U " TOTHE WESTERN CONTAINING SOLUTION OF ALL THt EXAMPLES AND QUESTIONS . E REFERENCE TO THE PAGES WHERE THEY STAND. TO WHICH IS ADDED, of the said district, have deposited in this office the title of a book, the right whereof they claim as proprietors in the words follow- ing, to wit: "A Key to the Western Calculator, containing the solution of all the examples and questions for exercise, with reference to the page* where they stand. To which is added, some Useful Rules. Design- ed chiefly to facilitate the labour of teachers; and assist such as have not the opportunity of a tutor's aid. By JOHN ARMSTRONG." In conformity to the act of the Congress of the United States, entitled, " An act for the encouragement of learning, by securing the copies of maps, charts and books, to the authors and proprietors of such copies, during the time therein mentioned." And also to the act, entitled, "Jin act supplementary to an act, entitled, "Jin act for the encouragement of learning, by securing tJie copies of maps, charts and books, to the author* and proprietors of such copies, during the time therein mentioned? and extending the benefits thereof to the arts of designing, engraving, find etc/ting historical and other prints. 13 Wm. WALKER, Clerk of the. Wettcrn District of Pcimsylvatua. CAJORI Numeration Addition - - - - ib. Subtraction ... 6 Multiplication 7 Division Federal Money - 12 Compound Addition - - 1 Compound Subtraction' - - 14 Compound Multiplication 1C Compound Division - - 18 Reduction Addition of Decimals - 3C Subtraction of Decimals 31 Multiplication of Decimals ib. Division of Decimals Reduction of Decimals - 33 , Single Rule of Three Direct 37 'Single Rule of Three Inverse 41 Double Rule of Three - 47 Practice - - - - 53 Tare and Tret - - C>0 Simple Interest - - 62 Compound Interest - - 68 Insurance - - - ib. Commission Brokage - - - ib. Buying and Selling Stocks 70 Rebate or Discount ib. Bank Discount - - 71 Equation of Payments Single Fellowship - - ib. Compound Fellowship - 74 Profit and Loss - - 75 Page. Barter .... 77 Exchange ... 78 Alligation 80 Reduction of Vulgar Fractions 82 Addition of Vulgar Fractions 86 Subtraction of Vulgar Fractions 88 Multiplication of Vulgar Frac. ib. Division of Vulgar Fractions 89 Rule of Three in Vulgar Frac. ib. Inverse Proportion - - 90 Involution ... ib. The Square Root - - 91 The Cube Root - - 96 Arithmetical Progression - 102 Geometrical Progression 103 Single Position 106 Double Position - - 107 Permutation - - 111 Combination - - - 112 Duodecimals ib. The Carpenters' Rule - 115 Measuring of Boards, &c. 117 Carpenters' & Joiners' Work 118 Bricklayers' Work 69 Masons' Work Plasterers' Work Pavers' Work - Painters' Work Glaziers' Work 72 Measurement of Ground Gauging Mechanical Powers Promiscuous Questions 120 ib. 121 122 123 124 ib. 126 ib. 127 antr = The sign of equality, and is pronounced equal. + The sign of addition, pronounced added to. The sign of subtraction, pronounced subtracted by. X The sign of multiplication, pronounced multiplied by* -r- The sign of division, pronounced divided by. : :: : The sign of proportion, pronounced is to 9 so is, to. & Sign of difference, pronounced the difference between. v/> or 5, The sign of the square root. ^y, or i* The sign of the cube root. T 2 , Denotes that the number 7 is to be squared. 8 3 , Denotes that the number 8 is to be cubed. -.- That is. - Vinculum, and ( ) parenthesis. 19; twelve added to 7 is equal to nineteen. -3=15; twenty-three subtracted by eight, equal fiftee 9x8=72; nine multiplied into eight equal seventy-tw 243= 3; twenty-four divided by three equal eight. Sometimes the division of one number by another, is e pressed in the manner of a vulgar fraction; thus % 7 = uy-seven divided by nine equal three. Multiplicati ?o frequently denoted by this mark . thus 7.1 W 2=:8 o twelve equal eighty-four 6 : 9 : : 8 : 1 2; as 6 is to 9 so is 8 to 1^. A OQ Z: .1 -.iifterence between A and Z. v'Si^rO, or 81i=95 the square root of 81 equals 9. 10 : 100 . I : 10; as 10 is to 100 that is as 1 is to 10. A vinculum is placed over several quantities to deno are to be considered as one simple quantity; 2 1 for the same purpose, JOSEPH B HAMMOND TO THE WESTERN CALCULATOR. NUMERATION. Page a Example 1. 16 2. 49 Page 9. 3. 385 4. 2610 5. 64536 6. 253842 7. 5600006 8. 90000305 9. 829006002 1. Five. 2. Seventeen. 3. Thirty-five. 4. Four hundred and fifty-eight. 5. Six thousand, eight hundred and twenty-nine. 6. Seventy -two thousand, three hundred and forty-eight. 7. Three hundred and eighty-four thousand, seven hundred and twenty-one. 8. Two millions, six hundred and eighty-three thousand, two hundred. 9. Fifty millions, six hundred and seventy-eight thousand and twenty-four. ADDITION. Ptgt 10, g. 1581366 4. 2284038452 6. 2968194198 8. 5075431465 10. 3644772*180651 1* 3. 2468727092 5. 2263764650 7. 5831S33360 9. 9750563355 11. 3847099705067 6 1 **125 V*: /:*?' , 2060 8009574 6 SO 11 804 5. 297 125 754 127 ' 245 si JBTRACTtOS, Application. fage 11. "' .dolls. >.*286 C 29 1)672 barrels dolls*/ 4. 60 for 480 75 675 220 2200 126 1386 10000 4000000 4018853 6. 1209 476 8017 3119221 S1112 7. 1811 21 481bbls.4741dols dolls, 8. George 3560 William 3240 Samuel 2850 Henry 2555 Thomas 2226 3832 3128923 1548 SUBTRACTION. Page 12. 14431 3. 44064062638 5. 56053355680 1. 1818 1777 41 Application* 2. 5648 3460 2188 4. 5390534442 6, OOOOOCC1 barrels 3. 1260 sells to A 520 B 435 sold in all 755 unsold 505 4. 6000 600 cents cents 145 35 penknife 75 25 slate 40 64 paper 30 apples given him 260 given away 154 154 left 106 MULTIPLICATION. MULTIPLICATION. Case 1. Page 13. 2, 24639576 4 3. 3675432568 8 98558304 5, 476324753 20403460544 Page 14. 6. 964703024 6 7. 74020005 8 2334123765 8.2901946808 9 26117521272 5788218144 592160040 9. 246354276 11 2709897036 Case 2. 2. Mult. 68523047653 by 2367 479661333571 411138285918 205569142959 137046095306 162194053794651 Mult, 5221 by 145 Page 15. 4. Mult. 23430 bv 230 26105 20884 5221 757045 Mult. 89536925 by 735 447684625 263610775 626758475 702900 4085 9, 3950 35550 7900 $ 1145,50 5388900 Mult. 78969587 by 5893 236897961 710693883 6317- 394829935 465346561391 10. 40 480 pence 5. Mult, 3800920 by 80750 190046000 2660644 30407360 306924290000 8. 1.15 75 575 805 $86.25 MULTIPLICATION 18292575 Case 3. Page 15. 2. Mult. 871075 by 21 3 3. Mult. 2453642 by 36 6 4. Mult. 43102 by 64 8 2613225 7 14721852 ' 6 344816 8 88331112 2758528 5. Mult. 23645 by 144 12 283740 12 3404880 Case 4. Page 16 6. Mult. 12071 by 99 9 108639 11 1195029 Mult. 3600 by 400 3. Mult. 44000 by 550000 4. Mult. 663000 by 60000 1440000 220 220 39780000006 24200000000 NOTE. Ex. 1. 200X10 =2000 2. 462X100 =46200 3. 879X1000=879000 dolls, 375 1. 15 2. 125 3. 32 cents 25 43 440 75 375 1280 30 500 128 dolls. 53,75 dolls. 140,80 4. 60 X 125=75,00 wheat 5. 100 6. 32 40x 85=34,00 rye 10 sub. 20X24=480 S4X 50=17,00 corn 1000 dimes 12x18=216 8 126,00 amount 10 paces 696 10000 cent* DIVISION. Page 17. 7- 7525 8. dolls. 250 A's g gio A 125 3 750 B 3000 C 37625 750 B's 15050 4 4000 altogether 7525 .. 3000 C's 940625 DIVISION. Case 1. 3, 4)4756394344 4. 5)97036142 5. 8)37846210 1189098586 19407228} 4730776f 6, 12)64381259 7. 6)3824966 8. 7)46825486 53651044! 6374941- 6689355! 9. 9)8297463813 921940423! Case 2. Page 13. divisor dividend quot. 4. 735)47989536925(65291886 quotien 3. 64)29687624(483869 4410 735 3889 3675. 'l470 6753 66*5 326459435* 195875759 457043209 408 384 1855476 2783214 247 192 556 512 29687616 8 rem. 29687624 proof 47989536925 proof "Add in the i 584 :>76 8 rem. 1386 rm'imlo" as you mul- 735 tiply when working the proof. 6519 5880 6392 5880 5125 4410 715 rem. 10 DIVISION. Page 18. 5. 2359)4fi7968967 ( 2084768 4718 2359 19996 18872 18762917 10423845 6254306 11248 4169537 9436 18129 16513 16166 14154 4917968967 proof 20127 18872 1255 rem. 7. 175296)19842712000 ( 113195 175296 231311 175296 560152 525888 342640 175296 679170 1018763 226392 56597ft 792373 113195 175296 19842712000 1673440 1577664 957760 876480 6. 671)5374608 ( 8009 5368 , 671 ~~6608 8018 6039 56069 48059 569 5374608 8. 108)5704392(52818 540 108 304 216 42-2552 528184 81280 Case 4. Page 19. 2, 6|00)876154 3. 8jO)2834|7 4. 146 54rgm. 354 27 rem. 883 5704392 864 199 108 912 864 48 16|00)1370|00(85 128 90 80 1000 rem. 3. 11000)8761029 4. 110000)8010000 NOTE. 1100)256154 or 256 quol. 54 rem, or 876 quot. 29 rem. or 80 quot. Application. I. 855)4275(5 boys 4275 DIVISION. 11 28)2072(74 196 112 112 Page 20. 3. 75;45000f60Q 450 00 5. Here begin with the 150 and work backwards, thus 5)150 50 12 subtract 2)18 9 Ans. 4. 8164 10 subtract 27)8154(302 81 54 54 6. 13440 dollars |=5360 eldest son's share -=2688 2d son's =2240 3d son's " |=1920 4th son's " J=1680 5th son's " 11888 sum of 5 shares then 13440 11888 subtract 1552 6th son^s share 72084J5190048C72 8. Because A has 10 cents per day more 504588 than B, and B 10 cents per day more than C, it is evident that A will get in all 8 dol- 144168 lars more than C, and B 4 dollars more 144168 than C, which together is 12 dollars, then from 228 take 12 Diyide by the number of persons 3)216 72 C's share consequently 76 B's share and 80 A's share 575;i437,50(250 1150 9. half Eagle 500 half a dollar 50 quarter dollar 25 575 cents 2875 2875 10.' Captain 6 shares Mate 4 " Seamen 40 " No. of shares 50 Then 5|0)455|0 dolls. 91 one seaman's sh, Now91x6=546Capt. share And 91x4=364 mate's share FEDERAL MONEY. FEDERAL MONEY. ADDITION. Page 22. 2. Dolls. 19520,43 5. Dolls. 04588,00$ SUBTRACTION. 2. Dolls. S25216,94| 3. Dolls. 1267S.53f MULTIPLICATION. Page 23, 2. Dolls. 15823,50 3. Dolls. 57408,79i DIVISION. 2. Dolls. 726,55 3. Dolls. 1034S,79& PROMISCUOUS QUESTIONS. E d d cm Quest. I. 2.5,0,0,0.0 6,2,0,0.0 8.0.0 7,5.5 31,3,5.5,5 Or, S SIS 55J cts. 3. Sugar g 39,87i Coffee 22,181 Tea 2,m S 64,181 5. Lent S 1000,00 Dolls. 2. 1055 in notes 260 gold 5650 silver 250 cents S 4~967,50 4. .From S 645,952 Take 350,00 . , . f 160,25 Received at sundry pay -ts. Received in all S 746,20| Unpaid g 253.791 S 295,65? 6. S 102,19 Mult, by 120 g 12262,80 7. Dolls. 4,50 16 3474 5211 173766 $2295,50 g 72.00 S. 5)6022,50 g 1204,50 COMPOUND ADDITION. 13 Page 23. 10, 18)252.90(14.05 11. 45)f>. 50(50 cts. 18 225 72 . 72 90 Page 24. 12. 25)15535,50(622(1. 22c. COMPOUND ADDITION. ENGLISH SIDNEY. Page 25. & 136739 6 3 3. 15725 11 Hi TROY WEIGHT. lb. oz. diet. gr. lb. or., dirt, gr, $. 22 7 6 6 3. 33 9 19 AVOIRDUPOIS WEIGHT. Page 20. T. cwt.qr.ll). oz. dr. -irt.qr.lb.oz.dr. 1. 128 12 1 18 14 10 2. 2921 3 9 S APOTHECARIES' WEIGHT. tit. oz. dr. sc. st. lb. oz. dr. sc. gr. 1. 24 4 5 2 16 2. 220 2 2 10 CLOTH MEASURE. Yds. qr. na E. FL qr. na. E. Fr. qr, rta. 1. 215 2 2. 173 S. 128 4 E Ln. qr. na. 4. 221 2 LONG MEASURE. Page 27. leg* m - f ur > P- yd- ft- 2W - be. . M. fur. yds. ft. in. 1. 207 27 5 6 1 4 2. 13 2 3 167 2 10 LAND MEASURE. Page 28. A. R. P. A. R P. A. R. P. 1. 161 1 29 2. 2134 2 23 3. 1999 2 32 CUBIC, OR SOLID MEASURE. ,Co.fl. in. T. ft. in. T. ft. in. I, 26 4 1407 2. 21 16 1542 S. 271 26 1294 14 COMPOUND BUBTRACTIOX. TIME. Page 29. F. mo. d. h. r.ii. sec. Y. d. h. rat e. 1. 104 6 13 3 27 2. 14 240 4 4 33 sis- de$. mi. sec. sig. ' /> 1. 9 16 9 17 2. 10 12 45 S3 LIQUID MEASURE. Page 30. 1. T. 20 hhd. gal. 15 qt. pi 3 1 2. r. 79 hhd. S<* 11 DRY MEASURE. 1. Lu. 175 P.. #. 3 2. Lu. 1553 P. 3 8*. 4 3. fctt. p. 41920 9* 7 COMPOUND SUBTRACTION. Page 31. T. c??f . qr*. ZZ;. 03. dr. mi /ur P. ft. in. It, 2. 156 19 22 2 6 5. 124 5 34 4 7 a iu. P. f D. h. mi. cc. 4 53 3 4 1 5. 83 21 44 S4 T. hhd. galqt.pt. sig.dcg. mi. *cc. JH. It. P 6. 29 2 . 47 3 t 7. 22 15 41 8. 408 2 22 Promiscuous Questions in Compound Addition and Subtraction. y(h. qr. ~na, . s. d. 1. No. 1, S6 3 2 2. 