UC-NRLF SB EbE 7M3 LIBRARY OF THK UNIVERSITY OF CALIFOI OH" QS*>t Received Accession No Jj~ 2 . Class No. i/ uou pumioiicu, AUO pages, puce 4B. ENGLISH ETYMOLOGY: A Text- Book of D-rn With numerous Exercises for the Use of Schools. B] DOUGLAS, Ph.D., Author of "Principles of English Gramma 5. Just ptiblished, 160 pages, price Is. PROGRESSIVE GEOGRAPHY. By DR JAMES DC An entirely New Work, showing the recent Changes on the Conti elsewhere, and embracing much Historical and other Informatio NEW EDITIONS OF NEW BOOKS: SPELLING AND DICTATION EXERCISES. For t of Schools. Py Dr JAMES DOUGLAS. 144 pages. Price Is. ' A good practical book, from which correct spelling and prom may he acquired." Athenccum. FIRST YEAR'S FRENCH COURSE. By CHARLES SCIINKIDI.K, Author of "French Conversation-Grammar," Price ls.6d. ** This work forms a complete course of French for beginn roinprrlirmls < iniminaticiil Kxcrcisc-s. with Rules; Reading with Notes; Dii-tation; Kxcrdsrs in Conversation; and a Voc of all the Word-< in the Jjpok. ny of any one- .or more of the above will bi post-free, by Oliver and Boyd, on receipt of PRACTICAL TREATISE ARITHMETIC, AEEANGED FOE PUPILS IN CLASSES. tlje Ise of BY ROBERT STEWART, LATE WRITING MASTER, DUNDEE PUBLIC SEMINARIES. E PROPOSED DECIMAL COINAGE. EDITION. OLIVER & BOYD, TWEEDDALE COURT. LONDON I SIMPKIN, MARSHALL, & CO. 1871. Price One Shilling and Sixpence bound. S73 A KEY" TO MR STEWART'S PRACTICAL TREATISE ON ARITHMETIC; containing Solutions at full length of all the Questions in that Work. By A LEX AND KB TKOTTER, Teacher of Math- ematics, etc., in Edinburgh. ISmo, 2s. bound. Also, STEWART'S FIRST LESSONS IN ARITHMETIC for JUNIOR CLASSES ; containing Exercises in Simple and Compound Quanti- ties, arranged to enable the Pupil to perform the Operations with the greatest Facility and Correctness. New Edition. 18mo, 6d. in stiff wrapper. KEY TO DITTO. ISmo, 6d 7 PREFACE. THERE is perhaps no branch of Education in which either activity or indolence may be more strikingly exemplified than in Arithmetic; for, by the manner in which it is taught, the pupil may either be kept in a state of inactivity and languor, or excited to energetic application. When taught in Classes, attention, activity, and emulation are secured; while, on the contrary, the solitary, individual mode of operation too frequently ex- hibits that listless and careless indifference which every experienced teacher must have observed. The various rules and examples in the following treatise are arranged rin^such a manner as appears best adapted to the simultaneous method of op- eration : and, to remove as far as possible the difficulties which occur in teaching Arithmetic, all superfluous matter, or questions tending to puzzle the learner, have been carefully excluded, a cir- cumstance not sufficiently attended to in many elementary school-books of this kind. CONTENTS. Page. Tables of Money, Weights, and Measures, - 3 Numeration Table, - 7 Initiatory Lessons in Addition, 8 Addition, - 9 Subtraction, ... 14 Multiplication, - 17 Division, - 2J Reduction, - - - - - 27 Exercises in Reduction, 32 Compound Addition, - 35 Compound Subtraction, - 46 Compound Multiplication, - - - 54 Compound Division, - - - - 61 Bills of Parcels, - - -70 Proportion, ... . 73 Compound Proportion, - - 79 Practice, - - 82 Vulgar Fractions, - - - - -88 Promiscuous Exercises in Vulgar Fractions, - 97 Decimal Fractions, - 98 Promiscuous Exercises in Decimal Fractions, - 106 Interest, - - - - - -107 Discount, - - - - - 112 Equation of Payments, - - - - 1 1 3 Commission and Brokerage, - - - 114 Insurance, - - - 114 Buying and Selling Stocks, - - 116 Profit and Loss, - - . 118 2 CONTENTS. Page Exercises in Commission and Brokerage, Insurance, Buying and Selling Stocks, and Profit and Loss, 120 Distributive Proportion, or Partnership, 121 Commercial Allowances, - - 123 Barter, 125 Exchange, - - - - - 126 Austria, - 126 Belgium; Canada, - 127 Denmark; France, - 128 Frankfort-on-the-Maine, - 129 ,, Hamburgh, - - 129 Holland, - - 130 India, - 130 Norway ; Nova Scotia, - 131 Portugal, - 131 Prussia; Eussia, - - - - 132 Spain; United States, - 133 Involution, _.-.-- 134 Evolution, - - 134 Position, - - 140 Double Position, - - 140 Arithmetical Progression, ... - 142 Geometrical Progression, 144 Duodecimals, ------ 146 Tonnage of Ships, - - 148 Compound Interest, - - 149 Annuities for a Time, .... 150 Annuities for Ever, ----- 151 Compound Interest and Annuities Tables, - 154 Promiscuous Questions, 158 Decimal Coinage, - 165 ANSWERS, 171 Questions for Examination, 201 MULTIPLICATION TABLE. 1 2 ~4~ 3 6 4 5 6 7 14 8 T6" 9 18 10 20 11 "22" 12 24 2 8 10 12 3 6 9 12 15 18 21 24 27 30 33 36 4 8 12 16 20 24 28 32 36 40 44 48 5 10 15 20 25 30 35 40 45 50 55 60 6 12 18 24 30 36 42 48 54 60 66 72 7 14 21 28 35 42 49 56 63 70 77 84 8 16 18 To" 24 32 40 48 56 64 72 80 88 96 9 10 27 36 lo" 45 54 63 72 81 90 99 no 108 120 30 50 60 70 80 90 100 11 22 33 44 55 66 77 88 "96" 99 110 121 132 12 24 36 48 60 72 84 108120 132 144 ARITHMETICAL TABLES. 1. STERLING MONEY. 4 farthings qrs=l penny d. 1 shilling /! pound or { sovereign = 1 guinea = 1 crown 4. AVOIRDUPOIS WEIGHT. 16 drams dr. = 1 ounce oz. 1 6 ounces = 1 pound Ib. 28 pounds = 1 quarter qr. 4 quarters = 1 hund. wt. cwt, 20 hund. wt. = 1 ton T. Ib, 7000 grains = 1 pound 14 pounds = 1 stone st. 8 stones = 1 cwt. 112 pounds = 1 cwt. This table is used for all articles ex- cept Gold, Silver, and Jewels. 12 pence 20 shillings 21 shillings 5 shillings 2. TROY WEIGHT <24 grains gr=l penny weighted 20 dwt =1 ounce oz. 12 ounces =1 pound Ib. 5760 grains =1 pound troy Gold, Silver, and Jewels are weighed by Troy Weight. 3. APOTHECARIES' WEIGHT.* 20 grains gr. = 1 scruple sc 3 scruples = 1 dram dr 8 drams = 1 ounce oz 12 ounces = 1 pound Ib Only used for medical prescriptions bat Apothecaries buy and sell by Avoir- dupois. In the British Pharmacopoeia (iS'tt), the oz. Troy of 480 grains has been abolished, while the Ib. avoir. oFTCOO grains, and the oz. avoir .f 437$ grains, have been adopted. 5. LINEAL MEASURE. 12 lines /t". = 1 inch in. 12 inches = 1 foot ft. 3 feet = 1 yard yd. 5 yards = 40 poles = 1 pole po. 1 furlong/wr 8 furlongs 1 I760yard0 f ~ 1 mile ml. ARITHMETICAL TABLES. 3 miles = 1 league le. 60 geographical miles, or 69J British miles = 1 degree } the earth's 360 degrees = v circumfer- The imperial chain = 66 feet in length, and is divided into 100 links ; each link being = 7.92 inches. A hand for measuring horses = 4 inches. A fathom for measuring depth = 6 feet. The length of a pendulum vibrat- ing seconds at London = 39.1393 inches. 6. CLOTH MEASURE. 2 inches = 1 nail nL 4 nails = 1 quarter yr. 4 quarters = 1 yard yd. 3 quarters = 1 Flemish ell Fl.e. 5 quarters = 1 English ellEn.e. 6 quarters = 1 French ell Fr.e. 37 inches = 1 Scotch ell So. e. 7. SQUARE OR LAND MEASURE. 144 sqre. inches = 1 square foot 9 sq. feet = 1 square yard 30 J sq. yards == 1 pole or perch 40 poles == 1 rood ro. 4 roods = 1 acre ac. 36 square yards of stone, brick, or slate work, are called a ROOD, in Scotland ; and 100 square feet of floor- ing, roofing, &c. are called a SQUARE in England. A Square mile = 640 acres. 8. CUBIC, OR SOLID MEASURE. 1728 cubic inches = 1 cubic foot 27 cubic feet 40 cubic feety of rough, or 60 > of hewn timber.] 40 cubic feet A barrel bulk : 1 cubic yard = 1 load to. : 1 ton rneas 1 . : 6 cubic feet 9. YARN MEASURE. Lint. 90 inches = 1 thread 120 threads = 1 cut 2 cuts =1 heer 6 heers = 1 hasp 4 hasps = 1 spindle Cotton. 54 inches = 1 thread 80 threads = 1 skein 7 skeins = 1 hank 18 hanks = 1 spindle 10. MEASURE OF CAPACITY. 2 pints pt. = 1 quart qt. 4 quarts = 1 gallon ga. 2 gallons = 1 peck pk. 4 pecks == 1 bushel bu,. 8 bushels == 1 quarter qi\ The imperial standard gallon con- tains ten pounds avoirdupois of dis- tilled water, at the temperature of sixty-two degrees of Fahrenheit's ther- mometer. It is used for wine, bee)-, ale, spirits, and all sorts of liquid? ; also for dry goods. It is = 277. 2H cubic inches. 11. ANGULAR MEASURE. 60 seconds* == 1 minute ' 60 minutes = 1 degree 30 degrees = 1 sign s. 12 signs = 1 circle ciro. 12. HAY AND STRAW WEIGHT. 36 Ibs. avoir. = 1 truss of straw 66 Ibs. = 1 truss of old hay 60 Ibs. = 1 truss of new toy 36 trusses = 1 load Hay sold between the beginning of June and the end of August, of tbaf year's growth, is reckoned new. ARITHMETICAL TABLES. 13. TIME. 60 seconds sec. I minute mi. 60 minutes = 1 hour ho. 24 hours = 1 day da. 7 days = 1 week we. 4 weeks = Ico.month/wo. 365 days, or 52 ? weeks and 1 day ?= 365J days =Uulianyear 366 days =1 leap year 865 days, 5 hours } 48min.50sec. * The year is divided into 12 calendar months, viz. January 31 days! July Sldays August 31 Septem.30 OctoberSl February28 March 31 April 30 May 31 June 30 Novem.30 Decem.31 The number of days in each month may be easily remembered from the following lines : Thirty days hath September, April, June, und November ; All the rest have thirty one, Excepting February alone, Which hath but 28 days cleat, And 29 in each leap year. In general if the year is divisible by 4, without leaving a remainder, it is leap year. 14. WOOL WEIGHT. 7 Ibs. avoir. =1 clove ct. 2 cloves, or 14 Ibs. =1 stone st. 2 stones =1 tod id. 6 tods =1 wey wy. 2 weys =1 sack sk. 12 sacks =1 last la. QUARTERLY TERMS. IN ENGLAND. Lady-day 25th March. Midsummer 24th June. Michaelmas 29th Sept. Christmas 25th Dec. IN SCOTLAND. Candlemas 2d Feb, Whitsunday 15th May. Lammas 1st August. Martinmas llth Nov. SIGNS IN ARITHMETIC. = (equal to) denotes equality + (plus) addition (minus) subtraction X (multiplied by) multiplication -r- (divided by) division V square root ^ cube root MISCELLANEOUS. Flour and Meal. 14 Ib. = 1 peck or stone of flour or meal ; 56 lb. = l bushel; 1401b. = l boll; 280 lb. = lsack; 196 lb. = l barrel. Bread. A peck loaf weighs 8 Ib. ; and a quartern loaf = 4 Ib. A barrel of beef= 200 Ib. ; of beer = 54 gal. ; of butter, 4 firkins = 224 Ib. ; of soap = 256 ib. ; of herring = 500 her. Ounce Thread. 36 inches = 1 thread ; 30 threads = 1 hank. Paper. 24 sheets = 1 quire ; 20 quires = 1 ream ; 21 quires = a London ream ; 2 reams = 1 bundle ; 10 reams = 1 bale ; 4 pages = 1 sheet folio ; 8 pages =1 sheet quarto; 16 pages = 1 sheet 8vo ; 24 pages = 1 sheet 12mo ; 36 pages = 1 sheet 18mo. A2 ARITHMETIC is the science or art which treats of numbers. It is divided into two parts, viz. THEORY and PRACTICE. The Theory explains the properties and rela- tions of numbers. The Practice applies the rules derived from the theory to the solution of questions. Unit or Unity is the number one. A Number is that which is composed of one or more units, as 1, 2, 3, 4. A whole number is that which represents one or more things, each of which is considered as a whole. An even number is that which can be divided by 2 without leaving any remainder, as 4, 6, 8. An odd number is that which, being divided by 2, always leaves unity for a remainder, as 5, 7, 9. An abstract number is that which denotes a num- ber of things in general, without application to any particular subject, as 8, 5. A concrete number is that which denotes any par- ticular thing, as 6 pounds, 4 feet. A prime number is that which cannot be produced by the multiplication of two other numbers, as 1 9, 23. A composite number is the product of two other numbers, as 24, which is the product of 3 and 8. A simple number is a number of one denomina- tion, as 68. A compound number is a number of different denominations, as 68 14s G^d. An integer is a whole number as distinguished from a fraction, as 4. 10, 26 A fraction is a parr, or division of a unit or integer, as J-. All numbers are expressed by the ten following figures : viz. 1, 2, 3,4, 5, 6, 7, 8, 9, 0. The cipher has no value of itself ; the other nine are called digits. Every figure has two values, viz. a single value, when it stands by itself, and a local value, from the position it holds among other figures. NOTATION is the art of representing any num- ber by means of the 9 digits. NUMERATION is the art of expressing any num- ber in words. NUMERATION TABLE. 435, 412, 154, 332, 222, 222. Quadrillions. Trillions. Billions. Millions. Thousands. Units. 6th. 5th. 4th. 3d. 2d. 1st Period. ROMAN NOTATION. M D C L X V I 1000. 500. 100. 60. 10. 5, 1. When the same letter is repeated twice or oftener, their values are added, as XX~20. When a numeral letter of less value is placed on the right of a greater, their values are added, as XV=15. When a numeral letter of less value is placed on the left of a greater, their values are subtracted, as!X=9 8 SIMPLE ADDITION. Sometimes thousands are represented by draw- ing a line over the top of the numerals, as ~X represents ten thousand, L fifty thousand, M one million. The fundamental rules of Arithmetic are, Addi- tion, Sub traction, Multiplication, and Division. INITIATORY LESSONS OR KEY TO ADDITION. To be added on the Book. 10 2 5 6 7 8 6171 3 8 7 7 8 8 8 2 10 2 5 4 3 2 4 3 7i2 3 3 2 2 2 8 10 2 5 6 7 8 6 7 8 8 7 7 5 8 8 5 >10 2 5 4 3 2 4 3 2 2 3 3 5 2 2 9 10 2 5 6 7 8 6 4 8 8 7 7 8 8 8 9 10 2 5 4 3 2 4 6 2 2 3 3 2 2 2 9 l.io 2 5 6 7 8 6 4 8 8 7 7 8 8 8 8 10 2 5 4 3 2 4 6 2 2 3 3 2 2 2 6 10 2 Il4 3 2 5 7 3 4 5 6 8 6 7 4 10 214 6 7 8 5 3 7 6 5 4 2 4 3 4 10 2 5 4 3 2 5 4 3 4 5 6 5 G 7 6 10 2 5 6 7 8 5 6 7 6 5 4 5 4 3 2 10 2 5 4 3 2 5 4 3 4 5 6 5 G 7 8 10 2 5 G 7 8 5 6 7 6 5 4 5 4 3 3 1012 5 4 3 2 5 4 3 4 5 G 5 6 7 7 10(2 5 6 7 8 5 6 7 6 5 4 5 4 3 5 10 2 5 4 3 2 5 4 3 4 5 6 5 G 7 5 10 2 5 6 7 8 6 7 8 8|7 7 8 8 8 9 1 2 3 4 5 67 8 9 10 11 12 13 14 16 16 9 SIMPLE ADDITION. ADDITION is that operation by which the sum of two or more given numbers is found. RULE. Write the given numbers under each other with units under units, tens under tens, &c., draw a line below them, then begin and add the units column ; put down the right hand figure, and carry the rest to the next ; continue doing so with each column to the last, under which place the whole sum. PROOF. The best test of the accuracy of the work is to begin at the top and add downwards ; then, if the sum is the same as when added upwards, the work is correct. 1. 2. 3. 4. 5. 6. 7. 8. 542 624 654 652 564 764 645 746 411 112 242 412 422 241 212 421 131 512 412 240 142 523 433 325 9. 10. 11. 12. 13. 14. 15. 16. 645 564 846 572 846 643 856 983 121 121 264 268 567 267 244 526 524 443 582 304 279 376 612 457 17. 18. 19. 20. 21. 22. 23. 24. 830 609 793 943 701 979 820 730 264 496 501 607 376 748 246 266 566 113 292 336 325_ 231 574 464 25. 26. 27. 28. 29. 30. 8504 6852 9467 7634 8520 6900 1686 4268 3968 6465 4786 1648 5898 3683 4786 9282 2895 6976 6818 2584 5499 1169 3734 5252 10 SIMPLE ADDITION. 31. 32. 33. 34. 35. 36. 37, 7003 9237 5042 8432 7032 5489 8467 5428 5428 1686 2689 5686 1855 2185 5676 5943 2869 5768 3006 6072 5768 1575 3809 3356 j)743 1346 3634 6282 38. 39. 40. 41. 42. 43. 44. 8794 8546 8916 9274 9042 9324 9846 5426 2758 2428 4216 4287 5684 2168 2684 4268 5678 3786 2467 2025 5684 3368 5788 6488 5058 4755 3640 7678 45. 46. 47. 48. 49. 50, 82645 52434 84236 83426 62342 84160 28678 38526 26843 52468 34686 36849 23850 17643 61343 68568 24897 52168 54937 49684 52348 17234 12602 20425 53967 13908 57393 30958 27656 47311 61. 52. 53. 54. 55. 56. 95648 63564 90423 84682 91519 84026 24854 36843 25846 55439 60285 34214 61485 89469 45742 28549 85226 68459 60037 12535 12357 36858 24342 31494 70794 16721 64577 29243 31234 49812 57. 68. 69. 60. 61. 62 43686 85673 94678 80751 98578 74923 12416 14341 12421 34876 16985 21468 87968 28569 48916 12345 23456 52842 12421 14215 14258 67890 78901 34386 31270 71332 82257 45875 81593 53455 SIMPLE ADDITION. 11 63. 64. 65. 66. 67. 68, 64501 90124 61248 90842 70354 84517 28694 14685 24746 24486 28898 56256 23059 92446 85297 89574 60857 43794 45684 14857 59614 35689 23468 79832 35807 75439 36502 66356 41456 28261 69. 70. 71. 72. 73. 74. 53678 84570 90243 54237 54270 ^72845 24783 28649 21487 36845 32846 29304 59528 34257 68524 24689 29597 87485 37487 21868 21687 56938 56845 26894 21046 34784 93428 49846 27084 37089 28895 55921 68756 17392 21424 43541 75. 76. 77. 78. 79. 80. 54268 98542 32042 98742 80234 78345 26942 65487 16857 45468 21685 26876 85427 24738 89428 29785 39426 34242 21216 58297 34687 34568 25898 56896 56349 25824 20432 22847 32429 21685 27326 33055 15185 53274 58549 51469 81. 82. 83. 84. 85. 86. 52456 93042 52876 87024 62901 50429 24690 27684 24687 23846 28745 28916 85468 84246 83214 78589 34285 74284 21346 28053 92326 21642 68194 26897 34582 36842 78560 34856 22843 91246 27766 65358 28189 63178 34156 21513 12 SIMPLE ADDITION. 87.35926 88.90214 89.5420890.8543291-50426 21864 26841 23456 56846 32845 38486 78901 94289 94684 83705 86768 57894 23456 23456 28769 26012 78923 78934 6527 3289 14062 63373 30752 28586 17581 92. 73110 93.97905 94.80377 95.8381496. 40188 40279 49171 12041 38187 13577 57684 80425 93020 74203 58467 24853 21689 24684 28576 21856 76929 38894 36956 24245 94684 67890 78368 34698 4684 3684 24216 20849 92438 3426 7938 32831 58734 68336 45627 26611 67.12345 98. 23426 99.92898100.36857101 .95859 9419 19303 90004 30068 11923 84203 30074 93042 28570 38400 1709 9531 50059 31375 60385 68918 18569 25678 23879 32867 96485 23456 90123 98589 98928 27694 7890 4567 2902 26892 2926 4123 2894 6789 83936 202.78580103.58042104.80205105.58240106.87654 23946 57896 74859 26856 53268 83597 26948 26926 97287 29827 26496 58689 39288 8695 3638 8568 42764 2428 7428 4580 2493 8898 3989 2596 64888 9389 5260 42230 8228 83569 54634 146 5346 31384 3438 SIMPLE ADDITION. 13 107. 51057 15285 96546 84397 73529 63429 42513 53497 32751 35772 108. 34832 11505 45798 75634 63279 32568 57639 95035 20572 23327 109. 83753 67364 98287 72689 63291 52389 47437 47897 24768 16389 no. 80904 44694 98256 78346 59630 32468 45908 53759 36736 36210 in. 95700 19533 94857 83042 63902 25793 63219 63472 93146 76167 112. 98902 87529 57698 93727 35195 26805 68769 92609 72163 11373 113. Add 8452+6854+9084+7685+1598 114. ... 82950+80847+5680+849+78+2108 115. ... 92857+84509+8467+523+26+4+8343 116. The building of St. Paul's in London cost 800,000 ; ^he Mansion House, 40,000; the Monument 13,000; Blackfriar's Bridge, 152,840 ; "Westminister Bridge, 389,000 : required the expense of these buildings ? 117. How much do the following sums of money amount to when added together: 8467, 348, 67, 280, 52, 93, 4 ? 118. A farmer laid out on oxen, 346 on horses, 568, on sheep, 984 on cows, 261 on labouring utensils, 241, how much did he lay out altogether ? 119. A Trading Company in taking stock, found that A. owed them 2487, B. 6271, C. 879, D. 976, E 8276, F. 49. and G. 3421, required the amount of their out- standing debts 120. Ten thou sand + fifteen thousand five hundred and sixty,-)- nineteen thousand and nineteen, 4- twenty-six thou- sand five hundred and ninety-five,-f-thirty-eight thousand and thirty-eight, -(-forty thousand and forty, -f-fifty-six thou- sand five hundred and two,-{- seventy thousand seven hundred and seventy-seven. SIMPLE SUBTRACTION. Subtraction is the method of finding the difference between two given numbers. The greater is called the minuend, the less the subtrahend. RULE. Write the less under the greater, with units under units, tens under tens, &c. Begin at the units, and take each figure in the subtrahend from the figure above it in the minuend, and set down the remainder ; but if any figure in the subtrahend be greater than the figure above it, add ten to the upper : subtract as before, carry one to the next, and proceed in the same manner with each figure to the end. METHODS OF PROOF. 1st. Add the difference to the subtrahend, and the sum is the minuend. 2d. Subtract the difference from the minuend, and it will give the subtrahend. Minuend. 94876859 78541903 Subtrahend. 22461234 23489769 Difference. 72415625 55051434 Proof. 94876859 78541203 1. 2. 3. 4. 84697598 92684567 85684976 95645864 21454264 21462434 41262344 42312341 5. 6. 7. 8. 76895689 84346746 60000000 81111111 24241362 31214232 42jt67483 34685716 9. 10. 11. 12. 92222222 63333333 74444444 65555555 18648346 24683768 26847367 24685694 SIMPLE SUBTRACTION. 15 13. 86666666 24684124 97777777 24684346 15. 38888888 15684376 16. 76546753 46546754 17. 85467130 24684346 18. 52456386 12456387 19. 85327637 14685468 20. 63004216 14682654 21. 810421642 268568586 22. 673124621 246854685 23. 71123426 21467567 24. 90023468 46894856 25. 261120042 146896568 26. 811240014 468427684 27. 62104264 46856847 28. 26846748 16864228 29. 56310046 41681684 30. 94210021 24689164 31. 62341042 14685468 32. 71124628 24685897 33. 64210068 16848468 34. 94200466 46816248 35. 81214240 24685684 36. 60421675 42864267 37. 24685742 18642795 38. 80246782 24684267 39. 65426976 24675489 40. 65468976 27684249 41. 80421685 46895427 42. 60042168 52869246 43. 82012654 42689256 44. 82768300 27685468 45. 50421642 42684977 46. 82164831 62847368 47. 64287021 19684278 48. 60021642 42684076 49. 82164832 62847368 50. 94287021 19684968 51. 84370582 25607854 52. 34268540 21846784 16 SIMPLE SUBTRACTION. 53. 90423406 24897485 64. 90426805 28563482 65. 50128501 21467184 66. 82124566 46858490 57. 32504168 14589276 58. 84230124 28568498 59. 92012342 24684268 60. 70423456 24346847 61. 60102012 21468946 62. 50423426 21684681 63. 28460752 24285426 64. 70123401 28416834 65. 80420126 28529364 66. 70423842 34684797 67. 82023468 21684974 68. 60012137 21459249 69. 84505742 27451270 70. 70142940 21457472 71. 54223020 23471773 72. 36234577 35797370 73. 90230420 53167931 74. 73429561 21685692 75. 80042730 21907173 76. 79323085 21960174 77. 78421915 24735251 78. 53290218 28460854 79. 90202057 27355723 80. 90320157 12705775 52003790 14281513 82. 70327510 34715709 83. 70214753 37517105 84. 25316705 23671731 85. 32572901 31753607 86. 85135791 75310705 87. 13857995 13578703 88. 35871935 10310305 SIMPLE SUBTRACTION.^ 89. Subtract 679515 from 98426758. 90. From 92467850 take 4678945. 91. Required the difference of 542762, and 1 million. 92. A person was born in the year 1769, how old was he in the year 1864, when he died ? 93. If a person's present age is 69 years, in what year was he born ? 94. Sir Isaac Newton was born in the year 1642, and died in 1727, how many years did he live P 95. Queen Victoria was born in the year 1819, how old was she when crowned in the year of 1838 ? 96. The Battle of Waterloo, was fought in the year 1815, how long is it since? 97. If a person was born in the year 1837, how old is he this year ? SIMPLE MULTIPLICATION. MULTIPLICATION is a short method of performing Ad- dition ; the number to be multiplied is called the multipli- cand', the number multiplied by is called the multiplier ; the number arising from the operation is denominated the product; both the multiplicand and multiplier are some- times c ailed factors. RULE I. "When the multiplier does not exceed 12, begin at the units place, and multiply each figure of the multiplicand by the multiplier, carrying by tens, as in Addition. If Ciphers on the right of either factors may be omitted in the work, but they must be written on the right hand of the product; hence to multiply by 10, 100, &c., annex the ciphers to the multiplicand, and the work is done. METHODS OF PROOF. 1st. Cast the 9s out of the factors separately, and multi- ply the remainders together : what remains, after casting the 9s out of this product, will be equal to what remains after casting the 9s out of the total product. 18 SIMPLE MULTIPLICATION. 2d. Multiply the multiplier by the multiplicand, and the product will be the same as before. 3d. Divide the product by the multiplier, and it will give the multiplicand. * 7. 4. 6. 3. 2. 8. 5. 9. 11. 10. 12. Proof. \ !/ 67497658 Multiplicand. 2\/7 4 Multiplier. /\ \ 269990632 1. 652489x2 16. 493857x8 31. 7812598x4 2. 485706x3 17. 742985x2 32. 6851253x10 3. 832149x4 18. 985367x3 33. 7638295x8 4. 659067x5 19. 534786x4 34. 3058724x6 6. 847562x6 20. 695767x6 35. 5964582x5 6. 793216x7 21. 782104x5 36. 6864695x4 7. 689473x8 22. 903785x7 37. 8543902x7 8. 954368x9 23. 546837x9 38. 9835493x12 9. 832749x10 24. 723572x8 39. 9031783x9 10. 648572x11 25. 785949x11 40. 8451637x9 n. 729604x12 26. 468264x8 41.7000662x9 12.468971x9 27. 498058x12 42. 9135246x9 13.756844x7 28. 868347x5 43. 4832934x8 U. 867139x6 29. 475784x3 44. 4832958x8 15. 973626x4 so. 294869x7 45. 4782514x8 r For exercising the Multiplication Table. SIMPLE MULTIPLICATION. 19 894390x8 78. 965483x9 110.622684x12 679485x7 79. 870359x3 111-854765x9 595318x7 so. 546978x4 112.482659x10 396453x9 Si. 935807x9 113.728952x11 894623x6 82.653829x7 114.540687x11 930147x6 83.549758x11 115.872545x11 897052x6 84. 756183x5 116.274389x11 946780x6 85. 923506x9 117.438920x9 678045x5 86. 537694x7 118.549764x7 987468x4 87. 793278x8 119.438587x10 842367x2 88. 785362x9 120.587327x20 842359x8 89. 230579x5 121.391872x30 967348x7 90. 846798x7 122.231975x40 349728x12 91. 458232x7 123.935017x50 928539x12 92. 847234x7 124. 654372x60 764987x12 93. 529702x7 125.902795x70 697543x12 94. 682394x8 126.753120x80 570389x2 95. 487869x8 127.921538x90 348386x4 96. 725493x8 128.379875x40 938547x5 97. 268397x8 129.765938x60 687539x3 98. 982428x9 130.639205x500 852807x11 99- 623589x9 131.782646x200 398456x6 100.438678x9 132.532768x300 859387x9 101.379427x9 133.739213x400 982705x8 102.628574x6 134.539642x500 768852x10 103. 934689x5 135.462359x600 349851x11 104. 568342x4 136.734986x600 938420x11 105.789575x3 137.654257x700 427285x11 106. 543857x2 138.367219x800 394669x11 107.972584x12 139.649538x700 697385x2 108,480729x12 140.536964x900 769521X5 109.542708x12 141.842784x100 20 SIMPLE MULTIPLICATION. RULE II. When the multiplier exceeds 12, place it under the mul- tiplicand, units below units, tens below tens, &c., then multiply the multiplicand by each figure of the multiplier, separately, observing to place tbe first figure of each pro- duct under the figure by which you multiply, then add the products. 1. 468x13 17. 467x524 33. 8467x 2009 2. 587x14 18. 785x746 34. 9046x 7005 3. 645x15 19. 923x237 35. 3897x 8006 4. 599x16 20. 5896x539 36. 5365x 5040 5. 978x18 21. 8524x426 37. 8769x 6080 6. 246x19 22. 3689x356 38. 9428x 2070 7. 389x21 23. 9205x172 39. 84687x 5096 8. 486x23 24. 3458x791 40. 90485x 7028 9. 584x34 25. 8674x581 41. 31854x 9054 10. 693x43 26. 5267x459 42. 80467X 2079 H. 924x56 27. 6082x671 43. 54825X 5200 12. 345x78 28. 5498x804 44. 68079x 76000 13. 678x95 29. 6207x306 45. 70584x 87000 4. 923x67 so. 8824x507 46. 51238x 60809 is. 456x89 31. 8837x908 47. 62459x 704080 16. 789x98 32. 5209x607 48. 78924x3200900 49. If a person's income be 32 per week, what is it per annum ? 50. How many letters are there in a book of 426 pages, and each page 1720 letters ? 61. The silk mill at Derby throws off at the rate of 24576 yards each minute, how many yards will be pro- duced in a year, working ten hours every lawful day ? 52. How many persons inhabit an island, containing 56 counties, each county 35 parishes, and each parish averaging 95 families of 7 persons each ? 21 SIMPLE DIVISION. Division is a compendious method of performing Sub- traction, or of finding how often one number is contained in another. The greater number is named the dividend, the less the divisor ; the number of times the greater contains the less is called the quotient, and what is over is named the re- mainder. RULE I. When the divisor does not exceed 12, divide from the left to the right, by finding how often the divisor is con- tained in the first, or first and second, figures of the divi- dend ; place the quotient under it ; prefix (in your mind) what remains to the next figure in the dividend"; divide a* before, and so on with each figure to the end, observing that the remainder must always be less than the divisor. METHODS OF PROOF. 1st. Cast the 9s out of the divisor and quotient, the product of which add to the remainder, if any after 9s are cast out, will be equal to what remains after 9s are cast out of the dividend. 2d. Multiply the quotient by the divisor, to the product add the remainder, if any, the sum will be the same as the dividend. 3d. Subtract the remainder from the dividend, divide the difference by the quotient, and the quotient thus found will be the same as the divisor. Dividend. Divisor, 5)46849762 Quotient, 9369952 Remainder. 6 46849762 Proof. 22 SIMPLE DIVISION. SHORT DIVISION. Dividends. Dividends. 1. 2)1677635778673 31. 3)2715465879013 2, 3)1034974921453 3. 4)2937546683543 32. 4)5276085452873 33. 10)7289057823895 4. 5)4848572413696 34. 7)9137085260617 6. 6)8596327406712 6. 7)7251588884967 7. 8)9046583713274 35. 9)3295763048573 36. 6)9176850297699 37. 5)5172375472897 8. 9)2054893553797 38. 12)3796210567425 9. 10)5736782367325 39. 11)2798004608673 10. 11)7805847299398 40. 12)7797936540021 11. 12)7948174264838 41. 12)2714301272673 12. 3)4980658115924 42. 12)9527460389425 13. 2)5145896372035 43. 12)274654789071 14. 5)9049168563976 44. 6)527792373272 15. 6)2786859648715 45. 5)122746585209 16. 5)2375868576459 46. 7)713751905429 17. 7)5782456875138 47. 3)273795183671 18. 7)4267381587428 48. 5)496225197947 19. 8)7139786246839 49. 7)125079152361 20. 11)2574854267836 so. 4)313458681427 21. 4)3713946854374 51. 3)565893542951 22. 3)1868439778719 52. 7)759679208329 23. 5)4837085728757 53. 9)126774593295 24. 2)1874378597518 54. 5)927732068017 25. 6)1251739768547 55. 6)793746398401 26. 5)7697857295714 56. 11)927824760319 27. 7)1674384654979 57. 12)576894374087 28. 2)1275283974289 58. 9)685478043187 29. 9)5214684264874 59. 9)376987643781 30. 7)2794936078757 GO. 9)548076090687 SIMPLE DIVISION. 23 61. Dividends. 9)607234271076 Dividends. 84. 5)68694372196 62. 2)703529850732 85. 6)91234764390 63. 3)932104296737 86. 7)83421676983 64. 4)478072347875 87. 8)43210642375 65. 7)923460820517 88. 9)92187684339 66. 7)268047683952 89. 10)64306581972 67. 7)904284230683 90. 11)52680908431 68. 7)428322050741 91. 12)34680647312 69. 5)984307975548 92. 2)70708968905 70. 6)684162230897 93. 3)61285429672 71. 7)807207849067 94. 4)92008540683 72. 8)912379385143 95. 10)85320175821 73. 8)453819758321 96. 100)32230913757 74. 8)312895494183 97. 20)35219753319 75. 8)32040876342 98. 90)25775231273 76. 10)50324567453 99. 70)13721271932 77. 11)97123452017 100. 70)72392379235 78. 11)98765432693 101. 200)70072719023 79. 11)34680509531 102. 300)37262977312 80. 11)92053214829 103. 700)10397951297 81. 2)94567408754 104. 500)50790210291 82. 3)72938679161 105. 900)92109211317 83. 4)48769487525 106. 100)62190180901 RULE II. LONG DIVISION. When the divisor is more than 12, draw a curve on the right and left of the dividend ; place the divisor on the left, then find how often it is contained in the first dividual ; subtract the product of the divisor and that quotient figure from this dividual, observing, as before, that the remainder must be less than the divisor ; annex to that remainder the next figure of the dividend for the next dividual, with which proceed as before, and continue in the same manner till the whole of the dividend is exhausted. 24 SIMPLE DIVISION. Dividends. Divisors.! 1. 4836375279U- 20 so 2. 69250765719-r- 2131. 3. 76742578723-=- 23132. 4. 85571489732-4- 3133. 5. 168864327513-1- 32 34. 6. 694745057317-4- 3435. 7. 563742379139-^- 5336. 8- 250765337192- 43s;. 9. 674257803713-f- 6538. 10.557148937729-1- 41 39. 11. 168864379517-4- 7640. 12. 976387629351- 6741. 13.850043967317-;- 7742. 14. 104589607923-r- 5343. 15. 205810579327 5744. 16. 785793007213-1- 9545. 17. 978548117326-4- 23 46. 18. 421116437127-4- 2647. 19. 100204467932-4- 3448. 20. 706312417539-7- 4649. 21. 875296387273-^- 57 50. 22. 200042567327-^- 6651. 23. 108379567323-4- 97 52. 24. 679760667346^. 91 53. 25. 854973107658-1. 52 54. 26. 527618785737-^. 82 55. 27. 497618707535-1- 87 56. 28. 324859645663^-12357. 29. 943759386578-^-13758. Dividends. Divisors. 7583805637-1- 267 1235579695^- 367 3765285072-f- 273 8546789634-4- 548 9065384351-4- 624 7223845809^- 783 9230428476-r- 841 3784674903-r- 958 7839546789-r- 462 6724385674^- 805 9843849265^- 907 8576984673^- 529 8457603253-^2345 8457603254H-3456 1234508423-^-5784 3948042874-1-6203 59384785464^-4380 27003214723-1-6080 98213407851-J-7109 69100824230-1-6842 87284385263^-2859 92845679560-^-1234 40872021347-1-4045 8412378560-4-31826 5876895394-4-68978 9425780945-4-92460 5216806412-.-54382 1247890436-1-67043 7419580461-1-20481 SIMPLE DIVISION. 25 RULE III. TO DIVIDE BY COMPOSITE NUMBERS. When the divisor is a composite number, divide suc- cessively by the component parts. If there be a remain- der from the last division, multiply it by the first divisor, and to the product add the remainder of the first divisor, if any ; the sum is the true remainder. Dividends. Dividends. 1. 7250975 - 14 9. 5970032 - - 35 2. 5477539 - - 16 10. 9112723 - - 36 3. 3752715 - - 18 11. 1092537 - - 42 4 7005731 - - 22 12. 9537235 - - 56 6. 7007537 - - 24 13. 2257253 - - 72 6. 7531953 - - 27 14. 7571325 - - 96 7. 7992759 - - 32 15. 9125937 - - 121 8. 7900015 - - 33 16. 9597523 - - 144 TO MULTIPLY BY COMPOSITE NUMBERS. Multiply successively by the component parts of the composite number. Multiplicands. 1. 5735790 x 16 2. 5709375 x 18 3. 2357537 x 21 4. 6123570 x 24 6. 5723790 x 32 6. 1397031 x 35 7. 8. 9. 10. 11. 12. Multiplicands. 7057351 x 5391375 x 7531575 x 8395170 x 36 42 56 96 1393597 x 121 3957130 x 144 TO MULTIPLY WHEN THE MULTIPLIER OR MULTIPLI- CAND CONTAINS A FRACTION. When the Multiplier contains a Fraction. Multiply the multiplicand by the upper figure of the frao- tion, and divide that product by the under figure, then multiply by the integers, and add the product to the quotient. 26 SIMPLE DIVISION. When the Multiplicand contains a Fraction. Multiply the upper figure of the fraction by the multi- plier, and divide that product by the under figure ; put down the remainder, if any, with the under figure of the fraction below it, and carry the quotient to the integers. 1. 84675678 x 2. 93742046 x 3. 84293481 x 4. 93428470 x 5* 61 ?i 81 5. 78493678 x 24f 6. 81576824 x 30J 7. 7847934^ x 5 8. 8410421f x 7 9. 1569847f x 9 10. 2134567J x 10 DIVISION. TO DIVIDE WHEN THE DIVISOR CONTAINS A FRACTION. Multiply both divisor and dividend by the under figure of the fraction, adding the upper figure to the product of the divisor, then divide, and if there be any remainder, di- vide it by the under figure of the fraction, and it will give the true remainder. 1. 84167696 -f- 2* 2. 46085748 -r- 5* 3. 87127680 - 2i 4. 84034769 - 71 5. 93482140 - 9 6. 84793480 - 304- 7. 58421686 - 24 8. 90285047 17f 9. 84210248 -j- 5U 10. 64230592 -f- 26* REDUCTION. REDUCTION is the method by which money, weights, and measures, are brought from one denomination to another. RULE I. To bring a number from a higher name (or denomina- tion) to a lower. REDUCTION. 