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 '<i. " O M 
 
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 ^>\^^'' 
 
 '. "o 
 
 ? I5« ■" 
 
 
 iJ. /' 
 
 ""'^.'; %'&^' 
 
 
 
 ■' --. c% -^r.^ 
 
 ^m**; 
 
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 "'^x/^r:^^^;^^' 
 
^ TO THE ^ M 
 
 EXERCISES IN ALGEBRA, 
 
 USEOF THE TEACHER 
 
 BY PRAXCIS J. GRUND, 
 
 Author of an " Elementary Treatise on Plane and Solid Geometr 
 "Elementary Treatise on Natural Philosophy," "Elements 
 of Chemistry,'' "Popular Lessons in Astronomy," 
 translator of " Meier Hirsch's Problems," etc. 
 
 icr;^ 
 
 ^1 
 
 f 
 
 BOSTON: 
 CARTER, HENDEE AND CO 
 
 1833. 
 

 
 .e^- 
 
 Entered according to act of Congress, in the year 1833, 
 
 By Francis J. Grund, 
 
 in the clerk's office of the District Court of Massachusetts. 
 
 /^:2^. 
 
 I. R. BUTTS, PRINTER, SCHOOL ST. 
 
C^ ' ^-^ 
 
 KEY 
 
 EXERCISES IN ALGEBRA. 
 
 SECTION I. 
 
 ANSWERS TO THE EXERCISES IN ADDITION, 
 
 SUBTRACTION, MULTIPLICATION, 
 
 AND DIVISION. 
 
 A. Answers to the Exercises in Addition. 
 
 a, ADDITION OF SIMPLE QUANTITIES. 
 
 1. Ans. 
 
 2 a. 
 
 2. A 
 
 ns. 14 a. 
 
 3. ' 
 
 
 6 6. 
 
 4. ' 
 
 ' 6/ 
 
 5. ' 
 
 
 a-\-b. 
 
 6. ' 
 
 ' 7a +10 6. 
 
 7. ' 
 
 
 0. 
 
 8. ' 
 
 ' 11 a. 
 
 9. ' 
 
 
 — a. 
 
 10. ' 
 
 ' a — b. 
 
 11. ' 
 
 
 8 a — 5 b. 
 
 12. ' 
 
 ' —7a + b 
 
 13. ' 
 
 
 — 17 a. 
 
 14. ' 
 
 ' — 12 a. 
 
 15. ' 
 
 
 8a + Sb. 
 
 16. ' 
 
 ' 6 a. 
 
 17. ' 
 
 
 -15/ 
 
 18. ' 
 
 ' —3 6. 
 
 19. ' 
 
 
 — 15 c. 
 
 20. ' 
 
 ' 5d. 
 
 21. * 
 
 
 0. 
 1 
 
 22. ' 
 
 ' 0. 
 
23. 
 
 Ans. 
 
 So. 
 
 24. Ans. 6. 
 
 25. 
 
 " 
 
 Qa 
 
 26. " 5 a + 4 6. 
 
 27. 
 
 '^ 
 
 10 6. 28. " _ — 6 + 4 6-. 
 
 29. 
 
 '' 
 
 4« 
 
 + 26 — 4rf. 30. " 3a — 46— c + f/. 
 
 31. 
 
 a 
 
 5 a 
 
 _j_ ft __ rf. 32. " 15 « — 19 6 — c. 
 
 33. 
 
 a 
 
 7 a 
 
 _6 6 — </. 34. " 2rt— 2 6. 
 
 35. 
 
 a 
 
 0« 
 
 -1-5 6. 36. '' 19 « + 18 6. 
 
 b. 
 
 ANSWERS 
 
 TO THE ADDITIOIV OF COMPOUND QUANTITIES. 
 
 
 ]. 
 
 Ans. 
 
 5rt_8c — 4 6. 
 
 
 2. 
 
 '' 
 
 8a — 86 + 4 c. 
 
 
 3. 
 
 (< 
 
 7^ + 5 6 — 13 c + 10. 
 
 
 4. 
 
 u 
 
 9 6, + 5c — 9/— 6^. 
 
 
 5. 
 
 (( 
 
 12 a + 11 6+ He— 17 f/. 
 
 
 6. 
 
 ii 
 
 17 a — 6 — 2 6 + 5 J. 
 
 
 7. 
 
 a 
 
 0. 
 
 
 8. 
 
 i( 
 
 7 a — 4a: — 5 i/ -\- 5 z — 10. 
 
 
 9. 
 
 u 
 
 rt __ 6 + 5 c + 3 f/. 
 
 
 10. 
 
 4« 
 
 ^a + b — Sc—id+g—f. 
 
 
 11. 
 
 (C 
 
 24a + 5c — 6fZ+e— / 
 
 
 12. 
 
 (( 
 
 11 «— 6 6 — f/+e. 
 
 B. Subtraction. 
 
 a. ANSWERS TO THE EXERCISES IN SUBTRACTION OF SIMPLE 
 QUANTITIES. 
 
 1. Ans. 0. 2. Ans. 4 a. 
 
 3 " 0. 4. " — 2 d. 
 
5. 
 
 Ans. 
 
 7 a — ob. 
 
 6. 
 
 Ans. 
 
 2 a. 
 
 7. 
 
 
 9 a. 
 
 8. 
 
 (< 
 
 11 a. 
 
 9. 
 
 
 2 a. 
 
 10. 
 
 i( 
 
 a + h. 
 
 11. 
 
 
 3 « + 2 6. 
 
 12. 
 
 a 
 
 — 12 a. 
 
 13. 
 
 
 — 4 a. 
 
 14. 
 
 (( 
 
 0. 
 
 15. 
 
 
 a. 
 
 16. 
 
 (( 
 
 — «. 
 
 17. 
 
 
 — 3 a + 5 Z» 
 
 18. 
 
 n 
 
 — 16. 
 
 19. 
 
 
 9. 
 
 20. 
 
 a 
 
 19. 
 
 21. 
 
 
 — 5. 
 
 22. 
 
 (( 
 
 — 16. 
 
 23. 
 
 
 0. 
 
 24. 
 
 (C 
 
 16. 
 
 b. ANSWERS TO THE EXERCISES IN" SUBTRACTION OF 
 COMPOUJVD QUANTITIES. 
 
 1. Ans, a — 2 k 
 
 2. *' a— 12 b. 
 
 3. '' 3f/+10/. 
 
 4. - « _[_ 5 6 + 9. 
 
 5. '' 5 « + 4 6 + 7 f/. 
 
 6. *^ 23 b. 
 
 7. " _/-j-8m — 6x — 3r/. 
 
 8. '^ _ 7 a + 2 c — 5 A + c. 
 
 9. " — « — 9 6 + 10 c + d — 3 ^^ 
 
 10. ^' _ 6 A + 6 A: + 7 + 2 Z. 
 
 11. ** 3A — 9Z+6A: + 7. 
 
 12. '' 3 A — 2a + 5 A:. 
 
 13. '' — 5a + 6 +c — e — 5/+7, 
 
 14. '* 2«. 
 
 15. " 7 6 + 5 a + 136 c — 2. 
 
 16. '' 9 a — 26 6 — 3^. 
 
KEY. 
 
 17. Ans. 
 
 3 c + 12 + 7 c. 
 
 18. ' 
 
 
 8 a~6 c—7 ^+5 e+c7 c—f—6g+7 k—2A 
 
 19. ' 
 
 
 27 a— 14 6. 
 
 20. ' 
 
 
 20« — 5 c —3/+ 5 a:. 
 
 21. ' 
 
 
 — 15« + 8 6 — c. 
 
 22. ' 
 
 
 _4«_13c-f8^ + 2c. 
 
 23. ' 
 
 
 28« + 66 — 5/— A. 
 
 24. ' 
 
 
 — a 4-4 6. 
 
 25. ' 
 
 
 7;^- 10 — X. 
 
 26. ' 
 
 
 7 a — 5 b. 
 
 27. ' 
 
 
 — 7 a + 5 6. 
 
 28. ' 
 
 
 2 6. 
 
 29. * 
 
 
 — « + 3 6. 
 
 30. ^ 
 
 
 — 2 a + 4 6. 
 
 C. MuJti])Ucation, 
 
 ANSWERS TO THE EXIRCISES IN MULTIPLICATION OF 
 SIMPLE QUANTITIES. 
 
 1. 
 
 Ans. 
 
 ah. 
 
 2. 
 
 Ans. 
 
 b a ox ah. 
 
 3. 
 
 
 ah. 
 
 4. 
 
 
 b a or ah. 
 
 5. 
 
 
 — ah. 
 
 6. 
 
 
 — ah. 
 
 7. 
 
 
 + ah. 
 
 8. 
 
 
 i2ab. 
 
 9. 
 
 
 — 42ac 
 
 10. 
 
 
 — 70 ah. 
 
 11. 
 
 
 — 7 ah. 
 
 12. 
 
 
 66 «x. 
 
 13. 
 
 
 a b c. 
 
 14. 
 
 
 ah c. 
 
 15. 
 
 
 a h c d e. 
 
 16. 
 
 
 S5 a a b c d e. 
 
 
 17 
 
 '. Ans. — 
 
 Aohhh del 
 
 xy. 
 
 
 18. " 85 a c e. 
 
KEY. 
 
 19. 
 
 20. 
 21. 
 22. 
 23. 
 24. 
 
 Ans. 42 a 6 df g. 
 
 12 abbe clef. 
 '■' 6 ab c ddf g. 
 " \2{iab edefgh. 
 " a a a a. 
 '' — a a a a. 
 
 b. ANSWERS TO THE EXERCISES IN MULTIPLICATION OF 
 COMPOUND QUANTITIES. 
 
 1. 
 
 2. 
 
 3. 
 
 4. 
 
 5. 
 
 6. 
 
 7. 
 
 8. 
 
 9. 
 10. 
 11. 
 12. 
 13. 
 14. 
 
 15. 
 
 16. 
 
 An 
 
 . 9 a e -\-6b c. 
 
 6 a d-{-6bd-\- 10 c d. 
 
 21 «/+ 14 bf— 35 f//+ 7 6/ 
 
 J6 ae-f-20 6 e— 4 rfe. 
 
 30a^-f-15 6^ — 25/^. 
 
 16bh — 24gh-\-8ch. 
 
 70 a b eld— 100 bb cd. 
 
 SO a a b d -^ 90 a b b c — 96 a a b cf. 
 
 a e -\- b c -\- a d -\- b d. 
 
 a a-{- 2 a b -^b b. 
 
 2ac-\-Sbc-}-2ad-{-3bd. 
 
 8ac-\- i2b c -{-I0ad-^l5b d. 
 ad -{- b d — c d — a e — b e -\- c e. 
 
 10 af— 15 6/— 35 r f — 12 a ^ + 18 6 ^ 
 
 + 42.^. 
 
 Uaf— 21 bf—56cf—7 df—4 ag 
 -\-6hg-\-\6eg^2dg—2ah-\-Sbh 
 
 + 8 c /i -f rf h. 
 
 8 «/— 12 6/— 32 c/— 4 ^/+4 ^/— 2« ^ 
 '-3 6 ^+8 e g-\.d g—ggAr2 a li—S b h 
 -Sch—dh+g/i. 
 
 (8 c 
 
b KEY. 
 
 17. Ans. 2111— SSI m — 27 1 + 22 7nm -{-99 m. 
 
 18. " 6aa-\-S5ab-{-9 ac-\-50bb-\-S0b c 
 
 _ 10 a/— 25 6/— 15 c/. 
 
 19. " 9aa-\-S6ab-{-9ac—\5ae-}-20bb 
 
 +30 6c — 50 6e + 3a/+2 6/+3c/— 5e/. 
 
 20. " aa — b b. 
 
 21. " 6aa + 7a6— 20 6 6. 
 
 22. '' a a — 2 « 6 + 6 6. 
 
 23. '' 6 a « — 29 a 6 + 35 6 6. 
 
 24. " 35aa6 6— 19« «6c— 28a6 6c — 24aacc 
 
 + 32 a 6 c c. 
 
 25. " 52 6 6 c c (Z + 80 6 6 c c e + 20 6 6 c ^ e 
 
 + 39 6 6 ccZ^— 30 b b d d e — \oQ b c d e 
 — 240 6 c e e + 120 6 fZ e e. 
 
 26. " 15«aa— 14 a «6-|-24 a6 6 — 76 6 6. 
 
 27. " aa-\-2ab-{-bb — cc. 
 
 28. " 4a«+14«6 — 2ac + 1266 + 6c — 20cc. 
 
 29. " 12aa<7aaa-[-20a«a«66-|-2Saaa666 
 
 — ISaaaabbb — 25 aabbbbb 
 — 20 abbbbbb-{- 20 a bbbbb 
 
 -\-\Qbbbbbb. 
 
 30. " 51 a a — 23 a 6 + 70 a c — 51 a ci +2 66 
 
 — 16 6 c + 6 6 fZ + 24 c c - 36 c r/. 
 
 31. " « a + 2a6 + 6 6 — c c — 2 c f? — f/rZ. 
 
 32. " 4 a a + 12 a 6 + 9 6 6 + 16 fl c — 20 a rZ 
 
 + 246c — 30 6 J + 16c c — 40 c^ + 25JfZ. 
 
 33. '^ 6 a/— 96/+3e c/+ 14 «a — 21 «6 
 
 + 7aec — 2acc-\-Sbcc — ccce. 
 
 34. " 3mm — 1 1 m /i + 17 m ;; p + 1 n w 
 
 — 28 u p p — 6 p p pp. 
 
KEY. 
 
 35. Ans. 3 m m m m — 4 m m n w -|- 2 ?« m, y ^-\-nnnn 
 
 + PPPP- 
 
 36. " 2 til m m m — 3 m m nn — '^m mpp -\-n n n n 
 
 -|- 2 71 n y p -\- p p p p. 
 
 Division. 
 
 a. ANSWERS TO THE EXERCISES IN DIVISION OF SIMPLE 
 QUANTITIES. 
 
 1. 
 
 Ans. 
 
 1. 
 
 3. 
 
 a 
 
 a a. 
 
 5. 
 
 (( 
 
 a a. 
 
 7. 
 
 (C 
 
 a 
 T' 
 
 9. 
 
 C( 
 
 a 
 
 11. 
 
 (I 
 
 — h. 
 
 13. 
 
 (( 
 
 3 6. 
 
 15. 
 
 (( 
 
 5« J. 
 
 17. 
 
 <c 
 
 2« 
 
 b ' 
 
 19. 
 
 (( 
 
 7 a 
 2 6' 
 
 21. 
 
 " 
 
 be 
 
 d ' 
 
 23. 
 
 (( 
 
 ^bc 
 
 2 ' 
 
 2. 
 
 Ans. 
 
 a. 
 
 4. 
 
 <4 
 
 a. 
 
 6. 
 
 (( 
 
 a. 
 
 8. 
 
 il 
 
 a 
 — 6"" 
 
 10. 
 
 (( 
 
 a 
 
 12. 
 
 il 
 
 + a. 
 
 14. 
 
 li 
 
 Q ab. 
 
 16. 
 
 11 
 
 5a 
 
 36* 
 
 18. 
 
 (( 
 
 Sa 
 
 b ' 
 
 20. 
 
 (( 
 
 be. 
 
 22. 
 
 il 
 
 4n 
 
 g ' 
 
 24. 
 
 <( 
 
 3 a 
 
 / 
 
KEY. 
 
 25. Ans. 
 
 S a a h f 
 
 26. Ans. 
 
 5 a h ffg 
 
 h. ANSWERS TO THE EXERCISES IJV DIVISION" OF COMPOUND 
 QUANTITIES. 
 
 1. Ans. 7 « c — 2 « f/ c — 3/+ 3 d. 
 
 2. 
 
 " 
 
 7 
 
 c — 
 
 -2d e — Sf-\-4d. 
 
 3. 
 
 (( 
 
 3f 
 
 "2"" 
 
 -de — 1-. 
 2a 
 
 4. 
 
 '' 
 
 6 c 
 d 
 
 2be 2 
 a 3/' 
 
 5. 
 
 (( 
 
 ' a 2 a 
 
 6. 
 
 " 
 
 3c 
 
 f ,^gh 
 
 6 
 
 b ~^ 4 a b' 
 
 7. 
 
 
 3 c 
 
 f , 3 ^ /. 
 
 a66 ' 4 a a b b' 
 
 
 8. 
 
 Ans. 
 
 a. 
 
 
 9. 
 
 
 C( 
 
 2 a. 
 
 
 10. 
 
 
 * 
 
 c +d- 
 
 
 11. 
 
 
 ' 
 
 2c-\-2d. 
 
 
 12. 
 
 
 ( 
 
 2a+Sb—5x. 
 
 
 13. 
 
 ( 
 
 c 
 
 a-Jf-b. 
 
 
 14. 
 
 
 ( 
 
 2 X X -\- 5 X — 7 . 
 
 
 15. 
 
 
 tc 
 
 2 a -^ 'S b -}- c. 
 
 
 16. 
 
 
 < 
 
 «_6-|_3c. 
 
 
 17. 
 
 
 i 
 
 a-\-2b — c. 
 
 
 IS 
 
 
 (( 
 
 4 a a -{-3 a— b. 
 
 
 19. 
 
 
 i 
 
 2a+5b—7 c + Sd. 
 
KEY. 
 
 20. 
 
 A 
 
 ns. 
 
 7 c — 13 J+ 24 6 — 5/. 
 
 21. 
 
 (( 
 
 
 2 « + 5 6. 
 
 22. 
 
 ( 
 
 
 3a— 46. 
 
 23. 
 
 a 
 
 
 12 xa; — 5x1/ — 3?/ 3/. 
 
 24. 
 
 (( 
 
 
 3a — 26 + 8 c. 
 
 25. 
 
 i 
 
 
 9 « _ 2 6 _j- c. 
 
 26. 
 
 I 
 
 
 — 5 « X + 12 X ?/. 
 
 27. 
 
 < 
 
 
 a + 6. 
 
 28. 
 
 i 
 
 
 2a+3 6. 
 
 29. 
 
 ( 
 
 
 4 a — 3 6. 
 
 30. 
 
 ( 
 
 
 «a-|-3a6-|-cc. 
 
 31. 
 
 e 
 
 
 a««-|-aa6+«66-[-66 6. 
 
 32. 
 
 ' 
 
 
 8aaa+12«a6 + 18«66 + 27 66 6. 
 
 33. 
 
 c 
 
 
 \6 a a a a — Sa«a6-)-46!a66 — 2 abbb 
 
 -\-hbhh. 
 
 34. 
 
 < 
 
 
 Zab — 5fm — l d. 
 
 C. ANSWERS TO THE PARTIAL DIVISIONS, PERFORMED IX 
 
 CASES WHERE THE DIVISOR IS NOT AN EXACT 
 
 NUMBER OF TIMES CONTAINED IN 
 
 THE DIVIDEND, 
 
 1. Ans. 1, and the remainder -\-b, or also 1 -f- 7. 
 
 2. '' I + 6, and the remainder -j- 6 6, or also 1 + 6 
 
 ^1—6 
 
 3. '* 1 -)- 6 -|- 6 6, and the remainder 6 6 6, or also 
 
 ,+5 + i6 + lAi. 
 
 4 « IJ^ b-}-bb-\-bbb-\-bbbb. 
 
 5 " 1+6 + 66 + 666+6 666 + 66666 + ... 
 
KEy. 
 
 h 
 
 10 
 
 6. " 1, and the remainder — b, or also 1 — t . 7 
 
 I -\- 
 
 7. " 1 — b, and the remainder -|- b b, or also 
 
 ' ' 1 -|- 6 
 9. *' \^h-\-bb — bbb-{-bbbb—bbbbb^ 
 
 10. " a J and the remainder is -|- a a. 
 
 11. '' a -\- a a, and the remainder -\- a a a. 
 
 12. *' a -^ a a -\- a a a -^ a a a a -^ a a o a a, and 
 
 the remainder a a a a a a. 
 
 13. " a ~\- a a -\- a a a -\- a a a a -\- a a a a a-\- 
 
 a a a a a a -\- . . . 
 
 14. " a, and the remainder — a a. 
 
 15. *' rt — a a -[- aaa, and the remainder — aaa a. 
 
 16. " a — a a-]- aaa — a a a a-\-a a a a a u"^ . . . 
 
 17. *' 1, and the remainder -[- 1. 
 
 18. " 1-1 , and the remainder -I . 
 
 19. - 14_1 + 1--J L__[_.... 
 
 ' a ^ aa a a a 
 
 20. " 1, and the remainder is — 1. 
 
 21. " 1 1 , and the remainder is . 
 
 aaa a a 
 
 22. " 1 — J-+i- L_i- 
 
 a a a aaa 
 
 23. " 1, and the remainder is + ^• 
 
 b . , . bb 
 
 24. " 1-1 , and the remainder is — . 
 
 ' c c 
 
KEY. 1 1 
 
 b . hh ^55 
 
 25. Ans. 1 -| 1 , and the remainder is 
 
 c c c f^ f. 
 
 J , _6_ &^ 6 /> 6 . 
 
