UC-NRlf 
 
 ■111 
 
 If 
 
 I 
 
%-b 
 
 &<? 
 
 ^ 
 
 IN MEMORIAM 
 FLOR1AN CAJORI 
 
* & *A 
 
 ..; * • \ 
 
CURRENT MONEY. 
 4 Farthings make ... 1 Penny 
 
 IS Pence 1 Shilling 
 
 20 Shillings, 1 Pound, or Sovereign 
 
 ARITHMETICAL TABLES. 
 
 NUMERATION. 
 
 Units 1 
 
 Tens 12 
 
 Hundreds 123 
 
 Thousands 1,234 
 
 Tens of Thousands ... 12,315 
 
 C. of Thousands 12 3,456 
 
 Millions 1 ; 2 5 4 , 5 6 7 
 
 X. of Millions... 1 2 ; 3 4 5 , 6 7 8 
 C. of Millions 125;456,789 
 
 PENCE. 
 d. 
 
 d. s. 
 
 20 are 1 
 24 ... 2 
 
 30 . 
 56 . 
 40 . 
 48 , 
 50 , 
 60 , 
 70 , 
 72 , 
 80 , 
 84 
 90 
 96 
 
 100 
 
 108 
 
 120 
 
 4 2 
 
 5 
 
 5 10 
 
 6 
 
 30 . 
 
 40 . 
 
 50 . 
 
 60 . 
 
 70 . 
 
 80 . 
 
 90 . 
 100 . 
 110 . 
 120 . 
 150 . 
 140 . 
 190 . 
 160 . 
 170 . 
 180 . 
 
 £. 3. 
 
 el 
 . 1 10 
 . 2 
 . 2 10 
 . 3 
 . 3 10 
 . 4 
 . 4 10 
 . 5 
 . 5 10 
 , 6 
 . 6 10 
 , 7 
 ,. 7 10 
 ,. 8 
 .. 8 10 
 ..9 
 
 40 poles I furlong 
 
 LONG MEASURE 
 3 barley-corns 1 inch 
 12 inches 1 foot 
 
 3 feet 
 6 feet 
 5* yds. 
 
 . 1 yard 
 .... 1 fathom 
 .... 1 pole 
 
 8 fur. 
 3 miles 
 69^ miles 
 
 MULTIPLICATION. 
 
 1 |. 2 | 3 | 4 | 5 | 6 | 7 | 8 I 9 \ 1 | 11 \ V . 
 
 2 | 4 | 6 | 8 | 10 1 12 | 14 [ 16 | 18 | 20 | 22 | 21 
 
 ,3 I 5| 9 ! '2 | 15 1 18 1 21 | 24 | 27 1 30 | 55 | 
 
 4 | 8 | 12 | 16 | 20 | 24 | 28 J 32 \^M 1 40 | 44 | 4 8 
 
 _ ^ P3~j 20} 25 | 50 I 31 | 40 | 
 6~| 12 ~j 18 j 24 1 50 1 36 \ 42 | 48 | 
 ZJ y I *L! 2 8 | 35 1 42 I 49 j 56 I 
 
 8 1 1 16 
 
 j!5J_50 |_55J_60 
 54 | 60~| 66~| 72 
 
 63 | 70 | 77| 84 
 
 | 4 j 48 | 56 | 6 4 j 72 { 80 j 88 | 96 
 
 9 "| 18 | 27 | 36 | 45 | 54 j 63 \ T> \ %\ \ 90 | 99 j 108 
 | 20 | 50~i"~40 T50 j 60 | 70 [ 89 ["^90 [ 100 | 110 j 120 
 
 1 1 | 22 | 33 | 44 1 55 | 66 | 77 | 88 [ 9.9 | 110 | 121 | 152 
 
 12 f 24 1 36 1 48 | 60 [ 72 | 84 | 96 | 108 | 120 | 152 | 144 
 
 . 1 mile 
 
 ... 1 league 
 ... 1 degree 
 
 LAND MEASURE. 
 
 9 feet make 1 yard 
 
 304: yds 1 pole 
 
 40 poles 1 rood 
 
 4 roods 1 acre 
 
 CLOTH MEASURE. 
 
 2^ inch, make 1 nail 
 1 quar. 
 1 Ft. ell 
 1 yard 
 1 En. ell 
 1 Fr. ell 
 
 4 nails 
 
 3 qrs. 
 
 4 qrs. 
 
 5 qrs. 
 
 6 qrs. 
 
 Note, this 
 table may 
 >e applied 
 todivision 
 by revers- 
 ing it: as 
 the2'sin4 
 are 2: the 
 2'sin6are 
 &c. 
 
 60 sec. 
 
 60 min. 
 
 24 hours 
 7 days 
 4 weeks 
 
 TIME. 
 
 ... 1 minute 
 1 hour 
 1 day 
 1 week 
 1 month 
 
 DRY MEASURE. 
 2 gall, make 1 peek 
 
 4 pecks 
 4 bushels , 
 8 bushels ., 
 4 quarters., 
 10 quarters ., 
 
 £ableg trf WtM)t* *ii* fHeaStire*. 
 
 OF A POUND, 
 OR SOVEREIGN. 
 
 s. d. £. 
 10 Oarel half 
 
 1 th:rd 
 
 5 ... 1 fourth 
 
 4 ... 1 fifth 
 
 5 4 ... 1 sixth 
 2 6 ... 1 eighth 
 2 ... 1 tenth 
 
 1 8 ... 1 twelfth 
 
 1 4 ... 1 fifteenth 
 
 1 5 ... 1 sixteenth 
 
 1 is 1 twentieth 
 
 OF A TON. 
 
 Cwt. T. 
 
 .10 are 1 half 
 
 | 5 ... 1 fourth 
 
 4 ... 1 fifth 
 
 2£... 1 eighth 
 
 2 ... 1 tenth 
 
 1 is 1 twentieth 
 
 OF A SHILLING. 
 d. *. 
 
 6 are 1 half 
 
 4 1 third 
 
 3 1 fourth 
 
 2 1 sixth 
 
 14 is 1 eighth 
 
 1 1 twelfth 
 
 OF A PENNY. 
 farth. d. 
 
 2 are 1 half 
 
 1 is 1 fourth 
 
 APOTHECARIES'. 
 
 20 gr. make 1 scruple 
 
 3 scr 1 dram 
 
 8 dr 1 ounce 
 
 12 oz 1 pound 
 
 OF A HUNDRED. 
 
 qr . lb. Cwt. 
 
 2 are 1 half 
 
 is 1 fourth 
 16 are 1 seventh 
 14 ... 1 eighth 
 
 TROY. 
 
 24 gr. make 1 dwt. 
 20 dwt. ... 1 ounce 
 12 oz. ... 1 pou: 
 
 AVOIRDUPOIS. 
 
 16 dr. make lost. 
 
 16 cm 1 ib- 
 
 14 lb I stone 
 
 28'b 1 quarter 
 
 4 qr 1 CWt 
 
 WOOL. 
 7 lb. make 1 clove 
 
 2 cloves 1 stone 
 
 2 stones 1 tod 
 
 6h tods 1 wey 
 
 2 weys 1 sack 
 
 12 sacks 1 last 
 
 ALE AND BEER. 
 pints make 1 quart 
 
 quarts 
 
 9 gallons ... 
 
 2 firkins ... 
 
 2 kilderkins 
 
 1 h barrel ... 
 
 rbarrels ... 
 
 3 barrels ... 
 
 1 gallon 
 1 firkin 
 1 kilder. 
 1 barrel 
 1 hhd. 
 1 punch. 
 1 butt. 
 
 VV 1 \ E, 
 2 pints make 1 quart 
 4 quarts 
 
 10 gallons 
 42 gallons 
 63 gallons 
 2 hhd*. 
 
 ■n 
 
 . i i 
 
 1 gallon 
 1 anker 
 1 tierce 
 1 hhd. 
 1 pipe 
 1 tun 
 
 1 bushel 
 1 sack I 
 1 quarter 
 1 chaldron 
 I last 
 
 SOLID MEASURE. 
 1728 inches 1 solid foot 
 27 feet ... 1 yard 
 
 COAL MEASURE. 
 3 bushels ... 1 sack 
 56 bushels ... 1 chaldron 
 
 CUSTOMARY 
 WEIGHT OF GOODS. 
 
 lb. 
 A firkin of butter is 56 
 
 A firkin of soap 64 
 
 A barrel of pot ashes 200 
 A barrel of anchovies 50 
 
 A barrel of soap 256 
 
 A barrel of butter ... 224 
 A fother of lead, 19 
 
 cwt. 2 qrs. or 2184 
 
 A barrel of candles... 120 
 A stone of iron or 
 
 shot 1 
 
 A gallon of train oil 7£ 
 
 A fagot of steel 120 
 
 A stone of glass 5 
 
 A seam of . glass 24 
 
 stone, or 120 
 
 A roll of parchment, 
 
 5 dozen skins 
 A barrel of figs from 
 
 nearly 96 to 360 
 
THE 
 
 TUTORS ASSISTANT; 
 
 BEING A. COMPENDIUM OF 
 
 PRACTICAL ARITHMETIC, 
 
 FOR THE USE OF 
 
 SCHOOLS, OR PRIVATE STUDENTS: 
 
 CONTAINING, 
 
 I. Arithmetic in Whole Numbers ; the Rules 
 of which are expressed in a clear, con- 
 cise, and intelligible manner ; and the 
 operations illustrated by examples work- 
 ed at length, and by numerous explan- 
 atory notes and observations ; with an 
 ample variety of Examples for the exer- 
 cise of Learners, calculated to initiate 
 them in the Knowledge of Real Business. 
 Also the New Commercial Tables, adap- 
 ted to the present Legislative regulations 
 of Weights and Measures, and the modern 
 practice of Trade. 
 
 II. Vulgar Fractions; explained in an 
 easy and familiar manner ; in the prac- 
 tice of which the most elegant and ab- 
 breviated modes of operation are pecu- 
 liarly inculcated. 
 
 III. Decimal Fractions: elucidated wit! 
 the utmost perspicuity; with Involu. 
 tion, Evolution, Position, Progression, 
 and the calculation of Interest and An- 
 nuities, on an extended scale. 
 
 IV. Duodecimals; or the Multiplication ol 
 Feet and Inches; with numerous ex- 
 amples for practice, adapted to the va- 
 rious business of Artificers. 
 
 V. Mensuration of Superficies; preceded 
 by plain and concise Geometrical Defini- 
 tions. 
 
 VI. A Collection of Questions, promiscu- 
 ously arranged; intended as recapitula- 
 tory exercises in the principal Rules of 
 Arithmetic. 
 
 VII. A Compendious System of Book-keep- 
 ing. 
 
 By FRANCIS WALKJNGAME, 
 
 REVISED, CORRECTED, AND ENLARGED 
 BY THE ADDITION OF SUPERFICIAL MENSURATION 
 
 AND A COMPENDIUM OP 
 
 BOOK-KEEPING BY SINGLE ENTRY; 
 By WILLIAM BIRKIN, ' , 
 
 Editor of the Improved Edition of Jones's Dictionary; 
 Author of the Rational English Expositor, <frc 
 
 PUBLISHED BY 
 
 JOHN AND CHARLES MOZLEY, 
 DERBY; 
 
 AND PATERNOSTER ROW, LONDUN 
 
Also published, price 3s. 
 
 A KEY 
 
 TO BIRKIN'S IMPROVED EDITION 
 OF WALKINGAME'S TUTOR'S ASSISTANT ; 
 
 Containing Answers to all the Examples in that popular work : with the full solutions, 
 '.n ail cases where they can be necessary or useful, exhibited in a plain, concise, and t 
 scientific manner ; 
 
 BY W. BIRKIN, 
 
 Editor of the Arithmetic, 
 
 Price 4d. ; or when bound with the Arithmetic or the Key, 3d. additional, 
 
 EXAMINING QUESTIONS IN ARITHMETIC, 
 
 ADAPTED TO BIRKIN*S IMPROVED EDITION 
 
 OF WALKINGAME'S TUTOR'S ASSISTANT, 
 
 ACCOMPANIED BY EXPLANATORY NOTES AND OBSERVATIONS'. 
 
 BY W. BIRKIN, 
 Editor of the Arithmetic and Author of the Key, 
 
Nil 
 THE EDITOR'S PREFACE. \S 
 
 The immensecirciilatioii of Walkin game's Tutor's Assistant. 
 even inTtiToriginal form, is sufficiently evinced by the very extensive 
 and uniformly increasing demand which the Proprietors of the present 
 Edition have for many years experienced. 
 
 To advance the utility of a work held in such high estimation 
 among Conductors of Schools ; by simplifying the Rules, correcting 
 and modernizing the antiquated phraseology, supplying deficiencies 
 where there was a paucity of Examples, and incorporating with its 
 original matter such emendations and additions as appear to be called 
 for by the present improved state of Arithmetical Science ; will, it 
 is presumed, be rendering an acceptable service to the Public. 
 
 Amongst the various improvements introduced in this Edition, 
 may be enumerated, a more intelligible elucidation of the system of 
 Notation ; of Direct, Inverse, and Compound Proportion, Practice, 
 Interest, Progression, &c. ; more perspicuous illustrations of the 
 theory and practice of Vulgar and Decimal Fractions, Evolution, 
 Duodecimals, &c. ; the substitution of the new Arithmetical and 
 Commercial Tables; the insertion of many additional Examples 
 (particularly in the elementary Rules) adapted to exercise and im- 
 prove the judgment of the Learner ; also of Rules for the particu- 
 lar cases in Profit and Loss, of Involution, of Theorems for the solu- 
 tion of all the possible cases in Arithmetical and Geometrical Pro- 
 gression, Superficial Mensuration, a number of useful Supplemen- 
 tal Questions, and a Compendium of Book- Keeping. 
 
 Such are the attempts that have been made to enhance the real 
 worth of this popular Treatise on Arithmetic. How far the intention 
 may have been judiciously or successfully executed, must be left to a 
 candid Public to determine. 
 
 A Vith a confident reliance, however, on the favourable consideration 
 of those whose judgment and experience most essentially qualify 
 them to discriminate between realities and specious pretensions to 
 improvement, and duly to appreciate the difficulties of such an 
 undertaking ; the work is respectfully submitted to a trial before the 
 tribunal of its legitimate judges, with an anxious and hopeful 
 anticipation of obtaining their verdict of approval. 
 
 Derby, February 7, 1827. 
 
 Advertisement to the Sixth Edition. 
 When the Editor first undertook the task of modernizing and im- 
 proving this work, with an anxious desire of rendering it more com- 
 mensurate with the progress of scientific information, and better 
 adapted to the present improved systems of instruction, thereby to 
 facilitate the arduous labours of the Teacher in the communication, 
 
 M306024 
 
ADVERTISEMENT. 
 
 IV 
 
 md of the Pupil in the acquisition of a branch of knowledge so use- 
 ul and indispensable ; he was limited materially in the execution 
 f his views by the necessity of conforming the whole of the ar- 
 angement as well as a considerable part of the methods of operation 
 that were susceptible of improvement, to those of Falconar's 
 Key to the original work. The same necessity compelled him to 
 retain every example contained in the old editions, however obso- 
 lete, useless, or absurd the subject might have become, from the in- 
 novations which time invariably introduces in the management of 
 commercial transactions. Hence the student was subjected to the 
 double inconvenience of referring, for some important parts of the 
 additional matter, to the Addenda in the Arithmetic, and to an Ap- 
 pendix to the Key. This impediment to improvement is now remo- 
 ved : for having constructed a new Key with correspondent im- 
 provements, he has been enabled to complete his intentions by the 
 regular incorporation of the newly introduced matter, by an ar- 
 rangement more rational and more consistent with the modern prac- 
 tice of instruction, by the expunction of some useless subjects, and 
 the substitution of others of more real utility. 
 
 Besides the emendations and additions enumerated in the prece- 
 ding Preface, the worked examples under each Rule are now exhibit- 
 ed in a more convenient form, considerably amplified, and illustra- 
 ted by copious explanatory notes ; — the theory and practice of Cir- 
 culating Decimals are comprehensively and clearly explained; — - 
 considerable improvements have been effected in that part which 
 treats of the Doctrine of Compound Interest and Annuities, the 
 Tables of which have been greatly extended ; and in order to cor- 
 rect the errors that abound in the Tables of this description con- 
 tained in several scientific publications which have been collated, 
 the accuracy of these has been verified by actual calculation. Some 
 useful additions have also been made in the Mensuration and Book- 
 keeping. 
 
 Three years have now elapsed since the Improved Tutor's As- 
 sistant was first issued from the press. That it has been greatly 
 approved, has been manifested by the most unequivocal and satis- 
 factory proof, the sale of several large impressions. Under such 
 auspicious symptoms of encouragement, therefore, the Editor feels 
 an increased confidence in the prospect of obtaining the sanction of 
 an enlightened public. 
 
 Derby, February I, 1830. 
 
a 
 
 THE AUTHORS PREFACE. 
 
 The public will, no doubt, be surprised to find there is another at- 
 tempt made to publish a book of ARITHMETIC, when there are 
 such numbers already extant on the same subject, and several of 
 them that have so lately made their appearance in the world ; but I 
 flatter myself, that the following reasons which induced me to com- 
 pile it, the method, and the conciseness of the Rides, which are laid 
 down in so plain and familiar a manner, will have some weight 
 towards its having a favourable reception. 
 
 Having some time ago drawn up a set of Rides and proper Ques- 
 tions, with their Answers annexed, for the use of my own school, 
 and divided them into several books, as well for more ease to my- 
 self, as the readier improvement of my scholars, I found them, by 
 experience, of infinite use ; for when a master takes upon him that 
 laborious (though unnecessary) method of writing out the Rules 
 and Questions in the children's books, he must either be toiling and • 
 slaving himself after the fatigue of the school is over, to get ready 
 the books for the next day, or else must lose that time which would // / 
 be much better spent in instructing and opening the minds of his 
 pupils. There was, however, still an inconvenience which prevent- 
 ed them from giving me the satisfaction I at first expected ; L e. 
 where there are several boys in a class, some one or other must wait 
 till the boy who first has the book finishes the writing of those rules 
 or questions he wants, which detains the others from making that 
 progress they otherwise might, had they a proper book of Rules and 
 Examples for each boy ; to remedy which, I was prompted to com- 
 pile one, in order to have it printed, which might not only be of use 
 in my own school, but in other schools, where the instructers wish 
 their scholars to make a quick progress. It will also be of great use 
 to such persons as have acquired some knowledge of numbers at 
 school, to make them the more perfect ; likewise to such as have 
 completed themselves therein, it will prove, after an impartial peru- 
 sal, on account of its great variety and brevity, a most agreeable and 
 entertaining Exercise Book. I shall not presume to say any thing 
 more in favour of this work, but beg leave to refer the unprejudiced 
 reader to the remark of a certain Author,* concerning compositions 
 of this nature. His words are as follow : 
 
 " And now, after all, it is possible that some who like best to tread 
 the old beaten path, and to toil at their business, when they may do 
 it with pleasure, may start an objection against the use of this well- 
 intended Assistant, because the course of Arithmetic is always the 
 same ; and therefore say, that some boys, lazily inclined, when they 
 see another at work upon the same question, will be apt to make his 
 operation pass for their own. But these little forgeries are soon de- 
 tected by the diligence of the Tutor : therefore, as different questions 
 to different boys do not in the least promote their improvement, so 
 neither do the same questions impede it. Neither is it in the power 
 of any master (in the course of his business) now full of spirits so- 
 
PREFACE 
 
 ever he may be, to frame new examples at pleasure in any Rule ; 
 but the same question will frequently occur in the same Rule, not- 
 withstanding his greatest care and skill to the contrary. 
 
 iC It may also be farther objected, that to teach by a printed book 
 is an argument of ignorance and incapacity ; which is no less trifling 
 than the former. He, indeed (if such a one there be) who is afraid 
 his scholars will improve too fast, will, undoubtedly, decry this 
 method : but that master's ignorance can never be brought in ques- 
 tion, who can begin and end it readily ; and, most certainly, that 
 scholar's non-improvement can be as little questioned, who makes a 
 much greater progress by this, than by the common method." 
 
 To enter into a long detail of every Rule would tire the reader, 
 and swell the preface to an unusual length ; 1 shall, therefore, only 
 give a general idea of the method of proceeding, and leave the rest 
 to speak for itself; which, I hope, the reader will find to answer the 
 title, and the recommendation given it. As to the Rules, they fol- 
 low in the same manner as the table of contents specifies, and in much 
 the same order as they are generally taught in schools. I have gone 
 through the four fundamental Rules in Integers first, before those of 
 several denominations : in order that they being well understood, 
 the latter will be performed with much more ease and despatch, ac- 
 cording to the rules shown, than by the customary method of dotting. 
 In Multiplication I have shown both the beauty and use of that ex- 
 cellent Rule, in resolving most Questions that occur in merchandi- 
 sing ; and have prefixed to Reduction several Bills of Parcels, which 
 are applicable to real business. In working Interest by Decimals, J 
 have added tables to the Rules, for the more ready calculating of An- 
 nuities, &c. and have not only shown the use, but the method of 
 making them : likewise a Table calculated for finding the Interest 
 of Money for any number of days, at any rate per cent, by Multipli- 
 cation and Addition only ; which may also be applied to the calcu- 
 lation of Incomes, Salaries, or Wages, for any number of days ; and 
 I may venture to say, I have gone through the whole with so much 
 plainness and perspicuity, that there is none better extant. 
 
 1 have nothing farther to add, but a return of my sincere thanks, 
 to all those gentlemen, schoolmasters, and others, whose kind appro- 
 bation and encouragement have now established the use of this book 
 in almost every school of eminence throughout the kingdom : but 1 
 think my gratitude more especially due to those who have favoured 
 me with their remarks ; though I must still beg of every candid and 
 judicious reader, that if he should, by chance, find a transposition of a 
 letter, or a false figure, to excuse it ; for, notwithstanding there has 
 been great care taken in correcting, yet errors of the press will inev- 
 itably creep in ; and some may also have slipped my observation : 
 In either of which cases, the admonition of a good-natured reader 
 vill be very acceptable to his 
 
 much obliged, 
 
 and most obedient humble servant, 
 
 F. WALKINGAME. 
 
CONTENTS. 
 
 Page 
 
 Numera tion and Notation 1 1 
 
 INTEGERS. 
 
 Simple Addition .... 14 
 
 Subtraction ... 16 
 
 Multiplication . . 18 
 
 Division .... 21 
 
 Arithmetical end Commercial 
 
 Tables 24 
 
 Definition 31 
 
 Reduction 31 
 
 Compound Addition . . • 35 
 
 Subtraction . . 39 
 
 Multiplication . 42 
 
 i 1 Division ... 46 
 
 Promiscuous Examples . . 47 
 
 Rule of Three Direct . . 51 
 
 . Inverse • . 54 
 
 Double Rule of Three . . 55 
 
 Practice 57 
 
 rare and Tret .... 63 
 
 Invoices, c]'c 65 
 
 Simple Interest j Commission, 
 
 Brokerage, Purchasing" 
 
 Stock, %c 68 
 
 Discount 71 
 
 Compound Interest ... 73 
 
 Equation of Payments . . 74 
 
 Barter .' 75 
 
 Profit and Loss .... 76 
 
 Fellowship 78 
 
 Alligation 80 
 
 Comparison of Weights and 
 
 Measures 83 
 
 Permutation 84 
 
 ^K^^VUhGAR FRACTIONS. 
 
 Definitions, $$c 84 
 
 Reduction 86 
 
 Addition 93 
 
 Subtraction ..... 93 
 
 Multiplication . . . . 94 
 
 Division 94 
 
 Rule of Three . ... 95 
 
 Double Rule of Three . . 96 
 
 DECIMAL FRACTIONS. 
 
 Definitions, fyc 96 
 
 A 
 
 Addition 
 
 Page 
 . 98 
 
 Subtraction .... 
 
 . 98 
 
 Multiplication 
 
 Division 
 
 . 98 
 . 99 
 
 Circulating Decimals 
 
 . 101 
 
 Reduction 
 
 . 103 
 
 Tables 
 
 . 105 
 
 Rule of Three .... 
 
 . 108 
 
 Exchange 
 
 Conjoined Proportion 
 Involution 
 
 . 109 
 . 112 
 . 114 
 
 Evolution 
 
 . 114 
 
 Biquadrate Root . . . 
 Roots of all Powers . 
 
 . 121 
 . 121 
 
 Single Position . • . 
 
 . 122 
 
 Double Position . . . 
 
 . 123 
 
 Progression Arithmetical 
 
 . 125 
 
 Geometrical 
 
 . 128 
 
 Simple Interest and Annui- 
 ties 132 
 
 Discount 140 
 
 Equation of Payments . .142 
 Compound Interest and An- 
 nuities 143 
 
 Discount 156 
 
 Duodecimals 159 
 
 superficial mensuration. 
 Geometrical Definitions . 164 
 Mensuration of a Parallelo- 
 gram, Ss'c 166 
 
 Triangle . . 167 
 
 Trapezium, S$c. 167 
 
 an Irregular 
 
 Figure 168 
 «— — — a Regular 
 
 Polygon 168 
 
 Circle , . 
 
 ,Area of . . . 
 
 , Inscribed square 
 
 Circular Arc . . . 
 
 Sector . . . 
 
 Segments . . 
 
 Ellipse 
 
 Collection of Questions 
 Supplemental Questions 
 Book-Keeping . . 
 
 169 
 170 
 170 
 171 
 171 
 172 
 173 
 174 
 178 
 183 
 
EXPLANATION OF 
 
 ARITHMETICAL SIGNS OR CHARACTERS. 
 
 -f. is the sign of Addition ; and is called plus, or more. 
 •— is called minus, or less, and denotes Subtraction. 
 
 X into, and denotes Multiplication. 
 
 _j_ by, and denotes Division. 
 
 as is the sign of Equality. 
 
 : : are the signs of Proportion. 
 J is the Radical Sign, or Sign of Evolution. 
 OS denotes the difference between two quantities when it is 
 uncertain which is the greater. 
 
 rations. 
 f)-f3=z9 signifies 6 plus 3 equal y : that is, 6 added to 3 equal 9. 
 7—4=3 signifies f minus 4 equal 3 : that is, 4 taken from 7 
 
 leaves 3. 
 4x3—12 is read 4 into 3 equal 12 : that is, 4 multiplied by 3 
 
 equal 12. 
 12 -£-4=3 is read 12 by 4 equal 3. But Division is more con- 
 veniently expressed in the form of a Fraction: 
 thus, ^ = 3 ; twelve divided by four e^wa/ three. 
 As 2 : 4 : : 8 : 16 ; As 2 are to 4, so are 8 to 16. 
 
 N / 1 6 z=z 4 ; the Square Root of 1 6 apa/ 4. 
 3 N '64 rr 4 ; the Cube Root of 64 equal 4. 
 
 A vinculum connects two or more terms which are to be 
 considered as forming one term or quantity. It is signified 
 by a line drawn over them ; or by parentheses including them. 
 A point is often used instead of the cross to denote Multi- 
 plication ; and in Algebraical Theorems, &c., the multiplica- 
 tion of the quantities denoted by different letters, is under- 
 stood by placing the letters together without any sign be- 
 tween them. Thus 2 . 5+7, denotes that 5 and 7 are to be 
 added, and the sum (12) to be multiplied by 2. Also, {ab+c) . 
 (n — 1 ) denotes that a and b are to be multiplied, c added to 
 the product, and the whole to be multiplied by one less than 
 the number or quantity represented by n. 
 
 The common sizes of Books are marked 
 
 Folio, of which 2 leaves make a sheet, ft'o. 
 
 Quarto, 4 4to. 
 
 Octavo, 8 8vo. 
 
 Duodecimo, 12 12mo. 
 
 Octodecimo,. ... 18 18mo. 
 
THE 
 
 TUTOR'S ASSISTANT; 
 
 BEING A COMPENDIUM 
 
 OF PRACTICAL ARITHMETIC. 
 
 Integers, or Whole Numbers. 
 
 Arithmetic is the science of numbers ; or the art of numeri- 
 cal computation. A whole number is a unit, or a collection of 
 units. 
 
 Numbers are expressed by ten written characters called 
 rigures, or digits : viz. 1 , 2, 3, 4, 5, 6, 7> 8, 9, which are 
 significant figures, all declaring their own values by the names ; 
 and the cipher, or nought (0) an insignificant figure, indicating 
 no value when it stands alone. 
 
 NUMERATION AND NOTATION. 
 
 A figure standing alone, or the first on the right of others, 
 denotes only its simple value, as so many units, or ones ; the 
 second is so many tens ; the third, so many hundreds, fyc. in- 
 creasing continually towards the left in a tenfold proportion. 
 
 Numeration is the art of reading numbers expressed in 
 figures ; and Notation, the art of expressing numbers by figures. 
 
 THE TABLE. 
 
 T3 *» 
 
 CCS CflO £ G •£ 
 
 P3 ^^^ fflr-^ &£*£■< ffiH£ 
 
 3 ; 5 7 8 , 6 9 5 ; 2 7 3 , 8 4 1 
 
 < ; 1 v ^ 1 
 
 Millions' period. Units' period. 
 This Table might be infinitely extended. 
 
12 NUMERATION. [THE TUTOR'S 
 
 Note. To read any Number. Divide it into periods of six figures 
 
 I J each, beginning at the right hand ; and each period into semi-periods 
 with a different mark,, for the sake of distinction. The first on the 
 ' right hand is the Units' period, the second the Millions' period, &c. 
 Beginning at the left, observe that the three figures of every com- 
 plete semi-period must be reckoned as so many hundreds* tens* and units ; 
 joining the word thousands when you come to the middle of the period, 
 and the proper name of the period at the end of it. 
 
 2. To express any given Number in Figures. Begin at the left, and 
 write the figures which denote (as so many hundreds, tens, and units) 
 the number in that semi-period ; and proceed thus with each succes- 
 sive semi-period, till the whole is completed ; placing a separating 
 comma in the middle of each period, or immediately after the thou- 
 sands, and a semicolon between the periods. But observe, that though 
 every semi-period but the first on the left must have its complete 
 number of three figures, that may be incomplete, and consist of only 
 one or two figures : also, where significant figures are not required in any 
 part of a number, no semi-period must be omitted, but the places 
 must be filled up with ciphers. 
 
 Example. Write in figures, seventy thousand four hundred bil- 
 lions, two hundred and ten thousand millions, and ninety-six. 
 
 First, write 70 (seventy) with a comma, these being thousands; 
 *hen 400 (four hundred) with a semicolon, denoting the end of the 
 period ; next, write 210 (two hundred and ten) and, because they are 
 thousands, put a comma after them, and then 000 (three ciphers, ther? 
 being no more millions) followed by a semicolon, to denote the com- 
 pletion of the period ; again, put 000 (three more ciphers, denoting 
 the absence of thousands) with a comma after them, and then 096, 
 (ninetv-six) which will complete the number : thus, 70,400 ; 210,000; 
 000,096. 
 
 EXERCISES IN NUMERATION AND NOTATION. 
 
 Ready or write in wards the following numbers. 
 
 •(1) 3 
 
 (13) 721 
 
 (25) 500050005 
 
 (2) 30 
 
 (14) 906 
 
 (26) 1010100 
 
 (3) 33 
 
 (15) 4294 
 
 (27) 11110101 
 
 (4) 300 
 
 (16) 94^94 
 
 (28) 499994949 
 
 (6) 303 
 
 (17) 294294 
 
 (29) 3584600987 
 
 (6) 330 
 
 (18) 3703 
 
 (30) 584610070840 
 
 (7) 333 
 
 (1.9) 703703 
 
 (31) 5846100708400 
 
 (8) 127 
 
 (20) 311311 
 
 (32) 37613590200116 
 
 (9) 172 
 
 (21) 113U3 
 
 (33) 5008000400000 
 
 (10) 217 
 
 (22) 131131131 
 
 (34) 601008000180070 
 
 (11) 271 
 
 (23) 708807780 
 
 (35) 37000000000075048 
 
 (12) 712 
 
 (24) 807078087 
 
 
 • The figures in parentheses refer to the Editor's Key to tktf 
 vork. See Advertisement on the first page. 
 
ASSISTANT.] 
 
 NUMERATION. 
 
 13 
 
 Express injigurcs the following numbers. 
 
 (1) Nine; ninety ; ninety-nine ; nine hundred ; nine hun- 
 dred and nine ; nine hundred and ninety ; nine hundred ana 
 ninety-nine. 
 
 (2) One hundred and eight; one hundred and eighty; 
 eight hundred and one ; eight hundred and ten ; one hun- 
 dred and sixteen ; one hundred and sixty- one ; six hundred 
 and eleven, 
 
 (3) One hundred and twenty-three ; one hundred and 
 thirty-two ; two hundred and thirteen ; two hundred and 
 thirty-one ; three hundred and twelve ; three hundred and 
 twenty-one. 
 
 (4) Two thousand five hundred and seventy-two. 
 
 (5) Seventy-two thousand five hundred and seventy-two. 
 
 (6) Five- hundred and seventy-two thousand five hundred 
 and seventy-two. ' 
 
 (7) Ten thousand nine hundred and ten. 
 
 (8) Nine hundred and ten thousand nine hundred and ten. 
 
 (9) One hundred and nine thousand nine hundred and one, 
 
 (10) One hundred and ninety thousand and ninety-one. 
 
 (11) Nine hundred and one thousand and nineteen. 
 
 (12) One hundred and fourteen millions, one hundred and 
 forty-one thousand four hundred and eleven. 
 
 (13) Four hundred and six millions, six hundred and four 
 thousand four hundred and sixty. 
 
 (14) Six hundred and forty m illion s, forty-six thou sand and 
 sixty-four. 
 
 (15) Seven millions, seventy thousand seven hundred. 
 
 (16) Seven hundred millions, seven thousand and seventy. 
 
 (17) Ten millions, one thousand one hundred. 
 
 (18) One hundred and one millions, eleven thousand one 
 hundred and ten. 
 
 (19) Twelve billions, seventeen thousand and nine mil- 
 lions, and eighty-nine. 
 
 (20) Seven thousand five hundred and four trillions, sixty 
 thousand millions, eight hundred thousand. 
 
 Roman Numerals. 
 
 I 
 
 1 
 
 One. 
 
 VI 
 
 6 
 
 Six. 
 
 XI 
 
 11 
 
 Eleven, 
 
 II 
 
 2 
 
 Two. 
 
 VII 
 
 7 
 
 Seven. 
 
 XII 
 
 12 
 
 Twelve. 
 
 III 
 
 3 
 
 Three. 
 
 VIII 
 
 8 
 
 Eight. 
 
 XIII 
 
 13 
 
 Thirteen. 
 
 IV 
 
 4 
 
 Four. 
 
 IX 
 
 9 
 
 Nine. 
 
 XIV 
 
 14 
 
 Fourteen. 
 
 V 
 
 5 
 
 Five. 
 
 X 
 
 10 
 
 T«» 
 
 XV 
 
 15 
 
 Fifteen. 
 
14 
 
 
 ADDITION OF INTEGERS. QtHE TUTOR'S 
 
 XVI 
 
 16 
 
 Sixteen . 
 
 CC 200 Two hundred. 
 
 XVII 
 
 17 
 
 Seventeen. 
 
 CCC 300 Three hundred. 
 
 XVIII 
 
 18 
 
 Eighteen. 
 
 CCCC 400 Four hundred. 
 
 XIX 
 
 19 
 
 Nineteen. 
 
 D 500 Five hundred. 
 
 XX 
 
 20 
 
 Twenty. 
 
 DC 600 Six hundred. 
 
 XXX 
 
 30 
 
 Thirty. 
 
 DCC 700 Seven hundred. 
 
 XL 
 
 40 
 
 Forty. 
 
 DCCC 800 Eight hundred. 
 
 L 
 
 50 
 
 Fifty. 
 
 DCCCC 900 Mine hundred. 
 
 LX 
 
 60 
 
 Sixty. 
 
 M 1000 One thousand. 
 
 LXX 
 
 70 
 
 Seventy. 
 
 MDCCCXXX 1830 One thou- 
 
 LXXX 
 
 80 
 
 Eighty. 
 
 sand eight hundred and 
 
 xc 
 
 90 
 
 Ninety. 
 
 thirty. 
 
 c 
 
 100 
 
 One hundred. 
 
 
 Note. A less numerical letter standing before a greater, must be 
 taken from it, as I before V or X, and X before L or C, &c. thus IV. 
 Four ; IX. Nine ; XL. Forty ; XC. Ninety, &c. And a less numer- 
 ical letter standing after a greater, is to be added to it, thus, VI. 
 Six ; XI. Eleven ; LX. Sixty ; CX. One Hundred and Ten. 
 
 All operations in Arithmetic are comprised under four 
 elementary or fundamental Rules : viz. Addition, Subtraction, 
 Multiplication, and Division. 
 
 ADDITION 
 Teaches to find the sum of several numbers. 
 
 Rule. Place the numbers one under another, so that units 
 may stand under units, tens under tens, &c. ; add the units, 
 set down the units in their sum, and carry the tens as so 
 many ones to the next row ; proceed thus to the last row, un- 
 der which set down the whole amount. 
 
 Proof. Begin at the top i 
 
 ind add the figures downwards: 
 
 if the sum is fc 
 right. 
 
 *(1)275 
 
 >und the same as before, it is presumed to be 
 
 (2) 1234 
 
 (3) 75245 
 
 (4)271048 
 
 110 
 
 7098 
 
 37502 
 
 325476 
 
 473 
 
 3314 
 
 91474 
 
 107584 
 
 354 
 
 6732 
 
 32145 
 
 625608 
 
 271 
 
 2546 
 
 47258 
 
 754087 
 
 352 
 
 6709 
 
 ■JL. ;- " 
 
 21476 
 
 279736 
 
 * Say 2 and 1 are 3, and 4 are 7, and 3 are 10, and 5 are 15, set 
 down 5 and carry 1 ; 1 and 5 are 6, and 7 are 13, and 5 are 18, and 7 
 are 25, and 1 are 26, and 7 are 33, set down 3 and carry 3 ; 3 and 3 
 are 6, and 2 are 8, and 3 are 11, and 4 are 15, and 1 are 16, and 2 are 
 18, set down 18: so the sum is 1835. 
 
 After practising a fcw examples, it will be better for the learner 
 
ArfSISTANT.J 
 
 ADDITION OF INTEGERS. 
 
 (5) 590046 
 
 (6) 
 
 3704 J 6 
 
 (7) 
 
 7S1943 
 
 73921 
 
 
 2890 
 
 
 56820 
 
 400080 
 
 
 60872 
 
 
 1693748 
 
 4987 
 
 
 998 
 
 
 300486 
 
 19874 
 
 
 47523 
 
 
 920437500 
 
 201486 
 
 
 9836 
 
 
 78632109 
 
 9882 
 
 
 26627 
 
 
 9408 175 
 
 15 
 
 (8) What is the sum of 43, 401, 9747, 3464, 2263, 314, 
 974? 
 
 (9) Add 246034, 298765, 47321, 58653, 64218, 5376, 
 9821, and 640 together. 
 
 (10) If A has £56. B £104. C £274. D £1390. E £7003. 
 F £1500. and G £998. ; how much is the whole amount of 
 their money ? 
 
 (11) How many days are in the twelve calendar months ? 
 
 (12) Add 87929, 135594, 7964, 3621, 27123, 8345, 35921, 
 2574, 64223, 42354, 3560, and 152165 together. 
 
 (13) Add 6228, 27305, 7856, 287, 7664, 100, 1423, 25258, 
 528, 3135, and 838. 
 
 (14) How many days are there in the first six months of 
 the year ; how many in the last six ; and how many in the 
 whole ? 
 
 (15) In the year 1832, how many days from the Epiphany 
 or Twelfth-day (Jan. 6th) to the last day of July ? 
 
 (16) In the common year how many days from each Quar- 
 ter-day to the next? That is, from Lady-day to Midsummer- 
 day, from thence to Michaelmas-day, from thence to Christ- 
 mas-day, and from Christmas-day to the ensuing Lady-day ? 
 
 (17) When will the lease of a farm expire, which was 
 granted in the year 1 799, for ninety-nine years ? 
 
 (18) A person deceased left his widow in possession of 
 £2500. His eldest son inherited property of the value of 
 £11340. To his two other sons he bequeathed a thousand 
 pounds each more than to his daughter ; whose portion ex- 
 ceeded the property left to her mother by £500. A nephew 
 and a niece had legacies of £525. each; a public charity 
 £105.; and his four servants the same sum to be divided 
 
 to add the figures without naming them. Thus, in adding the first 
 column of the above example, say 2, 3, 7, 10, 15; set down 5 and 
 carry 1, &c. 
 
 This method will tend both to quickness and precision. 
 
16 
 
 SUBTRACTION OF INTEGERS. [THE TUTOR'S 
 
 amongst them. What was the aggregate amount of his 
 property ? 
 
 (19) Tell the name and signification of the sign put between 
 the following numbers : and find what they are equal to, as 
 the sign requires? 
 
 1724 + 649 + 17 + 5400 + 12+999- 
 
 (20) Required the sum of forty-nine thousand and sixteen • 
 four thousand eight hundred and forty ; eight millions, seven 
 hundred and seven thousand one hundred ; nine hundred 
 and ninety-nine ; and eleven thousand one hundred and ten. 
 
 (2 1 ) When will a person born in 1 8 19, attain the age of 45 ? 
 
 (22) Henry came of age 13 years before the birth of his 
 cousin James. How old will Henry be when James is of age ? 
 
 (23) Homer, the celebrated Greek poet, is supposed to have 
 flourished 907 years previous to the commencement of the 
 christian era. Admitting this to be fact, how many years was 
 it from Homer's time to the close of the 1 8th century ; and 
 how long to A. D. 1827 ? 
 
 SUBTRACTION 
 
 Teaches to take a less number from a greater, to find the re- 
 mainder or Difference. 
 
 The number to be subtracted is the Subtrahend, and the 
 other is called the Minuend. 
 
 Rule. Having placed the Subtrahend under the Minuend 
 (in the same order as in Addition) begin at the units, and 
 subtract each figure from that above it, setting down the re- 
 mainder underneath. But when the lower figure is the 
 greater, borrow ten ; which add to the upper, and then sub- 
 tract : set down the remainder, and carry one to the next 
 figure of the subtrahend for the ten that was borrowed. 
 
 Proof. Add the Difference to the Subtrahend, and their 
 sum will be the Minuend. 
 
 (1) From 2714754 
 Take 1542725 
 
 (4) 
 
 271508300 
 72841699 
 
 (7) 
 
 1000000000 
 987654321 
 
 (2) 42087296 
 340961 87 
 
 (5) 
 
 375021599 
 278104609 
 
 (8) 
 
 274698 1340 
 109568 1539 
 
 (3) 45270509 
 32761684 
 
 (6) 
 
 400087635 
 9184267 
 
 (9) 
 
 666740825 
 1093481 72 
 
ASSISTANT.] SUBTRACTION OF INTEGERS. 17 
 
 (10) From 123456789 subtract 98765432. 
 
 (11) From 31147680975 subtract 767380799- 
 
 (12) Subtract 641 870035 from 1630054154. 
 
 (13) Required the difference between 240914 and 24091. 
 
 (14) How much does twenty-five thousand and four ex* 
 ceed sixteen thousand three hundred and ninety. 
 
 (15) If eighty-four thousand and forty-eight be deducted 
 from half a million, what will remain ? 
 
 (16) The annual income of Mr. Lemmington, senior, is 
 twelve thousand five hundred and sixty pounds. Mr. Lem- 
 mington, junior, has an income of seven thousand eight hun- 
 dred and eighteen pounds per annum. How much is the 
 son's income less than his father's ? 
 
 (17) George the Fourth, at his accession to the throne, in 
 J 820, was in the 58th year of his age. In what year was he 
 born, and how long had he reigned on the 29th of January, 
 1829, the anniversary of his accession ? 
 
 {IS) The sum of two numbers is 36570, and one of them 
 is twenty thousand and twelve : what is the other ? 
 
 (19) Thomas has 1 15 marbles in two bags. In the green 
 bag there are 68 : how many are there in the other ? 
 
 (20) Two brothers who were sailors in Admiral Lord 
 Nelson's fleet, were born, the elder in 1767, and the younger 
 JU 1775. What was the difference of their ages, and how old 
 was each when they fought in the battle of Trafalgar, in 1 805 ? 
 
 (21 ) Henry Jenkins died in 1670, at the age of 169. How 
 long prior to his death was the discovery of the continent of 
 America by Columbus, in 1 498 ? — Also, how many years have 
 elapsed from his birth to 1 827 ? 
 
 Example. From 32906547 subtract 8210468. 
 
 32906547 Minuend. Say 8 From 7 I cannot; borrow 10, and 7 
 
 8210463 Subtrahend, are 17, 8 from 17, 9 remain ; set down 9 and 
 
 24696079 Difference. carry 1 — 1 and 6 are 7, 7 from 4 I cannot; 
 
 QooTift^w p p borrow 10, and 4 are 14, 7 from 14, 7 ; set 
 
 ^JObW rrool. down ? aml carry L __ x and 4 are ^ & fr6m ^ 
 
 nothing ; set down (0) nought — from"6, 6 ; set down 6 — 1 from 
 I cannot ; but 1 from 10, 9 ; s<?t down 9 and carry 1. Proceed in like 
 manner to the end. 
 
 When the pupil is initiated m the practice by working an example 
 »r two, he may simplify the work by omitting to express some of the 
 particulars. Thus, in the preceding example, it will be sufficient 
 merely to say, 8 from 17, 9 ; set down 9 and carry 1 : 1 and 6 are 7, 
 7 from 14, 7 ; set down 7 and carry 1, &c. 
 
18 MULTIPLICATION OF INTEGERS. [_THE TUTORS 
 
 (22) Borrowed at various times, £644., £957-, £$0., 
 £1378., and £1293. ; and paid again the different sums of 
 £763., £591., £ll6l., £1000., and £847.— What remains 
 unpaid ? 
 
 (23) Explain the name and signification of the sign used ; 
 and work the two following examples. 
 
 10874 — 9999 51170 — 50049 
 
 (24) John is seventeen years younger than Thomas : how 
 old will Thomas be when John is of age ; and how old will 
 John be when Thomas is 50 ? 
 
 MULTIPLICATION 
 Teaches to repeat a given number as many times as there 
 are units in another given number. 
 
 The number to be multiplied is called the Multiplicand; 
 that by which we multiply is the Multiplier ; and the num- 
 ber produced by multiplying is the Product. 
 
 Rule. When the multiplier is not more than 12, multiply 
 the units' figure of the multiplicand, set down the units of the 
 product, reserving the tens ; multiply the next figure, to the 
 product of which carry the tens reserved : proceed thus till the 
 whole is multiplied, and set down the last product in full. 
 
 MULTIPLICATION TABLE. 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 2 
 
 4 
 
 6 
 
 8 
 
 10 
 
 12 
 
 14 
 
 16 
 
 18 
 
 20 
 
 22 
 
 24 
 
 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 
 
 24 
 
 32 
 
 40 
 
 48 
 
 56 
 
 64 
 
 72 
 
 80 
 
 88 
 
 96 
 
 9 
 
 18 
 
 27 
 
 36 
 
 45 
 
 54 
 
 63 
 
 72 
 
 81 
 
 90 
 
 99 
 
 108 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 60 
 
 70 
 
 80 
 
 90 
 
 100 
 
 110 
 
 120 
 
 11 
 
 22 
 
 33 
 
 44 
 
 55 
 
 66 
 
 77 
 
 88 
 
 99 
 
 110 
 
 121 
 
 132 
 
 12 
 
 24 
 
 36 
 
 48 
 
 60 
 
 72 
 
 84 
 
 96 
 
 108 
 
 120 
 
 132 
 
 144 
 
 • Example. Multiply 713097 by 4. 
 
 Say 4 times 7 are 28, aet down 8 and carry 2 ; 4 
 
 713097 times 9 are 36 and 2 are 38, set down 8 and carry 3 ; 
 
 4 4 times (nought) and 3 are 3, set down 3 ; 4 times 3 
 
 285*388 are **' set down ' an( * carry 1 ; 4 times 1 are 4 and 1 
 
 — - — . are 5. set down 5 ; 4 times 7 are 28. set down 28. 
 
4SSISTANT.] MULTIPLICATION OF INTEGERS. 1Q 
 
 (1) Multiply 25104736 by 2. 
 
 (2) Multiply 52471021 by 3. 
 
 (3) Multiply 792543752 1 by 4. 
 
 (4) Multiply 27104107 by 5. 
 
 (5) Multiply 23104759 by 6. 
 K 6) Multiply 7092516 by 7- 
 
 (7) Multiply 3725104 by 8. 
 
 (8) Multiply 4215466 by 9. 
 
 (9) Multiply 2701057 by 10.* 
 
 (10) Multiply 31 040171 by 11. 
 
 (1 1) Multiply 73998063 by 12. 
 
 (12) Multiply 780149326 by 3, 4, 5, 6, 7, 8, 9, and 10. 
 
 (13) Multiply 123456789 by 4, 5, 6, 7, 8, and 9. 
 
 (14) Multiply 987654321 by 9, 10, 11, and 12. 
 
 When the multiplier is between 12 and 20, multiply by 
 the units' figure in the multiplier, adding to each product the 
 ,ast figure multiplied.t 
 
 (18)2057165X16. 
 (19)6251721X17. 
 
 (20)9215324X18. 
 (21)2571341X19. 
 
 (15)5710592X13. 
 (16)5107252X14. 
 (17)7653210X15. 
 
 When the multiplier consists of several figures, multiply 
 by each of them separately, observing to put the first figure 
 of every product under that figure you multiply by. Add 
 fie several products together, and their sum will be the total 
 product. J 
 
 Proof. Make the former multiplicand the multiplier, and the 
 multiplier the multiplicand ; and if the work is right, the products 
 of both operations will correspond. Otherwise. A presumptive or 
 probable proof (not a positive one) may be obtained thus : Add to- 
 gether the figures in each factor, casting out or rejecting the nines in 
 the sums as you proceed ; set down the remainders on each side of a 
 cross, multiply them together, and set down the excess above the nines 
 
 • To multiply by 10, annex a cipher to the multiplicand, for the 
 product. To multiply by 100, annex two ciphers, &c. 
 Examples. 
 f Multiply 96048 by 15. 
 
 Say 5 times 8 are 40, set down and carry 4 ; 5 times 
 
 96048 4 are 20 and 4 are 24, and 8 are 32, set down 2 and 
 
 15 carry 3 ; 5 times and 3 are 3, and 4 are 7, set down 
 
 1440720 7 ; ^ tmies 6 are 30 » set down and carry 3 ; 5 times 9 
 
 are 45 and 3 are 48, and 6 are 54, set down 4 and carry 
 
 5 ; 5 and 9 are 14, set down 14. 
 J Multiply 76047 by 249. 
 76047 
 
 _J1? Proof. 
 684423 Product by 9. 
 
 304188 do. by 40. 6X6 
 152094 do. by 200. 
 
 18935703 Total product. 
 
MULTIPLICATION OF INTEGERS. 
 
 THE TUTOR S 
 
 in their product at the top of the cross. Then cast out the nines 
 from the product, and place the excess below the cross. If these two 
 correspond, the work is probably right : if not, it is certainly wrong. 
 
 (22) 271041071 X 5147. 
 
 (23) 62310047 X 1668. 
 
 (24) 170925164 X 7419. 
 
 (25) 9500985742 X 61879- 
 (26) 1701495868567 X 4768756. 
 
 When ciphers are intermixed with the significant figures in 
 the multiplier, they may be omitted ; but great care must be 
 taken to place the first figure of the next product under the 
 figure you multiply by.* 
 
 Ciphers on the right of the multiplier or multiplicand (if 
 omitted in the work) must be placed in the total product. t 
 
 (27) 571204 X 27009. 
 
 (28) 7561240325 X 57002. 
 
 (29) 562710934 X 590030. 
 
 (30) 1379500 X 3400. 
 
 (31) 7271000 X 52600. 
 
 (32) 74837000 X 975000. 
 
 A number produced from multiplying two numbers to- 
 gether, is called a composite number ; and the two numbers 
 producing it are called the factors, or component parts. When 
 the multiplier is a composite number, you may multiply by one 
 of the factors ; and that product multiplied by the other will 
 give the total product.^ 
 
 (33) 771039 X 35. 
 
 (34) 99.1563 X 32. 
 
 (35) 715241 x 56. 
 
 (36) 679998 X 132. 
 
 (37) 7984956 X 144. 
 
 (38) 8760472 x 9994 
 
 (39) 7039654 x 99999- 
 (40) A boy can point l6000pins in an hour. How many 
 
 can five boys do in six days, supposing them to work 1 
 clear hours in a day ? 
 
 Examples. 
 
 Multiply 31864 by 7008. 
 31864 
 7008 Proof. 
 
 6 
 4X6 
 6 
 
 254912 
 
 223048 
 
 223302912 
 
 f Multiply 63850 by 5200. 
 63850 
 
 £200 Proof. 
 
 1 
 4X7 
 1 
 
 12770 
 31925 
 
 332020000 
 
 t Multiply 63175 by 45. 
 63175 
 __5X9=45 
 
 315875 
 9 
 
 2842875 
 
 § For an abridged method of 
 multiplying by a series of nines, 
 see the Key. 
 
ASSISTANT.J DIVISION OF INTEGERS. 21 
 
 (41) If a person walks upon an average 7 miles a day, how 
 many miles will he travel in 42 years, reckoning 365 days to 
 a year ? 
 
 (42) Multiply the sum of 365, 9081, and 22048, by the 
 difference between 9081 and 22048. 
 
 (43) Required the continued product of 112, 45, 17, and 99. 
 Note. Multiply all the numbers one into another. 
 
 DIVISION 
 
 Teaches to find how often one number is contained in ano- 
 ther : or to divide a number into any equal parts required. 
 
 The number to be divided is called the Dividend; that by 
 which we divide is the Divisor ; and the number obtained by 
 dividing is the Quotient ; which shows how many times the 
 divisor is contained in the dividend. When it is not contain- 
 ed an exact number of times, there is a part of the dividend 
 left, which is called the Remainder. 
 
 Rule. When the divisor is not more than 12, find how often 
 it is contained in the first figure (or two figures) of the divi- 
 dend ; set down the quotient underneath, and carry the over- 
 plus (if any) to the next in the dividend, as so many tens ; 
 find how often the divisor is contained therein, set it down, 
 and continue in the same manner to the end. 
 
 When the divisor exceeds 12, find the number of times it 
 is contained in a sufficient part of the dividend, which may 
 be called a dividual; place the quotient figure on the right, 
 multiply the divisor by it, subtract the product from the di- 
 vidual, and to the remainder bring down the next figure of 
 the dividend, which will form a new dividual : proceed with 
 this as before, and so on, till all the figures are brought down. 
 
 Proof. Multiply the divisor and quotient together, add- 
 ing the remainder (if any) and the product will be the same 
 as the dividend. 
 
 (1) Divide 725107 by 2.* | (2) Divide 7?*0472 by 3. 
 
 * Example. Divide 7328105 by 4. 
 Divisor 4)7328105 Dividend. Say the fours in 7, once and 8 
 
 q.«c:;*nt Tl3?o~26--i Rem. over ? th ? *■» ™Jh**?: es J 
 
 a are 32 and 1 over ; the tours in 12, 
 
 3 times ; the fours in 8, twice ; the 
 
 7328105 Proof. f 0U rs in 1, and 1 over ; the fours 
 
 in 10, twice 4 are 8, and 2 over ; the fours in 25, six fours are 24 nxd 
 
 1 over. 
 
DIVISION OF INTEGER 
 
 [_THE 
 
 (3) Divide 7210416 by 4. 
 
 (4) Divide 7203287 by 5. 
 
 (5) Divide 5231037 by 6. 
 
 (6) Divide 2532701 by 7. 
 
 (7) Divide 2547325 by 8. 
 
 (8) Divide 25047306 by 9- 
 
 (9) Divide 70312645 by 10. 
 (10) Divide 12804763 by 11. 
 ^11) Divide 79043260 by 12. 
 
 (12) Divide 37000421 by 3, 
 
 5, 7, and 9- 
 
 (13) Divide 111111111 by 6, 
 and 12. 
 
 (14)7210473 -r-37.* 
 (15)42749467 -^-347. 
 ( 1 6) 734097 1 43 ~ 5743.+ 
 (17)1610478407 -r-547l6. 
 (18)4973401891 -5- 510834. 
 
 (19) 51704567874-f-4765043. 
 
 (20) 174537989^6123741 -r- 
 
 31479461. 
 (21)25473221 -^27100.$ 
 
 (22) 725347216 -*- 572100. 
 
 (23) 752473729 -fc- 373000. 
 (24)6325104997 -i- 2 15000. 
 
 9, 11, 
 
 When the divisor is a composite number, you may divide 
 the dividend by one of the component parts, and that quotient 
 by the other ; which will give the quotient required. But 
 the true remainder must be found by the following 
 
 Rule. Multiply the second remainder by the first divisor: 
 to that product add the first remainder, which will give the 
 true one. 
 
 (25) 3210473 -4- 27.§ 
 
 (26) 7210473 h- 35. 
 
 (27) 6251043 *. 42 
 
 (28) 5761034 -r- 54. 
 
 • Example. Divide 40855 by 29. 
 Dividend. 
 Divisor 29)40855( 1408 Quotient. 
 29 29 
 
 118 
 116 
 
 12672 
 2816 
 
 23 Remainder. 
 
 255 
 
 232 40855 Proof. 
 
 "23 
 
 -f- When the divisor is large, the quotient figures are most easily 
 found by trials of the Jirst fg are (or two) in the leading figures of tli6 
 dividend. 
 
 % Ciphers at the right of the divisor may be cut off, and as many 
 figures from the right of the dividend ; but these must be annexed 
 to the remainder at last. 
 
 § Example. Divide 314659 by 21. 
 21=7X3)314659 
 
 7)104886—1 1 
 
 - V =5X3+1 = 16 rem- 
 
 14983—5 J ^ 
 
ASSISTANT J DIVISION OF INTEGERS 23 
 
 A number may be divided by 10, 100, 1000, &c. by merely 
 cutting off one, two, three, &c. figures on the right : the 
 other figures are the quotient, those cut off are the remainder. 
 
 Thus 76390-f-l 0=7639; 238457-f-J 0=23845 and 7 rem. 
 And 4598653-7-1000=4598 and 653 rem. 
 
 (29) 65941089 -4-10. I (31) 18043329 -f- 10000. 
 
 (30) 7208465 ~- 100. (32) 7406572 -f- 1200. 
 
 (33) What is the difference between the 12th part 01 
 107724, and the 23rd part of 346610? 
 
 (34) If a ship bound to Jamaica set sail from Liverpool on 
 the 25th of January, 1828, and arrived at that island on the 
 8th of March, what was the velocity of her sailing per day 
 and per hour ; the distance being 4558 miles ? 
 
 Note. This is the direct distance. The circuitous course of the 
 ship would be considerably more. 
 
 (35) The period of Jupiter's revolution in his orbit round 
 the sun, which is the year of that planet, is 4330 of our days. 
 How many of our years, reckoning 365 days to the year, are 
 equal to five years of Jupiter ? 
 
 (36) I would plant 2072 elms in 14 rows, the trees in each 
 row 17 feet asunder : what length will the grove be? 
 
 (37) If a chest of oranges, 1292 in number, be distributed, 
 one moiety among 1 9 boys, the other among 1 7 girls : how 
 many will fall to the share of each ? 
 
 (38) The circumference of the earth's orbit, or annual path 
 round the sun, is about 596440000 miles. Supposing the 
 year to be exactly 865$ days, or 8766 hours, how many miles 
 in an hour, and how many in a minute, are we carried by 
 this motion ? 
 
 (39) Required the sum, the difference, the product, and 
 the quotient, of 3679 and 283: and also the quotient of the 
 product divided by the sum. 
 
 (40) The sum of two numbers is 4290 ; the less number 
 is 143 : what is their difference, product, and quotient; and 
 the quotient of the product divided by the difference ? 
 
 (41) The product of a certain number multiplied by 6Q4, 
 when 320 are added, is equal to 500000: what is that number r 
 
 (42) Allowing the earth to revolve on its axis in exact!;. 
 24 hours, and the circumference at the equator to be 248(> 
 miles; at what rate per hour and per minute are the inhabi 
 tants of that part carried round by the revolution ? Also, a 
 what rate are the inhabitants of London carried round, tlu 
 CTcumference in that latitude being 15480 miles? 
 
1— 
 
 It- -5 
 
 24 TABLES OF MONEY. [THE TUTOR* S 
 
 ARITHMETICAL AND COMMERCIAL TABLES. 
 
 STERLING MONEY. 
 
 4 farthings (qrs.) make 1 penny, d. 
 12 pence 1 shilling, s. 
 
 5 shillings 1 crown, cr. 
 
 20 shillings, 1 pound, or sovereign, £. 
 
 \d. denotes a farthing, \d. a halfpenny, and \d. three far- 
 mings. 
 
 Qrs. 4 = 1 penny. 
 
 48 = 12 = 1 shilling. 
 240 = 60 s 5 = 1 crown. 
 960 = 240 = 20 = 4 = 1 pound. 
 
 OBSOLETE COINS. 
 
 A guinea (weight 5 dwts. 94 grs.) value 21s. A moidore, 27*. A 
 pistole, 175. A mark, 13*. id. An angel, 10*. A noble, 6s. Sd. A 
 tester, 6d. A groat, id. 
 
 Notes. Gold is considered the standard metal ; and there is no al- 
 teration in the new coin, either in fineness or weight, from that of 
 former coinages ; 21 sovereigns being equal in weight to 20 guineas, 
 1869 sovereigns weigh exactly 40 lbs. troy. A sovereign is therefore 
 a little more than 5 dwts. 3J grs. (5 dwts. 3-274 gr*.) and a half sov- 
 ereign rather exceeds 2 dwts. 134 grs. (2 dwts. 13-637 grs.) The new 
 silver coin is of the same fineness as that of former coinages ; but 1 ft), 
 of silver is now coined into 66s. instead of 62*. as it was formerly, so 
 that one shilling now weighs 3 dwts. 15 T 3 T gr*., and other silver pieces 
 m proportion. 
 
 The mint value of gold is £3..17..10|. per ounce, and of silver 5s. 6d. 
 
 The standard for gold coin is 22 parts (commonly called carats) of 
 fine gold, and 2 parts (or carats) of copper, melted together. For sil- 
 ver coin, 1 1 oz. 2 dwts. of fine silver, alloyed with 1 8 dwts. of copper. 
 
 MONEY TABLE. 
 
 Farthings. 
 
 Farthings. 
 
