,transcriber's ,note3 ,this book includes extensive mathematical expressions and equations1 which can not always be easily represented in plain text4 ,the reader is encouraged to download the ,,html version of the text1 which represents the math more clearly4 ,for the plain text version1 the following conventions are used3 ,mixed fractions are represented by a dash with no spaces1 while subtraction is represented by a dash with spaces on either side4 ,for example3 #a-a_/b is one and one half4 #a - #a_/b is one minus one half4 ,the sideways-#h symbol for infinity is represented as infinity4 ,square1 cube1 and other roots are shown by raising a quantity to the appropriate fractional power4 ,for example3 #d7a_/b7 is the square root of #d4 x7#a_/n7 is the nth root of x4 ,extra parentheses have been added as needed to clarify the correct order of operations4 ,a ,,review ,,of ,,algebra ,,by ,,romeyn ,,henry ,,rivenburg1 ,a4,m4 ,,head ,,of ,,the ,,department ,,of ,,mathematics ,,the ,,peddie ,,institute1 ,,hightstown1 ,n4,j4 ,illustration ,,american ,,book ,,company ,,new ,,york ,,cincinnati ,,chicago ,,copyright1 #aiad1 ,,by ,,romeyn ,h4 ,,rivenburg4 ,,copyright1 #aiad1 ,,in ,,great ,,britain4 ,a ,,review ,,of ,,algebra4 ,e4 ,p4 #f ,,preface ,in most high schools the course in ,elementary ,algebra is finished by the end of the second year4 ,by the senior year1 most students have forgotten many of the principles1 and a thorough review is necessary in order to prepare college candidates for the entrance examinations and for effective work in the freshman year in college4 ,recognizing this need1 many schools are devoting at least two periods a week for part of the senior year to a review of algebra4 ,for such a review the regular textbook is inadequate4 ,from an embarrassment of riches the teacher finds it laborious to select the proper examples1 while the student wastes time in searching for scattered assignments4 ,the object of this book is to conserve the time and effort of both teacher and student1 by providing a thorough and effective review that can readily be completed1 if need be1 in two periods a week for a half year4 ,each student is expected to use his regular textbook in algebra for reference1 as he would use a dictionary1--to recall a definition1 a rule1 or a process that he has forgotten4 ,he should be encouraged to think his way out wherever possible1 however1 and to refer to the textbook only when forced to do so as a last resort4 ,the definitions given in the ,general ,outline should be reviewed as occasion arises for their use4 ,the whole ,outline can be profitably employed for rapid class reviews1 by covering the part of the ,outline that indicates the answer1 the method1 the example1 or the formula1 as the case may be4 ,the whole scheme of the book is ordinarily to have a page of problems represent a day's work4 ,this1 of course1 does not apply to the ,outlines or the few pages of theory1 which can be covered more rapidly4 ,by this plan1 making only a part of the omissions indicated in the next paragraph1 the essentials of the algebra can be readily covered1 if need be1 in from thirty to thirty-two lessons1 thus leaving time for tests1 even if only eighteen weeks1 of two periods each1 are allotted to the course4 ,if a brief course is desired1 the ,miscellaneous ,examples 7pp4 #ca to #ce1 #ej to #eb71 many of the problems at the end of the book1 and the ,college ,entrance ,examinations may be omitted without marring the continuity or the comprehensiveness of the review4 ,,romeyn ,h4 ,,rivenburg4 ,,contents ,,pages ,,outline ,,of ,,elementary ,,and ,,intermediate ,,algebra #g-ac ,,order ,,of ,,operations1 ,,evaluation1 ,,parentheses #ad ,,special ,,rules ,,of ,,multiplication ,,and ,,division #ae ,,cases ,,in ,,factoring #af1 #ag ,,factoring #ah ,,highest ,,common ,,factor ,,and ,,lowest ,,common ,,multiple #ai ,,fractions #bj ,,complex ,,fractions ,,and ,,fractional ,,equations #ba1 #bb ,,simultaneous ,,equations ,,and ,,involution #bc1 #bd ,,square ,,root #be ,,theory ,,of ,,exponents #bf-bh ,,radicals #bi1 #cj ,,miscellaneous ,,examples1 ,,algebra ,,to ,,quadratics #ca-ce ,,quadratic ,,equations #cf1 #cg ,,the ,,theory ,,of ,,quadratic ,,equations #ch-da ,,outline ,,of ,,simultaneous ,,quadratics #db1 #dc ,,simultaneous ,,quadratics #dd ,,ratio ,,and ,,proportion #de1 #df ,,arithmetical ,,progression #dg ,,geometrical ,,progression #dh ,,the ,,binomial ,,theorem #di ,,miscellaneous ,,examples1 ,,quadratics ,,and ,,beyond #ej-eb ,,problems--,,linear ,,equations1 ,,simultaneous ,,equations1 ,,quadratic ,,equations1 ,,simultaneous ,,quadratics #ec-eg ,,college ,,entrance ,,examinations #eh-hj ,,outline ,,of ,,elementary ,,and ,,intermediate ,,algebra ,important ,definitions ,factors2 coefficient2 exponent2 power2 base2 term2 algebraic sum2 similar terms2 degree2 homogeneous expression2 linear equation2 root of an equation2 root of an expression2 identity2 conditional equation2 prime quantity2 highest common factor 7,h4 ,c4 ,f472 lowest common multiple 7,l4 ,c4 ,m472 involution2 evolution2 imaginary number2 real number2 rational2 similar radicals2 binomial surd2 pure quadratic equation2 affected quadratic equation2 equation in the quadratic form2 simultaneous linear equations2 simultaneous quadratic equations2 discriminant2 symmetrical expression2 ratio2 proportion2 fourth proportional2 third proportional2 mean proportional2 arithmetic progression2 geometric progression2 ,s infinity ,special ,rules for ,multiplication and ,division #a4 ,square of the sum of two quantities4 7x y7#b4 #b4 ,square of the difference of two quantities4 7x - y7#b4 #c4 ,product of the sum and difference of two quantities4 7s t77s - t74 #d4 ,product of two binomials having a common term4 7x r77x m74 #e4 ,product of two binomials whose corresponding terms are similar4 7#cx #bt77#bx - #et74 #f4 ,square of a polynomial4 7m - n_/#c k7#b4 #g4 ,sum of two cubes4 7x#c y#c7_/7x y7 x#b - xy y#b4 #h4 ,difference of two cubes4 7x#c - y#c7_/7x - y7 x#b xy y#b4 #i4 ,sum or difference of two like powers4 7x#g y#g7_/7x y71 7x#e - y#e7_/7x - y71 7x#d - y#d7_/7x - y71 7x#d - y#d7_/7x y74 ,cases in ,factoring #a4 ,common monomial factor4 mx my - mz m7x y - z74 #b4 ,trinomial that is a perfect square4 x#b - #bxy y#b 7x - y7#b4 #c4 ,the difference of two squares4 7a7 ,two terms4 x#b - y#b 7x y77x - y74 7b7 ,four terms4 x#b #bxy y#b - m#b 7x y m77x y - m74 7c7 ,six terms4 x#b #bxy y#b - p#b - #bpq - q#b 7x y7 7p q77x y7 - 7p q74 7d7 ,incomplete square4 x#d x#b y#b y#d x#d #bx#b y#b y#d - x#b y#b 7x#b y#b xy77x#b y#b - xy74 #d4 ,trinomial of the form x#b bx c4 x#b - #ex #f 7x - #b77x - #c74 #e4 ,trinomial of the form ax#b bx c4 #bjx#b #gx - #f 7#dx #c77ex - #b74 #f4 ,sum or difference of two cubes4 ,see ,special ,rules1 #g and #h4 two like powers4 ,see ,special ,rules1 #i4 #g4 ,common polynomial factor4 ,grouping4 t#b p t#b q - #bmp - #bmq t#b7p q7 - #bm7p q7 7p q77t#b - #bm74 #h4 ,factor ,theorem4 x#c #agx - #ah 7x - #a77x#b x #ah74 ,h4 ,c4 ,f4 and ,l4 ,c4 ,m4 a#b #b;a - #c 7a #c77;a - #a74 a#b #g;a #ab 7a #c77;a #d74 a#d #bg;a a7a #c77;a#b - #c;a #i74 ,h4 ,c4 ,f4 a #c4 ,l4 ,c4 ,m4 7a #c77;a - #a77;a #d7;a7a#b - #c;a #i74 ,fractions ,reduction to lowest terms4 ,reduction of a mixed number to an improper fraction4 ,reduction of an improper fraction to a mixed number4 ,addition and subtraction of fractions4 ,multiplication and division of fractions4 ,law of signs in division1 changing signs of factors1 etc4 ,complex fractions4 ,simultaneous ,equations ,solved by addition or subtraction4 substitution4 comparison4 ,graphical representation4 ,involution ,law of signs4 ,binomial theorem laws4 ,expansion of monomials and fractions4 binomials4 trinomials4 ,evolution ,law of signs4 ,evolution of monomials and fractions4 ,square root of algebraic expressions4 ,square root of arithmetical numbers4 ,optional ,cube root of algebraic expressions4 ,cube root of arithmetical numbers4 ,theory of ,exponents ,proofs3 am x an a7m n72 7am7_/7an7 a7m - n72 7am7n a7mn72 a7mn77#a_/n7 am2 7a_/b7n 7an7_/7bn72 7abc7n an bn cn4 ,meaning of fractional exponent4 zero exponent4 negative exponent4 ,four rules ,to multiply quantities having the same base1 add exponents4 ,to divide quantities having the same base1 subtract exponents4 ,to raise to a power1 multiply exponents4 ,to extract a root1 divide the exponent of the power by the index of the root4 ,radicals ,radical in its simplest form4 ,transformation of radicals ,fraction under the radical sign4 ,reduction to an entire surd4 ,changing to surds of different order4 ,reduction to simplest form4 ,addition and subtraction of radicals4 ,multiplication and division of radicals a7#a_/n7 4 b7#a_/n7 ab7#a_/n74 7ab7#a_/n77_/7a7#a_/n77 b7#a_/n74 ,rationalization ,monomial denominator4 ,binomial denominator4 ,trinomial denominator4 ,square root of a binomial surd4 ,radical equations4 ,always check results to avoid extraneous roots4 ,quadratic ,equations ,pure4 x#b a4 ,affected4 ax#b bx c #j4 ,methods of solving ,completing the square4 ,formula4 ,developed from ax#b bx c #j4 ,factoring4 ,equations in the quadratic form4 ,properties of quadratics r#a -b_/#b;a 7b#b - #d;ac7#a_/b77_/7b;a74 r#b -b_/#b;a - 7b#b - #d;ac7#a_/b77_/7b;a74 ,then r#a r#b -b_/a4 r#a 4 r#b c_/a4 ,discriminant1 b#b - #d;ac1 and its discussion4 ,nature or character of the roots4 ,simultaneous ,quadratics ,,case ,i4 ,one equation linear4 ,the other quadratic4 #cx - y #ab1 x#b - y#b #af4 ,,case ,,ii4 ,both equations homogeneous and of the second degree4 x#b - xy y#b #ba1 y#b - #bxy -#ae4 ,,case ,,iii4 ,any two of the quantities x y1 x#b y#b1 xy1 x#c y#c1 x#c - y#c1 x - y1 x#b - xy y#b1 etc41 given4 x#b y#b #da1 x y #i4 ,,case ,,iv4 ,both equations symmetrical or symmetrical except for sign4 ,usually one equation of high degree1 the other of the first degree4 x#e y#e #bdb1 x y #b4 ,,case ,v4 ,special ,devices ,i4 ,solve for a compound unknown1 like xy1 x y1 7#a7_/7xy71 etc41 first4 x#by#b xy #f1 x #by -#e4 ,,ii4 ,divide the equations1 member by member4 x#d - y#d #bj1 x#b - y#b #e4 ,,iii4 ,eliminate the quadratic terms4 #dx #cy #bxy1 #gx - #ey #exy4 ,ratio and ,proportion ,proportionals mean1 third1 fourth4 ,theorems #a4 ,product of means equals product of extremes4 #b4 ,if the product of two numbers equals the product of two other numbers1 either pair1 etc4 #c4 ,alternation4 #d4 ,inversion4 #e4 ,composition4 #f4 ,division4 #g4 ,composition and division4 #h4 ,in a series of equal ratios1 the sum of the antecedents is to the sum of the consequents as any antecedent1 etc4 ,special method of proving four quantities in proportion4 ,let a_/b x1 a bx1 etc4 ,progressions ,development of formulas4 l ar7n - #a74 l a 7n - #a7;d4 ,s 7arn - a7_/7r - #a74 ,s 7n_/#b77;a l74 ,s 7rl - a7_/7r - #a74 ,s 7n_/#b7b;a 7n - #a7;d4 ,sinfinity 7a7_/7#a - r74 ,insertion of means ,arithmetical4 ,geometrical4 ,binomial ,theorem ,review of binomial theorem laws4 ,see ,involution4 ,expansion of 7a b7n4 ,finding any term by key number method4 r7th7 or 7r #a77th7 term method4 ,a ,,review ,,of ,,algebra ,,order ,,of ,,operations1 ,,evaluation1 ,,parentheses ,order of operations3 ,first of all1 raising to a power and extracting a root4 ,next1 multiplication and division4 ,last of all1 addition and subtraction4 ,find the value of3 #a4 #e 4 #bb - #be7a_/b7 _/ #e #bb 4 #h _/ #d - #b4 #b4 7#c x #f _/ #i7_/b - #bajj7a_/b7 _/ #e #d 4 #bc - 7#ad 4 #b7_/bh4 #c4 #i 4 #b _/ #f #c - #b 4 #db _/ #h7a_/c7 - #d 7#c 4 #bb7_/f4 ,evaluate3 #d4 7a#d - a#c b#c7_/7;a#b b#b7a_/b77 7ca7#a_/b7 a#c;bc7_/7abc71 if a #a1 b #b1 c #c4 #e4 t7#a_/c7 tm7#a_/c7 m7#a_/c71 if t #h1 m #bg4 #f4 7#bc #b;d a7#a_/b77_/7c;a b - cx - c7#a_/b77 77#c;c - d7x7_/7#g;ad - abc7#a_/b771 if a #e1 b #c1 c -#a1 d -#b1 x #j4 #g4 a - #e;b - a - 7#c;c - #c;b7 #b;c - #c7;a - #b;b - c71 if a -#c1 b #d1 c -#e4 7,yale47 ,simplify3 #h4 m - #bm - #cr - 7#dr - #bm74 #i4 #b;a - #e;d #c;c - 7a #b;d - #c;a #d;c74 #aj4 #c;c#b c7#b;a - #f;c - #c;a c - #d;a74 ,,special ,,rules ,,of ,,multiplication ,,and ,,division ,give results by inspection3 #a4 7g #a_/b k7#b4 #b4 7s - 7#bm7_/#c7b4 #c4 7#bv #cw77#bv - #cw74 #d4 7x #cts77x - #gts74 #e4 7#bl #c;g77#dl - #aa;g74 #f4 7a - 7#b;b7_/#c c - d7#b4 #g4 7x#c #hm#c7_/7x #bm74 #h4 7y#c - #bgk7#cm77_/7y - #ckm74 #i4 7c#e - d#e7_/7;c - d74 #aj4 7e#e d#e7_/7;e d74 #aa4 7x#d - y#d7_/7x - y74 #ab4 7x#d - y#d7_/7x y74 #ac4 7a - 4#jc77;a - 4#jjjg74 #ad4 7gn - #a_/b77;gn #c_/d74 #ae4 7t#g - v7#g_/b77_/7t - v7#a_/b774 #af4 7k#cb #a77k#af #a77k#h #a77k#d #a77k#b #a77k #a77k - #a74 #ag4 7a b7 7c d77a b7 - 7c d74 #ah4 7p - q r - s77p - q - r s74 #ai4 7#cm - n - l #br77#cm n - l - #br74 #bj4 7x #e77x - #b77x - #e77x #b74 #ba4 7a#b b#b - c - #b;d #c;e7#b4 #bb4 7s t - #bv_/#e #cw_/#f z#b7b4 #bc4 7x#e #cb7_/7x #b74 ,references3 ,the chapter on ,special ,rules of ,multiplication and ,division in any algebra4 ,special ,rules of ,multiplication and ,division in the ,outline in the front of the book4 ,,cases ,,in ,,factoring ,the number of terms in an expression usually gives the clue to the possible cases under which it may come4 ,by applying the test for each and eliminating the possible cases one by one1 the right case is readily found4 ,hence1 the number of terms in the expression and a ready and accurate knowledge of the ,cases in ,factoring are the real keys to success in this vitally important part of algebra4 ,,case ,i4 ,a common monomial factor4 ,applies to any number of terms4 #e;cx - #e;ct #e;cv - #ae c#b m #be c#c m#b #e;c7x - t v - #c;cm #e;c#b m#b74 ,,case ,,ii4 ,a trinomial that is a perfect square4 ,three terms4 x#b - #bxm m#b 7x - m7#b4 ,,case ,,iii4 ,the difference of two squares4 ,a4 ,two terms4 x#b - y#b 7x y77x - y74 ,b4 ,four terms4 x#b #bxy y#b - m#b 7x#b #bxy y#b7 - m#b 7x y m77x y - m7 ,c4 ,six terms4 x#b - #bxy y#b - m#b - #bmn - n#b 7x#b - #bxy y#b7 - 7m#b #bmn n#b7 7x - y7#b - 7m n7#b 7x - y7 7m n77x - y7 - 7m n74 ,d4 ,an incomplete square4 ,three terms1 and #dth powers or multiples of #d4 c#d c#b d#b d#d c#d #b;c#b d#b d#d - c#b d#b 7c#b d#b7b - c#b d#b 7c#b d#b cd77c#b d#b - cd74 ,,case ,,iv4 ,a trinomial of the form x#b bx c4 ,three terms4 x#b x - #cj 7x #f77x - #e74 ,,case ,v4 ,a trinomial of the form ax#b bx c4 ,three terms4 #bjx#b #gx - #f 7#dx #c77ex - #b74 ,,case ,,vi4 ,a4 ,the sum or difference of two cubes4 ,two terms4 x#c y#c 7x y77x#b - xy y#b72 x#c - y#c 7x - y77x#b xy y#b74 ,b4 ,the sum or difference of two like powers4 ,two terms4 x#d - y#d 7x - y77x#c x#by xy#b y#c72 x#e y#e 7x y77x#d - x#cy x#b y#b - xy#c y#d74 ,,case ,,vii4 ,a common polynomial factor4 ,any composite number of terms4 t#b p t#b q - t#b r - g#b p - g#b q g#b r t#b 7p q - r7 - g#b 7p q -r7 7p q - r77t#b - g#b7 7p q - r77t g77t - g74 ,,case ,,viii4 ,the ,factor ,theorem4 ,any number of terms4 x#c #agx - #ah 7x - #a77x#b x #ah74 ,,factoring ,review the ,cases in ,factoring 7see ,outline on preceding pages7 and write out the prime factors of the following3 #a4 #h;a7#ac7 am7#ab74 #b4 x#g y#g4 #c4 #dx#b #aax - #c4 #d4 m#b n#b - 7#a #bmn74 #e4 -x#b #bx - #a x#d4 #f4 x7#af7 - y7#af74 7,five factors47 #g4 7x #a7b - #ex - #bi4 #h4 x#d x#b y#b y#d4 #i4 x#d - #aax#b #a4 #aj4 x7#bm7 #b #a_/7x7#bm774 #aa4 x7#fm7 #acx7#cm7 #ab4 #ab4 #d;a#b b#b - 7a#b b#b - c#b7b4 #ac4 7x#b - x - #f77x#b - x - #bj74 #ad4 a#d - #h;a - a#c #h4 #ae4 p#c #gp#b #adp #h4 #af4 #ah;a#b b #fj;ab#b #ej;b#c4 #ag4 x#c - #gx #f4 #ah4 #bd;c#b d#b - #dg;cd - #ge4 #ai4 7a#b - b#b7b - 7a#b - ab7#b4 #bj4 a#b x#c - 7#h;a#b7_/7y#c7 - x#c #h_/7y#c74 #ba4 gt - gk gl#b xt - xk xl#b4 #bb4 7m - n77#b;a#b - #b;ab7 7n - m77#b;ab - #b;b#b74 #bc4 a#b - x#b - y#b b#b #b;ab #bxy4 #bd4 7#b;c#b #c;d#b7;a 7#b;a#b #c;c#b7;d4 #be4 7n7n - #a77_/7a 4 #b7 a7n - #b7 b#b 7n7n - #a77n - #b77_/7a 4 #b 4 #c7 a7n - #c7 b#c4 #bf4 7x - x#b7c 7x#b - #a7c 7#a - x7#c4 7,m4 ,i4 ,t47 #bg4 7#bgy#c7b - #b7bgy#c77h;b#c7 7#h;b#c7b4 7,princeton47 #bh4 7a#c #h;b#c77;a b7 - #f;ab7a#b - #b;ab #d;b#b74 7,m4 ,i4 ,t47 ,solve by factoring3 #bi4 x#c x4 #cj4 z#b - #dz - #de #j4 #ca4 x#c - x#b #dx - #d4 ,reference3 ,the chapter on ,factoring in any algebra4 ,,highest ,,common ,,factor ,,and ,,lowest ,,common ,,multiple ,define ,h4 ,c4 ,f4 and ,l4 ,c4 ,m4 ,find by factoring the ,h4 ,c4 ,f4 and ,l4 ,c4 ,m43 #a4 #cx#b - #cx1 #abx#b 7x#b - #a71 #ahx#c 7x#c - #a74 #b4 7x#b - #a77x#b #ex #f71 7x#b #cx77x#b - x - #f74 7,harvard47 #c4 x#b - y#b1 x#b y#b1 x#c y#c1 x#f y#f1 x#f - y#f4 7,college ,entrance ,board47 #d4 x#c x#b - #b1 x#c #bx#b - #c4 7,cornell47 #e4 x#e - #bx#d x#b1 #bx#d - #dx#c - #dx #f4 7,yale47 #f4 x#b a#b - b#b #b;ax1 x#b - a#b b#b #b;bx1 x#b - a#b - b#b - #b;ab4 7,harvard47 #g4 #bx#b - x - #ae1 #cx#b - #aax #f1 #bx#c - x#b - #acx - #f4 7,college ,entrance ,board47 #h4 7tv - v#b7c1 v#c - t#bv1 t#c - v#c1 v#b - #bvt t#b4 ,pick out the ,h4 ,c4 ,f4 and the ,l4 ,c4 ,m4 of the following3 #i4 #h7x#b y77#cb7 7t#b z77#ai7 7m - n#c77ad71 #ab7x#b y77#bc7 7t#b z77#da7 7m - n#c77ag71 #ah7m - n#c77aa7 7x#b y77#ci7 7t#b z77#cg74 #aj4 #ag;ax#c 7y z77#aj7 7y - x77#ai7 7x z77#bg71 #cd;a#b x#d 7y z77#aa7 7y - x77#ba7 7x z77#ac71 #ea;a#c x#e 7y z7#d 7x z77#cb7 7y - x77#bi74 ,reference3 ,the chapter on ,h4 ,c4 ,f4 and ,l4 ,c4 ,m4 in any algebra4 ,,fractions ,define3 fraction1 terms of a fraction1 reciprocal of a number4 ,look up the law of signs as it applies to fractions4 ,except for this1 fractions in algebra are treated exactly the same as they are in arithmetic4 #a4 ,reduce to lowest terms3 7a7 #cb_/bd2 7b7 7a#f - x#f7_/7;a#d - x#d72 7c7 7a b7#b - 7c d7#b_/7;a c7#b - 7b d7#b4 7,m4 ,i4 ,t47 #b4 ,reduce to a mixed expression3 7a7 #gef_/aa2 7b7 7a#c b#c7_/7;a - b74 #c4 ,reduce to an improper fraction3 7a7 #de-a_/h2 7b7 #i-aa_/ab qt42 7c7 a#b - ab b#b - 7b#c7_/7;a b74 ,add3 #d4 #e_/ah #g_/i #aa_/af #e_/h4 #e4 #e_/7a #bx7 - 7#cx7_/7#a - #bx7 7#d - #acx7_/7#dx#b - #a74 #f4 #a_/x7x - a77x - b7 #a_/;a7a - x77a - b7 #a_/;b7b - x77b - a74 ,multiply3 #g4 #gb_/aba x #ee_/ef x #gg_/ij4 #h4 7b - y7_/7a#c y#c7 x 7ca cy7_/7b#b by7 x 7b#f y#f7_/7;b#b y#b7 x b_/c4 ,divide3 #i4 7#ab_/be7 _/ 7#f_/ej74 #aj4 #a - 7ab7_/7a#b - ab b#b7 #a - 7ab7_/7a#b #b;ab b#b7 _/ 7a#c - b#c7_/7;a#c b#c74 7,yale47 #aa4 7x#d - y#d7_/7x#b - y#b7 _/ 7x y7_/7x#b - xy7 _/ 7x#b y#b7_/7x - y7 _/ 7x y7_/7xy - y#b74 7,sheffield47 ,simplify3 #ab4 7#dy7_/x - 7#aey#b7_/7x#b7 #d _/ #d - 7#afy7_/x 7#aey#b7_/7x#b7 x #c - 7#dx #bjy7_/7#bx #ey74 ,reference3 ,the chapter on ,fractions in any algebra4 ,,complex ,,fractions ,,and ,,fractional ,,equations ,define a complex fraction4 ,simplify3 #a4 7#c_/g #d_/e7_/7b - #c_/g 4 #d_/e74 #b4 7#b - #c_/b #b_/c7_/7e - #b_/c #c_/b74 #c4 #b - #b_/7a - #a_/a - #a_/7a #a_/b774 #d4 a_/7b#b7 - a_/b#b 7cb7_/7a - c_/b74 7,harvard47 #e4 ,if m #a_/7;a #a71 n #b_/7;a #b71 p #c_/7;a #c71 what is the value of m_/7#a - m7 n_/7#a - n7 p_/7#a - p78 7,univ4 of ,penn47 #f4 ,simplify the expression x y - #a_/x y - xy_/7x y77x#c - y#c7_/7x#b - y#b74 7,cornell47 #g4 ,simplify #a - 7#bxy7_/77x y7#b7_/a 7#bxy7_/77x - y7#b7 _/ 7#a - y_/x7_/7#a y_/x7#b4 #h4 ,solve 7#gy #i7_/d - y - 7#by - #a7_/i #g4 #i4 ,solve #b-a_/c - 7#b_/e77x#b #c7 7#ajx7_/#c #a - 7#bx#b7_/e4 #aj4 ,how much water must be added to #hj pounds of a #e per cent salt solution to obtain a #d per cent solution8 7,yale47 ,reference3 ,see ,complex ,fractions1 and the first part of the chapter on ,fractional ,equations in any algebra4 ,,fractional ,,equations #a4 ,solve for each letter in turn #a_/;b #a_/p #a_/q4 #b4 ,solve and check3 7#ex #b7_/c - 7#c - 7#cx - #a7_/b7 7#cx #ai7_/b - 77x #a7_/f #c74 #c4 ,solve and check3 7#a_/b77x - a_/#c7 - 7#a_/c77x - a_/#d7 7#a_/d77x - a_/#e7 #j4 #d4 ,solve 7after looking up the special short method73 7#cx - #a7_/cj 7#dx - #g7_/ae x_/#d - 7#bx - #c7_/7abx - #aa7 7#gx - #ae7_/fj4 #e4 ,solve by the special short method3 #a_/7x - #b7 - #a_/7x - #c7 #a_/7x - #d7 - #a_/7x - #e74 #f4 ,at what time between #h and #i o'clock are the hands of a watch 7a7 opposite each other8 7b7 at right angles8 7c7 together8 ,work out 7a7 and state the equations for 7b7 and 7c74 #g4 ,the formula for converting a temperature of ,f degrees ,fahrenheit into its equivalent temperature of ,c degrees ,centigrade is ,c 7#e_/i77;,f - #cb74 ,express ,f in terms of ,c1 and compute ,f for the values ,c #cj and ,c #bh4 7,college ,entrance ,exam4 ,board47 #h4 ,what is the price of eggs when #b less for #bd cents raises the price #b cents a dozen8 7,yale47 #i4 ,solve #b_/7x - #b7 #b_/7d - x#b7 #e_/7x #b74 ,reference3 ,the ,chapter on ,fractional ,equations in any algebra4 ,note particularly the special short methods1 usually given about the middle of the chapter4 ,,simultaneous ,,equations ,,note4 ,up to this point each topic presented has reviewed to some extent the preceding topics4 ,for example1 factoring reviews the special rules of multiplication and division2 ,h4 ,c4 ,f4 and ,l4 ,c4 ,m4 review factoring2 addition and subtraction of fractions and fractional equations review ,h4 ,c4 ,f4 and ,l4 ,c4 ,m41 etc4 ,from this point on1 however1 the interdependence is not so marked1 and miscellaneous examples illustrating the work already covered will be given very frequently in order to keep the whole subject fresh in mind4 #a4 ,solve by three methods--addition and subtraction1 substitution1 and comparison3 #ex y #aa1 #cx #by #a4 ,solve and check3 #b4 #ab,r#a - #aa,r#b b #ab;c1 ,r#a ,r#b #b;b c4 #c4 7r - s7_/#b #be_/f - 7r s7_/#c1 7r s - #i7_/b - 7s - r - #f7_/c #j4 #d4 ,one half of ,a's marbles exceeds one half of ,b's and ,c's together by #b2 twice ,b's marbles falls short of ,a's and ,c's together by #af2 if ,c had four more marbles1 he would have one fourth as many as ,a and ,b together4 ,how many has each8 7,college ,entrance ,board47 #e4 ,the sides of a triangle are a1 b1 c4 ,calculate the radii of the three circles having the vertices as centers1 each being tangent externally to the other two4 7,harvard47 #f4 ,solve #bx #cy #g1 x - y #a graphically2 then solve algebraically and compare results4 7,use cooerdinate or squared paper47 ,factor3 #g4 x#d #d4 #h4 #b;d7#aj7 - #ajbd;d4 #i4 #b7x#c - #a7 - #g7x#b - #a74 ,references3 ,the chapters on ,simultaneous ,equations and ,graphs in any algebra4 ,,simultaneous ,,equations ,,and ,,involution #a4 ,solve 7#c_/d7x - 7#e_/c7y #aa-a_/b1 