 26 
 
 c c c ' c c c ^ 
 1, and the remainder is — b. 
 
 28. •• 1 -^ + ^, and the remainder is -^. 
 
 c c 
 
 29. " 1-1+L*_L^ , 
 
 c ' c c c c c 
 
 30. " - + — + — + ^^'-|-&c 
 
 a ' aa ^ aaa^ a aaa^ 
 
 *^'- " — , and the remainder is — ^ 
 
 a 
 
 a 
 
 Of) u ^ ^ ^ b b c 
 
 "^^ — — — -1 and the remainder is -I . 
 
 a aa i ^^ ^^ 
 
 03 ,. _^^ ^ I ^ * ^ bbbc 
 
 a a a a a a a a a a ~^ ' ' ' ' 
 
 34. " 1, and the remainder is -j- 2 x. 
 
 35. ** 1 + 2 x -f- 2 .r .7:, and the remainder is 2 2 a; x. 
 
 36. - l+2x + 2xx + 2xxx+,... 
 
 SECTION II. 
 
 ANSWERS TO THE SUBTRACTION. MUL- 
 TIPLICATION AND DIVISION 
 OF POWERS. 
 
 1. Ans. The 2d power. 
 
 2. - The 2d power of X. 
 
 3. " The iM power of r/. 
 
 4. " The 3d power of T. 
 
12 
 
 KEY. 
 
 5. Ans. The 4th power. 
 
 6. '' The 4th power of 2:. 
 
 7. " The 11th power. 
 
 8. '' Of 10 such factors. 
 
 9. '' Of 15 such factors. 
 
 10. Ans. 14 such factors. 11. Ans. 7 factors, x^. 
 
 12. " 3 factors, x3. 13. " it is equal to 1 
 
 16. " 4- 17. " — . 
 
 a^ a 
 
 18 '' 1-1 19 '' 1-i 
 
 20. '< — = 1-, 21 ^ ^ 
 
 22 4* ' 2^ 8 
 
 41—4 
 
 22. '' ^= i. 23. '' 1 
 
 A. Addition of Poivtrs. 
 
 a. ANSWERS TO THE EXERCISES IN ADDITION OF SIMPLE 
 QUANTITIES, 
 
 15 cfi. 
 
 10x2 __ 11 a2 
 
 6 x2 — 9 7/2. 
 
 — 5 x2 — 12 X. 
 
 12 x2 ?/ X 3/2. 
 
 20x4. 
 13«1 
 
 1. 
 
 Ans. 
 
 6 a2. 2. 
 
 3. 
 
 
 x2 -f 3 63. 4. 
 
 5. 
 
 
 x2. G. 
 
 7. 
 
 
 — 2 x2 — 3 7/2. 8. 
 
 9. 
 
 
 11 x2+ 17x2?/. 10. 
 
 1. 
 
 
 6x2— 4x2y+8x. 12. 
 
 
 
 0. 
 
 (C 
 
 0. 14. 
 
13 
 
 15. 
 
 Ans. 
 
 7«- = 7xl = 
 
 7 
 a 
 
 16. 
 
 'i 
 
 «-='. 
 a^ 
 
 
 17. 
 
 (I 
 
 3.6-2 _|_ 4.52. 
 
 
 18. 
 
 (( 
 
 20.64 ^ 7.83. 
 
 
 b. ANSWERS TO THE EXERCISES IN ADDITION OF COMPOUND 
 QUANTITIES. 
 
 1. Ans. 4 a2 6-f 8a62 — c, 
 
 2. '* Sa^ b c —2a^b d — 6d. 
 
 3. " 8 a4 6 — 3 a2 ^2 c. 
 
 4. " 0. 
 
 5. *' 28 a3 6-2 c4 _ 8 & + 24. 
 
 6. " 10a36c2_«2 5_rf^5«52. 
 
 7. " 11 « 6 a;2 — 5 « +m. 
 
 8. " 2 ci 63 c + 2 a 62 c2 __ 4 «4 ^5 _ 4 „2. 
 
 9. " n a^b^d-\~a^b^ d— 15 d-^. 
 
 10. " 45 a2 53 _ 50 a2 63 ^-1 _|_ 4 ^2 _|_ 1 q ci. 
 
 S. Subtraction of Powers. 
 
 a. ANSWERS TO THE EXERCISES IN SUBTRACTION OF 
 SIMPLE QUANTITIES. 
 
 1. Ans. 0. 2. Ans. a^. 
 
 3. '* 0. 4. *' — 4 m4. 
 
 5. *' 6 «5 _ 4 55. 6. " 2 ^4. 
 
 7. *' 10 a3. 8. *' a2 _[_ 52. 
 2 
 
14 
 
 9. 
 
 Ans. 
 
 0. 
 
 
 10. 
 
 i( 
 
 «3. 
 
 
 11. 
 
 (( 
 
 — a2. 
 
 
 12. 
 
 (( 
 
 _ 3 fe2 _|_ 5 ^2 ; or 5 c2 - 
 
 - 3 62, 
 
 13. 
 
 <c 
 
 4«2. 
 
 
 14. 
 
 (( 
 
 _ ^2 _|_ ^,2 . or 62 _ ^2. 
 
 
 15. 
 
 << 
 
 — C2 — 6/2. 
 
 
 16. 
 
 (I 
 
 3 a2 — «. 
 
 
 i. ANSWERS IN SUBTRACTION OF COMPOUND QUANTITIES. 
 
 1. Ans. a2„^,2, 
 
 2. «' u «:i_6 62 + 2 c2. 
 
 3. " 4 a3+ 11 62-|-e2. 
 
 4. " 4«3 + 3a2_3tt62_5cfZ2c3-j-c//2. 
 5 u 3 ^2 _ 13 &2 ,12 _ 2 c3 c? — A;2. 
 
 6. " 10 «2 52c2_J_,/3y_4«2i2. 
 
 7. " 18 a^ 65 -|- 3 4- 3 « — (/. 
 
 8. *' 2 a2 _|_ 2 «2 62 c2 + 3 a 6^. 
 
 9. cc a3 54_ 11 ^2^,5 c_(/4. 
 
 10. " 2a4_)-8a3 62 + 4c ^2_^7e;_j_3«2, 
 
 il. u 2a-4 + 8a-3 62 + 4c-i J2 
 
 12. " 2 a-3 + a-"^ d-{- d^ e — ef. 
 
 C. Multiplication of Powers. 
 
 a. ANSWERS TO EXERCISES IN MULTIPLICATION OF SIMPLE 
 QUANTITIES. 
 
 1. Ans. a^. 2. Ans. a^. 
 
KEY. 
 
 15 
 
 3. Ans. 
 
 5. " 
 
 7. " 
 
 9. " 
 
 11. " 
 
 13. " 
 
 15. " 
 
 17. " 
 
 19. " 
 
 •21. " 
 
 23. " 
 
 25. " 
 
 27. " 
 
 29. " 
 
 31. " 
 
 33. " 
 
 35. '^ 
 
 36. " 
 
 37. " 
 39. " 
 41. " 
 43. '' 
 45. " 
 47. " 
 49. " 
 51. '' 
 
 53. " 
 
 54. « 
 
 56. " 
 
 a\ 
 
 
 4. 
 
 Ans. 
 
 a\ 
 
 a^. 
 
 
 6. 
 
 u 
 
 «6. 
 
 a^. 
 
 
 8. 
 
 a 
 
 a^ 
 
 a}^ 
 
 
 10. 
 
 (( 
 
 ai3. 
 
 3a'^- 
 
 
 12. 
 
 u 
 
 8 a^ 
 
 20 «8. 
 
 
 14. 
 
 (( 
 
 — 5a\ 
 
 — 5^7. 
 
 
 16. 
 
 a 
 
 12 a^. 
 
 3 a5. 
 
 
 18. 
 
 li 
 
 — 20 a9. 
 
 — a«. 
 
 
 20. 
 
 u 
 
 + ftS. 
 
 4«7. 
 
 
 22. 
 
 a 
 
 6«8. 
 
 6a^. 
 
 
 24. 
 
 u 
 
 — 3a9. 
 
 20^5 6. 
 
 
 26. 
 
 u 
 
 15 all. 
 
 105 «2i. 
 
 
 28. 
 
 ii 
 
 — 105 «2i. 
 
 — 48 «i9. 
 
 
 30. 
 
 it 
 
 48 ai7 ^,, 
 
 12 a^ h^. 
 
 
 32. 
 
 a 
 
 14 «-i2. 
 
 42 a-6. 
 
 
 34. 
 
 u 
 
 — 42 a-6. 
 
 42 a-6. 
 
 
 
 
 
 — 4S aO = 
 
 : 
 
 48 X 1 = 
 
 — 48. 
 
 12 . 25. 
 
 
 38. 
 
 Ans. 
 
 6.34. 
 
 18.46. 
 
 
 40. 
 
 (C 
 
 6.4-4. 
 
 20 . 4-6. 
 
 
 42. 
 
 u 
 
 — 60 . 4-2. 
 
 12 «4 h^. 
 
 
 44. 
 
 (( 
 
 50 a^ 69 c. 
 
 — 150«i2< 
 
 ^9C. 
 
 46, 
 
 ii 
 
 — 27 «i2 ^-9 c, 
 
 21 «3 59 c6. 
 
 
 48. 
 
 ii 
 
 21 «3 6-9 c6. 
 
 21 a3 &-9 c 
 
 ~6. 
 
 50. 
 
 ii 
 
 5 a^ ^12. 
 
 60 «6 ^13 ^ 
 
 
 52. 
 
 a 
 
 60 «6 ^,6. 
 
 7*11 /20 .^. 
 
 
 
 
 
 /1-3 /a X = 
 
 
 55. 
 
 a 
 
 /2t 
 
 3A-5/-4x2 
 
 
 57. 
 
 U 
 
 52 «4 66 c5. 
 
16 KEY. 
 
 58. 
 
 Ans. 
 
 52 a-2 6-6 c. 
 
 59. 
 
 Ans. 
 
 52 «-4 6-6 c-5. 
 
 60. 
 
 u 
 
 — 3 a4 65 c^ 
 
 61. 
 
 a 
 
 20a»6o c0=20. 
 
 62. 
 
 i< 
 
 — 39 68. 
 
 63. 
 
 (C 
 
 84 a-\ 
 
 64. 
 
 u 
 
 9a^b c2. 
 
 65. 
 
 u 
 
 12 a2 66 c. 
 
 66. 
 
 il 
 
 6 (a + a;)5. 
 
 
 
 
 b. ANSWERS TO THE EXERCISES IN MULTIPLICATION OF 
 COMPOUND Q,UANTITIES. 
 
 I. Ans. 2 a3_j_6a2. 
 
 3 ^3 _[_ 6 a2 _^ 3 rt 62. 
 6 a4 _ 9 «6 _ 3 «2 c. 
 
 4 ^3 _ 12 ^2 -f- 4 « 62. 
 
 — 4 «4 _|_ 12 g3 _ 8 a2 52, 
 
 4 ^4 5 _ 12 a3 62 _ 20 «2 63. 
 
 «5 57 c — 12 a4 65 c. 
 
 Q fj5 66 c2 — 15 a4 6 c8 + 27 a^ 63 c^. 
 
 56 a5 7,4 ;2 ^ J6 A4 /5 _ 24 a 10 l — S^h"^ I. 
 
 — 56 a-5 /i4 _ 16 /i4 /3 _!_ 24 a'l h^ /"i 
 
 + 24 7*4 Z-i. 
 
 2 «3 65 c2 fZ — 2 66 C3 ^4 j_|_ 6 6 c5 fZ. 
 2 a3 62 <Z — 2 63 c tZ4/+ 6 6-2 c^ d. 
 a2 _|- 2 a 6 + 62. 
 ^2 _ o « 6 4- 62. 
 
 «2— 62. 
 
 «3 _|_ 3 a 62 — a2 6 — 3 63. 
 a5 _ 2 a 63 — a4 6 + 2 64. 
 ftG _ 3 ^2 63 _- a4 62 _j_ 3 6^ 
 x3 _ 5 x2 — x + 14. 
 
 2. 
 
 u 
 
 3. 
 
 a 
 
 4. 
 
 u 
 
 5. 
 
 u 
 
 6. 
 
 (; 
 
 7. 
 
 u 
 
 8. 
 
 u 
 
 9. 
 
 a 
 
 10. 
 
 u 
 
 11. 
 
 (( 
 
 12. 
 
 (( 
 
 13. 
 
 (( 
 
 14. 
 
 a 
 
 15. 
 
 u 
 
 16. 
 
 u 
 
 17. 
 
 (( 
 
 18. 
 
 (< 
 
 19. 
 
 u 
 
 20. 
 
 4( 
 
KEY 
 
 17 
 
 21. 
 22. 
 23. 
 24. 
 25. 
 26. 
 27. 
 
 28. 
 
 29. 
 30. 
 
 31. 
 
 32. 
 33. 
 
 34. 
 
 35. 
 
 36. 
 
 37. 
 38. 
 
 Ans. 3 A:4 _ 26 P / _|_ 37 ^,2 p _ 14 /, p 
 6/5 4- 7/M — 65/3 12 _f_ i.2y2 p. 
 20 a^ — 88 a4 2; + 47 a^ x^ — 6 a^ x^. 
 «« — a^. 
 «5_|_32 55, 
 
 4.a^x^~6a^b'ix7/^— 6 aH^ x^ ij -\-9 b^ y3. 
 4 a9 a;4 _ 6 a^ 54 ^2 ^2 _ q ^4 ^,4 3.2 ^ 
 
 + 9 &8 j/4 
 21 a^ — 43 a6 ^, _|_ 159 «5 ^2 _ no ^4 53 
 
 + 104 a3 54 _ 32 a2 ^5. 
 1 x4 + « x3 — 22 a^ x2 -f- 4 a3 X + 7 a^. 
 7 aio — 25 a8 b^ + 48 ^e 54 _ 23 «4 ^,6 
 
 + 5 a2 ^,8. 
 
 a^ — 8 a' b +28 a^ b^ — 36 a^ b^ + 34 a? b^ 
 — 22 «2 66 _j_ 2 a 67 _|_ ^,8 
 
 a4 _J_ «2 ;^2 ^ ;^4, 
 
 120 a8 64 _ 101 «7 66 _j_ 4S a-2 js _|_2l ^6 b^ 
 
 — 18 rt-3 feio 
 120 a-8 64 _ 101 a-' b^ + 69 a-e 68 
 
 — 18 a-5 610. 
 6 2:5 ^12 _ 4 2:5 ^10 _|_ 18 ^3 ^lo _j_ jg ^9 ^8 
 
 — 12 z3 ^8 _j_ 36 ^.7 ^8 _[_ 24 2;7 3/6 _j_ 95 3.11 ^4. 
 
 78 «7 63 _j_ GO a4 64 _ 234 a^ 6^ _ 24 «3 65 
 
 — 180 a2 65 _ 91 «8 65 + 72 « 66 _ 70 aS 66 
 
 + 28a4 67. 
 78 «-8 63 — 174 a-5 64 — 269 a'^ 65 + 22 a b^ 
 
 + 20 «4 67. 
 10 «6 66 c4 + 3 a' 65 c7 + 14 ^n 68 c8 
 — 18 «8 64cio _|_ 21 ai2 67 cii — 30 «i» 6^ c^ 
 
 + 36 a^i 66 c8 — 42 a^s 6^ c^. 
 
 2* 
 
18 
 
 KEY. 
 
 39. Ans. 196 «!» c^ — 36 a"^ h^ c^ -\-\2a^h c^ — c^. 
 
 40. '' a4 __ 4 c6 f/8 _ 28 c^ d^ — 49 cl 
 
 X). Division of Potvers, 
 
 a. ANSWERS TO THE EXERCISES IN DIVISION OF SIMPLE 
 QUANTITIES. 
 
 1, 
 
 Ans. 
 
 1. 
 
 3. 
 
 << 
 
 aK 
 
 5. 
 
 £< 
 
 a. 
 
 7. 
 
 (( 
 
 ah c. 
 
 9. 
 
 li 
 
 — 2 a, 
 
 11. 
 
 C( 
 
 5a^b. 
 
 13. 
 
 u 
 
 — iab-^ 
 
 15. 
 
 (( 
 
 — 2 a^ 63^ 
 
 17. 
 
 <l 
 
 4r^6. , 
 
 19. 
 
 il 
 
 7a-6. 
 
 21. 
 
 a 
 
 9a-8. 
 
 23. 
 
 i( 
 
 C «12 
 
 rf ' 
 
 25. 
 
 (( 
 
 4 «9 66, 
 
 27. 
 
 (( 
 
 6Ar • 
 
 29. 
 
 11 
 
 5 64 c 
 3« • 
 
 31. 
 
 (( 
 
 6c2, 
 
 2. Ans. 
 
 1. 
 
 4. " 
 
 «2. 
 
 6. " 
 
 «'^. 
 
 a '' 
 
 «3 h C2. 
 
 10. '' 
 
 — 2 a. 
 
 12. " 
 
 — 4 a6. 
 
 14. '' 
 
 3a2, 
 
 16. '' 
 
 ^ la^h (I 
 
 18. " 
 
 x«^. 
 
 20. " 
 
 a-7. 
 
 22. " 
 
 5 a~^. 
 
 24. " 
 
 c a24 
 
 rf ' 
 
 26. '* 
 
 4«6. 
 
 28. '' 
 
 15 6^ 
 
 16 « ' 
 
 30. '' 
 
 — f a3 52. 
 
 32. '' 
 
 4 6' 
 
KEY. 19 
 
 ANSWERS TO THE EXERCISES IN DIVISION OF COMPOUND 
 QUANTITIES. 
 
 ] . Ans. Sab^— 5f. 
 
 2. 
 
 (( 
 
 4 «3 52 __ 4y2 _j_ 7 «3 6 ^. 
 
 3. 
 
 (( 
 
 2 x5 7 x^ 3b^x 
 a a2 1 2 • 
 x 7 3&2 
 
 4. 
 
 
 a^ 2a^x ^ a^ x 
 
 5. 
 
 <( 
 
 -2ax^-ax+l^. 
 
 6. 
 
 (( 
 
 -^ + ''-i' 
 
 7. 
 
 (( 
 
 ?/2 ^ 5 2:2 
 
 8. 
 
 (C 
 
 ^ -^27 + xyz. 
 
 9. 
 
 << 
 
 4 a:2 7/2^2 
 
 a;y s ' 2 
 
 10. 
 
 ii 
 
 3 64 5 c c2 
 2a "■ 2a2^,2 52 + « 
 
 11. 
 
 (( 
 
 c3. 
 
 12. 
 
 (( 
 
 « + 6. 
 
 13. 
 
 it 
 
 a^-{-2ab + b^. 
 
 14. 
 
 li 
 
 a^~\-2 ab + b^. 
 
 15. 
 
 (( 
 
 a^-\-S cfib-^Sab^ + b^. 
 
 16. 
 
 C( 
 
 Sa^ — 5a^b-{-2ab^. 
 
 17. 
 
 (< 
 
 «4 — 4 «3 ^,3 _j_ 6 ^2 56. 
 
 18. 
 
 i( 
 
 a2 _ 62, 
 
 19. 
 
 (t 
 
 a3 — 63. 
 
 20. 
 
 11 
 
 a4 + 64, 
 
 C2 
 
20 
 
 KEY. 
 
 21. Ans. 6 a;6 -|- flfS 2;5 — 7 <^4 ^4 
 
 22. 
 23. 
 24. 
 25. 
 26. 
 27. 
 28. 
 29. 
 30. 
 31. 
 32. 
 
 rt6 + 2 «4 ^2 _|. 4 «2 .4 _j_ 8 ;26. 
 
 a2 __ 5 ^ 5 _[_ 6 62. 
 2 c2 4- 3 6 c — 62. 
 2 c2 — 3 6 c + 62. 
 1 _|- a a; -|_ ^2 a;2 
 
 5 a3 62 c3 — 2 «2 62 c4 _ 3 a 62 c5 — 7 6 c^. 
 
 — a-5 a;2 _(- 7 «-i a;3 _|_ 8 q3 3^4, 
 
 a4 + 4 «^ X + 12 «2 ^,2 _[_ 1(5 ^ a;3 _[_ 16 3^4. 
 a7 _j_ «6 6 _[- rt5 62 _|_ «4 63 _[_ ^3 64 _[- a2 65 
 
 _[_ a 66 + 67. 
 
 C. ANSWERS TO THE PARTIAL. DIVISIOJV^ PERFORMED IN 
 
 CASES IN WHICH THE DIVISOR IS NOT AN EXACT 
 
 NUMBER OF TIMES CONTAINED IN 
 
 THE DIVIDEND. 
 
 1 . Ans. a — a X -^ a x^ — a x-^ -\- a x"^ ^ &c. . . . 
 '> '' a-}- ax-i^ax'^ -{-ax^+ ax"^ + &LC. . . . 
 
 a 
 
 — T + ^ ^ ^C- 
 
 c. . . . 
 
 a a a a 
 
 -+^ + ,:f + ^ + ^ 
 
 a ad a d^ a d^ 
 
 a , a d , a d^ . a d^ . „ 
 
KEY. 
 