 Pence. 
 
 Pence. 
 
 
 Pence. 
 
 Shillings. 
 
 qrs. d. 
 
 qrs. d. 
 
 d. s. 
 
 d. s. 
 
 4. 
 
 d. s. d. 
 
 s. £. ,. 
 
 4 are 1 
 
 32 are 8 
 
 36are3 
 
 20 are 1 
 
 8 
 
 160areI3 4 
 
 80 are4 
 
 6 ... 14 
 
 34 ... 84 
 
 48 4 
 
 30... 2 
 
 6 
 
 170... 14 2 
 
 90... 4 10 
 
 8 ... 2 
 
 36 ... 9 
 
 60 .. 5 
 
 40... 3 
 
 4 
 
 180...15 
 
 100... 5 
 
 10 ... 24 
 
 38 ... 94 
 
 72... 6 
 
 50.. 4 
 
 2 
 
 190.. .15 10 
 
 110... 5 10 
 
 12 ... 3 
 
 40 ...10 
 
 84... 7 
 
 60... 5 
 
 
 
 200... 16 8 
 
 120... 6 
 
 14 ... SJ 
 
 42 ...104 
 
 96... 8 
 
 70 .. 5 
 
 10 
 
 
 130... 6 10 
 
 
 16 ... 4 
 
 44 ...11 
 
 108... 9 
 
 80... 6 
 
 8 
 
 Shillings. 
 
 140... 7 
 
 18 ... A\ 
 
 46 ...114 
 
 120.. .10 
 
 90... 7 
 
 6 
 
 *. £. s. 
 
 150... 7 10 
 
 20 ... 5 
 
 48 ...1*. 
 
 132.. .11 
 
 100... 8 
 
 4 
 
 20arel 
 
 160... 8 
 
 22 ... 5| 
 24 ... 6 
 
 
 144... 12 
 
 110... 9 
 
 2 
 
 SO... 1 10 
 
 170... 8 10 
 
 Pence. 
 
 156.. .13 
 
 120.. .10 
 
 
 
 40... 2 
 
 180... 9 
 
 26 ... 64 
 
 d. s. 
 
 168.. .14 
 
 130... 10 
 
 10 
 
 50... 2 10 
 
 190... 9 10 
 
 28 ... 7 
 
 12 are 1 
 
 180.. .15 
 
 140... 11 
 
 8 
 
 60... S 
 
 200... 10 
 
 30 ... 74 
 
 24 ... 2 
 
 192.. .16 
 
 150... 12 
 
 6 
 
 70... 3 10 
 
 210.. .10 10 
 
ASSISTANT. J WEIGHTS AND MEASURES. 25 
 
 Note. When the units* figure is cut off from any number of shil- 
 lings, half the remaining figures will be the pounds. Thus, 2-56s.=* 
 £12. 16s. because half of 25=12 ; and the one over prefixed to the 6 
 gives 16*. 
 
 WEIGHTS AND MEASURES. 
 
 TROY WEIGHT. 
 
 24 grains (gr.) make 1 pennyweight, dwt. 
 20 penny weights . 1 ounce, . oz. 
 12 ounces . . 1 pound, . lb. 
 
 Grains. 24 *» 1 pennyweight. 
 i \ 480 — 20 = 1 ounce. 
 
 5760 - 240 = 12 = 1 pound. 
 ^\» Gold, silver, and gems, are weighed by this weight. 
 
 apothecaries' weight. 
 
 20 grains (gr.) make 1 scruple, ^ 
 
 3 scruples . . 1 dram, . . 5 
 8 drams ... 1 ounce, . . § 
 
 12 ounces . . 1 pound, . . lb. 
 
 Grains. 20 = 1 scruple. 
 i 60 = 3 = 1 dram. 
 
 480 a 24 = 8 = 1 ounce. 
 5760 = 288 = 96 = 12 = 1 pound. 
 This is used only in the mixing of medicines. 
 These are the same grain, ounce, and pound, as those in Troy Weight. 
 
 avoirdupois weight. 
 
 . \6 drams (dr.) make . . . . i ounce, . . oz. 
 
 \6 ounces 1 pound, . lb. 
 
 14 pounds 1 stone, . st. 
 
 28 pounds, or 2 stones ... 1 quarter, . qr. 
 4 quarters, or 8 st. or 112 ib . 1 hundred, cwt. 
 
 20 hundreds 1 ton, . . t. 
 
 Drams. 16 = 1 ounce. 
 
 256 = 16 = 1 pound. 
 3584 = 224 = 14 = 1 stone. 
 7168 » 448 = 28 == 2=1 quarter. 
 28672= 1792= 112= 8=4= 1 cwt. 
 573440 = 35840 = 2240 = 160 = 80 = 20 = 1 ton. 
 
 By this weight nearly all the common necessaries of life are weigh- 
 ed. A truss of hay =56 lb. and one of straw=36 lb. A load is 36 
 trusses. A peck loaf weighs 17 lb. 6 oz. 1 dr. In the metropolis, 8 lb. 
 are a stone of meat. A fother of lead is 19£ cwt. In some districts, 
 goods of various descriptions (as cheese, coal &c.) are sold by the 
 long cwt. or 1 20 lb. 
 
 B 
 
26 WRIGHTS Also mLASUREs. [THE TUTOR'S 
 
 WOOL. 
 
 When wool is purchased from the grower, the legal stone 
 of 14 lb. and the tod of 28 lb. are used. But in the dealing? 
 betw«cn woolstaplers and manufacturers., 
 15 pounds are . . 1 stone. 
 2 stones, or 30 lb. . 1 tod. 
 8 tods, or 240 lb. . 1 pack or sack. 
 
 COMPARISON OF WEIGHTS. 
 
 A grain is the elementary or standard weight. 
 1 ounce avoirdupois is . . 437§ grains. 
 
 1 ounce troy 480 
 
 1 pound troy 57^0 
 
 1 pound avoirdupois . . 7000 
 175 pounds troy=144 pounds avoirdupois. 
 175 ounces troyz^l92 ounces avoirdupois. 
 We may, therefore, reduce lbs. Troy into Avoirdupois by muiti- 
 alying them by 144, and dividing by 175, &c. 
 
 LINEAL, OR LONG MEASURE. 
 
 12 inches fin, J make 
 
 3 feet, or 36 inches 
 2 yards, or 6 feet . 
 5^ yards, or 16^ feet 
 
 4 poles, or 22 yards 
 
 1 foot, . . ft. 
 
 1 yard, . . yd. 
 
 1 fathom, . fa. 
 
 1 pole, rod, or perch, p. 
 
 1 land-chain,* ch. 
 
 40 poles, or 10 ch., or 220 yds. 1 furlong, . fur. 
 
 8 furlongs, or 1760 yards . 1 mile, . . m. 
 
 3 miles 1 league, . /. 
 
 Barlev -corns. 
 
 3 = 1 inch. 
 
 36 = 12 = 1 foot. 
 108 = 36 = 3 = 1 yard. 
 594 = 198 = 16| = 5 J = 1 pole. 
 23760 = 7920 = 660 = 220 = 40 = 1 furloLg. 
 190080=63360=5280 =1760 = 320 = 8 = 1 m^e. 
 Note. It is commonly supposed that the English inch was origi- 
 nally taken from three grains of barley, selected from the middle ot 
 the Mr pnd well dried. 
 A iweilLft part of an inch is called a line. 
 
 4 inches are a hand, used in measuring the height of horses. 5 
 feet are a pace. A cubit = 14 feet nearly. 
 
 This measure determines the length of lines. A line has the di- 
 mension of length only, without breadth or thickness. 
 
 • The chain consists of 100 links, each link being = 7 '92 inches. 
 
ASSISTANT.] TABLES OF MEASURES. 27 
 
 CLOTH MEASURE. 
 
 2j inches (in.) make . 1 nail, . . n. 
 4 nails, or 9 inches . . I quarter, . qr. 
 
 4 quarters 1 yard, . . yd. 
 
 5 quarters 1 English ell, E. e 
 
 A Flemish ell is 3 qrs. A French ell 6 qrs. 
 
 Used for all drapery goods. 
 
 SUPERFICIAL, OR SQUARE MEASURE. 
 
 144 square inches ( sq. in.) make 1 square foot, sq.ft. 
 
 9 square feet . . 1 square yard, sq. yd. 
 
 S0\ sq. yards, or 272 \ sq. feet 1 sq. rod, pole, or perch. 
 Also, in the measure of land, 
 40 perches make . . 1 rood, . r. 
 4 roods, or 4840 yards . 1 acre, . a. 
 
 10,000 square links . . 1 square chain, sq. c. 
 
 10 sq. chains, or 100,000 links 1 acre, . a. 
 
 640 acres .... 1 square mile, sq. m. 
 
 Inches. 144 = 1 foot. 
 
 1296 = 9 x= 1 yard. 
 
 39204 = 272J = 30J = 1 pole. 
 1568160 = 10890 = 1210 = 40 = 1 rood. 
 6272640 = 43560 = 4840 = 160 = 4 = 1 acre. 
 
 Roofing, flooring, &c. are commonly charged by the Square^ con- 
 taining 100 square feet. 
 
 By this measure is expressed the area of any superficies, or surface. 
 A superficies has measurable length and breadth. 
 
 CUBIC, OR SOLID MEASURE. 
 
 1728 cubic inches (in.) make . 1 cubic foot 
 
 27 cubic feet .... 1 cubic yard.* 
 40 feet of round timber, or 1 , , 
 
 50 feet of hewn timber J * 
 
 42 feet 1 ton of shipping. 
 
 A cord of wood is 4 feet broad, 4 feet deep, and 8 feet long, 
 being 128 cubic feet. 
 
 A stack of wood is 3 feet broad, 3 feet deep, and 12 feet long, 
 being 108 cubic feet. 
 
 This determines the solid contents of bodies. A solid has three 
 dimensions, length, breadth, and thickness. 
 
 A solid yard of earth is called a load. 
 
28 TABLES OF MEASURES, [THE TUTOR'S 
 
 IMPERIAL MEASURE. 
 
 This is the standard now established by Act of Parliament, 
 as a general measure of capacity for liquid and dry articles. 
 2 pints (pt.) make . . 1 quart, qt. 
 
 4 quarts ... .1 gallon, gal. 
 
 The imperial or standard gallon must contain 10 lbs. Avoir- 
 dupois Weight of pure water, at the temperature of 62° of 
 Fahrenheit's thermometer. This quantity measures 277^* 
 cubic inches ; being about one-fifth greater than the old wine 
 measure, one-thirty- second greater than the old dry measure, 
 and one-sixtieth less than the old ale measure. 
 
 In Dry Measure, 
 
 2 gallons (gaL) make . 1 peck, . pk. 
 
 4 pecks 1 bushel, b. 
 
 8 bushels 1 quarter, qr. 
 
 Corn to be stricken off the measure with a round stick, or roller. 
 Obsolete. A coom •=: 4 bushels ; a chaldron — 4 quarters ; a wev 
 rr 5 quarters ; a last — 2 weys. 
 
 Solid inches. 277 \ = 1 gallon. 
 
 554| =2=1 peck. 
 22] a = 8 = 4 = 1 bushel. 
 17744 = 64 = 32 = 8 = 1 quarter. 
 
 OF COALS, 
 
 3 bushels make . . I sack. 
 12 sacks, or 36 bushels 1 chaldron. 
 21 chaldrons ... 1 score. 
 
 All the measures used for heaped goods are to be of cylin- 
 drical form ; the diameter being at least double the depth. 
 The height of the raised cone to be equal to three-fourths of 
 the depth of the measure. 
 
 The old dry gallon contained 2 68 J cubic inches. 
 Note. The bushel, for measuring heaped goods, must be 17*81 
 inches in diameter, and 8*904 inches deep ; or if made 18 inches in 
 diameter, the depth will be 8*717 inches The cone to be raised 6*6 
 inches in height. 
 
 In wine and spirit measure, the old gallon contained 
 23 1 cubic inches. 
 
 63 gallons were . a hogshead, hhd. 
 
 2 hogsheads, or 126 gallons a pipe or butt. 
 4 hogsheads, or 252 gallons a tun. . | 
 
 More accurately, 277*274 cubic inches. 
 
ASSISTANT.] TABLES OF MEASURES. 20, 
 
 Some other denominations have been long obsolete ; as, an anker 
 UO gallons) ; a runlet (18 gallons) ; a tierce (42 gallons) ; a puncheon 
 (84 gallons). But casks of most descriptions are generally charged 
 according to the number of gallons contained. 
 
 Solid inches. 34§£ = 1 pint. 
 
 69 T 5 5 = 2 = 1 quart. 
 277J = 8 = 4=1 gallon. 
 17466f = 504 = 252 = 63 = 1 hogshead. 
 34933^ =1008= 504 = ] 26 = 2 = 1 pipe. 
 69867 = 2016 = 1008 — 252 = 4 = 2 = 1 tun. 
 
 In ale, beer, or porter measure, the old gallon contain- 
 ed 282 cubic inches; and measures of the following denomi- 
 nations have been in use : 
 
 A firkin, containing . Q gallons. 
 
 A kilderkin ... 18 gallons. 
 
 A barrel S6 gallons. 
 
 A hogshead .... 54 gallons. 
 
 A butt 108 gallons. 
 
 Cubic inches. 34 §| = 1 pint. 
 
 69 T \ = 2= 1 quart. 
 277± = 8= 4= ] gallon. 
 24954 = 72= 36= 9= 1 firkin. 
 49904 =144= 72= 18= 2=1 kilderkin. 
 9981 =288=144= 36= 4=2=1 barrel. 
 14971| =432=216= 54= 6=3=14=1 hogshead. 
 29943 =864=432=108=12=0=3 =2= 1 butt. 
 
 *rules for changing old measures to imperial. 
 
 Ale. Multiply by 60, and divide by 59 ; or add 5 ' § part. (True 
 within youoo P art °? tne whole.) 
 
 Or, multiply by 179, and divide by 176. (True, within Tuu i--_ part.*, 
 
 Dry. Multiply by 32, and divide by 33 ; or deduct ^ part. 
 (Error, less than ^^ part.) 
 
 Wine. Multiply by 5, and divide by 6, or deduct i part. (Error, 
 less than 5 ^o part.) 
 
 Or, multiply by 624, and divide by 749. (Error, less than S5g i__ . 
 part.) 
 
 *rules for changing imperial to old measures. 
 
 Ale. Multiply by 59, and divide by 60, or deduct ^ part. 
 
 Or, multiply by 176, and divide by 1*79. 
 
 Dry. Multiply by 33, and divide by 32, or add ^ part — That is, 
 add one peck in every quarter, one quart in every bushel, or half a 
 pint in every peck. 
 
 Wine. Multiply by 6, and divide by 5, or add | part. 
 
 Otherwise, Multiply by 749, and divide by 624. 
 
 * Examples applying to these Rules will be found in the Misc 
 neous Questions in the latter part of the book 
 
SO TIME. r THK TUTORS 
 
 TIME. 
 
 60 seconds (sec.) make . . . . 1 minute, . min. 
 
 60 minutes . • . .... 1 hour, . . hr. 
 
 24 hours 1 day,* . . d. 
 
 7 days 1 week, . . wk. 
 
 52 weeks, 1 day, 6 hours, or ) i T 1* 
 
 365 days, 6 hours .... J * ' $ * 
 
 365 days, 5 hours, 48 min. 51^ seconds The Solar year,t 
 
 100 years 1 century. 
 
 Seconds. 60 = 1 minute. 
 
 3600 = 60 = 1 hour. 
 86400 = 1440 == 24 = 1 day. 
 604800 = 10080 = 168 = 7=1 week. 
 31557600=525960=8766=365 d. 6h.=52 w. 1 d. 6 A.=l Julian year. 
 31556931=525948=8765=365 d. 5 h. 48 m. 51 4"= 1 Solar year. 
 
 The year is divided into 12 Calendar months; January, February 
 March, April, May, June, July, August, September, October, No- 
 vember, December. 
 
 And in each other thirty-one : 
 
 The days are thirty in September, 
 In April, June, and in November; 
 Twenty-eight in February alone, 
 
 But every leap-year we assign 
 To February twenty-nine. 
 
 The leap-years are those which can be exactly divided by 
 4; as, 1824, 1828, &c. Hence it appears that the year is ac- 
 counted 365 days, for three years together ; and 366 days in 
 the fourth : the average being 365\ days. (The Julian Year.) 
 
 Four weeks are frequently called a month; but in this 
 sense it is better to avoid the term. 
 
 Note. In all questions in this book, where the proposed or required 
 time consists of years, months, weeks, &c. allow 4 weeks to a month, 
 and 13 months to a year. 
 
 GEOMETRY. 
 
 60 seconds (") make . 1 minute, ' 
 60 minutes .... 1 degree. ° 
 S6() degrees . . . . ] circle. 
 Many highly important calculations in the mathematical 
 sciences are founded on this division of the circle. 
 
 In Astronomy, the great circle of the ecliptic (or of the zo- 
 diac) is divided into 12 signs, each 30°. 
 
 * A day is the time in which the earth revolves once upon its axis : 
 by law and custom it is reckoned from midnight to midnight ; but 
 the astronomical day begins at noon. 
 
 f The Solar, or true year, is that portion of time in which the earth 
 makes one entire revolution round the sun. 
 
ASSISTANT J REDUCTION. 31 
 
 In Geography, a degree of latitude, or of longitude on the 
 equator, measures nearly 69 1 1 ^ British miles. But a minute 
 of a degree is called a geographical mile. 
 
 ARTICLES SOLD BY TALE. 
 
 12 articles of any kind, are 1 dozen. 
 
 12 dozen 1 gross. 
 
 12 gross 1 great gross. 
 
 20 articles 1 score. 
 
 24 sheets of paper 1 quire. 
 20 quires . . 1 ream. 
 2 reams . . 1 bundle. 
 
 DEFINITIONS. 
 
 1. A number is called abstract, when it is considered simply, 
 or without reference to any subject ; as seven, a thousand, &c. 
 
 2. When a number is applied to denote so many of a par- 
 ticular subject, it is a concrete number ; as seven pounds, a 
 thousand yards, &c. 
 
 3. A denomination is a name of any particular distinctive part 
 of money, weight, or measure ; as penny, pound, yard, &c. 
 
 4. The association of a concrete number with its subject, 
 forms a quantity. 
 
 5. A simple quantity has only one denomination ; as seven 
 pounds. 
 
 6. A compound quantity consists of more denominations than 
 one ; as seven pounds five shillings. 
 
 REDUCTION 
 
 Is the method of changing quantities of one denomination 
 into another denomination, retaining the same value. 
 
 Rule. Consider how many of the less name make one of 
 the greater; and multiply by that number to reduce the 
 greater name to the less, or divide by It to reduce the less name 
 to the greater. 
 
 Examples. 
 
 Reduce £8..8..G£. into farthings. 
 
 The £8. being multiplied by 20, and the 8*. 
 
 added, make 168s. ; these being multiplied by 12, 
 
 and the 6d. added, make 2022rf. ; which being 
 
 multiplied by 4 and the 2 farthings added, make 
 
 8090 qrs. Ans. in the whole* $090 farthings. 
 
 £. s. 
 
 d. 
 
 8 8 
 
 6* 
 
 20 
 
 
 168*. 
 
 
 12 
 
 
 2022 d. 
 
 
 4 
 
 
3% REDUCTION. [[THE TUTOR'S 
 
 (1) In £12. how many shillings, pence, and farthings ? 
 
 Ans. £240,?. 2880d. 11520 qrs. 
 
 (2) In 3 1 1 520 farthings, how many pounds ? Ans. £324*.. 1 0. 
 
 (3) Change 21 guineas into farthings. Ans. 21168 qrs. 
 
 (4) In £l7»5..3^. how many farthings? Ans. 16573 qrs. 
 
 (5) In £25..14..1. how many pence? Ans. 6lo9d. 
 
 (6) Reduce 179^0 pence to crowns. Ans. 299 crowns. 
 
 (7) In 15 crowns how many shillings and sixpences ? 
 
 Ans. 75s. 150 sixpences. 
 
 (8) Change 57 half-crowns into threepences, pence, and 
 farthings. Ans. 570 threepences, 1710d. 6840 farthings. 
 
 (9) How many half-crowns, and how many sixpences, are 
 equivalent to £25.. 17-6. ? Ans. 207 half-cr. 1035 sixpences. 
 
 (10) Convert £ 17- 11. -9- into threepences. Ans. 1407 threep. 
 
 (11) Change £l0..13..10^. into halfpence. Ans. 5133. 
 
 (12) In 52 crowns, as many half-crowns, shillings, and 
 pence, how many farthings? Ans. 2 1 424 far 
 
 (13) Convert 17380 farthings into £. Ans. £l8..2..1. 
 
 (14) In 21424 farthings, how many crowns, half-crowns, 
 shillings, and pence, of each an equal number ? Ans. 52. 
 
 (15) Reduce 60 guineas to shillings, crowns, and pounds. 
 
 Ans. 1260s. 252 crowns, £63. 
 
 (16) Reduce 76 moidorest into pounds. Ans. £l02..12. 
 
 (17) How many shilling half-crowns, and crowns, an 
 equal number of each, are there in £556. ? 
 
 Ans. 1308 of each, and 2s. over. 
 
 (18) In 1308 crowns, as many half-crowns, and as many 
 shillings, how many pounds? Ans. £555..18. 
 
 (19) Seven men brought £15.. 10. each into the mint, to 
 be exchanged for guineas ; how many would they have ? 
 
 Ans. 103 guineas and 7s. over. 
 
 (20) In 525 American dollars, at 4*. 6d. each, how many 
 pounds sterling ? 4ns. £118..2..6. 
 
 Converse to the preceding Example. 
 In 8090 farthings, how many pounds ? 
 4)8090 grs. Dividing the farthings by 4, we obtain 2022d. 
 
 12)20224 d. anc * ^ over > which are farthings, because the re- 
 
 * * mainder is a part of the dividend. Divide 2022 
 
 2|0)16|8s. 6\d. by 12, and we obtain 168s. and 6d. over : these 
 Ans % £8..8..6£. shillings divided by 20, give £$. Ss. so that the 
 
 answer is £8..8..6|. 
 
 •J- 27 shillings. The moidore is current in Portugal, but not in 
 England. 
 
luudj T^dwcA^^?, 
 
 UK 
 
 ASSISTANT.] REDUCTION. 33 
 
 WEIGHTS AND MEASURES. 
 
 TROY WEIGHT. 
 
 (21) In 27 ounces of gold, how many grains ? Ans. 12960. 
 
 (22) Reduce 3 lb. 10 oz. 7 dwt. 5 gr. to grains ? Ans. 22253. 
 
 (23) In 8 ingots of silver, each ingot weighing 7 lb. 4 oz. 
 17 drvts. 15 gr. how many grains? Ans. 341304 gr. 
 
 (24) How many ingots weighing 7 lb. 4>oz. 17 drvts. 15 gr. 
 each are there in 341304 grains? Ans. 8 ingots. 
 
 APOTHECARIES* WEIGHT. 
 
 (25) In 27 lb. 7 3. 2 3. 1 B- 2 gr. how many grains ? 
 
 Ans. 159022 grains. 
 
 (26) In a compound of 9 5* 4 3. 1 9* how many pills of 
 £ grains each ? ^(fts. 916 jm7/*. 
 
 AVOIRDUPOIS WEIGHT. 
 
 (27) In 14769 ounces, how many cwt. ? 
 
 Ans. 8 crvt. #r. 27 lb. 1 oz. 
 
 (28) In 34 tons , 17 cw£. 1 qr. 19 /&• how many pounds ? 
 
 Ans. 78111 Z6. 
 
 (29) In 9 cwtf. 2 qrs. 14 Z6. of indigo, how many half stones, 
 and how many pounds? Ans. 154 half stones, 1078 /6. 
 
 (30) How many stones and pounds are there in 27 hogs- 
 Aeads of tobacco, each weighing neat 8f cwt. ? 
 
 Ans. 1890 stones, 26460 lb. 
 
 (31) Bought 32 bags of hops, each bag 2 cwt. 1 qr. 14 lb. 
 and another of 1 50 lb. how many cwt. are there in the whole ? 
 
 Ans. 77 cwt. 1 qr. 10 #>. 
 
 (32) In 27 cwt. of raisins, how many parcels of 1 8 lb. each ? 
 
 Ans. 168. 
 
 CLOTH MEASURE. 
 
 (33) In 27 yards, how many nails ? Ans. 432. 
 
 (34) In 75 English ells, how many yards ? 
 
 Ans. 93 yards, 3 qrs. 
 
 (35) In 24 pieces, each containing 32 Flemish ells, how 
 many English ells ? Ans. 4()0 English ells, 4 qrs. 
 
 (36) In 17 pieces of cloth, each 27 Flemish ells, how many 
 yards? Ans. 344 yards, 1 qr. 
 
 (31) In 911^ yards, how many English ells? Ans. 729. 
 (38) In 12 bales of cloth, each containing 25 pieces, of 
 15 English ells, how many yards? Ans. 5625. 
 
 LONG MEASURE. 
 
 (3Q) In 57| miles, how many furlongs and poles ? 
 
 Ans. 460 furlongs, 18400 poles. 
 B 5 
 
34 REDUCTION. [/THE TUTOR'S 
 
 (40) In 7 miles, how many feet and inches ? 
 
 Ans. 36960 feet, 443520 inches. 
 
 (41) In 72 leagues, how many yards? Ans. 380160 yards. 
 
 (42) If the distance from London to Bawtry be accounted 
 1 50 miles, what is the number of leagues, and also the num- 
 ber of yards, feet, and inches ? 
 
 Ans. 50 leagues, 264000 yards, 792000 feet, 9504000 inches. 
 
 (43) How often will the wheel of a coach, that is 1 7 feet in 
 circumference, turn in 100 miles? Ans. 31 058 {^ times round. 
 
 (44) How many barley-corns will reach round the globe, 
 the circumference being 360 degrees, supposing that each 
 degree were 69 miles and a half? Ans. 4755801 600. 
 
 See Table of Geometry, page 30. 
 
 LAND MEASURE. 
 
 (45) In 27 a. 3 r. 19 p. how many perches? Ans. 4459. 
 
 (46) A person having a piece of ground, containing 37 
 acres, 1 perch, intends to dispose of 15 acres: how many 
 perches will he have left ? Ans. 3521 perches. 
 
 (47) There are 4 fields to be divided into shares of 75 
 perches each ; the first field contains 5 acres ; the second 
 4 acres, 2 perches ; the third 7 acres, 3 roods; and the fourth 
 2 acres, 1 rood: how many shares will there be? 
 
 Ans. 40 shares, 42 perches rem. 
 
 (48) In a field of 9 acres and a half, how many garden? 
 may be made, each containing 500 square yards ? 
 
 Ans. 91, and 480 yards rem. 
 
 IMPERIAL MEASURE. 
 
 (49) In 10080 pints of port wine, how many tuns? 
 
 Ans. 5 tuns. 
 
 (50) In 35 pipes of Madeira, how many gallons and pints ? 
 
 Ans. 4410 gals. 35280 pints. 
 
 (51) A gentleman ordered his butler to bottle offf of a 
 pipe of French wine into quarts, and the rest into pints. 
 How many dozen of each had he? Ans. 28 dozen of each. 
 
 (52) In 46 barrels of beer, how many pints? Ans. 13248. 
 
 (53) In 10 barrels of ale, how many gallons and quarts? 
 
 Ans. 360 gals. 1440 qts. 
 
 (54) In 12480 pints of porter, how many kilderkins ? 
 
 Ans. 86 kil. \fir. 3 gals. 
 
 (55) In 108 barrels of ale, how many hogsheads ? Ans. 72. 
 
 (56) In 120 quarters of corn, how many bushels, pecks, gal- 
 lons, and quarts ? Ans. 96O bu. 3840 pics. 7 680 gal. 30720 qts. 
 
ASSISTANT."] COMPOUND ADDITION. 35 
 
 (57) How many bushels are there in 970 pints ? 
 
 Ans, 15 bu. 1 gal. 2 pis. 
 
 (58) In 1 score, 16 chaldrons of coals, how many sacks 
 and bushels? Ans. 444 sacks, 1332 bushels. 
 
 TIME. 
 
 (59) In 72015 hours, how many weeks? 
 
 Ans. 428 weeks, 4 days, 15 hours. 
 
 (60) How many days were there from the birth of Christ, 
 to Christmas, 1794, estimating 365\ days to the year ? 
 
 Ans. 655258^ days. 
 
 (61) Stowe writes, that London was built 1108 years 
 before our Saviour's birth. Find the number of hours to 
 Christmas, 1794? Ans. 25438932 hours. 
 
 (62) From July 18th, 1799, to April 18th, 1826, how 
 many days ? Ans. 9770^ days, reckoning S65\ days to a year. 
 
 (63) In a lunar month, containing 29 days, 12 hours, 44 
 minutes, 2 seconds and eight- tenths, how many tenth parts 
 of seconds? Ans. 25514428. 
 
 (64) How many seconds are there in 18 centuries, estima- 
 ting the solar year at 365 days, 5 hours, 48 minutes, 51 1 
 seconds? Ans. 56802476700 seconds. 
 
 COMPOUND ADDITION 
 
 Teaches to find the sum of Compound Quantities. 
 
 Rule. Add the numbers of the least denomination; divide 
 the sum by as many as make one of the next greater ; set 
 down the remainder (if any) and carry the quotient to tnose 
 of the next greater : proceed thus to the greatest denomination, 
 which add as in Simple Addition. 
 
 Proof. As in Simple Addition. 
 
 Example. Say 1, 2, 5, 7 farthings are 1 penny 3 far- 
 
 £. s. d. things; set down | and carry Id — 1, 10, 11, 
 
 15.. 7.. 4| 16, 20, 30, 40d. are 3*. 4d. ; "set down Ad. and 
 
 7..18..10} carry 3s — 3, 12, 20, 27, 37, 47, 57s. are £2. 17s. ; 
 
 11.. 19.. 5 set down 17s. and carry £2. The rest as in Sim- 
 
 6..10..11£ pie Addition. 
 
 4.. ().. 9| Ik Addition cf Money, the reduction of one 
 
 45~17 S denomination to the next greater is generally 
 
 .*_!! — 11__I done without the trouble of dividing, by the 
 
 knowledge previously acquired of the Money 
 
 Tables. 
 
86 
 
 
 COMPOUND ADDITION. 
 
 
 [the tutor's 
 
 
 MONEY. 
 
 > (1 > 
 
 (4) 
 
 (7) 
 
 (10) 
 
 £. 9. d. 
 
 £. s. d. 
 
 £. A d. 
 
 £. f. J. 
 
 2 13 54 
 
 15 3 
 
 21 14 7i 
 
 261 17 ll 
 
 7 9 H 
 
 54 17 1 
 
 75 16 
 
 379 13 5 
 
 5 15 41 
 
 91 15 111 
 
 79 2 41 
 
 257 16 7f 
 
 9 17 6i 
 
 35 16 If 
 
 57 16 5£ 
 
 184 13 5 
 
 7 16 3 
 
 29 19 Hi 
 
 26 13 8f 
 
 725 2 31 
 
 5 14 7| 
 
 91 17 31 
 
 54 2 7 
 
 
 359 6 5 
 
 (2) 
 
 (5) 
 
 (8) 
 
 (11) 
 
 £. *. d. 
 
 £. s. d. 
 
 £. «. d. 
 
 <£. *. d. 
 
 27 7 
 
 257 1 51 
 
 13 2 1| 
 
 31 1 11 
 
 34 14 10l 
 
 734 3 7f 
 
 25 12 7 
 
 75 13 1 
 
 57 19 2i 
 
 595 5 3 
 
 96 13 51 
 
 39 19 7J 
 
 91 16 
 
 159 14 7i 
 
 76 17 3\ 
 91 14 l| 
 
 97 17 31 
 
 75 18 7f 
 
 207 5 4 
 
 36 13 5 
 
 97 13 5 
 
 798 16 7i 
 
 54 11 71 
 
 
 24 16 31 
 
 (3) 
 
 , (6 ) 
 
 . (9) 
 
 (12) 
 
 £. *. d. 
 
 £. *. rf. 
 
 £. s. d. 
 
 £. s. d. 
 
 35 17 
 
 525 2 41 
 
 127 4 71 
 
 27 13 5i 
 
 59 14 10£ 
 
 179 3 5 
 
 525 3 10 
 
 16 12 101 
 
 97 13 101 
 
 250 4 71 
 
 271 
 
 9 13 01 
 
 37 16 81 
 
 975 3 51 
 
 524 9 1 
 
 15 2 101 
 
 97 15 7 
 
 254 5 7 
 
 379 4 01 
 
 37 19 
 
 59 16 01 
 
 379 4 5f 
 
 215 5 llf 
 
 56 19 If 
 
 1 
 
 HEIGHTS AN 
 
 D MEASURES. 
 
 TROY WE 
 
 IGIIT. 
 
 
 apothecaries' weight. 
 
 (13) 
 
 
 (14) 
 
 
 (15) 
 
 (16) 
 
 oz. dwt. gr. 
 
 lb. 
 
 oz. dwt. gr. 
 
 ft 
 
 >. 5- 3- B- 
 
 5- 3- 9-£r. 
 
 5 11 4 
 
 5 
 
 9 15 22 
 
 1 
 
 7 10 7 1 
 
 2 1 12 
 
 7 19 21 
 
 3 
 
 11 17 14 
 
 
 9 5 2 2 
 
 1 7 1 17 
 
 3 15 14 
 
 3 
 
 7 15 19 
 
 2 
 
 7 U 1 2 
 
 10 2 14 
 
 7 19 22 
 
 9 
 
 1 13 21 
 
 
 9 5 6 1 
 
 5 7 1 15 
 
 9 18 15 
 
 3 
 
 9 7 23 
 
 3 
 
 7 10 5 2 
 
 9 5 2 13 
 
 8 13 12 
 
 5 
 
 2 15 17 
 
 4 
 
 9 7 
 
 1 4 1 18 
 
 
 
 
 
 
 
 
 u i~ 
 
ASSISTANT.] 
 
 COMPOUND ADDITION. 
 
 SI 
 
 AVOIRDUPOIS WEIGHT. 
 
 (17) lb. oz. dr. 
 
 152 15 15 
 
 272 14 10 
 
 303 15 11 
 
 255 10 4 
 
 173 6 2 
 
 635 13 13 
 
 (20) yds. ft. in. 
 
 (18) crvt.qrs.lb. 
 
 25 1 17 
 
 72 3 26 
 
 54 1 16 
 24 1 16 
 17 19 
 
 55 2 16 
 
 (19) t.cwt. qrs. lb. 
 
 7 17 2 12 
 
 5 5 3 14 
 
 2 4 1 17 
 
 3 18 2 19 
 
 7 9 3 20 
 
 8 5 1 24 
 
 225 
 171 
 52 
 397 
 154 
 137 
 
 1 
 
 3 
 
 2 6 
 
 10 
 
 2 7 
 
 1 4 
 
 /I 
 
 — 
 
 _ 
 
 LONG MEASURE. 
 
 (21) lea. m.fur.po. 
 
 72 2 1 19 
 
 27 1 7 22 
 
 35 2 5 31 
 
 79 6 12 
 
 51 1 6 17 
 
 72 5 21 
 
 (22) 7W. y^r. j/cfa. 
 39 6 36 
 14 
 3 
 45 
 17 
 32 
 
 214 
 160 
 
 202 
 
 19 
 176 
 
 CLOTH M 
 
 (23) 
 
 yds. qrs. n. 
 
 135 3 3 
 
 70 2 2 
 
 95 3 
 
 176 1 3 
 
 26 1 
 
 279 2 1 
 
 EASURE. 
 
 — 
 
 (24) 
 E. e. qrs. 
 
 272 2 
 152 1 
 
 n. 
 1 
 
 2 
 
 79 
 156 2 
 
 1 
 
 
 19 3 
 154 2 
 
 1 
 
 1 
 
 
 LAND MEASURE. 
 
 (25) 
 a. r. p. 
 726 1 31 
 219 2 17 
 
 1455 
 
 879 
 438 
 
 3 14 
 
 1 21 
 
 2 14 
 
 (26) 
 
 a. r. p. 
 
 1232 1 14 
 
 327 19 
 
 131 2 15 
 
 1219 1 18 
 
 223 2 8 
 
 757 0! 236 9 
 
 IMPERIAL MEASURE. 
 
 (27) 
 hhds.gals.qts. 
 31 57 1 
 97 18 
 76 13 
 55 46 
 87 38 
 55 17 
 
 (28) 
 t. hkd.gals. qts. 
 
 14 
 19 
 17 
 15 
 
 54 
 97 
 
 3 <T 
 2 56 
 
 39 
 
 2 16 
 
 1 19 
 
 3 54 
 
 ALE AND BEER. 
 
 (29) 
 bar.Jir. gal. 
 
 25 2 7 
 
 3 5 
 
 2 6 
 1 8 
 
 3 7 
 5 
 
 17 
 
 96 
 75 
 96 
 15 
 
 (30) 
 /zM. gal. qts. 
 16 51 2 
 3 
 
 57 
 97 27 
 22 17 
 
 32 
 55 
 
 19 
 
 38 
 
38 
 
 COMPOUND ADDITION. 
 
 Qthe tutor's 
 
 DRY, 
 
 (31) 
 
 qrs. 
 
 b. p. 
 
 300 
 
 2 1 
 
 167 
 
 I 
 
 369 
 
 7 
 
 50 
 
 3 2 
 
 74 
 
 6 3 
 
 
 r. 
 
 b. 
 16 
 
 (32) 
 p. gal. qts. 
 
 2 1 2 
 
 21 
 
 
 
 1 
 
 3 
 
 7 
 
 3 
 
 
 
 
 
 15 
 
 1 
 
 1 
 
 2 
 
 3 
 
 2 
 
 
 
 1 
 
 
 
 (33) 
 
 
 (34) 
 
 w. 
 
 d. A. 
 
 TV. 
 
 e?. A. m. a-. 
 
 71 
 
 3 11 
 
 51 
 
 2 15 42 41 
 
 51 
 
 2 9 
 
 95 
 
 3 21 27 51 
 
 76 
 
 21 
 
 76 
 
 15 37 28 
 
 95 
 
 3 21 
 
 53 
 
 2 21 42 27 
 
 79 
 
 1 15 
 
 98 
 
 2 18 47 38 
 
 
 
 (35) A, B, C, and D, were partners in the purchase of a 
 quantity of goods : A laid out £7. half-a-guinea, and a crown ; 
 B, 49*. C, 54*. 6d. and D, 87 d. What was the purchase ? 
 
 Ans. £l3..6..3. 
 
 (36) A man lent his friend at different times these several 
 sums, viz. £63.— £25..15.— £32.-7.— £l5..14..10. and four 
 score and nineteen pounds, half-a-guinea, and a shilling. 
 How much was the whole loan? Ans. £236..8..4. 
 
 (37) Bought goods, for which I paid £54.. 17; for packing 
 13*. 8d; carriage £l..5..4 ; and expenses over making the bar- 
 gain 14*. 3d. What was the whole cost ? Ans. £57..10..3. 
 
 (38) A nobleman, previous to quitting town, wished to dis< 
 charge his tradesmen's bills. On enquiry he found that he 
 owed 82 guineas for rent ; — to his wine-merchant, £72. .5 ; — ■ 
 to his confectioner, £l2..13..4 ;— to his draper, £47..13..2 ;— 
 to his tailor, £ll0..15..6; — to his coach-maker, £l57-8 ; — to 
 his tallow-chandler, £8..17~9; — to his corn-factor, £l70..6.. 8; 
 —to his brewer, £52..17-0 ;— to his butcher, £l22..11..5;— 
 to his baker, £37..9-5 ; — and to his servants for wages, 
 £53.. 18. What money must he draw from his banker, in- 
 cluding £100. that he wished to take with him ? 
 
 Ans. £l032..17..3. 
 
 (39) A father was 24 years of age (allowing 13 months to 
 a year, and 28 days to a month) at the birth of his first 
 child ; between the eldest and next born was 1 year, 1 1 
 months, and 14 days; between the second and third were 2 
 years, 1 month, and 1 5 days ; between the third and fourth, 
 2 years, 10 months, and 25 days. When the fourth was 27 
 years, 9 months, and 12 days old, what age was the father? 
 
 Ans. 58 years, 7 months, 10 days. 
 
 (40) A clerk having been out collecting debts, presented 
 an account that A paid him £7-. 5. .2 ; — B £l5..18..6|; — 
 C£l50..13..2^;~D£l7..6..S;— -E 5 guineas, £ crown pieces. 
 
ASSISTANT.] COMPOUND SUBTRACTION. 
 
 39 
 
 4 half-crowns and 4«y. 2d ; — F paid him only twenty groats; — 
 G £76..15..9i;— and H £l21..12..4. How much was the 
 whole amount ? Ans. £396..7..6±. 
 
 (41 ) A nobleman had a service of plate, which consisted of 
 twenty dishes, weighing 203 oz. 8 dwts. ; 36 plates, 408 oz. 
 9 dwts. ; 5 dozen spoons, 112 oz, 8 dwts. ; 6 salts, and 6 
 pepper-boxes, 71 oz. 7 dwts. ; knives and forks, 73 oz. 5 dwts. ; 
 two large cups, a tankard, and a mug, 121 oz. 4>dwts.; a tea- 
 urn and lamp, 131 oz. 7 dwts. ; with sundry other small ar- 
 ticles, weighing 105 oz. 5 dwts. The weight of the whole 
 is required? Ans. 102 lb. 2 oz. 13 dwts. 
 
 (42) A hop-merchant buys 5 bags of hope, of which the 
 first weighed 2 cwt. 3 qrs. 13 lb.; the second, 2 cwt. 2 qrs. 
 
 1 1 lb. ; the third, 2 cwt. 3 qrs. 5 lb. ; the fourth, 2 cwt. 3 qrs. 
 
 1 2 lb. ; the fifth, 2 cwt. 3 qrs. 1 5 lb. He purchased also 
 two pockets, each pocket weighing 84 lb. I desire to know 
 the weight of the whole. Ans. 15 cwt. 2 qrs. 
 
 COMPOUND SUBTRACTION 
 
 Teaches to find the difference of Compound Quantities. 
 
 Rule. Subtract as in integers : but borrow (when there is 
 occasion) as many as are equal to one of the next greater de- 
 nomination : observing to carry one to the next for that which 
 was borrowed.* 
 
 Proof. As in Simple Subtraction. 
 
 (1) £. s. 
 From 715 2 
 Take 476 3 
 
 <4 
 
 MONEY 
 
 (2) £. . 
 316 i 
 218 ! 
 
 d. 
 
 (3) 
 
 £. 
 
 ,v. 
 
 d. 
 
 87 
 
 2 
 
 10 
 
 79 
 
 3 
 
 n 
 
 * Example. Subtract £54..17..9f. from £%S..W..1\. 
 
 £. s. d. 
 
 89.. 12.. % 
 
 54..17.. 9| 
 
 ~3-k7l4.. 9| 
 
 Because 3 farthings cannot be taken from 2, say 3 
 from 4, 1, and 2 are 3 ; set down 3 and carry 1 — 1 and 
 9 are 10, 10 from 12, 2, and 7 are 9 ; set down 9 and 
 carry 1 — 1 and 17 are 18, 18 from 20, 2, and 12 are 
 14 ; set down 14 and carry 1 to the pounds. 
 
40 
 
 COMPOUND SUBTRACTION. [THE TUTOR'S 
 
 (4) 
 £. s. d. 
 
 £. s. d. 
 
 (10) 
 £. s. d. 
 
 £. s. d. 
 
 3 15 l£ 
 
 321 17 l£ 
 
 527 3 5± 
 
 10 7 6 
 
 1 14 7 
 
 257 14 7 
 
 139 5 7| 
 
 9 19 7 
 
 (5) 
 £. s. d. 
 
 (8) 
 £, s. d. 
 
 £. s. d. 
 
 £. s. d. 
 
 25 2 5{ 
 
 59 15 Si 
 
 300 15 
 
 500 
 
 17 9 84 
 
 36 17 2 
 
 296 15 10 
 
 499 19 H| 
 
 £. s. d. 
 
 (9) 
 £. s. d. 
 
 (12) 
 £. s. d. 
 
 £. s. d. 
 
 37 3 4J 
 
 71 2 4 
 
 68 13 9 
 
 779 12 
 
 25 5 2| 
 
 19 18 7-f 
 
 44 19 101 
 
 689 13 6 
 
 16) 
 
 Borrowed 
 
 £. *. A 
 
 350 
 
 (17) £. s. d. 
 Lent 577 10 
 
 Paid at 
 
 different x 
 
 times 
 
 '26 5 
 73 10 6 
 41 9 8i 
 
 ^66 14 9 
 
 Received 
 
 at several 
 
 times 
 
 f 95 10 
 J 80 
 I 74 15 9 
 
 I 23 17 44 
 
 Paid in all 
 
 
 
 
 temainsto pay 
 
 
 WEIGHTS AND MEASURES. 
 
 TROY WEIGHT. APOTHECARIES' WEIGHT. 
 
 08) 
 
 lb. oz. drvt. gr. 
 52 1 7 2 
 39 15 7 
 
 (19) 
 lb. oz.dwt.gr. 
 
 7 2 2 r 
 5 7 15 
 
 (20) 
 
 ft). §. 3- 0. 
 5 2 10 
 2 5 2 1 
 
 lb 
 
 (21) 
 
 9 7 2 1 13 
 5 7 3 1 18 
 
 (22) lb. oz. dr. 
 35 10 5 
 29 12 7 
 
 AVOIRDUPOIS WEIGHT. 
 
 (23) cwt. qr. lb. 
 35 1 21 
 25 1 27 
 
 (24) t. cwt. qrs. lb. 
 21 1 2 7 
 9 11 3 15 
 
ASSISTANT.] COMPOUND SUBTRACTION. 
 
 41 
 
 LONG MEASURE. 
 
 (25) 
 
 yds. ft. in. 
 
 107 2 10 
 
 78 2 11 
 
 (26) 
 lea. mi.fur.po. 
 147 2 6 29 
 58 2 7 S3 
 
 CLOTH MEASURE. 
 
 (27) 
 
 (28) 
 
 yds. qrs, n. 
 
 j£. e. grs. n. 
 
 71 1 2 
 
 35 2 1 
 
 3 2 1 
 
 14 3 2 
 
 
 
 LAND T 
 
 IEASURE. 
 
 (29) 
 
 (SO) 
 
 «. 7*. p. 
 
 a. r. p. 
 
 17^ 1 27 
 
 325 2 1 
 
 59 37 
 
 279 3 5 
 
 IMPERIAL MEASURE WINE. 
 
 (SI) 
 hhd.gal.qts.pts. 
 
 47 47 2 1 
 28 59 3 
 
 (32) 
 
 tun hhd.gal. qts 
 
 42 2 37 2 
 
 17 3 49 3 
 
 ALE AND BEER. 
 
 (33) / ; _ (34) 
 
 «r. j£r. gal. 
 37 2 1 
 25 1 7 
 
 hhd.gal. qts. 
 
 27 27 1 
 
 12 50 2 
 
 CORN AND COAL. 
 
 (.35) 
 
 <?r. b. p. 
 
 65 2 1 
 
 57 2 3 
 
 * (37) #ro. two. rv. a. 
 79 8 2 4 
 23 9 3 5 
 
 TIME. 
 
 (38) h. m, sec. 
 24 42 45 
 19 53 47 
 
 (36) 
 
 sc. ch. sa. b. 
 
 3 16 1 
 
 2 12 2 1 
 
 f(39)yrs, m. d. 
 
 10 7 20 
 
 5 8 29 
 
 (40) When an estate of £300. per annum is reduced by the 
 payment of taxes, to 12 score and £l4..6. what are the taxes ? 
 
 Ans. £45.. 14. 
 
 (41) A horse with his furniture is worth £37»5; without 
 it, 14 guineas ; how much does the price of the furniture 
 exceed that of the horse ? Ans. £7-. 1 7. 
 
 (42) A merchant commencing trade, owed £750 ; he had 
 in cash, commodities, the stocks, and good debts, £l2510..7 ; 
 he cleared the first year by commerce £452..3..6. What was 
 he then worth ? Ans. £l2212..10..6. 
 
 (43) A gentleman left £45247. to his two daughters, of 
 
 In this example allow 4 weeks to a month, and 13 months to the 
 
 year. 
 
 t In this, reckon 30 days to a month, and 12 months to the year. 
 
42 COMPOUND MULTIPLICATION. [THE TUTOR'S 
 
 which the younger was to have 15 thousand, 15 hundred, and 
 twice £l5. What was the elder sister's fortune? 
 
 Arts. £28717- 
 
 (44) A tradesman being insolvent, called all his creditors 
 together, and found he owed to A £53..7»6 ; — to B £ 1 05.. 1 ; 
 — to C £34..5..2 ;— to D £28..l6..5;— -to E £l4..15..8 ;— to 
 F £ll2..9 ;-— and to G £l43..12..9. The value of his stock 
 was £212..6; and the amount of good book-debts was 
 £ll2..8..3 ; besides £21..10..5. money in hand. How much 
 would his creditors lose by taking the whole of his effects ? 
 
 Ans. The creditors lost £ 146.. 11. .10. 
 
 (45) My agent at Seville, in Spain, renders me the fol- 
 lowing account of money received for the sale of goods 
 Bent him on commission, viz. for bees' wax £37-1 5..4 ; stock- 
 ings £37»6..7 ; tobacco £l25..11..6; linen cloth£l42..14..8 ; 
 tin £l 15..10..5. He informs me at the same time, that he has 
 shipped, agreeably to my order, wines, value £250.. 15 ; fruit 
 £51..12..6; %s£l9..17..6; oil£l9..12..4; and Spanish wool, 
 value £ll5..15..6. How stands the balance of the account 
 between us? Ans. Due to the agent £28. .14.. 4. 
 
 (46) The great bell at Oxford, the heaviest in England, i» 
 stated to weigh 7 tons, 1 1 cwt. 3 qrs. 4 lbs. that of St. Paul'* 
 in London, 5 tons, 2 cwt. 1 qr. 22 lbs. and that of Lincoln, 
 called the Great Tom, 4 tons, 16 cwt. 3 qrs. 16 lbs. How 
 much is the aggregate weight of these three bells inferior to 
 that of the great bell at Moscow, which is 198 tons? 
 
 Ans. 180 tons, 8 cwt. 3 qrs. 14 lbs. 
 
 COMPOUND MULTIPLICATION 
 
 Is the method of multiplying Compound Quantities. 
 
 Rule. Multiply the least denomination ; reduce the product 
 and carry to the next as directed in Compound Addition ; 
 and the same with the rest. 
 
 When the multiplier is a composite number above 12, mul- 
 tiply (as before directed) by its component parts. For other 
 numbers, multiply by the factors of the nearest composite ; 
 adding to the last product, so many times the top line as will 
 supply the deficiency ; or subtracting so many times, if there 
 is an excess. 
 
ASSISTANT.] COMPOUND MULTIPLICATION 
 
 43 
 
 
 
 MONEY. 
 
 
 
 ••(1) 
 
 (2) 
 
 (3) 
 
 W 
 
 £. s. d. 
 
 £. *. d. 
 
 £. ft d. 
 
 £. ft £/. 
 
 35 12 7f 
 
 15 13 l£ 
 
 62 5 4J 
 
 57 2 4| 
 
 2 
 
 3 
 
 4 
 
 5 
 
 71 5 3^ 
 
 
 
 
 £. ft 
 
 <*. 
 
 £. ft d. 
 
 (5) 51 18 7i X 6. 
 
 (9) 135 13 6f X 10. 
 
 (6) 81 9 il| X 7. 
 
 (10) 79 16 7± X 11. 
 
 (7) 64 10 5 X 8. 
 
 (11)247 14 III x 12. 
 
 (8)118 6 4i X 9- 
 
 (12) 119 7 5f X 12. 
 
 £. ft d. 
 
 ft 
 
 d. £. s. d. 
 
 (IS) 096 xl8.t 
 
 (16)15 
 
 S±X35. 
 
 (19) 1 5 3X97. 
 
 (14) 1 2 6 X2(1J 
 
 (17) 7 
 
 2|X75. 
 
 (20) 6 4X43. 
 
 (15)0 7 8£x2i. 
 
 (18) 9 
 
 7 X37. 
 
 
 
 (2 1 ) What is the value of 127 lb. of souchong tea, at 12ft 3d, 
 per lb.? Ans. £77..15..9. 
 
 (22) 135 stones of soap, at 7ft 5d. per stone ? Ans. £50.. 1.. 3. 
 
 (23) 74 ells of diaper, at 1ft 4>\d. per ell? Ans. £5..1..9. 
 
 (24) 6doz. pairs of gloves, at 1ft 10c?. per pair? Aus. £6..12. 
 Note. When the fraction £, 4» or | is connected with the multiplier, 
 
 Aike fta//* the given price (or the price of one) for ^, Aa/f of that for J, 
 and for I, add them both together. § 
 
 * In this example, say twice 3 are 6, 6 farthings are \\d. set down 
 { -,d. and carry 1 ; twice 7 are 14 and 1 are 15, 15d. are Is. 3d. set down 
 3d. and carry 1 ; twice 12 are 24 and 1 are 25, 25*. are£1..5. set down 
 5s. and carry 1 ; twice 5 are 10 and 1 are 11, set down 1 and carry 1 ; 
 twice 3 are G and 1 are 7, set down 7. 
 
 s. d. £. s. d. 
 
 f 9.. 6 t 1- 2.. 6 
 
 2X9=18 8X3+2=26 
 
 19.. 
 9 
 
 9.. 0.. 
 3 
 
 £8.. 11.. Ans. 
 
 § Example. 
 
 What is the value of 
 11} lbs. of tea, at 10*. 9d. 
 per lb. ? 
 
 27.. 0.. 
 MultiplicandX2==2.. 5.. 
 
 £29,. 5.. Ans. 
 8. d. 
 
 IX 10.. 9 
 
 11 
 
 £5..18.. 3 =thevalueofll. 
 
 \ X 5.. 4i= ... do |. 
 
 2.. 8-|== ... do \. 
 
 £6.. 6.. 3| Ans. 
 
44 COMPOUND MULTIPLICATION. [THE TUTOR* 
 
 (25) What is the value of 25| ells of Holland, at 3s. 4|rf. 
 per ell ? Ans. £4..6..0f . 
 
 (26) 75^ lb. of hemp, at Is. 3d. per lb. ? 
 
 Ans. £4..14..4 i |. 
 
 (27) 19j yds. of muslin, at 4,y. 3d. per yd. ? Ans. £4..2..10| 
 
 (28) 35^ cwt. of raw sugar, at £4..15..6. per cwt. ? 
 
 Ans. £l6g.A0..3. 
 
 (29) 154^ cwt. of raisins, at £4..17..10. per cwt.? 
 
 Ans. £755..15..3. 
 
 (30) 1 \7\ gallons of gin, at 12s. 6d. per gallon ? 
 
 Ans. £73..5..7|. 
 
 (31) 85f cwt. of logwood, at £l..7..8. per cwt? 
 
 Ans. £ll8..12..5. 
 
 (32) 1 7f yards of superfine scarlet cloth, at £l..3..6. per 
 yard? Ans. £20..17..l£. 
 
 (33) 37 \ lb. of hyson tea, at 12*. 4>d. per ft). ? Ans. £23..2..(). 
 
 (34) 5()| cwt. of molasses, at £2.. 1 8.-7. per cwt. ? 
 
 Ans. £l66.A..7i. 
 
 (35) 87| ft), of Turkey coffee, at 4*. 3d. per lb. ? 
 
 Ans. £18.. 12.. 111. 
 
 (36') 120f cwt. of hops, at £4..7-.6. per cwt. ? 
 
 Ans. £528..5..7|. 
 
 When the multiplier is large, multiply the given quantity 
 (or price) by a series of tens, to find 10, 100, 1000 times, &c, 
 as far as to the value of the highest place of the multiplier ; mul- 
 tiply the last product by the figure in that place, and each 
 preceding product by the figure of corresponding value; that 
 is, the product for 100 by the number of hundreds, the product 
 for 10 by the number of tens, and the original quantity by the 
 units* figure, §c. The sum of the products thus obtained will 
 be the total product* 
 
 • Example. Multiply £7..U..9\. by 3645. 
 
 £. s. d. £. s. d. times 
 
 T..14.. 9^X5= 38..13..11|= 5 
 10 
 
 The product for 10 77.. 7..11 X4= 309..11.. 8 = 40 
 
 10 
 
 The product for 100 773..19.. 2 X6= 4643..15.. = 600 
 10 
 
 The product for 1000 7739..11.. 8 X3=23218..I5.. = 3000 
 
 Am. 2$mO.A5..T^=:~36U 
 
ASSISTANT.] COMPOUND MULTIPLICATION. 4>6 
 
 (37) 407 lb. of gall-nuts, at 3s. 9±d. per ft. ? Ans. £l7..S..9.\ . 
 
 (38) 729 stones of beef, at 7*. 7^. per stone? 
 
 Ans. £277..3..5! 
 
 (39) 2068 yards of lace, at 9s. 5|rf. per yard? 
 
 Ans. £977..19..10. 
 
 (40) What is the produce of a toll-gate in the course of the 
 year, if the tolls amount, on an average, to 11*. 7±d. per 
 day? Ans. £212..3..l|, 
 
 (41 ) How much money must be equally divided among 1 8 
 men, to give each £l4..6..8| ? Ans. £258..0..9. 
 
 (42) A privateer manned with 250 sailors captured a prize, 
 of which each man shared £l25..15..6. What was the value 
 of the prize? Ans. £31443..15. 
 
 (43) What sum did a gentleman receive as a dower with 
 his wife, whose fortune w r as a cabinet with two divisions* in 
 each division 87 drawers, and each drawer containing 21 
 guineas? Ans. £3836..14. 
 
 (44) A merchant began trade with £19118; for 5 years 
 together he cleared £1086. a year; and the next 4 years 
 £2715..10..6. a year; but the last 3 years he was in trade he 
 listd the misfortune to lose upon an average, £475..4..6. a 
 vear. What was his real fortune at the end of the 12 years ? 
 
 Ans. £33984<..8'..6. 
 
 (45) In many parts of the kingdom coals are weighed in the 
 wagon or cart upon a machine, constructed for the purpose. 
 If three of these draughts amounted together to 137 cwt. 
 2 qrs. 10 lb. ; and the tare, or weight of the wagon, was 13 
 cwt. 1 qr. ; how many coals had the customer in 12 such 
 draughts? Ans. 391 cwt. 1 qr. 12 lb. 
 
 (46) A certain gentleman lays up every year £294..12..6. 
 and spends daily £l,.12..6. What is his annual income? 
 
 Ans. £887.-15 
 
 WEIGHTS AND MEASURES. 
 
 (47) Multiply 9 lb. 10 oz. 15 dwts. 19 gr. by 9, 11, and 12. 
 
 (48) Multiply 23 tons, 9 cwt. 3 qrs. 18 lb. by 7, 8, and 9. 
 
 (49) Multiply 107 yards, 3 qrs. 2 nails, by 10, 17, and 29. 
 
 (50) Multiply 33 bar. 2J7. 3 gal. by 11, and 12. 
 
 (51) Multiply 110 miles, 6 fur. 26 poles, by 12, 13, and 39. 
 
 (52) A lunar month contains 29 days, 12 hours, 44 min 
 S seconds nearly. What time is contained in 13 lunar 
 months ? 
 
46 
 
 COMPOUND DIVISION. 
 
 [the TUTOR'S 
 
 COMPOUND DIVISION 
 
 Teaches to find any required part of a Compound quantity* 
 
 Rule. Divide the greatest denomination : reduce the re- 
 mainder to the next less, to which add the next ; divide that, 
 and proceed as before to the end. 
 
 When the divisor is above 1 2, the work must be done at 
 length: unless it is a composite number, for which observe 
 the directions in Simple Division. — Proof by Multiplication. 
 
 
 MONEY. 
 
 
 *(1) 
 
 £. s. d. 
 
 (2) 
 £. .9. d. 
 
 £. s. d. 
 
 £. s. d. 
 
 1)25 2 4 
 
 3)37 7 7 
 
 4)57 5 7 
 
 5)52 7 
 
 
 
 £. s. 
 
 d. 
 
 £. s. 
 
 d. 
 
 (5) 78 10 9i -*- 6. 
 
 (6) 25 19 7| -S- 7- 
 
 (7) 16 14 li ~ 8. 
 
 (8) 124 15 2\ -r- 9- 
 
 (9) 87 14 by 10. 
 
 (10) 68 by 11. 
 
 (11) 49 14 7 by 12. 
 
 (12) 496 8 6 by 12. 
 
 (13) 66 t 
 
 (14) 596 \ C A 
 
 (15) 564 4 
 
 ) 6| ~- 25. 
 5 7\ ~ 36. 
 t 6 -j- 63. 
 
 (16) 248 17 4 by 99- 
 
 (17) 928 12 8 by 110. 
 
 (18) 608 13 9 by 144 
 
 (19) Divide £1407.. 17»7. by 243. 
 
 (20) Divide £700791.-1 4..4. by 179*. 
 
 (21) Divide £490981..3..7|. by 31715. 
 
 (22) Divide £l9743052..5..7i. by 214723. 
 
 (23) If a man spend £257..2..5. in 12 months, what is that 
 per month? Ans. £2\..8..6\ T %. 
 
 (24) The clothing of 35 charity boys came to £57..3..7. 
 what was the expense of each boy? Ans. £l..l2..8if. 
 
 for nine pieces of cloth, what 
 Ans. £4,..2..11%. 
 
 (25) If I gave £37-.6..4| 
 was that per piece ? 
 
 * Example. 
 
 £. s. d. 
 6)27. .14..11£ 
 
 5..10..113 § 
 
 Divide X27..14.. 11£. by 5. 
 
 Say the fives in 27, 5 times 5 are 25 and 2 over ; 
 £2. are 40*. and 14 are 54, the fives in 54, 10 times 
 5 are 50 and 4 over; is. are 48d. and 11 are 59, 
 the fives in 59, 11 fives are 55 and 4 over; A>d. are 
 16 qrs. and 2 are 18, the fives in 18, 3 times five 
 are 15, and 3 over, or §• 
 
ASSISTANT.] PROMISCUOUS EXAMPLES. 4? 
 
 (26) If 20 cwt. of tobacco cost £27»5..4^ ; at what rate did 
 I buy it per cwt. f Arts. £l..7..3J§. 
 
 (2*7) What is the value of one hogshead of beer, when 120 
 hogsheads are sold for £l54..17..10? Ans. £l..5..9f j\%. 
 
