I^'l^ Ax. IMAGE EVALUATION TEST TARGET (MT-3) 1.0 I.I 1.25 IM 2.2 In 1^ 2.0 U III 1.6 ■** y] ^m A //A Photographic Sciences Corporation 23 WEST MAIN STREET WEBSTER, N.Y. 14S80 (716) 872-4503 .^ l^ w. CIHM/ICMH Microfiche Series. CIHM/ICMH Collection de microfiches. Canadian Institute for Historical Microreproductions / Institut Canadian de microreproductions historiques Technical and Bibliographic Notes/NoTas techniquas et bibliographiquas The Institute has attempted to obtain the best original copy available for filming. Features of this copy which may be bibliographically unique, which rray alter any cf the images in tha reproduction, or which may significantly change the usual method of filming, are checked below. G D □ □ n D Coloured covers/ Couverture de coulaur r~~] Covers damaged/ Couverture endommag^e Covers restored and/or laminated/ Couverture restaurde et/ou pelliculie □ Cover title missing/ Let! tre de couverture manque □ Coloured maps/ Cartes g^ographiques en couieur Coloured ink (i.e. other than blue or black)/ Encre de couieur (i.e. autre que bioue ou noire) I I Coloured plates and/or illustrstions/ Planches et/ou illustrations en couieur Bound with other material/ Relii avec d'autres documents Tight binding may cause shadows oi distortion along interior margin/ La re Mure serree peut causar de I'ombre ou de la distorsion le long de la marge ini^rieure Blank leaves added during restoration may appear within the text. Whenever possible, these have been omitted from filming/ II se peut que certaines pages blanches ajouties lors d'une restauration apparaissent dans le texte, mais, lorsque cela dtait possible, ces pages n'ont pas M film^es. Additional comments:/ Commentaires supplimentaires: L'Institut a microfilm^ le meilleur exemplaire qu'il lui a m possible de se procurer. Les details de cet exemplaire qui sont peut-dtre uniques du point de vue bibliographique qui peuvent modifier une image reproduite, ou qui peuvent exiger une modification dans la mithode normale de filmage sont indiquds ci-dessous. □ Coloured pages/ Pages de couieur r~T| Pages damaged/ Pages endommagdes Pages restored and/or laminated/ Pages restauriies et/ou pelliculias Pages discoloured, stained or foxed/ Pages dicclordes, tacheties ou piquees I I Pages detached/ n Pages ddtachees Showthrough/ Transparence Quality of prir Qualit4i inigale de ('impression IncSudes supplementary materia Comprand du materiel suppi^menvaire Only edition available/ Seule Edition disponible r~7\ Showthrough/ IVJ T I I Quality of print varies/ □ IncSudes supplementary material/ Comi nn Only edition available/ Pages wholly or partially obscured by errata slips, tissues, etc., have been refilmed to ensure the best possible image/ Les pages totaiement ou partiellement obscurcies par un feuillet d'errata. une pelure. etc.. cnt 4t6 film^es ^ nouveau de facon ei obtenir la meilleure image possible. This item is filmed at the reduction ratio checked below/ Ce document est filmi au taux de reduction indiqui ci-dessous. 10X 14X 18X 22X 26X 30X y_ 12X 16X 20X 24X 28X 32X Is J ifier ie ige The copy filmed here has been reproduced thanks L'exemplaire filmi fut reproduit grAce d la to the generosity of: g«ii«rosit6 de: Douglas Library Douglas Library Queen's University Queen's University The images appearing here are the best quality Les images suivantes ont 6t6 reproduites avec le possible considering the condition and legibility plus grand soin, compte tenu de la condition et of the original copy and in keeping with the de la netteti de l'exemplaire film6, et en filming contract specifications. conformity avec les cor^ditions du contrat de filmage. Original copies in printed paper covers are filmed beginning with the front cover and ending on the last pegs with a printed or illustrated impres- &ion, or the back cover when appropriate. All other original copies are filmed beginning on the first page with a printed or illustrated impres- sion, and ending on the last page with a printed or illustrated impression. The last recorded frame on each microfiche shall contain the symbol —^(meaning "CON- TINUED"), or the symbol V (meaning "END"), whichever applies. Les exemplaires originaux dont la couverture en papier est imprimie sont filmAs en commenpant par le premier plat et en terminant soit par la dernlAre page qui comporte une empreinte d'impression ou d'illustration, soit par le second piat, salon le cas. Tous les autres exemplaires oiriginaux sont filmis en commenpant par la premiere page qui comporte une empreinte d'impression ou d'illustration et en terminant par la darnidre page qui comporte une telle empreinte. Un des symboles suivants apparaitra sur la dernidre image de cheque microfiche, selon le cas: le symbols — ► signifie "A SUtVRE". le symbols y signifie "FIN". Maps, plates, charts, etc., may be filmed at different reduction ratios. Those too large to be entirely included in one exposure are filmed beginning in the upper left hand corner, left to right and top to bottom, as many frames as required. The following diagrams illustrate the method: Les cartes, planches, tableaux, etc., peuvent dtre film6s d des taux de reduction diffdrents. Lorsque le document est trop grand pour dtre reproduit en un seul clich6, il eat filmd d partir de I'angle sup^rieur gauche, de gauche d droite, et de haut en bas. en prenant le nombre d'images ndcessaire. Les diagrammos suivants illustrent la mdthode. ta ure, ] 1 2 3 1 2 3 4 5 6 I HBIVTUD AT rilK ;TEAM I'RIJ^S ESI Ani.ISHVrKS: t' of cow, I'LAHIC k CO. COLHORNK "fREFIT, TORON'TO, KEY TO ADVANCED ARITHMETIC FOR CANADIAN SCHOOLS. BY HARiNARD SMITH, M.A., St. Peter's College, Cambridge, AND ARCHIBALD McMURCHY, M.A., University College, Toronto. TORONTO: COPP, CLARK &. CO., 47 FRONT STREET. 1874. '~^Qfil03. Sq3^ l?7^ C3 630 694 V. . ^ KEY TO ADYANCED ARITHMETIC. ■*•> SIMPLE MULTIPLICATION. (1) 87298 46 523788 849192 4015708 (5) 840607 80 67248560 (10) 78847 8803 236541 C80776 630776 Ex. V. (p. 28.) (3) 16097 59 144873 80485 949723 694090141 (6) 175 189 1575 1400 175 33075 (13) 234578 18 1876624 234578 4222404 6298 769 56682 37788 44086 4843162 234578 29 2111202 469156 6802762 (3) 296897 83 890691 2375176 24642451 (9) 256073 5004 102428 128036 128137428 234578 53 703734 1172890 12432684 10^^263 8 924846 67 6473923 5549076 61964C82 KEY TO ADVANCED ARITHMETIC. 924840 95 4C24230 8323614 87860370 924846 430 27'MrmO 3699;j«4 397683780 2846067 206 17076402 5692134 580289802 2846067 1008 22768536 2846067 2846067 907 19922469 25614603 2808835536 2581382769 8409631 21711 8409031 8409031 58867417 8409G31 16819262 182581498641 8409631 7009 75686679 58867417 58943103679 8409631 8435 42048155 25228893 33638524 67277048 8409631 7980 672770480 75686679 58867417 (13) 1754 9306 10524 5362 15786 70935237485 -- 67108855380 16S22724 47506 4500 23753000 190024 213777000 149570 15790 13461300 104699 r 74785 14957 2361710300 554768 39314 2219072 554768 1664304 4992912 1664304 815085 20048 6520680 3260340 1630170 16340824080 OiO-< r\i Af\i KO MULTIPLICATION. 9 133450789 9b7«o4;j21 12;54;'>6789 240Ui;jr)78 49:3827150 61728;j945 7407407^4 804197523 987054;] 12 llllllliOl 121932031112035209 57298492093 700809050321 57298492093 114590985384 171895478070 286492403400 515080134228 458387941530 401089448844 40155302248305278754133 (14) 9487352 4731246 509241 12 37949408 18974704 9487352 28402050 00411404 37949408 44880990200592 38015733 400700005 190078000 328094393 200110124 15200292S 15232900283422580 574585014805 2837154309 5171370533785 1723750844595 2S;29380r-J325 57458501 4X()5 4022099304055 172375084-1505 4,590084918020 1149171229730 1630188053103049203385 4342760 599999 89084840 39084840 39084840 39084840 39084840 21713800 17376873 7399078 (15) 650090 3008 7600870: 900900! 139014976 5200720 684078885 I5JIS'' 1950270 6840?8885''' 150391848 I^SSS^o -^^^ I2l238?04 684763017903S85 'J005G51057240 128572831 324fii ft 10 (16) 12 17 8^ 13 ioi 19 1836 204 3876 KEY TO ADVAxVOED ARITHMETIC. 8781 3783 7563 30248 264G7 11343 14299742 3783 42809226 114397980 100098194 42899226 6565 6786 39390 52520 45955 39390 44550090 9898 356400720 400950810 356400720 400950810 54095923986 440956790820 (17) 20470 1030 614100 20470 21084100 2958 476 17748 20706 11832 1408008 ii (1) SIMPLE DIVISION. Ex. VI. (p. 36.) 0,7 ) 14683059 (543817 135 118 108 103 81 220 216 45 27 189 189 (2) 44 ) 817286228 ( 18574687 (S) ^ 59)54906734(930622 44 377 352 252 220 328 308 206 176 302 264 382 631 180 177 367 354 133 118 353 308 308 154 118 ~36 1>IVISI0X. 11 (4) 90)0848734753(71340987 138 90 337 288 393 384 947 804 835 768 673 673 (^^) 87 )70«0r»433( 814545 Oii> 120 87 o95 0-' 348 474 435 393 348 453 435 "l7 (^ 55)049305745(11805559 55 99 6S 443 440 305 275 307 275 324 275 495 495 (7) 133 ) 28894545 ( 234915 439 309 604 493 1135 1107 184 183 015 615 (8> 615)433418175(704745 4305 2918 2460 4581 4305 2707 2400 3075 3075 (9> 189)1074918(8803 1512 1029 1513 1171 1134 378 378 1> KEY TO ADVANCED AKITIIMETIC. M I (10) 7;U);ilssir 10 (40930 31 k; (ti) 907 )53081JJ71I ( 501803 ;247 '.oil O'w; 1 270 45.;.j n:};!1 8163 (12) 5016,0) 11! 111111111 1(22151337 10032 10S9 907 7827 7250 6714 5442 2721 2721 107lii 10032 7591 5016 25751 25080 6711 5010 16951 16951 15048 19031 15048 39831 35112 47191 (13) 141,0)82354608 7iO ^ 1035 1008 274 0,0 ( 757 5710070 (!4) ") ) 57380025 53025 37875 56>^12 53025 ( 7575 (l.T> 5406 ) 353008972662 ( ' 32436 5529947 144 1306 1290 28648 27030 16189 10812 37875 37875 1008 1008 53777 48654 51232 48654 41626 37842 25786 21624 37842 37842 v^628 DIVISION. 18 (16) 24C8 ) 5099G1567212 ( 243096269 10630 9872 (17) -^ 789,0000)2679953,4687(3396 7404 . . 3120 ^ ;: 3851)57111104051 (148302Ci 15447 14808 6393 4936 14561 12340 22212 22212 7625 7101 5243 4734 6094687 3851 18601 15404 ti971 30808 11630 11553 7740 7702 3851 3851 (19) ^^11 ) 10000000000000000 ( 9000900090009 iOOOO 9999 ^^^11 ) 10000000000000000 ( 900009000090 10000 9999 100000 99999 10000 9999 100000 99999 10 ;l!i u KEY TO ADVANCED AKITHMETIC. (20) 1646,00)6343945,67(3854 4938 14059 1S168 (21) 8914 90009)67157148872(746115 8^3Q 630063 6845 ~' 6584 26167 ! I N^ 415084 360036 (22) 550488 200563 ) 1320225292 ( 6084 ^"^^""^"^ 1203378 540054 104343 90009 14334. 90009 633383 450045 83337 1684729 1604504 *- 802252 802252 (88) 8496427 ) 7428927415293 ( 874359 67971416 63178581 59474989 1J7035925 33985708 30502172 25489281 60128919 42482135 76467843 76467843 (24) 79094451 ) 60435674536845 ( 764095 653661167 606955883 474566706 323691775 316377804 751397284 711850059 72 395472255 395472255 DIVISION. 15 (25) 5578 ) 65358547823 ( 11717^01 5578 9578 5578 40005 89046 J084 9594 5578 40167 39046 11218 11156 6223 6578 645 (27) 3854)152181255(39486 11562 (28) -~— 4093)143255(35 36561 12279 687637943 ) 3968901531620 ( 5771 3438189715 6307118166 4813465701 4936525652 4813465601 1230600510 687637943 542062567 15 34686 18752 15416 33366 30832 25335 23124 20465 20465 72 8 9 3211 203534191—7 25441773 91-7) 73-6) 55 2826863, with rem. 65. (29) 72 ) 203534191 ( 2826863 144 695 576 193 144 494 482 621 676 459 432 N0TB.-N08. 1, 2, 4, 6 should be done in the same way. 271 271 216 55 16 KEY TO ADVAi^CED ARITHMETIC. I II 24 11 (30) 78936 873 23G808 552553 631488 613 68911741 (32) (31) 855856651 2705 86783 ) 855853946 ( 9862 781047 204) 3060 (11 days and 156 miles 420 264 156 (33) ' 126 ) 4380000 ( 34761.90 378 600 504 900 883 over. 748069 694264 638054 520698 173566 173566 780 756 240 126 1140 1034 60 (34) 51 ) 7000000 ( 137254.90 51 190 153 ~i70 867 130 102 280 255 250 204 460 459 10 /. diff. of cost per mile = $137254.90 - $34761.90 = $102493 nearly. $11 6( 48:^ 241:^ 2894 Son' t DIVISIOX. (34) 1377 ) 94405914 ( G8559.13 8J?!)2 11785 IIOIG 7(599 G885 8141 6885 12^04 12898 17 352 ) 24777430 ( 70390.42 2464 1374 1056 3183 3168 1500 1408 920 704 216 1710 1377 3330 2754 576 $183U0n" ^'' ^^^^^^ = ^^0390.42-168559.12 nearly, = MISCELLANEOUS QUESTIONS. a) 603 48 4824 2412 28944 Ex. VII. (p. 38). I. (2) 90909 9181818 9000000 1053634 (3) icon's age=73 ypnrs-37 v"n----o/> „_.^„ a J --i^ o,jv.ai;3 — u;; years. 18 KEY TO ADVANCED ARITHMETIC. (4) loxJ.l,,3-7am-671o2-f-4+40734x2 =55721)5-^ 73174-10788+81408 =483831-16788+81468 =407033+81468 =548501. (2) 45 ) 18875 ( 415 180 67 45 11. 5,0 ) 7925,0 1585 ('5) 1396091 1111566 252047 330857 80857 124288 34816 101000 3431523 225 225 1 .-. in one year the deaths amount to .-. number of years requiretfitoo1)0^^^^^^^^^ 700409000000000000. Since (4) 494871 -94853=: 400018 45070- 3177= 41902,' 54;,12- 3987= -)325 l'G3+ 231= 1994 378 X 379=143641,' the expression is equal to 400018 + 41902- 50325-1994-H4Q«4i = 4419?0-50325- 1994+143641 ^^^ =391.595-19r4 + i436341 =389001+143641 = 5'^' iN umber required=528 x 30+44 = 19008+44 = iyi;o2. (2) 15683 9893 85)25575(300 255 75 MISCELLANEOUS QUESTIONS. HI. (3) 813215640 46536 19 62513 ) 813169104 ( 13008 62513 188039 187539 500104 500104 (4) (395456-2364)^HJ6;JS2iI^707; the expression is equal to (5) 8823-148-6+707 =8675-6+707 =8669+707 =9376. A, B and C score 108 runs, ^ and C' score 90 runs -4 and G score ^^ 51 runs ; ••• A scores (108-90) runs=18 runs ^scores (108-51) runs=:57 runs'. scores (iu«- (0) runs=33 runs. IV. Son' (2) a vvin De CoO+21) years=71 years (3) 100100101 ; "" ancUen" '"^ *^^ ^"^^^"«> one hundred and one thousand ■ 1840. KEY TO ADVANCED ABITHMETIC. 20 ii (5) Ans. 4549205. (1) 478 146 99 ) 4843 ( 48 396 2868 1913 478 883 793 Q1 V. (2) Number=163 x 430-^86 =70090-5-86 =815. 69788 (3) .-. in 120000 persons, 20000 speak English, 30000 speak French, 70000 speak English and French. His property=|(10000+15000-f-5500 x 4-^3750 x 3 + 4563+509) =1(10000+15000+33000+11250+4563+599) =$63412. (5) Divisor=(9281-373)-5-17 =8908-J-17 =524. VI. (1) MDLXm. and Tx. (3) Divisor=97+665+91=853; .-. dividend =853 x 665+97 =^567245+97 =567342. (4) Two hundred and seventy thousand, one hundred and foiir^io23r "" ^l^o^i-^and, seven hundred and eighty- 8 ) 10234 1279— 3 remainder; .*. C is the number required. 20 REDUCTION. 21 (5) Two years since eldest son's ao>e=: 5? -oo . .'. eldest son's age now=:31, youngest son's age= 52±?i _. ?! ..jg . daughter's age=60-(31+13) =60-44 =16. REDUCTION. Ex. VIII. (p. 57.) (1) $878.38 100 (2) £ 57 20 1140s. 12 13680^. (3) s. d. 8 4^ 12 100c?. 2 87828 cents. 201 half pence. (4) £ 8. 83 15 20 1675s. 12 d. 6i 20100d 4 $1027.87 100. 102787 cents. £ s. 15 13 20 312s. 13 B744d. £ 8. d. 1 3| 20 20s. 13 243 £ s. 393 30 7860s. 13 94331(f. 3 343 4 975(7. d. Hn425^. 188663 halfpence. 22 KEY TO Axj.^SCED ARITHMETIC. m i (5) 738 half-crowns. 30 22U0d. 4 570 crowns. 5 28508. 3 885G0g. (6) 5673542— g. 4 12 3,0 1418385— 2g. 11819,8— M 5909— 18.S. A)is: £5909 ISsrW. 8550 four penny-pieces. (7) B 25 8 200 half-crowns. 5 1000 six pences. 6 6000 d. (8) lbs. oz. dwt. grs. 59 7 14 19 12 715 oz. 20 24 14314 dwt. 24 4 6 2,0 12 1500 four pences. 37400157 grs. 9350039—1 155833,9—5 21 7791 n —19 dwt. 343555 grs. (9) 56332005 scruples. 20 6493—0 Ans. 6493 lbs. 19 dwt. 21 grs. 536 lbs. 12 24 2,0 12 1126640100 6432 oz. 8 281660025 —0 4694333,7—3 12 grs. 51456 drams. 8 154368 scs. 2347166 —17 dwt. 195597 — 2oz. Ana. 195597 lbs. Troy, 3 oz., 17 dwt, 12 grs. EEDUCTION. 23 10 (10) tons. cwt. qrs. lbs. 7 15 2 16 20 155 cwt. 4 622 qrs. 85 15566 16 249056 oz. 28 4 h 4. 2,0 (11) oz. 5838297—1 1459574—2 364893—1 91223—6 9 25 13031—3 qrs. 335,7—17 cwt. 162. 16 16 25 4 4 4 4 5 5 drs. 7563241—1 ) f 9 1890810—2 ) 472702—2 118175—3 29543—3 4 2,0 5908 3 14 18 1181—3 29,5.1 14,15 tons. cwt. qr. lbs. oz. dr. Ans. 14 15 1 18 14 9 tons. cwt. qr. lbs. drs. 33 17 3 27 15 20 677 4 2711 28 75935 16 1214960 oz. 16 19439375 drs. Ans. 162 tons. 17 cwt. 3 qrs. 25 lbs. 9 oz. 24 KEY TO ADVANCED ARITHMETIC. (12) lbs. oz. 17 2 12 206 oz. 8 1648 3 4946 sc. ®o Srs. apoth. grs. troy ^ „ . , . 34678=::34078 me gram being the same in each measure ' 4 I 34678—2 ) 24 4946 SG. 20 6 2,0 98920 grs. 8669—5 ) [22 grs. 144,4—4 dwt. 72 fj (18) mi. fur. po. miles. 8 7 8 573 8 31 fur. 40 1248 poles. 6240 624 6864 yds. 1760 34380 4011 573 1008480 yds. 3 Am. 72 oz. 4 dwt. 22 grs. (14) inches 1364428—4 in. 12 3 113702—2 ft. 3025440 ft. 12 11 4,0 8 37900 2 75800—10 half-yards. 36305280 inches. 74 mi. 3 fur. 4 yds. 8 ^ 689,0—10 po. 172 —4 fur. 21 595 fur. 40 lea. fur. po. yds. ft. in. Am. 7 4 10 5 2 4 33800 poles. 5i 119004 11900 130904 yds. 3 , (15) 4 lea. 2 mi. 2 in. 8 392712 ft. 12 14 mi. 1760 840 98 14 24040 yds. 8 73920 ft. 12 887042 in. 3 4712544 in. 24C40 yda. 2661126 barleycorns. J EEDUCnON. 25 22 2 11 (16) fur. yds. 7 200 220 1740—0 870—1 7d 2 yds. Am. 79 chains, 2 yds. 12 cub. span. cub. in. 6 1 = 69 18 117-9 in. 9 A718. 9 feet, 9 inches. (17) yds. qr. 84 1 4 337 qrs. 4 1348 na. Eug. ells. qr. 56 1 5 281 qrs. 4 1124 na. (18) Fr. ells. qr. 83 3 6 501 qrs. 4 2004 na. FI. ells. qr. 73 1 3 220 qrs. 4 880 na. (19) ac. ro. 35 2 4 142 ro. 40 5680 po. ac. ro. 56 2 4 226 ro. 40 9040 po. 30ir 271200 2260 273460 sq. yds, (20) ro. po. yds. 8 37 ^6 40 157 po. 30i 4736 39i 47751- sq. yds. «7 42977i sq. ft. 144 171908 171908 42977 36 6188724 sq. inches 26 KEY TO ADVANCED ARITHMETIC. ac. po. 8 30 4 12 ro. 40 510 po. 30i 15300 127J 15427^ sq. yds. 9 (21) ac. 15 4 ro. 3 63 ro. 25000 815000 120 4,0 4 po. 5000,0 1250 —2 312 Am. 312 ac. 2ro. 1575000 sq. links 138847A sq. ft. (22) c. yds. 29 27 203 58 1728 ) 158279 ( 91 c. ft. 15552 2759 1728 27 c. ft 3 9 91-1 30-3 3 10 eft. 783 c. ft. 1031 Ans. 3 c. yds. 10 c. ft. 1031 c. in. (23) c. yds. c. in. 17 1001 27 119 34 459 c. ft. 1728 3072 918 3213 459 1001 c. 3''ds. c. ft. 26 19 27 201 52 721 c. ft. 1728 5768 1442 5047 721 (24) galls. 563 4 2252 qts. 2 4504 pts. gills. 794153 c= in. 1245888 c. in. 2 4 365843—3 gills. 91460 45730—2 qts. 11432 Ans. 11432 galls. 2 qts. 3 gills. 1 5 6 BEDUCTION. 27 (25) bush. pka. 760 d 4 S04:j pks. 2 4 2 4 2875046 718911-3 qts. 6086 grails. 4 34^44 qts. (26) chald. 250 1500 750 0000 bush. (27) reams, quires. 56 19 20 1139 quh'cs. 24 4556 2278 350455—1 gal. I 89803-3 pks. A718. 89863 bus]). 3 ])ks. 1 gal. 2 qt3. pks. 186043—3 pks. 34 bush. , 1291 Ans. 1291 chald. 34 bush. 3 pks. 24 2,0 sheets. 52073— 1 ) — - — [ 17 sheets. 13018— 4 ) 2169 quires. 108 27336 sheets. ^'''' ^^^ '"'''"''^' ^ 'i'"^^^' ^'^ sheets. wks. 30 7 257 d. 24 1045 514 (28) dys. hrs. 5 17 mo. 1 30 hrs. sec. 23 59 6185 hrs. 60 371100 min. 60 30 days. 24 143 (?0 743 hrs. 60 44580 min. 60 6185 hrs. 22206000 sec. 743 h^s. 2674859 sec. riH KIOY TO AI»VAN<'ICI» A Kl IIIM lOTK (m iJ l:;:i(i:tsi (lt^sh!<),^ pinlM. :{(( \ I \) I OOSMOJH ii;5.'(>,;Vu biirrols. (:n) (•(» :\() \ i) rt'iuhrr llicic arc ;!l » .lannarv llicrc an* :»1 » lA'hiuarv tin re arc 'JS (IVM. his. 5^5 (J i)(ir)j ,lays. vrs. «lvH urn ) Aii!.>ust tlu>r(M'cmain 'Jd ;{ dys. 1 1 ScpttMnlMM- llicrc arc ;!<) It's. 8-15 418 To March M, (I a. m. ii (; 5()'jr) hiv. i.'Ol> U - (10 aoirMH) 1 (JO mil. IHODOOOO SCO HI 2i «» o 10 1 (Ml J Aom 2()2l)i 18 (iO Mr)r);i7i?o mui. m 87;{v;i.'a^()o sec. COiMPOU xN I ) Mi: I/n PLIO ATIOJ^. Ex. XL (p. (J7.) (4) S571) OS aOO IS i> tor lij 10 a7 !;{;({ \) lor 111 7oi7 «Ji i'ov'A £ i^. (f. ^r^'^\) oif 1501)18 t) lor 12 10 a7i;i7(J lor 141 osniU) 8i for 1 1 oTOUa U '^i lor 147 aUU745 8i lor ir)5 I 1 '"i ^■OMP(HINI) MlIi;iiru,jATION. 2U 10 2570 d. 10 ^5700 7i (or lo JO 2571)0 7i for 10 10 957000 ;{ for loo 2rm()0 « 4 lO.Mddl 5 for -100 257i)0();{ *» 1H05;{0 1 ^UorTO " loino jjioi-.i 3 for 100 10 for 1000 2 5I5H00({ n l^v^447 7i lor 474 77;}700 18 7;{70 2570 (5) ll>s. oz. (Iwls. crs. «tf 7 10 11 8 0011050 5 (7) 11)H. OZ. (I PH. 45 7 ;} for 3000 forJiOO lOi for ;jo Oil lor I S\ for 23yi sc. 2 «j);j 2 11 10 11»«. oz. (Uvts. trra HO 7 10 11 4 547 5 4 11 )H. oz. (Irs. 45 7 ;{ sc. 2 II •Jt^5 7 5 20 for 4 501 10 1 for 11 3111) 5 12 12 1br;m J{01l"o 2 for 00 91 7 1 for 2 *^'-(^- 3 1 iforGS (6) tons. cwt. qrs. lbs. oz 3 24 13 11 y OO -I K -^ /w'^ 1 (Continiiocl on next pago.) 000 1 2 Hi 'it , ji li 30 KEY TO ADVANCED ARITHMETIC. (Ex. 6 continued.) tons. cwt. qrs. lbs. oz. 3 24 33 (Ex. 8 continued.) yds. qr. na. 67 1 2 5 21 3 23 11 for 7 10 336 3 2 for 5 10 210 17 1 11 14 for 18 1 23 14 for 6 "O 3368 3 for 50 202 2 for 3 228 18 (9) fur. po. yds. ft 3 10 12 for 76 3570 3 2 for 53 m. 70 2 10 3 10 1 1 10 mi. fur. po. yds. ft. in. 70 2 10 1 11 2i 2 4 7 1 1 1 12 Oi 1 4 4 2 10 for 4 for 28 fori (10) ac. r. po. yds. ft. 16 3 38 27 2 11 186 3 27 26f 4 ac. r. po. yds. ft. 16 3 38 27 2 67 3 35 17ir 8 for 4 271 3 22 Hi 5 for 16 1903 36 23 8 for 112 (1.1) ac. r. p. 380 3 32 12 4571 24 ac. r. p. 880 3 32 10 3809 for 10 10 1 1 14 1 2 for 29 — 38095 for 100 2285 2 32 for 6 40380 2 32 for 106 (12) gals. qts. 57 3 10 577 2 gals. qts. 57 3 10 577 2 for 10 10 6775 for 100 (Continued on next page.) COMPOUND MULTIPLICATION. 