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Las diagrammas suivants illustrant la m*thoda. 1 2 3 4 5 6 ( ) ( ^ /.if '1 N ^ US MAI sol: LOVELL'8 SE RIES OF SCHOOL BlDOKS. KEY ' ■; TO THE NATIONAL ARITHMETIC- CONTAINING iil Mntims k ntarig all % iraWtms. DBSIQNBD FOR TEB USE OF TEACHERS AND PRIVATE STUDENTS. BY JOHN HERBERT SANGSTER, M.A., MATHEMATICAL MASTER AND LECTURER IN CHEMISTRY AND NATU3UL PHILOSOPHY IN THE NORMAL SCHOOL FOR UPPER CANADA. SECOND EDITION— CAREFULLY REVISED. fonlwal : PRINTED AND PUBLISHED BY JOHN LOVELL, SOLD BY ROBERT MILLER. MONTREAL; ADAM MILLER. TORONTO AND BV ALL BOOKSiiLLKRS. 1864. : -rt>iB«rtftitai|i4niMa»l|iir«n*ri, Entered, according to the Act of the ProvincM P«ri- . . the ,ear one thousand eight hundred n ^^ZX John Wi., in the Office of the Registrar^ thpi> ^ vince of Canada. registrar of the Pro- Txi6 PREFACE. It was the original intention of the Author to give, in the Key, merely a series of brief hints upon the Solutions of the more difficult Problems. He was led to modify this plan, and to issue the work in its present form, chiefly from the consideration that as there are in the country many young person.^ who, from various causes, are unable to avail them£:elves of the advice and assistance of a teacher, it would be a great boon to these to have access to a book to which they might refer with the certainty of having every doubt removed as to the correctness of their work and methods of solution. He offers the work to his fellow- teachers with the hope that they will accord it the same favorable reception that they have so kindly given to the National Arithmetic. Toronto, May^ 1861. CONTENTS. Pagb Reduction Descending , 7 Reduction Ascending g Reduction of Canadian Currency to Dollars and Cents. . 10 Multiplication U Division Ig Miscellaneous Exercises, Arith. page 118 22 Resolution of Numbers into their prime factors 29 Divisors of Numbers 29 Number of Divisors 30 Common measure of two or more Numbers , 31 Greatest Common Measure 32 Least Common Multiple. . . ^ 34 Reduction of Scales of Notation 36 Fundamental Rules in diflferent scales 41 Duodecimal Multiplication 42 Miscellaneous Exercises, Arith. page 149 43 Reduction of Fractions 50 Addition of Fractions 60 Subtraction of Fractions 65 Multiplication of Fractions 67 Division of Fractions 71 Miscellaneous Exercises in Vulgar Fractions *!& Reduction of Decimals 79 Addition of Decimals 85 Subtraction of Decimals 87 Multiplication and Division of Decimals » 88 Miscellaneous Exercises in Decimals 89 Miscellaneous Exercises, Arith. page 198 93 uimple Proportion ^ IO4 Compound Proportion m ^' CONTENTS. Conjoined Proportion ^^*"' Miscellaneous Exercises, Arith. page 222 ! lit Practice "^ Bills of Parcels ......!!..!!!!.!.'!." ^ ^^ Mfacellaneous Exercises, Arith. page 231..'..' .* Jf? Percentage ^ oommissL ....v*. i:;;; v.'. ''^ Brokerage |', , ' ". ^^^ Insurance '/^ ^^^ Custom House Business '.V. .*.*..' .*.'.' ^^^ Assessment of Taxes ..,,. ^^^ Simple Interest ....... ^^^ Compound Interest Discount ,[['' ^^^ Bank Discount ....'"...........'. * * * " ^^^ Equation of Payments '.., ^^^ Partnership ][' ^^* Profit and Loss ..,,..... ^^^ Barter. ..*.** ^^^ Alligation !!!.'!.*.'.*.'.*.'.!* ..[......[ ^^^ Reduction of Currencies *, '. ^ ^^ Exchange '.'.*.'.* .*..'..* ^^^ Arbitration of Exchange ^^^ Involution *."/.*..'* .'."** ^^^ Square Root [[ ^^^ Cube Root .....!!.!.!..."..*'. ^^^ Extraction of other Roots * ^^^ •Logarithms .......*!,... ^^^ Logarithmic Arithmetic .'.*.'.*..".'.*..* ^^^ Arithmetical Progression ' * * ? G».ometrical Progression Single Position ..,[ ^^^ Double Position .*.*.*.*.*.*.'!!!*..*.! ^^^ Compound Interest .*..*..*.' ^^^ Annuities at Simple Interest '.'-!!*..*!.'!!!.*!.'..'* 237 Annuities at Compound Interest ngs Examination Problems Arithmetical Recreations KEY TO NATIONAL ARITHMETIC. ExKROiSB 5— Page 50. (0 (2) (3) (4) (5) d £ £ s. • £ 8. £ 8. 23328 348 38 lOr 58 13 58 ^3 4 20 20 20 20 93312 f. 6960 8. IIQq. ins 8. 1175 8. 12 12 12 9240 d. 14076 d. 14076 d. 4 56304 f. (6) 0) (8) (9) £ 3, d. £ 8. d. cwt. qrs. lbs. cwt. qrs. lbs 59 13 6J 63 9 16 2 16 14 3 16 20 20 4 4 1193 8. 1260 8. 66 qrs. 59 qrs. 12 . 12 25 26 14322 d. 15129 d. 346 311 4 132 118 mn f. 1666 lbs. 1491 lbs. 8 KEY. OO) (U) (12) (13) 3 6 12 16 7 11 15 14 20 i 12 41 oz. 20 832 dwta. 24 12 86 20 366 8 [Nat. Abith. (14) yrs. d. h. m. 46 21 8 56 365 7300 dys. 8 fur. 24 40 lfll5 dwta. 29200 24 14600 3344 1664 V674 3830 1 75200 hrs 1600 160 251 276 — • 138 320 per. 61 16811 days. 24 18984 grs. 45974 grs. (15) sq per. 74 30i 2220 m (16) • (17) a. r. per. sq. miles. 46 3 12 767 67252 • 1760 yds. !?£^ 3 403472 hrs. 5280 ft. 60 24208376 min. (20) 640 (18) (19) cub. ft. pks. 767 767 1728 2 pks. 797 2 187 r. 40 30680 4602 2238i«i.,d.. 7492 per. 30i 224760 1873 226633 sq. yds. 6136 1534 gals 1594 gals 1534 4 4 5369 . 490880 sq. a. 767 6136 qts. 6376 qts. 1325376 cub. in. . " 12752 pts. (1) f. 4)32756 ExBRCiSB e—Page 51. (2) (3) P^' yds. 24)23547 5i)397024 2 2 11)794048 12)8189 d. 20)981 dwt. 3 grs. 20)682s. 5d. 12)49 oz. 1 dwt. 3 grs. x'TTo. K^«p--":„. 40)72186r.2hf-yd3.=lyd ^34 2s. 6d.lir 4 lbs. 1 oz. 1 dt. 3 g. smoZ f„r. .« . i J 325 m. 4 f. 26 r. 1 y. [Nat. Abith. £ZKR01Ba8 6, 0. K£T. 9 (4) sec. 60)28635 (6) lbs. 25)1666 (6) lbs. 25)1491 60)477 m. 16 sec. 4)66 qrs. 16 lbs. 4)69 qrs. 16 lbs. 7 hrs. 67 m. 15 sec. 16 cwt. 2 q. 16 lbs. 14 cwt. 3 q. 16 Ibg. O) (8) (9) grs. oz. cub. in. 24)115200 16)107520 1728)1674674 < 20)4800 dwt. 6720 lbs. 12)240 oz. (10) Fl.e. 767 3 969 ft 242 in. 4)2301 qrs. 20 lbs. (11) ft. 3)183810 575 yds. 1 qr. (12) cub. in. 1728)138297 (13) cub. ft. 128)67893 6O61270 yds. 2) 2 11)122540 27)80 ft. 57 in. 530 cords 53 c. ft. 2 c. yds. 26 c. ft. 67 c. in. 40)11140 per. 8)278 fur. 20 per. 3)34 m. 6 fur. 20 per. 11 lea. 1 m. 6 fur. 20 per. (14) (15) sec. 60)3561829 qts. 4)1597 (16) eft. 8)1000 60)59363 m. 49 sec. 2)399 gala. 1 qt. 125 cords. 24)989 h. 23 m. 49.S. 4)199 pks. 1 gal. 1 qt. 7)41 d. 5 h. 23 m. 49 s. 49 bush. 3 pecks 1 gal. 1 qt, 5 wHs. 6 days. 5 hrs. 23 min. 49 seo, 10 seconds. 60)10000 — _i— ' t *60)166' 40" 2o 46' 40" ESY. (18) sq. links. 10000)70000 [Nat. Abits. • (19) grs. 30)11521 (20) aq. ft. 9)26025 1 sq. ch. 3)676 scr. 1 gr. 8)192 dr. 1 gr. 12)24 oz. 1 gr. 2 lbs. 1 gr. 301)2891 yds. 6 ft. 4 ) 4 121)11564 quarter yards. 95 per. 69 quar. yds. 6 ft. = 40)95 per. 17 yds. 8 ft. 36 in. 2 r. 15 sq. p. 17 sq. y. 8 sq. ft. 36 sq. in. EsHROiSB 7— Page 53. (1) CO f^^^o^'^^^An^'^u^' -£29X406 =$116-00 IB. X ^U =r 140 " iQo V on o.cn Ud.= 5far.X 5^12=_2_^,«3id.=14far x^^?2= '-OSf £3 7s. lid. = 1342,V cts. £29 ISs. 3id.=$i79^ (3> Hid. = 45 far.x5-i-12 = 18Mts. (*) (B) f/' >f «"» = »"«-«0 18s. X 30 = $3-60 6d.= 34fai'xm3= 'To °i''-=34far.x5.M2=*.Hi *-69 1!is.6d.='i^ I8s 8Jd. =$3-74S [Nat. AfiiTB. EZEB0ISB8 6-13.] KEY. 11 (6) Xnx400 = $68-00 16s. >C 20= 3-20 53d.= 23far.x6-fl2 = 'Oh's £87X400 = $348-00 15s v20 — $3*00 £17 168. 5id. = $71-29^ lUd.=47far.X 5-5-12= -19^^ 153. lUd.=$3-19f^a- (9) (10) Jei6 X 400 = $64-00 £2 X 400=$8-00 68.x 20 = 1-20 9S.X20 = 1.80 2d.= 8far.X 5-J-12 = -034 lld.=44 far. X 5^-12= .18J £16 6s. 2d. = $65-23i £2 9s. lld.=$9-98i ExBROiSB 13— Page 90. (1) (2) (3) (4) 36=12X3 121 = 11X11 144=12X12 648=12X9X6 $169-78 12 2037-36 3 $6112-08 (5) 796342-3 11 8759765-3 11 96357418-3 (S) $33460 12 401520 12 $4818240 O) 735 12 8820 9 * 79380 6 476280 (8) 18 = 6X3 22 = 11X2 810=10X9X9 54 = 9X6 £ s. d. £ s. d. £ s. d. cwt. qrs. lbs. oz. 3 7 6 5 14 6i 3 4 7 11 3 14 7 6 11 10 9 20 5 3 62 19 m 2 32 5 10 9 107 4 15 6 60 15 125 19 U 290 12 6 9 642 1 4 10 2616 12 6 12 (9) 49 = 1 X 1 bush. pks. gal. qt. pt. 26 3 1 11 7 188 1 1 2 1 7 KEY. (10) 63 =: 9 X 7 yds. qrs. na. in. 2 2 2 2 9 24 2 7 [Nat. Arith. (11) 288 = 12 X 12 X 2 dys. hrs. min. sec. 5 17 33 11 12 1319 168 2 68 18 38 12 12 825 7 38 24 2 1650 15 16 48 (1) 83 = 3 + 10 X 8 •^ s. d. £ 8. d. 12 2 4X3= 36 7 10 ExBRCiSB 14— Pago 92. (2) 999 =r 10 X 10 X 10—1 ■£ s. d. 963 0| 10 121 3 4 X 8 = 969 6 8 , 1005 13 8 9630 7i 10 96300 6 3 10 (3) 963003 2 6 963 0| 962040 2 5i (4) 8m=8+10X74-10X10Xl-flOX10xiOX3 678=8+10X7+10X10X6 * °^ O* U.. liiioVk «a1. »_ 1 « • _ £ s d. £ s. d. 3 6 5iX8= 26 11 6 10 bu8h.pk.pnl. bush. pk. gal. 16 3 1X8= 135 10 33 4 4ix7= 232 10 7i 10 168 3 0x7= 1181 1 10 332 3 9 Xl= 332 3 9 10 ' 3321 17 6 X3=99e5 12 6 1687 2 0X6=10125 11441 1 10556 18 4i [Nat. Arith. EXBROIBXS 13-16.] KEY. 13 8 18 38 12 12 5 7 38 24 2 = 1181 1 : 10125 ■7071 556 42426 35355 35355 (5) 247 = 7+10x4+10Xl0xi m. tar. rds. yds. m. fur. rds. yds. 23 6 33 4 X7= 166 7 36 OJ 10 238 4 17 1JX4= 954 1 29 OJ 10 2385 4 12 4 X2 = 477l 25 2J 5892 2 10 3| (6) 721= 1+10X24-10X10X7 S. deg. min. sec. S. deg. min. sec. 3 16 30 45X1= 3 16 30 45 10 35 15 7 30X2= 71 10 15 355 1 15 0X7 = 2485 8 45 2559 25 30 45 ExEBCciSE 15 — Page 93. 0) 15607 3094 62428 140463 468210 3931476 48288058 (8) 39948123 6007 279636861 23968873800 239968374861 (9) 2778588 9867 19450116 1^671528 22228704 25007292 27416327796 (4) 3-2517 •023 97551 65034 Exercise 16 — Page 95. (5) (6) (7) 64-001 482000 3782-4 340 -37 '00917 2560040 192003 •0747891 21760-340 3374000 1446000 1 78340*00 264768 37824 340416 34'684608 (8) 87-96 220 175920 17592 19351-20 14 (1) 216 = 6X6X6 $83469 6 KEY. 600814 6 3004884 6 $18029304 (5) 116700 235 583500 350100 233400 27424500 ExEROISB 11-- (2) $61135'37 229 65021833 12227074 12227074 '■Page 100. • (3) 255226 143 765678 1020904 256226 $13999999'73 36497318 [ICAT. iLsns. (4) 176=11X8X2 203736 11 2241096 8 (6) 3721 73 11163 26047 271633 O) . 297=11 X 9 X 3 32000 11 362000 •9 3168000 3 17928768 2 35857536 (8) 35 = 7 X 5 9344000 7 65408000 5 « 327040000 9504000 (^) (10) 749=9+10X4+10x lOx 7 999998=1000000-2 ibs. oz.drs.scr.gr. lbs, oz.drs.scrs.grs. 123 4 7 2 17X9= 1110 8 7 1 13 10 1234 1 7 1 10X4 = 4936 7 6 . 10 12341 7 3 0X7=86391 3 5 1698732 1000000 1698732000000 3397464 1698728602536 92438 8 2 1 13 [Nat. jjtiTH. BxvBoien 17.] KBY. 16 (11) 640=5:10 X8X 8 bush. pk. gal. qt. pt. 123 1 1 1 1 10 1234 1 3 8 9873 3 8 78990 (13) 1143=34-10X4+10x10X1+10X10x10X1 yds. qrs. na. in. yds. qrs. na. in. 732 1X3= 23 23 03 10 79 1 X 4= 316 1 U 10 790 1 1 X 1= 790 1 1 10 7902 3 1X1= 7902 3 1 9032 3 2 (12) 88 •73 267 623 $64-97 (14) 1634*6789 635000 81728945000 49037367 98074734 1037957601-5 698728602536 $968-49 3-4 387396 290547 $3292'866 (15) $12183-6042 3292-866 968-49 $3292-866 3-7 23050062 9878598 $12183-6042 $16444-9602 16 KEY. ExERoisB 18— Page 110. [Nat. Abith. 6423)798965(124^6^3 16666 12846 28205 25692 2513 (11) 741)56789(76Hf 5187 4919 . 4446 473 (10) £ 8. d. 12)176 14 6 14 14 6| (12) 7894)6785158(859f|^5 63152 46995 39470 75258 71046 4212 (14) (13) 317)4728 16 2(14 I's 4/.V 429)$97896-64($228-l93i2 "^^^ 858 (15) 6)970763 317 1558 1268 ^90 20 5816 317 2646 2536 110 12 1322 1268 54 161793*8333+ (16) 9)71234 (17) 1209 47600)977076(2025g7fi 79141 858 3516 3432 84-6 42-9 41-74 38-61 313 95200 25076 [NAT. AbITH. EXSftCIBBS 18, 19.] KEY. 17 (18) lbs. oz. drs. ecr. grs.lbs. oz. drs. scr. gra. (20) 9712)7867674(8105^1^ 77696 498)7289 6 4 2 13(14 7 5 U\ll 498 2309 1992 12 3810 3486 324 8 2596 2490 106 3 320 20 6413 5976 437 (1) 25=5X5 5)3766 9807 9712 (19) £ 3. d. s. d. 954 317 487)157 16 7(6 5J..:ftV 150... 3 3x5+1=16 150JJ 3156 2922 234 12 2815 2435 380 1520 1461 59 (21) ra. Air. rds. m. tar. rds. 37)422 3 38(11 3 14 407 15 8 123 111 40 518 37 148 148 Exercise 19 — Page 112. (2) 42=7X6 7)26406 (3) 96=12X8 12)25431 5)753... 1 6)3772... 2 8)2119... 3 628... 4 4x7+2=30 628tS 264... 7 X 12+3=87 • 264f? (4) 24=12X2 £ 3. d. 12)24 17 6 2)2 1 5^ 1 81 B 18 KEY. [Nat. Abith. (6) 49 = 7 X 1 £ s. d. 7)T40 13 4 7)105 16 2i,..l (6) 56 = 8 X 7 £ 3. d. 8)547 12 4 7)68 9 OJ 0) 35 = 7 X 5 7)6789436 5)969919... 3 15 2 3g...-4V 9 15 61. ..A^ 193983... 4 4x74-3=31 193983^] (8) 147 = 7 X 7 X 3 7)753293 7)107613... 2 3)15373... 2 (9) 81 =9 X 9 lbs. oz. dwt. grs. 9)1798 6 11 9 9)199 10 1 6... 3 5124... 1 1X7x7+2x7 + 2 = 65 6124ifi/,- 22 2 9 0...6 6 X 9 + 3 = 57 22 Iba. 2 oz. 9 dwt. Ofl grs. Exercise 20 — Page 114. (1) £ s. d. £ s. d. 491 12 0^ ) 8968 13 7i 20 20 (2) m. fur. rds. m. fur. rds. 17 5 27 ) 1027 1 6 8 471937 ) 179373 12 2152483 4 8609934 (18HMf 471937 141 40 5667 ) 8 8217 40 328686 (58 28335 45336 45336 3890564 3775496 115068 [NAT. ABITH. EZBBOIBEB lD-21.] i;«i:. 19 5)969919... 3 193983... 4 .4xt+3=31 2 9 0...6 f 3 = 57 . 9 dwt. 0^ grs. ^e ti. 51 20 1140 12 13687 4 54750 ) d. £ 7i ) 171 20 3421 J2 41062 4 d. lOi dwt. grs, 6 9 ) 24 129 164250 (3 164250 (4) lbs 9 _ia 117 _20 2343 24 9384 4686 129)56244(436 516 464 387 ^ 774 774 oz. dwt. grs. 9 3 13 ExBapiSB 21— Page 115. (5) (6) a. r. per. a. r. per. 47-655 — 4-5 = 91 6 ) 2366 3 36 45)476-55(10-59 4 4 45 364 9467 40 40 14566 ) 378716 (26 29132 87396 87396 O) 756-98 -J- 76-73612 = 7673612)75698000 (9-864 + 69062508 26*6 22-5 405 4-05 (8) 47-5782975 h- 26-175 = 26175)47578-2975(1-8177 26175 6635492-0 6138889-6 496602-40 460416-72 '36185-680 30694-448 5491-232 21403.2 20940-0 463-29 261-75 201-547 183-225 18-3226 18-3225 IJf 20 KEY. [Nat. Aaith. (9) (10) 1 -T- 7-6346 = •75-347 4- 0-3829 = 76345) 10000-0(0'1309+ 3829)763470(196-7798 + 7634-5 2365-50 2290-36 75-1500 68-7105 3829 37057 3 4461 "25960 22974 2986-0 2680- 3 305-70 268-03 37-670 34-461 3-2090 3-0632 (11) •0002-f. 000000008= 8)200000 26000 -1458 ExBROiSB 22— Page 116. (1) 95)$3300000($34 736-8421 (2) 285 450 380 (3) dys. 28800)95270400(3308 86400 126)$3860000($30634-9206 378 700 665 350 285 650 570 80-0 76 '0 4-00 3*80 •200 •190 •100 •095 .005 88704 86400 230400 230400 days. 365i) 3308 4 4 yrs. days. 1461)13232(9 20^ 13149 800 756 440 378 620 504 116-0 113-4 4)83 20? 2^60 2-52 •800 •766 (4) •044 35781628)$! 14o012096($32 107344884 71663266 71563256 [Nat. Aaith. EZSBOIBIB 21,23.] EET. 21 (5) 27475271)$3Y64112127($13'r 27475271 (6) 9)$972 $108 0) 108)|972($9 972 101658502 (10) 82425813 (9) 1728)1000(578 oz. 792)340480(429^9 oz. 864-0 192326897 3168 136-00 2368 120-96 1584 (12) 192326897 (8) 294)$8526($29 588 (11) 19)4750(2501b3. 15040 m. fur. 7840 38 13-824 2646 33 2 . 7128 — 2646 8 95 1-216 — 712 95 266 Hi=a-5 40 (14) . bu8h.pk.gal.qt.pt.bush.pk.gal.qt.pt. 10640 297)729 1 1 1 1(21 1 2 Ifr 5i 694 63200 135 5320 4 1 155)58520(50 1'lVj 5775 770 50^7 B. d. 4 7JX7=1 12 6i 10 5997 9 53973 X2 C 51+1=2 6 51 3 18 Hi (6) 3t))285000000Q dvg. hra. 24)96000000(3958333 8 72 — yrs. - 230 365i)3958333(l0837 216 4 4 140 1461)15833332 120 200 192 80 73 80 72 1461 12233 11688 5463 4383 10702 10227 80 72 8 rem. 4)475 quarter days. days. hrs. 118i=118 18 Add 8 119 108 j7 /rs. 119 days, 2 hrs. £729X400 =$2916-00 17s. X 20 = 3-40 6id.=25 far. X 5+12= 10-/^ $2919-50,^^ (8) $10000 9876-23 24 KEY. (10) in. 12)7964327 12)663693-11 119 in. 55307- 9 9)65307 ft. 119 in. 30i)6145yd8. 2 ft. 119in. 4 4 [Nat. Abith (11) $729-43 16-70 976-81 * 9987-17 429-00 129-19 - $12268-30 121)24580 11)24580 203 p. 4i y.= 203 p. 4 yds. 2 ft. 36 in. Add 2 ft 119 in 11)2234-6 ) 40)203 p. 4 yds. 5 ft. 11 in. — > 17 qr. yds. — 203-1 > 4)5 rd. 3 p. 4 yds. 5 ft. 11 in. la. 1 r. 3 p. 4 yds. 5 ft. 11 in. (12) 429=94-10x2410x10X4 wks. dys. hrs. min. wks. dys. hrs. min. 6 4 3 17X9= 59 2 5 33 10 65 6 8 50X2= 131 5 17 40 10 659 16 20X4= 2636 2 17 20 2827 3 16 33 wks. 52)2827(54 yrs. 19 wks. 3 dys. 16 hrs. 33 min. 260 227 208 19 wks. [Nat. Abfth. EZXBOIBB 28.] KEY. 25 (15) (16) H tons. $136 324 $136X4: =544—95=449 20 1902 cwt. qr. lbs. 13 2 14 6480 2487 4 4 $9237- -ffi2487=*fi'7.Fin 54 25920 "^4|PM^W |"-"i- •^^^ • •^ V 25 25' (17) yds. ora. na. vds.ai's 1.. 2.. 4.. 5.. 10.. 20.. 25.. 50.. 100 30 KEY. [Nat. Abith, (2) 810=34X2X5. 1. .o. >«7. .Zi. «ol 1..2 1.. 3.. 9.. 27.. 81.. 2.. 6.. 18.. 54.. 162 1..5 1.. 3. 9.. 27.. ai.. 2.. 6.. 18.. 54.. 162.. 5.. 15.. 45.. 135.. 405. 10.. 30.. 90.. 270.. 810 = ^' '^;?' 'o;;^* ;^i-i2;a^^* •^^- •2^- -^o- -45. .54. .81..00..135.. ioz, . J70. .405. .810. (3) •» 920=2^X5X23. 1..2..4..8 1..5 — « 1-. 2., 4.. 8.. 5.. 10.. 20.. 40 1..23 1..2..4..8..5..10..20..40..23..46..92..184..115..230..460..920 = 1. .2. .4. .5. .8. . 10. .20. .23..40..46..92..115..184..230..460..920. (4) 25000=5«X23 ].. 5. .25. .125. .625. .3125 1..2..4..8 ^"^;nn^*;l!«-^^^-^^2^-2-10"50..250..1250..6250..4..20..lbO 500.. 2500.. 12500.. 8.. 40.. 200.. 1000.. 5000.. 25000 = 1..2..4..5..8..10..20..25..40..50..100..125. 200 250 non R'?fi lOfO. . 1250. . 2500. .3125. .5000. .6250. ! 12500. 25000 il ExKRCiSB 26--Page 128. (1) (2) 88200=23X32X58X7^ 3500=2^x53x7 3+1=4 2+1=3 2+1=3 3+1=4 ?+Jf? 1+1=2 ^-ri—o 3X4X2=24 4X3X3X3=108 [Nat. Abitu. ..45.. 135.. 405. .54..81..90..135. ..230..460..920 = 34..230..460..920. I ExKBOIBBS 25-27.] (3) 6336=2«X3^Xn 6+1='/ 2+1=3 1+1=2 7X3X2=42 (5) 49000=23 XS^X*?* 3+1=4 3+1=4 2+1=3 4X4X3=48 a) 75600=2 ♦XS'^X 5^X7 4+1=5 3+1=4 2+1=3 1+1=2 5X4X3X2=120 KEY. 81 (4) 824=2^X103 3+1=4 1+1=2 4X2=8 (6) 81000=2^X3^X5* 3+1=4 4+1=5 3+1=4 4x5x4=80 (8) 25600=2' 0X52 10+1=11 2+1= 3 11X3=33 J250..4..20..100. •00.. 25000 = 0..250..500..625. .25000. (2) =2^X53x7 hl=3 1-1=4 hl=2 K2=24 (1) 21= 7X3 18=2X3X3 27=3X3X3 36=4X3X3 3 is common to all. Exercise 27— Page 129. (2) 21= 3X "7 77=11X 1 42= 2X3X7 35= 5X 1 7 is common to all. (4) 82=41X2 118=59X2 146=73X2 2 is common to all. (3) 26= 2X 13 52=«2X2X13 91= 7x 13 143=11X 13 13 is common to all. 32 KEY. [NAT. Abith. (1) 296)407(1 296 ExERCiSB 28 — Page 130. (2) 308)606(1 308 111)296(2 222 74)111(1 74 »7)74(2 74 G. C. M. = 37. 198)308(1 198 110)198(1 110 (3) 74)84(1 74 88)110(1 88 22)88(4 88 G. C.M. = 22. 10)74(7 70 4)10(2 8 2)4 2 G. 0. M. = 2. (4) 1825)2555(1 1825 (5) 656)672(1 556 730)1825(2 1460 365)730(2 730 G. CM. = 365. 116)556(4 464 92)116(1 92 24)92(3 72 20)24(1 20 4)20(5 20 G. 0. M. =r 4. [NAT. Akitu, EXIKOISIS 28-80.] KEY. 83 I G. C. M. = 2. 20 r^ ^M mar - * U. Kj. M. =Z. 4. (1) 110)140(1 110 ExKRCiSK 29 — Page 131. (2) . 30)110(3 90 20)30(1 20 10)680 10)20 68 2 iTherefoie 10 is their G. C. M (3) [468)922(1 468 454)468(1 454 14)454(32 42 34 28 0)14(2 12 2)6 375 19 not divisible by 2, land therefore their G. 0. M. is 1, 1326)3094(2 2652 442)1326(3 1326 Also 4420 is divisible by 442 ; therefore it is their G.O. M.' 204)1190(5 1020 (4) 170)204(1 170 34)1445(42 136 85 08 34)170(5 170 I7)2006ril8 17 30 17 136 136 G. C. M.=17. 17)34(2 34 Exercise 30— -Page 132. (2) 56=23x7 84=2^X3X7 140=22X5X7 168=2^X3X7 The greatest factors which are coniraon are 2' and 7 • therefore the G. C. M.=2* x 7=28. C 84 KEY. [Nat. AbitiJ ! W (3) ■241920=28x3'XBXt 380160=2«X33x6xll 69120=29 X 3' X 5 103680=2" X3*X5 The greatest factors which are common are 2", 3 ^ and 5 ; therefore the G. 0. M.=28x 33x5=34560. (4) 10800=2* X 33 X5« 28040=2' XSxTOl 2160=2* X 33 X 5 The greatest factors which are common are 2^ and 5 ; therefore the G. C. M.=23x5=40. (2) 6=2X3 7=7 42=2X3x7 9=3* 10=2X6 630=2X3^X5X7 2 X 3* X 5x7=630. Exercise 31— Page 133. (3) 1=1 2=2 3=3 4=2* 5=5 6=2X3 7=7 8=23 9=3* 3*X23X5X7=2520. (5) (6) 670=2X5X67 8=2' 100=2«X6« 10=2X5 335=5X67 18=2X32 25=5* 27=3» 36=2^X32 22X5*X67=6700 44=2* X 11 396=2*X3*X11 2^X3-^X5X11 = 11880. (4) 6=2X3 9=3* 12=22x3 15=3X5 18=2X3* 21=3X7 30=2X3X5 2* X3«X 5X7=1260, [Nat. ARm^fcxBBoiBM 30-88.1 KEY. 35 re 2% 33 and 6; 5=34560. are 2^ and 5; =40. (4) 6=2X3 9=32 12=22x3 15=3X5 18=2X3« 21=3X7 30=2X3X5 «X3«X5X 7=1280,! (6) . =23 =2X5 =2x32 =3» =22X32 :2«X11 =22X32X11 <6X11 = 11880. (1) ExBRCiSK 32-~Page 134 (2) (3) 2)12. .10. .24 2)14. .21. .3. .2. .63 2)18. . 12. .39. .216. .234 2) 6.. 5. .12 3) 7. .21. .3. .1. .63 2) 3) 3. 6 7) 7.. 7..1..1..21 3) 1. 1.. 1..1..1, 3 3) 2X2X3X5X2=120 2X3x7x3=126 13) 9.. 6. .39. .108. .117 9.. 3. .39.. 54. .117 ) 3.. 1 . • lo . • 18 . . 39 ) 1.. 1..13.. 6.. 13 1.. 1..1 .. 6.. 1 (4) 2)8. .18. .15. .20. .70 4.. 9. 9. .15. .10. .35 2.. .15. . 5. .35 2.. 3. . 5. . 5. .35 2.. 3. . 1. . 1. . 7 2X2X3X3X13X6=2808 (5) 2)24. .16. .18. .20 2) 3) 12, 8.. 9. .10 1 6. . 4.. 9. . 5 3. . 2.. 9. . 5 1. . 2J. 3. . 5 2X2X3X5X2X3X7=2520 2x2X2X3X2x3x5=720, (6) 2)60.. 50.. 144.. 35.. 18 3) 3) 5) 2X2X3X3X5X5X 30.. 25.. 72.. 35.. 9 15. .25.. 36. . 35. . 9 5. .25.. 12. .35.. 3 5. .25.. 'X • • oO • • 1 1.. 5.. 4. . 7.. 1 0) 2)27. .54. .81. .14. .63 3) 3) 3) 7) 27. .27. .81.. 7. .63 9. . 9. .27.. 7. .21 3. . 3. • \J • 9 7. . 7 1 1. . 1. • • • 7. . 7 1. . 1. . 3.. 1. . 1 7=25200. 2X3X3X3X7X3=1134. Exercise 33— Page 136. 1300 10 (1) .200..1^IS|..^^..$IS|..^^..125 165 300X10=3000. ^ 21 12 <2> 20..60..1^..1^^..210..63..27 4.. 4 14..5^1.. 9 165X21X12=41580. 86 KEY. [Nat. AEim, (3) t 144 12 55 ^.. 132.. 1^4.. 60. 11 6 11 ^ 144X12X55=960 .96. .1728 40. ' 138. • ' ExKRCiSR 34 — Pag( (1) (2) (3) (4) 12)592835 5 f )3700 11)10000 6)1000000 12)49402.. e >)740..0 ll)tl 09. .1 82. .7 6)IG6GGG..4 12)411G..^ 5)148.. 11) 6)27777.. 4 12)343.. 5)29.. 3 7. .6 6)4G29..,^ 12)28.. 7 5)6.. 4 .571. 6)771. .3 2. .4 1..0 6)128.-3 2470^e 104300. 6)21. .2 3..;] 33233344 . « (B) 8)10000 12. 1 (6) 112345054321 0) 9)10000 (8) 2)300 8)1250. :0 2)1028804526.. 9 9)1111.. I 2)150.. 8)156.-2 12)85733710.. 6 9)123. 9)13. 4 2)75.. C 8)19.. 4 12)7144475..^ .6 2)37.-1 2. .3 12)595372.. c 1. 4 2)18. .1 23420. 12)49614.-4 14641 2)9.. 12)4134.. G ' 2)4.. 1 12)344.. 6 2)2.. 12)28. .8 1..0 « . • 4 248664e/69. ivyiOiiULf [Nat. ARiTn, :xBRCi8Eft 33-36.1 KKY. 87 728 ExSKOlSli! 35- -Page 139. n • (1) (3) (8) IX V ▼ IV 8)37704 7)444 7)4321 9)1212201 (4) C)1000000 8)4311. .5 8)480.. I 8)54.. 4 7)32.. 5 2. .3 235. 7)313. .5 7)21.. 6 1..4 1465. 9)2312 1..0 G)1666G6..4 9)1101. .0 6)27777.. 4 9)21. .0 C)4G29..3 6)771...') 61415. t < 1..0 10000. 6)128.. 3 E.XERCISB 36- —Pago 140. 6)21. .2 (1) (2) (3) (4) — IV III IX TI 3..;) 33233341 20212331 4 101202220 3 1522365 9 33233344 6 (8) — - - - )0 2)300 8 4 3 3 14 9 21 6 11.. I 2)150.. — — — — ^mm 34 10 128 128 23.. .4 2)75. .0 4 3 9 6 , ^^^^ V — "~~" 13. .6 2)37. .1 137 4 32 3 1154 9 771 , 6 1.. .4 3)18. .1 — / * 550 96 10389 4629 1641 2)9.. 2)4.. 1 4 2203 4 3 290 3 9 6 93507 9 27777 6 2)2.. ■ 8815 872 841568 166666 1..0 100101100 4 ?.5261 3 2618 6 lOOOOOQ 7854 ' WR I 38 KEY. [Nat. Arith . '(6) IX IX IX 3)132713 12)132713 8)132713 3)40834.. 3)13271.-1 12)10207. .9 12)682. .i 8)14757.. 1 8)1652. .0 3)4083.. 1 12)51. .8 8)184.. 6 3)1327.. 3../ 8)21. .5 3)408.. 1 2. .3 3)132. .2 3)40.. 2 3)13.. 3)4.. l.,l IX III XII VIII 132713 =. 11002210110 9 3 = 3tBt9 = 235601 12 8 12 110 9 997 9 8974 9 4 3 332 3 46 12 19 8 12 3 997 3 560 12 157 8 36 3 2991 3 8974 3 6730 12 1262 8 110 3 80769 den. ijenary. 10096 8 26923 3 BOTCy denary. 80769 [Nat. Abitb. H| EXKBCISK 8?.] KEY. ay 1 0) H XII XII XII xn 8)132713 H 9) t2t290 6) /2<290 4) *2/290 2) <2«290 8)14757..! 1 9)117978. .0 6)r85856..0 4)268683.. 2)515146.. 8)1652. .0 ■ 9)1624^ .2 6)34e4r..O 4)78180.. 3 2)268683.. 8)184. .6 ■ 9)20^. .4 6)69^9.. 5 4)lc0b0..O 2)134341.. I 8)21. .5 H 9)284. .2 6)1179.. 3 4)5913.. 2)78180.. 1 I 9)37. .1 6)233.. 3 4)1533.. 3 2)3/0<0..0 I 4. .7 6)46.. 3 4)439.. 3 2)lc060..0 I 6)9. .0 4)10c..l 2)c626..0 I 1..3 4)32.. 3 2)5913. .0 1 4)9.. 2 2..1 2)2 <67. 1 2)1633.. 1 viii H ?)877..:1 = 235601 ■ - 2)439.. 1 ^B .2)2U..l 1 2)10«..0 167 1 2)65.. 1 I 2)32. .1 1262 I t 2)17. .0 10096 ■ • 2)9.. 1 H 2)4.. 