 21 
 
 ANSWERS TO THE EXERCISES IN POWERS OF POWERS. 
 
 1. 
 
 Ans. 
 
 a\ 
 
 2. 
 
 Ans. 
 
 a^. 
 
 3. 
 
 (( 
 
 a}^. 
 
 4. 
 
 (( 
 
 4«4. 
 
 5. 
 
 {( 
 
 125 a^ 66. 
 
 6. 
 
 a 
 
 16 ai2 b^ 
 
 7. 
 
 i< 
 
 9 «4 54 c\ 
 
 8. 
 
 i( 
 
 g9. 
 
 9. 
 
 <( 
 
 + «1 
 
 10. 
 
 a 
 
 — a6. 
 
 11. 
 
 (( 
 
 + fl8. 
 
 
 
 
 12. 
 
 (( 
 
 _ alO^ _|_ «12^ _ 
 
 
 + « 
 
 16, respectively. 
 
 13. 
 
 <( 
 
 64 «6 59 c3. 
 
 14. 
 
 Ans. 
 
 , ~ 8 a6 69 c6. 
 
 15. 
 
 t( 
 
 — 64 a6 69 c9. 
 
 16. 
 
 u 
 
 «15 520 c25. 
 
 17. 
 
 i( 
 
 + 729 ai2 je. 
 
 18. 
 
 (( 
 
 16a4 66c8y2^2. 
 
 19. 
 
 iC 
 
 9«4. 
 
 20. 
 
 (( 
 
 8 «6 69. 
 
 21. 
 
 <c 
 
 + 4«4. 
 
 22. 
 
 li 
 
 — 8a6. 
 
 23. 
 
 ce 
 
 4096 «i2. 
 
 24. 
 
 (( 
 
 + «io. 
 
 25. 
 
 a 
 
 + 6-12. 
 
 26. 
 
 u 
 
 4096 a36. 
 
 27. 
 
 (( 
 
 ai20. 
 
 28. 
 
 'i 
 
 + a60. 
 
 29. 
 
 C( 
 
 — a-3. 
 
 30. 
 
 {( 
 
 + a-4. 
 
 31. 
 
 a 
 
 — «-io. 
 
 32. 
 
 <( 
 
 + a-12. 
 
 33. 
 
 (( 
 
 — a^\ 
 
 
 
 
 34. 
 
 i( 
 
 + 2-10 ^-10 ^ 
 
 1 
 
 
 
 
 1024 
 
 «io' 
 
 
 35. 
 
 it 
 
 + 2-«« = 34 
 
 
 
 
 36. 
 
 11 
 
 — 215 ai5 = — 
 
 3276^ 
 
 ^ «l-\ 
 
 
22 
 
 KEY 
 
 SECTION III. 
 
 EXERCISES I xN ADDITION, SUBTRACTION, 
 
 31 U L T I P L I C A T I O N, DIVISION, AND 
 
 RAISING TO POWERS OF 
 
 FRACTION S. 
 
 A. Addition. 
 
 a. ANSWERS TO THE EXERCISES IN ADDITIOK OF SIMPLE 
 QUANTITIES. 
 
 1. Ans. 
 
 a -\- 'il a -\- ^a oi 
 
 2. 
 
 li 
 
 96 
 
 2 c 
 
 
 3. 
 
 Ans. 
 
 9 a 
 '4 * 
 
 4. 
 
 (' 
 
 Hh «^. 
 
 
 5. 
 
 (( 
 
 A a2 — 4 63. 
 
 6. 
 
 i( 
 
 3a2 a2 
 b 62- 
 
 
 7. 
 
 u 
 
 9«3 
 64 ^'' 
 
 8. 
 
 (I 
 
 2 «3 7 «4 
 x3 + 5 ' 
 
 9. 
 
 (< 
 
 -V- «'• 
 
 10. 
 
 (C 
 
 1 
 
 2 « 
 
 (' 
 
 d + 
 
 11. 
 
 2 6 c 
 
 (( 
 
 a J -(- 6 c 
 6^ ' 
 
 2 6^/ 
 
 acd + b^d4-bc^- 
 
 13. '' -^j r— • 
 
 oca 
 
 Qacd + Sb^d + 2c^b 
 *'*• 6b cd 
 
 6acd+Sb^d + 2bc^ 
 ^^' \2bc d 
 
16. Ans. 
 
 17. 
 
 IS. 
 
 19. 
 
 20. 
 
 21. 
 
 22. 
 
 23. 
 
 24. 
 
 KEY. 23 
 
 i0acd-\-4:5 b^ (/ + 48 6 c^ 
 
 m be d 
 40 ^ f/ -|- 45 6 (/ + 48 c2 
 
 60b cd 
 b c -\^ a c — a b 
 
 ab c 
 66c + 4«c + 3«6 
 12 «6 c ■ 
 
 a b ef — b cf-\- c d e 
 
 cdcf 
 9 ad-\-Q b — 2c 
 3b c d ■ 
 
 ac fh + b^fh + bcdh + b cfg 
 bcfh ~~ "' 
 
 12 ft (Z e g- -[- 6 b ceg -\-4.bd^ g -{-3b def 
 !2 bdeg * 
 
 c df-\-b df+b cf-\-b c d^b c dfg-\-b c df h 
 b c df ' 
 
 b. ANSWERS TO THE EXERCISES IN ADDITION OF COMPOUND 
 QUANTITIES. 
 
 1. Ans. 7 ,, _ 2 7 c _ j_9 cl^3^a — 2^^ c — 4f d. 
 
 3. - 8i^a- J^6 + 7c-/. 
 
 4. - li«+1^^6. 
 
 * "^ 2 6 "f" 6/ "T" 6 a 6" 
 
 6 " « ■ 3 c n c llff 
 
 6 •" 2 ^ "^ ey "^ /r 
 
24 
 
 KEY. 
 
 '• '^"'' 2^+12/5+ 41^— 
 
 ,, 2a 2 3 6 c 71 rZe _/> A 
 3 "^ 20 ' 35 3~' 
 
 14 ^2 /i3 I 8 m3 ?i3 • 
 
 10. 
 
 ;;3 J g5 
 
 ^. Suhtr action. 
 
 a ANSWERS TO THE EXERCISES IN SUBTRACTION OF SIMPL] 
 QUANTITIES. 
 
 1. 
 
 Ans. 8f. 2. 
 
 Ans. — 16|. 
 
 3. 
 
 - - ItV 4. 
 
 " 6A- 
 
 5. 
 
 " f 6. 
 
 " - §1- 
 
 7. 
 
 - A«. 8. 
 
 " -A*- 
 
 9. 
 
 - 17 5=1^^6. 10. 
 
 " - li ft- 
 
 
 11. Ans. "^-^^ 
 6 d 
 
 
 
 6 df 
 
 -6<ie 
 
 
 .. 2«rf-36c 
 
 
 
 
 
 10 arf/— 15 6 c/— leirfc 
 
 20 6 f// 
 
25 
 
 ANSWERS TO THE EXERCISES IN COMPOUND SUBTRACTION. 
 
 1. Ans. 6|- « — 11 6 + 6i§ d. 
 
 s. 
 
 <( 
 
 f«+8iJ+3ifc~lifg. 
 
 4. 
 
 <( 
 
 SI a — 51 I — ^ m. 
 
 5. 
 
 (( 
 
 11 a^J^^a^b^c + i a2 b^ c^ — 
 
 6. 
 
 (( 
 
 |«-3+.-..+^f + |^. 
 
 7. 
 
 (( 
 
 2a^26 
 
 8. 
 
 t( 
 
 a ,3c 
 2b~^'2d' 
 
 9. 
 
 a 
 
 3 ab -]- m n ,4 e/ — ap 
 
 
 3cd ' 4^A 
 
 10. 
 
 {( 
 
 2«6 — « Sh^gh 
 
 
 4 c d ' 3 m w 
 
 11. 
 
 (( 
 
 a2 „ 1 69 — 1 
 
 a ' 6 • 
 
 12. 
 
 ii 
 
 c^d^—b f^g^-\-e h^ — k 
 cd ^ fg ^ h 
 
 C. Multiplication. 
 
 a. ANSWERS TO THE EXERCISES IN MULTIPLICATION OF 
 SIMPLE QUANTITIES. 
 
 1. Ans. f a. 
 
 2. " I « = 2f a. 
 
 3. " -V- a & = 13| a 6. 
 3 
 
26 KEY. 
 
 4. Ans. J^^- «/= 1| af. 5. Ans. — \ ah. 
 
 6. '' M « h. 7. '' — ^^""'^^^ . 
 
 in <c 1 7. 7 ahcd 
 
 10. *' ^ ah c d^= . 
 
 Jl. '^ ^^ ah clef. 12. '^ ^abcdef. 
 
 13. '' — 1. 14. " 
 
 15. '' -L. 16. " 
 
 17. " -J—. 18. ** 
 
 a 6' 
 
 1 
 
 ah c 
 
 2 a c e 
 
 Shdf 
 
 7 ah d^f 
 
 a c 
 Vd' 
 ace 
 
 hdf 
 
 21. '♦ : -' . 22. '' 3. 
 
 o 
 
 23. '• — . 24. - 1^. 
 
 25. " J:^. 26. " ^'^''■' 
 
 Sabfg icPbp 
 
 27 .< *'«" 28. " ?^^' 
 
 b. ANSWERS TO THE EXERCISES IN COMPOUND 
 MULTIPLICATION. 
 
 1. Ans. ^ « & — i 62 c ^ _4_ 5 elf. 
 
 2. " a^ f/ __ 2^8 a fZ2 e — ^1 a J/^. 
 
 3. - fe? + |6-|-. 
 
KEY. 
 
 27 
 
 4 - l_i_j-i- 
 
 6 15^18 
 
 26 cb 24 7i 
 
 20 « z zr- 
 
 d 5 
 
 6 
 
 8 
 
 9. 
 
 15. 
 
 J 6. 
 
 Sb7l~~ 10 fZ3 "T 28 ^ f/ "^ 2 * 
 l_ c 2Ji_ _4_ 
 
 i «^ + ti « & — f « c — I- 62 + I 6 c. 
 
 10. - ^a2___2^^6_^8_52. 
 
 11. - |«4__7_^^2^,2_|_i64. 
 
 12. '' 6f ft2 _ i6| ^ 5 _j_ 93 j2. 
 
 13. - |a9__3_^5_j_3^c— f 62+J_86c-~ic2, 
 
 14. " 5 X4 + X « x3 — -23- ft2 2;2 _|_ 7 or3 :C — f ft X' 
 
 i«2+lft6+l62__l^c2. 
 
 xa^ + ^ab + iac — iad + ib^ + ibc 
 
 Division. 
 
 a. ANSWERS TO THE EXERCISES IN DIVISION OF SIMPLE 
 QUANTITIES. 
 
 1. Ans. 
 
 ft 
 
 2 6* 
 
 2. Ans. ^. 
 
 3. ^^ 
 
 ft 
 
 4. " ^— 
 
 6 c 
 
 5. '' 
 
 df 
 e 
 
 6. - f^^ 
 h 
 
28 KEY. 
 
 7. 
 
 <( 
 
 6ab d 
 c 
 
 8. 
 
 (( 
 
 b 
 
 a 
 
 9. 
 
 a 
 
 b 
 a 
 
 10. 
 
 C( 
 
 2a 
 3 62- 
 
 IL 
 
 t( 
 
 ^ab 
 5cd^' 
 
 12. 
 
 (( 
 
 2/m2 
 
 13. 
 
 (I 
 
 2 
 7)1 n 
 
 14. 
 
 tc 
 
 
 15. 
 
 t( 
 
 9c 
 IQbd' 
 
 16. 
 
 (( 
 
 ^g- 
 
 17. 
 
 n 
 
 ^rg^- 
 
 18. 
 
 <c 
 
 1 
 
 2?' 
 
 19. 
 
 a 
 
 3a2 64 
 
 7c3 
 
 20. 
 
 li 
 
 5 
 « 
 
 21. 
 
 (( 
 
 36 
 
 2 a 
 
 22. 
 
 l( 
 
 5c3 
 4a2 6 
 
 23. 
 
 i( 
 
 9d 
 
 2ab^' 
 
 24. 
 
 (( 
 
 5cd 
 cfib' 
 
 25. 
 
 (I 
 
 6 
 
 26. 
 
 (( 
 
 2 b c\ 
 
 E. Answers to the Exercises in Reductions of 
 Fractions to the Lowest Term. 
 
 1. 
 
 Alls. 
 
 2a 
 3 6' 
 
 2. 
 
 Ans. 
 
 2«2 
 
 3c 
 
 3. 
 
 li 
 
 3«2 
 4 6-2' 
 
 4. 
 
 a 
 
 2 a 
 c 
 
 5. 
 
 ii 
 
 a;3 
 3 2/^- 
 
 6. 
 
 (( 
 
 2 
 
 2y 
 
KEY. 29 
 
 7. 
 
 Ans. 
 
 a 
 3" 
 
 8. 
 
 Ans 
 
 31a 
 
 66* 
 
 9. 
 
 (( 
 
 2« 
 3 6c2' 
 
 10. 
 
 (( 
 
 7a2' 
 
 11. 
 
 (C 
 
 1 
 2 a 6* 
 
 12. 
 
 (( 
 
 a 
 
 2T* 
 
 13. 
 
 '^ 
 
 4 
 
 5a' 
 
 14. 
 
 (C 
 
 a 
 
 15. 
 
 (C 
 
 1 
 xy 
 
 16. 
 
 « 
 
 1 
 
 xy 
 
 17. 
 
 (e 
 
 2a2«/2. 
 
 18. 
 
 (( 
 
 5 ah. 
 
 19. 
 
 (( 
 
 4a c d 
 
 20. 
 
 li 
 
 ^ab cd. 
 
 F. Answers to the Eivercises in Raising Fractions to 
 Powers. 
 
 1. 
 
 Ans 
 
 a2 
 
 
 2. 
 
 Ans. 
 
 9a2 
 16 62* 
 
 3. 
 
 (< 
 
 27 a6 
 64 69' 
 
 
 4. 
 
 t< 
 
 I 
 a2' 
 
 5. 
 
 a 
 
 +i- 
 
 
 6. 
 
 <( 
 
 1 
 
 
 9 a^ 610* 
 
 7. 
 
 U 
 
 1 
 8a6' 
 
 
 6. 
 
 C( 
 
 27 
 64/6' 
 
 9. 
 
 (i 
 
 , 16 a^ 
 
 "*" 81 
 
 64 
 
 10. 
 
 C( 
 
 64 
 
 27 a6' 
 
 11. 
 
 (( 
 
 ^62 
 
 
 12. 
 
 ;< 
 
 -r 512- 
 
 13. 
 
 ti 
 
 + """- 
 
 612 
 
 ■"ai2- 
 
 14. 
 
 ii 
 
 a4 66 
 
KEY. 
 
 15. Ans. 
 
 17. 
 
 19. 
 
 £?16 620 
 + --• 
 
 5-4 
 
 6-4 a-12 6-12 
 
 16. Ans. 
 
 18. " 
 20. '' 
 
 27 gQ 66 
 64 J12" 
 
 ^27_ 
 
 64^6 66" 
 4^4 66 
 
 + 
 
 9c6 # 
 
 SECTION IV, 
 
 OF ROOTS. 
 
 A. Of Square Roots. 
 
 a. ANSWERS TO THE EXERCISES IN THE EXTRACTION OF 
 SQUARE ROOTS OF NUMBERS. 
 
 ]. 
 
 Ans. 
 
 4. 
 
 2. 
 
 Ans. 
 
 10. 
 
 3. 
 
 (( 
 
 11. 
 
 4. 
 
 (< 
 
 16. 
 
 5. 
 
 (( 
 
 24. 
 
 6. 
 
 tc 
 
 64. 
 
 7. 
 
 cc 
 
 98. 
 
 8. 
 
 ii 
 
 247. 
 
 9. 
 
 ii 
 
 763 
 
 10. 
 
 ii 
 
 504. 
 
 11. 
 
 a 
 
 194. 
 
 12. 
 
 ii 
 
 950. 
 
 13. 
 
 a 
 
 7563. 
 
 14. 
 
 ii 
 
 8276. 
 
 15. 
 
 a 
 
 5083. 
 
 16. 
 
 ii 
 
 15367. 
 
 17. 
 
 ii 
 
 40093. 
 
 18. 
 
 ii 
 
 279433. 
 
 19. 
 
 u 
 
 37695. 
 
 20. 
 
 a 
 
 203975. 
 
 21. 
 
 i( 
 
 40005. 
 
 22. 
 
 ii 
 
 6950078. 
 
 23. 
 
 u 
 
 3476905. 
 
 24. 
 
 ii 
 
 2.23606. 
 
 25. 
 
 (< 
 
 3.6055. 
 
 26. 
 
 ii 
 
 4.69041. 
 
 27. 
 
 a 
 
 9.79795. 
 
 28. 
 
 ii 
 
 12 . 36931 
 
 29. 
 
 i( 
 
 10.04987... 
 
 30. 
 
 << 
 
 2.76586. 
 
 31. 
 
 C( 
 
 3 . 09838 . . . 
 
 32. 
 
 ii 
 
 3.90357. 
 
KEY. 
 
 31 
 
 33. 
 
 Ans. 
 
 : 23664 . . . 
 
 34. 
 
 Ans. 
 
 . 08882 . 
 
 35. 
 
 a 
 
 . 05477 . . . 
 
 36. 
 
 (( 
 
 . 11832 . 
 
 37. 
 
 (( 
 
 h 
 
 38. 
 
 (( 
 
 f. 
 
 39. 
 
 a 
 
 f. 
 
 40. 
 
 (( 
 
 if. 
 
 41. 
 
 ic 
 
 \h 
 
 42. 
 
 <i 
 
 M- 
 
 43. 
 
 a 
 
 m- 
 
 44. 
 
 (( 
 
 Iff. 
 
 45. 
 
 t( 
 
 ^m- 
 
 46. 
 
 u 
 
 mn- 
 
 47. 
 
 iC 
 
 1 . 32287 . . . 
 
 48. 
 
 li 
 
 1 . 24721 
 
 49. 
 
 <c 
 
 1 . 80277 . . . 
 
 50. 
 
 l( 
 
 3 . 41869 
 
 51. 
 
 (( 
 
 2 . 92575 . . . 
 
 52. 
 
 a 
 
 2.71313. 
 
 53. 
 
 ii 
 
 2.88203. 
 
 54. 
 
 (( 
 
 1.29099 
 
 55. 
 
 (C 
 
 0.89442. .. 
 
 ^Q. 
 
 (( 
 
 1 . 29099 
 
 57. 
 
 'i 
 
 0.93541. .. 
 
 58. 
 
 it 
 
 0.64549 
 
 59. 
 
 a 
 
 . 24253 . . . 
 
 60. 
 
 i( 
 
 . 54772 
 
 ANSWERS TO THE EXERCISES IN THE EXTRACTION OF THE 
 SQUARE ROOTS OF ALGEBRAIC QUANTITIES. 
 
 1. 
 
 Ans. 
 
 a. 
 
 2. 
 
 Ans. 
 
 a\ 
 
 3. 
 
 (( 
 
 a\ 
 
 4. 
 
 a 
 
 a\ 
 
 5. 
 
 a 
 
 2a\ 
 
 6. 
 
 ii 
 
 4 «62. 
 
 7. 
 
 '( 
 
 8 a^ b^. 
 
 8. 
 
 « 
 
 10 a3 R 
 
 9. 
 
 li 
 
 9 ab c. 
 
 10. 
 
 u 
 
 11 a^bc^. 
 
 11. 
 
 <« 
 
 16a2 58. 
 
 12. 
 
 ({ 
 
 12 X g^ z\ 
 
 13. 
 
 (( 
 
 3 a^ bp g\ 
 
 14. 
 
 (C 
 
 a 
 
 15. 
 
 i< 
 
 ab 
 
 16. 
 
 i( 
 
 2a 
 
 b ' 
 
 17. 
 
 a 
 
 3« 
 
 4 62- 
 
 18. 
 
 it 
 
 2ab 
 3 c2 d^' 
 
32 
 
 KEY 
 
 21. 
 
 23. 
 
 25. 
 
 27. 
 
 29. 
 
 31. 
 33. 
 35. 
 37. 
 39. 
 
 41. 
 43. 
 
 a ' 
 
 20. Ans 
 
 I 
 
 22. '' 
 
 Sab^' 
 
 2 
 3"«~6* 
 
 24. '' 
 
 ^a^. 
 
 26. " 
 
 a+b. 
 
 28. ^' 
 
 b 
 «-~2- 
 
 30. '^ 
 
 2;— 1. 
 
 32. " 
 
 /3 - 3 x\ 
 
 34. '' 
 
 x^-—i-ax. 
 «^ + &4. 
 _rt__2^ 
 T 3 c' 
 
 « + 6 _|- c. 
 
 3G. 
 
 33. 
 40. 
 
 42. 
 44. 
 
 1 
 
 2«' 
 J_ 
 
 3^* 
 
 tT' 
 
 « — b. 
 
 x+L 
 P+3x^. 
 
