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 ion 
 
PREFACE. 
 
 WHILE treatises on Arithmetic are already so numerous that 
 masters are perplexed in the choice, which to put before their 
 pupils, some apology may be needful for obtruding another work 
 on the same subject upon the public. 
 
 There was a time when such books presented the work of almost 
 every question at full length ; and when, of course, scarcely any 
 thing was left for the exercise of the scholar. Then it was, that 
 masters had the intolerable labour of writing the daily questions 
 for their pupils in their account books, or otherwise supplying the 
 defect by their own imperfect manuscripts. 
 
 As education became an object of more general regard, the evil 
 was proportionably felt, and a remedy was sought for ; and then, 
 instead of School Books, in which nothing was left for the learner, 
 others issued from the Press, in which he had nearly everything to 
 perform ; and which, with unfledged powers, he was bid to explore, 
 while altogether unequal to the task. 
 
 Hence, in the present day, there is scarcely any Tutor's Assist- 
 ant that has many of the operations given. The work of one 
 question, indeed, may be seen standing at the head of the rule ; 
 but it is often such a one as is little illustrative, either of the rule 
 itself or of the succeeding questions ; scarcely any other ray of 
 light has been shed to illuminate the path of the Tyro, though on 
 calculations widely different from the first example. 
 
 Nor has this been all ; the writers of these works seem to have 
 often put their invention upon the rack, to introduce, in every 
 rule, questions as useless as they are puzzling and intricate ; 
 not only beyond the learners power to work, but even to com- 
 prehend, though with the elucidations of a master. From what 
 motive so many writers on the subject have been thus misled, it 
 may be difficult to account, unless it were, to impress the public 
 with an idea of the profundity of their own scientific attain- 
 ments. 
 
 From whatever cause it may have originated, every experienced 
 teacher knows that the generality of scholars are scarcely able to 
 bring out the answer correctly to one question, in any rule, without 
 assistance from some source ; and where plagiarism is prevented, 
 their application to him is incessant, and he finds it needful, not 
 only to explain and illustrate, but frequently to work considerable 
 parts of each sum for his pupils. 
 
 Hence, their progress is not only very slow, but their compre- 
 hension very inadequate to what they are made to perform ; and 
 they often finish a rule without a sufficient knowledge of its 
 
. * -n Y- 
 
 iv Preface. 
 
 '* *** 
 
 \ p^itplpleg/ *T^9 .i$ J&$ present state of Arithmetic in schools, and 
 to the present unaccommodating systems it must be, in a consider- 
 able degree, attributed. 
 
 Every master who has numerous scholars to instruct, feels 
 its harassing effects ; and sees, without hopes of effectual relief, 
 the general incapacity of the pupils ; but a removal of the cause, 
 in any degree, is despaired of, or rather, never looked for ; it 
 seems never to have entered into a tutor's mind, that by the 
 very simple means here adopted, much of the incomprehensibility 
 of the scholar, and the ineffectual toil of the master may be 
 removed. 
 
 To remedy, then, in no inconsiderable degree, these defects, is 
 the purpose of this work ; to enable the young arithmetician to 
 understand what he is doing, and (by giving him sufficient examples, 
 at nearly full length to illustrate the rules) to bring all within the 
 compass of his powers. 
 
 It must also be considered, that the youths of the present 
 day commence the study of arithmetic earlier than in former 
 times ; the child at the age of seven or eight is now put upon 
 this important branch of knowledge, in which every succeeding 
 idea must be altogether new ; and at such a time to launch him 
 upon the ocean of unknown difficulties, with scarcely a gleam 
 of light to beam upon him, is to place him in a situation in 
 which even adult capacities can, unaided, scarcely explore their 
 way. 
 
 This may not be the case with every individual, but if one youth 
 in a thousand, or rather one in a million, should, with superior 
 powers and perseverance, attain his end, though comparatively 
 unassisted, it cannot argue against the use of this system, as 
 applicable to capacities generally, 
 
 Masters who have long struggled with the inconveniencies of 
 teaching by the present existing systems, but not so long as to 
 be wedded to their faults, and refuse relief, may be disposed to 
 try the effects of this now offered to their notice, and dedicated 
 to their service ; and the author presumes to hope, that, after 
 having not only critically examined, but often taught by, almost 
 every valuable arithmetical work, and connected therewith a wide 
 range of mathematical research, he cannot be incompetent to the 
 task. 
 
 Since this science has been so ably and fully developed, and 
 since, within little more than the last half-century, books on Arith- 
 metic have been multiplied, probably beyond that of any thousand 
 years preceding, little remains now to be done, by writers of 
 elementary works, but to select the purest principles, better to 
 arrange and methodize what has been already known, and to bring 
 down to the opening cap.icity rules which had been previously 
 enveloped in too much perplexity. 
 
 From the above remarks the following improvements may be 
 anticipated : 
 
 1st A complete lo<ly of Rides drawn up with clearness, and as 
 
1 'refacc. v 
 
 free as possible from technical forms of speech, for the learner's 
 more easy comprehension ; and also a Series of Examples, 
 selected with great care, accommodated to the present state of trade, 
 ample under each rule, and exhibiting every usual variety, for their 
 full illustration. 
 
 2nd. The omission of all such quaint and puzzling questions as 
 are too often found in such works, though in that place totally 
 useless, or above the learner's comprehension ; and supplying 
 their place with others more immediatly applicable to the purpose 
 of real transactions. All the abstruse sums in each rule are removed 
 as they ever should be, to the Appendix, as more fit for the 
 exercise or amusement? of his maturer powers. 
 
 3rd. Not only have the clearest illustrations of the rules been 
 carefully attended to, but the fitness of every example has had 
 its due consideration, by an arrangement that exhibits an easy 
 gradation ; by a careful association of such as are somewhat 
 similar, and by forming them into a kind of regular series. And 
 as each preceding question prepares the pupil for the succeeding 
 one, there will be found in this compendium, no abrupt 
 transitions from what is extremely easy to what is extremely 
 difficult. 
 
 4th. As in each rule of Arithmetic, questions are so various 
 that some examples can be no guide to the method of working 
 others, the work of the first of each series is given at full length 
 (or sufficiently so) as an example ; and the pupil, by comparing 
 that operation with the rule, will be enabled to work the succeeding 
 ones, generally, by the exercise of his own powers ; for when the 
 technical phraseology of the rule is of itself incomprehensible, an 
 appropriate example will illustrate it, and when difficulties occur, 
 they will be of such a nature as a few hints from the tutor will 
 remove. 
 
 5th. By these simple, though evident improvements, the tutor 
 will be released from much harassing and ineffectual toil ; and 
 the young arithmetician will be able to comprehend the principle 
 of all he does ; and as his capacity opens, he will be able to proceed, 
 through the whole course of Arithmetic, with a facility and pleasure 
 hitherto not often experienced. 
 
 Some persons, however, may be led, by the above pretentions, to 
 infer that the treatise must needs be superficial ; but this is by no 
 means the case each rule has had its usual full discussion. It 
 is easy, only because the difficulties have been so far solved, that 
 they are no longer incomprehensible. 
 
 There was a time when, from the paucity of books and instruc- 
 tors, it was thought that Arithmetic conld scarcely be thoroughly 
 gained, without absorbing all the faculties of a common mind, and 
 all the time of the boyish age ; when it left neither leisure nor 
 capacity for language and other liberal studies, There was a time, 
 too, when the first elements of arithmetic were often all that the 
 classical youth had inclination for ; or rather, when the science of 
 numbers was looked down upon as almost beneath his regard. 
 
vi Preface. 
 
 But in the present day of mercantile greatness, the world ia 
 awake to the importance of a union of these and other branches. 
 Arithmetic is now no longer thrown far into the background, but 
 on the contrary, even in scholastic establishments of eminence, it 
 is ever brought forward as an indispensable associate with classical 
 attainments ; and in such seminaries, where the variety of studies 
 leaves less time for each one, a work of this nature, which will so 
 materially accelerate the learner's progress, will prove, it is pre- 
 sumed, an important desideratum. 
 
 The Author is, after all, aware, that some persons, when they 
 hear of this publication, may expect to see some novel system 
 of magical effect, that will confer knowledge intuitively, without 
 juvenile exertion ; but this is no empyrical, treatise that 
 will supersede the necessity of research ; but it will enable the 
 learner to think it will bring his mind into a train for investi- 
 gation, and encourage him in the exercise of his juvenile powers. 
 He must still labour, but he will labour intelligibly, and not 
 flounder on, as by some systems, in blind confusion, uncertainty, 
 and disgust. 
 
 If the Author might be allowed to dictate to the teacher, he 
 would recommend him uniformly to enjoin upon his scholars not 
 only the full investigation of the first example that is worked at 
 the head of each series, comparing it carefully with the previous 
 rule, but also, in most instances, the .production of that sum at 
 full length on his slate, if not its transcription in his account book ; 
 the clear knowledge of which will lay open to him the principle of 
 the whole series. 
 
 With respect to the order in which the Author has placed VUL- 
 GAR AND DECIMAL ARITHMETIC, he is happy in its having received 
 the concurrence of several experienced masters. 
 
 Had this work been designed principally for adults, fractions 
 would most certainly have taken an earlier station : but children 
 are now uniformly put upon Arithmetic as soon as their faculties 
 can be made to apprehend even its very first elements. Therefore, 
 by first running through the most common rules, they will gain 
 those essential pre-requisites, & familiarity with the general modes 
 of calculation, and & facility in working; and they will have ample 
 time to do this before their capacities are fully equal to fractional 
 computations. 
 
 As a school book, therefore, the Author could not place 
 them earlier ; teachers, however, may transfer them to any 
 place that may best suit the age, capacity, and object of the 
 scholar. 
 
CONTENTS. 
 
 Page 
 
 ARITHMETIC 1 
 
 Numeration ib. 
 
 Simple Addition 3 
 
 Subtraction 6 
 
 Multiplication 9 
 
 Division 14 
 
 Compound Addition 18 
 
 Subtraction 29 
 
 Multiplication 33 
 
 Division 38 
 
 Bills of Parcels 42 
 
 Reduction 44 
 
 Rule of Three 52 
 
 Direct ib. 
 
 Inverse 59 
 
 --Double Rule of Three 63 
 
 Practice 66 
 
 - -Tare and Trett 74 
 
 Simple Interest 80 
 
 Commission 86 
 
 Brokerage 87 
 
 Purchasing of Stocks 88 
 
 Insurance 89 
 
 Compound Interest 91 
 
 Discount 95 
 
 Equation of Payments 98 
 
 Barter 100 
 
 Profit and Loss 103 
 
 Fellowship 107 
 
 Single ib. 
 
 Double Ill 
 
 Vulgar Fractions 114 
 
 Reduction of ib. 
 
 Addition 124 
 
 Page 
 VulgarFractionSjSubtractionl 26 
 
 Multiplication , 128 
 
 Division 129 
 
 Rule of Three Direct 130 
 
 Inverse... 132 
 
 Double Rule of Three 133 
 
 Decimal Fractions 134 
 
 Addition of 135 
 
 Subtraction 136 
 
 Multiplication ib. 
 
 Division 138 
 
 Reduction 140 
 
 Rule of Three Direct 147 
 
 Inverse... 149 
 
 Double Rule of Three ib. 
 
 Alligation Medial 1 50 
 
 Alternate 151 
 
 Partial 153 
 
 Total 154 
 
 Exchange 155 
 
 Arbitration of Exchanges... 171 
 
 Position, Single 173 
 
 Double 174 
 
 Progression, Arithmetical... 176 
 Geometrical ... 180 
 
 Permutation 186 
 
 Involution 187 
 
 Evolution 189 
 
 Square Root ib. 
 
 Cube Root 194 
 
 Biquadrate Root 198 
 Roots of all Powers ib. 
 
 Duodecimals 199 
 
 Appendix 203 
 
 AN IMPROVED SERIES OF ARITHMETICAL TABLES 206, 207 
 
EXHALATION 
 
 OF TUB 
 
 CHAEACTEES MADE USE OF IN THIS WOEK. 
 
 Although the characters may be found explained in their proper place 
 throughout the work, yet, for convenience, they are here brought 
 into one view. 
 
 + Plus, or more. The sign of Addition ; as, 2+4 j that is, 2 added 
 to 4, which is equal to 6. 
 
 Minus, or Utt. The iign of Subtraction ; as, 6 4 ; that is, from 
 6 take 4, or 6 minus 4, wliieh is equal to 2. 
 
 X Multiplied by. The sign of Multiplication ; as, 3x8 ; that is 3 is 
 to be multiplied by 8, which is equal to 24. 
 
 -r- Divided by. Tlie sign of Division ; as, 8 4-2 ; that is, 8 is to be 
 
 divided by 2, which is equal to 4. 
 
 Vff 2 or W & c - Numbers placed like a fraction, also denote divi- 
 
 sion ; the upper number being to be divided by 
 the lower. 
 
 : is to : : so is : to The sign of Proportion ; thus, as 2 : 4 : : 6 : 12 ; 
 that is, as 2 U to 4 so ii 6 to 12. 
 
 \/ or %/ Signs of the Square Root. 
 
 3 4 
 
 v/ and *J Signs of the Cube and Biquadrate Root. 
 
 8 2 , 8 8 , 8 4 . Signs of Involution, denoting that 8 is to be 
 
 squared, cubed, or raised to the 4th power. 
 
 = equal to. The sign of Equality; as, 2+3=5 ; that is, 2 
 
 added to 3, are equal to 5. 
 
 . ' . and ' . ' Ergo, or therefore, and because. 
 
 i.e. Id est, that is. 
 
 82 X 4 = 24. That is, 8 minus 2, multiplied by 4, are equal to 24. 
 
 8 2lxfi = 2. Thub is, the sum of 2 and 4 (6) taken from 8, u 
 
 equal to 2. 
 
 6~x~T+8 = 20. SU multiplied by 2 = 12, which add to 8 = 20. 
 
 6 x 2 + 8*= GO. Two added to 8 = 10, which multiplied by 6 - 6<X 
 
 4x5+7 Four multiplied by 5 are 20, to which add 7 = 27 
 
 8 and 27 divided by 8 are equal to 9. 
 
ARITHMETIC. 
 
 AEITHMETIC is the science of numbers' ; 'an4 te?rcb.ea 
 the art of computing by them. 
 
 The fundamental rules are, Notation (or Numeration), 
 Addition, Subtraction, Multiplication, and Division, from 
 which all the rest are derived : 
 
 OF NOTATION AND NUMEEATION. 
 
 NOTATION is the writing of numbers by figures ; and 
 NUMERATION is the art of reading figures correctly 
 
 The value of figures depends upon the place in which 
 they stand : which may be seen by the following table. 
 
 N.B. The first nine stands for nine units ; the nine in the second row 
 stands for nine tens, or ninety ; nine in the third row for nine hundrsd, 
 and nine in the fourth row for nine thousand, c. The same may be 
 observed of the increased value of every other of the above figures. 
 
 A nought or cypher lias no value of itself, but being placed on the right 
 hand of other figures, it increases their value in a tenfold proportion. 
 
 In the above table there are two whole periods of sir figures each ; but 
 for the more easily reading of large numbers, it is usual to subdivide 
 them into half-periods of three figures each. The first period has Units, 
 Tens, Hundreds, Thousands, Ten of Thousands, and Hundreds of Thou- 
 sands : the srcond period has Millions ; the third Billions, the fourth 
 Trillions, the fifth Quadrillions, as underneath. 
 
 
 
 n 3 OD 
 
 Js 
 
 I I I I ^11 
 
 AJLiJLjjJJjI 
 
 8676543219876543219876543 all's 2 t 
 B 
 
2 
 
 Numeration. 
 
 The pr$c$d&ig table is thus read. Nine hundred and eighty-seven 
 
 \ thousand; six hundred and fifty four quadrillion** three hundred and 
 
 twenty-one thousand^^nine hundred and eighty-seren trillions; six 
 
 . . .i^untk'e&ahdfjftr^our thousand, three hundred and twenty-one billions; 
 
 r.* *tjJQliuiiclred* and eighty-seven thousand, six hundred and fifty-four 
 
 " millions ; three hundred and twenty-one thousand, three hundred and 
 
 twenty-one. 
 
 EXAMPLES. 
 Write in WORDS, the following numbers. 
 
 Ex. 
 
 (1) 
 (2) 
 
 (3) 
 (4) 
 
 54 
 
 654 
 
 7,654 
 
 (5) 
 (6) 
 (7) 
 (8) 
 
 6,789 
 
 56,789 
 
 456,789 
 
 3,456,789 
 
 (9) 7,532,434 
 
 (10) 15,276,928 
 
 (11) 927,613,458 
 
 (12) 1,851,242,376 
 
 Write in FIGURES, the following numbers. 
 
 (13) Three hundred and sixty-five. 
 
 (14) One thousand eight hundred and twenty -three. 
 
 (1 5) Three hundred and twenty-four thousand, six hundred 
 
 (16) One million, three hundred and twenty-four thousand 
 
 (17) ITorty-four millions, five hundred and twenty -two 
 thousand, six hundred and ninety-seven. 
 
 (18) Nine hundred and eighty-seven millions, six hundred 
 
 (19) One hundred millions, one hundred thousand, and one 
 
 (20) One million, eight hundred and fifty-four. 
 
 (21) Five millions, one thousand, and twenty. 
 
 (22) Eighty-six millions, four hundred and thirty. 
 
 (23) One hundred and sixty-two millions, five hundred. 
 
 (24) One billion, one million, one thousand, and one. 
 
 THE KOMAIST NOTATION. 
 
 The Romans expressed numbers by the following letters, I. V. X. L. 
 C. D. M. which singly stood for, 1, 5, 10, 50, 100, 500, and 1000; 
 which were combined as follows : 
 
 I 
 
 1 IX 
 
 II 
 
 2 
 
 x 
 
 III 
 
 3 
 
 XI 
 
 IV or IIII 4 
 
 XII 
 
 V 
 
 5 
 
 XIII 
 
 VI 
 
 6 
 
 XIV 
 
 VII 
 
 7 
 
 XV 
 
 VIII 
 
 8 
 
 XVI 
 
 9 i XVII 
 
 10 
 
 11 
 
 12 
 13 
 14 
 15 
 16 
 
 XVIII 
 
 XIX 
 
 XX 
 
 XXX 
 
 XL 
 
 L 
 
 LX 
 
 17ILXX 
 18 LXXX 
 
 19 
 20 
 30 
 40 
 50 
 60 
 
 XC 
 
 C 
 
 CO 
 
 70 
 
 80 
 
 90 
 
 100 
 
 200 
 
 CCC 300 
 
 CCCC 400 
 
 DC 
 
 DCC 
 
 DCCC 
 
 DCCCC 
 
 M 
 
 MDCCCL 
 
 600 
 700 
 800 
 900 
 1000 
 
 One thousand eight 
 500 hundred and fifty. 
 
 From an inspection of the above table it is seen, that prefixing a letter 
 of o lower value to one of a higher, subtracts its value ; thus I prefixed 
 to V is four (IV). IX nine, XL forty, XC ninety, &c., and also annexing 
 a letter of lower value to one of a higher, increases its value, thus, VI 
 signifies six, and XI eleven, LX sixty, &c. &c. 
 
(3 ) 
 
 SIMPLE ADDITION. 
 
 SIMPLE ADDITION teaches the method of finding the 
 gum of two or more numbers. 
 
 ETJLE 1st. Place the numbers under each other, so that 
 units may stand under units, tens under tens, &c. 
 
 2nd. Add up the figures in the row of units, set down what 
 remains above the even tens, or if nothing remains, a cypher ; 
 and carry as many ones to the next row as there were tens. 
 
 3rd. Add up the other rows, in the same manner, and in 
 the last column set down the whole sum contained in it. 
 Proof. 
 
 1. Cut off the upper line, add up the rest as before, and 
 set the sum under the lower line. 
 
 2. Add this second sum to the upper line, and if it be the 
 same as that of the first addition, the work is right. 
 
 ADDITION AKD SUBTRACTION TABLE. 
 
 1 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 2 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 3 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 4 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 5 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 6 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 7 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 8 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 9 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 10 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 20 
 
 EXAMPLES. 
 
 Ex. (1) 2423 (2) 4140 (3) 2430 (4) 5250 
 
 3132 313 121 132 
 
 5344 3241 2562 3366 
 
 4210 1424 340 2644 
 
 2132 312 4213 322 
 
 1325 2430 2653 456<T 
 
 18566 
 16143 
 18566 
 
Simple Addition. 
 
 Ex. (5) 22471 
 
 (6) 21347 
 
 (7) 62316 
 
 (8) 13003 
 
 13163 
 
 2408 
 
 7705 
 
 3352 
 
 4235 
 
 43216 
 
 31232 
 
 51240 
 
 21377 
 
 3465 
 
 28101 
 
 5364 
 
 15426 
 
 26459 
 
 5364 
 
 26527 
 
 6132 
 
 17534 
 
 43429 
 
 1439 
 
 51595 
 
 3372 
 
 2357 
 
 18268 
 
 134399 
 
 
 
 
 (9) 42137 
 
 (10) 20104 
 
 (11) 74203 
 
 (12) 14721 
 
 1342 
 
 3210 
 
 4137 
 
 400 
 
 34035 
 
 14541 
 
 52314 
 
 43742 
 
 5260 
 
 5106 
 
 43173 
 
 4137 
 
 72043 
 
 15243 
 
 2425 
 
 52173 
 
 4605 
 
 432 
 
 13050 
 
 20736 
 
 13721 
 
 36715 
 
 7634 
 
 51342 
 
 
 
 (15) 14210 
 
 
 (13) 71403 
 
 (14) 12345 
 
 (16) 10342 
 
 54270 
 
 3456 
 
 22171 
 
 734 
 
 23152 
 
 84567 
 
 3413 
 
 26213 
 
 321 
 
 64278 
 
 47030 
 
 13473 
 
 37046 
 
 75359 
 
 5135 
 
 12305 
 
 54300 
 
 24678 
 
 13421 
 
 300 
 
 1021 
 
 2567 
 
 6342 
 
 76215 
 
 (17) 71426 
 
 
 
 
 (18) 47321 
 
 (19) 3219 
 
 (20) 75968 
 
 35751 
 
 13714 
 
 7946 
 
 68579 
 
 14935 
 
 6276 
 
 84297 
 
 42786 
 
 179 
 
 19395 
 
 421 
 
 34968 
 
 49214 
 
 4576 
 
 98468 
 
 71214 
 
 36348 
 
 67398 
 
 76529 
 
 12976 
 
 892 
 
 34264 
 
 1090 
 
 68798 
 
Simple Addition. 5 
 
 MISCELLANEOUS QUESTIONS. 
 
 (21) If a man was born in the year 1769, when was he 
 70 years of age ? Ans. 1839 
 
 (22) Add together a million, a thousand, and a hundred. 
 
 Ans. 1001100 
 
 (23) Add the following sums 98765 + 3240+567 + 310 
 and 1148. Ans. 104030 
 
 (24) Suppose a boy born in the yev;r 1852, when will he 
 be of age ? Ans. 1873 
 
 (25) Add the years before Christ (4004) to the year 1857, 
 and tell me the age of the world. Ans. 5861 years 
 
 (26) Add together the chapters in the several books of 
 the New Testament. Ans. 260 chapters 
 
 (27) In the year 1806 I took the lease of a house for 49 
 years, w r hen did the lease expire ? Ans. 1855 
 
 (28) How many days are there in the last 9 months of 
 the year ? Ans. 275 days 
 
 (29) Add the shillings in a florin, a crown, a half-sove- 
 reign, a sovereign, a guinea, and a six and thirty together. 
 
 Ans. 94 shillings 
 
 (30) A skilful boy won one day at marbles, of A. 13, of 
 B. 11, of C. 15, of 1). 21, of E. 9, of F. 19, and of GL 12, 
 how many did he win in all ? Ans. 100 marbles 
 
 (31) How many days from Lady-day till Michaelmas-day, 
 that is, from March 25 to September 29 ? Ans. 188 days 
 
 (32) What is the sum of one million and one, one thou- 
 sand and one, and one hundred and one ? Ans. 1001 103 
 
 (33) Add together 35040+32654 + 32697 and 98765. 
 
 Ans. 199156 
 
 (34) If I travel the first day 65 miles, the second 59, the 
 third 67, the fourth 41, the fifth 45, and the sixth 36, how 
 many miles do I travel in the six days ? Ans. 313 miles 
 
 (35) How many chapters are there in the first five books 
 of the Old Testament ? Ans. 187 chapters 
 
 (36) An American merchant shipped, at Liverpool, goods 
 to the following amount, which ho purchased in England : 
 namely, in Worcester to the amount of 570Z., in Kidder- 
 minster 340Z , in Birmingham 1600Z., in Wolverhampton 
 590Z., in Sheffield 900Z., in Leeds 1000Z., in Manchester 
 1500Z. : I demand the whole amount. Ans. 6500Z. 
 
 (37) Suppose London to contain 2,300,000 inhabitants, 
 Paris 1,000,000, Vienna 475,000, Petersburg 5-30,000, Edin- 
 burgh 162,000, Dubliu 250,000, what is the population of 
 She whole ? Ans. 4,717,000 
 
(6) 
 
 SIMPLE SUBTRACTION. 
 
 BY Subtraction we find the difference between any two 
 numbers. 
 
 HULE 1st. Place the less number under the greater, so 
 that units may stand under units, tens under tens, &c. 
 
 2nd. Begin at the right hand, that is, at the unit's place, 
 and take each figure in the lower line from the figure above 
 it, and set down the remainder. 
 
 3rd. If the figure in the lower line be the greater, add ten 
 to the upper line, and then take the lower figure from the 
 sum ; set down the remainder, and carry one to the next 
 lower figure, with which proceed as before. 
 
 PBOOF. Add the remainder to the last line ; and if the 
 sum be equal to the first, the work is right. 
 
 EXAMPLES. 
 
 From (1) 7463274 (2) 4673985 (3) 7900764 
 Take 2152143 1371702 4172413 
 
 Eemains 5311131 3302283 3728351 
 
 Proof 7463274 4673985 7900764 
 
 (4) 7574967 (5) 4768395 (6) 3472196 (7) 9372628 
 4261724 1426153 2461072 5130324 
 
 (8) 3746947 (9) 5728116 (10) 7052673 (11) 3710470 
 1729742 1348306 4872167 3402143 
 
 (12) 7421643 (13) 5301743 (14) 3000765 (15) 1213007 
 6917269 2700755 2999387 782714 
 
Simple Subtraction. 7 
 
 (16) 7502374 (17) 1426432 (18) 7421376 (19) 4712600 
 1924737 987654 4371498 841974 
 
 5577637 
 
 (20) 7421362 (21) 7142600 (22) 1000000 (23) 7111111 
 760493 847599 999999 600000G 
 
 (24) 7214000 (25) 9142174 (26) 4721610 (27) 1245671- 
 767284 407300 3091371 245675 
 
 (28) 9104002 (29) 4030300 (30) 7654321 (31) 7090900 
 9030001 1761073 765432 3241090 
 
 74001 
 
 (32) 6321736 (33) 9006654 (34) 7140091 (35) 1173000 
 239216 999999 26730 981911 
 
 (36) 8410824 (37) 4065410 (38) 1840842 (39) 9876023 
 231486 456789 236494 987134 
 
 (40) 4012348 (41) 9054102 (42) 5024102 (43) 6408230 
 124865 392047 1456789 2675864 
 
 (44) 3700046 (45) 8999994 (46) 7210841 (47) 8006678 
 143128 1045605 1486328 4118789 
 
8 Simple Subtraction. 
 
 MISCELLANEOUS QUESTIONS. 
 
 (48) If a child was born in the year 1794, how old \vas ho 
 in 1855 ? Am. 61 years old. 
 
 (49) Prom one hundred millions, take one hundred thou- 
 sand. Ans. 99900000 
 
 (50) How much is 9876 less than ten thousand ? Ans 124 
 
 (51) If London contains 2,300,000, and Paris 1,000,000 
 what is the difference of the population ? Ans. 1,300,000 
 
 (52) If from London to York be 196 miles, and from 
 London to Edinburgh, through York, 399 miles, what is the 
 distance from York to Edinburgh ? Ans. 203 miles 
 
 (53) If from London to the Land's End in Cornwall, 
 through Exeter, be 302 miles, and to Exeter be 176, what is 
 the distance from Exeter to the Land's End ? Ans. 126 miles 
 
 (54) Sir Isaac Newton was born in the year 1642, and he 
 died in 1727, how old was he ? Ans. 85 years 
 
 (55) The North American States were first peopled about 
 the year 1496, how many years had elapsed till their Inde- 
 pendence was acknowledged in 1783 ? Ans. 287 years 
 
 (56) A tradesman began business with 569Z., and in three 
 vears found himself worth 965Z., what had he gained in 
 trade ? Ans. 396Z. 
 
 (57) How much does 987654 exceed 123456? Ans. 864198 
 
 (58) A gentleman dying left 11,698Z. between his son and 
 daughter, the son's portion was 6349Z., what was the 
 daughter's ? Ans. 5349Z. 
 
 (59) How long did Homer live before Yirgil, if the era of 
 the former be fixed at 907 years before Christ, and the latter 
 only 70 years before that period ? Ans. 837 years 
 
 (60) A merchant had three debtors, A. B. and C. who 
 owed him 1760Z., A. and B. owed him 1111Z., what was C.'s 
 debt ? Am. 649Z. 
 
 (61) Noah's flood is said to have happened about the year 
 of the world 1656, and the birth of Christ in 4004, what is 
 the difference of time ? Ans. 2348 years 
 
 (62) What number must I add to 12345 to make it 98765. 
 
 Ans. 86420 
 
 (63) How much is X. whose age is 95, older than Z. who 
 is 59 years old ? Ans. 36 years 
 
 (64) Subtract the sum of 1236 and 7890 from 9876 added 
 fco 6789 ? Ans. 7539 
 
 (65) Take 50 millions from 50 billions. 
 
 Ans. 49999950000000 
 
(9) 
 
 SIMPLE MULTIPLICATION. 
 
 MULTIPLICATION is a short method of performing 
 Addition, and it teaches us to find what a number will 
 amount to, when repeated a certain numoer of times. It 
 consists of the following parts: 
 
 1st. The Multiplicand, or number to be multiplied ; 
 
 2nd. The Multiplier, or number by which you multiply 5 
 
 3rd. The Product, or number produced by multiplying. 
 
 CASE I. When the Multiplier does not exceed 12. 
 BULE. Begin at the right hand, and multiply every figure 
 in the multiplicand ; consider how many tens there are in 
 each product, the remaining units set dow r n under the figure 
 multiplied, and carry the tens as so many ones to the next 
 product. The last product is to be wholly set down. 
 
 THE MULTIPLICATION TABLE. 
 
 Twice 
 
 3 times 
 
 4 times 
 
 5 times 
 
 6 times 
 
 7 times 
 
 2 . . 4 
 
 2 . . 6 
 
 2 .. 8 2 . . 10 
 
 2 . . 12 
 
 2 . . 14 
 
 3 . . G 
 
 3 . . 9 
 
 3 . . 12 3 . . 15 
 
 8 ..18 
 
 3 . . 21 
 
 4 . . 8 
 
 4 . . 12 
 
 4 . . 16 4 . . 20 
 
 4 . . 24 
 
 4 . . 28 
 
 5 . . 10 
 
 5 . . 15 
 
 5 . . 20 5 . . 25 
 
 5 . . 30 
 
 5 . . 35 
 
 G . . 12 
 
 6 . . 18 6 . . 24 6 . . 30 
 
 6 . . 36 
 
 6 . . 42 
 
 7 .. 14 
 
 7 . . 21 
 
 7 . . 28 7 ... 35 
 
 7 . . 42 
 
 7 . . 49 
 
 8 . . 1G 
 
 8 . . 24 
 
 8 .. 32 
 
 8 . . 40 
 
 8 . . 48 
 
 8 . . 56 
 
 9 . . 18 
 
 9 . . 27 
 
 9 . . 36 9 . . 45 i 9 . . 54 
 
 9 . . 63 
 
 10 . . 20 10 . . 30 10 . . 40 10 . . 50 10 . . 60 
 
 10 . . 70 
 
 11 . . 22 11 . . 33 11 . . 44 11 . . 55 11 . . 66 
 
 11 . . 77 
 
 12 . . 24 
 
 12 . . 36 
 
 12 . . 48 
 
 12 . . 60 
 
 12 . . 72 
 
 12 . . 84 
 
 8 times 
 
 9 times 
 
 10 times 
 
 11 times 
 
 12 times 
 
 
 2 . . 16 
 
 2 . . 18 
 
 2 . . 20 1 2 . . 22 
 
 2 . . 24 
 
 denotes 
 lil.rae, or 
 
 3 .. 24 
 
 3 . , 27 
 
 3 . . 30 3 . . 33 
 
 3 . . 36 
 
 pounds. 
 
 4 . . 32 
 
 4 . . 36 
 
 4 . . 40 
 
 4 . . 44 
 
 4 . . 48 
 
 8, sol id i, or 
 
 o . . 40 
 
 5 . . 45 
 
 5 . . 50 5 . . 55 
 
 5 . . 60 
 
 s. 
 
 6 . . 48 
 
 6 . . 54 
 
 6 . . 60 6 . . 66 
 
 6 . . 72 
 
 d, denarii, or 
 pence. 
 
 7 . . 56 
 
 7 . . 63 
 
 7 . . 70 
 
 7 . . 77 
 
 7 . . 84 
 
 <jT*.qu*drantes, 
 
 8 . . 64 
 
 8 . . 72 
 
 8 . . 80 
 
 8 .. 88 
 
 8 . . 96 
 
 or fa things. 
 
 9 . . 721 9 . . 81 
 
 9 . . 90 9 . . 99 
 
 9 . .108 
 
 [ one larthing. 
 
 10 . . 80 10 . . 5)0 10 . .100 10 . .110 10 . .120 
 
 \ a halfpenny. 
 
 11 . . 88 11 . . Hi) ,11 . .110 11 . .121111 . .132 
 
 ? three far- 
 
 12 . . 96 12 . .108 12 . .120 12 . .132 
 
 12 . .144 
 
 things. 
 
 i 
 
 
 
 B 3 
 
LO Simple Multiplication. 
 
 EXAMPLES. 
 
 Multiplicand (1) 7421635 (2) 3764158 (3) 8742693 
 Multiplier 2 6 12 
 
 Product 14843270 22584948 104912316 
 
 Ex. (4) 7421635 (5) 3764158 (6) 8742693 
 
 234 
 
 (7) 9126543 (8) 2345674 (9) 6342157 
 
 567 
 
 (10) 716482539 (11) 987654321 (12) 864213579 
 8 9 10 
 
 (13) 471526389 (14) 234579876 (15) 371529684 
 11 12 12 
 
 (16) 4286925 (17) 7424394 (18) 3742845 
 234 
 
 (19) 9429545 (20) 8345974 (21) 9342487 
 567 
 
 (22) 718482599 (23) 987958824 (24) 884275579 
 8 9 10 
 
 (25) 474528389 (26) 254579878 (27) 377529884 
 11 12 12 
 
Simple Multiplication. 
 
 11 
 
 II. When the multiplier is any number between 
 RULE. Multiply by the unit figure, and to each product 
 add the remainder to be carried, and also the figure last 
 multiplied-. 
 
 Ex. (28) 4321835 (29) 1371429 (30) 7310286 
 
 14 15 16 
 
 60505690 
 
 (31) 372473673 
 17 
 
 (32) 571839264 
 18 
 
 (33) 987654321 
 19 
 
 III. When the multiplier consists of several figures. 
 
 RULE. Multiply the multiplicand by each figure of the 
 multiplier separately, beginning with the right-hand figure ; 
 and let the first figure of every product stand exactly under 
 the figure multiplied by. Add these products together, and 
 their sum will be the answer, or whole product required. 
 
 PEOOF. The common method is by casting out the nines, 
 but this is by no means infallible ; the best method is to 
 make the multiplier the multiplicand, and the multiplicand 
 the multiplier ; and if the product found from this operation, 
 be the same as before, the work is right. 
 
 EXAMPLES. 
 
 (34) 6375246 
 32 
 
 12750492 
 19125738 
 
 204007872 
 
 (35) 4563742 
 54 
 
 18254968 
 22818710 
 
 246442068 
 
 (37) 56347 
 681 
 
 (36) 4313247 
 98 
 
 34505976 
 38819223 
 
 422698206 
 
 (38) 25681 
 347 
 
 56347 
 450776 
 338082 
 
 38372307 
 
 179767 
 102724 
 77043 
 
 *8911307 
 
 * The method of proving multiplication sums, by casting out tft* 
 nines, the pupil will better understand by one minute's oral explanation, 
 than by a page of description. 
 
12 Simple Multiplication. 
 
 Ex. (39) 46572374 x 43 (40) 34583472 x 54 
 
 (41) 92416436 X 65 (42) 39321835 x 76 
 
 (43) 35732813 X 87 (44) 83742186 x 98 
 
 (45) 84213958 x 432 (46) 58236437 x 543 
 (47) 63857426 x 654 (48) 27948314 x 705 
 IY. When ciphers are intermixed with the figures in the 
 multiplier. 
 
 RULE. Omit the ciphers, and let the first figure of each 
 product be placed under its multiplier. 
 
 EXAMPLES. 
 
 Ex. (49) 5869374 (50) 50862470 
 
 3005 6009 
 
 29346870 457762230 
 
 17608122 305174820 
 
 17637468870 305632582230 
 
 (51) 7483952 x 4008 (52) 4372849 x 6004 
 V. When there are ciphers at the end of the multiplier or 
 multiplicand. 
 
 HULE. Multiply the significant figures in the multiplicand 
 by those of the multiplier, and place as many ciphers to the 
 right hand of the product as there are in both factors. 
 
 EXAMPLES. 
 (53) 8536274 (54) 842593700 
 
 40 8000 
 
 34L450960 6740749600000 
 
 (55) 87654321 x 30 (56) 4276958 x 900 
 
 (57) 24736849 X 6400 (58) 3869275 x 87000 
 VI. When the multiplier is the product of two or more 
 numbers or factors. 
 
 RULE. Multiply by one of the numbers, and that product 
 by the other, and so on ; the result will be the answer. 
 
 EXAMPLES. 
 
 (59) 5826347 x 25 (60) 4627538 x 84 
 5 12 
 
 29131735 55530456 
 
 5 7 
 
 145658675 388713192 
 
 (61) 9582374 X 30 (62) 5742983 X 45 
 
Simple BLultiplication. 13 
 
 MISCELLANEOUS QUESTIONS. 
 (G3) "What is the product of 123456 multiplied by 4321 ? 
 
 An*. 533453376 
 
 (64) A privateer of 284 men took a prize which amounted 
 to 95/. each man ; what was the value of the prize ? 
 
 Am. 26980Z. 
 
 (65) If a merchant began business with 5000Z. and retired 
 fifter 21 years' trading, in which he cleared on an average 
 1,8362. per year, what sum did he retire with ? Ans. 43556Z. 
 
 (66) Suppose a gentleman to lay by each year 365Z. what 
 will it amount to in 33 years ? Ans. 12045Z. 
 
 (67) How many trees are there in a plantation of 26 rows 
 of 1960 trees in each ? Ans. 50960 
 
 (68) The railroads of England are estimated at 8000 miles 
 in length, each mile costing 37,500Z. ; what was the cost of 
 the whole. Ans. 300,000,00p/. 
 
 (69) When the multiplicand is 98765 and the multiplier 
 43210, what is the product f Ans. 4267635650 
 
 (70) I planted 20 rows of potatoes, of 30 in each row ; how 
 many potatoes will they produce, supposing 7 to each root ? 
 
 Ans. 4200 
 
 (71) If a boy can point 16 thousand pins in an hour, how 
 many can lie do in 6 days, supposing him to work at them 
 9 hours each day ? Ans. 864,000 
 
 (72) How many miles will a person walk in a year, at the 
 rate of 15 miles per day ? Ans. 5475 
 
 (73) How much will a person spend in 7 years, at the 
 rate of 105. per day ? Ans. 25550s. = 1277 10*. 
 
 (74) What is the product of 49 times 12, multiplied by 
 25 times 11 ? Ans. 161700 
 
 (75) Multiply 12 dozen-dozen by half a dozen-dozen. 
 
 Ans. 124416 
 
 (76) In an army consisting of 189 battalions, each 450 
 men, how many effective soldiers ? Ans. 85050 
 
 (77) What is the product of one hundred millions, multi- 
 plied by one hundred thousand ? Ans. 10000000000000 
 
 (78) If 52 walnuts will fill a quart, how many will fill a 
 bushel containing 32 quarts ? Ans. 1664 walnuts 
 
 (79) A gentleman at his decease had 19 sons and daugh- 
 ters, and left his whole property equally among them ; when 
 divided, the portion of each was 9999Z. ; what was the sum 
 left ? Ans. 18998U 
 
 (80) A gentleman gave to his daughter, as her marriage 
 portion, a scrutoire in which were 9 drawers, in each drawer 
 were 4 divisions, and in each division 100 sovereigns ; what 
 was her fortune ? Ans. 3600J. 
 
14 
 
 SIMPLE DIVISION. 
 
 BY Division, we find how often one number is contained 
 in another of the same denomination : this is a short method 
 of performing subtraction. It consists of three parts : 
 
 The Dividend, or number to be divided ; 
 
 The Divisor, or number by which you divide ; and 
 
 The Quotient, or number arising from the divisor. 
 
 I. When the divisor does not exceed 12. 
 
 RULE. Draw a curve, and write the divisor on the left- 
 hand of the dividend ; then consider how many times the 
 divisor is contained in the first figure or figures of the divi- 
 dend, and set the quotient under it ; and for every unit 
 remaining after subtraction carry ten to the next figure of 
 the dividend. 
 
 PROOE. Multiply the quotient by the divisor, and to the 
 product add the remainder, if any ; the product will be equal 
 to the dividend, if the work is right. 
 
 EXAMPLES. 
 (1) Divis. 2)4682462 Divid. (2) 6)7296731 (3) 12)4684707 
 
 Quotient 2341231 1216121|- 390392A 
 
 2 6 12 
 
 Proof 4682462 
 
 7296731 
 
 4684707 
 
 (4) 2)62486284 (5) 3)42942369 (6) 4)72852696 
 
 (7) 5)15255105 (8) 6)86251734 (9) 7)38142171 
 
 (10) 8)16280168 (11) 9)18273783 (12) 10)71632987 
 
 (13) 11)10874216 (14) 12)14326871 (15) 12)17210543 
 
 II. When the divisor consists of several figures. 
 RULE 1. Draw a curve line on the right and left of the 
 iividend, and write the divisor on the left. 
 
Simple Itivmon. 
 
 2. Find how many times the divisor is contained in as 
 many figures of the dividend as are just necessary, and place 
 the number on the right for a quotient. 
 
 3. Multiply the divisor by this quotient figure, and having 
 subtracted the product from the above mentioned figures of 
 the dividend, bring down the next figure of the dividend, or 
 more if necessary, to the right of the remainder. 
 
 4. Divide the remainder, so increased, by the divisor, as 
 before, for the second figure of the quotient ; observing if 
 it goes times, to put a cipher, and bring down another 
 figure to the remainder. 
 
 5. Proceed with this result as with the former, and so on 
 till all the figures of the dividend are brought down. 
 
 PBOOF. As before. 
 
 (16) 
 
 65)123456( 
 65 
 
 EXAMPLES. 
 
 1899 (17) 
 65 
 
 543)7653927( 
 543 
 
 14095 
 543 
 
 584 
 520 
 
 645 
 
 585 
 
 9495 
 11394 
 21 
 
 2223 
 2172 
 
 42285 
 56380 
 70475 
 5192 342 
 
 123456 Proof. 4887 
 
 606 
 
 585 
 
 21 
 
 3057 
 2715 
 
 342 
 
 7653927 
 
 (18) 8731047 - 
 (20) 2526074 - 
 (22) 48408825 - 
 (24) 12345678 - 
 
 - 37 
 - 139 
 - 425 
 - 7304 
 
 (19) 3917253 - 
 (21) 12768423 - 
 (23) 68304137 - 
 (25) 56932073 - 
 
 - 64 
 - 543 
 - 876 
 - 9531 
 
 III. When ciphers are annexed to the divisor. 
 
 RULE. Cut off the ciphers from the divisor, and the same 
 number of figures from the right hand of the dividend ; then 
 divide the remaining figures of the dividend by the remain- 
 ing part of the divisor, and the result will be the answer. 
 To the remainder, if any, join those figures of the dividend 
 which were first cut off, and set the whole over the divisor, 
 for the fractional part. 
 
16 Simple Division. 
 
 EXAMPLES. 
 
 (26) 2,0)472163,9 (27) 12,00)27136,72 
 236081 H 2261 & 
 
 (28) 321,00)45686,46(142 (29) 543,00)270657,84(497 
 321 2172 
 
 1358 5305 
 
 1284 4887 
 
 "746 4187 
 
 642 3801 
 
 10446 remainder 38684 rein. 
 
 (30) 4712639 ~ 40 (31) 7294732 ~ 600 
 
 (32) 1398521 ~- 8000 (33) 3712659 ~ 1100 
 
 IV. Wlien the divisor is the product of two or more numbers. 
 ETJLE. Divide the given number by one of those parts, 
 and the quotient thus arising by another, and so on ; and 
 the last result will be the answer required. 
 (34) 24 f 6)7142168 -^ 24 (35) t 9)257169534-162 
 14)1190361-2 ) , 12 ) 6) 2857439-2 
 297590-1 J 8 
 
 1587461 
 
 The rem. = 1 X 6+2=8. The rem. 101=1 x 6 + 53 x 9 + 2 
 The Tutor will now teach his pupil how to value the re- 
 mainders, viz. by X the last remainder by the preceding 
 divisor, and to that product, add i\\Q first remainder, &c. 
 (36) 47321952 - 16 (37) 17304275 - 28 
 
 (38) 12347164 - 36 (39) 36924634 - 42 
 
 (40) 57132063 - 63 (41) 71285951 - 84 
 
 Y. What is often called the ITALIAN METHOD should 
 sometimes be taught, which is, to omit putting down the figures 
 resulting from multiplying the quotient with the divisor. 
 
 With this mode of operation, the first two long division 
 sums, page 15, will stand thus 
 
 (42) 65)123456 ( 1899 (43) 543)7653927 ( 14095 
 584 2223 
 
 645 5192 
 
 606 3057 
 
 21 342 
 
 As every Tutor knows this method, when the scholars have gained 
 more proficiency in arithmetic they may return and work the preceding 
 sums by it, under his instruction. 
 
Sim pie jji vision. 17 
 
 MISCELLANEOUS QUESTIONS. 
 
 (44) If 1664 walnuts will fill a bushel which contains 32 
 juarts, how many will there be to a quart ? Ans. 52 
 
 (45) If I planted 2520 potatoes in 84 equal rows, how 
 many were set in each row ? Ans. 30 
 
 (46) If the quotient be 49 times 12, and the dividend be 
 161700, what is the divisor ? Ans. 275 
 
 (47) Divide 12 dozen-dozen by half a dozen-dozen ? 
 
 Ans. 24 
 
 (48) Divide 10,000,000.000,000 by one hundred millions? 
 
 Ans. 100,000 
 
 (49) If a person walks 5475 miles in a year, how many 
 miles is that per day ? Ans. 15 
 
 (50) When the dividend is 4267635650, and the quotient 
 is 98765, what is the divisor ? Ans. 43210 
 
 (51) If the earth's distance from the sun be 95.000,000 
 miles, and a ray of light pass from the sun to us in 8 minutes, 
 at what rate does it fly per minute ? Ans. 11,875,000 miles 
 
 (52) If there are 4768 nuts in a bushel, how many are 
 there in a quart of 32 to the bushel ? Ans. 149 
 
 (53) A merchant cleared 38556Z. in 21 years ; in what pro- 
 portion is that per year ? Ans. 1836/. 
 
 (54) What is the quotient of 533453376 divided by 
 123456? Ans. 4*321 
 
 (55) A privateer of 284 men took a prize worth 26980/. 
 what was each man's share ? Ans. 95/. 
 
 (56) Suppose a gentleman in 33 years to lay by 12045Z. ; 
 how much is that on an average per year ? Ans. 365 /. 
 
 (57; The railroads of England are estimated at 8000 miles 
 in length, the whole costing 300,000,OOOZ. ; how much did 
 each mile cost. Ans. 37,500/. 
 
 (58) The number of young tr^es in even rows amounts to 
 50960, there were 26 rows ; I demand how many trees in 
 each row ? Ans. 1960 
 
 (59) Suppose a gentleman to leave 189,981/. equally 
 among his 19 sons and daughters ; I require tha portion of 
 each ? Ans. 9999/. 
 
 (60) Divide 10 times 48 by 10 times eight and forty ? 
 
 Ans. 4 
 
 (61) If those who live upon the equator, are carried by 
 the earth's diurnal motion 25,000 miles in 24 hours, what is 
 the movement per hour ? Ans. 1041 ^H 
 
 (62) Suppose the earth to move in its orbit at the rate of 
 72,000 miles in an hour (which is something greater than the 
 truth), what would be its rate every moment ? Ans. 20 milf.s 
 
(18) 
 
 COMPOUND ADDITION. 
 
 COMPOTJKD Addition is a method of collecting several 
 numbers of different denominations into one sum. 
 
 EULE 1. Place the numbers, so that those of the same 
 denomination may stand directly under each other, and draw 
 a line below them. 
 
 2. Add the numbers in the lowest denomination together, 
 and find how many units of the next higher denomination 
 are contained in their sum. 
 
 3. Set down the remainder, and carry the units to the 
 next higher denomination, which add up as before, and so 
 on to the end. 
 
 PBOOF. The method of proof is the same as in simple 
 addition. 
 
 OF MONEY 
 
 4 Earthings make 
 12 Pence . . . 
 20 Shillings. . . 
 
 marked. 
 
 Penny, d. 
 Shilling, s. 
 Pound, 
 
 ^ Earthing. 
 
 ^ Halfpenny. 
 
 f Three Earthings, 
 
 Farthings. 
 
 4 zz 1 Penny. 
 48 zz 12 rz 1 Shilling. 
 960 zz 240 zz 20 zz 1 Pound. 
 
 N.B. AMoidore,27a. A Sovereign, 20s. A Guinea, 21*. -A Mark, 
 ,3s. 4d. An Angel, 10s. A Noble, 6-. 8d A Crown, 5s. Half Crown, 
 is. Qd. 
 
 Farthing 
 Table. 
 
 The Pence Table. 
 
 The Shilling Talk. 
 
 qrs. d. 
 
 d. 
 
 s. d. d. s. d. 
 
 s. s. 
 
 s. s. 
 
 4 .. 1 
 
 20 
 
 ..1 8 90.. 7 6 
 
 20.. 1 
 
 140.. 7 
 
 8 .. 2 
 
 24 
 
 ..2 96.. 8 
 
 30.. 1 10 
 
 150.. 7 10 
 
 12 .. 3 
 
 30 
 
 .. 2 6100.. 8 4 
 
 40.. 2 
 
 160.. 8 
 
 16 .. 4 
 
 36 
 
 .. 3 0108.. 9 
 
 50.. 2 10 
 
 170.. 8 10 
 
 20 .. 5 
 
 40 
 
 .. 3 4110.. 9 2 
 
 60.. 3 
 
 180.. 9 
 
 24 .. 6 
 
 48 
 
 .. 4 0120.. 10 
 
 70.. 3 10 
 
 190.. 9 10 
 
 28 .. 7 
 
 50 
 
 .. 4 2 130.. 10 10 
 
 80.. 4 
 
 200.. 10 
 
 32 .. 8 
 
 60 
 
 .. 5 0132.. 11 
 
 90.. 4 10 
 
 210.. 10 10 
 
 36 .. 9 
 
 70 
 
 .. 5 10 140.. 11 8 
 
 100. 5 
 
 220.. 11 
 
 40 .. 10 
 
 72 
 
 .. 6 0144.. 12 
 
 110 . 5 10 
 
 230.. 11 10 
 
 44 .. 11 
 
 80 
 
 .. 6 8 150.. 12 6 
 
 120.. 6 
 
 240.. 12 
 
 *8 .. 12, 
 
 84 
 
 .. 7 0156.. 13 
 
 130.. 6 10,250.. 12 10 
 
Tables of Weights, $c. 2.9 
 
 TEOY WEIGHT. 
 
 24 Grains ((jr.) . . . make 1 Pennyweight diets. 
 20 Pennyweights , . . . 1 Ounce . . oz. 
 12 Ounces 1 Pound . . lb. 
 
 Grains. 
 
 24 zz 1 Pennyweight. 
 480 zz 20 zz 1 Ounce. 
 5760 zz 240 zz 12 zz 1 Pound. 
 
 By this weight, gold, silver, jewels, and precious stones, are weighed 
 It is also used in ascertaining the strength of liquors; and most othei 
 things of a fine or costly nature. 
 
 AVOIKDUPOIS WEIGHT. 
 
 16 Drains (dr.) . . . make 1 Ounce . . . oz. 
 
 16 Ounces 1 Pound . . .lb. 
 
 28 Pounds 1 Quarter . . . qr. 
 
 4i Quarters, or 112 Ibs. . 1 Hund. Weight cwt. 
 20 Hundred Weight ... 1 Ton . . . ton. 
 
 Drams. 
 
 16 zz 1 Ounce. 
 256 zz 16 zz 1 Pound 
 7168 zz 448 zz 28 zz 1 Quarter. 
 28672 zz 1792 zz 112 zz 4 zz 1 Hund. wt. 
 573440 zz 35840 zz 2240 zz 80 zz 20 zz 1 Ton. 
 
 Other Denominations in this Weight. 
 
 A Firkin of Butter . lb. 56 
 
 - Soap . 
 
 A Barrel of Anchovies . 30 
 
 Raisins. . 112 
 
 A Pother of Lead cwt. 19 J 
 
 A Stone of Iron Shot, lb. 
 or Horseman's Wt. . 14 
 A Truss of Straw ... 36 
 Hay. . 60 
 
 Old Hay . . 56 
 
 36 Trusses .... a Load 
 
 WOOL WEIGHT. 
 
 7 Pounds (lb.) 1 Clove . . . cl. 
 
 2 Cloves, or 14 lb 1 Stone . . . st. 
 
 2 Stone, or 28 Ibs 1 Tod . . . id. 
 
 6 Tods and a half .... 1 Wey . . . wy. 
 
 2 Weys 1 Sack . . . sa. 
 
 12 Sacks 1 Last . . . la. 
 
 And a Pack of Wool is 12 score, or 240 pounds. 
 
20 Tables of Heights, ffc. 
 
 APOTHECAKIES' WEIGHT. 
 
 20 Grains (gr.) . . . make 1 Scruple . sc. or 9 
 
 3 Scruples . .... 1 Dram . . dr. or 5 
 
 8 Drams 1 Ounce . . oz. or 5 
 
 12 Ounces 1 Pound . . Ib. or It) 
 
 Grains 
 
 20 zz 1 Scruple 
 
 60 =. 3zzl Dram 
 480 24 zz S = 1 Ounce 
 5760 zz 288 :=: 96 12 z= 1 Pound 
 
 The A.pothecaries mix their medicines by this weight ; but they buy 
 and sell their commodities by Avoirdupois Weight. 
 
 The pound and ounce in this weight are the same as those in Troy 
 Weight, but the smaller divisions are different. 
 
 CLOTH MEASUEE. 
 
 2Inches .... make 1 Nail nl. 
 
 4 Nails 1 Quarter of a Yard qr. 
 
 3 Quarters 1 Flemish Ell . . F. E. 
 
 4 Quarters 1 Yard .... yd. 
 
 5 Quarters. ..... 1 English Ell . . E. E. 
 
 6 Quarters 1 French Ell, . . Fr. E 
 
 Inches 
 2i zz 1 Nail 
 
 9 m 4 n: 1 Quarter 
 36 rz 16 =: 4 1 Yard 
 27 =: 12 3 1 Flemish Ell 
 45 20 = 5 = 1 English Ell 
 
 LONG MEASUEE. 
 
 3 Barley-corns (b. <?.) 
 
 1 Inch 
 
 in. 
 
 12 Inches ... 
 
 1 Foot 
 
 ft. 
 
 
 1 Yard 
 
 j l ' 
 yd 
 
 Feet or 2 yards .... 
 
 1 Fathom. . . . 
 1 Eod,Pole, or Perch 
 
 fatli 
 r. p. 
 
 40 Poles 
 
 1 Furlong 
 
 fur. 
 
 8 Furlongs 
 
 1 Mile 
 
 mi. 
 
 3 Miles 
 
 1 League . 
 
 lea. 
 
 GO Geographical Miles, or ) 
 G9|English Miles . . jj 
 
 1 Degree . dr-a. 
 
 or 
 
Talks of Weights, $c. 23 
 
 Barley C. 
 
 3 zz 1 Inch 
 
 36 zz 12 zz 1 Foot 
 108 zz 36 zz 3 zz 1 Yard 
 594 zz 198 zz 164 zz 5| zz 1 Pole 
 23760 zz 7920 zz 660 zz 220 zz 40 zz 1 Furlong 
 190080 zz 63360 zz 5280 zz 1760 zz 320 zz 8 zz 1 Mile 
 Note. 4 inches make a hand ; used only in measuring the height oi 
 horses. The origin of Long Measure is taken from a grain of Barley, 
 three of which, full-sized, make an inch. 
 
 SQUAEE OE LAND MEASUEE. 
 
 144 Square Inches .... 1 Square Foot 
 9 Square Feet .... 1 Square Yard 
 100 Square Feet . . . . 1 Sq. of flooring, roofing, &c. 
 30i Sq.Yds.or272iSq.Feet 1 Square Eod or Perch 
 40 Square Eods .... 1 Square Eood 
 4 Sq. Koods or 1GO Sq. I ^ Square Acre of Land 
 
 Eods, or 4840 Sq. Yds. j 
 640 Square Acres .... 1 Square Mile 
 
 30 Acres 1 Yard of Land 
 
 100 Acres 1 Hide of Land. 
 
 Inches 
 
 144 zz 1 Foot 
 
 1296 zz 9 zz 1 Yard 
 39204 zz 272 zz 30^ zz 1 Pole 
 15681GO zz 10890 zz 1210 zz 40 zz 1 Rood 
 6272640 zz 43560 zz 4840 zz 160 zz 4 zz 1 Acre 
 
 By this measure, all things that have length and breadth are measured; 
 as land, paving, plastering, roofing, tiling, flooring, plumbing, glazing, 
 &c. Land is measured by a chain, called Gunter's Chain, which is 4 
 poles, or 22 yards, or 66 feet long ; and consists of 100 links. Also 10 
 of these chains in length, and one in breadth, make an acre. 
 
 CUBIC OE SOLID MEASUEE. 
 
 1728 Cubic Inches 1 Cubic Foot 
 
 27 Cubic Feet 1 Cubic Yard 
 
 40 Feet of rough timber, or ) -, T , ^ 
 
 en -c< f i. . . 1 Load or Ton 
 
 50 Feet of hewn ditto . . ) 
 
 42 Cubic Feet 1 Ton of Shipping. 
 
 By this measure, stone, timber, and all work? that have 
 nreadth, and thickness, are measured 
 
22 Tables of Weights, t 
 
 WIJSTE MEASUEE. 
 
 4 Gills (gl.) 1 Pint . . . . pt. 
 
 2 Pints 1 Quart. . . . gt. 
 
 4 Quarts 1 Gallon . . . gal. 
 
 10 Gallons 1 AnkerofBrandy ank. 
 
 18 Gallons 1 Bundlet . . . run. 
 
 42 Gallons 1 Tierce . . . tier. 
 
 2 Tierces, or 84 Gallons . . 1 Puncheon . .pun. 
 63 Gallons or 52 J Imp. Gals. . 1 Hogshead . . hhd. 
 2 Hhds. 126 Gals, or 105 Imp. 1 Pipe or Butt . p. 
 2 Pipes, or 252 Gals, or 210 do. 1 Tun . . . . t. 
 Pints 
 
 2 ~ 1 Quart 
 8 zz 4 zz 1 Gallon 
 336 zz 168 zz 42 zz 1 Tierce 
 504 252 zz 63 zz 14= 1 Hogshead 
 672 zz 336 = 84 = 2 = 1J= 1 Puncheon 
 1008 zz 504 zz 126 = 3 = 2 zz Hzz 1 Pipe 
 2016 zz 1008 zz 252 zz 6 zz 4 zz 3~zz 2 zz 1 Tun. 
 A tun of Wine used to be reckoned about 18 cwt. Avoirdupois, the 
 wine gallon containing 231 cubic inches. By this measure all kinds of 
 spirits, as well as cider, mead, vinegar, oil, honey, &c., were measure*! 
 previously to the 1st of Jan. 1826. 
 
 N.B. The Pipe varies in different wines* thus Claret has 126 gallort 
 old measure; Madeira, 110; Vidonia, 120; Sherry, 130; Port, 138 
 Lisbon, 140 gallons, &c. 
 
 ALE AND BEER MEASURE. 
 
 2 Pints (pt.) 1 Quart qt. 
 
 4 Quarts 1 Gallon gal. 
 
 9 Gallons 1 Firkin* fir. 
 
 2 Firkins 1 Kilderkin Jcil. 
 
 2 Kilderkins or 4 Firkins . . 1 Barrel bar. 
 
 1 \ Barrel or 54 Gallons ... 1 Hogshead hhd. 
 
 2 Barrels 1 Puncheon pun. 
 
 2 Hogsheads 1 Butt but. 
 
 2 Butts 1 Tun t. 
 
 Pints 
 
 2 zz 1 Quart 
 
 8 zz 4 zz 1 Gallon 
 
 72 = 36 zz 9 zz 1 Firkin 
 
 144 zz 72zz 18zz 2 zz 1 Kilderkin 
 
 288 zz 144 zz 36 zz 4 zz 2 zz 1 Barrel 
 
 432 zz 216 zz 54 zz 6 zz 3 zz 14zz 1 Hogshead 
 
 576 zz 288 zz 72 zz 8 zz 4 zz 2 zz Hzz 1 Puncheon 
 
 864 zz 432 zz 108 zz 12 zz 6 zz 3 zz 2 - zz lzz 1 Butt. 
 
 * Formerly, while the wine gallon contained 231 cubic inches, the ale an'l b 
 crallon contained 282 cubic inches, and the gallon dry measure 2685, cubic ir.oa- 
 gure. But by Act 6 Geo. IV, c. T4, these distinctions are abolished. [See Tn'olas 
 %t the end of tfcui volume.] 
 
 :-ocr 
 tea- 
 
Tables of Weights, $c. 23 
 
 DKY MEASUEE. 
 
 2 Pints (pt.) .. 1 Quart qt. 
 
 2 Quarts 1 Pottle pot. 
 
 2 Pottles 1 Gallon gal. 
 
 2 Gallons 1 Peck 'pk. 
 
 4 Pecks 1 Bushel bu. 
 
 4 Bushels 1 Coomb co. 
 
 2 Coombs, or 8 bushels 1 Quarter qr. 
 
 4 Quarters 1 Chaldron ch. 
 
 5 Quarters 1 Wey or Load wey. 
 
 2 Weys 1 Last la. 
 
 Pints 
 
 8 zz 1 Gallon 
 16 zz 2 zz 1 Peck 
 64 zz 8 zz 4 zz 1 Bushel 
 256 zz 32 zz 16 zz 4 zz 1 Coomb 
 512 zz 64 zz 32 zz 8 zz 2 zz 1 Quarter 
 2560 zz 320 zz 160 zz 40 zz 10 = 5 zz 1 Wey 
 5120 zz 640 zz 320 zz 80 zz 20 zz 10 zz 2 zz 1 Last. 
 
 This measure is applied to all dry goods ; as Cora, Seeds, Eoots, Salt, 
 Coals, &c. 
 
 N.B. In the purchase of Coals, 3 bushels are 1 sack, and 12 sacks, or 
 36 bushels, 1 chaldron. 
 
 TIME. 
 
 60 Seconds 1 Minute mi. 
 
 GO Minutes 1 Hour ho. 
 
 24 Hours , 1 Day da. 
 
 7 Days 1 Week wk. 
 
 4 Weeks 1 Month mo. 
 
 13 Months, 1 day, 6 ho. or \ 
 
 12 Calendar Months, or > 1 Tear i/r. 
 
 52 Weeks ) 
 
 Seconds 
 
 60 zz 1 Minute 
 
 3600 zz 60 zz 1 Hour 
 
 86400 zz 1440 zz 24 zz 1 Day 
 
 604800 zz 10080 zz 168 zz 7 zz 1 Week 
 
 2419200 zz 40320 zz 672 zz 28 zz 4 zz 1 Month 
 
 31557600 zz 525960 zz 8766 zz 365zz 1 Year. 
 
 To know the days in each month, observe : 
 Thirty days hath September, April, June, and November, 
 February hath twenty-eight alone ; all the rest have thirty-one, 
 Except in Leap Year, at which time, February's days are twenty-nine. 
 