1 80 y 69 denai'y. 2)2.. (Continued on next page.) 1..0 40 XII IX KEY. (7 continued.) VI IV [Nat. Asith. II «r290^ 4712420= 130833500=21231330030= lOOllOllOlllllfiOOOlJOO. '12 9 6 If. 122 12 1474 12 17690 12 43 9 388 9 3494 9 212289 31450 12 9 2547468 283052 9 9 6 54 6 327 6 1965 6 11793 6 2547468 9 4 38 4 155 4 621 4 2487 4 70763- 6 9951 4 424578 6 39804 4 2547468 159216 4 636867 4 2547468 2 1243^ 2 2 4 2487 2 2 9 4975 2 2 19 9951 2 2 38 19902 2 2 77 39804 2 2 165 78608 2 2 310 159216 2 2 621 318433 2 2 243 63^867 2 1273734 • 2 2547468 [Nat. Abith. HlOllOlllllBOOOUOO. JEXBRCISES 36. 37,] KEY. Exercise 37- -Page 142. (0 (2) (3) (4) VI XII III Vlll 252 62/e)32c7572: l(62/c 201210 57264 252 31556 102221 675 41 544 2224 544 122024 (5) II 101 1001 nil 1011 1000 nil 10101 [1010100 16l€7 1059^ 58192 52512 21212 58801 58801 (6) VII 2143)142613(50-52544- 14111 1503-0 1411-1 61-60 43-16 15-410 14-111 1-2660 M635 VII 65432 43210 1444 65001 54321 326041 ■1022 354604 513354 434070 51117344 (8) XII 7/348 5«6/4 1/864 (9) (10) XII 34/7 6666 10010 11 il)1010100001(10010roV,J^y 100101 18536 18536 18536 101000 100101 18536 111 M36e296 w 1 i! 42 0) 4 ft. r 6" 10*" 9 7 11 11 KEY. ExBRCiSB 38— Page 146. [Nat. Arith. (2) 19 ft. 10' 3" 11 2 7 2 41 4 2 11"" 3'"" 4 2 11 3 2 8 4 11 10 8 1 6 -iHint 11 6 11'" 9"" 3 3 8 6 218 4 9 44 9 1 8 (3) 9'' Y"' 4"" 7 3 11'"" 2 4 10 7 3 4 222 8 5 (4) 9J in. =9' 9" 9' 9" — 5 7 4'" 3 5 8 4 9 3 5 10 11 8 8 7 ft. 4' 11" 3 2 2 (5) 4 6 8 1 2 9'" 10"" 1 2 9 10 22 2 9 23 6 9 7 10 (1) 15 ft. 1 2' BxBBCiSB 39 — Page 147. 2 6 16 17 6 8 (3) (2) 10 ft. XII 5 45-6 _- ^3 50 sq.ft. 7 1146 — c'rds.c.ft. 3870 128)350(2 94 256 398-46 2 94 cub.ft. il 8 0"= n|cub.ft.=ll cub.ft 1152 cub.in. cub.ft. 774-90=1096 9' XII 774;:;:1096 com. scale, [NAT. Abith. EXBBCIBBS 38-40.] KEY. 43 (4) 4 ft. 5i 20 1 21 sq. ft. 70 128)14T0(llgi cords 128 190 128 62 (5) XII 4-78 9-6 23 <0 3590 380^ 2-c 34492 7418 (6) 25 ft.=300 in. 20 " =240 " 2 ft. 6 in.= 30 " 8 4 32 2 64=8X8 cub. ft. <8'652=128 6' 5'' 2'" t8 duoden.= 128 den. 300 240 72000 30 8)2160000 128 — 64* £93X400 148.x 20 Exercise 40 — Page 149. (1) =$372-00 £276X400 = 2-80 19S.X20 8)270000 33750 =$1104'00 = 3-80 7id.=30f. X5~12 = -121 10id.=42 f. X5-M2= •I7i £93 14s. lid. =$374-92i £276 19s. lOid. =$1107'97J £275X400 =$1100-00 4S.X20 = -80 115d.=47 f. X5-M2= •19f2 £275 4s. Hid. :$ll00'99f2- (2) $729-18 710-50 166-78 374-92i 1107-97i 497-81 1100-99fjf $4688-16-/^ 576=6+10X 7+lOX 10X5 m.fur.per.yds.ft.in. m.fur.per.yds.ft.in. 47 6 17 4 2 7X0= 286 6 27 12 10 478 18 4 1 10X7= 3346 3 11 10 _ __^_ 5=23902 7 4780 4 4 2 4 14 3 2 27536 1 21 1 6 44 KEY. [Nai. Aeith. (3) 243000=2»X3«X5» 3+1=4 . 4+1=6 3+1=4 4X6X4=96 (4) 8)4234434 vin 5)713427 8)241110. .4 5)133721. .2 8) 13423.. 1 8)1024.. 1 8)32.. 3 2..1 VIll 713427 213114 600313 5)22303.. 2 5)3532.. 1 5)570.. 2 5)113.. 1 5)17. .0 3..0 V 30012122 4234434 20222133 (5) 79-342+.0000G378= 6378)7934300000(1243994-98275 6378 15503 13756 28060 • 25513 25480 19134 ^^3460 57402 60580 57402 31780 25512 . 6268'0 5740 •2 5i7 80 510 24 17- 560 12 756 4-8040 4- 4646 • 33940 • 31890 •0?050 EXEB^IBB 40.] KEY. 45 [Nai. Aeith. .00006378= )(1243994-98275 40 21 33 10 O) ^M-n. I?. 5|. ^1.11. ?. 1. ^.W t. 5( ^1.11. 5). 6|. 5. B.U. B. 11. 11. 5 11. SJ. 2. 5. 40X21X33X10=277200. (9) 9999993000=10000000000—7000. 64276-3427x10000000000=642763427000000 64276-3427x7000 = 449934398-9 642762977065601-1 IX (10) IX 5)78263 • I 1)78263 H 5) 15230.. 3 11)6430.-3 ^^1 5)2760.. 11)526.-6 H 5)511. .4 11)43.-0 I 5)102. .0 3-. 6 I 5)17. .3 • ■ 3..1 I IX V XI 1 7)78263 = 7)3130403 .3 = 7)36063 7)5640. . ■ 7)11160. .3 7)214200. H 7) 140 7.. 5 7)13220. .5 7)884. H -~— ^^1 7)177. .3 7)1101. .3 7)128. H 7)23.. 4 7)41. .4 7)U. H 3..0 3..0 3..0 46 KEY. [Nat. Abith. (12) £672X400 =$2688-00 7s.X20 = 1-40 7d.=28f.x5-j-12= -111 £672 7s. 7d. =$2689-5U (13 continued.) (13) 891)243000(272 1782 6480 6237 2430 1782 81)37800(466 - 324 648)891 (1 648 540 486 243)648(2 486 540 486 27)35100 162)243(1 162 ' 54)81(1 1300 64 81)162(2 — . 162 , 27)54(2 54 ^ Therefore G. C. M. = ! 27. (17) (18) (19) " £ s. d. 2)276000 178 16 45 6 ft. r 1" 9'" 10"" 97 15 Hi 2)138000 13 11 11 11 7 693 19 IH — 216 11 9i 678 14 7J 2)69000 • 5 3 8 7 5 6"" 2 2 197 13 Hi 117 6 5 2)34500 5 5 8 8 5 5 2 2 2 2 91 1 11 2)17250 80 10 5 7 10 2272 3i 3)8625 87 5)2875 5)575 5)115 1 1 3 10 8 10 10 I 2SX3X5*X23 [Nat. Abith. EXBBCISK 40.] KEY. 47 XII 7l3/96)7/c9-047(-011436 7l3<-96 IV 3333333 4 15 4 63 4 255 4 1023 4 4095 4 vm lOCOO 8 97^2<7 713-/96 8 8 266-4110 245-3720 64 8 21-05c00 19-3e846 612 8 3-862760 3-67 e490 4096 •1<3290 16383 / niintt ntnnti ■xn/rHfHf 2 2 8 10 10 (23) 74002702 -J- 144 = 513907 ft. 94 in. 513907 ft. -r 9 = 57100 yards 7 ft. 57100 yds. -7- 30i = 1887 per. 18^ yds. 1887 per. 18 yds. 2 ft. 36 in. Add 7 ft. 94 in. 40)1887 per. 19 yds. ft. 130 in. 4)47 r. 7 per. 19 yds. ft. 130 in. 11 a. 3 r. 7 per. 19 yds. ft. 130 in. (24) 1728 65 3 8 1728X65X3=336960. 48 KEY. [Nat. Arftb. I (25) 6 children will have 6 children's shares 4 women will have 4X2=8 " " 3menwillhave3x5x2=-30 " «' 3 men 4 w'n & 6 chin will have 44 children's sha. 4)$7894'1G ll)$1973-54 $179-41, 3 = child's share. $1 79-41 ,^1 X 2=$ 358-82 i-ij = woman's share. $358-82,'i,x5=$l794-12fii=inan'sshare. (26) 11 n UllllllU 1000000000 2 2 yds. ( 7 4 3 2 2 2 29 4 7 2 4- 2 117 2i i5 2 .31 2 8 2 16 2 t 235 29^ 264i 4 63 2 32 2 1057 127 2 64 2 • 255 2 128 2 511 2 250 2 (27) yds^ qrs. na. in. yds. qrs. na. in. 1 1 1 ) 729 3 2 1 2919 4 11679 23359 29195 262783 4 1051l5(99,<,Vr 9513 9985 9513 472 1023 512 [NAT. ARITB, Extoci8K40.] KEY. 49 3 shares li « ireii's slia. n; re. I's share. ihare. n. yds.qrs. na. in. 1 ) 729 3 3 1 4 2919 4 11679 23359 2919i 2G2783 4 1051l5(99,1,Vr 9513 9985 9513 472 (28) (29) (30) 762-4978 723426 lbs. oz. drs. scr. 63-423 938-9126141 129 63 4 7 2 22874934 722487-0873869 15249956 30499912 22874934 45749868 48359-8979694 65 7 I 1..2..4..8 1..7 (31) 1064=23X7X19. 1.. 2. .4. .8. .7. .14. .28. .56 1..19 1..2..4..8..7..l4..28..56..19..38..7o..l52..133..266..532.,1064= ll..2..4..7..8..l4..19..28..38..56..76 ,133..152..266..532..1064 (32) 30 ft. 6 in. = 366 in. 366 20 ft. 11 in. = 251 in. 251 2 ft. 7 in. = 31 in. — 366 1830 732 in. 31)91866(2963H 62 298 279 196 186 106 93 2963J K36=82t^«^^ yds. 13 50 KEY. GxcROiMFi 46 — Page 158. [Nat. Arits, (0 2X7X9X5X18 5X5X9X5X18 8X5X7X5X18 •2 ft 8 r( ft -— ^ M , , " ' ' "" ' 5X7X9X5X18 5X7X9X5X18 5X7X9X5X18 3X5X7X9X18 5X5X7X9X 5 11340 20250 25200 17010 7875 5X7X9X5X18'5X 7X9X5X18 2835o'2835o'2835o'28350 28350 (2) 8X13X14 12X11X14 5x11X13 145G 1848 715 *' '^*'~11X13X14'11X13X14' 11X13X14 2002 2002* 200.i (3) 6X11X13x7x2 4X 7X13X7X2 ' 7X11X13X7X2' 7X11X13X7X2 5X 7X11X7X2 4X 7X11X13X2 IX 7X11X13x7 7X11X13X7X2 7X11X13X 7x2 7xllXl3X 7x2 12012 5096 5390 8008 7007 " I > J > 14014 14014 14014 14014 14014 (4) 6X7X13 4X11X13 8X11X7 54t> 572 616 '' UxTxls' IIX 7X13' llX 7X13 lOOl'lOOl'lOOl (5) 5X7X5X11 4X6X5X11 4X6X7x11 ^, -f, h fr = 6X7X5X11 6x7x5Xll' 6X7X5X11 6x7x5X11 1925 1320 1848 420 231o' 23lo' 2310 2310 EXBRCISBS 4fi, 47.] KEY. 51 3 8X5XTX5X18 25200 noiO 78T5 145G 1848 715 7X13X7X2 11X13X7X2 7XllXl3Xt 54o 572 616 1 4X6X'7Xll i, h h -i = (6) 1X3X5X7 2X2X5X7 3X2x3x7 2X2X3X6 2X3X5X7 106 2X3X5X7' 2X3X5X7' 2X3x5x7* 140 126 60 210' 210 ' 210 210 ExERCiSB 47— Page 159. (1) The least common multiple of 5, 8, 6, 4, 15 is 120. The multiplier for both terms of the first fraction is '^t' =24; for the second J^F = 15 ; for the third ^" = 20 ; for the fourth ^4^=30 ; for the fifth ',V'' = 8. Multiplying by these numbers, we obtain -,'V'j, {KJ•^J, -j'^j^l, ^Vo- and -p.,\j. (2) A, h h -H, iil The least common multiple of 11, 3, 7, 77 and 33 is 231. The multiplier for both terms of the first fraction is - 'i = 21 ; for the second, -jp = 77 ; for the third, "I'-n: 33 ♦or the lourth, %V = 3 ; and for the fifth, ^^ = 7. Multiplying by these numbers, we obtaiu .^;i 7 ..!• 1.3 7. 3 7 2> 3' r>j 6) HJ 1 ii» I r>j \ I,) »0' The least common multiple of 2, 3, 5, 6, 8, 10, 15, 16 and 80 is 240. The multiplier for both terms of the first fraction is ^}^ = 120 ; for the second, '■^1'^ = 80 ; for the third, '^^^ = 48 ; for tlie fourth, -J" = 40 ; for the fifth, -^'^ = 30 ; for the sixth, ^fl^ = 24 ; for the seventlif V/i' = IG ; for the eighth, ^j^ = 15 ; and for Uio ninth, ^({^ = 3. wuitipiying by ines.; numofra, wt- obiain fj',',, i'*^'., Un. i'iW •i4(i) -i'4f)> ?iii> 240J ^"" 24o' 52 li KEY. (4) ■^- ^^- H, \l -' [Nat. Arith. The least common multiple of 5, 10, 25, 30, 45, and 60 is 900. The multiplier fcr both terms of the first fraction is ^"1^'° = 180 ; for the second, ''f,? = 90 ; for the third, %^^-^ — 36 ; for the fourth, £y>a = 30 ; for the fifth, 'fSi = 20 ; and for the sixth, "J^f = 15. Multiplying by these numbers, we obtain li^g-, Pl}, f^S, ^^^ 9 0) ^"*^ 9 00' (5) .1.1' ji- i-i -1- 2o> 15oj 40> 60' The last common multiple of 20, 30, 40 and 50 is 600. The multiplier for both terms of the first fraction is %^^r = 30 ; for the second, '^I'lP^ 20; for the third, ^^'^(fz: 15 ; and for the fourth, ^.^^= 12. Multiplying by these numbers, we obtain '^-Jg, 11?^, -^g^ and-g^fo. (6) h h h h h \h \h -li. The least common multiple of 2, 3, 4, 6, 8, 12, 16, and 24 is 48. The multiplier for both terms of the first fraction is ^/ = 24 ; for the second, '^Z = 16 ; for the third, ^ = 12 ; for the fourth, i? =8; for the fifth, ^8^=6 ; for the sixth, -f| = 4 ; for the seventh, U = 3 ; and for the eighth, || = 2. Multiplying by these numbers, we obtain |^, !|f, I'i, i% ||, -^i, i^^,and|'i. (7) D 1 .1 ■.!.. 'a.- ..'J- XI 71 i'Zi lot 2 7> as J 4 0' The least common multiple of 1, 12, 15, 27, 35 and 40 is Y560. The multiplier for both terms of the first fraction is '^^j^^ = 1080 ; for th<^ second, Mf" = *^30 ; for the third, ^^'^" = 504 ; for the fourth, ^^y^ = 280 ; for the fifth, ^t" = 216; for the sixth^ 7A.ja = i89. Multiplying by these numbers, we obtuin 'if,\\\], 'j^ui ylMlu, 552 0) To 6 II J ana 75feo. [Nat. Arith. Exercises 47, 48.] KEY. 68 (8) H, h h H, A, ih h U- The least common multiple of 15, 8, 3, 12, 11, 20, 1, and 35 is 9240. The multiplier for both terms of the first fraction is m^ = 616 ; for the second, ^'\^"z=: 1155 ; for the third, ^^ap. _ 3030 ; for the fourth, ''i^|"-=770; for the fifth, 93ta_840; for the sixth, -'-U^ =z 462 ; for the seventh, ^V^ = 1320 ; for the eighth, a|4a _. 264. Multiplyins' by these numbers, we obtain ^U, f}U, W/o"? ll5a,^§U,ini},?mand§«fe. V4" 4,ti/» and -J^2. jooj 600 •*"" 6T)0' 12, 16, and 24 is Exercise 48 — Page 160. (1) 4X3X 6X35 2520 foff ofAof3| = .1 1 1 7X5X11X72 27720 (2) 2X4X6X 8] X25 97200 'i of * of S of -*l- of 25 — 3X9X7X10.0X24 453600 l\- (3) 5 and 40 is 7560. 1-5-^-" = 504; for U ofWof 5^ = of f of A of |2 =: 21X 6 X77 = 7 35X11X30 / (4) 2X4x 3 Xl3 -,3.1- 2 6 6 43" 5X7X11X17 54 KEY. [Nat. Abith. ExKBCiaE 49 — Page 161. (1) 0f?0ff0fA=: 5X6X2X 3 5X<5X^X ? i "* Id 9X7X3X16 f)X7X^.Xll5 3Xtx4 -A. 84' «ini! (2) of ^- of ^i of -S- of H of I7 = 2X5X 18 X 6 X11X13 3X9X132X11 X13X1'7 2X5X l'8^ X I^XUXlS? 2X5 !5X?)Xl955XliXl!?Xl7 -33X17 33 1 !>.. — 3 6 1' (3) of -A- of 5^ = 2X 4 Xll 1 1 tXllX 2 55X 4 XU 7XliX ^ = *. (4) ^0fA0fU^0fT5^0fi?0fJii^ = ^ SI ixHxnvx !5^ X1!?X1^ IX 8 xn7x 50 X13X13 *- - — -- — ■• 9X13X200X169X17X 6 1 S^Xll?XSIS(f)Xl^5)Xl7x ^ -- — Ff' 17X3 1^ (5) T^of ^of ^of ^f of ?i of V- = 3 '2^ 3 XHX S| Xl?l?X?HX>tt 3X3 nx7xiS^xHtx^^x 7 7X7 3 X4X 9 X33X38X47 11X7X19X47X72X 7 = t(i' I JBXEBCISBS 49, 50.] KEY. 55 (6) 2 1^ ■I of A of M* = 4X3X154 4X3X1^^ 2X4X3 = 24. 7xnx 1 tXliX 1 1 ExERCiSB 50 — Page 162. (1) .11 m -U. 14X25 45X42 1^XS!< 9 9X3 (2) 1 1. 12 rl.7 113 18 IIX 18 12X143 11 X 1^ 15^X1^? 2 13 2X13 = i6' (3) ( 50 X 13X13 15 R3 7^ 7a :ia 78X5 5 X39 ^ X\5S| — 9 (4) 111 31- f 35. 13 35X 5 13X1 2X5 I > 121 9 f tJ.H > ) 3 X68 9 X4 7X3 — ¥04> :ltj> ? !• X 33X38X47 X47x72X 7 (5) 15i' h 3} If « ft fia -,■'< V 4 1 t) 7 ^ X ^ 47X1^ 15(X 7 1HX«|?' ^ X 3' 5 X^4 3 9 2' 1 47X2 7 — , ', , - h, 3ii, iV 3X9 3 5X2 56 KEY. . ENat. Abiie (6) 16| 6^ 17 21^ i figtt 3^ 1^ i^fl J Uf 13 18^' lOf 4| " a^' 1^' 5' ip ^ 10 1 31X111X3 \mx1 1X5 10 3151215 ^3' ^"i' "^7 l^^i = r'eT 6? 7oTe" '^' ^^'H,2.V„V 2 ExBRoiSB 51— Page 163. (1) - of — = /,y of a lb. 5 l(; (2) ^ ? 1 1 - of - of — of — = ? ^% 20 7x6X20 6 ^lii: (3) % 8^ 1 - of — of - = 9 4 1 9X2 = A wk. (4) 81 — of — of - of - = 81 11 ^ 6 11X4X5 = i^o-Eng.Ell. (5) _ of — of — = - of — of — = T?A 11 11 11 ^Vr- per. [Nat. Abxih EXBBO1SB8I60-62.1 KEY. (6) 10 '-» 4 11 ^XHX295X1 295 - of - f>f ai — of- — . - ,. .,,„ P ?X9^ 3 7 14 8 3X7X 14x^ 294 7 0) If , H,H,2-iV,-4V 3 4 1 I 1 3X SXl^Xl XI 3 — ^f_ofp nf— of — 19 n 2 40 4 1S|X17X2X40X^ 17X2; 67 — ly^T C. — = -nPtf u a. ^liii' #0- Eng. Ell. •f. per. ExBBCiSB 52— Page 164. (1) 14 4 2 4 - — of - of - of - = 7 (2 4 28 qrs. 26 2 4 8 na. .2i 18 13 5 na. in, fur. 9)8 (4) per. yds. ft. in. 35 3 2 £ s. d. 7)4 sq. m. a. r. pr.yds.ft. in. 113) 11 (62 1 8 4 2 79t4t 640 7040 a. 678 ■260 226 34 4 136 r. 113 23 40 920 per. 904 16 aoi 480 , 4 484 yds. 484 yds. 453 32 9 288 ft. 226 62 144 248 248 62 8928 in. 791 1018 1017 11 5| [Nat. Aritb, ExBROiSEfl 83, 54.] Iwt. grs; 3 8 r. pr. yds. ft. in. 1 8 4 2 ^9tJt KEY. ExKRCiSK 54— Page 16.5. 6 bus. 1 pk. 1 gal. 1 qt. 1 pt. = 411 ptg. 50 bush. = 3200 pts. And the required fraction is .,V..'o. (2) , 35 per. 9 ft. 2 in. = 7040 in. 1 fur. = 7920 in. The required fraction is TUR — jjs -. » (3) T hrs. 12 min. = 432 min. 1 daj = 1440 min. Therefore the fraction is AV,- — v"i- (4) 2 sq. yds. 2 ft. 120 in. = snoo in 3 sq. per. 13i yds. 1 ft. 72 in. = 135000 in' And the fraction is Teg8gu= -4^. (5) •7 oz. 7 drs. 2 scr. 14 grs. = 3834 grs. ^1 lbs. = 120960 grs. The fraction is T-i^^^=r||^^ = ^7 j_. (6) 9 min. 48 sec. = 588 sec. 1 day = 86400 sec. Tlie required fraction is .Hg^ = ^ig... O) 16 bush. 1 pk. 1 pt. = 1041 pts. 68 bush. = 4416 pts. Therefore the fraction is |J|^ = ^3^v^^ 69 60 KEY. (8) fNAT. AEITH. 3qrs. 3ina. = 15^=1^51 na. 1 Eng. ell = 20 na. And the fraction is — = f^n _ a< (9) 2CL 1 a •— 46 < 13 dwt. 7 gT3. = 319 grs. 1 lb. Troy = 5760 grs. The required fraction is ^Vb^V. (10) 4800 cub. ft. 54 cords = 6912 cub. ft. Therefore the fraction is ^^ = |^g =: fi| -. 25 Exercise 55— Page 167. (1) +*+IS + A = f^ = 2,v (2) tV + 11. + ^ + rk + H + ^= fl == 3^== 3^. 4f + llf + 16|+ 21f,+ 19. = 4 + 11 + 16 + 21 + 19 + (4) m 4?"i"i! ti'ft + 1^^^+11211 = 16 + 11+18+17+112+ (5) 4i + ^ + ., = 4 + i + a + y.^^^,^^^^_^ (6) [NAT. AEITH. I EX=«°'«=« 54, 66.J KEY. 61 - 3< ~ 46« fE = 6 + 21 + 19 + 73f. f 18+17+112+ + 3^1 = 17714. •(iVf + iV^ + n denominator in% + ii^s + (7) » ' "^ %t f ^^•'^"''■^^"ced to their least common denominator become ^tl + ^0 + 3f| = jgi;. -_ 2^;g. (8) Th f .. ^+6 + r+g + A. Those fractions when reduced to their least common denomi- na^r become ii^J + ^+ 5liS + 31^^ + 1^^" = ^V^VV = (9) . i + ^+i + ^ + ^ + |. These fractions when reduced to their least common denomi uator become U9. + VJ" -4- :qa a_ m ■ -o . „ "^"°""- 141' — ■'•1 4 (J- (10) ., 1^ + 47 + 21 + 19 = 103. ^l^^k^ IIP + vwv + vv^ + vvi^ + vv^ = 103+ie§=104-^/-^ (H) 941 + 2^1 = 943^7 (12) 1 + 16 + 1 1; -i :t Iff lit 3 fio 1 .)V 1 Jb* + ^ + 50 + 3 62 KEY. [Nat. Arith (13) } + 5 + 5 + 68i = 68 + (^ -f 2 + 7 4. i). <>8+ lfo^-~69i.|j. = 1' U' '11 (14) 173vV + 85 + 9Mi = 173 + 8 + 91 + (i + r. _L J . N 173 + 8 + 91 = 272. ^^-h.+la). 272 + 1,^J,(| = 273?«^ '^'^ (15) lil + 2i^+3^^+4^H = l+2 + 3+4+ (^§ + ^. 1 + 2 + 3+4= 10. + til;f:^^ + ^" = ^5^+ ^^^8 + ^^^n + mR= J7rt22 I 7 200 = 10 + 33«^ = 1332 4 • (16) ie + li + 11 + u + 11! = Vs^ = 3-A-: ''^ '^^ ■iV + (17) 7 4-lU + 18+26f + 79,,-= 7 + 11 + 18 + (i + ? + tV). •?+ 11 + 18+ 26+ 79 = 141. 141 + 1t^ = 142-,^,-. 26 + 79 + 104* (18) |0f?of^i=V«=3g. t+7,\ + 3i io + i,v,- = ii,7^^^.. 5 10 + (i + A + D. .1')- 1 4 TNat. Arith EXXRCXBM 56 ] KEY. ez = •?^' = I'tj.) f-^ + ^3). ^4. f (U + ?i! + 720(1 — 7200 = + is + A + IJ5. (19) 4i iV L.3. IS- IS X 18 3 X 7 = 7ft _ = lU. ^ on? of ,v of y _ 20S 7A A> 83X11 4X83 1X36X4X11 2X11X15X4 = I* = La If^. = 2; 1H+ 1VV+ 23 = 11 + I + 2 + (f + -■' + 2) = 14+(i-f 14+l-,V*r=15,M 1 4(l< 14 0' (20) 35 + lU + 14^11 = 3 + 11 + 14 + (5 4. J . 2,^ _ ^ + i+^' = n+4^ + {[;! = U = l;3. 28 + U3= 28 4. 2911 iof| = |, ^of« = p> J ofiofiofi=:j^^ 4 3 7; « (21) of 5 = ifij of§5 = ^gof i«of i ■^ + f + -,^. + A + 1.1 = f?io = f2U = ifm. .*!30 r«% + f JV°^ + V«Vo- + 1% + (22) 41i + 1055 + 3005 + 2413 +472i = 41 +105+300+ 2414- + 26+79+1 472 + (J + § + .^ 4. ^ _^ ^) 41 + 105 + 300 +241 + 472 = 1159. Jl!jt*"^^ + ^ = ^"+^^ + ^^^ + {n+.v^=ii^= W = 2U 1159 + 2^-2 = 1161?, (23) (i + A + K). I (A + ^^ + I) 92f, + 37A + 7| = 92 + 37 + 7 + (A- + ^%- + 4)= ^)=136+ 14 + .A + ?i = 'm + ?;l^ + 5?? ~ J - - .... J f J *^ — 136+i«B-^ = 137.?t^ _ J 43 — ^ » - tl , 64 10? V V 53 X 5 6X12 KEY. (154) » = U = 4^V fNAT. Artth. i of I = ,^,. 21J + 35^ + 4A + i'/ = 21 + 35 + 6 + (i + i) = mu, (25) W = of8. Vof,VofTofi!o. = Y = i8 " IOt^. + 6lf. 4- lli\U + IS = 10 4- 6 + 15 + 1 + f,i 4. 32 + 2H,^H==34|i:ii5 (1) Exercise 5G— Pago 169. A of a lb. 7 of an oz. A of a dr. ^ of a scr. oz. dr. scr. grs. 4 2 2 14^ 3 1 5^ 16f ^ of a yard I of an Eng. ell. ^ of a qr. • (2) qr. na. in. = 2 1 1/, = 2 l}-a = 3 O'il 4 6 2 (3) m. j of a yd. = 5f 7 of a ft. r= If i of an in.= j mvi fur. per. tV of a mile = 5 3 12 3 3 * 140 tV of a fur. v'a of a yd. (4) yds. ft. in. 3 1 6 1 2 1 2i«, 01^ 5 16 (6) . day hrs. min. i of a week = l is o i; of a day =r s '\ of an liour= 12 2 2 12 lofai; f of Ad. (6) s. = 2 •| of a 3 d. lOf 25 lu •^1 4S [Nat. Artth. W =1513'. ^ + 1 + (.^« + ■ (2) qr na. in. = 2 1 l-l, — 2 m = 3 or. 3 3 (4) iiyy rds. ft. in. 3 1 6 1 2 OIS 1 2A h% (6) s. d. : 2 lOf 2| A 3 Igi ezjuioifln 6(Hi7.] KIY. 0) 65 I of 218. = i of 6s. = f of jea 12s. 6d. = 2 iV of a X -- ltd. 1. d. 13 U 3 li S 3J 10 9-/V 3 12 m Exiioisi 57— Page 171. (1) 3 X 48 (2) ^of^ofn = — -^Hf. ,^^4-^^ = ^^^ + 17X11 _!? __ ^-^ _ 35X11 1§ - 1? = 0. ^ iii; 11 4X2 = ¥ = (3) JS^ "" ^^^° = ^^Wa, - 29tH§ = 981 + K3VL 29}n2 = 981?n)J - 29+5f«g = 952,4^ ' ' + '^'^ - (4) isHIS 69sV — 18t»^^ = 69y4*Af — 18*348 — fift a. i no 68Hit - 18^5$ = 50H2f = 5t;!lr '' "^ '''^' - (5) lOOi - 98 = lOOJ - 98 = 99 + 13 -- 9^ = 99,^,^ g. (6) 9f=s90i i'll=l=='^- 6--4S=6|-4f = 5V - 4f = 1§ 4^ = 5 + 18 _ 41 >- 0) s • ^jYVrfai-- m KET. (8) [Nat. Asith. ¥-iV = il^-il = f| fo^'^T'^\„*+J = A + ^ = i^ fofiJ = (10) v^' - = fr- 11 1^ H^ S|(l (11) 9^^ tT ? ^5)1 167 3X9xift7 ; ^~ ^~ X - X - x-x-x— = ^ 11 IJI t 1? 1 JJ^ ~=*V^=1127i. (12) la - X -- X — X f V i^ w X i'f X ? = ^X55 tl^xS) IS^xi^ 1 S^ ^X^ 7xiS( S(X!^ ""• X — X — = 101x5 !^ ^ 7x 101 — T^T- -x-x-x-- = (13) 19 2X2X19 ■jz: j!j. /' = lOf [Nat. Abith. - = V=13|. ¥ = 9?. EzBRonx 68.] 32 KEY. (14) 69 9 *? 11 ^i?«| 9X7X32 fx-x—x = 1^ 1!^ 11 — = «(LLfi — V^ = 403}. (16) 27 T )t ^ 7 X - X - X - = ^ «* 6 t !S = 27, TT* =^^^=1127i. 11 13 15 (16) 11X13X15 ? 1 1?!^ — X — X — = « 8 8 (17) '^ IS) $^ 11^ 49 - X ~ X — X — X ~ V V .. ^ ^* ^^ 191 8 4t!D^o^7:^:rx-x-x-x-x — = = ^'^^=268^. ^ 27 ?i ? 7X49X27X191 3X11X17 IS) S 11 Trftr^.^^~"X = I7fi.88&j. __ ifl^ = 4729§^^. m-^^- (18) 27 873 3^ 98J 2.} 128 iF^'lII^r^I^J^ '^^XS ^^x ^ 7X8 X ■ . X - 1^51 ^1 ^ 3 gxt^^ ^xt '^ 11 xn^~'^i-~^' (19) 1 $-x-x-x^ = nil 11 7 15 1S| 11X7 Biir 70 KEY* (20) •^51 ? of 8i X 1^ of 28 7i- i CNat. AMvm. 100 6tV Aof6| X iVof 24 15 ^ X — X - X 14? X — X H 9 Jl !Ji_^_>fJiofiofX--fofAofiof^ = !!x-X-X^x' i^, ^ 1 3X2X7X6 252^^ ^3 ^ ^ ^ ? 13X5 65 V ^X2 7X4 4X9 * 3X9 (13) ^X^ 3XS( ^X5| tX4 = 3 2X4 3 ^V -r - = - ^ ¥ 25 (14) 21X 2 5 X35 V (15) — X ^ x?^ %IX 2 = i. 113 H'x^4-?x V/x — = -~ i^i ^ 1 t l^ X - X - X X 9 3 107 2X1?^: 113X2X17 9X3X107 3ftil5! 2889 = l^ u (16) tKAT.ABt1». f 9 f 2X9 7X3 '^^7^.^ — ^ -^ — X — X ^X7 2 10X3 9XT 19 3 . , X - X - X — = 41 3X4 7 7X2 11X7 8X7 4 X4 -X— ^ t 9X^ 19 \xX i x\ 5 31X3X9X19X4 63612 ?XH tx^ 11X7 5X41X11 2255 H 19X3 f 11X5 ExBROiSH 61— Page 176. (1) = H. ^8 143. 6id.'-Hn=ri;8 143. 6|d. X ^^ =: £8 143. 6id. X 55 = £8 8s. 5id. 57 (2) « '^ X H = W- 1 m. 5 fur. 91 yds. 2 ft. -r- ^ii =: 1 m. 6 fur. 91 yds. 2 ft. X i% = . ax ^ m. 1 m. 5 fur. 91 yds. 2 ft. x 22 -' = 2 fur. 124 yds. 2 ft. 115 (3) 3 a. 3 r. 3 per. -r ? = 3 a. 3 r. 3 p. x f 6 a. 1 r. 5 per. (4) £7 16s. 2d. -^ ^ = £7 16s. 2d. X I : Jei7 Us. 4id. 3 a. 3 r. 3 p. X 5 3 £7 163. 2d. X 9 . i 6 [Kat. Aatlitt. I DxXHoisss 00-62.] C ~ X — : f i. 7X2 11x7 X ■ : 8X7 4 X4 81X^ ^ X^ tX^ IIXT 8iE^% KBt. m H 9 i* 1,3 ExBBorsB 62--Page 178. (1) ^'^ 76 W If"" 13X27 i 3. eid. X H =2 I. -f W = 1 m. . 2 ft. r. 3 p. X 5 3 "" is. 2d. X 9 4} f of32 T 3J 7 X35 13X27 I V^ I ^ 735 X - = = 846 «. 351 ?^? = 2f.V 6i i (2) f 2 21X13 If = 3. 8 7X13X19 42 7X13X19 42 8 42 ' 7X13X19 6 " Wki 76 ii H H 6 ii 81 T V T g KEY. (3) — 6'3 • T^ 220 8 220 9 55 8 U ~ 63 9 63 800 2000 420 = I ExBBoiSB 63— Page 180. (1) 100 ■2000 - AV. 36 2000 2000 — Too* (2) 2 ^ i — V(r- 160 2000 CNat. AmiTM. = |. — i'S' f of^ Off of i off of^, ^- X - X - X - X - X ~=r ,V ^ ^ ^ ^ !^ 45 (3) 61 X 65f cts. = ^ X ^P cts. = iVi^^ cts = $4.52^^. BxiBoiBia 82, 68J KEY. (0 77 3 + TV = VW + TVe = WV. (6) ^ + T^ + * + i = iV.r + iVa + 1 or J{ lV2'»Jf + V'^ = A'a = t«. = ii ^^2i 4i+6^^ (6) 4|H6f8 - X il!! X ^ - 2X64 ^ 128 *^ «x I 6xi|| - 7;^; - IT = '^* 1670:iVX12i^ts. =aijy 7 xiicts —110746, . Hence the longer = ? y q — a «f *!, u - « X 8 - 64 ft, , and longer = 136—64=72 1 (9) $1000 Sl3Wi+ $683.+ $4013A = »6831 + ».013-,3 $1000 r$1376i»,^ + l1 6 78 I ; KET. (10) [Hat. Akxth. (11) , 19JX|62=H«X$V = $iJ|a = $i34.I5f. (12) 376rt X $751 = m9 X $H^ = i^ijax ^ $28387-06i. 147J + 320^ 467n - 166M (13) * 147i^ + 320^V = 467^. 467i* 311gJ. — 166i = (14) 7X3X3 7(11 of I) IxlXl 1X2X4 — -71= 4-V= x,v *(3T") *x-x]P ^xigx^ ^ i + i+i IX^X^ iSi^ 111 hx^xi A+A+A il 5J -H— +- - ^ H H 4i I a if 2636 1 1 1 f^ V (15) = IWe- 17f^7^ = ip^V = H' X A = W=ra^|r i=2)l [Kat. Akitb. ExmoiBues, 06. ] KBY. (16) 79 lit n i ^r•^%t,%:^Jll = ^^^^ = w. T'«-ii=lftf*. I »4*+93i = 94,V + 93^S 187}J = iq,yi. T53 tD9 X M -f- ^;^/i = -! V !2 ^^ (17) M^ ^1? = l. 2f + 1 + 4= 2fff + 4« + 4_ 77 _ ?.'^'^ = 2Xt^/^ = — 2 = J,^ — ! — 683 I5«l< 10 '" 72 (18) i + i = i U+2i=:4A = 1§, 2,V-l.J=,i^ = ^. ^ ^SJ * 187 187 . . 4-722 = Am = 4{f = f t and -198 = ^f| = f .^j. fl X m- = m = -935. Exercise 76 — Page 196. (1) •m ^ ^V and -123 = Ul = ^h- ■r iVj = ^=V X W = f = -6. (2) 389-186 = 389^§0 = 38^?^^ ^nd 15-7 = 15^ = ip. '%W -^ H' = ^^7^ X Tk = Yi¥ = 24-6. (3) , -81664168350 = fiMi.§g^m- = mnmuh •47053 =: ^^U = IMU- mmmn -r ims = mumm x !h^ = %wm^^ ■«V: 1-736, v«*r>i' «;^'#. [Nat. Abiih r5. ^ = v<^. BximoiBBa 7ft*77,] •45= I (4) = TOr and '118881 89 3-823529411'';a058. (50 •6. 24-6. .2AfiJUyiPg3/: (1) iof?of,Vofl4=ixf XiVXi^ = ^=^,8, (2) '67 = ^ and 2'13 = 2^^ = J^gi. H X =^V- = -"gWd^ = 1-4445566778 + wk. (3) •678125 = 4 days 17 hours 65 minutes 30 7 4-746875 days. 24 2987500 1493750 17925000 hours. 60 66-600000 minutes. 60 ^A'AAAAAA .« J- seconds. (4) •92437 92 •92437= aim =1^8.' c 90 KEY. [Nat. Abith. Dissimilar. 67-234 98-713 91-03471234 (5) Similar. Similar and Coterminous. o7-2343434 zz: 07-23434343434 98-71371371 = 98-71371371371 91-03471234 = 9103471234234 Sum, = 256-98276949039 Dissimilar. Similar and Coterminous. 256-98276949039 = 256-98276949039 100-123466789 = 100-12345678945 Diflference = 156-80931270094 (6) 12) 9 in. 3) 2-75 ft. 5j) 2-916 yds. 2 2 11) 5-833 ■ ' ■ • 40) 36-5303 rds. 8) 5-913257 fur. •739157196 miles. 