 4 over, and the whole number of soldiers and officers was six times the number of guns and officers. But after a battle in which the disabled were one half of those left fit for duty, there lacked 4 of being 22 meri to every 4 guns. How many guns, how many officers, and how many soldiers were there ? Ans. 120 guns, 44 officers, 940 soldiers. 75. A car containing 5 more cows than oxen was started from Springfield to Boston. The freight for 4 oxen was $2 more than the freight for 5 cows, and the freight for the wdiole would have amounted to $30 ; but at the end of half the journey 2 more oxen and 3 more cows were taken into the car, in consequence of which the freight of the whole was increased in the proportion of 6 to 5. What was the original number of cows and oxen, and what was the freight for each ? 9 cows and 4 oxen. Freight for a cow, $2 ; for an ox, $3. 76. Two sums of money amounting together to $1600 were put at interest, the less sum at 2 per cent more than the other. ,If the interest of the greater sum had been in- creased 1 per cent, and the less diminished 1 per cent, the interest of the whole would have been increased one fif- teenth ; but if the interest of the greater had been increased 1 per cent while the interest of the other remained the same, the interest of the whole would have been increased one tenth. What were the sums, and the rates of interest? Ans. $1200 at 7 per cent ; $400 at 9 per cent. Ans. -j 244 ELEMENTARY ALGEBRA. It. A and B can perform a piece of work together in Hf days. They work together 10 days, and then B fin- ishes the work alone in 16§ days. How long would it take each to do the work ? 18. The Emancipation Proclamation of President Lincoln was promulgated on the let day of January in a year rep- resented by a number that has the following properties : the second (hundred's) figure is equal to the sum of the third and fourth minus the first ; or to twice the sum of the first and fourth ; the third is a third part of the sum of the four; and if 1818 is added to the number, the order of the figures will be inverted. What was the year ? Y9. A and B can do a piece of work in a days ; A and C in b days ; B and C in c daj'^s. In how many days can each do it ? • 80. A can do a piece of work in a days, B in J days, and C in c days. In how many days can A and B together do it ? B and C together ? A and C together ? All three together ? 81. A market-man bought some eggs for $0.28 a dozen, and sold some of them at 3 for 8 cents and some at 5 for 12 cents, receiving for the whole $6.24, and clearing $0.64. How many did he sell at each rate ? 82. One cask contains 56 liters of wine and 40 of water, and another 96 of wine and 16 of water. How many liters taken from each cask will make a mixture containing 52 liters of wine and 24 of water? 83. A and B are travelling on roads which cross each other. When B is at the point of crossing, A has 720 me- ters to go before he arrives at this point, and in 4 minutes they are equally distant from this point ; and in 32 minutes more they are again equally distant from it. AVhat is the rate of each? Ans. A's, 100; B's, 80 meters a minute. MISCELLANEOUS EXAMPLES. 245 84 Multiply x"^ by cc". 85. Multiply x^ by x'^, 86. Divide y-^ by y-^ . 87. Divide a!'-'' by a2+«. 88. Transfer the denominator of — f^s to the numerator. 89. Free _^ ^ _^ from negative exponents. X y z 90. Expand {—2a^)\ 91. Expand (a^h)'^. 92. Expand (—3 x-2y"')4. 93. Expand {x^ X ^")'. 94. Expand {x — \/^Y- 95. Expand (a^ — 2 5)^ 96. Expand {2x—yy. 97. Expand (2 a! — 3)^ 98. Expand {^a — 2b)\ 99. Find five terms of {x — y)-^. 100. Expand [2 — x — yf. 101. Expand (3 — « — 6 + c)^ 102. Find ^~^. 103. Find 104. Find \^ — lQx\ 105. Find the square root of ^— /'2a + 2:^—4) &/| 4-a2 — 4a -|-2ax + 4 — 4x + a:2. 106. Reduce ^'6 by ^Q. 113. Find the cube of S \/~2x. 114. Find the square root of 6/C^"3] 115. Multiply 6 + \/^ by 3 — \^J, 116. Expand (x^ — 2 ^/^)^ m. Expand (^^_y-^y. 118. Expand (^^-«-y. 119. The area of a rectangular field is 4 acres and 35 square rods ; and the sum of its length and breadth is equal to twice their difference. What are the length and breadth ? 120. Two travellers, A and B, set out to meet each otlier. They started at the same time and travelled on the direct road toward each other. On meeting it appeared that A had travelled 18 miles more than B, and that A could have travelled B's distance in 9 days, while it would have taken B 16 days to travel A's distance. How far did each travel? Ans. A, 72 miles; B, 54 miles. 121. Find three quantities such that the product of the first and second is a ; of the second and third, b ; and of the first and third, c. A , /«c , /ah , /be Ans. ±^/^, ±y/,^. ±y/- MISCKLLANKOUS EXAxMPLES. 