24 Addition of Money. 
 
 EXAMPLES IN- ADDITION OF MONEY. 
 
 s. d. 
 
 s. d. 
 
 5. d. 
 
 s. d. 
 
 (1) 24 4 7 
 
 (2) 37 7 4 
 
 (3) 174 19 11J 
 
 (4) 157 15 4| 
 
 15 7 3 
 
 29 .4 8J 
 
 26 15 4} 
 
 375 10 8 
 
 42 5 6 
 
 51 12 11 
 
 43 3 7} 
 
 39 12 11J 
 
 37 2 5 
 
 14 7 5} 
 
 59 13 5 
 
 4 16 10J 
 
 15 9 8 
 
 60 15 6 
 
 36 7 10} 
 
 700 
 
 73 6 4 
 
 92 3 10 
 
 28 9 3 
 
 73 19 10} 
 
 Sum 207 15 9 
 
 285 11 9} 
 
 369 9 7 
 
 1351 15 9 
 
 183 11 2 
 
 248 4 5} 
 
 194 9 7| 
 
 1194 4| 
 
 Proof 207 15 9 
 
 285 11 9} 
 
 369 9 7 
 
 1351 15 g\ 
 
 s. d. 
 
 . d. 
 
 s. d. 
 
 s. d. 
 
 (5) 72 3 7 
 
 '6) 27 3 7 
 
 (7) 76 7 4 
 
 (8) 37 4 7 
 
 44 4 3 
 
 98 7 4 
 
 54 3 7 
 
 19 7 8 
 
 73 7 8 
 
 35 5 5 
 
 32 9 2 
 
 70 3 9 
 
 26 2 2 
 
 73 9 8 
 
 12 8 4 
 
 59 6 7 
 
 94 6 9 
 
 85 8 6 
 
 34 7 7 
 
 46 9 8 
 
 57 5 2 
 
 42 6 3 
 
 56 8 2 
 
 39 8 9 
 
 67 4 3 
 
 87 8 9 
 
 99 4 6 
 
 68 8 8 
 
 
 
 . *. dJ. 
 
 . 8. d. 
 
 . s. d. 
 
 . s. d. 
 
 (9) 42 15 7 
 
 (10) 57 12 Hi 
 
 (11) 74 13 7| 
 
 (12) 98 17 6 
 
 35 2 4J 
 
 71 17 4 
 
 63 4 11J 
 
 24 13 11J 
 
 73 5 10 
 
 98 5 2f 
 
 79 6 5 
 
 37 7 8 
 
 98 15 3 
 
 35 13 6 
 
 42 15 21 
 
 45 18 10 J 
 
 32 17 2} 
 
 79 16 9 
 
 37 12 8 
 
 78 13 4| 
 
 14 5 6} 
 
 26 15 5* 
 
 98 8 5| 
 
 94 15 83 
 
 36 8 5^ 
 
 30 19 llf 
 
 98 11 9J 
 
 34 2 6 
 
 42 18 6 
 
 99 6 2 
 
 30 3 6 
 
 18 18 8 
 
 
 s. d. 
 
 s. d. 
 
 s. d. 
 
 s. d. 
 
 (13) 74 14 1\ 
 
 (14) 37 15 7} 
 
 (15) 47 14 4 
 
 (16) 73 19 10} 
 
 41 5 4 
 
 15 8 44 
 
 23 16 2 
 
 16 8 7 
 
 26 2 llf 
 
 91 4 2| 
 
 91 12 8 
 
 29 4 4 
 
 13 6 8 
 
 38 12 11 
 
 52 9 3 
 
 61 3 11 
 
 6.9 12 4 
 
 82 15 8 
 
 13 3 ?A 
 
 42 9 7| 
 
 92 8 10 
 
 24 3 4f 
 
 34 5 10} 
 
 94 2 6 
 
 87 4 3} 
 
 79 7 64 
 
 85 15 4 
 
 38 7 94 
 
 38 17 7| 
 
 33 2 9} 
 
 68 7 6} 
 
 55 6 10 
 
 65 5 4 
 
 66 19 101 
 
 89 9 7 
 
 87 14 4} 
 
Addition of Money. 
 
 25 
 
 
 
 8. 
 
 d. 
 
 
 
 8. 
 
 d. 
 
 
 
 8. 
 
 d. 
 
 
 
 8. 
 
 d. 
 
 (17) 12 
 
 13 
 
 ll 
 
 (18) 93 
 
 16 
 
 10 
 
 (19) 41 
 
 7 
 
 4 
 
 (20) 27 
 
 13 
 
 4 T 
 
 34 
 
 5 
 
 7A 
 
 37 
 
 4 
 
 8 
 
 35 
 
 11 
 
 7 
 
 53 
 
 
 
 1 
 
 56 
 
 7 
 
 3 
 
 15 
 
 3 
 
 3 
 
 78 
 
 14 
 
 9 
 
 29 
 
 8 
 
 3* 
 
 78 
 
 19 
 
 8 
 
 61 
 
 7 
 
 H 
 
 14 
 
 16 
 
 6| 
 
 16 
 
 3 
 
 7} 
 
 90 
 
 J 
 
 4 
 
 46 
 
 12 
 
 11} 
 
 62 
 
 19 
 
 
 91 
 
 17 
 
 6 
 
 87 
 
 14 
 
 Ti 
 
 82 
 
 13 
 
 
 27 
 
 17 
 
 3i 
 
 35 
 
 15 
 
 2 
 
 
 
 
 s. 
 
 d. 
 
 
 
 s. 
 
 d. 
 
 
 
 s. 
 
 d. 
 
 
 
 8. 
 
 d. 
 
 (21) 792 
 
 4 
 
 n 
 
 (22) 714 
 
 12 
 
 71 
 
 (23) 147 
 
 13 
 
 7^ 
 
 (24) 714 
 
 16 
 
 8 
 
 484 
 
 7 
 
 
 2 ! '8 
 
 14 
 
 l 
 
 216 
 
 19 
 
 41. 
 
 241 
 
 18 
 
 3 
 
 176 
 
 13 
 
 2J 
 
 314 
 
 11 
 
 4f 
 
 371 
 
 12 
 
 3-J- 
 
 376 
 
 13 
 
 in 
 
 508 
 
 17 
 
 H 
 
 672 
 
 16 
 
 
 498 
 
 4 
 
 6| 
 
 662 
 
 19 
 
 
 251 
 
 12 
 
 6* 
 
 485 
 
 ID 
 
 2 
 
 137 
 
 13 
 
 5 
 
 927 
 
 17 
 
 2 
 
 040 
 
 IS 
 
 
 141 
 
 13 
 
 3* 
 
 582 
 
 17 
 
 
 153 
 
 15 
 
 n 
 
 
 
 OF WEIGHTS AKD MEASUEES. 
 
 Troy Weight. 
 
 Ibs. oz. dwts. oz. dwts. gra. lbs.oz.dwts. oz. dwts. gra. 
 (25) 11 11 19 (26) 6 16 10 (27) 5 9 13 (28) 3 11 15 
 
 10 
 
 6 
 
 5 
 
 8 
 
 10 
 
 18 
 
 6 
 
 7 
 
 6 
 
 4 
 
 5 
 
 3 
 
 8 
 
 4 
 
 11 
 
 5 
 
 5 
 
 16 
 
 8 
 
 3 
 
 15 
 
 6 
 
 13 
 
 10 
 
 9 
 
 2 
 
 12 
 
 3 
 
 15 
 
 12 
 
 7 
 
 8 
 
 7 
 
 2 
 
 19 
 
 2 
 
 7 
 
 3 
 
 8 
 
 9 
 
 14 
 
 9 
 
 3 
 
 11 
 
 9 
 
 7 
 
 6 
 
 12 
 
 5 
 
 7 
 
 13 
 
 2 
 
 3 
 
 7 
 
 6 
 
 10 
 
 4 
 
 9 
 
 14 
 
 16 
 
 53 
 
 
 
 8 
 
 
 
 
 
 
 
 
 
 
 Avoirdupois 
 
 Weight. 
 
 tons 
 
 cwts. qr. 
 
 cwts. qr. 
 
 Ibs. 
 
 Ibs. 
 
 oz. 
 
 dra. 
 
 cwts. 
 
 qr. 
 
 Ibe. 
 
 (29) 
 
 13 
 
 1 
 
 (30) 16 
 
 1 
 
 15 (31) 18 
 
 13 
 
 5 
 
 (32) 3 
 
 
 8 
 
 7 
 
 6 
 
 
 
 14 
 
 3 
 
 7 
 
 13 
 
 9 
 
 8 
 
 6 
 
 2 
 
 9 
 
 5 
 
 3 
 
 3 
 
 12 
 
 
 
 6 
 
 15 
 
 11 
 
 13 
 
 2 
 
 
 
 7 
 
 8 
 
 9 
 
 2 
 
 15 
 
 2 
 
 5 
 
 11 
 
 10 
 
 2 
 
 5 
 
 3 
 
 5 
 
 9 
 
 11 
 
 
 
 18 
 
 1 
 
 13 
 
 12 
 
 8 
 
 7 
 
 4 
 
 3 
 
 6 
 
 4 
 
 15 
 
 3 
 
 10 
 
 
 
 12 
 
 10 
 
 4 
 
 8 
 
 8 
 
 
 
 i 
 
28 Talles of Weights, $c. 
 
 Apothecaries' Weight. 
 
 Ibs. oz, dra. oz. dra. scr. Ihs. oz. dra. dra. scr. gr. 
 
 (33) 6 10 7 (34) 5 5 1 (35) 5 10 3 (36) 1 1 12 
 
 8 
 
 5 
 
 6 
 
 4 
 
 3 
 
 2 
 
 3 
 
 9 
 
 7 
 
 3 
 
 
 
 11 
 
 4 
 
 8 
 
 4 
 
 2 
 
 G 
 
 
 
 2 
 
 6 
 
 6 
 
 4 
 
 2 
 
 6 
 
 5 
 
 11 
 
 5 
 
 6 
 
 4 
 
 2 
 
 
 
 5 
 
 
 
 7 
 
 1 
 
 16 
 
 6 
 
 9 
 
 2 
 
 3 
 
 2 
 
 
 
 4 
 
 11 
 
 3 
 
 6 
 
 2 
 
 10 
 
 9 
 
 7 
 
 3 
 
 7 
 
 7 
 
 1 
 
 3 
 
 8 
 
 2 
 
 2 
 
 
 
 9 
 
 42 
 
 5 
 
 3 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Cloth 
 
 Measure. 
 
 yds. 
 (37) 16 
 
 qrs. 
 3 
 
 nls. 
 2 
 
 E.E. 
 
 (38) 10 
 
 qrs. nls. 
 4 3 
 
 F.E. 
 (39) 7 
 
 qrs. 
 1 
 
 nls. 
 2 
 
 qrs. 
 (40) 2 
 
 nls. 
 3 
 
 in. 
 2 
 
 18 
 
 
 
 1 
 
 16 
 
 3 
 
 
 
 6 
 
 2 
 
 
 
 6 
 
 2 
 
 Sf 
 
 19 
 
 2 
 
 
 
 14 
 
 
 
 2 
 
 2 
 
 
 
 1 
 
 1 
 
 1 
 
 H 
 
 25 
 
 1 
 
 3 
 
 12 
 
 2 
 
 3 
 
 3 
 
 1 
 
 1 
 
 3 
 
 
 
 2^ 
 
 32 
 
 2 
 
 2 
 
 7 
 
 2 
 
 2 
 
 8 
 
 
 
 2 
 
 9 
 
 2 
 
 2 
 
 21 
 
 3 
 
 1 
 
 8 
 
 1 
 
 
 
 9 
 
 2 
 
 3 
 
 2 
 
 3 
 
 04 
 
 134 
 
 1 
 
 1 
 
 
 
 
 
 
 
 
 
 
 Measure. 
 
 mis. fur. po. lea. mis. fur. yds. feet. in. feet in. bar. 
 
 (41) 10 7 10 (42) 18 2 1 (43) 18 2 7 (44) 11 3 1 
 
 16 38 J711 27 16 350 
 
 12 6 9 13 5 32 8 24 7 2 
 
 15 4 16 16 1 6 45 1 11 390 
 
 20 2 13 72 2 7 61 2 10 70 8 2 
 
 19 7 39 16 1 4 12 1 9 10 4 1 
 
 95 7 15 
 
 Land Measure. 
 
 ac. r. po. ac. r. po. ac. r. po. ac. r. po. 
 
 (45) 12 2 11 (46) 16 3 19 (47) 8 1 15 (48) 909 
 
 16 3 15 12 7 9 2 10 827 
 
 24 25 13 1 18 7 3 16 718 
 
 18 1 18 16 10 1 3 13 516 
 
 30 3 32 15 2 16 2 2 20 3 3 10 
 
 16 2 8 16 3 30 6 1 30 625 
 
 119 1 29 
 
Compound Addition. 27 
 
 Wine Measure. 
 pip. hhda. gal. tunshhds.gal. hhds. gal. qts. 
 
 (19) 10 
 
 1 
 
 8 
 
 (50) 16 1 
 
 12 
 
 (51) 3 
 
 11 
 
 
 (52) 9 
 
 1 
 
 
 6 
 
 
 
 9 
 
 24 
 
 2 
 
 10 
 
 2 
 
 5 
 
 2 
 
 7 
 
 2 
 
 
 
 8 
 
 1 
 
 10 
 
 36 
 
 
 
 11 
 
 4 
 
 7 
 
 
 
 5 
 
 
 
 1 
 
 12 
 
 1 
 
 16 
 
 18 
 
 3 
 
 14 
 
 6 
 
 9 
 
 3 
 
 10 
 
 3 
 
 1 
 
 16 
 
 
 
 20 
 
 17 
 
 2 
 
 29 
 
 5 
 
 16 
 
 2 
 
 8 
 
 2 
 
 
 
 8 
 
 
 
 18 
 
 9 
 
 1 
 
 30 
 
 4 
 
 20 
 
 1 
 
 9 
 
 
 
 1 
 
 62 
 
 
 
 18 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ale 
 
 and 
 
 Beer Measure. 
 
 hhds 
 
 . gal. qts. 
 
 bar. 
 
 fir.j 
 
 
 fir. 
 
 gal. 
 
 pts. 
 
 bts. hhds 
 
 .gal 
 
 53) 7 
 
 16 
 
 
 
 (54) 8 
 
 3 
 
 8 
 
 (55) 16 
 
 8 
 
 7 
 
 (56) 3 
 
 1 
 
 30 
 
 10 
 
 7 
 
 
 
 9 
 
 1 
 
 7 
 
 21 
 
 7 
 
 2 
 
 2 
 
 
 
 16 
 
 15 
 
 9 
 
 2 
 
 7 
 
 
 
 5 
 
 34 
 
 5 
 
 5 
 
 7 
 
 1 
 
 35 
 
 24 
 
 14 
 
 1 
 
 6 
 
 1 
 
 6 
 
 18 
 
 4 
 
 6 
 
 8 
 
 1 
 
 10 
 
 8 
 
 21 
 
 
 
 5 
 
 2 
 
 4 
 
 27 
 
 8 
 
 3 
 
 9 
 
 1 
 
 8 
 
 9 
 
 35 
 
 3 
 
 10 
 
 3 
 
 2 
 
 16 
 
 2 
 
 4 
 
 5 
 
 
 
 4 
 
 74 
 
 50 
 
 1 
 
 
 
 
 
 
 
 
 
 
 Dry 
 
 Measure. 
 
 oha. bus. pks. 
 
 qrs. bus. ] 
 
 >ks. 
 
 bus. 
 
 pks. 
 
 gal. 
 
 las 
 
 . we. 
 
 qts, 
 
 (57) 31 
 
 12 
 
 3 
 
 (58) 10 
 
 7 
 
 3 
 
 (59) 6 
 
 1 
 
 1 
 
 (60) 3 
 
 1 
 
 
 21 
 
 18 
 
 1 
 
 11 
 
 
 
 1 
 
 5 
 
 2 
 
 
 
 2 
 
 
 
 2 
 
 35 
 
 13 
 
 2 
 
 15 
 
 5 
 
 
 
 4 
 
 3 
 
 
 
 9 
 
 
 
 3 
 
 42 
 
 14 
 
 
 
 8 
 
 6 
 
 2 
 
 3 
 
 
 
 1 
 
 6 
 
 1 
 
 
 
 17 
 
 10 
 
 2 
 
 7 
 
 3 
 
 1 
 
 2 
 
 2 
 
 1 
 
 5 
 
 1 
 
 2 
 
 21 
 
 35 
 
 3 
 
 9 
 
 
 
 3 
 
 1 
 
 1 
 
 1 
 
 4 
 
 
 
 4 
 
 169 
 
 32 
 
 3 
 
 
 
 
 
 
 
 
 
 
 Time. 
 
 yrs. mo. wks. 
 
 mo. wks. da. 
 
 wks. 
 
 da. 
 
 ho. 
 
 ho. min. 
 
 sec. 
 
 61) 6 
 
 3 
 
 2 
 
 (62) 11 
 
 2 
 
 6 
 
 (63) 6 
 
 6 
 
 18 
 
 (64)5 
 
 30 
 
 11 
 
 8 
 
 4 
 
 3 
 
 13 
 
 3 
 
 5 
 
 8 
 
 5 
 
 6 
 
 6 
 
 45 
 
 18 
 
 9 12 
 
 1 
 
 21 
 
 1 
 
 4 
 
 9 
 
 
 
 23 
 
 3 
 
 59 
 
 20 
 
 7 
 
 7 
 
 2 
 
 6 
 
 2 
 
 3 
 
 7 
 
 4 
 
 12 
 
 2 
 
 16 
 
 31 
 
 5 12 
 
 1 
 
 5 
 
 3 
 
 5 
 
 8 
 
 3 
 
 14 
 
 7 
 
 28 
 
 59 
 
 4 
 
 8 
 
 3 
 
 4 
 
 1 
 
 6 
 
 5 
 
 2 
 
 20 
 
 8 
 
 19 
 
 12 
 
 42 10 
 
8 Compound Addition. 
 
 MISCELLANEOUS QUESTIONS. 
 
 (65) Suppose the rent of my house 100Z. per annum;, 
 poor's rates, 15Z. 10s. ; window and house tax, 121. 15s. ; 
 other taxes and rates, 10Z. 5s. ; what is the whole sum ? 
 
 Ans. 1381. 10s. 
 
 (66) A gentleman's steward received for rents of A. 210Z. 
 (>s. ; of B. 169Z. 17s. ; of C. 150Z. 15s. ; of D. 260Z. 12s. ; of 
 E. 300Z. 10s. ; and of F. 751. 16s. 6d. ; what was the whole 
 sum ? Ans. 1167/. 16s. 6d. 
 
 (67) In taking an account of debts owing to me, I find 
 that Mr. W. owes me 161. 17s. 8$d. ; Mr. X. 271. 15s. 3R; 
 Mr. Y. 111Z. ; Mr. Z. 77Z. 17s. Qd. ; and there are other 
 small sums amounting to 35. 12s. 31 d. ; what is the whole 
 sum due to me ? Ans. 2691. 3s. O^d. 
 
 (68) Paid the carrier for the iollowing freights; viz. hops, 
 12 cwt. 2 qrs. lOlb. ; wool, 5 cwt. 2 qrs. ; teas, 1 cwt. 2 qrs. 
 14 Ib. ; sugars, 10 cwt. 1 qr. 10 Ib. ; and salt, weighing 
 2 qrs. 21 Ib. ; for how much weight was he paid ? 
 
 Ans. 30 cwt. 2 qrs. 27 Ib. 
 
 (69) Just received 4 parcels of cloth in the first parcel, 
 150 yds. 3 qrs. 2 nls. ; in the second, 120 yds. 1 qr. 1 nl. ; 
 in the third, 99 yds. ; and in the fourth, 305 yds. 3 qrs. ; 
 how many yards in the 4 parcels ? Ans. 675 yds. 3 qrs. 3 nls. 
 
 (70) Paid the following bills for the repairs of my house, 
 the mason's, 71. 3s. 6d. ; the bricklayer's, 9Z. 8s. 7d. ; the 
 carpenter's, 121. 13s. lOd. ; and the painter's, 15Z. 15s. ; 
 what did the whole of the repairs cost me ? 
 
 Ans. 45Z. Os. lid. 
 
 (71) The measurement of my Leasows' estate is as follows, 
 viz. the site of the house, garden, and fold, 1 acre 3 roods 
 20 poles ; the great orchard, 12 acres 2 roods 12 poles ; the 
 little close, 4 acres 3 roods ; the arable land, 30 acres 1 rood 
 19 poles ; the meadow land, 53 acres 2 roods, 30 poles ; and 
 the wood lands, 5 acres 2 roods ; how many acres in the 
 whole ? Ans. 108 acres 3 roods 1 pole 
 
 (72) A farmer sold at market, wheat to the amount of 
 187. 6s. 4d. ; barley, 121. 6s.; beans, 9Z. 8s. 9d. ; oats, 
 1 31. 15s. 3 d. ; and turnip seed, 10Z. 18s. 4d. ; what was the 
 whole amount ? Ans. 64Z. 14s. 8<7. 
 
 (73) Bought goods at Birmingham to the amount of 
 560Z. 15s. 6d. ; paid packing and porterage, II. 10s.; carriage, 
 *M. 6s. 8d. ; expenses of journey, 3Z. 8s. 6d. ; what did the 
 /roods stand me in ? Ans. 56SI. Os. 8d. 
 
( 29 ) 
 
 COMPOUND SUBTRACTION. 
 
 COMPOUND SUBTRACTION is the method of find- 
 ing the difference beuween any two given numbers of different 
 denominations. 
 
 RULE 1st. Place the less number under the greater, so 
 that the parts which are of the same denomination may stand 
 directly under each other ; then beginning at the right hand, 
 subtract each number in the lower line from that above it, 
 and set down the remainder. 
 
 2nd. When any of the lower numbers are greater than 
 the upper, increase the upper number by as many as make 
 one of the next higher denomination, from which take the 
 lower number ; set down the difference, and carry one to 
 the next number in the lower line, which subtract from that 
 above it, in the same manner as before. 
 
 PEOOP. As in integers. 
 
 EXAMPLES. 
 
 . s. d. . *. d. . s. d. 
 
 From (1) 12 9 6 (2) 27 12 9| (3) 19 10 7J 
 
 Take 8 5 4 13 4 3 4 16 9J 
 
 Remains 4 4 2 14 8 6 14 13 9f 
 
 Proof 12 9 6J 27 12 9} 19 10 7| 
 
 . s. d. . s. d. . 8. d. 
 
 (4) 126 17 8 (5) 473 14 9 (6) 276 10 6j 
 113 12 3 251 8 4 134 8 2} 
 
 s. d. s. d. s. d. 
 
 (7) 472 16 8J (8) 126 18 4| (9) 714 18 3$ 
 397 15 2 86 17 7 276 4 8$ 
 
 8. d. s. d. * d. 
 
 '10) 345 2 4 (11) 483 16 5 (12) 247 3 4 
 186 12 S 297 8 10J 185 17 8| 
 
Compound Subtraction. 
 
 *. d. 
 
 (13) 74 4 5 
 16 8 9 
 
 s. d. 
 
 (14) 9 11 5 
 3 15 7i 
 
 5. 
 
 (15) 22 8 
 15 12 
 
 d. 
 
 4 
 
 71 
 
 . d. 
 
 (16) 896 
 7 13 7| 
 
 s. d. 
 (17) 10 10 1 
 9 10 1J 
 
 s. d. s. d. 
 
 (18) 8 01 (19) 15 
 2 OJ 
 
 O. Mt 
 
 15 
 
 15 Of 
 
 d. 
 
 s. 
 
 (20) 7 7 7 
 499 
 
 Borrowed . 
 
 *. d. 
 
 . (21) 365 16 8| 
 
 50 9J 
 Paid at 20 10 
 different 30 15 6} 
 times. 37 10 8 
 100 
 50 12 10 
 
 Taid in all. . . 289 9 9} 
 Remains unpaid . 76 6 11 
 
 Lent 
 
 s. d. 
 . (22) 500 
 
 120 6 8 
 
 Received 50 9 4 
 
 at different 36 12 6 
 
 times. 100 
 
 50 10 
 
 75 16 6 
 
 Received in all 
 Remains . . 
 
 Or WEIGHTS AND MEASTJEES. 
 Troy Weight. 
 
 Ibs. oz.dwts. oz. dwts. gra. Ibs. oz.dwta. oz. dwts. gra. 
 
 (23) 30 7 8 (24) 12 7 16 (25) 15 8 6 (26) 8 13 9 
 
 9 8 12 5 10 18 10 11 4 7 17 
 
 20 10 16 
 
 Avoirdupois Weight 
 
 tons cwts. qr. cwts. qr. Ibs. Ibs. oz. dra. cwts. qr. Ibs. 
 
 (27) 13 7 2 (28) 17 1 17 (29) 25 312 (30) 5 2 13 
 
 8 12 3 7 1 27 16 11 5 4 2 20 
 
 4 14 
 
Uut/ijpound Sultr action. 
 
 31 
 
 Apothecaries Weight. 
 
 Iba. oz. dra. oz. dra. scr. Ibs. oz. dra. drs. scr. gr 
 
 (31) 13 5 6 (32) 971 (33) 934 (34) 12 2 15 
 
 087 522 661 5 17 
 
 
 12 
 
 * 
 
 7 
 
 
 
 
 
 
 
 
 
 yds. 
 
 35) 16 
 8 
 
 qr, 
 
 1 
 
 nls. 
 1 
 3 
 
 E.E. 
 
 (36) 10 
 5 
 
 Cloth Measure. 
 qrs. nls. F.E. 
 3 2 (37) 7 
 20 5 
 
 qrs. 
 2 
 2 
 
 nls. 
 1 
 2 
 
 qrs. 
 (38) 8 
 
 7 
 
 nls. 
 1 
 2 
 
 in. 
 
 H 
 oj 
 
 
 8 
 
 1 
 
 2 
 
 
 
 
 
 
 
 
 
 (39) 
 (43) 
 
 (47) 
 
 mis 
 24 
 17 
 
 .fur. 
 3 
 5 
 
 po. 
 29 
 
 Long Measure. 
 1 enrols, fur. yds 
 (40) 19 2 5 (41) 18 
 907 10 
 
 . feet in. 
 I 4 
 2 11 
 
 feet in 
 (42) 13 5 
 8 7 
 
 bar. 
 
 2 
 
 
 
 6 
 
 5 
 
 18 
 
 
 
 
 
 
 
 
 
 ac. 
 24 
 5 
 
 r. 
 2 
 1 
 
 po. 
 20 
 30 
 
 ac. 
 
 (44) 16 
 8 
 
 Land Measure. 
 r. po. ac. 
 3 13 (45) 18 
 03 10 
 
 r. 
 1 
 
 2 
 
 po. 
 12 
 
 18 
 
 ac. 
 
 (46) 12 
 2 
 
 r. 
 3 
 3 
 
 po. 
 16 
 
 27 
 
 19 
 
 
 
 30 
 
 
 
 
 
 
 
 
 
 $ 
 
 12 
 
 .hhd.gal. 
 1 18 
 10 
 
 Wine Measure. 
 tuns hhd.gal. hhd. 
 (48) 14 1 10 (49) 8 
 7 2 23 5 
 
 gal. 
 4 
 12 
 
 qts. 
 3 
 
 
 gal. 
 (50) 36 
 29 
 
 qts. 
 2 
 3 
 
 pts. 
 1 
 
 
 12 
 
 1 
 
 8 
 
 
 
 
 
 
 
 
 
 Ale and Beer Measure. 
 
 hhd. gal. qts. bar. fir. gal. fir. gal. pts. bts. hhd. gal. 
 
 (51) 12 11 2 (52) 16 2 5 (53) 19 5 5 (54) 14 1 53 
 
 7 24 3 932 987 8 50 
 
 4 40 3 
 
 Dry Measure. 
 
 cha. bus. pks. qrs.bus.pks. bus.pks.gals. las.wys.qrs. 
 
 (55) 15 16 3 (56) 18 1 (57) 13 1 (58) 11 1 3 
 
 8 25 54 2 721 704 
 
 6 27 3 
 
 Time. 
 
 yrs.mo.wks. mo.wks.dys. wks.dys ho. ho. mi. sec. 
 
 (59) 12 12 1 (60) 202 (61) 3 6 11 (62) 23 59 29 
 
 753 136 2 23 11 10 59 
 
32 Compound Subtraction. 
 
 MISCELLANEOUS QUESTIONS. 
 
 (63) From 100Z. take 99Z. 19s. llf d., what remains ? 
 
 Ans. 01, Os, 0\d. 
 
 (64) Take 146Z. Us. 9d. from 150Z. Ans. 3Z. 8*. 2? d. 
 
 (65) Subtract 372Z. 12s. 8f rf. from 4232. lls. 7Jrf. 
 
 ^zs. 50Z. 18s. 10! d. 
 
 (66) How much does 50 guineas exceed 25Z. 10s. G^d. ? 
 
 Ans. 26Z. 195. 5J& 
 
 (67) "What is the difference between 30Z. and 19 guineas? 
 
 Ans. 101. Is. 
 
 (68) If I have a bill to pay of 7Z. 17s. 6^., and I deliver 
 a 101. bank-note for that purpose, what change ought I to 
 receive ? Ans. 21. 2s. 6d. 
 
 (69) A person sends a note of 20Z. to discharge a bill, 
 and receives in change 4Z. 14s. 6d., what was the bill ? 
 
 Ans. 151. 5s. Qd. 
 
 (70) A servant's wages are 12 guineas per year, and 
 having received in part 77. 19s. 6d., what remains due ? 
 
 Ans. 4Z. 12s. 6d. 
 
 (71) Borrowed of my friend 50Z., and paid in part 
 37Z. 5s. 8^., how much remains to pay ? Ans. 121. 14s. 4<d. 
 
 (72) For my estate, which is 500Z. per year, I pay rates 
 and taxes to the amount of 56Z., what is my clear income ? 
 
 Ans. 444 1. 
 
 (73) Bought 24 yards of muslin, out of which I sold two 
 dresses of 8 yds. 2 qrs. 2 nls. each ; what quantity have I 
 left ? Ans. 6 yds. 3 qrs. 
 
 (74) The great bell at Oxford weighs 7 tons 11 cwt. 3 qrs. 
 4 lb., and that at St. Paul's, 5 tons 2 cwt. 1 qr. 22 Ib. ; how 
 much does the former in weight exceed the latter ? 
 
 Ans. 2 tons 9 cwt. 1 qr. 10 lb. 
 
 (75) A youth has served 5 yrs. 3 mo. 2 wks. 5 days of 
 his 7 years apprenticeship, how much longer has he yet to 
 serve ? Ans. 1 yr. 9 mo. 1 wk. 2 days 
 
 (76) Borrowed of A. B. 2000Z., and paid at 3 different 
 times 500Z. each, and at another time 265Z., what remains 
 due ? Ans. 235Z. 
 
 (77) Between the towns A. and B. there are two roads, 
 the one measures 16 miles 2 furlongs 10 poles, and the other 
 14 miles 7 furlongs 39 poles, what is the differf3nce ? 
 
 Ans. 1 mile 2 furlongs 11 poles 
 
 (78) Purchased 3 pipes 46 galls, of wine, and sold 1 pipe 
 and 120 galls, (allowing 136 gallons to the pipe) ; I demand 
 the quantity that remains ? Ans. 1 pipe 62 galls. 
 
(33) 
 
 COMPOUND MULTIPLICATION. 
 
 COMPOUND MULTIPLICATION is the method of 
 finding what any given number, of different denominations^ 
 will amount to, when repeated any proposed number of times. 
 I. When the multiplier does not exceed 12. 
 
 RULE. Set the- multiplier under the lowest denomination 
 of the multiplicand ; multiply separately each figure of the 
 multiplicand by the multiplier, and carry the several pro- 
 ducts, as they occur, to the next higher denomination. Set 
 down the several remainders, and carry the integers to the 
 next product ; with which proceed as before. 
 
 EXAMPLES. 
 
 s. d. s. d. s, d. 
 
 Multiply (1) 21 2 8$ (2) 42 10 4 
 
 by ... 2 3 
 
 (3) 354 8 6f 
 4 
 
 42 5 5 127 11 Of 1417 14 3 
 
 s. d. s. d. s. d. s. d. 
 (4) 16 5 8 (5) 9 10 7i (6) 13 8 6i (7) 10 12 7f 
 56 78 
 
 s. d. s. d. s. d. s. d. 
 
 (8) 19 64 (9) 17 9|- (10) 14 5 (11) 18 10^ 
 9 10 11 12 
 
 (12) 2 Ibs. at 5s. 6d. per Ib. 
 
 (13) 3 yds. at 4s. lOd. per yd. 
 
 (14) 4 ells at 6s. 2d, per ell. 
 
 (15) 5 pair at 3s. 9d. per pair. 
 
 (16) 6 doz. at Ss. 3d. per doz. 
 
 (17) 7hhds.atl4s.6J.prhhd. 
 
 (18) 8 Ibs. at 12s. 9d. per Ib. 
 
 (19) 9 cwt.at 135 Id. pr.cwt. 
 
 o 3 
 
 (20) 5 tons at 11. 4s. Gd. per 
 ton. 
 
 (21) 6 hhds. at 21. 3s. U. per 
 hogshead. 
 
 (22) 7 gals, at 51. 2s. 4d. per 
 gallon. 
 
 (23) 8 qrs. at 11. Qs. Sd. per 
 quarter. 
 
 (24) 9 bushels at II. Is. 9d. 
 per bushel. 
 
 (25) 10 Ibs. at 21 Us. Sd. 
 per Ib. 
 
 (26) 11 dozens at 3Z. 13s. 9d. 
 per dozen. 
 
 (27) 12 cwt. at 51. 16s. lid 
 per cwt. 
 
34 Compound Multiplication. 
 
 II. If the multiplier exceeds 12, multiply by any two 
 numbers which, multiplied together, will make the same 
 number. 
 
 (28) 16 Ibs. at 7s. Qd. per Ib. (29) 24 ells at 31. 4s. Sd. per ell. 
 
 4 3 
 
 1 10 9 14 
 
 4 8 
 
 600 77 12 
 
 (30) 16 bushels of barley, at 10s. 6d. per bushel ? 
 
 Ans. SI. Ss. Od. 
 
 (31) 18 yards of satin, at 12s. 8d. per yard ? 
 
 Ans. III. Ss. 9d. 
 
 (32) 20 gallons of ale, at 2s. Qd. per gallon ? 
 
 Ans. 21. 10s. Od. 
 
 (33) 32 Ibs. of tea, at 8s. Qd. per Ib ? Ans. 137. 12s. Od. 
 
 (34) 35 bushels of oats, at 5s. Qd. per bushel ? 
 
 Ans. 97. 12s. 6d, 
 
 (35) 36 ells of Holland, at 5s. 10J<7. per ell ? 
 
 Ans. 107. 12s. 3d. 
 
 (36) 48 gallons of porter, at Is. Sd. per gallon ? 
 
 Ans. 47. Os. Od. 
 
 (37) 60 reams of paper, at 21. 6s. per ream ? 
 
 Ans. 138Z. Os. Od. 
 
 (38) 72 ells of dowlas, at 2s. 3%d. per ell ? Ans. SI. 6s. Gd. 
 
 (39) 84 Ibs. of candles, at Is. I%d. per Ib. ? Ans. 47. 10s. 3d. 
 
 (40) 96 qrs. of barley, at 47. 14s. Qd. per qr. ? 
 
 Ans. 4537. 12s. Od. 
 
 (41) 100 Ibs. of butter, at Is. 9d. per Ib. ? 
 
 Ans. SI. 19s. 2d. 
 
 (42) 108 cwt. of cheese, at 37. 19s. Qd. per cwt. 
 
 Ans. 4297. 6s. Od. 
 
 (43) 120 pair of shoes, at 9s. Qd. per pair ? Ans. 577. Os. d. 
 
 (44) 121 Ibs. of tobacco, at 5s. 11^. per. Ib. ? 
 
 Ans. 367. Os. 11^. 
 
 (45) 132 qrs. of wheat, at 57. 7s. per quarter? 
 
 Ans. 7067. 4s. Od. 
 
 (46) 144 yards of cloth, at 3s. 9 $d. per yard ? 
 
 Ans. 277. 6s. Od. 
 
Compound Multiplication. 85 
 
 III. If no two numbers can be found whose product will 
 be equal to the given quantity, multiply, as before, by the 
 numbers that come nearest to it ; to this product add that of 
 the first line multiplied by the quantity remaining; the 
 sum of these will give the answer. 
 
 "When the quantity of |, 1, or f , or any other fractional, 
 parts are required, take parts of the given price, and add to 
 the former work ; or multiply by the upper figure, and divide 
 by the under. 
 
 (47) 26 yds. at 5s. 8%d. per yd. (48) 29f Ibs. at 4Z. Ss. Gd. per Ib. 
 5 4 
 
 1 8 6J= 5 17 14 
 
 5 7 
 
 7 2 8|=25 123 18 
 
 5 8i= 1 j 4 8 6 
 
 1^233 
 
 785 =26 1 2 14 
 
 131 12 
 
 (49) 29 firkins, at II. 7s. 9d. per firkin ? Ans. 40Z. 4s. 9d. 
 
 (50) 31 yards, at 11s. Wd. per yard ? Ans. 181. 6s. lOd. 
 
 (51) 34 ells, at 10s. 4^. per ell ? Ans. 171. 12s. 9<Z. 
 
 (52) 37 puncheons, at 91. 8s. 6d. per puncheon ? 
 
 Ans. 348Z. 14s. 6d. 
 
 (53) 41 firkins, at 21. Is. 8d. per firkin ? Ans. 851. 8s. 4<d. 
 
 (54) 43 yards, at 12s. 8d. per yard ? Ans. 271. 4s. 3d. 
 
 (55) 46 ells, at 16s. 4<d. per ell ? Ans. 371. 13s 3d. 
 
 (56) 51 puncheons, at HZ. lls. per puncheon ? 
 
 Ans. 589Z. Is. Qd. 
 
 (57) 93i butts, at 3Z. lls. Qd. per butt ? Jws. 333Z. 7s. 4$<?. 
 
 (58) 99i acres, at 3Z. 19s. per acre ? Ans. 393Z. Os. Qd. 
 
 (59) 103| firkins, at 3Z. 4s. per firkin ? Ans. 332Z. Os. Od. 
 
 (60) 11 If gallons, at U 7. 6rf. per gall, P^lws. 153Z.13s.l.W. 
 
 (61) 124| butts, at 4Z. 6s. 8/. per butt ? J[;2s. 538Z. 8s. 4rf. 
 
 (62) 135 i acres, at 4Z. 10s. per acre ? Ans. 609Z. 15s Od. 
 
 (63) 136-| firkins, at 4Z, 8s. per firkin? Ans. 601Z. 14s. Od. 
 
 (64) 147| gaUons, at 1Z. 12s. Qd. per gallon ? 
 
 Ans. 240Z. Is. 
 
3D Compound Multiplication. 
 
 OJT WEIGHTS AND MEASUHES. 
 
 Troy and Apothecaries' Weight. 
 
 Ibs. oz. dwts. oz. dwts. gr. Ibs. oz. dra. dra. scr. gr. 
 
 (65) 847 (66) 563 (67) 423 (68) 526 
 
 2345 
 
 16 8 14 
 
 Avoirdupois Weiyht. 
 
 tons. cwts. qrs. cwts. qrs. Ibs. Ibs. oz. dra. cwts. qrs. Ibs. 
 
 (69) 572 (70) 8 2 10 (71) 457 (72) 8 3 12 
 
 6789 
 
 32 5 
 
 Cloth Measure. 
 
 yds. qrs. nls. E.E. qrs. nls. F.E. qrs. nls. yds. qrs. nls. 
 (73) 10 3 2 (74) 6 1 2 (75) 800 (76) 12 1 2 
 10 11 12 
 
 (77) 
 
 108 
 
 3 
 
 
 
 
 
 
 
 
 mis. fur. 
 12 3 
 
 Long and Land Measure. 
 
 po. yds. feet in. feet in. b.co. acr. 
 12 (78) 827 (79) 9 11 2 (80) 24 
 345 
 
 r. po. 
 
 3 22 
 6 
 
 37 
 
 1 
 
 36 
 
 
 
 
 
 
 Wine, Ale, 
 
 and Beer Measure. 
 
 (81) 
 
 tuns 
 12 
 
 hhd 
 
 1 
 
 . gal. hhd. 
 2 (82) 9 
 
 7 
 
 gal. 
 2 
 
 qts. 
 3 (83) 
 8 
 
 bar. 
 11 
 
 fir. gat. fir. 
 3 2 (84) 15 
 9 
 
 gal. pts. 
 8 7 
 10 
 
 85 
 
 3 
 
 14 
 
 
 
 
 
 
 Dry Measure. 
 
 cha. 
 (85) 7 
 
 bus. 
 30 
 
 pks. qrs. 
 2 (86) 9 
 11 
 
 bus. 
 
 7 
 
 pks. 
 3 (87) 
 12 
 
 bus. 
 16 
 
 pks. gal. las. 
 3 1 (88) 24 
 
 2 
 
 wys. qrs. 
 1 4 
 3 
 
 
 86 
 
 11 
 
 2 
 
 
 
 
 
 
 Time. 
 
 yrs. mo. wks. mo. wks. dys. wks. dys. ho. ho. mi. sec. 
 
 (89) 17 12 3 (90) 23 3 6 (91) 12 6 23 (62) 23 6 59 
 
 4567 
 
 7T~12 
 
Compound Multiplication. 37 
 
 MISCELLANEOUS QUESTIONS. 
 
 (93) If 1 cwt. of cheese cost 21. 4s. Qd. what will 36 cwt. 
 cost ? Ans. 80Z. 2s. 
 
 (94) Bought 1 cwt. of tobacco for 16 guineas, I demand 
 the worth of 48 cwt. ? Ans. 806Z. 85. 
 
 (95) Gave 21. 5s. for 1 piece of cloth, I demand the worth 
 of 48 pieces ? Ans. 108Z. 
 
 (96) If a servant's wages be 7s. Qd. per week, what is that 
 per year ? Ans. 201. 3s. 
 
 (97) Bought a gallon of wine for 13s. 9 J., what will a pipe 
 (containing 126 galls.) come to at that rate ? 
 
 Ans. SQL 12s. Qd. 
 
 (98) If I lay up half a guinea per day, what will it amount 
 to in a year ? Ans. 19 1Z. 12s. Qd. 
 
 (99) If 1 Ib. of cheese cost Q^d. what wall 1 hundred 
 weight cost ? Ans. 3Z. Os. Sd. 
 
 (100) Gave lls. Qd. for a ream of paper, I demand the 
 price of 108 reams ? Ans. 62 Z. '2s. 
 
 (101) If 1 ell cost 6s. Sd. what will 63 ells cost ? Ans 21Z. 
 
 (102) Bought 1000 gallons of oil at 9s. per gallon, what 
 did it cost me ? Ans. 450Z. 
 
 (103) Sold a quarter of wheat, at 9s. lid. per bushel, 
 what did I receive for the 8 bushels ? Ans. 31. 19s. &d. 
 
 (104) What will 1 cwt. of sugar cost, at ll\d. per Ib. ? 
 
 Ans. 51. 7s. 4d. 
 
 (105) If my income be 1Z. 18s. 6 d. per day, what is that 
 per year ? Ans. 7021. 12s. Qd. 
 
 (106) Bought 1500 feet of deal, at lid. per foot, what 
 did it cost me ? Ans. 68Z. 15s. 
 
 (107) Gave 25Z. per acre for 200 acres of freehold land, 
 what did the estate cost me ? Ans. 5000Z. 
 
 (108) Suppose a person to clear by trade 392Z. per year, 
 on an average, for 50 years, what would be the amount ? 
 
 Ans. 19,600Z. 
 
 (109) If a person travel, on an average, 12 miles per day, 
 how many miles w^ould he travel in seven years.* Ans. 30,660 
 
 (110) What is the weight of 6 bars of silver, each weigh- 
 ing 3 Ib. 10 oz. 8 dwts. 16 gr. ? Ans. 23 Ib. 2 oz. 12 dwts. 
 
 (111) What does the painter charge for 1000 yards, who 
 is paid Q^d. per square yard ? Ans. 281. 2s. Qd. 
 
 (112) What sum was divided among 24 brothers and 
 Bisters, so that each had but 833Z. 6s. Sd. ? Ans. 20,OOOZ. 
 
 (113) Bought 120 gallons of cider at Is. 3d. per gallon, 
 what did the whole cost me ? Ans. 71. 10s. 
 
 * Not computing a leap year. 
 
( 38 ) 
 COMPOUND DIVISION. 
 
 COMPOUND DIVISION is the method of finding ho* 
 often one given number is contained in another of different 
 denominations. 
 
 I. When tlie divisor does not exceed 1 2. 
 BULE. Place the divisor on the left hand of the dividend. 
 Divide the highest denomination of the dividend by the 
 divisor, and write down the quotient ; reduce the remainder, 
 if any, into the next lower denomination, adding to it the 
 number that stands under the same denomination of the 
 dividend, which divide as before. 
 
 PBOOF. By compound multiplication. 
 
 EXAMPLES. 
 
 s. d. s. d. s. d. 
 
 (1) 2)46 8 6i (2) 3)36 12 9f (3) 4)84 16 Sft 
 
 23 4 3J 
 
 234 
 
 46 8 6J 
 
 s. d. .9. d. s. d. 
 
 (4) 5) 5 15 10 J (5) 6) 2 18 6J (6) 7) 9 14 7|( 
 
 s. d. s. d. s. d. 
 
 (7)8)9 8 7 j (6 over (8)9)8 6 4^(9)10)7 4 2|( 
 
 1 3 6f 
 
 II. If the divisor exceeds 12, divide by any two 
 numbers, which multiplied together, will make the same 
 number. 
 
 EXAMPLES. 
 
 5 ) 248 12 8 (3 over )275 17 7|( 
 
 5) 49 14 6|(2over ) 
 
 9 18 10| 
 (12) Divide 186/. 10*. 6 d. by 32. (13) Divide 168Z. IteAd. by 84 
 
Compound Division. 39 
 
 III. If the divisor cannot be exactly produced by small 
 numbers, divide after the manner of long division. 
 
 (14) Divide 396Z.12s.Gf d.by 365. (15) Divide 1000Z.by52. 
 
 3(55 ) 396 12 6| ( II 52 ) 1000 ( 191 
 
 365 52 
 
 ~~31 480 
 
 20 468 
 
 365)632(15. 12 
 
 365 20 
 
 ~267 52)240(4* 
 
 12 208 
 
 365)3210(8* 32 
 
 2920 12 
 
 290 52 ) 384 ( 7d. 
 
 4 364 
 
 365)1163(| 20 
 
 1095 4 
 
 68 52)80( 
 = 52 
 
 28 
 
 (1C) Divide 4Z. 11*. by 112 ? (17) Divide 62Z. 9*. Qd. by 53 ? 
 
 Ans.'9$d. Ans. II. 3s. 6f</. 33 
 
 (18) Divide 28Z.19.*.6<*. by 78 ? (19) Divide 36J.10*.8rf. by 67? 
 
 Ans. 7s. 5d. 12. Ans. 10 10f^. 31 
 
 (20) Divide 38/.ll*.6rf.by85 ? (21) Divide 112Z. by 71 ? 
 
 Ans. 9s. 0$d. 57. Ans. II. Us. 6^. 26 
 
 (22) Divide 124J. 10s. by 87 ? (23) Divide 200Z. by 105 ? 
 
 Ans. II. 8s. 7U 69 Ans. II. ISs. Id. 15 
 
 Or WEIGHTS AKD MEASUEES. 
 
 Troy Weight. 
 
 Ibs. oz. dwt. oz. dwt. gr. Ib. oz. dwt. 
 
 (24)2)5 9 6( (25)3)13 18 16 ( (26)4)11 10 12 ( 
 
 2 10 13 
 
 Avoirdupois Weight. 
 
 tonscwt. qr. cwt. qr. Ib. Ib. oz. dr. 
 
 (27)5)31 11 2(1 over. (28)6)8 3 21 ( (29)7)60 8 10 ( 
 
 661 
 
40 Compound Division. 
 
 Apothecaries Weight. 
 
 Ibs. oz. dr. oz. dr. sc. dr. so gr. 
 
 (30) 8)15 9 0( (31) 9)10 3 1( (32; 10)11 2 12( 
 
 1 11 5 
 
 Cloth Measure. 
 
 yds. qr. nls. E. E. qrs. nls. yds. qrs. nls. 
 
 (33) 11)11 3 3(4 (34)12)14 2 3( (35) 2)18 1 3( 
 
 10 
 
 Long Measure. 
 
 lea. mi. fur. mi. fur. po. ft. in. b.c. 
 
 (36) 3)8 2 6(1 (37) 4(9 6 18 ( (38) 5)18 11 2( 
 
 27 
 
 Land Measure. 
 
 ac. ro. po. ac. ro. po. ac. ro. po. 
 
 (39) 6)20 3 11(1 (40)7)18 1 32 ( (41)8)10 29 ( 
 
 3 1 35 
 
 Wine Measure. 
 
 tuns hhds gal. hhds. gal. qts. gal. qts, pt. 
 
 (42) 9)24 3 21 ( (43)10)36 2 3( (44)11)12 2 1( 
 
 Ale and Beer Measure. 
 
 butts hds. gal. hhds. gal. qts. fir. gal. qts. 
 
 (45J 12)18 12 ( (46)2)16 8 3( (47) 3)9 6 2( 
 
 Corn and Coal Measure. 
 
 qrs. bu. pks. cha. bus. pks. qrs. bus. pks. 
 
 (48)4)11 6 2( (49)5)60 30 3( (50)6)17 1( 
 
 Time. 
 
 yrs. mo. wks. mo. wks. da. ho. min. sec. 
 
 (51)7)17 0( (52)8)18 0( (53)9)19 0( 
 
Uivision of Money. 4,1 
 
 MISCELLANEOUS QUESTIONS. 
 
 (54) If 36 cwt. of cheese cost SOI. 2s. what will 1 c\\ ; 
 cost ? Ans. 21. 4s. Qd 
 
 (55) Bought 48 cwt. of tobacco for 1081/. 4.?. I demand 
 the worth of 1 cwt. ? Ans 221. 10s. Qd 
 
 (56) Grave 108/. for 48 pieces of cloth ; what is the value 
 per piece ? Ans. 21 5s. 
 
 (57) If a servant's wages be 201. 3s. per year, what is that 
 per week P Ans. 7s. Qd 
 
 (58) Bought a pipe of wine (126 gallons) for 86/. 125. Qd 
 what is the worth of one gallon ? Ans. 13s. Qd. 
 
 (59) What must I lay up per day, to save 150/. per ann.? 
 
 Ans. Ss. 2\d. 190 rem. 
 
 (60) If 1 cwt. of cheese cost 31 Os. Sd. what will 1 ]b. 
 cost? Ans.Q^d. 
 
 (61) Gave Q21. 2s. for 108 reams of paper, I demand the 
 price per ream ? Ans. lls. Qd. 
 
 (62) If 63 ells of Holland cost 211. what will 1 ell cost ? 
 
 Ans. Qs. Sd. 
 
 (63) What is oil per gallon, if 1000 gallons cost 450/. 
 
 Ans. Qs. 
 
 (64) Sold wheat at 31. 19s. 4<d. per quarter, what is that 
 per bushel ? Ans. 9s. lid. 
 
 (65) What is sugar per Ib. at 51. 7s. 4id. per cwt. 
 
 Ans. Hid. 
 
 (66) If ray income be 700Z. per year, what is it per day ? 
 
 Ans. 11 18s. 4M 35 
 
 (67) Bought 1500 feet of deal for 68Z. 15*. what is the 
 value per foot ? Ans. lid. 
 
 (68) G-ave 5000Z. for 200 acres of freehold land, what is 
 the value per acre ? Ans. 251. 
 
 (69) Suppose a person trading, to clear 19,6002. in 50 
 years, I demand his yearly increase ? Ans. 3921. 
 
 (70) If a person travel 4380 miles in a year, what is the 
 average per day ? Ans. 12 miles 
 
 (71) What does the painter charge per square yard, who 
 is paid 281. 2s. Qd. for 1000 yards ? Ans. 6V 
 
 (72) Divide 20,000/. equally among 24 brothers and sis- 
 ters. * Ans. 8331. Qs. Sd, 
 
 (73) Bought 120 gallons of cider for 71. 10s what is that 
 per gallon ? ^ ils ls 3(/ 
 
 (74) Sold 1 cwt. of tea for 327. 4s. what did 1 charge per 
 lb - Aug. 5s. 9d. 
 
 (75) What is wool per stone, if 77 stone cost 56Z. 15s 9d ? 
 
 Ans. 14s. Qd. 
 
BILLS OF PARCELS. 
 
 .No. 1. A MERCER'S BILL, 
 
 Mr. John Jones, Jan. 1st, 1847. 
 
 Bought of Thomas Perry. 
 . s. d. 
 
 4 Yards of Lawn at 2 6 per yard 
 
 12 Yards of Silk at 14 3 per yard 
 
 9 Yards of Cambric at 5 2 per yard 
 
 3 Yards of Velvet at 1 4 6 per yard 
 
 15 Yards of Brocade at 15 3 per yard 
 
 27 Yards of Lace at 12 2 per yard 
 
 42 18 3 
 
 No. 2. A STATIONER'S BILL. 
 
 Mr. Edwin Thomas, Feb. 1st, 1847. 
 
 Bought of James Dixon. 
 . s. d. 
 
 24 Reams of Demy at 2 12 6 per ream 
 
 75 Eeanis of wove Post ... at 2 per ream 
 
 27 Beams of Crown at 1 13 per ream 
 
 13 Beams of Hot Pressed, at 2 3 per ream 
 52 Reams of Foolscap ... at 1 5 6 per ream 
 70 Beams of thin Post ... at 1 8 9 per ream 
 
 452 8 6 
 
 A WINE MERGE ANSI'S BILL. 
 Mr. Abraham Jeynes, March 1st, 1847. 
 
 Bought of Henry Thomson. 
 . s. d. 
 
 1 Gallons of Sherry at 16 8 per gall. 
 
 7 Gallons of Oporto at 13 6 per gall. 
 
 13 Gallons of Claret at 12 per gall. 
 
 9 Gallons of Malaga at 11 3 per gall. 
 
 4 Gallons of Lisbon at 14 6 per gall. 
 
 15 Gallons of Brandy at 1 3 6 per gall. 
 
 46 8 11 
 
Sills of Parcels. 48 
 
 No. 4. A CAEPENTEE'S BILL. 
 
 Mr. Jonathan Johnson, April 1st, 1847. 
 
 Bought of James Lawson. 
 
 s. d. 
 
 57 Feet of wainscot Sashes .at 10 J per foot 
 
 6GO Feet of Ash at 3 per foot 
 
 79 Cubic feet of Oak at 4 2 per foot 
 
 136 Do. of framed Deal at 3 9 per foot 
 
 95 Do. of Oak at 5 6 per foot 
 
 9 Men's labour, for 153 days at 4 9 per day 
 
 115 1 7J 
 
 No. 5. A GEOCEE'S BILL. 
 
 Mr. Timothy Wall, May 1st, 1847. 
 
 Bought of Eliza Saunders. 
 
 s. d. 
 
 3 Ibs. of superfine sugar at 1 per Ib. 
 
 7 Ibs. of raw ditto , at 8 per Ib. 
 
 Ib. of black Tea at 7 4 per Ib. 
 
 | Ib. of London Soap at 11 per Ib. 
 
 3 Ibs. of Mould Candles at 10 per Ib. 
 
 7 Ibs. of Carolina Eice at 5 per Ib. 
 
 15 4 
 
 No. 6. A HOSIEE'S BILL. 
 
 Mr. Eginton, Jur,j> 1st, 1847. 
 
 Bought of Mark Eawson. 
 
 s. d. 
 
 16 Pair of Stockings at 4 6 per pair 
 
 9 Ditto of Worsted at 5 3 per pair 
 
 8 Ditto of Thread at 3 10 per pair 
 
 T2 Ditto of Men's Silk at 15 per pair 
 
 3 Ditto of Gloves at 3 6 per pair 
 
 6 Ditto of Cotton at 2 6 per pair 
 
 17 15 5 
 
( 44 ) 
 
 REDUCTION. 
 
 REDUCTION is the method of converting numbers from 
 one name or denomination, to another, without altering their 
 value. It is divided into REDUCTION DESCENDING, and EE- 
 DUCTION ASCENDING. 
 
 I. When the numbers are to be reduced from a greater 
 denomination to a less, it is called Reduction descending, and 
 is performed by multiplication. 
 
 RULE. Multiply. the given number by as many of the less 
 denomination as make one of the greater. 
 
 II. When the numbers are to be brought from a less de- 
 nomination to a greater, it is called Redaction Ascending, 
 and is performed by division. 
 
 RULE. Divide the given number by as many of the less 
 denomination as make one of the next greater. 
 
 N. B. Ascending and descending sums are proofs to each 
 other. 
 
 REDUCTION DESCENDING. 
 
 EXAMPLES. 
 (1) Reduce 36/. 12. 4&\d. into shillings, pence, farthings. 
 
 20 Proof 4 
 
 732 shillings 
 12 12 
 
 8788 pence 2 ,0 
 
 4 
 
 35153 
 
 73,2 
 
 35153 farthinas. 36 12 4 
 
 (2) Bring 30Z. Is. l|<J. into farthings ? Ans. 28853 qrs. 
 
 (3) In 100Z. 195 llfd. how many farth. ? Ans. 96959 qrs 
 
 (4) How many farth. in 111Z. 11s. llfd.? Ans. 107135 #r* 
 
 (5) In 47Z. 12s. lOd. how many farthings ? .^s. 45736 qrs. 
 
 (6) In 3456 shillings how many farthings ? ^/^. 165888 qrs. 
 
 (7) How many pence and hi 
 48Z. 12s. Z\d 
 20 
 
 ilf-penceare the 
 2 
 
 12 
 
 2,0 
 
 re in 48Z.12s., 
 23333 
 
 116664 
 
 972 
 12 
 
 97,2 2%d. 
 
 11666 pence 
 2 
 
 AQ. 1O Ol 
 
 (8) Reduce 76/. 16s. 8<f. into pence ? ^ws. 18440(/. 
 
 (9) In 430Z. 11s. lid. how many half-pence? Ans. 206686 
 
 (10) In 19Z. how many pence and half-pence 'tAns. 4560-9120 
 
 ( 11 ) lieduce 76Z. Os. 0|d. into half-pence ? Ans. 36481^.^?. 
 
 (12) Reduce 365Z. 8s. to half-pence ? Ans. 175392 
 
deduction. 43 
 
 (13) In 136 guineas, how many shillings, pence, and 
 farthings ? 
 
 136 Proof, 4 137088 
 
 _ 
 
 136 
 272 
 
 2856 shillings 
 12 
 
 12 
 21 
 
 34272 pence 
 4 
 
 34272 
 
 2856 
 136 guineas 
 
 137088 farthings 
 
 (14) Reduce 769 guineas into shillings, pence, and 
 farthings. Ans. 161495. 193788**. 7751 52/. 
 
 (15) In 128 guineas, 9s. 6d., how many farthings ? 
 
 Ans. 129480 
 
 (16) Bring 72 guineas and 12s. Qd. into sixpences. 
 
 Ans. 3049 
 
 (17) In 186 marks, each 13s. 4c?., how many groats ? 
 
 Ans. 7440 
 
 18) Reduce 738 half-guineas into quarter-guineas and 
 pence. Ans. 1476 qr. ^. 92988d. 
 
 (19) Bring 4325 guineas into seven-shilling pieces and 
 sixpences. Ans. 12975 pieces 181650 sixp. 
 
 (20) In 84Z. and a crown, how many crowns, half-crowns, 
 sixpences, and twopences ? 
 
 10110 
 
 84Z. 5s. or 1 cr. 
 4 
 
 337 crowns. 
 2 
 
 674 half-crowns. 
 o 
 
 Proof, 3 
 5 
 
 3370 sixpences. 
 
 337C 
 
 674 
 
 337 
 
 84/. 5s. 
 
 10110 twopences. 
 
 (21) In 120Z. 10s. how many crowns and pence ? 
 
 Ans. 482 crowns 28920 pence 
 
 (22) In 56Z. 2s. 6d. how many half-crowns and six- 
 ences ? Ans. 449 half-crowns 2245 sixpences 
 
 (23) Reduce 271. and a crown into shillings and three- 
 pences ? Ans. 545 shillings 2180 threepences 
 
 (24) In 360 crowns, how many sixpences and half-pence ? 
 
 Ans. 3600 sixpences 43200 half-pence 
 
 (25) Bring 100 half-crowns into threepences and farthings. 
 
 Ans. 1000 threepences 12000 Jarthings 
 
 (26) How many sixpences in 85?. 12s. Qd. ? Ans. 3425 s*>. 
 
Reduction. 
 
 (27) 
 
 pounds 
 
 REDUCTION ASCENDINQ. 
 In 36459 farthings, how many pence, shillings, and 
 
 farthings 
 36459 
 
 Proof, 
 
 . s. 
 37 19 
 20 
 
 759 
 12 
 
 9114 
 4 
 
 d. 
 
 6| 
 
 (28) 
 
 pounds 
 
 (29) 
 
 (30) 
 
 (31) 
 (32) 
 (33) 
 
 364 59 farthings 
 
 In 321457 farthings, how many pence, shillings, and 
 ? Am. S0364r/. 6697s. 334/. 17s. 
 
 Eeduce 100003 farth. into 's. Arts. 104. 3s. 
 Bring 21120 pence into shillings and 's. 
 
 Ans. 1760s. S8Z. 
 Change 4320 half-pence into sixpences and 's. 
 
 Ans. 360 sixpences 9/. 
 How many crowns and 's in 63840 pence. 
 
 Ans. 1064 crowns 266/. 
 In 9840 groats, how many shillings and 's ? 
 
 Ans. 32805. 164/. 
 
 (34) Cha 
 and guineas 
 4 
 
 3 
 
 / 7 
 1 12 
 
 age 241920 farthings 
 211920 farthings 
 
 into pence, threepences 
 
 Proof, 240 guineas 
 84 
 960 
 1920 
 
 60480 pence 
 
 20160 threepences 
 
 20160 
 3 
 
 2880 
 
 60480 
 4 
 
 241 920 farthings 
 
 240 guineas 
 
 (35) In 36288 half-pence, how many threepences and 
 guineas ? Ans. 6048 threepences 72 guineas. 
 
 (36) In 368172 pence, how many groats and guineas ? 
 
 Ans. 92043 gr. 1461 gum. 
 
 (37) Bring 48960 half-pence into sixpences and seven- 
 shilling-pieces. Ans. 4080 sixpences 291 pieces 3s. over. 
 
 (38) In 15246 twopences, how many seven-shillings-pieces 
 and guineas ? Ans. 363 pieces 121 guin. 
 
 (39) Change 2268 groats into shillings and guineas. 
 
 Ans. 756s. 36 guineas 
 
 (40) Bring 3024 threepences into quarter guineas and 
 guineas. Ans. 144 qr.-guin. 36 guin. 
 
Reduction. 47 
 
 ASCENDING AND DESCENDING. 
 (41) In 1050/. how many guineas ? 
 
 1050 Proof 1000 
 
 20 21 
 
 21) 21000 2,0) 2100,0 
 
 Ans. 1000 guineas 1050 
 
 (42) Change 840Z. into guineas ? Ans. 800 guineas 
 
 (43) Bring 8000 guineas into pounds? Ans. 8400 pounds 
 
 (44) In 80 moidores (each 27s.), how many 's ? 
 
 Ans. 1G8Z. 
 
 (45) How many moidores in 324Z. ? Ans. 240 moidores 
 
 (46) Change 182Z. into seven-shilling-pieces. Ans. 520 
 
 (47) In 18 pieces of 36 shillings each, how many moidores 
 of 27s. each ? Ans. 24 moidores 
 
 (48) In 4200Z. how many crowns, sixpences, half-guineas, 
 and guineas ? 
 
 4200 Proof, 4000 guineas 
 
 4 2 
 
 16800 crowns 8000 half -guineas 
 
 10 21 
 
 { | 168000 sixpences 1,0 
 
 56000 4 
 
 16800,0 sixpences 
 ~ 16800 crown* 
 
 8000 half-guineas 4200Z. 
 
 4000 guineas 
 
 (49) Bring 2800 guineas into half-guineas, sixpences, and 
 crowns ? ^trw. 5600 A. .117600 six. 11760 c?r. 
 
 (50) In 36Z. 15s. how many sixpences, half-crowns, and 
 crowns ? Ans. 1470 six. 294 h. c. 147 cro. 
 
 (51) In 888 quarter-guineas, how many pence and seven- 
 shilling-pieces ? Ans. 55944< 666 seven-shil. pieces 
 
 (52) In 1000 guineas how many crowns ? Ans. 4200 cr. 
 
 (53) Changp 420 crowns into guineas. Ans. 100 gain. 
 
