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 LIBRARY 
 
 UNIVERSITY OF CALIFORNIA. 
 
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 Received O^&A , , i8g /. 
 
 A ccessions No ^c^t&p3 Shelf No. ... 
 
 <,£♦ *3v> 
 
KEY 
 
 ELEMENTARY TREATISE 
 
 ON 
 
 ALGEBRA 
 
 BY THOMAS SHERWIN, A.M 
 
 Principal of the E.iglish High School, Boston. 
 
 tfUj* OF THB 
 
 f> * D* 
 
 . '>: 
 
 BOSTON 
 
 NICHOLS & HALL, 
 
 32 Bhomfield Skteet. 
 
O 3 * 
 
 Entered according to act of Congress, in the year 1842, 
 
 By Thomas Sherwin, 
 
 in the Clerk's Office of the District Court of Massachusetts. 
 
 ^ v 
 
 f/J 
 
JHI7BESIT7; 
 
 KEY. 
 
 [ In the first edition of the Algebra, the Articles are numbered erroneously from the com- 
 mencement of Section XXVI. To make that edition correspond with the Key, increase by 
 2 the number of each Article, commencing with the Section specified.] 
 
 SECTION I. 
 
 Art. 4. 
 
 1. Performed. 
 
 2. Horse $100. Saddle $10. 
 
 3. 6 bushels of each. 
 
 4. 11 of each. 
 
 5. 5 melons ; 10 pears ; 30 peaches. 
 
 6. 15 days. Man $20; boy $10. 
 
 7. 105, 210, and 315. 
 
 8. 25, 35, and 40. 
 
 9. 10 hours. One goes 80 miles, the other 70 miles. 
 
 10. 1st 8, 2d 16, 3d 24 guineas. 
 
 11. Wife $36000; elder son $27000; younger son $18000. 
 
 12. 120 cavalry ; 360 artillerymen ; 720 infantry. 
 
 13. A 8 days, 72 rods; B 16 days, 112 rods; C 80 days, 400 
 rods. 
 
 14. 8 yds, 16 yds, and 24 yds respectively. 
 
 15. Sheep $2; calf $4; cow $20. 
 
 16. Greater 14 galls; less 7 galls. 
 
 17. Beef 10 lbs; mutton 20 lbs. 
 
 18. 32 of each. 
 
 19. 80 guineas ; 240 crowns. 
 
 20. 10, 20, 30, 60 and 30 miles, on the respective days. 
 
KEY. IT 
 
 21. 8 bushels of each. 
 
 22. Cows $25, oxen $50 each. 
 
 SECTION fl. 
 
 Art. 7. 
 
 1. Performed. 
 
 2. 1st $23000; 2d $33000. 
 
 3. 73£ and 156f miles. 
 
 4. A's 15, B's 35, C's 50 years. 
 
 5. 1st 8 lbs ; 2d 22 lbs ; 3d 30 lbs ; 4th 60 lbs. 
 
 6. 6, 8, 10, 12, and 14 years respectively. 
 
 7. Black $30; green $32; blue $35. 
 
 8. A's 30, B's 20 years. 
 
 9. Each daughter $1000 ; younger son $2000 ; elder son 
 $5000. 
 
 10. Performed. 
 
 Art. 9. 
 
 1. 493 and 40o votes respectively. 
 
 2. 5 men ; 20 boys ; 180 girls. 
 
 3. 150 cavalry; 450 riflemen ; 1700 infantry. 
 
 4. A $2750; B $8250; C $22000; D $32000. 
 
 5. A $377; B $7; C$107. 
 
 6. 10 years in France; 60 in England; 10 in America. 
 
 7. 1500 English ; 900 Irish ; 500 French. 
 
 8. 50 eagles; 90 half-eagles ; 250 dollars. 
 
 9. Broadcloth 50 ; cassimere 144 ; silk 288 yards. 
 
 10. $2000. 
 
 11. A $10000; B $10600; C$20800; D $20300; E $30800 
 
 12. 80 oxen; 215 cows; 215 calves; 395 sheep. 
 
 13. 20 students; 55 merchants; 65 officers. 
 
Ill, IV KEY. 
 
 SECTION III. 
 
 Art. 11. 
 
 1. Performed. 
 
 2. Finer $6 ; coarser $4 per yard. 
 
 3. A's 50 ; B's 40 years. 
 
 4. 20 each. 
 
 5. Sum distributed 153d. Eldest 60d. ; 2d 42d. ; 3d 30d. ; 4tl 
 21c 5 
 
 3. 1st 10s. ; 2d 7s. ; 3d 8s. ; 4th 5s. ; 5th 3s. ; 6th 2s. 
 
 7. Smaller 40 galls. ; larger 80 galls. 
 
 8. Wheat 8s. ; rye 6s. per bushel. 
 
 9. 45 scholars in all. He wished to place 7 in a row. 
 
 10. 10 lbs; 95 cents. 
 
 11. 10 shillings. 
 
 12. $60. 
 
 13. 22 years. 
 
 14. 12 years ago. 
 
 15. 150 sheep. 
 
 16. 8 days ; 720 miles from the starting place. 
 
 17. Horse $200; chaise $275. 
 
 18. Wheat 8s. ; oats 3s. 
 
 19. $1000. 
 
 20. 125 miles. 
 
 SECTION IV. 
 
 Art. IS. 
 
 1st, 2d, 3d, performed. 
 
 4. 60 years old. 20 years in Germany ; 15 in France ; 24 in 
 England. 
 
 5. Whole estate $5040. He owed his creditors respectively 
 $630, $720, and $672. 
 
 I* 
 
6 KEY. V. 
 
 6. Whole mixture 32 oz. 12§oz. gold; 16 oz. silver; 3£ oz. 
 copper. 
 
 7. Whole estate $22500. Wife's portion $6500 ; eldest son's 
 $6500 ; 2d son's $5750 ; youngest son's $3750. 
 
 8. 80 years. 
 
 9. A's 40 ; B's 30 years. 
 10 30 inches. 
 
 11. 72. 
 
 12. $150. 
 
 13. $75. 
 
 14. 1st 30 cents ; 2d and 3d 15 cents each. 
 
 15. Whole number 30. The 1st had 10; 2d 6; 3d 4; 4th 10 
 
 16. 1st horse $100; 2d horse $150; chaise $300. 
 
 17. Income $1200 ; he spends $650. 
 
 18. 45 sheep. 
 
 19. 1st £56; 2d .£35; 3d £7. 
 
 20. 280 miles in all. 40 miles 1st day; 56 2d day; 70 3d day. 
 
 SECTION V. 
 
 Art. 13. 
 
 1st, 2d, 3d, performed. 
 
 4. A $300 ; B $200. 
 
 5. Man 40 ; woman 30 years. 
 
 6. Man's age 50 ; woman's 45 years 
 
 7. £50. 
 
 8. From A to B 40 miles , from B to C 60 miles ; from A to 
 C 100 miles. 
 
 9. 1st 105 yds; 2d 120 yds; 3d 135 yds. 
 
 10. Each has £150. 
 
 11. 100 yards. 
 
 12. Each has $140. 
 
 13. A $350 ; B $400 ; C $475. 
 
 14. 30 cows ; 69 sheep. 
 
VI, VII. KEY. 
 
 15. 840. 
 
 16. $1000. 
 
 1 7. Cider $4 ; beer $6 per barrel. 
 
 18. 54 and 60. 
 
 19. Shorter 20 rods; longer 40 rods. 
 
 20. Chaise $233£; horse $166§. 
 
 21. $100. 
 
 22. 1st car 18 miles; 2d car 20 miles per hour. 
 
 23. $150. 
 
 24. $110. 
 
 25. 139. 
 
 SECTION VI. 
 
 Art. 15. 
 
 1. Horse $150; chaise $250. 
 
 2. 80 gallons. 
 
 3. Greater 35 ; less 25. 
 
 4. Performed. 
 
 5. 20 tons. 
 
 6. $72. 
 
 7. Greater 75; less 72. 
 
 8. 100 gallons each. 
 
 9. 90 eggs. 
 
 10. Time past 40 years. 
 
 11. Lower, middle and upper parts respectively, 12, 16 and 20 
 feet. Whole pole, 48 feet. 
 
 12. 20. 
 
 v SECTION VII. 
 
 Art. 17. 
 
 1. liabcm. 3. 2labxy. 
 
 2. 845c. 4. Ziapqr. 
 
KEY. VIII, IX. 
 
 5. Slaghx 9. 21 mqs. 
 
 6. 156 a x y. 10. OOaqx. 
 
 7. Amnpq. 11. Performed. 
 
 8. 20 amp. 
 
 Art. 19. 
 
 1. Performed. 11. 12 a? h 6 n*x7 
 
 2. 54<z6 2 c. 12. 567a 3 r 2 * 7 y. 
 
 3. 301 a 4 66. 13. \Z2p 3 xSy 3 . 
 
 4. 46 a 1 m 7 x 14. 36 a 3 m 5 n-pP 
 
 5. 399 a 9 7?2 x. 15. 39z 3 # 3 z 4 . 
 
 6. llx*y 8 . 16. 170a 5 6 2 cz. 
 
 7. 72 a 3 b 3 c 3 (P. 17. 210a 5 6 3 c. 
 
 8. 146 tt* 9 ti 5 x 4 . 18. 54c 2 ^z 4 y 3 . 
 
 9. 125a 8 6 7 c 6 d 2 . 19. 28 a px^y\ 
 10. 162 i > 4 j 2 x 7 ?/ 9 . 20. 3a 7 m 5 z 4 y 7 . 
 
 SECTION VIII. 
 
 Art. 22. 
 
 1. 4a 2 6 — a& 2 . 
 
 2. 3a6c 2 . 
 
 3. 8 a b c 3 -J- a b 3 c — 6ab 3 — 4 m n. 
 
 4. — cp 3 r-{-7m! 2 ri 2 -\-7gh. 
 
 5. 2 77*71 r — 8p 3 q -j- 10 b c. 
 
 6. 27a; 2 — 8y 3 + 5m. 
 
 7. 7 « 3 ^? m — 3 ap m 3 -{- a mp^ -\- 6. 
 
 8. m rfip 3 r 4 + 12 ?rc 2 n — w 2 ti 2 ^ 3 r 4 . 
 
 SECTION IX. 
 
 Art. 25. 
 
 1. 20a + 26+16c + 4tf + 6. 
 
 2. 12a& + 4cd + 4ro 2 n + 3m. 
 
X. KEY. 9 
 
 3 18a& + 5crf + 5/w 2 + 4cm. 
 
 4. 13a 2 & 2 + 8m 2 — 18mrc -f-3& + 4zy. 
 
 5. 16 6c — 3a6 + 5ac+12crf + 4m» + 10aw. 
 
 6. 8a- S6 + c — 17c? + 25. 
 
 7. —8a + 6c—\09d + 3le — Wf+g. 
 
 8. 17 a 2 6— 13 a 6 2 — 10a6 + 18c 3 d — 10 c? 4 — 12m 2 /*-f 
 11 m w 2 + m ra. 
 
 9. 8a 3 + 14 6 3 — 7 6x 2 — 8cx 2 — 2 2 + 13r— 9Ay 2 — 2A-y s 
 — 5ky — 9 hy -\-7py 2 . 
 
 SECTION X. 
 
 Art. 27. 
 
 1st, 2d, performed. 
 