247 122. A and B invest in stocks. At the end of the year A sells his stocks for $108, gaining as much per cent as B invested ; B sold his for $49 more than he paid, gaining one fourth as much per cent as A. What sum did each invest? Ans. A, $45; B, $140. 123. Reduce 18x'- — 33:r — 40 = 0. 124. Reduce I X 6 336 125. Reduce g-.y-(?-.) = f Ans. y =1 — 2.1 ±3^1-4:9, or 5, or — f. 126. Reduce (x* — x" ^ 4)-^ + x' — 5704 + x>. Ans. x=r ±3, or ±2 V^, or ±^L±liz^, 127. Reduce \/2a! -j- 1 + 2>v/^ = yjtx^ 128. Reduce ^ — 2^/ — 3^ + 5 = 3^ — 2. 129. Reduce — yj x-\-1 y/ 130. Reduce "'., ^f + ^/ ^ ^ _ 2. 131. Reduce (^+A^£^l«f = . _ 3. \:r — v'r^— 16/ 132. Given \ ^ ^^^^'^ ^^1 , to find x and y. 133. Given |^ + ^ ^ ^^ l , to find a; and y. (^ZZ/ — 49) 134. Given \ x — y >-, to find x and y. ( 5 3:v— 75) 248 ELEMENTARY ALGEBRA. 136. Given i ' V >• , to find x and v. 136. Given f 5x^2^- 2x/= 875 1 ^^ ^^^ ^ ^^^ ( 3a:y =rl05i ^ 137. Given i ^/ — 4ary =: 96 > ^^ ^^^ ^ ^^ ( x2 + y2 = 25J ^ 138. Given •< ^ r , to find x and v. 139. Given 1'^'' + 2/' + ^ + y = 12| ^^ g^^ ^ ^^^ ^ Ans 1^===^' or4(-3±V21). (^ = 2, or i 140. l(-3:FV2i). Given |^ +2^ = ^U , to find x and u. (x*4- v' = 1921> ^ 141. A drover sold a number of sheep that cost him $297 for $7 each, gaining $3 more than 36 sheep cost him. How many sheep did he sell ? 142. A merchant sold a piece of cloth for $75, gaining as much per cent as the piece cost him. What did it cost him ? 143. A drover bought 12 oxen and 20 cows for $920, buying one ox more for $160 than cows for $66. What did he pay a head for each ? 144. A started from C towards D and travelled 4 miles an hour. After A had been on the road 6^ hours, B started from D towards C, and travelled every hour one fourteenth of the whole distance, and after he had been on the road as many hours as he travelled miles an hour, he met A. What'was the distance from C to D ? MISCELLANEOUS EXAMPLES. 249 145. A person bought a number of horses for $1404. If there had been 3 less, each would have cost him $39 more. What was the number of horses and the cost of each? 146. Find a number of four figures which increase from left to right by a common difference 2, while the product of these figures is 384. Ans. 2468. 147. A rectangular garden 24 rods in length and 16 in breadth is surrounded by a walk of uniform breadth which contains 3996 square feet. What is the breadth of the walk? Ans. 3 feet. 148. A square field containing 144 ares has just within its borders a ditch of uniform breadth running entirely round the field and covering 381.44 centares of the area. What is the breadth of the ditch ? Ans. 0.8 meter. 149. A and B hired a pasture into which A put 5 horses, and B as many as cost him $5.50 a week. If B had put in 4 more horses, he ought to have paid $6 a week. What was the price of the pasture a week? Ans. $8. 150. A father dying left $3294 to be divided equally among his children. Had there been 3 children less, each would have received $ 183 more. How many children were there ? 151. A merchant bought a quantity of tea for $66. If ho had invested the same sum in coffee at a price $0.77 less a pound, he would have received 140 pounds more. How many pounds of tea did he buy ? 152. Find two quantities such that their sum, product, and the sum of their squares shall be equal to one an- other. Ans. 4 (3 ± V"^^^) and i (3 q= \/^^^). 153. Find two numbers such that their product shall be 6; and the sum of their squares 13. 11* 250 ELEMENTARY ALGEBRA. 154. A and B talking of their ages find that the square of A^8«age plus twice the product of the ages of both is 3864 ; and four times this product, minus the square of B's age, is 3575. What is the age of each? Ans. A's, 42 ; B's, 25. 155. Find two numbers such that five times the square of the less minus the square of the greater shall be 20 ; and five times their product minus twice the square of the greater shall be 25. 156. A and B purchased a wood-lot containing 600 acres, each agreeing to pay $17500. Before paying for the lot, A offered to pay $20 an acre more than B, if B would consent to a division and give A his choice of situ- ation. How many acres should each receive, and at what price an acre ? Ans. A, 250 acres at $ 70 an acre ; B, 350 at $50. 157. A merchant bought two pieces of cloth for $175. For the first piece he paid as many dollars a yard as there were yards in both pieces ; for the second, as many dol- lars a yard as there were yards in the first more than in the second ; and the first piece cost six times as much as the second. What was the number of yards in each piece? Ans. In 1st, 10 yards ; in 2d, 5. 158. Two sums of money amounting to $14300 were lent at such a rate of interest that the income from each was the same. But if the first part had been at the same rate as the second, the income from it would have been $532.90; and if the second part had been at the same rate as the first, the income from it would have been $490. What was the rate of interest of each ? Ans. First, 7 per cent ; second, 7x(y per cent. 159. Divide 29 into two such parts that their product will be to the sum of their squares as 198 : 445. MISCELLANEOUS EXAMPLES. 