 (54) In 90 six-and-thirty shillings how many half-crowns 
 and pounds ? Ans. 1296 h. c. 162 J. 
 
 (55) In 780 nobles, how many groats, shillings, crowns, 
 arid pounds ? 
 
 780 Proof, 2607. 
 
 20 groats = 1 noble 4 
 
 15600 groats 1040 crowns 
 
 5200 shillings 
 
 1040 crowns 5200 shiUinys 
 
 260 pounds 
 
 3 
 
 2,0 ) 1560,0 groats 
 780 
 
48 Reduction. 
 
 (56) In 1364 guineas, how many groats, pence, and seven- 
 shilling- pieces ? 
 
 Am. 85932 gro. 343728^. 4092 seven-shill. pieces. 
 
 (57) How many French franc*, or livres of lOd. each, are 
 there in 1201. ? Am. 2880 F.fr. 
 
 (58) In 30 marks, each 13s. Ad. 9 how many pounds ? 
 
 Ans. 202. 
 
 (59) Change 2268 groats into threepences ? 
 
 Ans. 3024 threepences 
 
 (60) How many six-and-thirties in 30Z. 12s. ? Ans. 17 six. 
 
 (61) In 24 moidores of 27s. each, how many pieces of 36 >-. 
 each. .^rcs. 18 pieces 
 
 Troy Weight. 
 
 (62) In 3 Ib. 6 oz. 9 dwt. 2 gr. of gold, how many grains ? 
 03. dwt. gr. 
 
 3692 Proof, * 24 
 
 12 
 
 42 20 
 
 20 
 
 2037 
 
 849 
 
 12 I 42 
 
 2 grra. 
 
 i 
 
 9 dwts. 
 
 . 
 
 Slb.6oz.2dw.2g. 
 
 3398 
 
 1698 ( * The pupil may divide by 24 in Long 
 
 2o37~8 1 DiwMon t or by two figures in Short Division 
 
 ( as by 4 and 6, or by 3 and 8. 
 
 (63) Reduce 130 Ib. 10 oz. to penny-weights and grains. 
 
 Ans. 31400 dwts. 753600 gra. 
 
 (64) How many grains of gold are there in a cup weigh- 
 ing 8 oz. 4 dwts. ? Ans. 3936 gra. 
 
 (65) Bought 7 ingots of silver, each containing 22 Ib. 8 oz. 
 10 dwts. how many grains ? Ans. 915600 gra. 
 
 (66) How many pounds Troy, in 6530 penny-weights ? 
 
 Ans. 27 Ib. 2 oz. 10 dwts. 
 
 Avoirdupois Weight. 
 (57) How many Ibs. are there in 75 tons 12 cwt. 3 qrs. ? 
 
 ton* cwt. qrs. Proof. 
 
 75 12 3 28)169428(4)6051 
 
 20 108 -- 3 
 
 2,0)151,2 
 
 A 142 - 
 
 - - 140 7 5 t. )2 fvl 3 qr. 
 
 0051 _ 
 
 28 28 
 
 48408 28 
 
 12102 ^ ,. 
 
 _ 
 Ans. 169423 
 
Reduction. 
 
 49 
 
 (68) In 146 tons, how many quarters and Ibs ? 
 
 Ans. 11680 qrs. 327040 Ifa 
 
 (69) How many quarters in 111 tons 11 cwt. ? A. 8924 qn 
 
 (70) Eeduce 12 Ibs. 11 oz. 10 dr. into drams. A. 3258 </r<z 
 
 (71) In 6 tons cwt. 3 qrs. how many Ibs. ? Ans. 13524 Zfo, 
 
 (72) How many Ibs. are there in 98765 drams ? 
 
 Ans. 385 Ib. 12 oz. 13 dra. 
 
 (73) In 36540 drams, how many Ibs. and quarters ? 
 
 Ans. 142 Ibs. or 5 <?TS. 2 Zfo. 11 02. 12 dra. 
 
 Apothecaries 9 Weight. 
 
 (74) Eeduce 3 Ib. 4 oz. 6 dr. 1 sc. 17 gr. into grains ? 
 Ib. oz. dr. sc. gr. Proof. 
 
 oz. 
 4 
 
 sc. 
 
 1 
 
 17 
 
 _ 
 40 
 
 8 
 326 
 
 3 
 
 979 
 20 
 
 2,0 
 3 
 
 1959,7 
 
 979 
 
 326 
 
 40 
 
 17 gr. 
 1 sc. 
 6 dr. 
 
 3 Ib. 4 02. 6 dr. 1 sc. 17 #>. 
 
 grains. 
 
 (75) How many grains of rhubarb are there in 37 Ib. 2 oz. 
 7 dr. 2 sc. 12 gr. ? ^ns. 214552 yr. 
 
 (76) How many scruples are there in 321 Ib. 5 oz. 2 dr. 
 2 scr. of opium ? Ans 92576 ser. 
 
 (77) In 1 Ib. 1 oz, 1 dr. of Ipecacuhana, how many 
 drams ? Ans. 105 dra. 
 
 (78) In 40320 grains, how many scruples, drams, ounces, 
 and pounds ? Ans. 2016 scr. 672 dr. 84 oz. 7 Ib. 
 
 (79) In 120960 grains, how many ounces ? Ans. 252 oz. 
 
 (80) How many ounces and Ibs. in 3456 scruples ? 
 
 Ans. 144 02. 12 Ib. 
 
 Cloth JMeasure. 
 
 (81) In 36 yds. 3 qrs. 3 nls. how many nai. 1 :. md inches ? 
 
 yds. 
 
 36 
 
 4 
 
 147 
 4 
 
 591 
 
 2j 
 
 1182 
 147} 
 
 qrs. nls. 
 3 3 
 
 2* 
 9 9 
 
 Proof. 
 1329f 
 
 5319 
 
 inches. 
 
 591 
 
 < 
 
 147 
 
 3 nails. 
 
 36 yds. 3 37-5. 
 
I 
 
 50 Reduction. 
 
 (82) How many yards in 11616 nails ? Ans. 726 yds. 
 
 (83) In 365 Eng. ells 2 qrs. 3 nls. how many nails ? 
 
 Ans. 7311 nls. 
 
 (84) In 5008 nails, how many yards ? Ans. 313 yds. 
 
 (85) In 2000 nails, how many ells English? Ans. 100 E.E. 
 
 Long Measure. 
 
 (86) In 1760 yards, how many inches and barley-corns ? 
 
 1760 
 36 
 
 3 
 36 
 
 190080 
 
 63360 ( 1760 yards 
 36 
 
 63360 inches 
 3 
 
 272. &r.. 
 
 190080 b. corns 
 
 (87) How many barley-corns will reach 200 miles ? 
 
 Ans. 38016000 I. c. 
 
 (88) In 876 miles, how many yards and feet ? 
 
 Ans. 1541760 yds. 4625280 feet. 
 
 (89) In 18 miles 5 fur. 16 poles, how many yards ? 
 
 Ans. 32868 yds. 
 
 (90) In 792000 feet, how many leagues ? Ans. 50 leagues 
 
 (91) How many furlongs in 158400 feet ? Ans. 240 fur. 
 
 (92) In 95040 poles, how many leagues ? Ans. 99 lea. 
 
 (93) How often will a wheel 16 feet in circumference turn 
 round in 15 miles ? Ans. 4950 times 
 
 Land Measure. 
 
 (94) In 1069 acres, how many square roods, perches, and 
 yards ? 
 
 1069 
 4 
 
 30 5173960 
 4 4 
 
 4276 roods 
 40 
 
 121 
 4,0 
 4 
 
 20695840 ( 
 
 171040pofc*. 
 30 
 
 17104,0 
 
 5131200 
 42760 
 
 4276 
 
 
 5173960 yards 1069 acres 
 
 (95) In 2831 acres, how many yards ? Ans. 13702040 yds. 
 
 (96) How many perches are therein 736 ac. 3 ro. 12 po.? 
 
 Ans. 117892 perches 
 
 (97) Brincr 75 acres, 2 roods, and 30 perches, into yards. 
 
 Ans. 36G327| yds. 
 
 OS) In 6272640 square inches, how many square yards ? 
 
 Ans. 4840 yds. 
 (99) How many square inches in 36 square yards ? 
 
 Ans. 46656 inch. 
 
Reduction. 
 
 Wine, Ale, and Seer Pleasure. 
 (100) In 12 pipes 36 gal. of wine, how many pints ? 
 
 12 p. 36 gal. Proof. 8 
 
 Z26 
 1548 
 
 126 
 
 12384 
 
 1548(12 pip. 36 gal 
 1512 
 
 295 
 1 gal. 
 
 8 
 
 - 36 
 
 Ans. 12384 pints. 
 
 (101) In 5 tuns 1 hhd. 18 gal. how many quarts ? 
 
 Ans. 5364 qts. 
 
 (102) In 765 butts of beer, how many pints ? A.GQ() ( JGOp. 
 
 (103) Eeduce 79 hhds. to quarts. Ans. 17064 qts. 
 
 (104) In 6912 qts. of ale, how many hhds.? Ans. 32 hhds. 
 
 (105) How many kilderkins of 18 gals, each, are there in 
 2880 pints ? Ans. 20 Mid. 
 
 Dry Measure. 
 
 (106) In 36 bus. 3 pks. 1 gal. of wheat, how many gallons 
 and quarts ? 
 
 36 bus. 3 pics. 1 gal. Proof. 
 
 4 4 1180 
 1*7 pecks. 
 2 
 
 295 gallons 
 
 4 
 
 1180 quarts 36 i us . 3 p fo t i ffa i 
 
 (107) In 12 weys 3 qrs. 6 bus. of barley, how many bus. 
 and pecks ? Ans. 510 bus. 2040 pics. 
 
 (108) How many weys and bushels, in 72 lasts ? 
 
 Ans. 144 weys 5760 bus. 
 
 (109) In 33 bus. 3 pks. of oats, how many quarts and 
 pints ? Ans. 1080 qts.2I6Qpts. 
 
 (110) How many barleycorns will fill a bushel, supposing 
 9210 to fill a pint ? Ans. 589440 
 
 (111) In 71680 quarts, how many weys and lasts ? 
 
 Ans. 56 weys 28 lasts 
 
 (112) How many quarters of corn in 10,000 gallons ? 
 
 Ans. 156 qrs. 2 bus. 
 
 Time. 
 
 (113) In one year consisting of 365 d. 5 ho. 48 min. 49 sec 
 bow many seconds ? 
 
 365 d. 5/L 48m. 49 sec. Proof - 
 
 24 6,03155692,9 
 
 6,0 
 
 876.5 
 60 
 
 525948 24 
 
 60 
 
 52594,8 
 
 8765 
 
 - 49 sec. 
 -48mm. 
 
 81556929 seconds D 2 305 c?. 5 h 43 in, 4f 3. 
 
52 Proportion, 
 
 (114) In 63113858 seconds, how many days ? 
 
 Ans. 730 da. 11 ho. 37 min. 38 s<?o. 
 
 (115) How many days since the birth of Christ to Christ- 
 mas 1856 ? Ans. 677904 
 
 (116) How many days from May 1st till Nov. 1st ? 
 
 Ans. 184 days 
 
 (117) Prom March 20th till September 29th, how many 
 hours ? Ans. 4623 hours 
 
 (118) If you are now ten years old, how many minutes and 
 seconds have you lived ? A. 5259600 min. 315576000 sec. 
 
 PROPORTION, 
 OR, THE EULE OF THEEE, 
 
 Is so called, because by three numbers given we find a 
 fourth ; and it is either the Eule of Three Direct or Inverse. 
 
 THE EULE OE THEEE DIEECT 
 
 Teaches from three given numbers to find a fourth, which 
 shall have the same ratio to the second, as the third has to 
 the first. 
 
 EULE 1st. State the question ; that is, place the numbers 
 so that the first and third may be of the same name ; and the 
 second the same as tine fourth number required.* 
 
 2nd. Bring the first and third numbers into the same 
 denomination ; and the second into the lowest name men- 
 tioned. 
 
 3rd. Multiply the second and third numbers together, and 
 divide the product by the first ; and the quotient will be the 
 answer to the question, in the same denomination you left 
 the second number in : which quotient may be then brought 
 into any other denomination required. 
 
 The method of proof is by inverting the question. 
 
 EXAMPLES. 
 
 (1) If 7 yards of cloth cost 3Z. 10s. what will 65 yds. cost. ? 
 
 Proof. * 
 
 yds. 
 As 7 
 
 7 
 2,0 
 
 s. yds. 
 
 3 10 :: C 65 
 
 20 
 70 
 65 
 
 yds. 
 As 65 : 32 
 20 
 650 
 
 7 
 
 s. yds. 
 
 10 : : 7 
 
 4550 
 65,0 
 32 10s. Ans. 
 
 65)4550(70*. ^3 10*. 
 455 
 
 
 * If rightly placed, ike first and second terms will be ikeworlh of each other. 
 
or, the Rule of Three. 53 
 
 (2) If 11 Ib. of sugar cost 9s. 6d. what will 123 Ib. cost P 
 
 Ans. 51. 6s. 2d. 10 
 
 (3) Bought 75 gallons of brandy for 65 Z. ; I demand the 
 worth of 15 gallons ? Ans. 131. 
 
 (4) If 9 ells of cloth cost 61. 12s. what will 84 ells cost ? 
 
 Ans. 611. 12s. 
 
 (5) How much will 18 bushels of wheat cost, if 7 bushels 
 are worth 3Z. 4s. 6d. ? Ans. SI. 5s. 10\d. 1 
 
 (6) "What will 144 cwt. of cheese cost, if 9 cwt. be worth 
 231. 6s. 6d. ? Ans. 3731. 4s. 
 
 (7) If 24 Ib. of soap cost II. Is. 6d., what will 112 Ib. cost ? 
 
 Ans. 51. Os. 4>d. 
 
 (8) How many yards can I procure for 101. 8s. 4<d. at 
 
 the rate of 21. 12s. Id, Jor 7 yards ? 
 
 s. d. yds. s. d. 
 
 AS 2 12 i : r :: 10 s 4 
 
 20 20 
 
 52 208 
 
 12 12 
 
 625 2500 
 
 = 7 
 
 625)17500 (28 yd*. Ans. 
 1250 
 
 5000 
 
 (9) If 31. 10s. will buy 14 yards, what will 321. 10s. buy ? 
 
 Ans. 130 yds. 
 
 (10) Bought 9 ells for 61. 12s. 1 demand how many ells 
 61Z. 12s. will buy ? Ans. 84 ells 
 
 (11) How many hundred weight of sugar can I purchase 
 for 206Z. 4s. 3d. at 51. 5s. 9d. per cwt. ? Ans. 39 cwt. 
 
 (12) How many tons of iron can I procure for 121Z. 5s. 4>d. 
 at the rate of 212Z. 4s. 4d. per 7 tons ? Ans. 4 tons 
 
 (13) Grave 19s. 6d. for 8 bushels of coals, how many can 
 be bought for HZ. 14s.? Ans. 96 bush. 
 
 (14) How many yards of silk ribbon can be purchased with 
 56Z. at the rate of 3s. 4<d. for 9 yards ? Ans 3024* yds 
 
51 Proportion, 
 
 (15) Sold 13 J- yds. of velvet for 9Z. 8s. , how much must 
 2 yds. be sold for at the same rate ? 
 
 yds. 8. yds. 
 
 AS is j : 98 : : ai 
 __ 4 _ 20 __* 
 
 "54 ~~1SS ~~9 
 
 _ 9 
 
 64)1692(31*. 
 
 i??_ 1 11 4 4w*. 
 
 <7O 
 
 54 
 
 18 
 12_ 
 
 54)216"(4cZ. 
 216 
 
 (16) Purchased 7J- yards of broadcloth for 71. 10s. what 
 would 13| yds. of the same piece be worth ? Ans 13Z, 15s. 
 
 (17) Sold 15 J bushels of wheat for GZ. 19*. G&, what were 
 22 f bushels sold for ? Ans. 10Z. 4s. Qd. 
 
 (18) What are 38f yds. of cloth worth, if 46 yds. be 
 valued at 21. 6s. Gd. ? Ans. II. 18*. 9d. 
 
 (19) If 7f cwt. cost 3Z. 7s. Gdf. what will 541 cwt . cost ? 
 
 Ans. 231. 12s. 6cJ. 
 
 (20) Bought 27f bushels of oats for 6Z. 12s. 3d., what must 
 sell 9j bushels for at the same rate ? Ans. 21. 4s. Id. 
 
 (21) "What will 24| yards cost, if 18f yards are purchased 
 for 21 6s. ? Ans. 31. Os. 24 J. 99 
 
 (22) If 10 bushels of wheat cost 4Z. 15s. how many quar- 
 ters can I buy for 49Z. 8s. ? 
 
 s. bush. s. 
 
 AS 4 15 : 10 :: 49 s 
 
 20 _ 20 
 
 "95 988 
 
 10 
 
 95 
 
 ~380 13 qrs. Ans. 
 380 
 
 (23) Sold 3 pecks of potatoes for Is. 9^. how many sacks 
 of 2 bushels each, must I sell to receive 4Z. 4s.? Ans. 18 sa, 
 
 (24) How many pockets of hops of 2 cwt. each, can I buy 
 for 382Z. 8s. if 7 cwt. cost 47/. 16*. ? Ans. 28 pock. 
 
 (25) How many casks of raisins, each 2 cwt. 3 qrs., can be 
 bought for 77Z., if 3 qrs. of a cwt. cost 21 2s. ? Ans. 10 casks 
 
or, the Rule uf Three. 65 
 
 (26) Purchased 3 qrs. of a yard of Holland for 5s. 6d. , how 
 many yards may be bought for 61. 17s. 6d. ? Ans. 18$ yds. 
 
 (27) Laid out 101. 16s. Sd. in muslins, I demand the quan- 
 tity of English ells purchased at the rate of 6s. 6d. for 3 qrs. 
 of a yard ? Ans. 20 Eng. ells 
 
 (28) If 3 paces or common steps of a person be equal to 2 
 yds., how many yards will 180 of his paces make ? A. 120yds. 
 
 (29) If 12 oz. of pepper cost 2s. 6d. } what must be paid 
 for 1 cwt. 2 qrs. ? 
 
 02. s. d. cwt. qrs. 
 
 As 12 ; 26 : : 12 12 
 
 12 4 
 
 30 6 
 
 _28 
 
 168 
 
 12 
 
 2,0 
 
 80640 
 
 6720 pence 
 
 56,0 
 
 16 28 AMI. 
 
 2688 
 _ 30 
 
 80640 carried up. 
 
 (30) If 7 oz. of gold be worth 351. 2s. 6d., what is the 
 worth of 3 Ib. 8 oz. ? Ans. 220L 15s. Sd.2 
 
 (31) Bought 12 Ib. of butter for 14s. 6d. ; I demand the 
 worth of 1 cwt. 2 qrs. 7 Ib. ? Ans. 101. Us. 5d. 
 
 (32) "What will 12 dozen and 7 pair of stockings come to 
 if 5 pair cost 11 12s. 6d. ? Ans. 49Z. Is. 6d. 
 
 (33) Sold 3 cwt. 2 qrs, 18 Ib. of cheese, at the rate of 18s. 
 6d. for 24 Ib., what did I sell it for ? Ans. 151. 16s. 0d. 
 
 (34) Purchased 7 firkins and 5 gallons of porter for 
 51. 13s. 4J., the value of 6 gallons is required ? Ans. 10s. 
 
 (35) Bought tallow at 3s. Sd. per stone of 8 Ib., what is 
 the worth of 10 tons ? Ans. 5131 6s. 8d. 
 
 (36) If the carriage of 6 cwt. 2 qrs. 14 Ib. cost 21. 19s. 6J. 
 what should be paid for 1 ton 19 cwt. 3 qr. at the same rate ? 
 
 cwt. qr. Ib. s. d. ton cwt. qr. 
 
 As 6 2 14:2 19 6::1 19 3 742)3178728(12)4284 
 4 _20 20_ 2968 - 
 26" "59 39~ 2107 2,0)35,7 
 
 28 12 _4 1484 j 
 
 212 714 159 6232 
 
 53 28 5936 
 
 742 Ib. 1272 2968 
 
 318 2968 
 
 4452 
 
 714 = 
 
 3178728 carried up. 
 
60 Proportion, 
 
 (37) If 11 cwt. 3 qrs. 15 Ib. of butter cost 66/. 12s., i de- 
 mand the value of 2 cwt. 3 qr. 25 Ib. ? Ans. 16/. 13s. 3d. 3 
 
 (38) Bought 3 tons 12 cwt. 3 qrs. of sugar for 236Z. 9s. 
 what is the worth of 1 cwt. 3 qrs. 12 Ib. ? 
 
 Ans. 61. Os. 8 \d. 4824 rem. 
 
 (39) If 7 oz. 12 dwts. of gold be worth 34/. 10s., what is the 
 value of 21 Ib. 11 oz. 6 dwts. ? Ans. 1195/. 4s. lOJrf. 104 r. 
 
 (40) What will 28 Ib. 8 oz. of honey cost, if 31b. 8 oz. are 
 purchased for 4s. Sd. ? Ans. II. 18s. 
 
 (41) I demand the worth of 7 yds. qr. 3 nls. of cloth, if 
 35 yds. 3 qrs. 3 nls. be sold for 12/. 7s. Id. Ans. 21. 9s. 5d. 
 
 (42) If 4 oz. 15 dwts. of silver plate cost II. 11s. 6^., what 
 will 5 articles, each 3 oz. 12 dwts. cost ? A. 51. 19s. 4d. 65 r. 
 
 (43) How many dozen of table spoons can be manufactured 
 from 57 Ib. 8 oz. of silver, each doz. weighing 22 oz. 16 dwts. ? 
 
 oz. dwt. doz. Ib. oz. 
 
 22 16 : 1 : : 57 8 
 20 12 
 
 456 692 
 
 20 
 
 456) 13840 ( 30 doz'. 
 1368 
 
 160 
 12 
 
 456)1920(4xwmj 
 
 1824 
 
 Ans. 30 doz. 4 spoons 96 rem. 96 rem. 
 
 (44) If SI. 6s. 2 \d. will purchase 4 tons 9 cwt. of coals, what 
 quantity will 26Z. 9s. 6d. purchase ? Ans. 35 tons 12 cwt. 
 
 (45) If 18s. 6d. be the price of 5 qts. 1 pt. of wine, how 
 much will 101. 3s. 8|eJ. buy ? Ans. GOyts. Ipt. 
 
 (46) Suppose a servant's wages to be 10/. 16s. for42wks. 
 6 days, how long will he be earning 31. 12s. ? A. 14 wks. 2 d. 
 
 (47) If I pay three half-crowns for 10 Ibs. of butter, how 
 much can I have for 9 crowns and 9d. ? Ans. 61 Ibs. 
 
 (48) At a noble per week, how many months' board can I 
 have for 17Z. 6s. Sd. ? Ans. 13 months 
 
 (49) Gave 2^d. per Ib. for useful articles of old iron, what 
 weight did I buy for 251. 16s. Bd. ? Ans. 22 cwt. qr. 16 Ib. 
 
or, the Rule of TJtree. 57 
 
 (50) How many English ells can I procure for 360Z. 16s. Sd. 
 it the rate of 6s. Sd. per yard ? 
 
 s. d. 
 As 6 8 
 12 
 
 80" 
 
 yd. 
 
 4 
 
 s. d. 
 I'. 360 16 8 
 
 20 
 
 7216 
 
 12 
 
 86600 
 4 
 
 8,0 ) 34640,0 
 
 5 ) 4330 
 
 Ans. 866 Eng. ells. 
 
 (51) How many yards can I procure for 500Z. 10s., at the 
 rate of 7s. 6d. per Eng. ell? Ans. 1668 yds. 1 qr. 1 nl. 30 rem. 
 
 (52) How many tons of iron can be procured for 841Z. 6s. 8d. 
 at 371. 6s. Sd. per ton and half? Ans. 33 tons 16 cwt. qrs. Sib. 
 
 (53) How many cwt. of sugar at 5Z. 5s. 9d. per cwt. can 
 be purchased for 206Z. 4s. 3d. ? Ans. 39 cwt. 
 
 (54) If 15s. Sd. will purchase 1 yard, how many Flemish ells 
 can be bought for 71. 10s ? Ans. 12 ^.^. 2 qrs. 1 wJ.- -36 rem. 
 
 (55) What quantity of hops can I purchase for 300/. at 
 6/. 10s. Gd. per cwt. ? Ans. 45 cwt. 3 qrs. 25 /6s. 666 rem. 
 
 (56) If 3/. 10s. be paid a servant for 18 weeks' service, how 
 long will he be earning 8/. 15s. ? Ans. 45 weeks. 
 
 (57) If 7 pair of boots cost 121. 16s. what will 12 dozen 
 pair of similar articles come to ? 
 
 pr. s. pr. 
 
 As 7 : 12 16 : ; 144 N.B. 12 doz. = 144 
 
 256 
 144 
 
 7 ) 36864 
 
 3|cZ. 5 remainder 
 2,0 ) 526,6 
 
 2o3~~6s. 3|^. 5 Ans. 
 
 (58) Bought 10 doz. pair of silk stockings for 02/. what 
 will half a dozen pair be worth ? Ans. 47. 12s. 
 
 (59) Sold 12 doz. of wine for 28Z. 16s., what will 8 bottles 
 of the same be worth ? Ans. 11. 12s. 
 
 (60) What will a score of oranges cost, if 50 dozens be sold 
 for 31. 15s. ? Ans. 2s. Gd. 
 
 (61) If 35 yards of cloth will make 10 shirts, how many 
 can be made out of 7 pieces, each 27 yards ? Ans. 51 shirts 
 
 D3 
 
58 Proportion, 
 
 (62) I demand the value of 3 casks of raisins, each 2 c\vt. 
 3 qr. 14 Ib. at 4Z. 14s. Qd. per cwt. ? Ans. 40Z. 15s. Of d. 
 
 (63) Bought 12 cwt of sugar for 48Z. 15s. what is the 
 worth of 8 casks, each 3 cwt. 3 qr. 7 Ib. ? Ans. 123Z. 18s. 1 Id. 
 
 (64) If a servant's wages for 65 days come to 3Z. 18s. 
 what will be the amount of his wages for a year ? 
 
 days s. days 
 
 65 : 3 18 : : 365 
 
 20 
 
 78 
 
 365 
 
 65 ) 28470 (2,0) 43,8 
 260 
 
 "247 21 18s. Ans. 
 
 95 = 
 
 520 
 520 
 
 (05) If my rent and taxes amount to 5s. 7f d. per day, what 
 wilJ be the payment per year ? Ans. 103Z. Os. 8f (/. 
 
 (06) If I pay my landlord 100Z. per year, what is that per 
 day ? Ans. 5s. 5%d. 5 rem. 
 
 (67) Suppose my housekeeping to amount to 61. 17s., in 5 
 days, what will it cost me per year ? Ans. 500/. Is. 
 
 (68) If my horse stands me in 10s. Qd. for 6 days' keeping, 
 what will be the charge for a year? Ans. 32/. 13s. 11 \d. 
 
 (69) If my income be 1000Z. a year, what may I expend 
 daily, and lay by at the year's end 270Z. ? Ans. 21. 
 
 (70) What sum can I spend daily out of an income of 
 1200/. per annum, so as to lay by 50 moidores, 50 guineas, 
 and 50 sovereigns ? Ans. 21. 16s. 5^d. 15 rem. 
 
 (71) My year's rent of 350 acres of land is 300Z. and sundry 
 taxes and rates 36Z. 8s. what does my farm lie me in per acre'? 
 The rent . . 300 
 The taxes, &c. 36 8 
 acres acre 
 
 As 350 : 336 8 : : 1 
 
 20 
 
 350 ) 6728 ( 195. 350 ) 936 ( 2d. 
 
 350 700 
 
 3~228 23? 
 
 3150 4 
 
 78 350 ) 944 ( 2 Far. 
 
 12 700 
 
 350 ) 936 carried up. 244 remainder. 
 
 Am. 19s. 2\d. per acre, and 244 remainder. 
 
Inverse Proportion. 59 
 
 (72) If 100Z. in 12 months gain 51. interest, what will 3751. 
 gain in the same time ? Ans. 18/. 15s. 
 
 (73) A merchant failing for 10,0002. has in goods, debts, 
 &c. but 3760Z. what will that be in the pound ? 
 
 Ans. 7s. Gd. 24 rem. 
 
 (74) If a person earns 71. 10s. in three weeks, and lays by 
 the half of it, how much will he save per year ? Ans. 65Z. 
 
 (75) The rental of a parish is 3680/. and it has to pay ' 
 245/. 6s. 8d., how much is that in the pound ? Ans. Is. 4>d. 
 
 (76) Bought 420 galls, of oil for 761. 10s. 7d. of which 25 
 gallons were found damaged, how must I sell the remainder 
 so as neither to gain nor lose ? Ans. 3s. 10^. per. gall. 
 
 (77) "What is a quarter's rent of 350 acres of land, if 
 lls. 5s. 9J. per ann. be given for 9 acres ? A. 1091. 14s. 
 
 THE EULE OF THEEE INVEESE. 
 
 INVERSE PROPORTION is when more requires less, and 
 less requires more, i. e. two of the four proposed numbers 
 increase in the same proportion as the other two diminish. 
 
 EULE. State the question as in Direct ; and when needful 
 reduce the terms, as before : then multiply the first and 
 second terms together, and divide their product by the third; 
 the quotient will be the answer to the question, and will bear 
 such proportion to the second as the first does to the third. 
 
 The method of PROOF is by inverting the question. 
 EXAMPLES. 
 
 (1) If 12 men can reap a field in 18 days, how many days 
 will 36 men do it in ? 
 
 men days men Proof. 
 
 As 12 ; 18 : : 36 days men days 
 
 12 6 : 36 : : 18 
 
 6 
 
 36)216(6 days. Ans. 
 
 216 18)216(12mw 
 216 
 
 (2) Suppose 100 workmen to finish a piece of work in 96 
 days, how many are sufficient to finish it in 64 days ? 
 
 Ans. 150 men 
 
 (3) If 18 men can perform a piece of work in 28 days, 
 how many men can do it in a fourth part of the time ? 
 
 Ans. 72 men 
 
 (4) Suppose 120 men to complete a building in 15 months, 
 how many could finish it in 18 months ? Ans. 100 men 
 
60 Inverse Proportion. 
 
 (5) If I lend my friend 300Z. for 8 months, bow long 
 ought he to lend me 200/. to requite my kindness ? 
 
 mon, 
 
 300 ' 8 : : 200 
 
 300 
 
 2,00) 24,00 
 
 12 months. Ans. 
 
 (6) In what time will 336Z. gain 84Z. interest, when 280J. 
 will gain it in 6 years ? Ans. 5 years 
 
 (7) If 2501. gain ll/. 5s. interest in 12 months, what prin- 
 cipal will gain an equal sum in 8 months ? Ans. 3751. 
 
 (8) A lends B 75 1. 4s. for 9 months, how long ought B to 
 lend A 225Z. 12s. to requite his kindness ? Ans. 3 months 
 
 (9) How many pieces of 20 shillings value are equal to 
 300 pieces of 7s. each ? 
 
 s. pieces s. 
 
 As 7 : 300 : : 20 
 
 7 
 
 2,0 ) 210,0 
 
 Ans. 105 pieces of 20s. value. 
 
 (10) How many sovereigns, or pounds, are equal to a 
 thousand guineas ? Ans. 1050Z 
 
 (11) How many marks, each 160^. are equal to 186Z. 
 240d. each ? Ans. 279 marks 
 
 (12) How many nobles, each SOd. are equal to 1000 
 aagels, each I20d. ? Ans. 150 nobles 
 
 (13) In 72 sovereigns, how many pieces of 36*. each ? 
 
 Ans. 40 pieces 
 
 (14) How many yards of stuff 3 qrs. wide, are equal in 
 measure to 60 yards of 7 qrs. wide ? 
 
 qrs. yds. qrs. 
 
 7 : 60 : : 3 
 
 7 
 
 3)420 
 Ans. 140 yards. 
 
 (15) What must be the breadth of a court yard, which is 
 60 yards long, to be equal in measure to another that is 125 
 yar'ds long and 20 yards broad ? Ans. 50 yards broad 
 
Inverse Proportion. 
 
 01 
 
 (16) It' 12 inches long require 12 inches broad to make a 
 square foot, what length will 8 inches broad require ? 
 
 Am. 18 inches. 
 
 (17) How many yards of paper 27 inches wide, will hang 
 a room that measures 50 feet round and 9 feet high ? 
 
 Aw. 66 yds. 2ft. 
 
 (18) If 10,000 yards of 5 quarters wide will make coats 
 for 4,000 men, how many yards of shalloon of 3 qrs. wide 
 will line them ? Ans. 166t56f yds. 
 
 (19) If 220 yards in length, and 22 in breadth make an 
 acre, what must be the length when the breadth is 33yds. ? 
 
 Ans. 146 yds. 2ft. 
 
 (20) If for a certain sum 1 can have 15 cwt. 2 qrs. carried 
 fifty miles, what distance would 66 cwt. be carried for the 
 same money ? 
 
 Or thus, 
 
 cwt. qrs. 
 16 2 
 4 
 
 mil. 
 50 
 66 
 
 66 
 
 264)3300(121^. 
 264 
 
 660 
 
 528 
 
 cwt. 
 
 66 
 
 4 
 
 264 
 
 132 
 - 
 
 264 
 
 ~ 132 = 
 
 cwt. mil. 
 
 164 : 50 :: 
 
 2 33 
 
 33 132)1650(121 m. Ans. 
 132 
 
 330 
 264 
 
 66 
 
 66) = J 
 
 132 
 
 cwtf. 
 
 66 
 
 2 
 
 132 
 
 (21) If the carriage of 18 f cwt. for 56 miles come to 
 10s. 6J., how far can I have 129 J cwt. carried for the same 
 sum ? Ans. 8 miles 
 
 (22) If 14| cwt. be carried 100 miles for 365. , how many 
 Ibs. can I have carried 36 miles, for the same money ? 
 
 Ans. 44331 Ibs. 
 
 (23) If 27 men earn 13Z. 17s. in 2 days, how long will 12 
 men be earning the same ? Ans. 4% days 
 
 (24) If the penny loaf weighed 14 oz. when wheat was 4s. 
 per bushel, what niust it weigh when wheat is at 7s. per 
 bushel ? 
 
 s. oz. s. 
 
 As 4 : 14 : : 7 
 4 
 
 7)56(8 ounces, Ans. 
 
G2 Inverse Proportion. 
 
 (25) If a pasture serves 36 horses for 75 days, how many 
 horses would eat it in 25 days ? Ans. 108 horses. 
 
 (26) If a common field will feed 520 sheep 90 days, how 
 long may I turn out 600 sheep ? Ans. 78 days 
 
 (27) Suppose a hay-mow to be sufficient for 40 head of 
 cattle 18 weeks, how long would it serve 60 head of cattle ? 
 
 Ans. 12 weeks 
 
 (28) If 1,000 men, in a garrison, have provision for 6 
 months, how long would the same provisions last 1,500 men? 
 
 men mon. men 
 
 1000 : 6 :: 1500 
 
 1000 
 
 15,00) 6,000 (4 months, Ans. 
 60 
 
 (29) If a certain number of men can throw up an entrench- 
 ment in 9 days, when the day is 16 hours long ; what time 
 will it take when the day is 12 hours long ? Ans. 12 days 
 
 (30) If a person can perform a journey in 6 days, riding 
 9 hours each day, how long will it take him if he rides 12 
 hours a day p Ans. 4| days 
 
 (31) Travelled from London to York in 4 days of 12 hours 
 each, in how many days of 8 hours each can the same be 
 performed ? Ans. 6 days 
 
 (32) How many perches in length, with 12 in breadth, 
 must I receive in exchange for 40 perches in length and 18 
 in breadth ? 
 
 p. b. p. 1. p. b. 
 As 18 : 40 :: 12 
 
 18 
 
 12]~720 
 Ans. 60 perches 
 
 (33) There are two rooms, the floors of which ha?^ an 
 equal number of square feet ; the one is 50 feet by 30, the 
 other is 40 in length; what is the breadth ? Ans. 37 ft. 6 in. 
 
 (34) How many yards of paper, 3 qrs. wide, will cover a 
 chamber that is 60 feet round, and 10 feet 1| inches high ? 
 
 Ans. 90 yds. 
 
 (35) How much stuff 2 quarters wide, will face 15 yards 
 of silk, 3 qrs. wide ? Ans. 18 yards 
 
 (36) How many yards of brown drugget that is yard and 
 half wide, will cover a room that is 15 feet long and 14 feet 
 broad ? Ans. 15 yds. 1 foot 8 in. Or 15 yds. 2 qrs. nls. 2 in. 
 
(63) 
 
 COMPOUND PROPORTION, 
 
 OR THE 
 
 DOUBLE EULE OF THEEE. 
 
 IS so called because it is the method of resolving at one 
 operation such questions as by the common Eule of Three 
 would require two or more statings to be worked separately. 
 It teaches from five numbers given to find a sixth. Three 
 of the numbers contain a supposition, and the other two a 
 demand. 
 
 EULE 1st. Place two of the terms of supposition, one 
 above another, in the first place and that which is of the 
 same name as the term sought, must be put in the second 
 place. 
 
 2nd. Place the terms of demand one above another in the 
 ildrd place, in the same order as those in the first place. 
 
 3rd. The first and third term in every row will be of the 
 same name, and must be reduced to the same denomination : 
 and the middle term must be brought to the lowest denomi- 
 nation mentioned. 
 
 4th. Examine each row separately, using the middle term 
 as common to both, in order to know if the proportions be 
 direct or inverse ; by saying, if the first term give the second, 
 does the third require more or less. If direct, mark the first 
 term with an asterisk ; if inverse, mark the third term. 
 
 5th. Multiply the numbers marked for a divisor; and 
 those which are not marked for a dividend; and the quotient 
 will be the answer. 
 
 N.B. There is another method of stating questions in this rule, which, 
 though not so scientific as the former, is preferred by some teachers as more 
 easy for learners : for the use of such it is here subjoined. 
 
 RULE II. 1st. Let the principal cause of loss or gain, interest or 
 decrease, action or passion, be put in the first place. 
 
 2nd. Let that which betokeneth time, distance of place, and the like, 
 be put in the second place, and the remaining one in the third. 
 
 3rd. Place the other terms under their like, in the supposition. 
 
 4th. If the Hank falls under the third term, multiply the first and 
 second terms for a divisor and the other three for a dividend; and the 
 quotient will be the answer. But if the blank falls under the first or 
 ticnnd term, multiply the third and fourth terms for a divisor, and the 
 other three for a dividend ; the quotient will be the answer. 
 
 PKOOF. By two single rules of three. 
 
04 Compound ^Proportion. 
 
 EXAMPLES. 
 
 (1) If 6 men reap 18 acres of wheat in 5 days, how many 
 acres will 10 men reap in 12 days ? 
 
 By the second rule. 
 men days acres 
 
 5 :: 18 
 
 10 : 12 
 
 By the first rule. 
 men acres men 
 
 * 6 : 18 :: 10 
 
 * 5 days: : : 12 days 
 
 6 18 
 
 5 10 
 
 Divisor 30 180 
 
 12 
 
 3,0)216,0 
 
 Ans. 72 acres 
 
 (2) If 8 persons spend 100Z. in 4 months, how^ much will 
 20 persons spend in 6 months ? Ans. 375Z. 
 
 (3) If the carriage of 5 cwt. for 48 miles be 7s. 6d., what 
 will be the carriage of 15 cwt. for 24 miles ? Ans, lls. 3d. 
 
 (4) If 48Z. be the wages of 36 men for 9 days, what will 
 be earned by 12 men in 90 days ? Ans. 160Z. 
 
 (5) If a person travels 240 miles in 7 days, when the day 
 is 14 hours long, in how many days of 7 hours each will he 
 travel 120 miles ? 
 
 By the first rule. 
 miles days miles 
 
 * 240 : 7 : : 120 
 14 ho. : : : 7 ho.* 
 
 120 
 
 240 14 
 
 7 
 
 1680 
 Divisor 1680 7 
 
 By the second rule. 
 days hours miles 
 
 7 : 14 :: 240 
 : 7 :: 120 
 
 168,0) 1176,0 ( 7 days A ns. 
 1176 
 
 (6) If 14 men can dig 360 cubical yards of earth in 5 
 days, how many men can dig 144 cubical yards in 7 days ? 
 
 Ans. 4 men 
 
 (7) If 75 men can throw up an entrenchment in 5 days, 
 when the day is 12 hours long, in what time will 50 men do 
 it, when the day is 18 hours ? Ans. 5 days 
 
 (8) If a barrel of ale will last a family of 6 persons 2 
 months, how many persons would drink 9 barrels in a year ? 
 
 Ans. 9 persons 
 
Compound Proportion. 
 
 05 
 
 (9) Suppose 84 gallons of brandy will serve 220 seamen 
 8 days, how much will 380 seamen drink in 12 days ? 
 
 seamen galls. 
 
 * 220 : 84 
 
 * Sda.: 
 
 220 
 
 seamen 
 380 
 12 da. 
 
 seamen 
 220 
 380 
 
 days 
 
 12 
 
 galls. 
 
 84 
 
 1760 
 
 380 
 
 8 
 
 1520 
 JS040 
 
 31920 
 12 
 
 176,0 ) 38304,0 ( 217^% galls. 
 
 (10) If 1507. in 12 months gain 6Z. 15s., what will be the 
 interest of 7001. for 7 years ? Ans. 2201. 10s. 
 
 (11) If 7 horses eat 25 bushels of oats in 10 days, how 
 many horses will eat up 100 bushels in a fortnight ? 
 
 Ans. 20 horses 
 
 (12) If 6 horses plough 10 acres of land in 5 days, how 
 many horses will plough 16 acres in 12 days ? Ans. 4 horses 
 
 (13) How many bushels of wheat will serve 54 people 14 
 days, when 3 bushels will serve 6 people 21 days ? 
 
 Ans. 18 bushels 
 
 (14) Lent a friend 800Z. for 9 months at 51. per cent., how 
 long ought he to lend me 1250Z. at 4Z. per cent, to requite 
 my kindness ? 
 
 The truth of this operation may be 
 proved by the rule of Interest, thus : 
 8001 1250Z. 
 
 5 pr. c. 4 per cent. 
 mo. 
 
 
 800 
 
 By the first rule. 
 
 mo. 
 9 
 
 
 1250 
 4 
 
 ^ 
 
 1250 
 4 
 5000 
 
 800 
 9^ 
 
 7200 
 5 
 
 5,000)36,000 
 
 7 mo. -- = 7 mo. 6 da. 
 
 40(00 
 
 20 
 10 
 
 50(00 
 
 30Z. for 9 mo. 
 
 25 
 
 4 3 
 
 16 
 
 30 for 7 mo. 
 
 Hence it is evident that the 8001. for 9 
 months at 51. per cent, is 30. interest ; and 
 of 12501. for 7 months 6 days, at U. per 
 cent, amounts to the same. 
 
 (15) If 756 bricks, 14 inches long and 10 broad, will pave 
 a floor ; how many bricks would it take 15 inches long and 
 2 inches broad ? Ans. 784 bricks 
 
 (16) If 12 inches in length, 12 inches in breadth, and 12 
 in thickness, make a solid foot, what length of a plank that 
 is 6 inches broad and 4 inches thick will make the same ? 
 
 Ans. 72 inches 
 
(60) 
 
 PRACTICE. 
 
 PEACTICE is so called from its general use to all persons 
 concerned in trade and business ; it being a compendious 
 method of ascertaining the value of any quantity of goods 
 or other commodities. 
 
 All questions in this rule might be worked by Multiplica- 
 tion, or the Rule of Three ; but they are here more expedi- 
 tiously performed, by taking aliquot or even parts. 
 
 TABLES 03? ALIQUOT PAETS. 
 
 Of aPo 
 105. Qd. 
 6 8 
 5 
 4 
 3 4 
 2 6 
 2 
 1 8 
 1 4 
 1 3 
 1 
 
 und. 
 is | 
 i 
 
 f 
 
 ~G 
 } 
 
 TV 
 TV 
 
 iV 
 
 TV 
 ^ 
 
 Of a 
 Qd. 
 4 
 3 
 2 
 14 
 
 i 
 "I 
 
 T 0/a! 
 2 /art/*, 
 1 farthi'i 
 
 hilling. 
 
 IS | 
 
 1 
 
 1 
 
 TIT 
 l 
 
 "5T 
 
 A 
 
 Penny. 
 
 W ... \ 
 ig ... 1 
 
 Of 
 10 cwtf 
 5 
 4 
 
 01 
 
 I s 
 
 1 
 
 a To 7i. 
 
 is \ 
 
 "'. | 
 
 ::::: T J 
 
 Of a fiundrt 
 2 37-5. or 56 11 
 1 97- or 28 Ib 
 16 .. 
 14 .. 
 7 .. 
 4 .. 
 Of a Quarte 
 14 Ib. is 
 7 
 
 4 
 
 31 
 * 
 
 ;d. 
 .i 
 
 | 
 
 1 
 
 r. 
 i 
 
 f 
 
 I 
 
 Parts of a Ib. 
 8 02. is 
 4 J 
 2 * 
 1 ...... A 
 
 ETJLE I. When the price is less tlian a penny Divide the 
 given number by the aliquot parts that are in a penny ; then 
 by 12 and 20 for the answer. 
 
 (1) 
 
 iU 
 12 
 
 2,0 
 
 Ans. 
 
 3857 at i 
 
 
 fii 
 
 12 
 
 2,0 
 
 Ans. 
 
 5687 at i 
 
 (9) 
 
 (10) 
 ^lw 
 
 (11) 
 JLw 
 
 (12) 
 An 
 
 1 U 
 f|| 
 
 12 
 
 2,0 
 
 364S 
 
 5. 1U 
 
 307? 
 5. 9Z 
 
 7580 
 5. 23Z 
 
 7459 at j 
 
 3729J 
 1864f 
 
 964| 
 
 2843^ 
 
 8,0 4 
 
 23,6 111 
 
 559,4^ 
 
 d. 
 
 46,6 
 
 4 4J 
 
 LI 16 Hi 
 
 23 6 
 
 (2) 2794 at \ 
 Ans. 21. 185. 2\d. 
 
 (3) 4657 at 
 Ans. 4Z. 175. O^d. 
 
 (4) 6120 at I 
 Ans. 61. 7s. 6d. 
 
 (6) 3987 at 4 
 ^.^5. 8Z. 65. ~L^d. 
 
 (7) 6055 at 4 
 Ans. 121. 125. 3!<Z. 
 
 (8) 8317 at 1 
 ^l//5. 17Z. 65. 6^. 
 
 at | 
 85.0| 
 
 at f 
 125. 4, 
 
 atf 
 
 135.1 
 
Practice. 
 
 67 
 
 II. When tlie price is less than a shilling divide the 
 given number by the aliquot part or parts of a shilling, add 
 them together, and divide by 20 for the answer. 
 
 (1) Id. 
 
 T 
 
 2, 
 
 U3537atld. 
 01 W 3d. 
 
 (8) 2345 at 2$d. 
 Ans. 261. 17s. 4%d. 
 
 (21) 3219 at 6d. 
 Ans. 801. 9s. 6d. 
 
 
 
 14 17 3 
 
 
 
 (2) Id. 
 
 i 
 
 i . 
 
 4359 at Hd. 
 
 (9) 1342 at 3d. 
 Ans. 161. 15s. 6d. 
 
 (22) 2468 at 6%d. 
 
 1 
 
 i 
 
 363 3 
 
 90 9| 
 
 (10) 4320 at 3d. 
 
 (23) 5321 at 7^d. 
 
 2 
 
 ,0 
 
 45,4 Of 
 
 Ans. 581. 10s. 
 
 Ans. 1661. 5s. 7^d. 
 
 
 
 22 14 Of- 
 
 
 
 (3) IJrf. 
 
 11 
 
 5137atl^d. 
 
 (11) 5627 at 3%d. 
 Ans. 821. Is. 2%d. 
 
 (24) 6019at7JJ. 
 
 2 
 
 ,o 
 
 64,2 H 
 
 (12) 4396 at 3f d. 
 
 (25) 5068 at 8%d. 
 
 
 
 32 2 14 
 
 Ans. 68Z. 135. 9d. 
 
 Ans. 1791. 9s. lOd. 
 
 (4) l\d 
 
 1 
 
 5 
 
 6259 at If d. 
 
 (13) 5069 at 4d. 
 
 (26) 8271 at 9d. 
 
 1 O1A7O O^7 
 
 \ 
 
 i 
 
 782 4 
 130 4f 
 
 Ans. 841. 9s. 8d. 
 
 Aus. 3101. 3s. 3d. 
 
 
 2,0 
 
 91,2 9J 
 
 (14) 6908 at 4J. 
 
 (27) 4564 at 9f d. 
 
 (5) 2d. 
 
 1 
 
 4~> 12 9J- 
 - 7385 at 2d. 
 
 Ans. 1221. 6s. 7d. 
 (15) 8005 at k\d. 
 
 Ans. 1851. 8s. 3d 
 
 (28) 8765atlOid. 
 Ans 3741 Qs 9^d 
 
 
 2,C 
 
 123,6" 10 
 61 10 10 
 
 (16) 2759 at 4f 
 
 (29) 6213atlO|J. 
 
 16) 2d. 
 
 i 
 ^ 
 
 8479 at 2d. 
 
 Ans. 541. 12s. l^d. 
 
 (~\ 7^ 7Q^9 nf ^^ 
 
 Ans. 2781. 5s. 9%d. 
 dCh 50SB at 11 d 
 
 \ 
 
 i 
 
 f 
 
 1413 2 
 176 7f 
 
 ^/25. 165Z. 135. 4d. 
 
 Ans. 2741. 7s. 2d. 
 
 2 
 
 ,0 
 
 158,9 9| 
 
 
 /O1 \ ^7OOO 4- 1 1 I J 
 
 
 
 79 9 9| 
 
 (18) 6327 at 5Jd. 
 
 A TOO? O C\3 J 
 
 (31) 7328 at Ilia. 
 
 4 V* n Q A Q 7 1 (\n 
 
 V 
 
 i 
 
 9876 at 2 id. 
 1646 
 411 6 
 
 Ans. 1381. 85. Ojd. 
 
 (19) 3254 at 5Jd. 
 ^5. 74/. 115. 5d. 
 
 A.ns. 64<oL 1U5. 
 
 (32) 4537 at 11 \d. 
 Ans.217l.7s.ll l ^d. 
 
 2 
 
 ,o 
 
 205,7 6 
 
 ( 9 0) 4968 -it 5*d 
 
 (33) 97G5 at 11 -d 
 
 
 
 102 17 6 
 
 Ans. 1191. Os. 6d. 
 
 Ans. 4781. Is. 64d. 
 
68 
 
 Practice. 
 
 III. WTien the price is more than one shilling and less 
 tlian two take the aliquot part or parts for so much of the 
 given price as is more than a shilling, which add to the given 
 quantity, and divide by 20 for the answer. 
 
 (l)il Al 5364 at UJ 
 111 9 
 
 2,0 | 547,5 9 
 
 (14) 3d y 6547 at 1 3-| 
 |Ii!l686 9 
 
 409 2 
 
 (27) 6eZ. 1 3679 at 1 7i 
 1^. 1 1839 6 
 i 306 7 
 76 7| 
 
 2,0 | 859,2 11 
 
 273 15 9 
 
 2,0 | 590,1 8| 
 
 429 12 11J 
 
 295 1 8| 
 
 (2) 3620 at Is. 0d. 
 Ans. 1881. 10s. 10d. 
 
 (15) 3249 at Is. 4<7. 
 Ans. 2161. 12s. 
 
 (28) 4326 at Is. 7f/. 
 A. 3551. 19s. lQ\d. 
 
 (3) 5426 at Is. Of d. 
 Ans. 2881. 5s. ld. 
 
 (16) 7060 at Is. 4 $d. 
 Ans. 4781. Os. 5d. 
 
 (29) 5432 at Is. 8d. 
 Ans. 4521. 13s. 4<7. 
 
 (4) 6421 at Is. Id. 
 Ans. 3471. 16s. Id. 
 
 (17)6391atL?.4J<7. 
 Ans. 4397. 7s. l\d. 
 
 (30)6548atl.8f<7. 
 Ans. 5667. 2s. 7t/. 
 
 (5) 7536 at Is. lid. 
 Ans. 4161. Is. 
 
 (18)8325atls.4|6/. 
 Ans. 5811. Os. 3|<7. 
 
 (31)7464atls.94J. 
 Ans. 6607. 17s. G^/. 
 
 (6) 5897 at Is. l\d. 
 Ans. 3317. 14s. 1 J<7. 
 
 (19) 7510 at Is. 5d. 
 Ans. 531L 19s. 2d. 
 
 (32) 4263 at Is. 9 \d. 
 
 (7) 6230 at Is. If <7. 
 
 (20)423Satls.5|<7. 
 Ans. 3041. 12s. \\d. 
 
 (33) 6791 at U 10(7. 
 Ans. 6221. 10s. 2d. 
 
 (8) 4586 at Is. 2d. 
 Ans. 2671. 10s. 4d. 
 
 (21)6266atl*.5Je7. 
 Ans. 456Z.17s. lid. 
 
 (34)11 69atlff.lOJrf. 
 -4w*.1097.U*.!0|<7. 
 
 (9) 6329 at Is. 2W. 
 Ans. 3751. 15s. 8$d. 
 
 (22) 4326 at Is. 6f7 
 ^. 3197. 18s. 10J<7. 
 
 (35)5544atls.lO|J. 
 Ans. 5251. 10s. 6d. 
 
 (10) 7638 at Is. 2J<7. 
 Ans. 4611. 9s. 3d. 
 
 (23) 6007 at Is. 6 J. 
 -4n*. 4507. 10s. 6<7. 
 
 (36) 7590 at Is. lid. 
 Ans. 7271. 7s. 6d, 
 
 (Ll)4006atls.2fJ. 
 Ans. 2461. 4s. 0%d. 
 
 (24) 7805 at Is. 6\d. 
 Ans. 5937. 10s. 1^. 
 
 (37)4674atl*.liy. 
 A. 4527. 15 5 . 10^7. 
 
 (12) 7068 at Is. 3d. 
 Ans. 4411. 15s. 
 
 (25) 4265 at Is. 6iJ. 
 Ans. 3287. 15s. 2J<7. 
 
 ^;^9/293/. 15x! 
 
 (13) 4320 at Is. 3%d. 
 Ans. 2741. 10s. 
 
 (26) 3654 at Is. 7d. 
 Ans. 2897. 5s. 6d. 
 
 (39)4433atl*.ll?/. 
 Ans. 4381. 13s, 7%d, 
 
Practice. 
 
 69 
 
 lY. If the price le an even number of shillings, under 
 20 multiply the quantity by half the number, doubling the 
 first figure of the product for shillings, and the rest will be 
 pounds. 
 
 (1) 3643 at 2y. 
 1 
 
 (5) 3592 ab 10s. 
 5 
 
 (8) 3609 at 14*. 
 
 7 
 
 Atis. 364 6 
 
 Ans. 1790 
 
 Ans. 2526 6 
 
 (2) 3752 at 4*. 
 
 rAm<* T^O7 fto 
 
 Or thus : 
 10^. | i | 3876 at 10s. 
 
 (9) 8372 at 16-. 
 An? 66077 12s 
 
 
 Ans. 1938 
 
 
 (3) 6543 at 6*. 
 Ans. 19627. 18*. 
 
 (4) 7134 at 8s. 
 Ans. 28537. 12*. 
 
 (6) 3908 at 12s. 
 Ans. 23447. 16*. 
 
 (7) 7766 at 14.?. 
 Ans. 54367. 4s. 
 
 (10) 17654 at 18s. 
 Ans. 158887. 12*. 
 
 (11) 12346 at 18s. 
 Ans. 111117. 8s. 
 
 Y. If the price consists of odd shillings 1st. Multiply the 
 given quantity by the price, and divide by 20 for the answer. 
 Or, 2nd. Find the greatest even number as in the last rule ; 
 to which add -/oth of the given number for the odd shilling, 
 and their sum will be the answer. 
 
 By the first rule. 
 (1) 7462 at 7*. 
 7 
 
 (4) 7626 at 13s. 
 Ans. 49567. 18s. 
 
 By the second rule. 
 (8) 73641 at 17s. 
 8 
 
 2,0 | 5223,4 
 
 (5) 4258 at 15s. 
 
 58912 16 
 
 QQO 1 
 
 4ns 2611 14 
 
 Ans. 31937. 10s. 
 
 
 
 
 A m o fiOKQ/t 1*7 
 
 
 
 
 
 (2) 3264 at 9s. 
 Ans. 14687. 16s. 
 
 (3) 4689 at 11s. 
 Ans. 25787. 19s. 
 
 (6) 6384 at 17s. 
 Ans. 54267. 8s. 
 
 (7) 1234 at 19s. 
 Ans. 11727. 6s. 
 
 (9) 3258 at 17s. 
 Ans. 27697. 6s 
 
 (10) 1069 at 19s. 
 Ans. 10157. 11s. 
 
 YT. When the price is shillings and pence if they are the 
 aliquot part of a pound, divide the quantity by that part, 
 and the quotient will be the answer in pounds. 
 
 But 2ndly. If they are not an aliquot part, find first for 
 the shillings, then take parts for the pence, and add them 
 together. 
 
 3rdly. When the price is shillings, peuce, and farthings, 
 find for the shillings and pence as before, and for the far- 
 things, take parts from a preceding line. 
 
70 
 
 Practice. 
 
 (1) 3s. 4^ 
 
 I \ 
 
 | 3625at3s.4J. 
 
 (14) M. | i 
 
 f 6745 at 35 
 
 . ad. 
 
 
 
 
 
 Q 
 
 
 
 Ans. 60i 3 4 
 
 
 O 
 
 
 
 20235 
 
 (2) 5731 at 
 
 65. Sf*. 
 
 
 1686 
 
 
 
 Ans. 19101. 6s. Sd. 
 
 2,0)2192,1 3 
 
 (3) 2437 
 
 ' at 
 
 25. 6<*. 
 
 Ans. 
 
 1096 1 3 
 
 
 
 Ans. 304*. 12s. 6d. 
 
 (15) 6308 at 7s. 5d. 
 
 (4) 4675 at 5s. 
 
 Ans. 23391, 4s 
 
 Id. 
 
 
 
 Ans. 11681. 15s. 
 
 (16) 9085 at 11s. 6d. 
 
 (5) 6543 at 10*. 
 
 Ans. 52231. 17s 
 
 6d. 
 
 
 
 Ans. 3271*. 10s. 
 
 (17) 8712 at 15s. 9d. 
 
 (6) 1206 
 
 at 
 
 35. 4<f. 
 
 
 Ans. 68601. 
 
 14s. 
 
 
 
 ^775. 201*. 
 
 (18) 3240 at 18s. 8d. 
 
 (7) 9876 at 
 
 15. 4<f. 
 
 Ans. 3024*. 
 
 
 
 ^TIS. 658*. 85. 
 
 (19) 6267 at 19s. 6d. 
 
 
 
 
 Ans. 61101 65. 
 
 6d. 
 
 (8) 55. 
 
 i 
 
 4709 at 55. Sd. 
 
 (20) 6d. } 
 
 3866atlU 
 
 
 6d. 
 
 A 
 
 1177 5 
 
 
 11 
 
 
 2d. 
 
 J 
 
 117 14 6 
 
 
 42526 
 
 
 
 
 39 4 10 
 
 3^7. J 
 
 1933 
 
 
 
 
 
 ]L 
 
 966 6 
 
 
 
 
 
 1334 4 4 
 
 2 
 
 161 1 
 
 
 (9) 7890 
 
 at 
 
 7s. 6d. 
 
 2,0)4558,6 7 
 
 
 
 Ans. 29581. 15s. 
 
 ^rcs. 
 
 2279 6 7 
 
 
 (10) 6234 at 4s. Sd. 
 
 Ans. 1454*. 12s. 
 
 (11) 4327 at 5s. 9d. 
 
 Ans. 1244*. Os. 3d. 
 
 (12) 6432 at 10s. 
 
 ^725. 3484*. 
 
 (13) 6974 at 95. 6d. 
 
 Ans. 33121. 13s. Od. 
 
 (21) 4275 at 12s. 8|rf. 
 ^775. 2720Z. 17s. 
 
 (22) 2508 at 9s. 
 
 Ans. 12381. 6s. 6d. 
 
 (23) 4597 at 15s. tyd. 
 
 Ans. 35721. 5s. O^cl 
 
 1060 at 16s. 2\d. 
 
 Ans. 8571. 18s. 9d. 
 
 (25) 6324 at 18*. 3d. 
 
 Ans. 57771. 4*. 9d 
 
Practice. 
 
 71 
 
 TIL When the price is pounds and shillings ; or pounds, 
 shillings, pence, and farthings. 
 
 EULE. Multiply the quntity by the pounds, and proceed 
 for the shillings, pence, &c. according to the preceding rules ; 
 then these sums added together will give the answer. 
 
 6s. 
 
 Sd. U [ 547 at 7Z. 
 
 6s. Sd. 
 
 
 
 
 s. 
 
 rf. 
 
 
 7 
 
 
 (11) 
 
 10s. 
 
 \ 
 
 1396 at 119 11 f 
 
 
 
 
 
 5s. 
 
 \ 
 
 698 
 
 
 
 
 3829 
 
 
 
 4s. 
 
 
 349 
 
 
 
 182 6 
 
 8 
 
 
 6d. 
 
 | 
 
 279 4 
 
 
 
 
 
 
 4<d. 
 
 1 
 
 34 18 
 
 
 
 4011 6 
 
 8 
 
 
 i 
 
 1 2 
 
 i 
 
 23 5 4 
 81/1 f 
 
 
 
 
 
 
 3 
 
 6 
 
 1* b 
 
 
 
 
 
 
 
 
 191 
 
 
 (2) 
 
 456 at 8Z. 5s. 
 
 
 
 
 
 
 
 
 Ans. 
 
 3762Z. 
 
 
 
 
 
 2790 10 11 
 
 
 (3) 
 
 7960 at 91. 4s. 
 
 
 
 
 
 
 
 
 Ans. 73232Z. 
 
 (12) 
 
 3256 at 1Z. 15s. &\d. 
 
 (4) 
 
 4069 at HZ. 3s. 4c 
 
 
 
 
 An 
 
 s. 57721. 12s. 
 
 4/7. 
 
 
 Ans. 45437Z. 
 
 3s. 4J. 
 
 (13) 
 
 2568 at 1Z. 19s. 6R 
 
 (3) 
 
 897 at 12Z. 2s. 6d. 
 
 
 
 
 
 ^iws. 50771. 
 
 3s. 
 
 
 Ans. 10876Z. 
 
 2s. 6d. 
 
 
 
 
 
 
 (6) 
 
 5s. j ^ J 1287 at 3Z 
 3 
 
 . 7s. 6d. 
 
 (14) 
 
 4400 at 1Z. 13s. 4|d. 
 Ans. 7342*". 10s. 
 
 2s. 
 
 3861 
 6d. \ i | 321 15 
 
 
 (15) 
 
 6432 at 2Z. 17s. 7\d. 
 -4rcs. 18532Z. 4s. 
 
 
 160 17 
 
 6 
 
 (16) 
 
 432 
 
 at 
 
 SI. ISs. 4Jrf. 
 
 
 
 4342 12 
 
 6 
 
 Ans. 1584Z. 18s. 
 
 (7) 4685 at 6/. 9s. 4d. 
 
 Ans. 30296/. 6s. 8d. 
 
 (8) 2397 at 10Z. 15s. Wd. 
 
 Ans. 25867/. 12s. 6^/. 
 
 (9) 1234 at II. 14s. 7d. 
 
 Ans. 2133A 15s. 10^. 
 
 (10) 4538 at 51. 17s. 9d. 
 
 An*. 26717^. 9s. 6d 
 
 (17) 1006 at 4/. 11*. 
 
 s. 4596/. 3s. Sd 
 
 (18) 1436 at 51. 5s. 
 
 s. 7573Z. 8s. Id 
 
 (19) 326 at 61. 16s. 
 
 . 2230/. Os. 
 
 (20) 1281 at 77. 7s. 
 
 . 9455Z. 7s. 
 
72 Practice, 
 
 VIII. When the quantity is a whole number and a frac- 
 
 tion. 
 
 EFLE. Work for the whole number by the former rules ; 
 to which add \, g, or any other part of the price, and add as 
 before for the answer. 
 
 Or, for the fractional parts, multiply the price by the upper figure, 
 and divide by the under, as in Example the llth. 
 
 
 
 5. d. 
 
 
 
 
 
 s. 
 
 rf. 
 
 (1) 55. 
 
 l 
 
 628f 
 
 at 
 
 1 7 10 J 
 
 (11) 10s. 
 
 1 
 
 364 
 
 1 
 
 at 2 
 
 12 
 
 J* 
 
 25. Gd. 
 
 i 
 
 157 
 
 
 
 
 
 
 
 
 
 o 
 
 3d. 
 
 iV 
 
 78 
 
 10 
 
 
 
 
 728 
 
 
 8)7 
 
 17 
 
 g. 
 