1st. 6 2 4 6 - f2, 45" "l 3 2d. 5 10 9* " 3, 48 2 1 3d. 700 " 4, 52 3 4th. 8 10 6 5, G4 C 2 5th. 926 jards 247 21 ^36 6 l|coft Received ^22 10 6 Note dSlS 15 7i COMPOUND SUBTRACTION* 15 Page 32. lb. oz.dict. gr. Quest. S. Bought 26 9 10 Wrought up 18 16 10 Has left ib. 8 8 13 14 lb. os. dr. sc. gr. 4. Bought 6 10 6 2 Used 4 5 4 I 17 lb. 2 520 3 5. Charles born 1817 William born 1816 3 20 9 evening 1 15 6 morning Year 1 Cwt. qr. lb. 6. 1st 18 2 14 2d. 16 3 18 Sd. 22 24 4th. 24 1 2 5 15 7, foMfo c 7.3 yds. 40 sold and left. yds. Then from talc 40 Cwt 82 Or, 4 Ton, 2 cwt yds loforDandE. But hocnnsc E lias half ns muc'h ns D, and together have 15; therefore D has 10, and E has 5, 8, gats. bought 1 pipe =T26 2 hhds. S quarter casks -1 26 L 26 lea 330 bought 148 J sold and leaked gals. sold 1 hhd. 63 2 qr. casks 52 * fpipe 17 ed-} lihd. 11 I cask 5| gallons 148i 181 Heft 9. Bought E. Fr. qr. na. yds. qr. no. two first two last 2=14 1 2 3 2 = 14 1 2 2 5=12 2 3 2 3=12 2 3 yard* 54 2 40 2 13 a 2 yards left 16 COMPOUND MULTIPLICATION. COMPOUND MULTIPLICATION. Case 1. Page 33. T. cwt. qr. ft. oz. dr. 76. oz. diet, gr* 2. 146 -2 3 14 6 1 G. 71 10 17 7 lit pc. ql. ?. pi, deg. mi.ftar.p. 4. 199 30 5. ^3 33 ^ 1 6. 67 18 6 52 SI. R P. bit. pc. qt~ 1 140 8. 809 u 34 9. 533 1 yrs. in. w.d. 10. 76;- 11. 1508 '2 2 6 . s. d. Qaezt. 1. 2. 1 2 6| 5 9 10 2 SJ . A r, 7 . JE. 9. d. 4. 2 4 2i It 12 ^7~0~s] 6 10 6 Case 2. Page C4. r ication. T. cii-t. qr. l->. cz. c>. 1. 4 3 1 .16 8 10 by 36 6 5 1 15 3 12 product bj 6 130 21 76 8 product by 56 . .?. d T. cict.qr.fi>. 2. Mult. 1^> G 9 by 3. Mult. 21- 4 2 7 bj 43 G 12 V/6 29014 3 pro. by If 4 4 product by 24 1162 ID pro. by 4$ COMPOUND MULTIPLICATION. 17 Case 3. Page 34, d. h. mi. sec. Mult. 4 12 5 by 29 * 4 32 16 48 20 product by 4 7 223 21 S3 20 product by 28 8 4 12 5 " 1 days 257 1 50 25 product by 29 Case 4. Page 35. . s. d. 3. 1 2 3x7 10 11 2 oxl 10 72 10 ' HI 5 Oval, of 100 2 1126 :< < 10 7 15 9 ' < 7 145 value of 200 29 4 of 40 '130 3 S val. of 117 174 value of 240 . s. d. 4. 1 2 6x5 10 11 5 0X7 10 112 10 2 225 value of 200 78 15 4 of 70 5 12 G * of 5 309 7 6 value of 275 9* 13 COMPOUND DIVISION. COMPOUND DIVISION. Case 1. PageJS. . 8. d. ' T. cwt.gr. Ib. 2. Ans. 187 18 5| S. Ans. 15 GOT yds. ft. in f T. hhd. gal. qt. 4. Ans. *192 4} 5. An? 58 2 13 S{ w. d. h. mi. sec. 6. Ans. 3 16 16 53 Case 2. . 9. d. . 8. d. 2. 11J134 18 8 3. 12J984 4) 12 54 quot bj 11 12) 82 bj 12 3 1 4 by 44 ^6 1C 8bjl44 . 8. d. 4. 12;474 6; 39 10 quotient by 1 M 11 8 by 7 Case 3. Page 37. . s. d. . s. d. . s. d. 2, 34c)iO'^ 13 9 ( 1 a 9 3. 232 4 -- -,* -*. d. 64 524)4644(8 lCi] :>0 4192 S45)r23P<35. * 452 J2 524)5433(104. 524 34D):;105(9J, 193 3105 4 248 rem, COMPOUND DIVISION, ETC. 19 Page 37o &. 9. d. . 8. d. . i. d. . $. d. 4. 654)323(5 12 4J( 4 18 llf 5. 68)132 8(1 18 10 2616 63 620 64 20 20 054)13412(18*. 68)1280(18*. 654 6tf 5872 600 5232 544 640 56 12 12 654)7684(1 1& 68)680(10 ^u 23 179 SCO 2 20 447 S58 89'5 895 8. rf. 7X5 10 9. Mult, by bushels 1COO 10 cts. 5 .10x6 10 10000 500 18 4 SI05,00 8 15 in 200 days 1 15 * GO b 211 * 5 * 12 11 in 365 daj3 s. d. lit. cl c. (?. 11. s;47 12 10 J 155 XS 05 =276 75 sold for 135x1 6^=^19 S7.ipnmecost ^15 17 7i "S57 372 gain ^ 1 15 Si COMPOUND DIVISION, ETC. Page 38. dolls. ^ r - 1 9 9708 13. 17 8 2 9 5Y19416 716 wt. of 9 dolls. a *^ 3883 20 eldest son. oz. 39 wt of 45 dolls. 3)5824 80 rem. S 1941 60 other sons, each oz. dwt. gr. wt- lb - 4. 84 7 20 i 15 - 2 5 =2! 20 Mult, by 13^113 dwt.gr.gr. I50)l68r ( 11 6 + 3 over 150 11 6 standard 187 0+8 grains over 150 37 24 148 76 150)908(6 900 8 grains 16, 35X20 dolls. notes=700 dolls. 63 Eagles =630 284 dollars =284 642 half dollars =321 368 qr. dollars = 92 256xl2| cents = 32 Deposited Doll?. 2059 . Checks 560+820== 1380 Dolls, 6f9 COMPOUND DIVISION,' ETC. Page 38. dolls, eta. dolls, cti. 17. S6 yds. at 4 66=167,76 cost Add -29.56 gained Must sell all for Siyr,32 dolls, cts. d e. ' 4 yds. at 2 33 9,32 8yds. at 5 50= 44,00 12 yds. is sold for g53 3 32 dolls, efj. Now, From 56 yds. which must sell for 197 33 Take 12 yds. which brought 53 32 The diff. 24 yds. must sell for g!44 00 . doll*. But 24) 144 (6 dollars. cts. 18. 12^x5 = 62| beds. 62A x4=2 50 supper and* breakfast 75 for liquor 25 X5=l 25 for hay S5 12:}- From 6 dollars take S5,12, and the remainder is 87s cts. =875 mills , But 2:1=25)875(35 qts.=8j galls. And 6 dollars -7-5 travellers S 1,20=120 cents. Page 39. /*. m. h. m. minute* 19. 12 25xby5^ 62 5=3725 11 SOxby 9 = 103 30=6210 * Sum 9935mimites Now T minutes * Mult, by - 75 cents per day _ 49675 69545 h. mi. dolls, cts. MS. 8=480 ) 745125 ( 15 52 S+ REDUCTION. Page 39. gal qt. pt. 0. 5)1534 1 1 5)306 3 1 t galls. 61 1 1 PS A 1 B 2 C 6 ii. mi* 21. 9)114 45 127* 45rom, dolls. 9)180 20 A's share 20X2= 40 B's 40X3=120 C's DEDUCTION. MONEY. Page 41. 4)120506 farthings 12) 30126 \ S|0) 251JO 6| 6. 10 ^ *. d. 480 19 9 20 9619 12 Page 5. 10)260 centg 26 subtract 234 pence 7. 12)4658 2|0) 38|8 2 ^19 8 S 9) 115437 pence ^ add 1282631 cents 720 cents ^==72 subtract 648 pence 9)648 pence 72 add 720 cents 10. 2)24235 half pewt 12)1 211? J *4 REDUCTION. Page 42. 11. 216 French crowns Mult. 99 pence in a crown 3944 1944 12)2 1384 pence 2]0) 178J2 89/. 25. 13. 3752. 8 3)3000 $1000 12. . 8. 29 If 20 597 12 7164 = 796 add 7960 cents Or, 79 dolls. 60 cents 1. I TROY WEIGHT. 4) 11 5200 grains 6)28800 2|0) 480JO ( pennyweights 12/240 ounces Ib. 20 2. SO pounds 12 3. |0)4564|8 dwts 282oz. f 4)27260 grains \ 6) 6815 S|0)11S|5 20 12) 56 15 20 4lb. &oz. ISdwt. 20$r. 860 0&. 20 7200 dwfr. 24 172800 gr. Ib. oz. dwt. gr. 4. 4 3 15 12 56 ounces 20 1135dwts 24 4540 2272 S7260 graini. REDUCTION. Pas-e 42. o. dwt. gr. 8 6 24 198 gr. 24 spoons 1. 396 4752 grains AVOIRDUPOIS WEIGHT. 3 tons 20 60 cwt. 4 240 qr. 28 1920 480 6720 Ib. Tons 3. 5 20 100 4 400 gr. 28 11200 Ib. 16 179200 oz, 16 00 drams 2. f 4)2887200 (Jr, * (.4) 716SOO 16 1 79200 oz. (.4) 44800 23) 11200 Ib. 4) 400 qr. 2|0) 10|0 cwt. 5 tons. civt. qr. 4. 1 3 4 7 28 ,195 Ibs. Mult. 6 barrels 1176 Ibs. 2867200 dr. REDUCTION. Page 42. cwt. or. Ib. Ib. 5. 16 2 14 6. 8)2876 4) 102 20 66 28 25ctttf.2gr.20fl 532 133 1862lb. APOTHECARIES' WEIGHT. Ib. 1. 15 12 180oz. 8 1440 dr. 3 4320 scr. Page 43. Ib. . Ib. 2. 3 3. 2 4. 2!0)57600|0grai 12 < 12 3)28800 sc. 36 oz.- 8 283 dr. ou 8 8) 9600 dr. 16)240 drams 15 parcels 12)1200 oz. 100 Ib. 17280 gr. REDUCTION. 7 1. 50 yards 4 CLOTH MEASURE. Page 43. 2. 4)8642 nails 5)2160 2 nails 1000 qrs. 4 4000 nails. 524 Ells Fr. 6 4) 1944 qrs. 486 yards 260 miles 8 2080 40 83200 5J 416000 41600 457600 yds. 3 432 Ells E.Sna. 16 bales S6E.FI. 96 48 576 E. Fl. S 4)1728 qrs. 432 yards. LONG MEASURE. mi. fu. P. yds, ft. 2. 11 7 38 2 2 5 40 3838 5* 1372800 feet 12 16473600 inches 19192 1919 21111 3 63335 12 760020 2280060 b..c. 28 UCTION. Page 43. /,. fu. yds. ft. i C. 3)1267200 feet 4. 3 2 'no l i L 3 miles -:00 yards g 8 fur. s; 190 furlongs - - 74 GO; 240 miles 1493 4 dc^. 149 . lo 3 49171 12 inches SCO degrees round GO miles 21600 8 furlongs 172300 20 yards 38016000 Sfect 11404SOOO 12 inches 1368576000 LAND 3IEASURE. 25 acres 2. 4|0^ 1 7600(0 perch 4) 4400 roods 100 40 1100 acres perches DEDUCTION. Page 43. 4JOOJ6400|00 perches 4|0,) 160(0 each share 4) 40 roods 10 acres 10 acres 160 1600 perches 48000 400 48400 yards 9 435600 feet 144 62726400 inches CUBIC, OR SOLID MEASURE. 1. ft. feet. 128; 3200 ( 25 cords Page 44. ton. ft. feet. 2. 20X50^=1000 w. d. h. m. 8 2 6 20 7 58 24 238 116 1398 60 3. TOfE. ton. ft. cu. in cu. in. 30X40X1728=5=2073600 da. h. 365 6 in one year 24 1466 730 8766 60 83900 min. Page 45. 525960 60 31557600 sec. 10 years 315576000 yrs. days days hrs. 3. 1823X3651=665850 18 w. da. hr. mi. sec. 4. 1X7X24X60X60=604800 sec. 3* DECIMAL LIQUID MEASURE, Page 45. 4 tuns pints* 3 6 hhds. 8 pints S;4032 C3j 504 galls. 8 hhds. 38x63x8=19152 pints DRY MEASURE. t. 2, 315 8 2537 2 i>int3 ADDITION OF DECIMALS. 2)2196 8) 1093 4) 137 2 34 bu. Ipc.Sqt. 2. Page 47. Ans. 2923400,3687073 5. 283.604 4. ,24& 490006,003275 ,012 21,05 ,02 1,2 ,6 6200,3476 ,41$ ,5 Sum 4965 12,204875 Sum 1,791 6. 125,5 7. 5,4 220. 1000,000005 15,04 ,003S 15,072 100,004 844, 2,01 6000,00004 2,2 93680,0004 300. Sura 10142,582005 !oo Sum 100000,44444 Sum 1397, DECIMAL FRACTIONS. 31 SUBTRACTION OF DECIMALS. Page 47. 2. Ans. 685,49563.2 3. Ans. 8,3047 4. From 45,005 Take 23,65482 5. From 629,2 Take 200.002 Diff. 21,35018 Diff. 4#J,198 Page 48. 6. From 5, Take ,10438 7. From 2, Take ,00002 8. From 16, ' Take ,016 4,89562 1,99998 15,984 MULTIPLICATION OF DECIMALS. Mult. ,385746 by ,00463 1157238 2314476 1542934 ,00178600398 Mult. 158,694 by 23.15 793470 158694 476082 317383 367376610 6. Mult. ,024653 ,00022 ~49306 49306 7. Mult. 25,04 ,002 ,05008 ,00000542366 8. Mult. 645,003 ,000005 ,003225015 Contraction in Multiplication of Decimals. Page 50. 3. 23,463 multiplicand 43,2 multiplier reversed 46926 7039 938 54,903 5. 3,141592 multiplicand 8347,25 mult. rev. 1570796 62832 21991 1257 94 25 4. 234.216 multiplic. 543,2 mult. lev. 46843 7026 937 117 549,23 165,6995 $2 DECIMAL FRACTIONS. DIVISION OF DECIMALS. Page 50. 3. 23,7)65531,0(2756,16 4. 64,25)234,70525(3,653 m 19275 1792 41955 1659 38550 1331 34052 1J>:> 32125 1460 19275 14-22 19275 380 5. 3)10, 1422 8 rem. 6. ,9)9,0 7. ,00463)5001 78600398(,385746 1389 10 5970 3704 2660 2315 3453 3241 2129 1852 2778 2778 Page 51, 8. 2,46),2327898(,09463 9. ,09463),2327898(2,46 2214 18926 984 m 1476 56778 738 56778 738 DECIMAL FRACTIONS,. S3 .Page 51. 10. ,01S),000162(,009 162 Contraction in Division of Decimals. Page 52. 4, 1,346767)74,33373(55,193 5. 9 ; 3G^4C7}S7,07G326(9,297 67339 6994 2767 734 .! 260 914 135 843 125 71 121 65 4 6 4 6. 2,45)32,68744231(13,34 245 818 735 10 10 6,24),0046S72345(.00075 437 31 31 REDUCTION OF DECIMALS. Case 1. Page 53. * 2)1,0(,5 3. 4)3,00(,75 DECIMAL FRACTIONS. Page 53, 4. 8)7,000(,875 5. 25)1,00(,04 6. 60)5r,00(,95 r . 15)6,00(,40 cents Case 2, s. . d 9 2. 19-7-20=95 3. s'-M2=,25 < y ^. cwt.gr. or. 3-f-(12x20)= 5 12 5 5. 4 S=18--(20x4)=225to gr. lo . . . . 6. 14=70-f-(28x4)=,625cwt. ^r. ?2. wa. 7. 3 3=15-5-(4x4)= 5 9375jds. Or thm, 5. qr. A I 2, 6. Ib. 28 [ 14 7. net. 4 \ S cu;/. 20 I 4,5 4 I 2,5 ^r. 4 | 3,75 ,235 ton. cwt ,625 yds. ,9375 Case 3. Page 54. ^ ft. 2. ,75 20 shillings 15,00 ounces 8,4 -- 20 15. __ dwts. 8,0 8oz. Sdwts. *' ' 61 Jcwt 5, ,3375 acres _ 4 qr. 2,468 - rood. 1,3500 ^lb. 1^104 _ ^ _ per. 14,0000 OZ. j 1 rood 14 per. dr. 10,624 2qr.131b.lcz.10dr, DECIMAL FRACTIONS. Page 54, 6. tun. ,258 4 days. 7. ,761 24 hhd. 1,032 63 hrs. 18,264 60 gal. 2,016 mi. 15,840 60 Ihhd. 2 gal. sec. 50,400 18 hr. 15 mi. 50,4 sec. Ib. 9. 365,25 days in a year 8. ,7 ,3 12 days 109,575 oz. 8,4 24 20 hrs. 13,800 dwt. 8,0 60 8 oz. 8 dwt. min. 48,000 109 d. 13 h. 48m. day hr. hours T. cwf. qr. Ib. 10. ,41X24=9,84 11. ,17 ,19 ,17 ,7 ,16 20 hrs. 9,68 cwt. 3,59 60 4 min. 40,80 2,53 60 28 sec. 48,00 15,54 ' 9 h. 40 m. 48 sec. Scwt. 2qr. 15,541b. 36 DECIMAL FRACTIONS. Promiscuous Questions in Decimal Fractions Page 55. 2, ,36 ton 20 7^0 cwt. 4 28 quest. 1. Mult. ,09 by ,009 Prod. ,00081 3, ,9125 ounces 20 dwt. 18;Ix>0 24 gr. 6,OOJJ 18 dwt. 6gr. 4. S15)4,00(,0127 nearly oz. diet. gr. 5. 2 16 20=1364 grains And 1 pound=5760 grains Then, 5760)1S64,0(,236S 23040 5760 80ii,40 lb. 16 miles ' 6. ,1392 8 fur. 1,1138 40 per. 4,5440 27200 2720 yds. 2,9920 1 fur. 4 per. 3yds. 11. 1 doll.=100 cts. 3 15)300 6) 20 6 cents 7. 4;3,00f,75 8. 112;6,00(, 0535714 9. 365;i09,5(-,3 10. ,04X50X1 728 -=5456 12. hhds. ,875 63 2525 5250 gall. 55,125 4 ql. ,500 _2 pt, 1,000 55 gal. 1 pint SINGLE RULE OF THREE DIRECT. 37 Page 55. 14. 365,25 days in a year 15. ,73-i-(3x 5 25) that is >to ,75),73(,973 18,2625 24 yr. w. d. h. mi. 16. ,05=2 2 19 12 730500 j )V 'J^rOKA ILI * ,5=0 30 438,3000 hr. 60 2t0.2d ISA. 42m. * 26298,0 mi. A. R. P. 60 18. ,6X4X40X, 02=1,92 1577880 sec. T. h. gal. 17. , 4 ,3 ,8 cub. in. ft. in. 4 19. l-;-(i28x 1728) that is 1 ,9 221 184) l,OOOGGO(,000004-h 63 120,5 gall. 4 20. (28 deg. 48 mm.)