17-428571 sq. ft. =17JB|f5^sq. ft. = 17^ sq. ft.=: 100-8 sa. in. — 100-8 sq. in. Pifierence, = 17 sq.ft. 61^ in lOOJ 16 sq.ft. 104^^in. ^fflipisjB 77.] KEY. (8) 91 •91789772 917897 \ •9n897.>of.a=||8HS^Xf...^^g|Ji.-g la.3r. 13per. -^2 748. ViU^ 11 -28? 2 (9) 11-287:. Um-lli^ 1-04285 W*=HmV.= l A. (10) 47- 45 = YJ' and 1-76 = IJ^ = Vj/^. Dissimilar. 85-62 13-76432 (11) Similar. Similar and Coterminous. = 85-626 = 85.62626 = 13-76432 = 13-76432 Difference, (12) = 71-86193 :i3) -T34 ofa lb. =: 11-744 oz. 2ft 5t in - 901 in ss - •198ofanoz.= .]98 oz. ~ * "' == ^ '"' Difference, =17^ oz ^^'^ ^*- = ^^^ ^*' = ^^S in (14) 3-145 = 3^U = 3^« = V,3 and 4-29Y = 4M^, = 4H=W = Vs^* X \V =V,A,V = 13-5169533. ,%?"«»- IMAGE EVALUATION TEST TARGET (MT-S) 7a 1.0 1^128 |2.5 2.2 I.I 1.25 us 140 K^ U 1 1.6 Sciences Corporation 23 WEST MAIN STREET WEBSTER, N.Y. 14580 (716) 873-4503 ^ i i 92 KEY. [Nat. Abttb .8 40* (15) Here 40 = 2" x 6. Therefore the equivalent decimal will contain 3 places. 1^. " 24 = 2' X 3. Vfe. " 15 =: 6 X 3. ViV " 144 = 2* X 32. h-=i^ " 15 = 5 X 3, nV^i. " 3584 = 29 X t. <( « 3 1 4 1 9 Dissimilar. 81-6 61-126 328-23 6-624 (16) 81| = 81-6 and 328y =: 328-23. Similar 81-666 61-126 328-2323 • ■ 5-62462 Sum, (17) Similar and Coterminous 81-66666666 61-12666666 328-23232323 5-62462462 2 carried = 476-65028119 (4-4— 2-83 6-8 X3\ l-*64-2-629 ^'^^ / (1-61 4-296 + 20-4 2-25 n) 1 ) ■f + 2-8X2-27 1136 2^X2f^ 21^X2A 1-A VxA ^ (4?? ^ t) "^ *4 = (Hi X W) + = (i X M(')''+ V = V -f V = V = 9, [Kav. Abite ^i«B0f«E8 ft, 78.] jgjj^^ ExiBoisi 78— Page 198. §3 9)4312131 9)224322.. 8 9)12043.. 5 9)344. . 7 9)21. .0 1..2 (1) 3)4312131 3) 1234023.. 2 3)224322.. 2 3)41240.. 2 3)12043.. 1 3)2144.. 1 3)344.. 2 3)113. .0 3)21.. 3)3.. 2 1..0 V 8)4312131 8)242343.. 7 8)14022.. 2 8)1032.. 1 8)32.. 6 2..1 4312131 = 6 23 6 116 6 582 5 2911 5 14568 5 iz 120758 = 9 11 9 99 9 in 10200211222 £= 3 898 9 8087 9 3 3 11 3 33 3 99 3 299 3 898 3 2d95 3 8087 3 VllI 216127 8 17 8 142 8 1137 8 9098 8 72791 dec. 299 T2791 decimal. 24263 3 7!2901 d«» V2791 dec. u Kilt. tUAt. Aaittt. (3) 976'432 -^ -00000096 = 9'7643200000 -^ 96 and 96 = 12 X 8 12)97643200000 ' 8)8136933333-3 1017116666-6 (4) (21 + -5625 - 1-5 + -/,.) ^ Y (iT^r X J X 296 X tJt -r V) -r '9472947 (2|+T^-U+-iV)XA- V f Xt^t m x^X^Fx-HrT X^SrH-taSi H X 5 X 4^ X TirX^Xg^fS if If ¥ — IJS -6*" = 2f. 12 116 8 927 3 2783 20 66S60 (5) lbs. oz. dr. scr. lbs. oz. dr. scr. grs. 9 •? 7 2 ) 97 3 4 1 17 12 1167 8 9340 3 28021 20 660437(10j3^Yrf. 66660 3837 tl^At. A&l¥tt. d 96= 12 X 8. hX^rXUn titEUcm 78.] ItBY. (6) 16 yds. = 540 in. and 1 ft. = 84 in. 6 ft. = 72 in. and 4 ft. = 48 in. (640X84X13) ^ (72X48X13) = 589680 ~ 44928 544752 -i- 108 = 5044. = 644752. 0) 9 ft. 6' 4" 7'" 11 7 9 11 8 7 1 5 6 8 104 10 2 8 10"" 2'"" 6""" •9 5 3 8 1 6 111 9 7 4 (8) 4Hi-T^ m-^'i Wi (») = •— = — = \h^i-^ = 8iii^ W (10) pts. 2)782436 V7jn..42..27..21..33..14..t..ii..63.. 30 4)391218.. Opt. 2)97804.. 2 qt. 4)48902.. gal. 27 10 77X27X10 = 20790 = 1. c. mi 12225,, 2 pks. 12226 buEh. 2 pks. gal. 2 qts. &e KEY. [Kat. Afttftt, XII (11) IX 28e4)36«7942(1375M2 3762814 28e4 9e47 82^0 18679 17274 14054 11888 23882 23554 32^0 28e-4 5^80 55-^8 9 34 9 312 9 2810 9 25298 9 227683 9 2049151 (12) 150528 = 2»<'X3X7«. 10+1 = 11 1+1 = 2 2+1 = 3 11X3X2 = 66. 4-94 (13) (14) 2 wks. 2 dys. :s 16 dys. 728J = 8i + 2 X 10 + 7 X 10 XlO. •1234625 lbs. oz. dr. lbs. oz. dr. 16 27 4 3 X 8i = 10 7407750 1234625 272 9 14 X 2 10 1*9754000 dys. 24 2726 2 12 X 7 39016000 19508000 23-4096000 hrs. 60 231 11 9^ 545 3 12 19083 3 4 19860 2 di 24-5760000 min. 60 34 '5600000 = 34 ^ sec. [Kat. A&tttt, 'X3X72. 11 2 3 66. rTxlOxlO. )s. oz. dr. 31 11 9^ 15 3 12 ?3 3 4 30 2 9i ii^XEBGISS 78.] KEY. (15) 97 $430.27,V-.$271.52,V = $158-75.1 '" (16) •8 = «. -76 = 5|. .9123 3327—3 •003327 = 998000 9123—91 9900 = im vUlU = yflaTf* r^. (18) f('7632763xil)xUfmivn.f.....o\.. cf-25of96)-i-2 ,i of '6732467 4-^ Ma I X ^.i- X i X i g^ X ^ X j^ X ;. V .1 ;7irr7 '6^ h"- 10831 — o-r.hJ'. mn t 6 ti y r* G 98 KEY. INat. Abith, (19) 8 cbildren will have 8 children's shares. One woman will have 3 children's shares .'. 6 women will have 6 X 3 = 18 children's shares. One man will have 6 children's shares .*. 4 men will have 4 X 6 = 24 children's shares. 4 men, 6 women, and 8 child, will therefore have 50 child, shares. JESSO 33. lid. -f- 50 = £11 Os. OJd. = child's share. £ 11 Os. O^d. X 3 = .^33 Os. 2Jd. = woman's share. £ 33 Os. 2id. X 2 = £66 Os. 4|d. = man's share. (20) 16j2r + 19| + 23J 4- 129f = 16 + 19 4- 23 + 129 + (A + ^ + ! + f) = 187 + Sfo^ff = 190^0"^-. (21) 8100 = 2» X 3* X 5*. li«>>.»>7..Ai««Ol 1..2..4 1.. 3.. 9.. 27.. 81.. 2.. 6.. 18.. 54.. 162.. 4.. 12.. 36.. 108.. 324 1..5..25 1.. 3.. 9.. 27.. 81.. 2.. 6.. 18.. 54.. 162.. 4.. 12.. 36.. 108.. 324.. 5.. 15.. 45.. 135.. 405.. 10.. 30.. 90.. 270.. 810.. 20.. 60.. 180.. 540. . 1620. .25. . 75. . 225. . 675. .2025. . 50. . 150. . 450. . 1350 .. 4050.. 100.. 300.. 900.. 2700.. 8100. ' Therefore the divisors of 8100 are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 25, 27, 30, 36, 45, 50, 54, 60, 75, 81, 90,100, 108, 135, 150, 162, 180, 225, 270, 300, 324, 405* 450, 540, 675, 810, 900, 1350, 1620, 2025, 2700, 4050, 8100* INAT. Abith. I Esbrcisk 78.J (22) 2691) 1181 7/4 10764 men will have .36.. 108 .. 324 KEY, 1053)2691(2 2106 fi85)1053(l 685 468)585(1 468 2^)468(4 468 09 (23) 86c. 60)2551443 60)42524.. 3 24)708.. 44 Md, 12 h., 44 m.,1-sec' sec. 60)31556928 60)625948.. 48 24)8765.. 48 9828 is divisible by 117 .li7i«.. n Jf ^^< ••il7istheG. 0. AL 365d.,5h.,48m.,48'ef*' (24) J4 ft. 11 in. _ 38 miles H ^.^J^^^^- 2407680^ 179 IT 2407680 in. ' iW = 13450i^§. (25) One cubic follt^^^'l^l^^^^^^^^^^^^^ '-^'^t. -.uo..ei..ioi^-sS::r'^^^^^^ 134062J Iba. ' "~ ^^^^^^ = gals- in (26) ^^3 X 400 = lua -, «»rx. ^^^- X 20 = "»^J =^47far. X 6^12 = $292-00 3-40 ■19iV 93iV - 76H = 92H - 76H = lei^^^ - 4Aatt 106 61 -ft X >of KEY. (28) Hot Ah V X I (Nat. Arith. Ifofen-lO^ 135of6i fxfx^V X g X X I X 8 XV if^ X V ixA aTj"! ^15X3X31 16X2 9 X - X 5 ?t zi 6)91342 2X? Xl?tX2 til =-- 6AV (29) ZI 12)91342 5)19074.. 4 6)40 15.. 1 6)891. .0 6)184. .3 6)39.. 3 5)8. .2 1..8 12)83M..9 12)7'73..1 12)10.. 3 6.. 6 3X9X31 16X2X2X2 2)91342 2)46176..! 2)23093.. 2)115a..l 2)62*6.. 2)3153.. 2)1627.. 2)869.. 2)434. . 1 2)217.. 1 2)109. .0 2)5^.0 2)2*.. 1 2)15.. 2)8.. 2)4.. 2)2.. 1..0 (Nat. Aaith. X9X31 C2X2X2 2)91342 2)46176.. 1 2)23093.. 2)115/1. .1 2)62i6..0 2)3153.. 2)1627.. 2)869.. 2)434.. 1 2)217. .1 2)109. .0 2)6^.0 2)2/.. 1 2)15.. 2)8. .0 2)4.. 2)2.. 1..0 Ezinciss 78.] 11 91342 11 100 U 1103 11 12137 11 KEY (29 continued.) ▼ »x 13233014 60319 101 II 8 & 42 5 213 5 133509 dec. 1068 5 6340 5 26701 5 , 133608 deo. 12 n 12 927 12 11125 12 100000100110000101 2 2 2 2 8 2 133608 dec. 16 2 32 2 65 a 180 a SCO 260 2 621 2 1043 2 2086 2 4172 2 8344 2 16688 2 33377 2 66754 3 1183609 dec. 102 KEY. [Nat. Abith. (30) (31) 2)7680 = 2»X3X5 m. f. p. y. ft. in 2)3840 2)1920 2)960 2)480 2)240 2)120 2)60 2)30 3)15 1 72 3 7 2 1 7 8 579 fur. 40 23167 per. Si (32) $•47 X 97 5= $45-59. (88) (73X4Xll)-rl28=25^V $3-62JX25s='g=$90'96ii. 115837 11583} 127420} yda. 3 382262} ft. la 4587157 in. 12 ■t." 55045884 lines (34) 93*723 = 93U^ = mi^ and 29-4173 = 29|^fg = a§||.ja. 92786 $S|5((S{ 92786X111 11 293879 11X293879 'm^ = 3-185988 + (35) One bushel of oata weighs 34 lbs. .-. in 73429 lbs. there are 73429 -i- 34 = 2159|2 bushels. (36) ' In 719630 lbs. of wheat there are 719630 ^ 60 = 119935 bus. $1-80 X 11993^ = $21588-90, Or $1-80 per bushel = 3 cents per lb. 719630 X 3 = 2158890 cepts. = $21588-90. INat. Abitb, (32) 97 = $45-59. (33) )~128=25^!',. ,='4=$90'962i. £zsBoiu 78.J ' . > Kill 13879 }S. there are : 11993§ bus. )0. KEY. (38) 21389)180781(8 171112 103 (37) $72 14 + $93.76 = $165.90 $165.90X9 47 = $1571-0730 $1671.0730 ^11 = $142-8248+ 9669)21389(2 19338 2051)9669(4 8204 1465)2051(1 1465 686)1465(2 1172 293)586(2 Last divisor 293 = G. 0. M. (39) •"i^f. h ?, ^f, II, iV, J. 210; for il^! second 4""^!!^;^^ for the fourth, ^-K^ L t 7J']l ^Z I^' *^''^' '"'^^ = 3^0 ; sixth, »? . = ',,lt r-2 ' ''' *be fifth, mo ^ igg ^ » 1 ^Ji , for the seventh, Wa =1155 (40) ^' $11 X 17 = $1.87. $37* y 7Q~ «^,o. * Jl-87 + fj.laj + $31-461 +V>sIM.! = *"''^^'- 104 KEY, Nat. Arith ExERCiSB 84— Page 210. (1) 1-22 Baskets. $119^4^X87 11 ; 87 :: $13'42 : = $i06'14. 11 (2) 19 Cords. $^1^15X25 28 ; 26 :: $266 : = $237-60. 2 v«) days 1«|X83'60 $29-20 ; $83-60 :: 16 ; = 46^f days. 7-3 (4) •8 $n-!i(iiXl66 16 : 166 :; $12-80 : = $124-80. Hi (5) Feet. ft. 7X112 6; 112 :: 7 : =166fa (6) 9 Cows. days. S|!|X27 35 : 27 :: 99 : =48| day0„ 6 or Acres, bus. 9X48 \f . *v , , ;? , __ ouy UU0U. EXXBCISS 84.J KEY. 105 (8) Perciies. days. 2X$Q8 73; 803:: 2:- = 22 days. (9) Fails. lbs. 100X1128 l^c : 1128 :: 100 : ^=640^ lbs, 22 (10) •58 155 108: 405.-: ,.0.38 :!?!;!i>'«i=,s,.^. s s 16 : 1278 ::12": (11) 9 639 brls. t^Xl^tl^ = 6751 barrels. HI % (12) 16 Men. Acres lli(!5x3 11 : 3 :: 165 : = 45 ^^^g^ 11 (13) 125 Barrels, loaves ^^Qx 67 4 : 67 :: 250 : = 41871 loaves. 2 (14> Bushels, brls. 16 X??^ 190 : 38 :: 16 : m 9 3^ barrels. 106 K£T. [Nat. AxurB. (16) 6 Days. men S|()X12 15 : 12 :: 90 (16) D'. work. brls. 2x279 17 : 279 :: 2 = 72 men • • «i • 17 (1^) = 32i^ barrels. Botirs. miles. 1 : 24 :: 27 : 27 X 24 = 648 miles. (18) Cows. lbs. 30X23 7: 23 :: 30 ; = 98f Ibi. t A I U •■•• $9^50 ExiBciSB 85— Page 211. (1) 375 7 5|t^(^ %l 16 X — X -— = $42000. (2) %^ ? ]) s. Yard. s. 5 l i : i;: f : - X - x- = ^f = 2?d ^ ^ 1 (3) Tons. i: 8^ :: $7*49 $7-49X8 1-07 p-yi k X'H,^ 25 !^ - = X — X - = $80-25, Yards. 6^ : f :: $28*42 (4) •14 ? X - X — = $2-80. 1 ^k JSZXB0I6ZS 84-S6.J KEY. (6) 107 ^ 100: 472ii •• 981- !!i^^f^_. 98-8^6X472.44 100 IJJ = $467.12}. (8) ^ 2 (9) ioTfof3^.tof/otf^::^ofXof4t:i^°^^''«^^**^^«^ 15 224 i of I of 3^ •« «» ExraciM 86— Page 212. (1) 37sq.yds.4ft,120in.-=48648ia.,and9sq.yd8.2ft.= 11962in 2027 n9w'-'i8648 •• $3.B0 . '-^^X^W ^j'j^ . 4Hb48 .. ^'50 : = $14'245W- 498 108 KEY. (2) 12 lbs. 10 oz. rr 154 bz. Ounces. 1 : 154 :: $1-25 : 1-25 X 154 =: $1$2'50. [NAT. AXITB. (3) 10 jds. = 40 qrs, and 3 yds. 2 qrs. = 14 qrs. •IT 1 Quarters. I^-^(|XU 40 : 14 :: $3-40 : = $119. (4) 15 oz. 12 dwt. 16 grs. = 7604 grs., and 13 oz. 14 grs.= 6254 grs. •96 312T Grains. l^-^i^xiiX^^ 7504 : 6254 :: $3-80 : = $3*167 + 938 (6) 3 lbs. 1 oz. 11 dwt.=751 dwt. and 12 lbs. 6 oz. 4 dwt.=3004 dwt. 160 Dwt. $ H<)<^Xt^l 3004 : 751 :: 600 : = $160. (6) Barrels, h. m. s. 2 h. 46 m. 39 s. X %^ 64 : 24 :: 2 46 30 : = 1 br, 14 mi«, 9 iV. ASIT«. 6264 grs. Exjc&oiBK 86.] t3 yds. 3 qrs. 2 na. 1 in= KEY. 109 AU"jf,'T^!'^a'^v,vir^"-="'*'"'- 172id. *'^Vo'^" d. == i;i28 68. lOiid. ^^ (8) Ounces. 8^1b8. = i36f oz. 7 205 a 4, . 1361 . 8fi . _ X --- X^-- = ^U'=zm 93. Oil 16 (») Pages. 327 : 400 :: 166 : 62 11515 X 400 109 {004 dwt. (10) 46 a., 3 r., 14 p. = 7494 p., and 35 a., 2 r., 10 p. 60 » > F- Perches. £ Hijqx5690 7494 : 6690 ;; 100 : —1 .-£75 igg 3747 = 6690 p. mm. 17 Days, miles. 18 XHI^ 48 : 68 :: 12 : =17 j^^^g p^, ^^^^ no KEY. [^AT. ARITH. (12) 113 Shillings, lbs. 113 MS 3 m : 32, :: ,6J .• _. X _ X - = VW = 24 W. lb, 32 i (13) n493 X 1000 X 5 cub. ft. = 87465000 cub. ft. 192724 X 1000 X 4 cub. X = 770896000 cub. ft. 87465000 + 770896000 =858361000 cub. ft. Cubic feet. ton. 858361QQQ 9000 : 8683610C0 :: 1 : !!=: 953731 tons. (14) 50000x9000 = 450000000=:cub.ft. of gas in 50000 tons of coal Cubic feet. hour. 4 ; 460000000 :; 1 : ^^^^=U2500000h.= 12842 y.l 70 d. (15) lbs. lbs. lbs. lb. lb. 4+3 + 2 + 1 + i = 10ilbs. ibs. 11270 lOJ : 1127a;: 1 : — .-=io73,an(i31 lbs. remaining. lOJ \ (16) Yards. 180 miles = 180 X 1760 = 316800 yards, day. 3168^t) X 1 iOO : SiOSOO :■ I ; 3JC8 dys, or about 85 yrs. * WtJj iSxsBoiaus 86, 87.] KEY. Ill Emroisb 87— Page 216, (1) 120 : 90 bush. 6 ."14 hjorges. I 56 days 7 llj -= 7x14 = 98 days. (2) 28 : 32 ft. high. 8 .' 15 days. 63 men 9 3* «i?X8^X16 ^X^^ = 9X15 = 135 men. (3) 3 .' 45 length. 10 : 100 length. ) U •■ H width .1-- 3 lbs. : (4) 3XliXlOO ijxio - = 2xiiXlO = 25 1b8. 44 : 132 tons. 18 ; 5 days (6) 2 - 2 Jj ^..12horses:_.-_ = 2X6 = iohorse8. (6) to g 4 : 14 men. i 27x1^X10 . / . . * 13. ; — 1_«. . 7 ; 10 days. ) ^^^ 27x5 = 135s.=i£6 l.-Js. j 112 KEY. [Nat. Arith. 3:6 masters. 8:10 apprentices. 6:8 weeks 6:6J days per wk. 8 lMX5JXJiXlOx8{ ::$144 : — — ^ — ^440^ J?xl^X5xt« ( (8) 6 : 18 s.mak. :: 36 pairs of men's shoes : 9 3 1?I5X1^X5 4 : 6 weeks. ^ j^-^a 135 pairs men's and the women's = U = f of 135 = 90 pairs. 9 : 18 feet high. 4 : 6 dajs. \- : 390 miles. ) : 14 hours. } 130 : 390 miles. 1 10 : 60 oz. 22^ : 30 d oz. ) 10 : 5 compositors ' 1 : 14 hours. 20 : 40 sheets. 24 : 16 pages. 50 : 60 lines. 40 : 50 letters. (9) 3 2 15^X11^X6 12 men : = 3X2X6 = 36 men. (10) 2 3 3xlHX?5iq 3 days : = 3X2X3 = 18 days. (11) 60X30 i^q ^Q 1 2 , ■=~X — X— X — = 4X2=8d. 10X22i 1 1 1(| ^^ (12) 2 % 1^ ii«5x^xi^xJt«|xiex««isix5ii 16 days : .^ Hi|Xtx!ilSlx5i^X^(i|X^IS| 2 X 16 = 32 days* [Nat. Abith. KUftOMB 87.J KEY. (13'i 118 386 ; 240 men. fi •" 9 days. 10 : 12 hours. 6 ; 6 degrees. j" • • 70 yards ; fi;3yardswide ^JJXi^XUlXl^XHx? * • 2 yards deep J ^"^ 9X2X2 = 36yard8. t 5f 2 .^«lXS^«lX9xi!^XJfX?X2 6 : 12 horses. 4 : 9 months 1.5 • (14) 4 2 ■"l^='^^^^=^2acreB. (16) 25; 139 persons) 21 ^ :: 300 bush . y^^^^X^ j;^ =11676 bushels. 1 : 7 years. J 48 : 32 men. 36 : 864 feet long. 8 •■ 5 feet high. 4: 3 feet wide. (16) 3 2 l^^ 4days-'*il^^^*^^X^X^_ (17) 679: 22407 sold's. 336 ; 112 days. ►ld's.> -. i'' 702 bushels ; 234 33 .^«I^XJ?!^Jt«ltxU^ «|t!^X?9l| 234 X 33 = 7722 bushelB. (18) ? 13 : 494 suits. > 12X 19:27days. r**^^*^^'*' — 2 %^ 12X^S(^X27 \ H *if ^ ill = 648 tailors. 114 KBT. [Mat. Arith. (19) 17:40 head of cattle ) 6 a. 2 r. 10 p.x^()XS)l > :: 5a. 2 r. 10 p. :— - 30:61 days. 6 a. 2 r. 10 p. X 4 = 22 a. 1 r. (20) 30 5 20 : 100 ft. long : : 180 bricks : 6 : 4feet wide. ) %I^X^ 30 X 5 X 4 = 600 bricks. EzERCiSB 88.— Page 21 (1) A 7 cords 87 lbs. 19 barrels 92 days' work = 116 lbs. = 23 barrels = 34 days* Work 57 baskets peaches 31 baskets peaches = 24 dollars 12 dollars = 2 tons 35 tons = X cords WxJSltXl$X5(^X3lXl^X35 lll?X!^?X?^X^^X^4X2 ^ 2 :^ 2 31X36 2X2X2 = 11^ = 135§. (2) 6 lbs. tea = 29 lbs. sugar 17 lbs. sugar = 1 bushel 1 27 bushels = 4 tons 34 tons = 15 cows 1 i 29 cows = 1160 dollars 20 dollars 3 9 17 $Xl7xS7x94XJi$X5i^ j^Sixix^xi^ximq X 5 58 = X lbs. tea. 17X17X27 1 ' * 6X58 = Wo* = £Nat. Abith. ISxracjaw 87. 88.J KEY. (8) = 21 bush, potatoei 116 = 29 = 44 = 38 = 65 = nj " 11 bush, barlej ^^ " potatoes 11« " oats 141 " wheat 60 " peas 36 " clover sd. sr^ .^ « (4) « oats wheat peas rye clover sd. barley m 16 baskets pears = 29 turkeys 17 turkeva _ ^ , . ^ IV turkeys U days' work 3j loaves 1 lb. veal V92 cents * lbs. sugar = V days' work =*187 loaves = 4 lbs. veal = 11 cents = 63 lbs. sugar = 21 baskets pears 4 % J !!1!^2^!^ __ 11x7x21 iJXUx^iX?iXlXt5)JJ - —^ = ^^^ 404^^ / ^A = iiB 5B = 80 15 = 21D 111>= 5E 42 E = « A («) 2^^2a5>aix4_^ 7x15 WXl^X^l^^. ;- = -4* = 26i P'l' 110 7 barr«l8 flour 6 cords 46 cwt. £77 i^ ibeep 9 tons 3 !^ 7 X?XUx5i!5xS^X8 K£Y. (•) 23 cordi 11 cwt. £28 9 sheep : 8 tons X barrels flour [JSiJLTS. .Aritb. 3X7X5 8 — a = 13i. (7) 15 N. England = 20 New York 24 New York = 22i N. Jersey 30 New Jersey = 20 Canada 4807J Canada = a: N. England | ^ 6 ^ 96l| = 961J X 6 = 6769 8. = £288 9s. ^lilx^^ixxq ^ ? ^ EzBKOiSB 89.— Page 222. (1) 11 2 t It %^ 1515) l«| _ X — X -~ X X — = J* li 2(1^ IIS} )^^ VI 3 2 3 '4 m «> f w¥'- I: (2) £119 X 400 =$476-00 lf>s. X 20 = 3-20 6ii. = 26far.X5-rl2= -lOf .19 16s. 63d=?47Q30il 4rjr INat. Atura. £XKB0IRX8 88, 89.J KBY. 9 : 13 21 ; 27 7 ; iO : U : 15 : 9-rl3 21-r27 T-r 10 H-rl6 = '692 1 = -777 I = •7 \ = -733j 117 Hence 21 j 27 Is the greatest, and 9 : 13 the least. ' = £288 9s. Dissinilar. 70-23478 19 '134229*1 (5> Similar. ^6-234784 : 19 •134*2291 : Difference, Similar and CotGrminous. ^6 •234784784784784* 19' 134229122912291 = 67' 1005*55661872493 quinary, and 76o4||fr «nden«;! °''^' l20l4313,VMVo 0) 5-33 : 7.9 cubic inches.) oz ^ ok.^h i" q.9fi.t ^•254X7.9X1^220 1 : 1-220 spec. grav. 5*^'^^^ -— 31-362052 *^^ "T^S ~ 5-57052 oz. Ans. J^o.ducethefractlonalpart,.eauceb^^^ and denominatnv 118 E£7. gSAT. ABITB. (8) yds. qrs. na. in. yds. qrs. na. in. 17)63 3 2 1(3 3 OJi (9) 61 12 4 61 61 4 2 H 5i = V-rl7 = iJ. •913625 of an acre =s -913625X4840= 4421*945 sq. yds. 4421'945X$.6t=$2962-70+ (10) i of? off of 20 bush. X-5X-6Xj= J X f X f X^/^Xi X ? Xi=: ^l bush. = 1 bush. 2 pks. gal. 1 qt. (12) Whole amount of increase = 2571437— 1842265 = 72^172. 729172X100 1842265 : 100 :: 729172 : = 39 per cent. 1842265 (13) J of f of if -i of t of ^ = ^ -. A- = i\%^€' (14) ft. 11x7 100:7::ll: =foV H — /oV^lOftV 100 (15) 79 X 16 X JE -00163 = £2-06032 = £2 Is. 2|^fd. (16) 3 7 4:3 men ) ^}X3X1^X1?]5 10: 12 hours } :;2i days : = f ? — 3^ days* 20:35 acres) >[(xi(^X^C) 4 4 7347 100 Exxaoisa89.] KBT. 119 a of i^rX^02X'456> ^ (if of f of * of 61)= ^ SI 1 1^^ n ^ ^ 1 -X—X— X X — x-x-x--— t^' 37 5« 4 13 t - X - X — X - _ 2x3d 5X] (18) X -=4X13 = 63. (19) 50 barrels =125 yards 80 yards = 6 bales, 13 bales =3i hogsheads -' X hog8heads= looo barrels 6 3 125 ^^!5X^X3Jxi«|lil«| ^^X5{ISIX13 2 125X3X3J 2X13 (20) = 60f|. j3-4rx •0063-17-2345 — 7a4Ty ^a ,.,»,, ^347 63 333Q "^^^"'^~3Wo' = "lOO 1000^ ^ 513^'^'^'^^*^^'^^^^^^^^ = •02685659998H- (21) 2 roods 7 per. 4 yds. 3 ft. 117 in. = 3416481 io, and 7 acres = xuuuUJS. 3416481-1-43908480 = -0778+ 120 or. [Nat. Abith. (22 ^ of ^ of ^ of 70 miles = ^ milea = 5 -33333+1111168. •73 of 11 fur. = 8-03 fur. = 1*00375 mile. 5*33333 — 1-00376 = 4-32958 miles. (23) 274312 nonary = 167195 denary, 1101011010 = 858 denary, and •5555 septenary = 2000 denary. 167195 — 858 = 166337x2000 = 332674000. 332674000 denary = 764876837 nonary. = 10011110101000011001111010000 binary, = 11146453021 septenary. 275 38 18 (24) ^>i..5it5..18..1$q..!^(il^..!^H5 H 1H-* ^^'- 1S(.. 9 5( S^ 275X38X 18 = 188100 = 1. c. m* 10:6 weeks 6:5 days 11: 10 hours 2400:8742 feet long 18:20 feet wide 11:8 feet high (25) ^ 2914 2 men <5QXISX6XH5X^\^!^X^1S(XH :: 60: — — 5X2914X2 11X3X11 1«IX«}X11XW«IX181X11 3 =r HU^ = 803^?. (26) 172000 = 25 V K3 V d^ Jnovaaainff aa^U «v»»/v-«-* 1 -j ._J multiplying them together we obtain 6X4x2=48. [Nat. Abith. i33+mile8. mile. 63. ?58 denary, and '4000. lOlOOOO binary, 2914 2 IQX181X11 >nent by 1 and X2=48. £XXBOIR«89.] KEY. (27) 121 43 7 = 42J = 3|iL and 9-7123 = m^^ = g^^ - ^f^ X W^ = aj5§fp - 415.471137804. W^* 100 : 27 :: $73 '42 (28) 73*42 X 27 100 — = $19 •8234. 1..5..25 $73-42 -$19.8234 = $53-5966. (29) 6300 = 22 X 3« X 6« X 7. 1.. 5.. 26.. 2.. 10.. 50.. 4.. 20.. 100 1..3.. 9 1.. 5. .25. .2. .10. .50. .4. 12. .60. .300.. 9. .45.. 225.. 18 1..7 20, .100.. 3.. 15.. 75.. 6.. 30.. 150. .90. .450. .36. .180. .900 1.. 5. .25. .2. .10. .50. .4. 12. .60. .300. ..9. .45., 35.. 175.. 14.. 70. .350 210.. 1050.. 84.. 420.. 2100.. 63.. 315.. 1575 20. .100.. 3. .16.. 76.. 6.. 30. .150^ .252.. 1260.. 6300 Therefore the divisors of 6300 225. . 18. . 90. .450. .36. . 180. . 900. . 7 28.. 140. .700. .21. .105.. .525. ..42 . .126.. 630.. 3160 14,15,18,20,21,25,28,30,35,36.42.45 are 1, 2, 3, 4, 5, 6, 7, 9, 10. 12 ,36,42,45,50,60,63,70, 76,84, 300, 316, 360, 3150, 6300 2 Z\ 'It '''' '''' '''' '^'^ '''' 210, 225, 262, o 420, 450, 525, 630, 700, 900, 1050, 1260, 1675, 2100^ (30) Nf Nf 3} lbs. =: 3 Ibg., ^ of ^ of n of i of $1 = $^ i and J off of ^% of ^i of 90 lbs. = ^^s. ibg lbs. V-\W-: } 63 XW,? 2 \^%^ ^ f = - X rr x- = J3r^ft = $5-04. 26 i Bi !! I if- fiiil 122 KEY. [Nat. Akr». (31) 7 men will have 7 men's sliares. One woman has -pf of a man's share ; . X ■^r = 1^1 of a man's share. One child has f of ^\ = ^, of a man's share; . will have 11 X ^V = fof a man's share, fmen, 2 women, and 11 children will have 7 + ^ar + f = 8U men's shares. 2 women will have 2 11 children $2739-18 -^8?f = $325-99^^1 Vr of $325-99^? = $88-90j}f : f of $88 -90^1 = $25 '40^.1 : (33) ^ *-. , yds. ft. in. J of 6i yds. = Sgft yds. = 228 2 off of 8} ft. =10 ^oftVof7foin. = a : a man's share, a woman's share, a child's share. (34) Sum = 3 2 4 : «• 2 (5: — 13 : 8? ?i)?^i: 11 5 Hi VI = 104 : 5. (35) 23 bush. 2 pks. 1 gal. 1 qt. 1 pt. = 1515 pts. 1515 X 9000 X i = 4545000 in. = 71 miles 5 far. 34 per. 3 yds. (36) 4158 462 10395 VIII. 66 22 1155 165~~55 "~ (37 h h h ?• Here the common denominator is 2x3X5x7 = 322. The numerators of the fractions are, for the first, 1 x 3 X 5 x 7 = 151 ; for the second, 2 X 2 X 5 x 7 = 214 • for the third, 4 X 2 X 3 X 7 = 250 ; for the fourth, 2 x •> X 3 X !r ^V T\'^' T^^^"""' ^'*''^'°' ^''' *'^' "^^' Hi and ^^, which whon added together = m = 2#,, the numbers all tutougu wemg iu the ocienary scale. [Nat. Abtjh. ExiBOisu 89, 90.] KBT. 123 (38) n will have 2 •' 11 children re. ihare. ire. = 104:5. Its. 4 per. 3 yds. 3X5x7 le first, 1 X = 214; for X 2 X 3 X ih }M, and numbers all IT sheep = 6 cows 26 cows = 27J acres 12 acres = 13 horses 11 horses = 28 goats X goats = 68 sheep 2J X 28 = 70 goats (39) J 21 Yi ^ Vl'i 1? 28 (;^ ~-x — X — x~x— = ^^ ^<* 1% 11 1 27 24 36 21 10 3 : 54 dajs : 18 eel. ; 48 ft. 1. 28ft.w. 9 ft. d. 5 hrs. ^ <5 ^ 4 ? • 50X^^X1^X^I^X5^^XS(X.^ 50 men : ■ — ^tx^^X?Sx^iXllS|x5 ? <^ t ^ = 200 men. 4d. Id. 4 i ExBRcisii 90.— Page 226. (1) $•35X92647 = $32426-45. ^ ^ (2) * 8 d. 4746 17 = cost of 94937 pails at Is. 1582 5 8=" « « at4d. 395 11 5= « « u at Id. ^'6724 14 1 = cost of 94937 pails at Is. 5d. (3) (4) $-07J X 95972 = $7197-90 $28-80 X 62 = $1785-60. (S) (6) $'32J X 2310 = $750. 75. $.37| x 2117 = $793 -87i. 0) £ 8 d. 375 6 = price of 7506 pairs at Is. 187 13 = « « at 6d. 93 16 6 = « u at 3d. 23 9 1J= « u p^.3d i;680 4 7J = price of 7506 pairs at Is. H^ 6d. 3d. id. i 124 KM, IKat. Ahitk. (8) rpx $'m X 121? = $212,971. $3.07i X 2103 = $6466.721 (10) 10s. I I 63. 2s. 6d. Is. 3d. m. i i i 2096 3 £e28(i 0=;cogtof2096oz. at£3. 1048 = 624 = 262 = 131 = 13 2 = tt it u tt tt tt tt tt j£8266 2 = 2l- 25 £ 8. d. 6 1 U 10 1 6 4 31 7 n 6 6 10 8 10 4 9 14 15 10 $s •38 20 1-47 21 •25 64 •25 16-47 15 •12 .$136 •94 £ s. d. 6 2 7 10| 9 101 4 111 12 9 1 3 Oi 9 9 fixjtaolsu 81, 02 J kUT. BnaciBi 92^l*age 231, (2) 131 427.1 ^ .0000637 = 4271000000 ^ 637 = i 6704866.561 4.. 0) lOs. 68. 4s. 6d. 3d. 2d. id. Id. i i I £19 19 "m 19 £361 9 10 4 16 3 16 9 4 3 = cost of 19 tons at £19 = = = 6 = 9 = 2 = 4i = « tt It tt tt tt tt tt It tt tt tt tt tt tt at at at at at at at at 10 6 4 6 3 2 Ok Oi • ^379 19 n = costof I9ton3at£l9l9 1l3 19 cwt. 3 qrs. 27^ lbs. = 1 ton — 1 ih nn, • >« i»». im., J the 00, A j-j":rz I ' *"" " — 1127') J ., * "• — 44(rj Of jt;i9 19s. liM 11,(1. - i-,v;5^d. = i;i9 19s. lOfHIAd. ^379 19 7i = cost of 19 tons 19 19 10i-?ifl= tt — ^^°_ 19cwt.3qrs.27Hbs. ;3 13 5i ■•-*^~^,r#' 1 KEY. (4) • (Kat. Abith. Dissimilar. Similar. Similar and Coterminous. • • 13-123 = • • 73-723723 := . . • 73-723723723 11-342 = 11-3422 ^ 11-342222222 16-713 =r 1 • 16-7130 = 16-713000000 19-034 = 19-034034 = 19-034034034 ' 713-213437 = 713-213437 — • • 713-213437437 ! 12-345678 = 1 i 1' i! 1- 12-345678345 Sam : ^■^ 12.345678345 2 carried. 846-372095763 1 (6) Hence 5 : 7 is the greatest, and 9 : 13 least. 5: 7= 5-^ 7=: -714+ 9: 13= 9-i-13 = -692+ 12: 17= 12-M7 = -705+ 7: 10= -74-10= -7 6 ^ 9 1!