 1 1 j 
 
 i 
 
 7 
 
 17 
 
 
 2s 
 
 i 
 
 182 
 
 
 
 
 * 
 
 3 
 
 18 
 
 6 
 
 W. 
 
 I 
 
 36 
 9 
 
 8 
 
 2 
 
 
 
 19 
 
 8* 
 
 for the \ 
 
 13 
 
 11^ 
 
 * 
 
 8 
 
 i 
 
 2 c 
 
 
 
 
 for the 1 
 
 6 
 
 HJ 
 
 For | 
 
 19 6 
 
 i 
 
 
 
 876 
 
 6 
 
 4f 
 
 957 
 
 12 
 
 ^ 
 
 
 
 (2) 435J at 27. 125. 6rf. 
 
 S. 1143Z. 3s. Qd. 
 
 (12) 246f at 4Z. 105. Qd. 
 Ans. 1114Z. 165. 
 
 (3) 6081 at 3Z. 2*. 6J. 
 
 . 1900Z. 155. 7| 
 
 (13) 345f at 165. 6d. 
 
 Ans. 284Z. 185. B 
 
 (4) 439f at 4Z. 5*. 6J. 
 
 *. 1879Z. 185. 
 
 (14) 987f at 75. 
 
 Ans. 380Z. 
 
 (5) 5321 a t 51 Is. 4d. 
 
 Ans. 2696Z. 145. Bd. 
 
 (6) 276 J at 175. 6d. 
 
 Ans. 241Z. 185. 9rf. 
 
 (7) 426f at 185. 4d. 
 
 Ans. 391Z. 35. 9d. 
 
 (15) 105. 
 
 i 
 
 684f at 185. 
 
 55. 
 
 i 
 
 342 
 
 
 35. 4<J. 
 
 1 
 
 6 
 
 171 
 
 
 2J. 
 
 2 t 
 
 114 
 
 
 J 
 
 4 
 
 5 14 
 
 
 
 
 1 8 
 
 6 
 
 8 
 
 1 
 
 9 
 
 34 
 
 i 
 
 1 
 4 
 
 2 
 
 3 T 
 
 634 14 
 
 1 
 
 (8) 12341 a t 45. 
 
 5. 280Z. 55. 
 
 (9) 3211 a t 55. 10 J?. 
 
 Ans. 94Z. 85. 
 
 (10) 987f at 11*. llrf. 
 
 *. 588Z. 105. 
 
 (16) 189| at 155. 6d. 
 
 Ans. 146Z. 195. 2d. 
 
 (17) 365f at 1Z. 155. Gd. 
 
 Ans. 648Z. 195. Bd. 
 
 (18) 1000 at 2Z. 75. Gd. 
 
 Ans. 2375Z. 
 
Practice. 
 
 IX. When loth the price and the quantity are of several 
 denominations. 
 
 RULE. Multiply the price by the highest denomination, 
 and take parts for the lower denominations : then add them 
 together for the answer. 
 
 (1) At 51. 155. 6d. per cwt., what is the value of 75 cwt. 
 3 qrs. 21 Ib. of hops ? 
 
 s. d. 
 
 2 qrs. 
 
 i 
 
 5 
 
 15 
 
 6X3 
 
 
 
 
 
 
 
 6 X 12 
 
 + 3 
 
 = 75 
 
 
 
 34 
 
 13 
 
 = 6 
 
 
 
 
 
 
 
 12 
 
 
 
 
 
 
 
 
 qr. 
 
 Ib. 
 
 
 
 
 
 
 415 
 
 16 
 
 =72 
 
 
 
 
 
 
 
 17 
 
 6 
 
 6=3 
 
 
 
 
 
 Iqr. 
 14 Ib. 
 
 i 
 
 2 
 
 1 
 
 17 
 8 
 
 9=0 
 lOJzz 
 
 2 
 1 
 
 
 
 
 71b. | 
 
 
 
 14 
 
 5|= 
 
 
 
 14 
 
 
 
 7 
 
 2|= 
 
 
 
 7 
 
 Ans. 438 10 9= 75 3 21 
 
 (2) At 31. 18.9. 6d. per cwt,, what is the value of 36 cwt. 
 1 qr. 71b. of sugar ? Ans. 14*21. 10s. 6J. 
 
 (3) Sold 28 cwt. 3 qrs. 14 Ib. of cheese, at 21. 12s. 6d. per 
 cwt., what does it come to ? Ans. 751. 15s. ll^d. 
 
 (4) Bought 29 cwt. 1 qr. 11 Ib. of tea at 101. IBs. Sd. per 
 cwt., what was the cost of the whole ? Ans. 3201. 17s. 5^d. 
 
 (5) What is the value of 13 cwt. 3 qr. 4 Ib. of butter at 
 21. 18s. 4d. per cwt. ? Ans. 401. 4s. 2d. 
 
 (6) At 41. 16s. 9d. per cwt., what is the worth of 11 cwt. 
 
 qr. 14 Ib. of double refined sugar ? Ans. 53Z. 16s. 4d. 
 
 (7) What must I pay for 34 acres 2 roods 20 poles of land, 
 at 21. 11*. 6d. per acre ? Ans. SQL 3s. 2d 
 
 (8) Bought 37 qrs. 4 bush. 2 pecks of wheat, at 41. 16s. 6d. 
 per quarter, what does the whole cost ? Ans. 1811. 4s. 9d. 
 
 (9) At 11. 6s. 4d. per gallon, what will 17 gall. 2 qts. 
 
 1 pint of brandy come to ? Ans. 231. 4s. l$d. 
 
 (10) Sold 17 tons 5 cwt. 2 qrs. at 121. 10s. 4d. per ton, 
 how much did they realize ? Ans. 2161. 4s. 6d. 
 
 (11) Soap at 41. 2s. 8d. per cwt., what is the worth of 
 19 cwt. 3 qrs. 7 Ib. ? Ans. 811. 17s. lOd. 
 
 (12) Tobacco at 5L 16s. 8d. per cwt., what is the worth of 
 42 cwt. qr. 16 Ib. ? Ans. 2451. 16s. Sd. 
 
 (13) At 31. 14s. per cwt., what is the value of 18 cwt. 1 
 qr. 41b. of currants Y Ans. 671. 13s. 
 
TARE AND TRETT. 
 
 THIS Eule teaches the method of deducting such allow- 
 ances as are usually made by merchants and tradesmen in 
 selling their goods ; and the terms in general use are gross 
 weight, tare, trett, doff, suttle, and neat weight. 
 
 Gross weight is the whole weight of the goods, and of 
 that which contains them, whether box, barrel, bag, chest, 
 hamper, &c. 
 
 Tare is an allowance made to the buyer for the weight of 
 the box, barrel, &c. This is charged either at so much per 
 box, &c., or at so much per cwt., or at so much in the whole. 
 
 Trett is an allowance of 4 Ib. per 104 (i. e. a 26th part of 
 the whole) for waste, dust, &c. 
 
 Suttle weight is when part of the allowance is deducted 
 from the gross. 
 
 Clqff is an allowance (after tare and trett are deducted) 
 of 2 Ib. in every 3 cwt. (or 1 Ib. in every 168 Ib.) to make 
 the weight hold out when sold by retail. 
 
 Neat or net weight is the pure weight, when all allowances 
 are deducted from the gross weight. 
 
 EXAMPLES. 
 CASE I. When the tare is so much in the whole. 
 
 RULE. Subtract the tare from the gross, and the re- 
 mainder will be the neat weight. 
 
 (1) If the gross weight of several barrels of raisins be 
 130 cwt. 2 qrs. 18 Ib., and the tare be 3 cwt. 3 qrs. 24 Ib., 
 what is the neat weight ? 
 
 cwt. qrs. Ibs. 
 130 2 18 gross 
 3 3 24 tare 
 
 An*. 126 2 22 neat weight 
 
 (2) If the gross weight of several bags be 31 cwt. 1 qr. 
 10 Ib. and the tare be 3 cwt. 1 qr. 16 Ib., what is the neat 
 weight ? Ans. 27 cwt. 3 qrs. 22 Ib. 
 
 (3) If the gross weight of 20 frails of raisins be 11 cwt. 
 2 qrs. and the tare be 1 cwt. 3 qrs. 51b., what will be the 
 neat weight ? Ans. 9 cwt. 2 qr. 23 Ib. 
 
 (4) What is the neat weight of a quantity of goods, if the 
 gross is 1 ton 3 cwt. 3 qrs. 5 Ib. and the tare 7 cwt. 3 qrs. 
 16 Ib. ? Ans. 15 cwt. 3 qrs. 17 & 
 
Tare and Trett. 75 
 
 (5) In 17 barrels, weighing in the whole 3G cwt. 3 qrs, 
 16 Ib. gross, and tare in the whole 3 cwt. qr. 19 Ib. how 
 much neat weight ? Ans. 33 cwt. 2 qr. 25 Ib. 
 
 (6) In 7 frails of raisins, each weighing 3 cwt. 2 qr. 15 Ib. 
 gross, tare in the whole 3 qrs. 181b., how much neat weight? 
 
 cwt. qrs. Ib. 
 3 2 15 
 
 7 
 
 25 1 21 gross 
 3 18 tare 
 
 Ans. 24 2 3 neat weight 
 
 (7) In 8 barrels of figs, each 3 qrs. 27 Ib. gross, and tare 
 in the whole 2 qrs. 11 Ibs., how much neat weight ? 
 
 Ans. 7 cwt. 1 qr. 9 Ib. 
 
 (8) In 9 hhds. of nutmegs, each weighing gross 6 cwt. 
 3 qrs. IGlb., and tare in the whole 1 cwt. qr. 17 Ib., how 
 much neat weight ? Ans. 60 cwt. 3 qrs. 15 Ib. 
 
 (9) What is the neat weight of 20 casks of argol, weigh- 
 ing each 7 cwt. 2 qrs. 10 Ib., and tare in the whole 1 cwt. 
 3 qrs. 16 Ib. ? Ans. 149 cwt. 3 qrs. 16 Ib. 
 
 (10) The gross weight is 3 cwt. 1 qr. 11 Ib. per hhd., and 
 tare in the whole 182 Ib., what is the neat weight of 12 hhds,? 
 
 cwt. qr. Ib. 
 3 1 11 28)182(4)6 
 
 12 168 
 
 1 2 14 tare 
 
 40 20 gross 14 
 1 2 14 tare : 
 
 Ans. 38 2 6 neat weight 
 
 (11) If the gross weight of 5 loads be 3 tons 2 cwt. 1 qr. 
 per load, and the whole tare be 1760 Ib., how much neat 
 weight ? Ans. 14 tons, 15 cwt. 2 qrs. 4 Ib. 
 
 (12) What is the neat weight of 14 hhds. of tobacco, each 
 weighing 5 cwt. 3 qr. 12 Ib., tare in the whole 1260 Ib. ? 
 
 Ans. 70 cwt. 3 qrs. 
 
 II. When the tare is at so much per lag, barrel, boa: !fc. 
 
 ETJLE. Multiply the tare of each box, barrel, #e. by the 
 number of boxes, barrels, <fcc.,then subtract the product from 
 the gross, and the remainder will be the neat weight. 
 
 E2 
 
76 Tare and Trett. 
 
 (13) "What will be the neat weight of 7 bags of hops, weigh- 
 ing in the whole 12 cwt. 2 qrs. 9 lb., and tare 18 Ib. per bag ? 
 
 cwt qr. lb. lb. 
 
 12 2 9 18 
 
 1 14 tare 7 
 
 cwt. qr. lb. 
 
 Am. 11 1 23 neat weight 28 ) 126 ( 4 qr. rz 1 14 tare 
 - 112 
 
 14 
 
 (14) "What is the neat weight of 36 bales of silk, weighing 
 in the whole 74 cwt. qr. 16 lb., tare 17 lb. per bale ? 
 
 Ans. 68 cwt. 2 qrs. 20 lb. 
 
 (15) In 14 bags of pepper, weighing in the whole 9 cwt. 
 2 qrs. 13 lb. gross, tare per bag, 4 lb. 4 oz., how much neat 
 weight ? Ans. 9 cwt. qrs. 9 lb. 8 oz. 
 
 (16) What is the neat weight of 12 chests of sugar, each 
 weighing 14 cwt. 1 qr. 5 lb. gross, and tare 19 lb. per chest ? 
 
 cwt. qr. lb. lb. Shorter, thus : 
 
 14 1 5 19 cwt.qr.lb. 
 
 12 12 14 1 5 gr. of I chest 
 
 19 tare of ditto 
 
 171 2 4 gross 28)228(4)8 
 
 204 tare 224 14 14 n w. of I c. 
 
 204 tare 12 
 
 Ans.169 2 neat 
 
 169 2 neat weight 
 
 (17) What is the neat weight of 25 barrels, each weighing 
 gross 5 cwt. 3 qrs. 7 lb., and tare per barrel 2 qrs. 12 Ibs. ? 
 
 Ans. 130 cwt. qr. 15 lb. 
 
 (18) The gross weight of 21 hogsheads is 3 cwt. 1 qr. 8 lb. 
 per hogshead, and the tare is 3 qrs. 10 Ibs. per hogshead ; 
 what is the neat weight ? Ans. 52 cwt. qr. 14 lb. 
 
 (19) If there are 7 casks of goods, and the gross weight 
 of each cask is 4 cwt. 3 qrs. 16 lb. and the tare 1 qr. 21 lb. 
 per cask, what is the neat weight ? Ans. 31 cwt. qr. 21 lb. 
 
 III. When the tare is at so much per cwt. 
 
 BTTLE. Divide the gross weight by the aliquot part or 
 parts of a cwt., which subtract from the gross, the remainder 
 is neat. 
 
Tare and Tret I. 77 
 
 (20) What is the neat weight of 12 barrels of potash, each 
 weighing 287 Ib. gross ; the tare being 10 Ib. per cwt. ? 
 
 lb. 
 
 287 
 
 12 
 
 8/6. | ^ | 3444 gross 
 
 2 Ib. | i I 246 
 81J 
 
 307* tare 
 
 Ans. 3136 Ib. neat 
 
 (21) In 136 barrels of figs, each 126 Ib. gross, tare 12 lb- 
 per cwt., how many Ibs. neat ? Ans. 15300 II' 
 
 (22) How many Ibs. neat in 5 hhds., each 1 cwt. 3 qrs. 5 lb. 
 gross, the tare being 14 Ibs. per cwt. ? Ans. 879 1 lb. 
 
 (23) In 11 frails, each 3 qrs. 16 lb. gross, and tare 16 lb. 
 per cwt., how much neat weight ? 
 
 qrs. lb. 
 
 3 16 
 
 11 
 
 16 lb. | f | 9 3 8 gross 
 1 1 17 tare 
 
 Ans. 8 1 19 neat weight 
 
 (24) In 36 hogsheads, each 2 cwt. 3 qrs. 24 lu. gross ; and 
 tare 18 lb. per cwt., how much neat weight ? 
 
 Ans. 89 cwt. 2 qrs. 7 lb. 3 oz. 
 
 (25) "What is the neat weight of 21 casks, each weighing 
 2 cwt. 3 qrs. 18 lb. gross ; and tare 13 lb. per cwt. ? 
 
 Ans. 54 crot. qrs. 3J lb. 
 
 (26) In 33 parcels each weighing 2 cwt. 1 qr. gross ; and 
 tare 8 lb. per cwt., how much neat weight ? 
 
 Ans. 68 cnt. 3 qrs. 22 lb. 
 
 IY. When both tare and trett are allowed. 
 
 EULE. Find the tare as before, subtract it from the gross, 
 and call the remainder suttle : then divide the suttle by 26, 
 the quotient will be the trett; which subtract from the 
 suttle, the remainder will be the neat weight. 
 
78 Tare and Trett. 
 
 (27) In 112 cwt. 1 qr. 25 Ib. gross, tare 184 lb., trett 4 Ib, 
 per 104, how many pounds neat ? 
 
 cwt. qr. lb. 
 112 1 25 
 4 
 
 449 
 
 12597 gross 
 184 tare 
 
 26 | 12413 suttle 
 477 trett 
 
 Ans. 11936 lb. neat weight 
 
 (28) In 36 cwt. 2 qrs. 4 lb. gross, tare 36 lb., trett 4 lb. 
 per 104, how many Ibs. neat ? Ans. 3900 lb. 
 
 (29) What is the neat weight in Ibs. of 3 hhds. each weigh- 
 ing 2 cwt. 3 qrs. 21 lb., tare 38 lb. per hhd., and trett as 
 usual ? Ans. 839 lb. 
 
 (30) In 16 frails, each 3 qrs. 27 lb. gross, tare 7 Ibs. per 
 cwt., and trett 4 lb. per 104, how many Ibs. neat ? 
 
 Ans. 1601 lb. 
 
 (31) How much is the neat weight of 9 butts, each 7 cwt. 
 1 qr. 21 lb., tare 3 qrs. 14 lb. per butt, and trett as usual ? 
 cwt. qr. lb. qr. lb. Shorter, thus : 
 
 7 1 21 3 14 71 21 gross 
 
 9 9 3 14 tare 
 
 66 3 21 gross 7 3 14 tare 627 
 
 7 3 14 tare == 9 
 
 26 | 59 7 suttle 26 | 59 7 
 
 212 trett 212 
 
 Ans. 56 3 5 neat weight Ans. 56 3 5 
 
 (32) How much neat weight in 3 butts, each 3 cwt. 2 qrs. 
 8 lb., tare 1 qr. 26 lb. per butt, and trett 4 lb. per 104 ? 
 
 Ans. 8 cwt. 3 qrs. 18 lb. 
 
 (33) In 136 cwt. 2 qrs. 14 lb. gross, tare 12 lb. per cwt. 
 and trett 4 lb. per 104, how much neat weight ? 
 
 Ans. 117 cwt. I qr. 5% lb. 
 
 (34) "What is the neat weight of a hogshead which weighs 
 3 cwt. 3 qrs. 10 lb. ; tare 2 qrs. 8 lb. in the whole, and trett 
 as usual ? Ans. 3 cict. qr. 16 lb. 
 
Tare and Trett. ?9 
 
 V. When tare, trett ^ and cloff , are allowed. 
 RULE. Work for the tare and trett as before ; then 
 divide the remainder, or suttle, by 168 ; the quotient will be 
 clo/l which subtract from the suttle, the remainder will be 
 the neat weight. 
 
 N.B. Instead of dividing by 168 for the cloff, the mere 
 common and ready way is to multiply the cwts. suttle by 
 two, and divide the product by three, and the quotient will be 
 the pounds cloff. 
 
 (35) What is the neat weight of 7 hhds., each weighing 
 5 cwt. 2 qrs. 16 Ib. gross, tare in the whole 2 cwt. 1 qr. 8 lb. ; 
 trett 4 Ib. per 104, and cloff as usual ? 
 cwt. qr. Ib. 
 5 2 16 
 7 
 
 39 
 2 
 
 2 
 
 1 
 
 gross 
 8 tare 
 
 35 
 2 
 
 3|70 
 ~ 23 11. cloff 
 
 26 | 37 
 1 
 
 
 1 
 
 20 suttle 
 20 trett 
 
 1GS | 35 
 
 3 
 
 suttle 
 23 cloff 
 
 Am. 35 2 5 neat wt. 
 
 (3G) In 17 chests, each weighing, gross 4 cwt. 3 qrs., tare 
 in the whole 3 cwt. 3 qrs. 14 Ib., trett and cloff as usual, 
 how much neat weight ? Ans. 73 cwt. 1 qr. 26 Ib. 
 
 (37) In 25 cwt. 3 qrs. 16 Ib. gross, tare 16 Ib. per cwt., 
 trett 4 Ib. per 104, and cloff as usual, how much neat weight? 
 
 Ans. 21 cict. qr. 24 Ib. 
 
 (38) What is the neat weight of 14 barrels of molasses, 
 each containing 5 cwt. 1 qr. 12 Ib. gross, tare 14 Ib. Der 
 cwt., trett 4 Ib. per 104, and cloff 2 Ib. per 3 cwt. ? 
 
 cwt. qr. Ib. 
 5 1 12 
 
 10 2 24 
 
 7_ 
 
 14 Ib. | | | 75 gross 63 
 
 9 1 14 tare 2 
 
 26 |~65 2 14 tuttk 
 
 222^ trett 42=1 qr. 14 Ib. cloff 
 
 168 I 63 11 suttle 
 1 14 cloff 
 
 Ans. 62 2 25.^ neat weight 
 
 (30) What is the neat weight of 29 barrels, each 3 cwt. 
 2 qrs. 25 Ib. gross, tare 16 Ib. per cwt., trett 4 Ib. per 104, 
 and cloff us usual ? Ans. 88 cwt. 1 qr. 24 Ib. 
 
(80) 
 
 INTEREST. 
 
 
 INTEREST is the profit obtained by lending a sum o! 
 money for a certain time, and at a fixed rate. 
 
 INTEREST is EITHER SIMPLE OR COMPOUND. 
 SIMPLE INTEKEST 
 
 Is that which is reckoned on the principal only. 
 
 The PRINCIPAL is the money lent. 
 
 The RATE PER CENT, is the sum per cent, agreed on, to be 
 paid for the use of the principal per annum. 
 
 The AMOUNT is the principal and interest added together. 
 
 INTEREST is also applied to Commission, Brokerage, 
 Purchasing of Stock, and Insurance. 
 CASE I. To find the Interest of any sum of money for a year. 
 
 RULE. Multiply the principal by the rate per cent. ; that 
 product divided by 100 will give the interest required. 
 
 Or, take the aliquot part or parts with the given rate that 
 are in 100: viz, 5J.=A; 4Z.=A-; 2JZ.=A, &c. of 100. 
 EXAMPLES. 
 
 (1) What is the interest of 2521 10s. Qd. for a year, at 
 4 per cent per annum. 
 
 i | 252 10 6 Or thus : 
 
 1010 2 10*. I 1 1 10 S Oft 
 
 126 5 3 153 
 
 11/36 7 3 Am. 11 7 
 
 20 
 
 skill 1 1 27 
 12 
 
 pence 3/27 
 4 
 
 far. 1/08 
 
 (2) What is the interest of 384Z. 12s. lOd. for a year, at 
 51. per cent, ? Ans. 191. 4s. l\d. 
 
 (3) What is the interest of 7561. 10s. for a year, afc 4Z. per 
 cent. ? Ans. 30/. 5s. 2\d. 
 
 (4) What is the interest of 8561. for a year, at 3J per 
 cent. ? Ans. 291. 19s. 
 
Interest. 
 
 81 
 
 II. To Jind the Interest of any sum for several years. 
 
 ETTLE. Multiply the interest of one year by the number 
 of years given, and the product will be the answer. 
 
 Or, Multiply the principal by the rate and time ; and that 
 product divide by 100 for the answer. 
 
 EXAMPLES. 
 
 (5) What is the interest of 2851. 15s. for 3| years, at 51. 
 per cent. ? 
 
 285 15 Or thus : Then for the years, multiply, &c. 
 
 5 . | A I 285 15 
 
 14/28 15 14 5 9 
 
 20 = 
 
 s. 5/75 
 12 
 
 d. 9/00 
 
 (6) At 4J per cent, per annum, what is the interest of 
 450Z. 12s. for 5 years ? Ans. 101 1. 7s. 8d. 
 
 (7) What is the interest of 500 guineas for 7 years, at 3 J 
 per cent, per annum ? Ans. 1281. 12s. Qd. 
 
 (8) What is the interest of 1000Z. for 5J years, at 3 per 
 cent. ? Ans. 1571. 10s. 
 
 y 
 | 
 
 14 5 9 
 
 42 17 3 
 7 2 10 
 
 Arts. 50 1 
 
 (9) What is the interest of 
 3G5Z. 105. for 3 months, at 5Z. 
 per cent, per annum ? 
 365 10 
 
 5 3 TOO. | i | 18 5 6 
 
 18/27 10 
 20 
 
 *. 5/50 
 12 
 
 4 11 
 
 (10) What is the interest 
 of 1240Z. for 1 year and 10 
 months, at 4i per cent. ? 
 
 i I 1240 
 
 4960 
 620 
 
 55/80 
 20 
 
 6 mo. 
 4 mo. 
 
 55 16 
 27 18 
 IS 12 
 
 Ana. 102 C 
 
 d. 6/00 s. 16/00 
 
 (11) What is the interest of 256Z. 10s. Gd. for } a year, at 
 3f per cent. ? Ans. 4Z. 16*. 2d. 
 
 (12) What is the interest of 486Z. 15s. for 9 months, at 
 4} per cent. ? Ans. 161. 8s. 6d. 
 
 (13) What is the interest of 500Z. for 3 years and 8 months, 
 at 51. per cent. ? Aim. f JU. IJj*. Id. 
 
 3 
 
82 Interest. 
 
 (14) "What is the amount of 1000Z. for two years and 10 
 months, at 3 per cent. ? 
 
 m. 
 
 1000 
 3 
 
 30100 
 
 30 
 
 2 
 
 1000 principal 
 
 60 85 interest 
 
 15 
 
 10 Ans. 1085 amount 
 
 85 interest 
 
 (15) What is the amount of 500/. for 3 years 8 months, at 
 5 per cent, per annum ? Ans. 591Z. 13s. id. 
 
 III. When interest is required for any nuvriber of weeks. 
 
 RULE. Find the interest of the given sum for one year ; 
 then state As 52 weeks are to that interest, so are the 
 weeks given to the interest required. 
 
 EXAMPLES. 
 
 (16) "What is the interest of 348Z. 13s. 4d. for 25 weeks, 
 at 4 1 per cent, per annum ? 
 
 s. d. wlcs. s. d. wJcs. 
 
 i I 348 13 4 As 52 ; 15 13 9i : : 25 
 
 a 20 
 
 1394 13 4 313 
 
 174 68 12 
 
 15/69 3765 
 
 20 4 
 
 s. 13/80 15062 
 
 12 25 
 
 d. 9/60 52 ) 376550 ( 4 ) 7241 
 
 - rem. 18 12 ) 1810J 
 
 far. 2/40 
 
 2,0 ) 15,0 lO 
 
 Ans. 7 10 10| 18 rem. 
 
 (17) What is the interest of 237Z. 16s. Gd. for 20 weeks, 
 at 3^ per cent, per annum ? Ans. 31. 4s. Qd. 
 
 (18) Find the interest of 500Z. for 37 weeks, at 5 per 
 cent, per annum ? Ans. 171. 15s. 9J 
 
 (19) Lent 250Z. on a mortgage at 4J per cent, per ann., what 
 interest will be due for 48 weeks ? Ans. 10Z. 7s. 8d. 12 
 
Interest. 83 
 
 (20) I demand the interest of 750Z. 15s. for three years 
 and 12 weeks at 4| per cent per annum ? 
 
 8- wks. .8. d. wTcs. 
 
 i I 750 15 52 : 35 13 2* : : 12 
 
 52)427 18 6(8*. 4*. U. 
 \\ 375 7 6 416 
 
 187 13 9 jj 
 
 35/66 13 20 
 
 __ 52)"238~(4*. 
 . 13/21 35 15 2 i 208 
 
 ii _ 8- 10 
 
 -2/55 106 19 7 j 12 
 
 47 52 ) 366 ( 7cf. 
 
 _ 
 far. 2/20 115 4 2i4n. 364 
 
 (21) What is the interest of 680Z. for 2 years and 25 weeks, 
 at 5 per cent per annum ? Ans. 84Z. 6s. lid. 4 r. 
 
 (22) Lent 1250Z. on a mortgage at 4g per cent per annum, 
 what interest will be due for 7 years and 35 weeks ? 
 
 Ans. 431Z. 125. 2%d. 8 r. 
 
 (23) I demand the interest of 1000Z. for 5 years and 40 
 weeks, 1 at 4 per cent per annum ? Ans. 2302. 15. 4|c?. 24r 
 
 IY. When interest is required for any number of days. 
 
 RULE. Find the interest of the given sum for a year; and 
 then state As 365 days are to that interest, so are the days 
 given to the interest required. 
 
 EXAMPLES. 
 
 (24) "What is the interest of 370Z. 18s. 6d. for 150 days, 
 at 44 per cent per annum ? 
 
 . *. d. days . 8. d. days 
 i I 370 18 6 Aa 365 : 16 13 : : 150 
 4 20 
 
 1483 14 333 
 
 185 9 3 12 
 
 16/69 3 3 4005 
 
 20 4 
 
 .T3~83 16023 
 
 12 150 
 
 d79/99 305)2403450(4)6584 
 
 4 2190 12)1646 
 
 far. 3 '9 6 2134 2,0)13,7" 2 
 
 &c - 6 17 
 
 If the remainder (96) were incorporated with the stating (thus, 161 
 13s. 9$d. 96) the true answer would be a more. 
 
84 Interest. 
 
 (25) Find the interest of 680Z. for 250 days, at 5 per cent 
 per annum. Ans. 231. 5s. 9d. 15 rem. 
 
 (26) What is the interest of 365Z. 10s. for 280 days, at 3 
 per cent per annum ? Ans. 81. Ss. 2\d. 54 rem. 
 
 (27) "What is the amount of 1600Z. for 3 years and 73 days, 
 at 4 per cent per annum ? 
 
 . days . days 
 
 1600 64 As 365 : 64 :: 73 
 
 43 73 
 
 64/00 192 192 Or, 73 being one- 
 
 12 16 448 fifth of 3G5, only 
 
 divide by 5, 
 
 The interest 20416 365)4tT2(12/.16s. thus: 
 
 The principal 1600 5)64 
 
 The amount 1804 16 Ans. 12 16 
 
 (28) Borrowed on the 1st of January 1200Z. and paid it on 
 the 1st of August twelvemonth (1 year and 212 days), with 
 interest at 5 per cent per annum, I demand the amount ? 
 
 Ans. 1294Z. 16s. llfd. 125 rem. 
 
 (29) Required the interest of 250 guineas at 3| per cent, 
 for 7 years and 100 days ? Ans. 6(3Z. 16s. 7d. 
 
 (30) What is the interest of 1780Z. for 5 years and 120 
 days, at 3 per cent per ann. ? Ans. 331Z. 19s*. 7^d. 330 r. 
 
 (31) What is the amount of 500 guineas, at 4^ per cent per 
 annum, for 7 years and 73 days ? Ans. 695Z. 2s. Od 
 
 V. When the amount, time, and rate per cent, are given 
 to find the principal. 
 
 RULE. As the amount of 100Z. at the rate, and for the 
 time given, is to 100, so is the amount given to the princi- 
 pal required. 
 
 EXAMPLES. 
 
 (32) What principal being put to interest, will amount to 
 430/ 14s. in 4 years, at 4^ per cent per annum ? 
 
 4| X 4 18 + 100 118Z. the amount of 100 for the rate and time. 
 Then say As 118/. : 100J. :: 430/. 14*. 
 20 20 
 
 2360 8614 
 
 100 
 
 230,0 ) 861 40,0 ( 365 Ans. 
 708 ' 
 1534 
 
 HI 6 
 1180 
 1180 
 
Interest. 85 
 
 (33) What principal being put to interest will amount to 
 540?. in 5 years, at 4 per cent, per annum ? Ans. 450?. 
 
 (34) "What principal being put to interest for 7 years at 
 5 per cent, per annum, will amount to 708?. 15s. ? Ans. 525?. 
 
 VI. When the principal, rate per cent, and amount, are 
 given, to find the time. 
 
 B/ULE. As the interest of the principal for one year is to 
 one year, so is the whole interest to the time required. 
 EXAMPLES. 
 
 (35) In what time will 365?. amount to 430?. 14s. at 4 \ 
 per cent, per annum ? 
 
 s. d. yr. 8. 
 
 i | 365 As 16 8 6 : 1 :: 65 14 
 
 44 20 20 
 
 1460 328 1314 
 
 182 10 12 12 
 
 16/42 10 3942 3942)15768(4 yrs. Ans. 
 
 20 s. - 15768 
 
 430 14 amount - 
 
 s. 8/50 365 principal : : 
 
 - 
 
 - 65 14 interest 
 
 d. 6/00 - 
 
 (36) In what time will 450?. amount to 540?. at 4 per 
 cent, per annum ? Ans. 5 years 
 
 (37) In what time will 525?. amount to 708?. 15s. at 5 
 per cent, per annum ? Ans. 7 years 
 
 VII. When the principal, amount, and time are given, to 
 find the rate per cent. 
 
 RULE. As the principal is to the interest for the whole 
 time, so is 100?. to its interest for the same time. Divide 
 that interest by the time, and the quotient will be the rate 
 per cent. 
 
 EXAMPLES. 
 
 (38) At what rate per cent, will 365?. amount to 430?. 14s. 
 in 4 years' time ? 
 
 s. s. 
 
 430 14 As 365 : 65 14 : : 100 
 
 365 20 
 
 65 14 interest 1314 
 
 . 100 
 
 365 ) 131400 ( 360*. = 18Z. and 
 
 181. -7-1 yrs. := 4 per cent. 
 
86 Commission. 
 
 (39) At what rate per cent, will 450/. amount to 540Z. in 
 5 years' time ? Ans. 4 per cent 
 
 (40) At what rate per cent, will 525Z. amount to 708Z 155. 
 in 7 years' time ? Ans. 5 per cent. 
 
 COMMISSION. 
 
 COMMISSION is an allowance of so much per cent 
 from merchants to their factors, for the buying or selling of 
 goods. The term is also applied by bankers to drawing bills 
 and managing accounts. 
 
 EULE I. If the commission be above one per cent, mul- 
 tiply the principal by the rate per cent (as in interest), aud 
 divide by 100. 
 
 2nd. If under one per cent, divide the given sum by 100, 
 and take aliquot parts from the quotient, with the commis- 
 sion. 
 
 EXAMPLES. 
 
 (1) "What does the commission come to on 845 1. ISs. GJ. 
 at 3 1 per cent ? 
 
 s. d. 
 
 i 1 4 | 845 18 6 
 3 
 
 2537 15 6 
 
 i | | 422 19 3 
 
 211 9 7 
 
 100)31/72 4 4* 
 
 Ans. 31 14 
 
 (2) What must I allow my correspondent for disbursing 
 on my account, 389Z. 17s. 9d. at 2J per cent ? 
 
 Ans. SI. 15s. 5ltL 
 
 (3) What is the commission of 768Z. 1.2s. Gd. at 2J per 
 cent? Ans. 191. 4s. 3JJ. 
 
 (4) My correspondent writes me word that he has bought 
 goods on my account to the value of 890Z. 10s, 4<d. ; what 
 does his commission come to at 2| per cent ? 
 
 Ans. 24:1. 9s. 
 
Brokerage. 
 
 8? 
 
 (5) If I allow my factor f 
 per cent., what will be the 
 commission for disbursing on 
 my account 385L 10s. ? 
 
 s. d. 
 
 I 385 10 
 
 j 192 15 
 
 48 3 9 
 
 1,00)6,26 8 9 
 6 5 
 
 Ana. 
 
 (6) What will the commis- 
 sion of a country banker 
 amount to on 1650/. at f per 
 cent. ? 
 
 1,00)16,50 
 t 
 
 16 10 
 
 4 2 
 
 2 1 
 
 Or thus : 
 16 10 
 3 
 
 3 8 | 4J 10 
 
 Ans. 6 3 9 
 
 3 9 
 
 (7) "What will be the commission of 1000Z. at J per cent. ? 
 
 Ans. 21. 10s. 
 
 (8) What will a banker's commission for 7860Z. 16s. lOd. 
 amount to, at f per cent. ? Ans. 58Z. 19s. \\d. 
 
 (9) Suppose I allow my correspondent If for his commis- 
 sion, what will it amount to for disbursing on my account 
 758J. 18s. ? Ans. Wl. 8s. 
 
 BROKERAGE. 
 
 BROKEEAGE is a small allowance per cent, to a person 
 called a broker, for assisting merchants or factors in buying 
 or selling of goods. 
 
 RULE. The same as for Commission. 
 EXAMPLES. 
 
 (1) What is the brokerage 
 of 562Z. 10s. at 6s. Qd. per 
 
 cent. 
 
 8. 
 
 1,00)5,62 10 
 
 (2) If I allow a broker f per 
 cent., what will his brokerage 
 come to on 1456?. 12s. Qd. ? 
 
 s. d. 
 1,00)14,56 12 6 
 
 14 11 3| 
 G 
 
 7)87 7 10& 
 Ant. 12 i) 8 
 
 N.B. When aliquot p t arts are not 
 easily found, multiplying by the 
 upper figure and dividing by the 
 lower, will give the answer. 
 
88 
 
 Purchasing of Stocks. 
 
 (3) What is the brokerage of 487Z. 18s. at 12s. 6d. per 
 cent. ? Ans. 31. Os. 11 jrf. 
 
 (4) Find the brokerage of 1350Z. 16s. Sd. at 2s. 9d. per 
 cent. ? Ans. 1Z. 17s. If cZ. 
 
 (5) If I allow a broker f per cent., what will his brokerage 
 come to on 964Z. 14s. ? Ans. 51. 15s. 9d. 
 
 (6) A broker sold goods to the amount of 525Z. 12s., what 
 will his brokerage come to at 2f per cent.? Ans. 12Z. 9s. 7jd. 
 
 (7) If a broker sells goods to the amount of 1000 guineas, 
 what is his demand at If per cent. ? Ans. 18Z. 7s. Qd. 
 
 (8) What is the claim of a broker, at J per cent., on 
 1760Z. 12s. ? Ans. 4Z. 8s. 
 
 PURCHASING OF STOCKS. 
 
 STOCK is a name given to the money borrowed by govern- 
 ment ; and also to the property of our trading companies. 
 The rules for buying or selling shares in these stocks are as 
 follow : 
 
 RULE 1st. If the sum given is above par (i. e. above 100), 
 multiply the sum to be purchased by the excess above 100 ; 
 divide the product by 100, and add the quotient to the given 
 sum. 
 
 2nd. If the sum given is under par, multiply it ly the 
 price ; and that product divided by 100, will give the answer. 
 
 Or, 3rdly, Instead of multiplying, take parts for the whole 
 price. 
 
 EXAMPLES. 
 
 (1) "What is the purchase of 736Z. 10s. South Sea Stock 
 at 111 | per cent. ? 
 
 Or take the parts thus : 
 8. 
 
 
 736 
 
 s. 
 
 10 
 11 
 
 O i-l |i-i|w 
 i l 
 
 A 
 
 V 
 
 Or thus : 
 s. 
 736 10 
 73 13 
 7 7 
 1 16 
 18 
 
 9| 
 
 8101 
 184 
 
 92 
 
 10 
 2 6 
 1 3 
 
 736 10 
 
 73 13 
 
 9 4 
 
 18 
 
 H 
 4f 
 
 1,00)83,7713 9 Ans. 820 5 6 
 
 83 15 6 
 736 10 
 
 Ans. 820 5 6j 
 
 Ans. 820 
 
Insurance. 
 
 83 
 
 (2) Bought 782Z. 16^. 6d. Bank annuities, at 911 ?er cent., 
 \vlutt did it cost me ? 
 
 s. d. 
 
 782 16 6 
 
 7045 8 6 9 
 10 
 
 70454 5 90 
 
 782 166=1 
 
 97 17 Of = i 
 
 1,00 ) 713,34 18 6J 
 Ans. 713 6 llf 
 
 Or thus : 
 
 
 
 s. d. 
 
 50 
 
 782 
 
 16 6 
 
 25 
 
 1 
 
 391 
 
 8 2 
 
 10 
 
 A. 
 
 195 
 
 14 1$ 
 
 5 
 
 | 
 
 78 
 
 5 7f 
 
 1 
 
 -i 
 
 39 
 
 2 9| 
 
 k 
 
 i 
 
 7 
 
 18 64 
 
 
 19 6| 
 
 Ans. 713 6 
 
 (3) What is the purchase of 1340Z. 12s. East India stock, 
 at 110 per cent. ? Ans. 1478Z. Os. 2 Jrf. 
 
 (4) Sold 2365Z. 18s. 6d. India stock, at 104f per cent., 
 what sum did I receive ? Ans. 2475Z. 6s. 11 \d. 
 
 (5) Bought 758Z. 18s. Three per cent, consolidated annui- 
 ties, at 88 per cent. ? Ans. 669Z. 14s. 7d. 
 
 (6) What is the purchase of 1000Z. Consols, at 84f per 
 cent. ? Ans. 843Z. 15s. 
 
 (7) At 103f per cent., what is the purchase of 5620Z. Three 
 per cent, reduced annuities ? Ans. 5809Z. 13s. Qd. 
 
 INSURANCE. 
 
 1NSUEANCE is an allowance per cent, paid by the pro- 
 prietors of goods, &c., to certain persons or offices, who 
 engage to make good the loss of ships, houses, goods, &c., 
 which may happen by storms, fire, or other accidents : the 
 security the insurer receives, as well as the premium he pays, 
 is also called INSURANCE. 
 
 RULE. Multiply the sum to be insured by the rate, and 
 the product divided by 100 is the per centage to be paid. 
 Or, instead of multiplying, take parts for the rate. 
 
90 Insurance. 
 
 EXAMPLES. 
 (1) What is the insurance of 858Z. 125. Qd. at 15f per cent, f 
 
 *. d. 
 
 i|4| 658 12 6 
 
 15 
 
 Or thus, by taking parts of the rate : 
 
 9879 7 6 
 
 i | i | 329 6 3 
 
 82 6 6| 
 
 1,00)102,91 3j 
 
 Ans. 102 18 2 
 
 s. d. 
 658 12 6 
 
 65 17 3 
 
 32 18 7i 
 
 4 2 3| 
 
 : 102 18 2J 
 
 (2) "What is the insurance of 850/. at 12f per cent. ? 
 
 -4w*. 108Z. 75. 6d 
 
 (3) What is the insurance of a ship's cargo valued at 
 24630Z. at 18 per cent. ? Ans. 4494Z. 195. Qd. 
 
 (4) What is the insurance of 1000Z. at 6| per cent. ? 
 
 Ans. G3L 15s. 
 
 (5) What is the insurance of 1784Z. 12s. at If per cent.? 
 
 Ans. 281. 19s. llf d. 
 
 (6) What is the insurance of 364Z. 155. at 35. Sd. per cent., 
 and at 145. Sd. ? 
 
 s. d. 
 1,00 ) 3,64 15 
 
 3,-. 4rf. 
 id. 
 
 3 12 11J 
 
 12 2 
 1 2jr 
 
 5. 13 4 TCgarZi/ 
 
 1,00 ) 3,64 15 
 10s. i 3 12 11 
 
 1 16 5J 
 14 7 
 2 5 
 
 rf. Ans. 2 13 5 
 
 (7) What is the insurance of 584Z. 165. 6^. at 25. 9d. per 
 cent. ? Ans. 165. Of d. 
 
 (8) Required the insurance of 1234Z. at 35. 4d. per cent. ? 
 
 Ans. 21. Is. l\d. 
 
 (9) What is an underwriter to receive for insuring 5000Z. 
 at 10 guineas per cent. ?* Ans. 5251. 
 
 * When the premium is at so many guineas, work as for pounds, and 
 add a twentieth part to the answer. 
 
(01 ) 
 COMPOUND INTEREST 
 
 COMPOUND IJSTTEEEST is that which arises both from 
 the principal and interest ; that is, the interest due at each 
 payment is added to the principal, to bear interest for the 
 next payment. 
 
 EULE. Find the amount of the principal for the time of 
 the first payment, by Simple Interest. Call this amount the 
 principal for the second payment, and find its amount as 
 before ; and so on for the number of payments required. 
 
 2nd. Subtract the first principal from the last amount, 
 and it will give the compound interest required. 
 
 3rd. When the interest is half -yearly, quarterly, &c., find 
 the interest for one payment more than the number given ; 
 and take the parts from that payment, which add to the sum 
 before found. 
 
 EXAMPLES. 
 (1) "What is the amount of 2501. for 3 years at 5 per cent. ? 
 
 
 
 250 
 5 
 
 8. 
 
 250 0:=ls yr.'s prin, 
 12 10 1st yr.'s interest 
 
 8. 
 
 262 10 
 13 2 
 
 d. 
 
 Q~2nd yr.'s prin. 
 6 2nd yrs. int. 
 
 3?v/ 
 awi. 
 
 12/50 
 20 
 
 262 10 2nd yr.'s prin. 
 5 
 
 275 12 
 
 6 3rd! yr.'s prin 
 5 
 
 10/00 
 
 13/12 10 
 20 
 
 13/78 2 
 20 
 
 6 
 s. d. 
 275 12 6 
 13 15 7J 
 
 2/50 
 12 
 6/00 
 Or thus : * 
 
 15/62 
 12 
 
 7/50 
 
 29 8 li- 
 jjr.'* 
 
 s. d. 2/00 
 
 | 250 Q=:given principal 
 
 12 10 01 year's interest 
 
 | 262 10 0=27K/ year's principal 
 
 13 2 6 2nd year's interest 
 | 275 12 6 
 
 13 15 7k 
 
 Ans. 289 8 
 
 (2) "What is the amount of 384Z. 10s. for 5 years, at 4 per 
 cent, per annum ? Ans. 4G7Z. 16s. Od. 
 
 * N. B. When the interest is any ecual part of a hundred, as 5 per 
 cent, is -$, 4 per cent. -%, &c., the answer may be more expeditiously 
 found by successive divisions. 
 
92 Compound Interest. 
 
 (3) "What is the amount of 1000Z. for 3 years, at 5 per 
 cent, per annum ? Ans. 1157Z. 12s. Qd. 
 
 (4) Required the amount of 7501. 16s. Sd. for 4 years, at 
 4^ per cent, per annum ? Ans. 8951. 7s. l\d. 
 
 (5) What is the amount of 1001. for 3 years, at 3^ per cent. 
 per annum ? Ans. 110Z. 17s. 5d. 
 
 (6) What is the amount of 5701. 10s. for 3| years, at 4J 
 per cent, per annum ? 
 
 8. s. d. 8. d. 
 
 I | 570 10 570 10 596 3 5i 
 
 4 25 13 5| 26 16 6 
 
 2282 | 596 3 5J | 622 19 llf 
 
 285 5 4 4 
 
 25/67 5 2384 IS 9 2491 19 11 
 
 20 298 1 8 311 9 11} 
 
 45 26/82 15 5^ 28/03 9 10} 
 
 12 20 20 
 
 5/40 16/55 /69 
 
 4 12 12 
 
 1/60 6/65 8/38 
 
 - 4 4 
 
 2/62 1/55 
 
 8. d. 
 
 622 19 11| 
 28 8 
 
 * i 651 8 8. d. 
 
 4 i | 29 5 11 
 
 2604 28 7 6 5| 
 
 325 10 4 651 8 
 
 29/29 13 Ans. 658 7 If 
 
 20 
 
 5/93 
 12 
 
 11/16 
 
 (7) What is the amount of 500Z. for 4J years, at 4 per 
 cent, per annum ? Ans. 596Z. 12s. 6JdL 
 
 (8) What is the amount of 368Z. 12s. for 3f years, at 3 
 per cent, per annum ? Ans. 41U. 16s. 
 
Compound Interest. 93 
 
 (9) What is the compound interest of 400Z. for 2 years 10 
 months and 15 days, at 3 per cent, per annum ? 
 
 \ 
 
 
 
 | 400 
 3 
 
 1200 
 200 
 
 14/00 
 
 
 
 400 
 14 
 
 
 
 414 
 14 
 
 d. 
 
 9| 
 
 | 414 I] 428 9 
 
 1242 1285 9 4 
 207 214 4 lOf 
 
 14/49 
 20 
 
 9/80 
 12 
 
 9/60 
 2/40 
 
 14/99 14 
 20 
 
 19/94 
 12 
 
 11/31 
 1/25 
 
 s. d. 
 14 19 11 
 
 I 7 9 11| 
 \ 3 14 llf 
 1 17 5| 
 
 13 2 5 
 
 Amount, 441 12 2| 
 rw. 400 
 
 int. 41 12 
 
 Ans. 
 
 (10) "What is the compound interest of 100Z. for 3 years 
 4 months and 10 days, at 4 j per cent, per annum ? 
 
 Ans. 151. 19s. 4fd. 
 
 (11) What is the compound interest of 765Z. 10s. for 5J 
 years, at 4 per cent, per annum ? Ans. 184Z. 9s. 5id. 
 
 (12) What is the compound interest of 250Z. for 2 \ years, 
 at 4 per cent, per annum, payable half-yearly ? 
 
 s. d. 
 50 ) 250 the given principal 
 
 first half -year's interest 
 
 
 5 
 
 
 
 
 
 50) 
 
 255 
 5 
 
 
 2 
 
 , 
 , 
 
 50) 
 
 260 
 5 
 
 2 
 4 
 
 00 
 
 50 ) 
 50) 
 
 Ans, 
 
 265 
 5 
 
 6 
 6 
 
 \i 
 
 270 
 5 
 
 12 
 
 8 
 
 H 
 H 
 
 276 
 250 
 
 
 
 
 4, 
 
 j>; 
 
 , 26 
 
 
 
 4 
 
 second half-years principal 
 second half-year's interest 
 third half-year's principal 
 
 third half-year's interest 
 fourth half-year's principal 
 \\fourth half-year's interest 
 lyifth half -year's principal 
 Z^fifth half -year's interest 
 4 five half-years' amount 
 first principal, subtracted 
 
 N.B. 4 per cent, for 
 a year is -^5- of a hun- 
 dred consequently, 
 per half-year will be 3^. 
 [See the note to the first 
 sum in this rule.] 
 
 (13) Find the compound interest of 280Z. 10s. for 1 J year, 
 at 5 per cent, payable half-yearly ? Ans. 21Z. 11s. 4</. 
 
 (14) What is the compound interest of 760Z. 15s. for 2 yrs. 
 payable half-yearly, at 4 per cent, per ann. ? A. 62Z. 14s. 2<& 
 
9i Compound Interest. 
 
 (15) "What is the amount of 100Z. payable quarterly, sup 
 posing it to have been forborne 2 years, at 3 per cent. ? 
 
 
 
 8. 
 
 s. d. 
 
 s. d. 
 
 100 
 
 100 
 
 100 15 
 
 101 10 1| 
 
 3 per 
 
 cm. 15 
 
 15 1J 
 
 15 24 
 
 \ 3,00 
 
 100 15 
 
 amount 101 10 1 ami 
 
 !. 102 5 31 am* 
 
 /7K 
 
 3 
 
 3 
 
 3 
 
 //o 
 20 
 
 4)302 5 
 
 4 ) 304 10 3| 
 
 4 ) 306 15 11 
 
 15/00 
 
 /75 11 
 
 OA 
 
 3 /76 2 6| 
 
 OA 
 
 /76 13 11| 
 
 OA 
 
 
 15/11 
 
 15/22 
 
 15/33 
 
 
 12 
 
 12 
 
 12 
 
 
 1/35 
 
 2/70 
 
 4/07 
 
 
 4 
 
 4 
 
 -- 
 
 
 1/40 
 
 2/83 
 
 
 8. 
 
 d. 
 
 a. d. s. d 
 
 s. d. 
 
 102 5 
 
 3f 103 
 
 7| 103 16 1 
 
 104 11 7| 
 
 15 
 
 4 
 
 15 5i 15 6| 
 
 15 8| 
 
 103 7fom. 10316 1 ami. 104 11 7famf. 105 7 4 a 
 3333 
 
 1)309 1 11J 
 
 4)311 8 
 
 3 4)313 14 Hi 
 
 4)316 2 
 
 
 
 77 5 5| 
 20 
 
 /77 17 
 20 
 
 Of /78 8 81 
 20 
 
 /79 
 20 
 
 
 15/46 
 12 
 
 15/57 
 12 
 
 15/68 
 12 
 
 15/80 
 12 
 
 
 5/45 
 
 6/84 
 
 8/24 
 
 9/66 
 4 
 
 
 1/83 3/39 /99~j, nearly 2/64 
 
 a. d. 
 
 105 7 4 
 15 9i 
 
 . 106 3 1 \ amount of 2 i/ca?-s by quarterly paymtnts 
 
 (16) What is the amount of 50Z. payable quarterly, sup- 
 posing it to have been forborne 2J years, at 3J per cent, per 
 annum ? Ans. 5M. 10s. lV 4 d. 
 
DISCOUNT. 
 
 DISCOUNT is the allowance made to a person for pay- 
 ing money before it is due ; and is so much as that money, 
 if put to interest, would gain in the same time and at the 
 same rate. 
 
 Thus 100Z. present money, will discharge a debt of 105 1. 
 to be paid a year to come, rebate being made at 5 per cent. 
 
 The present worth, then, is the sum to be paid when the 
 discount is taken off. 
 
 E/ULE 1st. When the present worth is required ; say, As 
 the amount of 100 1. for the given rate and time is to 100Z., 
 so is the sum given to the present worth. 
 
 2nd. When the rebate or discount is required ; say, As the 
 amount of 100Z. for the given rate and tim^. is to its interest, 
 so is the given sum to its discount. 
 
 EXAMPLES. 
 
 (1) What is the present worth of 3GOZ. 10s. for 11 months, 
 at 6 per cent. ? 
 
 m. s. s. s. 
 
 61 6 Theu, As 105 10 : 100 :: 36010 
 
 20 20 
 
 130 
 
 t 2 2110 7210 
 
 10 100 
 
 s. d. 
 
 5 10 211,0)72100,0(341 14 U 
 
 100 633 
 
 105 10 amount oflQQl 880 
 
 == 844 
 
 360 
 211 
 
 149 
 Ans. 341Z. 14*. \\d. present worth 20 
 
 211)2980 (14*. IK 
 &c. 193 rem. 
 
 (2) "What is the present worth of 365 1. 10s. for 7 months, 
 at 44 per cent, per annum ? Ans. 356Z. 3s. 51 rem. 
 
 (3) "What is the present worth of 465Z, 12s. for 6 months, 
 at 3^ per cent, per annum ? Ans. 4*571. lls. 10^. 230 rem. 
 
96 Discount. 
 
 (4) Sold goods for 384Z. 15s. to be paid 10 months henc 
 what is the present worth at 6 per cent, discount ? 
 
 Ans. 366Z. 8*. 6fd. 45 rem. 
 
 (5) How much ready money can I receive for a note of 
 150Z. due 18 months hence, at 5 per cent. ? 
 
 Ans. 139Z. 105. 8^. 105 rem. 
 
 (6) What is the present worth of 2101. payable in a quarter 
 of a year, discounting at 4^ per cent. ? 
 
 Ans. 2071. IBs. 3J(J. 187 rem. 
 
 (7) What is the discount of 750Z. for 1 year and 9 months 
 at 4J per cent. ? 
 
 m. s. 
 
 4 10=1 year 
 
 25 s. d. 8. d. 
 2 6 Then, As 107 17 6 : 7 17 6 :: 750 
 20 20 20 
 
 7 17 6 
 
 100 2157 157 15000 
 12 12 12 
 
 107 17 6 
 
 25890 1890 180000 
 180000 
 
 Or, as 17*. 6d.=J of a . say 
 
 As 107| : 7| : 750 151200000 
 8 8 1890 
 
 63 2589,0 ) 34020000,0 ( 12 ) 13140 
 750 2589 
 
 2,0)109,5 
 
 863 ) 47250 ( 54Z. 15s. Ans. 813 &c. 
 15 rem. 540 rem. 
 
 Ant. 5415 
 
 (8) What is the discount of 120Z. 10s. Qd. for 1J year, at 
 4J per cent. ? Ans. 61. 8s. 4id. 1905 rem. 
 
 (9) Find the discount of 150Z. due 2 years hence, at 5 per 
 cent. ? Ans. 13Z. 12s. &%d. nearly. 
 
 (10) Sold goods to the value of 300Z. to be paid in 18 
 months, what would be the discount for present payment at 3^ 
 per cent. ? Ans. 14Z. 19s. Z\d. 1675 rem. 
 
 (11) What is the discount of 500Z. for 12 months, deduct- 
 ing 6 per cent. ? Ans. 28Z. 6s. 0^. 86 rem. 
 
 (12) Sold goods to the value of 1500Z. to be paid in 15 
 months, what must be the discount for present payment at 12 * 
 per cent. ? Ans. 2021. 14s. 0d. 1650 rem. 
 
Discount. 97 
 
 (13) "What is the present worth of 150Z. payable as follows 
 501. at 3 months, 501. at 6 months, and 50/. at 9 months, 
 discounting at 6 per cent. ? 
 
 
 
 4 1 
 
 s. 
 Then, As 101 10 : 
 20 
 
 2030 
 
 
 : 100 :: 50 
 20 
 
 1000 
 100 
 
 1 10 
 100 
 
 101 10 
 
 2030 \ 100000 
 
 8120 
 
 49Z. 5*. 2id. present worth 18800 
 
 18270 
 
 ^ _^____ 
 
 530 &c. 
 
 m. 
 
 6 | J | 6 Then. As 103 . 100 : : 50 
 
 50 
 
 o 
 
 100 103 ) 5000 ( 48Z. 10s. 10d. present worth 
 
 103 
 
 m. s. 
 6 Then, As 104 10 . 100 : : 50 
 20 1000 20 
 
 3 1000 
 
 1 10 2090 2090 ) 100000 ( 471 16s. lld. present 
 
 . worth 
 
 4 10 s. d. 
 100 49 5 24 
 
 48 10 10 
 104 10 47 16 ll| 
 
 Ans. 145 13 
 
 (14) "What is the present worth of 120/. 10s. due 1st May, 
 this being Jan. 1st, reckoning interest at 5 per cent ? 
 
 Ans. 118Z. 10s. 5id. 
 
 (15) Sold goods to the amount of 1000/. due ^ at 6 mouths 
 and J at 12 months ; required the discount at 10 per cent ? 
 
 Ans. 691. 5*. 3^. 
 
 (16) "What is the present worth of 750Z. payable one-third 
 at 4 months, one-third at 8 months, and one-third at 12 
 months, reckoning the discount at 7^ per cent ? 
 
 Ans. 714/. 11s. L/. 
 
(98) 
 
 EQUATION OP PAYMENTS. 
 
 
 EQUATION OF PAYMENTS is a rule for finding the 
 equated time to pay at one payment several sums due at 
 different times. 
 
 RULE. Multiply each payment by the time at which it 
 becomes due ; add the products together, and divide their 
 sum by the sum of the payments ; the quotient is accounted 
 the mean time.* 
 
 EXAMPLES. 
 
 (1) A owes B 560Z. ; of 
 which 2001. is to be paid at 
 3 months, 2001. at 5 months, 
 100/. at 6 months, and the 
 rest at 9 months ; I demand 
 the equated time for the 
 whole payment. 
 
 200x3= 600 
 
 200x5=1000 
 
 100x6= 600 
 Therest60x9= 540 
 
 56,0 ) 274,0(4mo. 26 days 
 224 
 
 50 
 30 
 
 56 ) 1500 ( 26 days 
 112 
 
 380 
 336 
 
 44 
 
 (2) Y owes Z a certain sum, 
 i of which is to be paid in 2 
 months, J in 3 months, in 
 4 months, ^ in 5 months, and 
 the rest in 6 months, I de- 
 mand the equated time. 
 
 N.B. We may suppose any sum. 
 Suppose 240Z. 
 
 =80x2 = 160 
 $=60x8=180 
 
 1=48x4=192 
 
 1=30x5 = 150 
 
 Therest22x6 = 132 
 
 240 ) 814 ( 3m. 11 days 
 720 
 
 94 
 30 
 
 24,0 ) 282,0 ( 11 day8 
 264 
 
 N.B. The months are multiplied 
 by 30, to bring them into days. 18 
 
 (3) I have to pay 35 Ql. at three payments, viz. : 120 at 3 
 months, 1502. at 6 months, and the rest at 9 months ; for 
 what length of time must a single note be given, to pay the 
 whole at once ? Ans. 5 mo. 21 days 144 rem. 
 
 (4) I have to receive 684Z. in notes as follow, viz. : 130/. at 
 2 months, ISO/, at 3 months, 3001. at 4 months, and 742. at 
 5 months, but preferring to have the whole in one note, for 
 what time must it be given ? Ans. 3 mo. 13 da. 648 rem. 
 
 * Though accounted the mean time, and deemed near enough for 
 business, yet if the learner's judgment be sufficiently mature, he may 
 be shown, that as some of the debt is paid before and some after it 
 is due, for the one interest is reckoned, but for the other only discount; 
 on which account the rule is not mathematically correct. 
 
Equation of Payments. 99 
 
 (5) A owes B a certain sum, \ of which is to be paid in 
 4 months, J in 6 months, and the rest in 8 months ; but they 
 agree that the whole shall be paid at one equated time : 
 what is that time ? Ans. 6 months. 
 
 (6) Bought goods to the value of 7501. which were to have 
 been paid for as follows : 2201. in 3 months, 350?. in 4 months, 
 and the rest in 6 months ; but afterwards agreeing to make 
 but one payment of the whole, I demand what that time 
 must be ? Ans. 4 mo. 5 days 450 rem. 
 
 (7) A debt is to be paid as follows : viz. J at 3 months, 
 \ at 4 months, 3 at 5 months, and ^ at 6 months, and the 
 rest at 7 months ; what is the equated time for the whole ? 
 
 Ans. 4 mo. 13 days 
 
 (8) I have one bill of 436/. 12s. 6d. payable in 75 days, 
 one of 284Z. 10s. 9d. payable at 66 days ; and one of '3351. 
 16s. &d. payable in 90 days ; if I receive one bill for the 
 whole, what must be the date ? 
 
 436 12 6x75 = 32746 17 6 
 
 284 10 9x66 = 18779 9 6 KB. When either the time or 
 
 335 16 8x90 = 30225 the debts are of differ entdenomina- 
 
 1056 19 11 81751 7 tion8 > as months, weeks, or days, 
 
 2Q 20 or & St d-, they may be reduced 
 
 to tlie same denomination, before 
 the several operations take place. 
 
 253679 253679 ) 19620324 ( 77 days. Ans. 
 
 (9) I have in my possession one bill for 123Z. 10s. 4cZ. due 
 in 55 days ; one for 99/. 8s. Qd. due in 60 days ; and one of 
 100/. due in 30 days ; at what date ought one bill to be 
 given for the whole sum ? Ans. 48 days 
 
 (10) Bought a quantity of goods to the value of 756/. 
 16s. 3d. for which I gave the following bills : viz. 1201. 10s. 
 at 90 days ; 2001. 14s. Qd. at 75 days ; 300?. at 60 days ; 
 and the rest at 30 days ; I demand the equated time for the 
 whole at one payment ? Ans. 63 days 
 
 (11) A debt of 15001. is to be paid as follows: viz. J at 6i 
 months : J at 12^ months ; and the rest in 1 year 6 mo. and 
 15 days : what is the equated time for the whole payment ? 
 
 1500 Note. 1 year 6 mo. 15 days = 18 | mo. 
 
 375 x 64 = 2437 10 Each line is multiplied thus: 
 
 750x124 = 9375 4 j 375 
 
 Therest375 x 18 = 6937 10 6 
 
 _ _ 
 150,0 ) 1875,0 0( 12mo. 15 days. Ans. 2250 
 
 187 10 
 2437 10 
 F 2 
 
100 
 
 Barter. 
 
 (12) A owes B 1000Z. to be paid os follows: 200Z. at 4 
 months ; BOOL at 8 months ; 2001. at 12 months ; 200J. at 15 
 months ; and the rest at the end of two years : the equated 
 time for one payment is required ? Ans. 11 months 
 
 (1.3) A person has owing to him 36/. 10s. to be paid in 3J 
 months ; 48. 12s. to be paid in 6J months ; and 100L pay- 
 able in 8 1 months ; what would be the equated time for the 
 payment of the whole ? Ans. 6 mo. 29 days 
 
 (14) A owes B a certain sum, of which ^ is to be paid in 
 4 months ; in 6 months ; J in 8 months ; in 10 months ; 
 and the rest in 12 months : I demand the equated time ? 
 
 Ans. 6 mo. 23 days 
 
 BARTER. 
 
 BY this rule traders are directed how to exchange one 
 commodity for another, so that neither party may sustain 
 
 RULE. Find the value of that commodity whose quantity 
 is given ; then find what quantity of the other at the rate 
 proposed, may be had for the same money. This is done by 
 dividing the value of the quantity exchanged, by the price of 
 a unit returned. 
 
 EXAMPLES. 
 
 (1) How many yards of 
 cloth at Gs. per yard, must be 
 delivered in barter for 99 Ib. 
 of tobacco, at 4s. per Ib. ? 
 
 Ib. 
 
 99 
 
 4 
 
 6)396 
 Ans. 66 yards of cloth. 
 
 N.B. Here the value of the to- 
 bacco is divided by the price of one 
 yard, which gives the answer. 
 
 (2) What quantity of cho- 
 colate at 4s. Qd. per Ib. must 
 be given in barter for 2 cwt 
 1 qr. 13 Ib, of tea at 7s. per 
 Ib. ? 
 
 cwt.qr. Ib. Ib.. ,,, n . 
 2 1 13 = 265 45 ' 6 ^ = 9m% 
 14 sixpences in 7*. 
 9)3710 8J 02. 
 28)412(4)14 
 28 
 
 132 
 112 
 
 3 2 20 3f 
 
 Ans. 3 cwt. 2 qr. 20 Ib. 3 oz. 
 