 «^+ 
 
 ^2. 
 
 .5+ 
 
 2/^. 
 
 a 
 
 1 
 
 3 a; — 5 a + -jr-. 
 
 45. Ans. 2 a:2 _|- 2 « :c + 4 62. 
 
 46. 
 
 (( 
 
 Sa--b + 5c+d. 
 
 47. 
 
 a 
 
 3 _|_ 2 a; — 7 a:2. 
 
 48. 
 
 (( 
 
 3:,2 ''^'' + bx. 
 
 49. 
 
 a 
 
 a+b 
 X — a 
 
 50. 
 
 (C 
 
 a — 2b 
 
 n 1 r> -.O' 
 
 x2 + 3 a2- 
 
KEY. 
 
 B. Cube Roots. 
 
 ANSWERS TO THE CUBE ROOTS OF NUMBERS. 
 
 33 
 
 1. 
 
 Ans, 
 
 . 2. 
 
 2. 
 
 Ans. 
 
 4. 
 
 3. 
 
 a 
 
 5. 
 
 4. 
 
 a 
 
 7. 
 
 5. 
 
 a 
 
 9. 
 
 6. 
 
 (( 
 
 10. 
 
 7. 
 
 (( 
 
 12. 
 
 8. 
 
 c< 
 
 24. 
 
 9. 
 
 (( 
 
 23. 
 
 10. 
 
 ;< 
 
 96. 
 
 11. 
 
 (I 
 
 74. 
 
 12. 
 
 (( 
 
 55. 
 
 13. 
 
 (( 
 
 108. 
 
 14. 
 
 (( 
 
 135. 
 
 15. 
 
 (( 
 
 223. 
 
 16. 
 
 u 
 
 SO. 
 
 17. 
 
 (( 
 
 106. 
 
 18. 
 
 i( 
 
 258. 
 
 19. 
 
 (( 
 
 368. 
 
 20. 
 
 (( 
 
 343. 
 
 21. 
 
 (< 
 
 401. 
 
 22. 
 
 ie 
 
 200. 
 
 23. 
 
 (( 
 
 420. 
 
 24. 
 
 (( 
 
 683. 
 
 25. 
 
 (( 
 
 698. 
 
 26. 
 
 (C 
 
 1854. 
 
 27. 
 
 (( 
 
 1936. 
 
 28. 
 
 (( 
 
 4820. 
 
 29. 
 
 a 
 
 4865. 
 
 30. 
 
 a 
 
 2667. 
 
 31. 
 
 11 
 
 2009. 
 
 32. 
 
 a 
 
 1.9129. .. 
 
 33. 
 
 <( 
 
 2 . 2S9 12 . . , 
 
 , 34. 
 
 i( 
 
 4.3444.. . 
 
 35. 
 
 (C 
 
 6.4392.. . 
 
 36. 
 
 u 
 
 8.1981... 
 
 37. 
 
 (( 
 
 8 . 8237 . . . 
 
 38. 
 
 a 
 
 1 . 7967 . . . 
 
 39. 
 
 i( 
 
 2 . 1897 . . . 
 
 40. 
 
 i( 
 
 4 . 6856 . . , 
 
 41. 
 
 a 
 
 3 . 0455 . . . 
 
 42. 
 
 i( 
 
 1 . 78668 . , 
 
 43. 
 
 <i 
 
 i- 
 
 44. 
 
 (( 
 
 *• 
 
 45. 
 
 (( 
 
 . 4622 . . . 
 
 46. 
 
 (( 
 
 f. 
 
 47. 
 
 a 
 
 h 
 
 48. 
 
 a 
 
 f. 
 
 49. 
 
 <( 
 
 ^l- 
 
 50. 
 
 (( 
 
 37^. 
 
 5L 
 
 a 
 
 . 8735 . . . 
 
 52. 
 
 (( 
 
 0.94103, 
 
34 
 
 KEY. 
 
 53. Ans. 0.92S31 . , 
 55. '^ 0.69336. 
 57. " 1 . 48124 . 
 59. " 1 . 56049 . 
 
 54. Ans. 0.79370 
 56. " 1 . 14471 
 58. *' 0.70949 
 60. '' 2 . 50222 
 
 h. ANSWERS TO THE EXTRACTION OF CUBE ROOTS OF 
 ALGEBRAIC QUANTITIES. 
 
 1. 
 
 Ans. 
 
 h. 
 
 2. 
 
 Ans. 
 
 m. 
 
 3. 
 
 (( 
 
 a\ 
 
 4. 
 
 ii 
 
 — 3 « 63. 
 
 5. 
 
 (< 
 
 5 «2 & c3. 
 
 6. 
 
 
 7 « 62 c4. 
 
 7. 
 
 a 
 
 a 
 
 ~~ T' 
 
 8. 
 
 
 2 a 
 3c2' 
 
 9. 
 
 (I 
 
 1 
 
 a 
 
 10. 
 
 - 
 
 1 
 
 'Sab' 
 
 11. 
 13. 
 
 
 1 
 
 12. 
 14. 
 
 61 
 
 abc^d^. 
 a + b. 
 
 15. 
 
 <( 
 
 a—b. 
 
 16. 
 
 le 
 
 x + 2. 
 
 17. 
 
 (( 
 
 x + l. 
 
 18. 
 
 11 
 
 2 « — 7 z. 
 
 19. 
 
 li 
 
 x^ — 2 ex. 
 
 20. 
 
 (( 
 
 2— a;5. 
 
 21. 
 
 (( 
 
 6 ^ d- 
 
 22. 
 
 i< 
 
 1 1 
 
 a b ' 
 
 23. 
 
 li 
 
 a + b + c. 
 
 24. 
 
 li 
 
 X2 + X + 1 
 
 C. ANSWERS TO THE HIGHER ROOTS OF SIMPLE QUANTITIES. 
 
 1. Ans. 2. 
 
 3. " a 
 
 2. Ans. 3. 
 4 '* 12. 
 
KEY. 
 
 35 
 
 5. 
 
 Ans. 
 
 25. 
 
 6. 
 
 Ans. 
 
 4. 
 
 7. 
 
 i( 
 
 5. 
 
 8. 
 
 (( 
 
 4 « 62. 
 
 9. 
 
 (I 
 
 3c3- 
 
 10. 
 
 u 
 
 1 
 
 2a2- 
 
 1. 
 
 it 
 
 3 «2 53 c. 
 
 12. 
 
 (< 
 
 2«2 
 
 6^ c4' 
 
 D. Addition, Subtraction, Multiplication, and 
 Division of Roots. 
 
 a. ANSWERS TO THE EXERCISES IN ADDITION OF ROOTS, 
 
 1. 
 
 
 2^a. 
 
 
 2. 
 
 (( 
 
 2/2. 
 
 3. 
 
 
 7/3. 
 
 
 4. 
 
 (( 
 
 6v/6. 
 
 5. 
 
 
 2/2. 
 
 
 6. 
 
 (( 
 
 2Jv/5. 
 
 7. 
 
 
 2/3- 
 
 -1/5. 
 
 8. 
 
 K 
 
 Ii\/3 = f\/f- 
 
 9. 
 
 
 3/«. 
 
 
 10. 
 
 a 
 
 4Jv/«. 
 
 11. 
 
 
 4/a3. 
 
 
 12. 
 
 u 
 
 J^3- 
 
 13. 
 
 
 ¥Vf. 
 
 
 14. 
 
 (i 
 
 3/3 + 3 /;}. 
 
 15. 
 
 
 2/«. 
 
 
 16. 
 
 li 
 
 4/4 — 3/3. 
 
 17. 
 
 
 t/i- 
 
 
 18. 
 
 i( 
 
 fil/2- 
 
 19. 
 
 
 /7. 
 
 
 20. 
 
 u 
 
 */9 + i/5. 
 
 21. 
 
 
 16/2- 
 
 -4/7. 
 
 22. 
 
 a 
 
 17 /2. 
 
 23. 
 
 
 11/2- 
 
 -5/6 + 5/5. 
 
 
 24. 
 
 
 5f/a- 
 
 -i/«6 
 
 -1^ 
 
 .1/4: 
 
 
36 KEY. 
 
 b. ANSWERS TO THE EXERCISES IN" SUBTRACTION. 
 
 3/«. 
 2 /a. 
 
 5v/3. 
 
 1. 
 
 Ans 
 
 5. ^3. 
 
 2. Ans 
 
 3. 
 
 u 
 
 |/a6. 
 
 4. " 
 
 5. 
 
 7. 
 
 
 
 6. " 
 
 a '' 
 
 9. 
 
 a 
 
 -v/2. 
 
 10. '' 
 
 11. 
 
 ti 
 
 12^7 — 3 ^6 + 1/11. 
 
 12. 
 13. 
 14. 
 
 (t 
 
 /a + 8 ^b. 
 7[/ab + ±y/b c. 
 
 1/5. 
 
 15. 
 
 (( 
 
 /« + 5^6. 
 
 
 16. 
 
 (( 
 
 5 v/«=^ + 2 /a2 5 
 
 — 9/c 
 
 C. ANSWERS TO THE ABBREVIATIONS AND TRANSFORMATIONS 
 
 1. Ans. 5/6. 2. Ans. 13/2. 
 
 3. 
 
 5. 
 
 7. 
 
 9. 
 11. 
 12. 
 
 5/3. 4. " —10/2. 
 
 21/3 = 1/3. 6. - 5i/3 = VV3. 
 
 — S/2. a '' 21/2. 
 
 5/2- 10. '' 1/2. 
 
 1/2. 
 
 I /2 = /2, (^ /^ being equal to ^ . ^ /2.) 
 
KEY. 
 
 37 
 
 d. MXTLTIPLICATIOW OF ROOTS, 
 ANSWERS TO THE MULTIPLICATION OF SIMPLE QUANTITIES. 
 
 1. 
 
 Ans. 
 
 2. 
 
 
 2. Ans. 3. 
 
 3. 
 5. 
 
 7. 
 
 
 /lO. 
 
 6/30. 
 
 a. 
 
 
 4. " 
 
 6. ' 
 
 8. ' 
 
 2/30. 
 
 42/12 
 
 6. 
 
 9. 
 11. 
 
 
 s/ah. 
 a c /6 d. 
 
 
 10. '^ 
 12. ' 
 
 5 /a c. 
 acd. 
 
 13. 
 
 C( 
 
 a. 
 
 
 14. ' 
 
 ' 6. 
 
 15. 
 
 
 ^ah. 
 
 
 16. ' 
 
 3 
 
 «c/6. 
 
 17. 
 
 
 a c /6 d, 
 
 
 18. ' 
 
 15 a 62, 
 
 19. 
 21. 
 23. 
 25. 
 
 
 15 /72. 
 45/3. 
 
 12 /f 
 
 /15. 
 
 20. ' 
 22. ' 
 24. ' 
 26. ' 
 
 12/4. 
 
 /A- 
 3/3. 
 
 27. 
 
 
 ae > 
 
 7 -1 ' 
 
 28. ' 
 
 ^ . /- 
 
 29. 
 
 30. 
 31. 
 
 32. 
 33. 
 34. 
 
 1, (because \/ - X /« = 1 y^-=/l. 
 
 1. 
 12. 
 
 I c |/ 6 c? ac /6 d 
 
 
 6/12- 
 1 
 
 8/15- 
 
38 KEY. 
 
 ' ,2 
 
 35. An. y/"^. 36. Ans. y/?^^. 
 
 40. " \/a. 
 
 41. " ab. 
 
 3 
 
 / \ ac 
 
 3 1 
 
 44. " v/4 = — • 
 
 45. " ^^s^=/J = i. 
 
 46. " -V- = 5, (because /^V X /3 = /2V-) 
 
 47. " l/«^ = \/«. 48. Ans. \/a h^ c. 
 49. " 12/30. 50. " i/18, 
 
 51. " /a 6 erf. 
 
 52. '' a 62 c c y/rf. 
 
 53. '' nab. 
 
 54. '* 4^a2fec2. 
 
 55. " ^abc^de^f. 
 
 ^^- ^' '\/^ = 
 
 57 " - 
 57. ^. 
 
 
KEY. 39 
 
 58. Ans. ^^ 
 
 r s/cP fy/c 
 
 59. " i-. 
 
 60. " §/§ = J^/l=l. 
 
 2. ANSWERS TO THE MULTIPLICATION OF COMPOUND QUANTITIES. 
 
 1. Ans, 3/2 + ^10. 2. Ans. 6/2 + 8/5. 
 
 3. " 6 + 4/10. 4. *' 3/3+/15— 6. 
 
 5. " 6 — 5 = 1. 
 
 6. '' 2 — /3__3=:--/3 — 1. 
 
 7. '' 3 — 17/6. a Ans. 42 — 13/6. 
 9. '' 3—2/2. 10. " —3 + 3/7. 
 
 11. " 24^. 12. '' 41. 
 
 13. " 1+/6. 14. '^ 75 + 11/15. 
 
 15. '' 5 + 2/6. 16. '' a^ — h. 
 
 17. '' a + 2 /« 6 + 6. 
 
 18. " c^ « — c d /« 6 + a c /a h — a dh. 
 
 19. " — 62. 20. Ans. 87 + 12 /42. 
 
 21. " 87 — 12/42. 22. " /4 — 4/9. 
 
 23. " /i-/^. 24. - y-^- 
 
 e. ANSWERS TO THE EXERCISES IN DIVISION. 
 
 1. Ans. 1. 2. Ans. /a^— :«. 
 
40 
 
 
 KEY. 
 
 5. An 
 
 s. 3 a. 
 
 6. Ans. 2 ^f . 
 
 7. '' 
 
 •\/'- 
 
 ^- " V\/l- 
 
 9. '' 
 
 v/«. 
 
 10. " /«. 
 
 11. " 
 
 2/a62_2j^/( 
 
 L 12. " ^a^b^-yab—ab 
 
 13. '' 
 
 v^:^- 
 
 - " V^' 
 
 15. " 
 
 np y/ 6 c 
 
 ^«' " v/^- 
 
 17. '' 
 
 /| = y/f=^/|.6 = i/6. 
 
 18. " 
 
 /I- 
 
 
 19. '' 
 
 fv/*- 
 
 
 20. " 
 
 |/| = lv/f = 
 
 = tVv/6. 
 
 21. '' 
 
 1. 
 
 3 
 
 22. Ans. ^a. 
 
 23. ' 
 
 3 
 
 24. '' 2 /«. 
 
 25. '' 
 
 2/«. 
 
 3 
 
 27. '^ 
 
 ^V/t- 
 
 ^^- " tV/^- 
 
 29. - 
 
 \/a^- 
 
 30. " /a3 : ^« = l/a^. 
 
 31. " 
 
 3 v/«2 6. 
 
 
 32. " 
 
 \/a^b^-yab = 
 
 = ^a^ b^. 
 
 33. " 
 
 
 34.A„s.2//^. 
 
KEY. 41 
 
 35. Ans. ^ » y^. 36. Ans. t /A 
 
 np \/ be ^ a 
 
 37. - v/f- ^^- " V/t- 
 
 39. - f/|. 40. - i^f. 
 
 ^. Fractional Exponents. 
 
 1. 
 
 Ans. 
 
 a*. 
 
 2. 
 
 Ans. 
 
 «t 
 
 3. 
 
 
 «». 
 
 4. 
 
 Cf 
 
 «f. 
 
 5. 
 
 
 ««. 
 
 6. 
 
 (S 
 
 a*. 
 
 7. 
 
 
 «*. 
 
 8. 
 
 (I 
 
 «i. 
 
 9. 
 
 
 a*=(/a5. 
 
 10. 
 
 
 a* = i/a^. 
 
 11. 
 
 
 a*=\/a7. 
 
 12. 
 
 (( 
 
 J^ =^a". 
 
 13. 
 
 
 a*='^a^. 
 
 14. 
 
 <( 
 
 a"«'''=/a". 
 
 15. 
 
 
 a^i =/«". 
 
 
 
 
 16. 
 
 
 a^ X «^ = «. 
 
 
 
 
 17. 
 
 
 a^=«^=^a^ 
 
 \^ 
 
 
 
 18. 
 
 
 a^=a^==/a= 
 
 3. 19. 
 
 Ans. 
 
 a' = t/a. 
 
 20. 
 
 
 aT^=/a. 
 
 
 
 
 21. 
 
 
 at:«t = ai = 
 
 Va. 
 
 
 
 22. 
 
 
 a ==:^« 
 
 
 
 
 4* 
 
42 KEY. 
 
 H. 
 
 23. Ans. d^^ :=\/a. 24. Ans. tti^—^^u 
 
 2 14 3. 2 1, 
 
 25. " «3 5^c3^ 26. " a^b^c^. 
 
 27. " a^A 28. " a^c?. 
 
 29. '' a^ b^ = /a5 54 __ ^^, ^454^^5 ^^^ 
 
 30. '' a^^ 51-1 ^ ^^17 ftio ^ ^ ^^5 510. 
 
 31. '' aibi = ^ab. 
 
 32. " a^b^: Jb^=za^b^=^ab. 
 
 F. Powers and Roots of Roots. 
 
 1. Ans. a. 
 3. " a 6. 
 5. " 5. 
 
 6* ==: ^63. 
 
 2. 
 
 Ans. 
 
 b. 
 
 4. 
 
 (( 
 
 2. 
 
 6. 
 
 (( 
 
 3. 
 
 8. 
 
 (( 
 
 a~^^ == ^ai2 ^ a^.. 
 
 9. " a^ = / «9 = ^«^.a = a"^ s/a, 
 
 10. '' }^a>^. 
 
 11. '' y/a4 = ^a3 . a = a /a. 
 
 3 i 4 
 
 12. " i/«^^ = a^. 13. Ans. a* == \^a. 
 
 14. " «^ = /a. . 15. " a^ = ^a3. 
 
 16. - J = v/«5. 17. '' «^'^=:{/a. 
 
 18, '' a^ = a3_,^«. 19. ^^ ^aS. 
 
 20. " v/«^. 21 " /«^. 
 
KEY. 
 
 43 
 
 22. Ans. ^a. 23. Ans. \^a. 
 
 24. " l/a. 
 
 Note. When the pupil has once understood the reason 
 why, in extracting the root of a root, we multiply the rad- 
 ical exponents, he need no longer write them in form of 
 fractional exponents ; but at once multiply the two or more 
 radical exponents. 
 
 G. Imaginary Quantities, 
 
 ADDITION. 
 
 1. Ans. 6 /^^^. 2. Ans. 3 \/^^^ 
 
 3. *' 
 
 7 /- 1 - ^3. 
 
 4. " 
 
 7/-l+^_4=7/-l+v/4.-l 
 
 
 = 7/-l+2^-l = 9/-l. 
 
 5. " 
 
 5 ^— 1 +/— 6 = 5 /— 1 4-^6. — 1 
 
 6. " 
 
 = 5/-l+/6/-l 
 
 
 SUBTRACTION. 
 
 1. Ans. 
 
 0. 2. Ans. 7^— 1. 
 
 3. " 
 
 — 7/—!. 4. '' —/—I. 
 
 5. *' 
 
 /-l + l. 
 
 6. " 
 
 /— 1 + 3 /— 3 — 16. 
 
 
 MULTIPLICATION. 
 
 1. Ans. 
 
 — 1. 2. Ans, 3 /— 1. 
 
 3. *' 
 
 a b \/— 1. 4. «f — 6. 
 
 5. *' 
 
 — \/ah. 6. " _cJ^a6. 
 
44 KEY. 
 
 DIVISION. 
 
 1. Ans. 1. 2. Ans. — 
 
 c 
 
 3. ^' -/~l. 4. - ---^^-1. 
 
 5. << _2/— 1. 6. '' 2. 
 
 SECTION V. 
 
 OF LOGARITHMS. 
 
 1 . Ans. The logarithm of a product, A B, is taken by 
 
 taking the sum of the two logarithms, log A -\-\og B. 
 
 A 
 
 2. Ans. The logarithm of a fraction —is taken, by sub- 
 
 tracting the logarithm of the denominator from the 
 logarithm of the numerator ; thus : log A — log B. 
 
 3. Ans. The logarithm of a power A^ is taken by mul- 
 
 tiplying the logarithm of the basis by the exponent ; 
 thus : 3 X log A. 
 
 3 
 
 4. Ans. The logarithm of a Root, ^A, is taken, by di- 
 
 viding the logarithm of the quantity under the rad- 
 
 ical by the radical exponent ; thus : — - — . 
 
 ANSWERS TO THE APPLICATIONS. 
 
 1. Ans. 5.5142847. 2. Ans. 5.6758471. 
 3. " 5.4908066. 4. " 10.4263942. 
 
KEY. 
 
 45 
 
 5. 
 
 Ans 
 
 . 0.09G9100. 
 
 6. 
 
 Ans. 
 
 , 0.7459666. 
 
 7. 
 
 iC 
 
 . 6690068. 
 
 8. 
 
 
 1 . 1972806. 
 
 9. 
 
 a 
 
 0.85S193. 
 
 10. 
 
 
 2.5654050. 
 
 11. 
 
 i( 
 
 2 . 2737376. 
 