 (28) Bought 72 yards of cloth for £85.. 6. What was the 
 price per yard ? Ans. £l..3..8£ f %. 
 
 (29) Gave £275..3..4. for 18 bales of cloth. What is the 
 price of one bale ? Am. £l5..5..8§ \\. 
 
 (30) A prize of £7257-3..6. is to be equally divided among 
 500 sailors. What is each man's share ? Ans. £l4..10..3£ f §|j. 
 
 (31) A club of 25 persons joined to purchase a lottery 
 ticket of £10. value, which was drawn a prize of £4000. 
 What was each man's contribution, and his share of the prize- 
 money ? Ans. each contribution 8s. and share of prize £l60. 
 
 (32) A tradesman cleared £2805. in 7^ years ; what was 
 his yearly profit ? Ans. £374. 
 
 (33) What was the weekly salary of a clerk who received 
 £266-.18..l£. for 90 weeks? Ans. £2..19»3|. 
 
 (34) If 100000 quills cost me £l87..17»l. what is the 
 price per thousand ? Ans. £l..l7»6f i 4 qV 
 
 WEIGHTS AND MEASURES. 
 
 (35) Divide 83 lb. 5 oz. 10 dwts. 17 gr. by 8, 10, and 12. 
 
 (36) Divide 29 tons, 17 cwt. qrs. 18 lb. by 9, 15, and If). 
 
 (37) Divide 114 yards, 3 qrs. 2 nails, by 10, and 16. 
 
 (38) Divide 1017 miles, 6 fur. 38 poles, by 11, and 49- 
 
 (39) Divide 2019 acres, 3 roods, 29 perches, by 26. 
 
 (40) Divide 117 years, 7 months, 26 days, 11 hours, 27 
 minutes, by 37. 
 
 PROMISCUOUS EXAMPLES. 
 
 (1) Of three numbers, the first is 2 15, the second 519, an d 
 the third is equal to the other two. What is the sum of 
 them all? Ans. 1468. 
 
 (2) The less of two sums of money is £40. and their dif- 
 ference £14. What is the greater sum, and the amount of 
 both ? Ans. £54. the greater, £94. the sum. 
 
 (3) What number added to ten thousand and eighty-nine, 
 will make the sum fifteen thousand and forty ? Ans. 4951. 
 
 (4) What is the difference between six dozen dozen, and 
 ftalf a dozen dozen ; and what is their sum and product ? 
 
 Ans. diff. 792, sum 936, product 62208. 
 
48 PROMISCUOUS EXAMPLES. [iHE TUTOR'S 
 
 (5) What difference is there between twice eight and fifty 
 and twice fifty-eight, and what is their product ? 
 
 Ans* 50 diff. 7656 product. 
 
 (6) The greater of two numbers is 37 times 45, and their 
 difference is 19 times 4: required their sum and product? 
 
 Ans. 3254 sum, 26*45685 product. 
 
 (7) A gentleman left his elder daughter £i500. more than 
 the younger, whose fortune was 1 1 thousand, 1 1 hundred, 
 and £ll. Find the portion of the elder, and the amount 01 
 both. Ans. Elder s portion £ 13611. amount £25722. 
 
 (8) The sum of two numbers is 360, the less is 144. 
 What is their difference and their product ? 
 
 Ans. 72 difference, 31104 product. 
 
 (9) There are 2545 bullocks to be divided among 509 men. 
 Required the number and the value of each man's share, 
 
 posing every bullock worth £9„14..6? 
 Ans. Each man had 5 bullocks, and £A8..12..6.Jbr his share. 
 
 (10) How many cubic feet are contained in a room, the 
 length of which is 24 feet, the breadth 14 feet, and the height 
 11 feet?* Ans. 3696. 
 
 (11) A gentleman's garden containing 9625 square yards> 
 is 35 yards broad: what is the length? Ans. 27 5 yards. 
 
 (12) What sum added to the 43rd part of £4429. will make 
 the total amount=£240 ? Ans. £137. 
 
 (13) Divide 20s. among A, B, and C, so that A may have 
 2s. less than B, and C 2s. more than B. 
 
 Ans. A 4s. 8d. B 6s. 8d. and C 8s. 8d. 
 
 (14) In an army consisting of 187 squadrons of horse, each 
 157 men, and 207 battalions of foot, each 560 men, how many 
 effective soldiers are there, supposing that in 7 hospitals there 
 are 473 sick? Ans. 144806. 
 
 (15) A tradesman gave his daughter, as a marriage portion, 
 a scrutoire, containing 12 drawers; in each drawer were six 
 divisions, and in each division there were £50. four crown 
 pieces, and eight half-crown pieces. How much had she to 
 her fortune? Ans. £3744. 
 
 (16) There are 1000 men in a regiment, of whom 50 are 
 officers : how many privates are there to one officer ? Ans. 1 9« 
 
 (17) What number must 7847 be multiplied by, to produce 
 3013248? Ans. 384. 
 
 * Multiply the three dimensions continua. ly together 
 
7*3 r' — sr f r" 
 
 ASSISTANT.] PROMISCUOUS QUESTIONS. 4<9 
 
 (18) Suppose I pay eight guineas and half-a-crown for a 
 quarter's rent, but am allowed 1 5s. for repairs ; what does 
 my apartment cost me annually, and how much in seven 
 years ? Ans. In one year, £3 1 . .2. In seven, £217-14. 
 
 (19) The quotient is 1083; the divisor 28604; and the 
 remainder 1788: what is the dividend? Ans. 30979920. 
 
 (20) An assessment was made on a certain hundred, for the 
 sum of £386.. 15.. 6. the amount of the damage done by a riot- 
 ous assemblage. Four parishes paid £37-.14..2. each; four 
 hamlets £31..4..2. each; and four townships £l8..12..6. each: 
 how much was deficient? Ans. £36.. 12. .2. 
 
 (21) An army consisting of 20,000 men, got a booty of 
 £ 12,000; what was each man's share, if the whole were 
 equally divided among them? Ans. 12s. 
 
 (22) A gentleman left by will, to his wife £4560;— to a 
 public charity, £572.. 10 ; — to four nephews, £750.. 10. each ; 
 — to four nieces, £375..12..6. each ; — to thirty poor house- 
 keepers, 10 guineas each ; — and to his executors 150 guineas. 
 What was the amount of his property ? Ans. £l0109..10. 
 
 (23) My purse and money, said Dick to Harry, are worth 
 12 s. 8d. but the money is worth seven times the value of the 
 purse : what did the purse contain ? Ans. lis. id, 
 
 (24) Supposing 20 to be the remainder of a division, 
 423 the quotient, and the divisor the sum of both, plus 19; 
 what is the dividend? Ans. 195446. 
 
 (25) A merchant bought two lots of tobacco, which weigh- 
 ed 12 cwL 3 qrs. 15 lb. for £ll4..15..6; their difference in 
 weight was 1 cwt. 2 qrs. 13 lb. and in price £7.. 15.. 6. Re- 
 quired their respective weights and value ?* 
 
 Ans. Greater weight 7 cwt. 1 qr. value £6l..5..6. 
 Less weight 5 cwt. 2 qrs. 15 lb. value £53.. 10. 
 
 (26) Divide 1000 crowns in such a manner among A, B, 
 and C, that A may receive 129 crowns more than B, and B 
 178 less than C. Ans. A 360 crowns, B 231, C 409- 
 
 (27) If 103 guineas and 7*. be divided among 7 men, how 
 many pounds sterling is the share of each ? Ans. £l5..10. 
 
 (28) A certain person had 25 purses, each purse contain- 
 ing 12 guineas, a crown., and a moidore, how many pounds 
 sterling had he in all ? Ans. £355. 
 
 9 Add the difference to the sum, and divide by 2 for the greater; 
 subtract the difference from the sum, and divide by 2 for the less. 
 
 c 
 
50 PROPORTION. [THE TUTOR S 
 
 (29) A gentleman, in his will, left £ 50. to the poor, and 
 ordered that ^ should be given to old men, each man to 
 have 5s. — \ to old women, each woman to have 9,s, 6d. — 
 } to poor boys, each boy to have 1*, — J to poor girls, each 
 girl to have $d. and the remainder to the person who dis- 
 tributed it : how many of each sort were there, and what re- 
 mained for the person who distributed the money ? 
 
 Ans. 66 men, 100 women, 2,00 boys, 222 girls ; 
 £2. .13. .6. for the distributor. 
 
 (30) A gentleman sent a tankard to his goldsmith, that 
 weighed 50 oz. 8 dwts. to be made into spoons, each weigh- 
 ing 2 oz. \6 drvis. how many would he have? Ans. 18. 
 
 (31) A gentleman has sent to a silversmith 137 oz. 6 dwts. 9 
 gr. of silver, to be made into tankards of 1 7 oz. 15 dwts. 10 gr. 
 each; spoons of 21 oz. 11 dwts. 13 gr. per dozen; salts, of 
 3 oz. 1 dwts. each ; and forks, of 2 1 oz. 11 dwts. 1 3 gr. per 
 dozen ; and for every tankard to have one salt, a dozen spoons, 
 and a dozen forks : what number of each will he have ? 
 
 Ans. Two of each sort, 8 oz. 9 dwts. 9 gr. over. 
 
 (32) How many parcels of sugar of 16 lb. 2 oz. each, are 
 there in 16 cwt. 1 qr. 15 lb. ? 
 
 Ans. 113 parcels, and 12 lb. 14 oz. over. 
 
 (33) In an arc of 7 signs, 14° 3' 53", how many seconds? 
 
 Ans. 806633". 
 
 (34) How many lbs. of lead would counterpoise a mass of 
 bullion weighing 100 lbs Troy?* Ans. 82 Ib.&oz. 9^ dr. 
 
 (35) If an apothecary mixes together 1 lb. avoirdupois of 
 white wax, 4 lbs. of spermaceti, and 12 lbs. of olive oil, how 
 many ounces, apothecaries* weight, will the mass of ointment 
 weigh, and how many masses of 3 drams each will it con- 
 tain ? Ans. the whole 247 oz. 7^% dr. and 66 1 of 3 dr. each. 
 
 PROPORTION. 
 
 Proportion is either direct, or inverse. It is commonly 
 called the rule of three ; there being always three num- 
 bers or terms given, two of which are terms of supposition ; and 
 the other is the term of demand : because it requires a, fourth 
 
 9 Bullion is the term denoting gold or silver in the mass. Lead is 
 weighed by Avoirdupois weight. See the Table of Comparison of 
 Weights - 
 
ASSISTANT.] RULE OF THREE DIRECT. 5] 
 
 term to be found, in the same proportion to itself, as that 
 which is between the other two. 
 
 General rule for stating the question. Put the term 
 of demand in the third place ; that term of supposition which 
 is of the same kind as the demand, the first ; and the other, 
 which is of the same kind as the required term, the second* 
 
 Also, the terms being thus arranged, reduce the first and 
 third (if necessary) into one name, and the second into the 
 lowest denomination mentioned. 
 
 the rule of three direct 
 
 Requires the fourth term to be greater than the second, 
 when the third is greater than the first ; or the fourth, to be 
 less than the second, when the third is less than the first. 
 
 Rule. Multiply the second and third together, and divide 
 their product by the first : the quotient will be the answer, 
 in the same denomination as the second.t 
 
 The following methods of contracting the operations in the Rule 
 of Three are highly important, and should never be lost sight of. 
 
 1. Let the first and third terms be reduced no lower than is necessary , 
 o make them of the same denomination. 
 
 2. Let the dividing term and cither (but not both) of the other terms be 
 divided by any number that will divide them exactly; and use the 
 quotients instead of the original numbers. 
 
 3. When it is conveniently practicable, work by Compound Mul 
 tiplication and Division, instead of redueing the terms. 
 
 * Some modern authors prefer placing the term of demand the 
 second, and that similar to the required term the third. This arrange- 
 ment will answer the purpose equally well, observing that those of 
 like kind must be reduced (if necessary) to the same name. 
 
 '\ The following General Rule comprehends both the cases of 
 Direct and Inverse Proportion under one head ; which is con- 
 sidered by many scientific men of the present day as a more syste- 
 matic arrangement. 
 
 Rule. The question being stated, and the terms prepared, con- 
 sider, from the nature of the case, whether the required term is to be 
 greater or less than the second, or term of similar kind : if greater, mul- 
 tiply that similar to the answer by the greater of the other two, and 
 divide the product by the less ; if less, multiply it by the less and 
 divide the product by the greater. In either case the quotient will 
 be the term required, in the same denomination as the similar term . 
 
 Note. It is evident that the above Rule will answer generally, 
 whether the term of demand is put m the second or third place. 
 
52 RULE OF THREE DIRECT. [THE TUTOR* S 
 
 (1) If one lb. of sugar cost 4|d. what will 54 lb. cost?* 
 
 (2) If a gallon of beer cost lOd. what is that per barrel ? 
 1 Ans. £l..lO. 
 
 (3) If a pair of shoes cost 4.?. 6d. what is the value of 12 
 dozen pairs ?t 
 
 (4) If one yard of cloth cost 15^. 6d. what will 32 yards 
 cost at the same rate? Ans. £24.. 1 6*. 
 
 (5) If 32 yards of cloth cost £24..l6. what is the value of 
 one yard ? Ans. 1 5s. 6d. 
 
 (6) If I gave £4.. 18. for 1 cwt. of sugar, at what rate did 
 I buy it per lb. ? Ans. 10±d. 
 
 (7) Bought 20 pieces of cloth, each piece 20 ells, for 12^. 
 6d. per ell, what is the value of the whole? Ans. £250. 
 
 (8) What will 25 cwt. 3 qrs. 14 lb. of tobacco come to, at 
 >5±d. per lb. ? Ans. £l87..3..3. 
 
 (9) Bought 27^ yards of muslin, at 6s. g^d. per yard, what 
 is the amount of the whole ? Ans. £$..5.. j ^. 
 
 (10) Bought 17 cwt. 1 qr. 14 lb. of iron, at 3\d. per lb. 
 what was the price of the whole ? Ans. £26..7-.0|. 
 
 (11) If coffee is sold for 5^d. per ounce, what will be the 
 price of 2 cwt. ? Ans. £82..2..8. 
 
 (12) How many yards of cloth may be bought for £21.. 
 ll..l^. when 3^ yards cost £2..14..3.? 
 
 Ans. 27 yards, 3 qrs. 1^ nail. 
 
 (13) If 1 cwt. of Cheshire cheese cost £l..l4..8. what must 
 I give for 3^ lb. ? Ans. Is. Id. 
 
 (14) Bought 1 cwt. 24 lb. 8 oz. of old lead, at Qs. per cwt. 
 what did the lead cost ? Ans. 10s. 11^ JJfrf. 
 
 (15) If a gentleman's income be £500. a year, and he spend 
 ] Qs. 4fd. per day, what is his annual saving ? Ans. £ 147-.3..4. 
 
 (16) If 1 4 yards of cloth cost 10 guineas, how many Flem- 
 ish ells can I buy for £283..17-.6. ? Ans. 504 Fl. ells, 2 qrs. 
 
 (17) If 504 Flemish ells, 2 quarters, cost £283..17..6. what 
 is the cost of 14 yards ? Ans. £l0..10. 
 
 lb. d. lb. pr. s. d. prs. 
 ' As 1 : 44 : : 54 f As 1 : 4..6 : : 144 
 4 18 12 
 
 "Is 4)972 qrs. 2..H..0 
 
 12 
 
 12)243 d. 
 
 20s. 3d.=z£l..0..3. Ans. 
 
 £32..8..0. Ans. 
 
ASSISTANT.] RULE OF THREE DIRECT. 53 
 
 (18) At the rate of <£l..l..8. for 3 lb. of gum acacia, what 
 must be given for 29 lb. 4 oz. ? Ans. £l0..1 1..3. 
 
 (19) If 1 English ell, 2 quarters cost 4>s. "d. what will 39| 
 yards cost at the same rate? Ans. £5..3..5\ I. 
 
 (20) If 27 yards of Holland cost £5.A2..6. how many 
 English ells can I buy for £100.? Ans. 384 ells. " 
 
 (21) If 7 3^ards of cloth cost 17-v. 8d. what is the value of 
 .5 pieces, each containing 27^ yards ? Ans. £l7..7-.0^ I. 
 
 (22) A draper bought 420 yards of broad cloth, at the rate 
 of 14s. lOfd. per ell English: what was the amount of the 
 purchase money ? Ans. £250..5. 
 
 (23) A grocer bought 4 hogsheads of sugar, each hogshead 
 weighing neat 6 cwt. 2 qrs. 14 lb. at £2..S..6. per cwt. what is 
 the value ? Ans. £64..5..3. 
 
 (24) A draper bought 8 packs of cloth, each pack contain- 
 ing 4 parcels, each parcel 10 pieces, and each piece 26 yards ; 
 at the rate of £4.. 16. for 6 yards: what was the purchase 
 money? Ans. £6656. 
 
 (25) If 24 lb. of raisins cost 6s. 6d. what will 18 frails cost, 
 each frail weighing neat 3 qrs. 18 lb.? Ans. £24..17..3. 
 
 (26) When the price of silver is 5s. per ounce, what is the 
 value of 14 ingots, each ingot weighing 7 lb. 5 oz. 10 dwts. ? 
 
 Ans. £313..5. 
 
 (27) What is the value of a pack of wool, weighing 2 cwt. 
 1 qr.19 lb. at 17*. per tod of 28 lb ? Ans. £8.A.M% f g. 
 
 (28) Bought 171 tons of lead, at £14. per ton; paid car- 
 riage and other incidental charges, £4.. 10. Required the 
 whole cost, and the cost per lb. ? 
 
 Ans. £2398.. 10. the whole cost, and the cost per lb. l^d. sfljfe. 
 
 (29) If a pair of stockings cost 10 groats, how many dozen 
 pairs can I buy for £43..5. ?. Ans. 21 doz. 7^ pairs. 
 
 (30) Bought 27 doz. 5 lb. of candles, at the rate of 5s. gd. 
 a dozen: what did they cost? Ans. £7. .17. .7%. 
 
 (31) A factor bought 86 pieces of stuff, which cost him 
 <£517..17-10. at 4s. lOd. per yard. How many yards wer 
 there in the whole, and how many English ells in a piece ? 
 
 Ans. 2143 yards ; and 19 ells, 4 qrs. 2|g nails, in a piece. 
 
 (32) A gentleman has an annuity of £896.. 1 7. W 7 hat m 
 he spend daily, that at the year's end he may lay up 2 
 guineas, after giving to the poor quarterly 10 moidores? 
 
 Ans. <£l..l4..8 ffo 
 
64 ,. RULE OF THREE INVERSE. [THE TUTOR 8 
 
 THE RULE OF THREE INVERSE 
 
 Requires the fourth term to be less than the second, when 
 the third is greater than the first ; or the fourth to be greater 
 than the second, when the third is less than the first. 
 
 Rule. Multiply the first and second together, and divide 
 their product by the third : the quotient will be the answer, 
 as before. 
 
 (1) If 8 men can do a piece of work in 12 days, in how 
 many days can 16 men do the same?* 
 
 (2) If 54 men can build a house in 90 days, how many 
 men can do the same in 50 days? Ans. 97^ men. 
 
 (3) If, when a peck of wheat is sold for 2s. the penny loaf 
 weighs 8 oz. ; how much must it weigh when the peck is worth 
 but Is. 6d. ? Ans. 1 Of oz. 
 
 (4) How many sovereigns, of 20s. each, are equivalent to 
 240 pieces, value 12s. each? Ans. 144. 
 
 (5) How many yards of stuff three quarters wide, are equal 
 in measure to 30 yards of 5 quarters wide ? Ans. 50 yds. 
 
 (6) If I lend a friend £200. for 12 months, how long ought 
 he to lend me £150. ? Ans. 16 months. 
 
 (J) If for 24s. I have 1200 lb. carried 36 miles, what weight 
 can 1 have carried 24 miles for the same money ? Ans. 1800 lb. 
 
 (8) If I have a right to keep 45 sheep on a common 20 
 weeks, how long may I keep 50 upon it ? Ans. 1 8 weeks. 
 
 (9) A besieged town has a garrison of 1000 soldiers, 
 with provisions for only 3 months. How many must be sent 
 away, that the provisions may last 5 months ? Ans. 400. 
 
 (10) If £20. worth of wine be sufficient to serve an ordi- 
 nary of 100 men, when the price is £30. per tun ; how many 
 will £20. worth suffice, when the price is only £24. per tun? 
 
 Ans. 1.25 men. 
 
 (11) A courier makes a journey in 24 days, by travelling 
 i2 hours a day : how many days will he be in going the same 
 journey, travelling 1 6 hours a day ? Ans. 1 8 days. 
 
 (12) How much will line a cloak, which is made of 4 yards 
 tf plush, 7 quarters wide, the stuff for the lining being but 
 $ quarters wide? Ans. 9^ yards. 
 
 m. d. m. ftVT2 
 
 [ As 8 : 12 : : 16 : = 6 days. Ans. 
 
 16 
 
ASSISTANT.J DOUBLE RULE OF THREE. 55 
 
 DIRECT AND INVERSE PROPORTION PROMISCUOUSLY ARRANGED. 
 
 (1) If 14 yards of broad cloth cost £9.. 12. what is the 
 purchase of 75 yards? Ans. £51..8..6f T 6 ? . 
 
 (2) If 14 pioneers make a trench in 1 8 days, in how many 
 days would 34 men make a similar trench ; working in both 
 cases, 12 hours a day? Ans. 7 days, 4 hours, 56^ T minutes. 
 
 (3) How much must I lend to a friend for 12 months, to 
 requite his kindness in having lent me £64. for 8 months ? 
 
 Ans. £42..13..4. 
 
 (4) Bought 59 cwt. 2 qrs. 21 lb. of tobacco, at £2..17-.4. 
 per cwt. what does it come to? Ans. £l71..2..1. 
 
 (5) A woollen draper purchased 147 yards of broad cloth, 
 at lis. 6d. per yard. Suppose that he sold it in pieces for 
 coats, each If yard, how much must he charge for each, so 
 as to gain £l6..10..9. by the whole? Ans. £l,.9..3f. 
 
 (6) If £100. gain £4.. 10. interest in 12 months, what sum 
 will gain the same in 18 months? Ans. £66.. IS. A. 
 
 (7) A draper having sold 147 yards of cloth, at the rate of 
 £l..g..3f. for If yard, found that he had gained £l6..10..9. 
 What did the whole cost him, and how much per yard ? 
 
 Ans. the whole £l06..11..6. and 14s. 6d. per yard. 
 
 (8) If £100. in 12 months gain £4..10. interest, in what 
 time will £66.. 13. .4. gain the same interest? Ans. 18 month*. 
 
 (9) If a draper bought 147 yards of cloth, at 14s. 6d. per 
 yard, and sold it in pieces for coats, each If yard, for 
 £l..9..3f. ; how much would he gain per yard, and by the 
 whole? Ans. c 2s. 3d. per yard, £l6..10..9. by the whole. 
 
 (10) If 1 cwt. cost £12.. 12. .6. what must be given for 14 
 cwt. 1 qr. 19 lb. f Ans. £l82..0..1l£ T f f . 
 
 (11) If £100. gain £4.. 10. in 12 months, what interest will 
 £375. gain in the same time? Ans. £l6..17-6. 
 
 (12) A regiment of soldiers, consisting of 1000 men, are 
 to have new coats, each to be made of 2^ yards of cloth, 5 
 quarters wide, and to be lined with shalloon of 3 quarters 
 wide. How many yards of shalloon will line them ? 
 
 Ans. 4166 yards, 2 qrs. 2| nails. 
 
 THE DOUBLE RULE OF THREE 
 
 Has Jive terms given, three of supposition and two of demand, 
 to find a sixth, in the same proportion with the terms of de- 
 mand, as that of the terms of supposition. It comprises two 
 
56 DOUBLE RULE OF THREE. £tHE TUTOR 
 
 operations of the single rule. — But it may comprise three, 
 Jour, or more operations of the Single Rule ; as there may be 
 seven terms given to find an eighth, or nine to find a tenth, &c. 
 In this respect it is unlimited ; and is therefore more properly 
 called compound proportion. 
 
 Rule 1. Put the terms of demand one under another in 
 the third place ; the terms of supposition in the same order in 
 the first place ; except that which is of the same nature as the 
 required term, which must be in the second place. 
 
 Examine the statings separately, using the middle term in 
 each, to know if the proportion is direct or inverse. When 
 direct, mark the first term with an asterisk : when inverse, 
 mark the third term. 
 
 Find the product of the marked terms for a Divisor, and 
 the product of all the rest for a Dividend : divide, and the 
 quotient will be the answer.* 
 
 IIule 2. (1) Of the conditional terms, put the principal cause of 
 action, gain or loss, &c. in the first place. (2) Put that which denotes 
 time or distance, &c. in the second, and the other in the third. (3) 
 Put the terms of demand under the like terms of supposition. (4) If 
 the blank falls in the third place, multiply the first and second terms 
 for a divisor, and the other three for a dividend. (5) But if the blan 
 is in the first or second place, divide the product of the rest by the 
 product of the third and fourth terms, for the answer. 
 
 Note. It will save much labour to write the terms of the Divi- 
 dend over, and those of the Divisor under a line, like those of a cow- 
 vound fraction, and to cancel them accordingly. See Reduction of 
 Vulgar Fractions, Case 6. 
 
 Proof. By two operations of the Single Rule of Three. 
 
 (1) If 14 horses eat 56 bushels of oats in 16 days, how 
 many bushels will serve 20 horses 24 days?t 
 
 * See also Supplemental Questions, Nos. 6 and 7. 
 + By Rule 1. 
 
 As 
 
 * 10 12 
 
 *l6j (24) U 
 
 n 
 
 i 
 
 x$4 
 
 120 busheis. 
 
 By Rule 2. 
 h. d. b. 
 14 : 16 : 56 
 
 20 • 24 : -- 
 
 r>y two smgK 
 $ 10 12 h. h. 
 
 „, ' ■ =120 
 
 ^ 4x ^ d. b. 
 
 3 stat 
 
 h. 
 
 :20; 
 
 d. 
 
 ings. 
 
 b. 
 :80 
 
 b. 
 
 % * (2) As 16:80: 
 i 1 w 
 
 : 24 : 
 
 120 
 
ASSISTANT.J PRACTICE. 57 
 
 (2) If 8 men in 14 days can mow 112 acres of grass, how 
 many men can mow 2000 acres in ten days ? Ans. 200 men. 
 
 (3) If £100. in 12 months gain £6. interest, how much 
 will £75. gain in 9 months ? Ans. £3..7..G. 
 
 (4) If £l 00. in 12 months gain £6. interest, what principal 
 will gain £3..7..6. in 9 months? Am. £75. 
 
 (5) If £100. gain £6. interest in 12 months, in what time 
 will £75. gain £3..7-6. interest ? Ans. 9 months. 
 
 (6) If a carrier charges £2..2. for the carriage of 3 cwl. 
 150 miles, how much ought he to charge for the carriage ot 
 7 cwt 3 qrs. 14 lb. 50 miles ? Ans. £l.. 16.^9. 
 
 (7) If 40 acres of grass be mown by 8 men in 7 days, how 
 many acres can be mown by 24 men in 28 days ? Ans. 480. 
 
 (8) If £2. will pay 8 men for 5 days' work, how much 
 will pay 32 men for 24 days' work ? Ans. £38..8. 
 
 (9) If a regiment of soldiers, consisting of 13()0 men, con- 
 sume 351 quarters of wheat in 108 days, how much will 1 1232 
 soldiers consume in 56 days? Ans. 1503/ 5 qrs. 
 
 (10) If 939 horses consume 351 quarters of oats in 168 
 days, how many horses will consume 1404 quarters in 56' 
 days? Ans. 11268. 
 
 (11) If I pay £14..10. for the carriage of 60 cwt. 20 miles, 
 what weight can I have carried 30 miles for £5..8„9* at the 
 *ame rate ? An s. 1 5 cwt 
 
 (12) If 144 threepenny loaves serve 18 men for 8 days, how 
 many fow-venny loaves will serve 2 1 men for 9 days ? 
 
 Ans. 189- 
 
 PRACTICE 
 
 Is so called from its general use among merchants and 
 tradesmen. 
 
 It is a concise method of computing the value of articles, 
 &c. by taking aliquot parts. 
 
 The General Rule is to suppose the price one pound, one shilling, 
 or one penny each. Then will the given number of articles, consid- 
 ered accordingly as pounds, or shillings, or pence, be the supposed value 
 of the whole ; out of which the aliquot pari or parts are to be taken for 
 the real price. 
 
 Note. An aliquot part of a number is such a part as being taken a 
 certain number of times will produce the number exactly : thus, 4 is an 
 aliquot part of 12 ; because 3 fours are 12. 
 
 c 5 
 
58 
 
 PRACTICE. 
 ALIQUOT PARTS. 
 
 [[the tutor's 
 
 Of a pound. 
 10 are 4 
 i 
 
 
 
 8 .. 
 
 l 
 
 So 
 
 
 
 6 .. 
 
 l 
 
 40 
 
 Of a shilling. 
 
 <L 
 
 
 s. 
 
 6 
 
 are 
 
 t 
 
 4 
 
 
 A 
 
 o 
 
 
 i 
 
 8 
 
 
 
 11 
 
 1 is 
 
 Of a penny. 
 
 2 qrs. are 4^« 
 1 gr. is |d. 
 
 0/a to/*, 
 cwtf. ton 
 
 10 are 
 
 5 
 
 4 
 
 2 3#r 12/5, 
 
 24 
 
 2 .. 
 1 
 
 To 
 l 
 5o 
 
 Of a cxvt. 
 
 qr. lb. cwt. 
 
 2, or 56 are 4 
 
 1, or 28 ... | 
 
 16 ... | 
 
 14 ... 4 
 
 0/"a quarter. 
 
 lb. 
 14 
 
 are 
 
 7 
 4 
 31 
 2 
 
 If 
 
 1 
 
 qr. 
 i 
 
 Of a lb. 
 oz. lb. 
 
 8 are £ 
 4 ... I 
 
 2 ... l 
 
 1 IS T ^ 
 
 OfaZk Troy, 
 oz. lb. 
 
 6 are | 
 4, &c. as in 
 the parts of 
 a shilling. 
 
 Of an oz. Troy. 
 
 The same as 
 the parts of a 
 £. changing 
 the names 
 from shillings 
 to dwts. 
 
 Of a dwt. 
 
 12 
 8 
 6 
 4 
 3 
 2 
 
 »4 
 
 i 
 
 diet. 
 
 k 
 
 1 
 
 i 
 
 Rule 1. When the price is less than a penny, call the given num. 
 er pence, and take the aliquot parts that are in a penny ; then divide 
 oy 12 and 20, to reduce the answer to pounds. 
 
 (1) iis^5704/6.at 
 12)1426 
 2|0)11|8„10 
 
 (2) 7695 at ±d. 
 Ans. £\6..0..7i. 
 
 (3) 5470 at id. 
 Ans. £ll..7..11. 
 
 (4) 6547 at %d. 
 Ans. £20..9..2£. 
 
 (5) 4573 at Id. 
 Ans. £l4..5..9|. 
 
 ***#. £5..18..10. 
 
 Rule 2. When the price is less than a shilling, call the given 
 number shillings, take the aliquot part or parts that are in a shilling, 
 add the quotients together, and divide by 20, as in the preceding 
 rule. 
 
 *(1) 7547 at Id. |r(2) 3751 at \{d. 
 Ans. £31..8..11.| Ans. £l9..10..8f. 
 
 (3) 54325 at l±d. 
 Ans. £339..10..7|. 
 
 Id. = ^ 7547*. 
 
 2i0)62|8..11 
 
 Ans. £31..8..11. 
 
 + U. = j^SUls. 
 
 I = \ 312..7 
 78..1| 
 
 2|0)39|0..8f 
 
 £l&.10..8f. 4m 
 
ASSISTANT.] 
 
 (4) 6254 at lfd 
 Ans. £45..12..Q^. 
 
 (5) 2351 at 2d. 
 Ans. £ig..U.AO. 
 
 6) 7210 at 2\d. 
 Ans. £67..H..10A. 
 
 PRACTICE. 
 
 (18) 2715 at 5^/. 
 ^»j. £59».7-9f • 
 
 (19) 3120 at 5^/. 
 J*w. £71. .10. 
 
 (20) 7521 at 5£d. 
 4**. £l80..3..9|. 
 
 59 
 
 (32) 9872 at gfd 
 -4wj. £359..! 8 .4. 
 
 (33) 5272 at 9<f. 
 Jrc*.£l97.-14. 
 
 (34) 6325 at 9±d. 
 Ans. £243..15..6j- 
 
 (7) 2710 at 2\d. 
 Ans. £28..4..7. 
 
 (21)3271 at 6d. 
 Ans. £81..15..6. 
 
 (8) 3250 at 2%d. 
 Ans. £37.A..g±. 
 
 (22) 7914 at Q\d. 
 Ans. £206..1..10A. 
 
 (85) 7924 at 9±d, 
 Ans. £313..13..2. 
 
 "(36) 2150 at 9f^~ 
 
 (9) 2715 at 3d. 
 Ans. £33..18..9. 
 
 (23) 3250 at 6±d. 
 Ans. £S8..0..5. 
 
 (37) 6325 at lOd. 
 Ans. £263.. 10.. 10 
 
 (10) 7062 at 3\d. 
 
 (24) 2708 at 6f d. 
 Ans. £76..3..3. 
 
 (38) 5724 at 10\d. 
 Ans. £244..9..3. 
 
 (11) 2147 SLtS^d. 
 Ans. £31..6..2^. 
 
 (25) 3271 at Id. 
 Ans. £95..8.. 1, 
 
 (39) 6327 at 10\d. 
 Ans. £270..4..3 j. 
 
 (12) 7000 at Sftf. 
 ^.£l09..7..6. 
 
 (26) 3254 at l\d. 
 Ans. £98..5..ll±. 
 
 (40) 3254 at 10±d. 
 Aits. £l42..7..3. . 
 
 (13)3257 at 4c/. 
 Ans. £54..5..8. 
 
 (27) 2701 at 7id. 
 A?is. £84..8..1 J. 
 
 7l4)2056at4^. 
 Ans. £36..8..2. 
 
 (28) 3714 at 7|rf. 
 ^wj.£119..18..7£. 
 
 (41)7291 at lOjrf. 
 Jwj. £326..11..fr^. 
 
 (42) 3256 at lid. 
 Ans. £l49»4..8. 
 
 (15) 3752 at 4^. 
 ^^. £70..7-.0. 
 
 (29) 2710 at 8d. 
 Ans. £go..6..8. 
 
 (43) 7254 at ll±d. 
 Ans. £340..0..7i. 
 
 (16) 2107 at 4fd. 
 ^wj. £41..14..Ql 
 
 (30) 3514 at 8^. 
 ^wj. £l20..15..10|. 
 
 (44) 3754 at 11^. 
 ifm, £l79..17..7- 
 
 (17) 3210 at 5d. 
 Ans.£66..17..6. 
 
 (45) 7972 at 1 If d. 
 Ans. £390..5..11. 
 
 (31) 2759 at 8\d. 
 Ans. £97. .l*..3\. 
 
 Rule 3. When the price is more than one shilling, and less than 
 two, take the part or parts for the excess above a shilling, add the 
 quotients to the given quantity, and reduce the whole to pounds as 
 Or, when convenient, take the aliquot part of a pound. 
 
 before. 
 
 *0)2106atl2^r/. 
 Ans. £l07..9.-10|. 
 
 (2) 3715 atl2|d 
 Ans. £l93..9«9i. 
 
 (3) 2712 at 12f ri. 
 Ans. £l44..1..6. 
 
 '-{f. 
 
 2106s. 
 "OT5..6) 
 43.J04 
 g| 0)214j9,.104 
 #& Xl07..9..l0i. 
 
 This example is w r orked by taking 
 y^, and then £ of that ; because a 
 farthing is ^ of a shilling ; which 
 is = T J 5 of £, or J of y's, because 4 
 twelves are 48. 
 
60 
 
 (4) 2107 at Is. Id. 
 A is. £ll4..2..7. 
 
 PRACTICE. 
 
 (19) 2750 at Is. 4fd. 
 Ans. £191..18..6± 
 
 Qthe tutor's 
 
 (34) 7104 at Is. 8^/. 
 Ans. £606.. 16. 
 
 (5) 3215 at Is. \\d. 
 Ans. £l77..9..10j. 
 
 (20) 3725 at Is. 5d. 
 ^rcs. £263..17..1. 
 
 (35) 1004 at Is. 8fd. 
 -4;w. £86..l6..1. 
 
 (6) 2790 at Is. l£d. 
 Ans. £l56..18..9. 
 
 (21) 7250 at Is. 5{d, 
 Ans. £521..!.. 10^. 
 
 (36) 2104 at Is. 9aL 
 Jws. £l84..2. 
 
 (7) 7904 at Is. lfd. 
 ^ws. £452..l6..8. 
 
 (22) 2597 at 1j.5£*/. 
 
 (37) 2571 at Is. 9^. 
 Jws. £227..12..9|. 
 
 (8) 3750 at Is. 2d. 
 Ajis. £218.. 15. 
 
 (23) 7210 at Is. 5$d. 
 Ans. £533..4..9^. 
 
 (38) 2 104 at Is. 9^. 
 Ans. £l 88.-9.. 8. 
 
 (9) 3291 at Is. 2^. 
 Ans. £l95..8..0|. 
 
 (24) 7524 at Is. 6d. 
 Ans. £564..6. 
 
 (10) 9254 at Is. 2^. 
 Ans. £559..1..11. 
 
 (39) 7506 at is. 9fd. 
 Ans. £680..4..7f . 
 (25) 7103 at Is. 6{d. (40) 1071 at Is. Wd. 
 Ans. £540..2..5f . Ans. £98..3..6. 
 
 (11) 7250 at Is. 2fd. 
 Ans. £445..! 1.. 5 j. 
 
 (26) 3254 at Is. 6±d 
 Ans. £250..l6..7- 
 
 (41)5200atls.l0-^/. 
 Ans. £482..!.. 8. 
 
 (12) 7591 at Is. 3d. 
 Ans. £474..8..9. 
 
 (27) 7925 at Is. 6%d 
 Ans. £6l9-.2..9j- 
 
 (42)2117atls.l0^. 
 Ans. £l98..9».4j. 
 
 (13) 6325 at Is. 3\d. 
 Ans. £401..18..0i. 
 
 (28) 9271 at Is. ^d. 
 Ans. £733..19..1. 
 
 (43)1007atls.l0f</. 
 Ans. £95..9..1j. 
 
 (14) 5271 at Is. 3±d. 
 Ans. £340..8..4^. 
 
 (29) 7210 at Is. 7id. 
 Ans. £578..6..Q^. 
 
 (44) 5000 at Is. lid. 
 Ans. £479-3..4. 
 
 (15) 3254 at Is. 3%d. 
 Ans. £2 13.. 10.. 10^. 
 
 (30) 2310 at Is. 7±d. 
 Ans. £187..13..9- 
 
 (45)2105atls.ll^ 
 Ans. £203..18..5^ 
 
 (16) 2915 at U.4d. 
 Ans. £l94..6..8. 
 
 (31) 2504 at Is. 7fd. 
 Ans. £206..!.. 2. 
 
 (46)1006atls.ll^/. 
 Ans. £98..10..1. 
 
 (17) 3270 at Is. 4^. 
 Ans. £221..8..1^. 
 
 (32) 7152 at Is. Sd. 
 Ans. £596. 
 
 (47)2705atls.ll|c?. 
 Jy?s.£267»13..7j. 
 
 (18) 7059 at Is. 4^, 
 Ans. £485..6..1i 
 
 (33) 2905 at Is. 8±d. 
 Ans. £245..2..2i. 
 
 (4S)5000atls.ll^. 
 Ans. £489..11..8. 
 
 Uule 4. When the price is an even number of shillings, the given 
 quantity may be multiplied by half that number, doubling the units* 
 figure of the product for shillings, and the rest of the product will be 
 pounds. Or take the aliquot part of a pound. 
 
 (1) 2750 at 2s. i (4) 1572 at 8s. (7) 5271 at 14s. 
 
 Ans. £275. I An s. £628..! 6. Ans. £368 9.. 14. 
 
 (2) 3254 at 4s. ^(5) 2~1 02 at 1 Os. (8) 3123 at l6s. 
 Ans. £650..! 6. Ans. £1051. Ans. £2498.. 8. 
 
 (3) 2710 at 6s. 
 
 Ans. £813. 
 
 (6) 2101 at 12s. 
 Ans. £1260.. 12. 
 
 (9) 1075 at 16s. 
 Ans. £860. 
 
ASSISTANT.] 
 
 (10) 1621 at 18.?. 
 Ans. £l458..18. 
 
 PRACTICE. 61 
 
 Note. At 2s. take the tenth, and at 10*. take 
 the half o£ so many £. 
 
 Rule 5. When the price is an odd number of shillings, work by 
 "Rule 4th. for the greatest even number, and add ^ of the given 
 quantity for the odd shilling — Or, take such parts" of a pound, as 
 will make the given price. 
 
 * (1) 3270 at 3s. I (4) 3214 at 9*. (7) 2150 at 15s. 
 Ans. £490..10.l Ans. £l446..6. Ans. £l6l2..10. 
 
 (2) 3271 at 5s. 
 Ans. £8 17- 15. 
 
 (5) 2710 at lis. 
 Ans. £1490.. 10. 
 
 (8) 3142 at 17*. 
 Ans. £2670.. 14. 
 
 \3) 2715 at Is. 
 
 Ans. £950.. 5. 
 
 (6) 3179 at ISs. 
 Ans. £2066..7- 
 
 (9) 2150 at 19s. 
 Ans. £2042.. 10. 
 
 Rule 6. When the price consists of shillings and pence, suppose 
 the given number to be pounds, and take such aliquot part, or the 
 
 sum of such aliquot pa it, as will make the given price Or, work 
 
 for the shillings as in the preceding Itules, and take parts for the 
 residue. 
 
 f (1) 2710 at 6s. 8d. 
 Ans. £903..6..8. 
 
 % (7) 271 Oat Ss. 2d, 
 Ans. £429- 1.. 8. 
 
 (13)7152atl7s.6fd. 
 Ans. £6'280..7. 
 
 (2) 3150 at 3s. 4>d. 
 Ans. £525. 
 
 (8) 7514 at 4.y. Id. 
 Ans. £l721..19..2. 
 
 (14)2510atl4s.7^i. 
 Ans. £\$32..l6..5\. 
 
 (3) 2715 at 2s. 6d. 
 Ans. £339..7~6\ 
 
 (9) 2517 at 5s. 3d. 
 Ans. £660..14..3. 
 
 (4) 7150 at Is. 8c?. 
 Ans. £595..l6..8. 
 
 (10) 2547 at 7*. 3\d. 
 Ans. £928..11..1Qi 
 
 (I5)3715at9<y.4i6/. 
 Ans. £l741..8..1^. 
 
 (16) 2572 at 13s. 7^/. 
 Ans. £l752..3..6\ 
 
 (5) 3215 at Is. 4d. 
 Ans. £214..6..8. 
 
 (11)3271 at 5s. §\d. 
 Ans. £943..l6'..4j. 
 
 (17)725 lat 14s. %\d. 
 Ans. £5324..19..0j. 
 
 (6) 7211 at Is. 3d. 
 Ans. £450.. 1 3.-9. 
 
 (I2)2103atl5s.4^. 
 Ans. £l6l6..13..7i. 
 
 (18)3210atl5s.7fd. 
 Ans. £25\\..3..\\d. 
 
 Rule 7. When the price consists of pounds, shillings, and pence, 
 multiply the given quantity by the number of pounds, and take ah- 
 quot parts for the residue — Or, work for the shillings as in the pre- 
 ceding Rules, &c — Or, wnen the given number of articles is not 
 large, work by Compound Multiplication. 
 
 s. s. 
 • 2= T >s 3270 
 
 1=4 327 
 
 163..10 
 
 s. d. £. s. d. £. 
 tt>..8=i2710 £2..6=|) 2710 
 
 Ans. £903..6..8. 8 =3o f ~"~338..15 
 
 90.. 6..Z 
 
 Ans. £490.. 10. 
 
 Ans. £429.. 1..8. 
 
* (1) 7215 at £7.A. 
 
 Ans. £51948. 
 (2) 2104 at £5..3.~~ 
 Ans. £10835..12. 
 
 PRACTICE. 
 
 (7) 2107 at £1..13. 
 Ans. £3476.. 11. 
 
 (8) 3215 at £4..6..8. 
 Ans. £13931..13 .4. 
 
 [the tuto 
 
 i(13)3210at£1..18..6f. 
 Ans. £6189..5..7^. 
 
 (14) 2157at£2..7 4j 
 Ans. £5109..7..10^_ 
 
 (3) 2107 at £2..8. 
 Ans. £5056..16. 
 
 (9) 2154at£7..1..3 
 Ans. £15212..12..6. 
 
 (4) 7156 at £5..6. 
 Ans. £37926.-16. 
 
 (10) 2701 at £2..3..4. 
 Ans. £5852..3..4. 
 
 (15) 142at£1..15..2f 
 Ans. £250..2..6i 
 
 (16) 95at£15..l4..7^. 
 Ans. £1494..7..4#. 
 
 (5)2710at£'2..3..7i!(li)2715at£1..17..2i 
 
 Ans. £5911. .3..9. | Ans. £5051..Q..7^ . 
 
 (6) 3215 at £1..17. j(12)2157at£3..15..2f 
 
 Ans. £5947..15. | Ans, " 
 
 (17) 37 at £1..19..5f. 
 Ans. £73..0..8#. 
 
 £8108..19..5^ 
 
 (18)2175at£2..15..4^. 
 Ans. £6022..0..7i. 
 
 Rule 8. When the given quantity consists of several denomina- 
 tions, multiply the price by the number of the highest, and take ali- 
 quot parts for the inferior denominations. 
 
 (1) At £3.. 17-6. per cwt. what is the value of 25 cwt. 
 2 qrs. 14/6. of soap ?t 
 
 (2) At £l..4..9. per cwt. what is the value of 17 cwt. 1 qr. 
 17 lb.? ' Ans. £21..10..8. 
 
 (3) Sold 85 cwt. 1 qr. 10 lb. of iron, at £l..7..8. per cwt 
 what is the value of the whole ? Ans. £ll8..1..0^. 
 
 (4) If hops are sold at £4*.. 5.. 8. per cwt. what must be given 
 for 72 cwt. 1 qr. 18 lb.? Ans. £310..3..2. 
 
 (5) What is the value of 27 cwt. 2 qrs. 15 lb. of logwood, 
 
 at £l..l..4. -per cwt.? 
 
 Ans. <£29..9»6i 
 
 (6) Bought 78 cwt. 3 qrs. 12 lb. of molasses, at £2..17.-9. 
 per cwt. what must I give for the whole ? Ans. £22 7.. 14. 
 
 (7) Sold 56 cwt. 1 qr. 17 lb. of sugar, at £2.. 15. .9. per cwt. 
 how much is the whole charge? Ans. £l57..4..4^. 
 
 (8) What is the value of 97 cwt. 15 lb. of currants, at 
 £3..17-.10. per cwt. ? Ans. £378..0..3. 
 
 (9) At £4.. 14.. 6. the cwt. what is the value of 37 cwt. 2 qrs. 
 13 lb. of raw sugar? Ans. £l77..14>.M. 
 
 s. 
 •4=1 7215 
 
 f2qrs.=z% £3..17..6 
 
 5X5=25 
 
 7 
 50505 
 
 19.. 7. .6 
 5 
 
 1443 
 £51948 Ans. 
 
 lb. 96..17..6 
 14=4 1.. 18.-9 
 9..8J 
 
 £99..5..ll±Ans. 
 
ASSISTANT J TARE AND TRET. 63 
 
 (10) Bought sugar at £3..14..6. the cwt. wnat did 1 give 
 for 15 cwt. 1 qr. 10 lb. ? Ans. £57..2..9- 
 
 (11) Required the value of 17 oz. 8 drvts. 18 grs. ot gold, 
 at £3..17..10^. per ounce. Ans. £67..17..H. 
 
 (12) At <£37..6..8. per cwt. the value of 1 cwt. 2 qrs. 
 10^ lb. of cochineal is required. Ans. £59.. 10. 
 
 (13) Required the value of 13 hhds. 42 gals, of Champagne 
 wine, at £ C 25..13..6. per hhd. Ans. £350..17..10. 
 
 (14) A gentleman purchased at an auction an estate of 
 149 a. 3 r. 20 ;;. at £54.. 10. per acre. What was the whole 
 purchase money, including the auction duty of 7d. in the £. 
 the attorney's bill for the deeds of conveyance, £33..6..8. and 
 his surveyor's charge for measuring it, at Is. per acre? 
 
 Ans. £8447..5..0i. 
 Rule 9. To find the price of 1 lb. at a given number of shillings per 
 
 CU't. 
 
 Multiply the shillings by 3 and divide the product by 7 ; the quo- 
 tient will be the price of 1 lb. in farthings.* 
 
 (1) What is the price of 1 lb. at 44s. 4<d. per cwt. ?f 
 
 (2) What are the respective prices per lb. at 86s. 4d. ; 91*. ; 
 and 11 6s. 8d. per cwt.? Ans. g^d., 9fd., and Is. Q^d. 
 
 Rule 10. It is sometimes expedient to change the price and the 
 quantity for each other. Thus 48 yards at 2s. 9d. will be equivalent 
 to 33 yards at 4s. ; because 2s. 9d. = 33d. and 4>s. = 48c?. 
 
 (1) What is the value of 72 yds. at 3s. 5d. and at 14«y. 7 d. 
 per yard ? Ans. <£l2..6\, and £52..l0. 
 
 (2) 80 yds. at 15.?. 3d. and at l6s. 8d. per yard ? 
 
 Ans. £6l., and £66..13.A. 
 
 (3) 42 lbs. at ll^Z. and at \s. 3\d. per lb. ? 
 
 Ans. £<2..Q..3. } and £2..13.A±. 
 
 TARE AND TRET. 
 
 Gross weigrit is the weight of any goods, together with that 
 of the package which contains them. 
 
 * Multiplying by 3 reduces the shillings to fourpences, and 7 four- 
 pences (or 2s. 4d.) are the value of 1 cwt. at 1 farthing' per lb. 
 f 44*. U. 
 3 
 
 7)133 
 
 19 farthings=43^ per lb. Am. 
 
64 TARE AND TRET. [THE TUTOR'S 
 
 Neat weight is that of the articles alone, or what remains 
 after the deduction of all allowances. 
 
 Tare is an allowance for the weight of the package. It is 
 either so much in the whole, or at so much per bag, box, bar- 
 rel, &c. or at so much in the cwt. 
 
 Tret is an allowance of 4 lb. in 104 lb. (or ^ part) for waste. 
 
 Cloffls an allowance of 2 lb. in 3 cwt. on some goods : but both these 
 are nearly obsolete. 
 
 Suttle is the remainder when any particular allowance has been de- 
 ducted. 
 
 Rule. When the Tare is at so much for each bag, &c. the 
 whole Tare may be found by multiplying by the number of 
 them. When it is at so much per cwt. take the aliquot parts 
 of the Gross for the Tare. Subtract the Tare from the Gross; 
 the remainder is the Neat : unless there is Tret allowed. 
 
 If Tret is allowed, it is ^ °f the Tare suttle, which being subtracted 
 from it, the remainder is the Neat. But if Cloff also is to be allowed, 
 the cwts. Tret suttle, multiplied by 2, and divided by 3, will be the 
 ibs. Cloff, which subtract to find the Neat. 
 
 (1) In 7 frails of raisins, each weighing 5 cwt. 2 qrs. 5 lb. 
 gross, tare at 23 lb. per frail, how much neat weight ?* 
 
 (2) What is the neat weight of 25 hogsheads of tobacco, 
 weighing gross 163 civL 2 qrs. 15 lb, tare 100 lb. per hogs- 
 head? Ans. 141 cwt. 1 qr. 7 lb. 
 
 (3) In 16 bags of pepper, each weighing 85 lb. 4 oz. gross, 
 tare per bag, 3 lb. 5 oz. how many pounds neat ? Ans. 1311 lb 
 
 (4) What is the neat weight of 5 hogsheads of tobacco, 
 weighing gross 75 cwt. 1 qr. 14 lb. tare in the whole 752 lb. f 
 
 Ans. 68 cwt. 2 qrs. 18 lb. 
 
 (5) In 75 barrels of figs, each 2 qrs. 27 lb. gross, tare in 
 the whole 597 lb. how much neat weight? Ans. 50 cwt. 1 qr. 
 
 (6) What is the neat weight of 1 8 butts of currants, each 
 8 cwt. 2 qrs. 5 lb. gross, tare at 14 lb. per cwt. ?f 
 
 cwt. qr. lb. cwt. qr. lb. 
 
 * 5..2.. 5 gross. f 8 - 2 - 5 
 
 23 tare. 9X2=18 
 
 X.L.IO neat of 1 frail. 767.3.. 17 
 
 7 
 
 tb. 
 
 Ans. 37.. 1.. 14 neat of the whole. 14=| 153..3.. 6 whole gross. 
 ' 19 1 .0..25| tare. 
 
 Ans. 13?..2..8§ neat. 
 
ASSISTANT.] 
 
 INVOICES. 
 
 65 
 
 (7) In 25 barrels of figs, each 2 cwt. 1 qr. gross, tare per 
 cwt. 16 lb. how much neat weight ? Ans. 48 cwt. qr. 24 lb. 
 
 (8) What is the neat weight of 9 hogsheads of sugar, 
 each weighing gross 8 cwt. 3 qrs. 14 lb. tare \6 lb. per cwt. ? 
 
 Ans. 68 cwt. 1 qr. 24 /6. 
 
 (9) In 1 butt of currants, weighing 12 cwt. 2 qrs. 24 /6. 
 gross, tare 14 lb. per cratf. tret 4 /&. per 104 /&. what is the neat 
 weight?* 
 
 (10) In 7 cwt. 3 qrs. 27 lb. gross, tare 36 lb. tret according 
 to custom, how many pounds neat ? Ans. 826 lb. 
 
 (11) In 1 52 cwt. 1 qr. 3 lb. gross, tare 1 lb. per cwt. tret 
 as usual, how much neat weight? Ans. 133 cwt. 1 qr. 12 lb. 
 
 (12) What is the neat weight of 3 hogsheads of tobacco, 
 weighing 15 cwt. 3 qrs. 20 lb. gross, tare 7 lb. per. cwt. tret 
 and cloff as usual ?t 
 
 (13) In 7 hogsheads of tobacco, each weighing gross 5 cwt 
 2 qrs. 7 lb.; tare 8 lb. per cwt. tret and clofFas usual, how much 
 neat weight ? Ans. 34 cwt. 2 qrs. 8 lb. 
 
 INVOICES, or BILLS OF PARCELS. 
 
 '1) Mrs. Bland, London, Sept. 1. 
 
 Bought of Jane Harris. 
 
 s. d. £. 
 
 6 per pr. 
 
 15 pairs worsted stockings at 
 1 doz. thread ditto 
 
 ^ doz. black silk ditto 
 1| doz. milled hose 
 
 2 doz. cotton ditto . 
 I / pairs kid gloves . 
 
 at 
 
 at 
 at 
 at 
 at 
 
 1830. 
 
 dr. 
 
 £21..18..4 
 
 lb. cwt. qrs. lb. 
 
 14;= J 12.. 2.. 24 gross. 
 ^ 1.. 2.. 10 tare. 
 
 4=^ u7. 0..14 suttle. 
 1..19 tret. 
 
 Am. 10.. 2..23 neat. 
 
 lb. cwt. qr. lb. 
 f 7= T ^ 15.. 3..20 gross. 
 3..27| tare. 
 
 26)14., 3..20± suttle. 
 2.. 8 tret. 
 
 14.. 1T124 suttle. 
 
 14X2-7-3= 9| cloff. 
 
 Ans. 14.. }.. 3 neas. 
 
66 
 
 INVOICES. 
 
 [the tutor's. 
 
 (2) Mr. Isaac Pearson, Derby, June 3, 1830. 
 
 Bought of John Sims and Son. 
 
 1 5 yds. satin ... at 9 
 
 18^ yds. flowered silk . at 17 
 
 12 yds. rich brocade . at 19 
 
 16A yds. sarcenet . . at 3 
 
 1 3 1 yds. Genoa velvet at 27 
 
 23 yds. lustring . . at 6 
 
 6 per yard 
 
 4 
 
 8 
 
 2 
 
 3 
 
 £. 
 
 d 
 
 £62..ll..9± 
 
 (3) Miss Enfield, Nottingham, June 4, 1830. 
 
 Bought of Joseph Thompson. 
 
 4 4 yds. cambric . . at 
 
 12^ yds. muslin . at 
 
 15 yds. printed calico at 
 
 2 doz. napkins . at 
 
 ells diaper ... at 
 
 ells dowlas . . at 
 
 14 
 35 
 
 *. 
 
 12 
 8 
 5 
 2 
 1 
 1 
 
 d. 
 
 6 per yard 
 
 3 
 
 4 
 
 3 each ... 
 
 7 per ell... 
 
 H 
 
 £. 
 
 d. 
 
 £17..I4..11 
 
 Received the above, 
 
 Joseph Thompson. 
 
 (4) Mrs. Mary Bright sold to the Right Honourable Lady 
 Anna Maria Lamb, 1 8 yards of French lace at 12*. 3d. per yd. 
 5 pairs of fine kid gloves at 2s. 2d. per pair, 1 dozen French 
 fans at 3s. 6d. each, two superb silk shawls at three guineas 
 each, 4 dozen Irish lamb at Is. 3d. per pair, and 6 sets of 
 knots at 2s. 6d. per set. — Please to make the Invoice for her. 
 
 Total amount <£23..14..4. 
 
 (5) Mr. Thomas Ward sold to James Russell Vernon, Esq. 
 1 7^ yards of fine serge at 3s. Qd. per yd. 1 8 yds. of drugget at 
 9s. per yd. 15| yds. of superfine scarlet at 22*. per yd. l6§ 
 yds. of Yorkshire black at 18*. per yd. 25 yds. of shalloon at 
 1*. Qd. per yd. and 17 yds. of drab at 17*. 6d. per yd. — Make 
 an Invoice of these articles. Total amount £60- 10.. 5 J. 
 
 (6) Mr. Samuel Green of Wolverhampton, sent to Messrs. 
 Wright and Johnson, agreeably to order, 27 calf skins at 
 3s. 6d. each, 75 sheep skins at 1*. 7d. 39 coloured ditto, at 
 
ASSISTANT.} INVOICES. Q'J 
 
 Is. Sd. 15 bucTv s»ans at \ls. 6d. 17 Russia hides at 10^. 7d. 
 and 125 lamb skins at 1*. 2^d. — Draw up the Invoice. 
 
 Total amount <£39..1..8^. 
 (7) Mr. Richard Groves sent the following articles to the: 
 Rev. Samuel Walsingham ; viz. 2 stones of raw sugar at 6^d. 
 per lb. 2 loaves of sugar. 15^ lb. at ll^d. per lb. a stone of 
 East India rice at S^d. per lb. 2 stone Carolina rice at 5d. per 
 lb. 15 oz. nutmegs at 5\d. per oz. and half a stone of Dutch 
 coffee at 1*. lOd. per lb. — Make a copy of the Invoice. 
 
 Total amount <£3..5..5f . 
 
 BILLS OF BOOK-DEBTS. 
 
 (8) Mr. Charles Cross, Chester. 
 
 To Samuel Grant, and Co., Dr. 
 
 1830. s. d. £. s. d. 
 
 April 14. Belfast butter, 1 cwt. at 6| per lb. 
 
 Cheese, 7 cwt • 3 qrs. 1 2 lb. at 56 long cwt. 
 
 May 8. Butter, £ firkin, 28 lb. at 5i per lb. 
 
 July 17. 5 Cheshire cheeses, 127$. at 6£ 
 
 Sept. 4. 2 Stilton ditto 15 lb. at 10^ 
 
 Cream cheese, 13 lb. at 8^ 
 
 Dec. 28. Received the contents, 
 
 Samuel Grant. 
 
 (9) Mr. Charles Septimus Twigg, Newark. 
 
 To Isaac Jones, Dr. 
 
 1829. s. d. £. s . <L 
 Oct. 22. Tares, 39 bushels at 110 per bush. 
 
 1830. Pease, 18 bushels at 30 4 per qr. 
 Feb. 18. Malt, 7 qrs. . at 63 6 per qr. 
 
 Hops, 2 cwt. 1 qr. at 1 5 per ft>. 
 Feb. 20. Oats, 6 qrs. . at 2 i£ per bush. 
 Beans, 17 qrs. . at 37 4 per qi 
 
 £U^U 
 
 1830, July 1. Received the above for Isaac Jones, 
 
 Thomas West. 
 
*- SIMPLE INTEREST. [THE TUTOR'S 
 
 SIMPLE INTEREST 
 
 Is the premium allowed for the loan of any sum of money 
 during a given space of time. 
 
 The Principal is the money lent, for which Interest is to 
 be received. 
 
 The Rate per cent, per annum, is the quantity of Interest 
 (agreed on between the Borrower and the Lender) to be paid 
 for the use of every £100. of the Principal, for one year. 
 
 The Amount is the Principal and Interest added together. 
 
 I. Tojind the Interest of any Sum of Money for a Year. 
 
 Rule. Multiply the Principal by the Rate per cent, and 
 that Product divided by 100, will give the Interest required. 
 
 Note. When the Rate is an aliquot part of 100, the Interest may 
 be calculated more expeditiously by taking such part of the Principal. 
 Thus, for 5 per cent, take ^ ; for 4 per cent, fo or £ of 1 ; for 2 pei 
 cent. ^ ; for 2\ per cent. ¥ l 5 ; for 3 per cent. 5 l & , plus ± of that ; &c. 
 
 This Rule is applied to the calculation of Commission, Bro- 
 kerage, Purchasing Stocks, Insurance, Discounting of Bills 
 &c* 
 
 II. For several Years. Multiply the Liter est of one yeai 
 by the number of years, and the product will be the answer. 
 
 For parts of a year, as months and days, &c. the Interest, 
 may be found by taking the aliquot parts of a year ; or by 
 the Rule of Three: and it is customary to allow 12 months 
 to the year, and SO days to a month.f 
 
 * To discount a Bill of Exchange is to advance the cash for it be. 
 iore it becomes due ; deducting the Interest for the time it has to run. 
 Bankers always charge Discount as the Interest of the sum. 
 
 ■f At the rate of 5 per cent, the interest of £1. for a year is Is. ,• oi 
 one penny for a month. Therefore, the principal X the number of 
 months, gives the interest in pence. 
 