251 160. What is the length and breadth of a rectangular field whose perimeter is 10 rods greater than a square field whose side is 50 rods, while its area is 250 square rods less than the area of the square field ? Ans. Length, 75 rods ; breadth, 30. 161. A rectangular piece of laud was sold for $5 for every rod in its perimeter. If the same area had been in the form of a square, and sold in the same way, it would have brought $90 less; and a square field of the same perimeter would have contained 272^ square rods more. What were the length and breadth of the field ? Ans. Length, 49 ; breadth, 16 rods. 162. A starts from Springfield to Boston at the same^ time that B starts from Boston to Springfield. When they met, A had travelled 30 miles more than B, having gone as far in If days as B had during the whole time ; and at the same rate as before B would reach Springfield in 5| days. How far from Boston did they meet ? Ans. 42 miles. 163. The product of two numbers is 90 ; and the dif- ference of their cubes is to the cube of their difference as 13 : 3. What are the numbers? 164. A and B start together from the same place and travel in the same direction. A travels the first day 25 kilometers, the second 22, and so on, travelling each day 3 kilometers less than on the preceding day, while B travels 14^ kilometers each day. In what time will the two be together again ? Ans. 8 days. 165. A starts from a certain point and travels 5 miles the first day, 7 the second, and so on, travelling each day 2 miles more than on the preceding day. B starts from the same point 3 days later and follows A at the rate of 20 miles a day. If they keep on in the same line, when will they be together ? Ans. 3 or Y days after 8 starts. 252 ELEMENTARY ALGEBRA. 166. A gentleman offered his daughter on the day of her marriage $1000; or $1 on that day, $2 on the next, $3 on the next, and so on, for 60 days. The lady chose the first offer. IIow much did she gain, or lose, by her choice ? 16*7. The arithmetical mean of two numbers exceeds the geometrical mean by 2 ; and their product divided by their sum is 3^. What are the numbers ? 168. A father divided $130 among his four children in arithmetical progression. If he had given the eldest $25 more and the youngest but one $5 less, their shares would have been in geometrical progression. What was the share of each ? 169. The sum of the squares plus the product of two numbers is 133 ; and twice the arithmetical mean plus the geometrical mean is 19. What are the numbers? 110. The sum of three numbers in geometrical progres- sion is 111 ; and the difference of the second and third minus the difference of the first and second is 36. What are the numbers ? 171. There are four numbers in geometrical progression, and the sum of the second and fourth is 60 ; and the sum of the extremes is to the sum of the means as 1 : 3. What are the numbers ? LOGARITHMS. 253 SECTION XXV. LOGARITHMS. 241. Logarithms are exponents of the powers of some num- ber which is taken as a base. In the tables of logarithms in common use the number 10 is taken as the base, and all numbers are considered as powers of 10. By Arts. 119, 120, 10°= 1, that is, the logarithm of 1 is 10^=10, " " 10 " 1 10^=100, " " 100 " 2 10^=1000, " " 1000 " 3 A^s- 6. If the logarithm of 0.5 is 1.8, what is the base 1 Ans. 32. 7. If the logarithm of 0.3 is 0.3, what is the base 1 EXPONENTIAL EQUATIONS. 266* An equation having the unknown quantity as an exponent, or an eocponential equation, may be solved by means of logarithms. For, if a' = ny by Art. 254, a; X log a =2 log n log n log a 1. Solve the equation 125'= 25. xX log 125 = log 25 _ log 25 __ 1.3979 2 , * ~ log 125^" 2:0969 ~" 3' ^' 2. Solve the equation 2048'= 16. 3. Solve the equation (0187)*= ^7. 268 LOGARITHMS OF NUMBERS. 1 N 1 2 3 4 5 6 7 8 9 PROPORTIONAL PARTS. 1 2 3 J^ 5 6 7 J_ 9 ^ 0000 043 086 128 170 212253 294 334 374 4» 8.3 12.4 16.6 20.7 24.8 29.0 331 37-3 11 414 453 492 531 669 607 645 682 719 755 38 7.6 "•3 '5-1 18.9 22.7 26.5 30.2 34-0 12 792 828 864 899 934 969|o04 „38 oT' j06 3-5 7.0 10.4 13-9 17.4 20.9 24.3 27.8 31.3 13 1139 173 206 239 271 303 335 367 399 430 3.2 6.4 9-7 12.9 16. 1 19-3 22.5 25-7 29.0 14 461 492 523 553 584 614644 673 703 732 30 6.0 9.0 12.0 15.0 180 21.0 24.0 27.0 15 1761 790 818 847 875 903 931 959 987 ^ 2.8 76 8.4 II. 2 14.0 ^8 19.6 22.4 252 16 •2041 068 095 122 148 175'201 227 263 279 2.6 5-3 7-9 10.5 132 1S.8 18.4 21. 