27 Multiply by as many of the lower name as make one oi the higher, and to the product add the number of the lower, if any. RULE II. To bring a number from a lower name to a higher. Divide by as many of the lower as make one of the higher. RULE III. When the higher name does not contain an exact num- ber of the lower. Reduce the given name to some lower one contained in that required, then divide by as many as will reduce it to the required name. RULE I. STERLING MONEY. 1. Reduce 426 to shillings. 2. Reduce ,481 to shillings and pence. 3. Reduce 728 to shillings, pence, and farthings. 4. Reduce 74 1.7 9^d to farthings. 5. Reduce 83 7 8d to halfpence. 6. Reduce 321 guineas to sixpences. RULE II. *1. In 8520 shillings how many pounds ? 2. In 115440 pence how many shillings and pounds ? 3. In 698880 farthings how many pence, shillings, and s ? 4. In 71893 farthings how many pounds ? 5. In 40025 halfpence how many pounds ? 6. In 13482 sixpences how many guineas ? TROY WEIGHT. 1. Reduce 8524 Ibs. to ounces. 2. Reduce 5698 Ibs. to ounces and dwt. 3. Reduce 674 Ibs. to ounces, dwt., and grains. * The second six questions in each table are the answers to the first six, and vice versa. 28 REDUCTION. 4. Reduce 29 Ibs. 3 oz. 5 dwt. to dwt. 6. Reduce 72 Ibs. 8 oz. 6 dwt. 16 gr. to grains 6. Reduce 65 Ibs. 9 grains to grains. 1. In 102288 ounces how many Ibs. ? 2 . In 1367520 dwt. how many oz. and Ibs. ? 3. In 3882240 grains how many dwt. oz. and Ibs. P 4. In 7025 dwt. how many Ibs. P 5. In 418720 grains how many Ibs. ? 6. In 374409 grains how many Ibs. ? APOTHECARIES' WEIGHT. 1 . Reduce 674 Ibs. to ounces and drams. 2. Reduce 236 Ibs. to ounces, drams, and scruples. 3. Reduce 365 Ibs. to ounces, drams, scruples, and grains. 4. Reduce 64 Ibs. 8 oz. 2 dr. 1 sc. to scruples. 5. Reduce 13 Ibs. 6 oz. 7 dr. 1 sc. 18 grs. to grains. 6. Reduce 66 Ibs. 6 grs. to grains. 1 . In 64704 drams how many ounces and Ibs. ? 2. In 67968 scruples how many dr. oz. and Ibs. ? 3. In 2102400 grains how many sc. dr. oz. and Ibs. ? 4. In 18631 scruples how many dr. oz. and Ibs. ? 5. In 78218 grains how many Ibs. ? 6. In 322666 grains how many Ibs. ? AVOIRDUPOIS WEIGHT. 1. Reduce 729 tons to cwt. 2. Reduce 572 tons to cwt. qrs. and Ibs. 3. Reduce 79 cwt. to qrs. Ibs. oz. and drams. 4. Reduce 23 tons, 16 cwt. 2 qrs. 13 Ibs. to ounces. 6. Reduce 4 tons, 17 cwt. 17 Ibs. 15 oz. to drams. 6. Reduce 27 cwt. 14 Ibs. to ounces. 1. In 14580 cwt how many tons ? 2. In 1281280 Ibs. how many qrs. cwt. and tons P 3. In 2265088 drams how many oz. Ibs. qrs. and cwt P 4. In 854096 ounces how many tons ? 6. In 2785776 drams how many tons ? 6 In 48608 ounces how many cwt P REDUCTION. 29 MEASURE OF CAPACITY. 1 . Reduce 26 qrs. 5 bus. 3 pecks to pecks. 2. Reduce 51 qrs. 6 bus. 2 pecks, 1 gll. to gallons. 3. Reduce 79 qrs. 7 bus. 3 pecks, I gll. 2 qts. to quartg. 4. Reduce 37 bus. 3 pecks, 1 gll. 1 pint to pints. 5. Reduce 82 qrs. 2 bus. 2 qts. 1 pint to pints. 6. Reduce 26 bus. 1 gll to pints. 1. In 855 pecks how many quarters ? 2. In 3317 gallons how many quarters ? 3. In 20478 quarts how many quarters ? 4. In 2425 pints how many bushels. 5. In 42117 pints how many quarters ? 6. In 1672 pints how many bushels ? LINEAL MEASURE. 1. Reduce 176 miles to furlongs and poles. 2. Reduce 128 miles, 6 furlongs to poles. 3. Reduce 76 miles, 5 furlongs, 26 poles to yards. 4. Reduce 29 miles, 7 furs. 12 pis. 2 yds. 2 ft. to inches. 5. Reduce 5 furlongs, 24 poles, 2 yards, 1 foot to lines. 6. Reduce 18 leagues, 1 mile, 3 furlongs, 18 poles to feet. 1. In 56320 poles how many furlongs and miles ? 2. In 41200 poles how many miles ? 3. In 135003 yards how many miles ? 4. In 1895352 inches how many miles P 5. In 533232 lines how many furlongs ? 6. In 292677 feet how many leagues ? SQUARE MEASURE. 1. Reduce 74 acres to roods. 2 Reduce 83 acres, 2 roods, 14 poles to poles. 3. Reduce 17 acres, 2 roods, 24 poles to square yards. 4. Reduce 43 acres, rood, 12 pis. 12 sq. yds. to sq. ft. 6. Reduce 26 acres, I ro. 32 pis. 14 sq. yds. 5 sq. ft. to ft. 6. Reduce 7 acres, 16 poles, 26 square yards to sq. yards. 1. In 296 roods how many acres ? 2. In 13374 poles how many acres ? 3. In 85426 square yards how many acres ? B2 30 REDUCTION 4. In 1876455 square feet how many acres ? 5. In 1152293 square feet how many acres ? 6. In 34390 square yards, how many acres ? CUBIC OR SOLID MEASURE. 1. Reduce 231 cubic yards to cubic feet. 2. Reduce 126 cubic yards to cubic inches. 3. Reduce 85 solid yards, 17 solid feet, to solid inches. 4. Reduce 64 loads of rough timber to solid feet. 5. Reduce 59 loads of hewn timber to solid inches. 6. Reduce 29 tons measurement to cubic feet. 1. In 6237 cubic feet how many cubic yards ? 2. In 5878656 cubic inches how many cubic yards ? 3. In 3995136 solid inches how many solid yards ? 4. In 2560 solid feet how many loads of rough timber ? 5. In 5097600 solid in. how many loads of hewn timber? 6. In 1160 cubic feet how many tons measurement. CLOTH MEASURE. 1. Reduce 27 yards, 2 qrs. 2 nl. to nails. 2. Reduce 45 yards, 3 qrs. nail, 1 inch to inches. 3. Reduce 36 yards, 1 inch to inches. 4. Reduce 71 English ells, 4 qrs. 3 nl. to nails. 5. Reduce 24 Flemish ells, 1 qr. nl. 1 in. to inches. 6. Reduce 75 French ells, 4 qrs. 2 nl. to nails. 1. In 442 nails how many yards ? 2. In 1648 inches how many yards ? 3. In 1297 inches how many yards ? 4. In 1439 nails how many English ells ? 5. In 658 inches how many Flemish ells ? 6. In 1818 nails how many French ells P YARN MEASURE. Cotton. 1. Reduce 26 spindles, 3 hks. 4 sks. 26 thds. to threads, 2. Reduce 7 sp. 12 hks. 6 sks. 39 threads to inches. 3. Reduce 3 sp. 15 hks. 4 sks. 50 threads to inches. 1. In 264106 threads how many spindles ? 2. In 4196826 inches how many spindles ? 3. In 2106540 inches how many spindles? REDUCTION. 31 Lint. 1. Reduce 28 sp. 3 hsp. 4 heers, to cuts. 2. Reduce 34 sp. 2 hsp. 3 heers, 1 cut, to threads. 3. Reduce 81 spindles, 26 inches, to inches. 1. In 1388 cuts how many spindles ? 2. In 199560 threads how many spindles ? 3. In 41990426 inches how many spindles ? TIME. 1. Reduce a Julian year to hours. 2. Reduce a leap year to minutes. 3. Reduce a solar year to seconds. 4. Reduce 181 days, 11 hours, 18 minutes, to minutes, 5. Reduce 246 days, 16 hours, 6 minutes, to seconds. 6. Reduce 168 days, 16 seconds, to seconds. 1. In 8766 hours how many Julian years ? 2. In 527040 minutes how many leap years ? 3. In 31556930 seconds how many solar years ? 4. In 261318 minutes how many days? 5. In 21312360 seconds how many days ? 6. In 14515216 seconds how many days ? ANGULAR MEASURE. 1. Reduce 8 signs, 16 26' to minutes. 2. Reduce 9 signs, 21 17' 14" to seconds. 3 Reduce 5 signs, 19 8' 12" to seconds. 4. Reduce 7 signs, 2 0' 15 /x to seconds. 1. In 16386 minutes how many signs? 2. In 1048634 seconds how many signs ? 3. In 608892 seconds how many signs ? 4. In 763215 seconds how many signs ? RULE III. 1. How many guineas in 426? 2. How many pounds in 460 guineas ? 3. How many crowns in 520 guineas ? 4. How many pounds in 639 crowns ? 5. How many pounds avoirdupois in 904 Ibs. troy ? 6. How many pounds troy in 246 Ibs. avoir. 6000 grains P 7. How many yards in 724 English ells ; 8. How many Flemish ells in 621 English ells? 32 REDUCTION. 1. How many pounds in 405 guineas, 15s? 2. How many guineas in 483 pounds ? 3. How many guineas in 2184* crowns ? 4. How many crowns in 159 15s? 5. How many pounds troy in 743 Ibs. avoir. 6040 grains ? 6. How many pounds avoirdupois in 300 Ibs. troy ? 7. How many English ells in 905 yards ? 8. How many English ells in 1035 Flemish ells ? EXERCISES IN REDUCTION. 1. Reduce 94 13 8jd to farthings. 2. Reduce 46 19 11 jd to half-pence. 3. In 489474 farthings" how many pounds ? 4. How many pounds are there in 840974 pence ? 5. How many guineas are there in 97874 shillings? 6. Reduce 48 19 6d tc sixpences. 7. Reduce 98 15 4d to fourpences. 8. In 879437 half-pence how many pounds ? 9. How many guineas are there in 8473 pounds ? 10. How many pounds are there in 9476 guineas ? 11. How many pounds troy are there in 98475 grains? 12. Reduce 58 Ibs. 13 dwt. to grains. 13. How many Ibs. troy are there in 78475 dwt. 14. How many tons are there in 784939 Ibs. avoirdupois? 15. In 96 cwt. 1 qr. 13 Ibs. how many ounces ? 16. How many grains are there in 843 Ibs. avoirdupois. 17. Reduce 29 qrs. 5 bus. 2 pecks, 1 gl. to pints. 18. In 846793 inches how many miles ? 19. In 846 miles, how many feet ? 20. How many inches are there in the circumference of the Earth, or in 360 degrees ? 21. How many square feet are there in 32 acres 3 roods ? 22. How many acres are there in 2029896 square feet ? 23. How many solid inches are there in 204 tuns measure- ment ? 24. Reduce 49 yards 3 qrs to inches. 25. In 47895 nails, how many English ells ? The second 36 questions in the Exercises in Reduction are the answers to the first 36, and vice versa. REDUCTION. 33 26. In 49 days, 14 minutes, 18 seconds, how many seconds? 27. In 849756 seconds how many days ? 28. Reduce 43 45' 45" to seconds. 29. How many degrees are there in 94876 seconds ? 30. How many Ibs. troy are there in 849 Ibs. avoirdupois? 31. How many Ibs. avoirdupois are there in 937 Ibs. troy? 32. How many dollars of 5s each are there in 8436? 33. How many francs of lOd each are there in 9465 ? 34. How many rubles of 3s 4d each are there in 4681 ? 35. How many 7s 6d are there in 846 15s ? 36. How many 4s 2d are there in 928 15s P EXERCISES IN REDUCTION. 1. In 90897 farthings how many pounds ? 2. In 22559 halfpence how many pounds f 3. Reduce 509 17 4d to farthings. 4. Reduce 3504 1 2d to pence. 5. In 4660 guineas 14s how many shillings f 6. In 1959 sixpences how many pounds 9 7. How many pounds are there in 5926 fourpences / 8. Reduce 1832 3 2d to halfpence. 9. In 8069 guineas lls how many pounds f 10. In 9949 16s how many guineas f 11. Reduce 17 Ibs. 1 oz. 3 dwt. 3 grains to grains troy. 12. In 334392 grains how many Ibs. troy 9 13. Reduce 326 Ibs. 11 oz. 15 dwt. to dwt. 14. Reduce 350 tons, 8 cwt. 1 qr. 15 Ibs. to Ibs. 15. In 172688 ounces how many cwts ? 16. In 5901000 grains how many Ibs. avoirdupois ? 17. In 15208 pints how many quarters 9 18. Reduce 13 miles, 642 yards, feet, 1 inch to inches 19. In 446C880 feet how many miles 9 20. In 1585267200 inches how many degrees f 21. In 1426590 square feet how many acres f 22. Reduce 46 acres, 2 ro. 16 poles, to square feet. 23. In 14100480 solid inches how many tons meas. ? 24. In 1791 inches how many yards f ' 25. Reduce 2394 English ells, 3 qrs. 3 nis. to nla. 34 REDUCTION AND COMPOUND ADDITION. 26. In 4234458 seconds how many days ? 27. Reduce 9 days, 20 ho. 2 minutes, 36 sec. to seconds. 28. In 157545"' how many degrees ? 29. Reduce 26 21' 16" to seconds. 30. How many Ibs. avoirdupois are there in 1031 Ibs. troy. 4440 grains ? 31 . How many Ibs. troy are there in 771 Ibs. 120 grs. avoir ? 32. How many s are there in 33744 dollars, of 5s each ? 33. How many s are there in 227160 francs, of lOd each P 34. How many s are there in 28086 rubles, of 3s 4d each ? 35. How many s are there in 2258 pieces of 7s 6d each ? 36. How many s are there in 4458 pieces of 4s 2d each ? PROMISCUOUS EXERCISES. 1. How many days are there between the 6th January and 4th April P 2. How many days are there between the 14th January and llth May? 3. How many days are there between 18th February and 6th June ? 4. How many days are there between 15th March and 15th July P 5. How many days are there between 22nd April and 19th August ? 6. How many days are there between 17th May and 29th March ? 7. How many weeks are there between 15th June and 5th February ? 8. How many weeks are there between 18th January and 4th December ? 9. How many shirts will a piece of linen make, which mea- sures 52 yards, allowing 2J yards each? 10. If a person step 2j feet at a time, how many steps will he take in walking 18 miles ? 1 1. How many times will a coach wheel, of 16 J feet in cir- cumference, turn round in going from London to York, the distance being 196 miles ? 12. How many doses, each 16 grains, are in a medicine 3 Ib. 6 oz. 1 sc. 1 2 grains ? 35 COMPOUND ADDITION. COMPOUND ADDITION is the method of adding numbers of different denominations. "Write numbers of the same denomination under each other ; find the sum of the right hand column, which divide by as many of that name as make one of the next higher ; place the remainder, if any, below the column added ; and carry the quotient to the next. Proceed in the same man- ner with the remaining denominations to the last, which add as abstract numbers. s. 64 14 13 16 of. *. d. 6i 96 12 4 8 26 14 6i *. d. 87 15 6i 24 17 8| 50 17 9[ I 69 17 9| 62 17 9 Sum 129 9 i | 193 4 8J 175 11 OJ 64 14 3 i 96 12 4i 87 15 64 Proof. 129 9 0; \ 193 4 8* 175 11 Oj i 2. 3. 4. s. d. s. d. . d. s. d. 6 12 4 9 12 8f 6 13 If 9 14 4j 2 14 8* 5 14 2} 3 16 4 7 18 6| 5 13 2 1 16 5} 8 15 6J 6 16 8 3 17 8* 3 18 6* 2 16 9f 1 15 lOj 5. 6. 7. 8. 9 17 8f 8 12 10J 8 15 5f 9 19 9 7 15 3 7 13 6J 6 17 11J 5 17 11^ 2 18 9J 9 6 8J 9 13 10J 9 15 3| 2 2 5f 19 4 1 17 6J 4 1 9f : 1 r** : 36 COMPOUND ADDITION. 9 10. 11, 12. s. d. s. d. s. d. s. rf. 6 16 8 4 14 8} 9 12 6* 8 13 9} 3 5 10} 19 10* 4 16 4 4 18 5 5 8 14 4 4 19 6 9 12 2 3 10 9f 3 14 9f 4 16 o* 3 15 4f 13 14 15, 16. 9 19 3f 5 13 ' 8 i 9 15 8f 2 14 8 i 6 12 9* 1 2 7 13 4* 1 19 5 8 8 5 7 13 2 2 7 8 3" 6 19 81 3 6 6} 4 10 9* 2 2 *t 15 3} 17 18. 19, 20. 5 14 8 9 17 10} 8 18 4} 6 16 8 2 15 7f 2 19 8 7 12 10 2 3 3 J 7 8 10 6 19 7 4 15 11* 4 12 7J 2 19 0} 6 18 2* 1 5 6} 4 13 4| 21 22. 23, 24. 99 12 8} 56 18 8} 96 12 8* 62 15 8 79 19 10 29 17 10 64 14 10 29 16 11 36 19 9* 39 18 9* 75 15 8f 64 17 H| 39 19 9} 29 17 11} 74 4 3 79 12 19 12 lOir 27 10} 31 17 10* 32 18 9 25 26. 27, 28. 39 12 8 i 92 19 7 56 18 4} 63 14 6f 24 16 26 16 8} 29 7 6 29 15 11} 76 18 'i 33 17 9* 52 9 n* 54 18 9* 39 12 10 29 12 4 63 2 62 13 10 14 15 iif 66 2 10J 27 10 10} 33 18 7J COMPOUND ADDITION. 37 29. 30. 31. 32. s. d. s. d. < s. d. s. d. C6 14 9 81 11 6 72 12 8f 90 12 6} 29 14 6} 29 18 llf 9 15 10i 8 19 llf 65 6 10 76 15 3 62 13 7f 67 8 4J 29 13 7} 6 15 10 29 811 889 67 3 51 12 6 62 16 9f 81 12 6 33. 34. 35. 36. 52 18 7f 62 18 4 9018 8f 71 12 11 29 5 8* 19 12 1H 2613 3 29 13 3 64 12 11 6417 8 62 12 7 35 17 8 8 13 llf 26 12 9 58 2 6f 61 10 10| 23 12 11 43 5 4 64 5 5f 41 19 7 37. 38. 39. 40. 46 13 7J 52 13 9f 90 19 6f 36 8 4 1919 6 8 18 lO-i 24 18 11 29 15 llf 61 12 11 9317 7 62 13 5f 62 19 8^ 6814 9 74 9 lOf 26 18 4 47 8 8 26 14 1} 43 14 llf 66 7f 6 12 4f 41. 42. 43. 44. 6419 9i 37 15 6f 96 19 8f 76 19 8f 37 7 8J 2418 7f 61 11 4 23 19 llf 23 13 llf 63 12 8 2413 7f 41 12 3 62 18 7f 61 13 10J- 6 9 llf 75 15 5f 1 15 5 815 9 2418 3 20 10 lOf 27 12 Of 12 16 10J- 35 8 4f 52 19 8| 38 COMPOUND ADDITION. 45 46. 47. 48. 8. d. 8. d. 8. d. 8. d. 34 16 4 27 8 4* 36 12 ^t 71 13 3 15 17 104 8 17 9* 3 18 11 26 14 8J 38 13 7 24 19 4* 27 15 9 38 18 8 18 8f 28 5 8* 74 15 o-y- 91 2 7 42 12 73 14 7 33 13 9 44 7 ^4 18 18 4 18 10 6f 32 13 Q i 44 18 6^ 49 50. 51. 52. 27 18 4*4 39 15 8 92 19 61. 52 13 8} 3 5 10 24 2 71 78 15 Of 49 19 9| 26 17 8} 19 18 9 34 5 7 85 5 7J 76 10 6 72 10 8J 34 6 7 85 5 7 24 12 6 i 15 13 i 14 4 5 f 2 13 104 53 54. 55. 56. 62 15 ' 9 56 3 9 93 15 6* 90 12 8 29 10 2 12 19 3t 7 8 Q 1 Q-i. 79 14 6 22 13 H 29 7 8 8 8 2 24 8 U 34 10 8 39 19 8* 34 12 OJ 72 18 * 33 5 7 44 4 5f 86 6 101 10 18 2 57 58. 59. 60. 29 14 ' 8 23 6 4 96 14 81 96 8 4J 16 9 2 8 5 75 6 f 64 13 36 7 4|. 76 5 6f 27 9 2 23 4 ?f 9 14 ot 5 12 9* 38 10 8f 24 6 4 13 5 6 15 1 4 21 7 10* 31 15 44 61 , 62. 63. 64. 97 14 9 46 10 4 65 17 8 49 18 4 46 5 8 39 3 9f 40 6 8 t 28 9 Oj 48 14 H 46 17 5 27 10 29 5 8 69 5 8* 37 10 2f 68 7 9? 64 3 5f 74 6' 4 8 2 4J 7 4 64 18 9J 51 9 1 7 6 6} 25 10 11J 21 9 3-3 COMPOUND ADDITION. 39 65 66 67. 68. #. d. 8. ' d. s. d. s. d. 73 2 7 92 8 8f 78 10 8* 81 2 8 64 10 8* 76 8 10 50 7 3 68 16 &i 76 7 2f 67 8 lOf 50 7 l 68 16 5 8 6 6 27 8 6 10 9 55 17 H 32 15 6f 54 6 8* 22 16 *f 34 9 2* 8 11 10* 15 19 lOf 28 3 6* 12 6 2| 69. 70. 71. 72. 48 7 9* 89 10 if 59 17 &i 74 8 If 25 8 54 10 34 18 Ti 24 10 8 43 7 &1 24 9 ** 82 7 3 5 3 21 93 16 4 78 3 9 76 14 8* 91 17 6 76 13 10| 26 17 Hf 52 17 3 23 14 iot 62 16 H 39 18 *t 28 17 10 78 18 6 36 19 8f 26 15 4 74 13 9* 24 13 8i 78 19 3 79 18 51 52 16 8 53 16 * 22 19 9* 35 9 6* 24 18 lOf 49 17 5f 73 74. 75. 76. 46 19 '8} 93 14 81 81 14 6* 85 12 4 27 18 4 26 18 4 29 17 10 26 18 Hi 56 13 10* 74 8 5* 94 11 ^ 68 13 8i 36 17 8 23 1 10 84 6 4 65 3 82 14 9 91 13 Hf 72 13 ni 57 18 T* 26 17 8J 52 17 8* 28 10 6 72 16 10 57 12 5 98 14 7 51 15 4f 86 14 81 92 13 Hi 29 16 8* 28 15 10 24 16 8 19 1 *i 66 16 *i 51 16 8J 58 13 4| COMPOUND ADDITION. 77. 78. 79. 80. s. d. s. & 8. d. 8. d. 78 17 H 93 18 4* 68 14 i 94 17 10 26 12 11* 25 17 i 27 15 4* 26 15 81 68 16 7* 58 16 8 89 17 11 57 19 Hi 92 14 6 24 13 10* 26 14 8* 76 18 * 25 19 8* 76 14 8* 37 16 5 27 13 4 76 6 8* 27 15 6 52 16 5* 53 18 74 27 8 71 52 13 61 26 18 9* 71 2 4 54 8 9* 26 18 9f 72 3 4 73 6 44 52 4 4* 68 7| 40 19 43 68 2 14 81 82. 83. 84. 37 18 104 84 18 4} 90 13 5 85 14 10* 22 4 11* 58 13 7 78 17 4* 54 18 73 26 18 6* 26 17 8* 26 18 6 18 11 T* 17 13 8i 78 14 9 52 17 8f 52 16 8 92 14 10 21 7 5* 29 15 7 36 14 74 87 18 4 76 18 10* 75 18 4* 76 7 10 76 13 7 23 14 8 16 7 7 23 5 84 15 13 10i 26 4 91 11 16 o* 30 16 H 85. 86. 87. 88. 76 15 n t 55 17 5 92 13 54 83 14 6* 53 15 4} 17 13 10* 73 13 11 55 13 7 78 15 4 33 17 3 76 12 11 73 6 61 35 5 10* 72 14 H 21 15 04 35 12 7 53 13 5 76 3 5 57 8 1 32 15 10} 71 18 4* 68 16 10} 52 13 7* 73 5 64 37 5 3* 53 13 7* 35 12 3* 75 13 35 13 5 37 15 7 73 15 51 13 5 23 6* 38 3 6J 18 19 9 28 11* COMPOUND ADDITION. 41 89. 90. 91. 92. 31 7 4i 91 15 8 73 15 3 75 15 4^ 16 5 4 27 6 8* 34 5 9i 27 13 4 21 15 6 63 5 3i 26 12 6i 28 13 5J 75 13 10 58 17 5i 65 15 7 34 5 3 57 13 5i 53 7 3 53 7 3 35 17 4f 75 15 3f 75 4 If 75 13 4 73 5 7* 53 14 75 3 5 53 15 7 53 6 3 53 13 Of 73 15 3J 563 33 5 7| 57 7 5 73 3 7j- 70 15 3J 75 13 9 15 2 0^ 64 8 ]1|- 39 9 5f 48 2 0| OF WEIGHTS AND MEASURES. TROY WEIGHT. 1. 2. 3. 12. 20. 24. lb. oz. dwt. gr. lb. oz. dwt. gr. lb. oz. dwt. gr. 24 8 13 4 64 8 5 7 83 11 2 4 18 4 18 12 27 9 14 15 26 9 12 18 36 7 12 19 63 8 19 8 51 6 14 15 26 10 15 20 36 8 12 17 58 3 18 9 17 3 7 5 27 10 17 14 27 8 19 17 6 3 14 16 36 10 10 16 57 1 9 10 APOTHECARIES WEIGHT. 4. 5. 6. 12. 8. 3. 20. lb. oz. dr. sc. gr. lb. oz.dr.sc.gr. lb. oz.dr.sc.gr. 68 5 3 2 14 73 9 3 2 16 81 2 7 15 13 6 5 1 18 54 4 5 2 19 26 9 2 1 12 39 8609 21 1 4 1 17 51 6 5 2 17 27 7 7 1 16 76 9 7 1 15 31 8 6 2 9 61 11 4 2 17 37 4 6 2 12 68 4 1 1 14 54 10 6 16 19 4 5 2 17 54 5 4 2 3 42 COMPOUND ADDITION. AVOIRDUPOIS WEIGHT. 7. 20. 4. 28. 16. ton.cwt.qr. Ib. oz. 89 14 2 34 27 8 3 21 2 52 15 1 17 9 61 17 1 18 8 28 9 3 23 5 62 5 2 10 2 8. ton.cwt.qr. Ib. oz. 62 8 2 27 3 16 11 1 18 10 74 9 3 22 8 53 7 2 23 14 46 9 1 14 13 45 17 1 8 9 9. cwt.qr. Ib. oz. dr. 91 2 10 12 12 56 3 16 10 14 29 1 23 11 13 48 2 24 6 15 61 17 13 8 34 2 22 1 14 MEASURE OF CAPACITY. 8. qi. bu. 56 6 28 4 73 5 79 5 51 3 28 2 10. 4. 2. pk.gl. 3 1 1 2 1 1 1 2 2 1 Vr- 20 01 2 3 1 1 1 qr.bu. 81 2 36 5 52 6 57 7 29 6 44 4 11. pk.gl.qt.pt. 1021 2130 1021 3120 2131 2031 qr. bu. 72 4 54 6 36 3 56 2 49 5 17 5 12. pk.gl.qt.pt 2121 3031 2120 3031 2121 3031 LINEAL MEASURE. 13. 8. 40. 5*. 3. ml. fu. po. yd. ft. 78 2 36 5 1 52 6 27 4 27 4 38 3 2 49 6 23 2 2 56 3 17 4 1 25 4 911 14. ml. fu. po. yd. ft. 52 4 28 5 1 26 6 31 4 93 7 16 2 2 92 5 12 3 2 64 3 12 4 25 5 37 1 1 16. ml. fu. po.yd.ft 52 3 32 5 1 26 5 21 3 2 69 7 19 2 1 68 2 17 3 2 47 6 29 4 1 25 6 11 1 2 COMPOUND ADDITION. 43 SQUARE MEASURE. 16. 17. 18. ac. ro. po. yd. ft. ac. ro. po. yd. ft. ac. ro. po. yd. ft. 82 2 27 28 4 72 2 37 22 5 85 3 36 26 5 27 1 36 19 5 46 3 21 15 4 56 2 28 19 7 64 3 28 17 6 65 2 36 18 8 64 1 18 23 6 29 3 8 22 3 27 2 18 23 7 56 2 17 18 4 51 2 39 21 7 63 1 24 16 6 47 2 9 15 3 55 31 88 25 3 16 71 29 1 S 67 CUBIC OR SOLID MEASURE. 19. 20. 21. c.yd. c.ft. c.in. c.yd. c.ft. c.in. c.yd. c.ft. c.in. 89 13 198 76 14 100 80 12 112 54 21 281 37 22 649 23 23 569 23 19 126 54 18 427 89 16 455 25 14 649 52 21 684 63 20 856 64 21 537 68 23 491 82 24 487 34 19 817 38 19 851 55 16 648 22. yd. qr. nl. in. 76 2 3 2 29 2 2 2 56 3 1 1 56 1 3 1 62 3 2 1 47 1 CLOTH MEASURE. 23. En.ell. qr. nl. in. 92 2 2 1 46 3 3 1 58 4 2 2 54 4 1 2 97 4 3 2 45 2 3 34. Fr.ell. qr. nl. in. 71 4 2 1 46 2 3 1 57 5 2 2 78 5 2 2 64 2 1 1 25 1 3 25. COTTON. sp. hk. sk. th. in. 57 4 6 71 26 24 16 1 37 41 79 3 6 58 49 56 12 5 52 27 YARN MEASURE. 26. LINT sp. hp.hr. ct. th. 53 2 4 1 29 26 2 5 64 54 3 3 1 27 97 3 4 87 27. sp. hp. hr.ct. th. 84 3 2 95 25 2 3 1 64 92 3 5 29 59 1 3 1 63 27 15 2 24 18 25 1 3 1 62 27 3 2 1 78 82 6 5 33 39 26 3 5 85 59 4 1 31 COMPOUND ADDITION. WOOL WEIGHT. 28. 12 2 6| 2 14 29. 30. la. sk.wy.td.st. Ib. la. sk.wy.td.st.lb. la. sk.wy.td.st. Ib. 24 3 1 4 1 8 86 81518 90 5 1 6 1 5 10 9 3 13 23 10 1 3 4 26 8 2 1 10 29 7 1 4 1 5 74 90416 57 4 1 4 12 71 7 1 4 27 51219 52 7 2 1 6 854141 7 64 70304 48 4 1 2 4 13 6 1 1 9 62 10 2 1 4 63 9 1 3 1 9 ANGULAR MEASURE. 31. 32. 33. circ. 8. ' ' circ. B. ' circ. s. ' ' 48 4152416 89 8 21 15 18 74 10 21 32 32 26 10 21 18 18 42 925 436 51 4281718 72 9241635 38 4171641 4911152436 49 5192424 29 7183428 72 9144142 37 8222222 16 5312736 56 7215837 21 524 554 46 10 26 10 42 23 5231514 TIME 34. 35. 36. yrs.*da.ho. m. se. yrs. da. ho. m. se. yrs. da. ho. m. se. 56 91 2 56 8 97 15 21 52 26 74 92 22 26 45 4787 5247 48 7 16 27 36 48'31 1852 1(7 56 93 4 33 8 51 6 5 39 52 37 13 14 38 27 7442 6276 58 4 7 6 8 72 54 9 33 53 36 85 7 85 64 9 29 53 43 49 87 7 27 48 9 321 321 49 8 5 25 50 26 61 3 34 29 365 days = 1 year. COMPOUND ADDITION. 45 USES OF COMPOUND ADDITION. 1. A debtor paid to A L46 18 4jd, to B L39 15 6|d, to C L93 15 6fd, to D L928 19 Hid, to E L5 13 7d, to F L51 19 8d, how much did he pay in all? 2. A gentleman's house-rent is L50 10s, land-tax L5 5, house duty L4 18 lld, income tax L2 18 6*d, po- lice tax L4 12 8d, poor's rates L3 16 8^d, water and city charges L3 15 9d, what is the amount of his rent and taxes ? 3. Find the rent of an estate of six farms, the first is rented at L369 18s, the second at L476 19 IQid, the third a* L269, the fourth at L176 15 9d, the fifth at L208 12s, and the sixth at L629 13 4d. 4. A gentleman owes his wine merchant L49 18 4d, draper L38 15 10 Jd, confectioner L12 13 lOd, coach- maker L246, brewer L16 13 8d, butcher L52 16 8|d, baker L31 14 9d, tailor L21 13 6|d, shoemaker L25 18 6fd, servant's wages L48 19s, how much money will be required to pay the accounts? 6. A merchant received the following sums: L246 18 10 Jd, L329 5 8^d, L385 19 6d, L478 19 8fd, L476 9s, L451 16 9d, L38 5s, how much did he receive in all? 6. What is the amount of L426 18 9|d, L128 14s, L539 17 lUd, L623 14 llfd, L529 18 IQid, L476 18 9d, L920 14 lOd? 7. A gentleman's silver pi ate weighs as follows, viz. : dishes and covers, 64 Ibs. 5 oz. 14 dwt. 18 grains; spoons and ladles, 28 Ibs. 10 oz. 17 dwt. 18 grs. ; a tea-pot, 4 Ibs. 8 oz. 10 dwt. 12 grs.; salts, 3 Ib. 4 oz. 17 gr.; trays, 10 Ib. 8 oz. dwt. 16 gr. ; candlesticks, 24 Ibs. oz. 18 dwt. grs. ; forks and other articles, 38 Ibs. 1 oz. 18 dwt. 15 grs. what was its total weight? 8. The produce of a sugar plantation was in the first year, 38 tons 7 cwt. 1 qr. 24 Ib.; next year it was 48 tons 16 cwt. 3 qrs. 19 Ibs.; in the third year 32 tons, 18 cwt. 15 Ibs.; in the fourth year 44 tons 1 qr. 23 Ibs.; and in the fifth year 35 tons 18 cwt. 18 Ibs., what was the total pro- duct? 9. A merchant imports 346 qrs. 4 bu. 3 pks. wheat, 428 C 46 COMPOUND ADDITION. qrs. 6 bu. 2 pk. barley, 748 qrs. 7 bu. 1 pk. oats, 268 qrs, 4 bu. 3 pks. pease, 86 qrs. 1 pk. beans, 126 qrs. 2 bu. 2 pk. rye, how much grain has he to pay freight for ? 10. A gentleman enclosed five fields, and found that the length of the wall surrounding the first was 2 fur 18 po. 4 yds. 2 ft.; the second 4 fur. 39 po. 3 yd. 1 ft; the third 3 fur. 3 yd. 2 ft. ; the fourth 1 fur. 3 po., and the fifth 3 far. 19 po. 4 yds. 1 ft, what length of wall was required to enclose the whole ? 11. A farm consisted of five fields, the first measured 34 ac. 2 ro. 26 po. 18 sq. yd.; the second, 26 ac. 1 ro. 24 po. 20 sq. yds., the third 49 ac. 29 po., the fourth 45 ac. 2 ro. 22 sq. yd., the fifth 19 ac. 2 ro. 15 po. ; how many acres were in the farm ? 12. A draper has in his shop, of black cloth 268 yds. 2 qrs % 2 nl., of blues 257 yds. 3 qrs. 2 nl. 2 in., scarlets 256 yds, 2 nl, 1 in., browns 125 yd. 1 qr. 2 nl., mixtures 261 yds., and of other colours 629 yds. 1 nl 1 in., how much has he in all P "3. A man who engaged to perform five pieces of work, fi- nished the firstin 28 days 8 ho. 34 min.; the second in 26 days 11 ho. 28 min.; the third in 19 days 13 ho.; the fourth in 38 days 17 min., and the fifth in 48 days 16 ho. 42 min,, in what time di'l he execute the whole? COMPOUND SUBTRACTION. COMPOUND SUBTRACTION is the method of finding the difference between two compound quantities. BULE. Place the less quantity under the greater, with numbers of the same name- under each other; subtract the under number of each denomination from the upper : but when the nnder number of any name exceeds the upper, subtract it from the number of that name which makes one of the next COMPOUND SUBTRACTION. 47 higher ; then add the upper to the difference ; cany one t * the next under name, and proceed in the same manner with each denomination to the last, which subtract as abstract, numbers. . 8. d. . s. d. . s. d. . s. d. 86 19 10} 84 16 8 71 13 4 95 16 8 61 13 9J 26 8 4 29 16 8* 21 12 4 25 6 1J 58 8 4 41 16 n 74 4 4 86 19 10} 84 16 8 71 13 4 95 16 8 1. 2. 3. 4. s. d. s. d. s. d. s. d. 53 17 6} 73 15 8 75 12 4 37 16 3| 35 18 3 53 5 94 52 15 Oj 31 14 6J 52 42 5. 10 23 16 6. 8 52 22 7. 16 64 51 36 8. 15 7i 92 27 9. 3 12 6i 83 26 10. 5 14 74 52 28 11. 15 17 24 7 63 47 12. 13 16 2} 7 61 46 13. 5 16 $ 8f 56 24 14. 13 15 2 84 67 46 15. 7 17 54 81 48 16. 1 13 7} 51 48 17. 17 16 6 7i 32 24 18. 11 14 34 8 24 8 19. 6 11 10 64 37 24 20. 3 14 1 8* 63 46 2.1. 4 13 84 7 62 24 22. 15 4 84 77 26 23. 8 12 54 8 71 64 24. 11 14 8* 48 COMPOUND SUBTRACTION. 25. 26. 27. 28. s. d. s. d. s. d. 8. d. 82 6 24 53 18 7 7l 12 64 83 17 5| 26 10 7 47 6 44 26 4 4 68 4 7| 56 27 29. 13 11 7 44 72 26 30. 14 18 HI 4 82 27 31. 4 13 24 6 32. 85 12 31 8 14 64 73 28 33. 4 16 If 8* 45 26 34. 16 18 5 54 64 26 35. 2 18 54 44 36. 52 17 26 8 6 104 81 26 37. 17 8 4 71 53 24 38. 8 17 6 5* 57 24 39. 4 17 5 9* 73 25 40. 7 18 104 5f 32 17 41. 5 18 64 84 63 28 42. 15 14 10 61 32 24 43. 7 8 64 4 83 28 44. 4 16 6* 4* 64 28 45. 14 16 4f 5 77 28 46. 5 14 6f 34 65 49 47. 3 16 84 10 55 26 48. 3 17 5 64 59 25 49. 6 5 4* 4-| 82 27 60. 17 15 6* a 56 27 51. 18 18 84 !i 67 26 52. 12 12 24 34 35 27 63. 16 15 34 H 63 37 54. 11 19 3 T 5f 85 57 65. 6 15 14 5 53 27 56. 9 10 34 5f 52 37 57. 7 17 ? 57 55 58. 3 17 3} 54 25 7 59. 18 17 24 6 38 8 60. 7 17 3J 3j COMPOUND SUBTRACTION. 71 26 61. 8. 4 17 d. 89 48 62. s. 5 12 d. B 62 48 63. 18 4 d. 9 6| 64. s. 82 15 28 15 d. 21 64 89 56 65. 15 7 74 Of 65 28 66. 17 13 9 54 85 45 67. 12 LOi 3 59 24 68. 19 11 7 74 91 24 69. 12 8 74 7 63 29 70. 16 11 7 71 85 46 71. 5 19 44 54 70 25 72. 6 14 41 89 24 73. 6 3 89 26 74 18 *0 Q i 92 57 75. 4 f 67 24 76. 17 01 4i 30 77. 4 75 78 'o H 57 79. 6f 53 80. 1 64 40 27 81. 2 16 24 9 63 56 82. 5 12 3 8* 80 26 83. 10 17 4 93 25 84. 12 15 24 3 80 37 85. 5 15 6 74 83 26 86. 10 12 3 5f 98 25 87. 11 17 n 66 25 88. 16 17 6j 54 75 27 89. 13 15 74 4 32 18 90. 3 19 7 11| 63 37 91. 12 17 3 88 65 92. 15 8 6 1 78 49 93. 10 18 v* 4J 99 21 94. 16 5 Of 77 24 95. 15 15 ^ 58 27 96. 2 14 5 50 COMPOUND SUBTRACTION. 97. 98. 99. 100. L s. d. s. d. a. d. 8. d. 52 4 5 81 1 6f 90 14 8f 58 12 9j 28 15 IQj. 29 13 8 29 19 4| 27 19 4 101. 102. 103. 104. 59 3 3 75 3 41- 93 10 If 92 7 3 28 17 9f 27 19 7J- 29 15 10 13 9 1| 105. 106. 107. 108. 72 5 9} 50 71 Oi 93 4 6J 29 14 9| 29 18 6f 27 5 3J- 25 19 9* 53 27 109. 5 19 1 H 93 29 110. 2 15 2| Ii 71 37 ill. 15 9 8 ?f 37 27 112. 5 13 ? 41 23 113. 12 15 4 79 21 114. 13 19 3f ?f 70 27 115. 13 15 1! 53 25 116. 5 17 4f 10} 72 27 117. 13 17 1 7 Of Of 9| 99 25 118. 1 18 3 53 28 119. 17 14 ? 71 37 120. 13 19 8 88 31 121. 5 1 19 75 17 122. 19 11 19 5i 97 37 123. 13 9 51 85 82 124. 19 17 4 53 31 125. 3 15 5 51 73 37 126. 3 5 4 50 17 127. 9 If 53 39 128. 5 17 Of 129. 130. 131. 132. 93 5 lOf 59 3 10J 90 17 3 83 18 5| 25 13 9 35 19 41 39 15 3 75 9 3| 51 COMPOUND SUBTRACTION OF WEIGHTS AND MEASURES. 1. lb. 02. dwt. gr. 84 2 15 12 26 8 18 16 TROY WEIGHT. 2. lb. oz. dwt. gr. 92 3 8 14 27 9 12 17 3. lb. oz. dwt. gr. 64 4 12 18 27 8 18 12 APOTHECARIES' WEIGHT. 4. 5. 6. lb. oz. dr. sc. gr. lb. oz dr. so. gr. lb. oz. dr. so. gr. 62 4 3 1 16 52 5 5 2 14 71 1 2 2 15 46 8 5 18 29 7 5 1 19 27 6 6 1 12 7. AVOIRDUPOIS WEIGHT. 8. 9. tonscwt.qr.lb.oz. dr. 62182151015 241232014 6 tonscwt.qr.lb. oz. dr. 82 6021 814 231221212 8 tonscwt.qr.lb. oz. dr. 9512216 815 231311810 5 MEASURE OF CAPACITY. 10. 11. 12. qr. bu.pk.gl.qt.pt. qr. bu.pk.gl.qt.pt. qr. bu.pk.gl.qt.pt 62 211 20 81 52011 55 72100 24 6 2 3 1 27 6 2 1 2 27 2 3 1 1 LINEAL MEASURE. 13. 14. 15. ml. fu. po. yds. ft. in. ml. fu. po. yd. ft. in. ml. fu.po. yd. ft. 76 4 27 3 1 8 92 2 16 4 1 5 83 3 26 5 1 29 5 36 2 2 9 27 6 28 2 2 8 27 6 32 2 2 SQUARE MEASURE. 16. 17. ac. ro. po. yd. ft. in. ac. ro. po. yd. ft. in. 52 1 27 29 6 84 42 1 12 28 2 19 28 2 31 17 8 25 28 2 24 13 4 37 18. ac. ro. po yd. ft. 62221 27 6 57 3 12 14 6 52 COMPOUND SUBTRACTION. CUBIC OR SOLID MEASURE. 19. 20. 21. c.yd. c.ft. c,in. c.yd. c.ft. c.in. c,yd. c.ft. c.in, 85 12 82 92 18 59 72 15 43 46 18 122 46 20 134 26 19 152 CLOTH MEASURE. 22. 23. yd. qr. nl. in. yd. qr. nl. in. 52 2 2 2 62 2 2 2 27 3 3 1 37 3 3 24. yd. qr. nl. io* 59 3 1 2 24 2 1 COTTON. 25, sp, lik. sk. th. in. 24 12 3 26 23 15 15 4 12 41 YARN MEASURE. LINT. 26. 27. sp.hp.hr,ct.th. in. sp.hp.hr.ct.th. in. 62 2 3 1 68 40 59 3 5 62 18 39 3 4 27 50 24 1 3 1 39 20 28. 12 2 6| 2 14 la. sk. wy.td.st. Ib. 64 4 6 1 10 27 10 1 4 12 WOOL WEIGHT. 29. la. sk. wy.td.st. Ib. 70 5 1 4 8 39 9 2 1 13 30. la. sk. wy.td.st. Ib. 54 7 5 9 49 8 1 3 1 11 31. ANGULAR MEASURE. 32. 33. 99 5 14 26 18 46 8 19 41 34 56 2 10 20 30 21 9 17 25 15 77 1 22 18 2 59 9 17 27 17 34. yr. da. ho. m. se. 61 240 19 28 14 37 121 20 46 52 TIME. 35. yr. da. ho. m. fie. 89 129 15 39 8 46 83 19 46 15 36. yr, da. ho. m. se, 50 329 8 5 6 45 136 23 46 30 ' 365 days = 1 year. COMPOUND SUBTRACTION. 53 USES OF COMPOUND SUBTRACTION. 1. Borrowed L846 13 2d, and paid in part L369 18 6|d, what is still due ? 2. Lent L528, and received back L129 14 6d, how much is due ? 3. A gentleman's income is L870 a-year, and he spends L523 16 10|d, how much does he save P 4. A gentleman's estate came to L68.426, he left to each of his three daughters L9000, to each of his two sons L 12. 000, and the rest to his eldest son, what was his por- tion ? 5. A bankrupt owed to one of his creditors L946 18 4 id, to another L678 14 7d, to a third L731 16 8d, to a fourth L826 17 9d, to a fifth L300 ; his effects were valued at L1927 13 8|d, how much was he deficient ? 6. A gentleman pays his grocer L126 13 4d, shoemaker L56 13 lid, tailor L85 14 9|d, wine-merchant L241 7 8|d, clothier L59 8 5|d, servants wages and taxes L256 13 8d; his income is L1000, how much does he save ? 7. Subtract id from L100. 8. Subtract 2d from L60. 9. What is the difference between 1 guinea and a half, and L10? 10. A silversmith bought 761bs. 8 oz. 13 dwt, 14 gr. of sil- ver, of which he made 49 Ibs. 9 oz. 4 dwt. 18 gr. into table spoons, how much has he remaining ? 11. A merchant bought 12 cwt. 2 qr. 14 Ibs. of tea, of which he sold 8 cwt. 24 lb., how much has he on hand ? 12. A merchant bought 26 tons of sugar, of which he sold 19 tons, 16 cwt. 1 qr. 15 Ibs., how much has he on hand ? 13. A piece of silk measured 124 yds. 2 qr., and there were sold at different times 92 yrds. 3 qr. 3 nls., how much re- mained unsold? 14. From London to Edinburgh is 398 miles, 3 fur. 20 po., and from London to York 200 miles, 5 fur. 25 po., how far is York from Edinburgh? 15. A gentleman's estate measures 4827 acres, and he has let of it in farms 3125 ac. 3 ro. 15 po. 16 sq, yds., how much remains in his own hands ? c2 COMPOUND MULTIPLICATION. COMPOUND MULTIPLICATION is the method of multi- plying a compound quantity by a simple number. RULE I. "When the multiplier does not exceed 12, place it under the lowest denomination of the multiplicand, then multiply and carry as in compound addition. s. d. s. d. s. d. 6 8 61 4 12 8* 8 16 2f 3 4 6 19 5 8i 18 10 9 52 17 4| 1. 13. 25. s. d. Mul. s. d. Mul. s. d. Mul. 24 12 Six 2 12 13 10|X 9 12 16 lOiXlO 2. 14. 26. 12 18 5x 3 41 16 ll^X 7 12 18 7X4 3. 15. 27. 53 19 6|X 4 53 9 9X 8 23 16 4!x 5 4. 16. 28. 41 16 SIX 5 21 12 6*X 4 34 7 Six 7 5. 17. 29. 82 18 6ix 6 43 8 IHXIO 50 18 4ix 2 6. 18. 30. 53 14 9iX 7 91 9 71X11 32 8 10 X12 7. 19. 31. 61 17 10IX 8 40 19 Six 2 42 8iX 3 8. 20. 32. 43 15 1HX 9 60 15 9iX 9 21 12 WhX 8 9. 21. 33. 24 19 2X 8 57 18 lOix 8 92 18 Six 7 10. 22. 34. 12 13 32-X 9 42 17 8x 7 80 17 10^X12 11. 23. 35. 45 14 41X12 59 12 7iXl2 71 15 9IX 9 12. 24. 36. 26 18 4iX 6 20 15 0X9 23 12 Six 6 B I V JB COMPOUND MULTIPLICATION. 55 37. 55. 73. s. d. Mul. s. d. Mul. s. d. Mul. 2 18 4tx 8 61 19 10tX 7 20 9 4ixlO 38. 56. 74. 2 8 4iX 8 85 12 10lx 7 20 8 31X10 39. 57. 75. 5 11 5ix 5 21 18 4lx 9 30 4 Six 7 40. 58. 76. 7 8 Xll 36 15 lllx 7 48 9 2iX 41. 59. 77. 2 15 6X6 62 12 4ixlO 36 8 0x 7 42. 60. 78. 5 8 4ix 9 24 17 81X12 124x2 43. 61. 79. 6 16 9IX 7 18 9 4x3 2 3 4iX 2 44. 62. 80. 4 4ixl2 26 17 lOix 5 356x8 45. 63. 81. 12 9ixlO 5 14 84X11 4 13 5iX 5 46. 64. 82. 134X2 22 16 4!x 8 2 17 6iX 8 47. 65. 83. 3 3 6fx 5 36 5 Six 7 5 3 4JX 2 48. 66. 84. 556X8 33 17 0X6 3 8 5!X 5 49. 67. 85. 4 13 5iX 5 9 17 lOix 2 396X9 50. 68. 86. 2 17 6iX 8 63 4 8iX 6 2 17 Six 7 51. 69. 87. 5 14 21X12 5 5 Six 3 1 15 10IX 3 62. 70. 88. 12 18 81X12 26 6 lOix 3 4 15 IHx 9 53. 71. 89. 6 14 7kX 7 22 18 9iX 9 8 17 6x9 64. 72. 90. 9 16 6lx 6 28 9 4iXlO 9 14 lOiX 9 56 COMPOUND MULTIPLICATION. 91. 98. 105. 1. s. d. Mill. L. s. d. Mul. L. s. d. MuL 6 18 IHx 9 2 17 4ixll 27 18 11 X 3 92. 99. 106. 5 15 6 XlO 3 16 lOiXll 14 13 Six * 93. 100. 107. 8 19 SiXlO 28 17 6ixl2 18 17 IGiX 5 94. 101. 108. 9 14 74X10 59 18 11 X12 21 19 6x6 95. 102. 109. 5 18 6 XlO 34 16 8iXl2 38 14 92X 7 96. 103. 110. 9 15 lOiXll 26 18 9 X12 12 11 8X8 97. 104. 111. 5 18 6 Xll 32 15 8iX 2 32 19 5iX 9 RULE II. To multiply by a composite number, multiply successively by the component parts. 1. Find the value of 24 yards cloth at 16s 8d each, 2. Find the value of 32 yards silk at 13s 6d 3. Find the value of 48 yards satin at 12s 6Jd 4. Find the value of 30 yards scarlet at 16s 9| 6. Find the value of 42 yards velvet at 18s 6. Find the value of 44 yards drab at 13s 7 7. Find the value of 64 yards calico at 2s 8. Find the value of 88 yards linen at 2s lOd 9. Find the value of 77 yards lace at 3s 8jd 10. Find the price of 84 Ibs. tea at 6s 9d 11. Find the price of 81 Ibs. green tea at 10s 5d 12. Find the price of 45 Ibs. fine tea at 12s 9|d 13. Find the price of 49 Ibs. Bohea at 11s 8d 14. Find the price of 80 Ibs. sugar at Is 2jd 15. Find the price of 120 sugar loaves at 12s 8d 16. Find the price of 121 gallons rum at 14s 3|d 17. Find the price of 132 gallons gin at L.I 10 8d 18. Find the price of J 44 gallons brandy at L.I 11 10|d. COMPOUND MULTIPLICATION. 