 12. 
 
 ec 
 
 0.8239087 — 1. 
 
 13. 
 
 a 
 
 0.7958800—1. 
 
 14. 
 
 
 0.0457575— 1. 
 
 15. 
 
 'C 
 
 0.698970 — 2. 
 
 16. 
 
 
 0.2099495— 1. 
 
 17. 
 
 (( 
 
 0.7106834—1. 
 
 18. 
 
 'C 
 
 0.6480628 — 1. 
 
 19. 
 
 (( 
 
 . 5440680. 
 
 20. 
 
 
 1 . 1014034. 
 
 21. 
 
 (( 
 
 . 1324838. 
 
 22. 
 
 
 0.8450980 — 1. 
 
 23. 
 
 << 
 
 0.5563025 — 2. 
 
 24. 
 
 
 0.8129134 — 3. 
 
 25. 
 
 a 
 
 0.7530480—4. 
 
 26. 
 
 
 1 . 247857. 
 
 27. 
 
 11 
 
 0.S5S5798 — 4. 
 
 28. 
 
 
 1.381037. 
 
 29. 
 
 <( 
 
 7.1568188.* 
 
 30. 
 
 
 18.8721901. 
 
 31. 
 
 (( 
 
 24 . 0823997. 
 
 32. 
 
 
 1 . 162920. 
 
 33. 
 
 i( 
 
 5 . 1516750. 
 
 34. 
 
 
 . 9943665. 
 
 35. 
 
 (I 
 
 12 . 1667597. 
 
 36. 
 
 
 . 358635 — 3. 
 
 37. 
 
 (C 
 
 . 142320 — 8. 
 
 38. 
 
 
 0.2518379—4. 
 
 39. 
 
 <( 
 
 . 3494850. 
 
 40. 
 
 
 1 . 065167. 
 
 41. 
 
 ii 
 
 0. 7101112. 
 
 42. 
 
 
 . 5230012. 
 
 43. 
 
 (( 
 
 . 9090787. 
 
 44. 
 
 
 0.0111394. 
 
 45. 
 
 (( 
 
 0. 13380,. 
 
 46. 
 
 
 . 0820045. 
 
 47. 
 
 '< 
 
 0.823908 — 1, which 
 
 num 
 
 ber correspondsrto 
 
 
 
 
 
 
 0.666... or" |. 
 
 48. 
 
 (C 
 
 . 5958482. 
 
 
 
 
 If logarithms to 7 decimals only are employed, the above an- 
 swers will not always be obtained. The last figure, therefore, may 
 vary from the results. ^ 
 
46 
 
 KEY. 
 
 ANSWERS TO THE ACTUAL CALCULATIONS OF SOME 
 
 NUMERICAL EXPRESSIONS BY MEANS OF 
 
 LOGARITHMS. 
 
 I. 
 
 Ans. 1.34590. 
 
 2. 
 
 Ans. 
 
 13.702. . 
 
 3. 
 
 '' 2.05528 . . . 
 
 4. 
 
 (I 
 
 2.4855 . . 
 
 5. 
 
 *' 0.9593 . . . 
 
 6. 
 
 et 
 
 1.1907. . 
 
 7. 
 
 " 1.9042 . . . 
 
 8. 
 
 <t 
 
 1.14605 . 
 
 9. 
 
 '' 1073742500. 
 
 
 
 
 [The incorrectness of this and the following answer is 
 occasioned by the impossibility of taking correctly the log- 
 arithms of numbers written with more than eight figures, 
 in tables, calculated only to 5, 6, or 7 decimals ; taking 
 therefore the corresponding numbers in the book, we must 
 add as many cyphers on the left, as the index requires.] 
 
 10. Ans. 2176760000. 11. Ans. 11.8632... 
 
 12. '' 11767.3. 13. " 3.1681 . . . 
 
 14. " 31.714... 15. '' 1.443779... 
 
 16. '' 0.0000305 . . . 
 
 17. " 0.0000000002328 . . . 
 
 18. " 0.05631 ... 19. Ans. 0.23256. . . 
 20. " 4.29980... 21. *^ 17.75783... 
 22. " 44879 ... 23. '^ 2.22818 . . . 
 24. " 8.22503 . . . 
 
KEY. 47 
 
 SECTION VI. 
 
 ANSWERS TO THE EXERCISES IN THE 
 BINOMIAL THEOREM. 
 
 1. Ans. a -\-h. 
 
 2. '' « + 6. 
 
 3. '^ ^2 _|_ 2 « 6 -f h^. 
 
 4. '' «2 _|_ 2 a 6 + hK 
 
 5. '^ a3 _|_ 3 «2 5 _|_ 3 « ^,e _|_ ^,3. 
 
 6. '' a3 + 3 a2 6 + 3 « ^>2 _|_ &3. 
 
 7. " o4 _|_ 4 ^3 6 _|_ 6 ^2 52 _[_ 4 « 53 ^ 54. 
 
 «4 _ 4 ^3 5 _[_ 6 «2 ^2 __ 4 ^ 53 _j_ 54, 
 
 a5_5^45_[_ J0a3 62_{0a2i3_^5^^,4__j5_ 
 
 9. '' a^^Q a^h-\- 15 «4 62 _|_ 20 a? b^ -\- 15 a^ 6^ 
 
 -f 6 a 65 + 66. 
 a6 — 6 a5 6 + 15 a^ h^ — 20 «:^ 63 _|_ 15 ^2 54 
 
 — 6 « 65 + 66. 
 
 10. i' a7 + 7 a6 6 + 21 «5 6^ + 35 «4 53 _|_ 35 ^3 54 
 
 + 21 a2 65 _|_ 7 ^ J6 _^ 57. 
 
 a^ — 7 a6 6 + 21 a^ 6^ — 35 a^ 6^ + 35 a? 6^ 
 — 21 a2 65 _|_ 7 ^ 5G _ 67. 
 
 11. " a8 _f_ 8 a7 6 -|- 28 a^ 62 _|- 56 a^ 6^ + 70 «4 ^4 
 
 + 56 a? 65 + 28 a^ 6^ + 8 « 6^ -)_ 68. 
 
 a8 „ 8 «7 6 4- 28 «6 62 _ 55 ^5 63 _f. 70 ^4 6^ 
 
 — 56 a3 65 + 28 «2 6G -_ 8 a ¥ ~f 68. 
 
48 
 
 KEY. 
 
 12. Ans. a9 + 9 a8 6 + 36 aU2 4- 84 a^ b^ + 126 a^ h^ 
 
 + 126 a4 ^5 _|_ 84 a? 6^ _J_ 36 ^2 ^,7 _j_ 9 « ^8 
 
 + 69. 
 
 «9 _ 9 ^8 ?; _j_ 3(5 ^7 i^ _ 84 «6 63-1-126 a^ M 
 
 — 126 a'l 65 _j_ 84 «3 6^ — 3G «2 57 _j_ 9 ^^ 58 
 
 — 69. 
 
 13. ^' «^o+lO «9 6 + 45 a8 fe2_^i20 a^ 63+210 «« fe4 
 
 + 252 a^ 65 + 210 a^ 6^ -[- 120 ^3 ^7 _^ 45 «2 58 
 
 + 10 « 69 + 610. 
 «io_io ft9 5_|_45 «8 1^—120 «' 63+210 a^ 6^ 
 — 252 a5 65 _|_210 «4 66_ 120 «3 ^,7 .^45 ^258 
 
 — 10 « 69 + 619. 
 
 14. '' l + 3x + 3a;2+a;3. 
 
 1 _ 3 a; + 3 2:2 — .t3. 
 
 15. " 1 +4a; + 6x2 + 4 x3 + x4. 
 
 1 _ 4 a; + 6 2;2 _ 4 a;3 + a;4. 
 
 16. ^« 1 + 5 a; + 10 x2 + 10 a;3 + 5 a;4 + x\ 
 
 1 __ 5 2; + 10 x2 ~ 10 x3 + 5 a;4 — x^. 
 
 17. <^ 32+240 x+720 x2+1080 x3+8I 2;4+243 x\ 
 
 32—240 x5+720 x^— 1080 ^3+810 x4_243 x^. 
 
 18. " 625 + 2000 X + 2400 x^ + 1280 x^ + 256 x^ 
 
 625 — 2000 X + 2400 x^ — 1280 a;3 + 256 xl 
 
 19. '' _i_ a;4 + 0:3 y + 6 a;2 ?/2 + 16 a; ?/3 + 16 3,4. 
 
 Jg a:4 — a:3?/ + 6 x2 y2 _ 16 x ?/3 + 16 i/\ 
 
 20. *' 243 + 810 x^ + 1080 x^ + 720 x^ + 240 x^ 
 
 + 32 xio. 
 
 243 — 810 x2 + 1080 x^ — 720 x^ + 240 x^ 
 
 — 32 xi9. 
 
49 
 
 21. Ans, d' c2 + 12 « 6 c ci + 4 6^^ d^, 
 
 () ^,2 c^ — 12 rt 6 c 6/ + 4 6-2 6/2. 
 
 22. '* « 4- 2 ^« 6 + 6. 
 
 a — 2 y^a b -\- b. 
 
 23. " a2 + G a 6 + 62 + (4 a + 4 6) y/a 6. 
 
 ^2 _|_ (5 ,^ ^, _|_ ^2 _ (4 rt _|_ 4 fe) ^a 6, 
 
 24. " «2 ^ _^ 2 « ^ y/6 r/ + c'^ J. 
 
 a^ 6 — 2 (i c /6 f/ -(- c^ d. 
 
 ANSWERS TO THE EXERCISES TN" THE APPLICATION' OF THK 
 
 GENERAL, f OMUL A TO THE EXTRACTION OF 
 
 IMPERFECT ROUTS. 
 
 I.Ans. l+|__+|^_-^-±&c... 
 
 3: x"^ x^ 5 .r"* 
 
 2 " I &LC . . . 
 
 2 8 l(i 128 
 
 3. 
 
 ^ 'S 9 ~ dl 24;i ~ 
 
 X x^ 5.r3 ]0j4 . „ 
 4 " 1 \- &c . . . 
 
 6. - l_^__3_^o__7_,.3__ri_:,4_&e.. 
 
 7. " /« + - 
 
 i/a 8 a \ya ' 1 G «- ^a 
 
 ■~128a^/a"'~*^^'-* ' 
 
50 KEY, 
 
 7. 2- X^ 
 
 8. Ans. v/« - .-7^ - g-;7T7^ - Tg-s-7-« 
 
 3 
 
 &/C . . 
 
 128 a^ \/a 
 
 '2a 8 a-^ ' JGa^ i28a^ 
 
 10. " a r T-T-T-^ — &;C . . . 
 
 2 a. 8 a-* ib £^-^ i'^8 a' 
 ,, ,, , a-3 j6 5x9 ] 0,-12 
 
 f_ xs __ 5.7 9 iOx^a 
 
 SECTION VI I. 
 
 A. Answers to the. Exercises in Arithmetical 
 Frogrissions, 
 
 1. Ans, The 14th term is 14, and the sum of all 14 terms 
 
 is 105. 
 
 2. " The 13th term is 38, and the sum of all the 
 
 terms is 2tJ0. 
 
 3. " The 20lh term is 41, and the sum of all 20 terms 
 
 440. 
 
 4. " The lOCth term is 1C3, and the sum of all the 
 
 terms is 5350. 
 
 5. " The 25th term is 125, and the sum of all the 
 
 terms is 1625. 
 
KEY. 51 
 
 6. Ars. The 10th term is IH, and the sum of all the 
 
 terms is 92^ 
 
 7. '' The 16th term is 10|, and the sum of all the 
 
 terms is 142. 
 
 8. ** The 10th term is 2f , and the sum of all the 
 
 terms is ]6|. 
 
 9. " The 12th term is ^i, and the sum of all the 
 
 terms is 15. 
 
 10. " The 100th term is 35^, and the sum of all the 
 
 terms is 1775. 
 
 11. '* The 2Gth term is ^, and the sum of all the 
 
 terms is G0^« 
 
 12. '' The ICth term isCf, and the sum of all the 
 
 terms is 45^. 
 
 13. ** The 14ih term is Gi, and the sum of all the 
 
 terms is 45^. 
 
 14. " The 32d term is C^^j, and the sum of all the 
 
 terms is 75^ 
 
 15. '' The 30th term is 5^-, and the sum of all the 
 
 terms is 1I4|-. 
 IG. " The Sth term is of, and the sum of all the 
 
 terms is 14f. 
 
 17. " The IGOth term is ICOO, and the sum of all the 
 
 terms is 50500- 
 
 18. ^- The SC'th term is 12^, and the sum of all the 
 
 terms is 325. 
 
 19. " The 22d term is 7J, and the sum of all the 
 
 terms is 8S. 
 
 20. '' The 10th term is 21}, and the sum of all the 
 
 terms is IGj. 
 21 „ " The 14th term is of, and the sum of all the 
 
 terms is 29f . 
 
53 
 
 KEY. 
 
 22. Ans. The 12th term is 10, and the sum of all the 
 
 terms is 54. 
 
 23. " The 2oth term is 63, and the sum of all the 
 
 terms is 825. 
 
 24. " The 40lh term is S^-, and the sum of all the 
 
 terms is 257^. 
 
 B. Answers to the Exercises in Geometrical Progression. 
 
 1. Ans. The 7th term is 64, and the sum of all the 
 
 terms is 127. 
 
 2. " The 1 0th term is 78732, and the sum of all the 
 
 terms is i 18(]9o. 
 
 3. '' The 9th term is 3276S0, and the sum of all the 
 
 terms is 4:J6905. 
 
 4. " The ICth term is 196S3, and t!ie sum of all the 
 
 terms is 29524. 
 
 5. " The 12th term is 83SS608, and the pum of all 
 
 the lerms is 11184810. 
 
 6. " The lOlh term is -^J-^, and the sum of all the 
 
 tnrms is l-|yj. 
 
 7. " The 6th term is yoVj, and the sum of all the 
 
 terms is ly^oV?-. 
 
 8. " The 8th term is -q\, and the sum of all the 
 
 terms is 3g^|-. 
 
 9. " The 9th term is t6^84> ^"<^ ^'^^ sum of all the 
 
 terms is 5^^^. 
 
 10. " The 7th term is 258f 3^^^ and the sum of all 
 
 the terms is 59!;^^^*^^. 
 
 11. " The 8th term is lC6|^f , and the sum of all the 
 
 terms is 307|f^. 
 
53 
 
 12. Ans. The 6th term is lfi|- and the sum of all the 
 
 terms is 19f^|. 
 
 13. " The 4th term is ^^, and the sum of all the 
 
 terms is 7§J-. 
 
 14. '' The 15th term is ^qVs' ^"^ ^^^ ^""^ ^^ ^^^ the 
 
 terms is 15|g||. 
 
 15. " The 12th term is -s^siVss* ^"^ t^® ^^"^ ^^ ^^^ 
 
 the terms is 10||f|-||. 
 
 16. " The 8th term is TVc^aj ^"d the sum of all the 
 
 terms is QUij^. 
 
 17. *' The 6th term is 15y%, and the sum of all the 
 
 terms is 41 j^g. 
 
 18. " The 4th term is ^, and the sum of all the terms 
 
 is 4|. 
 
 19. " The 4th term is f , and the sum of all the terms 
 
 20. '' The 6th term is g'^, and the sum of all the 
 
 terms is |f . 
 
 21. " The 4th term is g^^ and the sum of all the 
 
 terms is |^. 
 
 22. '' The 6th term is ^^Ve? ^^^ ^^^ ^""^ ^^ ^^^ ^^^ 
 
 terms is ^§f|. 
 
 23. *' The 6th term is 15 0^55 and the sum of all the 
 
 terms is t%9_o_6_, 
 
 24. " The 6th term is T^Ve' ^"d the sum of all the 
 
 terms is "^^^^q' 
 
 5* 
 
54 
 
 KEY. 
 
 SECTION VIII. 
 
 Equations. 
 
 a. SIMPLE EQUATIONS. 
 a. ANSWERS TO THE EQUATIONS WITH ONE UNKNOWN QUANTITY, 
 
 1. 
 
 Ans. 
 
 x= 12 — d. 
 
 2. 
 
 Ans. 
 
 x = 5. 
 
 3. 
 
 tt 
 
 x = 8. 
 
 4. 
 
 (< 
 
 a: = 6. 
 
 5. 
 
 i( 
 
 x = 5. 
 
 G. 
 
 (c 
 
 2; =4. 
 
 7. 
 
 (( 
 
 a:= 3. 
 
 8. 
 
 il 
 
 2; = G. 
 
 9. 
 
 (( 
 
 x = 7. 
 
 10. 
 
 ii 
 
 x== 3. 
 
 11. 
 
 (C 
 
 x= 10. 
 
 12. 
 
 " 
 
 a: == 3. 
 
 13. 
 
 (( 
 
 x = G. 
 
 14. 
 
 << 
 
 x = 4. 
 
 15. 
 
 (< 
 
 x = 4. 
 
 1(3. 
 
 a 
 
 X = 5. 
 
 17. 
 
 (( 
 
 a; = 5. 
 
 18. 
 
 a 
 
 x = n. 
 
 19. 
 
 li 
 
 x = G. 
 
 20. 
 
 (I 
 
 x = 4. 
 
 21. 
 
 ii 
 
 x = 5. 
 
 22. 
 
 i( 
 
 a: = 8. 
 
 23. 
 
 (( 
 
 x = 3. 
 
 24. 
 
 i( 
 
 x = 2. 
 
 25. 
 
 (( 
 
 X = 2. 
 
 2(3. 
 
 (i 
 
 2 = 4. 
 
 27. 
 
 a 
 
 X ==.]!. 
 
 28. 
 
 11 
 
 a; = yf y. 
 
 29. 
 
 (( 
 
 X = 00. 
 
 30. 
 
 a 
 
 x = 48. 
 
 31. 
 
 t( 
 
 x = 48. 
 
 82. 
 
 11 
 
 a; = 24. 
 
 33. 
 
 (( 
 
 X = 30. 
 
 34. 
 
 <( 
 
 X = 9. 
 
 35. 
 
 (( 
 
 j= 1. 
 
 3G. 
 
 il 
 
 a; = — 60. 
 
 37. 
 
 (( 
 
 x=li 
 
 38. 
 
 It 
 
 a:=14|. 
 
 39. 
 
 (( 
 
 X = 1391 
 
 40. 
 
 a 
 
 a: = 4. 
 
 41. 
 
 (( 
 
 x = 10A. 
 
 42. 
 
 (( 
 
 z = GGf . 
 
55 
 
 43. Ans. x- = 2^8. 44. Ans. x = -l]-\^-. 
 
 45. 
 
 (( 
 
 x= IK^^f. 
 
 46. 
 
 (( 
 
 2 = 2571.428... 
 
 47. 
 
 'i 
 
 x= 10.611... 
 
 4S. 
 
 (( 
 
 x= 2.0104.. . 
 
 49. 
 
 (i 
 
 x = G:^^.'J22... 
 
 50. 
 
 ii 
 
 a: = — 519.675... 
 
 51. 
 
 (t 
 
 h 
 a 
 
 .52. 
 
 (( 
 
 c 
 
 "-— « + 6- 
 
 53. 
 
 i( 
 
 d—c 
 a 
 
 54. 
 
 (( 
 
 d — c 
 
 55. 
 
 t< 
 
 ^ = iV. 
 
 56. 
 
 t< 
 
 _ 1 
 
 u 
 
 67. 
 
 (I 
 
 x = i. 
 
 5S. 
 
 K 
 
 a 
 
 59. 
 
 li 
 
 3 
 
 60. 
 
 (( 
 
 x == 1 — f a. 
 
 61. 
 
 {{ 
 
 a 
 
 6-2. 
 
 (1 
 
 x = 4. 
 
 03 
 
 (< 
 
 c 
 
 64. 
 
 <( 
 
 c-\-d 
 
 
 "- a-{-b + e' 
 
 65. 
 
 (t 
 
 X = 3. 
 
 06. 
 
 (( 
 
 a-\-b 
 c 
 
 67. 
 
 (I 
 
 r, -i-b+\ 
 '= cci ' 
 
 63. 
 
 (i 
 
 X = ^%. 
 
 69. 
 
 (( 
 
 d— 1 
 
 70. 
 
 (( 
 
 ac—2cr-bc 
 
 
 a — b — c 
 
 c — ad 
 
 71. 
 
 it 
 
 x= 16. 
 
 72. 
 
 (( 
 
 x = \'23. 
 
 73. 
 
 It 
 
 X = a^. 
 
 74. 
 
 ti 
 
 X = a^. 
 
 b. ANSWERS TO THE EQUATIONS WITH TWO UNKNOWN QUANTITIES. 
 
 1. Ans. z = 7, ?/ = 3. 2. Aiis. x = 10, y = A. 
 
56 
 
 KEV. 
 
 3. Ans. 
 
 X = 6|, 7/ = 61 
 
 4. " 
 
 ^ = ^h y==Qh 
 
 5. <' 
 
 X = 16, ij = 35. 
 
 6. " 
 
 a; = 4, y = 5. 
 