 3. Sab — 2c 2 + 36c. 
 
 4. _6ac- G«6 + 96c + 7cc?— 12awi.. 
 
 5. 8m — 62;— /— 3d — 8. 
 Art. 28. 
 
 1. 18a 2 z — G&c+IOz 2 — ax + b — 6. 
 
 2. lOaz 3 — 30a 2 z 2 + 49a 3 2; — 6a 4 . 
 
 3. 26a6 — ]86c 2 + 266 2 + c 2 — wz 2 z + 47. 
 
 4. 7a 2 zy — 86z 2 y + 30czy 2 — 29# 5 . 
 
 5. 18x 2 y 2 + 48zy + 46 2 c 2 + crf 3 + c 3 c? + 
 Art. 29. 
 
 1. am — (be — dm). 
 
 2. abc — a* 2 — (3 my — xy). 
 
 3. a + 6 — (c + d). 
 
 4. a 2_|_62^_2 a 5_( a _|_ & ). 
 
 5 4m 2 +12m7i + 97i 3 — (2m + 3n). 
 
 6. 4a6 + 7am-(x 2 -3f 2 ). 
 
 7. 7m« + x 2 — (pq — y' 3 ). 
 
io key. xi, xn 
 
 SECTION XI 
 
 Art. 31. 
 
 1st, 2d, 3d, performed. 
 
 4. 2<z 2 + 5a — 12. 
 
 5. a 4 + 4M. 
 
 6. 37 a 2 b c* — 12 a 2 e 4 + 22 a c 4 d* — 21 a 2 6 2 _ 2 6 a 6 c 2 d» 
 
 — 8c 4 d 4 . 
 
 7. 126 a 6 c 2 + 30 a 3 6 c 2 — 366 2 c 2 + 108<z 3 c zy 2 — 28a 3 cz 2 y 
 + 33a 3 c 3 + 72&cz# 2 +126cx 2 y — 246c 3 — 24 z 3 # 3 + 2 c 2 z 2 y 
 + 36 c 2 xy* — 3c 4 . 
 
 8. 3a6 6_i0a 5 6 2 + 17a 4 6 3 — I7a 3 6 4 + 8a 2 & 5 — ab 6 
 
 9. 4265 — 2176 3 c 2 — 116 2 c 3 + 86c 4 — St*. 
 
 Art. 33. 
 
 1. a3 + a»6 — 2a6 9 — ac» + 6c8. 
 
 2. a 4 — 4a 4 6 2 + 8a 2 6 3 — 46 4 . 
 
 3. m s — n 8 . 
 
 4. 2a 2 6 + 26 2 c + 2c 3 . 
 
 5. 7z 3 + z 2 y — 12zy 2 — 6?/ 3 . 
 
 6. 10acz-f-2a 2 c + 5a&c + 2z + 6a + 8&. 
 
 7. 4a&c + 2« 2 rf + 26 2 d 
 
 8. 4 a z 2 — 4azy + ac2 + 462;2 — 4 6 z y + 6 c 2 — a 3 c 2 w 
 
 — a c 2 x y -f- a 3 J 2 m -(- a d 2 z y — cPc 2 mz — c 2 z 2 y -f- a 2 d 2 m x 
 + d*x2y. 
 
 9. 13 6 3 c 2 z — 8 6 2 c 3 x 2 + 18 c 5 z 2 . 
 
 10. 1 12 b — 4 6 x + a 2 6 + 448 — 16 x + 14 a 2 + a 2 6 c — 
 7 a 2 c 2 . 
 
 SECTION XII. 
 
 Art. 40. 
 
 1. Performed. 4. 12 6 6 A 3 
 
 2. 4 a 2 ft c J. 5. 25 m 5 ?/ 6 . 
 
 3. 5a 2 6 3 cc/. 6. 31a 2 6 2 ?n 2 . 
 
XIII 
 
 KEY 
 
 11 
 
 7. 3 m 3 a 6 b A y. 
 
 8. « 2 /i 2 w 2 . 
 
 9. 15. 
 
 10. 9a 4 m 4 z 3 y. 
 
 11. Slxy*. 
 
 12. 2 a 4 6 2 . 
 
 13. 3mz 4 y 3 z. 
 
 14. 126 a 4 b 4 d 2 . 
 
 15. 37 mnp 1 . 
 
 16. 23 w 2 r 5 s. 
 
 17. 3z2y°. 
 
 18. 7 6 14 . 
 
 19. 79m 6 ra 5 . 
 
 20. 103 z. 
 
 SECTION XIII. 
 
 Art. 
 1. 
 2. 
 3. 
 4. 
 
 Art. 
 
 1. 
 
 2. 
 
 3. 
 
 4. 
 
 5. 
 
 6. 
 
 7. 
 
 8. 
 
 9. 
 10. 
 11. 
 12. 
 13. 
 
 Art 
 1. 
 
 V 8 ' 
 
 5. 3a 3 & 3 — 1. 
 
 6. Sx^y + l—Gx. 
 
 7. _2c + l+(3a2& 4 c» 
 
 49. 
 
 ab-\-m. 
 \y — 2 x -\- y~. 
 flm2_[_2»i 3 x — 3. 
 6rf— 46c — 262. 
 
 44. 
 
 a" 2 + 2 a m -f- w 2 . 
 
 a3 + 3a 9 x-|-3ax»+s». 
 
 m 3_3 m 2 z _|_3 mx 2__ z 3 # 
 
 2x3— 4x*+8i— Id 
 
 3„2_ a 6_26 2 . 
 4a 2 — 5a6 + 26 2 . 
 
 2z 2 — 3rtz + « 2 - 
 
 a^ + atz + azz + z 4 . 
 
 a 4 -|-4a 3 z+12rt 2 2; 2 + 16ax 3 +16z 4 . 
 
 x 4_ x 3_|_ x 2_ x _|_l. 
 
 — 5a 2 + 4a6 + 36 2 . 
 
 m 6 -f- m 5 x -\- 7ii 4 x l -\- m? x 3 + wi 2 z 4 + ■■ z 5 + z 6 - 
 
 5a 3 6 2 c 3 — 2a 2 6 2 c 4 — 3a6 2 c 5 — 7 6c6. 
 
 46. 
 
 i 8 + z 7 # + * 6 # 2 + z 5 y 3 + z 4 # 4 + z3 y 5 + x<2 y 6 + r y 1 + 
 
 2. 81a 4 +108a 3 & + 144a 2 6 2 + 192a6 3 + 256& 4 . 
 
12 KEY. XIV. 
 
 3. x 9 — x 8 y -\- x 7 y 2 — a; 6 y 3 -|- x 5 y 4 — x 4 y 5 -\- x 3 y G — z 2 y 7 -J- 
 xy 8 — y 9 . 
 
 4. 1 — m -\- m 2 — m 3 -{- m 4 — m 5 -\-m 6 — m 7 . 
 
 5. m 4 — m 3 n -j- m? n 2 — m n 3 -f- n 4 . 
 
 6. m 8 — m 7 -\-m 6 — m 5 -j- m 4 — m 3 -\- m 2 — m -(- 1. 
 
 m 12 
 
 7. 1 + m* + m 4 + nfi + m8 + m 10 + -£— -. 
 
 8. a — axy-\-ax 2 y 2 — ax 3 y 3 -\-ax 4 y 4 — ax 5 y 5 -] % 
 
 1-4-zy 
 Art. 47. 
 
 1. a 2 . 
 
 2. (*+l)(a + 6). 
 
 3. 2(x 2 + a; + l) (a + m). 
 
 4. (« + 6)( z a — zy + y 2 ). 
 
 6. 3m(i 3 -^ 3 ) (m 2 + n 2 ). 
 
 7. 5 (m 2 — 1) (x 4 — 2 3 _|_ X 2 _ x _j_ X ) # 
 
 SECTION XIV. 
 
 Art. 
 
 48. 
 
 
 1. 
 
 abcx 
 m 
 
 
 2. 
 
 b mx -\- 
 
 c mx 
 
 m* + 2 m n + w 2 
 
 5. ^ . 
 
 6 — c 
 
 12 a 4 + 25 a 3 b 
 
 6. 
 
 6 2 + 2«c 
 
 awi -|- bm-j- an -\-bn cfix — b 2 x-\-cfiy — V*y 
 
 6. — . 7. =— j 
 
 c b-\-c 
 
 Acxy-\-Sx 2 y 4a?c 2 — 4b 2 c' 2 
 
 a — x m 2 -f- 2 m n -f- ft 2 * 
 
 50^a:3 + 26fl3; 2 y+125«a; 4 + 65Q2:3 y 
 
 17 6 + 3 c. 
 
 10th and 11th performed. 
 
XIV. KEY. 13 
 
 Art. 40. 
 
 4a + 3xy 4 5 m 2 + 11 qx 2 — 93 
 
 --- i — ^ 6 
 
 12xy — 19x 2 y 2 
 
 n mt — rP + ab 
 
 3. ' . 7. 
 
 a — b 
 
 14ac + 256 2 + 43 _ 
 
 4. x : — zl — m ^ . C5« 
 
 x — y 
 36 c -\-4xy 
 
 a~+b ' 
 
 Sab 
 5(x 2 -*, 2 )' 
 7mx -j-Sb y 
 
 3a + 2 a 2 6 — 6 xy* 2 (a — b) 
 
 9 gx + 6y i 
 
 7 (m 5 — m i y-\-m 3 y' 2 — m 2 y 2 -\-my A — y 5 )' 
 15xy 
 
 10. 
 11. 
 
 2(a + 6)(z 4 — x 3 # + x 2 3, 2 — xy 3 -j-y 4 )' 
 a-\-b cm 
 3(c + d)' 
 
 12 Zac+xy 
 
 (m -j-n)(x 4 — x 3 y -\- z 2 y 2 — x y 3 -|- y 4 )' 
 13th and 14th performed. 
 
 15. a + b. 17. 3a 2 6c + 4xy. 
 
 16. ro 2 + 2mc + c 2 . 18. 46cm — 12z 2 y 
 
 Art. *50. 
 
 a? cm (fix^y* 
 
 1. • "• ?c ^ . 
 x y » w 
 
 cPm „ b&y — bcdy 
 
 2. — j— . 7. . 
 
 a x — t» 
 
 „ x 4 y 4 5a 2 c + 4«&c 
 
 3. — ~. 8. . 
 
 ac 2 ?( x — y) 
 
 7a*p*q' i g a 2 + 2ax + x 2 
 
 x?/ 3 2(«-|-??i) 
 
 3az-[-36z a 6c m 2 +a 2 x-f-6 2 c ffl 3-{-a 6 x 
 
 n ab (m~ -\- t J 
 
14 KEY. XV 
 
 SECTION XV 
 
 Art, til. 
 1. 
 
 a. if. t. s±» 
 
 116c x ~ry 
 
 3 26 2 y g a 2 + fl6 + 6 2 
 
 4 »t 2 — ra 2 * ' 36c — x y 
 
 4 17 o »*» + 8« 
 
 ' 6c + 4a:y' ' «6 + 74 2 ' 
 
 5. f±*. 10. 6a; - 7 
 
 llm + 4/i 2 
 11. Performed. 
 
 Art. 52. 
 