57 19. Find the price of 15 tons at L.2 13 6d each. 20. Find the price of 21 cwt. at L.I 12 8jd 21. Find the price of 25 tons at L.3 14 9|d 22. Find the price of 27 cwt. at L.I 19 2^d 23. Find the price of 33 qrs. at L.2 13 9Jd 24. Find the price of 35 roods at L.I 12 lOd RULE III. To multiply by prime numbers. Take the nearest composite number and multiply successively by its component parts, then multiply the multiplicand, by the difference and add the products. 1. "What is the price of 13 tons at L.4 16 8jd each? 2. What is the price of 17 tons at L.5 18 lOd 3. What is the price of 19 tons at L.8 10 6d 4. What is the price of 23 tons at L.2 15 8|d - 5. What is the price of 26 tons at L.9 18 7Jd 6. What is the price of 29 tons at L.4 13 lid 7. What is the price of 31 cwt. at L.3 18 4Jd 8. What is the price of 34 cwt. at L4 17 5f d 9. What is the price of 37 cwt. at L.5 18 9^d 10. What is the price of 38 cwt. at L.8 13 9d 11. What is the price of 39 cwt. at L.2 17 llfd 12. What is the price of 41 cwt. at L.5 14 8^d 13. What is the price of 43 cwt. at L.7 18 lOd 14. What is the price of 46 cwt. at L.9 17 6|d 15. What is the price of 47 cwt. at L.2 13 8^d 16. What is the price of 51 cwt, at L.4 18 6d 17. What is the price of 52 cwt. at L.2 4 lOd 18. What is the price of 53 cwt at L.6 13 ll|d 19. What is the price of 57 cwt. at L.2 18 7d 20. What is the price of 58 qrs. at L.I 14 8d 21. What is the price of 59 qrs. at L.2 16 4d 22. What is the price of 61 qrs. at L.7 13 6d 23. What is the price of 62 qrs. at L.3 13 3^d 24. What is the price of 65 qrs. at L.4 16 8d 25. What is the price of 67 qrs. at L.3 19 6d 26. What is the price of 69 qrs. at L.4 18 7d COMPOUND MULTIPLICATION. 27. What is the price of 73 acres at L.5 16 8d each ? 28. What is the price of 74 acres at L.8 4 7|d 29. What is the price of 75 acres at L.9 15 6 id ,, 30. What is the price of 76 acres at L.2 18 7d ,, 31. What is the price of 82 acres at L.3 4 8}d 32 What is the price of 87 acres at L.5 16 9^d 33. What is the price of 89 acres at L.I 12 6|d 34. What is the price of 93 hhds at L.4 13 11 id 35. What is the price of 95 hhds at L.2 16 8}d 36. What is the price of 98 hhds. at L.4 17 6id 37. What is the price of 103 hhds. at L.3 12 7d 38. What is the price of 107 hhds. at L.I 16 8d 39. What is the price of 111 hhds. at L.6 18 IQid MULTIPLY BY COMPOSITE & PRIME NUMBERS MIXED Find the value of Tons. s. d. Cwt. s. d. 1. 14 at 4 18 4^ each 16. 40 at 9 15 7| each 2. 15 at 6 13 10| , 17. 43 at 2 19 84 3. 16 at 3 16 Hi 18. 44 at 6 14 10 4. 17 at 6 12 8| > 19, 48 at 2 19 8} t 5. 18 at 4 17 7 , 20. 50 at 5 12 4 9 6. 19 at 2 14 8i , 21. 62 at 7 13 Ul w 7. 20 at 3 19 61 , 22. 60 at 3 17 6 M 8. 23 at 2 18 jrs 23. 57 at 9 18 51 9. 24 at 4 17 8 24. 60 at 2 17 6 4 M 10. 26 at 5 2 10} , 25. 63 at 8 14 10} w 11. 28 at 8 14 8i 26, 68 at 9 16 6* ,, 12. 32 at 2 1C 9| 27. 72 at 2 18 9 M 13. 47 at 1 4 5 28. 75 at 5 19 8f |f 14. 38 at 7 7 6 29. 84 at6 12 11 M 15. 39 at 8 14 10* 30. 100 at 9 15 8} .; RULE III. When the multiplier consists of several fi- gures, multiply by as many tens minus one as there are figures in the multiplier, then multiply the last product by the highest figure of the multiplier, the preceding pro- duct by the next inferior figure, and so on with the rest, then the sum of the products of these figures will be the answer. COMPOUND MULTIPLICATION. 59 EXAMPLE. Multiply 2 17 9Jd. by 3G5. Multiply 2 17 9^x5= 14 8 10^= 5 times. by 10 10times= 28 17 8^x6=173 63 = 60 times. 10 100 times=288 17 1 x3=866 11 3 =300_ times, 1054 6 4^=365 times. 1. Find the value of 694 yards cloth at 18s 6d per yard. 2. Find the value of 396 yards sail cloth at Is lO^d 3. Find the value of 432 yards linen at 2s 4^d 4. Find the value of 971 yards flannel at 2s 2*d ,, 5. Find the value of 841 yards cambric at 8s 4d 6. Find the value of 945 yards calico at lO^d 7. Find the value of 927 yards silk at 12s 8d 8. Find the value of 891 yards velvet at 19s 10*d 9. Find the value of 161 yards scarlet at 14s 6d 10. Find the value of 683 yards lace at 3s 10|d 1 1. Find the value of 304 yards satin at 11s 8*d 12. Find the value of 609 gallons whisky at 8s 9d per gall. 13. Find the price of 2964 gall, fine do. at 12s 8id 14. Find the price of 8001 gall. Glenlivet at 13s lOd 15. Find the price of 6890 gall, rum at 16s 8 id 16. Find the price of 8304 gall, brandy at 28s 4d 17. Find the price of 5965 gall, gin at 30s 2d 18. Find the price of 9567 gall, port wine at 12s 4 id ,. 19. Find the price of 2002 gall, sherry at 12s 10*d. 20. Find the price of 1002 gall. do. at 11s lid 21. Find the price of 1462 Ib. green tea at 10s 6|d per Ib 22. Find the price of 2102 Ib. do. do. at 11s 8d 23. Find the price of 1004 Ib. black do. at 6s 6d 24. Find the price of 1962 Ib. sugar at 8d 60 MULTIPLICATION OF WEIGHTS AND MEASURES. 1ROY WEIGHT. Ib. oz. dwt. grs. 1. 4 5 16 8 x 7 2. 5 10 13 16 x 40 3. 2 11 17 9 x 34 APOTHECARIES'. Ib. oz. dr. sc. gr. 4. 9 9 3 2 14 x 5 5. 2 5 18 x 22 6. 4 8 6 2 11 x 37 AVOIRDUPOIS. tons. cwt. qr. Ib. oz. 7. 2 10 1 16 llx 3 8. 6 14 3 24 14x35 9. 3 15 2 17 12x43 MEASURE OF CAPACITY. qr.bu.pk.gll.qrt.pt. 10. 5 7 2 1 2 1 x 8 11. 2 5 3 3 1 x 49 12. 8 4 1 1 1 1 x 53 LINEAL MEASURE, ml.fu. po. yd. ft. 13. 5 6 32 4 2 x 9 14. 6 4 26 3 1 x 56 15. 5 7 27 2 x 73 SQUARE MEASURE. ac.ro. po. yd. ft. 16 6 2 21 20 5 x 2 17. 3 1 35 15 3 x 64 18. 4 3 26 12 6 x 85 I CUBIC OR SOLID MEASURE, c.yd. c.ft. c.in. 19. 61 21 68 x 3 20. 24 17 100 x 70 21. 9 13 112 x 79 CLOTH MEASURE. yd. qr. nl. in. 22. 6 2 2 2 x 4 23. 5 3 3 x 80 24. 3 2 2 x 87 YARN MEASURE. (COTTON.) sp. hk. sk. th. in. 25. 6 12 4 35 24x 9 26. 4 13 5 61 32x96 (LINT.) sp. hp.hr. ct. th. in. 27. 8 3 4 1 64 10x 5 28.5 2 5 97 34x81 TIME, yr, *da. ho. mi. se. 29. 4 126 18 39 43x 11 30. 8 204 17 23 39x108 31.7 98214517x476 ANGULAR MEASURE, c. s. ' 32.8 7 25 45 45x 10 33. 9 6 7 30 26x144 34. 7 4 21 48 48x645 * oU5 dajs = 1 year 61 COMPOUND DIVISION. COMPOUND DIVISION is the method of dividing a com- pound quantity by a simple number. RULE. Divide the highest denomination of the given dividend by the divisor, and reduce the remainder, if there be any, to the next inferior denomination, adding the given number of that name : divide this as before, and proceed in the same manner to the lowest denomination. s. d. s. d. s. d. 2)67 3 41- 3)53 13 2J 4)78 15 2 " ~ ~~ 33 11 8J 17 17 8f 19 13 9J l. 13. 25. s. d. $. d. s. d. 42 12 8i+ 2 54 11 lli+ 7 99 5 4-2 2. 14. 26. 84 15 9 + 3 63 17 5i+ 6 68 2 9i 5 3. 1& 27. 78 12 61+ 4 29 8 7f+ 2 77 13 2 -16 4. 16. 28. 87 19 4i+ 5 65 Qi+ 4 26 9 6 6 5. 17. 29. 55 8 7i+ 6 41 1 8 + 7 88 4 7i- 7 6. 18. 30. 77 10 11 + 7 33 12 6f+ 3 69 15 2-9 7. 19. 31. 64 11 7J+ 8 61 10 9 + 9 79 15 7i- 5 8. 20. 32. 59 17 4J+- 9 58 6f+ 2 68 19 2 -11 9. 21. 33. 29 14 8J+10 69 1 81+11 74 5 9 - 6 10. 22. 34. 59 6 61+11 75 9 5i+ 5 69 12 101-10 11. 23. 35. 73 13 4 +12 68 71+10 76 11 6 + 4 12. 24. 36. 61 12 2-1-5 57 16 8 + 4 81 17 7f+ 9 62 COMPOUND DIVISION. 37. 55. 73. s. d. s. d. s. d. 64 8 11 + 3 90 16 8 + 2 568 + 3 38. 56. 74. 58 + 7 78 18 61+ 5 640 +10 39. 57. 75. 91 1 8J-5-12 96 17 8J+ 7 731 +11 40. 68. 76. 65 + 2 64 12 +11 429 9 9 + 2 41. 69. 77. 76 + 9 97 11 4 + 8 377 + 6 42. 60. 78. 33 8 6J+ 8 78 10 9 + 3 427 7 7 + 9 43. 61. 79. 68 1 5 +12 99 11 10i+ 7 500 8 +12 44. 62. 80. 85 15 3i+ 4 97 9 4 + 4 401 2 8i+ 5 45. 63. 81. 71 2 + 5 54 2 + 5 674 12 8J+ 3 46. 64. 82. 83 14 8i+ 7 268 1 6f+ 2 584 11 11 +12 47. 65. 83. 94 18 10i+ 3 347 6 li+ 9 648 14 4 + 9 48. 66. 84. 76 19 llf+ 4 499 11 8 +12 964 10 10 +11 49. 67. 85. 68 15 9 +11 248 7 + 9 428 5 4 + 5 50. 68. 86. 96 + 9 562 12 +10 820 + 3 51. 69. 87. 58 6 +12 920 8 Oi+ 5 428 6 +12 62. 70. 88. 69 8 + 6 827 10 Si-*.- 6 849 12 41+ 9 63. 71. 89. 9811 61+12 420 13 4 + 7 526 8 + 3 54. 72. 90. 87 4 + 8 98416 8i+ll 929 18 + 2 COMPOUND DIVISION. 63 RULE II. When the divisor is a composite number, divide successively by the component parts. 1. Find the value of 1 yd. of cloth at L26 16 8d for 24 yds. 2. Find the value of 1 yd. of silk at L18 14 4|d for 30 3. Find the value of 1 yd. of satin at L16 17 10dfor32 4. Find the value of 1 yd. of scarlet at L39 18 6d for 42 6. Find the value of 1 yd. of velvet at L41 17 S^d for 44 6. Find the value of 1 yd. of drab at L30 16 2d for 48 7. Find the value of 1 yd. of calico at L8 10 4d for 64 8. Find the value of 1 yd. of linen at L10 12 5d for 72 9. Find the price of 1 Ib. of tea at L28 16 4|d for 84 Ibs. 10. Find the price of 1 Ib. of green tea at L43 8 2d for 81 11. Find the price of 1 Ib.of sugar loaf at L7 12 8jd for 120 12. Find the price of 1 gall, rum at L78 19 4d for 121 gall. 13. Find the price of 1 gall, gin at L187 16 6d for 132 14. Find the price of 1 gall, brandy at L190 2 6d for 144 ,, 15. Find the price of 1 ton at L2 10 13 6d for 96 tons RULE III. LONG DIVISION. 1. Find the value of 1 qr. of wheat at L96 14 8d for 23 qrs. 2. Find the value of 1 qr. of oats at L51 18 6d for 34 3. Find the value of 1 qr. barley at L86 17 4d for 43 4. Find the value of 1 qr. pease at L108 13 9d for 52 5. Find the value of 1 qr. of beans at L142 15 6d for 67 6. Find the value of 1 qr. of rye atLlOO 14 8^d for 73 qrs. 7. Find the value of 1 ton at L978 for 47 tons 8. Find the price of 1 cvrt. at L789 for 75 cwts. 9. Find the price of 1 qr. when 85 qrs. cost L478 9s 10. Find the price of 1 ton when 92 tons cost L211 16 8d 11. Find the value of 1 acre at L6478 13 Od for 75 12. Find the value of 1 rood of land at L2158 19 6d for 86 13. Find the price of 1 cwt. of iron at L4785 16 9d for 243 14. Find the price of 1 qr. of tea at L3376 for 364 15. Find the price of 1 ton when 261 tons cost L6364 5 8d 16. What is the price of 1 cwt. when 475 cwt. cost L8738 6Jt COMPOUND DIVISION. 17. "What is the price of 1 rood when 287 roods cost L3493 5s ? 18. "What is the price of 1 acre when 172 acres cost L4807 19 6^d P 19. Find the value" of 1 ton when 342 tons cost L6782 14 1 Id 2G, Find the value of 1 ton when 527 tons cost L1000 8 4d 21. What is the price of 1 acre when 138 acres cost L2846 13 lOf d ? 22. "What is the price of 1 ton when 113 tons cost L1200 18 6d? 23. What is the price of 1 cwt. when 365 cwt. cost L2846 13 4d ? 24. What is the price of 1 ton when 472 tons cost L5287 18 9d? 25. What is the price of 1 qr. when 905 qrs. cost L9188 14 lOJd ? 26. What is the price of 1 cwt. when 571 cwt. cost L3009 !70dP 27. Find the value of 1 acre at L2108 12 9d for 608 acres 28. Find the price of 1 ton at L5209 13 9d for 52> tons 29. Fimi the price of 1 ton at L8207 14 7d for 934 tons 30. Find the value of 1 cwt. at L2842 17 8^d for 205 cwt. 31. Find the value of 1 qr. when 59 qrs. cost L826 16 4d 32. Find the price of 1 qr. when 26 qrs. cost L378 10 6d 33. Find the price of 1 qr. when 241 qrs. cost L926 4 Od 34. Find the price of 1 cwt. when 324 cwt cost L879 5 Id 35. Find the price of 1 ton when 426 tons cost L934 14 8d 36. Find the price of 1 cwt. when 51 cwt. cost L877 19 Od 37. What is the price of 1 furlong when 26 furlongs cost L926 14 8dP 38. What is the price of 1 acre when 62 acres cost L896 14 6id? 39. What is the price of 1 rood when 71 roods cost L700 13 OdP 40. What is the price of 1 pole when 111 poles cost L921 19 8d ? 41. What is the price of 1 yard when 222 yards cost L526 4 Od P 42. What is the price of 1 cwt. when 333 cwt. cost L940 16 Id ? 43. What is the price of 1 cwt. when 444 cwt. cost L777 ? COMPOUND DIVISION. 44. What is the price of 1 ton when 57 tons cost L934 5 6d ? 45. "What is the price of 1 yard when 555 yards cost L847 9 4|d ? 46. What is the price of 1 yard when 666 yards cost L920 13 8|d? 47. What is the price of 1 ton when 43 tons cost L528 4s ? 48. What is the price of 1 cwt. when 78 cwt. cost L634 ISSJd? 49. What is the price of 1 qr. when 365 qrs. cost L888 18 8d ? 50. What is the price of 1 cwt. when 424 cwt. cost L999 19 9d ? 51. What is the price of 1 ton when 521 tons cost L849 13 4d? COMPOUND DIVISION OF WEIGHTS AND MEASURES. TEOY WEIGHT. Ib. oz. dwt. gr. 1. 67 5 17 10-^- 2 2. 85 11 17 22-^-14 3. 96 8 15 19-23 APOTHECARIES'. Ib. oz. dr, sc. gr. 4. 89 6 5 2 18-5- 3 5. 95 8 7 13^-18 6. 70 10 3 1 16+34 AVOIRDUPOIS. tons cwt.qr.lb. oz. dr. 7. 68 14 2 16 10 lO-i- 4 8.56110221414+20 9.97 8 1 26 15 15-5-37 MEASURE OF CAPACITY. qr.bu.pk.gll.qt.pt. 10. 54 3 2 1 2 O-i- 5 11. 68 7 3 1 l-s-24 12. 84 4 1 1 3 l-s-43 LINEAL MEASURE. ml .fur. po. yd. ft. 13. 58 7 22 4 2 + 7 14. 65 5 36 3 -5- 32 15. 80 4 15 2 2 + 46 SQUARE MEASURE. ac.ro. po. yd. ft. in. 16. 54222 4216-1- 8 17. 60 3 30 13 7 28-35 is. 99 2 15 18 5 63-5-52 COMPOUND DIVISION. CUBIC OR SOLID MEASURE, c.yd. c.ft. c.in. 19. 420 14 146-*- 9 20. 846 21 100-5-42 21. 684 18 846-^58 CLOTH MEASURE. yd. qr. nl. in. 22. 246 2 2 2-;-10 23. 163 3 3 2-J-45 24. 346 3 0-^-62 YARN MEASURE. (COTTON.) sp. hk.sk. thr. in. 25. 60 16 4 31 32-f-ll 26. 72 10 2 54 30-^64 (LINT.) sp.hp.hr.ct.th. in. 27. 63 3 5 19 62-^-12 28. 90 2 4 1 54-T-56 TIME. yr, d. h. m. se. 29. 85 92 18 36 36+- 9 so. 72 55 20 44 44-^96 31. 82 36 21 45 45-^-98 ANGULAR MEASURE. c. s. ' ' 32. 70 9 16 34 34-4- 7 33. 63 6 15 25 21-^-81 34. 84 10 21 50 50+-86 MULTIPLICATION AND DIVISION, When the Multiplier or Divisor contains a fraction. MULTIPLICATION. To multiply when the multiplier contains a fraction. Multiply the multiplicand by the integer ; then multiply the multiplicand by the upper figure of the fraction, and divide that product by former product. the under, and add the result to the L. s. d. L. s. d. I. 2 4 84 X 2i 5. 3 15 8* X 22! 2. 5 16 10" X 3* 6. 8 17 9 X 26| 3. 6 18 4* X 5| 7. 9 13 6J X 264J 4. 7 13 8 X 8} 8. 10 9 8 X 365} MULTIPLICATION AND DIVISION. 67 DIVISION. To divide when the divisor contains a fraction. Multiply both divisor and dividend by the under figure of the fraction, adding the upper figure to the product of the divisor, then divide. L. s. d. i 624 6 4J 5J 2. 963 18 10 4J 3. 874 14 6 3f * 568 15 8 2f 6. 985 6 If L. s. d. 6. 968 5 6* -r- 7. 827 6 10" + 8. 468 17 9 -5- 46| 9. 879 13 Of -f- 17| 10. 940 18 6 -f- 26i USES OF COMPOUND MULTIPLICATION. 1. What is the price of 2 yards at 5s 6|d, per yard? 2. What is the price of 5 yards at 14s 8*d 3. What will 7 Ibs. of tea cost at 4s 10d per Ib? 4. What will 12 fibs of sugar come to at 6d per K> ? 6. Find the price of 9 pairs of shoes at 9s 8d per pair ? 6. Find the price of 11 hats at 19s 6d each. T. What cost 16 yards of linen at 2s 5d per yard? 8. What cost 17 yards of cloth at 18s 8|d per yard? 9. Calculate 18 weeks' wages at 16s 4d a-week ? 10. Calculate 19 days' wages at 3s 2d.a day? 11. What will 27 reams of paper cost at 26s 6d per ream ? 12. What will 31 Arithmetics cost at Is 6d each? 13. Required the cost of 42 Ibs. of coffee at 2s 2d each Ib. 14. Required the price of 46 Grammars at 2s 9d each. 15. Find the price of 48 Ibs. green tea at 8s 6d per Ib. 16. Find the price of 52 Ibs. cheese at lOjd per Ib, 17. Find the rent of 60 acres at 2 16 8d per acre. 18. What is the price of 68 handkerchiefs at 4s lO^d each ? 19. What cost 72 reams of paper at 23s 10|d per ream ? 20. What cost 79 stones of wool at 14s 8|d per stone? 21. What cost 81 quarters at 2 4 6d per quarter ? 22. Find the price of 95 yards of cloth at 12s 8d per yard. 23. If 1 yard of cloth cost 17s 5Jd whatwill 100 yards cost? 24. If 1 stone of wool cost 13s 8d, whatwill 106 stones cost ? 25. Calculate the price of 120 yards of muslin at 2s 2d per yard. 68 MULTIPLICATION AND DIVISION. 26. What cost 7 thousand quills at 15s 6d per thousand P 27. Find the rent of 260 acres at 2 18 8d per acre. 28. Find the cost of 368 cwt. at 9s lld per cwt. 29. Find the price of 39 tons at 1 11 6d per ton. 30. A man earns 21s 4^d a week, what has he a-year? 31. A person earns 5s 8d a day, how much has he a-year ? 32. A person spends 14 18 6d a-month, what does he spend a-year? 33. Find the price of 5* yards of cloth at 16s 4d each. 34. Find the price of 24| Ibs. of tea at 4s lOd each. 35. What cost 426 Ibs. of sugar at 8d per Ib? 36. What cost 31 1 yards of linen at Is lOd each ? 37. What cost 50 1 yards of cambric at 3s 6d each P 38. What is the price of 82| reams of paper at 18s 6d each ? 39. What is the price of lOOf yards of lace at 2s 9Jd 40. What is the price of 43f yards of calico at 2s 10d,, 41. What is the weight of 9 ingots of silver, each 2 Ib. 8 oz. 14 dwt. 16 grs ? 42. What is the weight of 24 guineas, each 5 dwt. 9 grs ? 43. What is the weight of 8 parcels of medicine, each 1 Ib. 8 oz. 5 dr. 1 sc. 16 grs ? 44. What is the weight of 26 hhds. tobacco, each 6 cwt. 1 qr. 18 Ibs ? 45. What length of road will a man make in 6 days, at 1 pole, 4 yds. 2 feet, 3 in. per day ? 46. How large is an estate consisting of 10 farms, each 84 acres, 3 roods, 26 poles, 28 yards ? 47. How much flour will a mill grind in 313 days, at 22 qrs. 3 bu. 2 pecks per day ? 48. How far will a man travel in 40 days, at 24 miles, 5 fur. 16 poles per day ? USES OF COMPOUND DIVISION. 1. Divide 841 13 5d among 5 men. 2. Divide 1000 into 7 equal shares. 3. Divide a prize of 847 13s among 10 seamen. 4. Divide 920 among 12 persons. 5. If 16 yards of cloth cost 18 10s, what is the price per yard ? 6. If 17 cwt. cost 42 13 10 id, what is the price of 1 cwt ? 7. If 18 tons cost 97, what is the price per ton ? COMPOUND DIVISION. 69 8. If 20 Ibs. of tea cost 4 12s, what is the price per ft P 9. "What cost 1 ft. of tea, when 26 Ibs. cost 5 13 8d? 10. What cost 1 cwt. when 30 cwt cost 48 ? 11. Find the value of 1 yard of cloth at 36 for 32 yards. 12. Find the value of 1 yard of silk at 54 6s for 46 yards. 13. Find the price of 1 yard of linen, at 10 11 4d for 88 yards. 14. Find the price of 1 ft. green tea, at 42 18s for 95 Ibs. 15. "What is the price of 1 gallon rum at 79 13s for 121 ? 16. What is the price of 1 gallon brandy at 191 5s for 145 gallons. 17. If 47 Its of tea cost 34 10 3|d, what is that per ft ? 18. A farm of 58 acres is let for L120 14 8d, what is the rent per acre? 19. What may a person spend a-week out of an estate of L1000 a-year P 20. A merchant in 12 years gained L10,476 how much did he gain annually? 21. What cost 1 ft. of tea when 65 Ibs. come to LI 6 10 7-d? 22. What cost 1 ft. of sugar, when 96 Rs. cost L4 4s ? 23. If 43 tons cost L58, what will 1 ton cost ? 24. If 100 yards of linen cost L12 10s, what will 1 yard cost ? 25. If 5i yards cost L6 14 8id, what is the price of 1 yard ? 26. If 2i tons cost L50 19 6d, what is the price of 1 ton ? 27. If 30^ cwt cost L59 16 4d, what will 1 cwt. cost? 28. If 15| yards of cloth cost L16 18s, what will 1 yard cost? 29. If 3 Ibs of soap cost Is 2d, what cost 1 ft ? 30. 4 yards cost L3 18 9|d, what is the price of 1 yard ? 31. A piece of cloth, at 6s 8d the yard, cost 28 16 8d, how many yards were in it ? 32. A gentleman gave 10 among some poor people, giv- ing each 5s 6d, how many poor were there ? 33. The revenues of an hospital amount to 6486, how many patients will it maintain, when each patient requires 8 14 6d? 34. If 7 ingots of silver weigh 12 Ib. 5 oz. 14 dwt. 6 grs., what is the weight of 1 ingot ? 35. If 12 guineas weigh 5 Ib. 5 oz- 6 dwt. 3 grs., what is the weight of 1 guinea ? D 70 COMPOUND DIVISION. 36. If 7 parcels of medicine weigh 11 Ib. 6 oz. 5 drs. 1 sc. 18 grs., what is the weight of 1 parcel ? 37. If 38 casks weigh 24 tons, 16 cwt. qr. 18 Ibs., what is the weight of 1 cask ? 38. Divide 4126 acres among 58 men. 39. Divide 85 qrs. 5 bu. 3 pks. 1 gl. 3 qts. 1 pt. among 7 persons, and give each an equal share. 40. Divide 246 days, 16 ho. 18 mi, 18 se. into 31 equal parts. BILLS OF PARCELS. A Bill of Parcels is an account of goods given when bought, shewing their quantity and price. 1. A B Bought of C D 4 yards lawn at 2s lOd per yard. 6 yards shalloon at Is Sd 8 yards linen at Is 6d. 10 yards serge at Is 8d 11 yards lace at 10s 9d 12 yards muslin at 6s 10 fd 6 yards cambric at 12s 6d per yard. 8 yards muslin at 7s 8d 4 yards satin at 8s 4d 7 yards sarsenet at 18s 6d 10 yards flowered silk at 16s 4u 12 yards velvet at 18s 6d 12 11 2. Bought of G H 34 4 6 BILLS OF PARCELS. 71 3. Bought of K- 14 yards nankeen at 3s 8d per yard, 15 yards cashmere at 12s 3d, 16 yards fine black cloth at 26s 4d, 18 yards fine blue cloth at 28s 2d, 22 yards superfine scarlet at 24s 8d, 24 yards drab cloth at 16s lOd, 105 10 1 4. M N Bought of P 20 yards green silk damask at 18s 4*d, 30 yards do. do. at 17s 8d 40 yards cambric at 10s 8|d, 60 yards black silk at 11s 4|d, 60 yards taffeta at 5s 6d, 100 yards brocaded satin at 17s 5jd, 198 9 2 5. Q R Bought of S T ^- 17 Ibs. green tea at 8s 7d 26 Ibs. black tea at 6s 6d 37 Ibs. ' aw sugar at 8 Jd 46 Ibs. coffee at Is 8d 59 Ibs. fine coffee at 2s 10d 68 Ibs. sugar at Is 2d 33 7 6. U V Bought of W X 69 gallons whisky at 7s 9d 74 gallons fine do. at 10s 6|d 87 gallons rum at 14s 8d 97 gallons fine do. at 18s 4d 98 gallons gin at 30s 6d 106 gallons brandy at 30s 8|d. 530 17 53 73 BILLS OF PARCELS. 7. Y - Z - Bought of A- 246 gallons Holland gin at 32s 6d 474 gallons Highland whisky at 9s 8|d 627 gallons grain do. at 6s 4d 764 gallons malt do. at 8s lOf 872 gallons brandy at 24s 2d 964 gallons fine do. at 32s 9d 3800 8 6 D 437 Ibs. hard soap a 379 Ibs. soft do. at 5d 264 Ibs. starch a 129 Ibs. raisins a 564 Ibs. rice at 3|d 699 Ibs. cheese at lOjd Bought of E - F -- G H 746 Ibs. raw sugar at 7d 839 Ibs. refined do. at Is 2*d 576 Ibs. tea at 6s lOd 698 Ibs. green tea at 12s 6*d 700 Ibs. starch at 8d 850 Ibs. soap 9. Bought of I 84 a Ibs. coffee at 2s 7d 96? Ibs. almonds at Is 3d 72 1 Ibs. raisins at 8jd 108* Ibs. cocoa at 2s 6d 25J Ibs. hops at Js 4d 36 L Ibs. cheese at 9d 10. Bought of M 753 9 N - 36 1 2? BILLS OF PARCELS. 73 11. P Bought of Q R- 246* reams thick post at 33s 4d 574f reams thin do. at 21s 9d 246* reams foolscap at 19s 2d 3 76 1 reams printing demy at 16s 8d 468J reams superline royal at 42s 8d 100| reams cartridge at 14s 6d 2,656 14 5 SIMPLE PROPORTION. SIMPLE PROPORTION is the method by which a fourth proportional number is found to three other given numbers, go that the third shall have to the fourth the same ratio the first has to the second. Ratio is the relation which one number bears to another with respect to magnitude. This relation can only exist between quantities of the same kind, thus : yards yards, . as 12 : 6 : : 8 : 4 Four numbers are proportional when the first contains the second, as often as the third contains the fourth, or when the product of the extremes is equal to the product of the means. The first and fourth terms are called the extremes. The second and third the means. Questions in this rule are of two kinds, viz. DIRECT and INVERSE. Direct proportion is when more requires more, or less requires less. Inverse proportion is when more requires less, or less requires more. RULE For stating and working the three given terms. Place, for the third term, that number which is of the same kind with that required, then consider from the nature 74 SIMPLE PROPORTION. of the question, whether the answer must be greater or less than that number. If greater, place the greater of the other two terms for the second, and the less for the first ; but if less, place the less for the second, and the greater for the first, with two points (:) between the first and se- cond, and four (: :) between the second and third terms. Reduce the two first terms to the same denomination, then multiply the second and third terms together, and divide that product by the first ; the quotient is the fourth proportional in the same denomination as the third term, and may be reduced to a higher name if necessary. PROPORTION DIRECT. If 2 yards 1 quarter of broad cloth cost 16 4d, what will 3 yards cost ? yards. qr. yards j. s. d. 2 1 : 3 :: J 6 4 4 4 12 9 12 9)15 16 1 15 li | PROPORTION INVERSE. If 40 men perform a piece of work in 90 days, in vrbat time will 12 men do the same ? men. men. days. 12 : 40 :: 90 40 12)3600 300 days 1. If 6 yards of cloth cost 4, what will 24 yards cost? 2. If 24 yards of cloth cost 16, what will 6 yards cost? 3. If 6 yards of cloth are bought for 4, how many yards may be bought for 16 ? 4. If 24 yards are bought for 16, how many yards may be bought for 4 ? 5. If 16 yard? cost 12 14 6d, what will 24 yards cost ? SIMPLE PROPORTION. 75 f). If 18 yards cost 16 18 lOd, what will 30 yards cost ? 7. If 20 yards cost 17 13 6*d, what will 15 yards cost ? 8. If 25 yards cost 14 16 8^d, what will 40 yards cost ? 9. If 30 yards of linen cost 3 1 4|d, what will 50 yards cost? 10. If 48 yards of calico cost 1 2 5*d. what will 96 yds. cost? 11. If90lbs.ofteacost161382d,whatwilll001bs.cost? 12. IfSOibs. of tea cost 14 13 6d, what will 200 Ibs. cost ? 13. If 150 Ibs. of sugar cost 3 4 8d, what will 70 Ibs. cost ? 14. If 140 Ibs. of soap cost 3 15 8d, what will 80 Ibs. cost ? 15. If 248 Ibs. of soda cost 1 11 6^d, what will 84 Ibs. cost ? 16. If 38 Ibs. of snuff cost 5 12 8d, what will 48 Ibs. cost ? 17. If 180 cwt. cost 6 18 9^d, what will 264 cwt. cost ? 18. If 428 cwt. cost 18 4 lO^d, what will 84 cwt. cost ? 19. If 24 qrs. cost 2 15 6*d, what will 184 qrs. cost? 20. If 60 qrs. cost 4 18 9d, what will 480 qrs. cost ? 21. If 60 yards of linen cost 8, what will 245 yards cost ? 22. If 30 yards of velvet cost 28, what will 25 yards cost ? 23. If 8 gallons of rum cost 7, what will 24 gallons cost? 24. If 95 cwt. of sugar cost 384, what will 57 cwt. cost ? 25. If 26 gallons of brandy cost 2 7, what will 38 gallons cost ? 26. If 28 gallons of rum cost 22, what will 16 gallons cost ? 27. If 240 gallons of whisky cost 104, what will 850 gallons cost f 28. If 260 cwt of sugar cost 642, what will 230 cwt. cost f 29. If 42 Ibs. of soap cost 21, what will 846 Ibs. cost? 30. If 500 Ibs. of tea cost 98, what will 1000 Ibs. cost f 31. How much wine may be bought for 396, if 90 gallons cost 72 9 32. How many yards of cloth may be bought for 426, when 6 yards cost 3 16s ? 33. How many gallons of wine may be bought for 396, i 90 gallons cost 72. 34. How many yards of broad cloth may be bought for 85, if 6 yards cost 5? 76 SIMPLE PROPORTION. 35. How many yards of linen may be bought for 124, if 4 yards cost 9s 6d. 36. How many Ibs. of tea may be bought for 83, when 3 Ibs. cost 12s 8d? 37. How many Ibs. of sugar may be bought for 75, if 2 Ibs cost 9id ? 38. How many Ibs. of soap may be bought for 98 16s, if 1 Ib. cost 6d? 39. How many cwt. of sugar maybe bought for 126 12s, if 3 Ibs. cost Is 2^d ? 40. How many tons of soap may be bought for 428, if 12 Ibs. cost 6s 4 |d? 41. How many cwt of tea may be bought for 267 10 8d, if 3 Ibs. cost 12s 4d? 42. If 6| yards of lawn cost 14s 8J--3, what will 94 yards cost? 43. If 13 yards of linen cost 1 2 4d, what will 7 yards cost? 44. If 5| yards of broad cloth cost 3 4 6|d, what will 100 yards cost? 45. If 28 Ibs. of tea cost 4, what will 101 Ibs cost? 46. If 6 1 tons cost 5 14s, what will 29 tons cost? 47. If 24 yards of cambric cost 12 7s, what will 12| yards cost? 48. If 6 yards 1 qr. of cloth cost 4 5 6|d, what will 10 yards cost? 49. If 18 yards 2 qrs of linen cost 1 5s, what will 120 yards cost? 50. If 8 Ibs. of tea cost 1 5 6d, what will 2 cwt. 1 qr. 18 Ibs cost? 51. If 5 Ibs. of sugar cost 2s 6d, what will 3 cwt. cost? 52. If 1 Ib. of soap cost 6d, what will 1 ton cost ? 53. If 8 Ibs of tea cost 1 12 4d, what will 4 cwt. 2 qrs. cost? 54. If 6 cwt. of sugar cost 16 6s, what will 3i Ibs cost ? 55. If 3 ounces of silver cost 18s 6d, what will 8 Ibs. 7 oz. 16 dwt. cost ? 66. If 6 quarters 2 bushels of wheat cost 21 13 6d, what will 2 quarters cost? 57. If 54 masons can build a house in 90 days, how many masons would it require to do it in 12 days? SIMPLE PROPORTION. 77 53. If 14 men could make a ditch in 18 days, in what time could 34 men do it ? 59. A ship was provisioned for a crew of 40 men for 4 months, how long would these provisions last, if the crew were reduced to 36 men ? 60. If 12 horses can subsist on a certain quantity of hay for 4 months, how long would 18 horses subsist on the same quantity ? 61. If 24 men finish a piece of work in 200 days, in what time would 60 men finish the same? 62. If 48 masons finish a house in 108 days in what time would 36 masons finish it ? 63. If a journey can be performed in 22 days, travelling 16 hours a-day, in how many days of 14 hours each will the same journey be performed? 64. If I lend a friend 140 for 6 months, how long should he lend me 240, to return the favour ? 65. A garrison had provisions when besieged for 30 days, at the rate of 24 ounces per day to each man ; what will be the allowance to each man per day, that the provisions may serve for 60 days ? 66. If 6| bushels of pease cost 12 7s, how much will 18 bushels cost ? 67. If 48 persons perform a piece of work in 120 days, in what time will 24 persons perform the same. 68. If 3 tons 14 cwt. 1 qr. of flax, cost 150, how much will 8 cwt, cost ? 69. If 62 masons build a house in 248 days, in what time will 88 masons do the same ? 70. How many yards of cloth may be bought for 94 if 3 yards cost 12s 8|d. 71. If 38 persons reap a harvest in 46 days, how many per- sons will reap it in 16 days ? 72. If 13 spindles 2 hasps of lint cost 1 8 2d, how much will 58 spindles cost? 73. If 3 spindles of lint cost 18s 4id, how much will 27 spindles, 2 hasps, 4 heers, cost P 74. If 2 spindles of lint cost 6s lO^d, how much will 15 spindles cost? 75. If 27 yards of cloth cost 16 12 8d, how much will 86 English ells cost? D2 78 SIMPLE PROPORTION. 76. If ] 8 Flemish ells of linen cost 2 4 9d, how much will 20 French ells cost? 77. If 21 French ells of cambric cost 8 9 6d, how much will 15 yards cost ? 78. If 16 English ells 3 qrs. of silk cost L4 13 8d, how much will 10 Flemish ells cost? 79. Find the price of a silver cup weighing 2 Ibs. 6 oz., at 11s 6d per ounce? 80. Find the price of a silver tea pot weighing 2 Ibs. 6 oz, at 11s 8^(1 per ounce. 81. Find the price of a silver snuff box weighing lib. 8 oz., at 12s 6d per ounce. 82. How much silver plate may be bought for L827 16s, when 6 ounces cost L3 9 3d ? 63. How many gallons of rum can be bought for L768, when 5 gallons cost L3 10s ? 84. A bankrupt's debt amounts to L2468, and his effects to L974, how much will his creditors receive per pound ? 85. A bankrupt's debt amounts to L6480, pays 8s 4d per pound, how much will his creditors receive ? 86. A bankrupt gives his creditors 6s lOd per pound, pays them for a discharge L1260, required the amount of his debt? 87. What is the height of a steeple, whose shadow is 163 feet 9 inches, when a shadow 5 feet 2 inches is pro- jected from a staff 6 feet high ? 88. "What is the height of a tree whose shadow is 100 feet 7 inches, when a shadow 5 feet is projected from a staff 4 feet high? 89. After seeing a flash of lightning, 12 seconds elapsed before the thunder was heard, required the distance sound moving at the rate of 1142 feet per second ? 90. Standing on the top of the Law at Dundee, I observed the flash of a cannon from a ship at the mouth of the Tay, 54 seconds elapsed before I t heard the report, required the distance ? 91. Observing the flash of a cannon fired by a ship in dis- tress at sea, 30 seconds elapsed before the report was heard how far is she off? SIMPLE PROPORTION. 79 92 . How much water must be mixed with 94 gallons of rum, at 16s per gallon, to reduce the price to 14s per gal- lon? 93. How much water must be mixed with 180 gallons of whisky, at 8s 6d per gallon, to reduce the price to 6s 6dper gallon? 94. Bought 120 gallons of brandy for 160, lost 20 gallons of it, required the rate per gallon I should sell the re- mainder, so as neither to lose nor gain by it ? 95. Bought 246 yards of broad cloth for 221, 8s, 42 yards of that got damaged, required the rate per yard I should sell the remainder, so as neither to lose nor gain by it. 96. If 40 acres of land worth 30 per acre, be exchanged for 56 acres of other land, what is this last valued at per acre ? 97. If a clerk have a salary of 100 a-year, commencing llth November, how much should he receive on leav- ing his situation on the 6th March following? 98. How much money lent at 4 per cent, will yield as much interest as 846 10s at 5 per cent? 99. If 32 yards of carpet, 4 quarters broad, cover a floor, how many yards 3 qrs. broad, will cover the same ? 100. How many yards of printed cloth, 5 quarters broad, will make curtains to a bed, when it requires 54 yard s 8 quarters broad ? COMPOUND PROPORTION. COMPOUND PROPORTION consists of several simple proportions united into one question, and has five, seven, or more terms given, to find another in proportion to them. RULE For stating and working the given terms. Write for the third term that number which is of the same kind with that required: then take any two terms which are of the same kind with each other, and state them as directed in simple proportion. Proceed in this manner with every remaining pair of terms, and place them directly 80 COMPOUND PROPORTION. under the former ; then multiply the product of the second terms by the third for a dividend, and divide hy the product of the first terms ; the quotient is the answer in the same name with the third term. CONTRACTION. When the first term and any other are divisible by the same number, they may be divided by that number to shorten the work. EXAMPLE. If 36 men build a house in 160 days, working 8 hours a-day, how many men will do the same in 80 days, working 12 hours per day ? days. days. men. 00 : m : : 36 1 2 n 3 2 _36 3)144 48 men. 1. If 16 men cut down 224 acres of oats in 42 days, how many acres will 12 men cut at that rate in 25 days P 2. If 16 men cut down 224 acres of oats in 42 days, in how many days will 12 men cut 100 acres ? 3. If 224 acres be cut by 16 men in 42 days, how many men will cut down 100 acres in 25 days f 4. If 60 men consume 20 bushels of flour in 35 days, how much will 40 men consume in 180 days ? 6 If 30 men consume 20 bushels of flour in 70 days, how long will 216 bushels maintain 40 men ? 6. If 84 masons build a house in 80 days, working 8 hours a-day, in how many days will 36 masons finish the same, working 10 hours per day ? 7. If 55 masons finish a wall in 36 days, working 12 hours COMPOUND PROPORTION. 81 a-day, how many masons would finish the same wall in 22 days, working 10 hours per day ? 8. If 26 cwt. he carried 130 miles for L10, how far will 13 cwt. be carried for L15 ? 9. If the carriage of 40 cwt. for 84 miles cost L12, hotf many cwt. may he carried 100 m les for L80 ? 10. If 24 horses in 6 days plough 16 acres of land, how many horses will plough 28 acres in 36 days ? 11. If 18 pioneers cut a trench 420 yards long, working 12 hours a-day, how many must be employed to cut a trench of 210 yards long, working 8 hours per 12. If a person perform a journey of 160 miles in 6 days, walking 8 hours a-day, in what time will he com- plete a journey of 280 miles, walking 16 hours per day? 13. If 4000 copies of a book of 6 sheets require 50 reams of paper hoTV many reams will 3000 copies of 18 sheets require ? 