^ V 54 -X — X — X — = = 54: 221. t 13 17 \l^ 221 (6) 1 acre = 160 rods, and 25 acres 2 roods 35 rods = 4115 rods. 40-25 823 rods. 8(IS|-^q X^n^ 160:4115 :: $80-50 : = $2070-3593. m 16 (8) $3'68i X 7439 = $27431 -31^. k^*" •». , K3:Enci8B&2.J KEY. 133 jmfi The G. C. M. of 135795 and 222210 is 12345 • when bo h terms of the fraction arc divided by 12345, it becomes ^ .4Hh Here 714235 and 999999 have no G. M • the fraction cannot be reduced. « «• w. m. , . . the h Ji!"!^^' ^^' ^' ^- ^- °^ ^^^^^^ *^*J 100000 is 3125 • when both terms ef the fraction are divided bv ^12/5. k reduced to U "i^wea by 3125, it becomes tJmf of th?fr,**;-*^' "■ °^""" "■* ''^'' " ^"^i "hen both itrms or tne fraction are dividpri hi, uno •* • 1 lowest terms, viz., -^i, ^ ^^^' '* ^ '^^^^^^ *<> ^^^ (10) rods. V iwo. ten 13 stone flour 38 cents 119 lbs bread 3 X IX 1 f 34i bu hels tu n PS =17 bushels potatoes t^h/l potatoes = 59J lbs.tea 6 lbs. tea = 111 stone flour = 360 cents 5= 12 lbs. bread = X bushels turnips 19 X ~X~ 2!^ __ 3X13X19 (11) m Hi — 2«si- 54 11 42 20 16; 3: 27 men 8 hours 77 floors 24 feet long 22 feet wide 5 coats paint : : 7 days : ^X^tx ^XVtxi^^X^^X!^ ^JXUX^^X^«)X1^X3 % tews. <^ V X 11 2X3 = 12f days. ^ ^ 134 KEY. [Nat. Abitb. iz. 12)'72342 12)5403.. 2 12)407.. 12)30.. t 2. .3 (13) IZ. 6)72342 IZ. 72342 9 65 9 588 9 6296 9 47666 ZII. = 23702 12 27 12 331 12 3972 12 47666 6)11806.. 2 6)1 731.. 6)264.. 4 6)40.. 4 6)6.. 1..0 VI. 1004402 6 6 6 36 6 220 6 1324 6 7944 6 4T66^ IZ. 8)72342 3)23713.. 2 3)7234.. 3)2371.. 1 3)723..1 3)237.. 3)72.. 1 • 3)23.. 2 3)7.. 2..1 m 2102101102 3 7 8 21 3 65 1765 3 3 196 5296 3 3 588 15888 3 3 1705 47§66 7at. Abith. !342 1713. .2 r234. .0 1371. .1 |723. .1 )237. .0 072. .1 • 3)23. .2 3)7. .0 2. .1 ra 102101102 1765 3 5296 3 15888 3 4T§66 E2UCBCISB 92.] If. ' mill 2 a 2 7 2 15 2 31 2 KEY. (14) • IV. rr. 100000 333333 100000 2 4 4 •■ — ^ 2 15 4 2 4 4 — — <» 4 63 16 2 4 4 i — _ 8 255 64 2 4 4 ^ 1023 256 2 4 4 1B6 63 Greatest. 32 Least. 4095 Greatest. 1024 Least. VI. 555555 6 36 6 215 6 1295 6 7775 6 vx. via. 100000 777777 6 8 — .. 6 68 6 8 36 ' 511 6 8 216 4095 6 8 1296 32767 6 8 VIII. 100000 8 8 8 64 8 612 8 4096 8 ' '1 ^i 46655 Greatest. 7776 Least. 262143 Greatest. 32768 (Continued on next page.) I ill t 136 KEY. [17at. Ajuth. (14 continued.) S3I. ZII. e e e ee e 12 • 100000 12 143 12 » 12 12 1727 .12 • 144. 12 20735 12 rreatest. 2 1728 12 ' 248831 12 20736 12 2985983 G 48832 Least. (15) - 1728<= 2« X 33 k 1.. 2. .4. .8. .16. 1 3. .9. .27 .32..64 1.. 2.. 4.. 8.. 16.. 32.. 64.. 3.. 6.. 12.. 24.. 48.. 96.. 192.. 9.. 18.. 36.. 72.. 144.. 288.. 576.. 27.. 54.. 108.. 216.. 432.. 864.. 1728. Therefore the divisors of 1728 are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 96, 108, 144, 192, 216, 288, 432,' 576, 864, 1728. 30 14 12 143 (16) 5^ ^ %.. X..^.. 3...55..11.. It. .13. .14 5i ^.. ? 11.. Vl3 11 "^ 30 X 14 X 12 X 143 = 720720 = 1. c. m. [Xat. Aritb. EXKBCISE892, 94.] KEY. 137 t. 432. .864.. 8, 9, 12, 16, L6, 288, 432, .13. .IH .13 Dissimilar. 97-91342 = 18 •133456*7 = (17) Similar. Similar and Coterminous. 97-913423 = 97-913423423423423 19-1234567 = 18-123456745674567 Difference = 79-789966677748855 (18) 20 ft r 19 ft .5 7" 1 1'" 8 6 U 391 1 "" ' " A .. = *' '": '''■ + i + rU + ^ +Trfe = ^^Wo^' sq. y ds. = 44 • 5 1 7 + • sq. yds. $i^-87i X 44-517 = $127-98+. (19) 916 acres 3 roods 17 per. 7 yds. = 443759U sq.yds., and 43 acres 1 rood 2 per. 17 yds. = 209407} sq yds 4437591i-h209407i=4437591-25 + 207407-5 = 21.191l7-f, ExaBoisH 94— Page 233 (1) (2) $742-10 X -05 = $37'-iOJ. $iooo x -11 = (3) $734-19 X •10 = $73'419. (4) $1624.50 X -875 = $1421-4375. $110. Ul >i, ii 138 KEY. 6) [Nat. Abitb $994*70 X •125=$124«3376. (6) $Tn.50 X '0875 = $68-03125, or $68 •03J. 0) $7135'80 X '0225 = $160»6556. (9) $7490 X "10 = $749 $7490 X -17 = $1273-30 (8) 2740 X •20 = 548. (10) $740 X '045 = $33-30 $1680 X -026 = $42-00 $7490 X •27 = $2022-30 $42'00 — $33*30 = $8-70 $7490 X -46 = $3445*40 (11) 729X -11 =80*19 (12) $763-22 X •25 = $190'8050 729— 80-19 = 648«81=648^oV- $847-16 X '16= 135-5456 $1234*17 X '0625 = 77-135625 Sum= $403-486225 (13) $17429-40 X -43 = $7494-64] $17429-40 X -37= 6448-87| (14) 68978 X •36 = 24832-08. (15) $13943-52 29800 X -17 = 5065 $17429 - 40 — $13943 • 52 = $3485 - 88. 29800 — 5066 = 24734 i EzxBOiSB 95—Page 235. (1) (2) $1000 X -045 = $45. $1678-30 X -0225,= $37-76175. $7893-87 [Nat. Ajbith (•03*. (8) X -20 = 648. (10) 15 = $33-30 15= $42-00 30 = $8-70 = $190-8050 = 135-5456 = 77-135625 = $403-486225 14) EXKKCiflBs 04.96. j KEY, IQQ (3) ■ (4) $7531.19 X •0375=$282. 419625. $508 '60 X •0125=$6-3575 (») (6) $7863-50 X •0175=$137.61125. $878-30 X •025=$21.9575 O) ' (8) ) $7193-16 X •03125=$224-78625. $6734'10X'17=$1144.797. (9) $7-13 X 718 X -0425 = $217-67195. (10) $1*85 X 3243 X •05625«= $867-7871876. EzBBOiSB 96— Page 236. (1) 16 = 24832-08. (15) X -17 = 5066 -5066 = 247341 $7893-87 X -02 = $157-8774. (2) $8000 X -00875 = $70. (3) $8643-22 X -0125 = $108-04025. (2) ; $37-76175. (4) $78963.80 X '00875 =$690-93326. (5) |1987-27 X -0375 = $74-522625. 140 EZXRCISBI KEY, [Nat AKiiaj ExBROiSH 97— -Page 237, (1) $20000 $19656 $21 $4000 -J- 1-0125 = $3950-61728 + = sum to be invested. $4000 — 3950-61728 = $49-38271 = commission, " $200 X ^2) dU stock ( I $7500 -f- 1-045 = $7177*03349 =; sum to be expended in Iace8.p21100 x $7500 — $7177-03349 :=: $322-96651 = commissioa. t (3) $8470 -r 1 -05 = $8066 -66| = sum to be invested. $8066 • 66 J -J^ $6 -40 =j 1260:^5^ barrels.' (4) $11000 ^ 1-00875 = $10904-584882 = sum to be invested. (5) $13000 -f. 1.045 = $12440-1913 + = sum to be invested, $13000 — $12440-1913 = $559-8086 + = commission. $12440-1913 + -f- $3-63 = 3427-0499 yds, 17500 X •( 16000 X '0 16400 X -0^ 136000 X •( ExEEOiSB 98— Page 238. (1) (2) $9000 ^ 0-83 = $10843-373. $8500 -J- Ml = $7657-65?5| $ggooo (3) $17500 -^ 1 -0125 = $17283-951 = amount to be invested. $l»283-951-f- 1-07 = $16153-2^ =stock. [Nat AbithI EzxRcisss 97-99.] KBY. (4) 141 $19656.01965 ^ 0-97 = $20263.937 = stock remitted. to be invested, nission. (5) $200 X 100 ::= $20000 == par value of 200 shares-. $1 stock costs $1,055. $1-055 X 20000= $21100 == cost of Stock, tpended ia Iace8.|?2n00 x -00875 — $184-625 = brokerage aissiou. I $21100 + $184,625 = $21284-625 = whole cost. , )e invested. Emboisb 99— Page 240.. to be invested. to be invested, ommission. (1) f7500X •0l75=r$i31*25. (3) poo X -01875 = $112.60 (5) 16400 X -0090 =r $57-60 (2) $8375 X -0075=: $62-8126. (4) $6000 X -0X17 = $58. 50. (6) $4500 X -0035 = $16.75. <'^ (8) 136000 X -03 = $1080. $27000 X -048^ X 4 = $5205-60 (2) I (9) ^ ^ (10) 1 = $7657.65« $39000 X -022 - ^868 ■ ■ «^-f-$868. '$I780ax •OO6=$80. (11) be invested. $12350 X '011x7 =5 $1235,' *1f [Nat. Asith. I Ezebozc 142 KEY « EzBBOisi 100— Page 241. (1) (2) $1 7000 ^ 965 = $1 7616 • 68. $22 760 -^ • 94 = $24202 '127. (3) (4) $16000-i-9775sc$15346'2685. $33000 -f- •9425=:$35013'2625. EuBOiSi 101— Page 243. 1347 X 6 = 6735 lbs. = gross weight 6735 X '06 = 404-1 lbs. tare. 6330*9 lbs. =z net at 3} cents per U = 6330*9 X -035 = $221-68. (3) 127 X 11 = 1397 lbs. = gross weight, 1397 X »03 = 41-91 lbs. = tare. 1355-09 lbs. = net at $-012 per lb, =.1355*09 X -012 .jt $16*26. (8) .29 X -13 = $16*77. (4) 31 X 207 = 6417 lbs. = gross weight 207 X 2|= 465i lbs. = tare. mii lbs. = BOt.At 5i cents per lb. sz 696U % \ *0675s; $942 '1968. [Nat. Abith. I ExsBasis 100-108.] I) = $24202-127. =$35013*2625. rll =6330 '9 lb»=s/1355«09 lb.s:596U x| (1) EBY. (») 214 X -47 = $100.68. ExiBoiSB 102-.page 243. 143 (2) $17429.80 X -21 =$3660.2580. $2920-16 X -075 = $219-012. <»> (4) $T1343.90X .25 = $i7836-725. $913-73 X .2=$182-746. (6) $14713-19 X -33 = $4855-3527. ExKBr-isa 103^Page 244, (1) $23900 4- 7142300 = $0-0033462 = rate per dollar. $•0033462 X 14729-50 =$49-2878 + (2) $100000 ^ 5793000 = $-017262 = rate per dollar. $•017262X18600 = $321^0732 (8> $100000 + 6793000 = $-017262 = rate per dollar. $•017262X7500 = $129,465: (4) $100000 + 5793000 « $-017262 ^ rate per dolla^ 144 KEY. INat. Abitb. ExBBOiSB 104— Page 262. (1) Here P == $723-19, r = -067, and t = 7 '32. Then I = Prt = 723-19 X '067 X 7-32 = $364-6813036. (2) Here P = 85719, r = -065, and / = 6J or 6'B. " Then .4 = 7^ (1 + rt) = $867-19 X 1-4225 = $1219-362775, (3) Here / = 11, and r = -726. Then n = /r + 1 = ll x '725 + 1 ^ 8-97B. (4) Here P = $654-32, /= $23456, and r = -07. I 234-56 Then t = — = _ 5-12112 or 5 years 1 niri3 d. Pr 654-32 X -07 (5) Here J = $1200, P = $700, and t = 6. ^~P 1200—700 Then r = = Pt = f = rate per ualt .*, 14f = 700 X 5 rate per cent, (6) Here n = 4, and r = -23. w^l 4—1 Then t = = = 13 years 16 dayg. t -23 <0 Here P = $270, 1 = $87 and r = -OT. J 87 Then ^ = — = — = 4 ) tira 7]/V monthi* Pr 270 X -07 [Nat. Abith. 4-6813036. £XSB0IM 104.] KEY. («) 145 ij?.°f ^=5680, /=Ui, and r=.ll. Then A=zP (1 + r/) = 680 x 2-265=$1640.30 19-352775, (9) Here ,/f=$2000,.* = 20, and r = -08. Then P = -A 2000 ^-j;^ = — = $769-23^ 976. rs 1 niriS d. <10) Here »: Then r : 21, and/ = 24. n— 1 21 — 1 24 — '^^i = 'ate per unit. .•. 83^ = rate per cent. flit ••. 14f = Here n Then/ (11) 23, and r 16. » — 1 23 — 1 •16 = 137J years tb8« (12) Hero P = $679-18, r = . 0776, and /= 11.73. Then/ = Pr/ = 679'18 x -0775 x U' 73 = |61 7-4255. (13) Here P = $950, ^ = $1763-42, and t = 10 ^ — P 1763-42 — 950 pt ...,.,,. = <^8562 = TAte per ui.it .*. 8-56? = rate per cent K -'^- ., 146 KEY, [Nat, Abith. (14) Here P = $666, ji = $1347-60, and r = -06. J — P 1347-60 — CG6 Then t = = = 17-054+ years, or 17 Pr 666 X -06 years 19 days. (18) Here P = $273, J = $100, and r = '09 1 100 Then t = = = 4-07 years =:e4 years 25 days. Pr 273 X -09 (16) Here P = $476-30, J = $500, and / i= 2. w4— P 500 —476-30 Then r = i = =j '0248 Pt 476-30 X 2 ,*. 2^§ = rate per cent. = rate per unit. (1») Here P = $749-49, /= $257, and * = 7. / '257 Then r = — = := '04898 Pt 749-49 X 7 .'. 4-898 = rate per cent. =. rate per unit. (18) Here J = $111M1, / = 11, and r = -11. ^ 1111-11 Then P = = = $502-7647.. li-rt 2-21 (19) P = £167 47, r = -ll, and t = 9. /= Prt = 167*47 X .U X 9 = i:i65-7953=i:i65 15s. lOtJfd. ^ [Nat. Asith. rs, or 17 3 25 days. ) per unit. ExBBOiMts 104, 106.] KjjY. * ExHRoiSB 105 — Page 253. 11 -7- 3 = 6i cents. 16^ 2 = 8 cat. = $0-08. (3) 9 years and 8 months = 116 months, and U6.^.3 = = $0-68. 147 68 cents (4) 16 years and 3 months = 195 months, and 195 ^ 2 = 97i cents = $0-97J. (8) 11 years and 7 months = 139 months, and 139.^ 2 = 69^ cents =$0-696. per unit. 17. )S. lOtJfd. (6) • 12 years and 5 months = 149 months, and 149 ^ 2 - 74^ = $0-745. 0) ji cents 3 years and 2 months = 38 months,- and 38 ^ 2 = 19 cents - interest of $1 for given rate and time. "" $0-19 X 279-40 ?= $53-086. I (8) - ..a.. „nu . muu.hs = ly months, and 79 -^ 2 = 39,^ cents - interest of $l for given rate and time "" $0-395X189.70 = $74.9315. 148 KEY. (9) [Nat. Arith. 3 years and 11 months =i*t months, and 4T -f- 2 = 23^ cci^t. ; interest of $1 for given rate and time. • $0-235 X 1463 = $343805. (10) 11 years and 1 month =133 months, and 133-f-2 = 66^^ cents: interest of $1 for given rate and time. $0-665 X 28967-50 = $19263-3875. ExBRCiSB 106— Page 254. (1) (2 2-^6 = 1 mill = $-0003. 7-f- 6 = 1^ mills = $0-001^. (3) (4) 11 -J- 6 = 1^ mills = $0-001^ 21 -i- 6 = 4} mills = $0-004^. (B) 47 -f- 6 = I'l mills = $0-00 7f. (6) 8-7-2 = 4 cents = $0-04. 12 H- 6 = 2 mills = $0-002 and $0-04 + $0-002 = $0,042 (7) 66 -2- 6 = 11 mills = $0011. 2 years 2 m'ths = 26 months, and 26 -^ 2 = 13 cents = $0-13. 19 H- 6 = 3^ mills = $0-003^ and $0-13 + $0-003^ = $0-133^. [Nat. Ajuth. • — ZSg- CGlit^i ;s ne. • I = 66^ cents= ae. 5. (2 lills = $0'001-^. (4) ills = $0-004 J. i2 = $0,042 sents = $0-13. 3^ = $0-133^. JiXERCISBS 105-107. J KEY. (9) 149 7 years 8 m'ths = 92 months and 92-9 .. (10) " '""' " '''""'' = "« T«"' -^ ^1= - 2 = lOH cent, = $1*075 " ^ a = 3§ mi„, = ,0-0038, .nd $1,076 + $0-0035 = $1-0785, (11) n-f-6z= 25 mills = $0- $0-765. 002g, and $0-755 + $0-002^ = $0-757g. ExKBciSB 107— Page 255. (1) Interest on $i for 7 months Interest on $1 for 17 days = $0-035 Therefore interest on $i for 7 months 17 days -ij^J Then $0-037^ X 917-30 = $34.^7047l6'^ (2) Interest on $i for 3 months Interest on $l for 13 days = $0-015 Therefore interest on $1 for 3 „,onths 13 days -i^ Then$0-0l7i X 842-50 = $1^462976 ^^^^^ (3) • Interest on $I for 2 vflar« n ^ ^i Interest on $1 for lo^s ^' = ^^'^ ^^ * — 1? Therefore interest on $1 for 2 years ben $0.i76i X 673-83 = $T5i llm'tha 10 /l-^a * 3766. ii^u i7e^ 160 EBT. (4) [Nat. Abith. Interest on $1 for 6 years 9 months = $0*405 Interest on $1 for 19 days = 3^ Therefore i terest on $1 for 6 years 9 m'ths 19 days = $0-408^ Then $0-408^ X 642*30 = $262-16545. (6) Interest on $1 for 5 years 5 months = $0-325 , Interest on $1 for Y days > = H Therefore interest on $1 for 5 years 5 months 7 days = $0'326^ Then$0-326i X 1427875 = $465-7252. (6) Interest on $1 for 4 years 7 months Interest on $1 for 16 days Therefore interest on $1 for 4 years 7 m'ths 16 days ' Then $0-2 77| X 709-63 = 197-040596. O) Interest on $1 for 7 years 7 months Intere&t on $1 for 22 days = $0-275 2^ •277« = $0-455 — O.J Therefore interest on $1 for 7 years 7 m'ths 22 days = $0-458 1 Then $0-458| X 2463-20=$1129-7877+$2463-20 =$3592-9877. (8) Interest on $1 for 9 years 9 months = $0-585 Interest on $1 lor 9 dayg ' = i^ Therefore interest on $1 for 9 years 9 months 9 days rr $0-586i Then $0o86i X 99d'09 s^ $086-494135. 4'AT. ABITH. EzxBoisxs 107,108.] KST. 151 (9) Interest on $1 for 3 years 4 months = $0.20 Interest on $1 for 27 days == 4L Therefore interest on $1 for 3 years 4 m'ths 27 days = $0^ Then $0-2045 x 68-70 = $14-04915. (10) Interest on $1 for 3 years Interest on $1 for 28 days = $0-18 = 4^ Therefore interest on $l for 3 years 28 days =$0-184| Then $0-184i^ X V42-63 = $137-139. (11) Interest on $1 for 7 years 4 months = $0-44 Interest on $1 for 11 days = jk Therefore interest on $1 for 7 years 1 m'ths 11 days =$0^ Then $0-441f X 200 =$88 366 + $200 = $288-366. (12) Interest on $1 for 9 years 3 months = $0-555 Interest on $1 for 9 days ~ ji Therefore interest on $1 for 9 years 3 months 9 days = i^T Then $0-5565 y »43-63=$413-830095+$743.63=$1157-460095! ExKRciSB 108— Page 256. (1) Interest on $1 at 6 per cent, for g?7en time = $o-526| Interest on $1234-56 at 6" per cent, for given time = $0 5261 X 1234-56 = $650-2016. ^ ^ Hence interest on $1234-56 at 7 per cent, for criv«n fin,n= ;ii)650-2016 4. one sixth of $650-2016 = $758-5685. 152 KEY. (2) [Nat. Arith. Interest on $1 at 6 per cent, for given time = $0-1265 Interest on $9876-54 at 6 per cent, for given time = $0-126§ x 9876-54 = $1252-67449. * '^g X Hence interest on $9876-64 at 3 per cent, for gi^en time - $1232-67449-r 2 = $626-337245. (3) Interest on $1 at 6 per cent, forgiven time=$0'216i Interest on $715-30 at 6 per cent. fOr given time = $0-216* x 715-30 = $154.98166. ^ 5||)154-98166 + one third of $154-98166 = $206-6422. Interest on $1 at 6 per cent, for given time=$0-14U Interest on $555-55 at 6 per cent, for given time = $0.14U x 555-55 = $78.51773. ^ ^Z'^nfllT Z ^'''*'' ** '2 P^' <^«^*- fo' S^^^ time = $78-51773X2=$157-03546 ; amount=$157-03546 +$555-55= $712-58546. (5) Interest on $1 at 6 per cent, for given time=$0-0l6S. ^ $m?rl5'.''' '''" ^?ll^-^.V* S P«r cent, for given time = $129-4425 — one sixth of $129-4425 = $107-86875 Amount =$107-86875 + $7766-55 = $7874-41875. (6) Interest on $1 at 6 per cent, for given time = $0-521^. Interest on $500 at 6 per cent, for given time = $0-521^ x 500 = $260-666|. ** Hence interest on $500 at 16 per cent, for given time = $1195 111 " "^-*^^^^^^J amount = $695.111+$500= [Nat. Arith. ElZXROISB 108.] KEY. 153 !6^ = $0-126g X ?en time = : $0-216f X iven time= 22. $0'14H X ren time = $0-016§ X 6n time = '5. n^ X 500 n time = I +$500= 0) Interest on $1 at 6 per cent, for given time = $0-206^. Interest on $576 at 6 per cent, for given time = $0'206i X 576 = $118-752. Hence interest on $576 at 5 per cent, for given time = $118-752 —one sixth of $118-752 = $98*96. (8) Interest on $1 at 6 per cent, for given time = $0151g. Interest on $2478-91 at 6 per cent, for given time = $0-15H X 2478-91 =: $376-38116. Hence interest on $2478-91 at 4^ per cent, for given time = $376-38116 — one fourth of $376-38116 = $282-285. (9) F"om May 9th to December 11th = 216 days. Interest on $1 at 6 per cent, for 216 days = $0-036. Interest on $780 at 6 per cent, for 216 days = $0-036 X 780 — $28-08. (10) Prom August 16th 1851 to June 19th 1852 = 30adays. Interest on $1 at 6 per cent, for given time = $0-051.'. Interest on $1830-63 at 6 per cent, for given time = $0-05U x 1830-63 = $93-97234. Hence interest on $1830-63 at 7 per cent, for given time = $93-97234 + one sixth of $93-97234 = $109-63439. (11) From September 3rd 1858 to January 9th 1859 = 128 days. Interest on $1 at 6 per cent, for given time = $0-021]. Interest on $6200 at 6 percent, forgiven time = $0-02 U x 6200 = $132-266. Amount = $132-266 + $6200 = $6332-266. 154 KEY. IN AT, ABrrH ExBBCiSB 109.— Page 2C8. From June 2nd to July 1 7th there are 45 days. Julj 17th to October 6th « 81 " " October 6th to December llth « 66 « " December llth to March 29th « 109 " " March 29th to October 7th " 192 « Whole sum $1217-30 for 45 days = $54778-50 for 1 day. 1st endorsement 207-80 . Balance $1009-50 for 81 days = $81769-50 for 1 day 2nd endorsement 209- 60 Balance $799-90 for 66 days =.$52793-40 for 1 day. 3rd endorsement 320-90 ^ 4th .!f °°° >^"^""' *■" '"' "^y" = «»22n.00 for I day. 4th endorsement 421-83 ^ Balance $57-17 for 192 days = $10976-64 for 1 day. Whole interest = that of $25252904 for 1 day! Interest on $252529-04 at 6 per cent, for 1 year = $15151-7424 Hence mterest for 1 day = $15151-7424 -f- 365 = $41 5116 ' Then interest due . =$41-5116 Balance on Note =$57-17 Principal and interest due = $98-6816 [Nat. Abith BXKBOISB 109.] KEY. 155 re 45 days. 81 " 66 " 109 " 192 « for 1 day. for 1 day. for 1 day. for 1 day. 'or 1 day. or 1 day. 15151-7424. $41-5116.* (2) . Prom 17th June to 6th September there are 80 days. 6th September to 7th December " 93 " 7th December to 11th June " ise 11th June to 7th February " 241 -■ ; II 7th February to 19th December " 315 u j 19th December to Ist May « ^33 Whole sum |7348.25 for 80 days = $587860-00 for 1 day. 1st endorsement 2463-80 "^ Balance $4884-45 for 93 days = $454253-85 for 1 day 2nd endorsement 392-20 ^ Balance $4492-25 for 186 days = $835558-50 for 1 day 3rd endorsement 982-20 ^' • Balance $35.10-05 for 241 days = $845922.05 for 1 day 4th endorsement 2842-90 ^' Balance $667-15 for 315 days = $210152-25 for 1 day 5th endorsement 317-23 * ""j-- Balance $349-92 for 133 days = $46539-36 for I day. Whole interest = that of $298028601 for 1 day. Interest on $2980286-01 at 8 per cent, for 1 year = $238422-8808 Hence interest for 1 day =^38422-8808^^ 365 = $6531'^^^^^ Then interest due Balance on Note = $6532133 = $349-92 Principal and interest due = $1003.1333 156 KEY. [Nat. ABiTtt $1800 108 $1908 114-48 $2022-48 121-3488 $2143-8288 128-629728 $2272-458528 136-347511 $2408-806039 1800 $608-806 = $700 49 $749 62'43 $801-43- 66-1001 $857-6- ^1 60-01.7107 $917-657207 64-229004 $981-786211 68-725034 $1060-611245 73-535787 $1124-047032 700 ExEBoiSB 110.— Page 259. Principal. Interest for Ist year. f^^rTJ f*''" o ^'*' = principal for 2nd year. Interest for 2nd year. Amount for 2 years = principal for 3rd year. Interest for 3rd year. Amount for 3 years = principal for 4tQ vear. Interest for 4th year, ' Amount for 4 years = principal for 6th year. Interest for 6th year. ^ Amount for 5 yea*ft. Given Principal. Compound interest required. (2) Principal. Interest for 1st half year. Amount for 1 half y. = principal for 2nd half y. Interest for 2nd half year. " ^ Amount for 1 year = principal for 3rd half y. Interest for 3rd half year. ^ Amount for l^ years = principal for 4th half y Interest for 4th half year. ^ Amount for 2 years = principal for 5th half v Interest for 6th half year. Amount for 2^ years = principal for 6th half y. Interest for 6th half year. ^ Amount for 3 years = principal for 7th half y Interest for 7th half year. Amount for 3^ years. Given Principal. $424047 _ Compound interest required. [Nat. ARtra 5nd year. 3rd year. 4ta year. 5th year. 2nd half y. fd half y. 4th half y )th half V Sth half y. •Zthhalfy EZESOIBB 110.] $673'40 20-202 KEY. (3) Principal. Interest for 1st quarter. 157 $693-602 20-80806 Amount for 1 quar. =; principal for 2nd quarter. Interest for 2nd quarter. $714.41006 Amount for 1 hnlf y. = principal for 3rd quart. 21432 3018 Interest for 3rd quarter. ^'^oJkP^^*^ Amount for 3 quarters = principal for 4th quar. 22-0752708 Interest for 4th quarter. ^^nJ'2^5^^^^ Amount for 1 year = principal for 5th quarter. 22-73 75289 Interest for 5th quarter. $780-6551615 Amount for 5 quarters = principal for 6th quar. 23-4196548 Interest for 6th quarter. $804-0748163 Amount for 3 half y. =z principal for 7th quarter. 24-122 2444 Interest for 7th quarter. $828-1970607 Amount for 7 quarters = principal for Sth qr 24-8459118 Interest for Sth quarter. $853-0429 = Amount for 2 years required. 673 40 Given Principal. $179-6429 = Compound Interest required. (4)' $860 34-4 Principal. Interest for 1st half year. Amount for 1 half year = principal for 2nd half y. Interest for 2nd half year. Amount for 1 year = principal for 3rd half year. $894-4 35-776 $930-1 76~ ^ ^_ 37-20704 Interest for 3rd half year $967-38304 Amount for 3 half years = principal for 4th half y 38-69532 Interest for 4th half year. $1006-07836 Amount for 2 years = principal for 5th half year 40-24313 Interest for 5th half year. $1046-32149 Amount for 5 half years = principal for 6th half y. 41-85285 Interest for 6th half year. $1088-17434 = Amount for 3 years required. $228-1743 = Compound Interest required. Nat. Abitb, ^^^ KEY. ExiRoisi lll—Page 261. O) By the table the am't of $1 at 6 per cent, for 11 years - Sl-flOfl^ Then $1.8983 X 875 = $1661.0125 = Amount "" ^ ' 8*^5 ■ Principal. $786-0125 = Interest. (2) By the table the am't of $1 for the given time and rate= $2-77247 Then $2-77247 X 643-98 = $1785-41523 = Amount. 643-98 Principal. $1141-43523 = Interest. (8) By the table the am't of $1 at 6 per cent, for 45 years= $13-76461 Then $13-76461 X -01 = $-l376i6 = Amount. •01 Principal. $•127646 = Interest. (4) By the table the am't of $1 for the given time and rate= $2-28793 ' Then $2-28793 X 78-2 = $178-916 = Amount. 78 2 Principal. $100-716 = Interest. (5) By the table the am't of $1 for the given rate and time=$2-40662 Then $2.40662 X 777-77 = $1871-7968 = Amount. 7V7-77 Principal. I $1094-0268 = Interest. Nat. ABiTfi. £ZXB0I8XS111,112.] KIBT. 159 a = $1-8983. 1. = $2-77247. luant. ncipal. tereat. (6) £iA 6s. 9d. = £44-2876. By the teble the am'tof XI at 6 percent, for 11 years = £1-8083 Tbon£l-8983x 44-2876 = £84-07096=:: £84 1 5 = Araouat. 44 5 9 Principal. £39 15 8 = Interest. O) £32 4s. 93<1. =£32-240625. By the table the amount of £1 for t riven lime and rate £1-26532. Then £1-26532 X 3? . .062;> ^ £40-7947076 = £40 15 103 no'riyr A.mount. 