 1. — 7. 3 + 7 * 
 
 6c»* 4^) -\-±q-\-bcp-\-bcq 
 
 * 6 2 — c 2 ' z 3 — y 3 ' 
 
 m + n 2m 2 + 3a; 
 4a? + 4z a y 6 2 + c 2 * 
 
 3a6 + 46 g « + ai y + ya 
 9fc 2 — 16 m 2 * « 2 
 
 x-f-y i « m3 — m 2 n -f- w n* — « 3 
 
 G. 
 
 m -\- ri x -|- y 
 
 *+y 
 
 « + 6 
 Art. 53. 
 
 L 3**** 5 a 3 6 3 
 
 2 ?M? 4 -L 
 
 * 7a6* 7atj* 
 
XVI. KEY. 15 
 
 7m 2 rc 8 2(x — y)(a + 6) > 
 
 ra 
 
 6. , 0. 
 
 2 (m 2 -f m n + rc 2 ) 
 
 4x-|-4y 5azy 
 
 s — y 
 21mrc* 
 
 SECTION XVI. 
 
 Art. 54. 
 
 1. Performed. 
 
 2. a(6 + c — 2m). 
 
 3. 2x(2a + y + 6). 
 
 4. 5x 2 (5 + 6y — 3m). 
 
 5. 3 3 a6c(3 + a6 + 2a 2 ). 
 
 6. 3 2 m 2 (llxy + 12pa + 2r )- 
 
 7. 2 2 .3a6(l+2c — 3x). 
 
 Art. 57. 
 
 1. 1, a, a 2 , 6, 6 2 , a 6, a 2 6, a6 2 , a 2 6 2 . 
 
 2. 1, a, a 2 , 6, a 6, a 2 6, 6 2 , ah* 2 , a 2 6 2 , c, ac, cfic, be, abc, 
 a 2 b c, 6 2 c, a 6 2 c, a 2 6 2 c, c 2 , a c 2 , a 2 c 2 , 6 c 2 , a b c 2 , a 2 b c 2 , 6 2 c 2 , 
 a6 2 c 2 , a 2 6 2 c 2 , c 3 , ac 3 , a 2 c 3 , 6c 3 , a 6c 3 , a 2 6c 3 , 6 2 c 3 , a6 2 c 3 , 
 a 2 6 2 c 3 . 
 
 3. 1, 3, 9, x, 3x, 9x, r« 3x 2 , 9x 2 , y, 3y,9y, xy, 3xy,9xy, 
 x» y, 3 x 2 y , 9 x 2 y , y 2 , 3 y 2 , 9 y 2 , x y 2 , 3 x y 2 , 9 x y 2 , x 2 y 2 , 3 x 2 y 2 , 
 9 x 2 y 2 , y 3 , 3 y 3 , 9 y 3 , x y 3 , 3 x y 3 , 9 x y 3 , x 2 y 3 , 3 x 2 y 3 , 9 x 2 y 3 . 
 
 4. 1, 2, 5, 10, 25, 50, m. 2m, 5m, 10m, 25m, 50m, m 2 , 2m 2 , 
 5 m 2 , 10 m 2 , 25 m 2 , 50 m 2 . 
 
 5. 1,2, 3, 6, 5, 10, 15, 30, 25, 50, 75, 150. 
 
 6. 1, 2, 4, 3, 6, 12, 5, 10, 20, 15, 30, 60, 13, 26, 52, 39, 78, 
 156, 65, 130, 260, 195, 390, 780. 
 
 7. 1,2, 4, a, 2a, 4a, 6, 2 6, 4 6, a6, 2 a 6, 4 a 6, a + 4, 
 2(a + 4),4(a + 4),a(a + 4),2a(a + 4),4a(a + 4), 6(a + 4) 
 26(a + 4), 46(a + 4),a6(a + 4),2a6(a-L4), 4a6(a-|-4). 
 
16 key. XVLI, XV III. 
 
 8. 1,5, a, 5a, 2ft + 5m, 5(26 + 5m), a(2ft + 5m), 
 5a(2ft + 5m). 
 
 SECTION XVII. 
 
 Art. *59. 
 
 1. 4a 2 ftc. 6 x _^ % 
 
 2 - 5a. • 7. z3 — 8a; — 3. 
 
 3- 15 a. 8. 2a + 2x. 
 
 4 - i» 2 + l. 9. a — 6. 
 5 x + 1. 
 
 SECTION XVIII 
 
 Art. 61. 
 
 1. Performed. 
 
 2. 36 a 3 m 4 ?/ 3 . 
 
 3. 150 a 3 m 3 a; 3 . 
 
 4. max 2 if. 
 
 5 (12aft(2ft + c), or 
 1 24 a ft 2 + 12 a b c. 
 27m 2 x 4 (l+2n 
 2? m 2 a; 4 + 54 m 3 s* 
 
 ( 27 m 2 a; 4 (1+2 mx), or 
 t 27 m 2 a; 4 + 54 m 3 s* 
 i{a + bf{a-b), or 
 < a 3 + o 2 ft — aft 2 — ft 3 . 
 
 8 i(« + 6)(«— &)(« 2 +«&+ft 2 ), 
 * a 4 + a 3 ft — aft 3 — ft 4 . 
 
XIX KEY. 17 
 
 Art. 63. 
 1. Performed. 10 
 
 SECTION XIX. 
 
 26 + 13m 2 
 
 144 a 
 
 2. I*. 11. ' 
 
 5y' " 3fli-2a 2 ' 
 
 3a 3 c 3am 2 
 
 t>. "^ ^. X/i, 
 
 2m 2 ' * a 2 — 2m 2 
 
 4. ^^ . 13. Performed. 
 15xy 2 
 
 5. = *. 14. * y 
 
 12 mxy' 2 x ~\~y 
 
 6. 4-. 15. a -±\. 
 t a 6 m a — o 
 
 „ aro + 35x %a 5(a-f6) 5a + 56 
 
 7. ^ . 16. V Q ' , or — ^ 
 
 8 . *y+ 2 . 17# « 2 +«*+* 2 
 
 3 x 2 y 2 ' a 
 
 x 
 
 9 25 an c^ 3(x-y) 
 
 5a6c ' 4(z 2 — iy + y' 2 ) 
 
 19. 9"^ 
 
 5 (x 3 — x 2 y-|-xy 2 — y 3 )' 
 
 iRT 64. 
 
 1. Performed. 5. 
 
 + n 
 2. J2-. ft »■ 
 
 2a6c* * 2xy + 3y 2 ' 
 
 3. JL. 7 . i 
 
 2xy' ' 2(x — y)' 
 
 c 
 4~^ 2 ' 
 
 8. * 
 
 1 
 
 9. 
 
 2* 
 
 3(x 2 + xy + y 2 )' 
 
IS 
 
 XX, XXI, 
 
 Art. ©*5. 
 4ab 4 
 
 1. 
 
 2. 
 
 SECTION XX. 
 
 9x 
 
 21c 
 
 48 S 3 ?/ 3 
 
 35 6 c m 2 * 
 
 12 s 3 + 9 & 2 s 2 
 76 + 9m 2 ' 
 J 8z 2 + 2z 
 4 ' 7— * 
 
 5. 
 
 6. 
 
 3a 3 — 3a 2 6 
 
 56 
 • 3s — 4 
 4 a x 2 — 6 a 
 
 5 c x 2 y 
 
 9. 
 10. 
 11. 
 12. 
 
 y 
 
 Slab ex — 21b ex* 
 
 
 a 
 
 • 
 
 4 
 
 ah c x -f- 46 c 
 
 z 2 
 
 3 
 
 a?m? — 3a?n 2 
 
 2; 
 
 3 
 
 a 2 m x 3 + 3 a 2 
 
 #z y 3 
 
 SECTION XXI, 
 
 Art. 68. 
 
 A w a 
 _ 29 a y + 35 x 
 
 24a6 2 -f95 3 s + 22 a 8 y 
 42 a 2 6 3 
 
 35y 
 
 6 2 + 9& 3 
 42 a 2 
 27 q* x 2 y + 6 a b p* + 2 p x y* 
 
 12 p 3 g 2 
 5 a x y z -|- 5 b x y~ -f- 33 s 2 y -f 117 a 
 
 225 x 3 w 2 
 17 a — 4 5 
 162 ' 
 
XXI. KEY. 
 
 63 m re 4 x y + 70 b c n 3 + 60 h x 
 
 525 77*3 w 4 
 
 150 m x* y 2 + 70 b c x \f + 35 c 2 y 2 + 56 a x» 
 8t 350x3^2 
 
 105 a 3 + 364 a 2 + 390 6 
 9 ' 910 a 
 
 io. *££ 
 
 1 X 4 
 
 i.i. 2 
 
 12. 
 13. 
 14. 
 15. 
 16. 
 17. 
 18. 
 19. 
 20. 
 21. 
 22. 
 
 1 — x 2 
 
 3 + 7a + 76 
 
 a* — b* ' 
 3a 2 6 + 6a 2 + 56 2 »i 
 
 4a& 3 + 8a& 2 
 14 6 — 9 am 
 
 12 m 2 
 3a 2 -}- life 2 
 
 2 
 
 196 
 
 a ex 
 
 /j2 — C 2* 
 
 4x2 + 3 
 
 17 
 
 3x * 
 
 «2_26 
 
 252 
 
 32 x — 3 ax + 7a 
 
 56 a 2 
 4a 2 x — Sax 2 — 6as 
 21a — 42x 
 
 4x 2 
 
 1 — X 4 
 
 b c + mjpx + m j>y 
 
2C key. XXII. 
 
 23 
 
 5a+36 
 4 * 
 
 9A 
 
 11 m 3 4 7 m n 2 — 2 m 2 w + ra 3 
 
 
 7m — 2w 
 
 25. 
 
 4« 2 — 2 6 2 
 
 a 2 + 6 2 
 
 „ fi 12 m 2 x y — 3 a -\- 7 b 
 
 4Lx y 
 27 5 h 3 n 3 x -\~ h 3 n 3 
 
 28. 
 
 x+l 
 7«6 + 4c 2 — 2m 3 
 
 m 
 
 29 9 x ~\-4b c — a 2 — ab — ax — bx 
 
 a-\-x 
 
 x -\-n 
 x — y 
 
 30. * 2 + y 2 — mx+my + x — y 
 
 SECTION XXII, 
 
 Art. 
 
 ©9. 
 
 
 3m 
 
 1. 
 
 ~sT 
 
 2. 
 
 7ac 3 
 3 ' 
 
 3. 
 
 4rc(x 2 + # 2 ) 
 
 m 2 
 
 4. 
 
 5c(a + 5) 
 2 
 
 5. 
 
 5m 
 
 6. 
 
 7. 
 8. 
 1). 
 
 a 3 a: 5 
 Wh*7 3 ' 
 
 May 
 
 bmx' 
 4bx 
 81a 3 mp' 
 W(x + y) 
 3 
 
 » * 
 
XXIII. KEY. 2l 
 
 9msV(l + 36) 
 1D ' 4(26 c + 6) * 
 31a(22m+j?) 
 429 6 jp 2 
 
 18 l " bc 
 
 1 1 
 
 4(*-3). 
 
 11 
 
 5 * 
 
 1° 
 
 9z — 3 
 
 
 X 
 
 13. 
 
 4m 
 ab' 
 
 14. 
 
 1 my ' 
 
 i* 
 
 2m 
 
 19. 
 
 20. 
 