1 237 17 304 330 356 380 405 430 455 480 504 529 2.5 S-o 7-4 9-9 12.4 »4-9 »7.4 19.9 22.3 18 553 577 601 625 648 672 695 718 742 765 2.3 4-7 7.0 9-4 II. 7 14.1 16.4 18.8 21.1 19 788 810 833 856 878 900 923 945 967 989 2.2 4-5 67 8.9 II. I 13.4 15^ 17^ 20.0 20 3010 032 054 076 096 118139 160 181 •201 2.1 4.2 6.4 8.5 10.6 TT, 14.8 17.0 19.1 21 22-2 243 263 284 304 324 345 365 385 404 2 4.0 6.1 8.1 10. 1 12.1 14.1 16.2 18.2 22 424 444 464 483 502 522 541 660 579 598 1.9 3-9 S.8 77 9-7 11.6 13-5 154 174 23 617 636 655 674 692 711729 747 766 784 1.8 3-7 5-5 74 9.2 11. 1 12.9 14.8 16.6 24 802 820 838 856 874 892 909 927 945 962 1.8 35 5-3 7-1 8.9 10.6 12.4 14.2 16.0 26 3979 997 7^ ,31 7» ^te o99 I'le l33 I-7 3-4 5- 1 6.8 8.5 10.2 11.9 136 rri 26 4150 166 183 200 216 232 249 265 •281 298 1.6 3-3 4-9 6.6 8.2 9.8 "•5 I3-I .4.8 27 314 330 346 362 378 393 409 425 440 456 1.6 3-2 4-7 6-3 7-9 9-5 II. I 126 14.2 28 472 487 502 518 633 548 664 579 594 609 ^■5 3-0 4.6 6.1 7.6 9-' 10.7 12.2 137 29 624 639 654 669 683 698713 728 742 757 1.5 _!:? 4.4 59 7-4 8.8 lOJ 11.8 13-3 30 4771 786 800 814 829 843 867 871 886 900 1.4 2.8 4-3 5-7 7-» "s^s 10.0 II. 4 12.8 31 914 928 942 955 969 983 997 0" 0^4 „38 1.4 2.8 4.» 5-5 6.9 8.3 9-7 II.O 12.4 32 6051 065 079 092 105 119132 146 159 172 1.3 2-7 4.0 5-3 6.7 8.0 9-4 10.7 12.0 33 185 198 211 224 237 250 -263 276 289 302 1-3 2.6 3-9 5-2 6.5 7.8 91 10.4 11.7 34 316 328 340 353 366 3_78 3_91 403 416 428 1-3 _2-5 ^ 5^ _6j J± 8.8 10. 1 "•3 35" 5441 453 465 m 490 502514 627 539 551 1.2 2-4 3-7 4.9 6.1 7-3 86 Ti II.O 36 563 575 587 599 611 623 635 647 668 670 1.2 2.4 36 4.8 5.9 7.1 8.3 9-5 10.7 87 682 694 705 717 729 740 1 752 763 776 786 1.2 2.3 3-5 4.6 5-8 6.9 8.1 9-3 10.4 38 798 809 321 832 843 856 866 877 888 899 11 2.3 3-4 4-5 56 6.8 79 9.0 10.2 89 911 922 933 944 — 955 966 977 988 999 0^0 _!i' 2.2 3-3 44 5-S 6.6 7-7 8.8 99 To 6021 031 042 053 064 075 085 096 107 117 I.I 2.1 3-2 4-3 5-4 t; 7-5 8.6 9-7 41 128 138 149 160 170 180 191 201 212 222 I.O 2.1 3-1 4-2 5-2 6.3 7-3 8.4 9-4 42 232 213 253 263 274 284 294 304 314 325 1.0 2.0 3» 4 > 51 61 7-2 8.2 9.2 43 335 346 355 365 375 385 396 405 415 426 1.0 2.0 3-0 4.0 5-0 6.0 7.0 8.0 9.0 44 435 444 464 464 474 484 '493 503 513 5-2'2 1.0 2.0 2.9 3-9 4-9 5-9 6.8 -Zi! as Is 6632 542 551 iel 671 580 590 599 609 618 1.0 1.9 2.9 .3.8 4.8 5 7 6.7 7.6 Ti 46 628 637 646 656 665 675 '684 693 702 712 0.9 1.9 2.8 3-7 47 5-6 6.5 75 8.4 47 721 730 739 749 758 767 776 785 794 803 0.9 1.8 27 3-7 46 5-5 6.4 7-3 8.2 48 812 821 830 839 848 857,866 876 884 893 0.9 1.8 27 3-6 4-5 5-4 6.3 72 8.1 49 902 911 920 928 937 946 955 964 972 981 0.9 1.8 2.6 3-5 4-4 5 3 6.1 70 79 jlo 6990 998 o07 ol6 7 ^y2 ^ ^ o67 0.9 »-7 2.6 3.4JTI 5-2 6.0 6.9 7-7 51 7076 084 093 101 no 118 126 135 143 152 0.8 1-7 25 3-4 4.2 S» 5-9 6.7 76 62 160 168 177 185 193 202 210 218 226 235 0.8 17 2.5 3.3 4.. 50 5.8 6.6 7-4 63 243 261 269|q67 276 284 292 300 308 316 0.8 1.6 2.4 3-2 41 49 S-7 6.5 7-3 54 324 332 340, 34S 366 36 4! 372 ^ 388 396 0.8 1.6 2.4 3-2 4.0 4.8 S-6 6.4 7.2 LOGARITHMS OF NUMBERS. 269 N 1 2 3 4 5 6 7 8 9 PROPORTIONAL PARTS. 1 2 3 4 5 6 7 8 9 55 7404 412 U9 427 m 443 47l 469 466 474 0.8 1.6 2.3 31 3-9 4-7 5-5 6.3 7.0 56 48-2 490 497 505 513 520 528 536 643 651 0.8 1-5 2-3 31 3-8 4.6 5-4 6.1 6.9 67 559 566 574 582 689 597 604 612 619 627 0.8 1-5 2-3 30 3-8 4-5 5-3 6.0 6.8 58 634 642 649 657 664 672 679 686 694 701 0.7 1-5 2.2 30 3-7 4-5 5-2 S-9 6.7 59 709 716 723 731 738 745 752 760 767 774 0.7 ^•5 2.2 2.9 J^ 4 4 5- 1 Ji! 6.6 ^ 778,! 789 796 803 810 818 825 ^ 839 846 0.7 1.4 2.2 2.9 3-6 4-3 S-o 5-7 ~6l 61 853 860 868 875 882 889 896 903 910 917 0.7 14 2.1 2.8 3-5 4-2 4-9 S.6 6.4 62 924 931 938 945 952 959 966 973 980 987 0.7 1.4 2.1 2.8 3-5 4.2 4-9 5-6 6.3 63 64 993 806-2 069 .0' 075 082 089 0^8 096 .36 102 109 o48 116 ,65 122 0.7 0.7 1.4 1-3 2.1 2.0 2.7 2-7 3-4 3-4 4.1 4.0 4.8 4-7 5-5 5-4 6.2 6.1 ^ 8129 136 Tli 149 156 162 169 176 182 189 0.7 1.3 2.0 2.7 3-3 4.0 ~6 5.3 "Zo 66 195 202 209 215 222 228 235 241 248 254 0.7 1-3 2.0 2.6 3.3 3-9 4.6 5-2 5-9 67 261 267 274 280 287 293 299 306 312 319 0.6 1-3 1.9 2.6 3-2 3-9 4.5 5- 1 5.8 68 325 331 338 344 351 357 363 370 376 382 0.6 1-3 1.9 2-5 3-2 3-8 4-4 5-1 57 69 388 395 401 407 414 420 426 432 439 445 0.6 1.2 1.9 2-5 31 3-7 4-4 5-0 1^ 70 8451 457 463 m 476 1^2 488 494 500 506 "oie 1.2 1.8 2.5 3-1 3-7 4-3 4-9 5-5 71 613 519 525 531 537 543 549 555 561 567 0.6 1.2 1.8 2.4 30 3-6 4-3 4.9 S-5 72 573 579 585 591 697 603 609 615 621 627 06 1.2 1.8 2.4 30 3-6 4-2 4I 5-4 73 633 639 645 651 657 663 669 675 681 686 0.6 1.2 1.8 2.4 30 3-5 4.1 47 5-3 74 692 698 704 710 716 722 727 733 739 745 0.6 1.2 1-7 2.3 2.9 3-5 4.1 4 7 5-2 75 8751 756 -m 768 774 779 785 791 797 802 ~6 1.2 1-7 2-3 2.9 3-5 4.0 T6 5-2 76 808 814 8» 825 831 837 842 848 854 859 0.6 I.I 1-7 2.3 2.8 3-4 4.0 4-5 S-i 77 865 871 8-6 882 887 893 899 904 910 915 0.6 I.I 1-7 2.2 2.8 3-4 3-9 4-5 5° 78 921 927 932 938 943 949 954 960 966 971 0.6 I.I 1-7 