Wr TO New Thomson WAWM08WM »WW» > y i ' ««M»$«)r.W)Mnf«HIMMM(MHM«^^ i>?-'aw«ass wi ( w > >'mii>T>t>\ iiw iw>wfflnry !t^^ OF THE university califor^^ THOMSON'S NEW SERIES OF MATHEMATICS. KEY TO NEW PRACTICAL imUU &n TEACH fi;^. JAMES B. THOMSON, LL. D., AUTHOR OF SERIES OF MATHEMATICS. NEW YORK: Clark & Maynard, Publishers, 5 Barclay Street. 1881. NOTE. ALL agree that the best kind of help for pupils in - Arithmetic and Algebra, is Self-hel]p ; that it is better for the learner not to know the answer to a problem, until he has tried his own ability to solve it. In a word, that it is better for him to solve a single example independ- ently, than a score by the help of a teacher or a Key. And yet it must be admitted that a majority of teachers desire a Key. This demand comes not only from young and inexperienced teachers, but from those whose char- acter and scholarship are above suspicion. They desire it, not because of their inability to solve the problems, nor because they shrink from labor. Their object is to save time, which they may devote to other branches of study. A well constructed Key will often disclose in a single minute the error in a pupil's work, which might consume half an hour of the teacher's time, if he were obliged to wade through a long operation. The plan of the work before us is to indicate in full the operations to be performed, and give the results ; omitting the minor details. It also contains many valuable sugges- tions as to the different methods by which certain problems may be solved. It is hoped teachers will find it adapted to theh- wants. UdociMOT IIBBJ • Copyright, 1877, by James B. Thomson. Electrotyped by Smith & McDougal, 82 Beekman Street. A ' O KEY. EXERCISES IN NOTATION. Images 12, 18, g. 4C -\- d -\- 7)1 — 5^: = ab. 3. 5C6/ + ^ = xy. 4. ^^ + 4?^i = c + oa — ^ax. 5. « — h ^ xy ^ 6mn. 6. a: — 1/ + 4« + ^ — ?» = cd + 15m. 2. The quotient of twice the product of a and h divided by X, plus a minus b, equals the quotient of a plus b divided by c, plus the product of a, x, and y minus four times the product of c and d. 3. The quotient of three times b plus c divided by 8, plus 3 times X, equals 3 times the product of c and d divided by a, plus the product of x, y, and z minus the quotient of c divided by d. 4. The quotient of 3 times a divided by 5, minus the product of a and x plus the product of b and c, equals the quotient of 4 times a minus b divided by x, plus the quotient of c times d divided by 4, minus 3 times x. 5. The product of a, b, and c minus x, divided by 3 times d, increased or diminished by 3 times x plus 5 times ?/, equals the product of c, d, and 7i plus x divided by twice a, minus the product of x and y. M57'?055 4 ALGEBRAIC OPERATlOIi^S. 6. The quotient of 4 times a into x into y divided by 5 times a, plus the quotient of a minus h divided by x, equals the quotient of x plus y divided by a^ minus the quotient of twice a plus d divided by 3 times c. ALGEBRAIC OPERATIONS. 4- 6. Page 15. Let X price of the apple, Then will 2iX " " '• orange. And 4.f 8 cents. Dividing by 4, X 2 cents, apple; ) , $x 6 cents, orange. ) And Let X value of the hay. Then will 4X " " " cow. And Sx I40. Dividing by And 5. 4X i$^2, cow. ) Let X one of the numbers, Then will ^x other number. And 4X — s^' Dividing by And 4, ^ } Ans. SX — 27. ) Let X C's number of peaches^ Then 2X — B's " " And 4X — A's " " " Adding, yx 28. Dividing by 7, X 4, C's number; ) And 2x — 8, B's " > Alls, li 4X — 16, A's " ) Let X son's age. Then will ^x father's age. And 4X 48. Dividing by 4, X 1 2 years, son's age ; \ ^x 36 '' father's age. ) And Ans, ALGEBRAIC OPF. R ATI OJSTS. Page 16. 8. Let X = B's share of gain, Then will 4X — A's " '' And S^ ^^ '$100. Dividing by 5, x ^ ^20, B's share of gain; And 4a; = ^3o, A's " a J. inSo Let a; = ist nnmber. Then will 2X = 2d '* And 32* = 3d " Adding, 6x = 90. Dividing by 6, x =: 15, ist; ) And 2X = 30, 2d; > Ans. 3^' = 45? 3^- ^ ee 1 o. Let X = value of the calf, Then will Sx — " " " cow. And ()X = ^6;^. Dividing by 9, x = ^7, price of calf; ) And Sx =z ^^6, " " cow. ) 1 1 . Let X = value of bridle. Then will 2X = " " saddle. And 21a; = " '^ horse. Adding 24X =: $126. Divid. by 24, x = -S5.25, worth of bridle; And 2x = $10.50, '•' " saddle; ]■ Ans. « 21X =z $110.25, " " horse. 12. Let X = daughters portion, Then 2X = son's " And gx = wife's " " I2rc = 836000. Therefore, x = $3000, daughter's portion ; And 2X =2 $6000, son's " ^ A71S. " gx = $27000, wife's " FOliCE OF THE SIGKS. ^3- Let X ist number, Then 3X 2d a And 4^ + 5 3d a (( Sx-\- 5 1877. Subt. 5 from each, Sx 1872. Therefore, X 234, 1 st number ; And SX 702, 2d a (( 4^ + 5 941. 3h. 4. 5^ — 2^. 5. 35- 6. 24. FORCE OF THE SIGNS. 8. dh — 7CX 4- $a. 9. bxy -\- cxy. Page 22. iSxy 7. 3a: + 27/ + fl;^». 10. II. h — a xy 4- a. + 22. ADDITION. 12. T,x + xy -\- 2ZX sy — ^x -h xy + 6yz, Ans. Z' i-^-y) X a — b ax — ay — hx -\- by Ans. d ~ d a = $, b = 4, c= 2, X = 6, y = S, and z = lo. ax 1 8 14. « H \- yz = s -\ h 80 =: 92, Ans. c 2 i^. -^ abtf 4- 2z= - + q6-\- 20 = 120, Ans. X — 2 ADDITION. Case If Page 24. ' 8. 45 «5^ 3. 2iab. 5- i5«^- 9. — ^gabx^y^. 10. 2g¥d7n^. 12. Jc = 4. 13- ^«/ = 5- Case II, Page 25, 16. Sx. 17. ^Z'C. 18. — 12b. 19. — 12?/. 20. — 2 772, 6. — 2T,bcd. 7. — i6x^y^. 21. 6ab + i4«^ + i5«^ + ^^cib — S'^ab', — 'jab — i2ab = — igab', Slab — igab — 32«A Since 32^/^ = 32, .-. ab = i, Ans. 22. bed — sbcd + 4bcd + 4bcd — sbcd = 75. Uniting, gbcd — 8&c^ = bed = 75, ^4fis. 1. 24rz + 2b — id. 2. 1677m — xy ■\- be. 3- 3^c , — ibc -\- xy — mn iibe EXAM Pagi PLES. 3 26. 4. 5«^> - ab Zah — 4a'b ab 4- 2Z gbc 4ab — ymn -\- 2z, Ans. i6bc -\- xy — mn, Ans. 8 5- 2>^y — xy -\- ah — i^y + ^ Zxy — xy izxy i^xy -\- ab + h, Ans. 7. 21 {a + b). 8. 19^(2; — y). ADDITIOK. 9. jaV^y- 10. 6\/«. II. \o^/x — y. f»ff= 141, ! II. 4^-}- 3^ + 2?/— 12 = 60 Uniting, gy z= 72 ^ = 8, ^7^s. 10 ADDITION. 12. Let a; = price of top, Then 3ic — 4 = " " ball, xind 4.1^—4 = 32 cents. Adding 4 to eacii side, 4X = 36 " X z=z g ce?its; ) , And 3^ — 4 = 23 " ) 13. Let X = price of bridle, Then 4^ — 5 = " " saddle. And s^ — 5 = I40. Adding 5 to each side, 52: = $45. X z= $9, bridle ; ] . And 4^' — 5 = %ij saddle, ) 14. Let X = sum spent in a.m., Then S'^ — 4 = " " " p- m. 4X — 4 = 100 cents. Adding 4 to each side, 4X =z 104 " X = 26c., A.M. ; ) , And ^x — 4 =z 74c., p. M., ) 16. 2x 4- 5'^' + 3^ — 10 = 130 Uniting, loip z= 140 a; = 14, Ans, Proof. 28 4- 70 + 42 — 10 = 130 17. 4X -{- ^x -{- 'jx — 12 = 86 Uniting, 14^^== 98 .*. X — 7? jfifiSt Proof. 28 + 21 + 49 — 12 m S6 18. loa; — 4a; + 93* — 25 = 155; .*. :c = 12, ^;i6\ Proof. 120 — 48 + 108 — 25 r= 155 19. 152; — 7.C — 2x — 60 =z 300 ; .'. X = 60, A71S. Proof. 900 — 420 — 120 — 60 = 300 20. 18.T — 4ic + a; — 75 =z 225 ; .*. x = 20, ^?i5. Proof. 360 — 80 + 20 — 75 = 225 SUBTRACTION. 11 SUBTRACTION. Page 31. 3. 14XUZ. 4. — 62ah. 5. ic^ab. 6. 2']xy. 7. ^yic. 8. T,']ax^. 9. 51^^^^. 10. — 44x'2?/^ 11. 2)^(v^h. 12. o. 13. _ 77///2.6-. 15. A debt is propei-ly regarded us a negative or minus quantity. Hence we have to take — $50 from 8100. $100 — (— $50) = $100 + ^550 = 't^i5o, Ans. 16. 15 c 10 17. 25°, $275 _|i45 Alls. 20, ^j Ans. 18. ^xy — 6a. 19. 13^^ + i6a7n. 20. i?>x^ + y'^ -\- 6a. 21. iyii-\-(l — X — 5?;z + 3?i. 22. <)cd — ah — 2^4-3^ + 47/. 23. i8wz — 23. 24. i2a;~ — 13a;. 25. \6ah + 13c + d. 26. a — b -j- c. Proof. — Tlie difference plus the subtrahend equals the minuend. rt — ^ + c = difference. b — c = subtrahend. a = minuend. 27. 6 (a + b). 28. g{a — b -\- x). 29. 5 (a + b). 30. — 7 (.^'2 _ 7y). 31- '1000 Then = the gain. Si 000 — $500 — $100 = $400, B's share. $500 + $100 = $600, A's " Ans. 32. If distances east be regarded as +, then those reckoned west must be considered as — . East longitude, West 23^ .37! 60°, Ans. 12 MULTIPLICATION Page 33. 34. (2b — c -\- d) x\ 35. {ah — c — d -\- x) y. 36. d^iy — h -\- c). 37. (cil) — ^c — d — m) X. Ty'^. {^ — alj ^ c — d) xy, 39. ^ac + bmc 2,ac — dc 2ac -\- hmc + dc (2a + Im -\- d) c, AnSo Page 34. 4. h — (c — d -{- m) z^h — c -\- d — m, Ans. 5. 5^; + ^ — «^ + 4d 6. 2a — [d -\- c — {x -{- y) — d] =^ 2a — {h -\- c — X — y — d) ^ 2a — 1) — c-\-x-{-y-\-d, Ans. 7. a — b-^-c — a-\-c + c — a-\-'b. MULTIPLICATION. Case If Page 36. l^fr- 2 4a^b^. 33- 2ia%^. 7. Sdmxy. 24. a^hf. 34. 4oc^x"y. 8. 6T,bcdxyz. 25. d^b'"^''. 35- 2MW. 9. ^6abxy. 26. Gx^z. 36. — Gx^. 10. 42acdx. 27. I W¥c\ 37. 21 dWc\ II. ^4bcdm. 38. 2%a^d^. 12. G^ad/xyz. 28. 3« — 9 ; r^^ 27 ; 7—9 18, yi;?.s. 39- xhfz^. 2: Page 37. Case Ilf Page 39. 13. Given. 29. 4X' 16; 3- Gacx^ + ^c^d. 14. — 4^abxy. ^ 4^ 256; 4. i^aWx—Gacdx 15. 42 a bed. 256 — 16 + 3fl!A 16. i^2abcxy. 240, Ans. 5- - ^d'bd + 6r/^2^ 17. — 4i4abcxy. — 2bdm^. 18. ()4^bcdxy. 30- 6x^y. 6. — 15^^^^ 31- — iSa'^b^c. + 20«2J2^'4- lOfl'^cS. MULTIPLICATION. 13 Case III, Page 40. I. 6. 8. 2a + b z^ + y 6ax + Tfhx -\- 2ay -\- by, Ans. 3^ + 41/ a — b 2,ax + ^ay — ^bx — ^by, Ans. 4b — c T,d — a i2bd — T,ccl — ^ah + ac, A^is. 6xy — 2 a b -\- c 6bxy — 2ab + 6c.r?/ — 2C'c, Ans. 3« + 4Z> — c ^ — y 2,ax 4- ^bx — ex — T,ay — ^by -\- cy, Ans, 5^ + 3^ 4- ^ a -\- b sax + 2>'^y + az + 5^.t + 3% + bz, Ans, ']cdx — 3rtJ 2m — ^n i^cdmx — 6abm — 2\cdnx -\- gabn, Ans. m Sabc 4- 4m 3^ — Ay 24abcx -\- i2mx — ^2abcy — i6my, Ans. 9, 10. Given. 1 1 . ;^abc"xyz''K 12. iiabchI"'+\ 13. a^X'-^". 14. ex (a + by. 15. 5c {a — by. 16. abc {x 4- y)"'+''. 17. —;^x{a + by. 14 MULTIPLICATION. Case Ill^Continued. l*age 41, 20. rt^ — ah A- W' a ■\- b «3 ^^2^ j^ ^^l2 ci^h (lb- + b^ 0? + ^^, Ans, 21. a^ ab + Z>2 • a^ + rtJ -f h^ a^ — a% -h aW a% aW + «&3 aW ab^ + Z*4 «4 ^ ^2^2 _^ ^4^ ^,i^. 22. X^ + .!• + I a;2 — X -f- I ^ _|- :^;3 _|. ^2 — X^ — X^ — X -- X^ -}- X -\- I x^ + x^ -\- I, Ans. 23- 3^2 _ 2xy + 5 X^ + 2iC^ — 6 3^:^ — 2.?-3^ +5^^ 6x^y 4x'^j/^ -\- loxy — 18:^2 + 120;^ — 30 2fX^ -f 47?y — \2,x^ — /\x^y^ + 22^^ — 30, Ans, 24. ^ax — 2ay 6ax + ^ay 24a^x^ — i2a^xy i2a?xy — 6cfy 24^^ — 6ahi^, Ans. MULTIPLICATION". 15 25- 29. 30- 31' d -i- hx d_-\- ex d'^'+ hdx + cdx -\- bcx^, Ans. 27. x + y ^ — y- x^ H- xy xy — y' 28. (V^ + I a -\- 1 a^ -\- I a^ — y% Ans. x^ + 2xy + «/2 ^ +^ a;^ + 22:2^ + xy'^ x^y + 2iC?/2 -\- y^ a^ -\- a^ -\- a -\- 1, ^?i.s. x-\- y -^ z x-Vy ^ z Q? -\- xy -\- xz ^y +y^+ y^ xz -\- yz -\- z- x^ + 2xy + 2XZ ■}- y^ + 2yz + z^, Ans. Page 45. Notes. — i. For the details in the following examples the teacher is referred to the formulas in the ])receding articles in the text. (Arts. 101-103.) 2. The learner should be able to write the following answers by means of the preceding formulas. 16 MULTIPLICATION^. 1. (a + i) {a + i) = «^ -\- 2a -]- I, A71S. 2. (2a + i) {2a + i) = 4(^^ + 4« + I, Ans. 3. (2a — b){2a — h) — 40? — 4ab + Ij% Ans. 4. {x + y) {x Jr y)=x^ -{- 2xy -\- if, Ans, 5. {x — y) {x — y) ^ x^ — 2xy + y% Ans. 6. {1 -\- x) (i — x) ^^ 1 — x^, Ans. 7. {iy^ — 2/) (7^^ — ^) = 49/ — 14^^ + ^^, ^ns, 8. (4m — 3^) (4^^? + 2>^i) = i6?m2 _ gn^^ Ans. 9. (a;2 — ij) {x- -{- y) =zx^ — y\ A ns, 10. (i — 72;) (i + 72;) = I — 492;^, Ans. 11. (42: — i) (42; — i) = 162;^ — 82: + I, ^?2S. 12. (5& + i) (5^ + i) = 25^^^ + loZ^ + I, Ans. 13. (i — a;) (i — 2:) = I — 22; + a;^, Ans. 14. (i + 22;) (i + 22;) = I + 42.' + ^x^, Ans. 15. (8^ — 3«) (8Z» — 3«) = 64^2 — 48«& + 9^2, ^?Z5. 16. (rtZ> + cf/) («& 4- cd) = aW + 2ahcd + c^t^^, vl^^s. 17. (3« — 2?/) (3<^ + 2?/) =ga^ — ^if, Ans. 18. {x^ + ?/) (.T^ — y) =1 x^ — y% Ans. 19. (2; — y'^) {x — y^) =:x^ — 2xy'^ 4- y\ Ans. 20. (2^2 + a;) (2^^ — x) = 4«^ — 2:^, A71S. PROBLEMS. Page 44, Let X — the number. Then 22: 3 ''■ Multiplying by 3, 2X —72. Dividing by 2, 2: — 36, A?is. Let X — No. of chickens Then 3^: 18. 4 Multiplying by 4. 3^ — 72. Dividing by 3, a; — 24 chickens, ^ MULTIPLICATION. 17 Page 43. 6. Let X = number. Then ^ _ ^ = 8. 3 2 Multiplying by 6, 4./; — 3a; = 48. X ^ 48, y4?2,S'. Proof. — = 32 — 24 = 8. 3 2 7. Let ^' = number of army. Then ^ = 840. 7 Multiplying by 7, s^ = 5^^°- Dividing by 3, x= i960, Ans. 8. Let - X = worth of yacht. Then ^ = 1^360. o Multiplying by 8, 3a; = $2880. Dividing by 3, ic = I960, Ans. o. Given — = 20. ^ 5 Multiplying by 5, 40; = 20 x 5. Dividing by 4, a; = 5 x 5 = 25, A)is. 10. Given — = 20. 4 Multiplying by 4, 5a; = 20 x 4. Dividing by 5, x = 4 x 4 = ^^> ^^^•'^• 11. Given — = 24. 7 Multiplying by 7, 3a; = 24 x 7 Dividing by 3, :c = 8 x 7 = 56, A71S. n- 4^ Q 12. Given — = 26. II MultipMng by 11, 4a' = 28 x n. Dividing by 4, rt = 7 x n = 77? ^^^^' 18 MULTIPLICATION. 13. Let Then Multiplying by 6, Dividing by 5, X =z number of apples. KX ^X =: 180. 2; = 36 apples, Ans. 4- Let X number of s Then SX 7 30 cows. Mult, and dividing, And X — 70 -f- 30 _ 70 sheep; 100 animals. 5- Let - X one part. Then SX 4 other part. And -ix ^+4 - 28. Multiply Uniting Dividing ing by 4, terms, by 7. 4X + s^ rx - X 112. 112. 16,) And Zx := 12. A7IS. Arts, 16. Let Then And X X = whole number of plums, — I — = number given away, X ^ =z 10, number left. 3 4 Multiplying by 3 and 4, i2x — 4X — T,x = 120. Uniting, dividing, etc., a; = 24 plums, Ans, 17. Let Then Mult, by 3 and 6, Uniting, dividing, X = number. %A^ JO - -\- = 21. 3 6 6x + 3.T = 378. X = 42, ^/i5. DIVISIOK. 19 t8. Let Then X = 11 umber, 19, 20. X X 4 6 Mult, by 4 and 6, 6a; Uniting and dividing, 4:/; =: 288. a: =: 144, Ans* Let Then And Mult, by 3, Dividing by 5, And Let Then X = one part, — = the other. 12; 4- — = 36. 3 5rr = 108. ^= 2if;| 22; ^ >• Arts, 7 = '4*- ) X = No. of bu. from one. ^ = " " the other. 7 And X , 3^^ 7 21. Mult. by 7, lore 147. Dividing by 10, a; — 14A bushels; ) Anc. 3^ 7 "10 < 1 DIVISION. CV?se J, Page 47 » 13- 7^. 21. C5. 3- 2ab. 14. Sd. 22. ;26. 4. s^y- 15- 61). 23- 4^. 5- 5- 16. Idf. 24. 2a; • 6. 2 a. 17. gag. 2/ 7- sab. 26. 8^5c2. 8. Sbc. C«^ ^e If Pafje 48. 27. Ga-^ 9- 47)171. 19. d\ 28. 5«^- 12. — 3O' 20. :^. « 29. 7^Y -47^5. 20 DIVISION. 30. abc. 31. 2abc, 32. ^x^yh. 34. i2clH^y. 35- a 36. 12X^Z^. 37. iirri^n. Case II, I*(ifje 40. 4. b^ + c3 + d\ 5- 3^ + 5- 6. 3^c — I 4- 4^'. 7. 2^?/^ 2 2/ 8. — 2a; + ?/. 9. y^ -{- z— I. 10. — 5«^ — 4b-\-6. 11. 3^^^ — 3^. 12. — 4x^ — ^cl^ + ax. 13. rt^ — 5« + 2Z». 14. I 4- 5« — gaJ. 15. 2^ — 4b — 5c. 16. 2 (6« + by^ + 3.r (ft + ^)^. 17. 9^?;— 9?/. 18. x{b — c) — a{b — c). 19. 3ff2 — 2«. 20. a — a^ -{- a^. Case III, Page 51, 3. The dividend is the square of the divisor. Hence the quotient i^ x -\- y, Ans. 4. The solution same as Ex. 3. a — b, Ans. 5. The dividend is the cM^e of the divisor. Hence, a^ — 2ab -\- W, Ans. 6. a -^ b ) ac -\- be -\- ad -\- bd {c -\- d, Ans. ac + be ad + bd 7. a -\- b ) ax -\- bx — ad — bd (x — d, Ans. ax + bx — ad — bd 8. X -{- 2y ) 2x^ + ']xy + 6y^ ( 2X -\- ^y, Ans, 2X^ + 4xy Sxy + 6?/2 9. By Art. 103, «2 _ j2 _ (^^ _^ j) (^ _ j)^ and (ft + b) {a — b) -r- (ft + Z*) = ft — J, ^?zs. 10. The soUition same as Ex. 9. x -\- y, Ans. DIVISION. 21 II. a — h) a^ — h^ {ci^ + cih + H^, Ans, a'b a%- -al^ aW- -¥ aW - W 12. 2a + 3 J ) 6^2 _j- iT^ab + ^W ( 3« + 2^, Ans, ist prod., 6^2 _|_ ^f^^^ 2d div., 4^^ + 6R 2d " \ab + 6^*2. 13- ^« — 3 ) rt^ — rt — 6 ( « + 2, ^y?s. ist prod., c^ — T^a. 2d div., 2a — 6. 2d " 2a — 6. 14. ( «^ — 3 w^^'^' + 3 «!.l'^ — X^^-^{(1 — .?;) := ft^ — 2ax-\-x^i ^ ^^^« Note, — The dividend is the ciLhe of the divisor. Hence the quotient is the square of the divisor. 15' 3^ — 6 ) 6:r4 — 96 ( 2^3 + 4a;2 -f 8ic + 16, ^^25. ist prod., dx^ — \2x^. 2d div., \2X^. 2d " \2x^ — 242:^. 3d " 24^^. 3d " 240^2 — 482-. 4th '^ 48a; — 96. 4tli " 48a; — 96. 16. .T + 2 ) a;2 4- 7^ + 10 {x -\- 5, ^?2s. ist prod., x^ + 2:?:. 2d div., 5a: + 10. 2d " ^x -\- 10. 17. .T — 3 ) rc^ — ^x + 6 ( .T — 2, ^/?s. ist prod., x^ — -^x. 2d div., — 2X ■\- 6. 2d " — 2a; -f 6. 18. (c^ — 2c:c + .^;^) -T- (c—x) = c— .T, ^47Z5. (Art. 102.) 19. {a^ -^ 2ab -\- b~) -^ (a + <^) = ci + b, Ans. (xA.rt. loi.) 20. 22 (« — b) -^ 11 {a — b) — 2 {a — b), Ans. 22 DIVISION". PROBLEMS. Page 51. 1. Let X = son's age. Then 5x — 4 = father's age. And 6x — 4 = 56 years. Adding 4 to each side, 6x = 60 "- Dividing by 6, x = 10 yrs., son; ) . And ^x — 4 = 46 " father. ) ^ 2. Let X = Frank's number, Then 3a; = John's " And 4X = 60 marbles. Dividing by 4, x = 15, Frank's number; ) . And 3:?; = 45, John's " 1 3. Let X = one, Then 5:?: = other. And 6x =72. And 5a; = 60. ) 4. Let a^ = No. given to one, Then 42; — 3 = " " " the other. And -5.^— 3 = 57 pears. Adding 3 to each side, 5^ = 60 " And 4X — 2, — 45. ) 5. Let ^ = No. ist had; Then 22; = " 2d " 3^ + 4 = " 3d " And 6:^ + 4 = 190 cents. Subtracting 4 from each side, 6x =186 " .'. X — 3^ cts. , And 2a: = 62 " )-Ans, « 3.7; + 4 := 97 " DlVISIOi^. 23 6. Let TbeD And And 8. II. 12, Page 52. X == number of cows, ^x ^ " " sheep, lort' =z 200. ic = 2o cows ; ) gx = i8o sheep. ( Ani>. Let Then And Subtracting 3, And Given Adding 3, Let Then And X = less, 3a; + 3 =: greater. 4^4- 3 = 57- 4.-^ = 54. •*• X — i.^'^? less . f i 3-*^ -h 3 = 43L greater, ) 2X -{- 4X + X — 3 = 60. IX = 6^. X ^= g, Ans, X = No. of hours each travels, 4X = Sx = 'jx = 35 miles, the given distance. X = ^ liours, Ans. "■ " miles A a a a J^ a a 10. Given i4« +7 = 119. Subtracting 7 from each side, 14a = 112. Given 20^ — 10 = 130. Adding 10 to each side, 20 J = 140. J = 7, ^?2S. Let Then And « X = No. bought of each fruit, 3^ = amount paid for pears, 4X = " " " oranges, 5x r^ '' " *•' bananas. 12a: = 60 cents. X :^ 5, Ans, 24 DIVISION^. 13. Let x = No. of hrs. it takes both to empty it- X Then — = part one will discharge ;n x honrs, 20 And - r= " other " " " ^?^s. ^x = 20, 3d " J X - A's number. 2X - B's a IX - C's a 6x 60. X _ 10, A's number; ) 2X _ 20, B's " yAns, IX — 30. C's '^ ) X ist part, 2X 2d (< 3^ 3d (( 6x — 48. X 8, ist part; ) 2X 16, 2d " >■ An&» 3^ — 24, 3d " ) a: — the number. f-== 23. ZX 18. Subtracting 5. 4 Multiplying by 4, 30; = 1 8 x 4. Dividing by ^, a; = 6 x 4 = 24, Ans, 26 FACTOKING. FACTORING Page 54, 5- (w + 2n) {m + 27i). 3- 2 X 3 X 3««&5. 6. (4a -h i){4a + i). 4- 2 X 2 X s^^'^^yV' 7. (7 + 5) (7 + 5). 5- 5 X Taaabbcc. 8. (2a — sb) (2a — 3&). 6. 3 X 7^2/^^^^;. 9- (y + i)(^+ 0- 7. I ^xxyyyz. lO. (, _,2)(i_,2). 8. 5 X ^ahhcxxx. II. ^x"^ _}_ ?^«) (a;"' + y"). 9- 7 X iiaahccd. 12. (20" — i) (2«" — i). lO. 5 X i;^77imnmix. 13- (a'^ + b^)(a^ + b^). 14. {ax^ + ?/) (ax^ 4- ?/). Case JTJ, Page 55, 4. ^ (2/ + ^ + 3^)- 5. 2a{x-\-y - 2z), 6. $bc [x — 2X — a). 7. Sdm (n — 3). 8. 7a (5m + 2x), 9. 2Td (bx— 2my). 10. 3«2(2j + 3c). 11. jaxyizx -+- 5). 12. 5(5 + 3^'^ — 4a^y)- 13. a;(i + ^ 4- ^^)- 14. 3 (a; + 2 — 32/). 15. 19^^(0; — i). Case III, Page 56, 3. {a + b){a-\-b). 4. {x ^y)(x — y). Case IV, Page 57 • 2. {a -\- x) {a — x). 3- (3^ + 42/) (3^ — 42/)- 4. {y + 2) (2/ - 2). 5- (3 + ^) (3 - ^)- 6. (a + i)(« — i). 7. (i + J)(i-^). 8. (5« + 4^) (5« — 4^). 9. (2.?; + y) (2X - 2/). 10. (i + 4«) (i — 4«). 11. (5 + 0(5- i)- 12. Oi-2 + 2/') (;r2 - 2/2). (Art. ] 13. {ax 4- Z»2/) {ax — by). 14. (m^ + n^) {m^ — n^). 15. (a'"+ ^^'O (^^'" — *")• 03-) FACTORING. 2? Case V, Page 59, 2. 7^ — 1 = {x— \) (a;2 -\- X -{- i), Alls. 3. X — y)x^ — y^{x^-\-x^y-\- oi?y^ -\- xhf -\- xa^ + ^^ ^^ — ^^ :i(^y — x^y'^ a^y"^ — x^y^ x^y^ x^y^ — x^y^ x^y^ — xi^ xy^ — l^ xy^ — y^ (x ^y){^ ■{• x^y 4- a^?/^ + x^y^ + xy^ -\- y^), Ans. Note. — As the last term in the dividend is not used till the final operation, there is no need of bringing it down every time. 4. x^ — I z= {x — i) (x -{■ i), Ans. 5. I — s^y^ = (i — 6?/) (i + 6y), Ans. 7. Z>2 — a;2 = (5 — x) (b + x)f Ans. 8. d^ — :^z= (d — z) (d^ + d^z + dz^ + z^), Ans. 9. a^—l^ = (a— h) («5 + a^h + a%^ + ct%^ + a¥ + h% A ns, 10. X— i)x^— i{a^-\-x^-\-X'\-i x^ — a^ 3^ «3_ X^- -X X — I X — I tr^ — I = [x — \) i^ -\- x^ •\- X -\- 1) , Ans, Note. — If the powers of i were expressed in the second factor, it would take this form : cc^ + x^i + xi^ + 1^. But as all powers of i are i, they may be omitted from the literal terms. (Art. 94, Note.) 38 DIVISORS. 11. I — a^= (i —a) (i -\- a -\- a- + a^ -\- a^ + a% Ans. 12. a^—i = (a—i)(a'^ + a^-\-a^-\-a^ + a^ + a^-\-a-j-i), Ans. 14. x^-{-y^ 15. «Hi 16. a^-\-i 17. 1+^^ 18. i-{-a^ 19. i+b^ rage 60. (x -{-y) {x^— x^y -\- xhj'^ — xy^ + y^), A ns. {a+i){a?' — a-\-i), Ans. {a-\-i){a^ — a^-\-a^ — «+i)j Ans. (i+?/)(i— 2/ + ^^), Ans. (i+ft)(i — a^a^ — a^ + a^), Ans. {iJ^-h){\—b + l^—¥ + ¥—h^-\-h% Ans. DIVISORS. Page 61. Page 63. Page 66 3. X. 4. 1). 3. 3«c. 6. X y. 5. ac. 4. 2«a;^. 7. a -{-h. 6. 2X. 5. 4«3a;V. 8. ^+2. 7. 'jm. 8. 6al). 6. 6ax^z^. 9- ^ + Z' 10. a^ — a — 2) a^ — :^a + 2 «2 _ ^7 — 2 (I Cancelling — 2, — 2a + 4 we have « — 2 ) a^ — a — 2(^4-1 a^ — 2a a — 2 a — 2 Dividing the greater quantity by the less, changing the signs in the remainder, and cancelling the factor 2, we have a — 2 for the second divisor. As this is contained in the first divisor without a remainder, it is the g •5 ;-i »: > C o § « o .2 • g c •i-H o PI o • .2 •f-i > o > 13 C8 a a • I-H o -4-^ 'd TS -a 'd 'w 'O C3 1-4 H M N N M CO CO u 1 C<0 rO -iJi • vO oo 00 r^ lO N M ^^ m ;-! 1-1 1— 1 M M ■^ M QJ + -- + 4- + + 4- »\ 0) ^^ ^ « 5^ 5^ ^ ^ « M r-| '3 fO CN lO ^ ^o o vO CO ^ s l-t CO ^ vo vO + + 1 + + H- + 5^ CO o 0^ "^^ '^^ "^ =^^ "^ •^c^ + ^ rt- a (N ^ sO o VO CO l-O p* CO M H- CJ CO CO c^ N lO lO ^ .2 'o + + h^ s c .1-* 1 ^ ^ 1 5^ CO CO ,0 03 ?-; ^ c^. « ^ o N CO lO lO % M M -- ^ -fcj r^ "^ + =1^ CO 1^ O ON 03 lO % CB O CO M o M 1 00 M 1 00 + + a .2 w ''3 + •>* CO 1^ CO 1 B m 1 N «5 ' >> M 1 s CJ <^ 5^ ^ X3 2 •rH c^ CO fO o S s ^ ^^ - CO -« rd ni ff o s a 03 u fcdD c3 ^, S 'd ^.^ a C ■^"' >• C4 N F • f-H ) •t^ i-t '-^ -e 'w c3 X ^w H C4 "p. :j3 w > • § 32 KEDUCTIOK OF FRACTIONS. REDUCTION OF FRACTIONS. (Jase I, Page 7-t. I 4. 5. sac d ' 6. J. 8. 9- 10. 'jabc^ • a — h • ^ + y ^ x — y zy — 3^ 7— a 2X — 2Z II. 12. X O? — 1/2 13- 14. 15- ^ — t I X ' + f a + I X a I I x + y Case II f Page 75. 2. a — X. 3- ^ — a 4. h — c. 5. ^ + c4- ic^ h—c 6. a — h. 7. ^ + 2 ah b — a 8. a ■\- X -\- a 9. 32; + I — a—x zy 4X Case III, Page 75» 4xy — b y 4- 5^? + a — c 2b I obcl -\- a — c "^ Yb a- + 2ab -\- b^ -{- 2X a -\- b 7. 8. 2. X -\- I i2ac — a -\- h 39^'^ 4- 3^ i2mx 3- 4. 5- 6. 24^2^:^ 4ff^ iMc^ 4- 24^6?^ a;2 — ?/2 dcv^xhf — Afbx^y 3«2 — 2b Page 77» 3' — ah ac 2\C^ 49« (im 5- 6. ir^ — ?/2 x^ — 2xy + ?/2 32^3 (.T + y) 8^2 (^ 4- «/)2* keductio:n' Of fractions 33 Case V, Page 7S, 2 ex 2b(l CpX 2dx' ac^y 2c^xy^ 2dx' 2dx 2l)xy 2(?X^ 5- 6. 2c^xy 2C? + 2r^^ 2(?xy ihx 7- 8. Zab + 3^^' 3^^^ + 3^^ x^ — 2xy 4- ?/^ ^^ — 11^ ' a;2 + 2xy 4- ^2 a^ — y^ c? + «^ 15a — 3 3« ' 3« ' ^(^a: 2ad he + b ~hd' ~bd' Id 2Wc—2y^d 2>C'^—T,ad ^b^c-ir^b^d $b^c — ;^b'^d 2bxy hz 4az 2bz ' 2hz'' 2bz ax 4- ay 6x + 6y 2X 4-2?/' 2X -|- 2^' 2X^ 4- 2?/'^ 12. 10. II. 2a; 4- 2y a' — 2ax 4- :^-2 d^ — x^ a^ 4- 2 ax 4- 2;2 a^ — x^ Case VI, Page 79, 2. Least common multiple = 4bcx. 4b^c^ y _ bxy a _ 2acx he 2b ~ 4bcx' X 4bcx^ 46" 4bcx 3. Least common multiple = :^abc. cd 3C%/ 2X 2bcx xy 2fixy ah ^abc^ ^a $abc^ ac ^abc 4. Least common multiple =: 1 2y. a _ 6ay b _ 4by c _ 2,cy x 2 i2?y' T, 122/' 4 12?