7#e_/h7x - 7#c_/b7y #aj-a_/d4 ,look up the method of solving when the unknowns are in the denominator4 ,should you clear of fractions8 #b4 ,solve #a_/x - #a_/y - #a_/z #a_/;a1 #a_/y - #a_/z - #a_/x #a_/;b1 #a_/z - #a_/x - #a_/y #a_/;c4 #c4 ,solve graphically and algebraically #bx - y #d1 #bx #cy #ab4 #d4 ,solve graphically and algebraically #cx #gy #e1 #hx #cy -#ah4 ,review3 #e4 ,the squares of the numbers from #a to #be4 #f4 ,the cubes of the numbers from #a to #ab4 #g4 ,the fourth powers of the numbers from #a to #e4 #h4 ,the fifth powers of the numbers from #a to #c4 #i4 ,the binomial theorem laws4 7,see ,involution47 ,expand3 7,indicate first1 then reduce47 #aj4 7b y7#g4 #aa4 7#b;a7_/#c - #ae4 #ab4 7x#b #b;a7#e4 #ac4 7x - y #bz7#c4 #ad4 ,a train lost one sixth of its passengers at the first stop1 #be at the second stop1 #bj of the remainder at the third stop1 three quarters of the remainder at the fourth stop2 #be remain4 ,what was the original number8 7,m4 ,i4 ,t47 ,references3 ,the chapter on ,involution in any algebra4 ,also the references on the preceding page4 ,,square ,,root ,find the square root of3 #a4 #a #afm#f - #djm#d #ajm - #hm#c #bem#b4 #b4 7a#b7_/7x#b7 7#f;a7_/x #aa 7#fx7_/a 7x#b7_/7;a#b74 #c4 ,find the square root to three terms of x#b #e4 #d4 ,find the square root of #ccg1efa4 #e4 ,find the square root of #ahbc4bi4 #f4 ,find to four decimal places the square root of #a4fgb4 7,princeton47 #g4 ,add #b_/7x - #a7c #a_/7a - x7#b - #b_/7a - x7 - #a_/x4 #h4 ,find the value of3 7#fd7a_/c7 4 #ab7_/bd _/ #b x #c - 7#b 4 #gb7_/7ad7 _/ #g x #a 7#a7a_/c7 4 #ag7_/7a 4 #ab7 - #d 4 #j4 #i4 ,simplify 7x y7#e 7x - y7#e7x y7#e - 7x - y7#e4 #aj4 ,solve by the short method3 #e_/7g - x7 - 7#b-a_/d7x - #c_/d - 7x #aa7_/h 7#aax #e7_/af #j4 #aa4 ,it takes #c_/d of a second for a ball to go from the pitcher to the catcher1 and #a_/b of a second for the catcher to handle it and get off a throw to second base4 ,it is #ij feet from first base to second1 and #acj feet from the catcher's position to second4 ,a runner stealing second has a start of #ac feet when the ball leaves the pitcher's hand1 and beats the throw to the base by #a_/h of a second4 ,the next time he tries it1 he gets a start of only #c-a_/b feet1 and is caught by #f feet4 ,what is his rate of running1 and the velocity of the catcher's throw8 7,cornell47 ,reference3 ,the chapter on ,square ,root in any algebra4 ,,theory ,,of ,,exponents ,review the proofs1 for positive integral exponents1 of3 ,i4 am x an a7m n74 ,,ii4 7am7_/7an7 a7m - n74 ,,iii4 7am7n a7mn74 ,,iv4 a7mn77#a_/n7 am4 ,v4 a_/bn 7an7_/7bn74 ,,vi4 7abc7n an bn cn4 ,to find the meaning of a fractional exponent4 ,assume that ,law ,i holds for all exponents4 ,if so1 a7#b_/c7 4 a7#b_/c7 4 a7#b_/c7 a7#f_/c7 a#b4 ,hence1 a7#b_/c7 is one of the three equal factors 7hence the cube root7 of a#b4 ,therefore a7#b_/c7 a#b7a_/c74 ,in the same way1 a7#d_/e7 4 a7#d_/e7 4 a7#d_/e7 4 a7#d_/e7 4 a7#d_/e7 a7#bj_/e7 a#d4 ,hence1 a7#d_/e7 is one of the five equal factors 7hence the fifth root7 of a#d4 ,therefore a7#d_/e7 a#d7a_/e74 ,in the same way1 in general1 a7p_/q7 ap7#a_/q74 ,hence1 the numerator of a fractional exponent indicates the power1 the denominator indicates the root4 ,to find the meaning of a zero exponent4 ,assume that ,law ,,ii holds for all exponents4 ,if so1 7am7_/7am7 a7m - m7 a#j4 ,but by division1 7am7_/7am7 #a4 ,therefore a#j #a4 ,axiom ,i4 ,to find the meaning of a negative exponent4 ,assume that ,law ,i holds for all exponents4 ,if so1 am x a7-m7 a7m - m7 a#j #a4 ,hence1 am x a7-m7 #a4 ,therefore a7-m7 #a_/7;am74 ,rules3 ,to multiply quantities having the same base1 add exponents4 ,to divide quantities having the same base1 subtract exponents4 ,to raise a quantity to a power1 multiply exponents4 ,to extract a root1 divide the exponent of the power by the index of the root4 #a4 ,find the value of #cb - #e x #dj #h7-b_/c7 #a7b_/e74 #b4 ,find the value of #h7-b_/c7 #i7c_/b7 - #b7-b7 #a7-b_/e7 - #gj4 ,give the value of each of the following3 #c4 7#cj7_/e1 #c_/7ej71 7#cj7_/7ej71 #cj x #e1 #c x #ej1 #cj x #ej1 #cj #ej1 #cj - #ej4 #d4 ,express #gj as some power of #g divided by itself4 ,simplify3 #e4 #af7a_/c7 4 #b7a_/b7 4 #cb7e_/f74 7,change to the same base first47 #f4 #b_/7h7-c777a_/e74 #g4 7xn77n #b7_/7x7n #a777x7n - #a774 #h4 7x #cx7#b_/c7 - #bx7#a_/c777c - #bx7-#a_/c7 #dx7-#b_/c774 #i4 7a#b;b7_/7c#b;d77#a_/b7 x 7c#c;d7_/7ab#c77a_/c7 x 7a7#a_/c7;c7_/7b7#a_/d7;d7#e_/ab77b4 #aj4 7a7-#d77_/7;b7-#b7;c77-#c_/d7 x 7a7-#a7;bc7-#c77a_/b77_/7;ab7-#a777a_/b7 4 #aa4 7a#b7a_/c77_/7;b7-#a77a_/d77 4 7c7-#c77a_/b77_/7;a7#a_/c77 4 7b7-#a_/d7;a7#a_/c77_/7;c7-#a777-f74 ,reference3 ,the chapter on ,theory of ,exponents in any algebra4 ,solve for x3 #a4 x7#b_/c7 #d4 #b4 x7-#c_/d7 #h4 ,factor3 #c4 x7#b_/c7 - #i4 #d4 x7#c_/e7 #bg4 #e4 x7#b;a7 - y7-#f74 #f4 a7#a_/c7 x7#a_/b7 - #c;a7#a_/c7 #ex7#a_/b7 - #ae4 #g4 ,find the ,h4 ,c4 ,f4 and ,l4 ,c4 ,m4 of a#b a7#c_/b7 b7#a_/b7 a7#a_/b7 b7#c_/b7 - b#b1 a#b - a7#c_/b7 b7#a_/b7 - a7#a_/b7 b7#c_/b7 - b#b4 #h4 ,simplify the product of3 7ayx7-#a777a_/b71 7bxy7-#b777a_/c71 and 7y#b;a7-#b7;b7-#b777a_/d74 7,princeton47 #i4 ,find the square root of3 #be;a7#d_/c7;b7-#c7 - #aj;a7#b_/c7;b7-#c_/b7 - #di #aj;a7-#b_/c7;b7#c_/b7 #be;a7-#d_/c7;b#c4 #aj4 ,simplify 7#b7n #b77_/7d7-n77 _/ 7#hn7_/7#bc77a_/e74 #aa4 ,find the value of 7#g 4 #acj _/ #g7_/7baj7 #cj x 7#dj 4 #gj7_/7g;a b7#j #h7-b_/c74 #ab4 ,express as a power of #b3 #hc2 #de2 #dc 4 #h7b_/c7 4 #af7c_/d74 #ac4 ,simplify 7x7a #a77_/7x7#a - a77a _/ 7xa7_/7x7#a - a777a - #a77a_/7c;a - #a774 #ad4 ,simplify 7x7#e_/b7 y7#d_/c77_/7z7-#e_/d77 4 7z#d7_/7x7-#c7 y7-#e_/c77 _/ 7y7-#b7 z7#a_/d77_/7x7-#a_/b777a_/e74 #ae4 ,expand 7a7#a_/b7 b7#a_/c77d1 writing the result with fractional exponents4 ,reference3 ,the chapter on ,theory of ,exponents in any algebra4 ,,radicals #a4 ,review all definitions in ,radicals1 also the methods of transforming and simplifying radicals4 ,when is a radical in its simplest form8 #b4 ,simplify 7to simplest form73 #b_/c7a_/b72 #a_/aa7a_/b72 #c_/e7a_/c72 #ce_/f7a_/b72 7#b;a_/b77#h;b#b7_/7bg;a77#a_/b72 #e_/7xn77#a_/bn72 7a b7#b 7-a#d7_/77;a b7#e77a_/c72 #bg7a_/b72 #ed7a_/c72 -#eabe7a_/b74 #c4 ,reduce to entire surds3 #bc7a_/b72 #bc7a_/d72 #fb7a_/c72 ab#b7a_/n72 -#cb7a_/c72 #c;a7a #b7_/7f;a#b77a_/c72 7a #by77a - #by7_/7a #by77#a_/b74 #d4 ,reduce to radicals of lower order 7or simplify indices73 a#b7a_/d72 a#c7a_/f72 #bg;a#c7a_/f72 #ha a#d x#h7a_/ab72 #ix#b y#d z#aj7a_/bn74 #e4 ,reduce to radicals of the same degree 7order1 or index73 #g7a_/b7 and #aa7a_/c72 #e7a_/c7 and #c7a_/d72 #g7a_/f7 and #c7a_/b72 xm7#a_/n7 and xn7#a_/m72 cy7#a_/x71 cz7#a_/y71 and cx7#a_/z74 #f4 ,which is greater1 #c7a_/b7 or #d7a_/c78 #bc7a_/c7 or #bb7a_/b78 #g4 ,which is greatest1 #c7a_/b71 #e7a_/c71 or #g7a_/d78 ,give work and arrange in descending order of magnitude4 ,collect3 #h4 #abh7a_/b7 - #bej7a_/b7 #gb7a_/b7 - #ah7a_/b74 #i4 #be_/c7a_/b7 7#a_/f7fj7a_/b7 #ae7a_/b7 #c_/e7a_/b74 #aj4 7m - n7#b;a7#a_/b7 7m n7#b;a7#a_/b7 - am#b7a_/b7 a7n - m7#b7a_/b7 - a7#a_/b74 #aa4 ,a and ,b each shoot thirty arrows at a target4 ,b makes twice as many hits as ,a1 and ,a makes three times as many misses as ,b4 ,find the number of hits and misses of each4 7,univ4 of ,cal47 ,reference3 ,the chapter on ,radicals in any algebra 7first part of the chapter74 ,the most important principle in ,radicals is the following3 7ab77#a_/n7 a7#a_/n7 b7#a_/n74 ,hence ab7#a_/n7 a7#a_/n7 4 b7#a_/n74 ,or1 a7#a_/n7 4 b7#a_/n7 ab7#a_/n74 ,from this also 7ab7#a_/n77_/7a7#a_/n77 b7#a_/n74 ,multiply3 #a4 #bd7a_/c7 by #cf7a_/c74 #b4 #b7a_/b7 by #c7a_/c74 #c4 #b7a_/d7 by #d7a_/f74 #d4 a x7#a_/b77a_/b7 by a - x7#a_/b77a_/b74 #e4 #b7a_/b7 #c7a_/b7 - #e7a_/b7 by #b7a_/b7 - #c7a_/b7 #e7a_/b74 #f4 -p_/#b 7p#b - #dq7#a_/b77_/b by -p_/#b - 7p#b - #dq7#a_/b77_/b4 ,divide3 #g4 #bg7a_/b7 by #c7a_/b74 #h4 #dah7a_/b7 by #ecb7a_/b74 #i4 #cab7a_/c7 by #f7a_/b74 #aj4 #c7a_/b7 by #c7a_/d74 #aa4 #faje7a_/b7 #ahdj7a_/b7 - #deab7a_/b7 by #cae7a_/b74 7,short division47 #ab4 #ajah7a_/c7 - #dfj7a_/c7 #eajj7a_/c7 by #ccj7a_/c74 ,rationalize the denominator3 #ac4 #b_/7c7a_/b772 #g_/7g7a_/b772 #e_/7be7a_/b772 #c_/7;a#b7a_/e772 #d_/7;a#c7a_/g774 #ad4 #b_/7b7a_/b77 #c7a_/b772 7a7#a_/b7 b7#a_/b77_/7;a7#a_/b7 - b7#a_/b772 #c_/7c - #c7a_/b774 #ae4 #c7a_/b7 #b7a_/b7_/f7a_/b7 #c7a_/b7 - #b7a_/b74 ,review the method of finding the square root of a binomial surd4 7,by inspection preferably47 ,then find square root of3 #af4 #e #bf7a_/b74 #ag4 #ag - #abb7a_/b74 #ah4 #g - #cc7a_/b74 ,reference3 ,the chapter on ,radicals in any algebra1 beginning at ,addition and ,subtraction of ,radicals4 ,,miscellaneous ,,examples1 ,,algebra ,,to ,,quadratics ,results by inspection1 examples #a-aj4 ,divide3 #a4 7x7#e_/ag7 y7#e_/ag77_/7x7#a_/ag7 y7#a_/ag774 #b4 7x - y7_/7x7#a_/c7 - y7#a_/c774 #c4 7m#b n#b7_/7m7#b_/c7 n7#b_/c774 #d4 7x - y#b7_/7x7#a_/c7 - y#b7a_/c774 ,multiply3 #e4 a7-#c_/d7 #b_/7m7#a_/b77b4 #f4 7,k7-#b_/g7 - g7-#aa_/be77b4 #g4 7r7#bs7 l7-#cm777r7#bs7 - l7-#cm774 #h4 a7-#b7 b7-#c7 - #a_/7;c#b7b4 #i4 7#c,kx #dt7-#c777c,kx - #gt7-#c774 #aj4 7#by7#b_/g7 - #dj,k#c77cy7#b_/g7 #ee,k#c74 ,factor3 #aa4 x7#b_/c7 - #fd4 #ab4 y7#c_/e7 #bg4 #ac4 b7#c_/b7 - #hm7-#a74 #ad4 #cp - #hp7#a_/b7 - #ce4 ,factor1 using radicals instead of exponents3 #ae4 #fj - #gc;b7#a_/b7 - #f;b4 #af4 #aem - #bmn7#a_/b7 - #bdn4 #ag4 a - b 7factor as difference of two squares74 #ah4 a - b 7factor as difference of two cubes74 #ai4 a - b 7factor as difference of two fourth powers74 #bj4 ,find the ,h4 ,c4 ,f4 and ,l4 ,c4 ,m4 of x#b xy7#a_/b7 - #by1 #bx#b #exy7#a_/b7 #by1 #bx#b - xy7#a_/b7 - y4 #ba4 ,solve 7short method7 7x - #g7_/7x - #h7 - 7x - #h7_/7x - #i7 7x - #d7_/7x - #e7 - 7x - #e7_/7x - #f74 #bb4 ,simplify 7ab_/c bc_/a ca_/b7_/7a_/bc b_/ca c_/ab7 x 77a b c7#b7_/7;ab bc ca7 - #b4 7,princeton47 #a4 ,solve for p3 #b7p - #c7 #abh4 #b4 ,solve for t3 t7#c_/b7 -#bg4 #c4 ,find the square root of #haad4djfd4 ,what1 then1 is the square root of 4#jjhaaddjfd8 of #haaddj4fd8 ,from any of the above can you determine the square root of 4#jhaaddjfd8 #d4 ,the ,h4 ,c4 ,f4 of two expressions is a7a - b71 and their ,l4 ,c4 ,m4 is a#b;b7a b77a - b74 ,if one expression is ab7a#b - b#b71 what is the other8 #e4 ,solve 7short method73 #e_/7g - x7 - 7#b-a_/d7x - #c_/d - 7x #aa7_/h 7#aax #e7_/af #j4 #f4 ,solve #b_/m - #c_/n #aj_/p -#c1 #d_/m #e_/p #f_/n #ae1 #a_/m - #a_/n #e_/p -#a_/b4 #g4 ,simplify #bab_/c7a_/b7 - #ed_/e7a_/b7 #fd-a_/f7a_/b7 - #ajc-a_/e7a_/b7 7#dj_/c7aa-a_/d7a_/b74 #h4 ,does #af x #be7a_/b7 #d x #e8 ,does #af #be7a_/b7 #d #e8 #i4 ,write the fraction #e_/7d #bc7a_/b77 with rational denominator1 and find its value correct to two decimal places4 #aj4 ,simplify 7p p#b - q7#a_/b7_/b77a_/b7 7p - p#b - q7#a_/b7_/b77a_/b7b_/p q7#a_/b74 7,princeton47 #a4 ,rationalize the denominator of #f7a_/b7 #c7a_/b7 - #cb7a_/b7_/f7a_/b7 - #c7a_/b7 #cb7a_/b74 7,univ4 of ,cal47 #b4 ,simplify #b7n #d7 - #b7bn7_/#b7b7n #c774 7,univ4 of ,penn47 #c4 ,find the value of #a #h7-x_/#c7_/7hx77#a_/b7 #aj7x - #b71 when x #b4 7,cornell47 #d4 ,find the value of x if x7#f_/e7 y#d1 y7#b_/c7 #i4 7,m4 ,i4 ,t47 #e4 ,a fisherman told