2.2 2.8 3-3 3-9 4-4 50 79 976 982 987 993 998 o04 o09 ol5 0^ o!i 0-5 I.I 1.6 2.2 2.7 3-3 3-8 4.4 4.9 80 9031 036 042 047 053 058 063 069 074 079 0.5 I.I 1.6 2.2 2.7 3-2 38 4-3 4.9 81 085 090 096 101 106 112 117 122 128 133 0-5 I.I 1.6 2.1 2.7 3-2 .3-7 4-3 4.8 82 138 143 149 154 159 165 170 175 180 186 0-5 I.I 1.6 2.1 2.6 32 3-7 4.2 4-7 83 191 196 201 206 212 217 222 227 232 238 o-S I.O 1.6 2.1 2.6 3-1 36 4.2 4-7 84 243 248 253 258 263 269 274 279 284 289 0-5 I.O 1.5 2.1 2.6 31 3-6 4.1 _4_6 85 9294 299 304 309 315 320 325 330 336 340 0-5 1.0 1-5 2.0 2-5 3.0 3-6 4.1 4.6 86 345 350 355 360 365 370 375 380 385 390 0.5 I.O 1-5 2.0 2-5 30 3-5 4.0 4-5 87 395 400 405 410 415 420 426 430 435 440 0-5 I.O 1-5 2.0 2-5 3-0 3-5 40 45 88 445 450 455 460 465 469 474 479 484 489 0.5 I.O 1-5 2.0 2-5 2.9 3-4 3 9 4.4 89 90 494 9542 499 547 504 552 509 513 562 518 623 566 571 528 576 633 581 538 586 0.5 0-5 I.O I.O 1-5 1.4 1.9 1.9 2.4 2.4 29 2.9 3-4 3-4 39 3.8 4-4 4-3 91 590 595 600 605 609 614 619 624 628 633 0.5 0.9 1-4 1.9 2.4 2.8 3-3 3-8 4-3 92 638 643 647 652 657 661 666 671 675 680 0.5 0.4 1.4 1.9 2.3 2.8 3.3 38 4.2 93 685 689 694 699 703 708 713 717 722 727 0-5 0.9 1.4 1.9 2-3 2.8 3-3 3-7 4.2 94 731 736 741 ^- 750 764 769 763 768 773 0-5 0.9 1.4 1.8 2-3 2.8 3-2 3-7 4.1 ~95 9777 782 786 791 795 800 805 809 ^4 818 0-5 0.9 1.4 Ts 2.3 2.7 3-2 3-6 4-1 96 823 827 632 836 841 845 850 854 869 863 05 0.9 1.4 1.8 2.3 2.7 3-2 3-6 4.1 97 868 872 877 881 886 890 894 899 903 908 0.4 0.9 ^1-3 1.8 2.2 2.7 3-1 3-6 4.0 98 912 917 921926 930 934 939 943 948 952 0.4 0.9 1-3 1.8 2.2 2.6 31 3-5 4.0 99 956 961 965 969 974 978 983 987 '991 '996 0.4 0.9 1-3 1-7 2.2 2.6 31 3-S 3.9 270 ELEMENTARY ALGEBRA. EXAMINATION PAPERS IN ALGEBRA FOR ADMIS- SION TO HARVARD COLLEGE. June, 1878. 1. Two workmen, A and By are employed on a certain job at different wages. When the job is finished, A receives $ 27, and Bj who has worked three days less, receives $ 18.75. If B had worked for the whole time, and A three days less than the whole time, they would have been entitled to equal amounts. Find the number of days each has worked, and the pay each receives per diem. 2. Find the value of x from the proportion / lO ^a^ \ 2 __ I ba^a'' 96- « V 3^6V • ^ — V4^a2.6» '' ^5 * Express the answer in its simplest form, free from negative and fractional exponents. 3. Simplify the expression ar^ -|- / a^ — y^ x^ — y"^ ar* + y« ^ — y I a^ + y a: + y "■ x — y 4. Write out the first five terms and the last five terms of 5. Find the value of x from the equations ax'\-hy=:.ly cy -\'dz^=.my *ex-\- fz =zn. EXAMINATION PAPERS. 271 6. Find the greatest common divisor and the least common multiple oi6x^+7x — 5aind2x^ — x^ + Sx— 4. 7. Solve the equation a;-)-13a4-36 a — 2b _ 5a — 36 — X x-\-2b September, 1878. 1. Three men, A, B, C, are tried on a piece of work. It is found that A and B together can do a certain amount in 12 hours ; B and C can do the same amount in 8 hours and 24 minutes ; and C and A can do the same amount in 9 hours and 20 min- utes. Find the time which each man would require to do the same amount singly. h — a fa — 2h Zx{a — h) \ ^Hh ~~ \x-\-b x' — b^J • 2. Simplify b^-\-3b^x-{-3bx^ + a^ _^ (x + bf a^ — b"^ ' ar^ + 6ic-f-62 3. Write out the first five terms and the last five terms of 4. Solve the equations, ^{x — y) =x — 4, a:y = 2 ^ + y + 2. 5. Solve the equation, x -\ — = 1 H jTa — * a^-\-3b^ 6. Find the value of x from the proportion, 3 a 262 '• ub^aK^b) -^^ Vv^Ie^) Find a result free from fractional and negative exponents, and in the most reduced form. 7. Find the greatest common divisor and the least common multiple of 4:X* — x^—6x — 9 and Sx^'-^2x^— Ix^— 6ic— 9. 272 ELEMENTARY ALGEBRA. June, 1879. 1. Solve the equation, 2ax — 46 bx — a 2abx bx — a 2ax — b'~ 2abx^ — (2a\r\-b^)x-\-ab* Eeduce the answers to their simplest forms. 2. Solve the equations, x ' ^-tV 4a:+32/=l. Stat& clearly what values of x and y go together. 3. Pind the value of x from the proportion 36 Mh"^ ^ _ 2(3qc)^ , 2^a 4 V c» • ^~ ^/{y'c^) ' be \ 4. Simplify the fraction 1 X V X — y a?' — y^ x^ -\- y^ 5. Find the greatest common divisor of 2ar»_3a:+land2ar» — x — 1. 6. Put the following question into equations : — A and B walk for a wager on a course of one mile (5280 feet) in length. At the first heat, A gives B a start of 45 seconds, and beats him by 110 feet. At the second heat, A gives B a start of 484 feet, and is beaten by 6 seconds. Required, the rates at which A and B walk. EXAMINATION PAPERS. 273 September, 1879. 1. Several friends, on an excursion, spent a certain sum of money. If there had been 5 more persons in the party, and each person had spent 25 cents more, the bill would have amounted to $ 33. If there had been 2 less in the party, and each person had spent 30 cents less, the bill would have amounted to only $ 11. Of how many did the party consist, and what did each spend 1 Find all possible answers. 