/' y i2y 3 i2y 4 i2y Least common multiple = 4b^c. 4abc^ 2cd _ Scd x^y _ bx'^y \2X \2y a^c 2cd 4b^c ' "b^c ab 4b'^c ' b^c 4bh' 4bc 4b^c 6. Least common multiple = 24^%. 3 iSa^c X 24.?: 2ab yjLG i6d^b 24a^c iSa^c 24d^c' X a^c 240^0 ' I 8 I 24^2^ 31 KEDUCTION OF FRACTIONS. 7. Least common multiple = 2'bc. 2a ac cd 2cd x^y 2xy 4b ~ 2bc' bc~ 2bc' hex 2hc 8. Least common multiple =: a^ — h\ a±h_ _ {a + bf a — b _ { a — bf a^ + ¥ 'a — b~ a? — W' a+b~ a^-l^' a^—W 9. Least common multiple = 6xy{x 4- y). 2 (x -\- y) _ 4xy {x -\- y) a _ 6a {x -\- y) 3(^ + y) ~~ ^^y i^ + yV ^y~ ^^y i^ + yY ab abxy 6{x-\-y)~ 6xy{x + y)' 10. Least common multiple = a^b^. d ad X __ hx aW ~ aW^ c^b ~ c^W 11. Least common multiple = aU^&d. X Wcdx m acdm y ab'^y ac~ab^^' ¥c~~(¥M' '^d'^ otMd 12. Least common multiple = xy^z. X _ xh a -{- b _ ayz + byz d _ dy^ y^~~ xyh' xy ~ xyH ' xz~ xyh 13. Least common multiple = iia^cx^. Ill -\- n 4cmx^ + 4cnx^ m — n 6acm — 6acn 3^2 ~ i2a^cx^ ' 2ax^ ~ i2a^cx" ' 4CX 1 2a^cx^ ADDITION OF FRACTIONS. 35 ADDITION OF FRACTIONS. Page SO, 'iac iiac Sac Kac 27«c . 3. ^ 1 1 h " - = — — , Ans, 2xy 2xij 2xy 2xy 2xy idxz indxz iidxz Adxz ^gdxz . Saoc saoc ^aoc ^aoc sabc 6. ^±±Al + ^Ji^l^l±±j]i,Ans. Page 81. 8. Given. m X 11 4S« + \2x -\- 2011 . 4 5 3 6o 2X I ru Sax 4- 6 4- gay . lo. 1 1- ^ = ^ , Ans. 3 2a 4 i.2a a X ah — ac -\- hx 4- ex . -\- c — c W" — &■ X 4- y X — y X -\- y -{- 2X — 2y -zx — y . 13. 14. 15- 2xy xy 2xy 2xy 2 ■\r X z-^ax _ 2a -{- 2ax+ z ^^^ y (^y ~ (^y ' ' a ah _ax — ay + abx + ahy . ; 1 — 5 ^ f A71S. X -\- y X — y x^ — y^ cd 2y^ hx _ ^cd^ + T,oxy + ^hdx^ . 7,x d s ~ 15^^^ ' a 2n 4- d mil — 2dn — d^ . 16. -, -, — = 7^^ , Ans. d 3^ z^'i * Fractions should be reduced to the lowest terms before reducmg them to a common denominator. (Art. 175, Note.) 36 ADDITION OF FRACTIONS. 17. 18. 19. 20. ad — am + dii am — diJ , — -\ = — -^ = ^ , Ans, y — m. — 7ny my — X — h nx — mx — hy H = , Ans. y m — n my — ny — 4 — 16 — 14 + 6 — 16 + = — 6, Ans» 4a 6c $711 _ 4adx -\- 6bcx — bd^n . o a ^x odx Page 81 — Continued. h d . hx -]- 2d . 2. a ■{ \- c + -=: a •\- c -{ , Ans, 2 X 2X a —X 3. ^ + Y + + d am — at/ — hx-\-hd . -L . — = ^ H z: — z. ' ^ m — y bm — by ns. 7 xy -\- z ^ b —c ^ xy + z . = 2td-{-a-\-b— c , Ans. 5- 5^ + T + a — y ^ 2a — by , 2b ns. Page 82. ^ 2a $bd + 2« . 8. a -{- b — 4y — ; -^ '^"« X •}- y X ■\- y — a^ . ^ " — — —^ — — , Ans. — a a a 10. 3^ + ?/ 4- a — b 3x^ — 2xy — y"^ -{■ (t — b x — y x-y , Ans. II. —a-\-si + X y ^x-y- a'^^eab -sb'^ ^ ^4,^^.. a — b a — b 12. 2a; + 2^4- a-^b 2X^-{-2Xii—2X — 2y-\-a-\-b . ziz. i Jx)lS% X—I X — I SUBTRACTION OF FRACTIONS. 37 SUBTRACTION OF FRACTIONS. rage 83. gabc a d — h ay — dm -\- hm . a 7. — m y my m \ yl m y my (^ -h 3d sa — 2d ^a -{- gd — 12a -f 8^/ ijd — ga . = — ^~ , Ans. 12 Pfifje 84:. m \ v I . h + d\ li li + d 10. — — -— = — + y I m y = bi^^!I^^ , Ahs. my h h — my . II. 711 — -— , Ans, y y my ISOTE. m = —^ • y hd + cli . = a -{ , — , Alls, cd h d — h 2a -{- h d — h 13. a -\ = 2 3 2 3 6a -[- 2>h — 2d -\- 2I _ 6a ■\- ^b — 2d . 38 MULTIPLICATION OF FKACTIONS. a c ad -[- ay — he -\- ex d — X d -\- y ~ (b — x) (d -\- y) ad -}- ay — be -\- ex bd — dx -^ by — xy^ ^ X 'id 2X xdti 2X-\-yl'u , 15. a — =:ij ^-^ = a :il^^ Arts. •^ y 2 2y 2y 2y X — 11 a — b x^ — y"^ — 10a + 10b , 16. ^ ; — =: — — ^- , Ans. 10 ^ ■\- y 10^ + To^ 2X3/2 3 _ (iX—iy-\-zc—2X-\-2y-\-(ia _ 4X—y-\-sc-\-6a — ^ — — — —:- — J JlTlS* 6 MULTIPLICATION OF FRACTIONS. Case If Page 85, 5. Cancel the denominator, h + 3^, A71S, 6. Cancelling, -- x IS == — , Ans. 4 2X — 7,11 , ,, 6ex—Qe'u-\-Adx — ^dy . 7. -^ X (3c + 2d) — - ^ -^ , . ^^, ^/2S. 8. ay. — X 6;^; = 2aZ>c, ^?is. IX a ■\- b Kx a -\- b . x^- = — 7—-, Ans, 202; + 2sxy I 4 + 5^ 10. Factor and cancel e. a 4- ab 2c^ 4- 2«'^J . X 2ac = — y , Ans, he -\- c b + I A 2X 4- ■? 11. — ^^ x'2^ = Sx^ ■{- I2X, Ans. 2 3 12. -— — X = (22' 4- 3) (^ — b) 5 I = 2r^j^ — 2bx -\- T,a ^— 2>b, Ans. MULTIPLICATION OF FRACTIONS. 39 13. -j X {d — x) ^ abc, Ans. it — X ^ „. a -\- b a -\- 1) . 14. Cancelling ^x, x 4a; = , Ans. 15. Cancelling 82? — 2. -^^ X (Sz — 2) = -, Ans, 40Z — 10 "^ ^ 5 4, 2B 4 17. Cancelling y — i, 3a; (^ + i) = ^xy ^- ^x, Ans. m^ X 4- z m^ X'^ — Z^ I X — z 18. Cancelling x -\- z, — ^ x — ; — = ;^ — ":v' ^'^^* Case II, Pages 86, 87, 3- 6xy. 4. dx ay 5. x^ — y^ ifz 4- \jZ^ 6. — X ^ — ft + 3:^ 8 8« + 24.-?; , Ans. 3a; (ft + ???) c 2i^x „ ft + ^ c^ d(a -\-h) . 8. — 5— X — = — '^ -i Ans. (r X ex 2X — y 6x — 2y 2x — y 2 (3a: — y) 4X y^ — 2xy ~ 2, Ax — y (2X — y) XX — y y — XX = ^ ^ = ^- ^ , Ans. , — 2xy 2xy 10 MULTIPLICATION OF FRACTIONS. 12. Change the signs in the multiplicand, then factor, and cancel. 4a — 2b 2b — 4a 2 (b — 2a) , P — 2ab ~ 2ab — b^~ b (2a — b)' 2 (b — 2a) (2a — b) _ b — 2a . b (2a — b) 6a ^ab ' 13. Reduce the mixed quantity to a fraction; then cancel the x. ax -{- a 4- b ax a^x 4- a^ 4- ab . X -T- = T J Ans. X by by 2:c XII -\- 2X x(y 4- 2) 14. X -\ = -^^— ^- — = ^^ ^ : y y y ^\y + 2) ^ (cc + ^) _ («/ + 2) (:r + y) y x^ xy xil 4- 2X 4- y^ 4- 211 . = ^ —-^ — ' — -, Ans. xy ^ X X y X xy x^ — ?/2 x^ 4- y"^ x^ — ?/* , — X —^ = TT^, Ans. X xy x^y offi c(^b 4- 2 a 16. Reducinff to a fraction, a -\ — ^ — -. ; ° ab ab c^b + 2«2 2ab , , . ab a^ Case III, Page 88. abc dx abclx , 2. — X — = , Ans. I cy y ad b -{• c abd -f acd . 3. — X = — — , Ans» I xy xy . MULTIPLICATION OF FRACTIONS. 41 ax m -\- n mx + ^^^ j A. — X = — J Ans. 1 4a 4 a^k AC _ 4ac + 4ch . 6. ^ X — = — ^-^, Ans, ^ y y 7. — —— X -^ — = --^, Ans, I X — 1 X — I 8. Cancel i -{- a. I — a^ yx , . . X — - — = 7a^ui — a)=^ IX — nax, Ans, I 1 + a ' ^ ^ ' ' ' 9. Cancel x 4- y. x^ — y'^ ac ac {x — y) acx — acy . X — 7 ; r — — } A71S, I si^ + y) 3 3 ri 1 7 a^ + ah 36' T.ac . 10. Cancel a + 0. x —r-^ — 77 = -— , Ans, ^ I 2{a -\-b) 2 ' a;2 4- I 2«!ic 2a:(^ -[- 2ax , 11. X —7 r — , Ans. I 3 0'^ — 3^' — 3 ^ , , 2:?;?/ [a — b) 4X Sx^y , 12. Cancel a — o. — ^^ x -^, — 7, = t> ^^^^• 1 a^—b^ a + b r. 1 3^(^ l) 2m 6ff??; . 13. Cancel x — i. --^ x -1, = , Ans. ^ I x^ — I a; + I 2ab + ^ a^y 2«^a:?/ H- ^rry . 14. x f-T = —T^—^ ^^^«- I 4fl + c> 4a -\- b 15. Change the order of the terms in the denominator, and cancel i + ?^ I — n^ I , X = I — n, Ans. 42 MULTIPLICATION OF FRACTIONS. EXAMPLES. Page 88. 3 xc — d %^x qcx — 2>dx . 4 2. Cancel ^ — i. - x = 33; (y + i), Ans. „ , „ x^y + 2x^ X -\- y (y-\-2)(x-]-y) 3. Cancel x^. -^-^ x — ^ = \^_^_ls__iju_ ^^ •^ xy x^ xy xy 2ax _ 3«5 _ 3^ 2)^0 _zc^ a ' ac c ' 2ab ~ 2^' 2X 35 3c . I C 2C> %o? Sy a . ^ %%y ^a 6' 2 3 o^ — W' a X . 6, X — — T X T = X, Ans, a a + a — Page 89. m^ X 4- z m^ . X — — = , Ans. *' x^ — z^ I X — z 8. ^^— X ^- = 6a^v^, Ans, y I 9. Cancel x — y. x+ji ^ x^-2xy+l^ ^ y)(x-y) =x^-y\ x — y I 10. Cancel ^z — 2. 2X + y 82; — 2 2x -^ y . ^- X = —^ , Ans, 40Z — 10 I 5 n JJIVISION^ OF FRACTIONS. 4 \ X / \y xj X xy x-y 20^ a^b -\- 2«2 a^{h -{- 2) ^ ab ~ ab ~~ ah ^ (2«2\ 2db cu^(h-\-2\ 2ab , . "^ab) ^ -«^ = ^J— ^ -«^ ^ ^* + 4' ^«*- 13. Cancel c + ^ and 2a. ^^ ^- X -, = 2a(c -\- a), Ans. 2a c -^ d V ' /J 14. Change the signs in the multiplier, and cancel 2x — y and 2. (Art. 166.) 2X — y 2y — 6x 2x — y^ 2 (y — 30;) y — ^x 4X 2xy — y'^ 4a; y (2X — y) 2xy 2bc i^ — bc+2bc ¥-{-bc b (b + c) 15. 4- T = r = "I = -^ ') J ^b—c b—c b—c b—c ' , 2bc b"^ -\- be — 2bc _b'^ — be _ b {b — c) ^ "^ V+~c ~~ b + c~~b-\-c~ b + c~ ' V-^b^cl ^V'b + cJ- b-c ^ b-{-V = l^, Ans» X — y ax X DIVISION OF FRACTIONS. Case I, Page 90, 6X^11 2X , 5. — ^ -^ 2>^y — — , Ans. 2«2 2a . 6. '- =: —r J Ans. sac 2>bc I ab\ b , \ c / 44 DIVISION OF FRACTIONS. 8. ax -\ := ax -\- xy, and (ax + xy) -^x- = ^ , Ans, aia -\- x) / , X (t . c? — (? f . a^ -\- ac ■\- & . 10. -J— '-(a — c) = 7—; J Ans. -\- c ^ -\- c 1 1. The numerator is a square. (x + yf . , . X -\- y . ~ — -^- -^ (x -{- y) = — -^ , A 7is. 12. — ^ — f ^(a + l) = f„, A71S, a — • a^ — tr lO II 12 Case II, Pages 91, 92» 3 times. 5- ab ay a^y . m X cax mhx ax ax^ . 6aby 2y 4y^ '10? 2 "KX . :^— X ■—= — f Ans. 4 ««/ 2y xy 2 2 . — ^— X — = , Ans. X — I xy X — I a — I ax c^ — a . X — = , Ans. X 2 t^x^y^ 2oahx 2X^y . ^—^ X = — - , Ans. loab isxy 3 1 2 (;?; 4- y) 2ab , . — z X —7^-, — \ —(i, A us. ab 4 (^ + //) 'ic^b^ a a^h , X -7—j = ; — tj A71S a + b (>ab 2a -\- zb DIVISION OP IHACTIONS. 45 X + 7/ 6bc 3^ + 31/ A 13. -Jyt X — = , ", Arts. '^ 2b^c^ axy aocxy X — a 2d X — a . 2X1/ Ay 8z/2 15. , X -^- = ~ , Anc. ^ ■\- y 3^^ 3^^ +. 3^^ 2X^lj'^ 211 ^Xlp . 16. ^ X — ^ = — ,^ , , ^^ rt + 6 3.^'?/^ 3az + 3152 a.-^ Z>a; hx^ . 17. — 5X — = -^,Ans. ' ax^ ax ci'^ Kb iSab b . 36aa loby 4ai/ 20. loby 3^ad_Ady . Case III, Page 92 — Continued, ex abchnii . 2. dby -T- -7- = ~ , Ans. ^ dm ex ax m 4- n mx + 7ix . 3. — X = , Ans. ^ 1 4a 4 a 4- X X ax 4- x^ . 4. — — X — = , A71S, I 5c so Kx — y Kx 2Kx^ — Kxy . ^ ^ y y 6. — -L- X -^; — = -^-^ , ^^S. I X 4- 1 a- + I 7. X -~ — = 3^ — 3«^; Ans, ' I I + « 46 DIVISION OF FRACTIONS. lo. II. 12. 13- « + I a — I a — 1 a -\- 1 a — h X + y x — y ' a-\-h x^ — y^ a — I x^-y x-\-y a — I a -\-b ' a 4- I a — I ' • a — I ' a -{- L a + I rt + I a — I a — I a^ + 2a + I a^ — 2a + I , Ans, a — h a -\- h c? —W- X — - — ^ X -\- y X — y 0? — y'^ , A71S, x^ — y^ ^-hy ^^ — xy^-\-x^y—y^ I a — a — X 4- y X — 1/ x^ — y^ , a—h a 4- b~ d^ — 1?' X y EXAMPLES. I. 2. 6. Page 94, iKahc I 5« . -^— X -^ = -^— , Ans. 4xyz $oc 4xyz SSicd ^ _i_ '^% ^^^^ 2^y gxy 24sxy 13a 2cd ca X X x — y_z^ 14b x-Yy a -\- h 2^xz , Ans. 42ah x^ — y^ 2Zxyz ^ A as. 36c K^ — y) G DIVISION OF FRACTIONS. 47 8. ^a a — X a . X = , Ans, 3 a ^ X a^ — x^ 4C^— Sc X -^ y _ 4c(c — 2) ^ ^ -\- y X -\- y (^ — 4 X ^ + y " {c ^ 2){c— 2) AC c -\- 2 , Ans, (P- — dx 4 id + a;) 10. ^ X ^ ' d (d — x) 4 (^ - '.) X ac -{-ax s{c — x) a {c -\- x) 3 (c — r) _ 4d {d^ — x^) 3a (c^ — x^) ' II. 2c2 a ■}- G X 2C a^ + c^ d^ — ac -{- (P' , Ans. 4 ia^ — x^) a — X 4 {a^ — 2ax -\- o?) 12. ^^^^ '- X —7 ^ =^-^ ^, Ans, X 3 (« + ^) 3^ 13 « — h X a -\- h ' {a + l){a-\-b) d^ —^ a^ -\- 2ah + IP {a 4- bf , Ans. 14 a; X — 1 X a; a; a;2 — I a; + I (a;+i)(a;+i) x^ + 2X-}-i , Ans, h(P + bed I + a 15- — — — — X be {c -\- d) I ■\- a X X -\- ax b{G ■{• d) X (i + «) b{c 4- d) = -, A71S, X 48 SIMPLE EQUATIONS. SIMPLE EQUATIONS. Payes 97, 9S, 3. Given i — c '\- x = a — d Transposing, x = a — b + c — d, Ans. 4. Given x -{- ab — c = a -\- b Transposing, x z= a -\- b — ab + c, Ans, 6. Given t,x -\- a — 6 =^ b — 4 -\- 2X, Transposing and uniting, x z= 2 — a -\- b, Ans. 7. Given x — 3-fc= 2X ^ a — b Transposing and uniting, — X =^ —b — c + «+3 ' Changing all the signs, x ^= b -\- c — a — 3, A71S. Note. — When it is more convenient, the unknown quantities may be transposed mentally to the second member and the known quan- tities to the first member. The correct result mav then be set down in regular order, and thus save the intermediate step of changing all the signs. 8. Given 2y -\- be — ad = y -\- 27n — 8 Transposing, y z= ad — be -\- 2m — 8, Ans. 9. Given 3rtZ» — y ■{- d =^ — 2?/ + 17 Transposing, ^ = 17 — T^ab — d, Ans, ic. Given 4c^-f 27— 4:^ + rZ = 2S — ^x-{-^bh Transposing, 4^6^+27 — 28 + ^'? — 3M = 4.^—3:?; Uniting, x = ^cd-\-d — 2fih — i, Aris. 11. Given J 4- c — 4.T = 32 + 5 — ^x -\- d Transposing and uniting, x =: 2,2 — c -\- d, Ans. Note — The 6 when transposed becomes minus, and cancels +h. Thus, &— 6 = o. 12. Given re + 4 — 2a; — 3 ■= 3.^ + 4 + 8 — ^x Transposing and uniting, x = 11, Ans. X E U N K N O W K QUANTITY. 49 Page 10(K 3. Given ^-+ 12 = ^+ i ^ 5 3 Multiplying by 5 X3, 9:^: +.180 = 20.T + 15 Transposing and uniting, ii£ = 165 X z^ 15, ^M5. 4. Given = 62; — 66 36 Multiplying by 6, 4a; — 2; — 36.?; — 66 x 6 Transposing and uniting, 33a; = 66 x 6 Dividing by coefficient, a; = 2x6 a; r= 12, Ans. 3 10 5 4.T + 6 350- — loa; >g^ 14a; 344 a: 24f, J7i5. 4« -5* 3^ X d r^- 4^ 3 5. Given — + ^ = 35 — a; Multiplying by 10, Transposing and uniting. Dividing by coefficient, 7. Given Multiplying by dx, 4ad — 5^f/ = — sbx Transposing, ^bx = ^bd — 4ad Dividing by coefficient, x =z , , Ans. yx 2b + c 8. Given ^x — ^— = a 5 10 Mult, by 10, 30a:: — 14a; = 10^ — 2b — c Uniting the terms, 16a; = 10a — 2b — c Dividing by coef., x = -, , Ans. 9. Given —X-] 1-^ — ^ 3 4 24 Multiplying by 24, — 24a:; + 8a; + i8a,' =15 Uniting, 237 = 15 Dividing, a: = 7^, Ai?s. 3 50 SIMPLE EQUATION'S. XX (* lo. Given « + J -f c = - + ~ + « -I- Z* + 24 5 Transposing, Dividing, 4^ 2X + X -- 4C 5 3^ 4C — 40 5 20c 5 ■4C X 1 6c 15' ^?iS EXAMPLES. Page 101, tjpiven X X x-\-- + ~ 2 4 14 Clearing of fractions, 4X -\- 2X -\- X 56 Uniting terms, ^x - 56 8, Ans. Given X 7X h 40 10 Clearing of fractions, 5a; + 10^ _ jx 4- 400 Transposing, ^x = 400 • • •// —~~. 50, Ans. Given AX rx 10 ^ -^ Transposing, X 10 3 Multiplying by 10, X 30, Ans. Given '' +6 X— 2 8 Transposing, 14 2 Multiplying by X — 2, 14 _ 20;— 4 Transposing, 2a: 18 • • X — 9, ^l?^5. OKE UN^KXOWX QUANTITY. 51 ^. 2X + I 5.. Given X -\ — = 10 Multiplying by 5, ^x + 2X -\- i = 50 Uniting, 70: = 49 X = 7, A)is. c 1^- 6a: + II ox — 20 6. Given 2X H ' = 18 + 5 4 Clearing of frac, ^ox-\-2dfXArAA = 360 + 45.T— 145 Transposing, 19a; = 171 X = 9, ^?iS. /y» /y» /v» 7. Given - H h - = 78 234 Clearing of frac., 6x-\-4X-\-^x = 78 x 12 Uniting the terms, i;^x = 78 x 12 a; = 6 X 12 = 72, A71S. on- 4^ — 6 _ x — 6 8. Given -^— ^ 8 = \- s 6 4 ^ Mult, by 12 (Lc. m.), Sx — 12 = 32: — 18 + 156 Transposing, 53; = 150 X = 30, ^7^5. ^ . 30; — 5 4a: — 8 9. Given ic 4- + ^^— ^ — = 1 2 2 Clearing of frac, 6x-\-gx — 1 5 + 4a; — 8 =z 72 Uniting terms, 19a; = 95 • • X — s, xLTliS* n- - 4a: + 8 10. Given zx— 16 =. - — — 3 Multiplying by 3, 6a; — 48 = 4a; + 8 Transp)Osing, 2a; = 56 a; = 28, Arts. ^. 2a; — 8 X + -12 X 11. Given 1 -^-^ + - = 30 4 2 ^3 ^ Clearing of fractions, 6x — 24 + 6.r -{- 192 + 4x = 360 Uniting the terms, 16a; = 192 a: = 12, A?i8. 52 SIMPLE EQUATIOi^S. 12. Given - + ^ z= i6 + - 2 6 Multiplying by 6, 3a; + 2a; = 96 + 2; Uniting terms, 4^: = 96 a; = 24, Ans, 13. Given 10 = 1- 2 36 Clearing of frac, 9^ + 3 — 60 = ^x -\- x — i Transposing, 42; = 56 a; = 14, A71S. 6x ^x 4X 10 6 ~ 15 14. Given tt + ^ = ^ — i A Eeducing to lowest terms, ~ — \- ^ =z 5 6 15 15 Mult, by 30 (1. c. m.), 18a; + 2^x = 8^ — 38 Uniting terms, $^x = — 38 Page 102. 16. Given ^f 6 -6-f 2X 6 X 6 + ^ Uniting terms. 2iC 3 8 Multiplying by 3, • • • 2X X 24 12, ^7i5. 17. Given Clearing of fractions, • • 4X 5 16a; X 7 -= - 15a; + 60 60, Ans. 18. Given ?x - -4 X -. + ' Uniting and mult, by 2 Transposing, 42; 3^ a; + 12 12 • • • X 4, ^?ic ^^ 9 3 Multiplying by 9, ^x =z gx — 6x -{- 168 Transposing, 2X = 168 x = 84, Ans. rage 103. 34. Given - — [- x = 25 4 2 Clearing of fractions, 3a: + 4X — 2X =: 100 Uniting terms, 5a; = 100 a: = 20, ^?25. 37 X 35. Given 80 = 4.?: ^ 2 6 Clearing of fractions, 480 — 242; — 3a: — :c Uniting terms, 20a; = 480 y% i» = 24, Ans. 4 56 SIMPLE EQUATIONS. ^. 2a: + I ^ + 3 36. Given =20: 3 4 Clearing of fractions, 8a: + 4 = 24a; — 3a; — 9 Transposing, 13a; = 13 •*• X — I, JxtlS' 37. Given 10 — 2a; = ^^ — ^— ^ ^- 3 3 Multiplying by 3, 30 — 6a; = 3a; + 4 — 24 + 36 Transposing, 92: = 14 ic = if, ^^5. 38. Given a; — 3 = 15 — ^ Uniting terms, x -{ =18 Multiplpng by II, iia; + a; + 4 = 198 Or, 12a; = 194 a; = 1 6 J, A71S, ^. a: + 8 a; 4- 6 ^9. Given a; + 2 = 3a; -I 4 3 Clearing of frac., i2a;+24 = 36a;4-3^ + 24 — 4.r — 24 Transposing, 23a; = 24 ^. -ix X — 4 X — 10 40. Given ^^ H = x — 6 42 2 Uniting terms, - — f- 3 z= a: — 6 4 Or, 3^ 3= ,• _ 9 4 Multiplying by 4, 3a: = 42* — 36 Transposing, x = 36, Ans, 0^'E UNKNOWN QUANTITY. 57 ^. IIX — I 5X — II X — I 41. Given = 12 4 10 Clearing of fnic, 55'"^'— 5 = 75^' — 165 — 6a; 4- 6 Transposing, 14X = 154 .*, X = II, Ans, 42. Given ' — = 120 5 ^o Multiplying by 10, Sx — "jx = 1200 X = 1200, Aiis. ^. 2a; -f- I 43. Given X — 20 = 5 Multiplying by 5, s^ — 100 = — 2.T — i Transposing, jx = 99 X = 14I, Atis. 44. Given = \- 12 —X 2 3 Clearing of frac., gx — 15 = S — 4X -{- 'j2 — 6x Transposing, 19:?; = 95 • • X S, jrLflSt /-^J • I ~~~ X 2X I — 'iX 45. Given — p 1- 10 = ^— 6 32 Clearing of fractions, i — a- + 60 = 42; — 3 + 9^ Transposing, 14a; =z 64 X = 4f, A71S. 46. Given = b « + I a — I Clearing of frac, ax — x — ax — x = a^ — b Uniting and factoring, — 2x = b{a^ — i) Changing signs and dividing, x = -(i — a^), Ans. 68 SIMPLE EQUATIOifS. 47. Given X 2 -\- X a — h a + b~ df^ —h^ Multiplying by a^ — ¥, ax -\- hx — 2a -\- 2h — ax -\- bx =z c Transposing, 2bx = 2a — 2b + c 2a — 2b -\- c Note. — This answer may be ex- pressed in various forms, as ; •V 2b X — a 7% c 2b' or X 2a -\- c - h or a c . X = --, -\ — 5 — I, Ans. 2b 48. Given Za -{-X X Multiplying by x, sa -\- x Transposing, 4X X = 6 5^ + 6 Za — 6 Za — 6 , Ans, 49. Given Clearing of fractions. Transposing, hx 2 2fix z ^bx = 6d -, bx a — — 3 6d — 2bx X =z 6d . -=r, Ans. 5^ 50. Given 8a = Multiplying by i + a;, 8a -\- Sax Transposing, Sax + x Factoring, ( i + Sa) x X = I — X 1 +x I — X I Sa I —Sa I Sa I + Sa A71S. OKE UKKNOWK QUANTITY. 59 51. Given x^ -\- j^x -\- ^ 4ab X -\- 2 Reducing to lowest terms, x -\- 2 Transposing 16^ a 4 a O' ar = 2, Ans. 2. Let Then And Let Then And Adding, 4. Let Then Adding, 5. Let Then And a a coat. Ans. PROBLEMS. rage 105. X = value of vest, 4X =z Sx = X = $S, vest; 4X = $32, coat, X = amount paid A, 2X = " " B, SX=" '' 0. 6x = $9000. X = I1500, A received; 2X = $3000, B ^x = $4500, C number of men. 2X = 22a; = " boys, a women. A7tS. 25a; =2 1000. X = 40 men ; 2X = 80 boys; }• Ans, 2 2X ■=. 880 women, X z= distance one runs, 2X = '^ the other runs. SX =z 120 miles. X =z 40 miles ; ) , o ,^ ^ Ans. 2X =z 80 '• f 60 . SIMPLE EQUATIONS. 6. Let 2X = number of barrels. Then loic = cost of one kind. ^x = " " the other kind. Adding, iSx ^ '$1200. Dividing by 9, 2X =z 133! barrels, A71S. 8. 10. II. Let Then Adding, iC a Let Then X = number ist receives, 2X = " 2d Sx_= "3d Sx = g6 pears. i?; = 12 pears, ist; 2X = 24 " 2d; SX = 60 " 3d, 12a; = length of post. 3:^ + 4X +12 =: 12a:. ^/^s. Uniting terms, SX = 12. a; ^ — 12 12a; = if^ == 2 8f feet, ^?is. Let 2oic = sum at first. Then 20a; — ^x—^x = I72. Uniting terms, 122: = $72. X = $6. 20:?: = -^120, A71S, Let a: =r B's share, Then 2X = A's " 3^ =: C'S " Adding, 6x =z I300. X = $50, B's share; 2.'?; = ^100, A's 3:?; = $150, C's Let X = age of wife, Then 2X = " " man. And 2a: + T 8 : .t 4- 1 8 : : 3 : 2 Changing to an equation, 4:r -|- 36 =z 3,?' + 54 Transposing, .t = 1 8 years, wife's age ; ) j ^ 2X = 36 '' man s a i( Ans. a ONE UNKNOWN QUANTITY. 01 12. Let X =z amount each invests. Then x + $1260 = {x — $870) 2. Or, X -\- ^1260 =^ 2X — $1740. Transposing, x = -S3000, Ans. 13. Let X =z one number, Then i^H- 25 = other " And 2 {2X + 25) = 114. Dividing by 2, 22; + 25 = 57. Transposing, 2X = 32. X ^:z 16, one number; ) . a; + 25 = 41, other " ) 14. Let 60a; = amount he had at first. Then 6ox — 20:6' — 15^ — 12a; — lox = $300. Uniting terms, ^x = I3C0. .-. X =1 $100. 60X =z $6000, Ans. Page 106, 15. Let X = the number. Then ^x — 17 = 22. Transposing, ^x = 39. X =:z 13, Ans. 16. Let X zr: number of days they worked. Then gx + 6x = 450. Or, 15a; = 450. ^ = 30 days, Ans. ly. Let X = No. of hours each is on the road. Then 40.x- ~ No. of miles one travels, 302: = " " '' the other travels. Adding, jox = 420 miles. X =z 6 hours. 4o,r = 240 miles; )^^^^._ ^ox =180 '' ) 02 SIMPLE EQUATIONS. i8. Let 32; one part, Then 4X other part. And yx . 28 inches. X 4 " 'ix 12 inches ; ) . 4X 16 " ) 19. Let X — Henry's money ; Then ^x . - Charles' money. And 8x $200. X - I25, Henry; | ^^^^ 'jx — I175, Charles, j 20. Let :c r= the time past midniglit. Then x -\- ^x =: 12. Clearing of fractions, 7a; + 3a; = 84. Or, 102; = 84. X =^ 8/0 hours, or 8 hr. 24 min. A. m., Ans, 21. Let a; = number of days. Then — = part A does in i day, 20 ^ — = " '« " X days. 20 -^ By conditions, ( 1 = i. •^ 20 30 40 Clearing of frac, 6a:* + 42; -f- 30; = 120. Uniting terms, 13^= 120. a; = 9^ days, Ans. 22. Let a; = number of each. Then loo.r = amount received for horses, 45a; = " " " cows, t^x ^ " " " sheep. Adding, 150.^ = I4800. a; = 32, the number of each, Ans. ONE UNKNOWN QUANTITY. G3 23. Let Then Adding, 24. 2X IX \ox X 2X SX Sx number ist receives. " 2d " 3d " 150 oranges. = 15 30 oranges, ist; 75 " 2d; 45 " 3^, Ans, Let Then X = price of 1 chicken. SX = 6x =z By conditions, i2x = And a 18:?; =z 102: = a a a a a " I goose. " I turkey. " 4 geese. " 3 turkeys. " 10 chickens. Adding, 40a; =: $10.00. X = $0.25, 2)r. of a chicken ; SX = I0.75, " " goose ; Ans. 6x = $1.50, ii a turkey, 25. Let X = length of fish, in inches. Then Addinor 4 m. = 48 in. = X . - m. = 2 a if a " liead, " tail, " body. to? X - + 52 m. = 2 ^ X inches. (( Mult, by 2, a; +104 in. = 2X X = 104 in. = 8 ft. 8 in., Ans. 26. Let Then Adding, X 20 + ^ one part, other part. 20 + 2X 2X i X : 20 + a; = 60, other " \ 100. 80. 40. one part; / Ans. G4 SIMPLE EQUATION'S. 27. Let •Then And 28. 30. X = less, a — X =^ greater. a — x X I Clearing of fractions, ad — dx =. ex. Transposing, Factoring, Dividing, ex + dx = ad. (c -\- d) X = ad. ad X = a — X = a — ad c + d ac -\- ad — ad c -\- d c -\- d ac + ad — ad ac j-livv*. ii.v>ixig, c -{• d c + d Less = ad ^ , ac : ^ ; Greater — -— ,, c -\- d' c -\- d' Let 12:?; — A's monev. Then 3^ ^ a 8a: — 1 " " a 2X — I'' " Adding, 13a; I1222. • • • X $94. 12a; In 28, Ans. Page 107, Let X — number lbs. of beef, Ans. Then Hence, 2X =r a " mutton. ii Adding, 25a: = cost of beef. 40a: = " " mutton. 65:^ z= $39, cost of both. X = 60 lbs. beef; ) 2a: = 120 " mutton, j A71S, ONE UNKNOWN QUANTITY. 65 31- Let X B's age, . Then 2X — A's " Hence, 2,x 2X + 15. • ^ _ 15 yrs., Bsage;) 2X — 30 " A's ^' S Let X C's age, Then x-\- s - B's " And a: + 8 — A's " ns. Z^' The sum of the ages, 3.^+13 = 110 yrs. Transposing, 3^ = 97 " X = 32 J yrs., C ; 1 ^-f 5 = 37t " B; fAni<. a; H- 8 = 4oi '' A, ) 33. Let X = votes for defeated candidate, Then x -]- i^o =z " " successful " Hence, 150 + 2a; = 2500. Transposing, 2X =z 2350. ic = 1175 votes, defeated c.;]. And 0:4-150 = 1325 " successful c, f 34. Let X = number of artillery, Then 3a; — 20 = " " cavalry, 3a; — 20 + 92 =: " " infantry. Hence, 'jx — 40 4- 92 = 1200. Transposing, ^x = 1148. X = 164, artillery; \ 33: — 20 = 472, cavalry; - Ans. 32; — 20 + 92 = 564, infantry, ) 35. Let X = B's share. Then x + lioo = A's X + $300 = C's Adding, 3^ + -^400 = $2000. Transposing, 3./; = $^1600. X = $533i, B's share; X + $100 = %6^sh ^'s " I Ans. X + 8300 = $833^ C's " G6 SIMPLE EQUATIONS, 36. Let Then Hence, 2,x share of one, 5^ a a the other. 2>x 1150.00. X $18. 75- 3^ I56. 25. one ; ] 5^ $93- 75. other, ) ^ 37. 38. 