 11 % 2 {m — p) 
 
 ox(m—l)' 
 56c 2 
 
 3x 2 (z- y)' 4(z+y) 
 
 SECTION XXIII 
 
 Art. 70. 
 
 1. Performed. 
 a 2 + 6 2 c 
 
 2. 
 
 a-\-bc 
 
 Sbcd — ad-\- be 
 
 6 + 6 dm 
 bem + c^m — 36 2 + 66c 
 8c — 46 + 6c? + cd ' 
 
 2a6 + 36c 
 
 5wi 2 + 3a + 12c* 
 a c 2 — ab c-\-bcd — ftd 
 a? — b* + bc ' 
 
 c' 
 
 76 2 c+ 150 ^ — 336 + 6 abd 
 Sad-\-7bc — 33 
 
 616 — 30 a 
 
 186 — 9fl' 
 
22 key. XXIV, XXV 
 
 SECTION XXIV. 
 
 Art. 74. 
 
 1. 1st 8; 2d 3. 
 
 2. 1st $12; 2d $16. 
 
 3. Cow $25 • ox $35. 
 
 4. A $12; B $20. 
 
 5. Potatoes 2s.; corn $1 per bushel. 
 
 6. Wheat 10 bushels ; rye 8 bushels. Sold 5 and 3. 
 
 7. Father's $1 ; son's 2s. 
 
 8. f 
 
 9. T V 
 
 10. He bought 200 lbs. of coffee and 300 lbs. of tea; he sold 
 160 lbs. of coffee and 200 lbs. of tea. 
 
 11. 200 eggs at 2 for a cent ; 100 at 5 for 3 cents 
 
 12. Smaller 75, larger 100 gallons. 
 
 13. Wheat 9s., oats 3s. per bushel. 
 
 14. Larger 160, smaller 80 gallons. 
 
 15. Upper part 15 feet ; lower part 25 feet. 
 
 16. A $500; B $700. 
 
 SECTION XXV. 
 
 Art. 75. 
 
 1st, 2d and 3d performed. 
 
 4. Flour $7, rice $8 per barrel; sugar $10 per box. 
 
 5. 1st 2 ; 2d 3 ; 3d 5. 
 
 6. A $2; B$0-75; C $050. 
 
 7. 1st $500; 2d $600; 3d $800. 
 
 8. A's 60, B's 80, C's 90 years. 
 
 9. 1st 5s. ; 2d 6s. ; 3d 7s. ; 4th 8s. 
 
 10. A $0-30, B $040, C $050, D $060 per lb. 
 11 Wheat $2, rye $1, barley $1£, oats $£, potatoes $± per 
 bushel. 
 
XXVI, XXVII. key. 23 
 
 SECTION XXVI. 
 Art. 76. 
 
 1st and 2d performed. 
 
 18. 
 
 425. 
 
 34. 
 
 1639. 
 
 3. 
 
 2100. 
 
 19. 
 
 90. 
 
 35. 
 
 561. 
 
 4. 
 
 10500. 
 
 20. 
 
 48. 
 
 36. 
 
 561. 
 
 5. 
 
 12600. 
 
 21. 
 
 880. 
 
 37. 
 
 121. 
 
 6. 
 
 4500. 
 
 22. 
 
 1011 
 
 38. 
 
 8470. 
 
 7. 
 
 35000. 
 
 23. 
 
 4. 
 
 39. 
 
 4. 
 
 8. 
 
 f 
 
 24. 
 
 —6. 
 
 40. 
 
 2. 
 
 9. 
 
 21. 
 
 25. 
 
 112. 
 
 41. 
 
 24. 
 
 10. 
 
 140. 
 
 26. 
 
 180. 
 
 42. 
 
 o. ; 
 
 • 11. 
 
 ». 
 
 27. 
 
 115. 
 
 43. 
 
 18. 
 
 12. 
 
 A- 
 
 28. 
 
 510. 
 
 44. 
 
 1. 
 
 13. 
 
 18. 
 
 29. 
 
 187. 
 
 45. 
 
 1. 
 
 14. 
 
 37. 
 
 30 
 
 33. 
 
 46. 
 
 2. 
 
 15. 
 
 47. 
 
 31. 
 
 — 33. 
 
 47. 
 
 f- 
 
 16. 
 
 100. 
 
 32. 
 
 3. 
 
 48. 
 
 — 1. 
 
 17. 
 
 220. 
 
 33. 
 
 187. 
 
 49. 
 
 h 
 
 The 
 
 riven. 
 
 ) remaining answers in this article 
 
 being indefinite, are not 
 
 SECTION XXVII, 
 
 4rt. 77. 
 
 
 1st and 2d performed 
 
 11th, 12th and 13th performed. 
 
 3. 60 and 90. 
 
 14. 9 oxen, 45 cows, 315 sheep. 
 
 4. 90 and 140. 
 
 15. Performed. 
 
 5. 540 and 660. 
 
 16. 2-f T days. 
 
 6. 11 and 16. 
 
 17. 3-^y hours. 
 
 7. 16 and 19. 
 
 18. Performed. 
 
 8. 33£ and 36£. 
 
 19. A $200; B$150; C £100 
 
 9. 17£ and 30f 
 
 W. A $54; B $27; C $18. 
 
 10. 32| and Gfy 
 
 21. $200; $150; $100; $50. 
 
24. KEY. XXVII. 
 
 22. A $128; B $80; C $64 
 
 23. Performed. 
 
 24. $691875. 
 
 tr 
 
 26. $75. 
 
 27. r=— . 
 
 tp 
 
 28. -055 or 5£ per cent. 
 
 29. t = — . 
 
 30. 7 years, 1 month. 
 
 31. Performed. 
 
 32. $8620625. 
 
 33. p = — — . 
 
 34. $250. 
 
 tp 
 
 36. *065 or 6£ per cent. 
 
 37. t = a -=P. 
 
 rp 
 
 38. 3£ years. 
 39 " TXT' 
 
 \-\-tr 
 
 40. ,£1 4s. 8Jd. 
 
 1 2 
 
 41. £ = - , doubled ; t = -, tripled, 
 
 r r 
 
 42. 16f years, doubled ; 33J years, tripled. 
 
 43. 20 years, doubled ; 40 years, tripled. 
 
 . . a an 
 
 44. 
 
 45. 1st — — ; 2d 
 
 46. 
 
 -\-n m -\- n 
 
 a am 
 
 \A-m-\-n' \-\-m-\-n i \-\-m-\-n 
 
XXVII. KEY. 25 
 
 bed — ab ac — bed 
 
 47. 
 
 48. 
 49. 
 
 c — 6 ' c — b ' 
 am-\-bmn an — bmn 
 
 m -f- n ' m-\-n 
 mnp 
 n — rri 
 
 amn 
 
 50. guineas. 
 
 mn — m — n ° 
 
 51. ^ dollars. 
 
 nq — mq — np 
 
 ~~ ab , 
 
 52. -, days. 
 
 6 — a 
 
 53. ±L. 
 
 5a g(4 + 6) g(3 + 26) a (2 + 36 ) a{l+ 4b) 
 54 - 6_i ; 6—1 ; 6 — 1 ; 6 — 1 ; 6—1 J 
 
 5ab r 
 
 6 — 1' 
 
 55. —— shillings. 
 
 ~ n t,t , c-4-bn . ...an — c . 
 
 56. Wrought — ~ — =- days : idle ; — r- days. 
 
 to a + 6 J ' a-\-b J 
 
 57. — ! — beggars. 
 
 , Q A , « + & B , « — b 
 
 58 As — - — years ; Bs years. 
 
 59. Boots dollars; shoes dollars. 
 
 ac — 1 ac — 1 
 
 be — 1 be — 1 
 
 ivt ap(m-\-n) . . an(p-A-q) 
 
 61. Numerator * v './ ; denominator l£_i_l/. 
 
 mq — np mq — np 
 
 __ __ ap(m-\-n) . an(p-\-q) 
 
 t)2. Numerator -*-* ! — L \ denominator vr ' -. 
 
 np — my rap — mq 
 
 3 
 
26 KEY XXVIII. 
 
 63. Particular answer to 61st f ; particular answer to 62d |£. 
 
 SECTION XXVIII. 
 
 Art. 84. 
 
 From the 1st to the 5th inclusive, performed. 
 
 6. This question as enunciated, if x represents the number, 
 
 gives 
 
 7x 5i . . 
 
 From this equation we find 
 
 x — _350. 
 
 Changing the sign of x in the original equation, we have 
 
 7x 5x . m 
 — : — — — -\- 5 ; or, by transposition, 
 
 To rectify the question, therefore, we enunciate it thus • 
 What number is that % of which exceeds y 7 ^ of it by 5 ? 
 
 7. This question gives the number of years =. — 40; and the 
 equation being modified accordingly, we find that the question 
 should have been : 
 
 How many years after marriage was his age to hers as 7 to 6* 
 
 x 
 
 8. If - represent the fraction, Hhe question gives 
 
 y 
 
 From these equations we find 
 
 as = — 3, and y = — 13. 
 
 Changing the original equations, in a manner similar to that 
 
 pursued in the solution of the 5th question, we have 
 
 % x 4-2 
 
 = i; and — - — =. -j- 5 ^; from which the modifica* 
 
 y + 2 -5 y 
 
 ♦ion to be made in the enunciation is manifest. 
 
XXIX. KEY. 27 
 
 9. Let x = the greater, and y = the less ; then, 
 
 x — y = 20, and 6x — 3y = 96. 
 These equations give 2; — 12, and y = — 8. 
 Changing the sign of y in the original equations, we have 
 
 x -\- y — 20, and 6 x + 3 y = 96 ; and the ques- 
 tion should be changed accordingly. 
 
 10. Let x and y represent the number of gallons, which flow 
 through the cocks A and B, respectively, in a minute. Then, 
 
 5x + 3y = 24, and7x + 5y:=32. 
 These equations give x = 6 gallons, and y =z — 2 gallons. 
 Changing the sign of y in the original equations, we have 
 
 5x — 3y = 24, and 7x — 5y =32. 
 Hence it appears that water flows out, instead of flowing in, 
 through the cock B. 
 
 11. A's$5000; B's$3000; C's — $2000. 
 
 Hence, either C is in debt $2000 ; or he possesses $2000, and 
 a certain number of times his estate is subtracted, in each case, 
 instead of being added. 
 
 SECTION XXIX. 
 
 Art. 85. 
 
 1. Let x represent his money in cents. Then 
 
 ^ + 40 = ^ + 49. 
 
 This equation gives 
 
 3x — 3x = 9; that is, (3 — 3)x or 0.x = 9; therefore, 
 
 T 9 
 
 X jj. 
 
 Hence, the question is impossible, or the value of x is infinite. 
 
 2. Let x = A's age ; then, x + 10 = B's, x + 20 = C's, and 
 
 1 — 34 = D's. We have, therefore, 
 
 s + 10 2z + 40 5x-170 
 
 2 +3 -\ e -ix-W. 
 
28 KEY. XXX. 
 
 This equation gives 
 
 12s— 12z or 0.z:=60 — 60; and 
 * = &. 
 
 Hence, the value of x is indeterminate; consequently, their 
 ages cannot be ascertained from the conditions. 
 
 3. If x represent the number killed to-day, and y the number 
 killed yesterday, we have 
 
 x y ; • . 3a; 
 
 These equations become, by multiplication and transposition, 
 
 3 x — 2 y = 30, and 3 x — 2 y = 10, in which the first 
 
 members are identical, while the second members are different. 
 