31 (12 continued.) gals. qts. 5775 for 100 2 11550 2887 404 14841 for 200 2 for 50 1 for 7 3 for 257 (13) Ids. qrs. bus. pks. 76 5 2 12 184 15 1 2 5 for 12 2 fori 199 1 7 2 for 13 lbs. qrs. bus. pks. 76 5 2 6 02 1 0for6 4 ^08 4 for 24 10 3G81 for 240. (14) yr. w. d. h. m. 5 6 18 14 11 1 13 4 8 34 yi". w. d. h. m. 6 18 14 10 5 1 4 14 9ft fnt. 1 A — ,. -i j^^. 10 11 24 3 23 20 for 100 yr. w d. h. m. 11 24 3 23 20 for 100 3 34 21 3 22 1 1 4 22 6 19 4 20 for 300 for 30 6 for 9 38 46 2 13 6 for 339 (15) tuns pi. hhd. gals. pt. 84 43 1 9 190 1 10 IforO 3 571 1 30 3 for 27 hhds. gals. pt. 84 43 1 10 846 53 2 for 10 10 8468 28 4 for 100 3 25405 5081 338 22 4 46 4 for 300 4 for 60 4 for 4 30825 10 4 for 364 (16) bar. gals. qts. pts. 43 14 1 1 4 173 17 2 0for4 9 1561 13 130 4 2 for 36 i for '6 1691 17 2 1 for 39 (Continued on next page.) 32 KEY TO ADVANCED ARITHMETIC. d. (16 continued.) bar. gab. qts. pts 43 13 11 10 433 25 3 for 10 10 4337 5 3 for 100 bar. 4337 gals. qts. pts. 3 for 100 5 30360 2603 173 2 10 17 3 3 2 for 700 for 60 for 4 33135 30 2 for 764 1 (17) d. H 10 10 7 11 for 10 62 7 5 6 for 60 6,} for 7 69 13 Oncost of lambs. £ 3 .«<. a. 2 Hi 10 31 9 4i for 10 7 150 6 156 £ 37 8. 2i 4 148 11 cost of cows. (18) 7 chests. 18 136 drawers. 8 1008 divisions. 185.35 1008 202 2525 $25452 5 7* for 70 8 Oil for 3 14 5i cost of sheep. £ S. 38 17 5 194 • 5 for 5 3 583 77 15 for 15 14 for 2 660 9c()stofhors( £ s. d. 69 13 Oncost of Iambs. 150 14 5i cost of sheep. 148 11 cost of cows. 660 9 cost of horses 18 i fi eyn("p,sps 1053 4 lOJ wliole outlay, COMPOUND DIVISION. COMPOUND DIYISIOK Ex. XII. (p. 69.) 33 (3) lbs. oz. dwt. sTs lh«» 20)459 4 4 33(15 169 145 24 13 292 ( 10 oz. 290 3 20 45 ( 1 dwt. 29 16 24 406(14grs. 29 116 116 (4) lbs. oz. drs. sc. 68)15511 3 6 2(228 loo 191 136 551 544 ~ 13 87 ( 1 oz. 68 19 8 158 ( 2 drs. 136 22 3 68 ( 1 sc. Am. 2281bs.,loz.,2drs.,lsc. Arts. 15 lbs., 10 oz., 1 dwt., 14 grs. £ s. d. 754)1288 1 8(£1 754 534 20 754 ) 10081 ( 145. 754 3141 3010 125 125 12 < shares the tourth person must Jiave 2i sliares': therefore, dividing liie whole into (24+6+3+1) or 33 , 108 "^^ "33""'' dollars, therefore, G, 12, 36, 144 are the numbers. IV. 12^'^ 1st casejj is selling price of a stamp. 13 2d case— is selling price of a stamp. .'. gain in first case x ll=gain in 2d case x 12. (3) Value of each: 95x20 95 100 "5" = 19. (5) 20864 ,^„ ~228~~ "^ gallons bought, 25920 180 ' :144 gallons sold. (6) ac. ro 24 17 3 25 ro. 40 1000 po. 163-144=19 gallons leaked. (7) dvs. hrs. hrs 365 6=8766 8766 1851 8766 438,']0 70128 8766 16225866 hours. 44 KEY TO ADVANCED ARITHMETIC. V. (2) . miles. barleycorns. barleycorns. 25000=(35000 x i;(JO x 3 x 12 x 3)=4752jJ00D00. (3) 86 X 12=1032 cents, cost of one dozen. ••• ^¥3¥:=40i-!} dozen. (4) It makes 2i vibrations in one second ; therefore, in 24 hours it makes 24 x GO x 60 x 2^=216000 vibrations. (5) 247 ) 1859.56 ( $3.48 741 1185 988 (6) £ c. m. £ f. c. m. (23 4 6) X 12=276 5 5 2 (18 l)x6 =108 6 amount =384 5 5 8 1976 1976 therefore, $3.48=price per gallon with duty. 6 13.48 .58 duty per gall. VI. (1) £ 8. d. 2793461 ) 130524465 4 6 ( £46 11173844 18786025 16760766 2025259 20 2793461 ) 40505184 ( 14s. 2793461 12570574 11173844 1396730 12 2793461 ) 16760766 ( M. 167G0766 12570574 Ann. £ lu 14.-*. M. MISCELLANEOUS EXAMPLES. 4$ (2) 1 mi., 467 yds., 1 ft.=6G82 ft., and ^4^=13 ft.=4 yds., 1 ft (3) Half a ton =1000 lbs.; .-. (1000 X 11 cents =11000 cents: .'. he gains 10-$90=$20. (4) galls- s. £ 8. d. 10 cost 10 X 13 = 6 15 •' 15xl4i=10 17 6 18 *'*18x 151=14 3 6 48 31 1 3 5 6 33 6 6 .'. he must sell the mixture at , ,.£33 6s. Gd. rate of ~^^ =15«. 6d. (5) By the question, 16 women's shares=48 children's ,, , 13 men's shares =73 children's' (dO+48+73) children=150 children ' • f; '^^ ^ o= 13.96, woman's share. $1.33x6=17.93, man's share. (6) £3 2«.4-£l l5.+145.=£3 17«.=77«., £1636 55.=33735«- • therefore, he will have ?f5=4675^3^^ ^^^ ^^^^^^ ' (1) VII. 37 1 1853, 1856, 1860 bein^ leap years Sesfdays ; ^^^^ '"^ ^^^ ^^ years=(10 x 365+3) days= " Sfiutes^^ "" ^^^ min.=(87673 x 60) min.=5360320 (3) tons. cwt. qrs. lbs. mi. fur. po. n 425 15 3 13 28 L4i 18 IS 1$ 14i 1361 4 38 340 3 48 5 46 KEY TO ADVANCED ARITHMETIC. t (3) 110 yards are contained 16 times .'. 5 X 16=80 feet=26| yards. in 1760 yards (4) Number of seconds =?5eWS_?W>. 192000 therefore, number of days- ^Q^QQ QyQQ QQOO , 200000000 o-fif^af^c^f^ ..^..L ^^^ X 60 X 60 X 24-192 x 6 x 6 x^4 2o0000 00 6250000_781250 390625 390625 IB^.L''^'''^ 192 X 27-24 X 27-12 x27-"324 =1205§ti (5) $480.60-^45=: $10.68, share of each. $10.68 X 20= $213.60, share of 20. |480.60-$213.60=$267 to be divided among 15; $267-^13=$17.80, share of each. (6) $60480 share of third son. 7560 68040 share of second son. 24000 92040 share of eldest son ; $60480+$68040H-$92040=$220560,sum left by the father. ton 1 (3) cwt. 3 VIII. qr. lbs. 1 = 2325 Am. 7 lbs., 6 oz., 4 drs. 315 ) 2325 ( 7 2205 120 16 1930 ( 6 1890 30 16 480(0 24 MISCELLANEOUS EXAMPMES. 47 (3) 7000 per hour. 9 63000 per day. 67 (4) In 1852 there were 360 days Kow $9.63 X 366=13534 58 • .••$3034.584- $200=13724.58, his income. 441000 378000 2,0 ) 422100,0 £211050 iu 67 days. (5) (•) £ t c. 896 5 4 391 5 3 23 9 m. 7 8 6 O £ 896 391 f. 5 5 c. 4 3 m. 7 8 £1311 1 8 1 O mills. 23096 248 £505 9 184768 92384 46193 (6) 5727808=£5727,8f.,0c..8m. . . A'yu+418=£108, whole cost of house. (1) IX. ^an.^l77o'l!^^^^^^^ f3^,% ^^^ ^^^ween was nit a leap year! ' '^''^' '"''" ^^^^ y.-y^ .-. u^ , ooo X ly-f-l71) davs— 20'7.^,'ii rUva Now (30000-29755) days=245 days ^ and reckoning back from Jan. iflTTO weTnd that th.^ day required was May 1, 1769 ' ^^'^* ^^^ 48 KEY TO ADVANCED ARITHMETU', (3) ^^^ 528000 ^^^^ "" ^^^^ "" ^^ feetr.528000 feet ; •'• ~Q~=52800 revolutions of fore- wheel, 528000 Yg— =33000 revolutions of hind-wheel, 52800 33000 19800 Ans. (3) 4i miles=(4i x 1760 x 3) feet,=r23760 feet 1142 ) 23760 (20 2284 920 •". 20-,9i\S-=20H^ number of seconds required. . (4) 1+2+34-4+5+0+7+8+9+10+^ -12=78 •'. 3x78= 156 strikes in 24 hours- also (1+2+3 + 4) X 24=10 x 24=240 chimes in 24 hnnr« . /. (156+240) X 365=396 x 365=14l5Ttimes in ar ' (5) 16 mi.=(16 X 1760 x 3) ft.=84480 ft 550 *' (110 X 3i) ft.=^ ft.=275 ft. he walks per minute. ,84480 16896 1536 „„ to w-Ke'ScI^ """'"• "' ' ''''■' ' "■-■■ 12 -«• (6) " (SaSpLj^^?^^'^'''^^^^^ received by each man weeklv Jfu77^TTrl^^^^'^'-=''^^S'' «^'^^" tJ^^ "^^n weekly ^' noQL Vn^'^)^- =6930d=wae-es of the boys weeklv • • ■ ^S,-^f ^^)^-=^^^20^- t"t^i «f weekly wagTs; '"' f2 ^•=(89i0xl3)«.=ll5830«.=£5791 10«. annual wages. MISCELLANEOUS EXAMPLES. 49 X. (1) 2oz., 16dwts.=56dwts 100 oz., 16 dwts.=2010 dwts. therefore number of spoons=?^=?5?:=3e (2) ^^ '^ Now 1848 wa?a leapTe:' V^ZTfn O" V<=""- were ^ -^ ' meietoie, in the 5 years there ^'''^"08^4^''^'''^ '''''='''' ^'y^- ^""'^ 1836 ^=m^=^^'' IW. mq. what he may spend daily. (3) 19 years-(i9 x 365i) davs-i?ii-??5i i„ uu^i-; udys ^ — lunar months. __19xl461 27759 (4) Italent=2t9000grs.=^?:??2?oz. ^ 24x20^^-' therefore value of a talent =?^^5?^x^0 ^^ 34x20xl06^~^^*''^^- (6) 11.27 x56=$71.13 also |366.88+$17.04=$283 92 .-. $383.92-$7112-«IKof9on fi' • .• to sell the remaind^r^^ ' ^^'^ ^'^^^ ^^^ ^^^^^ lie has *313.80-T-133=$1.60. (6) ' Vamed.*'' """^'^^' "^ men=number of pence the men 20 times the number of mpn-T,,,n,hP" nf ,.- •• earned. i^moex ot pence ine women 'V^?d.*'' """''^^ of men=number of pence the boys I 1 \1 i 50 KEY TO ADVAXCED ARITHMETIC. .*. 02 times the number of men = number of pence earned by all ; also (£7 lo. 33 ) 55 ( 1 33 22)33(1 22 ill' 11)22(3 22 •. 11 is the G. C. M. of 319 and 407 ; — 11 11 333 ) 504 ( 1 333 171 ) 333 ( 1 171 i'i 162 ) 171 ( 1 162 9 ) 162 ( 18 — .-. 9 is the G. 0. M. of 333 and 504 ; 72 _ :?;j.qvr)04 ir57«'>o 73 .-. L. C. M.= ^ '^'^ ■'" 9 = 18648. 9 LEAST COMMON MULTIPLE. 61 (7) '99 ) 2061 C 3 2397 564 ) 799 ( 1 564 235)564(3 470 94)235(3 188 47)94(3 94 47 is the G. C. M. of 2961 and 799 ; L.C.M.=-^^^^^^96 2365839 47 (8) 7568 ) 9504 ( 1 7568 47 r50337. 1936)7568(3 5808 1760 ) 1936 ( 1 ••. 176 is the G. C. M. of 7568 and 9504 • 1!!? L. C. M.=^5^.^^Zi5?52Z2_408fi.o 176 ) 1760( 10 176 -■ *"«»^^- 1760 176 . (9) 4662 ) 5476 ( 1 4662 814) 4663 ( 5 4070 592) 814 ( 1 592 222)592(2 444 •■■ 74 is the G. C. M. of 4662 and 5476 • ..L C M-^^^^"^^^"'^ 25539112 • ^^ 74 ='~~^A =344988. 148)222( 1 148 74 74 )14.9f 3 MS 62 KEY TO ADVAA'CED ARITHMETIC. ill £ fd \,tu (10) 6327 ) 28997 ( 3 18981 5016 ) 6337 ( 1 5016 1311 ) 5016 (3 3933 1083) 1311 ( 1 1083 228 ) 1083 ( 4 912 171 )228( 1 171 .:. 57 is the G. C. M. 6327 and 28097 ; 57 ) 171 ( 3 0/ o7 (11) 5415)30105(5 27075 3030) 5415 ( 1 3030 2385 ) 3030 ( 1 2385 645)2385(3 1935 450 ) 645 ( 1 450 195 )450( 2 390 3 60)105(1; .*. 15 is the G. C. M. of 5415 anri 30105 : —1 In 15 ''^' h. LEAST COMMON MULTIPLE. (12) \rm'-i ) 21489 ( 1 15863 5620) 15863 ( 3 11253 4611 )5626( 1 4611 = 11754483. 29 - 39 63 1015 ) 4611 ( 4 4060 551 ) 1015 ( 1 551 4G4 ) 551 (1 461 29 is the G. C. M. of 15863 and 21489 • ^^ ^ 435 ^ ^ L. C. M-lggg_3> <31489 _ 340880207 ' _ 29)87(3 87 2 (13) 12, 8, 9 2 6, 4, 9 3 3, 3. 9 3 2 3 (14) 8, 12, 16 6, 8 12 3 L.C.iVi.:^3x3x3x3x3=72. ^ 3, 1, 3, 4 2 .•.L.C.M.=2x3x3x 3x3=48. (15) 3 I 6, 10, 15 3, 15 1. 5. 3 3 (16) 8, 12, 20 6, 10 o 1' 1, 1 h (J, i\I. =2x3x5=30. .-. L. C. M.=3 X 2 X 2 X 8 X 5=130. f in Iff;! I m i m Mi 64 KKY TO ADVANCED ARITIIMETIC. ri7) 27, 24, 15 9, 8, 5 L. C.M.=3y 9x8x5=1080. 19 (19) 19, 29, 38 1, 29, 2 .: L. C. M.r_-19x29x3 = 1102. 2 (18) 12, 51, 68 2 6, 51, 34 3 3, 51, 17 17 1, 17, 17 1, 1, 1 L. C. M.=3 X 2 X 3 X 17=204. II 2 (20) 24, 48, C4, 193 2 12, 24, 32, 96 2 6, 12, 16, 48 2 y, 6, 8, 24 2 3, 3, 4, 12 2 3, 3, 2, 6 3 1, 3 1, 1, 1, 1 L. C. M.=2x2x2x2x2x3 X 3=192. 2 63, 12, 84, 14 2 63, 6,42, 7 3 63, 3,21, 7 7 21, 1, 7, 7 3, 1, 1, 1 .-. L. C. M.=2x 2x3x7x3 =252. 3 5 (23) 5, 7, 9, 11, 15 5, 7, 3, 11, 5 1, 7, 3, 11, 1 .-. L. C. M.=3x 5x7x3x11 =3165. (23) 2 3 5 1, 1, 4, 5 L. C. M.=2x 3x5x4x5 =600. 6, 15, 24, 25 3, 15, 12, 25 1, ^ 4, 25 LEAST COMMON MULTIPLE. 2 (34) 13, 18, 30, 48, 60 3 6, 9, 15, 24, 80 8 5 3, 0, 15, 12, 15 1, 3, 5, 4, 5 1, 3, 1, 4, 1 3 3 5 7 (35) 15, 35, 63, 72 5, 35, 31, 24 5, 3). 7, 8 1, 7, 7, 8 1, 1, 1, 8 65 .L.aM.=:2x2x3x5x3x4 /. L. C. M.Ux3>;5x7x8 =2520. 2 9, 12, 14, 210 3 9, 6, 7, 105 7 3, 2, 7, 35 3, 3, 1, 5 •••^•C. M.=2x3x7x3x2x5 2 (37) 54, 81, 63, 14 3 37, 81, 63, 7 3 9, 37, 21, 7 3 3, 9, 7. 7 7 1, 3, 7, 7 1, 3, 1, 1 •.L.C. M.=2x3x3x3x7x3 — llo4. 2 2 2 3 5 (38) 34, 10, 32, 45, 25 12, 5, 16, 45, 25 6. 5, 8, 45, 25 3, 5, 4, 45, 25 1, 5, 4, 15, 25 1. t 4, 8, 5 .-. LC. M.=2x2x2x3x5x4 X 3x5=7200. 2 2 3 (39) 1, 3, 3, 4, 5, 6, 7, 8, 9 1, 1, 3, 3, 5, 3, 7, 4, 9 1, 1, 3, 1, 5, 3, 7, 3, 9 1, 1, 1, 1, 5, 1, 7, 2, 3 L.C. M-2x2x3x5x7x2 X 3=2520. no KKY TO AI)VANCI:D ARITHMETIC. I ll :l , V (30) (31) 7, 8, 9, 18, 21, 72, Hi 2 12, 20, 24, 54, 81, 63, 14 7. 4, 9, 9, 12, 30, 72 7, 2, 9, 9, 6, 18, 36 7, 1. 9, 9, 3, 9, 18 7, 1, 3, 3, 1, 3, 6 , i^, ~v, iw-r, <^-r, wj., ut^, 0, 10, 12, 27, 81, 63, 7 3, 5, ., 27, 81, 63, 7 1, 5, 2, 9, 27, 21, 7 1, 5, 2, 3, 9, 7, 7 7. 1. 1, 1, 1, 1. 1 1 ■» » •'I L. C. M.=2x2x2x3x3x7 X 2=1008. (32) 2. 1, 7, 7 1, 5, 2, 1, 3, 1, 1 I.. C. M.=2x2x 3x3x3x7 X 5x3x3=22680. 8 13 225, 255, 289, 1023, 4095 75, 85, 289, 341, 1365 25, 85, 289, 341, 455 T), 17, 289, 341, 91 1, 17, 289, 341, 91 1, 17, 289, 341, 13 1 1, 17, 289, 341, 1 1, 1, V 41, L. C. M. : X 3 X .') X 5 X 7 X 13 X 17 X 17 X 341=2017790775. vulgah fractions. Ex. XXIV. (p. 95.) ^oTK.— In tlio followinsr E.Kampks, common factors are strnck out of numerator and denominator. (8) ^,fl2iof^of''^of%f9=!of?5oftof^of^of9 y " - () 8 « 2 o 8 7x25x4x5x3x9 175 '9x3x5x3x3x2x4 8' , G3, 14 ., 63, 7 ., 03, 7 r, 21, 7 ). 7, 7 ^>. 7, 7 18 VULGAll FRACTI0X3. (9) o 10 4 10 27 18x3xl0x4">a0^^ _.5 X Tx^lj^x 9x3x8 x 4 x 2 14 18x3x5x2x4xox 2 x"3xl)~l5' (10) 67 |of?of5of70|oflofllofl47 5 -3 ^0 ^032 3 _5x^_x^x G33x^x 18 X 147 7 X b X 7 x ai _ 3x3x79 x3x36399 2xli -~22"- nek out of Ex. XXV. (p. 96.) In each of the followin- Examples we divide the numerator •nd denominator by their G. C. M. Thus in (1) l=~=l (dividing by 4, the G. C. M.) ; or, as we shall write it in the following Examples, 4 4-f-4 1 8 8-f-4~2' (13) ,825 j25-;-3 275 2709~270y-:-3~9U3' (15) 324 324 4- .'50 a 612 G12-j-a6~ir (17) 5l84_5m-f-1738 3 6912~6912-j^l728'^4' (14) 630 .630^_35 936 93G-f-18~52' (16) j>3G^_ 936-V-8 117 23G8~23G8T8~296* (18) ?:yi^3444--28__123 35,jfJ 355G-h28~"127" 1 1 68 f 'i ? KEY TO ADVAyClSD AUITIIMKTIC. (19) 7845 78154-15 523 067tlO U0780-4-15~045a' (31) 625 625-^125 5 0000 9000-^125~72• (23) lGa2_lCa2-^90 17 21)70~21)70-^im""ijT (25) 430 1 _ 4*.50 1 -f- n _ 25^ 95807 ~ 95807V1 7 ~ mil' (27) 6093_0093^;3_2031 i)17'4'~917ff3""i3058' (29) 2519j4_251^n4^12597_3 881 79 " 88 1 79 -^ 12597 ~ 7* (31) n4m_ 1141354-7609_15 220GG1 ~~220Gai^7()09~29" (20) 2472_2472-|-24_103 82G4"'a2G4-*-24""ia0' (22) _81___81-4-3 _J7_ 4872" 4872-^3" 1G24* (24) 102G5 _ 102G5-}-2053 _ 5 14371 "14371 +2053 ~r (20) 55247_55247-fJ01 _547 7484 1 ~ 74841 -h 101 ~ 7'4l' (28) 10812 10812-*-12 901 22800~22800-f-12~1900* (30) 374192_3741 92-^28784 13 575G80~575G80-^28784~20' (32) 128352_ 128352-1-1 3036_^ 238308 ~"2a83G84-18336~i3' Ex. XXVI. (p. 98.) (1) 12 4 j^, - , ^. 30 is the L. C. M. of the dcuominators ; .'. fractions become -L^15 ?_^J? i^ 1'^ 20 24 2x15' 3x10' 5x0' ^^30' 30' 30* (3) 2 7 5' 8' 40 is L. C. M. of dpnominotor;^' . r *' 1 2x87x5 ..fractions become—, — , or 16 35 40' 40* VITLGAR FRACTIONS. 2 3 5 (3) ■^ II u ;j. 4. Q- 12 is L. C. M. of denominators; 2x4 32^3 nx2 8 9 10 3x4' 4x3' 6x3' ^^13' 13' 13- .*. fractions become ^— - - - x^, 3x4' 4x3' 6x3' ^^T5i' iq« 7^- (4) * ^, :^j. 27 is L. C. M. of denominators; .-. fractions become?^ A ^. ^ 5 9x3' 37' ^^37' 37" (5) 3 ^ 11 7' 14' 38" ^^ '^ ^•^- ^- ^J- of denominators ; ••. fractions become ~ l^l 11 13 10 11 r 28' ^^ 9«' o5. 1 3 5 (0) 7x4' 14x3' 38' ^^38' 38' 38" o- 4. 9- 06 is L. C. M. of denominators ; .-. fractions become i^l? ?Jll ''5><4 18 37 on 3x18' 4x0' (HTi'^rg- - |. 7 n 17 8' 13' 18- '^^^^^•<^-^I- of denominators; .-. fractions become— ILl? 1*^x4 63 66 68 8x9' 13x6' irx~4'°^73' 72' 70- (8) "' A ^ 13 13' 16' 34- ^^^^^•C'-^^. of denominators; .-. fractions become A^i. '^x 3 13x3 30 31 9« 12x4' i6x-3' 3rx-3'«r48' 48' I' r,^ 10' 15- 2^ is L. G. M. of dci-iorainalors ; •■•fractions become '^-^ ?^3 14x2 25 27 28 6x5' IOx-3' iVxl'^rg^, 30' i 69 I Ipl 70 KEY TO ADVANCED AIUTHMETIC. (10) 2 2 5 7 5' 3' 9' 10' ^^ ^^ ^' ^' ^^' "^ denominators ; .'. fractions become 2x18 2x;]0 5x10 7x9 36 60 50 63 5 x 18' 3 X iX)' 9 X 10' 10 x 9' ^^ 90' 90' 90' 90* (11) 2 3 5 7 24 is L. C. M. of denminators ; 3' 4' 6' 8' .". fractions become 2x8 3x0 5x4 7x3 16 18 20 21 3 x'8' 4x0' 0x4' 8 X 3' ^^ 24' 24' 24' 24* hb ."■'' 1 1 i \ i' ■!il. ■. 0003 is L. C. M. of denominators; (12) r 11' 13' 3" .•. fractions become 1x420 4x273 7x231 2x1001 429 1092 1617 r, or 7x429' 11x273' 13x231' 3x1001' 2002 3003' 3003' 3003' 3003" (13) 3 85 14 5' 80' 200' 400 is L. C. M. of denominators; .•. fractions become 3 X 80 35 X 5 14 X 2 240 175 28 5 X 80' 80 X 5' 200 x 2' ^^ 400' 400' 400* (14) 12' 7' 63' 84" ^^'^ '^^ ^' ^' ^^' ^^ denominators ; .•, fractions become 7x31 n^x'T) 20xJ 13x3 147 216 80 39 12x21' 7x3(r 03x4' 84"x 3' ^^ 252' 252' 252' 252" VULttAU FHACTIONS. 71 (15) 7 ^ 13 _3 1 »' ir 1«' 25' ao' ^^^^sL. CM. of denominators; fractions bccomo '^~ j11^^ 13x23 3x18 1x11 0x14' 11x30' 18x22' 22x18' iwTII* ^j. 308 180 280 51 11 390' 390' 3J>0' 390' 396' (16) 17 5^ 3 17 3* 8' G' 14' 28' 3"2- ^^^ ^^ L. C. M. of denominators ; .*. fractions becoino 1x224 7x84 5x112 9x48 3x24 17x21 or ; 3x224' 8x84' 0x112' 14x48' 281724' 32l^» 224 588 560 432 72 357 672' 072' 072' 072' 072' 672* 3 4 (17) 7 8 16 31 3' 9' 2r' 81' 243' 7^" "^^^ is L. C. M. of denominators ; .'. fractions become 2 X 243 4x81 7 x 27 or -- __-_!_ S><9 16x3 31 .■ix243' 9x81' 27x27' 811^9' 243^' 279' 486 324 ISO 72 48 31 729' 729' 729' 729' 729' 729' 9 (18) 9 9 9 10' 100' luoo' 10000- lOOOOisL. C. M. ofdenominatore; .•. fractions become 9x1000 0x100 0x10 9 10x1000' 100x100' 1000x10' 10000' ^OjIO 000 90 or ---— _1'1"1. _ '" lOOOi)' inn/in' ma, 9 lOUOO' 10000' ioooo' lOOOO' rfT'^ .11: 11 : 73 KEY TO ADVANCE^ ARITHMETIC. (19) 31 17 la J_ 5 60' 90' 25' 105' 9* .•. fractions become 31x105 17x70 13x253 G300 is L. C. M. of denominators ; 1 X 60 5 X 700 or 60x105' 90x70' 25x252' 105x00' 9x700' 3255 1190 3276 jGO_ 8500 6300' 6300' 6300' 6300' 6300' (30) 31 11 53 3 54' 28' 63' 12' .•. fractions become 31x14 11x27 53x12 756 is L. C. M. of denominators ; 3x63 434 297 636 189 54 X 14' 28 X 27' 63 x 12' 12 x 63' 756' 756' 756' 756' — or (1) Ex. XXVII. (p. 98.) 90 is L. C. M. of denominators ; 3 8 7_ 5' 9' 10' .*. fractions become 3x18 8x10 7x9 or 54 80 63 5x18' 9x10' 10x9' "'90' 90' 90' .•. in order of magnitude tlie fractions stand thus, 8 7_ 3 9' iO' 5* 24 is L. C. M. of denominators ; (3) 13 6 7 2' 4' 6' 8' .•. fractions become 1x12 3x6 5x4 7x3 or 12 18 20 21 2x12' 4x6' 6x4' 8x3'"^* 24' 24' 24' 24' .'. in order of magnitude tlie fractions stand tlius, 7 5 3 1 8' 6' 4' 2* VULGAll FfJ ACTIONS. 73 (3) 1 .3 7 4 ft 3 5°^ 8' 12' 3 ^^7'^' 40' .*. fractions become _7 8 12' 7' 840isL. C. M. ofden"; A^ 7x70 8x120 63 490 960 ^ 40x21' 12x70' 7xf20'"^840' 840' 840' •• in order of magnitude the fractions stand thus, 4 6 7 13 3°^ 7' 12' 5 ^^8* 5^ 12' (4) A 1? '1^ 16' 21' '60' ^^ ^' ^' ^- of (denominators ; .*. fractions become 5x140 3x105 10x80 31x28 or 13x140' 10x"l05' 21x80' 001728' 700 315 800 868 1680' 1680' 1080' 1680 ' .-. in order of magnitude the fractions stand thus, 31 10 _5^ J^ 60' 21' 12' 16" (5) ? 1 1 1 21 7' 13' 22' 11' 20' ^^^2 i^ L. C. M. of denominators ; .". fractions become J>.x91 1^183^ 31_>^77 22x91' 11x182' 26>r77' or— ^ -•- -'" 1456 1617 3 X 286 7x 154 7x286' 13xT5l' 858 1078 819 2002' 2002' 2002' 2002' 2002 .". in order of magnitude tlic fractions stand thus, 21 £ ^ 3 9 26' 11' 13' 7' 22' 74 KEY TO ADVANCED ARITHMETIC. I' (6) 3 5 3 3 11. It 15 19 1 ^ of y of 4, ^^ of - of r>, ^ of^ of 41, ^, or ^, -^^-, j^, -^. 3G9G is L. 0. :^I. of dcnoniinalors; .'. fractions become 15 X 204 G X 3;j0 19 xJTT l_x 1848 ^ 3960 3016 14G3 1848 _ 14x'3(r4' iTx^JiaO' 48 X 77' 3 x'i848' °^' 3090' 3090' 3090' 3696 ' .". iu order of niiignilude the fractions stand thus, |of^of4, lof^of5, y, Jof^of4|. (7) ? ?Z 1. _L ^" ifiO is L. CM. of denominators; 8' 33' 10' 10' 40* .•. fractions become 3x20 27x5 _9.ni' 10x10' iOxiO' 40x4' 60_ 135 _1)0 \\2 108 _ 100' 100' 100' i()0' 100' in order of magnitude the iractions stand thus, or 15 4' 27 _7^ 27 9^ 3 32' To' 40' 10' 8" (8) o 2 '> 15 ^^ ?i 3^, ^ of 9i^, or ^ , ^-, yg. 420 is L. C. M. of den(»uiiiators; .-. fractions become 15x105 1^x140 94xJ3 , m5 1400 1138 l^io^' "iT^TlO ' 35 X 12' ^^' 4^20 ' 430 ' 420' ' .-. in order of magnitude the fractions stand thus, ^, 3Jr, ^of9|. (9) 6 13 5 29 6 13 13 5 29 1 i . — or ■ — — — — . 7' 28' ^' 8' 50' 7' 28' 9' 8' 50 504 is L. C. 3T. of denominntors; .-. fractions become 0x72 13x18 13x^56 5x^6^ 29x9 7lv~73' 28x18' 'd'xSO' 8x03' 56x9' ^ 432 234 728 rri5 201 _ ^^'504' 504' 504' 504' 504' .'. iu order of magnitude the fractions stand thus, , 6 5 29 13 I9, 7' 8' 56' 38' VULGAR FRACTIONS. (10) 5^' 11' 1«' 23' -jg' ^^^ 's L. G. M. ofiionominatoi-s; .'. Ihictioiis become 8x_44 3x36^ 7x^33 9 xJ18^ 5x11 J)x44' ilxao' r8x32' 3:3"x 18' a6>riT' or^ y|8 154 10^ 00 iiiW 3JiG' 390' 3i)G' ;]9{j' ••. in order of magnitiule the fractions stand thus, 8 ^ ^ 3 5 9' 23' 18' Tv m (11) 76 M m 401 700 51 113 30 401 700 7«' 153' 1-' 448' 748' «^ 76' 103' 38' 448' 748- 1591744 is L. C. M. of denominators ; .-. fractions become 5l_x 2^944 113x10473 JJ0x4l888 401x3553 700x2128 76x30944' 153 x l(m3' 8^x41888' 4-48x3553' 748» ^^.1068144 118133^0 1G33G33 1434753 1489G00 1591744' 1591744' 1591744' 1591^44' 1591744' .