14. If the expense of 16 horses for corn be L16 for 4 months, how many horses will consume the value of L28 in 8 months ? 15. If the interest of L100 for a year be L3 10s, what is the interest of L480 for 246 days ? 16. If L16 be the interest of L480 for 9 months, what is the interest of L 100 for a year ? 17. What is the interest of L846 for 218 days, at 5 per cent, per annum ? 18. What is the interest of L427 for 27 weeks, at L5 per cent, per annum ? 19. How long must L846 be out at interest to gain L140 15s, at 4J per cent per annum ? 20. If a man can travel 460 miles in 16 days, of 8 hours each, how many miles will he travel in 80 days, of 6 hours each ? 21. If 14 ounces of wool are sufficient for 3J yards of cloth, 6 quarters wide, how much wool will be required to make 240 yards, 4 quarters wide ? 22. If 14 men consume 46 Ibs. of beef in a week, how many men will 12,460 Ibs. serve for 7 weeks P COMPOUND PROPORTION. 23. If 8 compositors set up a work of 7 sheets in 12 days, in what time will 4 compositors set up a work of 12 sheets. 24. Borrowed from a banker 600 for 216 days, at 5 per cent., how long ought he to retain 750 of my money when he allows me only 4 per cent ? 25. If 12 tailors can make 20 suits of clothes in 4 days, how many suits can 40 men make in 14 daysP 26. A person engaging to make a mile of a railroad in 58 days, for that purpose employed 52 men, but found that, at the end of 24 days, he had only fi- nished 500 yards, how many additional men must he employ to have it finished in the stipulated time ? 27. A wall, 800 yards long, was to be built by 14 men in 30 days ; but, after working 12 days, they find that only 220 yards of the wall are completed : how many men must now be employed, at the same rate of working, to finish the wall at the appointed time ? PRACTICE. PRACTICE is an expeditious method of calculating the value of any number of articles, when the price of one is given. TABLE OF ALIQUOT PAUTS. Ol' a Pound. Of a Shilling. Of a Cwt, Of a Ib. Troy. 10s = A- 5s = J 4s = | 6s 8d = ^ 3s 4d = fc 2s 6d = I 2s = !>* Is 8d = ^ Is = gV 8d -A 6d = A 4d = & 3d = B 'o Of a Guinea. 10s 6d = 7s = *, 6d = A 4d =| 3d =| 2d =i lid = I Of a Penny. 2 qr. = 1 qr. = J 16 Ib. = A 14 Ib. = | Of a qr. 71b. =i 4 Ib. = A 3J Ib. = * 6 oz. = 4 oz. = A 3 oz. =| Of an Ounce. 10 dwt.= 4 dwt.= I 5 dwt.= | 2 dwt.= ^ PRACTICE. 83 CASE I. When the price is at shillings. RULE. Take aliquot parts of a pound, and the answer will be pounds. 1. Find the price of 4684 yards at 10s each, 2. Find the price of 5879 yards at 5s 3. Find the value of 6485 yards at 4s 4. Find the value of 96/8 yards at 6s 8d 5. Find the value of 4264 yards at 3s 4d 6. Find the value of 5953 yards at 2s 6d 7. Find the value of 6004 yards at Is 8d 8. Find the value of 5469 yards at 2s CASE II. "When the price is at pence. RULE. Take parts of a shilling, and the answer will be shillings ; then bring them to pounds. 9. Find the price of 4678 Ibs. at 4d each. 10. Find the price of 8467 Ibs. at 3d 11. Find the price of 6780 Ibs. at IJd 12. Find the price of 5849 Ibs. at 6d 13. Find the price of 8468 Ibs. at 8d 14. Find the price of 6297 Ibs. at 9d 15. Find the price of 8768 Ibs. at lOd 16. Find the price of 5489 Ibs. at d CASE III. When there are pounds in the pnce. RULE. Multiply the quantity by the pounds, then take parts for the other denominations. 17. Find the price of 4678 cwt. at L2 10s each. 18. Find the price of 8469 cwt. at L3 6 8d 19. Find the price of 4784 cwt. at L4 1 8d 84 PRACTICE. 20. Find the price of 8769 cwt. at L2 6 6d each. 21. Find the price of 8794 cwt. at L3 12 6d 22. Find the price of 4689 cwt. at L3 17 6d 23. Find the price of 8496 cwt. at L4 18 4d 24. Find the price of 2468 cwt. at L4 7 8 Jd CASE IV. When the difference between the price and 20s is a part of a pound. RULE. Value for the difference, then subtract the result from the given quantity. 25. Find the price of 3648 yards at 15s each. 26. Find the price of 4847 yards at 16s 27. Find the price of 6808 yards at 16s 8d 28. Find the price of 6960 yards at 17s 6d 29. Find the price of 6848 yards at 18s 30. Find the price of 5009 yards at 13s 4d 31. Find the price of 6206 yards at 18s 4d 32. Find the price of 5200 yards at 19s CASE V. When the price is at an even number of shillings. Rui/E. Multiply the quantity by half the price, and double the right hand figure of the product for shillings, the other figures of it are pounds. 33. Find the price of 8769 yards at 2s each. 34. Find the price of 5206 yards at 4s 35. Find the price of 8769 yards at 6s 36. Find the price of 6246 yards at 8s 37 Find the price of 5684 yards at 12s 38. Find the price of 6296 yards at 14s 39. Find the price of 4672 yards at 16 a 40. Find the price of 8763 yards at 18g PRACTICE. 85 CASE VI. When there is a fraction in the quantity. RULE. First find the value of the quantity at the given price, then find the value of the fraction by multiplying the given price by the upper figure of the fraction, and dividing by the under. 41. Find the price of 4678i cwt. at L2 5 6 each. 42. Find the price of 684? cwt. at Ll 4 8 43. Find the price of 84081 cwt. at L3 89 44. Find the price of 4009i cwt. at L4 13 9 45. Find the price of 5486 cwt. at L5 1 7 6 46. Find the price of 8290| cwt. at L6 18 8 47. Find the price of 5468 1 cwt. at L7 11 8* 48. Find the pi ice of 4684 1 cwt. at L3 16 9J CASE VII. hen the quantity consists of more than one denomination. RULE. Multiply the price by the highest denomina- tion, then take parts for the rest. 49. *Find the price of 16 cwt. 2 qr. 14 Ib. at L6 14 8 eh.c. 60. Find the price of 20 cwt. 1 qr. 16 Ib. at L2 17 6| , 51. Find the price of 25 cwt. 3 qr. 8 Ib. at L4 16 9 , 52. Find the price of 6 Ib. 6 oz. 10 dwt. at Ll 15 4 , Ib. 53. Find the price of 3 Ib. 8 oz. 12 dwt. at L3 10 81 , 54. Find the value of 5 Ib. 10 oz. 14 dwt. at L2 19 6 , 55. Find the price of 14 acres, 3 rds. 20 pis. at L4 8 7 ac. 66. Find the price of 16 acres, 2 roods, 30 poles, at L3 16 8 per acre 57. Find the price of 20 acres, 3 roods, 35 poles, at L5 6 9^ per acre 58. Find the price of 13 tons, 15 cwt. 2 qrs. at L4 12 6 per ton 69. Find the price of 17 days, 4 hours, 10 minutes, at LO 4 6 per day's work of 12 hours each In the answers in this case, fractions less than a farthing are omit- ted m addition. 86 PRACTICE. EXERCISES ON THE FOREGOING RULES IN PRACTICE. 60. Find the value of 4634 yards at 6d each. 61. Find the value of 5789 yards at 9d 62. Find the value of 6874 yards at 6| 63. Find the value of 3274 yards at 4 64. Find the value of 5769 yards at 10 65. Find the value of 6274 yards at 1 1 id 66. Find the value of 3320 yards at llfd 67. Find the value of 5700 yards at 5d 68. Find the value of 4678 yards at Is 2d 59. Find the value of 5760 yards at 2s 7d 70. Find the value of 3008 yards at 3s Ifd 71. Find the value of 4689 yards at 13s lOd 72. Find the value of 5746 yards at 14s 8|d 73. Find the value of 6764 yards at 15s l|d 74. Find the value of 5684 yards at 15s 7d 75. Find the value of 2700 yards at 16s 4d 76. Find the value of 6009 yards at 16s 9d 77. Find the -value of 5206 yards at 17s 5d 78. Find the value of 9409 yards at 17s 7d 79. Find the value of 6293 yards at 18s 3^d 80. Find the value of 3769 yards at 19s Hd 81. Find the value of 2553 yards at 19s 5d 82. Find the value of 6830 yards at 19s lOf d 83. Find the value of 8425 yards at LI 2s 84. Find the value of 6829 yards at Ll 6s 85. Find the value of 5240 yards at Ll 9 6 86. Find the value of 8796* yards at L2 8 8 87. Find the value of 4674| yards at L2 11 9 88. Find the value of 3748 yards at L2 12 4 89. Find the value of 40001 yards at L3 13 7 90. Find the value of 9868 yards at L3 14 1 ,. 91. Find the value of 4685 yards at L3 14 11 92. Find the value of 18462 yards at L2 18 10. \ 93. Find the value of 2768| yards at L3 19 11$ 94. Find the value of 4678 g yards at L4 19 lia 95. Find the value of 4268 \ tons at L2 5 6 96. Find the value of 5849$ cwt, at Ll 18 4 87 PROMISCUOUS EXERCISES ON THE FOREGOING RULES. 1. Find the price of 8497 pencils at |d each 2. Find the price of 2168 pens at d each 3. Find the price of 3526 pen holders at fd each 4. Find the price of 2167 books at 2d each 6. Find the price of 3759 bottles at 3fd each 6. Find the price of 2975 knives at 5|d each 7. Find the price of 9568 Ibs. of sugar at 6| ( 1 per Ib. 8. Find the price of 5279 Ibs of soap at 7d per Ib. 9. Find the price of 4387 Ibs. of sugar at lOd per Ib. 10. Find the price of 9759 Ibs. of tea, at 4s 8d per Ib. 11. Find the price of 2148 Ibs. of green tea at 7s per Ib. 12. Find the price of 3478 yards of cloth at 13s each 13. Find the price of 2222 yards of linen at 2s 4d each 14. Find the price of 3333 yards of cloth at 19s each 15. Find the price of 4444 cwt. at Ll 6s each 16. Find the price 50451 tons at L2 19 6d each 17. Find the price of 1 yard of linen, when 13 yards cost Ll 12 6 18. Find the price of 1 Ib. of sugar when 248 Ibs. cost L6 14 4 19. If 61 yards of cloth cost L4 8 7^d, what will 100 yards cost? 20. If 80 men perform a piece of work in 248 days, in what time will 120 men do the same. 21. If 1 yard of cambric cost lls 4^d, what will 58| yards cost ? 22. If 24| yards of lawn cost L9 18 41, what will 1 yard cost ? 23. If 1 yard of sheeting cost Is 2d what will 426| yards cost ? 24. If 58f yards of broad cloth cost L42 16 81, what will 1 yard cost P 25. The length of the London and Bristol railway is 120 miles, the expense of its construction L2, 550,300 what is the average price per mile ? 26. The sun is distant from the earth 91,718,000 miles, how long would a cannon ball be in reaching that distance with the velocity of 8 miles in a minute ? 88 VULGAR FRACTIONS. 27. The productive property of Scotland is L318, 100,000, the unproductive L5 1,100, 000, public property, L3,000,000, how much would that be to each indi- vidual, estimating the population at 2.365,807 ? 28. Magna Charta was signed by King John in the year 1215, how long is it since? 29. The planet Uranus is distant from the sun 1900,000,000 miles, how long would a balloon be in going over that space, travelling at the rate of 60 miles an hour? 30. Saturn is distant from the earth 811 millions of miles, ho^w long would a cannon ball be in reaching that dis- tance, going with a velocity of 8 miles in a minute? 31. The surface of the planet Jupiter contains 24884 mil- lions of square miles, how many inhabitants would it contain, allowing 280 to a square mile? 32. England contains 32,268,000 acres of cultivated land, of which 3| millions are in wheat; 4* millions in barley, oats, pease, &c. ; 2,400,000 in green crops ; 2,100,000 fallow, 18,000 pleasure grounds, 17 mil- lions pasture, 1.200,000 in hedges, copses, and woods ; and the rest in roads, highways, water- courses and railroads, required the number of acres contained in these P VULGAR FRACTIONS. U l'* I I 2 * I 1% I T 4 o 1 I 5 o I T% ' l ? o I 1*0 I T 9 o I \%- A fraction is a quantity which represents a part of a unit, or integer. A simple fraction consists of two numbers, the numerator and the denominator; the denominator shews into how many equal parts the unit is divided, and the numerator the number of those parts taken. 3 numerator. 8 denominator. A proper fraction is when the numerator is less than the denominator, as T 7 5 . An improjier fraction is when the numerator is equal to, or greater than the denominator, as |, or J. A compound fraction is a fraction of a fraction, of . VULGAR FRACTIONS. 89 A mixed number consists of a whole number with a frac- tion annexed to it, as 6|. A complex fraction is that which has a fraction or mixed number, either in its numerator or denominator, or in both, A common measure is any number that will divide the numerator and denominator of a fraction without leaving a remainder. A common multiple of two or more numbers, is the smallest number which can be divided by each of the given numbers without a remainder. NOTE 1. To multiply a fraction by an integer : multi- ply the numerator by the integer, and retain the same de- 6x4 24 nominator; thus, ^ multiply by 4 is = 2. To divide a fraction by an integer: multiply the de- nominator by the integer, and retain the same numerator ; o 3 thus, divided by 4 is g 4 = g - 3. If the numerator and denominator of a fraction be either multiplied or divided by the same number, its value is not altered. REDUCTION. CASE I. To reduce a fraction to its lowest terms. RULE. Divide the numerator and denominator by their greatest common measure, which is found by dividing the greater term by the less, and the last divisor by the last re- mainder, continually, till nothing remains ; the last divisor is the greatest common measure, NOTE. A fraction ending with an even number is di- visible by 2 ; also one ending with 5 or is divisible by 5. Reduce ^|, i|, 433, if|f, |if, JJIS, T ^p 5 , rfAfc, !iio> IIIoS? to tne i r lowest terms. CASE II. To reduce an improper fraction to a \\hole or mixed number. RULE. Divide the numerator by the denominator. 90 VULGAR FRACTIONS. Reduce y, V, l $ 8 V, 3 IS Wi VoS 'f 8 , 'I , S?> to whole or mixed numbers. CASE III. To reduce a mixed number to an improper fraction. RULE. Multiply the whole number by the denominator of the fraction ; to the product add the numerator, under which place the denominator. NOTE. To reduce an integer or whole nnmber to a fraction, place 1 for its denominator. Reduce 2$, HI, 55, 201, 41{j, 68y, 52^, 21, 24, 6, to improper fractions. CASE IV. To reduce a whole number to a fraction having a given denominator. RULE. Multiply the whole number by the given deno- minator for the numerator, under which place the denomi- nator. 1. Reduce 4 to a fraction whose denominator is 8. 2. Reduce 5 to a fraction whose denominator is 9. 3. Reduce 7 to a fraction whose denominator is 11. 4. Reduce 2 to a fraction whose denominator is 12. 5. Reduce 9 to a fraction whose denominator is 16. 6. Reduce 1 1 to a fraction whose denominator is 20. CASE V. To find the least common multiple of several numbers. RULE. Arrange the given numbers in one line, and di- vide the greatest number of them by any number that will divide them without a remainder ; place the quotients and the numbers not divided in a line under the former ; divide this line as before, and so on, till all the quotients are 1 ; then multiply all the divisors together for the least common multiple required. Required the least common multiple of 2, 14, 18. and 24 ; of 3, 9, 12, 15 and 27 ; of 8, 14, 28, and 32 ; of 32, 44, 52, 13, and 48 ; of 76, 38, 152, 448, and 748. CASE VI. To reduce a complex fraction to a simple one. RULE. Reduce both the numerator and denominator to a simple fraction, then multiply the numerator of each of VULGAR FRACTIONS. 91 these fractions by the denominator of the other, for the simple fraction. 21 34 44 6 3 5 Reduce -~ -f -*- to simple fractions. 4, 11, 8, 7, 6$, 6i, CASE VII. To reduce a compound fraction to a simple one. RULE. Multiply all the numerators together for the numerator, and all the denominators for the denominator of the simple fraction. NOTE. Whole or mixed numbers must he reduced to improper fractions. 1. Reduce | of f to a simple fraction. 2. Reduce i of | of f to a simple fraction. 3. Reduce | of | of.} of | to a simple fraction. 4. Reduce 1 of 5 of 4 of -^ to a simple fraction. 5. Reduce of IJ of 2 of ^ to a simple fraction. 6. Reduce T ' s of 4 of 2 of ^ to a simple fraction, CASE VIII. To reduce a fraction from one name to another. RULE. If from a lower name to a higher, multiply the denominator by as many of the lower as make one of the higher ; if from a higher to a lower, multiply the numerator by as many of the lower as make one of the higher. OR, Make the numerator of the same name as the given fraction, and the denominator of the same name as the re- quired fraction, then reduce both to the same denomination by common reduction. 1. Reduce f of a farthing to the fraction of a pound. 2. Reduce penny to the fraction of a pound. 3. Reduce | penny to the fraction of a guinea. 4. Reduce farthing to the fraction of a crown. 5. Reduce T 2 r grain to the fraction of a Ib. troy. 6. Reduce i grain to the fraction of a Ib. avoirdupois. 7. Reduce f Ib. to the fraction of a ton. 8. Reduce | yard to the fraction of a mile. 9. Reduce f pole to the fraction of an acre. 10. Reduce f nail to the fraction of a yard. 92 VULGAR FRACTIONS. 1 . Reduce ^-^ to the fraction of a farthing. 2. Reduce ^ to the fraction of a penny. 3. Reduce yy 1 ^ guinea to the fraction of a penny. 4. Reduce 9^ crown to the fraction of a farthing. 5. Reduce ST^SO lb. troy to the fraction of a grain. 6. Reduce T3 ^oo lb. avoir, to the fraction of a grain. 7. Reduce T o&go ton to tne fraction of a lb. 8. Reduce ^^-g mile to the fraction of a yard. 9. Reduce s | 5 acre to the fraction of a pole. 10. Reduce g 3 5 yard to the fraction of a nail. I 1. Reduce f pound to the fraction of a guinea. 2. Reduce f crown to the fraction of a . 3. Reduce | lb. troy to the fraction of a lb. avoirdupois, 4. Reduce 2 yard to the fraction of an ell English. 5. Reduce f ell Flemish to the fraction of a yard I 1. Reduce if guinea to the fraction of a pound. 2. Reduce T \ to the fraction of a crown. 3. Reduce f f lb. avoir, to the fraction of a lb. troy. 4. Reduce f ell English to the fraction of a yard. 5. Reduce T 9 yard to the fraction of an ell Flemish. CASE IX. To Reduce a quantity to a fraction of any denomination. R,ULE. Reduce the given quantity to the lowest name it contains for the numerator, and the proposed integer to the same name for the denominator ; then reduce the frac- tion to its lowest terms. 1. Reduce 4s 8d to the fraction of a pound. 2. Reduce 5s 6Jd to the fraction of a pound. 3. Reduce 18s ( J|d to the fraction of a pound. 4. Reduce 16s lljd to the fraction of a pound. 5. Reduce 4 oz. 14 dwt. to the fraction of a lb. troy. 6. Reduce 5 oz. 16 dwt. 18 grs. to the fraction of a lb. 7. Reduce 4 cwt. 2 qrs. 16 lb. to the fraction of a ton. 8. Reduce 4 bu. 2 pk. 1 gall, to the fraction of a qr. 9 Reduce 3 ro. 17 poles to the fraction of an acre. VULGAR FRACTIONS. 93 10. Reduce 2 qr. 1 nl. to the fraction of a yard. 1 1. Reduce 6 hours, 31 minutes, 24 J seconds, to the frac- tion of a day. 12. Reduce 14 24' 16^" to the fraction of a sign. CASE X. To find the value of a fraction. RULE. Divide the numerator by the denominator, as in Compound Division, the quotient is the value of the fraction. 1. Find the value of fo of a pound. 2. Find the value of f g of a pound. 3. Find the value of | of a pound. 4. Find the value of yf of a pound. 5. Find the value of T y ff of a Ib. troy. 6. Find the value of if | of a Ib. troy. 7. Find the value of if of a ton. 8. Find the value of f | of a quarter. 9. .Find the value of f of an acre. 10. Find the value of T 9 g of a yard. 11. Find the value of T 4 yVsVs of a day. 12. Find the value ofi|^si fa sign. ADDITION. RULE. Reduce the given fractions to a common deno- minator, which is done by multiplying the numerator and denominator of each fraction by all the other denominators : then add the numerators, arid under their sum write the common denominator. NOTE. Compound fractions must be reduced to sim- ple fractions before they are brought to a common deno- minator. 9. Add | + f + i + | 10. ... j + | + p _^| 12. ... *f + + i + | 13. ... 5 + i + i + a + L 14. ... i otf + | of i + | 15. ... of f + $ of $ + | of | 16. ... f of f + f of J + & of i E 1. Add f + 1 + 1 2- -. i + + 3. 4. 5. VULGAR FRACTIONS. NOTE 1. When mixed numbers are given, find the sum of the fractious as before, to which add the integers. 21. Add 62f+824f+lU 22. 620|+ 59J+ 8j 23. - 68f+100|+39f+f 24. 62|+ 42A+841 + ! NOTE 2. "When fractions of different names are given, find their values, (by Case X.) and add as in Compound Addition. Add 17. Add 18. 20f+ 241+ 19. 5242+ 8i+79f 20. 5. i + T ' r + f sh. 26. f + | sh. + } guin. 27. A + A + | guin. 28. | Ib. + I oz. + A dwt. 29. 4 1 ton + f cwt. + I qr. 30. 6| ac. + 5 j ac + | ro. 31. 2 J yd. + \\ yd. + f qr. 32.6| day-f | ho. + fmi. 33. A person borrowed at one time 36|, at another time 27 f, at another time 17| sh. ; how much did he borrow in all ? SUBTRACTION. RULE. Reduce the fractions to a common denominator, then find the difference of the numerators, under which write the common denominator. Subtract 5- T 9 T- I 6- f- i 7- it A 8. it | 10. if 11- i 12. A NOTE. In mixed numbers first subtract the fractions, then the whole numbers ; but if the under numerator exceed the upper, subtract it from the common denominator, to the remainder add the upper numerator, under their sum write the common denominator, and carry one to the whole number in the lower line. 13. 14. 15. 69f 13| 80s _ 56| 1C. 17. 18. 601 148/ t 650 231^3 203 108ft 19. 101 8} 20. 95 f 7f 21. 56f 9 VULGAR FRACTIONS. 95 22. Find the difference between |f of a pound and | of a guinea. 23. Find the difference between f of a guinea and f of a shilling. MULTIPLICATION. RULE. Multiply all the numerators together for the numerator, and all the denominators together for the deno- minator. Multiply 1. 2. 3. 6. A x f 7. fx* 8. fx| 9. fxi 10. X 12. T 2 5 x 1% iq _2 v 8 T7 * 3o3 14. $ of f x f of | 15. f of | x j of | NOTE. (In multiplication and division,) when there are integers and mixed numbers, reduce them to improper fractions. 16. 17. 18. |X4 |x7 6xf DIVISION. RULE. Invert the divisor, and proceed as in multiplication. Divide 1. I s ? ;** 7. 1 +-I- 13. 16i- -I*- 2. T*S S 4* 8. | -T- |. 14. 18| - -3f. 3. 1 6 T 'I' 9. T 3 T ~^~ iV 15. 29|- 4. T \ -I- 10. 1 of A ~ f 16. 59i- -6|." 5. A 11. l-^i^l 17. 102f - -14- 6. i -I 2 , 12. i of | -f- } of f 18. 59 7i. 96 VULGAR FRACTIONS. PROPORTION. RULE. State the terms as directed in simple propor- tion ; reduce them to simple fractions, and the two first to the same denomination ; then invert the first term, and multiply the three fractions together ; the result is the answer in the same denomination as the third term. 1. If J of a yard of cloth cost |, what will f of a yard cost ? 2. If | Ib. cost f, what will f Ib. cost? B. If | cwt. cost f , what will j cwt. cost ? 4. If J T ton cost ii, what will ton cost. 5. If | yards cost T 7 g , what will 2^ yards, cost ? 6. If | of a ship cost 846, what will f of the same ship cost ? 7. If 2i tons cost 3J, what will 6J tons cost? 8. If l| cwt. cost li, what will 101 cwt. cost? 9. If 8} yards cost 5 1, what will 2j yards cost ? 10. If Ifc cwt. cost 2^, what will ton cost? 11. What cost 84| yards, when of a yard cost -fo? 12. What cost 19 1 tons when f of a ton cost | ? 13. If | Ib. tea cost T y<>, what will T *y Ib. cost ? 14. If i Ib. of coffee cost >, what will 4 J Ib. cost ? 15. If J cwt. of sugar cost T 7 S , what will 16 cwt. cost? 16. If I yard of broad cloth cost T %, what will 27| yards cost? 17. If 2 yards cost 2| what will f yards cost? 18. If | gallons of rum cost 13s 6Jd, what will 9J gallons cost? 19. If 2| gallons O f whisky cost 15s 8jd, what will 200 gallons cost? 20. If the value of f ship is 921|, what will come to ? 21. If 6 f cwt. of sugar cost 18 2 4d, how much may be bought for 32 8 2d P 22. How much wine may be bought for 84 7 ld, if 4$ gallons cost 2 14s ? 23. How many yds. of cloth may be bought for 43 12 8d, if 2 yards cost 1| ? 97 PROMISCUOUS EXERCISES IN VULGAR FRACTIONS. 1. Reduce f f f to its lowest terms. 2. Reduce f f to its lowest terms. 2 1 3. Reduce? to a simple fraction. 4. Reduce f of * of f of f to a simple fraction. 6. Reduce of | of 6 of 8 to a mixed number. 6. Reduce 1 of a farthing to the fraction of a pound. 7. Reduce | of a grain to the fraction of a Ib. troy. 8. Reduce 55^3 of a ton to the fraction of a Ib. 9. Reduce gl ^ ? of a guinea to the fraction of a penny. 10. Reduce f of a pound to the fraction of a guinea. 11. Reduce 14s 8jd to the fraction of a pound. 12. Reduce 4s 2d to the fraction of a shilling. ] 3. Reduce 4 cwt. 2 qrs. 16 Ibs. to the fraction of a ton. 14. Find the value of f of a pound sterling. 15. What is the value | of a shilling ? 16. What is the value of f of a cwt. ? 17. What is the value of | of a yard ? 18. Add + | + a + T 2 T and | together, 19. Add | of |, | of |, i of f , and of f. 20. Add 681, 372, 246|, and 8. 21. Find the amount of f of a pound, | of a guinea, and | of a shilling. 22. From f of a pound take f of a shilling. 23. Subtract 24 from 126|. 24. Whether is f or | the greater value ? 25. Whether is | or fo the greater value, and what is the difference ? 26. Whether is f or ^ T the greater value, and what is the difference ? 27. Multiply 6| by 8. 28. Multiply | and f of a together, and find the value of the product. 29. Divide 146f by 2|. 30. Divide 627 1 into 11 f shares. 31. If | Ib. of tea cost <, what will T 9 T cwt. cost ? 32. If | Ib. of sugar cost of a shilling, what will { cwt. cost? y8 PROMISCUOUS EXERCISKS. 33. If cwt. cost 14s 8d, what will | of a ton cost ? 34. If 1^ buy of a ton, how much can be bought fot 6|? 35. A bankrupt's effects amount to f of his debts, what is that per pound ? 36. A person having f of a ship, sells ^ of his share fo* 328, what was the whole ship worth ? 37. The number 100 may be expressed by a certain digit repeated four times : quere, the digits, and manner of placing them ? DECIMAL FRACTIONS. A fraction whose denominator is 1 with ciphers after it is a decimal fraction^ the unit being divided into 10, 100, &c. equal parts. The numerator only is used by placing a point as many figures from the right of it as there are ciphers in the deno- minator; thus, T Vu 5 o = '125; T ^o = -0004. Decimals are either terminate or interminate. A terminate or finite decimal is one which extends onlj to a limited number of places, as .5, .125, &c. Interminate decimals are such as could extend ad infi- nitum, and are divided into four classes. 1st, A pure repeater is a continual repetition of the same figure, and is distinguished by a point over It, thus, .3 2d, A mixed repeater is composed of a finite part and a pure repeater, and is distinguished by a point over the repeating figure, thus, .326 3d. A pure circulate is a continual repetition of more figures than one, and is distinguished by points over the first and last figures, thus, ,841 4th, A mixed circulate is composed of a finite part and a pure circulate, and is distinguished by points over the first and last figures of the circulate, thus, .45361 Circulates are similar or coterminous when they begin and end at the same distance from the decimal point, thus, .8127 and .3654 DECIMAL FRACTIONS. 99 ADDITION. To add terminate decimals. RULE. Write down the numbers, so that the decimal points may stand directly under one another, then add and place the decimal point in the sum directly under the other points. 1. 2. 3. 246.684 859.38 847-30876 367.547 47.865 529.0075 84.95 98.4839 438.258 8.549 427.274 79.8463 36.48 93.0482 369.08 7.9573 9.005 42.53848 4. Add 246.042 + 52.4285 + 8468.32 + 23.47685 5. Add 6058.62 + 567.3045 + 42.3248 + 524,00768 6. Add 402.008 + 348.0065 + 52.042 + 9.40672 7. Add 608.241 + 74.3864 + 93.567 + 830.5552 + 8.2761 8. Add 9246.815 + 563.42 + 85.065 + 3.774 + 97.3553 9. Add 574.0835 + 43.367 + .81204 + 354.0064 + ,65042 SUBTRACTION. To subtract terminate decimals. RULE. Write the less below the greater, so that the decimal points may stand directly under one another, then subtract, and place the decimal point in the remainder directly under the other points. 1. Subt. 84.0765-61,4878 2. Subt. 142.8421 84.65 3. Subt. 319.0428 148.65 4. Subt. 121.84-29.6345 5. Subt, 84.02736.8496 6. Subt. 100.1047.21685 7. Subt, 4087.3436.5847 8. Subt. 325.849124.00846 9. Subt, 46.084- 32.02468 10. Subt. 92.2348-Hl9.36~ 100 DECIMAL FRACTIONS. MULTIPLICATION. To multiply when both factors are terminate. RULE. Multiply as in whole numbers ; then point off from the right of the product as many figures for decimals as there are decimal places in both factors; if the product has not so many places, supply the defect by prefixing ciphers. 1 Mult 425.61 by 2.42 2. Mult. 32.8104 by 1.52 3. Mult. 69.429 by .465 4. Mult. 4.94893 by .068 5. Mult. 7.5436 by .0014 6. Mult. .93685 by .009 7. Mult. 21.4567 by .0425 8. Mult. 523.004 by 2.087 9. Mult. 84.748 by 100 10. Mult. 694.86 by 1000 DIVISION. To divide when the divisor is terminate. BULE. Divide as in whole numbers ; then point off from the right of the quotient for decimals as many figures as the number of decimal places in the dividend exceed those in the divisor ; if the quotient has not so many, sup- ply the defect by prefixing ciphers. Divide 1. 437.68745 by 4.21 2. 943.76859 by 5.64 3. 627.89476 by .846 4. 4586.7693 by .659 6. 8.5274630 by 524.6 6 527.68982 by 13.49 7. 904.23845 by 4.26 8. 843.91051 by .481 9. 45.862089 by 36.5 10. 8.42/8502 by .425 11. 684.39765 by 100 12. .73028794 by 1000 REDUCTION. CASE I. To reduce a vulgar fraction to a decimal. RULE Annex ciphers to the numerator, and divide by the denominator until the division terminates or repeats; then point off from the right of the quotient as m;my figures as there were ciphers annexed. 3. 4. 6. DECIMAL FRACTIONS. Reduce to decimals 6. 7. 8. 9. 10. A 11. 12. 13. 14. 15. 101 16. 17. 18. 19. 20. !f I 6 ! A CASE II. To reduce a terminate decimal to a vulgar fraction. RULE. "Write the given decimal for the numerator of the fraction, and for the denominator 1, with as many ci- phers annexed to it as there are figures in the given decimal, which reduce to its lowest terms. 1. 2. 3. 4. .5 .25 .2 .4 Reduce to vulgar fractions. 5. 6. 7. 8. .125 .875 .75 .4375 9. 10. 11. 12. .021875 .046875 .00375 .00025 CASE III. To reduce a pure repeater, or circulate to a vulgar fraction. RULE. Write the given decimal for the numerator, and for the denominator as many 9s as there are figures in the given decimal, which reduce to its lowest terms. Reduce to vulgar fractions. 1. .3 6. .90 9. .148 2. .6 6. .63 10. .063 3. .1 7. .48 11. .14634 4. .2 8. .36 12. .428671 E 2 102 DECIMAL FRACTIONS. CASE IV. To reduoe a mixed repeater, or circulate, to a vulgar fraction. RULE. Subtract the finite part from the whole decimal for the numerator, and for the denominator write as many 9s as there are repeating figures, to which annex as many ciphers as there are finite places, which reduce to its lowest terms. Reduce to vulgar fractions. 1. .75 5. .3409 9. .03248 2. .83 6. .5681 10. .2357*8 3. .416 7. .1136 11. .0046296 4. .24*7 8. .0185 12. .0065006 ADDITION. CASE I. To add repeaters. RULE. Extend the repeating figure in each number, one place beyond the longest finite decimal, then add and carry at 9 in the first column. 1. Add 5G9.3 + 486.413 + 65.8142 + 653.84251 2. Add 837.247 + 59.346 + 9.52386 + 735.00862 3. Add 359.00873 + 92.0421 + 138.2467 + 365.8246 4. Add 210.3 + 194.21 +.850743 + 900.08165 6. Add 546.83 + 629.124 + 548,372 + 84.9687 6. Add 200.121 + 361.0083 + 42.62 + 8.34 + .9685 CASE II. To add circulates and repeaters. RULE. Make them first similar and coterminous by extending the circulates and repeaters to as many places DECIMAL FRACTIONS. 103 beyond the longest finite part, as there are units in the least common multiple of the numher of figures in the several circles, then add and carry to the right hand column the tens from the first column on the right of the longest finite part. 1. Add .4225 + .6537 + .9845 +.5286 2. Add 8.2038 + 9.0468 + 7.36548 + 43.4683 3. Add 30.62085 + 6.3028 4- 29.00642 + 365.6 4. Add 32.80567 +.4268 + .93478 + 6.1281 5. Add 81.0048164 + 3.20*5 + 5.07426 +5.85 6. Add 39.0034 + 6.0526 + 82.08257*8 + 9,5218 To subtract repeaters. RULE. Extend the repeaters one place beyond the long- est finite part, and borrow 9 on the right when necessary. 1. Subt. 498.046- 126.821 2. Subt. 520.20458.3 3. Subt. 931.382438.6 4. Subt. 562.8713.49683 5. Subt. 820.042121.8461 6. Subt. 450.84168.58 To subtract circulates. JEluLE. Make the circulates similar as in addition, and if the first figure in the subtrahend on the right of the long- est finite part be greater than the one above it, add 1 to the right hand figure of the subtrahend before subtracting. Subtract 1. 68.247215.1345 2. 92.384618.674371 3. 3.04268 .5167 4. 32.7826427.85735 5. .846732 .087342 6. 32.5027626.0426875 To multiply when the multiplicand is a repeater or circulate. RULE, "When a repeater, carry at 9 on the right of each product, and add as directed for repeaters ; when a 104 DECIMAL FRACTIONS. circulate, to the right of each product, add the carriage that will arise from the left of the circle ; then make them simi- lar, and add as directed in addition. 1. Mult. 4.6046 by .24 2. Mult. 9.7823 by 4.25 3. Mult. 24.576 by 84.8 4. Mult. 85.002 by 9.69 5. Mult. 8.4764 by .548 6. Mult. 93.095 by 2.34 7. Mult. 67.895 by 6.57 8. Mult. 7.5426 by 8.94 To Multiply when the multiplier is a repeater or circulate. RULE. Reduce the multiplier to a vulgar fraction, then multiply by the numerator and divide by the denomi- nator. 1. Mult. 4.6782 ly .3 2. Mult. 85.276 by 2.6 3. Mult. 943.78 by .33 4. Mult. 6.3287 by .75 5. Mult. 58.427 by .416 6. Mult, 32.582 by .247 7. Mult 67.583 by 3.63 S Mult. 8.4735 by 4.16 To divide when the divisor is a repeater or circulate. RULE. Reduce the divisor to a vulgar fraction, then multiply by the denominator, and divide by the numerator. 1. Divide 4678.9353 by .3 2. Divide 25.387925 by .6 3. Divide 700.00873 by .3 4. Divide 98,756231 by .16 5. Divide 90.253643 by .247 6. Divide 678.47984 by .5*1*7 7. Divide 39.002829 by 2.3 8. Divide 907.24676 by 3.27 REDUCTION. To reduce lower denominations to decimals of higher. RULE, If the given number be simple, annex ciphers, and divide by as many of that name as make one of the higher, then point off as many figures as ciphers; but if it DECIMAL FRACTIONS. 105 be a compound, begin at the lowest, and reduce it to the next higher name ; to this decimal prefix the next higher denomination, reduce this decimal to the next higher, and so on to the required name. *1. Reduce 19s 6|d to the decimal of a pound. 2. Reduce 16s 8^d to the decimal of a pound. 3. Reduce 17s 8d to the decimal of a pound. 4. Reduce 2s 6|d to the decimal of a pound. 5 Reduce 9s 10|d to the decimal of a pound. 6. Reduce 2 14 8d to the decimal of a pound. 7. Reduce 4 oz. 7 dwt. 6 grs. to the decimal of a Ib. tr. 8. Reduce 7 oz. 6 dr. 2 sc. 8 grs. to the decimal of a Ib. 9. Reduce 6 cwt. 2 qrs. 14 Ib. to the decimal of a ton. 10. Reduce 6 fur. 36 poles to the decimal of a mile. 1 1 . Reduce 4 bu. 2 pk. 1 gal. 4 pts. to the decimal of a quarter. 12. Reduce 2 ro. 35 po. to the decimal of an acre. 13. Reduce 3 qr. 2 nl. to the decimal of a yard. 14. Reduce 1 hour, 45 m. 45 sc. to the decimal of a day. 15. Reduce 21' 46" to the decimal of a degree. 