 7. " 
 
 X = 5, ?/ = 4. 
 
 8. '' 
 
 x = 32, ?/= — 21. 
 
 9. " 
 
 x = 6, 7/ = 4. 
 
 10. " 
 
 ^ = 6J§, y==12/^. 
 
 11. " 
 
 .T = 88f , ij = 17|. 
 
 12. *' 
 
 2;= 24, y=18. 
 
 13. '' 
 
 ^ = ^> y = u- 
 
 14. '' 
 
 z= 12.6550, 3/ = 6.0750. 
 
 15. " 
 
 a: = —0.278 . . 7/ = — 4.984 . . . 
 
 16. '' 
 
 a 4- b a — b 
 
 17. '' 
 
 b — c b d — a c 
 
 
 "" a — d' ^ d — a ' 
 
 18. '' 
 
 f^ — ^g dc—ag 
 — af—bd' ^ — db—af 
 
 10 " 
 
 be a c 
 
 1 i7. 
 
 '^ — a_|_6' ^ — a + 6* 
 
 20. " 
 
 bd d 
 "" ab-\-ac' '^~b~{-c' 
 
 21. " 
 
 ab d-\- a c c bed — bee 
 
 
 '^ 6 + c ' ^ 6+c 
 
 22. •' 
 
 ^= 2a '^= 26 • 
 
 23. " 
 
 a: = 3, 2/ = 7, 2=16. 
 
 24. '' 
 
 a; = 4, ?/=3, ^ = 2. 
 
 25. '' 
 
 X = 17, 3/ = 22, 2 = 45. 
 
KEY. 
 
 57 
 
 26. Ans. T = 22f, ?/ = ;]5f , s = — 7f 
 
 27. " x = S, y = !0, z = (l 
 
 a -U b — c a — 6-4-c 
 2S. '' ^=—4: . !/ = :r^— ' 
 
 6 — a-\- c 
 
 z = . 
 
 re — h f of — c d 
 
 ae — a ^ at — b a 
 
 _aUl —fir) — (] {bl — c g ) 
 li {a e — d) 
 z = n, y=4, 2 = 5. 
 
 ^ ^>1 34 J, 7162 2 lQ-4_5_ 
 
 x= 13, 7/ =24, z=G2. 
 ^ = 1' ^ = -7, z = m^. 
 X == 12, ?/ = 25, c = (3. 
 X = 4, ?,' = 3, z = 2. 
 
 30. 
 31. 
 32. 
 33. 
 34. 
 35. 
 36. 
 
 37. 
 
 
 u -\~ — c ^ a — -\- c 
 
 b -^ c — a 
 
 38. " This problem is undetermined; because after 
 
 eliminating u, two equal equations of the forms 
 3 X -f- 2 ?/-[-% = 20 are olnained. The piob- 
 lem therefore admits of a variety of solutions. 
 
 39. " X = J, ?/ = 1, 2 = J-, w = 0. 
 
 40. ** X = 7, ?/ = 5, r =3, M = 1. 
 
 B. Quadrafic Equations. 
 1. Ans. X = d= 6. 2. Ans. x = ± i>. 
 
58 
 
 KEY. 
 
 3. 
 
 Ans 
 
 ■>. x = ^4. 
 
 4. 
 
 a 
 
 X = 3, x = — 9. 
 
 5. 
 
 i« 
 
 X = 1 2, X =~6. 
 
 6. 
 
 (( 
 
 X == 9, X = 5. 
 
 7. 
 
 (( 
 
 X = j)j, X =: h. 
 
 8. 
 
 (I 
 
 X r= S," X== — 21. 
 
 9. 
 
 (( 
 
 ^=4, x = --\^. 
 
 10. 
 
 (( 
 
 X = v?j 2' ^X' 
 
 n. 
 
 12. 
 13. 
 
 <e 
 
 X = 71, a: = _ 101. 
 
 x=lx = -l 
 X = 1, x = 0. 
 
 14. 
 15. 
 16. 
 
 (( 
 
 2: = 3, x^3-^. 
 ^=h x = ~±. 
 z = 6, 2== — 7. 
 
 17. 
 
 le 
 
 a- =5, 3;— — 4J. 
 
 18. 
 
 (( 
 
 :^ = 221, :^:= 18f. 
 
 19. 
 
 <t 
 
 X = — 21, X = 51. 
 
 20. 
 
 (< 
 
 :/: = ()f, a: = 31. 
 
 21. 
 
 (( 
 
 .-=12, X=:~5. 
 
 22. 
 
 (( 
 
 x = 9, x=-Gl 
 
 23. 
 
 (( 
 
 X = 3, a; = j . 
 
 24. 
 
 (C 
 
 a- = f, :r = f. 
 
 25. 
 
 i( 
 
 .T = 7, 07 = r>3. 
 
 20. 
 
 ti 
 
 2:= J, 2: = — |. 
 
 27. 
 28. 
 
 
 ^=^, :^ = 1^ = 1A 
 X = (>■ '?: = :/■ = S 
 
 29. 
 
 .==i±V^. 
 
 whence 
 
 2- = 1.3099 ... or a: = — I.7GS2 
 
59 
 
 30. Ans. a; = 0.25 ± i/-^.oo25, whence x = 1.8507 . . . 
 
 orx = — 1.3507.. . 
 
 31. *« x= 1.3699. . . orar = — l.'/(K3'2 . . . 
 
 32. " i = 2.5974 ... or x = — I. '3474 . . . 
 
 33. •' X = I + /=U, or X = I — \/=^9 ; there- 
 
 fore in both ca^es imaginary. 
 
 34. '' x = 1 + v/^^^, «'• ^ = ' — V/^- ' ^hei*^- 
 
 fore in both cases imaginary, 
 
 = _. 21.253 , . . 
 
 35. 
 
 (( 
 
 X = 6.537 ... or X = 
 
 36. 
 
 <( 
 
 2: = 9, :r = l--^.. 
 
 37. 
 
 (( 
 
 x= 10, x = — |. 
 
 38. 
 
 (( 
 
 x = 5|, x = 5. 
 
 39. 
 
 (( 
 
 _ d ^ h_ 
 
 X c ' ^~" a' 
 
 40. 
 
 (( 
 
 . ag ag 
 
 SECTION IX. 
 
 PROBLEMS FOR S [ M P L i: EQUATIONS WITH 
 O iS E UNKNOWN QUANTITY. 
 
 A. Comparison of the Unknown (Quantity icith one 
 or more known ones. 
 
 a. ANSWERS TO THE PROBLEMS IN WHICH THE UNKNOWN 
 QUAKTITY IS DiiTERMlXMED BY A MULTIPL-E. 
 
 1. Ans. $1G0. 2. Ans. $8. 
 
 3. " $1. 4. " $2. 
 
 5. " p. G. '* $5. 
 
60 
 
 KEY. 
 
 7. Ans. $0,000. 
 
 8. " 1 spent glO and retained ^W. 
 $30. 
 
 A was worth $1^9-%, but owed $l,578if 
 B owed $4,73G|f ; and 
 
 C's age is 2'2f years ; 
 R's age is 45^ years ; and 
 A's age is 91f years. 
 12. " $3,000. 
 
 9. 
 10. 
 
 11 
 
 b. ANSWKRS TO THE PROBT.EM3 I IV WHICH THE UNKMOWK 
 QUANTITY IS DETEKM I N KD BY ITS PARTS. 
 
 1. 
 
 Ans. 
 
 I3;>, Ans. 1 
 
 53. 
 
 2. 
 
 Ans. 
 
 25, Ans. s$12,000 
 
 3. 
 
 (( 
 
 <;o. 
 
 
 4. 
 
 
 4.^ yards. 
 
 5. 
 
 (( 
 
 $-iO. 
 
 
 6. 
 
 
 s$nO. 
 
 7. 
 
 it 
 
 $128. 
 
 
 8. 
 
 
 $15. 
 
 9. 
 
 (( 
 
 #l->0. 
 
 
 10. 
 
 
 2-;^ men. 
 
 11. 
 
 (( 
 
 8'J years. 
 
 
 12. 
 
 
 ps. 
 
 13. 
 
 '' 
 
 1-2 o'clock. 
 
 
 14. 
 
 
 $u. , 
 
 15. 
 
 (( 
 
 $-M,(](m. 
 
 
 IG. 
 
 
 g()0. 
 
 17. 
 
 (( 
 
 534 iihds. 
 
 
 18. 
 
 
 $54 apples. 
 
 C. ANSWEKS TO THE PROBLEMS IN WHICH THE UNKNOWN 
 
 QUA,\TiTY IS DKTFRMINKD BY BOTH IIS MULTIPLES 
 
 AND PARTS. 
 
 1. Ans. Tlie amount of the bill was $30 : A had $10, 
 
 and B $(J0. 
 
 2. " $20. 3. Ans. 24. 
 
61 
 
 4. Ans. $2,000. 5. Ans. $1,500. 
 
 6. " $l,724.13ffcts. 7. '' $1,000. 
 
 8. " A had 4 sheep, B 12, and C 3. 
 
 9. " $13,540. 10. Ans. $1,750. 
 
 d. ANSWERS TO THE PROBLEMS IN" WHICH THE UNKNOWNT 
 QUANTITY IS DETERMINED BY COMPARISON TO ITSELF. 
 
 1. Ans. $li-. 2. Ans. f 
 
 3. " Ahad24,Bhadl2. 4. '' $9. 
 5. '' $12,000. 6. '' $15,000. 
 
 7. " $15,000. 8. " 36,000. 
 
 9. '' The father is 60 years old, and the son is 20 
 years old. 
 
 10. " B has $4,800, and A $6000. 
 
 11. " There were 16 children, and 108 apples. 
 
 12. " 31 workmen. The sum was £1, 5s. lid. 
 
 13. " 40 pounds. 
 
 14. " The gentleman called for an article that was 
 
 $80 per cwt. and the merchant had two other 
 articles, $70 per cwt. and $60 per cwt. 
 
 15. " The price of the house is $10,000. The num- 
 
 ber of his debtors is 32, and he must exact 
 $312.50 from each. 
 
 16. " 5 yards. 
 
 €. ANSWERS TO THE PROMISCUOUS PROBLEMS BELONGING TO 
 b, C, d, WITH SEVERAL ADDITIONAL CONDITIONS. 
 
 1. Ans. 16 lbs. 2. Ans. 10|-. 
 
KEY. 
 
 3. 
 
 Ans 
 
 1. 12. 
 
 4. Ans. 20 years. 
 
 5. 
 
 (( 
 
 605 bushels 
 
 6. " 2 miles. 
 
 7. 
 
 fi 
 
 #20. 
 
 a *' $150. 
 
 9. 
 
 t( 
 
 71t^^. 
 
 10. « A $150, B $40, C$50. 
 
 1. 
 
 ft 
 
 $37^. 
 
 12. '' 24. 
 
 B. Dividing a known Quantity into two or more 
 unknown parts. 
 
 a. ANSWERS TO THE PROBLEMS IJV WHICH THE PARTS ARE 
 IMMEDIATELY DEDUCED FROM THE WHOLE dUANTITY. 
 
 1. Ans. One receives $500, and the other $2,000. 
 
 2. " One receives $15, and the other $135. 
 
 3. " A's share is $54, B's share is $S, C's share is 
 
 $10f. 
 
 4. " In 20 hours. 
 
 5. '^ One is to have $250, and the other $450. 
 
 6. " The steward is to receive $1,000, the valet de 
 
 chambre $800, the cook $400, and each of 
 the 4 lackeys $200. 
 
 7. " S. 8. Ans. 9 of each kind. 
 9. " 9 days. 10. *' 6 days. 
 
 11. " A must be paid for 1 ,800 feet, and B for 2,000 feet. 
 
 12. " 1,240 men, infantry, and the same number of 
 
 cavalry. 
 
 0. ANSWERS TO THE PROBLEMS IN WHICH THE PARTS OF 
 
 THE UNKNOWN QUANTITY ARE DEPENDING UPON 
 
 ONE ANOTHER. 
 
 1. Ans. One has $1,800, and the other $:?,600. 
 
KEY. 
 
 63 
 
 2 Ans. One receives $500, and the other ^^2,000. 
 
 3. " $2,160 in notes, and $480 in specie. 
 
 4. " 159f , and 677f 
 
 5. " 26 tons of tea, 52 tons of coffee, and 182 tons of 
 
 sugar. 
 
 6. " 38 students, 152 merchants, and 76 officers. 
 
 7. " 200 cavalry, 1,800 foot soldiers, and 600 artillery. 
 
 8. " 229|f on horseback, SOS^V by water, and 
 
 2,007f I on foot. 
 
 9. " 45f, and 191f . 
 
 10. « A's share is $53^-, B's $26f , C's $13^, and D's $6|. 
 
 11. *< The first receives $40, the second $80, and the 
 
 third $160. 
 
 12. '' A's share is $6|, B's $13^, C's $26f, and D's 
 
 $53^. 
 
 13. " A receives 5 guineas, B 10 guineas, C 30 guineas, 
 
 and D 120 guineas. 
 
 14. *' C and D's shares are $600 each, B's share 
 
 $1,200, and A's $2,400. 
 
 15. '' B's share is $706.20, A's is $353.10, and C's is 
 
 $117.70. 
 
 16. " A's share is $2,240, B's share is $1,120, C's 
 
 share is $560, D's share is $280. 
 
 17. " A's share is $l,246i*y, B's share is $2,492i-8-j-, 
 
 C's share is $830ia. 
 
 18. *' A's share is $180, B's share is gl20, C's share 
 
 is $60. 
 
 19. " A's share is $280, B's share is $140, C's share is 
 
 $70, D's share is $70. 
 
 20. *' A's share is $100, B's share is $200, C's share 
 
 is $300, D's share is $600, E's share is 
 $1200. 
 
64 
 
 ANSWERS TO THE QUESTIONS BELONGING UNDER THE HEAD 
 
 OF a AND h, WITH ONE OR MORE ADDITIONAL. 
 
 CONDITIONS. 
 
 1. Ans. A had $50, and B $49. 
 
 2. " A's share is $24,500, and B's $29,500. 
 
 3. '' One had 112, and the other 147. 
 
 4. " A put in $54^, B put in 45f . 
 
 5. " One is 56, and the other 40. 
 
 6. '^ The most needful is to receive $68, and the 
 
 other $32. 
 
 7. *' The oldest is to receive $240, and the youngest 
 
 brother $760. 
 
 8. " One is to receive $766f , and the other $433^. 
 
 9. '' The father is 69, and the son 31 years old. 
 
 10. " This problem is indefinite, and admits of an infi- 
 nite number of answers. 
 IJ. " One is to receive 250 lbs, and the other 110 lbs. 
 
 12. *' The first is to receive 30^ lbs, the second is to 
 
 receive 34^ lbs, and the third is to receive 
 3.]i lbs. 
 
 13. *' B receives $33^, A 851^, and C $15^. 
 
 14. " The youngest receives $366f , the second son 
 
 $566f , and the third $666f . 
 
 15. " The widow's share is $4,000, each of the sons 
 
 receives $1,000, and each daughter $500. 
 
 16. *' 22 men, 18 women, and 50 children. 
 
 17. " A 2,480, B 2,204, and C 3,316. 
 
 18. " $160. 
 
 19. " The whole sum is $38,400. A's share is $16,200, 
 
 B's share is $11,800, C's share is $10,400. 
 
 20. '« $7,200. 
 
KEY. 65 
 
 d. ANSWERS TO THE PROBLEMS COlVTAIJf I3VG THE PROGRESSIVE 
 DIVISION" OF A KNOWN QUANTITY. 
 
 1. Ans. 8, 9, 10, 11. 
 
 2. « ftJ- 7-i- 6J- 5i 4J- 3J- 
 
 3. « $160, $180, $200, $220, $240. 
 
 4. " On the first day he must drink y^- of a bottle, on 
 
 the second -f>Q, on the third ^-^^ on the fourth 
 y^g, and so on. 
 
 5. " $171f 6. Ans. $2. 
 7. " $3,000. 8. *' $500. 
 
 9. " $4,000. 10. " by $240. 
 
 e. ANSWERS TO THE DIVISIONS OF A GIVEN NUMBER IN A 
 GEOMETRICAL RATIO. 
 
 1. Ans. A's share is $900, B's $1,500. 
 
 2. " 2621, and 157J-. 
 
 3. " A's share $266f, B's $933^. 
 
 4. " A 135, B 297, C 432 roods. 
 
 5. " A 144, B 240, C 210 men. 
 
 6. " A $270, B $360, C $510. 
 
 7. " A $3,200, B $4,800, C $6,000, D $7,000. 
 
 C. Comparison and Determination of numbers hy 
 Addition and Subtraction. 
 
 a. ANSWERS TO THE PROBLEMS IN WHICH ONE QUANTITY 
 
 IS MADE EQUAL TO ANOTHER BY CONTINUED 
 
 ADDITION OR SUBTRACTION. 
 
 1. Ans. 100 times. 2. Ans. 8 times 5. 
 
 6* 
 
66 
 
 KEY. 
 
 3. " 45 days. 4. " 66| hours. 
 
 5. '' 30 games. 6. « 20 times. 
 
 6. ANSWERS TO THE PROBLEMS IJV WHICH TWO QUANTITIES 
 
 ARE MADE EQUAL, TO ONE ANOTHER BY ADDING OR 
 
 SUBTRACTING FROM BOTH. 
 
 1. Ans. 10 days. 
 
 2. " 8 hours. 
 
 3. " 5y\ minutes, or at Sjx minutes past I o'clock. 
 
 4. The equation is 200 -j- 40 a; = 90 x. Ans. 4 years. 
 
 5. The equation is 990 -f- 220 x=400 x. Ans. 5^ years. 
 
 6. There are 200 bills in each pocket. The amount 
 
 of the first therefore is $400, and that of the se- 
 cond $000. 
 
 7. The equation is 45 -|- 2 x = 60 -j- ^ ; whence the 
 
 Ans. 15 years. 
 
 8. The equation is 60 — 3 a; = 50 — 2 x ; and the 
 
 Ans. 10 weeks. 
 
 9. The equation is 240 + 60 z = 80 x ; and the 
 
 Ans. 12 hours. 
 
 10. The equation is 48 + a; = 2 X (10 -f- x), 
 
 or 48 4- ^ = 20 + 2 a; : 
 Ans. 28 years. 
 
 11. The equation is 40 -f- ^ = 30 -|- 3 x. Ans. 5 years. 
 
 12. The equation is 30 + a: = | X (20 + x). Whence 
 
 :r = 20. 
 
 13. The equation is 20 -f- 2 a: = f X (36 + 2 x), 
 
 or 20 + 2 ar = 27 + U a;. 
 And the Ans. 14. 
 
 14. The equation is 100 + x = 80 + 2 x. Ans. $20. 
 
KEY. 67 
 
 15. The equation is GO + x = 40 + 2 x. 
 
 Ans. 20 gentlemen with their ladies. 
 
 16. The equation is 3 x + 12 == 2 3; + 24. 
 
 Ans. A had 12, and B 3(5. 
 
 17. The equation is f x + 2000 = f (x -f 2000), 
 
 or 15 X + 40000 == IH x -f 32000 : 
 Ans. B's fortune is $8000, A's fortune is $6000. 
 
 18. The equation is x — 75 = 2 (fa? — 75), 
 
 or 5 X — 375 = 8 X — 750 : 
 Ans. 125, 100. 
 
 C. ANSWERS TO THE PROBLEMS IN WHICH TWO QUANTITIES 
 
 ARE MADE EQUAL TO ONE ANOTHER, BY SUBTRACTING 
 
 FROM THE ONE AND ADDING TO THE OTHER. 
 
 1. The equation is 7 — x== 3 -|" ^- Ans. 2. 
 
 2. The equation is 100 — 5 x = 50 -{- 5 x. Ans. 5. 
 
 3. The equation is 6() — 2 x = 54 -}- 2 x. Ans. 3 games. 
 
 4. The equation is 300 + 3 x = 1000 — 4 x. 
 
 Ans. 100 days. 
 
 5. The equation is 200000 + x = 2f (lOOOOO — x), 
 
 whence 800000 + 4 x = 1100000— 11 x. 
 Ans. 20,000. 
 
 6. The equation is 8 x — 400 = 5 (x -|- 40). whence the 
 
 Ans. 200 sovereigns, and 1,600 crowns. 
 
 7. The equation is 3 a? — 8 = 5 (x — 8). whence 
 
 Ans. 16 ladies and 48 gentlemen. 
 
 8. The equation is 56 -j- x == -U- (24 — x), 
 
 or 224 + 4 x=264 — 11 x. 
 Ans. 2| lbs. 
 
 9. The equation is 3 x + 15 = 4 (x — 15). 
 
 Ans. A has $225, and B $75. 
 
d. ANSWERS TO THE PROBLEMS IN WHICH ONE QUANTITY 
 
 IS MADE EQUAL TO ANOTHER BY ADDING OR 
 
 SUBTRACTING. 
 
 1. This problem is again undetermined; because the 
 
 number of the garrison is not given. 
 
 2. The equation is 125 a; -f 75 (20 — x) = 2100. 
 
 Ans. 12 men and 8 women. 
 