-. in order of magnitudo the fractions stand thus, 700 401 113 51 1-1. 748' 448' 153' 70* 15 T' (12) '^h ^ofO^, ?of^f^,or^, ^, »* 1 ' 7 9 5' 4' 3' 35' 03' 1'300 is L, C. M. of denomin:irors; .-. fractions become; 1^x315 10x430 91x30 8x30 or 4x315' 3x430' .35x30' 4735 4300 3384 100 1300' 1300' 1300' 1300'' 03 X 30 In order of magnitude the fracti ons stand thus. T. ^h "of 9? "f.-fj :6 KKY TO ADVANCED ARITHMETIC. 1)1 1 1' I I'i 3 7 2 1 1 ' YT^, T, ., nnii .. 12 is L. C. M. of denominators; ,•. tViicti«;ii3 become :ix:) 7_ 3 X 4 1x2 U<._C) . ^ ^ 8 2 J5_ 4 X a' 12' '3x4' g"x2' 2x"G' *^^' 12' 12' 12' 12' 12 ' 3 1 .'. '- is the gi'Ciitesl fraction and - the least. 4 ' - o (14) 11 29 n 7 47 12' 80' 18' Tg' 48" 720 is L. C. M. of denominators ; fractions become llxGO 29 X 24 17xjlO 7x^5 4 7x15 12xG0' 30x24' 18^40' 1G"x45' 48x15' GGO G9G G80 315 705 or 720' 720' 720' 720' 720' 47 . 7 -- 13 tiie greatest fraction and — the least. 48 * - IG ADDITIOJS^ OF YULGAE FEACTIONS. Ex. XXVIII. (p. 100.) (1-) 2 5 _;^_2x4 5x2 7_8 10 7_25_ 3"^6"^12~;rx4'^G'x'2^i2~i2'^r2"^i2~"12~ "'^* (14) 3 2 1 8x7x3 2x5x3 1x5x7 G3 30 35 5 7 3 7x5x3 5x7x3 ^3x5x7~105^105 105 128 105' 12 3- ■•■IDS* (ir)) 2"*"5"^l6~2x5'^5x2"^10~i0+T0'^10-T0~^' 5 1 ADDiriOX OF VULGAIl FKACTIONS. (10) _02ll, 1x3 7 10 3 7 20 5 77 12^8~^24-13x2"^8ir3+24 (17) 2^3. 7 2x84 24^24 "^ 24~:i4~G' 8 12 0x24 «x 15^ 12 X 10-120 ■^i20 + i20=i20=l' 'U (18) 'Ui J_'^x7x9,4x4 4'^7'*"9-4 + x9 7x4x7 x7x9^7x4xj> "^9x4177 189 144 196 529 =>T^ + (19) 15. 3 1x21 5x7 252 "^252 "^252-252 3x2 21 35 6 62 =2/A. 2"^6-^21-2x21-^6^+2rx"^=4^-+43+4^=~=l|i=:lif (20) l-i-i-ulo-l-l^^O, 1x20.1x15 ►+5+7+?= 2 3^4^5-2 x30-^3ir20+4iri5 + 5 ;+■ 1x12 xl2 ^30 20 15 12_77 60 "^00 "^60 "^ 60-60"=^^* (21) i4._?._f*^45 6x7 9x9 ^ + 7P + oP = 7 45^35-7x4o^45x~7 + 35ir9 6 ^225 42 81_ 348 31o^315"^315-315=^"^^'^«"=liVy. (22) + 2f+13A=15+s+s+-=154- — -4-1^1^. 3x7 2 ' 7^10 x3o'^7xl0'^10x7 =15+55,1^^21 06 '^+70-^70+70=^'^+70=^^5* (23) ?+|of 1 + 9^0=9+?+-+ 5-94-'^ ^12 4^4 3 053-^^ -^5+15+20-^ + 5^12 + 15174+20^ _o,36 16 9 60 ' 60 60~'-^ . 01 + r-:=10 00 r (■ ■Hi I Ill 78 KEV TO ADVANCED ARITHMETia 3 (24) ^^i«f^SH^=^-f5;^^=5+!-,U-? 11) 19 5"^3"^iy =5 + _^^1^38 1^05 3x10 190 + lUO "^luiF _. 38 05 20 (25) 7 4 1 11_7^ 4x10 Ixa li 5 3"^G"^JiO-5x6"^3x 10+6175 + 30 iuo^iuo"^iyo' x5 11 =5+J|=5m. ^42 40 ^ 11_98 30 + 30"+30"^30~30'^^"**'f (26) 10 2 18 5 2 9 5 14 + 15 "^70-7+" 15 "+35-7 (27) ^_5xl5 2x7 9x3 xlS^l 5x7 + 35 x3 _75 , 14 27 116 = '-,-77^ + 105^105^105~105 — Irinr. |+I + 4+l-.t^42 7x21 4x30 2x10 5 10^7+21-5 x43 + 10^r2r-+77.SO+- 7x30^21x10 _ 168 147 120 20 455 210"^210+210'+2T0=2l0~^'^'^-^*- 1 7 „ (2S) ^4_1_1^15., ^x 30 5 X 40 Ox 18 8 12'^9"^20-8x45"^12ir30+(rx-4?]^- =57rn+:T:77\ + 40^20x18 10 200 163 617 360^360 ^300 "^360-360 ~"««n~ls6"i (29) 1 4 4 ,+6H,ofJ-6+i + t+^4=6+^+i 5 + 5+-21=^+-5+2l=^+l+21='^-^ (30) 3 100U64f+^ of 701=100+64+430+ ( 2 5 3 5+9^+5 since ^ of 701: 2103 --g— 420Jj,==584+-+-=: 5 '9 =585i ADDITION OF VULGAR FKACTIONS. n (31) 26li4-174i+| of 10^=261 + 174+8+ U'^-^3 J 4 4 ( since - of 10A=il?l_35 \ =443 + -+^.443 + Ul,.444+f4=444f. 6'C' :444f. (32) 387i+885J + 394i+?of3704 . ■ * (»' since ^ of 3704=^^= =387+285+394+1481 + ?4-Uia.^ 2^4'^3"^5 =14813-), =:2547+lll!5. liil'V^^S^^ 3x12 2 X 30-^4 X 15+3^20+5^13 =2547+??4.15. 20 36 60+60+60 + 00 =2547+^=2547+1*1=2548^ (33) ^11000+1^4.110^.11 10221 10000 '=iS5o=^"i^"Wo-. (34) U+l!..29^47 59 t8^18"^30+48+60 ^n^ 17x40 29x24 47x15 59x12 18 ^4v o0xjj4 48x15 '60x12 ^660+_6^+G96 + 705 + 708 3449 720 = - - = 45.69 720 ''^- n so KEY TO ADVANCED ARITIl^fETIC. I! (35) 1 1 3 20 88 (5814 is L. C. M. of tlcnominntors) l_x^2907 lx64G J}^S42 '>\)xW,l 33 x 171 '3 X 2!K)7"^9 X (i4G'^17 x 842 ^8« x 153"^34'xl7T 2007+040 + 1020 + 4487 -r 5618 14G59 5«14 5bl4 =2g^H. I : (30) ,. ,. , 11 21 10 10 20 11 =^^^^ + 17+84+51+51-^08+12 =8 ILll^x— ^- —^A ^1 1 1x17 ~ +17 X 12 + 81 x0"^51x4 + 08~x"8 + i2l7T7 204 204 5 8 (37) 2j + 3|+4|+5KC?=3+3+4+5+6+^+?+|+5 + ? o 4 5 o 7 =20 + =20 + 2 X 140 + 3 X 105+4 x 84 +5 x 70+6 x 60 _ -__ 280+315 + 330+350 + 300 430 -20+^'^=20+3fii=231U. (38) 1 ^ 00 ^^+^^^ + 15+^"'+28+7 «^ 5 ,.,»,, 4 1 13 5 9 2 ^^+"+'+7 + 10 + 15+10+28 + 85 =11 4x240 + 1x1084-13x112 + 5x105 +9x60+2 x 48 108"0 - 1 1 _!- ^1!.^ 1 1^1? +^4^>6 + 525 + 540 + 90 + ■" 1080 -ll + j^=ll+2i3-^=13H. ADDITION Oi^' VULGAIi FRACTIONS. i (39) 5iH-|of ^of 3i + 9A-f|of |of4 =•'5+2 + 9+1 + ^?+— 4-1 n, iviJO H —17-4-1 la 1043 (40) S Of i.^^+| or «.3* of 15 Of 5^ IS „,, 3^ ^^ 1 ^^ ^_^ ^ -iO 1/s 34 __900 + Gl-i-rO+5 1030 120 ~ 12^ -«m-. 120 '•1 I (41) 270i-f-650A +5000^ + 53„i+ 1-/„- =270+650+oOOO+53+l+?-i.-i.l 4 1 ^ ^4+20 + 4+5+20 =5974+Hl_ii±_5 + ^fi+l ,.., 40 20 =^^'-^+2o=^>m+2=5076. -c.+^ of ^4-'i+^ of ^- ^4.7.^1 189 3 7^ 8+31 __ 3xlO+7 j<^8-KJt X 4+ 180 1-28 _48-f 50+124+ 1 41 128 --™3.3 3 — o 128- iH' 8'i KKY TO ADVANCKI) AIIITIIM ETIO, ' ilV i ' E I II Sl[P>TIlACTION OF VULGAR FRACTIONS. (10) E.v. XXIX. (p. 102.) _., , 17x1:] 7x13 _ 221-84 „,,, (11) 1x8 1x2 3-2 10 24 lOxtJ 21x2 48 (12) 42-30A = 42-hl-31.^.=.lH-l-A^ll+l?_A=ll,if. 00 0^ (13) i^/^*-i/A=io+i+;;^-2^ _ 701 16 _ 761 48 ,_,,, - ^'^"^'72U-24^.-^^+72y-72U-^^^^^- (14) 90,';\ -25f,^=9()+l + -^--26M=G4+^fJ_ii 111 111 125 111x125 111x125' 1J«75 (15) 21-li5^=21 + l-24i3 = 19+^||?-^-19H-^5^^=10HI. 2iib 298" 298 (16) 127 -* of 14 =125 -4=: 121. SUBTIIACTION OF VULG. VK FIUCTIONS. (17) 5x3 1x4 S G :31+_1'^_ 15-4 HxS 6x4' ^^+ii-=3iH. 24 13 5 (18) 56 28 -~ of l,L=:i:^_l_y 3x4 13-13 1 oU 14 oG 14x4' 5G 56' (19) 3 2 of * of I-? of ?=l_.^^l^l_l_x5_9-5 4 4 9 (20) 4 10 G 10x3"~6 5 no 30~15' A 7 «f 9 -^- 6 «^ Si-g of I of 11 of 1,^ 3_1^2x^ 1x3 4-3 1 "6~'~6' ^ ^ i3x2 2xS (31) ? of 1-'^ ^ 4 8 °' 10 ^of-i 1 Jt 1^37 4x4 27-10 11 9 ^1 4 27 4<27-3rx"4=-Tnsr-T7^» 108 -108' ^ of 1-1 of -1^1_1_.1J14 1x3 4-3 1 . •• difference = il _ i _ il__ J J^_? _ 1 1 - 8 8 3 108 3U 108 i^Gxa— 108-=-~108=27- (33) .*. requiredresult=3+— _'^-0 4.95 5x8 ^ 95-40 " 9 72 9x8' 72 (23) lli+85=19+^+^^^jj^C_^7^ 3 ' 9-"''"^9+9=19+ 13 9 ' .*. result=: r. :19+i^-9|^-_10+li^_19_x9 ! II ^ft a4 KEY TO ADVANCED ARITHMETIC. i Difference of two first fractions , =5H-3?=3+S-''-!-|i??=^3 + l?l.-A*=34''^ 32x7 7x33 224 224* Sum of 8"^14"^C~« X 21 "^14 X 12"6 x 28 105 + GO +38 193 1G8 168 .1.2.fi.. . -••168 1 1, 07 12^ o 97 25 ,^ 97x3 25x4 .-. result=3A^4— lA%=2+i^-T— =2 224 168 224 X 3 ' 168 X 4 __ 2 91-100 _,, ~"'^' 072 ~^^"'*' ' 3 (25) c, -. ,. , . 9 27 9 27+18 45 Sum of fractions=l-/,+--=-+-=-^^=^-^, difference of fractious= 27-28 9 26 20" 45 9 Now r^^TTT X 5 ; .•. sum =5 times difference. 26 26 ' MULTIPLICATION OF VULGAR FRACTIONS. Ex. XXX. (p. 104.) (10) I - 3 ^„ . o 1x2x43x3 43 5 of ^xof of 3= - 8 d 2x3x8 — _— ni -8~ "• 1 2 (11) ^of3,xl^of|of? P'^'^'^zl^ X ^^9 X 3_ 4 X 3 V 11 X 2 X 17 X 3 X 13 X 3 ' 13 X 3 X 33 X 5i X 8 ~13 X 3 X 3 X 11 X 3 x 17x4^72 =1. OF VLLGAR FRACTIOJJfS. ,% (12) 15^^^-^^^f ^x^lof37x.of3jof^^ ^nx2,5x 1x7x1309x25x1 isx^-ixTcnri^ysiTs-x-ri =5 ^I^i5i^xllx7xl7x5x'i (13) 3 8 *'^^''^' ^f lo^3 of 3,4x1 of l?=l^_?3j^«4x2GGx3xlO «x«x2x7x;jx3->r3-^uVlI3^1^^^^. (14) 5A of 3i of A of Q4 V 3 ^9 117^^ 34Xj^of-of 1^x19 ^07x23_x 8x^4x3x9x4x19 Iyx4xll7xiurxlidir3~~ ^?7xJ0^4ji2x2xl7x3x9x2x2xlQ l^x.x2l0JlU3ir2l^7|TxT^^ 1 ^^'^^ 5xix^x|x5^l^3x3x4x5 1 ^-45 G~2x3x4~x5Mi=6-. 1'^ IG 11 70 9. ,. ,^ 64^27^10 >^.:i^x?,^=:J.:!illlx 11x72x21 10 080 ^5 G4x27xl(rxli85x85 ^xlrix_n_x4x2x9x3x7 ^ -^ - X.. J X 11x0x7x5x17 ._3 ^ 5x5x17-42^ ii S(\ KEY TO ADVANCED ARITHMETIC, k 07) IH x2j of IjVx^x — x5^ of 49x4; -•"^^-^ 8x44x33x19x111x49 x 3 ~ Td X 3 xIjTx 42 x'32«"x22 x 70 4x8x2x3x2x11 x4x3xllx 19 x3x 3'/ x7x7x3 5x3x3x37x2x3x7x8x41x3x11x4x19 2x 4x11 x 7_(nG 5:c41 ~''M -Q._i_ (18) ?x'>«x3^x5^x6 ' _'L^ 11x38x97x1165 gX.^xa..xo,,x6Tg4- GxIl^liiriy^liM _ 5x 11x2 x19 x 97 x llO.jS x 11C5_5825_ ~2T3 x 4 X 11 xlu x 2 X U7~2 x 3 x 4-~24~~ "* III , ft itt (19) !r> 17 116^21^153 8;^ 136 4 _ 9 5x17x87 x126x4x7 ^ 1 90 ^ 7 ^ "~\ 1 «) X 21 X 153 xT90 x 7 /- 5 5xl9x V7 X 32^2_9_x 9x2x7x4x7 _ 1 '4 X 29 X 3'x7 X 9 x l7 x 19 x'2^5 x 7 x 5~5' DIVISION OF VULGAR FRACTIONS. Ex. XXXI. (p. 100.) (10) 3iof3Mf^-.75.?;^xl5x^x4 15x5x3 1 1 4x3x2 xuxl5~4x3~12' DiVlSIO.y OF vu^o^^ FliACTlO.VS. 87 4x4 33 *x4> V 1 1 n ^>^^'<^x2xox 19x3x11 -S^F^io^l^'^- (1-j) IVo(luctof'?.i.uul;u.=''^^'51 155 " t\ "^ -""IT — - — • ^ » ly » M'loliont of 2' bv 'U~^^ ^ 30 '-' oi 31 1. ' ^'^1 31 X hi' '" ui,. iind —-. J(ix3J 31 X IG' 4«0,-) "490 320 49(i' I i 88 KEY TO ADVANCED ARITHMETIC. i 3 4 3 4 (16) 8 8 3x4x2 2 H~15~4^15~4x3x5' i 1 ; 1 (17) 14 14 1 27 27 14 8 7x2x3 2 2i~7~27^7~yx3x7~'9 * 3 (18) 35 2U 12 35 9 5x7x3x3 7x3 21 ^, ■~5~12''5" 3x4x5 (19) 5 9 9 6 n A 0^_2x3xllx9_27_ lo'^ll^lO" 11x2x5 ~5^ ^" 99 40 (20) 15i_l._^ 1 _20x2_2 20~29~3 ^20"3x20~3" (21) 56 56_J__5() 9_ 4xl4x9 _ l|-14-l ""U- 14 ~'^*'' 9 133 it< 1 1 L 1 1 i 139 10 14 (22) 139 14 139x2x7 139x7 973 ^,. 10 '■ 25" 2 X 5 X 25 "■ 5 X 25 "125 * *'" 11 iiEDvcnoi, or inactions. REDUCTION OF PRACTIOKS. 89 (3) Ex. XXXII. (p. 108.) 8 Ql ^ ^ '"^'"y ^^^^«-3 tons, 8 cwt., 2 qrs., 7 lbs (4) 14 ^f ra «f $21=.$3 GG2- ^ ^^ 3 ^^ ^ ^''^ 3-. O., \^ IMAGE EVALUATION TEST TARGET (MT-3) EXAMPLES IN FRACTIONS. 97 QUESTIONS AND EXAMPLES IN FRACTIONS. (2) 5 XXXIV. (p. li;}.) I. 2i 31 H% 1 31 31 7 17 7 -» U 5 * o 4 2 -5 4 «' ^ ^ 4 ■ 8~M-^17~r7; i 8 -47" 8x17 im 136 ,.«. 527-113 415 nmerence=: — -— — = — =:2i ^s- 130 136 '•**'• Also sum of 5i+- of 3*+?-^- o 4 7 (3) 43 21 43 X 3 X 7 301 ~6 "" 4 ~ 2x3x4 -"T"^'^^- 47 («^ 3-i - of ^^^ *^ nf 0-^'^ . P 2^ 345 V ) 0| Jft 01 U7-T-— r Ot y = -.-^ of - X 7- o4,j 12o 7 47x0 .?ii!Z^^•^.•'>x•'^><23 8_x23_184 o fxlT ~ 21 5x25^' 7 ''47i:^i^='rx¥"=¥r=^^>- {') £ 15 17 4 ^^_^lrl.-l_l ^_15_^'^ l'^x4 4x2 7 4 8 4f 4i^2i-30 17'^5-4x30~5~17"^7x5"8~5'^35 7 4 2 _ 7x 35-4 x7 x8+8x8 245-224+04 85 -17 8 X 35 280 •■>UA'~ ft/'* I S I I m KKV TO ADVAXCEO AUITHMETTO. I') 13 10 13 2197 2197-27 4itx4^-l \'i 13 O O 27 IG!) 9 _21T0 9 _ 217x9 217 27 "^ iG0~9 X 8 X 10" 48 ~ •* "' 27 109-9 9 (^•) 3 ^ ,-o,_l__q, 16 339+16 355 .,„ ' + 17} 112 + 1- 16 113 ^ iia ~ii3 I; :p (4) , 2j--l-f^0_l2_4 3 + 5 + 7~io"~;/ \r 2 4 . , ,, 70 84 90 Now^, ,, ^, arc equivalent to j^^, - -; 2 + 4 + 6 •*• :z—R rry ^'^^^ '^<^ between the greatest and least of the o-i~0+ I (■ .' 2 4 6. 2 . ,, , , ^6 1 tactions -, - -, since - is the least, and - o 7 o < the greatest. {' By the question, the smaller number 11x7-4x3 :20fi-15,'s=5 + 105 II. -=5- 77-12 105 =5A^=5H. (2) Wy (he question, the number=41i— 19^=21^. By the question, 4 4 7 4 24 4 nu,nbe,.=3iof--3,^,of-3=^><5^jgof± 7 4 10 13 7x4x2x5x13 91 . 2 5 i4 4 2x5x24x4 ""24' P D (•) 9;?2(^ _ 93208 -f-1 22 _ 76 4 13786 ~i3tS()-^ 1^3 '"113 05469 _ 95409-^789 _ 1 21 35o?s4~85978i^789 "456* QUESTIOXS A.VD EXAMPLES IN I'lcACTIONS. (4) 5 3 (> 4 4^« 4 4^6 _19_^ 8._41^10_x :3x 254 -8x39 -41 x 25 13 25 31) " 23x'6d _75() + 312 -1025 37 975 ~ "y?o 7 1 O 3iof oMf;^-' of -'- = lI^U^_ _£>_ 1309 5 9 3 12 5x2x9 3xl2"~ 90 ~36 2618-25 2593 =: =: . — 14. la. 180 180 ■~""^' \19 13/ \ 3/ V3^5/ 19xl3-:i''T^ 45 3 8^_^ ■I9xl3''8''i5~24r 19x13 "3 15 / 3 . 4^ . GA S_ 14 _ 3 X 41 X 2 X 3 X 2 X 37 X 7 (i\ :l nf '^^ ..f "'"•' ^ <« 41 6 „ 74 7 2x7x3x3x3/xllx2x41'~ir (5) Product of 2h and 2|=??x-=i'iZ 15 8 5 Difference of 2i and 2i='^-^~'^J _1^1 5 4 "90 0,0-20' = 16. 5 4"20 20""20 quotient=i^-^-^-4^7x20 (2) 5 '20- 5x7 III. (•) 275i+62,V, + 1031i- + ^ of 4150f =4999+ 1,11.1 3 8 40+11+24-1-45 .., 120 =275 + 62 + 1031+3631 + -^ + Jl, 1 .'^ 3^120^5'^8 "120 =4909+ .^^iooft u 1 ^5000 90 i t 101) KEi' ru ADVANCiiU AUiTlIAlETIC. ^' 51 • »5 "^ 207 ■ * ' "51 X 1 12 X 207 x 3(>~ 18* ^^ d} y ■^2"^4i~lG 36^1()"^32~lG~4'^l«'^iG _ 5-44-20 + 2 32_^ ~ 16 ~1G '' ^^ 4i+3r4-3i~7f+3~8''6l'^3''3~Gl'^9~^*^' (3) --- oi 4=-, and . of 5=t. 4^n^ 5-^5 «^^^*' • =?xlx--? of 41^1ii^^l^^_3x7_21 4 11 5*5 4 4xllx5x2x41~llx~3-23' 1_1 l_JL__15-10+8-0 23-16 8 12^15 20~ 120 27968__279 68-4-64 437 37a7(J~i[7370H-()4~5b4' 120 _1_ 120* ]^ofa+5i)+^of iof(7-2|)-i =^of0K-^-of iof 4|-i^lxl^ + 5xlx^^ 1 1^ 6 27 ^ 3 13'' 2 +6^27'' 5"~3 -L _gg. 1^ 81+23-54 50 _25 "2 102 3 162 ~lG2~8l* 10 5 20 140+35-40 3 "^6~2T 42" 5_4" ~ a.l-v 6 7 ~l2" 135 ,^, ■=-jj-=12-A-. (3) Number=3i-| of (^+J-^h^1).^_3 ,, (15±0_^2+5) 13 4 .?x^J!^13_17_105-17_178 4 45 4 00~ 60 ~60~^^""' ,„,! „„,, , .,. 64 1 1 1 1 1 ami uuuiucr— X ~ X y - — -^ 315^4^6^8-315^3=945- 103 KKY TO ADVANCED AUITUMETIC. I! «>i ii (4) 1 2 After paying away - of my mouey, - remains; o b after paying away ^ of - of my money, ?-? of |, 3 2 or ^ remams ; after paying away j of - of my money, --— , or -j remains 3 13' (5)' Sum of fractions=5J-+5i=10fi, difference of fractions 12 144'" '"'12" .•. quotient is 144 times as great as product 12' 1 127 1 •• product=rlOf, X :r^=-rjr, quotient=10A-^i-=127 ; ■'n V. (1) 2+3 4 + 3i 4 7i 5 15 5 7 7 44-5 ■ 5+5i~8 • 10,;~9 ' 21""9 0~9" 1 5 Sum of - , If, and '- = l + l+^+'^=.l+?l±18±35^1^74^ 3_7_58 o r, .. ... -r^^ ^^21~21' 2 ' 7 ' 6 42 viT r 4 , 3 10-9 7 58 7 29 difference of -^ and -r-=: =— • • nrodnrl— -- x —— — • 15 " 20 GO GO' • • P^«<^'ici_2j X g^_y^ , .-. quotient=:^-^ il of m-^ x 1? x ^-A ^ 90* 18 ^"'~90''ll''29"ir (2) ^^ V2 5/"\3 i)~ 10 • "12 ~-10'''7~T~"'d'~ 35^. e (QUESTIONS AND EXAMPLES IN FRACTIONS. 103 ~_^— A il_l ^ ^''^ 5> ir> 9 784-729 m IC^ 4i) 54 10:3 4i) 1 (;:J 10:3 ~49 4 1 j;] -^4,7 vr='^r~r= 9'^2 14 9"^14 U 9 14 81 40 _81^4"9" 4_3 ~ 28-27" 9 7 9x7 55 >: 9 re 7 55 55 81x49 9x7~(J3' ^^ 5 ^^ ra~(ii ^^ 2l)+7 ^^ W =1?-1 of 1^4-^ of '^-^^ ^^ '^ 26-19+60 67 An A "* on "Try 01 7 — TTt— ;t;;:+ t= r^ = — 40 4 "' 20-^7 "' 4-40"80'^4 3 6 (^ ?, 8 1 3 9 5 2 3 1 9 80 ttO" 27 3 864 + 3 867 40 1280" 1280 ~1280' ill I Six 5 '=^ X w '=,> X >^ 3x17 5x3x17 255 31+1 ^'1?,1 a "170+12- "2x182 -364* ^ 4i 3 ^17 CO =; of 10 8 3 ■^4«^2l 14+21 2 , 9 5 2^ 3 "^ 14-C «^ I5 0+4 42 ^___y 63_13x3 39 3_1 2J^-7-42'' 20-2^20-40' 7 9 6':J (T) llf-7-,V 4f^ 328 22 _ 8x41x2x11 41 3^+5 A- 8^i '^7 ^iy2-7^^1^^172Tl2=84• (3) Slim of I ^.ncl??=i+li+??-'53_4, 4' G 12 12 + 12 + 12-12-^' 2. •. - is the fraction required. i I* 104 KEY TO ADVANCED ARITHMETIC. (4) 3 5 4 7 . 1 . , 135 150 100 120 8' 12' & 20- "^'^ ^^"'^'"^^"* *^ 300' 300' 300' 300' 280 .'. sum of the greatest and least =.-t7;7,, sum of the other two 300' 285 .._ 1 = -^; .-. differences— . (5) 13 8 The man gives away ^ of ^ or — ; -6 o 10 •. he has left 8 1G~10 10~16' ,. 11 li'", VI. (2) ,,,4^12-9 3 .... 2 18 81 <*> 5^f2-3«fn-^5«fl'^=5-5l+85 2x51-18x5 + 81x3 103-90+248 255 255 255 255" :1. ?l±^^ K-^ 3t-_123 ll_x_2_41 23_G3_ ^^ 4i+5ir"^10i~"9,^o"^10.i^~189"^21 x 3~G3"''03~C3~ __ j 5x2x27 ) ^ j 1 + 840 ) _15 "(7x9x2) < 21 )7 21_45 7 ^841 "841" .4s 2^.2f,-_/20J^\ /29 ION ^ ' 2} ""8 A - Vll 13/ • Vll 87/ ~ 2x5 11x3 ^ X —r/r~=o. 11 10 (3) 2 . 5 „ 81 - , , r; of ;; Ot ^tt of IJ-: 9 141 2 X 5x9 x3x3x2 x 3_ 2 x 2 x 3 12 - ________ 3 X 9 X 3 X 47 X 5 QUESTIONS AND EXAMPLES IN FIIACTIONS. (5) 3i^3J,._28 31_14x31 404 20} 4U 83 8 2 105 ". sum of fractions __434 30_4G4 ■"45 "^45" 45' diCFercnce of fractious =^-^-^-=^. 45 43 45 ' difference of these results=-i=^=U 45 3 Number=3!J5-''y-^==y^_?!?_200._9I5_qi_p. 35 ai 420 420 420~420~28~ '^;, and numbers --• x 2A'=— x — — -(Ji 15 ^'' 15''lG~20- VII. 0) (2) ■•4- 5^ 7_1^3 m_2x^_3 51^ 14 4 '^^+14 =:1?4. 5i_y_l , :51_li?+l375_1424 25"^14<^3^9x0_5^5x9 75 lof^of- ^^'^ 2x3x7x«- 2x"0~-T' ' o 81 • •• dincrcnce roQnire(l=:?-^i^_''''^_?i51--1875 386 100 4 ~ (4) lo-^a-^^^J^^ + ^O+irj + lo 77 3'^^"^4+i>- ^ 100" ~100~*^**' 2^ 43 ^ ^x^ of -of 8 = 40 W -(iO' "^^-ii?.^ '^ ^ 3 X 3 X 8 129 o_77^120--77_43 (iO GO ~g6' '3x3xl0x3x8x5~50 ' 128 • •. fraction rcquirccl^—^. I^-Jl 9}l O0x99-1G50' 1 Whole numbcr-(^+|^ + .f ■) of number=126; .*. TN'iiole number -1^ of number=126; .•.4-yOfnumber=12C; .-. A ofnumber^-6; .-. number=340: eadM r' T T"'"'^ '' ' '^'' ""'"''''^ ''''' 2^ ' ^^--^^ others, each U. Also by question, 21 number sco-pd by lastrr-lSG- therefore number scored by him=:6 ; therefore, m,mber scored by the rcmaming four, cach=30. MULTIPLICATION OF DECIMALS. lor MULTIPLICATEON OF DECIMALS. Ex. XXXVIII. (p. m ) (1) 8-8 42 76 153 159-6 (2) 417 •417 2919 417 1G68 173 889 2053 •0031 2053 C15G •00G3G13 •916 4 07 0412 8GG4 3-72812 (4) 81-4033 •0378 6517056 5702424 2443896 30793089G or81-4633x0378 814G32 378 ' 10000 ^ 1 30703080G 10000 ^ 10000 '100000000 -3 07930896. (5) 2735 7^70071 2735 19145 19145 19145 2106144185 or 2735 x 770071 2735 770071 X 100 100000 210G144185 ,_ =Toooooo(r=^^^"^i44ia5. (6) •04375 -0754 17500 21875 30625 •0032987o0 or 04375 x ^0754 4375 754 X -t:-— r 100000 10000 _ _3398750 __ "lOOOonoorirt- '00339875. i Mi 108 KEY TO ADVANCED ARITHMETIC, (7) •0046 7-85 230 308 322 •030110 (8) •00S40 •C0324 1G02 2538 •000027410'l (9) •314 •0021 314 628 •0006594 (10) •009 •00846 54 36 7.3 •00007614 (11) •000207 6056 55242 46035 55242 •055757502 or ^0046 X 7-85 __46_ 783 "lOOUO^lOO 36110 lOOO'JOO = •03611. or -OOSIG >: 00324 846 324 100000 lOUOOO 27}<04 loouooTuo-oo='^^^^^^4^«'^- or -314 X -0021 louo ^ ioouo 6504 10000000 = 0006594. or -009 X -00846 9 846 X 1000 lOOUOO " 10001)0000 "="^^^^^^^^' or •009207x6 056 _ 9J207_ 605G "lOOdOOO^lOJO 55757502 lUuOOOOOOO = -055757599. 1% MULTIPLICATION OF DECIMALS. (12) •00984 29 8532 1896 •37492 or -00948 y 29 __048_ 29 100000 ^ T 27492 100000 n7.= -27492. 109 (13) 1 •01 •01 •001 •00001 100 •00100 or 1 X -01 X -001 X 100 xlOO = lx-A-x-l- 100 1000 ._100^ 1 "iooooo~iooo~"^^^- (14) 7-6 •071 76 632 •5396 21 539G 10792 113316 29 1019844 226632 32-86164 (15) •007 700 4900 760-3 147*"" 294 343 3725-47 •00416 2235282 372547 1490188 15-4979553 100000 15497955i* no KEY TO A1)VA^XED AKITIIMETIC. DIVISION OF DECIMALS. Ex. XXXrX. (p. 126.) ^^) 5-16 ) 10-836 ( 21 10 32 516 516 10-836^5-t6=l?-^^?^l!^_10836 1 21 1000 516—510" "" io=io=^"^' •S81 ) 34-96818 ( 91-78 34 29 678 381 2971 2667 3048 3048 34-96818-f.-381=?i^?^ v 1^00_3496818 1 9178 100000 "^ 38r— 38r-^roo=-ioo=^^*^®- (2) 1003) 025075 (-025 2006 t 5015 5015 •025075-^1•003= J??J1 ^ 1000^25075 1 1000000 1003- iooT ^ 1000=1000 •0012 ) 02916 ( 24-3 24 25 ^ =77^=025. ■^ 51 48 •0291fi^oniQ^J?ll_ 100000 X 86 36 * 100()0_2916 1 949 13 -l2~^ 10= 1^=24-3. t ■^ DIVISION OF DECIMALS. Ill (3) 27 ) -00081 ( -00003 •Si •OO081-*-27= 81 J__ 100000 37~100000 4-735 ) 1-770890 ( 374 14205 = 00003. 35039 33145 18940 18940 1 77089-^4-735=?^^?^ xi55?_ll'0890 1 374 „^, ^""'"^ '-";— /«''7y7^XT7T7n^ = 77.7^ = '374. 100000 •'4735- 473r"''lOOO=ioW (4) •1 ) 10 ( 10 10 •01 ) 100 (100 100 •0001 ) 10000 (100(?0 10000 1-^-1 = 1x^^=10. 1-^-01 = 1 xl??=100. 1-^-0001=1 xll«22:.ioooo. 1 (5) •126)31-500(250 252 630 630 31 -5-^1^6-?^ x^QQQ glgOO, 10 "" 126 •32 ) 5-2000 ( 16-25 32 80 — 64 200 192 160 160 80 126 :250. 5-2-f-32='^^ 15? 10 32 _52000^JI_ 1025' ^ 100^ m '' 32 112 KEY TO ADVANCED AKITIIMETIC. (6) •0625)32170000(51472 3125 920 C25 2950 2500 4500 4375 3217-i-0C25 _32i;^x 10000 ~ 025 _ 321 70000 ~ 025 =51472. 1250 1250 C250 ) -0321700000 ( 0000051472 32170000 •03217-^G250-^,V^-x-^: 100000 0250 (7) 025 10000000000 51472 Tooooodoooo" '^^^^^^'^• 81-34 ) 4G3C38 ( -057 4 0070 5G938 66938 A-AQrQQ . Qt.9i 463638 100 A 4 63638-^81 -34= ~;;7:^ X —--=--— --057. 100000 8134~1000' (21) 8-7 ) 32-50000 ( 3-7350 261 640 609 310 261 490 435 550 523 325-^8-7=-5x- 10 87 ?250000 87 ^ 10000 37356_ 10000 :3-7356. . . . I 550 28 "ih DIVISION OP DECIMALS. 113 1-7) 02000 (0117 17 30 17 130 119 11 •013)10000000(76 9230 91 90 78 120 117 80 30 2G 40 39 \f>^-7-l 4 — : V — 100 17 2000 1 17 ^10000 .117. . . ■ 10000 = 0117.,.. i.-niq-l 1000 l-.013_^x — 10000000 13~ 7692;]0. ^ ioooo 10000 7C 9230 .... i (22) •0063 ) 00938400 (1-4895 63 252 600 567 564 501 330 315 GOO 1 03 ) 51846-734000 ( 508301313 510 846 816 807 308 134 102 320 320 806 140 102 880 306 74 •009384-- 0063 ^_938^ 10000 1000000 "" G3~ _938400 1 (y^i ^ 10000 14895. . . . =~ioo(r-'=i*4895... 51846-734-f-l 03 _5I840734 100 1000 ""102 _518 167:^000 1 loO ^10000 50SC0I313. . . 10000 =508301313. if It4 KEY TO ADVANCED ARITHMETIC. (•33) •033 ) 7;5M() J)(i K)000 ( 320911-4782 7380 964-023 09 48 40 209 207 73809C4 1000 X 26 as 34 34 23 110 92 J 80 161 3-42 ) 6 500000 ( 10005 342 190 184 60 46 14 1000 23 ■3809640000 1 ~ X 23 3209114782 10000 10000 =3209114782.... ^.. ^ ._ 65 100 6o-5-3'42=— X- 10 342 3080 3078 2000 - 1710 6500000 2000 19 ) 25 0000 ( 13157 19 342 10000 ^^ ,„ 250000 25-j-19= — -- — X =19005.... 1 19 10000 60 57 30 19 13157. 