-=-360 deg. 482,0 qt. ^at * s > o mi. mi. %Ipints 21600)1728,00(,08 SINGLE RULE OF THREE DIRECT. Page 57. lb. d. C. lb. T Quest. 2. As 112 : 12,32 :: 16 : - /6. cts. lb. cwt. 3. As 1 : 36 :: 336 (=3) : 120 dolls. 96 cts. yds. yds. yds. yds. yds. 4. 23 -f 24 + 25+27=99 then yd. cts. yds. As 1 : 72 :: '99 : 99X72=71 dolls. 28 cts. * This answer is obtained by reckoning 12 months to the year, 4 weeks to the moath, 7 days to a week, &c. But at 52 weeks to the year, the answer will be 2w. 4d. 4h. 18rai. And at 365 days, it will be 2w. 4d. 5h. 48mi. the true Answer. S3 SINGLE RULE OF THREE DIRECT. Page 57. Ib. cts. Ib. 5. As 4 : 48 :: 512 (4 cwt. 2 qr. 8 Ib.) : 61 dolls. 44 cts. Ib. cts. Ib. 6. As 1 : 8 : : 128 : 10 dolls. 24 cts. pair d. c. pair 7. As 114 (=9i doz.) : 68,40 :: 3 : 1 doll. 80 cts. bu. d. c. bn. 8. As 20 : 9,60 :; 3 : 1 doll. 44 cts. cts. yd. d. cts. 9. As 75 : 1 :: 16,50 ; 22 yards. c. qr. Ib. d. cts. oz. cts. 10. As 32080 (=17 3 17) : 320,80 :: 6 : 6 Page 58. oz. Ib. dol. oz. QTv/t 5 d. c. 11. As 116,4 (=9,7) : 97 :: 1,5 : ~=l 25 acres dolls. acres 12. As 125,5 : 627,5 . 1 : 5 :: 4,75 : 23 doll. 75 cents. gal. dolls, cts. gal. 13. As 1,5 : 4 50 :: 378 (=1,5 tuns) : 1134 dolls. d. cts. d. cts. d. cts. d. cts. 14. First 1 66 + 1 97-f2 31=5 94 the price of I ream of each sort. Then say d. CtS. of each sort. rf. cts. As 5 94 ; 1 :: 528 66 : 89 reams of each sort. Ib. T. d. Ib. qr. Ib. 15. As 2240 (=1) : 224 :; 42 (=1 14) : 4 dolls 20 cts. d. c. fibl. d. c. 16. As 5 50 : 1 :; 1402,50 : 255 barrels da. d. cts. da. 17. As 365 : 1186,25 :: 1 : 3 dolls. 25 cents. SINGLE RULE OF THREE DIRECT. Pag! 58. da. d. cts. da. 18. As 1 : 2 25 :: 365 : 821 dolls. 25 cents, the um he spends in a year. Now, 821 dol. 25 cts.+S78 dol. 75 cts.=1200 dolls. T. cwt. qr. Ib. Ib. 19. 4 10 1 12=10120 Ib. cts. Ib. Then, as 112 : 1,12 :: 10120 : 101 dolls. 20 cents 20. 4ft. 6 in. = 54 inches $ of 54=27 add 81 27 2187 19683 solid inches eu. in. cts. cu. in. Then, as 1728 : 110 :: 19683 : 12 dolls. 53 cts. nearly in. in. in. in, 21. (28+14) X14X3,5=205S cubic inches cu. in. cts. cu. in. d. cts. m. Then, as 1728 : 190 :: 2058 : 2 26 2>+ . s. s. T. cwt. qr. Ib. Ib. 22. Oneton=2240lb. 22 8=448 & 203 9 3 3=455815 Ib. s. Ib. s. Now, as 2240 : 448 v 5 : 1 : : 455815 : 9116S-=4558 3s. d. cts. yds. d. cts. yds. 23. As 11 25 : 5 that is, as 225 : 1 : : 850 50 : 378 in all And, as 18 pieces : 378yds. :: 1 piece : 21 yards hf. yds. yds. d. . s. d. hf. yds. d. s. 24. As 25(=12i) : 450 (= 1 17 6) :: 2 : 36=3 ft. ft. ft. in. 25. As 7 : 4 :: 218 9 : 125 feet 40 SINGLE RULE OF TRHEE DIRECT, A. R.P. 6. 476 5 28=76308 perches. Then say P. d. c. P. As 76308 : 4292 S2i :: 160 : 9 dollars da. cts. da. d. cts. 7. As 1 : 214 :; S65 : 781 10 spends dolls, cts. Then, from 1333 00 annual income take 781 10 yearly expense S551 90 he will save Page 59. bu. d. c. bu. 8. As 321 : 240,75 :: 1 .- 75 cents na. yds. cts. na. qr. na. 29. As 24(=U) : 250 :: 6(==1 2) ; 62| cents gal. gal. gal. 50. 120i + 124+126|=37U gallons gal. d. s. d. gal. pence . s. d. As 1 : 66(=5 6) :: 371$ : 24502^=102 1 10i 7ui. da. mi. 31. 12 x 5 =60 the distance that A has gone before B starts mi. mi. 16 12=4 miles B gains on A per day mi. da. mi. Then, as 4 : 1 :; 60 ; 15 days ,. d. s. d. . pence . 32. As 1 : 150(=12 6) ;; 1000 : 150000=625 men bbls. men bbls. bbls. S3. As 365 : 75 :: 500 : 54. This is. properly a question belonging to the rule of Three Inverse, stated thus, cts. 3. cts. As 575 : 360 :: 250 then, S75X5GO ff/f . -- =540 acres, 250 cts. SIKGLE RULE OF THREE INVERSE.' 41 Page 59. mi. hrs. mi. deg. mi. mi. mi. fur. S5, As 1440(=24) : 25020(=360x69i) :: 1 : 17 3 SINGLE RULE OF THREE INVERSE, Page 60. men days men men days men Quest. 2, As 60 : 100 :: 20 3. As 65 : 4 :: 5 60 4 < 2lO)GOO|0 5)260 300 days 52 days per. days per. fiy04 4. As 6 : 24 :: 9 (=6+3) : :i =16 days cwt. miles cwt. 1 y 1 *)() 5. As 1 : 150 :: 6 : - =25 miles o ft. wide yd?, long ft. wide QA V ^Ofl 6. As 80 : 300 :: 60 : : r. 1. r. w. r.l fiOv^n fiv^ft 7. As 80 : 30 :: 70 : ? ^p~34r. 4ft. 8|m. foot feet feet 1 f}(\ 8. As 1 : 12 :: ,75 :~ ~ =16 feet 3 7o _ydi. yds. yds. T1v40 1 9. As ,75 : 42,5 :: 1,25 : 'I^pff.,25,5 yards ^0 men months men Iflvd ^ 10. As 10 : 4,5 :: 15 (=10+5) : ' .=3 months dolla. yrs. dolls. fin v 1 lobugh C 60 horses days 203 36 days Then _io X 3x2==60 bushels. Or thus, hories days bu. As 18 : 20 :: 10 Then ;= 60 bushels. 60 : 36 18 *20 Or, by two statings, horses bu. horses As 18 : 10 :: 60 : S3| bushels. days bu. days bushels And, as 20 : 33] :: 36 : 60 as before. 48 DOUBLE RULE OF THREE. Page 66. Stated thus: Or thus, 3. Men 7? ^ 1K 5 21 mei1 men daw Ibs, days 14 5 '' Sdays 71456 21 S Then =4x3x3=36 pounds / x J. * Or by two stating*, men men Ib. Ib. As 7 : 21 :: 56 : 168, and days days Ib. As 14 : 3 :: 168 : 36 pounds. Stated thus, Or thus, 4. Students 81 dolls. f!2 stud. stu. mo. dolls, I S84 1 8 6384 i months 6 J (,10 mo. 1210 Then, 384 g 1 x S 6 Xl =48x2xlO=960 dollars. Or by two statin gs, stu. stu. dolls. dolls. As 8 : 12 :: 384 : 576, and mo. mo. dolls. As 6 : 10 :: 576 : 960 dollars. Stated thus, Or thus, 5. Cwt. 201 dolls. f40 cwt. cwt. mi. dolls. I 25 ] 20 5025 miles 50 J (.lOOmiles 40100 Then, 25X40XlQ -25XX2=100 dollars. 20X30 Or by two statin gs, cwt. cut. dolls, dolls. As 20 : 40 : : 25 : 50, and miles milds dolls. As 50 : 100 : : 50 : 100 dollars. Stated thus, Or thus, 6. Dolls. 7001 dolls, f 400 dolls. dolls., mo. dolls. L 14 \ 700 614 months 6J ^60 mo. 40060 The., =2x4xlO=80 dollars. DOUBLE RULE OF THREE. 49 Page 66. Or by two stntlngs, dolls, dolts. dulls, dolls. mo. wo. dolls. As 700 : 400 :: 14 : 3 and, As 6 : 60 :: 8 : 80 d. Statement, Or thus. 7. Men 41ro(lsfS men men days rods I 12 < 4 612 days 6 J (.24 days 8 24 Then, ^ -=2x8x6=96 rods. 4x6 Or by two stalings, men men rods rods da. days rods As 4 : 8 :: 12 : 24 and, As 6 : 24 :: 24 : 96 rods. Inverse Proportion. }") days f 1 rl Stated thus, Quest. 2. Men 4 inverse") days f 16 men inverse Dolls. 24 J 139* dollars. Or thus, men days dolls. 4 3 24 16 384 Then, 5X4x384 =3x4xl = 1 16x24 Or by two statings, dolls, dolls, days days men men days days As 24 : 384 :: 3 : 48 and, As 16 : 4 :: 48 : 12 & Dolls. 241 men f 93 dollars men days dolls. y 4 < 431.04 day 3 inverse J (/inverse 16 days. 16 96 ^. 4X96X3 4X4X3 Then, = - -=3 men. 24X16 16 Or by two stating?, dotts. dolls, men men days days mm *nen As 24 : 96 :: 4 : 16 and, As 16 : 3 :: 16 : 3 5 50 DOUBLE RULE OF THREE. Page 67. Stated thus, Or thus, 4. Acr. 841 men f 100 acres men days acra [ 7 < 712 84 inv. days 12 J (^5 days inverse 5 100 84X5 1X1 Or by two statings, acres acres men men days day$ men mm As 84 : 100 :: 7 : 8 and, As o : 12 :: 81 : 20 Stated thus, Or thus, 5. Men inv. 71 days f 20 men inv, men days atrn \ te \ 712 84 acres 84 J (^ 100 acres 100 12X7X100 1X1X5 Then, - = - =:> days 20x84 1X1 Or by two statings, crcs days acres days men men day* As 84 : 12 :: 100 : 14f and, As 20 : 7 :: 14f : 5 daji Stated thus, 6. Inverse 200 lb.~) miles"! 20200 Ib. inverse. I 40 L 40 cts. J J 6060 cents. Ibs. miles cts. '.-*-. Or thus, 20040 40 20200 06060 mi. Ib. cts. ^ 40X200X6060 CA ., Then, - - =60 miles. 20201b.x40c. Or by two statings, els. mi. cts. miles As 40 : 40 : : 6060 : 6060, and . Ib. Ib. miles As 20203 : 200 :: C060 : 60 miles. Stated thus, Or thus, 7. 82001 w.da. f 500 dolls. men w. d. dott*. 1 22 6<{ 522 6200 inv. men 5 J / 12 men inv, 12 - 300 Then, & 6rfq.)x300X5 jl4 weeks 2 d 200X12 DOUBLE RULE OF THREE. Page 67. Or by two statings, dolls, dolls, w. d. w. d. As 200 : SOO : : 22 6 : 34 2 and, men men w. d. w. d. As 12 : 5 :: 34 2 : 14 2 Promiscuous Questions. . Stated thus, Or thus, 12 oxen> acres C24 oxen o^e;z daysacret 8 days 5 10 48 days 12 810 2448 _, 10x24x48 1on Then, -^-=120 acres. Or by two statings, 0#. oa?. acres acres days days acres As 12 : 24 :: 10 : 20 and, As 8 : 48 :: 20 : 120 acres wt. wt. ivt. . 8000 4500=5500 and 9 days 6 days=S days 4500 cwt.> horses { 3500 cwt. inrcrse 6 days 3 18 (3 days inverse horses days cwt. Or thus, 18 -6 4500 3 35CO _, 18x3500x6 . Then, =28 horses. 4500X3 Or by two statings, twt. cwt. hor. hor. days days hor. hor. As 4500 : 3500 :: 18 : 14 and, As 3 : 6*:: 14 : 28 cwt. hhd. cwt. cwt. bbl. cwt. 3. 12x9=108, and 2,5x50=125. Stated thus, C wt. toi. dolls. 108 cwt. dolls. C 125 cwt. Or thus, 108-460100 0railes5 100 ^300 miles 125-300 Then 100xl5x50 108x60 > o uuiia* i i UT^Y uems Or by two statings, mil. mil. dolls. dolls. As 60 : 300 :: 100 : 500, and cwt. cwt. dolls. dolls, d*. As 108 : 125 : : 500 : 578 70-Jf 5 DOUBLE RULE OF THREB Page 67. yds. $r S o- 5 qrs. S 1 ? 4 qrs. 4540 4. 3 yds. lib. 545yds. O, s. S 1 " 3x5 By two statings, yds. yds. Ib. Ib. qrs. qrs. Ib. #. As 3 : 45 :: 1 : lj and, As 5 : 4 :: 15 : 12 Or thus, 5. 240 miles 7 days C720 miles mil. dayshr*. inverse 12 hours : 27 dajfc 6. 16,5 feet! doll. cts. fSO feet 1,5 feet L 1 25 < ^ feet 1 footj (_ 4.5 feet 1.25X30X26X4.5 d !K c ' s - . The "' I6,oxl,oxi =1 -^Tr 7. S feetl f SCO feet lor^ 4 feet ll corcN 10 feet high 4 feetj [36 feet broad Then, ^OOX10X36 =5G , cord3 , ? X4X4 8. 10 : yards ^lb. ^ 100 yards 6 qrs. 3 3 ^ 3 qrs. ,_. 3X100X3 _ rhen^ =15 pound. Page 63. 9- Inv. 24 men ^ f 6 re en iir !'>ns; (dnysj - ; '' liigh f SU ^ 6 feet i G ft. thick J L 4 ^ e ' thick iien, 80x24x20x6x4_ 4 PUACT1CE. 5$. \ . Pago G8. \ 9 persons) doll*. C. \4 (=9+5) persons 5 months 5 450 ( 8 months 450X14X8 Then - 11. 8 persons! doUx. f!2 (=S-f4) persons 1 month > 11] ^ 2,5 fjr^atly Hbridjjed. Thus, in the above example. T oh-erve tlmt 4^,5 goes into 45 twice, without a remainder; that 17, :i is oo>ita!n:jd twice, i 34,6; and that SX8,2 goes once into 6X 1*2,3. Now, by dividing the numerator aud " . . ; denominator by these divisors, we obtain -- ... , ;r --- as above 54 3. 201 yds, 4,20 ^ 4020 804 8844,20 4. 2210 yards X 1,10 cents 82431,00 5. 2,415 X 421 yards, 2415 4830 9660 1016,715 FRACTION Pago 6& Or, 201 yds. 4 804 cts. is|= 40,2 cts. Or, 10 Or, cts. 20 20 10 ms. 5 Or, 101 6 dolls, nets. 5ms. 6. 625 25 3125 1250 8156^5 >t 7. 8275 8844,20 2210 yards 221 2431 dollars 421 2 842 84,2 84,2 4,21 2,105 1016,715 Or thus, cts. 25 is | 625 8156,25 ms 33100 33100 36410,0 8. 8275 5 Or, 364 ddls, 10 cts. 41375 mills. A ,., Or, 41 dolls, 37 cts. 5 ms. PRACTICE* Case 2. Page C& cwt. qr. Ib. Ib. , 4 I 14=490 1 = 122,5 Or, 122 dolls. 50 cts. civt. qr. Ib. Ib. cwt. qr. Ib. Ib. S, 12 2 13 = 1413 4. 14 2 7=1631 X 2 X 7 Ib. 3)2826 5x8=40) 11417 S 942 8285,4*5 Application. Page 70. cwt. Ib. hhds. Quest. 1. 12,5X112X6X^=3150 dollars. 2. 60 ton 3. 12, 650 feet X 3 X 10 5)180 126,500 $isj = 6.325 tonEag. 36 fisj= 3,1625 60X2= 120 |is^ 1,58125 156 Eagles 137,56875 Or, 1560 dolls. Or, 137 dolls. 56 cts. 8| ma. Case 3. dolls, cts. dolls, cts, 2. 10 94 3. 13 41 Mult, by 17 cwt. Mult, by 5 cwt. 7658 67,05 1094 1 qr. is i=3,352| f qr, is one hal f 547 1 is one half g735 70,402J 14 Ib. is one half 13675 4 is one seventh 3907 70 dolls. 40 cts. 2J ms, I is one fourth 976 $19(5,0408 Or, 196 d. 4 cts. PRACTICE. Page 70. dolls, cts. m. 105,035 16 If), is 4= 2. 14357+ lh. is = ,oG794 + 1 Ib. is - = ,13597 + 107,58048 + 107 dolls. 58 cts.&c. Case 4. Page 71. s. d. s. d Or, 4f 3 yards 6 8 is I ) 4*3 vds. at 6 8 88| cts. 5784 5784 s. d. s. d. 420| S; S 4 is J) 397yds. at 3 4 4 Or, 597 yards 1588 1588 Or, 159,25 4. 1591/6. at 15. 8r/. 1850 ) 159 5 value at l. 31850 - f= 3538| 5 5 S35,3888| Or, S35 S8| 67 10*. is i Page 71. 658 at IS*. JE394 16 Or, 658/Z>. 39480 658 81052,80 12* 6. 745 at 16*. 55. is 5 372 10 l5.i*-J | 186 5 37 5 j96 Q Or, 745 yards S 1589,331 7. Or, 105. is i 069 at 195. llrf. 969 5 I 4 ' i 8d.' i 964 19 3 ^tO'-t 1U 242 5 4845 193 16 5814 32 6 1938 8 1 6 = 5S8J 4r\ n u y R9i7S 9..