32 4 9 J Princ^al. £8 11 1 = Interest. $13-76461. I. = $2-28793* =$2-40662 at. >al. It. EzBBOisB 112— Pago 262. (1) Amount of $1 for 7 years at 4 per cent. = $1-31593. $7439-87 -i- 1-31593 = $5653-697. (2) Amount of $1 at 5 per cent, for 20 years = $2-6533. $9193-90 -r 2-6533 = $3465-031. (3) £595 10s. 2»d. =£595-51 Amount of £l at 6 per cent, for 3 years = £1-19102. £595-61 -f 1-19102 = £500. (4) Aiavuxxu Oi .pi nu u per yuut. lor 7 years = $1-50363. $7111-11 -i- 1-50363 = $4729-296. ' it 160 KEY. <») [Nat. Arith. £268 Os. 4^d. = £268-02. Amount of £l at 5 per cent for 6 years = £1*3401. £268-02 -^ 1-3401 = £200. , ExKRCiSB 113— Page 263. (1) Here A = $932, r z=: -04, and t = 1. Whence 1 + rt = 1-04. Ji 962 Then P = , == = 1925. 1 + rf 1-04 (2) Here A z= $2202, r = -06, and t = 5-75. Whence 1 + rf = 1-345. A 2202 Then P = = = $1637-174. 1 + rt 1-345 (3) Here A = $1003-50, r = -06, and * = a year. Whence 1 + rt = 104. ^ 1003-50 Then P = l + rt 104 (4) = $964-9038. I Here ./? = $716, r = -08, and / = f,- year. Whence 1 + r^ = 1-04|. A 716 Then P = = = $684-0764. 1 + rt I04f [Nat. Arith. ExiEKoiSKS 112,118.] Ksr. (B) 161 Here ^ = $,3,3.50, . == oe5, and t =H year. iThence 1 + rf=l.022>». ^ I Then P -^ -^ _ 1342-50 (6) Here ^ = $2400,.= -05, and ..^gg ^. Whe.;eI + „ Then p - '^ _ 2400 (7) fie.^ = $2202,. = .05, and. = .75 year. Whence, +., = 1-0376. ^ $2203 ^ 1 .0275 = $2122-40963+ = p„,ent w„„l. $220a - $3122.40963+ = $T9:;9036 = pLrut' (8) Here A =$4360, r =-06, .nd t = I,V Whence I + rt = 1 .085 Then P = -jf_ _ <3«» » + rt ~ r^ = $«18.43317. (9) Her. ^ = $1647, r = .06, .nd ^= H year. Whence l+rt~ 1-065. ^ - i: 1 + r« - i:^ = »1«61.13744. (10) Bere A=$20«6, r =.o«, .nd t = 3,V Whence 1 + W = i .ju Then P ^ _!. _ ="<'» '11 182 EET. [Nat. Abita. (11) Here A = $2070«90, r = '05, and t = l^V Whence 1 + r« = 1-07^. A 2070-90 Then P = — = — = $1918 '9806. 1 + ri 1-07H $2070 — $1918*9806 = $151*019 = Discount required. (12) Here A = $970 '63, r = '08, and * = iJ year. Whence 1 + r/ = 1 -07 ^. A 970*63 Then P = = ^ = $904*313. 1 + rf 1*07^ (13) Here in first case^^ = $1512, r,= *07, and* = '5 year. Whence l+rf= 1-035. A 1512. Then P = = = $1460*8695. \-\-rt 1-035 Also A = 1512, r = -07, and t = 1. Whence 1 + r« = 1 -07, A 1512 Then P = = = $1413*0841. 1 + r* 1-07 $1460-8695 +$1413-0841 = $2873*9536= Present worth of whole amount. $3024— $2873-9536 = $150*0464= Discount required. (14) Here in first case A = $440, r = •08, and * = 1*25. Whence l + r/ = l-l. A 440 Then P = = = $400. 1 + r* 1-1 In second cnte A = $896, r = *08, and * = 1*5. Whence 1 + r<=l-12. A 80e feen P a = = $800. J. "T /* i.- i.a $400 + $800 = $1200. i... .- [Nat. ABira. mce 1 + rf = as -9806. t required. rhence l-{-rt 04*313. ^ar. Whence [60*8695. + r«=sl.07, 1*0841. sent worth of it required. 26. Whence Whence 1 + to«oi8K« Z13-116., KBY. ig3 ExBBOisa 114— Page 265. (1) «0-1580 J X 986 = $155-8701. (a) J0.0225X 640 = $14.6488. (3) Here the time the note has to run is 94 daTs- $1000 X == 1600 X 1 = 1500 600 X 3 = 1800 700 X 5 = 3500 1400 X 7 = 9800 6200 5200) 16600(3/^ months. 15600 1000) MOO) " tP^ [Kit. Abith. •004f, and this : present worth n. KXBROISM 115, H7.J KEY, (ft) 165 Six months ^-- l.^h Jan^^^^^^ I5th Jul,, and from Ist Jul, o; ,^ , *<* 15th July there are 14 days. Six .ouths fro?;;^ et^^Tlt^: Z '' ''''• to firh ci T r . September, aad from Ist July 9,v .u , '''^'''^'P*e«»ber there are 67 days Su months from 8tk June — «fh n u fa «fi T. r ^ December, and from 1st July to 8th December there are 160 days. 1^ f 1 4^ months. i $3750 X 14 3000 X 40 : 2400 X 67: 2250 X 160 : ; 62500 120000 160800 360000 11400 11400)693300(603i days 684000 9300 1 11400) "" Therefore the note must be made payable on the 61st day from the Ist of July, which is the 31st of August. 10 )0 )0 )Q 30(82^^ months. )0 '11 = )0) ^'s BxsBoiSB 117.— Page 369. (1) Whole stock :A's stock:: whole profit :A's profit. rvu 4.- «.. m'' X 3000 Thatis,$4300r$3000::$lli7: $779..0'>^ ., . 4300^ — ^^^^^^024.= A's sh- 11117-, $779.302 + =:|337.69T:^B's share. if-? ■.': ail. 166 KEY. a) [Nat. AsiTB. Whole stock =5: $6470 + $3780 + $98Q0 =? $30110. Whole stock : A's stock :: whole profit ; A's profit, 7890 X 6470 Thatis,$20ll0:$6470::$7890: — =$2538-453+=:4'3sb. 20110 Again, whole stock : B'a stock :: whole profit ; B's profit, 7890X3780 ^ That J3,$20110 ; $3780 : : $7890 : — ^^~;^i 483-053+=B'3 sh, 20110 Lastly, whole stock : O's stock : : whole profit : O's profit. 7890X9860 That is,$30UO;$,'i 860: :$I390: ^=$3868'493-H=0's zh, 20U0 m (3) Whole stock : B's stock : : whole gain : B's gain. 80.x 120 That is, $320 : $120 : : $80 : = $30 = B's gain. 320 Again, whole stock : G's stock : : whole gain : G's gain. 80 X 200 That is, $320 : $200 :: $80 : = $50 = O's share, 320 (4) i- / 20 bole stock : B's stock :: whole gain : B's gain. 728 X 1200 That is, $2800 : $1200 :: $728 : = $312 = B's gain. 2800 Again, whole stock : G's stock :: whole gain : C'-, gain. 728 X 1600 That is, $2800 : $1600 : $728: =|.l].fi -z^ O's gain, 3800 kT. AjUTB. £ZBBOI0X 117.] KEY. 167 110. ofit, I profit, ■|-=B'3 sh, 3 profit. (6) Whole stock : B's stock : : whole amount to be diyided : B's share m, 100X2 That is, $3 : $2 : : $100 : = $66-665 = B's share. 3 Again, whole st'k : O's st'k : : whole amo't to be divided : O's sh'e lOOXl That is, $3 : $1 : : $ioo : . = $33-33^ = O's share. .1 (6) i ^HOOcilfioO :: Xlioo :ll!!ili!!=£392f= B's share, 1400 ^eilOO — i;392§ = je707f = Cs share. am. B's gain. 's gain. [!'s share. (7) casks, casks. 180 x 200 900 : 200 :: 180: - = 40 casks ='3's loss. 900 180X300 900:300 :: I8O : = 60 casks = O's loss 900 180— (40 + 60) = 80 casks = D's loss. 7, ."I (8: .in. B's gain. '3 gain. O's gain, 100X800 $1800 : $800 :: $100 : = $44-44^ = B's snare. 1800 100X600 $1800: $600 :: $100: = $33-33 l = C's share 1800 $44-44 + $33-33^ = $77-775, aiid $100 —$77-775 = $'><>. 003 — rv» «--5=- 168 KEY. (Nat. Abitb. (9) 120X1 •6j1 :: 120 : = 20 6 120X2 6:2 :: 120 : = 40 6:3 :: 120 6 120X3 6 (10) = 60 Whole 1093 = $900 — $540 = $360. 8: 1 8 $360 $360 360 8 = $45 = B'sl0Sg. 360X2 8 = $90 = O's loss. $45+90 =$135, and $360—135 = $225 = D's loss. $12 : $6 :: $1320 $12 : $4 :: $1320 $12 : $2 :: $1320 (11) 1320X6 12 1320X4 12 1320 X 2 12 (12) ' = $660 =B's gain. = $440 = C's gain. = $220 = D's gain. :86 + £29 =£64, and £110— £64 = £46 = D's profit. D's profit : B's profit : : D's stock : B's stock. Thati8,£46:£35 :: £1090; 1090X35 46 .=£829 6s. llHd. = B'S3t. Again, D's profit : O's profit :: D's stock : O's stock. Thati5,£46:£29 :: £1090 1090X29 46 £687 33. S^|d. = C'3St. EXBB0I8K8 117, 118.1 KEY. 169 ExKROiSB llS.—Page 271. (1) $357 X 5 = $1785 for one month) 371 X 7 = 2597 for one month > = $6076 for one month 154 X 11 = 1694 for one month > ' 347-20X1785 $6076 : $1785 :: $347-20 : = $102 ^ 6076 347-20x2597 $6076 : $2597 :: $347-20 : = $148-40 6076 347-20X1694 $6076 : $1694 :: $347-20 : = $96-80 6076 (2) 40 X 6 = 240 for one month > ' 30 X 5 = 150 for one month > = 440 for one month. 50 X 1 = 50 for one month J 160X240 440 : 240 :: $160 : = $87-27^-; B's share. 440 160X150 440 : 150 :: $160 : = $54-54-i6i; 'a share. 440 160 X 50 440 : 60 :: $160 : = $18-18^«f; D's share. 440 (3) £150 X 6 = i;900 for one month ) 200 X 3 = 600 for one month > =£3500 for one month 125 X 16 = 2000 for one month ) £291 133. 4d. X900 £3500 : £900 :: £291 133. 4d.: = £75 3600 £291 13s. 4d.X600 £3500 : £600 :: £291 13s. 4d.: = £50. £3500 : £2000:: £291 138. 4d 3500 £291 13s. 4d.X 2000 3500 -=£166 13s. 4d, 170 KEY. (4) {Sat, Ajutb $4000 X 12 =$48000 for one month 3000 X 15= 45000 for one month 5000 X 8 = 40000 for one month i= $133000 for one month f?65X 48000 $133000 : $48000 ;: $esb : =?240; B's share. 133000 665X45000 $133000 : $45000 :: $665: =$225; O's share. 133000 665v/innno $133000 ; $40000 :: $665:- :^ $200 ; D's share. 133000 (6) 66 X 12 = 672 for one day ) 64 X 15 = 960 for one day > = 3072 for one day. 80 X 18 = 1440 for one day ) 320X672 3072: 672:: $320: =$70-.: rent to be paid by Ist troop. 3072 320X960 3072: 960:: $320: ^=$100= «< i( " 2nd " 3072 320X1440 3072: 1440:: $320: =$150= " " « 3rd « .^072 0) Sum of profits = 240 + 800 + 400 = $1440. Whole profit : A'spioEt :: Whole sl'^ = 9-89 or | nearly 10 per cent. <16) $4 • 25 X 723 = $3072 • 75 = price of 723 yds. /© $4 - 25. $3072.75 X -07 = $215 -0925 = amount for Insurance. $3072-75 X -22 = $676 -005 = amount for ad valorem duty. Then whole cost = $3072 -75+ $215-0925 4.$23-70 + $2W -f $3 - 16 4- $676 • 005 = $3993-4075. Whole gain =$5270 — $3993-4075 = $1276-5925. That is, $3993-4075 gains $1276-5925 of $1276-5925 - T^fi^Wo^j of a $. gains -ryyljW cucu ^iy^u per ceo% = ^^VsTfV'fi'? ^ iw^^an'yQua or 82 per cent. *'^ ^liOsonBS 121-123.] KEY. EzBBoiSR 122— Page 278. It5 (1) Losa on $1 is 4 cents, or for every $1 paid I receive $9-96. Hence cost = $24 • 60 v- 0-96 = $25-625. (2) Loss on $1 is 10 cents, or for every $1 paid be receivea $0.90. Hence cost ==-$2360 -J- -90 = ${!622 -22. $1 loses tTTi-Vo'o (3) Gain on $1 is 11 cents, or for every $1 paid he receives $Ml. Hence cost = $7400 -i- Ml = $6666 -666. ! = Wj^oie sum $1 gains Ty^i^j "fta_9.89 orj (4) Gain on $1 is 17 cents, or for every $1 paid he receives $1-17. * ^ ^ 3789-40x100 $117 ; $100 :: $3789-40 : — - = $3:^38 -803- 117 (5) Loss on $1 is 13 cents, or for every $1 paid I receive $0'87, 2740X100 $87 : $100 :: $2740 : =:$3149'425. 87 M 4 ExBBCiSB 123— Page 279. (1) $2 gains 60 cents. rt -k - ^ 2-OOxIO Hence $0-60 ■ $0-10 :: $2-00 : =40 cent! 60 11& r 'y-^ or. (2) [Kat. Attrra. $2*00: $2*80 :: $2*50 2-50X2. SO 2*00 •=$3*60. J (3) 8 cents gain 6 cents in 9 months. 5X6 Hence 9 mo's : C mo's : : 5 cents : = 3J = gain for 6 mo's. 9 / 3JX12 8 cts. : 12 cts, :: 3J : ' ■ ' = 5 cts. gain on 12 cts. for 6 mo'g. 8 Therefore 12 + 5 = 17 cents = his selling price. 36i (4) l'85X-55 $1-60 ; $1-85 :: $0'65 : = $0-6359375 =■ what L 1-60 ought to get in order to sell at the same profit as K. But L only gets 60 cents, therefore K has the advantajre. 70 yds. of cloth at $1-85 = $1-85 X 70 = $129 -50. $129-50 + $-60 = 2 15^. (5) 6 tons of butter at $102 =± $102 X B = $510 lOi tonsoftallowat$135 =$135 X 10J = $1417-50 Total yalue =$1927-50 Deduct ready money, $600 • 30 $1327-20 11327 •20-f-$4* 20::= 316 barrels. [Nat. Abith. EXXBOXSXB 128,121.] KBT. 177 EzxBcisi 124— Page 281. >3'60. iin for 6 mo^9. 2 cts. for 6 mo'g. 375 =■ what L fit as K. idvantajre. = $129'50. = $510 = $1417-50 = $1927-50 By, $600-30 $1327-20 (1) 7 oz. X 22 = 154 carats. 12} " X 21 = 262j « 17 "X 9 = 153 « 36} 3Gi)569i " 2 2" 73)li39(15f^carat8. 73 409 366 44 (2) 2 gallons (& 148. 1 " /® 12s. 2 « /a 9s. 4 « /® 8s. 8 (3) 15 bushels /® $1-20 30 " /®$l-50 60 « /3>$1-10 83 *• /a) $1-75 188 $18*00 $45-00 $66-00 $145-25 188)$274-25($1' 458 188 86-2 75-2 1105 9-40 • 1-650 1-604 ^^Aa = 288. = 128. = 18s. = 32s. 9)90s. lOl. 1 i .178 KEY. X4) [Nat. Asith. 12 lbs (d> 60 cents = 600 cents. 16 " (cb V2 u = 1152 a 22 " /a>65 u = 1430 II 18 " (Sb 86 i( = 1630 II 100 " (cb ^2 u = 4200 II 168 168)8912 cents (63^1 cents. 840 612 504 168) il^ EzaBOisB 125— Page 283. (1) 'Prices. Differences. Prices. 125 = 160 — 35 140 — 15. 15 + 110 25 + 100 = 126 Prices. Differences; Prices. 126 =^ 160 — 35 ^ 15 + 110 i 140 — 15 '^■^ 25 + 100 > = 125. Ant. 35 bush. td> $1.10, 15 (Sb $1-60, 15 /5)$l,and 25 fcb $1-40. 36 bush fcb $1 •00, 15 (Sb $1-40, 15 ® $1 • 10, and 25 (d) $1'60. ■A IJiAT. jLmTH. ^1 cents. = 125 = 126. and 25 /© $1 -40. and 25 fa> $1'60. EZBBCIBJM 124, 125J (2) 179 Prices. Diflfereuces. Pricefl. 50 — 60 — 16- 5. 3 + 42 ^+38 = 45. 15 + 30 J Prices. Differences. Prices. 45 3 + 43 + 38 1=45. /60— S-'V^ V , ^15 + 30 ^ns. 15 quarts ^ 42 cents, 3 /© 60 cents, 5 ^ 38 cents 5 cents, and 7 + 15 = 22 /S) 60 cents. Tall ZflT^k ' V'J'''''^ ' ^ ^2 cents, 15 ceais, ana 7 + 15 = 22 /® 60 cents. (3) Prices. Differences. Prices. /® 30 ^30 12i =- = 12i. ( 14-li- J j^ns. i lb. fS> 18 cents, I /® n cents, i rS> 16 cents 21 /^ 15 cents, 2i ^ 14 cents, 5i + 44 + 31 -i^i IVo '. •u _i_ II _ ^ /^ ,/x » T «j t- di — i3i /® 12 cents, and -i + 1 j = 4 /2) 10 cents. ' (4) Prices. DJfferoneaSo Prices. 13 - j.~ 3_}.7 12 ! 180 i KST. EzBBOiSB 126 — Page 284. [Nat. A2UTH. (1) By Case I we find that 17 quarts f8) 31 cents, 6 fd) 16 cents, 6 f(i 19 cents, and 6 ^ 23 cents will make a mixture worth 25 cents per quart. 6 X 87 Therefore 17 qts. : 87 qts. :: 6 qts. : = 30 jf quarts fd) 17 16 cents, and as there are 6 lbs. at each of the other prices, the same statement may be used, and the answer Is therefore 30|^ quarts i® each price. (2) To produce a mixture worth 75 cents per bushel, we require 45 bushels fS) 80 cents, 5 (Q) 37 cents, and 5 /S) 68 centd. 5 X 70 Therefore 45 bush. : 70 bush. :: 5 bush. : = 7^ bush. 45 oats fS) 31 cents. 5X 70 45 bueh. : 70 bush. :: 5 bush. : = 11 bush. 45 barley fS) 68 cents. (3) To produce a mixture worth Is. per lb,, we require IJ lbs. fa) 16d., U fd) 14d., and 6 ^ lOid. Then IJ lbs. ; 50 lbs. :: IJ lbs. : 50 lbs. brass fdb 14d. IJ lbs. ; 60 lbs. :: 6 lbs. : 200 lbs. pewter fa) lO^a. (4) '' By Case I we find that 1 oz. of 20 carats fine, 1 of 21 carats fine and 3 of 23 carats fine, will make a mixture 22 carats fine. Then 1 oz. : 30 oz. :: 1 oz. : 30 oz. of 21 carats fine. 1 oz. : 3U oz. : r 3 oz. : 90 oz. of 23 carats fine. INAT. A.UTH. EXKBOISES 126, 127,1' KEY. 181 • ExHBoiSH 127— Page 286. (1) To produce a mixture worth $1-40 per lb., we require 20 lbs ^$1.00 40 ^$1.20, 40 ^$,.60, and 2o'/®$1.80. But ali of these added together, will make 120 lbs. lbs. lbs, lbs. lbs, 168 X 20 Therefore 120 ; 20 :: 168 = 28 lbs., the require(' 120 quantity z® $1'00. 168 X 40 120 : 40 :: 168 : = 56 lbs., the required IaO quantity /a $1.20. 120 : 40 168 X 40 ^^^ • — TZT" ~ ^^ ^^^•' ***® required quantity/® $1.60. f-- 168 X 20 - — 28 lbs., the required uo quantity /a $ I'SO. (2) T« produce a mixture worth 4s. 4d, per ' I. we require 10 lbs ® 5s, and 8 tS> Ss. 6d. But these added together make 28 lbs lbs. lbs. lbs. lbs. n^u ^ 27x10 Therefore 18 : 10 ;• 27 • ^K lUc *x. . , XM .. n . — 15 lbs., the xequired quan- 18 tity of tea (cb 5s. 27 X 8 18 ; 8 :: 27 : . . = 12 ibs., the required quan- tity of tea (a> 3s. 6d. 182 KBT. (3) [I^AT. AbITR. / 70 : 20 70 : 10 63 63 To produce a mixture worth $2-70 per gallon, we require 20 gallons ^ $2 '40, 10 (cb $2-60, 10 (d> $2-80, and 30 fob $2-90. But all of these added together will make 70 gallons. Therefore gals, gals gals. gals. 63X20 - - = 18 gallons, the required quantity 70 * ->. J of brandy/® $2-40. 63X10 = 9 gallons, the required quantity 70 ■' of brandy/® $2 '60. 63X10 - = 9 gallons, the required quantity of brandy /© $2-80. 63X30 — -— = 27 gallons, the required quantity of brandy /® $2.90. 70 : 10 :: 63 70 : 30 63 ExEROiSH 128 — Page 289.. 0) 1974-80 X f = £740-55 = £740 lis. (2) 765-43 X f = £306-172 = £306 3s. 5^d. (3) 8172-19 X i= £2043-0475 = £2043 Os. 11^. EzEBoiSB 129— Page 289. (1) £743 18s. lid. = £743-91583 and 743-94583 -J- fj, sa |2479-8194. Wat. Artth. | Exxboisks MT-lSl.J KB7. 18a s require 20 fd> $2-90. Therefore id quantity (2) £119 9b. 8id. = £119-484375 and 119-484375 -f- 1 = $318-626. £473 17s. IJd = £473 '8572916, and 473-8672916 -^ ^^ = $2030-816964. d quantity d quantity 1 quantity GzsROisie 130— Page 290. (1) 1006-90 -^ 4-867 = £206-88309 = £206 11b. 73d. (2) 916-87 -f 4-867 = £188-38504 = £188 7s. 8id. (») 2114-81 -5- 4-867 = £434-52023 = £434 lOs. 4Jd. d. £2043 lis. 3d. = £777 78. 7d. = SxawjiSB 131— Page 290^ (0 £2043-5625 and 2043-6626 X 4 '867 $9946-01868. (2) £777-37916 and 777-37916 X 4-867 $3783-50437. 479-8194. (3) £557 19s, 5id. = £557-972Qlfi unA KK^T.^Ttaic w A.o^tr _ $2715-65418. CVat. Amm. $16786'25- X 6-04 = 84697 francs 66 oentimei. Commercial value of the marc banco = 35 centi. Add 1 per cent 35 3535 Then $0-3635 X 4000 =: $1414. $36678 X 1-0225 = $36480 •755. The p-cr value of 1 ruble =: 75 . cents. Deduct 2 per cent 15 Then $0-735 x 2560 = $1881-60. Old commercial par of £1 sterling = $4,444 = $4-44444 Add 8 per cent .35555 $4-79999 ^6 V BiBRoisi 133 — pagg 29a, lU £1 = 420 J. 19]d. = 1 franc. 300 franc =r 60 ducatg. 1 ducat = 360 mRrayedlfc * = £1000. M^'?XlXt|^X360X1000 Ik 42id. : £1000 :: ara marayudis : — = 1664138 marayedi 272X10* ex. -tJXl2 ' ^xioooxsxir 4^, -1536000 mararedis by direct exchange. ^^'''31)27138 '^'-^^36000 = 28138 mararedia. 8)827 reals 20 mararedis 103 piastres 3 reals 20 maravedis. (3) Old coramercial par of £1 sterling = $4-444 10 which add 10 per cent, of itself = -4444 «,-.«, ^ ^^ f®'"*^^ . IMAGE EVALUATION TEST TARGET (MT-3) 1.0 ^^ 1^ 12.2 I.I « 112.0 11.25 ! IIIIIB 1.4 mil 1.6 1^ -►« Photographic Sciences Corporation 23 WEST MAIN STREET WEBSTER, N.Y. 1 4580 (716) 872-4503 v % V ■^ \ \ c^ ^ m ^^^^ 4b. '/. % o ^ v^^ -^^"' 186 KBT* [Kav. Aaxob 18 cents. = 1 franc, 25 franca. = 240d. I80d. = 3 milrees, 5 milrees. =18 marcs ban. 1200 marcs ban=; a: 60 5 X = X^^ix^xiH = $376 = circuitous exchange or sum be pays for 1200 marks. 1200 X '35 = $420 = direct exchange or sum paid for 1200 marks, $420 — $376 = $45 =;z gain hy circuitous ezcua;nge. EZIBOI 13»' SxiROisi 134— Page 298. 0) (3)» «=3X3X3X3X3«243. (2) (20)»o = 40 X 20 X 20 X 20 X 20 X 20 X 20 X 20 X 20 X 20 = 10240000000000. (3) (106)« « 1-06 X 1-06 X 1-05 X 1*05 X 1*05 X 1*06 = 1*340095640625. (4) (5) (ft)« = f X § X t X * X t == AVA. ^ {(' {( {(5! • • • 19536 16 84)353 886 882)176 176 PAtt AxBtm ExxBCira8l8»437.] KET. 187 5 txiH 200 marks. lid for 1200 ixcii&nge. ) X 20 X 20 EziBOiSB 135— Page 29ft. (1) 4* X 4* X 4» X 4' =r 4« ♦ * ♦ » ♦ ' « 4i«. (2) (3) 13X1 «5- 13« = 131* ' « =1 139. (38)8 «i 33 « » =a 3". (4) {(7* X t3)-^(7«x7«)}6= {(7* ♦9)-:.(7* ♦»)}*•=. 1 7^ -^ 7* } « = (7^ - *)6 = (73)» = 79 » « « 7i». (5) {(53 X 6* X 611 X 69)^(53 X5* X 5'X6')}8=. (6«T- 17J3 __ (510^3 _, 510 « a 5^ 530^ 1»06 = EzsBoiaa 137— Page 304i^ ^. 1481 ^A. (1) 195364(442. 16 84)363 336 882)1764 1764 (2) • • •0676<'26 4 46)276 276 (-> 984064(992 81 189)1740 1701 1982)3964 3964 188 W KBT. UTat. Ajutm, . 6*0000000000(2*23606 42)100 •84 443)- 1600 •1329 4466)27100 26796 44t206)304000(r 2683236 356764 ' (6) 60.487129(^-777 49 147)1148 1029 1647)11971 10829 15547)114229 108829 6400 (8) •600000000000(-70n06 1407)10000 9849 14141)15100 14141 1414206)9690000 8485236 1104764 vo V979226629761200Z(2834t5249 48)397 384 662)1392 1124 6644)26826^ 22576 56437)425062 395409 664945)2965397 2824725 •0000012321(-00111 €1649502)14067261 11299004 21)23 21 66495044)276825720 225980176 221)221 221 664950489)5084554401 6084554401 BzsRozsxs 187-189.] KEY. IgO ExsRoiSB 138— Page 304. (1) 1 = i and Vf=i. (3) 5f = 5-142857142867 and V5-142857142857 = 2*267786. (4) Ui 5= -4033457249 and V -4033457249 = -63609. (6) ^H = 13-2 and V"i3^2 = 3-63318 Embowb 139--Page 306. • 11333311(2626 46)433 411 552)2233 1434 6546)46611 46611 33233344(4344 24 123)523 413 1304)11033 10024 13124)100544 100644 190 KET. [Nat. Abith. £x (8) (*) 4234-101230(43-412 31 • • • • 133)1134 1004 1414)130-10 122-21 14231)2'3412 1-4231 142322)-413130 •340144 22431 888888-8880(888*88 11 178)1788 1601 1878)18788 16801 18878)1887-88 1688*01 188878)188-8780 168-8801 18-8878 (6^ 248664e /69(S4373 21 U)386 354 '.■i Y83)3264 2809 /867)657e t 62etl ^8723)281969 281969 BTi 17i EzBKOiSB 140>-Page 30Y. (1) 100« 60« 10000 3600 [Nat. Abitb. £zB«oisxg 189, liO. ] 60« = 2500 803 _ 5400 KEY, ca) Sum = 8900 and V8900 = 94-34 nearly (3) I 242 = 576 -^ 2 == 288 ^^^ ^^ ^ ^^.^^^ (4) 36« z= 1296 20* = 400 Difference = 896 and V896 = 29-933. (6) 40*= 1600 142 196 DiflTerence = 1404 aad V1404 = 37-469. 402 = 1600 262 = 676 Difference = 924 and V924 = 30-397 37-469+ 30-397 =^ 67-866 and 67-866 + 3 = 22-622. (6) 1760 sq. yds. = 15840 sq. ft. and V15840 = 125-867. (7) V141376 = 376. (8) 3« =9 3*=- 9 B8Sx = 18 and y'I3 = 4*24264. 191 192 KBT. (») OX AX. Abzth. 16« = 12« = : 266 :144 • Siim = 400 and V400 = (10) :20 3' + 3* + 3« = 27 and V2T= (11) 6196. (iV)» = rh and (1)« = 1. 450 Then yi^ : 1 :: 450 : — = 46000. (12) 1 sq. acre = 160 sq. pe rche s . 16 -^ 3-1416 = 60-929463 V50-929462= 7-136. and EXIBOIBB 141— -Page 311. (1) 62712728317(3973 27 3«X300r 3X9X30= 9«= 2700 810 81 35712 3591 32319 ^ 39»X300= 39X7X30= 7» 456300 8190 49 3393728 464539 3251773 , 397«X 300=47282700 397x3X30= 36730 3«= 9 47318439 141966317 141965317 Nat. Ajotb. )-929462 and 73 m 111.] c» • 1963125(126 I 1*X300= 300 1X2X30= 60 2«= 4 868 — i. 364 12B 12«X 300=43200 12X5X30= 1800 6»= 25 226126 45026 225126^ m • • • 1*X300= 10*X30?= 10X2X30= 22— 300 30000 600 4 30604 1076890626(1025 1 "76 76890 102«X300=3121200 102X5X30= 16300 6«= 25 3136525 61208 16682626 16682626 K 194 VXja.Jam, CD 8«X300=: ld200 8X8X30= 1920 8'*= 64 "697864103('88T 612 185864 • 21184 88«X300=: 88X7X30= r= 2323200 18480 49 234l728t 169472 16392108 1639210a (»> 4«X300=: 4800 4X6X30= 720 6*= 36 5566 46* X 300=634800 46X8X30= 11040 8«= 64 645904 102503'232(46*a 64 38603 33336 6167*232 6167*232 I 1 m^ iTBxaiiiin 185 W • • 664«X 300=95428800 664X2X30= 33840 2«= 4, 179637 •068283(06*42 las 5»X300= 6X6X30= 6«= 7600 900 36 64687 8436 60616 66«X300= 66X4X30= 4»= 940800 6720 16 3981 '068 947536 3790-144 100 '826288 86462644 180*826288 0) ^*X300= ^X8X30= 8»= 14700 1680 64 483* 736625(7*85 343 140*736 16444 V8«X 300=1825200 78X6X30= 11700 6«= 26 283e 26 8*184625 • 196 Wf. {V4».Aam. •636056' 'M 612 «:■ •i-^' 8*X300=rl9200 8X6X30=:: 1440 6«=. 36 124056 20676 ft 124066 BxiBOiai 142.— Pago 8U, (1) )V= •1062631578944- and ;V' 105263157894= '4721. (a) ■^=5 '1764705882354- and ^-176470588235 == '6609. (3) } of 2| ={ = -8333333334- and {^833333333= •941. (4) 28% is 28'76 and ^2^75 = 3'06ai (B) I653«=3; 1653X31 82^ = 32 • 72 and ■;'32 • 72 = 3 • 198. 014*. Aim. 16 i= •4721. Izs •6609. 333= 'S*!. fli mm} my. lift ExiBcwt 143.— Pa^* 818. One milUon = 83133344 geiuiry. 2«=-4x300=S 2000 JX30=a00X4= 400 4«= 24 2424 248=1 104 X 300=332000 24X30=1200X4= 5200 ^ 4«= 24 1*X300= 1X30X6= 6»= 16*=304X300= 16X30=620X5= 6*= 341224 300 220 44 664 111400 3220 31 165«=32571X300= 166X30=6370X3= 3«= 12015300 20350 11 12035661 I653«=327207lx 300=1205625300 1053 X 30=54010X2= 130020 2«= 4 98. 1205755324 33233344(244. 12 21233 14544 2246344 2246S44 6131271 •000000(165-32. 6131 4270 641271 114651 600115 41164*000 36131 •423 3022 •356000 2413 ' 732650 406-422130 198 VXA!g»Jkxam, <3) 10121013 102000000 IX 1000 ixixioo * 11«= =121X1000: 11X100=1100X2- 2* 1000 100 1101 121000 2200 11 200211 tt2«=21021Xl000= 21021000 1120«=2102100X?000= 1120X100=112000X1: 1«: 11201«=S211010101X1000: 11201X100=^1120100X2: 2«: 2102100000 112000 1 2102212001 n } :211010101000 = 10010200 = 11 211020111211 1 |112'012=uroot. 2221 1101 1120012 1101122 11120-102 11120-102000 2102-212001 2010-112222000 1122-111000122 U1'001221101 ■^!«>W»^. VSAX*JLxam, 012 102000000 12'012=::root. 012 122 120-102 120*102000 102-212001 )10- 112222000 ll!zicBot8all&] lll'00122I101 KIT* (4) 199 teteet»00\)000(e1't2, 92« «»=nx3oo= eX30=290x^=: 72= 26300 1 18eeet 17301 41 i^'tli e7«~«221X300= 2966300 «7X30=2/90X<= 24«60 l'= 84 298e324 «7*'^(|)a :: $120 : Ans,=$614d. (4) (Y0)« : (fifa)« :: 180 lbs* : Am. , 843000 : aAi|o^(ii ;. igo : Ans.ss 180 X »^§F^i X I,^Wo= 1015-1 Ibi. ' (5) »733 = 921167317 453 = 91125 623 = 238328 30» = 2V000 803 — 512000 203 -; 8000 921.167317 — (91125 + 238328 + 27000 + 512000 + 8000) 920290864 and ^920290864 = 972 •69. («) 8 feet 3 inches = 99 inches, 3 feet =r 36 inches, and 2 feet 7 • inches = 31 inches. 