a yarn about a fish he had caught4 ,if the fish were half as long as he said it was1 it would be #aj inches more than twice as long as it is4 ,if it were #d inches longer than it is1 and he had further exaggerated its length by adding #d inches1 it would be #a_/e as long as he now said it was4 ,how long is the fish1 and how long did he first say it was8 7,m4 ,i4 ,t47 #f4 ,the force ,p necessary to lift a weight ,w by means of a certain machine is given by the formula ,p a b,w1 where a and b are constants depending on the amount of friction in the machine4 ,if a force of #g pounds will raise a weight of #bj pounds1 and a force of #ac pounds will raise a weight of #ej pounds1 what force is necessary to raise a weight of #dj pounds8 7,first determine the constants a and b47 7,harvard47 #g4 ,reduce to the simplest form3 #d_/b7n #b77a_/n72 ax7a7-#a7x - ax7-#a77_/x7#b_/c7 - a7#b_/c74 #h4 ,determine the ,h4 ,c4 ,f4 and ,l4 ,c4 ,m4 of 7xy - y#b7c and y#c - x#by4 7,college ,entrance ,board47 #a4 ,simplify 7a - #hm7_/7a7#a_/c7 - #bm7#a_/c77 - #b;a7#a_/c7m7#a_/c74 #b4 ,simplify1 writing the result with rational denominator3 7a7#a_/b7 7#a7_/7x7-#a_/b77b - 7#a7_/7;a7-#a_/b77 - x7#a_/b7b7 _/ 7x a#b x#b7a_/b774 7,m4 ,i4 ,t47 #c4 ,find #g - #dh7a_/b77a_/b74 #d4 ,expand 7a#c7a_/b7 - b#e7a_/b77e4 #e4 ,expand and simplify 7#a - #bc7a_/b7 #cb7a_/b77b4 #f4 ,solve the simultaneous equations x 7-#a_/b7 #by7-#a_/b7 #g_/f1 #bx7-#a_/b7 - y7-#a_/b7 #b_/c4 7,yale47 #g4 ,find to three places of decimals the value of 7a b77-#a_/c7_/7aa;a b#b77a_/f7 4 7a#c - b#c77-a_/b7_/7;a - b77#a_/b77a_/b71 when a #e and b #c4 7,columbia47 #h4 ,show that 7#aj - #de7a_/b77_/7e #ce7a_/b77 is the negative of the reciprocal of 7#aj #de7a_/b77_/7e - #ce7a_/b774 7,columbia47 #i4 ,solve and check #e_/cx #b7a_/b7 #cx #b7a_/b7 #cx - #a7a_/b74 #aj4 ,assuming that when an apple falls from a tree the distance 7,s meters7 through which it falls in any time 7t seconds7 is given by the formula ,s 7#a_/b7;gt#b 7where g #i4h71 find to two decimal places the time taken by an apple in falling #ae meters4 7,college ,entrance ,board47 ,excellent practice may be obtained by solving the ordinary formulas used in arithmetic1 geometry1 and physics orally1 for each letter in turn4 ,,arithmetic p br i prt a p prt ,,geometry ,k 7#a_/b7 bh ,k bh ,k 7a#b7_/d #c7a_/b7 ,k 7#a_/b7 7b b'7 h ,k pi ,r#b ,c #b pi ,r ,k pi ,r ,l ,s #d pi ,r#b ,v pi ,r#b ,h ,v 7#a_/c7 pi ,r#b ,h ,v 7#d_/c7 pi ,r#c ,s 7pi ,r#b ,e7_/7#ahj7 ,c_/7,c'7 ,r_/7,r'7 ,k_/7,k'7 7,r#b7_/7,r'#b7 ,,physics v gt s 7#a_/b7 gt#b s 7v#b7_/7b;g7 ,c ,e_/,r ,e 7wv#b7_/7b;g7 e 7#d,pl#c7_/7;bh#c m7 ,e 7mv#b7_/7b7 t pi l_/g7#a_/b7 ,f 7m,v#b7_/7r7 mh 7mv#b7_/7b;g7 ,r gs_/7g s7 ,e 7#dn#bl#bw7_/7g7 ,c 7#e_/i77;,f - #cb7 ,,quadratic ,,equations #a4 ,define a quadratic equation2 a pure quadratic2 an affected 7or complete7 quadratic2 an equation in the quadratic form4 #b4 ,solve the pure quadratic 7#g7_/7c,s#b7 - 7#aa7_/7i,s#b7 #e_/f4 ,review the first 7or usual7 method of completing the square4 ,solve by it the following3 #c4 x#b #ajx #bd4 #d4 #bx#b - #ex #g4 #e4 7x - #a7_/b #b_/7x - #a7 #b-a_/b4 #f4 ax#b bx c #j4 ,review the solution by factoring4 ,solve by it the following3 #g4 x#b #hx #g #j4 #h4 #bdx#b #bx #ae4 #i4 #c #ajx - #cx#b4 #aj4 -#g #fx - x#b4 ,solve1 by factoring1 these equations1 which are not quadratics3 #aa4 x#d #af4 #ab4 x#c #h4 #ac4 x#c x4 ,review the solution by formula4 ,solve by it the following3 #ad4 #ex#b - #fx #h4 #ae4 7#a_/b77x #a7 - 7x_/#c77bx - #a7 -#ab4 #af4 x#b #d;ax #ab;a#b4 #ag4 #cx#b #brx #br#b4 ,solve graphically3 #ah4 x#b - #bx - #h #j4 #ai4 x#b x - #b #j4 ,reference3 ,the chapter on ,quadratic ,equations in any algebra 7first part of the chapter74 #a4 ,solve by three methods--formula1 factoring1 and completing the square3 x#b #ajx #bd4 ,review equations in the quadratic form and solve3 #b4 x#d - #ex#b -#d4 #c4 #bx7-#b77a_/c7 - #cx7-#a77a_/c7 #b4 #d4 7x #c7_/7x - #c7 #f #e7x #c7_/7x - #c77a_/b74 7,let y 7x #c7_/7x - #c77a_/b7 and substitute47 #e4 #cx#b - #dx #bcx#b - #dx - #f7a_/b7 #ba4 #f4 x#b #ex - #e 7#f7_/7x#b #ex74 ,solve and check3 #g4 x #g7a_/b7 #cx - #b7a_/b7 7#dx #i7_/7cx - #b7a_/b774 #h4 x#b - #e7a_/b7 #f_/x#b - #e7a_/b7 #e4 #i4 7#ajw7_/7#ajw - #i7a_/b77 - #ajw #b7a_/b7 #b_/7ajw - #i7a_/b774 ,give results by inspection3 #aj4 7a7#a_/b7 b7#a_/b777;a7#a_/b7 - b7#a_/b774 #aa4 7#aj #ai7a_/b77a_/b777aj - #ai7a_/b77a_/b774 #ab4 ,how many gallons each of cream containing #cc butter fat and milk containing #f butter fat must be mixed to produce #aj gallons of cream containing #be butter fat8 #ac4 ,i have #f in dimes1 quarters1 and half-dollars1 there being #cc coins in all4 ,the number of dimes and quarters together is ten times the number of half-dollars4 ,how many coins of each kind are there8 7,college ,entrance ,board47 ,reference3 ,the last part of the chapter on ,quadratic ,equations in any algebra4 ,,the ,,theory ,,of ,,quadratic ,,equations ,i4 ,to find the sum and the product of the roots4 ,the general quadratic equation is ax#b bx c #j4 7#a7 ,or1 x#b 7b_/a7x c_/a #j4 7#b7 ,to derive the formula1 we have by transposing x#b 7b_/a7x -c_/a4 ,completing the square1 x#b 7b_/a7x b_/#b;a#b 7b#b7_/7d;a#b7 - c_/a 7b#b - #d;ac7_/7#d;a#b74 ,extracting square root1 x b_/#b;a -b#b - #d;ac7#a_/b7_/7b;a74 ,transposing1 x -b_/#b;a - b#b - #d;ac7#a_/b7_/7b;a74 ,hence1 x -b - b#b - #d;ac7#a_/b7_/7b;a74 ,these two values of x we call roots4 ,for convenience represent them by r#a and r#b4 ,hence1 r#a -b_/#b;a b#b - #d;ac7#a_/b7_/7b;a74 r#b -b_/#b;a - b#b - #d;ac7#a_/b7_/7b;a74 ---------------------------------------- ----- ,adding1 r#a r#b -7#b;b7_/7#b;a7 -b_/a4 7#c7 ,also1 r#a -b_/#b;a b#b - #d;ac7#a_/b7_/7b;a74 r#b -b_/#b;a - b#b - #d;ac7#a_/b7_/7b;a74 ---------------------------------------- --- ,multiplying1 r#a r#b 7b#b7_/7d;a#b7 - 7b#b - #d;ac7_/7#d;a#b7 7b#b - b#b #d;ac7_/7#d;a#b7 7#d;ac7_/7#d;a#b7 c_/a4 7#d7 ,hence we have shown that r#a r#b -b_/a1 and r#a r#b c_/a4 ,or1 referring to equation 7#b7 above1 we have the following rule3 ,when the coefficient of x#b is unity1 the sum of the roots is the coefficient of x with the sign changed2 the product of the roots is the independent term4 ,,examples3 #a4 x#b - #ix #ba #j4 ,sum of the roots #i4 ,products of the roots #ba4 #b4 #cx#b - #gx - #ah #j4 ,sum of the roots #g_/c4 ,product of the roots -#f4 #c4 -#bax #ag - #dx#b4 ,sum of the roots #ba_/d4 ,product of the roots -#ag_/d4 ,,ii4 ,to find the nature or character of the roots4 ,as before1 r#a -b_/#b;a b#b - #d;ac7#a_/b7_/7b;a71 r#b -b_/#b;a - b#b - #d;ac7#a_/b7_/7b;a74 ,the b#b - #d;ac7#a_/b7 determines the nature or character of the roots2 hence it is called the discriminant4 ,if b#b - #d;ac is positive1 the roots are real1 unequal1 and either rational or irrational4 ,if b#b - #d;ac is negative1 the roots are imaginary and unequal4 ,if b#b - #d;ac is zero1 the roots are real1 equal1 and rational4 ,,examples3 #a4 x#b - #dx #b #j4 b#b - #d;ac7#a_/b7 #af - #h7a_/b7 #h7a_/b74 ,therefore3 ,the roots are real1 unequal1 and irrational4 #b4 x#b - #dx #f #j4 b#b - #d;ac7#a_/b7 #af - #bd7a_/b7 -#h7a_/b74 ,therefore3 ,the roots are imaginary and unequal4 #c4 x#b - #dx #d #j4 b#b - #d;ac7#a_/b7 #af - #af7a_/b7 #j7a_/b74 ,therefore3 ,the roots are real1 equal1 and rational4 ,,iii4 ,to form the quadratic equation when the roots are given4 ,suppose the roots are #c1 -#g4 ,then1 x #c1 ,or1 x - #c #j1 x -#g4 x #g #j4 ------------------- ,multiplying to get a quadratic1 7x - #c77x #g7 #j4 ,or1 x#b #dx - #ba #j4 ,or1 use the sum and product idea developed on the preceding page4 ,the coefficient of x#b must be unity4 ,add the roots and change the sign to get the coefficient of x4 ,multiply the roots to get the independent term4 ,therefore3 ,the equation is x#b #dx - #ba #j4 ,in the same way1 if the roots are #b #c7a_/b7_/g1 #b - #c7a_/b7_/g1 the equation is x#b - 7#d_/g7x #a_/di #j4 ,find the sum1 the product1 and the nature or character of the roots of the following3 #a4 x#b - #gx #ab #j4 #b4 #ix#b - #fx #a #j4 #c4 x#b #bx #igcd #j4 #d4 #af #e_/x #ag_/7x#b74 #e4 7x - #h7_/7x - #c7 x4 #f4 7x #g77x - #f7 #gj4 #g4 x#b - x7#b77a_/b7 #c4 #h4 pr#b qr s #j4 ,form the equations whose roots are3 #i4 #e1 -#c4 #aj4 #b_/c1 #e_/c4 #aa4 c d1 c - d4 #ab4 -#c1 -#e4 #ac4 #b - -#c7a_/b7_/e4 #ad4 #h_/c 7#b_/c7cg7a_/b71 #h_/c - 7#b_/c7cg7a_/b74 #ae4 -#b - -#b7a_/b7_/b4 #af4 ,solve x#b - #cx #d #j4 ,check by substituting the values of x2 then check by finding the sum and the product of the roots4 ,compare the amount of labor required in each case4 #ag4 ,solve 7x - #c77x #b77x#b #cx - #d7 #j4 #ah4 ,is e7#dz7 #b;e7#cz7 e7#bz7 #b;ez #b e7-#bz7 a perfect square8 #ai4 ,find the square root 7short method73 7x#b - #a77x#b - #cx #b77x#b - x - #b74 #bj4 ,solve 7#a4bx - #a4e7_/7a4e7 74#dx #a7_/74bx - 4#b7 74#dx #a7_/74e74 #ba4 ,the glass of a mirror is #ah inches by #ab inches1 and it has a frame of uniform width whose area is equal to that of the glass4 ,find the width of the frame4 ,,outline ,,of ,,simultaneous ,,quadratics ,simultaneous ,quadratics ,,case ,i4 ,one equation linear4 ,the other quadratic4 #bx y #g1 x#b #by#b #bb4 ,,method3 ,solve for x as in terms of y1 or vice versa1 in the linear and substitute in the quadratic4 ,,case ,,ii4 ,both equations homogeneous and of the second degree4 x#b - xy y#b #ci1 #bx#b - #cxy #by#b #dc4 ,,method3 ,let y vx1 and substitute in both equations4 ,,alternate ,,method3 ,solve for x in terms of y in one equation and substitute in the other4 ,,case ,,iii4 ,any two of the quantities x y x#b y#b xy x - y x#c y#c x#c - y#c x#b xy y#b x#b - xy y#b given4 x y #e1 x#b - xy y#b #g4 ,,method3 ,solve for x y and x - y2 then add to get x1 subtract to get y4 ,,case ,,iv4 ,both equations symmetrical or symmetrical except for sign4 ,usually one equation of high degree1 the other of the first degree4 x#e y#e #bdb1 x y #b4 ,,method3 ,let x u v and y u - v1 and substitute in both equations4 ,special ,devices ,i4 ,consider some compound quantity like xy1 x - y7#a_/b71 xy7#a_/b71 x_/y1 etc41 as the unknown1 at first4 ,solve for the compound unknown1 and combine the resulting equation with the simpler original equation4 x#b y#b xy #f1 x #by -#e4 ,,ii4 ,divide the equations member by member4 ,then solve by ,case ,i1 ,,ii1 or ,,iii4 x#c - y#c #aeb1 x - y #b4 ,,iii4 ,eliminate the quadratic terms4 ,then solve by ,case ,i1 ,,ii1 or ,,iii4 xy x #ae1 xy y #af4 ,,simultaneous ,,quadratics ,solve3 #a4 x y #g1 x#b #dxy #eg4 #b4 #bx#b #df y#b1 xy y#b #ad4 #c4 x#b y#b #be1 x y #a4 #d4 x#d y#d #b1 x - y #b4 #e4 x#c y#c #bh1 x y #d4 #f4 x#b y#b xy - #ab #j1 x y #d4 #g4 #bxy - x #by #af1 #cxy #bx - #dy #aj4 #h4 7#cx - #by77#bx - #cy7 #bf1 x #a #by4 #i4 #dx#b #cxy #by#b #ah1 #cx#b #bxy - y#b #c4 #aj4 x#e y#e #bdb1 x y #b4 #aa4 x - y x - y7#a_/b7 #f1 xy #e4 #ab4 #dx#b - x y #fg1 #cx#b - #cy #bg4 #ac4 x - y - x - y7#a_/b7 #b1 x#c - y#c #bjdd4 7,yale47 #ad4 x#b xy x #ad1 y#b xy y #bh4 7,princeton47 #ae4 x#b y#b #ac1 y#b #d7x - #b74 ,plot the graph of each equation4 7,cornell47 #af4 x#b y#b xy #cg1 x y xy - #ag4 7,columbia47 ,in grouping the answers1 be sure to associate each