-;* 55. i 3715 at 95. 4iJ. Or, 3715 4 T P28 15 4d- T\ 743 18575 Gl IS 4 7^50 7 14 91 3715 . 4lc 1741 8 84643,73 rRACTICB. Page 71. 9. 4567 at 195. ll|e/. Or, 4567 lOt. j _ 2664 5 | 2283 10 4 i 1141 15 27402 Sd. i 913 8 7402 2 | 152 4 8 9134 1 i 1 i 38 1 2 10 7 *== 507$ 9 10 5| 812153,29! 19*. 11 Id 4557 9 Cas e 2. Page 73 . . 5. 1, 00 gave 8261,00 paid cwt. qr. Ib. 5. 4 2 26 12 56 3 4 gross 450 Qqr. is 5 4,50 Sib. is T ^ 4 06+ 14 6 | 1,12J 4 2 3+ 7 4 | 56| 6 9 tare ' 8456,341 Cwt 50 2 23 neat cwt. Ib. cwt. qr. Ib. 6. 6=672 gross 7. 120 3 = 13524 gross 100 tare 177 tare 572 13347 jV 5538 22 tret 5^= 513,346 tret Ib. 550 neat 12833,654 ,73 cents Ibs. cts. Ib. CO 5f) cts - As 550 : 8250 :: 1 : ~ =15 38500962 89835578 9368,56742 Or, 9368 dolls. 56 cts. 7 ms. 6 62 SIMPLE INTEREST Page 74. cwt. qr. ft. 8. 6 12 16 dollars 9 50 cwt 59 1 2 4 gross 800 16ft. is *~ 1 27 \qr. is 4 14ft. 4 | 4 2 1 TV 2 3 7 * 5 1 i ' j- 0,14f 9 2 tare < 807.142. Cwt. 50 1 22 neat /6. cict.qr.lbs bags Ib. 9. 325(=2 3 17) X27=8775 gross 877aft.X T Vf =1018 tare 7757 7757-7-26= 298 tret 7459 lb.=66ctf . 2^r. 1 1ft*. Ibs. ct. dolls, cts. And 7459X32=2386,88 SIMPLE INTEREST. Case 1. Page 75. dolls, dolls, yr. cts. 4. 200x6x2=2400=24 dollars Page 76. doUs. dolls, yr. cts. dolls, cts. 5. 1260X7X4=35280=352,80 dolls. dolls, yrs. cts. dotts. cts. 6. 560X8X3=13440=134,40 d. cts. d. d. cts. Now 1S4,40 + 560=*694,40 d. di yrs. cts. cts. 7. 4520X5X6=135600=1356,00 d. cts. d. And 1356,00+4520=5876 dollar! SIMPLE INTEREST. 63 Page 76. dolls, d. yrs. cts. d. d. d. 8. 400X6^X2=5200=52 dolls. And 52+400=452 cts. ms. 5 dolls, dott& y-rs. cents d. cts. 9. 4925x7sX9=332437;|=3324 37 dolls, dolls. cts. dolls, cts. 10, 2500X71 = 19375=193 75 d. c. d, d. cts. And 193 75+2500=2693 75 Case 2. Page 77. dolls, c. ms. d. cts. ms. 4. 1260 50 5 Now 283 61 3 + 71 1260 50 5 8823535 1544 11 8 amount 6302525 cents 9453,7875 interest for 1 year 28361,3625 Or, 283 dolls. 61 cts. 3+ms. interest for three years dolls, cts. dollars cts. d. yrs. d. cts.ms. 5. 630 50 6. 3462 84 X3^x4=450 169.2 ink 8 1274646X73X3=286 79535 int. 50 4400 int. for 1 year 163 37385 cliff. 5 232 20 int. for 5 years 650 50 principal dolls, cts. ms. Or, 163 37 3,85 g882 70 amount 7. 3 ) 3422 25 dolls, cts. ms. 1140 75'3 T f 0=68 44 5 int. for 1 year 68 44 5 interest for one year 1209 19 5 first payment. 64 SIMPLE INTEREST* Question seven continued v dolls, cts. dolls, cts, d.c. m. yr. 1140 75 d. c. m. 1140 75 68,44,5X2=136 8 9 int. for 2 yrs. 68,44,5x3= 205 335 31277 64 2d payment 3d pay't 1346 085 d. c. ms Now, 1209 19 5 first payment 1277 64 second < 1346 08 5 third < $3832 92 whole amount Case 3. Page 7S. 4. 18 months Then, 724 dollars 4 per cent 6 12 ) 72 S43.44 interest 4,00 principal 6 rate for the time S767 44 amount 5. 15 months d.c. m. 7k per cent Then, 9 375 dollars 12)11-1 375000 9,375 rate for the time 18750 22,50000 Or, 22 dollars 50 cents. 6, 4 me dolls. Gi per cent. Then, 1260 12)26 [ rate for the time 210 30 cents. 'SIMPLE INTEREST. 65 Case 4. Page 80. dolls, 'days ( const v ! div.) d. cts. 5. (12000X40) ~- 52K = 92 06 nearlj dolls, days (c. d) d. c?s.. ms. 6, (8400X2p)-7-73CO=23 01 3 d. cts, days (c. d.) r: (517 90X8 l)-r-6083=7 dolls. 15 cts. 1m. d. cs. days (c. d.) 8. (73 41X27)-7-GO=33cts . cfc. days i'c. d ) d c. m. 9. (225 24X40)-r-{>083=l 48 1 dolls, days (c. d) d. c. m. 10. (1QO;X80)^6083=15 78 1 dollars cts, dc,ys (:. d.) d. cte. m. 11. (2962 19X254)~608S=123 68 8 d. cf5. dvjs (/; '') d c. m. 12. (1733 97 Xl05i) -7-6083=^9 07 5 Case 5. Page 81. i/?'5. d. d. d. 2. (12X6)4100-- -=172 Then, dolls. dolls, O^^OvlDO d" 118 - As 172 : 100 :: 2750 : -'-= yrs. d. d. d. 3, (5X6) + 1CO--130 Then, dolls. doll?. dolls. TOPvlfin dolls. As 130 : 100 :: 728 : -^=560 130 2/rs. d. d. d. 4. (4 XTijy- 100=130 Then, dolls. dolls. foil-.-. IhCOv/lOO dolls. As 130 : 100 :: 1638 : iL:l_=l260 JL oU i/r*. d. cs. d. d. c^. 5. (5X5 75)+ 100=: 128 75 Then, doll*, eta. dolls. ii->lb. OOOXlOO do ^ s< cta< As 128 75 : 100 :: 2000 : ^^^-- = 1553 S9 SIMPLE INTEREST. ^ Case 6. Page 82. clolls. dolls. dolls. ^0800 dolls. 2. 858 amount As 650 : 280 :: 100 : -=32 650 principal Then 52 d.~-4 yrs.=8 per cent S208 interest dolh. dolls. 115200 dolli, '52 araount As 1600 : 1152 :: 100 : 5 =72 1800 prin. SI 152 interest. Then 72 dolls, -r- 12 yrs.=6 dollars cts. dolls. dolls, cts. dolls. Ogn on dolls. 4. 800,80 amount As 640 : 220,80 :: 100 : - =34| 640 3 00prin. 8220,80 interest Then 34| d. -r-6 yrs.=5| per cent, dolls. dolls. dolls. d. gin doJk 5, 20100 amount As 12000 : 8100 :: 100 : =67! 12000 prin. S 8100 interest. Then 67 d,-^15 yrs.=4| per cent. Case 7. Page 83. dolls. dolls. 2. 650 910 amount 8 650 principal 852,00 int. for 1 year $260 whole interest dolls, year dolls. 2()0 Then, as 52 : 1 :: 260 : -=5 years (lotts. dolls. >. 1600 2080 amount 6 160Q principal &96,00 int. for 1 year S 480 whole interest dolls^ year dolls. 4gQ Then, as 95 : 1 :: 480 : =5 years SIMPLE INTEREST* Case 8. Page 83. ' 2. 640^. 3*. 6flf.=2845| dollars 5 8142,26-i- interest for 1 year 3 426, 78^ interest for 3 years 2845, 22f principal 83272,000f amount* . -s. cl 3. 1374 1 9 rate per cent. 5-f . s. d. 77 5 10 in'st. for 1 yn 38 12 11 < for | yr. I is* 6870 8 9 11518 9 int. for 1J yr, 687 10i 171 15 2| 77,29 4 10 20 5.5,84 12 . 10,18 Case 9. Page 85- & 1820, January. falls. 182V- April 1. 1 \ paid 700 Principal Mult, by 1800 dollars days 455 equal 819000 1822~\Jan. 1. 1100 j)aid 400 Mult. 275 equal 302500 700 Jul}A 1,- paid 500 Mult. 181 equal 102500 \ 200 \ 214 16 6 Am6uat\due $414 16 6 Mult. 184 equal 3680ft 6,0; 1285000 Interest $ 214,766 *The answer givcA with the question is obtained by recsoninj $4,44 to be a pound sterling, instead of $4,44 68 INSURANCE. COMPOUND INTEEEST. Page 87. tabular number dolls. d. cts. 3. 1,27628X1500 = 1914 42 tabular no. d. d. c. m. 4. 2,54035X4500=11431 57 5 amount subtract 4500 principal S6931 57 5 interest tabular no. d. d. cts. 5. 3,20713 X650=*2084 63 4,5 mills yrs. yrs.- 6. ', 21 4=17 years, on interest Then, tabular no. d. d. cts. 2,69277X8000=21542 16 amount d. cts. sons But 21542 16-7-3=7180 dollars 72 cents INSURANCE. Page 88. dolls. 2, dolls. doBs. 1650 3. 4500 15| per cent. 25 per cent 8250 22500 1650 90 825 ii0.*i.nn 8255,75 100 dollars 25 dollars=75 doMars, Then? dolls. dolls. dolls. . . . As 75 : 100 :: 4500 : = i 5 ** 100 dolls. 9 dolls. =91 doll?. Thetf k dolls, dolls. dolls. 560DO dolls ' cls \Ag 91 : 100 : 560: -^=615 38$f y JL 4. BROKAGE. COMMISSION, Page 88, dolls. 1260 6 per cent. 875,60 dolls. 26342 3 per cent, 790,26 Page 89. 100 dollars. As 103 : 100 :: 4120 dolls. 2550 4 per cent 5. 8102,00 dolls. 6422 3 per cent. 4)19266 41^000 ~' =4(100 dolls, dolls, hir dollars 4000 Then, as 8 j 1 : : 4000 : --- =500 barrels d. cts. .50 3} percent 4876,50 541,83| 5418,33 Or, 54 dolls. 18] cts. 3. dolls. 1868 \ per cent 3736 934 846,70 4. 560 dollars 6 per cent 833,60 cents TO REBATE OR DISCOUNT. BUYING AND SELLING STOCKS Page 90. dolls. 2. 1500 110 per dolls. Or, 1500 cent. -^0 = 15 dolls. 3. 1686 X 128 per cent. 81650,00 dolls. 4. 25000 108 S1650 per cent. 13488 3372 1686 82158,08 827000 d. d. d. 5. 1260 6. 9254 7. 1518 90 per cent. 84 per cent. S3? per cent 81124 S7773,36 REBATE OR DISCOUNT. Page 91. mo. d. mo. d. 2. As 12 : 8 :: 18 : 12 int. for the time Then, d. d. d. d. 2464x100 *L- As 112 : 100 :: 2464 : 2200 Because ^^ =2200 mo. d. mo. d. S. As 12 :-5i :: 8 : 3 f interest for the time d. d cts. d. cts. Then, aslOSf : 100 :: 1857 50 : 1791 80/ T T 1 85750X100X3 ^ cts ' For - rr - =1/91 4. (7X2) + 100=114 dollars doiu. doiis. doiis. 650X14 *V. Then, as 114 : 14 :: 650 : =79 82|4 REBATE OR DISCOUNT. 71 Page 9J. mo. mo. d. d. 5. As 12 : 4f :: 8 : 3= a rate for the time dolls. dolls. dolls. 5150X100 dN 9 - Then, as 103 : 100 :: 5150 : - - =5000 p. w. lUo> Again, dolls. dolls. As 100 : 99 (=1001) :: 5000 : 4950 Page 92. Note 1st. mo. mo. dolls, dolls. 3. As 12 : 18 :: 8 : 12 rate for the time dolls, dolls. dolls. dolls. Then, as 112 : 12 :: 7280 : 780 the discount dolls, mo. cents Again, 7280x9=65520 interest at 6 per cent. i =21840 interest at 2 < * $873,60 interest at 8 per cent, Subtract g780,00 discount Difference 8 93.60 Note 2d, dolls. dolls. %. 1650X3=4950=-49 dollars 50 cents. 3. 2464 dollarsx8=~19712 cents discount But 2464 dollars 19712 cents=2266 dollars 88 cents BANK DISCOUNT. + Page 93. days dolls. 2. =1 dollar 41-f cents 60 Or, 17X25 =1 dollar 4 If cents days dolls. 3. ( 90+ ? X600 = 9 dollars 40 cents ou Or, 47da y sx600dolls =9 dollars 40 cent. 30 72 SINGLE FELLOWSHIP. Page 93. days dolls, cis. (604-4) XI 260 40 f _ , 4. v - 1 i- - = 13 dollars 44|| cents. uu ~ 32X1260 40 Or, - - -- =13 dollars 44ff cents. days dolls, cts. (60-f4)x2649 75 rt0 , :, ... 5. i -- - =28 dollars 26 cents 4 mills 60 Or, =28 dollars 26 cents 4 mills bO EQUATION OF PAYMENTS. Page 94. dolls, mo. do/Zs. mo. 2. 500X2= 1000 3. OOX 5=1000 1000x5= 5000 SOOX 8=2400 1500X8=12000 500X10=5000 - - - -- mo. da. 3000 ) 18000 (6 mo. 1000 ) 8400(812 18000 4. 1400 dollars 1000 dollars=400 dollars d d. mo. Then, as 400 : 1000 :: 1 : 2J months SINGLE FELLOWSHIP. Page 95. d. d. d. 2. A 7500+B 6000+C 4500=18000, Then, dolls. dolls. dolls, dolls. f7500 : 2250 A*s gain As 18000 : 5400 . 10 : 3 : : < 6000 (_4500 d. cts. d. cts. 5. A 1291 23+B500 37+C 709 40+228=2729 d. d. cts, d. d. Now, as 2729 : 2046 75 :: JOO : 75 d. cts. d. cts. : 963 Then, as 100 : 75 :: \ 500 37 : 375 Or, 4:3 :: ^ 709 40 : 532 05 171 dolls. : 2250 A*s gain "] : 1800 B's 4 I : 1350- C's * J d. cts. ( 1291 23 : : 75 :: \ 500 37 : : 3 : : \ 709 40 : / 228 00 : SINGLE FELLOWSHIP. 73 Page 95. dolls, J. R. P. dolls. A. R. P. f ISO : 175 2 34j 1 -4L As 600 : 585 2 34 :: < 195 : 190 1 17^ > (.225 : 19 2 22| J Page 96. barrel*. IU. bbl. f 960 : 4001 5. As 2160 : 900 v 12 5 : : < 720 : 300 [ ^480 : 200 J dolls. d. d. dolls, d. d. 6, A 1260+B 840=2100, And 825275=550 d. d. d- d. C 1260 : 330 A'se;aia? Now, as 2100 : 550 : : < C 840 : 220 B's ' 5 B's gain. B's stock. d. d. And, as 220 : 840 v 11 : 42 : : 275 : 1050 C's stock dolls. dolls, dolls. 7. (80dx)-f 40=1640. whole gain, And d. d. d. d. A's 140-f B's 260-fC's 300=700 Then, 800 dollars -700 dollars=100 dollars D's stock. dolls. dolls dolls. Mis K Now, as 800: 1640 S : LlOO : 205 D's 8, d. cts. cattle dolls. As SOO : 96 v 25 . CIS. f 80 : 25 60 A pays 1 . R < 100 : 32 00 B 4 L ' (.120 : 38 40 C 4 J 9. First 2+3+5=10 dollars, Then ddlh. r /* . dolls, dolls. AB 10 : 5000 . 1 : 500 f2 : 1000 A gets") -! 3 : 1500 B < {> ' 15 : 2500 C < J I COMPOUND FELLOWSHIP. Page 95. 10. 20 dollars+85 dollars=105 dollars whole rtock d. d. d. d. Now, as 105 : 63 :: 20 : 12 A'sgain d. d. d. d. And 63 (21 + 12)=63 33=30 C ? s gain d. d. . doii,. ton*. T21 :35Bputin 7 Then, as 12 : 20 :: 2 C 30 : 50 C put in } COMPOUND FELLOWSHIP. Page 97. d. mo. d. mo, 2. 600X8=4800 1200x10=12000 200 600 800x8=6400 600X 6= 360D A 11200 B 15600 1000X12=12000 11200 A 800 15600 B . 19200 C 1800X 4= 7200 460CO C 19200 dolls, f 112CO : 560 dolls. A> sham Now, as 46000 : 2300 ::< 15600 : 780 dolls. B's * That is '20 : 1 [ 19200 : 960 dolls. C'i 4 mo. mo. dolls, dolls. 3. As 9 : 12 :: 600 : 800 Page 98. mo. mo. d. 4. As 8 : 12 :: 1800 : 2700 doikrs mi. mi. ml, mi. 5. A 600-f B 600-f C 300+ D 100=1800 miles T600 : 60 dolls. A pajs milts dolls. j fiftn . < R * As 1600 : 160-,' 10 : 1 SsooJsQ C * I 100: 10 * D * PROFIT AND LOSS, 75 Page 98. 5 oxenx4 months =22i 8 X5 =40 Sum 121 . doiis. cts. U>. Then, as 44S .- 22 40 :; 1 .' 5 cents per Ib, w BARTER. 77 Page 100. dolls. dolls. 11. As 120 : 100 :: 17-28 : 1440 whole cost yards. dolls. yd. Then, as 360 : 1440 : : 1 : 4 dollars 1& 84 9084=90 cents; Then, dolls, cts. dolls. dolls. As 4 : 90 : : 1000 : 225 d. d. cts. 13. As 120 : I :: 150 : 1 25 per bushel d. cts. lu. d cts. Now, as 1 25 : 1 :: 57 50 : SO bushels 14. 