99 X 36 X 31 = 110484 and ^ll0484 = 47*9843. ♦ After the first has wound off her portion,. there will remain | of the thread. tVAt. ExiBoxiM 144*146.] KST. m Then the whole ball : part remaining : : uaoo of diameter of whole ball : cube of diameter of part remaining. That is, 1 : i : : 33 : a:3,and hence x = 3 x -^frr 3 x JfTu = -90856 X 3 = 2-72568 = diameter of the ball after the first has wound off her portion. Similarly after the second has wound off her portion there will remain i of the ball, and after the third has taken her nor- tion, ioftheball. ^ Hence 1 : i :: 3» : ar», whence a: =r 3 x W=i 3 X ^T = 3 X -79370 = 2.38110 = diameter after the second has taken her portion. 1 : i : : 33 : a;3, whence a: = 3 x Vr= 3 x ^':2T= 3 y •62996= 1-88988 = diameter after the third has taken her 00^- tion. ^ Hence 1st takes off 3 — 2. 72568 = -27432 inches 2nd « « 2.72568 — 2-38110= -34458 « ' '. 3rd « « 2.38110—1-88988= -49122 **•» " " remaining 1.88988 ({ <( + 8000) = nd 2 feet 7 9843. ill remain } BzBBOiSB 145— Page 316, (1) .V19987173376 = 141376, and Vi41376 = 376. ; (2) 1^308915776 = 676, and V676 =s 26. (3) V40353607 ;= 343, and -^343 = 7. I f ^387420489 = 729, ^739 = 9, and Vr= 3. 202 KE7. USWr.Axam. («) ^134217728 =512, ^^512 = 8, and ^8 =2. EziBOiSB 148— Page 321. (1) The mantissa of the logarithm of 8193 (the first four digits) : '913443, and the next following mantissa is '913496. Then from -913496 Subtract.. '913443 Difference, 53 ; and 53X217 (remaining digits of given n,\imber) = 11501, from which we cut off three digits, since we * nii'ltiplied by a number of three digits, and since the highest digit cut off is not less than 5, weadd unity to the part retained, which gives us 12. Then mantissa of logarithm of first four digits .913443 Add, 12 Mantissa of logarithm of given number^ .913455 To which attach the characteristic 6 and required logarithm =: 6-913455. The mantissa of the logarithm of 7392 (the first four digits) = •868762, and the next following mantissa is '868821. Then from -868821 Subtract.. -868762 Difference, 59 ; and 59 X 45 (remaining digits of given number) = 2655, from which we cut off two digits, since we multiplied by a number of two digits, and since the highest digit cut off is not less than 5, we add unity to the part retained, which gives us 27. Then mantissa of logarithm of first four digits, .'868762 Add, 27 Mantissa of logarithm of given number, /Continned on next nair©,^ ^868789 [KjlT.Abiiu :=2. rat four digits) H3496. iigits of given igits, since wc :e the highest > part retained, igits .913443 12 .913455 1 logarithm = 3t four digits) 18821. Iigits of given gits, since we 3 highest digit part retained, 9, .'868762 2T ^^^EXBcnsEs 146*148.] KBT. (1 continued.) 208 To which attach the characteristic 1 and reqnircd logarithm = 1-868789. ^The mantissa of the logarithm of 8437 (the first four digits) -"Then froV"926'r'' '""'""^ "^^'^^^^ "" "'''''' Subtract.. '926188 Difference, 52 ; and 52 x 42 (remaining digits of given number) = 2184, from which we cut off two digits, since we multiplied by a number of two digits, and since the highest diirit whicf bl^comes'2" ''^"^ '' "^ ''' '^'^^^^ ^° '^^ P«' -Gained, Then mantissa of logarithm of first four digits -926188 -A-dd, ' 22 Mantissa of logarithm of given number, -926210 To which attach the characteristic land required logarithm = 1.926210. (2) The mantissa of the logarithm of 2345z= following mantissa is -370328. Then from -370328 Subtract.. -370143 •370143, and the next Difference, 185; and 185 X 64 = 11840, from which we cut off two digits, since we multiplied by a number of two digits, which gives us 118. Then mantissa of logarithm of 2345 =r -37014? 118 Add, .«868789 Mantissa oflogarithm of given number = -370261 To which attach the characteristic 4 an required logMithm,=: 4-370261. (Oontinned on next page.) 204 KIT. tSAT.AMmtr (2 continued.) The mantissa of the logarithm of 1007 S; 003G29, and the next following mantissa is '003461. Thenflrom '003461 Subtract.. '003029 Difference, 432 ; and 432 X 013 = 8616, from which we cut off three digits, since we multiplied by a number of three digits, and since the highest digit cut off is not less then 5, we add unity to the part retained, which gives us 6. ''^hen mantissa of logarithm of 1007 = '003029 Add, 6 Mantissa ot logarithm of given number '003035 To which attach the characteristic 3, and required logarithm »= T- 003035. (2) » Mantissa of logarithm of 6237 • 719083 Difference from column D = 83 ; and 83 X 6 = 498 from which we cut off 1 digit and add so And also attach the characteristic 1, and required logarithm = > 1.719133 Mantissa of logarithm of 1294 '111934 Difference from column D = 335 ; and 335 X 76 = 25460 from which we cut off two digits and add, 255 And also attach the characteristic 2 and' required logarithm ^S. s«s:s:s««««e»»»«»se*»esssss««ss 2* \}SS^fi tSAv. Amrat^ t029, and the Dm which we iber of three )s then 5, we 103029 6 03036 logarithm B3 JBZXBPXtB 148.] 206 Mantissa of logarithm of '0004713 = -673297 P. P. corresponding to -00000009 = 83 P.P to -000000008 == 74 Sum,=s -6733874 Therefore required manUssa = -673387 and required logarithm = 4-673387. Mantissa of logarithm of 9136000 : P. P. corresponding to 700 P.P. P.P. u to to 10 = 2 = '960756 33 5 9 n,. , . Sum, =5 -96078959 Therefore required mantissa = -960790 and^equired logarithm = 6-960790. ■719083 »8 3d .. 1.719133 .. -111934 d, , 2-1 255 (5) Mantissa of logarithm of 4-23400 P. P. corresponding to 20 P.P. "to 9 Sum, = -6267802 Therefore required logarithm is 0-626780, Mantissa of logarithm of 763-1 P. P. corresponding to -02 P. P. " to -009 ^- P- " to -0008 ^•^* " to -00009 •882581 11 61 46 40 Sum, = -882597600 Therefore required loturithro is 2-882598, 206 KBY. ExiBOUB 149.— Page 323. (0 Oiren logarithm, *137139 Next lower in table, -137037 = log. of 1371. UTat. Amto. Difference 102, Tabular difference = 316. Then 1020000-7-316 gives 3227 for digits in 6th, 6th, 7th, and 6 th places. Hence the digits of the natural number are 13713227 ; and since the characteristic is 4. i.e., one less than the number of digits to the left of the decimal point the required number is 13713-227. Given logarithm, '718134 Next lower in table, '718086 = iog. of 6225. Difference, 48, Tabular difference = 83. Then 48000.^83 gires 578 for digits in 6th, 6th, and 7th places. Hence the digits of the natural number are 5225578, and since the characteristic is 0, i.e., one less than the number of digits to the left of the decimal point, the required number is 6*225578. Given logarithm, '635421 Next lower in table, '635383 = log. of 4319. Difference, 38, Tabular difference = 101. Then 38000 -r gives 376 for digits in 6th, 6th, and 7th places. Hence the digits of the natural number are 4319376, and since the characteristic is 4, i.e., one more than the number of ciphers between the decimal point and the first figure to the right, the required number is '00Q4319376. ^^ [Nat. Amitu, Sznoiui48.] 207 (2) Giren log. -921686 = log. of 8350. And since the characteristic is 2, i.e, one less than the num- ber of digits to the left of the decimal point, the required number is 835. Given logarithm, -922165 Next lower in table, -922154 = log. of 8359. Difference = n, Tabular difference = 52. Then 11000 -^ 52 gives 211 for digits in 5th, 6th, and 7th places. Hence the digits of the natural number are 8359211; and since the characteristic is T, i.e., one more than the number of' ciphers between the decimal point and first figure to the right the required number is -8359211. ' (8) Given logarithm, Next lower in table, Difference, = Highest P. P. not greater than 67 = •407968 •407901 = log. of 2558. 67 51 corresponds to 3 for 5th place. 160 Highest P. P. not greater than 160 = 153 corresponds to 9 ■ for 6th place. 70 Highest P. P. not greater than 70 = 68 corresponds to ■ 4 for 7th place. 2 Therefore digits of required number are 2558394 ; and since the characteristic is 5, there must be six digits to the left of the decimal point. Hence required number is 255839-4. (Gontinued on next page.) i:f . m ■mf' (3 coBtinaed.) HUx. A»im Oiren logarltlntty Next lower in table, •408866 •408240 s= log. of 2660. Difference, = 146 Higliest P.P. not greater than 146 = 136 corresponds to 8 ' " in 6th place. 100 Highest P.P. not greater than 100 = 86 corresponds to 6 — — — in 6th place. 160 Highest P.P. not greater than 160 = 136 corresponds to 8 — — — — in 7th place. ' ■ ' 140 Highest p.p. not greater than 140 = 136 corresponds to. 8 in 8th place. Therefore digits of required number are 25608588 ; and since the characteristic is 7, there mast be eight digit? to the left ol the decimal point. Hence required number is 26608588. Given logarithm, Next lower in table, Pifference, = Highest P.P. not graater than 61 =: ♦416369 •416308 =: log. of 2608. 61 49 corresponds to 3 in 6th place. 12 Therefore digits of required number are 26083 ; and since tho characteristic is 3, there must be two ciphers between the deci- mal point and first figure. TTAncA rAnnirAil nnrnhnr in 'flOSAORS. >g. of 2560. responds to 8 6th place. •responds to 6 a 6th place. sponds to 8 rth place. sponds to. 8 1 8th place. 588 ; and since t3 to the left of log. of 2608. rresponds to 3 in 5th place. ; and since tho ween the deci- • KEY, » • (4) Given logarithm, Next lower in table, •877777 •8777T4 = log. of 7647. Difference, = There is no P.P. not jrreater than ^ « "v greaier man 3 o corresponds to in 6 th place. 30 Highest P.P. not greater than 30 = 29 corresponds to 6 in - 6th place. fO Highest P.P. not greater than 10 = 6 corresponds to 1 - in 7 th place. 40 Highest P.P. not greater than 40 = 35 corresponds to 6 " in 8th place. ' 60 Highest P.P. not gre.,„ ,)..„ 60 = -46 correspond, to ' — " 8 in 9th place. 4 Therefore digits of required number are 754705168 ; and since tt dtr ;r:: ' '' ''-'- ^^-^ '- «- '-'^- -^^« ^- &en<5e I'eqnired number is 75470.5168. Given logarithm. Next lower in table. Difference) ts: TT High.,t P.P. a„, greater th.n 98 = 93 correspond, to 8 in 5th place. •555555 •655457 = log. of 3593. Therefore digits of required number are 35938 ; and since the ^:^!^!^ " '' ^^«- --' »>« 0- 23 83 Am. 725-8033 tt. 212 KET, (Kat. Abitil Logarithm of 437-89 « • 62-735 2-64136S 1-797610 Difference = -843855 =r logarithm of 6*98 Logarithm of 93-217 " -0007132 (8) 1-969495 4-853211 Difference = 5*116284 6-116276 f . logarithm of 1307000 8 = 2-4 Ans. 130702-4 (4) Logarithm of 23 = 1-361728 " 189 = 2-276462 u 2-748 = 0-439017, Sum = 4-077207 Logarithm of 9836267 = 6-992786 4-077207 Difference = 2-915579 j2-9l555S = logarithm of 823-300 2L = 39 Ana. 823-339 .^.'j EziBOXSl 163.— Page 326. (1) Logarithm of 6 = 0-698970* Then 698970 X 6 s^ 3494850 ==: logarithm of 3l2l [Nat. Abitil 6-98 of 1307000 2-4 .ns. 130702-4 )f 323-300 39 r t I a. 823-339 EXXBOISX0 162-164.] KBT. (3) 218 Logarithm of 1.073 rs '030600. Then -039600 x 6 =;= -183600 = logarithm of 1-6261. (3) Logarithm of -0279 = 2'-445604. Then 2'-445604 X 4 = f 782416 = logarithm of -00000060592. (4) Logarithm of 1-111 = -045714. Thea -045714 X U = -502854 = logarithm of 3-1831, of 3121 EzBBOiBB 154.— Page 327. (1) Logarithm of 913426000 = 8-960673. 8-960673 -r 7 = 1-2800961 = logarithm of 19-0588. (2) Logarithm of 1-61342 = -207747. •20V747 -Ml = -01868609 = logarithm of 1-0444. (3) Logarithm of -000007139 = 6-853637 =r 10*+ 4-853637. (10 + 4-853637) -f- 6 = 2-970727 = logarithm of -0934817. (4) ^ Logarithm of -002147 ="3-331832 ="7 + 4-331832. ("7 -j- 4.331832) ^ 7 = T-6188331 = logarithm of -41576, m m^' |K4lT. Asavn. y>. SziBOm 15G.— Page 328. V (1) 14000 = 7 X 2 X 1000 ,\ log. 14000 = (log. 7) + (log. 2) + (log. 1000). Log. 7 = 0-845098 Log. 2 = 0-301030 Log, 1000 = 3 Sum, = 4-146128 := log. 14000 4-» =7« -r 10 .-. log. 4-9 = (log. 7) X 2 — (log. 10). Log. 7 :s 0'845098 X 2 = 1*690196 Log. 10 s 1 DiSTdreace = -690196 = log. 4-9 •00196 = 49 X 4 -r 100000 rs 7« X 2« -^ 100000 .-. log '00196 = (log. 7) X 2 -f (log. 2) X 2 — (log. 100000). Log. 7 =s 0-845038 X 2 = 1.690196 Log. 2 =s 0.301030 X 2 = 0.602060 Sum r= 2-292256 Log. of 100000 = 5 and 2*292256 — 5 = 3-392266 = log of .00196. Since 6 = 10 -r 2, the logarithm of 6 = log. 10 — log. 2=1 » 0'301<»30 = 0* 698970. 1750 = 6« X 7 X 10 .-. log. 1750 = (log. 5) X 3 -f (log. 7) + (log. 10) Log. 6 =; 0*698970 X 2 = 1-397940 Log. 7=5 -845098 Log, 10 s 1 8ani| = 3-243033 == log. of 1 750^ X428'67142*g =4 X lOOQQ ,\ log. U28-67M28 = (log. j) r^ IMT. lOOOG. JCvnoim 166.] KEY, (1 continued.) 215 Log. I = (log. 1) — (log. 7) = — 0-845098 Log. 10000 M 54902 4 .-. log. of 1428-571428 = sum = 3154902 •00000112 = 2*X 7 -f 100000000 .-. log. -00000112 — (log. 2) X 4 + (log. 7) — (log. 100000000). Log. 2 = 0-301030 X 4 = 1-204120 Log. 7, = 0-845098 Sum = 2-049218 = and log. 100000000 = 8 2-049218 — 8 = 6-049218 = log. -00000112 3'0625 = t? r. log. 30625 = (log. 49) — (log. 16) = (log. 7) X 2 - (log. 2) X 4. '^ ^ ^ Log. 7 = 0-845098 X 2 = 1-690196 Log. 2 = 0-301030 X 4 = 1-2U4120 Difference = 0-486076 = log. of 3-0625. (2) 49lr =V=3* X 11 X i .-. log. 49i=(log. 3) X 2 + (log 11) •4- (log. i). Log. 3 = 477121 X 2 = 0-954242 Log. 11 — 1-041393 Log. J = 1-698970 Sum = 1-694605 = log. of 49^. 363 = 11« X 3 .-. log. 363 =(log. 11) X 2 + (log. 3). Log. 11 == 1-041393 X 2 =2 082786 Log. 3 =0-477121 Sum = 2-559907 = log. of 363. Log. -5 ori = 1-698970, and by altering the characteristic wo get 0-698970 for log. of 6. H (Continued oa next pfige.) 216 KEY. (2 oontinued.) [Nat. Abitb. 4-09 = 4iV = *} = 3' X 5 -4- 11 .-. log. 4-09 = (log. 3) X 2 + (log. 5) ~ (log. 11). V 8 / A -« Log. 3 =: -477121 X 2 = 954242 Log. 5 zz -698970 1-653212 Log. 11 = 1-041393 and 1-663212 — 1-041393 = 0-611819 = log. of 4 09. 2 4 = 2< = V = H X 2 -f- 9 .-. log. 2-4 = (log. 11) + (log. 2) — (log. 3) X 2. 6 ^ -r V K Log. 2 = (log. 10) — (log. 5) =: 1 — 0-698970 = 0-301030. Lo^. 11 = 1 041393 Log. 2= 0-301030 1-342423 Log. 3 = 0-477121 X 2 = 0-954242 and 1-342423 — 0-954242 = 388181 = log. of 2-4. 392-72 = 392vSr = ^^^ = 2* X 3^ x 10 -f 11 .-. log. 392-72 = (log. 2) X 4 + (log. 3) X 3 + (log. 10) — (log. 11). Log. 2 = 0-301030 X 4 =: 1-204120 Log. 3 = 0-477121 X 3 = 1431363 Log. 10 =r 1 Sum = 3-635483 Log. 11 = 1-041393 and 3-635483 — 1-041393 = 2-594090 = log. of 392.72. 293333i = a8c^fi'^Q =2' X H X 10000 -'r 3 .-. log. 293333.V = (log. 2) X 3 + (log. 11) + (log. 10000) -, (log. 3). Log. 2 = 0-301030 X 3 = 0-903090 Log. 11 = 1-041393 Log. lOOOO = 4 Sum = 6-944483 {Coniinaeq on next page,) IK AT. AbZTH. I EZXR0ISE8 155, 156.] = (log. 3) X 2 4242 3212 = 0-611819 = og. 11) + (log. ' = 0-301030. KEY (2 continued.) 21T Log.3 =: 0-477121 and 5-944483 -- 0-477121 = 6.467362- log. of2'i3333i. »"^^A ~ 407J62 — 19-965^ 113 X 5 X 3 -J- moo .-. log. 19-965 = (log. H) v 3 + (log. 5) -f. (log. 3) - (log. 1000). ^ ^ ^ ^ Log. :i r= 1041393 X 3 = 3124179 }'°S' 5 =0-698970 ^^8' ^ =0.477121 Sum = 4-300270 Log. 1000 = 3 and 4-30D270 - C ^ l-^00270 = log. of 19-965. ExBRcxsB 156— Page 336. 13 — 0-954243 .-. log. 392-72 >) — (log. 11). 20 63 83 = 2-594090 = log. 293333^ )) — (log. 3). (1) Here we have given the first term 4, the number of terms 17 and ^h^ sum of the series 884, to find /, the last term. 2r 884x2 7'hen / =r — — «j zr — — ■ n 17 4::=^0t-- t?='00. (2) H(?re .ve have given the first teru 21, the last term 497 and ttie unuibor of terms 41, to find the common difference. I — a 497 --21 7^endz= n— 1 ■ -— 17£ — lis. 11 9 41-1 """^ '' -^^'^ ^ :b) Here we have given a, I, and d, to find -t, and since a = 12 / = 96, and rf = 6, we have ' I — a 96—12 + ^^' -»+ 1 e= V + l — .J4+ 5 — J5, n = d 6 ■*;. ■■■-..av^jip 2ia or. plA- ABxn, I £3^5^ ii) Here we have gWen /, ^6"j2=: _3 + Vl00200r-fr(-2"ir3)! — _ 3 + V1002001 = — 3 + 1001 =5 998. (16) Here we have given s, n, and /, to find a, and since « n = 18, and I = 120, we have 2» 2 X 1125 <» = — — ^ = 120 = 135 — 120 n 18 = $1125, = 6, Here = 4, .= {2 riere and n z (KAT.AMTH. I £ZXB0I8|166. d since a =: 83500 1000 X 996 167000—1000 id since « = 360, 2— (2X2) 2X2 + 60 + (i)« = days = 16 days, ids. since « = 83500, 6 + 2000 + (2 — 3)« = 998. EET. (IT) 221 Here we have giren a, /, and n, to find d, and since a = 6. t=27J, andn= 11 weiiave d = since* = $1125, - 120 = 6, / — a 27J — 6 22} n — 1 11 — 1 10 (18) Here we have a, d, and n given, to find *, and since to deposit one stone he must walk 6 yards, and the distance travelled for eadi^succeeding stone is 5 yards, therefore a = 5, d = 6, and Then,= {2a+ (n- l)rf}j = {2x6 + (220-i)6}sp . = { 10+ (219X6)} 110 S 1105 X 110 =3 121550 yard3= 69i\, miles. (19) Here we have s, n, and / given, to find a, and since * = 39840. ft = 16G, and i = 478, we have ' 2s n 2 X 39840 166 — 4T8 =s 480 -^ 478 = 2. (20) Here we havd «, a, and d given, to find «, and since n =i 12 = 4, and d == 2, we have ' ,= {2a+(n-l)d}« = {2X4+(12-1)2} V=/8+(lix2)!6= 30X6 = 180. (21) tiere we hAire given a, I, and n, to find »* and « s= i t and n = 24. ' = 24^ Theu* = (a -{- l)" = (l-f.24)V S: 25 X 12 = 300« 2^ KWtl CVat. Asm, ^JBiiRouB 167-.PAftt 843./ (1) Here n =; 11, o = £1024, and r= 11. Then f£=ar"— 1=1024 x (*)i" = 10; r~l i-1 I £176099 = whol« fortune. 1 («) Here a =s 7, Z= 1240029 and s =: 1860040. «^-a 1860040—7 Then r s = , ^ _, jLJ«floM •— / 1860040—1240039 ^^^^^ S3. (») Here n =s 12, a = £l, and / = £20^48. »=- J-i— a (2X2048)— 1 ^_j — ^33j = 4096 ^ i = je4095. iHere r ; Then« (4) » I, n = 8, and Z = 106f^. /(r"-!) _10 6H|X[(g)'»^lJ i^|?TAxV/ (r— i)r»~i 25 X 6306 aoTjfi.. [Nat. Asxnr, I £znoini ist PSi^^ = £59049 iXAoaa 'as. ^ KSt 228 Herea=l,n=r7,aiidrss:8. a(r''~i) lx(37— 1) Then r— 1 3^1 ^j/fl-ioaa. («) fiere ct a i, / =, ioOT7696, and » « 10. Then Z^— a"-i Z^^^ (10077696)Tiil^jTto- (10077696) "V-_ 1-n.iT V(I0077696)r^^l ^7^1^-1 6--.1 60466176-1 ^10077696 — 1 ^216 .- 1 6—1 12093335. }a. Here 3072, and « ^ ei8«. 6138 — 6 « — a « — / 6138— 807a fiii-s. (» Here f ^2, n « n, and 4 « 20470. !nieal«:!2l!!!!ll (2~i)X20470X2io — 1 a^i— 1 •i* 10240. 20470X1024 2047 224 KIT. tKAT. Amnm. w Here a Tben« ll.y n *!■ 12, And r » 3. «(>-»— 1) lx(2i«— 1) r— 1 2 — 1 «> i;204 158. 4

^vib B' Vl9683 3. (13) Here tf =» 6, f « 2, and n = 7. Then/= ar^-^ «: 6 x 2^- » « 6 X 2« =. 6 X 64«i 320. Here a Tbeof (18) 6, / =» 327680, and r « 4. rl—a (327680X4)— 6 = ^axjTU ss 4^0903, •9 ' — ' » CVat. Amnm. Mmmmiiu;!.} 226 A mm 40901. "" yJ7~ ~ =lW4ff744073709B51616gr. 18446744073709551616^(7680 X 64) = 37529996894754 ba.h $1 -80 X 37529996894764 = |562 94995343 131 167295 fat. ^19683 «, 3, e4»320w <15) Herer = 3, n = 10, and » = 295240. Then ^_ C>-J)"-"2 _(^X2!!i^Q ^ 3'_2 X 195240 X 1 9683 Hew a =^1, i a: fi04|^ iMT « -c U, Then « == ^'^--Wt 2048"^ -liW V^(2048)»TZr 2i»~l II V^2048^ a-i = 2»---l =s 4095. ^ s= 4^6905* H) Here « .= 6,r: =^4|«nd.n .•55 9. • Then / = «.•» = 5X49-» = -5X4» = vXo555S = 3276S0. 226 KEY. |Bax»Abixb. BzEBcnsi 158.'>-Page 344. (1) Here fl = ^, and r = g. Then « = 1— r 1-1 (2) Here a = 4, and r = |, Then < = 1-4 <8) Hereo = ^^,andr = T^. Then s s a 1— r 1 — rfv -^s <*) Here a = ^q% and r = Tjfjnr. Then « = T^m 1— r l-nri irc m«( = mu Wir ISaz* iAJBIXB. = f. ExBBoisjca 168, 168J B^y 1 BxiEoiSB 159,— Pago 345. (1) ^ Siace there are 9 means and 2 extremes the nnmber of I— a 92—2 = n = 9. 227 terms Then d = »— I 11—1 l8tterm = 2;2nd = 2 + 9 = li; 3rd-- 114-9-0,.^,. =;:„tr^'' -=-+3=33, eT.'its%-/i';;^ And series fa 2, II, 20, 29, 38, 47, 66, 65, 74, 83, 92. = 8. = «. = ttiit ^ Since then, are 4 mean, and 3 extremes the nnmber of term, Thenrf = tl ^ ^ ^.,^, ^ ^, 41^ + 83- = 60. 3- T Ofi ^ 4ig , and 6th = And series is 7, 15|, 24^, 32^, 4Xf, 60. (3) Since there are 3 means and 2 extremes the nnmber of IS 10. terms Then r = (l^^ = ^^^^^^.-, ^ ^^^j ^ ^ -'it?"".^ *"'" ■' ""^ = *"' '^ * = 2048 ; 3rd = 2048 X i - 1024 ; 4th = 1024 X J = 512 ; 5th = 612 X i = 256, an! 80 on And the means are 2048, 1024, 612, 256, 128, 64, 32, .-d 16. 228 KEY. [ITat. A&itb- (4) Since there are 7 means and 2 extremes the number of terms is 9. (I -V .1. — y* = (2a/4-}as4)?^ = (1679616)^ = 6. 1st term = 14 ; 2nd = 14 X 6 = 84 ; 3rd = 84 X 6 r= 504 ; 4th = 504 X 6 = 3024; 5th = 3024 X 6 = 18144, and so on.' And the means are 84, 604, 3024, 18144, 108864, 653184, and 3919104. ExxROiSB 160.— Page 347 Assume 4 to be the number of men. Then 2X4 = 8 = number of women. And 8X3 = 24 = number of children. 6d. X 4 = 24d. = amount received by the men, 4d. X 8 = 32d. = " «' <« women. 2d. X 24 = 48d. = « «« «< children. Sum, = 104d., but it-should, by question, = 78d, 78 X 4 Then 104 : 78 :: 4 : = 3 = number of men. 104 3x2=:6=:uumber of women, and 6X3=18=number of child- [ren. (^) Assume £% to be the price of the harness. Then £8 X 2 = 16 = price of horse. And £84.j£l6 =£24X2 = 48 = " chaise. Sum, = £ 72, but it should by question = £60. 8 X 60 Then £72 : £60 :: £8 : = £6 13 4 = price of harness. 72 £6 13 4 X 2= 13 6 8= »' horse. £6 13 4 4- £13 6 8 = £20 X 2 •= 40 = « chaise. [HAT.AiiTH. I EiaikJXawlSJtlflOj EST, 229 number of terms (1679616)^ = 6. = 84 X 6 rr 504 ; 3144, aad so on. 864, 653184, and the men. " women. *' children, tion, =: 78d. r of men, aumber of child- [ren. of the harness. 8. le. question = £60, price of harness, " horse. ** chaise. (3) Assume 20 as G's age. Then 20 X 3 = 60 = B's age. And 60 X 2 = 120 = A's age. Sum = 200, but by question it should = 140. ►nu 20 X 140 Then 200 : 140 :: 20 : = u = C's age. 200 14 X 3 z= 42= B's age, and 42 X 2 =84= A's age. (4) Assume 100. One fourth of 100 = 25 and remainder— 100 — 25 — 75 One fifth of 75 = 15 and remainder = 75 - 15 = 60, but it ihould by the question = 72. • «^ 100 X 72 Then 60 ; 73 :: loo : = 120. 60 («) A can do the work in 7 days .-. he will do I of it in 1 day ^ " " 6 « ... « 1 « u " « g « . (I 1 It Then all working together will do f + ^ + j ~ |^ j i J'l ^ay. Therefore to do the whole work it will take them — =1^.= H8? days. (6)» m A and B working together can do it in 10 days .-. they will do 1 of it in 1 day. A can do it in 15 days .-. he will do ^, of it in 1 day. Therefore iS — i^s = 3^ = amount done by B in 1 day. Tnea if he does ,^, in 1 day, it will take him 30 days to do the wnole. *ThemodeOfWOrkll1l*t.hoannno£.M«.,»l ..._.. . ... canuot trouble anv nno? ir»r-..Tu "ZCi"" "iJ H"3"«yH " so simple mat they them by staple walysiV. " ^*" «»«etow been tUougbt advis&la to worfc 230 EET. [Nat. AaitH. Exnozfs The first pipe empties the whole of it in 1 hour. The second pipe empties i of it in 1 hour. The third pipe empties J of it in 1 hour. Then all these pipes running together will emptj 1 + J + f = V itt 1 hour. Therefore to empty the cistern it will take 1 -f »^ = ft hours. i;> (8) Assume 84 One third of 84 = 28 One sixth of 84 = 14 One seventh of 84 = 12 Sum = 54, but by question it should = 27. 