10000 110 95 150 — 133 110 (24) ■01257 ) 176432 76 ( 1403G0r90930 176432 -76 -^01 257 1257 2400 1257 -=1-3157.... 5073 5028 17643276 100000 X 100 1257 4527 3771 7568 7542 2400 11430 11313 11700 11313 3870 3771 176432760000000 1 1257 ^ioOOO 140360190930.... 10000 =14036019 0930. . . . " 990 .. PIVISIOX OF DECIMALS. 653549G3)745M;H')0(0011 116 911(1^830 65354063 745713454-6535496-2=^i?!5 . 10 10000 '05354963 .745713450 1 n 65354963 lOOOO^TooOO""'^^^^' ' ' (25) 2-9 ) 37-24000 ( 12-8413 29 82 58 244 233 120 116 37-24-2-9 3724 10 "100 ""29 3724000 1 40 29 110 • 87 40 27-53 ) 071900 ( 0026 * 5506 29 ^iooOO ^128413^^^ 10000 =12-8413,... ►719-^-27-53= iiL^ 100 16840 16518 26 ' 10000 10000 "3753 = 0036.... (26) 157 ) -0039203 ( 0000186 157 1350 1356 943 943 •0039203 -- 157 __39303 1 lOUOOOOO "" 157 _ • 186 -10000000== ■^^^^^^^• 1-57) -0029303 (00186 •0029302-j-l-o7=-l2?5?_ ^ 100 186 10000000 ^ i57~iooooo'='^^^^^- 116 KEY TO AI)VA^XI':r) AUrniMETIC. (37) 1053125 ) 5005 OOnOOOOO ( •0025G25« i]S)()(>:i50 1()!)S7500 9r(;5(;25 # 12218750 11718750 Upoooo •i)()250 10!);57500 'J7(i5(125 5005 -t- 1953125 _ 500500000000 ~~ 11)53125 1 ^ 100000000 ___2r)(5250^ "loJooo'ooo = 00250250. 11718750 11718750 105-3125) 5005 ( 250250 1 60•05-^195•3125^ll5?^ x J""^^-^^^^^^ , lUO 1953125 1953125 1000000 = •250256. "ioouooo •0001953125 ) 05005 ( 250 250 •O5005-^0001953125 = -''^^^^ x l^^^-^??^^^!^ 100000 1953125 _5005000000()0 1 _.2'5C25G^ " 1953125 ""lOOO" 1000" -^^^'^^^ (38) 1 . 17 3 17 109 218 ^318. '^^ ^^ 5"^25~3'^25^"50"100 •0005)21800(4300; 428 5 5253 _ 1 0^^xJ x ,5x 51x103 51 jx70 616 ^ 4 ^ 18i90~ 103 x 5 x '4 x 1 70 x 107 ~ 170" 17^00 3 .-. 31 •008-^ •3=103-30; ~10~'^' ifta ^^ ^"^ .,n,i .-r- . "^^ -7575x0 45450 ^,^^^ m%^ VULGAR FKACTIOXS EXPRESSED AS DECIMALS. 117 Ex. XL. (p. 129.) 4 I 300 20 too •25 8 (5100 25] « (5 180 10 8 2 •75 2 000 (2) 128 •ao fi I G60 lG-5 4i2i 103125 (100 200-^ ( 2 570 57 •515625 f 125 2-85 250^ (4: 4 4 4 •171875 .-. .4?i.9.=rGl71875. 13 3-25 •8125 •203125 ■05078125 •0250 10 2 I 50CJ0 •625 19 1.90 •3125 95 125, 540 15 108 216 •432 170 34 68 1-36 •00625 240 flO 4 I 6 57 5-7 f4 4 512^4 4 13 1-425 •2375 300 Vo •1875 •046875 oiins/o (Continued on nextj^age.j 00585^375 118 KEY TO ADVANCED ARITHMETIC. (3 continiiod.) _588__ 1 1 'r^_20-.')3 _4;70£_ -O^OS^ _ 18810^ 78125 "" 15020 ~ yl'ilo'"" 025 "" 125 ~ 25 = 0075204; .•.^7J«.- 15 0075204. 037033 (4) f JL_ -^ _3625 ^ 512~8x512"~ 512 = 007080078125. •453125 050040625 04 8 VUl+i=^-^±-^±i=g=?4^=-84376. 3 ' 4 ' 10 ' 32' (6) "32" 8 r4^'^«^-(r4^rl,t)o=i(rloo=-^^^- (7) ?4--061 = -G+0Gl = -GGl. o (8) 1 ^__L_ ^Q+8— '^ _23_2-3_ 2''"5 8~ 40 ~40"" 4 ~ (9) 47i llf_47r.25 47 94 75"" 94 "^4x75" 2x30 47-625 4-7625 6 =•79375. (10) 7-75 9 2^ ?0 7-75 . 2 . 20 ^^21^^31=-^°^ 2-5^^31 "9 7-75x5x0x20 7-75x10 775x2 15-5 ax 2x25x31 0x31 31 31 -^ M t» CIRCULATING DECIMALS. 119 (11) 5-5- ;--75of ?of7i=5~+75x?x^''^ 040 078125 10 80 2 + -75x9=5-0078125 + G-75=ll-7578125 (12) ^25+110"^ '^^iooo+al-^5-^io"^^^iooo"^3rxl =3-16+-3+81007+2=8G497. (13) 247 . 1512 17 5 + -200 7 11 108 "7^ 10^62-5 _1729 168 17_x_9 ~ 5 "^ 12~"'' G8~ 200 1 il5 10"^ 625 =845-8+14+|J+200-7+ — ■ 4 2a 345-8+14+2-25+200-7+-176r=o62'926. Ex. XLI. (p. 132.) 9 I 50 (1) 11 i 200 '00. . . . •1818.... 37) 1000 (027.... 74 260 259 •4285714. . . . (3) 17 1-7 =•566. 308 73-6 8177.... 30~ 3 16_177. . . . 81~ '' 495 ~ 99 ^ = •197530864; 11 1 •74343....; 333 ) 52000 ( -156 ; 333 52 • • 15gg^=15156. 1870 1665 2050 2050 1998 52 I I \ 120 KEY TO ADVANCED ARITHMETIC. (3) 3231 3231 29-37272. 7-3431818. 3520" 352 32 = •91789772. 3307 ) 902-000000 ( '285714, 6734 28800 20930 19240 10835 8 902 /. 7^^=7-285714; 17 -017 -00154 24050 23509 4810 3307 14/130 13408 902 (4) 9708 ) 83-000000000 ( -008497133 V 78144 48500 39072 94880 87912 69680 68370 13040 9708 32720 29304 r 34100 29304 4850 99000" 99 =•00017. 9 24~g=24008497133; '700 ' 7 =1701857142; 139808 279736 833325 100005 .5594-72 33333 KM ^|» 4- I CIRCULATING DECIMALS. 121 33333 ) 5594-7200000 ( -1078433 33333 226142 199998 261440 233331 281090 2066. 4 144260 133332 /.2|||=21078433. 109280 99999 92810 66666 92810 26144 (5) 19 ) 100 ( 05263, 95 50 38 120 114 60 57 3 *3 ) 100 f 043478, 92 hence -,\= 05203-,%; .•.!%= -15789 A; " A =0526315789,%; .-.-,%= -473684210,. A= •052631578947368421, 80 69 110 92 180 161 100 hencej{i-=-043478A-; ••• /,-=-260869i|; " iS-=043478260869H; H=565217391304....; ,-. i^=-0434782608695052173913. 190 184 a I 122 KEY TO ADVANCED ARITiniETIC. 29 ) 100 ( -03448, 87 130 116 140 116 240 232 8 31 ) 100 ( -osasos, 93 hence-/9 = -03448:/9; .-. -Ar= -27 oSCy.P. i ; -/., = 0344827080..% ; . •. o% = -20089650 1 7, V ; ify- = 03448275802069(555 1 7^}y ; .-. /y = -2413793103 ; « -/5= 034482758020689055 1724137931. 70 63 80 63 180 155 350 hence ;fr= 033358^,^,-; .', -;^='064516:fr ; -L._ a I 032258004510/, ; .-. -3^-=*13903. " 3^=032258064516129. 250 248 (C) •7- - • ' 9' • 7—0 7 90 ~90' "" 990 227-3 _225_ 35 _ 5 ■990~110~33 (7) 4- ■rQ6- ''^^^-^'^ -.''"^78_280 '"^^ 990 ~990""495' 135 45 15 ■\OK 1 5 999~333~111""37' •363= 363-36_237_ 79 900 ~ 900 "300" •00185= (8) 185 37 • • 24- 8 99900"" 19980 ' "^^-^099-*5"'>. 1007 153 1'>3P,_1'J» 1001 ' 99000 "99000" i2;!75~1375' 333" 333 ' 17 4- 17 t.. CIIJCULATIN(i DECIMALS. 123 (9) • i, 142857 15873 5291 481 'yro;^" •397916 = 39791G-a9791 ^58125 71625 14325 •o8214-.J8.)'r=r yOUUOO ~90U000~r80000 2805 __ 573 _ 191 .^00 ~1440~4b0' 42600275 36000 382142857_-382_3J2.142475_ '999999000 " ~ 999999G00 ~ 1 1 flllOOO ^ 386002j^ _ 772{m __ 154101 _ 514G7 10101000 ~ 2020200 ~ 4040 JO ^ 1 MOSO" (10) •307692==^ -^^?P-^^^i_l??i_1036 999999 333333 ""30303~3367' •6:J07692=:^^^?^^~-?^-5i^_^^^'^^<5^ o-.>.);J30 9999990 -9099090 oooo .1[H142_21238 10019 41 '303030 "33670 10835-65 2-7857142=2'^-?5iy'ii:Z_«!557^_o5!519045_^338095 9999990 ~^9909990~^33333S0~"303030 _o ^^^^•'^ ol5873 101010 (11) 20202 ^.5291^^^4811x11 11_39 ■~6734 "'481x14 14-14' •342753= •03132132 342753-312 34241J _1 14137 999000 - 99900T) " 333000 ' 3132132-3132 3129000 1043 99900000 8'020S3=8-^4?~~-8 ^^"^ 99900000 ~ 33300' /5 =8i = 90000 385 9000O~^18000~^3600 -8-11 720 '48-48" (12) ^r60806=85?5§0G2:«080_ 54726__ 9121 90000 -^"90000 -^''15000 _ 1275000 + 91 21 1284121 15000 ~ 15000 ' (Continued on nest page,) I 134 KEY TO ADVANCED ARITHMETIC. (12 continued.) Q n;i,o-,.;_ J>428571-G_ 64285G5_ 714285 d04:.«o i 1 -^- J^yyy^^^ '-^yiiuu990-'^rn UIO _ 79:;(;5x9^_„9 _5J. ~ 793.5 xl4~''l4~ 14' .n-^nAA««n<\- ,,..22095-220 ,^,^ 21875 127-00022090=121 --^-^^-^127----^ _io^_i3^^__1o^ 875 _ 175 ~ ' 19800000"" ' 3960000 792000 =12'; 35 158400 =127 40233G7 3108U 3 1680 * Ex. XLII. (p. 134.) (1) 2-418418418 1-1GGG6G667 3-009009009 •735444444 24-042 (3) 234666GG6GG67 9-9288888889 •0123456789 ■0044004400 456- ■ 31-371538538 .-. .4?i-'J'r) 360-86 75 330 7o75x 366=:--^-^ X -_=_x~ _ 75x33 _ 5 _2-5 _ . "99x90~3x6~ir~'^"' (6) •406 x62=156;;40^^^^ 366x63 132x63 900 7564 75 6j ' 300 ~ 3 900 ; 35-313; 800 835 X •36z=835 x ^^J^l^=7n x 4=300. (7) 7'58x48-3=7t5x48i: 677x14598165 90x3 "90x3 98165 1090-73 • • 368 X -6=363 X (8) 2 736 3 :245-3 125 H45 X -4297=3^^ x i^-??:.?llliil???-l??6?403_ 990 9990 990 X 9990- 9890100 -•^^^^^^- 20fx.84=l|!x|=l«^il!.l^^.n-45. (9) 195-03+-4=48-75 ; •1759^+05=-''^^— X ^^-?I?^^^^ 99900 "5-99900T5" _75tJ _3n3x37_203_ • 1110 ~ 30x37 -~J-~^'^^- ,r* — r- 126 KEY TO ADVANCED AKITHAIETIC. (10) 90 16 ~ 8 --Q-— ^03 75; 13-2-f-5G=13§-oJ=-l? X -=-=2-3 (11) 411-3519-H5«rG45=411??i^=58^ _ 4109406 9DU0 4109406 9990 ""587058"" 587058"'^''' 2-16595-^ •04:=3ll??5 x ?2-i?:l^_48734_ 90000 ^ 4 -4 X lOOO-lOOO"-^^ ^^* » •6559903-f-48-76=r 555^^344 K)0 9U90000 4876 _ 1344x48 76^ 1344 _ 99900 X 487G'~liy9~~"^^^** • * * REDUCTION OF DECIMALS. Ex. XLIII. (p. li?6.) (4) league. •875 3 2-625 mi. 1760 days. 25384375 24 21537500 10768750 37500 4375 625 12 9225000 Lrs. 60 1100 000 yds. Ans. 2 mi., 1100 yds. 55 3500 m. 60 gi 00 soc. Ans. 2 days , 12 iirs., 55 m , 21 src, (Continued on next pa^c.) 1 X T KEDUCTUXV OP DECIMALS. C. (4 continued.) lbs Troy. •6 12 - 7-2 oz. 20 . 40 dwts. Am, 7 oz., 4 dwts. (6) cwt. •85076 4 cwt. •07325 4 3-40304 qrs. 25 •29300 qrs. 25 201520 80608 146500 58600 1007600 lbs. 16 7-32500 lbs. 16 1 21600 oz. s., 10 lbs,, 1-2: 5.20000 oz. 16 oz. Arts. 7 lbs., 5-2 oz. mile. •045 8 •360 fur. 40 14-400 po 5^ 2000 200 2 200 yds. 36 7-200 in. Ans, 14 po., 2 yds., 7-2 in. 1^7 i\ 128 KEY TO ADVANCED ARITHMETIC. (6) tons. 4-1G525 20 3-30500 cwt. 4 (7) qrs. 2-40875 8 3-75000 busli. 4 1-22000 qrs. '35 3-00000 pks. Am. 2 qrs., 3 bus., 3 pks. 5-50000 lbs. IG 8 00000 oz. Ans. 4 tons., 3 cwt., 1 qr., 5 lbs-, 8 oz. cwt. 3-625 4 2-500 qrs. 25 12-500 lbs. An;i. 3 cwt., 2 qrs., 12 lbs., S^oz.* Ans. acre. •05 4 •20 ro. 40 8 00 po. Ans. 8 sq. poles. lbs. Troy. 3-8343 12 100116 oz. 20 •2320 dwts. 24 9280 4640 5-5680 grs. lbs., 10 oz., 5 508 grs. lbs. 4-106 346 24636 16424 12318 1420-676 lbs. 16 Ans. 1430 lbs., 10 816 oz. -14 cwt., 20 lbs., 10 81(1 oz. ^ 1 REDUCTION OF DECIMALS. 129 \S. TS. (8) ac. 3-8375 (9) fur. j25 40 3 3500 ro. 40 37 000 po. Am. 37 poles. 140000 po. Ans. 3 ac, 3 ro., 14 po. gall. 3-5 lun. mo. •34375 28 18 280 275000 68750 35 630 gall. Ans. 63 gallons. (11) ro. 2-25 14 jOO 225 9-62500 days. 24 250000 125000 15-00000 hrs. Ans. 9 days, 15 hours. (13) sq. ft. 4-751 31-50 ro. 40 2000 po. 25 23755 9502 Am. 7 ac, 3 ro., 20 po. j'-ards. 20396 2290 118-775 sq. ft. 144 3100 3100 1835640 40793 40792 775 111-600 sq. in. Ans. 13s.yds.,ls. ft.,lll'6s.in. (Continued on next page.) 4670-6840 yds. 3 oz. 20520 ft. A:%s. 2 m., 1150 yds., 2053 ft. I — -^ 130 KLY TO ADVANCED ARITHMETIC. (12 continued.) miles. 30091)4:] i3 19993G0yds 3 401988G mi. 8 2-998080 It. 13 •159088 fur. 40 11970960 in. C'3C3520 po. 5i 1817G00 1817G0 1 -999000 yds. Ans. 4 mi., G pa, 1 yd., 2 ft., 11-97690 in. (13) 345 •383 of '1^1=-^^-/-- "f 'J^l^^ljo-o «f ^l=38i cents; 9U0U0 90000 4ri04— 4fi0 •4094 of 1 lb. Trov=:^^-~-- of lb. Troy 9000 -^ 4225 „ ^ 169x12 ^90O0"^-^^'"^'=-— 360-^^- =— -- oz.— 5 oz., 13 dwts,, 16 grs. oO (14) .^.A .^r. 5740-574 ,.^^ 51GGx27.s. _ -_^^ •5740 of 37..=-^^^^-^- of 07s.=-^^^=15s. 3-976^. ; •138 of 10.s\ G^.=r— ^_;r— of 12Gdf.= — --- — d. =—d.=ls. 5A(f. ; 40 20 of 5«.=::21j of 5>;.=23 of 5-9.= --^5.= 13s. 4d. REDUCTION OF DECIMALS. 131 (1.5) 4-05 of li Rq. y(ls.= AcAr x -^ sq. yds. 4-.V X ^j sq. yds.=.-- sq. yds.^G sq yds., 108 sq. in. ; •163 of 2i miles=^|^":i? or 2^ miles /147 ry\ 147 ., =120 miles=3 fur., 10 po., 3 yds., 3 ft. ; 4 90 of 4 days, 3 lirs -4§^ of 99 hrs. 30G + 90 ,^^, =—99 — oi 99 hrs.=48G brs.=:30 days, hrs. (16) 3243 of 2i acrc3= ( 3—— x ^') ncreq \ 900 2/ *^^^^'^- - (^3H-g X -J acrcs=^^ across 8, l.^t acres ; •Q^318 /9;il8-93 9;HH) 25 5\ •5681 " V 9i»000 50Sl_5a 12 3/ -^ /9225 _1 41 ^ ^25 "^ 12 ^ ^ j ^^'^-'^"^^ij^ ^^''^^^'^^-^^^ ^'^"^ . '^^ »»»• (17) •777 "^"77 20 8. d. 15 0(118 8 Gf)48 i.">or>5io>'. Ans. 7 66480^;. cents. •70323 4 8 5'T25^4 281202 ^».»'i 200 7s. 8AH)^3*,rf.=93-i,Vu^./^ -^.V.r'j^^f?. ; 10s. Gd.=md. ; therefore fraction = 931943 fyo. 7317 130x10000" 10000 = •7317. (15) The fractions ,^^7= 13125 ; 2,0 20 Q-m I •33 r*^ i. II 13G KEY TO ADVANCED ARITUilETIC. (10) The fnictiourri ^--300122. . . .; 2 qrs,==(2x8x4) pks., 7 therefore fr;iclioa= •875 -109375 8x2x«x4 8x« •013671875. (17) ^ of a guinea =95. and £2=40^., 9 ' tiicrcforo fraction = ' - = -225 ; 1 40 -^ j,\hjj of a year ■= jz^^^^ days, . P • 7x78 ... therefore fraction = T7wwr=='^ll^ (IB) f of -ij of 40 yds.= ~-^-^yds.=y yds. i of 2 miles:= --?--- yds.=(.352 x2) yds., 5 r^ 3 therefore fraction=^-^^-^=^;^=^^=-00243. .. ; /I '^ \ 7 i of 3i sq. ycis = (4 '< oj sq y^'^—j3 sq- y^^s- ; 2 ac, 1 ro.=0 ro.=(9 x 40 x CQi) sq. yds.=(9 x 10 x 121) sq. yds., ^7 therefore fraction.— - — r:-Tri~wi ^trfi'.Tn^"^'^^^^^ 8 X U x 10 X 121 8/i-^U (19) ;] X 40 , 8 , t of 4| hrs.^r^^— hrs. = .-^ hrs; 305^ days= 5 X 9 1401x24 brs.r=(14GlxG)hr.s., therefore fn.ction = ^- J.— = j3^^=-000304. , . .. /lO , A , 320 , Zh qvs.=( — ^ ^ '^ -i) 1^'^s. ---- pKS. ; o[ '^02.") thereCorc fraction-;^— ="-—-- = '005025. [ o,dV 40 1875. 13...; yds., REDUCTION OF DECIMALS. 137 (20) 8 lbs., 6 oz. Tro}— 43 oz.=(43 x 20 x 24) grs. ; 10 lbs. av. =(10x7000) grs., 43 X 20 X 24 G X 20 X 24 2880 therefore lraction=- 10x7000 ~ 10000 -lUOOU"'^^^^ i oz. av.= (j X i X 7000) grs.=^ grs; i oz. Troy = (- x 20 x 24^ grs. = 160 grs. ; therefore fraction =-J-i5 ?l^l2.-.r,Annryr^ 3 X 1G0~ 40 " '^^"'^75. (21) iday+|hour4-^ofGhours=(^>i?f4+^4x ^^^.^^208 and 1 wcckrr? x 34 liours; ' therefore fraction = —^^?— ^ ^98 -837777 15x34x7 3x13x7" 7 = 11825396. (33) -83 of $1.93 = $L593G, 05 of $5.04= $0.2545, ISof $l-20=$2-16! .-. value =$4. 008145; .Vac.ion=«i=l,90a.... (23) 6icwt.=r23qrs., therefore 33 qrs. + 3 135 qrs.=25-135 qrs., and a ton=r(30x4)qrs.; therefore fraction=^f^-^-^==?:^i?5.-.3140fio. 20 X 4 8 ~ '^140030. 0) 12 2,0 (24) GO • K 50 and 3c. 5m. =£025 20 •025 .*. 6(^.=2c. 5 m. •500» 13 6000d 138 KEY TC ADTANCED ARITHMETIC. V. i O 12 2.0 100 •833 •0416 10d=4c. If m.=4c. Ifm. (3) 4 13 20 45 2,0 •375 •01875 • \4id .=lc. 8f m. e) 2,0 5-0 •25 • '. 58.: =3fl. 5 c. 2,0 60 105 •525 .-. 10s. 6iy '697916 .-. 15s. 4K =7 f. 6 c. 9-7916 m. 15-39589;k 13 4'750000f?. 4 {") 13 2,0 8-16 f. and 7 £ f. c. and 2 7 9 3-000000(7 c. m. £ 3 4=734 14.68 20 .•.14s. ('■-) 12 •734 8-16d=7f. 3 c. 4 m. 11088 14-G80S. 13 8-160f? m. £ 6i =37962 2,0 15-924 20 2-7963 15 •924.9 I'i £2. 15s. 11088o ' 1 74-,V~108.57 29G1-7 x 10857 * 11 x 2901 10 17 _3 40 X 10 X 11 X 3 961_ 17 x 4 x 5 x 10x11 x 141 x 3 x 7 ~ 7 X 10857 X 80 X 17"" 7 x 141 x'^7 x 2 xAxlOxlT~ 5x3 15 7-5 ,^i,,^,j =773=7^=-=^-^^^4^^^^' 91853-^87-5(>=01?5?^87S=?l???x.^^ '™'' ^ 97620 '874791 999 "90~ 999 7881~111 x7881 =1-048. ...=1-05 nearly. •000700409; (1) 121345 24269 10000 ~ 2000 ' II. •0032546. (2) Tliree hundred and ninety-seven thousand and eight, and four hundred and five thousand and nine millionths; 397008405 '009 ; 307008405009. Three hundred and ninety-seven millions, eight thousand four hundred and five, and nine thousandths. Three hundred and ninety-seven, and eight millions four hundred and five thousand and nine thousand millionths. (3) ^=•75 •09375 246 '5 5 .5 43GG25 •87325 •17465 125^ decimal required -03493 IT M 1 D ^h i 144 (4) ror> 10 KEY TO ADVANCED AlilTIIMETlC. 8727588 525 105 1020^ r 9 •0<.)G97;]3 11025 _.11035_2205_4il lOOU ~20U ~"40' •0008 X -004 -0000012 •000 -000 ' 1 9 2^_C35+225-208 G-i3 iG'^400 025 ""' o 10 •0107748 •0053874 •0005;]874 10000 lOOOO' :-0642. 0) (5) 23 ) 21 DC -91304.... 207 70 09 30 23 1 •^1 70 100 93 '••10000«^23=='^^^^^^^^4- O (2K0)H.f3^-i)=^?-^?!^lZil?-?«_??l_o.^io w-.-n»r,— ,» .V -8/28 2x2i~2r"~ 9 -'^^^^■ O 4^4 -I- - 4-4+ -G 5 5 I 0(04 3000 • /5 1 7-oVo+-75--125 8-125--125~8~"^^^* 4 8 »-00l : 2 000333 t) 3000873 10 0013 QUESTIONS AND EXAMPLES IN DECIMALS. 145 (0) {3133-458 ) 7823-G572 ( 2-497G09G088 63G4916 15587412 12529832 C0575800 28192122 2383G780 2192720G 19095740 18794748 30099200 30099200 28192122 19070780 18r94748 27G03200 25059GG4 254353G0 25059GG4 375G9G i (1) 130021 ) G841197 ( -57 G00105 840147 840147 III. •047 ) 594-270000 ( 12G44043. . . 47 •0130021 ) G84-1197000 ( 57000 124 94 302 282 207 188 190 188 200 188 ' 120 94 (') C) (3) 21 190 36 •015 X 2-1 _ 1000 10 15x21x1000 3x3 9 1000 7 4 1000x10x35" 10 ~"10~"^' 31-04 5 -0025 2 100 350-4 foo" _625 50U0(J-G2.') " 493 V5 ~ 98 10000 10000 34600 G920_1384 75~1975 •7007. e 140 KKY TO ADVANCED AltlTIlMKTIC. It •> 4 5^100^4 lOOUO _ 74 ;5 X 2()!)a 7400 + 8088 15488 ami 100 ' 10000 ~ 10000 l_5^^_l{)^fi_0')8 ioooo~"i5^o~«^' 10000 =1-5488, ^^ U-J ^o + ^V =80 ^10=24=12 =-2»l«- (3) 5-81x-458;3=52U-^ 45S3-458 570x4125 576x375 " ' SK)00 99 X 9000 ~ 9 x 9000 04 X 375 24000 8 IJOOO' a 133 102 999900 102x9090 9000 ~ 113-i--006l32=U2 111 .> '^'^j 999900 90 X 132 9x12 102x1010 12 17 x 505=8585, 91 13 5^=yj). and since 80=2x2x2x10, the fraction will be re- ducible to a terminating decimal. (4) 1.^.4-3,^+3^+4^, =10+|+^=10+y + l=10M^. A^^-iin, 1 ■,',=! -083333 4-A- =4-148148 . (5) He walks (00-13 95) milos=4605 miles in three days, 8 ! 4r>-05 he walks 15 -'5 miles each day. re- . QUESTIONS AND EXAMPLES IN DECIMALS. (6) 1fi He sells 17 of 1875 or i^ of -1875: therefore, be has left (i-^t^) of 1875 =- of 'lS75=~'^-^^-^^^^^-JJi—^ 90 90xl0000~6xl0U00~6x80~240* 147 IV. (1) 1-23 •123 •0123 •00123 123^ 124-36653 12436653 31-457=31 457 1)99' 457 '^?4_3i457_3,14 ,4662-2385 =1- 4995 457 999 2377 4995' 100000 (2) 3006005 ; three hundred thousand, six hundred and five-tenthi (3) 5 X -05=25; 1-5 x -75=1125; 2625-^5 =525; therefore, in order of magnitude they stand thus 1-5 X -75; 2-625^5; 5x-05. (4) •0147 X •333=^- X -=— =-0649 • 91)90 3 9990 ' ■'«««^-'«^»=«|-i«5^=S-i«i^==^^*'^''^^ 30000 X 640 476x11 5336 = 006545; 10000x80 "800000" (Continued on next page.) us KEY TO AI>VA\(;KI) .\ UITHMETIC. (i COD ti lined.) •24.J ) 1;}-27I)0 0UU( 542000 1225 1029 980 4J)0 490 000 -ftiAoni . Q Q '014004 X 25 -00 1(150 x 25 ^^ ■O14904-T-3A = -^ ::^ -. = -000184 x 25-0046; bl 305 ) 0100100(20020 GIO 010 CIO u 305000 ) OlOniOOOO ( -02002 G 10000 G 10000 01 0000 El ; (5) •904:58 •904;j7532 •90437532 •90437 ■000004GS ^00000532 nnd since of iliese difTcrcnccs -00000408 is tlie less, it follows that the statement in the question is correct. To fmd an approximate vahie of the expression. Jl 1_ 1 1_ ) 4 '^ib 3x5^ "^5x5^ Ixb'^'^^^' y~239' 1 1 •*' -009 o '3x5^ 3x25 3 -""~^'^'' _J_ _ -008 00032 1 000064 5 X 5^- -5-x-2.5^ ~5-= ^^^^^^'^ rx^=T>rr = 000002 nearly; the last result shows us that we may neglect the rest of lln series, and we s^et 16x I -2 -002667+ 000064- 000002 [ -~ =(16 X 197395)- •01673=315832-01673=3^14159. QUESTIONS AND EXAMPLES IN DECIMALS. 149 (6) After tJ\e first sale lie has rcmainino' '8'5 or ^ of the estate ; therefore, after the secoiul sale he has A-A^ of L7_12 17 13 3 V V) °^ 20-17 ^^ 20=20=5' V. (1) 8978_1705G __35912 71834 143G48 3125- 025 125 /wtl 5 ■=2-87296 ; . G 29 39-8 6 '^7=8-^-T2-"7 8fx(3i-|) ^29x^l^^_809 112-375 _56-1875 8x12x7 8x2x7~ 2xT~- — 7 — (2) ,, 4;255x;032_4;255x-0lGx2 4-255x2 851 •OOOIG - •0i6^0r~="~^0l =8-0207857143. — Tqj— 851. ^G0-f-30 + 20j^+22 30 + 18+lO+G 137 90 90 120 ^ G4 _137x3__4ll _102-75 12-84875 4x64~4x64-~Gr~'= 8 =1"60546875. (j|- of 75i-3L)+(2-5G25 + 7i) ^-{~ of 35-2-3125 )4-(2-5G25+7-25)=9x3-2-3'125 =28-8-3125+9-8125=25-675+9-8125=35-4875. +9'8i25 150 KEY TO ADVANCED ARITHMETIC. 534 100 36 1000 3x1000 3-5-lYO: < 00— u 900 178 99 '/2 ~9 X 99 X 2 500 ~3x99" 45-45 ~3x9 _505 _i.g83goi. 3 (4) 1=1'00000000 y L-oogooooo 1 _ 1x2 •50000000 1 1x2x3 •16666667 i 1 •04166667 1x2x3) <4~ 1 1x2x3x4: <5 •00833333 1 •001 88889 1x2x3x4x5x6 1 1x2x3x4x5x6 x7 •00019841 1 •00002480 1 x2x3x4x5x6x7 x8 1 ^^^ •00000275 1x2 x3x4x 5x6x7x8 x9 1 '00000027 1 x2x3x 4x5x6x7x8x9x 10 lx2x 1 3x4x5x6x7x8x9xl0x U" •00000002 .-. snm=2^71828181 1/3 3^ _L.?iii^ J_^ 03"^\ lO'' "^1x2'' 10^ "^1x2x3"^ lOV 103 =i^"0- 3 00000/ 100 10000"^ 1000000 ^Jl_ / iooooo- 3Q oo + 60 4-i\ ""103 ^V lOOOOO ' J 97061 97061 1000 X 100000" 100^00^00 = 00097061. QUESTIONS AND EXAMPLES IX DECIMALS. 151 (6) 111-454 ) 8833450000 ( 7925-7 miles nearly 780178 1031670 1003086 285840 222908 (6) •12r.55556 416363636 9-45777778 13-74696970 629320 557270 720500 VI. (1) 40404 •030303 121212 121212 121212 •111 345 111 13-24362412 •345x-iii-=^2x iiLv^^ 4-3 9U9 1000 4'3 9xl00'x43'~3xl0irx43 115 ^ =3^=0089147.... 6593 ) •048l:]4899G3 ( -0730091 4j1.')1 19838 10779 59996 5!)337 6593 0593 205 2 05 22-55 (3) •006593 ) 04813489963 ( 730091 20 5 18-45 . 22^55 _22_55_ 451 18-45 ~1845~369' 162 KEY TO ADVANCED ARITHMETia (3) 123-48 _ 12;j48 _1029_ 343 _ 49^ _ 1033-3 ~103320~b()10~28t6~410' 36-59 5 _3r)r)9r)_ 13 X 381 5_ 2S15_ ,30^ 67980 """5798 ~ 13 x 440 ~ 440 "" *'^^' tin. (4) •375 X -375-025 x -025 -140025- •000625_j4_14_2 ~-35~35~5' < 375-025 n3 10 •375-025 3 + 777,— "1 1 Si andll3)10 00(-14159.... 113 470 453 180 113 .-. 3f,A,-=3-14159 nearly; 12931-129 •1293131 99000 12803 G401 "99000"" 49500" 670 565 1050 1017 33 (5) ^■'^^■+8'^3;5-^ + ^72 -1^^— '^' therefore, to make this equal to 3, we must add 52 13 0-5 73~18~' 9 — ■70 (0) Dividena=21i x •1C= 35 X -15 35 X -05 175 13 •4oV0. QUESTIOXS AND EXAMPLES I.X DECIMALS. 163 VIT. (1) In 1 day A and B can do J of the work • " A and C " | " « " B and C " J " « .-. 2 xV and 2 B and 3 C do ?+ ^ + 1-? . ••. A, B and C do f ; ' .