value of x with the corresponding value of y4 #ag4 ,the course of a yacht is #cj miles in length and is in the shape of a right triangle one arm of which is #b miles longer than the other4 ,what is the distance along each side8 ,reference3 ,the chapter on ,simultaneous ,quadratics in any algebra4 ,,ratio ,,and ,,proportion #a4 ,define ratio1 proportion1 mean proportional1 third proportional1 fourth proportional4 #b4 ,find a mean proportional between #d and #af2 #ah and #ej2 #abm#bn and #cmn#b4 #c4 ,find a third proportional to #d and #g2 #e and #aj2 a#b - #i and a - #c4 #d4 ,find a fourth proportional to #b1 #e1 and #d2 #ce1 #bj1 and #ad4 #e4 ,write out the proofs for the following1 stating the theorem in full in each case3 7a7 ,the product of the extremes equals etc4 7b7 ,if the product of two numbers equals the product of two other numbers1 either pair etc4 7c7 ,alternation4 7d7 ,inversion4 7e7 ,composition4 7f7 ,division4 7g7 ,composition and division4 7h7 ,in a series of equal ratios1 the sum of the antecedents is to the sum of the consequents etc4 7i7 ,like powers or like roots of the terms of a proportion etc4 #f4 ,if x 3 m 33 #ac 3 #g1 write all the possible proportions that can be derived from it4 ,see 7#e7 above4 #g4 ,given rs #afam2 write the eight proportions that may be derived from it1 and quote your authority4 #h4 7a7 ,what theorem allows you to change any proportion into an equation8 7b7 ,what theorem allows you to change any equation into a proportion8 #i4 ,if xy rg1 what is the ratio of x to g8 of y to r8 of y to g8 #aj4 ,find two numbers such that their sum1 difference1 and the sum of their squares are in the ratio #e 3 #c 3 #ea4 7,yale47 ,reference3 ,the chapter on ,ratio and ,proportion in any algebra4 ,an easy and powerful method of proving four expressions in proportion is illustrated by the following example3 ,given a 3 b c 3 d2 prove that #c;a#c #e;ab#b 3 #c;a#c - #e;ab#b #c;c#c #e;cd#b 3 #c;c#c - #e;cd#b4 ,let a_/b r4 ,therefore a br4 ,also c_/d r4 ,therefore c dr4 ,substitute the value of a in the first ratio1 and c in the second3 ,then 7#c;a#c #e;ab#b7_/7c;a#c - #e;ab#b7 7#c;b#cr#c #e;b#cr7_/7#c;b#cr#c - #e;b#cr7 b#cr7#cr#b #e7_/;b#cr7#cr#b - #e7 7#cr#b #e7_/7cr#b - #e74 ,also 7#c;c#c #e;cd#b7_/7c;c#c - #e;cd#b7 7#c;d#cr#c #e;d#cr7_/7#c;d#cr#c - #e;d#cr7 d#cr7#cr#b #e7_/;d#cr7#cr#b - #e7 7#cr#b #e7_/7cr#b - #e74 ,therefore 7#c;a#c #e;ab#b7_/7c;a#c - #e;ab#b7 7#c;c#c #e;cd#b7_/7c;c#c - #e;cd#b74 ,axiom #a4 ,or1 #c;a#c #e;ab#b 3 #c;a#c - #e;ab#b #c;c#c #e;cd#b 3 #c;c#c - #e;cd#b4 ,if a 3 b c 3 d1 prove3 #a4 a#b b#b 3 a#b c#b d#b 3 c#b4 #b4 a#b #c;b#b 3 a#b - #c;b#b c#b #c;d#b 3 c#b - #c;d#b4 #c4 a#b #b;b#b 3 #b;b#b ac #b;bd 3 #b;bd4 #d4 #b;a #c;c 3 #b;a - #c;c #h;b #ab;d 3 #h;b - #ab;d4 #e4 a#b - ab b#b 3 7a#c - b#c7_/;a c#b - cd d#b 3 7c#c - d#c7_/;c4 #f4 ,the second of three numbers is a mean proportional between the other two4 ,the third number exceeds the sum of the other two by #bj2 and the sum of the first and third exceeds three times the second by #d4 ,find the numbers4 #g4 ,three numbers are proportional to #e1 #g1 and #i2 and their sum is #ad4 ,find the numbers4 7,college ,entrance ,board47 #h4 ,a triangular field has the sides #ae1 #ah1 and #bg rods1 respectively4 ,find the dimensions of a similar field having #d times the area4 ,,arithmetical ,,progression #a4 ,define an arithmetical progression4 ,learn to derive the three formulas in arithmetical progression3 l a 7n - #a7;d1 ,s 7n_/#b77;a l71 ,s 7n_/#b7b;a 7n - #a7;d4 #b4 ,find the sum of the first #ej odd numbers4 #c4 ,in the series #b1 #e1 #h1 4441 which term is #ib8 #d4 ,how many terms must be taken from the series #c1 #e1 #g1 4441 to make a total of #bee8 #e4 ,insert #e arithmetical means between #aa and #cb4 #f4 ,insert #i arithmetical means between #g-a_/b and #cj4 #g4 ,find x1 if #c #bx1 #e #fx1 #i #ex are in ,a4 ,p4 #h4 ,the #gth term of an arithmetical progression is #ag1 and the #acth term is #ei4 ,find the #dth term4 #i4 ,how can you turn an ,a4 ,p4 into an equation8 #aj4 ,given a -#e_/c1 n #bj1 ,s -#e_/c1 find d and l4 #aa4 ,find the sum of the first n odd numbers4 #ab4 ,an arithmetical progression consists of #ba terms4 ,the sum of the three terms in the middle is #abi2 the sum of the last three terms is #bcg4 ,find the series4 7,look up the short method for such problems47 7,mass4 ,inst4 of ,technology47 #ac4 ,b travels #c miles the first day1 #g miles the second day1 #aa miles the third day1 etc4 ,in how many days will ,b overtake ,a who started from the same point #h days in advance and who travels uniformly #ae miles a day8 ,reference3 ,the chapter on ,arithmetical ,progression in any algebra4 ,,geometrical ,,progression #a4 ,define a geometrical progression4 ,learn to derive the four formulas in geometrical progression3 ,i4 l ar7n - #a74 ,,ii4 ,s 7arn - a7_/7r - #a74 ,,iii4 ,s 7rl - a7_/7r - #a74 ,,iv4 ,sinfinity 7a7_/7#a - r74 #b4 ,how many terms must be taken from the series #i1 #ah1 #cf1 444 to make a total of #efg8 #c4 ,in the ,g4 ,p4 #b1 #f1 #ah1 4441 which term is #dhf8 #d4 ,find x1 if #bx - #d1 #ex - #g1 #ajx #d are in geometrical progression4 #e4 ,how can you turn a ,g4 ,p4 into an equation8 #f4 ,insert #d geometrical means between #d and #igb4 #g4 ,insert #f geometrical means between #e_/af and #eabj4 #h4 ,given a -#b1 n #e1 l -#cb2 find r and ,s4 #i4 ,if the first term of a geometrical progression is #ab and the sum to infinity is #cf1 find the #dth term4 #aj4 ,if the series #c-a_/c1 #b-a_/b1 444 be an ,a4 ,p41 find the #igth term4 ,if a ,g4 ,p41 find the sum to infinity4 #aa4 ,the third term of a geometrical progression is #cf2 the #fth term is #igb4 ,find the first and second terms4 #ab4 ,insert between #f and #af two numbers1 such that the first three of the four shall be in arithmetical progression1 and the last three in geometrical progression4 #ac4 ,a rubber ball falls from a height of #dj inches and on each rebound rises #dj of the previous height4 ,find by formula how far it falls on its eighth descent4 7,yale47 ,reference3 ,the chapter on ,geometrical ,progression in any algebra4 ,,the ,,binomial ,,theorem #a4 ,review the ,binomial ,theorem laws4 7,see ,involution47 ,expand3 #b4 7b - n7#g4 #c4 7x x7-#a77e4 #d4 a_/x - x_/a#f4 #e4 x_/#by - xy7#a_/b7e4 #f4 7x#b - x #b7c4 #g4 7#b;b#b7a_/c77_/7y7 7#cy7#a_/b77_/7;b#c7d4 #h4 7a b7n an na7n - #a7;b n7n - #a7_/7a4b7 a7n - #b7;b#b n7n - #a77n - #b7_/7a4b4c7 a7n - #c7;b#c n7n - #a77n - #b77n - #c7_/7a4b4c4d7 a7n - #d7 b#d 4444 ,show by observation that the formula for the 7r #a7th term n7n - #a77n - #b74447n - r #a7_/a4b4c4d 444 r a7n - r7br4 #i4 ,indicate what the #igth term of 7a b7n would be4 #aj4 ,using the expansion of 7a b7n in 7#h71 derive a formula for the rth term by observing how each term is made up1 then generalizing4 ,using either the formula in 7#h7 or 7#aj71 whichever you are familiar with1 find3 #aa4 ,the #dth term of a #a_/;a7#cj74 #ab4 ,the #hth term of 7#a xy7#a_/b777ac74 #ac4 ,the middle term of 7#b;a7#c_/d7 - ya7#a_/c777aj74 #ad4 ,the term not containing x in x#c - #b_/x7#ab74 #ae4 ,the term containing x7#ah7 in x#b - a_/x7#ae74 ,reference3 ,the chapter on ,the ,binomial ,theorem in any algebra4 ,,miscellaneous ,,examples1 ,,quadratics ,,and ,,beyond #a4 ,solve the equation x#b - #a4fx - 4#bc #j1 obtaining the values of the roots correct to three significant figures4 7,harvard47 #b4 ,write the roots of 7x#b #bx77x#b - #bx - #c77x#b - x #a7 #j4 7,sheffield ,scientific ,school47 #c4 ,solve #bbx #b7a_/b7 #bx #a7a_/b7 7#abx #d7_/7hx #ha_/b74 7,yale47 #d4 ,solve the equation ,v 7,h_/#c77;,b x ,bx7#a_/b77 for x1 taking ,h #f1 ,b #h1 and ,v #bh2 and verify your result4 7,harvard47 #e4 ,solve x 3 y #b 3 #c1 x#b y#b #e7x y7 #b4 #f4 ,solve #bx#b - #dx #cx#b - #bx #f7a_/b7 #ae4 7,coll4 ,ent4 ,board47 #g4 ,find all values of x and y which satisfy the equations3 x7#a_/b7 y7#a_/b7 #d1 #a_/x #a7a_/b7 - x7#a_/b7 - #a_/x #a7a_/b7 x7#a_/b7 y4 7,mass4 ,inst4 of ,technology47 #h4 ,if alpha and beta represent the roots of px#b qx r #j1 find alpha beta1 alpha - beta1 and alphabeta in terms of p1 q1 and r4 7,princeton47 #i4 ,form the equation whose roots are #b #c7a_/b7 and #b - -#c7a_/b74 #aj4 ,determine1 without solving1 the character of the roots of #ix#b - #bdx #af #j4 7,college ,entrance ,board47 #aa4 ,if a 3 b c 3 d1 prove that a b 3 c d a#b b#b7a_/b7 3 c#b d#b7a_/b74 7,college ,entrance ,board47 #ab4 ,given a 3 b c 3 d4 ,prove that a#b b#b 3 7a#c7_/7;a b7 c#b d#b 3 7c#c7_/7;c d74 7,sheffield47 #ac4 ,the #ith term of an arithmetical progression is #a_/f2 the #afth term is #e_/b4 ,find the first term4 7,regents47 ,solve graphically3 #a4 x#b - x - #f #j4 #b4 x#b #cx - #aj #j4 #c4 ,find four numbers in arithmetical progression1 such that the sum of the first two is #a1 and the sum of the last two is -#ai4 #d4 ,what number added to #b1 #bj1 #i1 #cd1 will make the results proportional8 #e4 ,find the middle term of #c;a#e 7b7#c_/d77_/7b7h4 #f4 ,solve 7x #a7_/7cx #b7 7#bx - #c7_/7cx - #b7 - #a - #cf_/7d - #ix#b74 7,princeton47 #g4 ,a strip of carpet one half inch thick and #bi-f_/g feet long is rolled on a roller four inches in diameter4 ,find how many turns there will be1 remembering that each turn increases the diameter by one inch1 and that the circumference of a circle equals 7approximately7 #bb_/g times the diameter4 7,harvard47 #h4 ,the sum of the first three terms of a geometrical progression is #ba1 and the sum of their squares is #ahi4 ,what is the first term8 7,yale47 #i4 ,find the geometrical progression whose sum to infinity is #d1 and whose second term is #c_/d4 #aj4 ,solve #dx #dcx#b - #gx #c7a_/b7 #cx#b - #cx #f4 #aa4 ,solve #bx#b #cxy - #ey#b #d1 #bxy #cy#b -#c4 #ab4 ,two hundred stones are placed on the ground #c feet apart1 the first being #c feet from a basket4 ,if the basket and all the stones are in a straight line1 how far does a person travel who starts from the basket and brings the stones to it one by one8 ,solve graphically2 and check by solving algebraically3 #a4 x#b y#b #be1 x y #a4 #b4 x#b - #cx - #ah #j4 #c4 x#b #cx - #aj #j4 ,determine the value of m for which the roots of the equation will be equal3 7,,hint3 ,see page #dj4 ,to have the roots equal1 b#b - #d;ac must equal #j47 #d4 #bx#b - mx #ab-a_/b #j4 #e4 7m - #a7x#b mx #bm - #c #j4 #f4 ,if #b;a #c;b is a root of x#b - #f;bx - #d;a#b #i;b#b #j1 find the other root without solving the equation4 7,univ4 of ,penn47 #g4 ,how many times does a common clock strike in #ab hours8 #h4 ,find the sum to infinity of #b_/7b7a_/b771 #a_/7b7a_/b771 #a_/7bb7a_/b771 4444 #i4 ,solve x_/#b #f_/x#b - #fx_/#b #f_/x #h #j4 #aj4 ,find the value of the recurring decimal #b4badbad4444 #aa4 ,a man purchases a #ejj piano by paying monthly installments of #aj and interest on the debt4 ,if the yearly rate is #f1 what is the total amount of interest8 #ab4 ,the arithmetical mean between two numbers is #db-a_/b1 and their geometrical mean is #db4 ,find the numbers4 7,college ,entrance ,exam4 ,board47 #ac4 ,if the middle term of #cx - 7#a7_/7bx7#a_/b77d is equal to the fourth term of #bx7#a_/b7 #a_/bx#g1 find the value of