 Or, take the parts of a year for the months, out of as many shillings 
 'iS there are pounds in the principal. 
 
 Thus, to tind the interest of £40..10. for 2 months, say 40|rf. X 2 
 = 8 Id. = 6s. 9d. ; or, 2 months being i of a year, 40.?. 6d. - 1 - 6 = 
 6s. 9d. Ans. 
 
 For days, take the aliquot parts of a month. The interest for days, 
 
 . at 5 per cent, may also be found by multiplying the principal by the 
 
 number of days ; and the product divided by 365 will give the answer 
 
 in shillings ; or divided by 7300 (=365X20) will give the answer in 
 
 pounds. 
 
ASSISTANT.] SIMPLE INTEREST. 69 
 
 (1) What is the interest of £375. for a year, at £5. per 
 cent, per annum?* 
 
 (2) What is the interest of £945.. 10. for a year, at £4. per 
 cent, per annum ? Ans. £37..l6..4f. 
 
 (3) What is the interest of £547.-153 at £5. per cent, per 
 annum, for 3 years? Ans. £82..3..3. 
 
 (4) What is the interest of £254..17»6. for 5 years, at £4. 
 per cent, per annum ? Ans. £50..19»6. 
 
 (5) What is the amount of £556.. 13. A. at £5. per cent 
 per annum, in 5 years ? Ans. £695..l6..8. 
 
 Note. Commission and Brokerage (commonly called Brokage) are al- 
 lowances of so much per cent, to an agent or broker, for buying or 
 selling goods, or transacting business for another. 
 
 (6) My correspondent informs mje that he has bought 
 goods to the amount of£754..l6. on my account, what is his 
 commission at £2|. per cent. ? Ans. £l8..17..4§. 
 
 (7) If I allow my factor £3f . per cent, for commission, 
 what will he require on £876..5..10 ? Ans. £32..17„2^. 
 
 Note. Stock is a general term to designate the Capitals of our 
 Trading Companies ; or to denote Property in the Public Funds ; which 
 means the Money paid by Government for the interest of the National 
 Debt. The quantity of Stock is a nominal sum, for which the owner 
 receives a certain rate of interest while he holds the same. 
 
 (8) At£ll0^. percent, what is the purchase of £2054.. 1 6. 
 South Sea Stock? Ans. £2265..8..4. 
 
 (9) At £l04§. per cent. South Sea annuities, what is the 
 purchase of £l797»14.? Ans. £l876..6..11{. 
 
 (10) At £96'f. per cent, what is the purchase of £577-19 
 Bank annuities ? Ans. £559»3..3f . 
 
 (11) At £l24§. per cent, what is the purchase of £758.. 
 17..10. India stock? Ans. £945..15..4£. 
 
 (12) What sum will purchase £1284. of the 3 per cent. 
 Consols, at £59 J. per cent. ; including the broker's charge of 
 \ % or 2*. 6d. per cent, on the amount of stock ? 
 
 Ans. £770..7..Hf. 
 
 f £.375 
 
 5 Better thus : 
 
 £. £. 
 
 5^ J, 875 
 
 £. 18)75 
 20 
 
 s. 15100 Am * ^ 18 " 15 ' 
 
 Ans. £18.15. 
 Cutting the two figures in the above divides the number by 100 
 see Division, p. 23. 
 
70 SIMPLE INTEREST. [_THE TLTOH's 
 
 (13) If I employ a broker to buy goods for me, to the 
 amount of £2575..17»6. what is the brokerage at 4s-. per 
 cent. ?* 
 
 (14) What is the broker's charge on a sale amounting to 
 £7105..5..10. at 5s. 6d. per cent. ? Ans. £l9..10..9|-. 
 
 (15) What is the brokage on goods sold for £975..6'..4. 
 at 6s. 6d. per cent. ? Ans. £3..3.A±. 
 
 (16) What is the interest of £257..5..1. at £4. per cent, per 
 annum, for a year and three quarters? Ans. £l8..0..1^. 
 
 (17) What is the interest of £479-5. for 5± years, at £5 
 per cent, per annum ? Ans. £l25..l6..0§. 
 
 (18) What is the amount of £576..2..7- in 1{ years, at 
 £4|. per cent, per annum? Ans. £764".1..8^. 
 
 (19) What is the interest of £%59.>13..5. for 20 weeks, at 
 £.5. per cent, per annum? Ans. £4..19..10^. 
 
 (20) What is the interest of £2726..1..4. at £4|. per cent 
 per annum, fo> 3 years, 154 days? Ans. £419..15..6^. 
 
 (21) Compute the interest of £155. for 49 days, and for 
 146 days, at £5. per cent, per annum ? 
 
 Ans. £l..0..9^. and £3..2..0. 
 
 (22) What will a banker charge for the discount of a bill 
 of £76.. 10. and another of £54. negotiated on the 18th of 
 May ; the former becoming due June 30, and the latter July 
 13; discounting at £5. per cent. ? Ans. 8s. lid. and 8s. 3d. 
 
 When the Amount, Time, and Bate per cent, are given, to t find 
 the Principal. 
 
 Rule. As the amount of £ 100. at the rate and for the time 
 given, is to £100., so is the amount given, to the principal 
 required. 
 
 (23) What principal being put to interest will amount to 
 £402.. 10. in 5 years, at £3. per cent, per annum ?t 
 
 (24) What principal being put to interest for 9 years, will 
 amount to £734.. 8. at £4. per cent, per annum ? Ans. £540. 
 
 8. £. s. d. 
 • 4^z\ 2575..17.. 6 
 
 £. 5\15.. 3.. 6 
 20 
 
 £. 
 
 : 100 
 
 03 
 
 12 
 
 42~ 
 4 
 
 7|68" 
 
 £. s 
 : : 402.. 10; 
 
 
 s. 3J0S 
 
 Ans. X5..3..0J. 
 £. 
 •f £SX5-r-100=£ll5. As 115', 
 
 £. 
 
 : 350 Ans. 
 
ASSISTANT. J DISCOUNT. 71 
 
 (25) What principal being put to interest fbr 7 years, at 
 £o, per cent, per annum, will amount to £334.. 1 6.? 
 
 Ans. £248. 
 
 When the Principal, Rate per cent, and Amount are given, to 
 Jind the Time. 
 Rule. As the interest for 1 year, is to 1 year, so is the 
 whole interest, to the number of years. 
 
 (26) In what time will £350. amount to £402..! 0. at £3. 
 per cent, per annum ?* 
 
 (27) In what time will £540. amount to £734.. 8. at £4. 
 per cent, per annum ? Ans. 9 years. 
 
 (28) In what time will £248. amount to £334.. 1 6. at £5. 
 per cent, per annum ? Ans. 7 years. 
 
 When the Principal, Amount, and Time are given, to Jind 
 the Rate per cent. 
 
 Rule. As the principal, is to the whole interest, so is £100. 
 to its interest for the given time. Divide that interest by the 
 number of years, and the quotient will be the rate per cent. 
 
 (29) At what rate per cent, will £350. amount to £402.. 10. 
 in 5 years ?t 
 
 (30) At what rate per cent, will £248. amount to £334..l6. 
 in 7 years ? Ans. £5. per cent. 
 
 (31) At what rate per cent, will £540. amount to £734.. 8. 
 in 9 years ? Ans. £4. per cent. 
 
 DISCOUNT 
 
 Is the abatement of so much money, on any sum received be- 
 fore it is due, as the money received, if put to interest, would 
 gain at the rate, and in the time given. Thus £]00. present 
 money would discharge a debt of £105. to be paid a year 
 hence, Discount being made at £5. per cent. 
 
 £. 
 
 * 350X3 
 
 - =£10.. 10. the interest for 1 year. 
 
 100 
 
 £402.. 10 £350. =£52.. 10. the whole interest. 
 
 As £10..10 : 1 year : : £5$i..l0 : 5 years. Ans. 
 f As £350 : £52..lb : : £100 : £l5=the interest of £100. for 5 
 years. Then 15-r-5=£3. the rate per cent. 
 
/2 discount. [the tutor's 
 
 Rule. As £l00. with its interest for the time given, is to 
 that interest ; so is the sum given, to the Discount required. 
 
 Also, As that Amount of £100. is to £100. so is the given 
 sum, to the Present worth. 
 
 But if either the Discount or the Present worth be found by 
 the proportion, the other may be found by subtracting that 
 from the given sum. 
 
 (1) What are the discount and present worth of £386..5 
 for 6 months, at £6. per cent, per annum ?* 
 
 (2) How much shall I receive in present payment for a 
 debt of £357-10. due 9 months hence; allowing discount at 
 £5. per cent, per annum? Ans. £344..11..6£ ||. 
 
 (3) What is the discount of £27 5.. 10. for 7 months, at £,5. 
 per cent, per annum? Ans. £7..l6..1f 2W 
 
 (4) What is the present worth of £527..9»1« payable in 7 
 months, at £4^. per cent, per annum ? 
 
 Ans £514..13..10^ r?f{\%. 
 
 (5) Required the present worth of £875..5..6. due in 5 
 months, at £4^. per cent, per annum ? Ans. £859»3..3f 4-^75 • 
 
 (6) What is the present worth of £500. payable in 10 
 months, at £5. per cent, per annum ? Ans. £480. 
 
 (7) How much ready money ought I to receive for a note 
 of £75. due in 15 months, at £5. per cent, per annum ? 
 
 Ans. £70..11..9 T V 
 
 (8) What will be the present worth of £ 150. payable at 3 
 instalments of four months ; i. e. one third at 4 months, one 
 third at 8 months, and one third at 12 months, discounting 
 at £5. per cent, per annum ? Ans. £l45..3..8^. 
 
 (9) Of a debt of £575.. 10. one moiety is to be paid in 3 
 months, and the other in 6 months. What discount must be 
 allowed for present payment, at £5. per cent, per annum ? 
 
 Ans. £10..11..4|. 
 
 • 6 
 
 m, 
 
 -\ £6 
 100+3= 
 £. 
 As 103 
 
 £. 
 
 103=amount of £10C 
 
 £. £. s. 
 : 3 : : 386.. 5 £. 
 
 3 386.. 
 
 >. in 6 months. 
 5 
 
 
 
 
 103)1158.. 
 
 15( 
 
 11.. 
 
 5 discount. 
 
 
 
 
 1133 
 
 
 375.. 
 
 present worth. 
 
 
 25 
 20 
 
 
 
 
 
 '03)515=5*. 
 
ASSISTANT.] COMPOUND INTEREST. 73 
 
 (10) What is the present worth of £500. at £4. per cent, 
 per annum, £100. being to be paid down, and the rest at two 
 6 months P Ans. £488..7..8£. 
 
 (11) Bought goods amounting to £l09..10. at 6 monuis' 
 credit, or £3^. per cent, discount for prompt payment. How 
 much ready money will discharge the account ?* 
 
 Ans. £l05..13..4| 
 
 Note. The Rule to find the present worth of any sum of money 
 is precisely identical with that case in Simple Interest in which the 
 Amount, Time, and Hate per cent, are given to find the Principal* 
 See page 70. 
 
 COMPOUND INTEREST 
 
 Is that which arises from both the Principal and Interest : 
 that is, when the Interest of money, having become due, and 
 not being paid, is added to the Principal, and the subsequent 
 Interest is computed on the Amount. 
 
 Rule. Compute the first year's interest, which add to the 
 principal :*' then find the interest of that amount, which add 
 as before, and so on for the number of years. Subtract the 
 given sum from the last Amount, and the remainder will be 
 die Compound Interest. 
 
 (1) What is the compound interest of £500. forborne 3 
 years, at £5. per cent, per annum ?t 
 
 (2) What is the amount of £400. in 2>\ years, at £5. per 
 cent, per annum, compound interest ? Ans. £474..12..6|. 
 
 (3) What will £650. amount to in 5 years, at £5. per cent, 
 per annum, compound interest ? Ans. £829..11..7^. 
 
 (4) What is the amount of £550.. 10. for 2>\ years, at £6. 
 per cent, per annum, compound interest ? Ans. £675. .6.. 5. 
 
 (5) What is the compound interest of £764. for 4 years and 
 9 months, at £6. per cent, per annum ? Ans. £243..! 8. .8. 
 
 * The discount in cases of this sort is so much per cent, on the sum, 
 without regard to time. It is, therefore, computed as a year's 
 interest. 
 
 So 
 
 £500 5*5 £551.. 5 
 
 25 27..11.. 
 
 5 l 5 525 amount in 1 yr. 578..16.. 3 amount in 3 years. 
 
 26.. 5 500.. 0.. principal subtract. 
 
 551.. 5 do. in 2 yrs. £78^1 b\. 3 Ans. 
 
74 EQUATION OF PAYMENTS. [/THE TUTOR'S 
 
 ' (6) What is the compound interest of £57 .10.. 6. for 5 
 years, 7 months, and 1.5 days, at £5. per cent, per annum ? 
 
 Ans. £18..3..8£. 
 (7) What is the compound interest of £259- 10. for 3 
 vears, 9 months, and 10 days, at £4^. per cent, per annum ? 
 
 Ans. £46..1<M0£. 
 
 2|00)14|20 
 
 Ans. 7i 2 o months. 
 
 EQUATION OP PAYMENTS 
 
 Is when several sums are due at different times, to find a 
 mean time for paying the whole debt ; to do which, this is 
 the common 
 
 Rule. Multiply each term by its time, and divide the sum 
 of the products by the whole debt ; the quotient is accounted 
 the mean time. 
 
 £. 
 
 (1) A owes B £200. whereof 40x 3= 120 
 £40. is to be paid at 3 months, 60X 5= 300 
 £60. at 5 months, and £100. at 100x10=1000 
 10 months: at what time may 
 the whole debt be paid together, 
 without prejudice to either? 
 
 (2) B owes C £800. whereof £200. is to be paid at 3 
 months, £l00. at 4 months, £300. at 5 months, and £200. 
 at 6 months ; but they agree that the whole shall be paid 
 at once : what is the equated time ? Ans. 4 months, 18 J days. 
 
 (3) A debt of £360. was to have been paid as follows : 
 viz. £120. at 2 months, £200. at 4 months, and the rest at 5 
 months ; but the parties have agreed to have it paid at one 
 mean time: what is that time ? Ans. 3 months, 13 J days. 
 
 (4) A merchant bought goods to the value of £500. to pay 
 £l00. at the end of 3 months, £150. at the end of 6 months, 
 and £250. at the end of 12 months; but it was afterwards 
 agreed to discharge the debt at one payment : required the 
 time. Ans. 8 months, 12 days. 
 
 (5) H is indebted to L a certain sum, which is to be paid 
 at 6 different payments, that is ^ at 2 months, J at 3 months, 
 | at 4 months, \ at 5 months, \ at 6 months, and the rest at 
 7 months ; but they mutually agree that the whole shall be 
 paid at one equated time : what is that time ? Ans. 4^ months. 
 
 (6) A is indebted to B £120. whereof^ is to be paid at 3 
 months, ^ at 6 months, and the rest at 9 months : what is the 
 equated time of the whole payment? Ans. 5\ months. 
 
ASSISTANT.] IARTER. J& 
 
 BARTER 
 
 Is the exchanging of commodities. 
 
 Rule. Compute, by the most expeditious method, the 
 ralue of the article whose quantity is given : then find what 
 quantity of the other, at the rate proposed, may be had for 
 the same money. 
 
 Note. Sometimes one tradesman, in bartering, advances his goods 
 above the ready money price. In this case, it will be necessary to 
 proportionate the other's bartering price to his ready money price, by 
 the Rule of Three. 
 
 (1) What quantity of chocolate at 4?. per lb. must be ex- 
 changed for 2 cwt. of tea, at 9«?- per lb. ?* 
 
 (2) A and B barter: A has 20 cwt. of prunes, at 4a 7 . per 
 lb. ready money, but in barter will have 5d. per lb. and B 
 has hops worth 32s. per cwt. ready money : what ought B 
 to charge his hops, and what quantity must he give for the 
 20 cwt. of prunes ?t 
 
 (3) How much tea at Qs. per lb. can I have in barter for 
 4 cwt. 2 qrs. of chocolate, at 4s. per lb. ? Ans. 2 cwt. 
 
 (4) A exchanges with B 23^ cwt. of cheese, worth 52s. 6d. 
 per cwt. for 8 pieces of cloth containing 248 yards, at 4>s. 4rf. 
 per yard ; the difference to be paid in money. Who receives 
 the balance, and how much? Ans. A receives <£7..19,.l. 
 
 (5) How much ginger at 15\d. per lb. must be excnangecl 
 for 3\ lb. of pepper, at \3\d. per lb. 9 Ans. 3 lb. Iff oz. 
 
 (6) How many dozen of candles, at 5s. 2d. per dozen, 
 must be bartered for 3 cwt. 2 qrs. 16 lb. of tallow, at 37s. 4a. 
 per cwt. ? Ans. 26 dozen, 3f | lb. 
 
 (7) A exchanges with B 60S yards of cloth, worth 14s. per 
 
 * 224X9=2016*. the value of the tea. 
 As 4*. : 1 lb. : : 2016*. : 504 lb. of chocolate. Ans. 
 •f As 4d. : 5d. : : 32*. : 40*. the price per cwt. to be charged ft> 
 
 the hops. 
 
 20 cotf.=:2240 lb. 
 5 
 
 11200J. the value of the prunes. 
 
 As 40*. : I act. :: 11 200 J. : *— ° =23 cwt. lqr. 9 L§lb. An; 
 12 480 
 
 480& 
 
76 PROFIT AND LOSS. [THE TUTOR** 
 
 yard, for 85 ctvt. 2 qrs. 24 lb. of bees' wax, and £l25..12. in 
 cash. What was the wax charged per cwt. ? Ans. £3.. 10. 
 
 (8) A barters with B 320 dozen of candles at 4.?. 6d. per 
 dozen, for cotton at 8d. per lb. and £30. in cash. What was 
 the quantity of cotton ? Ans. 1 1 cwt. 1 qr. 
 
 (9) How much cotton, at 1*. 2d. per lb. must be given for 
 114 lb. of tobacco, at 6d. per lb. ? Ans. 48? lb. 
 
 PROFIT and LOSS 
 
 Is a Rule by which we discover the gain or loss in the buy- 
 ing and selling of goods ; and which enables us to adjust the 
 prices of articles, so as to gain or lose so much per cent. &c. 
 The questions are solved by the Rule of Three, or Practice. 
 The prime cost means the purchase money: therefore 
 The prime cost -! p ?* tfie kr' ° T \ e ^ uai ^ ae se ^* n £ V vice * 
 
 The selling mice minus i H* pr ^ e C0St T al * he gain# 
 ±ne selling price minus j the gdn ^^ the prime ^ 
 
 The selling price phis the loss equal the prime cost. 
 
 Gain or loss per cent, means so much on £100. purchase money, o\ 
 prime cost: therefore, when £20. per cent, are gained, £120. is tlu. 
 selling price per cent. ; when £20. per cent, are lost,, £80. is the selling 
 price. 
 
 Case 1. Given, the prime cost and the selling price of an integer or 
 quantity, to find the gain or loss per cent. 
 
 As the prime cost given : the gain or loss : : £100. : the gain or 
 loss per cent. 
 
 Case 2. Given, the prime cost as before, with a proposed gain or loss 
 per cent, to find the selling price. 
 
 As £100. = { or £ i? °b P minus e th^L } : : the***** : theselling price. 
 Case 3. Given, the selVmg price of an integer or quantity, and the 
 gain or loss per cent, to find the prime cost. 
 
 £ iZ: S$£ & } ■ *«■ ■ • the ** ** ■■ «* f*- «*• 
 
 Case 4. Given, the selling price of an integer, and the gain per cent. 
 to find the gain per cent, at some other proposed price. 
 
 As the selling price : £100. plus the gain : : the proposed price : the 
 selling price per cent, from which deduct £100. for the gain per cent, re- 
 quired. 
 
 Secondly, To find another selling price, at a different gain per cent. 
 
 As £100. plus the gain : the selling price: : £100. plus the proposed 
 gain : the selling price required. 
 
 A much greater variety of cases may occur ; but it is presumed 
 that the student who attains £ \e knowledge of these, will easily 
 comprehend the rest. 
 
kSSISTANT.] PROFIT AND LOSS. 77 
 
 (1) If 1 yard of cloth costs 11$. and is sold for 12s. 6d. 
 adiat is the gain per cent. ?* 
 
 (2) If 60 ells of Holland cost £l 8. what must 1 ell be sold 
 for to gain £8. per cent. ?t 
 
 (3) If 1 lb. of tobacco cost l6d. and be sold for 20d. what 
 is the gain per cent. ? Ans. £25. 
 
 (4) If a parcel of cloth be sold for £560. gaining £12. 
 per cent, what is the prime cost ? Ans. £500. 
 
 (5) If a yard of cloth be bought for 13$. 4c?. and sold again 
 for 16$. what is the gain per cent. ? Ans. £20. 
 
 (6) If 1 12 lb. of iron cost 27$. 6d. what must 1 cwt. be sold 
 for to gain £l5. per cent. ? Ans. £i..ll..7^. 
 
 (7) If 375 yards of cloth be sold for £490. at £20. per 
 cent, profit, what did it cost per yard ? Ans. £l..l..9^ Jf f. 
 
 (8) Sold 1 cwt. of hops for £6..15. at the rate of £25. per 
 cent, profit. What would have been the gain per cent, if 
 they had been sold for £8. per cwt. ? Ans. £48..2.. 11^ 
 
 (9) If 90 ells of cambric cost £60. how must I sell it per 
 yard to gain £18. per cent. ? Ans. 12$. 7 ^%d. 
 
 (10) A plumber sold 10 fothers of lead for £204..15. and 
 gained after the rate of £l2..10. per cent. What did it 
 cost him per cwt. ? Ans. 18s. 8d. 
 
 (11) What was the profit on 436 yards of cloth, bought at 
 8$. 6d. and sold at 10$. 4c?. per yard ? Ans. £39..19»4. 
 
 (12) Bought 14 tons of steel at £69. per ton, which was 
 -etailed at 6d. per lb. What was the loss sustained ? 
 
 Ans. £182. 
 
 (13) Bought 124 yards of linen for £32. How should the 
 ame be retailed per yard, to gain £15. per cent. ? 
 
 Ans. 5s. llj%\d. 
 
 (14) Bought 249 yards of cloth at 3s. 4c?. per yard, and 
 etailed the same at 4$. 2c?. per yard. What was the whole 
 rain, and how much per cent. ? % 
 
 Ans. £l0..7-6. profit, and £2 5. per cent. 
 
 cost gain cost mov^ 
 
 • As lit. : 1*. 6a\ : : £100 : LlZ^L =£13..12..8£ if. Ans. 
 
 2 2_ 22 
 
 step. 22 3 sixp. 
 
 cost s. price cost inqvm 
 + As £100 : £108 : : £18 : -i^_=£19..8..94 A the selling 
 1 100 
 
 ■rice. And £l9..8..9|-r-60:=6s. 5%d. the price per elL 
 
 X For the solving of this question, see Cases 1 and 2. 
 
78 FELLOWSHIP. [[THE TUTOR'S 
 
 FELLOWSHIP, or PARTNERSHIP 
 
 Is a rule by which any number or quantity may be divideu 
 into certain proportionate parts. It is applied to determine 
 the respective shares of gain or loss of the several partners in 
 a company, in proportion to their respective shares of the 
 capital employed as a joint stock : also in the division of com- 
 mon lands, and other cases of a similar kind. 
 
 FELLOWSHIP WITHOUT TIME. 
 
 Rule. As the whole stock, is to the whole gain or loss ; so 
 is each individual share, to the correspondent gain or loss. 
 
 Proof. The sum of the shares will be equal to the whole 
 gain or loss. 
 
 (1) A and B join in trade. A puts into stock £20. and B 
 £40 ; they gain £50. What is the share of each ?* 
 
 (2) A, B, and C joined in trade ; A put in £20 ; B £30 ; 
 and C £40; and they gained £l80. What is each man's 
 part of the gain? Ans. A £40. B £60. C £80. 
 
 (3) Four persons, B, C, D, and E formed a joint stock ; F 
 put in £227 ; C £349 ; D £115 ; and E £439 ; they gained 
 £428. Required each person s share of the gain. 
 
 Ans. B £85..19..6f T \%. C£l32..3..9 ,*&. 
 D £43..11..1f ffi. E£l66..5..6i t?f 
 
 (4) D, E, and F entered into partnership. D's stock wa. 
 £750 ; E's £460 ; and F's £500 ; and at the end of 12 months 
 they had gained £684. What is each man's particular share 
 of the gain ? Ans. D £300. E £184. and F £200. 
 
 (5) A tradesman is indebted to B £275..14 ; to C £304..7 * 
 to D £152; and to E £l04..6; but upon his decease his 
 estate is found to be worth but £675.. 15. How must it be 
 divided among his creditors ? 
 
 Ans. B's share £222..15..2— 6584. Cs £245.. 18.. 1^—1 5750. 
 D's £122..16..2|— 12227. and E's £84..5..5— 15620. 
 (6) A, B, C, and D are the principal partners of a trading 
 company, of which A provided J of the capital, B \, C J, and 
 
 • 20+40=60 
 a «n en \ 20 : £16 .. 13 .. 4=A's share. 
 AS ou : su : : | 4Q . 88 „ 6..8 «B'a share, 
 
 50.. 0..0 Proof: 
 
ASSISTANT.] FELLOWSHIP. 7S 
 
 D J. At the end of 6 months their portion of the profits 
 amounts to £100. What will be the amount of each person's 
 share? Ans. A £35..1..9 if. B £ 26..6..3f A- 
 
 C£21..1..0j ff. and D £17..10..101'/t 
 
 (7) Two persons joined in the purchase of an estate yield- 
 ing £1700. per annum, for £27200. whereof D paid £15000. 
 and E the rest : some time after, they sold it for 24 years 
 purchase. What was each person's share?* 
 
 Ans. D £22500. ££18300. 1 
 
 (8) D, E, and F form a joint capital of £647- Their re- 
 spective shares are in proportion to each other as 4, 6, and 8 ; 
 and the gain is equal to D's stock. Required each person's 
 stock and gain. 
 
 Ans. D's stock £l43..15..6§ gain, £31..19..0/ T . 
 E's . . . 215..13..4 . . . 47..I8..62V 
 F's . . . 287..11..1|. . . 63..18..0/ T . 
 
 (9) D, E, and F joined in partnership ; the amount of their 
 stock was £100 ; D's gain was £3 ; E's £5 ; and F's £8 ; 
 what was each man's stock ? 
 
 Ans. Us stock £l8..15. E's £31..5. and F's £50. 
 
 FELLOWSHIP WITH TIME. 
 
 Rule. As the sum of the products of each person's money 
 and time, is to the whole gain or loss ; so is each individual 
 product, to the corresponding gain or loss. 
 
 (1) D and E enter into partnership ; D puts in £40. for 
 three months, and E £75. for four months, and they gain 
 £70. What is each man's share of the gain ?t 
 
 (2) Three tradesmen joined in company ; D put into the 
 joint stock £l95..14. for three months ; E £l69-.18..3. for 5 
 months; and F £59-1 4.. 10. for 11 months: they gained 
 £364.. 18. What is each man's share of the gain ? 
 
 Ans. D's £l02..6..4— 5008. E's £ 148.. 1.. 11—482802. 
 and F's £ll4..10..6J— 14707. 
 
 (3) Three merchants join in company for 18 months : D 
 puts in £500. and at 5 months' end takes out £200. at 10 
 
 • The sale of a property for so many years' purchase, is understood to 
 be, for so much present money as the annual rent or value X that 
 number of years. 
 
 f 40X3=120 . . . . 7 |o . . I 120 : £20=D's shar e. 
 
 75X4=300 -™ * ' (300 : 50= E's share. 
 
 420 70 Proof. 
 
80 ALLIGATION. [THE TUTOR'S 
 
 months' end puts in £300. and at the end of 14 months takea 
 out £130 ; E puts in £400. and at the end of 3 months £270 
 more, at 9 months he takes out £140. but puts in £100. at 
 the end of 12 months, and withdraws £99. at the end of 15 
 months ; F puts in £900. and at 6 months takes out £200 
 at the end of 11 months puts in £500. but takes out that and 
 £100. more at the end of 13 months. They gain £200. 
 Required each man's share of the gain ? 
 
 Ans. D£50..7..6— 21720. E £62..12..5±— 29859- and 
 
 F £87..0..0f— 14167. 
 
 (4) D, E, and F, hold a piece of ground in common, for 
 
 which they are to pay £36..10..6 : D puts in 23 oxen 27 
 
 days ; E 21 oxen 35 days ; and F 16 oxen 23 days. What is 
 
 each man to pay of the said rent ? 
 
 Ans. Z>£l3..3..1i— 624. E £l 5.. 11. .5—1 688. and 
 F £7..15..ll—U36. 
 
 ALLIGATION 
 
 Is a rule by which we ascertain the mean price of any com- 
 pound formed by mixing ingredients of various prices ; of 
 the quantities of the various articles which will form a mix- 
 ture of a certain mean or average value. It comprises four 
 distinct cases. 
 
 Case 1. Alligation Medial. The various quantities 
 and prices being given, to find the mean price of the mixture. 
 
 Rule. Multiply each quantity by its price, and divide the 
 sum of the products by the sum of the quantities.* 
 
 (1) A grocer mixed 4 cwt. of sugar, at 56s. per crvt. with 
 7 cwt. at 43s. per cwt. and 5 cwt. at 37s. per cwt. What is 
 the value of 1 cwt. of this mixture ? Ans. £2..4..4^. 
 
 (2) A vintner mixes 15 gallons of canary, at 8s. per gallon, 
 with 20 gallons, at 7*. 4a?. per gallon ; 10 gallons of sherry, 
 
 Example. 
 *A farmer mixed 20 bushels of s. s. 
 
 wheat, at 5s. per bushel, and 36 20 X 5 = 100 
 
 bushels of rye, at 3*. per bushel, 36 X3= 108 
 
 with 40 bushels of barley, at 2#. 40 X 2 = 80 
 
 per bushel. What is the worth "gg 9 6)2"88(3,. Anu 
 
 of a bushel of this mixture? — /*ww ™>*. 
 
ASSISTANT."] ALLIGATION. 81 
 
 at 6s. Sd. per gallon ; and 24 gallons of white wine, at 4s. 
 per gallon. What is the worth of a gallon of this mixture ? 
 
 Ans. 6s. 2\d. £g. 
 
 (3) A maltster mixes SO quarters of brown malt, at 28s. 
 per quarter, with 46 quarters of pale, at 30s. per quarter, 
 and 24 quarters of high dried ditto, at 25s. per quarter. 
 What is one quarter of the mixture worth ? 
 
 Ans. £l..8..2±d. T %. 
 
 (4) A vintner mixes 20 quarts of port, at 5s. 4c?. per 
 quart, with 12 quarts of white wine, at 5s. per quart, 30 
 quarts of Lisbon, at 6s. per quart, and 20 quarts of moun- 
 tain, at 4<s. 6d. per quart. What is a quart of this mixture 
 worth? " Ans. 5s. 3fd. f g. 
 
 (5) A refiner melts 12 lb. of silver bullion, of 6 oz. fine, 
 with 8 lb. of 7 oz. fine, and 1 lb. of 8 oz. fine ; required the 
 fineness of 1 lb. of that mixture ? Ans. 6 oz. 18 dwt. \6 gr. 
 
 Case 2. Alligation Alternate. The various prices 
 being given, to find the quantities which may be mixed, to 
 bear a certain average price. 
 
 Rule. Arrange the given prices in one column, with the 
 proposed average price on the left. 
 
 Link each less than the average with one greater. 
 
 Place against each term the difference between that with 
 which it is linked and the mean ; and the respective differ- 
 ences will be the quantities required. 
 
 Note. Questions in this rtae admit of a great variety of an- 
 swers, according to the manner of linking them : also by taking 
 other numbers proportional to the answers found. 
 
 (1) A vintner would mix four sorts of wine together, of 
 18dL 20c?. 24c/. and 28c?. per quart, what quantity of each 
 sort must he take to sell the mixture at 22c?. per quart ?* 
 
 (2) A grocer would mix sugar at 4c?. 6d. and 10c?. per #. 
 
 •18- 
 
 22; 
 
 20—, 
 
 '24— — 
 28—1 
 
 Answer. Proof. 
 2 of lSd. = S6d. 
 6 of 2Qd. — 120 
 4 of 2\d. = 96 
 2 of 28 d. = .56 
 
 li 14)308 
 
 or thus, Proof. 
 
 18 6 of lSd. = lOSd. 
 
 22 
 
 20- 
 24_ 
 
 28- 
 
 22d. 
 d5 
 
 2 of 20d. = 40 
 2 of 2M. = 48 
 4 of 28rf. = 112 
 
 14 
 
 14)308 
 
 22d. 
 
82 ALLIGATION. [THE TUTOR'S 
 
 so as to sell the compound for 8d. per lb. What quantity 
 q f each kind must he take ? 
 
 Ans. 2 lb. at 4c?. 2 lb. at 6d. and 6 lb. at lOd. 
 
 (3) How much tea at 16s. 14s. 9s. and 8s. per lb. will 
 compose a mixture worth 10s. per /6. ? 
 
 Ans. 1 Z6. dtf l6s. 2 lb. at 14s. 6 lb. at Qs. and 4 lb. at 8s. 
 
 (4) A farmer would mix as much barley, at 3s. 6d. per 
 bushel, rye at 4,?. per bushel, and oats at 2s. per bushel, as 
 will make a mixture worth 2s. 6d. per bushel. How much 
 of each sort ? Ans. 6 b. of barley, 6 of rye, and 30 of oats. 
 
 (5) A tobacconist would mix tobacco at 2s., Is. 6d., and 
 Is. 3d. per lb. so that the compound may be worth is. 8d. 
 per lb. What quantity of each sort must he take ? 
 
 Ans. 7 lb. at 2s. 4 lb. at Is. 6d. and 4t lb. at Is. 3d. 
 
 Case 3. Alligation Partial. This is similar to Case 2> 
 except that one of the quantities is limited. 
 
 Rule. Link the prices, and place the differences as before. 
 
 Then, as the difference opposite to that whose quantity is 
 given, is to each other difference ; so is the given quantity to 
 each required quantity. 
 
 (1) A tobacconist intends to mix 20 lb. of tobacco at 15c?. 
 per lb. with others at l6d. 18d. and 22d. per lb. How many 
 pounds of each sort must he take to make one pound of th<* 
 mixture worth 17<f. ?* 
 
 (2) How much coffee, at 3s. at 2s. and at Is. 6d. per lb. 
 with 20 lb. at 5s. will make a mixture worth 2s. 8d. per lb. ? 
 
 Ans. 35 lb. at 3s. 70 lb. at 2s. and 10 lb. at Is. 6d. 
 
 (3) A distiller would mix 10 gallons of French brandy, at 
 48s. per gallon, with British at 28s. and spirits at 16s. per 
 gallon. What quantity of each sort must he take to afford 
 it for 32s. per gallon ? Ans. 8 British, and 8 spirits. 
 
 (4) What quantity of teas at 12s. 10s. and 6s. must be 
 mixed with 20 lb. at 4s. per lb. that the mixture may be 
 worth 8s. per lb. ? Ans. 10 lb. at 6s. 10 lb. at 10s. 20 lb. at 12s. 
 
 • 15 
 
 22- 
 
 Answer. 
 
 Proof. 
 
 
 lb. 
 
 lb. 
 
 5 20 lb. at Ud. 
 
 = 300rf. 
 
 As 5 : 1 : 
 
 : 20 
 
 % 
 
 1 4 lb. at ISd. 
 
 = 64>d. 
 
 
 
 
 1 4 lb. at \8d. 
 
 m 72d. 
 
 As 5 : 2 : 
 
 : 20 
 
 : 8 
 
 2 8 lb. at 22d. 
 
 = 17 6d. 
 
 
 
 
 Ae*36 lb. : 
 
 6\2d. 
 
 : : 1 Jft. : 
 
 17 d. 
 
 
ASSISTANT.] COMPARISON OF WEIGHTS AND MEASURES* b*d 
 
 Case 4. Alligation Total. This is also similar to Case 
 2, except that the whole quantity of the compound is limited. 
 
 Rule. Link the prices, and place the differences as before. 
 
 Then, As the sum of the differences, is to each particular 
 difference; so is the quantity given, to each required 
 quantity. 
 
 (1) A grocer has four sorts of sugar at 12 d. Wd. 6d. and 
 4>d. per lb. and would make a composition of 1 44 lb. worth 
 8d. per lb. What quantity of each sort must he take ?* 
 
 (2) A grocer having 4 sorts of tea at 5s. 6s. 8s. and Qs. 
 per lb. would have a composition of 87 lb. worth 7s. per lb. 
 What quantity must there be of each sort ? 
 
 Ans. 14^ lb. of 5s. 29 lb. of 6s. 29 lb. of 8s. and 14£ lb. of 9s. 
 
 (3) A vintner having 4 sorts of wine, viz. white wine at 
 l6s. per gallon, Flemish at 24s. per gallon, Malaga at 32s. 
 per gallon, and Canary at 40s. per gallon ; would make a 
 mixture of 60 gallons worth 20s. per gallon. What quan- 
 tity of each sort must he take ? 
 
 Ans. 45 gallons of white wine, 5 of Flemish, 5 of Malaga, 
 and 5 of Canary. 
 
 (4) A jeweller would melt together four sorts of gold, of 
 24, 22, 20, and 15 carats fine, so as to produce a compound 
 of 42 oz. of 17 carats fine. How much must he take of 
 each sort ? Ans. 4 oz. o/24, 4 oz. of 22, 4 oz. of 20, and 30 oz. 
 cf 15 carats fine. 
 
 COMPARISON OF WEIGHTS AND MEASURES. 
 This is merely an application of the Rule of Proportion. 
 
 (1) If 50 Dutch pence be worth 65 French pence, how 
 many Dutch pence are equal to 350 French pence ?t 
 
 (2) If 12 yards at London make 8 ells at Paris, how 
 many ells at Paris, will make 64 yards at London ? Ans. 42|. 
 
 • 12- 
 -,10- 
 
 4- 
 
 Answer. Proof. #>. lb. 
 
 48 at 12d. = 576 As 12 : 4 : : 144 : 48 
 24 at lud. = 240 As 12 : 2 : : 144 : 24 
 24 at 6d. = 144 
 48 at 4>d. = 192 
 
 Sum 12 144 144)1152(8^. 
 
 + As 65 : 50 3500 
 
 or, as 13 : 10 : : 350 : — =269A* Afi*. 
 
84 VULGAR FRACTIONS. [jTHE TUTOR'S 
 
 (3) If 30 lb. at London make 28 lb. at Amsterdam, how 
 many lb. at London will be equal to 350 lb. at Amsterdam ? 
 
 Ans. 375. 
 
 (4) If 95 lb. Flemish make 100 lb. English, how many lb. 
 English are equal to 275 lb. Flemish ? Ans. 289 I 9 g . 
 
 PERMUTATION 
 
 Is the changing or varying of the order of things. 
 
 Tojind the number of changes that may be made in the 
 position of any given number of things. 
 
 Rule. Multiply the numbers 1, 2, 3, 4, &c. continually 
 together, to the given number of terms, and the last product 
 will be the answer. 
 
 (1) How many changes may be rung upon 12 bells ; and 
 in what time would they be rung, at the rate of 10 changes 
 in a minute, and reckoning the year to contain 365 days, 
 6 hours ? 
 
 1 X2 X3X4X5X6X7X8X9X10XHX12 = 479001600 
 changes, which -r- 10= 47900160 minutes = 91 years, 
 26 days, 6 hours. 
 
 (2) A young scholar, coming to town for the convenience 
 
 of a good library, made a bargain with the person with 
 
 *rhom he lodged, to give him £40. for his board and lodging, 
 
 luring so long a time as he could place the family (consist- 
 
 ig of 6 persons besides himself) in different positions, every 
 
 day at dinner. How long might he stay for his £40. ? 
 
 Ans. 5040 days. 
 
 VULGAR FRACTIONS. 
 
 DEFINITIONS. 
 
 1. A Fraction is a part or parts of a unit, or of any whole 
 number or quantity ; and is expressed by two numbers, 
 called the terms, with a line between them. 
 
 2. The upper term is called the Numerator, and the lower 
 term, the Denominator. The Denominator shows into how 
 
ASSISTANT.] VULGAR FRACTIONS. 85 
 
 many equal parts unity is divided ; and the Numerator is 
 the number of those equal parts signified by the Fraction.* 
 
 3. Every Fraction may be understood to represent Di- 
 vision ; the Numerator being the dividend, and the Denomi- 
 nator, the divisor A 
 
 Fractions are distinguished as follows : 
 
 4. A Simple Fraction consists of one numerator and one 
 denominator : as f, \\, &c. 
 
 5. A Compound Fraction, or fraction of a fraction, con- 
 sists of two or more fractions connected by the word of: as 
 \ of | of T 7 2 , &c. This properly denotes the product of the 
 several fractions. 
 
 6. A Proper Fraction, is one which has the numerator 
 less than the denominator : as ^, f , f, { J, &c.J 
 
 7. An Improper Fraction is one which has the nume- 
 rator either equal to, or greater than the denominator: as |, 
 
 5 8 15 Sirn f 
 
 8. A Mixed Number is cornposed of a whole number and 
 i fraction, as If, 17^, 8Ji, &c. 
 
 9. A Complex Fraction has a fractional numerator or 
 denominator : but this denotes Division of Fractions, Hi us, 
 
 2 . . 8 
 
 ~ y two-thirds divided by five-sixths, — eight divided by one 
 
 d two thirds. 
 
 * In the fraction Jive-twelfths ( T 5 2), the Denominator 12 shows that the 
 unit or whole quantity is supposed to be divided into 12 equal parts : so 
 that if it be one shilling, each part will be one-twelfth of 1*. or one 
 penny. The Numerator shows that 5 is the number of those twelith 
 parts intended to be taken : so j% of a shilling are the same as 5 
 j pence ; ^ of a foot, the same as 5 inches. 
 
 f The fraction T 5 5 signifies not only T 6 5 of a unit, but 5 units di- 
 
 \ vided into 12 parts, or a twelfth part of 5 : and it is obvious that^w 
 
 twelfth parts of one shilling (or five pence) is the same as one twelfth 
 
 part of Jive shillings. This mode of considering Fractions removes 
 
 many of the student's difficulties. 
 
 X A proper fraction is always less titan unity : thus j wants one fourth , 
 and \\*wants one-twelfth of being equal to 1. But an improper fraction 
 is equal to unity when the terms are equal, and greater than unity when 
 the numerator is the greater. 
 
 Thus I, or \\ % or }|, is each =1 ; and |=1J, f =2, f |=3 5 V 
 
86 VULGAR FACTIONS. [/THE TUTOR* 
 
 10. A Common Measure (or Divisor) is a number that 
 will exactly divide both the terms. When it is the greatest 
 number by which they are both divisible, it is called the 
 Greatest Common Measure. 
 
 Note. A prime number has no factor, except itself and unity. 
 A multiple signifies any product of a number; and is therefore 
 divisible by the number of which it is a multiple : thus 14, 21. 28, 
 &c. are multiples of seven* Also 14 is a common multiple of 2 and 
 7 ; 21, of 3 and 7, &c. 
 
 REDUCTION 
 
 Is the method of changing the form of fractional numbers 
 or quantities, without altering the value. 
 
 Case 1. To reduce a fraction to its lowest terms. 
 
 Rule. Divide both the terms by any common measure 
 that can be discovered by inspection; which will produce 
 an equivalent fraction in lower terms. Treat the new frac- 
 tion in a similar manner ; repeating the operation till the 
 lowest terms are obtained.* 
 
 When the object cannot be accomplished by this process, 
 divide the greater term by the less, and that divisor by tho 
 remainder, and so on till nothing remains. The last divisor 
 will be the greatest common measure ; by which divide both 
 terms of the fraction, and the quotients will be the lowest 
 terms, 
 
 (1) Reduce f£g to its lowest terms. Ans. 5 6 j. 
 
 (2) Reduce §§§ to its lowest terms. Ans. T 5 T 2 T . 
 
 (3) Reduce £f § to the least terms. Ans. ^. 
 
 (4) Reduce f §§ to the least terms. Ans. f |. 
 
 (5) Abbreviate f$f f as much as possible. Ans. f. 
 
 (6) Reduce Jfffj to its lowest terms. Ans. 4. 
 
 * This first method of abbreviating fractions is, when practicable 
 always to be preferred : and in the application of it, the following 
 observations will be found exceedingly useful. 
 
 An even number is divisible by 2. 
 
 A number is divisible by 4, when the tens and units are so ; and by 
 8, when the hundreds, tens, and units are divisible by 8. 
 
 A number is a multiple of 3, or of 9, when the sum of its digits is a 
 multiple of 3, or of 9. 
 
 A 5 or a in the units' place, admits of division hy 5 ; one cipher 
 admits of division by 10, two by 100 &c. 
 
ASSISTANT.j REDUCTION. 87 
 
 (7) Reduce ffgtffb to the lowest terms. Ans. |, 
 
 (8) What are the lowest terms of |Jf | ? Ans. \. 
 
 Case 2. To reduce an improper fraction to its equivalent 
 number. 
 
 Rule. Divide the upper term by the lower. 
 
 This is evident from Definition 3. 
 
 (1) Reduce x f 9 to a mixed number. 2 f 9 =18f. Ans. 
 
 (2) Reduce ^ 9 to its equivalent number. Ans. 13$. 
 CS) Reduce 2 -J 5 to its equivalent number. Ans. 27jj, 
 
 (4) Reduce 1 % J 5 to its equivalent number. A?is. 56±?,. 
 
 (5) Reduce 3 § p to its equivalent number. Ans. 183$. 
 
 (6) Reduce ^f 1 to its equivalent number. Ans. 714 1. 
 
 Case 3. To reduce a mixed number to an improper fraction.* 
 
 Rule. Multiply the whole number by the denominator oi 
 the fraction, and to the product add the numerator for the 
 numerator required, which place over the denominator. 
 
 Note. Any whole number may be expressed in a fractional form, 
 by putting 1 for the denominator: thus 11 = y. 
 
 (1) Reduce 18f to the form of a fraction. t 
 
 (2) Reduce 56 J § to an improper fraction. Ans 
 
 124; 
 
 A number is a multiple of 11, when the sum of the 1st. 3rd. 5th. &c. 
 digits = that of the 2nd. 4th. 6th. &c. digits, after retrenching tne 
 elevens contained in each. 
 
 A multiple of both 2 and 3, is, of course, a multiple of 6 ; and a 
 multiple of 3 and 4, may be divided by 12. 
 
 AM prime numbers, except 2 and 5, have 1, 3, 7, or 9, ill the units 
 place : all others are composite. 
 
 Examples. 
 
 (1) Reduce |||g to the least 
 terms possible. 
 -H10 -j-9 -~2 
 
 i?»0 — 126 —- 14 -— .7 Ans. 
 1G20 162 18 »* ^ 1 " ,0# 
 
 (2) Reduce \\%\% to the lowest 
 terms. 
 _^_ 5 _i-3 -i_ 11 
 
 I j 5 4 O == 2 5 8 —- fl 3_6 7 6 
 
 81 54 J 4389 1463 13 3* 
 
 greatest com. meas. 19)57(3 
 57 
 
 Now, because we cannot easilj 
 discover a common measure, pro- 
 ceed thus : 
 
 76)133(1 then 19)76 . 
 
 qa — • = $. Alt*. 
 
 11 19)133 
 
 57 )76(1 
 57 
 
 This is the cdnverse of Case 2. 
 t 18f«E2SL±JL„ lit, Ans. 
 
88 VULGAR FRACTIONS. [THE TUTOR'S 
 
 (3) Reduce 183/ T to an improper fraction. Ans. 3 §f 8 . 
 
 (4) Reduce 13§ to its equivalent fraction. Ans. % 9 . 
 
 (5) Reduce 27 § to its equivalent fraction. Ans. 2 | 5 . 
 
 (6) Reduce 514^ to a fractional form. i4fit. 8 f § 9 . 
 
 Case 4. To reduce a fraction to another of the same value, 
 having a certain proposed numerator or denominator. 
 
 Rule. As the present numerator, is to the denominator ; 
 so is the proposed numerator, to its denominator. Or, as 
 the present denominator, is to the numerator ; so is the pro- 
 posed denominator, to its numerator. 
 
 (1) Reduce f to a fraction of the same value, whose nu- 
 merator shall be 12. As 2 : 3 : : 12 : 18. Ans. ff. 
 
 (2) Reduce f to a fraction of the same value, whose nu- 
 merator shall be 25. Ans. §£-. 
 
 (3) Reduce f to a fraction of the same value, whose nu- 
 merator shall be 47. a 47 
 
 65*-. 
 
 (4) Reduce § to a fraction of the same value, whose de- 
 nominator shall be 18. Ans. if. 
 
 (5) Reduce f to a fraction of the same value, whose de- 
 nominator shall be 35. Ans. §f . 
 
 (6) Reduce § to a fraction of the same value, whose de- 
 nominator shall be 19- a 16§ 
 
 Case 5. To reduce complex and compound fractions, to a 
 simple form. 
 
 Rule. For a complex fraction, reduce both terms to simple 
 fractions : then by inverting the lower fraction, they may 
 be considered as the terms of a compound fraction. And to 
 reduce a compound fraction, arrange all the numerators above 
 a line, and the denominators below, with the signs of mul- 
 tiplication interserted : divide all the upper and lowei 
 terms that are commensurable,* cancelling them with a dash, 
 and placing their quotients above and below them respec- 
 tively. Do the same with the quotients : then the products 
 of the uncancelled numbers will give the single fraction in its 
 lowest terms.f 
 
 * That is, having a common divisor. 
 •J" This rule is of the highest importance as tending to expedite the bu- 
 
ASSISTANT.] REDUCTION. 
 
 
 352 
 
 (1) Reduce — — to a simple fraction. 
 ' 48 
 
 Ans. £ §. 
 
 235 
 
 (2) Reduce -to a simple fraction. 
 
 38 
 
 ^»j. ^5. 
 
 47 
 (3) Reduce -^— to a simple fraction. 
 
 -4/w. f . 
 
 89 
 
 (4) Reduce to a simple fraction. Ans, f . 
 
 44$ 
 
 (5) Reduce § of f of f to a single fraction. Ans. ±. 
 
 (6) Reduce § of f of £J to a simple fraction. Ans. ^%. 
 
 siness of computation, by abbreviating to the utmost all fractional ex- 
 pressions, as we proceed. 
 
 Examples. 
 
 71 
 (1) Reduce the complex fraction — - to a simple form* 
 
 8 i 
 
 rrl 22 
 
 17 
 
 (2) Reduce T 9 o of § of f of T \ to a simple fraction. 
 
 3 111 
 
 JU of I of 2 of T \ = 73 2 t — 77 = A. ^w«?. 
 
 10 3 * T * J0X0X$Xj£ 1S 
 2 14 2 
 
 (3) Reduce the annexed fractional expression to its proper quan- 
 tity. 
 
 16 lll,r 9 - 7 2 6_8, £» Ifi 1452 _7* 98 Ag? 
 
 ilof ill* of ^of-2SL of B|{ = ^X-i^-X-^-X -1L-X — 
 11 108 a § 24ft 64 U ^ 8 if 3_ 2 _o 6 4 
 
 1 7 
 
 11 11? U 
 
 1 %n % $ & x 00 £. g 
 
 __Ux UWx lxfllx %x HxUxWti _ 77 *; 
 
 ""WX *0$xWx«xMjrxW0XWx W"~82"" 5i 
 1 $ I t ft fty f 8 
 
 1 10 4 1 
 
 'cs £2..8J*. = £2..8..1^. ^»J. I 
 
90 VtJIGA FRACTIONS |_TH£ TUTOR'* 
 
 (7) Reduce U of-illof 28 to a simple fraction. 
 
 sins, gyg. 
 
 (8) Reduce f of y \ of 1 1 1 to a single fraction. Ans. T %. 
 
 (9) Reduce T y F of 37j of 5 to its equivalent number. 
 
 ll i dns. 112 2 \. 
 
 (10) Reduce — of 2 f 6 of -|- to its equivalent number. 
 
 14 7 J/w. 7|. 
 
 Case 6. To reduce a fractional quantity of a given denomi- 
 nation, to an equivalent fraction of another denomination 
 Rule. Consider what numbers m ould reduce the greater 
 denomination to the less ; then to reduce to a greater name, 
 multiply the denominator by those numbers, and to reduce 
 to a less name, multiply the numerator : the compound thus 
 produced, when reduced to a simple form, will be the frac- 
 tion required. 
 
 (1) Reduce | of a penny to the fraction of a pound.* 
 
 (2) Reduce jd. to the fraction of a crown. Ans. ^ cr. 
 
 (3) Reduce § drvt. to the fraction of a lb. troy. Ans. ^J^ lb. 
 
 (4) Reduce f lb, avoirdupois to the fraction of a cwt. 
 
 Ans. T $3 cwt. 
 
 (5) Reduce tj&jj of a pound to the fraction of a penny, f v 
 
 (6) Reduce £y£g. to the fraction of a penny. Ans. fd. 
 
 (7) Reduce ^J of a pound troy to the fraction of a pen- 
 ny-weight. Ans. | drvt. 
 
 (8) Reduce T |^ cratf. to the fraction of a /£. Jrc.?. f lb. 
 Case 7. To find the proper value of a fractional quantity. 
 Rule. Reduce the numerator to such lower denomination 
 
 as may be necessary, and divide by the denominator ; abbre- 
 viating as much as possible in valuing the remainders. 
 
 Note. It is evident, from Definition 3, that this Case is precisely 
 that of Compound Division. 
 
 (1) Reduce f of a pound sterling to its proper value.J 
 
 5 8X12X20 T92 ° 
 
 . - , 7X20X12 7X12 ta A 
 
 T T¥57S 1Q20 96 
 
 $ £|== 3 X 20 ==3X5==1 ^ ^ 
 
ASSISTANT.] REDUCTION. 91 
 
 (2) Reduce §s. to its proper value. Ans. 4d. 3% qrs. 
 
 (3) Reduce f of a lb. avoirdupois to its proper value. 
 
 Ans. 9oz.2%dr. 
 
 (4) Reduce J cwt. to its proper value. Ans. 3 qrs. 3\lb. 
 
 (5) Reduce § of a lb. troy to its proper value. 
 
 Ans. 7 oz. 4 c?wte. 
 
 (6) Reduce |f of an ell English to its proper value. 
 
 Ans. 2 qrs. 3± nails. 
 
 (7) What is the value of £| JJJ ? Ans. 19s. 10^ J. 
 
 (8) Reduce fff of a mile to its proper value. 
 
 Ans. 6 fur. 105 yds. 
 
 (9) Reduce |f of an acre to its proper value. 
 
 Ans. 1 a.2r. 3 J per. 
 
 (10) Find the value of $£f f J cratf. ^ras. 1 c/r. 22 /6. §§f 
 
 Case 8. To reduce any given quantity to the fraction of a 
 greater denomination. 
 
 Rule. Reduce the given quantity (if compound) to the 
 lowest denomination mentioned, that it may assume a sim- 
 pie form : then multiply the denominator as in Case 6. 
 
 (1) Reduce 15s. to the fraction of a pound sterling. 
 
 15s.=£^=£^. Ans. 
 
 (2) Reduce 4c?. 3} qrs. to the fraction of a shilling. Ans. f. 
 
 (3) Reduce 9 oz. 2f dr. to the fraction of a lb. avoirdu- 
 pois. Ans. f lb. 
 
 (4) Reduce 3 qrs. 3 J lb. to the fraction of a cwt. 
 
 Ans. $ cwt. 
 
 (5) Reduce 7 oz. 4 dwts. to the fraction of a /6. troy. 
 
 ^/w. f lb. 
 
 (6) Reduce 2 grs. 3\ nails, to the fraction of an English 
 ell. Ans. | e//. 
 
 (7) Reduce 14s. 6^d. T 2 X to the fraction of a £. 
 
 Ans. £^ T . 
 
 (8) Reduce 4c?. 1\ J cjts. to the fraction of a crown. 
 
 Ans. fifo cr. 
 
 (9) What fraction of an acre are 3 roods, 32 perches ? 
 
 Ans. 1% a. 
 
 (10) What part of a shilling are § of 2c?. Ans. \s. 
 
 Case 9. To find the least common multiple of two or mon 
 numbers. 
 
 Rule. Arrange the given numbers in a line, (omitting any 
 one that is a factor of one of the others) and divide any two 02 
 more of them by a common divisor, placing the quotients anc : 
 
92 VULGAR FRACTIONS. [_THE TUTOR'S 
 
 undivided numbers below ; proceed with these in the same 
 manner, and repeat the process till there remain not any two 
 numbers commensurable : the continued product of the di- 
 visors, quotients, and undivided numbers, will be the least 
 common multiple. 
 
 (1) Required the least common multiple of 2, 3, 4, 5, 6, 
 
 7, 8, 9, and 10.* 
 
 (2) Find the least number divisible by 3, 4, 5, 6, 7, and 
 
 8. Ans. 840. 
 
 (3) What is the least common multiple of 2, 3, 4, 5, 6, 7> 
 8, 9, 10, 11, and 12 ? Ans. 27720. 
 
 Case 10. To reduce fractions to a common denominator. 
 
 Rule 1. Multiply each numerator into all the denomina- 
 tors, except its own, for a numerator ; and all the denomina- 
 tors for a common denominator. Or, 
 
 Rule 2. Find the least common multiple of the denomina- 
 tors, which will be the least common denominator. Divide 
 this by each denominator, and multiply the several quotients 
 by the respective numerators for the required numerators. 
 
 (1) Reduce § and f to a common denominator, t 
 
 (2) Reduce \, f , and §, to a common denominator. 
 
 Ans - l%> U> and ii > or i> h and !• 
 
 (3) Reduce |, f , ^, and f , to a common denominator. 
 
 AnS 3^£ £§£ £fii /7W/7I20 
 -<* /M » ¥40> 840' 840' una ¥40* 
 
 (4) Reduce f , ^, and ^, to a common denominator. 
 
 s i Ans. H,U,and\%. 
 
 (5) Reduce T 5 T , $, and — of 2, to a common denominator. 
 
 *■* v4w* 2L1_5_ JBJ5JL /7M/7 53 9 
 
 (6) Reduce l£, 2 J, and ^ of 1^, to a common denomi- 
 nator. Ans. jjfjj, ! B Y, 0wd|g. 
 
 * 2 and 4, being factors of 8, 3 a factor of 9, and 5 a factor of 
 10, may be omitted. Thus, 
 
 2 )6, 7, 8, 9, 10 Then 2X3X7X4X3X5 = 42 X 60 
 
 3)3, 7, 4, 9, 5 =2520, the least number divisible 
 
 I rj 4 3 5 by all the given numbers. 
 
 a. 2 v 7 = 14 ) 
 
 4 x 4 _ 16 f numerators. Ans. i| and || 
 
 4 X 7 = 28 the denominator. 
 
ASSISTANT.]] SUBTRACTION. $3 
 
 ADDITION. 
 
 Rule. Reduce the given fractions to a common denomina- 
 tor, over which place the sum of the numerators. 
 
 (1) Add | and f together. |+f — H+Jf =S f =1 2 8 t- ^*- 
 
 (2) Add |, f , and 
 
 (3) Add J, 4$, and" f .* 
 
 (4) Add 7| and f together. 
 
 (5) Add f , and § of f . 
 
 (6) Add 5§, 6§, and 4£. 
 
 (7) Add If, 3*, and £ of 7. 
 
 (8) Add -^ of 6£, and | of 7£. 
 
 (9) Add 1 of 9f, and | of4|. 
 
 Fractional quantities may be reduced to their proper 
 values, and the sum found by Compound Addition. 
 
 (10) Add § of a pound to f of a shilling. Ans. 8s. 4td. 
 
 (11) Add id. I*, and <£§. Ans. 14j. 
 
 (12) Add J lb. troy, \ oz. and § oz. Jtw. 7 oz. 19 dwte. 
 20 gr. 
 
 (IS) Add f of a ton to § of a ctttf. 
 
 ^fw^. 12 cwt. 1 qr.l^ lb. 
 
 (14) What is the sum off of £l7..7-.6tf., f of £i§. and 
 § of a crown ? ^rc,y. £l3..0..2^. 
 
 (15) Add j of 3 a. I r. 20 |?., f of an acre, and f of 3 
 roods, 1 5 perches. Ans. 3 a. 2 r. 33\ p. 
 
 SUBTRACTION. 
 
 Rule. Reduce the given fractions to a common denomi- 
 nator, over which place the difference of the numerators. 
 
 When the numerator of the fractional part in the subtra- 
 hend is greater than the other numerator, borrow a fraction 
 equal to unity, having the common denominator ; then sub- 
 tract, and carry one to the integer of the subtrahend. 
 
 (1) From f take f. f— f=f $—«==&. Ans. 
 
 (2) From § take f . 
 
 (3) From 5§ take & of |. 
 
 (4) From f f take f of ^. 
 
 (5) From }§ take 4 of §. 
 
 (6) From 64^ take § of f . 
 
 (7) From 15| take 12 5 7 D . 
 
 (8) Subtract ff from If. 
 
 (9) Subtract J^f from \ of 9. 
 
 Fractional quantities may be reduced to their proper 
 values, as directed in Addition. 
 
 * When there are integers among the given numbers, first find 
 the sum of the fractions ■, to which add the integers. 
 
 Thus in Ex. 3, 1+1=11 then t+i-A-fftHft ; and 4-Hf- 
 
 4 if. Ans. 
 
94 VULGAR FRACTIONS. [_THE TUTOJl's 
 
 (10) From f of a pound take f of a shilling. Ans. 7s. l±d. 
 
 (11) From If.?, take § of tyd. Ans. Is. 3d, 
 
 (12) What is the difference between f of £l 5 \. ; and 
 j% of £l§§. ? ^w*. 2 d. $\ qrs. 
 
 (IS) Subtract § c/ztf. from 4 to?*. Ans. 10 c/ztf. 2 ^. 10| lb. 
 
 (14) From f of 5 lb. troy subtract § of 3^ oz. 
 
 Ans. 3 lb. 2 oz. 1 dwt. 2| gr. 
 
 (15) Subtract 7Mo furlongs from 1^ mile. Ans. 4>fur. 
 
 ) yds. 
 
 MULTIPLICATION. 
 
 Rule. Prepare the given numbers (if they require it) by 
 the rules of Reduction : then multiply all the numerators 
 together for the numerator of the product, and all the de- 
 nominators for the denominator. 
 
 (1) Multiply | by |. |Xf=A. ^ns, 
 
 (2) Multiply J by §. 
 