 Hence, the two conditions are incompatible with each other. 
 
 By subtracting and canceling the terms containing y, we have 
 
 3 x — 3 x = 20, or . x = 20, and x = %<*. 
 Or, by subtracting and canceling the terms containing x, we 
 have 
 
 2y — 2y = 20, or . y = 20, andy = % -. 
 The problem, therefore, is impossible; or the values of x and 
 y are infinite. 
 
 Other forms for the answers may be obtained by different 
 methods of elimination ; but they are all essentially the same, 
 because they are all infinite. 
 
 SECTION XXX 
 
 Art. 86 
 
 1. 28. 
 
 2. 53. 
 
 3. 78. 
 
 4. 29. 
 Art. 87. 
 
 1. 37. 
 
 2. 49 
 
 5. 
 
 36. 
 
 
 6. 
 
 93. 
 
 
 7th and 8th 
 
 performed. 
 
 3. 
 
 19. 
 
 
 4. 
 
 351. 
 
 
X3 
 
 :i, xxxn. 
 
 KEY. 
 
 
 5. 
 
 709. 
 
 9. 
 
 1197. 
 
 & 
 
 109. 
 
 10. 
 
 12325. 
 
 7. 
 
 190. 
 
 11. 
 
 30021. 
 
 8. 
 
 1049 
 
 12. 
 
 50309. 
 
 29 
 
 SECTION XXXI 
 
 Art. 
 
 89. 
 
 1. 
 
 t- 
 
 2. 
 
 h 
 
 3. 
 
 * 
 
 A.RT. 
 
 90. 
 
 1. 
 
 1- 
 
 2. 
 
 *-> 
 
 3. 
 
 A-- 
 
 Art. 
 
 93. 
 
 1. 
 
 1-4142 +. 
 
 2. 
 
 51962—. 
 
 3. 
 
 5-8095—. 
 
 4. 
 
 12 1370 +. 
 
 5. 
 
 58676—. 
 
 6. 
 
 180537 +. 
 
 4. 
 
 tf 
 
 5. 
 
 M- 
 
 6. 
 
 Mr 
 
 4. 
 
 A+. 
 
 5. 
 
 B+. 
 
 6. 
 
 e-. 
 
 7. 
 
 0-8528 +. 
 
 8. 
 
 0-8292 — . 
 
 9. 
 
 0-8660 +. 
 
 10. 
 
 003937 +. 
 
 11. 
 
 0-039 15+. 
 
 12. 
 
 201008 +• 
 
 ECTION XXXII. 
 
 Art. 93. 
 
 1. A's 21, B's 27 years. 
 
 2. A 10, B 12 miles per hour. A went 1 )0, B 120 miles. 
 
 3. 5 yards and 6 yards. 
 
 4. 10 rows ; 12 trees in a row ; 960 bushels. 
 
 5. 30 miles. 
 
 6. 50-662 -f inches. 
 
 3* 
 
30 KEY. XXXIII 
 
 7. 12-649 + rods. 
 
 8. 43-818— rods. 
 
 9. 16. 
 
 10. Greater 60; less 15. 
 
 11. Length 30, breadth 25, height 20 feet. 
 
 12. 37 yards in each ; prices 42s. and 32s. per yard. 
 
 13. 30 persons; $150 each. 
 
 14. Greater 57 ; less 45. 
 
 15. $150. 
 
 SECTION XXXIII 
 
 Art. 95. 
 
 1. By the 2d statement, 9 yards. 
 Art. 97. 
 
 1st and 2d performed. 
 
 3. 15 rods. 
 
 4. 9 rods ; $3 per rod. 
 
 5. Greater 90 ; less 85. 
 
 6. Length 100, breadth 60 rods. 
 
 7. 10 sheep ; 13 calves. 
 
 8. 40 hours ; 4 miles per hour. 
 
 9. $100. 
 
 10. 80 feet. 
 
 11. Coffee $0-10, tea $0*50 per pound. 
 
 12. 4 rods. 
 
 13. 225. 
 
 14. 5 sons. 
 
 15. 12 fowls. 
 
 16. A's stock $100. 
 
 17. 2 rods. 
 
 8. 10 gallons. 
 
XXXIV, XXXV, XXXVI. key. 
 
 SECTION XXXIV 
 
 Art. lOO. 
 
 1. Performed. 6. 335. 
 
 2. 67. 7. 364. 
 
 3. 39. 8. 909. 
 4 9L 9. 3009 
 
 5. 77. 
 
 SECTION XXXV. 
 
 Art. 101. 
 
 1. i- 
 
 2. T V 
 
 4. 
 5. 
 
 23 
 
 27* 
 
 ag. or 4}. 
 
 3. T V 
 
 
 
 Art. 102. 
 
 i. H+- 
 
 3. 
 
 AV+- 
 
 2. iM— 
 
 
 
 Art. 103. 
 
 1. 1-2599+ or 1260— . 
 
 2. 19129+ or 1913— . 
 
 9. 
 10. 
 
 0-956 +. 
 0-7688+ or 0769 
 
 3. 4-8629+ or 4863—. 
 
 11. 
 
 1-957 +. 
 
 4. 11447+ or 1145—. 
 
 12. 
 
 2-3339 + or 2334 
 
 5. 2951 +. 
 
 6. 0-292 +• 
 
 7. 2-3129+ or 2-313—. 
 
 8. 10-790+. 
 
 13. 
 14. 
 15. 
 16. 
 
 1542+. 
 0183 +. 
 2044 +• 
 
 2-823+. 
 
 SECTION 
 
 XXXVI 
 
 Art. 104. 
 
 
 
 1. 4, 6 and 10. 
 
 
 
 2. Length 9, breadth 5, depth 7 feet. 
 
32 key. XXXVII, XXXVIII 
 
 3. 5 and 10. 
 
 4. Length 30, depth 8, breadth 22 feet. 
 
 5. 16 feet. 
 
 6. 18 feet. 
 
 7. 6 and 12. 
 
 8. 22-071+ inches. 
 
 9. Side of the base 31*072+ feet; altitude 124288+ feet. 
 
 10. Radius 2 feet; length 7 feet. 
 
 11. 1-8837+ feet. 
 
 SECTION XXXVII, 
 
 1. 
 
 49 a 2 m 4 . 
 
 2. 
 
 64 6 4 c 2 z 8 . 
 
 3. 
 
 225 a 8 m 6 p u . 
 
 4. 
 
 128z 14 3/ 21 . 
 
 5. 
 
 &26 c 78^91. 
 
 6. 
 
 1024 630 C20J90. 
 
 7. 
 
 Performed. 
 
 8. 
 
 p mn q mt . 
 
 9. 
 
 Performed. 
 
 10. 
 
 Z m p mX q my . 
 
 11. 
 
 81 »V. 
 
 12. —32x^35. 
 
 13. — 343aH 6 c3^. 
 
 14. 64 a 6 m 18 rc 12 jp 30 ? 42 x 6 y* 
 
 15. Performed. 
 
 16. ^ 
 17. 
 18. 
 
 125m 6 rc 3 
 216 &lp 
 1 
 
 81p 8 x 
 
 8- 
 
 
 SECTION XXXVIII 
 
 Art. 
 
 106. 
 
 1. 
 
 2. 
 3. 
 4. 
 5. 
 
 (6m — n-\-pY 
 
 (b + c + d)™. 
 
 (4a6 + 4zy) 14 . 
 
 (x-yf* 
 
 (a + 6) 2 (a — 6)6 
 
XXXIX. key. 33 
 
 6. 243 (x — y) 15 (p— qf 5 . 
 
 7. S(a + b + c)^. 
 
 8. a*m*\c — d)* m (z + yy\ 
 
 9. (a + 6+c) 3OT . 
 
 10. ( a +bf n {c — df n . 
 
 11. (z + 2#r. 
 
 12. (a + x) m? (n — x) mq . 
 13th and 14th performed. 
 
 (b + c)* 
 
 15. 
 16. 
 17. 
 18. 
 
 (ra + rc) 6 
 
 (i> + ?) 9 * 
 
 l 6 (ft + &) 4 (s — y) 8 
 
 81 (C + d) 32 ' 
 
 a 7( m ^_ w )Ql( a: _y)35 
 
 V( r + S y*(a + b + c)2 1 ' 
 
 Art. 107. 
 
 1. »i 2 — 2»ix + x 2 . 
 
 2. a 3 c + 3a 2 &c + 3a6 2 c + 6 3 c. 
 
 3. o 2 »z-|~2«^»i + ^ 2 ^ + 2acm-|-26cm-|-c 2 /M+a a w f- 
 2a6»+6 2 w + 2acra+26cn + c 2 w. 
 
 4. a 3 x 2 -\- 2 a 3 a: y -f- a 3 ?/ 2 . 
 
 5. 4c 2 +12cfl? + 9d 2 . 
 
 G. 
 
 c 3_j_ 3c 2^^.3 c ^_|. rf 3- 
 
 SECTION XXXIX. 
 
 Art. 113. 
 
 1. a 7 + 7a 6 6 + 21a 5 6 2 + 35a 4 6 3 + 35a 3 6 4 + 21« 2 6 5 ^ 
 7 a 6 6 + b\ 
 
 2. x 10 + 10x 9 y + 45x 8 y 2 + 120x7y 3 + 210xV + 252x 5 y 
 + 210 x 4 3/6 _|_ 120 x 3 if + 45 x 2 y 8 + 1 x y» + y ™. 
 
 3. m 5 — 5 ra 4 w + 10 m 3 w 2 — 10 m 2 n 3 + 5 m n* — n 5 . 
 
34 key. XXXIX. 
 
 4. 6 11 + 116 10 c + 55&9c 2 + 165& 8 c 3 + 330 6 7 c 4 + 462# c 5 
 + 462 b 5 c 6 + 330 M c 7 + 165 b 3 c 8 + 55 6 2 c 9 + 1 1 6 c 10 + c 11 . 
 
 5. a;l8 — 13 x 12 y + 78 x 11 y* — 286 x 10 y 3 + 715 x 9 y 4 — 
 
 1287 z 8 y5_|_ 1716 z 7 yi_ me 2 6y7_j_ 1 287z 5 y 8 — 715x 4 y 9 + 
 286 x 3 y 1 ° — 78 x 2 y^ + 13 a; y 12 — y 13 . 
 
 6. Performed. 
 
 7. x5 + 10x 4 y + 40x 3 y 2 + 80x 2 y 3 + 80xy 4 + 32y5. 
 
 8. 216 a 3 + 540 a 2 x + 450 a x 2 + 125 x 3 . 
 
 9. Performed. 
 
 1 0. 32 a 5 + 240 a 4 x + 720 a 3 x 2 + 1080 a 2 x 3 + 810 a x 4 + 
 243x5. 
 
 11. 64 x 3 — 144x 2 y + 108x3/ 2 — 27y 3 + 48«x 2 — 72axy 
 + 27ay 2 + 12 a 2 x — 9 a 2 y + a 3 . 
 
 12. a 3 + 3 a 2 b + 3 a 6 2 + b 3 + 3 a 2 c + 6 a 6 c + 3 5 2 c + 
 3 a 2 d + 6a&d + 3 6 2 d+3a c 2 +6acd + 3ad? + 3b c 2 + 
 66c^+36fZ 2 + c 3 + 3c 2 ^ + 3crf 2 + ^. 
 