x4 7,m4 ,i4 ,t47 ,,problems ,linear ,equations1 ,one ,unknown #a4 ,a train running #cj miles an hour requires #ba minutes longer to go a certain distance than does a train running #cf miles an hour4 ,how great is the distance8 7,cornell47 #b4 ,a man can walk #b-a_/b miles an hour up hill and #c-a_/b miles an hour down hill4 ,he walks #ef miles in #bj hours on a road no part of which is level4 ,how much of it is up hill8 7,yale47 #c4 ,a physician having #ajj cubic centimeters of a #f solution of a certain medicine wishes to dilute it to a #c-a_/b solution4 ,how much water must he add8 7,a #f solution contains #f of medicine and #id of water47 7,case47 #d4 ,a clerk earned #ejd in a certain number of months4 ,his salary was increased #be1 and he then earned #dej in two months less time than it had previously taken him to earn #ejd4 ,what was his original salary per month8 7,college ,entrance ,board47 #e4 ,a person who possesses #ae1jjj employs a part of the money in building a house4 ,he invests one third of the money which remains at #f1 and the other two thirds at #i1 and from these investments he obtains an annual income of #ejj4 ,what was the cost of the house8 7,m4 ,i4 ,t47 #f4 ,two travelers have together #djj pounds of baggage4 ,one pays #a4bj and the other #a4hj for excess above the weight carried free4 ,if all had belonged to one person1 he would have had to pay #d4ej4 ,how much baggage is allowed to go free8 7,yale47 #g4 ,a man who can row #d-a_/c miles an hour in still water rows downstream and returns4 ,the rate of the current is #b-a_/d miles per hour1 and the time required for the trip is #ac hours4 ,how many hours does he require to return8 ,simultaneous ,equations1 ,two and ,three ,unknowns #a4 ,a manual training student in making a bookcase finds that the distance from the top of the lowest shelf to the under side of the top shelf is #d ft4 #f in4 ,he desires to put between these four other shelves of inch boards in such a way that the book space will diminish one inch for each shelf from the bottom to the top4 ,what will be the several spaces between the shelves8 #b4 ,a quantity of water1 sufficient to fill three jars of different sizes1 will fill the smallest jar #d times1 or the largest jar twice with #d gallons to spare1 or the second jar three times with #b gallons to spare4 ,what is the capacity of each jar8 7,case47 #c4 ,a policeman is chasing a pickpocket4 ,when the policeman is #hj yards behind him1 the pickpocket turns up an alley2 but coming to the end1 he finds there is no outlet1 turns back1 and is caught just as he comes out of the alley4 ,if he had discovered that the alley had no outlet when he had run halfway up and had then turned back1 the policeman would have had to pursue the thief #abj yards beyond the alley before catching him4 ,how long is the alley8 7,harvard47 #d4 ,a and ,b together can do a piece of work in #ad days4 ,after they have worked #f days on it1 they are joined by ,c who works twice as fast as ,a4 ,the three finish the work in #d days4 ,how long would it take each man alone to do it8 7,columbia47 #e4 ,in a certain mill some of the workmen receive #a4ej a day1 others more4 ,the total paid in wages each day is #cej4 ,an assessment made by a labor union to raise #bjj requires #a4jj from each man receiving #a4ej a day1 and half of one day's pay from every man receiving more4 ,how many men receive #a4ej a day8 7,harvard47 #f4 ,there are two alloys of silver and copper1 of which one contains twice as much copper as silver1 and the other three times as much silver as copper4 ,how much must be taken from each to obtain a kilogram of an alloy to contain equal quantities of silver and copper8 7,m4 ,i4 ,t47 #g4 ,two automobiles travel toward each other over a distance of #abj miles4 ,a leaves at #i ,a4,m41 #a hour before ,b starts to meet him1 and they meet at #ab3jj ,m4 ,if each had started at #i3ae ,a4,m41 they would have met at #ab3jj ,m4 also4 ,find the rate at which each traveled4 7,m4 ,i4 ,t47 ,quadratic ,equations #a4 ,telegraph poles are set at equal distances apart4 ,in order to have two less to the mile1 it will be necessary to set them #bj feet farther apart4 ,find how far apart they are now4 7,yale47 #b4 ,the distance ,s that a body falls from rest in t seconds is given by the formula ,s #aft#b4 ,a man drops a stone into a well and hears the splash after #c seconds4 ,if the velocity of sound in air is #ajhf feet a second1 what is the depth of the well8 7,yale47 #c4 ,it requires #bjjj square tiles of a certain size to pave a hall1 or #cabe square tiles whose dimensions are one inch less4 ,find the area of the hall4 ,how many solutions has the equation of this problem8 ,how many has the problem itself8 ,explain the apparent discrepancy4 7,cornell47 #d4 ,a rectangular tract of land1 #hjj feet long by #fjj feet broad1 is divided into four rectangular blocks by two streets of equal width running through it at right angles4 ,find the width of the streets1 if together they cover an area of #gg1ejj square feet4 7,m4 ,i4 ,t47 #e4 7a7 ,the height y to which a ball thrown vertically upward with a velocity of #ajj feet per second rises in x seconds is given by the formula1 y #ajjx - #afx#b4 ,in how many seconds will the ball rise to a height of #add feet8 7b7 ,draw the graph of the equation y #ajjx - #afx#b4 7,college ,entrance ,board47 #f4 ,two launches race over a course of #ab miles4 ,the first steams #g-a_/b miles an hour4 ,the other has a start of #aj minutes1 runs over the first half of the course with a certain speed1 but increases its speed over the second half of the course by #b miles per hour1 winning the race by a minute4 ,what is the speed of the second launch8 ,explain the meaning of the negative answer4 7,sheffield ,scientific ,school47 #g4 ,the circumference of a rear wheel of a certain wagon is #c feet more than the circumference of a front wheel4 ,the rear wheel performs #ajj fewer revolutions than the front wheel in traveling a distance of #fjjj feet4 ,how large are the wheels8 7,harvard47 #h4 ,a man starts from home to catch a train1 walking at the rate of #a yard in #a second1 and arrives #b minutes late4 ,if he had walked at the rate of #d yards in #c seconds1 he would have arrived #b-a_/b minutes early4 ,find the distance from his home to the station4 7,college ,entrance ,board47 ,simultaneous ,quadratics #a4 ,two cubical coal bins together hold #bhj cubic feet of coal1 and the sum of their lengths is #aj feet4 ,find the length of each bin4 #b4 ,the sum of the radii of two circles is #be inches1 and the difference of their areas is #abepi square inches4 ,find the radii4 #c4 ,the area of a right triangle is #aej square feet1 and its hypotenuse is #be feet4 ,find the arms of the triangle4 #d4 ,the combined capacity of two cubical tanks is #fcg cubic feet1 and the sum of an edge of one and an edge of the other is #ac feet4 7a7 ,find the length of a diagonal of any face of each cube4 7b7 ,find the distance from upper left-hand corner to lower right-hand corner in either cube4 #e4 ,a and ,b run a mile4 ,in the first heat ,a gives ,b a start of #bj yards and beats him by #cj seconds4 ,in the second heat ,a gives ,b a start of #cb seconds and beats him by #i-e_/aa yards4 ,find the rate at which each runs4 7,sheffield47 #f4 ,after street improvement it is found that a certain corner rectangular lot has lost #a_/aj of its length and #a_/ae of its width4 ,its perimeter has been decreased by #bh feet1 and the new area is #cjbd square feet4 ,find the reduced dimensions of the lot4 7,college ,entrance ,board47 #g4 ,a man spends #eci for sheep4 ,he keeps #ad of the flock that he buys1 and sells the remainder at an advance of #b per head1 gaining #bh by the transaction4 ,how many sheep did he buy1 and what was the cost of each8 7,yale47 #h4 ,a boat's crew1 rowing at half their usual speed1 row #c miles downstream and back again in #b hours and #dj minutes4 ,at full speed they can go over the same course in #a hour and #d minutes4 ,find the rate of the crew1 and the rate of the current in miles per hour4 7,college ,entrance ,board47 #i4 ,find the sides of a rectangle whose area is unchanged if its length is increased by #d feet and its breadth decreased by #c feet1 but which loses one third of its area if the length is increased by #af feet and the breadth decreased by #aj feet4 7,m4 ,i4 ,t47 ,,college ,,entrance ,,examinations ,,university ,,of ,,california ,,elementary ,,algebra #a4 ,if a #d1 b -#c1 c #b1 and d -#d1 find the value of3 7a7 ab#c - #c;cd#b #b7c;a - b77c - #b;d74 7b7 #b;a#c - #c;b#d 7#d;c#c d#c77d;c#b d#b74 #b4 ,reduce to a mixed number3 7#c;a#d - #d;a#c - #aj;a#b #da;a - #bh7_/7;a#b - #c;a #d74 ,simplify3 #c4 7a #b7_/7;a#b #c;a - #dj7 - 7b - #b7_/7;ab - #e;b #c;a - #ae74 #d4 #a - 7#b - #c;b - #b;c7_/7a #b7 _/ 7a#b - #d;c#b #i;b#b #f;ab7_/7#b;a#b a - #f74 #e4 ,a's age #aj years hence will be #d times what ,b's age was #aa years ago1 and the amount that ,a's age exceeds ,b's age is one third of the sum of their ages #h years ago4 ,find their present ages4 #f4 ,draw the lines represented by the equations #cx - #by #ac and #bx #ey -#d1 and find by algebra the cooerdinates of the point where they intersect4 #g4 ,solve the equations bx - ay b#b - ab1 y - b #b7x - #b;a74 #h4 ,solve 7#bx #a77cx - #b7 - 7#ex - #g77x - #b7 #da4 ,,colorado ,,school ,,of ,,mines ,,elementary ,,algebra #a4 ,solve by factoring3 x#c #cjx #aax#b4 #b4 ,show that #a - 7a#b b#b - c#b7_/7b;ab7#b 7a b c77a b - c77a - b c77b c - a7 _/ #d;a#b;b#b4 #c4 ,how many pairs of numbers will satisfy simultaneously the two equations #cx #by #g1 x y #c8 ,show by means of a graph that your answer is correct4 ,what is meant by eliminating x in the above equations by substitution8 by comparison8 by subtraction8 #d4 ,find the square root of #bbc1gbh4 #e4 ,simplify3 7a7 #a_/c7a_/b7 #ab7a_/b7 - #c_/d7a_/b74 7b7 7--#c-d7a_/b77a_/b77d4 #f4 ,solve the equation 4#jcx#b - #b4bcx #a4ajge #j4 #g4 ,how far must a boy run in a potato race if there are n potatoes in a straight line at a distance d feet apart1 the first being at a distance a feet from the basket8 ,,columbia ,,university ,,elementary ,,algebra ,,complete ,,time3 ,,three ,,hours ,six questions are required2 two from ,group ,a1 two from ,group ,b1 and both questions of ,group ,c4 ,no extra credit will be given for more than six questions4 ,group ,a #a4 7a7 ,resolve the following into their prime factors3 7#a7 7x#b - y#b7b - y#d4 7#b7 #ajx#b - #gx - #f4 7b7 ,find the ,h4 ,c4 ,f4 and the ,l4 ,c4 ,m4 of x#c - #cx#b x - #c1 x#c - #cx#b - x #c4 #b4 7a7 ,simplify x_/y y_/x - #b_/a_/x #a_/y x_/y y_/x #b_/a_/x - #a_/y4 7b7 ,if x 3 y 7x - z7#b 3 7y - z7#b1 prove that z is a mean proportional between x and y4 #c4 ,a crew can row #aj miles in #ej minutes downstream1 and #ab miles in an hour and a half upstream4 ,find the rate of the current and of the crew in still water4 ,group ,b #d4 7a7 ,determine the values of k so that the equation 7#b k7x#b #bkx #a #j shall have equal roots4 7b7 ,solve the equations x#b - xy y#b #g1 #bx - #cy #j4 7c7 ,plot the following two equations1 and find