 (3) Multiply 48 J by 13$. 
 
 (4) Multiply 430 J by ISf. 
 
 (5) Multiply if byf off 
 
 (6) Multiply I off by f 
 
 (7) Multiply 5f by L 
 
 (8) Multiply 24 by | , 
 
 (9) Multiply! of 9 by 2. 
 
 (10) Multiply £3..15..9i £ by T 9 T of 5. Ans. £l5..9..Hf T \. 
 
 (11) Multiply 3^| miles by f of iff. Ans.Sm.2f 
 I88f #d.?. 
 
 (12) Required the product, in square feet, of 14* ft. 7 n*. 
 by 8 ft. 9 in. Ans. 127 Jf *?./' 
 
 DIVISION. 
 
 Rule. Prepare the given numbers (if they require it) by 
 lie rules of Reduction ; then invert the divisor, and proceed 
 is in Multiplication. 5 " 
 
 j) Divide fo by ft 
 
 (2) Divide U by f 
 
 (3) Divide 672 j^ by 13f 
 
 (4) Divide 7935f§ by 18f 
 
 (5) Divide 16 by 24. 
 
 (6) Divide T \ by 4^. 
 
 (7) Divide |f J by & of ^ 
 
 (8) Divide 9J by ^ of 7. 15 
 
 (9) Dividef by 1 off off 
 
 (10) Divide | of l6byf off 
 
 * A number inverted becomes the reciprocal of that number ; which 
 is the quotient arising from dividing unity by the given number : thus 
 1-7-7=4, the reciprocal of 7 ; l-f-|=|, the reciprocal of f. 
 
 3 1 
 
 + 9 _i_ 3 __ v — $ Arte 
 
 4 i 
 
ASSISTANT] RULE OF THREE- Q$ 
 
 %l\ S^ e £ll ^ Z b ^ V <*& Arts. £3..l7..10l |. 
 
 ( 2 ) ^de 1,. 4ldf by I of V . Ans. 6d. |f j \rs. 
 
 (13) Divide 3 ™. 24-if & by # of l£, in the fraction of a 
 cw*. ,• and value the quotient. Ans. 1 cw/. 1 or. 15* lb 
 fttVn^?* must ^..U..6. b * multiplied by, to produce 
 
 THE RULE OF THREF 
 
 Rule. Prepare the terms, previous to stating, so tnat nu 
 subsequent Reduction will be necessary : then, having stated 
 the question, as previously directed, invert the dividing term, 
 and the continued product of the three will be the answer. 
 
 (1) If f of a yard cost <£§. what will T 9 of a yard cost ?* 
 
 (2) If I yd. cost £§. what will \\ yd. cost? Ans. 14*. 8d. 
 
 (3) If f of a yard of lawn cost 7*. 3d. what will 10J yards 
 cost? Ans. £4..19..10i §. 
 
 (4) If | lb. cost f s. how much will f s. buy ? Ans. 1 j 1 * /';. 
 
 (5) If 48 men can build a wall in 24^ days, how many 
 men can do the same in IQ2 days ? Ans. 6-^ men. 
 
 (6) If | of a yard of Holland cost ££. what will 12§ ells 
 Cost at the same rate? Ans. ,£7..0..8§ f. 
 
 (7) If 3^ yards of cloth, that is 1 J yard wide, be sufficient 
 \o make a cloak, how much that is | of a yard wide, will 
 make another of the same size ? Ans. 4| yards. 
 
 (8) If 12 <| yards of cloth cost 15*. Qd. what will 48^ yards 
 cost at the same rate? Ans. £3..0..9^ 2 V 
 
 (9) If 25f*. will pay for the carriage of 1 cwt., 145^ miles, 
 now far may 6^ cwt. be carried for the same money ? 
 
 Ans. 22 2 ^ miles. 
 
 (10) If i 9 o °f a cw t' cost <£l4. 4*. what is the value of 7^ 
 cwt.? Ans. £118..6..8. 
 
 (11) If I lb. of cochineal cost £l..5. what will 06-^ lb 
 come to? A?is. £6l..3.A. 
 
 (12) How much in length that is 7 A inches broad, wil 
 make a foot square ? ^(w*. 20 T f j inches. 
 
 (13) What is the value of 4 pieces of broad cloth, each 27 f 
 yards, at 15|*. per yard ? Ans. £85..14..3^ f. 
 
 yd. £. yd, J 4 £. 
 * A *f : I : : A : jX^x|=f= 15*. ^*. 
 
 2 2 J 
 
Q6 DECIMAL FRACTIONS. 
 
 (14) If a penny white loaf weigh 7 oz. when a bushel of 
 wheat costs 5s. 6d. what is the bushel worth when a penny 
 white loaf weighs but 2^ oz. ? Arts. 15s. 4>d. 3} qrs. 
 
 (15) What quantity of shalloon that is f of a yard wide will 
 line 7i yards of cloth, that is 1 ^ yard wide ? Ans. 1 5 yards. 
 
 (16) Bought 3\ pieces of silk, each containing 24§ ells, at 
 6s. Of d. per ell. How must I sell it per yard, to gain £5. 
 by the bargain? Ans. 5s. $\d. fff. 
 
 THE DOUBLE RULE OF THREE. 
 
 (1) If a carrier receive £%fa for the carriage of 3 cwt. 
 150 miles, how much ought he to receive for the carriage of 
 7 cwt. 3\ qrs., 50 miles ? Ans. £l..l6..9» 
 
 (2) If £100. in 12 months gain £5\. interest, what princi- 
 pal will gain £3§. in 9 months? Ans. £85..14..3£ f. 
 
 (3) If 9 students spend £l0^. in 18 days, how much will 
 20 students spend in 30 days ? Ans. £39..18..4§f . 
 
 (4) Two persons earned 4|s. for one day's labour : how 
 much would 5 persons earn in 10^ days, at the same rate ? 
 
 Ans. £6..1..4f i 
 
 (5) If £50. in 5 months gain £2^. what time will £l3^ 
 require to gain £l T ^. ? Ans. 9 months. 
 
 (6) If the carriage of 60 cwt., 20 miles, cost £14^. what 
 weight can I have carried 30 miles for £5 T 7 5 . ? Ans. 15 cwt. 
 
 DECIMAL FRACTIONS. 
 
 In Decimal Fractions the unit is supposed to be divided 
 into tenths, hundredths, thousandth parts, &c. consequently 
 the denominator is always 10, or 100, or 1000, &c. 
 
 In our system of Notation, the figures of a whole number 
 follow each other in a decimal (or tenfold) proportion. Hence, 
 the numerator of a decimal Fraction is written as a whole 
 number, only distinguished by a separating point prefixed to 
 it. Thus -5 for ^%, -25 for T %%, -123 for T \%%. 
 
 The denominator is, therefore, not expressed ; being always 
 understood to be 1, with as many ciphers affixed, as there are 
 places in the numerator. 
 
 The different values of figures will be evident in the 
 annexed Table. 
 
ASSISTANT.] DECIMALS. 97 
 
 Integers. Decimal parts, 
 
 7654321 • 23456 7, &c. 
 
 PagoagS.' g. £ o b £ ~ 
 
 
 oo* ST>B 
 
 B" 
 
 2 P B - B 
 3 £ B C r* £ 
 P» &G.B B*E 
 
 JTT3 B* * B>B 
 
 
 § 1 
 
 From this it plainly appears that the figures of the decimal 
 fraction decrease successively from left to right in a tenfold 
 proportion, precisely as those of the whole number* 
 
 Ciphers on the right of other decimals do not alter their 
 value: for "te$j, -20=1^, -200=^% are all equal. But 
 one cipher on the left diminishes the value ten times, two 
 ciphers, one hundred times, &c. for -02:=^ § 5 , '002=^-^^, &c. 
 
 A vulgar fraction having a denominator compounded of 
 2, or 5, or of both, when converted into its equivalent decimal 
 faction, will be finite : that is, will terminate at some certain 
 number of places. All others are infinite; and because they 
 have one or more figures continually repeated without end, 
 they are called Circulating Decimals. The repeating figures 
 are called repetends. 
 
 One repeating figure is called a single repetend ; as -222, 
 &c. ; generally written thus, -2'; or thus, •$. But when 
 more than one repeat, the decimal is a compound repetend ; 
 as -36 36, &c, or -142857 142857, &c. These may be written 
 - s 36', and -'142857'; or -j80, and -J4285jf. 
 
 Pure repetends consist of the repeating figures alone ; but 
 mixed repetends have other figure* before the circulating de- 
 cimal begins : as -045', '96 s 35 4'. 
 
 Finite decimals may be considered as infinite, by making 
 ciphers to recur, which do not alter the value. 
 
 Circulating decimals having the same number of repeating 
 figures are called similar repetends, and those which have an 
 unequal number are dissimilar. Similar and conterminous 
 repetends begin and terminate at the same places. 
 
 * The first, second, third, fourth, &c. places of decimals are called 
 primes, seconds, thirds, fourths, &c. respectively ; and decimals are read 
 thus: 57-57 fifty -seven, axi&five, seven, of a decimal; that is, fifty- 
 seven, and fifty-seven hundredths. 206*043 two hundred and six, 
 and nought, four, three ; that is, 206, and forty-three thousandths. 
 
 E 
 
98 DECIMALS. £T11E TUTOR'S 
 
 ADDITION. 
 
 Rule. Place the numbers so that the decimal points may 
 stand in a perpendicular line: then will units be under units, &c. 
 according to their respective values. Then add as in integers. 
 
 (1) Add 72*5+32*071+2*1574+371*4+2*75. 
 
 (2) Add 30-07+2-0071 +59'432+7-l. 
 
 (3) Add 3-5+47-25+927-01+2-0073+1-5. 
 
 (4) Add 52*75+47*21+724+31*452+*3075. 
 
 (5) Add 3275+27-514+1-005+725+7-32. 
 
 (6) Add 27-5+52+3-2675+-5741+2720. 
 
 SUBTRACTION. 
 
 Rule. Place the subtrahend under the minuend with the 
 decimal points as in Addition ; and subtract as in integers. 
 
 (1) From -2754 take -2371 
 
 (2) From 2*37 take 1*76. 
 
 (3) From 271 take 215*7. 
 ' v 4) From 270*2 take 75*4075. 
 
 (5) From 571 take 54*72. 
 
 (6) From 625 take 76*91. 
 
 (7) From 23*415 take *3742. 
 
 (8) From -107 take -0007. 
 
 MULTIPLICATION. 
 
 Rule. Place the factors, and multiply them, as in whole 
 numbers ; and in the product point off as many decimal 
 places as there are in both factors together. When there are 
 not so many figures in the product, supply the defect with 
 ciphers on the left. 
 
 (1) Multiply 2*071 by 2*27. 
 
 (2) Multiply 27*15 by 24*3.* 
 
 (3) Multiply -2365 by *2435. 
 
 (4) Multiply 72347 by 23*15. 
 
 (5) Multiply 17105 by *3257. 
 
 (6) Multiply 17105 by -0327. 
 When the multiplier is 10, 100, 1000, &c it is only removing 
 
 the separating point in the multiplicand so many places towards 
 the right as there are ciphers in the multiplier: thus, *578 X 10 
 = 5*78, *578 X 100 = 57*8, -578 X 1000 = 578, and -578 X 10000 
 =5780. 
 
 CONTRACTED MULTIPLICATION. 
 
 Rule. Write the multiplier under the multiplicand in an inverted 
 order, the units' figure under that place which is intended to be re- 
 tained in the product. 
 
 * The 2nd. example may be multiplied in two products, first by 
 3, and that product by 8 for 24. The 3rd, 6th, 7th, and 12th may 
 be contracted in a similar way. 
 
 (7) 
 
 27*35 X 7*70011. 
 
 (8) 
 
 57*21 X *0075. 
 
 (9) 
 
 •007 X *007. 
 
 (10) 
 
 20*15 X *2705 
 
 (11) 
 
 *907 X *002/ 
 
 (12) -3409803 X -0016218. 
 
ASSISTANT. J DIVISION. 99 
 
 In multiplying, begin with that figure of the multiplicand which 
 stands over the multiplying figure, rejecting all on the right of 
 that ; and set down the first figures of all the products in a perpen- 
 dicular row. 
 
 Increase the first figure of each product by carrying to it what 
 would arise from multiplying the two next rejected figures on the 
 right, at the rate of one from 5 to 14 inclusive, two from 15 to 24, 
 three from 25 to 34 inclusive, &c. 
 
 Note. If perfect accuracy as far as the last decimal figure be de- 
 sired, it will be eligible to find one figure more in the product than is 
 actually wanted, 
 
 (13) Multiply 384-672158 by 3'683, and let there be 
 only four places of decimals in the product.* 
 
 (14) Multiply 3-141592 by 527438, retaining only 4 places 
 of decimals in the product. Arts. 165-6995. 
 
 (15) Multiply 238-645 by 8217*5, retaining only the inte- 
 gers in the product. Ans. 1961064. 
 
 (16) Multiply 375-13758 by 16*7324, and reserve only 
 one place of decimals ; and again, reserving three places. 
 
 Ans. 6ll6'9, and 6276'951. 
 
 (17) Multiply 3953756 by '75642, retaining only 4 places 
 of decimals. Ans. 299*0700. 
 
 DIVISION. 
 
 Divide as in integers ; and the first figure of the quotient 
 will be of the same value as that figure of the dividend which 
 stands over the units in the first product of the divisor : so 
 that the point must be placed accordingly ; ciphers being 
 prefixed, when necessary. 
 
 Note 1. After proceeding through the dividend, to ascertain if 
 the quotient is correctly pointed, observe that the decimal places in 
 the divisor and quotient 'together, must equal in number those of 
 the dividend. 
 
 2. When there are fewer decimal places in the dividend than 
 in the divisor, equalise them by affixing ciphers ; and the quotient, 
 to tJiat extent, will be a whole number. 
 
 * Contracted method. Common method. 
 384-672158 384'672158 
 386-3 3-683 
 
 "Tl540165 11540)16474 
 
 T7264 
 
 2308033 307737 
 
 307738 2308032 
 
 11540 11540164 
 
 1416-7476 1416-7475 57914 
 
 948 
 74 
 
100 DECIMALS [_THE TUTOR'S. 
 
 3. Ciphers may be subjoined to the decimal part of the dividend, 
 ©r brought down as if they were subjoined ; in order to continue the 
 operation to any degree of exactness desired. 
 
 (1) Divide 217*75 by 65. 
 
 (2) Divide 709 by 2*574. 
 
 (3) Divide 125 by -1045. 
 
 (4) Divide 48 by 144. 
 
 (5) Divide 5714 by 8275. 
 
 (6) Divide 715 by -3075. 
 
 (7) 7382-54 ~ 6*4252. 
 
 (8) -0851648 — 428. 
 
 (9) 267*15975 -r- 13-25. 
 
 (10) 72-1564 -T--1347. 
 
 (11) 85643*825 ~ 6-321. 
 
 (12) 1 -r- 3*1416. 
 
 To divide by 10, 100, 1000, &c. remove the separating point in 
 the dividend so many places towards the left, as there are ciphers in 
 the divisor, and the thing is accomplished. 
 
 Thus 5784- -r- 10 = 578-4, 5784 ~ 100 == 57-84, 5784 -r- 1000 = 
 5-784, 5784 -r- 10000 = -5784. 
 
 (13) 3719-H 10. (15) 130-7-^- 1000. 
 
 (14) 3-74—100, (16) 34-012-r-lOOOO. 
 
 CONTRACTED DIVISION. 
 
 Ascertain the value of the first quotient figure : from which it 
 will be known what number of figures in the quotient will serve 
 the purpose required. Use that number of the figures in the divisor, 
 (rejecting the others on the right) and a sufficient number of the divi- 
 dend, to find the first figure of the quotient; make each remainder 
 a new dividual, and for each succeeding figure reject another froi# 
 the divisor : but observe to carry to each product from the reject 
 ed figures as in Contracted Multiplication. 
 
 Note. When there are fewer figures in the divisor than the num- 
 ber wanted in the quotient, proceed by the common rule till those in 
 the divisor are just as many as remain to be found in the quotient, 
 and then use the contraction. 
 
 (17) Divide 70*23 by 7 "9863, to three places of decimals.* 
 
 (18) Divide 721*17562 by 2257432, to the extent of only 
 three places of decimals in the quotient. 
 
 (19) Divide 25*1367 by 217*35, to the fourth decimal. 
 
 .... * Contracted Method, 
 
 Common Method. 
 
 7-9863)70-230(8- 
 
 793 
 
 7-9863)70-2300(8-793 
 
 63 890 
 
 
 638904 
 
 6340 
 
 6339 
 
 60 
 
 5590 
 
 
 5590 
 
 41 
 
 
 
 
 
 750 
 
 
 749 
 
 190 
 
 719 
 
 
 718 
 
 767 
 
 
 
 
 
 SI 
 
 
 30 
 
 4230 
 
 24 
 
 
 23 
 
 9589 
 
 
 
 
 
 7 
 
 
 6 
 
 4641 
 
ASSISTANT.] CIRCULATING DECIMALS. 101 
 
 (20) Divide 51*47542 by '123415, to the second decimal. 
 
 (21) Divide 27104 by '3712, the integral quotient only 
 
 CIRCULATING DECIMALS. 
 
 To reduce a circulate to a vulgar fraction. 
 
 Rule. 1. For a pure repetend, make the circulating figures 
 the numerator, to as many nines for the denominator, 
 
 2. For a mixed repetend, subtract the finite part from the 
 whole, and make the difference the numerator ; the denomina- 
 tor to which will consist of as many nines as there are repe- 
 tends, with as many ciphers subjoined as there are finite 
 figures. 
 
 Examples. 
 
 (1) Reduce •!', -3', •{# -\)1', and -'142857' to their equiva- 
 lent vulgar fractions. 
 
 •l'ssfc -5'=f=i; •9 , = { = 1;W=A; and -'142857 
 
 14 2 8 5 7 15873 — 5291 — 48 1 — 1 
 
 9SS999 — TlTTl — 57037 — 33S7 — T' 
 
 (2)Reduce •03 N 45 / and3 , 5'126' to equivalent vulgar fractions 
 •03*45' = 3 i|=2=//^ =I f §,=#,. 
 
 S^lg g-rr;^ 1 ^ 35 7r >^^^p P $tf&. Or thus; 
 
 3-5'126'=3 + 5126 ~ 5 =3fifS=3i¥iV 
 9990 
 
 In Addition and Subtraction of Circulating Decimals, 
 make them similar and conterminous, and carry to the figures 
 on the right whatever would arise from the repetends bein^ 
 continued. 
 
 Note. In all cases, when the repetend is 9, make it a cipher, and 
 add 1 to the next figure : for -999, &c. = 1. 
 
 In Multiplication, carry to the product of the right 
 hand figure what would arise from the product of the repe- 
 tends continued ; and in finding the sum of the products, ob- 
 serve what is directed in Addition. 
 
 In Division, it is only necessary to observe that the ope- 
 ration may be carried on with the repeating figures of the 
 dividend, to any extent required. 
 
 Note. When the Multiplier or the Divisor is a circulate, the most 
 convenient method is, to change it into a common fraction. 
 
102 decimals. [_the tutor's 
 
 Examples. 
 (3) What is the sum of 25- v L42857 / , 10'3ty)'> 12-035', and 
 
 4*02 N 567'? 
 
 Similar. Similar and conterminous, 
 
 25* V L42857' = 25-142857' = 25*14^85714' 
 
 10-3W = 10-3^09090' = 10-39 > 090909 / 
 
 12-035' = 12*03^55555' = 12-03^555555' 
 
 4-02W = 4-02^567567' = 4*02 N 567567' 
 
 Sum 51-59^99746' 
 
 r (4) What is the difference between 567* W and55•0 > 9729 , ? 
 Also, between 57, and 49*8^53' ? 
 567^367' = 567-3 x 67367367S67S / 57' 
 
 55-0^9729' ess 55-0^972997299729' 4>9'8'53' 
 
 difference 512-2700676373943' diff. 7-lW 
 
 (5)Multiply 65-316'by -753. 
 
 65-31& 
 
 •753 
 
 195950 
 
 3265833' 
 
 45721666' 
 
 product 49-183450 
 
 (6) Multiply 13-^45' by 3*36'. 
 
 13- N 45' 
 
 37 
 
 94 N 18' 
 403 x 63'__ 
 
 ii)497 - y sF 
 
 product 45-256198, &a 
 
 (7)Divide 150*9\)45'by 33. 
 3)150-9^045^ 
 11)50^615' 
 
 4-5728637' quotient. 
 
 (8) Divide 17*8054' by 3'6\ 
 7. 
 53-4163' 
 
 S-ff=SJ = S| = V- 
 
 17-8054'x T \ = 
 
 4-856 N 03' quotient. U 
 
 ? 
 
 (9) What are the equivalents to -004^354' and 65*00063^648 '? 
 
 Arts. % %%v and 65 7 £^. 
 
 (10) What is the sum of 57*575 + 3*59 N l63' + 210-16' + 
 •06^759' ? Ans. 271-397'057674235892'. 
 
 (11) Required the difference between 36*30M<5207' and 
 47'280 N 43'. Ans. 10*975 N 9135982268'. 
 
 (12) Multiply 4- v 428571' by 347; and 17*0^54' by 6*^148'. 
 
 Ans. 1536-714285'; and 104*85^387205'. 
 
 (13) Divide 1536'7l4285' by 347; and 104*85^87205' by 
 6*148'. Ans. 4**428571'; and i7*0\54'. 
 
ASSISTANT.] REDUCTION. 103 
 
 REDUCTION. 
 
 To reduce a Vulgar Fraction to a Decimal. 
 
 Rule. Add ciphers to the numerator, and divide by the de^ 
 nominator ; the quotient will be the decimal fraction required 
 
 (1) Reduce \, ^, j, and §, to decimals. 
 
 Ans. -25, '5, -75, and -375 
 
 (2) Reduce ^, J, \ § and \, to decimals. 
 
 Ans. -3', -2, -'142857', and -1'. 
 
 (3) Reduce fo to a decimal. Ans. -1^23076'. 
 
 (4) Reduce ±a of $§ to a decimal. Ans. -6 N 043956'. 
 
 To reduce a given quantity to the Decimal of any denomination 
 required. 
 
 Rule. Reduce those of the lowest denomination to decimal 
 parts of the next superior, on the left of which place the 
 given quantity of that denomination; reduce this to the 
 next, and proceed as before, till it is of the denomination re- 
 quired. 
 
 (5) Reduce 5s.* 9s. and l6s. to the decimals of a pound. 
 
 Ans. £-25, £-45, and £-8. 
 
 (6) Reduce 8s. 4>d. to the decimal of a £. Ans. £-416'. 
 
 (7) Reduce 16s. 7fd. to the decimal of a <£.t 
 
 (8) Reduce 19*. 5±d. to the decimal of a £. Ans. £-972916'. 
 
 (9) Reduce 12 grains to the decimal of a lb. troy. 
 
 Ans. lb. -002083'. 
 
 (10) Reduce 12 J § drams to the decimal of a lb. avoirdupois. 
 
 Ans. lb. -047668+. 
 
 (11) Reduce 2 qrs. 14^ lb. to the decimal of a cwt. 
 
 Ans. cwt. -62723+. 
 
 (12) Reduce 2 furlongs, l6l^ yards to the decimal of a 
 mile. Ans. -341 761^6' mile. 
 
 (13) Reduce 5{§ pints to the decimal of a gallon. 
 
 A?is. *741 6' gal. 
 
 (14) Reduce 4^ gallons of wine, to the decimal of a hogs- 
 head. Ans. •0 V 714285 / . 
 
 f 4| 3-00 qrs. 
 * 20)5-00*. 12 
 
 £•25 Ans. 
 
 20 
 
 7-75 d. 
 
 16-64583' *. 
 
 £•83229 16' Ans. 
 
jQ4 DECIMALS. [/THE TUTOR* S 
 
 (15) Required the mixed decimal number equivalent to 
 
 £3..9..1f//o^ . Ans - ^3-45471. 
 
 (16) Express 7 weeks, 3 days in the decimal of a year. 
 
 Ans. yr. *1 42465+- 
 To find the proper value of a Decimal Fraction of any Integer. 
 Rule. Multiply the given decimal by the proper number to 
 reduce it to the next inferior denomination, pointing off the 
 given number of decimals in the product ; reduce these to 
 the next, and so on to the lowest ; and the whole numbers on 
 the left (being collected together) will be the value required. 
 
 A decimal of a £. may be thus valued by inspection. Double the 
 tenths for shillings, and call the number in the second and third, far- 
 things, abating one above 12, and two above 37. But if the second is 5, 
 or upwards, call the 5 one shilling, and reckon only the excess above 
 five with the third. 
 
 By reversing these directions, any given sum in shillings, &c. may 
 be expressed in the decimal of a £ — Thus, half the shillings are 
 tenths, and an odd shilling, 5 hundredths ; the rest (in farthings) add 
 into the second and third places, increasing one above 11 farthings, 
 and two above 36. 
 
 (17) What is the value of -8322916 of a £.* 
 
 (18) Reduce £'740596 to its proper value. 
 
 Ans. 14*. 9^.2*97216 qrs. 
 
 (19) What is the value of -082084 of a lb. troy ? 
 
 Ans. 19 drvts. 16*80384 grains. 
 
 (20) What is the value of -4909375 lb. avoirdupois ? 
 
 Ans. 7 oz. 13*68 drams. 
 
 (21) What is the value of £-19895 ? Ans. 3s. llcL2*992 qrs. 
 * (22) What is the value of *625 of a cwt 9 Ans. 2 qrs. 14 lb. 
 
 (23) What is the value of -071428 of a hogshead of wine ? 
 
 Ans. 4fgal. 1'999S56 qts. 
 
 (24) What is the value of -0625 of a barrel of beer ? 
 
 Ans. 2 gallons, 1 quart. 
 
 (25) What is the value of *142465 of a year? 
 
 Ans. 51-999725 days. 
 
 *£. -8322916 
 
 
 
 
 
 
 20 
 
 
 
 
 
 By inspection. 
 
 *. 16-6458320 
 12 
 
 
 
 
 
 £. s. d. 
 •8 = 16..0 
 •032 — 0..7| 
 
 d. 7-749984 
 4 
 
 •832 = 16..7| 
 
 qrs. 2-999936 
 
 Ans. 
 
 s. 
 16. 
 
 d. 
 
 very 
 
 nearly. 
 
SISTANT.] 
 
 DECIMALS. 
 
 105 
 
 Decimal Tables of Coin, Weight, and Measure. 
 
 TABLE I. 
 
 Sterling Money. 
 £l. the Integer. 
 
 s. 
 
 19 
 18 
 17 
 16 
 15 
 14 
 13 
 12 
 11 
 10 
 
 dec. 
 95 
 
 9 
 
 85 
 
 8 
 
 75 
 
 7 
 
 65 
 
 6 
 
 55 
 
 5 
 
 s. dec. 
 9 '45 
 
 •4 
 
 •35 
 
 •3 
 
 •25 
 
 •2 
 
 •15 
 
 •1 
 
 •05 
 
 table iii. 
 
 Troy Weight. 
 
 1 lb. the Integer. 
 
 Ounces the same 
 
 as Pence in Table n 
 
 6d. 
 5 
 
 4 
 3 
 2 
 
 1 
 
 •025 
 
 •020833 
 
 •016666 
 
 •0125 
 
 •008333 
 
 •004166 
 
 Sqrs. 
 
 2 
 
 1 
 
 •003125 
 
 •0020833 
 
 •0010416 
 
 TABLE II. 
 
 Ekg. Coin. Is. 
 
 Long Meas. 1 Foot 
 
 the Integer. 
 
 Sqrs, 
 
 2 
 
 1 
 
 •0625 
 
 •041666 
 
 •020833 
 
 drvts. 
 10 
 
 9 
 8 
 7 
 6 
 5 
 4 
 3 
 2 
 1 
 
 Pence or 
 Inches. 
 
 6 
 
 5 
 
 4 
 
 3 
 
 2 
 
 1 
 
 Decimals. 
 
 •5 
 
 •416666 
 
 •833333 
 
 •25 
 
 •166666 
 
 •083333 
 
 12 gr, 
 
 11 
 
 10 
 
 9 
 8 
 7 
 6 
 5 
 4 
 3 
 2 
 1 
 
 Decimals. 
 •041666 
 •0375 
 -033333 
 •029166 
 •025 
 •020833 
 •016666 
 •0125 
 •008333 
 •004166 
 
 •002083 
 •001910 
 •001736 
 •001562 
 •001389 
 •001215 
 •001042 
 •000868 
 •000694 
 •000521 
 •000347 
 •000173 
 
 1 02. the Integer. 
 
 Penny- weights the 
 same as Shillings 
 in the first Table. 
 
 E 5 
 
 Grains. 
 12 
 11 
 10 
 
 9 
 8 
 7 
 6 
 5 
 4 
 3 
 2 
 1 
 
 Decimals. 
 •025 
 •022916 
 •020833 
 •01875 
 •016666 
 •014583 
 •0125 
 •010416 
 •008333 
 •00625 
 •004166 
 •002083 
 
 TABLE IV. 
 
 Avoir. Weight. 
 1 cwt. the Integer. 
 
 Qrs. 
 
 Decimals. 
 
 3 
 
 •75 
 
 2 
 
 •5 
 
 1 
 
 •25 
 
 14/fo. 
 
 13 
 
 12 
 
 11 
 
 10 
 
 9 
 8 
 7 
 6 
 5 
 4 
 3 
 2 
 1 
 
 8 oz. 
 
 7 
 
 •125 
 
 •116071 
 
 •107143 
 
 •098214 
 
 •089286 
 
 •080357 
 
 •071428 
 
 •0625 
 
 •053571 
 
 •044643 
 
 •035714 
 
 •026786 
 
 •017857 
 
 •008928 
 
 •004464 
 •003906 
 
106 
 
 DECIMALS. 
 
 [the tutor's 
 
 Decimal Tables of Coin, Weight, and Measure. 
 
 6oz. 
 
 •003348 
 
 5 
 
 •002790 
 
 4 
 
 •002232 
 
 3 
 
 •001674 
 
 2 
 
 •001116 
 
 1 
 
 •000558 
 
 3 
 4 
 
 •000418 
 
 1 
 2 
 
 •000279 
 
 1 
 4 
 
 •000139 
 
 TABLE V. 
 
 Avoir. Weight. 
 1 lb. the Integer. 
 
 Ounces. 
 8 
 7 
 6 
 5 
 4 
 3 
 2 
 1 
 
 Decimals. 
 •5 
 
 •4375 
 •375 
 •3125 
 •25 
 •1875 
 •125 
 •0625 
 
 8 dr. 
 
 7 
 
 6 
 
 5 
 
 4 
 
 3 
 
 2 
 
 1 
 
 •03125 
 
 •027343 
 
 •023437 
 
 •019531 
 
 •015625 
 
 ^011718 
 
 •007812 
 
 -003906 
 
 TABLE VI. 
 
 Liquid Measure. 
 1 Tun the Integer. 
 
 Gallons. 
 10G 
 90 
 
 Decimals. 
 •396825 
 •357142 
 
 80 g. 
 
 70 
 
 60 
 
 50 
 
 40 
 
 30 
 
 20 
 
 10 
 
 9 
 
 8 
 
 7 
 
 6 
 
 5 
 
 4 
 
 S 
 
 2 
 
 1 
 
 4* pts. 
 3 
 2 
 1 
 
 •317460 
 •277777 
 •238095 
 •198412 
 •158730 
 •119047 
 •079365 
 •039682 
 •035714 
 •031746 
 •027777 
 •023809 
 •019841 
 •015873 
 •011904 
 •007936 
 •003968 
 
 •001984 
 •001488 
 •000992 
 •000496 
 
 1 Hogshead the 
 Integer. 
 
 Gallons. 
 30 
 20 
 10 
 
 9 
 
 8 
 7 
 6 
 5 
 4 
 3 
 2 
 1 
 
 Decimals. 
 •476190 
 •317460 
 •158730 
 •142857 
 •126984 
 •111111 
 •095238 
 •079365 
 •063492 
 •047619 
 •031746 
 •015873 
 
 3pts, 
 2 
 
 1 
 
 •005952 
 •003968 
 •001984 
 
 table vii. 
 
 Measures. 
 
 Liquid. Dry. 
 
 1 Gal. 1 Qr. 
 
 the Integer. 
 
 Pts. 
 
 4 
 3 
 
 2 
 
 1 
 
 s 
 4 
 1 
 2 
 1 
 4 
 
 Dec. 
 
 •5 
 
 •375 
 
 •25 
 •125 
 09375 
 0625 
 •03125 
 
 Busk. 
 4 
 3 
 
 2 
 1 
 3 p. 
 
 Decimals. 
 •0234375 
 •015625 
 •0078125 
 
 Qr. Pks. 
 3 
 2 
 1 
 
 •005859 
 •003906 
 •001953 
 
 3 pts. 
 
 table viii. 
 
 Long Measure. 
 
 1 Mile the Integer. 
 
 Yards. 
 
 1000 
 900 
 800 
 700 
 600 
 
 Decimals. 
 •568182 
 •511364 
 •454545 
 •397727 
 •340909 
 
S1STANT.J 
 
 DECIMALS. 
 
 10? 
 
 Decimal Tables of Coin, Weight, and Measure. 
 
 500yd. 
 
 400 
 
 300 
 
 200 
 
 100 
 
 90 
 
 80 
 
 70 
 
 60 
 
 50 
 
 40 
 
 30 
 
 20 
 
 10 
 
 9 
 8 
 7 
 6 
 5 
 4 
 3 
 2 
 1 
 
 2ft. 
 
 1 
 
 6 in. 
 3 
 
 2 
 
 1 
 
 •284091 
 •227272 
 •170454 
 •113636 
 •056818 
 •051136 
 •045454 
 •039773 
 •034091 
 •028409 
 •022727 
 •017045 
 •011364 
 •005682 
 •005114 
 •004545 
 •003977 
 •003409 
 •002841 
 •002273 
 •001704 
 •001136 
 •000568 
 
 •0003787 
 •0001 894 
 
 •0000947 
 •0000474 
 •0000315 
 •0000158 
 
 TABLE IX. 
 TIME. 
 
 1 Year the Integer. 
 
 Months the same as 
 Pence in Table n, 
 
 Days. 
 
 300 
 
 200 
 
 100 
 
 90 
 
 Decimals. 
 •821918 
 •547945 
 •273973 
 •246575 
 
 80 d. 
 
 70 
 
 60 
 
 50 
 
 40 
 
 30 
 
 20 
 
 10 
 
 9 
 
 8 
 
 7 
 
 6 
 
 5 
 
 4 
 
 3 
 
 2 
 
 1 
 
 •219178 
 •191781 
 •164383 
 •136986 
 •109589 
 •082 192 
 •054794 
 •027397 
 •024657 
 •02 191 8 
 •019178 
 •016438 
 •013698 
 •010959 
 •008219 
 •005479 
 •002739 
 
 1 Day the Integer. 
 
 \2hrs. 
 
 11 
 
 10 
 
 9 
 
 8 
 
 7 
 
 6 
 
 5 
 
 4 
 
 3 
 
 2 
 
 1 
 
 30 m. 
 
 20 
 
 10 
 
 9 
 
 8 
 
 7 
 
 6 
 
 5 
 
 4 
 
 3 
 
 2 
 
 I 
 
 •5 
 
 •458333 
 
 •416666 
 
 •375 
 
 '333333 
 
 •291666 
 
 •25 
 
 •208333 
 
 '166666 
 
 •125 
 
 •083333 
 
 •041666 
 
 •020833 
 
 013888 
 
 •006944 
 
 •00625 
 
 •005555 
 
 •004861 
 
 •004166 
 
 •003472 
 
 •002777 
 
 •002083 
 
 •001388 
 
 •000694 
 
 TABLE X. 
 
 Cloth Measure. 
 1 Yard the Integer. 
 
 Qrs. tha MUfc* as 
 
 Table iv. 
 
 Nails. I Decimals. 
 3 '1875 
 
 2 -125 
 
 1 -0625 
 
 TABLE XI. 
 
 Lead Weight. 
 A Foth.the Intger. 
 
 Hund. 
 10 
 
 9 
 8 
 7 
 6 
 5 
 4 
 3 
 2 
 1 
 
 Decimals. 
 •512820 
 •461538 
 •410256 
 •358974 
 •307692 
 •256410 
 •205128 
 •153846 
 •102564 
 •051282 
 
 3 qrs. 
 
 2 
 
 1 
 
 •038461 
 •025641 
 •012820 
 
 1 lbs. 
 
 13 
 
 12 
 
 11 
 
 10 
 
 9 
 8 
 7 
 6 
 5 
 4 
 3 
 2 
 1 
 
 ) -0064102 
 •0059523 
 •0054945 
 •0050366 
 •0045787 
 •0041208 
 •0036630 
 •0032051 
 •0027472 
 •0022893. 
 •0018315 1 
 •0013736 
 •0009157 
 •0004578 
 
108 DECIMALS*. [THE TUTORS 
 
 THE RULE OF THREE. 
 
 (1) If 26$ yards cost £3..l6..3. what will 32{ yds. cost ?• 
 
 (2) If 7| yards of cloth cost £2..12..9. what will 140$ 
 yards of the same cost ? Ans. £47-.l6..3$. 
 
 (3) If a chest of sugar, weighing 7 cwt. 2 qrs. 14 lb. cost 
 £36..12..9. what will 2 cwtf. 1 qr. 21 /&. of the same cost ? 
 
 Jx*.£ll..U.£$. 
 
 (4) What will 326£ lb. of coffee be worth when 1$ lb. is 
 sold for 3s. 6d. ? Ans. £38..1..3. 
 
 (5) What is the value of 19 oz. 3 dwts. 5 grs. of gold, at 
 £2..19. per oz. ? Ans. £56..10..5..2-3 qrs. 
 
 (6) Wbat is the charge for 82 7f yards of painting, at 10^d. 
 per yard ? Ans. £36.A..3..1'5 qrs. 
 
 (7) If I lent my friend £34. for | of a year, how much 
 ought he to lend me for T 5 2 of a year ? Ans. £51, 
 
 (8) If f of a yard of cloth, that is %\ yards broad, make 
 a garment, how much of § of a yard wide will make a similar 
 one? Ans. % yds. 1*75 nail. 
 
 (9) If 1 oz. of silver is worth 5s. 6d. what is the price of a 
 tankard that weighs 1 lb. 10 oz. 10 dwts. 4 grs. ? 
 
 Ans. £6..3..9..2*2 qrs. 
 
 (10) What is the value of 15 ctvt. 1 qr. 19 lb. of cotton, at 
 15d. per lb. f Ans. £l07..18..9- 
 
 (11) If 1 cwt. of currants cost <£2..9..6. what will 45 cwt. 
 3 qrs. 14 lb. cost at the same rate ? Ans. <£ll3..10..9§. 
 
 (12) Bought 6 chests of sugar, each 6 cwt. 3 qrs. at <£2..l6. 
 per cwt. What do they come to ? Ans. <£ll3..8. 
 
 (13) Bought a tankard for £l0..12. at the rate of 5s. 4>d. 
 per oz. What was the weight? Ans. 39 oz. 15 dwts. 
 
 (14) Gave £l87»3..3. for 25 cwt. 3 qrs. 14 lb. of coffee : at 
 what rate did I buy it per lb. ? Ans. Is. 3^d. 
 
 (15) Bought 29 lh. 4 oz. of snuff for £l0..11..3. W T hat is 
 the value of 8 lb. ? Ans. £l..l..8. 
 
 (16) If I give 1$. Id. for 3$ lb. of rags, what will be the 
 value of 1 cwt. ? Ans. £l..l4..8. 
 
 • As 26-5 
 
 £. 
 
 yds. 
 
 £. 
 
 3-8125 : 
 
 : 32-25 ; 
 
 1 4-63974 
 
 32-25 
 
 
 
 26-5) 1 22*953125(4-6S974=£4.. 1 2..9J. Am. 
 
ASSISTANT.] EXCHANGE. 109 
 
 EXCHANGE 
 
 Is the act of bartering the money of one place for that of 
 another ; by means of a written instrument called a Bill of 
 Exchange. 
 
 The operations in this Rule consist in finding the quantity of one 
 tort of money that will be equal to a given sum of the other, according 
 to the existing Course of Exchange. 
 
 Par of Exchange signifies the equality in the intrinsic value 
 of two sums of money of different countries ; and shows how 
 much of the one is worth a constant sum (or piece of coin) 
 of the other. 
 
 Course of Exchange is the comparative value between the 
 money of two different countries at any particular time; 
 which often fluctuates above or below the Par. 
 
 Agio is a difference of so much per cent, in the value of 
 the Bank-money and the Current-money of some foreign coun- 
 tries, the former being of superior value. 
 To change Foreign Money into British Sterling Money, or Ster- 
 ling into Foreign; according to a given Course of Exchange. 
 
 Rule. As the quantity of Foreign mentioned in the given 
 course of exchange, is to the quantity of Sterling ; so is any 
 other sum of the Foreign, to its corresponding value in 
 Sterling money. 
 
 And by mutually changing the words Foreign and Ster- 
 ling, the Rule will serve for changing Sterling into Foreign 
 money. 
 
 I. FRANCE. 
 
 Accounts are kept at Paris, Lyons, and Rouen, in livres, 
 sols, and deniers, and exchange is made by the ecu, or crown 
 =4tf. 6d at par. 
 
 Table. 12 deniers make 1 sol. 
 20 sols ... 1 livre. 
 3 livres ... 1 ecu, or crown. 
 (1) How many crowns must be paid at Paris, to rece ; ve 
 in London £180. exchange at 4*. 6d. per crown?* 
 
 s. d. cr. £. 
 ■As4..6 : 1 : : 180 
 2 40 
 
 9 sixp. 9)7200 sixp. 
 
 800 crowns. Ans. 
 
110 EXCHANGE. [THE TUTOR'S 
 
 (2) How much sterling must be paid in London, to receive 
 in Paris 758 crowns, exchange at 4$. 8d. per crown? 
 
 Ans, £l76..17»4. 
 
 (3) A merchant in London remits £l76..17..4. to his cor- 
 respondent at Paris ; what is the value in French crowns, at 
 4s. 8c?. per crown? A?is. 758 crowns. 
 
 (4) Change 725 crowns, 17 sols, 7 deniers, at 4*. 6^d. 
 per crown, into sterling money. Ans. £l64..14..0£. \\\. 
 
 (5) Change £l 64.. 14..0^. sterling into French crowns, ex- 
 change at 4*. 6^d. per crown. 
 
 Ans. 725 crowns, 17 sols, 7j^y deniers. = 
 
 II. SPAIN. 
 
 Accounts are kept at Madrid, Cadiz, and Seville, in dollars, 
 rials, and maravedies, and exchange is made by the piece of 
 eight:=4s. 6d. at par. 
 
 Table. 34 maravedies make 1 rial. 
 
 8 rials .... 1 piastre, or piece of eight. 
 1 rials .... 1 dollar. 
 
 (6) A merchant at Cadiz, remits to London 2547 pieces of 
 eight, at 4.9. 8d. per piece, how much sterling is the sum ? 
 
 Ans. £594.. 6. 
 
 (7) How many pieces of eight, at 4s. Sd. each, will answej 
 a bill of £594..6. sterling? Ans. 2547. 
 
 (8) If I pay here a bill of £2500., for what Spanish monej 
 may I draw my bill at Madrid, exchange at 4s. 9 \d. per piece 
 of eight? Ans. 10434 pieces of eight, 6 rials, 8 § § mar. 
 
 III. ITALY. 
 
 Accounts are kept at Genoa and Leghorn, in livres, sols, 
 and deniers, and exchange is made by the piece of eight, or 
 dollar zz 4s. 6d. at par. 
 
 Table. 12 deniers make 1 sol. 
 20 sols . . 1 livre. 
 
 5 livres . . 1 piece of eight at Genoa. 
 
 6 livres . . 1 piece of eight at Leghorn. 
 
 N. B. The exhange at Florence is by ducatoons ; at Venice 
 by ducats. 
 
 Table. 6 solidi make 1 gross. 
 24 gross . 1 ducat. 
 
 (9) How much sterling money may a person receive in 
 London, if he pay in Genoa 976 dollars, at 4s. 5d. per dollar? 
 
 Ans. £215..10..8. 
 
ASSISTANT.] EXCHANGE. Ill 
 
 (10) A factor has sold goods at Florence, for 250 duca- 
 toons, at 4$. 6d. each : what is the value in pounds sterling ? 
 
 Ans. £56..5. 
 
 (11) If 275 ducats, at 4,?. 5d. each, be remitted from Venice 
 to London, what is the value in pounds sterling ? 
 
 Ans. £60..14..7. 
 
 (12) A traveller would exchange £60..14..7. sterling for 
 Venice ducats, at 4^. 5d. each : how many must he receive ? 
 
 Ans. 275. 
 
 IV. PORTUGAL. 
 
 Accounts are kept at Oporto and Lisbon, in reas, and ex- 
 change is made by the milrea =6$. 8^d. at par. 
 Table. 1000 reas make 1 milrea. 
 
 (13) A gentleman being desirous to remit to his corres- 
 pondent in London, 2750 milreas, exchange at 6s. 5d. per 
 milrea, for how much sterling will he be creditor in London ? 
 
 Ans. £882..5..10. 
 
 (14) A merchant at Oporto remits to London 4366 mil- 
 reas, 183 reas, at 5s. 5f d. exchange per milrea: how muck 
 sterling must be paid in London for this remittance ? 
 
 Ans. £ll93..17..6..3-0375 qrs. 
 
 (15) If I pay a bill in London of £1193.. 17-6..3 -0375 qrs, 
 what must I draw for on my correspondent in Lisbon, ex- 
 change at 5s. 5§e?. per milrea? Ans. 4366 milreas, 183 reas. 
 
 V. HOLLAND, FLANDERS, AND GERMANY. 
 
 At Antwerp, Amsterdam, Brussels, Rotterdam, and Ham- 
 burgh, some accounts are kept in pounds, shillings, and 
 pence, as in England ; others in guilders, stivers, and pen- 
 nings : exchange with London, at from 33s. to 36s. or 38s 
 Flemish per pound sterling. 
 
 Table. 8 pennings make 1 groat. 
 
 2 groats, or 16 pennings 1 stiver. 
 20 stivers 1 guilder, or florin. 
 
 also 12 groats, or six stivers make 1 schelling. 
 20 schellings, or 6 guilders 1 pound. 
 
 (16) Remitted from London to Amsterdam, a bill ot 
 £754.. 10. sterling: how many pounds Flemish is the sum, 
 the exchange at 33s. 6d. Flemish per pound sterling? 
 
 Ans. £l263..15..9. Flemish. 
 
 (17) A merchant in Rotterdam remits £ 1263.. 15..9. Flem- 
 ish to be paid in London, how much sterling money must he 
 
112 CONJOINED PROPORTION. £ THE TUTOR'S 
 
 draw for, the exchange being at 33s. 6d. Flemish per pound 
 sterling? Ans. £754..10. 
 
 (18) If I pay in London £852..12..6. sterling, how many 
 guilders must I draw for at Amsterdam, exchange at 34 schil- 
 lings, 4^ groats Flemish per pound sterling ? 
 
 Ans. 879^ guild. 13 stiv. 1 gr. 6^ pennings. 
 
 (19) What must I draw for in London, if I pay in Amster- 
 dam 8792 guild. 13 stiv. 14^ pennings, exchange at 34 schel- 
 lings, 4^ groats per pound sterling? Ans. £852..12..6 
 
 To convert Bank Money into Currency ; and the contrary. 
 
 As 100 : 100 plus the agio : : the Bank-money : the 
 Currency. 
 
 As 100 plus the agio : 100 : : the Currency : the Bank- 
 money. 
 
 (20) Change 794 guilders, 15 stivers, Current money into 
 Bank florins, agio 4| per cent. 
 
 Ans. 76l guilders, 8 stivers, 11 f§f pennings. 
 
 (21) Change 76 1 guilders, 9 stivers Bank, into Current 
 money, agio 4f per cent. 
 
 Ans. 794 guilders, 1 5 stivers, 4 T 5 S pennings. 
 
 VI. IRELAND. 
 
 The par of Exchange, long established with Ireland, was 
 £108..6..8. Irish=£l00. English. That is, £l..l..8. Irish 
 =,£1. English; or 13c?. Irishrrl^. English. 
 
 But the English and Irish currency are now assimilated. 
 
 (22) A gentleman remitted to Ireland £575..15. sterling: 
 what would he receive there, the exchange being at £10. per 
 cent. ? Ans. £633..6..6. 
 
 (23) What would be paid in London for a remittance of 
 £633..6..6. Irish, exchange at £10. per cent ? Ans. £575..l5. 
 
 CONJOINED PROPORTION; or COMPOUND 
 ARBITRATION OF EXCHANGE 
 
 Is the method of comparing the coins, weights, or measures 
 of one country with those of another, when the comparison 
 is to be made through the medium of those of other coun- 
 tries. 
 
 Case 1. When it is required to find how many of the Jlrst 
 sort mentioned are equal to a given quantity of the last. 
 
ASSISTANT.] CONJOINED PROPORTION. 113 
 
 Rule. Place the terms alternately, antecedents, and con- 
 sequents, in two columns, left and right The last term, be- 
 ing an antecedent, will stand on the left, 
 i Divide the product of the antecedents by the product of 
 the consequents for the answer. 
 
 Proof. By as many single statements as the question re- 
 quires. 
 
 (1) If 20 lb. at London make 23 lb. at Antwerp, and 155 lb. 
 at Antwerp make 180 lb. at Leghorn, how many lb. at London 
 aKe equal to 72 lb. at Leghorn?* 
 
 (2) If 12 lb. at London make 10 lb. at Amsterdam, and 
 100 lb. at Amsterdam 120/6. at Thoulouse, how many lb. 
 at London are equal to 40 lb. at Thoulouse ? Ans. 40 lb. 
 
 (3) If 140 braces at Venice be equal to 156 braces at Leg- 
 horn, and 7 braces at Leghorn equal to 4 ells English, how 
 many braces at Venice are equal to 1 6 ells English ? 
 
 Ans. 25/g. 
 
 (4) If 40 lb. at London make 36 lb. at Amsterdam, and 90 
 lb. at Amsterdam make 116/6. at Dantzic, how many lb. at 
 London are equal to 130 lb. at Dantzic ? Ans. 112/g. 
 
 Case 2. When it is required to find how many of the last 
 sort mentioned are equal to a given quantity of the Jirst. 
 
 Rule. Place the antecedent and consequent terms as before. 
 But the last term, being a consequent, will stand on the right. 
 Divide the product of the consequents by that of the ante- 
 cedents, 
 
 (5) If 12 lb. at London make 10 lb. at Amsterdam, and 
 100 lb. at Amsterdam 120 lb. at Thoulouse, how many lb. at 
 Thoulouse are equal to 40 lb. at London ? Ans. 40 lb. 
 
 (6) If 40 lb. at London make 36 lb. at Amsterdam, and 90 
 lb. at Amsterdam 116/6. at Dantzic, how many lb. at Dant- 
 zic are equal to 122 lb. at London? Ans 141 Jf. 
 
 * Antecedents. Consequents. 
 
 20 lb. London = 23 lb* Antwerp. 
 155 lb. Antwerp = 180 lb. Leghorn. 
 72 lb. Leghorn = how many London ? 
 
 !0XJ55XJM_!?4O_ ^ *t 
 
 23 X JL$0 23 
 
114 EVOLUTION. |_THE TUTOR'S 
 
 INVOLUTION 
 
 Is the method of finding the powers of numbers. 
 
 Any number is the Jlrst power of itself, and the root of all 
 its powers: and when the root is multiplied by itself the 
 product is the second power ; the second multiplied by the 
 Jlrst produces the third, &c 
 
 The second power is commonly called the square, and the 
 third power, the cube. 
 
 Numbers called indices, or exponents, are placed on the 
 right a little above the line, to denote the respective powers. 
 Thus, 3 2 signifies the square, or second power, of 3 ; the small 
 figure 2 being the index, or exponent. 
 
 To involve a number to any power. 
 
 Rule. Multiply the given number (or root) by itself con- 
 tinually one time less than the index of the power : that is 
 once for the second, twice for the third power, &c. 
 
 Observe, that any two or more powers multiplied together, 
 will produce a power whose index is the sum of their indices. 
 Thus, the seventh power is the product of the fourth and the 
 third ; because the sum of the indices, 4-|-3=7. 
 
 ILLUSTRATIONS. 
 
 The first power of 3 is 3 l , or 3. 
 The second power of 5 is 5 2 =5x5=25. 
 The third power of 4 is 4 5 =4x4x4=64. 
 The fourth power of -05 is -05 4 =-05x-05X'05X'05= 
 •00000625. _ 
 
 The fifth power of § is || 5 =§xf xf Xf xf^V 
 
 (1) Required the squares of 43, 2174, 4*3, and -2174. 
 
 Ans. 1849, 4726276, 18-49, and -04726276. 
 
 (2) Cube 111, 1-11, |, and 2|. 
 
 Ans. 1367631, 1*367631, §|, and 18^. 
 
 (3) Involve 9 to the ninth power. Ans. 387420489- 
 
 (4) Find the third, fifth, and eighth powers (without find- 
 ing the fourth, sixth, and seventh) of 1 *7. 
 
 Ans. 4*913, 14-19857, and 69*75757441. 
 
 (5) What are the third and sixth powers of -05 ? 
 
 Ans. -000125, and -000000015625. 
 
 EVOLUTION 
 
 Is the method of extracting the roots of powers. It is there- 
 fore the reverse of Involution : by referring to which it will 
 
ASSISTANT.] SQUARE ROOT. 17 5 
 
 be obvious, that the Square Root of a number multiplied by 
 itself, will produce that number ; and that the Cube Root, 
 multiplied twice by itself, will produce the number (or power) 
 of which it is the root. 
 
 Note. The roots of complete powers are called rational; and those 
 which cannot be completely extracted, are called surds, or irrational 
 roots: thus ^4 = 2, is rational; but J 5 is a surd. The surd roots 
 may, however, be found to any extent proposed. 
 
 SQUARE ROOT. 
 
 Rule. 1. Place points over the units, hundreds, &c. so as to 
 form periods of twojlgures each. 
 
 2. From the first period on the left, subtract the greatest 
 square contained in it ; put the root on the right as a quotient ; 
 annex the succeeding period to the remainder, and call that 
 number the Resolvend. 
 
 3. Divide the resolvend, exclusive of the units, by double 
 the root ; annex the quotient to the root, and also to the 
 right of the divisor to complete it : then multiply the divisor 
 by that quotient figure, and subtract the product from the 
 resolvend. 
 
 4. The remainder, with the next period joined, will form 
 a new resolvend ; and double the root, a new divisor ; with 
 which proceed as before. 
 
 Note 1. When the number of figures in the Integer is uneven, the 
 first period will consist of but one figure. When there is an odd 
 number of decimals, a cipher must be added to complete the periods. 
 
 2. When the figures of the whole number are exhausted, periods 
 of ciphers may be used at pleasure, to continue the extraction in 
 decimals. In all cases, the root will consist of as many figures as 
 there are periods, whether integral or decin-al. 
 
 Roots. 1. 2. $. 4. 5. 6. 7. 8. 9. 
 Squares. 1. 4. 9- 16. 25. 36. 49. 64. 81. 
 
 (1) What is the square root of 119025 ?* 
 
 (2) What is the square root of 106929 ? A?is. 327. 
 
 (3) What is the square root of 22071204 ? Ans. 4698. 
 
 119025(345 the root. 345 
 
 9 345 
 
 64)290 1725 
 
 256 1380 
 
 685)3425 1035 
 
 3425 Proof 119025 
 
116 EVOLUTION. [THE TUTOR'S 
 
 (4) What is the square root of 2268741 ? Ans. 1506*23+. 
 
 (5) What is the square root of 7596796 ? Ans - 2756*228+. 
 
 (6) What is the square root of 4-372594 ? Ans. 2*091+. 
 
 (7) What is the square root of 2*2710957? Ans. 1*50701+. 
 
 (8) What is the square root of -000327S4 ? Ans. -01809+. 
 
 (9) What is the square root of 1*270059 ? Ans. 1*1269+. 
 
 To find the Roots of Fractional Numbers. 
 
 Rule. When the terms of a Fraction are complete, power s y 
 extract their roots for the corresponding terms of the root. 
 
 When they are surds, find an equivalent fraction, by mul- 
 tiplying both terms by the denominator ; or by the least number 
 that will make the denominator a square. Then divide the 
 root of the numerator by the root of the denominator for the 
 answer. — Or, reduce the fraction to a decimal, and extract 
 its root. 
 
 Mixed numbers may either be reduced to their equivalent 
 fractions ; or into a decimal form. 
 
 (10) What is the square root of ffff f Ans. §. 
 
 (11) What is the square root of T 9 2 *V? 6 5 ? Ans. f . 
 
 (12) What is the square root of 51 f| ? Ans. 7}. 
 
 (13) What is the square root of 27 T % ? Ans. 5\. 
 
 (14) What is the square root of 9f § ? Ans. 31. 
 
 (15) What is the square root of fj| ? Ans. -89802+. 
 
 (16) What is the square root of ||j ? Ans. *93309+. 
 
 (17) What is the square root of 85{$ ? Ans. 9*27+. 
 
 (18) What is the square root of 8f ? Ans 2*9519+. 
 
 To find a mean proportional between any two given numbers. 
 Rule. Extract the square root of their product. 
 
 (19) What is the mean proportional between 3 and 12 ? 
 
 ^3x12=^36=6 the mean proportional. Ans. 
 
 (20) What is the mean proportional between 4276 and 
 842? Ans. 1897*4+. 
 
 To find the side of a square equal in area to any given surface. 
 
 Rule. Extract the square root of the given area for the 
 side of the square sought. 
 
 (21) If the content of a given circle be 160, what is the 
 side of the square equal? Ans. 12*64911. 
 
 (22) If the area of a circle is 750, what is the side of the 
 square equal ? Ans. 27*38612. 
 
ASSISTANT.^ SQUARE ROOT 117 
 
 The area of a circle given, to find the diameter. 
 
 Rule. As 355 : 4*52, or, as 1 : 1-273239 : : the area : the 
 square of the diameter : — or, multiply the square root of the 
 area, by 1*12837, and the product will be the diameter. 
 
 (23) What length of cord must be tied to a cow's tail, the 
 other end fixed in the ground, to enable her to eat just an 
 acre of grass, and no more ; supposing the cow and tail to 
 measure 5^ yards ? Ans. 33*75 yards. 
 
 The area of a circle given, to find the circumference. 
 
 Rule. As 113 : 1420, or, as 1 : 12*56637 : : the area : 
 the square of the circumference : — or, multiply the square 
 root of the area by 3* 5449, and the product will be the cir- 
 cumference. 
 
 (24) When the area is 12, what is the circumference ? 
 
 Ans. 12*279- 
 
 (25) When the area is 160, what is the circumference ? 
 
 Ans. 44*839. 
 
 Two sides of a right-angled triangle being given, to find the 
 third side. 
 
 Case 1 . The base and perpendicular being given, to find the 
 hypotenuse. 
 
 Rule. The square root of the sum of the squares of the 
 base and perpendicular, is the length of the hypotenuse. 
 
 Case 2. The hypotenuse and perpendicular being given, to find 
 the base. 
 
 Rule. The square root of the difference of the squares of 
 the hypotenuse and perpendicular is the length of the base, 
 
 Case 3. The base and hypotenuse being given, to find the per- 
 pendicular. 
 
 Rule. The square root of the difference of the squares of 
 the hypotenuse and base is the height of the perpendicular. 
 
 (26) The top of a castle from the ground is 45 yards high, 
 and it is surrounded with a ditch 60 yards broad : what length 
 must a ladder be to reach from the outside of the ditch to the 
 top of the castle ? Ans. 75 yards. 
 
us 
 
 EVOLUTION 
 
 [/THE tutor's 
 
 Base 60 yards. 
 
 (27) The wall of a town is 25 feet high, and is surrounded 
 by a moat of 30 feet in breadth : required the length of a lad- 
 der that wiK reach from the outside of the moat to the top of 
 the wall ? A?is. 39'05feeL 
 
 N. B. These two questions may be varied for examples to the 
 second and third cases. 
 
 (28) In an army consisting of 33 1776 men, how many must 
 be in rank and file to form a solid square ? Ans. 576. 
 
 (29) A certain square pavement contains 48841 equai 
 square stones. How many are contained in one of the sides f 
 
 Ans. 221. 
 
 CUBE ROOT. 
 
 Rule. 1. Point every third Jlgure of the given number, 
 beginning at the units' place ; find the greatest cube in the 
 first period, and subtract it therefrom ; put the root in the 
 quotient, and bring down the figures in the next period to 
 the remainder, for a Resolvend. 
 
 2. Multiply the square of the root found by 300, for a Divu 
 sor, and annex to the root the number of times which that is 
 contained in the Resolvend, 
 
 3. Add 30 times the preceding figure (or figures) multiplied 
 by the last, and the square of the last, to the divisor ; and 
 multiply the sum by the last, for a Subtrahend : subtract it 
 from the Resolvend, and repeat the process as far as necessary.* 
 
 • The subjoined Theorems (deduced from Problem 91, page 266, Ew.~ 
 erson's Algebra) are very convenient approximations for the Cube Root. 
 
ASSISTANT."] CUBE ROOT. 119 
 
 Note. As the units must always be pointed, there will be some- 
 times only one or two figures in the first period — The decimals must 
 always consist of so many figures as will constitute complete periods, 
 as in the Square Root — Also, what is observed in Note 2, Square 
 Hoot, will hold good in this ltule. 
 
 Roots. 1. 2. 3. 4. 5. 6. 7- 8. 9- 
 Cubes. 1. 8. 27. 64. 125. 216. 343. 512. 729. 
 
 (1) What is the cube root of 99252847 ?* 
 
 (2) What is the cube root of 38901? ? Ans. 73. 
 
 (3) What is the cube root of 5735339 ? Ans. 179- 
 
 (4) What is the cube root of 32461759 ? Ans. 319- 
 
 (5) What is the cube root of 84604519 ? Ans. 439- 
 
 (6) What is the cube root of 27054036008 ? Ans. 3002. 
 
 (7) What is the cube root of 673373097125? Ans. 8765. 
 
 (8) What is the cube root of 12*977875 ? Ans. 9,-35. 
 
 (9) What is the cube root of -001906624 ? Ans. -124. 
 
 R, the required root, nearly. In which n denotes the given number 
 and r, an assumed root found by trial. 
 
 The second formula, which is more convenient than the other, be- 
 cause it contains no higher power than the square of r, may be thus 
 expressed. 
 