39- Page 108, Let a; = price of one horse. Then %6i6—x = " " the other horse. Hence, 5:?; = 6 x $616 — 6:?;. Transposing, i lic = 6 x '^616. = 6 X I56 = $336, price of one ; \ . i56i6 - a; = I280, " " other, j ^^' X = Let Then And Adding, X = age of youngest, a; + 2 = " " next older, ri; + 4 = " " eldest. 3^ + 6 = 48. 2; = 14 yrs., youngest; a; + 2 =r 16 " next; a; + 4 = 18 " eldest, Suppose A and B are the messengers. Let X =z days B travels, And X -{- s = " A '* Then 652: = distance B ' " And 502; + 250 = " A " Hence, 50:?; + 250 = 65;?:. 15a; = 250. X =: i6|days, A?is. Ans. Proof. 50 x i6| + 250 = 1083 J miles. 6^ X 16 = 1083I <( ONE UNKNOWN QUANTITY 07 40. 41. 42. 43- Let X number. Then (^ + 75) * — 250. Dividing by |, ^ + 75 625. Transposing, X - 550, Ans, Let ^x one part. Then ^x other part. Adding, 2,x — 48. X := 6. 3^ = 18, Ans. Let Then 12.7; =r quantity. 6rc 4- 4^ + 32^ = 13^ = «• Therefore, And Let Then Adding, a X = — 13 12a . 12a; = , Ans, 13 Src = A's acres, jx = B's " 12a; = 540. X — 45. Sx =1 225 acres, A's share; ) yx = 315 " B's <( Ans, 44. Let .^ = number of hours. Consider the cistern a unit, or i. Since in i hr. i faucet will empty J of it, another -^q, another t^, in x hours, all Therefore, will empty - H 1 ^ "^ 6 10 12 XXX 6 10 12 Clearing of fractions, Uniting, » • lox -{- 6x -{- sx = 60. 2iX ^=z 60. X :=^ 2^ hours, 4^?'S 68 SIMPLE EQUATIONS. 45. Let X — I = ist part. Then x -\- 2 — 2^ " And - = 3d '' 4X = 4th '' '>^'^'^' Or, dividing, x = 3x2. X =z 6. Then x~i = 5, istpart;\ a; + 2 = 8, 2d '• / 4X = 24, 4th '^ / 46. Let X =L number. Then 6a; + 12 = 66, the sum. Transposing, 6x = 54. X ■= (), Ans. 47. Let X = whole Xo. of sheep. Then x — 7 = remaiDdei'. $94 a; cost of one. A - -94 ^ — 7 Ana -^ X = ^20. a; 4 Clearing of frac, 94.T — 65S = 80a:. Transposing, 14a: = 658. X = 47 sheep, A}is. 48. Let 4x = income. Then 3.? = A's expenses i yr. And 3a; + I50 = B's expenses i year. By conditions, 20X -i- $100 =r 15a; + 6250. Transposing, 5a: = -Si 50. .-. X = $30. ^a; = §120, A71.S, ONE UNKNOWN QUANTITY. 69 raffc 109. 49. Let X ^ number of minutes required. Considering the volume of the cistern unity, or i, it follows: If tlie supply pipe will fill it in 20 min., in i min. X it will fill o^n, and in x min., — of it. In like manner, the ^^ 20 X discharge pipe will empty ^ of it in i minute, and — in X minutes. Then • ^-^= 1, 15 20 Multiplying by 60, ^x — 3.T = 60. X =. 60 min., Ans, Or, let X = number of hours required to empty it. Since one empties ^ of the cistern while the other fills ■^ of it in I minute, the former gains ^ per minute in the discharge. If to gain -^ requires 1 minute, to gain ^ must require 60 times i minute, or 60 minutes. ^or, ^ ' i% '■' I min. : :i- min. ic = 60 minutes, A}is. 50 Let X = number. Then mx — nx = d. Factorinar and dividing, x = , Ans, 51. Let $x = No. leaps of Greyhound. Then 4X z= " " Hare takes after G. starts. 50 + 4.T = whole distance H. goes. x\gain, 2 leaps of G. = 3 leaps of H. Hence, i leap " = i of 3 " '* — 3. — 2 And 3.^ leaps Qj. J ^ = ^3^^- X = 16. i2oic = r^72o, Ans, 66. Let 12:?; = sum. Then 6x — $30 = A's share. 4x — lio = B's " 30;+ 18 = C's " Adding, 13.1— $32 =12:^. X = I32. I2X = $384, the sum divided; 6a;— $30 ^ $162, A's share; ( . 4:r — lio = $118, B's " 32; + $8 = $104, C's " 67. See Book. Prif/e 111. 6S. Let 3a; and 4X = the numbers. Then 3:^ + 4 : 4^^ + 4 - 5 ' 6 Changing to an equation, 18:2; + 24 = 2o.r + 20. Transposing, 2:r = 4. 3^ = 6 ; 4:?; = 8, 69. Let . 3:?; = greater. Then 2X = less. Adding, 50: = 5760. .-. X = 1 152. 3.'^ = 3456, greater; 2x = 2304, less, Ans. Ans. 70. 7i. 72. X E r N K N W X QUANTITY. 75 Let Then Aud By conditions, Or, Transposing, Let Then X 9 + •'■ 9 — X g -\- X 9 + ^<^ . 3^ the rate of the current, crew's rate down stream. up a 2 (9 — X). 18 — 2X. X ^ $ miles an hour, A71S. 600X = length of rod 30a; = 202: =: 122: := loa; = red, orange, yellow, green, blue, indigo. 302 inches. a u i( a a a a a a a — 302 a Hence, 6oox — 1472^ Uniting terms, 453^*= • /y» — . • • %Mj — 600a; = 400 in. = 2>zi f^v ^ns. 302 2. ¥TT = 4 m. Let Then 24a; ^ whole number of kings. 2)X = kings of one name. 6x = " " another name. a ic (( <( a a Hence, ZX = 2X =: 24a; — 19Z = 5 kings. . « X ^—* X . ii Hence, there were 8, 6, 3, and 2 kings of each name, respectively, Ans, 73. Let X Then x -\- 1 And x^ -\- 2X -{- I — 0? Uniting terms, 2X X + I u one number, other 15- 14. = 7; = 8, Alls. 76 SIMl>LEEQtATlOKS. 74. (See Prol). 51.) Let X = No. of leaps of G. Then 80 + J2: = '' '' D. And X leaps of G. = 2X leaps of D. By the conditions, 80 + f ;2: = 2a;. Multiplying by 2, 160 + 3:^ = ^x. ic = 160. ic := 240 leaps, ylws. 75. The steamers will meet in any number of days which is a common multiple of 20 and 25, and tlie time of their first meeting in New York will be the I. c. ni, of these numbers, which is 100 days. Again, since the first makes i trip in 20 days, in 100 days she will make as many trips as 20 is contained times in 100, or 5 trips. In like manner, the second will make 4 trips. Let X = number of miles ist sails. Then i trip : 5 trips : : 6000 m. : x m. X = 30000 m., ist sails; \ AX ( And — = 24000 " 2d " } Ans, 5 I Time of meeting = 100 days, 7 Or thus: Since the ist makes a trip 5 days sooner than the 2d, it is plain she will make 5 trips while the other makes 4 ; now 4 times 25 days =100 days. Again, the second will sail f as many miles as the i st in the same time. Let X = number of miles ist sails. And ^x = " " 2d " Then ^x = 6000 miles. X = 30000 m., ist sails; ) And fx = 24000 '• 2d " \ Ans. Time of meeting = (20 x 5) or (25 x 4) = 100 da. / TWO U is KNOWN QUANTITIES. 71 SIMULTANEOUS EQUATIONS. TWO UNKNOWN QUANTITIES. Page 114:. 2. Given x -\- y =^ 12 (i) And a; — 1/ + 4 = 8 (2) From (i), x=i2—y (3) " (2), x= 4^y (4) Equating (3) and (4), 12 — y = 4 -{- y Transposing, 2?/ = 8 Substituting 4 for ?/ in (i), x ==z S, ) 3. Given ^x -{- 2y = 48 (i) And 2x — ^y — 6 (2) From (i), 2: = 4^1:1^1 (3) " (3), .^^ (4) Equating (3) and (4), 2 48 — 2y _ 6 + 3^ 3 2 Clearing of fractions, 96 — 4y = 18+97/ Transposing, 13^ = 78 Substituting value of «/ in (2), a; =: 12, ) Given a: + y ~ 20 (i) And 2X -{- ^y = 42 (2) From (i), x=2o — y (3) « (2), ^ = ^~^ (4) Equating (3) and (4), 20 — y = ^^—'-^^ Multiplying by 2, 40 — 2?/ =r 42 — 3?/ Substituting 2 in (i), a: — i8, j ^^* 78 SIMPLE EQUATIONS. 5. Given 4X -\- ly — 13 (i) And 3X -^ 2y =z 9 (2) From (i), X = i^-^M (3) 9 — ^y 3 13 — 3y _ 9 —_^y " (3), ^ = -T^. W Equating (3) and (4), Clearing of fractions, 39 — 9?/ = 36 — Sy 2/ = 3 j I ^1^5. Substituting 3 for ?/ in (2), 2: = i, f 6. Given 3^' + 2^ — ^^^ (I) And X + sy — 191 (2) 118 — 2W From (i), X - ^ ^ (3) " (2), ^ — 191 sy (4) 118 — 2«/ Equating (3) and (4), 191 52/ — " ^ Multiplying by 3, 573 — 15?/ = 118 — 2U Transposing, 13?/ = 455 ••. ^ = 35 ; ) ^^^^^ Substituting 35 for y m {2), x = 16, ) Given 4X -{- sy — 22 (i) And 7a; + 3?/ = 27 (2) 22 — Ky , . From(i), x = ^^-^ (3) " (a). ^ = -^^ (4) Equatnig (3) and (4), = Clearing of frac., 154 — 35// = 108 — 12?/ _ Transposing, 23?/ =: 46 y = 2 ; ) . Substituting 2 for ?/ m (3), a; :^ 3, ) TWO UJS'KJSTOVVK QUAXTiTIES. 79 Case II f Page 111 — Continued. 9. Given rr + 3// = 19 (i) And ^x — 2y = 10 (2) From (i), X =19-3?/ (3) Substituting value oix in (2), 5(19-3^) — 2y=io (4) Keducing, 95 — 157/ — 2?/ = 10 Transposing, etc., 17?/ = 85 2/ = 5 ' I ^ Substituting 5 for y in (3), a; = 4, f 10. Given - + ^ = 7 (i) 23 ^ ^ And - + ^ zrr 8 (2) Mult, (i) by 6, 2>x-^2y — ^2 (3) Mult. (2) by 6, 2.T + 3^ = 48 (4) From (3), a; = ^^^— (5) Substituting value of .r in (4), '-^^ + 3. = 48 (6) Mult. (6) by 3, 84 — 4?/ + 9?/ = 144 Uniting terms, 5?/ = 60 y =12'/ Substituting 12 for y in (5), a: = 6, IT. ^ns. Given 2^ + 3?/ — 28 (0 And 3a: + 27/ _ 27 (2) From (i), ^ 28 - 31/ 2 (3) Subst. in (2), 84- 2 97/ — - + 2y 27 Mult, by 2, 84- - 9i/ + 4^ - 54 Uuitiug terms, 5^ - 30 ci__i - J • J _^ J • _- ^ c > • ?/ 6 ; I 80 SIMPLE EQUATIONS. 12. Given 4X -j- y = 4$ (i) And 52: H- 2?/ = 56 (2) From (i), y = 4s^4X (3) Siibst. in (2), 5a? -|- 86 — 82: = 56 Uniting terms, $x =z 30 cc = I o ; I Substituting 10 for x in (3), ?/ =: 3, f 13. Given ^x -\- S = yy (i) And 5?/ + 32 := 7.-1; (2) From (i), y = ^^ (3) Subst. m (2), -^ — -!— ^ + 32 r= yrc 7 Mult, by 7, 25:?; + 40 4- 224 = 49a; Transposing, 24:?: = 264 Substituting 1 1 for x in (3), y = g, ) 14. Given 4.?; + 5?/ — 22 (i) And 7^ + 32/27 (2) ^ / X 22 — t;?/ From (i), a; ^ ^ ^ 4 (3) Substituting value of x in (2), 154 352/ , ,„ „ 4 Clearing of fractions, 154 — 352/ + 12?/ = 108 Transposing, 232/ =r 46 ?/ = 2 ; ) Substituting 2 for ?/ in (3), a: = 3, j ^ TWO UNK]S"0\VN QLAM'ITIES. 81 Case Illf Page 116, 17. Given sx + 4?/ = 29 (i) And 7x + iiy = 76 (2) Multiplying (i) by 7, 21X -\- 28^ = 203 (( (2) by 3, 21a; + 33^ = 228 Subtracting, 5y - 25 Substituting 5 for y in y - (l), X- ^ ' I Ans. 3. i 18. Given 9^ — Ay — 8 (I) And 13^ + iy - loi (2) Multiplying (i) by 7, 6^x — 28y 56 Multiplying (2) by 4, Adding, S2X + 2Sy — 404 iiSX 460 Substituting 4 for x in X (I), y "^ ' i Ans. 7, ) 19. Given z^-iy - 7 (i) And \2x-\- sy — 94 (2) Multiplying (i) by 4, Subtracting, i2x — 2Sy = 28 (3) 33y - 66 Substituting 2 for y in y - (l), X - ' [ Ans. 7, S 20. Given SX+ 2y — 118 (i) And x + sy - 191 (2) Multiplying (2) by 3, 3^ + 15// — 573 (3) Bringing down (i). ZX-\- 2y - 118 Subtracting, i32/ = 455 ^ = 35 j [ j^^^g^ Substituting 35 for 2/ in (i), 2; = 16. ) 82 SIMPLE EQUATIONS. ti. Given 4^ + 5?/ (i) And ^x + 2>y 27 (2) Multiplying (i) by 3, 122^ + 15^ - 66 Multiplying (2) by 5, 35.6- -f 1M_~ 135 Subtracting, 23a; — 69 • • • Substituting 3 for x in (i), X y - 2 ' [ ^^^^• EXAM PLES. I. Given 2x -\- 7,y . . 23 (i) And 52; — -2y - 10 (2) Multij^lying (i) by 2, 4^ + 6?/ — 46 (2) by 3, 15a; - . 61, 30 Adding, 190; 76 • • • Substituting 4 for x in (2), X y - 5. ) 2. Given 4% + y - 34 (I) And 4y -\- X 16 (2) Multiplying (2) by 4, 42; + i6ij — 64 (3) Subtracting (1) from (3), ^by - 30 Substituting 2 for ?/ in (2), 2'= :'U„.. :c := 8, ) Given 32 + 4?/ _ 27 (i) And SX + sy - 34 (2) Multiplying (i) by 3, 9^ + 12?/ _ 81 • (3) (2) by 4, 202: +121/ 136 Subtracting, 11^ - 55 Substituting 5 for x in (3), y - 3>) TWO UNKNOWN QUANTITIES. 83 Given 2X-^ VJ --^ 34 (I) And 5:/; + 9^ _ 5 I (2) Multiplying (i) by 5, lo-'' + zsy — 170 Multi]jlyii]g (2) by 2, lox + 18^ 102 Subtracting, ny 68 Substituting 4 for y in {i), a: = 3, f ^ 5. Given ^x + 7y = 43 (i) And iia; + 97/ = 69 (2) Multiplying (i) by 11, 55X + 77?/ = 473 (2) Ijy 5. 55-^- + 45^ ^ 345 Subtracting, 32?/ = 128 Substituting 4 for 2/ in (i), x = 3, j 6. Given 8a; — 21?/= t^t, (i) And 6:?;+ 35?/ = 177 (2) Multiplying (i) by 3, 24X — 63?/ = 99 " (2) by 4, 24:^ + 140?/ = 708 Subtracting, 2037/ = 609 y = ^'•'^ Ans Substituting 3 for ?/ in (i), a; = 12, f 7. Given 21?/ -f 20.T = 165 (i) And 77?/ — 3o.f = 295 (2) Multiplying (i) by 3, 63^ + 6o.i- = 495 (2) by 2, 154^ — 602'= 590 i( Adding, 217^ =: 1085 ?/ = 5 ; ) Substituting 5 for ?/ in (i), a? = 3, f " 84 SIMPLE EQUATIONS. 8. Given iiic — loij _ 14 (l) And sx -\- iy — 41 (2) Multiplying (i) by 7, 77^ — 70?/ _ 98 " (2) by 10, 50X + 70?/ 410 Adding, 1272: 508 • • • Substituting 4 for a; in (i), y - 3^ ) Page 117. Given 6y — 2x = 208 (i) And — 4?/ + iojC = 156 (2) Multiplying (i) by 5, 30?/ — lor?; =: 1040 Adding, 26?/ =1196 Substituting 46 for ?/ in (2), :r = 34, f 10. Given 4^ + 3^ - (i) And 5^ 7^ - 6 (2) Multiplying (1) by 5, 20X + 15^ — no (2) by 4, 202; — 28?/ 24 Subtracting, 43^ - 86 Substituting 2 for y in y - (2), X = ' [ Ans, 4, ) IT. Given 3^-sy - 13 (0 And 2x ^ ry - 81 (2) Multiplying (i) by 2, 6x — 10?/ 26 (2) by 3, Subtracting, 6X -\- 211/ 243 3Ky - 217 Substituting 7 for ?/ in y - (0, X = .6; } •^-- TWO LNKNOWK QUANTITIES. 85 12. Given 5^ iy - ZZ (I) And iix + i2y 100 (2) Multiplying (i) by ii, ss^-iry - Z^l (2) by 5, SSX + 6oy - 500 Subtracting, 137^ - 137 y - Substituting i for y in (i), x 8,' } ""'''' 13. Given ^ ,y 5^6 18 (I) And X y 2 4 21 (^) Multiplying (i) by 30, 6:^^+5^- 540 (3) (2) by 4, 2X y - 84 (4) (4) by 3, 6x 2>y — 252 (5) Subtracting (5) from (3) ^y 288 Substituting 36 for y in y - (4), X — 36;) 60, f Ans. 14. Given 1 6a7 + lyy 500 (I) And ijx— sy — no (2) Multiplying (i) by 3, 482; + 51?/ _ 1500 " (2) by 17, 289a: — 51?/ 1870 Adding, 337-^ — 3370 Substituting 10 for re in (2), y - 10; ) 20, j Ans, 15. Given Sx + y — 42 0) And 2X + 4// — 18 (2) Multiplying (2) by 4, %x + i6z/ — 72 (3) Subtracting (i) from (3) 15// - 30 y - Substituting 2 tor y in (2), x 5. ( ns. 86 SIMPLE EQUATlOI^rS. i6. Given 2X' + 4^ — 20 (0 And 4^' + 5^ - 28 (2) Multiplying (i) by 2, Subtracting, 4a; + 8?/ 40 (3) 3!/ - 12 Substituting 4 for y in (0, ^ - X 2. j ns. 17- Given A-^ + 3«/ - 50 (■) And 3^ 3.?/ — 6 (^) Adding, 7^ _ 56 Substituting 8 for x in (2), X y - 8;l 4 ??5. i8. Given z^ + sy - 57 (I) . And 5^ + 32/ — 47 (2) Multiplying (r) by 5, I SX + 25?/ _ 285 (2) by 3, Subtracting, I 5^ + 9y = 141 i6y _ 144 Substituting 9 for y in (I), «/- A, ) /zs. 19. Given ^ ,y 2 3 7 (■) And - H- - 3 4 5 (2) Multiplying (i) by 2, 211 14 (3) (2) by 3, ^ + ?- 15 (4) Subtracting (3) from (4), 3?/ 2,?/ 4 3 I Clearing of fractions, 9y-^ = 12 • Substituting!: 12 for y in (3) y - X 6, ) ' ins. TWO UNKNOWN QUANTITIES. 87 20. Given And 2X -{- y — so X y - 4- - — ^ 67 ^ (I) (2) nx 6+y ^5 '3) ^x 2X ^ 15 12X — 7.6' 90 X 18: ) A Multiplying (2) by 7, Subtracting (3) from (i), Clearing of fractions, .T =r 18 ; ) Substituting 1 8 for .r in (i), ^ = 14, j PROBLEMS. 1. Let X =z one number, y = other " Then x + y =70 (i) And X — y = 16 (2) Adding 2a; = S6 Substituting value of a: in (i), ?/ = 27, ) 2. Let ic = price of a lemon, And ?/ = " " an orange. Then Sx -\- 4y = 56 cents. (i) And ^x + Sy = 60 " (2) Mult, (i) by 2, 16a; + 8^ = 112 " (3) Subt. (2) from (3), 13^ =52 cents. a; = 4 cts., lem. ; ) Substituting 4 for a: in (i), y = 6 '- or'ge, f ^ 3. Let .T = votes cast for one. And y — " " " the other. Then ^ + ^ =:z 375 (i) And X — y = gi (2) Adding, 2X — 466 X = 233, V. for one; | Subst. value a; in (i), y — 142, " other, \ 88 SIMPLE EQUATIONS. 4. Let X = greater part, And y = the less. Then a; -f- ?/ = 75 (i) And ^x — ^y— 15 (2) Multiplying (i) by 3, z^ + ZV = 225 (3) Subt. (2) from (3), loy = 210 y =z 21, less; Subst. value of «/ in (i), a; = 54, greater .! Ans, 5. Let X = price of a horse, And y =z ^^ " cow. Then gx -^ yy = 1 1200 (i) And 6a;+ 13?/ =r I1200 (2) Mult, (i) by 2, i8i?;+i4?/ = $2400 (3) « (2) by 3, i8:^; + 39^ ^ $3600 (4) Subt. (3) from (4), 25?/ = I1200 y = $48, pr. of cow; | ^^^^ Subst. value y in (i), x = I96, " horse, ) 6. Let X = No. of gentlemen, And y = " ladies. Then «/ — 1 5 = ladies who remained, a; — 45 = gent. " " And X = 2 (?/ — 15) (i) 5(^-45) = 2/- 15 (2) From (i), X =z 2y — ^o (3) " (2), s^ — 22s = y - 15 (4) Substituting value of x in (4), 5 i^y— 30)— y = 210 Transposing, gy = 360 ^ = 40 ladies ; ) ^^^^^ Subst. 40 for y in (3), x = 50 gentlemen, f TWO L.NKNOWN QUANTITIES. 89 7. Page 118. Let X — ist, And y 2d. Then 5 a; + 2?/ 19 (>) And Tx — 6y 9 (2) Multiplying (r) by 3, 15a; + 6?/ r= 57 Adding, 22a; 66 Substituting 3 for a; in (2), ?/ = 2, f 8. Let X = No. of men in one army, And . 2/ =: " " the other. Then re + ?/ = 21110 (i) And 20; + 3?/ = 52219 (2) Mult. (i)by2, 2X -\- 2y ^^ 42220 Subtracting, y — 9999, one army; ) ^^^^^ Substituting in (i), x = 11 1 11, other " ) 9. Let x = digit in tens' place, And y = " " units' ^• Then x -\- y = 11 (i) And iz^ 4- 13 = 3^ (2) From (i), X — 11 —y (3) Substituting value of x in (2), II — ?/+ 13 = 3^ Transposing, 4?/ = 24 ?/ = 6, units' digit. Subst. 6 for «/ in (3), a; = 5, tens' '' The number = 56, Ans. 10. Let cc =: A's share, and y = B's share. Then x+ y = 8570 (i) And sx + sy = ^2350 (2) Mult, (i) by 3, 3a; + 3j!/ = 81710 Subtracting, 2?/ = $640 2/ = ^320, B's share; ) ^^^.^ Subst. 320 for y in (i), a; = $250, A's " f DO SIMPLE EQUATIONS. II. Let - = the fraction. Then = -, or 3Z + 3 = ?/ (i) y 3 And z^ - , or 4X = 1/ + I (2) y + 1 4 Substituting the value of y in (2), we have 4:^; = 32; + 3 + I Uniting, x = 4 2/ = 15 And — , Ans. y 15 12. Let 6x A's money, • And 8?/ - B's " Then • 6x -{- y I1200 (>) And a; + 8?/ — $2550 (2) Mult. (i) by 8, 48:?; + 8?/ I9600 Subtracting, 47a; $7050 X = I150 6a; = I900, A's m. ; ) . In (i), y = I300, and Sy = $2400, B's " \ 13. Let X and y denote the numbers. Then x — y =z 14 And X -\-' y — 4 8 Adding 2a; =62, •*• ^ = 3^ ; [ ^^^^ Subtracting, 2?/ == 34, .*. ?/= 17, j 14. Let it' = price of the house, y -- a « garden. Then x + y ^ $8500 y = — 12 Substituting, x -\- -- — 8500 Mult, by 12, etc., 17a: = 8500 x 12 a; =: I6000, house; ) . And ^ = $2500, garden, \ TWO unknow:n^ qua duties. 91 15- i6. Let 40; one part, And 6y the other. Then 4^ + 6// — 50 (i) And 3'^ + 5^-40 (2) Mult. (i)by3, 122; 4- 18?/ 150 " (2) by 4, 12a; -f 2oy 160 Subtracting, 2y - 10 (3) Mult. (3) by 3, 6^ 30, one; )^^^ 4ic 20, other, J Substituting in (0, Let a? — A's share, And y — B's " Then X + y — $1280 (i) Anc. ^x — gy (2) From (2), 7 Substituting, 9y 7 4- ?/ — $1280 ^W5. Clearing of frac., gy + iy = $1280 x 7 Uniting terms, i6?/ = $1280x7 Dividing, ?/ = $560, B's share; Subst. vahie ?/ in (i), a; = $720, A's " 17. Let X = age of elder. And ?/ = " " younger. Then x— y z= 10 (i) And x—is = 2{y— 15) (2) Subt. (i) from (2), ?/— 15 = 2y — 40 Transposing, ^ = 25 yrs., y'nger; ] ^^^^^ Subst. 25 for ?/ in (i), ic = 35 " elder, j 18. Let X = value of ist horse. And y = " " 2d " Then a: 4- 50 r=: 2?/ or x = 2y — 50 (i) And y + 50 =: ;r — 15 or x :=z ?/ 4- 65 Equating the values of x, 2y — 50 = y + 65 Transposing, y =z $115, value 2d ; ) Subst. value y in (1), x =^ $iSo, " ist; j 92 SIMPLE EQUATIONS. rage 119. 19. Let X = distance steamer goes, And y = ^' sliip " Then 20 + ?/ = a; (i) Their respective distances for the same period of time will be m the ratio of 8 to 7. Hence, forming a proportion, X : y :: S : y Reducing to an equation, yx ^ 8y Dividing by 7, ' x = — (2) Substituting in (i), 20 -\- y =z -^ Clearing effractions, 140 -\- jy =: Sy y = 140 miles, ship ; ) From (2), X = = 160 ^' steamer, j 20. Let 6x =z greater. And 6y r= less. Then 3a: + 2?/ = 13 (i) And 2X — $y = o (2) From (2), 2X = 7,y Or, X = M (3) Substituting in (i), — -i- 2?/ = 13 Multiplying by 2, <^y -^r ^y = 26 y = 2 Substituting ?/in(3), a;= 3 6x =18, greater; ) . 6y =: 12, less^ j TWO UNKKOWN QUANTITIES. 93 21. Let ^x -^ part hft, And 6y - " carried away. Then X -^ y — 28 feet. (I) And 15a; -36^ - 12 " (^) Mult. (0 by I (2) from 5. (3), ' 15Z + 152/ - 420 " (3) Subt. 5^2/ - 408 feet. • • y - 8 " And 6y — 48 " (4) Subst. 8 for y in (i), 1 X 20 " And 3^ - 60 " (5) Adding (4) and (5), 3a; + 6?/ = 108 feet, Ans. 22. Let 2x = the lady's age, And y = number of verses. Then y z= x — 2 ( i ) And 2X + y = 43 (2) Substituting, 2:^ + 0;— 2 = 43 Uniting terms, $x = 45 X — 15 2x = 3oyrs., her age ;)^^^^^ Subst. 15 for a; in (i), y = 13 yerses, f 23. Let X = greater, And y — less. Then x — y = 20 (i) X And - = 3 (2) y Fro-m (2), X = sy (3) Substituting in (i), 3?/ — y ^= 20 ?/ = 10, less; I . From (3), X — 30, greater, j 04 SIMPLE EQUATIONS. 24. Let X = No. of oxen, and y = No. of colts. Then 65:?; + 25?/ = 720 (i) And 25a; + 6sy = 1440 (2) Mult, (i) by 13, 845:^ + 325?/ = 9360 " (2) by 5, 125:2; + 3252/ = 7200 720ic = 2160 .-r = 3 oxen ; Subst. in (i) and transposing, y = 21 colts, Ans, 25- Let X digit in tens' place. And y " " units ' « Then ^ + y -\- 7 — 3^ (0 And lore -\- y — 18 loy 4- i? (?) Reducing (i), y — 2X'-'j (3) " (2), X y - 2 Substituting, X — 2iC + 7 2 X 5 From (3), y - z lo:?; + y 53, A71S, 26. Let . 6x A's entire ca ipital, And Sy B's " (( Then 6ir + 5?/ I9800 (I) And 5z _ 4?/ (?) Or, X ^y 5 ' Substituting, ^ + 5?/ 19800 Multiplying by 5, and uniting terms, 49?/ = ^9800 X 5 y = $1000 And 5?/ = $5000, B's; I ^^^^ Subst. in (i) and transp., 6x = I4800, A's, ) TWO UNKNOWN QUANTITIES. 95 27. Let X = part of purse i guinea fills, And y = " " I c'ollar " Then Anc. 6x + 19?/ I 5^ + 4y - U 0) Multiplying (i) by 5. 30X -f 95y - 5 (2) by 6, 302; + 24^ — If Subtracting, 7i«/ — 5 - ,/ I y 21 34 _ 2T - _ 71 - 2T From (2), 5^ + A - H Transposing, 5-^ — 6 3 X — ^j 4 ~ 2T A Now if I guinea fills ^^3 of the purse, it is plain that 63 guineas will fill f |, or the entire purse. Reasoning in like nianner, we find 21 silver dollars will likewise fill the purse. Hence, Ans. $21, or 6;^ guineas. 28. Let X = greater, And y = less. Then .r 4- ?/ = « And X = ny Substituting, ny -\- y = a Factoring and dividing, y = ; J Ans. And X = n+ i' 96 SIMPLE EQUATION'S. THREE OR MORE UNKNOWN QUANTITIES. Page 121, 2. Giyen And Adding Mult, (i Adding Mult. (4 " (6 Adding (2) and (3), ) ^y 2, (2) and (5), ) by 4, ) by 5» (7) and (8), 5^ - 32/ + 22; 2,x-\- 2y — 4Z 2X -h 5«/ 102; — 6?/ + 42; 13-^ — 4«/ 8a; + 2oy 65a: — 2oy 28 15 24 39 56 71 156 355 73a; Substituting 7 for x in (4), " val. X and y in (2), a; =: y = (0 (3) (4) (5) (6) (7) (8) 3. Given And Adding (2) to (3), Mult, (i) by 2, " (2) by 3, Adding (5) to (6), Dividing (4) by 3, Subt. (8) from (7), From (8), a (2), 2^ + 5^ — 3^ = 4 AX — zy-\-2Z— 9 5a: + 6?/ — 2Z ^n 18 9^ + 32/ 4a: + loy — 62; 12a; — ()y -\- dz — 27 = 8 = 27 16a; + 3^ + .?/ 35 9 13a; = 26 a; (I) (2) (3) (4) (5) (6) (7) (8) = ^ : }• ^?i5. THREE OR MORE UNKXOWN QUAXTITIES. 97 4. Given 2X -{- ^1/ — 4^ = -^ (0 X — ly + z^ — 6 (2) And 3a; — 2?/ + 52; = 26 (3) Adding (i) to (2), ;^x -\- y — z =1 26 (4) Subt. (4) from (3), i^y — 6z = o Or, y—2z=: o (5) Bringing down (i), 2x -\- ^y — 4z = 20 (i) Mult. (2) by 2, 22: — 4^ + 6;z = 12 Subtracting, ^y — loz = 8 (6) Mult. (5) by 7, 7y — 14Z = o Subtracting, 42; = 8 2; — 2 ; j From (5), «/ = 4; >-4/^s. " (2), x = S, ) 5. Given 5a; + 2?/ + 42 = 46 (i) 32; + 2^ + 2; = 23 (2) And 10^ + 5^ + 4^ = 75 (3) Subt. (i) from (3), 5^ + 3^ = 29 (4) Mult. (2) by 4, i2:z; + 8?/ 4- 4^ = 92 (5) Subt. (3) from (5), 2X 4- sy ^J7 (6) « (6) " (4), 3^ From (6), ^ = 3 ; j- ^47i5. " (2), 6. Given a; 4- ?/+;;= 53 (i) x-^ 2y + 3^ = 105 (2) And >^ + 3^ + 4^ == 134 (3) Subt. (i) from (2), y 4- 2Z = 52 (4) '' (2) " (3), ^+ ^= 29 (5) " (5) " (4), From (5), 98 SIMPLE equatio:n^s. 7. Given zx -h 42; 57 (i) 2y z _ 1 1 (2) And 5^ + 3i/ - 65 (3) >lulti plying (2) by 4, 2>y — ^z 44 (4) Adding (i) to (4), 3^ + 8^ - lOI (5) Multiplying (5) by 5, 15:?; + 40?/ 505 (6) (3) by 3, 15a: + 9^ - 195 Subtracting, z^y — 310 ^ - 10; > 9; h^ From (2), z ins. " (i). X 7 J 8. Given X y z « + - + - 234 62 (I) X y z - + - + - 3 4 5 47 (4) And 4'^5 6 - 38 (3) Clearing of frac. 6a; + 4?/ + 32 _ 744 (4) 202; + 15?/ + 122 _ 2820 (5) 15a: + 127/ + 10:2; 2280 (6) Mult. (4) by 4, 2/\x + 16?/ 4- 125; 2976 (7) Subt. (5) from (7), 4-1' +2/ - 156 (8) Mult. (5) by 5, ] [Oo:r + 75?