84X27 Then 64 : 27 ;: 84 : = 42. 54 Ai (9) Ail »> mills working together will grind 7+5 + 4+3 + 1 = 20 bujhels in 1 hour. Therefore to grind 500 bushels it will take them 500 + 20 = 25 hours. (10)» One pipe fills ^ of the cistern in I hour, and the other empties ^ of it in 1 hour. Then j^ — ,^8 = sS = part of the cistern filled in 1 hour when both are left op .n. And if s of it is filled in 1 hour, the whole will be fiUed in 1 ' — = 36 hours. Sun Therefore Asf I of 8 ^ See note on page 227. [▲T. ABXtB. EzsBOZfwl60,l61.] SET. 281 1 + i + f : A hours. Id = 27. + 3+1 + 20 = ihe other n 1 hour fiUed in EzKBOisi 161.— Page 352. (I) AflBi«ne60 for father's age, then 15 = aon'i. g 6 6)55 JJ 11 ' 10 ~ 1 Assume 100 for father's age, then 25 = son's. 5 6 6)95 19 20 +r 20 Errors. Assumed numbers. — 1 X 100 = 100 + 1 X 60 = 60 Sum of errors = 2 Sum of products = 160 Therefore result required = 160 ^ 2 = 80 = father^ age, and i of 80 = 20 = son's age. (2) Assume 80 34 46 8 138 80 68 i of 80 =: 20 + 38 Assume 44 34 10 8 30 44 — 14 I of 44= 1] (Continnod on next page.) —25 ton KBT. (SAt, AittB (2 eonllnaedO Errors. Ai^ataed nambers. — 26 X 80 = 2000 + 38 X 44 = 1672 piimoferrors = 63 — 3572 Therefore result required = 3672 -J- 63 « 68f » (4) and 7 2 X 7= iT and 3 2X3=^6 Assume 18 One half of 18 ==T U — 6 Assume 22 On»half-of 22 =Tl + 6 Errors. Assumed numbers. --S X 22 =: 110 ±_± X 18 == 90 Sum of errors = 10 Sum of products =loo Then 200 -^ 10 « 20 = one number, and 26 — 20 «=3 other number. A. Suppose 8 22| 180 132 8)4r +T 6 36 72 i^8)36 (5) B. 6 9 12 15 18 21 24 27 132 A. B. Suppose 6 6 221 9 • — 12 136 15 81 18 -1 ■ 21 6)64 — 8 "tt" 81 10 BiffeBOin 4tl] NOTB.— F Qumbttra an lilxainple 1] It may, h( Suppose A Then 9 + fir 6+1 = 7, Suppose A Then 11 + fin 7+1 = 8, 8 — 6 as 3 » difiSirence of errorg. £if«&€fin«l«l.) Ksr. 238 0> Assame 30. iof30 = 15; Jof30 = 7i: / iof30=r6; and ^ of 30 = 5: ' 15 X TJ X 6 X 51=3375: 3376 - 69982 = - 3623-4 = error. Assume 60. iof60 = 30; Jof60 = 15; 30 X 15 X 12 X 10 =a 54000. 54000 - 6998^ = -f 47001-6 = error. 30* =: 810000, and 60* = 12960000 -- 3623-4 X 12960000 =46959264000 + 47001-6 X 810000 =38071296000 Sum = 50625 Sum = 85030560000 85030560000 -i. 50625 = 1679616 4tli root = square root of square root. V1679616 = 1296, and Vr296 = 36 = required number. lixample 11 * ^^ '°*** °' *^« 'l'^°««'^*' ««e Arith. page 363, It may, however, perhaps be clearer from the foUowing Ulustratioii : ljetx=z the number required. X X X X X* Then —x-x-x- = — =: 2 4 6 6 240 .-. X* =. 1679616 .-.« =V 1679616=: 88. (8) Suppose A had 9s. at first. Then 9+1 = 10; 10 -f 2 = 5; 5 + i - g -. nrsv* 6 + 1 = 7, but should = 9 — 1 = 8. „ Error =7 — 8 = — i SuppcseAhadlls. atflrat. Then 11 +^1=12; 12^2^656+1 = 7 = 7+1 = 8, but should = u — . i = jq. Error = ft --= 10 = 1— 2 (Continned on next page.) what fi had at what B had at 284 7-f-l 7+1 (8 continaed.) Errors. — 2 X 9 = 18 — I X 11 =» 11 Diff. = 1 diflf. = T 7 = shillings A had at first. 8;8-^2=4;4+l=6=, shillings B had at first <») Assume 24 and 6. V4-V + V = 24. 24 — 9 =.+ 15=, Assume 20 and 10. V.+ *»^ + V = 20. = 9. error. Y + i 20—16 of 10 + i^ = +6 = Errors. + 15 X 20 + 5X24 = 300 = 120 = lo. error. Diff. = 10 diff. = 180 180 + 10 = 18 = one number. 30 — I8=sl2 = other number. (10) Suppose 1st horse to be worth £20. 20 + 50 = 70 ; 70 + 2 = JC35 = value of 2nd horse. 35 -f 50 = 85, but it should equal 60, i. e. (20 X 3). Then 60 — 85 = — 25 = error. ' v ' . Suppose 1st horse to be worth £60. £60 + £50 =£110; £110 + 2 = £55 = worth of 2nd horse. 55 + 50 = 105, but it should equal 180, i.e. (60 X 3). 180 — 105 = + 75 = error. Errors. + 75 X — 25 X 20 60 1500 1500 V 3000 + 100 £30 + £50 Sam= 100 Sum= 3000 ' £30 = value of 1st horse. £80 ; £80 + 2 =3 £40 = value of 2nd horse. ExXftOIB Suppose 11 X 4 11 X 6 = Suppo 12 X 4 = 12 X 6 = E] Diff. = Here P = Then ^ = = 2-1 Henc Here n = Then t i 130'6J Here ji = Then t ] ▲nABITB BizasotsKs 161, 162.] KEY. 235 d ttt first 10. :20. ^^ = 15. = error. -- 72 , 72 — 12 «= 60 = pence he had. 60 — 54 = -i- 6 « error. Errors. + 6 X 11 =6C + 4 X 12 = 48 Diff. := 2 diff. = 18, and 18 ^ 2 = 9 =, number of beggars. ExKRoiSH 162.— Page 357.. (1) Here P = $713-29, r = -045, and t = 14 Thca^==P(l + ,y,orlog..4=Iog.>+iog.(i + ,)x/ = 2-8532G7 + (.019116 X 14) - 3-120891 i ^ofL Hence amount = $1320-96. d horse. horse. (2) Here n = 7, r = -015. -,, __ loff- «• -845098 - = _ 130.698 payments, and log. (I + r) '006466 130-698 -i- 4 = 32-674 years = 32 years 8 months 2 days. (3) Here J ^ $111M1, p = m-n^ ^nd r =» -08. Then t J^flfjI^^L^ _ 3^045757 - 2045753 1-000004 ~~ ' " " 111. — log. (1 + r) -033424 -033424 = 29-918 years = 29 years 11 months. / 236 (4) (Kay. AftiTit. Here J = $3333-33, P = $222-22, and i « 120. I "^ lo«f- -* — log. P Then r =«| l ; or log. (1 + r) = V P I 3-522878 — 2-346'783 1-176095 120 120 10228, r = -0228, and rate per cent. « 2/5, (5) = '009800 7« Hence 1 -f r Here n = 2 and r =z -07. log. n. 0-301030 Then* = -= log. (I -f r) 0029384 2 months 28 days. 10-2446 years =« 10 yra. («) Here v4 = $100, r =: -0225, and ^ « 28. Then P = , or log P = log. w4 — log. (1 + r) X ^ (l + r)« Log. P = 2 — (0009664 X 28) = 2 — 0-270592 s= 1-729408. Hence P «= $53'63. (T) Here P = $2468- 13, r = -0375, and « = 26. Then .i = P (1 + r)*, or log. ,4 = log. P + log. (1 -f r) X *• Log. J = 3-392368 -f (00 15988 X 26) = 3-392368 + 0-415688 =: 3-808056. Hence ji = $6427-705. (8) Here A = $7137-40, r =a -0425, and t = 22. A Then P — , or log. P = log. -fl— log. (1 f r) X ^ ♦ „ (1 + ^)* Log. P = 3-853540 — (0018076 X 22) = 3-853540 — 0*397672 = 3-455868. Hence P= $2856-723. (Nat. AftiTtt. .P nee 1 + r JSzxBoiSBS 162, 168.J KBT. (16) 287 Here n = Id, and r = '0525, log. n 1 -278764 Then * = . = , , _ 57.5445 payminta = log. (1 + r) 0022223 28-7722 yeara =5 28 f^m 9 W0«tlj8 a days. I =i 10 yra. f r) X /. : 1-729408. 1 + r) X ^. + 0-415688 — 0*397672 ExiBOjBB l63.^Page 360, (1) • Here r =5 '03, « =s 600, j1 = 8366. Formula IV. / = V? — +(2~r)8S-.(2-.r) 2r 8 X -03 X 8366 600 f (2-.03)^-(2-. •03) 2 X '03 1 2007-6 i 500 + 3-8809 > — 1-97. •06 V(4-0152 + 3-8809) — 1-97 -06 2-81 — 1-97 -84 V7-8J61 — 1*97 •06 84 •06 = — - = '=14 paymenta= 7 years. •06 6 (2) Here a = 112*50, r = -015, t =: 44. Formula I. jiz= at ll (*-l) = 112^50 X 44/1 + (■ (44—1) X -015 ■7 •015V I = 4950 X 1-3225 = $6646-375. 288 KEY. [Nat. AftiTB. Here a = 300, A = 1680, and / = 6. Formula III. ^^ ^(-^"^ O, 2 (1680 ^ (300 x j)| at (t—l) 300 X 6 (5 — 1) _ 2 (1680 - 1500) __ 2 X 180 360 300 X 5 X 4 6000 ""eooo ~" .V Rate per cent = -06 x 100 = 6. (4) Here J = 2080, r = '04, and / = 16. Formula II. a = 2 X 2080 /{2 -f. (f— 1) r] 16 {2 + (16 -. 1) .041 4160 4160 - ^ 4160 41600 16 X {2 + (16 X -04)} 16 X 2-6 416 "" 416" - $100 = 1 payment or rent for half a year, hence yearly rent = $100 x 2 = $200, 90 •04 X\ EzBROisx 164 Page 368. (0 i Here r = -04, and v = $3000. Then a = vrz:^ 3000 x '04 = $120» Here a == 663, and v = 11260 a 563 Thonr = -. = «^o = -05, and hence rate 11260 per cent. =: 6. Here r* [Nat. AsiTQ. I ExiBoioiB 163, 164.] Then v > KBT. (3) 06, and $ zs 14. 15 m r (I + r)« 05 X (106)'* log. f = log. 75 — f(log. 1-05 X 14) + loir. -05) « 1-876061 - (0- 021189 X_14 + log. -05) ^ -= 1-875061 — (0-296646 + 2-698970.) = 2-879445. ^ .-. V =. nat. number corresponding to the logarithm 2-879446, which is $757-608. (4) Herea = $90, r =, -04, / =: 12, , == 7, and .:»+t= 19. Formula VIII. » = J!. 5 ««, __ 2. I 1 1 i __90r 1 •04 } (1 -04 9000 04)1^ (l-04>'» ) 4 > i-eoiOl 2-10fi8!«( _ 2350 X (-624605 -474649) L 2260 X '149966 ^ =$337-401. (8) Here a =,1600, and r =: -06. „ , ,_ « 1500 160000 Formula IX. »=-.= :=. ^$30000 r -06 5 — 20 X 1600 or 20 years' purchase. («) Here a = 22, » = 308-64166, and r =» -04. „^ _ ^og. n — log. (a — rr> Then Formula VII.* ^ ' , log. (1 + r) ^log. 22 — log. (22 — 308-6416 X -04) log. (1 04) „ 1-342423 — log. (9-65425) 1-342423 — 0-984707: " ■■■ 0-017033 0-357716 357716 0017033 17033 0017033 = 2i + \ , ^0 «ET. Here a as 154, t mz 19, and r =: '05. Formula Y. v es [Mat. JifauB, r I XI + ry ) 1 ) 15400 C 1 ) > = X ^1 > •06) i» > 6 t 2-5269 ^ 154 ( 1 ) 15400 r= — xn V = — X ^1 — •05 ( (106) i» > 6 ( 2- , = 3080 X (1 39674) = 3080 X '60426 «= $1861 -12 4 (8) Here j1 = 600, t r= 40, and r =» -0376. ^r 600 X -0375 Formula II. a » 22-6 (l + r).-l 22-5 (10375)*0— J 2260000 4-36034 —1 ;;= Je6-6957 3-36034 336034 £6 138. 10id+. (9) Here a = 8, A » 187*315625, and r » '03. log. (Jr -{-a) — log. a Formula IV. t =»-^ log. (1 + r) log. (187-315625 X • 03 + 8) — log 8 log. 103 log (5-61946875 + 8) — • log 8 log 1-03 log. 13-61946875 — log. 8 1 134160 — '903090 log 1 -03 0-231070 -012837 231070 .18. 0-01283^ 12837 (10) Here a = 74, r = -04, and t = 30 Formula I. A = a { (1 + r) *- 1) } 74 X { (l-04)3o -. i } r -04 74 7400 = X (3*24332 — 1) := X 2-24332 = $4150- 142 •04 4 By Table, page31|. Amount of $1 for 30 years, at 4 per cent. jerj.. =:SRR-n8494 \Then $5&-08494 X 74 «$4150-28. k / \ fiziBOl $75«0 : D is to : fon $6139-81 an( A is to ] $4604'85 $1 $1 A and therefore A by hi he can do hour. Til riy is con Then 192 • £179 14 $718-94582 $1497) Kat. April. 2-5269 > >1861 -12 4 [)0 X -03^5 0376)*O-.j t090 4)">— 1 } 04 : $4160-142 4 per cent. Bx»0Ma. 184,164.] ^ nir_ 2^^ BuBoiBi 163— Page 867. BXAMINATION PEOBLBlia riBST SB5II8. (2) D « to have one third as much as A, B, and together there Aist„ha,.,m.„.^.^,:,jrr^'r'r°; a, $292-91 $4604-85 ~ $m.9l = $4311-94 = three times 0. share, $4311.94 -^ 3 = $1437-311 = C's share $1628.2U + $111-11 = $163932^ = A's share. (3) thi^^^ ''°'?''* *°^'**'''* '*" ^° *be work in 96 hours therefore in one hour they will do ^V of it ' hein'd'^X'of U 'T^T^ !!! '"" ^ r ^^^^ '"^ ' ^-^ hour. Therefore e ^II reotrajta^rh ' ^ "' '° " ^"^^ Ti. is contained ti.es In t^; oT^TiIT = iT. '' " Then 192 ^14 :^ 13^ days. *• «' ^ "J" xk = 192 hours. (4) *.fo'!' '*'• ®''^- == ^^^^'^^^ =^ 1718-94583. . ^Ao?o " ''''''''' ^ $71894683333.3 -^ 48 = $1497803819'4444. . » . « as Y7 30 36 (5) ^J|..18..30..tt..6l5..27 R <>««» -•• 10 433 2 14 14 X 8 10 3465 9 16 16 4332 3 6 20 X 7 ss 30326 11 20 33943 4 8 14^ (9) Here a = 1, and r^^. Then S =s Si 1 — r 1 — 1 (10) f 1 — = 2. i i of I of 192 8 8 =1 64 *5— — 2 « 64-f — s 64 -^ - f ^5 2 -2- ! i^ 64 X -il' S± 1291-. BXKBO DLOTereni A is to h) $897-43 - equal 7 lbs. wb 6 « ryi 13 « oai 27 " bucl 24 " barl 11 " peas X " pota 9 AT. ASXTH. 48 years nonary » grg. 2» 1$ 20 64 4- — BxBBcisi 165.] KEY. (11) Logarithm ofl29I40163 = 8- 11 loei 8-111061 - 17 =.477121 =logarithm of 3 (12) 24d Suppose 48 18 66 84 — 18 Suppose 36 , 18 54 63 — 9 Errors. Assumed numbers. ^18 X 36 = 648 -• 9 X 48 = 432 . Difference of errors -^ 9 9)216 = sum of products, 24 SBCOID ^^klEB. (13) A is to have $69-18 + sis^^o'l^^^^! Js' "T ^^.f <^-„ ''' «at^i,:^CBS1'= '^'-- *° ^e divided |221.|u"+^$1stl^S;o\^=B^^^^ $291-07 + $93-40 = $384-47 = A's (14) Tibs. T;rheat = 9 lbs. rye ^ C " rye = 8 « oats J^ " °? =21 "buckwheat ^7 buckwheat= 20 " barley i 24 " barley ±= 26 '« peas '^ 11 "peas =35 "potatoes X " potatoes = 16 « t^heat (I 8 3 J Jins. 4 X 2 X 35 X 16 4480 l? = ^i ^^ = 5Q 55^ = !?t ?«iX «^X9XllX5iqxl£ = C's stock. 6371 (24) i: + i + iV 4- -iV = -iV(r = l^ = part of the cistern filled in one hour when the four pipes are left open. i + i- + i 4- i = -iii^ = 4rt = part of the cistern emptied in one hour when the four are left open. ^0 "— H = ii = part of the cistern which remains filled after the eight pipes have been left open for one hour. And if ■iS- of the cistern are emptied in oae hourj it will take 1 -^ lu = ^'^$ hQ\u:9 to empty the whole of it,' * IlfAT. AbITB SixxBozas 166.] KIY. 247 : 4=100000 $170000 I -of the pro* 3=$3492-06 $6746*03 1 share. IS ; $225 3 = $1871 : $125 = $537^ =A's stock. = B'3 stock. : G's stock. irn filled in emptied in filled after ir. And if I tillfA 1 JL TEUO SSBIB8. (26) As often as the first receives 4 the second receives 3, there- fore as often as the first receives 6 the geoond receives 4i Then6+4i + 7= I7J. loaves. 2310 X 6 1 7J : 6 : : 2310 ; =s792 loavesi^number the first receires, m 2310X4J 17J:4J;;2310: =594 '< >« 17J 2310 X 7 X7i:7 ::2310: .=924 « r^ u u m second ^< third (( (27) To produce a mixture worth 8 cents a pound, we require 4 lbs. fd) 12 cents, 4 z® 4 cents, 1 /Q 5 cents, and 3 /® d cents, or 3 lbs /® 13 cents, 1 fQ 4 cents, 4 ^ 5 cents, and 4 /© 9 cents. Ibs.lb3.lbs. Ibs.lbs.lha. Then 4: 72:: 4: 72 lbs./® 4 cts. or 3:72::l:24 lbs. /® 4 cts. 4:72:: l:i8 lbs. ta> 5 cts. 3:72::4;96 lbs. z® 5 cts, 4:72::3:54 lbs. /® a cts. 3:72:;4:96 lbs. /® 9 gts. (28) Here d s=$4444'44, r « -0444, and t =4'3| •tf $4444-44 $4444*44 Then P = ■ .j, . *oNor >,^. , I , = ■ ss $3725*764 l+rt l+C0444X4-3^) 1.19289. ^'^^^'^ '*>*• (29) $1*00 - $0-0225 = $0*9775. $33470 ^ 0-9775 « $24010*23. (30) Here ^ = $7493-47, r = -07, and t « 8. ThenP= -1 ^ ^493* 47 ^7493*47 li-it l+(-07x8)"' 1-56 = $4803*5064 us JLEY, t^AT. A&ITH, (31) $17460 -ir 1-03125 =« $16930.909 =- sum to be invested. 46930-9.09 -i. 2-95 = 5739-29 yda. cloth. 16930-909 X -021 = $423-27272 = ad valorem duty. $17460 -I- $1347-90 + $47940 -f $169-83 + $42327272 =- $19880-40272 = whole cost; $25000 — $19880-40272 = $5119-59728 = whole gain. „, ^ 5119-59728 X 100 Then $19880-40272 : $100 : : $51 19-59728 : . = 25-76 = 25i per cent. 19880-40272 V. 134234 6 8 6 44 6 222 6 1113 6 6569 den. (32) III. 21122021 3 1 3 22 8 68 3 206 3 618 3 VIII. = 12701 8 10 8 8? 8 696 8 6669 den. XII. 3281 12 38 12 464 12 5669 den. 1656 3 n 5569 den. (33) ? of 4J of — of I'g of I of £43 18s. Hid. £43 18s. lljd, == $175-79^. (Ccuuuucu vu xi&xi page.) Nat. Abjth, ted. lin. X 100 0272 XII. 3281 12 38 12 64 12 69 den. EZXBOIBX 165.] KIT. \ 249 I. iijd. = '^ (33 continued.) ? of I of ~ of -iV of I of $115191 == - of- of- of - of 1 fib ^ 2 1 1^ ji of f 175-t9^ = J of $175-79^ =. $263-6875. 1 3|of of -56 of 1-75 of 6 J times $97-18 = 17i I ^^<>^~ of if?a of 11^ of 61 times $97-18; 6| times $9718 = $631-67. 7 n 1 ?5 % m n^ 49 — of— of — of — of $631-67 « of $631-67 9 ?^ m im 9X25 25 = i!*A of $631-67 l!*,^ of $631-67 = $137-5636. Then $263.6875 — $137-5636 = $126-1239 = difference. (34) iV = 1_ -r 13 .-. log. -^ = log. I — log. 13 = — 1-113943 = 2-886057. 19-5 = 3 X 13 X 6 -f. 10 .-. log. 19-5 = log. 3 + log. 13 + log. 5 — log. 10. log. 3 = 0-477121 log. 5 = log. 10 — log. 2 ^1—0-301030 .-. log! 5 = 0.698970 Sum = 2-290034 Prom which take log. 10 = 1 Rem. = 1-290034 =log. 19-5. 1125 = 53 X 3« .-. log. 1125 = (log. 5) X 3 + (log. 3) X 2. log. 5 = 0-698970 X 3 = 2096910 log. 3 = 0-477121 X 2 = 0954242 Sum = 3-051152 =^ log. of 1125, (Continued on next page. ) 250 Km, IIXJlT. Abits. (34 coatinned.) 28-16 = 281 « 4* « 13» ^ 6 .-. log. 28-16 =■ (log. 13) X 2 — (log. 2 + log. 3.) log. 13 » 1-113943 X 2 s» 2'227886 (log. 2 -f log. 3) =» (0-301030 4- 0-477121) =^ 0-778151 Diff. «. 1-449735 ?= k)g. 28-16, 65000 = 13 X 6 X 1000 .-. log. 65000 « log. 13 + log, 6 + log. 1000. log. 13 « 1-113943 log. 6 B= 0-698970 log. 1000 B= 3 Sum = 4812913 » log. of 65000. log. -0005 => log. 5 with characteristic changed to — 4 »= 4-698970. 152-1 = 3« X 13« ^ 10 .-. log. 152-1 ». (log. 3) X 2 + (log. 13) X 2 — log. 10. log. 3 =0-477121 X 2 «0-954242 log. 13 == 1-113943 X 2 SB 2227886 Sum -= 3182128 From r'licL take log. 10 = 1 Diflf. =1 2-182128 ^ Ipg. 152-1 8-112 = 2* X 132 X 34- 1000 .-. log. 8112 » (log. 2) X 4 + (log. 13) X 2 + log. 3 — log. 1.000. log. 2 =»=0-30 1030 X 4= 1-204120 log. 13 — 1-113943 X 2 = 2-227886 log. 3 « 0-477121 Sum = 3-909127 From whiQli take log. 1000 = 3 Diff. s? 0-909127 = log. 8-112. Fax. Abits. . 13) X 2 27886 78151 j 49735 i + log, — 4 3) X 2 g. 152-1 g. 2) X 4 8-112. SZXBOIBB 166.] KBT, 251 (W) t*x 300 B> 21000 tX8 X 3flr r=> 1800 8» SB 64 t 8« X 300 ^ 8 X * X 30 tBt*X 300 ^ 8 ^ X 2 X 30 28 22854 = 2454000 » 22800 » 84 2476884 249^61000 64500 4 2499e6504 XII. 87l«e/-72 {t8*tt 6e4 119tet 159768 2035272a le/372£4 617428000. 4977^^/08 3e8391«4 / (86) 6"h*ir4"7+5 year8=-U- of lifetime+5 year3=aage at birth of son, H — (i 8+5)=" J 8 of his life time — 5 years » time he lived after birth of son. II of father's life time — 5 years — 4 years =» age of sou = j father's age. ] J of father's life time — 9 years =. i father's age. /. 9 years is the diflfereace between U and J* of father's age. .-. 9 years is equal to r/g of father's age. If 9 years is -^ of his age, ^V will be the J of 9 which is 3 years. If jV is 3 years, If or the whole age wUI be 3X28 =« 84 years. Or by Position. Assume 42 for father's age at death, the son's age =-21. 6 + iV + 1 + 5 =1 Jl 4- 5 ; ^ir of 42 = 16i and I6i + 5 s. 21i = age of father when son was born. ,-. heiived after birth of his son 42 21| = y (Continued on next page. ) 252 pVAT. Asm, (36 coathined.) 26 years. By the question he lived 21 f 4 = The error 25 — 20i =» — 4j. Assume 98 for father's age, then son's age «. } of 98 -49 • + Vi- + 1 + 5 =. • 1 + 5; ii of 98 :=. 38J,.aad 38i + 6 1-431 = age of father at birt^ of son. .-. he lived after birth of his son 98 — 43J =. 64| years. But by the question he lived. 49 + 4 years. « 63 years ' Then 63 -^ 54i = + i j =, error. Errors. — 4i X 98 =- 441 • + li X 42 =« 63 Sum = 6 604 604 -5- 6 =. 84 =- fathers age. - 6»3-f-« nu 63 8 607 40 fair S (37) per. yds. ft. in. 7 3 a. 7 fur. 40 303 6i per. 23 yds, 34 20287 6i 1618} 161} 101438 10143} I670i 3 111581} 3 6010} 12 60129 334U6} 12 401696$ . (pdotmued on Bd3(t jMigo,) UXat. AMm. In«oi«il0i.] KBT. (37 cotttlnaed.) 60129)4016965(66*80578 tImM 360774 -49. + 6 =-43 J ars. rg. 409225 360774 484510 481032 rdsk 347800 > 300645 i 471550 ' 430903 6U6470 481032 6*3 ^ •000000274 274)6300000000(22992700'7299270ai' 648 258 820 648 % 2720 2466 2540 2466 ''jM 740 648 '2466 > 1920 1918 2540^ 2466 - HS« 2000 1918 740 548! 1 1920 1 1918 1 200 remalndeiv ^ 82Qi §48 1 ■1 I • 254 XET. (39) IKat. Aaitn. I yds. : e^yda. :: $|p : f x |? >e I? ^^55^ $6,482. (40) /= PH. = $423r'ri X 'OGB X i.er-= $460-0034205. (41) ^^—P $100a — $m'30 325'TO i'r $674-30 x -085 57-3155 6 years 8 months 6-7288 days. (42) By Table, pap:e 260, th(» amount of $1 for 14 paymeutg nt 4 per cent, ia $1-73168. ^ /wwnw «t 4 Then $1-73168 x 8l3-7l=$l409.0853328=Amount. Subtract 813-71 Difference = 595-3 753328=5lntereat. (43) $300 X = Too X 4 =- 2800 T60 X 7 =r 5260 850 X 9 = 7660 400 X 13 =- 5200 1300 X 10 » 24700 4300 ) 46600 (10 montha 19-^ days. 4300 2600 30 78000 =2 day» 4500 35000 84400 {m ^xeaoiuiMl65.] KEt. 255 (44) 23 per cent, of $4200 =a -,ajJ^ of 4200 =. $906-00, and $4200 -- $956-00 = $3234-00. E has half as much as A, B, C, and D togother ; therefor© E has one-third of $3234-00, which is $1078-00. Deducting E's share, $1078, from $3234, the whole sum to be dmded, there remains $2156 to be divided among A, B, C and D. Now D ^ets a certain amount; C gets $42-ll' more than D ; B gets $61-S4 (42-11 + 19-23) more than D ; and A gets $78-44 (61-34 + 17-10) more than D. Together thej get, then, four times D's share, to.:;ether with $42-11 + $31-34 ■+• $78-44, or, in ether words, four times D's share, together with $181-89. That is, four times D's share, together with $181-89 is equal to $2156. Hence $2156-00-$l81.89=»$1974-ll =.four times D's share. Then $1974-11 ^ 4 -$493-5275 =D's share. Add 42 a Sum $533-6375 ^C'aahara, Add 19*23 Sum $554-8675 Add 17-10 ' B's share. P& Sum $571-9675 =* A'fl share* $3786-80 (45) 3786-80 378680 !+'•* 1+1- 76» 2-76 (46) 276 ■ i=: $1372-02898 + .|C3^^27j) X ^e^-^ of -i42857}-^8| times (§+]+U {(•'TSX -12345 ^??§) + p + 93 +17,V} -f- 27-4922077 J(3?g~2fg) X ^g--?-f of ? }^V X m-h\'a+ H-^\Vo) { \»M ^ 89000 -TT-f T 7 "P 9« +• i'l'f j (Continued on next page.) ~- 2r49Z2077 i ! I • rV— 10|t) — (-4 4- g 4- -9— })}-r- (-8378 ^ f of 31) •632263*2 X i of 9i -i- (^ of 4^ of ,V of SSj^ToA) fEES X j X \^'r (} X V X 1^ X HF X t4t) H X if^f8 X V- f8^ X i X ^4^ X ! X ■A-'f r X 2i! ,t X 'V fU X sUt X -L^ VoV X V X ^ X ^V X ^Ut X ^ 14 X 37 5 kxtx Y 14 X 37 6 6 X 3T 4 = «f=-2A. (64) Bach child gets I child's share, .-. 17 children get 17 ahareg. Bach woman gets three times a child's share, .-. 4 women get 12 shares. Bach man gets six times a child's share, .•. 3 men get iS shares. And together they get 47 times a child's share. Therefore $7200 -i- 47 — $153-19-^V = a child's share. $153-19-4VX 3 =$459-5714. $153-194V X 6 = $919-14if. (65) 26400 = 2» X 5« X 127. Addioj? nnity to each index mi multiplying the resultSi we get 4 x 3 X 2 = 24. iT«.Ammt. £zxsoisal6B. XB7. (66) 261 9* f of 4i of--, of J of £3 168. Hid. = JL X -- 1 k n to X ;$15-3y^ =r 6 times $16-39^ = h of 42 of 19i $92-35. i of 120 = 40- 6 = 35 gal. = water i 264 KEY, plAT. Aain 132 IBTSNTH SaRIU. (13) • • • • . 401241 •34;d4 (422*aa 8] ) 412 314 1342 ) 4341 3234 13443 ) ^0234 101434 140012) 330024 330024 (74) Suppose father's age = 60, the son's age now = 60 -^ 5 = 12. and son's age fo.r years ago = 12 - 4 = 8. But the son's ag^four years ago should, by the question, have been 60 ~ * — 8f. Therefore 8 — 8f = — f = error. Suppose father's age = 35 ; then soT,'a age now = ' ^ 6 = T, and age four years ago . .; 1 4 = 3 But son's a<(e four years ago should, by question, have been 35 -7-1 = 5. Therefore 3 — 6 = ^2 = error. Errors. — 2 X 60 = 120 — 4 X 35 = 20 \ diff. 1? dlff. ="lOO 100 4- Jf s; to 5= father's and aon'a act* — Tn jl « _ i ^ EZXB0I8B 166.] KEY. (T5) 265 J J 99900 * 9900= ; Wl5 nm__ 195025 I mm ^IT"^ Is^ = 223 82460mS , (76) ^°^^9880fi°cf '^?,^^^*-3^2 is 7.988089 ^---«-----.X-ro%72947e4.37a. (77) Assume 43J for the greater number n:3i:: 43* : — -J= 21 the less ; 43J ~ 21 = 22i but i? should = 30 , Therefore error = 22J — 30 = — 71 Assume 72i for the greater number •* • ^ = 35 = the less ^ 2i — 35 = 3- but it should = 30 -therefore err =^ 3^ , 30 ^ _,_ ^^ Errors. + n X 43i == 326i — 7J X •72i ^ 5 131 iSum=i5 Sum =='770 ' 870^ 15 « 68 greater ^. 58 X 3i Assi 35 Assume 16 Assume 9 '. «. n. n OB) ?!^. 16, 18, 2§ . 62 SB. Mfi 1 ^ 31 3i - X 16 X 9 X 81 156J240. MQ KBT. (79) Here a ■■ 1, d ^ e, n (Wax. Asnm. 101, f- I 2a+(,i-i)d|L {2 X 1 + (101-1)X6 ? 1!1 2 ^ '2 101 602 X 101 i« (2 + 600) ^ 30401. 2 a (80) s 4 6 1^ U, ?5 11? ij ^i 1T?X4XB 228f ^ ^l« ni !J5( ^;j 3 7X7X11X3 1617 ^^*' 7 11 Y (82) (jffH- 4\h + 3y^l6n) y .'54} -f. If) X 35 times >U2857 i -97 X -24378 x (1^,- X 4,Vr } X(4.^-2,V) {{onn-ietmx^} -;-j,i) x35xf t " II X UlU X H X ^4^^ X (4i»«-V - 2:^) iU X A- X ^V X 3/1 X I I i<^ X-iWaXU XVA^X ?ff (83) Suppose the hour hand moves over 4 minutes, then since the minute band moves 12 limes as fast, it will have travelled over 48 minutes. But in order to overtake the hour hand, the minute hand must traverse the entire circle, 60 minutes, plus the 4 mi- nutes we hnve supposed the hour hand to have moved forward t .. 64 minutes. Then 48 should equal 64, for we should find the same number by each process ; 48 — 64 = — I6 error Suppose hour hand moves over 6 minutes, the minate hand moves over 6 X 12 = 72 minutes. But minute hand moves over 60+ 6 = 66 minutes. Theft 72 — 66 = + 6 error. (Continued on novt mm t Ktjiboji 120 -f. 2 bem =»6 Log. .'. Log. 31 .-. Log. .'. Log. • Log. -y iB 1- Log. !