 N.B. As the divisor must be 
 brought into the lowest name 
 mentioned (sixpences) so must the 
 dividend. 
 
Barter. 101 
 
 (3) How much cloth at 7s. Qd. per.,y^r(J, *nr<Ast be given in 
 b irter for 84 reams of paper, at 1'. 12s^ Qd. per ream ? 
 
 -4ns. 3G4 yds. 
 
 (4) How much cheese at 21. 7s. Qd. per cwt. must be bar- 
 tered for 20 cwt. of hops, at 51. Us. 7d. per cwt. ? 
 
 Ans. 47 cwi. 
 
 (5) How much tobacco at 6Z. 18s. 6 d. per cwt is eq ual in 
 value to 5 cwt. 3 qr. 14 Ib. of snuff, at 4s. Qd. per Ib. ? 
 
 Ans. 21 cwt. 1 qr. 14 Z. 756 rem. 
 
 (6) How many dozens of wine, at 21. Ss. 4<d. per dozen, 
 must be received in exchange for 3 puncheons of rum, at 
 60Z. 8s. 4<d. per puncheon ? Ans. 75 dozens 
 
 II. When part of the value is returned in cash, and the 
 remainder in goods. 
 
 HTJLE. Deduct the cash from the value of the given com- 
 modity; and then work for the remaining commodity as 
 before. 
 
 (7) How many yards of velvet, at 9s. 8d. per yard, must I 
 give, with 261. Ss. Sd. in cash, for 50 gallons of Geneva, at 
 \8s. Qd. per gallon ? 
 
 105. 
 
 U. 
 
 4s. 
 Qd. 
 
 50 . *. d. 
 
 19 16 4 
 25 20 
 
 10 
 
 10 9s.8d 396 
 
 1 5 12 12 
 
 46 5 116 ) 4756(410oH*. Ans. 
 
 Deduct the Cash 26 8 8 464 
 
 19 16 4 116 
 
 = 116 
 
 (8) A sold to B 30 cwt. of rice, at 21. 4s. per cwt. for 
 which B returned him 15Z. 11s. 8d. in cash, and the rest in 
 serges, at 4s. 2d. per yard ; how many yards did A receive ? 
 
 Ans. 242 yds. 
 
 (9) Bought 12 quarters of wheat, at 21. 16s. per quarter, 
 for which I paid in cash 13Z. 12s. and the remainder in beans 
 at 5s. per bushel, how many bushels had I to return ? 
 
 Ans. 80 lush. 
 
 (10) A and B barter. A has 84 galls, of brandy at 18s. Qd. 
 per gallon, for which B gives him 30Z. in money, and the 
 rest in raisins at 9d. per Ib. ; what quantity of raisins must 
 A receive ? Ans. 1272 Ib.or 11 cwt. 1 qr. 12 Ib. 
 
102 Barter. 
 
 
 (11) A has a quantity of pepper, weighing neat 1800 Ib. 
 at IQd. per lb v .for which Z gives him SOL in money, and the 
 ,i?est in goods at-9^. pe^ib.'; how many pounds weight must A 
 receive ? Ans. 1578if Ib. or 1579 Ib. nearly 
 
 (12) Eeceived in barter 1200 yards of linen at 3s. 4J. per 
 yard, and returned 84 Ib. of tea at 6s. Sd. per Ib. and the 
 rest in wine at 40s. per dozen ; the quantity of wine is re- 
 quired.* Ans. 86 doz. of wine 
 
 III. The rate of one commodity being given, to find now 
 ike other should be rated. 
 
 RULE. Divide the value of the one commodity by the 
 quantity of the other. 
 
 (13) Bartered 2 pieces of cloth containing 64 ells, at 7s. Gd. 
 per ell, for 4 yards of velvet ; I demand what the velvet was 
 rated at per yard ? 
 
 fa 
 
 2s. 6d. 
 
 64 24 
 
 20 
 
 16 
 
 8 4,0)48,0 
 
 24 Ans, 12s. per yard 
 
 Received of A 12 cwt. 2 qrs. of cheese, at 21. 12s. IcL 
 per cwt. and returned him as an equivalent a tierce (42 
 galls.) of rum ; I demand the value of the rum per gallon ? 
 
 Ans. 15s. 6d. 
 
 (15) X sends to T 260 yards of drugget, and receives in 
 return 4 cwt. 3 qrs. of hops, at 4Z. 2s. Gd. per cwt. ; the price 
 of the drugget per yard is required ? Ans. Is. Qd. f/- per yd. 
 
 (16) C delivered 84 gallons of brandy, at 25s. per gallon, 
 to D, for 450 yards of cloth ; what was the cloth per yard ? 
 
 Ans. 4s. Sd. per yd. 
 
 IY. When the ready money price of one commodity has 
 been raised in barter, to find how to raise the other in pro- 
 portion. 
 
 RULE. As the ready money price of the one commodity 
 is to its bartering price, so is the ready money price of the 
 oilier to its bartering price. 
 
 * Ex. 12. In this case we deduct the 281. for the tea, as wo did before 
 for cash. 
 
Profit and Loss. 103 
 
 (17) A has wines at 48s. per doz. ready money, but in 
 barter advances it to 54s. ; B has brandy at"24s. per gallon 
 ready money ; how much must B raise his brandy per gall. 
 to be equivalent to A's ? 
 
 A* 48 : 24 :, 54 
 54 
 
 96 
 120 
 
 48)1296(27*. per gall. An*. 
 96 
 
 336 
 336 
 
 (18) A tradesman has velvets at 10s. Gel. per yard ready 
 money, which he raises to 12s. in bartering with me for 
 Welsh flannels, which. I sell at Is. Qd. per yard ready 
 money ; how must I rate them per yard in barter to be an 
 equivalent to the velvet ? Ans. 2s. per yard 
 
 (19) Y has linen cloth worth 2s. 6 d. per ell ready money, 
 but in barter he will have 3s. ; Z has broad cloth worth 
 II. 5s. per yard ready money; at what price ought the broad 
 cloth to be rated in barter ? Ans. 30s. per yard 
 
 (20) A merchant with whom I bartered tea for sugar, 
 raised his sugar from Is. to 13| d. per Ib. ; what ought 1 to 
 have charged him for tea which I sold, ready money, for 
 5s. Qd. per ib. ? Ans. 6s. 2d. 
 
 PROFIT AND LOSS. 
 
 PBOFIT and LOSS is a rule that discovers what is 
 gained or lost in the buying or selling of goods ; and also 
 teaches how to raise or fall the price, so as to make a giver 
 gain or lose by them. 
 
 This rule has several variations, but the questions are 
 mostly performed either by the rule of Proportion or Prac- 
 tice. 
 
 EXAMPLES. 
 
 CASE I. To find the WHOLE GAIN* or LOSS on any quan- 
 tity f goods. 
 
104 
 
 ^Profit and Loss. 
 
 (1) Bought 19 cwt. 3 qrs. 14 Ibs. of cheese at 21. 18s. Qd. 
 per cwt., and sold it out at 31. 3s. per cwt., what was the 
 profit on the whole ? 
 
 Or by Proportion. 
 Ib. s. d. cwt. qrs. Ib. 
 As 112 : 4 6 : : 19 3 14 
 4 
 
 79 
 28 
 
 2226 
 54 
 
 s. d. 
 3 3 q 
 2 18 6 S 
 
 4 6 
 
 Then 
 rs. s. 
 
 H* 
 
 by Practice. 
 6 gain per cwt. 
 
 13 
 
 6 
 6 
 
 Igr. 1 
 14 Ib. \ 
 
 4 1 
 4 
 
 9 
 r * 
 i 
 
 
 
 zz 18 cwt. 
 6 zz 1 cwt. 
 3 zz $ 
 
 c|zzl4Z6. 
 
 Ans. 4 9 5 gain in the whole. 
 
 112) 120204(12)1073 
 )28(i 2,0)8,9 
 
 Ans. 4 9 
 
 (2) Bought 7 Ibs. of tobacco for II. Ss. Qd. and sold it for 
 II. Us. 6d. 9 what was the gain per cwt ? Ans. 21. 8s. 
 
 (3) If butter be bought at 9%d. per Ib. and sold at Is. per 
 Ib., what would be gained by 2 cwt. 1 qr. 7 Ibs. at that rate ? 
 
 Ans. 21. 13s. 11 \d. 
 
 (4) Purchased 1000 yards of cloth at the rate of 4s. 6d. 
 per yard, and sold the whole at 56*. 9d. per yard, what was 
 the whole gain ? Ans. 621. 10s. 
 
 (5) Paid 56Z. for one ton of steel, which I retailed at Q^d. 
 per Ib., what was the profit or loss by the sale of 10 tons ? 
 
 Ans. 46Z. 13s. 4<d. gain 
 
 II. To find the SELLING PEICE of goods, at a certain 
 
 GAIN. 
 
 (6) At what price must I sell raisins per cwt. which cost 
 21. 10s. per cwt. to gain 12 per cent. : and also at what rate 
 per Ib. ? 
 
 Then, for the price per Ib. say, 
 
 Ib. s. Ib. 
 
 As 112 : 2 16 : : 1 
 
 First, for the price per cwt. say, 
 
 s. 
 
 As 100 : 2 10 : : 112 
 
 Or thus : 
 
 20 
 
 50 
 112 
 
 1,00)56,00 
 
 2 10 
 5 
 1 
 
 s. 
 2 16 
 
 20 
 
 56 
 12 
 
 2nd Ans. 
 
 2/. 16*. per cwt. 112 ) 672 ( 6d. per Ib. 
 
 56s.zz2Z 16*. per cwt. 1st Ans. 
 
Profit and Loss. 105 
 
 (7) Bought soap at 70s. per cwt. ; at how much per Ib. 
 must I retail it, to gain 10 per cent, profit ? Ans. S^d. 
 
 ($) Purchased cotton stockings at 4s. 2d. per pair ; how 
 must I sell them per pair to gain 20 per cent, profit ? 
 
 Ans. 5s. 
 
 (9) If 107 Flemish ells 1 qr. of Cambric cost 641. 8s. how 
 must I sell it per yard to gain 15 per cent. ? Ans. 18s. 4^d.\ 
 
 (10) Bought sugar at 31. 18s. Gd. per cwt. ; how must I 
 retail it per cwt. to gain 12 per cent. ? Ans. 41. 7s. lid. 6 
 
 III. To sell at a CERTAIN LOSS. 
 
 (11) Bought 120 yards of drugget for 91. 10s. which I find 
 much damaged ; how must I sell it per yard, so as to lose 
 80s. by the whole ? 
 
 s. 
 
 Cost . . . 9 10 
 To lose . . 1 10 
 
 As 120 yds. : 80 : : 1 yd. 
 
 20 
 
 12,0)16,0(1$. 4</. per ycvrd. Ans. 
 12 
 
 4 
 
 (12) I gave 38Z. for 2 cwt. 2 qrs. cP tobacco ; but becoming 
 damaged, at what rate must I sell it per Ib. to lose 101. by 
 the whole ? Ans. 2s. 
 
 (13) Sold bacon at 7JtZ. per Ib. which I bought at 9f d. ; 
 what shall I lose by the sale of 3 cwt. 2 qrs. 12 Ib. ? 
 
 Ans. 3Z. 15s. 9d. 
 
 (14) A quantity of tea cost me 6s. 8^. per Ib. but proving 
 damaged, how must I sell it per Ib. to lose 10 per cent. ? 
 
 Ans. 6s. 
 
 (15) Lost 14 per cent, on pepper, which I bought at 2.?. 2%d. 
 per Ib. how did I sell it per Ib. ? Ans. Is. lO^.nro 
 
 IV. To find the gain or loss per CENT. 
 
 (16) If rum cost 15s. Qd. per gallon, how should it be sold 
 per gallon to clear 15 per cent. ? 
 
 s. d. d. 
 
 100 : 15 6 :: 115 12)213 90 Or thus 
 
 12 4 
 
 17 9| 3/60 
 
 186 
 
 115 Ans. 17*. 
 
 15 6 
 
 
 
 17 9| 
 
 1,00)213/90 ^7 
 
 F 3 
 
106 Profit and Loss. 
 
 (17) How much is gained per cent, at the rate of Is. 8d. 
 in the ? Am. SI. 6s. 8d. 
 
 (18) If 2s. Gd. is gained in a guinea, how much is .that 
 per cent. ? Ans. III. 18s. Id. 2 \ 
 
 (19) Bought coffee at 2s. 2d. per Ib. and sold it at 2s. S^d. 
 per Ib., required the gain per cent. ? Ans. 251. 
 
 (20) If I sell cheese at G%d. per Ib. which cost me l\d. 
 per Ib., what do I lose per cent. ? Ans. 16Z. 2s. 7d. nearly. 
 
 Y. To find the PRIME COST of GOODS. 
 
 (21) If 350 yards of cloth be sold for 210Z. 12s. at 20 per 
 cent, profit, what did it cost per yard ? 
 
 s. yds. s. yd. 
 
 As 120 : 210 12 : : 100 Then, As 350 : 175 10 :: 1 
 20 Or shorter, thus : 20 
 
 4212 
 
 iOO 
 
 210 12 35,0)351,0(105, 
 
 35 2 350 
 
 12,0)42120,0( 175 10 1 
 
 48 
 
 2,0)351,0 Ans. 10*. O^/. 13 rem. 
 
 85)48(1 
 175 10 first cost. 35 
 
 13 rem. 
 
 (22) Sold wines at 58s. per dozen, by which I cleared 1C 
 per cent. ; required the prime cost per dozen ? Ans. 50s. 
 
 (23) rfold broad cloth at 2Gs. 4d. per yard, by which I 
 gained 12 per cent. ; required the prime cost per yard ? 
 
 Ans. II. 3s. Gd. ^ 
 
 (24) Sold a fother of lead (19 \ cwt.) for 18Z. and gained 
 after the rate of 20 per cent. ; what did it cost me per cwt. ? 
 
 Ans. 15s. 4*d. 18 rem. 
 
 (25) Sold a pipe of wine (120 gallons) for 95Z. 12s. and 
 gained 201. by the bargain ; 1 demand the prime cost pe* 
 gallon? Ans. 12s. 
 
 VI. Promiscuous examples. 
 
 (26) Sold 1 cwt. of hops for 3Z. 16s. Gd. at the rate of 20 
 per cent, profit ; what would have been the gain per cent., if 
 I had sold them at 4Z. 9s. 3d. per cwt. ? 
 
 s. d. 
 493 
 8 16 6 125. 9d. of 3 16 6 = | of 120 = 20, and 20 + 20=40 per cent 
 
 12 9 
 
Single Fellowship. 
 
 107 
 
 The usual way. 
 s. d. s. d. 
 As 3 16 6 : 120 : : 493 
 20 1071 20 
 
 The same by two 
 As 120 : 3 16 6 
 
 918 
 100 
 
 statings. 
 : : 100 
 
 s. d. 
 493 
 339 
 
 76 918)128520(140 89 
 12 1 
 
 12,0)9180,0 
 
 156 
 
 
 12)765 
 2,0) 6,3 9 
 339 
 
 rp| 1Pn 
 
 918 = J071 
 = 140 = 
 100 
 
 Ans. 40 per cent. 
 
 Or the truth of this stating may 
 be demonstrated by the adjoining 
 operations in two statings. 
 
 s. d. 
 :: 1 5 
 
 306cZ. 
 (4W.Ans. 
 
 s. d. 
 As 3 3 9 : 100 
 306 
 
 765)30600 
 
 (27) Bought goods at 7|J. per Ib. and sold them at the 
 rate of 4<L 10s. 5d. per cwt. ; what was the gain per cent. ? 
 
 Ans. 25 per cent. 
 
 (28) Purchased goods at 21. 16s, per cwt. and sold them 
 again retail at l\d. per Ib. ; what was the gain per cent. ? 
 
 Ans, 25 per cent. 
 
 (29) If when I sell cloth at 4s. 6d. per yard, I gain 12 
 per cent., what will be the gain per cent, when it is sold for 
 6s. per yard ? Ans. 49Z. 6s. Sd. 
 
 (80) Bought 96 gallons of porter for 51., but by accident 
 16 gallons of it were lost; how must I sell the remainder 
 per gallon so as neither to get nor lose ? Ans. Is. 3d. 
 
 FELLOWSHIP. 
 
 FELLOWSHIP or PAKTNERSHIP is a rule by which 
 merchants, &c., trading together in company with a joint 
 stock, ascertain their proper shares of the gain or loss, in 
 proportion to their stock. 
 
 By this rule a bankrupt's estate may be divided among his 
 creditors ; legacies are also adjusted by it, when there is a 
 deficiency of assets or effects. 
 
 Fellowship is either with or without time. 
 
 FELLOWSHIP WITHOUT TIME, 
 On, SINGLE FELLOWSHIP, 
 
 Is when the calculations are made in proportion to the 
 shares only, without any regard to time. 
 
108 Single Fellowship. 
 
 RULE. As the ivhole stock is to the whole gain or loss, so 
 is each man's share in stock to his share of the gain or loss. 
 ^ PROOF. Add all the shares together, and if the work be 
 right, the sum will be equal to the given gain or loss. 
 
 EXAMPLES. 
 
 (1) Three persons trade together ; A puts in 5GO/., B 750?., 
 and C 1000Z., and they gain 1200Z. ; what is each person's 
 share of the profit ? 
 
 
 
 A 500 Then, as 2250 : 1200 : : 500 
 
 B 750 500 
 
 C 1000 . 
 
 2250 ) 600000 ( 266?. 13s. 4d. 
 2250 A.'* share 
 
 
 
 As 2250 : 1200 : : 750 As 2250 : 1200 : : 1000 
 750 1000 
 
 2250 ) 900000 ( 400J. B's share 2250 ) 1200000 (533Z. 65. 8d. 
 . C's share 
 
 Proof. 
 
 A 1 s share 266 13 4 
 
 's share 400 
 
 C's share 533 6 8 
 
 1200 = to the given gain 
 
 (2) Two merchants trade together; A puts into stock 
 120Z. and B 240Z., and they gain 1501. ; what is each person's 
 share of the profit ? Ans. A 50Z., J9.'100Z, 
 
 (3) A, B, and C enter into partnership ; A puts in 762Z. 
 10^., B 850Z. 155., and C 910Z. 12s., and in one year they 
 gain 536Z. 10s. ; I demand each person's share of the gain ? 
 Ans. A 162Z. Is. 8%d. 37846. B 180Z. 16s. lOJrf. 43153. 
 C193Z. 11s. 4^.-19955. 
 
 (4) Four merchants, "W, X, T, Z made a stock ; "W put 
 in 1000Z., X 2000Z., Y.3000Z., Z 4000Z.; by trading they 
 gained in 3 years 5000Z. ; I demand each person's gain ? 
 
 Ans. 7F500Z., JT1000Z., T1500Z , and Z 2000Z. 
 
 (5) Four persons traded and gained 950Z., which was to 
 be so divided that their shares might be to each other as 1, 
 2, B, and 4 respectively ; what had each to receive ? 
 
 s. the \st 95Z., the 2nd 190Z., the 3rd 285Z., the kth 380Z. 
 
Single Fellowship. 109 
 
 (6) A merchant at his decease owed to D 1261. 12s., to 
 E 241Z. 105., to P 3501. 15s., to G 4701. 10s., and to II 550Z 
 - 13s. ; but he left property to the amount of only 580Z. ; how 
 much may each creditor receive ? 
 
 8. 9 . 
 
 D .. 126 12 As 1740 : 580 : : 126 12 
 
 E .. 241 10 20 2532 20 
 
 F .. 350 15 . . . 
 
 G .. 47010 34800 34800 ) 1468560 (421 4$. D's 2532 
 
 H .. 550 13 __ , 8hart ___ 
 
 1740 
 
 8. s 
 
 As 1740 : 580 : : 241 10 
 
 20 4830 20 
 
 4830 
 
 As 1740 : 580 : : 380 15 
 20 7015 20 
 
 7015 
 
 34800) 2801400 (802. 10. E's share. 34800) 4068700(1162. 185. 4d. 
 
 f"s share. 
 
 As 1740 : 580 : : 740 10 
 20 9410 20 
 
 9410 
 
 As 1740 : 580 : : 550 13 
 20 20 
 
 34800 ) 5457800 ( 1562. 16s. 8d. 34800 11013 
 
 ' G's share. 580 
 
 Proof. 34800)6387540(1832. 11*. 
 
 D ... 42 4 U's share. 
 
 E ... 80 10 
 
 F... 116 18 4 
 
 G ... 156 16 8 
 
 H ... 183 11 
 
 580 
 
 (7) A bankrupt is indebted to P 384Z. 12s., to Q 786Z. 15s., 
 and to B, 850/. 13s., and his estate is worth but 1348Z. ; if 
 the whole were divided how much would each creditor 
 receive ? Ans. P 256Z. 8s. Q 524Z. 10s. E 5G7Z 2s. 
 
 (8) A ship worth 8000Z. being entirely lost, of which ^ 
 belonged to A, to B, J to C, and J to D ; what loss will 
 each sustain supposing 2000Z. of her to have been insured ? 
 
 Ans. A 7501. B 7501. C 15001. D 30001. 
 
 (9) Pour merchants, E, P, Gr, and H freight a ship with 
 300 tons of wine : E loaded 95 tons, P 100, Gr 110, and H 
 the rest; in a storm the seamen threw 72 tons overboard; 
 how much must each sustain of the loss ? 
 
 13. -F20. G 22. Hilton*. 
 
110 Single Fellowship. 
 
 (10) A person ignorant of numbers left 3000}. among his 
 4 children, and ordered that A should have J, B ^, C 1, and 
 D | ; what will be the just share of each, according to the 
 Intention of the donor ? 
 
 Ans. A 1052Z. 12*. 7.U-90. B 780Z. 9.?. 5^. 210. 
 C G31Z. 11*. Gf 225. D 526Z. 0*. 3 J 45. 
 
 (11) Four persons join in the purchase of a house and 
 premises for 1000Z. ; Q paid |, E J, S J, and T the remain- 
 der ; but the house and premises being afterwards burnt 
 down, and only 7501. insured, I demand what each subscribed 
 and what each lost ? 
 
 
 
 1000 purchased 
 
 1000 750 insured 
 
 333 6 8^1 m , 250 total loss 
 
 250 ol 
 
 S i 300 f Q i S3 6 8^ 
 
 Trem.21613 4j ty each. R ^ 6 2 10 I The loss 
 
 S % 50 I of each. 
 
 1000 Proof. T rem. 54 3 4 J 
 
 250 Proof. 
 
 (12) A gentleman leaves by will to A 1000Z., to B 950Z., 
 to C 800Z., and to D 750Z., but his effects are found to amount 
 only to 2900Z. ; how much will each legatee have to receive ? 
 
 Ans. A 828Z. 11*. 5d. 20. B 7871. 2s. lOirf. 5. 
 C6G2Z. 17^. 1| 30. D 621Z. 8s. CJ 15. 
 
 (13) Purchased a ship for 3700Z. ; A paid 1000Z., B 1500Z., 
 and C the rest ; they afterwards sold her for 4500/. ; re- 
 quired the gain of each ? 
 
 Ans. A 21GZ. 4<s. 3J d. 21. B 324Z. 6*. 5| 13. 
 0259Z. 95. 2\d. 3. 
 
 (14) A and B venturing equal sums of money, cleared by 
 trade 550Z. ; by agreement A was to have 8 per cent, on 
 account of the time he spent in the execution of the project, 
 and B was to have only 5 per cent. ; what was A allowed for 
 his trouble ? Ans. A 120Z. 18s. 5^d.2 rem. 
 
 (15) X, T, and Z join their stocks in trade, the amount of 
 their stock is 1500/. in the proportion of 3, 4, and 5 to each 
 other ; what is each man's stock ? 
 
 Ans. JT375/., T500/., and Z G25L 
 
Double Fellowship. Ill 
 
 FELLOWSHIP WITH TIME, 
 On, DOUBLE FELLOWSHIP, 
 
 IS when the different shares are employed for different 
 terms of time. 
 
 RULE 1st. Multiply each man's stock and time together. 
 
 2nd. Add together the several products thence arising. 
 
 3rd. Then say, As the sum of these products is to the 
 whole gain or loss, so is each man's particular product to his 
 share of the gain or loss. 
 
 PEOOF. As in Fellowship without Time. 
 
 EXAMPLES. 
 
 (1) Three merchants join in company ; A puts into stock 
 5G5Z. for 6 months, B 400Z. for 9 months, and C 3007. for 
 10 months, and they gained 6G07. ; what is each man's share 
 of the gain ? 
 
 
 
 65 x 6=3390As 99 90 : 660 ;; 3390 
 400 x 9=3600 3390 
 
 300 y. 10=3000 
 
 999,0)223740,0(2232. 195. 3d. 405 r. 
 
 9990 
 
 405 rem. 
 
 As 9990 ; 600 ; ; 3600 
 3600 
 
 Ans. 
 
 s. d. 999,0)237600,0(237*. 165. 9d 324 r. 
 
 A. . 223 19 3J405 
 
 B. . 237 16 9 324 324 rem. 
 
 C. . 198 3 114270 
 
 As 9990 ; 660 : : 3000 
 
 660 Proof. 3000 
 
 IST.B. The sum of these 999,0)198000,0(1982. 35. 
 
 remainders, being equal to 
 the divisor 999, is conse- 270 rem. 
 
 quently equal to a farthing. 
 
 (2) D and E enter into partnership ; D puts into stock 
 7507. for 15 months, and E 6007. for 18 months, and they 
 gained 3G07. ; the share of each is required ? 
 
 Ans. D 1837. 13s. 5\d. 1170. E 17G7. 6s. G\d. 1035 
 
 (3) Three merchants trade together ; A puts in 1207. for 
 9 months, B 1007 for 16 months, and C 1007. for 14 months, 
 and they gained 1507. ; hew must it be divided ? 
 
 Ans. A. 397. Us. l\d. 201. B 587. 16*. 6*d. 210 
 (7517. 9s. 43^, 312 rem. 
 
112 Double Fellowship. 
 
 (4) Two persons put 2000Z. into trade ; their stock is in 
 the proportion of 3 to 2, i. e. A puts in 1200Z. and B 800J. 
 A leaves his money in the concern 18 months, and B 27 
 months ; what profits belong to each supposing they gain 
 7001. ? Am. 350Z. each 
 
 (5) X, Y, and Z hold a piece of ground, in common, for 
 which they agree to pay 221. 10s., X puts in 35 oxen for 
 30 days, Y 25 oxen for 35 days, and Z 40 oxen for 25 days ; 
 what has each man to pay of the rent ? 
 
 s. 
 
 35x30=1050 As 2025 : 2210 :: 1050 
 
 25x35= 875 20 
 
 40x25=1000 
 
 450 
 
 2925 1050 
 
 2925)472500(2,0)16,1 
 
 2475 rem. 82. 1*. 6J& 1575 
 
 Aa 2925 22 10 : : 675 
 
 20 
 
 450 
 875 
 
 2925)393750(2,0)13,4 
 
 1575 rem. 61. 14t. 7{d.I575 
 
 As 2925 I 22 10 ; ; 1000 
 
 Ans. 20 
 s. d. 
 
 .8 1 6 2475 450 
 
 . 6 14 7i 1575 1000 
 . 7 13 10 1800 
 
 2925)450000(2,0)15,3 
 
 1800 rem. 71. 13*. lOrf. 
 
 (6) Three graziers hired a piece of pasture land for 50Z. ; 
 A put in 30 sheep for 3 months, B 25 for 3J months, and 
 C 20 for 4 months ; what is each person'.s proportion of the 
 rent ? Ans. A 111. 9*. Qd. 360. 1GZ 19*. 9|^. 350. 
 
 C 15Z. 10^. 8d. 320. 
 
 (7) A, B, and C hold a pasture, in common, for which 
 they pay 30/. In this pasture A had 40 oxen for 76 days, 
 
Double Fellowship. 113 
 
 B had 3G oxen for 50 days, and C had 50 oxen for 90 days ; 
 
 what had each to pay ? Ans. A 91. 15*. 3\d. 818 
 
 B 51. 15*. 1\d. 300. C 141. 9s. Of ^.750 
 
 (8) Two troops of horse rented a field, for which they 
 were to pay 75Z. ; one of the troops sent 84 horses for 28 
 days, and the other 60 horses for 35 days ; how much of the 
 rent had each troop to pay ? 
 
 Ans. The 1st troop 391 12*. 5J d. 3276 
 The 2nd 351. 7*. 6^. 1176 
 
 (9) Three merchants, D, E, and F, trade with a common 
 stock of 5000Z. ; D gains 230/. in 9 months, E 250Z. in 10 
 months, and F 300?. in 12 months ; what was each person's 
 particular stock ? 
 
 230 x 9 = 2070 As 8170 : 5000 :: 2070 
 250x10 = 2500 2070 
 
 300x12 = 3600 
 
 817,0 ) 1035000,0 ( 1266Z. 16. 7d. 548 
 
 8170 
 
 548 rem. 
 
 AS 8170 : 5000 ; : 2500 
 
 2500 
 
 817,0 ) 1250000,0 ( 1529Z. 19*. 9ef. 204 
 
 204 rem. 
 Ans. 
 
 . *. d. AS 8170 : 5000 :: seoo 
 
 7>..1266 16 7 548 3600 
 
 E.. 1529 19 9 204 
 
 P . . 2203 3 7J 65 817,0 ) 1800000,0 ( 2203Z. 3*. 7fd. 65 
 
 5000 Proof. 65 rem. 
 
 (10) The joint stock of 3 tradesmen was 1800Z. ; K gained 
 ;300/. in 18 months, L 350Z. in 21 months, and M. 4001. in 
 2 years ; I demand how much was the stock of each ? 
 Ans. K 4341. 17*. llf d. 795. L 5911. 18*. lid. 1020. 
 
 M 7731 3s. Id. 420. 
 
 (11) Three merchants join in trade ; A puts in 560/. for 3 J 
 years, B 700Z. for 3 years, and C 800/. for 2 years, but by 
 misfortune they lost goods to the value of 525/. ; what must 
 each man sustain of the loss ? Ang. A 159Z. 18*. \\d. 3150 
 
 185/. 12*. ld. 52G5. C 1791. 9s. 8 Jrf. 4155 
 
Vulgar Fractions. 
 
 FRACTIONS. 
 
 FRACTIONS are a part or parts of a unit, or of any whole 
 quantity expressed by a unit. They are divided into two 
 sorts, Vulgar and Decimal. 
 
 VULGAR FRACTIONS. 
 
 
 A VULGAR, or COMMON FRACTION is so called 
 because any number may be its denominator, and is repre- 
 sented by two numbers with a line between them, as \, , f . 
 The upper number is called the numerator, and the lower or 
 under one the denominator. 
 
 The denominator shows how many parts the unit is divided 
 into ; and the numerator, how many of these parts are to be 
 taken. 
 
 There are four kinds of Vulgar Fractions : Simple, Com- 
 pound, Mixed, and Complex. 
 
 A simple or single Fraction has only one numerator, and 
 one denominator, as, |, f, ff, V"; when the numerator is less 
 than the denominator it is termed a proper fraction, as f , 
 i.| . when the numerator is equal to, or greater than the 
 denominator, it is called an improper fraction, as -, - 8 *-. 
 
 A compound fraction is the fraction of a fraction, and is 
 known by the word of, as J of \ of f , &c. 
 
 A mixed number, or fraction, is composed of a whole 
 number and a fraction, as 4, G, 84f, &c. 
 
 A complex fraction has a fraction, or a mixed number, for 
 
 its numerator or denominator, or both as ~ 7 -f f &c. 
 
 o, (t, iz, yf, 
 
 Note. Any whole number may be expressed like a fraction by writing 
 1 under it as a denominator ; thus, 6, 18, 240, may be written, y> 2 T > 
 &c. 
 
 REDUCTION OF VULGAJR FKACTIONS. 
 
 REDUCTION of Vulgar Fractions is the method of chang- 
 ing them from one form or denomination to another, 
 without altering their value ; in order to prepare them for 
 Addition, Subtraction, Multiplication, and Division. 
 
Vulgar Fractions. 115 
 
 CASE I. To reduce fractions of different denominations 
 lo others of equal value, having a common denominator. 
 
 RULE 1st. Multiply each numerator into all the denomi- 
 nators except its own, for a new numerator ; and all the 
 denominators for a common denominator. Or, 
 
 2ndly. Multiply the common denominator by the several 
 given numerators separately, and divide the product by the 
 several denominators j the quotients will be the new nume- 
 rators. 
 
 EXAMPLES. 
 
 (1) Eeduce f and to a common denominator ? 
 
 By Rule IL 
 
 3x5 = 15 common denominator, lil*.- 12 J *** numerators ' 
 
 Ans. "B- and 
 
 (2) Eeduce and | to a common denominator. 
 
 Ans. 
 
 (3) Eeduce f and f to a common denominator. 
 
 Ans. f f and 
 
 (4) Eeduce , f, and A to a common denominator. 
 
 Ans. m, *, and 
 
 (5) Eeduce f , f , f , f , and 2 a common denominator. 
 3x9x6x8x1 = 1296^ 
 2x7x6x8x1= 672 
 
 4x7x9 x8xl = 2016 
 
 new numerators. 
 
 5x7x9x6x1 = 1890 
 
 2x7x9x6x8=6048j 
 
 7x9x6x8x1 = 3 02 4 common denominator. 
 
 Ans. Hff, sVA, t*if, HH, 
 
 (G) Eeduce f , i, , and 4, to a common denominator. 
 
 ^s. ff, H, fg-, an J 
 
 (7) Eeduce f, i, 7, f, and 3, to a common denominator. 
 
 Ans. /A, 3 Vb, Wo*, H*, w^ V 
 
 (8) Eeduce , f, f , and 4, to a common denominator. 
 
 Ans. 
 
 II. To reduce fractions to their lowest terms. 
 
 EULE 1st. Divide both the numerator and denominator 
 of the fraction by awy number that will divide them ivithout 
 a remainder ; and these again in the same manner till no 
 number greater than unity will divide them ; and the last 
 fraction will be in its lowest terms. Or, 
 
 2nd. Find a common measure by dividing the greater term 
 by the less, and that divisor by the but remainder, and so 
 
116 Reduction of 
 
 on till nothing remains: the last divisor is the common mea- 
 sure : then if the numerator and denominator of the given 
 fraction be divided by this common measure, it will reduce 
 it to its lowest terms. 
 
 N. B. When fractions have ciphers to the right hand, 
 they may be cut off, as f f (H- 
 
 EXAMPLES ~by the 1st Rule. 
 
 (9) Eeduce J$-H& to its lowest terms. 
 Divisors, 6) 7) 4) 3) 
 
 HHI*==W=H=A-=i Ans. 
 
 (10) Eeduce if ff to it lowest terms. Ans. 
 
 (11) Eeduce if f % to its lowest terms. Ans. -$ 
 
 By the 2nd Rule. 
 
 (12) Eeduce Hf* to its lowest terms. 
 180,0)432,0 (2 
 
 360 
 
 Then36)ttSI*(=&^w. 
 72)180(2 
 
 144 
 
 The common meas. 36) 72 ( 2 
 
 72 
 
 (13) Eeduce iff to its lowest terms. Ans. H 
 
 (14) Eeduce sVeo- to its lowest terms. Ans. i 
 
 (15) Eeduce AVo- to its lowest terms. Ans.-s 
 
 (16) Eeduce ffjg- to its lowest terms. Ans. -fy 
 
 (17) Eeduce %%% to its lowest terms. Ans. f 
 
 III. To reduce a mixed number to an equivalent improper 
 
 fraction. 
 
 ETJLE. Multiply the whole number by the denominator 
 of the fraction, and to that product add the numerator for 
 a new numerator, under which place the denominator, and 
 it will form the fraction required. 
 
 EXAMPLES. 
 
 (18) Eeduce 64-A- to an improper fraction. 
 
 64 
 11 
 
 Or it may be expressed thus, 
 
 704 64 X 11 +8 = 712, new numerator. 
 
 New numerator 712 
 
 Ans. 
 
 11 
 
 64x11+8 712 
 Or thus, 64^ = ^ -= -^- 
 
Vulgar Fractions. 117 
 
 (19) Reduce 84 ^ to an improper fraction. Ana. i ^T i 
 
 (20) Eeduce 96|1 to an improper fraction. Ans. -HI- 2 -. 
 
 (21) .Reduce 100 \\- to an improper fraction. Ans. 
 
 (22) Eeduce 346^0 to an improper fraction.^ras. 
 
 (23) Eeduce 27 ^ to an improper fraction. Ans. 
 
 IV. To reduce an improper fraction to a whole or mixed 
 number. 
 
 EULE. Divide the upper term by the lower, and the quo- 
 tient will be the whole or mixed number required. 
 
 EXAMPLES. 
 
 (24) Eeduce Vi 3 - to its proper terms. 
 
 11 ) 712 Or, expressed thus : 
 
 712-7-11 =64^^*. 
 ns. Or more technicality thus : 
 
 (25) Eeduce ^Y^to its proper terms. Ans. 1064 A 
 
 (26) Eeduce ^H L to its proper terms. Ans. 84 A or 84|. 
 
 (27) Eeduce ^ff 2 - to its proper terms. Ans. 96fJ. 
 
 (28) Eeduce Hi- 5 - to its proper terms. Ans. 100H . 
 
 (29) Eeduce Hi* 1 - to its proper terms. Ans. 346 T !v. 
 
 Y. To reduce a compound fraction to a single one. 
 
 EULE. 1st. If any of the proposed quantities be either 
 whole or mixed numbers, reduce them to improper fractions 
 by Case 3rd. 
 
 * 2ndly. Multiply all the numerators together for a new 
 numerator, and all the denominators for a new denominator : 
 then reduce the new fraction to its lowest terms. 
 
 EXAMPLES. 
 
 (30) Eeduce % of t of | to a simple fraction. 
 
 -| X X f=i 4 A = A- Ans. in its lowest term. 
 
 (31) Eeduce -f of -f of & to a simple fraction. 
 
 Ans. 5^ = A- 
 
 (32) Eeduce A of f of & of f to a simple fraction. 
 
 Ans. ffr\. 
 
 (33) Eeduce if of H% to a simple fraction. Ans. iV*. 
 
118 Reduction of 
 
 EXAMPLES, with whole or mixed numbers. 
 
 (34) Eeduce f of 2 f of 8 of *- of 6| to a simple fraction. 
 First prepare the fractions 2f = V ; 8=f ; 6$ = -. 
 
 then f x V X I- X V- X V- = A f H- = 275 J^s. 
 
 (35) Eeduce i of f of 3f of 9 to a simple fraction. 
 
 Am. **** = 114. 
 
 (36) Eeduce A of 7 of 5f of 12 to a simple fraction. 
 
 Ans. ^ 
 
 N.B. If the same figures are found both in the nume- 
 rator and the denominator, they may be struck out of each. 
 Note also, if in the numerator and denominator there are 
 such numbers as the same figure will divide, the quotients 
 may be used instead of them. 
 
 (37) Eeduce f of f of f of | to a simple fraction. 
 
 1235 5 
 _ X -X-X- = 
 ? ? 4 6 24 
 
 (38) Eeduce f of f of A of ft to a simple fraction. 
 
 2 
 
 3 4 8 ?4 3x2 2 1 
 _ x x X = - - = = - 
 
 4 $ ffl 36 ffl 12 6 
 
 12 
 
 (39) Eeduce -f of H of A of ff to a simple fraction. 
 
 3 
 tf 7 ?? 3 , 
 
 EX x x = - -* 
 
 7 ;? ;; ^p 8 
 
 8 
 
 (40) Eeduce f of A of i of A to a simple fraction. 
 
 2689 2x6 1x6 1 A 
 Or thus - X X - X = -- = - = Ans. 
 8 11 9 12 11X12 11x6 11 
 
 VI. To reduce tJie fraction of one denomination to the fraction 
 of another , BUT GEEATEB, retaining the same value. 
 
 EIJLE. Eeduce the given fraction to a compound one, 
 and that to a single one ; that is, multiply the denominator 
 by all the denominations, from that given to the one sought. 
 Thus one farthing, reduced to the fraction of a , would be 
 Jof A of A = 9 ioofa. 
 
Vulgar factions. 119 
 
 EXAMPLES. 
 (41) Reduce -J of a penny to the fraction of a . 
 
 -2 Of -iV Of -/Q- = 
 Or tllUS J * 
 
 (42) Eeduce of a penny to the fraction of a . 
 
 (43) Eeduce | of a shilling to the fraction of a . Ans. -&. 
 
 (44) Eeduce | of a penny to the fraction of a shilling. 
 
 Ans. A- 
 
 (45) Eeduce of a penny to the fraction of a guinea. 
 
 Ans. ToVjf- 
 
 (46) Eeduce f of a shilling to the fraction of a moidore. 
 
 __ Am. &. 
 
 (47) Eeduce of a dram to the fraction of a ton. 
 
 Of ^6 Of iS Of ~/- s - Of J Of - 2 V = 3440640 = 688128 A.HS. 
 
 (48) Eeduce | of a Ib. to the fraction of a hundred weight. 
 
 Ans. T^T. 
 
 (49) Eeduce f of a grain to the fraction of a Ib. Troy. 
 
 Ans. 8 - 6 Vo. 
 
 (50) Eeduce j of a pint of wine to the fraction of a hhd. 
 
 Ans. 
 
 (51) Eeduce f of a yard to the fraction of a mile. 
 
 Ans. 
 
 (52) Eeduce f of a second to the fraction of a week, 
 
 806400' 
 
 VTI. To reduce the fraction of one denomination to tn^ 
 fraction of another , BUT LESS, retaining the same value. 
 EULE. Multiply the numerator by all the denominations, 
 
 from that given to the one sought, for a new numerator, and 
 
 place it over the given denominator. Eeduce the new frac- 
 
 tion to its lowest terms. 
 
 EXAMPLES. 
 
 (53) Eeduce fA-tr of a to the fraction of a farthing. 
 eW X 20 X 12 X 4 = f&\% = A- = qfafarth. Ans. f 
 
 Orthus:^ 8 -o = 1X 2 2 ^ 2 ** = -^ = i of a forth. 
 
 (54) Eeduce -&-f - - of a pound to the fraction of a penny. 
 
 Ans. f. 
 
120 Reduction of 
 
 (55) Eeduce -V of a shilling to the fraction of a farthing. 
 
 Ans. J of a farthing. 
 
 (56) Eeduce - 8 \- of a moidore to the fraction of a shilling. 
 
 Ans. f . 
 
 Eeduce irnoei-o of a ton to the fraction of a dram. 
 5 x 20 X 4 x 28 X 16 x 16 = H4t*i*= when reduced, = . 
 
 (58) Eeduce TOT of a yard to the fraction of a nail. Ans. . 
 
 (59) Eeduce -ri u of a wey to the fraction of a peck. Ans. \. 
 
 (60) Eeduce -rrhro of a Ib. Troy to the fraction of a grain. 
 
 Ans. f. 
 
 VIIII. To reduce a fraction of one denomination to another 
 
 of the same value, having either the numerator or deno- 
 
 minator of the required fraction given. 
 
 EULE 1. When the new numerator is given, say, As the 
 numerator of the given fraction is to its denominator, so is 
 the new numerator to its denominator. 
 
 2nd. When the new denominator is given, say, As the 
 denominator of the given fraction is to its numerator, so is 
 the new denominator to its numerator. 
 
 EXAMPLES. 
 
 (61) Eeduce f to a fraction of the same value, whose 
 numerator shall be 9. 
 
 Say, As 3 : 5 :: 9 to 15. i. e. 6 -^L = 15. Ans. A. 
 
 (62) Eeduce f to a fraction of the same value, whose nume- 
 rator shall be 12. Ans. ?-. 
 
 (63) Eeduce f to a fraction of the same value, whose nume- 
 rator shall be 45. Ans. -{-. 
 
 (64) Eeduce -&- to a fraction of the same value, whose deno- 
 minator shall be 44. 
 
 Say, As 11 : 6 :: 44 to 24. Ans. f J. 
 
 (65) Eeduce f to a fraction of the same value, whose deno- 
 minator shall be 21. Ans. -H. 
 
 (66) Eeduce | to a fraction of the same value, whose deno- 
 minator shall be 81. Ans. f. 
 
 IX. To reduce a complex fraction to a single one. 
 
 ETJLE. If the numerator or denominator be whole or 
 mixed numbers, reduce them to improper fractions ; then 
 multiply the numerator of the upper fraction into the denomi- 
 nator of the lower for a new numerator : and the denominator 
 
Vulga/r Fractions. 121 
 
 of the upper into the numerator of the lower, for the nev, 
 denominator ; which reduce to its lowest terms. 
 
 EXAMPLES. 
 (67) Eeduce ^ to a simple fraction. 
 
 (G8) Eeduce -- 3 to a simple fraction. Ans. \ t 
 
 i fi 
 
 (69) Eeduce to a simple fraction. 
 
 16 V 6 16 x 5 80 numerator. 
 
 24 == a "|> " 1 x 124 124 denominator. 
 
 Ans. 
 
 14 
 
 (70) Eeduce to a simple fraction. 
 
 (71) Eeduce y~ to a simple fraction. 
 33 x 3 
 
 (72) Eeduce -| to a simple fraction. Ans. | ? 
 
 X. To reduce fractions to their proper quantities in money, 
 
 weights, or measures. 
 
 ETTLE. Multiply the numerator by the common parts of 
 the integer, and divide by the denominator. 
 
 EXAMPLES. 
 
 (73) What is the value of | of a 
 5 
 20 
 
 Or thus : = * -- = = 12 * Gd ' An9 ' 
 
 t. Gd. 
 
 (74) What is the value of f of a ? Ans. 16s. 
 
 (75) What is the value of J of a shilling ? Ans. 4 \d. 
 
 (76) What is the value of J-f of a ? ^W5. 11^. ll/.-i 
 
 (77) What is the value of 1 " 2 of a guinea ? ^w. 12s. 3d. 
 
 (78) What is the value of i U of a ? Ans. G 
 
122 Reduction of 
 
 (79) Reduce f of a ton to its proper quantity 
 5 
 
 20 
 
 -- Or thus : 5 x 20=100 and -f- 8 = 12 cwt. 2 art. Ans. 
 8) 100 
 
 C W ~2 2 q rs. 
 
 (80) Eeduce & of a Ib. Troy to its proper quantity. 
 
 Ans. 6 oz. 15 dwts. 
 
 (81) Eeduce & of a yard to its proper quantity. 
 
 Ans. 3 qrs. 
 
 (82) Eeduce ^A of a bushel to its proper quantity. 
 
 Ans. 1 peck 1 gtf . A < 
 
 (83) Eeduce of a chaldron to its proper quantity. 
 
 Ans. 15 bus. 
 
 (84) Eeduce A of a day to its proper time. 
 
 Ans. 11 Jio. 12 
 
 XI. jT0 reduce money, weights, and measures to fractions. 
 
 ETJLE. Eeduce the given quantity to the lowest denomi- 
 nation mentioned, for a numerator ; and the specified integer 
 or whole number into the same name for a denominator. 
 This fraction reduced to its lowest terms, will be the answer 
 required. 
 
 EXAMPLES. 
 
 (85) Eeduce 4s. S^d. to the fraction of a 
 s. d. s. 
 
 4 8 20 = II. 
 
 12 12 
 
 56 240 ji^fi Us. An? 
 
 4 . 960 480 An*. 
 
 226 numerator. 960 denominator 
 
 (86) Eeduce 12s. 8J^. to the fraction of a . 
 
 ^ws. MB- = 
 
 (87) Eeduce 6 oz. 15 dwts. to the fraction of a Ib. Troy 
 
 (88) Eeduce 12 cwt. 2 qrs. to the fraction of a ton Ans. f . 
 
 (89) Eeduce 16 bus. 2 pecks to the fraction of a chaldron 
 of coals. Ans. ^A = ii 
 
 (90) Eeduce 16| cwt to the fraction of a ton. Ans. f& 
 
Vulgar Fractions. 123 
 
 (01) Hedtice 9|^. J to the fraction of a shilling. 
 d. d. 
 
 OB 12 
 
 4 4 
 
 39 48 
 
 3 3 
 
 118 numerator 144 denominator 
 
 (92) Eeduce 65. 8J<7. f to the fraction of a . 
 
 (93) deduce 8 oz. 61 dr. to the fraction of a Ib. avoirdu- 
 pois ? Ans. ff-f. 
 
 (94) Eeduce 2 qrs. 3J nails to the fraction of a yard. 
 
 Ans. f|. 
 
 (95) Eeduce 6 days 6 ho. 15 nun. to the fraction of a 
 week. ^ Ans. * 
 
 (96) Eeduce 2 roods 16J poles to the fraction of an acre. 
 
 XII. To reduce a fraction of an integer to an equivalent 
 fraction of another integer, differing in value. 
 
 EULE. Multiply the numerator of the fraction by the 
 integer in its next lower denomination ; and the denominator 
 by the value of the integer sought, in the same denomination, 
 and it will produce the answer required. 
 
 EXAMPLES. 
 
 (97) Eeduce of a . to the fraction of a guinea. 
 
 I of V- = ^ of a sliill. ; and V X A = 1% = \\ofaguln. 
 Or thus : f of ^ of A = -&V = a f- ^$- as before. 
 
 (98) Eeduce f of a guinea to the fraction of a . 
 
 f of V of A = WV = Is 0/a . ^*. 
 
 (99) Eeduce f of a guinea to the fraction of a moidore. 
 f of V of A = iVa = * J- = f = i o/ moidore. Ans. 
 
 (100) Eeduce A of a crown to the fraction of a seven- 
 shilling piece. -jfr of f of -f = H ^to. 
 
 (101) Eeduce f of a yard to the fraction of an ell English. 
 | of t of i = if = f of an English ell. Am. 
 
 (102) Eeduce f of a barrel to the fraction of a hogshead 
 of beer, 
 
 f of V- of A- = Hf = iofa hogshead. Ans, 
 
 (103) Eeduce f of 6s. S l 2 d. to' the fraction of 10s. 
 6s. 8^d. = 322farth. and 10s. = ISQfarth. 
 
 Then, | of H* of Tio = tHJ = ill- 
 
 G2 
 
121 Addition of 
 
 ADDITION OF VULOAE REACTIONS. 
 
 CASE I. Bring compound fractions, if any, to single ones. 
 
 2ndly. .Reduce these fractions to a common denominator, 
 by Case 1st, and add all the numerators together, under 
 which place the common denominator. 
 
 N.B. When large mixed numbers are to be added, 
 reduce only the fractional part for a common denominator, 
 and add the whole numbers separately. 
 
 EXAMPLES. 
 
 (1) Add \ , f , and together. 
 
 2x4x648 
 3x3x6=54 
 5x3x4 = 60 
 
 162 
 
 = 2 ff = 2* Am. 
 3X4X6 = 72 
 
 (2) Add f , f, f, and f, together. Ans. 
 
 (3) What is the sum off and A ? Ans. Iff. 
 
 (4) Eequired the sum of Hi and &. 
 
 Am. 1-AVoV or H. 
 
 (5) What is the sum of -f , f , -A-, f, and -f ? Ans. 2|fHi. 
 
 (6) Add 1 off, H, and f of A- together. 
 
 Tnen the simple fractions are -J, V, a ^d -ff. 
 
 Therefore 1 X 6 X 77 = 462 
 11 X 3 x 77 = 2541 
 18 x 3 X 6 = 324 
 
 3327 
 
 - = 2 T VA=2Jtt 
 3 X 6 X 77 = 1386 
 
 (7) What is the sum of t, 2f , &-, and ^ of f . 
 
 (8) Eequired the sum of f of --, 3^, f , and | of f . 
 
 (9) Add ; of I of f , f of 6, U, I- of |, and f together. 
 
Vulgar Fractions. 125 
 
 (10) Add 12J, 16|, and 26J together. 
 The fractional parts are 5, 
 
 Therefore 1x3x4 = 12 
 2x2x4 = 16 
 3x2x3 = 18 12 
 
 - 16 
 
 46 26 
 
 - = H*= 1 
 
 2x3x4 = 24 
 
 (11) Add 10*-, 9?, 12f together. Ans. 33H* 
 
 (12) What is the sum of ISA and 56 fl ? ^/?s. 75 l *& 
 
 (13) Bequiredthesumoff,85i,|ofi,and9i-. ^1. 95ff 
 
 II. "When the fractions are of various denominations, 
 reduce them to their proper quantities, and add their sums. 
 
 Or 2ndlj. The fractious may be first reduced to the same 
 integer, and added together, before being reduced to the 
 proper quantity. 
 
 EXAMPLES. 
 
 (14) Add -J- of a guinea, f of a , and 1 of a shilling 
 together. 
 
 *. d. 
 
 1 5 7 2 7| 
 
 21 20 12 12 6 
 
 10 
 8)21 8)100 8)84 
 
 16 Ans. 
 2s. *l\d. 12 6 10 
 
 s. 
 Or thus : ^ of a guinea = V and | of a . =. 
 
 s. 
 Then, V + H 2 " + I = H- 8 - = 16*. ^ws. a* fo/bre. 
 
 (15) Required the sum of -f of a guinea, f of a , and 
 I of a shilling. Ans. II. 10s. Qd. 
 
 (16) What is the sum of A of a , ^ of a shilling, and 
 | of a penny ? -4rcs. 8.?. 4|</. 
 
 (17) Add A of a crown, !* 4 of sevep shillings, and f of a 
 guinea together. Ans. 14s 
 
126 
 
 Subtraction of 
 
 (18) What 
 quarter, and f 
 
 20 
 
 is the sum of { 
 of f of a Ib. ? 
 i or 1 qr. of cwt. 
 
 } of a ton A of a cwt. f of a 
 
 3 I of | = A = | of a Ib. 
 
 28 
 
 5)80 
 
 
 8) 84 
 
 16 cwt. 
 
 
 10^ Ib. 
 
 And 16 cwt. + lqr. + 10 ta. + . = 16 cwtf . 1 qr. 11 Z6. 4w. 
 
 (19) Add f of a Ib. Troy, f of an ounce, and f of a dwt. 
 together. Ans. Soz. 8 dwts. 8 grs. 
 
 (20) [Find the sum of 5- of a mile, J of a yard, and J of 
 a foot. -4w*. 440 yJs. ] /. 9 in. 
 
 (21) Required A of a chaldron of coals, | of a bushel, 
 and | of a peck. Ans. 24 bus. 3^ />c&s 
 
 (22) Add A of a week, f of a day, and i of an hour 
 together. Ans. 1 day, 18 /zo. 12 min. 
 
 SUBTEACTION OF VULaAH PEACTIONS. 
 
 Reduce the fractions, if needful, to a common 
 denominator, as in Addition : then subtract the less nume- 
 rator from the greater, and place the remainder over the 
 common denominator. 
 
 2nd. When the lower fraction is greater than the upper, 
 subtract the numerator of the lower fraction from the com- 
 mon denominator, and to that difference add the upper 
 numerator, carrying one to the unit's place of the lower 
 whole number. 
 
 N. B. This is the principle upon which farthings are subtracted in 
 money ; suppose it be required to subtract 4f d. from 6%d. we should take 
 the numerator 3 from 1 saying, 3 from 1 you cannot, but 3 from 4 (the 
 common denominator) leaves 1, and 1 = ; put down , and carry one to 
 the whole number. 
 
 EXAMPLES. 
 
 (1) From f subtract f. 
 5 x 6 = 30 
 3 X 7 = 21 Or thus : -g ^ 1 = -&. Ans. 
 
 9 Ans. 
 5 X 7 = 35 
 
fractions. 
 
 127 
 
 .(2) Prorn^ offtake?, Ans. A- 
 
 (3) Eequired the difference off and A. Ans.'H. 
 
 (4) Subtract iVo from -. Ans. 0. 
 
 (5) What is the difference between & and of f of | r 
 
 (6) From 1G A take 8 A- And from 6f take 4f. 
 
 9 x 12 = 108 num. 
 7 x 11 = 77 www. 
 
 11 x 12 = 132 den. 
 
 Then, from 1 6f 
 take 
 
 2x4 = 8 num. 
 3x3 = 9 num. 
 
 3 x 4 =12 den. 
 
 Then, from 
 
 (7) Eequired the difference between 12f and 8|. Ans. 4f. 
 
 (8) Subtract llOf from 250|. .4*1*. 140 ./ . 
 
 (9) What is the difference between f of | of f and 24 ? 
 
 Ans. 1>4. 
 
 (10) Erom 185 take 67f . ^5. 117| 
 
 "When the fractions are of several denominations, 
 reduce them to their proper quantities, or to fractions of the 
 same integer, and subtract as before. 
 (11) Prom J of a take f of a shilling. 
 
 3 
 12 
 
 4)36 
 9d. 
 
 Or thus, -f of a shilling = -^ of a . 
 Then the fractions would be $ and - 
 
 1 x 80 = 80 num. 
 
 3x3=9 num. 
 
 Then, from 6*. 8d. 
 take 9 
 
 5s.lld.Ar, 
 
 3 x 80 = 
 
 (12) Prom | of a take | of a guinea. Ans. 9s. 
 
 (13) Prom g of a Ib. Troy take | of an ounce. 
 
 Ans. 1 oz. 12 dwts. 12 y/*s. 
 
 (14) Prom A of a chaldron take | of a bushel. 
 
 ^/<5. 23 bus. 1 jc>ec^, 1 gal. 
 
 (15) Prom lo weeas taKe 6| days Ans. 12 w. c?. 16 A. 
 
128 Multiplication of 
 
 MULTIPLICATION OF YULGAE FRACTIONS. 
 
 RULE 1. Prepare the fractions, if needful, by the rules 
 of Reduction ; then multiply all the numerators together 
 for a new numerator, and all the denominators for a new 
 denominator. 
 
 2. AVhen any number, either whole or mixed, is mul- 
 tiplied by a fraction, the product will be always less than the 
 multiplicand, in the same proportion as the multiplying frac- 
 tion is less than the unit. 
 
 EXAMPLES. 
 
 (1) Multiply f , ?-, and f together. 
 
 f X f X | = AS Ans. 
 
 (2) Multiply -A- by f, and $ by A. Ans f?- and &. 
 
 (3) Multiply f, f, |, and f- together. 
 
 T B f o = oVo = Tf T. 
 
 (4) Multiply | of f by J of t of f . 
 
 (5) Multiply -?- of f by A of |. 
 
 (6) "What is the product of of A and ^ of U 
 
 N.B. If the same figures are found in the numerator as 
 in the denominator, they may be left out in multiplying ; or 
 if any figure in one line will divide a number in the other, it 
 may be done, and the work will be abbreviated. Thus, ia 
 the following figures 
 2 
 
 2 $ 4i & J2 2x2 4 1 
 
 -of -of -of -of may be abbreviated thus : - -=-=- 
 ^ ^ $ 24 24 24 6* 
 
 (7) Multiply tV of i | by If of {. Am. -f*. 
 
 (8) Multiply A of f-| by of of J. 
 
 (9) Multiply 4|, 1 T 3 9 -, and 3| of 8 together. 
 
 41 = V ; IsV = ; 3^ of 8 = V off. 
 Then V- X f I X Y- X { = 134 rfy. .4^. 
 
 (10) "What is the product of f, f , 6|, 5f, and 12 ? 
 
 . 993 
 
Vulgar Fractions. 129 
 
 (11) How many yards of cloth in 12 f pieces, each con- 
 taining 26 yards? Ans. 337|. 
 
 (12) How many Ibs. arc there in 9J parcels, each contain- 
 ing llf Ibs. ? Ans. 111|. 
 
 (13) How many Ibs. are there in 7 sugar loaves, each 
 weighing 12| Ibs. ? Ans. 
 
 DIVISION OF YULaAE FEACTIONS. 
 
 EULE. Prepare the fractions, if needful, by the former 
 rules, then invert the divisors, and proceed as in multiplica- 
 tion. 
 
 EXAMPLES. 
 (1) Divide H by H- 
 
 = 1A- Ans. 
 
 (2) Divide H by f . Ans. 1-A-. 
 
 (3) Divide If by f . Ans. 1-A- 
 
 (4) Divide f off by f of*. 
 
 i X I- X f X i- = W = 2?t = 2H ^*- 
 
 (5) Divide f of f by A of f. Ans. 1 A- 
 
 (6) Divide f of A A- An *- A- 
 
 (7) Divide 5} by 6|. 
 
 5 j. = AJL an d 6f = V- 
 
 Then V- -^ V = V X A = tt - 4715 - 
 
 (8) Divide 3J by f . ^w^. ff-. 
 
 (9) Divide 4i by 8|. -4*w. H. 
 
 (10) Divide f of 9 by 7. Pirst, 9 = f and 7 = |. 
 
 foff--T-i = *XfX \ = \\Arn. 
 
 (11) Divide 17| by 8. Ans. 2}J- 
 
 (12) Divide 654 $ by 9. -4w*. 72 [f. 
 
 (13) "What part of 54 is of 9. 
 
 * + f of ^ = V- X f X * = V a = 10 ^w#. 
 
 (14) Divide 72 by f of 9. ^?s. 12. 
 
 (IT)) Divi.il- J Of 1C 1)V - Of ,S. J;/x. 30. 
 
180 Rule of Three. 
 
 With abbreviations. 
 
 (16) Divide f of f by ? of f . 
 
 ix^xZxi^S^S 
 
 (17) Divide of 12 by if of 24. Ans. J. 
 
 (18) Divide f of A by A off. Ans. 1H = H- 
 
 THE EULE OF THREE DIEECT 
 IN VULQAE EEACTIONS. 
 
 RULE. Prepare the fractions (if needful) as in the pre- 
 ceding rules, and state the question as in the Eule of Three 
 in whole numbers. Then invert the first term (being the 
 divisor), and proceed as in multiplication. 
 
 Lastly, reduce the new fraction to its proper quantify for 
 the answer. 
 
 EXAMPLES. 
 
 (1) If f of a yard cost of a what will f of a yard cost ? 
 
 yd. yd. 3 
 
 As f : | : : f 20 
 
 5)60 
 
 fxfxf = n = J- = 1|Z.= 1Z. 12^. AM. 12*. 
 
 (2) If A of a Ib. cost f of a shilling, what will f of a Ib. 
 cost ? Ans. 2s. 6d. 
 
 (3) If * of a shiUing will buy A of a Ib. Troy, what will J- 
 of-a shilling buy ? Ans. 1 oz. 4 dwts. 
 
 (4) If i of f of a Ib. be worth f&l. what are 6 Ib. worth ? 
 
 Ans. 21. 7s, Gd. 
 
 (5) If 3 ells cost A of a , what will 10J ells cost ? 
 First, 3J = f eZfo. 10| = AL e n 8% 
 
 ells. ells. 
 
 Then, As : A : : V 
 
 3 
 
 ? 8 ?/ 8x3 = 24 
 
 And,; X 12 X ? 12 12' ~ 
 
 (6) Bought 10| Ibs. of butter for 12|s. I demand the wort 
 of 16 i Ibs.? Ans. 19*. 3d 
 
 (7) Sold 8| Ibs. of cheese for 6/2*., what is the worth of 
 12jf Ibs. ? Ans. 9s. 
 
Vulgar Fractions. 131 
 
 (8) If 7 Ib. cost II. 6s. Set. what will 12j Ibs. cost ? 
 First 7 lb.=l II. II. Qs. 8<J.=lR=iZ. 12|=V #. 
 
 Then, AsiZft. : |Z. :: A < 1 Z. 
 f X f X V =="iy 3 -==H= V- = 2R = 2Z. 8^. 6^. 3 
 
 (9) If 3| yards cost 2Z. 13s. 4d., what will 18 & yards cost : 
 
 Am. 14Z. 14^. 3Jrf. H. 
 
 (10) Bought TT Ibs. for 7s. 6J., I demand the worth ol 
 Ibs. at the same rate ? .^s. 13^. 11 d 
 
 (11) If If of a will buy 3 Ibs. 8| oz. (troy), what will 
 5|ofa buy? 
 
 First, 1*=*Z. 3 Z&s. 8J o.=3H Z5*.=!f Zft. 5}=V= a /Z. 
 
 Z6. 864)10235(11 tt. 
 
 Then, As f : :: V 864 
 
 s X -If X V-==H*i A =ll^- 10 a*. 3 cfotf . 1 yr.f If 1595 
 Ans. 
 
 731 
 12 &c. 
 
 (12 If f of a shilling will buy 3| yards, what will f of a 
 buy ? Ans. 76 yds. 
 
 (13) If 3i cwt. of sugar cost 12 &Z., what will 1| cwt. 
 cost at that rate ? Ans. Gl. 9s. 
 
 (14) If 3 yards of broad cloth cost 3|Z., what will 4 
 pieces cost, each 2G| yards ? 
 
 yds. yds. 
 
 First, 3=?- 3J=V 26f=H I 
 
 . . 
 