2. Solve the equations, 2x-\- 4y + 272 = 28, 7x— 3y — 152=3, 9a;— lOy— 332=4. 3. Solve the equation, x-\-3b 3 6 __ a + 36 I 7 8a^—Uab ' 4a^ — 96^ (2a+ 36) (x — 36) Reduce the answers to their simplest forms. 4. Calculate the sixth term of / ^a _ V2 Y Reduce the answer to its simplest form, cancelling all common factors of numerator and denominator, performing the numerical multiplications, and giving a result which has only one radical sign and no negative or fractional exponents. 5. Simplify the fraction 2a; — / 8a;« — / ' 2^ + / 8;«;8 + i/« 6. Find the greatest common measure and the least common multiple of 4 ^6 _|_ 14 rt a;* — 18 a^ic^ and 24 a^^ + 30 a«^ + 126 a*. 274 ELEMENTAIIY ALGEBRA. July, 1880. 1. Reduce to its simplest form x + I-^-^ 2x+l 2. Divide 60^" + ^"— 19:r^ + 2n_j_ 20af« + » — 7a:"» — 4a:^-" by 3a;2«_5ar"+4. 3. Find the fourth term of [^—'^ — — ) , reducing it to its simplest form. 4. Find the greatest common measure and the least common multiple of 2a;5 — 11^2_9 .^^^ ^ x^ -\- U x^ -\- ^\. 5. A man walks 2 hours at the rate of 4J miles per hour. He then adopts a different rate. At the end of a certain time, he finds that if he had kept on at the rate at which he set out, he would have gone three miles further from his starting- point ; and that if he had walked three hours at his first rate and half an hour at his second rate, he would have reached the point he has actually attained. Find the whole time occupied by the walk and his final distance from the starting-point. 6. Solve the equation •a &(2a;-j-l) + b(2x — l) a{x*—l) (2x— l)(x+l) ' (2x— l)(a;— 1) Reduce the answers to their simplest forms. September, 1880. 1. Reduce to its simplest form as one fraction /^^+ ?/ _i_ -r^ + //^\ _^ ( x — / / __ ^ — ?/ ^\ ' V — ^'^ x^ + fj * V+y *'" + .vV* EXAMINATION PAPERS. 275 2. Find the greatest common measure and the least common multiple of 3. Find the sixth term of ( —^ 6 <^6^ j , reducing the literal part of the term to its simplest form, and the numerical part into its prime factors. 4. A reservoir, supplied by several pipes, can be filled in 15 hours, every pipe discharging the same fixed number of hogs- heads per hour. If there were 5 more pipes, and every pipe discharged per hour 7 hogsheads less, the reservoir would be filled in 12 hours. If the number of pipes were 1 less, and every pipe discharged per hour 8 hogsheads more, the reservoir would be filled in 14 hours. Find the number of pipes and the capacity of the reservoir. 5. Solve the equation 2x-]-\ _ Zx-\-\ _ 1 A _ 2\ b a^ X \b aj ' Reduce the answers to their simplest forms. June, 1881. 1. What are the factors of x^-\- i/^1 Reduce to its simplest ^ 7/2 y form the product of o ■ 3 and — r r P a? 2. Solve the equation - — : ; — = 1 H 1 — * 1 -f-a + ^ ^ a ' X 3. Find the square root of 1 _|_ y'Te^ + 10;;c"* + 12 V^ + 9^"*. 276 ELEMENTARY ALGEBRA. 4. What is meant by the expression a? 1 5. Solve the equation ^x — 8 — ^x — 3 = ^/x, 6. A man rows down a stream, of which the current runs 3^ miles an hour, for 1§ hours. He then rows up stream for 6^ hours, and finds himself two miles short of his original starting- place. Find his rate and the distance he rowed down stream. 7. Find the 4th and the 14th term of (2 a — hf. EXAMINATION PAPERS. 277 EXAMINATIONS FOR ADMISSION TO YALE COLLEGE. June, 1878.. L (a) Reduce — — — 5 5 to its lowest terms. (b) Multiply a-^b^ by -^ ; and divide a'^b^ by ~. 2. Solve the equations : 7a; — 6 x — 5 x {«) 35 6a;— 101 5 ... 7*4-9 / 2a:— 1\ 3. (a) Solve the equation — — --|-7|=8. 2 3 (6) It is required to find three numbers such that the product of the first and second may be 15, the product of the first and third 21, and the sum of the squares of the second and third 74. 4. Find the sum of n terms of the series 1, 2, 3, 4, 5, 6, &c. 5. By the binomial theorem expand to five terms (a^ — b^)-k September, 1879. 1. Add together ^^s^^._^^.y ^^e J_ ^,y 2a*(a* + xi 2. (a) Multiply together ^y' 3, ^^3, and ^^3. (b) Divide 9 m^ (a — 6)i by 3 m (a — b)i. 278 ELEMENTARY ALGEBRA. 3. Solve the equations , . ^ X — 4: , 5x + U 1 (6) --|_- = a; _4._==^,. __(__ — c. X y X z y z 4. Solve the equation 15 72 — 6a; 2. « 2a:2 5. Find the sum of 20 terms of the series 1, 4, 10, 20, 35, &c. 6. By the binomial theorem expand to 4 terms (a) (l-A)-i; (6) (a^-ar")*. July, 1880. 1. (a) Divide __j + -^ by -— ^ - ^-p^, and re- duce the quotient to its simplest form. if) Find the greatest common divisor of x^— 6x2 — 8a: — 3 ^nd 4x»— 12a: — 8. 2. (a) Find the sum of 6 V^47^ 2 'V^2rt, and \^^\ {h) Reduce to its simplest form the product (a:-l - A/^)(a:-l + V^) (a:- 2 + \/^)(ar- 2 - a/^). 3. Solve the equations (a) \{2x _ 10) - Ti^(3a: - 40) = 15 — ^(57 - x) ; (6) ._1 + -A_ = 0; W ar' — 1 = ^+i. EXAMINATION PAPERS. 