/ + 60Z 14100 (9) " (6) by 6, ^ox + 72?/ -h 60Z 13680 Subtracting, io:r + 3^ = 420 Mult. (8) by 3, 12a; +32/ — 468 Subtracting, 2a; — 48 • • X — 24; ) From (8), y - 60; V A7IS. " (3), z — 120, ) THREE OR MORE UXKXOVrK^ Ql:A^■TITlES. 99 12. Pofje 122. Giyen to -\- X -\- Z 10 (i) X -\- y -\- z 12 (2) tv + x^ y 9 (3) And w -\- y -\- z II (4) Adding, ^w -\- Z^ -\- Zy -^ 2,^ 42 (5) Dividing (5) by 3, w + x + y + z 14 (6) Subtracting each given equation from (6), we have, w=2; x =z T^; y = 4; and z =z ^, Ans. 3. Given - + ' ^ X y 6 I I 7 2^ 2; 12 (I) (2) And II 3 a; 2; 4 (3) Adding, 2 2 2 26 a; ?/ ^ 12 (4) Dividing / \ 1 III 13 (4) by 2, - + - + - -^ ^ ' -^ X y z 12 (5) Subtracting each given equation from (5), we have. 16 14 — ._ • — • X 12 ' ?/ 12 ' I z _ 3 . ~ 12' Hence, X -. 2\ y — z\ PROBLEMS. z = = 4, Ans, # Taqe 123. I. Let X age of ist, y — " " 2d, And z - " " 3d, 100 SIMPLE EQUATION'S. Then x + 1/ 27 (0 X + z 29 (2) And y + ^ 32 (3) Adding, tx -\- 2y + 2Z 88 (4) Dividing (4) by 2, x + y + z — 44 (5) y- " And z — " Then . 7^ + 13^ — I205 I4:r + 52; I300 And 123/ + 20;2 I140 Mult, (i) by 2, 142: -f- 26?/ 410 Subtracting equations (3), (2), and (i) from (5), we have a; = 12 yrs.; ^ = 15 yrs. ; and z = ij yrs., ^ws/ 2. Let (c r= price of a calf, " sheep, " lamb. (I) (2) (3) (4) Subt. (2) from (4), 26?/— 52; = no (5) Mult. (5) by 4, 1041/ — 202; = 440 (6) Adding (3) to (6), 116?/ = 580 «/ = $5, pr. of sheep ; ) From (5), z= $4, '' Vdmhs;)- Ans, " (i), X = $20, '' calves, ) 3. Let a; = ist number, «/ = 2d " And 2 = 3d ** Then J» + 2/ = ^3 (0 X -\- z =z 16 (2) And y -\- z = ig (3) Adding, 2a; + 2^/ + 22; = 48 Dividing by 2, x -{- y + z =^ 24. . (4) Subt. (3) from (4), a; = 5, istr; \ " (2) « (4), ^ = 8, 2d; V^^^5. " (i) " (4), 2;= II, 3d, ) THREE OK MORE UNKNOWN QUAXTITIES. 101 Let 6x = number of men in ist, yj = " " " 2d, (i) (3) (4) (5) (6) (7) (8) And 2Z = a a " 3d. Then 6iP + 32/ + 22; 3^ + 2/ = 1905 2Z — 60 And ig (i) and (2), Z -\- 2X — sy 165 Addii gx + 4y 1845 Mult. (3) by 2, 4X — 6y -\- 2z — —330 Subt. (5) from 1 (0, 2x-\- gy = 2235 Mult. (6) by 9, 18a; + 8it/ _ 20115 ti (4) by 2, acting (8) from iSx + Sy = 3690 Subtr ' (7), 73^ - 16425 ••• y = 225 From (6), X - 105 « (2), z = 300 6x — 630, men in ist; 1 sy 675, " " 2d ; ^ Ans. 2Z — 600, " " 3d, ) 5, Let X = price of ist, y = '' " 2d, And z = " " 3d. Then 122;+ izy 4- 142 25 (0 10.^ + 17?/ + 112 24 (2) And 6x -\- i2y -\- 6z 15 (3) Mult. (3) by 2, 12a; + 24?/ +122 30 (4) Subt. (i) from (4) iiy — 2Z — 5 (5) Mult. (2) by 6, 60X -\- lozy + 662 144 (6) '' (i) by 5, 60a; + 651/ -1- 702 125 (7) Subt. (7) from (6), 3iy 4Z - 19 (8) Mult. (5) by 2, 22?/ — 4Z 10 Subtracting, ^sy - 89.00 \ V — 80.60, 2d; ) From (5), z — -^0.80, 3d;[ Ans, " (3)^ X $0.50, i§t, ) 102 SIMPLE EQUATIONS Let X number minutes it takes A, y " • B, And - " 0. Now if A can fill it in x minutes, A can fill - of it in i minute, X B " i< I ~y a i( , ii And « a I z a (C J <{ Then And I I I X y 'JO T I I X z S4 I I I y z 140 (1) (3) Adding, 2 X 2 2 28 y z 840 2 60 Dividing by 2, • I X III y z 60 (4) Subtracting (3) from (4), Or, I 140 — 60 X 8400 I _ I X ~ 105 Clearing of fractions, x= io5min., A. Subt. (2) from (4), - = -,- ^- = , ^ ^ ^ ^^ y 60 84 60 X 84 A I 24 _ I Or, - = 7 ^- = y 00 X 84 210 Clearing of fractions, ,y = 210 min., B. Subtracting (i) from (4), etc., z = 420 'min,, G. A fills it in 105 minutes ; B " " 210 " y A71S, " '• 420 " THREE OR MORE UNKNOWN QUANTITIES. 103 Denote the parts by to, x, y, and z. Then w + x -\- ij -]- z — %()o (i) 10 -{- 2 =z 2]! (2) x—2^2y (3) And ; = 2^ (4) Multiplying (4) by 2, ^ = 4i/ (s) Adding (5), (3), and (2), lo^x^z = Sy (6) Subtracting (6) from (i), Or, ii ii From (2), w = 18, ist; f ^^^^ " (3), • :r = 22, 2d; r ''*• " (5), 2 = 40, 4th J 8. Let X = A's distance, y=:B's And z = C's Then a; + // + 2; = 62 (i) a: = 42+ 2;^ (2) And 20; + sy = i^^ (3) Subtracting (2) from (i), 3^ + 52^ = 62 (4) From (3), 2:r + 3^ — 172: = o (5) Mult. (2) by 2, 2X — \y — 82: =^ (6) Subt. (6) from (5), ly — 9^ = o (7) Multiplying (4) by 7, 21?/ + 352: = 62 x 7 " (7) t)y 3, 2iy — 27z= o Subtracting, 622; = 62 x 7 .-. =7 miles, C's distance; j From (7), «/ = 9 " B's " I ^?^s. '' (2), .T = 46 '• A's " ) 104 GENERALIZATION^. Let 4X . _ A's n loney, 2y — B's a And 3^ — C's a Then 4x + y — $100 . (•) 21J -\- Z — $100 (2) And 32; + a; _ $100 (3) Mult. (3) by 4, 4X -\- 12Z — 400 (4) Subt. (i) from (4), I2Z — y 300 (S) Mult. (5) by 2, (2)7 - 2y 4- 240 600 (6) Adding (6) and 252; _ 700 z $28 And 32 — $84. C's;) From (4), 4X $64, A's;[ Ans, " (2), zy — $72, B's, ) GENERALIZATION. rages 124, 125. 1. a; = y = — ^ = 3 chickens, Ans, b IS 480 . . 2. X = — ^- = 30 rods, Ans, 16 576 . 3. X = -— =: 12, ^??5. 48 0! 61320 _ ^, . 4. .^' — — =r ^i — — ciyL years, C s age, ^W5. 5. a; = ~ — „ = 7 feet, ^w*-. ^ 9x8' 62730 , 6. X z^ ^^ = 34, Ans. 41 X 45 s + ^ 392 + 18 (/ =: = = $)2 05 2 8 Ans, GENEKALIZATION". 105 rages 126-127. o ^ IS75 + 347 ^ . ., , \ o. g = = I961, As part; j / Ans, I = 157i_p_34Z = $614, B's " 2150 4- -546 , \ 9' 9 = — ^- -^-^^^ = 1248 votes; j . ) Ans. , 2150 — 346 ^ / «^ 8 X 12 96 10. X = — -—7 = — — — - = — = 44 days, Ans, a -\- b 8 -I- 12 20 ^ -^ ' 9 X 15 135 . , 11. X = ^— — ^ = -^ = 5I hours, A?is. 9 + 15 24 ^^ 40 X 50 2000 . , . 12. .T = — ^^ = = 2 2# hours, Afis, 40 4-50 90 13. ;j = br = 748 X .09 = $67.32, ^?i5. 14. p = 45385 X .20 = 9077, ^;?s. 15. ^^ = 2763 X .375 = 11036.12^, Ans. 16. p =z 1587 X .37 = 587.19 bushels, Ans. b 2700 17. /• = 4r = i^r = -^^f, ^«6'. 2C5.2 18. r = ^- = .40, .l7i5. 03 291 19. r = -|- = .60, A71S. 485 r .25 *^ 106 INVOLUTION. 21. h = 37- TageH 130-133, -^ i=r I37500, J;Z5. 22. ^ 23- <-? f2)6 ^ 64^12^,^2. 10. {abcdY = a^b^cfd^. lis VOLUTIOX. 107 12 15 Page 13S. The 5tli power of {a + hf = {a + ly^. The 2(1 power of (^^ + Z*)" = {a + Z-)"-". The ?/th power of {x — y)"' — (x — i/)'"". The nth power of (:<; + ?/)- = (:t' + ?/)-". 16. The 2d power of (a^ + Z-^) — (^^3 ^ ^3^2. 17. The 3(1 power of (a^Vi^) = aWi^. ^aI)'^Y__ 27^3^6 \ 2a I /2a^cy_ 8«3 \6a^¥c^ hjtl^V-_ 49«^ ^' \ xy'' ' ~ x^'y"'"' ' or x"y'^ 26. (x -\- 2y -{- 2)3 z=x^ -\- 6x^y +6.i;2+ i2a;«/^+ 24;/-?/+ 12:?: + 8z/3 + 24?/'^' + 24?/ + 8. Prt2 + c2— 2a^+2r^6'— 2Z>»3 — Q* 11 14. ¥-^W = t^. 15- i2a%~'^c -r- ^d^b~h^ =z 4abc~K .'-5 c ^ =. c ^. ax a = -^ , Ans. y xhj /-4 a _ ay . ~, — — ^ — 1 A )IS' 4 h by ad~^ a . 26. — ^- =: -,^-7; , ^^^5. iC^ rZ5.r2 27, —^— , Ans. ax"" a no EVOLUTION". EVOLUTION. Pages 14S, 140. 13. as. 5 14. x^. 15- y'- 16. ai = a-^, Ans. 17. a- = a'^, Ans. 18. r/s r= a'^, Ans. 19. Js — ^-8^ A7IS. 20. 2;^ = 2:1'^, A71S. 21. «/T ^ Z/*"^, ^^5. 3. \/r/ = a^, Ans. 4. vV or a = «T^ yi?^5. 5. '\/4xy = 4a:r3?/3, ^7is. 6. '\/Sa^O^ = 2asb% Ans. 7. \^ 2'jabc ^ TfCt-^b^c^, Ajis. 8. 'v/i6a4'" = 9. \^yc\v^^ 2a'"\ Ans. :^sa'!>x^, Ans. 10. Vs^ci^b'^ = 6^2^, ^«5. 2^x-^y^, Ans. : S(fib% Ans. (13)^x^1/^, Ans. = 'jx^y^ Ans. == T^a^b^, A71S. 11. V 2.1'^^'^ = 12. \/64«!^<^® = 13. \/isxy =z 14. V49^y 15. ^ 2^aW 16. 49^^ 7a;' = -^ , Ans, 64/ 8i/ Page 152 1 X -{- 2. a — I. I + X. X + ^ a 3- I 2* b X + -' 2 Page 153. x^^2xy-[-y^-\-2xz-\-2yz + z^ ( x-\-y-\-Zi Ans. x? 2X-\-y ) 2xyAr\f 2X+2y-^z) 2XZ -\- 2yz + z^ 10. a' ^■^4ab-\-4b^-\-2a—4b+i (« — 2^+1, A?is. 2a— 2b ) — 4ab + 4b^ 2a — 4^4- 1 ) 2a — 4^+1 EVOLUTION 111 II. a* + 4U^b-\-4^—4a^—Sh-{-4 ( ^^4-2^ — 2, Ans. a" 2a^-\-2b ) 4C(?b-\-4lP' 2(1'^ -\- 4b — 2 ) — 4d- — 8Z'H-4 12. I — 4^' + 4¥ + 2x—4l>^x -\-x- ( i—2U'-\- Xy Ans. 13- 2-2^/2) _4/,2 + 4^4 2 — 4l^-\-X ) 2X — 4Wx-\-x^ 4a^— 161^-^2401^— i6a-T 4 ( 2^^ — 4^f + 2, ^W5. 4«4 4^2 _4f^ ) —i6a^-\-24a^ 4«2 — 8« + 2 ) 8rt^ 8^2. i6rt + 4 i6rt + 4 14. a a^ 2_ 4 2 « , Ans. 2a -n- 7 . ^' 4 — «J H 4 15. 2rc IT X^ 2+^ X^ 9 I -, Ans, y ^ f y X 2 4- — Note.' — 2-= = — 2 x — = — -, second term of root ; y 2.1* ^ Also, — X— ^=— 2; and — "^x--^ — • y X X X x^ 112 UEUUCTION OF RADICALS. REDUCTION OF RADICALS. Case I, Page 156. 6. aVlf' 7. 2aV2b. 8 9 10 II 12 13 14 15 16 17 6^/36 X 7«^^ 3r^V 2C. 2Ifl^\/l —3^. 3«V^^- 6«\/i3^- Vi584a^=:A/i44«^x 11 2 3 4 5 6 7 8 9 10 II 12 13 3/ ft^Z'^c^ 27 V 27 (« — h)\ V{a — If. Va""\ 2. Case III, Page 158, a^ ^ a^ =^ (a^)^ ; Ans. 3I _ 34 = 3 = (125)^, ) ^2 =i a^ =1 {a^^; \ . »t == 66 =: (t296)«, ' ns. V5 = V'S^ = V^i5625; ^2 = A^73 = v^8, ADDITION OF RADICALS, 113 6. 'v^^ and 1/1252;^. 7. \^64a^ and \/4.a\ 8. V^« and V^. 9. (^")2"'' and (^2)2"". 10. \/{a + ^)2 and Via — b)K 11. v^(ic — yf and v^(.c + ?/p. 3- 4- Case JF, Pagre 159, (3^)^ and (42)^ (aio)^ and {b^^)k ^ 4 3-^^ = S X ^ = 4; 4 ^ I 3 _ I 4 4 3 ' 4~ 3 3~ 9 Hence, (a^)^ and (S^)^, ^?is. 6. (a^)^ and (^tI)!, 7. (««)" and (^'^/'. 6. 7- ADDITION OF RADICALS. Page 160. 'v/12 2^3 5. '\/20 = 2^/5 A/27 _ 3\/3 V48 4V3 5\/3, ^ns. 2A/5+4V3. ^^^5- 2^/W - 2bVb i^M = 3a Vb {2b + 3a) a/^, ^W5. aVsct^b = cVzjnb = a^Vsab 3C\/3ab {a^ + 3^) V3«f^. -^^^5- 114 SUBTRACTION OF RADICALS, 8. lO. II. 12. 13. gxy 2a (9a; + Za) \^ za, Ans. 3V^54 = 9V^2 4'V^I28 := 161^2 25 ^^2, Ans, 7^/243 = 7V81 X 3 = 63A/3 5V363 = 5V121 X 3 = 55V3 1 18 A/3, Jws. a \/Sib = gaVb ^aV49b = 2ia\/i ^oa\/b, A71S* sV^ = s^^^y 4x^vy + 5^a/^'^j ^^^s. SUBTRACTION OF RADICALS. 2. Page 161, 4\/ii2 = = 16^/7 1/448 = . 8a/7 8^7, J7^ V480 _ W30 4A/63 I2a/7 '41/30 — I2V7> ^^?^« MULTIPLICATION OF KADICALS. 115 4V320 = 32^/5 — 5V30 = — 20^/5 52^/5? ^^^5- 2IiC a/^ (212: — 10) Vctx, Ans, 6. s\^a + b — 3V^fl^ + <^ = 2VCI + ^, ^^^s* 8. 3'V^25oJ^^ = i^b'\/2hx 2'V^54^% = ^hV~2bx <^l)'^2lx, Ans, 10. 5^1 = 4V3 = ¥V3 2^/1 = IV3 = f V3 i|a/3j -4ws. MULTIPLICATION OF RADICALS. rage 162. 4. 5^/18 = 15^2 II. abx. < 3'\/2o 6v5 6. V«^ — b^' 9o\/io, v4;z6'. 7. ^/acxy. s. «i — V«^; Va^ X Vc — «V«f, Ans. BO. 1 J!L aj» it'"'" 7^"' mil / J 7V^4 X 3\^4 = 2i\^T6 = 42^/2, J/?,9. J2. 130!, 116 INVOLUTION OF KADICALS. 13. 14. 16. 17. 18. 19. 6. 4ax. 2a/| X 2VI = 4VA = V5, ^^«. Wi X sVi = i2'\/3\ = 2V5, A71S. 00 4. (m + ^^)^ X (m + ny^ = (in + nY', (m + w)3 = (pi 4- 7?) V m + n, Ans, /gad 2db Af/ DIVISION OF RADICALS. Page 164, or, = aVs(i' (Art. 306.) 5. sVb^' 6. (r«2 _^ a;)i 7. i2(fl5?/)i 8. 3&\/i?J. i5« Wl, 10. 2« v^. 2 I II. n n n (a + b)^, Ans. 12. 3^/25:^= 153;^/^, ^^?*% 13. ^/x — y, (Art. 128.) 14. 8 a/8 = 8^/4 X 2 = 16^/2, ^/i^. 15. 2^256 = 2 X 16 = 32, ^WA-. INVOLUTION OF RADICALS Page 164, 3. «i 4. (3A/2a:)2 _ ^ >< 2a; = 18:?;, ^?J5. 5. Sa. 6. ( a/2:c) — - X 2.r a/2^ = - V^, A?is. \2 / b 4 4y - — 1 = (2xVcty == 8a^3y^^ ^^^^^ EVOLUTIOK OF HADICALS 117 8. [sV I = = 9^^ -^^is. y ~ 9 9. CI? + 2(1 Vy + y, (Art. 266). 2 3 4 5 6 7 8 9 10 II 12 EVOLUTION OF RADICALS. Pafje 165. 2,^/a^ = 3^«, Ans. isV^y)^ = {Vgxy)^ = "s/^xy, Ans. (2hV^)^ = (a/8^)^ = \/Sb% Ans. (128^^)^ = yz'^a^p = 20^^, Afis. "Va^b^ = cM^, Ans. (4«2y^)i^ = (V32«^)^" = ^/ 120.^ = \/2a, Ans, i i i REDUCING A RADICAL TO A RATIONAL QUANTITY. Case I, Page 166, 3211 4. = -; a% Ans, 3 3 3 5- ^3^3 X a^c^ = ac; hence ^^^c^^ Ans, 6. (« + h)^ X (rt + b)^ = (a + b); hence (aj 4-^)5,^^5. 118 REDUCTIOK OF RADICALS. 7. ^/ aWc — ah^/ac itb\Uic X ^/ (ic — c^hc. Hence ^ac, Ans. 8. v(i«J + 2/)^ X \x-\-i) =z'x-\-y. Hence Vx-\-y,Ans, 9. \^{a + Z?)2 X Vci-^b = a + b. Hence \^a~i- d, Ans. 10. Va -{- b -\- c X Va ^b + c = a -{-!)-}- c. Hence Va + b -\- c, Ans. 3 4 5 6 7 8 9 10 Case JJ, Paae 167. X — 4^9. {V9 — V6) (V9 -h V6) =: 9 — 6 = 3, ^?^5. V; — Vci. (6 — Vs) (6 + Vs) = 3^ — 5 = 3h -4.ns, Vs^^ + Vs^' (Va — Vs) {Va + Vs) = « — hi ^^«- 3A/^ — a/8. 4^2^ + 5 a/^. Case III, raf/e 168, ^/x yx^ ^ 3/- ^ ~37^^ — — IT" J -4?^5. yc ,yc^ ^ , a/o; + a/v a/^ 4- A^V X ■\- 2 Vxy ± 1/ . 6. — / X — = p = — ^ ^i-^ , ^^5. A/ic — Vy yx 4- A^^ ^ — y X ^/a + y^o xiy^a 4- a/c) . X -— = — = — ^ -, Ans, a/« — Vc Va + A^c ^ RADICAL EQUATIOXS. llO 8 I , I — V s _ I — A/ 3 _ I — A/ 3 I + a/s I — A/3 ^_~ 3 — 2 V3 — I ^ =r — , Ans. 2 Note. — Changing the signs of both terms of the fraction does not alter its value. (Art. 166.) _V3__ ^ 3 + V s _ 3V3 + 3 _ 3 (A/3 + 3 — V3 3 + A/3 9 ~- 3 6 A/3+1 . = , A?is. 2 RADICAL EQUATIONS. Par/e KiO. 4. Given a -\- \/x -\- c ^ d Transj)osing, V '^' = d — a — c Involving, ic = (fZ — a — cY, Ans. 5- Given Involving, \^x -j- 2 3 X + 2 — 27 Given Transposing, Dividing, Involving, X 25, An 6. 3VX — 4 + 5-71 3VX — 4 2\ 1 A/a; — 4 1 ^ 4-11 7. Given V4 "■' Dividing, Vh » Involving, 5 = '. X 256, ^^s. 120 EADICAL EQUATIONS. 8. Given ^/2X + 3 — 6 — 13 Involving, Transposing, 2^" + 3 — 6 2197 2X 2200 X 1 100, Ans, 9- Given V(^ — 4 3 Involving, Given X — 4 — 27 X 31, Ans, lO. 2\/^ — 5 _ 4 Dividing, Involving, Vx — 5 2 X — s — 1^ X 21, Ans, II. Given 5\/7 30 Involving, etc. 2; --30 X _ 252, ^WS. Given rage 170. 12. i-\-V2ax-}-b „ Multiplying, etc., \/2ax-\-b =z b^—a Involving, 2ax-\-b = b^—2ab^ + a^ Transposing, 2ax = b^—2a¥-\-a^—b b^—2ab^-\-a^ — b . X ■= ,Ans. 2a n- y — (^y "^y 14. Given ^ — — - = — -' \/y y Clearing, etc., y (y — ay) = y Dividing by ?/, y — ay ■=. \ Factoring, «/ (i — «) = i Dividing, y = , Ans, RADICAL EQUATIONS. 121 15. Given x + V«- + x- — 20.^ Clearing of frac, x\/ci^ + x'-\-a^-]-x^ — 2^2 Transposing, x^/c^ + a:;2 ^2 _ ^^ Involving, a^x'^ -\- x^ — a^ 2a^x^ + x^ Uniting, ^a^x^ a^ Dividing by a^, y ^2 - ''' Extracting root. • 3 X aVi, A71S. 17. Given c + 12 2 -\- -y/x Involving, ic + 12 4 -f ^^X + X Transposing, 4\/x 8 Dividing, Vx 2 Squaring, VI X 4, Ans. 18. Given X a/^ + 2 2 + V5X Then Vs^ + 10 _ 2 + Vs^^' Squaring, Sx + 10 — 4 + Ws^ + S-'^ Uniting, 4V5-^' 6 Dividing, Vs^ 2 Involving, . 5^ 1 19. Given *C/ — -■■ 2 0? "^^ /^t) • ^/x X — ax ^ Vx Clearing of fractions, x = x(x — ax) Dividing hy x, (i — a) x = i — ^ J • • X — « jfxii/Ot I — a 133 PUKE QUADRATICS. PURE QUADRATICS. Page 17 S. 2. Given 32;^ — 5 = 70 Transposing, etc., rc^ == 25 Extracting root, ^ = ± 5. 3. Given 9^2 _j_ g _ ^^^ -}- 62 Transposing, etc., a:^ = 9 X =z ±i 3, Ans. 4. Given 53^2 + 9 = 2x^ + 57 Transposing, etc., a;^ = 16 5. Given 6^2 + 5 = 4^2 ^ 55 Transposing, etc., x^ =^ 25 X =: ±^ 5, ^^S. 6. Given ^ h 35 = 3^'^ + 7 4 Mnltiplying by 4, 53^2 -f- 140 = 120;^ + 28 Transposing, etc., .^^ _ 15 X := ±i 4, Ans. ^ . 2a;2 + 8 a:^ — 6 7. Given = 1- 1; ' 10 10 ^ Multiplying by 10, 2x^ -j- S ^ x^ — 6 -\- ^o Transposing, etc., x^ = ^6 X =z ± 6, Ans, 8. Given X X 4 42 a; Multiplying by 4X, x^ = 2.7^ — 16 Transposing, rc^ = 16 a; = ± 4, ^7ic^. t>tJKE QUADRATICS. 123 9- Given X 2 X s 2 X 3 X Multiplying by 6x, 3.^2 4- 1 2 — 2:c^ + 1 8 Transposing, x^ - 6 X ± V6, A71S lO. Given 2X^ + 12 SX^— 37 Transposing, x^ 49 X ±7, A71S. II. Given 7^^ — 7 — 3^^ + 9 Transposing, etc.. x^ 4 X it 2, Ans, 12. Given «V - «4 Dividing by a% X ± «, ^ws. 13- Given (ic -1- 2)2 4a; + 5 Or, ic^ + 4X -\- 4 4^ + 5 Transposing, X^ = I a; ±1, Ans, 14. Given 6x^ — 12 iC'' — I 4 Multiplying by 4, 4x^ — 4 6x^—12 Transposing, etc., x^ 4 .T ±2. Ans. 15. Given a- (2^ + 9) 30^ + 6 30 10 Multiplying by 30, 2X^ + 9^' — 90^+18 Transposing, etc.. ic2 9 iC _ ± 3, ^725. 16. Given — 4- 5 , 5 _8 — X 4 + X 3 Clearing of fractions. 60 + l^X + 60 15.'?: — 128 82^2 Transposing, etc.. x^ - I X — ± ij Ans. 124 PtJR'E QUADEATICS. Page 17 S — Continued. ^. ax^ (a — 2) 17. Given ^, = I ^x ' I -\- X Clearing of frac, a^x^ — 2ax^ = i — x^ Transposing, a^x'^ — 2ax^ -}- x^ =z i Factoring, (a^ — 2« + i)rr2 = i Dividing, x^ = -z ° «-* — 2« + I I « — l' — r 4^ Dividing by 2, and squaring. 19. Given 2a/^^ — 5 = ^ 3 ,2 - 4^' i^ — 5 = 9 Multiplying by 9, 92:^ — 45 = 4^^ Transposing, etc., a:^ = 9 20. Given 2^/-'^^ — 4 = 4V<'«^ — i Dividing by 2, and sq., x^ — 4 =z 40,^ — 4 '• Transposing, x^ = 40^ X = ±: la, Ans, 21. Given V^ -{- c = Mult, by denominator, x^— c^ = d^ Transposing, x^ ^= c^ -\- d^ X = ±,'\/ d^ -\- (P, Ans, 22. Given A / = vx Squaring, etc., ^x^ — i ^ x^ Transposing, etc., x"^ := \ • • X — ^t 2 > -^ '^i5» PUKE QUADRATICS. 125 23. Given Squaring, etc., Transposing, 24. Given Vic — z= \/x + a a h^ = x^ = x^ — a^ «2 + 0^ X = ±Va^ + P, A ins. 24 Va: + 10 Clearing of fractions, 24 Squaring, 576 Transposing, x^ =::: 'S/ X — I O V^^ — 100 : J?' — 100 : 676 X = ± 26, A71S, PROBLEMS Page 174, 1. Let Then Clearing of fractions, 2, Let Then Multiplying by 4, X =: the number. XX - X - = 108 3 4 x^ = 1296 X = -t 36, Ans. X zzz the number. 25-— = 9 4 a;2 ^ 64 Let Then ic =: ± 8, Ans. X = No. of rods on one side. a;2 =3 1600 sq. rds., area. (Ax. 10.) X = 40 rods, Ans. Note. — It is advisable for tlie pupil to represent the area by a diagram, in this and like problems. Let Then X = length of side of square. x^ = 50x18 = 900 rods, area. X — ^o rods, Ans, 126 PURE QUADRATICS. 8. Let 2X one, Then Sx the other. By conditions. ] [ox^ 360 Dividing by lo, x^ ^6 • • • X ±6 And 2X 12;) . ]■ Ans. 5^ — 30. 1 Let X No. of dollars. Then a^- -7 — 29 Transposing, x^ — 36 • • • X 16, Ans. Let X number. Then X 8 X X - X 5 I 16 Clearing of fractions, x^ 6400 • • • X 80, Ans. Let x less, Then 4X greater. And 4X^ 900 Dividing by 4, x^ 225 And • • • 4:^ 60, ) Let X - number of yards. Then 40^ -i r X price. 40J- '-X 81 2X A A 81 And — 2X 3 ' 54 Mult., etc., X^ 729. (Art. 378.) • • • X 27 yards; ) And 81 I1.50, price, i AFFECTED QUADRATICS. 127 10. Let X = number. Then ^- 12 = i8o 4 Transposing, etc., x^ = 256 ,\ X =: ± i6j Ans* 11. Let X = length of side. Then x^ = area of bottom. 6x^ = capacity in cubic feet. By conditions, 6x^ = •^ 1728 77 77 Cancelhng, x^ = -y^^-^^^ x^ = (77Y, and 2' = 77 feet, Ans, 12. Let X = number. Then {x -\- 10) {x — 10) = 156 Or, x^ — 100 r= 156 Transposing, x^ = 256 i?; =: ± 16, A71S. AFFECTED QUADRATICS. First Method, Page 178. 6. Given t,x^ — 24J" = — 36 Dividing, x^ — ^x ^ —12 Completing square, x^ — 8a; + 16 = 4 Extracting root, ;i: — 4 = ±2 2: = 6 or 2, yl7Z5. 7. Given ^x^ — ^ox = 45 Dividing by 5, x^ — 8rc = 9 Comp. sq., x^ — 8a: +16 =: 9 + 16 = 25 Extracting root, a* — 4 = ±5 2- = 4 ± 5 = 9 or —I, J??^, 128 AFPECTED QUADRATICS. 8. Given x^ — dax = d Oomp. sq. , x^ — 6ax -\- ga^ ^n d -{- ga^ lo. II. Extracting root, x — $a ^ ±^\/d-{-ga^ ^ = S^t dz V d + ga^, Ans. Given 2X^ — 22a; — 120 Dividing by 2, a;2 — iia; — 60 Comp. sq., x^ — iia:+ 121 := 60+ ^r = H"- Extracting root, rv. II -4-19 ^2 ±2 /». II _4_ 19 • • •*' '2" ~2~ 15 or — 4, Ans, Given x^ — 1 40 i2,x Transposing, x^ — 132; 140 Comp. sq., x^ — 13^ + ^ = 140 + 1J9 ^ 72, Extracting root, x-i^- +-V- . /v. 13 1 27 • • . 3 _|_ 5 •*' — ^ TV. 2 *. X — 4 or — I, Ans. Given 4 {x^ — i) 4X — I Or, 42;2 — 4a; _ 3 (See Ex. 19.) X I J or — ^, A71S. Given (2X — 3)2 _ SX Eeducing, 4X^ — 20a; — 9 By 3d method, ^x^ — - 20a; + 25 — 9+25 16 Extracting root. 2X — S — - 4 Whence, 2x — s - 4 X — 4^ or J, Ans. Given 14 3^—2 '^ X— I Mult, by ^ — I, etc., ^X^ — ^X 12 By 3d meth., 363^2 — - 602- + 25 — 144 + 25 — 169 Extracting root. 6x — s + 13 Whence, 6x — s - 13 X 3 or — i|, ^^^s. Given ^x 14 — .r : 14 Mult, by 2: 4- I, etc., 4^2 — g^^- 28 By 3d meth., 64:^:2- -i44a: + 8i 448 + 81 529 Extrncting root. 8ic — 9 H- 23 Whence, 8a: — 9 23 a; = 4 or — ij, J«s. AFFECTED QUADEATICS. 137 25. Given x^ 4- ~ = ^- 25 5 Transposinsr, x^ — ~zzz — ^ 5 25 ^^ By 2d method, x = f ± V— AT^ Reducing, a; = f ± | a; = -| or I, ^4?25. 26. Given x^ -{- - = - 2 2 By 2d method, x = -i± ViT^ = - i ± V^ Reducing, ^ = —I ±i X =z i or ~ I, A71S. 27. Given x^ — 2nx = tv^ — n^ By 2d method, x = n ±_ Vrn^ — n^ + tv^ ,; X =. n ±: m, A ns. 28. Given 9(i-^^x-ia X if Clearing of fractions, 9^2 _ g^^ = x^ ^ ^ax Transposing, x^ — ^ax = 9^ — c^nh By 2d method, x — ^± \J ^h'^ _ ^ah + ^ Reducing, ^ = f ± ^g:Z36^+^^ Or, i. = ^ ± ^^' - 3^ 2 -^ 2 X =1 ih or 3« — 35, ^^. 138 AFFECTED QUADRATICS. EQUATIONS OF THE QUADRATIC FORM. Fiifje 183. 4. Given a:^ + 8 = 6i«;2 Transposing, By 2d method, ^- 6a;2 — — 8 x^ _ 3 - V- 8-9 Reducing, x^ 3 ± I Uniting, a^ — 4 or 2 Extracting root, X — i 2 or -- V2, A71S. 5- Given od^ - By 2d method, - 2X^ X^ - 3 — I - V3 + I Reducing, X^ 1 + 2 Uniting, X^ = 3 or — I Extracting root, X = ± V3 or -f- V— I, Ans. 6. Given x^ — yx^ -. Transposing, x^ ja^ Dividing by x^. a? =z y Extracting cube root. X Vl i«9i4-, Ans. 7. Given 2 +4 ^' Multiplied by 2, By 2d method. ^^\-i. X -i±VA + TV Reducing, /v. I _U 2 X — \ or — J, Ans. 8. Given \/x^ + i\/x = I By 2d method, v^- -i±Vi + T% Reducing, ^:r = - 1 + i Uniting, Vii^ i or — 2 Involving, X \ or — 8, ^1??*', AFFECTED QUADRATICS. 130 9. Given 4X -{- a^^/x + 2 = 7 Transposing, 4V^' + 2 = 7 — 4a; Squaring, 16a; + 32 = 49 — 562:+ i62;2 Transposing, 162;^ — 72::?; = — 17 By 3d me., 2562'^ — ii52a;-|-(36)^ = — 272+1296 ^ 1024 Extracting root, 16a: — 36 = ± 32 Whence, 16a; = 36 + 32 a; = 4I: or J, ^;^5. ^. V4^+2o 4 _ V^ 10. Given — — ~— 4 + \x vx Clearing of fractions and squaring, ^x^ + 20X' = 256 — 32:?; + x^ Transposing, ^x^ -\- ^2X = 256 By 3d meth., 92:2+ 156.^+ (26)2 r= 768 + 676=1444 Extracting root, 30; + 26 = ±38 Whence, 3X = — 26 + 38 a; = 4 or —21 J, Alls. PROBLEMS. Page 184, 1. Let X = One, Then 12 — x ^ The other. And 12a; — a:^ = 32 Or x'^ — i2x = — 32 Comp. sq., x^— i2X + 36 = —3^ + 3^ (^I't. 335.) Extracting root, 2; — 6 = +2 Whence, a:: = 8 or 4 ; ) . 12 — ic = 4 or 8, ) 2. Let X = Cost, Then - — = Rate per cent, 100 And — = Percentage lost. joo 140 , AFPECTED QUADRATICS. By conditions, x — = 24 •^ 100 Clearing, 1002: — .t^= 2400 Changing signs, x^ — looa: = — 2400 By 2d method, x = 50^^—2400 + 2500 a; = 50 ± 10 = $60 or $40, Ans. 3. Let X = one number, And y = the other. Then x -{- y =z 10 (i) And xy = 24 (2) Squaring (i), a^ -\- 2xy + y^ = 100 (3) Mult. (2) by 4, 4xy = 9 6 i4) Sub. (4) from (3), x^ — 2xy -^ y^ = 4' (5) Extracting root, x — ?/ = ±2 (6) Bringing down (i) x + y ^ 10 Adding (6) and (i), etc., a; == 6 or 4; | Subtracting (6) from (i), etc., y = 4 or 6, f Note. — Let tlie pupil compare this solution with that of Problem i, in which one unknown quantity is used. 4. Let X = no. of sheep. Then — == price of each. X 80 80 And — = h I X X -{- 4 Clearing of fractions, 80a: + 320 = 80:2; -{- x^ -^4X Rejecting 802: from each, x^ + 43; = 320 Completing square, x^ + 4X + 4 — 324 Extracting root, x -{- 2 = ±18 Whence, x = — 2 ± 18 X = 1 6 sheep] AT 80 „ , \Ans, And — = $5 each, J X ^ ' The problem will not admit of a negative result. There- fprQ the Ans. 16 sheep, at $5 a head. AFFECTED QUADRATICS. 141 Let X =: the number. Then 2x^ ^65 — 32' Transposing, 2x^ + 3X = 65 Dividing ])y 2, x'^ -\- ^x zzz ^ Completing square, etc., a; = — J ± V-^^ +Ti TF r^ 3 + 23 = 5 or — 6 J, A71S. 6. Let X = No. of scholars, Then = " " oranges each received. X ^ A 1 144 144 And -— = -^^- + I X x -\- 2 Clearing of frac, 144a; + 288 = 1443; + x^ -\- 2X Or a? + 2x = 288 By 2d method, x z= —i ±^ A/288 + i a; = — I 2b 17 = 16 or — 18. Note. — The second value of x, being negative, cannot be applied to children. It may be observed that negative values do not apply to concrete numbers. Hence the Ans. 16 scliolars. 7. Let X = share of one, Then 50 — .t = " " the other. And 50.T — x^ =^ 600 Changing signs, x^ — 50.