5000 galls, at 48 cents=2400 dellars Interest for 2 months = 32 ' Costs 82432 to pay cash 5000 galls, at 50 cents =82500 in two months Diff. 8 68 gamed by borrowing the money and paying the cash* head dolls, dolls. 15, 12 ) 240 ( 20 the cost per head cts. Ib. dolls. Ib. C.qr.lb. Then, as 4 : 1 : : 20 : 500=4 1 24 BAR TEN. Page 101. 2. 320 bushelsxgl 20=g384 Subtract 160 cash . 8224 cts. Ib. Then, as 20 : 1 : : 224 dollars : 11201k cts. cts. bu. 3, As 70 : 125 : : 28 : 50 bushels cts. cts. m. cts. 4, As 250 : 319 : : 23i : 29 9|, or 30 nearly d Ct8, ft Ct9, 5, As 100 : 120 : : 1 25 : 1 50 EXCHANGE* Page 102. d. rU. d. ct. d. cts, o gf) y 1 Ort 4- ct*. auu As 2 50 : 2 80 :: 1 20 : c * *~ =1 34 4 cts. cts. bu. As 99 : 90 : : 240 : 218 S * T bushels ets. cts. cts. d. cts. And, as 90 : 95 :: 99 : 1 04 5 mills oz. if. cts. oz. Ib. oz. As 16 : 1 28 :: 420 (=26 4) : S3 dolls. 60 cts : cts. Ib. d. cts. Ib. Then, as 6 : 1 : : S3 60 : 560=5 cwt eta. cts. eta. 9. As 22 : 25 : : 28 : 110X31 g r Ctfc Then, as 100 : 110 : : 31 T fl T cts. : - ^=35 10. 250 barrelsx6 dolls. 25 cts. =1562 dolls. 50 cts. Ib. cwt. cts. dolls, cts. And 448 (=4) at 30=134 40 64 Ib. at gl 75 = 112 00 ^25 7. at g6 =150 00 Cash 206 10 S602 50 do 11$. cts. dslls.cts. dolls. d. bbl. d 6W. ; 562 50602 50=960 Then, as 8 : 1 : : 960 : 120 EXCHANGE. Case 1. Page 104, d. 8. d. s. d. . . 96 (=8) : 90 (=T 6) : : 256 : 240 Or, 256 ^=16 subtract 8240 EXCHANGE. Page 104. d. s. d. d. B. d. L. L. *. d, 3. As 90 (=7 6) : 56 (=1 8):: 1500: 933 6 8 ^ ' . ^1=933 6 8 Answer as before D g. s. . . 4. As 6 : 8 ;: 180 : 240 Or, .180 i- = 60 240 as befor d. s. d. d. * . . 5. As 56 (=4 8) ; 72 (=6) ::280 : 360 Or, .280 f = 80 add 360 as before d. s. d. s. d. . . s. d. 6, As 72 (=6) . 56 (=4 8) ;: 304 ; 236 8 10| Or, 304 = S3 15 6f 270 4 5k 33 15 6-| 236 8 lOf Case 2. Page 105, . s. d. shillings s. s. dolls, cffo 4, 468 19 6=9379,5 Then, 9379,56=1563 25 . s. pence d. d. dolls, cts. 5. 750 10=180120 Then, 180120-7-90=2001 33{ Case 3. Page 108. . x 9, 540 15=540,75 Then, 540,75X84,10=32217 07i 6. 4 dolls, 1.0 cts ALLIGATION Page 108. . cl cfs. d. cts. 7. 320 X 4 44 = 14-20 80 at par 12 is = 177 60 add S139S40 Page 109. 8. S3259S4 10=^794,873, Then, . . . As 108 : 100 :: 79-1,8:8 : 735,9999, or 736 nearlj Otherwise, S. (L 34,10X108 L 1 7 " 3< ALLIGATION. . 20 Ihs, so * 40 " Case 1. , at 10 cents 00 at 15 " = 450 at2J ;i =1000 90 cts. 1650 cte. /^. c^c Now, as 90 : 1630 :: 1 : 18| 3. 10 bushels at 150 cents=-1500 20 ^ M 160 =-3^00 SO " at 170 " =5100 60 9SOO bu. d. bu. cts. Now, as 60 : 9S :: 1 .- 16S| P^.ge 110. 4 4 ounces at 7 c^nts=SOO 8 " at CO " =480 12 780 02. r'y OZ. Ct3+ Now, as 12 . 780 :t 1 : 65 ALLIGATION. Case 2. Page 110. gallons at 3 dollars ("3\\ 1+2 3 gallons at 3 dolla K5/) i =.1 ' at 5 4 (^ ' 1 =1 * atG ' 40 5 5 20// 15 + 5=20 Or, Or, 1 1 4 6 bushels of rje 6 * of corn 24 6 of oats x| /1 iuo) Cfe 4. 1st. f!20 )20.at.l20M&ri20\ 1 10+20=30 at 120 110\|10 110| inn ! 110A I =20 110 u l 90/110 90 L 80 J 20 80 100 I 90 )| =20 90 80 /J 20+10=30 80 f Ci5. Cf4T. 3d. f!20 v lOat'UOU/A. ( 120 N =10atl20 Iix-L/v 1 1/ ui i -cu i Tt/t i i -oiy ~^=ii/ai> ixu 110)v20 110J inn ) 110\) X 10 + 20^=30 110 90 ; )20 90 j J J 90/ / )20+10=f=SO 90 so /in 80 1 C 80 ' =plO 80 10 + 20=30 at 120 cents = 10 110 120+10=30 90 =20 80 120 N =20 at 120 cents 10\\ |10+20=30 110 90/1 I =10 90 /20 + 10=30 80 10+20=30 at 120 centa 120+10=30 90 y 20 + 10=30 80 Case 3. Page 112. ass 4 ' \ w =4 ~ )-30\)| 26+14 + 8=48 given, ', as 48 : 24bu : : 4 : 2 bushels of each sort H REDUCTION OF VULGAR FRACTION3. Page 112. S. / \16 22 < 18 / 20 * 24 SB 2 r= 2 = 2 quantity girea = 2 2 : 10 Ib. of alloy 2 : 10 of 16 2 : 10 of 20 t34 : 170 of 24 Cue 4. 2. 2 16 2 : 5 oz. of 15 carats fin* The*, 16: 40::< J; * f JJ | ,' O : 25 of 29 * > -C r ^0/5. C 5 58 s galls, of wine ^0/1 Then, as 6 : 70 :: ^1 llf galU. ofwiter a REDUCTION OF VULGAR FRACTIONS. Case 1. Page 114. Now, 72 )&(=} 8. 144)216(1 144 torn, measure 72)144(2 144 5. 75 ) 125 ( 1 75 (8) (9) 144 18 Or, -- = =* as before 216 27 . Then, 25 ) fa 50)75(1 (5) (5) Or, f/ T =if =*f as before eoau measure 25 ) 50 ( OP VULGAR FRACTIOUS. 83 Page 114, 4. 4800)10800(2 Then, 1200 ) 9600 com. measure 1200)'4800(4 Or, T $%**=% as befora 4800 5. 91)117(1 91 Then, 13) T Vr.(l 26)91(3 78 com. measure 13) 26 ( 2 26 5. 9876)88884(9 Therefore, 9876 ) \V 6 y (t=| 88884 Case 2. Page 115. 12x17-1-15 2. ! = 2 T y > fraction required 183x214-5 3. = If- 8 fraction required 514x164-5 8229 '' 4. _ 3=3 B *** traction required ' _ 68425x443 ' . . 03 r 5. _^ -=.27^70, fraction required . Case 3. x ' %. 3848-~21=:1832 5 r equivalent mixed number 3. 2465-~7=352| equivalent mixed number 4. 96 1-7-17= 56 f 9 T equivalent mixed number 5. 8229-7-16=514/5 equivalent mixed number Case 4. Page 116. ft. 3x5x6= 90 first numerator 4x4x6= 96 second numerator 5x4x5 = 100 third numerator And 4x5x6=120 common denominator Therefore, T \<^, ^ and {fg, are the fractioni reqmrtd KIDUCTION OF VULGAR FRACTIONS, 5. Page 116. 1X5X15X 9= 675 first numerator SX3X15X 9=1215 second ' 4X3X 5X 9= 540 third 5XoX 5X15 = 11*25 fourth * And 5x5xl5x 9=2025 common denominator Therefore, - 2 V/ 5 -, |li ? &% and -Jiff are the frac. required Case 5. 2. 4)3 8 13 20 5)3 2 35 1215 How, 4x3x2x5=120 common denominator Then, 1 20 com. denominator 8 Divisors < j0 L20 Hence, 5. 40x2=80") 15X5=45 I XT 10X4=40 ^ew numerator* 6x5=50j 45 <10 ovi/1 30 T20' T20 " I1U T35" Page 117. 3)3 5 15 9 5)1 553 1113 Now, 3x5x3=45 common denominator Then, 45 com. denominator Divisors <| J 19 Her 15X1 = 15") 9X3=27 ! 5x5=25j ice - - 7 - 1 2 New nu Case 6. S. 7X4X 9=252 num. 8x6x10=480 denom. Or, 3 7x4x# 7x5 Hence, REDUCTION OF VULGAR FRACTIONS. 85 Page 117. 3. 5x4x3= 60 num. 5x^X^_ 5 9X8X4=288 denom. *' # $ 4~~3x2x4 ^T ffpnrp 60 _ 5_ 32 Ai.CIlL"j -s-o ~o ~~"'2'X 4. 5x4x11=220 num. 5x>Cl'l 5x11 9x7x12=756 denom. '9 7 $ 9x7x3~~ f ^^ tlencej Y5"s" == Y'^"5 Case 7. 4X1X1 5x"20xT2 ==7 " 2 ~^ ==3 ^ fraction required Page 118. /10X1X1 1 4. 11X 6 Case 8. 1X12X20 A , .. 2 ' 30^:^===!^-=^ fraction required 1x63x8 3. -ITXT; =T?|=TI fraction required 728xlXl 1 X 24x60 = T||| == "TT fraction required' Case 9. 2. 12 ounces=l pound 3 5)36 7 ounces 4 pennyweights Page 119. 3. 8 furlongs=l mile 4. 24 hours^l day 4 5)32 10)72 6 furlongs 16 perches 7 hours 12 min. 8 35 ADDITION OF VULGAR FRACTIONS. Page 119. 5. 100 cents=l dollar 4 5)400 80 cents Case 10. 3, 31 galls. 2 qts. 63 galls,= 1 hhd. 4 4 126 quarts 252 quarts Wherefore, v||=l is the fraction required cut. qr. Ib. cwt. 4. 6 2 18f- 20=1 ton 4 4 26 80 28 28 216 240 53 5 746 6720 thirds of a pound 2240 thirds of a pound Wherefore, ff f= J is tne fraction required Case 11. Page 120. g. 20) 17,00 (,85 ADDITION OF VULGAR FRACTIONS. Case 1. 25 15+25+45+55_ 140 _ 60 *" ADDITION OF VULGAR FRACTIONS*^ 87 Case 2. Page 121. Given denominators 40 com. denominator 2. 2)2 458 2)1 2 5 4 1152 And 2 5 8 20 first numerator 10 second 4 8 third < 5 fourth ' Then, 2x2x5x2=40 common denominator 20+10+8 + 5 Wherefore, = 14= 1*3- 40 Given denominators 840 com. denominator 3. 5)567815 Then, 5 3)16783 6 2)1 2 7 8 1 15 168X4=672 140x5=700 120X6=720 105X3=315 56x8=443 11741 - Now, 5X3X2X7X4=*840 com. denoni 2855 sum Wherefore, ^ Case 3. Page 122. 3. oH= Wherefore, 4. of| = ^ TK. tT-..'! 108 + 105+40+60 Then, Lastly, Case 4. ft. in. h. m. 2. of a yard= 16 3. I of a day== 8 | of a foot =08 i of an hour=0 30 Sum 2ft. 2 in. Sum 8 h. 30 mi. 8S MULTIPLICATION OF VULGAR FRACTIONS. Page 122. d. h. m yds. ft. in* 4. ofaweek=28 5. | of a mile=1540 k of a day =060 fofayard= 020 i- of an hour 50 $ of a foot = 009 Sum 2 d. 14 hrs. SO mi. 1540y. 2ft 9in> SUBTRACTION OF VULGAR FRACTIONS Page 123. 1 1 _ q 4 11 J5_-1 1 9 _ A1 y _ 2 _ 1 T? 4 T i 1 2 ~ i~2 5* 05 _ 04 . .-"IS M^ 6 MJ^j .-..-I. ..,_418-gl_ r-T ^'32 '. 4T3T 8. l-/ 5 -6| = 12^ 6-^=5-}| 9. 13i 8if=13/ f 8if=- 10. 10 1-&=IOJ{[ U&=5 11 1Q5 7 _ 1O75 77 _ 1Q163 11. iy ri TJ ^i OT6 ^ T g- T io T6T /. /i. ??iz. 13. 7 weeks=7 000 9-^days=l 2 16 48 Diff. 5w. 4d. 7h. 12 min. MULTIPLICATION OF VULGAR FRACTIONS. Page 124. 4 _ 12 _ 3 -"40 TO" 5, ^yx^V^ 6. I of 9=V Then, 2 J X| RULE OF THREE IN VULGAR FRACTIONS. 89 Page 124. DIVISION OF VULGAR FRACTIONS. Page 125. 5. H-^4 T 8 o-|xA=M-==i 5 6 6. 1-4=1x1=^ 8. 5205^-= 26 f 26 , and f of 91 = 3 4 RULE OF THREE IN VULGAR FRACTIONS. 2. As'f ton. : 4 | 3 dolls. (=164 dolls.) : f ton. n X 493 1479 dolls - ct3 ' Then, ^X-Y-X-= -=211 28| '19 dolls. 5. } of $=2 5 oS Then, As ^ : 171 :: 1 7 yd. dolls. yds. A$ v ff . dolls. 4. As 1 : f :: 42 : ^-^=35 5. f off- A; Then say, dolls. ' Ai* : 312 :: 1 : Page 126. 6. ll=!,79!= 2 8 , and3|=V 7 Now, as|6z/. : 2 J 7 c/5. :: ^bu. : ^ 238 17 Because, -gX Xy== 4 f^ 6 =202 r V 7. As 17535 : | :: 60120 : | 60120 8* _)LUTIONV L\TE2!SE PROPORTION. Page 126. . 6 1 ,cwt. \/cwt. and 22^=^-1 miles cwi. cwt. mi. miles -, as 1 : y :: ^y 53 1 ;^ r ?*=- =145| miles S. As 16 : 12 :: 1 | 8 (=37|) : 28} days, 47 3 Because, = '|>=28j days 4 4. 100| do!ls.= 3 f 2 dolls. 6f mo.=\ month:? 5} years=V jears=46 months mo. mo. dolls. Then, as 46 : 2 T :: 3 f 2 : 14||f dollars /<. //. /if. /?. 5. Asl : I :: V (=20i) : 1/ Because, X X== 1 f 3 =17| feet / J?4 P / 6. 20| yards^V yards 1| jard=f yards Now, as | yd. : J- yd. : : % l yd. : 34 J. yards Because, ^X^X- 7 = 2 | 5 =34-J- yards i 2 INVOLUTION, OR RAISING OF POWERS. Page 123. 2. (35) 4 = 35X35XS5X35=1500625 3. l03 3 = SQUARE ROOT. 91 Page 128. 4. (,029) 5 =,029X,029X,029X,029X,029 =,000000020511149* 5. l 4 =l SQUARE ROOT. Page 129. 2. 100929(327 root required 3. 4762969(2187 root required 9 4 62)169 41)78 124 41 647; 4529 428)3729 4529 3424 4367)30569 30569 4. 43046721(6561 root required 5. 337420489(19683 root req. 36 1 125) 704 29)237 625 261 1306) 7967 333)2642 7836 2316 13121) 13121 3928) 32604 13121 31424 39363) 118089 118089 6. 22071204(4698 root req. 7. 36372961(6031 root req. 16 36 86) 607 1203 03729 516 3609 929) 9112 12061) 12061 8361 12061 9388) 75104 75104 "The answer given with the question is the 4th power of ,0 9& SQUARE ROOT. Page 129. 8. 2268741 (1506,23+ root req. 9. 7596796(2756,228-r-root req. 1 4 25) 126 47; 359 125 329 3006J 18741 545; 3067 18036 2725 30122 ) 70500 5506; 34296 60244 33036 301243; 1025600 55122; 126000 903729 110244 121871 remainder 551242; 1575600 1102484 5512448) 47311600 44099584 Page 130. *? 3^12016 remainder 10. 9712,718051(98,55^1001 req. 11. 3,17218120(1,78106+1001 req 81 1 188)1612 27)217 1504 189 1965)10871 348) 2821 9825 2784 19705; 104680 3561) 3781 98525 3561 197103; 615551 356206) 2202000 591309 2137236 24242 remainder 64764 remainder 15. 4795,2573l6;69,247-Hootreq. 13. ,00008836 f,0094 root req. 36 81 129; 1195 184; 736 1161 736 1382; 3425 2764 13844; 66173 55376 138487; 1079710 969409 110301 HOOT. 93 Square root of vulgar fractions, Page 130. 1. The greatest common divisor is 576 TA\2304 - 4. ThATl A / 4 - 2 S'OJ-STFT "9? en ? v -3 "3 2. The greatest common measure is 169 )!lt^f; Now, v^f=* 46656(216 denom, 4 41)066 41 426)2556 2556 , root required 3. 15825(125 num. 1 22)056 ' 44 ,245)1225 1225 That is 4. 5. 6. 1 1 2. =,755 Then, ^/, 75=, 86602 Jf , 8706739526 Then, v%8706739526=,93S09 |-| , 52439024 39 Then, ^/,524S9024S9=,72414 Square root of mixed numbers. Page 131. 71^-= is.! 9 Then. ^/ 1 -^ 9 =^?=6 l '49 49 LUC11, \ ? - U 7 Then, ^^^=^=71- 9|f/=\V Then, v/.W==3-} Surds. % /7 I \=v/7 5 S18I81'&c. =2,7961 + &c. ^=2,9519+ &c. =9,27+ 94 SQUARE ROOT. Any two sides of a right angled triangle given to find the third tide. Page 132. ' 1. 36X36=1296 2. 45x45=2025 27X27= 729 60x60=3600 Sum 2025 Sum 5625 Now, ^2025=45 feet Then, v/5625=75 yards 3. 45X45=2025 AC squared 4. 75x75=5625 27X27= 729 AB squared 45x45=2025 Diff. 1296 EC squared Diff. 3600 Then, v/1296=36 feet Then, y/3600=60 feet To find the side of a square equal in area to any given. sup erf tecs. Page 133. 