279 4. Four immbers are in arithmetical progression ; the product of the first and third is 27, and the product of the second and fourth is 72. What are the numbers 1 5. By the binomial theorem expand to 4 terms, (a) (1-6)-^ (b) (x'-y')K September, 1880. 1. (a) Eequired, in its simplest forhi, the quotient of a* — X* _ a^ X -{- x^ €? — 2ax -\- x^ ' ci? — dt? (6) Find the greatest common divisor of 6a;2— 17x+ 12 and 12a;2 _ 4a; — 21. 2. Find the sum of ^l6, ^^, — v'dsB, ^^192, — 7'V^9. 3. Solve the equations 2 a- 1 :r 4- 2 (a) 5x_^^^ + l==3:. + ^ + 7; (6) 3;z:2_j_ |o^_57_0; / s a;2 ;r , 1 1 (c) — = — ^^ 3 10 "^6 5 4. Find three geometrical means between 2 and 162. 5. By the binomial theorem expand to 4 terms. 280 ELEMENTARY ALGEBRA. July, 1881. 1. Free from negative exponents (Aa~^b^ar*)~*. ^ 2ar 15 2. Keduce to lowest terms a:2-|-10a; + 21 3. Factorw^— 2ra''4-?i; ar* — 1 ; a^ — nV; a;« + /. 2 4. Make denominator rational of — =:^ —> >v/5 — a/2 5. Multiply v^ — 2 + ^^^Z by V^ + 2 — V^Ts. 6. Solve — , 5 __ 3 a: 4-1 _ 1 X c^ 4 7. Solve 01? — xy= 153 ; x -\- y =^\, 8. By the Binomial Theorem expand to four terms 9. Sum the infinite series 1 -|- - -|- - -|- &c. EXAMINATION PAPERS. 281 EXAMINATIONS FOR ADMISSION TO AMHERST COLLEGE. June, 1878. , -r^. ., a* — m* - a^ -\- am 1. Divide ^ j :, by '■ ^2 I 2. Reduce —, r to its lowest terms. ab — I _l_ ^ 3. Given b — , = : to find x. 1 — X 4. 2x-{-37/=2'd; 5x'-'2i/=10; findxandi/. 5. Find the cube root of a^ VI? 6. Divide 4 V^Iac by 2 VSa. 7. A father's age is twice that of his son ; but 10 years ago it was three times as great. What is the age of each 1 8. If 1 be added to the numerator of a fraction, its value is ^ ; and if 1 be added to the denominator, its value is J. What is the fraction 1 September, 1878. 1. Resolve 1 — 36 y'^ into two factors. 2. Find the least common multiple of 9a^, 12a^x^, and 24:ax^i/. -. , - , a , — X -4- d 3. Find the sum of a; -|- r- and b in 4. Divide —. by — ' — - 282 ELEMENTARY ALGEBRA. I X 5. Given 8a = -. — j — , to find x. I "f- X 6. Reduce 7 v9a^ — 27 a^b to its simplest form. 7. Find the square root of 4a* — 12 a^ -\- 5 a^ -\- 6 a -\- I, 8. What numbers are those whose difference is 20 and the quotient of the greater divided by the less is 3 1 June, 1879. 1. Reduce 3 a — (2 a — [a -|- 2]) to its simplest form. 2. Find the greatest common divisor oi2a^ — 7ar*-|-5j; — 6 and3a.^ — 7;r2 — 7a;4-3. 1 — X 1 -\- X 3. Reduce — — to its simplest form. + l^x ' l+x 4. Resolve a^ — Ifi into four factors. 11^ 3 5. Given 7x = 3a: + 7; find x. 6. A crew can row 20 miles in 2 hours down stream, and 12 * miles in 3 hours against the stream. Required the rate per hour of the current, and the rate per hour of the crew in still water. 7. Extract the square root of dx* — l2a^-\-l6x^— Sx -{- i. 8. Express 2 x^i/^ — 3 ar" V* — x~*ir^ with positive exponents. 9. Divide af-^hjx * and reduce the quotient to its sim- plest form. 10. Multiply a-\-b ^{— 1) by a — 6 ^(— 1). EXAMINATION PAPERS. 283 September, 1879. 1. Interpret a^ j aP ; a~^ ; ai 2. Multiply a — 6 by c — c?, and deduce the rule that " like signs give -j- ^^^ unlike give — ." 3. Separate into prime factors 3m*x — 3n*x. 4. Explain the reason of the following equations : a^'a'' = «"* + "; a"" -i- a"" = a*"-" ; (a"*)" = a"«. ^ 2x — 9 x—3,x 25 — 3x ^ ^ 5. ^^ ^+-=-^— ;findx. 6. A left a certain town at the rate of a miles an hour, and in n hours was followed by B at the rate of b miles an hour. In how many hours did B overtake A 1 7. The sum of two numbers is s and the difference is d. What are the numbers 1 Show from the result that if from the greater of two numbers you subtract one half the sum, the remainder will be one half the difference. 8. Find the square root of x* — 4a^y -|- 6 x^y^ — 4:xy^ -\- ^. 9. Simplify 7 ^(a2»62"»«c^). 10. Subtract 3 ^a« from 6 ^a\ June, 1880. 1. Reduce (a + 6 — c) \x-\-y — (a -|- J -|- c) {x -\- y)\ to its simplest form. 2. Resolve c^ — W into its prime factors. 3. Divide -3-^ by 66Mc£2 ' h^ddPe 284 ELEMENTARY ALGEBRA. , ^x -\- 2a X — 5a ^ ,, , 4. ^ = 5 a ; nnd x, A o 5. What numlDer multiplied by m gives a product a less than n timea the number ] 6. 5^-[-3y=19, 7a; — 2y = 8; finda;andy. 7. Find the square root oi a -\- la^x^ -\- x. 8. Find the square root of 81a^a;~^ylr"i. 9. Free _^ ^ from negative exponents, and reduce the c ~~~ ct result to its simplest form. 10. Multiply a 4- 6 ^T^l hj a — b V^^l. September, 1880. 1. Factor 81 a» — l. 2. Find the greatest common divisor of a' — a^x -\- Sax^ — 3x^, and a^ — 5aa: -|- ix^. 3. Reduce a -\- b to a fractional form. ' a — b find X and y. 5. Find the square root of 9a;* — 12a:8 + I6x^ — 8a; + 4. 6. Find the sum of ^/3aH and VS¥b. 7. Multiply 5ai by 3 ah. 8. Find the cube of a — bxh. EXAMINATION PAPERS. 285 9. Find the cube root of (x -\- y) \^x -f- y. 10. Simplify «-+A^ + ^-^A^p. June, 1881. 