 13. a 6 + 12 a 5 6 — 6 a 5 c + 60 a 4 6 2 — 60 a 4 b c + 15 a 4 c 2 
 + 160 a 3 b 3 — 240 a 3 6 2 c + 120 a 3 6 c 2 — 20 a 3 c 3 + 240 a 2 b 4 
 
 — 480 a 2 6 3 c + 360 a 2 6 2 c 2 — 120 a 2 6 c 3 + 15 a 2 c 4 + 192 a b 5 
 
 — 480 a 6 4 c + 480 a 6 3 c 2 — 240 a 6 2 c 3 + 60 a 6 c 4 — 6 a c 5 + 
 64 &6 __ 192 65 c _|_ 240 b 4 c 2 — 160 b 3 c 3 + 60 6 2 c 4 — 12 6 c 5 
 + C e. 
 
 14. a$ + 5 a 4 b + 10 a 3 ^ + 10 a?b 3 + 5 ab 4 + b$ — 10 a 4 c 
 _40a36 c _ 6 0a 2 6 2 c— 40a6 3 c— 15a 4 d— 60a 3 6d — 90a 2 6 2 d 
 
 — 60«6 3 rf — 15 6 4 d + 40a 3 c 2 + 120 a 2 6 c 2 + 120«6 2 c 2 + 
 40b 3 c* + 120 a 3 cd + 360a* bed + 360 ab*cd + 120b 3 cd + 
 90 a 3 d? + 270 a 2 6 d 2 + 270 a 6 2 d 2 + 90 b 3 d> — 80 a 2 c 3 — 
 360a 2 c 2 rf — 540 a 2 cd 2 — 270 a 2 d 3 — 160a6c 3 — 720a6c 2 d 
 — 1080 a6 erf 2 — 540 a 6 d 3 — 80 6 2 c 3 — 360 6 2 c 2 rf— 540 6 2 c^ 
 
 — 270& 2 d 3 + 80ac 4 + 480ae 3 rf+1080ac 2 d 2 + 1080acd 3 
 + 405 a d 4 + 80 b c 4 + 480 bc 3 d + 1080 b c 2 <P + 1080 b c d 3 
 + 405 b d 4 — 32 c 5 — 240 c 4 rf — 720 c 3 d 2 — 1080 c 2 rf 3 — 
 8l0crf 4 — 243^5. 
 
 Erratum. In 2d line of 14th, insert — 10 b 4 c. 
 
XL, XLI, XLII. key 
 
 SECTION XL 
 
 Art. 115. 
 
 1. 5. 
 
 35 
 
 4. 
 
 73. 
 
 5. 
 
 211 
 
 6. 
 
 13. 
 
 2. 12. 
 
 3. 73. 
 
 SECTION XLI. 
 
 Art. 117. 
 
 1. ±a?m*z. 7 3s 3 y 
 
 2. ±8*y. • ' 4m 4 * 
 
 3. 7a 3 p</ 2 . 8 2q2fe3 
 
 4. _9a 2 6c 4 . 5z# 2 
 
 5. i2a 2 6 4 c3. 3a 2 6 3 c 
 
 2a ' 4x 4 ^ 5 ' 
 
 6 ' *** 
 
 Art. 119. 
 
 In this article the answers are indefinite, and, therefore, the, 
 
 are not given. 
 
 SECTION XLII. 
 
 Art. 130. 
 
 
 
 1. 3x 2 + 2x + 5. 
 
 5. 
 
 2x 2 + 2ax + 46 2 . 
 
 2. a 2_|_6«6 + 96 2 . 
 
 6. 
 
 X 3-f-2x 2 — Z + 2. 
 
 3. 3x 3 — 2x 2 + x— 1 
 
 7. 
 
 2x2 + ^ + 5 . 
 
 4 3a 2 — 2a& + 56 3 . 
 
 
 ^ 2 ■ 
 
 Art. 131. 
 
 
 
 1. 3a + 3x. 
 
 4. 
 
 2a 2 — 6 3 . 
 
 2. 2x 3 + 5a 2 . 
 
 5. 
 
 3x + 2y: 
 
 3. 5c 2 — 2yz. 
 
 
 
36 key XLII1, XLI* 
 
 SECTION XLII1. 
 
 i2. if d (o bd) i 
 
 RT, 
 
 122. 
 
 1. 
 
 SaWs/Wab. 
 
 2. 
 
 4 c 2 (5 a 6)* 
 
 3. 
 
 4. 
 5. 
 
 3 a 3 6 2 y/Td*. 
 
 3bc\/5^ 
 
 Performed. 
 
 3 
 
 6. 2a^/3« — b. 
 
 3. 
 
 13. £c^/2a 2 62. 
 
 14. £b(6abrf. 
 
 a 
 
 15. f t/3£ c . 
 4c K 
 
 7. a(26 2 + a 2 6 c )* «>. g- c y/ -3; 
 
 2 6 2 
 
 8. «^/l + 6 3 . 17 7a % / ZScfib—lab 
 
 17 Ifl /^L^ 
 
 5 my 20m- 
 
 9. Aafotf—\. bm \ 20in»— 10w» 
 
 10. 24i/6«36_3. ,0 i^« y/ 5b+lQa* 
 
 8 ' 3 c K c w + 2 m 2 * 
 
 11. -^afc 
 
 Art. 123. 
 
 1. y/9a 2 6z 2 . 5. (12 a 2 6)2. 
 
 2 , / 20 6 c 6 (125 1 y)* 
 
 • — ? — - 7. (le^ft^ + aa^^i 
 
 4 X 8(a— 6) /27a3f«2w 
 
 SECTION XL1V 
 
 Art. 125. 
 
 1. 4 a 2 (3 z)*. 3. 5(5)*. 
 
 2. 14(2)*. 4 23(3)*. 
 
XIJV. KEY. 37 
 
 5. 
 
 8 (2)* 
 
 6. 
 
 8(2)*. 
 
 7. 
 
 5a$ + &ab* 
 
 8. 
 
 ii(l#. 
 
 9. 
 
 fa (10)* 
 
 10. 
 
 £(l3 + 2c)(ab)} + (xy)i. 
 
 11. 
 
 (2)* 
 
 12. 
 
 2x(3a)*. 
 
 13. 
 
 4 a (2 6)*. 
 
 14. 
 
 2a(3a& 2 )* 
 
 15. 
 
 »(«)*. 
 
 16. 
 
 (5)*. 
 
 17. 
 
 ( a j)i_|_(10 c — 7) (m)* 
 
 18. 
 
 T y(3 a &)* + ,& (25 m)* 
 
 19. 
 
 21 a 62. 
 
 20. 
 
 10a* 6* A or 10a 6 c (a 6 \* 
 
 21. 
 
 3 ro 2 a" c** 
 
 22. 
 
 75 x^ y* , or 75 y (x 5 y )* 
 
 23. 
 
 Performed. 
 
 24. 
 
 30 (10)* 
 
 25. 
 
 Performed. 
 
 26. 
 
 8 (216)* 
 
 27. 
 
 140. 
 
 28. 
 
 m-\-n 
 
 a »• 
 
 29. 
 
 mq + np 
 
 15 a »? . 
 
 30. 
 
 (648000)^. 
 
38 key XLIV 
 
 81. (a + b)&. 
 
 32. 12a(c — d)^ 
 
 33. 12 a2 6 (« + &)*(«— y jM 
 
 34. 35(m + «)£(c — d)&. 
 
 35. 4. 
 
 36. 3—17(6)* 
 
 37. 41. 
 
 38. a*. 
 
 39. J 6* A 
 
 40. 2a&b$c. 
 
 41. !«*&*<£ 
 
 42. 6. 
 
 43. 15. 
 
 44. 8(2)*. 
 
 n — m 
 
 45. a mn . 
 
 46. Ja* p 6** 
 
 47. « (2,* 
 
 48. (« — &)*. 
 
 49. lto>*! 
 
 a 
 
 50. (« + »)*(«-«« )* 
 
 3m 
 
 51. 4a%bK 
 
 52. 125 a^c. 
 
 53 8 ^ 
 
 27x 2 y 3 
 54. A (3)*. 
 
XLIV. key. 38 
 
 288 £ b 5 c 10 
 
 55. 
 
 m 
 
 56. (a + 6)*. 
 
 57. 27 (z — y)i 
 
 59. 32 (** — **)*(* — «)*. 
 
 59. a* b"^ c m . 
 
 60. a^ P" c* <P\ 
 
 61. 8 ( fl + & ) 1 
 
 27 ( c _^ 
 
 62 4(x + y)*( C -rf)* ^ 
 25(m-f 7i)£(z— y)* 
 
 63. a* 6* 
 
 64. 3c£b*. 
 
 65. 2* a* 6* 
 
 66. 3*a*6*c*. 
 
 i J I JL 
 
 67. 10 m a* m z m y®* 
 
 JL JL JL 
 
 68. 6 m a« m b sm 
 
 69. (a + 6)* 
 
 70. 4 (a — 6)*(c — d)*. 
 
 71. 2a(m-n)^(c-^. 
 
 72. 3* (« -*)*(* + «*)* 
 
40 KEY. XLV. 
 
 
 SECTION XLV 
 
 Art. 132. 
 
 
 1. 
 
 7\/% 
 
 2. 
 
 Q\/b: 
 
 3. 
 
 (6« + 5)v/S 
 
 4. 
 
 8^47 
 
 5. 
 
 43^/37 
 
 6. 
 
 w r °- 
 
 7. 
 
 191 yU 
 
 8. 
 
 ¥&% 
 
 9. 
 
 i(3« + 4^) v /I 
 
 10. 
 
 -13^3. 
 
 11. 
 
 8^2. 
 
 12. 
 
 8|/T— y^a 
 
 13. 
 
 «&(3a + 5)^/2^6. 
 
 14. 
 
 (a — c) i/5 c. 
 
 15 
 
 /a(fl</c + 6ce-6^ 2 )\ /F - 
 
 X(J. 
 
 V b*de /**'' 
 
 16. 
 
 sy% 
 
 17. 
 
 4v/tT 
 
 18. 
 
 2yU 
 
 19. 
 
 va 
 
 20. 
 
 2^5. 
 
 21. 
 
 Ay^ 
 
 22. y/2. 
 
XLV- 
 
 kit. 
 
 41 
 
 23. 
 24. 
 25. 
 
 26. 
 
 27. 
 
 28. 
 
 29. 
 30. 
 31. 
 32. 
 
 33. 
 
 34. 
 
 35. 
 
 36. 
 
 37. 
 38. 
 39. 
 
 40. 
 
 41. 
 
 42. 
 43. 
 
 44. 
 45. 
 
 46. 
 
 47. 
 
 48. 
 49. 
 
 2 a (2 a — z)\/5x. 
 2a (4 — a) fa 
 * (fa -[/*)• 
 
 tW 75 ' 
 
 8y/2. 
 
 £V/M-rVv/6 
 
 12. 
 
 4. 
 
 2. 
 
 6a^/6c. 
 
 6 — 
 
 5 a 2 c y'a c. 
 
 30^/10. 
 
 6^432. 
 
 4a 2 — 96. 
 
 21 + 18/3" + 28 fa + 24 ^"108. 
 
 l-V/5. 
 
 6 c ^/a 13 . 
 
 12. 
 