16. Reduce 26 18' 22" to the decimal of a degree. To find the value of a decimal.' RULE. Multiply the given decimal by as many of the next lower name as make one of the given name. From the right of the product point off as many figures for deci- mals as there are places in the given decimal, which reduce to the next lower name ; proceed in this manner to the last denomination. The figures remaining on the left will express the required value. *1. Find the value of .978125 . 2. Find the value of .834375 . 3. Find the value of .884375 . 4. Find the value of .127083 . 5. Find the value of .4947916 . 6. Find the value of 2.73 . 7. Find the value of .3635416 Ib. troy. 106 DECIMAL FRACTIONS. 8. Find the value of .65416 Ib. apothecaries. 9. Find the value of ,33125 ton. 10. Find the value of .8625 mile. 11. Find the value of .5859375 quarter. 12. Find the value of .71875 acre. 13. Find the value of .875 yard. 14. Find the value of .0734375 day. 15. Find the value of .3627 degree. 16. Find the value of 26.3061 degree. PROPORTION IN DECIMAL FRACTIONS. 1. If 24.31 yards of cloth cost L12.125, what will 236.038 yards cost ? 2. If 14 1 cwt. cost L.2 7 6|d, what will 94| cwt cost ? 3. If 23^ cwt. of sugar cost L42 16 8|, what will 9| tons cost? 4. How many yards of linen may be bought for L16 15 6d, when 6| yards cost lls 6|d ? 6. How many Ibs. of tea may be bought for L26 18 6|d, when 3i Ibs. cost 13s 8 ^d ? 6. How many yards of cloth may be bought for L68 19 6d, when 2*9* yards cost L5 17 8|d P 7. If 23 yards 3 qrs. of linen cost L2 5 lOfd, what will 7 yards 2 qrs. cost ? 8. If 4 Ibs. 6 oz. 15 dwt. of silver cost L14 13 6Jd, what will 18 Ibs. 3 oz. 15 dwt. cost ? PROMISCUOUS EXERCISES IN DECIMALS. 1. Add 84.7684 + 423.482 + 9.004 + 68,208465 2. Add 864.3 + 92.84686 + 426.00*7 + 9.8456943 3. Add 926.86 + 46.84678 + 428.8436 + 42.848 4. Subtract 8457.004 246.834795 5. Subtract 946.0847 365.8423958 6. Subtract 847.846 245.09268 PROMISCUOUS EXERCISES. 107 7. Multiply 278.5679 by 4.825 8. Multiply 43.0857 by 2.124 9. Multiply 51.08465 by 3 10. Divide 8.12045674 by 46.26 11. Divide 428.9846 by 6 12. Reduce to a decimal fraction. 13. Reduce 26| to a decimal fraction. 14. Reduce 18 1 to a decimal fraction. 15. Reduce 18s 6|d to the decimal of a pound. 16. Reduce 19s lO^d to the decimal of a pound. 17. Find the value of .834375 18. Find the value of .9927083 19. Reduce 16 cwt. 2 qr. 16 Ibs. to the decimal of a ton. 20. Find the value of .833125 ton. 21. If 21 1 yards of cloth cost 12 14 8|d, what will 123$ yards cost? 22. If 28 1 cwt. cost 5 18 4d, what will 161 C wt. cost P 23. If 37lbs. of tea cost 7 3 6d, what will 12 Ibs. cost ? 24. How many yards of linen may be bought for 24 16 6d, when 16| yards cost 15 8jd. 25. How many Ibs. of sugar may be bought for 5 13 S^d, when 2 1 Ibs. cost Is 0|d ? INTEREST. INTEREST is an allowance of a certain rate per cent, per annum, given by the borrower to the lender, for the use of money. Principal is the money lent. Interest is the rate per cent, agreed upon. Amount is the sum of principal and interest. Interest is either Simple or Compound; Simple, when it is calculated upon the original principal alone, Compound, when at stated intervals the interest itself is converted into principal. (For Compound Interest, see page 149.) 108 SIMPLE INTEEEST. CASE I. To find the interest of any sum of money for any num- ber of years. RULE. Multiply the principal by the number of years, and by the rate per cent., and divide the product by 100. 1. Find the interest of 789 for 8 years, at 5 per cent. 2. Find the interest of 688 for 12 years, at 4 per cent. 3. Find the interest of 1264 for 9 years, at 3 per cent. 4. Find the interest of 2408 for 5 years, at 2J per cent. 6. Find the interest of 1000 for 6 years, at 3} per cent. 6. Find the interest of 824 16 4d, for 3 years at 4J per cent. 7. Find the interest of 948 15 9d. for 6 years, at 3| pei cent. 8. Find the interest of 364 19 8d, for 14 years, at 2| per cent. 9. Find the interest of 126 13 lOd, for 28 years, at 3| per cent. 10. Find the interest of 210 19 6d, for 7 years, at 2j per cent. 11. Find the amount of 33 13 4.1, for 10*. years, interest at 5 per cent. 1?. Find the amount of 26 16 lid, for 67 years, interest at 5 per cent. CASE II. To find the interest for months. RULE. Multiply the principal by the number of months, and by the rate per cent, and divide the product by 1200. Take parts of a year and divide by 100. 13. Find the interest of 374 16 8d, for 5 months, at 3 per cent. 14. Find the interest of 628 14 6d, for 8 months, at 2 per cent. SIMPLE INTEREST. 109 15. Find the interest of 900 10 8d, for 11 months, at 3| per cent. 16. Find the interest of 276 1 4d, for 2^ months, at 4 per cent. 17. Find the interest of 576 8 lOd, for 7J months, at 2 per cent. 18. Find the interest of 278, for 2 years and 4J months, at 3 per cent. 19. Find the amount of 246, for 91 months, interest at 5 per cent. 20. Find the amount of 324 for 1^ month, at 5 per cent. CASE III. To find the interest for days. Multiply the principal by the number of days, and by the rate per cent., and divide the product by 36,500. 21. Find the interest of 459 for 168 days, at 2 per cent. 22. Find the interest of 743 12 4d, for 142 days, at 4 per cent. 23. Find the interest of 379 16 6d, for 258 days, at 2| per cent. 24. Find the interest of 145 13 8d, from 4th June to 16th October, at 3 per cent. 25. Find the interest of 362 15 9d, from 6th May to 8th September, at 2 per cent. 26. Find the interest of L724 18s from 3d January to 20th August, at 5 per cent. 27. Find the amount of L85 16s, from 8th October to 14th March following, at 4 per cent. CASE IY. To find the interest when partial payments are made. RULE. Multiply the principal and the successive bal- ances by the number of days between the times of payments, add the products, and find the interest by Case III. 28. Find the interest on a bill of L854, due 8th June, of which L240 were paid 16th August, L169 October 110 SIMPLE INTEREST. 4, L238 January 20, and the balance March 8, at 4 per cent. 29. Fin the interest on a bond of L1000, due 16th March, of which L324 were paid 3d May, L166 July 18, L102 December 2, and the balance January 6, at 4J per cent. 30. Find the interest on a debt of L1264, due 1st Janu- ary, of which L499 were paid 4th March, L365 May 24. LI 00 July 18, and the balance October 20, at 5 per cent. CASE Y. To find the interest on accounts current. RULE. Take the sums in the order of their dates, and when two Drs. or two Crs. follow each other, add them, but when a Dr. an Cr. follow each other, subtract the less from the greater, and prefix their titles to the sum or dif- ference. Multiply the several balances by the number of days between the transactions, as in last case, and place the products of Drs. in one column, and those of Crs. in another, then add each column ; multiply each sum by its rate per cent., and find the interest of the difference by Case III. 31. Find the interest on the following Account to 31st December, allowing 5 per cent, when the balance is due to the bank, and 3.| per cent, when due to A.B. A. B's account-current with the Dundee Union Bank. Drawn on the Bank by A. B. ( Paid to the Bank by A. B. Dr. Cr. Feb. 24, To cash L826 March 18, By cash L300 May 8, To cash 131 June 28, By cash 727 Aug. 15, To cash 400 Sept. 6, By cash 564 Sep. 27, .Nov. 2, To cash To cash 348 408 Oct. 21, Dec. 10, By cash By cash 322 68 32. Find the interest on the following Account to 31st December, allowing 5 per cent, when the balance is due to the bank, and 4 per cent, when due to C. D. SIMPLE INTEREST. Ill C. D.'s account-current with the Bank of Scotland. Drawn on the Bank by C. D. | Paid to the Bank by C D. Dr. Cr7~] March 2, To cash L428 April 16, By cash L355 Mar. 29, To cash 500 June 8, By cash 839 May 4, To cash 118 July 25, By cash 456 Aug. 7, To cash 800 Oct. 13, By cash 422 Aug. 28, To cash 169 Nov. 26, By cash 166 EXERCISES IN SIMPLE INTEREST. 1. Find the interest of L978 17s for 3j years at 4 per cent. 2. Find the interest of L671 18 4d for 5J years at 3| per cent. 3. "What will L846 16s amount to in 6| years at 4J per cent? 4. What will L978 18 6d amount to in 8J years at 3| per cent ? 5. Required the interest of L789 for 7 months at 3 per cent. 6. Required the interest of L677 14 6d for 10^ months at 5 per cent. 7. What is the interest of L2468 for 287 days at 3 per cent ? 8. Required the interest of L976 14 8Jd for 312 days at 4j per cent. 9. What is the interest of L621 from 6th May to 19th December, at 4| per cent? 10. Find the interest on L726 18 6d from 17th June to 15th January, at 4^ per cent. 11. What is the amount of a bill of L468, wWch lay at in- terest from 2d January to 18th September, at 5 per cent? 12. Required the interest, at 5 per cent., on a bond of L2000, due 4th January 1840, of which L426 were paid 3d March, L375 June 18th, L529 October 22d, L231 December 4th, L128 on the 5th March following, and the balance on the 14th September. 112 DISCOUNT. DISCOUNT is an allowance made for the payment of money before it becomes due, The present value of a sum of money due at a future period, is that which, being put to interest for the same time and rate, would amount to the sum then due. True method of finding the Discount. RULE I, As the amount of L100 for the given rate and time: Is to the interest of JL100 for the same time : : So is the given sum : To the discount ; Which, subtracted from the given sum or debt, leaves the present value. 1. Find the discount and present value of L400, due 3 years hence, at 4 per cent. 2. "What is the discount and present value of L364, due 2 years hence, at 3^ per cent ? 3. "What is the discount and present value of L568, due 5 years hence, at 5 per cent? 4. Required the discount and present value of L840, due 8 months hence, at 4i per cent. 6, Find the discount and present value on a bill of L628, due 18 months hence, at 41 per cent. 6. Required the discount and present value on a bill of L248, payable 146 days hence, at 5 p. c. interest. Common method of finding the Discount. RULE II. Find the number of days the bill has to run, reckoning from the day it is discounted to the day it is pay- able, to which add three days of grace, then find the inter- est on the given sum. The answer thus found is called Discount* NOTE. Subtract the discount from the sum of the bill, the difference gives the proceeds, or present value. 7. 'A. bill of L400, dated 4th August, at 4 months, was When the rate per cent, is not stated, 5 per cent, ig always understood. DISCOUNT. 113 discounted 10th August ; find the discount and pre- sent value. 8. A bill of L200, dated 4th August, at 4 months, was discounted 12th September ; find the discount and proceeds. 9. A bill of L378, dated 14th March, at 3 months, was discounted 14th April j find the discount and pro- ceeds. 10. Find the discount and proceeds on a bill for L846, dated 1st January, at 8 months, which was discount- ed 3d June following. EQUATION OF PAYMENTS. EQUATION OF PAYMENTS is the method for ascertain- ing the time at which several debts, due at different periods, may be settled at one payment. "RULE. Multiply each debt by the time it has to run, then divide the sum of the products by the sum of the debts, the quotient will be the time required. 1. If L300 is payable in 3 months, L460 in 4 months, and L500 in 6 months, when may the whole be paid at once ? 2. If a debt of L268 be payable at 2 months, another of L364 at 3 months, and L444 at 10 months, when may the whole be paid at once ? 3. A owes toB L168 due in 30 days, L64 due in 40 days, and L97 due in 60 days, when may the whole be set- tled at one payment P 4. M is indebted to R in the sum of L628, which was to be paid thus, L100 at the end of 1^ years, L266 at the end of 2^ years, L134 at the end of 3 years, and the rest at the end of 4 years, at what time ought the whole to be dischaiged in one payment ? 5. Delivered to a banker the following bills, viz. A. B's bill for LI 00, due in 20 days, M. B's for L264, due in 30 days, and M. H's for L420, due in 60 days ; at how many days should he grant me a bill for the whole ? 114 COMMISSION AND BROKERAGE. COMMISSION and BROKERAGE are allowances of a cer- tain rate per cent, to hankers, agents, or brokers, for tran- sacting the business of others. RULE I. When the rate exceeds Ll per cent, multiply the sum by the rate per cent, and divide the product by 100, as done in interest. RULE II. "When the rate is under Ll percent., work the question by Simple Proportion, or find the result by Practice, and divide that by 100. 1. Find the commission on L847, at 2 per cent. 2. Find the commission on L479 10 6d, at 1J per cent. 3 Find the brokerage on L496 14 8d, at 2j per cent. 4. Find the brokerage on L648 16 4d, at 2s 6d per cent. 5. Find the commission on L573, at 6s Sd per cent, 6. "What does a broker receive for selling stock to the amount of half a million, at 2s 6d per cent ? 7. "When a factor is allowed 10s per cent, for commission, what should he charge for transacting business to the amount of L6SOO ? 8. My agent writes me that he has transacted business on rny account to the amount of L8560, what commission is he entitled to, at 2^ per cent P 9. If an agent transact business to the amount of L64,896 per annum, and is allowed 2j per cent, what is his in- come, supposing he loses by bad debts L548 ? INSURANCE. INSURANCE is a contract by which the insurer engages to repay losses sustained by the insured for a certain per centage on the sum insured. INSURANCE. 115 The insurer is the party who undertakes the risk. The insured is the party protected by the insurance. Premium is the sum paid to the insurer. Policy is the paper or parchment containing the contract of insurance. NOTE. The duty on policies for sea insurances, upon any voyage whatever, varies from 3d. to 4s. per cent, according to the rates of premium on the sums insured. The examples given below are at Is. and 4s. respectively. Policy duty is charged on the fractions of 100, as if it amounted to 100 ; thus 342 pays the same duty as 400. CASE I. To find the premium and policy. RULE. Calculate as in Commission and Brokerage. 1. Find the insurance on 600 at 2 per cent. 2. Find the expense of insuring 84~0 against risk by fire, premium 2s per cent. 3. Find the expense of insuring 460 on a house, premium 5s per cent. 4. Find the expense of insuring 600 on furniture, &c. premium 2s 6d per cent. 5. Find the expense of insuring 840 on goods from Dundee to London, at 1 per cent, and policy, Is. 6. Find the expense of insuring 980 on a ship and cargo from Glasgow to Jamaica at 3 per cent, policy 4s, and commission ^ per cent. CASE IT. To find how much must be insured to cover a given sum. RULE. Subtract the sum of the premium, policy, and commission per cent, from 100 then say, as the dif- 116 INSURANCE. ference is to 100, so is the given sum, to the sum to be insured. 7. How much must be insured to cover 400, premium 1^ guineas, policy 2s p. c., and commission 10s per cent. ? 8. How much must be insured to cover 600, premium 2 guineas, policy 3s p. c., and commission 10s per cent. ? 9. How much must be insured to cover 700, premium 2 j guineas, policy 4s, and commission 10s per cent. ? BUYING AND SELLING STOCKS. The STOCKS are debts contracted by Government for defraying the expense of the nation. Stock is also the name given to the Capital of a Bank, Railway, Insurance or Mercantile Company. NOTE. -Stock is bought and sold through the medium of brokers, who receive ^ per cent for every quantity of stock which they buy or sell. N. B. The allowance for brokerage is omitted in the following questions : CASE I. To find the value of any quantity of stock. RULE. Multiply the quantity of stock by the rate, and divide the product by 100. 1. Find the value of 1260 three percent, consols, at 87f per cent. 2. Find the value of 1000 three per cent. red. at 88| per cent. 3. Find the value of 840 four pen cent, consols, at 97| per cent. BUYING AND SELLING STOCK. 117 4. Sold 640 India stock at 240 per cent., what did I receive P CASE II. To find how much Stock may be bought for a given sum. RULE. Multiply the quantity of Stock by 100, and divide the product by the rate. 5. How much stock at 86| per cent, will 680 purchase P 6. How much of four per cents, at 80J per cent, will 840 purchase ? 7. How much of India stock at 188 J per cent, will 1000 purchase P 8. How much of bank stock at 160 per cent, will 960 purchase ? CASE III. To find the rate of interest arising from money invested in the stocks. RULE. Multiply the interest or dividend by 100, and divide the product by the selling price. 9. What interest will be obtained from vesting money in the 3 per cents., when selling at 79f per cent. ? 10. What interest will be obtained from vesting money in the 4 per cents., when selling at 84 ^ per cent. ? 11. What interest arises from money vested in India stock, when the price is at 230, the dividends being 10 per cent. ? V2. What interest arises from money vested in bank stock, when the price is at 180, the dividends being 8 i er cent. ? F us PROFIT AND LOSS. Profit and Loss is the rule by which merchants can ascertain how to buy and sell, so as to gain or lose a cer- tain rate per cent, &c. CASE I. The prime cost and selling price given; to find the gain or loss per cent. RULE. As the prime cost is to the gain or loss, so is 100 to the gain or loss per cent. 1. Bought cloth at 14s 6d per yard, and sold it at 18s 2d ; what was the gain per cent. ? 2. Bought tea at 6s per pound, and sold it at 7s lOd; what was the gain per cent. ? 3. Bought a cwt. of soap for 2 4s, and sold it at 10s per cwt. profit ; what was the gain per cent. ? 4. Bought cloth at 6s 4d, and sold it at Is 2d per yard loss ; what was the loss per cent. ? 6. Bought tea at 6s 8d per lb., but getting damaged, was obliged to sell it at 5s ; what was the loss per cent. ? CASE II. The prime cost and gain or loss per cent, given, to find the selling price. RULE. As 100 is to 100 with the rate per cent, added to it, if gain, or subtracted if loss, so is the prime cost to the selling price. 6. Gained 9^ T per cent, by cloth, which I bought at 7a 4d; what did I sell it at? 7. Gained 15 per cent, by brandy, which I bought at 1 4s per gallon ; what did I sell it at ? PROFIT AND LOSS. 119 8. Bought rum at 12s 6d per gallon, what must I sell it at to gain 23 r \ per cent. ? 9. Bought tea at 6s per lb., at what must I sell it, by being obliged to lose 16| per cent. ? 10. Lost OJy per cent, by cloth which I bought at 22s per yard, what did I sell it at ? 11. Bought coffee at 2s 6d per lb., at what must I sell it per cwt, to gain 20 per cent. ? CASE III. The selling price, and the gain or loss per cent, given, to find the prime cost. RULE. As 100 with the rate per cent, added if gain, or subtracted if loss, is to 100, so is the selling price to the prime cost. 12. If I gain 15 per cent, on cloth which 1 sold at 10s, what was the prime cost ? 13. Gained 20 per cent, on rum, which I sold at 14s 6d, what was the prime cost ? 14. Lost 25 per cent, on coffee, which I sold at 2s 6d, what was the prime cost ? 15. Lost 20 per cent, on tea, which I sold at 6s, what was the prime cost ? 16. Sold a quantity of cloth at 13s 6d per yard, by which I cleared 20 1 per cent., what did I pay for it per yard ? 17. Sold green tea at 12s, on which I gained 26 T 6 g per cent., what did I pay for it per lb. ? CASE IV. When two selling prices, and the gain or loss per cent, on one of them is given, to find the rate per cent, on the other. RULE. As the price whose rate per cent, is given, is to the other price, so is 100 with the rate per cent, added, or subtracted, to a 4th number, from which subtract 100 if 120 PROFIT AND LOSS. pain, but which subtract from 100 if loss ; the remainder will be the required rate. 18. By selling cloth at 6s I gained 12 per cent., what did I gain per cent, by selling it at 8s ? 19. By selling tea at 5s 6d per Ib. I gained 15 per cent., what did I gain per cent, by selling it at 7s ? 20. Sold hats at 16s, I lost 14s per cent., what do T lose or gain by selling at 18s ? 21. Sold wheat at 3 10s a quarter, I lost 20 per cent., what do I lose or gain by selling at 4 a quarter? 22. Sold paper at 20s a ream, and lost 12 per cent. ; whaf do I lose or gain by selling at 24s a ream ? CASE V. When the price is to be advanced, so as to allow any proposed discount or credit. RULE. As 100 minus its discount is to 100, so is tha jirice to be received to the advanced price, 23. How must I sell sugar to have 9Jd perlb. after allow- ing a discount of 7f per cent. ? 24. How must I sell green tea to have lls a Ib., and al- low a discount of 12 per cent. ? 25. How must I rate broad cloth to obtain 24s a yard, and allow a discount of 10 per cent. ? 26. Bought flax at 48 a ton, how must I sell it to gain 10 per cent., and allow a discount of 4 per cent. ? 27. Bought wheat at 38s a quarter, how must I sell it to gain 15 per cent., and allow a discount of 6 per cent. ? EXERCISES IN COMMISSION & BROKERAGE, INSURANCE, BUYING AND SELLING STOCKS, AND PROFIT & LOSS. 1. Find the commission on goods amounting in value to 2476, at 2s 6d per cent. H w S9T r Tf T* EXERCISES IN COMMISSION, ETC. 121 2. Find the commission on 978, at 5s 6d per cent 3. Find the insurance on a house valued at 786, at 3s per cent. 4. Find the expense of insuring a ship and cargo valued at 2487, at 2 per cent. 5. Find the value of 5768 bank stock, at 140 per cent. C. Find the value of 2478 of the city of Edinburgh bonds, sold at 75 per cent. 7. How much stock, at 87^ per cent, will 8426 purchase ? 8. How much Railway stock, at 130 per cent, will 4836 purchase ? 9. "What interest will be obtained from vesting money in the 3 per cents, when selling at 84 i per cent ? 10. "What interest may be obtained by vesting money in the Edinburgh city bonds, when selling at 75 per cent., dividend 3 per cent P 11. Bought cloth at 12s 6d per yard, and sold it at 14s 2d, what was the gain per cent ? 12. Bought tea at 3s 4d per lb., and sold it at 4s 8d, what was the gain per cent ? 13. Bought tea at 4s per lb., what must I sell it at, to gain 16 per cent? 14. If I gain 15 per cent on cloth which I sold at 12s, what was the prime cost ? 15. Lost 20 per cent on cloth, which I sold at 14s per yard, what was the prime cost ? 16. How must I sell tea to have 5s per lb., after allowing a discount of 8 per cent ? 17. How must I sell cloth to have 14s per yard, and allow 10 per cent, discount ? DISTRIBUTIVE PROPORTION, OR PARTNERSHIP. Partnership is when more persons than one unite in trade, and agree to divide the profits or losses arising from the business, in proportion to their respective shares of the capital. 122 DISTRIBUTIVE PROPORTION. RULE. As the sum of the several shares is to each particular share, so is the gain or loss to the respective shares of it ; OR As the sum of the given numbers is to each particular number, so is the number to be divided to the required parts. PROOF. The sum of the answers is equal to the num- ber to be divided. 1. Three merchants, A, B, and C, enter into partnership A's share is 300, B's 400, and C's 500, they gained 240 ; what must each receive of the gain ? 2. Three men, A, B, and C, enter into trade, A's share is 620, B's 480, and C's 840, they gained 400 ; what must each receive of it ? 3. Four merchants freight a ship to Riga, A's share of it is 540, B's 250, C's 436, and D's 324, they gained 280 ; what must each receive of it ? 4. Four merchants in company lost 168, A's share of the capital is 420, B's 566, C's 724, and D's 650 ; required the loss of each. 6. A's share of a ship is 98, B's 86, C's 100, D'* 140, and E's 120; damages at sea are sustained to the extent of 100; how much of this sum must each proprietor lose ? 6. Divide 864 acres among A, B, C, and D, according to their rents, which are as follows : A's 240, B's 280, C's 308, and D's 122. 7. A testator bequeathed to A 260, to B 488, to C 622, and to D 500 ; but at his death the net amount of his property was only 1243; how much of this sum should each legatee have received ? 8. A bankrupt owes to A 126, to B 104, to C 98, to D 249, to E 84, and to F 97 : his money and effects amount to 508 ; how much can he pay per pound, and what is the just dividend to each of his creditors ? DISTRIBUTIVE PROPORTION. 123 When the unequal shares of the several partners remain in company for unequal times. RULE. Multiply each share by the time of its continu- ance in trade, then say, as the sum of the products is to each particular product, so is the gain or loss to the respec- tive shares of it. 9. Three merchants enter into partnership; A puts in 488 for 4 years, B 625 for 6 years, and C 800 for 2 years, they gained 468 ; what must each receive of the gain ? 10 A, B, and C pay money into one common stock ; A pays 240, which lies for 6 months ; B 343 for 8 months, and C 468 for a year ; they gained 200 ; what must each receive of it ? 11, Three graziers rent a field for 30, into which A put 32 cows for 2 months, B 24 for 6 months, and C 20 for 8 months ; what part of the rent must each pay? 12. Three farmers rent a field of grass for 42 ; A puts in 48 sheep for 4 months, B 50 for 2 months, and C 30 for 3 months ; what part of the rent must each farmer pay ? COMMERCIAL ALLOWANCES, OR TARE AND TRET, Are certain deductions made from goods which are weighed in the chest, barrel, or whatever contains them. Gross Weight is the weight of both goods and packages. Tare is an allowance granted to the buyer for the weight of the barrel, &c., containing the goods, and is deducted from the gross weight. Tret is an allowance of 4 Ibs. on 104 Ibs., or 5 < g on goods liable to waste, and is deducted after the tare ; it is now nearly discontinued, or allowed for in the price. 124 COMMERCIAL ALLOWANCES. Draft is an allowance on some goods for loss in retailing them, and is deducted before the tare. NOTE. After subtracting the tare from the gross weigtit, the remainder is called tare suttle ; and after subtracting the tret, the remainder is called tret suttle ; and what remains after all the deductions are made, is called neat or net weight. RULE. Subtract the tare from the gross weight, and from the tare suttle deduct ^ part, the remainder, which is the tret suttle, will be the neat weight. NOTE. In calculating commercial allowances, remain- ders less than ^ of a Ib. are rejected, but when , or more, they are considered as 1 Ib. 1. Find the neat weight of 4 chests of tea, each 2 cwt. 1 qr. 24 Ibs. tare 24 Ibs. per chest, deducting also the usual allowance for tret. 2. Find the neat weight of 12 hhds. sugar, each 18 cwt. 2 qrs. 14 Ibs., tare 26 Ibs. per hhd., also deducting the usual tret. 3. Find the net weight of 20 barrels figs, each 3 cwt. 3 qrs. 18 Ibs., tare 36 Ibs, per barrel, also deducting tret. 4. Find the tret suttle of 15 bags of cotton wool, each 2 cwt. 3 qrs. 22 Ibs., tare 1 Ib. per bag, and tret the usual allowance. 5. What is the net weight of 468 cwt. 1 qr. 21 Ibs. allow- ing for tare, Sec., 16 Ibs. per cwt. ? 6. What is the neat weight of 24 bags of coffee, each 4 cwt. 2 qrs. 14 Ib., tare, &c., 14 Ibs. per cwt., and find the price of it at 2s 2d per Ib. ? 7. How many gallons net, (1\ Ibs. to a gallon,) are in 10 casks oil, each 3 cwt. 2 qrs. 8 Ibs. allowing for tare 16 Ibs. per cwt. and what is the price of it at 2s 4d per gallon P 125 BARTER Barter is the method of exchanjing goods without profit to either party. RULE Find the value of the goods given away ; then find what quantity of other goods may be pur- chased for that money. 1. Plow much tea at 6s. 6d. per Ib. should be given in barter for 142 yards of linen, at 3s. per yard ? 2. How many yards of cloth at 14s. per yard should be given in barter for 20 cwt. of sugar, at 7d. per Ib. ? 3. How many gallons of rum at 16s. per gallon should be given for 84 gallons of whisky, at 7s. 6d. per gallon ? 4. Exchanged 86 yards of broad cloth at 19s. 6d. per yard, for Irish linen at 3s. 4d. per yard, how much linen should I receive ? 5 Exchanged 159^ yards of muslin at 8s. lOd. per yard, for Holland gin at 26s. per gallon, how much gin should I receive ? 6. Exchanged 97 \ cwt. of sugar at 9|d. per Ib. for cloth at 18s. 4c^d. per yard, how much cloth should I receive ? 7. Exchanged 124 yards shirting muslin at 2s. 6d. per yard, for ICO yards of printed cotton at Is. 4d. per yard, and the remainder in ribbons at Is. 2d. per yard ; how many yards of ribbon should I receive ? 8. Exchanged 68 1 yards of velvet at 22s. per yard for 85 yards cambric at 12s. 4d. per yard, and the remainder in stockings at 2s. 6d. a pair ; how many pairs of stockings should I receive ? F2 126 EXCHANGE. EXCHANGE is the method of finding- how much of the money of one country is equal in value to any proposed sum of the money of another country. The Par of Exchange is the intrinsic value of the money of one nation, compared with that of another nation, which is estimated by the quantity of pure gold or silver. The Course of Exchange, is the current value allowed for the money of one country, when reduced to the money of any other country. This is seldom at par, but is continually varying, according to the circum- stances of trade. Agio is the difference between bank and current money, and also between the intrinsic and circulating value of foreign coins. Usance is the usual time allowed by merchants and bankers to pay bills of exchange. Days of Grace are the days allowed for paying bills after their term is expired, before diligence is used. All calculations in Exchange may be performed by Proportion, and often by Practice. AUSTPvIA. In Austria, the money of account is now the new florin =10U kreusers = Is. ll^d. Par of exchange with London 10 fl. 36 kr. per 1. 100 convention or old florins = 105 new florins. The usance of bills is 14 days after acceptance; 3 days of grace are allowed on bills drawn at more than 7 days' sight or date. EXCHANGE. 127 1. How much sterling is equivalent to 375 kreusers 20 florins, exchange at 10 fl. 30 kr. per 1 ? 2. How much money of Austria is equivalent to 427, 17s. 6d., exchange at 10 fl. 36 kr. per 1 ? 3. Reduce 475 fl. 20 kr. to 's sterling, when the exchange is at 10 fl. 20 kr. per 1. 4. Reduce 750, 12s. 6d. into Austrian money, ex- change at 10 fl. 25 kr. per 1. BELGIUM. In Belgium, the general monetary unit is the French franc of 100 centimes == 9|d. The par of exchange with London from gold coins is 25 francs 22 4 cents, per 1, from silver coins 25 fr. 57 c. per 1. The usance of bills from London is 1 month's date; no days of grace are allowed. 1. How much sterling is equivalent to 750 francs 50 cents., exchange at 25 francs 20 cents, per 1 ? 2. How much money of Belgium is equivalent to 575, 2s 6d., exchange at 25 fr. 2 c. per 1 ? 3. Reduce 495, 17s. 6d. into Belgian money, ex- change at 25 fr. 25 c. per 1. 4. Reduce 4785 francs 20 cents, into sterling money, exchange at 25 fr. 24 c. per 1. CANADA. In Canada, money is reckoned in dollars of 100 cents. ; also in , s. d. Halifax Currency. 1 currency = 4 Spanish dollars, each dollar being called 5s. instead of 4s. 2d., its average value ; hence 100 sterling = 120 currency. 128 EXCHANGE. 1. How much sterling is equivalent to 15, 12s. 6d. Halifax currency ? 2. How much Halifax currency is equivalent to 75, 17s. 6d. sterling ? 3. In 105, 16s. 8d., how many dollars at 4s. 2d. each ? 4. How much sterling is equal in value to 486 dollars 66 cents., at 4s. 1 id. per dollar ? DENMARK. In Denmark, accounts are generally stated in rigs- bank dollars = 6 marcs = 96 skil lings = 2s. 2|d. ; in some houses they are kept in Hamburgh marcs banco : 200 II. D. = 300 M. Bco. Par of exchange with London 9 R. D. 10 sk. per 1 ; there is no established usance, but 8 days of grace are allowed. 1. How much sterling is equal to 250 R. D. 16 sk., exchange at 9 R. D. 8 sk. per 1 ? 2. In 199, 12s. 6d., how much Danish money, ex- change at 9 R. D. 10 sk. per 1 ? 3. How many R. D. are equal to 525 M. Bco. ? 4. How many M. Bco. are equal to 750 II. D. ? FRANCE. In France, the money of account is the franc = 10 declines =100 centimes = 9|d. Par of exchange with London as under Belgium. The usance of bills on London is 30 days' date ; no days of grace are allowed. EXCHANGE. 129 1. How much sterling is equal to 846 francs, ex- change at 25 francs 20 cents, per 1 ? 2. In 684, how many francs, exchange at 25 fr. 22 c. per 1 ? 3. In 1000, how many francs, exchange at par ? 4. In 8475 francs, how many 's at par ? FRANKFOIIT ON THE MAINE. In Frankfort, accounts are stated in florins of 60 kreusers = Is. 8d. Par of exchange with London 120 florins 34 kr. per 10. The usance of bills not payable at the fairs is 14 days' sight ; 4 days of grace on bills at more than 4 days' sight or date only. 1. How much sterling is equal to 4756 fl. 30 kr., exchange at 120 fl. 30 kr. per 10 ? 2. In 527, 3s. 4d., how many florins, exchange at 120 fl. 32 kr. per 10? 3. In 575 fl. 20 kr., how much sterling, exchange at 120 fl. 30 kr. per 10? 4. In 575, 12s. 6d., how many florins, exchange at 120fl. 24 kr. per 10 ? HAMBURGH. In Hamburgh, accounts are stated in marks of 16 schillings or 192 pfennings; 3 marks = 1 dollar. Money is here distinguished into banco and currency, the agio varying about 25 per cent. ; a mark current = Is. 2d., a marco banco = Is. 5d., agio being 25 per cent. Par of exchange with London is 13 M. Bco. 10i sch. per 1 ; usance of bills from London, 1 month's date ; days of grace are now discontinued. 1. How much sterling is equal to 325 marcs cur. 8 sch., exchange at 13 M. Bco. 8 sch. per 1, agio 25 per cent. ? 130 EXCHANGE. 2. How much Hamburgh currency is equal to 377, 10s., exchange at 13 M. Bco. 10 sch. per 1, agio 24 per cent. ? 3. In 722, 10s., how much Hamburgh currency, exchange at 13 M. Bco. 8 sch. per 1, agio 25 per cent. ? 4. Reduce 961 marks cur. to sterling, exchange at 13 M. Bco. 9 sch. per 1, agio 24 per cent. HOLLAND. In Holland, the monetary unit is the florin or guilder of 100 cents, or 20 stivers = ls.8d. ; par of exchange with London, 1 2 florins for 1. Usance of bills from London, 1 month's date ; days of grace are now in disuse. 1. How many florins are equal to 476, 19s. 6d., ex- change at 11 fl. 95 cents, per 1 ? 2. How much sterling is equal to 567 fl. 75 cents., exchange at 11 fl. 96 c. per 1 ? 3. Reduce 67, 18s. 9d. to florins, exchange at 11 fl. 90 cents, per 1. 4. Reduce 1725 fl. 80 c. to sterling, exchange at 12 florins per 1. INDIA. In India, the monetary unit is the current or Com- pany's rupee of 16 annas = 192 pice = Is. 10d. ; 100 sicca rupees = 116 cur. rupees; 100,000 rupees are termed a lac, 10 millions a crore, and 100 crores a mas. 1. How much sterling is equal to 375 cur. rupees 8 annas at Is. 10|d. per cur. rupee ? 2. How many cur. rupees are equal to 97, 17s. 6d. at Is. 10 Jd. per cur. rupee ? 