1. ^160=12,6491106 x/2025=*45 3. v/750=27,SS612r9 To find the diameter of a circle of a given proportion, larger or less than a given one. 1. 4X4=16, and 16x4=64 Then, S9 310-25 _fi41 62106)176736 64207) 482198 372636 ' 449449 C21127) 4410002 612145; :'>'J74940 4347889 3210725 6211341) 6211341 6481001) 64^1501 6211341 6421501 CUBE ROOT. Page 138. 2. 2X2X3=12. . 13824(24 root required square of 4 16 8 2X4X30= 40 5824 complete divisor 1456x4= 5824 3. 7X7X3-=147. . 573248(72 root required square of 2= 04 343 7X2X30= 420 30248 complete divisor 15124X2=30248 4. 1X1X3=3.. 5735539(179 root required square of 7= 49 1 1X7X30=210 - 4735 complete divisor 559X7=3913 17X17X3=867.. 822339 square of 9= 81 17X9X30= 4590 complete divis. 91371X9=822339 CUBE ROOT. 97 Page 138 5. 4x4x3=48 . . 84604519(439 root required square of 3= 09 64 4X3X30 360 2C604 complete divisor 5169x3=15507 4SX43X35547.. 5097519 square of 9= 81 43X9X30= 11610 comp. divisor 566391 X 9=5097519 6. 3X3X327.. 27054036008(3002 root defective divisor 2700 . . 27 defective divisor 270000 . . square of 2 ' 04 054036008 306X2x30= 18000 complete divisor 27018004x254056008 7. 4X4X3=48.. 122615327232(4963 root square of 9= 81 64 4X9X30=1080 58615 ' complete divisor 5961 x9= 53649 1080 9 2 X2= 162 4966327 dividend defective divisor* ^203 . . square of 6 36 49X6X30= 8820 complete divisor 729156x6= 4374936 subtrahend 8820 : * 6 2 X2= 72 591391232 dividend defective divisor 738048 . . square of 8= 64 496X8X30= 119040 complete divisor 73923904x8=591391232 subtrahend * Defective divisors, after the first, may be more concisely found by addition, thus: To the last complete divisor, add the number which completed it, with twice the square of the last figure in the root; the sum will be the next defective divisor. 6 98 CUBE ROOT. Page 138. 8. 2X2X3= 12.. 2206981oi25(2805 root square of 8= 64 8 2X8X30= 480 14069 complete divisor 1744x8= 13952 480 8 2 x2= 128 117810125 dividend defective divisor 235200.. square of 5= 25 280X5X30= 42000 complete divisor 23562025x5=117810125 subtrahend 9. 6x6x3=108.. 219365327791(6031 defective divisor 10800.. 216 square of 3= 09 ~~S365327 60X3X30= 5400 complete divisor 1085409x3= 3256227 5400 square of 3x2= 18 109100791 defective divisor 1090827.. square of 1= 01 603X1X30= 18090 |109 : 1 00791 XI =109100791 root rtquind 10. 8X8X3=192.. 673373097125(8765 square of 7= 49 512 8x7X30= 1680 T61373 complete divisor 209*9x7= 146503 1680 14870097 twice the sq. of 7= 98 defective divisor 2^707.. square of 6= 36 87X6X30= 15660 complete divisor 2286396x6= 13718376 15660 1151721125 twice the sq. of 6= 72 defective divisor 2302128., square of 5= 5 876X5X30= 131400 complete divisor 230344225X5== 115172 11 25 CUBE ROOT. 99 Page 138. 11. 2X2X3=12.. square of 3= 09 2X3X30= 180 complete divisor 1389x3= 180 twice the sq. of 3= 18 defective divisor 1587. . square of 5= 25 23X5X30= 3450 complete divisor 16175x5=810875 12,9778f5(2,35 root 8 4977 4167 810875 12. X2X3= square of 5= 2X5X30= 25 300 complete divisor 1525 X5* 300 twice the sq. of 5= 50 defective divisor 1875 . . square of 1 = 01 25X1X30= 750 complete divisor 188251x1 = 750 twice the sq. of 1= _ 2 defective divisor 189003.. square of 6= 36 251X6X300= 45180 complete divisor 18945516x6= 15926,972504(25,16+ 8 7926 7625 301972 188251 113721504 113673096 48408 rem. 36155,027576(33,06 27 9155 8937 13. 3x3x3= 27.. square of 3= 09 3X3X30= 270 complete divisor 2979x3= 33X33X5=3267.. "21802/576 defective divisor 326700 . . square of 6= 36 330X6X30= 59400 ; / complete divisor 32729436X.6 =196376616 21650960 rem. 100 CUBE ROOT. Page 139. 14. 5x3x3= 27.. ,0532582/9(,S76-froot square of 7= 49 27 3X7X30= 630 6258 complete divisor 3379x7= 23653 37X37X3=4107.. 2605279 square of 6= 36 37X3X30= 3330 complete divisor 414066x6= 2484396 ' 120883 remainder 15. 1 xl X3=3 -. . ,001906624(,124 root square of 2= 04 1 1X2X30 60 906 complete divisor 364x2= 728 12X12X3=432.. 178624 square of 4= 16 12X4X30= 1440 complete divisor 44656x4=178624 16. ,OOQ000729(,Q09 root required 729 17. 1X1X3=3., 2,(l,25+roet square of &= 04 1 1X2X30= 60 1000 complete divisor 364x2= 728 12X12X3=432.. JJOO square ' 1X30= 1800 complete divisor 40035x5*= 225125 \46875 remainder CUBE ROOT., 101 Page 139. To extract the cube root tf&wlgw 'frty&fa 1. If =4ff Then, ^|f Hr root required Then > ^AV=I root required Then > ^2T=I root required Surds. 4< ^4 = ^571428571 &c. =,829+root required 5. ^/|=^/ 5 666 &c. = 5 873+root required' 6. ^/|=^/ 5 555 &c. =,822+root required To extract the cube root of a mixed number. 3. Surds. Page 140. 4. -\^7y= ^7,2=1,93-}- root required 5. ^/8f = ^8,7142857 &c. =2,057+root required 6. ^/9|=:^/9,166 &c. =2 ? 092+root required To find the side of a cube that shall be equal to any given solid, as a globe, a cone, tyc. 1, ^10648=22 2. ^389017=73 Note. Then, -^24 =2,8 845 feet =2 feet 10,614 inches . 12 3 XS=12X12X12X3=5184 Then, ^5184=17,306 inches ARITHMETICAL PROGRESSION Page 140. 1. As 6- : 3 3 v 216 : 27 :: 32 Ib. : 4 Ib. 3. 288x216x48=2935984 Then, ^72985984= 144 3, i !/ 3S9017=73 Then, 73X73=5329 Page 141, 4. Because there is two half feet in a foot -- 1 -^ =|=4 solid half feet in half a solid foot And then, 41=3 solid half feet 5. 2X2X2=8 sis inch cubes. And because there are three times 4 inches in a (ot we find that there is 3x3x3=27 four inch cubes in a solid foot ARITHMETICAL PROGRESSION. Case 1. Page 142. 2. (201 X3)+3=(19X3)+3=60 last term Then, (60+S)x Y= 6 3Xlr=6 dollars 30 cents 3. Here 4 is the common difference And (100 1X4)+4=(99X4) +4=400 last term Thea. (400 + 4) X 1 ! =404X50=20200 yards But, 20200 yards=ll ir/iles 3 furlongs 180 yards Page 143. 4. (ToITixlO} + 20=90--f-20=110 dollars last term Then, (110-j-20)X ^=130x5=: 650 dollars the \O amount. And 650-7-1055 dollars per annum ~~oITix 10; -f 10~999G-rlG 10000 cents, or 100 dollars for the last acre And (100CO-f-10)x 10 s 00 =50050 dollars, whole cost GEOMETRICAL PROGRESSION. 105 Case 2. Page 143. 2. =|$=2 the common difference Then, 20,22,24,26,28, &c. are their ages respectiyely Page 144. ^g 3 3. - ff=5 miles daily increase And (58+3) XV 2 61x6=366 miles whole distance GEOMETRICAL PROGRESSION. Page 146. 2. Indices 1 2 S 4 Ratio 2, 4, 8, 1 6, powers of the ratio ^6 In this question the 256:=8th power number of terms is 1 S 3d power 2048=1 1th power of the ratio X 1 the first term [progression 2048=: 12th or last term of the X 2 the ratio 4096 Subtract 1 the first term Ratio 2l=l)409o 84095 the sum of the series 3. Indices 12345 Ratio 2, 4, 8, 15, 32, powers of the ratio (54 Here the number of 98 terms i* 15 1024=1 Oth power 16=4th power I(j;i84=14lh power of the ratio. And because the first term oi' the progression is 1, the 14th power of the ratio is the ia^t, or 15th term of the progression TheBj Z. 32 767 Aillinp=163&e. T*. 104 GEOMETRICAL PROGRESSION. Page 146. 4. Indices 1234 Ratio 4, 16, 64 256 powers of the ratio 256 1536 Here the number of 1280 terms is 12 512 655S6=:Sth power 64=3d power 262144 393216 4194304 llth power of the ratic which is also the 12th or last term of the progression. Then, ( 4194 ^Xlhz^55924 05 sold for And 12X20240 00 cost 855684 05 gain 5. Indices 123456 7 Ratio 2, 4, 8, 16, 32, 64, 128, powers of the rati 128 Here the number of 1024 terms is 32 256 128- 16384=14th power 26384=. do. 65536 131072 49152 98304 16384 268435456 28th power 8~3d power 2147483648=3lst power of the ratio i And because the first term of the progression is 1, it is also the 32d or last term of the progression Then, (2^-483648 X 2 )-l =4a949 29 5 mg 2 1 Ratio GEOMETRICAL PROGRESSION, 1.05 Page 148. 6. Indices 12345 Ratio 3, 9, 27, 81, 243 powers of the ratio 243 729 972 486 59049=10th power 59049= do 531441 236196 5314410 295245 34867B4401=20th power 59049=10th power 31381059609 13947137604 313810596090 17433922005 205891 132094649=30th power 3=single power 617673396283947=*31st power of the ratio, or S2d term of the progression Then , -- =926510Q94425 Ratio 3 1 ' (4 32 l). 1844674^073709551615 3 6148914691236517 dollars 20 cents 5 mills (3 30 ~- 1) 2X205891132094648 ___ __ 205891132094648 pins Which at 1000 for a farthing amounts to . 214469929 5 S cost of 30 yards at ^100^ 3000 gain .214466929 5 3 SINGLE POSITION. SINGLE POSITION. Page 148. S. Suppose A paid 12 dollars Then B 4 < And C 1 < 17 sum dolls. T12 : 240 A paid Now, as 17 : 340 :: < 4 : 80 B < i 1 : 20 e 4. Suppose 60 {IS Now, as 37: 60:: 148: ^==240 d. J = 10 S7 5. Suppese 12 = 6 1= 4 Then, as 2 : 12 : : 26f dolls. : 160 dolls. 10 left 2 6. Suppqse A's age 6 Then,B's=6xl|=9 AndC's=(6+9)x2 1 V=31| *tu?^ r 6 : 6,5 : 95 :: -{9 : C31,5: . - . 12 A's Then, as 46,5 : 9$ :: ^ 9 : 18 B's 63 C's 7. Suppose 1 to be the number Then, I 1= 17 -=A difference DOUBLE POSITION. 107 Page 148. 8. Suppose 1 the harness -3EPH B HAMMOJsl) then ' horse and 6 * chaise harness horse chaise y dolls, f 1 : 40 dollars harnei Then, as 9 : 360 ; : 1 2 : 80 dollars horse 1 6 : 240 dollars chaise 9. Suppose 60 10. Suppose 12 5=- 4 1=20 1=15 A spends 8 |=12 2 =10 B spends 16 57 12 B sinks 4 dolls, dolls. dolls, dolls. Now, as 57 : 60 :: 228 : 240 Then, as 4 : 12 :: 50 : 150 DOUBLE POSITION. Page 150. 2. Suppose 8 Again, suppose 40 A spends 7 A spends 35 per year 30 30 B spends 37 per year B spends 65 per year 8 8 B spends 296 in 8 years B spends 520 in 8 yrs. 8X8=* 64 40X8=320 B is indebted 232 B is indebted 200 40 40 192 error too little 160 e. too lit, 192 error X40=7680 !60errorX 8=1280 6400 difference of the products 192160=32 the difference of the errors 108 DOUBLE POSITION. Page 150. Question 2d continued. Then, 640032=200 dollars their income. And 200 - 2 =200 25=175 dolls. A spends per annum Also, 175+30=205 dollars B spends per annum S. Suppose A's 12 Again, suppose A's 40 Then B's is 15 Then B's is 43 And C's * 19 And C's 47 46 130 100 100 Error 54 too little Error SO too much 54X40=2160 30X12= 360 2520 sum of the products And 54 + 30=84 sum of the errors Then, 2520-i-S4=30 A's share 30+ 3=33 B's share 33+ 4=37 C's share 4. Suppose A paid 100 Again, suppose A paid 200 then B 4 1100 then B 1200 and C * 1200 and C ' 1400 2400 2800 10000 10000 error 7600 error 7200 Errors are alike, i. e. both too little 7600X200=1520000 7200X100= 720000 Diff. of the products, 800000 76007200=400 difference of the errors Then, 800000400=2000 dollars A paid Aad 2000+1000=3000 dollars B paid Al69_2000+3000=5000 dollars C paid DOUBLE POSITION. 109 Page 150. 5, Suppose first horse 10 Again, sup. first horse 820 50 50 2)60 2)70 The second horse 30 Then, second horse 35 50 50 80 10X3=30 85 20X3=60 Error too little 50 error too little 25 50X20=1000 25x10= 250 50 25 Diff, of the prod. 750 DifF. of the errors 25 Then, 75025=30 dollars first horse & ^o dollars second horse 6. Suppose 20 body Again, suppose 50 bod j Then, 2 /+ 9 = 19tail Then, 3 ^ +9=24 tail 9 head 9 head Head and tail =28 Head and tail 33 body=20 body SO error 8 too little error 3 too little 50x8=240 8 3=5 difference of the errors 20x3= 60 Then, 1805=36 inches the body Diff. of pro. 180 And^-f9=27 tail 9 * head 72 in.=6 feet 10 110 DOUBLE POSITION. Page 150. 7. Suppose he wrought days, and was idle Then, 20 days X 20 cents =400 20 ' xlO =200 He would receive 200 cents 500 error too little 300 Again, suppose he wrought 25 days, and wag idle 15 Then, 25 daysXSO cents=500 15 k " xlO < =150 He would receive 350 500 error 150 too little 25X300=7500 20x150 = 3000 300 150=150 difference of tke error* Diff. of pro. 4500 Then, 4500150=30 days, wrought And --10 days, idle 8. Suppose each had 300 dollars ;}-J- 3 !=300 + 75=375 And 5)X2=75X2=150 Difference 225 error ppose each had 400 dollars IVn. 400 + 4 J = 400 -H = 500 And (400 i25)x2=rl75x2=35Q Difference 150 err . like Now, 4 OCX 225 =90000 X 150=45000 DifF. of the products 45000 A n <] :3J 150 -=75 difference of the errors Then, 45GO-r-75=600 dollars PERMUTATION. HI Page 150. 