 Then, As f- : ^ :: *i* 24 ) 3317 ( 138Z. is. 2d. 
 
 i X V x H i=a 3 F=138Z. 4s. 2 J. 
 
 (15) Bought 3J pieces, each 25 f- ells, at 6,*. 3<7. per ell, 
 what did it cost me ? Ans. 30Z. Os. 7d. 3 \. 
 
 (16) If one ell Flemish cost 4 AS., what will 7 yards corne 
 to ? Ans. 2Z 4s. 2rf. 
 
132 Vulgar Fractions. 
 
 THE RULE OF THREE INVERSE 
 IN VULGAR FRACTIONS. 
 
 RULE. Proceed in all respects as in the preceding rule ; 
 except that in Inverse Proportion the third term (being the 
 divisor) must be inverted. 
 
 EXAMPLES. 
 
 (1) If 12 men in 10 1 days can mow 120 \ acres, in how 
 many days will 15 men do the same ? 
 
 And supposing they work 16 hours per 
 
 day, then 16 
 men. days. men. x by 2 
 
 V ' V '' V 5)32(6Aow. 
 
 2 rem. 
 
 y x V x T *5 = VT? = V = 81- days = 8 days 6 hours. Ans. 
 
 (2) If G men will mow 18J- acres in 3| days, how many 
 men will do the same in 10| days ? Ans. 2 men. 
 
 (3) If a traveller performs a journey in 6 days, when the 
 days are 10| hours long, how many days will he require of 
 15| hours each ? Ans. 4 days. 
 
 (4) If 28 men can build a house in 50f days, how many 
 men could do the same in 12 {\ days ? Ans. 112 men. 
 
 (5) How many pieces of cloth, at 34fs. per piece, are to 
 be given for 136 J pieces at 50|s. a piece ? Ans. 198 ^ pieces. 
 
 (G) A lends E 751?. for months ; how long ought E to 
 lend A 225 1 1. to requite his kindness ? 
 
 First, 75J =^J A 9= f- 225f = -HP- 
 Then, As H* : f : : H** 
 A t* X f x rA-fc =--8-! = 
 
 (7) If the penny loaf weighs 12 J ounces, when wheat is 
 4fs. per bushel, what must it weigh when wheat is at Gfs. 
 per bushel ? Ans. 8 oz. 
 
 (S) Tn what time will 33GJ7. gain 84 jZ. interest, if 280 J7. 
 will g;iin it in G years ? 
 
Vulgar Fractions. 133 
 
 (9) If 4 men in 12 f days mow a field of barley of 39 acres, 
 in how many days will 18 men do the same ? Ans. 2% days. 
 
 men. days. men. 
 
 f X V* X A = W = 2J- 
 
 (10) How much in length that is 8| broad, will be equal in 
 measure to another piece 12f broad and 24 J long ? 
 
 Ans. 37| yards. 
 
 (11) "What quantity of shalloon that is 3 quarters wide, will 
 line 12 j yards that is 2^ yards wide ? Ans. 38J yards. 
 
 THE DOUBLE EULE OF THREE IN 
 VULGAR FKACTIONS. 
 
 RULE. Having prepared the fractions as before directed, 
 state the question and work as in whole numbers (see page 
 64). 
 
 EXAMPLES. 
 
 (1) If 6 men can reap 18 1 acres in 5| days, how many 
 acres will 10 men reap in 12 J days ? 
 
 First, 18 1 acres = -*-/ acres. 5 \ = V da. 12 \ = Y fo. 
 
 mew. acre*. men. 
 
 Then*f : V- :: * 
 
 days. days. 
 
 #11 _ , .. AA 
 
 2 2 
 
 1 75 10 2 25 18759 9375 
 6 X 4 X I X fl X - -- 
 
 (2) If the carriage of 3J cwt. for 20f miles cost 2%s., what 
 will the carriage of 10 cwt. cost, being carried G2^ miles ? 
 
 Ans. 1Z. 1$, 
 
 (3) If 150Z. in 12 months gain 7JZ. interest, what principal 
 will gain 5/. in li years ? A?is. 66/. 13s. 4<d. 
 
 (4) If 12 men in 9^ days mow 100 acres, how many men 
 can ino'A 150 acres in ls days? Ans. men. 
 
( 134 ) 
 
 DECIMAL FRACTIONS. 
 
 NUMERATION. 
 
 DECIMAL FRACTIONS are so called from the Latin 
 decem, ten, because their denominators are 10, 300, 1000, 
 and so on, in a tenfold proportion; the number of noughts 
 being always equal to the number of figures in the numerator, 
 
 thim 3. 3.5. 12.3. J.3.2.1 jPrp 
 
 uuu.0 i Qj 100J lOOOj 1OOOOJ tJLVy. 
 
 And as it is evident that the denominator of a decimal frac- 
 tion is always known, it is never set down ; the numerator 
 being distinguished from the whole number by a comma 
 prefixed. Hence 7 A, 12 A^o, 36 A a <M are expressed as 7,5 ; 
 12,25 ; 36 ; 365, &c. ; and read thus : Seven and five-tenths, 
 twelve and 25 hundredths, thirty-six and 3G5 thousandths 
 
 Decimals express the parts of a unit, in the same manner 
 as whole numbers express the number of units ; which may 
 appear plainer by the following table : 
 
 Whole Numbers. Decimal Parts. 
 
 76 54 321 ,2 3 45 67 
 
 IvsjII&JIJsTFt 
 
 P | 3 I oooooo 
 
 1 I I F H W g * g 
 
 P P ' S S o H H t 
 
 I I 'Urn 
 
 From the above table it may be observed 1st. That as 
 whole numbers increase in a tenfold proportion to the left 
 hand, decimal parts decrease in a tenfold proportion to the 
 right hand. 
 
 2nd. Cyphers placed lefore decimal parts decrease their 
 value, by removing them farther from the comma or unit's 
 place ; thus ,5 is 5 parts of 10, or A ; but ,05 is 5 part of 
 100, or T o 5 ,005 is 5 parts of 1000, or oVo, &c. 
 
 3rd. Cyphers after decimal parts do not alter their value, 
 for .5 ,50 ,500 &c., are each but A of the unit. 
 
 Decimals are of different kinds, and are termed Jinife or 
 terminate; infinite or interminate; single repetends or re- 
 curring ; compound repetends or circulate ; also pure, similar, 
 and dissimilar repetends, &c. 
 
Decimal Fractions. 135 
 
 A FINITE or TERMINATE DECIMAL, is that which ends at 
 some certain number of places ; but an INFINITE or INTER- 
 MINATE has no end. 
 
 A RECURRING DECIMAL is that in which one or more figures 
 are continually repeated. 
 
 A SINGLE REPETEND, or recurrlnc] decimal, is one in which 
 the same figure is repeated, as ,3333, &c. 
 
 A COMPOUND EEPETEND, or circulate, is when more than 
 one figure is repeated, as ,275275275, &c. 
 
 PURE REPETENDS have no figures in them but what are 
 repeated, as ,364364, Ac. 
 
 MIXED REPETENDS have significant figures between the 
 repetend and the decimal point, as ,645321321, &c. 
 
 SIMILAR REPETENDS begin at an equal distance from the 
 decimal point ; and DISSIMILAR REPETENDS begin at dif- 
 ferent places from the decimal point. 
 
 ADDITION OE DECIMALS. 
 
 RULE. Write down the numbers so that the decimal points 
 may stand in a line directly under each other, and find their 
 sum as in whole numbers. 
 
 EXAMPLES. 
 
 (1) Add 36,432 + 532,1234 (2) Add 3,005 + 100,6 + 
 + 26,019 + 160,0876 + 31,976 + 4,35 + 7,0865 + 
 86,2015 + 8,9765 together. ^4321, together. 
 
 36,432 
 
 532,1234 
 
 26,019 
 
 160,0876 
 
 86,2015 
 
 8,9765 
 
 849,8400 
 
 3,005 
 100,6 
 31,976 
 4,35 
 
 7,0865 
 4321, 
 
 4468,0175 
 
 (3) Add 73,143+46,32 + 7,905+46,731 + 500,5001 to- 
 gether. Ans. 674,5991 
 
 (4) Add 234,6 + 3210 + 54,321+ 9,001+7,1324 + 1000 
 together. Ans. 4515,0544 
 
 (5) Add 46,5094 + 6,123 + 8564, + ,6982 + 360,063 + 
 1231,2312 together. Ans. 10208,6248 
 
136 
 
 Decimal Fractions, 
 
 SUBTRACTION OF DECIMALS. 
 
 (8) From 3960,0076 
 Take 578,900864 
 
 HULE. Place the numbers as in Addition, having the 
 decimal points under each other ; then subtract as in whole 
 numbers. 
 
 EXAMPLES. 
 
 (1) From 36,243 take 4,8941 
 Take 4,8941 
 
 Ans. 31,3489 difference 
 
 (2) From 650,231 take 9,765 
 
 (3) From 32,402 take 15,43 
 
 (4) From 132,3461 take ,987 
 
 (5) From 2341, take ,2341 
 
 (6) From 10, subtract ,324 
 
 (7) From ,00623 take ,00094 
 
 Ans. 3381,106736 differ. 
 
 (9) From 679,2 take 6,792 
 
 (10) From584,312take58,4312 
 
 (11) From 234,5 take 2,3567 
 
 (12) From 4300, take ,0003 
 
 (13) From ,3205 subtr. ,27981 
 
 (14) Froml2,3123takel,23123 
 
 MULTIPLICATION OF DECIMALS. 
 
 RULE 1. Place and multiply the factors as in whole 
 numbers. 
 
 2ndly. Point off as many figures from the product for 
 decimal places, as there are decimals in both the factors. 
 
 3rdly. If there should not be so many figures in the pro- 
 duct, supply the defect by prefixing ciphers. 
 EXAMPLES. 
 
 (1) Multiply 243,21 by 35,6 
 243,21 
 35,6 
 
 145926 
 121605 
 72963 
 
 8658,276 
 
 (2) Multiply 34,56 by 2,437 
 
 (3) Multipl'y 27,16 by 467,1 
 
 (4) Multiply 2000, by ,203 
 
 (5) Multiply 1209,6 by 6,032 (12) Multiply ? 321 by ,321 
 
 (6) Multiply 254, by ,032 
 
 (7) Multiply 376, by ,0025 
 
 (8) Multiply ,0234 by ,123 
 ,0234 
 ,123 
 
 702 
 468 
 234 
 
 ,0028782 
 
 (9) Multiply ,2345 by ,0015 
 
 (10) Multiply ,0123 by ,0152 
 
 (11) Multiply ,071 by ,124 
 
 (13) Multiply 3,42 by ,001 
 
 (14) Multiply ,2631 by ,0054 
 
 When any number is to be multiplied by 10, 100, 1000, &c., it is done 
 by only removing the separating point in the multiplicand, so many places 
 towards the right hand, as there are ciphers in the multiplier, thus, 
 5.43-2x10 = 54.32; and 5,432x100 = 543,2 and 543,2x1000^5432; 
 
 an. I o.lUJx l 
 
Decimal Fractions. 187 
 
 CONTRACTED MULTIPLICATION OF DECIMALS. 
 
 When only a certain number of decimal places are to be 
 retained. 
 
 RULE 1st. Write the multiplier under the multiplicand 
 in a contrary order to what is usual ; and to ascertain their 
 exact position, consider how many decimal places are to be 
 retained in the product, and place the unites place of the 
 multiplier under the last decimal in the multiplicand which 
 is to be reserved. 
 
 2ndly. In multiplying, reject all the figures that stand to 
 the right of the figure you are multiplying by ; and place the 
 figures of each line so that the right hand figures may form 
 an even column. 
 
 Srdly. But when you multiply the rejected figures, carry 
 1 at 5 and upwards ; at 15 and upwards carry 2 ; at 25 and 
 upwards carry 3 ; at 35 and upwards carry 4, &c. 
 
 EXAMPLES. 
 
 (15) Multiply 32,542163 by 23,5463, and retain only 4 
 places of decimals in the product. 
 
 Contracted way. 
 
 32,542163 32,542163" 
 
 3645,32 23,5463 
 
 6508433 97,626489 
 
 976265 195252978 
 
 162711 130168652 
 
 13017 162710,815 
 
 1952 97626489 
 
 98 6508432J6 
 
 766,2476 766,2475326469 
 
 (16) Multiply 276,4301 by 64,3265, and leave only 2 places 
 of decimals in the product. Ans. 17781,79 
 
 (17) Multiply 3124,0651 by 123,456, and leave only 2 
 places of decimals in the product. Ans. 385674,57 
 
 (18) Multiply 2,41342 by 5432,1, and leave 5 places of 
 decimals in the product. Ans. 13109,93878 
 
 (19) Multiply 4,252603 by 63,8549, and leave only 4 
 places of decimals. Ans. 271,5495 
 
138 Decimal fractions. 
 
 (20) Multiply 3242,431 by 324,5164, having no decimal 
 fractions in the product. 
 
 3242,431 3242,431 
 
 4615,423 324,5164 
 
 972729 1,2969724 
 
 64849 19454586 
 
 12970 3242431 
 
 1622 16212155 
 
 32 12969724 
 
 19 6484862 
 
 1 972729i3 
 
 1052222 1052222,0353684 
 
 DIVISION OF DECIMALS. 
 
 ETJLE 1st. Divide as in whole numbers ; and point off 
 from the quotient as many figures for decimals as the divi- 
 dend has more than the divisor. 
 
 2nd. If the dividend has not so many places of decimal 
 parts as are in the divisor, or if there be a remainder, then 
 annex cyphers to the dividend, and carry on the quotient 
 further to any extent required. 
 
 3rd. If the quotient, when carried, have not the number 
 of decimal parts wanted, the number must be made up by 
 putting a cypher or cyphers on its left hand. 
 
 4th. If both the divisor and dividend have the same number 
 of decimal parts, the quotient will be a whole number. 
 
 EXAMPLES. 
 
 (1) Divide 42,576 by 1,2. 
 
 1,2)42,576 
 
 35,48 
 
 (2) Divide 6,7342 by 1,1 
 
 Ans. 6,122 
 
 (3) Divide 67,342 by 11 
 
 Ans. 6,122 
 
 (4) Divide 425,84 by 8 
 
 Ans. 53,23 
 
 (5) Divide 5,25784 by 120 
 
 120)4,25784 
 
 ,035482 
 
 (6) Divide 67,837 by 1,1 
 
 Ans. 61,67 
 
 (7) Divide 8,97658 by ,7 
 
 Ans. 12,8236 + 
 
 (8) Divide 463,26 by ,6 
 
 Ans. 772,1 
 
Decimal Fractions. 
 
 139 
 
 (9) Divide 425,76 by ,12 
 ,12 ) 425,76 
 
 3548, 
 
 (10) Divide 32684. by ,8 
 
 Ans. 40855. 
 
 (11) Divide 9876. by 7 
 
 Ans. 1410,8 + 
 
 (12) Divide 3254. by 11. 
 
 Ans. 295,8 + 
 
 (13) Divide 3264,36 by ,12 
 
 ,12) 3264,36 
 
 27203, 
 
 (14) Divide 21321,9 by .9 
 
 Ans. 23691 
 
 (15) Divide 342361,2 by ,6 
 
 Ans. 570602 
 
 (16) Divide 1000,02 by ,07 
 
 Ans. 14286, 
 
 KB. "When the divisor is either 10, 100, 1000, 10000, 
 &c. it is only necessary to remove the decimal point so many 
 places towards the left hand, as there are cyphers in the 
 divisor. 
 
 Thus, 9876. ~ 10 = 987,6 12,34 - 10 = 12,34 
 
 5432. ~ 100 = 54,32 43,21 - 100 = ,4321 
 
 3540, ~ 1000 = 3,540 534,2 ~ 1000 = ,5342 
 
 DIVISION. 
 
 (23) Divide 6978,6 by 3,24 
 3,24 ) 6978,600 ( 2153,8 Ans. 
 648 
 
 (17) Divide 73,89785 by 42,5 
 
 42,5 ) 73,89785 ( 17,387 Ana. 
 
 425 
 
 3139 
 2975 
 
 1647 
 1275 
 
 3728 
 3400 
 
 3285 
 2975 
 
 310 
 
 (18) Divide 72^3456 by 54,32 
 
 Ans. 1,3318 + 
 
 (19) Divide 13,426 by 12,4 
 
 Ans. 1,0827 + 
 
 (20) Divide ,5982 by 126. 
 
 Ans. ,004747 + 
 
 (21) Divide 12,54 by 6789. 
 
 Ans. ,00184 + 
 
 (22) Divide 34574. by 3162 
 
 Am. 10,931+ 
 
 498 
 324 
 
 1746 
 1620 
 
 1260 
 972 
 
 2880 
 2592 
 
 288 
 
 (24) Divide 3765,4 by ,284 
 
 Ans. 13258 + 
 
 (25) Divide 11269,8 by 1,116 
 
 Ans. 10098,3 + 
 
 (26) Divide 76, by ,1254 
 
 Ans. 606 
 
 (27) Divide 467,2 by 11,4 
 
 Ans. 40,982 + 
 
 (28) Divide 925G3. by 7,214 
 
 Ans. 12831. + 
 
140 Decimal Fractions. 
 
 CONTEACTED DIVISION OF DECIMALS. 
 
 ETILE 1st. Consider how many of the left hand figures 
 of the divisor are equal to the number of whole numbers 
 and decimals that are wanted in the quotient, and with 
 these divide ; omitting one figure of the divisor at each suc- 
 ceeding operation. 
 
 2nd. In multiplying the figures left out in the divisor, 
 carry 1 at 5, and upwards ; carry 2 at 15, and upwards ; 3 at 
 25, &c. 
 
 EXAMPLES. 
 
 (29) Divide 36,45413 by 3,54621, and leave only 3 places 
 of decimals. 
 
 By ike common way. 
 
 35,4621 ) 36,45413 ( 10,279 3,54621 ) 36,45413000 ( 10,279 
 
 35462 354621 
 
 .992 ..992030 
 
 709 709242 
 
 283 2827880 
 
 248 248 2347 
 
 .35 .3455330 
 
 32 31 91589 
 
 3 .263741 
 
 (30) Divide 735,851 by 42,5, and leave only 3 places of 
 decimals. Ans. 17,314 + 
 
 (31) Divide 72,3456 by 54,32, and leave 4 places of 
 decimals. Ans. 1,3318 + 
 
 (32) Divide 6978,6 by ,324, and leave 1 place of decimals. 
 
 Ans. 21538,8 + 
 
 EEDUCTION OF DECIMALS. 
 
 CASE I. To reduce a Vulgar Fraction to a Decimal of 
 the same value. 
 
 EULE. Annex cyphers to the numerator, and divide by 
 the denominator, continuing the operation as far as may be 
 needful ; the quotient will be the decimal fraction required. 
 
Decimal Fractions. 
 
 EXAMPLES. 
 
 (1) Reduce f to a decimal. 
 
 8)3,000 
 
 ,375 Ans. 
 
 (2) Reduce to a decimal. 
 
 Ans. ,25 
 
 (3) Reduce \ to a decimal. 
 
 .4ws. ,5 
 
 (4) Reduce f to a decimal. 
 
 Ans. ,75 
 
 (5) Reduce 7f to a decimal. 
 
 9)6,0000 
 
 Ans. 7,6666 &c. 
 
 ,6666 &c. 
 
 (6) Reduce 8 if- to a decimal. 
 
 Ans. 8,333;* &c. 
 
 (7) Reduce 110 ft-to a decimal. 
 
 Ans. 110,5714 &c. 
 
 (8) Reduce 158 M- to a de- 
 cimal. Ans. 158,066298 
 
 (9) Reduce -J- of f- of f to 
 a decimal. 
 
 f*, 8 v/ 3 120 5 
 X ^ X T 'Sins' If 
 
 9)5,0000 
 
 ,5555 &c. ^4w*. 
 
 (10) Reduce -A- of H to a 
 decimal. .^rcs. ,171875 + 
 
 (11) Reduce A- of J to a 
 decimal. ^b*s. ,02 
 
 (12) Reduce f of 8 to a 
 decimal. Ans. 
 
 II. 
 
 (13) Reduce -f-^/9 to a decim. 
 8469 )368,0000( ,04345 &c. Ans. 
 33876 
 
 .29240 
 25407 
 
 38330 
 33876 
 
 .44540 
 42345 
 
 2195 &c. &c. 
 
 (14) Reduce - 6 W 4 - to a de- 
 cimal. Ans. ,050854 -f 
 
 To reduce money , weights, measures, fyc., to equivalent 
 decimals. 
 
 RULE 1st. If the quantity given be but of one denomina- 
 tion, divide it by as many as make one of the denomination 
 sought, and the quotient will be the answer required. 
 
 2nd. If the quantity given be of different denominations, 
 reduce it to the lowest name mentioned for a dividend ; then 
 reduce the integer into the same denomination for a divisor ; 
 the quotient will be the decimal required. 
 
 Or 3rdly. Write the given numbers perpendicularly under 
 each other, for dividends ; then divide each line (beginning 
 with the uppermost) by that figure which will raise it to the 
 next superior name ; and the last quotient will give the 
 decimal required. 
 
112 
 
 Decimal Fractions. 
 
 By 'Rule 1st. 
 (15) Eeduce 5s. to the de- | (21) Eeduce Qd. to the de- 
 
 cimal of a . 
 20)5,00 
 
 ,25 Ans. 
 
 (16) Eeduce 7*. to the de- 
 cimal of a . Ans. ,35 
 
 (17) Eeduce 10<J. to the de- 
 cimal of a shilling. A. ,83jJ + 
 
 (18) Eeduce Id. to thedeci- 
 mal of a shilling. A. ,583 + 
 
 (19) Eeduce 3 forth, to the 
 decimal of a penny. Ans. ,75 
 
 (20) Eeduce 1 forth, to the 
 decimal of a penny. Ans. ,25 
 
 cimal of a . 
 In a = 240)9,0000 
 *{* = & Q375 Aru. 
 
 (22) Eeduce Sd, to the de- 
 cimal of a . Ans. ,0333 + 
 
 (23) Eeduce 10 d. to the de- 
 cimal of a crown. A. ,1666 + 
 
 (21) EeduceSfarth.tothe 
 decim. of a shilling. A. ,0625 
 
 (25) Eeduce to the deci- 
 mal of a crown. A. ,004166 + 
 
 (26) Eeduce 16s. 10 J<?. to the decimal of a pound. 
 By Rule the 2nd. Or, by Rule the 3rd. 
 
 20*. 
 
 12 
 
 240 
 4 
 
 960 
 
 12 
 
 202 
 4 
 
 960)810,00000(, 84375 Ans. 
 7680 
 
 12 
 20 
 
 2,0 
 
 10,500 
 
 16,87500 
 ,84375 Ans. 
 
 420 &c. 
 '27) Eeduce 3s. 4J to the decimal of a . Ans. ,16875 
 
 (28) Eeduce 18s. llf d.to the decimal of a .Ans. ,9489583 
 
 (29) Eeduce 16s.6]d. to the decimal of a .Ans. 9 82601166 
 
 (30) Eeduce 3s.7|d to the decim. of a guinea.^. ,172619 
 
 (31) Eeduce 3s. Qd. to tho decimal of a crown. Ans. ,7 
 
 (32) Eeduce 2s. 10* d. to the decim. of a moidore. ^.,107253 
 
 (33) Eeduce 12 dwts. 20 gr. to the decim. of an ounce Troy. 
 First, 12 dwts. 20 gr. = 308 grains. and 1 ounce = 480 grains. 
 
 Whence f| = ,641666+ Ans. 
 
 (34) Eeduce3qrs.l61bs. to thedecim. of acwt.J., 892857 + 
 
 (35) Eeduce 2 feet 4 in. to the decimal of a yard. A. ,777 + 
 
 (36) Eeduce 1 mile 2 fur. to the dec. of aleague.^.,41660 + 
 
 (37) Eeduce 8 oz. 10 dwts. to the decimal of a Ib. Troy. 
 
 Ans. ,7083 + 
 
 (38) Eeduce 4 qts. 1 pt. to the decimal of a barrel. (36 galls.) 
 
 Ans. ,03125 
 
 (39) Eeduce 3 qts. 1 pt. to the decimal of a gallon. A. ,875 
 
 (40) Eeduce 7J inches to the decimal of a foot. Ans. ,625 
 
 (41) Eeduce 75 days to the decimal of a year. Ans. ,20547 
 
Decimal Fractions. 
 
 143 
 
 III. To find ilie value of a decimal fraction. 
 KULE. Multiply the decimal by the number of pnrts in 
 the next inferior denomination, cutting off the decimals from 
 the product : then multiply the remaining decimals by the 
 next inferior denomination, and cut off the decimals as 
 before ; thus proceeding to the least known parts of the 
 integer; and the several figures cut off on. the left hand will 
 be the answer. 
 
 EXAMPLES. 
 
 (42) "What is the value of 
 ,568 of a sterling ? 
 
 
 ,563 
 20 
 11,300 
 
 12 
 
 4,32 
 
 4 
 
 1,28 Ans. 11s. 4\d. ,28 
 
 (43) "What is the value of 
 ,3625 of a ? Ans. 7s. 3d. 
 
 (44) What is the value of 
 ,485 of a ? Ans. 9s. Sd. 
 
 (45) What is the value of 
 ,086 of a moidore ? A. 2s. 3*d. 
 
 (46) What is the value of 
 ,3188 of a shilling ? 
 
 8,188 
 
 12 
 
 8,8256 
 
 4 
 
 3,3024 
 Ans. 3j d. ,31)24 
 
 (47) What is the value of 
 ,6376 of a shilling ? 
 
 Ans. lid. ,6048 
 
 (48) What is the value of 
 ,3225 of a shilling ? Ans. 3f d. 
 
 (49) What is the value of 
 ,75 of a penny ? Ans id. 
 
 (50) What is the value of 
 ,8638 of a ton? 
 
 cwt. 17,2760 
 
 qr. 
 
 1,104 
 28 
 
 Ib. 2,912 
 16 
 
 oz. 14,592 
 
 16 
 
 dra. 9,472 
 
 (51) What is the value of 
 ,8625 of a ton ? 
 
 Ans. 17 cwt. 1 qr. 
 
 (52) What is the value of 
 ,0087 of a Ib. troy ? 
 
 Ans. 2 dwt. 2 gra. 
 
 (53) What is the value of 
 ,305 of a pipe of wine ? 
 
 pipe. 
 
 ,305 
 
 126 
 
 1830 
 
 3660 
 
 gall. 3M30 
 
 4 
 
 quart 1,72 
 
 2 
 
 pint 1,44 
 
 (54) What is the value of 
 ,6789 of a mile ? 
 
 Ans. 1194 y^. 2ft. 7 in. 
 
 (55) What is the value of 
 ,58 of a year ? 
 
 Ans. 211 da. 16 lio. 48 min. 
 
( 144 ) 
 
 Decimal Tables of Coins, Weights, and Measures. 
 
 TABLE T 
 
 Farth. 
 
 Decimals. 
 
 Grains. 
 
 Decimals. 
 
 
 
 
 3 
 
 ,0625 
 
 12 
 
 ,025 
 
 ENGLISH COIN. 
 
 1 the Integer. 
 
 1 
 1 
 
 ,041666 
 ,020833 
 
 11 
 10 
 
 ,022916 
 ,020833 
 
 sh. 
 
 , 
 
 sh. 
 
 , 
 
 
 9 
 
 ,01875 
 
 19 
 18 
 
 ,95 
 ,9 
 
 9* 
 
 8 
 
 ,45 
 ,4 
 
 TABLE III. 
 
 8 
 
 7 
 
 ,016666 
 ,014583 
 
 17 
 
 ,85 
 
 7 
 
 3Z. 
 O 
 
 TROY WEIGHT, 
 
 6 
 
 ,0125 
 
 16 
 
 ,8 
 
 6 
 
 ',3 
 
 1 lb. the Integer. 
 
 5 
 
 ,010416 
 
 15 
 
 ,75 
 
 5 
 
 ,25 
 
 Ounces the same as 
 
 4 
 
 ,008333 
 
 14 
 
 ,7 
 
 4 
 
 ,2 
 
 Pence in the la&t Ta- 
 
 3 
 
 ,00625 
 
 13 
 
 ,65 
 
 3 
 
 ,15 
 
 lk. 
 
 2 
 
 ,004166 
 
 12 
 11 
 
 ,6 
 ,55 
 
 2 
 1 
 
 ,1 
 
 ,05 
 
 Penny- 
 
 Decimals. 
 
 
 ,002083 
 
 10 
 
 ,5 
 
 
 
 we^ 
 
 ,041666 
 
 
 
 
 
 
 9 
 
 ,0375 
 
 TABLE IV. 
 
 Pence. Decimals. 
 
 6 ,025 
 
 
 8 
 
 ,033333 
 
 A.VOIRDU. WT. 
 
 5 ,020833 
 4 ,016666 
 
 7 
 6 
 
 ,029166 
 ,025 
 
 1 12 lb. the Integer. 
 
 3 ,0125 
 
 
 5 
 
 ,020833 
 
 Qrs. 
 
 Decimals. 
 
 2 ,008333 
 
 4 
 
 ,016666 
 
 3 
 
 ,75 
 
 1 ,004166 
 
 3 
 
 ,0125 
 
 2 
 
 ' 5 r 
 
 
 
 
 2 
 
 ,008333 
 
 
 
 Farth. Decimals. 
 
 i 
 
 Ofvf 1 ac\ 
 
 
 ,2o 
 
 3 ,003125 
 
 JL 
 
 , \J\Jtfc LUO 
 
 Pounds 
 
 Decimals* 
 
 2 ,0020833 
 
 Grains. 
 
 Decimals. 
 
 14 
 
 ,125 
 
 1 ,0010416 
 
 12 
 11 
 
 ,002083 
 ,Cui910 
 
 13 
 12 
 
 ,116071 
 ,107143 
 
 TABLE II. 
 
 10 
 9 
 
 ,001736 
 ,001562 
 
 11 
 10 
 
 ,098214 
 ,089286 
 
 ENG. COIN, IS. 
 
 8' 
 
 ,001389 
 
 9 
 
 ,080357 
 
 LONG MEASURE, 
 
 7 
 
 ,001215 
 
 8 
 
 ,071428 
 
 1 foot the Integer. 
 
 6 
 5 
 
 ,001042 
 ,000868 
 
 7 
 6 
 
 ,0625 
 ,053571 
 
 Pence 
 and 
 Inches 
 
 G 
 
 Decimals. 
 
 4 
 3 
 2 
 
 ,000694 
 ,000521 
 ,000347 
 
 5 
 
 4 
 3 
 
 ,044643 
 ,035714 
 ,026786 
 
 5 
 
 ^416666 
 
 1 
 
 ,000173 
 
 2 
 
 ,017857 
 
 4 
 
 ,333333 
 
 1 02. the Integer. 
 
 1 
 
 ,008928 
 
 3 
 2 
 
 ,25 
 ,166666 
 
 Pennyweiahts the 
 same as Shillings in 
 
 Ounce*. 
 
 8 
 
 Decimals. 
 ,004464 
 
 1 
 
 ,083333 
 
 the first Table. 
 
 7 
 
 ,003906 
 
( 145 ) 
 
 Decimal Tables of Coins, Weights, and Measures. 
 
 6 
 5 
 4 
 3 
 2 
 1 
 
 ,003348 
 ,002790 
 ,002232 
 ,001674 
 ,001116 
 ,000558 
 
 80 
 70 
 60 
 50 
 40 
 30 
 20 
 10 
 9 
 8 
 7 
 6 
 5 
 4 
 3 
 2 
 1 
 
 ,317460 
 
 ,277777 
 ,238095 
 ,198412 
 ,158730 
 ,119047 
 ,079365 
 ,039682 
 ,035714 
 ,031746 
 ,027777 
 ,023809 
 ,019841 
 ,015873 
 ,011904 
 ,007936 
 ,003968 
 
 Pints 
 
 3 
 2 
 1 
 
 Decimals. 
 ,005952 
 ,003968 
 ,001984 
 
 TABLE VII. 
 
 MEASURES. 
 LIQUID. DRY. 
 
 1 Gallon, 1 Quarter, 
 the Integer. 
 
 t Oz. 
 3 
 2 
 
 1 
 
 Decimals. 
 ,000418 
 ,000279 
 ,000139 
 
 TABLE V. 
 
 AVOIRDU. WT. 
 
 1 lb. the Integr. 
 
 rt*. 
 4 
 3 
 2 
 1 
 
 Dec. 
 ,5 
 ,375 
 ,25 
 ,125 
 
 Bus. 
 4 
 3 
 2 
 1 
 
 Ounces. 
 
 8 
 7 
 6 
 5 
 4 
 3 
 2 
 1 
 
 Decimals. 
 ,5 
 
 ,4375 
 ,375 
 ,3125 
 ,25 
 
 ,1875 
 ,125 
 ,0625 
 
 Pints 
 4 
 3 
 2 
 1 
 
 Decimals. 
 ,001984 
 ,001488 
 ,000992 
 ,000496 
 
 Q.pt. 
 3 
 2 
 1 
 
 Dec. 
 
 ,09375 
 ,0625 
 ,03125 
 
 Peck. 
 
 3 
 
 2 
 1 
 
 Decimals. 
 ,0234375 
 ,015625 
 ,0078125 
 
 Q.M. 
 3 
 2 
 1 
 
 A 
 
 Hogshead 
 the Integer. 
 
 Draws. 
 
 8 
 7 
 6 
 5 
 4 
 3 
 2 
 1 
 
 Decimals. 
 ,03125 
 ,027343 
 ,023437 
 ,019531 
 ,015625 
 ,011718 
 ,007812 
 ,003906 
 
 Decimals. 
 ,005859 
 ,003906 
 ,001953 
 
 Pints 
 
 3 
 2 
 
 1 
 
 Galls. 
 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 
 
 TABLE VIII. 
 
 LONG MEASURE. 
 
 1 Mile the Integer. 
 
 TABLE VI. 
 
 LIQUID MEASURE. 
 
 1 Tun the Integr. 
 
 Yards. 
 1000 
 900 
 800 
 700 
 600 
 
 Decimals. 
 ,568182 
 ,511304 
 .454545 
 ,397727 
 ,340909 
 
 0. ills 
 100 
 90 
 
 Decimal**. 
 
 ,396825 
 ,357141 
 
Decimal Tables of Coins, Weights, and Measures. 
 
 500 
 400 
 300 
 200 
 100 
 90 
 80 
 70 
 60 
 50 
 40 
 30 
 20 
 10 
 9 
 8 
 7 
 6 
 5 
 4 
 3 
 2 
 1 
 
 ,284091 
 ,227272 
 ,170454 
 ,113636 
 ,056818 
 ,051136 
 ,045454 
 ,039773 
 ,034091 
 ,028409 
 ,022727 
 ,017045 
 ,011364 
 ,005682 
 ,005114 
 ,004545 
 
 70 
 60 
 50 
 40 
 30 
 20 
 10 
 9 
 8 
 7 
 6 
 5 
 4 
 3 
 2 
 1 
 
 ,191781 
 ,164383 
 ,136986 
 ,109589 
 ,082192 
 ,054794 
 ,027397 
 ,024657 
 ,021918 
 ,019178 
 
 TABLE X. 
 
 CLOTH MEASURE. 
 
 1 Yard the Integer. 
 
 Qrs. the same as 
 Table 4. 
 
 Nails. 
 3 
 2 
 1 
 
 Decimals. 
 
 ,1875 
 ,125 
 ,0625 
 
 ,016438 
 ,013698 
 ,010959 
 ,008219 
 
 TABLE XL 
 
 LEAD WEIGHT. 
 
 A Foth. the Integer. 
 
 ,005479 
 ,002732 
 
 ffund. 
 10 
 9 
 8 
 7 
 6 
 5 
 4 
 3 
 2 
 1 
 
 Decimals. 
 ,512820 
 ,461538 
 ,410256 
 ,358974 
 ,307692 
 ,256410 
 ,205128 
 ,153846 
 ,102564 
 ,051282 
 
 ,003977 
 ,003409 
 ,002841 
 ,002273 
 ,001704 
 ,001136 
 ,000568 
 
 1 Day the Integer. 
 
 Hours. 
 12 
 11 
 10 
 9 
 8 
 7 
 6 
 5 
 4 
 3 
 2 
 1 
 
 Decimals. 
 
 
 ,458333 
 ,416666 
 ,375 
 ,333333 
 ,291666 
 ,25 
 
 Feet. 
 2 
 1 
 
 Decimals. 
 ,0003787 
 ,0001894 
 
 ,208333 
 ,166666 
 ,125 
 
 Qr. 
 2 
 1 
 
 Decimals. 
 ,025641 
 ,012820 
 
 Inches. 
 6 
 3 
 1 
 
 Decimals. 
 ,0000947 
 ,0000474 
 ,0000158 
 
 ,083333 
 ,041666 
 
 ound. 
 14 
 13 
 12 
 11 
 10 
 9 
 8 
 7 
 6 
 5 
 4 
 3 
 2 
 1 
 
 Decimal*. 
 ,0064102 
 ,0059523 
 ,0054945 
 ,0050366 
 ,0045787 
 ,0041208 
 ,0036630 
 ,0032051 
 ,0027472 
 ,0022893 
 ,0018315 
 ,0013736 
 ,0009157 
 ,0004578 
 
 TABLE IX. 
 
 TIME. 
 
 1 Year the Integer. 
 
 Months the same as 
 pence in the 2nd Table. 
 
 Min. 
 30 
 
 20 
 10 
 9 
 8 
 7 
 6 
 5 
 4 
 3 
 2 
 1 
 
 Decimals. 
 ,020833 
 ,013888 
 ,006944 
 ,00625 
 ,005555 
 ,004861 
 ,004166 
 ,003472 
 ,002777 
 ,002083 
 ,001388 
 ,000694 
 
 Days. 
 365 
 300 
 200 
 100 
 90 
 80 
 
 Decimals. 
 1,000000 
 ,821918 
 ,547945 
 ,273973 
 ,246575 
 ,219178 
 
Decimal Fractions. 14.7 
 
 THE ETJLE OF THEEE IN DECIMALS. 
 
 EXAMPLES. 
 
 (1) If 34f yards cost 51 12s. 6^., what will 104| yards 
 
 cost? 
 
 First, f=,75 12*. 6rf.=,625. i = ,25 
 
 yds. yds 
 
 Then, As 34,75 : 5,625 : : 104,25 
 104,25 
 
 34,75)586,40625(16,875 
 3457 20 
 
 23890 17,500 
 20850 12 
 
 30406 &c. 6,0 Ans. 16Z. 17*. 6d. 
 
 (2) If 12 J yards cost SI. 6s. 8d., what will 18| yards cost 
 at the same rate ? Ans. 51. 
 
 (3) Bought 24 Ib. 4 oz. of tea for 8Z. 16*., what will 60 Ib. 
 10 oz. cost at that rate ? Ans. 221. 
 
 (4) If 1 cwt. 2 qrs. of coffee sell for 30Z. 15s., what will 
 3 cwt. 3 qrs. sell for ? Ans. 76Z. 17s. Qd. 
 
 (5) What is the price of a tankard that weighs 32 oz. 5 
 dwts. at the rate of 5s. 4<d. per oz. ? Ans. 81. 12s. 
 
 (6) If 12f Ib. of butter cost 10s. 4d., what will be the 
 worth of 1 cwt. ? Ans. 4Z. 10s. 9^ d. 
 
 (7) If I buy 12| yards for 10 guineas, how many Flemish 
 ells can I purchase for 52 1. 10s. ? 
 
 First, 12| yds.-^yds and^-of \ o4 = ^ = 16f Hem. 
 o p 1 o b 
 
 2 
 elh := 1 0,83333 Ac. and 107. 10s.=10 } =10,5Z. 52Z. 10s.=52,5Z. 
 
 F.E. 
 
 Then, As 10,5 : 16,8333 : : 52,5 
 52,5 
 
 841666 
 
 3366666 
 
 84166666 
 
 Fl. 09. 
 
 10,5)88374999(84,1066 -H 
 
 Or shorter, thus- 
 Finding the third term 
 to be just 5 times the 
 first, I need only multi- 
 ply by 5. 
 
 16,8333 
 5 
 
 840 
 
 Fl. Ells. 84,1666 
 Ac. 
 
 n 2 
 
Decimal Fractions. 
 
 (S) ITow many yards of Holland can be bought for 32Z. 
 15s. tW if 3 yards cost II. Is. Qd. ? 
 
 ./iftS. 106 y</5. 2 0TS. 3 ftZs. 
 
 (9) How many yards of muslin may be purchased for 
 271 13s. 4d. at the rate of 65. Sd. per yard ? Ans. 83 yds. 
 
 (10) How many gallons of brandy may be bought for 50 
 guineas, if 7| gallons cost 8|Z. ? Ans. 46| galls. 
 
 (11) If 12^. 9fi?. will pay the carriage of 12 cwt. 3 qrs. from 
 London to Bath, what weight can be carried for 11. 16s. 
 10 \d. ? Ans. 36 cwt. 3 qrs. 14 Ibs. 
 
 (12) If a chest of sugar, weighing 6 cwt. 3 qrs. 7 Ib. cost 
 75Z. 16,9. 8d., what will 3 cwt. 1 qr. 21 Ib. be worth ? 
 
 First, 6 cwt. 3 qrs. 7 lb. = 6,8125 cwt. 75Z. 16s. 8rf. = 75,8333 + 
 
 3 cwt. 1 qr. 21 II. = 3,4375 cwt. 
 
 cwt. cwt. 
 
 Then, As 6,8125 : 75,8333 ; ; 3,4375 
 3,4375 
 
 3791666 
 5308333 
 2274999 
 3033333 
 2274999 
 
 6,8125)260,67697896(38,2645 
 204375 20 
 
 563019 5,2900 
 
 545000 12 
 
 18019&C. 3,4800 
 
 =. 4 
 
 Ans. 88J. 5. 3Jd. 1,92m) = 
 
 Bought a quantity of tobacco, which was found to 
 1 cwt. 3 qrs. 14 Ib. 6 oz. for 21Z. 0*. 9rf., I demand 
 the worth of 2 qrs. 14 Ibs. 2 oz. 
 
 (14) What is the worth of 1 Ib. 8 oz. 10 dwts. of go-Id, at 
 3Z. 4s. Qd. per ounce ? 
 
 (15) If If of a will buy 3 Ib. 8} oz. avoirdupois what 
 will 5 1 of a buy ? Ans. 11 Ib. 40*. 7 dr. + 
 
 (16) If 6 cwt. 2 qrs. 14 Ib. cost 21 19*. 6rf., what will 
 1 ton 19 cwt. 3 qrs. cost? 
 
Decimal Fractions. 14>9 
 
 INVERSE PEOPOETIOJST IN DECIMALS. 
 
 (1) If the carriage of 12 cwt. 3 qrs. of goods for 100^ 
 miles cost II. 5s. 6^., how much ought I to have carried 75| 
 miles for the same money ? 
 
 cwt. qrs. 
 First, 12 3 = 12,75 lOOf = 100,375 mi. 75 = 75,75 mi. 
 
 miles. cwts. miles. 
 
 Then, as 100,375 : 12,75 : : 75,75 
 100,375 
 
 75,75 ) 1279,78125 ( 16,8948 = 16 cwt. 3 qrs. 1C Ib. 3 02. 
 150 rem. 
 
 (2) If 18f yards of carpeting that is 1| yards broad will 
 cover a room, how much that is 1 yard broad will cover the 
 same ? Ans. 28,125 = 28| yards 
 
 (3) If I lend my friend 100Z. for f- of a year, how much 
 ought he to lend me A of a year to requite my kindness ? 
 
 Am. 1071. 2s. I0d. + 
 
 (4) How much stuff that is | of a yard wide, will line 27| 
 yards of cloth that is wide ? 
 
 Ans. 38,85 = 38 yds. 3 qrs. 1 nl. + 
 
 (5) If 530Z. 155. will gain 171. 10s. in 10 J- months, what 
 principal will gain an equal sum in 15 months ? 
 
 Ans. 365Z. Ss. 8d. 
 
 (6) If 4 men in 12f days will mow a field, in how many 
 days will 18 men do the same ? Ans. 2,8333 + day* 
 
 DOUBLE BULB OF THEEE IN DECIMALS. 
 
 If 24J bushels of flour be sufficient for 20 men 10J days, 
 how many men will consume 148 J bushels in 20| days ? 
 24 = 24,75 10J = 10,25 148^ = 148,5 20 J = 20,1 
 
 bus. men. bus. Or thus : 
 
 * 24,75 : 20 :: 148,5 men. days. lus. 
 
 days. days. 20 : 10/25 : : 24,75 
 
 10,25 : : : 20,5 : 20,5 : : 148,5 
 
 20 x 148,5 x 10,25 
 
 Then 24,75x 20,5 = 60 men. Ans. 
 
 N.B. If more questions be wanted in this rule, tl ey may be taken frorc 
 the Double Rule of Three in Vulgar Fractious, and worked decimally. 
 
( 150 ; 
 ALLIGATION. 
 
 ALLIGATION is a rule that teaches either to find the 
 value of any compound ; or how to mix things of different 
 values, so as to ascertain the quantities. The whole may be 
 comprised in four cases, viz. : 
 
 MEDIAL, ALTERNATE, PARTIAL, and TOTAL. 
 CASE I. ALLIGATION MEDIAL Is when the rates and 
 quantities of the several ingredients are given, to find the 
 value of the mixture. 
 
 RTJLE. Multiply each quantity of the mixture by its rate ; 
 then divide the sum of the products by the sum of the 
 quantities, and the quotient will give the rate of the mixture 
 required. 
 
 EXAMPLES. 
 
 (1) A tobacconist would mix 60 Ib. of tobacco at 3s, per 
 Ib. with 70 Ib. at 3s. 6d., 75 Ib. at 4s., and 80 Ib. at 4s. 6J. 
 per Ib. : what will 1 Ib. of the mixture be worth ? 
 
 d. d. 
 60 X 36 = 2160 285 ) 13020 ( 12)45 
 
 70 x 42 = 2940 1140 
 
 75 X 48 = 3600 3*. ! 
 
 80 x 54 = 4320 1620 = 
 
 285 13020 
 
 1425 
 
 195 
 4 
 
 285 ) 780 ( 
 rem. 210 
 
 Or thus : 
 
 s. d. 
 
 60 x 3 = 180 
 70 x 3 6 = 245 
 75 X 4 = 300 
 80 x 4 6 = 360 
 
 285 285)1085(8911} 
 
 (2) A farmer would mix 12 bushels of wheat at 6s. Gd. per 
 bushel with 10 bushels of barley at 4s. Qd. per bushel, and 
 8 bushels of rye at 3s. Qd. per bushel ; what is the worth of 
 a bushel of this mixture ? Ans. 5s. O^d. $ 
 
 (3) A vintner makes with a compound a pipe of wine, viz. 
 36 galls, at 12s. per gall, with 40 galls, at 13s. per gall, and 
 50 galls, at 14s. per gall. ; what will a gallon of this mixture 
 be worth ? Ans. 13s. ld. J 
 
 (4) A maltster mixes 20 bushels of high-dried malt at 5s. 6d. 
 per bushel with 15 bushels of pale at 5s. per bushel, and 12 
 bushels at 4s. 9d. per bushel ; what is the value of 1 bushel 
 ot this mixture ? Ans. 5s. l$d. -A 
 
Alligation. 151 
 
 (5) A flour dealer mixes 15 bushels of fine flour at 9s. 6d. 
 per bushel, and 18 bushels at 10s. 4J. per bushel, with 20 
 bushels of seconds at 7s. per bushel, and 24 ditto at 6s, Sd. 
 per bushel ; I demand the worth of a bushel of this mixture? 
 
 Ans. Ss. l$d. -H. 
 
 (6) A composition is made of 18 Ib. of tea at 5s. 6d. per Ib. 
 with 20 Ib. at 5s. Qd., 24 Ib. at 6s. 3d., and 16 Ib. at 6s. 6d. 
 per Ib. ; what is the worth of 3 Ib. of this mixture ? Ans. 18s.* 
 
 CASE II. ALLIGATION ALTERNATE is when the rates 
 of several things are given, to find what quantity of each must 
 be taken, to make a mixture of a certain mean value. 
 
 BULE 1st. Place the rates of the ingredients under each 
 other ; and place the mean rate on the left hand of them. 
 
 2nd. Link the several rates together, so that one greater 
 than the mean rate may be joined to one that is less. 
 
 3rd. Take the difference between each price and the 
 mean rate, and place it opposite to the rate to which it is 
 linked. 
 
 4th. If only one difference stand against any rate, that 
 difference will be the answer ; but if more than one, their sum 
 will be the answer. 
 
 PBOOF. By Alligation Medial. 
 EXAMPLES. 
 
 (1) A grocer would mix sugar at 10^., 9^., 7d., and 6^. per 
 Ib. to make a mixture worth Sd. per Ib. ; how much of each 
 sort must he take ? . 
 
 d. Ib. d. 
 10 1 ... 2 at 
 
 .... 2 at 6 
 
 ON 91 
 
 8) 7 J 
 
 Or they may be linked thus : 
 d. Ib. d. 
 
 10 ... lat: 
 
 9 , ... 2 at 9 I 2nd-\- 
 
 7 1 ... 2 at 7 [ Ans. 
 
 Q 1 ... 1 at 6J 
 
 (2) A tobacconist would mix tobacco at 3s. 6^., 3s., 2s., 
 
 and l&d. per Ib. ; what quantity of each must he take to 
 
 make a mixture worth 2s. 6d. per Ib. ? 
 
 Is* Ans. 12 Ib. at 3s. Gd., 6 Ib. at 3s., 6lb. at 2s., 12 Ib. at l&d. 
 
 2nd Ans. 6 Ib. at 3s. 6d., 12 Ib. at 3s., 12 Ib. at 2s., 6 Ib. at l&d. 
 
 * Where the worth of more than 1 Ib. &c. is wanted, find the value of 
 1 Ib. &c. as before, and multiply it by the number of Ibs. &c. required. 
 
 *t* Note Questions in this rule admit of different answers, according 
 to the manner of linking them. Also, instead of so many Ibs. each, 
 they may be reduced to ounces each, or increased to cwt*. each, or any 
 quantity whatsoever in like proportion. 
 
152 Alligation. 
 
 (3) A maltster has several sorts of rnalt, viz. 4s., 5s., 6s., 
 and 6s. Gd. per bushel ; how much of each sort must be 
 taken to make a mixture worth 5s. Gd. per bushel ? 
 
 1st Ans. 12 bus. at 4s., 6 1. at 5s., 6 b. at 6s., 18 b. at 6s. Gd. 
 
 2nd Ans. 6 bus. at 4s., 12 b. at 5s., 18 b. at 6s., 6 b. at 6s. Gd. 
 
 (4) What quantity of raisins of the sun at 7^d. and G^d. 
 per lb., with Malagas at 5^d. per lb., must be mixed together 
 to sell at Gd. per lb. 
 
 Or reduce the pence to farth. thus : 
 
 d. lb. d. 
 
 - , 
 n 
 J -J 
 
 Ans. 
 
 qrs. lb. d. 
 
 30 
 
 24 ) 26 
 22 
 
 -- 
 "! 
 J - 
 
 ...... 2 at 
 
 6 + 2 = 8 at 
 
 (5) How much rye at 5s. per bushel, barley at 4s. and oats 
 at 3s. per bush., will make a mixture worth 3s. Gd. per bush. ? 
 
 Ans. G bus. at 5s., 6 at 4s., and 24 at 3s. per bus. 
 
 (6) A victualler had ale at 166?., 12 J., and Sd. per gallon ; 
 how much of each sort must he take to sell at Wd. per gall.? 
 
 Ans. 2 gal. at 16^., 2 gal at ~L2d., and 8 gal. at Sd. per gal. 
 
 (7) A tea dealer has several sorts of tea, viz. at lls., 9s., 
 8s., and 7s. per lb. ; how much of each sort must be used 
 that the whole quantity may be afforded at 10s. per lb. ? 
 
 1ft v 
 10 > 
 
 s. 
 
 lb. s. lb. 
 
 Proof. 
 
 8. S. 
 
 in __ 1 + 2 + 3 = 6 at 11 per lb.\ 6 at 
 
 9J .. 1 at. Q f . 1 nf. 
 
 11 = 3 6 
 9 = 9 
 8 = 8 
 
 81 1 
 
 lat 8 ( Ans - lat 
 
 
 1 at 7 ) 1 at 
 
 7 = 7 
 
 
 9 
 
 9)4 10 
 
 
 
 10s. per lb. 
 
 (8) How many ounces of gold of 22, 18, 17, and 16 carats 
 fine must be mixed, so that the composition may be 20 carats 
 fine? 
 
 Ans. 9 oz. of22 carats, 2 oz. of IS, 2 oz.ofl?, and 2 oz. 
 oflG carats fine. 
 
 (9) How much wine at 7s., 8s., 9s., and 16s. per gallon, 
 must be mixed together, to make a mixture that may be sold 
 at 10s. per gallon ? 
 
 Ans. 6 galls, (or any equal quantity, more or less) of each 
 
Alligation. 153 
 
 CASE III. ALLIGATION PARTIAL is when one of its 
 ingredients is limited to a certain quantity. 
 
 RULE 1st. Take the difference between each price ana 
 the mean rate, as before. 
 
 2ndly. State, As the difference of that commodity whose 
 quantity is given is to the rest of the differences severally 
 so is the quantity given, to the several quantities required. 
 EXAMPLES. 
 
 (1) A farmer would mix 54 bushels of wheat at 7s. Qd. 
 per bushel, with rye at 4s. 6d. and barley at 5s. 3d. per bushel, 
 to make a mixture worth 6s. per bushel. 
 
 90"h 18 + 9x27 As 27 : 18 :: 54 
 
 72)54J 18 54 
 
 63J 18 
 
 72 
 90 
 
 27)972(36^9. 
 
 N.B. As both the lines are the 
 same (viz. 18) one stating will serve 
 for both. 
 
 (2) A distiller would mix 30 gallons of French brandy at 
 24s. per gallon, with English at 12s. and spirits at 8s. per 
 gallon ; what quantity of each must be taken to be afforded 
 at 16s. per gallon ? 
 
 Ans. 30 at 24s. per gal. ; 20 at 12s.; and 20 at Ss. per gal. 
 
 (3) A grocer mixes 24 Ib. of fine tea at 18s. per Ib. with 
 others at 13s. and 12s. per Ib. to make a mixture worth 15s. 
 per Ib. ; what quantity of each does he take ? 
 
 Ans. 24 Ib. at 18s.; 14| Ib. at IBs.per Ib.; and 14| at 12s. per Ib. 
 
 (^4) How much rum at 10s. Qd., 12s. 6d., and ISs. per gal. 
 must be mixed with 18 gallons at 16s Qd. per gallon, to 
 make a composition worth 15s. per gallon ? 
 
 Proof. 
 bus. s. d. 
 54 at 7 6 
 36 at 4 6 
 36 at 5 3 
 
 126 
 As 126 : 37 16 : 
 
 8. 
 
 = 20 i 
 = 82 
 = 99 
 
 37 1(5 
 
 : 1 to 6*. 
 Proof. 
 
 198+N 36 
 ,00x126] 30 
 180 >150J I 54 
 
 2161 18 
 
 As 36 : 30 : : 18 to 15 galls. Ans. 
 
 36 : 54 : : 18 to 27 galls. galls. *. d. 
 36 : 18 : : 18 to 9 galls. 13 at 16 6 per g. 
 
 The prices are here reduced 15 at 10 
 
 to pence, for the greater con- 27 at 12 
 
 venience in working. 9 at 18 
 
 (5) A tobacconist would mix 56 Ib. of tobacco at 3s. per 
 Ib. with others at 3s. 9d., 4s. 3d., and 4s. 6d., to make a com- 
 position worth 4s. per Ib.; how much of each must he take r* 
 1st Ans. 28 Ib.at 3s. Qd'; 28 Ib. at 4s. 3d. ; $ 112 Ib. at 4s. 6,1. 
 2ndAns. 112 Ib. at 3s. 9d.; 224 Ib. at 4s. 3d.; $ 56 Ib.at 
 
154 Alligation. 
 
 (6) A mealman mixes 60 bushels of flour at 10s. 6d. with 
 others at 9s., 8s., and 7s. 6d., to make a mixture worth 9s. 6d. 
 per bushel : what quantity of each does he take ? 
 
 Am. 15 bushels of each, at 9s., 8s., and 7s. 6d, 
 
 CASE TV. ALLIGATION TOTAL is when the whole of the 
 ingredients is limited to a certain quantity. 
 
 ETJLE 1st. Take the difference between each price and 
 the mean rate as before. 
 
 2ndly. State, As the sum of the differences is to each par- 
 ticular difference^ so is the quantity given to the quantity 
 required. 
 
 EXAMPLES. 
 
 (1) A brewer has ale at 12d., I0d., and 8^. per gallon, and 
 he would make a composition of a hogshead (54 gallons) to 
 sell for 9d. per gallon : how much of each must he take ? 
 
 12- -, 1 
 
 9)10-, 1 
 
 gJJ 3 + 1= 4 
 
 gallant. . d. 
 
 As 6 : 1 : : 54 to 9 at 1*. = 9 
 
 6 : 1 : : 54 to 9 at lOd. = 76 
 
 6 : 4 :: 54 to 36 at Sd. = 24 
 
 And 54 galls, at 9d. are 206 Proof. 
 
 (2) A druggist who has drugs of 8s., 5s., and 4s. per lb., 
 would make a composition of 11.2 lb. worth 6s. per lb. ; what 
 quantity of each must he take ? 
 
 Ans. 48 lb. of 8s.; 32 lb. of 5s.; and 32 lb. of&s.per lb. 
 
 (3) A goldsmith has several sorts of gold : viz. some of 24 
 carats fine ; some 22, and some 18 carats fine, with which 
 he would make a compound of 30 oz. of 20 carats fine ; I 
 demand how much of each sort he must take ? 
 
 Ans. 6 oz. of 24 carats ; 6 oz. of 22 ; and 18 oz. of 18 
 carats fine. 
 
 (4) A person has raw sugars at 12 d., 7d., Qd., and 5d. per 
 lb., with which he makes a composition of a quarter of a 
 cwt. worth Sd. per lb. what quantity of each does he take ? 
 
 lb. oz. d. 
 
 As 18 : 6 : : 28 to 9 5 T 6 at 12 
 18 : 4 : : 28 to 6 Sfg- at 7 
 18 : 4 : : 28 to 6 3 j$ at 6 
 18 : 4 : : 28 to 6 3f at 5 
 
 28 lb. 
 
Exchange. 155 
 
 (5) A wine merchant has four sorts of wine, viz., Canary 
 at 14s. per gallon, Malaga at 13s., Khenish at 11s., and 
 Oporto at 10s. per gallon ; and he is desirous of making a 
 composition of a pipe (126 galls.) to sell for 12s. per gal. ; 
 the quantity of each is required ? 
 
 1st Am. 42 of Canary, 21 of Malaga, 21 of Rhenish, 
 
 and 42 of Oporto. 
 2nd Ans. 21 of Canary, 42 of Malaga, 42 of Rhenish, 
 
 and 21 of Oporto. 
 
 (6) I have teas at 4s., 5s., 7s., and 9s. per Ib. and I would 
 make a mixture of ^ a cwt. (56 Ib.) to sell at 6s. per Ib. ; 
 what quantity of each will be required ? 
 
 Is* Ans. 24 Ib. at 4s., 8 Ib. at 5s., 8 Ib. at 7s., 16 Ib. at 9s. 
 2nd Ans. 8 Ib. at 4s., 24 Ib. at 5s., 16 Ib. at 7s., 8 Ib. at 9s. 
 
 EXCHANGE. 
 
 EXCHANGE is bartering the money of one place for 
 that of another, by means of a Sill of Exchange ; and the 
 rule teaches how to find what quantity of one kind of money 
 will be equal to a proposed quantity of another, according to 
 the course of exchange. 
 
 The course of exchange is the value agreed on by mer- 
 chants, and is almost daily fluctuating above or below the 
 par of exchange. 
 
 The par of exchange is always fixed and certain ; it being 
 the intrinsic value of the money of one place compared with 
 that of another. 
 
 Agio is a term used in some countries abroad, especially 
 in Amsterdam and Italy ; and denotes the difference between 
 Sank money (usually called Banco) and current money ; the 
 former being something finer than the latter. 
 
 Usance is a certain time allowed by one country to 
 another, for the payment of bills of exchange. 
 
 Days of grace are a certain number of days allowed for 
 payment, beyond the time specified in the bill. 
 
 Questions in Exchange are performed either by the Eule 
 of Three or Practice. 
 
156 
 
 Exchange. 
 
 1. ENGLAND WITH FEAKCE. 
 
 In France, before the Revolution, accounts were kept in 
 livres, sols, and deniers, and they exchanged with England 
 by the crown Tournois. But at present they are kept ir 
 francs and centimes, and they exchange by the franc. 
 
 12 deniers make 1 sol, or sou = 1 half-penny. 
 
 20 sols 1 livre = lOd. nearly. 
 
 3 livres 1 ecu, or crown Tournois. 
 
 Also, 
 
 10 centimes make 1 decime. 
 
 10 decimes, or 100 cents 1 franc. 
 
 Exchange at par 25 francs 20,8 centimes per sterling. 
 EXAMPLES. 
 
 (1) How many crowns must 
 be paid in Paris, to receive in 
 London 540Z., Exchange at 
 4s. Qd. per crown ? 
 
 d. cr. . 
 
 As 54 : 1 :: 540 
 240 
 4ns 
 
 (2) A merchant in Paris re- 
 mits to his correspondent in 
 London 2400 crowns, at 4s. 6^. 
 each ; what is the value in 
 sterling ? 
 
 cr. d. cr. 
 
 As 1 : 54 : : 2400 
 54 
 
 54)129600(2400 cr. 
 108 
 
 12) 129600 
 
 216 
 
 Oi 
 
 2,0)1080,0 
 
 
 540 Ans. 
 
 
 
 (3) How much sterling must be paid in London to receive 
 in Paris 1000 crowns, exchange at 54J<f. per crown ? 
 
 Ans. 2271. Is. Sd. 
 
 (4) A merchant in London remits 227Z. Is. Sd. to his 
 correspondent in Paris ; what is the value in French crowns 
 at 54g< per crown ? Arts. 1000 crowns. 
 
 (5) Change 566 cr. 17 sols, 5J den. at 55d. per crown, 
 into sterling ; what is the sum ? Ans. 129Z. 15,?. 6d 
 
 (6) Change 389Z. 6s. 6d. sterling into French crowns, 
 exchange at 55d. per crown. 
 
 Ans. 1698 crowns, 52 sols, 4 $%denicrs. 
 
Exchange. 
 
 157 
 
 2. SPAIN.* 
 
 In general, they keep their accounts in Spain in piastres, 
 rials, and maravedis, and exchange by the piastre or peso. 
 
 34 maravedis make 1 real or rial. 
 
 8 reales or rials 1 piastre, peso, piece of eight, 
 
 or dollar. 
 
 4 piastres 1 pistole of exchange. 
 
 375 maravedis 1 ducat. 
 
 EXAMPLES. 
 
 (1) Spain draws on London 
 for 2354 piastres 4 rials, ex- 
 change at 3s. 4i\d. per piastre, 
 how much sterling is the sum ? 
 
 piastre s. d. piast. rials 
 1 : 34^.: 2354 4 
 8 8 
 
 1 O fnvt"h 
 
 (2) In 350Z. 105. \d. ster- 
 ling, how many rials, &c., 
 exchange at 3s. 3J d. per piece 
 
 of eight ? 
 
 s. d. piastre s. d. 
 As 3 3^ : 1 : : 350 10 9| 
 12x2" 8 20x12x2 
 
 8 18836 
 162 
 
 79 rials 8 168259 
 8 
 
 8 ) 3051432 
 
 79 ) 1346072(17038 
 
 4 ) 381429 
 
 70 
 34 
 
 12 ) 95357| 
 
 79 ) 2380 (30 mara 
 
 2,0 ) 794,65 
 
 10 rem 
 
 Ans. > 397 6 5J 
 
 Ans, 17038 rials 30 marav. 
 
 (3) In 9876 piastres 4 rials of plate, how many sterling, 
 exchange at 42d. per piastre ? Ans. 1728Z. 7s. 9d. 
 
 (4) How many piastres should I receive for 1728Z. 7s. 9d. 
 exchange at 42^. per piastre ? Ans. 9876 pias. 4 rials 
 
 * Note They have t'ico kinds of money in Spain, called plate and vellon; 
 *The real of plate or silver 4f d. nearly ; and the real vellon = 2^ nearly : 
 hence 17 reales of plute = 32 reales vellon. In exchanges with 
 England, plate ouly is used. When the dollar or peso of exchange with 
 London = 37f d., then the peso fuento or Spanish dollar = 50|rf. nearly. 
 
 In some parts of Spain accounts are kept in rials and maravedis 
 vellon, and exchange by the ducat. The ducat is worth 4a. ll\d. The 
 piastre 3s. 7cZ. at par. 
 
158 
 
 Exchange. 
 