from the graphs the approximate values of their common solutions3 x#b y#b #be1 #dx#b #iy#b #add4 #e4 ,two integers are in the ratio #d 3 #e4 ,increase each by #ae1 and the difference of their squares is #iii4 ,what are the integers8 #f4 ,a man has #eci to spend for sheep4 ,he wishes to keep #ad of the flock that he buys1 but to sell the remainder at a gain of #b per head4 ,this he does and gains #bh4 ,how many sheep did he buy1 and at what price each8 ,group ,c #g4 7a7 ,find the seventh term of a #a_/;a7#ac74 7b7 ,derive the formula for the sum of n terms of an arithmetic progression4 #h4 ,a ball falling from a height of #fj feet rebounds after each fall one third of its last descent4 ,what distance has it passed over when it strikes the ground for the eighth time8 ,,cornell ,,university ,,elementary ,,algebra #a4 ,find the ,h4 ,c4 ,f43 x#d - y#d1 x#c - xy#b x#by - y#c1 x#d #bx#by#b - #cy#d4 #b4 ,solve the following set of equations3 x y -#a1 x #cy #bz -#d1 x - y #dz #e4 #c4 ,expand and simplify3 #bx#c - #a_/x#g4 #d4 ,an automobile goes #hj miles and back in #i hours4 ,the rate of speed returning was #d miles per hour faster than the rate going4 ,find the rate each way4 #e4 ,simplify3 7x #a7_/7x - #a7b - #b 7x - #a7_/7x #a7b _/7x #a7_/7x - #a7b - 7x - #a7_/7x #a7b4 #f4 ,solve for x3 7#bx #c7_/7x - #a7 - #f #e_/7x#b #bx - #c74 #g4 ,a1 ,b1 and ,c1 all working together1 can do a piece of work in #b-b_/c days4 ,a works twice as fast as ,c1 and ,a and ,c together could do the work in #d days4 ,how long would it take each one of the three to do the work alone8 ,,cornell ,,university ,,intermediate ,,algebra #a4 ,solve the following set of equations3 x y -#a1 #bz #ew #a1 x #cy #bz -#d1 x - y #dz #dw #e4 #b4 ,simplify3 7a7 #f - #bj7a_/b77a_/b74 7b7 #a x#b #a7a_/b7_/a x#b #a7a_/b7 x#b4 #c4 ,find1 and simplify1 the #bc;d term in the expansion of 7#bx#b7_/7c7 - #c_/d7bh74 #d4 ,the weight of an object varies directly as its distance from the center of the earth when it is below the earth's surface1 and inversely as the square of its distance from the center when it is above the surface4 ,if an object weighs #aj pounds at the surface1 how far above1 and how far below the surface will it weigh #i pounds8 7,the radius of the earth may be taken as #djjj miles47 #e4 ,solve the following pair of equations for x and y3 x#b y#b #d1 x 7#a #b7a_/b77y - #b4 #f4 ,find the value of #a #h7-x_/#c7_/7hx77#a_/b7 #aj7x - #b71 when x #b4 #g4 ,from a square of pasteboard1 #ab inches on a side1 square corners are cut1 and the sides are turned up to form a rectangular box4 ,if the squares cut out from the corners had been #a inch larger on a side1 the volume of the box would have been increased #bh cubic inches4 ,what is the size of the square corners cut out8 7,see the figure on the blackboard47 ,,harvard ,,university ,,elementary ,,algebra ,,time3 ,,one ,,hour ,,and ,a ,,half ,arrange your work neatly and clearly1 beginning each question on a separate page4 #a4 ,simplify the following expression3 #a_/;a #a_/7;b c7_/#a_/;a - #a_/7;b c7 #a 7b#b c#b - a#b7_/7b;bc74 #b4 7a7 ,write the middle term of the expansion of 7a - b7#ad by the binomial theorem4 7b7 ,find the value of a#g;b#g1 if a x7#b_/g7y7-#c_/b7 and b 7#a_/b7 x7-#a_/g7y7#a_/b71 and reduce the result to a form having only positive exponents4 #c4 ,find correct to three significant figures the negative root of the equation #a - #b_/7x #a7 #dx_/7x #a7b #j4 #d4 ,prove the rule for finding the sum of n terms of a geometrical progression of which the first term is a and the constant ratio is r4 ,find the sum of #h terms of the progression #e #c-a_/c #b-b_/i 4444 #e4 ,a goldsmith has two alloys of gold1 the first being #c_/d pure gold1 the second #e_/ab pure gold4 ,how much of each must he take to produce #ajj ounces of an alloy which shall be #b_/c pure gold8 ,,harvard ,,university ,,elementary ,,algebra ,,time3 ,,one ,,hour ,,and ,a ,,half #a4 ,solve the simultaneous equations x y a b1 7y b7_/7x a7 a_/b1 and verify your results4 #b4 ,solve the equation x#b - #a4fx - #j4bc #j1 obtaining the values of the roots correct to three significant figures4 #c4 ,write out the first four terms of 7a - b7#g4 ,find the fourth term of this expansion when a x7-#a7 y7#a_/b77a_/c71 b #ixy7-#d77a_/f71 expressing the result in terms of a single radical1 and without fractional or negative exponents4 #d4 ,reduce the following expression to a polynomial in a and b3 7#f;a#c #g;ab#b #ab;b#c7_/7c;a#b - #e;ab - #d;b#b7 - #a_/c_/ai;b - 7#e;a #d;b7_/7#ai;a#b74 #e4 ,the cost of publishing a book consists of two main items3 first1 the fixed expense of setting up the type2 and1 second1 the running expenses of presswork1 binding1 etc41 which may be assumed to be proportional to the number of copies4 ,a certain book costs #ce cents a copy if #ajjj copies are published at one time1 but only #ai cents a copy if #ejjj copies are published at one time4 ,find 7a7 the cost of setting up the type for the book1 and 7b7 the cost of presswork1 binding1 etc41 per thousand copies4 ,,harvard ,,university ,,elementary ,,algebra ,,time3 ,,one ,,hour ,,and ,a ,,half #a4 ,find the highest common factor and the lowest common multiple of the three expressions a#d - b#d2 a#c b#c2 a#c #b;a#b b #b;ab#b b#c4 #b4 ,solve the quadratic equation x#b - #a4fx #j4c #j1 computing the value of the larger root correct to three significant figures4 #c4 ,in the expression x#b - #bxy y#b - #db7a_/b77x y7 #h1 substitute for x and y the values x 7u v #a7_/b7a_/b71 y 7u - v #a7_/b7a_/b71 and reduce the resulting expression to its simplest form4 #d4 ,state and prove the formula for the sum of the first n terms of a geometric progression in which a is the first term and r the constant ratio4 #e4 ,a state legislature is to elect a ,united ,states senator1 a majority of all the votes cast being necessary for a choice4 ,there are three candidates1 ,a1 ,b1 and ,c1 and #ajj members vote4 ,on the first ballot ,a has the largest number of votes1 receiving #i more votes than his nearest competitor1 ,b2 but he fails of the necessary majority4 ,on the second ballot ,c's name is withdrawn1 and all the members who voted for ,c now vote for ,b1 whereupon ,b is elected by a majority of #b4 ,how many votes were cast for each candidate on the first ballot8 ,,massachusetts ,,institute ,,of ,,technology ,,algebra ,a ,,time3 ,,one ,,hour ,,and ,,three ,,quarters #a4 ,factor the expressions3 x#c x#b #bx4 x#c x#b - #dx - #d4 #b4 ,simplify the expression3 #a - 7b#b7_/7;a#b7a - 7ab - b#b7_/7;a#b77;a#d7_/7;a#c b#c7 4 7a - b7_/7a#b b#b74 #c4 ,find the value of x #a x#b7a_/b71 when x 7#a_/b7;a_/b7#a_/b7 - b_/a7#a_/b74 #d4 ,solve the equations3 7#gx #f7_/aa y - #af 7#ex - #ac7_/b - 7#hy - x7_/#e1 #c7cx #d7 #ajy - #ae4 #e4 ,solve the equations3 ,a ,c #b1 -,a ,b ,c ,d #a1 #b;,a - ,b #b;,c ,d #e1 ,b ,d #a4 #f4 ,two squares are formed with a combined perimeter of #af inches4 ,one square contains #d square inches more than the other4 ,find the area of each4 #g4 ,a man walked to a railway station at the rate of #d miles an hour and traveled by train at the rate of #cj miles an hour1 reaching his destination in #bj hours4 ,if he had walked #c miles an hour and ridden #ce miles an hour1 he would have made the journey in #ah hours4 ,required the total distance traveled4 ,,massachusetts ,,institute ,,of ,,technology ,,algebra ,b ,,time3 ,,one ,,hour ,,and ,,three ,,quarters #a4 ,how many terms must be taken in the series #b1 #e1 #h1 #aa1 444 so that the sum shall be #cde8 #b4 ,prove the formula x -b - b#b - #d;ac7#a_/b7_/7b;a7 for solving the quadratic equation ax#b bx c #j4 #c4 ,find all values of a for which sqa is a root of x#b x #bj #b;a1 and check your results4 #d4 ,solve x#b #cy#b #aj1 x - y #b1 and sketch the graphs4 #e4 ,the sum of two numbers x and y is #e1 and the sum of the two middle terms in the expansion of 7x y7#c is equal to the sum of the first and last terms4 ,find the numbers4 #f4 ,solve x#d - #bx#c #cx#b - #bx #a #j4 7,,hint3 ,divide by x#b and substitute x #a_/x z47 #g4 ,in anticipation of a holiday a merchant makes an outlay of #ej1 which will be a total loss in case of rain1 but which will bring him a clear profit of #aej above the outlay if the day is pleasant4 ,to insure against loss he takes out an insurance policy against rain for a certain sum of money for which he has to pay a certain percentage4 ,he then finds that whether the day be rainy or pleasant he will make #hj clear4 ,what is the amount of the policy1 and what rate did the company charge him8 ,,massachusetts ,,institute ,,of ,,technology ,,algebra ,a ,,time3 ,,two ,,hours #a4 ,simplify m #a_/m#b n #a_/n#b mn #a_/mn#b - m #a_/mn #a_/nmn #a_/mn4 #b4 ,find the prime factors of 7a7 7x - x#b7c 7x#b - #a7c 7#a - x7#c4 7b7 7#bx a - b7#d - 7x - a b7#d4 #c4 7a7 ,simplify 7xq7_/7xr77q r7 7xr7_/7xp77r p7xp_/xq7p q74 7b7 ,show that 7x#a_/7n#a77a_/n77_/7x#a_/7n#b7a_/7n#a77 x7#a_/n7 4 x#a_/7n#b7_/x#ba_/7n#a74 #d4 ,define homogeneous terms4 ,for what value of n is xn y7#e - n_/#b7 x7n #a7 y7#bn - #f7 a homogeneous binomial8 #e4 ,extract the square root of x7x - #b7a_/b777x - #h7a_/b777x - #ah7a_/b77 #d4 #f4 ,two vessels contain each a mixture of wine and water4 ,in the first vessel the quantity of wine is to the quantity of water as #a 3 #c1 and in the second as #c 3 #e4 ,what quantity must be taken from each1 so as to form a third mixture which shall contain #e gallons of wine and #i gallons of water8 #g4 ,find a quantity such that by adding it to each of the quantities a1 b1 c1 d1 we obtain four quantities in proportion4 #h4 ,what values must be given to a and b1 so that 7#c;a #b;b #ag7_/b1 7#b;a - #c;b #be7_/c1 #d - #e;a - #ac;b may be equal8 ,,mount ,,holyoke ,,college ,,elementary ,,algebra ,,time3 ,,two ,,hours #a4 ,factor the following expressions3 7a7 a7#c_/d7 - b7#c_/d74 7b7 x#b y#b z#b - x#b z - y#b z #a4 7c7 #af7x y7#d - 7#bx - y7#d4 #b4 7a7 ,simplify 7a#b b#b77;b#d7_/7;b#b - a#b7 - a#b_/;a_/7a b7 b_/7a - b74 7b7 ,extract the square root of x#d - #bx#c #ex#b - #dx #d4 #c4 ,solve the following equations3 7a7 #a_/x #a_/y #e1 #a_/7x#b7 #a_/7y#b7 #ac4 7b7 x#b - #ex #b #j4 7c7 #bgx #a7a_/b7 #b - #ccx7#a_/b74 #d4 ,simplify3 7a7 #ged7a_/c7 #bef7a_/f7 #dcb_/7-bej77a_/c74 7b7 #a_/7;a - b77b - c7 #a_/7;c - a77b - a74 7c7 ,find #ai - #hc7a_/b77a_/b74 #e4 ,plot the graphs of the following system1 and determine the solution from the point of intersection3 x - #by #j1 #bx - #cy #d4 #f4 7a7 ,derive the formula for the solution of ax#b bx c #j4 7b7 ,determine the value of m for which the roots of #bx#b #dx m #j are 7i7 equal1 7ii7 real1 7iii7 imaginary4 7c7 ,form the quadratic equation whose roots are #b #c7a_/b7 and #b - #c7a_/b74 #g4 ,a page is to have a margin of #a inch1 and is to contain #ce square inches of printing4 ,how large must the page be1 if the length is to exceed the width by #b inches8 #h4 7a7 ,in an arithmetical progression the sum of the first six terms is #bfa1 and the sum of the first nine terms is #big4 ,find the common difference4 7b7 ,three numbers whose sum is #bg are in arithmetical progression4 ,if #a is added to the first1 #c to the