9. Suppose the parts Again, suppose the parts to be 8 and 7 to be 10 and 5 Then, Sx 4= 32 Then, 10 X 4=40 7X16=112 5x16=80 80 error 40 error Errors alik. 10X80=800 80 40=40 difference of the errors 8X40=320 Then, 48040=12 the greater DIff.ofpro. 480 And 1512=3 the less Page 151. 16. Suppose 1 o'clock Then, ( T 2 3- of ll)c 1== ^ 13 ^ ^ rror i \j Agaia, suppose it to be two of the clock/5 0(5 OQ Then, (ft of 10)c*s 2=- -^ = T *^ error 1 o Errors unlike Now, T 'jX2=if, and &xl& Then, }+& =fj ^ e sum ^ tfc And TT+l*? 3 *!! ^ e sum ^ t^ e errors Lastly, ^^1=11x11=14=1 hour 36 minu PERMUTATION. 2. changes 1x2X3X4x5x6x7x8x9x10x11x12=479001600 10)479001600 changes daythrs. ^)J7900160 365 6=8766) 798336(91 years 78894 9396 8766 days w. d. 24) 630(26=3 5 48 144 6 hours. Ans. 91 years 3 weeks 5 days 6 hours 1 IS COMBINATION. DUODECIMALS* Page 151. 5. 1X2X3X4X5X6X7=5040 days. Then, 5040-7-365=13 years 295 days Page 152. 4. 1X2X3X4X5X6X7X8X9=362880 5. 1X2X3X4X5X6X7X8X9X10X1 1X12X13X14X15 X16X17X 18X19X20X21X22X23X24X25X26= 403291461 126605635584000000 COMBINATION. 4 11 7 0^X23X00X^X00X19 5 ' 13 5 5X29X27X13X5X23=5852925 ADDITION OF DUODECIMALS, Page 153. m. w "' '"' 2, S780 9479 SUBTRACTION OF DUODECIMALS, Page 154. 2. 916 feet 8 inches 1 second 8 thirds 2 fourths //. in. " '" ft. in. " '" "" S- From 35 920 From 803 3 4 00 Take 24 10 55 Take 70 3 7 10 ifi: 10 10 87 Diff. 732 11 8 1 S DUODECIMALS. 113 MULTIPLICATION OF DUODECIMALS. Case 1. Page 155. ft. in. //. in. ft. to." 2. 9 5 3. 7 10 4. 846 3 11 8 11 2 7 4 877 722 2 9 6 28 3 62 8 4 10 7 6 1fi Q O square ft 36 10 7 sq. ft. 69 10 2 A VJ y \J square feet 21 10 5 feet in. 5, 5 7 Then, 150 cents I 10 10 square feet 4 7 10 2 in. - ( S- 1500 5 7 8 " - 25 2 " 2 \ 81 square ft. i 10 2 10 3& ft. in. 7 10 6 8 5 4 19 10 3 11 18 2 2 59 6 77 8 2 3 Then, 233 square feet 14 cents 3262 - A 832,62^ 6 square feet 10* 114 DUODECIMALS Sf Case 2. Page 156. 2, ft. 76 in. 7X1 . //. in. S. 127 6x2 //. in. 4. 184 8X7 459 6 6 10 12 1275 2216 3 6 9 10 1378 11475 22160 in. 76 7 in. 255 in. 1292 8 6 is 4 ; |= 38 3 6 6 4 4is|= 42 6 6is= 924 11772 6 i ?n. fret 23545 sq/ft. 1518 10 10 Page 157. 6. 6 in. is feet in. 38 11 10 389 2 7 2724 2 19 5 6 6 in. is square feet 2750 1 4 1434 = 9 10| 9)1463 Wk 162 yds. 5 ft. 10|in, 8. fest in. feet 54 9=54,75 58 6= 58,5 27375 43SOO 27375 9)3202,875 355,875 yards 15 ceiita 1 10 6 22 6 20 3 9 24 4 6 853,33125 square feet 44 8 3 Page 157. ft. in, ft. in. ft. in. ft. in. " 10, (2 10)X(2 4)x(l 9) = 11 6 10 (2 10)X(2 6)X(1 3)= 8 10 3 (3 6,X(2 2)x(l 8) = 12 7 B (2 10) X (2 8)X(1 9) = 1S 2 8 (2 10)X(2 6)x(l 9) = 12 4 9 "' (2 lb)x(2 8)X(1 8)=12 11 6 8 71 7 8 8 Or, 71,6435 square feet ft. dolls. sq. feet. Then, as 40 : v 2 : 1 : : 71,6435 : 835,82+ To find a ship'* tonnage. Page J58. 3, 4 . THE CARPENTER OR SLIDING RULE. Page 160. 1st. To multiply numbers together. 2. Set 1 on B to 12 on A, then against 16 en B stands 192 the product on A. 3. Set 1 on B to 35 on A, then against 19 on B stands 665 the product on A. 4. Set 1 on B to 54 on A, then against 270 on B stands 14580 the product on A. 2d. Division of numbers by the Carpenters* rule. . Set the divisor 19 on B to the dividend 665 on A, then against 1 on B stands 35 the quotient on A. 3. Set 27 on B to 396 on A, then against 1 on B stands 14,6 the quotient on A. 116 Page 160. 4. Set 42 on B to 741 on A, then against 1 on B stands 17,6 the quotient on A. 5. Set 24 on B to 7680 on A, then against 1 on B stands 320 the quotient on A. Sd. To square, numbers by the Carpenter? rule. Page 161. 2. Having set 1 on C to 10 on D, then against 30 on D stands 900 on C. 3. Having the rule set as above, against 55 on D jou will find 1225 its square on C. 4. Also against 40 on D stands 1600 its square on C. 4th. To find a fourth proportional to three numbers. 2. Set the first term 25 on B to 75 the second term on A, then against 100 the third term on B stands 300 the fourth term on A. 3. Set 27 on B to 20 on A, then against 73 on B stands 54 on A. 5th, To extract the square root of any number by the Carpenters' rule. 2. Having set 1 on C to 10 on D, against 529 on C stands 23 its root on D. 3. Being set as in the last example, against 900 on C stands 30 the root on D. 4. The rule set as before, against 300 on C stands 17,3 the root on D, MEASURING OF BOARDS, ETC. 117 MEASURING OF BOARDS AND TIMBER. Page 162. 1st To find the superficial content of a board or plank. foot in. 2. 12 Or, by the sliding rule. 12 6 As 12 on B : 14 on A:: 12 ft. 6 in. on B : 14 ft. 7 in. on A*. ==70l= 2 f 3 inches feet V feet 4x12 And 171 feet= 3 2 5 feet Then, \Vxyx3 cts.=^~^= 9 || 5 =r3d. 09f cti. 2d. Having the breadth of a board or, plank in inches j to find how much in length will make afoot, fye. Page 163. 5. 144-r-23=6,26 inches S. 1| yards=1944 square inches Then, 1944-7-26=74,7692 inches=6,2307 feet 3d. To find the solid content of square or four sided timber. Page 164. 15 in. Xl5in.xl8 ft. 144~~ 25 in. X9in. 3 - 5" **tf 1!l ches, the quarter girt 17in.xl7in.x20ft. Then, ^ =40,1388 feet, &c. *The operation by the sliding rule is omitted after this, it beinr presumed that the direclions in the W. Calculator make it superflu- ous to work more examples. 116 cAmpfwraRS 5 AND JOINERS' WQKK. Pa&e 164. SSin.-flOin. - 2 - tea mc es the mean breadth And - ^- - -=*18 inches the mean bread th 8 4th. To find the solid content of round timbtr. Page 165. 14X2 2X4 Then, 2x2x24=96 feet. Page 166. 5. 24 inchea=2 feet Then, ftetXS fcetx IJB feet=7S feet. 4. nft4in.+2ft.8in. 2X4 Then, 1,75 ft. X 1,75 ft. X21 ft.=64,S125 feet 5. 24 x (|) 2 =24X4 =96 14ix(Y 5 ) 2 = l4,5x, 62015625-= 8.9922+ 17!x( 6 ;T) 2 =17,25x2,4649 =42,5195 sum 147,51 17 feet CARPENTERS* AND JOINERS' WORK. Page 168. fc (5Sft.6in.)x(47ft.9in.) 53,5x47,75 ,?!:**. 2L = - = >^o o40>oO 100 100 , (36ft Sin.) x(16 ft. 6 in.) 36,25X16,5 , --- . = - s 100 100 Thtti, 5 ? 98125 8quaresx4 d. 50 c.=26 d. 91 c. 5,625 1 CARPENTERS.' AM) JOSSERS* WORK. 168'. 4. 35 ft. X30 ft. =1050 square feet (4ft. 6in.)x6ft.= 4,5x6=27 square ft. fire place (I0ft.6in.)x8ft.=10,5x8=84 < stair* 111 feet Then, 1050 "" m =9,39 squares 100 , Lastly, 9,39 squares X 3d. 75c.=35d. 21 c. 2,3 mr. /?. in. ft. in. ft. ft. sq. feet 5 (82 6)x(12 3) 8a,5Xlg4 = 1Q 1Q 6gg 100 100 (91ft 9in.)x(llft. 3 in.) 91,75x111 "~ioo~ "Too^ squares dolls, cts* d, ctf. milli Then, 10,321875x4 50=46 44 8,4375 ft. in. 8, 2)30 6 Page 169. 15 3 45 9~45,75 feet. And 52 feet 8 in.=52f feet Then, 1^ pf 24,095 squares Lastly, 24,095 sq.xHO cts.=S3 dolls. 7S cts. 5 raille feet in. 9. 20 6 i = lO 3 30 9=30,75 feet. And 40 feet 6 incfees=4Q,5 feet Then, ? ' 7 *^' 5 lg,45375 squares Lustly, 12,45375 squaresx225 cts. =28 dollars 2 centa-f Page 170. iquare ftet 11. (137 ft. 6 in.)x(16 ft. 3 in.) =137,5x164 =2234,375 Then, 2234,375 feet-^9=248 yards 2,375 fet 10 MASONS 9 WORK* Page 170. !. (69 ft. 9 in.)x(6 ft. 3 in.) =69, 75x6^=435,9375 I work=217,96875 9)653,90625 square yards 72,65625 13. (83ft. 8in.)x(12ft. 6 in.)=83fx 12,5=1045,831 (7ft. 8in.)x(2ft. 6in.)xli=7fx2,5xl,5=r= 8,75 wind. (3ft. 6in.)X 7ftx =3,5x7X,5 = 12,25door 9)1086,831 square yards 120 5 7|f Then, 120,7if J ds - x80 cts.=96 dolls. 60 cts. 7\\ mills ft. (57 BRICKLAYERS' WORK. Page 173. t. in. ft. im. half brick (57 S)X(24 6)X5~57,25X24,5X5=7013,125 Then, 3)7013,125 9)2337,708^ standard thickness 25 9,74 5 -f- square yards Or, 2337,7081272^=8,5866 rods feet in. feet in. 5. (245 9)X(16 6)X5=245,75X16,5X5=?20274,575 And 20274.S75 -r-3=6758,125 standard thickness Then, 6758?125 =24 rods 3 quarters 19 ? 945 feet 272,25 4. (45 ft. x V ft - x5)-f-3=562,5 standard thickness Then, 562,5-^-272,25=2 rods 18 feet MASONS? WORK. Page 175. feet in. feet in. *. (53 6)X(12 3)x2==53,5Xl2|x2=lS10,75fcet Then, 1310,7524,75=52,9595 rods PLASTERERS' WORK. Page 175, ( 107 ft. 9 in.; x (20 ft. 6inJ=*2208 feet 10J inches (112ft. S in J x16 ft. 6inJ 1 1 2, 25 X 1 6, 5 "* / _ ~63~ ~~6 5. fa ft. 7 in.;x(l ft. 10 in.;=f.|x||= T^V^VV feet 7^7x 1 Then, V'/ X30 cts,=:^~ =-318-1 cts. =8 dolls. 18| cts. PLASTERERS'* WORK. Page' 177. (141 6;x(U 3)_141.5X11,25_^^ 9 9"" ' ' 3. ^22 ft. 7injxf 13 ft. 11 iuJ=V/ x/= 4 i-!i 7 * feet But, <*' ^et= Then, 4 T 5 2 2 A 7 X 15 cts.= 22 ^3V 5=5doil s- 23 cts. 4. (30 ft+14 ft. 6 in.j X2== 60 feet round the room 69 ft.X(10 ft. 4 in.j =713 sq. feet in the walls (4 ft. 4 in) x4 ft = 17 ft. 4 inches fire place (S ft. 2 in.) X6 ft. X2i= 38 ft , two windows Sum 55 ft. 4in.=55^ feet 713 ft 55l=657f Lastly, 657|-7-9^73/ f 5. (14ft 5 in.+lS ft. 2in.)x2=55 ft 2 inches round ft. in. " Then, (55ft 2 in. )X (9 ft. 3 in)=510 3 6 7ftx4ft= 28 9)482 3 6 Rendering 53 j. 5 ft. 3 in. 6 s. ft. in. in. ft. in. ^ 14 5 less 10=13 7 And 15 ft. 2 in. 10 in.= 12 ft. 4 in. TheD , (!iM 11 22 PAVERS' WOHK. Page 177. feet in. feet tn. (105 6)X(275 5)_105,5 X 275 T 9 feet 9 Then, S22S,4}-- sq. yds. X12 cts. = 387 d. 41 c. 9|ra /ee tn. feet 7* (30 ft. 6 in.-f 24ft. 9 in.)x2=110 6 = 110,5 round 110,5 ft. XlO ft.= 1105 square feet in the walla 30,5 X24J < = 754,875 in the ceiling 9) 1859,875 206,652| square yards 2 cents Or, 4 dollars 13 cents 3 mills. PAVERS' WORK. Page 178. (35. ft. 4in)x(8 ft 3in.)_35^x8,5 9 9 Then, 32,3|- sq. yds.x28 cts. =9 dolls. 06 cts. 8| mills 5. (27ft. 10in.)x(14ft. 9in.)=27JXl4,75=410,54J g.f. 410,54-J ^9=45,6l|^ square yards Then, 45,61^4x38 cts. = 1733,S9|f cts.= 17 S3 S,9-f 4. 45 feet 5 feet 3 inches=S9 feet 9 inches broad Then, 56| square vardsXS6 cts. = 13 23 278i < " * X30 ' =83 47i Sum 896 70i 123 PAINTERS' WORK. Page 179. ft. in. " 2. (74ft. 10 in.) X (11 ft. Tin.) = 866 9 10 (7ft 6in.)x(3 ft. 9 in.)= 28 I 6 door (6ft 8 in.)x(3ft 4in.)x5= 111 1 4 windows (22 ft. 8in.)x(l ft. 2 in.)x5 = 132 2 8 breaks 1138 3 4 (6 ft. 9 in.) X5 feet= 3390 chimney feet 1104 6 4=1104f| And 1104i-7-9=122ff square yards Then, 122fff s. yds.x8<| cts.=10 dolls. 43 cts. If J ms. 5. (20 ft+14ft. 6 in.)x2=69 feet round 69 ft X(10 ft. 4 in.) =69x10^=713 square feet (4 ft 4 in.)x4ft=~4^x4= 17| feet fire, place (3ft. 2in.)x6ftx2=3x6x2=38 feet windows Sum 55| 5 6574 * =73^ 2 T square yards.* Page 180. ft. in. " 4. (24ft. 6in.Xl6ft. 3in.)x2=1039 16 (3 ft. 6 in.) X7 feet = 2460 door (7ft 9in.)x(Sft 6tn.)x2= 54 3 w. shut. 24 ft X(l ft 3 in.) X2= 60 breaks 1177 10 6 (5ft 6in.)x5feet= 27 6 1150 4 6 9)iiaoj 127 Then, 127|Jx6 cts. =7 dolls. 66 cts. 9-J- ms, *Tbe same as question 4, plasterers' work. 124 MEASUREMENT OF GROUND, GLAZIER WORK Page 181. 2. (7ft. 5 in.)x(3 ft. 5 in.) =24 square feet 9 in. 3 secJ 3. (6 ft. 1 1 in. +5 ft. 4 in.-f 4 ft. 3 in.) X(3 ft. 6 in.) X7= 16,5X3,5X7=404,25 square feet Then, 404,25 sq. ft, Xl4 cts. =58 d. 61 cts. 6,25 ras. 4. Then, 104,6875 sq. ft. xlO cts. =10 d. 46 cts. 8| ms. 5. (14 ft. 6 in.)x(4 ft. 9 in.)=68 sq. ft. 10 in. 6 sec. 6. (7ft 10in.+6ft. 8in.+5ft 4in.)x(3ft. Ilin.)x9= 19fx3l|-x9=-^XT^X0= 5 V 3 =699| square feet > $#1 2 4 Then, 699| square feetx!4 cents=97 dollars 87* cents MEASUREMENT OF GROUND. 1st. To find the content of a square piece of ground. Page 162. 35x2=70 perches Then, 70x70=4900 square feet=30 A. 2 R. 20 P. sq. perches Jl. R* P- 3. 16| x 16| ; = 16,5 X 16, 5 =-72, 25== I 2 32i 2d. To find the content of an oblong piece of ground, called a parallelogram. Page 183. 120 perches X 84 perches 2: =63 acres perches MEASUREMENT OF GROUND. 125 Page 183. ! 40^0 ;=5 8SOO =1()5 16X16 272i But, 105 per. 213|=2 rods 25 perches 213| feet 3d. To find the content of a triangular piece of ground. Page 184. 2. 75 XV =75X25=1875 per.=ll A. 2 R. 35 P. 3. 120X 8 /=120x42=5040 per.=31 acres 2 roods A. R. dolls. dolls, cts. Then, (31 2)x45=31,5x45=1417 50 4. 140 ft.x 7 2ft.=140X35=4900 square feet Then 4900-7-9=544 yards 4 feet. 4th. To find the content of apiece of ground, in the form of an oblique parallelogram. Page 185. 