 72y/23328 
 
 24 ^rpr^. 
 
 4 2 
 
 iV/«) 2l (7) 14 6 6 . 
 
 2. 
 4* 
 
*2 
 
 Ki jr. 
 
 XIV 
 
 50. 4\/b. 
 
 52. 12^/27 
 
 53. f^/T0. 
 
 54. ^ 
 
 55. |^/a 2 6 2 . 
 
 56. 
 
 si«v/4*v* 
 
 57. 
 
 a 2 + 2«y/& +6 
 
 
 a 2 — 6 
 
 58. 
 
 1(10-^/2). 
 
 59. 
 
 KV/3-1). 
 
 60. 
 
 ^ 
 
 61. 
 
 26^5? 
 
 62. 
 
 9^/a6. 
 
 63 
 
 64^/aar. 
 
 64. 
 
 a 4 5 c ^/6T 
 
 65. 
 
 «3 63 m io ^££ 
 
 66. 
 
 x m y m i/a m b m 
 
 67. 
 
 \/ x y- 
 
 68. 
 
 »vA 
 
 69. 
 
 *v« 2 . 
 
 70. 
 
 3. 
 
 71. 
 
 V/(« + *) 10 or (a-f ** 
 
 72. 
 
 2^^ 
 
 73. 
 
 18 
 
 3^/a6 
 
XLVI. key. 43 
 
 74. 4^/^ 78. \/(a + b)K 
 
 75. 2y/a6c 2 . 79. \/{a — 6) 9 . 
 
 76. 2/^6. 80. \/x + y. 
 
 77. (/W 81. V?9(a + 6) 
 
 SECTION XLVI 
 
 Art. 134. 
 
 1. 7m 7 n- 5 x 2 . 
 
 2. 12«-6 6-5 c 5. 
 
 3. 30a*&M. 
 
 4. 100 a*6 3 A 
 
 5. 1. 
 
 „ 9 a 2 c w 3 
 
 6 - "I— 
 
 7. 14a 3 7» _1 r" 1 xy. 
 
 ^ 49 a -3 b~~ J c~ 3 m~ 3 z 2 ?/ 
 
 8 ' 5 "• 
 
 4 ( C - «*)(« + 6)- 
 
 J. g . 
 
 49a 3 6(x + 3/) 2 (m — n)(m-f w)- 1 ^ — <f) "^ 
 
 10. 5 
 
 (T 2 m- 1 
 
 IX. 
 
 2 " 
 
 12. 
 
 5a 3 2r 9 y" 2 
 
 3 
 
 13. 
 
 a 5 6 -1 c~~ 1 mz 1 
 
 4 
 
 14. 
 
 aa 3 ^ 3 ^^- 1 ^. 
 
 15 
 
 7a- 1 &- 2 c- 2 2f« 
 
44 kev. XLVII, XLV1II 
 
 1 6 ZHa + b)(x + y)(c-J)r^ 
 
 m 
 
 22 ar* (2 c — 3rf)-3 (m + n)-i (x— y)-i 
 i/. - . 
 
 18. 9a- 2 6-*c6. 
 
 19. 64«~ 6 6 6 c- 9 , 
 
 20. 16a 2 6-*c-*. 
 
 21. 1000 m-£ x- 1 y\ 
 
 22. 4a-i&-2 C 3. 
 
 23. 2«- 3 6c" 2 . 
 
 24. 3a*b-± c-*d~*. 
 
 25. 3 a* ft-**-* 
 
 26. S^a-^a^y-*™- 1 * 5 . 
 
 SECTION XLVII, 
 
 Art. 136. 
 
 1. Performed 
 
 2. 20. 
 
 3. 51. 
 
 SECTION XLVIII. ' 
 
 Art. 137. 
 
 1. 16. 4. 12 J. 
 
 2. 5jh 5. 6. 
 
 3. 5 "and 15. 6. 1st term 10 : 3d 18 
 
L, LI KEY. 
 
 45 
 
 SECTION L. 
 
 Art. 143. 
 1. Performed. 
 
 3. n = g , S_ 22 
 
 2S-m , 2(S— ara) 
 
 3. / J I* / , \ 
 
 w w(n — 1) 
 
 4 - w - a + r a - 2£— (J+a)' 
 5. Performed. 
 
 _^_ t=v /(2^~rf)2+8^ _ d— 2a=fc\/(2a— d)«+8f/ig 
 6 - l - 2 ;W "" 2d 
 
 2S—dn(n—l) . 2£ + c/n (w — 1) 
 
 7. a = = ; * — o~ . 
 
 2w ^« 
 
 v ■ 2rcZ— dn(n — 1) 
 
 8. a=Z— d(7i — 1); S=z jp ^. 
 
 2S—nl ' 2(n/— fif) 
 
 9. or j d=— -r-. 
 
 n n(n — 1) 
 
 _ <?=Fy/(rf+2Z) 2 — 8dS __ <7-|_27dLy/(^+2/)2— 8</£ 
 10. «_ ^ ;w ~ 2d 
 
 Let the learner verify the preceding formulae, by making a = 1, 
 dz=z 2, w = 4, / = 7, and fife 16. 
 
 SECTION LI. 
 
 Art. 1 l*%. 
 
 1. 78 strokes. 
 
 2. 7 = 65; £=425 
 
 3. Z=70; £=1360. 
 
 4. Z:=48£; S=903f. 
 
 5. J=26f; £=414f. 
 
 6. The progression is 20, 25, 30, 35, 40, 45, 50, 55. 
 
46 key. LII, LII1. 
 
 7. The progression 
 
 8. The progressior 
 to 41 inclusive. 
 
 i is 6, 6f , 74, 8, 8§, 9J, 10. 
 
 i is 5, 7, 9, 11, &c, all the odd numbers 
 
 9. 300 strokes. 
 
 
 10. 950 trees. 
 
 
 11. 31700 feet. 
 
 
 12. 1608£. 
 
 13. 8=ian\ 
 
 
 14. 8 days. 
 
 15. 5 days and 10 days ; 55 miles and 80 miles. 
 
 16. 12, 17, 22. 
 
 
 17. 4,6 and 8. 
 
 
 18. 14, 12, 10, 8 and 6. 
 
 19. 531. 
 
 
 
 SECTION LII. 
 
 Art. 1 •51. 
 
 
 1. Performed. . 
 
 
 .n—i 
 
 3. a=J =T ;S= 
 
 4. a^^-V: 
 
 n— l n— l 
 
 £> 7i—l n— I • 
 
 \/l — y/tf 
 
 'a-'.te-ij" 
 
 sy-i(?-i) 
 
 gr n — 1 ' £ n — 1 
 
 Let the learner verify the preceding formulae, by making a=.% 
 q — S f w = 3, Z= 18, and £=26. 
 
 SECTION LIU 
 
 Art. 152. 
 
 1. 7=640; £=1275. 
 
 2. z = ¥ ' T ; ^sr=iM- 
 
IJV ****« * 7 
 
 i 
 
 3. 2, 10 and 50. 
 
 4. $4095. 
 
 5. Performed. 
 
 6. 3, 15, 75, 375 and 1875. 
 
 7. 7, 35 and 175. 
 
 8. 8, 24 and 72. 
 
 9. Any progression in which the ratio if %. 
 
 10. 3, 5 and 7. 
 
 11. 6. 
 
 12. 10. 
 
 13. 7 937+. 
 
 14. 178 535+. 
 
 15. 2. 
 
 16. 6. 
 
 17. 21. 
 
 18. 91*. 
 
 19. Progression 2, 6, 18, 54, 162. 
 
 20. Progression 5, 30, 180, 1080. 
 
 21. Progression 3, 15*, 75, 375, 1875, 9375, 46875. 
 
 22. Progression 1, 2, 4, 8, 16, 32, 64, 128, 256, 512. 
 
 SECTION LIV. 
 
 \rt. 153. 
 1st to 5th inclusive, performed. , g „ or * / jq 
 
 6. 626. ' ' ^ 
 
 7. 10, or -2. 19 25 ' or49 « 
 
 8. 4. 20. ±8, or rhy/— 1331 
 
 9. f. 21. 729, or (—V-) 6 - 
 
 11. 81. 22 ' 3, or ^/— 41. 
 
 12th to 15th inclusive, performed. 23. 2, or — 3. 
 
 16. 2, or— 3. 24. 25, or 81. 
 
 17. zkv/^oriy/ITe. 25. (£ ± /-t^' 
 
48 kbt. LV 
 
 26. 27, or (— «)* 
 
 28. 81, or (— J£)4. 
 
 29. 4, or y. 
 
 30. 4, or 1. 
 
 •( 
 
 ±v/6 2 + 4«c — 6\* 
 
 31. 
 
 2a 
 
 32d and 33d performed. 
 
 34. 9, or — 12. 
 
 35. 10, or— 2. 
 
 36. l,or lzkt/60. 
 
 37. 6, or — 9. 
 
 38. 9, or 5 + ^25. 
 
 *> - . dry/377 — 1 
 rf9. 5, or — o, or — E — 
 
 Atl Q i 5 ±4/1329 
 40. 3, or — £, or ^ . 
 
 SECTION LV. 
 
 Art. 151. 
 
 1. Performed. 
 
 2. K==^,y = 4ior«==2,3rs=5. 
 
 3. x=18, y = 3; or z=12£, y~— 2£. 
 
 4. x = 2,y = 3; or i = - 46,y = 15. 
 
 5. z = 4, y = 2; orz = — 4, y = — 2. 
 6th to 11th inclusive, performed. 
 
 12. z = 6,y=3; or x = — 6, y — — 3. 
 
 13. a;=7,y=:3; orx = 3,y = 7. 
 
 14. x = 3, y = 5; a? = — 3, y = — 5; x = 5, y = 3 ; or 
 * = — 5,y = — 3. 
 
LV. KEY. 49 
 
 15. z = 9, y = S; or x = — 9, y = — 3. 
 
 16. zz=49, y = 25. 
 17.' z = 625, y = 16. 
 
 18. z = 5, y = 4; or x=. — 5,y = — 4. 
 
 19. x = 7, y = 3 ; or z = — 3, y =z — 7. 
 
 20. z = 1, y = 2; x = — 1, y = — 2 ; z := 2, y = 1 ; or 
 x = — 2,y = — 1. 
 
 21. z = 2,y = 4;z=:4,y = 2;z = 3 + v/ 21 ^=--3— \/21; 
 or z = 3 — ^/2l, y = 3 + ^/21. 
 
 22. z = 1, y = 2 ; or z = 2, y = 1. 
 
 23. z = 4, y = 9 ; or z = 9, y = 4. 
 
 24. z = 8, y = 2 ; or z = 2, y = 8. 
 25th and 26th performed. 
 
 27. z = 20, y=16. 
 
 28. z = 5, y = 3. 
 
 29. z z= 8, y = 6 ; z = — 8, y = — 6 ; z = 6, y = 8 ; or 
 z = — 6, y = — 8. 
 
 30. z=14, y = 12. 
 
 31. z = 27, y = 148£ ; or z = 6, y = 33. 
 
 32. x = 5, y = ±3. 
 