1. Eemove the parentheses and reduce to its simplest form a — \a — [a — (a — x)]j. 2. Find the greatest common divisor of 8a^-j-2a; — 3 and 120^34-10^^ — 4. 3. Resolve ar^^ — y^^ into its simplest factors. Ou X X 4. Reduce -5 -I 7—- — • to its simplest form. x^ — l'ic + 1 1 — ^ 6. Find the square root of ^a^— 12^/+ \Qa?y^ — UxUf -\- iy^ -\-\&xf. 6. y — a=z2{x — h) ; y — h z=z2{x — a) ; find x and y. 7. Multiply Y4a by 'v/G^. 8. A wine-merchant has two kinds of wine which cost 90 cents and 36 cents a quart, respectively. How much of each must he take to make a mixture of 100 quarts worth 50 cents a quart 1 9. Divide xl -^ x\ -^ Q, hy xh ^ 2. — 3 — ) *^ ^*^ simplest form. 11. \i a '.h =. c : d, prove a -\~h -.a — h =. c -\' d : c — d. 12. 3;r2 — 4a;=7; find^ir. 286 ELEMENTARY ALGEBRA. 13. Insert three geometrical means between 13 and 208. 14. Demonstrate the two fundamental formulae of Arithmeti- cal Progression. 15. What is the Binomial Theorem? Give the first four terms of {a -\- xy^. EXAMINATION PAPERS. 287 ENTRANCE EXAMINATIONS TO DARTMOUTH COLLEGE. 1878. 1. Define term, factor, coefficient, exponent, power, root, equa- tion. What is the degree of a term ] When is a polynomial homogeneous 1 2. Write the following without using the radical sign : 3. Write the following without using negative exponents : V 4. Multiply a — 6 V — 1 by « + ^ V — 1« Also a — 6 V — 1 by a 4" ^ V — 1- 5. Raise a — 6 v — 1 to the 3d power. Simplify the radi- cal (a^ — 2 a^ft + a i^)*. 6. Solve ^^^^ — ^^-^ = 6. Also4-, + <^^° + c=:0. a -\- X a — X X ^ .. X — 1 X — 2 X -\-\ .. ah — (a — x^ 1 Also — — = — ^- — . Also ^ '- = -. 2 3 6 ah + {a — x)^ a 1880. 1. Define Algebra, factor, coefficient, exponent, fraction, equa- tion. 2. Write the following without using the radical sign : 288 ELEMENTARY ALGEBRA. 3. Write the following without using negative exponents : 4. Write, in the simplest form, the values of 5. Find the product of ^/ab X — "^J X (— a*a:i) X 2 a6 ; also of (2 + V^^) (2 — V^^). 6. Solve X , X , ar— 2 .301 ^^, ^ ar — 2 b = ; also — - — + — — = .001 a; + .6 — a + 1 a - 1 ' 5 ' .5 ' .05 1881. 1. Define term, factor, coefficient, exponent, power, root, equation. 2. What is the degree of a term 1 When is a polynomial homogeneous 1 3. Write the following without using the radical sign : \^; v^?; Va' + 62 — 2a6. 4. Write the following without using negative exponents : x~^: ax-^ : ^ b-' 5. Multiply a — c V — 1 hy a ~\- b V — 1- 6. Raise 6 — a V — 1 to the 3d power. Simplify the radical \/(^ — 2<^b -^ cb\ EXAMINATION PAPERS. 289 7. Solve r-^ = a. a-\- y a — y X —l X — 2 ^4-1 8. Solve — — — =- G a X 9. Solve ;^ — 6;r« + c = 0. 290 ELEMENTARY ALGEBRA. EXAMINATIONS FOR ADMISSION TO BROWN UNIVERSITY. 1878. c — 1 c — \ 5. Reduce to a simple fraction. c + l 6. Divide a into two parts, such that m times one shall be n times the other. 7. If 4 be subtracted from both terms of a fraction, the value will be ^ ; and if 5 be added to both terms, the value will be ^. What is the fraction 1 8. Given a/o:.- — 9 + Va; + 1 1 = 10, to find x. ^ ^. rt , 3^ — 6 _ 3a: — 3 - , 9. (jriven zx -\ =z bx — , to nnd x. '2 X — 6 1879. ^ ... 1 x-\-\ . ^«+^+ 1 2. Multiply {a + bf by (a + 6)?, and Divide {a + 6) Va^TT'l by {a — i) a/«* + 2 a + 1, giving answers in simplest forms. 3. Given , "'<" + -> , - r^^,t-h, = ''^i^^ {a — b){x — a) (a — b) {x — 6) or — or to find X. 4 4. Given ^'1 -\- x -\- ^/x =. — , to find x. a/2 +ar EXAMINATION PAPEKS. 291 5. Divide the number s into two such parts, that if m^ be divided by the second, and this quotient multiplied by the first, the product is the same as if 7i^ be divided by the tirst and the quotient multiplied by the second. 1880. 1. Find the H. C. D. of 6^:3 _ g^^2 _^ 2fx and l'2x^ — I5x,>/ + 3/. 2. Given ax-^bf/=zc and mx =zni/ -\-dj to find x and y. 3. Extract the cube root of _ 99:z;3 _ 9^.5 _j_ -^8 _^ (34 _ 144^ _^ i^q^ _|_ 39^4^ 4. Given — — ^ -=- =^ . ■—, to Imd x. 5. Two pipes, A and B, will fill a cistern in 70 minutes, A and C in 84 minutes, and B and C in 140 minutes. How long will it take each to fill it alone % 6. Given V^ + x -\- \/x = — , to find x. V5 + x 7. A gentleman bought two pieces of silk which together measured 36 yards. Each cost as many shillings per yard as there were yards in the piece, and the cost of the pieces were to each other as 4 to 1. Required the number of yards in each piece. 8. Given x -\-\/5x -{- 10 = S, to find x. 9. Given sr^ — ^0 = 50, to find x. 10. Given a^ -\- xi/ = 15 and x^ — ?/^ = 2, to find x and y. 14 DAY USE 1 RETURN TO DESK FRQM WHIGHi^l^?ROWED This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. \I PAY l!^.F mv.iiiQ. SUMMER SESSIONS (Wc IL, /fH 7 ' LD 21-50m-8,'r.7 (.08481810)476 General Library University of California Berkeley 541 Hi 2 UNIVERSITY OF CALIFORNIA LIBRARY