^ = — 600 By 2d method, x = 2^ ±: V— 600 + 625 X = 830, one; \^^^^^ And ^o — X =z $20, other, ) ^ 8. Let X = one, Then 100 — x = the other. And loo:?; — a;^ = 2400 Changing signs, a^ — loox = — 2400 By 2d method, x = SoiV — 2400 + 2500 .T = 60 ; ) , Ana 100 — X = 40, ) 142 AFFECTED QUADRATICS. 9. Half the perimeter (i|-^ = 64), or 64 rods, is equal to the sirm of Ibe length and breadth. Let X = the length, Then 64 — x =^ " breadth. And 64X — x^ =^ 1008 sq. rds., area. Changing signs, x^ — 64X = — 1008 By 2d method, x = 32^^—1008+1024 X = 36 rds., I'gth ; ) 64-x = 2S " brmh, j 10. Let X = number in file. Then ic + 60 = " " rank. And x^ + 60a; = 1600 men. By 2d method, x =z — 30 ± a/i6oo + 900 Keducing, x = —30 + 50 a; = 20 men in file; ) . X + 60 = 80 " " rank, f Page 185, II. Let X ~ number of lambs. Then -— = cost of each. X I TX And s\'^> ^^ — = ^™t* received for them. By conditions, 50 = — -' ' 2 X Clearing of frac, \ix^ — loo^r =100 By 3d method, 4842:^ — 44oo:r + 1 0000 = 14400. (Art. ^^6.) Extracting root, 22^ — 100 = ±120 Transposing and dividing, a; = 10 lambs, Ans. AFFECTED QUADRATICS. 143 12. het ^ ^ one number, 'X'hexi ^ — X — the other. Clearing of frac, etc., x^ — 4X = — 4 By second method, x = 2 ; |. ^^^ And 4 — a; = 2, p SECOND SOLUTION. Denote the numbers by x and y. Then x -\- y = 4 (i) And - + - = I (2) X y Clearing (2), x + y — xy (3) Subtracting (3) from (i), o — 4 — xy Or, xy — 4 (4) From (i) and (4) find the value ot x — y, thus: x + y = 4 (5) Squaring ( I ), x^-\-2xyi-y^ = 16 (6) Mult. (4) by 4, 4^y = 16 (7) 'Subt. (7) from (6), x^—2xy-\-y'^ = o Extracting root, x — y = o Or, x = y (8) Substituting in (i), 2x = 4 .T = 2 From (8), y = 2- ^^^s. 2 and 2. 13. Denote the numbers by ic and ?/. Then x -{- y = s (i) And a::^ _p ^3 _ 5^ ^2) Dividing (2) by (i), x^—xy-\-y'- = 13. (Ax. =;.) (3) Squaring (i), x^-{-2xy + y'^ = 25 (4) Subt. (3) from (4), zxy =112 (5) 144 AFFECTED QUADRATICS Or, xy — 4 Whence, y = l (6J Substituting in (i), ^ + 1 = 5 Clearing, etc.. a^ — 5* — — 4 X 1 ± a/- By 2d method. -4 + ¥ ^' = I ± f = 4 or I Substituting in (6), «/ = i or 4. A71S. 4 and I. This problem may be solved with one unknown quantity, thus, Let a; = one number. Then 5 — a; = the other. And 125 — 75a: -f 15^:2 — oi?-\-x^ = 65 Transposing, etc., i^x^ — ^^x ■= — 60 Or, x^ — ^x =: — 4 Completing sq., etc., ic = f± V— 4+^- a: = 4 or i And 5 — .T = I or 4 Ans. 4 and i. i4. Let ir = No. of yards in the width. Then a; + i = " " " length. I acre = 4840 square yards. 3 acres = 14520 " " Length x width = a:^ + a: = 14520 By 2d method, x = — I ± V14520 + ^ Reducing, x = —^ ± V^-^ Or, x= -i± ^fi X = 1 20 yards, width ; ) . X -\- I = 121 " length, f AFFECTED QUADRATICS. 145 15. Let X = B's rate, Then. x + i — A's " - — = time it takes B to go. 300 X -\- 1 a a A <.' And 3£? ^ ^o^ X X -\- 1 Dividing liy lo, clearing, etc., .1-2 -j- ic = 30 By 2(1 metli., x ^= — | + V30 + J Reducing, 2; = — } ib V" a; = 5 miles, B's rate ; And X -\- I = 6 " A' s " Ans. 16. Let X = No. that B relieves, Then a; + 40 = " "A " = sum B gives to each, " A " " X 1200 a; + 40 T, ,.,. 1200 1200 By conditions, = h 5 •^ ' X a- + 40 ^ -rx' • T 1 240 240 Dividmor by t: -^^ = \- 1 Clear, of frac, 240.^ + 9600 ^ 240.^ + x- + 40.T Transposing, a;^ -|- 40a: = 9600 By 2d method, ic = — 20 ± 1/9600 + 400 Reducing, x. = —20 + 100 X = 80 AVhence, ic + 40 = 120 80 relieved by B ; ) . And 120 " " A, ) 146 AFFECTED QUADPtATICS. 17. Let X =: one part. Then 48 — ir = the other. And x^ — 48:r = — 252 Completing square, etc., Ans. 42 and 6. BY TWO UNKNOWN QUANTITIES. Denote the parts by a; and y. Then X -\-y — 4S (I) xy — 252 (2) Squaring (i), {x + yf — 2304 (3) Multiplying (2) by 4, /^xy 1008 (4) Subtracting, {x — y)^ 1296 (5) Extracting root. X — y ±36 (6) X -\- y 48 (0 Adding (i) to (6), Subt. (6) from (i). etc., X 42 or 6 ; ) . etc., y 6 or 42. ) ns. i8. Let X number A bought, And y - " B -^ Then i X + y — 10 (i) 12 A • T • — cents A paid apiece. X (^) 12 y (3) By conditions. 12 12 — — — + I X y (4) Clearing of frac, i2y 12.1; -|- a:?/ (5) Taking value of x from (i), 12?/ = 12(10—//) f y(io-y) Eeducing, i2y z= 120 — 127/ + 10^ — y^ Transposing, y"^ + 14?/ =120 By 2d method, y ^ — 7 ± ^169 Eeducing, y = — 7 ± 13 y =z 6 lemons, B ; ^ . From (i), X = 4 " A, J AFFECTED QUADEATICS. 14t BY ONE UNKNOWN QUANTITY. Let X = No. A bought, Then ^o — x = " B " _ _ . 12 12 By conditions, — = h i •^ ic lo — X Clearing of frac, etc., x^—$4x = — 120 By 2d method, x = ly ± V— 120 +289 a; = 17 + 13 = 4, A; J And 10 — X = 6, B, ) 19. Let ^ = the breadth. Then 24 — .t = the length. By conditions, 24X — x^ — 35(24— 2x) Eeducing, x^ — 94a!; = — 840, etc. A71S. 14 ft. length; 10 ft. breadth. BY TWO UNKNOWN QUANTITIES. Let X = the length, And y = the breadth. Then x -\- y = 24 ft., or half the perimeter. (i) xy = 35 (a; — y), the area. (2) Mult, (i) by 35, 35^ + 352/ = ^4° (3) Transposing (2), 3 5-^—35^ — ^y (4) Subt. (4) from (3), 70?/ = 840 — xy Takingval. of a:from(i), joy = 840 — (24 — y) y Eeducing, Toy = 840 — 24^ + y^ Transposing, etc., y^—g^y = — 840 By 2d method, ^ = 47 ± V— 840 + 2209 ^ = 47 ±37 y = 10, breadth ; ^ ^^ And X = 14, length ^Am 148 AFFECTED Q U A D II ATICS . 20. Let X ■=. No. of rows. Then a; + 3 = " " trees in each row. And a;2 + 3:^ = 1 80 By 2d method, x =: — J ± V180 + \ Eeducing, ^ = — f ± ¥ OJ = 12 rows ; l 4 . And ic + 3 = 15 trees in each, j " 21. Let a; = digit in tens' place, Then 7 —:?;=: " " units' " And i^^ + (7 — "x^^ = 29 Eeducing, x9- — 7a; = — 10 Completing square, etc., Arts, 52. BY TWO UNKNOWN QUANTITIES. Let loa; + «/ = number. Then a; + ?/ = 7 (i) And a;2 4- ^2 _ 29 (2) Squaring (i), x^ + 22:?/ + «/2 z= 49 (3) Subt. (2) from (3), 2xy = 20 (4) Combining (2) and (4), x — y = $ (5) (5) " (0. ^= 5 y = 2 (See solution of Ex. 17.) iox-}-y = 52, Ans. 22. Let x = No. in the party, Then x + ^o z= contribution required of each. And x^ -\- ^ox =z 1000 By 2d method, x = —15^^/1000 + 225 Keducing, --^ = — 15 ± 35 a; = 20 persons, Ans. AFFECTED QUADRATICS'. 149 Page 186. 23- Let X — the less, Then * I20 X — " greater. And (x — ' \ X -4 120 Reducing, x^ + 2X 8o, etc. Ans, 8 ant. 15- BY TWO UNKNOWN QUANTITIES. Let X = the greater, and y =: the less. Then xy = 120 (i) And (y + 2) {:c — 3) = 120 (2) From (2), xy-^2x—^y — 6 = 120 (3) From (i), xy ^ 120, and a; = (4) From (3), 120 + ^ — 3?/ = 126 (5) Clearing, etc., 3?/^ -f 6?/ == 240 Dividing by 3, y^ -\- 2y =z 80 By 2d method, «/ = — i ± V^ Reducing, ?/ = —1+9 7/ = 8 or — 10. less ; ) j , From (i), a; = 15 or — 12, greater, f 24. Let x = one, Then 36 — x = other. By the conditions, ;^6x — x'^ = 80 (36 — 2x) Reducing, x^ — ig6x = — 2880 Completing sq., etc., x = 98 + ^—2880 + 9604 Or X = g8±V6'j24 And a: = 98 ± 82 X = 16 : ) And 36 — a" =r 20, 3 ^ 150 AFFECTED QUADRATICS. 25. Denote the numbers by x and y. Then x -\- y = ^s (0 And xy \ x^ -\- y'^ \: 2 '. ^ (2) Changing (2) to an equation, 2X^ + 2«/2 = ^xy (3) Squaring (i), x^ + 2xy -{- y^ = 5625 (4) Transposing and mult, 22:2+2?/^= 11 250 — 4xy (5) Equating (3) and (5), s^y — 1 1250— 42:2/ (6) Reducing, xy = 1250 (7) Combining (4) and (7), x — ?/ = 25 (8) Combining (i) and (8), x = ^o ; ) . Substituting in (8), ?/ = 25, f BY ONE UNKNOWN QUANTITY. Let X rr: oue number, Then 75 — a; = the other. And 75rc — x^ \ x^ -\- (75 — .t)^ : : 2 : 5 Placing the product of the extremes equal to the product of the means and reducing, x^ — 75a; =r — 1250, etc. 26. Denote the numbers by x and 146 — x. Then V146 — x — \/x z= 6 Transposing and involving, 1 46 — :?; = 36 + 1 2 Vx + a? Transposing and dividing, s^ —X = e^/x Squaring, 3025 — 11 ox + a.'^ = 2>^x Transposing, x?' — 146.?' =r — 3025 Completing sq., etc., x = 73±V— 3025 + 5329 Reducing, •'^ = 73 ± 48 .T = 121 or 2c; ; ) , And 146 — X = 25 or 121, AFFECTED QUADRATICS. 151 27. Let X = circumference of fore-wheel, And y — " " hind " mi '?600 3600 ^ , ^ Then — ^ \- 60 (i) X y ^ ' And — = h 40 (2) Dividing (i) by 60, -^ = — + i Clearing of fractions, Soy = 602^ + xy Factoring and dividing, x = - — ^ (3) Dividing (2) by 40, —^ — = — 1- i 60 -i-y 90 ^ + 3 y + 3 Clearing of fractions, 90^ + 270 = gox-^27o-\-xy-\-sx-{-sy + g Transposing, etc., ^ly — 9 = 93^ + ^y Factoring and dividing, x = — ~ (4) Equating (3) and (4), ^ °"^ == -^- — ^ Clearing, 558o?/ + 6o^^ =: 5220^—540 + 87^/2—9?/ Transposing, etc., 27?/^ — 369/y = 540 By 3d method, 4X272?/2— 4x 27 X 369^ + 369^ = 540 X 108 + 136161 Extracting root, 54?/ — 369 = ± Vi9448i = d:44i Whence, 54^ = 810 ?/ = 15 ft.; ) T^ . , 60 X 15 60 X 15 ,, \ Ans. From (3 , .?; = :; = = 12" \ ^^ 60+15 75 ) BY ONE UNKNOWN QUANTITY. Let X = revolutions of hind-wheel, Then 60 -\- x z=z " " fore-wheel, And = circumference of hind-wheel, x And J^^^ _ ,, u fore-wheel. 69 -^ X 152 AFFECTED QUADRATICS. T» XI J'^- 3600 3600 By the conditions, -r^ = —7 — 40 Clearing of frac, etc., 0^24-2460:?; = 648000 By 2d method, x = —1230 + ^648000 + 1512900 Reducing, x = — 1230 ± 1470 :?; =: 240 ^ =: 15 ft. hind-wheel; } Ans. = 1 2 ft. fore-wheel. 60 + iC 28. Let X = one number. And X -\- 16 = the other. By the conditions, x^ -\- i6x =z ^6 Completing square, etc., x ^= — 8 ± ^36 + 64 X ^= 2 or — 18; ) . And a: + 16 = 18 or — 2, i 29. Denote the numbers by x and y. Then x -\- y ii 4 (I) And - + - - a; ?/ 3^ ¥ (2) Multiplying (2) by xy, x -^ y — i(>.nj 5 (3) I Oxn Equating (i) and (3), f Multiplying by j, 4xy 5 (4) Squaring (1), x^ -\- 2xy + if 16 9" (5) Subt. (4) from (5), etc., x—y I (6) Combinine^ (1) and (6), x f:> And . y = \, ) AFFECTED Q U A D K A T I C S . * 153 BY ONE UNKNOWN QUANTITY. Let X = one number, Then f — ^ = the other. And - + —^— = ¥ X 4 — 3^ Clear, of frac, etc., 1 2^:^— 1 6x == — 5, etc. 30. Let X = the less number. Then x -\- 15 = " greater " And ^_i5f ^ ^ 2 Clearing of frac, etc., a^+ 15 = 2X^ Transposing, etc., x^ — io; — ^ Completing square, etc., ^ = i ± V^- + Reducing, a; = ^l ± V Or, X = l±-' 121 I_I "4 2: = 3 less; ) . And a; + 15 r= 18 greater, f 3 1 . Let X = her age. Of' Then - + yx —12 = 2 Clearing of frac, etc., x+zVx = 24 Completing square, etc, Vx = — i ± 5 Uniting, Vx = 4 or — 6 Involving, x =z t6 or $6. Hence her age is 16, or 36 years, Ans. "Note. — The minus sign must be placed before the radical in the first equation when the second value is taken. 32. Denote the length by x and the breadth by 1/. Then 2 (.^ + ?/) = 96 (i) And xy = 1o(x-y) (2) For solution, see that of Problem 19, above. 25 8 rods, length \ \ a ^ 20 " breadth, ) X A's age, 120 X B's a 154 AFFECTED QUADRATICS. 33' Let And By conditions, (x — 3) | [-2| = 120 Multiplying, etc., 2X^ + II4;^; = 1202; +360 Transposing, etc., x^ — 3-^ = iSo Oonii)leting scp, x^ — 32; + f = 1 80 + | = ^ J^ Extracting root, x — ^ = di -j- ^ = I ± -Y-, or 15 years, A's age; ii5= 8 « B's " ^'^'"- X 34. Let X = No. of lbs. pei^per bought for 8 crowns. Then - = price of pepper per lb. X 80 X - = = sum paid tor ])epper. XX ' ^ ^^ X — 14 = lbs. saffron for 26 crowns. 26 =. price of saffron per lb. X — 14 ^6 X = ^^ — = sum paid for saffron. X — 14 X — 14 And 1 — — = 188, by conditions. X X — 14 '' Clearing of fractions, 640a; — 8960 + 936:?; = i88ic2 — 2632.^' Transposing, i88a;2 — 4208a; = — 8960 Dividing by 4, 47X"2 — 1052a; = — 2240 By 3d, 94^^—1052 = iV — 2240 X 188 + (1052)2 Eeducing, 94a; — 1052 = iV — 421120 + 1106704 " 942;— 1052 = +V685584 " 94a; — 1052 =: ±_ 828 Whence, 94a: =. 1880 a; = 20 lbs. p. for 8 c, Ans. SIMULTANEOUS QUADRATICS. 155 SIMULTANEOUS QUADRATIC EQUATIONS. Page ISS, 2. Given cr^ + y2 -_ 25 (i) And X -^ y — J (2) Squaring (2), x^ + 2x1/ + ?/2 = 49 (3) Snbt. (i) from (3^ 2xy = 24 (4) - (4) " (i),x^-2xy + f= I (5) Extrcicting root, x — ?/ = ± i (6) Combining (2) and (6), ^ = 4 or 3 ; jj^ And ?/ =r 3 or 4, \ 3. Given x^ -\- y'^ = 'j^ (i) And X -\- y := 12 (2) Squaring (2), x^ -f- 2a:'?/ + ?/2 = 144 (3) Subt. (i) from {3), 2xy = 70 (4) " (4) " (i), x^-2xy^-f =4 (5) Extracting root, x — y =z ^ 2 (6) Bringing down (2), x -\- y ^ 12 Combining (6) and (2), .t = 7 or 5 ; | . And y =z ^ or 7, ) 4. Given x^ — y- =28 (i) And X — ^ = 2 (2) Dividing (i) by (2), x ^ y z= 14 (3) Ans. Combining (3) and (2), ^ = 8; And y zzz 6, Given x^ -\- y"^ =^ 244 (i) And y — x ^ 2 (2) Squaring (2), if — 2xy + ^r^ = 4 (3) Subt. (3) from (i), 2xy = 240 (4) Adding (4) to (i), x^ + 2xy + if = 484 (5) Extracting root, ^ -^ !f =^ dz ^^ (6) Bringing down (2), — x -{- y =z 2 Combining (6) and (2), x =z 10 or— 12;) And y = i2or — io_, j 156 SIMULTANEOUS QUADRATICS. Given 2,x^ — if — 251 (i) And ^ + 4^ — 38 (2) From (2), ^ — 38 — 4«/ Squaring, x^ 1444—304^+16^^ Multiplying by 3, 3^'- = 4332 — 912^ + 487/2 (3) Substituting in (i), " 4332 — 9i2?/ + 48?/2— «/2 = 251 Transposing, etc., ^yif — gj2y = —4081 By 3d method, (47)y— 47 X9I2^/ + (456)2 = — 191807 + 207936 =: 16129 Ext. root, 47?/— 456 = ± 127 Whence, 47j/ = 456 ±127 y = 12JI or 7; I From (2), using 2d value of y. ) 7. Given Sx^ + 52/2 =728 (i) And 61/ — X =: 15 (2) From (2), X = 6^—15 (3) Squaring, x^ = ^6f — iSoi/ + 225 Multiplying by 8, Sx^ = 2887/2 — 1440?/ + 1800 Substituting in (i), 2887/2— 1440?/+ 1 800 + 5 7/2 =728 Transposing, etc., 2932/^ — 1440^ = — 1072 By 3d method, (293)27/2—1440x293?/ + (720)2 — —314096 + 518400 = 204304 Ext. root,. 293?/ — 720 =r ±452 Whence, 293?/ = 720 + 452 y = 40Y i^i;) X = g, ^ Ans. Dropping 2d value of y from (3). ) SIMULTANEOUS QUADRATiC'S. 157 Page 189. 9. Given x + y ^= g (i) And a:^ + ^3 _ ig^ (2) Div. (2) by (i), x^ — xij-\- f — 21 , (3) Squaring (i), x^ -\- 2xy -\- y^ =. ^i Subtracting, 3^2/ = 60 (4) And xy =20 (5) Subt. (5) from (3), x^—2xy-\-y'^ = i (6) Extracting root, ^ — ^ = ± i (7) Bringing down (i), x -}- y ^ g Combining (7) and (i), re = 5 or 4; ) And ?/ = 4 or 5, ) 10. Given x — y = 2 (i) And a^ — y^ =z 98 (2) Div. (2) by (i), a:2-(-a;?/ + 2/2 = 49 (3) Squaring (i), a:;^ — 2xy -{- y^ = 4 (4) Subt. (4) from (3), 3a:?/ = 45 And xy =15 (5) Adding (5) to (3), etc., x -\- y = ± S (6) Bringing down (i), x — y = 2 Combining (6) and (i), :c = 5 or — 3 ; ) ^^^^^ And - y = s or —5, f " 11. Given 3^^ — jy^ = — i (i) And 4xy = 24 (2) From (2), a; = - (3) Squaring and mult, by 3, ^x^ = — ^ •J _ _ Q Substituting in (i), — ^ 7^^ = — i Clearing, 108 — "jy* ■= — y^ Transposing, • iy^ — y^ ~ 108 By 3d method, 1/2 = 4 or — 3^ Dropping negative value, y =z 2 \ ] From (3), a: = 3, j 158 SIMULTAN-EOUS QUADRATICS. 12. Ans. Given :v^ — xy + if =19 (i) And xy = 15 (2) Siibt. (2) from (i), x^ — 2xy -f y^ =4 (3) Extracting root, x — ?/ = di 2 (4) Adding (1) to 3 times (2), x^ + 2xy + ?/2 = 64 (5) Extracting root, ^ + «/ = ± 8 (6) Combining (6) and (4), ic = di 5 ; | And y = ±2,, f 15. Given x -^ y = z'] (i) And xy =: 1^0 (2) Squaring (i), ic^ _|_ 2xy -{- y^ = 729 Multiplying (2) by 4, 4a:*?/ =720 Subtracting, x^ — 2xy -\- y^ = g (3) Extracting root, ^ — .V = ih 3 (4) Bringing down (i), x -{- y z= 2^ Combining (4) and (i), x =z 1$ or 12 And y ^ 12 or 15, 16. Given x — y= 14 (i) And xy = 147 (2) Squaring (i), x^ — 2xy -{- y^ z= ig6 (3) Multiplying (2) by 4, 4xy = 5 88 (4) Add. (4) to (3), X^ + 2Xy + ?/2 r:r 784 (5) Extracting root, ^ + i/ = ± 28 (6) Bringing down (i), x — y =1 14 Combining (6) and (i), a; == 21 or — 7 ; ) And ?/ = 7 or — 21, ) 17. Given o:^ — ?/t — 3 (i) And a:T 4- yT — 7 (2) Combining (i) and (2), ic^ — 5 And y^ = 2 Involving, ^ = 625 ; | ^^^^^^ And y = 16, \ ^ Aiis. SIMULTAKEOUS QUADRATICS. 159 1 8. Given ^y"^ -\- x^y-' =12 (i) And x^y -\- xy'^ = 6 (2) Factoring (i), xhf {x -{- y) = 12 (3) " (2), xy{x-i-y) =6 (4) Dividing (3) by (4), xy = 2 (5) (4) by (5), x+y = 3 (6) Combining (5) and (6), x — y— ±1 (7) Combining (6) and (7), re = 2 or i; ) . AT ( Ji-HS, And y z^ I ov 2, ) PROBLEMS. Page 191. 1. Let x = greater, And y = less. Then x — y = 4 ( 1 ) And 3? — y^ =z 448 (2) Divid. (2) by (i), x^-\-xy-\-y^ = 112 (3) Squaring (i), x^ — 2xy -\- y^ =: 16 (4) Subtracting (4) from (3), 327/ = 96 Or xy = 32 (5) Add. (3) and (5), x^+2xy^y'^ — 144 (6) Extracting root of (6), x -\- y z=z ±12 (7) Combining (i) and (7), a: = 8 or — 4; ) , And ?/ = 4 or — 8, i 2. Let a; = wife's age, Then a; + i = man's '• And ic^ -f a; = 930 By2dmethod, x — —\ ±1^930 + J—— J + V^ .-. ir = — |±-%i = 30 years, wife's age; ) And a; + I = 31 " man's " f 21 IGO SIMULTANEOUS Q U A D K A T I C S . 3. Let X = greater, and y = less. Then (x -{- y)x — 180 (i) And {x — y)y=i6 (2) From (i), x^ -{- xy =z 180 " (2), xy -7/2=16 Assuming, x = py Substituting, p^y^-\- py^ = iSo (3) And py^ — y^ =16 (4) ■^ , K 180 / , From (3). y=^l^, (5) " (4), y'^jzi-, (^) ■^ . , X 1 /^^ 16 180 Equating (5) and (6), ^— -^ = ^^^y^ Clearing, etc., \p^ + 4i^ = 45i^ — 45 Transposing, ^i9- — 4ip = — 45 By 3d, 64p^ — 6^6p -\- (41)2 = '—7204-1681 = 961 Extracting root, 8;;— 41 = ±31 Whence, 8;? =: 41 ± 31 Dropping 2d value, P = 9 (?) From (6), y"^ = 2 Extracting root, y — ±^2 ; | ^^^^^ And py, oy x — 9 A/2, \ 4. Let X = number of rows. And «/ = " " trees in each. Then xy — 1000 (i) And X — y = 15 (2) Squaring (2), x^ — 2xy + ?/2 = 225 (3) Multiplying (i) by 4, 4^ ^ := 4000 (4) Adding, x^ + 2ir?/ + «/^ = 4225 (5) Extracting root, .r + ?/ = ± 65 (6) Combining (6) and (2), x *- y — 15 (2) X = 40; y = 25, ^?z.<;. SIMULTANEOUS QUADRATICS. 161 5. Let X =^ length, And y = breadth. Then xy = g6o (i) And X — y = 16 (2) Multiplying (i) by 4, 4xy = 3840 (3) Squaring (2), x^—2xy-\-y'^ = 256 (4) Add. (3) to (4), x^ + 2xy + f = 4096 (5 ) Extracting root, a; -f // == ±64 (6) Combining (2) and (6), x =z 40 yards; ) . > Ans. y = 24 \ This problem may be solved by one unknown quantity thus : Let X = breadth. Then a; -f- 16 = length, And x^ -\- i6x = 960, area. By 2d method, x = — S ±: V960 + 64 Reducing, x =:z — 8^32 a; = 24 yds.; a: + 16 = 40 yds., A718. 6. Denote the numbers by x and y. Then x^ -t y^ — .(.c + y) = 78 (i) And ^^ + ^* + y = 39 (2) Adding (i) to (2) x 2, x^ + 2xy -{- y^ -\-x + 1/ = 156 (3) Or *(x + 2/)2 + (^ + ^y = 156 By 2d method, x -\- y — — |± 1/156 4- \ Eeducing, x + y ^=^ — i ± -¥■ Dropping 2d value, x -\- y ^=: 12 (4) Substituting (4) in (2), xy =27 (5) ^ Combining (4) squared and (5) x 4, x^ — 2xy-\-y'^ = 2>^ (6) .Extracting root, .9- — _?/ = ^h 6 (7) Combining (4) and (7), x — 9 or 3 ; ) ^^^^^ And , _?/ = 3 or 9, f ^ * Consider {x + y) as a single quantity. (Art. 69.) 163 SIMULTANEOU.S QUADEATICS. 7. Let X and y denote the numbers. Then x^ + «/^ + '^^' + ^ = 188 (i) And ic?/ = 77 (2) Adding (i) to (2) x 2, {x + yf J^ {x + y) — 342 (3) By 2d method, x -\- y ^ — -J ± a/342 + J Eeducmg, a; + «/ = 18 (4) Subt. (4) from (i), x^ -\- y'^ =1 i^o (5) Subtracting (2) x 2 from (5), x^ — 2xy 4- ?/2 = 16 (6) Extracting root, ^— ^=±4 (7) Combining {4) and (7), ;t' = 11 or 7; ] nd 2/ = 7 or II, j 8. Let a; = length, and y = breadth. Then 2{x + y) = 100 (i) And xy == $Sg (2) From (i), x-^y = s^ (3) Comb. (2) and (3), comp. sq., extract, root, etc., x — y =: ± 12 (4) Combining (3) and (4), ic = 31 rods; ) . And 7/ m 19 " ) 9. Let :?: and «/ denote the numbers. Then xy = 28 (i) And a;2 + ?/2 — 65 (2) From(i), 2xy = s^ (s) Adding, comp. sq., etc., x -{- y — ^ n (4) Subt., comp. sq., etc., ^ — «/ = ± 3 (5) Combining (4) and (5), x = ±7;) And y = ±A^ ) 10. Let X = number on side of greater square. And y z= '' « '^ less " Then x^ + y^ =z 1154 (i) And X — y = 2 (2) SIMULTANEOUS Q U A D 11 A T I S . 163 Subt. (2)2 from (i), 2xy =1150 (3) Add. (i) and (3), comp. sq., etc., x + y = ±48 (4) Combining (2) and (4), a; = 25 ; j ^^^^ And ^ = 23, f This problem may be solved by one unknown quantity: Let cfi -\- {x -\- 2)2 =1154 Uniting, comp. sq., etc., q: = 23 ; ) And a; + 2 = 25, j 11. Denote the numbers by a; and «/. Then ^U = zi^ + y) (0 And x^ -\- i/^ — 160 (2) Adding (i) x 2, ar^ -f 2x1/ -\- y"^ =z 160 -{- 6(x -\- y) (3) Transp., etc., (x-\-yY — 6{x-{-y) = 160 Comp. sq., 2d meth., etc., .'^ + ^ ^ 3 ± 13 Dropping negative value, x-\-y ^ 16 (4) Substituting in (i), xy = 4S (5) Combining (4) and (5), a: — «/ = ± 8 (6) (4) " (6), ^ = 12 or4; / ^^_ And y =: 4 or 12, f 12. Denote the numbers by a- and 7/. Then xy =z 6{x — y) (i) And a:^ -f 1/2 z= 13 (2) Subt. (i) X 2, x^ — 2xy + ?/2 = 13 — 12 (:c — y) Transposing, etc., (x-7jY + i2(x-y) = 13 Completing square, etc., x — y == — 6 ± V13 + 36 Dropping negative val., x — ^ = i (3) Substituting in (i), xy = 6 (4) Combining (2) and (4) and extracting root, •r + y= ±5 (5) (3) " (5)? a; = 3 or - 2;)^^^^ And ^ = 2 or — 3, I 164 RATIO. 3- 4- 5- 14. 15- RATIO. Page 195, 40 I 40 : 160 — --— = -, Ans. 160 4 1:81=-, Ans. o 64 : 320 = — = -, Ans, 320 5 6. 8^2 : 4fi^ = — z=z 2a, A71S, 4a J J iKabc , 7. isabc : sab — -^^— = 3c, ^ws. 8. 500 : 50 = - — z= 10, Ans, 9- 75 : 600 ::. -/^ = \ , Ans. 10. 35 : 35 X 4 2r<^ -^^ — = - , Ans, 35 X 4 4 II. 26^2 : 4ff rr: == - , ^?i. 15 : ij, Ans. 8:25 = 2^ ; V4 : V^5 = i, or il .'. fs < If, Ans, Let ic = tlie consequent, 56 Then X = S; 8.T = 56; X 7, yl^S. 26. Let X = the antecedent, X Then - = 14; 7 X = 98, ^?^5. PROPORTION. Page 204. 2. Let :» = the first term. Then x : ^ :: 6 : 12 And 12a; = 48 X =z 4, ^^.9. 3. Let a; = the third proportional. Then 25 : 400 : : 400 : x And 25a; = 160000 a; r= 6400, Ans. 4. Let X = the mean proportionaL Then g -. x : : x i id And a;2 r= 144 ic = 12, ^4ws. Let Then And Equating (i) Reducing, And X = the greater, and y = the less. X : y :: x-\-y : 42 X : y :: i?:;— ?/ : 6 and (2), (Theorem 9), x-\-y : x.—y :: 42 : 6 x-\-y '. X — y :: 7:1 ^ + y — i^ — iy (2) Transposing, etc., X . - ^y (3) 166 PROPORTION. Substituting in (2), ~ ' y '-'- ~ —y \ ^ o o 4 y Div. ist couplet by ?/, - : i :: - : 6 (Th. 6) Mult, antecedents by 3, 4:1 : : ^ • 6 y =^ 24, less: I ^^^^^ Substituting in (3), x = :^2, gr., f 6. Let X = one part, and 18 — a; = the other. Then x^ : (18— a;)^ :: 25 : 16 (Th. 12) Extracting root, a; : 18 — x :: 5:4 Eeducing, 4X = 90 — 5^ r?; z= 10; ) And i8-a:= 8, \^'''' 7. Let ic = the greater, and 28 — x = the less. Then -— : :: 32 : 18 28 — it' X Reducing, gx^ = 16 (28 — xY Extracting root, (Th. 12), 32; = 4 (28 — x) Transposing, jx = 112 And X = 16:) . 28 — ^ = 12, J 8. Denote the numbers by x and y. Then xy = 24 (i) And x^—y^ : {x—yY :: 19 : i (2) Dividing hj x — y, x^-\-xy-{-y^ : x^ — 2xy + y^ :: 19 : i By Theorem 8, ^xy : {x—yY :: 18 : i " " 6, xy : (x—yY :: 6:1 " " I, 6 (x — yY = ocy Subst. value of xy, 6 (x — yY = 24 Eeducing, x — y =^ 2 (3) Combining (i) and (3), a; + ?/ = 10 (4) a (3) " (4), - = oa^^, = 4> ) And y I>t?OPORTtON-. 16'^ Denote the numbers by x and y. Then x-\-y \ x — y :: 9 : 6 (i) And '^^—y ' xy :: 1 : 12 (2) Reducing (i) by Theorem (6), x-\-y : X — y 1:3:2 By Theorem 7, 2X : x — y : : 5:2 I, 4^ = 5 {^ - y) (3) Eeducing (2) by Th. i, xy = 12 (x — y) (4) Dividing (3) by (4), etc., sy = 48 iz; = 48, f ri ns. Substituting in (3), 10. Let X = the length in rods, And y = " breadth " Then the area, xy = 860 x 160 (i) And X : y :: 4S : 32 (2) Reducing (2), 322; = 43?