 33. Performed. 
 
 34. z = 3,y = l; z = -.3 ? y = — 1; z = 8 tf& y = ^/J ; 
 orz = — 8^/i, y = — [/h 
 
 35. z = 4, y = 3; x = — 4,y==— -3; a: = 1 1 V ^7 y = 
 — 12\/£"; orz= — ll\/^y = 12V^ 
 
 36. x=l,y = 2; z = — 1, y = — 2; z = |, y = — V > 
 or z = — f, y— V- 
 
 37. z = 4,y = 2; z = — 4,y = — 2; iaB-^f=^j 
 
50 
 
 KEY. 
 
 LVI, LVI] 
 
 SECTION LVI. 
 
 Art. 155. 
 
 1. 2048. 
 
 2. 65536. 
 
 3. 1048576. 
 
 4. 4194304. 
 
 5. 131072. 
 
 6. 4194304. 
 
 7. 32. 
 
 8. 32. 
 
 9. 16. 
 
 10. 1024. 
 
 11. 16. 
 
 12. 64. 
 
 13. 32768. 
 
 14. 16384. 
 
 15. 65536. 
 
 16. 1048576. 
 
 17. 1048576. 
 
 18. 1048576. 
 
 19. 32. 
 
 20. 8. 
 
 21. 8. 
 
 22. 16. 
 
 23. 8. 
 
 24. 8. 
 
 SECTION LVII. 
 
 Art. 162. 
 
 
 
 
 
 1. 31048284. 
 
 
 6. 
 
 41041671. 
 
 
 2. 4-7581015. 
 
 
 7. 
 
 0-4392806. 
 
 
 3. 21048284. 
 
 
 8. 
 
 2-5228783. 
 
 
 4. 34456042. 
 
 
 9. 
 
 2-6389892. 
 
 
 5. 5-5152060. 
 
 
 10. 
 
 20467815 
 
 
 Art. 163. 
 
 
 
 
 
 1. 30. 
 
 
 6. 
 
 1-63472. 
 
 
 2. 2000. 
 
 
 7. 
 
 0-3048. 
 
 
 3. 3155888. 
 
 
 8. 
 
 00593562. 
 
 
 4. 1309534. 
 
 
 9. 
 
 00016891. 
 
 
 5. 221754-6. 
 
 
 10. 
 
 00002155. 
 
 
 Art. 167. 
 
 
 
 
 
 1. 1-8593785, 
 
 or 9 8593785, 
 
 3. 
 
 4-5789370, or 
 
 6-5789370 
 
 2. 3-7604225, 
 
 or 77604225. 
 
 4. 
 
 T-8239087, or 
 
 9-8239087 
 
LVIII. 
 
 KEY. 
 
 51 
 
 5. 
 
 1-2179083, 
 
 or 9-21 
 
 79083. 
 
 8. 
 
 26596926. 
 
 *>• 
 
 0-7262335. 
 
 
 
 9. 
 
 2U628630. 
 
 7. 
 
 0-5682017. 
 
 
 
 10. 
 
 1-3322956, or 93322956 
 
 Art. 
 
 168. 
 
 
 
 
 
 1. 
 
 0-273603. 
 
 
 
 5. 
 
 000003. 
 
 2. 
 
 00536312. 
 
 
 
 6. 
 
 0-772481. 
 
 3. 
 
 00002. 
 
 
 
 7. 
 
 00676142. 
 
 4. 
 
 000000004' 
 
 H435. 
 
 
 8. 
 
 000274074. 
 
 SECTION LVIII, 
 
 Art. 
 
 169. 
 
 
 
 1st to the 13th inclusive, 
 
 performed. 
 
 
 14. 
 
 151-807. * 
 
 33. 
 
 3-48428. 
 
 15. 
 
 77190. 
 
 34. 
 
 0-422838. 
 
 16. 
 
 324-324. 
 
 35. 
 
 0-77992. 
 
 17. 
 
 00035991. 
 
 36. 
 
 0-99285. 
 
 18. 
 
 7-71707. 
 
 37. 
 
 0-867362. 
 
 19. 
 
 301182. 
 
 38. 
 
 071254. 
 
 20. 
 
 3581242. 
 
 39. 
 
 0-80818. 
 
 21. 
 
 4577-626. 
 
 40. 
 
 11 1204. 
 
 22. 
 
 499336. 
 
 41. 
 
 0-59905. 
 
 23. 
 
 5932-52. 
 
 42. 
 
 0-4809. 
 
 24. 
 
 47701902. 
 
 43. 
 
 172002. 
 
 25. 
 
 0187295. 
 
 44. 
 
 Performed. 
 
 26. 
 
 933333. 
 
 45. 
 
 23774. 
 
 27. 
 
 71197. 
 
 46. 
 
 11291. 
 
 28. 
 
 5554569. 
 
 47. 
 
 1-9699. 
 
 29. 
 
 768575. 
 
 48. 
 
 Performed. 
 
 30. 
 
 00000000243. 
 
 49. 
 
 705. 
 
 31. 
 
 165-3817. 
 
 50. 
 
 30-787. 
 
 32. 
 
 2-236068. 
 
 
 
 51. Progression 7, 86394, 10662, 1316, 16242, 20 
 
52 KEY. LIX. 
 
 Art. 170. 
 1. Performed. 
 
 2 S — a log. 7 — log, a , 
 
 * q ~W^~V W — log. (£—«) — log. (S— /)"*" 
 
 c + ^( g — 1) log.(a + £ g — .9 ^— lo g<g 
 
 ? log. q 
 
 4. fl = 7 g -g( g -n; >l = lo g- / - lo g'(^^±jg? + 1 
 
 log. q ~ 
 
 5. £=122S5; ti=12. 
 
 6. q — 3; n = 8. 
 
 7. Z=177147; n = 10. 
 
 8. a = 2; w = 8. 
 
 SECTION LIX. ' 
 
 Art. 171. 
 
 1st to 5th inclusive, performed. 
 
 6. $569-801. 
 
 7. $194867. 
 
 8. $33324. 
 
 9. $1829-71. 
 
 10. $250. 
 
 11. 14-137 years, or 14 years, 1 month, 29 days 
 
 12. 0-06085, or 6 T V per cent, nearly. 
 
 13. 11-8956 years, or 11 years, 10 months, 22 days 
 
 14. 18854 years, or 18 years, 10 months, 7 days. 
 
 15. 14-2067 years, or 14 years, 2 months, 14 days 
 
 16. $644-781. 
 
 17. 83310. 
 
 18. 003205. 
 
 19. 2691 years, or 26 years, 10 months, 28 days. 
 Art. 17S. 
 
 1. Performed. 2. $3558-094 
 
LX. KEF. 
 
 SECTION LX. 
 
 Art. 173. 
 
 1st 2d and 3d performed. 
 
 , 4. $7686-22. 
 ' 5. $7857-98. 
 
 6. $1169 84. 
 
 7. 19387 years, or 19 years, 4 months, 19 days. 
 
 53 
 
 MISCELLANEOUS QUESTIONS 
 
 1. A $948, B $1896, C $2844, D $4740. 
 
 2. 3 yards at 9s. ; 5 yards at 7s. 
 
 3. 8 years. 
 
 4. $871|. 
 
 5. 60 years. 
 
 6. 55. 
 
 7. 4 days. 
 
 8. 20. 
 
 9. Breadth 30, length 40 rods. 
 
 10. 10 eagles ; $30. 
 
 11. A $60; B $48. 
 
 12. $1000. 
 
 13. $50. 
 
 14. ,£400, .£500, .£700. 
 
 15. 800 men. 
 
 16. $20. 
 
 17. £280. 
 
 18. 14 beggars ; 32s. 
 
 19. 6£. 
 
 20. 40, 20 and 12. 
 
 21 Wheat 10s., rye 6s. per bushel 
 
 22. / T . 
 
 23. 23. 
 
 24 21 and 16. 
 5* 
 
54 KM. 
 
 25. 17 and 8. 
 
 26. 24s. 
 
 27. 53. 
 
 28. A $20; B $50. 
 
 29. Greater 79; less 35. 
 
 30. A .£300; B £350. 
 
 31. Length 20, breadth 18 feet 
 
 32. A$l; B$l£; C $2. 
 
 33. 15 and 12. 
 
 34. 10 and 8. 
 
 35. 12 and 19. 
 
 36. 8 and 15. 
 
 37. 40 sheep. 
 
 38. A .£10 ; B .£20 ; C .£30 j D £40. 
 
 39. A 6, B 5 miles per hour. 
 10. 93. 
 
 41. 8 hours. 
 
 12. 6 hours, or 31 hours. 
 
 13. 3 and 243. 
 
 44. A 204, B 200 acres. 
 
 45. 5 and 4. 
 
 46. 36 and 24. 
 
 47 Breadth 100, length 150 feet 
 
 48. 12 and 10. 
 
 49. Breadth 40, length 50 rods. 
 
 50. 16 and 9. 
 
 51. 6 and 4. 
 
 52. 18 and 6. 
 
 53. 3 and 5. 
 
 54. 5 and 9. 
 
 55. 81 scholars. 
 
 56. 23. 
 
 57. Depth of 1st 4 feet, that of 2d 5 feet. 
 
 58. 8. 
 
 59. 25. 
 
 60 Court 30 by 19 feet ; height of fence 4 feet 
 
KEY 55 
 
 61. 12 yards. 
 
 62. 150 miles. 
 
 63. 25. 
 
 64. A $200; B $300. 
 
 65. 9 and 15. 
 
 66. 741. 
 
 67. A $81 ; B $41 ; C $21 ; D $11 ; E $6. 
 
 Explanation. 
 
 Let v, w, x, y t and z represent the money of A, B, C, D, and 
 E, respectively, when they began to play ; and let S represent 
 the known amount of their money, which is 5 . 32 or $160. 
 
 In the first game A loses the whole amount minus his own 
 share, or S — v ; he will therefore have left v — (S — v), or 
 2 v — 8. After this his money is doubled four times ; he will 
 then have 32 v — \QS\ this must be equal to $32 or \S. 
 Hence, 
 
 32 v — 16 S= £ S. from which we have 
 
 . B, in the first game, doubles his money ; he will then have 
 2m>; in the second game he loses S — 2 w y and will have left 
 Aw — 8. Afterwards his money is doubled three times ; he will 
 then have 32 w — 8 8. Hence, 
 
 32m>-8£ = |#, and^ = i^z=$41. 
 In like manner, for C's money we find 
 
 32 x — 48=18, and 1 = ^=: $21; 
 
 for D's money, 
 32 
 and for E's money, 
 
 32y-2 S= J S, and y= 1^ = 111; 
 
 32z-S=iS >a ndz = ^ = 
 
56 
 
 KEY 
 
 68. 
 
 A 40, B 60 eggs. 
 
 69. 
 
 10 and 14. 
 
 70. 
 
 4 and 2. 
 
 71. 
 
 4, 6, and 11. 
 
 72. 
 
 Depths 4 feet and 5 feet 
 
 73. 
 
 5 and 3. 
 
 74. 
 
 6. 
 
 75. 
 
 16. 
 
 76. 
 
 $50. 
 
 77. 
 
 7 and 3. 
 
 78. 
 
 7, 5 and 3. 
 
 79. 
 
 1, 3, 5, ami 7. 
 
 80. 
 
 2, 6, and 18. 
 
 81. 
 
 10, 20, 40, and 80. 
 
 82. 
 
 .£94 18s. Od. 1 qr. 
 
 83. 
 
 $491585. 
 
 TJBI78E.SIT 
 
UNIVERSITY OF CALIFORNIA LIBRARY