 (5) In 8768 rials of plate, 
 how many > sterling, ex- 
 change at 40|<#. per piece of 
 eight ? 
 
 p. eight d. rials 
 
 1 : 40i : : 8768 
 
 8 ) 355104 
 
 12 ) 44388 
 
 2,0 ) 369,9 
 Ans. 184 19 
 
 (6) In 184Z. 195. sterling, 
 how many rials of plate, ex- 
 change at 40|d. per piece of 
 
 eight ? 
 
 d. p. eight 8 . 
 
 40 : 1 : : 184 19 
 
 162 
 
 4 
 
 p.eig. 
 
 162 ) 177552 ( 1096 
 
 rials 8768 
 Ans. 8768 rials. -- - 
 
 (7) In 2345 1. how many rials, exchange at 50d. per piastre 
 or peso ? Ans. 90048 rials 
 
 (8) In 90048 rials how many sterling, exchange at 50d. 
 per piastre ? Ans. 2345Z. 
 
 (9) In 67530 rials vellon, how many rials of plate ? 
 Multiply by 17, and divide by 32 (see note 1). A. 35875 ri. T'V- 
 
 (10) In 35875 iV rials of plate, how many rials vellon ? 
 
 Ans. 67530 rials vellon 
 
 3. PORTUGAL. 
 
 Accounts are kept in Portugal in reas and milreas, and 
 the exchange is by the milrea, at from 60d. to 68d. sterling. 
 Its value at par is 5s. 7%d. 
 
 400 reas make 1 crusado. 
 
 1000 reas, or 2J crusadoes 1 milrea. 
 
 (1) If a bill be drawn from 
 Lisbon, of 5432 milreas 346 
 reas, at 6s. 6d. per milrea, 
 what is the value in English 
 money ? 
 1000 : 6s. Qd. : : 5432,346 
 1000 
 
 (2) If a bill be drawn from 
 London of 1765Z 105. 2$d. 
 how many milreas at 6s. 6d. 
 each are equal in value to the 
 sum ? 
 
 s. d. mil. s. d. 
 66:1:: 1765 10 2| 
 
 5432346 
 
 312/ar&. 1694891 far. 
 
 mil. reas. 
 312 ) 1694891 ( 5432 842fH 
 
 1,000 ) 35310/249 249 
 1 
 
 2,0)3531,0 d. 2/988 
 
 
 
 
 1765 10 2 ^952 
 
Exchange. 
 
 159 
 
 (3) In 1000 milreas 100 reas, how many pounds sterling, 
 exchange at 5s. 5d. per milrea ? Ans. 2701. 17s. 2\d. 
 
 (4) How many milreas must be given for 7581. 8s. Gd. 
 sterling, exchange at 5s. 4<%d. per milrea ? 
 
 Ans. 2822 mil 46 re. 66. 
 
 (5) Reduce 1234 crusadoes 67 reas, into sterling, exchange 
 at 67d. per milrea ? Ans. 138Z. 16s. lOJrf. 90. 
 
 (6) In 138Z. 16s. 10$d. how many crusadoes, exchange at 
 67^. per milrea ? Ans. 1234 cru. 66 reas. 18 rem. 
 
 4. ITALY. 
 
 At GENOA and LEGHORN some keep their accounts in 
 piastres or pezzos, soldi, and denarii ; and others in lires, 
 soldis, and denarii; and they exchange by the piastre or 
 pezzo, which is equal to 4s. Qd. at par. 
 
 The course of exchange is from 47 d. to 58 d. sterling per 
 piastre or pezzo. 
 
 12 denarii make 1 soldi. 
 
 20 soldi 1 lire, piastre, or pezzo. 
 
 Exchange from 45 d. to 54<d. per piastre. 
 At VENICE in dollars, soldi, and denarii, and exchange 
 by the ducat and piastre. 
 
 12 deniers d'or make 1 sol d'or. 
 
 20 sols d'or 1 ducat. 
 
 Also, some here keep their accounts in ducats, grossi, and 
 Boldi. 
 
 5 soldi make 1 grossi. 
 
 24 grossi 1 ducat. 
 
 Agio from 20 to 30 per cent. 
 
 EXAMPLES. 
 
 (1) In 368Z. 12s. Qd. ster- 
 ling, how many piastres of 
 Grenoa, exchange at 4fi^d. per 
 piastre ? 
 d. piastre. 8. d. 
 47 j : 1 :: 368 12 6 
 
 (2) How much sterling mo- 
 ney is equal to 2345 pezzos 
 10 soldi of G-enoa, exchange 
 at 50f per pezzo ? 
 
 pezzo. d. pezzo sol. 
 1 : 50| :: 2345 10 
 
 95 95)176940(1862 jo. 
 
 20 8 20 
 
 50 
 20 
 95)1000(10 soldi. 
 
 20 403 46910 
 403 
 
 20 ) 1890473,0 
 
 50 
 
 8 ) 945236^ 
 
 12 
 
 12)118154!=' 
 
 95)600(6 denarii. 
 
 2,0 ( 984,6 2d. 
 
 80 
 
 492 6 2J 1| 
 
1GO 
 
 Exchange. 
 
 (3) If I pay in London 500Z. sterling, for how many 
 piastres may I draw my bill to be paid at Leghorn, exchange 
 at 4s. 5d. per piastre ? Ans. 2264 pias. 3 soldi. 1 rem. 
 
 (4) How much sterling money is equal to 8765 lires 
 15 soldi of Leghorn, at 4s. 2 d. per piastre ? 
 
 Ans. 186Z. 3s. 
 
 (5) How many ducats banco 
 at Venice are equal to 789Z. 
 15s. sterling, exchange at 48 d. 
 per ducat banco ? 
 
 d. due. s. 
 
 As 46 : 1 : : 789 15 
 
 48)189540(3948 d. 
 
 36 
 20 
 
 48)720(15 50/5. 
 
 Ans. 3948 due. 15 sol. 
 
 (6) How much sterling mo- 
 ney is equal to 896 ducats 
 12 sols 6 deniers, banco mo- 
 ney of Venice, exchange at 
 4s. 5%d. sterling per ducat ? 
 due. s. d. due. sol. den. 
 1 : 4 5J : : 896 12 6 
 
 20x12 
 
 240d.214/ar. 
 
 215190 
 214 
 
 24,0)4605066,0(4)19187* 
 
 18 rem. 12;47969J 
 2,0)399,7 5 
 
 199 17 51 
 
 (7) London remits to Venice 385Z. 16s. sterling, exchange 
 at 4s. 4d. per ducat banco, how many ducats banco will he 
 receive at Venice ? Ans. 1780 due. 12 sol. 3 If . 
 
 (8) In 1960 ducats 10 sols 9 den. banco at Venice, how 
 many . sterling, exchange at 49 \d. per ducat banco ? 
 
 Ans. 404Z. 7s. 2d. fa 
 
 (9) In 1243 due. 16 sol. 10 den. current money, how many 
 pounds sterling, exchange at 4Qd. per ducat banco, and agio 
 20 per cent ? 
 
 due. d. 
 
 1st. As 1 : 49 
 
 s. 
 
 2nd. As 120 : 100 
 
 due. sol. den. s. 
 
 1243 16 10 to 253 1958 rem. 
 
 8. s. d. 
 253 19 to 211 12 6 Ans. 
 
 Note. They have three sorts of money at Venice : the Banco, the cur- 
 rent money, and the paoli : the banco is that in which they keep their 
 accounts ; the current is the standard of their coin ; and the paoli is used 
 for the purchase. 
 
^Exchange. 
 
 5. HOLLAND, FLANDEBS, AND GERMANY. 
 
 They keep their accounts at Amsterdam, Botterdam, 
 Antwerp, Brussels, various parts of Germany, &c., some in 
 guilders, stivers, and pennings ; others in pounds, shillings, 
 and pence, as in England, but distinguished by the name of 
 Flemish ; and they exchange by the sterling. 
 
 8 pennings make 1 grote or penny Flemish. 
 
 2 grotes, or pence 1 stiver. 
 
 6 stivers 1 schilling. 
 
 20 stivers 1 guilder, or florin. 
 
 2J florins', 1 rix-dollar. 
 
 6 guilders, or florins ... 1 pound Flemish. 
 Or thus, for s. and d. Flemish, as in England : 
 
 12 grotes or pence 1 schilling. 
 
 20 schillings 1 pound Flemish. 
 
 Exchange from 33s. 6d. to 36s. 6d. Flemish, per > ster- 
 ling, and Agio from 3 to 6 per cent, for current. 
 
 To change Flemish money into sterling. 
 RULE. As the given rate of exchange is to 1Z. sterling, 
 so is the given Flemish to the sterling required. 
 
 To change sterling money into Flemish. 
 RULE. As 1Z. sterling, is to the given rate of exchange? 
 so is the sterling given to the Flemish sought. 
 
 EXAMPLES. 
 
 (1) Remitted from London to Amsterdam a bill of 1250Z. 
 15s. sterling ; how many pounds Flemish is the sum, the ex- 
 change being at 34s. 6d. Flemish per > sterling ? 
 
 s. s, d. s. 
 
 As 20 : 34 6 : : 1250 15 
 
 12 
 414 
 
 20 
 
 25015 
 414 
 
 20)1035621,0 
 
 12)517810Jfci 
 
 2,0)4315,010 
 
 2157 10 
 
 Or by Practice, thus : 
 
 10s. 
 4s. 
 6d. 
 
 s. d. 
 
 1250 15 
 
 625 7 6 
 
 250 3 
 
 31 5 44 
 
 2157 10 IQ^Ans. 
 
 Note. There are two sorts of money in these countries, bank money and 
 current ; the difference between them is from 3 to 6, or even more thuu 
 8 per cent, in favour of the bank money. 
 
1G2 
 
 Exchange. 
 
 (2) Eotterdam remits 2157Z. 10s. 10^. Flemish, to be 
 paid in London ; how much sterling money can he draw for 
 exchange being at 34s. Qd. Flemish per sterling ? 
 
 Ans. 1250Z. 15s. 
 
 (3) If I pay in London 500Z. sterling, how many guilders 
 must I draw for at Amsterdam, exchange at 33s* Qd. Fle- 
 mish per sterling ? Ans. 5025 guilders 
 
 (4) If I pay at Amsterdam 5025 guilders, what must I 
 draw for in London, exchange at 33s. Qd. Flemish per 
 sterling ? Ans. 6001. 
 
 (5) In 365Z. 15s. Qd. sterling, how many Dutch rix-dollars, 
 exchange at 35s. 4<d Flemish per sterling ? 
 
 Ans. 1550 rix-dol. 44 stiv. 4 penn. | -f-<ro* 
 
 To change Sank into Current Money. 
 
 BULE. As 100 is to 100 with the agio added to it, so is 
 any given sum banco to the current required. 
 
 To change Current Money into Sank. 
 
 KFLE. As 100 with the agio added to it is to 100, so is 
 any given sum current to its value in bank money. 
 
 (6) Change 835 guilders 
 12 stivers bank, into current 
 money, agio 4f per cent. 
 
 guil. gu. sti. pe. guil. sti. 
 As 100 : 104 12 8 :: 835 12 
 20 20 20 
 
 (7) Change 874 guild. 4 sti. 
 14 penn. current money into 
 bank money, exchange at 4| 
 per cent. 
 guil. guil. guil.sti. pe. 
 As 104| : 100 : : 874 4 14 
 Or, As 104| : 874 4 14 : : 100 
 8 20 8 
 
 2000 2092 16712 
 16 
 
 837 17484 800 
 16 
 
 279758 
 800 
 
 33480 
 16712 
 
 8000)559517760 
 
 837 ) 223806400 ( 267391 
 
 16)2797581760 rem. 
 
 133 rem. 
 
 2,0 ) 1748,4 14 pennings. 
 
 16 ) 267391 
 
 2,0 ) 1671,2 nearly. 
 
 Ans. 874 guil. 4 sti. 14 pen. lank. 
 
 (8) How much current mon< 
 banco, the agio being 5 per ce 
 (9) How much bank monej 
 the agio being 4 per cent. ? 
 
 Ans. 835 guil. 12 sti. cur. 
 jy can I have for 2345 guilders 
 nt. ? Ans. 2462J guilders. 
 r can I receive for 5421 florins, 
 Ans. 5212 It florins 
 
Exchange. 163 
 
 (10) In 1456 Flemish current, how many sterling, the 
 agio being 4 per cent., and the exchange 34s. Gd. Flemish 
 per sterling ? 
 
 
 
 1st. As 104 
 
 
 
 : 100 
 
 
 : : 1456 to 1400 Flemish. 
 
 3. 
 
 2nd. As 34 
 
 2 
 
 stcrt, 
 : 1 :: 
 
 
 : 1400 
 20x2 
 
 69 69 ) 56000 ( 81U 12s. nearly. 
 
 KB, 34s. 6d. = 34s. 
 
 (11) In 438Z. 7ys. sterling, how many rix-dollars current, 
 agio 4|, exchange at 36s. Gd. Flemish per sterling ? 
 
 s. ster. s. d. s. 
 
 1st. As 20 : 36 6 : : 438 7f to 800 
 
 
 
 2nd. As 100 : 104| 800 to 837 current. 
 
 3rd. For the rix-dollars, the table says, 2^ florins = 1 rix-dol. ; 
 and 6 flor. = R Flem. 
 
 2 1 
 Hence the proportion is =^-. Therefore, multiply the 837Z. by 12, 
 
 6 
 and divide by 5 = 2008f rix-dollars. Ans. 
 
 6. HAMBUBGH. 
 
 At Hamburgh the accounts are kept in marks, sol (or 
 schilling) lubs,* and pfennings ; and they exchange by the 
 pound Flemish, as in Holland. 
 
 8 pfennings make 1 schilling or penny. 
 
 12 pfennings 1 groschen. 
 
 6 sol-lubs (or 12 d. Flemish, ) - , 
 
 > or72deniers) ....5 ! so1 g ros - 
 
 16 schillings 1 mark = Is. 2 \d. nearly. 
 
 2 marks, or 32 schillings . . 1 Hambro' dollar. 
 
 3 marks (or 48 sol-lubs) .... 1 rix-dollar. 
 
 6 marks banco,or7 1 marks cur. 1 ducat = Ss. lid. 
 
 7g marks , 1 liver gross, or Flemish. 
 
 T\\e par of exchange is 35s. 6?d. Flemish for II. sterling; 
 and the course of exchange is from 32s. to 36s. Flemish per 
 
 * Sol-lub, schilling or schillings lub, means the same. The word lub 
 (so called from Lubeck, where it was first coined) is now falling into 
 disuse, and the word Uambro' substituted. 
 
164 
 
 Exchange. 
 
 sterling. Agio from 18 to 20 per cent, for currency; and 
 from 30 to 35 per cent, for light coin.* 
 
 EXAMPLES. 
 
 (1) In 3021 rix-dollars 35 
 sol-lubs, bow many ster- 
 ling, exchange 36s. 4d. Fle- 
 mish per sterling. 
 
 s. d. rix-d. sol-lubs. 
 36 4 : 1 :: 3021 35 
 12 48 
 
 436 
 
 24173 
 
 12087 
 
 145043 
 2 
 
 436)290086( 665 6 
 
 204 rein. 
 
 (2) How many marks must 
 be received at Hainbro' for 
 250/. 15s. sterling, exchange 
 at 34s. 6d. per sterling. 
 
 s. s. d. s. 
 
 20 : 34 6 : : 250 15 
 12 20 
 
 4H 
 5015 
 
 5015 
 
 2,0 ) 207621,0 
 
 32) 103810 J 
 
 Ans. 3244 marks 2 den. 
 
 (3) Eeduce 4321 marks 12 schill. into sterl. exchange 
 at 345. 4d. Flemish per sterling ? Ans. 335Z. 13s. 4f d. i4. 
 
 (4) Eeduce 335Z. 13s. 4f d. sterling, into marks and sol- 
 lubs banco, exchange at 34s. 4d. Flemish per sterling. 
 
 Ans. 4321 marks 11 schill. 11 pfenn. 348. 
 
 (5) In 665Z. 6s. S^d. sterling, how many rix-dollars, ex- 
 change at 36s. 4d. Flemish per sterling ? 
 
 Ans. 3021 riX'dol. 35 schil. Od. 1 pfen. fig. 
 
 (6) Eeduce 8766 marks current into sterling, ex- 
 change at 35s. 3d. Flemish per sterling, and agio 20 per 
 cent. 
 
 marks, marks. 
 1st. As 120 : 100 : : 8766 to 7305 banco. 
 
 s. d. . marks. 
 
 2nd. As 35 3 : 1 : : 7305 
 12 32 
 
 423d 
 
 423 ) 233760 ( 552?. 12s. 5|cZ. A us. 
 63 rem. 
 
 * The different moneys of Hamburgh are, 1st, bank money; 2nd, specie; 
 3rd, the gold ducat; 4th, light coin; 5th, currency. 
 
Exchange. 1G5 
 
 (7) In 552Z. 12s. 6d. sterling, how many marks current, 
 exchange at 355. 3d. Flemish per sterling, and agio 20 
 per cent. ? 
 
 d. s. d. F. d. 
 
 1st. As 240 : 35 3 : : 552 12 6 to 974 Flem. 
 
 
 
 2nd. As 100 : 120 : : 974 
 
 7 J marks. = 11. Flem. 
 
 7305 
 120 
 
 1,00)8766,00(8766 marls. Ans. 
 
 (8) How much sterling money will a bill of 1830 rix-dol- 
 lars current amount to, exchange at 35^. Qd. Flemish per 
 sterling, and agio 18 per cent ? 
 
 Ans. 349Z. 9s. 8f d. 1177 rem. 
 
 (9) How many Hambro' marks must be received for a bill 
 of 349 1. 9s. Qd. sterling, exchange at 35s. 6d. Flemish per 
 sterling, and agio 18 per cent ? marks pf. 
 
 Ans. 5490 2518. 
 
 7. POLAND AND PRTJSSIA. 
 
 At Dantzig and Konigsberg accounts are kept in florins, 
 gross (groshen) penins, and exchange by the gross ; 270 of 
 which are supposed equal to 11. Flemish, and 110 to a rix- 
 dollar at Hamburgh. Exchange is made with Poland and 
 Prussia by way of Holland. 
 
 The course of exchange is from 240 to 295 gross per 
 Flemish. 
 
 6 penins make 1 shelon. 
 
 18 penins 1 gross. 
 
 18 grossi 1 oort. 
 
 30 grossi 1 florin, or Polish guilder. 
 
 3 florins, or 90 grossi 1 rix-dollar. 
 
 2 rix-dollars 1 gold ducat. 
 
1GG 
 
 Exchange. 
 EXAMPLES. 
 
 (1) In 1000Z. sterling, how 
 many Prussian florins, ex- 
 change 270 grossi per Fle- 
 mish, and 34s. &d. Flemish 
 per sterling ? 
 
 s d. 
 
 1st. As 1 : 34 4 :: 1000 
 
 412 
 1000 
 
 412000 Fl. pence, 
 d. Flem. groesi. d. Flem. 
 2nd. As 240 : 270 : : 412000 
 412000 
 
 240)111240000 
 
 3,0 ) 46350,0 gross. 
 Ans. 15450 Pruss.flor. 
 
 (2) How many sterling 
 for 3456 rix-dollars 40 grossi, 
 exchange at 280 Polish grossi 
 per Flemish, and 33s. 4td. 
 Flemish per sterling ? 
 
 gross. d.Fl. r-doll. gro. 
 lst.As 280 : 240 : : 3456 40 
 90 
 
 311080 
 240 
 
 d.Fl. 
 
 280)74659200(266640 
 s. d. d. 
 
 2nd. As 33 4 : 1 :: 266640 
 
 d. 40,0 ( 26664,0 ( 
 
 666240 
 20 
 
 Ans. 666Z. 12*. 4,00 ) 48,00 ( 12*. 
 
 (3) Change 1760 florins into sterling money, 275 Polish 
 grossi being equal to the Flem. and 34s. 4<d. Flem. to 
 one sterling. Ans. 111Z. 16s. lO^d. -^/V. 
 
 (4) In 111Z. 165. lOf d. sterling, how many Polish guilders, 
 exchange at 275 Polish grossi per Flem. and 34s. 4J. 
 Flem. per sterling ? Ans. 1760 Pol. guilders 143 rem. 
 
 9. SWEDEN. 
 
 They keep their accounts in Sweden in copper dollars 
 and oorts, or in silver dollars, and exchange by the copper 
 dollar. 
 
 The par of exchange is one sterling for 34f copper 
 dollars ; and the course of exchange is from 40 to 50 copper 
 dollars per sterling. 
 
 8 penins make 1 runstychen. 
 
 8 runstychens 1 copper marc. 
 
 3 copper marcs 1 silver marc. 
 
 4 copper marcs 1 copper dollar. 
 
 9 copper marcs 1 Caroline. 
 
 3 copper dollars 1 silver dollar. 
 
 3 silver dollars 1 rix-dollar. 
 
 2 rix-dollars 1 ducat. 
 
Exchange. 
 EXAMPLES. 
 
 107 
 
 (1) In 480Z. 55. 6d. ster- 
 ling, how many copper dol- 
 lars, &c., exchange at 48 
 copper dollars, per ster- 
 ling ? 
 
 c.d. s. d. 
 As 1 : 48J :: 480 15 6 
 
 240d. 193 
 
 115386 
 193 
 
 240)22269498(4)92789 
 
 138 rem. 
 
 23197J 
 
 (2) In 75G ducats, how much 
 sterling money, exchange at 
 42 J copper dollars per 
 sterling ? 
 
 c.d. due. 
 As 42| : 1 : : 756 
 
 2 IS cop. d. = lduc. 
 
 85 
 
 13608 
 2 
 
 Ans. 
 
 85) 27216 (320/. 3s. 9d. 
 15 rem. 
 
 (3) In 73SZ. sterling, how many copper dollars, exchange 
 at 48 copper dollars per sterling ? 
 
 .Ans. 35424 copper dollars. 
 
 (4) Reduce 320Z. 3s. Qd. sterling into rix-dollars, &c., ex- 
 change at 42^ copper dollars per sterling ? 
 
 Ans. 1512 rix-dollars^ nearly. 
 
 (5) In 1034 silver dollars 
 16 run. 6 pen. how many 
 sterling, exchange at 49 cop- 
 per dollars, per sterling ? 
 
 c. d. s. dol. run. pen. 
 As 49 : 1 : : 1034 16 6 
 
 32 3 
 
 1568 
 
 12544 
 
 3102 
 32 run. = 1 cop. dol. 
 
 99280 
 
 1 2544 ) 794246 ( 63/. 6s. 4rf. Ans. 
 
 416 rem. 
 
 (6) In 158Z. 165. 8d. ster- 
 ling, how many silver marcs, 
 exchange 48 copper dollars 
 per sterling ? 
 
 d. 
 
 240 
 
 c. dol. 
 : 48 
 4 
 
 3)192 
 
 s. d. 
 
 158 16 8 
 
 88120* 
 
 64 
 
 sil. marcs. 
 
 s. marcs 64 240)2439680(10165^. 
 
 8 rem. 
 
 N.B. 3 cop. dol. are = 4 silver 
 marks (see the Table). Hence, a sil. 
 mark is f of a cop. dol. 
 
 8. DENMARK. 
 
 In Denmark accounts are kept in rix-dollars, marrs, and 
 schillings ; and the exchange from 4 to 5 rix-dollars per 
 sterling. 
 
 16 schillings make 1 marc. 
 
 6 marcs 1 rix-dollar. 
 
168 Excl 
 
 EXAIU 
 (1) How many sterling 
 in 1765 rix-dollars Danish, 
 exchange at 4s. 9d. per rix- 
 dcilar ? 
 
 r.-dol. s. d. r.-dol. 
 As i : 4 9 : : 1765 
 
 iange. 
 
 [PLES. 
 
 (2) In 133Z. Us. 4id. ster- 
 ling, how many Danish marcs, 
 exchange at 45. 8%d. per rix- 
 dollar ? 
 
 s. d. r.-dol. s. d. 
 As 4 8 : 1 : : 133 14 4 
 
 57 
 1765 
 
 113 half-pence. 64184 half -p. 
 
 12)100605 
 
 113) 64184 ( 568 rix-dollars. 
 6 
 
 Ans. 3408 
 
 2,0)338,3 9 
 
 419 3 9 Ans. 
 
 (3) In 3408 Danish marcs, how many sterling, ex- 
 change at 4s. S^d. per rix-dollar ? Ans. 133Z. 145. 4td. 
 
 (4) How many Danish marcs in 419Z. 85. 9d. sterling, ex- 
 change at 45. 9d. per rix-dollar ? Ans. 10590 marcs. 
 
 10. RUSSIA. 
 
 In Russia they keep their accounts in rubles and copecs ; 
 and exchange by the ruble. 
 
 The par of exchange is 45. Gd. ster. per ruble ; and the 
 course of exchange is from 4Sd. to GOd. per ruble ; or from 
 48 to 59 stivers per ruble by way of Hamburgh and Am- 
 sterdam. 
 
 3 copecs ....... , .......... make 1 altin. 
 
 10 copecs ........................ 1 grievener. 
 
 25 copecs ........ . ............... 1 polpolitin. 
 
 2 polpolitins .................. 1 poltin. 
 
 2 poltins ........................ 1 ruble. 
 
 2 rubles ........................ 1 ducat. 
 
 EXAMPLES. 
 
 (1) In 500Z. 175. ster. how 
 many rubles, exchange at 45. 
 5d. ster. per ruble ? 
 
 s. d. 
 As 4 5 
 
 53 
 
 rub. 
 : 1 
 
 s. d. 
 
 500 17 
 
 120204 
 
 53 ) 120204 ( 2268 rubles. Ans. 
 
 (2) In 1896 rubles, how 
 many ster. exchange at 
 45. Qd. per ruble ? 
 
 rub. s. d. rub. 
 
 As 1 : 4 6 : : 1896 
 
 7584 
 948 
 
 Ans. 426Z. 12*. 2,0 ) 853,2(426 12s. 
 
Exchange. 169 
 
 (3) What is the value of 1572 rubles and 60 copecs, at 
 4?. 9%d. per ruble ? Ans. 376Z. 15s. 4%d. 
 
 (4) How many rubles, &c., must be received in Peters- 
 burgh for 376Z. 155. 4^., exchange at 4s. 9|J. per ruble ? 
 
 Ans. 1572 rubles, and 60 copecs 
 
 (5) London remits to Pe- 
 tersburgh 725 1. sterling, ex- 
 change at 34s. 6d. Mem. per 
 sterling, and the exchange 
 from thence 50 stivers per 
 ruble, how many rubles must 
 be received in Petersburgh ? 
 
 s.d. s.d. 
 1st. As 1 : 34 6 :: 725 to 1250 12 6 
 
 stio. rub. s. d. 
 2nd.As50 : 1 :: 1250 12 6 
 . 20x12 
 
 100 10,0)3001,50 
 
 Ans. 3001 rub. 50 cop. 
 
 (6) Archangel remits to 
 London, per bill of exchange, 
 5000 rubles 4 CrT)ecs, ex- 
 change 122 copecs per rix- 
 dollar of 50 stivers, and 34s. 
 7d. Elem. per ster. ; how 
 much sterling is the sum ? 
 
 cop. stiv. rub. cop. stiv. 
 1st. 122 : 50 : : 5000 4 to 204919 
 
 s. d. stivers. 
 2nd. 34 7 : 1 : : 204919 
 
 12 2 
 
 - - s. d. 
 
 415 415 ) 409839 ( 987 11 3j 
 
 125 rem. 
 
 11. IBELAND. 
 
 In Ireland accounts are kept in pounds, shillings, and 
 pence, Irish, divided as in England. The par of exchange 
 is 108 JZ. (108Z. Qs. Sd.) Irish, for 100Z. sterling ; or 1Z. Is. Sd. 
 Irish per sterling. The course of exchange is from 5 to 
 12 per cent, according as the balance of trade favours Ire- 
 land or England. 
 
 EXAMPLES. 
 
 (1) London remits to Ire- 
 land 786Z. 125. sterling, what 
 money must be received there, 
 exchange at 10^ per cent ? 
 
 s. 9. 
 
 As 100 : 110 10 : : 786 12 
 
 2000 
 
 2210 
 15732 
 
 15732 
 
 200,0 ) 3476772,0 ( 2,0 ) 1738,3 
 Jft = A 869 3 10J 
 
 (2) Dublin remits to Lon- 
 don 15002. Irish, how much 
 sterling must Ireland be cre- 
 dited for, exchange at 12 per 
 cent ? 
 
 
 As 112 
 
 
 
 100 
 1500 
 
 
 1500 
 
 112 ) 150000 ( 1339Z. 5*. 
 32 rem. 
 
170 
 
 Exchange. 
 
 (3) Purchased in Ireland goods to the amount of 869Z. 
 3s. 10%d. Irish, how much money must I pay in London to 
 settle the account, exchange at 10^ per cent ? 
 
 Ans. 786Z. 125. sterling 
 
 (4) London remits to Dublin 1339Z. 5s. 8$d. sterling ; for 
 how much Irish must London be credited, exchange at 12 
 per cent ? Ans. 1500Z. Irish 
 
 12. AMEEIOA AND THE WEST INDIES. 
 
 Accounts are kept in these places in pounds, shillings, 
 and pence, as in England. Here is no fixed par in conse- 
 quence of the fluctuation of the agio : the money is called 
 currency, and, on account of the scarcity of cash, 51. sterling 
 is worth 71. of the currency of the West Indies. 
 
 EXAMPLES. 
 
 (1) America is indebted to 
 England 850Z. 18s. sterling; 
 with how much currency will 
 England be credited in Ame- 
 rica, exchange at 35 per ct. ? 
 
 s. 
 
 As 100 : 135 : : 850 18 
 
 17018 
 135 
 
 1,00 ) 22974,30 
 
 rem. 30 
 12 
 
 d. 3/60 
 4 
 
 ^/40 
 s. d. 
 
 2,0 ) 2297,4 = 1148 14 
 
 (2) London receives from 
 Jamaica 1756Z. 10s. currency, 
 what will be its value here 
 in sterling, exchange at 30 per 
 cent? 
 
 s. . d. 
 
 130 : 100 : : 1756 10 to 1351 3 0| T 9 3 
 
 Or thus, as the proportion in frao- 
 
 8. 
 
 Multiply 1756 10 
 by 10 
 
 and divide by 13)17565 
 
 gives the Ans, 1351 3 Of ^ 
 
 (3) Bought goods in the West Indies to the amount of 
 486Z. 155. sterling, how much is that in their currency, ex- 
 change at 25 per cent ? Ans. 608Z. 8s. 9d. 
 
 (4) London receives a bill of exchange from Philadelphia 
 for 750Z. sterling, for how much currency was London 
 indebted, exchange being at 33 J per cent ? Ans. 10007. 
 
ARBITRATION OF EXCHANGES. 
 
 THE course of exchange varying, as we have seen, between 
 one nation and another, as the balance of trade varies ; this 
 rule teaches how to draw upon, or remit, to, foreign places, 
 in a way the most profiable. 
 
 This is done by finding such a rate of exchange between 
 any two places, as shall be in proportion with the rates 
 assigned between each of them and a third place. 
 
 Arbitration of exchange is either Single or Compound. 
 
 SINGLE ARBITEATIOK 
 
 Single Arbitration is when the course of exchange between 
 one place and two others is given, to find the rate between 
 them.* 
 
 EXAMPLES. 
 
 (1) If the exchange between London and Petersburgh be 
 50d. per ruble, and between London and Amsterdam be 
 34s. 5d. Flem, per sterling, what is the par of arbitration 
 between Petersburgh and Amsterdam ? 
 
 d. 8. d. d. 
 
 As 240 : 34 5 : : 50 
 413 
 50 
 
 24,0 ) 2065,0 ( 86 A per ruble. Ans. 
 192 
 
 145 
 
 144 
 
 1 
 
 (2) London exchanges on Amsterdam at 33s. 9d. per 
 sterling, and on Paris 40^. sterling per ecu, what is the par 
 of arbitration between Amsterdam and Paris ? Ans. Q7^d. 
 
 (3) If the exchange between Amsterdam and Paris be 58d. 
 per crown, and between Amsterdam and London 34s. 4<d. 
 per sterling, required the arbitrated par between Paris 
 and London ? Ans. 33fd. /& sterl.per ecu. 
 
 (4) If Amsterdam remits to Cadiz at 39^. per piastre, and 
 to London at 35s. 5%d. per sterling, how stands the par 
 of arbitration between Cadiz and London ? Ans. 22 nearly 
 
 (5) If London remits to Lisbon at 5s. 3J. per milrea, and 
 to Paris at 56d. per ecu, what is the par of arbitration be- 
 tween Lisbon and Paris ? Ans. G7 sol. per milrea 
 
 * This rate is termed either the arbitrated price, arbitrated par, or par 
 arbitration. 
 
 12 
 
172 Compound Arbitration 
 
 (6) London exchanges with Amsterdam at 33s. 9d. per 
 sterling, and with Genoa at 4<8d. per pezzo, what is the par 
 of arbitration between Amsterdam and Genoa ? 
 
 Ans. 81 Flem. pence per pezzo of Genoa 
 
 COMPOUND ARBITRATION. 
 
 COMPOUND ARBITRATION is when the exchange is to be 
 conducted through several places ; and the rule shows how 
 much a remittance will amount to at the last place. This 
 amount is termed the arbitrated price, or par of arbitration 
 between ike first and last place. 
 
 Questions of this kind are solved either ly the Rule of 
 Proportion, or by arranging the terms into antecedents and 
 consequents. 
 
 RULE 1st. Place the antecedents in one column, and the 
 consequents in another; observing that the second antecedent 
 must be of the same kind with ihefirst consequent ; and the 
 third antecedent of the same kind with the second conse- 
 quent, &c. 
 
 2. The first antecedent and the last consequent must bo 
 of the same kind. 
 
 3. Multiply the antecedents together for a divisor, and 
 the consequents for a dividend, and the quotient will be the 
 arbitrated price of exchange. 
 
 EXAMPLES. 
 
 (1) A merchant in London intends to remit 1000/. to 
 Cadiz in Spain, by way of Holland, at 36s. Flemish per 
 sterling, thence to France at 56d. Flemish per ecu ; thence 
 to Venice, at 100 ecus for 60 ducats ; thence to Cadiz at 360 
 maravedis per ducat ; how many piastres of 272 maravedis 
 each, will the 1000Z. sterling amount to in Cadiz ? 
 Antecedents. Consequents. 
 
 II. sterling, 36s. Flem ....... 432 d. Flem. 
 
 56 Flem ......................... 1 ecu. 
 
 100 ecus ....................... , 60 ducats. 
 
 1 ducat ........................ 360 maravedis. 
 
 272 marav ...................... 1 piastre. 
 
 How many piastres == 1000Z. ? 
 
 ..... 432x60x360x1000 
 Omitting the units, we have by the rule - 56x100x272 - 
 
 And further reduced,* gives X t* * - = 6 1 26 rfy piastres. Ans. 
 
 * The operation may be abridged, by dividing any of the antecedents 
 and constituents by their common measure. See under N.B. Case 5th, 
 Vulgar Kiactioni, page 118. 
 
Position. 173 
 
 (2) If 100 Ibs. at London are equal to 96 Ibs. at Amster- 
 dam, and 135 at Amsterdam equal to 1G8 at Toulouse, how 
 many Ibs at Toulouse are equal to 175 Ibs. at London ? 
 If 100 Ibs. London = 96 Ibs. Amsterdam. 
 
 135 Ibs. Amsterdam = 168 Ibs. Toulouse. 
 How many Ibs. at Toulouse = 175 Ibs. London. 
 24 56 7 8 
 
 3136 
 
 - 9 lbs - l 
 
 POSITION. 
 
 POSITION, sometimes termed the BTTLE OE FALSE, is a 
 rule which, by supposed numbers, enables us to find the real 
 ones. It consists of two parts, single and double. 
 
 SINGLE POSITION. 
 
 SINGLE POSITION is when only one supposed number is 
 requisite to obtain the answer. 
 
 HTJLE 1st. Suppose any jit number, and work with it as 
 if it were the true one.* 
 
 2nd. Then say, As the result of this ivorJc is to the true 
 total, so is the supposed number to the true one required. t 
 EXAMPLES. 
 
 (1) A schoolmaster being asked how many scholars he 
 had, said, if I had as many more, half as many, and a quarter 
 as many, I should have 330 ; how many had he ? 
 
 Proof. 
 
 Suppose 60 Then, As 165 : 330 : : 60 120 
 
 As many more 60 60 120 
 
 as many. ... 30 60 
 
 iasmany 15 165) 19800(120 Ans. 30 
 
 165 330 
 
 (2) A person, after spending ^, J, and of his money, had 
 . left ; how much had he at the first ? Ans. 1000Z. 
 
 * Any supposed number will produce the true answer ; but for con- 
 venience in working, those numbers are to be preferred, from which all 
 the part* can be taken without remainders. Some, however, recommend 
 the number 1 to be made the constant supposition. 
 
 f Only those questions belong to SINGLE POSITION, whose parts are 
 certain proportions of the suppositions, or of some power or root of 
 their suppositions. 
 
174 Double Position. 
 
 (3) A gentleman bought a chaise, horse, and harness for 
 72Z. ; the horse came to twice the price of the harness, and 
 the chaise to thrice the price of the horse ; what did he give 
 for each ? Ans. For the harness, SI. ; for the horse, 1GZ. / 
 
 for the chaise, 48Z. 
 
 (4) A, B, and C purchased a ship for 3300?. ; A paid a 
 certain sum, B paid twice as much as A, and C four times 
 as much as B ; how much did each man pay ? 
 
 Ans. A paid 300Z. ; 600Z. / and C 2400Z. 
 
 (5) A man meeting a maid driving a flock of geese, said, 
 Where are you going, sweetheart, with these 50 geese ? she 
 replied, I have not 50, but if I had \ as many more, a third 
 and a quarter as many more, I should have 50 ; how many 
 had she ? Ans. 24 geese. 
 
 (6) Lent a sum of money to receive 6 per cent, per annum 
 simple interest, and at the end of ten years received for 
 principal and interest 1000Z. ; what was the sum lent ? 
 
 Ans. 625Z. 
 
 Suppose 100 Then, As 160 : 1000 : : 100 to 625/. 
 1 yrs. int. at 6/. per ct. 60 
 
 160 
 
 (7) Borrowed a sum of money, for which I paid 5Z. per cent, 
 per ann. simple interest ; in 7 years it amounted to 810Z. ; 
 what was the sum borrowed ? Ans. 600Z. 
 
 DOUBLE POSITION. 
 
 DOUBLE POSITION is when two supposed false numbers 
 are requisite to obtain the true answer. 
 
 RULE 1st. Take any two convenient numbers, and pro- 
 ceed with each according to the nature of the question, 
 noticing how much each result differs from the true result, 
 and call this difference its error. 
 
 2nd. Place each error against its respective position, and 
 multiply them cross-ways. 
 
 3rd. If the errors are alike, that is, both greater or both 
 less than the given number, take their difference for a divisor, 
 and the difference of their products for a dividend. But if 
 unlike, that is, one too much, and the other too little, take 
 their sum for a divisor, and the sum of their products for 
 a dividend ; the quotient will be the answer 
 EXAMPLES. 
 
 (1) A, B, and C would divide 1200Z. between them so 
 that B may have 100Z. more than A, and C 100Z. more than 
 B ; how much must each have ? 
 
Double Position. 175 
 
 Suppose A had ...... 500 Suppose again A... 400 
 
 Then B would have 600 Then B...500 
 
 AadC ............... 700 And C... 600 
 
 1800, too much by 600 1500, too much by 800 
 
 Here the errors are of one Tcind : therefore, by the rule, 
 
 sup. error. Proof. 
 
 500 v.600 600 A's share 300 
 
 X sub. 300 B's ...... 400 
 
 400 300 C's ...... 500 
 
 - - 300 divisor. , - 
 
 240000 150000 Shorter, thus... 1200 1200 
 
 sub. 150000 B extra ...... Too 
 
 C . .. 200 
 
 8,00)00,00 
 
 300 A 's share. 900-J-3 = A's share, &c. 
 
 (2) Divide 500Z. between two persons, A and B, so that 
 A may have 120Z. more than B ; what is each person's share ? 
 
 Ans. A 310Z. and B. 190/. 
 
 (3) A, B, and C built a house which cost 1500/. of which 
 A paid a certain sum, B paid 2001. more than A, and C 
 paid 2001. more than B ; what sum did each pay ? 
 
 Ans. A paid 3001.; B 5001.; O 7001. 
 
 (4) A person dying, bequeathed to three of his friends 
 600^ which he had in his chest, in this manner, to the first 
 a certain portion, to the second half as much more, wanting 
 10Z. and to the third double the sum, wanting 30Z. ; what 
 sum did each receive ? 
 
 Ans. The 1st, 142R; the 2nd, 203R; the 3rd, 254* Z. 
 
 (5) A, B, and C are indebted to D in certain sums ; A's 
 and B's debts united amount to 500/., B's and C's to 700Z., 
 and A's and C's to 600L; what is each man's particular debt? 
 Suppose A's... 190 Suppose agaiu, A's... 220 
 
 Then B's... 310 Then B's.. .280 
 
 And C's... 390 And C's... 420 
 
 Therefore A and C 580, too little Therefore A and C 640, too much 
 
 by 20. by 40. 
 
 Here the errors are unlike therefore, by the rule, 
 sup. 190 vJ20 too little. 
 
 220 40 too much. 
 
 Proof. 
 
 , , S 20 4400 7600 A's debt 200+B's 300 = 500 
 
 ia f 40 7600 B's ...... 300+ C's 400 = 700 
 
 C's ...... 400 + A's 200 = 600 
 
 ,0 ) 1200,0 
 
 200 Ant. A't debt. 
 
176 Progression. 
 
 (6) Three persons discoursed concerning their ages ; said 
 A, I am 20 years of age ; said B, I am as old as A, and 
 half of C ; and said C, I am as old as you both ; the age of 
 each person is required. 
 
 Ans. A 20, J5 60, and C 80 years of age. 
 
 (7) A man left his estate to his 3 sons thus ; to the eldest 
 one half, wanting 501. ; to the second one third ; and to the 
 youngest the rest, which was 10Z. less than the share of the 
 second ; I demand the sum left, and each son's part ? 
 
 Ans. The sum left was 360Z., of which the eldest had 130Z., 
 the second 120Z., and the third 110Z. 
 
 (8) A gentleman bought a house, orchard, and garden, for 
 1000Z. ; he paid three times the price of the garden for the 
 orchard, and 30Z. more ; and four times the price of the 
 orchard for the house, and 40Z. more ; what was the value of 
 each? 
 
 Ans. The garden 50Z. 12s. 6d.; the orchard 181Z. 17s. Qd. ; 
 and the house 767Z. 10s. 
 
 PROGRESSION* 
 
 Consists of two parts, AEITHMETICAL and GEOME- 
 TRICAL. 
 
 AEITHMETICAL PEOGKESSION 
 
 Is when & series of numbers increases or decreases regu- 
 larly, by the continual adding or subtracting of the equal 
 numbers ; as 1, 2, 3, 4, 5, 6, &c., are in Arithmetical Pro- 
 gression by the continual addition of one ; and 11, 9, 7, 5, 
 3, 1, by the continual decreasing or subtracting of two. 
 
 The numbers which form the series are called the terms 
 of the Progression, of which there are five 
 First term. 
 The last term. 
 The number of terms. 
 The common difference. 
 The sum of all the terms. 
 
 Any three of these being given, the other two may DO 
 found. 
 
 * The most useful parts of Progression, as far as the Rule relates to 
 common arithmetical purposes, are comprised in this treatise ; but as it 
 is for the exercise of the juvenile capacity, to whom Algebra is unknown, 
 it has been thought proper to divest it of all symbolic characters. 
 
Progression. 
 
 177 
 
 CASE I. The first term, the last term, and the nwtnber 
 of terms given to find the sum of all the terms. 
 
 EULE. Multiply the sum of the two extremes by half tlie 
 number of terms, or, multiply half the sum of the extremes 
 by the whole number of terms ; the product is the answer. 
 
 EXAMPLES. 
 
 (1) How many strokes do 
 the clocks of Venice strike in 
 24 hours ; where they strike 
 from one to 24 ? 
 
 To the first term 1 + 24= 25 
 Then, 25 multiplied by half the 
 number of hours. 
 
 Or 25x12=300 the Answer. 
 
 (3) The first term of an 
 arithmetical progression is 5, 
 the last is 74, and the number 
 of terms 24 ; what is the sum 
 of the series ? 
 
 (2) If 100 stones are placed 
 in a right line 3 yards asun- 
 der, and the first 3 yards from 
 a basket, what length of 
 j ground will a man go over 
 who gathers them up one by 
 one, returning with each to 
 the basket ? 
 
 To fetch the first he will walk 6 
 yards, and the last 600 ; hence, the 
 first term is 6, and the last 600. 
 Therefore 6 + 600=606 
 
 Multiplied by 50= % the No. 
 - of terms. 
 
 5 + 74=79, and 79 x 12=948 Ans. 
 
 Ans. He will walk 30300 yds. = 17 
 miles 38 yds. 
 
 (4) What debt may be discharged in a year by weekly 
 payments in arithmetical progression, of which the first 
 payment is 3s. and the last 105s. ? Ans. 140Z. 8s. 
 
 (5) A mercer bought 30 yards of silk, and gave for the 
 first yard 3s. and for the last 3Z. Is. increasing in arithmeti- 
 cal progression ; what did the whole cost him ? Ans. 48 1. 
 
 CASE II. The first term, the last term, and the number 
 of terms, given to find the common difference. 
 
 RULE. From the last term subtract the first ; the re- 
 mainder divided by the number of terms, less one, gives the 
 common difference. 
 
 EXAMPLES. 
 
 (6) The extremes are 1 and 
 24, and the number of terms 
 24 ; required the common dif- 
 ference ? 
 
 No. of terms 24 From last term 24 
 Subtract 1 Sub. the first 1 
 
 Divisor 23 23 
 
 23 23 ( 1 the com. diff An*. 
 
 (7) The first term is 6, the 
 last term 600, and the number 
 of terms 100; what is the 
 common difference ? 
 
 No. of terms 100 From last term 600 
 Subtract 1 Sub. the first 6 
 
 Divisor 99 594 
 
 99 ) 694 ( 6 the com. diff. Ans. 
 
178 
 
 Arithmetical 
 
 (8) The first term of an arithmetical progression is 5, the 
 last is 74, and the number of terms 24 ; what is the common 
 difference ? Ans. 3. 
 
 (9) The first and last terms of an arithmetical series are 3 
 and 105, and the number of terms 52 ; what is the common 
 difference ? Ans. 2. 
 
 (10) Bought 30 yards of cloth, and gave for the first yard 
 3s. and for the last 61s. ; what is the common difference of 
 the price of each yard ? Ans. 2. 
 
 CASE III. The first term, the last term, and the com- 
 mon difference, given to find the number of terms. 
 
 RULE. From the last term subtract ihe first ; divide the 
 remainder by the common difference, and the quotient in- 
 creased by one, gives the nwnber of terms. 
 EXAMPLES. 
 
 (11) A person travelling 
 went 4 miles the first day, and 
 increased every day by 5 
 miles, till at last he went 64 
 miles in one day ; how many 
 days did he travel ? 
 
 The last term 64 
 Subtract the first 4 
 
 Com. diff. 5 ) 60 
 
 12 
 Add 1 
 
 Ans. 13 number of 
 
 (12) A man being asked 
 how many sons he had, said 
 the youngest was 5 years old, 
 and the eldest 29 ; and that 
 he increased one in his family 
 every 3 years ; how many bad 
 he? 
 
 The last term 29 
 
 Sub. the first 5 
 
 Com. diff. 3 ) 24 
 
 8 
 Add 1 
 
 Ans. 9 sons. 
 
 MIA y a. ___^ 
 
 (13) If the extremes be 5 and 74, and the common dif- 
 ference 3, what is the number of terms ? Ans. 24. 
 
 (14) If the first and last terms of an arithmetical progres- 
 sion are 3 and 105, and the common difference 2, what is 
 the number of terms ? Ans. 52. 
 
 (15) Bought cloth, and gave for the first yard 3s. and for 
 the last 61s., the common difference of the price of each yard 
 is 2s. ; required the number of yards. Ans. 30 yds. 
 
 CASE IY. The last term, the number of terms, and the 
 common difference, given to find the first term. 
 
 RULE. Multiply the number of terms, less one, by the 
 common difference, the product subtracted from the last term 
 gives the first. 
 
Progression. 
 
 EXAMPLES. 
 
 (16) A person in 13 days 
 travelled to a certain place, 
 every day' s journey increasing 
 the former by 5, and the last 
 he went was 64. miles ; what 
 was the first ? 
 
 NumberoftermslS From 64 
 Less one 1 Take 60 
 
 12 
 
 Common diff. 5 
 
 60 
 
 Ans. first 
 term. 
 
 (17) A person had 9 sons, 
 his family having increased 1 
 every 3 years ; the eldest was 
 29 years of age : what was the 
 age of the youngest ? 
 
 Number of terms 9 
 Deduct 1 
 
 Common diff. 3 
 24 
 
 From 29 
 Subt. 24 
 
 Ans. 5 
 
 Ans. 4 miles, the }st day's journey. Ans. the youngest was 5 yrs. old. 
 
 (18) If the last term be 74, the number of terms 24, and 
 the common difference 3, what is the first term ? Ans. 5. 
 
 (19) If the last term be 105, the number of terms 52, and 
 the common difference 2, what is the first term ? Ans. 3. 
 
 (20) Purchased 30 yards of cloth, and gave for the last 
 yard 61s., the common difference of the price of each yard 
 was 2s. ; how much was given for the first yard ? Ans. 3s, 
 
 CASE Y. The common difference, the number of terms, 
 and the sum of all the terms given, to find the first term. 
 
 RULE. Divide the sum of all the terms by the number of 
 terms, and from the quotient subtract halfoi the product of 
 the common difference, multiplied by the number of terms 
 less one, for the answer. 
 
 EXAMPLES. 
 
 (21) A man is to receive 
 540/. at 12 several payments, 
 each exceeding the former by 
 6 ; what will be the first pay- 
 ment ? 
 
 12)540 
 
 12 
 1 
 
 3=^ the com. 
 
 Ans.lZ first pay diff. 
 
 ment. 
 
 (22) A person had 9 sons, 
 his family having had an in- 
 crease of 1 every 3 years, and 
 the amount of all their ages 
 was 153 ; what was the age of 
 the youngest ? 
 
 9)153 9 
 
 1 
 
 != c. diff. 
 
 Ans. 5 age of the 
 
 youngett. 12 -8=4. 
 
 (23) If the common difference be 2, the number of terms 
 52, and the sum of all the terms 2808, what will be the first 
 term ? Ans. 3. 
 
180 
 
 Geometrical. 
 
 CASE VI*. The first term, the number of terms, and 
 the common difference given, to find the last. 
 
 RULE. Multiply together the number of terms and the 
 common difference ; from that product subtract the common 
 difference ; to that remainder add the first term, and it will 
 give the last. 
 
 EXAMPLES. 
 
 (24) A person received a 
 snm at 12 several payments, 
 the first was 12 1. and each 
 succeedingpayment exceeded 
 the former by 6, what was the 
 last payment ? 
 
 Number of terms 12 
 Common diff, 6 
 
 mult. 
 
 The first term 
 
 66 
 12 
 
 add. 
 
 Ans. 78 the last term. 
 
 (25) The first term of an 
 arithmetical series is 5, the 
 number of terms is 24, and 
 the common difference is 3 ; 
 what is the last term ? 
 
 Number of terms 
 Common diff. 
 
 24 i 
 3 I 
 
 mnlt. 
 
 Ans. 
 
 (26) "What is the last number of an arithmetical progres- 
 sion beginning with 3, and continuing by the increase of 2 
 to 30 places ? Ans. 61. 
 
 GEOMETRICAL PROGRESSION 
 IS when any rank or series of numbers increases by one 
 common multiplier, or decreases by one common divisor : as, 
 1, 2, 4, 8, 16, 32, &c. increase by the constant multiplication 
 of 2 : and 81, 27, 9, 3, 1, , i, &c. decrease by the constant 
 division or ratio of 3. 
 
 In Geometrical Progression, the same five things are to be 
 observed, as in Arithmetical ; viz. 
 Theftrst term. 
 The last term. 
 The number of terms. 
 The equal difference or ratio. 
 The sum of all the terms. 
 
 Any three of these terms being given, the otJiers may be 
 found. 
 
 * This rule might be extended to ten cases, each containing two pro- 
 positions ; but some teachers will deem what has been already introduced 
 nwre than sufficient, till the pupil enters upon Algebra. For the Alge- 
 braic Formulae, see Nicholson and Rowbotham's Algebra. 
 
Progression. 181 
 
 Note 1st. As the last term in a long series of numbers is very tedious 
 to come at by continual multiplication, so, for the more readily finding 
 it out, a series of numbers is made use of in Arithmetical Proportion, 
 called Indices, beginning with a unit, whose common difference is one : 
 also, whatever number of indices you may make use of, set as many 
 numbers, in Geometrical Proportion, under them, thus : 
 
 1.2.3.4.5.6.7.8 Indices* 
 
 2 . 4 . 8 . 16 . 32 . 64 . 128 . 256 Numbers in Geometrical Proportion. 
 
 2nd. But if the first term in Geometrical Proportion be different from 
 the ratio, the indices must begin with a cypher, thus : 
 
 0.1.2.3.4.5.6.7 Indices. 
 
 L 2 . 4 . 8 . 16 . 32 . 64 . 128 Numbers in Geometrical Proportion. 
 
 3rd. When the indices begin with a cypher, the sum of the indices 
 made choice of, must always be one less than the number of terms given 
 in the question ; for one in the indices stands over the second term, and 
 two in the indices over the third, &c. 
 
 4th. Add any two of the indices together, and that sum will agree 
 with the product of their respective terms ; thus, 
 
 In either table of indices 2 + 5 = 7 & 2 + 4 = 6 
 So in the Geometrical Proportion 4 x 32 = 128 & 4 x 16 = 64. 
 
 5th, If any number of terms be continued in Geometrical Progres- 
 sion, the product of the two extremes will be equal to the product of any 
 two means, equally distant from the extremes, as in 2, 4, 8, 16, 32, 64, 
 where 2x64 = 4x32 = 8x16, each product being 128. Also, if the 
 number of terms be odd, then the square of the mean will be equal to 
 any two terms equally distant, as in 1, 3, 9, 27, 81, 243, 729, where 
 1x729 = 3x243 = 9x81= the square of the mean 27, the product of 
 each, being 729. 
 
 6th. The common multiplier, or divisor, is called the ratio : thus in 
 
 2, 4, 8, 16, 32, &c., the ratio is 2, because each succeeding term is in- 
 creased by multiplying by 2; and in 81,. 27, 9, 3, 1, ^, &c., the ratio is 
 
 3, because each succeeding term is decreased by dividing by 3. 
 
 CASE I. Given the first term, the last term, and the 
 common ratio, to find the sum of the series. 
 
 B/TJLE. Multiply the last term by the ratio, and from the 
 product subtract the first term ; the remainder divided by 
 the ratio, less one, will give the sum of the series. 
 
 * By the help of these indices, and a few of the first terms, in any series 
 of Geometrical Progression, any term, whose distance from the first 
 term is assigned, however remote, may speedily be obtained, without 
 producing all the intermediate terms. 
 
182 Geometrical 
 
 EXAMPLES. 
 
 (1) The first term of a series 
 in geometrical progression is 
 2, the last term is 13122, and 
 the ratio 3 ; what is the sum 
 of the series ? 
 
 13122 last term. 
 Multiplied by 3 ratio. 
 
 39366 
 Subtract 2 first term. 
 
 Ratio 3-1=2)39364 
 
 (2) Sold 12 bushels of wheat, 
 and received for the first bushel 
 one farthing, and for the last 
 1048576 farthings ; the ratio 
 of each bushel is 4, what were 
 the 12 bushels sold for ? 
 
 1048576 last term. 
 4 ratio. 
 
 4194304 
 
 1 first term. 
 
 Ratio 4 -1 = 3)4194301 
 
 s. d. 
 
 Ans.farth. 1398101 = 1456 7 
 
 (3) The extremes of a geometrical series are 20 and 
 10,000, and the ratio is 2, what is the sum of the series ? 
 
 Ans. 19980. 
 
 (4) A thresher worked at a farmer's 24 days during the 
 winter months, and received for his first day's work 2 barley- 
 corns, for the second 4, for the third 8, &c., doubling them 
 each day, and for the last day's work 16777216 barley-corns ; 
 the sum of the series is required. Ans. 33554430 larley-c.* 
 
 CASE II. G-ive the first term, the number of terms, and 
 the common ratio, to find the last term. 
 
 The last term might be obtained by a long series of continual multipli- 
 cation, but to avoid so tedious a process, observe the following rules, 
 
 1. When ike first term is equal to ratio. 
 
 RULE 1st. Find a few of the leading terms, over which 
 place their indices. 
 
 2nd. Find what figures of these indices, when added to- 
 gether, will give the index of the term wanted. 
 
 3rd. Multiply the numbers standing under such indices, 
 into each other ; and the last product will be the term re- 
 quired. 
 
 EXAMPLES. 
 
 (5) A man agrees for 20 fat oxen, to pay only the price of 
 the last, reckoning 3 farthings for the first, 9 farthings for 
 the second, &c., trebling the price to the last (the common 
 ratio being 3), what must he give ? 
 
 * Supposing 493447 barleycorns to fill a bushel, the answer is equal 
 to 68 bush., which, at fo. per bushel, would amount to 17/. sterling. 
 
Progression. 183 
 
 1.2.3.4.5.6. 7 Indices. 
 3 . 9 . 27 . 81 . 243 . 729 . 2187 Terms in Oeomet. Prop. 
 Take any of the indices, which added together make 20, and multiply 
 by the terms underneath. 
 
 Or take 5+5 + 5+5 that is 
 
 Thus, 7 + 7+6 = 20, therefore, 
 Multiply 2187 = 7 
 by 2187 = 7 
 
 4782969 
 and by 729 = 6 
 
 5 + 5: 
 
 = 10, and 10 + 10 = 20 
 243 = 5 
 243 = 5 
 
 59049 = 10 
 59049 = 10 
 
 farth. 3486784401 = 20 3486784401=20 
 
 Which when divided by 4, U\ and 20, will give 3632067Z. Is. 8|d/or 
 the Answer. 
 
 (6) "What is the last terra of a geometrical series, having 
 12 terms, of which the first term is 2, and the ratio 2 ? 
 
 Ans. 4096. 
 
 (7) A draper sells 21 yds. of cloth, the first y^ard for 3^., 
 the second for 9d., the third for 27^., &c., in a triple propor- 
 tion geometrical ; I demand the price of the last yard ? 
 
 Ans. 43584805Z. 05. 3d. 
 
 2nd. When ihejirst term is not equal to the ratio. 
 
 BTJLE 1st. Write down a few of the leading terms, as be- 
 fore, andplace over them their indices, beginning with a cypher. 
 
 2nd. Find what figures of the indices, when added toge- 
 ther, less one, will give the index to the term wanted. 
 
 3rd. Multiply the numbers standing under suchindices, into 
 each other; observing to divide every pro duct by the first term. 
 EXAMPLES. 
 
 (8) The first term of a geo- 
 metrical series is 5, the ratio 
 3, and the number of terms 
 15, what is the last term ? 
 
 1 , 
 15. 
 
 45 
 
 3.4, 
 
 135 . 405 , 
 
 1215 = 5 
 
 1215 = 5 
 
 5 
 
 1215 
 
 5)1476-225 
 
 295245 
 
 405 = 4 
 
 5)119574225 = 14 
 
 23914845 Ans. 
 
 (9) A man agrees to pur- 
 chase 22 yards of velvet, at 3 
 farth. for the first yard, 6 farth. 
 for the second, 12 farth. for 
 the third, &c., (the ratio being 
 2) geometrical proportion ; 
 what was the charge for the 
 last yard ? 
 
 0.1.2.3.4.5. 6 . 7 
 3.6.12.24 .48.96.192.384 
 Then as one less 22=21, take three 
 sevens, 
 884x884 
 
 = 49152 
 
 3 
 49152x384 
 
 aiid = 6291456/07-^. 
 
 3 the Ans. 
 
 divided by 4, 12, and 20 = 
 12*. Ans. 
 
184 Geometrical 
 
 (10) A sum of money is to be divided among 9 persons ; 
 the first is to have 30Z., the second 60Z., the third 120/., &c., 
 what will the last receive ? Am. 7680Z. 
 
 (11) A moneyed man, ignorant of numbers, and unjust in 
 the distribution of his property, left 12 sons, and bequeathed 
 his estate thus ; to his executor 50Z., to his youngest son 
 double that sum, and each son was to exceed the next 
 younger by as much more ; what was the eldest son's por- 
 tion ? Ans. 204SOOZ. 
 
 CASE III. The first term, the number of terms, and the 
 ratio given, to find the sum of all the terms. 
 
 HTJLE. Find the last term, as before, then subtract the 
 first from it, and divide the remainder by the ratio less one ; 
 to the quotient of which add the greater, and it will give the 
 sum required. 
 
 EXAMPLES. 
 
 (12) A servant agreed to serve his master 12 months, for 
 a farthing the first month, 3 farthings the second month, 9 
 farthings for the third, &c., (the common ratio being 3) ; 
 what did his wages amount to ? 
 
 0.1.2.3.4. 5 . 6 Indices. 
 
 1 . 3 . 9 . 27 . 81 . 243 . 729 Terms in Geomet. Proportion. 
 
 729 = 6 177147 = 11 
 
 243 = 5 1 
 
 2187 ratio 3-1=2) 177146 
 
 2916 
 
 1458 88573 
 
 the last term 177147 
 177l47 = llA or last term. 
 
 k. = 276/. 15s. 
 
 (13) A man bought a horse, and by agreement was to give 
 a farthing for the first nail, a halfpenny for the second, a 
 penny for the third, &c., (the common ratio being 2) the 
 number of nails was 32 ; what was the price of the horse ? 
 
 Ans. 4473924Z. 5s. 3%d. 
 
 (14) One new year's day a gentleman married, and re- 
 ceived of his father-in-law a sovereign, with a promise that 
 it should be doubled on the first day of every month for one 
 year ; what was the lady's portion ? Ans. 4095 1. 
 
 (15) A person agrees to purchase 15 yards of lace, for 3 
 pins the first yard, 9 pins the next, &c., in treble proper- 
 
Progression. 
 
 185 
 
 tion; reckoning 100 pins for a halfpenny, what is the 
 amount ? 
 
 Here, the first term 3, being = the ratio 3, proceed according to Case 
 II. Rule 1st, to place 1 in the indices over the first term, thus 
 
 1.2.3.4.5. 6 
 
 3 . 9 . 27 . 81 . 243 . 729 
 
 Then, as the indices 4 + 5+6=15. "VVe multiply the respective terms 
 81 x 243=729= 14348907 pins, which at 100 for a halfpenny = 29 81 18s. 
 8^d. Ans. 
 
 (16) A goldsmith, when his customer refused to give 4-01. 
 per Ib. for gold, offered to sell it him at a farthing for the 
 first ounce, a penny for the second, &c., in quadruple pro- 
 portion geometrical, to which he agreed ; I demand what the 
 Ib. of gold cost him ? Ans. 5825Z. 8s. 
 
 CASE IV. The first term, the last term, and the sum of 
 the series being given to find the ratio. 
 
 ETJLE. From the sum of the series subtract the first 
 term, and divide the remainder by the difference between 
 the sum of the series and the last term, and the quotient 
 will be the ratio required. 
 
 EXAMPLES. 
 
 (17) If the extremes of a 
 series be 2 and 13122, and the 
 sum of the series be 19682, 
 what is the ratio ? 
 
 19682 
 13122 
 
 19682 sum of the series. 
 2 first term to be sub. 
 
 6560 ) 19680 ( 3 the ratio. 
 19680 
 
 (18) Sold wheat, and re- 
 ceived for the first bushel 1 
 farthing, and for the last 
 1048576 farthings; and for 
 the whole 1398101 farthings ; 
 required the ratio ? 
 1398101 1398101 
 1048576 1 
 
 349525 ) 1398100 ( 4 the ratio. 
 1398100 
 
 (19) A person bought 10 acres of land, and gave for the 
 first acre 3d. for the last 59049^. and for the whole 88572^. ; 
 what was the equal difference or ratio ? Ans. 3. 
 
 (20) A gentleman received as his wife's portion, in one 
 year 4095 1. by monthly payments in geometrical progression; 
 his first receipt was \.l. and his last 2048/. ; what was the 
 ratio ? Ans. 2. 
 
( 186 ) 
 
 PERMUTATION. 
 
 PEEMUTATION is the changing of the relative position 
 of things, so that no two may have all their parts placed 
 twice in the same situation. 
 
 Thus, the figures 1, 2, 3, may have six different positions, viz. 123, 
 132, 213, 231, 312, and 321. 
 