/ And x = ^^ (3) 32 ^"^^ 4"??/^ Substituting in (i), ^^^^- — 860 x 160 o T^ 1 . „ 8S0 X 160 X 32 Reducino- ?/^ = -^^ "" -^ 43 Cancelling 43, y'^ = 6400 x 16 Extracting root, ?/ = 80 x 4 = 320 r. breadth; Substituting in (3), x = 430 r. length. 1 1. Let X = side of one, Then 10 + .'c = " " the other. And x^ : (x i- lo)'-^ : : 4:9 By Theorem 12, ic:a;+io :: 2:3 " '*' I, 3:c == 2:?; + 20 a; = 20 rods ; ) , * -. .. ' \ A71S. And 10 + a; = 30 " ) 1G8 ARITHMETICAL PROGRESSION". 12. Denote the numbers by x and y. Then xy = 135 (i) And x^ — y^ : {x — yf By Theorem 6, x -\- y : x — y " " 7, 2T : 2; — y 4 : I 4 : I 5 ' I (6 « " I, . 2a; = 5^ — 5?/ And X = -^ (2) 3 Substituting in (i), ^^ =r 135 Reducing, 2/^ = 81 Extracting root, y = g\ \ Substituting in (2), 2; = 15, i 13. Denote the numbers by x and y. Then ^?/ = 320 (i) And x^— y^ : (x — yY : : 61:1 By Theorem. 6, x^-\-xy-\-y^ : x^ — 2xy-\-y'^ :: 61 : i " 2>, ^xy : x^ — 2xy -\- y^ : : 60 : i " I, ^xy ^ 60 (x^—2xy-{-y'^) Dividing by 3. 20 {x — yY = xy Subst. val. of xy, 20 (x — yY = ^20 Dividing by 20, (2; — ^)^ = 16 Extracting root, x — y =z 4 (2) Combining (i) and (2), x -{- y = s^ (3) " (2) " (3), .-r = 20 ; ) . And y = ^^, ) ARITHMETICAL PROGRESSION. Page 207- 2. / = « ± (^ — i) ^ = 25 + (9 — 1) (— 2) =3 25 — 16 = 9, Ans, 3. ?= 12 + 14 X 4 = 68, .1^5. 4. ? = I — 12X2 = — 5' ^^^^* 5. Z r= J -f 8 X -J = ij, ^1/iS. ARITHMETICAL PROGRESSION. 169 6. I =^ I — 9 X .01 = .91, Ahs. 8. ^ :^ I + 14 X 3 = 43? ^n^' 9. Z = 31 — 8 X 2 = 15, A71S. 10. Z = I + 29 X i^ = 44h ^*^^- 11. Z = a; + 24 X 2a; = 4gx, Ans. 12. Z = 2a H- (/i — i) 3a = 3«7i — a, Ans, Page 208. 2. s ■= X /i ; therefore, 2 s = ^ ^ X 50 = 762J, JW5. 3. s = -^ X 9 = 216, Ans, 4. s = ^ - X 35 = 1400? ^^i5« 2 5. 5 = — ' — X 17 = 25i, A)is, 2 6. ? = 2 + 19 X 3 = 59 ; and s ■=. X 20 = 610, Ans. 2 7. / = I + 24 X i = 13 ; and s = X 25 = 175, Ans. 2 8. Z = 75 — 14 X 3 = 33 ; and 75 + 33 ^ 15 = 810, ^?2s. 2 1. ? = 25 + II X 3 = 58, Ans. 2. ? = 58 + 44 X 5 = 278, yl?k -'— x n 2 2 e ..^_ fjiy^ I fl'}'). Multiplying by 2, etc., a = — • 10. Given d, I, and s, to find a, the first term. Equating Formulas (5) and (8), I — a _ 2s ~d~ "^ ^ ~ «+l Clearing of fractions, 12 _ ^^2 _|_ ^f/ -I- f^/ = 2^.§ Transposing, a^ — da ^ 1^ + dl — 2ds d I d? Completing sq., etc., « = - ± A / 'l?-\-dl-\ 2ds Changing the form, « = - ± A / (Z + - ) 2ds 11. Given a, 1, and ., I3i^ 20f, 2%^ 35J, 42|, 50, 57-J, 64J, 72, J725. AKITHMETICAL PliOGRESSlON". 175 / — a 1 08 — 12 11. d =. =r — 9.6 w 4- I 10 ^ Hence the series: 12, 2 1.6, 31.2, 40.8, 50.4, 60, 69.6, 79.2, 88.8, 98. j|, 108, Aiii^. 12. I = 100 — 14 X 5 = 30; and 100 + "^O . ^ — ^ X 15 = 975. ^'^5- 14. Let X = the second nnniber, And y = the common difference. Then, x^y + x-{-x-{-y = 15; or 32:= 15, a; =5. (i) And {x—yy + x^+{x + yY = 49S (2) Expanding and reducing, 30.^4-62:^'^ = 495 Substituting value of .r, etc., 30?/^ =120 y = ±2 Hence the numbers: 3, 5, and 7, Aiis, 15. Since he lias to pass over the ground twice for each marble, the problem requires us to find twice the sum of the series. By Formula (i), I = a -i- (n — 1) d I z= I -f 99 r= 100 By Formula (2), s = x 7i 2 (i + 100) 100 , 25 = -^ = 10 100 yds. == 51 miles nearly, Aus. Page 214. 16. By Formula (2), 2s = — x 12 = 156, Ans. 17. By Formula (i), 7= 10 + 24 x 20 = I4.90 « « (2), s = ;^- X 25 8 = $2.50 X 25 = $62.50, Ans, 176 ARITHMETICAL PROGRESSION. 1 8. By Formula (2), s = — "tJ-l x 365; s = 183 X 365 = I667.95, ^^^« T „1„ *^ A 19. By Formula (2), s = x 24 = 300, Ans. 20. By Formula (i), ^ =: .06 + 19 x .06 = I1.20, int. ; li.oo + I1.20 = $2.20, amount, Ans, 21. Denote the numbers by a — d r a And a -^ d Then 30^ = 120; a = 40, And 3^2 + 2d^ = 5600 d ■=. 20 Hence, a — d =z2o,a = 40, and a + d = 60, ^ws. 22. Let n = No. of days the 2d travels, Then 30 + ion = " " miles " ist " By Formula (i), I = 4 -\- {n — i) / = 3 + ?^ By Formula (2), s = - — xn = dis. 2d goes. Equatnig, -5^--^^— = 30 + ion Mult, by 2, fn-j-n^ = 60 + 2on Transpos., n^—i$n = 60 Comp. sq., etc., n = ^ ± \/6o + -3^ 71 =: 16.61+ days, A71S. 23. Denote the numbers by ff — 3^?, a — d, a -\- d, and a + 3d, Square of ist, a^ —3f^(^ + 9^^ " " 4th, r?^ 4- 6 flfr/ 4- 9^<^^ ' Sum, 2^2 + i8(?2 = 4500 (i) ARITHMETICAL PROGRESSION". 177 Square of 2d, a^ — 2 ad + d^ « " 3d, a^ -{- 2ad+ ^ Sum, 20? + 2f/2 = 4100 (2) Subtracting (2) from (i), i6d^ = 400 d := ^ Substituting in (2), 2^2 — 4050 «^ = 45 Hence, cj — 3^/ = 30, a — d = ^o, ) « -}- ^ = 50, « + 3fZ = 60, j ^ 24. Let ?z = No. of days B travels. By Formula (i), I = 7 -}- {71 — 1) 2 = distance A goes the last day. ? = 5 + 2W By Formula (2), s = ^ — x n = 6n + n^, 2 Hence, g -\- 6n -{- 71^ = No. of miles A travels. By Formula (i), I = 11 + 71 — i = 10 -{- n = distance B goes the last day. ■D 17 1 /\ II^-Io + ?^ 21?? 4- 7^2 By Formula (2), 5 = ' — x n = — 2 2 = B's distance. Whence, g + 671 + 71^ = 2 Reducing, w = 3 or 6 days, A7is, 25. By Formula (i), l=io — 2oxi = ^ 10 -I- -ip- " " (2), 5 = — ?- X 21 = 140, Ans. 2 26. By Formula (i), ? = I + 59 X 3 = $178 last payment; ) By For. (2), .9 = i±-il5 x 60 = $5370 debt, ( ^^^' 178 GEOMETRICAL PKO GRESiSI K. GEOMETRICAL PROGRESSION. rage 210. 1. I ;=. af'~^ = 5 X 2^ =: i6o, Ans, 2. Z = 2 X 3"^ = 4374? ^ns, 3. Z = 72 X (4)4 = 4i ^^^s. 4. Z = 5 X 4^ = 320, Ans, 5. ^ = 7 X 2^ = 112, Ans, 6. ^ = 10 X (— 5)^ = — 31250, A71S, rage 217- Ir — a 2000 X 5 — 8 „ . 2. s = = — = 2498, Ans, r — 1 4 5000 X 10 — 9 , 4. 5 = £_ 5QOQ ^ "^ ~ ^ = 3333 1 1? ^^«- 3 20 X 6 — 15 . c. s = = 21, ^?i5. ^ 5 12 X 4 — 25 , . 3 rage 218, I. ? := «r"~^ = 3 X 10* = 30000, Ans» 2. ? = 5 X 5^ = 15625, ^W5. / 256 . 3. « = ^1 = -^ = 2, ^^is. 4. « = -^ = 3, ^;^s. 5- " = (it = iP?f = ^' ^- 6. r = f-^ — y = 1/25 X 25 = 5, Anci, GEOMETEICAL PROGKESSION. 179 I. S = Page 21i). Ir — a io8 X 3 — 2 r= l6l, A71S. r — I 2 2. « = /r — -s (r — i) = 54 X 3 — 8o X 2 = 2, A7is. ^' r 5 s — a 15624 — 4 . ^ s — I 15624 — 12500 Ir — a 150 X 6 — 5 . ^ 5. 6' = -^^--- = -'-- ^ = 179. ^4^.5. r 10 FORMULAS. Given ??, r, and *', to find a. By Formula (i), I = ccr"-'^ Multiplying by r, Ir = af' Substituting in For. (6), a = ar" — (r — i) s Transposing, etc., «r" — a= {r — i) s {r-i)s /•" — I Factoring, etc., a = 10. Given I, n, and s, to find a. Equating Formulas (4) and (8), sj-a _ //\„4r s — I \al 'Invohnng and clearing of fractions, a{s — aY-' = lis — iy-K 11. Given «, 71, and s, to find I. Transp. For. (10), I (s — I}"-^ = a (^ — ay-K 180 GEOMETRICAL PKOG R ESS lOl^". 12. Given n, r, and s, to find I. Equating Formulas (3) and (9), I _ (r — i)j a»n— 1 /pll T /^» i\ g^n—\ Multiplying by r-\ I = ^^-^-"1 13. Given a, I, and s, to find n. By Formula (10), a{s — «)«-' = l{s — 0»-i By logarithms. log.a-^\og.{s—c(){u—i) = log./ + log. (s—l)(n — i) Transposing, log. {s—a)(7i—i)—\og. (s—l){7i — i) = \og.l—\og.a Factoring, etc., log. I — log. a 71 — I = log. {s — a) — log. {s — I) r^ . log. / — log. a , Transposing, « = to^,-Tr^^T^g.-(J^7) + '' 14. Given a, r, and s, to find 71. By Formula (9), a = —r^- Kemoving denominator, ar" — a = (r — i)s Transposing, at" = «+(/• — 1)5 By logarithms, log. a + log. r X /^ = log. [rt -\- (r — i) s] Transp., etc., 7i = ^~^- YogT^"^ ^ — GEOMETllICAL f ROG R ESS J ]^. 181 15. Given I, r, and s, to find n. By Formula (12), I = ^, _ ^ Removing denominator, Ir"" — I = (r — i) sr"^~^ Transposing, In — (r — 1) sr"~^ = I Factoring, [Ir — (r — i)s]r"'^ = I Dividing, r"~^ = j . r- °' Ir — (r—i)s By logarithms, log. r X {n— i) = log. I — log. [/r — (r — i) s] Dividing and transposing, _ log. / - log. [Ir - {r - 1)5] log. r 16. Given a, w, and 5, to find r. By Formula (9), a = ~^P~^ Removing denom,, «r" — a = rs — s Transposing, ar"^ — rs ■= a — s s s Dividing, y" r = 1 — -• 17. Given /, n, and s, to find n ^^' j\ ^/•"■~* By Formula (12), 1= ,,n _ j — Removing denom., Irn — I =z (r — i) sr"~^ Transposing, Zr" — (r — i) sr"~^ = / Or, Ir — sr" + sr"-^ = I s I Factoring, etc., r" -f- j r""^ = , __ -» 18. Given a, n, and r, to find s. By Formula (9), « = - ^ _ — Removing denom., a (r« — i) = {r — i) s ^. .,. «(r''- I) Dividmg, s = -_ • 182 GEOMETRICAL PROGEESSION. 19. Given /, Uy and r, to find s. By Formula (12), I = fr~'^^'^-- r" — I Removing denom., Ir'^ — I = (r" — r"~^) s Ir" — I Dividinsf s = 20. Given cj, Z, and ^i, to find s. By Formula (4), r = ^ — T^ — a /a Substituting in For. (2), 5 =r —-^ n — 1/ Multiplying both terms by '^V^? _ v/" — ^ V«" v? — v« 2, T — f-V"^^ = (128 -^ i)^ = V^6 =r 4. ^?^5. \, 2, 8, 32, 128. I. By Formula (2), 5 = Substituting, PROBLEMS. Ir —a r — I 2916x3 — 6 . = ~ = 437 1 J A71S, 2, By Formula (18), 5 = , or s = Substituting, s = ?-— :i:Aill_ =, 2 rrm 16 40 .(.,.. GEOMETRICAL PKOGKESSION. 183 By Fomiulu (i), I — ar""' = i x 3^^; ^ '^ r — I 2 2 (243)^ — I . = ^^^^ = 7174453. ^^^•^'• 2 2il2 4. By Formula ( I ), I = i x (f)"; « " (2), s = = '-^ — I— ^ ^ r — I — ^ — ,17 3 5 1 A )iq 5. By Formula (i), / = nf'-^ — 2 x 3^^—9565938, Ans. 6. By Formula (i), / = 3 x 3^^ = 43046721, Ans, rage 222, 8. By Formula (i), / = «r"~^ / = I X 2^1 = 2048 By Formula (2), s = 2048 X 2 — I . . s = — ^^— = '^4095? A?is. 9. By For. (i), Z = i x 3^ = 19683, « « /2^ „^ 19683x3 -I ^ 59048 ^ ^' 3 — 1 2 = I295.24, entire cost; ) $196.83, last cow, \ 10. By Formula (i), ? = i x 2^ = 512 (2), s = 5iiJX_l^I =: $10.23, ^l;i5. 2 — I 11. By the conditions, a-^ar + ar^ =: 26 (i) And «2 ^ ^^2 ^ cf2)A — 364 (2) Transposing ar in (i) and squaring, a^ H- 2rtV2 + rt'H = 676 — 52ar + «2r2 Reducing, a^ + a^r^ + a^/"^ = 676— 52«r (3) 184 GEOMETKICAL PKOGRESSION. Eq. (2) and (3), 676— 52fljr = 364 ar = 6 (4) ^ = r (5) Substituting in ( I ), - + 6 -{- 6r = 26 Reducing, r^ — ^r = — 1 Completing square, etc., ?' = 3 From (5), a = 2 Hence, a = 2, ar = 6, ar^ = 18, Ans, 31 12. By Formula (i), I =z i x 2^ = 2 (2), S = ^'' ^ ^ ~ ' := 232 _ I ^ 2 — I =z $4294967.295, Ans. 13. By the conditions, a -\- ar -\- ar'^ = 130 And ar -j- ar^ + ar^ = 390 Factoring, «(i + r + r^) = 130 " «r (i + r + r^) = 390 Dividing, ^' = 3 Substituting, ^(1 + 3+9) = 130 « = 10, 057* = 30, «r2 = 90, ar^ = 270, ^^6\ rage 223, 14. By the conditions, a -\-ar-\-ar'^ =1 210 (i) And « — ar^ =z 90 (2) ^'•"'n (')' « = t:^^:^ (3) Equating (3) and (4), — Reducing, 7 — 7/*^ = 3 H- 3?' + 3^^ Completing square, etc., ^* = | Substituting in (4), a = 120 Hence, a = $120, ar == $60, ar^ = $30, ^W5. GEOMETRICAL PROGRESSION. 185 15. By the conditions, a-\-ar-}-ar^ + ar^ = 46S (i) And ar + ar^ + ar^ + ar* = 2340 (2) Factoring, a (i -{- r -\- r^ -\- 7-^) = 468 (3) « ar (i + r + r^ + r^) =: 2340 Dividing, ^ = 5 Substituting in (3), 1560! = 468 a = 3 Hence the numbers are 3, 15, 75, 375, 1875, Ans. 16. Denote the shares by — , cc, y, and — • y a; Then — + '^' + 2/ + "|- = 7oo (i) And ^^^ : y—x :: 37 : 12 (2) Clearing (i), Adding, 2X^y -\- 2xy^ = 2X -y -f 2^^^ (A x. 2) - (2: + yY =: -jooxy + 2iry {x -\-y) (3) Reducing (2), ^— : ^/-a: :: 37 : 12 Dividing by {y — x), (Theorem 6), y^ + xy + x^ 37 : 12 xy Multiplying by xy, (Theorem 6), y^ -\- xy -{- x^ : xy :: 37 : 12 Or, (Th. 7), (y + x^ : xy :: 49 : 12 (4) Multiplying hj y -\- x, (Theorem 6), (y + :r)3 : xy :: 4g{y + x) : 12 Or, (, + ,)s = 49^(£±1) (5) Equating (3) and (5), , 4QXU (x + y) •jooxy + 2:^7/ (.r + y) = '^^—-^ Reducing, 8400 + 24 {x + ?/) = 49 {x 4- y) And ^ + y = 33*^ (6^ 186 GEOMETRICAL PROGRESSION. Substituting in (4), ^^xy =z (Z2>^Y x 1 2 xy =z (48)2 X 12 Multiplying by 4, /^y = (48)3 = 110592 (7) Squaring (6), (x + yY = (ss^f = 112896 (8) Subt. (7) from (8), {y — xY = 2304 Extracting root, y — a; = 48 (9) Combining (6) and (9), y -\- x ^ 336 y = I192; x = I144; — := I108; / A71S. y ^ = I256, * 17. By Formula (i), ar"~^ =z I "Substituting, loooor'' — 1464. Extracting square root, loor^ =121 r = fj, or I.I, A71S, « " " lor = II ^ ?/ 18. Denote the numbers by ~, x, y, and — • '/ X J*2 Then + y And -^4- X y2 '-\-y' + \, 85 Assume 2^4- i/ _ 5 a ^2/ - ^ Then .T^ 4- v'^ '^^ — "^p And ^ _|_ yZ ^3 ^^p From (i), X^ if' • ^'' (^)^ ^+J = «5 ^+V^ (0 (2) (3) (4) INFINITE SERIES. 18T Squaring (3), --^ + -^ = 225— 30.S + 6-2-2;; (5) Equating (4) and (5), 85 — 5^ + 2]) = 225 — 305 + 5^ — 2p Transposing, 85 + 47; = 225 — 305 + 26^ (6) From (3), Q^ + y^ = ^5P — sp Or, s^ — 2,sp = ISP — sp Substituting in (6), 85 + ^7^— = 225 - 305 + 2S'2 Removing denominator, etc., 35*2 -j- ijs = 210 Comp. sq., etc., s = 6 or ct-\-i/ = 6 (8) Subistituting in (7), ;; = 8 or xy =z S (9) Comb. (8) and (9), x—y =2 (10) « (8) " (10), x = 4. y = 2, x^ ifi And — = 8, and -^ = i. Hence, 8, 4, 2, i, Ans. y ^ INFINITE SERIES. Pnge 227, 4. I — X ) I H- .1' ( I + 2a; + 2x^ -\- 2x^ H- 2x^, etc., Ans. I — .r 2X 2X — 2X'^ 2X^ 27?' — 2.1*3 2;7'3 20^3 _ zx^, etc. 188 INFI]SriTE SERIES. rage 228. x^ — y^\ if y^ if , . 2X — ^- 2X — i/2 -f+^ 20; — -^ X 8a^ 4X^ — ^, + f + y^ 4^^ Sic^ 64^;^ 2x^t^y^^ '^ X ^T" 167^ 1_ __^_ , etc. 8. I -^ I ( I + * — i + tV» etc., Ans. 2 + 1) I 2+I-i)-i i-i + A 2 + 1-I + A) ^ — ^4? ^^^c. 10. The exponents of x decrease by 2. (Art. 271.) The coefficients are found by means of the Binomial Formula. (Art. 270.) ist coef. = I 2d " =n = \ n ^^ __ i X — I i_ 2 2 2«4 3d <<; = W X ., ,, M— IW— 2 I % 4th " = 7^ X X = — — X — 2 3 2-4 3 2'/\'(> IKI^I^ITE SERIES. 189 Sth coei. ^ 71 X X X 234 X -^= ^5, etc. • 2'4«6 4 2-4'6'8 Hence we have, (:^ + j,)i = r. + 5:^ - ?-'-i^' + 3^""-! - 3J^, etc. ^ ^' 2 2-4 2-4-6 2-4-6-8 Or, transferring x to the denominator (Art. 279), ^a;+2/j -^+2^; 2.42:3+2.4.62:5 2.4.6.82:7' ^^^• II. Find the first five terms as in the preceding sohition. ,, , . n — I 71 — 2 n — X 71 — 4 6th term = ?j x x — — x x 2345 _ ^'S ^ i _ 3-5-7 2.4'6.8 5 2.4.6.8.10 rage 231. I. The ratio may be found by dividing any term by the preceding one. a I I s = = — y = „ = li Ans, __ I _ ^ _ 2 J I I 3. s = — ^ = Y = I, Ans. 1 ^^ -ij -5^ 2 2 X 1 _2 _ 2 _ .1 4. 5 = ; 2 = 1= ^h ^^«» 1 — 3 3 _ _^_ _ * _ 5. S = —^2 = Y = 2, ^W5. 6. s = ^ = 1=9, Ans, 4 4 ^ 7. S = 3 = 2 = 10. ^?i«. too LOG ARITiiMg. 8. 5 = 9. s = 3 TO I — TO 3 10 9 10 = -'^ = h Ans. 6 To 6 TO I -TO I — 9 — 3' 9 TT7 2 ^^5. 10. S = i^(,_i)^l^('i:=_L)^ a \ a/ a \ a I T flf - X — - — a a — I II. s =. 10 4 10 — , ^;i5. a — I 50 rods, Ans, LOGARITHMS. rage 235. I. Log. 7 = 0.84510, A71S, 2. « 9 — 0.95424, Ans, 3. « 108 = 2.03342, Ans. 4. a 176 — 2.24551, Ans. 5- ce 1990 — 3.29885 Log. 223 X 9 1999 : 200 3.30085, Ans. 6. « 0-95 = 1.97772, Ans, 7. <( 0.0125 — 2.09691, Ans. 8. a 0.0075 — 3.87506, Ans. 9- a 16.40 "^^ 1. 21484 264 X 5 — 132 Log. 16.45 = 1.21616, Ans 0. (( 185.0 — 2.26717 235 X 3 — 70 Log. 185.3 — 2.26787, Ans, LOGARITHMS. 191 11. Given 2.17231 z=z log. 148.7, Ans. 17026 301 ) 20500 ( 68 12. Given 1.2526 1 = log. 17.89, Ans. 25042 249 ) 21900 ( 87 13. Given 3.27715 = log. 1893, Ans, 27646 235 ) 69000 ( 293 14. Given 2.30963 = log. 204, Ans. 15. Given 4.29797 = log. 19858.29, Ans, 29667 223 ) 1300000 ( 5829 16. Given 1. 14488 = log. c.1396, Ans, 14302 322 ) 18600 ( 57 17. Given 2.29136 = log. 0.01956, Ans. 29003 223 ) 13300 ( 59 18. Given 3.30928 = log. 0.002038, Ans, 3 0750 212 ) 1780 ( 8 Page 237. 2. Log. 109.0 = 2.03743 416 X 3 = 125 '• 14.10 = 1. 14922 301 X 7 = 211 3. 1 900 1 = loo. 1548.86, Ans. 18752 281 ) 249000 ( 886 193 LOGARITHMS. 3- Log. 1.460 .16435 301 X5 151 " 1.340 — .12711 322 X7 225 .29522 — log. T.973, Ans. 447 223 ) 750 ( 3 4. Log. .074 ^z 2.86923 " 1500 = 3.17609 2.04532 = log. Ill, Ans, 6. Log. 12.40 = 1.09342 349 X 8 = 279 1.09621 *' 0.16 = 1. 20412 1.89209 = log. 78, Ans, Log. .041; = 2.65321 " 1.20 = .07918 2.57403 = log. .0375, Ans. 8. Log. 1.380 = .13988 322 X I = 32 . 14020 " .096 = 2.98227 1.15793 = log- 14-38, Ans, 534 301 ) 2590 ( 8 Log. — 128 = 2.T0721 " —47 = 1. 67210 .43511 = log. 2.723, Ans, 457 158 ) 540 (3 loCtAKithm^. 19^ lO. Log. 186 2.26951 a — 0.064 2.80618 3-4^333 — log- 2906.3, Ans, 240 147 ) 930 ( 63 Log. — 0.156 zzz 1-19313 a —0.86 1-93450 1.25863 log. .1814, Ans, 768 235 ) 950 ( 4 Log. — 0.194 — 1.28780 a 0.042 — 2.62325 .66455 = log. —4.619, Ans, 370 93 ) 850 ( 9 rage 239, 14. Log. .135 = 1. 13033 4 4.52132 = log. .0003321, Ans, 15. Log. 1.42 X 10 = 1.5229 = log. 33-335. ^ns, .52244 16. Log. 1.230 = 349 X 4 = 130 ) 4600 ( 35 .08991 140 .09131 25 45655 18262 : log. 191 -77. 2.28275 = Ans, 103 223 ) 17200 ( 77 104 LOGARITHMS. i8. Log. 143.0 = 2.15534 301 X 2 = 60 3 ) 2.155 94 .71865 = log. 5.23, Ans, 19. Log. 1.62 z= .20952 Dividing by 6, .03492 = log. 1.0836, J ?25. 342 416 ) 15000 ( 36 20. Log. 1540 = 3.18752 281x9 = 253 8 ) 3-19 005 •39875 = log- 2.504, A^is. 794 171 ) 810 ( 4 21. Log. 1876 -T- 10 = .32732 = log. 2.124, Ans. rage 240. 23. Log. .001624 = 3.21058 Div. by 6, 6)64-3.21058 1-53509 = log. .342 + , Ans. 24. Log. .01449 =r 2.16107 Div. by 7, 7 ) 7+5.1610 7 1.73729 = log. .546+, ^^i5. 25. Log. .0001236 = 4.09200 Divid. by 8, 8)8 + 4.09200 1.5 1 150 = log. .324+, Ans. 27. Log. 1.07 X 4 = -11752 « 1500 = 3 -17^09 Adding, 3.29361 = log. $1966.05, Ans. feUSlKESS FORMULAS. 19d 28. Log. 1.05 X33 = .69927 "' 370 =: 2.56820 Adding, ^ 3.26747 = log. $1851.27 4-,^?25. BUSINESS FORMULAS. Par/e 24^'>. 2. p = cr = $4370 X .08 = $349.60, A71S. P ^500 i. A 3. r = - = r= .20 ; 20 per cent, ^;^s. ^ c 2500 ' -^ $^00 4- ^' = ^^Yg^Q — -161; 16J per cent, .4??5. (Art. 238.) 7; I67.48 „ . . 5. c = - = = $5269.92, Arts, *» r .25 ^ ^ 6. c = '- = -^^ = I12600, Ans, r .i2i 7. s = c{i + r) = $750 X 1. 15 — 1:862.50, Ans. 8. s = c{i —r) = ^960 X .87I = $840, ^7is. s $S40 i^ 10. c = = -^^^^ = I600, A71S I — r .90 12. ^ = — - = 16J; 16 1 years, Ans, 13. ^ = — = 10; 10 years, Ans. 14- ** = 7 = o = 'i^l; 12J per cent, ^^25. 15. r = J = — = .02i; 2f per cent, Ans. 17. a=z 2^{i + r)" = I1500 X (1.05)6 = -^2010.14, ^;25. 18. a = $2000 X (1.03)6 = ^2388.05, A71S. 19. a = ^5000 X (i.oi)^ = I5414.28, A?is. 196 BUSINESS FORMULAS. „ s I3600 I + nr 1.30 / V v). 1 ' ;> J;?s-. $3600 — I2 769.23 = I830.77, discount, 22. P = = $6000, present worth ; / , 1.30 >Ans, cl = I7800 — I56000 = $1800, discount, / T1 S I23OO „ ^ . 26. tZ = I2500 X .15 = I375; I2500 — $375 = $2125, cash value, Ans. c (i + r) $1.7'; X 1.20 . I — <:/ .90 ^^^ ^j.qo X 1. 21; .„ ^ 29. m = — ^ = $S.Ss + , Ans. .92 ^r I6000 X .06 31. M =^— = ^2 T~ir^- = 5MI P^r cent, ^?Z5. *^ c I6000 + I180 *^^"^ -^ 32. E —^= l^^^ ^ '^^ = i2i per cent, A71S, C ^1000 — «|)200 33- -^ = IsoooVs'/o" ^ 9A per cent, ^«.. 34. R = — ^r '- — = 6 per cent, Massachusetts; qpIOO „ $100 X .08 ., . /%! • Hence the Ohio bonds are preferable, Ans. s $215000 . ^ 36. a = = — = $24630.54, invest. ; . . ^ 14-r 1.015 t o j-tj >Ans, $25000— $24630.54 = $369.46, commission (i+r)"— I (1.07)*— I. .3108 „ 38. a =z ^-I— ^ s = ^^ — -^ $300 = -^ — $300 ^^ r .07 "^ .07 = $1332, A71S, BUSINESS FORMULAS. 197 (l. 05)1*'— I . .629015,, . , 39. a — ^— ^^ $500= — ^ — ^•^500=16290.15, Ans. (i +r)«_ I {1.05)5— I 40. s = a-T- - — -—^ = $5000 -^ ^ ^^ *• " .05 250 . = —z = ^905.80, Ans. .276 -' ^ (i +r)"— I . (1.10)5 — I 41. § = fl5 -i- -i ^^ = $20000 -r- r .10 = I20000 -T- .61 = $3278.69, Ans. (i.o6)i0— I I1800 , 00. 42. S = $30000-^-^^ — = = I2 278.48, ^?2S. ^ .06 .79 (1.07)^ — I,, ."iioS., „ ^„, 43. 5 = ^ — ^-^ '^650 r= -^ $650 = I2886, Ans. .07 .07 Note. — The answers will vary slightly according to the number of decunals used in the solution. 44. , = (' + '•)' - ' a = il£^'^ $880 = :4i8 ^33^ r .06 .06 = $6130.67, Ans. (I.OO'^ I ^ •407 >, A. ^ ^ A 45. ,*? — ^^ — ^ I340 = -^-^$340 = $2767.60, Ans. .05 .05 47. ^ = ^ — ^^$525 = --^%25 = $2336.25, Ans, .04 '04 49. P = - =z — ~- = S14166.67, A71S, r .06 ' 51. P = ^ [(i + r)-" - (i +r)-'-^] P = ^-^[(i.o6)-e-(x.o6)-] .06 ^''^ _ Pr _ $3840 X .05 53- «^ - I _^+ r)-- ~ I -(1.05P0 $3840 X .05 .7687 = I249.77, Ans, 198 IMAGINARY QUAIS^TITIES. IMAGINARY QUANTITIES. rages 265, 206, 2. + V—x X — V— y = + V^ X V— I X — Vy X V— I = — Vxy X — I = + V^y, ^ns, 3. V— 9 X a/— 4 r= 1/9 X V— I X A/4 X V— I = 6x — 1 = — 6, ^>i5. 4. V— 2 X a/iS = a/2 X a/— I X a/i8 = A/36 X a/— ^i = 6a/— I, Ans, 5. a/— a? X Vy — Vx X a/— I X \^y = a/^^?/ X a/— I = a/— xy, Ans, V — X \/x X a/ — I A 7. — ....^ = — = j==^ = I, Ans. V — X Vx X a/ — I Vy Vy \ y a/cc a/^ / ^ 4 9. — - — = — = — = \/ — ' ^^^^^* V—y Vy X a/ — I V ~ 2^ ioa/— 14 io\^i4 X a/— I /- 4 10. — ;=^ = — 7^ 7=^=- = ^^2, Ans. 2a/— 7 2A/7 X y — I cV — I c . II. -;zr= = ^, ^?Z6-. fh/- T ^ NEGATIVE SOLUTIONS. 199 IMPOSSIBLE PROBLEMS. Page 20S, 2, Let a; = the number. Then = ik 5 4 Clearing of fractions, 4X — 5:^ = 300. It is impossible that 4X should be greater than $x. 3- Let X one part. And y the other. Then X + y — S And xy 18 (0 Now the product will be greatest when the parts are equal. Making y equal to x, the equations become, 2^=8 (3) :^^ — 18 (4) From (3)* X = 4 (5) Squaring (5), a:^ = 16 (6) Equations (4) and (6) are contradictory. Hence, the problem is impossible. NEGATIVE SOLUTIONS. Page 209. Let X = the number of years. Then 36 + :c = (20 + x) 2 Uniting terms, x ^ —4, Am, 200 hor:n^er's method. HORNER'S METHOD. rffge 27 s. 3. ^ + 3^^ + 5^ — 178. A B C D a b cde I + 3 + 5 = 178 ( X = 4. 5388, ^yi». 4 28 132 7 33 46 = !)' B^G'=h 4 44 42.375 46-^77 =.5 II 77=.C' 3.625 = D" 4 7.75 2.797377 15 = 5' 84.75 .827623 = D" .5 8 .749942 15-5 92.75 = C" .077681 1 •5 •4959 .074994 16 932459 .5 .4968 16.5 = £ V 93.74 27 = C" •03 16.53 •03 16.56 4. p;3 + 9.^-2 _ - jx - 2200. •. A B C D abed 5 + 9 -7 — 2200 ( a'=7. 1073536, ^;is. 35 + 308 2107 44 + 301 93 = 1)' 35 553 86.545 79 854 = c 6.455 - B" 35 11-45 6.144311215 114 = B' 865.45 .3106887S5 = D" •5 11.5 .263570321 114- 5 876.95 = C" .047118464 .5 115 .808745 .043928386 877.758745 .003190078 •5 115.5 = B" .80899 878.56717315= =(7 .002635703 '" .000554375 ■035 .C00527140 115.535 .000027235 .035 115-57 TEST EXAMPLES FOR IlEVIEW. 201 5. 2'3 + x^ + .r = 100. A B. C D ahcde I +1 + 1 = 100 (4.264429 + _4 20 84 5 21 16 = D' 4 36 11.928 9 57 = t7' 4.072 = D' _4 2.64 3.788376 13 = B> 59-64 .283624 = U" 13.2 2.68 .256071744 134 62.32 = G" .027552256 = i>»^ 13.6 = B" .8196 .025631441984 13.66 63.1396 .001920814016 = iV" 13.72 8232 .001281682442 13.78 = B" 63 9628 = C" .000639131574 13.784 -055136 -000576757098 13-788 64.017936 .000062374476 13.792 = fi«^ .055152 137924 64.073088 = C'^" 13-7928 .00551696 64.07860496 .00551712 64.084122 o|8 = iT = 4.264429 + , Ans. I. 2. TEST EXAMPLES FOR REVIEW. Prff/e 274:, 6« + 4« X 5 + 8a-T-2 — 3« + i2« X 4 = 75«, Ans. (8.'C + 3a:)5+4.^-f-7 — (5.^4-93;) ^7 = 57a;+7, ^/^s. (sax — «^ + 4c^) — {2ax — 4^?^ + 2cd) = 3«'^' + 3«^ + 2C(h Ans. 4bc + [T,cd — (2.?;?/ — ??iM) 5 + Tfic] 202 TEST EXAMPLES FOE REVIEW, 5. See Book, Article 75, Prin. 4. 