 Divide the given number by three times the assumed root, and 
 >Tom the quotient subtract & of the square of the assumed root: 
 the square root of the remainder, added to half the assumed root, 
 will give the root required. See also the method of extracting any 
 root by approximation. 
 
 • 99252847(463 the root. 
 4 3 = 64 
 
 4* X 300=4800)35252 resolvend. 
 
 720=4X30X6 
 36=6* 
 4800 divisor. 
 
 5556 
 6 
 
 333S6 subtrahend. 
 
 46 a X300=634800)1916847 resolvend. 
 
 4140=46X30X3 
 9=3 2 
 6 34800 
 
 "638949 
 3 
 
 1916847 subtrahend. 
 
120 EVOLUTION [THE TUTOR'S 
 
 (10) What is the cube root of 3615502756? Ans. 33-06+ . 
 
 (11) What is the cube root of 33-230979937 ? Ans. 3*215+. 
 
 (12) What is the cube root of 15926*972504 ? Ans. 26-16+ 
 
 Tojind the Roots of Fractional numbers. 
 
 Rule. When the terms of a fraction are complete powers, 
 extract their roots for the corresponding terms of the root. 
 
 When they are surds, if both terms be multiplied by the 
 square of the denominator, an equal fraction will be produced, 
 the denominator of which will be a cube. Then divide the 
 root of the numerator by the root of the denominator for the 
 answer. — Or, the fraction may be reduced to a decimal, and 
 its root extracted. 
 
 Mixed numbers may be reduced as in the Square Root. 
 
 (13) What is the cube root of §|g ? Ans. f . 
 
 (14) What is the cube root of £%% ? 
 
 (15) What is the cube root of lgjf ? 
 
 (16) What is the cube root of 31 J& ? 
 
 (17) What is the cube root of 405 T 2 ^ i 
 
 (18) What is the cube root of f ? 
 
 (19) What is the cube root of § ? 
 
 (20) What is the cube root of 7j ? 
 
 (21) What is the cube root of 9£ ? 
 
 (22) What is the cube root of 8f ? 
 
 (23) A water cistern in the form of a cube contains 60 cubic 
 feet, 143 inches: what is the length of the side? 
 
 Ans. 47 inches. 
 
 (24) There is an excavation made for a cellar equal in 
 length, breadth, and depth; which required 4913 cubic feet 
 of earth to be dug out. What is the length of the side ? 
 
 Ans. 17 feet. 
 
 (25) There k a building of cubic form, which contains 
 389017 solid feet : what is the superficial content of one of 
 its sides ? Ans. 5329 sq.feet. 
 
 Between two numbers given, to find two mean proportionals. 
 
 Rule. Divide the greater extreme by the less, and the cube 
 root of the quotient multiplied by the less extreme gives the 
 less mean ; multiply the said cube root by the less mean, and 
 the product will be the greater mean proportional. 
 
 (26) What are the two mean proportionals between 6 and 
 162? Ans. 18 and 54. 
 
 (27) What are the two mean proportionals between 4 and 
 108? Ans. 12 and 36. 
 
 
 Ans. f . 
 
 
 Ans. 2|. 
 
 
 Ans. 31. 
 
 ? 
 
 Ans. 7f . 
 
 Ans. 
 
 •8298265+. 
 
 Ans. 
 
 •8220707+. 
 
 Ans. 
 
 1-930978+. 
 
 Ans. 
 
 2-092845+. 
 
 Ans. i 
 
 J-0578352+. 
 
assistant/] roots of all powers. 121 
 
 To find the side of a cube, equal in solidity to any given solid, 
 as a globe, cylinder, prism, cone, fyc. 
 
 Rule. The cube root of the solid content given, is the side 
 of a cube of equal solidity. 
 
 (28) If the solid content of a globe is 10648, what is the 
 side of a cube of equal solidity ? Ans. 22. 
 The side of a cube being given, to find the side of a cube that 
 
 shall be double, treble, fyc. in quantity to the cube given. 
 
 Rule. Cube the side given, and multiply it by 2, 3, &c. 
 the cube root of the product will be the side sought. 
 
 (29) There is a cubical vessel, whose side is 12 inches, ami 
 it is required to find the side of another vessel, that will con- 
 tain three times as much. Ans. 17*30699 inches. 
 
 BIQUADRATE ROOT. 
 
 Rule. Extract the square root of the given number, and 
 then the square root of that square root ; which will be the 
 biquadrate root required. 
 
 (1) What is the biquadrate root of 531441 ? Ans. 27 
 
 (2) What is the biquadrate root of 33362 176 ? Ans. 76. 
 
 (3) What is the biquadrate root of 5719140625 ? Ans. c 21o 
 
 A general rule for extracting the roots of all powers. 
 
 1. Prepare the given number, by pointing it into periods of 
 two figures each for the square root, three for the cube root, &c. 
 
 2. Find the first figure of the root, and subtract its power 
 from the first period. 
 
 3. Bring down the first figure in the next period to the re- 
 mainder, and call that the dividend. 
 
 4. Involve the root to the next inferior power to the given 
 one, and multiply it by the index of the given power, for a 
 divisor. 
 
 5. Find a quotient figure by common division, and annex 
 it to the root ; then involve the whole root to the given power 
 for a subtrahend, which subtract from the first two periods. 
 
 6. To the remainder bring down the first figure of the next 
 period for a new dividend ; find a new divisor and a new sub- 
 trahend as before ; subtract from three periods, and proceed 
 thus to the end. 
 
122 POSH ion. [the tutor's 
 
 Otherwise. To find ANY ROOT by approximation. 
 Rule. Let g denote the given number or power; n, the 
 index of the power; a, an assumed power nearly equal tog; 
 r, its root, and R, the required root. 
 
 _„ (n + 1) a + (n — 1) s 
 
 Then, as ^ 1 p ^ : a v> g : : r : R v> r ;* 
 
 which difference or correctional number, being added or sub- 
 tracted (as required) will give R : and by repeating the pro- 
 cess, any degree of accuracy may be obtained. 
 
 (1) What is the square root of 141376? t 
 
 (2) What is the cube root of 53157376 ? Ans. 376. 
 
 (3) What is the fourth root of 19987173376 ? Ans. 376. 
 
 (4) Required the fifth root of 2508'4746l56l4240625. 
 
 Ans. 4-785. 
 
 (5) Required the sixth root of 3*1416. Ans. 1-2102014- 
 
 SINGLE POSITION 
 
 Is the method of using one supposed number, and working 
 with it as the true one, to find the real number required. J 
 
 Rule. As the result from the supposition, is to the tru* 
 result ; so is the supposed number, to the true one required 
 
 Proof. Add the several parts together, according to tht 
 conditions of the question. 
 
 (1) A schoolmaster being asked how many scholars he had, 
 said, " If I had as many, half as manyj and one quarter as 
 many more, I should have 88." How many had he ?§ 
 
 • This fcr tie Cube Root will be, As 2a + g : a <x g : : r : R & r. 
 
 f 141376(376 the root. 
 9 
 3X2 = 6)51 dividend. 
 37 2 = 1369 subtrahend. 
 
 37X2 = 74)447 dividend. 
 276 2 = 141376 subtrahend. 
 
 X Questions belonging to this Rule have the results proportional 
 o their suppositions : the conditions requiring the number sought to 
 be increased by the addition of itself, or of some known multiple or 
 part thereof; or to be diminished by the subtraction of such part. 
 
 § Suppose he had 40 Then 40+40+20+10=110. 
 
 And, as 110 : 88 : • 40 : --?- •-— -= 09 Av . 
 
ASSISTANT. J POSITlO 123 
 
 (2) A person who had a certain number of antique coins, 
 said, If the third, fourth, and sixth parts of the number were 
 added together, they would make 54. How many had he ? 
 
 Ans. 72. 
 
 (3) A chaise, a horse, and harness, cost £6'0 ; the horse 
 being double the price of the harness, and the chaise double 
 the price of the horse and harness. What was given for 
 each? Ans. Horse £l3..6..8. harness £6\.13..4. chaise £40. 
 
 (4) What sum of money will amount to £300. in ten 
 years, at £6. per cent, per annum, simple interest ? 
 
 Ans. £187.-10. 
 
 (5) A, B, and C, dividing a quantity of goods, which 
 cost £120. mutually agreed that B should have a third part 
 more than A, and C a fourth part more than B. What must 
 each man pay? Ans. A £30. B £40. C £50. 
 
 (6) A gentleman bought a house, with a garden, and a 
 horse in the stable for £500. He paid four times the price of 
 the horse for the garden, and 5 times the price of the garden 
 for the house. What were their respective prices ? 
 
 Ans. horse £20. garden £80. house £400. 
 
 DOUBLE POSITION 
 
 Requires the use of two supposed numbers to find the true one 
 required* 
 
 Rule. Work with the two supposed numbers, and mark 
 the errors in the results with + or — > according as they 
 exceed^ or fail short of the true result : then place the errors 
 against their respective positions, and multiply them cross- 
 wise. 
 
 If the errors be of like kinds, i. e. both greater, or both less 
 than the given number, take their difference for a divisor, and 
 the difference of the products for a dividend. But if unlike, 
 take their sum for a divisor, and the sum of their products 
 for a dividend : the quotient will be the answer.t 
 
 * Questions belong to this Rule which require the addition or sub- 
 traction of a number, &c, which is not any known part of the number 
 equired. The results are, therefore, not proportional to their sup- 
 positions. 
 
 •f The following Rule will, in some cases, be found more eligible : 
 
 Multiply the difference of the supposed numbers by the less error ; 
 
 and divide the product by the difference of the errors when they are of 
 
124 position. Qtue TUToR s 
 
 (1) A, R, and C, would divide £200. among them, so 
 that B may have £6. more than A, and C £8. more than B. 
 How much must each have ? * 
 
 (2) A man had two silver cups of unequal weight, having 
 one cover to both of 5 ounces. Now if the cover is put on 
 the less cup, it will double the weight of the greater ; and 
 put on the greater cup, it will be thrice as heavy as the 
 less. What is the weight of each ? 
 
 Ans. S ounces the less, and 4 the greater. 
 
 (3) Three persons conversing about their ages; says K, 
 c * My age is equal to that of H, and ^ of L's ; and L says, 
 " I am as old as both of you together." Required the ages of 
 K and L ; H's being 30. Ans. K 50, and L 80. 
 
 (4) D, E, and F, playing at cards, staked 324 crowns ; but 
 disputing about the tricks, each man seized as many as he 
 could : E got 1 5 more than D ; and F got a fifth part of both 
 their sums added together. How many did each person get ? 
 
 Ans. D 127& E 142^, and F 54. 
 
 (5) A gentleman meeting with some ladies, said to them, 
 " Good morning to you, ten fair maids." " Sir, you mistake," 
 
 like kinds, or by their sum, when unlike : the quotient will be a co\ 
 rectional number ; which being added to the nearest supposition when de . 
 fective, or subtracted from it, when excessive, will give the number re- 
 quired. 
 
 £. £. 
 
 1st. Suppose A's share = 40 2nd. Suppose A's share = 70 
 
 then B's = 46 then B's = 76 
 
 and C's = 54 and C's = 84 
 
 Sum 140 
 
 
 Sum 230 
 
 Therefore the error is Here the 
 — 60, or 60 too little. much. 
 
 error is + 30. or 30 too 
 
 sup. err. 
 
 40 y. 60 
 
 70 * 30 
 
 A 
 B 
 C 
 
 £. 
 60 
 66 
 
 4200 1200 
 divisor. 120 ° 
 
 74 
 200 Proof. 
 
 60+30=9j0)540j0 dividend. 
 
 
 
 £60 -= A's share. 
 
 
 
 Or, by the Rule in the Note. 
 70 — 40 X 30 30 X 30 
 
 .. . ■- = — = 10. the c 
 
 orre 
 
 ctional number. 
 
 60 + 30 90 " 
 
 Then 70 — 10 = 60 = A's share, as before. 
 
ASSISTANT. J PROGRESSION. 125 
 
 answered one of them, " we are not ten : but if we were three 
 times as many as we are, we should be as many above ten as 
 we are now under." How many were they ? Ans. 5. 
 
 ARITHMETICAL PROGRESSION. 
 
 An Arithmetical Progression, is a series of numbers increasing 
 or decreasing uniformly by a continued equal difference. Thus, 
 
 ' J J ' j ' o C ' > are increasing Arithmetical Series. 
 
 P 1 9 8 A* n /i i are decreasing Arithmetical Series. 
 
 Observe, that the terms of the first series are formed by adding 
 successively the common difference 1, and the second by the common 
 difference 3. The terms of the third and the fourth diminish con- 
 tinually by the subtraction of 1 and 4 respectively. 
 
 In an odd number of terms, the double of the mean (or 
 middle term) is equal to the sum of the extremes, or of any 
 two terms equidistant from the mean. Thus, in 1, 2, 3, 4, 5, 
 the double of 3 = 1 + 5=2 + 4 = 6. 
 
 In an even number of terms, the sum of the two means is 
 ifqual to the sum of the extremes, or of any two equidistant 
 terms. Thus, in 2, 4, 6, 8, 10, 12; 6 + 8 = 2 + 12 = 4 
 + 10 = 14. 
 
 To give Theorems or Rules for the solution of the various 
 cases, the terms are represented by symbols, or letters. 
 Thus, let a denote the less extreme, or least term, 
 
 z the greater extreme, or greatest term, 
 
 d the common difference, 
 
 n the number of terms, and 
 
 s the sum of all the terms. 
 
 Any three being given, the others may be found. 
 Note. The twenty cases in this Rule may be resolved by the follow- 
 ing Theorems. 
 
 2s s 
 
 a=z—(n—l )d= 2= \d(n—l )==!</+ ^(^+ z ) *— 2ds. 
 
 2s 
 
 z=a+(n—l)d= «=- HK»— l)=*/^d—ay+2ds— i* 
 
 z—a__ s—an 2 _ nz—s 2 ___ (z+a) . (z— a) 
 
 ' n — i n . i n n — 1 n 2s ■ 
 
1*6 
 
 PROGRESSION. [THE TUTOR'S 
 
 2 — a 
 •> j 
 
 2 s id—a+J^d—ay+vds 
 
 "- d ' 
 
 a-\-z ~~ d 
 
 jd+z—j Qd+zy—Zds 
 d 
 
 a+z z — «+^ _, — 
 
 ^\n{ci+z)——T--' — ~2 — —in.2a+d(n—l)=±n.2z—d(n—l). 
 
 Moreover, when the least terra a—nothing, the Theorems be- 
 come, z=zd(?i — 1), and s=z±nz. 
 
 Case 1. The two extremes, and the number of terms being given, 
 to find the sum. 
 
 Rule. Multiply the sum of the extremes by the number 
 of terms, and half the product will be the answer.* 
 
 ( 1 ) How many strokes does the hammer of a clock strike 
 in 12 hours ?t 
 
 (2) A man bought 17 yards of cloth, and gave for the first 
 yard 2,?. and for the last 10^. What was the price of the 17 
 yards? Ans. £5..2. 
 
 (3) If 100 eggs be placed in a right line, exactly a yard 
 from each other, and the first a yard from a basket, how far 
 must a person travel to gather them all up singly, and return 
 with every egg to put it into the basket ? 
 
 Ans. 5 miles, 1300 yards. 
 
 Case 2. The same three terms given, to find the common 
 difference. 
 
 Rule. Divide the difference of the extremes by the number 
 of terms less 1 ; and the quotient will be the answer. 
 
 (4) A man had eight sons, whose ages were in arithmetical 
 progression ; the youngest being 4 years old, and the eldest 
 32. What was the common difference of their ages? J 
 
 (5) A man travelling from London to a certain place, 
 went 3 miles the first day, and increased every day by an 
 equal excess, making the twelfth day's journey 58 miles. 
 
 . 
 
 * The learner should find each of these cases among the preceding 
 Theorems. Thus, the present Rule will be found designated by 
 $ =z \n(a -\- z\ &c. 
 
 f 12+ 1 X 6 = 13 X 6 = 78. Ans. 
 
 £32— 4 -f- 8 — 1 = 28 — 7 = 4 years. Ans. 
 
ASSISTANT.] PROGRESSION. 127 
 
 What was the daily increase, and how far did he travel in 
 12 days? 
 Ans. 5 miles daily increase, the whole distance $66 miles. 
 
 Case 3. The two extremes and the common difference being 
 given, tojind the number of terms. 
 
 Rule. Divide the difference of the extremes by the common 
 difference, and the quotient increased by unity is the number 
 sought. 
 
 (6) A person travelling into the country, went 3 miles the 
 first day, and increased every day 5 miles, till at last he went 
 58 miles in one day. How many days did he travel? Ans. 12. 
 
 (7) A man being asked how many sons he had, said, that 
 the youngest was 4 years old, and the eldest 32 ; and that 
 his family had increased one in every 4 years. How many 
 had he ? Ans. 8. 
 
 Case 4. The greater extreme, the number of terms, and Ike 
 common difference being given, to find the less extreme. 
 
 Rule. Multiply the commcn difference by the number of 
 terms less 1 ; subtract the product from the greater extreme, 
 and the difference will be the less extreme. 
 
 (8) A man went from London to a certain town in the 
 country in 10 days ; every day's journey exceeding the former 
 by 4 miles, and the last being 46 miles : what was the first ? 
 
 Ans. 10 miles. 
 
 (9) A man took out of his pocket at 8 several times, so 
 many different numbers of shillings, every one exceeding the 
 former by 6, the last being 46 : what was the first ? Ans. 4. 
 
 Case 5. The common difference, the number of terms, and the 
 sum being given, to find the less extreme. 
 Rule. Divide the sum by the number of terms : from the 
 quotient subtract half the product of the common difference 
 into the number of terms less 1 ; and the remainder will be 
 the less extreme. 
 
 (10) A man is to receive £360. at 12 several payments, 
 each payment to exceed the former by £4. and is willing to 
 bestow the first payment on any one that can tell him what 
 it is. What will that person have for his pains ? Ans. £8. 
 
 Case 6. The less extreme, the common difference, and the number 
 of terms being given, tofnd the greater extreme. 
 
 Rule. Multiply the number of terms less 1 bv the com- 
 
128 PROGRESSION. QtHE TUTOR'S 
 
 mon difference; to this product add the less extreme, and 
 the sum will be the greater extreme. 
 
 (11) What is the last number of an arithmetical progres- 
 sion, beginning at 6, and continuing by the increase of 8 to 
 20 places? Ans. 158. 
 
 GEOMETRICAL PROGRESSION. 
 
 A Geometrical Progression is a series of numbers increasing 
 or decreasing uniformly by a common ratio ; that is, by the 
 continual multiplication or division of some particular num- 
 ber. Thus, 
 
 1, 2, 4, 8, 16, 32, &c. is an increasing Geometrical Series, 
 in which the terms are formed by multiplying successively 
 by the ratio 2. 
 
 81, 27, 9* 3 y 1, ^, &c. is a decreasing Geometrical Series, in 
 which the terms are formed by dividing successively by the 
 ratio 3. It is evident that either of these may be continued 
 without end. 
 
 In an odd number of terms, the square of the mean is equal 
 to the product of the extremes, or of any two terms equidistant 
 from the mean. Thus, in 3, 6, 12, 24, 48 ; 12 X 12 — 3 X 48 
 =6X24 = 144. 
 
 In an even number of terms, the product of the two means 
 is equal to the product of the extremes, or of any two equidis- 
 tant terms. Thus, in 32, 1 6, 8, 4, 2, 1 ; 8 X 4 = 32 X 1 = 
 16x2 = 32. 
 
 To give Theorems, or Rules expressed in symbols, for the 
 solution of the various cases, as in Arithmetical Progression, 
 let a denote the less extreme, 
 z the greater extreme, 
 
 r the ratio, 
 
 n the number of terms, and 
 
 s the su?n of all the terms. 
 
 Any three being given, the others may be found. 
 
 Note. The twenty Theorems following solve all the possible cases 
 in Geometrical Progression. 
 
 t« T n-i * * Log. z— log. a 
 
 iheor. I. r ±= — ; or, — ^ - \-l=m. 
 
 a Log. r 
 
 z 
 * In this case, if the quotient of — be divided continually by r, til! 
 
 nothing remains ; the number of divisions -j- 1 will give n. 
 
ASSISTANT.] PROGRESSION. 129 
 
 rT • rz — a z — a 
 
 II. — =zs; or, z-\ -~s 
 
 r — 1 r — 1 
 
 \"a"/ n ^ 1==r ' or > ^og. z — 1°§* a + n — l =Log. r. 
 
 z — a __ $ — a 
 IV. z+ —s. V. =r 
 
 / 2 \ _J_ *— 2 
 
 IT)"- 1 -! 
 
 VI. / s a \ n "" 1 -— 2 . from which n may be found as in 
 V s — z J a ' Theorem I. 
 
 L og- 2 — log. a n __! 
 
 or, = -r— r f r+l=w. VII. ar =zz. 
 
 Log. (s — a) — log. (s — z) 
 
 viii. *£=!>*,. ix. ^- i y+ a = u-j^=.. 
 
 r — 1 r r 
 
 „ (r—l)s Log. (r— l)s+a— log. a 
 
 X. r = hi; or, - z=m. 
 
 a Log. r 
 
 XI. zxs — z| =z«x* — «| n "" ,* whence 2 may be found by 
 
 ST n S a 
 
 Double Position. XII. r — ; whence r may be 
 
 a a 
 
 found in the same manner. 
 
 z z(r n — 1^ 
 XIII. --a. XIV. ^- if=» 
 
 XV. rz— (r— l)s=a. XVI. r^ss ~ ; or, 
 
 rz — (r — \y 
 
 Log. 2 — log. rz — (r — 1)* 
 Log. r 
 
 XVII. aXs=ST l =zx^r l . XVIII. r 
 
 r n — If 
 
 u 
 
 From the two preceding, a and r are to be found by Dou- 
 ble Position. 
 
 YTV (r—l)s r n__ r ii-l 
 
 XIX. —=za. XX. X* = 2. 
 
 r n — x r n — 1 
 
 In a Geometrical series decreasing ad infinitum, a becomes 
 2= 0, and 11 is infinite, or greater than any assignable number. 
 
 f5 
 
130 PROGRE&blON. [THE TUTOR'S 
 
 Hence the three following will exhibit all the various cases 
 of such a series. * 
 
 r rz , z z * tt i. < r — 1 ) TTT ,. s 
 \.s=z =:zA =: . Il.z= . lll.r= , 
 
 r — 1 r — 1 2 *• s — 2 
 
 2 
 
 r 
 Note. In these cases, when the ratio is a proper fraction, r must be 
 taken =z the reciprocal of the fraction. Thus, when the ratio is f , 
 
 r = 5- 
 
 Case 1 . The less extreme, the ratio, and the number of terms, 
 
 being given, to Jlnd the greater extreme (or any remote term J 
 
 without producing all the intermediate terms. 
 
 Rule. 1. When the least term is equal to the ratio. Write 
 down a few of the leading terms of the series and over them 
 the arithmetical series, 1, 2, 3, 4, &c. as indices or exponents. 
 Find which of the indices added together will give the index of 
 the term sought ; and the continual product of the terms 
 standing under those indices, will be the term sought. 
 
 2. When the least term is not equal to the ratio. Write 
 down the leading terms as before^ and over them the in- 
 dices, 0, 1, 2, 3, 4, &c. Examine which of these added toge- 
 ther will give an index one less than the number of the term 
 sought ; multiply the terms under such indices into each 
 other, dividing the product of every two by the first term, and 
 the last quotient will be the term required.* 
 
 Otherwise. By Theorem VII. 
 
 (1) A man agrees for 12 peaches, to pay only the price of 
 the last ; reckoning a farthing for the first, and a halfpenny 
 for the second, &c. doubling the price to the last. What must 
 he give for them ?t 
 
 (2) A farmer who went to a fair to buy some oxen, met 
 with a drover who had 23 ; for which he asked him £l6. a 
 piece. — After a great deal of dodging between the parties, it 
 was finally agreed that the farmer should pay the price of the 
 last ox only, reckoning a farthing for the first, and doubling 
 it to the last. How much would they cost him ? 
 
 Ans. £4369.. LA 
 
 * If the least term is unity, there will (of course) be no division. 
 
 THere a = 1, r = 2, and n = 12 ; a and r being unequal. 
 
 Indices, 0, 1, 2, 3, 4. t _, . , , ] 
 
 G*om. series, 1, t t *!> s) 16./ Then * + * + 3 = 1 1 - n -1. | 
 
 Hence 16X16X8 = 2048 qrs. = *. And 2048 qrs. = £2..2..8. Ans \ 
 
ASSIST PROGRESSION. 131 
 
 (3) A sum of money is to be divided among 8 persons, 
 the first to have £20. the second £60. and so on in triple pro- 
 portion. What will the last have ?* Ans. ,£43740. 
 
 (4) A gentleman dying left nine sons, to whom and to his 
 executors he bequeathed his estate in the manner following : 
 to his executors £50., to his youngest son twice as much as 
 to the executors, to the next, double that sum, and so on to 
 the eldest. What was his fortune ? Ans. £25600. 
 
 Case 2. The less extreme, the ratio, and the number of terms 
 being given, tojlnd the sum of all the terms. 
 Rule. Find the greater extreme as before, and divide the 
 difference between the extremes by the ratio less 1 : to the quo- 
 tient add the greater extreme, for the sum required. This is 
 Theorem II. Or, by Theorem VIII ; without finding z. 
 
 (5) A young man conversant with numbers, agreed with a 
 gentleman to serve him twelve months, provided he would 
 give him a farthing for his first month's service, a penny for 
 the second, and 4d. for the third, &c. What did his wages 
 amount to ?t Ans. £5825..8..5^. 
 
 (6) A man bought a horse, and by agreement was to give 
 a farthing for the first nail, three for the second, &c. Now 
 supposing there were 8 nails in each of his four shoes, what 
 was the price of the horse? Ans. £966 11 4681 693.. IS. A. 
 
 (7) A person whose daughter was married on new-year's 
 day, gave her husband 1.?. towards her portion; prom- 
 ising to double the sum on the first day of every month during 
 the year. What was her portion ? Ans. £204.. 15. 
 
 (8) A laceman, well versed in numbers, agreed with a 
 gentleman to sell him 22 yards of rich brocaded gold lace, 
 for 2 pins the first yard, 6 pins the second, &c. in triple pro- 
 portion. What was the price of the lace, valuing the pins 
 
 * Indices 0, 1 , 2, 3. ) m1 a , , . _ , 
 
 Gedm. series 20, 60, 180, 540. } Then 3 + S + 1 = 7 = « - 1. 
 
 Hence 540x540x60 —540x27x3=43740 = z. 
 
 20X20 
 Otherwise, ar n - 1 zi:20x3 7 =4374<0=z. 
 
 + Here a = 1 , r = 4, and z = 12. Therefore * am 
 J67T721S 5S9M05 4 -' 
 
 3 
 
[32 SIMPLE INTEREST. [THE TUTOR'S 
 
 at 100 for a farthing ? Also, what did the laceman gain, sup- 
 posing the lace to have cost him £7. per yard ? 
 
 Ans. The lace sold for £326886. .0..9. Gain £326732..0..p. 
 
 Case 3. The first term, and the ratio, being given, to find the 
 sum of an infinite decreasing series. 
 Rule. Divide the square of the first term by the difference 
 between the first and second.* 
 
 (9) What is the sum of the circulating decimal '9', or the 
 series T 9 S -f- too "T* to 9 oo"> & c * continued ad infinitum ? Ans. 1. 
 
 (10) Required the sum of the infinite series i+ J-h J 
 &c. ; also of the series J + g +$V* & c * Ans. 1, and \, 
 
 (11) Suppose a body to be put in motion by a force which 
 gives it a velocity of 10 miles the first minute (or any given, 
 space of time) 9 miles in the second equal space, and so on in 
 the ratio of T % ; how many miles would it pass over, if con- 
 tinued in motion for ever? Ans. 100 miles. 
 
 SIMPLE INTEREST, by decimals. 
 
 To give Theorems for the solution of the different cases in 
 Simple Interest, let p denote the principal, r the ratio, t the 
 time (in years) i the interest, and a the amount. 
 
 Note. The Ratio is the interest of £l. for one year, at the rate 
 per cent, proposed, and may be found by Proportion : thus, at £&. 
 per cent, per annum, sav, 
 
 As £100 t £3 1 1 £l s £-05, the ratio. 
 
 Therefore the ratio at 
 3 per cent, is . . . -03 |4£ per cent, is . . .-045 
 
 34 -035 \5 -05 
 
 * -04 '5i -055, &c. 
 
 Case 1 . When the principal, rate per cent, and time are given, to 
 find the interest. 
 
 Rule. Multiply the principal, ratio, and time together, 
 and the product will be the interest required. 
 That is, pit = L 
 
 (1) What is the interest of £945.. 10. for 3 years, at £i 
 per cent, per annum ?t 
 
 * See the third formula, Theorem I. for infinite series, page ]30. 
 + prt =£945-5X-05X3=£l41-825. = £141..16..6. Ans. 
 
ASSISTANT.] SIMPLE INTEREST. 1.33 
 
 (2) What is the interest of £547»14. at £4. per cent, per 
 annum, for 6 years ? Ans. £131..8..1 1..2-08 qrs. 
 
 (3) What is the interest of £796..15. at £*£. per cent, per 
 annum, for 5 years? Ans. £l79..5..4^. 
 
 (4) W T hat is the interest of £397..9»5- for 2± years, at £3\. 
 per cent, per annum? Ans, £34..15..6..3*55 qrs. 
 
 (5) What is the interest of £554.. 17.. 6. for 3 years, 8 
 months, at £4<|, per cent, per annum ? Ans. £91..11.A'05d. 
 
 (6) What is the interest of £236..18..8. for 3 years, 8 
 months, at £5t|. per cent, per annum ? 
 
 Ans. £47..15..7..2-293 qrs. 
 When the interest is for any number of days only. 
 - Rule. Multiply the interest of £l. for a day, at the given 
 rate, by the principal and the number of days ; and the pro- 
 duct will be the answer. 
 
 The Interest of £i. for one day, 
 
 at £4= -00010958904 
 41 =-00012328767 
 5 =-00013698630 
 5^ — -00015068493, &c. 
 
 At£2percent.= £-00005479452* 
 2£ = -00006849315 
 
 3 = -00008219178 
 
 S| = -00009589041 
 
 (7) What is the interest of £240. for 120 days, at £4. per 
 cent, per annum ? t 
 
 (8) What is the interest of £364..18. for 154 days, at £5. 
 per cent, per annum ? Ans. £7.. 13.. 11^. 
 
 (9) W T hat is the interest of £725..15. for 74 days, at £4. 
 per cent, per annum ? Ans. £5. 17.. 8^. 
 
 (10) What is the interest of £100. from the 1st of June, 
 1826, to the 9th of March following, at £5. per cent, per 
 annum? Ans. £3..l6..11|. 
 
 Case 2. When p, r, and / are given, to find a. 
 Rule, prt + p = a. 
 
 (11) What will £279-.l 2. amount to in 7 years, at £4|. 
 per cent, per annum ?$ 
 
 (12) What will £320..17- amount to in 5 years, at £34. 
 per cent, per annum? Ans. £376..19-.H..2-8 qrs. 
 
 * The table is formed thus : 
 
 As 365 days : £-02 : : 1 day : £-00005479452, &c. 
 
 t '00010958904 X 240 X 120 = £3-156164352= £3..3..1|. Ans. 
 
 t 279-6 X -045 X 7 + 2796 = £367-674 = £367..13..5..3'04 qrs. 
 
134 SIMPLE INTEREST. [THE TUTOR'S 
 
 (13) What will £926..12. amount to in 5^ years, at £4. 
 per cent, per annum ? Ans. £l 130..9..0..1'92 qrs. 
 
 (14) What will £273..18. amount to in 4 years, 175 days, 
 at £3. per cent, per annum ? Ans. £310..14..1..3*3512 qrs. 
 
 Case 3. When a, r, and / are given, to find p. 
 
 (15) What principal, being put to interest, will amount to 
 £367..13..5..3*Q4 qrs. in7 years, at £4^. per cent, per annum ?* 
 
 (1 6) What principal will amount to £376..19..11..2'8 qrs. 
 in 5 years, at £3^. per cent, per annum ? Ans. £320.. 17. 
 
 (17) What principal will amount to £ll30..9..0..1*92 qrs, 
 in 5^ years, at £4. percent, per annum? Ans. £926.. 12. 
 
 (18) What principal will amount to £310..14..1..3*3512 
 qrs. in 4 years, 175 days, at £3. per cent, per annum ? 
 
 Ans. £273..18. 
 
 Case 4. When a, p, and / are given, to find r. 
 
 a — p 
 
 Rule. = r. 
 
 pt 
 
 (19) At what rate per cent, per annum will £279-12. 
 amount to £367..13..5..3-04 qrs. in 7 years ?t 
 
 (20) At what rate per cent, per annum will £320.. 1 7. 
 amount to £376..\9..ll..2 m 8 qrs. in 5 years? 
 
 Ans. £3^. per cent. 
 
 (21) At what rate per cent, per annum will £926.. 12. 
 amount to £U30..9"0..l'9 c Z qrs. in 5| years? 
 
 Ans. £4. per cent. 
 
 (22) At what rate per cent, per annum will £273.. 18. 
 amount to £310..14,.l.. 3*3512 qrs. in 4 years, 175 days? 
 
 Ans. £3. per cent. 
 Case 5. When a, p, and r are given, to find t 
 
 ■ a — p 
 
 Rule. — s= U 
 
 pr 
 
 • -045 X 7 -f 1 = 1-315; then 367*674 -f- 1-315 = £279-6 = 
 £27 9.. 12. Ans. 
 
 t ^±=**>* = ™™L _ . 045> or m _ ceDt Ani _ 
 
 *79-fi X 7 1957-2 
 
ASSISTANT.] SIMPLE INTEREST. 135 
 
 (23) In what time will £279-12. amount to £367..13..5.. 
 3*04 qrs. at £4^. per cent, per annum ?* 
 
 (24) In what time will £320..17- amount to £376.. 19.. 11.. 
 2*8 qrs. at £3^. per cent, per annum ? Arts. 5 years. 
 
 (25) In what time will £926.. 12. amount to £ll30..9..0.. 
 1 *92 qrs. at £4 per cent, per annum ? Ans. $\ years. 
 
 (26) In what time will £273.. 18. amount to £310..14..1.. 
 3*3512 qrs. at £3. per cent per annum ? 
 
 Ans. 4i years, 175 days. 
 
 ANNUITIES. 
 
 An annuity is a yearly income or rent. Perpetual Annuities 
 are those which are to continue for ever ; Terminable Annui- 
 ties are to cease within a limited time ; and Life Annuities 
 are to continue during the term of life of one or more per- 
 sons. 
 
 The Amount of Annuities in Arrears. 
 
 Let u denote the annuity, r, t, and a, as before. 
 Case 1. Given, u, r, and t, to find a. 
 
 . (L hS + 1 ) X *« = a. 
 
 Rule 
 
 (27) If a salary of £ 150. be forborne 5 years, at £5. per 
 cent, per annum, what will be the amount ?t 
 
 (28) If £250. yearly pension be forborne 7 years, what 
 will it amount to at £4. per cent, per annum? Ans. £1960. 
 
 (29) There is a house let upon lease for 5^ years, at £60. 
 per annum, what will be the accumulated rent, allowing in- 
 terest at £4^. per cent, per annum? Ans. £363..8..3. 
 
 (30) Suppose an annual pension of £28. remain unpaid 
 for 8 years, what would it amount to at £5. per cent, per 
 annum ? Ans. £263..4. 
 
 Note. When the annuity is payable half-yearly, or quarterly, « 
 will denote a single payment, r, the interest of £l. for that interval 
 of time, and t, the number of payments. 
 
 (31) If a salary of £ 150. payable every half-year, remain 
 unpaid for 5 years, what will it amount to in that time, al- 
 lowing interest at £5. per cent, per annum ? Ans. £834..7..6. 
 
 • 367-674 — 279-6 = 88-074, and 279-6 X -045 = 12-582; then 
 8S-074 ~ 12-582 = 7 years, Ans. 
 
 + / 4X -05 
 
 ( 2 + 1) X 5 X 150 =(2 X -05 + 1)X 5 X ' «> - * X 
 
 5 X 150= £825. Ang^ 
 
)36 SIMPLE INTEREST. [jTHE TUTOR'S 
 
 (32) If a salary of £ 150. payable every quarter, were left 
 unpaid for 5 years, what would it amount to in that time at 
 £5. per cent, per amium ? Ans. £83g.. 1.. 3. 
 
 Note. It may be observed by comparing the results of the 27th 
 31st, and 32nd examples, that half-yearly payments are more advan- 
 tageous than yearly, and quarterly more than half-yearly. 
 
 Case 2. When a, r, and t are given, to find u. 
 
 2 a 
 
 Rule. - — ; r — = u. 
 
 (r.(t—l) + 2)xt 
 
 (33) If a salary amounted to £825. in 5 years, at £5. per 
 cent, per annum, what was the salary ?* 
 
 (34) If a house has been let upon lease for 5^ years, and 
 the amount in that time is £363..8..3. at £4^. per cent, per 
 annum, what is the yearly rent ? Ans. £60. 
 
 (35) If a pension amounted to £ig60. in 7 years, at £4. 
 percent, per annum, what was the pension? Ans. £250. 
 
 (36) Suppose the amount of a pension was £263.. 4. in 8 
 years, at £5. per cent, per annum, what was the pension ? 
 
 Ans. £28. 
 
 (37) If the amount of a salary, payable half-yearly, be 
 £834..7..6. in 5 years, at £5. per cent, per annum ; what is 
 the salary per year ?t Ans. £150. 
 
 (38) If the amount of an annuity, payable quarterly, was 
 £839..1..3. in 5 years, at £5. per cent, per annum ; what was 
 the annuity ? Ans. £ 1 5 0. 
 
 Case 3. When u, a, and t are given, to find r. 
 
 (a — tit) X 2 
 
 Rule. —, ~ = r. 
 
 (t-l)xuf 
 
 (3[)) If a salary of £150. per annum amounts to £825. in 
 5 years, what is the rate per cent. ?J 
 
 (40) If a house has been let upon lease for 5i years, at 
 £60. per annum, at what rate per cent, would it amount to 
 £363..8..3 ? Ans. £4|. per cent. 
 
 825 X 2 1650 1650 
 
 =£150. Ans. 
 
 (•05X4 + 2) X 5 2-2X5 11 
 
 f See Note, p. 135. 
 
 X (825 — . 150 X 5) X 2 __ 825 — . 750 _ 75 1 
 
 4~X~150 X~5 ~2~X150X5 ~"l5o"x~To "~ 20 ~~ ' 0S 
 
 = r .• therefore the rate is £5. per cent. 
 
SSiSTAN'l.] SIMPLE INTEREST. 13? 
 
 (41) If a pension of £250. per annum amounts to £1960. 
 in 7 years, what is the rate per cent. ? Ans. £4. per cent. 
 
 (42) Suppose the amount of a yearly pension of £28. be 
 £263..4. in 8 years, what is the rate per cent, per annum ? 
 
 Ans. £5. per cent. 
 
 (43) If a salary of £ 150. per annum, payable half-yearly, 
 amount to £834..7..6. in 5 years, what is the rate per cent. ?* 
 
 Ans. £5. per cent. 
 
 (44) If an annuity of £ 150. per annum, payable quarterly, 
 amount to £839.. 1.. 3. in 5 years, what is the rate per cent. ? 
 
 Ans. £5. per cent. 
 Case 4. When a, a, and r are given, to find t. 
 
 J 8r-+(2-r) 2 — (2— r) 
 _ u 
 
 R ULE, — t. 
 
 2r 
 
 1 2 a 
 
 Or, put i = m ; then J ( l m z\ — m — /. 
 
 r \ ur ' 
 
 (45) In what time will a salary of £150. per annum, 
 amount to £825. at £5. per cent. ?+ 
 
 (46) If a house is let upon lease at £60. per annum, till it 
 amount to £36*3..8..3. at £4^. per cent, per annum, for what 
 term of years was it let ? Ans. 5\ years. 
 
 (47) If a pension of £250. per annum having been for- 
 borne a certain time, amount to £1960. at £4. per cent, how 
 long has been the time of forbearance ? Ans. 7 years. 
 
 (48) In what time will a yearly pension of £28. amount to 
 £263..4. at £5. per cent, per annum ? Ans. 8 years. 
 
 (49) If an annuity of £150. per annum, payable half- 
 yearly, amounted to £834..7.6. at £5. per cent, what time 
 was the payment forborne.* Ans. 5 years. 
 
 (50) If a yearly pension of £150. payable quarterly, 
 amounts to £S39..l*.S. at £5. per cent, per annum, what has 
 been the time of forbearance? Ans. 5 years. 
 
 * See Note, p. 135. 
 
 f J 8 X -05 X -— + (2 — '05)2 — (2 — -05) 
 2 X '05 
 
138 
 
 SIMPLE INTEREST. 
 
 [the tutor's 
 
 A Table by which the Interest of any sum from £1, to £30000. may 
 be easily computed, for any number of days, at any rate per cent. 
 
 No. 
 
 £7 
 
 s. 
 
 w 
 
 qrs. 
 
 | No. 
 
 s. 
 
 d. 
 
 qrs. 
 
 \No. 
 
 qrs. 
 
 •30000 
 
 82 
 
 3 
 
 10 
 
 0-11 
 
 200 
 
 10 
 
 11 
 
 2-03 
 
 1 
 
 2-63 
 
 20000 
 
 54 
 
 15 
 
 10 
 
 2-74 
 
 100 
 
 5 
 
 5 
 
 3-01 
 
 0-9 
 
 2-37 
 
 10000 
 
 27 
 
 7 
 
 11 
 
 1-37 
 
 90 
 
 4 
 
 11 
 
 0-71 
 
 0-8 
 
 2-10 
 
 9000 
 
 24 
 
 13 
 
 1 
 
 3-23 
 
 80 
 
 4 
 
 4 
 
 2-41 
 
 0-7 
 
 1-84 
 
 8000 
 
 21 
 
 18 
 
 4 
 
 1-10 
 
 ' 70 
 
 3 
 
 10 
 
 0-11 
 
 0-6 
 
 1-58 
 
 7000 
 
 19 
 
 3 
 
 6 
 
 2-96 
 
 60 
 
 3 
 
 3 
 
 1-81 
 
 0-5 
 
 1-32 
 
 6000 
 
 16 
 
 8 
 
 9 
 
 0-82 
 
 50 
 
 2 
 
 8 
 
 3-51 
 
 0-4 
 
 1-05 
 
 5000 
 
 13 
 
 13 
 
 11 
 
 2-68 
 
 40 
 
 2 
 
 2 
 
 1-21 
 
 0*3 
 
 0-79 
 
 4000 
 
 10 
 
 19 
 
 2 
 
 0-55 
 
 30 
 
 1 
 
 7 
 
 2-90 
 
 02 
 
 0-53 
 
 3000 
 
 8 
 
 4 
 
 4 
 
 2-41 
 
 20 
 
 1 
 
 1 
 
 0-60 
 
 0-1 
 
 0-26 
 
 2000 
 
 5 
 
 9 
 
 7 
 
 0-27 
 
 10 
 
 
 
 6 
 
 2-30 
 
 0-09 
 
 0-24 
 
 1000 
 
 2 
 
 14 
 
 9 
 
 2-14 
 
 9 
 
 
 
 5 
 
 3-67 
 
 0-08 
 
 0-21 
 
 900 
 
 o 
 
 9 
 
 3 
 
 3-12 
 
 8 
 
 
 
 5 
 
 1-04 
 
 0-07 
 
 0-18 
 
 800 
 
 2 
 
 3 
 
 10 
 
 0-11 
 
 7 
 
 
 
 4 
 
 2-41 
 
 0-06 
 
 0-16 
 
 700 
 
 1 
 
 18 
 
 4 
 
 1-10 
 
 6 
 
 
 
 3 
 
 3-78 
 
 0-05 
 
 0-13 
 
 600 
 
 1 
 
 12 
 
 10 
 
 2-08 
 
 5 
 
 
 
 3 
 
 VIS 
 
 0-04 
 
 0-11 
 
 500 
 
 1 
 
 7 
 
 4 
 
 3-07 
 
 4 
 
 
 
 2 
 
 2-52 
 
 0-03 
 
 0-08 
 
 400 
 
 1 
 
 1 
 
 11 
 
 0-05 
 
 3 
 
 
 
 1 
 
 3-89 
 
 0-02 
 
 0-05 
 
 300 
 
 
 
 IG 
 
 5 
 
 1-04 
 
 2 
 
 
 
 1 
 
 1-26 
 
 0-01 
 
 0-03 
 
 The above Table is thus constructed: as 365 days : £l. : : 1 daj 
 : 2-63 qrs. &c. Hence it appears that the several tabular sums are 
 those which answer to the respective numbers of days, at the rate / 
 £l. per year. 
 
 In a similar Table in Reefs Cyclopcedia, there are no fewer than 16 
 errors. In Dr. Hutton's Table, Arith. page 84, 1 2th edition, there 
 is one error. The above may be depended on as accurate. 
 
 Rule. Multiply the principal by the rate, both in pounds ; 
 and the product by the number of days : divide the last pro- 
 duct by 100, collect from the Table the several sums answer- 
 ing to the several parts of the quotient, and the aggregate 
 amount will be the Interest required. 
 
 Example 1. What is the interest of £370..10. for 220 days, 
 at <£4|. per cent, per annum ? £. s. d. (jrs. 
 
 370'% 
 4-5 
 
 7s~525~ 
 14820 
 
 1667*25 
 220 
 
 5334500 
 333 450 
 
 3667J95-00 
 
 Against 3000 stands 8 
 
 600 1 
 
 60 
 
 7 
 
 0*9 
 
 0*05.... 
 
 3667-957... "To" 
 
 4 2-41 
 
 10 2-08 
 
 3 1-81 
 
 4 2*41 
 2-3? 
 0-13 
 
 "II "Ml Ans. 
 
 True to the last decimal. 
 
ASSISTANT.] SIMPLE INTEREST. 139 
 
 Example 2. Taking Ex. 8. page 133, we have 
 364-9 
 
 5 £. s. d. qrs. 
 
 TgoZT Against 2000 stands 5 9 7 0*27 
 
 800 2 3 10 0-11 
 
 9 5 3-67 
 
 7^980 .y 1-84 
 
 273675 0-03.... 
 
 0-08 
 
 2 
 
 809173'Q 2809-73 7 13 11 1'97 An*. 
 
 Example 3. What is the interest of £17.. 10. for 117 days 
 at £4f . per cent, per annum? 
 
 IT'5 
 
 4*75 <£. s. d. qrs. 
 
 ~^T Against 90 stands 4 11 0*71 
 
 122 5 7 4 2-41 
 
 700 0-2 0-53 
 
 J 0-05 0*13 
 
 83*125 
 
 117 97-25 5 3 378 An*. 
 
 581875 
 9143 75 
 
 97|25-625 
 
 To find the amount of a yearly income or salary, fyc. for a 
 number of days. 
 
 Multiply the number of pounds per year by the number 
 of days ; collect the Tabular sums answering to the product, 
 as before, and their aggregate will be the answer. 
 
 Example. What will a person receive for 45 days, at the 
 rate of £ 105. per annum? 
 
 105 £. s. d. qrs. 
 
 45 Against 4000 stands 10 19 2 0*55 
 
 -505 700 1 18 4 1-10 
 
 420 20 1 1 0-60 
 
 5 3 1-15 
 
 4795 
 
 J — 4725 12 18 10 8-40 Ans. 
 
 Note. Any of the preceding examples of interest for days, in page 
 133, or examples 20 and 21, page 70, may be worked by this method. 
 
140 DISCOUNT. [THE TUTOR'S 
 
 A Table showing the number of days from any day in. the month 
 to the same day in any other month, through the year. 
 
 To 
 
 s 
 
 ctf 
 *5 
 
 .fa. 
 
 n 
 
 .a' 
 
 Ok- 
 
 < 
 
 a. 
 
 0) 
 
 d 
 
 ■■9 
 
 
 be 
 
 ex, 
 
 J/J 
 
 
 > 
 
 
 
 i 
 
 
 "Jan. 
 
 365 31 
 
 59 
 
 90 
 
 120 
 
 1.51 
 
 181 
 
 212 
 
 243 
 
 273 
 
 304 
 
 334 
 
 
 Feb. 
 
 334 
 
 365 
 
 28 
 
 59 
 
 89 
 
 120 
 
 150 
 
 181 
 
 212 
 
 242 
 
 273 
 
 303 
 
 
 Mar. 
 
 306 
 
 337 
 
 365 
 
 31 
 
 61 
 
 92 
 
 122 
 
 153 
 
 184 
 
 214 
 
 245 
 
 275 
 
 
 Apr. 
 
 275 
 
 306 
 
 334 
 
 365 
 
 30 
 
 61 
 
 91 
 
 122 
 
 153 
 
 183 
 
 214 
 
 244 
 
 
 May- 
 
 245 
 
 276 
 
 304 
 
 335 
 
 365 
 
 31 
 
 61 
 
 92 
 
 123 
 
 153 
 
 184 
 
 214 
 
 i\ 
 
 June 
 
 214 
 
 245 
 
 273 
 
 304 
 
 334 
 
 365 
 
 30 
 
 61 
 
 92 
 
 122 
 
 153 
 
 183 
 
 8 
 
 July 
 
 184 
 
 215 
 
 243 
 
 274 
 
 304 
 
 335 
 
 365 
 
 31 
 
 62 
 
 92 
 
 123 
 
 153 
 
 
 Aug. 
 
 153 
 
 184 
 
 212 
 
 243 
 
 273 
 
 304 
 
 334 
 
 365 
 
 31 
 
 61 
 
 92 
 
 122 
 
 
 Sept. 
 
 122 
 
 153 
 
 181 
 
 212 
 
 242 
 
 273 
 
 303 
 
 334 
 
 365 
 
 30 
 
 61 
 
 91 
 
 
 Oct. 
 
 92 
 
 123 
 
 151 
 
 182 
 
 212 
 
 243 
 
 273 
 
 304 
 
 335 
 
 365 
 
 31 
 
 61 
 
 
 Nov. 
 
 61 
 
 92 
 
 120 
 
 151 
 
 181 
 
 212:242 
 
 273 
 
 304 
 
 334 
 
 365 
 
 30 
 
 
 [Dec. 
 
 31 
 
 62 
 
 90 
 
 121 
 
 151 
 
 182J212 
 
 243 
 
 274 
 
 304 
 
 335 
 
 365 
 
 DISCOUNT. 
 
 Let s represent the sum to be discounted, r the ratio, t the 
 time (in years) and 4? the present worth. 
 
 Case 1. Given s, r, and t, to find p. 
 
 Rule. 
 
 rt + 1 
 
 = p. 
 
 (1) What is the present worth of £357-10. to be paid c 
 months hence, at £5. per cent, per annum?* 
 
 (2) What is the present worth of £275..10. due 7 months 
 hence, at £5. percent. per annum? Ans. £267..13..10*152d. 
 
 (3) What is the present worth of £875.. 5..6. due 5 months 
 hence, at £4^. per cent, per annum ? 
 
 Ans. £859..3..3..3-01824 qrs. 
 
 (4) How much ready money can I receive for a note of 
 £75. due 15 months hence, at £5. per cent, per annum ? 
 
 Ans. £70.. 11. .9-1 76U. 
 
 S57-5 -*- '05 X -75 -f 1 = 344-5783 =, £344..11..6..3-1(J8 qrs. Ans. 
 
ASSISTANT.] DISCOUNT. 141 
 
 Case 2. When p, r, and t are given, to find s. 
 Rulk. prt -\- p =s .v. 
 
 (5) If the present worth of a sum of money due 9 months 
 hence, allowing £5. per cent, per annum, be £344.. 11. .6.. 
 3*1 68 qrs. what was the sum due?* 
 
 (6) A person owing a certain sum, payable 7 months hence, 
 agrees with the creditor to pay him down £267..13..10*152d. 
 allowing £5. per cent, per annum, for present payment : 
 what is the debt ? Ans. £275.. 10. 
 
 (7) A person receives £859»3..3.. 3-01 824 qrs. for a sum 
 of money due 5 months hence, allowing the debtor £4^. per 
 cent, per annum, for present payment : what was the sum 
 due? Ans. £87 5..5.S. 
 
 (8) A person paid £70..! 1.. 9*1 764d. for a debt due 15 
 months hence, being allowed £5. per cent, per annum, for 
 the discount. How much was the debt ? Ans. £75. 
 
 Case 3. When s, p, and t are given, to find r. 
 
 s — p 
 Rule. 
 
 pt 
 
 (9) At what rate per cent, per annum, will £357.. 10. 
 payable 9 months hence, produce £344..11..6..3*l68 qrs. for 
 present payment ?f 
 
 (10) At what rate per cent, per annum, will £275. .10. 
 payable 7 months hence, be worth £267..13..10*i52d. for 
 present payment ? Ans. £.5. per cent. 
 
 (1 1) At what rate per cent, per annum, will £875..5..6\ 
 payable 5 months hence, produce the present payment of 
 £859..3..3..3-01824 qrs. f Ans. £4£. per cent. 
 
 (12) At what rate per cent, per annum, will £75. pay- 
 able 15 months hence, produce the present payment of 
 £70..1 1..9*1764c?. Ans. £5. per cent. 
 
 Case 4. When s, p, and r are given, to find /. 
 
 Rule. ±HL = t. 
 pr 
 
 9 344-5783 X -05 X -75 + 344-5783 == £357..10. Am. 
 
 f 357-5 — 344-5783 At „. _■' 
 
 1 = -05 or £5. per cent. Ans. 
 
 344-5733 X -75 
 
142 EQUATION OF PAYMENTS. £THE TUTOR'S 
 
 (13) The present worth of £357-10. due at a certain time 
 to come, is £344..11..6..3*l68gr*. at £5. per cent, per annum : 
 in what time should the sum have been paid without any 
 discount ?* 
 
 (14) The present worth of £275.. 10. due at a certain time 
 to come, is £%67-.l3.A0'152d. at £5. per cent, per annum. 
 In what time should the sum have been paid without dis- 
 count? Ans. 7 months, 
 
 (15) A person receives £859..3..3..3'01824 qrs. for £875,. 
 5..6. due at a certain time to come, allowing £4^. per cent, 
 per annum discount. In what time should the debt have 
 been discharged without any discount ? Ans. 5 months. 
 
 (16) I have received £70..11..9*1764d. for a debt of £75. 
 allowing the person £5. per cent, per annum, for prompt 
 payment. When would the debt have been payable without 
 discount? Ans. 15 months. 
 
 EQUATION OF PAYMENTS. 
 
 To find the equated time for the payment of a sum of money due 
 at several times. 
 
 Rule. Find the present worth of each 
 payment for its respective time, by Case 1. s = p. 
 
 Discount, page 140, thus : rt -J- 1 
 
 Add all the present worths together; s f — p' 
 call their sum //, and the sum of all the ? r ~ ~ e > fye 
 payments/: then by Case 4. Discount, F equated time. 
 p. 141.t 
 
 » 357-5 — 344-5783 _ r n .. 
 
 = '15 — 9 months. Ans. 
 
 344-5783 X -05 
 
 f The above is Kersey's Rule. It produces a result something less 
 than the precise truth ; but sufficiently accurate for anv purpose of real 
 utility. The only Rule that is strictly true for the equation of two pay- 
 ments at Simple Interest , is that given by Malcokn ; which is founded on 
 the principle, that the interest of the money withheld after it becomes 
 due, ought to be equal to the discount of that which it paid before it is 
 due. But Malcolm's Rule, though it has been simplified in expression 
 by Bonnycastle and others, and is capable of further simplification 
 than I have yet seen in print, is at best very operose, and may be re- 
 garded as Mr. Keith iustly ooserves, as a useless curiosity. Editor. 
 
ASSISTANT.] COMPOUND INTEREST. 143 
 
 1 ) D owes E £200. whereof £40. is to be paid at three 
 months, £60. at 6 months, and £100. at 9 months: at what 
 rime may the whole debt be paid together, discount being 
 allowed at £5. per cent, per annum ?* 
 
 (2) D owes E £800. whereof £200. is to be paid in 3 
 months, £200. at 4 months, and £400. at 6 months : but 
 they agree to have the whole paid at once, allowing discount 
 at the rate of £5. per cent, per annum. The equated time 
 is required. Ans. 4 months, 22 days. 
 
 (3) E owes F £1200. which is to be paid as follows: 
 £200. down, £500. at the end often months, and the rest at 
 the end of 20 months ; but they agree to have only one pay- 
 ment of the whole, discounting at £.3. per cent, per annum. 
 The equated time is required. Ans. 1 year, 1 1 days. 
 
 COMPOUND INTERESTS 
 
 The same symbols are adopted in this as in Simple Interest, 
 and denote the same things ; except that the ratio (r,) which 
 in Simple Interest denotes the interest of £i., signifies in this 
 Hule the amount of £l. for a year. It may be thus found by 
 Proportion. 
 
 As £100 : £105 : : £l : £l*05, thereat £5 per cent 
 per annum. The ratios are therefore, 
 
 at 3 per cent 1*03 
 
 Si V03 
 
 4 1-04 
 
 at 4^ per cent 1*045 
 
 5 1-05 
 
 54 1-055, &c 
 
 • 40 60 100 
 
 — — - = 39-5061 ; _^__ = 58-5365 ; f * 96-3855 ; 
 
 1-0125 1*025 1*0375 
 
 then 200 — 39-5061 + 58-5365 -f- 96-3855 == 5-5719 ; 
 ,„j 5-5719 
 
 ana — = .57315 = 6 months, 26 days. Ans. 
 
 194-4281 X -05 ' " 
 
 *j* The law of England does not allow the lender to receive Com- 
 pound Interest for his money, when the receipt of the Interest has 
 been forborne. But in the granting or purchasing of Annuities, 
 Leases, &c. either immediate or in reversion, it is customary, and 
 indeed necessary, to compute them on the principles of Compound 
 Interest ; for otherwise the calculation would involve most egregious 
 injustice and absurdity. 
 
144 COMPOUND INTEREST. [THE TUTOR'S 
 
 A Table of Ike Amount of £l. for years. 
 
 yrs. 
 
 3 per Cent. 
 
 3% per Cent. 
 
 4 per Cent. 
 
 4^ per Cent. 
 
 5 per Cent' 
 
 1-0300000 
 
 1-0350000 
 
 1-0400000 
 
 1-0450000 
 
 1-0500000 
 
 2 
 
 1-0609000 
 
 1-0712250 
 
 1-0816000 
 
 1-0920250 
 
 1-1025000 
 
 3 
 
 1-0927270 
 
 1-108/179 
 
 1-1248640 
 
 1-1411661 
 
 1-1576250 
 
 4 
 
 M 255088 
 
 1-1475230 
 
 1-1698586 
 
 1-1925186 
 
 1-2155062 
 
 5 
 
 6" 
 
 1-1592741 
 1-1940523 
 
 1-1876863 
 
 1-2166529 
 
 1-2461819 
 
 1-2762810 
 
 1 •2292553 
 
 i -2653190 
 
 1 -3022601 
 
 1-3400956 
 
 7 
 
 1-2298739 
 
 1-2722793 
 
 1-3159318 
 
 1-3608618 
 
 1-4071004 
 
 8 
 
 1-2667701 
 
 1-3168090 
 
 1-3685690 
 
 1-4221006 
 
 1 4774554 
 
 9 
 
 1-3047732 
 
 1-3628974 
 
 1-4233118 
 
 1-4860951 
 
 1-5513282 
 
 10 
 
 1-3439164 
 
 1-4105988 
 
 X -4802443 
 
 1-5529694 
 
 1-6288946 
 
 if 
 
 1-3842339 
 
 1-4599697 
 
 1-5394541 
 
 1-6228530 
 
 1-7103394 
 
 12 
 
 1-4257609 
 
 1-5110687 
 
 1-6010322 
 
 1-6958814 
 
 1-7958563 
 
 >3 
 
 1-4685337 
 
 1-5639561 
 
 1-6650735 
 
 1-7721961 
 
 1-8856491 
 
 14 
 
 1-5125897 
 
 1-6186945 
 
 1-7316764 
 
 1-8519449 
 
 1-9799316 
 
 15 
 
 1-5579674 
 
 1-6753488 
 
 1-8009435 
 
 1-9352824 
 
 2-0789282 
 
 If? 
 
 1 -6047064" 
 
 1*7339860" 
 
 1-8729812 
 
 20223702 
 
 2-1828746 
 
 17 
 
 1-6528476 
 
 1-7946756 
 
 1-9479005 
 
 2-1133768 
 
 2-2920183 
 
 18 
 
 1-7024331 
 
 1-8574892 
 
 2-0258165 
 
 2-2084788 
 
 2-4066192 
 
 19 
 
 1-7535060 
 
 1-9225013 
 
 2-1068492 
 
 2-3078603 
 
 2-m&m , 
 
 20 
 21* 
 
 1-8061112 
 
 1-9897889 
 
 2-1911231 
 
 2-4117140 
 
 2-6532977 
 
 1-8602946 
 
 2-0594315 
 
 2-2787681 
 
 2-5202412 
 
 2-7859626 I 
 
 22 
 
 1-9161034 
 
 21315116 
 
 2-3699188 
 
 2-6336520 
 
 2-9252607 
 
 23 
 
 1-9735865 
 
 2-2061145 
 
 2-4647155 
 
 2-7521663 
 
 3-0715237 
 
 24 
 
 2 0327941 
 
 2-2833285 
 
 2-5633042 
 
 2-8760138 
 
 3-2250993 
 
 25 
 
 2-0937779 
 
 2-3632450 
 
 2-6658363 
 
 3-0054345 
 
 3-3863549 
 
 26 
 
 21565913 
 
 2-4459586 
 
 2-7724698 
 
 3-1406790 
 
 3-5556727 
 
 27 
 
 2-2212890 
 
 2-5315671 
 
 2-8833686 
 
 3-2820096 
 
 3-7334563 
 
 28 
 
 2-2879277 
 
 26201720 
 
 2-9987033 
 
 3-4297000 
 
 3-9201291 
 
 29 
 
 2-3565655 
 
 2-7118780 
 
 3-1186514 
 
 3-5840365 
 
 4-1161356 
 
 30 
 31 
 
 2-4272625 
 2-50008~03~ 
 
 2-8067937 
 
 3-2433975 
 
 3-7453181 
 
 4-3219424 
 
 2-9050315 
 
 3-3731334 
 
 3-9138574 
 
 "1T5380395" 
 
 32 
 
 2-5750827 
 
 30067076 
 
 3-5080587 
 
 4-0899810 
 
 4-7649414 
 
 33 
 
 2-6523352 
 
 3-1119423 
 
 3-6483811 
 
 4-2740302 
 
 5-0031885 
 
 34 
 
 2-7319053 
 
 3-2208603 
 
 3-7943163 
 
 4-4663615 
 
 5-2533479 
 
 35 
 
 2-8138624 
 
 3-3335904 
 
 3-9460890 
 
 4-6673478 
 
 5-5160153 
 
 36 
 
 2-8982783 
 
 3-4502661 
 
 4-1039325 
 
 4-8773785 
 
 5-7918161 
 
 37 
 
 2-9852266 
 
 3-5710254 
 
 4-2680898 
 
 5-0968605 
 
 6-0814069 
 
 38 
 
 3-0747834 
 
 3-6960113 
 
 4-4388134 
 
 5-3262192 
 
 6-3854773 
 
 39 
 
 3-1670269 
 
 3-8253717 
 
 4-6163660 
 
 5-5658991 
 
 6-7047511 
 
 40 3-2620378 3-9592597 4-8010206 
 
 5-8163645 
 
 7-0399887 
 
 These tabular numbers are the successive powers of r; thus, 
 1-05 2 = 1-1025, &c* 
 
 * The amount of £1. in t years, is the last term of an increasing geo- 
 metrical series, of which thejirst term = the ratio, and the number of 
 terms ~ t ; because the first year's amount is identical with the ratio : ami 
 as 1 : r : : r : r 2 = the amount in 2 years ; as 1 : r : : r 2 : : r 3 = the 
 amount in 3 years, &c. The successive amounts, r, r 2 , r s , &c. are 
 evidently in geometrical progression, and the amount in t years J? 
 