• 93^ •'. Log. 9; X Simple Int< Amount Cc = $70( Int. ,267 '4 ltT.iiioi.M M6.J KEY, i(83contmaed.) ' ErroM. — 16 X 6 = 96 + 6 X 4 = 24 Sum 22 Sum 120 t!!r *'' "'*'• ^ °''°"*'' P*^^^'' «^«' t'y the hour hand hence space passed over by the minute hand == 6^ X Vi - 65/r mm. == i hour 6^^,- mxt, = time. .(84) 3850000= 6 X 7 X 11 X 10000. .'. Log. 3850000 = loir 6 4- Incr "rj.'^^ n ■ i = 0.698970 A^t5i^e|-|tr4?3ji^ 3181-81 = 31-81 X 100 = 31-^, x 100 = 3^,a ^ loo. .-. Log- 3181.81 = log. 5 + log. 7 + log. 1000- loir 11 . Lncr .nnS2?2J^^ = 2 X 7 X 11 -^ 10000000 . . Log. 0000154 = log. 2 + log. 7 + log. U - log. loooOOOO « 0-301030 + 0.845098 + 1-041393 -7=1 187521 '-'• $ ro^i9lrJl- 1-^86^.11 ^:>2=iW^--- * 1-571428=: If OS y. 9317 = 93U ^ 100 = n»xt-i- 100. X 3 + 0-845098 — 2 = 1-969277. - I Eighth Sbbiis. (85) Simple Interest = Prt = $700 x -045 X 3 = $94-50 Amount Compound Interest = P (, 4. r)« ^^ $700 x ? 045)» = $700 X 1-H116 = $798-814-1700 = $98-8li=Comp $98,814 -=$94-50 =: $4'5i4. KEY. (Nat. Abitb, (86) X's gain = tV, »nd Z'b = J j .-. Y's gain = 1 — (^V + |) X's gain is •^\- for 3 months, therefore for 1 month it is ^. Y's gain is -J^i for 9 months, " " " y^j. Z's gain is | for 4 months, " " " i. ^ : 3>^ :: $3024 : $3024 X -fff X f = $6'72 =i. X's stock, i : igg ;; $3024 : $3024 x i^s X f = $1120 = Y's stock. (87) f X V15 -H (iJ)» = I X V^-^.(i)» = I X f X A=iV. (88) 4« = 15 X 300 4 X 3 = 12 X 30 3« = 4800 360 9 5169 43« = 1849 X 300 = 564700 43 X 2 = 86 X 30 = 2580 2« = 4 557284 432* = 186624 X 300 = 55987200 432 X 1 =432 X 30 = 12960 1« = 1* 80677568161 (4321 cube rt 64 16677 15507 1170568 1114668 56000161 66000161 66000161] (89) J+6) 4 lbs. at 8d. ^ 1 lb. at 4d. > Make a mixture of 6 lbs. At 7d, 1 lb. at 6d. ) 112 X 4 « : 112 :: 4 : = 74| at 8d. 6 (Contianed on next jwgtb ) Assume 4< Since 1st - And 1st -\ ^3rd And 2nd - + 3rt And 3rd •{ + 2nd Assume 48 Since 1st -\ And 1st + 4- 3rd) And 2nd 4 + 3rd) And 3rd -f- + 2nd; — 2 — 1 Diff.=T 957^. (89 continued.) A 112 X 1 tt:112::i: =i8iat4(I. 6 112 X 1 6;m::i: =l8|at(id.' 6 (90) Assume 40 as the sum of the three numbers Since l8t + 2nd + 3r(i = 40 "^™''®"- ^""^i' 3rt) L ^""^ "^ ^'^^ == ^* ••• ^ <2nd ^"'ilrt) t^^^*' +* ^^ - 3*4 •;.: •fVisV*- '"' + '''^ = '' ^~ « -.181+ 2nd = 8 Adding, 2 X (1st + 2nd + 3rd) = 29 o . ,. ••• Ist + 2nd + 3rd = 141 But the sum should equal 40. Hence 14J — 40 = — 251. Assume 48 as the sum of the three numbers Since Ist + 2nd 4- 3rd = 48 "°°»oe"- ^""^i'srV- U "^ ^'^^ == ^^'' * ^2"^ And 2nd + Mlsr+Vrii'^'VrVHl'sV**'"^"*"''^"'^ + 3rd) = 14...T........ ••MJst And 3rd + i (ist + 2nd) = 34 V * 'i V'lVt ' "^ == " + 2nd) =14 ^« . . » (ist ^ "^-A* •. Ist4-2nd = 18| Adding, 2 X (1st + 2nd + 3rd)=67| .-. 1st + 2nd + 3rd = 33^ But the ►nm should equal 48. Vr^nT ^^^ - 48 = - 14^ = error, errors. — 25i X 48 = 1224 — 14^ X 40 = 566J Diff. = lij Diff: = ^ «57i -f ui = 58 =: the sum of the three namberg. (Continued on nextpagt.) m kEf. [Kat. Axttn, (90 continned.) I8t 4. i t^nd 4. 3rd) =: 34 .-. i (2nd + 3rd) =iS 68 — 34 = 24 .-. 2nd 4- 3rd = 48. 2nd + J (1st 4- 3rd) = 34 .-. | (1st + 3rd) = 68 ^ 34 = 24 .-. 1st + 3rd =r 36. \ll t 0°^ t o 'f = ^^' ^^'^ 2nd + 3rd = 48 .-. Ist = 10. 1st + 2nd +-3rd = 58, and 1st + 3rd = 36 .♦. 2nd = 22. ^na 4- 3rd = 48, and 2nd = 22 .'. 3rd = 26. (91) 4 means -f 2 extremes = 6 terms. / ~- a 4Q _ I OA Formula IX, p. 333. d = - = - — =3— — 74 n— 1 6 — I 6 1,8J, 16?,243.32J,40. (92) » = 1860040, i = 1240029, and r = 3. Formula XI, p. 340. o = r/ — (r — 1) , = 1240029 v % -(ax 1860040) s= 3720087 ~ 3720080 =J7. ^ (93) ^ *^ n petce.^'*" "'"'* ^^ ^"'"''^ •*• ^ ''PP*««+ 3Jpfear3C08t ^^ ^sf pente^ ^'*" ''°'* ^^ ^®°''^ •** * *PP^®* + *^ P**'« «o^' Subtract and 2i — ig pears cost lid. — 8jd. That IS, j^ of a pear cos^s 2id. If iV cost ,id, +1 will cost i„'*d.= 3d. * 6 apples + 7 pears cost 33 pence, and 7 pears cost 21d • fl apples cost 12d. and 1 apple costs 2d. ' ' (94) V ixixfx -- XfX^Xi t 19 ^ Bf i$t ^ 1( B 19 6=1 x-x-x-x-.x-xl = — -i-. * 3 ^J4 5? «i ^ 2X4X3X2 3 . — 19 — 48 A.T. Aann, • 34: = 24 34 = = 24 8t=: 10. ndc 22. !6. 029 X 3 iara cost sars cost !ld. .-. 6 -=i! fiziBcn^ii 166.1 (06) m ^''=illJo""^-"^2« - « of 2nd r«.,. = 130 ...2nd *'° = m 50.' ""• - ^^^ ••• ^ o^ ^«' -m. .= $70 ... 1st rem. $87-60 =J of original 8um-$50 • 1 of nr?»}«-i .-. originai sum = $137.60 x* 2 = $m. ^""* =^^37-50 (96) « = 60,»-i7,andrf=.4. Formula VI, p. 333. « = [ 2 a + (n-. i)^ M. = 13X60+ (n-.l)x4^1?:=(I20+64)xi! _ 184 X 17 2 ; ■ = ^1564 = sum received for 17 years. Formula I, p. 333. / — « u. /„ ,x ^ - ou ^- b4 « $1^4 = ^^ggg ^^^ j^^^ yearT Ni.vTn Skbws, (98 $3649-3932. (99) 2)177408 2)88704 ■■ nil iri 2)44352 2)138e 2)22176 3)693 2)11088 3)231 2)5544 7)77 2)2772 JX »• X 3« X r X n. 2T2 EST. fr->. UTat. Asixa^ (100) Formula HI, page 354, r rr-jp- 1 .-. r + 1 sjp- Log. (r + 1) = (log. ^ ^ log. P) ^ < That is, log. (r + 1) = (log uumi _ jog. 704) ^^ 11 = (4.046757 — 2-847573) -^ u' -,, , =1-198181 -Ml =.0108925 Thci-efor^r + 1 == natural number corresponding to the loga- rithm 0-108926 which is 1-285 Since r+ l = 1-285, r = .285 = rate per unit and rate per cent. = -286 x 100 = 28^. (101) ^ ^ ^8 tV> ii or the whole will equal » x 13 = UT. (102) 3 gal. + 4 gal. + 7 gaL =: 14 gal. 292 V 3 Hence 14 gal. ; 292 gal. :: 3 gal. : —^ = 62f of Ist kind. 14 292 ^ A 14 gaL : 292 gaL - 4 gaL : ^=83? gal. of-2d " 14 292 X 7 14 gal. ; 292 gaL :; 7 gal. : ^ 14a ^aL of 3d^' 14 (103) Then £m : jcsoo :: £\ : irsoo x i x f ? = S= i;i94 16s. l|?d. ^m : £509 :: jCj : £500 x i x f^ = t=jei29 178. 4fH. £m : JE500 :: £i ; £500 x i x ?? = = £97 88. Offd. £lll : £500 ;; £^ : £500 X ^ X f ? =: =;£7T m, f^m, 01 % Tab Hence | % Fori 10( •06 100 6 Since ea( Also, 5 m Hence the hours ia e( £15000 I Thtsn Forn 1 —ioi 77 Vli)33" £10000 77 6»: £7500 77 •!««.- £6000 ( 77 1 • li^CDtOIBBl(S6.J KEY, (104) 278 By Formula V, page 361, t, = 1 5 j ]_ > ^ (I — 0'261795) 100 , = — X A L V- •»« \ (I -OB)"] - 10000 6 = -X 0.73820^ ^^^-^^ 6 ^"^38205 = = $1230.34 8 (105) Also, 6 men worHr.,, i k "^^ ^ 25. > «iBa worKmg 1 hour per dav for t> ^» ^ : ^^^ fo"- 12 days make up 5 x 12 Hence they will each have tnV . ""' ''*'^*°"^^^^«°>anyhoarsperdaya3 60 = 10 hours. fe«« B contained times in 24 x 25 hours, i.. !lf.'' (106) ' «• ■ ^'' 'onn„,.„, :,;:."rJ!!r;;.ii- (107) 63 * 717==- ^*«29 days. * a-06)2^ -^"Sr.l-Iog.icc X 23^0-,(» (v:i5308 X 28 • 'iL-m).7 - «^«' "n.ber <.,.e.poncih,, to the Wh.k. l'*17968, wh'ch fs 0-201786 274 KEY. (108) [l^AT. ASI3^< Lo^r, . '" '.° '"""" ■"" "■"' • »"" »f olothe, = for , mootha he received £2 ,3s. 4d. and the eaU „f ..ithe, ••■ for 5 months he wo»ld hare received the difference 1: t^een £B and £2 13s. 4d., which is £5 es. g^'^"""" "«- Hence for 1 month he ^ould have received £5 Os 8d • , wh.ch ,s£l ,s. 4d.,aud hence his,va,,es for thevearwiuld bave been, in money alono, £1 is 4d v lo , 7 T.T Therefo. the^.nit of clothes 'was valued Vk^^'^t Tbtth SflBiBa. (109) It It *r \l '''^ '': ^'^"''^^ - ^«' ^V will « 48^13^3^ " ^iH — i?> it, or the whoie number = 3rj X U = 44-V 6' : 83 :: eOO : (110) 600 X 83 600 X 512 = 1422«2 lbs. 6^ 216 (See Art. 33, sec. X.) ("1) Part of ball remaining after 1st has taken off her share = 3 Then u-^iole ball : remainder:: cube of diameter of whole • cube or diameter of remainder ' 1:3:: 5' : x3 .,ence x =^l l>n2?= ^qT^mnT^ 4.549 .-.Part taken onr by 1st = 5 in. --4.543 iL-. 0-458 in After 2od had taken off her port_^on J of the ball remained . . Part tak'^n off by 2nd=: 4-543 ~3-9G8 = 0-574 in After 3rd had taken off her siha^e there remained i of th« ball ''' U'rf! 'f' ""IT '^^^'' '''=-■ ^'^^- 3 - 149 in. • - Parttakenoffby3rd^3•963_3.]4e»_a.«^a;„.^..- jBemaxncle^ - 3 . 14& - part taken off by 4th. ^ ' """"" J'ixicaoisBies.J KEY. 276 72734 62831 88033 87625 6570.2385*52(71.118 =.5q.rt. o4 3070-0 2468-4 500-50 .371-36 318-130 111-067 7-0520 6-2831 161) 170 151 1621) 18-23 15-21 15221) 3-0285 1-5221 162228)1 •4o'g"457 1*360051 •036^1* •028480 •024684 •003685 NoTB.— Unless tha (113) 1st . r $60 X 48 = $4;; ; , ' $800 V 40 f '''' ' '"°°**> 1 Sum = $43380 I = $43280 for 1 month. I X 42 ^ 75^ for 1 ^onth | « $104400 for V Si,«,_» f 1 month. Sum =$104400 {ContiDued on next page.) iS7t) KBT. CVat. ijun. (113 continned). ' $400 X 48 = $19200 ] < $500 X 42 = 21000 $500 X 30 = 18000 $500 X 30 = 15000 3rd i $500 X 24= 12000 $500 X 18 = 9000 $500 X 12 = 6000 > ss $103200 for 1 month. $500 X 6 = 3000 t Sum = $103200 ^ $900 X 40 = $30000 " $900 X 34 = 30G00 $900 X 28 = 25200 $900 X 22 = 19800 * 4th $900 X 16 = 14400 $900 X 10 = 9000 $900 X 4 = 3G00 y = $138600 for 1 month. Sum = $138600 $43280 # 104400 4 years at $1 • 25 per day 103200 ;= $1 • 25 X 4 X 365 = $1825 = share of 3th, 138600 $389480 for une month. $20000 — $1825 r= $ 1 8175 = sum to be divided among the four. $389480 : $18175 :: $43280 : $2019-651 = share of 1st. $389480 : $18175 :: $104400 : $4871-803 = " 2ad. $389480 : $18175 i: $103200 : S4815-805= " 3rd. $389480 : $18175 :: $138(300 : $3467-739 = " 4th. (114) „__! 16 — 1 15 Simple Ti 1! 500 f .05 .05 SS "" — = 300 years. 5 log. n Compound Interest, formula V, p. 354. tmt — ~ _ log. (1 + r) loff. 16 1-204120 1204120 ■~ = jc- *= 66 -82 7 years. log; 1-05 0-021189 21189 i BxtmoxaM 166.] KHT, (115) 277 ■$: X 920' - SDvoTf. ' ""^ '' ^2°^ '^"les. it ai6) 3H X 22 ^ nfr ''''' "''^''^ '^ ^"^^ "^^'»- i«r ! ^^ ^"®' = ^^^«3 Cleared bj comnanrof 22 n,«n 165 acres — 77ii acres — ATio o '-""ipany or J2 men. Pany Of 26 men ~ " """ = """' «'»"«■ "^ ""- '" rr;:r;r ^ -"" "^■' ''" ^-^ -p-^ »<• - '""=z:zr ' ""■ °"" """ """' «"^ *«^ - 3 Each man clears 3$? acres, and recoils $28-661 for it • a^ •ore coat of 1 acre = $28 .66, -;. 3; > = $84ft ' "" «/ Hence less == ^121 =; n. (118) Formula V, page 248, ^=P (l+.O=$1200XI.95=$234a.00. (119) 34 : 496 9 : 11 7; 4 1 1 465 : 33 7i • • 3^ : 51 n : 3j Si '9 (Continued on next pagt. 278 KEY. CKat. Abcth. (119 oontimied.) 11 4 (?i:i5 ^^ t 11 1 1 1 1 — x~x-x — x-x-x — X— x-x-x X — X -= 11 X 4 X a= 1?2 days. 11 \ (120) 9i A+ B+ = a Bj+ 0+ D= U A + 0+ D= tH A+ B-Jh D= ^5 3A + 3B+30 + 3D = i6V .•.A+ B+ XJ+ D= §5 A + B+C + D=^J A+B+0 =^ A+B+O + D =^5 •••A=i8'=i- A.fK + + D=§5 A +0 + D=:|g ••.B='^ = ^ A + B+o + rr=i|5 A+B +D=i^ .*. = J 3. — J. to * 6ff •• 7 • |- X H X f? = ^? = D's true share which is therfefore = ^9 of $6213 = $1090. ¥o •' ^0 :.' -^ : ^ Xfjii X .f 7=!?- = A's true share which is there- fore = §^ of $6213 = $2180. ^l : H :: i : i X &S X f<^ = l^ = B's true share which is therefore = -}^ of $6213 = $1635. • n :: ^ : i y U X ^ = H = O's true share which is therefore = ^f of $6213 = $1308. ■w>*;?f:v:ryv ''xaBoiio^ 185.1 KEr. 279 (122) ^1^ mam. =: 861 dg». • 291 -iai — q^a w = 47 . den. ^ ^"^ ' ^" «^^ 291 Then 713 --. underu = 8613 j^ ^,^ ^ 861^ den. "''"" rloZ :^ ''^ ' ^"^^ ^-' = ^^^ ^- ^ ^00000 ^. Then 12m, „,V =411AV.^«. (123) 3|of2}of7-of£i = .iofi,i-ofVa^ofi:i 9? of 38. of ir=V'i ofVi ofW-i'Li ~ "^^^ ^ ^"'•''- i:58 08. 8,Vod: = '^22.^.iUd = 32414-56. * ^124) 34 : 90 2i : 4^ 12' H i . 9^ 4V :: 139J ;« 3^ ; 2^ (Contiimed ou next ptgei) lllj .Jip-'tf- IMAGE EVALUATION TEST TARGET (MT-3) k A ^^ .^4g. fi"'^^ f^ AV «.., C V.% ^ m 1.0 I.I 11.25 IM no 1 2.5 1^ yii •"-3*1 2.2 lis IM II 2.0 1.4 lli'-^ PhotDgraphic Sdences Corporation 23 WEST MAIN STREET WEBSTER, N.Y. 14580 (716) 873-4503 m \ \\ A V O^ '^ V o % .^t. I/. .<$> Mg ^ i 280 111 KEY. 094 oontinaed.) Max* Asnvr »2 2X4X5 xT""*!^^'*^^^''- I S182 is ^ of bnjrjQgP price .-. $182 -s- 91 - «2 - ^ ^ w« t";/u ! r '/" ^^'^^•^ ^^ --^ recL.e $1.07 for e^ery $l the goods cost ; but thej cost him $200 ther. fore he must sell for $1 -07 x 200 = $214 (126) «— 1 llj — 1 10-6 1050 •06 "" ToT *"* V 'Ofir- 111 1.060693 0.02a30d Sh&ple Interest t = r = 176 years. Compound Interest f^ —^^'*^ _ _^ 1060698 ^*»«- (!+'•> l«g.l«06 (127) An acre contains 4 roods = lei sq. perches. .-. 160 ^ I5i = 10 la perches = length. • (128) 36 yards = 32 metres ...1 yd. = 3^ of a metre. 69 .. mUes =: 69,V^x -p yards = 69,^ X xz^ X ^ metre. 217 {^() _ ni^ V^i^ 32 t WJ 1-or for 0, there- 1050 =: - 6 .060693 KBT. m (12a) Formula XXll, p. 340. r = fL] "'^.^ /^^^®^ * / \ f fienc« means are 9, 27, 81, 243, 729, 2187, J ^ 6561. (130) •V>rmula XXI, p. 344. t=:^ -_][ ^ _ 28 (131) Part rerna,!^, after 1st ^ ^ • t .. 60« , X* ; whence x = V?6oT~v-^ /~ - Hence Ist ground oif 60 - 51-96 - 80A f », Whence. V3600ln= 30/2 = 1-4142 X SO = 42-426. Hence 2nd ground off 61 ■9R-.40Mo« Igafnea -- 2Ib. 1 half guinea = lo^g. 1 crown =s 53^ 1 hflif crown = 2'g. 1 shilling =5 ig, Siipence =3 Js. Sam a 40|f. laso j00guinea3=2l00 8hriHn«. 2^0« - 40^ = «, ein.es fnd re- mainder, 69 haif-shiilingg. 282 KEY. VWILVTH 8IBIM. [Nat. Abztb. (133) 3 2 4 » 2J 2 10 3 4 11 9 17 561 ' 4i s"" n s"*!? 88 8 4 4 561 200 ^ 51^1 — :--:: $12^,: $i23Vx--x— = — x-x — ^^^ 1^ IT 8 ^^ It 1^ % = $200. (134) By Formula III, page 354,r = V/> — 1 .-. r + l =V/i .-. Log. (1 4- r) = (log. ji — . log. P) ^.f = (log. 1679-40 — . log. 700-90) 4- 5* = (3-225154 — 2-845656) -^ 5*. ^ = 0-379498 ~- 5 = 0075899. .♦. I + r = tiat. num. corrcs-?onding to the logarithm 0-075899 which is M9, . •. r = -19 = rate per unit, and hence rate per cent. = 19. (135) Having paid 10 per cent, ho had 90 per cent, remaining. ■\%^S or ^g of his salary = $1250, . •. -ji^ = Xi^io. == $i38|, If $138| = i»tf, the whole = $138f X 10 = $1388-888. (136) 21 children receive 21 times a child's ahare 21 women " 42 " « 21 men " 63 « «« Together they receive 126 " £3 133. 6d. — 126= 7d. 7d. X 2 = Is. 2d. 7d. + Is. 2d. = Is. 9d. (I a child's share. ; a woman's share. a man's share. 4T. Axrtm. •X (137) A gets 1 time A's share B " 1 « A's « « 2 « A's « D « 4 « A's « Together thej got 8 times A's share. $25 + $251 1= ^'T ^'^ ^'^-^ ' $25 = B's share. ' $25 + $25 = $50 = Cs share ; $25 + $25 + $50 = $100 = D^s share. 1^. VI 1^18 (138) i of 2-62074 i of 2-44948 Dififercnce =: -87358 = -81649 = -05709 0-075899 ence rate ?. $138|. 888. 6347 -of- of— of— of— z= ^ ^ 1!^ 11 23 253* ■^2i = n*^-'^=a3^4iXA = 2308. (140) 9' = 81 X 300 9 X 6 = 64 X 30 6« 24300 1620 36 884736 (96= cube root. 729 155736 re. lare. s. 25956 155736 »595m}=: 95951-2576. (Continued on nay t nave « tu KBIT. JXjlt. Asm. • • • • (140 oontinued.) ^5951. 2576 (309 • 76 = square root. 9 ^09) 5951 6481 309«76 (17«6= 172 = fourth root. 1 618«7)470-25 433-09 619'46)37l676 37-1676 27)209 189 84-6)2076 2076 (141) 260 800 400 600 $520X250 1450: 250:: $520: = $89B = contrib.on l3t village. 1450 $520X300 1450: 300:: $520: =$107i5= » 2iid « 1450 $520>:400 1450:400:: $520:- =$143i2= « 3rd « 1450 $520X500 1450 : 500 : : $520 : =$179^?^ = « 1450 4th « (142) By Table on p. 362, the amount of $1 for 34 payments at 3 per cent. = $57-73018. $57-73018 X 260 = $15009-84. a {(l + r)*-!} By Formula I, page 361, j1z=» r 26000 , 26000 X 1- 731835 -— X(a'73l865-1) = = $15000.41 JEZSBOIiXlQQJ KBT. (143) 2S6 By Fomral. IX, p. 333, d = = „. Heoce the series « 2, nj, 32J, «i, 632, and W PonnulH I p. 333. / = a + (,_i)rf = 3^. = 3 + (8X4) = 3 + 32=35.. ^ '^* Formula VI, p. 333. • =| 2a + („_,), |^ 207 = (6+824) X -"^ — !!2^ X 207 a' =-66905. (144) BtraTelr4 miles per-dayfaster than A, and trill th« * the circumference of the i.Iand in rTl T^" ^^'''^'^"^ «»^ n . , » «u 4 — 18^ days. C-travels 10 miles per dav fastp,. ♦». of I8i days. ^ ' **' ^**™® Other multiple Similarly C cannot be with A excent Rt ♦!,. ^ . . of some other multiple of 7" days. ' ^^^ ^"^"^ *^' Therefore and B will both be with A fn. ♦», ^ the lapse of a number of IT **•* ^"* *^*°'« '^^^ mon multiple ofTs/ La 7^ ''""""^ '"^ ''' "^^^ ^• The greatest common factor of 18* and 7.3, is s^J. Hence the I. c. m. of 7^and IB* is !iilI!L ,,, 31 s — ^°* = auabeF of daja when A, B. and n ^n g,.^. */■ 286 KEY. P7at. Abitb ARITHMETICAL REOREATIOI^S. 1. The third of 6 = 2, and the fourth of 20 = 6 or «"20>.. 3.__8y 20XJ 2J X 11 J • = -H. I*afftly, what part of Sis ^JT 7: ii> •• or Jn», i : » = _l X 5 X 11 9X7 «^. 3. 99f. 4. J of 2d. = gd. Then |d. is what part of 3d.? Jns. |. 6. lid for a herring and a half is at the rate of Id. per her- ring; hence 11 herrings will cost lid. 6. 12 apples = 21 pears = 1 cents. If 12 apples cost 7 cents, what will 100 apples <508t? 100 X 7 12 : 100 :: » : , ^ 5,, „„^ - , 12 8. The h.rdles are arranged so as to form . rectangular en. elosnr, having 49 hurdle, on each side and oof on each each end. and will thn, double the size of the eaclo- Bure. £xzaoi8B 166.1 or. 287 10. 33| 11. XIII; rub out the lower half, and there remains the expression VIJI = 8 "' "'^Lll^' ^-^"^^^ '-' -' -^*^ ^* ^- the 2nd^S^ep : Again fill the S-gallon cask out of the S-gallon "l'r^/his:n^ '"''"^^^ ^"' out of the ..gallon cask. Th.s will leave one gallon in the latter. 4th btep : Empty the 5.gallon cask into the ^-gallon cask 5th Step. Pour the one gallon out of the 3 gallon cask into the 5.gallon cask. ^ ^ 6th Step : Pill the 3.gallon cask out of the S-gallon caak and empty it into the 5-gallon cask. ^ ^' The following diagrams show this more clearly • ^'' ®'^P- 2nd Step. * 3rd Step. 4th Step. Bth Step. <"'h Step. 288 XET. PXAT. AMITU' 13. The heary line in the aecompanTing figure shows how the board is to be cut. "" T ^ rr MB iM n "■ 1 - - - ~ "' — ir 1 ^ Jj , _j ± 8 3 4 1 6 9 6 7 2 IB. Weigh out 7 lbs. as often as possible and there will remain 2 lbs. ; add two four pounds and one seven pounds to this, and the sum will be 17 lbs., the share of one. Weigh 7 lb3. as often as possible out of the remaining 34 lbs. and there will remain 6 lbs., to which add 7 lbs. and 4 lbs., and the sum will be 17 lbs., the share of the second. The remaining 17 lbs. will be the share of the third. 16. The hurdles are, in t!;e first case, placed 12 on a side and one on each end, and then they inclose a space represented . by 12 squares whose area is, by the question, 40 square yards. If two hurdles be taken away there will remain 24 and if these be placed in the form of a square, each side containing 6 hurdles, they will enclose a space represented by 3G squares of the same size as the former. Fence they now inclose three times as much space as before, i. e. three times 40 square yards, or 120 square yards. 17. He takes the goose to the remote bank and leaves it there returning, he next carries over the fox, which he loaves but takes the goose back with him. He now leaves the goose on the first bank, and carries over the oats which he allows to remain on the remote bank with the fox, and returns for the goose. 18. The following diagrams exhibit the solution of this problem ; ^^«CTMHeB.J Kmr. m ;::^^-^-tt.^i^^ I 1 ■i f «W T, ^'"""""Xrw^^r'^ "-n-Jto which iMt filled. Tj, ,^, !,.•„''" °«« •«'»» the coll «U«d «ft,r till, mettod : ^"*"' »' » cell, ta , rfd, (Ooj •rmatu n. .... X m XB7. [NAt.iUx^ ZTZi 80 39 48 1 10 19 28 88 47 7 9 18 27 29 46 6 8 17 26 36 37 6 14 16 2fi 34 86 45 18 IS 24 33 42 44 4 21 23 32 41 43 3 12 22 81 40 49 2 11 20 !- ',. 31. Half-a-dozen dozen =: 6 X 12 = 72/ Six dozen dozen = d X 12 X 12 = 864. 864— 72 = 792. jin$. 22. The toUovring shows the mode of performing this. It will be observed that the two side counters are merely moved one counter higher when the other two are taken away. " 23. This problem admits of the following two solutions : ^ IST SotUTIOW. >o® Persons. Ftill bottles. Hf.-full bottlea. JBmptj bottles. iBt a 3 ' a 2nd „ 18, S i 8 8rd . , 8 '.K i-.-illt^.')- ».ld 7, 7 » Bath person has 3| bottles of wine and T bottles. — aiiD Solution. iio:.' 2nd d '^di 0.1 t'^iii J 3 sfa; 3ifd I ' 1 1 7 T Bach person, as before, has 71)ottle8 and 3i bottles of wine. »o« 0' 07 *™™»l«-I ""' •■ KIT. £^1 bo« , beloagipg to hlo, who ow«d the 5 botti:.. H.»c* o/I? 1)*°*^ *^® former 1 shilling^. - '-i'^'^'^ J«t'nx c * «*"in^8, »?,<>; od 36. ^ns. 1 lb., 3 lbs., 9 lbs., and 27 IbrfT "^ ^ C -:• £ x " S^>r 1 lb, .= 1 lb. ; 2 lbs. = 3 lbs. -. 1 lb , i • ^i T » ^'ta ^"I..^ """• = »»<»• + 3 lb.., I8,b,.= » lbs.+ 3 lbs. + 1 lb. , 14 lb,. = 3T lb.. _ (Slli + 31b. lb». + 1 lb.), 18 lb.. = « lb.._ » ,b,. , Ac, 4,. "• <* "■ L?J^' .'° -' '°""' ""'* <"'*««iS" point., in. „.„,. r«d to he potM from wSicl. the first riarU^m ibMto the pomt fro» which the .econd .t«rted, »o!^^«' " , ,J?iBS it the first conntir is )if ^Tpt\ in 1 hour, and to fill the reservoir thej require 1 -J' iVV ;=; W = 4 hours 43 min. X6ff seo. SC9ii, 7he jpenon who t^nks of the numbers must proceed as follows i He must multiply the ]st by 2 and add 6 to the product ; be must nej(t multiply this sum by 6 and add the secoi^d number to the product; he must next multiply this result by 10 and add the third number to the product; lastly, he must subtract 250 and name the remainder. The three digits of the remainder will be the three numbers thought of, and will he in the order in whioh they were thought of. ' The reason is obvioua: let^. sc Ist^ fr.ss2nd, and c =r 3rd number thought of. a X 2 + 5 = 2a + 5. (2 « + P) X 6 + ft ==: 10 « + ft -f 36,r (10 ^ + ft ^ 26) X 10 + c =5 100 a + IQ o-f c + 250. (100 a -h 10 ft + C + 2«0) r- 260 = 100 • + 10 ft + c =* ._a in hundreds' plaoe, ft in feens' plaoe, and c in units' place. 80« Since each main possesses 63 square rods of l^nd more thauc hiftson, we-mnst form three pairs of numbcaniy such that the; d|fil»renoe of their squares shall be 63. The difi!^rence of the squares of two numbers is equal to their su^ multiplied by their d^renoe, and h9nce 63 must bo; divided into two faotora in three djatinctways, thus : 63 = 63 X 1 = 21 X 3=;SX 7, If sum,= 63 and diffbrenoe =s 1, the numbers are 32 and 31.. If sumzs^l and difference = 3, the numbers are 12 and 9. If sum ='9 i^d difference = V, the numbers are 8 and 1. Hence the squares of Jones, Brown, and Smith, aie respee- tlTcly ^2 rods, 12 rods, and 8 rods on the side, and the son's squares are respectively 31, 9, and 1 yards on the side. Jones' piece was 23 rods lougec on each side than Tom's, and since the difference between 32 and 9 is 23^ we may con- clude that Jones' square was 32 rods, to the side, and Tom'* V roas on, A giae. ■sn 2*- ■^^KOisa l«.j SET. 293 « side, and H.„r, pie<»,„d """*'"' " '^"^ Hence Tom w„ Brown's „B, H„„ wa. ami.1,. wasJonea-fon '"■*"' "^ amth's bob, and Ifed 0=4, and »=.6 »'•— -^le^ a,.» — 3^ whit.,. Lja'l wtt • "^ "= ' """"=• - '■ = 3 Jin.. B7.ab.,.c«ngtSi:n;'',ft'dS^'°'"°-''' der from the next hiffhi»r m«u- i !"*»*" ®' *^e remain- digit crossed ^^ « ^^' '^ ' ^*»" ^«*««^ine the Thus suppose you select 117 a. ««««w "^ . .,^ ijou write. .. , .Ji9886|vT^ r ,0 = :>: Suppose Jie.writea 614«1 i ■ -^«,_ You write . 38548 5 ^^ W999 Suppose he Writes $§00) "^^ 38548$ — W9999 Yonwritr Suppose he writes 11 ^ Toii#rit* 999981 =^®^* Stipposi lie writes 99999 I ^J^^ Ywiwril* :r: 00900 f r^ ^^???> Bom^ 499995. 1 -=:9999$X4 oiU ou-ri:~-jjo:) !)ii 'y. iiiOOl^?. ■■■yr rroV: .'.Z^r^':lTz -::. ~ ' ' :::.'j)h/;07 jacjir ; T -r C f r^ <^ *• ^ -•' '-fjtriid ,ir.VD dxI-* ••*^ A^^^^, , "r\ t '^ • • -; r . — '% .?/■ ,., ,. .?■ /-, ■? .. ,. «;? .?»»igri.' VXAie. >Da that tach tigits. Then siashown as If ;3nt»ii o::.': sju -iii noiia^j eaowxit '' ~; f '■li'-C ''■" .■-.t-ilil-u '■-^^i . .'.T^oJ .4-ooco 1 I .onifl . O J. C' .1. 5." xnrr.fircr oi'ji:;.;