3. How many sicca rupees are equal to 750, 12s. 6d. at Is. 10] d. per cur. rupee? 4. What is the value of a lac of sicca rupees at Is. H)| d. per cur. rupee ? EXCHANGE. 131 NORWAY. In Norway, accounts are kept in paper species dollars of 5 marks or orts, or 120 skillings ; the silver species dollar = 2 Danish rigsbank dollars = 4s. 5d., which is now also the value of the bank paper dollar. 1. How much sterling is equal to 120 dollars 3 marks, exchange at 4s. 4d. per dollar ? 2. How much money of Norway is equal to 59, 5s. fid. exchange at 4s. 4^d. per dollar ? 3. Reduce 375 doL 2 marks 20 sk. to sterling at 4s. 5d. per dollar. 4. Reduce 124, 4s. 6d. to dollars at 4s. 5d. per dollar. NOVA SCOTIA. In Nova Scotia, accounts are stated in , s. d. cur- rency ; 16s. sterling = 20s. currency. 1. How much currency is equal to 105, 7s. 6d. sterling ? 2. How much sterling is equal to 366, 6s. 8d. cur- rency ? 3." Reduce 527, 12s. 6d. ster. to currency. 4. Reduce 425, 13s. 4d. cur. to sterling. PORTUGAL. In Portugal, accounts are stated in reas, 1000 of which make a milrea = 56d. ; 400 reas = 1 crusado, 6400 reas=ljohanese; amillion of reas, 1000$000, are termed a conio. Usance of bills from London, 30 days' sight ; the days of grace are 15 on inland bills, 6 on foreign bills when accepted. 1. Reduce 2496 milreas 120 reas into sterling money, exchange at 56 d. per milrea. 132 EXCHANGE. 2. Reduce 576, 18s. fid. into Portuguese money, exchange at 58d. per milrea. 3. Find the value of a crusado at 5Gd. per milrea. 4. Find the value of conto of reas at 56d. per miirea. PRUSSIA. In Prussia, accounts are stated in thalers or dollars of 30 silver groschen or 360 pfennings = 2s. 10|d. Par of exchange with London, 6 Prus. dol. 27 sil grosch. per 1. The usance of bills on Berlin and Dantzic is 1 4 days' sight, and 3 days of grace are allowed. 1. How much sterling is equal 760 dol. 20 s. groschen, exchange at 6 dol. 25 gros. per l ? 2. How much Prussian currency is equal to 562, 18s. 9d., exchange at 6 dol. 28 grosch. per 1 ? 3. Reduce 495 dol. 10 gros. to sterling money, ex- change at 6 dol. 24 gros. per 1. 4. Reduce 197, 2s. 6d. to Prussian currency, ex- change at 6 dol. 28 grosch. per 1. RUSSIA. In Russia, the integer of account is the silver ruble of 100 copecs = 37 |d. Par of exchange with London, 6 rubles 40 copecs per 1. 350 copecs in paper=100 copecs in silver; hence a paper ruble = 10i{d. 10 days of grace are allowed for bills after date, 3 for bills after sight. The Julian Kalendar or old style is still used. 1. Reduce 46'8 into rubles, exchange at 3s. 2d. per ruble. 2. Reduce 5550 rubles into sterling money, exchange at 37d. per ruble. EXCHANGE. 133 3. How many rubles are equal to 476, 12s. 6d., exchange at 6 rub. 36 cop. per 1 ? 4. How much sterling is equivalent to 4950 rub. 20 cop. exchange at 6 ru. 40 cop. per 1 ? SPAIN. In Spain, accounts are generally stated in reals of 34 maravedis vellon or 100 centavos ; 20 reals de vellon = 1 duro or hard dollar, valued at par at 4s., from 100 reals vellon being reckoned 1. 1. How much sterling is equal to 75 dol. 15 reals 20 centavos, exchange at 48'5d. per duro ? 2. How much Spanish money is equal to 375, 12s. 6d., exchange at 48 -4d. per duro ? 3. Reduce 120 dol. 10 reals 50 centavos into sterling money, exchange at 48d. per duro. 4. Reduce 75, 17s. 6d. into dollars, exchange at 4s. per duro. UNITED STATES. In the United States, the integer of account is the dollar of 100 cents. = 4s. lid. at par. The par of ex- change with London is 4-86 1 dollars per 1. In ex- change the dollar is assumed to be 4s. 6d., that is at a premium of 9 4 per cent. Bills on Europe are com- monly drawn at 60 days' sight, and 3 days of grace are allowed. 1. How much sterling is equivalent to 596 dol. 75 c., exchange at 4'86 dollars per 1 ? 2. How many dollars are equivalent to 575, 17s. 6d., exchange at 4-86 1 dollars per 1 ? 3. Reduce 375 dol. 95 c. to sterling, exchange at 4s. 6d. per dollar, premium 9| per cent. 4. Reduce 99, 12s. 6d. to dollars at 4s. 6d., premium 94 per cent. 134 INVOLUTION. INVOLUTION is the method of finding the power of a given number. A power is the product arising from the multiplication of the given number by itself the proposed number of times. The required power is generally expressed by a small figure placed on the right hand corner of the given number, called the index, or exponent ; thus, 6 2 , 6 3 , 6 4 . 1 st power or root 2d power or square 3d power or cube 125 36 49 G4 81 216343512729 Any power of a proposed number is found by multiply- ing that number continually by itself, till the number of factors is equal to the number of units in the index. 1. What is the square or second power of 4 ? 2. What is the cube or third power of 3 ? 3. What is the biquadratic or fourth power of 5 ? EVOLUTION. EVOLUTION is the method of finding the root of a given number. Square root is that of which the given number is the square or second power. TO EXTRACT THE SQUARE ROOT. RULE. Divide the given number into periods of two figures each, beginning from right to left in integers, and from left to right in decimals. Find the greatest square number in the left hand period ; place its root in the EVOLUTION. 135 quotient, and subtract the square number itself from that period; to the remainder annex the next period for the dividend, double the quotient figure for the first part of the divisor, and see how often it is contained in the figures of the dividend to the left of the units' place, then pat the result on *he right of both divisor and quotient, multiply the divisor by this figure, and subtract the product from the dividend ; to this remainder annex the next period for a new dividend. To the last divisor add the last figure of the root for a new divisor, with which proceed as before, and so on till the periods are exhausted. If there be a remainder, the root may be carried out decimally, by annexing two ciphers at each step of the process. NOTE. "When the square root of a vulgar fraction is required, extract the root of the numerator and that of the denominator separately, for the respective terms of the root ; but if it cannot be done exactly, reduce the fraction to a decimal, and extract the root. EXAMPLE. Find the square root of 54756. 43 464 4756(234 147" 129 1856 1856 Find the square root of the following numbers : I. 18769 11. 123369.5376 2. 69696 ]2. 7590.417129 3. 149769 13. 256.416169 4. 33431524 14. .000235483 5. 32821441 15. 2. 6. 91604041 16. 3 7. 6031936 17. 6 8. 4937284 18. 7 9. 1194877489 19. 35 10. 7334552364 J - t/ * 39 20. ^Va 136 EVOLUTION. APPLICATION OF THE SQUARE BOOT. C In any right angled triangle ABC, the square of the hypotenuse A C, is equal to the squares of the base A B, and perpendicular B C. 1. Given the base of a right angled triangle 90 feet, and the perpendicular 67.5 feet, find the length of the hypotenuse. 2. Given the hypotenuse of a right angled triangle 112.5 feet, and the base 90 feet, find the perpendicular. 3. Given the perpendicular of a right angled triangle 67.5 feet, and the hypotenuse 112.5 feet, find the base. 4. Find the length of a ladder that will reach from the edge of a ditch which is 120 feet broad, (surrounding a fort,) to the top of the fort, the height of which is 90 feet. 6. "Wanting to find the height of a rock, I stood 86 feet from the bottom, and found that the distance from the place where I stood to the top of the rock was 120 feet ; find its height. 6. A ladder 42 feet long is so placed that it reaches a window 34 feet from the ground on one side of the street, and without moving it at the foot, will do the same to a window 20 feet high on the other side ; find the breadth of the street. EVOLUTION. 137 To find the mean proportional between two given ex- tremes. RULE. Multiply the two numbers together, and ex- tract the square root of the product for the mean propor- tional. 7. Find the mean proportional between 16 and 36. 8. Find the mean proportional between 7 and 135. 9. Find the mean proportional between 6 and 24. To find the side of a square equal in area to any given superficies. RULE. Extract the square root of the given area for the side of the square. 10. The area of a circle is 5776, required the side of a square equal in area thereto. 11. A gentleman has a field of an irregular form, con- taining 10 acres, which he wants to exchange for a square field of the same extent ; required the side of the square in poles. 12. Required the side of a square equal in surface to an irregular figure, containing 40 acres. CUBE ROOT. CUBE ROOT is that of which the given number is the third power. RULE. Divide the given number into periods of three figures each, beginning from, right to left in integers, and from left to right in decimals. Find the greatest cube num- ber, in the left hand period ; place its root for the quotient, and subtract the cube number itself from that period : to the remainder annex the next period for a dividend, to which find a divisor by multiplying the square of the quo- tient found by 300, then consider how often it is contained in the dividend, and place the number of times in the quo- tient. 138 EVOLUTION. Multiply the former part of the quotient by this last fi- gure, and by 30 ; write the product under the divisor, and under it place the square of the last figure in the quotient; the sum of these three numbers is the complete divisor, which multiply by the last quotient figure ; subtract the product from the dividend, and to the remainder annex the next period for a new dividend, to which find a divisor as before, and proceed in the same manner till all the periods are exhausted. If there be a remainder, the cube root may he carried out decimally, by annexing three cyphers at each step of the process. EXAMPLE. Find the cube root of 12812904. 4\128*12904(234 2 2 x300 =1200\_8 2 x 3x30 = 180 \ 4812 32 == 9\ 1389 x 3 \ 4167 23 2 x 300 = 158700 \ 645904 23x4x30= 27GO \ 4 16 \ 161476 x 4 \ 645904 Find the cube root of the following numbers : 1. 46656 9. 2773.505125 2. 49836032 ! 10. .0024689745 3. 94818816 ! 11 oo 4. 194104539 \* 5. 306182024 6. 13858588808 j 14 * 5* 7. 150483876759 16 * 4*13 8. 1879.080904 ] G * EVOLUTION. 139 APPLICATION OF THE CUBE ROOT. To find the mean proportional between two given ex- tremes. Multiply each extreme by the square of the other; then extract the cube root of the products for the two mean proportionals. 1. Find the two mean proportionals between 5 and 320. 2. Find the two mean proportionals between 2 and 16. 3. Find the two mean proportionals between 6 and 162. Given the dimensions of a solid body, to find the dimen- sions of a similar one that will be greater or less. RULE. Multiply the cube of the given dimensions by the number of times the required solid is to be great- er or less than the given one, then extract the cube root of the product for the answer. NOTE. "When the dimensions of two solids, and the weight of one of them are given, to find the weight of the other, say, as the cube of the given dimension of the solid f which the weight is known, is to the cube of the other given dimension, so is the given weight to the required weight. 4. There is a cubical vessel whose side is 12 feet, required the side of a similar vessel that will contain three times as much. 5. The lineal side of a cubic piece of marble is 36 feet, find the side of a similar one 4 times as large. 6. Given a cubical vessel whose side is 9 feet, find the side of a similar vessel that will hold half as much. 7. If a globe of brass 4 inches diameter weighs 9 Ibs., find the weight of another globe, the diameter of which is 12 inches. 8. If a globe of 6 inches diameter weigh 16 Ibs., required the diameter of another weighing 84 Ibs. 9. The solidity of a sphere is 18191.447 inches, find the lineal side of a cube of equal solidity. 140 POSITION. POSITION is the method of resolving a class of questions which do not come directly under any of the other rules of Arithmetic, and is divided into SINGLE and DOUBLE. Single Position is when one supposition only is employed in the calculation. RULE. Suppose any number at pleasure, and try if it answers the conditions of the question ; if it does it is the answer, if not, say as the result of this operation is to the given number, so is the supposed number to the answer. 1. Find a number, to which if you add of itself, the sum will be 120. 2. Find a number, to which if you add and of itself, the sum will be 240. 3. A person after spending and J of his money had 72 left, what had he at first. 4. A young gentleman was left a fortune, \ of which he spent in gambling, J among his companions, ^ on a house and furniture, ^ on a stud of horses ; he then finds he has only 4240 remaining, what was his fortune ? 5. A man being asked his age, said, if to my age you add i and fc of itself, the number will be 87, what was his age? 6. A , B, and C, purchased a house for 800, of which A was to pay double of B, and B three times as much as C, what should each pay? DOUBLE POSITION. DOUBLE POSITION is when two suppositions are em- ployed in the calculation, and their results are not propor- tional to the supposed numbers. RULE. Suppose any numbers at pleasure, and wort with each of them as the question directs ; find the differ- DOUBLE POSITION, 141 ence between these results and the result in the question, and mark the error of each result with the sign -f or , according as it is an error of excess or of defect ; then mul- tiply the first supposition by the second error, and the se- cond supposition by the first error. When the signs are both + or both , divide the difference of these products by the difference of the errors, the quotient is the answer ; but when one of the signs is + and the other , divide the sum of the products by the sum of the errors, and the quo- tient is the answer. EXAMPLE. How many guineas and half-crowns will pay a bill of 72, the number of pieces of both kinds being 132 ? Shills. I. Sup- "1 90 guineas = 1890 pose J 42 half-cr. = 105 2. Sup- 1 36 guineas pose J 95 half-cr. Shills. = 756 = 240 pieces 132. result 1995 72 = 1440 132 72 guineas == 63 tialf-cr. 9 72 996 = 1440 4-555 90 x 444 = 39960 36 x 555 = 19980 ~999 )59940 60j 721 132 444 1. What number is that which being multiplied by 6, the product increased by 8, and that sum divided by 8, gives 64 ? 2. What number is that which being multiplied by 6, and the product diminished by 12, one half of the re- mainder is 36 ? 3. Three persons, A, B, and C, playing at cards for 324, disagreed about the game, and the money being on the table each seized as much as he could ; B got 15 more than A, and C J of both their sums, re- quired how much each got. G 142 DOUBLE POSITION. 4. A gentleman engaged a footman for 60 days on condi- tion that he should have 5s for every day he attended, and forfeit 2s 6d every day he was absent ; at the end of the engagement he received 9, how many days was he absent ? ARITHMETICAL PROGRESSION. ARITHMETICAL PROGRESSION is when a rank or series of numbers increases or decreases by the conti- nual addition or subtraction of some number called a com- mon difference, thus, 1, 2, 3, 4, 5, 6, is an increasing arithmetical series, where the common difference is 1 , and 13, 10, 7, 3, is a decreasing series, where the common difference is 3. The first and last terms of an arithmetical series are called extremes, and the intervening terms are called the means. CASE I. When the extremes and the number of terms are given, to find the sum of the series. RULE. Multiply the sum of the extremes by the number of terms, and half of the product is the sum of the series. 1. The extremes of an arithmetical series are 1 and 147, and the number of terms 18, what is the sum of the series ? 2. If the greatest extreme of an arithmetical series be 28, the least 2, and the number of terms- 14, required the sum cf the series. 3. A merchant was in business for 24 years, the first year he cleared 40, and the last year 600, how much did he clear in all, supposing his gains to be in arith- metical progression ? ARITHMETICAL PROGRESSION. 113 4. If 100 small stones be placed in a straight line, one yard from each other, and the first one yard from a basket, how far would a man travel to bring them, one by one to the basket? CASE II. When the extremes and the number of terms are given, to find the common difference. RULE. Divide the difference of the extremes by one less than the number of terms, the quotient is the common difference. 5. In an arithmetical series the extremes are 2 and 46, and the number of terms 12, what is the common dif- ference ? 6. If the extremes be 6 and 234, and the number of terms 20, what is the common difference? 7. A man has 9 children, the youngest is 5 years old and the eldest 29, they increased in arithmetical pro- gression, what is the common difference of their CASE III. ^/V hen the extremes and common difference are given, to find the number of terms. RULE. Divide the difference of the extremes by the common difference, and add 1 to the quotient for the number of terms. 8. If the extremes of an arithmetical series be 4 and 24, and the common difference 2, what is the number of terms ? 9. If the extremes be 1 and 49, and the common differ- ence 4, required the number of terms. 10. A man going a journey, travelled the first day 6 miles, and increased his journey every day 4 miles till his last day's travel, which was 38 miles, how many days did he travel? 144 ARITHMETICAL PROGRESSION. CASE IV. When one extreme, the number of terms, and common difference are given, to find the other extreme. RULE. Multiply the common difference by one less than the number of terms, this product added to the less extreme gives the greater; or substracted from the greater, leaves the less. 11. The less extreme of an arithmetical series is 4, the number of terms 12, and the common difference 2, what is the greater extreme? 12. The greater extreme of an arithmetical series is 26, the number of terms 12, and the common difference 2, what is the less extreme? GEOMETRICAL PROGRESSION. GEOMETRICAL PROGRESSION is when any series of num- bers increases or decreases by a common multiplier or divisor: thus, 1, 3, 9, 27, 81, is an increasing series, and 81, 27, 9, 3, 1, is a decreasing series; 3 is the multiplier and divisor, or common ratio, CASE I. When the extremes and common ratio are given, to find the sum of all the terms. RULE. Divide the difference of the extremes by one less than the ratio, and to that quotient add the greater extreme for the sum of all the terms. 1. The extremes of a geometrical series are 3 and 2187, the ratio 3, what is the sum of all the terms? 2. The extremes of a geometrical series are 6 and 2841, the ratio 8, what is the sum of all the terms? 3. The extremes of a geometrical series are 10 and 6000, the ratio 6, required the sum of the series. GEOMETRICAL PROGRESSION. 145 CASE II. When the extremes and the common ratio are given, to find the numher of terms. RULE. Divide the greater extreme by the less, and find what power of the ratio is equal to the quotient. The in- dex of this power increased by 1 will be the number of terms. 4. The extremes of a geometrical series are 3 and 2187, the ratio 3, required the number of terms. 5. The extremes of a geometrical series are 2 and 8192, the ratio 2, required the number of terms. 6. The first and last terms of a geometrical series are 4 and 65,536, the ratio 4, required the number of terms. CASE III. When the least extreme, ratio, and number of terms are given, to find the greatest extreme, or any distant term. RULE. Raise the common ratio to a power less by unity than the number of the required term; multiply that power by the least term, the product is the greatest. 7. Given the least term 3, and the ratio 2, required the 10th term. 8. Given the least term 5, the ratio 3, and the number of terms 8, required the greatest. 9. A merchant bought 14 yards of cloth at the price of the last yard, reckoning Id. for the first, 2d. for the second, and so on, in geometrical progression; required how much he paid. CASE IV. When the first term and the ratio in any decreasing geometrical progression are given, to find the sum of the series produced, ad infinitum. RULE. Divide the square of the first term by the differ- ence between the first and second. 146 GEOMETRICAL PROGRESSION 10. Find the sum of $ + + | + -fa, &c. ad infinitum. 11. Find the sum of ^ + T oo + io 4 oo & c. ad! infniturn,, 12. If a body move ^ of a mile in the first second, T 3 in the second, T5 3 oo i n the third, and so on for ever, how far will it go? 13. Suppose a cannon ball to be discharged by a force which carries it 10 miles in the first minute, 9 miles in the second minute, and so on, in the same ratio, what distance would it go? 1 4. Suppose a body to move fa of a mile in the first second, T <5 in the second, ^^ in the third, and so on for ever; how far will it go? DUODECIMALS. DUODECIMALS, OR CROSS MULTIPLICATION, is chiefly used by artificers in computing the contents of their work. RULE. Place denominations of the same name under each other, and multiply each denomination of the mul- tiplicand, (beginning at the lowest,) by the feet in the multiplier, and place each product under that denomina- tion of the multiplicand from which it arises, always car- rying at 12; then multiply by the inches, and write each product a place farther to the right hand, and so on with each denomination of the multiplier to the lowest, always ^lacing the products a place farther to the right. Thus, Feet multiplied by feet give sq. feet Feet multiplied by inches give twelfths of s. ft. Feet multiplied by parts give 144ths of s. ft. Inches multiplied by inches give sq. inches. Inches multiplied by parts give 12ths of s. in. Parts multiplied by parts give 144ths of s. in. DUODECIMALS. 147 12 fourths 12 thirds 12 seconds 12 inches or primes = 1 foot, = I third. = 1 second or part, =. 1 inch or prime. MULTIPLY. feet in feet in. feet in. pts. feet n. pu. 1. 14 6 X 4 3 20. 52 5 6 X 3 4 2. 15 8 X 5 2 21. 42 1 4 X 4 1 3 3. 26 8 X 5 6 22. 24 8 x 9 5 1 4. 18 9 X 3 8 23. 58 1 5 x 6 8 5. 38 9 X 6 7 24. 30 1 9 Xl2 11 6. 42 5 X 4 9 25. 42 8 5 x 8 4 8 7. 49 10 X 8 5 26. 21 7 3 x 8 10 5 8. 28 4 X 4 10 27. 52 8 4 x 6 2 7 9. 57 11 X 9 8 28. 63 9 x 7 9 10. 41 3 X 4 5 29. 82 1 7 x 3 4 8 11. 63 7 X 10 2 30. 25 11 xlO 9 2 12. 28 6 X 4 10 31. 61 2 5 Xl2 6 9 13. 87 3 X 2 11 32. 91 3 1 x 2 4 14. 31 8 X 9 33. 84 2 10 X 6 1 3 15. 26 11 X 12 9 34. 36 5 9 x27 1 6 16. 21 7 X 8 4 35. 41 9 8 X31 9 11 feet in. pts. feet in. pts. 36. 22 6 8 X22 6 8 17. 39 3 10 X 831 37. 86 11 X43 3 3 18. 28 1 4 X 568 38. 35 2 4 X35 2 4 19. 26 1 8 X 648 39. 84 3 8 X26 8 9 NOTE. When there are yards in the question, reduce them to feet, and perform as above. The area of a rectangular surface is found by multiply- ing the length by the breadth. The content of a rectangular solid is found by multiplying the length by the breadth, and that product by the depth. The solid content of round timber is found by multi- plying the square of J of the mean circumference by the length. oar 148 DUODECIMALS. EXERCISES. 1. Find the area of a board 12 fee*- 4 inches long, and 2 feet 6 inches broad. 2. Find the area of a board 14 feef 3 inches long, and 1 foot 9 inches broad. 3. How many superficial feet in a board 18 feet 2 inches by 2 feet 11 inches ? 4. How many superficial feet in a room 18 feet 8 inches by 12 feet 10 inches ? 5. How many square yards in a room 37 feet 6 inches by 30 feet 4 inches ? 6. How many solid feet in a log of wood 26 feet 8 inches long, 3 feet 2 inches broad, and 2 feet 1 inch deep ? 7. How many cubic feet in a stone 13 feet 9 inches long, 2 feet 11 inches broad, and 1 foot 9 inches deep ? 8. How many solid feet in a tree 26 feet 3 inches long, and its mean girth 6 feet? 9. How many cubic yards in a mound of earth 268 feet long, 135 feet broad, and 58 feet deep ? 10. What is the difference of the areas of the floors of two rooms, the one 42 feet 8 inches by 30 feet 2 inches, the other 28 feet 5 inches by 19 feet 7 inches ? 11. The canal which joins the Forth and Clyde is 27 miles long, 36 feet broad, and mean depth 7 feet, required the number of cubical yards of excavation j* 12. How many imperial gallons will a cistern contain whose inside dimensions are 20 feet 4 inches, 18 feet 2 inches, and 12 feet 10 inches ? An imperial gal- lon contains 277.274 cubic inches. TONNAGE OF SHIPS. To find the tonnage of ships. RULE. Multiply together the length of the keel by the length of the midship beam, and that product by one-half the length of the midship beam, both taken within the ves- sel ; this product divided bv 94 gives the number of tons. TONNAGE OF SHIPS. 149 1. What is the tonnage of a ship whose keel is 160 feet, and midship beam 30 feet 1 2. Find the tonnage of a ship whose keel is 140 feet, and midship beam 26 feet. 3. If a ship's keel be 1 00 feet, and the midship beam 22 feet, what is her tonnage ? 4. Required the tonnage of Noah's ark, whose length was 300 feet, breadth 50 feet, and depth 30 feet? 42 solid feet make a ton of shipping. COMPOUND INTEREST. COMPOUND INTEREST is an allowance, not only for the use of the principal, but also for the use of the interest after it becomes due, which is added to the principal, and the amount becomes a new principal for the next term. CASE I. To find the amount and interest of any sum, for any number of years, compound interest. RULE. Multiply the amount of 1 for the given time and rate per cent, from Table l,by the given principal, and the product is the required amount, from which subtract the principal, and the difference will be the interest. 1. Find the amount of 600 for 8 years at 5 per cent, com- pound interest. 2. Required the amount of 847 for 10 years at 4J per cent. 3. If 365 : 10s were lent for 16 years at compound interest, at the rate of 3| per cent, what would the amount and interest come to? 4 A legacy of 841 : 17 : 6d, was left to a boy 8 years old, to be improved at compound interest, at 4 per cent, till he was 21 years of age, how much had he then to receive ? G2 150 COMPOUND INTEREST. CASE II. To find the present value of any sum, due at the end of any number of years, at a given rate, compound interest. RULE. Multiply the present value of 1, for the given rate and time, from Table II, by the given sum, and the product will be the present value. 5. Find the present value of 4000 due 10 years hence, at 5 per cent per annum, reckoning compound interest. 6. What sum lent at compound interest, at the rate of 4^ per cent, will amount to 2800 in 15 years? 7. What principal lent at compound interest, at the rate of 3^ per cent, per annum, will, at the end of 8 years, amount to 1000? 8. What principal put to interest for 6 years, at 4 per cent. per annum, will amount to 1280: 10:6? ANNUITIES. An Annuity is any periodical income, payable either yearly, half-yearly, quarterly, or at any other equal inter- val. An annuity is said to be in possession when it is payable immediately: to be in reversion when the payment will not commence until some given period has elapsed. When an annuity is not limited with respect to time, but to continue for ever, it is called perpetuity. ANNUITIES FOR A TIME. CASE. I. To find the amount of an annuity at compound interest. RULE. Multiply the amount of 1 annuity taken from Table III. for the given time and rate per cent, by the an- nuity ; the product is the amount required. ANNUITIES. 151 1. What will an annuity of 60 amount to in 16 years at 5 per cent, compound interest 2 2. To what sum will an annuity of 100 for 19 years amount, improved at the rate of 4 per cent, compound interest ? 3. What will an annuity of 80, payable yearly, amount to in 12 years, reckoning compound interest at the rate of 4 4 per cent ? CASE II. To find the present value, or sum of money that will purchase an annuity. RULE. Multiply the present value of 1 annuity, taken from Table IV. for the given rate and time, by the annuity, the product is the amount required. 4. Find the present value of an annuity of 100 to continue 16 years at 4 per cent, compound interest. 5. Find the present value of an annuity of 50 for 10 years at the rate of 5 per cent, per annum. 6. What is the present value of a pension of 10 to conti- nue 15 years at 3^ per cent, per annum \ CASE III. To find what annuity may be purchased for a given sum. RULE. Divide the given sum by the present value of an annuity of 1, taken from Table IV. for the rate and time, the quotient is the annuity required. 7. What annuity, to continue 20 years, will 1248: 15s. purchase, compound interest at 4^ per cent.? 8. What annuity will 1000 purchase, to continue 18 years, at 5 per cent. ? 9. A Gentleman wants to sink 1500 for an annuity, to continue 20 years ; how much ought he to receive per annum, as an equivalent, allowing compound interest to both parties at 4 per cent. ANNUITIES FOR EVER. In annuities for ever, or freehold estates, there are to be considered the annuity or yearly rent, the price or pre- sent value, and the rate of interest. 152 ANNUITIES. CASE I. To find the value of an annuity or estate, the rent and rate of interest being given. RULE. As the rate of 1 (Table I.) is to 1, so is the rent to the value required. Ex. The yearly rent of an estate is 350 ; how much ready monev is it worth at 4 per cent. ? .04 : 1 : : 350 : 8750 Ans. 1. The yearly rent of an estate is GOO ; what is it worth in cash, allowing interest at 4| per cent. 2. What is the value of a perpetual annuity of 100, dis- counting at 5 per cent 1 3. The yearly rent of the estate of Greenfield is 800 ; what is its present value, allowing interest at 3| per cent. ? CASE II. To find the rent or annuity when the value and rate of interest are given. HULE. As 1 is to its rate of interest (Table I.), so is the value to the rent or annuity. Ex. A Gentleman, by purchasing an estate for 8750, had 4 per cent, for his money. Required the rent yearly. 1 : -04 : : 8750 : 350. 4. A Gentleman purchases an estate for 10,000, and has 5 per cent, for his money. Required the yearly rent. 5. Find the yearly rent of an estate, bought for 6000, at the rate of 4| per cent, interest. CASE III. To find the rate of interest when the value and rent are given. RULE. As the value is to the rent, so is 100 to the rate per cent. Ex. An estate of 350 yearly rent is bought for 8750 ; what rate of interest has the purchaser for his money ? 8750 : 350 :: 100 : 4 per cent. 6. An estate of 1 000 is bought for 16,000; what rate has the purchaser for his money ? 7. A Gentleman bought an estate for 60,000, and the yearly rent of it is 3300 ; what rate per cent, has he lor his money I ANNUITIES. 153 CASE IV. To find how many years' purchase should be given for an estate, to allow the purchaser a certain rate per cent. RULE. Divide 100 by the given rate per cent. 8. A gentlemen wishes to purchase an estate, and have 4 per cent, for his money; how many years' purchase should he offer? 9. A person wishes to have 6| per cent, for his money in purchasing an estate ; how many years' purchase should he offer? 10. A person wishes to have 7^ per cent, for money invest- ed in houses ; how many years' purchase should he offer? CASE V. To find the rate of interest, when the number of years' purchase at which an estate is bought or sold is given. RULE. Divide 100 by the number of years. 11. A gentleman gives 25 years' purchase for an estate; what interest has he for his money? 12. A person gives 20 years' purchase for property; what interest has he for his money? 13. A person gives 21 years' purchase for feus; what in- terest has he for his money? EXERCISES. 1. What is the value of a perpetual annuity of 60, dis- counting at 5 per cent? 2. The yearly rent of an estate is 1000; what is it worth in cash, allowing 4^ per cent? 3. A gentleman purchases an estate for 12,000, and has 4 per cent for his money; required the yearly rent? 4. An estate of 500 a-year is bought for 8000; what rate has the purchaser for his money? 5. A person wishes to have 6| per cent, for his money in purchasing an estate ; how many years' purchase should he offer? 6. A gentleman gives 25 years' purchase for an estate, what interest has he for his money? COMPOUND INTEREST AND ANNUITIES. TABLE I. AMOUNT of 1 in any Number of Years. Yrs. Sper cent. 3\percent. 4 per cent. ^percent. 5 per cent. j 2 3 4 5 1*03 1-0609 1-092727 1-125508 1-159274 035 071225 108717 147523 187686 1-04 1-0816 1-24864 1-169858 1-216652 1-045 1-092025 1-141166 1-192518 1-246181 1-05 1-1025 1-157625 1-21550G 1-276281 6 7 8 9 10 1-194052 1-229873 1-26677 1-304773 1-343916 229255 272279 316809 362897 410598 1-265319 1-315931 1-368569 1-423311 1-480244 1-30226 1-360861 1-4221 1-486095 1.552969 1-340095 1-4071 1-477455 1-551328 1-628894 11 12 13 14 15 1-384233 1-42576 1-468533 1-512589 1-557967 459969 511068 563956 618694 675348 1-539454 1-601032 1.665073 1-731676 1-800943 1-622853 1-695881 1.772196 1-851944 1-935282 1710339 1795856 1-885649 1-979931 2-078928 16 17 18 19 20 1-604706 1-652847 1-702433 1-753506 1-806111 733986 794675 857489 922501 989788 1-872981 1-9479 2-025816 2-106849 2-191123 2-02237 2*113376 2-208478 2-30786 2.411714 2-182874 2-292018 2-406619 2-52-695 2-653297 21 22 23 24 25 26~ 27 28 29 30 1-860294 1-916103 1-973586 2-032794 2-093777 2-059431 2-131511 2-206114 2-283328 2-363245 2-278768 2-369918 2-464715 2-563304 2-665836 2-520241 2-633652 2752166 2-876013 3-005434 2-785962 2-92526 3-071523 3-225099 3-386354 2-156591 2-221289 2-287927 2-356565 2-427262 2-445958 2.531567 2-620172 2-711878 2-806793 2-772469 2-883368 2-998703 3-118651 3-243397 3-140679 3-282009 3-4297 3-584036 3745318 3-555672 3*733456 3-920129 4-116135 4-321942 40 50 3-262037 4-383.006 3-959259 5-584926 4-80102 7-106683 5-816364 9-032636 7-039988 11-467399 | COMPOUND INTEREST AND ANNUITIES. 155 TABLE II. PRESENT VALUE of 1 due at the end of any Number of Years. Yrs. I 2 3 4 3 per cent. ^percent. 