ASSISTANT.] COMPOUND INTEREST. 145 
 
 Case 1. When p, r, and / are given, to find a. 
 Rule. pr % = a. Or, log. rx< + l°g» V — 1°&- a * 
 
 Or by the Table. Multiply the tabular amount of £1. by the princi- 
 pal, and the product will be the amount required. 
 
 (1) What will £225. amount to in 3 years, at £5. per 
 cent, per annum ?* 
 
 (2) What will £200. amount to in 4 years, at £5. per 
 cent, per annum ? Ans. £243..2..0..1*2 qrs. 
 
 (3) What will £450. amount to in 5 years, at £4. per cent, 
 per annum? Ans. £547.-9.. 10..2-0538368 qrs. 
 
 (4) What will £500. amount to in 4 years, at £4^. per 
 cent, per annum ? Ans. £596..5..2*232075d. 
 
 Case 2. When a, r, and t are given, to find p. 
 
 a 
 
 Rule — - z=zp. Or, log. a — log. r x t = log. p. 
 
 r* 
 
 (5) What principal being put to interest will amount to 
 £260.-9.. 3§ . in 3 years, at £5. per cent, per annum ?t 
 
 (6) What principal being put to interest will amount to 
 £243..2..0..1*2 qrs. in 4 years, at £5. per cent, per annum ? 
 
 Ans. £200 
 
 (7) What principal will amount to £547..9-.10..2-0538368 
 rs. in 5 years, at £4. per cent, per annum ? Ans. £450. 
 
 = r* ; because the index always corresponds with the time. By re- 
 ferring to Theorem VII. Geometrical Progression, it will also be seen 
 that such last term = r X r*-' = r*, when a = r. 
 
 The immense increase of money accumulating at Compound In- 
 terest for a long period, is sufficient to astonish the human mind 
 and to stagger the credibility of persons who are not in some degree 
 conversant with the properties of Geometrical Progression. The 
 amount of a farthing, placed out at Compound Interest at the com- 
 mencement of the Christian era, and continued to the conclusion of 
 the eighteenth century, would be 144035 quintillions of pounds. But 
 of the magnitude of this sum, spoken of in the abstract, no just con- 
 ception can be formed. When, however, by a further calculation, we 
 nave ascertained that to coin such a quantity of money (were it pos- 
 sible) in sovereigns of the present weight and fineness, would require 
 60,308170 solid globes of gold, each as large as the earth, we are ena 
 bled to entertain a more adequate idea of the sum whose vastness, 
 without having recourse to this adscititious assistance, placed it al- 
 most beyond the reach of our limited understandings. 
 
 The amount at Simple Interest for the same period, would be only 
 l#. 10f<7. Editor. 
 
 * l-05« X 225 = 1-57625 X 225 = 260*465625 = £260..9..3|. Ans. 
 260-465625 _ 26 0-46562 5 _^ 9 , ^ 
 1-05' 1-157625 G 
 
146 COMPOUND INTER [/THE TUTOR'S 
 
 (8) What principal will amount to £596..5..2*232075d. in 
 4 years, at £4i. per cent, per annum ? Ans. ,£500. 
 
 Case 3. When p, a, and t are given, to find r. 
 
 p rt , the root of which being extracted 
 
 p will give r. 
 
 Or, log. a — log. p ~ t=. log. r. 
 
 (9) At what rate per cent, per annum, will £225. amount 
 to £260..9..3f. in 3 years?* 
 
 (10) At what rate per cent, per annum, will £200. amount 
 to £243..2..0..1'2 qrs. in 4 years ? Ans. £5. per cent. 
 
 ( 11 ) At what rate per cent, per annum, will £450. amount 
 to £547..9» 10.-2*0538368 qrs. in 5 years ? Ans. £4. per cent 
 
 (12) At what rate per cent, per annum, will £500. amount 
 to £596-2593003125 in 4 years? Ans. £4^. per cent. 
 
 Case 4. When p, a, and r are given, to find t. 
 
 a which being continually divided by •" till 
 
 Rule. — = r* nothing remains, the number of the divi- 
 P sums will be equal to t. 
 
 Or, log. a — log. p -*- log. r = /.+ 
 
 (13) Jn what time will £225. amount to £260.-9..3f. * 
 £5. per cent, per annum ?J 
 
 (14) In what time will £200. amount to £243..2'025s. a. 
 £5. per cent, per annum ? Ans. 4 years. 
 
 (15) In what time will £450. amount to £547-.9-10» 
 2*0538368 qrs. at £4. per cent, per annum ? Ans. 5 years. 
 
 (16) In what time will £500. amount to £596*2593003125 
 at £4^. per cent, per annum ? Ans. 4 years. 
 
 THE AMOUNT OF ANNUITIES IN ARREARS. 
 
 Note, u represents the annuity, pension, or yearly rent ; a, r, and t 
 as before. 
 
 260-465625 , •«",.*-•. * ^t o~ 
 
 • ss 1-157C25, and 3 J 1*157625 = 1*05, or £o. per cent. 
 
 225 V Arts. 
 
 t In all cases of this nature, t cannot be found without Logarithms 
 unless it be a whole number. 
 
 225 1*05 1*05 
 
 - - = 1 ; the number of divisions being three, which gives the time 
 1 u ' 5 sought = 3 years. Ans. 
 
ASSISTANT.] COMPOUND INTEREST. 147 
 
 A Table of the amount of £l. annuity for years. 
 
 yrs. 
 
 3 per Cent. 
 
 Z\per Cent. 
 
 4 per Cent. 
 
 4| per Cent. 
 
 5 per Cent. 
 
 1 
 
 1-0000000 
 
 1-0000000 
 
 1-0000000 
 
 1-0000000 
 
 1-0000000 
 
 2 
 
 2-0300000 
 
 2-0350000 
 
 2-0400000 
 
 2-0450000 
 
 2-050000Q 
 
 3 
 
 3-0909000 
 
 3-1062250 
 
 3-1216000 
 
 S- 137026.0 
 
 S-152500C 
 
 4 
 
 4-1836270 
 
 4-2149429 
 
 4-2464640 
 
 4-2781QM 
 
 4-3101 25q 
 
 5 
 
 5-3091358 
 
 5-3624659 
 
 5-4163226 
 
 5-4707097 
 
 5-5256312 
 
 6 
 
 6-4684099 
 
 6-5501522 
 
 6-6329755 
 
 6-7168916 
 
 6-8019128 
 
 7 
 
 7-6624622 
 
 7-7794075 
 
 7-8982945 
 
 8-0191517 
 
 8-1420084 
 
 8 
 
 8-8923361 
 
 9*0516868 
 
 9-2142263 
 
 9-3800135 
 
 9-5491088 
 
 9 
 
 10-1591062 
 
 10-3684958 
 
 10-5827953 
 
 10-8021141 
 
 11-0265642 
 
 10 
 11 
 
 11-4638794 
 12-8077958 
 
 11-7313932 
 
 12-0061071 
 
 12-2882092 
 
 12-5778924 
 
 13-1419920 
 
 13-4863514 
 
 13-8411786 
 
 14-2067870 
 
 12 
 
 14-1920297 
 
 14-6019617 
 
 15-0258055 
 
 15-4640316 
 
 15-9171264 
 
 13 
 
 15*6177906 
 
 16-1130304 
 
 16-6268377 
 
 17-1599130 
 
 17-7129827 
 
 14 
 
 17-0863243 
 
 17-6769865 
 
 18-2919112 
 
 18-9321091 
 
 19-5986318 
 
 15 
 
 18-5989140 
 
 19-2956810 
 
 20-0235876 
 
 20-7840540 
 
 21-5785634 
 
 16 
 
 20-1568814 
 
 20-9710298 
 
 21-8245311 
 
 22-7193364 
 
 23-6574916 
 
 17 
 
 21-7615878 
 
 22-7050158 
 
 23-6975123 
 
 24-7417066 
 
 25-8403662 
 
 18 
 
 23-4144354 
 
 24-4996914 
 
 25-6454128 
 
 26-8550834 
 
 28-1323845 
 
 19 
 
 25-1168685 
 
 26-3571806 
 
 27-6712293 
 
 29-0635622 
 
 80-53 90037 
 
 20 
 
 26-8703745 
 
 28-2796819 
 
 29-7780785 
 
 31-3714225 
 
 33-0659539 
 35-719251^ 
 
 21 
 
 28-6764857 
 
 30-2694708 
 
 31-9692016 
 
 33-7831365 
 
 22 
 
 30-5367803 
 
 32-3289023 
 
 34-2479697 
 
 36-3033777 
 
 38-5052142 
 
 23 
 
 32-4528837 
 
 34-4604139 
 
 36-6178885 
 
 38-9370297 
 
 41-4304749 
 
 24 
 
 34-4264702 
 
 36-6665284 
 
 39-0826040 
 
 41-6891960 
 
 44-501 9986 1 
 
 25 
 
 36*4592643 
 
 38-9498569 
 
 41-6459082 
 
 44-5652098 
 
 47-7270985 
 
 26 
 
 38-5530422 
 
 41-3131019 
 
 44-3117445 
 
 47-5706443 
 
 51-1134534 
 
 27 
 
 40-7096335 
 
 43-7590605 
 
 47-0842143 
 
 50-7113233 
 
 54-6691261 
 
 28 
 
 42-9309225 
 
 46-2906276 
 
 49-9675829 
 
 53-9933329 
 
 58-4025824 
 
 29 
 
 45-2188502 
 
 48-9107996 
 
 52-9662862 
 
 57-4230329 
 
 62-3227115 
 
 30 
 
 47-5754157 
 
 51-6226776 
 54-4294713 
 
 56-0849376 
 
 61-0070694 
 
 66-4388471 
 
 31 
 
 50-0026782 
 
 59-3283351 
 
 64-7523875 
 
 70-7607895 
 
 32 
 
 52-5027585 
 
 57-3345028 
 
 62-7014685 
 
 68-6662449 
 
 75-2988290 
 
 33 
 
 55-0778412 
 
 60-3412104 
 
 66-2095272 
 
 72-7562259 
 
 80-0637704 
 
 34 
 
 57-7301764 
 
 63-4531527 69-8579083 
 
 77-0302561 
 
 85-0669589 
 
 S5 
 
 60-4620817 
 
 66-6740130 73-6522246 
 
 81-4966176 
 
 90-3203068, 
 95-8363221 
 
 36 
 
 63-2759441 
 
 70-0076034 
 
 77-5983136 
 
 86-1639654 
 
 37 
 
 66-1742224 
 
 73-4578695 
 
 81-7022461 
 
 91*0413439 
 
 101-6281382. 
 
 38 
 
 69-1594490 
 
 77-0288949 
 
 85-9703359 
 
 96-1382044 
 
 107-7095451) 
 
 39 
 
 72-2342324 
 
 80-7249062 
 
 90-4091493 
 
 101-4644236 
 
 114-0950224 
 
 40 
 
 75-4012593 
 
 84-5502779 
 
 95-0255153 
 
 107-0303227 
 
 120-7997735 
 
 Note. The preceding Table is formed thus : the first year's amount 
 is £1. ; and 1 X 1*05 -j-1 =2-05, the second year's amount ; 2*05 X 1*05 
 +1 m 3-1525, the third year's amount, &c. 
 
143 COMPOUND INTEREST. [THE TUTOR'S 
 
 Case 1. When u> t, and r are given, to find a. 
 
 Rule. x w = «. 
 
 r — 1 
 
 Or, by the Table, Multiply the tabular amount of £l. an- 
 nuity by the given annuity, and the product will be the 
 amount required. 
 
 (17) What will an annuity of £50. per annum, amount to 
 in 4 years, at £5. per cent, per annum ?* 
 
 (18) What will a pension of £45. per annum, payable 
 yearly, amount to in 5 years, at £5. per cent, per annum ? 
 
 Ans. £248..13.?..3-27 qfs. 
 
 (19) If an annual salary of £40. be forborne 6 years, at 
 £4. per cent, per annum, what is the amount ? 
 
 Ans. £265..6..4..2-257756l6 qrs. 
 
 (20) If an annuity of £75. payable yearly, be omitted to 
 be paid for 10 years, what is the amount at £5. per cent, per 
 annum ? Ans. £943..6..10'0656d.+ 
 
 Case 2. When a, r, and t are given, to find m. 
 
 r — * 
 
 Rule. — X ci z=.u. 
 
 r t — 1 
 
 (21) What annuity, being forborne 4 years, will amount *o 
 £2 15. .10.. 1^. at £5. per cent, per annum ?t 
 
 '32) What pension, forborne 5 years, will amount to £248.. 
 13.Y..3-27 qrs. at £5. per cent, per annum ? Ans. £45. 
 
 (23) What salary, being omitted to be paid 6 years, will 
 amount to £265..6..4..2*257756l6 qrs. at £4. per cent, per 
 annum ? Ans. £40. 
 
 (24) If the payment of an annuity, being forborne 10 
 years, amount to £943..6..10*0656d. at £5. per cent, per an- 
 num, what is the annuity ? Ans. £75. 
 
 Case 3. When u, a, and r are given, to find L 
 
 JLJ! L x 50 = (1-21550625 — 1) X 1000 = 215-50625 = 
 
 •° 5 £215..10..14. Am. 
 
 Or, by the table thus, 4-310125 X 50 = £21 5-50625, as before. 
 , -05 X 215-50625 -05 X 215-50625 
 
 1-05*.— I -21550625 
 
 ■ = 05 X 1000 = £60. Am 
 
 I 
 
ASSISTANT.] COMPOUND INTEREST. 14Q 
 
 which being continually divi- 
 
 P ( r — a , 1 t clecl by r till nothing remains, 
 
 u the number of divisions will be 
 
 equal to t. 
 
 (25) In what time will £50. per annum amount to £215.. 
 10..1^. at £5. per cent, per annum, for non-payment?* 
 
 (26) In what time will £45. per annum amount to £248.. 
 13.v..3*27 qrs. allowing £5. per cent, per annum, for forbear- 
 ance of payment ? Ans. 5 years. 
 
 (27) In what time will £40. per annum amount to £265.. 
 6..4..2*257756l6. qrs. at £4. per cent, per annum? 
 
 Ans. 6 years. 
 
 (28) In what time will £75. per annum amount to £943.. 
 6.. 10*0656d. allowing £5. per cent, per annum, for forbear- 
 a n ce of payment ? Ans. 1 years. 
 
 Note. The examples relating to the Present Worth of Annuities at 
 Simple Interest are now expunged from this work ; because, being en- 
 tirely useless, except as a mere arithmetical exercise, it is presumed that 
 the judicious teacher will prefer the substitution of other matter of 
 more real utility which is introduced to supply their place. The The- 
 orems are, however, retained in a Note, page 150; in order that the 
 ingenious student who may wish to calculate any example both ways, 
 may have an opportunity of indulging his curiosity, and of compa- 
 ring the true and the false results. That the principle of computing 
 their value by Simple Interest is erroneous and absurd, will be manifested 
 by the following observations. 
 
 The present worth of an annuity of £150. to continue only 40 years, 
 calculated at £5. percent, per annum, Simple Interest (by Theorem 
 1, Note, page 150) would be £3950. But this sum, put out at the 
 same rate, will produce £197..10. annual interest (or £47.-10. a year 
 more than the proposed annuity) jor ever. If computed on the true 
 principle (by the Theorem, Case 1, page 151) the present value is 
 £2573..17..3. 
 
 The present value of any perpetual annuity (great or small) computed 
 at Simple Interest, is an unlimited, or infinite sum. But by using Com- 
 pound Interest, we shall obtain a rational result. For instance, an an- 
 nuity of £150. to continue for ever, will (by Case 1, Perpetual Annui- 
 ties, page 154) at £5. per cent, be worth £3000. purchase: which, it 
 is evident, is the sum that mil yield £150. annual interest. Editor. 
 
 0-5 X 215-50625 
 « — (- 1 = '001 X 215-50625 + 1 = 1-21550625 ; 
 
 >rhich being continually divided by 1-05, the number of divisions 
 will be 4, the years req-iired. Ans 
 
150 COMPOUND INTEREST. [THE TUTOR'S 
 
 THE PRESENT WORTH OF ANNUITIES.* 
 
 A Table of the present worth of £l. annuity for years. 
 
 yrs. 
 
 3 per Cent. 
 
 3^ per Cent. 
 
 4 per Cent. 
 
 4^ per Cent. 
 
 5 per Cent. 
 
 0-9708738 
 
 0-9661836 
 
 0-9615385 
 
 0-9569378 
 
 0-9523810 
 
 o 
 
 1 '9134697 
 
 1-8996943 
 
 1-8860947 
 
 1-8726677 
 
 1-8594105 
 
 3 
 
 2-8286114 
 
 2-8016370 
 
 2-7750910 
 
 2-7489643 
 
 2-7232481 
 
 4 
 
 3-7170984 
 
 3-6730792 
 
 3-6298952 
 
 3-5875256 
 
 3-5459506 
 
 5 
 
 4-5797072 
 
 4-5150524 
 
 4-4518223 
 
 4-3899766 
 5-1578723 
 
 4-3294768 
 
 6 
 
 5-4171915 
 
 5-3285530 
 
 5-2421368 
 
 5-0756922 
 
 7 
 
 6-2302830 
 
 6-1145440 
 
 6-0020546 
 
 5-892" r 008 
 
 5-7863735 
 
 8 
 
 7-0196922 
 
 6-8739556 
 
 6-7327448 
 
 6-5958859 
 
 6-4632129 
 
 9 
 
 7-7861089 
 
 7-6076866 
 
 7-4353315 
 
 7-2687903 
 
 7-1078218 
 
 10 
 Tl" 
 
 8-5302028 
 
 8-3166054 
 
 8-1108956 
 
 7-9127180 
 
 7-7217351 
 
 9-2526241 
 
 9-0015511 
 
 8-7604765 
 
 8-5289167 
 
 8-3064144 
 
 12 
 
 9-9540040 
 
 9-6633344 
 
 9*3850735 
 
 9-1185806 
 
 8-8632518 
 
 13 
 
 10-6349553 
 
 10-3027385 
 
 9-9856476 
 
 9*6828522 
 
 9-3935732 
 
 14 
 
 11-2960731 
 
 10-9205203 
 
 10-5631227 
 
 10-2228251 
 
 9-8986412 
 
 15 
 
 11-9379350 
 
 11-5174109 
 
 11-1183872 
 
 10-7395455 
 
 10-3796583 
 
 12-5611019 
 
 12-0941168 
 
 11-6522954 
 
 11-2340148 
 
 10-8377698 
 
 17 
 
 13-1661183 
 
 12-6513206 
 
 12-1656686 
 
 11-7071912 
 
 11-2740665 
 
 18 
 
 13-7535129 
 
 13-1896817 
 
 12-6592967 
 
 12-1599916 
 
 11-6895872 
 
 19 
 
 14-3237989 
 
 13-7098374 
 
 13-1339391 
 
 12-5932934 
 
 12-0853210 
 
 20 
 ~2T 
 
 14-8774747 
 
 14-2124033 
 
 13-5903260 
 
 13-0079363 
 
 12-4622105 
 12-8211529 
 
 15-4150240 
 
 14-6979742 
 
 14-0291596 
 
 13-4047237 
 
 22 
 
 15-9369165 
 
 15-1671248 
 
 14-4511150 
 
 13-7844246 
 
 13-1630028 
 
 23 
 
 1 6-4436083 
 
 15-6204104 
 
 14-8568413 
 
 14-1477747 
 
 13-4885741 
 
 24 
 
 16-9355420 
 
 16-0583675 
 
 15-2469628 
 
 14-4954782 
 
 13-7986420 
 
 25 
 
 ~26 
 
 17-4131476 
 
 16-4815145 
 
 15-6220796 
 
 14-8282088 
 
 14-0939448 
 
 17-8768423 
 
 16-8903522 
 
 15-9827688 
 
 15-1466113 
 
 14-3751855 
 
 27 
 
 18-3270314 
 
 17-2853644 
 
 16-3295854 
 
 15-45130^7 
 
 14-6430338 
 
 28 
 
 18-7641082 
 
 17-6670187 
 
 16-6630629 
 
 15-7428734 
 
 14-8981274 
 
 29 
 
 19-1884546 
 
 18-0357669 
 
 16-9837143 
 
 16-0218884 
 
 15-1410737 
 
 30 
 
 19-6004414 
 
 18-3920453 
 
 17-2920330 
 
 16-2888884 
 
 15-3724511 
 
 31 
 
 20-0004286 
 
 18-7362757 
 
 17*5884933 
 
 16-5443908 
 
 15-5928106 
 
 32 
 
 20-3887656 
 
 19-0688654 
 
 17-8735512 
 
 16-7888907 
 
 15-8026768 
 
 33 
 
 20-7657919 
 
 19*3902081 
 
 18-1476454 
 
 17-0228619 
 
 16-0025493 
 
 34 
 
 21-1318368 
 
 1 9-7006842 
 
 18-4111975 
 
 17-2467578 
 
 16-1929041 
 
 35 
 36 
 
 21-4872202 
 
 20-0006611 
 
 18-6646130 
 
 17-4610122 
 
 16-3741944 
 
 21-8322526 
 
 20-2904938 
 
 18-9082817 
 
 17-6660404 
 
 16-5468518 
 
 37 
 
 22*1672355 
 
 20-5705254 
 
 19-1425785 
 
 17-8622396 
 
 16-7112874 
 
 38 
 
 22-4924617 
 
 20-8410873 
 
 19-3678639 
 
 18-0499900 
 
 16-8678928 
 
 39 
 
 22-8082153 
 
 21-1024998 
 
 19-5844845 
 
 18-2296555 
 
 17-0170408 
 
 40 
 
 23-1147721 
 
 21-3550723 
 
 19-7927735 
 
 18-4015842 
 
 17-1590865 
 
 Theor. I. — — 
 
 • Present Worth of Annuities at Simple Interest. 
 r (*_ l)-f 2 _ 1r+ 1 
 
 2 tr -J- 2 
 
 — Xtu 
 
 II. 
 
 tr. (t — 1) -r 2).* 
 
 X2pz=z 
 
ASSISTANT. J COMPOUND INTEREST- 151 
 
 Note. The above table is thus formed : £1. ~- 1-05 = -9523810. 
 the present worth of the first year ; this -f- 1*05 = -9070295, which ad- 
 ded to the first year's present worth, gives 1*8594105, the present 
 worth of 2 years ; then, *9070295 -f- 1*05, and the quotient added to 
 1-8594105 = 2-7232481, the present worth of 3 years, &c. 
 
 Case 1. When u, t, and r are given, to find p, the present worth. 
 
 Rule. (« — "^ )-*"(»• — l ) — P- 
 
 Or, by the Table* Multiply the tabular present worth for 
 the time given, by the given annuity, and the product will be 
 the present worth required. 
 
 (29) What is the present worth of an annuity of £30. to 
 continue 7 years, at £5. per cent, per annum ?* 
 
 (30) What is the present worth of a pension of £40. per 
 annum, to continue 8 years, at £5. per cent, per annum? 
 
 4ns. £258..10..6..3'264< qrs. 
 
 (31) What is the present worth of an annual salary of 
 £35. to continue 7 years, at £4. per cent, per annum? 
 
 Ans. £210..1..5'04<£ 
 
 (32) What is the yearly rent of £50. to continue 5 years^ 
 tvorth in ready money, at £5. per cent, per annum ? 
 
 Ans. £2l6..9-.5..2*08 qrs. 
 
 Case 2. When p, t, and r are given, to find u 
 
 Rule. l — K - '- = u. 
 
 r* — 1 
 
 (33) If an annuity be purchased for £l73..11..10'08d. tc 
 
 m (fa — P )X2 
 
 <2/>— (f— lj U ).t" 
 
 IV, 
 
 1 / P . 1 \ /2 p v 
 
 Put T ~^V + t) = m ; then «/ V= + »•) 
 
 ru ' ' 
 
 For Annuities in Reversion, it is only necessary to observe the Rules 
 for Reversionary Annuities at Compound Interest, and to calculate* 
 (according to the directions therein given) by the Theorems for Sim- 
 ple Interest. 
 
 30 30 
 
 • 30 — jt— = 30 njofT = 30 " 21 ' 3204 = 8-6796 and 
 
 8-6796 -T- -05 = £173-592 = £l73..11..10-08d. Ans. 
 Or, hj the Table, 5-7863735 X 30 = £173-591205. Ans. 
 
1 152 COMPOUND INTEREST. [THE TUTOR'S 
 
 be continued 7 years, at £5. per cent, per annum, what is 
 the annuity ?* 
 
 (34) If £258..10..6..3'264 qrs. be paid down for a salary 
 8 years to come, at £5. per cent, per annum, what is the 
 salary ? Ans. £40. 
 
 (35) If the present payment of £210.. 1.. 5 *04d. be required 
 for a pension for 7 years to come, at £4. per cent, per annum, 
 what is the pension ? Ans. £35. 
 
 (36) If the present worth of an annuity 5 years to come, 
 be £2l6..9..5..2 0S qrs. at £5. per cent, per annum, what is 
 the annuity ? Ans. £50. 
 
 Case 3. When u, p, and r are given, to find L 
 
 which being continually divided 
 
 Rule. = *•' by r till nothing remains, the num- 
 
 p + u pr b er j divisions will be equal to /. 
 
 (37) How long may a lease of £30. yearly rent be had for 
 £173. .11. .10-OSd. allowing £5. percent, per annum to the 
 purchaser ?t 
 
 (38) If £258. 10..6..3-264 qrs. is paid down for a lease of 
 £40. per annum, at £5. per cent, per annum, how long i? 
 the lease purchased for ? Ans. 8 years. 
 
 (39) If a house is let upon lease for £35. per annum, and 
 the lessor disposes of the lease for £210..1..5*04(/. allowing 
 after the rate of £4. per cent, per annum ; what term of the 
 lease remains unexpired ? Ans. 7 years. 
 
 (40) For what time is a lease of £50. per annum pur- 
 chased, when present payment is made of £21 6.-9.. 5. .2*08 
 qrs. at £5. per cent, per annum ? Ans. 5 years. 
 
 ANNUITIES, &C. IN REVERSION. 
 
 To find the present worth of annuities in reversion. 
 
 Rule 1. Find the present worth of the annuity,/^ the time 
 of its continuance, as if it were to commence immediately ; 
 by Case 1, page 151. Then find what principal will amount 
 to that sum in the given time before the annuity commences ; 
 
 * 173-592 X 1*4071 X '05 — -4071 = 12-213 -7- -4071 = £30. Ans. 
 Or, by the Table, 173-592 -i- 5-7863735 = £30. Ans. 
 
 + 173-592 -f 30 — (173-592 X 1-05) = 203*592 — 182-2716 = 
 21-3204 ; and 30 -r- 21-3204 = 1-4071 ; which being continually divi- 
 ded by 1*05, the number of divisions will be 7 : therefore t = 7 
 years. Ans. Or, by the Table; 173-«92 -*- 30 = 5-7864 : and re- 
 ferring to the column of 5 per -jfwfc, we find the number 5-7S63735 
 against 7 vears. 
 
ASSISTANT.] COMPOUND INTEREST. 153 
 
 (by Case 2. Compound Interest, page 145) which will be the 
 present worth. 
 
 Itule 2. Find the present worth of a similar annuity supposed to 
 commence immediately, and continue during the whole period ; and also 
 the present worth of the same till the time when the reversionary an- 
 nuity actually commences ; and the difference of these two will be the 
 present value required. 
 
 Note. When calculating by the Table, this is the most eligible 
 method. 
 
 Rule 3, Find the amount of the annuity at the time of its cessation 
 (by Case 1. page 148;) and the present worth of that amount (being 
 found by Case 2. Compound Interest, page 145) will be the value re- 
 quired. 
 
 (41) What is the present worth of a reversion of a lease of 
 <£40. per annum, to continue for 6 years, but not to com- 
 mence till the end of 2 years, allowing £5. per cent, per an- 
 num, to the purchaser ?* 
 
 (42) What is the present worth of a reversion of a lease of 
 £60. per annum, to continue 7 years, but not to commence 
 till the end of 3 years, allowing £5. per cent, per annum, to 
 the purchaser? Ans. £299.A8..2'l6d. 
 
 (43) A house is let at £30. per annum, on a lease, of which 
 4 years are yet unexpired, and which the lessee is desirous of 
 renewing at the same rental, to continue 7 years beyond the 
 term of the present lease. What will the lessor expect as a 
 bonus for such a renewal of the lease, considering the house to 
 be worth double the present rent ; and allowing interest for 
 the money now advanced, at £5. per cent, per annum ? 
 
 Ans. <£l42..l6..3..M52 qrs. 
 
 To find the annuity in reversion, which a given sum will 
 purchase. 
 
 Rule. Find the amount of the given sum for the time before 
 the annuity commences ; by Case 1, Compound Interest, page 
 145, which will be the value of the annuity at its commence- 
 ment. 
 
 Call this value p, and then find the annuity as in Case 2, 
 page 151. 
 
 (44) What annuity to be entered upon 2 years hence, and 
 
 40 40 
 
 * 4 °-Foi; =40 -Woo^ =40 -^^ 
 
 and 10-15139 -f- -05 = 203*0278: then 203-0278 -r- 1-05 2 = £184- 
 1522 = £184..3..0..2-112 qrs. Ans. 
 
 g5 
 
154- COMPOUND INTEREST. l_THE TUTOR'S 
 
 then to continue 6 years, maybe purchased for £l84..3..0.. 
 2*1 12 qrs. at £5. per cent, per annum ?* 
 
 (45) The present worth of a lease, taken in reversion for 
 7 years, but not to commence till the end of three years, is 
 £299-1 8..2*16gl allowing £5. percent, per annum, to the 
 purchaser : what is the yearly rent ? Ans. £6'0. 
 
 (46) There is a lease that has yet 4 years to run, and the 
 lessee has purchased the reversion of a renewed lease, at the 
 same rental of £30. per annum, for the term of 7 years, com- 
 mencing at the expiration of the present lease ; for which he 
 has paid down £l42..l6..3..1*l 52 qrs. What increase of rent 
 is reckoned on the property, according to this contract, allow- 
 ing £.5. per cent, per annum, for present payment? Ans. £30. 
 
 PERPETUAL ANNUITIES ; OR FREEHOLD ESTATES. 
 
 Case 1 - When u and r are given, to find p, the present 
 worth, or purchase money. 
 
 u 
 
 Rule. s± p.t 
 
 r— 1 1 
 
 (47) What is the worth of a freehold estate of £50. yearlj 
 rent, allowing £5. per cent, per annum, to the buyer ? J 
 
 (48) What is a real estate of £140. per annum, worth in 
 present money, allowing £4. per cent, per annum, to the 
 purchaser? Ans. £3500. 
 
 (49) What must the purchaser give for a freehold estate 
 of £437.. 10. yearly rent, so as to make £3^. per cent, per an- 
 num by the investment of his capital ? Ans. £ 12500. 
 
 Case 2. When p and r are given, to find u. 
 
 Rule, p x r — I ~u. 
 
 (50) If a freehold estate is bought for £1000. what must 
 
 * 184-1522 X 1*1025 = 203*0278; then 
 203-0278 X 1-3400956 X '05 
 
 '3400956 
 
 203-0278 X 1 + (203-0278 X -3400956) 203-0278 
 
 : 3^0956 " X * 05 = -3400956 
 
 + 203-0278 X '05 = 800-0005 X -05 = £40. Ans. 
 
 Or, by the Table; 184-1522 ■*- (6-4632129 — 1-8594105) = 184- 
 1522 -*- 4-6038024 = £40. Ans. 
 
 •f-This rule is deduced from the formula in page 151: for in Annui- 
 ties continuing for ever, t is infinite, and the subtractive quantity « 
 -5- r* = ,- therefore the theorem assumes the above form. 
 t 50 ~- '05 = £1000. Ans. 
 
AS&ISTAN'.* .] COMPOUND INTEREST. i55 
 
 be the yearly rent, to pay the purchaser £5. per cent, per 
 annum interest for his money ?* 
 
 (51) If an estate be sold for £3500. what is the yearly 
 rent, allowing to the purchaser £4. per cent, per annum ? 
 
 Ans. £140. 
 
 (52) If a freehold estate is bought for £12500. and will 
 yield the purchaser £3^. per cent, per annum, what is the 
 yearly rent ? Ans. £437.-10. 
 
 Case 3. When p and u are given, to find r. 
 
 u 
 Rule. (- 1 = r. 
 
 P 
 
 (53) If an estate of £50. per annum is bought for £1000 
 what is the rate per cent, per annum ?t 
 
 (54) If a freehold estate of £140. per annum is sold for 
 £3500. what interest will it pay to the purchaser ? 
 
 Ans. £4. per cent. 
 
 (55) If an estate in perpetuity of £437- 10. per annum is 
 sold for £12500. what interest will it pay to the purchaser ? 
 
 Ans. £S\. per cent. 
 
 freehold estates in reversion. 
 
 To find the present worth of a freehold estate in reversion. 
 
 Rule. Find the value of the estate, supposing it were to 
 come into immediate possession, as in Case 1, page 154. Then 
 suppose that value (p) to be a, and find what principal will 
 amount to a, in the time to come, previous to possession, by 
 Case 2, Compound Interest, page 1 45. Such principal will 
 be the present value. 
 
 (56) What must be paid down for the purchase of a free- 
 hold of £50. per annum, to be entered upon 4 years hence, 
 allowing the purchaser at the rate of £5. per cent, per an- 
 num for his purchase money ? J 
 
 (57) What must be paid down for the reversion of a real 
 estate of £200. per annum, so as to pay the purchaser £4. 
 per cent, per annum, for his capital ; supposing 2 years to 
 elapse before the estate comes into possession ? 
 
 Ans. £4622.. 15..7- 1*76 qrs. 
 
 9 1000 X *05 = £50. Ans. 
 50 
 f + 1 = 1*05 = £5. per cent. Ans. 
 
 $50-7- -05 = 1000 ; then 1000 -r- 1*2155 = £822-7067 = £322. 
 14..!.. 2-432 qrs, Ans. 
 
156 COMPOUND INTEREST. [THE TUTOR'S 
 
 (58) A freehold producing £280. annual rent is to be dis- 
 posed of, with a reserve of the next 3 years' rent to the pre- 
 sent proprietor. What is it worth in ready money, allowing 
 £3^. per cent, per annum to the purchaser ? 
 
 Ans. £7215..10..9..3'36 qrs. 
 To Jind the yearly rentofaii estate in reversion; having its 
 present value given. 
 
 Rule. Find the amount of the given present value, in the 
 time before possession : thus, pr* =z a. Then consider that 
 amount to be the present value (p) of the perpetual annuity, 
 and find the annuity thus : p X (r — 1 ) = u. 
 
 (59) What must be the rent of a freehold property, to come 
 into possession 4 years hence, for which £822..14..1..2*432 qrs. 
 is paid down ; allowing the purchaser £5. per cent, per 
 annum ?* 
 
 (60) A freehold estate is sold for £4622..15..7..1*76 qrs. 
 the vendor reserving to himself the first two years' rent. Re- 
 quired the annual value, to pay the purchaser £4. per cent. 
 per annum for his capital ? Ans. £200. 
 
 (61) A freehold estate has been purchased for £721 5.. 10.. 
 9..3-S6 qrs. the possession of which is not to be given up til 1 
 after the expiration of 3 years. What must be the annudt 
 rent, to pay the purchaser at the rate of £3^. per cent, per 
 annum? Ans. £280. 
 
 DISCOUNT ; ON THE PRINCIPLES OF COMPOUND INTEREST.t 
 
 Note. The following Table is constructed by the continual divi- 
 sion of 1 by the ratio(r) : thus 1 ~~- 1*05 = '9523810, the first year's 
 present worth; then -9523810 ~ 1*05 = -9070-295, the second year's 
 present worth ; and -9070295 4- 1-05 = -8638376, the third, &c 
 
 •822-70625 X 1-2155= 1000; then 1000 X -05 =£50. Ans. 
 
 •f- This is merely a repetition of the various cases in Compound 
 Interest. For instance, to find the present worth of any debt due 
 some time hence, is precisely the same operation as finding what 
 principal will amount to that sum in the given time ; and this obser- 
 vation will equally apply to the identity of the other cases. The 
 entire omission, therefore, of Discount (at Compound Interest ) arrang 
 ed under that specific head, would be no detriment to the learner. 
 It is, however, retained here, for the sake of those who may think 
 some repetition of the subject desirable. Editor. 
 
SSISTANT.] 
 
 COMPOUND INTEREST. 
 
 157 
 
 A Table of the present worth of £l. due any number of years 
 hence, from 1 to 40. 
 
 Yrs. 
 
 3 per Cent. 
 
 dhper Cent. 
 
 4 per Cent. 
 
 4| per Cent. 
 
 5 per Cent. 
 
 1 
 2 
 3 
 4 
 5 
 
 •970873S 
 •9425959 
 •9151417 
 •8884870 
 •8626088 
 
 •9661836 
 •9335107 
 •9019427 
 •8714422 
 •8419732 
 
 •9615385 
 •9245562 
 •8889963 
 •8548042 
 •8219271 
 
 •9569378 
 •9157299 
 •8762966 
 •8385613 
 •8024510 
 
 •9523810 
 •9070295 
 •8638376 
 •8227025 
 •7835262 
 
 6 
 7 
 8 
 9 
 10 
 
 •8374843 
 •8130915 
 •7894092 
 •7664167 
 •7440939 
 
 •8135006 
 •7859910 
 •7594116 
 •7337310 
 •7089188 
 
 •7903145 
 •7599178 
 •7306902 
 •7025867 
 •6755641 
 
 •7678957 
 •7348285 
 •7031851 
 •6729044 
 •6439277 
 
 •7462154 
 •7106813 
 •6768394 
 •6446089 
 •6139133 
 
 11 
 
 12 
 13 
 14 
 15 
 
 •7224213 
 •7013799 
 •6809513 
 •6611178 
 •6418619 
 
 •6849457 
 •6617833 
 •6394041 
 •6177818 
 •5968906 
 
 •6495809 
 •6245970 
 •6005741 
 •5774751 
 '5552645 
 
 •6161987 
 •5896639 
 •5642716 
 •5399729 
 •5167204 
 
 •5846793 
 •5568374 
 •5303214 
 •5050680 
 •4810171 
 
 16 
 17 
 18 
 19 
 20 
 
 •6231669 
 •60501 64 
 •5873946 
 •5702860 
 •5536758 
 
 •5767059 
 •5572038 
 •5383611 
 •5201557 
 •5025659 
 
 •5339082 
 •5133732 
 •4936281 
 •4746424 
 •4563869 
 
 •4944693 
 •4731764 
 •4528004 
 •4333018 
 •4146429 
 
 •4581115 
 •4362967 
 •4155207 
 •3957340 
 •3768895 
 
 21 
 22 
 23 
 24 
 25 
 
 •5375493 
 •5218925 
 •5066918 
 •4919337 
 •4776056 
 
 •4855709 
 •4691506 
 •4532856 
 •4379571 
 •4231470 
 
 •4388336 
 •4219554 
 •4057263 
 •3901215 
 •3751168 
 
 •3967874 
 •3797009 
 •3633501 
 •3477035 
 •3327306 
 
 •3589424 
 •3418499 
 •3255713 
 •3100679 
 •2953028 
 
 26 
 27 
 28 
 29 
 30 
 
 •4636947 
 •4501891 
 •4370768 
 •4243464 
 •4119868 
 
 •4088377 
 •3950122 
 •3816543 
 •3687482 
 •3562784 
 
 •3606892 
 •3468166 
 •3334775 
 •3206514 
 •3083187 
 
 •3184025 
 •3046914 
 •2915707 
 •2790150 
 •2670000 
 
 •2812407 
 •2678483 
 •2550936 
 •2429463 
 •2313774 
 
 31 
 32 
 33 
 34 
 35 
 
 •3999872 
 •3883370 
 •3770263 
 •3660449 
 •3553834 
 
 •3442304 
 •3325897 
 •3213427 
 •3104761 
 •2999769 
 
 •2964603 
 •2850579 
 •2740942 
 •2635521 
 •2534155 
 
 •2555024 
 •2444999 
 •2339712 
 •2238959 
 •2142544 
 
 •2203595 
 •2098662 
 •1998725 
 •1903548 
 •1812903 
 
 36 
 37 
 38 
 39 
 40 
 
 •3450324 
 •3349829 
 •3252262 
 •3157536 
 •3065568 
 
 •2898327 
 •2800316 
 •2705619 
 •2614125 
 
 •2525725 
 
 •2436687 
 •2342968 
 •2252854 
 •2166206 
 •2082890 
 
 •2050282 
 •1961992 
 •1877504 
 •1796655 
 1 -1719287 
 
 •1726574 
 •1644356 
 •1566054 
 •1491480 
 •1420457 
 
158 COMPOUND INTEREST. [THE TUTOR'S 
 
 Case 1. To find the present worth of any sum due after a 
 certain period. 
 
 Rule. The same as in Case 2, Compound Interest ; 
 considering a as the debt whose present value is required. 
 
 (1) If £344..14..9..1'92 qrs. be payable in 7 years time, 
 what is the present worth, discount being made at £5. per 
 cent, per annum ?* 
 
 (2) A debt of £409..9*00992^ payable 4 years hence, is 
 agreed to be paid in present money : what sum must the 
 creditor receive, discounting at £4. per cent, per annum ? 
 
 Ans. £350. 
 Case 2. To find the debt whose present worth is given. 
 Rule. See Case 1, Compound Interest. 
 
 (3) If £245. be received for a debt payable 7 years hence, 
 allowing £5. per cent, per annum to the debtor for present 
 payment ; what is the debt ?t 
 
 (4) There is a sum of money due at the expiration of 4 
 years ; but the creditor agrees to take £350. in ready money, 
 allowing £4. per cent, per annum, discount. What was the 
 debt? Ans. £409..9«00992*. 
 
 Case 3. When the rest are given, to find the time. 
 Rule. See Case 4, Compound Interest. 
 
 (5) A person receives £245. now for a debt of £344.. 14.. 9.. 
 1*92 qrs. discounting at £5. per cent, per annum: in what 
 time was the debt payable ?J 
 
 (6) There is a debt of £409..9'00992,y. due a certain time 
 hence ; but £4. per cent, per annum being allowed to the 
 debtor for the present payment of £350. it is required to find 
 in what time the sum was to be paid. Ans. 4 years. 
 
 Case 4. When the rest are given, to find the rate per cent. 
 
 Rule. As in Case 3, Compound Interest. 
 
 (7) The present worth of £344.. 14..9.. 1*92 qrs. payable 
 
 * 344-7395 -4- 1*4071 =£245. Ans. 
 
 Or, by the Table, -7106813 X 344-7395 = £245. Ans. 
 
 f 245 X 1*4071 = £344-7395 = £344.. 14..9.. 1-92 qrs. Ans. 
 
 X 344-7395 -4- 245 == 1-4071 ; the continual divisions of which by 
 1 -05, will be 7 = the number of years. Ans. 
 
ASSISTANT.] DUODECIMALS. 15Q 
 
 7 years nence, is £245. at what rate per cent, per annum is 
 discount allowed ?* 
 
 (8) There is a debt of £409..9*00992*.. payable in 4 years, 
 but it is agreed to take £350. present payment. Required 
 the rate of discount. Ans. £4. per cent. 
 
 EQUATION OF PAYMENTS AT COMPOUND INTEREST. 
 
 Rule 1. Find the present worth of each payment respec- 
 tively ; and add them together for the whole present worth : 
 then the time in which that present worth will amount to 
 the sum of the debts will be the true equated time required. 
 
 2. Find the amount of each debt from the time of its be- 
 coming due till the time of the last payment, and add the 
 respective amounts and the last payment into one sum. 
 Then find the time in which the sum of the debts would 
 amount to that sum of the amounts : subtract this from the 
 time of the last payment, and the difference will be the true 
 equated time. 
 
 (1) Required the true equated time for the payment of a 
 debt of £400. of which £320. is now due, and the rest at 
 the end of 5 years ; reckoning compound interest at the rate 
 of £5. per cent, per annum ? Ans. '90714 years. 
 
 (2) If £100. will become due one year hence, and £104. 
 three years hence, what is the true equated time for payment 
 of the whole, allowing compound interest at £4. per cent, 
 per annum ? Ans. 2 years. 
 
 (3) If a person will have to receive £200. at the end of 3 
 years, and £S0. more at the end of 5 years, in what time 
 ought he to receive the whole at one payment, allowing £5. 
 per cent, per annum, compound interest ? 
 
 Ans. 3' 5 5 18 years. 
 
 DUODECIMALS 
 
 Are so named from the integer of each denomination con- 
 taining twelve of the next inferior. They are in general use 
 among artificers for computing the quantities of their mate- 
 rials and labour ; both in Superficial and Solid Measure.* 
 
 * 344-7395 -r- 245 = 1-4071 ; and V 1-4071 sz 1-05 ; which gives 
 £5. per cent. Ans. 
 
 f For a clear and intelligible exp 7 anation of the different Measures, 
 see the Tables, page 26, &c. 
 
160 DUODECIMALS. [THE TUTOR S 
 
 12 inches (' ) make 1 foot. 
 
 12 seconds (" ) 1 inch, or prime. 
 
 12 thirds ('") 1 second, &c. 
 
 To multiply duodecimal!^. 
 Rule. 1. Under the multiplicand write the corresponding 
 terms of the multiplier. 
 
 2. Multiply by the feet in the multiplier, observing to 
 carry one for every twelve, from each lower denomination to 
 the next superior. 
 
 3. In the same manner multiply by the inches in the multi- 
 plier, setting the result from each term one place farther to 
 the right. 
 
 4. Proceed in like manner with the remaining denomi- 
 nations, and the sum of the products will be the total product. 
 
 Note 1. Length and breadth multiplied together produce the area 
 of a superficies; and this multiplied by the thickness, produces the 
 solid content of a body. 
 
 2. It is generally more eligible to take aliquot parts out of the 
 multiplicand for the inches., &c. in the multiplier. 
 
 
 A 
 
 i 
 
 ft. 
 
 F W 
 
 
 A 
 
 1 
 
 n N » FT u 
 
 (1) Mult. 
 
 7 
 
 9 
 
 by 3 
 
 6* 
 
 
 
 
 
 (2) Mult. 
 
 8 
 
 5 
 
 by 4 
 
 7 
 
 Ans. 38 
 
 6 
 
 11 
 
 (3) Mult. 
 
 9 
 
 8 
 
 by 7 
 
 6 
 
 a. 
 
 72 
 
 6 
 
 
 
 (4) Mult. 
 
 8 
 
 1 
 
 by 3 
 
 5 
 
 a. 
 
 27 
 
 7 
 
 5 
 
 (5) Mult. 
 
 7 
 
 6 
 
 by 5 
 
 9 
 
 a. 
 
 43 
 
 1 
 
 6 
 
 (6) Mult. 
 
 4 
 
 7 
 
 by 3 
 
 10 
 
 a. 
 
 17 
 
 6 
 
 10 
 
 (7) Mult. 
 
 7 
 
 5 
 
 9 by 3 
 
 5 3 
 
 a. 
 
 25 
 
 8 
 
 6 2 3. 
 
 (8) Mult. 
 
 10 
 
 4 
 
 5 by 7 
 
 8 6 
 
 a. 
 
 79 
 
 11 
 
 6 6. 
 
 (9) Mult. 
 
 75 
 
 7 
 
 by 9 
 
 8 
 
 a. 
 
 730 
 
 7 
 
 8 
 
 (10) Mult. 
 
 97 
 
 8 
 
 by 8 
 
 9 
 
 a. 
 
 854 
 
 7 
 
 
 
 (11) Mult. 
 
 57 
 
 9 
 
 by 9 
 
 5 
 
 a. 
 
 543 
 
 9 
 
 9 
 
 (12) Mult. 
 
 75 
 
 9 
 
 by 17 
 
 7 
 
 a. 
 
 1331 
 
 11 
 
 3 
 
 (18) Mult. 
 
 87 
 
 5 
 
 by 35 
 
 8 
 
 a. 
 
 3117 
 
 10 
 
 4 
 
 (14) Mult. 
 
 179 
 
 3 
 
 by 3S 
 
 10 
 
 a. 
 
 (i960 
 
 10 
 
 6 
 
 (15) Mult. 
 
 259 
 
 2 
 
 by 48 
 
 11 
 
 a. 
 
 12677 
 
 6 
 
 10 
 
 (16) Mult. 
 
 257 
 
 9 
 
 by 39 
 
 11 
 
 a. 
 
 10288 
 
 6 
 
 3 
 
 (17) Mult. 
 
 311 
 
 4 
 
 7 by 36 
 
 7 5 
 
 a. 
 
 11402 
 
 2 
 
 4 11 lh 
 
 (18) Mult. 
 
 321 
 
 7 
 
 3 by 9 
 
 3 6 
 
 a. 
 
 2988 
 
 2 
 
 10 4 6, 
 
 ft. in. 
 
 
 in. 
 
 Otherwise, 
 
 Proof by Decimals. 
 
 
 *7 9 
 
 
 6 Z 
 
 = ^7 9 
 
 
 
 7-75 
 
 
 
 3 6 
 
 
 
 3 
 
 
 
 3-5 
 
 
 
 23 3 
 
 
 
 23 3 
 
 
 
 3875~~ 
 
 
 
 3 10 6" 
 
 
 
 3 10 6 
 
 c 
 
 2325 
 
 /**, 
 
 
 27 I 6 
 
 27 1 6 
 
 27-125 sq. 
 
 
ASSISTANT.] DUODECIMALS. l6l 
 
 Glazing, Mason's flat work, and some parts of Joiners' 
 work, are computed at so much per square foot. 
 
 Painters', Plasterers', Pavers', and some descriptions of 
 Joiners' work, are estimated by the square yard. 
 
 Roofs, Floors, Partitions, fyc. by the square of 100 feet. 
 
 Bricklayers' work by the square rod, containing 272^ feet. 
 
 (19) A certain house has 3 tiers of windows, 3 in a tier, 
 the height of the first tier being 7 feet 10 inches, the second 
 6 feet 8 inches, and the third 5 feet 4 inches ; and the breadth 
 of each window is 3 feet 1 1 inches. What will the glazing 
 cost at 14*/. per square foot?* 
 
 (20) What is the price of 8 squares of glass, each measur- 
 ing 4 feet 10 inches long, and 2 feet 11 inches broad, at 4|d. 
 per square foot ? Ans. £l.,18..9. 
 
 (21) What is the value of 8 squares, each measuring 3 feet 
 by 1 foot 6 inches, at 7f d. per square foot ? Ans. £l..3..3. 
 
 (22) What is the price of a marble slab, 5 feet 7 inches 
 long, and 1 foot 10 inches broad, at 6s. per square foot? 
 
 Ans.£3..1..5. 
 
 (23) W T hat will be the expense of ceiling a room, the 
 ^ngth of which is 74 feet 9 inches, and the width 1 1 feet 6 
 mches, at 3s. I0^d. per square yard? Ans. £l8..10..1. 
 
 (24) What will the paving of a court-yard cost, at 4f d. 
 per square yard, the length being 58 feet 6 inches, and 
 the breadth 54 feet 9 inches ? Ans. £7..0..1 0. 
 
 (25) The circuit of a room is 97 feet 8 inches, and the 
 height 9 feet 10 inches : what is the charge for painting it, 
 at 2s. 8fd. per square yard ? Ans. £ 14.. 11. .2. 
 
 (26) What is the expense of a piece of wainscot 8 feet 
 3 inches long, and 6 feet 6 inches broad, at 6s. 7±d. per 
 square yard? Ans. £l..iy..5. 
 
 ft, in. 
 
 • 7 10 
 
 6 8 
 
 5 4 
 
 19 10 the whole height. 
 
 3 11 
 
 in. ft. in. 
 
 6 =|19 10 
 
 11 
 
 in. 218 2 
 3=4 9 11 
 
 s. 4 U 6 " 
 
 3 
 
 I ==" 55 233 6 at 14d. 
 
 11 9 the whole breadth, 
 vali 
 
 rf, 
 
 2=z| £1113 the value at U 
 1 18 10 the value at 2d. 
 le of 6" = 04 
 
 
 £13 11 10A Arts. 
 
l()2 ODECIMALS. [jHE TUTOR'S 
 
 (27) What will the paving of a court-yard cost, at 3.9. 2d. 
 per square yard, the length being 27 feet 10 inches, and the 
 breadth 14 feet 9 inches? Ans. <£7..4..5. 
 
 (28) A certain court-yard is 42 feet 9 inches in front, and 
 68 feet 6 inches long ; a causeway the whole length, and 5 
 feet 6 inches broad, is laid with Purbeck stone, at 3s. 6d. per 
 square yard, and the rest is paved with pebbles, at 3s. per 
 square yard. What is the expense ? Ans. £49..17..0^. 
 
 (29) What will the plastering of a ceiling cost, at lOd. per 
 square yard, supposing the length 21 feet 8 inches, and the 
 breadth 14 feet 10 inches ? Ans. £l..9..9- 
 
 (30) What will the wainscoting of a room cost, at 6s. per 
 square yard, supposing the height of the room (including the 
 cornice and moulding) is 12 feet 6 inches, and the compass 83 
 feet 8 inches : the three window shutters each 7 feet 8 inches 
 by 3 feet 6 inches, and the door 7 feet by 3 feet 6 inches ? 
 The shutters and door being worked on both sides, are reck- 
 oned work and half- work. Ans. £36.. 12..2^. 
 
 (31) In a piece of partitioning 173 feet 10 inches long, 
 and 10 feet 7 inches in height, how many squares? 
 
 Ans. 18 squares, 39 feet, 8' 10". 
 
 (32) A house of three stories, besides the ground floor 
 measuring 20 feet 8 inches by 1 6 feet 9 inches, is to be floorer 
 at £6.. 10. per square : there are 7 fire-places, two of which 
 measure 6 feet by 4 feet 6 inches each, two others 6 feet by 
 5 feet 4 inches each, two others 5 feet 8 inches by 4 feet 8 
 inches each, and the seventh, 5 feet 2 inches by 4 feet ; and 
 the well-hole for the stairs is 10 feet 6 inches by 8 feet 9 
 mches. Wnat will the whole amount to ? 
 
 Ans. £53..13..3±. 
 
 (33) If a house measures within the walls 52 feet 8 inches 
 in length, and 30 feet 6 inches in breadth, the roof being of 
 a true pitch ; what will it cost roofing at 10.?. 6d. per square ? 
 
 Ans. £l2..12..11f. 
 
 Note. A roof is said to be of a true pitch, when the rafters are f 
 of the breadth of the building. In this case, therefore, the breadth 
 must be accounted equal to the breadth and half-breadth of the 
 building. 
 
 (34) What will the tiling of a barn cost, at 25.?. 6d. per 
 square ; the length being 43 feet 10 inches, and the breadth 
 27 feet 5 inches on the flat, the eave boards projecting 16 
 inches on each side ? Ans. £24..9..5|. 
 
 Not* Bricklayers commute their work at the rate of a brick and 
 
ASSISTANT.] DUODECIMALS. l66 
 
 a half thick ; therefore, if the thickness of a wall is more or less, it 
 must be reduced to the standard thickness by multiplying the area 
 of the wall by the number of half bricks in the thickness, and dividing 
 the product by 3. 
 
 *{S5) If the area of a wall is 4085 feet, and the thickness 
 two bricks and a half, how many rods does it contain of the 
 standard thickness ? Ans. 25 rods, 8 feet. 
 
 (36) If a garden wall is 254 feet in compass, 12 feet 
 7 inches high, and 3 bricks thick, how many rods does it 
 contain? A?is. 23 rods, 136 feet. 
 
 (37) How many rods are there in a wall 62| feet long, 
 4 feet 8 inches high, and 2^ bricks thick ? 
 
 Ans* 5 rods, l67feeL 
 (3S) The end wall of a house is 28 feet 10 inches in 
 length ; the height of the roof from the ground is 55 feet 8 
 inches; and the gable (or triangular part at the top) rises 42 
 courses of bricks, reckoning 4 courses to a foot. The wall 
 to the height of 20 feet, is 2^ bricks thick, 20 feet more, 2 
 bricks thick, and the remaining part a brick and half thick ; 
 and the gable is 1 brick thick. What is the charge for the 
 
 vhole wall, at £5..l6. per rod ? Ans. £48..13..5^. 
 
 To multiply several figures by several, and obtain the product in one line only. 
 
 Rule. Multiply the units of the multiplicand by the units of the 
 multiplier, set down the units of the product, and carry the tens ; 
 text multiply the tens in' the multiplicand by the units of the mul- 
 tiplier, to which add the product of the units of the multiplicand mul- 
 tiplied by the tens in the multiplier, and the tens earned ; then mul- 
 tiply the hundreds in the multiplicand by the units of the multiplier, 
 adding the product of the tens in the multiplicand multiplied by the 
 tens in the multiplier, and the units of the multiplicand by the hun- 
 dreds in the multiplier ; and so proceed till you have multiplied the 
 multiplicand all through, by every figure in the multiplier. 
 
 Multiply . . 35234 
 by . . 52424 
 
 Product 1847107216 
 
 EXPLANATION. 
 
 First, 4X4=16, that is, 6 and carry 1. Secondly, (3 X 4) -4> 
 (4 X 2) and 1 that is carried zs 21, set down 1 and carry 2. Thirdly, 
 (2 X 4) + (3 X 2) + (4 X 4) + 2 carried = 32 ; that is, 2 and car- 
 ry 3. Fourthly, (5 X 4) + (2 X 2) + (3 X 4) -f (4 X 2) -f- 3 car- 
 ried |= 47 ; set down 7 and carry 4. Fifthly, (3 X 4) -f (5 X 2) -f 
 (2 X 4) -f- (3 X 2) + (4 X 5) -j- 4 carried s= 60 ; set down and car- 
 ry 6. Sixthly, (3 X 2) + (5 X 4) -f- (2 X 2) + (S X 5) -f 6 carried 
 
 * In this and the three following examples, the rod is considered 
 =: 272 feet. 
 
164 
 
 MENSURATION OF SUPERFICIES. [THE TUTOR* S 
 
 := 51 ; set down 1 and carry 5. Seventhly, (3 X 4) + (5 X 2) -f (2X 
 5) -}- 5 carried = 37 ; set down 7 and carry 3. Eighthly, (3 X 2) -f 
 (5 X 5) + 3 carried — 34 ; set down 4 and carry 3. Lastly, 3X5 
 -f- 3 carried = 18 ; set down 18, and the work is finished. 
 
 MENSURATION OF SUPERFICIES. 
 
 GEOMETRICAL DEFINITIONS. 
 
 Geometry is the science which investigates the nature and 
 properties of lines, angles, surfaces, and solid bodies. 
 
 A point has no parts or magnitude. 
 
 A line has length only, without breadth or thickness. 
 
 A line drawn wholly in the same direc- 
 tion, or the shortest distance between two 
 points, is a right or straight line. That 
 which continually changes its direction is 
 a curve. 
 
 Parallel lines preserve the same distance 
 from each other throughout ; and there- 
 fore would never meet, though infinitely 
 produced. 
 
 An angle is the degree of inclination 
 of two lines, or the opening between them 
 when they meet in a point; which is 
 called the angular point. 
 
 When a line meeting another inclines 
 not either way, but makes equal angles on 
 each side, those are called right angles ; 
 and the lines are perpendicular to each 
 other. Thus, the angle ADC =. the an- 
 gle BDC.* a" 
 
 An oblique angle is either acute or obtuse. An acute angle 
 is less than a right angle, as BDE ; and an obtuse angle, 
 greater than a right angle, as ADE. 
 
 / 
 
 * When more than two lines meet, forming several angles at tht 
 same point, it is necessary to designate each angle by three letters, 
 placing that which is at the angular point in the middle. Thus the 
 angle BDC is that formed by the lines BD and CD. 
 
ASSISTANT.] MENSURATION OF SUPERFICIES. i(J5 
 
 A superficies or surface, is a space contained within lines, 
 and has two dimensions, length and breadth. 
 
 A solid is a space or body bounded by surfaces, and has 
 three dimensions, length, breadth, and thickness. 
 
 A triangle is a superficies bounded by three lines. A 
 quadrangle, or quadrilateral, is bounded by four lines. 
 
 A right-angled triangle has one right angle ; (Fig. page 
 118;) an obtuse-angled triangle has one obtuse angle ; and an 
 acute-angled triangle has all its angles acute. 
 
 An equilateral triangle has the three sides (and consequently 
 the three angles) all equal to each other. 
 
 An isosceles triangle has two equal sides. 
 
 A scalene triangle has all the three sides unequal. 
 
 A parallelogram is a quadrangle having the opposite sides 
 equal and parallel.* When the angles are right ones, it is 
 called a rectangle?* And a rectangle having all its sides equal 
 is Si squared 
 
 A rhombushas all its sides equal ; but oblique angles. 11 
 
 A rhomboid has oblique angles, and only its opposite sides 
 equal. 
 