-. C does ^- J, or - ; /. C does the work iu 12 days ; B does --^, or- .-. B " « 4 " ; A T '^ 1 •'5 Adoes^-^,„r_; .-.A ( 1 4,^ 5. — = value of 6 cwt. : value of GO cwt. :valu(! of 2 qrs. : value of 1 qr. :value of 7 lbs. = value of 4 lbs. value of 1 lb. £407 1 (ij ?(7.=vulueofG0cwt.,3qr8.,121bg. iii 166 KKY TO ADVAXCKD AKITII.METIC. (17) 2qrs.=i cwt. 141l)s.-i()f3(ivs. 2 lbs. = i of 14 lbs. £ s. d. J] 7 G o o 10 2 G 1 13 9 8 r)i =vuluc of 1 cwt. —value of 3 c^Yt. =valuu of 2 qrs. = v:ilue <>r 14 U)3. 1 2i ?<7.=vtUuoof2 lbs. £12 5 lOi ^«/. = valuoof3cwt.,2qr9.,lGlbs. til (18) 1 fi.=^aofiy'^i'^"i 1 ft.— ^ oCl yaril 6in. = .^ ofl toot. 4in.=|^ ofl foot. (10) 81bs. = -iVofacwt. subtracting 2 lbs. = J of 8 lbs. £ . «> 15 7^1 147 10 5 Al,l 1 4.//-4 I! r MISCELLANEOUS tiUESTIOXS AND EXAMPLES. 157 Jibs. i in. 10 Iba. (20) l4Ibs.=ioflqr. £ 1 a. 17 12 23 12 3 10 22(5 2 15 G 4.V 229 17 18 10^ =cost of 1 qr. :cost of 12 qrs. (21) =cost of 120 qrs. or 30 cwt = C()st of 2 qrs. =C()st of 30 cwt., 2 qrs. =C()^l of 14 lbs. £2;J0 10 iik iry.=costof30cwt.,2qrs.,141bs. 6dwts.=iofloz. £ 8. d. s. 1 dwt. = ^of5dwt 12grs. = ^ofl dwt. 4grs.=^of 12grs. lgr.=:]Lof4grs. 5 10 3 17 6 5 4 7 1 n-V Ol •>J n ■h .Z- 4 8 :value of 1 oz. = value of 3 oz. — vnUic of 15 oz. = viiliie of 5 (hvts. = vaIiio of 1 dwt. = value of 12 grs. =:v;ilne of 4 grs. = value of 1 gr. (24) £4 9 51- i^7.=val.ofl5oz.,0dwts.,17gra. 24 lbs. @ 12 cts. = $2.8S 7} lbs. @ 75 cts.= 5.Hli- 4f lbs. @ 32 cts.= 1.40 .35 5 lbs. @ 7 cts.= 20 ^L ii)s. @ Hi cts.= 2.35} 17^ lbs. (i^ 19 cls.-=: 3.32J $lG.12i- MISCELLANEOUS QUESTIONS AND EXAMPLES. Ex. XLVII. (p. 157.) I. (1) 34'17-i-3i = 34-17x4 l3(Vr.8 13 13 -=10-51384015, 158 KEY TO ADVANCED AIlITIIMETia (3) Value in £=£}~~^~==£Qi2=£0i. is. ^H23 iiH (3) 111 ?._3 I-l ^-^-.ror (4) ^oflliecstate=$4818.50 i of the estates 9G3.70 the whole estates C745.90 therefore i of the estate= 1349.18 % of the estates 2G98.36 I El. i n i (5) 17 cents hi the $ is -,hh of any debt ; therefore amount received =,J^i?o of $17658= |3001.86. (6) He sells } of i^tj of -,^4- of the estate ; therefore = — rr-Ti ^^ the estate =£'---^ ; 7x8x ]4 8 .905x7x8x14 therefore the estate =£' 5 X 3 X 8' therefore ji of A^ of the estate=£'— ^— - — — - 5 X 3 X 5 X 16 ^^193 x49 ^^9457^£236U=£336. 8s. Cd 0x8 40 (7) The wife earns as much as 2 children, the man earns as much as children ; therefore (2 + G+3) x (what each child earns)= $24.75; therefore each cliild earns -,i,- of $2475=$2.25; therefore the msm earns $2 25 x 6=$13.50; I t MISCELLANEOUS QUESTIONS AND EXAMPLES. 159 (8) 25 fr. 56 cent. =3550 cent. ; therefore 1 florm=-,if of255G cent. =213 cent. =2 fr. 13 cent i (9) If 5 men in 6 weeks cam $405 1 man in G weeks ei>rns 81 1 man in 1 week earns 13.50 4 men in 1 week earn 54 therefore number of weeks=$5404-$54=10 weeks. II. '^ (1) 1 franc=-,ii of £l.=r|g of 1 shilling=|5 of a shiUing. (3) 33 1 yard = -J metre, 00 1760 yards=?i^il^ metres, therefore f of ^ of a mile=^i^'i?^^^ metreT' 45050 , ^„«^,„ =-7,y- metres=lG68H metres. (3) For 28 cattle they pay $192; 193 therefore for 1 they pay $-^— ; lOO y ft therefore for 8 cattle A should pay $ ;:^ -=$if&=$54f, 28 and for 9 cattle B pays $'-:i^=$61f, 28 -I 09 y 1 1 and for 11 cattle C pays $^— -=$75f. «8 160 KEY TO ADVANCED AUITIIMETK'. ft te s 4 (4) 80 12 a75 10 •3125 13-5 ) 0'.]Kr> ( 0025 2.")0 •oai25 (5) $288000 has to produco $12000; l^'OfO 1 therefore $1 must produce ^O8~y^o0^^24^^^^ ^^^' Circumrcreuce of the AYheel=(.") x 3"1415'.)) feet ; therefore number of revohitions made by the ^vheel in 10 10 k 1700x3 10r,(}0 ._,., Sx^l-lloi) 314loU 0) s. d. £ s. (f. 4 9x7,'j-.l 14 0.^ =vahic ofuheat 5 8xOL--2 8 OJ =vahie of m:dt 2 4x05=0 14 O5 -=vuhio of oats £t 17 l',^=:rent. Also, to find what he would pay in decimal coinage, wc have 36-! (4 i 1700 (0 12 2,0 4-2.-) 1472 17122G851 4-850134250 therefore he would pay £4, 8 11., 5 c, G-1C425U m. A can do -|V part of tlie work in 1 hour, B can do -i'.> part of the work in 1 liour; A and B will do (,^,+-,^2) <>i* H i" 1 hour; there foi'e they will complete' the work in f f hours or 5-A- hour: MISCELLANEOUS QUESTIOIsS AND EXAMPLES. 1^ (9) The first set iu 1 hour die: 4fl of a load o I U the second set in 1 hour dig -,^3 of a load ; therefore hoth sets in 1 hour dig(f g+rj), or f J of a load; therefore tliey dig 100 loads ir (lOO-j-^J) hours, i. e. iu ^'i^ hours, or 7i.?f hours. [il« ive ' 36 4 (1) [ons cwt. qrs. lbs. 28 4 3 III. 2 qrs.i=^ cwt. 14 lbs. =i of 2 qrs. $36.10 15 9 7 1 21 542.40 18.08 9.04 4.53 15 2 21 $574.04 (3) 25 1-00 12 8-04 ' 3 207 22-^ ( 11 18G-89 93-445 . 8 495 and 8-495 chains=8 chairs, 4 chainlets, 9 links, 5 linklet». (3) 25i francs— £1=45 pauls, G baiocchi=:45v,r pauls; therefore 1 franc = ( --r-^'— ■ ) pauls; 459 X '^ X '"*0 therefore 20 francs= -~- -'^_--— pauls=3G pauls. 10 X ol ^ 162 KEY TO ADVANCEO AUITIIMETIO. (4) £25143 produces £831 2; lUercfore £1 produces £(SPj^2-f-2."5143) -^K 10 "" 20115) '60-^"^-' and, therefore, £1155 P-iy^ (115^x8)i.=92jd=£3 175. lA $4800 g.iins $lo'2 in 7 inonths, 4800 •' 1 *' -- months, 1 " 1513 " 1 — j.-i:^ — months, y X i.)i3 1512 " 97,^ " ^-4?.- . X ~- months=^ months =5 months. m I (6) Averai^o / ()7 \ l(>ngth of y<^':^i'=('>^'^ + ^-^j ^^='ys (-05 + —-) days=3G524'25 days. (T) 15 horses-f-148 slicop cat as much as (:} X 84+148) sheep, or 400 sheep; 10 horses+lo2 sheep cat as much as (•3 X 81+ 1;')-3) sheep, or 000 sheep; 75" Then 400 sheep in 1 day cost £-~, P>n;3 1 1 m 1 8 8 « x> '"0 X 4 X 400' t( £ 101 x_8_ ;r>r4T4oo' 100x3x800 ^101 » X 4110 = £- :£50.10«. MISCKILANEOUS QUESTIONS AND EXAMPLES. 168 (8) A does ^ of the work in 1 hour, B does -i\ of the work in 1 hour; A, B and C do — or | of the work iu 1 hour; therefore U Iocs in 1 hour (J— i— j^i) or -J^, in 5 hours C does J of the work, auu iu 9 hours B does -^V, i. e. i of the work. (9) The capital but in by both persons=$(3400 + 4080)=$0480; therefore §lS^=first person's share of profits; tlierefore first sh;u*e=|^^ x 18O0=$fJ0«J, Becoud share=$1800— $0GG3 = $liy3i. IV. iths. I.10«. (1) 729x37==3G973. (2) I of |0.34-.|-of $1.20+^ of $5.04=.(^^V?|-Vf )cts. 37(«4 therefore fraction — 37G34 1 151 _ 11-5l 10.3 X 130 X 100~535 x 100" 535 2;}03 -4^04 •ir);54(; • • =-3|]3-=-^^—=—y— =0319308095. Area of b;jse=:^ circumference x ^ diameter =^ X 3; diameter x i diameter=i x 3j- x 2 ft. x ^ x 3 ft. 1 X 23 X 3 x 3 , 33 =-3^yx3-^^-^^=y^^-f^-5 (00 \ y xis) cub. ft.; therefore cost=("^~ l^\s.=£dC^ liis. 8.^d }g. 164 KEV TO ADVAXCED ARITHMETIC. (4) The men do tlie work in (12x6) hours=73 hours ; therefore, if the day k 8 hours long, they will take f^j days=9 days. (•5) s. d. £ s. d. 4 a xl0=2 2 « = value of tea 1 0^x18=1 3 3 = value of coffeo 4^x23=0 8 1 J -value of sugar 71x10=0 10 4 = value of candles 24 -i p 4 4 25 (722 n : entire cost 05 0= share of each person. (C) (2375Jx40K=05030^Z.=:£395 IDs. 2d., and 2s. o^d x 1000000=(~ x 100000o')(7. • =(27875000K =£110145 IGs. 8(^. (T) Area covered in each revolution =(0^ x 2\) sq. ft, area covered in all the revolutions = Ao x 12 X J X ^j sq. ft.=(195 x 9) sq. ft.=rl95 sq. yds. (8) C's shares J B's share, A's sharc=:!J- B's share; theref(n-e f B's share + B's share +f B's share=$1400; therefore M B's share=$1400; *, r Til , ^1400x13 ^,„^ therefore B's share^f — -- — = |480, C's sharerrg of $480= $320, A's share =5- of $480 =$000. t MISCELLANEOUS QUESTIONS AND EXAMPLES. 165 (n) Take 3 g-allons oftlio mixture. Tlie.se consist of 2 parts brandy and 1 part rum, and tliey cost tiie merci.ant ($10.80+$3.52) or $13 33 He then sells them for |.">.4()x;], or |16.20; therefore he gains on 3 gallons of the mixture $2.88- tiierelore ou each gallon lie gains ^^ cts.=96 cts. ' na> V. 0) 650974--1472=074 with remainder 446. (3) 45' f 35' ^- ^' ^- ^^ ^1^» ' and the fractions become — 1 5_ll5 j!jij!_ 42 225 81 45x7' 7x45' 35xy'^^'8l5' 315 ^315' and their sum^l?±?H+ll_?18_ ,, 315 ~3I5~" '"'"• (S) 7000 loz,avoird.=-~m.s Trov- i""" fAf^,, rr. 1 (i ^ ^-IQx 20^24^021^ ^^^^ lbs.Troj 7000 l(i X 20 X 24102^40 "^ ^^^''^ sovereigns 70 X ISflO ^10";n>ir24lU2T4 ^^v^re^Sns 21805 '"0144 s"vcrcigns=32ni sovereigns. (4) Value for 1 yearr^(37500000x45),9., value f or ; year=£il_ 27500000 X 45 4x20 =£(1718750 X 9)=£154G8750. 166 KEY TO ADVANCED ARITHMETIC. (5) The first set in 1 hour mow ^ acres, the second set in 1 hour mow g acres; therefore toL^ether in 1 hour they mow H acres; therefore tliey mow 44 acres in 15 hours, aud 11 acres in \^ hours, or 3J liours. (6) He docs the work in (8^ x G) hours, or 51 hours; therefore if he has to finish it in 5 days, he must work each day -'g hours, or 10 hrs., 12 min. (7) 1 man=Y woman, 11 mcn + 7 women~(^-y- + 7j women: 170 women. Now, by the question, 1 woman does the work in (U x 17) days; . /ll X tT\ , ereforc 170 women do the work m (^ ^"o~/ ^^^' thert 170 11x17x7 therefore -^ women do the work in — ^^^ — days ; 7/ therefore the time required is t^ days, or 7^- days Debts amount to $58^5.12 and the estate is worth $4377.45. If the bankrupt has $1 he pays ^^-^ of debts. 437745 ... he has $4377.45 he pays ^^^^^^ of debts; therefore lie pays .75i:?H6 in the $. Also. A receives tllin of $2475 =$1856.72-4%^ $t953.00 = $t405.572V,Vj as B C $1406.52=:$105:).152liH MISCELLANEOUS QUESTIONS XND EXAMPLES. 167 (9) 430 thalers=(420 x 3) sliillino-s =£ 420x8 430x3x24 20 20 francs =1513 francs. (1) VI. £10 17s. e^d x87G4=£95335 17s. 9d. (2) For every 55 cents in his assets he owes 100 cents ; therefore his tlebts=~ of his assets=?5 of $3603 53060 11 =$4733-,^. (3) True lenjUi = 10| ycls.-(10| x §) ia.=10^ yds.-7 in, =10 yd. 1] in. (4) 6 oz. of almonds cost ^-^ cts.=31 cts.. lb ' i lb. of raisins cost — ^~ cts.=8i cts. ; therefore whole cost=(3|+8i) cts.=12 cts. 5cwt., 3qrs., 14 1bs.=589 1bs; therefore price =3534 cts., 3534 cts.,-1178 cts.=3356 cts.; therefore price of each lb.=?5?2? cts. =4 cts. ((5) The sugar costs liini he pays for expenses he has to gain 589 $54.80 2.74 8.23 therefore he must sell it for $05.76 Nowl0cwt..3Qrs..811bs — mofilha opfi *«(: 'r/!_^ef^/» -x_ therefore price of eacli Ib.=J||?cts.=6ct8. 1G8 KEY TO VDVANCED ARITHMETIC. (7) Length of polc=(35 x 12; in. =420 in. Now in 24 hrs. the snail creeps up 15 in. ; therefore in (24 x 2G) hrs. the snail creeps up (15 x 26) in., or ai)0 in. ; therefore he has (420-390) in., or 30 in. to get up. And he goes over 1 in. in — hrs. _. . . 12x30, and over 30 ni. m — -^ — hrs. ; - «« /■ 12 X 30\ therefore he reaches the top in ^^24 x 26 H — gp" ) ^^• orin(624 + llK)lii's^^^035Jf^. (8) £ s. d. Wages of 3 foremen weekly =660 " 10 shopmen weekly =10 10 " 5 assistants weekly =476 total amount of wages=21 3 6 weekly profits =54 6 5 difference =33 2 11 10 331 9 2 for 10 weeks 5 1657 5 10 for 50 weeks 66 5 10 for 2 weeks annual income=1723 11 8 for 1 year annual outgoings= 723 11 8 net proflt= 1000 VII. (1) 2x6 f of i of 6 dollars=q^=^ therefore fraction 100 fj 23750" 475 = 0042. 1" r )in., rs. mSCELLANEOUS QUESTIONS AND EXAMPLES. 161) (2) £5 6«. S^d.=5i01q., £85 On. 4d.=SlGlGq.; Of p-i f therefore number= -=1G 5101 Again, (4 + 3 + 1) times tlio third part=£04 13s., or 7 times the lliird part-£:J4 13.9.; therefore tiiird part- £4 19s., second part= £9 18s., first part=£19 IGs. (3) days. 36535 rrleno-th of civil year 365-243364 =trno lenj-th of year •0077oO=:aniinal defect. lyfyoi^ Since, therefore, r~ -- ' 1000000 of 1 day is tlie defect cacli year, number of years required- (1) 1000000 135000 7730 907 ~ ^67- 1 man mows 000 acres in (15 x 17) days, , 15 X 17 1 man mows I acre in - — — • davs 300 ^ ' 27 men mow 167 acres in -—M^^ clavs 300x37 -^ 3839 , ~ 540 ^^''^ys='>«4fl days. (5) 6 times the work may be done in 12 days by 130 men; therefore times the work may be done in i? days by 1300 (6) J0f^0fi = i; therefore vakie of my share was $6000 and ^ of tlie ship's value is $5000 therefore value of part remaining is $1000 Iff zx 170 KEY TO ADVANCED ARITHMETIC. (7) .675 Since $000 b:is $075 to meet it, $1 has $;tt-: or f to meet it, also claim of third creditor will be $(900-125-375), or $400; therefore he receives f of $400 or $G00. (8) $13 X number of workmen in first class= wages of first class, $130 X number of workmen in first class= wages of feccond class, $88 X number of workmen in firsS class— wages of third class ; therefore $131 x number of workmen in first class=wage3 of all =$847; 847 therefore number of workmen in first class=r7rT=7; therefore number of workmen=:7+14 + 77=98. (9) £ francs. l=25i francs, cents. 25 25 400=(10000 + 100:1=10100 = 126 25 i = 8 41f therefore £405 Qs. 8fZ.= 10234 VIII. (1) 661 13 lbs. oz. = l()5 oz. 2 dwls.= I'lT of 1 oz. 1 d\vt. = i^of 1 dwt. £ •> 5 d. 10 32 10 0= 10 rCOSt of 10 OZ. 325 195 16 5 0= 0= 0= rCOSt of 100 OZ. -cost of 00 oz. =co3t of 5 oz. 536 5 6 0= 6= 3:r =cost of IGT oz. -cost of 2 dvvts. -cost of 1 dwt. £536 14 9~value required. MISCELLANEOUS QUESTIONS AND EXAMPLES. 171 (2) i'l lis. Gid = ]510r?., 2 florins=:<)6|7.; 96 Ihereforo nuinber=-~rr loZ^ (3) 3 kreiUzers:=:l penny, 1 krcutzer=: of ^1 5x1^ Lzers=^ -~ of a giiil(len=^-- of a guiklen. okrcutzers= (4) 12if. 4(^. is i?, 0f£l; therefore'he first receives ^ of £29G=JL'183 10^. 8^., the deficiency is then £113 9s. Ad.; and since 3s. 9d. is -i\ of £1, he next receives ,3. of £113 9s. 4d.=z£21 5,s. 6d. ; therefore in all he receives £203 10,9. 2d. (5) 207 12 ft., 4i in. = 148' in.=~in., Imile=(1760x3xl2) in.; therefore fraction 297 1760x3x12x2 9x11x3 o '11x160x3x12x2 1280' and 3 -375 -04687.5 1280 ~ 160 '20 •00234375. («) Income tax=(200 x 7)^.^=2800 halfpence, and he has to save 3 halfpence on each lb.; therefore number of lbs. orsiigar=;;^=U33^ ii,y.. o therefore he must use 9 c\v(., 33;V lbs. of suo-f gar. I\ 172 KEY TO ADVANCED ARITHMETIC. (7) B's work in 34 hours=A's work in 20 hours; tlierefore A could do j tlie work iu (12 + 20) L Airs, or in 32 hours ; therefore C docs i the work in (f x 32) hours, or in 57| hours. (8) •3 of £1 = 1 of £l=i of £1=6.'*. Sd. (9) $14400 X 52=$748800 annual earnings $3723.40 X 52=$10;}G1G.80 annual expenses 10 943410.80 94241.08 848175.12 115880.08 $732295.04 net profit IX. (1) 9375 1522842 '37r'^~710 =25 + 2118=2143. (3) 3 2 0_ 5~ 3 ~ ' ti) 15 _^_ 1 _100 18^) " 375 "75" 35 ~ 25 (3) •04. I 6 « 4 -2 4 — ») 34 and since 8=2 x 2 x 2, the fraction will become a finite d'-j^imal. A-ain, |^| of $1=V' of $1 = -^^ of $l = fo of $1=70 cts. .. MISCELLANEOUS QUESTIONS Ax\D EXAMI>LES. 173 (4) (Ir-i) of the term=13i clays; therefore ^ of the terin=13i rtays; therefore the terin=(13i x 6) days=80 days. (5) 4 12 3,0 20 6-5 17-541"6 £19-877083 Ans. £19 8f. 7c. 7m. with •083m. over, and -083111.=??^:? m. =^ m -i m 900 • 900 ""12 '"• (6) s. d. Sd.x2^=0 6i= price of milk 3«. x-iV=0 2 =pnce of lemon Is. X i=0 U=price of 2 eggs 13s. xi=rl 7i=priceofrum 24s. 8d. X -i^^ =1 Gi = price of brandy 4 = whole cost. (7) The English hen while catini,^ a pint of barley lays 10 eggs, and these are equivalent to 15 Cochin China eggs. Conse- quently the English hen lays 15 eggs to the other hen's 12 eggs, and is the more economical layer. (8) C.^0 X 3 X H X 4) cub. ft.=3G0 cub. ft.rrcontent of first trench, (30 ^ 3 V 91 V. r:\ p„u ff 2025 , „ v-Ji, .. .1 ,_4 X o) CUD. lt.= ^ CUD. it.=C()ntent of second trench, (Continued on next pujje.) II m 174 KEY TO ADVANCED AlilTIIMETIC. (8 continu d.) cub. ft. lirs. men. therefore since 360 are dug in 30 b} 72 1 i( 30 300 1 (( 1 ., 72x30 360 1 (( 135 ,, 72 X 30 135 X 300 2025 2 (( 135 ^^ 72x30x2025 2 X 135 X 360 of ' men 3x 2025 2025 405 niuei ~ 135 "" 45 ~ 9 «M[^ (9) Wages of men =$(420 x 14-40)= I therefore wages of boys= $(7200-6048)= $1152' iir20 therefore number of boys=— ;^^=160. ■ X. (1) 3 4-061= -6 4- 061 =-661; 02- 003= 017; 0672-?- 006= 11-2. (3) 3 11 2 6~33 12~132' and 132 'A7"* 8/ "132x7x8"" 2464" 132^ V7 8y~132x (3) 3x5 4 12 20 7-5 2,0 10625 £25-53125 therefore £25 lOs. 7if?. = £25 5fl. 3c. lim.=255fl 3c. Urn. ik MISCELLAXKOUS QUESTIONS AND EXAMPLES. 175 '4^ Number of yartls= 34560 53 =064^. (4) men. daysi. Since 6 in -^ earn $90 1 in y earns $15 1 in 1 earns $2 10 in 1 earn $20 47 10 in — earn $235. (6) Gas consumed in 1 hour=(10 x 4 x fiO x 60) cub. in. ; therefore . ust= H^;:; ^^^ ^\Jl^^. -.^. , i;2»x 1000 432 *— 432*' ^ =^.=6^. f < I (7) $1.20 89 $82.80 90 price 3 pks. Also 8 qrs., 6 bnsh., 2 pks.=70i pks. ; thernfore price (423-T-70i)«. $83.70 (8) men. cnb yds. lirs. Since 36 dig 72 x , x 12 in 8 x 16, 8 X 10 X BO 1 digs in 72 X 18 X 12' 32 dig 64 X 27 X 1 9 in ^^/5..^^i.x_64 x_27^x_18 ;J2 X 72 X 18 X 13 tlierefore number of ho-.rs - "^ ^iL^-^ ^^ ^g x 16 x 9- io X 1'^ therefore number of day3=- ^ ^^ ^ ^ -2 x 4 x 3=24. M m. SE f ■ I 170 KEY TO ADVANCED ARITHMETIC, (0) Area of slu'(!t-(CG x 30) sq. incljcs ; thercloie each sheet will give a slr-ip (00 x .']0) linear inches in length and 1 rmear inch l)roa(l ; therefore number required = J^oOOOx I7()()x;ixl3 GO X no :2ol)00 X 10 X 2=800000 sheets. R! V fi XL (t) V3^f='O83rO07G0917293233, value=.$24.81-,Vj. (2) If $1 pays a rate of 2 cents, $3070 pay a rate of (3070 x 2) cents =$01.40. ('5) If 2i lbs. cost 20 cts., 1 lb. will cost 8 cts., .•dso 2 tons, 10 cwt,., 17 lbs. =5017 lbs.; therefore price required=(rjG17 x 8) eta = $449.36. . (4) It must be sold for $30.90; 3096 therefore price per 11). = i^rr.^TTi cts. =-3090 cts, 10000 ■00003551130= 3551130-35511 '"9!)0()000000(r ' therefore value in inches: ^351 5025_x 1 00^^ . _ 2250()0 Qp00 lOUOOOUOOO ^'^•-— w.,>;ta-;^ in =2-25in.=2iin. (0) 3515025x1700x3 x 12 99^00(4)00000 0{)OQpOO ^ looopoo^oo ID. 5^.=G0d and2I(?.=2-4^.; therefore 5.s.=^-^-^=25, 2,9. Gff.=iof 5.s\=12-5, [(Continued on next page.) MISCELLANEOUS QUESTIONS AND EXAMPLES. 177 [ir «^ cntg (G continued.) 18-^ of 5j».=5, GfZ.=i of U=2ry^ 4d=^ of l«.=l-6, 3rf.=:iofU = l-25. Also, -jAj omd.z=Ud.=Ad. ; therefore in this unit of money one halfpenny=-£ (7) 4s. 3H=5-42fr.,or?Prf. 543 5 fr.; 109 . , ,, . - 543 X 5 „ therefore lcZ.==^^^—fr.; 543 X 5 X 340 therefore £1 = fr. 1084 ' 4a 258 X 100 Also, when unit of money is ^d., , ,, 358 6 358x25 „,^ ., a dollars: --^^.-:.-^^-^=315 units, 258 X 100 X 25 543 X fr.=25-/jfr. „ P /258 542\ 6 258 x 1 a franc= i j i _: — _ \ 5 100/ 35 5x5 215 X 100 10750 '543 371 :39iH units. (8) A and B do | daily, B and C do | daily ; therefore, taking these successively from the work done by A, B and C daily, we have A's work daily=i— ^, C's work daily^J— ^ ; therefore A antl C do diiily i— 1 -f i— i-, or /,- of the work ; therefore A and C do it in V days, or 5^ days. (9) (20 X 13)(Z X number of sovereigns— value of the sovereigns in pence, (13 X 3)(Z. X number of sovereigns = value of the shillings in pence, id. X number of sovereigns^ value of the pence; therefore (240+30 +4)d x munbcr of sovereigns = value of coins in pence ; therefore 380f/. x number of soverei!rns=(280 x 30 x 12)d\ 2S0 X '^O X 13 there.'bre number of sovereigns=-'^ — ^-"^ =240, number of s]iiirmgs=240x 3=720, number of pence=:240 x 4 ~i)G0. I i ^ 178 KKY I'O ADVANOKI) AKITIIMKTIC. KILE OF THREE. Kx. XI.IX. (p. 177.) (15) Id. : £\0 \s. l]d.::£\ ■ lialfycMi-'s iiioomo; 524 1 5 tlK'rcroic li.'.ll yciir's inconior-JL; ^ ; tlu'rcloi'c \ car's iii<'(>ino~£(>})0. :£345; liOt the Hcholar do ll;c above qiioslion as Tollows: Since 7(/. tax irivcs I'l income; .'. Ir/. tax nivcs Ij^ income; .-. S-l 15,-/. tax -Ives .€« V * = -€;M5. The same method ma|' be adopted in every case. (17) M53 els. ; lj<15()0 :: $1 : amonnt of debts; .1 <• . • 1 45(>000 ..,^,^ , therefore amount requu'ed— - . - = |3047-,V. (10) 400 jic., 2 ro., '20 po. : 1 ac. :: $1201.87.^ : rent p^r acre; (10 X 210; (54100 x^ 100x2IO;}75 ^ -„ therelore rent per ii(^i*e = - ,.^ ,/\,\ ,,- cts.=$3. (24) LoMS on .€1 cU'bt~4 fl,, 2 e., 5 m. ; therelore £1 : .CI 17!);} 5 II. ::425 m. : loss required; thercrore l,,ssrrlllL^"''*---l*m.r=£50l2 2 11., 3 c, 7i m. (25) 2i lbs. : 2 tons, lO ewt., 17 lbs. :: 10c/. : ]iriee required; (!2Si) X 10x2 tiierefore price rc(niired=: -— — -^ ^-d=£104 IQs. 4d. 5 (25)) \m ao., n ro., ;]5)l i)o. : 2 ae. :: .