4 per cent. 44 per cent. 5 per cent. 970873 942595 915141 888487 862608 966183 93351 901942 871442 841973 961538 924556 888996 854804 821927 956937 915729 876296 838561 802451 952381 907029 863837 822702 783526 6 7 8 9 10 837484 813091 789409 766416 744093 8135 785991 759411 733731 708918 790314 759917 73069 702586 675564 767895 734828 703185 672904 643927 746215 710681 676839 .644608 613913 11 12 13 14 15 722421 701379 680951 661117 641861 684945 661783 639404 617781 59639 64958 624597 600574 577475 555264 616198 589663 564271 539972 51672 584679 556837 530321 505067 481017 16 17 18 19 20 623166 605016 587394 570286 553675 576705 557203 538361 520155 502565 533908 513373 493628 474642 456387 494469 473176 4528 433301 413642 458111 436296 415520 395734 37688ft 21 22 23 24 25 537549 521892 506691 491933 477605 48557 46915 453285 437957 423147 438833 421955 405726 390121 375116 396787 3797 36335 347703 33273 358942 341849 325571 310067 295302 26 27 28 29 30 463694 450189 437076 424346 411986 408837 395012 381654 368748 356278 360689 346816 333477 320651 308318 318402 304691 29157 279015 267 281240 267848 255093 242946 231377 40 50 306556 228107 252572 179053 208289 140712 171928 110709 142045 087203 156 COMPOUND INTEREST AND ANNUITIES. TABLE III. AMOUNT of 1 per annum in any Number of Years. Yrs. 34 per cent. 4 per cent. 44 per cent. 5 per cent. 1 2 3 4 5 I- 2-035 3-106225 4-214942 5-362465 1- 2-04 3-1216 4-246464 5-416322 1- 2-045 3-137025 4-278191 5-470709 1- 2-05 3-1525 4-310125 5-525631 6 7 8 9 10 6-550152 7779407 9-051686 10-368495 11-731393 6-632975 7-898294 9-214226 10-582795 12-006107 6-716891 8-019151 9-380013 10-802114 12-288209 6.801912 8-142008 9-549108 11-026564 12-577892 11 12 13 14 15 13-141991 14-601961 16-11303 17-676986 19-29568 13-486351 15-025805 16-626837 18-291911 20-023587 13-841178 15-464031 17*159913 18-932109 20-784054 14-206787 15-917126 17-712982 19-598632 21-578563 16 17 18 19 20 20-971029 22-705015 24-499691 26-35718 28-279681 21-824531 23-697512 25-645412 27-671229 29-778078 22-719336 24-741706 26-855083 29-063562 31-371422 23-657491 25-840366 28-132384 30-539003 33 065954 21 22 23 24 25 30-26947 32-328902 34-460413 36-666528 38-949856 31-969201 34-247969 36-617888 39-082604 41-645908 33-783136 36-303377 38-93703 41-689196 44-56521 35-719251 38-505214 41-430475 44-501998 47*727098 26 27 28 29 30 41-313101 43-75906 46-290627 48-910799 51-622677 44-311744 47-084214 49-967583 52-966286 56-084937 47-570644 50-711323 53-993333 57-42;5o: > ,;} 61-007069 51 113433 54-669126 58-402582 62-322711 66-438847 40 50 84-550277 130-99791 95-025515 152-667083 107-030323 178.503928 120-799774 209-347996 COMPOUND INTEREST AND ANNUITIES. 157 TABLE IV. PRESENT VALUE of 1 per annum for any Number of Years. Yrs. I 2 3 4 5 S^ per cent. "Imefss" 1-899694 2-801637 3-673079 4-515052 4 per cent. "b^eTsss" 1-886094 2-775091 3-629895 4-451822 4 per cent. ~6 7 9569 : 3T 1-872667 2-748964 3-587525 4-389976 5 per cent. "~95238~ 1-85941 2-723248 3-54595 4-329476 6 7 8 9 10 5-328553 6-114543 6-873955 7-607686 8-316605 5-242136 6-002054 6-732744 7-435331 8-110895 5-157872 5-8927 6-595886 7-26879 7-912718 5-075692 5-786373 6-463212 7-107821 7721734 11 12 13 14 15 16 17 18 19 20 9-001551 9-663334 10-302738 10-92052 11-51741 8-760476 9-385073 9-985647 10-563122 11-118387 8-528916 9-11858 9-682852 10-222825 10-739545 8-306414 8-863251 9-393573 9-89864 10-379658 12-094116 12-65132 13-189681 13-709837 14-212403 11-652295 12-165668 12-659297 13-133939 13*590326 11-234015 11-707191 12-159991 12-593293 13-007936 10-837769 11-274066 11-689586 12-08532 12-46221 21 22 23 24 25 14-697974 15-167124 15-62041 16-058367 16-481514 14-029159 14-451115 14-856841 15-246963 15-622079 13-404723 13-784424 14-147774 14-495478 14-828208 12-821152 13-163002 13-488573 13-798641 14-093944 26 27 28 29 30 16-890352 17-285364 17-667018 18-035767 18-392045 15-982769 16-329585 16-663063 16-983714 17-292033 15-146611 15-451302 15-742873 16-021888 16-288888 14-375185 14-643033 14-898127 15-141073 15-372451 40 50 21-355072 23-455617 19-792774 21-482184 18-401584 19-762007 17*159086 18-255925 158 PROMISCUOUS QUESTIONS. 1. The distance of Mars from the Sun is one hundred and forty-four millions, eight hundred and twenty-four thousand miles, and Neptune's distance is two billions seven hundred and ten millions, one hundred and fourteen thousand miles more than Mars; how far distant from the Sun is Neptune ? 2. What is the difference between twice twenty-five, and twice five, and twenty ? 3. The art of printing was discovered in the year 1449 how many years have elapsed since ? 4. How many strokes does the hammer of a clock strtke in a year of 365 days ? 5. How many times does the wheel of an engine, 15 feet in circumference, revolve between London and Edin- burgh, the distance being 400 miles? 6. How many seconds have elapsed since the birth of Christ to Christmas 1864 ? 7. After seeing a flash of lightning, 24 seconds elapsed before the thunder was heard; required the distance, sound moving at the rate of 1142 feet per second. 8. How many pounds troy in 4 cwt.? 9. How many yards in 840 English ells, and find the price at Gs. per yard? 10. The price of 64| yards of broad cloth is 70:4:6d; required the price of 1 yard. 11. The price of 1 yard of linen is 2s. Gd; what is the price of 40 Flemish ells? 12. Received 110 yards of cloth at 8s. 6d. per yard, for 184 Ib. of tea; required the price of the tea per Ib. 13. The bill lor a public dinner came to 16: 4s, and each person in the company paid 9s; how many dined? 14. How much can a person spend daily out of an income of 246 per annum, after paying 60 for rent and taxes? 15. How many inches are there in the circumference of the earth, which is 360 degrees, and each degree 69J miles? PROMISCUOUS QUESTIONS. 159 16. A gentleman's income is 360; what should be his daily expenses to save 150 per annum? 1 7. Find the weight and price of 9 hhds. tea, each 7 cwt. 2 qrs. 14 Ibs., at 5s. 4d the Ib. 18. What is the weight and price of 20 hhds. sugar, each 18 cwt. 2 qrs. 4lbs., at 5|d per Ib.! 19. Divide 10:4:8d among 4 boys, 5 women, and 6 men, and give each woman double the share of a boy, and each man thrice that of a woman; what will each receive? 20. A question was debated in the House of Commons, and when it came to a division, 462 members were present, 325 voted for the one side; how many voted for the other, and what was the majority? 21. A steeple projected a shadow of 200 feet, when a staff 4 feet high projected 6 feet of shadow; required the height of the steeple. 22. A bankrupt owes his creditors 6428; how much will he pay them at 6s. 4^d per pound? 23. How many cwt. will 24681 sovereigns weigh, when one of them weighs 5 dwt. 3.274 grains? 24. Required the weight of a shilling, 1 Ib. of silver being coined into 66 shillings. 25. How many crowns, half-crowns, and sixpences, and of each an equal number, are in 40? 26. If 4^ gallons of water are mixed with 16 gallons of brandy, at 28s. per gallon; what is the mixture worth per gallon? 27. Find the commission on 846:14:8d, at 2 per cent. 28. If a chest 6 feet long, 5 feet deep, and 4 feet wide, con- tain 22 bolls of grain, how many bolls will a chest 24 feet long, 12 feet deep, and 10 feet wide contain? 29. Find the neat weight and price of 42 hhds. of sugar, each 16 cwts. 1 qr. 18 Ibs., tare 26 Ib. per hhd. 3 and tret 4 Ibs. on 104 Ibs., at 5^d. per Ib. 30. After a debate in the House of Commons, when there we re 460 members present, the question was carried by a majority of 36; how many members voted on each side? 31. A's share of a ship was |, of which he sold ; what re- mains of his share? 160 PROxMlSCUOUS QUESTIONS. 32. If 6 ounces of gold cost 2 pounds sterling, what will oz. cost? 33. Find the interest and amount of 568.875 for 6 years, at 2.5 per cent. 34. A solid foot of stone was 36 inches long, 3 inches thick; required its breadth. 35. A, B, and C, join in an adventure, by which they clear 648; how much of that sum should each re- ceive, their capital being 2000, of which A advanced 846, B 754, and C 400? 36. When the barometer stands at 30 inches, the weight of the atmosphere on every square inch of surface is 14.75 Ibs.; now, if a person's body contain 16 square feet, how many tons of air must he sustain? 37. How many dollars of 4s. 6d. are there in 846? 38. How many pounds sterling are there in 4879 francs, at lOJd. each. 39. How many rubles, of 3s. 4d. each, are there in 927? 40. Bought cloth at 12s. 9d. per yard, and sold it at 15s; what was the gain per cent.? 41. By selling cloth at 9s. 7d. I lost 8 per cent; at what price did I buy it? 42. Sold rum at 12s. 6d, by which I gained 15s. per cent; what is gained by selling it at 15s.? 43. When I sold yarn at 2s. 7^d per spindle, 1 gained 8J per cent; what was the prime cost? 44. A Stock-holder wishes to know what rate of interest he has for his money in the Four per cents, when they are at 85 per cent. 45. What rate of interest may a person have by investing money in the Dundee Union Bank, when selling at 168 per cent, and the dividend at 8 per cent? 46. What is the difference of f of a guinea, and jj of a crown? 47. A has J share of a flax mill, B , C |, and D the re- mainder; required his share. 48. A gentleman's estate was divided among his three sons; the first gets 10,000, the second f of the first, and the third f of the second; what was the value of the estate! PROMISCUOUS QUESTIONS. 161 49. A fraudulent balance has one of its arms 12 inches long, and the other 11 inches; what weight suspended from the shorter arm will balance 22 Ibs. suspended from the longer arnf? 50. If a balance be 20 inches long, and 12 Ibs. suspended from the one arm balance 14 Ibs. suspended from the other, what is the length of each arm? 51 . How much will it cost to have a court-yard at 6s. lOd. per foot, the length of it being 28 feet 4 inches, tha breadth 14 feet 6 inches? 52. What is the solid content of a block of marble, 10 feet 5 inches long, 6 feet 8 inches broad, and 2 feet 4 inches thick? 53. If 3600 men be placed in the form of a square, how many will be in a side? 54. There are two pipes by which water flows into a cis- tern, one of them can fill the cistern in 4 hours, and the other in 6; in what time would it be filled by both running together? 55. How long would a person take to count one billion of sovereigns, at the rate of 100 per minute, for 10 hours a-day, and 313 working days in a year? 56. A person being asked what o'clock it was, answered it is | of | of f of | of T 7 5 of 24 hours; what o'clock was it? 57. A bankrupt's debts amount to 16,128, and his effects to 11,088; how much can he ofier his creditors per pound? 58. If a bankrupt's effects pay 6s. 4d. per pound, how much will B receive, to whom he owes 847? 59. How many feet of timber in a board 16 feet 4 inches long, the breadth at one end 20 inches, and at the other 15? 60. Required the solidity of a tree 40 feet long, and 60 inches in girth. 61. Reduce 246:18:6| to half-farthings. 62. In 98,467 half-farthings how many guineas? 63. If a grocer gain 8^d. on a Ib. of cinnamon valued 3s 8d, how much will he gain on 100 value? 64. How long will a horse be in going round a race-course. 162 PROMISCUOUS QUESTIONS. at the rate of 20 miles an hour, if he has performed it in 7| minutes at the rate of 30 miles an hour? 65. Sold James Watt, Merchant, New York, 24,847 yards of bagging, at 6|d. per yard; how much sterling money should he send me, when the exchange is at 8 i per cent? 66. In a shoal of herrings, 6 miles in length, 2^ in breadth, and 240 yards deep, how many herrings, allowing 16H to a solid foot? and how many casks would they fill, each containing 820? 67. A cistern has three cocks, A, B, and C; when opened singly, A empties the cistern in 60 minutes, B in 80, and C in 100 minutes: in what time would the cistern be emptied if the 3 cocks were set open at once? 68. A wall is 36 feet high, and a ditch before it is 27 feet wide: required the length of a ladder that would reach to the top of the wall from the opposite side of the ditch? 69. Sold John Mitchell, Merchant, New York, 8,476 yards of canvass, at 5|d. per yard; how much sterling money should he send me, when the exchange is at 10| per cent? 70. An American dollar contains 371 grains of pure sil- ver; what is its value if 11 oz. 2 dwt. of pure silver be worth 65s? 71. A herring and a half for three halfpence; how many for lid? 72. How many deals, of 12 feet long 6 inches broad, are required for a floor 20 feet by 16? 73. How long would a steam carriage take to go round the globe, which is of 360 degrees, and each degree 69* miles, the carriage travelling at the rate of 30 miles an hour? 74. Mr John Sinclair bought of Mathew Shepherd 364 yards bagging at 6d. per yard, 436 yards of canvass, at7d.,436 yards of broad cloth at!8p.,568^ yards of Victoria tartan at 2s., 97 yards of velvet at 26fs, per yard. Required the amount. ,*5. Mr John Smith bought of "Thomas Miller 121 J yards of silk at 13s. per yard, 95 yards of satin at 12s., PROMISCUOUS QUESTIONS. 163 365 yards of flowered silk at 15 is., 167| yards of velvet at 1|, 27 f yards of rich brocade at !&< Required the amount. 76. Mr James Denham bought of George Inches 87 Ib. of green tea at 12^s. per Ib., 57| Ib. imperial tea at I6s., 2G8 Ib. bohea at 8s., 824 black tea at 4fs., 876| Ib. of sugar at 8 id., 13f Ib. of coffee at 2^8. per Ib. Required the amount. 77. Mr Robert Pram bought of William Stewart 34 reams thick post at 24 |s. per ream, 86| reams foolscap at 12^s., 142 reams printing demy at 16|s., 19f reams superfine royal at 36 s., 15| reams cartridge at 8|s. per ream. Required the amount. 78. Mr Robert Glenday bought of James Mitchell 16i pieces osnaburghs, each 36 yards, at 7|d. per yard; 2i pieces Irish linen, each 26 yards, at Ifs.; 18| pieces checks, each 28 yards, at H^d.; 12| pieces sheeting, each 34 yards, at ll|d per yard. Required the amount. 79. Mr Thomas Wise bought of William Ritchie 24 J pieces of bagging, each 35 4 yards, at 6|d. per yard; 17^ pieces of canvass, each 45 f yards, at 5d.; 34 pieces sheeting, each 36^ yards, at Ifs.; 9| pieces osnaburghs, each 46^ yards, at 6|d.; 12 pieces linen, each 48 yards, at l|s. per yard. Required the amount. 80. Mr William Malcolm bought of David Hean 4 hhds. sugar, each 8 cwt. 2 qr. 16 Ib., at 25 |s. per cwt. ; 3 boxes soap, each 9 cwt. 2 qr. 26 Ib., at 6^d. per Ib.; 5 barrels raisins, each 2 cwt, 2 qr. 14 Ib., at 1 f per cwt.; 6 bags coffee, each 2 cwt. 1 qr. 15 Ib., at 2|s. per Ib.; 8 bags ginger, each 4 cwt. 3 qr. 14 Ib., at 4^d. per Ib.; 3 hhds. tobacco, each 3 cwt. 3 qr. 13 Ib. at 4|s. per Ib. Re- quired the amount. 81. Mr James Brown bought of William Kennedy 6 chests tea, each 6 cwt. 2 qr. 14 Ib., at 4|s. per Ib.; 10 hhds. sugar, each 6 cwt. 2 qr. 18 Ib., at 6d. per Ib.; 9 chests soap, each 8 cwt. 1 qr. 15 Ib., at 7^d. per Ib.; 5 barrels raisins, each 2 cwt. 1 qr. 8 Ib., at 1| per cwt.; 4 casks prunes, each 14 cwt. 17 Ib., at 1 $ per cwt.; 6 cannis- ters snuff, each 25 1 Ib., at 3s. per Ib.; 8 boxes tea, each 3 cwt. 19 Ib., at 5is. per Ib. Required the amount. 164 PROMISCUOUS QUESTIONS. 82. Eliza Cruickslianks bought of Helen Smith 126 yards silk, at 12?s. per yard; 59 yards satin, at 9s.; 87$ yards flowered silk, at 14 |s.; 93i yards velvet, at 18s.j 26f yards velvet tartan, at lf; 238| yards scarlet, at 1| per yard. Required the amount? 83. Grace Pullar bought of Mary Mitchell 59 yards fine serge, at 3s. per yard; 81 f yards rich brocade, at 13s.; 53| yards shalloon, at 19d.; 228 yards drab, at 12*5.; 94 1 yards royal tartan, at 4|s.; 25| yards scarlet, at 1| per yard. Required the amount. 84. Eliza Ramsay bought of Eliza Neish 26 J yards cambric, at 8s. per yard; 85 yards printed calico, at 2^3.; 52 yards jaconet muslin, at 4s.; 126| yards long lawn, at 3s.; 73 yards fine linen, at 2|s.; 76 yards tartan velvet, at 18s.; 53 yards black satin, at 8,|s. Required the amount. 85. A bond for 2000 was due 18th March 1858,~bf which 560 were paid on 16th June 1860, 320 were paid 20th September 1862, and 470 were paid on 17th August 1863. The account is to be settled 12th March 1866; what will then be due? 86. Borrowed on bond 18th October 1860 4000 ; of which I paid 10th August 1861 628, on 5th January 1863 1260, and on the 12th May 1865 1650. The ac- count is to be settled 20th July 1867 ; what sum will then be due ? 87. Borrowed on bond from the Eastern Bank of Scotland on the 10th August 1859 3000, of which I pa : d 18th September 1860 897, on 18th March 1862 1268. and on 14th January 1864 1100. The account is to be settled 20th January 1866, what sum will then be due? 88. Borrowed upon the Estate of Crookmains, 18th May 1860, the sum of 10,000, of which I paid, 16th July 1862, 4268 ; and on 20th September 1864, 2876. The account is to be settled 14th December 1867, what sum will then be due? * Interest at 5 per cent. Fractions below pence are omitted. The above dates contain leap years. DECIMAL COINAGE. 165 DECIMAL COINAGE. PART I. THE system of Decimal Coinage, which is most likely in course of time to be substituted for the present system in this country, is exhibited in the following table: TABLE OF DECIMAL MONEY. 10 mils = lcent(c.) =TO = 2 l d - 100 mils = 10 cents = 1 florin (fl.} = T ^ = 2s. 1000 mils = 100 cents =10 florins =1 =20s. 500 mils = 50 cents = 5 florins = 10s.; 250m. = 25c. = 2fl. 5c. = 5s. ; 125m. = Ifl. 2c. 5m. = 2s. 6d. ; 50m. = 5c. = Is. ; 25m. = 2c. 5m. = 6d., etc. In the system now in use, the pound is divided into 960 equal parts or farthings ; in the proposed system, it would be divided into 1000 equal parts or mils, as is shown in the above table. Calculations in money would then be per- formed as in Simple Numbers, the pounds being separated by a point from the florins, cents, and mils, thus : 57, 5fl. 2c. 5m. would be written decimally 57-525 75, 3c. 5m. 75-035 66, 4fl. 66-400 99, 8m. 99-008 Write in the decimal form, 1. 27 2fl. 2. 36 4 3. 47 8 4. 59 9 13. 36-450 14. 63-504 15. 73-628 5. 49 5fl. 2c. 6. 57 8 1 7. 75 3 8. 44 2 9. 5 8fl. 2c. 5m. 10. 6 2 4 11. 7 2 8 12. 4 2 Express in 's, florins, etc., 16. 42-500 17. 24-004 18. 48-612 19. 7-856 20. 8-207 21. 10-820 22. 17-040 23. 25-813 24. 52-108 H 166 DECIMAL COINAGE. ADDITION AND SUBTRACTION. EULE. Write the sums decimally under each other, then add or subtract as in Simple Numbers. Ex. Find the sum of 497, 2fl. 5c. 6m., 386, 8fl. 5c., and 540, 9fl. 6m. ; also from 536, 8fl. 2m., take 378, 9fl. 8c. 6m. (1.) 497-256 (2.) 536-802 386-850 378-986 540-906 Difference, 157-816 Sum, 1425-012 1. 2. 3. 75 6fl. 2c. 5m. 745 2fl. Oc. 5m. 695 7fl. 7c. 5m. 57 4 8 6 475 5 6 965 8 6 63 8 7 457 6 7 956 6 8 36 7 8 547 9 596 7 54 8 6 754 2 8 5 965 8 5 4 45 5 574 6 78 695 5 48 I 5. 6. 625 8fl. 5c. 8m. 731 8fl. 5c. 2m. 597 6fl, 6c. 6m. 265 7 9 371 8 9 6 957 4 4 4 256 9 4 173 7 2 9 579 8 8 8 526 4 9 7 713 2 9 7 759 6 5 652 7 7 7 137 3 4 3 975 5 6 562 8 05 317 4 33 795 70 7. 8. 9. 222 2fl. 2c. 2m. 132 Ifl. 3c. 2m. 343 3fl. 4c. 3m. 333 333 144 1 4 4 729 9 2 7 444 444 196 6 9 1 504 5 4 555 5 5 5 297 7 2 9 706 7 6 666 6 6 6 405 4 5 216 1 2 6 777 7 77 750 75 300 03 10. 11. 12] 412 8fl. 7c. 2m. 471 2fl. Oc. 4m. 563 7fl. 5c. Om. 227 9 86 174 3 09 365 8 76 13. 14. 15. 704 On. Oc. 2m. 317 Ofl. 7c. 4m. 256 7fl. Oc. 8m. 407 2 6 5 173 6 8 78 7 8 9 DECIMAL COINAGE. 167 16. 17. 18. 512 2fl. Oc. Om. 807 Ifl. 5c. 2m. 980 Ofl. Oc. Ora. 125 3 75 389 7 6 378 05 MULTIPLICATION AND DIVISION. RULE. Multiply or divide the sum expressed deci- mally as in Simple Numbers, and point oft' three figures from the right of the result. Ex. 75, 4fl. 7c. 5m. X 37 ; and 976, Ifl. 7c. 5m. -f- 25. (1.) 75-475 (2.) 25 < 5)976-175 -f- 25 37 1 5)195-235 528325 226425 Quotient, 39-047 Product, 2792-575 19. 57, 8fl. 7c. 5m. X 35 31. 729, 5fl. 3c. 6m. - - 16 20. 75, 6fl. 7m. X 48 32. 580, 3fl. 6c. 5m. - - 27 21. 99, 7c. 5m. X 66 33.873,3fl. 7c. 6m. - - 49 22. 55, 7m. X 108 34. 754, 7fl. 4m. - 72 23. 59, 6fl. 7c. 5m. X 17 35.1638,6fl.8c.4m.- -108 24. 68, 7fl. 4c. X 23 36.1423,9fl.2c.8m.- -121 25. 89, 5fl. 2c. 8m. X 39 37. 339, 3fl. 9c. 7m. - - 19 26. 98, 6fl. 7m. X 68 38. 634, 5c. 6m. - 29 27. 104, 2fl. 5c. 4m. X 153 39.1414,5fl.2c.9m.- - 61 28. 175, 3c. 5m. X 227 40.1714,6fl.3c.6m.- -133 29. 157, 7m. X 343 41.1532,8fl.9c.5m.- -195 30. 163,4fl.4c.8m. X 365 42.790,9fl.5c.5m. - -865 43. A man's house rent is 36, 7fl. 5c. annually; his taxes are 9, 2fl. 7c. 5m. ; servant's wages, 8, 7fl. 5c. 2m. ; his household expenses, 75, 6fl. 2c. 5m. ; and he saves 20, 5fl. Ic. 2m. : what is his income ? 44. A merchant received from A, 75, 3fl. 7c. ; from B, 67, 8fl. 7c. 5m. ; from C, 84, 7fl. 2c. 4m. ; D, 89, 7c. 8m. ; E, 92, 4fl. 6m. ; F, 98, 7m. : how much did he receive in all ? 45. Borrowed 476, 2fl. 7c. 5m., and paid in part 378, 4fl. 8c. 8m. ; what is still due ? 46. A gentleman's income is 555, 5fl. 5c., and his 168 DECIMAL COINAGE. expenses amount to 375, 6fl. 8c. 4m. ; how much does he save ? 47. A bankrupt owes to one of his creditors 156, 7fl. 8c. 4m. ; to another, 187, 2fl. 5c. ; to a third, 198, 3fl. 7c. 5m. ; to a fourth, 209, 8c. 4m. ; to a fifth, 232, 6fl. 6c. 6m.; to a sixth, 253, 7c. 5m.; to a seventh, 305, 8fl. 8m. ; to an eighth, 345, 5fl. 8c. ; his effects amount to 1525, 2fl. 2c. 5m. : how much is he deficient ? 48. What is the value of 14 Ihs. of sugar, at 2c. 5m. per Ib. ? 49. What is the price of 23 reams of paper, at 1, Ifl. 2c. 5m. per ream ? 50. Calculate 26 weeks' wages, at 8fl. 2c. 4m. per week. 51. What is the rent of a house for a year, at 2, 8fl. 7c. 2m. per week ? 52. What is the value of 39 tons of coal, at 7fl. 5c. 2m. per ton ? 53. Divide 131, Ifl. Ic. 2m. equally among 27 men. 54. If 23 cwt. of sugar cost 51, Ifl. 7c. 5m. ; how much is that per cwt. ? 55. How many yards of cloth, at 3fl. 7c. 5m., can be purchased for 46, 8fl. 7c. 5m. ? 56. The revenues of a hospital are 2506, 6fl. 3c. 6m. ; how many patients will it maintain, if each requires 20, 7fl. Ic. 6m. ? 57. In how many days would a person save 45, 3fl. 7c. 5m., by laying aside Ifl. 2c. 5m. daily? 58. How many gallons of brandy, at 2, Ifl. 2c. 5m. per gallon, can be bought for 159, 3fl. 7c. 5m. ? 59. How many poor people could be relieved out of 55, 8c. 8m., if each receives 1, 2fl. 5c. 2m. ? 60. A bankrupt owed his creditors 7528, and paid them 4705 ; how much was it per pound ? 61. A and B gain together 45, 3fl. 6c. 9m.; A and C, 51, 7fl. 5c. 2m.; and B and C, 56, 8fl. 7c. 3m.; what is the gain of each ? DECIMAL COINAGE. 169 PART II. To reduce '$, sh. and d. to the proposed system mentally. RULE. Divide the shillings by 2 for florins, then the farthings in the pence and farthings increased by their 24th part give the cents and mils. NOTE. When the shillings are odd, 5 cents must be added to the result obtained by the rule. Ex. Reduce 14, 15s. 7d. to the proposed system. Here 14, 15s. 7d. = 14-781^ = 14'781 = 14-781,25, or 14, 7fl. 8c. IJm. Reduce from the present to the proposed system, 1. 4s. 6d. 5. 14s. 8d. 9. 2, 7s, 6d. 13. 17, 11s. 1 2. 9 6 6.17 4 10. 3 18 7 14. 24 2 6 3.10 9 7. 18 4 11. 5 11 8i 15. 26 15 6 4.15 3 8. 19 10| 12. 8 8 9 16. 30 14 8 To express 's,/. c. and m. in 's, sh. and d. mentally. RULE. Divide the florins and cents by 5 for shillings, then to the remainder (if any) annex the mils, and from this subtract its 25th part; the result is farthings, from which the pence may be found by dividing by 4. Ex. Express 5, 7fl. 7c. 5m. in 's, sh. and d. Here 5, 7fl. 7c. 5m. = 5-775 = 5, 15s. 6d. Reduce from the proposed to the present system, 17. -525 18. -875 19. -750 20. -625 21. 22. 23. 24. -274 986 729 507 25. 4-841 26. 7-958 27. 8-692 28. 7-666 29. 15-833 30. 21-333 31. 27-816 32. 35-659 PRACTICAL EXERCISES. 1. What should a man receive for 292 days' service, at 45, 8fl. 7c. 5m. per annum ? 2. What should be paid for 129 quarters of wheat, when 29 quarters cost 65, 6fl. 5c. 6m. ? 3. What is the commission on 555, 7fl., at 2 per cent? 4. What part of 21, 2fl. 4c. 8m. is 5, 8fl. 4c. 3im. ? 5. Required the interest on 755, Ifl. 7c. 5m. for 4 years, at 2 J per cent per annum. 170 DECIMAL COINAGE. 6. What is the amount of 420, 6fl. 7c. 5m. for 4 years 4 months, at 4 per cent, per annum? 7. What is the interest of 1469, Ifl. 2c. 5m. for 512 days, at 3 per cent, per annum ? 8. What should be paid for 3 chests of tea, each con- taining 2 cwt. 1 qr. 14 Ibs., when 3 qrs. 21 Ibs. cost 18, 3fl. 7c. 5m. ? 9. If the quartern loaf costs 3c. 7^m., when wheat is at 3, 5fl. per quarter, what should it cost when wheat is at 2, 9fl. 4c. per quarter? 1 0. How many yards at 3fl. 7c. 5m. per yard should be given for 435 yards, at 7fl. 2c. 5m. per yard? 11. Find the present value of 486, 6fl. 7c. 5m. due 9 months hence, at 4 per cent, per annum. 12. What discount should be allowed for present payment of 276, 4fl. 8c. 8m. due 2 ye. 4 mo. hence at 3 per cent, per annum ? 13. In what time will the interest of 225, 8n. 4 2c. 5m. pay a debt of 12, 4c. 4m., at 4 per cent, per annum ? 14. Find (by practice) the rent of a farm containing 525 ac. 3 ro. 25 per., at 4, 8fl. 6c. 4m. per acre. 15. What is the difference between the interest of 455, 6fl. 7c. 5m. for 4 years, at 2 \ per cent., and the discount upon the same due 4 years hence at 2^ per cent. ? 16. Bought goods for 325, 6fl. 4c., how should they be sold to gain 7 per cent. ? 17. If 137, 5fl. Ic. 2m. amounts to 165, Ic. 4-4m. in 5 years, what rate per cent, per annum is charged for the loan ? 18. The sum of the incomes of two persons is 798, 5fl. 8c. 6m., and the difference is 52, 8fl. 6c. 4m. ; what is the income of each ? 19. What is the income of a gentleman, who after paying taxes at the rate of 3c. 7m. per 1, has remain- ing 469, 2fl. 4c. 5-7m. ? 20. Divide 256, 8fl. 7c. 2m. among 4 persons, so that J of the 1st, of the 2d, \ of the 3d, and of the 4th, may each make the same sum. 21. By selling an article for 21, 2fl. 6c. 1m., 5 per cent, is lost ; what was its prime cost ? ANSWERS. The Answers to the Simple Rules, Compound Rules of Money Addition, and Subtraction of Weights and Mea- sures, are published separately. ANSWERS to the Promiscuous Exercises at the end of Reduction. 1. 88 days 2.117 3.108 , 4. 122 days 5.119 6.316 7. 33 wks. 4 days' 10. 42240 steps 8.45 5 9. 231 shirts' 11. 62720 times 12. 1232 doses USES OF COMPOUND ADDITION. 1. 1167 2. d. [6. 3646 18 2J 2 8fl7. I741bs.4oz. Idwt 1 cwt.12 75 17 7|J8. 200 ton; 3.2130 18 11 1, Oqrs*. 15 Ibs. 4. 545 4 3J 9. 2005 qrs. 1 bu. 5. 2407 14 7 |10. 15 fur. 1 po. 5 yd 11.175 acres, 1 rood, 15 po. 292 yds. !. 1798 yds.0qr.2nl. If in. 13. 161 days, 1 hour, 1 minute. USES OF COMPOUND SUBTRACTION. 1. 2. 3. . . 476 14 7i 398 346 5 5* 3 4. 17426 5, 1556 13 8 6.173 8 lill.4cwt.lqr. 18 Ibs. 7. 99 19 11| 12. 6 tons, 3 cwt. 2 qr. 13 Ib. 8. 5919 9 13. 31 yds. 2 qrs. 1 nl. 9. 8 8 6 14. 197 mis. 5 fur. 35 po. 10.261b.lloz. 8dwt.20grs. 15. 1701 ac. ro. 24 po. 141 s. yds. 172 ANSWERS. COMPOUND MULTIPLICATION. RULE IJ. BY COMPOSITE NUMBERS. s. d. s. d. s. d. 1. 20 9. 14 7 If 17. 202 8 2. 21 12 10. 28 8 9 18. 229 13 3. 30 2 11. 42 3 9 19. 40 3 li 4. 25 3 IJ 12. 28 16 61 20. 34 6 5^ 5. 39 1 4 13. 28 11 8 21. 93 10 3j 6. 29 19 6" 14. 4 15 22. 52 18 7, 7. 8 1 4 15. 76 23. 88 14 8. 12 9 4 16 86 11 9| 24. 57 9 2 EULE III. BY PRIME NUMBERS. s. d. s. d. s. d. 1. 62 16 11J 14. 454 7 10A 27. 425 16 8 2. 101 2 15. 126 3 3j 28. 609 3 9A 3. 162 3A 4. 64 1 9| 16. 251 5 7 17. 116 11 4 29. 733 4 Of 30. 222 12 4 5. 258 3 8| 18. 355 10| 31. 265 6 1 6. 136 3 7 19. 167 5i 32. 607 19 Of 7. 121 9 74 20. 100 11 10 33. 144 18 02 8. 165 14 3} 21. 166 6 1^ 34. 436 16 21 9. 219 14 6j 22. 468 3 6 35. 269 7 3 10. 330 2 6 23. 227 2 9 36. 477 17 11. 113 1 2J 24. 314 3 4 37. 373 16 1 12. 235 3 0| 25. 266 9 3 38. 196 7 9A 13. 341 9 10 26. 340 2 3 39. 770 12 9 ANSWERS. 173 RULE IV. BY COMPOSITE AND PRIME NUMBERS MIXED. s. d. s. d. s. d. I. 68 17 3 11. 244 11 3 21. 400 4 9 2. 100 8 &k 12. 90 18 22. 232 10 3. 61 11 13. 67 7 7 23. 565 10 11* 4. 95 16 4| 14. 280 5 24. 172 10 5. 87 16 6 15. 340 19 3| 25. 550 17 1 6. 51 19 51 16. 391 5 10 26. 668 3 5 7. 79 10 5 17. 128 6 65 27. 211 10 8. 67 8 10* 18. 296 12 8 28. 448 19 8* 9. 117 4 19. 143 6 29. 558 5 10. 133 14 9 20. 280 16 8 30. 978 10 10 RULE Y. s. d. s. d. > s. d. 1. 641 19 9. 116 14 6 17. 8997 4 2 2. 36 14 3 10. 132 6 7i 18. 5909 12 33 3. 51 6 11. 177 13 19. 1286 14 4. 106 ,4 Of 12. 266 8 9 | 20. 597 6 5. 350 8 13. 1883 7 6 | 21. 770 11 11 6. 40 7 8* 14. 5534 6 22. 1230 10 11 7. 587 2 15. 5748 16 101 , 23. 328 7 10 8. 884 10 Pi 16. 11764 0" | 24. 67 8 10; MULTIPLICATION OF WEIGHTS AND MEASURES. TROY WEIGHT. APOTHECARIES. Ibs. oz. dwt. grs. Ibs. oz. dr. sc. gr. 1. 31 4 14 8 4. 48 11 3 I 10 2. 235 7 6 16 5. 45 2 4 1 16 3. 101 7 10 18 6. 175 3 5 1 7 H 2 174 ANSWERS. AVOIRDUPOIS, tons cwt. qrs. Ibs. oz. 7. 7 11 22 1 8. 236 4 2 10 9. 162 13 1 7 4 MEASURE OF CAPACITY. qr. bu. pk. gl. qr. pts. 10. 47 5 2 1 11. 133 7 3 1 12. 453 2 1 3 1 LINEAL MEASURE. ml. fa. po. yd. ft. 13. 52 5 15 3 '4. 368 5 9 4j 2 5. 435 37 3 SQUARE MEASURE. ac. ro. po. yd. ft. 16. 13 1 3 10| 1 17. 222 32 13 3 18. 417 3 5 I7i 6 19. 20. 21. SOLID MEASURE. c. ft. 9 6 c. yd. 185 1724 c. m. 204 88 749 6 208 CLOTH MEASURE. yd. qr. nl. in. 22. 26 2 3 li 23. 475 24. 309 1 1 0* YARN MEASURE. (Cotton) sp. lik. sk. th. in. 25. 60 5 4 79 26. 457 13 72 48 (Lint) sp. hp. hr. ct. th. in. 27. 44 2 5 1 80 50 28. 463 2 5 1 87 64 TIME. yrs. dy. hr. mi. se 29. 47 299 13 16 53 30. 924 210 6 34 12 31.3460 359 11 14 52 ANGULAR MEASURE. C. 8. ' " 32. 86 6 17 37 30 3J. 1371 1 2 24 34. 4769 29 36 COMPOUND DIVISION. PART II. s. d. s. / o/o9j5 IDS. 8. 23 14 84 1 38. 3793 5 |lbs. 9. 6 2 3| | 39. 56 cwt. qr. 14 5 Ibs, 10. 2 4 11 40. 7 tons, 3 cwt. 1 qr. 11. 18 10 9 5 I 16y$ Ibs. 12. 36 13 9 41. 11 cwt. 2 qr. 13f 13. 1 10 2 T 8 S 42. 10 9 5| 5 S 3 14. 2 3 3 43. 12 10 T , 15. 10 8$ If 44. 54 18 6| 2 16. 7 2 4 T 3 45. 1 9 64 ] 17. 10 3 2* ft 46. 25 16 6| 18. 3 11 ~4 iV? 47. 6 11 2 5 5 19. 21 5 7* 1 48. 6 16 lOJ I 20. 39 10 49. 8 2 If f 21. 32 13 4 50. 43 74 22. 23 6 8 51. 880 23. 21 52. 66 24. 230 8 53. 95 19 7| T> T 25. 39 9 2| ft 54. 1 74 If 26. 12 11 5| f 55. 32 1 6$ 27. 368 6 8 56. 6 18 8| If 28. 567 18 >| ft 57. 405 masons 29. 423 58. 7ft days 30. 196 59. 4| months ANSWERS. 179 60. 2 months 81. 12 10 61. 80 days 82. 1191bs. 6 oz. 9 awt. 62. 144 days 2 grs. 5 9 7 4 7 63. 25 7 days 83. 1097 7 gallons 64. 3 months 84. 7s 10 |ff 65. 12 ounces 85. 2700 66. 33 11 1 ff 86. 3687 16 la || 67. 240 days 87. 190/ T feet 68. 16 3 2| 5 y 7 88. 80 T ' 5 feet 69. 174ft days 89. 2 miles, 1048 yards 70. 443 |f yards 90. 11 miles, 1196 yards 71. 109i persons 91. 6 miles, 860 yards 72. 6 1 Oa if 92. 13 f gallons 73. 8 9 3a I 93. 55 T 8 3 gallons 74. 213 94. 1 12 75. 66 4 6e 5 T 95. 1 1 8i {ft 76. 4 19 6i | 96. 21 8 6| f 77. 4 8 f 97. 31 10 1 || 78. 1 13 101 g s g 98. 1058 2 6 79. 17 6 3 99. 42 1 yards 80. 17 10 7* 100. 32 1 yards COMPOUND PROPORTION. 1. 100 acres 15. 11 6 51 |aj 2. 25 days 16. 4 8 10 | 3. 12 men 17. 25 5 3i |f 4. 68 f bu. 18. 11 1 8 ft 5. 567 days 19. 3f f | years. 6. 1491 days 20. 1725 miles 7. 108 masons 21. 640 ounces 8. 390 miles 22. 54ljif 9. 224 cwt. 23. 41 7 days 10. 7 horses 24. 216 days 11. 13 pioneers 25. 233 suits 12. 5i days 26. 40 1|| additional mfca 13. 112 reams 27. 24| men 14. 14 horses 180 ANSWERS. PRACTICE CASE I. 27. 5673 6 8 s. d. 28. 5215 1. 2342 29. 6163 4 2. 1469 15 30. 3339 6 8 3. 1297 31. 6688 16 8 4. 3226 32. 4940 5. 710 13 4 6. 744 2 6 CASE V. 7. 500 6 8 s. d. 8. 546 18 33. 876 18 34. 1041 4 CASE II. 35. 2630 14 s. d. 36. 2498 8 9. 77 19 4 37. 3410 8 10. 105 16 9 38. 4407 4 11. 42 7 6 39. 3737 12 12. 146 4 6 40. 7886 14 13. 282 5 4 14. 236 2 9 CASE VI. 15. 365 6 8 s. d. 16. 11 8 8| 41. 10643 11 9 42. 8445 10| CASE III. 43. 28903 7 2i s. d. 44. 18792 19 4i 17. 11695 45. 32234 3 4 18. 28230 46. 57479 18 8 19. 19534 13 4 47. 41483 7 7* 20. 20387 18 6 48. 17982 7i 21. 31878 5 22. 18169 17 6 CASE VII. 23. 41772 s. d. 21. 10823 4 2 49. Ill 18 10 60. 68 13 6 CASE IV. 51. 124 18 2J 8. d. 62. 11 11 1} 25. 2736 63. 13 2 9i 26. 3877 12 54. 17 10 7} ANSWERS. 181 s. d. s. d. 55. 65 17 8 75. 2205 56. 63 19 4A 76. 5045 1 l\ 57. 111 19 3 77. 4538 19 7i 58. 63 14 9 78. 8272 1 7 59. 3 18 5 79. 6755 9 5$ 80. 3604 2 1$ EXERCISES. 81. 2478 10 9 60. 115 17 82. 6794 8 6 61. 217 1 9 83. 9267 10 62. 193 6 7i 84. 8877 14 63. 67 19 6 85. 7729 64. 252 7 10 86. 21405 8 6 65. 294 1 10^ 87. 12096 2 7 66. 162 10 10 88. 9815 14 7$ i 67. 130 12 6 89. 14717 17 101 i 68. 272 17 8 90. 36555 3 8 6S\ 750 91. 17552 10 1 | 70. 473 2 8 92. 5436 2 6 71. 3243 4 6 93. 11069 12 3| | 72. 4231 13 9 94. 23387 2 64 | *3. 5115 5 6 95. 9710 16 9 74. 4434 14 1 96. 11211 4 5 J PROMISCUOUS EXERCISES. 1. 8 17 OJ 14. 3166 7 2. 4 10 4 15. 5777 4 3. 11 4J 16. 15010 7 3 4. 20 6 3| 17. 026 6. 54 16 4* 18. 6J 6. 71 6 6| 19. 68 3 5$ ft 7. 249 3 4 20. 165 days 8. 153 19 5 21. 33 9 81 | 9. 182 15 10 22. 8 la ff 10. 2277 2 23. 25 15 7| \ 11. 751 16 24. 14 6|||| 12. 2260 14 25. 21,252 10s 13. 259 4 8 26. 21 ye. 296 d. 15 h. 10 m. 182 ANSWERS. 27. 157 6 si 28. Pr. ye. 1215. 25. 3614y. 334 d. lOh. 40 m. 30. 192 y. 319 d. 7 h. 20 m. 31. 6,967,520,000,04)0 32. 1,300,000 acres. VULGAR FRACTIONS. CASE I. CASE IV. ' T ,H, H.-IMMftt#U*P) (2) (3) (4) (5) (6) !t> # III- V, Y, H, II, W, W. CASE V. 504,540,224,13728,1591744 CASE VI. I, H, if, It, T 5 , II- CASE VII. (1) (2) (3) (4) (5) (6) CASE II. 52 T V, 21, 24, 6. CASE III. W, V, V, !, 68||, The questions in the second half of Case VIII are the answers to the first half, and vice versa. The questions in Case X are the answers to the ques- tions in Case IX, and vice versa. ADDITION. 3. Iy 5 3 4. 1! 5. \m 6. 2J 7. i?a 8. 10. 11. 12. iiii 13. 2^ 14. 1 T 7 5 15. NOTE I. 17. 102 T 7 3 18. 53ft 19. 612| 20. 688J 21. 899^ 22. 6881 23. 24. NOTE II 25. 1 10 11 3 |f 26. 16 22 ft 27. 1 11 8 lb. oz. dwt. gr 28. 11 1 645 ANSWERS. 183 tons. cwt. qr. Ib. 29. 4 12 3 22 MULTIPLICATION, 1. A 14. ft 30. ac. ro. po. 12 1 30 2. 3. if A 15. T | f 4. NOTE. yd. qr. nl. 5. i 16. 3 31. 4 1 3 6. 3 17. 31 7. l 18. 4^ da. ho. m. sc. 8. 3 19. ]9i ?2. 33. 6 13 56 51f 64 18 8 & 9*. 10. 32 20. 63| 21. 11 SUBTRACTION. 11. 12. T/*3 22. 13 23. 6| 1. $| NOTE. 13. 'ify 24. 23| 9 l ^ *?SA JN 3. }y 1O. Oo^ 14. 46^f DIVISION. 4. If 15. 24fJ 1. ? 1- 5 7 ? 5. tl 16. 452 A 2. u U. 4 5 6. i 17. 418 T 3 3. 1! 12. | 7. III 18. 94/ s 4. M 13. 12| 8. 19. 92^ 5. A 14. 5| 9. | 20. 87^| 6. 15. 7 T V 5 10. II 21. 46| 7. 27 16. 10|| 11. 22.01510a 2 8. 2f 17. 77/8 12. 1$ 23. 017 2 | 9. 1* 13. 7{| PROPOETION OF VULGAR FRACTIONS. I. 2. 3. 4. 5. s. ct. 163 16 2 10 5 1 15 6. 966 17 1| | 7. 9 1 Of f 8. 11 5$ $ 8 8| & 9. s. d. 10, 23 17 HH? 11. 60 9 6 12. 24 17 9U 13. 6 2U 14. 7 54 15. 46 4 16. 28 10 101} 17. 17 18. 7 3 If? 184 ANSWERS. s. 19. 62 16 20. 11,94 4 d. T s u reduce days to seconds ? How do you bring days to How do you bring p<>uud avoirdupois to pounds trov? How do you bring yards to years ? English elis ? How do you bring hours to years ? How do you bring seconds to days? How do you bring Englinh all! to yards? How do you bring French ells to yards ? FOR EXAMINATION. 205 How do you bring Flemish ells to yards? 35. What is COMPOUND ADDI- TION ? Repeat the rule. 46. What isCo.Mi OUND SUBTRAC- TION? R-peat the Rule. 04. What is COMPOUND MULTI- PLICATION ? Repeat the rule. How do you mnltiply a com- pound number when the multiplier is a composite number ? How do you multiply when the multiplier consists of several figure* ? fil. What is COMPOUND DIVI- SION ? Repeat the rule. 63. How do you divide when the divisor is a c omposite, number? 66. How do you mult ply when the multiplier contains a fraction? 67. How do you divide when the divisor contains a fraction ? 70. What i^ a bill of parcels ? 