 All other quadrilaterals are trapeziums* : but a trapezium 
 Ihat has two sides parallel, is called a trapezoid. 
 
 The base { of a figure is the side on which it is supposed to 
 stand ; and a line drawn from the vertex, or opposite angle, 
 perpendicular to the base, is the altitude/ or perpendicular 
 height. 
 
 Right-lined plane figures of more than four sides are called 
 polygons. A polygon of five sides (or angles) is a pentagon ; 
 one of six, a hexagon, &c. Vide Table, page 168. 
 
 A circle* is a plane figure, contained under one curve line, 
 called the circumference ; which is in every part equidistant 
 from the centre, or middle point within it. The circle con- 
 tains more space than any other plane figure of equal compass. 
 
 A straight line passing through the centre, and meeting 
 the circumference in two opposite points, is called the du 
 ameter ; h and half the diameter, or the distance from the 
 centre to the circumference, is the radius,- 
 
 An arc of a circle is any portion of the circumference. 11 
 
 a Figs. 1, 2, and 3. b Fig. 2. c Fig. 1. d Fig. 3. e Fig. 5. 
 
 f The line AB, Figs. 3 and 4 is the base, and CD, the altitude. 
 
 « Fie. 7 h The line AB, Fig. 7. I AC, or BC. k AD, or ADB, Fig. 8. 
 
166 
 
 MENSURATION OF SUPERFICIES. L,THE TUTOR'S 
 
 A c Fiord is a right line joining the extremes of an arc.' 
 
 The versed sine is part of the diameter cut off by the 
 chord." 1 
 
 A segment is a space contained between an arc and its 
 chord." A semicircle is a segment, the chord of which is the 
 diameter. 
 
 A sector is bounded by an arc and two radii: when the 
 two radii are at right angles, it is a quadrant, or fourth part 
 of the circle. 
 
 The circumference of every circle is supposed to be divi- 
 ded into 360 equal parts, called degrees ; and each degree into 
 60 equal parts, called minutes, &c. 
 
 The measure of an angle is determined by the number of 
 degrees in the arc of a circle contained between the two lines 
 forming the angle, described round the angular point as a 
 centre. Thus the angle ACB (Fig. 8.) is an angle of so many 
 degrees as are contained in the arc AB. Hence a right angle 
 contains 90 degrees. 
 
 An ellipse (or regular oval) is a plane figure bounded by a 
 curve called the circumference, returning into itself, and de- 
 scribed from two points, called the foci, or focuses, in the 
 transverse (or longest) diameter. The shortest diameter, 
 which intersects the transverse at right angles, is called tit 
 conjugate. The diameters are also called axes. 9 
 
 MENSURATION. 
 
 Problem 1. To find the area of a Parallelogram ; whether it 
 be a Square, an oblong Rectangle, a Rhombus, or a Rhomboid. 
 
 Rule. Multiply 
 the length by the 
 altitude or perpen- 
 dicular breadth : 
 the product will 
 be the area. 
 
 D 
 
 D 
 
 Fig. 1. 
 
 Fig. 2. 
 
 B 
 
 (1) The base of the largest Egyptian 
 pyramid is a square, whose side is 693 
 teet. Upon how many acres of ground 
 does it stand ? 
 
 (2) Required the area of a rectangu- 
 lar board, whose length is 12^ feet, and 
 breadth 9 inches. 
 
 (3) What quantity of land does a 
 
 Fig., 
 
 A C 
 
 1 AB. or AO Fir R. m DE. "AKBDA. « Fig. 9. p Fig. 10. 
 
ASSISTANT^] MENSURATION OF SUPERFICIES. 
 
 107 
 
 rhombus contain, the base of which is 1490, and the perpen- 
 dicular breadth 1 280 links ? 
 
 Problem 2. Tojind the area of a Triangle. 
 
 Rule. Multiply the base by the alti- 
 tude, and half the product will be the area. 
 
 (1) Required the number of square 
 yards in a triangle, whose base is 49 
 feet, and height 25^ feet. 
 
 (2) What is the area of the gable 
 of a house, the base or distance between 
 the eaves being 22 feet 5 inches, and 
 the perpendicular from the ridge to the 
 middle of the base, 9 feet 4 inches ? A D B 
 
 Rule 2. When the three sides only are given. — From half 
 the sum of the sides subtract each side severally: multiply the 
 half sum and the three remainders continually together ; and 
 the square root of their product will be the area. 
 
 (3) The three sides of a triangular fish-pond, are 140, 
 336, and 415 yards respectively. What is the value of the 
 land which it occupies, at £225. per acre ? 
 
 problem 3. Tojind the area of a 
 Trapezium, or a Trapezoid. 
 
 Rule. Divide the trapezium 
 into two triangles by a diagonal: 
 multiply the diagonal by the sum 
 of the two perpendiculars falling 
 upon it ; and half the product 
 will be the area. 
 
 That is, 
 
 DE + BF x AC 
 
 = the area. 
 
 For a trapezoid. Multiply the sum of the two parallel 
 sides by the perpendicular distance between them ; and half 
 the product will be the area. 
 
 (1) How many square yards of paving are in the trapezi- 
 um, whose diagonal is 65 feet, and the perpendiculars 28 and 
 
 33$ feet ? 
 
 (2) Find the area of a trapezium whose south side is 2740 
 links, east side 3575, west side 4105, and north side 3755 
 Links ; and the diagonal from the south-west to the north-east 
 angle 4835 links. 
 
1-68 
 
 MENSURATION OF SUPERFICIES. |_THE TUTOR'S 
 
 (?) Required the area of a trapezoid whose parallel sides 
 are v 20i feet and 12{ feet respectively; the perpendicular 
 
 distance being lOf feet? 
 
 (4) How many square feet are in a board, whose length 
 is 12i feet, and the breadths of the two ends 15 inches and 
 1 1 inches respectively ? 
 
 Problem 4. To find the area of an Irregular Figure. 
 Rule. Divide it, by drawing diagonals, into trapeziums 
 and triangles. Find the area of each, and their sum will be 
 the area of the whole. 
 
 1 . Required the content of the 
 irregular figure abcdefga, in 
 which are given the following 
 diagonals and perpendiculars : 
 namely, ac = 5*5 
 
 fd == 5'2 
 gc = 4*4 
 cm =z 1*3 
 Bn = 1-8 
 go =z 1*2 
 Ep = 0-8 
 Dq = 2*3 
 
 Problem 5. To find the area of a Regular Polygon. 
 
 Rule 1. Multiply the perimeter (or sum of the sides) by 
 the perpendicular drawn from the centre to one of the sides ; 
 and half the product will be the area. 
 
 Rule 2. Multiply the square of the side by the correspond- 
 ing tabular area, or multiplier opposite to the name in the 
 following table ; and the product will be the area. 
 
 No. of 
 sides. 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 Names of Polygons. 
 
 Trigon, or Equilateral Triangle 
 
 Tetragon, or Square 
 
 Pentagon 
 
 Hexagon 
 
 Heptagon 
 
 Octagon 
 
 Nonagon 
 
 Decagon 
 
 Undecagon 
 
 Duodecagon 
 
 Areas, or 
 Midtipliers. 
 
 0*4330127 
 1-0000000 
 1-7204774 
 2-59807^2 
 3-6339124 
 4-8284271 
 6-1818242 
 7-6942088 
 
 9'3656399 
 11-1961524 
 
ASSISTANT.] MENSURATION OF SUPERFICIES. l6Q 
 
 (1) Required the area of a regular pentagon whose side is 
 25 feet. 
 
 (2) Required the area of an octagon whose side is 20 feet. 
 
 Problem 6. To find the Diameter or Circumference cf a Circle 
 the one from the other.* 
 
 Rule. As 7 : 22t) 
 or, as 113 : 355% >' 
 or, as 1 : 3'14l6§ J 
 diameter : the circumference ; and reversing the terms will 
 find the diameter. 
 
 (1) Required the circumference of a circle whose diameter 
 is 12.f 
 
 * To find the proportion which the circumference bears to the 
 diameter, and thence to find the area of a circle, is a problem that has 
 engaged the anxious attention of mathematicians of all ages. It is 
 now generally considered impossible to determine it exactly; but 
 various approximations have been found, some of which have been 
 carried to so great a degree of accuracy, that in a circle as immense 
 in magnitude as the orbit of the planet Saturn, the diameter of which 
 j3 about 158 millions of miles, we are enabled to express the circum- 
 ference (the diameter being given) so nearly approximating to the truth, 
 as not to deviate from it so much as the breadth of a single hair. 
 The three approximations in the Rule are those in general use. 
 
 •j- This is the ratio assigned by Archimedes, a celebrated philoso- 
 pher of Syracuse, who flourished about two centuries before the 
 Christian era. It answers the purpose sufficiently well when par- 
 ticular accuracy is not required. 
 
 X This was discovered by Metius, a Dutchman, about two centuries 
 since. It is a very good approximation, agreeing with the truth to 
 the sixth figure. 
 
 § This is an abridgment of the celebrated Van Ceulen's proportion, 
 who was nearly contemporary with Metius. By a patient and lost 
 laborious investigation, he determined it truly to 36 places of figures. 
 (3-141598, &c.) But it has been since extended to considerably more 
 
 than 100 places This proportion is extremely convenient, from the 
 
 circumstance of the first term being unity ,• which saves the labour o| 
 division, in finding the circumference of any other circle whose diame- 
 ter is given. It is not quite so accurate as the preceding. 
 
 % As 7 : 22 : : 12 : 22 X 11 = 264> — 37-714285 
 
 ott v 19 >thecircum- 
 
 or, as 113 : 355 : : 12 : ~ — = 37-699115 ( ference. 
 
 113 
 
 ,r, as 1 : 3*1416 : : 12 : 3-1416 X 12 = 37-6992 
 
 
170 MENSURATION OF SUPERFICIES. L THE TUTOR'S 
 
 (2) What is the circumference when the diameter is 45 ? 
 
 (3) What is the diameter of a column whose circumference 
 is 9 feet 6 inches ? 
 
 (4) If the circumference of a great circle of the earth (as 
 the equator) were exactly 25000 miles, what would be the 
 diameter ? 
 
 Problem 7. To find the area of a Circle. 
 Rule 1. The area is equal to a fourth part of the product 
 of the circumference into the diameter ; or, the product of 
 half the circumference into half the diameter. 
 
 1 X 3*14l6 
 Therefore, when the diameter is 1, the area = 
 
 = '7854 ; whence we have 
 
 Rule 2. Multiply the square of the diameter by *7854 ; 
 and the product will be the area. 
 
 Rule 3. Multiply the square of the circumference by 
 •07958 for the area. 
 
 (1) Required the area of the circle proposed in Example 1, 
 Problem 6.* 
 
 v 2) Find the area of the circle proposed in Example 2 
 Problem 6. 
 
 (3) What is the area of tfte end of a roller whose diamete\ 
 is 2 feet 3 inches ? 
 
 (4) Required the area when the circumference is 8^ feet. 
 
 Problem 8. To find the side of a square inscribed in a circle. 
 
 Rule. Multiply the radius by 1*4142 (that is by */2) or, 
 multiply the diameter by '707 l.t 
 
 (1) Find the side of the square inscribed in the circle 
 whose diameter is 12. 
 
 (2) What is the side of the square inscribed in a circle 
 whose diameter is 6 feet 5 inches ? 
 
 • 37 ' 6 " 2 X — = 37-6992 X 3= 113-0976) 
 
 4 > the area, 
 
 or, 122 X/7854 = 12 X 12 X -7854 == 113*0976 J 
 T The following Rules exhibit the principal relations between the 
 :itcie and its equal square, or inscribed square. 
 J. The diameter X -8862269 ) ^ > A c , 
 
 2. The circumfer. X -2820948 } == the Slde of an e 4 ual S( l uare - 
 S. The diameter X -7071068) 
 
 4. The circumfer. X'2250791 >- —the side of the inscribed square 
 
 5. The are* .. -6366 i^" 
 
MENSURATION Ob' SUPERFICIES. 
 
 173 
 
 Problem 9. To find the length of a circular arc 
 Rule 1. From 8 times Fig. 8. jp 
 
 the chord of half the arc 
 subtract the chord of the 
 whole arc, and \ of the 
 difference will be the 
 length of the arc, nearly. 
 That is AD x 8— AB 
 *-3=arcADB. F C <* 
 
 . Note. Half the chord of the whole arc, the chord of half the arc, 
 and the versed sine, are sides of a right angled triangle ; any two of 
 which being given, the third may be found as directed in page 117. 
 
 Rule 2. Multiply the number of degrees in the arc by the 
 radius, and the product by # 01745, for the length of the arc 
 
 (1) The chord of the whole arc is SO, and the versed sine 
 8 : what is the length of the arc ? 
 
 (2) What is the length of the arc when the chord of the 
 half arc is 10*625, and its versed sine 5 ? 
 
 (3) Required the length of an arc of 12° 10', the radius 
 being 10 feet. 
 
 Problem 10. To find the area of a Sector of a circle. 
 
 Rule 1. Multiply the length 
 of the arc by the radius, and 
 half the product will be the 
 area. 
 
 Rule 2. As 360° : the de- 
 grees in the arc : : the area of 
 the circle : the area of the 
 Sector. 
 
 Fig. 9- 
 
 (1) Required the area of the sector, when the radius is 15, 
 and the chord of the whole arc 18 feet. 
 
 (2) What is the area of a sector whose arc is 147° 29', and 
 the radius 25 ? 
 
 (3) Required the area of a sector whose radius is 20 feet, 
 and the versed sine 1 foot 9 inches.* 
 
 6. The side of a square Xl'414214— the diameter ) of its circum- 
 
 7. The side of a square X4'442883n:the circumf. j" scribing circle. 
 
 8. The side of a square X 1*1 2837 9=z the diameter \ of an equal cir- 
 
 9. The side of a square X3*544908— the circumf. ( cle. 
 
 * By the properties of the circle, the versed sine X the remaining 
 part of the diameter = the square of half the chord of the arr ; whence 
 all the requisites may be found. 
 
172 MEN SURATION of SUPERFICIES. [THE TUTOR'S 
 
 (4) What is the area of the sector, when the chord of half 
 tts arc is 14 feet 2 inches, and the versed sine 6 feet 8 inches ?* 
 Problem ll. To find the area of a circular Segment. 
 
 Rule 1. Find the area of the sector ; and also the area of 
 the triangle formed by the chord and the two radii of the 
 sector : their difference, when the segment is less than a 
 semicircle, or their sum, when it is greater, will be the area 
 of the segment. 
 
 Rule 2. Divide the height of the segment by the diameter, 
 and find the quotient in the column of heights in the follow- 
 ing table. Multiply the corresponding area by the square of 
 the diameter, for the area of the segment.t 
 
 Table of the Areas of Circular Segments. 
 
 <<£. 
 
 Area 
 
 f«a? 
 
 Area 
 
 ><z. 
 
 Area 
 
 -5i 
 
 Area 
 
 
 of 
 
 H> 
 
 of 
 
 'M 
 
 of 
 
 
 of 
 
 8 
 
 01 
 
 Segment. 
 
 
 Segment. 
 
 &3 
 
 Segment. 
 
 ft 
 
 Seg7nent. 
 
 •00133 
 
 •14 
 
 •06683 
 
 •26 
 
 •16226 
 
 '39 
 
 '28359 
 
 •02 
 
 •00375 
 
 •15 
 
 •07387 
 
 •27 
 
 •17109 
 
 •40 
 
 •29337 
 
 •03 
 
 •00687 
 
 •16 
 
 •08111 
 
 •28 
 
 •18002 
 
 •41 
 
 •30319 
 
 •04 
 
 •01054 
 
 •17 
 
 •08854 
 
 •29 
 
 •18905 
 
 •42 
 
 -3i304< 
 
 •05 
 
 •01468 
 
 •18 
 
 •09613 
 
 •30 
 
 •19817 
 
 •43 
 
 •32293 
 
 •06 
 
 •01924 
 
 •19 
 
 •10390 
 
 •31 
 
 •20738 
 
 •44 
 
 •33284 
 
 •07 
 
 •02417 
 
 •20 
 
 •11182 
 
 •32 
 
 •21667 
 
 •45 
 
 •34278 
 
 •08 
 
 •02944 
 
 •21 
 
 •11990 
 
 '33 
 
 •22603 
 
 •46 
 
 •35274 
 
 '09 
 
 •03501 
 
 •22 
 
 •12811 
 
 •34 
 
 •23547 
 
 •47 
 
 •36272 
 
 •10 
 
 •04088 
 
 •23 
 
 •13647 
 
 '35 
 
 •24498 
 
 •48 
 
 •37270 
 
 •11 
 
 •04701 
 
 •24 
 
 •14494 
 
 '36 
 
 •25455 
 
 .49 
 
 •38270 
 
 •12 
 
 '05339 
 
 •25 
 
 '15355 
 
 -37 
 
 •26418 
 
 •50 
 
 •39270 
 
 -13 
 
 •06000 
 
 
 
 •38 
 
 •27386 
 
 
 
 (1) What is the area of a segment, when the chord of the 
 whole arc is 60, and the chord of half the arc 37^ ? 
 
 * When the half chord (see AE, Fig. 7) cf the arc is found by the 
 properties of a right angled triangle, then AE 2 = the versed sine 
 (DE) X the remaining part of the diameter; whence the diameter 
 (and consequently the radius) will be known. 
 
 •j- When there is a remainder (or fraction) after the second quotient 
 figure, in dividing the height by the diameter ; having taken out the 
 area answering to the two figures, add to it such fractional part of the 
 difference between that and the next succeeding area, for the sake of 
 greater accuracy. 
 
ASSISTANT.] MENSURATION OF SUPERFICIES. 173 
 
 (2) What is the area of a segment whose height is 18, and 
 jhe diameter of the circle 48 ? 
 
 (3) Required the area of a circular segment whose height 
 is 2, and chord 20. 
 
 (4) What is the area of the segment of a circle whose radius 
 is 24, the chord of the whole arc 20, and the chord of haL 
 the arc 10*2 ? 
 
 (5) If the radius of a circle is 10 feet, what is the area or 
 the segment whose chord is 12 feet? 
 
 Problem 12. To find the circumference of an Ellipse, the 
 transverse and conjugate diameters being given. 
 Rule. Multiply the square Fig. 10. 
 root of half the sum of the C 
 squares of the two diameters ^^-— j . 
 by 3' 141 6, and the product /^ \ >v 
 will be the circumference / \ 
 nearly.* r ""f yB 
 
 (1) What is the circum- \. [ y 
 ference of an ellipse whose ^^^^^\^^^^^ 
 Sransverse diameter is 24, J) 
 
 jtnd conjugate 18? 
 
 (2) The two axes of an ellipse are 60 and 45 yards respec- 
 tively : what is the circumference ? 
 
 Problem 13. To find the area of an Ellipse. 
 
 Rule. Multiply the product of the axes by -7854, for 
 the area. 
 
 (1 ) Required the area of an ellipse whose axes are 35 and 25. 
 
 (2) What will be the expense of trenching an elliptic gar- 
 den, whose axes are 70 and 50 feet, at 3fd. per square yard ? 
 
 (3) Required the area of the ellipse in Grosvenor Square, 
 London ; the transverse diameter being 8*40 chains, and the 
 conjugate 6*12 chains. 
 
 Problem 14. To find the area of an Elliptic Segment , the 
 base being parallel to either axis. 
 
 Rule. Divide the height of the segment by that axe of 
 which it is a part, and find, in the Table of Circular Segments 
 a versed sine equal to the quotient. 
 
 * If the half sum of the two diameters be multiplied by 3-1416, the 
 product will give the circumference sufficiently near for most practical 
 purposes. 
 
174 A COLLECTION OF QUESTIONS. [jTHE TUTOR'S 
 
 Multiply the corresponding tabular area and the two axes 
 continually together, and the product will be the area re- 
 quired. 
 
 (1) What is the area of an elliptic segment cut off by a 
 line (called a double ordinate) parallel to the conjugate diam- 
 eter, at the distance of 36 yards from the centre ; the axes 
 being 120 and 40 yards respectively ? 
 
 (2) Required the number of square yards in the segment 
 of an ellipse, cut off by an ordinate parallel to the transverse 
 diameter ; the height being 5 feet, and the two axes 35 and 
 2.5 feet respectively. 
 
 A COLLECTION OF QUESTIONS. 
 
 (1) What is the value of 14 barrels of soap, at 4^d. per lb 
 each barrel containing 254 lb, f Ans. £66..13..6. 
 
 (2) A and B joined in partnership ; A put into the join< 
 stock £320. for 5 months, and B £460. for 3 months : they 
 gained £100. What is each man's share of the gain ? 
 
 Ans. As £53..13..9f 3§. and B's £46..6..2/ 5 V 
 
 (3) How many yards of cloth, at 17*. 6d. per yard, can J 
 have for 13 cwt. 2 qrs. of wool, at 14c?. per lb. ? 
 
 Ans. 100 yards, 3 J qrs. 
 
 (4) If I buy 1000 ells of Flemish linen for £90. at what 
 price must I sell it per English ell, to gain £10. by the 
 whole ? Ans. 3s. 4>d. per ell. 
 
 (5) A has 648 yards of cloth, at 14*. per yard, ready 
 money, but in barter will have l6s. B has wine at £42. per 
 tun, ready money : what must he charge it per tun in barter, 
 and what quantity must be given in exchange for the cloth ? 
 
 Ans. £48. per tun. and the quantity, 10 tuns, 3 hhds. 12 § gals. 
 
 (6) A jeweller sold jewels to the value of £1200. for which 
 he has received in part 876 French pistoles, at Ids. 6d. each. 
 How much more is due to him ? Ans. £477..6. 
 
 (7) An oilman bought 417 cwt. 1 qr. 15 lb. gross weight 01 
 :rain oil, tare 20 lb per cwt. how many neat gallons were 
 there, allowing 7f lb. to a gallon? Ans. 5120 gallons. 
 
 (8) If I buy cloth at 14s. 6d. per yard, and sell it at l6s. 
 $d. what is the gain per cent. ? Ans. £l5..10.A Q %. 
 
 (9) Bought 27 bags of ginger, each weighing gross S4| lb 
 tare If lb. per bag, tret as usual, what is the value at S\d 
 per lb. ? Ans. £76..13..2£. 
 
ASSISTANT.] A COLLECTION OF QUESTIONS. 375 
 
 (10) If § oz. cost ?* what will £ lb. cost ? Ans. 17 s. 6a\ 
 
 (11) If £ of a gallon cost f of a £. what will § of a tun 
 cost ? Ans. £105. 
 
 (12) A gentleman who spends one day with another 
 £l..7..10^. lays up at the year's end £340. What is his an- 
 nual income ? Ans. £848..14..4^. 
 
 (13) What is the difference in ounces, between 13 fothers 
 of lead, and 39 boxes of tin, each box weighing 388 lb. f 
 
 Ans. 2 1 2 1 60 ounces, 
 
 (14) A captain, commanding a crew of 160 mariners, Cap- 
 tured a prize worth £1360. The captain was allowed one- 
 fifth, and the rest was equally divided among the sailors. 
 What was each man's share ? 
 
 Ans. The captain had £272. and each sailor £6..l6.« 
 
 (15) At what rate percent, will £956. amount to £1314.. 
 10. m 7\ years, at simple interest ? Ans. £5. per cent. 
 
 (16)* A has 24 cows worth £3.. 12. each, and B 7 horses 
 worth £13. each. How much will make good the differ- 
 ence, in case they interchange their droves of cattle ? 
 
 Ans. £4.. 12. 
 
 (17) A man left £120. to be given to three persons, A, B, 
 *nd C ; B to have twice as much as A, and C as much as A 
 end B ; what was the share of each ? 
 
 Ans. A £20. B £40. and C £60. 
 
 (18) £1000. is to be divided among three men, in such a 
 manner, that if A has £3. B shall have £5. and C £8. How 
 much will each man have ? 
 
 Ans. A £187»10. B £312..10. and C £500. 
 
 (19) A piece of wainscot is 8 feet 6^ inches long, and 2 feet 
 9f inches broad : what is the superficial content ? 
 
 Ans. 24 feet 0' 3'[ 4'" 6"". 
 
 (20) A garrison of 360 men, who had originally six months' 
 provisions, having endured a siege of 5 months, without ob- 
 taining any relief or fresh supply, wish to know how many 
 men must depart, that the provisions may suffice for the resi- 
 due 5 months longer ? Ans. 288 men. 
 
 (21) The less of two numbers is 187; the difference 34 
 The square of their product is required. Ans. 1 707920929- 
 
 (22) A butcher sent his man with £21 6. to a fair to buy 
 cattle ; he bought oxen at £ll. cows at 40s. colts at £l..5. 
 and hogs at £l..l5. each, and of each a like number. What 
 was the number of each? Ans. 13 of each sort, and £8. over. 
 
 (23) What number added to llf will produce 36|f I ? 
 
 Ans. 24fi§. 
 
176 A COLLECTION OF QUESTIONS. [jTHE TUTORV 
 
 (24) What number multiplied by f will produce H^V? 
 
 Ans. 26ff. 
 
 (25) What is the value of 179 hogsheads of tobacco, each 
 weighing 13 cwt. at £2..7-.l. per cwt. ? Ans. £5478..2..11. 
 
 (26) My factor informs me that he has bought goods on 
 my account, of the value of £500.. 1 3..6. What will his com- 
 mission come to at £3^. per cent. ? Ans. £l7..10..5..2|f qrs 
 
 (27) If J of 6 were three, what would £ of 20 be ? " 
 
 Ans. 7|. 
 
 (28) Reduce 3 qrs. 14 lb. to the decimal of a cwt. 
 
 Ans. *875 cwt. 
 
 (29) How many lb. of sugar, at 4|eZ. per lb. must be given 
 in barter for 60 gross of inkle, at 8*. 8d. per gross ? 
 
 Ans. 1386% lb. 
 
 (30) If I buy yarn for Qd. per lb. and sell it again for 13|<f. 
 per lb. what is the gain per cent. ? Ans. £50. 
 
 (31) A tobacconist mixes 20 lb. of tobacco at Qd. per lb. 
 with 60 lb. at 12c?. per lb. 40/6. at I8d. per lb. and 12 /6. at 
 
 oer lb. what is a pound of the mixture worth ? 
 
 Ans. ls.2ld.j\. 
 
 (32) What is the difference between twice eight and twenty 
 and twice twenty-eight ; also between twice five and fifty, 
 and twice fifty-five ? Ans. 20, and 50. 
 
 (33) Whereas a noble and a mark just 15 yards did buy ; 
 How many ells of the same cloth for £50. had I ? 
 
 Ans. 600 ells. 
 
 (34) A broker bought for his principal, in the year 1720, 
 £400. South Sea stock, at £650. per cent, and sold it again 
 when it was worth but £130. per cent. What was the whole 
 loss ? Ans. £2080. 
 
 (35) C has candles at 6s. per dozen ready money, but in 
 barter will have 6s. 6d. per dozen ; D has cotton at Qd. per 
 lb. ready money : what price must the cotton be charged in 
 barter; and how much must be exchanged for 100 dozen of 
 candles ? Ans. The cotton at 9f <£ per lb. and the quantity, 
 
 7 cwt. qrs. 1 6 lb. 
 
 (36) If a clerk's salary is £73. a year, what is that per 
 day ? Ans. 4.y. 
 
 (37) B has an estate of £53. per annum, and pays 5s. lOd. 
 to the subsidy : what must C pay, whose estate is worth 
 £100. per annum ? Ans. lis. 0/~d. 
 
 (38) If I buy 100 yards of riband, at 3 yards for a shilling, 
 and 100 yards more at 2 yards for a shilling, and sell the 
 
ASSISTANT.] A COLLECTION OF QUESTIONS. 177 
 
 whole at the rate of 5 yards for 2 shillings ; whether do I 
 gain or lose, and how much ? Ans. lose 3s. 4>d. 
 
 (39) From what number must f be deducted, that the 
 remainder may be ^ ? Ans. |g. 
 
 (40) A farmer wishes to mix rye at 4>s. a bushel, barley at 
 3.?. and oats at 2s. How much must he take of each to sell 
 the mixture at 2*. 6d. per bushel ? 
 
 Ans. 6 of rye, 6 of barley, and 24 of oats, 
 
 (41) If f of a ship is worth £3740. what is the value of 
 the whole ? Ans. £9973. .6..8. 
 
 (42) Bought a cask of wine for £62..8. at 5s. 4d. per gallon. 
 How many gallons were there ? Ans. 234. 
 
 (43) A dit sipated young fellow in a short time got through 
 I of his fortune ; he then gave £2200. for a commission in 
 the army : his profusion continued till he had no more than 
 880 guineas left ; which was fo of his money after the com- 
 mission was bought. What was his fortune at first ? 
 
 Ans. £10450.' 
 
 (44) A sum of money is to be divided amongst four men, 
 so that the first shall have \, the second \, the third \, and 
 the fourth the remainder, which is £28. What is the sum ? 
 
 Ans. £112. 
 
 (45) What is the amount of £1000. in 5^ years, at £4|. 
 percent, per annum, simple interest ? Ans. £l26l..5. 
 
 (46) Sold goods amounting to the value of £700. at two 
 4< months. XVhat is the present worth, at £5. per cent, 
 per annum, simple interest ? Ans. £682..19"5^ - t \V T . 
 
 (47) A room 30 feet long, and 18 feet wide, is to be 
 covered with painted cloth. How many yards of three- 
 quarters wide will cover it ? Ans. 80 yards. 
 
 (48) Betty told her brother George, that though her mar- 
 riage portion took £19312. out of her family, it was but § of 
 two years' rent. What was his annual income ? 
 
 Ans. £l6093..6..8. 
 
 (49) A gentleman having 50^. to pay among his labourers 
 for a day's work, gave to every boy 6d. to every woman 8c?. 
 and to every man 1 6d. There was an equal number of each 
 description. What was that number? Ans. 20 of each. 
 
 (50) What is the solid content of a stone that measures 
 4 feet 6 inches long, 2 feet 9 inches broad, and 3 feet 4 inches 
 deep ? Ans. 41^ solid feet. 
 
 (51) What does the pay of a ship's crew, consisting of 
 640 sailors, amount to for 32 months' service, each man's pay 
 being 22*. 6d. per month ? Ans. £23040. 
 
 h 5 
 
ITS SUPPLEMENTAL QUESTIONS. QrHE TUTOR S 
 
 (52) A traveller would change 500 French crowns, at 4 s. 
 6d. per crown, into sterling money; but he must pay a half- 
 penny per crown for change. How much sterling money 
 will he receive ? Ans. £ 1 1 1..9..2. 
 
 (53) B and C traded together, and gained £100; B put 
 in £640 ; C put in so much that he was entitled to £60. of 
 the ^ain. What was C's stock ? Ans. £960. 
 
 (54) From what principal sum did £20. interest arise in 
 one year, at the rate of £5. per cent, per annum ? Ans. £400. 
 
 (5 ) How many French pistoles, at 17s. 6d. each, are equal 
 to 672 Spanish guilders, at 2*. each ? Ans. 76f . 
 
 (56) Out of 7 cheeses, each weighing 1 cwi. 2 qrs. 5 lb. 
 how many allowances for seamen may be cut, each weighing 
 5 oz. 7 drams? Ans. 3563|f. 
 
 (57) If 48 taken from 120 leaves 72, and 72 taken from 
 91 leaves lQ f and 7 taken from thence leaves 12, what num- 
 ber is that, out of which, when you have taken 48, 72, 19, 
 and 7, leaves 12 ? Ans. 158. 
 
 ^58) A farmer, unskilled in numbers, ordered £500. to 
 be divided among his 5 sons, thus : " give A, says he, ^, B £, 
 C J, D ^, and E \ part." Divide this equitably among them 
 according to the father's intention. 
 
 Ans. A £152..10..4 iff. ££ll4./7..6f ffg 
 C£91..}0..0| #fc. £>£76..5..0iff§. £ £65..7..2i ¥ V S . 
 (59) When first the marriage knot was tied 
 Between my wife and me, 
 My age did hers as far exceed, 
 
 As three times three does three , 
 But when ten years, and half ten years, 
 
 We man and wife had been, 
 Her age came then as near to mine, 
 As eight is to sixteen. 
 Quest. What was each of our ages when we married ? 
 Ans. 45 years the man, 15 the woman. 
 
 SUPPLEMENTAL QUESTIONS. 
 
 (1) How many gallons of the imperial standard measure 
 are respectively equal to a hogshead of wine, and a hogshead 
 of ale, old measure ; and what was the difference between 
 the two hogsheads in cubic inches ? 
 
 (2) What quantity of the old ale measure would corres- 
 pond to 54 gallons of the imperial standard ? 
 
ASSISTANT.] SUPPLEMENTAL QUESTIONS. 179 
 
 (3) How many gallons of the old wine measure are equal 
 in quantity to 63 gallons, imperial measure ? 
 
 (4) Reduce 15 quarters, 3 bushels, 1 peck, old measure, to 
 its equivalent in the imperial standard measure. 
 
 (5) A lady who was asked the time of the day, said that 
 it was between three and four : but being desired to name 
 the exact time, she replied ; " the minute hand is advanced 
 half an hour precisely before the hour hand." Required the 
 exact time. 
 
 (6) If 7 inen can build a wall 40 yards long, 4 feet high, 
 and 2 feet thick in 32 days ; how many men will build a wall 
 240 yards long, 6 feet high, and 3 feet thick, in 8 days ?* 
 
 (7) The weight of a certain bar of iron 2 feet long, 3 
 inches broad, and 1 inch thick, is 20 lbs. What is the weight 
 of a bar cf similar quality which is 7i feet long, 4^ inches 
 broad, and 3\ inches thick? 
 
 (8) A person who had five-ninths of a mine, made his 
 younger brother a present of half his share, and sold half the 
 remainder to his cousin John, who soon after purchased ^ of 
 the younger brother's share ; but now offers to dispose of 
 half his interest in the mine for £ 150. Estimating at the 
 same rate, what is the value of the whole mine, and of each 
 brother's share ? 
 
 (9) A, travelling from London to Manchester, and B, 
 from Manchester to London, set out at the same time. They 
 meet at the end of six days, A having travelled 3 miles a day 
 
 * Questions of Compound Proportion in which the terms are numer- 
 ous may be solved by Rule 1, for the Double Rule of Three ; but 
 the following method is more convenient. 
 
 Rule. Arrange the terms of the first cause and effect in one line, 
 and the corresponding terms of the second cause and effect exactly under 
 them ; supplying the place of the term sought with an asterisk^ and 
 conneccing the contrary causes and effects by cross lines. Multiply 
 continually the terms of each cause and the other effect : divide the 
 product arising from the full number of terms, by the product of those 
 with which the blank term is connected, and the quotient will be the 
 answer. 
 
 Solution oftlie above example. 
 1 
 
 men days buM y. long ft.h. ft. th. 4 6 3 
 
 If 7 : 32 v 40 : 4 : 2 > 7X^X^X0X3 
 
 • : 8 X 240 , 6 : 3f : £X^X*X;5 ^ 
 
 a wall. 1 111 An *~ 
 
180 SUPPLEMENTAL QUESTIONS. [THE TUTOR'S 
 
 more than B. At what rate did each go, the distance being 
 186 miles? 
 
 (10) A coach which runs the whole distance in 31 hours, 
 starts from London at the same time that another which does 
 it in 2 1 hours, starts from Manchester. Required the num- 
 ber of hours elapsed, and the distance travelled by each when 
 they meet. 
 
 (11) A load of corn was sold for £ 15. at a loss of 15 per 
 cent. What should it have been sold for to gain as much per 
 cent, as the corn cost ? 
 
 (12) Two men purchased a grinding stone 42 inches in 
 diameter for a guinea • of which the first paid twelve shil- 
 lings. They agree that the first shall use it till his share is 
 worn down. What will be the diameter when the second 
 receives it ? 
 
 (13) If A and B together do a piece of work in 7f days, 
 which A alone would accomplish in 12^ days ; in what time 
 would B do it himself? 
 
 (14) A person lent £400. and agreed to receive in return a 
 yearly payment of £50. for 13 years. Whether would he 
 gain or lose thereby, reckoning Compound Interest at £5. 
 per cent, per annum ? 
 
 (15) By selling a horse for £50. I gained one-fourth of what 
 he cost me ; but the whole cost (including the expense of his 
 keep) was one-fourth more than the original purchase. How 
 much did I give for him, what did I expend in keeping him, 
 and what did I gain per cent. ? 
 
 (16) It has been found by experiment, that sound is con- 
 veyed through the air at the rate of 1142 feet in a second. 
 How far distant is the cloud, when 7f seconds elapse between 
 seeing the flash of lightning and hearing the thunder ? 
 
 (17) What is the height of a tower that projects a shadow 
 75*75 yards long, at the same time that a perpendicular staff 
 3 feet high, gives a shade of 4*55 feet in length ? 
 
 (18) A bankrupt owes £2580. and the value of his effects 
 is £846. and the amount of recoverable debts £358.. 12. be- 
 sides which he has an unexpired lease that has 1 3 years to 
 run, valued at £l2. a year more than the stipulated rent. 
 If the lease be disposed of for present money, allowing 
 Compound Interest at £5. per cent, per annum ; and if the 
 working of the commission and other expenses amount to 
 £472 ; what will his creditors have in the pound, provided 
 they allow him £150. to recommence business ? 
 
 (19) A youth aged 12 years, having had bequeathed to him 
 
ASSISTANT.] SUPPLEMENTAL QUESTIONS. 181 
 
 an annuity of £50. for 1 2 years, to commence when he comes 
 of age ; the executors think it will be more advantageous to 
 exchange this for an annuity to commence immediately, and 
 continue till he is 21 ; to enable them to give him some edu- 
 cation and a trade. What will be such annuity, £100. being 
 reserved at the conclusion to set him up in business ? 
 
 (20) There is an island 73 miles in circumference; to travei 
 round which three pedestrians all start at the same time : 
 A travels 5 miles a day, B travels 8, and C 10 miles a day. 
 In how many days will they all come together again, and 
 how many circuits will each have made ? 
 
 (21) What will a banker charge for discounting a bill of 
 £52.. 10. on the 7th of April ; the bill being due on the 19th 
 of May? 
 
 (22) My agent in London having, advised me, that he has 
 purchased goods on my account to the amount of £756.. 10. 
 at six months credit, or £7|. per cent, discount for prompt 
 payment ; if I send a remittance of £400. to be paid down 
 on account, after deducting out of it his charge for commis- 
 sion at £2^. per cent. ; what will remain to be discharged at 
 die end of six months ?* 
 
 (23) If I insure a house for £250. at the annual charge of 
 1 *Z. per cent, and the furniture, &c. for £150. at the rate 
 of %s. 6d. per cent. ; what shall I have to pay yearly to the 
 Insurance office, including the duty paid to Government of 
 |, or 2s. 6d. per cent. ? 
 
 (24) If 12 oxen will eat 31 acres of grass in 4 weeks, and 
 21 oxen will eat 10 acres in 9 weeks, how many oxen will 
 eat 24 acres in 18 weeks, allowing the grass to grow uni- 
 formly ? Newton. 
 
 (25) A bath is supplied with water by two cocks ; from 
 one of which it may be filled in 40 minutes, and from the 
 other in 50 minutes : a discharging cock will empty it (when 
 filled) in 25 minutes. If all the three be opened at the same 
 time, in what time will the bath be filled, supposing the 
 influx and efflux to be uniform ? 
 
 (26) A person who had spent two-thirds of his money at 
 one place, and half the remainder at another, found that he 
 had £32..12. left. How much had he at first ? 
 
 * See Note to Example 11, Discount, page 73. 
 
182 SUPPLEMENTAL QUESTIONS. ]_THE TUTOR 
 
 (27) The length, breadth, and height of a room are 9, 6, 
 and 4 yards respectively. What is the longest right line that 
 can be taken within that room ? 
 
 (28) What must be the length of a cord with which a horse 
 may be tethered to a certain point in a straight fence, so as 
 to allow him the liberty of grazing to the extent of 1 rood ; 
 supposing that he can reach 2 yards beyond the tether ? 
 
 (29) A person playing at cards, lost three nights succes- 
 sively. The first night he lost half his sovereigns and half a 
 sovereign besides ; the second night he lost half the remain- 
 der and half a sovereign more ; and the third night half the 
 remainder and half a sovereign more, which reduced his 
 stock to twenty. How many sovereigns had he at first ? 
 
 (30) The month of July, 1828, was remarkable, both in 
 England, and several parts of the Continent, for excessive 
 rains. At Derby, the quantity collected in the pluviameier 
 (or rain-guage) between the hours of nine, A. M. of the 9th. 
 of that month, and six the following morning (an interval of 
 21 hours) was 3'59 inches: to the evening of the 15th. it 
 amounted to 7| inches ; and by the conclusion of the 29th, 
 an interval of 21 days, of which 10 only were very rainy, 
 the total depth of water collected was 11 J inches. How 
 many hogsheads of 54 and 63 imperial gallons respectively 
 fall on an acre of ground to amount to the depth of one inch ; 
 and how many hogsheads of each kind fell on the surface of 
 an acre during each of the three several intervals above men- 
 tioned ? 
 
 (31) A person who occupies a piece of ground for which 
 he pays a rent of £l0. per annum, wishes to take it upon a 
 lease for forty years, with the obligation of laying out upon 
 it during the present year £600. in the erection of a building, 
 which is to be left in good tenantable condition at the termi- 
 nation of the lease. The question is, how much will be a fair 
 annual rent for the lessee to pay, during the term of continu- 
 ance of such tenure, admitting the ground rent paid at present 
 to be a fair one ; and supposing the customary interest of 
 money to be at the rate of £5. per cent, per annum ? Also, 
 supposing interest to be at £4. per cent, per annum ? 
 
 (32) What will be the expense of covering and guttering 
 a roof with lead at 18«y. per cwt . the length of the roof be- 
 ing 43 feet, and the breadth or girt over it 32 feet ; the gut- 
 tering 57 feet long and 2 feet wide : the lead for the former 
 being 9*831 lb. and for the latter 7*373 lb. to the square foot? 
 
A COMPENDIUM OF 
 
 BOOK-KEEPING, 
 
 BY SINGLE ENTRY; 
 
 /'ntended for the purpose of initiating Youth in the LEADING 
 PRINCIPLES of that important Branch of Science 
 
 T300K-KEEPING is the art of recording pecuniary or commercia 
 -*-* transactions in a regular and systematic manner. 
 
 The science of Book-keeping admits of innumerable varieties of method : but its general 
 principles are invariable. These being well understood, the knowledge of any particular 
 system, adapted to the peculiar concerns of any counting-house, will be easily acquired. 
 
 Single Entry, being the most simple and concise, is the method usual- 
 ly adopted in retail business. 
 
 The General Rule to be observed in every system of Book-keep- 
 ing, is, 
 
 To make any person Debtor (Dr.) for money or goods which He re- 
 ceives from me, and to make him Creditor (Cr^^r whatever I receive 
 from him. 
 
 The books usually kept in Single Entry, are, the Day-Book, the 
 £ash-Book, the Ledger, and the Bill-Book. 
 
 The Day-Book, when a person commences business, begins with an 
 inventory of the existing state of his affairs : after which are entered, 
 in the regular order of time, the daily transactions of Goods bought 
 <md sold. 
 
 The Cash-Book contains the particulars of all Money transactions. 
 It is ruled in a folio form : on the left-hand page, Cash is debited to all 
 sums received, and on the right, Cash is credited by all sums paid. The 
 Balance (or quantity which the Dr. side exceeds the Cr.) shows the 
 amount of Cash in hand. This should be ascertained weekly, and in 
 some concerns daily, in order to prove if it corresponds with the real 
 Cash in hand. 
 
 In the Ledger are collected the dispersed accounts of each per^cS 
 from the Day-Book and Cash-Book, and entered in a concise manner 
 in one folio ; the sums in which he is Dr. being arranged on the left- 
 hand, and those in which he is Cr. on the right-hand page of the folio : 
 ' so that the Balance of his account (the difference between the Dr. and 
 Cr. sides) may always be easily ascertained by inspection. The trans- 
 ferring of accounts from the Day-Book and Cash-Book to the Ledger, 
 is called posting. 
 
 In many trades it is found convenient to keep the accounts of Goods Sold at the former 
 end, and those of Goods Bought at the latter end of the Ledger. But in concerns of mag- 
 nitude, two Ledgers are more convenient; one for Goods Sold, and the other, called the 
 ** Bought Ledger," for Goods Bought. 
 
 In the BILL-BOOK are copied the particulars of all Bills of Exchange, whether Receiv- 
 able or Payable. The former are those which come into the Tradesman's possession, and 
 are drawn upon someother person ; the latter are those which are drawn upon and accepted 
 Dy him — Printed Bill-Books may be had of any Bookseller. 
 
 Note. In the following transactions, Bills Receivable are considered as Cash : but many 
 Accountants do not enter them as such, till Cash has been actually received for them. 
 
( 184) 
 MEMORANDUMS 
 
 OP THE TRANSACTIONS STATED IN THE FOLLOWING BOOKS. 
 
 Jan. 5. Received from Allen, Wild, and Co. of Leeds, on credit, 3 
 pieces of super blue cloth, each 36 yds. at 25s. 6d. per yard ; — 
 and 2 pieces of narrow brown, 84 yds. at 4s. 9d. 
 
 8. Sold Bernard Mason 2 st. raw sugar, at 9^d. per lb. — 3± lb. green 
 tea, at Ss. 6d. — and 3f yds. blue cloth, at 28s. 
 
 9. Bought of Samuel Fletcher, 1 cwt. I qr. 5 lb. Kent hops, at £5,.7. 
 per cwt. — and 1^ cwt. of Worcester, at £5. .11. .6 ; — six months' 
 credit, or 5 per cent, discount for cash. — Paid him Bill, No. 1, 
 £24..3..9. and received from him a cheque on Smith and Co. 
 Bankers, for the difference, including discount. 
 
 10. Bought of Simmonds and Co. Liverpool, 2 cwt. yellow soap, at 
 76s. — 12 doz. dip candles, at Ss. 6d. — and 4 doz. mould do. at 
 11*. 3d. 
 
 14. Sold William Tomlinson 7 yds. narrow cloth, at 5*. 3c?. — and 
 15 yds. calico, at S^d. 
 
 19. Sold Hazard and Jones l£ st. yellow soap, at 9d. per lb. — ^ st 
 mottled, at 9±d. — 9 lb. candles, at*9c?. — and 3 lb. mould do. at I2<|c?. 
 Paid Cash for Bill, No. 1, W. Holmes, £45..10. 
 
 23. Received from J. Sanderson, goods as per invoice £7. .3. .6. 
 
 28. Sold Hazard and Jones 17^ lb. loaf sugar, at Is. Id. — 12 lb. ra^ 
 do. at lOd. — 1^ lb. Congou tea, at 7s. 6d. — and ^ lb. Hyson, at 12s. 
 
 31. Paid the balance due to J. Herdson, cash £37. .5. .6. abatement 
 4c?. — Sold him on credit 10 lb. hops, at 13d — and half a ream of 
 cap-paper, at 7c?. per quire. 
 
 Feb. 1. Sold W. Tomlinson 2 st. yellow soap, at 9c?. per lb. — 6 lb. 
 mould candles, at Is. lc?. — and 16 lb. lump sugar, at 12^c?. Re- 
 ceived of him cash for account due, including the present goods, 
 £4.. 13. abatement 3^d. 
 
 Paid Bernard Mason the balance due, cash £832 ; abatement 
 5s. 2\d. 
 
 4. Received of James Taylor \ yr's. interest on £70. due this day, 
 £1..15. 
 
 5. Sold W. Tomlinson 1 piece super blue cloth, 36 yds. at 27s. 6d. 
 — for Bill at one month. 
 
 8. W. Tomlinson paid me a Bill on Jones and Co. London, due 
 May 10, £60. which should have been at one month. — Charged 
 him discount, 10s. — Paid, agreeably to his order, the difference to 
 J. Sims, in cash, £10. 
 
 10. Bought of J. Sanderson, cheese 25 cwt. 3 qrs. 17 lb. at £3. .2.. 6. 
 per cwt. — Paid his whole account in cash, £88. abatement Is. 8-fc?. 
 Accepted Allen, Wild, and Co.'s Bill at two months, drawn 
 Jan. 3, £80. 
 
 12. Received of Hazard and Jones's assignee, cash £2.7..2|. for 
 composition on £3.. 15. .6. at 12.?. &/. in the £. 
 
( 185 ) 
 
 DAY-BOOK, (pagei) 
 
 Inventory, January 1st, 1830. 
 
 I have in Ready Money 
 Bills Receivable , No. L, on S. Johnson, due 
 29th instant ... 
 
 cwt. qr. lb. 
 Tea, 3 chests, gross wt. 3 1 17 
 tare 2 7 
 
 Duty included on 2 3 10 at 6s. 2d. per lb, 
 
 cwt. qr. lb. 
 Raw Sugar, 2 hhds. weighing 27 3 18 neat, 
 
 at £3.. 14.. 8. per cwt. 
 James Taylor owes me on bond, dated Aug. 
 
 4th, 1826, with interest at £5. per cent, per 
 
 annum - 
 
 I owe as follows : 
 John Herdson, a balance of accounts 
 Bernard Mason, for purchase of my house bv 
 
 auction, to be paid 1st Feb. next £800 \ 
 
 Duty on do. at £5. per cent. - 40 j 
 Bills payable, viz. No. 1, W. Holmes's bill, to 
 
 H. Williams or order, accepted by me, due 
 
 1 9th instant - 
 
 £. 
 
 1500 
 
 24 
 
 98 
 
 I 104 
 
 Allen, Wild, and Co. Leeds Or. 
 
 •fBy 3 pieces superfine blue cloth, each 36 
 yds. at 25s. 6d. per yd, - 
 2 pieces narrow brown, 84 yds. at 4s. 9d 
 Wrappers - 
 
 70 
 
 1796 8 
 
 Bernard Mason 
 +To 2 st. raw sugar, at 9\d. per lb. 
 3 J lb. green tea, at 85. 6d. 
 3| yds. blue cloth, at 28s. 
 
 Dr. 
 
 37 
 
 840 
 
 45 
 
 922 
 
 157 
 
 5 
 
 
 10 
 
 15 
 
 137 14 
 
 19 19 
 
 5 
 
 18 
 
 
 
 1 
 
 10 
 
 
 10 
 
 2 
 
 H 
 
 
 
 14 
 
 H 
 
 « This column contains the figures of reference to the folio of the Ledger in which any 
 account is posted. They should be written in red ink. 
 
 f The preposition By is always put before any article for which a person is credited. 
 
 £ The preposition To is generally placed before articles for which a person is debited* 
 but some Book-keepers omit it. 
 
186 
 
 day-book, (page 2) 
 
 1 R^O Tnn 
 
 i* 
 
 
 Samuel Fletcher Cr. 
 cwL qr. lb, £. s. d. 
 By Kent hops, 1 1 5 at 5 7 
 Worcester do. 1 2 at 5 11 6 
 Six months' credit, or £5. per cent, discount 
 for present payment - - . 
 10 
 
 6 
 
 8 
 
 18 
 
 7 
 
 6 
 3 
 
 2 
 
 15 
 
 5 
 
 9 
 
 
 Simmonds and Co. Liverpool Cr. 
 By Yellow soap, 2 act. at 76*. 
 1 2 doz. candles, at 8s. 6d. 
 4 doz. mould do. atlU. 3d. 
 
 14 
 
 7 
 5 
 2 
 
 12 
 2 
 5 
 
 
 
 
 
 2 
 
 14 
 
 19 
 
 
 
 
 William Tomlinson Dr. 
 To narrow cloth, 7 yds. at 5s. 6d. 
 calico, 15 at Os. 8$d. 
 
 19 
 
 1 
 
 
 18 
 10 
 
 6 
 
 3 
 
 2 
 
 9 
 
 H 
 
 
 Hazard and Jones Dr. 
 To l\ st. yellow soap, at 9d. per lb. 
 \ st. mottled do. at 9\d. 
 9 lb. candles, at 9d. 
 3 lb. moulds, at Is. 0\d. 
 
 °3 
 
 ! 
 i o 
 
 ; o 
 
 i o 
 
 15 
 5 
 6 
 3 
 
 9 
 
 H 
 
 9 
 
 H 
 
 S 
 
 ! i 
 
 11 
 
 2 
 
 s 
 
 James Sanderson Cr. 
 By goods as per Invoice* 
 °8 
 
 i 
 
 ! 7 
 
 3 
 
 6 
 
 
 Hazard and Jones Dr. 
 To 17-§Z&. loaf sugar, at Is. Id. 
 12 lb. raw do. at lOd. 
 Ij-lb. Congou tea, at 7*. 6d. 
 •| lb. Hyson, at 12$. 
 
 31 
 
 
 
 
 
 
 18 
 
 10 
 
 9 
 
 6 
 
 
 
 4i 
 
 
 
 s 
 
 2 
 
 4 
 
 4 
 
 
 John Herdson Dr. 
 To Hops 10 lb. at Is. Id. 
 
 i ream cap paper, at Id. per quire 
 
 - Feb 1 
 
 
 
 
 10 
 5 
 
 10 
 10 
 
 1 
 
 
 
 16 
 
 8 
 
 
 William Tomlinson Dr. 
 To 2 st. yellow soap, at 9J. per/6. 
 6 lb. mould candles, at 1*. Id. 
 16 lb. lump sugar, at ls.O±d. - 
 
 1 
 
 
 
 
 1 
 
 6 
 
 16 
 
 
 6 
 8 
 
 2 
 
 2 
 
 4 
 
 2 
 
 * Many tradesmen keep an Invoice Booh, into which all invoices of goods purchased are 
 regularly transcribed. Some paste the invoices themselves within a blank book. The ac- 
 count of Allen, Wild, and Co. (D. B. p. 1.) is supposed to be a copy from their invoice. 
 Some make the Invoice Book serve also for the, " Bought Ledger." This last method, 
 when the nature of the tra e will admit, is well worth adopting. 
 

 day-book, (page 3) 
 
 187 
 
 1 
 
 James Taylor Dr. 
 To half' a year's interest on £70. at £5. per 
 cent, per annum 
 
 5 
 
 1 
 
 15 
 
 
 
 2 
 
 William Tomlinson Dr. 
 To 1 i?te superfine blue cloth, 36 yds. at 
 
 27*. 6d. - 
 For bill at one month. 
 10 
 
 49 
 
 10 
 
 
 
 3 
 
 James Sanderson Cr. 
 cwt. qr. lb. £. s. d. 
 By cheese* 25 3 17 at 3 2 6 per cwt. 
 
 80 
 
 18 
 
 *i 
 
 2 
 
 Allen, Wild, and Co. Leeds Dr. 
 To my acceptance of their Bill at 2 mon. 1 
 drawn »rd Jan. B. P. Book, No. 2. J 
 
 JO 
 
 80 
 
 
 
 
 
 
 Oafo' Purveyance, in Partnership with 
 
 «/. Herdson.'f Dr. 
 To Cash, for Oats purchased by me % 
 
 Do. do. - - - by J. Herdson 
 Do. for warehouse room, &c. 
 
 Cr. 
 
 By Cash received for Oats sold - 
 
 Do do. - - - byJ.H. 
 
 £. s. d. Dr. 
 
 To Profit, % (42..13..5£) being my share \ 
 J. H. i (42..13..5-!) being his share J 
 
 26 
 449 
 
 1 
 
 11 
 
 
 
 13 
 
 
 3 
 8 
 
 3 
 
 477 
 
 4 
 
 11 
 
 
 507 
 55 
 
 9 
 2 
 
 2 
 8 
 
 3 
 
 562 
 
 11 
 
 10 
 
 3 
 
 85 
 
 6 
 
 11 
 
 
 John Herdson Dr. 
 To Cash advanced to him on Oats' concern 
 Oats sold and received for by him 
 
 Cr. 
 
 By Oats purchased by him 
 his share of profit 
 
 433 
 55 
 
 17 
 2 
 
 
 
 8 
 
 1 
 
 488 
 
 19 
 
 8 
 
 
 449 
 42 
 
 
 13 
 
 3 
 
 54 
 
 1 
 
 49i 
 
 13 
 
 H 
 
 * This is supposed to be bought by the long cwt. (120 lb.) agreeably to the general prac- 
 tice. 
 
 f Vide Example of a Partnership Concern, p. 189. 
 
 % In this concern, the entries of Cash are not made in the Cash-Book ; this being only a 
 general statement of transactions supposed I o have taken place before the opening of 
 these Books. 
 
138 
 
 CASH-BOOK. 
 
 S| O || o <o Ci 
 
 o 
 
 O O O -j 
 
 =1 
 
 ■J. 00 , , O V) i> 
 
 r- 
 
 O O O CM 
 
 21 
 
 ^ v> t- co 
 J <N I I n< tt cm 
 
 00 
 
 00 
 
 CM O CC *0 
 
 CO r-l CC CO 
 
 CO *o 
 
 21 
 
 f-i W 00 CO 
 
 c 
 o 
 
 cq 
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 t/2 
 
 
 OS 
 
 •s2*2 
 
 • !2 : : 
 
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 C- M oi G 
 
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 z2 73 . 5 c 
 
 ?! § 
 
 n ° 
 
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 Ci 
 
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 * ^ « 
 
 s 
 
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 J| § 
 
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 c ^ o J 
 
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 fd 
 
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 CO 
 
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 ^° 
 
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 <M 
 
 
 . o 
 
 CM 
 
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 CM 
 
 § 1 
 
 CM 
 
 
 
 •^ -i CM 
 
 ■fc 
 
 cZi 
 
 
 • £ 
 dp 
 
 • o 
 
 CD 
 
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 D CU 
 
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 rv 
 
 • g ■ * g g 
 
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 f-l sj 
 
LEDGER jNDEX. 
 
 18' 
 
 INDEX TO THE LEDGER* 
 
 | A Allen, Wild, and Co. 2 
 
 N 
 
 J} Bills Payable 
 
 2 
 
 Q Oats' Purveyance 
 
 3 
 
 c 
 
 P 
 
 D 
 
 Q 
 
 E 
 
 R 
 
 J? Fletcher Samuel 
 
 2 
 
 g Stock 
 
 Simmonds and Co. 
 Sanderson James 
 
 1 
 2 
 3 
 
 G 
 
 JJ Herdson John 
 Hazard and Jones 
 
 1 
 3 
 
 r p Taylov James 
 Tomhnson Wm, 
 
 1 
 
 2 
 
 I J 
 
 uv 
 
 K 
 
 w 
 
 L 
 
 XY 
 
 ]fyj Mason Bernard 
 
 1 
 
 Z 
 
 * The Ledger has an Alphabetical Index, showing at one view, in what folio any per. 
 son's account may be found. 
 
 EXAMPLE OF A PARTNERSHIP CONCERN. 
 
 John Herdson and I have been engaged in a joint concern as purveyors of oats for the 
 army. I have purchased oats for the joint stock to the amount of £26..11. J. H. has paid 
 for oats purchased by him, £449..0..3. I have received for oats that have been disposed 
 of, £507.. 9-2.; and my partner has received from the same source, £55.. 2..8. 1 have 
 advanced to him at different times £433..17« ; and have paid for warehouse room and 
 other sundry expenses £1..13..8. — From these general heads, collected from the particu- 
 lars recorded in the Granary Book, it is required to state the transactions. 
 
 It may perhaps be interesting to the learner, to be informed, that the above was a real 
 occurrence ; for an accurate statement of which the writer was some time since applied 
 to by the parties concerned. 
 
190 
 
 LEDGER, (folio 1) 
 
 tS °> 
 
 1 
 
 o o 
 
 o 
 
 O 
 
 1 QC 
 
 
 €> 
 
 gj CO 
 
 Vi o 
 
 W5 
 
 wj 
 
 CO 
 
 
 O 
 
 CO 
 
 r- O 
 
 l> 
 
 CO 
 
 Oi 
 
 
 o 
 
 CO 
 
 -°.6 
 
 c>33 
 
 3 
 
 T3 Oi 
 S3 2 
 
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 3-& Ct3 
 
 3-2 o « 
 
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 .s ° c S 
 
 i££ eg .a 
 
 
 5^3 i 
 
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 lit 
 
 £§- 8 
 
 CJ 4J 
 
 .8 'SB 
 
 45 ^ 
 
 5 a 
 
 Cm CD 
 O 3 
 
 <"> -5 
 
 a 
 
 w 
 a 
 
 p 
 w 
 
 
 
 8*" 
 
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 o 
 
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 nfc* 
 00 
 
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 o 
 
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 o »o 
 
 
 U5 
 
 CO Oi t- 
 
 <fl 
 
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 o 
 
 ^ Oi ^ 
 
 g- 
 
 1> 
 
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 CO 
 
 O00H 
 
 CO 
 
 Oi 
 
 ** CM 
 
 CO 
 CO 
 
 o 
 
 00 
 
 CO 
 
 •H 00 
 
 3* " 
 
 . 0* CO c2 
 
 <3 
 
 S7oc& D?. 
 
 To sundries, amount of 
 my debts 
 Balance account 
 
 Note. Stock is a term used to re- 
 present the Owner of the Books. 
 This account shows what he is 
 worth at the commencement of bu- 
 siness: and, when a general balance 
 is taken, will enable him to discover 
 the value of his property at that 
 time, and the gain or loss attending 
 trade. It cannot be carried on by 
 
 James Taylor Dr, 
 To money on bond 
 half a year's interest 
 
 John Herdson Dr, 
 To cash £37..5..6. abat. 4d. 
 
 To sundries 
 
 Do. on Oats' concern 
 
 Balance 
 
 Bernard Mason Dr. 
 To sundries 
 Cash £832. abat. 5s. 2$<J. 
 
 1830 
 Jan. 1 
 
 Feb. 12 
 
 1830 
 Jan. 1 
 Feb. 4 
 
 
 •-5 
 
 cn 
 
 
 1830 
 Jan. 8 
 Feb. 1 
 
 
LEDGER. (follO 2) 
 
 191 
 
 ■Jf? 
 
 
 co 
 
 C5 CO 
 
 o 
 
 o 
 
 He* 
 O 00 
 
 CO 
 
 O 
 
 
 it® 
 
 o 
 
 00 
 
 «5 CO 
 
 o> 
 
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 <43 
 
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 1-4 
 
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 « 
 
 c3 c3 c5 
 
 Contra Cr. 
 By Holmes's bill No. 1 
 
 Allen, Wild, and Co.'s bill, 
 No. 2 
 
 <6 
 
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 6. 
 
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 1830 
 Jan. 1 
 
 Feb. 10 
 
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 •-5 
 
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 Bills Payable Dr. 
 To Cash 
 
 Balance 
 
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 11 
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192 
 
 LEDGER. (foilO 3) 
 
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