£051 lO.s. lOf?. : rent required; ,, , , . , 2 x ;120 X 228478 , 1142;50 " J --r:12S0(/. = £5flN. 8(/. KULK OF THREE. nif (30) 27 bus., 2 pkH. : IGi bus. ::.i:i() 7.s. 2i 27=146; therefore 365 : 146:: $224 : wages required; ,, „ ' . , 140x35 . therefore wages required = — ^r,r— guineas = $89.60. r^ (43) 12. f ,- of I of -,V : j| of I :: $1600 : vahie required ; ^, - , -1 Jl x9x 17x5x4x2x1600 therefore value required =$ * ^ 3x2x3x4x17x5 = $13200. (44) 48 yds. : 60 yds. :: 7s. M. : price required ; therefore price required= — ^x '~ ^' (45) 97 cts. : $1 :: $7838.12 : income required ; , , /• . ,«. < 838 1 2 therefore income =$ 97 $808051t^ (46) From noon on Monday to 10 hrs., 15 min. A. M. on Saturday there are 118i hrs. ; therefore 24 : 1181:: 3' 10" : gain required ; therefore guin=-~^^- sec. = 15' 36/8"; therefore time indicated by the watch will be 10 hrs., 15' + 15' 30 fa" + 10', or 10 hrs., 40' 36-A". BULE OF TIIKEE 181 (47) 1 : 8-Ui6::22^ ft. : circumference required; therefore circumference required=( 3-1416 x — ) ft =70 ft., 8.233 in. . I (48) 25f : 40 :: 3 cwt. : weight required ; therefore weight required==i^^ cwt.=:4,V3- cwt. (49) 14 : 365 :: $20 : annual expenditure; therefore annual expenditure=$5?5^-^=:|521f ; therefore income must be $721 f. 14 (50) 175 guineas : i;!20:: £6 17.9. 9^^. : tax required; therefore t^x=^' ' ^'^'% =.£4 10, (51) 1 lb., 10 oz., 10 dwts. : J ///. .< $20.70 : price required ; therefore price per oz.=$— -|?i??=$i.33. (52) 4^ days : 9 days :: 7^ hours : number of hours required ; therefore number of hours=?4i4— =15. 9x2 (53) Cost of silver when manufactured will be ($15.84 + 36 cts.+44 cts ), or $16.64 per lb. ; therefore 1 lb. : 7 lbs., 7oz., 10 dwts. :: $16.64 -. nrirr^ r^^A . therefore price required = $l??^l?:l[i = i j o^ss u 182 KEY TO ADVANCED ARITHMETIC. (54) 365 : 63:: $3935 +$90 : sum required; . , ^63x4015 -.,,^ therefore sum requu'ea=$ — ^ — =$09.1. r !i i (55) Since 1 man=3 boys and 1 woman=2 boys, the question is llie same as the following : if (45+24 + 9) boys do the worli in 50 days, in what time will (27+30 + 18) boys do 4 times as much? therefore 75 : 78 :: 50 days : days required for same work ; . , . , 78x50 _ therefore days required for same work= ^^^z— =02; therefore days required to do 4 times as much=208. (56) Tax=$40; therefore he has $711.75 to spend annually; therefore 305 : 1 :: $711.75 : daily expenditure ; 711 75 therefore sum required=$ „ —$1.95. (57) Since 3 cows eat as much as 7 horses, 49 7 cows eat as much as — horses ; o therefore 3+^ :7::29 : days required; o therefore d ays = ^r:^ — = 10 J. 58 (58) 440 Area of room =— sq. yds. ; therefore ^ sq. yd. : — - sq. yds. :: 1 yd. : niiinber of yds. 4 required ; therefore number of yds. required =- 4 . . A A/\ 3x9~' :65 ^i^'- r KULE OF TIIKEE. 183 (HO) Cost of eggs ^ (."50 + ,03 h)d. ~ Q^d. , also 5 : 200 :: 2d. : selling price ; 200 X ^ therefore eggs are sold for — p-^74~ ■ ^=4166f. ir [!i J 184 KEY TO ADVANCED ARITHMETIC. (04) 2i fur. : 240000 miles:: 5 oV.. : -weight required ; ., . . , , . , 240000 X 5 X 3 X 8 tliercfore weiq:lit required = ©z. 5 :^(240000 X 16) oz. =240000 lbs. (05) (3 X 3-141 6) ft. : 4 miles :: 1 : number of revolutions ; 4x1700x3 therefore number rcvolutious=:- (GO) 3x31410 7 — »'^'±\J^i)>if. 8 oz. : '75 ton :: -SGOSs. : price required ; 15 X 100 X 10 X 0525 8 therefore prit:c required = $- = $(8000 X -0525)= $157.50. (GT) •4583.9. : £01 126-. :: -0025 lbs. : wcii^ht required; therefore weight required=-^^^-^^lbs.=lG84§t3 lb. •4583 (G8) 4' 10" : GO' :: 1 day : number of days required; therefore number of days=^j^-.= 14dcays., 9 hrs., 80 min. ; therefore the time will be on Monday fortnight at hrs., 36 min. P. M. (00) (2-5 X 3) sq. ft. : (20 x 27) sq. ft. :: 1 yd. : number yds. required ; 20 X 27 therefore number of yards =^--- — =72. i'o (70) 060 galls. : 1 gall. :: $3072 : price required ; therefore price required =$-i^= $3.20. JIULE UF THREE. 185 lb. iin. (Tl) 13 A soldier's daily all()\vanco=-^ lbs. = l| lbs.; 8 therefore 1 : 3GG :: (^850 x '^ lbs. : weisrht required ; therefore weight required =—-^i55i!2 lb. ~40G050 Ibs^ (72) ; 1600 men will have provision enough for 2000 men for 80 days; therefore 1000 : 2000 :: 80 days : number of days required j therefore number of days=?!^~ = ioo. IGOO (7S) 20 sq. poles : (10000 x IGO) sq. poles :: (:] x 52) cts. : yearly income- therefore yearly income^^J}^!!}lll![^^Ji^ ^ts. = $124800. (T4) 20 1 oz. : •3GS2201G lbs. ::.€4180.")S:{ : value required; therefore value required = ,12x-;{GS2201Gx37T0G25 =£■ (75) iJOOOUO 850 men will have enough provisions to last iOOO men for 68 days ; therefore 850 : 1000 :: 08 days : time required ; "^ therefore time required=-^^^^ "" ^''^ 850 days=80 da vs. (TG) 1 ac. : 182-3 ac. :: £-,*''- : rent required ; therefore rent required =::£'!^^il--£2 11 10,9. M. o X 4 (77) |(57G+124-204)=,|882; therefore 2 tons, 3 cwt, 3 qrs. : 1 cwt. :: |882 : price required ; therefore price per cwt-^S-i^'^iT'-ji-oo c «l IH ' f 186 KEY TO ADVANCED ARITHMETIC (78) Area of piece=(2 x 1 4i x 13, S) sq. yfls. ; therefore 40 ^ sq. yds. : ^2 x 11^ x ^) sq. yds. :: 1 yd. : number of yds. required ; ♦a ,• 1* r 1 3x117x171x2 tbereiore number of yds.= — — — - — 1-7: — =\)t' ol X O X io (TO) 3-75 yds. =00 nulls, and 08 yds., 3 qrs., 8 nails-ClO nails; therefore (jO : 019:: $;}.825 : priee required; ,OlOx:5-82r) there fure price =^- m -::::$39.4G120. t 1 COW eats as much as horses, 9 cows eat as much as y horses; 54 81 therefore 4 + -- : 18+--:: 1 : relative size of second field; 7 X ''07 ''07 therefore size of second neld==-y^y=^ size first field. (81) A reaps --, and B — in 1 hour; therefore together they reap ^-{-.t,., or— in 1 hour; 55 ' GO' 30 10 therefore -,: : 1 " t day : days required ; oO . , 30 ,, therefore days requn-ed= --=3. r DOUBLE RULE OF THREE. Ex. L. (p. 188.) 6 ac. : 15 ae. ).^ ^^^,^^ . j^^^rnijei. of men required ; rs. : 13 hr.s. \ 14 hrs. 7 X 15 X 13 . ^ therefore number of men= — - — z-. — =15. O X 14 •^M DOUBLE KULE OF THREE. 187 f2) 9 $75 : $78.75 ) „ , » days : 20 dav^ \ "^^" * """^^'^^ of men required; ,, ^ ,1. 20x315x3 „ tiicrelore number of mcnr= — 7— tt: — =7. DxaOO (3) 7 horses : 10 liorses ) jn ^ 1 ,. ■, . ■, 96 bus. : 00 bus. p- ^^ "^""y^ '■ nw'nber of days reqmred ; 41 r T ri 60x10x42 ^„ therefore number of days= — ;^—^7. — =06. 7x96 (4) 5 sacks : 15 sacks ) oaa it r i- ■, -,• . •, 2 days : days j ^^" sokliers : number of soldiers required; 15 X 6 X 800 therefore number required=-^ =7200. 2x0 (•'5) horses : 8 horses ) w ~ , , ,, , , , . , 13 days : 11 days j "^* ' o"Ji^"cr "f bushels required; 11 x 8 X 17 therefore number of busliels= =19.\. i (0) 16 liorscs ; 12 horses > ..-,oqa 1 .. • , 8 days : 5 days l " ^^' ' ^™^^^^^^ ^^ '"icres required ; therefore number of acres=— ?-^— — =000. 10x8 (7) 36 cwt., 23 lbs. : 11 cwt £1 5 c. : £o2r. 5 therefore number of miles = 23 lbs. : 11 cwt. ) -,0 -i £I 5 c. : £0 2f. 5 c j '" ^^des : miles required ; 1100x5-25_xJ_2_ ,^ >/iO*> ■»•'*';* 6 2 3* 1-50 X 3023 (8) 53 miles : 124 miles ) o . , ^ 130.24 : $15.12 C " ^^ " '^^^•^^^cr of cwt. required ; 15-02 X 124x8 therefore number of cwt. 53 X 3024 :9 19 6 i- / 188 KEY TO ADVANCED ARITHMJiTIC. hi i h ■i (9) 5 men : 7 men ) *ir^A • i 11 mo : 4 mo. ( " ^^^^^ ' ^""^ required ; therefore sum required =$ — - — -~ — = $784. 5x11 (10) ilOO • \ 49A [ *'^ months : number of months required; ,, . ^, . , 1000x41)5x5 _ therefore months requu'ed=-7rT= — r^^;T;r-=ll. 225 X 1000 (11) 3 horses : 2 horses lorses : 2 horses ) aq^ • i 7 mo.: iinio.r-'^S^^^"^^^^^"'^"^^^' therefore sum required =$ 2x11x84 3x7 $30 [ ''^^ ^^*' * "^^'S^^*^ required; (12) 100 miles : 160 miles $3.85 : ,. e . 1 , . T 100x30x10 therefore weight required^ — qrs. 100 X o'85 =59 cwt., 227^7 lbs. * (13) 7 men : 1 man ) i i , . , 345f sq. yds. : G ac. p ' ^ ^^^- = ^^^^"^^ required ; therefore number of hours required =-^-^^'^--—10 (14) 12 men : 20 men ) o i ^^ c ^ • i S900 • S51500 " * weeks : number of weeks required ; ., P 1 p 1 20x1500x3 „, tlierelore number of wecks-= — —^ — —^~ — =8|. 12 x 900 (15) 1 cwt., 3 qrs., 21 lbs. : 2^ tons ) ^-, ^ , ^ . , 52X miles : 46^ miles \ ''''^^'- ^^' ' ^"^^ required; ,, ,. , . , 5 X 2240 X 93 x 209 x \ , „^^ therefore cost requ)red= — -— - — rr^-- — vr,-r — d=£20. 2 X 2 X 2 1 7 X 209 t DOUBLE JLE v I*' T'lUEE. 189 (16) 8 men : 10 men } 7i ac. : 9 ac <" therefore ■ Uoiii"s : numbers required ; 10 V 0x30x3 ^, ihv r of hourb— ^-_ =54; >- X 15 54 there'' )re iiumbi r of days--, j hours. i so men : 25 men ) o i v n • i 16 days : 34 days [ "^^'''^- '' ""'"^'^' ^^ ^^""^^ required; 8 X 25 X 34 therefore number of hours =-—, — ;-— =10. oO X 10 (18) 17ac.,3ro.,2po.:26ac,2ro.,23po. ) ..^^^^^ ^^ , mtreq'd; I ac. I ae. ) ., X. . . -. 4203x0x9415, „_^ ^ ^, therefore rent required= — - — jt-tt^^ — d.=£oO 8s. Qd. ^ 7 x 3843 (19) 1500 conies : 5000 copies } nn i ■ •, 11 sheets : 25 sheets [ •• ^^ ^"^^"'^ ' '^"^^^^ ^'^^"^^ required ; ^. . ^ c 5000x35x60 ^_ therefore number of reams = — — — ■— — — =500. 11 X loOO ' t (20) 7 men : 5 men 12 hrs. : 14 hrs. i oi t ^ f -, . , 800 ft. : 1800 ft. I •* ^^ "^y^ • ^""^^^^ of days required ; 700 ft." !• 900 ft! . 41 P t, i. 1 5x14x1800x960x7 therefore number of days=--— — ^ — -- --=9 7 X 13 X 8U0 X 700 x 3 (31) 1500 men : 1000 men 6f oz. : 16 ) oz \ ''^ weeks : number of weeks required ; ., P 1. ^ , 1000x]6x5x;5 therefore number of weeks= — ^' lauo x 30 ^ c{\y. ^Z^ ^^,0. IMAGE EVALUATION TEST TARGET (MT-3) // '^^C /. & ^ t/j 1.0 I.I 2.5 12.2 u 1^ ^ »£ 12.0 !.3 L25 iU 1.6 P^. <^ ? c^: ^# ? *>V ''. ^. y € m Photographic Sciences Corporation 23 WEST MAIN STREET WEBSTER, N.Y. 14580 (716) 873-4503 M V' <5P ..W i^. .'•& *UL n^H 1 :.,,;i 190 KEY TO ADVANCED AKlTllMETlC. (22) 60 masons : 20 masons \ 10 hrs. : 7 lirs. / ^2 ft • ^^ f^' i '' ^^ ^^^^ ' ^^^^^^ ^^" ^^^y^ required ; ft J 14 ft. 10 therefore number of days=?^'ili^ 5 00x4x 16x12 CO X 10 X 50 X 2 X 14 -=64. (23) V days : 24 days ) .^ , ^ 1 day : 7 days j" •• ^^ men : number of men required ; therefore number of men=:-'*--^^-^?^^-35o. (24) 100 yds. : 1000 yds. ) 20 ft.: 1(3 ft. f ^ "^ 4 ft. : 6 ft. ( ••■ ^^"^ ^^^ •■ number of men required ; 30 lirs. : 48 hrs. ) therefore number of men=155?^.^li!ii8 x 125__ 100x20x4x:j0 --^^OO. (25) 5 ft. :12ift. ) II J{- : ^1 [[• [ ••• 7500 lbs. : weight required ; therefore weight required=r?^iiL^-i^i^lli.^^O A,y . .^ 2x2x4x5x15x5 47 tons , 17 cwt, 66 lbs. (26) |96 : $345.60) ,^^ , . I 1.20 : $ 1.08 [ •• ^^" '"^^* • ""mt)er ol men required ; therefore number of men=:^^'^'^^^ "",.^=324. - lbs. 120 X m (27) Here 25 horses eat as mucii as 40 ponies. Also for $205.15 we can buy, at 55 cts. a bushel, 373 bushels; therefore 40 ponies : 13 noiiics / ,., , 15 qrs. : ^^ qrs. j" *• ^"^ ^^^y^ ■ nuraber days req'd , 12 X :573 X 64 therefore number of days— - 40xl.j>?M~-^'>^^^" m^ DOUBLE RULE OP THREE. 191 X (28) 18 in. : iii in. \ " ^''^ ^^•''- ^f?- = cost required ; 473x30x2x14380 therefore cost= 4x85x18 d=£333 5«. 2M' (39) 134men : ()S men ) .....^ o o >^^ i, .^ rs. : 03 hrs. \ •• (IK^ x 3 x 3 x 4) cub. ft 55 hri therefore number cub. ft. = number of cub. ft. required ; 110x3x3x4x02x03 55 X 124 =3268. (30) 3-35 lbs. : 47-5 lbs. ) o . $1.14 : $i.GO J" •• ^ ^^^' ' P^'^^^ required ; therefore price required=lZl^^cts.=$1.59-A^,V 3'3oxl-14 J8i»- 03 ft 114 (31) Ift. : 14ft.) «, -, .40 : $21.00 j" ''^^ "• • iiumber of ft. required; therefore number of ft.=~^^'^^^.^.^-8. 03 X 14-40 (32) 21 in. : 27 in. ) a,1t^r,t, . . , 1) cts. : 8 cts. C "• $l"-35 : cost required; therefore cost=— ^-^^-^^110.35. 24x9 (33) 7 men : 4 men ) ooa i , « , 10 days : 43 days \ '' ^^" >''^'^- ' """^^er of yds. required ; therefore number of yds.=— ''--^=:1320. 7x10 (34) 3i ft. : 2i ft. ) 7iin. : 8 in. - :: 10 ft. : 1 -380 lbs. w 111. r 2038 lbs. ) number of feet required; therefore number of ft.=?-^^??Ji^_^?_^=,i.sjH 13x15x1380x4 192 KEY TO ADVANCED ARITHMETIC. (35) Here 12 ox(!n eat as iniich as 28 sheep, 9 " '• •' 21 slieep ; tUereibre (28+3G) slieep : (21 + 12) sheep | ^^n -.« . .. 8 days : 28 days f -^20.16 : cost req'd; therefore cost=|!Hlil28_><_2(m^^^^^_^^ 03x8 (3G) Here 1 man does f ofa woman's work ; therefore (3 + 1) women : (|+2) women } .^ , 1 woman : 4 women j" * " "^^^ number of days requu*ed ; therefore number of days— ' ^/ ^^ ——35 4x2 (37) 142-2 miles : 508-6 miles ) .^ i 8 4 hrs. : 101G4hrs. \ ''^^ ^^'^y^ ' ^^umber of days required; therefore number of days^^^^^lJil^l^^^ig.gg^ 8-4 X 142-2 (38) i^. ; 18,2,^. ISls. : 5 y^ r -"^/u- lbs. • number of lbs. required; therefore number of lbs.=?:^"-5?-^-^^-^iiH 20 X 92 X 15 X 4 -=49-3. 10 (39) 9 people : 8 people ) 7 mo. f- : 12 mo. : a) : $7803.40 t cost in Shillings; therefore cost=$ 8 X 7 X 3 X 17862-4 ^ -9^2-^~-=^344G.08. ^ 2q'd; red; ired; <^ (9) £ 8. d. 236 6 8 3 709 20 l-80«. (10) £ .<». d. 98 15 10 197 11 49 7 8 11. 2-46 19 20 7 9 29,-?. 12 4-75r?. 4 300'/. £ l.TM (10) .s\ d G 3 8 12274 20 10 14-:0 12 lOSO SIMPLE INTEREST. SIMPLE INTEREST. Ex. LI. (p. 194.) 193 1-80S. 12 9-60ff. 4 2-40?. £ 8. d. and 7 1 9i 2i 14 3 7 I 3 10 lOJ i £17 14 6 £ 8. d. q. and 2 9 4^ m. 1050625 8 (13) £1 4 8i 8405 000 interest for 1 year. 504 30 interest for 6 years 1050-625 principal. 1554-925 amount. £ .9. d. 122 14 10< n- 122 14 10^ 01 < 5^ 15 6 10/.r V.Yd 9 2.^\ 15;J4 3 o. •Hi 191 LEY TO ADVANCED ARITHMETIC. I (21) t |U) Now from Mawh 10, 1850, to January 23, 1851, there 8 wcri' ;5i;{ (lays; 11 :5.20 IVIUIWIU IIIIUICSL IdllllHiVI^ — .jj^A *'l ^,10.»wV = $2.74^^, and iiniount= $43,747'!!. (22) 7 23.152") o Now ink rest lor 1)5 djxys=3\^; of $23.4536 thcrc'ioro $44,005 2.15 *2 47.055 interest. 320.75 44.9050 (23) £ n. 84 10 (1 207 20 1-40 12 3(57.80 aniount. Now from Aii^-. 10 to October 21 there are 72 days ; thcri'Yoro interest reqnired =r;,^i\or£2 0.s-. 4-8rf. =:^S,s'. 2^/. nearly; therefore amount ==.£U4 18.s. 2d. nearly. 4-80 Ex. LII. (p. 195.) (1) $132 : $150:: $100 : sum required ; . 1 J50xl00 ^,.„A^ therefore sum required=:$ — -,- — =$113.63. lo2 (2) $540 : $100:: ^j of $028.80 : rate required; . T ^100xO'38.80 ^„ therefore rate required = $ — rjo ~«T — ~^ {■\) Interest on $350 for 1 year:=$24.50: therefore $24.50 : $98:: 1 year : time required; therefore tune required =-—>- years— 4 years. SIMPLE INTEREST. 192 <. (4) Interest for 1 year=? of $0S.n025= $19,537^; therefore $;l33.o0 : $100:: $19.5^ f . j.^te required; therefore rate requirea=$--^^ ~?:?2.?5.-*« Interest on $143.50 for 1 yei\r-$9.07*; therefore $9.97^ : |0'J.75:: 1 year : time required; therefore time required=j^j^~ years =10 years. (6) $157 : $100::-a»6- of $335.50 : rate required; therefore rate required=$^^^''~^''''''^ (7) $Gi : $87.75:: $100 : sum required; therefore sum required =$?^^:^-^-?5j^== $500 li> * (8) $100 amounts in 3.} years to $134^; therefore $1341 : $1014.G7i:: $100 : sum required; therefore sum required=$?^'^fi^.'ll5!?.-|si5 157 X 25 249 (0) £113 : £387 7^. 7id. :: £100 : sum required ; therefore sum required=£l!^i^5?-^-151!-£345 17, 6. :: 1 year : time required ; therefore timfi rpmiimrl— 1^'^ vfire — --^ vf^ni-c — c)(jc) >^^rs '^j >ears. 196 KEY TO ADVANCED ARITHMETIC. (11) £936 13s. 4d : £100:: -/g of £330 14.-?. Oid. : rate required; ♦1 „. r * -1 plOOx 21 1874x8 therefore rate required=r£— ^---_____:=:£4g.^ 224bU0 X 39 (12) Interest on $125 for 1 year=$C.25; tlierefore |G.25 : $125 :: 1 year : time required ; 125 X 20 tlierefore time rcquired=:~^^ years--^20 years. (13) £100 in 10 3''ears amounts to £135 ; therefore £135 : £100 :: £425 19s. 4^ : sum required ; therefore sum required =-5?ii|f?ZIfg.- £3 15 lOs. 8d. loo Again, interest on £315 10s. 8d. for 1 year, at 3^ per cent., will be £11 Os. 10-48d, and £453 lls.-7d-£425 19s. ^d.=£27 12s. 2^^ the interest to be acquired ; therefore £11 Os. lOiSd. : £27 12s. 22^. :: 1 year : time required ; therefore time required=~y^— years=2i years. (14) Interest on $250 for 1 year =$15 ; therefore interest on $250 will amount in 6 years to $90. Aga*i|$100 in 4 years, at 10 per cent, will acquire $10 interest; therefore $40 : $90:: $100 : sum required; therefore sum required =$—^-^=$235. 40 (1) COMPOUND INTEREST. Ex. LIIL (p. 198.) $2000 6 120.00 $2120 6 127.20 $120 interest for 1st year 127.20 interest for 2d year $247.20 interest required. «lf COMPOUND INTEREST. 197 (2) $800=lst principal $8oG= 2d prin. $915.92=3d prin. ^ 7 ,1 . ^-^-^^ $64.1144 therefore amount=$(800+56+59.92 + G4 1144^ =$980.0344. -rv'x.xx'i^) (3) $270=lst principal $291.60 2d prin. .•. $21.60 . S 23.328 21.60 23.3280 (4) $44,928 interest $690=lst principal $738.30=2d prin. $789,981 =3d prin ^ 7 (.5) |230.75==lst principal $244.595:.2d prin. $259.2407=3d prin ^_ G 13.8450 14fi7s;7n TZIT". *K« . 14.0/0^0 15.554442 4m8TGm^'^'''^ (6) $415.50= 1st principal $444.58a=2d principal 29.0850 31.12095 $475.70595=:3d prin. $509.0053665 =4th prin. 35.630375655 33.2994165 therefore total interest . =$(29.08. . . . +81.12. . . . +33.30+35.63 V-$12Q iq simple interest=$29.085x4=r $116 34- ^•'''' '• •''-^129.13, liiereforedirterence=$129.13-$H6.34=$12.79. 198 KEY TO ADVANCED ARITHMETIC. (7) 1st payment=$130 2d " - IB'5 20 8d " = 140.C08 4th " = 140.2:3232 5tli " = 152.0s2G128 Cth " XT*,T=$5 20 XTiu= 5.408 xjU= 5.02433 xtU= 5.8492928 xtJj,7= 0.083304512 15aiG5917312 x jU= 6:3 2003009248 therefore compound iQterest= $34.491o3400448 f 1700.50= 1st prin. $1901.34=:2d prin. $2053.4472 = 3d prin. 8 8 o 140.840 152.1073 104.275770 therefore interest for i year= $82.137888 ; therefore amount=$(17e0.50+140.84+152.1073+82.137888) = $2135.58+ (9) $230=lst principal $240.10 7 2o0 16.10 470.10=2d principal $470.10 33.337 230 Ans. $739,427 33.3270 (10) |416=lst prin. 6 $440.90=2d prin. Comp. int.=$24.96 20.4576 34.96 26.4570 51.4176 simple iut.=49.92 Ans. $1.4976 (11) /j of $13333 =$666.65, K- of 13999.65 = 099.9825 Kof 14699.6325 = 734.981625 S-of 15434.614125 = 77173070625, VJof 16200.34483175= 810.31724....; therefore compound interest= $3083.663 simple interest= 3333.3o Ans. $350,413 COMPOUND INTEriEST. 199 48 48 Jd prin. 137888) 37 37 IM.96 26.4576 51.4176 =49.92 , $1.4976 (13) $100, at compound interest, will amount in 3 years to $114 49- therefoi-c $114.40 : $100:: $100 : sum required; ' ' therefore sum iequirea=$~ili^-^=$87.34. . . (14) $100, at compound interest, wiR amount in 2 years to $116 64- tliereibre .$110f : $100 :: |2G4 : sum required; therefore sum required=$l^-l?i!i:=226.33. (15) £ 256 116.64 £ s. d. 11 10 4-8 3 1024 128 11-52 20 10-405. 13 4-80d £34 11 2 4 simple interest £ s. d. 267 10 4-8=2d principal 4i 1070 1 72 133 15 2-4 1203 16 9-6 20 £ s. d. 279 11 2 016=3d principal ^'1^'' ^ ^ 9-216r?. 1118 4 139 15 8064 7008 12-58 20 3072 11 -eo^. 12 £ s. d. ■'q^ 1? 9 24672 compound interest 2* ^ * 4 simple interest £ 1 11 6 84672 7'23072(?. and 13 2,0 Ans. 6-84073 11-57056 0-( ervn^-rfcr^ 200 KEY TO ADVANCED ARITHMETIC. 11 1 DISCOUNT. Ex. LIV. (p. 202.) (t) $107 : |321 :: $100 : present worth ; 321 X 100 therefore present worth=$ — — — =$300. (2) $106 : $251.50 :: $100 : present worth ; therefore present wofth=$^^^ — ~z =$237.32. . . . (3) $104 : $083.28 :: $100 : present worth ; ., ^083.28x100 ^„_ therefore present worth =$ :^-^^ =$657. 104 (4) $103.50 : 944.92 :: $100 : present w^orth ; 944 50 X 100 therefore present worth =$ — 1Q35Q ^ -$912.965. . . . (5) $103 : $463.50 :: $100 : present worth ; ., ^463.50x100 ^.^^ therefore present worth=$ irrr^ =$450. (6) £103H : £390:: £100 : present worth ; X, „390x 100x24 ^„„^.^ ^,, therefore present worth =£ ^^7777 =£375 15a. O^d. 2491 (7) $104.50 : $856.96 :: $100 : present worth ; .-, r ^ n <»856.96xl00 ^00^ therefore present worth = $ — 7777-^ — = $824. 104.5 (8) $101 : $1252.40:: $100 : present worth; therefore present worth =$^?5?^— =$1240. (9) $101.75 : 1250 :: $100 : present worth ; ., ^1250x100 ^.aoann therefore present worth=$— ^77-^-^-^- =$1228.50. 101. It/ DISCOUNT. 201 (10) £105i : £2110:: £100 : present worth; tlierefore present worfh.-£^llg l\^-^_ je^qoq (H) iill i:i05 : £37oJ :: £100 : present worth ; therefore present worth =£??-^J^l|0^£3C2 4.. 5id. ^. (12) £ 120 :£D 18:: £100: present worth; therefore pi-escnt worth =£?^^^1^^^^^,65, (13) 120 $108^^ • $500:: $100 : present worth; therefore present worth -$''^^^ ^.,!?^>~ -^$462 47 (14) 10379 £112 0.. 8-T« : £2197:: £100 : present worth; ,527280 X 100 ' therefore present Avorlh^£' 20990-736 ^^527280x100x1000 15G25 20999730 =£— g— ="£1953 2«. Bd. (15) ^ $i02:$64::$2: discount; therefore discounts $?