73. What is SIMPLB PROPOR- TION ? What is a ratio? When are four numbers pro- portional ? What are the first and last terms called? What are the second and third called ? How many kinds of ques- tions are in this rule ? How many terms are given in each question in pro- portion ? Repeat the rule for stating and working the three ffiven terms. 79. What is COMPOUND PROPOR- TION ? Repeat the rule for stating and working the given terms. 82. What is PRACTICE ? 83. When the price is at thil- ling?, how"do you proceed ? When the price is at pence, how do you proceed ? "When there are pounds in the price, how do you pro- ceed ? 84. When the diffetence be- tween the price and 20s. is a part of a pound, how do you do ? When the price is an even number of shillings, how do you proceed ? 85. When there is a fraction in the quantity, what is to be done? When the quantity consists of more than one denomi- nation, how So you pro- ceed? 88. What is a FRACTION ? What is a simple fraction ? What is a proper fraction ? What is an improper frac- tion ? 89- What is a compound frac- tion ? What is a mixed number? What is a complex fraction ? What is a common measure ? How do you multiply a frac tion by an integer ? How do you divide a fraction by an integer ? If the numerator and deno- minator of a fraction are either multiplied or divided by the same figure, is the value of that fraction al- tered ? How do you reduce a frac- tion to its lowest terms ? 90. How do you reduce an im- proper fraction to a whole or mixed number ? How do you reduce a mixed number to an improper fraction ? How do you reduce a whole number to a fraction, have 206 QUESTIONS ing a given denomina. tor? How do you reduce a com- plex fraction to a simple one ? SI. How do you reduce a com- pound fraction to a simple one? Ho\v do ynu reduce a fraction from one name to another ? 92. How do you reduce a quan tity to a fraction of any denomination? 93. How do you reduce fractions to a common denomina- tor ? How do you find the value of a fraction? How do you add fractions ? 94. How do you add mixed numbers? When fractions of different names are given, how do you add ? How do you subtract frac- tions ? How do you subtract mixed numbers? 95. How do you multiply frac- tions ? When there are integers and mixed numbers, how do you proceed ? How do you divide fractions ? 96. How do you perform ques- tions in proportion of frac- tions ? 88 What is a DECIMAL FRAC- TION ? Is the denominator of a deci- mal fraction used ? How are decimals distin- guished ? What is a terminate decimal ? What are intermiuute deci- mals ? What is a pure repea'er ? What is a mixed repeater t What is a pure circulate ? What is a mixed circulate ? When are circulates simi-i Urf 99. How do you add terminate decimals? How do you subtract termi- nate decimals ? 100. How do you multiply wlina both factors are termi- nate ? How do you divide when the divisor is terminate ? 101. How do you reduce a vulgar fraction to a decimal ? How do you reduce a terrci. n>ite decimal to a vulgar fraction ? How do you reduce a pure repeater or circulate to a vulgar fraction ? Ho\v do you reduce a mixed repeater or circulate to a vulgar fraction ? 102 How do you add repeaters ? 103. How do you add circulates and repeaters ? How do you subtract repeat- ers? How do you subtract circu- lates ? 104. How do you multiply when the multiplicand is a re- peater or circulate ? How do you multiply wJion the multiplier ia a repeater or circulate ? How do you divide when the divisor is a rtpeater or circulate? How do you reduce lower denominations to decimals of higher? 10.5. How do you fine! the value of a decimal? 107. What is INTEREST ? What is the principal ? Whatistheiutt-rest? What is the amount ? What is simple interest ? 103. How do you find the interest for years ? How do you find the interest for mouths ? id9 Howdoyou find the interest for days ? FOR EXAMINATION. 207 How do you find the interest when partial payments are made ? 1 10. How do you find the interest on accounts current? 112. What is DISCOUNT? What does the present value mean ? What is the true method of finding the discount ? What is the common me- thod ? 113. What is EQUATION OP PAY- MENTS? Repeat the rule. 114. What is commission and bro- kerage? How do you find the re-ult. when the rat exceeds 1 per cent? Ho\v do you find the result, when the rate is under 1 per cent? What is insurance? 11% Who is the insurer? Who is the insured? What is the premium? What is the policy? What is the duty on policies for sea insurances for voy- ages, from oue part of the United Kingdom to an- other? What is the duty on foreign voyages? How is policy duty charged? How do you rind the pre- mium and policy? How do you find how much mu>t be insured, to cover a given sum ? 116 What are the stocks ? How is stock bought and sold? How do you find the value of any quantity of stock ? 117. How do you find how much stock may be bought for a given sum ? How d- you find the rate of interest arising from money invested iu the stocks ? 118. What is profit and loss ? When the prime cost and selling price are given, how do you find the gain or loss per cent ? When the prime cost and gain or loss per cent are given, how do you find the selling price ? 119. When the selling price and the gain or loss per cent are given, how do you rind the prime cost? When two selling prices French Grammar .20 Latin and Greek. Ainsworth's Latin Dictionary 23 Cicero's Orationes Selectee 24 Cato Major, De Officiis....24 Clyde's Greek Syntax 21 Dymock's Caesar and Sallust 22 Edin. Academy Class-Books : Rudiments of Latin Language... 21 Latin Delectus 21 Rudiments of Greek Language.. .21 Greek Extracts 21 Ciceronis Opera Selecta 21 Selecta e Poetis 21 Ferguson's (Prof. )Gram. Exercises 24 Latin Delectus 24 Ovid's Metamorphoses 24 Fergussou's (Dr) Xenophon's Ana- basis 23 Greek Gram. Exercises 23 Homer's Iliad, with Vocab. 23 Geddes' (Prof.) Greek Grammar... 21 Greek Testament, by Duncan 23 Hunter's Ruddiman's Rudiments .22 Sallust, Virgil, & Horace 22 Livy, Books 21 to 25 22 Latin Testament, by Beza 23 Macgowan's Latin Lessons 22 Mair's Introduction, by Stewart... 23 Massie's Latin Prose Composition 22 M'Dowall's Csesar and Virgil 22 Melville's Lectiones Selectae 22 Neilson's Eutropius 22 Stewart's Cornelius Nepos 23 Veitch's Homer's Iliad 23 German. Fischart's New German Reader... 24 Logic. Port-Royal Logic (Prof. Baynes')24 School Registers. Pupil's Daily Register of Marks. 17 School Register of Attendance, Absence, and Fees 17 Geometrical Drawing. Kennedy's Grade Geometry 17 Messrs Oliver and Boyd were awarded Medals for their Educa- tional Works by Her Majesty's Commissioners of the London International Exhibition, and by the Jurors of the Paris Uni- vfersal Exhibition. EDUCATIONAL WOBKS. ENGLISH BEADING, GEAMMAE, ETC. IN the initiatory department of instruction a valuable series of works has been prepared by DK M'CULLOCH, formerly Head Master of the Circus- Place School, Edinburgh, now Minister of the West Church, Greenock. DR M'CULLOCH'S SERIES OF CLASS-BOOKS. These Books are intended for the use of Schools where the general mental culture of the pupil, as well as his proficiency in the art of reading, is studiously and systematically aimed at. They form, collectively, a progressional Series, so constructed and graduated as to conduct the pupil, by regular stages, from the elementary sounds of the language to its highest and most complex forms of speech; and each separate Book is also progressively arranged, the lessons which are more easily read and understood always taking the lead, and preparing the way for those of greater difficulty. The subject-matter of the Books is purposely miscellaneous. Yet it is always of a character to excite the interest and enlarge the knowledge of the reader. And with the design of more effectually promoting his mental growth and nurture, the various topics are introduced in an order con- formable to that in which the chief faculties of the juvenile mind are usually developed. That the moral feelings of the pupil may not be without their proper stimulus and nutriment, the lessons are pervaded throughout by the religious and Christian element. DR M'CULLOCH'S READING-BOOKS FOR SCHOOLS. FIRST READING-BOOK, ijd. Do. Large Type Edition, in two parts, price 2d. each. Do. In a Series of Sheets for Hanging on the Wall, Is. ; or on Roller, Is. 8d. SECOND READING-BOOK, 3d. THIRD READING- BOOK, containing simple Pieces in Prose and Verse, with Exercises. Now Printed in Larger Type, 1 Od. FOURTH READING-BOOK, containing only Lessons likely to interest. With SYNOPSIS OF SPELLING, . . . Is. 6d. SERIES OF LESSONS in Prose and Verse, .... 2s. COURSE OF ELEMENTARY READING in SCIENCE and LITERATURE, compiled from popular Writers, 39 Woodcuts, 3s. MANUAL OF ENGLISH GRAMMAR, Philosophical and Practical ; with Exercises ; adapted to the Analytical mode of Tuition, Is. 6d. Oliver & Boyd's New Code Class-Books. STANDARD BEADING-BOOKS. By JAMES COLVILLE, M.A., Senior English Master George Watson's College-Schools, Lauriston, Edinburgh, one of the Educational Institutions of the Merchant Company. The following are already published: PRIMER : Being Spelling and Reading Lessons introductory to Standard I. (Illustrated.} 36 pages. ld. FIRST STANDARD READING-BOOK ; with Easy Lessons in Script. (Illustrated.} 95 pages. 4d. in stiff wrapper, or 6d. cloth. SECOND STANDARD READING-BOOK; with Dictation Exercises, partly in Script. (Illustrated.} 108 pages. 4d. in stiff wrapper, or 6d. cloth. THIRD STANDARD READING-BOOK; with Dictation Exercises, partly in Script. 144 pages, strongly bound. 8d. AEITHMETIC. By ALEXANDER TROTTER, Teacher of Mathematics, etc., Edinburgh; Author of "Arithmetic for Advanced Classes," etc. PART I., embracing Standards 1 and 2. 36 pages. 2d. Answers, 3d. PART II., embracing Standards 3 and 4. 36 pages. 2d. Answers, 3d. PART III. in Preparation. STANDARD GEOGRAPHIES. By W. LAWSON, F.R.G.S., St Mark's College, Chelsea; Author of "Geo- graphy of the British Empire." GEOGRAPHICAL PRIMER, embracing an Outline of the Chief Divisions of the World. Adapted to Standard IV. 36 pages. 2d. GEOGRAPHY OF ENGLAND AND WALES; with a Chapter on Railways. Adapted to Standard V. 36 pages. 2d. LAWSON'S ELEMENTS OF PHYSICAL GEOGRAPHY. Adapted to the Requirements of the New Code. 90 pages. 6d. The following Works will also be found adapted to the Requirements of the New Code: REID'S RUDIMENTS OF MODERN GEOGRAPHY, with 36 pages of information on Counties and Railways, . . . Catalogue, page 10 DOUGLAS'S PROGRESSIVE GEOGRAPHY, a New Work, . . 9 LENNIE'S GRAMMAR, with Analysis of Sentences, . . . . 6 DOUGLAS'S GRAMMAR, with Analysis of Sentences, ... 6 REID'S GRAMMAR, with Analysis of Sentences, .... 5 HUNTER'S SCHOOL SONGS, with Music, 17 ENGLISH READING, GRAMMAR, ETC. 5 OUTLINES OF ENGLISH GRAMMAR AND ANALYSIS, For ELEMEMTARY SCHOOLS, with EXERCISES. By WALTER SCOTT DAL- GLEISH, M.A. Edin., lately one of the Masters in the London Interna- tional College. 8d. KEY, Is. DALGLEISH'S PROGRESSIVE ENGLISH GRAMMAR, with EXERCISES. 2s. KEY, 2s. 6d. From Dr JOSEPH BOSWORTH, Professor of Anglo-Saxon in the University of Oxford; Author of the Anglo-Saxon Dictionary, etc., etc. "Quite a practical work, and contains a vast quantity of important information, well arranged, and brought up to the present improved state of philology.. I have never seen so much matter brought together in so short a space." DALGLEISH'S GRAMMATICAL ANALYSIS, with PKO- GRESSIVE EXERCISES. 9d. KEY, 2s. DALGLEISH'S OUTLINES OF ENGLISH COMPOSI- TION, for Elementary Schools. With Exercises. 6d. Key, 4d. DALGLEISH'S INTRODUCTORY TEXT-BOOK OF ENGLISH COMPOSITION, based on GRAMMATICAL SYNTHESIS; con- taining Sentences, Paragraphs, and Short Essays. Is. DALGLEISH'S ADVANCED TEXT-BOOK OF ENGLISH COMPOSITION, treating of Style, Prose Themes, and Versification. 2s. Both Books bound together, 2s. 6d. KEY, 2s. 6d. A DICTIONARY OF THE ENGLISH LANGUAGE, con- taining the Pronunciation, Etymology, and Explanation of all Words authorized by Eminent Writers. By ALEXANDER REID, LL.D., late Head Master of the Edinburgh Institution. Reduced to 5s. DR REID'S RUDIMENTS OF ENGLISH GRAMMAR. Copious Exercises have been introduced throughout; together with a new Chapter on the Analysis of Sentences ; while the whole work has been revised and printed in a larger type. 6d. DR REID'S RUDIMENTS OF ENGLISH COMPOSITION. 2s. KEY, 2s. 6d. ENGLISH GRAMMAR, founded on the Philosophy of Language and the Practice of the best Authors. With Copious Exer- cises, Constructive and Analytical. By C. W. CONNON, LL.D. 2s. 6d. Spectator. " It exhibits great ability, combining practical skill with philosophical views." CONNON'S FIRST SPELLING-BOOK. 6d. 6 ENGLISH READING, GRAMMAR, ETC. LENNIE'S PRINCIPLES OF ENGLISH GRAMMAR. Comprising the Substance of all the most approved English Grammars, briefly defined, and neatly arranged; with Copious Exercises in Parsing and Syntax. New Edition; with the author's latest improvements, and an Appendix in which Analysis of Sentences is fully treated. Is. 6d . THE AUTHOR'S KEY; containing, besides Additional Exercises in Parsing and Syntax, many useful Critical Remarks, Hints, and Observations, and explicit and detailed instructions as to the best method of teaching Grammar. 3s. 6d. ANALYSIS OF SENTENCES; being the Appendix to Lennie's Grammar adapted for General Use. Price 3d. KEY, 6d. THE PRINCIPLES OF ENGLISH GRAMMAR; with a Series of Progressive Exercises, and a Supplementary Treatise on Analysis of Sentences. By Dr JAMBS DOUGLAS, lately Teacher of English, Great King Street, Edinburgh. Is. 6d. DOUGLAS'S INITIATORY GRAMMAR for Junior Classes, printed in larger type, and containing a Supplementary Treatise on Analysis of Sentences. 6d. DOUGLAS'S PROGRESSIVE ENGLISH READER. A New Series of English Reading Books. The, Earlier Books are illus- trated with numerous Engravings. FIRST BOOK. 2d. I THIRD BOOK. Is. I FIFTH BOOK. 2s. SECOND BOOK. 4d. | FOURTH BOOK. Is. 6d. | SIXTH BOOK. 2s. 6d. DOUGLAS'S SELECTIONS FOR RECITATION, with In- troductory and Explanatory Notes; for Elementary Schools. Is. 6d. DOUGLAS'S SPELLING AND DICTATION EXERCISES. 144 pages, price Is. Athenaeum. "A good practical book, from which correct spelling and pronunciation may be acquired." DOUGLAS'S ENGLISH ETYMOLOGY : A Text-Book of Derivatives, with numerous Exercises ; for the Use of Schools. 168 pages, price 2s. Now ready. SHAKSPEARE'S KING RICHARD II. With Historical and Critical Introductions ; Grammatical, Philological, and other Notes, etc. Adapted for Training Colleges. By Rev. Canon ROBINSON, M.A., late Principal of the Diocesan Training College, York. 2s. WORDSWORTH'S EXCURSION. THE WANDERER. With Notes to aid in Analysis & Paraphrasing. By Canon ROBINSON. 8d. ENGLISH READING, OBJECT-LESSONS, ETC. 7 HISTORY OF ENGLISH LITERATURE; with an OUTLINE of the ORIGIN and GROWTH of the ENGLISH LAN- GUAGE. Illustrated by EXTRACTS. For Schools and Private Students. By WM. SPALDING, A.M., Professor of Logic, Rhetoric, and Metaphysics in the University of St Andrews. Continued to 1870. 3s. Gel. Spectator. " A compilation and text-book of a very superior kind. . . The volume is the best introduction to the subject we have met with." POETICAL READING- BOOK; with Aids for Grammatical Analysis, Paraphrase, and Criticism; and an Appendix on English Versification. By J. D. MORELL, A.M., LL.D., Author of Grammar of the English Language, etc.; and W. IHNE, Ph.D. 2s. 6d. STUDIES IN COMPOSITION : A Text-Book for Advanced Classes. By DAVID PRYDE, M.A., Head-Master of the Edinburgh Merchant Company's Educational Institution for Young Ladies. 2s. Recently Published. ENGLISH COMPOSITION FOR THE USE OF SCHOOLS. By ROBERT ARMSTRONG, Madras College, St Andrews; and THOMAS ARMSTRONG, Heriot Foundation School, Edinburgh. Part I., Is. 6d. Part II., 2s. Both Parts bound together, 3s. KEY, 2s. ARMSTRONG'S ENGLISH ETYMOLOGY. 2s. ARMSTRONG'S ETYMOLOGY for JUNIOR CLASSES. 4d. SELECTIONS FROM PARADISE LOST; with NOTES adapted for Elementary Schools, By Rev. ROBERT DEMAUS, M.A., late of the West End Academy, Aberdeen. Is. 6d. DEMAUS'S ANALYSIS OF SENTENCES. 3d. SYSTEM OF ENGLISH GRAMMAR, and the Principles of Composition. With Exercises. By JOHN WHITE, F.E.I S. Is. 6d. MILLEN'S INITIATORY ENGLISH GRAMMAR. Is. EWING'S PRINCIPLES OF ELOCUTION, improved by F. B. CALVERT, A.M. 3s. 6d. Consists of numerous rules, observations, and exercises on pronunciation, pauses, inflections, accent, and emphasis, accompanied with copious extracts in prose and poetry. RHETORICAL READNIGS FOR SCHOOLS. By WM. M'DOWALL, late Inspector of the Heriot Schools, Edinburgh. 2s. 6d. OBJECT-LESSON CARDS ON THE VEGETABLE KINGDOM. Set of Twenty in a Box. 1, Is. HOW TO TRAIN YOUNG EYES AND EARS: Being a MANUAL OF OBJECT-LESSONS for PARENTS and TEACHERS. By MARY ANNE Ross, Mistress of the Church of Scotland Normal Infant School, Edinburgh. Is. 6d. 8 HOUSEHOLD ECONOMY, GEOGRAPHY AND ASTRONOMY. HOUSEHOLD ECONOMY : a MANUAL intended for Female Training Colleges, and the Senior Classes of Girls' Schools. By MAR- GARET MARIA GORDON (Miss Brewster), Author of " Work, or Plenty to do and how to do it," etc. 2s. Athenceum "Written in a plain, genial, attractive manner, and consti- tuting, in the best sense of the word, a practical domestic manual." SESSIONAL SCHOOL BOOKS. ETYMOLOGICAL GUIDE. 2s. 6d. This is a collection, alphabetically arranged, of the principal roots, affixes, and prefixes, with their derivatives and compounds. OLD TESTAMENT BIOGRAPHY, containing notices of the chief persons in Holy Scripture, in the form of Questions, with references to Scripture for the Answers. 6d. NEW TESTAMENT BIOGRAPHY, on the same Plan. 6d. FISHER'S ASSEMBLY'S SHORTER CATECHISM EXPLAINED. 2s. PART I. Of what Man is to believe concerning God. II. Of what duty God requires of Man. GEOGKAPEY AND ASTKONOMY, IN compiling the works on these subjects the utmost possible care has been taken to ensure clearness and accuracy of statement. Each edition is scrupulously revised as it passes through the press, so that the works may be confidently relied on as containing the latest information accessible at the time of publication. SCHOOL GEOGRAPHY. By JAMES CLYDE, LL.D., one of the Classical Masters of the Edinburgh Academy. With special Chapters on Mathematical and Physical Geography, and Technological Appendix. Corrected throughout. 4s. Athenaeum." We have been struck with the ability and value of this work, which is a great advance upon previous Geographic Manuals. . . . Almost for the first time, we have here met with a School Geo- graphy that is quite a readable book, one that, being intended for advanced pupils, is well adapted to make them study the subject with a degree of interest they have never yet felt in it. ... Students pre- paring for the recently instituted University and Civil Service examinations will find this their best guide." DR CLYDE'S ELEMENTARY GEOGRAPHY. Corrected throughout. Is. 6d. In the Elementary Geography it has been endeavoured to reproduce that life-like grouping of facts geographical portraiture, as it may be called which has been remarked with approbation in the School Geography. GEOGRAPHY AND ASTRONOMY. A COMPENDIUM OF MODERN GEOGRAPHY, POLITICAL, PHYSICAL, and MATHEMATICAL : With a Chapter on the Ancient Geo- graphy of Palestine, Outlines of Astronomy and of Geology, a Glossary of Geographical Names, Descriptive and Pronouncing Tables, Questions for Examination, etc. By the Rev. ALEX. STEWART, LL.D. Carefully Revised. With 11 Maps. 3s. 6d. GEOGRAPHY OF THE BRITISH EMPIRE. By WILLIAM LAWSON, St Mark's College, Chelsea. Carefully Revised. 3s. PATTT I. Outlines of Mathematical and Physical Geography. II. Physical, Political, and Commercial Geography of the British Islands. III. Phy- sical, Political, and Commercial Geography of the British Colonies. LAWSON'S ELEMENTS OF PHYSICAL GEOGRAPHY, adapted to the requirements of the New Code. 90 pages, 6d. Now Ready. EDINBURGH ACADEMY MODERN GEOGRAPHY. Carefully Revised. 2s. 6d. EDINBURGH ACADEMY ANCIENT GEOGRAPHY. 3s. AN ABSTRACT OF GENERAL GEOGRAPHY, compre- hending a more minute Description of the British Empire, and of Pales- tine or the Holy,Land, etc. With numerous Exercises. For Junior Classes. By JOHN WHITE, F.E.I. S., late Teacher, Edinburgh. Carefully Rovised. Is. ; or with Four Maps, Is. 3d. WHITE'S SYSTEM OF MODERN GEOGRAPHY; with Outlines of ASTRONOMY and PHYSICAL GEOGRAPHY ; comprehending an Account of the Principal Towns, Climate, Soil, Productions, Religion^ Education, Government, and Population of the various Countries. With a Compendium of Sacred Geography, Problems on the Globes, Exercises, etc. Carefully Revised. 2s. 6d. ; or with Four Maps, 2s. 9d. AN INTRODUCTORY GEOGRAPHY, for Junior Pupils. By Dr JAMES DOUGLAS, lately Teacher of English, Great King Street, Edinburgh. Carefully Revised. 6d. DR DOUGLAS'S PROGRESSIVE GEOGRAPHY. An entirely new work, showing the recent changes on the Continent and elsewhere, and embracing much Historical and other Information. 160 pages, Is. Now Ready. DR DOUGLAS'S TEXT-BOOK OF GEOGRAPHY, con- taming the PHYSICAL and POLITICAL GEOGRAPHY of all the Countries of the Globe. Systematically arranged. 2s. 6d.; or with ten Coloured Maps, 3s. Carefully Revised. 10 GEOGRAPHY AND ASTRONOMY. FIRST BOOK OF GEOGRAPHY; being an Abridgment of Dr Reid's Rudiments of Modern Geography ; with an Outline of the Geography of Palestine. Carefully Revised. 6d. This work has been prepared for the use of young pupils. It is a suitable and useful companion to Dr Reid's Introductory Atlas. RUDIMENTS OF MODERN GEOGRAPHY. By ALEX. REID, LL.D., late Head Master of the Edinburgh Institution. With Plates, Map of the World. Carefully Revised. Is.; or with Five Maps, Is. 3d. Enlarged by 36 pages of extra information regarding the Counties and principal Railways of the United Kingdom. The names of places are accented, and accompanied with short descrip- tions, and occasionally with the mention of some remarkable event. To the several countries are appended notices of their physical geography, productions, government, and religion; concluding with an outline of sacred geography, problems on the use of the globes, and directions for the construction of maps. DR REID'S OUTLINE OF SACRED GEOGRAPHY. 6d. This little work is a manual of Scripture Geography for young persons. It is designed to communicate such a knowledge of the places mentioned in holy writ as will enable children more clearly to understand the sacred narrative. It contains references to the passages of Scripture in which the most remarkable places are mentioned, notes chiefly historical and descriptive, and a Map of the Holy Land in provinces and tribes. MURPHYS BIBLE ATLAS of 24 MAPS, with Historical Descriptions. Is. 6d. coloured. Witness. " We recommend this Atlas to teachers, parents, and indivi- dual Christians, as a comprehensive and cheap auxiliary to the intelligent reading of the Scriptures. EWING'S SYSTEM OF GEOGRAPHY. Carefully Revised. 4s. 6d. ; with 14 Maps, 6s. Besides a complete treatise on the science of geography, this work con- tains the elements of astronomy and of physical geography, and a variety of problems to be solved by the terrestrial and celestial globes. At the end is a pronouncing Vocabulary, in the form of a gazetteer, containing the names of all the places in the work. ELEMENTS OF ASTRONOMY: adapted for Private i ii.su uction and Use of Schools. By HUGO REID, Member of the College of Preceptors. With 65 Wood Engravings. 3s. REID'S ELEMENTS OF PHYSICAL GEOGRAPHY; With OUTLINES of GEOLOGY, MATHEMATICAL GEOGRAPHY, and ASTRONOMY, and Questions for Examination. With numerous Illustrations, and a large coloured Physical Chart of the Globe. Is. GEOGRAPHY AND ASTRONOMY, HISTORY. 11 REVISED EDITIONS OF SCHOOL ATLASES. A GENERAL ATLAS OF MODERN GEOGRAPHY; 29 Maps, Coloured. By THOMAS Ewrxo. 7s. 6d. SCHOOL ATLAS OF MODERN GEOGRAPHY. Maps 4to, folded 8vo, Coloured. By JOHN WHITE, F.E.I.S., Author of "Abstract of General Geography," etc. 6s. WHITE'S ELEMENTARY ATLAS OF MODERN GEO- GRAPHY. 4to, 10 Maps, Coloured. 2s. 6d. CONTENTS. 1. The World; 2. Europe; 3. Asia; 4. Africa; 5. North America; 6. South America; 7. England; 8. Scotland; 9. Ireland; 10. Palestine. A SCHOOL ATLAS OF MODERN GEOGRAPHY. 4to, 16 Maps, Coloured. By ALEXANDER REID, LL.D., late Head Master ot the Edinburgh Institution, etc. 5s. REID'S INTRODUCTORY ATLAS OF MODERN GEO- GRAPHY. 4to, 10 Maps, Coloured, 2s. 6d. CONTENTS. 1. The World; 2. Europe; 3. Asia; 4. Africa; 5. North America; 6. South America; 7. England; 8. Scotland; 9. Ireland; 10. Palestine. H I S T E T, THE works in this department have been prepared with the greatest care. They will be found to include Class-books for Junior and Senior Classes in all the branches of History generally taught in the best schools. While the utmost attention has been paid to accuracy, the narratives have in every case been rendered as instructive and pleasing as possible, so as to relieve the study from the tediousness of a mere dry detail of facts. A CONCISE HISTORY OF ENGLAND IN EPOCHS. By J. F. CORKRAN. With Maps and Genealogical and Chronological Tables, and comprehensive Questions to each Chapter. 2s. 6d. * # * Intended chiefly for the Senior Classes of Schools, and for the Junior Students of Training Colleges. In this History of England the writer has endeavoured to convey a broad and full impression of its great Epochs, and to develop with care, but in subordination to the rest of the narrative, the growth of Law and of the Constitution. He has summarized events of minor importance; but where illustrious characters were to be brought into relief. ^ where the story of some great achievement merited a full narration, he hns occupied more space than the length of the history might seem to justify; for it is his belief that a mere narration of the Deeds of England in her struggles for liberty and for a high place among the nations of the world, is more fertile in instruction to youth, and more stimulating to a healthy and laudable ambition than any other mode of treating our past Seceut events have been treated with more than usual fulness. 12 HISTORY. HISTORY OF ENGLAND FOR JUNIOR CLASSES; with Questions for Examination. Edited by HENRY WHITE, B.A. Trinity College, Cambridge, M.A. and Ph. Dr. Heidelberg. Is. 6d. Athtnceum. "A cheap and excellent history of England, admirably adapted for the use of junior classes. Within the compass of about a hundred and eighty duodecimo pages, the editor has managed to give all the leading facts of our history, dwelling with due emphasis on those turn- ing points* which mark our progress both at home and abroad. The various changes that have taken place in our constitution are briefly but clearly described. It is surprising how successfully the editor has not merely avoided the obscurity which generally accompanies brevity, but invested his narrative with an interest too often wanting in larger historical works. The information conveyed is thoroughly sound ; and the utility of the book is much increased by the addition of examination questions at the end ot each chapter. Whether regarded as an interesting reading-book or as an instructive class-book, this history deserves to rank high. When we add, that it appears in the form of a neat little volume at the moderate price of eighteeupence no further recommendation will be necessary." HISTORY OF GREAT BRITAIN AND IRELAND; with an Account of the Present State and Resources of the United Kingdom and its Colonies. With Questions for Examination, and a Map. By Dr WHITE. 3s. Athenaeum. "A carefully compiled history for the use of schools. The writer has consulted the more recent authorities : his opinions are liberal, and on the whole just and inlpartial : the succession of events is developed with clearness, and with more of that picturesque effect which so delights the young than is common in historical abstracts. The book is accom- panied by a good map. For schools, parish and prison libraries, workmen's halls, and such institutions, it is better adapted than any abridgment of tJie kind we know." HISTORY OF SCOTLAND FOR JUNIOR CLASSES; With Questions for Examination. Edited by Dr WHITE. Is. 6d. HISTORY OF SCOTLAND, from the Earliest Period to the Present Time. With Questions for Examination. Edited by Dr WHITE. 3s. 6d. HISTORY OF FRANCE ; with Questions for Examination, and a Map. Edited by Dr WHITE. 3s. 6d. Athenaeum. "We have already had occasion to speak favourably of Dr White's 'History of Great Britain and Ireland.' The perusal of the present work has t^'ven us still greater pleasure. . . . Dr White is remarkably happy in combining convenient brevity with sufficiency of information, clearness of exposition, and interest of detail. He shows great judgment in apportioning to each subject its due amount of consideration." OUTLINES OF UNIVERSAL HISTORY. Edited by Dr WHITE. 2s. HISTORY. 13 DR WHITE'S ELEMENTS OF UNIVERSAL HISTORY, On a New and Systematic Plan. In THREE PARTS. Part I. Ancient History; Part II. History of the Middle Ages; Part III. Modern History. With a Map of the World. 7s. ; or in Parts, 2s. 6d. each. This work contains numerous synoptical and other tables, to guide the researches of the student, with sketches of literature, antiquities, and manners during each of the great chronological epochs. OUTLINES OF THE HISTORY OF ROME ; with Ques- tions for Examination. Edited by Dr WHITE. Is. 6d. London Bevlew. "This abridgment is admirably adapted for the use of schools, the best book that a teacher could place in the hand of a youthful student." SACRED HISTORY, from the Creation of the World to the Destruction of Jerusalem. With Questions for Examination. Edited by Dr WHITE. Is. 6d. ELEMENTS OF GENERAL HISTORY, Ancient and Modern. To which are added, a Comparative View of Ancient and Modern Geography, and a Table of Chronology. By ALEX. ERASER TYTLER, Lord Woodhouselee, formerly Professor of History in the University of Edinburgh. New Edition, with the History continued. With two large Maps, etc. 3s. 6d. WATTS' CATECHISM OF SCRIPTURE HISTORY, and of the Condition of the Jews from the Close of the Old Testament to the Time of Christ. With INTRODUCTION by W. K. TWEEDIE, D.D. 2s. SIMPSON'S HISTORY OF SCOTLAND; with an Outline of the British Constitution, aud Questions for Examination at the end oi each Section. 3s. 6d. SIMPSON'S GOLDSMITH'S HISTORY OF ENGLAND; With the Narrative brought down to the Middle of the Nineteenth Century. To which is added an Outline of the British Constitution. With Questions for Examination at the end of each Section. 3s. 6d. SIMPSON'S GOLDSMITH'S HISTORY OF GREECE. With Questions for Examination at the end of each Section. 3s. 6d. SIMPSON'S GOLDSMITH'S HISTORY OF ROME. With Questions for Examination at the end of each Section. 3s. 6d. 14 WRITING. ARITHMETIC. AND BOOK-KEEPING. , AKITHMETIO, AND BOOK-KEEPING. THIS section will be found to contain works in extensive use in many of the best schools in the United Kingdom. The successive editions have been carefully revised and amended. ARITHMETIC ADAPTED TO THE NEW CODE, in Three Parts. By ALEXANDER TROTTER, Teacher of Mathematics, etc., Edinburgh. Parts I. and IL, embracing th* first four Standards, are now Beady. Each containing 361 pages, 2d., stiff wrapper. Book of Answers sold separately. Part III. in Preparation. PRACTICAL ARITHMETIC FOR JUNIOR CLASSES. By HENRY G. C. SMITH, Teacher of Arithmetic and Mathematics in George Heriot's Hospital. 64 pages, 6d. stiff wrapper. Answers, 6d. From the Eev. PHILIP KELLAND, A.M., F.R.SS. L. & E., late Fellow oj Queens' College, Cambridge, Professor of Mathematics in the University of Edinburgh. " I am glad to learn that Mr Smith's Manual for Junior Classes, the MS. of which I have examined, is nearly ready for publication. Trusting that the Illustrative Processes which he has exhibited may prove as efficient in other hands as they have proved in his own, I have great pleasure in recommending the work, being satisfied that a better Arithmetician and a more judicious Teacher than Mr Smith is not to be found." PRACTICAL ARITHMETIC FOR SENIOR CLASSES; Being a Continuation of the above. By HENRY G. C. SMITH. 2s. Answers, 6d. KEY, 2s. 6d. %* The Exercises in both works, which are copious and original, have been constructed so as to combine interest with utility. They are accompanied by illustrative processes. LESSONS IN ARITHMETIC FOR JUNIOR CLASSES. By JAMES TROTTER. 66pages,6d. stiff wrapper; or 8d. cloth. Answers, 6d. This book was carefully revised, and enlarged by the introduction of Simple Examples of the various rules, worked out at length and fully explained, and of Practical Exercises, by the Author's son, Mr Alexander Trotter, Teacher of Mathematics, etc., Edinburgh; and to the present edition Exercises on the proposed Decimal Coinage have been added. LESSONS IN ARITHMETIC FOE ADVANCED CLASSES; Being a Continuation of the Lessons in Arithmetic for Junior Classes. Containing Vulgar and Decimal Fractions; Simple and Compound Proportion, with their Applications; Simple and Compound Interest; Involution and Evolution, etc. By ALEXANDER TROTTER. New Edition, with Exercises on the proposed Decimal Coinage. 76 pages, 6d. in stiff wrapper ; or 8d. cloth Answers, 6d. Each subject is also accompanied by an example fully worked out and minutely explained. The Exercises are numerous and practical. WRITING, ARITHMETIC, AND BOOK-KEEPING. 15 A COMPLETE SYSTEM OF ARITHMETIC, Theoretical and Practical; containing the Fundamental Rules, and their Application to Mercantile Computations ; Vulgar and Decimal Fractions; Involution and Evolution; Series; Annuities, Certain and Contingent. By Mr TROTTEB. 3s. KEY, 4s. 6d. %* All the 3400 Exercises in this work are new. They are applicable to the business of real life, and are framed in such a way as to lead the pupil to reason on the matter. There are upwards of 200 Examples wrought out at length and minutely explained. INGRAM'S PRINCIPLES OF ARITHMETIC, and their Application to Business explained in a Popular Manner, and clearly Illustrated by Simple Rules and Numerous Examples. Remodelled and greatly Enlarged, with Exercises on the proposed Decimal Coinage. By ALEXANDER TROTTER, Teacher of Mathematics, etc., Edinburgh. Is. KEY, 2s. 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Exemplified in one set of Books. Is. 6d. A Set of Ruled Writing Books, expressly adapted for this work, Is. 6d. SCOTT'S FIRST LESSONS IN ARITHMETIC. 6d. stiff wrapper. Answers, 6d. SCOTT'S MENTAL CALCULATION TEXT -BOOK. Pupil's Copy, 6d. Teacher's Copy, 6d. COPY BOOKS, in a Progressive Series, By R. SCOTT, late Writing-Master, Edinburgh. Each containing 24 pages. Price : Medium Paper, 3d. ; Post Paper, 4