 3. The equation is x -\- 6 x -\- x = 72. Ans. 9 days. 
 
 X 20 X 
 
 4. The equation is — - -| — = 3f , 
 
 4 u 
 
 or 3 :i + 40 — 2 X = 45. 
 Ans. 5 masters and 15 journeymen. 
 
 X 44 X 
 
 5. The equation is — — |- 
 
 4 ' 16 
 or 4 a; -|- 44 — x = 45. 
 Ans. 12 quarter dollars, and 32 4pences. 
 
 6. The equation is 30 x = 20 (S — a;) + 90. 
 
 Ans. 5 were above 15 years old, and 3 were 
 under 15. 
 
 7. The equation is ^^^ x = ^^^ (1000 — a;) -|- 4, 
 
 or 4 a; = 5000 — 5 x + 400. 
 Ans. $600 at 4 per cent, and $400 at 5 per cent. 
 
 8. The equation is 75 x = 25 (24 — x), 
 
 or 3 X = 24 — X. 
 Ans. he worked 6 days for his master, and 18 days 
 for other persons. 
 
 9. The equation is 70 x = 20 (27 — x), 
 
 or 7x=2 (27 — x). 
 Ans. 6 men and 21 women. 
 
 10. The equation is 6 x + 5 (50 — x) = 276. 
 Ans. 26 and 24. 
 
KEY. 69 
 
 11. The equation is 4 a: — 3 (36 — a;) = :]2, 
 
 or 4 a: — 108 + 3 X = 32. 
 Ans. 20 and 16. 
 
 12. The equation is 3 2; = 5 (40 — x). Ans. 25 and 15. 
 
 D. Answers to the Promiscuous Examples for the 
 Ejcercise of the Learner, 
 
 1. Ans. 46 years. 
 
 2. " 57 years. 
 
 3. « $6000. 
 
 4. " Undetermined ; because it is not stated what my 
 
 property is. 
 
 5. " 25 lawyers, and 10 physicians. 
 
 6. " $600. 7. Ans. $30. 
 
 8. '' 320. 9. '* 35 feet. 
 
 10. They will meet in 20 hours. A will have made -^^^ 
 
 and B y^j of the whole distance. 
 
 11. X being the number of hours, the equation is 
 
 8 a; — 5 x = 24, 
 whence the number of hours = 8. The distance 
 was 64 + 40 :== 104 miles. A has come 64 miles, 
 and B 40 miles. 
 
 12. The equation is x — {1 x -\- ^ x -{- ^ x) = 9, 
 
 or 12 X —6x — Sx-—2x= 108. 
 Ans. 108 yards. 
 
 13. The equation is 6 a; — 4200 = 2400. 
 
 Ans. $1100 per annum. 
 
 14. The equation is f a; -f- J. a; == a; +8^, 
 
 or 4 X 4- 3 X = 6 a; + 25. 
 
 Ans. §J25. 
 
 15. The equation is 5 x -f- 3 x = 360. Ans. 45 hours. 
 
70 
 
 KEY. 
 
 16. The equation is x- — (^-^-^ x -\- ^ x -\- ^ x -\- ^x) = 975, 
 
 or40 2; — 4a; — 8a: — Sx— ]0x = 39000. 
 Ans. $3000. 
 
 17. The equation \s x — {^x -^-^x -\-l x-{- ^x) = 9^0, 
 or 120a;— 30 x — 24:X—20x=z 15 0^ = 111600. 
 
 Ans. $3000. 
 
 18. The equation is ^ x -\- ± x -\- ^ x -\- \2 = x, 
 
 or 6x+5 x + 3x4- 180 = 15 a;. 
 Ans. The whole number of the company was 180; 
 the number of gentlemen = 7'2, the number of 
 ladies = GO, the number of boys = 36, and the 
 number of girls = 1*2. 
 
 19. Let the whole gain be x. Then you will have the pro- 
 
 portions, 
 
 22000 : 10000 : : a; : |pao x= y5_ x= A's gain ; 
 added 22000 : 12000 :: I : ifooo ^_6_^= B's gain ; 
 whence, by the last condition, we have the equation 
 y\ X — -f>^ X = 800. Ans. The whole gain is 
 $8800. A's share is $4000, B's share $4800. 
 
 20. The equation is x -|- 20 = 3 x. Ans. 10 years. 
 
 21. The equation is x -{- x -\~ ^ x -^ ± x -{- ^ x -\-2 = 150, 
 
 or 24 X + 6"x + 4 X 4- 3 X = 1776. 
 Ans. 48. 
 
 22. The equation is 42 + x = 5 (24 — x), 
 
 or 42 + X = 120 — 5 x. Ans. 27. 
 
 23. The equation is 2 a; -)- 2 = 76. Ans. 37 years. 
 
 24. The equation is x — i x = 3, 
 
 or f X = 3. 
 Ans. one was$l3i, and the other only $10^. 
 
 25. The equation is f^ x -|- 6f = x, (because f + f 
 
 = U) ; whence || = 6|, or 9x = 270a7 = $30. 
 Ans. the watchmaker values his gold watch at $30. 
 
KEY. 71 
 
 26. The equation is 2 a: + ^ a; — 6 == 50. Ans. J$22f , or 
 
 $22.40 cents. 
 
 27. The equation is ~ + | a: -}- -^ = H i^ 
 
 or 2 a: + 3 a: + 4 a; = 90. Ans. 10. 
 
 28. The equation is —-)--— -|_ 30 = j;^ 
 
 or 2 2: + a^ -f 120 = 4 x. Ans. 120. 
 
 29. The equation is 
 
 2 2: + i z + f a: + I 2; + 3-9^ a; + 1 = 100, 
 or 40 X + 10 a^ + 15 2; + 16 2; 4. 18 2: = 1980. 
 Ans. 20. 
 
 30. The equation is 400 -J- 50 x = 90 x. Ans. 10 days. 
 
 31. They shared wrong; because the 8 loaves having 
 
 been divided equally among the three persons, A 
 received 2f loaves ; he lost therefore only i loaf, 
 while B, in receiving 22 loaves, lost 2^^ = x loaves! 
 The division therefore must be made according to 
 the loss, which is as 1 : 7. A receives 1 piece, and 
 B receives 7 pieces. 
 
 32. The equation is |- + ^ + ^ + 318 = 2:, 
 
 or 40 2: + 15 X + 12 a^ + 38160 = 120 x. 
 Ans. $720. 
 
 33. The equation is | X | 2; = 60, 
 
 or ^\ X = 60. Ans. $250. 
 
 34. The equation is f 2; — -i- 2: = 70, 
 
 or 9 2: — 2 2; = 420. Ans. 60 years. 
 
 35. A's age being x, the equation is 2; -f- 2; -j- 2f + 2; 
 
 + 4i = 72; whence A's age = 21f years, 
 B's age = 24 1 years, and C's age = 26 "years. 
 
72 KEY. 
 
 36. The equation is 15 a; — 12 a; = 27. Ans. 9. 
 
 37. The equation is 6 x — x = 45. Ans. A has J54, and 
 
 B has p, 
 
 38. The equation is ^2: -f-f ^+i ^ + ^^6 = a;; whence 
 
 the Ans. = $945. 
 
 39. The equation is 19 (x + 9) + 19 a; = 323, 
 
 or 19 .T+ 171 +19 a; =323: 
 Ans. A travelled 22 miles, B travelled 13 miles. 
 
 40; The equation is x-}- ^ x -}- ^ x = 280, 
 or 4x-\-2x-{-x = 1120. 
 Ans. A bought 160 bushels, B bought 80 bushels, 
 C bought 40 bushels. 
 
 41. The equation is Of 2; — 12 = 300, 
 
 or 39 a: — 48 = 1200. Ans. 32 years. 
 
 3 a: -I- 15 
 
 42. The equation is ~ — - -\- 6 = x, 
 
 or 3 a: + 15 + 36 = 6 :r. Ans. $17. 
 
 43. The equation is ^ 4 == 15, 
 
 or 7 .r -j- 3 — 8 = 30. Ans 5. 
 
 44. The equation is (^^-HL^pHi-? = 23, 
 
 or 20 a; — 12 + 2 = 230. Ans. $12. 
 
 4r m^ • .OX 24 , , ^ 
 
 45. The equation is 1- ]2 = x, 
 
 or 5 a: — 24 + 78 = 6 a^. Ans. 54. 
 
 46. The equation is 
 
 r X — 100 + ^ X — 80 + i X — 60 -\-l x=zx, 
 or f§ a: + |o :, 4- ^2 ^ + 1^ ^ _ 60 ^ == 240. 
 
 Ans. the whole amount of his fortune is $847xV- 
 
KEY. 
 
 73 
 
 A receives $S2S^\, B receives $202tV, C receives 
 $109yV, D receives $21 Hf 
 
 47. The equation is i a; — ^ « = 17. The number of 
 
 sheep of their friend is 204 ; A has 68 sheep, and 
 B has 51. 
 
 48. The equation is f a; — | x = 400, 
 
 or 18 a? — 14 X =25200. Ans. $4200. 
 
 49. The equation isa: + 2; + 2+22;+6 = 96, 
 
 or 4 a; = 88. 
 Ans. Alexander was 24 years old, Hephaestion was 
 22 years old, Clytus was 50 years old. 
 
 50. The equation is 20 ~{- x = 2 {^ -^ x), 
 
 or 30 -j- ^ == 1 + 2 a:. Ans. in 29 years. 
 
 51. The equation is 50 + a? == 2 (24 + x), 
 
 or 50 + a: = 48 -[- 2 X. Ans. in 2 years. 
 
 52. The equation is x + x + 150 + a; + 300 = 1200, 
 
 or 3x = 750. 
 
 Ans. C's share is $250, B's share is $400, A's 
 share is $550. 
 
 53. The equation is 200 + i^ + TV^-|~B^=='2-^> 
 
 or 1200 -}- 12 X + 6 x*+ 10 x = 30 x. 
 Ans. $600. 
 
 54. The equation is x + 23 + x + x + 23 = 100. 
 
 Ans, 18 years. 
 
 55. The equation is ^:^- x — -y- x = 36. Ans. They sailed 
 
 18 hours. One sailed 135 miles, and the other 99 
 miles. 
 
 56. The equation is x + x + 12 -f- x + 28 =400. Ans. 
 
 A's loan is $120, B's loan is $132, C's loan is $148 
 7 
 
74 
 
 57. The equation is j^ X 500 + ^q X 600 = 125, 
 
 or 20 X -^30 x= 125. Ans. 2^ years. 
 
 58. The equation is x -\- ^ x -\-2x = 1170. A's loan 
 
 is $270, B's is $360, C's is $540. 
 
 59. The equation is x + a; + 16 + a: + 28 + x + 36 
 
 = 400 : A's share is $80, B's share is $96, C's 
 share is $108, D's share is $116. 
 
 60. The equation is 9 x — 18 = 48. The first receives 
 
 $7|, the second $6f , the third $6|, the fourth $5f , 
 the fifth 5f , &c. 
 
 61. The equation is formed in the following manner : 
 
 f X — 1000 = end of the first year ; 
 
 ^ {^^x— 1000) — 1000 J 
 
 == -\^-- X — 4 oj)_o — _3 o_o_o ( end of the 2d year ; 
 
 16 r 7000 S 
 
 I (J_6 a; _ 7P_0_0) _ 1000 ^ 
 
 = ^fx — 2_8 00 _ 9 p_o ( end of the 3d year. 
 
 64 n- 37000 ) 
 
 Consequently, f f a: — ^--O-^.o ,_ o j.^ 
 
 or 64 x— 11 1000 = 54 X. Ans. 811.100. 
 
 62. The equatiou is 50 x -[- 75 x == 500. Ans. 4 days. 
 
 63. The equation is 8 x -f 7 (20 — x) = 144, 
 
 or Sx 4- 140 — 7x= 144. 
 The men were 4 in number, and the women 16. 
 
 64. The equation is 4| x + 2i = 5 x — 10, 
 
 or 39 X + 20 = 40 X — 80. 
 
 Ans. 100 lbs. 
 65 The equation is 
 X -f. 4 X — 300 + 12 X — 1100 + 24 X — 2300 =8600, 
 
 or 41 x = 12300. 
 
 D's share is $300, C's share is $900, B's share is 
 
 $2500, A's share is $4900. 
 
75 
 
 66. The equation is formed in the following manner : 
 
 (x — 10) — 2 :^ what he breaks, 
 
 x—lO — [^{x— 10) — 2] = 
 
 X — 10 — {^ X ^ 8 —2) = 
 
 X — 10 — |-3; + 8 + 2 = |a:= what he has 
 
 left. Consequently ^ x -\-5S= x — II. Ans. 80 
 
 eggs. 
 
 67. The equation is 30 -[- 6 a; = 8 a?. Ans. 15 days. 
 
 68. The equation is formed in the following manner : 
 
 2 a; — 2 = what she had after leaving Jupiter's 
 
 temple ; 
 
 4ix — 4 — 2 = 4a; — 6= what she had after 
 
 leaving the temple of Apollo ; 
 
 whence 4 x — 6 = 2 x. Ans. She had 3 drachms at 
 
 first. 
 
 69. The equation is 3^ a? + 5^ x = 80 — 28 = 52, 
 
 or 21 a;4-31a; = 312. Ans. 6 hours. 
 
 70. The equation is f x — f x = 8, 
 
 or 24 X — 21 X = 448. Ans. 149^ yards. 
 
 71. The equation is x + yV^ x = 2000. Ans. $1600. 
 
 72. The equation is x + y?^% x = 900. Ans. $500. 
 
 73. The equation i.s 4 (x — 1 ) z= 3 x -|- 6. There were 
 
 10 soldiers, and the amount of money $36. 
 
 74. The equation is 90 -J- \^~ x= s^- x, 
 
 or 1080 = 80 a? — 75 x. Ans. 216 hours. 
 
 75. The equation is6a:-|-3a: + 75 = 225. Ans. $l6f 
 
 is the price of the new wine per hhd, and 4 If of 
 the old wine. 
 
 76. The equation is 800 a? + 700 a; = 6000. Ans. In 4 
 
 minutes. 
 
76 KEY. 
 
 77. The equation is 800 x = 6000 + 700 x. Ans. in 60 
 
 minutes, or 1 hour. 
 
 78. The equation is 61 -|- a? =: 3 (13 + x.) Ans. In 11 
 
 years. 
 
 79. The equation is formed in the following manner : 
 
 r ^00 
 
 ^200+ ^^ =h^ + 100, = the valet de 
 '^ chambre's ; 
 
 ;r — i.T— 100 , ,^^ :r + 3800 
 
 ^ h 400 = —^ = the cook's ; 
 
 5 ' 10 
 
 consequently ^ x -{- 100 -{- ^ '"' \- 500 == x. 
 
 Ans. $2450. 
 
 80. The equation is 15 j: = 10 a; -f- 80. Ans 16 yards of 
 
 the first and 24 yards of the second. 
 
 81. The equation may be formed as follows : 
 
 ^ X -{- 1 = end of the first game ; 
 
 = end of 2d game ; 
 
 J- — j _|_ 3 = 4 a^j 
 
 or, 27 X + 42 4- ^-^^^^ ^• 
 Ans. $131 
 
 82. The equation is 12 a,- — IS = 9 .t + 12. Ans. She 
 
 wishes to buy 10 yards, and has 85. 6cL 
 
 83. The equation is JJ- ^ — i^ = ^^0. Ans. 583^ yards. 
 
 84. The equation is 70 + ^^^ x X 500 = -^^^xX 480, 
 
 or 70 -|- 20 a? = 24 x. Ans. in 17| years. 
 
 85. Ans. $108. 
 
 86. The equation isx-\-2x-^l+4:X-\-S=^ 102. 
 
 The first was 14 years old, the second was 29 years 
 old, the third was 59 years old. 
 
KEY. 77 
 
 87. The equation is 18 a? + ISar + 30 = 228. Ans. The 
 
 sugar is 6d per lb, and the coffee Sd. 
 
 88. The equation is a? + 28 = 5 a?. He was 35 years, 
 
 and his daughter 7 years old. 
 
 89. The equation is formed as follows : 
 
 2 X — 800 = end of 1st year ; 
 4 a: — 1600 — 1600 = 4 a; — 3200 = end of 2d 
 
 year ; 
 and consequently 8 a? — 6400 — 2400 = 0. Ans. 
 $1100. 
 
 90. The equation is 200 a: — 700 = 150 a; + 1100. 
 
 Ans. $6500. 
 
 91. The equation is 2 a: + 1^ (120 — ar) = If X 120, 
 
 ' or 2 a? + ISO — f ar = 200. 
 Ans. 40 bottles at $2, and 80 bottles at $1,50. 
 
 92. Ans. 8 days. 
 
 93. The equation is 480 -|- 40 a^ = § X 40 x, 
 
 or 1440 + 120 a; = 320 x. Ans. 7| days. 
 
 94. The equation is ^'^% x + ^\% (lOOOO — x)= 1080, 
 
 or 12 a; + 100000 — 10 aj = 108000. 
 Ans. In the one he has $4000, and in the other 
 $6000. 
 
 95. X = number of gallons of first cask, ^x = number of 
 
 gallons of 2d cask, | a?= number of gallons of 3d 
 cask ; consequently, x =: ^ x -\- ^ x -\- SO. Ans. 
 the first cask contains 135, the 2d cask contains 
 45, and the third 60 gallons. 
 
 96. This problem is similar to problem 31. Ans. A must 
 
 pay 10s, and B Is. 
 
 97. The equation is 1.25 x — 7.50 = 0.90 x + 13.50. 
 
 7* 
 
78 
 
 Ans. 60 lbs. is the weight, and the prime cost 
 $1.12^- cents per lb. 
 
 98. The equation is 5x— I6=2{x + 24). 
 
 Ans. B had |21i and A had $106f . 
 
 99. The equation is ^ — j^^ (24000— o^) = 600, 
 
 or 7 a: — 120000 + 5 x =^ 60000. 
 Ans. 15000 at 7 per cent, and $9000 at 5 per cent. 
 
 100. The equation is | a? — x = 3000. Ans. The one 
 
 2000 feet, and the other 5900 feet. 
 
 101. The equation is y -| — = 10. Ans. 28, 18. 
 
 102. The equation is formed thus : 
 
 X = first, 2 X + 1 = 2d, 
 
 6 x + 6 = 3d ; hence 9 x -f 7 = 70. 
 Ans. 7, 15, 48. 
 
 103. The equation may be formed thus : 
 
 Let X = the hire of 1 man in pence, 
 
 ^ 328 — 12x ^ ,. ^ 
 
 then the hu-e oi one woman, 
 
 o 
 
 (328 pence being the whole pay). 
 Again ; from condition 2d, 
 
 370 — 12 X 
 
 — = hire or 1 woman. 
 
 328— 12a: 370— 12x 
 consequently = — ; 
 
 a 11 
 
 whence ISd. = the hire of a man, I4c?. = the 
 hire of a woman. 
 
 104. This problem is similar to problem 91. 
 
 Ans. 66f gallons at $1,60, and 133^ gallons at 
 $1.00. 
 
 105. Let X = the whole fortune ; 
 
KEY. 79 
 
 r 1 no 
 
 100 + —Y^ = 90 + iV ^ = share of 1st, 
 
 200 + ^^=:-55^P^-=171+,4^a;=share 
 ^^ of 2d. 
 
 and because they were to share equally, 
 
 whence 9000 +10 3;== 17100 + 9 x. 
 
 Ans. His whole fortune was $8100. The share of 
 
 a child $900, and the number of children 9. 
 
 106. Ans. 1001 cubic feet from the larger, and 440 from 
 
 the smaller. 
 
 107. The equation is 50 -|- a; = J| x. Ans. 700 paces. 
 
 108. The equation is 27 -f f | a; == x. Ans. 189 throws. 
 
 109. Ans. A must receive £5, B only IO5, and C nothing ; 
 
 because in either case he shares ^ of the gain or 
 loss. 
 
 110. The equation is 1200 =500 + 1 B. 
 
 whence B's share = 280, A's share = 720, C's 
 share = 840, D's share = 360. 
 
 111. The equation is 2 x + 24 = 3 x. 
 
 The first is worth $24, and the 2d $64. 
 
 4 a; -4- 400 
 
 112. The equation is + =63. 1 49. Ans. $700. 
 
 o 
 
 113. Ans. ^. 
 
 114. The equation is f + a X — = f, 
 
 62; ^ ^ 
 
 Ans. by 3L part. 
 
 115. The equation is 2a; — 2 = x -\- 1. 
 
 Ans. 3 girls and 4 boys. 
 
80 KEY. 
 
 1 16. Let X = the cover, 
 
 6 -f- J X = the 2d cup, whence 
 6 -f f 2; = 36. 
 
 Ans. The cover weighs 20 ounces, and the 2d cup 
 16 ounces. 
 
 117. Ans. 5600 feet. 
 
 118. The equation is 3 x +3850 — 5 a: = 2350. 
 
 A has $250, B has $320. 
 
 1 19. The equation is 9 z + 250 — 5 x = 400. 
 