6. See Book, Article 91. 7. See Book, Article 91. 8. Given £^ _ (a; + 8) = ^ + - - i7f 3 9 7 Uniting terms, — — x =^ — ^ — 8 3 3 Multiplying by 3, etc., .t = 8, ^4;i6'. ., , 4^;^ X -16 7.1 9. Given ^ — -. \- 2x — ^-^ X ^^- 5532 Reducinpr, ()x = — x — 3 2 a* = 31, ^??,S'. 10. 3^% — 65V — c^Z =: c (sb- — 6I)'^c — cd), Ans. 11. 32% — gxh — iSx^yz = 3^^(y — 3^ —6yz), Ans. 12. a2» — Z^2« — (^« -I- ^") (<2'^ _ 5"), Ans. 13. 8(7 — 4 = 2 X 2 (2 Ans, a^ — 6^ a — b 21. ()X^y^ + i2a:?/2; + 4^^ = (32'^ + 2^) {^xy + 22), J/^s. 22. 952 — 6Jc + 6*^ = (3^ — c) (3^ — c), Ans. 23. See Book, Article 231. 24. Let 40; = length of fence. Tlien X = Xo. of acres. Reducing area to sq. rods, x^ = iGox X =1 160 And 4x = 640 rods, A}is» 204 TEST EXAMPLES FOR KEVIEW 25. Let Then And By conditions, Multiplying by 9, 26. Given Multiplying by i — x, Transposing, 29 30- 2X = entire distance. X --■ I J = hours of ascent, a; -i- 4I = " " descent. 2X 2X 3 9 ^ 8ic = 117 2X = 2gl miles, A71S, , I -\-x =z O I — X h — hx =z I -{- X bx -\- X =z b ^ I X = 27. See Book, Article 103. 28. See Book, Article 104. (^2 _ ^2) (^ ^ y) b+ I , A?is. (x + y) {x - y) {x + y) (x^ + 2xy 4- 2/^) {x — y) {x + yf {x — y) = I, ^4/^5. a'^ — ¥ _ {a^ J^ h\{a^--W) (rt2 _ 2ah + ^2) («2 + ^2) - («"_ If (^2 ^ J2) a , ^?i5. 31. Multiply the terms of the second fraction by i—c?. ■a'' 32. Let i + rt^ J — 2ct?-\-a^ $a^—a'^ I— a'' I— a' ■«'' , Aus, X = number. 33' tjfl ry* ry* Then x -\ 1 -= 154 Clearing of fractions, 60a: + 15-i' + i2:r — 10^ = 154 X 60 Uniting terms, 77.^ = 154 x 60 x =z 120, Ajis. Let X =: amount each had. Then a; — I30 = 2 (.r — 1540) X = $50, Ans. TEST EXAMPLES T O R R E \' 1 E W . 205 Pafje 276. 34. Let X = No. of hours. Then 24 + Sx = distance ship sails, And i2x — " privateer. By conditions, 1 20; = 24 + Sx. X = 6 hours, Ans. ^. Sx + ru . . 35. Cliven —J- = 7 (U And H^-^-y^o (2) Clearing of fractions, Sx-\-$y = 4g (3) And —3'^' + 5// = o (4) Mult. (3) by 5, 40^'+ 151/ = 245 " (4) by 3, — 9^+15 ^ ^ Q Subtracting, 49a; =245 X = S> I ^^^^^ Substituting 5 for x in (4), ?/ = 3, f ^6. Given a; = ^^ f- 5 (i) And 4^ -— = 3 (2) Clearing of fractions, yx — i/ =: ;^^ (3) And —x-]-i2y=ig (4) Mult. (4) by 7, - 7^ + 84^/ = 133 (5) Adding (3) and (5), 83?/ = 166 Substituting 2 for y in (i), a; = 5, ( " 37. Let a = the distance ; m = rate of one; n = rate of the other ; And X = the required time. Then inx •\- nx ^ a . .*. X = — — — . Hence, the m + 11 Rule. — Divide the given distance hy the sum of the rates ; the quotient will le tke time of meeting. 206 TEST EXAMPLES FOE REVIEW. 38. Let Then And 39- 40. X By conditions, Multiplying by 16, Let Then And 12 + 4 X 12—4 16 "^8 X + 2X SX ', 2X X y i» — 6 X -\- 6 2X = entire distance, time down stream, a up (( = 8 2/ + 6 Clearing of frac, etc., 2X — 1/ = And 4X — sy = Mult. (3) by 2, 4X — 2ij z = Subtracting (4) from (5), Substitut. 18 for ^ in (3), Hence, Let Then And From (i), y = X = X y~~ X = y = x-\- y \ ij : : ^2 _ ^2 — = T28 = 128 = 85! miles, Ans. = the fraction, I 2 ^ 3 4 6 — 6 12 18 12 a; 2Sr 12 . the greater, " less. 8 : 3 49 5|/ 3 49 Sub. value of x in (2), — - — / = i6?/2 = 441 4// = 21 // = -V-=5iless; ) Sub. -y for 2/ in (3), 2: = ^r^SJ gr., \ Multiplying by 9, Extracting root, 0) (3) (4) (5) (0 (3) (4) Ans. TEST EXAMPLES FOR REVIEW. 207 41, 42. 43- Given I02' + 6y 76 (I) 4?/ — 2Z 8 (^) And 6x + 8;2 - 88 (3) Multiplying (2, by (4), i6y — 8^—32 (4) Adding (3) and (4), 6x + 161J 120 (5) Multiplying (5) by 5, 302' + 80^ 600 ((') (i)by3, 2,ox + i8_^ 228 Subtracting, 62y — 372 y -(>]) Substituting 6 for y in (i), x A-'->\ A\ US, « 6 " ^ ' ' (2), ;2 8, ) Given 2:t' -{- sy + z 24 (I) 30; + ?/ + 22 26 (2) And .T H- 2^/ + 32 — 34 (3) Multiplying (i) by 3, 62: + 9// + 32 _ 72 (4) (2) by 2, 6x + 27/4-42 — 52 (5) Subtracting (5) from (4), 72/ — 2 _ 20 (6) Multiplying (3) by 3, 3.!' + 6^ + 92 102 (7) Subtracting (2) from {7), sy 4- 7^ = 76 (8) Multiplying (6) by 7, 492/ — 72; = 140 Adding, 541/ _ 216 ^ — 4;) :n(6), 0-8;U; Substituting 4 for y \ ns. " these values in (i), x 2, ) Let ic — A's money, y — B's " And z - C's " Then x-\- y + z — $180 (0 a; — ^ + 2; I60 (2) And .-c + ^ - 2 _ - (3) Subtracting (2) from (l), 2y — 't^I20 ?/ $60 208 TEST EXAMPLES FOR KEVIEW. Adding (i) aud (2), 2X -^ 2Z = I240 (4) " (2) " (3), 2X-^= $60 4 Subtracting, — = $180 4 z = I80, C's; ) Subst. I80 for in (4), x = I40, A*s ; >■ Ans, y = %o, B\ ) 44* Let X =■ circumference of fore-wlieel, Then x -^ s = " " hind " A J 240 240 And -— z=z — \- 40 X X -\- $ Clearing of fractions, etc., x^ -\- ^x = iS * Comp. sq., etc , a; = 3 meters, circ. f. wheel; | , And x-j- s = 6 '' " h. " f 45. Let X = side of one. Then x -\- 2 = " "the other. Difference of cubes, 6x^ 4- 1 2:^' + 8 = 488 Eeducing, x^ + 22^ = 80 Comp. sq., etc., x = 8 ft., side of one ; ) . And a: + 2 == 10 " " other, j Page 277. 46. See Book, Article 232. 47. Let jx = one part," And iicc = the other. Then I8.^' = 126 .'. x = J } A71S. no: = 77, TEST EXAMPLES FOK REVIEW. 209 48. Let X = number of meters. rm ^120 $120 Then h -So = X ^ X — ^ Dividing by .50, clearing of fractions, and reducing, 0? — S.r ==1920 Comp. square, etc., a; = 48 meters, Ans. 49. See Book, Article 472. s = c(i + v) = $175 X 1.25 = $218.75; $218.75 -^ 89 — $2,457, A)is. 50. See Book, Article 490. 51. Let a; = price of horse, Then a; + ^100 = " carriage. And a:+ioo : x :: x : $0 Changing to an equation, X^ — ^OX r= 5000 Comp. square, etc., x = $100, horse; And X 4- $100 = $200, carriage. Ans. 52. See Book, Article 247. 53. Let _x — share of younger, Then a; + 35 = " elder. Adding, 2X + 35 = 165 Transp., etc., x = 65 hectares, younger; ) a; + 35 = 100 " elder,' \ ^^^^' 4- Let X the number Then 3a; X — 40 _ 5 1 2 Transposing, etc., 7.r 182 • • 9 X _ 26, Ans. •210 TEST EXAMPLES FOR REVIEW. 55. Let X =z price of a sheep. ?/ = " " lamb. Then 6x -{- ry = $ji (i) 4X -^ 8i/ = 1^64 (2) Mult. (2) by 6, 24^-f "4% = ^384 " (i) by 4, 24ic + 281/ t= ^^284 Subtracting, 2oy = lioo Substituting 5 for ^ in (2), x = $6, \ 56. Ijet X = No. who voted for one, Then 0^4-271= " " " the other. And 2X -h 271 = 1425 Transposing, 2X = 1154 :c = 5 7 7 for one ; And a; 4- 271 = 848 " the other, 57. Let X = C's age, Then 3a; = B's " And 6x = A's " Adding, 102; = 150 X = 1$ years, C's age; ^x = 45 " B's " I Ans. 6x = 90 " A's " 58. \/243 = V81 X 3 = gVs, ^^is. 59- Ans. ^Jif + af- = Vy'(i + a) = yVi+a, ^ns. Page 278, 60. x^ = (x^)^; y^ = {y^Y, Ans, 3/- 61. 3 (f/ — &) = V 27 {a — ^)^ ^w."?. 62. Let X =: price per bushel. Then I'jx — it^x =z $3.60 Or, 4X = I3.60 iv = $0.90, ^4?^6'. » TEST EXAMPLES FOR REVIEW. 211 ^3' 64. Let X Then x -\- S And 2x 4- 8 Transposing, x X + 8 amount lost on second, first. i( (S 2r<- $ I o on the second ; \ . $18 " first, f 71S. I 3 4 Let X = men first employed. Then, by conditions, 6x : io(.?;-|-i2) ; Changing to an equation, etc., 8;r =120 a* = 15 men, Ans. 65. Let X number in the party. Then Sx amount they paid. And Sx — 7 (x 4- 4) Eeducing, a; = 28, Ans. 66. Let X wages of a woman. And «/ — " " boy. Then Bx + 6y — $72 (i) 6x -\- iiy »^8o (2) Mult. (2) by 4, 24a; + 44_^ $320 " (Obys, 24.T 4- 18^ — $216 Subtracting, 26^ I104 ••• y - ^4, boy; \ ^^^^ fovym{i), x $6, woman, f Substituting 4 • Given 67. X i7x 3^/17^'. Hence, 9, ^ws. 68. V'-C +12 V« + 12 Squaring, etc., ;r a. Ans. 69. Given Multiplying by Vy, etc.. Factoring and dividing, y a/// y ((y «/ Vy I y «i/ y I I — a' ^ ;? A71S. — I5, each, 24 ; And 79- Let lox + y number. Then iox-{-y - 9i^' + U) (I) And iox-\- y — 6:^ — loy + x (2) From (i), X - 8y (3) " (2), gx — gy =z 63 (4) Dividing (4) by 9, x — y -_ 7 Substituting, y - 1 From (3), X — d> Hence, loa; -{- y — 81, Ans» 80. Let 3^ — distance one goes. Then 'JX " the other goes. « And 102; — 150 miles. • • • X 3^ ^x - 15 " - 45 miles, one; ) ^^^^ . - 105 " other, j TEST EXAMPLES FOTl liEVTEW. 2l5 8r, Let X — Xo. of B's acres, Thou X ^ lo — " A's " And $2800 , „ -r,, — — - _ price per acre of B s, X a ^2800 ^, ,, ,, ^,^ a; + 10 tt 2800 2800 X X 4- ^o Clear, of frac, 28002; + 28000 = 2 8oo.T-f-5:?:2-}-5oa: Transposing, etc., x^ -\- lox =: 5600 Completing square, etc., x^ —5 + ^/5600 + 25 Reducing, x = — 5 ± 7 5 x = 'JO acr. B; And a; + 10 = 80 " A, Ans. 82. {Vx + V7) ( V'^ — V7) = ir — 7. Hence, a/^'— V7?^'^^' ^Z' { Vs^ - V3^) ( Vs^ + Vs^) = 3-^' - zy- Hence, V^ + VzU' -^ ^^■^' 84. Given V^V^ = _-£±.A^ Multiply, by denom., Z^ + v^ = r? + 3 Transposing, ^/^^' = rZ + 3 — //^ Squaring, x = d^-i-6d— -21^(1 -\-g — 6b^-\-¥, Ans. 85. Let iP = clerk's salary. Then loa; = mayor's ^' And no; r= 1^13200 .r = I J 200, clerk; ) . And 10.T = I12000, mayor, ) 216 TEST EXAMPLES FOR REVIEW. S6. Denote the numbers by x and y. Then x -{- ij \ x — y :: 8:6 And X — // : xy : : i : 36 By Theorem i. And Sub. 7?/ for ^ in (2), Dividing by y. Equating (i) and (3), x X =z ^y xy = ^6x — s^y iy^ = 252?/ — 36?/ jy = 216 = 216, ) (0 (3) 87. 88. rage 280, Denote the numbers by x and y. Then xy And x^ — y^ : {x — yY By Th. 6, x^-\-xy-\-y'^ : x^—2xy + y^ By Theorem 8, 3:2:?/ : (x — yY " " 6, iT?/ : (:r — ?/)^ Vahie of xy from (i), 4S : {x — yY By Theorem 6, 4 : {x — yY a a (I a 12, Combining (i) and (3), " (3) " (4), x — y x-y x + y 48 (i) 37 : I (2) 37 : I 36 : I 12 : I 12 : I I : I I : I (3) 14 (4) iK = 8; «/ = 6, .4^Z5'. ic Let a; = price per dozen ; — = price of one, And By conditions.. X 144 12-2 = -— 12 X 144 := No. for 1 2 cents. Dividing (i) by 2, J 44 X -{- I . 72 + 2 (I) + 1 (2) Clearing of fractions, 72:^ + 72 Transposing, x^ -\- x Completing square, etc., x r — —1-1- U. »t- — 2 ni 2 Reducing, x + I 72a:: + .^2 + a: 72 - i ± V^!^ 8 cents, Ans. TEST EXAMPLES POR REVIEW. 5i]'- 89. Let X = Xo. of clavs tliev travel. Then 8^ = distance one goes, And jx = " other ^' By the problem, 15a; = 150 miles, X =z 10 davs, Ans. 90. Let X =z A's income. Then 3:^ = B's " And 4x = I1876 .-. Of = $469, A; I ^^^^ And $x = $1407, B, j 91. Let X = cost of cow. Then 4X = " horse, And ^x = I250 X = $50, cow; I And 4:?; = I200, horse, j 92. Let X = rate per hour he rode, 24 Then -^ =r hours spent in riding, X And -2^ :r= 8 = « « walking. By the problem, 24 — + 8 = 12 a; ic = 6 miles per hour, Ans, 93. Let X z=z the length, And ^ = " width. Then 2a; + 2?/ =: 320 (i) And ./•// = 6000 (2) Dividing (i) by 2, a; + ^ = 160 (3) Squaring (3), x^-\-2x?j-]-y^ = 2^600 (4) Mult. (2) by 4, 4:r7/ = 24000 (5) Subtracting, x^—2xy-\-y' = 1600 (6) Extracting root, x — // = 40 (7) Combining (3) and (7), .r = 100 ft. length; ) ^ And ^ = 60 ft. width, j 318 TEST EXAMPLES FOR REVIEW. 04. d = = — ^-^ = 44- Hence tlie series ^^ m +1 6 ^ 3^ lh 1 2 J, 17, 2 1 J, 26i 31, ^?i5. 95. ? = «^ + (^^ — i) fZ = 1 + 49 X i = 25 ; « + ^ 4+25 ^ T ^ S = X ?i =: X 50 = 637I, ^^25. 96. Let 2: = No. pair bought; r A71S And 3^ + I zi: 185 " " the other, j 108. Let X = time by one ; y = time by the other. Then - = part one does in i day, X - = " the other does in i day. Since both together can do the whole work in 1 6 days, they do J of it in 4 days and -^ of it in i day. Hence, By the conditions, - + - = — (i) •^ X y 16 V / And 36 ^ 3 /^x y A Dividing by 3, etc., ?/ = 48 days. Subst. 48 for y in (i), ^ + ^ = ~ /. X = 24 days, one ; y =: 4S days, other, AnSo ' TEST EXAMPLES FOR REVIEW. 221 109. Let X = digit in tens' place, And y = *' " units' *' Then loa: ^- y = number. By the conditions, — ' ^ = ^i (0 And lo.T 4- y + 9 = lo?/ + a; (2) From (i), 40a; + 4/y = 9.^7/ (3) Uniting and dividing (2) by 9, xi- I = 7J (4) Substituting value of ^ in (3), 40a; + 4a; + 4 = gx^ + 9a; Transposing, Dividing by 9, Completing square, etc.. Reducing, And From (4), Hence, .... . = 0= = iff = r,rs = s. Hence the series, 2, 6, 18, 54, 162, 486, Ans. III. Let a; = No. of hats. x^ — •35^'^' — 4. 35 y. 4 ^-«±^ -r — 3 s .1 . 37 •*^ 18 i 18 a; 4 y = 5 + ?/ _ 45, ^W5. 1 etc. '4 1225 9 "^ i82 • • • loa; '1369 i82 Then 80 X price paid apiece. And 80 X 80 h I .r + 4 Clearing of fraction s, 80a' + 320 So.r + a'2 f 4a: Transposing, etc., X 320 Completing square, 2 + A/320--4 Reducing, X ? + 18 f • a; :^ 16 hats, .i?k^. 222 TEST EXAMPLES FOR REV^IEW. 112. ? = «r"~^ = 2x5^^ = 97656250, Ans. 113. Let X = length of shortest. Then 5:?; + 3^ + ic = 90 feet. Uniting terms, gx = 90 " r?; = 10 feet; A71S. 114. 3^ — 30 " And Sx — 50 " Let 3^ + 19 — 3d, Then 2X 2d, 4a; + II ist And 9a; + 30 _ 219 Transposing, gx 189 X 2J 4:?;+ II 95, ist; 2X — 42, 2d; And 3X + 19 _ 82, 3d, Ans, 115. Denote the numbers by x, \/xy, and y. Then x + \/xy + ^ = 14 (i) And a^ + a;«/ + ^^ = 84 (2) Transposing and squaring (i), x^ + 2xy + y^ = 196 — 28Vxy + a;^ (3) Subt. (2) from (3), rr^ == 112 — zSVxy + .-c^ Transposing, etc., V^y = 4 (4) Involving, xy =: 16 And 3x1/ = 48 (5) Subtracting (5) from (2), x^ — 2xy -1- ?/2 = 36 Extracting root, x — y = ^ (^) Subst. (4) in (i), X + y — 10 (7) Comb. (6) and (7), x ^z S y = 2 Vxy = 4 Hence the numbers, 8, 4, and 2, Ans. TEST EXAMPLES FOE REVIEW. 223 ii6. Let X = time the coffee would last the wife. The man drinks i lb. in 4 weeks. His wife " 1 " " x " Hence he " - " " i week. 4 And she " - " " i " X He " - " "3 weeks. 4 She « - " " 3 " X And by the problem, ^ + - = i Clearing of fractions, 3:^ + 1 2 =43: Transposing, a; = 1 2 weeks, Ans. 117. Let X = sum in ist purse, y =z " « 2d " Then x + y =^ ^300 (i) And ic — 30 = ^ + 30 (2) From (2), X — y -^ 60 ^ (3) Subst.in(i), y + 6o-\-y = soo Transposing, 2y = 240 y = $i2o, 2d;)^^^^^ From (3), X = !i5i8o, ist, ) 118. Let X = cost of the cloth. Then = rate per cent. 100 X X X — = percentage gained. 100 And X -{ = 39 100 Clearing of fractions, x^ + IOO.Z; = 3900 Comp. sq., etc., x = — 50 ± ^3900 -t- 2500 Reducing, x = — 50 ± 80 X = 830, Ans* 224 TEST EXAMPLES FOR REVIEW. 119. Let And Then And 122. X = No. of acres in one part, 2/ = '^ *' the other part X -\- y =z 100 (i) 202; + 30^ = 2450 (2) Mult, (i) by 20, 20.T + 20?/ = 2000 Subtracting, etc., From (i), ?/ = 45 acres; , ic = 55 120. / = ar"~^ = : 2X3^^; Jr — a s — — r — I 2X3^^- ~ 2 14348906, Ans, 121. Let X No. of pine, And y — '' " hemlock. Then X + ?/ - 300 a;2 : ?/- 25 : 49 Product means = product extremes, Extracting sq. root, 7:?; = SV 7 Subst. in (i), 5?/ ;VIult. by 7, etc., 12^ 2100 From (i), Let By conditions, Reducing, Transposing, y = 175. ^'^"^^^^^^l^w.. X =i 125, pine, f X = No. of men. 2X = 3i(-* — 150) 3^ = 5'^— 750 2X = 750 o; = 375 n^en, ^?i5. APPENDIX. I. Given. 2. CUBE ROOT. Page 284:, fl3 3«2 ) 3^2^-1-3^5^24-^3 x^-\-6x^-\-i2x-\-2> ( x-\-2, Ans. x^ 3^' ) 6.^•2-|-I2:^:-f-8 3.^'2 -f- 62; -f- 4 ) 6.T~ 4- T 20; 4- 8 3^;^ ) — 6:?;2^ 4- i zxy"^ — Zif 5' i2a^ — 2 6. 8ft3_48flr2_|_^6^_54 ( 2rt— 4, Ans, W i2«2 ) — 48rt2 4-96r«— 64 4«4-i6 ) —48^2 4-96^—64 27«^— 54«^-^' + 36«.^-^— 8.r3 ( 2,a — 2X, Ans, 2765^ 2 7 «M — 5 Aft'^x + 3 6«:r2 — S.r^ 1 8a2' 4- 4-2^^ — 5 4«''^ + 3 6«^ — 83^ 226 CUBE EOOT. ^ O OJ •^ • I— 1 P 1^ O -+J m a> • f—t • O • r— t P o Q •l-H •i—i f-H P rH H 1 rzi h-t • 1— 1 o o -- O g OJ ^ • I— 1 i-H O CO VO • I— 1 o 1 • P^ 1 OD CQ a ^3 • 1— 1 1— 1 H + 'r-t • r— * CO + CO 1— + M (N a "^ e ro ro ro fO + CO 52 O 00 M 1 I Ttf ■>!)< e C lO M M M + + in in « SS VO VO + + VO VO + + ;h ^ m OQ • r-H • (— ( • >■ k ?-i • r—i • f— 1 ?; O 02 P P O CO • 1— ( • l-H o • r— 1 P OH • 1—1 P •f-H P r— H s • 1—1 B • • 1— 1 CD 1— 1 O o Q 1 CO + CO ^ P O CO 1 "^ + 1 CO 1— 1 5^ « •\ o3 VO Ov HH -* 5^ CO + 4- + Eh « ro rO ^x f ^ + + -- CO CO 1 PR •?. "^ ■"^ in in 1 CO VO VO ^ H 1 1 1 VO VO 1 1 <9 2 1^ CO CO 1^ CO 1 1 VO 1 1 VO VO 1^ VO 1 VO 1 00 1^ 00 1^ On 1^ VO 1^ CO 1^ CO + -- — CO CO CO + 1 T 1 to CO CO 1 1 1 . I factori:n-g. 22? FACTORING. 1. x^ — gx -\- 20 := {x — ^) {x — 5), Ans. Extract the sciuarc root of tlie first term for the first term of the factors ; resolve tlie third term into two factors, whose sum is tlie coefficient of the second term, and thes:/ will be the second terms of the re(|uired factors. Thu» ^/x^ — X', 20 = {— s) X (— 4) ; — 5 — 4 = — 9- 2. a^ -|- 7^ — 18 = (« + 9) {a — 2), Ans. 's/ifi =z a\ — i8=— 2 X9; 9 — 2 = 7. 3. a^ — \T,a -{- 40 = {a — 8) (« — 5), Ans. ^^=a', 40 = (-8)(-5); -8-5 = -13. ^. 2al)c^ — i^adc — ()oab =: 2ad {c^ — 7c — 30) = 2al) (c — 10) {c + 3), A71S. ^Jc^ — C — 30=— 10x3; — 10 + 3 = — 7- 5. xHp — 2XIJ + I = {xij — i) {xi/ — i), Ans. \/xhf — xy; i = (— i) (— i) ; — i — i = — 2. 6. 2>x^ — 32?/2 = 8 (.r^ — ^if) (Art. 131) = 8 (.1- + 2?/) (.T — 2y), Ans, 7. a;2 + y'^z 4- 2/»?2 = ^"^ + ^ (i/ + '^0 2' -^^^' Note. — It sometimes occurs, as in this example, that only a portion of a polynomial can be factored when there is no factor common to all the terms. 8. i2a^x — Mhj + 4^2 = 4« (3^^^ — 2^y + 2), Ans. 9. Extract the cube root and thus find one of the three equal factors : {a — x) (a — x) (a — x), Ans, 10. 1 — a^ = {i + d^) (i + a) (i — a), Ans. (Art. 131.) 228 G. C. D. OF POLYXOMIALS. II. I + 2rt ) I + 8«3 ( I _ 2« + 4«2 (Art. 132.) I 4- 2a — za — 2a — 40^ 4^2 -f 2>a^ Hence (i + 2c«) (i — 2rt + /^cfi), Ans. 12. «6 _ ^4^.2 _ (^3 + §2^^.) (^3 _ ^2^^.), ^^^5. (Art. 131.) G. C. D. OF POLYNOMIALS l*(i*je '> (Art. ioi.) Z»2 ' a^ — h^ a? + W + 2ah (a-\-hf a + l) 3' 3 + 5 + 7 15 15 a — b b — ab be ^4_^4 a^_a^j^ab^—b^ xa — 2X 2a x — a ^ -j I • X X a ' 3«^ — 2ax — 2rt^ — x^ -\- ax Ans. ax a 2 tt*^ •v ax , Ans. c c - +- a -\- ac ac ac — be + ab — ac -\- be — ab -\- abc abc And From Take abc abc . -zr- = I, Ans. abe abc^ a -\- zh — 3ff — h + d -h 2 d -^ b — 2 a -\- 4/1 -\- b-sd Or — 12a -{- 24h -\- b — ^d , Ans. 8. FRACTIONS. 231 X 6. From $x -f- t Take 2X — h x-h 3'-^ + A + X X — h h ' c ' ^ ex -\- bx — ¥ . Or 3^ + f^ ' ^^^^' X 7. From a -\- X -\- x^ — y^ Take a — x -\- X + y 2X 4- X X — y x^ — y^ x^ — y^ J Or 2X -{- -T-^— T,, A^is. x^ — y^ Multiply the terms of the 2d fraction by x — y. a — I a Or I — a; 1 -\- X a a — I i ^2 — 2(1 + I - a^ «2. — a I - — 2ft A71S. a2 a ' I T + X ^2 ' (Art. 183.) X I I x^ y I X I — X^ I — x^ r. X — I 1 — X 1 . Or, ■„ = — 7 — ^-7 r = , Ans. I — x^ {1 + x) {1 — X) I -\- X 2o2 MULTIPLICATION OF FRACTIONS. MULTIPLICATION OF FRACTIONS. a ~h a -\- b — a -j- b 2b I. I 2 + a -{- b ~ a -\- b ~ a -\- b^ lb 2a — 2b -\- 2b 2a a — b a — b ~ a — b'* 2b 2a Aab , X , — -~ To? ^>is. a -\-b a — b if — b^ i" + ^ ^x 20 2b T^X ZX 4a Sab gx^ Sab QX'^ , gx^ Sab 7'2 /lOM/ _i_ /J/2 X'^ — 2X1/ -\- ir X -\ n , . , . I ^ — y J I \ ^ iji = x- — \f, Ans. Cancel x — ij from iinmerator and denominator. 7& 2a^ — 4«^ 2«2 — Sa^ 21b 'jb 2C^ (l — 20) 2^2 ( I -^ 20) (l — 2r/) 2 1^*- Cancelino' common factors, etc., we have 7-, Ans, 3 -f 6« _^_ X ^^ ~ ^^ r= ^^ X (a-4-y)(^-- y) ^ + ?/ ab X + // «^ a {x — y) . = — ^ — ^ , Ans. D I ^' I S I O N OF FRACTIONS. '2'^3 I «— I I a — I = 5b(a + i), Ans. i5^< - 30 ^ 3«^_ _ 3 (5^ — 10) ^ 3^^ 2« 5« — 10 2« 5a — 10 = — , Ans. 8. ^ a;y xy X + // x-y ^y + y^ — ^y ^ ^ji -y'^ + ^y X + y X y -~- X -^ — = — I f- , Ans, X '\- y X — y x^ — y^ I. n + a — DIVISION OF FRACTIONS. Page 285. 2a «' — 3« + 2« _ ^^^ — ^ . rt + (f 3 a — z «^ — 3' 2« rt^ . — 3« — 2^ (v^ — 5^ ^ n 3 ^ — 3 « - 3 ' 2« 2^ ^^■■^ — « y . 2y — 2' 2y sy X ^ ^ 2 {y i) 2y 3 ■4' ^nS. ah + ¥ 9 h a — J) b (a H- b) v^ a — b a^ — ¥ ' (a b) («2 Jr ab + b^) '^ b a + b a^ + ab-\r^' - I a 1 — a -\- a -\- a^ i + «2 ^2 I + a ] [ — a ~ 1 — a^ I J ii>.+ a \ (i + a^f I + a^ ' (i a^Y I «2 ^.2)2 1 — aJ + a^f I ^2 ^/iS • l^rtflre 286. 3« 2« — 2 sa 2a — 2 X — — -, A71S, 2a 4 2« a — I X 8. 3 _ 3^ + a: . 4^ + // _ 4^ ^ 4 _ i6.r yj^y_ 3 4 3 5^ i5.y a« x^ — «/^ — I x^ ^ -\- y . .?/ + a?2 — y^ y x^ — ?/2 5+ I .T^ — ?/2 rc2 r y x^ — y^ X .r^ — ?y2 y J^ 10. EQUATIONS. 235 I I a ab^ If -\- \ b- — b -\- I I ab^ ' b (&+i)(^-^+ I) ^ h a^2 , Ans, EQUATIONS. Piuje 286. 1. Let X =z price of house. Then $850 — x= " « biirn. And 5a: = 12 X $850 — i2.f Transposing, 17a; = 12 x $850 Dividing, a; = 12 x S50 = $600, house ; And $850 — x — $250, barn, 2. Let 12^ = No. of C's acres, Then 8./; = '' A's " And gx = " B's '" Then 29^' =145 Dividing, x = 5 8.1' = 40 acres, A's ; ) gx := 45 " B's ; V Ans. i2:r = 60 '• C's, ) Ans. 3. Let 6x = No. liters cask holds, Then 2X = '• '*^ in it before leakage. X =z 21 liters, 6.C = 126 •• A us, 230 EQUATION' S. 4. Given XX ~ — 4 = ~ — — '^ 4 3 12 Clearing of fractions, 36rr — 3:^ + 12 — 48 = 2o:r + 56 — 1 Uniting terms, 1 32: =: 91 ic = 7, ^?Z5. 5. Given 3^jL9^7^±i_ii±4^+6 ^ 235 Clearing of fractions, 45-^' + 135 = 70^' + 50 — 96 — 24a; H- 180 Uniting terms, ;r = i, ^^^5. ^. X ■\- % X — 6 6. Given — — y x ^^ x -{- 2 4 3 Cancel .f and clear of fractions, 3.1' 4- 24 — /^x + 24 = 24 Uniting terms, a; = 24, Ans. ^ . a: + 8 ^; — 6 7. Given X — 2 = .t H 4 3 Cancel ^ and clear of fractions, — 24 = 3.r + 24 — 4:^ + 24 Uniting terms, ^ = 72, ^;i5. 8. Given 2X -\- \ (— x "' X (-^) 3 ^ 4 Clearing of fractions, 8a; + 4 + 3.1' -f 9 = I2X Uniting terms, ^ = 13, yl';i6'. 9. 2^ = If X 7 = ff = 7^3^X1 o'c'^^<^^'^ P-^i- -1'^*^ ^ce Art. 247, lO. EQL'ATIOyS. 23T Let 27.T + £200 r= stock. 50 =: annual expense. 2^x + £150 gx + 50 = gain ist year. a a 36.T + £200 = snm 50 r= expense. 36:6- + £150 12a; + 50 = gain. 48^; + £200 = snm 2d year. 6^x + £200 = " 3d " By conditions, 64.^ + £200 = 54,>- + £400 Transposing, lo.r := £200 X = £20 And 273^ + £200 = £740, Ans. rage 287. I. Given ^^^ — - — - = 26 3 , - 6a; 6?/ And ^ ' }• Ans, = 9. ) (I) = (2) 23 From (i), 4rr 4- 6^ = 78 (3) " (2), 9a; = 6// (4) Substituting in (3), 4.'?; + 9:^ = 78 .*. .«• = 6 And from (4), y 2. Given - + -^ = 8 (i) 32 ^ ^ And - -{-^ — -] (2) 23 Multiply (i) by 18, 6x + 9// =144 (3) " (2)byT2, 6x + 4;/ — 84 (4) Subtracting, 57/ = 60 ?/ = 12 ; ) Substituting in (4), x ^=z 6. i 238 £Qi 'AT I 0:^5*^. Given ZX ^ _ 42^ (I) And 2X — 211 4 '^ ' u) Multiply each by 4, SX 21J — 36 (3) And 2X — 2y 16 (4) Subtracting, ' > Ans. y - 12, \ Substituting in (4), Given 2 3 (I) And X y 3 4 (^) Multiply (2) by 36, \2X -\- 9?/ Tfic " (i) by 24, \2X -\- Zy 24b Subtracting, Substituting in (i), y _ 36r — 24^ ; [ ic 181^ — 24c, \ ^W5. Let X Price paid for harness, Then dfic .. .. .. i^iorse, ^ And 3^ - " '' buggy. Hence, ^x $400 ic = I50, harness ; J 4X = $200, horse ; ^ Ans. ZX = I150, buggy, ) 6. Let lo.'T + ?/ = The Number. Then loa; + 3/ + (loy + x) = 121 (i) And loa; + ?/ — (lo?/ -^ x) ^ 9 (2) From (i), 11.7; + ii?/ = 121 Or a; + ?/ = II (3) From (2), X — y ^ i (4) Combining (3) and (4), x z= 6 And y = 5 Hence, loa:; + ?/ = 65, Ans. E Q U A T I X S . 230 7. Given x -^ y — z = o (1) And X -\- z — y ^= 2 (2) And y + ^ — a: = 4 (3) Adding, x -\- y -\- z = 6 (4) Subtracting (3) from (4), 2X = 2 (2) " (4), (I) " (4), And Ans. 8. Given f + - = ^ (i) And I + I = ^ <^) And ^ + ^ =r I (3) . -. T 2a; 2?/ 22 / X Adding, T'^ r~^'d "^ ^ ^^^ 2 ?/ 22 Subtracting 2 times (3), -^ + — = 2 2.7; 211 Subtracting 2 times (2) from (4), ^^ = i cc 2 " (I) - (4), -I = I And A71S. (C (C (C A ns. 240 EQUATIONS. 