*il?= $1.35.^, (16) $100:. $1380:- $0: discount^ therefore discount=$l^^=$78.ii^j. (17) Ii02i:$107i::$2i: discount; therefore discount^^^^il^^^g.ei.^^ (18) 205 X 4 X 2' 1102. $125.40:: $3: discount; therefore discount=$?^-?:f?.^_2 ^^^ ja *>02 KEY TO ADVANCED ARITHMETIC. (19) $102 H» : $487::$2H : discount; ^487x35 riK^ooA4o therefore cliscoant== $— -i;;^-= ^Id.SOijVV. 1285 (20) $102.50 : $340 : : $2.50 : discount ; 340 X 2 5 therefore discount^ $-^^^j^-^- = $8.29H. (21) £104 : £3G40 : : £4 : discount ; 3640 X 4 therefore discount=£'— t^t:; — =£140. 104 (22) £106^ : £8l32''o::£Gi : discount; , ^10269x19x3 p969 „.^f,^ therefore discount-£-----g-^g-=£^=£48 9«. (23) $11U •• $250|::$1H ■ discount; J003x34 ^rtp-KOQi tliereforc discount =$-j^-gT- =ij,45.0^r6V- (24) $102 : $102:: $2 : discount; ^102x2 ^^ therefore discount=$— 7;:7-=$2. 102 (25) £100 : £ai0::£lfA'u- : interest on £640; tlK'-cfore intcrest:-£^|-^^-^=£9 17.9. 7VHf*. 100 X 14b0 Affain, £101 ,^/^ : £649:: £l-,V6'd : discount; ^649 X 2223 x 1460 oniA. q tiiii ^ therefore discount=£-^^^-^^3^^-£9 Us. Qjmhd. therefore banker's gain =2.9. ll(f., nearly. A1 AA (26) : $100:: §21- tlicrerefore discount Mint = ^ 'f ^00 '3"6892 :$1.06^5t^. + STOCKS. ^100: |o45::.^14: interest; ihereibro iiiterest=$'-^~~ = $70.30. Again, ^114 : $021.30:; $14 : discount; tiierefore discount = $-^L---^_— |7G.30. 203 114 (38) ■^ of t!ie sum is tlie price of one volume at the end of a year, and Jj of the sum is the present price; therefore ^— ^=.^- of given sum is the allowance mndi' per volume ; therefore allowance on 5 volumes=a^r=6 of the sum; therefore 1 : 100 :: ^ : rate of discount ; therefore rate of discounts J-g^^iGj per cent. (39) £105 : £100 :: £2-^- : cash price ; therefore cash price=£|^^i?r^£i=£2 6s. 8d. (8) STOCKS. Ex. LV. (p. 207.) pM^ : $2353 :: $100 : stock required ; therefore stock required =:$?H2-!ii?.5^_|o(;oo. lol (9) i'iOC : £3277 :: £100 : stock requirocl ; therefore slock required=.£':il!J^^12?:^£3091 10s. 2l^rf. (14) £ 100 : £1000 :: £97J- : value required ; therefore value required =£^^'1^^^':^= $972 10.. (15) £!00 : £215;V ::£18S : vulun rc(.i,i,.p(^. Ihcrefore value requircd=£-- 4307x168 2x100 :£4018 It.v. 7^/. 204 KEY TO ADVANCED ABITIIMETIC. I I 5 (22) jP94i : £3500:: £3 : income reqiiired ; i. . ^'3500 x3x4 r>^HA r, -.iKij therefore income =£ rrxu: =:£lll 8s. Ihtkd. 377 (2G) £3 : £87 :: £74 1 : sum required; 87 X 597 therefore sum required— £ , —£2164 25. Qd. 3x8 (27) £4 : £37^ :: £93^ : sum required ; 75 X 747 therefore sum required =£ ^^ ^ =£875 75. Ofd. (30) fi)l)i : $ 4x2x8 interest required ; therefore interest=$^^^^—=$8.27^H. _:_" 773 (31) $103 : $100 :: $7 : interest required ; therefore interest=$ -— — -^=$G.79-l^oV lUo (32) ^U05 : $7927i:: $7.88 : net income; ^'79^^7*'5 X 7-88 -.^^, _ . thcreiore net mcome=$ -7- =-$594.94. 105 (33) .•fOO : $100 :: $7 : rate per cent. ; therefore rate per cent. = $7.77^; jj;80 : $100 :: $7 : rate per cent. ; therefore rate per cent.=$8.75 ; th eref ore ad vantage= $8.75 - $7.77^ = .971. (34) ^S : $7:: : price of stock ; 100x7 therefore price of stock— $ — - — =$871. 8 $87i : $1200:: $100 : quantity of stock; .. ^ . , ^1200x100x2 ^.n^ii therefore quantity of stock=$ :j-^ = $1371?. 175 (35) $i(53 : $157 :: $5500 : amount required ; therefore amount^ $^-^^!-— = $5207.54-,^;'^^. STOCKS. 205 (36) 1100: $4000:: $8 .-whole gain; therefore whole gain=$i?5?;^=|320. t37) 100 1163 : $9000 :: $6 : loss required ; therefore loss =:$?L^2A?^ $331 ggx^, (38) £3 : £3^ :: £89^ : price required ; therefore price =£l-^^^£ 104 8.. 4d^ (39) $92 : $400 :: $3^ : half-yearly income; therefore incomer=$15?4_7^|i5 g^ix (40) 2x92 $100 : $5000 :: $6 : 1st income ; therefore 1st income=$300- $100 : $5000 :: $160 : value of stock : ' therefore value of stock =$55?5_^l^^|800O, and $107: $8000:: $4: income; 100 therefore 2d mcomo=$^-^^=p^QQ^y. theiefore difference in his income = $300 -$299-iJy=$j-^^. (41) $100 : $5000 :: $4 : income from $5000 stock ; therefore 1st income =$5^5^^^= $200. <$106 : $5000:: $3 : income from investment; therefore 2d iucome=$'^?^-$i4i,5ofi.a 100 (42) $100 : $7000:: $4 : income; therefore income=$280; $100 : $7000:: $105 : value of stock; therefore valueL$7350 (43) £98i : £981 ::£3000 •. quantity of 3{. per cent, stock; therefore quantity of stock=£illL"_z!i[?^£4_^^ , . ;J9;Jx8 — -'^"^J^T, (Continued on next page.) 1 806 KEY TO ADVANCED ARITHMETIC. (43 contlniicfl.) £100 : £2729t^t :: 3 J : income from 8. J per cents. ; therefore 2a income=£ i'5:^?!;^>il^=£95 10.. S;'M. and 1st i.icome=i;51???-iJ?=£90 X\)\J therefore alteration in his incomes £ 5 10 3,«/,- (44) In the C. B. of C. stock at 101, $1 gives $, Jy interest, In the Q. B. stock at lOG, $1 gives $^, or $^}j ; and since tJt is greater than j^j, the former is the better investment. (45) (6 X 2) cts. = 12 cts.rrincome tax on $100 stock ; therefore net income on $100 stock=r$G.OO-. 12=15.88- therefore $5.88 : $000 :: $104 : sum required ; therefore sum ix^quircd^ $55.^-^^11=: $10G12.24U. ^4C) £83 : £1037i - £100 : quantity of 3 per cent, stock ; therefore quantity of 3 per cent, stock =£^~^^J£0_£ior;fl. 83x2 "^^-"'"t £96 : £84:: £1250 : quanfUy of 4 per cent, stock; therefore quantity of 4 per cent. stock=£^-^=£1093J • 9G * ' therefore £100 : £1250 :: £3 : 1st income, or 1st income=£37 10«., £100 : £10933 •*: £4 : 2d income, or 2d inc()nic=£43 15«. J therefore dilferencc of income=£43 156\ -£37 10.v.=£G 5s. (47) £1031 : £100 :: £1054 : present worth ; therefore present worth r=£—--^^^'^^ ^ ^=£1GOO- 827 ' therefore £9G : £1G00 :: £100 : quantity of stock required; thcreforcquantityofstock=£--]ii— =£1066 Ids. id. APPLICATIONS OF THE TERM PER CENT. 207 (48) In 13 years the dividend on £100 stock, at 3 per cent was £(3x13) .-.£39; .-. £:]!) : £3081 :: £100 : amount, of stock ; .*. stock=:£7900, and £100 : £7900 :: £791 : value of t^tock ; .-. value=£G310 2s. Gd. (49) £79^ : £1911 :: 100 : stock bouirht; therefore stock bough t-:r £3400, and stock must be sold for£1911 + £l;-)0 = £20()l; therefore £3400 : £100:: £2001 : price of stock; thcretbre pricc = £85J; therefore to pay the brokerage the price must be £(855--f-i), or£8G. (50) Income from South-Sea annuities =£300, £100 : £1*10::£10000 : amount of 3^ stock; therefore amount of 2} stocks:: £11000; therefore income from 3| stock =£(110x30:= £375; therefore loss of income from accepting this stock=:£25. Again, £93 : £10000:: £3 : income from investment in consols;' therefore income from consols=£322 lU. 7;', [(?. ; therefore gain by investing in consols=£32 11«. 7iH (51) £100: £4000000 ::£i : saving; tlierefore saving =£30000, £100 : £4000000 :: £of : loss of fundholders ; therefore loss=£ 4000000 X 4.' , — =£225000. 100x8 APPLICATIONS OF THE TERM PER CENT. Ex. LVI. (p. 21G.) (1) 100 : i :: 56394 : percentage required ; .-. percentage rcquired=?5^^=?^iii .^.isr-gs, 100 : fr: 56394 : percentage required; /•(I'lOi V A OQ107 • percentage required = il^47^:='~^- = 352 •402.-) 100 bO 208 KEY TO ADVANCED ARITHMETIC. fi ! (3) 96 : 100 :: 15 : number required ; 100 X 15 l*") .*. number required = 1— 1=_:::1— 15-G25 81 : 100:: 19 : number required ; , ., 100x19 1900 ^„,, „„ ,_ number requircd= — ^- — =-^;:^=23^,} =23-456. . » . 81 81 (7) If 27 per cent, leaked away, 73 per cent, remained in the cask; .'. 100 : 2005:: 73 : number of gallons remaining; gallons remaining: (8) 200 > X 73 140305 100 100 : 1463-65. 100 : 7500 :: 112i : bushels required ; /. bushels requu-ed=Zi>50^??2=15?L3=8437-5. 100 X 2 2 (9) I * By selling 15-75 oz. he gains -25 oz, ; .'. 1575 : '25 :: 100 : gain per cent. ; .-. gain per cenl=M^^:^J^-m-^ 15-75 ~1575~03--^G3* (14) 14804 : 100:: 1588 : rate per cent, of mortality in Small-pox; .-. rate required=500^588^39700^ ^ 14804 3701 ••• 2422 : 100:: 211 : rate per cent, of mortality in Scarlet Fever; rate required = 100x211 10550 2422 1211 =8-71. 17226 : 100 :: 1799 : rate per cent, of mortality in both sicknesses- .-. rate rcq„irca=l«iiJ^=?2?55-io.44 17226 8013 (15) Since 8175124-7707401-407723, and 8175124-0515794=1659330, and 7767401-6515794=1251607, 767401 : 407723:: 100 : increase per cent, in the 1st ten years; .•. increase per cent. =^5ZZ??^J^- 5-24 '^767401 VVi (Continued on next page.) ' APPLICATIONS OF THE TERM PER CENT. J 209 (15 continued.) 8175124 : 1659330 :: 100 : decrease per cent, in the last ten years ; , , 1009330x100 __ .'. decrease per cent.=^ — ^tti^ft^a — =20-29 8175124 7767401 : 1251007:: 100 : decrease per cent, in the twenty years; , , 1251007x100 ,,,, .'. decrease per cent.=: — ^^^.^^ - — = 10-11 (16) 7707401 T>^ 1 «• * A i>^^ 10000.00 xlOU ,^,^„„« Population at end of first year= -— ^=1015000, Population at end of second ycar=——^i^^= 1030225, in^n'^'>f» V 101-1 Population at end of third year=:^^:i:^ ^ ^ =1045678-375. (17) Of the age of 18 there are 3 scholars. Between 15 and 18 there are ^ ^^„^^=19, t( 12 10 15 12 (( (( 100 10 x 380 100 35 X 380 =38, 100 =133, and under 10 there will be 383— (3 + 19+38 + 133) =383-193=190. (21) Since 1 ton=(20x 100) lbs. =2000 lbs. 100 : 49-85G :: 2240 : number of lbs. of oxygen ; .-. numbei of lbs. oxygen=^?^?^?522. ==997.13. 100 : 432G5 :: 2240 : number of pounds of carbon ; , „ , 43-205x2000 ^^^ ^ .-. number lbs. carbon = -— =8G5-3, and number of lbs. hydrogen =2000 -(997-12 +8G5-3)=137-58. (33) Cost of w]ieat=$(13G00 x 1.05)=|14280. 1 * u 1 1 . ■> 13000x21 „,^ number of bushels wasted= ^=340: 100 (Continuixl on next page.) 210 KEY TO ADVANCED AllITIIMETIC. '? (23 continnod.) number of biislicLs left to sell = i;jr)U()-340= 13260; of which ho sells p;irt f(jr $-^^^—=^425.00. f 1-1 1 n *r ^20x13300x1.05 ^^^„,^^ of which he sells part for $ =$2784.00. of which he sells the rest for $llili2:^'-"_:ri=$3978. .-. he sells his wheat for $(7425.00+2784.00+3978) =$14188.20; .'. he loses $(li280-14188.20)=$91.80. (24) If M and 7^ represent the number of males and females respect- ively, 101-8x(.lf+7^) ' 100 95-4 X M — =increased number of males and females, 100 100-8 X F =decreased namber of males, 100 = increased number of females, .p, 95-4x3/ 109-8 xF 1018 x (.If+i^^ Then — — 1 r= ■: 100 ^ 100 100 .-. 95-4 X 3/+ 100-8 x F=101-8 x3/+101-8x ii^; .-. (109-8-101-8) X F=(101-8-95-4) x M; .-. 8 i^=G-4 X J/, or 80 F=U 3f, or 5 F=4: M, and.-. M : F::5:4. (35) £05 ; £100:: 56'. : cost price; , . 100x5 . o-, 7 .-. cost price =—— — s.=i)S. 3-,^cZ. Again, 100 : 104.^:: 5.*?. 3-|\ff. : required price; . , 209x1200 .•. price requirea= 100x2 xlJ d.=GQd.=:5s. Qd. (20) Expense of sale=-,Li of $1.G0=.08; .-. whole outliy=$l.G0 + .03=$1.68, and $100 : $125 :: $1.03 : selling price; selling pricc=$ 125x1.08 = $2.10. APPLICATIONS OF THE TERM PER CENT. 21] (27) £92 : £100 ::£25i: cost price; .'. cost price =£l^-l^i_£07a3 .=192s. x no. of qrs. ; 288x20 „„ .-. no. of qrs.=-^T7:^-=i3U. 19!^ (36) By <.lie sale tlie person receives ^-^ of cost of watcb. If he had received 101 more he would had ^^ ofcost of watch ; 100 95 cost of watch =:;-^^ cost of watch +$3, 100 1^^ 6 or — cost of watch:=$3 ; .'. watch costs $50. 100 Again, since the duty was 25 per cent., 125 • 100 :: $50 : price received by French maker ; ^100 X 50 ^.f. .-. maker receives $ ^,-^.~ or ^w. EQUATIONS OF PAYMENTS. Ex. LVII. (p. 330.) CT) $^4^=$604, the amount to be paid in money, ^1512 _ ^^53^ ^^^ amount to be paid in wheat, $1812-|1057=$755 to be paid in barley. Now $1.51 : $1.75 :: $004 : amount to be paid m wlieat, 453 X 175 ^f,-^^ .-. value of wlieat=$— ^^---i"^-'- Also, 75i cts. : 85 cts. :: $453 : amount to bs 1 ^755x85 ^.-,. . .-. value of barley=$-;:n^--=l>5'^'- laid in barley wUolc rent= $604+ $525 + $850= $1979. EQUATIONS OP PAYMENIS. «. 310 \ 20 . bince — - — s. IS paid in each kind of corn, 810x20 "3x"5ir~°"'^^^^ of quarters of wheat, 810x20 "sTi^"""'"^^^ ^^ quarters of barley, 810x20 Tx22~~"""^^^^ of quarters of oats, .*. rent in second c;ise= ^/'3102<20_xW 310x^q^<^ 31^ V 'Sxoli "^ yx32 + 3x"22 — j** 213 _810x_20/G4 44 24\ a V5G + ij2"^22;''- _310x20/8 11 12\ 3 V7'^8+il/- 6200 2223 3 '^7x8x11' =7458-727-5. =£372 185. lid. nq. (5) Ex. LVIII. (p. 266.) . . weight of gold=12 30A, .*. weight of silvers 3 30-,^,-, . weightofcopper= 2 50\\, Sum of parts=3+4+ 15=22; therefore cwt. 22: 15::18: weight of gold; 22 : 4 : : 18 : weight of silver ; 22 : 3 :: 18 : weight of copper ; (8) Number of sharcs=3+G + 3=12 ; .-. 12 : 3 :: £13000 : eldest son's share ; •■• eldest son's share=£3250 0«. Od. 12 : 2 :: £13000 : a younger son's share; 'V-, .< ■ -~ J .s- ' '-'••" - aiiaic — ^oi^jLUO 108, 4NGE. Ex. LIX. (p. 235.) (1) Value of thalerrz $-727; n-jO 707 OA-j .-. sum required=$-^ x ^ x |^=$447'14863. (2) 1 mil. : 4750-280 mil. :: G4J^. : sum required; .-. sum required^-— ?^?51^.=r£127l 135. d^%d. by enactment £0 st. = $40: £1271 13s. 9 A-od = ^^¥u°o^^^^. ; sum: 30520549 100" ^^^* !o-^^2-iO^^^^^^'li- (3) 47K : £346 15.s. M.::! pias. : vnisbiji of piastres required ; 1 "1,91 5-'' ^ o .*. number pias. \)5 :1346 pias., G^f rials. EXCHANGE. 215 (4) 809;2x258__-„. ^ ^^0^ -^3190.']0=no. grs. pure gold in |10 (eagle.) TVx2lil-99;30=3.319936=^grs. pure gold in $1 1869 S0V8. arc coined Ironi 40 lbs. Troy. Btaudard gold which is .-. weight of 8ov.= 123-27447, and weight of pure gold in 8ov. = 11300159; F 6 1" -•. 2319930 : 11300159:: $1 : par of exchange; par of e.\' cli-inge = $4.87. (•5) 412'5x-,2o=,i7(-25=numbcr grs. pure silver in |1. 480=number grs. in 1 oz. Trov 37x480_..^ 4-^—- 444=:no. grs. pure silver in 1 oz. Troy of standard. 371-25 : 444::$1 : $1.19*^ rvalue of 5s. Ud - .-. 5/?. nd. : 20.v:: ^Vl^ .. par of exchange; .'. par of exchange=$4-67|g|3J. (0) ^^1^/'='^^-^=''''^^^^ grs- pure silver in $i, and (aa above) 444 grs. are worth pure silver in 5.9 Ig^ ^ 172-8 : 444 :: |^ : .|1.08=,valiie of 5.9. l^d. 58. lid. : 20s. :: {^1.28 ; par of exchange; .-.par of exchanges $5.02. Avl (7) 36«. 3^. : 9k/. :: £1 : value of ruble via Amsterdam ; .-. value of riible=?L'i?f?^ -50-fi-rf And in the direct way the ruble costs 50d • .'. the direct way is the cheaper. ' ' (B) ^5'U fr. : .500 fr. :: £1 ; value required ; ••• value rGquired.=£^=:£i9' 10.. 7id. - Again, 1 ree=i^ francs; .-. 400 rees=| francs; .-. Si,. Flemisb=| francs; .-. U Flemish=| francs; ••• £l-35.v. Flemish=?l^i^^ francs=25 francs. 216 KEY TO ADVANCED ARITHMETIC. (0) On 1st supposition 1 franc=£— ; .-. 1 ruble=£— . 23 23* On 2d supposition 1 franc=£l; .-. 1 rub]e=£i. .-. broker gains £(^— __^ on each ruble. .'. he gains £11 5s. (10) Income in England £90= $438 at par. £3000 stock @ 97=£2910 sterling, £2910=$^'^ X \U X 2910=$13968. Income in Canada=$|aa ^ 139G8 x 7^7j=$918.08H. .-. differeuce=$918-08f^-$438=$480-08n. SQUARE ROOT. Ex. LX. (p. 241.) (5) 104 1083 10862 ^ • • • • 29506624 ( 5432 25 450 416 5345344(2312 4 ^■^ 3466 3249 (- 21724 21724 43 461 4622 • • • • 134 129 553 461 (6) 14356521 ( 3789 9 9244 9244 236144689 ( 15367 67 748 7569 535 469 6665 5984 68121 68121 25 136 125 303 1114 909 i)066 20546 18396 30727 215089 215089 ."^ SQUAKE KOOT. 217 282429536481 ( 531441 25 103 I 324 309 106.1 10624 106284 1062881 1529 1061 46853 42496 435764 425136 1062881 1062881 26 328 33607 282475249 ( 16807 182 . 156 2647 2624 235249 235249 4160580062506 ( 2039750 ^ 403 I 1605 1209 4069 40787 407945 39680 36621 305906 285509 2039725 2039725 00 (8) 167-9616 ( 12-96 22 67 44 249 2396 2241 2586 15516 15516 28-8369 ( 5-37 25 450 416 (7) 2956d6240606 ( 543200 104 1083 10882 2172' 0000 ?03 I 383 309 1067 7469 7469 3466 3249 5764801 ( 240-1 4 21724 21724 44 I 176 176 4801 4801 4801 U '^ KEY TO ADVANCED AKITHMETIC. 218 (9) -3486784401 ( 59049 109 -• • • • 25 •00203401 ( -0451 16 11804 118089 986 981 57844 47216 85 901 1062801 1062801 434 435 901 901 39-15380329 ( 6-2573 36 122 315 344 1345 7138 6225 12507 91303 87549 25143 375429 375429 (11) 44 4803 5774409 ( 3403 4 177 176 14409 14409 5.774409 ( 2-403 4 44 4803 177 176 14409 14409 (10) •042849 ( -207 4 (12) 12088808379b25 ( 347'6905 9 407 2849 2849 •00139876 ( -0374 9 67 744 498 409 2976 2976 64 687 6946 69539 09."i3805 308 256 5288 4809 47968 41076 629237 625761 34769025 347000-5 SQUARE liOOT. 219 240398012416 ( 490-304 (14) 23.3-66o6o6o6 ( 15 16 89 I 803 801 (15-3492... 25 I 135 125 9803 98004 (13)16(4 16 29801 29409 3922416 3922416 303 8064 30689 306982 1060 909 15100 12256 i66000O0O( 1-2649.. 22 ! 60 44 240 1600 1476 ^524 12400 10096 25289 230400 227601 2799 ■^G{'4: •0i66060()(-1264... OO 240 60 44 1600 1470 12400 10096 23^ 284400 276001 839900 613964 225936 ■l60()0d0()(-3162... 9 61 100 61 626 3900 3756 6322 14400 12644 1756 •Oi ( .1 •50000000 ( -7071 49 ^ •• 1407 10000 9849 14141 1 15100 14141 959 230 KEY TO ADVANCED ARITHMETIC. 506060606 ( 2-2360. 4 42 100 84 443 1600 1329 4460 27100 26790 4472 31400 (15) -0004 ( -02 4 379-86400000 (19-4901... 29 279 261 384 1886 1536 3889 35040 35001 389801 390000 389801 199 •00081000 ( -0284. 4 48 564 410 384 2600 2256 344 (16) vm=^^lJ-=u. Vl53l^gi|j^37-2021... =12-4007. . . . Vi=:NJ-333=-5773.... kit 87 2209 ( 47 16 609 009 189 980i ( 99 81 1701 1701 [2209 47 •''N 9801 "99* E 1 (17) — - N5-\j /3x5\ \5x5/ Vl5 3-8729... 5 5 =•7745... N17 -\jV 17x17/ - Vl7 412310... 17 V2i= i'2-5 =1-5811 \K. N4J- i/n n \9 17 ■=-8819, •2425.. (18) CUBE ROOT. 221 '5 031 •04 1 5040 , i3T=^\|~2T"= '^^^Orris^giQ. . . t:^_ (225" 25 ^^ 88 9606 96132 23-10060606( 4-8062... 16 710 704 GOOOO 57636 236400 192244 44156 124 4200060606 (6-4807.. 36 600 496* 1288 I 10400 10304 129607 960000 907249 I • 52751 CUBE ROOT. Ex. LXI. (p. 348.) (5) 25i23959i ( 631 3x62=108 3 X (60)2=. 10800 3x60x3= 540 32= 9 11349 3 34047 3 X (63)2=11907 3 X (630)2=1190700 3x630x1= 1890 1^= _1 1192591 35239 34047 1193591 1192591 1 •f 2'i2 KEY TO ADVANCED ARITHMETIC. I^ i ! 3 X 3»= 27 3 X (80)2=2700 3 X (300)2=270000 [3 X 300 X 5= 4500 52= 25 274525 5 1372625 (6) 3x2^: 3 X (20)2=1200 3x20x5= 300 52= 25 1525 5 7625 :12 3 X (8)2=27 3 X (30)2=2700 3x30x6= 540 62= 36 28372625 ( 305 ^^^6 27 6 19656 1372625 3 X (36)2=3888' 3 X (360)2=388800 3 X 360 X 4= 4320 42= 16 393136 4 1372625 1572544 17173512 ( 258 8 48228544(364 27 21228 19656 1572544 1572544 9173 3 X (25)2=1875 3 X (250)2=187500 3 X 250 X 8= 6000 82= 64 7625 1548512 103564 8 1548512 1548512 1548512 3 X 62=108 3 X (60)2=10800 3x60x3= 540 32= 9 11349 3 34047 259694072 ( 638 216 43694 3 X (63)2=11907 3 X (630)2=1190700 3 X 630 X 8= 15120 82= 64 34047 9647072 1205884 8 9647072 9647073 9647072 CUBE ROOT. 223 3x9»=843 3 X (90)2=24800 3x90x7= 1890 7^= 49 36239 7 183673 (7) 926859375 ( 975 729 197859 3 X (97)2=28227 3 X (970)2=2822700 3x970x5= 14550 25 52= 183673 2837275 14186375 5 14186375 14186375 14186375 27054036008 ( 3003 27 3x32= 27 3 X (30)2= 27QQ 3 X (300)2=270000 3 X (3000)2=27000000 3 X (3000) X 2= 18000 22=_ 4 27018004 2 54036008 54036008 54036008 219365327791 ( 6031 216 3x62= 108 o ,...o^^(W= 10800 3 X (600)2=1080000 3 X 600 X 3= 5400 9 32= 1085409 3 3256227 3365327 3256227 3 X (603)2=1090827 3 X (6030)2=109082700 3x6030x1= 18090 1'=-- _1 109100791 109100791 I 109100791 . 224 KEY TO ADVANCED AEITHMEnC. (8) •389017 ( -73 343 *, 3x72=147 4 X (70)2= 14700 3x70x3= 630 32= 9 15339 3 46017 46017 3x32=27 3 X (30)2=2700 3x30x1= 90 40017 ^'- 1 ^791 32-461759 ( 3"19 37 5461 2791 3 X (31)2=2883 3 X (310)2=288300 3x310x9= 8370 92= 81 296751 9 2670759 2670759 2670759 1 » 95443-993 ( 45'7 64 3x42=48 ' 3 X (40)2=4800 3x40x5= 600 52= 25 31443 5425 5 27125 27125 3 X (45)2=6075' 3x(450y^=607500 8 X 450 X 7= 9450 72= 49 4318993 616999 7 4318993 4318993 CUBE BOOT. •00i906624 ( -124 1 3x12=3 3 X (10)2=300 3x10x2= 60 22=_4 364 __2 728 906 728 178624 3 X (12)2=432 3 X (120)2=43200 3x120x4= 1440 42= 16 44656 4 178624 225 178624 178624 3x92=243 3 X (90)2=24300 3x90x7= 1890 72= 49 26239 7 183673 (9) •000912673 ( '097 729 183673 3x22=12 3 X (20)2=1200 3x20x9= 540 92= 81 •000024389 ( '029 8 183673 1821 9 16889 3006000006 (1-442... 16389 16389 3x12=3 3 x (10)2=300 ^x 10x4=120 42=J6 436 J1744 3 X (14)2=588 3 X (140)2=58800 3x140x4= 1680 42=__16 60496 4 241984 2000 1744 256000 3 X (144)2=62208 3 X (1440)2=0220800 3x1440x2= 8640 22= 4 14016000 241984 1401G000 0:>29444 2 12458888 12458888 1557112 220 KliY TO ADVANCED AHITIIMETIC- v?0()OOUUUO(OG«... 