2. (80X24)~160=12 acres 5th. To find the content of a piece of ground bounded by four sides, none of which are parallel or equal. Page 186. 120 X (48+24) 2. ~- -=4320 perches=27 acres 6th. To find the area of a piece of ground lying in a circle, or ellipsis. Page 187. perches A. R. P. 2. 30X20X, 7854=471, 24=2 3 31,24 3. 160xl60x,7854=20106,24 square feet But, 20106,24-7-9=2234,02| square feet. 11* MECHANICAL POWERS. GAUGING. Page 169. 2. 3+f(24 18) = 18+4==22 mean diameter 4 Then, ^| 6 9|| gallons -' 3. o S -L|(36 28)=28-f5;;=S3f mean diameter 20 _ . X S S ! X 40 1 00 X 1 00 X TnPii ___ __ 200000 . ineB ^ 294 9x " TT ' 2 " ? 147 4. 154-|(18 15)=15 + 2=17 mean diameter \~ \t 17 x2Q TheC ' ' 359 ^ ^ V =23 *^ Salons 5. 18-f-|(22 18)=18 + 2|=0| mean diameter 4 MECHANICAL POWERS, 1st. THE LEVER. Page 190. As 1 inch : 20 inches : : 5 Ib. : 100 pounds the answer 2d. THE WHEEL AND AXLE. Page 191 As 60 inches,' =5 lb-j : 6 inches :: 10 Ib.: 1 Ib. PROMISCUOUS QUESTIONS- PROMISCUOUS QUESTIONS. Page 192. Quest. 1. 2578+2578=5156 2. 14676 14 f 76 =14676 3669=11007 3. 1468 f461 + 5Si; = 1468 1042=426 4. | of 100 cents =33^ cents iof i=j-of 100cts.=16 2 ' Sum 50 cents 5. 1080-r-45=24 the number required 6. 7. Suppose one side of the square to consist of 100 men Then, 100 2 +231 =10231 the whole number of men And 101 2 44=10157 the whole of the men 74 error too little Again, suppose the side of the square to have 120 men Then, 120 2 + 231=-14631 the whole number of men And 121 2 44=14597 the whole number of men 34 error too little Errors alike. (12QX74>- flOOX34; Then, l ~ ^= 5 || = 137 men on one side 7* ' o i Lastly, (137; 2 +251, or (13SJ 2 44=19000 men 8. 113X147 21 3 =16611 9261=7350 9. f of t=f= then, as : 1260 d. :: I : 5040 dolls. 10. 35002500=1000 dollars Then, 2500 dollars ) dolls. C 100 dollars 8 years $ 1000 1 year 40 128 PROMISCUOUS QUESTIONS* Page 193. 210x3 11. As : 210 :: 1 : - =8315 elder brother's por. Now, 315X5X^=1890 dollars 12. (650 130;x400=520 d. x4 dolls =2080 dolls- 13. 17+8+46+f20x2;=lll years 14. 1000 f350+400;=250 dollars C's share dolls, dolls. dolls, dolls. Then, as 250 : 500 . as 1 : 2 : : 350 : 700 A put in And, as 1:2:: 400 dolls. : 800 dolls. B's cloth is worth 15. | of 2720=544 dollars the captain's share gT'gQ 544 Then, = 2 f y 5 6 =13 d. 60 cts. a sailor's share lOU 16. As 6 : 100 :: 972 dolls. : 16200 dolls, her portion And, as | : 16200 dollars :: 1 : 27000 dollars Then, 27000-r-3=9000 d. int. of the father's estate 1 year dolls. dolls. Lastly, as 6 : 100 :: 9000 : 150000 the father's estate 17. ( 4ft. 6 inj x(2 ft. 9 in.; X(3 ft 4 in.;=41 sq. ft. S in. 18. 1+ 18 Then, as *V : 12 :: 1 : 12x18=216 feet Page 194. 19. r-y =2l| yean o SO. As ,y =1 of |; : 1710 : : 1 : 1^^=3800 dolls. 21. 62 2 ( 6 2 3 J 2 =3969X992,25=3938240,25 product And C63xV^r 63 + 6 2V= 198 V 94,5=1890 diff. mi. day miles da. h. min. 22. As 68 f=34x2; : 1 :: 2000 : 2 Jf=29 9 52{f PROMISCUOUS QUESTIONS, 129 Page 194. Ib. times Ib. C. qr. Ib. ftS^V^ 23. 6:2:: 885(=7 3 17) : ^-^==295 times M. Suppose 1, then 1-fff sum And, as 4 : Jfe: 20 : 40 26. |-15 28. Suppose the number to be 6 Then, 6x3 8=-10 the whole of his money And, 6x2+3=15 the whole of his mpney 5 error Again, suppose 10 to be the number Then, 10x3 8=22 his money And, 10x2-{-3=23 his money 1 error Errors alike, _ T 5X106X1 Now, - - sss^cali children D" ' ' '1 29. Inverse 100 dollars 7 year C 500 dollars inverse 6 dollars 3 1 \ 500 .dollars Then, 10,0^6 g ft J 30. Suppose the number to/be 100 Then, 1 f + 15 + l -0o + 10 =:io8i the members But, 1081100=81 error too little Again, suppose 200 members in all Then, 2 | + 15 + 2 -f 10*19 If the members But, 200191f=85 error too much Errors unlike. Lastly, 130 PROMISCUOUS QUESTIONS Page 194. SI. 360 deg. x69 miles= 25020 miles round the earth Then, as 20 miles : 1 day :: 25020 : 25 2o 20==: 1251 days Lastly, 1251 days-r-365| days=3 years 155? days Page 195. 32. 100X3|= 325 150X4= 675 204X51=1173 454 )2 173(4 months 23-||f days da. w. da. w. S3. As 7 : 1 : : 1 : \ A can do alone in one day And, as 12 : 1 :: l : ^ the part B can do in a day Then, f + T V = ""gT-^B the part of the work that they can do in a day, working together Lastly, as jj work : 1 day : : 1 work : 1|=:4^ days 34. (U05 7 1)X400^,40710042265625X400 1,051 ,05 =3256,80338125=3256 dollars 80 cents 3 milla TTio. dolls, mo. d. S5. ' As 12 : 5 :: 4 : If and 100+l|=101f Then, doiis. doiis. i nn v ^nn v ^ As s | 5 (-101|) : 100 :: 700 : ^ --=688 52 4|f 01 4.04 36. f+^^-^ii and !_}!= ft C's parts 7 Then, asJJ.: f !40::f :^ 8 . i - * And, asfj- : 140 :: $ : P === 3 ^ =30545^^ paid _ xu *RULE. A= 5 - - . Here w=the annuity R=* lv I the ratio or amount of .1, or gl, in one year as in com- pound interest; 1= the time for which the annuity is ia arrear. A=the amount of annuity at the end of t years* PROMISCUOUS QUESTIONS. 131 Page 195. 57. Suppose the number to be 12. Then, 12x4 + 16=64 his money And, 12x612=60 his money 4 error too little Again, Suppose 20 to be the number of beggars Then, 20X4+16= -96 his money "And 20X6 12=108 his money 12 error too much Errors unlike. Th8D , = - 14 beggars 38. As 18 : 1 :: 1 : ^ the part B and C can do in a day working together. And, as 11 : I :: 1 : ^ T the part of the work they can do in a day, all working 18 _ i[ Now, VT rV^ 1Q o ' ^TSF tne P art of the work that A can do in a day Lastly, as ^ : 1 :: 1 : 1 | 8 = 28 days 3 hrs. 25f min* 39. 20 congress each spends 4=5 dollars 30 merchants each spends y=6 ' 4 lawyers each spends |=3 ' 24 citizens each spends 7\=2 < 16 d. dolls. dolls. C 5 : 60 paid by the congress 40. is 16 : 192 _ r . ~j .. ) 6 : 72 ) 3 : 36 ( 2 : 24 merchants lawyers citizens < Pagel96.il 28X20 = 14X10X2= 560 square inches 280 ' 280= 1 1 acres PROMISCUOUS QUESTIONS. Page 196. 41. 200 acres=S2000 perches=6 40 perches =a. The rule is,* \X(| 2 +&) I the short side, and^/( 2 +&)+| the long side Thus, V^-i +32000)- 4 2=160 the short side And, ^/(^ +32000) + ^=200 the longer side 42. Suppose the side of the square meadow to be 1 ; Then, N /l 2 + l 2 = v /2=l,41421S6 its diagonal And, 1,4142136 1=,41421S6 what the diagonal is more than the side Now say, as ,41421 36 : 1 :: 20 : 48.28425 the side of the required square in perches. Lastly, square the side, and we get the contents 2331, 37 perches=14a. 2r. llp.f L. 43. As 18 : 1 :: 24000(=100) : 2 \ P- = ^oo S q Uare ft. And, 4o 3 oo^ 8g 3 =: 40 i oo x _i__iooo feet perpendicular Then, ,e of the equal sides; consequently ^ . . Lastly, as 1 : 3d ;: 1372 : 4802=20 2 48. Here v/39 2 -15 2 =V 1 oil 225 = 36 faet standing Then, 36+39=75 feet the whole height 49. Here work backwards Thus, 4x9=216; ^/2l6 - 6: Cx^-=8; 8 2 =64; 64 4=r60; 6D.-r-5=12; 12 X 8965 lastly, 96+7103. Page 197. 50. Here (lie several portions of wine drawn off were 50, remains 450 50X450 . 50X450 450 3 rema 12 154 PROMISCUOUS QUESTIONS. Page 197. Question 50 continued, 5QX450 2 . '450* 50 X 450 s 450 s " S " " 500 2 50X450 3 . 450 3 50X45G 3 450 4 500^ 500^ 50X450 4 450 4 50X450 4 450 5 - - _ 500 4 500' 500* 450* 184581 500000 Consequently, 500 295^ 9 o :=2 04|-| ( 1 r gall. of water* * Analytically. Put 500=c, 50= c . c 2 c* wine left after the third drawing; and universally the quantity of ~^the ouantity of wine drawn c out at the i drawing. And , _ J Xi c c '. remainino; after the t drawing, Ey substituting the values of c, d, ami t, in this formula, we get (00 50)^450^^ u f . the game ag aboye 4 * PROMISCUOUS qUESTIONS, 135 Page 197. 4X4X8~128~ 52. 1817, April 1st. principal 1864 dollars Oct. 15th 1864 doll, mult by 197 days=367208 Paid 225,50 1818 Jan. 10th 1638,50 X 87 =142549,5 Paid 188,86 May 16th 1449,64 X 126 =182654,64 Paid 585 Aug. 1st. 864,64 X 77 = 66$?f,28 Interest 126,49 , . 6|0)75898]9,42 Balance 991,13 '- Interest 126,49 S6x6ix8 2047-1 53. -i28" = 15 '2-f- cords days dolls, cts, day 54. As 365 : 356 34 :: 1 : 97 cents 6f mills cts. doll, dolls, cts. dolls, cts. 55. As 18| : 1 :: 2564 95| : 13679 76 56. Inverse 8 men > days \ 64 men inverse 820,50 5 30 { 100 dollars 25 30X^X^0000 30X25 41 57. Inverse 34 men ) months C 86 men inverse 1 bridge 5 6 ^2 bridge months 135 PROMISCUOUS QUESTIONS* Page 197. 58. As 4 : 6 : : r : 10^= 10,5 dollars C must pay v A pays o dollars and B 7 dollars. And, 5 + 7+10=22i=2-2,5 dollars dolls, dolls, f 5 : g 53 S3 3 A receives Lastly, as 22,5 : 240 : : 4 7 : g 74 66f B ' (,10,5 : S 112 00 C * Page 198. dolls, cts. 7775. dolls, cts. dells, cts. 59. As 100 : 7 S :: 8564 20 : 6 25+ tax 60. f4 galls. 2 qt.) (2 galls. 2 qt. 1 pt.) = l gall. 3 qt 1 pt.=15 pints gained, or remaining in the vessel at the end*ftf one hour. And, 84 gallons is 676 pints Then, as 15 pts. : 1 hr. : : 676 pints : 45 hours '4 minutes P. ft. ft. *q.J*t 61. One acre!60xl6,5xi6,5=160x272|=4356a Then, 4356036=1210 feet=73 perches 50 men bushels C 40 men 12 90 days 63. Bor IX 6= 6 cents Women 3 X 8=24 ; Men 6X16=96 c 126 cents f 6 : 90 cents the boys get Then, as Or: -{.24 : S60 cents the women get [96 : 1440 cents the men get Consequently, 90 6=15 boys "1 -H 8=45 women t And 1440-4- 16=90 men J G4. 7X4J 8x3 = SlJ 24=7i miles apart in one day 1 day : 7\ mi. : : 13 days : 97d miles apart PROMISCUOUS QUESTIONS. 137 Page 198. 65. 962X6=5772 1 635 XS =5722^ 320 4081 1282X6=7692 12261x5=6131,25 A 13464 B 11853,75 A 13464 B 11853,75 25317,75 Then, dolls, cts. ms. dolls, ds. C 13464 : 258,79,5^14 As 25317, 75: 486 64:: 11853,75 : 227,84,4^|ff Proof g486,64 5280 3 X2 3 66. Here 2500x6x5280X 3 44 1056 75 25 =4956^- hours 206 days 12 hours 9 min. 36 seconds 67. 45 j ft 9| in. =546 in.9j in.=536^ inches And 9| in. 6| in. =2| inches gained in a day Now, 536^~-2j = 1 % 73 X T 4 T = a ^ 6 =195 T 1 r days That is, it will be within 9| inches of the top in 195^ dayi. Consequently it will get to the top in 196^ days. tree. PROMISCUOUS QUESTONS* Page 198. 69. Suppose 200 dolls. Again, Suppose 300 dollare 94 94 106 206 ioM 05= 53 lent of 206= 103 lent J of 106= 21 \ clothes of 206= 4l clothes 94 debts 94 debts sum gone g238 sum gone T 'oGf200=lSO & of 300=270 1 1 error too much error 5 1| too much Errors alike, 200 X 3 1 1300 X 1 1$ 63603540 Then > 3ij-iii 1 Page 199. Then, as T V : 84 :: 1 : 84X12=1008 71. Here 3 : 4ft. :: 3 : 12 feet the semidiameter of the circle that the outer wheel makes. Consequently 12x2=24 feet the diameter of the outer circle. And, 24 4x2=16 feet the diameter of the inner circle Lastly, 24X3,1416=75,3984 feetbj the outer wheel And, 13x3,141650,2653 feet by the inner wheel PROMISCUOUS QUESTIONS. Page 199. 72. Here* 1 | 5 =one of the equal sides, 1 | 5 =half the sum of the sides ^75 o^n And '| 5 l $ 5 =- = 1 | 5 one of the equal rem. Then, V/ 125* . 15625 . 15625X1,7320508 "6T"^ /3== ~^r x/3 ~T6~ 751,75816 square feet. Or, by RULE n,* thus; 1253= 1 | 5 is one of the equal sides of the triangle, and because the triangle is equilate- ral, and consequently equiangular, the included angle of any two sides is 60, whose natural sine is ,8660254 Hence, 1 | 5 X 1 | 5 X, 866 | 254 =751,7581596 square feet * Here data is given to find the sides. The rule for solving this question, and all of a similar kind, is by mensuration, as follows: 1. From half the sum of three sides subtract each side severally. 2. Multiply the half sum and the three remainders con- tinually together, and the square root of the product will be the area required. RULE. ii. Any two sides of a triangle being multiplied together, and the product again by half the natural sine of there included angle, will give the area of the triangle. That is, AC xCBx natural sine of the angle C= twice area. JOHNSTON & STOCKTON, Have just published the Sixth Edition of the BY THE REV. JOSEPH STOCKTON. 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