9. Given iv -\- x -\- y ^=: 6 (i) w -^ X ^ z — ^ (2) w + ^ + . = 8 (3) '^ + Z/ f :^ = 7 (4j Adding, 3^6; + 3^ + 3.^ + 3^ = 3° Or w -\- v -\- f/ -{- z = 10 (5) Subtracting (4) from (5), iv =: 3 (3) " (5), ^ = 2 (2) " (5), y = I (i) - (5), z = 4 10. Griven xy = 600, or y = — - (i) 300 / X a^z = 300, or z =z (2) yz = 200 (3) Substituting in (3) the values of y and z, 600 -200 ?/5; = X - — = 200 Reducing, x^ = 900 Extracting root, * ic. = 30 ) From (i), ?/ = 20 >■ A71S. From (2), 2; = 10 ) ir. Given ic -f •-^- =: 10 (i) 48 ^ '^ X y z - + - + -== 22 2 234 4 +^+- = 33 (3) Multiplying (i) by 16, i6x ^ ^y — 2Z — 304 (4) (2) '' 12, 6X + 4?/ 4- 32 =: 264 (5) (3) '' A, X -\- 4y + 2Z — 132 (6) Subtracting (5) from (4), lo.^; — 52 = 40 (7) (6) " (5). 5-^ + ^ = 132 (8) Multiplying (8) by (5), 25.T -{- s^ = 660 (9) Add (7) and (9), 35;?; = 700 .-. x — 20 ) From (8), 2 = 32 > Ans. (3). y ^ 12^ . a EQUATIONS. 241 12. Given w -\- 50 x, 01- .1: — lU 50 (l) :C + 120 =: 3^', Ol' 3i/ — X 120 (2) y -\- i 20 — 2z, or 24; — 11 — 120 (3j ^ + 195 = 3^'-'r or 7,1'' (2) Adding and reducing, x = 29, ist ; ) Substituting in (]), // = 19, 2d: ^A (2), Z = IT. 3d. ^ ns. a 11 242 gekeralizatio:n". GENERALIZATION. raye '^SS, en 5 X 6 30 , . 1. X ^ = ^, -— — = lo hours, Ans. p — c S — s 3 General Problem. — Given two objects starting at dif- ferent times and moving at different rates of speed, to find the time required by one object to overtake the other. Rule. — Multiply the rate of the first by the number of hours, or periods of time, between starting, and divide the product by the difference between the rates. abc . 2 X 5 X 10 ^ , 2. X = — — = r= i± days, Ans. ab + ac + be 10 + 20 + 50 General Problem. — Given the times required for n forces separately to produce a result, to find the time required by the united forces to produce the same result. EuLE. — Divide tlie product of the numbers denoting the time required by each force, by the sum of the differ- ent products of the same numbers taken n — i in a set. ^. Formula x ^=^ - \ an s X 4400 „ . , ax = -- = ^^^— — liooo, A's; 5 22 ^ b?l 7 X 4400 „ -r>, ( . bx = — — -^-^ = I1400, B's; \Ans, CM 10x4400 dh n^ ex =z z= = $2000, C S, S 22 General Problem. — The proportions being given, to divide a number into proportional parts. Rule. — Divide the nnmber by the sum of the propor- tions and multiply tlie ((uotient by each of the proportions in turn; the several products are the ])arts re([uired. GENEllALIZATION. 243 4. Formulas, „ am (n — i) 9 x 9 (3 — i) F r=z ^^- — ^ — ^-^ =27 years; m—n 9—3 , . a{n-i) 9x2 ^^^^^♦ S = -^ = ^—7 — — 3 years. vi — n 6 General Problem. — F is now m times older than S ; in a periods F will be n times older than S : to lind the age of each. EuLE — I. Divide the product of the given interval, first multiple, and second multiple less one, by the difference between the multiples ; the quotient is the age of the older. 2. Divide the product of the given interval and the second multiple less one, by the difference between the multiples ; the quotient is the age of the younger. EXPANDING POWERS OF BINOMIALS I. Powers of 2a, Za^ + 4^^ + 2^ -f i '' —Zh, I — 3^* + 9^2 _ 27^3 Coefficients, 1331 {2a — TfiY = 8a^ — 2,^o?b + 54«J^ — 27^^^ Ans. 2. Powers of ^x, 812^+ 2^01^ + gx^ + 30: + i " " 21J, I + 2Z/ + 4?/2 + 8?/3 +162/ Coefficients, i 4 6 4 i (3a: + 2yy =z Six^-^2i6x^i/-{-2i6x^y^-{-()6xij^-\- i6ij^yA7is, 3. Powers ofi, 1+T + 1+ i -f-i " " 3^> I + 3« + 9^^ + 27^3 4- Sia'^ Coefficients, 146 41 (i + 3^^)^^ = I + T2C/ + 54^2 _^ jo8a^ + Sia^, Ans. 244 POWERS. 4. The expoiiciits decrer,se and iiicreuse by 2. (x^ -j- ij'^Y = -''^ + 3^^i/^ + 3'^"'^"^ + l/^} Alts. 5. Powers of a, a^ -\- c^ + rt^ + i Coefficients, ^ 2) Z ^ (a + cr^Y = «^ + 3« + 3<^"~^ + «"^ ^^5. 6. Powers of «2, «^2_|_^io_p ^8_|_ ^6_|_ ^^4_|_ ^^2_|_ j '^ " — 2rt, 1 — 2rt + 4rt^— 8r^3_[_ j6^4_22rt5^64«6 Coefficients, i. d 15 20 15 6 i. (^2 — 201)^ = rt^^ — i2a^i4-6ort^^ — i6ort^ + 24o«^ — ig2a'^-\-64a% Page 288. 1. «Z^:r'^ X a^ == akf:^, A71S. 2. a^^~%~^ X a'Wx'^ = a^'^x"^, Ans. 3. ic-'" X .^~" = a;~'"-% Ans. 4. 2/"^ X y^ ^ "if or I, ^?^5. 1. 6«~^ -^- 2>(^~^ — 2rt~3, ^W5. 2. ^-^hc-^ -^ 4aWc^ = 2cr^I)-^c-^\ Ans. 4. (a + x)-" -=-(« + ^)-'" = ((1 + x)", Ans. rage 280. Transfer by changing the sign of the exponent. (Art. 279.) I. ^y^ = xhj^^ 3. ^^ = 3 X 2-^tv^.-i. z^ -^ 2y~z MULTIPLICATION^ OF RADICALS. 245 UNITING RADICALS. rufje 28U. 1. A/48 + V29 + V243 == 4V3 + Z^3 + 9 V3 = 16 V 3, -4/is. 2. V54^ — a/96x" + \/24.i' = 3^6.1- — ^\/Gx -\- 2'\JGx = ^/dx, Ans, 2'\/\b — 2\/^b = ^\/2b = V^2^ 3^2 J, vl^is. 4. a;A/25a:2c + V36a:^c = ^x^Vc + 6x^\/c 5. 'v/Sof*^^-^ — V^oa^x = 4«2v'5.r — 2a^^x, Ans. 6. 3V'i282;^^;2 — 4:?:'\/t6?/2 = \2X'\/ 2yz — %x^/ 2yz = ^xy 2ijz, Ans. . MULTIPLICATION OF RADICALS. rage 289. 1. (rt + 2/)'^^ X (Z* + //,)»^ = a/(^ + h) {a + ?/), ^^is. 2. 4 + 2V2 2 — V 2 8 + 4A/2 — 4 1/2 —4 8 — 4, or 4, ^7^s. 3. {x + 2/)^ X (x + ?/)^ = (.1- + y)^ X (.f + y)' — ('^' + ^)^ ^ns, 246 REDUCTION OF RADICALS 4. 2>^^/d + y X Wa = 3b\/{cl + yy x 4\^a'' = i2b\/a" (d + yY, Ans, I. DIVISION OF RADICALS. raf/e 289. (a%)m _^ («ic)»* = ( — I , A71S. 2. 2/^x^ay -T- 6\^a = 4^Vy, Ans. 3. \/l6(i^ — I2a^iC -^ 2« = 2«'\/4« — 3a; -^ 205 =1 Vvi — 3X, Ans. 4. {b + ?/)^ -^ (5 + y)^ = (b -j- y)K Ans. 5. 4aVcib -^ 2^ac = — -'a / — = 2a\ / 2 \ ac V ~Vbc, Ans. REDUCING RADICALS TO RATIONAL QUANTITIES. rar/e 2S9. I. 2\/a — Vl, Ans. 3. A/3, ^W5. See Arts. 103, 317. 4. V5 + V^j -4^5. 5. 4V2.T -f sVy? ^i?^5. RADICAL EQUATIOXS. 247 6. v^ — '\/6 + 2, Ans, This factor is obtained by trial. Multiply V3 + V2 + 1 By \' 2 — V6 + 2 ^/6 -h 2 + V2 — a/6 — 3a/2 — 2A/3 2 + 2 a/2 -f 2 A/3 4 We multiply first by A/2 to rationalize the A/2 in the given denominator. The product contains two radicals, a/6 and A/2, which may be made to disappear by multiply- ing by a negative quantity; we therefore try — a/ 6, and the result, 2 — 2 A/2 — 2 A/3, has two negative radicals, the same as in the denominator; and we make these disappear by multiplying by + 2. I. RADICAL EQUATIONS. ratjv 2SU. Given ^/x + I — a/i I + X Squaring, X + 2 \/x +1 l\ -\- X Reducing, Vx 5 X 25, Ans. 2. Given V^ + 18 — A/5 = Vx—'] Squaring, x + 18 — iV'^x + 90 + 5 = .r — 7 Reducing, \/$x + 90 = 15 Squaring, 5a; + 90 = 225 Transposing, 5:6' =135 /, ;?; = 27, ^?zs. 2^8 QUADRATICS. 3. Given, ^/29■ —11=5 Squaring, x- — 11=25 Transposing, :7;2 = 36 Extracting root, 2: = 6, AiiSo 4. Given = ^ ^3"+^ V 3 + ^c Clear of fraction, 6 = 34-0; Transposing, a; = 3, Ans. 5. Given (13 + V23 + ^2)1 _ ^ Squaring, 13 + V23 -f ?/2 = 25 Transposing, A/23 + // =12 Squaring, 23 + ?/2 = 144 Transposing, ^"^ = 121 y ■= 11, Ans, 6. Given 2\^ = V^' + 3« Squaring, ^a = :r + T,a Transposing, x = rif, Ans. QUADRATICS. Page 290. ^ . -zx — ^ ^x — 6 I. Given ^x — ^ ^ = 2X + ^^ Transposing and clearing of fractions, 6x2 _ i8r — 6.T + 6 = ^x^ — i5.r -f 18 Uniting terms, 3.^^ — 9.r = 12 Reducing, ^2 _ ^2. — 4 By Art. 334, x = i± \U + i Reducing, ^' = i ± » » X =.4 or — I, Ans, QUADRATICS. 249 2. Given X 100 — 9.T \x' ^ Cleiiriiig-, dOfX - - 100 + ()X 12X^ Traiispijsiug. 12.^^ — 73.6' — 100 Divieling, ^2 737. 10 X 'i2-^ -12 Second nietliodj 7. 731 a/ I00_i_5329 1 •*' 2 4 m V 12 i^ 5 7^ Reducing, /v 7 3 12 3 •^ 24 ^ 24 .'. X 4 or 2-jl2^ ^^^^^• Given 3 2 24 Mnltipl3'ing by 2, X ^ i^ Second method. -7-3 I _1_ /»/ I 1 1 X 4 ^ V 16 "T IS" Extracting root , X ^/\, Ans. Given V4X 4-2 4 — Vx 4 + ^x Vcc Clearing, 2X H- 2'\/x 16 — :r Transposing, 2\^X 16 — 3a; Squaring, 4:c 256 — 96a; + 9a;2 Or ( ^a:^ — 100.?; — 256 Dividing, ^2 I J. 2 5 6 Second method, ')■ 50_j_a/ 256 1 2500 •^^ 9 It A' 9 1 81 Eeducing, ^ 50 _i_ 14 ^ — "9' 1 "9^ .'. X 7^ or 4, ^726\ Let ,r greater number. And y less number. Then .T — 7/ 12 (i) And rc^ + ^2 _ 1424 (2) Squaring (i). X? — 2Xij -^ If — 144 (3) Subtracting (3) from (2), 2./V/ 1280 (4) Add (2) and (4), x^ + 22:?/ Extract root, x Combining with (i), x We have And + if = 2704 + ^ = 52 — ?/ = 12 X = 20; ) 250 QUADKATICS. 6. 8. Denote the numbers by x and y. Then x -\- y = 6 And x^ 4- ?/3 = 72 Dividing (2) by (i), x^ — Q:y + // = 12 Squaring (i), x^- -\- 2xy 4- y'^ = 36 Subtracting (3) from (4), yxy = 24 Or xy =: S Subt. (5) from (3), a;^ — 2xy -^ y^ = 4 Extract root, a; — ^ = 2 Combine with (i), x -{- y =^ 6 By addition, x =z 4. By subtraction, y = ^ Denote one part by x. Then the other part =r 56 — ic (•) (3) (4) (5) ;[ Ans. By condition, 562; — x^ -— 640 Changing signs. , x^ — S6x — 640 By 2d method, X 28^^ — 640 + 784 Reducing, X 28 + 12 And S6 —X — 16, ) Let X B's hourly progress. Then a; + 3 ^ A's And — - — B's time on the road. X And '5° ^ 81 _ A's " " x + 3 By condition. 150 1 25 150 X -\- 3 3 X Reducing, 6 I 6 ^ + 3 3 ^' Clearing, 18a; + x^ + 3;?: = 18a; + 54 Or, x' + 3.r = 54 By 2d method, x = — J ± ^54 + f Reducing, x = — | + -^^ .*. X = 6 miles, B's rate : ) , And a; + 3 = 9 " A's " f QUADRATICS. 251 Let X = greater number, And 1/ := less ii umber. Then X — t/ = 6 ( ! ) And 2y^ -\- 4j = x^ (2) From (1), ^~' — ■^6 -f 1 2_^ -f if Substituting, 2Xp + 47 = 36 + 12^ + ^2 Transposing, //^ — 1 2^ := — 1 1 By 2d method, ij — d ±_ V^Ti + 36 Eeducing, y = n, less, From (i), X =1 ' ^ II, less, ) 17, greater, f 10. Denote the length and l)readtli by x and y. Then ic + ^ = 42 (i) And 2;// =: 432 (2) Squaring (i), x^' -{- 2xy + //- =1764 Multiplying (2) by 4, 4^?/ =1728 Subtracting, X'^ — 2xy -\- y^ =z ;^6 Extract root, x — y = 6 Combining with (i), x -{- y = 42 By addition, x ==24 ft., len! ; ^ By subtraction, y z= iS ft., bre., f ^ II. Let X = A's age;' And ?/ = B's age. T 20 Then xy = 120, or..r = (i) And (x ~ s){y + 2) = 120 Or 2;^ 4- 2X — 3// — 6 = 120 (2) -n • 240 By substi., 120 + -^ — 31/ = T26 Reducing, ^2 _|_ 2^ — 80 By 2d method, //=: — !+ a/So + i Reducing, ^ = 8 yrs., B's age ; ) ^^^^^ From (i), ^ — 15 yrs., A's age, f 252 QUADRATICS. 1 2. Given \^x^ + V^^ 6\/x Sffuaring, Dividing by x, Extracting root, X^ -{- 2X^ -{- X^ X^ -\- 2X^ -\- X'^ X^ -j- X X X — 36^' =: 6 By 2d Method, Reducing, - 4 + V6 + i 1 4_ 5 2 =1= 2 * ,\ X 2, Ans. 13. Given X + V^ + 6 — 2 + ^Vx + 6 Transposing, X — 2 2Vx + 6 Squaring, x^ — 4^' + 4 — 4;r 4- 24 Uniting terms. x^ 8x X 20 By 2d method. 4 -- V20 + 16 Or X 4 + 6 « • . */y — 10, or — 2, J??5. 14. Let Then x X No. lbs. pepper for Cio. + 60 " ginger for £20. And — Price X of pepper per pound. Ari'rl — 20 O'l n nr£iv ^^ xxllU. X + 60 gingei Bv conditions, 80 x |-ioox ^"^ 6c ^^ + 6o ^5 Reducing, i6o- 400 + 60 '3 Clearing, Uniting and divi i6o;6' + 9600 + ding, .i2 4- X — - X - ' 400.?' rr: 132--+ 780.^ 2 2 v 9 6 "13-'^ 13 By 2d method, Reducing, I I _l_ /t/9600 1 I 2 I 00 13 -^ 13 1 "13^ [10 13 70 13' 13 2: = 20 10 And price of pepper per lb. — = ]^ — i£, or tos.; A71S << (( gmger ii 20 2: + 60 ^(7 ^ i"^j 01' 5^*' COLLEGE EXAMINATION PROBLEMS. 253 COLLEGE EXAMINATION PROBLEMS. Pafje :>U1, Art, J^V. • ('5-'-?)-(-^^); 5^ 5c ex — a -\- 5 [rx — a -\- u) 2. (a^ — b^) -^ (a — h) = a^ + a% + al^ + b^, Ans. See Art. 129, Prin. i. 3. Given x 4- ^^ ^ = 12 -^ 2 3 Clearing of fractions, 6x-\-gx — 15 = 72 — 4.T + 8 Uniting terms, 19:^ = 95 4. Multiply 3^45 — 7V5 = 2 A/5 By VTf + 2V9f = 17VJ And lyVy X 2V5 = 34, ^4w.*?. l7i i?! 17-2- > 5. «^03 -^ a-iO^ = ci^b^^, Ans. Subtract the exponent of the divisor from that of the dividend. 6. xy~'^ -^ a.%~3 — x^y~^, Ans. 7. Given 2>^^ + ix — 9 = 76 Reducing, .r"^ + |:?! := -^ By 2d method, x = — J + \/ ^ + J Reducing, :r — _ i _[- j.fi .-. a; = 5, or — 5I, Ans. I,- — 7^ 8 / x^ 3"^ X — 5 4 1 3 Vj + I 9^ X — I 3 7 • • • X 1 : h ^1' - 5 ^7 Ans. 254 coLLECxE examinatio^nt problems. 8. Given ^.t? - Transposing, etc., By 2(1 method, Eeducing, Denote the numbers by x and y. Tlien X \ y w X ^ y '. dt2 (i) And ' X \ y \ \ X — ^ : 6 (2) From (i), (Art. 378.) 42.C =z xy -\- y'^ From (2), 6x = xy — y^ (3) Subtracting, Or x = ^ (4) 36.^ 2lf X 18 t 3 18 y' I y 18 I Substituting in (3), Dividing by y^, Reducing, y = 24, less ; ) ^^^^ From (4), X = 32, gr., f 10. S = — ^ , = ^^ = - = li .4??.s'. (Art. 435.) ^ ^ I T 3 11. {a + ^)i2 = r/i^ + i5fti4j + 105^13^2^ etc., Ans. 12. (i + .T2)-i = I _ - -f ^- — ^ , ^?z.s. (Art. 270.) 5 25 125 ^ = — i; /t' I T S" I J 1 ^ w X = -i X — ^ — - =r -J X -J = 2^; 2 2 w— I n — 2 - —4—2 w X X = -h X 233 3 V IT I .T I. COLLEGE EXAMINATION' PROBLEMS. r Ans. And from (i), y — — ^, (^r — 9, \ 2X — y 21 ^ 2.i' — 2 I (I) 2a;2 4- ?/2 _ 153 (2.) 258 COLLEGE EXAMINATION PROBLEMS. rage 292, Art. 544, 4. Let X =z No. of yards. Then — = price per yard. By conditions, — = — ; 1- i -^ X X -\- 2 Clearing, iSoiC 4- 360 = 180a; -\- x^ + 2X Or x^ + 2X = 360 By 2d method, x = — i ± Vs^^i = — I ± 19 /. a; = 1 8 yards ; \ And ^ = $5, price, 5. (a — by^ = aP — \20>^h + (i(ia^%^ — 2 2oa^h^^A9S<^^^^ — ']g2CVh^ +924«^Z'^— 792rt^^' + 495«4Z'^ — 22oa%'^ ^6(}a^^^—i2al)^^-^l^\ A?is. 6. 4^2 _ gy2 — (2flr + 3^) (2fl! — 3?/), Jw5. (Art. 103.) «5 /a . / a^ , /2«^ 6fl^ = ^^2, Ans. 8. Given ic + 2?/ = 7 (i) And 2X + 3// =12 (2) Mult, (i) by 2, 2X 4- 4 // = 14 (3) Snbt. (2) from (3), ^ = 2 ; ) ^^^^^^ Substituting in (i), a; = 3, f ^. a; — ^ X X — 10 10. Given -\ — = 12 32 3 Clearing of frac, 2ic — 10 + 3.6' = 72 — 2^ + 20 Uniting terms, 7:?: = 102 X — 14^, Ans, COLLEGE EXA3[INATI0N PHODLEMS. 259 Page 292, Art, ^45. I. From T.x A — 7 subtract x — ' 2h c . X X — a cx-{-2h ix — a) Ans. 2X -\ — r H , or 2X + x^ 2b c ' ' 2bc I — y- I x \ X — ,-^T, X 1 1 H ), or X + x- \ \ — XI 1+2/ ^ + (i +a:)(i-.r )(i + ^) (i - ij) ^ \ -y ^^^^ {\ -^ y)x{\ ^ x){\ —x) X ' c& — \d^h + ZaW + ^¥ { a^ — 2ab — 2b^, Ans. a' 2a^ — 2ab ) — 4a^b — 4d^b + 4aW 2^2 — ^ab — 2b^) — 4nW + Ub^ + 4^^ ,. From 2^/320 = 8^/5 Take 3'V^4o = 6^/5 2V^5, ^7i5. 1 tI 5. a'^b^ ~ a^b"^ = cr'^b^^, Ans. 6. Given ^* + 4.^^ = 12 By 2d method, x^ ■=^ — 2 ± A''^i2 4- 4 Eeducing, .t^ = — 2 ^fc 4 ^= 2, or — 6 Extracting root, a: = ± A/2, or±\/ — 6, J?kv. Given x^ — a; V3 = :z^ — ^Vs Transposing-, x^ — (1 + A^)-^' = — 2 By 2d methcd. 3 ■" V 2 4 260 COLLEGE EXAMIN^ATION PROBLEMS. Eeduciug, * x — ^—^ — - ±_ i 2 2 8. Denote the numbers by x and y. Then x -\- y ^=^ 2a (i) And ^2 _|_ ^2 — 2^ (2) Squaring (i), x^ + 2xy + «/2 = 4«^ (3) Subt. (2) from (3), 2:?;?/ = 4^^ — 2^ (4) " (4) " (2), a;^ — 2xy + ?/2 = 4^ — 4fl'2 Extracting root, x — y^ ±. 2\/b — n^ (5) Combining (i) and (5), r*:: = « ± V^ — c^ And y = a^^ ^h — c^. Ans. : 2 : 3 (i) : 2 : 5 (2) : 5 • 3 9. Denote the numbers by x and y. Then x — y : x -[- y And a; — ^ : .t// From (i), (Th. 7) 2x : x -\- y Theorem i, 6x ^^ <^x -\- <^y Or X =1 sy (3) From (2), (Th. i) 5a; — 5?/ = 2a;?/ Substituting value of x, 2oy := loy'^ Reducing, ?/ = 2 ' From (3), ' X =. 10, 10. r = Q"""' = (iA2)i ^ ^87 = 3. See Art. 407. Hence the ceries 2, 6, 18, 54, 162, .4?is, ' ' Alls. COLLEGE E X A M I K^ A T I X PROBLEMS. 2G1 il. __^.^ = 7)1 ix- + ^2) 2 zz: — I I -f I ; 'y/x' + «2 X \ J- J /I «2 ^^/4 (^ — ^^, + 7r~- ^7? etc.), Ans. \x 2 r^ 8:?:^ 48.1-7 ' /' See Art. 270. y^ = — ^ ; w — I _ J — i — I _ J 3 _ i ^i X ^ _ — 2 X ^ — —2 X — :f — %; w — I n — 2 - — I — 2 n X X == f X — 233 — 3 V 5 — 15 rage 292, Art. 546. 12.?'— 102 I2(.T^ — 16) 000^ 1. -f— = — ^7 r- = 4X^ + 8x^-\- 16.T+ 32, Ans. 3'" — 30' — 2) 2. Divide cc^ H 7 by y — m „ , . x^ ia — h) + x^ ah — yn (a — t) Keducmsr, — ^ S '• S a — a — -P^. . x^((i — Z*) + ^3 Dividme^, -~ j- tt, Ans. ^ ab — m\a — b) ^. 3a; — II S^ — S 91 — IX 3. Given 21 + — = • „ ^ 4- ^ -- 16 8 2 Clear, of frac., 336 + 3.^ — 11 = lo.r — 10 + 776 — 56:^ Uniting terms, 49X = 441, x = 9, ^ ;?,-?. i 2 3 _ 4 6 7 . 4. a-a^a^a » = flrso^ ^«,9. i + l + i-f=" 3o±4CH^5^8 ^ 6 7^ exponcnt. ^,. 11; 72—62: 5. Given -^ — r— = 2 X 2X^ Clearing of frac, t,ox — 72 + 62' = ^x^ Redncing, x^ — gx = — 18 By 2d method, x = ^ ± V— 18+"^ .'. X = ^ + ^ — 6, or 3, Ans. 262 COLLEGE EXAMINATION PROBLEMS. 6. See Art. 394. d = I — a 50 - I = 7. m +1 6 -f I Hence the series, I, 8, 15, 22, 29, 36, 43, 50, ^^5. /'ttr/e ;^.9.j?, Art. 546. 7. EuLE. — Divide the product of the natural numbers from m down to 7n — w + i inclusive,, by the product of the natural numbers from i to 71 inclusive. -CI ' ^ w . .(m — n-\-\) 8x7x6x5 . Formula, G — ^ = = io,Ans. 1 ' 2 ' 2i ' ' ' ' n 1x2x3x4 Note. — m, denotes the whole number of letters, and n the number of letters taken in a set. 8. (, _ J)-} = L f, _ *^ .1 = - I + — + "^ + -Vo, etc. (« _ J) T ^ -_.+ -_ 4- -A_ + __^^^^ etc., «T 4^1 32«T 1280^"^ 9. Denote the smaller by x. ) Ans, Tlien By Theorem i, Or 10. Given And Subtracting, Substituting, a; : 150 — .t : : 7 : 8 Sx — 1050 — yx 15.2- z=: 1050 .-. X = 70, smaller ; 150 — .T = 80, greater, 5-'^ H- 2// = 29 — X -\- 21/ =z — I 6x ■= 30 I^ Ans. A71S, COLLEGE EXAMTNAflOX PROBLEMS. 2G3 ruf/e V?.'>.V, Art. 547. 1. Given [- I z= o 2 + // 2/ — 2 loy — 20 — lo — 5// + 3// — 12 = o Uniting terms, 3?/ -f 5?/ == 42 Dividing, // + |^ = 4 By 2(1 method, ?/= — 6^± Vi4 + |f Keducing, /y = — f ± -¥" ••• y = 3? <^>i* — 4f, ^^'^^• 2. c!^ ^ rt^ -- rt '^ ; rt~i -^ r<^ = /^r^. Sec Art. 396. Hence, a, a^, a~ ^ are part of a geometrical series of which the ratio r = cr^, Ans. 3. Let 20: = first number, And 47/ = second number. Then x -i- ^y = 11 ( i ) And a; + 3y = 6./' — 4y (2) From (2), X — — (3) Substituting in (i), ^^ + 3?/ = 11 Or 22?/ = 55 ^y = 5 ^ = I From (3), a; = I X I .-. 4// =1 10, second ; ) . And 2.T 1= 7, urst, ) 4. Powers of 2a, 256^8+ 128^?'' -f- 64'/^ .. 4rt~ + 2rt 4- I Powers of — , 3 h y^ ¥ W h^ 3 9 729 21 - Coefficients, I 8 28 .. 28 8 (- - -3)' loz^a'!) \']()2n%^ 112(1%^ i6ah'' ¥ . 3 9 729 2187 6561 264 COLLEGE EXAMIN^ATIOK PROBLEMS. 5. (p -W = {a + 1)) {(I - b) ; «2 — 2ab + />^ = {a — hy. Hence the (j. e. d. is a — b, Ans. 6. a^ — x^ =(« + :?;) {a — x)\ Hence I. c. in. is 4^^ — 42;^, Ans, \7,a — 2()b 7 J — 21a gh — iia i3«2_42fl',^ + 2 9^^ — 2'iab-\-2ia^-\-']b'^-]-()b — 11a 5 (« - <^j2 34f/2 — 63r/Z> 4- 36^^ + gb -^ iia , or 5 (« - ^)' 8. Given ^/x -\- a = Vx + ^ Squaring x -\- a =z x -\- 2a\/x + «2 Eeducing, 2^/x =: i — a Squaring, 4.T = i — 2a -f- a^ I — 2« + «2 Par/e 293, Art. 54S. a^4-ah' — ax^ — x^ (a-\-xY(a — x) I. 9 — ^ = ;— ^ — (-7 i = a+x, Ans. a?—x? [a-{-x) [a — x) 2. ar%'^ X a-W -^ -^— r = —-, or -^ , or « 3 h^, 7ic — 6 :r — ^ X Given ■^- 7 ^— = - 35 6x — loi 5 Multiplying by 35, jx - 6 - j^^^ = 7^ Transposing, etc., 362: — 606 + 352' — 175 = o Uniting terms, yix = 781 .-. X = ii" Atis. COLLEGE EXAMINATION PROBLEMS. 265 n- 7^' + 9 / 2./; — i\ 4. Given — (x 1 — 7 4 V 9 / Clearing of fractions, 63^ + 81 — 28.^—4 = 252 Keducing, 35^ = ^75 Of 00 5. Given + yf = 8 Trans, and mult, by 2, x^— ^x ^= ^ By 2d method, x = I ± Vi -\- ^ Eeducing, rr = | ± | .-. X = i^, or — f, A71S, 6. Denote the numbers by x, y, and z. Then xy = 15 Or ^ = 'I (i) And xz = 21 21 / X Or -'^^ == V (2) And / + 2;2 _ y^ (3) Equating (i) and (2), 15 _ 21^ y ~ z -^ 21 ^ 441 Substituting in (3), \- z^ — 74 ° 441 Or 666z^ = 74 X 441 .-. z^ =: 4.g Extracting root, 2; 1= 7, 3d ; ) Substituting in (2), ic = 3. ist ; )- Ans, Substituting in (i), y = 5, 2d, ) 266 COLLEGE EXAMIKATIOJS" PROBLEMS. Page 2i)4, Art. 548. 7. I z= a -\- {n — 1) d z=z I -\- (n — i) x i = ■?^; See Art. 388. a-\-l i-\-n n-{-n^ . S = xn = xn = — , Ans. 222 See Art. 389. «. K -.,-' = ;■(.-!)-' = - ( I + — -3 H -6 + -^9 + -T2' etc.) J3 2^ 14^9 ^5^2 30* "^ 96?^ "^ 8i«io "^ 243^ I + t:::! + r::^ + ^7::rn + 7^3' etc., Ans. n = — J ; ?i — I , — i — I 2 . ^2- — ^ 2 3 >< 3 9 > n—1 n — 2 _ — J — 2 14 . /t- X X — g A A Q — 233 ^1 J n—1 71 — 2 /^ — 3 ^z X X X 3 14 w 3^ 3 — I4v 5 — _3_5 — — ^T ^ ] — ^T-^ ^ — 24 5' 4 9. A/300 + A/75 = 10 A/3 + 5a/3 = 15 V3, ^^2^. 10. Given - -\ — = -7- 7 ^ + S Clearing of frac., x^ + 5^ + 147 = 23a: +115 Transposing, x^ — i8ic = — 32 By 2d method, x = g ± a/— 32 + 81 Or X =^ 9 ± 1 .'. X = 16, or 2, Ans. Page 294, Art. 549. I. a/i8«^ + VJoa%^ — yi^bV2ab ± scibV2ab, Ans. collegeexami:n'atiox problems. 267 2, Multiply 2 A/3 — V — 5 By. 4V3 — 2V— 5 54 — 4V — 15 4\/— 15 + 2 X — 5 (Arts. 312, 514.) 14 — 8\/ — 15, Ans. ^— I 23 — a; 4 + ^ 3. Given = 7 — •^ 7 5 4 Clearing, 2o.^ — 20 + 644 — 28a: = 980—140—352; Uniting terms, 2'jx =216 .-. X =z "8, Ans. n- ^' — 3 ^ — 4 7 4. Given = 2V re — 2 X — I Clearing of fractions, 2o{x^—^x-\-^—x^ + 6x—8) — {x^ — 3.T + 2) 7 Or 40X — 100 = ']X^ — 2i:6' + 14 Trans, and divid., x^ — ^-f-x — — ^^^ By 2d method, x Reducing, ^ = t4 ± t^ T~" By 2d method, x = ii±V—^^ + WT■ C.J , -^ — I4-i-I4 .*. o; = 6, or 2f, ^4;?5. o A ^ 2. ^^^ — T ^ - — 9 5 n — I n — 2 , i — 2 w X X = — i X - — =: — i X — I = ^ ; 23- 3 (a^—h^y = f^ ^^^ -—i-\' etc., Jw5. ^ ^ 3ft^ 9«=' 8ifr 2G8 COLLEGE EXAMIN'ATION PROBLEMS. 7. (a^ — 20^1 + aWfi = a/(^^ — 2ab + 1?) a = (a — b) Vci, Ans. 8. Let Xy y and z be the times required by A, B, and 0. . Then i + i = ^4^ = ^ (:) Part A and B do in i day ; X y Part B and C do in i day ; Part A and C do in i day ; Adding and reducing, - + - + - = i§ (4) Part A, B and C do in i day ; Subtracting (2) from (4), Part A does in i day; Subtracting (3) from (4), Part B does in i day ; Subtracting (i) from (4), Part C does in i day. Clearing of fractions, a: == 10 days, A's time ; I I y 7 30 I I X z I . 6 /s I I X y + I z 10 3^0 I X A • I y i I z A (2) (3) ;) y zzi 6 days, B's time 2; = 15 days, C's time. ,' Ans. If all united do \ in i day, it will take 3 days to finish. 3 = First number ; \ 5 = Second number ; ,- A71S. 7 =r Third number, ) See solution of Ex. 6, under Art. 548. COLLEGE EXAMINATION PROBLEMS. 2G9 Page 294, Art. 550, I. Given - + - = 2 (i) (3) I + I — 2 X y I I H- - 3 X ;2 I I — + — 3 y z Add and div. by 2, -i 1- - = 4 (4) '' X y z Subtracting (3) from (4), - = i, or a; = i ; « (2) " - = I, or y = i\) Ans. «• (i) " - = 2, or 2 = I ;2 3 8 — ^ ^—11 2. Given 7; = ^ — X 3 12 Clearing of fractions, 36 — 256 + 642- — 4.1-2 — i(^x — x^ — 88 Uniting and divid., x^ — 15.?' = — 44 By 2d method, x = -\^- ±a/ — 44 + H^ Reducing, x = -^/ ± | .♦. X =^ II, or 4, ^w.9. 3. Factoring the given quantities, we have x'^ -^ 4X — 21 = {x + 7) {x — 3) X2 _ X- S6 ^ {X + 7) (^^ - 8) Hence f/- c ^Z. = a; + 7 ; ) And I. c. m. = {x + l){x — 3){x — ^)^ Ans. = .,-3 _ 4.^2 _ 53:r + 168, ) 270 COLLEGE EXAMINATION^ PROBLEMS. 4. Let .r = No. of days B requires ; Then a; + 10 = " A " By condition, - H = T2J part A and B do in i day. X X -^ 10 ^ "^ Clearing of frac, 12.T+ 120 + i2.r = x- 4- 102: Or x^ — 14X = 120 By 2d method, ' x =^ j ± ^120 + 49 Eeducing, ^ = 7 ± 13 .*. a; = 20 days, B's time ; ) And a: + 10 = 30 days, A's time, f See Arts. 270, 271. Powers of y, y^-\- y^-\- y'^-\- y -f i Powers of 3, 13 9 27 81 Coefficients, 146 41 ^ = (y + 3)' = ^4+ 12^/3 + 54^2+ io8?/ + 8i —x^ = —{y + 3)^ = — y^— 9?/— 27?/— 27 2X^ = 2(?/ + 3)2=z 2/+ 12?/+l8 — 3 = - 3 ;r4— a:3 4-2.r2_3 — ^4_|_ ji^3_|_4y2/24- g^y-^-GgyAns .^ m 35912 M577055 QA153 T485 Educ. Lib .