 To find all the variations of position that can take place 
 in any given number of things : 
 
 B-ULE. Multiply into each other, successively, as many 
 of the numbers 1, 2, 3, 4, &c., as there are things to be varied, 
 and the last product will give the nuniber of permutations. 
 EXAMPLES. 
 
 (1) How many changes may 
 be rung on 6 bells ? 
 
 2 
 8 
 
 (5 
 
 4 
 
 84 
 
 _J 
 120 
 . _6 
 A ns. 720 change*. 
 
 (2) For how many days can 
 7 persons be placed in dif- 
 ferent positions round a table 
 at dinner ? 
 
 1 
 
 2 
 3 
 6 
 4 
 
 a! 
 
 j> 
 
 120 
 __6 
 
 720 
 
 Or 1x2x3x4x5x6=720 Ans. 
 
 Ans. 5040 days. 
 
 (3) How many different ways can 7 notes in music be 
 varied ? Ans. 5040. 
 
 (4) How many permutations can be made of any 9 letters 
 of the alphabet ? Ans. 362880. 
 
 (5) How many transmutations can be made of the letters 
 in the word Britannia ? Ans. 362880. 
 
 (6) A scholar wishing to reside with a gentleman whose 
 family consisted of five persons besides himself, offered him 
 30Z. for his board, for only so long as they could be all 
 seated differently every day at dinner : this being accepted, 
 how long did he continue ? Ans. 5040 days. 
 
 (7) How many transpositions can be made of the follow- 
 ing words, " Die quibus in terris, tres pateat coeli spatium 
 non amplius ulnas ?" Ans. 39916800. 
 
Involution. 187 
 
 (8) I demand how many changes may be rung upon 12 
 bells ; and also how long they would be in ringing them but 
 once over, suppose 24 changes to be rung in one minute, 
 and the year to consist of 365 days and one quarter ? 
 
 Ans. The number of changes is 479001600 ; the time is 
 37 year s, 49 weeks, 2 days, 18 hours. 
 
 (9) Seven gentlemen travelling met at an inn, and being 
 pleased with each other's company, and with their host, 
 offered him 501. if he would board them so long as they 
 could sit every day at dinner with him in a different order, 
 to which he readily consented ; I demand how long they 
 stayed, and how many different positions they sat ? 
 
 Ans. The number of positions was 40320 ; and the time 
 they stayed was 110 yrs. 142^ days. 
 
 The preceding rules of Progression, together with this of Permutation, 
 and those of Combinations, and Composition of Numbers, might be greatly 
 extended, by many interesting questions, not merely as subjects of 
 curiosity, but of real utility ; but they may be solved much more easily 
 and neatly by symbolic characters, when the student arrives at Algebra ; 
 a study which, if he has a taste for, will afford him a high source of 
 entertainment, and reward him for the research. See Progression in 
 " Nicholson and Rowbotham's Algebra." 
 
 INVOLUTION. 
 
 INVOLUTION is multiplying any number by itself, and 
 that product by the same number, and so on to any assigned 
 number of places. This is also termed The raising of 
 Powers.* 
 
 Any number may be called tine first power ; the product of 
 that number multiplied by itself, is called the second power, 
 or square ; if this be multiplied by the first power again, the 
 product is called the third power or cube : and if by the same 
 again, the product is called the fourth power or biqiiadrate. 
 
 Thus, suppose 3 to be the first power, then 3x3 gives 9, the second 
 power, or square ; and 9x3 gives 27, the third power, or cube ; and 
 27x3 = 81, the fourth power, or biquadrate. The small number denoting 
 the power, is called the index or exponent of that power ; thus 3 2 is the 
 square or second power; 3 s the cube or third power, &c. 
 
 * This rule, though not found in some treatises, is a useful prelimi- 
 nary to the Square and Cube Roots, &c. 
 
188 Involution. 
 
 EXAMPLES. 
 
 (1) What is the square of 
 
 (3) Eequired the 9th power 
 
 24? 
 
 of 2. 
 
 24 
 
 2 = 1st power. Or thus, 2=lst. 
 
 24 
 
 2 2 
 
 93 
 
 4 = 2nd power. 4 = 2nd. 
 
 48 
 
 2 4 
 
 576 Ans. 
 
 8 = 3 r d power. 16 = 4^. 
 
 
 
 2 16 
 
 (2) What is the square and 
 
 16-4 h ~96 
 
 cube of 64 ? 
 
 2 16 
 
 64 
 
 
 64 
 
 32 = 5th power. 25G = 8th. 
 
 ______ 
 
 2 2 
 
 256 
 
 
 384 
 
 64 = 6th power. 512 = 9 th. 
 
 
 
 2 
 
 4096 the square. 
 
 When a power 
 
 64 
 
 123 = Tth power, higher than a 
 
 1 
 
 2 cube is wanted ; 
 
 16384 
 
 the operation 
 
 24576 
 
 256 = 8th power, maybe shorten- 
 21 i 
 
 262144 the cube. 
 
 ed, as above. 
 
 K1 O QfJt nnnniif,,*. 
 
 (4) "What is the square of 144 ? Ans. 20736. 
 
 (5) What is the cube of 72 ? Ans. 373248. 
 
 (6) Required the third power of 36. Ans, 46656. 
 
 (7) It is required to find the fourth power of 24 ? 
 
 Ans. 331776. 
 
 (8) What is the biquadrate of 48 ? Ans. 5308416. 
 
 (9) What is the 6th power of 7 ? Ans. 117649. 
 
 (10) Eequired the 9th power of 3. Ans. 19683. 
 
 (11) Wbat is the square 
 and cube of 602 ? 
 
 6,02 
 6,02 
 
 1204 
 3612 
 
 36,2104 
 
 6,02 
 
 724808 
 21744240 
 
 218,167208 Ans. 
 
 (12) What is the square 
 off? 
 
 (13) What is the cube of 
 
 Hance, 
 
 Ans. 
 
 
Evolution. 189 
 
 (14) What is the 4th power of ,08 ? Ans. ,00004096. 
 
 (15) What is the 5th power of ,74 ? Ans. ,2219006624. 
 
 (16) What is the 6th power of 4,2 ? Ans. 5489,031744. 
 
 (17) Required the 7th power of \ ? Ans. ri-g-. 
 
 (18) Required the cube of 2 ? Ans. 12if . 
 
 (19) E/equired the biquadrate of f ? Ans. i^nr. 
 
 (20) What is the 5th power of 1,1 ? Ans. 1,61051. 
 
 (21) What is the 6th power of 2,01 ? A. 65,944160601201. 
 
 (22) What is the 7th power of 1^ ? Ans. 
 
 EVOLUTION. 
 
 EVOLUTION, the reverse of Involution, is the method of 
 finding the root of any number ; as the square-root, the cube- 
 root, &c., and hence called the extraction of roots. 
 
 The root of any number or power, is such a number, as 
 being multiplied into itself & given number of times, produces 
 that power. 
 
 Thus, 3 is the square root of 9, because 3x3 = 9; and 4 is the cube 
 root of 64, because 4x4x4 = 64. Also, 2 is the biquadrate root of 16, 
 because 2x2x2x2 = 16* 
 
 EXTRACTION OP THE SQUARE EOOT. 
 
 Extracting the Square Hoot of any number, is finding 
 such a number as, being multiplied once in itself, wil 1 
 produce the given number. 
 
 Rule 1st. Begin at the unifs place, and point the give*, 
 numbers into periods of two figures each. If the figures cor* 
 sist of whole numbers and decimals, the whole numbers mus? 
 be pointed from right to left, the decimals the contrary way 
 
 * The power of any given number may be found exactly ; but there 
 are many numbers from which the root cannot be exactly obtained, as 
 the square root of 5, 7, 10, &c., because no two numbers multiplied into 
 themselves will give 5,7, 10, &c. ; although, by means of decimals, we 
 may attain to any degree of exactness. 
 
 Roots are often denoted by writing ^/ before the power, with the index 
 
 2 
 against it ; thus the square root of 24 is described by ^/ 24, or only 
 
 /v/24, without the 2 ; for 2 is always meant when no index is written. 
 The cube root of 24 is expressed thus, *J 24. Sometimes the roots are 
 expressed with a small figure above ; as 24 T is the square root of 24, 
 and 125 T is the cube root of 125. 
 
190 
 
 Square Root, or 
 
 2nd. Find the greatest square number that is contained 
 in the first period towards the left-hand ; placing the square 
 number under the first point, and set its root in the quotient. 
 
 3rd. Subtract the square number from the first point; 
 and to the remainder bring down the two figures under the 
 next point, for a dividend. 
 
 4th. Double the quotient, and place it for a divisor on 
 the left-hand of the dividend ; see how often it is contained 
 in the dividend (exclusive of the unit's place) and put the 
 answer in the quotient, and also on the right-hand of the 
 divisor. 
 
 5th. Multiply the divisor by the last figure put in the 
 quotient, and subtract the product from the dividend ; to 
 the remainder bring down the next period, and proceed thus 
 till all the periods are brought down. 
 
 6th. If any thing remain, add two cyphers thereto, and 
 repeat the work, and for every two thus added, you will have 
 one decimal in the root. 
 
 Eoots 1.2.3.4.5.6.7.8.9 
 Squares 1 . 4 . 9 . 16 . 25 . 36 . 49 . 64 . 81 
 
 EXAMPLES. 
 
 (1) What is the square root 
 
 of 54756 ? 
 
 64756(2*4 root. 
 4 
 
 43)147 
 129 
 
 464 ) 1856 
 1856 
 
 (2) Required the square 
 root of 321489 ? 
 
 321489 ( 667 root. 
 25 
 
 (3) "What is the square 
 root of 1234,56 ? 
 
 1234.56(35,1363 + 
 9 
 
 65)334 
 325 
 
 701). . 956 
 701 
 
 7023 ) 25500 
 21069 
 
 70266 ) .443100 
 421596 
 
 702723 ) .2150400 
 2108169 
 
 . .42231 
 
 106). 714 
 636 
 
 1127;. 7889 
 7889 
 
 Ans. 35,1363 -\-tJie required root. 
 
 (4) "What is the square root of 7056 ? Ans. 84. 
 
 (5) What is the square root of 9216 ? Ans. 96. 
 
Evolution. 191 
 
 (6) What is the square root of 119025 ? Am. 345. 
 
 (7) What is the square root of 459684 ? Ans. 678. 
 
 (8) What is the square root of 27394756 ? Ans. 5234. 
 
 (9) What is the square root of 18671041 ? Ans. 4321. 
 
 Note. When the given number consists of a whole number 
 and decimals together, make the number of decimals even (if 
 they are not so), by adding cyphers to them, so that a point 
 may fall on the unit's place of the whole number* 
 
 (10) What is the sq. root of 4712,81261 ? Ans. 68,649 + 
 N.B. See the 3rd example that is worked at length. 
 
 (11) What is the sq. root of 3,1721812 ? Ans. 1,78106+ 
 
 (12) What is the sq. root of 761,801261 ? Ans. 27,6007 + 
 
 (13) What is the sq. root of 9712,718051 ?Ans. 98,553 + 
 
 (14) What is the sq. root of ,0007612816 ? Ans. ,02759 + 
 
 (15) What is the sq. root of 4,0000671 21 ? Ans, 2,000016 + 
 
 CASE II. To extract the Square Root of a Vulgar 
 Fraction. 
 
 EULE. Reduce infraction to its lowest terms ; then ex- 
 tract the square root of the numerator for a new numerator, 
 and the square root of the denominator for a new denominator. 
 
 If the fraction be a STJRD (i. e. a number whose root can- 
 not be exactly found), reduce it to a decimal, and extract 
 the root from it. 
 
 EXAMPLES. 
 
 (1) What is the square root of 3 Hi ? Ans. |. 
 
 3044 ) 6849 ( 2 
 
 6088 Com. measure 761 ) ||| = | lowest term*. 
 
 .761 ) 3044 (4 4(2 num. 9 ( 3 den. or */ f = . 
 
 3044 4 9 
 
 (2) What is the square root of ** P . 
 
 (3) What is the square root of iMt P Ans. i' t 
 
 SURDS. 
 
 (4) What is the square root of tiff ? 
 |f=,5116279069, the square root of which is ,71528 -f 
 
 (5) What is the square root of f H ? Ans. ,87447 + 
 
 (6) What is the square root of Hi ? Ans. ,72414+ 
 
 CASE III. To extract the Square Boot of a Mixed 
 
 Number. 
 
 RULE 1st. Reduce the fractional part of the mixed 
 number to its lowest terms, and then the mixed number to 
 an improper fraction. 
 
192 Square Root. 
 
 2nd. Extract the roots of the numerator and denomina- 
 tor for a new numerator and denominator. 
 
 If the mixed number given be a surd, reduce the frac- 
 tional part to a decimal, annex it to the whole number, and 
 extract the root from the whole. 
 
 EXAMPLES. 
 
 (7) "What is the square root of 5f-/- 2 ? 
 
 5M = 5|f = ^ 441 ( 21 num. 81 ( 9 den. V=2f=21-. 
 
 _81 
 41 ) .41 
 
 Jl Ans. 21. 
 
 (8) What is the square root of V7 && ? -^*- 4J. 
 
 (9) What is the square root of 37f|- ? Ans. 6j. 
 
 SURDS. 
 
 (10) What is the square root of 8| ? 
 8f - 8,6 8,60 ( 2,9325 + An*. 
 
 460 
 441 
 
 512 ) 1900 
 1749 
 
 15100 &c. 
 
 (11) "What is the square root of 76ff- ? Ans. 8,7649 + 
 
 (12) "What is the square root of 7 A ? Ans. 2,7961 + 
 
 CASE IY. To find a Mean Proportional between two 
 
 given Numbers. 
 
 RULE. Multiply together the two given numbers, and 
 extract the square root of the product ; which root will be 
 the mean proportional sought. 
 
 EXAMPLES. 
 
 (13) What is the mean proportional between 4 and 9 ? 
 
 4 x 9 36. Then, 36 ( 6 the mean proportional. Ans. 6. 
 36 
 
 (14) What is the mean proportional between 8 and 18 ? 
 
 Ans. 12 
 
 (15) What is the mean proportional between 12 and 48 ? 
 
 Ans. 24. 
 
 (16) Required the mean proportional between 15 and 35? 
 Ans. 22,912 ~r 
 
 7'A.SE Y. To find the side of a Square equal in area to 
 
 any given Superficies. 
 
 RULE. The square root of any given superficies will be 
 the side of the square sought. 
 
Evolution. 
 
 193 
 
 EXAMPLES. 
 
 (17) If the area of a given triangle be 9876 yards, I demand 
 the side of a square equal in area thereto ? 
 9876 (91.378 + 
 81 Am. 99,378 + 
 
 189 ) 1776 
 1701 
 
 ...7500 &c 
 
 (18) If the area of a given circle be 961, what is the side 
 of a square equal in area ? Ans. 31. 
 
 (19) If the area of a given circle be 1000, what is the side 
 of a square equal in area ? Ans. 31,6 + 
 
 (20) If an oval fish-pond contain 1 acre (=4840 square 
 yards) ; required the side of a square fish-pond of equal 
 dimensions. Ans. 69,57 -{-yds. 
 
 CASE VI. Any two sides of a right-angled triangle 
 given, to find the third side. 
 
 RULE. If the hypothenuse or longest side be required. 
 The square root of the sum of the squares of the base and 
 perpendicular ', will be the hypcthenuse sought. But */ either 
 of the other two sides be wanted, extract the square root of the 
 difference of the squares of the given sides, for the answer. 
 EXAMPLES. 
 
 (21) The top of a tower from the ground is 36 yards, and 
 surrounded with a moat 20 yards broad ; what must be the 
 length of a ladder to reach from the outside of the moat to 
 the top of the tower ? 
 
 36x36 = 1296 
 20x20= 400 
 
 1696(41. 18 
 16 
 
 81 ) . . 96 &c 
 
 I 
 
 The Base 27 
 Ans. 41 \ yds. nearly. 
 
 (22) The two shortest sides of a right-angled triangle are 
 27 and 36 yards ; required the length of the hypothenuse ? 
 
 Ans. 45 yards. 
 
Cube Root. 
 
 (23) The base of a right-angled triangle is 80, and the per- 
 pendicular 40 feet ; required the length of the hypothenuse ? 
 
 Ans. 50 feet. 
 
 (24) A river, 30 feet in breadth, flows round the base of a 
 tower, and if a line of 50 feet will reach from the opposite 
 bank to the top of the tower, what is its height ? 
 
 Here 50x50 = 2500 the line> or hypothen. squared. 
 Subt. 30 x 30 = 900 the river, or base do. 
 
 1600 ( 40 the height of the tower. 
 16 
 
 ..00 Ans. 40 feet. 
 
 (25) If from the opposite bank of the river to the top of 
 the tower be 50 feet, and the height of the tower be 40 feet, 
 what is the breadth of the river ? Ans. 30 feet. 
 
 MISCELLANEOUS QUESTIONS. 
 
 (26) If an army were placed rank and file (that is, in the 
 form of a square) each side having 356 men, how many men 
 would the square contain ? 
 
 356 x 356=126736 men, the Ans. 
 
 (27) If each side of a square pavement contains 120 feet 
 how many square feet are contained therein ? 
 
 Ans. 14400 sq.feet. 
 
 (28) A kitchen garden which is to contain 4 acres (=19360 
 sq. yards), is to be a complete square; the length of each 
 side is required. Ans. 139, 14 yards, or 139 yds. Oft. 5 in. -{- 
 
 (29) How long must a ladder be to reach a window 36 feet 
 high, when the bottom stands 15 feet from the building ? 
 
 Ans. 39 feet. 
 
 (30 Two ships sail from the same port ; the one sails north 
 24 leagues, the other west 18 leagues ; the distance from 
 each other is required. Ans. 30 leagues distant. 
 
 N.B. The courses of the 2 ships are as the base and perpendicular of 
 a right-angled triangle hence the distance will be equal to the hypo- 
 thenu.e. 
 
 EXTRACTION OF THE CUBE EOOT. 
 
 Extracting the Cube Eoot is finding out a number which, 
 eing multiplied by its square, will produce the given 
 number. 
 
 RULE 1st. Begin at the unit's place and point the given 
 numbers into periods of three figures each ; towards the left 
 hand in whole numbers, and towards the right in decimals. 
 
 2nd. Find the greatest cube in the first left-hand period, 
 
Evolution. 195 
 
 and subract it therefrom, put the root in the quotient, and 
 bring down the figures in the next period to the remainder 
 for a resolvend. 
 
 3rd. To find a divisor, square the quotient, and multiply 
 it by 3, See how often it is contained in the resolvend, 
 rejecting the units and tens, and put the answer in the quo- 
 tient. 
 
 4th. To find the subtrahend. 1st. Cube the last figure in 
 the quotient. 2nd. Multiply all the figures in the quotient 
 ~by 3, except the last, and that product by the square of the 
 last. 3rd. Multiply the divisor by the last figure ; adding 
 their products together, gives the subtrahend, which subtract 
 from the resolvend. To the remainder bring down the next 
 period, and proceed as before. 
 
 Boots 1.2.3.4.5.6.7.8.9 
 Cubes 1 . 8 . 27 . 64 . 125 . 216 . 343 . 512 . 729 
 
 EXAMPLES. 
 
 (1) What is the cube root of 12812904 ? 
 12812904 ( 234 Ans. 
 
 Square of 2 X 3 = 12 divisor. ) 4812 resolvend. 
 27 = cube o/3. 
 
 54 =2 x 3 x by square of 3, i. e. 9. 
 36 = divisor x by 3. 
 4167 subtrahend. 
 
 Square of 23 X &=divisor 1587) 645904 resolvend. 
 
 61 = cube of 4. 
 
 1104 = 23 x 3 x fy sq. of 4, i. *. 16. 
 6348 = divisor x 4. 
 645904 subtrahend. 
 
 CASE II. Another method of Extracting the Cube Root* 
 
 RULE 1st. Find by trials the nearest cube to the given 
 number, and call it the assumed cube. 
 
 2nd. Say, as twice the assumed cube added to the given 
 number, is to twice the number, added to the assumed cube, 
 so is the root of the assumed cube to the root required, 
 nearly. 
 
 EXAMPLES. 
 
 (2) "What is the cube root of 64484 ? 
 
 Here the nearest root that is a whole number is 40, the cube of which 
 is 64000. Therefore, 
 
 * For a general and easy method of extracting the roots of all powers, 
 see " Nicholson and Rowbotham's Algebra." 
 K 2 
 
19G Oule Root. 
 
 Assumed cube 64000 64484 
 
 2 2 
 
 128000 128968 
 
 Given number 64484 64000_ 
 
 Then say, As 192484 : 192968 : : 40 
 
 40 
 
 192484 ) ,718720 ( 40,1 + Ana. 
 769936 
 . .193600 
 192484 
 . .1116 &c. 
 
 (3) "What is the cube root of 13824 ? Ans. 24. 
 
 (4) What is the cube root of 110592 ? Ans. 48. 
 
 (5) What is the cube root of 884736 ? Ans. 96. 
 
 (6) What is the cube root of 1860867 ? Ans. 123. 
 
 (7) What is the cube root of 14886936? Ans. 246. 
 
 (8) What is the cube root of 8120601000 ? Ans. 2010. 
 
 (9) What is the cube root of 64964808000 ?Ans. 4020. 
 
 When the given number consists of a whole number and 
 decimals together, make the decimals consist of either 3, 6, 9, 
 &c. places, by adding ciphers thereto, if needful. 
 
 (10) What is the cube root of 7612,812161 ?Ans. 19,67+ 
 
 (11) What is the cube root of 61218,00121 ?Ans. 39,41 + 
 
 (12) What is the cube root of 7121,1021698 ? Ans. 1 9,238 + 
 
 CASE III. To extract the Cube Hoot of a Vulgar 'Fraction. 
 
 EFLE 1st. Eeduce the fraction to its lowest terms ; then 
 extract the cube roots of its numerator and denominator, for 
 a new numerator and denominator. 
 
 2nd. But if the fraction be a surd, reduce it to a decimal, 
 and then extract the root from it. 
 
 EXAMPLES. 
 
 (13) What is the cube root of AVs ? Ans. f . 
 Here tWs = aS the cube of which is f . 
 
 (14) What is the cube root of J*f- ? Am. f , 
 
 (15) What is the cube root of AVo ? Ans. 5 . 
 
 SUEDS. 
 
 (16) What is the cube root of | ? Ans. ,763 + 
 Here = ,444444444 the cube root of which is ,763 + 
 
 (17) What is the cube root of f ? Ans. ,949 + 
 
 (18) What is the cube root of \ ? Ans. ,693 + 
 
Evolution. 107 
 
 CASE IV. To extract the Cube Root of a Mixed Number. 
 
 RULE 1st. Reduce the fractional part to its lowest terms, 
 and then the mixed number to an improper fraction ; then 
 extract the cube roots of the numerator and denominator for 
 a new numerator and denominator. 
 
 2nd. But if the mixed number be a surd, reduce the 
 fractional part to a decimal, annex it to the whole number 
 and extract the root from it. 
 
 EXAMPLES. 
 
 (19) What is the cube root of 578^ ? Am. 8 J. 
 578 *= 18 5 y 5 the cube root of which is y = 8^. 
 
 (20) What is the cube root of 42! ? Ans. 3.J. 
 
 (21) What is the cube root of 5iU Ans. l. 
 
 SHEDS. 
 
 (22) What is the cube root of 8 & ? -4n*. 2,013 + 
 
 8 A = 8 J = 8,166660666 the cube root of which is 2,013 + 
 
 (23) What is the cube root of 7 ? Ans. 1,966 + 
 
 (24) What is the cube root of 91 P Ans. 2,13 + 
 
 CASE V. Between two numbers given, to find two Mean 
 
 Proportionals. 
 
 BTTLE. 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 pro- 
 portional. 
 
 EXAMPLES. 
 
 (25) Find two mean proportionals between 8 and 512. 
 8 ) 512 ( 64 the cube root of which is 4. 
 
 then 4 x 8 = 32 thelest mean. \ . 
 
 64 and 4 x 32 = 1 28 the greater mean. J *"* 
 
 The truth of which may be proved thus : 
 
 As 8, the less extreme : 32, the less mean : : 128, the greater mean : 512 
 the greater extreme. 
 
 (26) What are the two mean proportionals between 7 and 
 189 ? Ans. 21 and 63. 
 
 (27) Find two geometric means between 5 and 1715 P 
 
 Ans. 35 and 245, 
 
 CASE VI. To find the side of a Cube that shall be equal in 
 
 solidity to any given solid. 
 
 RULE. The cube root of the solid content of the given 
 body will be the side of the cube required. 
 
198 Biquadrate Root. 
 
 (28) The solid content of a given cylinder is 18G08G7 
 inches ; required the size of a cube that is equal in area 
 thereto ? Ans. 123. 
 
 PEOMISCTJOUS QUESTIONS. 
 
 (29) If a cubical piece of stone contains 46656 solid feet, 
 what is the superficial content of one of its sides ? Ans. 36. 
 
 (30) If a cubical piece of timber be 3G inches long, 36 
 inches broad, and 36 inches deep, how many cubical inches 
 does it contain ? Ans. 46656. 
 
 (31) How many solid feet of earth must be dug out, to form 
 a cellar 16 feet in length, breadth, and depth ? Ans 4096. 
 
 (32) The content of a globe is 3375 inches, what is the side 
 of a cube of equal dimensions ? Ans. 15 inches. 
 
 (33) There is a cube whose side is 4 feet : I demand the 
 aide of another cube whose solid content is treble the former ? 
 
 Here 4 cubed is 64 ; which trebled = 192 ; the cube root of which is 
 5,76 feet + Ans. or rather more than 5 feet 9 inches. 
 
 EXTEACTION OP THE BIQUADEATE EOOT. 
 
 EULE. First extract the square root of the given num- 
 ber ; then extract the square root of that square root for the 
 biquadrate root. 
 
 EXAMPLES. 
 
 (1) What is the biquadrate root of 1G77721G ? Ans. 64. 
 
 First, 16777216 ( 4096 square root. Then, 4096 ( 64 Ans. 
 
 IQ 36 
 
 809 ).. 7772 124 )T496~ 
 
 7281 496 
 
 8186)49116 
 49116 
 A ns. 64 the biquadrate root. 
 
 (2) What is the biquadrate root of 5308416 ? Ans. 48. 
 
 (3) What is the biquadrate root of 84934656 ? Ans. 96. 
 
 TO EXTEACT THE EOOTS OF ALL POWEES. 
 A general Rule,givenby Wm. Mountaine, Esq., JF.J2.S. 
 
 EULE 1st. Prepare the given number for extraction, by 
 pointing off from the unit* s place, as the root required directs. 
 
 2. Find the first figure of the root by trial, and subtract 
 the power from the given number. 
 
 3. To the remainder bring down thejlrs t figure in the next 
 period, and call it the dividend. 
 
Duodecimals. 199 
 
 4. liiTolve the root to the next inferior power to that 
 which is given ; 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. 
 
 6. Involve the wliole root into the given power for a sub- 
 trahend ; and subtract it from as many points of the given 
 power as are brought down. 
 
 7. To the remainder bring down the first figure of the 
 next period, for a new dividend. 
 
 8. Find a new divisor, as before, and proceed in like man- 
 ner till the whole is finished. 
 
 EXAMPLE. 
 What is the cube root of 115501303 ? 
 
 115501303 ( 487 the root. Aits. 
 64 = 4 3 
 4 2 x 3 = 48 ) 515 dividend. 
 
 48 3 = 110592 subtrahend. 
 4 8 2 3 = 691 2)49093 dividend. 
 487 3 = 1155010303 subtrahend. 
 
 DUODECIMALS. 
 
 DUODECIMALS, or Cross Multiplication, is a rule much 
 used by workmen and artificers, for finding the contents of 
 their works.* 
 
 12 fourths ("") 1 third 
 
 12 thirds 1 second 
 
 12 seconds 1 inch or prime ' 
 
 12 inches or primes 1 foot (ft.) 
 
 HTJLE 1st. Under the multiplicand write the correspond- 
 ing denominations of the multiplier ; that is, set feet under 
 feet, inches under inches, &c. 
 
 * It is called Duodecimals, because the feet, inches, &c., are divided 
 into twelve parts: and Cross Multiplication, because the factors were 
 formerly multiplied cross way*. 
 
 Note Feet multiplied by feet give feet. 
 
 Feet multiplied by inches give inches. 
 Feet multiplied by seconds give seconds. 
 Inches multiplied by inches give seconds. 
 Inches multiplied by seconds give thirds. 
 Seconds multiplied by seconds give fourths. 
 
200 
 
 Duodecimals. 
 
 2nd. Multiply each term in the multiplicand, beginning at 
 the lowest by the feet in the multiplier, write each product 
 under its respective term ; observing to carry one for every 
 12, from each lower denomination to its next superior. 
 
 3rd. Multiply in the same manner with the inches : and 
 set the product of each term one remove farther to the right- 
 hand, and carry one for every 12 as before. 
 
 4th. Work in like manner with the seconds, &c.. and the 
 sum of the lines will be the product required. 
 
 EXAMPLES. 
 
 (1) Multiply 8 feet 9 inches 
 by 4 ft. 6 inches. 
 ft. in. 
 8 9 
 
 (2) Multiply 
 by 3 ft. 5 in. 6''. 
 789 
 356 
 
 7 
 
 ft. 
 
 8 in. 9" 
 
 
 4 6 
 
 
 
 
 
 
 23 
 
 2 3 
 
 r- x by 3ft. 
 
 35 = x 4 ft. 
 4 4 6"= x 6 in. 
 
 
 3 
 
 279 
 3 10 4 ( 
 
 j ./ 
 = x by 5 Ml. 
 5""=xby 6". 
 
 39 4 6 Ans. 
 
 
 26 
 
 891 
 
 5 A 
 
 Ins. 
 
 
 
 feet 
 
 in. 
 
 feet 
 
 in. 
 
 feet 
 
 in. 
 
 // 
 
 
 (3) 
 
 Mult. 8 
 
 6 
 
 
 by 
 
 5 
 
 9 
 
 
 Am. 48 
 
 10 
 
 6 
 
 
 (4) 
 
 Mult. 6 
 
 4 
 
 
 by 
 
 6 
 
 2 
 
 
 ... 39 
 
 
 
 8 
 
 
 (5) 
 
 Mult. 7 
 
 5 
 
 
 by 
 
 3 
 
 6 
 
 
 ... 25 
 
 11 
 
 
 
 
 (6) 
 
 Mult. 12 
 
 3 
 
 
 by 
 
 7 
 
 6 
 
 
 ... 91 
 
 10 
 
 6 
 
 
 (7) 
 
 Mult. 14 
 
 6 
 
 
 by 
 
 9 
 
 3 
 
 
 ... 134 
 
 1 
 
 6 
 
 
 (8) 
 
 Mult. 3 
 
 4 
 
 6 
 
 by 
 
 2 
 
 4 
 
 3 
 
 ... 7 
 
 11 
 
 4 
 
 ]/"(}"" 
 
 (9) 
 
 Mult. 4 
 
 6 
 
 9 
 
 by 
 
 3 
 
 6 
 
 4 
 
 ... 16 
 
 1 
 
 1 
 
 9 
 
 (10) 
 
 Mult. 5 
 
 9 
 
 3 
 
 by 
 
 4 
 
 3 
 
 8 
 
 ... 24 
 
 10 
 
 1 
 
 11 
 
 (11) 
 
 Mult. 7 
 
 8 
 
 10 
 
 by 
 
 6 
 
 9 
 
 6 
 
 ... 52 
 
 6 
 
 5 
 
 11 
 
 (12) 
 
 Mult. 9 
 
 
 
 11 
 
 by 
 
 1 
 
 2 
 
 3 
 
 ... 10 
 
 9 
 
 4 
 
 9 
 
 (13) 
 
 Mult. 18 
 
 2 
 
 
 
 by 
 
 4 
 
 3 
 
 6 
 
 ... 77 
 
 11 
 
 7 
 
 
 (14) 
 
 Mult, 20 
 
 3 
 
 9 
 
 by 12 
 
 2 
 
 3 
 
 ... 247 
 
 6 
 
 8 
 
 5 3 
 
 N.B. 
 
 The 1st question may be proved by 
 
 the five following methods. 
 
 By Cross Mult. 
 
 By Practice 
 
 . By 
 
 Vulgar Fractions. 
 
 By Decimals. 
 
 A 
 
 in. 
 
 
 
 ft. in. 
 
 
 
 
 
 
 
 
 8 
 
 9 
 
 
 i 
 
 8 9 
 
 
 
 ^r 
 
 y = 8f = \ 5 
 
 
 
 8,75 
 
 4 
 
 6 
 
 
 
 4s 
 
 
 
 T 
 
 f = 4|=V 8 
 
 
 
 4,5 
 
 32 
 
 0=4x8 
 
 
 35 
 
 
 VxV-W- 
 
 89/*. 4 in. 6" 
 
 ft. 32,375 
 12 
 
 3 
 
 4x9 
 
 OA o v a 
 
 
 
 
 
 
 in 4,500 
 
 
 (J o X O 
 
 4 6=9x6 
 
 
 39 
 
 4 6 
 
 
 
 
 
 
 
 12 
 "6.0 
 
 39 4 6 
 
 A nd lastly by whole numbers, thus 8ft. 9 in. ^ 1 05 in . and 4 ft. 6 in. = 54 in. 
 Therefore 105x54 = 5670 square inches; wtiicli, divided by 144, gives 
 39/J. 4 in. 6". 
 
Duodecimals. 201 
 
 AETTF ICEE' S work is computedb y different measures, viz. 
 
 1st. Glazing and mason's flat work by the foot. 
 
 2nd. Painting, plastering, paving, &c., by the yard. 
 
 3rd. Partitioning, flooring, roofing, tiling, &c., by the 
 square of 100 feet. 
 
 4th. Brickwork, <fec., by the rod, or 16^ feet, the square 
 of which is 
 
 I. Glazing, Mason's Work, fyc., ly tlie Foot. 
 
 EXAMPLES. 
 
 (15) What will be the expense of glass for a window that 
 measures in the clear 9 feet 8 in. in height, and 4 ft. 3 in. in 
 width, at 2s. 3d. per foot. Ans. 4Z. 12s. 
 
 9 S 
 4 3 
 
 83 
 
 41 feet. 
 
 2 
 
 10 3 
 
 250 2\=for 1 inch. 
 
 41 1 4 12 
 
 (16) "What must I give for 3 marble slabs, measuring in the 
 whole 12 feet 9 inches in length, and 1 foot 3 inches in 
 breadth, at 7s. Gd. per foot ? Ans. 51. 19s. 6JJ. ,5. 
 
 (17) If a pane of glass be 2 ft. 6 in. long, and 1 ft. 9 in. 
 broad, how many feet does it contain ? Ans. &ft. 4 in. 6". 
 
 II. Painting, Plastering, Paving, $fc., by the Yard. 
 EULE. Divide the square feet by 9, for the answer in 
 square yards. 
 
 (18) What will the ceiling of a room come to, that measures 
 30 ft. 9 in. in length, and 20 ft. 6 in. in breadth, at Is. 3d. 
 per square yard ? 
 
 ft. in. 
 
 30 9 1*. 3d = 
 
 4 7 6 and 
 
 000^ for the 4 in. 6 
 
 -- 
 
 630 Then for the value of 4 in. 6" 
 
 70 yds. 4 in. 6" =^ of a sq. yd. 
 
 (19) A room is to be painted, that measures 88 ft. 6 in. 
 about, and 10 ft. 9 in. in height ; what will it come to at 
 3s. 4d. per yard ? Ans. 171. 12s. 4^. ,332. 
 
 (20) What will the paving of a court-yard come to at Is. 3d. 
 per ft. whose length is 120 ft. and width G5 J ft. A. 491Z. 5s. 
 
 (21) A piece of wainscoating round a room is G3 ft. 8 in. 
 and height 3 ft. 9 in. ; what will it come to, at 5s. Qd. per 
 yard? Ans. 71. 5s. 10^7. 333. 
 
202 Duodecimals. 
 
 III. Partitioning, Roofing, fyc., ~by tlte square of 100 feet. 
 (22) "What will the tiling of a house cost at 27s. Qd. per 
 square, each side of the roof being 40 feet by 15 ? 
 40 1,00 ) 12,00 
 
 15 
 
 5s. 
 
 600 one side. o 
 
 ^2 8. {* 1 10 
 
 12 sq. oflQQfcet. 
 3 
 
 =&o^A sides. 
 
 (23) There is a range of houses, the whole length of whose 
 root is 324 ft. 6 in. and the whole breadth 32 ft. 9 in. ; what 
 will the whole come to at 265. Sd. per square ? 
 
 Ans. 14U 13^. ll\d. 4. 
 
 IV. Bricklayer's Work, $c., by the rod of '272^ feet. 
 Bricklayers always value their work at the rate of a brick 
 and a half thick ; and if the wall be more or less, it must be 
 reduced to that thickness ; thus : 
 
 RULE. Multiply the area by the number of half bricks, 
 and divide by three. 
 
 (24) How many square rods are there in a wall 84| fee 
 long, 12 feet 6 inches in height, and 2 bricks thick ? 
 
 ft. in. rods. 
 
 84 6 1056 3 272 = 272,25) 1760,41666(6,466 + Jrw*. 
 
 12 6 5 = 2% bricks 163350 
 
 foIT~~0 3)5281 3 126916 
 
 42 3 1760 5 = 1760,41666 + 
 
 1056 3 180166, &c. 
 
 (25) If the area of a wall be 14085 feet, and the thickness 
 one brick and a half, how many rods does it contain ? 
 
 Ans. 51,73 rods, or 51f nearly. 
 
 (26) A brick building is 50 feet long and 25 feet wide (con- 
 sequently measuring 150 feet round) ; the cellars and founda- 
 tion are 15 feet deep and 2| bricks thick ; the ground floor 
 12 feet in height and 2 bricks thick ; the chamber floor 10 
 feet in height and a brick and thick ; and the attic floor 9 
 feet in height, and one brick thick : the number of square 
 rods of brickwork is required ? 
 
 First, the cellars, &c. 2nd, the ground floor. 
 
 150x15 = 2250 150x12 = 1800 
 
 _5 = i bricks. 4=2 brick* 
 
 3)JL125Q 3)7200 
 
 ~ 3750 "2400 
 
 3rd, chamber floor 1 50 X 1 = 1 500 at a 4th attic floor, 
 
 brick and i, the standard thickness. 150x9 = 1 350 
 
 2=1 , 
 Then 3750 + 2400 + 1500 + 900 = 8550 feet 
 
 of brickwork, which being divided by o;^uu 
 
 272J, or by 272,25 gives 31,4 + rods. Ans. 
 
( 203 ) 
 
 APPENDI X. 
 
 MISCELLANEOUS QUESTIONS. 
 
 (1) Write down in figures, one hundred billions, one hundred mil- 
 lions, one hundred thousand, one hundred, and one. 
 
 (2) Write down in figures, nine hundred and eighty-seven billions, six 
 hundred and fifty -four millions, three hundred and twenty-one thou- 
 sand, one hundred and twenty -three. 
 
 (3) Write down the year 1857 in the Roman method of notation by 
 letters. 
 
 (4) If the distance from London to Jamaica be 1330 leagues, how 
 many miles per day will a ship go that makes the voyage in 50 days ? 
 
 (5) In a printed book that shall contain 200 pages, each page 45 lines, 
 and in each line 43 letters, how many letters will it take to compose the 
 whole book ? 
 
 (6) What number is that which, multiplied by 1234, will make the 
 product 1522756? 
 
 (7) What number added to the cube of 25, will make the sum equal 
 to the square of 125 ? 
 
 (8) If Moses was born when Aaron was 15 years old, how old would 
 Aaron be when Moses was 80 ? 
 
 (9) A. horse in his harness is worth 45Z. and out of it 35 guineas ; how 
 much is the price of the harness less than that of the horse ? 
 
 (10) The sum of two numbers is 560, the least of them is 144 ; what 
 is their product and the square of their difference ? 
 
 (11) There are two numbers, the greater of them is 14 times 40, and 
 their difference is 19 times 9 ; their sum and product are required? 
 
 (12) My purse and money, said Dick, are worth a mark, but the 
 money is worth seven times the purse ; what did the purse contain ? 
 
 (13) A captain and ICO sailors took a prize worth 1360Z , of which the 
 captain had \ for his share, and the rest was equally divided among the 
 sailors ; what was each man's share ? 
 
 (14) How many minutes have elapsed since the birth of Christ to the 
 year 1857 inclusive, allowing the year to consist of 365 days 5 hours 
 49 minutes ? 
 
 (15) Divide 100 shillings between A, B, and C, so that A may have 
 5s. less than B, and C 5s. more than B ? 
 
 (16) How long would it take to count one hundred millions of money, 
 at the rate of 100/. per minute ? 
 
 (17) From January 1st, 1800, to July 1st, 1856, how many days, 
 reckoning the year to consist of 365^ days ? 
 
 (18) Bought a pipe of wine (containing 136 gallons) for 80Z., but in tha 
 conveyance it leaked 18 gallons; what shall I gain or lose by the pipe, 
 if I sell the remainder at 12*. Qd. per gallon ? 
 
 (19) A can do a piece of work in 14 days, B alone in 12 days ; if both 
 work together, in what time will it be finished ? 
 
201 Miscellaneous Quest ions. 
 
 (20) Supposing 32 bricks will pave a yard square, how many will it 
 take to pave a passage 25 feet long and 7 feet wide ? 
 
 (21) If the cock of a large cistern will empty it in 29| minutes, how 
 many such cocks will empty it in 4 minutes and -| ? 
 
 (22) If 12 ells of cloth f wide cost 51. 3s. 6d., what will 36 ells of the 
 same stuff cost if 5 qrs. wide ? 
 
 (23) A wall that is to be built to the height of 21 feet, was raised 
 7 feet by 6 men in 8 days, how many men must be employed to finish 
 the wall in 4 days at the same rate of working ? 
 
 (24) If I pay Is. for 7 Ibs. of bread, when corn is worth 65. per bushel, 
 what must I pay for 10^ Ibs. when corn is 46. the bushel ? 
 
 (25) If a person spends as much in four months as he gains in three, 
 how much can he lay by annually with an income of 150?, a year ? 
 
 (20) What quantity of water must I add to a pipe of wine (126 galls,) 
 value 70?., to reduce it to 9s. per gal. ? 
 
 (27) A company at a tavern spent 71. 4s., and each of them had as 
 many shillings to pay as there were persons hi company ; how many 
 persons were there 1 
 
 (28) Sold goods for 500?. to be paid for thus : 100/. down, and the 
 rest at two four months (that is at four mouths, and \ at 8 months), 
 what is their present worth, discounting at 51. per cent. ? 
 
 (29) A factor takes 1?. per cent, for his commission ; what must he 
 receive for 743?. 17s. 3d? 
 
 (30) What is the amount of 1000?. for 5\ years, at 4| per cent, simple 
 interest ? 
 
 (31) Two men depart from one place, and both go the same road ; 
 the one travels 15 miles a day, and the other 22 miles ; how far are they 
 distant at the fortnight's end, both resting on Sundays ? 
 
 (32) The 4 sides of a room measure 150 feet in length, and the height 
 is 12 feet ; how much paper 2 feet 3 inches wide will cover it, and what 
 is the value at 8d. per yard ? 
 
 (33) A man kept a one-horse chaise, value 50?. with two horses of 
 unequal value ; when the younger horse was put to the chaise, their 
 value was double that of the older horse, and when the older was in, 
 their value was treble that of the younger ; what was the value of each ? 
 
 (34) A servant at market purchased for half a guinea, an equal num- 
 ber of fowls at 9c?. each, rabbits at 6d, pigeons at 4d., and larks at 2cZ. 
 each ; how many of each had he ? 
 
 (35) If by selling goods at 2s. 9d. per Ib, I clear 50?. per cent., what 
 do I clear per cent, by selling them at 3s. per Ib. ? 
 
 (33) Bought 127 pieces of cloth, for which I delivered 3589 ells of 
 Holland, at 7s. lie/, per Eng. ell ; what did the cloth cost per piece? 
 
 (37) Divide 1000?. among three men, so that for every 3?. which A 
 had, B shall have 4?. and C 51. ; how much must each receive ? 
 
 (38) Two merchants enter into partnership for 18 months : A puts 
 into stock at first 1000/. and at the end of 12 months takes out 200?. ; 
 B puts in at first 700?. and at the end of nine months puts in 300?. 
 more ; at the expiration of the time they find they have gained 750?., 
 what is each man's share ? 
 
 (39) Two merchants trade in company; the first advanced 640?. and 
 took | of the gain ; how much did the other advance ? 
 
 (40) What is { the | of? 
 
 (41) What part of 4d is a third part of 3d 
 
 (42) What number is that of which 12 is { of it? 
 
 (43) What must be paid for & of a ship that is valued at 2500?. ? 
 
 
Miscellaneous Questions. 205 
 
 (44) Shipped for Jamaica 550 pair of stockings at Us. Gd. per pair, 
 460 yards of stuff at 14c/. per yard ; in return for which I received 
 46 cwt. 3 qrs. of sugar at 24s. ()d. per cwt., and 1570 Ibs. of indigo at 
 2s. 4d. per Ib. ; what remains due to me of my adventure ? 
 
 (45) If a tower which was originally 384 feet high, had, through a 
 convulsion of nature, a sixth part at the base surrounded with earth, 
 and an eighth part above with water, how much in height is visible ? 
 
 (46) If I lend my friend 800Z. for 9 mouths at 5 per cent., what sum 
 at 4 per cent, for 7 months, should he lend me, to requite my 
 kindness ? 
 
 (47) From a marble slab 20 inches broad, what distance from the end 
 must I cut a piece that shall measure 4 square feet ? 
 
 (48) A person bought 150 eggs at the rate of 3 for a penny, and 150 
 at the rate of 5 for a penny ; what does he get or lose by selling them 
 all out at 8 for 2d. ? 
 
 (49) What will the tiling of a stable cost at 155. 6cZ. per square, the 
 length 45 feet 6 inches, and the breadth of the building 30 feet, the 
 eave-boards projecting 16 inches on each side? 
 
 (50) How many ducats must I deliver at Venice, to receive at London 
 1781. 2s. exchange at 4s. 4d. per ducat? 
 
 (51) A garrison of 1000 men can allow each man 14 ounces a day for 
 12 weeks ; now suppose them reduced to 750 men, how much must 
 each man have per day to last them 18 weeks ? 
 
 (52) If the third of 6 were 3, what would the fourth of 60 be ? 
 
 (53) If a regiment of 1000 soldiers consume 256 quarters of wheat in 
 148 days, how many soldiers will consume 64 quarters in 74 days ? 
 
 (54) Suppose a person who possessed a f share of a copper mine, to 
 sell f of his share for 1500., what was the value of his f share at that 
 rate, and also the worth of the whole mine? 
 
 (55) A hundred hurdles may be so placed as to enclose 200 sheep, and 
 with 4 hurdles more, the fold may be made to hold 600 ; how is this to 
 be done ? 
 
 (56) A garden wall 1000 feet in circuit, was raised 12 feet above, and 
 sunk 4 feet below the surface ; the 4 feet below 2 bricks thick, the first 
 6 above a brick and a half, and the upper 6 one brick thick, how many 
 rods of brickwork did the wall contain ? 
 
 (57) If the distance between the earth and sun be 95 millions of miles, 
 and between the earth and moon 240 thousand miles, how far are the 
 sun and moon asunder in an eclipse of the sun ? and how far also ia an 
 eclipse of the moon ? 
 
 ** Many other Miscellaneous Questions, somewhat more difficult of 
 solution, are inserted at the end of the Key, for the exercise of tU cento* 
 
ARITHMETICAL TABLES. 
 
 NUMERATION TABLE. 
 
 02 
 
 MULTIPLICATION TABLE. 
 
 9 
 
 .2 
 
 N 
 
 
 twice 
 
 5 times 
 
 8 times 
 
 11 times 
 
 ^ c 
 r^ 
 
 i a 1 
 
 
 2 are 4 
 
 2 arelO 
 
 2 are 16 
 
 2 
 
 are 22 
 
 S O CT" 1 " 
 
 S ^ * s 
 
 3 .. 
 
 . 6 
 
 3 ...15 
 
 3 . 
 
 24 
 
 3 
 
 ... 33 
 
 o r; 
 
 1 2-3 4 
 
 4 .. 
 
 . 8 
 
 4 ...20 
 
 4 ... 
 
 32 
 
 4 
 
 ... 44 
 
 <D <+ 
 
 j 'S e^ 3 ""S 
 
 5 .. 
 
 .10 
 
 5 ...25 
 
 5 ... 
 
 40 
 
 5 
 
 ... 55 
 
 r I ^ 
 
 6 .. 
 
 12 
 
 e 
 
 > ...30 
 
 6 ... 
 
 48 
 
 6 
 
 ... 66 
 
 f 
 
 I S 9 o : 
 
 [ 3 ^ 
 
 7 .. 
 
 .14 
 
 7 ...35 
 
 7 ... 
 
 56 
 
 7 
 
 ... 77 
 
 W E- 
 
 S W EH EH t 
 
 
 8 .. 
 
 .16 
 
 8 ...40 
 
 8 ... 
 
 64 
 
 8 
 
 ... 88 
 
 9 8 
 
 7654 321 
 
 9 .. 
 i n 
 
 .18 
 
 O A 
 
 9 ...45 
 
 9 ... 
 
 72 
 
 9 
 
 ... 99 
 
 
 PENCE TABLE. 
 
 10 .. 
 
 11 .. 
 
 .20 
 .22 
 
 10 ...50 
 11 ...55 
 
 10 ... 
 
 11 ... 
 
 80 
 88 
 
 10 
 11 
 
 ... 110 
 ... 121 
 
 dL 
 
 5. rf. 
 
 d. 
 
 s. d 
 
 1.2 .. 
 
 .24 
 
 12 ...60 
 
 12 ... 
 
 96 
 
 12 
 
 ... 132 
 
 20 a 
 
 re 1 8 
 
 90 a 
 
 v* 7 
 
 
 
 
 
 
 
 
 
 24 . 
 
 . 2 
 
 96 ... 8 
 
 3 times 
 
 6 
 
 times 
 
 9 times 
 
 12 times 
 
 30 . 
 36 . 
 40 . 
 48 .. 
 50 .. 
 60 .. 
 70 .. 
 72 .. 
 80 .. 
 84 .. 
 
 . 2 6 
 . 3 
 . 3 4 
 . 4 
 . 4 2 
 . 5 
 . 5 10 
 . 6 
 . 6 8 
 . 7 
 
 100 . 
 108 . 
 110 . 
 120 . 
 130 . 
 132 . 
 140 . 
 144 . 
 150 . 
 156 . 
 
 ..84 
 ..90 
 ..92 
 ..10 
 ..10 10 
 ..11 
 ..11 8 
 ..12 
 ..12 6 
 ..13 
 
 2 are 6 
 3 ... 9 
 4 ...12 
 5 ...15 
 6 ...18 
 7 ...21 
 8 ...24 
 9 ...27 
 10 ...30 
 
 11 ...S3 
 
 2arel2 
 3 ...18 
 4 ...24 
 5 ...30 
 6 ...36 
 7 ...42 
 8 ...48 
 9 ...54 
 10 ...60 
 1 fifi 
 
 2 are 18 
 3 ... 27 
 4 ... 36 
 5 .. 45 
 6 ... 54 
 7 ... 63 
 8 ... 72 
 9 ... 81 
 10 ... 90 
 11 ... 99 
 
 2 are 24 
 3 ... 36 
 4 ... 48 
 5 ... 60 
 6 ... 72 
 7 ... 84 
 8 ... 96 
 9 ... 108 
 10 ... 120 
 11 las 
 
 SHILLING 
 
 FARTH. TAB. 
 
 12 .. 
 
 36 
 
 12 
 
 ...73 
 
 12 ...108 
 
 12 
 
 ... 144 
 
 TA 
 
 BLE. 
 
 
 
 
 
 
 
 
 
 
 
 
 d. 
 
 /. s. d. 
 4 are 1 
 
 4 times 
 
 7 
 
 times 
 
 10 times 
 
 Characters. 
 
 20 are 1 
 
 8 .. 
 
 . 2 
 
 2 are 8 
 
 2arel4 
 
 2 are 
 
 20 
 
 = equal. 
 
 30 .. 
 
 . 1 10 
 
 12 .. 
 
 . 3 
 
 3 ... 
 
 12 
 
 3 
 
 ...21 
 
 3 ... 
 
 30 
 
 less. 
 
 40 .. 
 
 . 2 
 
 16 .. 
 
 . 4 
 
 4 ... 
 
 16 
 
 4 
 
 ...28 
 
 4 ... 
 
 40 
 
 + plus. 
 
 50 .. 
 
 . 2 10 
 
 20 .. 
 
 5 
 
 5 ... 
 
 20 
 
 5 
 
 ...35 
 
 5 ... 
 
 50 
 
 X multiply 
 
 60 .. 
 
 . 3 
 
 24 .. 
 
 6 
 
 6 ... 
 
 24 
 
 6 
 
 ...42 
 
 6 .. 
 
 60 
 
 -T- divide. 
 
 70 .. 
 
 , 3 10 
 
 28 .. 
 
 7 
 
 7 ... 
 
 28 
 
 7 
 
 ...49 
 
 7 ... 
 
 70 
 
 : is to. 
 
 80 .. 
 
 4 
 
 32 ... 
 
 8 
 
 8 ... 
 
 32 
 
 8 
 
 ...56 
 
 8 ... 
 
 80 
 
 : : so is. 
 
 90 .. 
 
 4 10 
 
 36 ... 
 
 9 
 
 9 ... 
 
 36 
 
 9 
 
 ...63 
 
 9 ... 
 
 90 : to. 
 
 
 100 .. 
 
 5 
 
 40 ... 
 
 10 
 
 ...40 
 
 
 
 ...70 
 
 ...100 % quarter. 
 
 110 .. 
 
 5 10 
 
 44 ... 
 
 11 
 
 1 ...44 
 
 1 
 
 ...77 
 
 1 ...110' 1 ha 
 
 If. 
 
 120 .. 
 
 6 
 
 48 ... 
 
 1 
 
 2 ...48 
 
 2 
 
 ...84 |12 ...120 | 3 quarters 
 
 ADDITION TABLE. 
 
 and 
 
 2 and | 3 and 
 
 4 and 
 
 5 and 
 
 6 and 
 
 7 and 
 
 Sand 
 
 9 and 
 
 10 and 
 
 Iare2 
 
 Iare3 
 
 Iare4 
 
 1 are5 
 
 Lare6 
 
 Iare7 
 
 lareS 
 
 1 are9 
 
 1 arelO 
 
 larell 
 
 2... 3 
 
 2... 4 
 
 2... 5 
 
 2... 6 
 
 2... 7 
 
 2... 8 
 
 2 ... 9 
 
 2. ..10 
 
 2 ...11 
 
 2 ..12 
 
 3... 4 
 
 3... 5 
 
 3... 6 
 
 3... 7 
 
 5... 8 
 
 3... 9 
 
 3. ..10 
 
 3. ..11 
 
 3 ...12 
 
 3 ...13 
 
 4... 5 
 
 4... 6 
 
 4... 7 
 
 4... 8 
 
 4... 9 
 
 4. ..10 
 
 4. ..11 
 
 4. ..12 
 
 4 ..13 
 
 4 ...14 
 
 5... 6 
 
 5... 7 
 
 5... 8 
 
 5... 9 
 
 5. ..10 
 
 5. ..11 
 
 5.. .12 
 
 5. ..13 
 
 5 ...14 
 
 5 ...15 
 
 6... 7 
 
 6... 8 
 
 6... 9 
 
 6. ..10 
 
 6. ..11 
 
 6. ..12 
 
 6 ..13 
 
 6.. .14 
 
 6 ...15 
 
 6 ...16 
 
 7... 8 
 
 7... 9 
 
 7.. .10 
 
 7.. .11 
 
 7. ..12 
 
 7.. .13 
 
 7.. .14 
 
 7.. .15 
 
 7 ...16 
 
 7 ...17 
 
 8... 9 
 
 8. ..10 
 
 8. ..11 
 
 8. ..12 
 
 8. ..13 
 
 8. ..14 
 
 8.. .15 
 
 8. ..16 
 
 8 ...17 
 
 8 ...18 
 
 9. ..10 
 
 9. ..11 
 
 9. ..12 
 
 9. ..13 
 
 9. ..14 
 
 9. ..15 
 
 9. ..16 
 
 9. ..17 
 
 9 ...18 
 
 9 ...19 
 
 I 
 
ARITHMETICAL TABLES. 
 
 TROY WEIGHT. 
 Gold, Silver, and Jewels, are 
 weighed by this Table. 
 '4 Grains 1 Pennyweight 
 
 DRY MEASURE. 
 Thus were measured all dry goods 
 2 Pints 1 Quart. 
 
 2 Quarts 1 Pottle. 
 
 
 Penny weights... 1 Ounce. 
 12 Ounces 1 Pound. 
 
 2 Galls, or 8 Quartsl Peck. 
 4 Pecks 1 Bushel. 
 
 AVOIRDUPOIS WEIGHT. 
 Bread, Groceries, with all coarse 
 Articles, are weighed by this 
 Table. 
 16 Drams 1 Ounce. 
 
 8 Bushels 1 Quarter. 
 36 Bushels 1 Chaldron of 
 
 Coals. 
 N.B. Of other articles, 32 Bushels 
 make a Chaldron. 
 
 LONG MEASURE. 
 3 Barleycorns ...1 Inch. 
 4 Inches 1 Hand. 
 
 16 Ounces 1 Pound. 
 
 28 Pounds 1 Quarter. 
 
 4 Quarters . 1 Hundred wt. 
 
 12 Inches 1 Foot. 
 
 20 Hundred wt. ...1 Ton. 
 
 3 Feet 1 Yard. 
 
 6T?i__f I TTo-fVinTn 
 
 APOTHECARIES' WEIGHT. 
 Medicines are mixed by this Table 
 20 Grains 1 Scruple 9 
 
 5 Yards 1 Rod or Pole. 
 
 40 Poles 1 Furlong. 
 
 8 Furlongs 1 Mile. 
 3 Miles... 1 League. 
 
 8 Drams 1 Ounce J 
 12 Ounces 1 Pound tt> 
 
 69 Miles .1 Degree on the 
 Equator. 
 N.B. A Hand is 4 Inches, and a 
 Fathom 2 Yards. 
 
 
 CLOTH MEASURE. 
 2 Inches 1 Nail 
 
 SQUARE MEASURE. 
 144 Square Inches 1 Square Foot. 
 9 Square Feet ...1 Square Yard. 
 30i Square Yards 1 Square Pole. 
 40 Square Poles... 1 Square Rood. 
 4 Square Roods ..1 Square Acre. 
 640 Square Acres ..1 Square Mile. 
 
 4 Nails 1 Quarter of a Yd 
 
 4 Quarters 1 Yard 
 
 5 Quarters 1 Ell English. 
 
 WINE MEASURE. 
 All Liquors, except Ale and Beer, 
 were measured by this Table. 
 2 Pints 1 Quart 
 
 SOLID OR CUBIC MEASURE. 
 1728 Cubic Inches 1 Cubic Foot. 
 27 Cubic Feet ...1 Cubic Yard. 
 231 Cubic Inches 1 Gall, of Wine. 
 282 Cubic Inches 1 Gall, of Ale. 
 2150 Cubic Inches 1 Bush, of Malt. 
 
 4 Quarts 1 Gallon 
 
 10 Gallons 1 Anker 
 
 18 Gallons 1 Rundlet 
 
 2O-fl11ons 1 Tia^rto 
 
 63 Gallons 1 Hogshead. 
 
 TIME. 
 
 84 Gallons 1 Puncheon. 
 
 
 2 Hogsheads 1 Pipe. 
 2 Pipes 1 Ton. 
 
 24 Hours 1 Day. 
 7 Days .... 1 Week. 
 
 ALE AND BEER MEASURE. 
 2 Pints . 1 Quart 
 
 4 Weeks 1 Month. 
 12 Calendar Months, or 365 Days 
 and 6 Hours 1 Year. 
 Thirty days hath September 
 
 4 Quarts 1 Gallon 
 
 9 Gallons 1 Firkin 
 
 Vpril, June, and November; 
 February hath twenty-eight alone, 
 And all the rest have thirty-one, 
 Except in Leap-year, at which time 
 rebruary's days are twenty-nine. 
 
 2 Firkins 1 Kilderkin 
 
 2 Kilderkins 1 Barrel. 
 51 Gallons 1 Hogshead 
 
 2 Hogsheads 1 Butt. 
 
" An A ct for establishing Uniformity of Weights and Measures/ 7 passed 
 in June, 1824, and its operations commenced Jan. 1, 1826. 
 
 By this Act the distinction between the Ale, Wine, and Corn gallon 
 is abolished, and an Imperial gallon is established, as well for liquids 
 as for dry goods, not measured by heaped measure ; this gallon must 
 contain precisely "10 pounds, avoirdupois weight, of distilled water, 
 weighed in air, at the temperature of 63 of Fahrenheit's thermometer, 
 the barometer standing at 30 inches." The Act prescribes the scientific 
 modes of determining the principal measures, in case they should be 
 lost. By this Act 
 
 The pound Troy contains 5760 grains. 
 
 The pound Avoirdupois contains 7000 grains. 
 
 The Imperial Gallon contains 277'274 cubic inches. 
 
 The Corn Bushel, eight times the above. 
 
 With respect, therefore, to A le, Wine, and Corn, it will be useful to 
 possess a TABLE OF FACTORS, 
 
 For converting Old Measures into New, and the contrary. 
 
 
 By Decimals. 
 
 Vulgar Fractions, nearly. 
 
 Corn 
 
 Measure. 
 
 Wine 
 Measure. 
 
 Ale 
 Measure. 
 
 Corn 
 Measure. 
 
 Wine 
 Measure. 
 
 Ale 
 
 Measure. 
 
 To convert Old Measures ") 
 to New J 
 
 96943 
 
 83311 
 
 101704 
 
 H 
 
 * 
 
 60 
 3TT 
 
 
 To convert NewMeasures ") 
 to Old j 
 
 1 03153 
 
 1-20032 
 
 98324 
 
 1! 
 
 1 
 
 tt 
 
 
 N.B. For reducing the Prices, these numbers must be all reversed. 
 
 The subjoined Tables will serve to facilitate computations : 
 Comparison between the Old WINE Comparison between the Old BEER 
 Measures and those of the New Measures and those of the New 
 
 Imperial Standard. 
 
 Old Wine Measures. New Standard. 
 
 Imperial Standard. 
 
 Old Beer Measure. New Standard. 
 
 A Gill is equal to 000 083 
 Half Pint 000 166 
 
 A Gill is equal to 00012* 
 Half Pint 000 2 3 
 
 Pint 000 333 
 
 Pint 0010 7 
 
 Quart 1 266 
 
 Quart 1 13 
 
 1 Gallon 030 265 
 
 1 Gallon 1 54 
 
 10 Do. or Anker 810 258 
 18 Do. or Rundlet 14 3 1 387 
 42 Do. or Tierce 34 3 1 370 
 63 Do. or Hogshead ... 52 1 1 355 
 84 Do. or Puncheon.... 69 8 1 310 
 126 Do. or Pipe 104 3 1 311 
 252 Do. or Tun 209 3 1 222 
 
 9 Do. or Firkin 901 0-91 
 18 Do. or Kilderkin .. 18 1 182 
 36 Do. or Barrel 36 2 364 
 54 Do. or Hogshead ..54 3 1 145 
 72 Do. or Puncheon .. 73 1 3-27 
 108 Do. or Butt 109 3 291 
 
 The New Standards being only about l/60th 
 part more than theOld Beer MeMWtt.will scarcely 
 affect the retail prices. 
 
 The New Standards being about l-5th larger 
 than the Old Wine Measure*, will occasion an 
 advance of about twopence halfpenny in every 
 shilling on the old price. 
 
 Comparison between the old DRY Me 
 
 Stanc 
 
 Old Dry Measure. New Standard, 
 gills & 
 
 A Gill is equal to 0-97 
 Half Pint 00000 194 
 Tint 3-88 
 
 asures and those of the new Imperial 
 lard. 
 
 Old Dry Measure. New Standard. 
 
 8 Do. or Quarter 73000 1 41 
 32 Do. orChaldronSl 1 164 
 36 Do.orChal- 34 3 i o 1 2 34 
 
 Quart 00001 375 
 
 dron of Coals 
 40 Do. or Wey ... 38 3 1 3 6 
 80 Do. or Last ... 77 2 1 1 2 li* 
 
 The New Dry Measures being about l-32nd part 
 larger than the Old, may naturally be expected 
 to occasion an advance of about 8 pr cent, upon 
 the old prices. 
 
 Gallon 0003132 
 2 Galls, or 1 Peck 0013124 
 1 Bushel 03130 017 
 
 2 l*o. or Strike ... 1 31 20 035 
 4 Do. or Coomb ... 3 3100 0-70 
 
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