 Ans. 37i gallons at 9s., 12|- gallons at 55. 
 
 120. This problem is similar to many preceding ones. 
 
 Ans. A lost $2, B lost $6i, C lost $11, D lost $8|, 
 E lost$12x 
 
 121. Ans. 13^ oz. of 14 carets fine, 6f oz. of 8 carets fine. 
 
 122. Ans. 6| bottles. 
 
 123. The equation is 420 + 12 x = 325. Ans. 8|. 
 
 124. Ans. 30 lbs. 
 
 125. " 8 fourpences, and 9 quarter dollar pieces, 
 
 126. '' 10 lbs. 
 
 127. " li^ hours. 
 
 128. " 48 minutes. 
 
 129. The equation is|a; + |x-|-fx = 756. 
 
 Ans. I373W tiays. 
 
 130. Ans. 20 cubic inches. 
 
 131. Ans. The company consisted of 100 persons ; the 
 
 sum which was to be collected was $50, and the 
 contribution of each 50 cents. 
 
 132. The watch costs $120, and he sold 80 tickets. 
 
 133. The equation is a;— 6 = 4 (^^q\ 
 
KEY. 81 
 
 or a; — 6:= -^—24. 
 o 
 
 The father is 54 years old, and the son is 18 years 
 
 old. 
 
 134. Ans. $240. 
 
 135. The equation is | a; = f (9800 — x). 
 
 Ans. One had $4800, and the other had $5000. 
 
 136. The equation is 70 a; — 70 = 70 — 28 x, 
 
 whence x = Iff = number of sheets written in 
 one hour, and as he works 28 hours in a week, he 
 will write 28 X Iff = 40 sheets. 
 
 137. The equation is Y- — ^x-\-x = 2d. 
 
 A has $19^, andB $14^-. 
 
 138. The equation is4x-f-4x + 2|a:== 52^-. 
 
 Ans. 5 yards of the best kind, 10 yards of the 
 second kind, 20 yards of the third kind. 
 
 139. The equation isx-ffx-f- fa: + 2x = l]70, 
 
 or 10 x + 12 X + 15 X + 30 x = 11700 ; 
 whence D's gain $174|f-, C's gain $209ff, B's 
 gain |261ff , A's gain $349^f 
 
 140. Ans. 360 feet. 
 
 141. The equation is 5 x — 20 = 20 — x. Ans. 6f . 
 
 142. The equation is U^l+}^ _ 1725+_30x 
 
 56 84 
 
 or 4200 + 48 X = 3450 + 60 x. 
 Ans. 62J cents. 
 
 143. The equation is f (| x -}- 50) + 70 = 120, 
 
 or i. X 4- 37 1 + 70 == 120. Ans. $25. 
 
 144. Ans. $275. 
 
 145. This problem is solved in the same manner as the 
 
 preceding, (144). Ans. 80. 
 
82 KEY. 
 
 146. 
 
 Ans. 
 
 9i months. 
 
 147. 
 
 (( 
 
 2^ months. 
 
 148. 
 
 (( 
 
 The first instalment is due in 7| months, the 
 second 7| months after that, and so on. 
 
 149. 
 
 {( 
 
 $7936. 
 
 150. 
 
 K 
 
 In 3§ months. 
 
 151. 
 
 i( 
 
 $7722. 
 
 152. 
 
 (( 
 
 A receives $208^f , B lOS^f , C 182^f. 
 
 153. 
 
 (( 
 
 A contributed $2450, B $3675, C $6425. 
 
 154. 
 
 « 
 
 A $3450, B $3770. 
 
 155. 
 
 IC 
 
 The rate of interest is 8 per cent, per annum, 
 or f per cent, per month ; and it would take 
 6 children 10 months to spend at the same 
 rate $1650. 
 
 156. 
 
 u 
 
 $110|. 
 
 157. 
 
 (( 
 
 40 eggs. 
 
 158. 
 
 (( 
 
 31. 
 
 159. 
 
 This problem has already occurred before. 
 Ans. $11000. 
 
 160. 
 
 We have the proportion -\3. : .^ : I : x. Ans. -^^ of an 
 hour. 
 
 161. The equation may be formed in the follov^^ing manner : 
 
 ex — 1000 at the end of the first year, 
 
 II a; _ 2200 at the end of the 2d year, 
 
 2i§ X — 3640 at the end of the 3d year, 
 
 which must be equal to f x + 200. Ans. $30000. 
 
 162. The equation is (3f x — 60) 2^ — 30 = 0, whence 
 
 the number is = 21. 
 
 163. Ans. 1975 men. 
 
KEY. 
 
 83 
 
 164 Ans. $355. 
 
 165. The equation is Jj x X tV ^ = ^j 
 
 or 1^^ x=\. Ans. 144 oxen. 
 
 166. The equation is /_. x -f- 50 = a?. Ans. The third 
 
 cask contains 120 gallons, the 2d cask contains 90 
 gallons and the 1st cask contains 70 gallons. 
 
 167. Ans. The first contains 140 gallons, the 2d contains 
 
 60 gallons, the 3d contains 45 gallons, the 4th 
 contains 80 gallons. 
 
 SECTION X. 
 
 Answers to the Questions in Simple Equations with two 
 or more unknown Quantities. 
 
 1. Ans. 40 and 30. 
 
 2. " The first $180, the second gl20. 
 
 3. *' One contains $20, and the other $30. 
 
 4. The two equations are x -f- 100 z= y — 100, 
 
 y-f 100=2 07—100. 
 Ans. A has $500, B $700. 
 
 5. Ans. The first is worth $24, the second $64. 
 
 6. The two equations are x -\- y = 570, 
 
 3x-\-5 7/ =2350. 
 Ans. A has $250, B $320. 
 
 7. The two equations are a? + 2 = y — 2, 
 
 y-f2 = 2a: — 4. 
 Ans. One had 10, and the other 14. 
 
84 KEY. 
 
 8. The equations are 2 a? -|- 5 y = 31, 
 
 7 a; -)- 4 y = 68. Ans. 8 and 3. 
 
 9. The equation is re -j- 4 = 3|^ y, 
 
 i/-\-8 = ^x. Ans. 48 and 16. 
 
 10. The equations are x — 6 =3^ y, 
 
 Ans. The father 36, and the son 15 years. 
 
 11. The equations are a? -|- y = 9800, 
 
 -I x = I y, 
 Ans. A has $4800, and B $5000. 
 Remark. The same sum has been solved before, with one 
 unknown quantity. 
 
 12. The equations are f y -{- a? = 600, 
 
 ^x-\~I/ = 600. 
 Ans. A has $240, and B $480. 
 
 13. The equations are a: -[- i y = 1200, 
 
 y -f 1 a? = 2500. 
 Ans. A had $906if , B $2348|f . 
 
 14. The two equations are ^ x = ^ 7/, 
 
 Ans. A had $16, and B $20. 
 
 15. The equations are | a? = | ?/, 
 
 Ans. One had $18, and the other $16. 
 
 16. Ans. At 41 and 5^ per cent. 
 
 17. The two equations are ^ y — 96 = | x. 
 
 The first weighs 720 lbs, the 2d 512 lbs. 
 
 18. Ans. The first pipe discharges 15, and the second 6 
 
 buckets ; both will require 1 hours to fill the 
 cistern. 
 
KEY. 85 
 
 19. The two equations are x -f- ?/ == 500, 
 
 20 a: 4- 3/= 1694. 
 Ans. 326 sovereigns, 174 shillings. 
 
 20. The two equations are 4 x -j- 4 i/ = 28, 
 
 4 X + 5 ?/ = 33. 
 The price of an orange was 2 cents, and that of a 
 lemon 5 cents apiece. 
 
 21. The two equations are 60 x + 40 ?/ = 3000, 
 
 3 X = 4 y. 
 The price of the coffee was 33§ cents per lb, and 
 that of the sugar 25 cents per lb. 
 
 22. The two equations are 8 a; -f 9 y = 1846, 
 
 20 0^ + 16 y = 3640. Ans. 62 
 cents and $1,50. 
 
 23. The two equations are 1 5 a; -f- 33 y = 39^, 
 
 24 X + 55 ij = 65. 
 Ans. The Silisian ell is to the Brabant ell as 5 to 
 6. The Leipzig to the Brabant as 9 to 11, the Si- 
 lisian to the Leipzig as 55 to 54. 
 
 24. The French mile is to the German as 3 to 5; the 
 
 English to the German as 23 to 106; and the 
 French to the English as 318 to 115. 
 
 25. The two equations are a; -[- 50 — 8 = ?/-{- 2, 
 
 3:f Cr + 2)=y+50. 
 Ans. The first horse is worth §30, and the 2d §70. 
 
 26. 
 
 Ans. ^\, 
 
 
 
 
 
 
 27. 
 
 The two 
 
 equations 
 
 are 
 
 X — 
 I/ — 
 
 3 
 
 "3~" 
 5 
 
 h 
 ■i- 
 
 Ans. yg-. 
 
 28. Ans. A has lent $10000, B $22600, C $13000; A 
 at 4 per cent, B at 5, and C at 6 per'cent. 
 
 8 
 
86 
 
 29. Ans. There were 11 persons in the company, and 
 
 each spent SO cents. 
 
 30. The second equation is {x — 2) (y — 3) = a: y — 145. 
 
 Ans. 29 lines per page, and 32 letters in a line. 
 
 31. Ans. The wheat is ^5, and the rye §^2,75 per barrel. 
 
 32. " The first holds 22, and the second 10 gallons. 
 
 33. " The best wine is §^1,12 per gallon, and the worst 
 
 80 cents. 
 
 35, The two equations are z -|- y = 120, 
 Ans. 74 lbs. of tin, and 46 of lead. 
 
 36. 
 
 Ans. 112 lbs of silver, and 36 lbs of copper. 
 
 37. 
 
 *' 14.77 . . of gold and 5.22 ... of silver> 
 
 38. 
 
 "' 2 and 10. 
 
 39. 
 
 " 5 and 8. 
 
 40 
 
 '' 3 and 6. 
 
 41. 
 
 " 4 and 16. 
 
 42. 
 
 The three equations are x -}- J/ = 54, 
 
 
 2/ + ^ = 109, 
 a; 4- 2 = 85. 
 Ans. He himself is 18 years old, his father is 38 
 
 
 years old, and is grandfather 62. 
 
 43. 
 
 The two equations are .t — 7 = 3 y — 21, 
 a;4-7 = 2y + 14. 
 Ans. The age of the father is 49, and that of his 
 
 
 son 21. 
 
 44. The equations are a; -[- f y = 2190, 
 y + 1^=2190, 
 %x-\- z = 2190. 
 Ans. A $1530, B $1540, C $1170. 
 
KEY. S7 
 
 45. Ans. A has $200, B $360, C $840. 
 
 46. " In the first were $120, in the second $380, and 
 
 in the third $500. 
 
 47. *' A $980, B f 1540, C $2380. 
 
 48. '' 20, 28, and 50. 
 
 49. *^ A $400, B $640, C $780. 
 
 50. *' They spent $6^ ; A has $5, and B $6. 
 
 51 . " The 15 carats fine weighs 8 lbs, the 10 carats 16, 
 
 and the 9 carats weighs 10 lbs. 
 
 52. " The barrel of wheat is worth $7, the barrel of 
 
 rye $6, and the barrel of barley $4. 
 
 53. " The coffee 75 cents, the sugar 50 cents, and the 
 
 tea $2,00. 
 
 54. " A $52, B $28, C $16. 
 
 55. ** 37, 25, 21. 
 
 56. " In the first there were $70, in the 2d $52, and 
 
 in the 3d $40. 
 
 57. " 30, 48, 50. 
 
 SECTION XI. 
 
 PROBLERiS WHICH LEAD TO QUADRATIC 
 EQUATIONS. 
 
 A. Answers to the Problems which lead to pure 
 (Quadratics, 
 
 1. Ans. 12. 2. Ans. 14. 
 
 3. '* 24. 4. '' 72. 
 
 5. ♦' 120. 6. " 224. 
 
KEY. 
 
 7. Ans. 18. 8. Ans. 50, and 15. 
 
 9. " 85 and 76. 10. '' 12 persons. 
 
 11. " Undetermined ; because it is not stated what he 
 
 paid for them. 
 
 12. '' $6480. 
 
 13. Undetermined. 
 
 14. The equation is 2J X j^ = a;, 
 
 5 X 
 
 or .-tttttt: = 1- Ans. 20 merchants 
 2000 
 
 15. Ans. $2631 66 cents, nearly. 
 
 16. " Of the first 15, of the 2d 20, and of the 3d 70 lbs. 
 
 17. The equation is -L3 a.2 _ i25 = 125 — — , 
 
 or 26 2;2_ 1250 = 1250—5 x^ 
 or 31 x2 = 2500. 
 Ans. 8.98 . . . lbs. 
 
 18. The equation is | x^ = 2352. Ans. 42. 
 
 B. Auswei's to the Problems leading to Mixed (Quad- 
 ratic Equations. 
 
 1. The equation is a;^ + 8 a; = 240. Ans. 12 and 20. 
 
 2. The equation is 2^ -f- 59 z = 1200. Ans. 16 persons. 
 
 3. The equation is a;^ -]- 6 2; = 91. Ans. 7 broad and 
 
 13 long. 
 
 4. The equation is x^ — x z= 306. Ans. 18. 
 
 5. Ans. 48. 
 
 6. The equation is x^ -f- 6 a: = 135. Ans. He paid $9 
 
 a day, and stayed 15 days. 
 
 7. Ans. 42. 
 
KEY. 89 
 
 8. The equation is x^ = 36 x-{- 832. Ans. 16 oxen. 
 
 x^ 
 
 9. The equation is — + x = 119. Ans. $120. 
 
 10. Ans. $40, $72, and $80. 
 
 11. " 12 pieces. 
 
 12. '' 54. 
 
 13. The equation is a;^ -{- ] x — • 2475 = 900000. 
 
 Ans. 945. 
 
 14. The equation is3a:2-j-6a;-f-5= 245. 
 
 Ans. The first worked 8 days, and received Ss. per 
 day, the 2d worked 9 days, and received 9s. per 
 day, and the 3d worked 10 days, at Ws. per day. 
 
 15. The equation may also be 1^ + 5 = ^^^^ 
 
 ^ X ^ .T — 40' 
 
 or x^ — 40 X = 9600. 
 The first had 120 men, and the 2d had 80. Each 
 soldier in the first company received $10 ; and each 
 soldier in the 2d company received $15. 
 
 16. The equation is 4 x^ -f 4 a; = 1680. Ans. 20 roods 
 
 broad, and 84 rods long. 
 
 17. The equation is a;^ — - 11 a: -j- 30 == 3 ^^ 
 
 or a;2 — 14 x = — 30. 
 Ans. $11,358 .. . or $2.642 . . . 
 
 18. Ans. 69.53 . . . lbs. 
 
 19. ** He is 35, and his brother 36 years old. 
 
 24 24 
 
 20. The equation is — = j^yr f- 1. Ans. 8 men, and 
 
 a; 20 — x ' 
 
 12 women. 
 
 21. Ans. $80. 
 
 oo rpi .• • 100 , 100 ,^ 
 
 22. The equation IS —_-)-__ r= 43a. Ans. 3 and 10, 
 
 •^ I • X 
 
90 KEY. 
 
 23. Ans. ^300. 
 
 24. *' $20. 
 
 o- mu . . 9x 4.X+ J20 
 20. The equation IS _^-^ = ^r . 
 
 Ans. The distance between C and D is 150 miles. 
 B travelled 60 miles, and C 100 miles. 
 
 2G. Ans. 15 and 16. 27. Ans. 12 and 20. 
 
 28. 
 
 30. 
 
 32. 
 
 33. 
 
 34. 
 
 36. 
 
 15 and 17. 29. « Into 4 and 12. 
 
 1296. 31. *' 561. 
 
 One $200, and the other pOO. 
 
 One $1200, and the other $800. 
 
 9 and 15. 35. Ans. 862. 
 
 654. 37. *' 6. 24, 96. 
 
APPENDIX 
 
 CONTAINING PROBLEMS IN COMPOUND IN 
 
 TEREST AND ANNUITIES, FOR THE USE 
 
 OF LOGARITHMS. 
 
 
 ANSWERS TO THE 
 
 APPLK 
 
 NATION 
 
 rs OF 
 
 THE FOMULAS. 
 
 1. 
 
 Ans 
 
 !. $1343.90 nearly. 
 
 17. 
 
 Ans 
 
 1. $5128.80 nearly. 
 
 2. 
 
 iC 
 
 $2653.36 
 
 (( 
 
 18. 
 
 << 
 
 $771.89 cents. 
 
 3. 
 
 (I 
 
 $7401.30 
 
 (( 
 
 19. 
 
 (( 
 
 $644.61 nearly. 
 
 4. 
 
 (( 
 
 $10955.45 
 
 (( 
 
 20. 
 
 ii 
 
 $6755.65 '' 
 
 5. 
 
 tt 
 
 ^24005.00 
 
 <( 
 
 21. 
 
 u 
 
 $310.86 
 
 6. 
 
 (( 
 
 $34050.84 
 
 
 22. 
 
 a 
 
 $12424.20 cents. 
 
 7. 
 
 <( 
 
 $3385.55 nearly 
 
 23. 
 
 (e 
 
 $1884.44 cents. 
 
 8. 
 
 (( 
 
 144S9276 
 
 
 24. 
 
 it 
 
 $27919.80 nearly. 
 
 9. 
 
 'i 
 
 6890190000 
 
 
 25. 
 
 a 
 
 $6246 
 
 10. 
 
 a 
 
 $35917.10 
 
 
 26. 
 
 CC 
 
 is given in the book. 
 
 11. 
 
 <c 
 
 $49744.60 
 
 
 27. 
 
 Cl 
 
 a little over 3 per 
 
 12. 
 
 (I 
 
 $24924.12 
 
 
 
 
 cent. 
 
 13. 
 
 ((. 
 
 $21673.30 
 
 
 28. 
 
 << 
 
 $3683.48 cents. 
 
 14. 
 
 (( 
 
 $148.59 
 
 
 29. 
 
 u 
 
 $3350.37 nearly. 
 
 15. 
 
 ll 
 
 $136.72 
 
 
 30. 
 
 (( 
 
 $3322 
 
 IG. 
 
 (C 
 
 $22.08 
 
 
 31. 
 
 (( 
 
 $228 
 
\)'^ 
 
 
 APPENDIX. 
 
 
 
 32. 
 
 Ans. a little over 13f 
 
 48. 
 
 Ans. in between 23 and 
 
 
 
 per cent. 
 
 
 
 24 years. 
 
 33. 
 
 it 
 
 a little over IS per 
 cent. 
 
 49. 
 
 (( 
 
 in a little over 4 
 years. 
 
 34. 
 
 C( 
 
 nearly 28^ per ct. 
 
 50. 
 
 '( 
 
 nearly 27 years. 
 
 35. 
 
 i( 
 
 a little over 31 per 
 cent. 
 
 51. 
 
 (( 
 
 in a little over 22 
 years, at 5 per ct. 
 
 36. 
 
 i< 
 
 by a little more than 
 4^- per ct. 
 
 
 
 and in about 19 
 years, at 6 per ct.l 
 
 37. 
 
 <( 
 
 little over 2 per ct. 
 
 52. 
 
 (( 
 
 in between 13 and 
 
 38. 
 
 a 
 
 <( (( J^ u 
 
 
 
 14 years. 
 
 39. 
 
 ii 
 
 <C CC O iC 
 
 53. 
 
 (( 
 
 in a little over 20 
 
 40. 
 
 a 
 
 a little more than 
 
 
 
 years. 
 
 
 
 247 millions. 
 
 54. 
 
 <i 
 
 in about 88 years. 
 
 41. 
 
 i( 
 
 is given in the book. 
 
 55. 
 
 (( 
 
 nearly 4 per cent. 
 
 42. 
 
 (C 
 
 between 11 and 12 
 years. 
 
 56. 
 
 (( 
 
 a little over 4f per 
 cent. 
 
 43. 
 
 (( 
 
 is between 13 and 
 14 years. 
 
 57. 
 
 (C 
 
 between 16 and 17 
 years. 
 
 44. 
 
 (( 
 
 between 40 and 41 
 
 58. 
 
 <( 
 
 a little over 9 years. 
 
 
 
 years. 
 
 59. 
 
 (( 
 
 $3621.29 nearly. 
 
 45. 
 
 iC 
 
 between 17 and 18 
 
 GO. 
 
 (< 
 
 $20429.50 cents. 
 
 
 
 years. 
 
 61. 
 
 i( 
 
 nearly 11 per cent. 
 
 46. 
 47. 
 
 
 between 17 and 18 
 
 years. 
 
 in a little over 14 
 
 62. 
 
 (( 
 
 nearly 28 per cent. 
 
 years. 
 
 "h-i^ 
 
 '*£,, 
 
 '<^ZJ 
 

 
 o > 
 
 
 n-' 
 
 
 
 ^' 
 
f/A/o %,,^^ 
 
 -P^ v' 
 
 
 ■^^^.^ 
 
 
 
 
 o.^- 
 
 
 ^-/l^