second1 and #aa to the third1 the sums will be in geometrical progression4 ,find the numbers4 7c7 ,derive the formula for the sum of n terms of a geometrical progression4 #i4 7a7 ,expand and simplify 7#b;a#b - #cx#c7g4 7b7 ,for what value of x will the ratio #g x 3 #ab x be equal to the ratio #e 3 #f8 ,,university ,,of ,,pennsylvania ,,elementary ,,algebra ,,time3 ,,three ,,hours #a4 ,simplify3 7a x7_/7a - x7 - 7a - x7_/7a x7 _/ 7#d;ax7_/7a#b - x#b74 #b4 ,find the ,h4 ,c4 ,f4 and ,l4 ,c4 ,m4 of #aj;ab#b7x#b - #b;ax71 #ae;a#c;b7x#b - ax - #b;a#b71 #be;b#c7x#b - a#b7b4 #c4 ,a grocer buys eggs at #d for #g;c4 ,he sells #a_/d of them at #e for #ab;c1 and the rest at #f for #aa;c1 making #bg;c by the transaction4 ,how many eggs does he buy8 #d4 ,solve for t3 7t #d;a b7_/7t a b7 - 7#dt - a - #b;b7_/7t a - b7 -#c4 #e4 ,find the square root of a#b - 7#c_/b7;a7#c_/b7 - 7#c_/b7;a7#a_/b7 7#da_/af7;a #a4 #f4 7a7 ,for what values of m will the roots of #bx#b #cmx -#b be equal8 7b7 ,if #b;a #c;b is a root of x#b - #f;bx - #d;a#b #i;b#b #j1 find the other root without solving the equation4 #g4 7a7 ,solve for x3 #bx - #c;a7#a_/b7 #cx - #b;a7#a_/b7 #c;a7#a_/b74 7b7 ,solve for m3 #a - 7#a7_/7b - m7 #a_/7m #b7 7m - #f7_/7d - m#b74 #h4 ,solve the system3 x#b #by#b #ag2 xy - y#b #b4 #i4 ,two boats leave simultaneously opposite shores of a river #b-a_/d mi4 wide and pass each other in #ae min4 ,the faster boat completes the trip #f-c_/d min4 before the other reaches the opposite shore4 ,find the rates of the boats in miles per hour4 #aj4 ,write the sixth term of x_/7#by#b7a_/c77 - 7y7#a_/b77_/x#i without writing the preceding terms4 #aa4 ,the sum of the #b;d and #bjth terms of an ,a4 ,p4 is #aj1 and their product is #bc-dg_/fd4 ,what is the sum of sixteen terms8 ,,princeton ,,university ,,algebra ,a ,,time3 ,,two ,,hours ,candidates who are at this time taking both ,algebra ,a and ,algebra ,b may omit from ,algebra ,a questions #d1 #e1 and #f1 and from ,algebra ,b questions #a 7a71 #c1 and #d4 #a4 ,simplify 7a#c a#b;b ab#b7_/7;a#b - #c;ab - #d;b#b7 _/ 7a#b #f;ab - #g;b#b7_/7;a#b #h;ab - #i;b#b7 4 7a#c - b#c7_/7;a#b - #g;ab #ab;b#b74 #b4 7a7 ,divide a7#e_/b7 ab7#c_/b7 b7#e_/b7 - #b;a7#a_/b7;b#b - a7#c_/b7;b by a7#c_/b7 - b7#c_/b7 a7#a_/b7;b - ab7#a_/b74 7b7 ,simplify 7#a7_/7x7-#a7 y7-#a7 4 7x7#a_/d7y7#a_/b77c #a4 #c4 ,factor3 7a7 7x#b #cx7#b - 7#bx - #f7b4 7b7 a#b ac - #d;b#b - #b;bc4 #d4 ,solve #a_/7x #a7 - 7#a7_/7x - #a7 - 7#a7_/7x - #c7 7#a7_/7x - #e7 #j4 #e4 ,solve for x and y3 mx ax my - by1 x - y a b4 #f4 ,the road from ,a to ,b is uphill for #e mi41 level for #d mi41 and then downhill for #f mi4 ,a man walks from ,b to ,a in #d hr42 later he walks halfway from ,a to ,b and back again to ,a in #c hr4 and #ee min42 and later he walks from ,a to ,b in #c hr4 and #eb min4 ,what are his rates of walking uphill1 downhill1 and on the level1 if these do not vary8 ,,algebra ,b #a4 ,solve 7a7 7x #a7_/7x - #b7 7#bx #a7_/7x #a7 7#cx #c7_/7a - x7 #j4 7b7 #bx #g7a_/b7 #cx - #ah7a_/b7 - #gx #a7a_/b7 #j4 7c7 #f_/7x#b #bx7 #e - #bx - x#b4 #b4 ,solve for x and y1 checking one solution in each problem3 7a7 #bx #cy #a1 #f_/x #a_/y #b4 7b7 x#b x y1 y#b #cy - x4 #c4 ,a man arranges to pay a debt of #cfjj in #dj monthly payments which form an ,a4 ,p4 ,after paying #cj of them he still owes #a_/c of his debt4 ,what was his first payment8 #d4 ,if #d quantities are in proportion and the second is a mean proportional between the third and fourth1 prove that the third will be a mean prop4 between the first and second4 #e4 ,in the expansion of #bx #a_/cx#f the ratio of the fourth term to the fifth is #b 3 #a4 ,find x4 #f4 ,two men ,a and ,b can together do a piece of work in #ab days2 ,b would need #aj days more than ,a to do the whole work4 ,how many days would it take ,a alone to do the work8 ,,algebra ,,to ,,quadratics #a4 ,simplify 7ab7-#b7;c#b77a_/b7 4 7a#c;b#b;c7-#c777a_/c7 7a#f7_/7;b77#a_/c74 #b4 ,simplify a_/7a - b77a - c7 b_/7b - c77b - a7 c_/7c - a77c - b74 #c4 ,factor 7a7 x#d - #ajx#b #i4 7b7 x#b #bxy - a#b - #b;ay4 7c7 7a b7#b 7a c7#b - 7c d7#b - 7b d7#b4 #d4 ,find ,h4 ,c4 ,f4 of x#d - x#c #bx#b x #c and 7x #b77x#c - #a74 #e4 ,solve x_/7x - #b7 7x - #i7_/7x - #g7 7x #a7_/7x - #a7 7x - #h7_/7x - #f74 #f4 ,the sum of three numbers is #ea2 if the first number be divided by the second1 the quotient is #b and the remainder #e2 if the second number be divided by the third1 the quotient is #c and the remainder #b4 ,what are the numbers8 ,,smith ,,college ,,elementary ,,algebra #a4 ,factor e7#bx7 - #b e7-#bx71 x7#ab7 - #h1 x#b - x - y#b - y1 #ah;a#bx#b -#bd;axy - #ajy#b4 #b4 ,solve #g #dx #cbx#b #ex #g7a_/b77a_/b7 - #c #j4 #c4 ,the second term of a geometrical progression is #cb7a_/b71 and the fifth term is #c_/af4 ,find the first term and the ratio4 #d4 ,solve the following equations and check your results by plotting3 x#b y#b - xy #g1 x y #d4 #e4 ,solve #a_/7x#c7 #a_/7y#c7 #bdc_/h1 #a_/x #a_/y #i_/b4 #f4 ,in an arithmetical progression d -#aa1 n #ac1 s #j4 ,find a and l4 #g4 ,expand by the binomial theorem and simplify3 7#bx7_/7y#c7 - 7y#d7_/7x#e -#f7a_/b77e4 #h4 ,the diagonal of a rectangle is #ac ft4 long4 ,if each side were longer by #b ft41 the area would be increased by #ch sq4 ft4 ,find the lengths of the sides4 ,,smith ,,college ,,elementary ,,algebra #a4 ,find the ,h4 ,c4 ,f4 of #hx#c - #bg1 #cbx#e - #bdc1 and #fx#c - #ix#b #dx - #f4 #b4 ,solve3 7a7 7#bx #e77-e7 #ca7bx #e77-e_/b7 #cb4 7b7 7x - #a77a_/b7 7#cx #a77a_/b7 #d4 #c4 ,a farmer sold a horse at #ge for which he had paid x dollars4 ,he realized x per cent profit by his sale4 ,find x4 #d4 ,find the #acth term and the sum of #ac terms of the arithmetical progression 7#b7a_/b7 - #a7_/b1 7#b7a_/b77_/b1 7#a7_/b7b7a_/b7 - #a71 4444 #e4 ,the difference between two numbers is #dh4 ,their arithmetical mean exceeds their geometrical mean by #ah4 ,find the numbers4 #f4 ,expand by the binomial theorem and simplify #c;a7-#b7 - a_/-#b7a_/b7e4 #g4 ,solve3 #a_/x #a_/y #c_/b1 #a_/7x#b7 #a_/7y#b7 #e_/d4 #h4 ,solve the following equations and check the results by finding the intersections of the graphs of the two equations3 x#b #dy1 x #by #d4 ,,vassar ,,college ,,elementary ,,and ,,intermediate ,,algebra ,answer any six questions4 #a4 ,find the product of #a #b;a_/#c - 7#e;a#b7_/7f7 and #b - #c;a_/#d 7a#b7_/7c74 #b4 ,resolve into linear factors3 7a7 #dx#b - #be2 7b7 #fx#b - x - #ab2 7c7 a#b;b#b #a - a#b - b#b2 7d7 y#c 7x - #c7y#b - 7#cx - #b7y #bx4 #c4 ,reduce to simplest form3 7a7 z_/7#a_/x - #a_/y7 y_/7#a - y_/x7 - x_/7#a - x_/y74 7b7 -7x#c77a_/b77a_/c7 x 7#dy7-#c777a_/b74 #d4 7a7 ,divide x7#c_/b7 - x7-#c_/b7 by x7#a_/b7 - x7-#a_/b74 7b7 ,find correct to one place of decimals the value of #e7a_/b7 #g7a_/b7_/b - #c7a_/b74 #e4 7a7 ,if a_/b c_/d1 show that 7a#b c#b7_/7;b#b d#b7 ac_/bd4 7b7 ,two numbers are in the ratio #c 3 #d1 and if #g be subtracted from each the remainders are in the ratio #b 3 #c4 ,find the numbers4 #f4 ,solve the equations3 7a7 7x #a7_/7b7 - #c_/x x_/#c - 7#e - x7_/7#f74 7b7 #aax#b - #aa-a_/d #ix4 7c7 x#b - #by#b #ga1 x y #bj4 #g4 ,a field could be made into a square by diminishing the length by #aj feet and increasing the breadth by #e feet1 but its area would then be diminished by #baj square feet4 ,find the length and the breadth of the field4 ,,vassar ,,college ,,elementary ,,and ,,intermediate ,,algebra ,answer six questions1 including ,no4 #e and ,no4 #g or #h4 ,candidates in ,intermediate ,algebra will answer ,nos4 #e-i4 #a4 ,find two numbers whose ratio is #c and such that two sevenths of the larger is #ae more than one half the smaller4 #b4 ,determine the factors of the lowest common multiple of #cx#d 7x#c - y#c71 #ae 7x#d - #bx#by#b y#d71 and #ajy 7x#d x#by#b y#d74 #c4 ,find to two decimal places the value of #d;a7-#b_/e7 b#j;ab7-#a77a_/b77a_/b71 when a -#cb and b -#h4 #d4 ,solve the equations3 #bx #ey #he1 #by #ez #ajc1 #bz #ex #eg4 #e4 ,solve any #c of these equations3 7a7 x#b #dd - #aex #j4 7b7 #b_/x - x_/#e x_/#bj - #bbc_/cj4 7c7 x#b #hx - #dx#b #cbx #ab7a_/b7 #ba4 7d7 #e_/7x #a7 #h_/7x - #b7 #ab_/7dj - #bx74 #f4 ,the sum of two numbers is #ac1 and the sum of their cubes is #iaj4 ,find the smaller number1 correct to the second decimal place4 #g4 ,the sum of #i terms of an arithmetical progression is #df2 the sum of the first #e terms is #be4 ,find the common difference4 #h4 ,explain the terms1 and prove that if four numbers are in proportion1 they are in proportion by alternation1 by inversion1 and by composition4 ,find x when 7#c x7_/7#c - x7 7#dj x#c7_/7dj - x#c74 #i4 ,find the value of x in each of these equations3 7a7 #gx7#a_/d7 - #cx7#a_/b7 #b4 7b7 7x#b #b77e_/b7 #c_/x#b #b7a_/b7 #dx#b #h4 ,,yale ,,university ,,algebra ,a ,,time3 ,,one ,,hour ,omit one question in ,group ,,ii and one in ,group ,,iii4 ,credit will be given for six questions only4 ,group ,i #a4 ,resolve into prime factors3 7a7 #fx#b - #gx - #bj2 7b7 7x#b - #ex7#b - #b7x#b - #ex7 - #bd2 7c7 a#d #d;a#b #af4 #b4 ,simplify #e - 7a#b - #aix#b7_/7;a#b - #dx#b7 _/ #c - 7a - #ex7_/7a - #bx74 #c4 ,solve #b7x - #g7_/7x#b #cx - #bh7 7#b - x7_/7#d - x7 - 7x #c7_/7x #g7 #j4 ,group ,,ii #d4 ,simplify #b7a_/b7 #bc7a_/b7_/b7a_/b7 - #ab7a_/b71 and compute the value of the fraction to two decimal places4 #e4 ,solve the simultaneous equations x7-#a_/b7 #by7-#a_/b7 #g_/f1 #bx7-#a_/b7 - y7-#a_/b7 #b_/c4 ,group ,,iii #f4 ,two numbers are in the ratio of c 3 d4 ,if a be added to the first and subtracted from the second1 the results will be in the ratio of #c 3 #b4 ,find the numbers4 #g4 ,a dealer has two kinds of coffee1 worth #cj and #dj cents per pound4 ,how many pounds of each must be taken to make a mixture of #gj pounds1 worth #cf cents per pound8 #h4 ,a1 ,b1 and ,c can do a piece of work in #cj hours4 ,a can do half as much again as ,b1 and ,b two thirds as much again as ,c4 ,how long would each require to do the work alone8 ,,yale ,,university ,,algebra ,b ,,time3 ,,one ,,hour ,omit one question in ,group ,i and one in ,group ,,ii4 ,credit will be given for five questions only4 ,group ,i #a4 ,solve 7x a7_/7x b7 7x b7_/7x a7 #e_/b4 #b4 ,solve the simultaneous equations x#by#b #bhxy - #dhj #j1 #bx y #aa4 ,arrange the roots in corresponding pairs4 #c4 ,solve #cx7-#c_/b7 #bjx7-#c_/d7 #cb4 ,group ,,ii #d4 ,in going #gejj yd4 a front wheel of a wagon makes #ajjj more revolutions than a rear one4 ,if the wheels were each #a yd4 greater in circumference1 a front wheel would make #fbe more revolutions than a rear one4 ,find the circumference of each4 #e4 ,two cars of equal speed leave ,a and ,b1 #bj mi4 apart1 at different times4 ,just as the cars pass each other an accident reduces the power and their speed is decreased #aj mi4 per hour4 ,one car makes the journey from ,a to ,b in #ef min41 and the other from ,b to ,a in #gb min4 ,what is their common speed8 ,group ,,iii #f4 ,write in the simplest form the last three terms of the expansion of 7#d;a7#c_/b7 - a7#a_/b7 x7#a_/c77h4 #g4 7a7 ,derive the formula for the sum of an ,a4 ,p4 7b7 ,find the sum to infinity of the series #a1 -#a_/b1 #a_/d1 -#a_/h1 4444 ,also find the sum of the positive terms4