2i0 3x(60)3=10H0() :jxOOxG= 1080 0*= m _ 7141)0 840U0 3x(C0)»=130C8 3 X (000)*= 1300800 3x000x0^- 178->0 9'^== 81 7149G ll\504000 1324701 9 1192;i309 135040^0 11932309 581091 •030000000 ( -310. . . 27 3x3'=27 3 X (30)'-' -2700 3 X 30 X 1 = 90 !■'= 1 2791 3 X (31)2=2883 3000 3791 209000 (10) 3 WS\ 2 \|\27y~3' \\i}m)~ n|\343/~7' 44 000000000 ( 3-540. . . 27 3x32=37 3 X (30)2=2700 3x30x5= 450 52= 25 3175 5 15s;5 17000 3 X (35)2=3075 3 X (350)2=307500 3x350x4= 4200 42= 10 15S75 1725000 371710 4 1480804 3 X (354)2=370048 8 X (3540)2 = 37094800 3 X 3540 X (>= 03720 02 = m 37758550 6 220551330 1735000 1480804 23813000 220551836 11584004 CUBE ROOT. (11) \im,A)=^^^(^^^,,, 227 Since 50653 ( 37 27 3x3'=r27 3x(30)*^=27()0 8x30x7= 030 7'^= 49 8379 7 23053 23053 23653 V(7i)=V(7-2); unci 7-20{)O0000d ( 1 930. 1 3x12=3 3 X (10)2=300 3x10x9=270 92=_81 051 9 6b59 0200 5859 341000 3 X (19)2=1083 3x(190)2=10>s;}()0 3x190x3= 1710 32=_ 9 11 do] 9 3 330057 3 X (193) -'=11747 341000 3;i0057 10943000 3004150000 ( 1-442. 3 X 12=3 3 X (10)2=300 3x10x4=120 4'=J10 430 4 1744 2004 1744 3 X (14)2=588 I 2001.50 3 x (1 44)2=62208 3 X (140)2=58800 | 3 x (1440)2=(5220800 3x140x4= 1080 42= TUI 00496 4 241984 3x1440x2= 22= 8040 4 241984 18106000 0229444 2 12458888 18166000 12458888 5707112 228 KEY TO ADVANCED ARITUilETIC. I (12) 000100000 ( 64 •046.... 3x42=48 3 X (40)2=4800 x40x6= 720 62= 36 36000 5556 6 33336 33336 2664 1257-728 157-216 19052 4913 ^^°^^ 16384 ~ 2048 ~ 256 ~ 64 3 /1257-728\ 3 /4 913\ 17 nJv. 16384 )-'\\ 64 )' 4 " ^" (18) 233-744896 ( 616 210 3x62=108 3 X (60)2=10800 3x60x1= 180 12= 1 17744 10981 10981 3x(61)2r.lll63 3 X (610)2=1110300 3 X 610 X 6= 10980 62= 36 6763896 1127316 6 6763896 6763896 1^ V(233-744896 x -008) =616 x -2=1-232. £10481 Is. 4d. (14) Number of cubic inclies m mass= — YOsTM. — 2515456 =20123-648. (Continued on next page.) *iu_ CUBE ROOT. S39 (14 continued.) 26l2:V64H ( 372 iD.chc3=cdgc of cube 8 3x2^=12 3x(20)»=-1200 3x20x7= 420 7^^= 49 1669 7 11683 8x(27)«=2187 3 X (270)2=218700 8x270x2= 1620 2'^= 4 220324 2 440648 12123 11683 440648 ) 440648 (15) 3 X 3^=27 3 X (30)2=2700 3x30x7= 630 V= 49 3379 7 50653 ( 37 27 23t,d3 28653 .-. area=(37x37) sq. ft.=1369 sq. ft. (16) 56 cub. ft, 568 cub. iu. =97336 cub. iu. 97336 ( 46 mches=3 ft., 10 iu. 64 3 x 42=48 3 X (40)2=4800 3x40x6= 720 62= 36 5556 6 33336 33336 33336 230 (8) KEY TO ADVANCED AKITHMETIC. SCALES OF NOTATION. Ex. LXII. (p. 251.) 2064 312 4150 2064 6234 650410 9294 344 30d4 30fl4 23340 2704054 57304 ; 675 468 701 468 2r310 85r ; 734 354001 513354 434070 3117 2368 5484 2r3568 51117344 6rl3 814 574007 e7r8; 2rT9 23448 6rl3 46894 88f00 086r8 986r8 1 f'AQA 475r968 29r9G580 (4) 0541 ) 1438231 ( 1456 6541 44012 36134 64551 45665 55530 55536 2rT9 ) 29r96580 ( e7r8 27fr3 lrf35 18433 27028 250^6 20320 20320 ■ H 13)201003(10233 13 110 32 120 111 83 82 4331 ) 24510503 ( 3403 21433 30335 30204 13103 13103 SCALES UF NOTATION. 2IU (r>) 45 5404 12041424 ( 2504 4 404 401 34434 34434 (0) 7 1838 7 361—1 37—3 .-. 1838 5 2 = (5331);, (7) G3re 13 74 12 g !98 12 10787 32^75731 {C)2Te 30 103 104r 1058f 3f7 304 ^357 r404 f5331 t5331 47010370 ( 40 135 761 687 1U4 0303 5757 11183 43570 43570 13 13 13 13 80108 0083 550 40 11 11 1000 90— r 8—2 1000-(82r),i 3-r .-. 80198=(3r4f3),2, 13 13 13 34533 1534—3 54—4 .-. (34533)o=(3r43),2 .1 •■»(;> KEY TO AUVA.NClCn AUITIIM laii.'. 7 1 054321 (8) 12 23784 7 7 7 ( OioS— 5 UW3f— 3 12 [ 1730—4 12 i:;2— 9 2858—3 12 7 7 10—2 4321 313—5 21—0 478—0 U—'3 0—9 (23784)<, = (9294),« 1—4 (4321)8 = (1405)t 11—4 11 11 1— G 2304 104— r 2—7 C2304)«::r(27r),i 12 12 12 3250 100—4 11— 1 0—8 .'. (3250)t = (814),2 (054321) i«= =(10430335)7 (9) 12 12 8978 810-2 12 75-1 C— r .-. (8978).,=(0ri2).« on 2 814 23448 6rl2 40894 475r908 Ex. "^ XIIL (p. 255.) 25 X . . ..1x0 /25\ 25x0 ,, , , 1 x() .. •.(TiS), ,, = (1141)6: /ir,\ 3,Txl3 „, „ , 1 xl3 .7),„ = c:),.,(5^)^^=-„,,-=8k=8+-^3-=8+4i .-. (7f,E1,„ = (7-8.|),,: (37),„ = (101)«,(,','i) =ll;,^^-ll|=3+'^=3+3 + i&c. (37if^,o = (10l332)e: '• !'>1 513—5 21—0 1—4 G5)t o -1 -r 12) 1 i ■' E» SCALKS OK NOTATION. 233 (a7),o=(:5i)i2, a^) I (I 10x13 ■■'"27 =::3!)=3f 1_xt2 =r3-|-lxi &c. (37i?),o-:(31-314) 11 I 2 (1)40) ,,. = (4201),, (n),/- 77-=-'^'^-3-h 15x_0 It" =3 4r),^ &c.; (010iUio--=('1204-3M ..), (040), ,, = (004) 1SJ> (',') = 1 1 X 13 17 :7i?-=7 + 13x13 17 ■.7+{)^ &c (040 11),,.^-- (004-703...) I »• /135X \173H/ 1 1) 135x0 1738' 2\) 1'> 135 X =0+:^^=rO+ir' =0+2^ 2H8 288 ()2 + v,=r:()238ifcc. 48 Vl738/.„ (•03334..), I II 135x12 J 25 1738 125x12 "144 =0rA==0r5 Vl738/,n V173M;,„ /^ox / CUBTC MEASURE. Ex. LXIV. (p. 264.) (1) Circ.=4ffft-x¥=4yas.,2ft 8in.: numhcr of rev. = 10i mis. -^4 yds., 2 ft., 8 m.=3690. (2) Arca=(^Jxf =176H sq. ft. = 19 sq. yds., 5 ft., 113V in. 997920 22 " (3) 15,1 mls.=997920 in. ; .-. diameter =-^2(Kr^T =2 yds., 2-1 b jn^ diameters J s'^x 4x1=124-096.... yds. ^^> 22 77 . _121 r.. (1) Circumferenceof circle made by horse=yx-g- ft.--^ n-, ioi 35 120 distance passed over in an liour=-|- x f x "y ^^• =24200 ft.=8000| yds., .,,.,„„_Qqoo-de • • difference=8800yds.-8066tyds •=733 yds., 1 ft. O No. of sq. yd3.=(12^-8^) x ¥ ft.=27 sq. yds., 8 it, 6H in. :.. r SQUARE A^^iJ CUBIC MEASUKE. 2137 (5) Circiin)rercncc=2(52 ft., in.) x ^^=330 ft.=110 yds. ; .-. c(ysl=110x 84 cts. — $l)2.4U: greater space =(14^- 7-') x V yds. =402 sq. yds. (G) Hypotlieniiso= y{2P+\^'^)-^:M] ft., in. : ]iypothenuse - v{4:-2o^-2-5o^)-S ch., 40 Iks. (7) Sideofsq.== Vi380=^7 229yds., radius= ^1386-^-^ -31 yds. ; .-. diifereuce=37-229. . . .yds.-21 yds. = 16-229. . . .yds. 7^ 5 (18) 7iin.=j|ft.=^ft.; .-. length=?x?ft.=4ft., 9i in. 1 o (19) (') Area of a pq.=:diag. squared -^2 =735 links X 735-: -3 =540225-^2=2701121 links =2 ac., 2 8045 roods. (3) If acres= 175000 links; required length in links=175000-T-500=3 ch., 50 Iks. (25) (0 Wt. in oz.=6 x 3 x | x 1000=&c. O Depth in inches=277-25 x 121G-^-2432=ll ft, Gi%\- in. (26) C) No. of limes=^ x ~ x ~?-r-2150=365-? 5 5 125 C^) Solid contents on floor =^ x -^ x ^ x 14 in. 7 2 2 =550875 inches ; .-. no. of bushels=55C875-?-2218-192=251*049. 01 01 oo e) IIeioht=2218-102-v-^x^xy in. = 7-087 in. (27) C) Area of a flag-stone=23-804 sq. ft : perimeter of court=2 x (137-31 + 21'9 f 125-79)=57() ft., area of cloister=57n x 12-45 sn, ft.. = 7171 "7 «n ft number of flag-Stones =71 71-2 pq. ft. ^23-804=300. i;i:y to advanced akitiimetio. Ir •:| C) Side of sqnun; =81-24 ft.; .-. perimeter of moat=2 x (81-24+00 + 81-24) ft. =504-1)0. cubic C(mtent8=r)04-9G x 7i ft. =8787-2 ; .-. number of gull<)ns="-Y-t -,^7^^=3155002. ^ 2/r2<4 (29) (') Area of tlie court =50 x 80 s(i. yds. =4000, yards uudcr ;xrn-s=(50— ())x(80— G)=8256, yards iii W!ilk«=4000 sq. yds.— 3200 sq. yds. =744 sq. yds. 3250 @, 8.<(. = €4S8 8*., 744 sq. yds. =0090 sq. ft., 0090 @ 20d=£558; .-. total cost=£1040 8«. («) ^x2l ^2=24c. feet, 4x2^xH=15c. ft.: 24-j-15=li 1^ X 50=80, DO. of books iu larger box, total number of books=50 + 80= 180, number left is 150 - 130=20. „ ,, „, 81x22. , 81x22^ ^ (31) C) 92 X 3^= — ;r— inches =sr-T-r; feet. 81 X 22 252 7x144 =31, ft. =3 ft., U in. 7x144' 81 (32) 113 X 108= J 2204, | x 12204=8130 ^5^.]0Ucb.yds.: 8130: 1:: 600 tons (000 tons =1344000 lbs.) lbs. in c. ft=~i;^=105im=165*. 81 00 (33) (') 2 ft., 7 in.=2A ft. = fi ft.=radius _31 31 22 55 1 _581405_ 9809 ^- y^^'— 13 "^ 12 "" 7 "^ 2 "" 27" 27210 "27210" 135108000 277274 (2) 2 ft., 8 in.=2i ft.=f ft. ^ , 8 8 22 7 1728 no.ofgals. = -x3X^x-x-^^-;^: =487^ nearly. (•') Number of cubic feet obtained by the surface sinking ^ , 8 8 22 1408 onefoot=-x^^x-=— , number of gallons=— ^— 00 X 62^=139-28 EX AMIX ATIOX QUESTIONS. 239 OG. nkiug EXAMIiNATION PAPERS. I. (p. 2G8.) (1 ^oi=.S5.2r), £1-85 =51^5.40, £? sl^£U c'y=^3•08^ sum=!{;(5-25 + 5-40-2-08|)=$8-5(J^ (2) Let 1 represent C's capticily for work p{!r hour; tlieu 1^ and 2 will represent B's and A's respectively. The comparative work of each will be as follows : 9x6x8x1 =:432==C's 2x6x7x1^=126 7x4x2x2=112 2x6x5x11= 90 2x6x10x2=240 4x6x3x1^=108 2x6x4x2= 48 lx6xllxH=_99 A's=400:' B's=423 : total mimber=400 +423+432=1255 ; .-. A's share will be -,^2^^- of $125.25=$40, B's= ^42.30, C's=|43.20. (3) Insurance by steamer on $780 @ 1^$^= $11.70, insurance by sailing vessel on $780 @ 45$^=$37.05: 11 13 13 1 freight by steamer =— x -q- x "T ^ In ^ $19.20= $37.56H, 11 13 13 1 freight by sailing vessel =-- x — x -j- ^ tk ^ $6=$11.61i ; therefore total cost by steamer is $49.26^f, by sailing vessel $48.66^, difference = 59 H cts. in favor of sailing vessel. (1) £1 4s. 4^^ currency =$4,871; .*. shilling st.=24| cts (5) 4 2 34 x+^=5r=part of filled portion of tank made by the two 7 o do pipes ; .'. time reqmred=l-s-|| lirs.=l-3\- hrs. (6) 30 X 120=3600 sq. ft.; no. of sq. ft. in an acre=43560 ; .-. Pnce=$~-^ X 100=$36300. 240 KKY TO ADVANCED AllITiniETir. s i (8) 15 gninoas \vcit,Mi 1020 £^rs. ; .-. there are ITOO & 100 v^vmus pure ^:ol(l and alloy respectively in l.> guineas, and 1700 jrrains pure g()ld=: — ^^^ grs. =382-40 -rains alloy; hence 15 guinea^ or 315.s.=(:i8240+100) grains alloy =38400=-!;^? lbs. avoir.=:V!? lbs.; 1 lb. av<)ir.=:— ^_--.9. = £2 17.s. oVgrZ. lU.' (^n) Quantity of pulp ni trough = 10-- x 3 x ^^^=-40" c. ft., 1 "^OT '^ 1 "^07 of which only -~^ x ~=-^,- c ft. available for making •^ 48 5 120 paper. 1 11 , 11 ., Area of one shcet=^^^^ x - sq. ^.=5^-713 sq. ft. ; , , 1397 11 ••• ^^"Stli of PaP^'^'=-48--57oirT2 =731.-)^ inches=l mile, 3 fur., 3 per., 1 yd., 3 ft., 8| in. (10) 150 lbs. @ 14 cts.=:$21.00 39 lbs. @ cts.=: 2.34 tot:d costr=23.34 12.V ^ on $2;) 34^:2.9175; .-. whole cost is |20-2575. 189 lbs. wer.' hoii-'ht; the grocer gains 1 lb. in every 63 lbs.; .-. 189 liis. jiiv, sold for 192 lbs. : 192 1bs. @ 25 cts.= $48.00; .-. gain is $(48-26'257u)=$21-7425 on $20-2575, or 83 per cent., nearly. (11) By ten figures ; by 10 x 10=100. (12) £4 4,s'. 11 ,^.r?.=£VA^=value of 480 grs., or -h lb. Troy; , , . o. 1 440 ,, 110 .-. ^veight in pure gold of 4-1=^ x ^^^ It^-slTigGO lbs. Troy ; 110 144 .'. Avoirdupois v',;iglit of sovereign ^--^^—x - 3x1809 175 352_ ' 21805 lb. pure gold. EXAMTNATIOX QUESTIONS. 241 (2) TT. (p. 2T0.) _8 1263 _ 8983 _g 923^ ~3^r020~1020~' 1020" (3) A few words are omitted from the question in the text ; it sliould read as follows: the popu'ation is 18422G5 souls, and the revenue from customs is $3595754 by an average duty of 13^ per cent. If the duty be raised to 20 per cent, and the consumption fall off one-tenth ; how much is the average taxation per head altered ? .359575400^^23^^^^32 imports, (4) m 9 ^ of $38700032 =$25889428.80 imports; revenue from $25880428.80 @ 20,^ =$51 77885.70; .-. difference is $(5177885.76-3595754) = $1582131.7C; . .- . u 1. ^ ^1582131.70 .*. taxation has been alterci ^—TE^TK-dTT^— lo4»Ji0a =86 cts., nearly. •034695 -J- -000241 = 143-902. . . . •084 of a mile=147 yds., 2 ft.,0-24 inches. (5) From January 1st, 1862, to 15th April, 1863, is 409 days; . ^ ^ „400 409x7 „., _ ,-._- .-. interest=£-7r-x ^-^_ -=£11 19s. lOW- o oOoOO (6) (a) 8642. Ans. (b) -1809.... 74633 (7^ -075386: '990000' 4100 The expression=— -=-891110628. . . . (8) The interest and protesting charges amount to $55 ; be- sides this there are $24 at interest for 6 mos. ai 2 per cent, per month, and $31 for 3 mos. at 2^ per cent, per month ; interest on these two items=$5.205; .'.total interest for 6 mos. =$60,205: and .*. for year = $120.41 ; .-. rate per cent, per annum =$30.10. 342 Ki:V It) ADVANCliU AUITIIMETIC. (0) The price of the metal a+. first was $GG.40; /. profit on the spoons wjis $23.00: in the second place the metal cost $74-972, hence to make the same profit he must sell the spoons for $(74-972+ 23-00) =: $98-573, or each spoon for $24(543. r 10) 0000279-^300000= -00600000093 ; 234ir)9 X 00839= 1-90093. II) Friictiou = I 6 5700~2304 12) Fraction =- 101 309* III. (p. 271.) 1 ) |i.= 03125, |== -170470588235294; (2) (a) $5958.75. (/>) $22.75, nearly. (o; If d=disconnt, m=amount, n=timc and r=ratc per cent. ; mnr ^l) then d=7 , or 5=(5— i)n ; .-. n: 1+nr ^ X ^ = — sq. f t. 23 c. yds. , = 021 c. f t. ; :^§ years. .,. 391 ,^12„, - . ^^5 ,12 ^108 • • ^^i-^-F^i^n ''■ ^^"- •'"^'•=-'^10-1^17=^ '5) A mile=1700 yards, acre=4840 sq. yds. 50 miles=88000 yards; amount received ^.88000x22 ,^ „,^^^, =£ — iiTiTx — X 55 =£22000 ; 4840 ' .-. v.due of field=£22000-i- £3000= £25000 : -umber of roods in field=25000-7-10=2500=G40 acres; .'. side of field is one mile long. ( 0) £120 discount on £1720 for 21 months is at the rate of £4^ per cent, per annum. amount of bill=£35e S-h\-8. + (£356 dhh) x t^ x ^ :£35GiM (-s"«)= 12 200 £305 10.^ EXAMINATION QUEf^TIONS. (7) In a crown, half-crown, shilling, sixpence anil penny liien- are 100 pence ; therefore nuniher of each=£ll 7«. Ir^. -1-109=25. (8) $100 ^i\(X buys $205 treasury notes, which is in ratio of 20 : 41 ; .-. discount=:tl of $100=$51-4^f per cent. (9) AmoinU of American silver bought witli $100=$|f x 100 — $104^, in New York $100=$180 in greenbacks; 0'*5 9 •*• ^lOti^^l-^x ^=$187.50 greenbacks. Gold falls to 150; .-. $150=1100, and $18T.50=$187.50x 2 gold=$12o; hence gain on $100 is $25, and therefore we have $25: $120:: $100: $480. (10) Income from £1500 at given rate=r £45= $219.15 : £1500 stock=£1470= $7158.90, which being invested in stock at 105 gives $0818, income trom which at 6 p.r cent, is $409.08; gain in income=$409.08-$219.15=$189.8:>. (12) Gold at 250, 40 cts, only will be got for a $1 greenback ; .'. 39 cts. : 40 cts. :: 250 : 25Gif ; raise=Ci§. IV. (p. 272.) (1) Eagle weighs 258 grs. ; .-. pure gold in an eagle=2322 grs. 1869 sovereigns weigh 40 lbs.; .-. pure gold in sovereign =11300159 grs. valueofasovereign=$i^'^-^^^^»^l« =$4*80G. ...=$4.86? 1 1) From the question 40 Spanish dollars=£9; 40 y : 4.86^ :: 100 : rate of exchange ; .-. rate of excliange=4-8G| x 100^^= 109^. C2) ^of 4^. 76^.+^ of U 5H-^ of 5s.=U 11^^. (3) If the denominator be composed of p cyphers and q nines, and the numerator consist of p+q figures ; then the fraction will be such as required. 1 .---. 1-65 — 07^00 ir-« < 'jy^o ; 13 ixr4x---i. If' Ml i KEY TO ADVANCED AinTllMKTlC. (4) Loss on each l)iishel=:2318-2124--r:94, or in fraction of a bnshol------ ^- '. t.liat propt^i'tion of tho whole rent; therefore loss on rentr=£1075 x i^j^r^^'^ ^^'- ^''^'''^• (5) The number of ^-raiTls in each parcel, taken order, is 485640, 2Gl2r,2, 22;i(;4; the G. C. M. of Avhich is 7G, and by divid- ing 7G into each of the above numbers the number ot par- cels is obtained. (6) 37-OGO X 24=:880-G5G inches in a chain, (88l)G5G)' X 10=7914877-08;3aG sq. in. in an acre ; .-. sq. in. in 4:2 aeres=3;'.242487530112 =:2n08r)0l>-07848 6q. ft.: number of eq. ft. in 55 acrcs=2395800 ; .-. dillerence=8729'M)2l52 sq. ft. (7) Let A represent number of lbs. of tea bought, price of^A@ 82 cts. = $-A; . , „ $^;:iA-$t90_ 41 A^- 11400, .-. price of a lb. = -,-r * ^^^ \ 17 price of ~ @ 85 cts.=$-^^A ; .r,3A_.^4lA-11400 ^"40'-* ""■50" .41 A , ,J7A GO 120 ^3/41A-11400\ n.R;oi (3) $604. 16|. 248 Ki:V ro ADVANCKl) AKITIIMETIC. H i I i ( w 'i: |!! (4) 5-7061. (5) ^^^ (G) 9. (7) 4(34- ^ 95^99* ' (8) 1U.4,V?. (9) 113 men. (10) £3258 4.s. 3^8 W- (11) .$27150541 n, ■-- '^493 £078 186'. 3-25 d. (12) 9000' (1) (2) (3) (6) (10) (12) (1) (4) (6) (8) XI. (p. 279.) $3423.07,H(. Solid content of gold bar is '827328 c. in., solid content of silver bur is 9-801020 c. in. ; .-. ratio=827328 x |^^ : 9861020 x y^-^, or gold bar : silver bar = l : 647 10-325, 7-74375. (4) 6-519. (5) $82.89.... . (7) 427. (8) 19-/o. (9) 16. 122-64. . . •037399.. lbs. £28-12-,^3Vf'?. (11) 1-00352. 1011 lbs. 92°, 14', 4rh'rhh%V (9) (11) f (3) (5) (6) XII. (p. 280.) (2) 5-,Wri"- (3) £4.33 lis. 2-3\(?. (5) 15o8ti miles. 23-855.... per cent. (7) .$2094-38^. 16-8 geogTaphical miles- U)-4G statute miles, 365-25 days =31557000 seconds, 31557600x19-46 miles=^61411()890 miles =: circumference of circle made by the earth: 614110896 miles^3-i4159=lD5477734-52. . . .miles; .-. radius of circle described by the earth =r-.195477734'52. . .mUes-^2=97738867-26. . .miles. .-. distance required=37140769-56. 2|inches. (10) 41484375. £)5i>«.llgK (13) Ji^99-627i. XIIL (p.C;U.) 11 . lof { U -2083; 12044/f. •00983 ; 5-19. , = -.; 8 ro., 20 per. £2U8.m (7) £G;£im (8) 30 men. EXAMINATION <2l'ESTlONS. 24^ (9) (13) (1) (5) (9) (11) 35 (10) £;o-. (\[) £7 105.; £18. XIV. (p. 283.) (3) -668. (a) 11401; 89. (4) 30. a) (2) ^(3) (6) (8) (10) (11) (13) (1) (3) (4) 100 333" SOA. (6) Udays. (7) 33/y. (8) ^0■9S,^. 69 per ceut, nearly. (10) $5.43i 83| cents. One solution Is 1 lb. at $1, 1 lb. at 90 cents and 4 lbs. at 80 cents. A large number ol solutions can be obtained. XV. (p. 383.) 1 lir., 41', 4G1" ; £1 68. id ; £3970 Us. 5d. 443 feet. ; no. of times is greatest integer in 10x8x40x5^x3 ^^ 443 •001; 13; 1304. 8643; -1809. (5) £31 7.9. nd. (7) I3s. lO^i^d; 13 yds., 3 nls., 1 A\ i^- 7 9 36 men. (9) g; ^ 2 ro., 20 per. ; U. lUd ; 3 per., 2 ft., 11 98848 in. ; 7Jc7. ??. ^? ■ -001083; -00035. 1 ' 1 ' £110 Ss. 8kH ; £600. (13) 69-5(584. . . .yds. XVI. (p. 284.) 16^ ; 19 30" (3) -16875; '07. 23.12. 3i^8-3_v:7^11t3Vl4^ .^4^ ^^ "17-483... £25+ 10 per cent, of £35=r£27 10s.t=entire selling price, 38 -8 =30= no. of gallons sold, 30 men. 30 .rrl8s. 4<:?.=selling price per gallon. IB' i ! f f ' i; I 250 KEY TO ADVANCED ARITHMETIC. (5) Gold bg. at i per cent, disc, £100 is bought for £99^ ; .-. £99i=$484.565, add 3.634 for freight, $488.20 cost of £100 in gold, |497.77=cost of a bill of cxch. for £100, . $9.57= difference on $488.20 ; .-. difference per cent. = -r— — - x 9-57=l-9, or 2, nearly. 4oo"/©U _5-3+ V2( V5+ V3)-2 V2 yS (6) vs V5+ V3 3V3 ~ V5-H V3+ v2~ 2v2( V5- V3+ v2) _ 4/2^5- 1/24/3 + 2 _ V2( vr>- 4/3+ V2) 1 ~2V2( V5- V'S+ V2) "^2 V2( v5- ^^3+ V2)~2" (n 2+ V 2+ V~2) "N 2+ V 2+ V~2 =N 2+ i^ 2+ V~2]2-hi/ 2+ V~2-3[ =N 2^ V 2+ V~2 N 3+ V~2-2 4/ 2+ V~2 = \G-^2 .v''2-4 V 2+ ^^+3 V 2+ V~2+ V~2 [^"2 + 71-2(2+4/^) =^\i 2- V 3-h V^+ 1^2 V 2+ V^ ~nJ 2+ V^ 2+ V~2(V'^-l) = n] 2+ V/ 2-t- V~2 V 3-2 V^ =n| 2+ V 2- \^. (8) Discovint on £567 for one j'^ear at given rate is £• .'. time required =:^j-^years=l^ years. •2 I) 9 .^ p,_40 n2i_4J) 2^ . rfQin 40 235 ,^10 40 225 379 379 ^, . ^^ .-. £18 m9.=-x,-^xl82^=^X3-^x^=-^=$94.75. (10) -et^:!^, 14.^. 5103 209' yO' •JO EXAMINATION QUESTIONS. dsi H; nearly. 2) 1 '-2' 103 309' : $94.75. (') •9385714 = -92857142 •82i42857= -82142857 •48 = -48484848 2-87 - 2-87878787 511363636 (3) •163x-66=-0099173553.... (*) l•6l5873-^l•63690476i=l-i5?--^l5?5??^ 999999 999999900 169312 111111100 111111 272817433 (11) -00776 clays=error made each year; years this error will amount to a daj m- (13) (1) (3) •00776 =128|f years. 10 men. 21164 18975 •. Vf is the greater. XVII. (p. 285.) 21160 26312' 26312' 263i2 difference of 1st two: 2189 fraction required: 26312' difference of 2d two: 2189 . 2185 '2631» 2185' (3) 1600 ounces, 4840 6. yds. (4) 3'04; 30-4. (5) £9 5s. 2/o-d (6) $254.60. (7) 4 women=3 men ; .-. 2 women=l^ men 5 boys=:3 men ; . -. 3 boys=l|^ men 6 girls=:3 men ; .-.4 gir]s= 2 men 1 man-i-2 women + 3 b()3's + 4 girl8=6-i% men 3 men do it in 00 days 1 " " 180 « 6^ .i.r " (( 180 ^—-28^ days. I ^- V, lil •> i !' ^52 KEY TO ADVANCED Alin'HMETIC. (8) Difference=£3 9.^. 3|(f. (9) £315 lOs. Sd. (10) £12 5s. XYIII. (p. 28G.) y^^ ^ '' 2013G 11 o (ior.A)4"K4-«D4°^n4<'^l ,37 12 367 924' 1 5+- 34- 4+, "X 2 I G8 ■2+'H-i5r o a"f3-^fo)HA \ /6 ,.7 „9\ 11 Hli"^8"^2)=i35- ,,, 13 1 11 1 ^^ 21 ""2 14^3 13-11 2 16 1 13 1~1(>-13~3' til 21 '^2 14 '^3 (2) (1) £4^r=£4 3«. 4?., lH.9.=lls. 4d., 7-,%(?.=7H ; .-. £m-lHs. + 7Adr=£4 15s. 3H (») 1416 ac, 2 ro., 16 p. 226656 po. x 8 ^of(4ac.,3ro.,27p.) ^1813248_po._23^^^ 787 po. (3) 787 po. ^of U 9(?.__ f of21 _81 '64' 35. 4c?. 40 (4) (») 2-7=2-777 •913 1-864 C») ~ - 00044 405 •0112 '' 112 (2) 91-78 •381 9178 73424 27534 34-96818 =•0396482 142857. EXAMIXATIOX PAPERS. 90t 55 4 2 . • . . 23 110 2-27-*-l-ll3G=2A-5-l-i^%^ , y X ^^=2. C'; 1 ^zz , L- 1 1,2 =: •5 1 1,2,3 r= •1G666666 1 1,2,3,4 •0416G668 1 1,2,3,4, 5 rTT ■00833333 1 1 9 '-i 1, 0P 331 =£46 14. C. >< i^^^r^^ 13»- OHWic^- IGO (7) 120: 111::100 : £93 10.v. (8) £1000 •03 100 8 £3000= Ist income. HOOrranit. of stock held in 2d case. •04 , £3300=2d income. £2= difference of incomes. -^ A 6d. is d. L 2d case.